I'm not suggesting that any of the following will
definitely be your specific problem here, but unless you make sure that all of this lot is good
before you start looking too much further, you might needlessly be chasing your own tail down an endless rabbit hole searching for a problem that's actually at least partly due to one or more of these basic things not being up to speed - and it's really not hard or expensive to do any of this, so I'd suggest you check all this stuff out
FIRST! 
Did your 'new battery' get a good charge before installation - at least 8 hours on a proper 'charger' not just a tender?? Is the 'new' battery one of the current crop of OEM/factory supplied but '
less than adequate' excuses for a battery? Your Spyder
NEEDS at least a 350 CCA 21 A/hr battery or you'll have problems sooner or later; and since the latest factory issued batteries are often these cheaper, lower capacity 300CCA batteries that
might be able to work for a short while,
IF you're lucky, you really need to check that too! And as always, check to make sure all the terminals, cables, lugs, and earth/ground points are clean, tight, and making good contact on shiny metal?
These Spyder things are really power hungry, and even if you've got a brand new 350+ CCA 21+ A/hr good quality battery that's just been installed, many brand new batteries don't get a good charge by the retailer/supplier/dealer
before being installed, so they might be potentially capable of working well for you, but the 'barely adequate' charge time they got to boost the surface charge up to 12v might not be sufficient to do anything but show there's some charge there on a meter or load test. And even new batteries can fail too, so you might want to get your battery load tested to make sure that it still has at least 12v under starting load - anything less might be OK for less power hungry machines, but not for our Spyders!!
As for the cables, lugs, battery terminals, and all the earth/ground points onto the frame - if even
just one of these isn't clean, tight, and making good contact on shiny metal, even the best of batteries might not be able to get that '
better than 12v' to where it needs to go!! There's one frame earth/ground point onto the frame that's hidden behind the frunk deep inside the hole behind the LH Front wheel, in the middle of the bike and right alongside the DPS (the DPS is the electrical power steering unit, one of the largest power users on your Spyder!) which is often an issue on many Spyders, so you need to check that, and while you're at it, check the lugs on the end of the cables too - on mine, the frame end lug there was loose on the end of the cable and I had an intermittent but complete power loss as a result whenever that cable moved as I took a corner!
Aaaand, on those early Spyders, where the battery is sorta tucked away on its side under the seat instead of behind the frunk's rear wall; and where the earth/ground cable securing bolt tightens the ground lug onto the frame just in front of the battery, that nutsert/thread/whatever you've got there is prone to stripping the thread &/or captive nut out of the frame, leaving you with a poor 'main earth' for the battery even tho the bolt in that earth point might seem to be tight cos it's tight on the cable ending lug, but you also need to check/make sure that the lug is actually secured properly into bright metal on the frame and the bolt/lug/earth isn't just floating loosely in the breeze despite that really tight bolt!
So check all those before you get too far down that rabbit hole - and if you haven't already, even if it might just be on the off chance, you might want to check & cycle the Kill Switch a few times too?! If we are truly honest about it, and even (or maybe 'especially'??) if we've been riding for years, there's not too many of us who
HAVEN'T been caught out by that bloody anachronistic Kill Switch at least once!!
Good Luck! I do hope it's one of these simple to find and resolve issues.
