# Properties

 Label 950.2.bg.a Level $950$ Weight $2$ Character orbit 950.bg Analytic conductor $7.586$ Analytic rank $0$ Dimension $1200$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.bg (of order $$90$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$1200$$ Relative dimension: $$50$$ over $$\Q(\zeta_{90})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$1200q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1200q + 18q^{11} + 90q^{15} + 12q^{20} - 60q^{22} - 60q^{23} - 96q^{25} - 36q^{29} - 60q^{33} + 30q^{35} - 120q^{45} - 60q^{46} - 60q^{47} + 660q^{49} - 24q^{50} + 120q^{51} + 36q^{54} + 36q^{55} + 48q^{56} - 36q^{59} - 30q^{60} - 144q^{61} - 150q^{64} + 120q^{65} + 96q^{66} + 48q^{69} - 60q^{70} - 120q^{71} - 48q^{79} - 96q^{81} - 30q^{83} + 78q^{84} - 192q^{85} + 60q^{87} - 72q^{89} - 36q^{90} + 60q^{92} - 96q^{94} - 24q^{95} - 180q^{97} - 240q^{98} - 192q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −0.898794 0.438371i −1.80064 + 2.88162i 0.615661 + 0.788011i −1.98988 1.01998i 2.88162 1.80064i −1.84599 1.06578i −0.207912 0.978148i −3.74633 7.68112i 1.34136 + 1.78906i
9.2 −0.898794 0.438371i −1.54722 + 2.47606i 0.615661 + 0.788011i 1.80261 + 1.32311i 2.47606 1.54722i 0.146440 + 0.0845473i −0.207912 0.978148i −2.42190 4.96562i −1.04016 1.97941i
9.3 −0.898794 0.438371i −1.48382 + 2.37460i 0.615661 + 0.788011i 1.05677 1.97059i 2.37460 1.48382i −0.246693 0.142428i −0.207912 0.978148i −2.12191 4.35057i −1.81367 + 1.30790i
9.4 −0.898794 0.438371i −1.22036 + 1.95299i 0.615661 + 0.788011i −0.755334 + 2.10463i 1.95299 1.22036i 1.23161 + 0.711072i −0.207912 0.978148i −1.00977 2.07033i 1.60150 1.56051i
9.5 −0.898794 0.438371i −1.21964 + 1.95183i 0.615661 + 0.788011i 0.512699 2.17650i 1.95183 1.21964i 1.49771 + 0.864704i −0.207912 0.978148i −1.00701 2.06468i −1.41492 + 1.73147i
9.6 −0.898794 0.438371i −0.891893 + 1.42733i 0.615661 + 0.788011i −0.847823 + 2.06911i 1.42733 0.891893i −4.31713 2.49250i −0.207912 0.978148i 0.0733240 + 0.150336i 1.66905 1.48804i
9.7 −0.898794 0.438371i −0.838664 + 1.34214i 0.615661 + 0.788011i 1.22007 + 1.87388i 1.34214 0.838664i 0.0845996 + 0.0488436i −0.207912 0.978148i 0.217124 + 0.445170i −0.275136 2.21908i
9.8 −0.898794 0.438371i −0.716423 + 1.14652i 0.615661 + 0.788011i −2.00197 0.996049i 1.14652 0.716423i −1.22683 0.708310i −0.207912 0.978148i 0.513875 + 1.05360i 1.36272 + 1.77285i
9.9 −0.898794 0.438371i −0.654190 + 1.04692i 0.615661 + 0.788011i 2.23588 0.0292423i 1.04692 0.654190i 3.55795 + 2.05418i −0.207912 0.978148i 0.647030 + 1.32661i −2.02241 0.953861i
9.10 −0.898794 0.438371i −0.443586 + 0.709885i 0.615661 + 0.788011i 0.161877 2.23020i 0.709885 0.443586i −3.38490 1.95427i −0.207912 0.978148i 1.00794 + 2.06659i −1.12315 + 1.93353i
9.11 −0.898794 0.438371i −0.401035 + 0.641790i 0.615661 + 0.788011i −2.22848 + 0.184025i 0.641790 0.401035i 1.40727 + 0.812487i −0.207912 0.978148i 1.06405 + 2.18162i 2.08362 + 0.811502i
9.12 −0.898794 0.438371i −0.0925144 + 0.148054i 0.615661 + 0.788011i 2.22820 + 0.187404i 0.148054 0.0925144i −3.43865 1.98531i −0.207912 0.978148i 1.30175 + 2.66899i −1.92054 1.14522i
9.13 −0.898794 0.438371i 0.123716 0.197987i 0.615661 + 0.788011i −2.12450 + 0.697499i −0.197987 + 0.123716i −0.938580 0.541889i −0.207912 0.978148i 1.29122 + 2.64739i 2.21525 + 0.304411i
9.14 −0.898794 0.438371i 0.145245 0.232441i 0.615661 + 0.788011i −1.06342 + 1.96701i −0.232441 + 0.145245i 2.84810 + 1.64435i −0.207912 0.978148i 1.28218 + 2.62886i 1.81808 1.30176i
9.15 −0.898794 0.438371i 0.281867 0.451082i 0.615661 + 0.788011i 1.08365 + 1.95594i −0.451082 + 0.281867i −1.73157 0.999724i −0.207912 0.978148i 1.19109 + 2.44209i −0.116554 2.23303i
9.16 −0.898794 0.438371i 0.324851 0.519870i 0.615661 + 0.788011i 1.89925 1.18020i −0.519870 + 0.324851i 2.84170 + 1.64066i −0.207912 0.978148i 1.15038 + 2.35862i −2.22440 + 0.228181i
9.17 −0.898794 0.438371i 0.543822 0.870297i 0.615661 + 0.788011i −0.314385 2.21386i −0.870297 + 0.543822i −0.435827 0.251625i −0.207912 0.978148i 0.853439 + 1.74981i −0.687924 + 2.12762i
9.18 −0.898794 0.438371i 0.613064 0.981108i 0.615661 + 0.788011i 1.45906 1.69444i −0.981108 + 0.613064i −2.05896 1.18874i −0.207912 0.978148i 0.728388 + 1.49342i −2.05419 + 0.883345i
9.19 −0.898794 0.438371i 0.750998 1.20185i 0.615661 + 0.788011i −0.805035 2.08613i −1.20185 + 0.750998i 2.85027 + 1.64561i −0.207912 0.978148i 0.434672 + 0.891210i −0.190936 + 2.22790i
9.20 −0.898794 0.438371i 1.25757 2.01253i 0.615661 + 0.788011i −0.556445 + 2.16573i −2.01253 + 1.25757i −1.28276 0.740599i −0.207912 0.978148i −1.15369 2.36542i 1.44952 1.70261i
See next 80 embeddings (of 1200 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 929.50 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner
25.e even 10 1 inner
475.bg even 90 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bg.a 1200
19.e even 9 1 inner 950.2.bg.a 1200
25.e even 10 1 inner 950.2.bg.a 1200
475.bg even 90 1 inner 950.2.bg.a 1200

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bg.a 1200 1.a even 1 1 trivial
950.2.bg.a 1200 19.e even 9 1 inner
950.2.bg.a 1200 25.e even 10 1 inner
950.2.bg.a 1200 475.bg even 90 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(950, [\chi])$$.