# Properties

 Label 276.2.e.a Level $276$ Weight $2$ Character orbit 276.e Analytic conductor $2.204$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$276 = 2^{2} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 276.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.20387109579$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} + 4q^{6} + 4q^{8} - 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 4q^{2} + 4q^{6} + 4q^{8} - 24q^{9} + 8q^{16} - 4q^{18} - 4q^{24} - 24q^{25} + 40q^{26} - 32q^{29} - 36q^{32} + 16q^{41} - 32q^{48} + 40q^{49} - 12q^{50} - 40q^{52} - 4q^{54} + 24q^{58} - 40q^{62} + 48q^{64} + 16q^{69} + 72q^{70} - 4q^{72} + 16q^{77} + 24q^{81} - 40q^{82} - 64q^{85} + 44q^{92} + 16q^{93} + 72q^{94} + 44q^{96} - 52q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1 −1.36293 0.377393i 1.00000i 1.71515 + 1.02872i 2.63738i 0.377393 1.36293i −3.76288 −1.94939 2.04936i −1.00000 −0.995329 + 3.59456i
91.2 −1.36293 0.377393i 1.00000i 1.71515 + 1.02872i 2.63738i 0.377393 1.36293i 3.76288 −1.94939 2.04936i −1.00000 0.995329 3.59456i
91.3 −1.36293 + 0.377393i 1.00000i 1.71515 1.02872i 2.63738i 0.377393 + 1.36293i 3.76288 −1.94939 + 2.04936i −1.00000 0.995329 + 3.59456i
91.4 −1.36293 + 0.377393i 1.00000i 1.71515 1.02872i 2.63738i 0.377393 + 1.36293i −3.76288 −1.94939 + 2.04936i −1.00000 −0.995329 3.59456i
91.5 −0.588134 1.28612i 1.00000i −1.30820 + 1.51282i 2.28672i −1.28612 + 0.588134i −3.41490 2.71506 + 0.792755i −1.00000 −2.94099 + 1.34490i
91.6 −0.588134 1.28612i 1.00000i −1.30820 + 1.51282i 2.28672i −1.28612 + 0.588134i 3.41490 2.71506 + 0.792755i −1.00000 2.94099 1.34490i
91.7 −0.588134 + 1.28612i 1.00000i −1.30820 1.51282i 2.28672i −1.28612 0.588134i 3.41490 2.71506 0.792755i −1.00000 2.94099 + 1.34490i
91.8 −0.588134 + 1.28612i 1.00000i −1.30820 1.51282i 2.28672i −1.28612 0.588134i −3.41490 2.71506 0.792755i −1.00000 −2.94099 1.34490i
91.9 −0.279557 1.38631i 1.00000i −1.84370 + 0.775104i 1.27568i 1.38631 0.279557i 2.49131 1.58995 + 2.33924i −1.00000 −1.76848 + 0.356624i
91.10 −0.279557 1.38631i 1.00000i −1.84370 + 0.775104i 1.27568i 1.38631 0.279557i −2.49131 1.58995 + 2.33924i −1.00000 1.76848 0.356624i
91.11 −0.279557 + 1.38631i 1.00000i −1.84370 0.775104i 1.27568i 1.38631 + 0.279557i −2.49131 1.58995 2.33924i −1.00000 1.76848 + 0.356624i
91.12 −0.279557 + 1.38631i 1.00000i −1.84370 0.775104i 1.27568i 1.38631 + 0.279557i 2.49131 1.58995 2.33924i −1.00000 −1.76848 0.356624i
91.13 0.714279 1.22058i 1.00000i −0.979610 1.74366i 3.78153i 1.22058 + 0.714279i −1.02234 −2.82799 0.0497743i −1.00000 −4.61564 2.70107i
91.14 0.714279 1.22058i 1.00000i −0.979610 1.74366i 3.78153i 1.22058 + 0.714279i 1.02234 −2.82799 0.0497743i −1.00000 4.61564 + 2.70107i
91.15 0.714279 + 1.22058i 1.00000i −0.979610 + 1.74366i 3.78153i 1.22058 0.714279i 1.02234 −2.82799 + 0.0497743i −1.00000 4.61564 2.70107i
91.16 0.714279 + 1.22058i 1.00000i −0.979610 + 1.74366i 3.78153i 1.22058 0.714279i −1.02234 −2.82799 + 0.0497743i −1.00000 −4.61564 + 2.70107i
91.17 1.11292 0.872582i 1.00000i 0.477201 1.94224i 0.970352i −0.872582 1.11292i −4.31859 −1.16367 2.57796i −1.00000 −0.846712 1.07993i
91.18 1.11292 0.872582i 1.00000i 0.477201 1.94224i 0.970352i −0.872582 1.11292i 4.31859 −1.16367 2.57796i −1.00000 0.846712 + 1.07993i
91.19 1.11292 + 0.872582i 1.00000i 0.477201 + 1.94224i 0.970352i −0.872582 + 1.11292i 4.31859 −1.16367 + 2.57796i −1.00000 0.846712 1.07993i
91.20 1.11292 + 0.872582i 1.00000i 0.477201 + 1.94224i 0.970352i −0.872582 + 1.11292i −4.31859 −1.16367 + 2.57796i −1.00000 −0.846712 + 1.07993i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.e.a 24
3.b odd 2 1 828.2.e.f 24
4.b odd 2 1 inner 276.2.e.a 24
8.b even 2 1 4416.2.i.d 24
8.d odd 2 1 4416.2.i.d 24
12.b even 2 1 828.2.e.f 24
23.b odd 2 1 inner 276.2.e.a 24
69.c even 2 1 828.2.e.f 24
92.b even 2 1 inner 276.2.e.a 24
184.e odd 2 1 4416.2.i.d 24
184.h even 2 1 4416.2.i.d 24
276.h odd 2 1 828.2.e.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.e.a 24 1.a even 1 1 trivial
276.2.e.a 24 4.b odd 2 1 inner
276.2.e.a 24 23.b odd 2 1 inner
276.2.e.a 24 92.b even 2 1 inner
828.2.e.f 24 3.b odd 2 1
828.2.e.f 24 12.b even 2 1
828.2.e.f 24 69.c even 2 1
828.2.e.f 24 276.h odd 2 1
4416.2.i.d 24 8.b even 2 1
4416.2.i.d 24 8.d odd 2 1
4416.2.i.d 24 184.e odd 2 1
4416.2.i.d 24 184.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(276, [\chi])$$.