# Properties

 Label 276.2.c.a Level $276$ Weight $2$ Character orbit 276.c Analytic conductor $2.204$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.20387109579$$ Analytic rank: $$0$$ Dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q - 3q^{6} - 9q^{8} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q - 3q^{6} - 9q^{8} - 2q^{9} + 4q^{10} + 14q^{12} - 4q^{13} + 12q^{14} + 4q^{16} - 14q^{18} - 14q^{20} + 2q^{22} + 22q^{23} + 22q^{24} - 18q^{25} + 27q^{26} + 12q^{27} + 6q^{28} - 24q^{30} - 20q^{32} - 8q^{33} - 6q^{34} - 8q^{35} + 3q^{36} - 4q^{37} + 22q^{38} - 24q^{39} - 4q^{40} - 38q^{42} - 56q^{44} + 8q^{47} + 17q^{48} - 14q^{49} + 20q^{50} + 16q^{51} - 19q^{52} - 54q^{54} - 18q^{56} + 12q^{57} + 3q^{58} - 72q^{59} + 64q^{60} + 12q^{61} + 63q^{62} - 20q^{63} + 3q^{64} - 18q^{66} - 20q^{68} + 40q^{71} + 48q^{72} - 4q^{73} + 28q^{74} + 48q^{75} + 26q^{76} - 46q^{78} - 84q^{80} + 10q^{81} - 29q^{82} - 8q^{83} + 76q^{84} + 8q^{85} + 28q^{86} - 48q^{87} - 30q^{88} - 26q^{90} + 12q^{93} - 13q^{94} + 32q^{95} + 18q^{96} - 4q^{97} + 64q^{98} + 20q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1 −1.40657 0.146847i 0.169868 + 1.72370i 1.95687 + 0.413102i 2.76583i 0.0141905 2.44945i 3.82694i −2.69181 0.868418i −2.94229 + 0.585602i −0.406156 + 3.89033i
47.2 −1.40657 + 0.146847i 0.169868 1.72370i 1.95687 0.413102i 2.76583i 0.0141905 + 2.44945i 3.82694i −2.69181 + 0.868418i −2.94229 0.585602i −0.406156 3.89033i
47.3 −1.37110 0.346521i 1.70706 0.293173i 1.75985 + 0.950233i 2.20059i −2.44214 0.189562i 3.71987i −2.08365 1.91269i 2.82810 1.00093i 0.762550 3.01723i
47.4 −1.37110 + 0.346521i 1.70706 + 0.293173i 1.75985 0.950233i 2.20059i −2.44214 + 0.189562i 3.71987i −2.08365 + 1.91269i 2.82810 + 1.00093i 0.762550 + 3.01723i
47.5 −1.15961 0.809507i −1.20859 + 1.24069i 0.689397 + 1.87743i 3.47795i 2.40584 0.460365i 0.968466i 0.720358 2.73516i −0.0786444 2.99897i 2.81542 4.03307i
47.6 −1.15961 + 0.809507i −1.20859 1.24069i 0.689397 1.87743i 3.47795i 2.40584 + 0.460365i 0.968466i 0.720358 + 2.73516i −0.0786444 + 2.99897i 2.81542 + 4.03307i
47.7 −0.565709 1.29614i −1.40231 1.01662i −1.35995 + 1.46647i 0.662888i −0.524382 + 2.39270i 4.60297i 2.67009 + 0.933083i 0.932961 + 2.85124i −0.859194 + 0.375001i
47.8 −0.565709 + 1.29614i −1.40231 + 1.01662i −1.35995 1.46647i 0.662888i −0.524382 2.39270i 4.60297i 2.67009 0.933083i 0.932961 2.85124i −0.859194 0.375001i
47.9 −0.469504 1.33400i 1.64481 0.542779i −1.55913 + 1.25264i 2.34886i −1.49631 1.93934i 0.796368i 2.40305 + 1.49177i 2.41078 1.78553i −3.13338 + 1.10280i
47.10 −0.469504 + 1.33400i 1.64481 + 0.542779i −1.55913 1.25264i 2.34886i −1.49631 + 1.93934i 0.796368i 2.40305 1.49177i 2.41078 + 1.78553i −3.13338 1.10280i
47.11 0.133777 1.40787i −1.73013 0.0815319i −1.96421 0.376681i 3.66772i −0.346238 + 2.42490i 2.47568i −0.793084 + 2.71496i 2.98671 + 0.282122i 5.16368 + 0.490656i
47.12 0.133777 + 1.40787i −1.73013 + 0.0815319i −1.96421 + 0.376681i 3.66772i −0.346238 2.42490i 2.47568i −0.793084 2.71496i 2.98671 0.282122i 5.16368 0.490656i
47.13 0.456308 1.33857i 0.784130 1.54439i −1.58357 1.22161i 1.42397i −1.70948 1.75433i 1.12562i −2.35780 + 1.56229i −1.77028 2.42200i −1.90608 0.649767i
47.14 0.456308 + 1.33857i 0.784130 + 1.54439i −1.58357 + 1.22161i 1.42397i −1.70948 + 1.75433i 1.12562i −2.35780 1.56229i −1.77028 + 2.42200i −1.90608 + 0.649767i
47.15 0.725653 1.21385i 0.173813 + 1.72331i −0.946854 1.76167i 1.59003i 2.21796 + 1.03954i 4.71351i −2.82548 0.129022i −2.93958 + 0.599066i 1.93005 + 1.15381i
47.16 0.725653 + 1.21385i 0.173813 1.72331i −0.946854 + 1.76167i 1.59003i 2.21796 1.03954i 4.71351i −2.82548 + 0.129022i −2.93958 0.599066i 1.93005 1.15381i
47.17 1.04620 0.951558i −1.08652 1.34888i 0.189073 1.99104i 0.289949i −2.42026 0.377313i 1.62347i −1.69678 2.26295i −0.638952 + 2.93117i −0.275904 0.303345i
47.18 1.04620 + 0.951558i −1.08652 + 1.34888i 0.189073 + 1.99104i 0.289949i −2.42026 + 0.377313i 1.62347i −1.69678 + 2.26295i −0.638952 2.93117i −0.275904 + 0.303345i
47.19 1.27564 0.610520i −0.436111 + 1.67625i 1.25453 1.55761i 3.87509i 0.467061 + 2.40455i 0.468259i 0.649379 2.75287i −2.61961 1.46206i −2.36582 4.94323i
47.20 1.27564 + 0.610520i −0.436111 1.67625i 1.25453 + 1.55761i 3.87509i 0.467061 2.40455i 0.468259i 0.649379 + 2.75287i −2.61961 + 1.46206i −2.36582 + 4.94323i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.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.c.a 22
3.b odd 2 1 276.2.c.b yes 22
4.b odd 2 1 276.2.c.b yes 22
12.b even 2 1 inner 276.2.c.a 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.c.a 22 1.a even 1 1 trivial
276.2.c.a 22 12.b even 2 1 inner
276.2.c.b yes 22 3.b odd 2 1
276.2.c.b yes 22 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{11} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(276, [\chi])$$.