Properties

 Label 276.2.c.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q + 9q^{8} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + 9q^{8} - 2q^{9} + 4q^{10} - 7q^{12} - 4q^{13} - 12q^{14} + 4q^{16} + 13q^{18} + 14q^{20} + 2q^{22} - 22q^{23} - 30q^{24} - 18q^{25} - 27q^{26} - 12q^{27} + 6q^{28} + 34q^{30} + 20q^{32} - 8q^{33} - 6q^{34} + 8q^{35} - 36q^{36} - 4q^{37} - 22q^{38} + 24q^{39} - 4q^{40} + 26q^{42} + 56q^{44} - 8q^{47} - 22q^{48} - 14q^{49} - 20q^{50} - 16q^{51} - 19q^{52} + 22q^{54} + 18q^{56} + 12q^{57} + 3q^{58} + 72q^{59} - 28q^{60} + 12q^{61} - 63q^{62} + 20q^{63} + 3q^{64} + 60q^{66} + 20q^{68} - 40q^{71} - 36q^{72} - 4q^{73} - 28q^{74} - 48q^{75} + 26q^{76} + 11q^{78} + 84q^{80} + 10q^{81} - 29q^{82} + 8q^{83} - 38q^{84} + 8q^{85} - 28q^{86} + 48q^{87} - 30q^{88} + 84q^{90} + 12q^{93} - 13q^{94} - 32q^{95} - 45q^{96} - 4q^{97} - 64q^{98} - 20q^{99} + O(q^{100})$$

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 −1.33491 0.466911i −1.38398 1.04144i 1.56399 + 1.24657i 0.588622i 1.36124 + 2.03643i 0.538069i −1.50575 2.39431i 0.830813 + 2.88266i 0.274834 0.785760i
47.2 −1.33491 + 0.466911i −1.38398 + 1.04144i 1.56399 1.24657i 0.588622i 1.36124 2.03643i 0.538069i −1.50575 + 2.39431i 0.830813 2.88266i 0.274834 + 0.785760i
47.3 −1.27564 0.610520i 0.436111 1.67625i 1.25453 + 1.55761i 3.87509i −1.57971 + 1.87204i 0.468259i −0.649379 2.75287i −2.61961 1.46206i −2.36582 + 4.94323i
47.4 −1.27564 + 0.610520i 0.436111 + 1.67625i 1.25453 1.55761i 3.87509i −1.57971 1.87204i 0.468259i −0.649379 + 2.75287i −2.61961 + 1.46206i −2.36582 4.94323i
47.5 −1.04620 0.951558i 1.08652 + 1.34888i 0.189073 + 1.99104i 0.289949i 0.146820 2.44509i 1.62347i 1.69678 2.26295i −0.638952 + 2.93117i −0.275904 + 0.303345i
47.6 −1.04620 + 0.951558i 1.08652 1.34888i 0.189073 1.99104i 0.289949i 0.146820 + 2.44509i 1.62347i 1.69678 + 2.26295i −0.638952 2.93117i −0.275904 0.303345i
47.7 −0.725653 1.21385i −0.173813 1.72331i −0.946854 + 1.76167i 1.59003i −1.96571 + 1.46151i 4.71351i 2.82548 0.129022i −2.93958 + 0.599066i 1.93005 1.15381i
47.8 −0.725653 + 1.21385i −0.173813 + 1.72331i −0.946854 1.76167i 1.59003i −1.96571 1.46151i 4.71351i 2.82548 + 0.129022i −2.93958 0.599066i 1.93005 + 1.15381i
47.9 −0.456308 1.33857i −0.784130 + 1.54439i −1.58357 + 1.22161i 1.42397i 2.42509 + 0.344899i 1.12562i 2.35780 + 1.56229i −1.77028 2.42200i −1.90608 + 0.649767i
47.10 −0.456308 + 1.33857i −0.784130 1.54439i −1.58357 1.22161i 1.42397i 2.42509 0.344899i 1.12562i 2.35780 1.56229i −1.77028 + 2.42200i −1.90608 0.649767i
47.11 −0.133777 1.40787i 1.73013 + 0.0815319i −1.96421 + 0.376681i 3.66772i −0.116665 2.44671i 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 −0.116665 + 2.44671i 2.47568i 0.793084 2.71496i 2.98671 0.282122i 5.16368 + 0.490656i
47.13 0.469504 1.33400i −1.64481 + 0.542779i −1.55913 1.25264i 2.34886i −0.0481753 + 2.44902i 0.796368i −2.40305 + 1.49177i 2.41078 1.78553i −3.13338 1.10280i
47.14 0.469504 + 1.33400i −1.64481 0.542779i −1.55913 + 1.25264i 2.34886i −0.0481753 2.44902i 0.796368i −2.40305 1.49177i 2.41078 + 1.78553i −3.13338 + 1.10280i
47.15 0.565709 1.29614i 1.40231 + 1.01662i −1.35995 1.46647i 0.662888i 2.11098 1.24248i 4.60297i −2.67009 + 0.933083i 0.932961 + 2.85124i −0.859194 0.375001i
47.16 0.565709 + 1.29614i 1.40231 1.01662i −1.35995 + 1.46647i 0.662888i 2.11098 + 1.24248i 4.60297i −2.67009 0.933083i 0.932961 2.85124i −0.859194 + 0.375001i
47.17 1.15961 0.809507i 1.20859 1.24069i 0.689397 1.87743i 3.47795i 0.397138 2.41708i 0.968466i −0.720358 2.73516i −0.0786444 2.99897i 2.81542 + 4.03307i
47.18 1.15961 + 0.809507i 1.20859 + 1.24069i 0.689397 + 1.87743i 3.47795i 0.397138 + 2.41708i 0.968466i −0.720358 + 2.73516i −0.0786444 + 2.99897i 2.81542 4.03307i
47.19 1.37110 0.346521i −1.70706 + 0.293173i 1.75985 0.950233i 2.20059i −2.23896 + 0.993503i 3.71987i 2.08365 1.91269i 2.82810 1.00093i 0.762550 + 3.01723i
47.20 1.37110 + 0.346521i −1.70706 0.293173i 1.75985 + 0.950233i 2.20059i −2.23896 0.993503i 3.71987i 2.08365 + 1.91269i 2.82810 + 1.00093i 0.762550 3.01723i
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.b yes 22
3.b odd 2 1 276.2.c.a 22
4.b odd 2 1 276.2.c.a 22
12.b even 2 1 inner 276.2.c.b yes 22

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

Hecke kernels

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