Properties

 Label 637.2.bc.a Level $637$ Weight $2$ Character orbit 637.bc Analytic conductor $5.086$ Analytic rank $0$ Dimension $24$ CM no Inner twists $8$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.bc (of order $$12$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$6$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} + 8 q^{8} + 4 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24 q + 4 q^{2} + 8 q^{8} + 4 q^{9} + 8 q^{11} - 8 q^{15} - 16 q^{16} + 8 q^{18} - 32 q^{22} - 8 q^{29} + 16 q^{32} - 12 q^{37} - 40 q^{39} - 12 q^{44} - 24 q^{46} - 56 q^{50} + 12 q^{53} - 16 q^{57} + 44 q^{58} - 44 q^{60} + 40 q^{65} - 60 q^{67} + 28 q^{72} + 48 q^{74} + 88 q^{78} + 4 q^{79} + 92 q^{81} + 24 q^{85} - 36 q^{86} + 48 q^{92} + 28 q^{93} + 24 q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.16746 0.312819i −1.96915 1.13689i −0.466951 0.269594i −0.224373 + 0.837372i 1.94326 + 1.94326i 0 2.17009 + 2.17009i 1.08504 + 1.87935i 0.523892 0.907408i
31.2 −1.16746 0.312819i 1.96915 + 1.13689i −0.466951 0.269594i 0.224373 0.837372i −1.94326 1.94326i 0 2.17009 + 2.17009i 1.08504 + 1.87935i −0.523892 + 0.907408i
31.3 0.550552 + 0.147520i −1.06729 0.616199i −1.45071 0.837565i −0.377817 + 1.41003i −0.496696 0.496696i 0 −1.48119 1.48119i −0.740597 1.28275i −0.416015 + 0.720559i
31.4 0.550552 + 0.147520i 1.06729 + 0.616199i −1.45071 0.837565i 0.377817 1.41003i 0.496696 + 0.496696i 0 −1.48119 1.48119i −0.740597 1.28275i 0.416015 0.720559i
31.5 1.98293 + 0.531325i −1.57586 0.909822i 1.91766 + 1.10716i 0.737409 2.75205i −2.64141 2.64141i 0 0.311108 + 0.311108i 0.155554 + 0.269427i 2.92446 5.06531i
31.6 1.98293 + 0.531325i 1.57586 + 0.909822i 1.91766 + 1.10716i −0.737409 + 2.75205i 2.64141 + 2.64141i 0 0.311108 + 0.311108i 0.155554 + 0.269427i −2.92446 + 5.06531i
411.1 −1.16746 + 0.312819i −1.96915 + 1.13689i −0.466951 + 0.269594i −0.224373 0.837372i 1.94326 1.94326i 0 2.17009 2.17009i 1.08504 1.87935i 0.523892 + 0.907408i
411.2 −1.16746 + 0.312819i 1.96915 1.13689i −0.466951 + 0.269594i 0.224373 + 0.837372i −1.94326 + 1.94326i 0 2.17009 2.17009i 1.08504 1.87935i −0.523892 0.907408i
411.3 0.550552 0.147520i −1.06729 + 0.616199i −1.45071 + 0.837565i −0.377817 1.41003i −0.496696 + 0.496696i 0 −1.48119 + 1.48119i −0.740597 + 1.28275i −0.416015 0.720559i
411.4 0.550552 0.147520i 1.06729 0.616199i −1.45071 + 0.837565i 0.377817 + 1.41003i 0.496696 0.496696i 0 −1.48119 + 1.48119i −0.740597 + 1.28275i 0.416015 + 0.720559i
411.5 1.98293 0.531325i −1.57586 + 0.909822i 1.91766 1.10716i 0.737409 + 2.75205i −2.64141 + 2.64141i 0 0.311108 0.311108i 0.155554 0.269427i 2.92446 + 5.06531i
411.6 1.98293 0.531325i 1.57586 0.909822i 1.91766 1.10716i −0.737409 2.75205i 2.64141 2.64141i 0 0.311108 0.311108i 0.155554 0.269427i −2.92446 5.06531i
460.1 −0.531325 1.98293i −1.57586 + 0.909822i −1.91766 + 1.10716i 2.75205 0.737409i 2.64141 + 2.64141i 0 0.311108 + 0.311108i 0.155554 0.269427i −2.92446 5.06531i
460.2 −0.531325 1.98293i 1.57586 0.909822i −1.91766 + 1.10716i −2.75205 + 0.737409i −2.64141 2.64141i 0 0.311108 + 0.311108i 0.155554 0.269427i 2.92446 + 5.06531i
460.3 −0.147520 0.550552i −1.06729 + 0.616199i 1.45071 0.837565i −1.41003 + 0.377817i 0.496696 + 0.496696i 0 −1.48119 1.48119i −0.740597 + 1.28275i 0.416015 + 0.720559i
460.4 −0.147520 0.550552i 1.06729 0.616199i 1.45071 0.837565i 1.41003 0.377817i −0.496696 0.496696i 0 −1.48119 1.48119i −0.740597 + 1.28275i −0.416015 0.720559i
460.5 0.312819 + 1.16746i −1.96915 + 1.13689i 0.466951 0.269594i −0.837372 + 0.224373i −1.94326 1.94326i 0 2.17009 + 2.17009i 1.08504 1.87935i −0.523892 0.907408i
460.6 0.312819 + 1.16746i 1.96915 1.13689i 0.466951 0.269594i 0.837372 0.224373i 1.94326 + 1.94326i 0 2.17009 + 2.17009i 1.08504 1.87935i 0.523892 + 0.907408i
619.1 −0.531325 + 1.98293i −1.57586 0.909822i −1.91766 1.10716i 2.75205 + 0.737409i 2.64141 2.64141i 0 0.311108 0.311108i 0.155554 + 0.269427i −2.92446 + 5.06531i
619.2 −0.531325 + 1.98293i 1.57586 + 0.909822i −1.91766 1.10716i −2.75205 0.737409i −2.64141 + 2.64141i 0 0.311108 0.311108i 0.155554 + 0.269427i 2.92446 5.06531i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 619.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
13.d odd 4 1 inner
91.i even 4 1 inner
91.z odd 12 1 inner
91.bb even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bc.a 24
7.b odd 2 1 inner 637.2.bc.a 24
7.c even 3 1 91.2.i.a 12
7.c even 3 1 inner 637.2.bc.a 24
7.d odd 6 1 91.2.i.a 12
7.d odd 6 1 inner 637.2.bc.a 24
13.d odd 4 1 inner 637.2.bc.a 24
21.g even 6 1 819.2.y.h 12
21.h odd 6 1 819.2.y.h 12
91.i even 4 1 inner 637.2.bc.a 24
91.z odd 12 1 91.2.i.a 12
91.z odd 12 1 inner 637.2.bc.a 24
91.bb even 12 1 91.2.i.a 12
91.bb even 12 1 inner 637.2.bc.a 24
273.cb odd 12 1 819.2.y.h 12
273.cd even 12 1 819.2.y.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.i.a 12 7.c even 3 1
91.2.i.a 12 7.d odd 6 1
91.2.i.a 12 91.z odd 12 1
91.2.i.a 12 91.bb even 12 1
637.2.bc.a 24 1.a even 1 1 trivial
637.2.bc.a 24 7.b odd 2 1 inner
637.2.bc.a 24 7.c even 3 1 inner
637.2.bc.a 24 7.d odd 6 1 inner
637.2.bc.a 24 13.d odd 4 1 inner
637.2.bc.a 24 91.i even 4 1 inner
637.2.bc.a 24 91.z odd 12 1 inner
637.2.bc.a 24 91.bb even 12 1 inner
819.2.y.h 12 21.g even 6 1
819.2.y.h 12 21.h odd 6 1
819.2.y.h 12 273.cb odd 12 1
819.2.y.h 12 273.cd even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.