# Properties

 Label 637.2.x.b Level $637$ Weight $2$ Character orbit 637.x Analytic conductor $5.086$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{2} + 12q^{4} - 16q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 4q^{2} + 12q^{4} - 16q^{8} - 16q^{9} + 20q^{11} - 8q^{15} + 12q^{16} - 64q^{18} + 4q^{22} + 12q^{23} + 4q^{29} + 64q^{32} + 4q^{37} + 36q^{39} - 48q^{43} - 84q^{44} - 108q^{46} - 44q^{50} + 12q^{51} - 36q^{53} - 92q^{57} + 44q^{58} + 28q^{60} + 28q^{65} + 64q^{67} + 84q^{71} + 4q^{72} - 24q^{74} + 148q^{78} + 40q^{79} - 56q^{81} + 36q^{85} + 108q^{86} + 24q^{92} - 24q^{93} + 84q^{95} - 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −2.28713 0.612835i 1.20226i 3.12335 + 1.80327i 3.08075 0.825486i −0.736784 + 2.74971i 0 −2.68981 2.68981i 1.55458 −7.55197
19.2 −2.28713 0.612835i 1.20226i 3.12335 + 1.80327i −3.08075 + 0.825486i 0.736784 2.74971i 0 −2.68981 2.68981i 1.55458 7.55197
19.3 −0.813723 0.218036i 1.43324i −1.11745 0.645157i 1.38386 0.370805i −0.312498 + 1.16626i 0 1.96000 + 1.96000i 0.945832 −1.20693
19.4 −0.813723 0.218036i 1.43324i −1.11745 0.645157i −1.38386 + 0.370805i 0.312498 1.16626i 0 1.96000 + 1.96000i 0.945832 1.20693
19.5 1.33353 + 0.357317i 1.07207i −0.0814361 0.0470171i −2.77200 + 0.742756i 0.383068 1.42963i 0 −2.04421 2.04421i 1.85067 −3.96194
19.6 1.33353 + 0.357317i 1.07207i −0.0814361 0.0470171i 2.77200 0.742756i −0.383068 + 1.42963i 0 −2.04421 2.04421i 1.85067 3.96194
19.7 2.26733 + 0.607529i 2.76026i 3.03963 + 1.75493i 1.53921 0.412430i 1.67694 6.25841i 0 2.50608 + 2.50608i −4.61904 3.74046
19.8 2.26733 + 0.607529i 2.76026i 3.03963 + 1.75493i −1.53921 + 0.412430i −1.67694 + 6.25841i 0 2.50608 + 2.50608i −4.61904 −3.74046
80.1 −0.433802 1.61897i 0.637748i −0.700831 + 0.404625i 0.520288 1.94174i −1.03250 + 0.276656i 0 −1.41124 1.41124i 2.59328 −3.36932
80.2 −0.433802 1.61897i 0.637748i −0.700831 + 0.404625i −0.520288 + 1.94174i 1.03250 0.276656i 0 −1.41124 1.41124i 2.59328 3.36932
80.3 0.0302180 + 0.112775i 2.59871i 1.72025 0.993184i −0.456951 + 1.70537i 0.293070 0.0785278i 0 0.329103 + 0.329103i −3.75328 −0.206131
80.4 0.0302180 + 0.112775i 2.59871i 1.72025 0.993184i 0.456951 1.70537i −0.293070 + 0.0785278i 0 0.329103 + 0.329103i −3.75328 0.206131
80.5 0.239080 + 0.892257i 1.14107i 0.993087 0.573359i 1.02497 3.82526i 1.01813 0.272807i 0 2.05537 + 2.05537i 1.69796 3.65816
80.6 0.239080 + 0.892257i 1.14107i 0.993087 0.573359i −1.02497 + 3.82526i −1.01813 + 0.272807i 0 2.05537 + 2.05537i 1.69796 −3.65816
80.7 0.664504 + 2.47996i 2.69629i −3.97660 + 2.29589i −0.103104 + 0.384791i 6.68671 1.79170i 0 −4.70528 4.70528i −4.27000 −1.02278
80.8 0.664504 + 2.47996i 2.69629i −3.97660 + 2.29589i 0.103104 0.384791i −6.68671 + 1.79170i 0 −4.70528 4.70528i −4.27000 1.02278
215.1 −0.433802 + 1.61897i 0.637748i −0.700831 0.404625i −0.520288 1.94174i 1.03250 + 0.276656i 0 −1.41124 + 1.41124i 2.59328 3.36932
215.2 −0.433802 + 1.61897i 0.637748i −0.700831 0.404625i 0.520288 + 1.94174i −1.03250 0.276656i 0 −1.41124 + 1.41124i 2.59328 −3.36932
215.3 0.0302180 0.112775i 2.59871i 1.72025 + 0.993184i 0.456951 + 1.70537i −0.293070 0.0785278i 0 0.329103 0.329103i −3.75328 0.206131
215.4 0.0302180 0.112775i 2.59871i 1.72025 + 0.993184i −0.456951 1.70537i 0.293070 + 0.0785278i 0 0.329103 0.329103i −3.75328 −0.206131
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 570.8 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
91.w even 12 1 inner
91.bd odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.x.b 32
7.b odd 2 1 inner 637.2.x.b 32
7.c even 3 1 91.2.bc.a 32
7.c even 3 1 637.2.bb.b 32
7.d odd 6 1 91.2.bc.a 32
7.d odd 6 1 637.2.bb.b 32
13.f odd 12 1 637.2.bb.b 32
21.g even 6 1 819.2.fm.g 32
21.h odd 6 1 819.2.fm.g 32
91.w even 12 1 inner 637.2.x.b 32
91.x odd 12 1 91.2.bc.a 32
91.ba even 12 1 91.2.bc.a 32
91.bc even 12 1 637.2.bb.b 32
91.bd odd 12 1 inner 637.2.x.b 32
273.bs odd 12 1 819.2.fm.g 32
273.bv even 12 1 819.2.fm.g 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bc.a 32 7.c even 3 1
91.2.bc.a 32 7.d odd 6 1
91.2.bc.a 32 91.x odd 12 1
91.2.bc.a 32 91.ba even 12 1
637.2.x.b 32 1.a even 1 1 trivial
637.2.x.b 32 7.b odd 2 1 inner
637.2.x.b 32 91.w even 12 1 inner
637.2.x.b 32 91.bd odd 12 1 inner
637.2.bb.b 32 7.c even 3 1
637.2.bb.b 32 7.d odd 6 1
637.2.bb.b 32 13.f odd 12 1
637.2.bb.b 32 91.bc even 12 1
819.2.fm.g 32 21.g even 6 1
819.2.fm.g 32 21.h odd 6 1
819.2.fm.g 32 273.bs odd 12 1
819.2.fm.g 32 273.bv even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.