# Properties

 Label 637.2.x.a Level $637$ Weight $2$ Character orbit 637.x Analytic conductor $5.086$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

# 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: $$28$$ Relative dimension: $$7$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{2} - 6q^{4} + 6q^{5} - 12q^{6} - 4q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{2} - 6q^{4} + 6q^{5} - 12q^{6} - 4q^{8} - 12q^{9} + 12q^{10} + 2q^{11} - 8q^{12} + 10q^{15} - 2q^{16} + 6q^{17} - 4q^{18} + 8q^{19} + 36q^{20} - 8q^{22} - 6q^{23} - 12q^{24} - 24q^{26} - 8q^{29} + 38q^{31} - 20q^{32} - 18q^{33} - 12q^{34} + 54q^{36} - 16q^{37} + 28q^{39} - 48q^{40} - 18q^{41} + 48q^{43} - 6q^{44} - 12q^{45} + 18q^{46} + 42q^{47} - 12q^{48} + 10q^{50} + 12q^{51} + 28q^{52} + 12q^{53} + 30q^{54} + 6q^{55} + 12q^{57} + 62q^{58} + 6q^{59} + 16q^{60} + 36q^{62} - 2q^{65} - 66q^{66} - 4q^{67} - 30q^{68} - 42q^{69} - 42q^{71} - 38q^{72} - 14q^{73} - 6q^{74} + 20q^{75} - 52q^{76} - 62q^{78} + 4q^{79} - 12q^{80} + 12q^{81} + 108q^{82} + 66q^{83} - 54q^{85} - 30q^{86} - 42q^{87} + 30q^{89} + 72q^{90} - 156q^{92} + 14q^{93} - 6q^{95} - 18q^{96} - 62q^{97} - 36q^{99} + O(q^{100})$$

## 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 −2.38212 0.638288i 0.168862i 3.53505 + 2.04096i 2.38343 0.638637i −0.107782 + 0.402250i 0 −3.63054 3.63054i 2.97149 −6.08525
19.2 −1.56673 0.419805i 0.513926i 0.546366 + 0.315445i −1.47457 + 0.395109i −0.215749 + 0.805186i 0 1.57027 + 1.57027i 2.73588 2.47612
19.3 −0.670702 0.179714i 3.13022i −1.31451 0.758931i 0.0359277 0.00962681i −0.562545 + 2.09944i 0 1.72723 + 1.72723i −6.79826 −0.0258269
19.4 0.263976 + 0.0707322i 2.50625i −1.66737 0.962657i 1.43012 0.383199i −0.177273 + 0.661591i 0 −0.758543 0.758543i −3.28130 0.404621
19.5 0.369709 + 0.0990633i 0.914861i −1.60518 0.926751i −3.58134 + 0.959617i −0.0906291 + 0.338232i 0 −1.04293 1.04293i 2.16303 −1.41912
19.6 1.34471 + 0.360315i 1.44853i −0.0536242 0.0309600i 0.643078 0.172312i 0.521927 1.94786i 0 −2.02975 2.02975i 0.901760 0.926843
19.7 2.14116 + 0.573722i 1.10837i 2.52336 + 1.45686i 2.92938 0.784926i −0.635898 + 2.37321i 0 1.43221 + 1.43221i 1.77151 6.72261
80.1 −0.629770 2.35033i 1.97095i −3.39540 + 1.96034i −0.0608458 + 0.227080i 4.63238 1.24124i 0 3.30464 + 3.30464i −0.884626 0.572032
80.2 −0.521585 1.94658i 1.44369i −1.78508 + 1.03062i −0.849184 + 3.16920i −2.81025 + 0.753004i 0 0.0872533 + 0.0872533i 0.915773 6.61202
80.3 −0.320827 1.19734i 2.22531i 0.401352 0.231720i 0.674321 2.51660i −2.66446 + 0.713939i 0 −2.15924 2.15924i −1.95199 −3.22957
80.4 −0.127046 0.474142i 2.44579i 1.52338 0.879524i 0.931242 3.47544i 1.15965 0.310728i 0 −1.30475 1.30475i −2.98191 −1.76616
80.5 0.0745816 + 0.278342i 1.06594i 1.66014 0.958482i −0.133809 + 0.499383i 0.296695 0.0794993i 0 0.798123 + 0.798123i 1.86378 −0.148979
80.6 0.470172 + 1.75471i 3.06997i −1.12588 + 0.650030i −0.288156 + 1.07541i −5.38690 + 1.44342i 0 0.899098 + 0.899098i −6.42473 −2.02252
80.7 0.554474 + 2.06932i 0.0197323i −2.24261 + 1.29477i 0.360406 1.34505i 0.0408324 0.0109410i 0 −0.893066 0.893066i 2.99961 2.98319
215.1 −0.629770 + 2.35033i 1.97095i −3.39540 1.96034i −0.0608458 0.227080i 4.63238 + 1.24124i 0 3.30464 3.30464i −0.884626 0.572032
215.2 −0.521585 + 1.94658i 1.44369i −1.78508 1.03062i −0.849184 3.16920i −2.81025 0.753004i 0 0.0872533 0.0872533i 0.915773 6.61202
215.3 −0.320827 + 1.19734i 2.22531i 0.401352 + 0.231720i 0.674321 + 2.51660i −2.66446 0.713939i 0 −2.15924 + 2.15924i −1.95199 −3.22957
215.4 −0.127046 + 0.474142i 2.44579i 1.52338 + 0.879524i 0.931242 + 3.47544i 1.15965 + 0.310728i 0 −1.30475 + 1.30475i −2.98191 −1.76616
215.5 0.0745816 0.278342i 1.06594i 1.66014 + 0.958482i −0.133809 0.499383i 0.296695 + 0.0794993i 0 0.798123 0.798123i 1.86378 −0.148979
215.6 0.470172 1.75471i 3.06997i −1.12588 0.650030i −0.288156 1.07541i −5.38690 1.44342i 0 0.899098 0.899098i −6.42473 −2.02252
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 570.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.w 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.x.a 28
7.b odd 2 1 91.2.w.a 28
7.c even 3 1 91.2.ba.a yes 28
7.c even 3 1 637.2.bd.b 28
7.d odd 6 1 637.2.bb.a 28
7.d odd 6 1 637.2.bd.a 28
13.f odd 12 1 637.2.bb.a 28
21.c even 2 1 819.2.gh.b 28
21.h odd 6 1 819.2.et.b 28
91.w even 12 1 inner 637.2.x.a 28
91.x odd 12 1 637.2.bd.a 28
91.ba even 12 1 637.2.bd.b 28
91.bc even 12 1 91.2.ba.a yes 28
91.bd odd 12 1 91.2.w.a 28
273.bw even 12 1 819.2.gh.b 28
273.ca odd 12 1 819.2.et.b 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.w.a 28 7.b odd 2 1
91.2.w.a 28 91.bd odd 12 1
91.2.ba.a yes 28 7.c even 3 1
91.2.ba.a yes 28 91.bc even 12 1
637.2.x.a 28 1.a even 1 1 trivial
637.2.x.a 28 91.w even 12 1 inner
637.2.bb.a 28 7.d odd 6 1
637.2.bb.a 28 13.f odd 12 1
637.2.bd.a 28 7.d odd 6 1
637.2.bd.a 28 91.x odd 12 1
637.2.bd.b 28 7.c even 3 1
637.2.bd.b 28 91.ba even 12 1
819.2.et.b 28 21.h odd 6 1
819.2.et.b 28 273.ca odd 12 1
819.2.gh.b 28 21.c even 2 1
819.2.gh.b 28 273.bw even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$.