# Properties

 Label 637.2.bd.b Level $637$ Weight $2$ Character orbit 637.bd 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.bd (of order $$12$$, degree $$4$$, 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q + 4q^{2} + 6q^{3} + 6q^{4} - 12q^{6} - 4q^{8} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q + 4q^{2} + 6q^{3} + 6q^{4} - 12q^{6} - 4q^{8} + 6q^{9} - 6q^{10} - 4q^{11} + 16q^{12} + 10q^{15} - 2q^{16} + 6q^{17} + 2q^{18} - 22q^{19} + 36q^{20} - 8q^{22} + 6q^{23} + 30q^{24} - 8q^{29} + 30q^{30} - 34q^{31} + 10q^{32} + 30q^{33} - 12q^{34} + 54q^{36} + 26q^{37} - 8q^{39} - 18q^{41} + 48q^{43} + 12q^{44} + 18q^{45} - 42q^{46} - 36q^{47} - 12q^{48} + 10q^{50} - 2q^{52} - 24q^{53} + 6q^{55} + 12q^{57} - 16q^{58} - 48q^{59} - 26q^{60} - 30q^{61} + 36q^{62} - 26q^{65} + 14q^{67} + 30q^{68} - 42q^{69} - 42q^{71} - 8q^{72} - 26q^{73} - 6q^{74} + 20q^{75} - 52q^{76} - 62q^{78} - 8q^{79} - 18q^{80} - 6q^{81} - 54q^{82} + 66q^{83} - 54q^{85} + 48q^{86} + 42q^{87} + 6q^{88} - 30q^{89} + 72q^{90} - 156q^{92} - 34q^{93} + 18q^{94} + 6q^{95} + 84q^{96} - 62q^{97} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1 −0.573722 2.14116i −0.959879 0.554186i −2.52336 + 1.45686i −2.14446 2.14446i −0.635898 + 2.37321i 0 1.43221 + 1.43221i −0.885755 1.53417i −3.36131 + 5.82195i
97.2 −0.360315 1.34471i 1.25446 + 0.724265i 0.0536242 0.0309600i −0.470766 0.470766i 0.521927 1.94786i 0 −2.02975 2.02975i −0.450880 0.780947i −0.463421 + 0.802669i
97.3 −0.0990633 0.369709i −0.792292 0.457430i 1.60518 0.926751i 2.62172 + 2.62172i −0.0906291 + 0.338232i 0 −1.04293 1.04293i −1.08152 1.87324i 0.709559 1.22899i
97.4 −0.0707322 0.263976i −2.17048 1.25313i 1.66737 0.962657i −1.04692 1.04692i −0.177273 + 0.661591i 0 −0.758543 0.758543i 1.64065 + 2.84169i −0.202311 + 0.350412i
97.5 0.179714 + 0.670702i 2.71085 + 1.56511i 1.31451 0.758931i −0.0263009 0.0263009i −0.562545 + 2.09944i 0 1.72723 + 1.72723i 3.39913 + 5.88747i 0.0129135 0.0223668i
97.6 0.419805 + 1.56673i 0.445073 + 0.256963i −0.546366 + 0.315445i 1.07946 + 1.07946i −0.215749 + 0.805186i 0 1.57027 + 1.57027i −1.36794 2.36934i −1.23806 + 2.14439i
97.7 0.638288 + 2.38212i 0.146239 + 0.0844309i −3.53505 + 2.04096i −1.74479 1.74479i −0.107782 + 0.402250i 0 −3.63054 3.63054i −1.48574 2.57338i 3.04263 5.26998i
293.1 −2.06932 + 0.554474i −0.0170886 0.00986613i 2.24261 1.29477i 0.984647 0.984647i 0.0408324 + 0.0109410i 0 −0.893066 + 0.893066i −1.49981 2.59774i −1.49159 + 2.58351i
293.2 −1.75471 + 0.470172i 2.65867 + 1.53499i 1.12588 0.650030i −0.787258 + 0.787258i −5.38690 1.44342i 0 0.899098 0.899098i 3.21237 + 5.56398i 1.01126 1.75155i
293.3 −0.278342 + 0.0745816i −0.923129 0.532969i −1.66014 + 0.958482i −0.365574 + 0.365574i 0.296695 + 0.0794993i 0 0.798123 0.798123i −0.931889 1.61408i 0.0744896 0.129020i
293.4 0.474142 0.127046i 2.11812 + 1.22290i −1.52338 + 0.879524i 2.54420 2.54420i 1.15965 + 0.310728i 0 −1.30475 + 1.30475i 1.49096 + 2.58241i 0.883082 1.52954i
293.5 1.19734 0.320827i −1.92717 1.11265i −0.401352 + 0.231720i 1.84228 1.84228i −2.66446 0.713939i 0 −2.15924 + 2.15924i 0.975997 + 1.69048i 1.61479 2.79689i
293.6 1.94658 0.521585i −1.25027 0.721843i 1.78508 1.03062i −2.32001 + 2.32001i −2.81025 0.753004i 0 0.0872533 0.0872533i −0.457887 0.793083i −3.30601 + 5.72618i
293.7 2.35033 0.629770i 1.70689 + 0.985473i 3.39540 1.96034i −0.166234 + 0.166234i 4.63238 + 1.24124i 0 3.30464 3.30464i 0.442313 + 0.766108i −0.286016 + 0.495394i
440.1 −0.573722 + 2.14116i −0.959879 + 0.554186i −2.52336 1.45686i −2.14446 + 2.14446i −0.635898 2.37321i 0 1.43221 1.43221i −0.885755 + 1.53417i −3.36131 5.82195i
440.2 −0.360315 + 1.34471i 1.25446 0.724265i 0.0536242 + 0.0309600i −0.470766 + 0.470766i 0.521927 + 1.94786i 0 −2.02975 + 2.02975i −0.450880 + 0.780947i −0.463421 0.802669i
440.3 −0.0990633 + 0.369709i −0.792292 + 0.457430i 1.60518 + 0.926751i 2.62172 2.62172i −0.0906291 0.338232i 0 −1.04293 + 1.04293i −1.08152 + 1.87324i 0.709559 + 1.22899i
440.4 −0.0707322 + 0.263976i −2.17048 + 1.25313i 1.66737 + 0.962657i −1.04692 + 1.04692i −0.177273 0.661591i 0 −0.758543 + 0.758543i 1.64065 2.84169i −0.202311 0.350412i
440.5 0.179714 0.670702i 2.71085 1.56511i 1.31451 + 0.758931i −0.0263009 + 0.0263009i −0.562545 2.09944i 0 1.72723 1.72723i 3.39913 5.88747i 0.0129135 + 0.0223668i
440.6 0.419805 1.56673i 0.445073 0.256963i −0.546366 0.315445i 1.07946 1.07946i −0.215749 0.805186i 0 1.57027 1.57027i −1.36794 + 2.36934i −1.23806 2.14439i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 587.7 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.bc 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.bd.b 28
7.b odd 2 1 637.2.bd.a 28
7.c even 3 1 91.2.ba.a yes 28
7.c even 3 1 637.2.x.a 28
7.d odd 6 1 91.2.w.a 28
7.d odd 6 1 637.2.bb.a 28
13.f odd 12 1 637.2.bd.a 28
21.g even 6 1 819.2.gh.b 28
21.h odd 6 1 819.2.et.b 28
91.w even 12 1 91.2.ba.a yes 28
91.x odd 12 1 91.2.w.a 28
91.ba even 12 1 637.2.x.a 28
91.bc even 12 1 inner 637.2.bd.b 28
91.bd odd 12 1 637.2.bb.a 28
273.bv even 12 1 819.2.gh.b 28
273.ch 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.d odd 6 1
91.2.w.a 28 91.x odd 12 1
91.2.ba.a yes 28 7.c even 3 1
91.2.ba.a yes 28 91.w even 12 1
637.2.x.a 28 7.c even 3 1
637.2.x.a 28 91.ba even 12 1
637.2.bb.a 28 7.d odd 6 1
637.2.bb.a 28 91.bd odd 12 1
637.2.bd.a 28 7.b odd 2 1
637.2.bd.a 28 13.f odd 12 1
637.2.bd.b 28 1.a even 1 1 trivial
637.2.bd.b 28 91.bc even 12 1 inner
819.2.et.b 28 21.h odd 6 1
819.2.et.b 28 273.ch odd 12 1
819.2.gh.b 28 21.g even 6 1
819.2.gh.b 28 273.bv even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{28} - \cdots$$ $$T_{3}^{28} - \cdots$$