# Properties

 Label 637.2.bb.a Level $637$ Weight $2$ Character orbit 637.bb 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.bb (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.

 $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{2} + 6q^{3} + 6q^{5} + 12q^{6} - 4q^{8} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 2q^{2} + 6q^{3} + 6q^{5} + 12q^{6} - 4q^{8} + 6q^{9} + 6q^{10} + 2q^{11} + 8q^{12} + 10q^{15} + 4q^{16} + 12q^{17} + 2q^{18} - 14q^{19} - 36q^{20} - 8q^{22} + 18q^{24} - 24q^{26} - 8q^{29} - 30q^{30} + 4q^{31} + 10q^{32} + 12q^{33} + 12q^{34} + 54q^{36} - 10q^{37} - 20q^{39} - 48q^{40} + 18q^{41} + 48q^{43} - 6q^{44} + 6q^{45} + 24q^{46} + 6q^{47} + 12q^{48} + 10q^{50} - 12q^{51} + 26q^{52} + 12q^{53} + 30q^{54} - 6q^{55} + 12q^{57} - 46q^{58} - 42q^{59} + 10q^{60} - 30q^{61} - 36q^{62} + 28q^{65} - 66q^{66} - 10q^{67} + 42q^{69} - 42q^{71} + 46q^{72} - 40q^{73} + 12q^{74} + 40q^{75} + 52q^{76} - 62q^{78} + 4q^{79} - 30q^{80} - 6q^{81} + 54q^{82} - 66q^{83} - 54q^{85} - 18q^{86} - 6q^{88} - 72q^{90} - 156q^{92} + 20q^{93} + 18q^{94} + 66q^{96} + 62q^{97} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
227.1 −1.56744 + 1.56744i −0.959879 + 0.554186i 2.91373i 0.784926 2.92938i 0.635898 2.37321i 0 1.43221 + 1.43221i −0.885755 + 1.53417i 3.36131 + 5.82195i
227.2 −0.984398 + 0.984398i 1.25446 0.724265i 0.0619199i 0.172312 0.643078i −0.521927 + 1.94786i 0 −2.02975 2.02975i −0.450880 + 0.780947i 0.463421 + 0.802669i
227.3 −0.270646 + 0.270646i −0.792292 + 0.457430i 1.85350i −0.959617 + 3.58134i 0.0906291 0.338232i 0 −1.04293 1.04293i −1.08152 + 1.87324i −0.709559 1.22899i
227.4 −0.193244 + 0.193244i −2.17048 + 1.25313i 1.92531i 0.383199 1.43012i 0.177273 0.661591i 0 −0.758543 0.758543i 1.64065 2.84169i 0.202311 + 0.350412i
227.5 0.490988 0.490988i 2.71085 1.56511i 1.51786i 0.00962681 0.0359277i 0.562545 2.09944i 0 1.72723 + 1.72723i 3.39913 5.88747i −0.0129135 0.0223668i
227.6 1.14693 1.14693i 0.445073 0.256963i 0.630890i −0.395109 + 1.47457i 0.215749 0.805186i 0 1.57027 + 1.57027i −1.36794 + 2.36934i 1.23806 + 2.14439i
227.7 1.74384 1.74384i 0.146239 0.0844309i 4.08193i 0.638637 2.38343i 0.107782 0.402250i 0 −3.63054 3.63054i −1.48574 + 2.57338i −3.04263 5.26998i
362.1 −1.56744 1.56744i −0.959879 0.554186i 2.91373i 0.784926 + 2.92938i 0.635898 + 2.37321i 0 1.43221 1.43221i −0.885755 1.53417i 3.36131 5.82195i
362.2 −0.984398 0.984398i 1.25446 + 0.724265i 0.0619199i 0.172312 + 0.643078i −0.521927 1.94786i 0 −2.02975 + 2.02975i −0.450880 0.780947i 0.463421 0.802669i
362.3 −0.270646 0.270646i −0.792292 0.457430i 1.85350i −0.959617 3.58134i 0.0906291 + 0.338232i 0 −1.04293 + 1.04293i −1.08152 1.87324i −0.709559 + 1.22899i
362.4 −0.193244 0.193244i −2.17048 1.25313i 1.92531i 0.383199 + 1.43012i 0.177273 + 0.661591i 0 −0.758543 + 0.758543i 1.64065 + 2.84169i 0.202311 0.350412i
362.5 0.490988 + 0.490988i 2.71085 + 1.56511i 1.51786i 0.00962681 + 0.0359277i 0.562545 + 2.09944i 0 1.72723 1.72723i 3.39913 + 5.88747i −0.0129135 + 0.0223668i
362.6 1.14693 + 1.14693i 0.445073 + 0.256963i 0.630890i −0.395109 1.47457i 0.215749 + 0.805186i 0 1.57027 1.57027i −1.36794 2.36934i 1.23806 2.14439i
362.7 1.74384 + 1.74384i 0.146239 + 0.0844309i 4.08193i 0.638637 + 2.38343i 0.107782 + 0.402250i 0 −3.63054 + 3.63054i −1.48574 2.57338i −3.04263 + 5.26998i
423.1 −1.72056 1.72056i 1.70689 0.985473i 3.92067i −0.227080 0.0608458i −4.63238 1.24124i 0 3.30464 3.30464i 0.442313 0.766108i 0.286016 + 0.495394i
423.2 −1.42500 1.42500i −1.25027 + 0.721843i 2.06123i −3.16920 0.849184i 2.81025 + 0.753004i 0 0.0872533 0.0872533i −0.457887 + 0.793083i 3.30601 + 5.72618i
423.3 −0.876516 0.876516i −1.92717 + 1.11265i 0.463441i 2.51660 + 0.674321i 2.66446 + 0.713939i 0 −2.15924 + 2.15924i 0.975997 1.69048i −1.61479 2.79689i
423.4 −0.347096 0.347096i 2.11812 1.22290i 1.75905i 3.47544 + 0.931242i −1.15965 0.310728i 0 −1.30475 + 1.30475i 1.49096 2.58241i −0.883082 1.52954i
423.5 0.203761 + 0.203761i −0.923129 + 0.532969i 1.91696i −0.499383 0.133809i −0.296695 0.0794993i 0 0.798123 0.798123i −0.931889 + 1.61408i −0.0744896 0.129020i
423.6 1.28453 + 1.28453i 2.65867 1.53499i 1.30006i −1.07541 0.288156i 5.38690 + 1.44342i 0 0.899098 0.899098i 3.21237 5.56398i −1.01126 1.75155i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 509.7 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.ba 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.bb.a 28
7.b odd 2 1 91.2.ba.a yes 28
7.c even 3 1 91.2.w.a 28
7.c even 3 1 637.2.bd.a 28
7.d odd 6 1 637.2.x.a 28
7.d odd 6 1 637.2.bd.b 28
13.f odd 12 1 637.2.x.a 28
21.c even 2 1 819.2.et.b 28
21.h odd 6 1 819.2.gh.b 28
91.w even 12 1 637.2.bd.a 28
91.x odd 12 1 91.2.ba.a yes 28
91.ba even 12 1 inner 637.2.bb.a 28
91.bc even 12 1 91.2.w.a 28
91.bd odd 12 1 637.2.bd.b 28
273.bv even 12 1 819.2.et.b 28
273.ca odd 12 1 819.2.gh.b 28

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

## Hecke kernels

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