Properties

Label 91.2.ba.a
Level $91$
Weight $2$
Character orbit 91.ba
Analytic conductor $0.727$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.ba (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{2} - 6q^{3} - 6q^{5} - 12q^{6} - 6q^{7} - 4q^{8} + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{2} - 6q^{3} - 6q^{5} - 12q^{6} - 6q^{7} - 4q^{8} + 6q^{9} - 6q^{10} + 2q^{11} - 8q^{12} - 20q^{14} + 10q^{15} + 4q^{16} - 12q^{17} + 2q^{18} + 14q^{19} + 36q^{20} - 6q^{21} - 8q^{22} - 18q^{24} + 24q^{26} + 2q^{28} - 8q^{29} - 30q^{30} - 4q^{31} + 10q^{32} - 12q^{33} - 12q^{34} - 20q^{35} + 54q^{36} - 10q^{37} - 20q^{39} + 48q^{40} - 18q^{41} - 10q^{42} + 48q^{43} - 6q^{44} - 6q^{45} + 24q^{46} - 6q^{47} - 12q^{48} - 50q^{49} + 10q^{50} - 12q^{51} - 26q^{52} + 12q^{53} - 30q^{54} + 6q^{55} + 54q^{56} + 12q^{57} - 46q^{58} + 42q^{59} + 10q^{60} + 30q^{61} + 36q^{62} + 54q^{63} + 28q^{65} + 66q^{66} - 10q^{67} - 42q^{69} - 88q^{70} - 42q^{71} + 46q^{72} + 40q^{73} + 12q^{74} - 40q^{75} - 52q^{76} - 62q^{78} + 4q^{79} + 30q^{80} - 6q^{81} - 54q^{82} + 66q^{83} + 104q^{84} - 54q^{85} - 18q^{86} - 6q^{88} + 72q^{90} + 26q^{91} - 156q^{92} + 20q^{93} - 18q^{94} - 66q^{96} - 62q^{97} - 56q^{98} - 36q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −1.56744 + 1.56744i 0.959879 0.554186i 2.91373i −0.784926 + 2.92938i −0.635898 + 2.37321i 2.25305 + 1.38700i 1.43221 + 1.43221i −0.885755 + 1.53417i −3.36131 5.82195i
45.2 −0.984398 + 0.984398i −1.25446 + 0.724265i 0.0619199i −0.172312 + 0.643078i 0.521927 1.94786i −2.46519 0.960657i −2.02975 2.02975i −0.450880 + 0.780947i −0.463421 0.802669i
45.3 −0.270646 + 0.270646i 0.792292 0.457430i 1.85350i 0.959617 3.58134i −0.0906291 + 0.338232i 1.30385 + 2.30217i −1.04293 1.04293i −1.08152 + 1.87324i 0.709559 + 1.22899i
45.4 −0.193244 + 0.193244i 2.17048 1.25313i 1.92531i −0.383199 + 1.43012i −0.177273 + 0.661591i −2.15474 1.53528i −0.758543 0.758543i 1.64065 2.84169i −0.202311 0.350412i
45.5 0.490988 0.490988i −2.71085 + 1.56511i 1.51786i −0.00962681 + 0.0359277i −0.562545 + 2.09944i −0.176775 + 2.63984i 1.72723 + 1.72723i 3.39913 5.88747i 0.0129135 + 0.0223668i
45.6 1.14693 1.14693i −0.445073 + 0.256963i 0.630890i 0.395109 1.47457i −0.215749 + 0.805186i 0.0531605 2.64522i 1.57027 + 1.57027i −1.36794 + 2.36934i −1.23806 2.14439i
45.7 1.74384 1.74384i −0.146239 + 0.0844309i 4.08193i −0.638637 + 2.38343i −0.107782 + 0.402250i −2.04541 + 1.67818i −3.63054 3.63054i −1.48574 + 2.57338i 3.04263 + 5.26998i
54.1 −1.72056 + 1.72056i −1.70689 0.985473i 3.92067i 0.227080 0.0608458i 4.63238 1.24124i 1.27342 2.31914i 3.30464 + 3.30464i 0.442313 + 0.766108i −0.286016 + 0.495394i
54.2 −1.42500 + 1.42500i 1.25027 + 0.721843i 2.06123i 3.16920 0.849184i −2.81025 + 0.753004i −0.111396 + 2.64341i 0.0872533 + 0.0872533i −0.457887 0.793083i −3.30601 + 5.72618i
54.3 −0.876516 + 0.876516i 1.92717 + 1.11265i 0.463441i −2.51660 + 0.674321i −2.66446 + 0.713939i 2.20101 1.46818i −2.15924 2.15924i 0.975997 + 1.69048i 1.61479 2.79689i
54.4 −0.347096 + 0.347096i −2.11812 1.22290i 1.75905i −3.47544 + 0.931242i 1.15965 0.310728i 0.701045 + 2.55118i −1.30475 1.30475i 1.49096 + 2.58241i 0.883082 1.52954i
54.5 0.203761 0.203761i 0.923129 + 0.532969i 1.91696i 0.499383 0.133809i 0.296695 0.0794993i −2.60732 0.449339i 0.798123 + 0.798123i −0.931889 1.61408i 0.0744896 0.129020i
54.6 1.28453 1.28453i −2.65867 1.53499i 1.30006i 1.07541 0.288156i −5.38690 + 1.44342i −0.978346 2.45822i 0.899098 + 0.899098i 3.21237 + 5.56398i 1.01126 1.75155i
54.7 1.51485 1.51485i 0.0170886 + 0.00986613i 2.58954i −1.34505 + 0.360406i 0.0408324 0.0109410i −0.246373 + 2.63426i −0.893066 0.893066i −1.49981 2.59774i −1.49159 + 2.58351i
59.1 −1.72056 1.72056i −1.70689 + 0.985473i 3.92067i 0.227080 + 0.0608458i 4.63238 + 1.24124i 1.27342 + 2.31914i 3.30464 3.30464i 0.442313 0.766108i −0.286016 0.495394i
59.2 −1.42500 1.42500i 1.25027 0.721843i 2.06123i 3.16920 + 0.849184i −2.81025 0.753004i −0.111396 2.64341i 0.0872533 0.0872533i −0.457887 + 0.793083i −3.30601 5.72618i
59.3 −0.876516 0.876516i 1.92717 1.11265i 0.463441i −2.51660 0.674321i −2.66446 0.713939i 2.20101 + 1.46818i −2.15924 + 2.15924i 0.975997 1.69048i 1.61479 + 2.79689i
59.4 −0.347096 0.347096i −2.11812 + 1.22290i 1.75905i −3.47544 0.931242i 1.15965 + 0.310728i 0.701045 2.55118i −1.30475 + 1.30475i 1.49096 2.58241i 0.883082 + 1.52954i
59.5 0.203761 + 0.203761i 0.923129 0.532969i 1.91696i 0.499383 + 0.133809i 0.296695 + 0.0794993i −2.60732 + 0.449339i 0.798123 0.798123i −0.931889 + 1.61408i 0.0744896 + 0.129020i
59.6 1.28453 + 1.28453i −2.65867 + 1.53499i 1.30006i 1.07541 + 0.288156i −5.38690 1.44342i −0.978346 + 2.45822i 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 89.7
Significant digits:
Format:

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 91.2.ba.a yes 28
3.b odd 2 1 819.2.et.b 28
7.b odd 2 1 637.2.bb.a 28
7.c even 3 1 637.2.x.a 28
7.c even 3 1 637.2.bd.b 28
7.d odd 6 1 91.2.w.a 28
7.d odd 6 1 637.2.bd.a 28
13.f odd 12 1 91.2.w.a 28
21.g even 6 1 819.2.gh.b 28
39.k even 12 1 819.2.gh.b 28
91.w even 12 1 637.2.bd.b 28
91.x odd 12 1 637.2.bb.a 28
91.ba even 12 1 inner 91.2.ba.a yes 28
91.bc even 12 1 637.2.x.a 28
91.bd odd 12 1 637.2.bd.a 28
273.bs 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 13.f odd 12 1
91.2.ba.a yes 28 1.a even 1 1 trivial
91.2.ba.a yes 28 91.ba even 12 1 inner
637.2.x.a 28 7.c even 3 1
637.2.x.a 28 91.bc even 12 1
637.2.bb.a 28 7.b odd 2 1
637.2.bb.a 28 91.x odd 12 1
637.2.bd.a 28 7.d odd 6 1
637.2.bd.a 28 91.bd odd 12 1
637.2.bd.b 28 7.c even 3 1
637.2.bd.b 28 91.w even 12 1
819.2.et.b 28 3.b odd 2 1
819.2.et.b 28 273.bs odd 12 1
819.2.gh.b 28 21.g even 6 1
819.2.gh.b 28 39.k even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(91, [\chi])\).