Properties

Label 91.2.w.a
Level $91$
Weight $2$
Character orbit 91.w
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.w (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.

\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{2} - 6q^{4} - 6q^{5} + 12q^{6} + 2q^{7} - 4q^{8} - 12q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q - 2q^{2} - 6q^{4} - 6q^{5} + 12q^{6} + 2q^{7} - 4q^{8} - 12q^{9} - 12q^{10} + 2q^{11} + 8q^{12} - 20q^{14} + 10q^{15} - 2q^{16} - 6q^{17} - 4q^{18} - 8q^{19} - 36q^{20} - 2q^{21} - 8q^{22} - 6q^{23} + 12q^{24} + 24q^{26} - 18q^{28} - 8q^{29} - 38q^{31} - 20q^{32} + 18q^{33} + 12q^{34} - 2q^{35} + 54q^{36} - 16q^{37} + 28q^{39} + 48q^{40} + 18q^{41} - 4q^{42} + 48q^{43} - 6q^{44} + 12q^{45} + 18q^{46} - 42q^{47} + 12q^{48} + 8q^{49} + 10q^{50} + 12q^{51} - 28q^{52} + 12q^{53} - 30q^{54} - 6q^{55} - 24q^{56} + 12q^{57} + 62q^{58} - 6q^{59} + 16q^{60} - 36q^{62} - 38q^{63} - 2q^{65} + 66q^{66} - 4q^{67} + 30q^{68} + 42q^{69} + 68q^{70} - 42q^{71} - 38q^{72} + 14q^{73} - 6q^{74} - 20q^{75} + 52q^{76} - 62q^{78} + 4q^{79} + 12q^{80} + 12q^{81} - 108q^{82} - 66q^{83} - 56q^{84} - 54q^{85} - 30q^{86} + 42q^{87} - 30q^{89} - 72q^{90} - 42q^{91} - 156q^{92} + 14q^{93} - 6q^{95} + 18q^{96} + 62q^{97} + 112q^{98} - 36q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.38212 0.638288i 0.168862i 3.53505 + 2.04096i −2.38343 + 0.638637i 0.107782 0.402250i 0.932289 + 2.47605i −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 1.27657 2.31741i 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 −1.16683 + 2.37456i 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 2.63370 0.252224i −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 −2.28025 + 1.34181i −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 2.61524 + 0.400640i −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 −2.64470 + 0.0746479i 1.43221 + 1.43221i 1.77151 −6.72261
24.1 −2.38212 + 0.638288i 0.168862i 3.53505 2.04096i −2.38343 0.638637i 0.107782 + 0.402250i 0.932289 2.47605i −3.63054 + 3.63054i 2.97149 6.08525
24.2 −1.56673 + 0.419805i 0.513926i 0.546366 0.315445i 1.47457 + 0.395109i 0.215749 + 0.805186i 1.27657 + 2.31741i 1.57027 1.57027i 2.73588 −2.47612
24.3 −0.670702 + 0.179714i 3.13022i −1.31451 + 0.758931i −0.0359277 0.00962681i 0.562545 + 2.09944i −1.16683 2.37456i 1.72723 1.72723i −6.79826 0.0258269
24.4 0.263976 0.0707322i 2.50625i −1.66737 + 0.962657i −1.43012 0.383199i 0.177273 + 0.661591i 2.63370 + 0.252224i −0.758543 + 0.758543i −3.28130 −0.404621
24.5 0.369709 0.0990633i 0.914861i −1.60518 + 0.926751i 3.58134 + 0.959617i 0.0906291 + 0.338232i −2.28025 1.34181i −1.04293 + 1.04293i 2.16303 1.41912
24.6 1.34471 0.360315i 1.44853i −0.0536242 + 0.0309600i −0.643078 0.172312i −0.521927 1.94786i 2.61524 0.400640i −2.02975 + 2.02975i 0.901760 −0.926843
24.7 2.14116 0.573722i 1.10837i 2.52336 1.45686i −2.92938 0.784926i 0.635898 + 2.37321i −2.64470 0.0746479i 1.43221 1.43221i 1.77151 −6.72261
33.1 −0.629770 + 2.35033i 1.97095i −3.39540 1.96034i 0.0608458 + 0.227080i −4.63238 1.24124i 2.26239 1.37172i 3.30464 3.30464i −0.884626 −0.572032
33.2 −0.521585 + 1.94658i 1.44369i −1.78508 1.03062i 0.849184 + 3.16920i 2.81025 + 0.753004i −1.41817 + 2.23356i 0.0872533 0.0872533i 0.915773 −6.61202
33.3 −0.320827 + 1.19734i 2.22531i 0.401352 + 0.231720i −0.674321 2.51660i 2.66446 + 0.713939i 2.64022 0.170970i −2.15924 + 2.15924i −1.95199 3.22957
33.4 −0.127046 + 0.474142i 2.44579i 1.52338 + 0.879524i −0.931242 3.47544i −1.15965 0.310728i −0.668469 + 2.55991i −1.30475 + 1.30475i −2.98191 1.76616
33.5 0.0745816 0.278342i 1.06594i 1.66014 + 0.958482i 0.133809 + 0.499383i −0.296695 0.0794993i −2.03333 1.69280i 0.798123 0.798123i 1.86378 0.148979
33.6 0.470172 1.75471i 3.06997i −1.12588 0.650030i 0.288156 + 1.07541i 5.38690 + 1.44342i 0.381837 2.61805i 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 80.7
Significant digits:
Format:

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

Hecke kernels

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