Properties

Label 91.2.k.b
Level $91$
Weight $2$
Character orbit 91.k
Analytic conductor $0.727$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.2346760387617129.1
Defining polynomial: \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + (\beta_{6} + \beta_{4} + \beta_1) q^{3} + ( - \beta_{11} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{4} + ( - \beta_{10} + \beta_{8} + \beta_{3}) q^{5} + ( - \beta_{11} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 1) q^{6} + (\beta_{11} - \beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + (\beta_{6} + \beta_{4} + \beta_1) q^{3} + ( - \beta_{11} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{4} + ( - \beta_{10} + \beta_{8} + \beta_{3}) q^{5} + ( - \beta_{11} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 1) q^{6} + (\beta_{11} - \beta_{7} + \beta_{4} + \beta_{2} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{8} - \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_1) q^{10} + (\beta_{10} - \beta_{8} - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{11} + (\beta_{10} - 2 \beta_{9} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_{11} + \beta_{10} + 2 \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2}) q^{13} + ( - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 1) q^{14}+ \cdots + ( - 2 \beta_{11} - 4 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 8 \beta_{4} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - 8 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{3} - 8 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} - q^{9} + 12 q^{10} + 12 q^{11} - q^{12} - 2 q^{13} + 4 q^{14} - 12 q^{15} + 16 q^{16} - 34 q^{17} + 3 q^{18} + 9 q^{19} - 3 q^{20} + 21 q^{21} - 15 q^{22} - 6 q^{23} + 15 q^{24} - 5 q^{25} - 6 q^{26} + 12 q^{27} - 9 q^{28} - q^{29} + 11 q^{30} + 18 q^{31} - 6 q^{33} - 6 q^{35} - 13 q^{36} + 19 q^{38} - 4 q^{39} - q^{40} - 6 q^{41} - 8 q^{42} + 11 q^{43} - 33 q^{44} - 15 q^{47} + 19 q^{48} - 3 q^{49} + 18 q^{50} + 4 q^{51} - 7 q^{52} - 8 q^{53} - 15 q^{55} + 27 q^{56} - 24 q^{58} - 30 q^{60} + 5 q^{61} + 41 q^{62} - 30 q^{63} + 2 q^{64} + 21 q^{65} - 34 q^{66} + 15 q^{67} + 22 q^{68} + 7 q^{69} + 3 q^{70} + 30 q^{71} + 57 q^{72} + 42 q^{73} + 66 q^{74} - 2 q^{75} - 45 q^{76} - 19 q^{77} + 44 q^{78} - 35 q^{79} - 63 q^{80} + 14 q^{81} + 5 q^{82} - 12 q^{84} - 21 q^{85} - 57 q^{86} - 20 q^{87} - 14 q^{88} - 7 q^{91} - 66 q^{92} + q^{94} - 4 q^{95} + 21 q^{96} - 3 q^{97} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256 ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288 ) / 224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664 ) / 224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88 ) / 28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48 ) / 28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464 ) / 112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056 ) / 112 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + 2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 2\beta_{2} + 3\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta _1 - 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-\beta_{4}\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
1.32725 0.488273i
−1.38488 0.286553i
0.655911 + 1.25291i
−1.18541 + 0.771231i
0.874681 1.11128i
1.21245 + 0.727987i
1.21245 0.727987i
0.874681 + 1.11128i
−1.18541 0.771231i
0.655911 1.25291i
−1.38488 + 0.286553i
1.32725 + 0.488273i
2.58860i −0.259233 0.449005i −4.70085 −1.39608 + 0.806027i −1.16229 + 0.671051i 1.06153 2.42346i 6.99143i 1.36560 2.36528i 2.08648 + 3.61389i
4.2 1.37905i −1.44060 2.49520i 0.0982074 −0.697972 + 0.402974i −3.44101 + 1.98667i 0.0699870 + 2.64483i 2.89354i −2.65067 + 4.59109i 0.555723 + 0.962541i
4.3 0.180824i 0.913006 + 1.58137i 1.96730 −2.32670 + 1.34332i 0.285950 0.165093i −2.46263 0.967177i 0.717383i −0.167162 + 0.289532i 0.242904 + 0.420723i
4.4 0.499987i −0.424801 0.735776i 1.75001 −0.902810 + 0.521238i 0.367878 0.212395i 2.63491 0.239300i 1.87496i 1.13909 1.97296i −0.260612 0.451393i
4.5 1.34523i −1.02505 1.77544i 0.190366 3.08979 1.78389i 2.38837 1.37893i −2.44601 + 1.00849i 2.94654i −0.601462 + 1.04176i 2.39973 + 4.15646i
4.6 2.30327i 0.736680 + 1.27597i −3.30504 0.733776 0.423646i −2.93889 + 1.69677i −0.357777 2.62145i 3.00585i 0.414604 0.718115i 0.975769 + 1.69008i
23.1 2.30327i 0.736680 1.27597i −3.30504 0.733776 + 0.423646i −2.93889 1.69677i −0.357777 + 2.62145i 3.00585i 0.414604 + 0.718115i 0.975769 1.69008i
23.2 1.34523i −1.02505 + 1.77544i 0.190366 3.08979 + 1.78389i 2.38837 + 1.37893i −2.44601 1.00849i 2.94654i −0.601462 1.04176i 2.39973 4.15646i
23.3 0.499987i −0.424801 + 0.735776i 1.75001 −0.902810 0.521238i 0.367878 + 0.212395i 2.63491 + 0.239300i 1.87496i 1.13909 + 1.97296i −0.260612 + 0.451393i
23.4 0.180824i 0.913006 1.58137i 1.96730 −2.32670 1.34332i 0.285950 + 0.165093i −2.46263 + 0.967177i 0.717383i −0.167162 0.289532i 0.242904 0.420723i
23.5 1.37905i −1.44060 + 2.49520i 0.0982074 −0.697972 0.402974i −3.44101 1.98667i 0.0699870 2.64483i 2.89354i −2.65067 4.59109i 0.555723 0.962541i
23.6 2.58860i −0.259233 + 0.449005i −4.70085 −1.39608 0.806027i −1.16229 0.671051i 1.06153 + 2.42346i 6.99143i 1.36560 + 2.36528i 2.08648 3.61389i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.k.b 12
3.b odd 2 1 819.2.bm.f 12
7.b odd 2 1 637.2.k.i 12
7.c even 3 1 91.2.u.b yes 12
7.c even 3 1 637.2.q.g 12
7.d odd 6 1 637.2.q.i 12
7.d odd 6 1 637.2.u.g 12
13.e even 6 1 91.2.u.b yes 12
13.f odd 12 2 1183.2.e.j 24
21.h odd 6 1 819.2.do.e 12
39.h odd 6 1 819.2.do.e 12
91.k even 6 1 inner 91.2.k.b 12
91.l odd 6 1 637.2.k.i 12
91.p odd 6 1 637.2.q.i 12
91.t odd 6 1 637.2.u.g 12
91.u even 6 1 637.2.q.g 12
91.x odd 12 2 8281.2.a.cp 12
91.ba even 12 2 8281.2.a.co 12
91.bd odd 12 2 1183.2.e.j 24
273.bp odd 6 1 819.2.bm.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 1.a even 1 1 trivial
91.2.k.b 12 91.k even 6 1 inner
91.2.u.b yes 12 7.c even 3 1
91.2.u.b yes 12 13.e even 6 1
637.2.k.i 12 7.b odd 2 1
637.2.k.i 12 91.l odd 6 1
637.2.q.g 12 7.c even 3 1
637.2.q.g 12 91.u even 6 1
637.2.q.i 12 7.d odd 6 1
637.2.q.i 12 91.p odd 6 1
637.2.u.g 12 7.d odd 6 1
637.2.u.g 12 91.t odd 6 1
819.2.bm.f 12 3.b odd 2 1
819.2.bm.f 12 273.bp odd 6 1
819.2.do.e 12 21.h odd 6 1
819.2.do.e 12 39.h odd 6 1
1183.2.e.j 24 13.f odd 12 2
1183.2.e.j 24 91.bd odd 12 2
8281.2.a.co 12 91.ba even 12 2
8281.2.a.cp 12 91.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 16T_{2}^{10} + 88T_{2}^{8} + 197T_{2}^{6} + 172T_{2}^{4} + 36T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + 88 T^{8} + 197 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + 14 T^{10} + 17 T^{9} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{12} + 3 T^{11} - 8 T^{10} - 33 T^{9} + \cdots + 121 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + 6 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} - 12 T^{11} + 41 T^{10} + \cdots + 85849 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} - 18 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} + 17 T^{5} + 96 T^{4} + 198 T^{3} + \cdots + 19)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 9 T^{11} + T^{10} + 234 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{6} + 3 T^{5} - 50 T^{4} - 259 T^{3} + \cdots + 793)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + T^{11} + 87 T^{10} + \cdots + 16072081 \) Copy content Toggle raw display
$31$ \( T^{12} - 18 T^{11} + \cdots + 241274089 \) Copy content Toggle raw display
$37$ \( T^{12} + 147 T^{10} + 7164 T^{8} + \cdots + 123201 \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} - 159 T^{10} + \cdots + 389707081 \) Copy content Toggle raw display
$43$ \( T^{12} - 11 T^{11} + \cdots + 418898089 \) Copy content Toggle raw display
$47$ \( T^{12} + 15 T^{11} + 92 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$53$ \( T^{12} + 8 T^{11} + 102 T^{10} + \cdots + 289 \) Copy content Toggle raw display
$59$ \( T^{12} + 553 T^{10} + \cdots + 35582408689 \) Copy content Toggle raw display
$61$ \( T^{12} - 5 T^{11} + 100 T^{10} + \cdots + 3157729 \) Copy content Toggle raw display
$67$ \( T^{12} - 15 T^{11} + \cdots + 5708255809 \) Copy content Toggle raw display
$71$ \( T^{12} - 30 T^{11} + \cdots + 639230089 \) Copy content Toggle raw display
$73$ \( T^{12} - 42 T^{11} + \cdots + 484396081 \) Copy content Toggle raw display
$79$ \( T^{12} + 35 T^{11} + \cdots + 65086724641 \) Copy content Toggle raw display
$83$ \( T^{12} + 463 T^{10} + \cdots + 402363481 \) Copy content Toggle raw display
$89$ \( T^{12} + 430 T^{10} + \cdots + 145033849 \) Copy content Toggle raw display
$97$ \( T^{12} + 3 T^{11} - 176 T^{10} + \cdots + 1681 \) Copy content Toggle raw display
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