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Properties

Label 91.10.a.b
Level $91$
Weight $10$
Character orbit 91.a
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,10,Mod(1,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4945 x^{11} - 8694 x^{10} + 9009530 x^{9} + 27431200 x^{8} - 7320118704 x^{7} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + ( - \beta_{5} + 13) q^{3} + (\beta_{2} - \beta_1 + 253) q^{4} + (\beta_{5} + \beta_{4} + 4 \beta_1 - 204) q^{5} + ( - \beta_{6} + 9 \beta_{5} + \cdots - 82) q^{6}+ \cdots + ( - \beta_{10} - \beta_{9} + \cdots + 7301) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + ( - \beta_{5} + 13) q^{3} + (\beta_{2} - \beta_1 + 253) q^{4} + (\beta_{5} + \beta_{4} + 4 \beta_1 - 204) q^{5} + ( - \beta_{6} + 9 \beta_{5} + \cdots - 82) q^{6}+ \cdots + (1130 \beta_{12} - 9236 \beta_{11} + \cdots - 356807842) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 26 q^{2} + 163 q^{3} + 3286 q^{4} - 2640 q^{5} - 995 q^{6} - 31213 q^{7} - 6738 q^{8} + 94756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 26 q^{2} + 163 q^{3} + 3286 q^{4} - 2640 q^{5} - 995 q^{6} - 31213 q^{7} - 6738 q^{8} + 94756 q^{9} + 42588 q^{10} - 107493 q^{11} + 157399 q^{12} - 371293 q^{13} + 62426 q^{14} - 469556 q^{15} + 1033802 q^{16} + 50812 q^{17} - 2994615 q^{18} + 479470 q^{19} - 1834962 q^{20} - 391363 q^{21} - 5474013 q^{22} - 984639 q^{23} - 12496965 q^{24} + 4519039 q^{25} + 742586 q^{26} + 5965117 q^{27} - 7889686 q^{28} - 3441800 q^{29} + 25168012 q^{30} - 2185751 q^{31} - 2746342 q^{32} + 34793355 q^{33} - 966694 q^{34} + 6338640 q^{35} + 23974587 q^{36} - 31532363 q^{37} - 51039796 q^{38} - 4655443 q^{39} + 27446642 q^{40} - 38029287 q^{41} + 2388995 q^{42} - 65479740 q^{43} - 64795239 q^{44} - 190647152 q^{45} - 68737615 q^{46} + 18884785 q^{47} - 43918333 q^{48} + 74942413 q^{49} - 295918964 q^{50} - 97799092 q^{51} - 93851446 q^{52} - 37670088 q^{53} - 420784337 q^{54} - 11739604 q^{55} + 16177938 q^{56} - 119447794 q^{57} - 351819004 q^{58} - 86030686 q^{59} - 1421949708 q^{60} - 413609773 q^{61} + 21747651 q^{62} - 227509156 q^{63} - 611561502 q^{64} + 75401040 q^{65} - 154290083 q^{66} + 121596783 q^{67} - 613335382 q^{68} - 1089108303 q^{69} - 102253788 q^{70} - 900222116 q^{71} - 1897573017 q^{72} - 586910355 q^{73} - 688661251 q^{74} - 1466887131 q^{75} - 180912510 q^{76} + 258090693 q^{77} + 28418195 q^{78} - 590012173 q^{79} - 1724662122 q^{80} - 58178363 q^{81} + 145984865 q^{82} + 94283256 q^{83} - 377914999 q^{84} - 1689818164 q^{85} + 13901738 q^{86} + 1073171888 q^{87} - 1814132379 q^{88} - 1154652750 q^{89} + 2671175016 q^{90} + 891474493 q^{91} + 670826733 q^{92} - 5057835587 q^{93} - 2961146369 q^{94} - 3377803464 q^{95} - 4898921405 q^{96} - 2173622401 q^{97} - 149884826 q^{98} - 4653424330 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 4945 x^{11} - 8694 x^{10} + 9009530 x^{9} + 27431200 x^{8} - 7320118704 x^{7} + \cdots + 20\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3\nu - 761 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 22\!\cdots\!19 \nu^{12} + \cdots - 72\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!73 \nu^{12} + \cdots - 34\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 79\!\cdots\!17 \nu^{12} + \cdots + 10\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!27 \nu^{12} + \cdots + 88\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23\!\cdots\!29 \nu^{12} + \cdots - 18\!\cdots\!00 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25\!\cdots\!51 \nu^{12} + \cdots + 67\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!21 \nu^{12} + \cdots - 38\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33\!\cdots\!31 \nu^{12} + \cdots - 99\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 99\!\cdots\!63 \nu^{12} + \cdots + 30\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 61\!\cdots\!17 \nu^{12} + \cdots + 38\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3\beta _1 + 761 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{9} + \beta_{7} + \beta_{6} + 35\beta_{5} + 5\beta_{2} + 1301\beta _1 + 1992 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 28 \beta_{12} - 7 \beta_{11} - 2 \beta_{10} + 14 \beta_{9} - 15 \beta_{8} + 68 \beta_{6} + 504 \beta_{5} + \cdots + 990049 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2188 \beta_{12} + 437 \beta_{11} - 170 \beta_{10} + 2178 \beta_{9} - 143 \beta_{8} + 2384 \beta_{7} + \cdots + 5948395 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 71862 \beta_{12} - 8643 \beta_{11} - 5186 \beta_{10} + 42452 \beta_{9} - 40431 \beta_{8} + \cdots + 1436682267 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4247846 \beta_{12} + 1389631 \beta_{11} - 442966 \beta_{10} + 4188552 \beta_{9} - 544573 \beta_{8} + \cdots + 14560466981 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 149564998 \beta_{12} - 273967 \beta_{11} - 10132634 \beta_{10} + 98306868 \beta_{9} + \cdots + 2239687654803 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8072971890 \beta_{12} + 3289863579 \beta_{11} - 905266814 \beta_{10} + 7864885676 \beta_{9} + \cdots + 32333660188601 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 295436293550 \beta_{12} + 30213911145 \beta_{11} - 18648168330 \beta_{10} + 210532129340 \beta_{9} + \cdots + 36\!\cdots\!35 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15324028691962 \beta_{12} + 6991751289131 \beta_{11} - 1727402662910 \beta_{10} + 14756530605364 \beta_{9} + \cdots + 68\!\cdots\!73 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 575705378564254 \beta_{12} + 108757475936609 \beta_{11} - 34518487594874 \beta_{10} + \cdots + 63\!\cdots\!91 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−39.9033
−34.7365
−31.6351
−28.9147
−8.24674
−5.16491
−3.18070
0.714333
12.3946
25.2754
32.2577
37.1005
44.0395
−41.9033 259.004 1243.89 −2349.43 −10853.1 −2401.00 −30668.6 47400.0 98448.7
1.2 −36.7365 −181.146 837.568 2263.64 6654.66 −2401.00 −11960.2 13130.9 −83158.3
1.3 −33.6351 −109.950 619.318 −1358.15 3698.16 −2401.00 −3609.64 −7594.10 45681.6
1.4 −30.9147 166.293 443.721 268.017 −5140.90 −2401.00 2110.83 7970.34 −8285.69
1.5 −10.2467 −61.4030 −407.004 −1449.57 629.180 −2401.00 9416.80 −15912.7 14853.4
1.6 −7.16491 −126.235 −460.664 1492.86 904.459 −2401.00 6969.05 −3747.85 −10696.2
1.7 −5.18070 162.037 −485.160 849.478 −839.466 −2401.00 5165.99 6573.07 −4400.89
1.8 −1.28567 −259.576 −510.347 −2321.41 333.729 −2401.00 1314.40 47696.7 2984.56
1.9 10.3946 229.479 −403.953 −347.024 2385.34 −2401.00 −9520.95 32977.8 −3607.17
1.10 23.2754 5.80529 29.7447 2107.58 135.120 −2401.00 −11224.7 −19649.3 49054.9
1.11 30.2577 148.920 403.526 −657.246 4505.98 −2401.00 −3282.17 2494.28 −19886.7
1.12 35.1005 65.6740 720.042 −1365.06 2305.19 −2401.00 7302.37 −15369.9 −47914.1
1.13 42.0395 −135.904 1255.32 226.307 −5713.33 −2401.00 31248.8 −1213.14 9513.82
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.a.b ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.a.b ✓ 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 26 T_{2}^{12} - 4633 T_{2}^{11} - 115196 T_{2}^{10} + 7759190 T_{2}^{9} + \cdots - 84\!\cdots\!96 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots - 84\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 2401)^{13} \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T + 28561)^{13} \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 36\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots + 21\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots + 34\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 27\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 41\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 51\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 22\!\cdots\!40 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 40\!\cdots\!68 \) Copy content Toggle raw display
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