Properties

Label 91.10.a.c
Level $91$
Weight $10$
Character orbit 91.a
Self dual yes
Analytic conductor $46.868$
Analytic rank $0$
Dimension $14$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 4752 x^{12} + 9346 x^{11} + 8576824 x^{10} - 26923636 x^{9} - 7450416552 x^{8} + 31594524240 x^{7} + 3232668379296 x^{6} + \cdots - 24\!\cdots\!40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{4} + 12) q^{3} + (\beta_{2} - 6 \beta_1 + 171) q^{4} + (\beta_{6} + \beta_{2} - 13 \beta_1 + 213) q^{5} + ( - \beta_{9} + \beta_{4} - 25 \beta_1 + 35) q^{6} + 2401 q^{7} + ( - \beta_{9} + 3 \beta_{6} - 4 \beta_{4} - \beta_{3} + 5 \beta_{2} - 139 \beta_1 + 3102) q^{8} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 15 \beta_{4} - 4 \beta_{2} + \cdots + 10292) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{4} + 12) q^{3} + (\beta_{2} - 6 \beta_1 + 171) q^{4} + (\beta_{6} + \beta_{2} - 13 \beta_1 + 213) q^{5} + ( - \beta_{9} + \beta_{4} - 25 \beta_1 + 35) q^{6} + 2401 q^{7} + ( - \beta_{9} + 3 \beta_{6} - 4 \beta_{4} - \beta_{3} + 5 \beta_{2} - 139 \beta_1 + 3102) q^{8} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{7} + 3 \beta_{6} - 15 \beta_{4} - 4 \beta_{2} + \cdots + 10292) q^{9}+ \cdots + ( - 7344 \beta_{13} + 22104 \beta_{12} + 48191 \beta_{11} + \cdots + 361745986) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 27 q^{2} + 163 q^{3} + 2389 q^{4} + 2964 q^{5} + 471 q^{6} + 33614 q^{7} + 43263 q^{8} + 144129 q^{9} + 126524 q^{10} + 81825 q^{11} + 157399 q^{12} - 399854 q^{13} + 64827 q^{14} + 163856 q^{15} + 166361 q^{16} - 44922 q^{17} - 826396 q^{18} + 171756 q^{19} + 3899724 q^{20} + 391363 q^{21} + 917579 q^{22} + 1930479 q^{23} + 2992373 q^{24} + 8222344 q^{25} - 771147 q^{26} + 4139125 q^{27} + 5735989 q^{28} - 3799608 q^{29} - 5918004 q^{30} - 4392203 q^{31} + 3135663 q^{32} + 17499977 q^{33} - 20071132 q^{34} + 7116564 q^{35} + 2121398 q^{36} + 29198909 q^{37} - 44208366 q^{38} - 4655443 q^{39} + 134932928 q^{40} + 48410973 q^{41} + 1130871 q^{42} + 52650242 q^{43} - 14827353 q^{44} + 99215088 q^{45} - 34410455 q^{46} + 160580841 q^{47} + 227620515 q^{48} + 80707214 q^{49} + 149462949 q^{50} + 57114360 q^{51} - 68232229 q^{52} + 80753796 q^{53} + 301368833 q^{54} + 328919412 q^{55} + 103874463 q^{56} + 151101102 q^{57} + 335044204 q^{58} + 442445502 q^{59} + 561078360 q^{60} + 270199089 q^{61} + 543824517 q^{62} + 346053729 q^{63} + 223643137 q^{64} - 84654804 q^{65} + 317483345 q^{66} + 92500909 q^{67} + 255771204 q^{68} + 292017029 q^{69} + 303784124 q^{70} + 84383796 q^{71} + 1456696818 q^{72} + 367274315 q^{73} + 1091659407 q^{74} + 1154152501 q^{75} + 674789222 q^{76} + 196461825 q^{77} - 13452231 q^{78} + 434861545 q^{79} + 2644363752 q^{80} + 644207518 q^{81} + 634104331 q^{82} + 1013603934 q^{83} + 377914999 q^{84} + 1103701048 q^{85} + 2514069096 q^{86} + 1039292304 q^{87} + 1071310221 q^{88} + 1069739706 q^{89} - 1271572324 q^{90} - 960049454 q^{91} + 2301673917 q^{92} - 933838861 q^{93} + 2025486277 q^{94} + 2504029998 q^{95} - 116199027 q^{96} + 2839636281 q^{97} + 155649627 q^{98} + 5063037274 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 4752 x^{12} + 9346 x^{11} + 8576824 x^{10} - 26923636 x^{9} - 7450416552 x^{8} + 31594524240 x^{7} + 3232668379296 x^{6} + \cdots - 24\!\cdots\!40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 679 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 64\!\cdots\!57 \nu^{13} + \cdots + 10\!\cdots\!40 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!69 \nu^{13} + \cdots - 57\!\cdots\!92 ) / 91\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!31 \nu^{13} + \cdots + 27\!\cdots\!00 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!17 \nu^{13} + \cdots + 47\!\cdots\!00 ) / 38\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 67\!\cdots\!27 \nu^{13} + \cdots - 12\!\cdots\!00 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 38\!\cdots\!01 \nu^{13} + \cdots + 29\!\cdots\!80 ) / 30\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!67 \nu^{13} + \cdots + 31\!\cdots\!88 ) / 91\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 22\!\cdots\!11 \nu^{13} + \cdots - 45\!\cdots\!40 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 91\!\cdots\!79 \nu^{13} + \cdots + 31\!\cdots\!36 ) / 18\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!09 \nu^{13} + \cdots - 42\!\cdots\!80 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 77\!\cdots\!61 \nu^{13} + \cdots + 92\!\cdots\!60 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 679 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 3\beta_{6} + 4\beta_{4} + \beta_{3} + \beta_{2} + 1139\beta _1 - 1068 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{13} - 5 \beta_{12} - 10 \beta_{10} + 2 \beta_{9} - 7 \beta_{8} + 5 \beta_{7} + 40 \beta_{6} + 4 \beta_{5} - 448 \beta_{4} + 2 \beta_{3} + 1703 \beta_{2} - 2712 \beta _1 + 774355 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 57 \beta_{13} + 139 \beta_{12} - 16 \beta_{11} + 100 \beta_{10} + 2598 \beta_{9} + 199 \beta_{8} + 25 \beta_{7} - 5378 \beta_{6} + 96 \beta_{5} + 2290 \beta_{4} + 2200 \beta_{3} + 4171 \beta_{2} + 1555516 \beta _1 - 1359331 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22473 \beta_{13} - 8539 \beta_{12} - 482 \beta_{11} - 28010 \beta_{10} + 8992 \beta_{9} - 19875 \beta_{8} + 11609 \beta_{7} + 113712 \beta_{6} + 4906 \beta_{5} - 1243376 \beta_{4} + \cdots + 1058849655 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 166839 \beta_{13} + 450971 \beta_{12} - 29406 \beta_{11} + 389990 \beta_{10} + 5351348 \beta_{9} + 463823 \beta_{8} - 62865 \beta_{7} - 8126240 \beta_{6} + 266430 \beta_{5} + \cdots - 806964759 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 45212397 \beta_{13} - 12253027 \beta_{12} - 675838 \beta_{11} - 59179538 \beta_{10} + 24047884 \beta_{9} - 41169943 \beta_{8} + 21878741 \beta_{7} + 233781560 \beta_{6} + \cdots + 1584160438431 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 373693479 \beta_{13} + 1029825399 \beta_{12} - 43167866 \beta_{11} + 981276858 \beta_{10} + 10194694092 \beta_{9} + 804596651 \beta_{8} - 327648225 \beta_{7} + \cdots + 1051406974421 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 84497221453 \beta_{13} - 17004072115 \beta_{12} - 496041614 \beta_{11} - 112706667682 \beta_{10} + 53523605148 \beta_{9} - 76892259527 \beta_{8} + \cdots + 24\!\cdots\!07 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 761884384903 \beta_{13} + 2053769756647 \beta_{12} - 64990791498 \beta_{11} + 2057998443082 \beta_{10} + 18702580757268 \beta_{9} + 1272899982587 \beta_{8} + \cdots + 53\!\cdots\!09 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 152576903255957 \beta_{13} - 23489568476763 \beta_{12} + 195383404914 \beta_{11} - 204409231580338 \beta_{10} + 109580433247180 \beta_{9} + \cdots + 40\!\cdots\!67 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14\!\cdots\!79 \beta_{13} + \cdots + 14\!\cdots\!37 \) Copy content Toggle raw display

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
41.8127
33.1611
26.5220
21.0054
19.1573
15.6702
4.59170
−1.08476
−12.8857
−14.8710
−25.1291
−31.6981
−34.2541
−40.9975
−39.8127 138.193 1073.05 553.584 −5501.81 2401.00 −22336.8 −585.818 −22039.7
1.2 −31.1611 −183.696 459.017 −1660.96 5724.18 2401.00 1651.02 14061.3 51757.3
1.3 −24.5220 116.541 89.3306 −1444.79 −2857.83 2401.00 10364.7 −6101.17 35429.2
1.4 −19.0054 −267.441 −150.796 1106.90 5082.81 2401.00 12596.7 51841.8 −21037.1
1.5 −17.1573 244.102 −217.626 2666.11 −4188.13 2401.00 12518.4 39902.6 −45743.4
1.6 −13.6702 −25.3649 −325.126 344.296 346.743 2401.00 11443.7 −19039.6 −4706.60
1.7 −2.59170 254.265 −505.283 −1903.58 −658.979 2401.00 2636.49 44967.7 4933.51
1.8 3.08476 −166.247 −502.484 1085.59 −512.832 2401.00 −3129.44 7955.00 3348.80
1.9 14.8857 54.8703 −290.415 1171.15 816.786 2401.00 −11944.5 −16672.2 17433.4
1.10 16.8710 −26.4361 −227.369 −2366.09 −446.004 2401.00 −12473.9 −18984.1 −39918.3
1.11 27.1291 −167.275 223.990 −1059.06 −4538.03 2401.00 −7813.47 8297.99 −28731.4
1.12 33.6981 231.670 623.565 224.065 7806.84 2401.00 3759.53 33987.9 7550.59
1.13 36.2541 −166.815 802.361 2128.44 −6047.74 2401.00 10526.8 8144.30 77164.9
1.14 42.9975 126.635 1336.79 2118.33 5444.99 2401.00 35463.9 −3646.62 91082.8
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.a.c 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 27 T_{2}^{13} - 4414 T_{2}^{12} + 102102 T_{2}^{11} + 7541636 T_{2}^{10} - 138358068 T_{2}^{9} - 6416361792 T_{2}^{8} + 83326151472 T_{2}^{7} + 2851128216192 T_{2}^{6} + \cdots - 38\!\cdots\!08 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 27 T^{13} + \cdots - 38\!\cdots\!08 \) Copy content Toggle raw display
$3$ \( T^{14} - 163 T^{13} + \cdots - 24\!\cdots\!20 \) Copy content Toggle raw display
$5$ \( T^{14} - 2964 T^{13} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 2401)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 81825 T^{13} + \cdots - 15\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T + 28561)^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 44922 T^{13} + \cdots - 57\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{14} - 171756 T^{13} + \cdots - 42\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( T^{14} - 1930479 T^{13} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + 3799608 T^{13} + \cdots + 52\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{14} + 4392203 T^{13} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{14} - 29198909 T^{13} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} - 48410973 T^{13} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{14} - 52650242 T^{13} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} - 160580841 T^{13} + \cdots + 78\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{14} - 80753796 T^{13} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{14} - 442445502 T^{13} + \cdots - 44\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{14} - 270199089 T^{13} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{14} - 92500909 T^{13} + \cdots - 38\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{14} - 84383796 T^{13} + \cdots - 17\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{14} - 367274315 T^{13} + \cdots - 19\!\cdots\!22 \) Copy content Toggle raw display
$79$ \( T^{14} - 434861545 T^{13} + \cdots - 16\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{14} - 1013603934 T^{13} + \cdots + 34\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{14} - 1069739706 T^{13} + \cdots + 68\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{14} - 2839636281 T^{13} + \cdots - 70\!\cdots\!42 \) Copy content Toggle raw display
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