Properties

Label 91.8.a.e
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} + \cdots + 59402280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 7) q^{3} + (\beta_{2} + 82) q^{4} + ( - \beta_{4} - \beta_{3} - 3 \beta_1 + 87) q^{5} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 28) q^{6} + 343 q^{7} + ( - \beta_{8} - \beta_{4} - 15 \beta_{3} - 86 \beta_1 + 60) q^{8} + (\beta_{11} - 3 \beta_{4} + 12 \beta_{3} + 2 \beta_{2} + 20 \beta_1 + 901) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 7) q^{3} + (\beta_{2} + 82) q^{4} + ( - \beta_{4} - \beta_{3} - 3 \beta_1 + 87) q^{5} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 28) q^{6} + 343 q^{7} + ( - \beta_{8} - \beta_{4} - 15 \beta_{3} - 86 \beta_1 + 60) q^{8} + (\beta_{11} - 3 \beta_{4} + 12 \beta_{3} + 2 \beta_{2} + 20 \beta_1 + 901) q^{9} + ( - \beta_{11} + \beta_{9} - 3 \beta_{4} + 5 \beta_{3} + 7 \beta_{2} - 106 \beta_1 + 609) q^{10} + (\beta_{8} + \beta_{6} + 3 \beta_{4} + 16 \beta_{3} + 8 \beta_{2} + 46 \beta_1 + 992) q^{11} + ( - \beta_{11} - 3 \beta_{9} + 3 \beta_{8} - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \cdots - 106) q^{12}+ \cdots + ( - 1598 \beta_{11} + 779 \beta_{10} + 1018 \beta_{9} + \cdots + 803049) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9} + 6668 q^{10} + 12168 q^{11} - 183 q^{12} + 26364 q^{13} + 2058 q^{14} - 28790 q^{15} + 85914 q^{16} + 82710 q^{17} - 44965 q^{18} - 10302 q^{19} + 141318 q^{20} + 28126 q^{21} - 97457 q^{22} + 98376 q^{23} - 519981 q^{24} + 272736 q^{25} + 13182 q^{26} + 306652 q^{27} + 338198 q^{28} + 350592 q^{29} + 231528 q^{30} + 55092 q^{31} + 114420 q^{32} + 609912 q^{33} + 812002 q^{34} + 351918 q^{35} + 1472143 q^{36} + 376310 q^{37} + 2825424 q^{38} + 180154 q^{39} + 2169290 q^{40} + 1387272 q^{41} + 105987 q^{42} + 568708 q^{43} + 3392031 q^{44} + 3556226 q^{45} - 1736829 q^{46} + 1359444 q^{47} + 4151249 q^{48} + 1411788 q^{49} + 3983712 q^{50} + 2709260 q^{51} + 2166242 q^{52} + 2061780 q^{53} + 2196651 q^{54} - 2112846 q^{55} + 78204 q^{56} + 2359902 q^{57} + 670268 q^{58} + 395964 q^{59} - 1052376 q^{60} + 444006 q^{61} + 2854353 q^{62} + 3739386 q^{63} + 12026858 q^{64} + 2254122 q^{65} - 4605681 q^{66} - 3094010 q^{67} + 4668954 q^{68} + 3839892 q^{69} + 2287124 q^{70} + 5694366 q^{71} - 9780585 q^{72} + 7052346 q^{73} - 4436259 q^{74} - 16288696 q^{75} - 3051830 q^{76} + 4173624 q^{77} + 678873 q^{78} + 4304160 q^{79} + 3807018 q^{80} - 6689556 q^{81} - 4733665 q^{82} + 2704554 q^{83} - 62769 q^{84} + 9301878 q^{85} + 1510998 q^{86} + 16231802 q^{87} - 70453923 q^{88} - 10986042 q^{89} - 12851300 q^{90} + 9042852 q^{91} - 16505451 q^{92} - 47230934 q^{93} - 24306151 q^{94} - 21839424 q^{95} - 86512741 q^{96} - 24462382 q^{97} + 705894 q^{98} + 11555078 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} + \cdots + 59402280000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 209 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!97 \nu^{11} + \cdots + 16\!\cdots\!40 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!33 \nu^{11} + \cdots + 34\!\cdots\!20 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 67\!\cdots\!59 \nu^{11} + \cdots - 91\!\cdots\!60 ) / 25\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44\!\cdots\!49 \nu^{11} + \cdots - 11\!\cdots\!80 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\!\cdots\!39 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 93\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!61 \nu^{11} + \cdots - 34\!\cdots\!40 ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!13 \nu^{11} + \cdots + 39\!\cdots\!68 ) / 12\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!89 \nu^{11} + \cdots - 14\!\cdots\!68 ) / 42\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!93 \nu^{11} + \cdots + 47\!\cdots\!40 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 209 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{4} + 15\beta_{3} + 3\beta_{2} + 345\beta _1 + 312 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{11} + 4 \beta_{10} - \beta_{9} + 6 \beta_{8} - \beta_{7} - 5 \beta_{6} + 13 \beta_{5} + 5 \beta_{4} + 177 \beta_{3} + 500 \beta_{2} + 1371 \beta _1 + 71412 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{11} + 6 \beta_{10} - 60 \beta_{9} + 636 \beta_{8} + 44 \beta_{7} - 32 \beta_{6} + 102 \beta_{5} + 444 \beta_{4} + 10958 \beta_{3} + 2840 \beta_{2} + 138759 \beta _1 + 229516 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3024 \beta_{11} + 2494 \beta_{10} - 1240 \beta_{9} + 5248 \beta_{8} - 214 \beta_{7} - 3574 \beta_{6} + 11534 \beta_{5} - 1600 \beta_{4} + 150228 \beta_{3} + 237831 \beta_{2} + 859638 \beta _1 + 28454111 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 30290 \beta_{11} + 8156 \beta_{10} - 57708 \beta_{9} + 337579 \beta_{8} + 31150 \beta_{7} - 37726 \beta_{6} + 111982 \beta_{5} + 146303 \beta_{4} + 6630709 \beta_{3} + 1943243 \beta_{2} + \cdots + 151525518 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1739412 \beta_{11} + 1232150 \beta_{10} - 962609 \beta_{9} + 3561774 \beta_{8} + 125333 \beta_{7} - 2084039 \beta_{6} + 7571523 \beta_{5} - 2609795 \beta_{4} + 99880753 \beta_{3} + \cdots + 12316639262 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20964960 \beta_{11} + 6343594 \beta_{10} - 39535284 \beta_{9} + 172280246 \beta_{8} + 17399558 \beta_{7} - 29329626 \beta_{6} + 85364640 \beta_{5} + 38780526 \beta_{4} + \cdots + 93516704558 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 920147760 \beta_{11} + 573136640 \beta_{10} - 634693664 \beta_{9} + 2189379316 \beta_{8} + 166855192 \beta_{7} - 1151225688 \beta_{6} + 4445406756 \beta_{5} + \cdots + 5613426256161 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12687632436 \beta_{11} + 4070868840 \beta_{10} - 23812818640 \beta_{9} + 87449732997 \beta_{8} + 9157045524 \beta_{7} - 19286669780 \beta_{6} + \cdots + 55368631574220 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
23.0670
19.0596
15.8833
10.7733
10.5927
0.0691404
−4.06937
−4.08138
−12.7078
−13.1706
−18.9614
−20.4545
−22.0670 71.8770 358.952 25.2300 −1586.11 343.000 −5096.40 2979.31 −556.750
1.2 −18.0596 −64.0898 198.149 13.5880 1157.44 343.000 −1266.87 1920.50 −245.394
1.3 −14.8833 −5.24717 93.5131 −195.689 78.0952 343.000 513.280 −2159.47 2912.49
1.4 −9.77328 −73.6575 −32.4830 542.246 719.876 343.000 1568.45 3238.43 −5299.52
1.5 −9.59274 50.3591 −35.9794 255.731 −483.082 343.000 1573.01 349.039 −2453.16
1.6 0.930860 23.4478 −127.134 −494.625 21.8266 343.000 −237.493 −1637.20 −460.427
1.7 5.06937 91.5295 −102.301 −15.8892 463.997 343.000 −1167.48 6190.65 −80.5484
1.8 5.08138 −25.2372 −102.180 250.262 −128.240 343.000 −1169.63 −1550.08 1271.68
1.9 13.7078 50.2321 59.9043 544.453 688.573 343.000 −933.443 336.267 7463.26
1.10 14.1706 −35.5734 72.8048 −268.857 −504.095 343.000 −782.147 −921.532 −3809.85
1.11 19.9614 56.3144 270.459 1.53059 1124.12 343.000 2843.68 984.313 30.5528
1.12 21.4545 −57.9549 332.295 368.020 −1243.39 343.000 4383.05 1171.77 7895.68
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 91.8.a.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.e 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 6 T_{2}^{11} - 1243 T_{2}^{10} + 6942 T_{2}^{9} + 561506 T_{2}^{8} - 2852052 T_{2}^{7} - 113159964 T_{2}^{6} + 520830672 T_{2}^{5} + 9905539840 T_{2}^{4} + \cdots - 1109222885376 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 6 T^{11} + \cdots - 1109222885376 \) Copy content Toggle raw display
$3$ \( T^{12} - 82 T^{11} + \cdots + 28\!\cdots\!40 \) Copy content Toggle raw display
$5$ \( T^{12} - 1026 T^{11} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 343)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 12168 T^{11} + \cdots - 14\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( (T - 2197)^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 82710 T^{11} + \cdots - 63\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{12} + 10302 T^{11} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} - 98376 T^{11} + \cdots - 21\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{12} - 350592 T^{11} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 55092 T^{11} + \cdots - 45\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{12} - 376310 T^{11} + \cdots - 54\!\cdots\!40 \) Copy content Toggle raw display
$41$ \( T^{12} - 1387272 T^{11} + \cdots - 22\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{12} - 568708 T^{11} + \cdots - 19\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{12} - 1359444 T^{11} + \cdots - 42\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{12} - 2061780 T^{11} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} - 395964 T^{11} + \cdots + 96\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{12} - 444006 T^{11} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + 3094010 T^{11} + \cdots - 14\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{12} - 5694366 T^{11} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{12} - 7052346 T^{11} + \cdots - 57\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{12} - 4304160 T^{11} + \cdots + 98\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{12} - 2704554 T^{11} + \cdots + 12\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{12} + 10986042 T^{11} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + 24462382 T^{11} + \cdots - 41\!\cdots\!60 \) Copy content Toggle raw display
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