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Properties

Label 91.8.a.b
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} - 22500802 x^{2} + 176026849 x + 176334338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - 3) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 43) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_{2} - 3 \beta_1 - 19) q^{5} + ( - \beta_{8} + \beta_{6} + \beta_{3} - 15 \beta_1 + 84) q^{6} - 343 q^{7} + (2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + \cdots + 114) q^{8}+ \cdots + (\beta_{8} - 4 \beta_{7} - \beta_{6} + 5 \beta_{4} - 11 \beta_{3} + \beta_{2} - 10 \beta_1 + 366) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - 3) q^{3} + (\beta_{3} - \beta_{2} + \beta_1 + 43) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_{2} - 3 \beta_1 - 19) q^{5} + ( - \beta_{8} + \beta_{6} + \beta_{3} - 15 \beta_1 + 84) q^{6} - 343 q^{7} + (2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + \cdots + 114) q^{8}+ \cdots + (11614 \beta_{8} - 8073 \beta_{7} - 12235 \beta_{6} + \cdots + 5624142) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} - 26 q^{3} + 393 q^{4} - 181 q^{5} + 703 q^{6} - 3087 q^{7} + 1197 q^{8} + 3211 q^{9} - 5124 q^{10} - 9826 q^{11} - 20919 q^{12} + 19773 q^{13} + 1715 q^{14} - 20346 q^{15} + 31113 q^{16} - 22766 q^{17} - 12978 q^{18} - 17769 q^{19} - 44204 q^{20} + 8918 q^{21} - 203553 q^{22} - 49103 q^{23} + 52737 q^{24} + 227466 q^{25} - 10985 q^{26} + 103624 q^{27} - 134799 q^{28} - 487455 q^{29} - 287992 q^{30} - 63843 q^{31} - 587099 q^{32} - 314392 q^{33} - 576240 q^{34} + 62083 q^{35} - 1514926 q^{36} - 796926 q^{37} - 766702 q^{38} - 57122 q^{39} - 2887296 q^{40} - 1567546 q^{41} - 241129 q^{42} - 277899 q^{43} - 1281195 q^{44} - 1650593 q^{45} - 1907445 q^{46} + 1077367 q^{47} - 1110835 q^{48} + 1058841 q^{49} - 267459 q^{50} - 3054368 q^{51} + 863421 q^{52} - 7322659 q^{53} - 3355387 q^{54} - 2613324 q^{55} - 410571 q^{56} - 3751946 q^{57} - 2992332 q^{58} - 169804 q^{59} - 2754416 q^{60} - 6352284 q^{61} + 6001087 q^{62} - 1101373 q^{63} + 1657017 q^{64} - 397657 q^{65} - 5962713 q^{66} + 921120 q^{67} + 5615224 q^{68} - 5202780 q^{69} + 1757532 q^{70} + 3786654 q^{71} + 2229758 q^{72} + 5792889 q^{73} - 1991961 q^{74} + 145628 q^{75} - 2806026 q^{76} + 3370318 q^{77} + 1544491 q^{78} + 3464037 q^{79} + 15422512 q^{80} - 5010363 q^{81} - 12539943 q^{82} + 6834945 q^{83} + 7175217 q^{84} + 3880662 q^{85} - 7977524 q^{86} + 3727078 q^{87} + 7013709 q^{88} - 20408371 q^{89} + 34910060 q^{90} - 6782139 q^{91} - 3544371 q^{92} + 3121742 q^{93} + 61343967 q^{94} + 3360807 q^{95} + 23547905 q^{96} + 41644125 q^{97} - 588245 q^{98} + 50754068 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} - 22500802 x^{2} + 176026849 x + 176334338 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6015827 \nu^{8} + 20549143 \nu^{7} + 4991944185 \nu^{6} - 10970858567 \nu^{5} - 1276673073927 \nu^{4} + \cdots - 881196617033550 ) / 10791021610368 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6015827 \nu^{8} + 20549143 \nu^{7} + 4991944185 \nu^{6} - 10970858567 \nu^{5} - 1276673073927 \nu^{4} + \cdots - 27\!\cdots\!10 ) / 10791021610368 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23932735 \nu^{8} + 207426657 \nu^{7} - 19557244969 \nu^{6} - 180490638053 \nu^{5} + 4805504970323 \nu^{4} + \cdots + 19\!\cdots\!74 ) / 21582043220736 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25387529 \nu^{8} + 253465957 \nu^{7} + 16934639179 \nu^{6} - 118481098757 \nu^{5} - 2968338922149 \nu^{4} + \cdots + 16\!\cdots\!46 ) / 10791021610368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97509171 \nu^{8} + 1324540219 \nu^{7} + 60465028597 \nu^{6} - 697004753079 \nu^{5} - 9709920887303 \nu^{4} + \cdots - 56\!\cdots\!62 ) / 21582043220736 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28587983 \nu^{8} - 424374023 \nu^{7} - 17815070701 \nu^{6} + 242290842287 \nu^{5} + 2849696573915 \nu^{4} + \cdots + 804364226993614 ) / 5395510805184 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 114544149 \nu^{8} + 615032077 \nu^{7} + 83581078923 \nu^{6} - 310190501481 \nu^{5} - 17860948796465 \nu^{4} + \cdots - 13\!\cdots\!18 ) / 21582043220736 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 3\beta _1 + 170 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{8} + 3\beta_{7} + 2\beta_{6} + \beta_{5} + 4\beta_{4} - \beta_{3} - 2\beta_{2} + 313\beta _1 + 369 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 16\beta_{7} - 38\beta_{6} + 47\beta_{5} + 3\beta_{4} + 420\beta_{3} - 466\beta_{2} + 1365\beta _1 + 53267 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 830 \beta_{8} + 1708 \beta_{7} + 1252 \beta_{6} + 706 \beta_{5} + 1970 \beta_{4} - 284 \beta_{3} - 592 \beta_{2} + 114811 \beta _1 + 174164 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1316 \beta_{8} - 6312 \beta_{7} - 19064 \beta_{6} + 28520 \beta_{5} + 4964 \beta_{4} + 173513 \beta_{3} - 176957 \beta_{2} + 546579 \beta _1 + 19568550 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 294670 \beta_{8} + 788851 \beta_{7} + 603418 \beta_{6} + 386369 \beta_{5} + 897256 \beta_{4} - 30045 \beta_{3} - 13922 \beta_{2} + 44657045 \beta _1 + 71334873 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1418771 \beta_{8} - 1673588 \beta_{7} - 7584502 \beta_{6} + 13920203 \beta_{5} + 3739871 \beta_{4} + 71823548 \beta_{3} - 65854578 \beta_{2} + 219482481 \beta _1 + 7634119271 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−20.0538
−11.7379
−10.5479
−5.28525
−0.950186
3.64558
10.6825
17.5192
20.7278
−21.0538 −33.4331 315.264 271.446 703.895 −343.000 −3942.62 −1069.23 −5714.98
1.2 −12.7379 −48.2951 34.2551 −506.573 615.180 −343.000 1194.12 145.416 6452.69
1.3 −11.5479 14.0702 5.35382 200.806 −162.481 −343.000 1416.30 −1989.03 −2318.88
1.4 −6.28525 79.3185 −88.4957 −293.470 −498.536 −343.000 1360.73 4104.43 1844.53
1.5 −1.95019 −76.8097 −124.197 57.0015 149.793 −343.000 491.831 3712.72 −111.163
1.6 2.64558 25.8257 −121.001 466.249 68.3242 −343.000 −658.753 −1520.03 1233.50
1.7 9.68252 60.1290 −34.2488 −158.582 582.200 −343.000 −1570.98 1428.50 −1535.48
1.8 16.5192 −52.3583 144.884 210.689 −864.917 −343.000 278.914 554.387 3480.42
1.9 19.7278 5.55266 261.185 −428.566 109.542 −343.000 2627.45 −2156.17 −8454.65
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.b ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.b ✓ 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 5 T_{2}^{8} - 760 T_{2}^{7} - 3814 T_{2}^{6} + 169652 T_{2}^{5} + 935392 T_{2}^{4} - 11208672 T_{2}^{3} - 60001504 T_{2}^{2} + 92525632 T_{2} + 316890112 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 5 T^{8} - 760 T^{7} + \cdots + 316890112 \) Copy content Toggle raw display
$3$ \( T^{9} + 26 T^{8} + \cdots - 62487429836832 \) Copy content Toggle raw display
$5$ \( T^{9} + 181 T^{8} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 343)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 9826 T^{8} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T - 2197)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 22766 T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{9} + 17769 T^{8} + \cdots - 66\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( T^{9} + 49103 T^{8} + \cdots + 39\!\cdots\!59 \) Copy content Toggle raw display
$29$ \( T^{9} + 487455 T^{8} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{9} + 63843 T^{8} + \cdots + 22\!\cdots\!51 \) Copy content Toggle raw display
$37$ \( T^{9} + 796926 T^{8} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{9} + 1567546 T^{8} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{9} + 277899 T^{8} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{9} - 1077367 T^{8} + \cdots - 14\!\cdots\!23 \) Copy content Toggle raw display
$53$ \( T^{9} + 7322659 T^{8} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{9} + 169804 T^{8} + \cdots - 27\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{9} + 6352284 T^{8} + \cdots - 56\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{9} - 921120 T^{8} + \cdots + 15\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{9} - 3786654 T^{8} + \cdots - 73\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{9} - 5792889 T^{8} + \cdots + 69\!\cdots\!79 \) Copy content Toggle raw display
$79$ \( T^{9} - 3464037 T^{8} + \cdots - 13\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{9} - 6834945 T^{8} + \cdots - 22\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{9} + 20408371 T^{8} + \cdots + 84\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{9} - 41644125 T^{8} + \cdots + 94\!\cdots\!47 \) Copy content Toggle raw display
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