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Properties

Label 91.8.a.d
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{3} - \beta_1 - 10) q^{3} + (\beta_{2} + 36) q^{4} + ( - \beta_{6} - \beta_{3} - 2 \beta_1 + 23) q^{5} + (\beta_{8} + \beta_{7} - \beta_{6} - 6 \beta_{3} + \beta_{2} + 16 \beta_1 + 104) q^{6} - 343 q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 35 \beta_1 + 39) q^{8} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{6} + 6 \beta_{3} + 13 \beta_{2} - 97 \beta_1 + 1253) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{3} - \beta_1 - 10) q^{3} + (\beta_{2} + 36) q^{4} + ( - \beta_{6} - \beta_{3} - 2 \beta_1 + 23) q^{5} + (\beta_{8} + \beta_{7} - \beta_{6} - 6 \beta_{3} + \beta_{2} + 16 \beta_1 + 104) q^{6} - 343 q^{7} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 35 \beta_1 + 39) q^{8} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{6} + 6 \beta_{3} + 13 \beta_{2} - 97 \beta_1 + 1253) q^{9} + (\beta_{8} + \beta_{6} + 2 \beta_{4} - 36 \beta_{3} + 6 \beta_{2} + 19 \beta_1 + 241) q^{10} + ( - 5 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 16 \beta_{3} + \cdots + 37) q^{11}+ \cdots + ( - 2481 \beta_{9} - 7434 \beta_{8} - 23047 \beta_{7} + \cdots - 4252001) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9} + 2548 q^{10} + 451 q^{11} - 16241 q^{12} - 21970 q^{13} + 1029 q^{14} + 27184 q^{15} + 11897 q^{16} - 8654 q^{17} + 159348 q^{18} + 10130 q^{19} - 82012 q^{20} + 34643 q^{21} - 57863 q^{22} - 52155 q^{23} - 49227 q^{24} + 47190 q^{25} + 6591 q^{26} - 155171 q^{27} - 123823 q^{28} + 520154 q^{29} + 1070236 q^{30} + 692605 q^{31} + 149835 q^{32} + 436053 q^{33} + 1059060 q^{34} - 77518 q^{35} + 2843742 q^{36} - 20511 q^{37} + 1905286 q^{38} + 221897 q^{39} + 636320 q^{40} + 355049 q^{41} - 379015 q^{42} + 1256772 q^{43} - 687913 q^{44} + 1259926 q^{45} + 4043075 q^{46} + 1260721 q^{47} + 1128551 q^{48} + 1176490 q^{49} + 609035 q^{50} + 1411976 q^{51} - 793117 q^{52} + 928854 q^{53} + 6642607 q^{54} + 3423196 q^{55} - 99813 q^{56} + 3014966 q^{57} + 1612588 q^{58} + 3144446 q^{59} + 7738848 q^{60} + 6322923 q^{61} + 6545331 q^{62} - 4200721 q^{63} - 6629943 q^{64} - 496522 q^{65} - 14343317 q^{66} + 3944507 q^{67} - 1787356 q^{68} - 148281 q^{69} - 873964 q^{70} + 6032248 q^{71} + 9760866 q^{72} + 1248533 q^{73} - 8263279 q^{74} + 1573413 q^{75} + 1788254 q^{76} - 154693 q^{77} - 2427685 q^{78} - 14947605 q^{79} - 9147616 q^{80} + 25716334 q^{81} - 6987095 q^{82} - 14177784 q^{83} + 5570663 q^{84} - 11788444 q^{85} + 8748840 q^{86} - 29484448 q^{87} - 15390723 q^{88} + 6734836 q^{89} + 5994972 q^{90} + 7535710 q^{91} - 24493215 q^{92} + 17307847 q^{93} - 22760149 q^{94} - 9329708 q^{95} - 36488483 q^{96} - 12365397 q^{97} - 352947 q^{98} - 43198042 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 164 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2710730315 \nu^{9} - 10591077029 \nu^{8} - 2080232699020 \nu^{7} + 8256593138470 \nu^{6} + 475250169039696 \nu^{5} + \cdots - 20\!\cdots\!52 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3194242487 \nu^{9} - 1907850873301 \nu^{8} + 3542382315514 \nu^{7} + \cdots - 23\!\cdots\!96 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8702011532 \nu^{9} + 219416775281 \nu^{8} - 6637241612701 \nu^{7} - 166817327091610 \nu^{6} + \cdots + 29\!\cdots\!04 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 77448321017 \nu^{9} - 118851619051 \nu^{8} + 58643694893350 \nu^{7} + 91889448816290 \nu^{6} + \cdots - 21\!\cdots\!04 ) / 66\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27147386411 \nu^{9} + 204615595672 \nu^{8} - 20771638781257 \nu^{7} - 155247362966420 \nu^{6} + \cdots + 25\!\cdots\!24 ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19136930095 \nu^{9} - 96570802877 \nu^{8} + 14586243188762 \nu^{7} + 73109374683374 \nu^{6} + \cdots - 12\!\cdots\!48 ) / 733783735522432 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 131832796597 \nu^{9} + 712892783321 \nu^{8} - 100446640439600 \nu^{7} - 544641329504038 \nu^{6} + \cdots + 10\!\cdots\!56 ) / 33\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 164 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} + 291\beta _1 - 39 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{9} - 7 \beta_{8} - 8 \beta_{7} + 14 \beta_{6} + \beta_{5} + 2 \beta_{4} - 33 \beta_{3} + 383 \beta_{2} - 132 \beta _1 + 47819 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{9} - 49 \beta_{8} + 446 \beta_{7} + 888 \beta_{6} + 387 \beta_{5} + 428 \beta_{4} - 1683 \beta_{3} - 733 \beta_{2} + 96370 \beta _1 - 34217 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1077 \beta_{9} - 4935 \beta_{8} - 5062 \beta_{7} + 7492 \beta_{6} + 659 \beta_{5} + 942 \beta_{4} - 13063 \beta_{3} + 134191 \beta_{2} - 62784 \beta _1 + 15912781 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12079 \beta_{9} - 34639 \beta_{8} + 172474 \beta_{7} + 325824 \beta_{6} + 136231 \beta_{5} + 160342 \beta_{4} - 1061051 \beta_{3} - 238425 \beta_{2} + 32895336 \beta _1 - 14490419 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 303237 \beta_{9} - 2550843 \beta_{8} - 2497490 \beta_{7} + 3250664 \beta_{6} + 313483 \beta_{5} + 351318 \beta_{4} - 4270631 \beta_{3} + 46699871 \beta_{2} - 22807272 \beta _1 + 5459291269 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7698359 \beta_{9} - 17800439 \beta_{8} + 65138942 \beta_{7} + 115752360 \beta_{6} + 47493695 \beta_{5} + 58790582 \beta_{4} - 527514763 \beta_{3} + \cdots - 5176466335 \) 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
19.2053
17.3557
9.01767
8.95987
3.66228
−4.83483
−6.50047
−6.95530
−17.8475
−19.0627
−19.2053 20.2953 240.842 −234.137 −389.776 −343.000 −2167.15 −1775.10 4496.66
1.2 −17.3557 −89.1399 173.221 −9.67118 1547.09 −343.000 −784.843 5758.93 167.850
1.3 −9.01767 −31.1400 −46.6816 514.920 280.810 −343.000 1575.22 −1217.30 −4643.38
1.4 −8.95987 −21.0756 −47.7208 −55.0053 188.835 −343.000 1574.43 −1742.82 492.840
1.5 −3.66228 42.3449 −114.588 −251.753 −155.079 −343.000 888.424 −393.907 921.992
1.6 4.83483 62.4820 −104.624 314.530 302.090 −343.000 −1124.70 1717.00 1520.70
1.7 6.50047 −52.7234 −85.7438 295.835 −342.727 −343.000 −1389.44 592.761 1923.06
1.8 6.95530 −32.1557 −79.6238 −381.581 −223.652 −343.000 −1444.09 −1153.01 −2654.01
1.9 17.8475 86.1795 190.532 250.298 1538.09 −343.000 1116.04 5239.91 4467.19
1.10 19.0627 −86.0671 235.388 −217.435 −1640.67 −343.000 2047.10 5220.54 −4144.91
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.d ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.d ✓ 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 3 T_{2}^{9} - 816 T_{2}^{8} - 2298 T_{2}^{7} + 213848 T_{2}^{6} + 507132 T_{2}^{5} - 19919976 T_{2}^{4} - 24331248 T_{2}^{3} + 727257184 T_{2}^{2} + 56397312 T_{2} - 7335224320 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 3 T^{9} + \cdots - 7335224320 \) Copy content Toggle raw display
$3$ \( T^{10} + 101 T^{9} + \cdots + 39\!\cdots\!08 \) Copy content Toggle raw display
$5$ \( T^{10} - 226 T^{9} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T + 343)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 451 T^{9} + \cdots - 10\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( (T + 2197)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 8654 T^{9} + \cdots - 90\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{10} - 10130 T^{9} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{10} + 52155 T^{9} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{10} - 520154 T^{9} + \cdots + 97\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{10} - 692605 T^{9} + \cdots - 12\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{10} + 20511 T^{9} + \cdots + 31\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{10} - 355049 T^{9} + \cdots + 73\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{10} - 1256772 T^{9} + \cdots - 50\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{10} - 1260721 T^{9} + \cdots + 68\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{10} - 928854 T^{9} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} - 3144446 T^{9} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} - 6322923 T^{9} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} - 3944507 T^{9} + \cdots + 65\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{10} - 6032248 T^{9} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{10} - 1248533 T^{9} + \cdots - 50\!\cdots\!06 \) Copy content Toggle raw display
$79$ \( T^{10} + 14947605 T^{9} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{10} + 14177784 T^{9} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{10} - 6734836 T^{9} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + 12365397 T^{9} + \cdots + 60\!\cdots\!54 \) Copy content Toggle raw display
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