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: |
\( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \)
|
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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} - 164 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 2710730315 \nu^{9} - 10591077029 \nu^{8} - 2080232699020 \nu^{7} + 8256593138470 \nu^{6} + 475250169039696 \nu^{5} + \cdots - 20\!\cdots\!52 ) / 66\!\cdots\!88 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 3194242487 \nu^{9} - 1907850873301 \nu^{8} + 3542382315514 \nu^{7} + \cdots - 23\!\cdots\!96 ) / 66\!\cdots\!88 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 8702011532 \nu^{9} + 219416775281 \nu^{8} - 6637241612701 \nu^{7} - 166817327091610 \nu^{6} + \cdots + 29\!\cdots\!04 ) / 11\!\cdots\!48 \)
|
\(\beta_{6}\) | \(=\) |
\( ( - 77448321017 \nu^{9} - 118851619051 \nu^{8} + 58643694893350 \nu^{7} + 91889448816290 \nu^{6} + \cdots - 21\!\cdots\!04 ) / 66\!\cdots\!88 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 27147386411 \nu^{9} + 204615595672 \nu^{8} - 20771638781257 \nu^{7} - 155247362966420 \nu^{6} + \cdots + 25\!\cdots\!24 ) / 16\!\cdots\!72 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 19136930095 \nu^{9} - 96570802877 \nu^{8} + 14586243188762 \nu^{7} + 73109374683374 \nu^{6} + \cdots - 12\!\cdots\!48 ) / 733783735522432 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 131832796597 \nu^{9} + 712892783321 \nu^{8} - 100446640439600 \nu^{7} - 544641329504038 \nu^{6} + \cdots + 10\!\cdots\!56 ) / 33\!\cdots\!44 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} + 164 \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} + 291\beta _1 - 39 \)
|
\(\nu^{4}\) | \(=\) |
\( 3 \beta_{9} - 7 \beta_{8} - 8 \beta_{7} + 14 \beta_{6} + \beta_{5} + 2 \beta_{4} - 33 \beta_{3} + 383 \beta_{2} - 132 \beta _1 + 47819 \)
|
\(\nu^{5}\) | \(=\) |
\( - 9 \beta_{9} - 49 \beta_{8} + 446 \beta_{7} + 888 \beta_{6} + 387 \beta_{5} + 428 \beta_{4} - 1683 \beta_{3} - 733 \beta_{2} + 96370 \beta _1 - 34217 \)
|
\(\nu^{6}\) | \(=\) |
\( 1077 \beta_{9} - 4935 \beta_{8} - 5062 \beta_{7} + 7492 \beta_{6} + 659 \beta_{5} + 942 \beta_{4} - 13063 \beta_{3} + 134191 \beta_{2} - 62784 \beta _1 + 15912781 \)
|
\(\nu^{7}\) | \(=\) |
\( - 12079 \beta_{9} - 34639 \beta_{8} + 172474 \beta_{7} + 325824 \beta_{6} + 136231 \beta_{5} + 160342 \beta_{4} - 1061051 \beta_{3} - 238425 \beta_{2} + 32895336 \beta _1 - 14490419 \)
|
\(\nu^{8}\) | \(=\) |
\( 303237 \beta_{9} - 2550843 \beta_{8} - 2497490 \beta_{7} + 3250664 \beta_{6} + 313483 \beta_{5} + 351318 \beta_{4} - 4270631 \beta_{3} + 46699871 \beta_{2} - 22807272 \beta _1 + 5459291269 \)
|
\(\nu^{9}\) | \(=\) |
\( - 7698359 \beta_{9} - 17800439 \beta_{8} + 65138942 \beta_{7} + 115752360 \beta_{6} + 47493695 \beta_{5} + 58790582 \beta_{4} - 527514763 \beta_{3} + \cdots - 5176466335 \)
|
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 | 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 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} + 3 T^{9} + \cdots - 7335224320 \)
$3$
\( T^{10} + 101 T^{9} + \cdots + 39\!\cdots\!08 \)
$5$
\( T^{10} - 226 T^{9} + \cdots + 31\!\cdots\!00 \)
$7$
\( (T + 343)^{10} \)
$11$
\( T^{10} - 451 T^{9} + \cdots - 10\!\cdots\!20 \)
$13$
\( (T + 2197)^{10} \)
$17$
\( T^{10} + 8654 T^{9} + \cdots - 90\!\cdots\!44 \)
$19$
\( T^{10} - 10130 T^{9} + \cdots + 12\!\cdots\!44 \)
$23$
\( T^{10} + 52155 T^{9} + \cdots + 40\!\cdots\!12 \)
$29$
\( T^{10} - 520154 T^{9} + \cdots + 97\!\cdots\!12 \)
$31$
\( T^{10} - 692605 T^{9} + \cdots - 12\!\cdots\!20 \)
$37$
\( T^{10} + 20511 T^{9} + \cdots + 31\!\cdots\!80 \)
$41$
\( T^{10} - 355049 T^{9} + \cdots + 73\!\cdots\!40 \)
$43$
\( T^{10} - 1256772 T^{9} + \cdots - 50\!\cdots\!12 \)
$47$
\( T^{10} - 1260721 T^{9} + \cdots + 68\!\cdots\!72 \)
$53$
\( T^{10} - 928854 T^{9} + \cdots - 10\!\cdots\!00 \)
$59$
\( T^{10} - 3144446 T^{9} + \cdots + 58\!\cdots\!00 \)
$61$
\( T^{10} - 6322923 T^{9} + \cdots + 74\!\cdots\!00 \)
$67$
\( T^{10} - 3944507 T^{9} + \cdots + 65\!\cdots\!80 \)
$71$
\( T^{10} - 6032248 T^{9} + \cdots + 18\!\cdots\!24 \)
$73$
\( T^{10} - 1248533 T^{9} + \cdots - 50\!\cdots\!06 \)
$79$
\( T^{10} + 14947605 T^{9} + \cdots + 31\!\cdots\!48 \)
$83$
\( T^{10} + 14177784 T^{9} + \cdots + 14\!\cdots\!56 \)
$89$
\( T^{10} - 6734836 T^{9} + \cdots - 24\!\cdots\!00 \)
$97$
\( T^{10} + 12365397 T^{9} + \cdots + 60\!\cdots\!54 \)
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