Newspace parameters
| Level: | \( N \) | \(=\) | \( 9065 = 5 \cdot 7^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9065.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.3843894323\) |
| Analytic rank: | \(1\) |
| Dimension: | \(17\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{17} - \cdots)\) |
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| Defining polynomial: |
\( x^{17} - x^{16} - 23 x^{15} + 23 x^{14} + 209 x^{13} - 205 x^{12} - 971 x^{11} + 907 x^{10} + 2497 x^{9} + \cdots + 14 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{16}\) 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
\( x^{17} - x^{16} - 23 x^{15} + 23 x^{14} + 209 x^{13} - 205 x^{12} - 971 x^{11} + 907 x^{10} + 2497 x^{9} + \cdots + 14 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 8907 \nu^{16} + 3003 \nu^{15} - 212375 \nu^{14} - 81971 \nu^{13} + 2014126 \nu^{12} + 929475 \nu^{11} + \cdots - 1202107 ) / 41143 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13900 \nu^{16} - 834 \nu^{15} + 330336 \nu^{14} + 25519 \nu^{13} - 3124085 \nu^{12} + \cdots + 608988 ) / 41143 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 14779 \nu^{16} - 6519 \nu^{15} - 347650 \nu^{14} + 135980 \nu^{13} + 3249407 \nu^{12} + \cdots - 375292 ) / 41143 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 20929 \nu^{16} + 6150 \nu^{15} + 487886 \nu^{14} - 133717 \nu^{13} - 4510141 \nu^{12} + \cdots + 246922 ) / 41143 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 20929 \nu^{16} - 6150 \nu^{15} - 487886 \nu^{14} + 133717 \nu^{13} + 4510141 \nu^{12} + \cdots - 205779 ) / 41143 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 25776 \nu^{16} + 4838 \nu^{15} - 586074 \nu^{14} - 107385 \nu^{13} + 5261013 \nu^{12} + \cdots - 1046079 ) / 41143 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 26967 \nu^{16} + 4142 \nu^{15} + 639443 \nu^{14} - 68329 \nu^{13} - 6030270 \nu^{12} + \cdots + 1075006 ) / 41143 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 32817 \nu^{16} + 3791 \nu^{15} + 749758 \nu^{14} - 98288 \nu^{13} - 6758869 \nu^{12} + \cdots + 126024 ) / 41143 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 35534 \nu^{16} + 9388 \nu^{15} + 828879 \nu^{14} - 192934 \nu^{13} - 7660739 \nu^{12} + \cdots + 751884 ) / 41143 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 49888 \nu^{16} - 4639 \nu^{15} + 1158141 \nu^{14} + 112297 \nu^{13} - 10667147 \nu^{12} + \cdots + 2172650 ) / 41143 \)
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| \(\beta_{13}\) | \(=\) |
\( ( - 53706 \nu^{16} + 18172 \nu^{15} + 1244626 \nu^{14} - 406359 \nu^{13} - 11417488 \nu^{12} + \cdots - 2527 ) / 41143 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 56802 \nu^{16} + 31152 \nu^{15} + 1304907 \nu^{14} - 702493 \nu^{13} - 11832075 \nu^{12} + \cdots - 786049 ) / 41143 \)
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| \(\beta_{15}\) | \(=\) |
\( ( 68557 \nu^{16} - 34561 \nu^{15} - 1587937 \nu^{14} + 779818 \nu^{13} + 14555978 \nu^{12} + \cdots + 959162 ) / 41143 \)
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| \(\beta_{16}\) | \(=\) |
\( ( 78874 \nu^{16} + 1441 \nu^{15} - 1821626 \nu^{14} - 37827 \nu^{13} + 16664273 \nu^{12} + \cdots - 2215472 ) / 41143 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 5\beta _1 - 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{13} - \beta_{11} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 8\beta_{2} + \beta _1 + 15 \)
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| \(\nu^{5}\) | \(=\) |
\( -\beta_{15} - 2\beta_{13} + \beta_{10} + \beta_{9} + 10\beta_{7} + 9\beta_{6} + 31\beta _1 - 10 \)
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| \(\nu^{6}\) | \(=\) |
\( - \beta_{16} + 10 \beta_{13} - \beta_{12} - 12 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} + \cdots + 88 \)
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| \(\nu^{7}\) | \(=\) |
\( \beta_{16} - 13 \beta_{15} - \beta_{14} - 24 \beta_{13} - \beta_{12} + \beta_{11} + 12 \beta_{10} + \cdots - 76 \)
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| \(\nu^{8}\) | \(=\) |
\( - 15 \beta_{16} + 2 \beta_{15} + \beta_{14} + 81 \beta_{13} - 12 \beta_{12} - 109 \beta_{11} + \cdots + 555 \)
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| \(\nu^{9}\) | \(=\) |
\( 18 \beta_{16} - 124 \beta_{15} - 15 \beta_{14} - 217 \beta_{13} - 15 \beta_{12} + 17 \beta_{11} + \cdots - 528 \)
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| \(\nu^{10}\) | \(=\) |
\( - 161 \beta_{16} + 35 \beta_{15} + 18 \beta_{14} + 620 \beta_{13} - 107 \beta_{12} - 901 \beta_{11} + \cdots + 3655 \)
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| \(\nu^{11}\) | \(=\) |
\( 214 \beta_{16} - 1058 \beta_{15} - 157 \beta_{14} - 1781 \beta_{13} - 155 \beta_{12} + 197 \beta_{11} + \cdots - 3532 \)
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| \(\nu^{12}\) | \(=\) |
\( - 1502 \beta_{16} + 411 \beta_{15} + 214 \beta_{14} + 4669 \beta_{13} - 861 \beta_{12} - 7144 \beta_{11} + \cdots + 24781 \)
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| \(\nu^{13}\) | \(=\) |
\( 2131 \beta_{16} - 8573 \beta_{15} - 1429 \beta_{14} - 14008 \beta_{13} - 1385 \beta_{12} + 1942 \beta_{11} + \cdots - 23256 \)
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| \(\nu^{14}\) | \(=\) |
\( - 12996 \beta_{16} + 4069 \beta_{15} + 2127 \beta_{14} + 35022 \beta_{13} - 6627 \beta_{12} + \cdots + 171548 \)
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| \(\nu^{15}\) | \(=\) |
\( 19296 \beta_{16} - 67586 \beta_{15} - 12133 \beta_{14} - 107854 \beta_{13} - 11524 \beta_{12} + \cdots - 152447 \)
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| \(\nu^{16}\) | \(=\) |
\( - 107476 \beta_{16} + 36711 \beta_{15} + 19192 \beta_{14} + 262542 \beta_{13} - 49956 \beta_{12} + \cdots + 1206120 \)
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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 |
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−2.73034 | −1.31730 | 5.45475 | 1.00000 | 3.59668 | 0 | −9.43265 | −1.26471 | −2.73034 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43819 | 1.64223 | 3.94477 | 1.00000 | −4.00408 | 0 | −4.74171 | −0.303065 | −2.43819 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76952 | 2.60485 | 1.13121 | 1.00000 | −4.60935 | 0 | 1.53734 | 3.78527 | −1.76952 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.62951 | 0.786801 | 0.655295 | 1.00000 | −1.28210 | 0 | 2.19121 | −2.38094 | −1.62951 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.39989 | −3.04811 | −0.0403202 | 1.00000 | 4.26700 | 0 | 2.85621 | 6.29096 | −1.39989 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.06032 | −0.385754 | −0.875718 | 1.00000 | 0.409024 | 0 | 3.04919 | −2.85119 | −1.06032 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.635391 | −1.02963 | −1.59628 | 1.00000 | 0.654216 | 0 | 2.28504 | −1.93987 | −0.635391 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.193652 | −2.10787 | −1.96250 | 1.00000 | 0.408193 | 0 | 0.767346 | 1.44311 | −0.193652 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −0.154938 | 1.82884 | −1.97599 | 1.00000 | −0.283356 | 0 | 0.616032 | 0.344640 | −0.154938 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.846570 | 0.429369 | −1.28332 | 1.00000 | 0.363491 | 0 | −2.77956 | −2.81564 | 0.846570 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.860727 | 3.44167 | −1.25915 | 1.00000 | 2.96234 | 0 | −2.80524 | 8.84510 | 0.860727 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.04553 | −1.71638 | −0.906859 | 1.00000 | −1.79453 | 0 | −3.03922 | −0.0540515 | 1.04553 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.66207 | 0.197771 | 0.762469 | 1.00000 | 0.328708 | 0 | −2.05686 | −2.96089 | 1.66207 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.84982 | 1.68709 | 1.42185 | 1.00000 | 3.12081 | 0 | −1.06948 | −0.153738 | 1.84982 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 1.97186 | −3.27267 | 1.88824 | 1.00000 | −6.45325 | 0 | −0.220383 | 7.71036 | 1.97186 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.04084 | −0.493323 | 2.16504 | 1.00000 | −1.00680 | 0 | 0.336816 | −2.75663 | 2.04084 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.73432 | −0.247596 | 5.47652 | 1.00000 | −0.677008 | 0 | 9.50592 | −2.93870 | 2.73432 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(37\) | \( +1 \) |
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 | 9065.2.a.p | ✓ | 17 |
| 7.b | odd | 2 | 1 | 9065.2.a.q | yes | 17 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9065.2.a.p | ✓ | 17 | 1.a | even | 1 | 1 | trivial |
| 9065.2.a.q | yes | 17 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9065))\):
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\( T_{2}^{17} - T_{2}^{16} - 23 T_{2}^{15} + 23 T_{2}^{14} + 209 T_{2}^{13} - 205 T_{2}^{12} - 971 T_{2}^{11} + \cdots + 14 \)
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\( T_{3}^{17} + T_{3}^{16} - 29 T_{3}^{15} - 27 T_{3}^{14} + 311 T_{3}^{13} + 261 T_{3}^{12} - 1578 T_{3}^{11} + \cdots + 7 \)
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\( T_{11}^{17} + 19 T_{11}^{16} + 81 T_{11}^{15} - 553 T_{11}^{14} - 4943 T_{11}^{13} - 229 T_{11}^{12} + \cdots + 1236544 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{17} - T^{16} + \cdots + 14 \)
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| $3$ |
\( T^{17} + T^{16} + \cdots + 7 \)
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| $5$ |
\( (T - 1)^{17} \)
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| $7$ |
\( T^{17} \)
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| $11$ |
\( T^{17} + 19 T^{16} + \cdots + 1236544 \)
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| $13$ |
\( T^{17} + 11 T^{16} + \cdots + 36608 \)
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| $17$ |
\( T^{17} + 13 T^{16} + \cdots - 9284864 \)
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| $19$ |
\( T^{17} + \cdots - 1469997566 \)
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| $23$ |
\( T^{17} - 4 T^{16} + \cdots - 2115298 \)
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| $29$ |
\( T^{17} + 5 T^{16} + \cdots - 74887168 \)
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| $31$ |
\( T^{17} + 30 T^{16} + \cdots + 90586496 \)
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| $37$ |
\( (T + 1)^{17} \)
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| $41$ |
\( T^{17} + \cdots - 703029248 \)
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| $43$ |
\( T^{17} - 8 T^{16} + \cdots + 18606592 \)
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| $47$ |
\( T^{17} + \cdots + 4475365939 \)
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| $53$ |
\( T^{17} + \cdots + 223821363318272 \)
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| $59$ |
\( T^{17} + \cdots - 1764811976746 \)
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| $61$ |
\( T^{17} + \cdots - 9283118649064 \)
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| $67$ |
\( T^{17} + \cdots + 110314715766784 \)
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| $71$ |
\( T^{17} + \cdots + 40473310647476 \)
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| $73$ |
\( T^{17} + \cdots - 3310413712384 \)
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| $79$ |
\( T^{17} + \cdots + 457261413496576 \)
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| $83$ |
\( T^{17} + \cdots - 3281445661696 \)
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| $89$ |
\( T^{17} + \cdots + 13605670582144 \)
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| $97$ |
\( T^{17} + \cdots + 4203581231104 \)
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