Newspace parameters
| Level: | \( N \) | \(=\) | \( 487 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 487.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.88871457844\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{16} - 7 x^{15} - 5 x^{14} + 131 x^{13} - 132 x^{12} - 977 x^{11} + 1666 x^{10} + 3671 x^{9} + \cdots - 3825 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 7 x^{15} - 5 x^{14} + 131 x^{13} - 132 x^{12} - 977 x^{11} + 1666 x^{10} + 3671 x^{9} + \cdots - 3825 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 4 \nu^{15} + 19 \nu^{14} + 47 \nu^{13} - 308 \nu^{12} - 165 \nu^{11} + 2009 \nu^{10} + 35 \nu^{9} + \cdots - 1851 ) / 21 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13 \nu^{15} - 81 \nu^{14} - 25 \nu^{13} + 1043 \nu^{12} - 1331 \nu^{11} - 4326 \nu^{10} + 10178 \nu^{9} + \cdots - 13075 ) / 35 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 7 \nu^{15} + 49 \nu^{14} + 5 \nu^{13} - 707 \nu^{12} + 939 \nu^{11} + 3839 \nu^{10} - 7567 \nu^{9} + \cdots - 465 ) / 15 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 7 \nu^{15} - 49 \nu^{14} - 5 \nu^{13} + 707 \nu^{12} - 939 \nu^{11} - 3839 \nu^{10} + 7567 \nu^{9} + \cdots + 555 ) / 15 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 2 \nu^{15} + \nu^{14} - 125 \nu^{13} + 287 \nu^{12} + 1416 \nu^{11} - 4564 \nu^{10} - 6118 \nu^{9} + \cdots + 21145 ) / 35 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 16 \nu^{15} + 132 \nu^{14} - 120 \nu^{13} - 1666 \nu^{12} + 4037 \nu^{11} + 6727 \nu^{10} + \cdots + 11895 ) / 35 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 8 \nu^{15} - 206 \nu^{14} + 935 \nu^{13} + 1393 \nu^{12} - 14811 \nu^{11} + 7994 \nu^{10} + \cdots - 89370 ) / 105 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 19 \nu^{15} + 62 \nu^{14} - 1310 \nu^{13} + 1484 \nu^{12} + 16182 \nu^{11} - 31598 \nu^{10} + \cdots + 144195 ) / 105 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 149 \nu^{15} - 923 \nu^{14} - 895 \nu^{13} + 14749 \nu^{12} - 8538 \nu^{11} - 94528 \nu^{10} + \cdots + 104865 ) / 105 \)
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| \(\beta_{12}\) | \(=\) |
\( ( - 2 \nu^{15} - 16 \nu^{14} + 220 \nu^{13} - 232 \nu^{12} - 2811 \nu^{11} + 5884 \nu^{10} + \cdots - 31035 ) / 15 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 2 \nu^{15} + 16 \nu^{14} - 220 \nu^{13} + 232 \nu^{12} + 2811 \nu^{11} - 5884 \nu^{10} - 13888 \nu^{9} + \cdots + 31065 ) / 15 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 4 \nu^{15} + 38 \nu^{14} - 65 \nu^{13} - 419 \nu^{12} + 1508 \nu^{11} + 1003 \nu^{10} - 9539 \nu^{9} + \cdots + 7390 ) / 5 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 151 \nu^{15} + 922 \nu^{14} + 1160 \nu^{13} - 15701 \nu^{12} + 5547 \nu^{11} + 109592 \nu^{10} + \cdots - 184005 ) / 105 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{12} + \beta_{2} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + \beta_{12} + \beta_{6} + \beta_{5} + 7\beta_{2} + 8\beta _1 + 13 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{15} - \beta_{14} + 9 \beta_{13} + 7 \beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 10 \)
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| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{15} - \beta_{14} + 14 \beta_{13} + 10 \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_{9} + \cdots + 67 \)
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| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{15} - 10 \beta_{14} + 70 \beta_{13} + 43 \beta_{12} + 2 \beta_{11} + 10 \beta_{10} + 16 \beta_{9} + \cdots + 81 \)
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| \(\nu^{8}\) | \(=\) |
\( 32 \beta_{15} - 13 \beta_{14} + 139 \beta_{13} + 75 \beta_{12} + 16 \beta_{11} + 14 \beta_{10} + \cdots + 380 \)
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| \(\nu^{9}\) | \(=\) |
\( 132 \beta_{15} - 74 \beta_{14} + 532 \beta_{13} + 260 \beta_{12} + 39 \beta_{11} + 78 \beta_{10} + \cdots + 607 \)
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| \(\nu^{10}\) | \(=\) |
\( 355 \beta_{15} - 114 \beta_{14} + 1216 \beta_{13} + 513 \beta_{12} + 182 \beta_{11} + 137 \beta_{10} + \cdots + 2293 \)
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| \(\nu^{11}\) | \(=\) |
\( 1220 \beta_{15} - 490 \beta_{14} + 4044 \beta_{13} + 1583 \beta_{12} + 486 \beta_{11} + 572 \beta_{10} + \cdots + 4382 \)
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| \(\nu^{12}\) | \(=\) |
\( 3403 \beta_{15} - 848 \beta_{14} + 10031 \beta_{13} + 3385 \beta_{12} + 1801 \beta_{11} + 1173 \beta_{10} + \cdots + 14463 \)
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| \(\nu^{13}\) | \(=\) |
\( 10708 \beta_{15} - 3074 \beta_{14} + 30857 \beta_{13} + 9748 \beta_{12} + 5008 \beta_{11} + 4148 \beta_{10} + \cdots + 31076 \)
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| \(\nu^{14}\) | \(=\) |
\( 30280 \beta_{15} - 5779 \beta_{14} + 80244 \beta_{13} + 21998 \beta_{12} + 16518 \beta_{11} + \cdots + 94309 \)
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| \(\nu^{15}\) | \(=\) |
\( 90924 \beta_{15} - 18712 \beta_{14} + 236118 \beta_{13} + 60667 \beta_{12} + 46690 \beta_{11} + \cdots + 218861 \)
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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 |
|
−2.10546 | −1.54239 | 2.43298 | −0.121792 | 3.24745 | −3.20952 | −0.911627 | −0.621033 | 0.256429 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.10279 | −3.16730 | 2.42172 | 3.71978 | 6.66017 | 3.38749 | −0.886796 | 7.03181 | −7.82191 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.07454 | 2.32654 | 2.30370 | 1.98734 | −4.82650 | −3.05697 | −0.630041 | 2.41281 | −4.12281 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.80312 | −0.869745 | 1.25125 | −0.139670 | 1.56826 | 2.94775 | 1.35009 | −2.24354 | 0.251842 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.22075 | 2.87556 | −0.509776 | −3.27010 | −3.51033 | 0.627241 | 3.06380 | 5.26883 | 3.99197 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.03114 | 0.439542 | −0.936755 | 4.17391 | −0.453228 | 2.25691 | 3.02820 | −2.80680 | −4.30388 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.672138 | −1.46680 | −1.54823 | −3.64004 | 0.985895 | −2.34760 | 2.38490 | −0.848487 | 2.44661 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.670372 | −3.19701 | −1.55060 | −1.69409 | −2.14319 | 0.588925 | −2.38022 | 7.22087 | −1.13567 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.44349 | −1.48569 | 0.0836744 | 3.49515 | −2.14458 | 3.42362 | −2.76620 | −0.792727 | 5.04523 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.72465 | 1.53488 | 0.974421 | 4.08045 | 2.64714 | −4.44601 | −1.76877 | −0.644129 | 7.03736 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.76531 | 2.88221 | 1.11633 | −2.07593 | 5.08800 | 1.23587 | −1.55995 | 5.30713 | −3.66466 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.17402 | −2.90748 | 2.72637 | 0.701983 | −6.32093 | 0.0299776 | 1.57915 | 5.45345 | 1.52613 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.41686 | −0.523806 | 3.84121 | −1.14731 | −1.26597 | 5.17054 | 4.44996 | −2.72563 | −2.77289 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.49691 | 0.498388 | 4.23455 | 0.0587589 | 1.24443 | −0.265202 | 5.57946 | −2.75161 | 0.146716 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.56721 | 2.37003 | 4.59059 | −1.19418 | 6.08437 | −4.36698 | 6.65060 | 2.61703 | −3.06571 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.75110 | −1.76693 | 5.56855 | 4.06573 | −4.86099 | −0.976032 | 9.81744 | 0.122026 | 11.1852 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(487\) | \( -1 \) |
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 | 487.2.a.d | ✓ | 16 |
| 3.b | odd | 2 | 1 | 4383.2.a.f | 16 | ||
| 4.b | odd | 2 | 1 | 7792.2.a.t | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 487.2.a.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 4383.2.a.f | 16 | 3.b | odd | 2 | 1 | ||
| 7792.2.a.t | 16 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(487))\):
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\( T_{2}^{16} - 7 T_{2}^{15} - 5 T_{2}^{14} + 131 T_{2}^{13} - 132 T_{2}^{12} - 977 T_{2}^{11} + \cdots - 3825 \)
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\( T_{3}^{16} + 4 T_{3}^{15} - 27 T_{3}^{14} - 120 T_{3}^{13} + 254 T_{3}^{12} + 1398 T_{3}^{11} + \cdots - 1224 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 7 T^{15} + \cdots - 3825 \)
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| $3$ |
\( T^{16} + 4 T^{15} + \cdots - 1224 \)
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| $5$ |
\( T^{16} - 9 T^{15} + \cdots + 72 \)
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| $7$ |
\( T^{16} - T^{15} + \cdots - 632 \)
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| $11$ |
\( T^{16} - T^{15} + \cdots - 50616 \)
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| $13$ |
\( T^{16} - 4 T^{15} + \cdots - 26544200 \)
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| $17$ |
\( T^{16} - 38 T^{15} + \cdots - 3212856 \)
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| $19$ |
\( T^{16} - 7 T^{15} + \cdots + 57166587 \)
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| $23$ |
\( T^{16} - 8 T^{15} + \cdots - 33132600 \)
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| $29$ |
\( T^{16} - 6 T^{15} + \cdots - 17301 \)
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| $31$ |
\( T^{16} + 10 T^{15} + \cdots - 3147649 \)
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| $37$ |
\( T^{16} + \cdots - 476550392568 \)
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| $41$ |
\( T^{16} + \cdots - 10411698483 \)
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| $43$ |
\( T^{16} + \cdots + 1074595624 \)
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| $47$ |
\( T^{16} + \cdots - 1443891048 \)
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| $53$ |
\( T^{16} + \cdots + 60642302967 \)
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| $59$ |
\( T^{16} + \cdots + 216910227192 \)
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| $61$ |
\( T^{16} + \cdots + 38879870388355 \)
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| $67$ |
\( T^{16} + \cdots - 1094760181 \)
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| $71$ |
\( T^{16} + \cdots + 32941340595 \)
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| $73$ |
\( T^{16} + \cdots + 454215094299 \)
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| $79$ |
\( T^{16} + \cdots - 10950003598648 \)
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| $83$ |
\( T^{16} + \cdots + 247711920327 \)
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| $89$ |
\( T^{16} + \cdots + 1291850272305 \)
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| $97$ |
\( T^{16} + \cdots + 19558912908721 \)
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