Newspace parameters
| Level: | \( N \) | \(=\) | \( 591 = 3 \cdot 197 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 591.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.71915875946\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{14} - 24 x^{12} + 220 x^{10} - 958 x^{8} - 4 x^{7} + 2002 x^{6} + 28 x^{5} - 1792 x^{4} - 15 x^{3} + \cdots - 26 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 24 x^{12} + 220 x^{10} - 958 x^{8} - 4 x^{7} + 2002 x^{6} + 28 x^{5} - 1792 x^{4} - 15 x^{3} + \cdots - 26 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 48068 \nu^{13} - 115577 \nu^{12} - 937256 \nu^{11} + 2161255 \nu^{10} + 7005751 \nu^{9} + \cdots - 21240222 ) / 4218661 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 48684 \nu^{13} - 85814 \nu^{12} - 1137083 \nu^{11} + 2511925 \nu^{10} + 9332822 \nu^{9} + \cdots - 19085557 ) / 4218661 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 86549 \nu^{13} - 52119 \nu^{12} + 2031265 \nu^{11} + 1559682 \nu^{10} - 18070456 \nu^{9} + \cdots - 19474129 ) / 4218661 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 87900 \nu^{13} + 165334 \nu^{12} - 1750432 \nu^{11} - 4146352 \nu^{10} + 12291918 \nu^{9} + \cdots + 21275402 ) / 4218661 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 131182 \nu^{13} + 195183 \nu^{12} + 2545922 \nu^{11} - 4083822 \nu^{10} - 17585393 \nu^{9} + \cdots + 23101924 ) / 4218661 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 154463 \nu^{13} - 169494 \nu^{12} + 4023552 \nu^{11} + 2975594 \nu^{10} - 39439683 \nu^{9} + \cdots + 1367041 ) / 4218661 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 197509 \nu^{13} - 120244 \nu^{12} - 4715091 \nu^{11} + 2983508 \nu^{10} + 42663680 \nu^{9} + \cdots + 8683646 ) / 4218661 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 246481 \nu^{13} + 136584 \nu^{12} - 5836024 \nu^{11} - 2887515 \nu^{10} + 52591393 \nu^{9} + \cdots - 16188223 ) / 4218661 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 246481 \nu^{13} + 136584 \nu^{12} - 5836024 \nu^{11} - 2887515 \nu^{10} + 52591393 \nu^{9} + \cdots - 16188223 ) / 4218661 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 414793 \nu^{13} + 4118 \nu^{12} + 9756788 \nu^{11} - 62040 \nu^{10} - 87020029 \nu^{9} + \cdots + 21470374 ) / 4218661 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 429001 \nu^{13} - 24910 \nu^{12} + 10366275 \nu^{11} + 67950 \nu^{10} - 95274680 \nu^{9} + \cdots + 4591061 ) / 4218661 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{10} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} + 8\beta_{2} - \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} - \beta_{12} + 9\beta_{11} - 9\beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 38\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{13} - \beta_{12} + \beta_{11} + 11 \beta_{10} + 11 \beta_{9} + 2 \beta_{8} - 10 \beta_{7} + \cdots + 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{13} - 14 \beta_{12} + 69 \beta_{11} - 71 \beta_{10} + 11 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \cdots + 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( 91 \beta_{13} - 14 \beta_{12} + 15 \beta_{11} + 90 \beta_{10} + 93 \beta_{9} + 29 \beta_{8} - 81 \beta_{7} + \cdots + 606 \)
|
| \(\nu^{9}\) | \(=\) |
\( 118 \beta_{13} - 139 \beta_{12} + 508 \beta_{11} - 546 \beta_{10} + 89 \beta_{9} - 34 \beta_{8} + \cdots + 75 \)
|
| \(\nu^{10}\) | \(=\) |
\( 682 \beta_{13} - 140 \beta_{12} + 159 \beta_{11} + 661 \beta_{10} + 716 \beta_{9} + 292 \beta_{8} + \cdots + 3943 \)
|
| \(\nu^{11}\) | \(=\) |
\( 936 \beta_{13} - 1209 \beta_{12} + 3690 \beta_{11} - 4153 \beta_{10} + 644 \beta_{9} - 383 \beta_{8} + \cdots + 927 \)
|
| \(\nu^{12}\) | \(=\) |
\( 4890 \beta_{13} - 1231 \beta_{12} + 1484 \beta_{11} + 4587 \beta_{10} + 5273 \beta_{9} + 2531 \beta_{8} + \cdots + 26194 \)
|
| \(\nu^{13}\) | \(=\) |
\( 6968 \beta_{13} - 9853 \beta_{12} + 26674 \beta_{11} - 31322 \beta_{10} + 4437 \beta_{9} - 3620 \beta_{8} + \cdots + 9523 \)
|
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 |
|
−2.73586 | −1.00000 | 5.48491 | −2.56045 | 2.73586 | 2.13954 | −9.53420 | 1.00000 | 7.00501 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.58738 | −1.00000 | 4.69451 | −0.0129831 | 2.58738 | −4.50255 | −6.97171 | 1.00000 | 0.0335922 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.09820 | −1.00000 | 2.40243 | 2.80027 | 2.09820 | 4.46840 | −0.844386 | 1.00000 | −5.87552 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.94322 | −1.00000 | 1.77609 | −2.46899 | 1.94322 | 2.99013 | 0.435099 | 1.00000 | 4.79779 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.02711 | −1.00000 | −0.945042 | 4.10142 | 1.02711 | −2.75116 | 3.02489 | 1.00000 | −4.21262 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.756413 | −1.00000 | −1.42784 | −4.20254 | 0.756413 | −3.69759 | 2.59286 | 1.00000 | 3.17886 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.200273 | −1.00000 | −1.95989 | 2.68647 | 0.200273 | 5.17201 | 0.793058 | 1.00000 | −0.538026 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.340520 | −1.00000 | −1.88405 | 0.479903 | −0.340520 | −4.78860 | −1.32260 | 1.00000 | 0.163417 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.507888 | −1.00000 | −1.74205 | −3.87524 | −0.507888 | 1.74487 | −1.90054 | 1.00000 | −1.96819 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.30329 | −1.00000 | −0.301447 | −1.26824 | −1.30329 | 2.79646 | −2.99944 | 1.00000 | −1.65288 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.56512 | −1.00000 | 0.449610 | 3.41005 | −1.56512 | 1.36753 | −2.42655 | 1.00000 | 5.33715 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.38358 | −1.00000 | 3.68147 | 2.48122 | −2.38358 | −1.21276 | 4.00794 | 1.00000 | 5.91421 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.61158 | −1.00000 | 4.82036 | 1.65704 | −2.61158 | −0.888628 | 7.36560 | 1.00000 | 4.32748 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.63646 | −1.00000 | 4.95092 | −3.22792 | −2.63646 | 4.16234 | 7.78000 | 1.00000 | −8.51027 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(197\) | \( -1 \) |
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 | 591.2.a.e | ✓ | 14 |
| 3.b | odd | 2 | 1 | 1773.2.a.h | 14 | ||
| 4.b | odd | 2 | 1 | 9456.2.a.bd | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 591.2.a.e | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 1773.2.a.h | 14 | 3.b | odd | 2 | 1 | ||
| 9456.2.a.bd | 14 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 24 T_{2}^{12} + 220 T_{2}^{10} - 958 T_{2}^{8} + 4 T_{2}^{7} + 2002 T_{2}^{6} - 28 T_{2}^{5} + \cdots - 26 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(591))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - 24 T^{12} + \cdots - 26 \)
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| $3$ |
\( (T + 1)^{14} \)
|
| $5$ |
\( T^{14} - 55 T^{12} + \cdots - 1136 \)
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| $7$ |
\( T^{14} - 7 T^{13} + \cdots + 970664 \)
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| $11$ |
\( T^{14} - 4 T^{13} + \cdots - 4126904 \)
|
| $13$ |
\( T^{14} - 8 T^{13} + \cdots + 28307456 \)
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| $17$ |
\( T^{14} - 4 T^{13} + \cdots + 20324944 \)
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| $19$ |
\( T^{14} + \cdots - 253550756 \)
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| $23$ |
\( T^{14} + 3 T^{13} + \cdots + 991232 \)
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| $29$ |
\( T^{14} - 7 T^{13} + \cdots + 145408 \)
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| $31$ |
\( T^{14} + \cdots + 6682722304 \)
|
| $37$ |
\( T^{14} + \cdots - 189846502 \)
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| $41$ |
\( T^{14} - 23 T^{13} + \cdots - 16893952 \)
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| $43$ |
\( T^{14} + \cdots - 26585850316 \)
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| $47$ |
\( T^{14} + \cdots - 56345575424 \)
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| $53$ |
\( T^{14} + \cdots - 232574697472 \)
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| $59$ |
\( T^{14} + \cdots - 11351179264 \)
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| $61$ |
\( T^{14} + \cdots + 41225752726 \)
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| $67$ |
\( T^{14} + \cdots - 43054686208 \)
|
| $71$ |
\( T^{14} + \cdots - 15405571072 \)
|
| $73$ |
\( T^{14} + \cdots - 5194379264 \)
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| $79$ |
\( T^{14} + \cdots - 13915068710912 \)
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| $83$ |
\( T^{14} + \cdots - 111499268096 \)
|
| $89$ |
\( T^{14} + \cdots + 3407458444 \)
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| $97$ |
\( T^{14} + \cdots + 6809860552 \)
|
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