Newspace parameters
| Level: | \( N \) | \(=\) | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2366.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(139.598519074\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{12} - 6 x^{11} - 162 x^{10} + 827 x^{9} + 8829 x^{8} - 37248 x^{7} - 176166 x^{6} + 653969 x^{5} + \cdots - 7489 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 13^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} - 162 x^{10} + 827 x^{9} + 8829 x^{8} - 37248 x^{7} - 176166 x^{6} + 653969 x^{5} + \cdots - 7489 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 36\!\cdots\!75 \nu^{11} + \cdots + 21\!\cdots\!47 ) / 27\!\cdots\!86 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 91\!\cdots\!12 \nu^{11} + \cdots + 35\!\cdots\!21 ) / 93\!\cdots\!62 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 14\!\cdots\!78 \nu^{11} + \cdots - 37\!\cdots\!75 ) / 93\!\cdots\!62 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 64\!\cdots\!84 \nu^{11} + \cdots + 67\!\cdots\!35 ) / 27\!\cdots\!86 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 25\!\cdots\!73 \nu^{11} + \cdots + 16\!\cdots\!81 ) / 93\!\cdots\!62 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 58\!\cdots\!41 \nu^{11} + \cdots + 40\!\cdots\!28 ) / 13\!\cdots\!43 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 40\!\cdots\!96 \nu^{11} + \cdots + 16\!\cdots\!91 ) / 93\!\cdots\!62 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 13\!\cdots\!41 \nu^{11} + \cdots + 23\!\cdots\!52 ) / 27\!\cdots\!86 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 64\!\cdots\!82 \nu^{11} + \cdots - 21\!\cdots\!97 ) / 93\!\cdots\!62 \)
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| \(\beta_{10}\) | \(=\) |
\( ( 28\!\cdots\!27 \nu^{11} + \cdots + 12\!\cdots\!87 ) / 39\!\cdots\!98 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 58\!\cdots\!43 \nu^{11} + \cdots + 77\!\cdots\!70 ) / 39\!\cdots\!98 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{9} + \beta_{7} - \beta_{3} - 13\beta_{2} + 5 ) / 13 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{9} + 13\beta_{8} + 14\beta_{7} - 13\beta_{6} - 5\beta_{5} - 13\beta_{4} + 7\beta_{3} - 26\beta_{2} + 375 ) / 13 \)
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| \(\nu^{3}\) | \(=\) |
\( ( - 13 \beta_{11} - 13 \beta_{10} + 11 \beta_{9} + 39 \beta_{8} + 214 \beta_{7} - 39 \beta_{6} + \cdots + 683 ) / 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 52 \beta_{11} + 182 \beta_{10} + 815 \beta_{9} + 897 \beta_{8} + 1292 \beta_{7} - 884 \beta_{6} + \cdots + 22714 ) / 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 1040 \beta_{11} - 949 \beta_{10} + 3800 \beta_{9} + 3822 \beta_{8} + 19849 \beta_{7} - 4628 \beta_{6} + \cdots + 78583 ) / 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 8970 \beta_{11} + 20449 \beta_{10} + 74143 \beta_{9} + 65754 \beta_{8} + 105022 \beta_{7} + \cdots + 1610697 ) / 13 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 73541 \beta_{11} - 37232 \beta_{10} + 430799 \beta_{9} + 343746 \beta_{8} + 1622004 \beta_{7} + \cdots + 7699398 ) / 13 \)
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| \(\nu^{8}\) | \(=\) |
\( ( 890448 \beta_{11} + 1877720 \beta_{10} + 6378629 \beta_{9} + 5075733 \beta_{8} + 8504217 \beta_{7} + \cdots + 122171152 ) / 13 \)
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| \(\nu^{9}\) | \(=\) |
\( ( - 4642417 \beta_{11} + 838370 \beta_{10} + 42073011 \beta_{9} + 30351737 \beta_{8} + 127442447 \beta_{7} + \cdots + 704780794 ) / 13 \)
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| \(\nu^{10}\) | \(=\) |
\( ( 76712844 \beta_{11} + 164868665 \beta_{10} + 543256891 \beta_{9} + 403112684 \beta_{8} + \cdots + 9589498965 ) / 13 \)
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| \(\nu^{11}\) | \(=\) |
\( ( - 250840486 \beta_{11} + 387470538 \beta_{10} + 3911626134 \beta_{9} + 2650689509 \beta_{8} + \cdots + 62391945745 ) / 13 \)
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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 |
|
2.00000 | −8.26945 | 4.00000 | −7.23331 | −16.5389 | 7.00000 | 8.00000 | 41.3838 | −14.4666 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −8.14613 | 4.00000 | 21.2016 | −16.2923 | 7.00000 | 8.00000 | 39.3594 | 42.4033 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −7.91788 | 4.00000 | 2.25379 | −15.8358 | 7.00000 | 8.00000 | 35.6928 | 4.50758 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −4.59438 | 4.00000 | −18.7970 | −9.18877 | 7.00000 | 8.00000 | −5.89164 | −37.5941 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | −2.82544 | 4.00000 | 7.85430 | −5.65088 | 7.00000 | 8.00000 | −19.0169 | 15.7086 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | −2.47655 | 4.00000 | −14.9661 | −4.95311 | 7.00000 | 8.00000 | −20.8667 | −29.9321 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 0.608442 | 4.00000 | −16.3018 | 1.21688 | 7.00000 | 8.00000 | −26.6298 | −32.6037 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 2.02464 | 4.00000 | −0.214291 | 4.04929 | 7.00000 | 8.00000 | −22.9008 | −0.428582 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.00000 | 2.14065 | 4.00000 | 9.18764 | 4.28131 | 7.00000 | 8.00000 | −22.4176 | 18.3753 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.00000 | 3.88678 | 4.00000 | 4.25233 | 7.77356 | 7.00000 | 8.00000 | −11.8929 | 8.50466 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.00000 | 6.05572 | 4.00000 | 0.131944 | 12.1114 | 7.00000 | 8.00000 | 9.67172 | 0.263889 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.00000 | 9.51360 | 4.00000 | −9.36911 | 19.0272 | 7.00000 | 8.00000 | 63.5086 | −18.7382 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 2366.4.a.bf | yes | 12 |
| 13.b | even | 2 | 1 | 2366.4.a.bc | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2366.4.a.bc | ✓ | 12 | 13.b | even | 2 | 1 | |
| 2366.4.a.bf | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):
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\( T_{3}^{12} + 10 T_{3}^{11} - 142 T_{3}^{10} - 1561 T_{3}^{9} + 5332 T_{3}^{8} + 71630 T_{3}^{7} + \cdots + 10125416 \)
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\( T_{5}^{12} + 22 T_{5}^{11} - 559 T_{5}^{10} - 14298 T_{5}^{9} + 53450 T_{5}^{8} + 2223165 T_{5}^{7} + \cdots + 128850184 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{12} \)
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| $3$ |
\( T^{12} + 10 T^{11} + \cdots + 10125416 \)
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| $5$ |
\( T^{12} + \cdots + 128850184 \)
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| $7$ |
\( (T - 7)^{12} \)
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| $11$ |
\( T^{12} + \cdots - 14\!\cdots\!43 \)
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| $13$ |
\( T^{12} \)
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| $17$ |
\( T^{12} + \cdots + 24\!\cdots\!04 \)
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| $19$ |
\( T^{12} + \cdots - 28\!\cdots\!88 \)
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| $23$ |
\( T^{12} + \cdots - 16\!\cdots\!19 \)
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| $29$ |
\( T^{12} + \cdots + 15\!\cdots\!21 \)
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| $31$ |
\( T^{12} + \cdots + 78\!\cdots\!64 \)
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| $37$ |
\( T^{12} + \cdots - 94\!\cdots\!09 \)
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| $41$ |
\( T^{12} + \cdots - 56\!\cdots\!88 \)
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| $43$ |
\( T^{12} + \cdots + 26\!\cdots\!61 \)
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| $47$ |
\( T^{12} + \cdots + 65\!\cdots\!84 \)
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| $53$ |
\( T^{12} + \cdots - 29\!\cdots\!07 \)
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| $59$ |
\( T^{12} + \cdots + 95\!\cdots\!16 \)
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| $61$ |
\( T^{12} + \cdots + 32\!\cdots\!36 \)
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| $67$ |
\( T^{12} + \cdots + 26\!\cdots\!11 \)
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| $71$ |
\( T^{12} + \cdots + 85\!\cdots\!83 \)
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| $73$ |
\( T^{12} + \cdots + 42\!\cdots\!16 \)
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| $79$ |
\( T^{12} + \cdots + 38\!\cdots\!03 \)
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| $83$ |
\( T^{12} + \cdots + 34\!\cdots\!88 \)
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| $89$ |
\( T^{12} + \cdots + 32\!\cdots\!92 \)
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| $97$ |
\( T^{12} + \cdots - 42\!\cdots\!64 \)
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