Newspace parameters
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(132.713684003\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{12} - 2 x^{11} - 8734728401 x^{10} + 62781909607608 x^{9} + \cdots - 20\!\cdots\!72 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{32}\cdot 3^{7}\cdot 5 \) |
| 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
\( x^{12} - 2 x^{11} - 8734728401 x^{10} + 62781909607608 x^{9} + \cdots - 20\!\cdots\!72 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( 12\!\cdots\!43 \nu^{11} + \cdots - 18\!\cdots\!24 ) / 36\!\cdots\!80 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 30\!\cdots\!78 \nu^{11} + \cdots - 10\!\cdots\!56 ) / 36\!\cdots\!80 \)
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| \(\beta_{4}\) | \(=\) |
\( ( - 14\!\cdots\!57 \nu^{11} + \cdots + 69\!\cdots\!76 ) / 45\!\cdots\!80 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 42\!\cdots\!31 \nu^{11} + \cdots - 37\!\cdots\!48 ) / 63\!\cdots\!60 \)
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| \(\beta_{6}\) | \(=\) |
\( ( - 39\!\cdots\!23 \nu^{11} + \cdots + 15\!\cdots\!32 ) / 18\!\cdots\!40 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!75 \nu^{11} + \cdots - 68\!\cdots\!52 ) / 13\!\cdots\!40 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 45\!\cdots\!29 \nu^{11} + \cdots - 13\!\cdots\!32 ) / 36\!\cdots\!80 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 90\!\cdots\!41 \nu^{11} + \cdots + 28\!\cdots\!92 ) / 52\!\cdots\!40 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 16\!\cdots\!66 \nu^{11} + \cdots + 76\!\cdots\!64 ) / 36\!\cdots\!80 \)
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| \(\beta_{11}\) | \(=\) |
\( ( 37\!\cdots\!05 \nu^{11} + \cdots - 32\!\cdots\!24 ) / 52\!\cdots\!40 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} - 73\beta_{2} - 10773\beta _1 + 1455789887 \)
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| \(\nu^{3}\) | \(=\) |
\( - 74 \beta_{11} + 259 \beta_{10} - 2125 \beta_{9} - 1978 \beta_{8} - 4534 \beta_{7} + \cdots - 15691616973127 \)
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| \(\nu^{4}\) | \(=\) |
\( 212149 \beta_{11} - 4614365 \beta_{10} + 18483962 \beta_{9} - 34251493 \beta_{8} - 30390937 \beta_{7} + \cdots + 44\!\cdots\!46 \)
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| \(\nu^{5}\) | \(=\) |
\( - 420644395625 \beta_{11} + 1013543162164 \beta_{10} - 9791749692961 \beta_{9} - 10151464525270 \beta_{8} + \cdots - 10\!\cdots\!56 \)
|
| \(\nu^{6}\) | \(=\) |
\( 39\!\cdots\!43 \beta_{11} + \cdots + 16\!\cdots\!38 \)
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| \(\nu^{7}\) | \(=\) |
\( - 19\!\cdots\!43 \beta_{11} + \cdots - 52\!\cdots\!62 \)
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| \(\nu^{8}\) | \(=\) |
\( 33\!\cdots\!59 \beta_{11} + \cdots + 62\!\cdots\!70 \)
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| \(\nu^{9}\) | \(=\) |
\( - 84\!\cdots\!87 \beta_{11} + \cdots - 24\!\cdots\!86 \)
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| \(\nu^{10}\) | \(=\) |
\( 21\!\cdots\!11 \beta_{11} + \cdots + 25\!\cdots\!10 \)
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| \(\nu^{11}\) | \(=\) |
\( - 36\!\cdots\!71 \beta_{11} + \cdots - 10\!\cdots\!10 \)
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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 |
|
512.000 | −59086.7 | 262144. | 3.23908e6 | −3.02524e7 | −1.16357e8 | 1.34218e8 | 2.32897e9 | 1.65841e9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 512.000 | −57023.3 | 262144. | 492994. | −2.91959e7 | 1.44828e7 | 1.34218e8 | 2.08940e9 | 2.52413e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 512.000 | −18402.2 | 262144. | 6.39215e6 | −9.42193e6 | −6.96146e7 | 1.34218e8 | −8.23620e8 | 3.27278e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 512.000 | −13811.6 | 262144. | 4.16497e6 | −7.07152e6 | 1.33908e8 | 1.34218e8 | −9.71502e8 | 2.13247e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 512.000 | −2772.78 | 262144. | −2.35715e6 | −1.41966e6 | −9.02490e7 | 1.34218e8 | −1.15457e9 | −1.20686e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 512.000 | 1089.74 | 262144. | −5.97399e6 | 557947. | 1.67528e8 | 1.34218e8 | −1.16107e9 | −3.05869e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 512.000 | 2990.75 | 262144. | −3.82422e6 | 1.53126e6 | −1.74601e8 | 1.34218e8 | −1.15332e9 | −1.95800e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 512.000 | 28973.9 | 262144. | −7.58248e6 | 1.48346e7 | 1.05167e8 | 1.34218e8 | −3.22775e8 | −3.88223e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 512.000 | 38008.3 | 262144. | 8.70394e6 | 1.94602e7 | 1.18129e8 | 1.34218e8 | 2.82368e8 | 4.45642e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 512.000 | 41932.2 | 262144. | 1.58401e6 | 2.14693e7 | −1.31195e8 | 1.34218e8 | 5.96051e8 | 8.11014e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 512.000 | 48345.8 | 262144. | 3.03871e6 | 2.47531e7 | −610691. | 1.34218e8 | 1.17506e9 | 1.55582e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 512.000 | 65393.8 | 262144. | −460887. | 3.34816e7 | 1.45013e8 | 1.34218e8 | 3.11409e9 | −2.35974e8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 58.20.a.d | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.20.a.d | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 75638 T_{3}^{11} - 6112554341 T_{3}^{10} + 558238507035708 T_{3}^{9} + \cdots - 11\!\cdots\!00 \)
acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(58))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 512)^{12} \)
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| $3$ |
\( T^{12} + \cdots - 11\!\cdots\!00 \)
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| $5$ |
\( T^{12} + \cdots - 33\!\cdots\!00 \)
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| $7$ |
\( T^{12} + \cdots + 59\!\cdots\!00 \)
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| $11$ |
\( T^{12} + \cdots - 28\!\cdots\!56 \)
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| $13$ |
\( T^{12} + \cdots + 48\!\cdots\!32 \)
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| $17$ |
\( T^{12} + \cdots - 67\!\cdots\!44 \)
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| $19$ |
\( T^{12} + \cdots - 12\!\cdots\!00 \)
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| $23$ |
\( T^{12} + \cdots + 57\!\cdots\!36 \)
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| $29$ |
\( (T - 14507145975869)^{12} \)
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| $31$ |
\( T^{12} + \cdots - 54\!\cdots\!16 \)
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| $37$ |
\( T^{12} + \cdots - 10\!\cdots\!56 \)
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| $41$ |
\( T^{12} + \cdots + 77\!\cdots\!48 \)
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| $43$ |
\( T^{12} + \cdots - 25\!\cdots\!00 \)
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| $47$ |
\( T^{12} + \cdots + 40\!\cdots\!36 \)
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| $53$ |
\( T^{12} + \cdots + 17\!\cdots\!96 \)
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| $59$ |
\( T^{12} + \cdots - 35\!\cdots\!00 \)
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| $61$ |
\( T^{12} + \cdots + 13\!\cdots\!12 \)
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| $67$ |
\( T^{12} + \cdots - 30\!\cdots\!88 \)
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| $71$ |
\( T^{12} + \cdots + 19\!\cdots\!52 \)
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| $73$ |
\( T^{12} + \cdots - 29\!\cdots\!32 \)
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| $79$ |
\( T^{12} + \cdots - 53\!\cdots\!80 \)
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| $83$ |
\( T^{12} + \cdots + 23\!\cdots\!88 \)
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| $89$ |
\( T^{12} + \cdots - 86\!\cdots\!00 \)
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| $97$ |
\( T^{12} + \cdots + 10\!\cdots\!76 \)
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