Newspace parameters
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(162.096859686\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{12} - 2 x^{11} - 85606746065 x^{10} + 168391612240800 x^{9} + \cdots - 43\!\cdots\!52 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{32}\cdot 3^{8}\cdot 5^{3}\cdot 7^{3} \) |
| 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
\( x^{12} - 2 x^{11} - 85606746065 x^{10} + 168391612240800 x^{9} + \cdots - 43\!\cdots\!52 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( 18\!\cdots\!15 \nu^{11} + \cdots - 81\!\cdots\!84 ) / 19\!\cdots\!84 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 85\!\cdots\!79 \nu^{11} + \cdots - 67\!\cdots\!64 ) / 19\!\cdots\!84 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 45\!\cdots\!75 \nu^{11} + \cdots - 89\!\cdots\!12 ) / 48\!\cdots\!96 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 18\!\cdots\!43 \nu^{11} + \cdots + 10\!\cdots\!36 ) / 19\!\cdots\!84 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 80\!\cdots\!80 \nu^{11} + \cdots + 13\!\cdots\!88 ) / 48\!\cdots\!96 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 63\!\cdots\!65 \nu^{11} + \cdots - 59\!\cdots\!96 ) / 27\!\cdots\!12 \)
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| \(\beta_{8}\) | \(=\) |
\( ( 24\!\cdots\!99 \nu^{11} + \cdots + 77\!\cdots\!68 ) / 97\!\cdots\!92 \)
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| \(\beta_{9}\) | \(=\) |
\( ( - 49\!\cdots\!02 \nu^{11} + \cdots - 22\!\cdots\!20 ) / 19\!\cdots\!84 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 11\!\cdots\!99 \nu^{11} + \cdots + 72\!\cdots\!40 ) / 27\!\cdots\!12 \)
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| \(\beta_{11}\) | \(=\) |
\( ( - 27\!\cdots\!88 \nu^{11} + \cdots + 38\!\cdots\!76 ) / 65\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 100\beta_{2} - 2915\beta _1 + 14267791447 \)
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| \(\nu^{3}\) | \(=\) |
\( 7561 \beta_{11} - 1125 \beta_{10} - 2021 \beta_{9} - 21086 \beta_{8} + 19802 \beta_{7} + \cdots - 42058630100737 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 779856207 \beta_{11} + 924968916 \beta_{10} + 408649242 \beta_{9} + 1273908189 \beta_{8} + \cdots + 30\!\cdots\!36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 326545026012123 \beta_{11} - 24810469334091 \beta_{10} - 95520937244745 \beta_{9} + \cdots - 20\!\cdots\!40 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 34\!\cdots\!78 \beta_{11} + \cdots + 70\!\cdots\!73 \)
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| \(\nu^{7}\) | \(=\) |
\( 11\!\cdots\!83 \beta_{11} + \cdots - 99\!\cdots\!67 \)
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| \(\nu^{8}\) | \(=\) |
\( - 11\!\cdots\!49 \beta_{11} + \cdots + 17\!\cdots\!22 \)
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| \(\nu^{9}\) | \(=\) |
\( 37\!\cdots\!65 \beta_{11} + \cdots - 42\!\cdots\!78 \)
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| \(\nu^{10}\) | \(=\) |
\( - 39\!\cdots\!40 \beta_{11} + \cdots + 44\!\cdots\!23 \)
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| \(\nu^{11}\) | \(=\) |
\( 11\!\cdots\!41 \beta_{11} + \cdots - 16\!\cdots\!29 \)
|
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 |
|
−1024.00 | −170180. | 1.04858e6 | 3.62705e7 | 1.74265e8 | 7.30085e7 | −1.07374e9 | 1.85010e10 | −3.71410e10 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1024.00 | −139963. | 1.04858e6 | −4.15192e7 | 1.43322e8 | 1.19518e8 | −1.07374e9 | 9.12935e9 | 4.25157e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1024.00 | −138261. | 1.04858e6 | 2.20569e6 | 1.41580e8 | −7.26501e8 | −1.07374e9 | 8.65586e9 | −2.25862e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1024.00 | −88740.7 | 1.04858e6 | −1.23697e7 | 9.08704e7 | 6.92430e8 | −1.07374e9 | −2.58545e9 | 1.26666e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1024.00 | −84712.1 | 1.04858e6 | 3.01193e7 | 8.67452e7 | 1.46527e9 | −1.07374e9 | −3.28421e9 | −3.08421e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1024.00 | 7985.21 | 1.04858e6 | 1.71413e7 | −8.17686e6 | −1.67627e8 | −1.07374e9 | −1.03966e10 | −1.75526e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1024.00 | 19224.7 | 1.04858e6 | −1.85439e7 | −1.96861e7 | −1.12639e9 | −1.07374e9 | −1.00908e10 | 1.89890e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1024.00 | 60416.0 | 1.04858e6 | −4.59497e6 | −6.18660e7 | 3.08972e7 | −1.07374e9 | −6.81026e9 | 4.70525e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1024.00 | 108158. | 1.04858e6 | −2.17418e7 | −1.10753e8 | 4.76282e8 | −1.07374e9 | 1.23770e9 | 2.22636e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1024.00 | 149856. | 1.04858e6 | 3.75050e7 | −1.53452e8 | −3.34207e8 | −1.07374e9 | 1.19964e10 | −3.84051e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1024.00 | 153038. | 1.04858e6 | 1.09284e7 | −1.56711e8 | −1.37709e9 | −1.07374e9 | 1.29603e10 | −1.11906e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1024.00 | 164246. | 1.04858e6 | −3.39994e6 | −1.68188e8 | 1.31679e9 | −1.07374e9 | 1.65164e10 | 3.48154e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 58.22.a.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.22.a.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} - 41066 T_{3}^{11} - 84833805237 T_{3}^{10} + \cdots - 93\!\cdots\!20 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(58))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1024)^{12} \)
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| $3$ |
\( T^{12} + \cdots - 93\!\cdots\!20 \)
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| $5$ |
\( T^{12} + \cdots + 54\!\cdots\!00 \)
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| $7$ |
\( T^{12} + \cdots - 10\!\cdots\!80 \)
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| $11$ |
\( T^{12} + \cdots - 51\!\cdots\!48 \)
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| $13$ |
\( T^{12} + \cdots + 21\!\cdots\!76 \)
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| $17$ |
\( T^{12} + \cdots - 17\!\cdots\!76 \)
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| $19$ |
\( T^{12} + \cdots - 28\!\cdots\!44 \)
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| $23$ |
\( T^{12} + \cdots + 53\!\cdots\!88 \)
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| $29$ |
\( (T - 420707233300201)^{12} \)
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| $31$ |
\( T^{12} + \cdots - 32\!\cdots\!44 \)
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| $37$ |
\( T^{12} + \cdots - 69\!\cdots\!84 \)
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| $41$ |
\( T^{12} + \cdots - 15\!\cdots\!76 \)
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| $43$ |
\( T^{12} + \cdots - 19\!\cdots\!00 \)
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| $47$ |
\( T^{12} + \cdots - 58\!\cdots\!00 \)
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| $53$ |
\( T^{12} + \cdots - 30\!\cdots\!84 \)
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| $59$ |
\( T^{12} + \cdots - 93\!\cdots\!00 \)
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| $61$ |
\( T^{12} + \cdots - 33\!\cdots\!72 \)
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| $67$ |
\( T^{12} + \cdots - 46\!\cdots\!04 \)
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| $71$ |
\( T^{12} + \cdots + 38\!\cdots\!00 \)
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| $73$ |
\( T^{12} + \cdots + 17\!\cdots\!64 \)
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| $79$ |
\( T^{12} + \cdots + 34\!\cdots\!96 \)
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| $83$ |
\( T^{12} + \cdots - 38\!\cdots\!08 \)
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| $89$ |
\( T^{12} + \cdots - 16\!\cdots\!52 \)
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| $97$ |
\( T^{12} + \cdots + 13\!\cdots\!52 \)
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