Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 62 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(47.1312366529\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 12647052800494929984742x - 89415598046843247001853947608260 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{5}\cdot 5^{3}\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 12647052800494929984742x - 89415598046843247001853947608260 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3840\nu - 1280 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 819200\nu^{2} - 8687707743724800\nu - 6906977102774068526419462400 ) / 1614111 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1280 ) / 3840 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 1614111\beta_{2} + 2262423891595\beta _1 + 6906977102776964429000704000 ) / 819200 \)
|
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 |
|
1.07374e9 | −2.39027e14 | 1.15292e18 | 2.71171e21 | −2.56654e23 | 1.04689e26 | 1.23794e27 | −7.00393e28 | 2.91168e30 | |||||||||||||||||||||||||||
| 1.2 | 1.07374e9 | 2.33056e14 | 1.15292e18 | −3.38038e21 | 2.50242e23 | 5.70486e25 | 1.23794e27 | −7.28582e28 | −3.62966e30 | ||||||||||||||||||||||||||||
| 1.3 | 1.07374e9 | 6.23367e14 | 1.15292e18 | 3.16817e21 | 6.69335e23 | −9.12886e25 | 1.23794e27 | 2.61413e29 | 3.40180e30 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 2.62.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.62.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 617396096502588 T_{3}^{2} + \cdots + 34\!\cdots\!64 \)
acting on \(S_{62}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1073741824)^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 34\!\cdots\!64 \)
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| $5$ |
\( T^{3} + \cdots + 29\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 54\!\cdots\!12 \)
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| $11$ |
\( T^{3} + \cdots - 27\!\cdots\!28 \)
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| $13$ |
\( T^{3} + \cdots + 34\!\cdots\!44 \)
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| $17$ |
\( T^{3} + \cdots + 37\!\cdots\!32 \)
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| $19$ |
\( T^{3} + \cdots - 69\!\cdots\!00 \)
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| $23$ |
\( T^{3} + \cdots - 21\!\cdots\!76 \)
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| $29$ |
\( T^{3} + \cdots + 72\!\cdots\!00 \)
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| $31$ |
\( T^{3} + \cdots + 37\!\cdots\!32 \)
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| $37$ |
\( T^{3} + \cdots + 50\!\cdots\!72 \)
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| $41$ |
\( T^{3} + \cdots + 68\!\cdots\!12 \)
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| $43$ |
\( T^{3} + \cdots + 96\!\cdots\!84 \)
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| $47$ |
\( T^{3} + \cdots + 24\!\cdots\!92 \)
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| $53$ |
\( T^{3} + \cdots + 13\!\cdots\!64 \)
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| $59$ |
\( T^{3} + \cdots - 74\!\cdots\!00 \)
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| $61$ |
\( T^{3} + \cdots + 13\!\cdots\!72 \)
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| $67$ |
\( T^{3} + \cdots + 29\!\cdots\!32 \)
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| $71$ |
\( T^{3} + \cdots + 87\!\cdots\!52 \)
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| $73$ |
\( T^{3} + \cdots + 15\!\cdots\!24 \)
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| $79$ |
\( T^{3} + \cdots - 96\!\cdots\!00 \)
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| $83$ |
\( T^{3} + \cdots + 59\!\cdots\!04 \)
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| $89$ |
\( T^{3} + \cdots + 46\!\cdots\!00 \)
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| $97$ |
\( T^{3} + \cdots + 22\!\cdots\!92 \)
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