Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 42 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(31.9415011369\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - x^{2} - 14982256920x + 433388802120300 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{5}\cdot 7 \) |
| 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
\( x^{3} - x^{2} - 14982256920x + 433388802120300 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 4 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 72\nu^{2} + 3124008\nu - 719149373520 ) / 5 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 4 ) / 12 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{2} - 260334\beta _1 + 719148332184 ) / 72 \)
|
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 |
|
−1.34440e6 | −3.48678e9 | −3.91623e11 | 1.06877e14 | 4.68762e15 | −3.99469e17 | 3.48285e18 | 1.21577e19 | −1.43685e20 | |||||||||||||||||||||||||||
| 1.2 | −467196. | −3.48678e9 | −1.98075e12 | −2.77079e14 | 1.62901e15 | 3.80292e17 | 1.95278e18 | 1.21577e19 | 1.29450e20 | ||||||||||||||||||||||||||||
| 1.3 | 1.52221e6 | −3.48678e9 | 1.18107e11 | 2.08853e14 | −5.30763e15 | −2.54249e16 | −3.16760e18 | 1.21577e19 | 3.17919e20 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
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 | 3.42.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.42.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.42.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.42.a.a | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 289380T_{2}^{2} - 2129531401728T_{2} - 956096849530847232 \)
acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots - 95\!\cdots\!32 \)
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| $3$ |
\( (T + 3486784401)^{3} \)
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| $5$ |
\( T^{3} + \cdots + 61\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 38\!\cdots\!00 \)
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| $11$ |
\( T^{3} + \cdots + 28\!\cdots\!68 \)
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| $13$ |
\( T^{3} + \cdots + 39\!\cdots\!16 \)
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| $17$ |
\( T^{3} + \cdots - 10\!\cdots\!28 \)
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| $19$ |
\( T^{3} + \cdots + 13\!\cdots\!80 \)
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| $23$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
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| $29$ |
\( T^{3} + \cdots - 18\!\cdots\!00 \)
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| $31$ |
\( T^{3} + \cdots + 28\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 31\!\cdots\!00 \)
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| $41$ |
\( T^{3} + \cdots - 28\!\cdots\!00 \)
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| $43$ |
\( T^{3} + \cdots - 47\!\cdots\!96 \)
|
| $47$ |
\( T^{3} + \cdots - 32\!\cdots\!28 \)
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| $53$ |
\( T^{3} + \cdots + 61\!\cdots\!00 \)
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| $59$ |
\( T^{3} + \cdots + 43\!\cdots\!20 \)
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| $61$ |
\( T^{3} + \cdots - 59\!\cdots\!68 \)
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| $67$ |
\( T^{3} + \cdots + 10\!\cdots\!52 \)
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| $71$ |
\( T^{3} + \cdots + 22\!\cdots\!32 \)
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| $73$ |
\( T^{3} + \cdots + 30\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots - 72\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 66\!\cdots\!24 \)
|
| $89$ |
\( T^{3} + \cdots + 55\!\cdots\!20 \)
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| $97$ |
\( T^{3} + \cdots + 35\!\cdots\!72 \)
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