Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.9834127149\) |
| 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: |
\( x^{3} - x^{2} - 51006696x - 44860596480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{5} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 51006696x - 44860596480 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36\nu^{2} - 47538\nu - 1224144870 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{2} + 7923\beta _1 + 1224160716 ) / 36 \)
|
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 |
|
−41490.5 | 4.78297e6 | 1.18459e9 | 2.21062e10 | −1.98448e11 | 8.25035e11 | −2.68740e13 | 2.28768e13 | −9.17198e14 | |||||||||||||||||||||||||||
| 1.2 | 9153.03 | 4.78297e6 | −4.53093e8 | 7.75011e7 | 4.37787e10 | 2.36726e12 | −9.06117e12 | 2.28768e13 | 7.09370e11 | ||||||||||||||||||||||||||||
| 1.3 | 43707.4 | 4.78297e6 | 1.37347e9 | 1.33092e10 | 2.09051e11 | −2.46838e12 | 3.65655e13 | 2.28768e13 | 5.81709e14 | ||||||||||||||||||||||||||||
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 | 3.30.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.30.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.30.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.30.a.c | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 11370T_{2}^{2} - 1793148768T_{2} + 16598475030528 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 16598475030528 \)
|
| $3$ |
\( (T - 4782969)^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 22\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 48\!\cdots\!00 \)
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| $11$ |
\( T^{3} + \cdots - 45\!\cdots\!72 \)
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| $13$ |
\( T^{3} + \cdots - 10\!\cdots\!04 \)
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| $17$ |
\( T^{3} + \cdots + 51\!\cdots\!92 \)
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| $19$ |
\( T^{3} + \cdots + 17\!\cdots\!80 \)
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| $23$ |
\( T^{3} + \cdots + 75\!\cdots\!00 \)
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| $29$ |
\( T^{3} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 88\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots + 76\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 11\!\cdots\!96 \)
|
| $47$ |
\( T^{3} + \cdots + 12\!\cdots\!52 \)
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| $53$ |
\( T^{3} + \cdots + 89\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots - 37\!\cdots\!80 \)
|
| $61$ |
\( T^{3} + \cdots - 84\!\cdots\!28 \)
|
| $67$ |
\( T^{3} + \cdots + 11\!\cdots\!72 \)
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| $71$ |
\( T^{3} + \cdots - 14\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots + 84\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 73\!\cdots\!64 \)
|
| $89$ |
\( T^{3} + \cdots + 55\!\cdots\!20 \)
|
| $97$ |
\( T^{3} + \cdots + 75\!\cdots\!52 \)
|
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