Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 32 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.2631398457\) |
| 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: |
\( x^{3} - x^{2} - 147804080x + 172607524800 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{5}\cdot 5 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 147804080x + 172607524800 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
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| \(\beta_{2}\) | \(=\) |
\( 36\nu^{2} + 63018\nu - 3547318938 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 10503\beta _1 + 3547297932 ) / 36 \)
|
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 |
|
−78765.0 | −1.43489e7 | 4.05644e9 | 3.27434e10 | 1.13019e12 | −1.81576e13 | −1.50359e14 | 2.05891e14 | −2.57903e15 | |||||||||||||||||||||||||||
| 1.2 | 4529.33 | −1.43489e7 | −2.12697e9 | −8.32197e10 | −6.49909e10 | −4.52546e12 | −1.93604e13 | 2.05891e14 | −3.76930e14 | ||||||||||||||||||||||||||||
| 1.3 | 66609.6 | −1.43489e7 | 2.28936e9 | 1.01099e11 | −9.55776e11 | −3.04721e12 | 9.45035e12 | 2.05891e14 | 6.73414e15 | ||||||||||||||||||||||||||||
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 | 3.32.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.32.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.32.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.32.a.c | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 7626T_{2}^{2} - 5301561600T_{2} + 23763162267648 \)
acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 23763162267648 \)
|
| $3$ |
\( (T + 14348907)^{3} \)
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| $5$ |
\( T^{3} + \cdots + 27\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 25\!\cdots\!60 \)
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| $11$ |
\( T^{3} + \cdots + 15\!\cdots\!56 \)
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| $13$ |
\( T^{3} + \cdots - 32\!\cdots\!04 \)
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| $17$ |
\( T^{3} + \cdots + 32\!\cdots\!64 \)
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| $19$ |
\( T^{3} + \cdots + 45\!\cdots\!00 \)
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| $23$ |
\( T^{3} + \cdots - 54\!\cdots\!40 \)
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| $29$ |
\( T^{3} + \cdots - 15\!\cdots\!00 \)
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| $31$ |
\( T^{3} + \cdots - 90\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 18\!\cdots\!20 \)
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| $41$ |
\( T^{3} + \cdots + 12\!\cdots\!20 \)
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| $43$ |
\( T^{3} + \cdots - 53\!\cdots\!92 \)
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| $47$ |
\( T^{3} + \cdots + 10\!\cdots\!76 \)
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| $53$ |
\( T^{3} + \cdots - 15\!\cdots\!60 \)
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| $59$ |
\( T^{3} + \cdots - 58\!\cdots\!00 \)
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| $61$ |
\( T^{3} + \cdots - 13\!\cdots\!32 \)
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| $67$ |
\( T^{3} + \cdots + 94\!\cdots\!64 \)
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| $71$ |
\( T^{3} + \cdots + 26\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots - 58\!\cdots\!00 \)
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| $79$ |
\( T^{3} + \cdots + 33\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 98\!\cdots\!16 \)
|
| $89$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 59\!\cdots\!84 \)
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