Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 36 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.2785391901\) |
| 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: |
\( x^{3} - x^{2} - 1847580440x + 20051963761200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{5}\cdot 5 \) |
| 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
\( x^{3} - x^{2} - 1847580440x + 20051963761200 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36\nu^{2} + 585984\nu - 44342125900 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 23\beta_{2} - 97664\beta _1 + 44341930572 ) / 36 \)
|
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 |
|
−314887. | −1.29140e8 | 6.47940e10 | 2.58564e12 | 4.06645e13 | 8.89060e14 | −9.58334e15 | 1.66772e16 | −8.14184e17 | |||||||||||||||||||||||||||
| 1.2 | 41241.5 | −1.29140e8 | −3.26589e10 | 2.39670e12 | −5.32594e12 | −9.42639e14 | −2.76395e15 | 1.66772e16 | 9.88435e16 | ||||||||||||||||||||||||||||
| 1.3 | 186315. | −1.29140e8 | 3.53648e8 | −2.21366e12 | −2.40608e13 | 5.41817e14 | −6.33585e15 | 1.66772e16 | −4.12439e17 | ||||||||||||||||||||||||||||
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 | 3.36.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.36.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.36.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.36.a.c | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 87330T_{2}^{2} - 63970719552T_{2} + 2419568332406784 \)
acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 24\!\cdots\!84 \)
|
| $3$ |
\( (T + 129140163)^{3} \)
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| $5$ |
\( T^{3} + \cdots + 13\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 45\!\cdots\!00 \)
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| $11$ |
\( T^{3} + \cdots + 12\!\cdots\!48 \)
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| $13$ |
\( T^{3} + \cdots + 20\!\cdots\!72 \)
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| $17$ |
\( T^{3} + \cdots + 44\!\cdots\!56 \)
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| $19$ |
\( T^{3} + \cdots + 10\!\cdots\!20 \)
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| $23$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
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| $29$ |
\( T^{3} + \cdots + 16\!\cdots\!20 \)
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| $31$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots + 29\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 27\!\cdots\!08 \)
|
| $47$ |
\( T^{3} + \cdots + 89\!\cdots\!16 \)
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| $53$ |
\( T^{3} + \cdots - 31\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots + 73\!\cdots\!60 \)
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| $61$ |
\( T^{3} + \cdots + 36\!\cdots\!72 \)
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| $67$ |
\( T^{3} + \cdots + 53\!\cdots\!36 \)
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| $71$ |
\( T^{3} + \cdots - 14\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots - 99\!\cdots\!00 \)
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| $79$ |
\( T^{3} + \cdots + 39\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 21\!\cdots\!52 \)
|
| $89$ |
\( T^{3} + \cdots + 28\!\cdots\!40 \)
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| $97$ |
\( T^{3} + \cdots - 29\!\cdots\!44 \)
|
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