Newspace parameters
| Level: | \( N \) | \(=\) | \( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3654.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.1773368986\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.3241536.1 |
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|
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| Defining polynomial: |
\( x^{5} - 10x^{3} - 2x^{2} + 15x - 6 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 10x^{3} - 2x^{2} + 15x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 9\nu^{2} - 4\nu + 8 \)
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| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} + 2\nu^{3} - 20\nu^{2} - 22\nu + 22 \)
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| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + 2\beta_{2} + 7\beta _1 + 2 ) / 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 9\beta_{2} + 2\beta _1 + 28 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −3.47489 | 0 | −1.00000 | −1.00000 | 0 | 3.47489 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −1.86045 | 0 | −1.00000 | −1.00000 | 0 | 1.86045 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | −0.317476 | 0 | −1.00000 | −1.00000 | 0 | 0.317476 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | 2.38713 | 0 | −1.00000 | −1.00000 | 0 | −2.38713 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | 3.26568 | 0 | −1.00000 | −1.00000 | 0 | −3.26568 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(29\) | \( -1 \) |
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 | 3654.2.a.bi | ✓ | 5 |
| 3.b | odd | 2 | 1 | 3654.2.a.bj | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3654.2.a.bi | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 3654.2.a.bj | yes | 5 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3654))\):
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\( T_{5}^{5} - 16T_{5}^{3} + 52T_{5} + 16 \)
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\( T_{11}^{5} + 5T_{11}^{4} - 24T_{11}^{3} - 104T_{11}^{2} + 144T_{11} + 336 \)
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\( T_{13}^{5} - 5T_{13}^{4} - 48T_{13}^{3} + 248T_{13}^{2} + 156T_{13} - 788 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} - 16 T^{3} + \cdots + 16 \)
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| $7$ |
\( (T + 1)^{5} \)
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| $11$ |
\( T^{5} + 5 T^{4} + \cdots + 336 \)
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| $13$ |
\( T^{5} - 5 T^{4} + \cdots - 788 \)
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| $17$ |
\( T^{5} + 8 T^{4} + \cdots - 224 \)
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| $19$ |
\( T^{5} - 7 T^{4} + \cdots + 1344 \)
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| $23$ |
\( T^{5} + 5 T^{4} + \cdots + 448 \)
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| $29$ |
\( (T - 1)^{5} \)
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| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 432 \)
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| $37$ |
\( T^{5} + 7 T^{4} + \cdots - 1744 \)
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| $41$ |
\( T^{5} + 12 T^{4} + \cdots + 32848 \)
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| $43$ |
\( T^{5} - 8 T^{4} + \cdots - 896 \)
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| $47$ |
\( T^{5} + 15 T^{4} + \cdots - 164 \)
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| $53$ |
\( T^{5} - 3 T^{4} + \cdots - 20272 \)
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| $59$ |
\( T^{5} - T^{4} + \cdots + 29724 \)
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| $61$ |
\( T^{5} - 22 T^{4} + \cdots + 13216 \)
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| $67$ |
\( T^{5} - 13 T^{4} + \cdots - 71232 \)
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| $71$ |
\( T^{5} + 8 T^{4} + \cdots + 20992 \)
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| $73$ |
\( T^{5} - 13 T^{4} + \cdots - 27604 \)
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| $79$ |
\( T^{5} - 12 T^{4} + \cdots - 28416 \)
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| $83$ |
\( T^{5} + 11 T^{4} + \cdots + 67052 \)
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| $89$ |
\( T^{5} - 4 T^{4} + \cdots - 49072 \)
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| $97$ |
\( T^{5} - 43 T^{4} + \cdots - 3148 \)
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