Newspace parameters
| Level: | \( N \) | \(=\) | \( 2610 = 2 \cdot 3^{2} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2610.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.8409549276\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.27004.1 |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 3x + 1 \)
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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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 6x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 6\nu - 1 \)
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| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} + 2\nu^{2} + 12\nu - 4 \)
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| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 6 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 1 \)
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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 | 1.00000 | 0 | −4.90338 | 1.00000 | 0 | 1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | 0.375781 | 1.00000 | 0 | 1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | 1.37888 | 1.00000 | 0 | 1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 1.00000 | 0 | 3.14871 | 1.00000 | 0 | 1.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( -1 \) |
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 | 2610.2.a.z | yes | 4 |
| 3.b | odd | 2 | 1 | 2610.2.a.y | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2610.2.a.y | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 2610.2.a.z | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2610))\):
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\( T_{7}^{4} - 18T_{7}^{2} + 28T_{7} - 8 \)
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\( T_{11}^{4} - 18T_{11}^{2} + 28T_{11} - 8 \)
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\( T_{13}^{4} - 2T_{13}^{3} - 36T_{13}^{2} + 128T_{13} - 96 \)
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\( T_{19}^{4} - 6T_{19}^{3} - 26T_{19}^{2} + 56T_{19} + 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( T^{4} - 18 T^{2} + \cdots - 8 \)
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| $11$ |
\( T^{4} - 18 T^{2} + \cdots - 8 \)
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| $13$ |
\( T^{4} - 2 T^{3} + \cdots - 96 \)
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| $17$ |
\( T^{4} - 8 T^{3} + \cdots + 128 \)
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| $19$ |
\( T^{4} - 6 T^{3} + \cdots + 32 \)
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| $23$ |
\( T^{4} - 10 T^{3} + \cdots - 2400 \)
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| $29$ |
\( (T - 1)^{4} \)
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| $31$ |
\( T^{4} - 4 T^{3} + \cdots - 216 \)
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| $37$ |
\( T^{4} - 44 T^{2} + \cdots + 320 \)
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| $41$ |
\( T^{4} - 10 T^{3} + \cdots + 4832 \)
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| $43$ |
\( T^{4} - 72 T^{2} + \cdots - 128 \)
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| $47$ |
\( (T - 4)^{4} \)
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| $53$ |
\( (T - 2)^{4} \)
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| $59$ |
\( T^{4} - 8 T^{3} + \cdots + 608 \)
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| $61$ |
\( T^{4} - 6 T^{3} + \cdots - 32 \)
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| $67$ |
\( T^{4} + 6 T^{3} + \cdots + 32 \)
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| $71$ |
\( T^{4} - 4 T^{3} + \cdots + 256 \)
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| $73$ |
\( T^{4} + 4 T^{3} + \cdots - 96 \)
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| $79$ |
\( T^{4} - 8 T^{3} + \cdots - 216 \)
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| $83$ |
\( T^{4} - 4 T^{3} + \cdots - 1032 \)
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| $89$ |
\( T^{4} - 2 T^{3} + \cdots - 96 \)
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| $97$ |
\( T^{4} + 12 T^{3} + \cdots - 10464 \)
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