Newspace parameters
| Level: | \( N \) | \(=\) | \( 260 = 2^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 260.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.6997931514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 39x^{2} - 3x + 54 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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
\( x^{4} - x^{3} - 39x^{2} - 3x + 54 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{3} + 8\nu^{2} + 288\nu + 24 ) / 3 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 10\nu^{2} - 90\nu - 240 ) / 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 6\beta _1 + 318 ) / 16 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{3} - 5\beta_{2} + 150\beta _1 + 510 ) / 16 \)
|
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 |
|
0 | −20.3241 | 0 | −25.0000 | 0 | 247.310 | 0 | 170.068 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.05861 | 0 | −25.0000 | 0 | 30.2740 | 0 | −233.645 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 6.56458 | 0 | −25.0000 | 0 | −98.2905 | 0 | −199.906 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 28.8181 | 0 | −25.0000 | 0 | −53.2936 | 0 | 587.483 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(13\) | \(1\) |
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 | 260.6.a.e | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1040.6.a.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.6.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1040.6.a.n | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 12T_{3}^{3} - 576T_{3}^{2} + 2224T_{3} + 11760 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(260))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 12 T^{3} + \cdots + 11760 \)
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| $5$ |
\( (T + 25)^{4} \)
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| $7$ |
\( T^{4} - 126 T^{3} + \cdots + 39219072 \)
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| $11$ |
\( T^{4} + \cdots + 17981197728 \)
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| $13$ |
\( (T + 169)^{4} \)
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| $17$ |
\( T^{4} + \cdots - 225649370352 \)
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| $19$ |
\( T^{4} + \cdots + 1701937111680 \)
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| $23$ |
\( T^{4} + \cdots - 5400469005552 \)
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| $29$ |
\( T^{4} + \cdots + 20405734721200 \)
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| $31$ |
\( T^{4} + \cdots + 284966019561600 \)
|
| $37$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots - 12\!\cdots\!68 \)
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| $43$ |
\( T^{4} + \cdots - 10\!\cdots\!48 \)
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| $47$ |
\( T^{4} + \cdots + 67\!\cdots\!16 \)
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| $53$ |
\( T^{4} + \cdots + 30\!\cdots\!56 \)
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| $59$ |
\( T^{4} + \cdots - 78\!\cdots\!20 \)
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| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!28 \)
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| $67$ |
\( T^{4} + \cdots - 68\!\cdots\!80 \)
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| $71$ |
\( T^{4} + \cdots - 31\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots - 22\!\cdots\!20 \)
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| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!88 \)
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| $83$ |
\( T^{4} + \cdots + 46\!\cdots\!28 \)
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| $89$ |
\( T^{4} + \cdots - 89\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots + 49\!\cdots\!68 \)
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