Newspace parameters
| Level: | \( N \) | \(=\) | \( 3080 = 2^{3} \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3080.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(24.5939238226\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.116404.1 |
|
|
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| Defining polynomial: |
\( x^{4} - 10x^{2} - 2x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} - 10x^{2} - 2x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 8\nu - 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 8\beta _1 + 1 \)
|
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 | −3.06153 | 0 | 1.00000 | 0 | −1.00000 | 0 | 6.37299 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.05492 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.88714 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.33659 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.21354 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.77987 | 0 | 1.00000 | 0 | −1.00000 | 0 | 4.72768 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 3080.2.a.p | ✓ | 4 |
| 4.b | odd | 2 | 1 | 6160.2.a.bt | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3080.2.a.p | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 6160.2.a.bt | 4 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3080))\):
|
\( T_{3}^{4} - 10T_{3}^{2} + 2T_{3} + 12 \)
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\( T_{13}^{4} - 10T_{13}^{2} + 2T_{13} + 12 \)
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\( T_{17}^{4} - 6T_{17}^{3} - 54T_{17}^{2} + 238T_{17} + 636 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 10 T^{2} + \cdots + 12 \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( (T + 1)^{4} \)
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| $11$ |
\( (T + 1)^{4} \)
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| $13$ |
\( T^{4} - 10 T^{2} + \cdots + 12 \)
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| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 636 \)
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| $19$ |
\( T^{4} + 8 T^{3} + \cdots + 80 \)
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| $23$ |
\( T^{4} - 12 T^{3} + \cdots + 8 \)
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| $29$ |
\( T^{4} - 10 T^{3} + \cdots + 832 \)
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| $31$ |
\( T^{4} + 6 T^{3} + \cdots - 32 \)
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| $37$ |
\( T^{4} - 12 T^{3} + \cdots - 16 \)
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| $41$ |
\( T^{4} - 20 T^{3} + \cdots - 128 \)
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| $43$ |
\( T^{4} - 14 T^{3} + \cdots - 1968 \)
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| $47$ |
\( T^{4} - 102 T^{2} + \cdots + 2104 \)
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| $53$ |
\( T^{4} - 4 T^{3} + \cdots + 128 \)
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| $59$ |
\( T^{4} + 6 T^{3} + \cdots - 1000 \)
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| $61$ |
\( T^{4} + 12 T^{3} + \cdots - 1396 \)
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| $67$ |
\( T^{4} + 10 T^{3} + \cdots - 2656 \)
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| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 1920 \)
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| $73$ |
\( T^{4} - 10 T^{3} + \cdots + 596 \)
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| $79$ |
\( T^{4} + 6 T^{3} + \cdots + 13928 \)
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| $83$ |
\( T^{4} - 22 T^{3} + \cdots + 128 \)
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| $89$ |
\( T^{4} + 2 T^{3} + \cdots + 22464 \)
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| $97$ |
\( T^{4} - 416 T^{2} + \cdots + 38144 \)
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