Newspace parameters
| Level: | \( N \) | \(=\) | \( 906 = 2 \cdot 3 \cdot 151 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 906.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.23444642313\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2777.1 |
|
|
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| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} + x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 4x^{2} + x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
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 |
|
−1.00000 | 1.00000 | 1.00000 | −3.74002 | −1.00000 | −3.91423 | −1.00000 | 1.00000 | 3.74002 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 1.00000 | 1.00000 | 1.40464 | −1.00000 | −0.275008 | −1.00000 | 1.00000 | −1.40464 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 1.00000 | 1.00000 | 1.60135 | −1.00000 | −0.907129 | −1.00000 | 1.00000 | −1.60135 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.00000 | 1.00000 | 2.73403 | −1.00000 | 4.09637 | −1.00000 | 1.00000 | −2.73403 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(151\) | \( +1 \) |
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 | 906.2.a.k | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2718.2.a.y | 4 | ||
| 4.b | odd | 2 | 1 | 7248.2.a.ba | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 906.2.a.k | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2718.2.a.y | 4 | 3.b | odd | 2 | 1 | ||
| 7248.2.a.ba | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(906))\):
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\( T_{5}^{4} - 2T_{5}^{3} - 11T_{5}^{2} + 33T_{5} - 23 \)
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\( T_{7}^{4} + T_{7}^{3} - 16T_{7}^{2} - 19T_{7} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( T^{4} - 2 T^{3} + \cdots - 23 \)
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| $7$ |
\( T^{4} + T^{3} - 16 T^{2} + \cdots - 4 \)
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| $11$ |
\( T^{4} - 11 T^{3} + \cdots - 328 \)
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| $13$ |
\( T^{4} + 6 T^{3} + \cdots - 1 \)
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| $17$ |
\( T^{4} - 12 T^{3} + \cdots - 176 \)
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| $19$ |
\( T^{4} + 3 T^{3} + \cdots + 162 \)
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| $23$ |
\( T^{4} - 11 T^{3} + \cdots - 776 \)
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| $29$ |
\( T^{4} - 14 T^{3} + \cdots - 1451 \)
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| $31$ |
\( T^{4} + 3 T^{3} + \cdots + 2456 \)
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| $37$ |
\( T^{4} + 8 T^{3} + \cdots + 592 \)
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| $41$ |
\( T^{4} - 8 T^{3} + \cdots + 2801 \)
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| $43$ |
\( T^{4} - 3 T^{3} + \cdots - 694 \)
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| $47$ |
\( T^{4} - 3 T^{3} + \cdots + 412 \)
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| $53$ |
\( T^{4} + 2 T^{3} + \cdots - 64 \)
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| $59$ |
\( T^{4} - 12 T^{3} + \cdots - 256 \)
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| $61$ |
\( T^{4} + T^{3} + \cdots - 74 \)
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| $67$ |
\( T^{4} + 11 T^{3} + \cdots + 1592 \)
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| $71$ |
\( T^{4} + 5 T^{3} + \cdots + 2008 \)
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| $73$ |
\( T^{4} + 16 T^{3} + \cdots - 13024 \)
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| $79$ |
\( T^{4} - 9 T^{3} + \cdots + 446 \)
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| $83$ |
\( T^{4} - 25 T^{3} + \cdots - 6368 \)
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| $89$ |
\( T^{4} - 8 T^{3} + \cdots - 583 \)
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| $97$ |
\( T^{4} + 11 T^{3} + \cdots - 2998 \)
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