Newspace parameters
| Level: | \( N \) | \(=\) | \( 3776 = 2^{6} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3776.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(30.1515118032\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2777.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} + x + 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1888) |
| 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
\( x^{4} - x^{3} - 4x^{2} + x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 2\nu - 4 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 5 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} + 5\beta _1 + 7 ) / 2 \)
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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 |
|
0 | −3.42194 | 0 | 1.42194 | 0 | −0.865790 | 0 | 8.70967 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.84662 | 0 | −0.153382 | 0 | 4.28324 | 0 | 0.409999 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.325837 | 0 | −2.32584 | 0 | 3.24216 | 0 | −2.89383 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.94272 | 0 | −3.94272 | 0 | −3.65960 | 0 | 0.774163 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(59\) | \( +1 \) |
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 | 3776.2.a.be | 4 | |
| 4.b | odd | 2 | 1 | 3776.2.a.bj | 4 | ||
| 8.b | even | 2 | 1 | 1888.2.a.j | yes | 4 | |
| 8.d | odd | 2 | 1 | 1888.2.a.g | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1888.2.a.g | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 1888.2.a.j | yes | 4 | 8.b | even | 2 | 1 | |
| 3776.2.a.be | 4 | 1.a | even | 1 | 1 | trivial | |
| 3776.2.a.bj | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3776))\):
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\( T_{3}^{4} + 3T_{3}^{3} - 5T_{3}^{2} - 11T_{3} + 4 \)
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\( T_{5}^{4} + 5T_{5}^{3} + T_{5}^{2} - 13T_{5} - 2 \)
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\( T_{7}^{4} - 3T_{7}^{3} - 17T_{7}^{2} + 39T_{7} + 44 \)
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\( T_{11}^{4} + 2T_{11}^{3} - 16T_{11}^{2} - 8T_{11} + 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 4 \)
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| $5$ |
\( T^{4} + 5 T^{3} + \cdots - 2 \)
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| $7$ |
\( T^{4} - 3 T^{3} + \cdots + 44 \)
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| $11$ |
\( T^{4} + 2 T^{3} + \cdots + 32 \)
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| $13$ |
\( T^{4} - 4 T^{3} + \cdots - 16 \)
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| $17$ |
\( (T - 2)^{4} \)
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| $19$ |
\( T^{4} + 3 T^{3} + \cdots + 4 \)
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| $23$ |
\( T^{4} - 12 T^{3} + \cdots - 64 \)
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| $29$ |
\( T^{4} + 7 T^{3} + \cdots - 254 \)
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| $31$ |
\( T^{4} - 2 T^{3} + \cdots - 64 \)
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| $37$ |
\( T^{4} + 22 T^{3} + \cdots + 368 \)
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| $41$ |
\( T^{4} + 5 T^{3} + \cdots + 46 \)
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| $43$ |
\( T^{4} + 6 T^{3} + \cdots + 6112 \)
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| $47$ |
\( T^{4} + 10 T^{3} + \cdots + 128 \)
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| $53$ |
\( T^{4} + 5 T^{3} + \cdots - 2 \)
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| $59$ |
\( (T + 1)^{4} \)
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| $61$ |
\( T^{4} + 22 T^{3} + \cdots - 496 \)
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| $67$ |
\( T^{4} - 10 T^{3} + \cdots - 352 \)
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| $71$ |
\( T^{4} - 4 T^{3} + \cdots + 2048 \)
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| $73$ |
\( T^{4} + 18 T^{3} + \cdots - 16 \)
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| $79$ |
\( T^{4} - 5 T^{3} + \cdots + 2972 \)
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| $83$ |
\( T^{4} - 2 T^{3} + \cdots - 32 \)
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| $89$ |
\( T^{4} - 2 T^{3} + \cdots + 3184 \)
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| $97$ |
\( T^{4} - 34 T^{3} + \cdots + 16 \)
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