Newspace parameters
| Level: | \( N \) | \(=\) | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5202.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.5381791315\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{16})^+\) |
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|
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| Defining polynomial: |
\( x^{4} - 4x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 306) |
| 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 \(\nu = \zeta_{16} + \zeta_{16}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
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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 | −1.84776 | 0 | −1.53073 | −1.00000 | 0 | 1.84776 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | −0.765367 | 0 | 3.69552 | −1.00000 | 0 | 0.765367 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | 0.765367 | 0 | −3.69552 | −1.00000 | 0 | −0.765367 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | 1.84776 | 0 | 1.53073 | −1.00000 | 0 | −1.84776 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(17\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5202.2.a.bs | 4 | |
| 3.b | odd | 2 | 1 | 5202.2.a.bv | 4 | ||
| 17.b | even | 2 | 1 | inner | 5202.2.a.bs | 4 | |
| 17.e | odd | 16 | 2 | 306.2.l.b | yes | 4 | |
| 51.c | odd | 2 | 1 | 5202.2.a.bv | 4 | ||
| 51.i | even | 16 | 2 | 306.2.l.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 306.2.l.a | ✓ | 4 | 51.i | even | 16 | 2 | |
| 306.2.l.b | yes | 4 | 17.e | odd | 16 | 2 | |
| 5202.2.a.bs | 4 | 1.a | even | 1 | 1 | trivial | |
| 5202.2.a.bs | 4 | 17.b | even | 2 | 1 | inner | |
| 5202.2.a.bv | 4 | 3.b | odd | 2 | 1 | ||
| 5202.2.a.bv | 4 | 51.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5202))\):
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\( T_{5}^{4} - 4T_{5}^{2} + 2 \)
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\( T_{7}^{4} - 16T_{7}^{2} + 32 \)
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\( T_{23}^{4} - 16T_{23}^{2} + 32 \)
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\( T_{47}^{2} - 8T_{47} + 8 \)
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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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| $5$ |
\( T^{4} - 4T^{2} + 2 \)
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| $7$ |
\( T^{4} - 16T^{2} + 32 \)
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| $11$ |
\( T^{4} - 32T^{2} + 128 \)
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| $13$ |
\( (T^{2} + 8 T + 14)^{2} \)
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| $17$ |
\( T^{4} \)
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| $19$ |
\( (T^{2} + 8 T + 8)^{2} \)
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| $23$ |
\( T^{4} - 16T^{2} + 32 \)
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| $29$ |
\( T^{4} - 100T^{2} + 1250 \)
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| $31$ |
\( T^{4} - 16T^{2} + 32 \)
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| $37$ |
\( T^{4} - 20T^{2} + 98 \)
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| $41$ |
\( T^{4} - 20T^{2} + 2 \)
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| $43$ |
\( T^{4} \)
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| $47$ |
\( (T^{2} - 8 T + 8)^{2} \)
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| $53$ |
\( (T^{2} - 16 T + 62)^{2} \)
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| $59$ |
\( (T^{2} - 16 T + 56)^{2} \)
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| $61$ |
\( T^{4} - 52T^{2} + 578 \)
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| $67$ |
\( (T^{2} - 200)^{2} \)
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| $71$ |
\( T^{4} - 272 T^{2} + 16928 \)
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| $73$ |
\( T^{4} - 292 T^{2} + 21218 \)
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| $79$ |
\( T^{4} - 80T^{2} + 1568 \)
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| $83$ |
\( (T^{2} - 16 T - 8)^{2} \)
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| $89$ |
\( (T^{2} - 8 T - 82)^{2} \)
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| $97$ |
\( T^{4} - 324 T^{2} + 13122 \)
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