Newspace parameters
| Level: | \( N \) | \(=\) | \( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5200.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(41.5222090511\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.108664.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 20x^{2} + 42x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2600) |
| 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
\( x^{4} - x^{3} - 20x^{2} + 42x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 14\nu - 14 ) / 6 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} - \nu^{2} + 16\nu - 10 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} - \beta _1 + 12 \)
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| \(\nu^{3}\) | \(=\) |
\( -3\beta_{3} - 3\beta_{2} + 17\beta _1 - 22 \)
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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 | −1.56155 | 0 | 0 | 0 | −1.30550 | 0 | −0.561553 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.56155 | 0 | 0 | 0 | 3.30550 | 0 | −0.561553 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.56155 | 0 | 0 | 0 | −3.20531 | 0 | 3.56155 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.56155 | 0 | 0 | 0 | 5.20531 | 0 | 3.56155 | 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 | 5200.2.a.cl | 4 | |
| 4.b | odd | 2 | 1 | 2600.2.a.z | ✓ | 4 | |
| 5.b | even | 2 | 1 | 5200.2.a.ck | 4 | ||
| 20.d | odd | 2 | 1 | 2600.2.a.ba | yes | 4 | |
| 20.e | even | 4 | 2 | 2600.2.d.p | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2600.2.a.z | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 2600.2.a.ba | yes | 4 | 20.d | odd | 2 | 1 | |
| 2600.2.d.p | 8 | 20.e | even | 4 | 2 | ||
| 5200.2.a.ck | 4 | 5.b | even | 2 | 1 | ||
| 5200.2.a.cl | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5200))\):
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\( T_{3}^{2} - T_{3} - 4 \)
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\( T_{7}^{4} - 4T_{7}^{3} - 17T_{7}^{2} + 42T_{7} + 72 \)
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\( T_{11}^{4} - 3T_{11}^{3} - 17T_{11}^{2} - 3T_{11} + 18 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} - T - 4)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 72 \)
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| $11$ |
\( T^{4} - 3 T^{3} + \cdots + 18 \)
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| $13$ |
\( (T + 1)^{4} \)
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| $17$ |
\( T^{4} + 3 T^{3} + \cdots + 2 \)
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| $19$ |
\( T^{4} + T^{3} + \cdots + 304 \)
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| $23$ |
\( T^{4} - 3 T^{3} + \cdots - 32 \)
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| $29$ |
\( T^{4} - 15 T^{3} + \cdots - 1532 \)
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| $31$ |
\( T^{4} - 10 T^{3} + \cdots + 604 \)
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| $37$ |
\( (T^{2} + 2 T - 16)^{2} \)
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| $41$ |
\( T^{4} + 9 T^{3} + \cdots + 4864 \)
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| $43$ |
\( T^{4} - 5 T^{3} + \cdots + 144 \)
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| $47$ |
\( T^{4} - 18 T^{3} + \cdots - 304 \)
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| $53$ |
\( T^{4} + T^{3} + \cdots - 166 \)
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| $59$ |
\( T^{4} + 6 T^{3} + \cdots - 1024 \)
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| $61$ |
\( T^{4} + 7 T^{3} + \cdots + 8 \)
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| $67$ |
\( T^{4} - 25 T^{3} + \cdots - 7138 \)
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| $71$ |
\( (T^{2} - 18 T + 64)^{2} \)
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| $73$ |
\( T^{4} + 13 T^{3} + \cdots + 8 \)
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| $79$ |
\( T^{4} - 3 T^{3} + \cdots - 32 \)
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| $83$ |
\( T^{4} + T^{3} + \cdots + 628 \)
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| $89$ |
\( T^{4} - 27 T^{3} + \cdots - 3200 \)
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| $97$ |
\( T^{4} - 10 T^{3} + \cdots + 1216 \)
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