Newspace parameters
| Level: | \( N \) | \(=\) | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2352.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(138.772492334\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.57516.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 24x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 21) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 24x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + 2\nu - 17 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 16 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 33 ) / 2 \)
|
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 | −3.00000 | 0 | −17.8732 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −5.56140 | 0 | 0 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 12.4346 | 0 | 0 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 2352.4.a.cg | 3 | |
| 4.b | odd | 2 | 1 | 147.4.a.m | 3 | ||
| 7.b | odd | 2 | 1 | 2352.4.a.ci | 3 | ||
| 7.d | odd | 6 | 2 | 336.4.q.k | 6 | ||
| 12.b | even | 2 | 1 | 441.4.a.t | 3 | ||
| 28.d | even | 2 | 1 | 147.4.a.l | 3 | ||
| 28.f | even | 6 | 2 | 21.4.e.b | ✓ | 6 | |
| 28.g | odd | 6 | 2 | 147.4.e.n | 6 | ||
| 84.h | odd | 2 | 1 | 441.4.a.s | 3 | ||
| 84.j | odd | 6 | 2 | 63.4.e.c | 6 | ||
| 84.n | even | 6 | 2 | 441.4.e.w | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.e.b | ✓ | 6 | 28.f | even | 6 | 2 | |
| 63.4.e.c | 6 | 84.j | odd | 6 | 2 | ||
| 147.4.a.l | 3 | 28.d | even | 2 | 1 | ||
| 147.4.a.m | 3 | 4.b | odd | 2 | 1 | ||
| 147.4.e.n | 6 | 28.g | odd | 6 | 2 | ||
| 336.4.q.k | 6 | 7.d | odd | 6 | 2 | ||
| 441.4.a.s | 3 | 84.h | odd | 2 | 1 | ||
| 441.4.a.t | 3 | 12.b | even | 2 | 1 | ||
| 441.4.e.w | 6 | 84.n | even | 6 | 2 | ||
| 2352.4.a.cg | 3 | 1.a | even | 1 | 1 | trivial | |
| 2352.4.a.ci | 3 | 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):
|
\( T_{5}^{3} + 11T_{5}^{2} - 192T_{5} - 1236 \)
|
|
\( T_{11}^{3} + 35T_{11}^{2} - 1368T_{11} + 9564 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} + 11 T^{2} + \cdots - 1236 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 35 T^{2} + \cdots + 9564 \)
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| $13$ |
\( T^{3} + 62 T^{2} + \cdots - 18452 \)
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| $17$ |
\( T^{3} + 48 T^{2} + \cdots - 112896 \)
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| $19$ |
\( T^{3} + 202 T^{2} + \cdots + 233804 \)
|
| $23$ |
\( T^{3} + 216 T^{2} + \cdots - 1580544 \)
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| $29$ |
\( T^{3} - 53 T^{2} + \cdots - 824976 \)
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| $31$ |
\( T^{3} + 95 T^{2} + \cdots - 11823 \)
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| $37$ |
\( T^{3} - 262 T^{2} + \cdots - 49152 \)
|
| $41$ |
\( T^{3} + 244 T^{2} + \cdots + 300384 \)
|
| $43$ |
\( T^{3} + 360 T^{2} + \cdots - 18269746 \)
|
| $47$ |
\( T^{3} + 210 T^{2} + \cdots + 5119128 \)
|
| $53$ |
\( T^{3} - 393 T^{2} + \cdots + 33169392 \)
|
| $59$ |
\( T^{3} - 1143 T^{2} + \cdots + 100468944 \)
|
| $61$ |
\( T^{3} - 70 T^{2} + \cdots + 84631000 \)
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| $67$ |
\( T^{3} - 628 T^{2} + \cdots - 27993002 \)
|
| $71$ |
\( T^{3} + 318 T^{2} + \cdots + 28535976 \)
|
| $73$ |
\( T^{3} + 988 T^{2} + \cdots - 143207118 \)
|
| $79$ |
\( T^{3} + 861 T^{2} + \cdots - 193956337 \)
|
| $83$ |
\( T^{3} - 519 T^{2} + \cdots + 47916036 \)
|
| $89$ |
\( T^{3} + 1766 T^{2} + \cdots - 13004544 \)
|
| $97$ |
\( T^{3} + 19 T^{2} + \cdots - 44776452 \)
|
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