Newspace parameters
| Level: | \( N \) | \(=\) | \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8624.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(68.8629867032\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1016.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 6x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 308) |
| 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
\( x^{3} - x^{2} - 6x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
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 | −2.85577 | 0 | 4.15544 | 0 | 0 | 0 | 5.15544 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.321637 | 0 | −3.89655 | 0 | 0 | 0 | −2.89655 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 2.17741 | 0 | 0.741113 | 0 | 0 | 0 | 1.74111 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 8624.2.a.cj | 3 | |
| 4.b | odd | 2 | 1 | 2156.2.a.j | 3 | ||
| 7.b | odd | 2 | 1 | 1232.2.a.r | 3 | ||
| 28.d | even | 2 | 1 | 308.2.a.c | ✓ | 3 | |
| 28.f | even | 6 | 2 | 2156.2.i.m | 6 | ||
| 28.g | odd | 6 | 2 | 2156.2.i.k | 6 | ||
| 56.e | even | 2 | 1 | 4928.2.a.ca | 3 | ||
| 56.h | odd | 2 | 1 | 4928.2.a.bx | 3 | ||
| 84.h | odd | 2 | 1 | 2772.2.a.s | 3 | ||
| 140.c | even | 2 | 1 | 7700.2.a.y | 3 | ||
| 140.j | odd | 4 | 2 | 7700.2.e.p | 6 | ||
| 308.g | odd | 2 | 1 | 3388.2.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 308.2.a.c | ✓ | 3 | 28.d | even | 2 | 1 | |
| 1232.2.a.r | 3 | 7.b | odd | 2 | 1 | ||
| 2156.2.a.j | 3 | 4.b | odd | 2 | 1 | ||
| 2156.2.i.k | 6 | 28.g | odd | 6 | 2 | ||
| 2156.2.i.m | 6 | 28.f | even | 6 | 2 | ||
| 2772.2.a.s | 3 | 84.h | odd | 2 | 1 | ||
| 3388.2.a.o | 3 | 308.g | odd | 2 | 1 | ||
| 4928.2.a.bx | 3 | 56.h | odd | 2 | 1 | ||
| 4928.2.a.ca | 3 | 56.e | even | 2 | 1 | ||
| 7700.2.a.y | 3 | 140.c | even | 2 | 1 | ||
| 7700.2.e.p | 6 | 140.j | odd | 4 | 2 | ||
| 8624.2.a.cj | 3 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):
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\( T_{3}^{3} + T_{3}^{2} - 6T_{3} - 2 \)
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\( T_{5}^{3} - T_{5}^{2} - 16T_{5} + 12 \)
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\( T_{13}^{3} + 12T_{13}^{2} + 34T_{13} - 8 \)
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\( T_{17}^{3} + 2T_{17}^{2} - 46T_{17} - 156 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} + T^{2} - 6T - 2 \)
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| $5$ |
\( T^{3} - T^{2} + \cdots + 12 \)
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| $7$ |
\( T^{3} \)
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| $11$ |
\( (T + 1)^{3} \)
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| $13$ |
\( T^{3} + 12 T^{2} + \cdots - 8 \)
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| $17$ |
\( T^{3} + 2 T^{2} + \cdots - 156 \)
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| $19$ |
\( T^{3} + 2 T^{2} + \cdots - 16 \)
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| $23$ |
\( T^{3} - 7 T^{2} + \cdots + 72 \)
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| $29$ |
\( T^{3} - 6 T^{2} + \cdots - 24 \)
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| $31$ |
\( T^{3} + 9 T^{2} + \cdots - 146 \)
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| $37$ |
\( T^{3} - 7T^{2} + 32 \)
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| $41$ |
\( T^{3} - 2 T^{2} + \cdots + 156 \)
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| $43$ |
\( T^{3} - 4 T^{2} + \cdots + 32 \)
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| $47$ |
\( T^{3} + 4 T^{2} + \cdots - 96 \)
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| $53$ |
\( T^{3} - 2 T^{2} + \cdots + 96 \)
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| $59$ |
\( T^{3} + 23 T^{2} + \cdots + 402 \)
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| $61$ |
\( T^{3} + 14 T^{2} + \cdots - 916 \)
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| $67$ |
\( T^{3} - 5 T^{2} + \cdots + 8 \)
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| $71$ |
\( T^{3} - 5 T^{2} + \cdots - 312 \)
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| $73$ |
\( T^{3} - 14 T^{2} + \cdots + 244 \)
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| $79$ |
\( T^{3} + 14 T^{2} + \cdots - 936 \)
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| $83$ |
\( T^{3} - 4 T^{2} + \cdots + 1248 \)
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| $89$ |
\( T^{3} - 19 T^{2} + \cdots + 1284 \)
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| $97$ |
\( T^{3} + 15 T^{2} + \cdots + 4 \)
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