Newspace parameters
| Level: | \( N \) | \(=\) | \( 2312 = 2^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2312.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.4614129473\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
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| Defining polynomial: |
\( x^{2} - 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 136) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 | −2.82843 | 0 | −1.41421 | 0 | 2.82843 | 0 | 5.00000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 2.82843 | 0 | 1.41421 | 0 | −2.82843 | 0 | 5.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 2312.2.a.k | 2 | |
| 4.b | odd | 2 | 1 | 4624.2.a.t | 2 | ||
| 17.b | even | 2 | 1 | inner | 2312.2.a.k | 2 | |
| 17.c | even | 4 | 2 | 2312.2.b.a | 2 | ||
| 17.d | even | 8 | 2 | 136.2.k.d | ✓ | 2 | |
| 51.g | odd | 8 | 2 | 1224.2.w.e | 2 | ||
| 68.d | odd | 2 | 1 | 4624.2.a.t | 2 | ||
| 68.g | odd | 8 | 2 | 272.2.o.a | 2 | ||
| 136.o | even | 8 | 2 | 1088.2.o.b | 2 | ||
| 136.p | odd | 8 | 2 | 1088.2.o.q | 2 | ||
| 204.p | even | 8 | 2 | 2448.2.be.i | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.2.k.d | ✓ | 2 | 17.d | even | 8 | 2 | |
| 272.2.o.a | 2 | 68.g | odd | 8 | 2 | ||
| 1088.2.o.b | 2 | 136.o | even | 8 | 2 | ||
| 1088.2.o.q | 2 | 136.p | odd | 8 | 2 | ||
| 1224.2.w.e | 2 | 51.g | odd | 8 | 2 | ||
| 2312.2.a.k | 2 | 1.a | even | 1 | 1 | trivial | |
| 2312.2.a.k | 2 | 17.b | even | 2 | 1 | inner | |
| 2312.2.b.a | 2 | 17.c | even | 4 | 2 | ||
| 2448.2.be.i | 2 | 204.p | even | 8 | 2 | ||
| 4624.2.a.t | 2 | 4.b | odd | 2 | 1 | ||
| 4624.2.a.t | 2 | 68.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\):
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\( T_{3}^{2} - 8 \)
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\( T_{5}^{2} - 2 \)
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\( T_{7}^{2} - 8 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} - 8 \)
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| $5$ |
\( T^{2} - 2 \)
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| $7$ |
\( T^{2} - 8 \)
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| $11$ |
\( T^{2} - 8 \)
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| $13$ |
\( (T - 4)^{2} \)
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| $17$ |
\( T^{2} \)
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| $19$ |
\( (T - 8)^{2} \)
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| $23$ |
\( T^{2} - 8 \)
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| $29$ |
\( T^{2} - 18 \)
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| $31$ |
\( T^{2} - 72 \)
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| $37$ |
\( T^{2} - 50 \)
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| $41$ |
\( T^{2} - 2 \)
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| $43$ |
\( (T - 8)^{2} \)
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| $47$ |
\( T^{2} \)
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| $53$ |
\( (T + 12)^{2} \)
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| $59$ |
\( (T - 8)^{2} \)
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| $61$ |
\( T^{2} - 2 \)
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| $67$ |
\( (T - 12)^{2} \)
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| $71$ |
\( T^{2} - 72 \)
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| $73$ |
\( T^{2} - 18 \)
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| $79$ |
\( T^{2} - 8 \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( (T - 8)^{2} \)
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| $97$ |
\( T^{2} - 18 \)
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