Newspace parameters
| Level: | \( N \) | \(=\) | \( 3648 = 2^{6} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3648.k (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(29.1294266574\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 912) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} - \nu + 3 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 5\nu + 3 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} - 2\beta _1 + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times\).
| \(n\) | \(1217\) | \(1921\) | \(2053\) | \(2623\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2431.1 |
|
0 | 1.00000 | 0 | −2.37228 | 0 | − | 2.52434i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||
| 2431.2 | 0 | 1.00000 | 0 | −2.37228 | 0 | 2.52434i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 2431.3 | 0 | 1.00000 | 0 | 3.37228 | 0 | − | 0.792287i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 2431.4 | 0 | 1.00000 | 0 | 3.37228 | 0 | 0.792287i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3648.2.k.h | 4 | |
| 4.b | odd | 2 | 1 | 3648.2.k.g | 4 | ||
| 8.b | even | 2 | 1 | 912.2.k.g | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 912.2.k.h | yes | 4 | |
| 19.b | odd | 2 | 1 | 3648.2.k.g | 4 | ||
| 24.f | even | 2 | 1 | 2736.2.k.o | 4 | ||
| 24.h | odd | 2 | 1 | 2736.2.k.n | 4 | ||
| 76.d | even | 2 | 1 | inner | 3648.2.k.h | 4 | |
| 152.b | even | 2 | 1 | 912.2.k.g | ✓ | 4 | |
| 152.g | odd | 2 | 1 | 912.2.k.h | yes | 4 | |
| 456.l | odd | 2 | 1 | 2736.2.k.n | 4 | ||
| 456.p | even | 2 | 1 | 2736.2.k.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 912.2.k.g | ✓ | 4 | 8.b | even | 2 | 1 | |
| 912.2.k.g | ✓ | 4 | 152.b | even | 2 | 1 | |
| 912.2.k.h | yes | 4 | 8.d | odd | 2 | 1 | |
| 912.2.k.h | yes | 4 | 152.g | odd | 2 | 1 | |
| 2736.2.k.n | 4 | 24.h | odd | 2 | 1 | ||
| 2736.2.k.n | 4 | 456.l | odd | 2 | 1 | ||
| 2736.2.k.o | 4 | 24.f | even | 2 | 1 | ||
| 2736.2.k.o | 4 | 456.p | even | 2 | 1 | ||
| 3648.2.k.g | 4 | 4.b | odd | 2 | 1 | ||
| 3648.2.k.g | 4 | 19.b | odd | 2 | 1 | ||
| 3648.2.k.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 3648.2.k.h | 4 | 76.d | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3648, [\chi])\):
|
\( T_{5}^{2} - T_{5} - 8 \)
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\( T_{31}^{2} + 6T_{31} - 24 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( (T^{2} - T - 8)^{2} \)
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| $7$ |
\( T^{4} + 7T^{2} + 4 \)
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| $11$ |
\( T^{4} + 7T^{2} + 4 \)
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| $13$ |
\( T^{4} + 28T^{2} + 64 \)
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| $17$ |
\( (T^{2} - 5 T - 2)^{2} \)
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| $19$ |
\( (T^{2} - 8 T + 19)^{2} \)
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| $23$ |
\( T^{4} + 76T^{2} + 256 \)
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| $29$ |
\( T^{4} + 112T^{2} + 1024 \)
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| $31$ |
\( (T^{2} + 6 T - 24)^{2} \)
|
| $37$ |
\( T^{4} + 28T^{2} + 64 \)
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| $41$ |
\( (T^{2} + 48)^{2} \)
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| $43$ |
\( T^{4} + 87T^{2} + 36 \)
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| $47$ |
\( T^{4} + 19T^{2} + 16 \)
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| $53$ |
\( T^{4} + 112T^{2} + 1024 \)
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| $59$ |
\( (T + 4)^{4} \)
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| $61$ |
\( (T^{2} + 5 T - 2)^{2} \)
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| $67$ |
\( (T^{2} - 4 T - 128)^{2} \)
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| $71$ |
\( (T - 4)^{4} \)
|
| $73$ |
\( (T^{2} - 9 T - 54)^{2} \)
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| $79$ |
\( (T^{2} - 2 T - 32)^{2} \)
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| $83$ |
\( (T^{2} + 12)^{2} \)
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| $89$ |
\( (T^{2} + 176)^{2} \)
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| $97$ |
\( (T^{2} + 176)^{2} \)
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