Newspace parameters
| Level: | \( N \) | \(=\) | \( 1425 = 3 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1425.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.3786822880\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{7})\) |
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| Defining polynomial: |
\( x^{4} - 3x^{2} + 4 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 285) |
| 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
\( x^{4} - 3x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu ) / 2 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 5\nu ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 3 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 2 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).
| \(n\) | \(476\) | \(1027\) | \(1351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 799.1 |
|
− | 2.64575i | − | 1.00000i | −5.00000 | 0 | −2.64575 | 3.64575i | 7.93725i | −1.00000 | 0 | ||||||||||||||||||||||||||||
| 799.2 | − | 2.64575i | 1.00000i | −5.00000 | 0 | 2.64575 | 1.64575i | 7.93725i | −1.00000 | 0 | ||||||||||||||||||||||||||||||
| 799.3 | 2.64575i | − | 1.00000i | −5.00000 | 0 | 2.64575 | − | 1.64575i | − | 7.93725i | −1.00000 | 0 | ||||||||||||||||||||||||||||
| 799.4 | 2.64575i | 1.00000i | −5.00000 | 0 | −2.64575 | − | 3.64575i | − | 7.93725i | −1.00000 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1425.2.c.i | 4 | |
| 5.b | even | 2 | 1 | inner | 1425.2.c.i | 4 | |
| 5.c | odd | 4 | 1 | 285.2.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 1425.2.a.p | 2 | ||
| 15.e | even | 4 | 1 | 855.2.a.g | 2 | ||
| 15.e | even | 4 | 1 | 4275.2.a.u | 2 | ||
| 20.e | even | 4 | 1 | 4560.2.a.bo | 2 | ||
| 95.g | even | 4 | 1 | 5415.2.a.s | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 285.2.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 855.2.a.g | 2 | 15.e | even | 4 | 1 | ||
| 1425.2.a.p | 2 | 5.c | odd | 4 | 1 | ||
| 1425.2.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 1425.2.c.i | 4 | 5.b | even | 2 | 1 | inner | |
| 4275.2.a.u | 2 | 15.e | even | 4 | 1 | ||
| 4560.2.a.bo | 2 | 20.e | even | 4 | 1 | ||
| 5415.2.a.s | 2 | 95.g | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1425, [\chi])\):
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\( T_{2}^{2} + 7 \)
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\( T_{7}^{4} + 16T_{7}^{2} + 36 \)
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\( T_{11}^{2} - 6T_{11} + 2 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 7)^{2} \)
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| $3$ |
\( (T^{2} + 1)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 16T^{2} + 36 \)
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| $11$ |
\( (T^{2} - 6 T + 2)^{2} \)
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| $13$ |
\( T^{4} + 32T^{2} + 4 \)
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| $17$ |
\( (T^{2} + 16)^{2} \)
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| $19$ |
\( (T - 1)^{4} \)
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| $23$ |
\( T^{4} + 88T^{2} + 144 \)
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| $29$ |
\( (T^{2} + 2 T - 62)^{2} \)
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| $31$ |
\( (T - 6)^{4} \)
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| $37$ |
\( T^{4} + 16T^{2} + 36 \)
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| $41$ |
\( (T^{2} + 14 T + 42)^{2} \)
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| $43$ |
\( T^{4} + 32T^{2} + 4 \)
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| $47$ |
\( T^{4} + 88T^{2} + 144 \)
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| $53$ |
\( T^{4} + 128T^{2} + 64 \)
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| $59$ |
\( (T^{2} + 12 T + 8)^{2} \)
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| $61$ |
\( (T^{2} + 12 T + 8)^{2} \)
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| $67$ |
\( T^{4} + 256T^{2} + 9216 \)
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| $71$ |
\( (T^{2} - 4 T - 24)^{2} \)
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| $73$ |
\( (T^{2} + 100)^{2} \)
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| $79$ |
\( (T^{2} - 8 T - 96)^{2} \)
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| $83$ |
\( (T^{2} + 36)^{2} \)
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| $89$ |
\( (T^{2} - 18 T + 18)^{2} \)
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| $97$ |
\( T^{4} + 176T^{2} + 1444 \)
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