Newspace parameters
| Level: | \( N \) | \(=\) | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3675.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.3450227428\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{24})^+\) |
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| Defining polynomial: |
\( x^{4} - 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 735) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.93185 | 1.00000 | 1.73205 | 0 | −1.93185 | 0 | 0.517638 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.517638 | 1.00000 | −1.73205 | 0 | −0.517638 | 0 | 1.93185 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0.517638 | 1.00000 | −1.73205 | 0 | 0.517638 | 0 | −1.93185 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1.93185 | 1.00000 | 1.73205 | 0 | 1.93185 | 0 | −0.517638 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3675.2.a.bu | 4 | |
| 5.b | even | 2 | 1 | 3675.2.a.bs | 4 | ||
| 5.c | odd | 4 | 2 | 735.2.d.f | ✓ | 8 | |
| 7.b | odd | 2 | 1 | 3675.2.a.bs | 4 | ||
| 15.e | even | 4 | 2 | 2205.2.d.t | 8 | ||
| 35.c | odd | 2 | 1 | inner | 3675.2.a.bu | 4 | |
| 35.f | even | 4 | 2 | 735.2.d.f | ✓ | 8 | |
| 35.k | even | 12 | 2 | 735.2.q.c | 8 | ||
| 35.k | even | 12 | 2 | 735.2.q.d | 8 | ||
| 35.l | odd | 12 | 2 | 735.2.q.c | 8 | ||
| 35.l | odd | 12 | 2 | 735.2.q.d | 8 | ||
| 105.k | odd | 4 | 2 | 2205.2.d.t | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 735.2.d.f | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 735.2.d.f | ✓ | 8 | 35.f | even | 4 | 2 | |
| 735.2.q.c | 8 | 35.k | even | 12 | 2 | ||
| 735.2.q.c | 8 | 35.l | odd | 12 | 2 | ||
| 735.2.q.d | 8 | 35.k | even | 12 | 2 | ||
| 735.2.q.d | 8 | 35.l | odd | 12 | 2 | ||
| 2205.2.d.t | 8 | 15.e | even | 4 | 2 | ||
| 2205.2.d.t | 8 | 105.k | odd | 4 | 2 | ||
| 3675.2.a.bs | 4 | 5.b | even | 2 | 1 | ||
| 3675.2.a.bs | 4 | 7.b | odd | 2 | 1 | ||
| 3675.2.a.bu | 4 | 1.a | even | 1 | 1 | trivial | |
| 3675.2.a.bu | 4 | 35.c | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):
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\( T_{2}^{4} - 4T_{2}^{2} + 1 \)
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\( T_{11}^{2} - 12 \)
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\( T_{13} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4T^{2} + 1 \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( (T^{2} - 12)^{2} \)
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| $13$ |
\( (T + 4)^{4} \)
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| $17$ |
\( (T + 4)^{4} \)
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| $19$ |
\( T^{4} - 28T^{2} + 4 \)
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| $23$ |
\( T^{4} - 52T^{2} + 484 \)
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| $29$ |
\( (T^{2} - 4 T - 44)^{2} \)
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| $31$ |
\( (T^{2} - 54)^{2} \)
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| $37$ |
\( T^{4} - 112T^{2} + 64 \)
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| $41$ |
\( (T^{2} - 72)^{2} \)
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| $43$ |
\( T^{4} \)
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| $47$ |
\( (T + 6)^{4} \)
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| $53$ |
\( (T^{2} - 54)^{2} \)
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| $59$ |
\( T^{4} - 112T^{2} + 64 \)
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| $61$ |
\( T^{4} - 84T^{2} + 36 \)
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| $67$ |
\( T^{4} - 208T^{2} + 7744 \)
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| $71$ |
\( (T^{2} - 8 T - 92)^{2} \)
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| $73$ |
\( (T^{2} + 8 T - 32)^{2} \)
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| $79$ |
\( (T^{2} + 16 T + 52)^{2} \)
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| $83$ |
\( (T + 6)^{4} \)
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| $89$ |
\( T^{4} - 256T^{2} + 4096 \)
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| $97$ |
\( (T^{2} + 24 T + 96)^{2} \)
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