Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.75274723129\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{35})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 17x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 17x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 8\nu ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{2} + 8\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 74.1 |
|
−5.91608 | −8.87412 | − | 1.50000i | 19.0000 | 0 | 52.5000 | + | 8.87412i | − | 72.0000i | −17.7482 | 76.5000 | + | 26.6224i | 0 | |||||||||||||||||||||||
| 74.2 | −5.91608 | −8.87412 | + | 1.50000i | 19.0000 | 0 | 52.5000 | − | 8.87412i | 72.0000i | −17.7482 | 76.5000 | − | 26.6224i | 0 | |||||||||||||||||||||||||
| 74.3 | 5.91608 | 8.87412 | − | 1.50000i | 19.0000 | 0 | 52.5000 | − | 8.87412i | − | 72.0000i | 17.7482 | 76.5000 | − | 26.6224i | 0 | ||||||||||||||||||||||||
| 74.4 | 5.91608 | 8.87412 | + | 1.50000i | 19.0000 | 0 | 52.5000 | + | 8.87412i | 72.0000i | 17.7482 | 76.5000 | + | 26.6224i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.5.d.c | 4 | |
| 3.b | odd | 2 | 1 | inner | 75.5.d.c | 4 | |
| 5.b | even | 2 | 1 | inner | 75.5.d.c | 4 | |
| 5.c | odd | 4 | 1 | 75.5.c.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.5.c.f | yes | 2 | |
| 15.d | odd | 2 | 1 | inner | 75.5.d.c | 4 | |
| 15.e | even | 4 | 1 | 75.5.c.d | ✓ | 2 | |
| 15.e | even | 4 | 1 | 75.5.c.f | yes | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.5.c.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 75.5.c.d | ✓ | 2 | 15.e | even | 4 | 1 | |
| 75.5.c.f | yes | 2 | 5.c | odd | 4 | 1 | |
| 75.5.c.f | yes | 2 | 15.e | even | 4 | 1 | |
| 75.5.d.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.5.d.c | 4 | 3.b | odd | 2 | 1 | inner | |
| 75.5.d.c | 4 | 5.b | even | 2 | 1 | inner | |
| 75.5.d.c | 4 | 15.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 35 \)
acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 35)^{2} \)
|
| $3$ |
\( T^{4} - 153T^{2} + 6561 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 5184)^{2} \)
|
| $11$ |
\( (T^{2} + 33635)^{2} \)
|
| $13$ |
\( (T^{2} + 40804)^{2} \)
|
| $17$ |
\( (T^{2} - 5915)^{2} \)
|
| $19$ |
\( (T + 367)^{4} \)
|
| $23$ |
\( (T^{2} - 1260)^{2} \)
|
| $29$ |
\( (T^{2} + 134540)^{2} \)
|
| $31$ |
\( (T + 318)^{4} \)
|
| $37$ |
\( (T^{2} + 565504)^{2} \)
|
| $41$ |
\( (T^{2} + 2570435)^{2} \)
|
| $43$ |
\( (T^{2} + 964324)^{2} \)
|
| $47$ |
\( (T^{2} - 1428140)^{2} \)
|
| $53$ |
\( (T^{2} - 5544140)^{2} \)
|
| $59$ |
\( (T^{2} + 560)^{2} \)
|
| $61$ |
\( (T + 3728)^{4} \)
|
| $67$ |
\( (T^{2} + 55398249)^{2} \)
|
| $71$ |
\( (T^{2} + 45326540)^{2} \)
|
| $73$ |
\( (T^{2} + 157609)^{2} \)
|
| $79$ |
\( (T - 6108)^{4} \)
|
| $83$ |
\( (T^{2} - 176715315)^{2} \)
|
| $89$ |
\( (T^{2} + 22456035)^{2} \)
|
| $97$ |
\( (T^{2} + 64673764)^{2} \)
|
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