Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(23.4288769113\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{31})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 15x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - 15x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 7\nu ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 23\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 30 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 30 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{2} + 46\beta_1 ) / 4 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 15.1355i | − | 27.0000i | −101.084 | 0 | −408.659 | − | 198.897i | − | 407.384i | −729.000 | 0 | ||||||||||||||||||||||||||
| 49.2 | − | 7.13553i | 27.0000i | 77.0842 | 0 | 192.659 | − | 112.897i | − | 1463.38i | −729.000 | 0 | ||||||||||||||||||||||||||||
| 49.3 | 7.13553i | − | 27.0000i | 77.0842 | 0 | 192.659 | 112.897i | 1463.38i | −729.000 | 0 | ||||||||||||||||||||||||||||||
| 49.4 | 15.1355i | 27.0000i | −101.084 | 0 | −408.659 | 198.897i | 407.384i | −729.000 | 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 | 75.8.b.e | 4 | |
| 3.b | odd | 2 | 1 | 225.8.b.o | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.8.b.e | 4 | |
| 5.c | odd | 4 | 1 | 75.8.a.d | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.8.a.f | yes | 2 | |
| 15.d | odd | 2 | 1 | 225.8.b.o | 4 | ||
| 15.e | even | 4 | 1 | 225.8.a.l | 2 | ||
| 15.e | even | 4 | 1 | 225.8.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.8.a.d | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 75.8.a.f | yes | 2 | 5.c | odd | 4 | 1 | |
| 75.8.b.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.8.b.e | 4 | 5.b | even | 2 | 1 | inner | |
| 225.8.a.l | 2 | 15.e | even | 4 | 1 | ||
| 225.8.a.u | 2 | 15.e | even | 4 | 1 | ||
| 225.8.b.o | 4 | 3.b | odd | 2 | 1 | ||
| 225.8.b.o | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 280T_{2}^{2} + 11664 \)
acting on \(S_{8}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 280 T^{2} + 11664 \)
|
| $3$ |
\( (T^{2} + 729)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 52306 T^{2} + 504227025 \)
|
| $11$ |
\( (T^{2} + 4612 T - 3456108)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 89618352089809 \)
|
| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} - 23934 T - 791814895)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} - 346508 T + 29988616500)^{2} \)
|
| $31$ |
\( (T^{2} + 129178 T - 25626949575)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + 340928 T - 186336929520)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 81\!\cdots\!41 \)
|
| $47$ |
\( T^{4} + \cdots + 37\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots - 1475873866620)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 8423736487399)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 25\!\cdots\!01 \)
|
| $71$ |
\( (T^{2} + 2607296 T - 284467184496)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 9582807148800)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 16\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} + \cdots + 45203910693120)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 18\!\cdots\!81 \)
|
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