Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.e (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(38.6276877123\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 9\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 32.1 |
|
−11.0227 | + | 11.0227i | −99.2043 | − | 99.2043i | 269.000i | 0 | 2187.00 | 0 | −8608.73 | − | 8608.73i | 19683.0i | 0 | ||||||||||||||||||||||||
| 32.2 | 11.0227 | − | 11.0227i | 99.2043 | + | 99.2043i | 269.000i | 0 | 2187.00 | 0 | 8608.73 | + | 8608.73i | 19683.0i | 0 | |||||||||||||||||||||||||
| 68.1 | −11.0227 | − | 11.0227i | −99.2043 | + | 99.2043i | − | 269.000i | 0 | 2187.00 | 0 | −8608.73 | + | 8608.73i | − | 19683.0i | 0 | |||||||||||||||||||||||
| 68.2 | 11.0227 | + | 11.0227i | 99.2043 | − | 99.2043i | − | 269.000i | 0 | 2187.00 | 0 | 8608.73 | − | 8608.73i | − | 19683.0i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
| 15.e | even | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.10.e.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 75.10.e.c | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 75.10.e.c | ✓ | 4 |
| 5.c | odd | 4 | 2 | inner | 75.10.e.c | ✓ | 4 |
| 15.d | odd | 2 | 1 | CM | 75.10.e.c | ✓ | 4 |
| 15.e | even | 4 | 2 | inner | 75.10.e.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.10.e.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 75.10.e.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 75.10.e.c | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 75.10.e.c | ✓ | 4 | 5.c | odd | 4 | 2 | inner |
| 75.10.e.c | ✓ | 4 | 15.d | odd | 2 | 1 | CM |
| 75.10.e.c | ✓ | 4 | 15.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\):
|
\( T_{2}^{4} + 59049 \)
|
|
\( T_{7} \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 59049 \)
|
| $3$ |
\( T^{4} + 387420489 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 78\!\cdots\!24 \)
|
| $19$ |
\( (T^{2} + 1073950851856)^{2} \)
|
| $23$ |
\( T^{4} + 58\!\cdots\!84 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T + 8247368)^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + 19\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + 13\!\cdots\!44 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T - 197894882)^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} + 17\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{4} + 53\!\cdots\!64 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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