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: | \(\Q(i, \sqrt{601})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 301x^{2} + 22500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{4} + 301x^{2} + 22500 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 151\nu ) / 150 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 151 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 151 \)
|
| \(\nu^{3}\) | \(=\) |
\( 150\beta_{2} - 151\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 15.7577i | 27.0000i | −120.304 | 0 | 425.457 | 34.4284i | − | 121.278i | −729.000 | 0 | ||||||||||||||||||||||||||||
| 49.2 | − | 8.75765i | − | 27.0000i | 51.3036 | 0 | −236.457 | 1338.43i | − | 1570.28i | −729.000 | 0 | ||||||||||||||||||||||||||||
| 49.3 | 8.75765i | 27.0000i | 51.3036 | 0 | −236.457 | − | 1338.43i | 1570.28i | −729.000 | 0 | ||||||||||||||||||||||||||||||
| 49.4 | 15.7577i | − | 27.0000i | −120.304 | 0 | 425.457 | − | 34.4284i | 121.278i | −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.d | 4 | |
| 3.b | odd | 2 | 1 | 225.8.b.n | 4 | ||
| 5.b | even | 2 | 1 | inner | 75.8.b.d | 4 | |
| 5.c | odd | 4 | 1 | 15.8.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.8.a.e | 2 | ||
| 15.d | odd | 2 | 1 | 225.8.b.n | 4 | ||
| 15.e | even | 4 | 1 | 45.8.a.i | 2 | ||
| 15.e | even | 4 | 1 | 225.8.a.t | 2 | ||
| 20.e | even | 4 | 1 | 240.8.a.p | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.8.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 45.8.a.i | 2 | 15.e | even | 4 | 1 | ||
| 75.8.a.e | 2 | 5.c | odd | 4 | 1 | ||
| 75.8.b.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.8.b.d | 4 | 5.b | even | 2 | 1 | inner | |
| 225.8.a.t | 2 | 15.e | even | 4 | 1 | ||
| 225.8.b.n | 4 | 3.b | odd | 2 | 1 | ||
| 225.8.b.n | 4 | 15.d | odd | 2 | 1 | ||
| 240.8.a.p | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 325T_{2}^{2} + 19044 \)
acting on \(S_{8}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 325 T^{2} + 19044 \)
|
| $3$ |
\( (T^{2} + 729)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 2123366400 \)
|
| $11$ |
\( (T^{2} - 3448 T - 29376048)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 66\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 82\!\cdots\!56 \)
|
| $19$ |
\( (T^{2} - 49584 T - 20483920)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + 181772 T - 2060549340)^{2} \)
|
| $31$ |
\( (T^{2} - 304232 T + 22433068800)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} - 631172 T + 30837469380)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 55\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots + 24\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 29\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots - 3349911332400)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 1579956747716)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!36 \)
|
| $71$ |
\( (T^{2} + \cdots + 2634564492864)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 48\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots + 4381741411200)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 63\!\cdots\!84 \)
|
| $89$ |
\( (T^{2} + \cdots + 68134385955780)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 66\!\cdots\!16 \)
|
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