Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(137.416565508\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 7285x^{2} + 13264164 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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
\( x^{4} + 7285x^{2} + 13264164 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 3643\nu ) / 3642 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 10927\nu ) / 1214 \)
|
| \(\beta_{3}\) | \(=\) |
\( 6\nu^{2} + 21855 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 21855 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3643\beta_{2} + 32781\beta_1 ) / 6 \)
|
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 |
|
− | 659.106i | 6561.00i | −303349. | 0 | 4.32440e6 | − | 1.59855e6i | 1.13549e8i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||
| 49.2 | − | 65.1063i | − | 6561.00i | 126833. | 0 | −427163. | 2.28730e7i | − | 1.67913e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||
| 49.3 | 65.1063i | 6561.00i | 126833. | 0 | −427163. | − | 2.28730e7i | 1.67913e7i | −4.30467e7 | 0 | ||||||||||||||||||||||||||||||
| 49.4 | 659.106i | − | 6561.00i | −303349. | 0 | 4.32440e6 | 1.59855e6i | − | 1.13549e8i | −4.30467e7 | 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 | 75.18.b.c | 4 | |
| 5.b | even | 2 | 1 | inner | 75.18.b.c | 4 | |
| 5.c | odd | 4 | 1 | 3.18.a.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 75.18.a.b | 2 | ||
| 15.e | even | 4 | 1 | 9.18.a.c | 2 | ||
| 20.e | even | 4 | 1 | 48.18.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.18.a.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 9.18.a.c | 2 | 15.e | even | 4 | 1 | ||
| 48.18.a.h | 2 | 20.e | even | 4 | 1 | ||
| 75.18.a.b | 2 | 5.c | odd | 4 | 1 | ||
| 75.18.b.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.18.b.c | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 438660T_{2}^{2} + 1841439744 \)
acting on \(S_{18}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 1841439744 \)
|
| $3$ |
\( (T^{2} + 43046721)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $11$ |
\( (T^{2} + \cdots + 24\!\cdots\!72)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 73\!\cdots\!44 \)
|
| $17$ |
\( T^{4} + \cdots + 55\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + \cdots - 18\!\cdots\!20)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 50\!\cdots\!80)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots + 26\!\cdots\!40)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots + 36\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + \cdots - 17\!\cdots\!80)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 19\!\cdots\!56)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 74\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + \cdots + 45\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $89$ |
\( (T^{2} + \cdots - 16\!\cdots\!60)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 65\!\cdots\!16 \)
|
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