Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(209.608008215\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 154\nu - 1756122 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 711\nu^{2} - 2080768\nu - 978048758 ) / 22 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 154\beta _1 + 1756122 \)
|
| \(\nu^{3}\) | \(=\) |
\( 22\beta_{3} - 711\beta_{2} + 2190262\beta _1 - 270553984 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1487.97 | 59049.0 | 116903. | 0 | −8.78631e7 | 1.26833e9 | 2.94655e9 | 3.48678e9 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −625.272 | 59049.0 | −1.70619e6 | 0 | −3.69217e7 | −3.78633e8 | 2.37812e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1293.36 | 59049.0 | −424379. | 0 | 7.63714e7 | −1.27960e9 | −3.26124e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 1716.88 | 59049.0 | 850541. | 0 | 1.01380e8 | 6.24477e8 | −2.14029e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.22.a.h | 4 | |
| 5.b | even | 2 | 1 | 15.22.a.e | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 75.22.b.h | 8 | ||
| 15.d | odd | 2 | 1 | 45.22.a.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.a.e | ✓ | 4 | 5.b | even | 2 | 1 | |
| 45.22.a.g | 4 | 15.d | odd | 2 | 1 | ||
| 75.22.a.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.h | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 897T_{2}^{3} - 3210438T_{2}^{2} + 1891862624T_{2} + 2065963121664 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 897 T^{3} + \cdots + 2065963121664 \)
$3$
\( (T - 59049)^{4} \)
$5$
\( T^{4} \)
$7$
\( T^{4} - 234577504 T^{3} + \cdots + 38\!\cdots\!00 \)
$11$
\( T^{4} - 31491830256 T^{3} + \cdots + 28\!\cdots\!64 \)
$13$
\( T^{4} - 27017977768 T^{3} + \cdots + 26\!\cdots\!84 \)
$17$
\( T^{4} - 2946095028888 T^{3} + \cdots + 31\!\cdots\!56 \)
$19$
\( T^{4} + 24270353300752 T^{3} + \cdots + 34\!\cdots\!00 \)
$23$
\( T^{4} + 10350924920928 T^{3} + \cdots - 97\!\cdots\!00 \)
$29$
\( T^{4} + \cdots - 51\!\cdots\!00 \)
$31$
\( T^{4} + \cdots + 15\!\cdots\!00 \)
$37$
\( T^{4} + \cdots + 50\!\cdots\!00 \)
$41$
\( T^{4} + \cdots - 70\!\cdots\!00 \)
$43$
\( T^{4} + \cdots - 16\!\cdots\!04 \)
$47$
\( T^{4} + \cdots - 21\!\cdots\!44 \)
$53$
\( T^{4} + \cdots + 34\!\cdots\!00 \)
$59$
\( T^{4} + \cdots + 97\!\cdots\!00 \)
$61$
\( T^{4} + \cdots + 27\!\cdots\!76 \)
$67$
\( T^{4} + \cdots + 28\!\cdots\!56 \)
$71$
\( T^{4} + \cdots - 17\!\cdots\!44 \)
$73$
\( T^{4} + \cdots - 99\!\cdots\!00 \)
$79$
\( T^{4} + \cdots - 12\!\cdots\!00 \)
$83$
\( T^{4} + \cdots - 79\!\cdots\!56 \)
$89$
\( T^{4} + \cdots + 85\!\cdots\!00 \)
$97$
\( T^{4} + \cdots - 16\!\cdots\!24 \)
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