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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 16\nu^{2} - 1100\nu - 2828475 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 157\nu^{4} - 456377\nu^{3} + 37318593\nu^{2} + 36666498918\nu - 909462022508 ) / 4004 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 157\nu^{4} + 712633\nu^{3} - 66403649\nu^{2} - 112675632118\nu + 2940276712412 ) / 4004 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 571\nu^{4} - 544465\nu^{3} - 249005991\nu^{2} + 48958029442\nu + 11600260895639 ) / 1001 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 275\beta _1 + 2828750 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 454\beta_{2} + 4870675\beta _1 + 781801849 ) / 64 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 88\beta_{5} + 121\beta_{4} - 231\beta_{3} + 1628146\beta_{2} + 751841447\beta _1 + 3444795464265 ) / 64 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6908 \beta_{5} + 237687 \beta_{4} + 338183 \beta_{3} + 156769854 \beta_{2} + 856594358333 \beta _1 + 266494147996751 ) / 32 \)
|
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 |
|
−2300.75 | 59049.0 | 3.19629e6 | 0 | −1.35857e8 | 2.10374e8 | −2.52885e9 | 3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1570.86 | 59049.0 | 370456. | 0 | −9.27578e7 | −1.39580e9 | 2.71240e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −594.286 | 59049.0 | −1.74398e6 | 0 | −3.50920e7 | 9.49356e8 | 2.28273e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 340.910 | 59049.0 | −1.98093e6 | 0 | 2.01304e7 | −6.73159e8 | −1.39026e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1464.04 | 59049.0 | 46252.4 | 0 | 8.64499e7 | 1.40408e8 | −3.00259e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2568.95 | 59049.0 | 4.50235e6 | 0 | 1.51694e8 | −5.58323e8 | 6.17885e9 | 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.i | ✓ | 6 |
| 5.b | even | 2 | 1 | 75.22.a.j | yes | 6 | |
| 5.c | odd | 4 | 2 | 75.22.b.i | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.22.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 75.22.a.j | yes | 6 | 5.b | even | 2 | 1 | |
| 75.22.b.i | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 92 T_{2}^{5} - 8482448 T_{2}^{4} - 2069443264 T_{2}^{3} + 15258494066176 T_{2}^{2} + \cdots - 27\!\cdots\!28 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 92 T^{5} + \cdots - 27\!\cdots\!28 \)
$3$
\( (T - 59049)^{6} \)
$5$
\( T^{6} \)
$7$
\( T^{6} + 1327143454 T^{5} + \cdots - 14\!\cdots\!75 \)
$11$
\( T^{6} + 78310112516 T^{5} + \cdots - 99\!\cdots\!52 \)
$13$
\( T^{6} + 116029746338 T^{5} + \cdots + 45\!\cdots\!97 \)
$17$
\( T^{6} + 5382900513068 T^{5} + \cdots - 85\!\cdots\!96 \)
$19$
\( T^{6} - 6969638997622 T^{5} + \cdots - 69\!\cdots\!75 \)
$23$
\( T^{6} + 295754895311052 T^{5} + \cdots - 72\!\cdots\!00 \)
$29$
\( T^{6} + \cdots - 38\!\cdots\!00 \)
$31$
\( T^{6} - 492586859482706 T^{5} + \cdots + 43\!\cdots\!25 \)
$37$
\( T^{6} + \cdots - 26\!\cdots\!00 \)
$41$
\( T^{6} + \cdots + 16\!\cdots\!00 \)
$43$
\( T^{6} + \cdots - 29\!\cdots\!31 \)
$47$
\( T^{6} + \cdots - 14\!\cdots\!04 \)
$53$
\( T^{6} + \cdots + 61\!\cdots\!00 \)
$59$
\( T^{6} + \cdots - 36\!\cdots\!00 \)
$61$
\( T^{6} + \cdots + 14\!\cdots\!41 \)
$67$
\( T^{6} + \cdots - 96\!\cdots\!51 \)
$71$
\( T^{6} + \cdots + 54\!\cdots\!84 \)
$73$
\( T^{6} + \cdots + 27\!\cdots\!00 \)
$79$
\( T^{6} + \cdots - 37\!\cdots\!00 \)
$83$
\( T^{6} + \cdots - 16\!\cdots\!32 \)
$89$
\( T^{6} + \cdots - 85\!\cdots\!00 \)
$97$
\( T^{6} + \cdots - 80\!\cdots\!91 \)
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