Newspace parameters
Level: | \( N \) | \(=\) | \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 7500.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(59.8878015160\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.324000000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 300) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 2 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 3\nu \) |
\(\beta_{4}\) | \(=\) | \( \nu^{4} + 4\nu^{2} + 2 \) |
\(\beta_{5}\) | \(=\) | \( \nu^{5} + 5\nu^{3} + 5\nu \) |
\(\beta_{6}\) | \(=\) | \( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \) |
\(\beta_{7}\) | \(=\) | \( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 3\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{4} - 4\beta_{2} + 6 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 \) |
\(\nu^{7}\) | \(=\) | \( \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times\).
\(n\) | \(2501\) | \(3751\) | \(6877\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1249.1 |
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0 | − | 1.00000i | 0 | 0 | 0 | − | 4.78339i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
1249.2 | 0 | − | 1.00000i | 0 | 0 | 0 | − | 0.511170i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
1249.3 | 0 | − | 1.00000i | 0 | 0 | 0 | 0.547318i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1249.4 | 0 | − | 1.00000i | 0 | 0 | 0 | 0.747238i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1249.5 | 0 | 1.00000i | 0 | 0 | 0 | − | 0.747238i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1249.6 | 0 | 1.00000i | 0 | 0 | 0 | − | 0.547318i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
1249.7 | 0 | 1.00000i | 0 | 0 | 0 | 0.511170i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1249.8 | 0 | 1.00000i | 0 | 0 | 0 | 4.78339i | 0 | −1.00000 | 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 | 7500.2.d.d | 8 | |
5.b | even | 2 | 1 | inner | 7500.2.d.d | 8 | |
5.c | odd | 4 | 1 | 7500.2.a.d | 4 | ||
5.c | odd | 4 | 1 | 7500.2.a.g | 4 | ||
25.d | even | 5 | 2 | 1500.2.o.a | 16 | ||
25.e | even | 10 | 2 | 1500.2.o.a | 16 | ||
25.f | odd | 20 | 2 | 300.2.m.a | ✓ | 8 | |
25.f | odd | 20 | 2 | 1500.2.m.b | 8 | ||
75.l | even | 20 | 2 | 900.2.n.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
300.2.m.a | ✓ | 8 | 25.f | odd | 20 | 2 | |
900.2.n.a | 8 | 75.l | even | 20 | 2 | ||
1500.2.m.b | 8 | 25.f | odd | 20 | 2 | ||
1500.2.o.a | 16 | 25.d | even | 5 | 2 | ||
1500.2.o.a | 16 | 25.e | even | 10 | 2 | ||
7500.2.a.d | 4 | 5.c | odd | 4 | 1 | ||
7500.2.a.g | 4 | 5.c | odd | 4 | 1 | ||
7500.2.d.d | 8 | 1.a | even | 1 | 1 | trivial | |
7500.2.d.d | 8 | 5.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 24T_{7}^{6} + 26T_{7}^{4} + 9T_{7}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(7500, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} + 1)^{4} \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 24 T^{6} + 26 T^{4} + 9 T^{2} + \cdots + 1 \)
$11$
\( (T^{4} + T^{3} - 4 T^{2} - 4 T + 1)^{2} \)
$13$
\( T^{8} + 45 T^{6} + 290 T^{4} + \cdots + 25 \)
$17$
\( T^{8} + 84 T^{6} + 2186 T^{4} + \cdots + 73441 \)
$19$
\( (T^{4} - 5 T^{3} - 25 T^{2} + 95 T + 145)^{2} \)
$23$
\( T^{8} + 89 T^{6} + 2346 T^{4} + \cdots + 961 \)
$29$
\( (T^{4} - 4 T^{3} - 94 T^{2} + 571 T - 599)^{2} \)
$31$
\( (T^{4} + 9 T^{3} + 6 T^{2} - 126 T - 279)^{2} \)
$37$
\( T^{8} + 96 T^{6} + 2546 T^{4} + \cdots + 32761 \)
$41$
\( (T^{4} - 70 T^{2} + 240 T - 155)^{2} \)
$43$
\( T^{8} + 354 T^{6} + 40151 T^{4} + \cdots + 5621641 \)
$47$
\( T^{8} + 224 T^{6} + 16146 T^{4} + \cdots + 2627641 \)
$53$
\( T^{8} + 96 T^{6} + 2546 T^{4} + \cdots + 32041 \)
$59$
\( (T^{4} + T^{3} - 139 T^{2} + 41 T + 2341)^{2} \)
$61$
\( (T^{4} + 2 T^{3} - 96 T^{2} + 398 T - 449)^{2} \)
$67$
\( T^{8} + 334 T^{6} + 30891 T^{4} + \cdots + 6046681 \)
$71$
\( (T^{4} + 20 T^{3} + 50 T^{2} - 875 T - 3875)^{2} \)
$73$
\( T^{8} + 426 T^{6} + 42111 T^{4} + \cdots + 2653641 \)
$79$
\( (T^{4} - 3 T^{3} - 186 T^{2} + 1278 T - 1359)^{2} \)
$83$
\( T^{8} + 84 T^{6} + 2006 T^{4} + \cdots + 7921 \)
$89$
\( (T^{4} + 15 T^{3} + 30 T^{2} - 90 T + 45)^{2} \)
$97$
\( T^{8} + 396 T^{6} + \cdots + 18139081 \)
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