Newspace parameters
Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 750.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.98878015160\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} + 1 \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} + 2\nu \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} - 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} - 2\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(251\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
499.1 |
|
− | 1.00000i | − | 1.00000i | −1.00000 | 0 | −1.00000 | − | 1.23607i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||
499.2 | − | 1.00000i | − | 1.00000i | −1.00000 | 0 | −1.00000 | 3.23607i | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||
499.3 | 1.00000i | 1.00000i | −1.00000 | 0 | −1.00000 | − | 3.23607i | − | 1.00000i | −1.00000 | 0 | |||||||||||||||||||||||||||||
499.4 | 1.00000i | 1.00000i | −1.00000 | 0 | −1.00000 | 1.23607i | − | 1.00000i | −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 | 750.2.c.b | 4 | |
3.b | odd | 2 | 1 | 2250.2.c.b | 4 | ||
4.b | odd | 2 | 1 | 6000.2.f.a | 4 | ||
5.b | even | 2 | 1 | inner | 750.2.c.b | 4 | |
5.c | odd | 4 | 1 | 750.2.a.c | ✓ | 2 | |
5.c | odd | 4 | 1 | 750.2.a.f | yes | 2 | |
15.d | odd | 2 | 1 | 2250.2.c.b | 4 | ||
15.e | even | 4 | 1 | 2250.2.a.c | 2 | ||
15.e | even | 4 | 1 | 2250.2.a.n | 2 | ||
20.d | odd | 2 | 1 | 6000.2.f.a | 4 | ||
20.e | even | 4 | 1 | 6000.2.a.d | 2 | ||
20.e | even | 4 | 1 | 6000.2.a.y | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
750.2.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
750.2.a.f | yes | 2 | 5.c | odd | 4 | 1 | |
750.2.c.b | 4 | 1.a | even | 1 | 1 | trivial | |
750.2.c.b | 4 | 5.b | even | 2 | 1 | inner | |
2250.2.a.c | 2 | 15.e | even | 4 | 1 | ||
2250.2.a.n | 2 | 15.e | even | 4 | 1 | ||
2250.2.c.b | 4 | 3.b | odd | 2 | 1 | ||
2250.2.c.b | 4 | 15.d | odd | 2 | 1 | ||
6000.2.a.d | 2 | 20.e | even | 4 | 1 | ||
6000.2.a.y | 2 | 20.e | even | 4 | 1 | ||
6000.2.f.a | 4 | 4.b | odd | 2 | 1 | ||
6000.2.f.a | 4 | 20.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 12T_{7}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{2} \)
$3$
\( (T^{2} + 1)^{2} \)
$5$
\( T^{4} \)
$7$
\( T^{4} + 12T^{2} + 16 \)
$11$
\( (T^{2} - 5 T + 5)^{2} \)
$13$
\( T^{4} + 23T^{2} + 121 \)
$17$
\( T^{4} + 27T^{2} + 81 \)
$19$
\( (T + 6)^{4} \)
$23$
\( T^{4} + 75T^{2} + 625 \)
$29$
\( (T^{2} + 3 T - 59)^{2} \)
$31$
\( (T^{2} - 11 T + 19)^{2} \)
$37$
\( T^{4} + 63T^{2} + 81 \)
$41$
\( (T^{2} + 2 T - 4)^{2} \)
$43$
\( T^{4} + 47T^{2} + 1 \)
$47$
\( T^{4} + 135T^{2} + 2025 \)
$53$
\( T^{4} + 12T^{2} + 16 \)
$59$
\( (T^{2} + 15 T + 55)^{2} \)
$61$
\( (T^{2} - 180)^{2} \)
$67$
\( T^{4} + 27T^{2} + 121 \)
$71$
\( (T^{2} - 80)^{2} \)
$73$
\( (T^{2} + 144)^{2} \)
$79$
\( (T^{2} - 9 T + 9)^{2} \)
$83$
\( T^{4} + 268 T^{2} + 13456 \)
$89$
\( (T^{2} - 14 T + 44)^{2} \)
$97$
\( T^{4} + 348T^{2} + 5776 \)
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