Newspace parameters
Level: | \( N \) | \(=\) | \( 760 = 2^{3} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 760.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.06863055362\) |
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: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{4} \) |
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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( 2\nu^{3} + 4\nu \) |
\(\beta_{3}\) | \(=\) | \( 4\nu^{2} + 6 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} - 6 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{2} - 2\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(381\) | \(401\) | \(457\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
609.1 |
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0 | − | 3.23607i | 0 | −1.00000 | + | 2.00000i | 0 | − | 4.47214i | 0 | −7.47214 | 0 | ||||||||||||||||||||||||||
609.2 | 0 | − | 1.23607i | 0 | −1.00000 | − | 2.00000i | 0 | − | 4.47214i | 0 | 1.47214 | 0 | |||||||||||||||||||||||||||
609.3 | 0 | 1.23607i | 0 | −1.00000 | + | 2.00000i | 0 | 4.47214i | 0 | 1.47214 | 0 | |||||||||||||||||||||||||||||
609.4 | 0 | 3.23607i | 0 | −1.00000 | − | 2.00000i | 0 | 4.47214i | 0 | −7.47214 | 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 | 760.2.d.b | ✓ | 4 |
4.b | odd | 2 | 1 | 1520.2.d.c | 4 | ||
5.b | even | 2 | 1 | inner | 760.2.d.b | ✓ | 4 |
5.c | odd | 4 | 1 | 3800.2.a.k | 2 | ||
5.c | odd | 4 | 1 | 3800.2.a.q | 2 | ||
20.d | odd | 2 | 1 | 1520.2.d.c | 4 | ||
20.e | even | 4 | 1 | 7600.2.a.w | 2 | ||
20.e | even | 4 | 1 | 7600.2.a.be | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
760.2.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
760.2.d.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
1520.2.d.c | 4 | 4.b | odd | 2 | 1 | ||
1520.2.d.c | 4 | 20.d | odd | 2 | 1 | ||
3800.2.a.k | 2 | 5.c | odd | 4 | 1 | ||
3800.2.a.q | 2 | 5.c | odd | 4 | 1 | ||
7600.2.a.w | 2 | 20.e | even | 4 | 1 | ||
7600.2.a.be | 2 | 20.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 12T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 12T^{2} + 16 \)
$5$
\( (T^{2} + 2 T + 5)^{2} \)
$7$
\( (T^{2} + 20)^{2} \)
$11$
\( T^{4} \)
$13$
\( T^{4} + 12T^{2} + 16 \)
$17$
\( T^{4} + 48T^{2} + 256 \)
$19$
\( (T + 1)^{4} \)
$23$
\( (T^{2} + 4)^{2} \)
$29$
\( (T + 2)^{4} \)
$31$
\( (T^{2} + 12 T + 16)^{2} \)
$37$
\( T^{4} + 108T^{2} + 1936 \)
$41$
\( (T^{2} - 16 T + 44)^{2} \)
$43$
\( T^{4} + 72T^{2} + 16 \)
$47$
\( T^{4} + 168T^{2} + 1936 \)
$53$
\( T^{4} + 172T^{2} + 5776 \)
$59$
\( (T^{2} - 12 T + 16)^{2} \)
$61$
\( (T^{2} - 20)^{2} \)
$67$
\( T^{4} + 12T^{2} + 16 \)
$71$
\( (T^{2} + 12 T + 16)^{2} \)
$73$
\( T^{4} + 48T^{2} + 256 \)
$79$
\( (T^{2} - 4 T - 16)^{2} \)
$83$
\( T^{4} + 232T^{2} + 1936 \)
$89$
\( (T^{2} + 4 T - 76)^{2} \)
$97$
\( T^{4} + 348T^{2} + 5776 \)
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