Newspace parameters
Level: | \( N \) | \(=\) | \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 1560.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(92.0429796090\) |
Analytic rank: | \(1\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).
\(n\) | \(391\) | \(521\) | \(781\) | \(937\) | \(1081\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
961.1 |
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0 | 3.00000 | 0 | − | 5.00000i | 0 | − | 8.00000i | 0 | 9.00000 | 0 | ||||||||||||||||||||||
961.2 | 0 | 3.00000 | 0 | 5.00000i | 0 | 8.00000i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1560.4.g.a | ✓ | 2 |
13.b | even | 2 | 1 | inner | 1560.4.g.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1560.4.g.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
1560.4.g.a | ✓ | 2 | 13.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 64 \)
acting on \(S_{4}^{\mathrm{new}}(1560, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 3)^{2} \)
$5$
\( T^{2} + 25 \)
$7$
\( T^{2} + 64 \)
$11$
\( T^{2} + 144 \)
$13$
\( T^{2} + 78T + 2197 \)
$17$
\( (T + 36)^{2} \)
$19$
\( T^{2} + 15376 \)
$23$
\( (T + 42)^{2} \)
$29$
\( (T - 76)^{2} \)
$31$
\( T^{2} + 30276 \)
$37$
\( T^{2} + 17424 \)
$41$
\( T^{2} + 125316 \)
$43$
\( (T + 380)^{2} \)
$47$
\( T^{2} + 9216 \)
$53$
\( (T + 202)^{2} \)
$59$
\( T^{2} + 583696 \)
$61$
\( (T + 882)^{2} \)
$67$
\( T^{2} + 252004 \)
$71$
\( T^{2} + 26896 \)
$73$
\( T^{2} + 470596 \)
$79$
\( (T + 280)^{2} \)
$83$
\( T^{2} + 440896 \)
$89$
\( T^{2} + 1822500 \)
$97$
\( T^{2} + 2328676 \)
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