Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(18.5856016518\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{5}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 105) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−0.236068 | 0 | −7.94427 | 5.00000 | 0 | −7.00000 | 3.76393 | 0 | −1.18034 | ||||||||||||||||||||||||
1.2 | 4.23607 | 0 | 9.94427 | 5.00000 | 0 | −7.00000 | 8.23607 | 0 | 21.1803 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(5\) | \(-1\) |
\(7\) | \(1\) |
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 | 315.4.a.l | 2 | |
3.b | odd | 2 | 1 | 105.4.a.d | ✓ | 2 | |
5.b | even | 2 | 1 | 1575.4.a.n | 2 | ||
7.b | odd | 2 | 1 | 2205.4.a.be | 2 | ||
12.b | even | 2 | 1 | 1680.4.a.bd | 2 | ||
15.d | odd | 2 | 1 | 525.4.a.o | 2 | ||
15.e | even | 4 | 2 | 525.4.d.k | 4 | ||
21.c | even | 2 | 1 | 735.4.a.m | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.4.a.d | ✓ | 2 | 3.b | odd | 2 | 1 | |
315.4.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
525.4.a.o | 2 | 15.d | odd | 2 | 1 | ||
525.4.d.k | 4 | 15.e | even | 4 | 2 | ||
735.4.a.m | 2 | 21.c | even | 2 | 1 | ||
1575.4.a.n | 2 | 5.b | even | 2 | 1 | ||
1680.4.a.bd | 2 | 12.b | even | 2 | 1 | ||
2205.4.a.be | 2 | 7.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 4T_{2} - 1 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 4T - 1 \)
$3$
\( T^{2} \)
$5$
\( (T - 5)^{2} \)
$7$
\( (T + 7)^{2} \)
$11$
\( T^{2} - 92T + 2096 \)
$13$
\( T^{2} - 8T - 7204 \)
$17$
\( T^{2} - 44T - 9196 \)
$19$
\( T^{2} + 108T - 464 \)
$23$
\( T^{2} - 320T + 23600 \)
$29$
\( T^{2} - 236T + 13204 \)
$31$
\( T^{2} + 60T - 51120 \)
$37$
\( T^{2} - 204T + 7524 \)
$41$
\( T^{2} + 44T - 31516 \)
$43$
\( T^{2} - 136T - 77296 \)
$47$
\( T^{2} + 400T - 101120 \)
$53$
\( T^{2} + 16T - 36916 \)
$59$
\( T^{2} - 464T + 47344 \)
$61$
\( T^{2} + 684T + 81684 \)
$67$
\( T^{2} - 736T + 944 \)
$71$
\( T^{2} - 740T + 41680 \)
$73$
\( T^{2} - 424T - 29476 \)
$79$
\( T^{2} + 408 T - 1129664 \)
$83$
\( T^{2} + 608T + 57136 \)
$89$
\( T^{2} - 1332 T + 380836 \)
$97$
\( T^{2} + 2448 T + 1461196 \)
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