Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{17}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - x - 4 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 35) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). 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 |
|
−1.56155 | 0 | 0.438447 | −1.00000 | 0 | −1.00000 | 2.43845 | 0 | 1.56155 | ||||||||||||||||||||||||
1.2 | 2.56155 | 0 | 4.56155 | −1.00000 | 0 | −1.00000 | 6.56155 | 0 | −2.56155 | |||||||||||||||||||||||||
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 | 315.2.a.e | 2 | |
3.b | odd | 2 | 1 | 35.2.a.b | ✓ | 2 | |
4.b | odd | 2 | 1 | 5040.2.a.bt | 2 | ||
5.b | even | 2 | 1 | 1575.2.a.p | 2 | ||
5.c | odd | 4 | 2 | 1575.2.d.e | 4 | ||
7.b | odd | 2 | 1 | 2205.2.a.x | 2 | ||
12.b | even | 2 | 1 | 560.2.a.i | 2 | ||
15.d | odd | 2 | 1 | 175.2.a.f | 2 | ||
15.e | even | 4 | 2 | 175.2.b.b | 4 | ||
21.c | even | 2 | 1 | 245.2.a.d | 2 | ||
21.g | even | 6 | 2 | 245.2.e.h | 4 | ||
21.h | odd | 6 | 2 | 245.2.e.i | 4 | ||
24.f | even | 2 | 1 | 2240.2.a.bd | 2 | ||
24.h | odd | 2 | 1 | 2240.2.a.bh | 2 | ||
33.d | even | 2 | 1 | 4235.2.a.m | 2 | ||
39.d | odd | 2 | 1 | 5915.2.a.l | 2 | ||
60.h | even | 2 | 1 | 2800.2.a.bi | 2 | ||
60.l | odd | 4 | 2 | 2800.2.g.t | 4 | ||
84.h | odd | 2 | 1 | 3920.2.a.bs | 2 | ||
105.g | even | 2 | 1 | 1225.2.a.s | 2 | ||
105.k | odd | 4 | 2 | 1225.2.b.f | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.2.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
175.2.a.f | 2 | 15.d | odd | 2 | 1 | ||
175.2.b.b | 4 | 15.e | even | 4 | 2 | ||
245.2.a.d | 2 | 21.c | even | 2 | 1 | ||
245.2.e.h | 4 | 21.g | even | 6 | 2 | ||
245.2.e.i | 4 | 21.h | odd | 6 | 2 | ||
315.2.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
560.2.a.i | 2 | 12.b | even | 2 | 1 | ||
1225.2.a.s | 2 | 105.g | even | 2 | 1 | ||
1225.2.b.f | 4 | 105.k | odd | 4 | 2 | ||
1575.2.a.p | 2 | 5.b | even | 2 | 1 | ||
1575.2.d.e | 4 | 5.c | odd | 4 | 2 | ||
2205.2.a.x | 2 | 7.b | odd | 2 | 1 | ||
2240.2.a.bd | 2 | 24.f | even | 2 | 1 | ||
2240.2.a.bh | 2 | 24.h | odd | 2 | 1 | ||
2800.2.a.bi | 2 | 60.h | even | 2 | 1 | ||
2800.2.g.t | 4 | 60.l | odd | 4 | 2 | ||
3920.2.a.bs | 2 | 84.h | odd | 2 | 1 | ||
4235.2.a.m | 2 | 33.d | even | 2 | 1 | ||
5040.2.a.bt | 2 | 4.b | odd | 2 | 1 | ||
5915.2.a.l | 2 | 39.d | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T - 4 \)
$3$
\( T^{2} \)
$5$
\( (T + 1)^{2} \)
$7$
\( (T + 1)^{2} \)
$11$
\( T^{2} + T - 4 \)
$13$
\( T^{2} - 5T + 2 \)
$17$
\( T^{2} - 5T + 2 \)
$19$
\( T^{2} + 6T - 8 \)
$23$
\( T^{2} - 2T - 16 \)
$29$
\( T^{2} + T - 38 \)
$31$
\( T^{2} \)
$37$
\( (T - 6)^{2} \)
$41$
\( T^{2} + 2T - 16 \)
$43$
\( T^{2} - 10T + 8 \)
$47$
\( T^{2} - 5T - 32 \)
$53$
\( T^{2} - 2T - 16 \)
$59$
\( (T - 4)^{2} \)
$61$
\( T^{2} - 6T - 144 \)
$67$
\( T^{2} - 4T - 64 \)
$71$
\( (T + 8)^{2} \)
$73$
\( T^{2} + 8T - 52 \)
$79$
\( T^{2} + 9T + 16 \)
$83$
\( (T + 4)^{2} \)
$89$
\( T^{2} + 6T - 8 \)
$97$
\( T^{2} + 9T - 86 \)
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