Newspace parameters
Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 480.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.83281929702\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.1649659456.5 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | no (minimal twist has level 120) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + \nu^{6} - 4\nu^{2} - 8 ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{6} - \nu^{5} + 2\nu^{4} - 2\nu^{3} + 8 ) / 4 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} + \nu^{6} + 8\nu^{3} + 4\nu^{2} - 16 ) / 8 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{7} - \nu^{6} - 4\nu^{2} + 8\nu + 8 ) / 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} - 4\nu^{3} + 4\nu^{2} + 8 ) / 8 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + \nu^{5} + 2\nu^{4} - 2\nu^{3} + 4\nu^{2} + 8\nu + 8 ) / 4 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{7} + \nu^{5} - 4\nu^{4} + 2\nu^{3} - 12 ) / 4 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} - 2\beta _1 + 4 ) / 4 \) |
\(\nu^{4}\) | \(=\) | \( ( -3\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 - 4 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{7} + 3\beta_{6} - 9\beta_{5} - 3\beta_{4} - 5\beta_{3} - 5\beta_{2} - 2\beta _1 + 4 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( -3\beta_{7} + \beta_{6} + 5\beta_{5} - \beta_{4} + \beta_{3} - 7\beta_{2} + 10\beta _1 + 12 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( 7\beta_{7} + 3\beta_{6} - \beta_{5} - 3\beta_{4} + 3\beta_{3} + 11\beta_{2} + 14\beta _1 + 20 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(421\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
431.1 |
|
0 | −1.48716 | − | 0.887900i | 0 | 1.00000 | 0 | − | 0.797253i | 0 | 1.42327 | + | 2.64089i | 0 | |||||||||||||||||||||||||||||||||||||
431.2 | 0 | −1.48716 | + | 0.887900i | 0 | 1.00000 | 0 | 0.797253i | 0 | 1.42327 | − | 2.64089i | 0 | |||||||||||||||||||||||||||||||||||||||
431.3 | 0 | −0.751690 | − | 1.56044i | 0 | 1.00000 | 0 | 4.28591i | 0 | −1.86993 | + | 2.34593i | 0 | |||||||||||||||||||||||||||||||||||||||
431.4 | 0 | −0.751690 | + | 1.56044i | 0 | 1.00000 | 0 | − | 4.28591i | 0 | −1.86993 | − | 2.34593i | 0 | ||||||||||||||||||||||||||||||||||||||
431.5 | 0 | 0.520627 | − | 1.65195i | 0 | 1.00000 | 0 | − | 1.92736i | 0 | −2.45790 | − | 1.72010i | 0 | ||||||||||||||||||||||||||||||||||||||
431.6 | 0 | 0.520627 | + | 1.65195i | 0 | 1.00000 | 0 | 1.92736i | 0 | −2.45790 | + | 1.72010i | 0 | |||||||||||||||||||||||||||||||||||||||
431.7 | 0 | 1.71822 | − | 0.218455i | 0 | 1.00000 | 0 | − | 3.64426i | 0 | 2.90455 | − | 0.750707i | 0 | ||||||||||||||||||||||||||||||||||||||
431.8 | 0 | 1.71822 | + | 0.218455i | 0 | 1.00000 | 0 | 3.64426i | 0 | 2.90455 | + | 0.750707i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 480.2.b.b | 8 | |
3.b | odd | 2 | 1 | 480.2.b.a | 8 | ||
4.b | odd | 2 | 1 | 120.2.b.a | ✓ | 8 | |
5.b | even | 2 | 1 | 2400.2.b.f | 8 | ||
5.c | odd | 4 | 2 | 2400.2.m.d | 16 | ||
8.b | even | 2 | 1 | 120.2.b.b | yes | 8 | |
8.d | odd | 2 | 1 | 480.2.b.a | 8 | ||
12.b | even | 2 | 1 | 120.2.b.b | yes | 8 | |
15.d | odd | 2 | 1 | 2400.2.b.e | 8 | ||
15.e | even | 4 | 2 | 2400.2.m.c | 16 | ||
20.d | odd | 2 | 1 | 600.2.b.f | 8 | ||
20.e | even | 4 | 2 | 600.2.m.c | 16 | ||
24.f | even | 2 | 1 | inner | 480.2.b.b | 8 | |
24.h | odd | 2 | 1 | 120.2.b.a | ✓ | 8 | |
40.e | odd | 2 | 1 | 2400.2.b.e | 8 | ||
40.f | even | 2 | 1 | 600.2.b.e | 8 | ||
40.i | odd | 4 | 2 | 600.2.m.d | 16 | ||
40.k | even | 4 | 2 | 2400.2.m.c | 16 | ||
60.h | even | 2 | 1 | 600.2.b.e | 8 | ||
60.l | odd | 4 | 2 | 600.2.m.d | 16 | ||
120.i | odd | 2 | 1 | 600.2.b.f | 8 | ||
120.m | even | 2 | 1 | 2400.2.b.f | 8 | ||
120.q | odd | 4 | 2 | 2400.2.m.d | 16 | ||
120.w | even | 4 | 2 | 600.2.m.c | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
120.2.b.a | ✓ | 8 | 24.h | odd | 2 | 1 | |
120.2.b.b | yes | 8 | 8.b | even | 2 | 1 | |
120.2.b.b | yes | 8 | 12.b | even | 2 | 1 | |
480.2.b.a | 8 | 3.b | odd | 2 | 1 | ||
480.2.b.a | 8 | 8.d | odd | 2 | 1 | ||
480.2.b.b | 8 | 1.a | even | 1 | 1 | trivial | |
480.2.b.b | 8 | 24.f | even | 2 | 1 | inner | |
600.2.b.e | 8 | 40.f | even | 2 | 1 | ||
600.2.b.e | 8 | 60.h | even | 2 | 1 | ||
600.2.b.f | 8 | 20.d | odd | 2 | 1 | ||
600.2.b.f | 8 | 120.i | odd | 2 | 1 | ||
600.2.m.c | 16 | 20.e | even | 4 | 2 | ||
600.2.m.c | 16 | 120.w | even | 4 | 2 | ||
600.2.m.d | 16 | 40.i | odd | 4 | 2 | ||
600.2.m.d | 16 | 60.l | odd | 4 | 2 | ||
2400.2.b.e | 8 | 15.d | odd | 2 | 1 | ||
2400.2.b.e | 8 | 40.e | odd | 2 | 1 | ||
2400.2.b.f | 8 | 5.b | even | 2 | 1 | ||
2400.2.b.f | 8 | 120.m | even | 2 | 1 | ||
2400.2.m.c | 16 | 15.e | even | 4 | 2 | ||
2400.2.m.c | 16 | 40.k | even | 4 | 2 | ||
2400.2.m.d | 16 | 5.c | odd | 4 | 2 | ||
2400.2.m.d | 16 | 120.q | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{23}^{4} + 2T_{23}^{3} - 44T_{23}^{2} - 188T_{23} - 192 \)
acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - 4 T^{5} - 2 T^{4} - 12 T^{3} + \cdots + 81 \)
$5$
\( (T - 1)^{8} \)
$7$
\( T^{8} + 36 T^{6} + 384 T^{4} + \cdots + 576 \)
$11$
\( T^{8} + 48 T^{6} + 672 T^{4} + \cdots + 256 \)
$13$
\( T^{8} + 52 T^{6} + 880 T^{4} + \cdots + 9216 \)
$17$
\( T^{8} + 52 T^{6} + 816 T^{4} + \cdots + 4096 \)
$19$
\( (T^{4} - 2 T^{3} - 36 T^{2} + 72 T - 32)^{2} \)
$23$
\( (T^{4} + 2 T^{3} - 44 T^{2} - 188 T - 192)^{2} \)
$29$
\( (T^{4} - 64 T^{2} + 112 T - 48)^{2} \)
$31$
\( T^{8} + 140 T^{6} + 4544 T^{4} + \cdots + 9216 \)
$37$
\( T^{8} + 228 T^{6} + 17072 T^{4} + \cdots + 746496 \)
$41$
\( T^{8} + 64 T^{6} + 1344 T^{4} + \cdots + 16384 \)
$43$
\( (T^{4} - 12 T^{2} + 4 T + 16)^{2} \)
$47$
\( (T^{4} - 14 T^{3} - 20 T^{2} + 788 T - 2208)^{2} \)
$53$
\( (T^{4} + 8 T^{3} - 24 T^{2} - 128 T + 144)^{2} \)
$59$
\( T^{8} + 160 T^{6} + 672 T^{4} + \cdots + 256 \)
$61$
\( T^{8} + 208 T^{6} + 11392 T^{4} + \cdots + 36864 \)
$67$
\( (T^{4} - 16 T^{3} - 52 T^{2} + 828 T + 2896)^{2} \)
$71$
\( (T^{4} + 12 T^{3} - 64 T^{2} - 992 T - 2304)^{2} \)
$73$
\( (T^{4} + 4 T^{3} - 112 T^{2} - 624 T - 656)^{2} \)
$79$
\( T^{8} + 108 T^{6} + 2048 T^{4} + \cdots + 9216 \)
$83$
\( T^{8} + 152 T^{6} + 7952 T^{4} + \cdots + 1032256 \)
$89$
\( T^{8} + 384 T^{6} + 45056 T^{4} + \cdots + 1048576 \)
$97$
\( (T^{4} - 4 T^{3} - 176 T^{2} - 784 T - 784)^{2} \)
show more
show less