Newspace parameters
Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 480.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.83281929702\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | 6.0.839056.1 |
Defining polynomial: |
\( x^{6} + 6x^{4} + 8x^{2} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{5} \) |
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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 6x^{4} + 8x^{2} + 1 \)
:
\(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + \nu^{2} + 4\nu + 2 \)
|
\(\beta_{3}\) | \(=\) |
\( 2\nu^{4} + 8\nu^{2} + 3 \)
|
\(\beta_{4}\) | \(=\) |
\( -\nu^{5} - 5\nu^{3} + \nu^{2} - 4\nu + 2 \)
|
\(\beta_{5}\) | \(=\) |
\( 2\nu^{5} + 12\nu^{3} + 14\nu \)
|
\(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{2} - 4 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{2} - 3\beta_1 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -4\beta_{4} + \beta_{3} - 4\beta_{2} + 13 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( -5\beta_{5} - 6\beta_{4} + 6\beta_{2} + 11\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | 1.00000 | 0 | −2.11491 | − | 0.726062i | 0 | 4.05705i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
49.2 | 0 | 1.00000 | 0 | −2.11491 | + | 0.726062i | 0 | − | 4.05705i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
49.3 | 0 | 1.00000 | 0 | 0.254102 | − | 2.22158i | 0 | − | 2.64265i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
49.4 | 0 | 1.00000 | 0 | 0.254102 | + | 2.22158i | 0 | 2.64265i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||
49.5 | 0 | 1.00000 | 0 | 1.86081 | − | 1.23992i | 0 | 0.746175i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||
49.6 | 0 | 1.00000 | 0 | 1.86081 | + | 1.23992i | 0 | − | 0.746175i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
40.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.d.b | 6 | |
3.b | odd | 2 | 1 | 1440.2.d.f | 6 | ||
4.b | odd | 2 | 1 | 120.2.d.b | yes | 6 | |
5.b | even | 2 | 1 | 480.2.d.a | 6 | ||
5.c | odd | 4 | 2 | 2400.2.k.f | 12 | ||
8.b | even | 2 | 1 | 480.2.d.a | 6 | ||
8.d | odd | 2 | 1 | 120.2.d.a | ✓ | 6 | |
12.b | even | 2 | 1 | 360.2.d.e | 6 | ||
15.d | odd | 2 | 1 | 1440.2.d.e | 6 | ||
15.e | even | 4 | 2 | 7200.2.k.u | 12 | ||
16.e | even | 4 | 2 | 3840.2.f.m | 12 | ||
16.f | odd | 4 | 2 | 3840.2.f.l | 12 | ||
20.d | odd | 2 | 1 | 120.2.d.a | ✓ | 6 | |
20.e | even | 4 | 2 | 600.2.k.f | 12 | ||
24.f | even | 2 | 1 | 360.2.d.f | 6 | ||
24.h | odd | 2 | 1 | 1440.2.d.e | 6 | ||
40.e | odd | 2 | 1 | 120.2.d.b | yes | 6 | |
40.f | even | 2 | 1 | inner | 480.2.d.b | 6 | |
40.i | odd | 4 | 2 | 2400.2.k.f | 12 | ||
40.k | even | 4 | 2 | 600.2.k.f | 12 | ||
60.h | even | 2 | 1 | 360.2.d.f | 6 | ||
60.l | odd | 4 | 2 | 1800.2.k.u | 12 | ||
80.k | odd | 4 | 2 | 3840.2.f.l | 12 | ||
80.q | even | 4 | 2 | 3840.2.f.m | 12 | ||
120.i | odd | 2 | 1 | 1440.2.d.f | 6 | ||
120.m | even | 2 | 1 | 360.2.d.e | 6 | ||
120.q | odd | 4 | 2 | 1800.2.k.u | 12 | ||
120.w | even | 4 | 2 | 7200.2.k.u | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.d.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
120.2.d.a | ✓ | 6 | 20.d | odd | 2 | 1 | |
120.2.d.b | yes | 6 | 4.b | odd | 2 | 1 | |
120.2.d.b | yes | 6 | 40.e | odd | 2 | 1 | |
360.2.d.e | 6 | 12.b | even | 2 | 1 | ||
360.2.d.e | 6 | 120.m | even | 2 | 1 | ||
360.2.d.f | 6 | 24.f | even | 2 | 1 | ||
360.2.d.f | 6 | 60.h | even | 2 | 1 | ||
480.2.d.a | 6 | 5.b | even | 2 | 1 | ||
480.2.d.a | 6 | 8.b | even | 2 | 1 | ||
480.2.d.b | 6 | 1.a | even | 1 | 1 | trivial | |
480.2.d.b | 6 | 40.f | even | 2 | 1 | inner | |
600.2.k.f | 12 | 20.e | even | 4 | 2 | ||
600.2.k.f | 12 | 40.k | even | 4 | 2 | ||
1440.2.d.e | 6 | 15.d | odd | 2 | 1 | ||
1440.2.d.e | 6 | 24.h | odd | 2 | 1 | ||
1440.2.d.f | 6 | 3.b | odd | 2 | 1 | ||
1440.2.d.f | 6 | 120.i | odd | 2 | 1 | ||
1800.2.k.u | 12 | 60.l | odd | 4 | 2 | ||
1800.2.k.u | 12 | 120.q | odd | 4 | 2 | ||
2400.2.k.f | 12 | 5.c | odd | 4 | 2 | ||
2400.2.k.f | 12 | 40.i | odd | 4 | 2 | ||
3840.2.f.l | 12 | 16.f | odd | 4 | 2 | ||
3840.2.f.l | 12 | 80.k | odd | 4 | 2 | ||
3840.2.f.m | 12 | 16.e | even | 4 | 2 | ||
3840.2.f.m | 12 | 80.q | even | 4 | 2 | ||
7200.2.k.u | 12 | 15.e | even | 4 | 2 | ||
7200.2.k.u | 12 | 120.w | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{3} - 4T_{13}^{2} - 16T_{13} + 56 \)
acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( (T - 1)^{6} \)
$5$
\( T^{6} - T^{4} + 8 T^{3} - 5 T^{2} + \cdots + 125 \)
$7$
\( T^{6} + 24 T^{4} + 128 T^{2} + \cdots + 64 \)
$11$
\( T^{6} + 32 T^{4} + 96 T^{2} + 64 \)
$13$
\( (T^{3} - 4 T^{2} - 16 T + 56)^{2} \)
$17$
\( T^{6} + 36 T^{4} + 368 T^{2} + \cdots + 1024 \)
$19$
\( T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024 \)
$23$
\( T^{6} + 92 T^{4} + 2304 T^{2} + \cdots + 16384 \)
$29$
\( T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544 \)
$31$
\( (T^{3} - 8 T^{2} - 4 T + 64)^{2} \)
$37$
\( (T^{3} - 8 T^{2} + 8)^{2} \)
$41$
\( (T^{3} + 2 T^{2} - 100 T + 56)^{2} \)
$43$
\( (T^{3} - 64 T + 64)^{2} \)
$47$
\( T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024 \)
$53$
\( (T^{3} + 12 T^{2} + 32 T - 8)^{2} \)
$59$
\( T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776 \)
$61$
\( T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536 \)
$67$
\( (T^{3} - 64 T - 64)^{2} \)
$71$
\( (T^{3} + 8 T^{2} - 80 T - 128)^{2} \)
$73$
\( T^{6} + 384 T^{4} + 34560 T^{2} + \cdots + 16384 \)
$79$
\( (T^{3} + 8 T^{2} - 4 T - 64)^{2} \)
$83$
\( (T^{3} + 8 T^{2} - 64 T - 448)^{2} \)
$89$
\( (T^{3} + 10 T^{2} - 164 T - 1384)^{2} \)
$97$
\( T^{6} + 336 T^{4} + 28416 T^{2} + \cdots + 262144 \)
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