Newspace parameters
Level: | \( N \) | \(=\) | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 480.w (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.83281929702\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | 8.0.1698758656.6 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} + 18\nu^{5} - 8\nu^{4} + 105\nu^{3} - 72\nu^{2} + 248\nu - 64 ) / 64 \) |
\(\beta_{5}\) | \(=\) | \( ( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{7} - 6\nu^{6} + 10\nu^{5} - 92\nu^{4} - 15\nu^{3} - 358\nu^{2} - 120\nu - 336 ) / 64 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} - \beta_{6} + \beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta _1 - 10 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} - 5\beta_{4} - 13\beta_{3} - 5\beta_{2} - 13\beta_1 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -9\beta_{7} + 9\beta_{6} - 17\beta_{4} - 27\beta_{3} + 17\beta_{2} + 27\beta _1 + 74 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} + 37\beta_{4} + 149\beta_{3} + 37\beta_{2} + 149\beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 89\beta_{7} - 89\beta_{6} + 201\beta_{4} + 235\beta_{3} - 201\beta_{2} - 235\beta _1 - 650 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} - 325\beta_{4} - 1565\beta_{3} - 325\beta_{2} - 1565\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).
\(n\) | \(31\) | \(97\) | \(161\) | \(421\) |
\(\chi(n)\) | \(-1\) | \(\beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
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127.1 |
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0 | −0.707107 | − | 0.707107i | 0 | −1.52773 | − | 1.63280i | 0 | 2.16053 | − | 2.16053i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
127.2 | 0 | −0.707107 | − | 0.707107i | 0 | 2.23483 | − | 0.0743018i | 0 | −3.16053 | + | 3.16053i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
127.3 | 0 | 0.707107 | + | 0.707107i | 0 | −1.19663 | + | 1.88893i | 0 | −1.69230 | + | 1.69230i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
127.4 | 0 | 0.707107 | + | 0.707107i | 0 | 0.489528 | − | 2.18183i | 0 | 0.692297 | − | 0.692297i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
223.1 | 0 | −0.707107 | + | 0.707107i | 0 | −1.52773 | + | 1.63280i | 0 | 2.16053 | + | 2.16053i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
223.2 | 0 | −0.707107 | + | 0.707107i | 0 | 2.23483 | + | 0.0743018i | 0 | −3.16053 | − | 3.16053i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
223.3 | 0 | 0.707107 | − | 0.707107i | 0 | −1.19663 | − | 1.88893i | 0 | −1.69230 | − | 1.69230i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
223.4 | 0 | 0.707107 | − | 0.707107i | 0 | 0.489528 | + | 2.18183i | 0 | 0.692297 | + | 0.692297i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 480.2.w.c | ✓ | 8 |
3.b | odd | 2 | 1 | 1440.2.x.q | 8 | ||
4.b | odd | 2 | 1 | 480.2.w.d | yes | 8 | |
5.b | even | 2 | 1 | 2400.2.w.j | 8 | ||
5.c | odd | 4 | 1 | 480.2.w.d | yes | 8 | |
5.c | odd | 4 | 1 | 2400.2.w.i | 8 | ||
8.b | even | 2 | 1 | 960.2.w.e | 8 | ||
8.d | odd | 2 | 1 | 960.2.w.f | 8 | ||
12.b | even | 2 | 1 | 1440.2.x.r | 8 | ||
15.e | even | 4 | 1 | 1440.2.x.r | 8 | ||
20.d | odd | 2 | 1 | 2400.2.w.i | 8 | ||
20.e | even | 4 | 1 | inner | 480.2.w.c | ✓ | 8 |
20.e | even | 4 | 1 | 2400.2.w.j | 8 | ||
40.i | odd | 4 | 1 | 960.2.w.f | 8 | ||
40.k | even | 4 | 1 | 960.2.w.e | 8 | ||
60.l | odd | 4 | 1 | 1440.2.x.q | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
480.2.w.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
480.2.w.c | ✓ | 8 | 20.e | even | 4 | 1 | inner |
480.2.w.d | yes | 8 | 4.b | odd | 2 | 1 | |
480.2.w.d | yes | 8 | 5.c | odd | 4 | 1 | |
960.2.w.e | 8 | 8.b | even | 2 | 1 | ||
960.2.w.e | 8 | 40.k | even | 4 | 1 | ||
960.2.w.f | 8 | 8.d | odd | 2 | 1 | ||
960.2.w.f | 8 | 40.i | odd | 4 | 1 | ||
1440.2.x.q | 8 | 3.b | odd | 2 | 1 | ||
1440.2.x.q | 8 | 60.l | odd | 4 | 1 | ||
1440.2.x.r | 8 | 12.b | even | 2 | 1 | ||
1440.2.x.r | 8 | 15.e | even | 4 | 1 | ||
2400.2.w.i | 8 | 5.c | odd | 4 | 1 | ||
2400.2.w.i | 8 | 20.d | odd | 2 | 1 | ||
2400.2.w.j | 8 | 5.b | even | 2 | 1 | ||
2400.2.w.j | 8 | 20.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} - 24T_{7}^{5} + 132T_{7}^{4} + 320T_{7}^{3} + 512T_{7}^{2} - 1024T_{7} + 1024 \)
acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 1)^{2} \)
$5$
\( T^{8} + 2 T^{6} - 16 T^{5} + 2 T^{4} + \cdots + 625 \)
$7$
\( T^{8} + 4 T^{7} + 8 T^{6} - 24 T^{5} + \cdots + 1024 \)
$11$
\( T^{8} + 36 T^{6} + 388 T^{4} + \cdots + 1024 \)
$13$
\( T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 141376 \)
$17$
\( T^{8} + 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 64 \)
$19$
\( (T^{4} + 4 T^{3} - 36 T^{2} + 128)^{2} \)
$23$
\( T^{8} - 192 T^{5} + 2832 T^{4} + \cdots + 4096 \)
$29$
\( T^{8} + 196 T^{6} + 12868 T^{4} + \cdots + 2458624 \)
$31$
\( (T^{2} + 16)^{4} \)
$37$
\( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 4426816 \)
$41$
\( (T^{4} - 4 T^{3} - 36 T^{2} + 128)^{2} \)
$43$
\( T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 10240000 \)
$47$
\( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 4096 \)
$53$
\( T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 8111104 \)
$59$
\( (T^{4} - 8 T^{3} - 98 T^{2} + 1136 T - 2848)^{2} \)
$61$
\( (T^{4} - 12 T^{3} - 84 T^{2} + 1120 T - 1600)^{2} \)
$67$
\( T^{8} - 24 T^{7} + 288 T^{6} + \cdots + 36192256 \)
$71$
\( (T^{2} + 32)^{4} \)
$73$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 795664 \)
$79$
\( (T^{4} + 8 T^{3} - 144 T^{2} + 2048)^{2} \)
$83$
\( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 6885376 \)
$89$
\( T^{8} + 584 T^{6} + \cdots + 154157056 \)
$97$
\( T^{8} - 32 T^{7} + 512 T^{6} + \cdots + 258064 \)
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