Newspace parameters
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.cx (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.74922894553\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
Coefficient field: | 8.0.12960000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + 13\nu ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} - 4\nu^{4} + 8\nu^{2} + 4\nu - 3 ) / 4 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{6} + 4\nu^{4} - 8\nu^{2} + 4\nu + 3 ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( 3\nu^{7} - \nu^{6} - 8\nu^{5} + 24\nu^{3} - \nu - 13 ) / 8 \) |
\(\beta_{6}\) | \(=\) | \( ( -3\nu^{7} - 3\nu^{6} + 8\nu^{5} + 8\nu^{4} - 24\nu^{3} - 16\nu^{2} + 9\nu + 1 ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 15\nu ) / 8 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{2} + 3 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta _1 + 1 \) |
\(\nu^{4}\) | \(=\) | \( ( -4\beta_{4} + 3\beta_{3} - 3\beta_{2} ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( 6\beta_{7} - 5\beta_{6} + 5\beta_{5} - 5\beta_{2} + 5 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -4\beta_{6} - 4\beta_{5} + 4\beta_{3} - 9 \) |
\(\nu^{7}\) | \(=\) | \( ( -13\beta_{3} - 13\beta_{2} + 16\beta_1 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).
\(n\) | \(181\) | \(271\) | \(577\) | \(641\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(\beta_{1} - \beta_{7}\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
223.1 |
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0 | −1.34425 | − | 1.09224i | 0 | 0.133975 | + | 2.23205i | 0 | −2.71028 | + | 0.726216i | 0 | 0.614017 | + | 2.93649i | 0 | ||||||||||||||||||||||||||||||||||
223.2 | 0 | 1.71028 | − | 0.273784i | 0 | 0.133975 | + | 2.23205i | 0 | 0.344250 | − | 0.0922415i | 0 | 2.85008 | − | 0.936492i | 0 | |||||||||||||||||||||||||||||||||||
367.1 | 0 | −1.09224 | + | 1.34425i | 0 | 1.86603 | + | 1.23205i | 0 | −0.726216 | − | 2.71028i | 0 | −0.614017 | − | 2.93649i | 0 | |||||||||||||||||||||||||||||||||||
367.2 | 0 | −0.273784 | − | 1.71028i | 0 | 1.86603 | + | 1.23205i | 0 | 0.0922415 | + | 0.344250i | 0 | −2.85008 | + | 0.936492i | 0 | |||||||||||||||||||||||||||||||||||
463.1 | 0 | −1.09224 | − | 1.34425i | 0 | 1.86603 | − | 1.23205i | 0 | −0.726216 | + | 2.71028i | 0 | −0.614017 | + | 2.93649i | 0 | |||||||||||||||||||||||||||||||||||
463.2 | 0 | −0.273784 | + | 1.71028i | 0 | 1.86603 | − | 1.23205i | 0 | 0.0922415 | − | 0.344250i | 0 | −2.85008 | − | 0.936492i | 0 | |||||||||||||||||||||||||||||||||||
607.1 | 0 | −1.34425 | + | 1.09224i | 0 | 0.133975 | − | 2.23205i | 0 | −2.71028 | − | 0.726216i | 0 | 0.614017 | − | 2.93649i | 0 | |||||||||||||||||||||||||||||||||||
607.2 | 0 | 1.71028 | + | 0.273784i | 0 | 0.133975 | − | 2.23205i | 0 | 0.344250 | + | 0.0922415i | 0 | 2.85008 | + | 0.936492i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
36.f | odd | 6 | 1 | inner |
180.x | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 720.2.cx.a | ✓ | 8 |
4.b | odd | 2 | 1 | 720.2.cx.b | yes | 8 | |
5.c | odd | 4 | 1 | inner | 720.2.cx.a | ✓ | 8 |
9.c | even | 3 | 1 | 720.2.cx.b | yes | 8 | |
20.e | even | 4 | 1 | 720.2.cx.b | yes | 8 | |
36.f | odd | 6 | 1 | inner | 720.2.cx.a | ✓ | 8 |
45.k | odd | 12 | 1 | 720.2.cx.b | yes | 8 | |
180.x | even | 12 | 1 | inner | 720.2.cx.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.2.cx.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
720.2.cx.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
720.2.cx.a | ✓ | 8 | 36.f | odd | 6 | 1 | inner |
720.2.cx.a | ✓ | 8 | 180.x | even | 12 | 1 | inner |
720.2.cx.b | yes | 8 | 4.b | odd | 2 | 1 | |
720.2.cx.b | yes | 8 | 9.c | even | 3 | 1 | |
720.2.cx.b | yes | 8 | 20.e | even | 4 | 1 | |
720.2.cx.b | yes | 8 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 6T_{7}^{7} + 18T_{7}^{6} + 36T_{7}^{5} + 23T_{7}^{4} - 36T_{7}^{3} + 18T_{7}^{2} - 6T_{7} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} + 2 T^{7} + 2 T^{6} - 8 T^{5} + \cdots + 81 \)
$5$
\( (T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25)^{2} \)
$7$
\( T^{8} + 6 T^{7} + 18 T^{6} + 36 T^{5} + \cdots + 1 \)
$11$
\( (T^{4} + 6 T^{3} + 10 T^{2} - 12 T + 4)^{2} \)
$13$
\( T^{8} - 900 T^{4} + 810000 \)
$17$
\( (T^{2} - 4 T + 8)^{4} \)
$19$
\( (T^{4} - 64 T^{2} + 64)^{2} \)
$23$
\( T^{8} - 18 T^{7} + 162 T^{6} + \cdots + 14641 \)
$29$
\( T^{8} - 62 T^{6} + 3843 T^{4} + \cdots + 1 \)
$31$
\( (T^{2} - 6 T + 12)^{4} \)
$37$
\( (T^{4} + 4 T^{3} + 8 T^{2} - 112 T + 784)^{2} \)
$41$
\( (T^{4} + 12 T^{3} + 123 T^{2} + 252 T + 441)^{2} \)
$43$
\( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 38416 \)
$47$
\( T^{8} - 30 T^{7} + 450 T^{6} + \cdots + 1500625 \)
$53$
\( (T^{4} + 8 T^{3} + 32 T^{2} - 176 T + 484)^{2} \)
$59$
\( T^{8} + 64 T^{6} + 3612 T^{4} + \cdots + 234256 \)
$61$
\( (T^{2} - 5 T + 25)^{4} \)
$67$
\( T^{8} - 42 T^{7} + 882 T^{6} + \cdots + 25411681 \)
$71$
\( (T^{4} + 256 T^{2} + 14884)^{2} \)
$73$
\( (T^{4} + 14400)^{2} \)
$79$
\( T^{8} + 64 T^{6} + 3612 T^{4} + \cdots + 234256 \)
$83$
\( T^{8} + 18 T^{7} + \cdots + 141158161 \)
$89$
\( (T^{4} + 482 T^{2} + 57121)^{2} \)
$97$
\( (T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324)^{2} \)
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