Newspace parameters
Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 360.bd (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.87461447277\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\(\beta_{1}\) | \(=\) | \( \zeta_{24}^{3} + \zeta_{24} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{24}^{4} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{24}^{6} + \zeta_{24}^{2} \) |
\(\beta_{4}\) | \(=\) | \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) |
\(\beta_{5}\) | \(=\) | \( \zeta_{24}^{5} - \zeta_{24}^{3} \) |
\(\beta_{6}\) | \(=\) | \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24} \) |
\(\beta_{7}\) | \(=\) | \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{7} + \beta_{5} + 2\beta_1 ) / 3 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( ( \beta_{4} + \beta_{3} ) / 3 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( -\beta_{7} - \beta_{5} + \beta_1 ) / 3 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( \beta_{2} \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 3 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( ( -\beta_{4} + 2\beta_{3} ) / 3 \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( -2\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_1 ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).
\(n\) | \(181\) | \(217\) | \(271\) | \(281\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
59.1 |
|
−1.22474 | + | 0.707107i | −0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | −0.448288 | + | 2.19067i | 2.12132 | + | 1.22474i | −1.22474 | − | 2.12132i | 2.82843i | −1.50000 | + | 2.59808i | −1.00000 | − | 3.00000i | ||||||||||||||||||||||||||
59.2 | −1.22474 | + | 0.707107i | 0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 1.67303 | − | 1.48356i | −2.12132 | − | 1.22474i | −1.22474 | − | 2.12132i | 2.82843i | −1.50000 | + | 2.59808i | −1.00000 | + | 3.00000i | |||||||||||||||||||||||||||
59.3 | 1.22474 | − | 0.707107i | −0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0.448288 | − | 2.19067i | −2.12132 | − | 1.22474i | 1.22474 | + | 2.12132i | − | 2.82843i | −1.50000 | + | 2.59808i | −1.00000 | − | 3.00000i | ||||||||||||||||||||||||||
59.4 | 1.22474 | − | 0.707107i | 0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | −1.67303 | + | 1.48356i | 2.12132 | + | 1.22474i | 1.22474 | + | 2.12132i | − | 2.82843i | −1.50000 | + | 2.59808i | −1.00000 | + | 3.00000i | ||||||||||||||||||||||||||
299.1 | −1.22474 | − | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | + | 1.73205i | −0.448288 | − | 2.19067i | 2.12132 | − | 1.22474i | −1.22474 | + | 2.12132i | − | 2.82843i | −1.50000 | − | 2.59808i | −1.00000 | + | 3.00000i | ||||||||||||||||||||||||||
299.2 | −1.22474 | − | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | + | 1.73205i | 1.67303 | + | 1.48356i | −2.12132 | + | 1.22474i | −1.22474 | + | 2.12132i | − | 2.82843i | −1.50000 | − | 2.59808i | −1.00000 | − | 3.00000i | ||||||||||||||||||||||||||
299.3 | 1.22474 | + | 0.707107i | −0.866025 | + | 1.50000i | 1.00000 | + | 1.73205i | 0.448288 | + | 2.19067i | −2.12132 | + | 1.22474i | 1.22474 | − | 2.12132i | 2.82843i | −1.50000 | − | 2.59808i | −1.00000 | + | 3.00000i | |||||||||||||||||||||||||||
299.4 | 1.22474 | + | 0.707107i | 0.866025 | − | 1.50000i | 1.00000 | + | 1.73205i | −1.67303 | − | 1.48356i | 2.12132 | − | 1.22474i | 1.22474 | − | 2.12132i | 2.82843i | −1.50000 | − | 2.59808i | −1.00000 | − | 3.00000i | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
40.e | odd | 2 | 1 | inner |
45.h | odd | 6 | 1 | inner |
72.l | even | 6 | 1 | inner |
360.bd | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 360.2.bd.a | ✓ | 8 |
5.b | even | 2 | 1 | inner | 360.2.bd.a | ✓ | 8 |
8.d | odd | 2 | 1 | inner | 360.2.bd.a | ✓ | 8 |
9.d | odd | 6 | 1 | inner | 360.2.bd.a | ✓ | 8 |
40.e | odd | 2 | 1 | inner | 360.2.bd.a | ✓ | 8 |
45.h | odd | 6 | 1 | inner | 360.2.bd.a | ✓ | 8 |
72.l | even | 6 | 1 | inner | 360.2.bd.a | ✓ | 8 |
360.bd | even | 6 | 1 | inner | 360.2.bd.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.2.bd.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
360.2.bd.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
360.2.bd.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
360.2.bd.a | ✓ | 8 | 9.d | odd | 6 | 1 | inner |
360.2.bd.a | ✓ | 8 | 40.e | odd | 2 | 1 | inner |
360.2.bd.a | ✓ | 8 | 45.h | odd | 6 | 1 | inner |
360.2.bd.a | ✓ | 8 | 72.l | even | 6 | 1 | inner |
360.2.bd.a | ✓ | 8 | 360.bd | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 6T_{7}^{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - 2 T^{2} + 4)^{2} \)
$3$
\( (T^{4} + 3 T^{2} + 9)^{2} \)
$5$
\( T^{8} + 8 T^{6} + 39 T^{4} + 200 T^{2} + \cdots + 625 \)
$7$
\( (T^{4} + 6 T^{2} + 36)^{2} \)
$11$
\( (T^{2} - 9 T + 27)^{4} \)
$13$
\( (T^{4} + 6 T^{2} + 36)^{2} \)
$17$
\( (T^{2} - 27)^{4} \)
$19$
\( (T + 5)^{8} \)
$23$
\( (T^{4} - 8 T^{2} + 64)^{2} \)
$29$
\( (T^{4} + 72 T^{2} + 5184)^{2} \)
$31$
\( (T^{4} - 54 T^{2} + 2916)^{2} \)
$37$
\( T^{8} \)
$41$
\( (T^{2} - 3 T + 3)^{4} \)
$43$
\( (T^{4} - 81 T^{2} + 6561)^{2} \)
$47$
\( (T^{4} - 50 T^{2} + 2500)^{2} \)
$53$
\( (T^{2} + 2)^{4} \)
$59$
\( (T^{2} + 3 T + 3)^{4} \)
$61$
\( (T^{4} - 54 T^{2} + 2916)^{2} \)
$67$
\( (T^{4} - 9 T^{2} + 81)^{2} \)
$71$
\( (T^{2} - 18)^{4} \)
$73$
\( (T^{2} + 9)^{4} \)
$79$
\( (T^{4} - 54 T^{2} + 2916)^{2} \)
$83$
\( T^{8} \)
$89$
\( (T^{2} + 108)^{4} \)
$97$
\( (T^{4} - 9 T^{2} + 81)^{2} \)
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