Newspace parameters
Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 390.y (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.11416567883\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.303595776.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
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 in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) |
\(\beta_{4}\) | \(=\) | \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} + 13\nu ) / 48 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{6} - 13 ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{4} + 5\beta_{2} \) |
\(\nu^{5}\) | \(=\) | \( -\beta_{7} + 15\beta_{3} \) |
\(\nu^{6}\) | \(=\) | \( -16\beta_{6} - 13 \) |
\(\nu^{7}\) | \(=\) | \( 48\beta_{5} - 13\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).
\(n\) | \(131\) | \(157\) | \(301\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1 - \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
139.1 |
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−0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −1.50000 | − | 1.65831i | 0.500000 | + | 0.866025i | 2.00626 | − | 1.15831i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.469882 | + | 2.18614i | |||||||||||||||||||||||||
139.2 | −0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −1.50000 | + | 1.65831i | 0.500000 | + | 0.866025i | −3.73831 | + | 2.15831i | − | 1.00000i | 0.500000 | + | 0.866025i | 2.12819 | − | 0.686141i | ||||||||||||||||||||||||||
139.3 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −1.50000 | − | 1.65831i | 0.500000 | + | 0.866025i | 3.73831 | − | 2.15831i | 1.00000i | 0.500000 | + | 0.866025i | −0.469882 | − | 2.18614i | |||||||||||||||||||||||||||
139.4 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −1.50000 | + | 1.65831i | 0.500000 | + | 0.866025i | −2.00626 | + | 1.15831i | 1.00000i | 0.500000 | + | 0.866025i | −2.12819 | + | 0.686141i | |||||||||||||||||||||||||||
289.1 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −1.50000 | − | 1.65831i | 0.500000 | − | 0.866025i | −3.73831 | − | 2.15831i | 1.00000i | 0.500000 | − | 0.866025i | 2.12819 | + | 0.686141i | |||||||||||||||||||||||||||
289.2 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −1.50000 | + | 1.65831i | 0.500000 | − | 0.866025i | 2.00626 | + | 1.15831i | 1.00000i | 0.500000 | − | 0.866025i | 0.469882 | − | 2.18614i | |||||||||||||||||||||||||||
289.3 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −1.50000 | − | 1.65831i | 0.500000 | − | 0.866025i | −2.00626 | − | 1.15831i | − | 1.00000i | 0.500000 | − | 0.866025i | −2.12819 | − | 0.686141i | ||||||||||||||||||||||||||
289.4 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −1.50000 | + | 1.65831i | 0.500000 | − | 0.866025i | 3.73831 | + | 2.15831i | − | 1.00000i | 0.500000 | − | 0.866025i | −0.469882 | + | 2.18614i | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
65.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 390.2.y.g | ✓ | 8 |
3.b | odd | 2 | 1 | 1170.2.bp.g | 8 | ||
5.b | even | 2 | 1 | inner | 390.2.y.g | ✓ | 8 |
5.c | odd | 4 | 1 | 1950.2.i.bb | 4 | ||
5.c | odd | 4 | 1 | 1950.2.i.be | 4 | ||
13.c | even | 3 | 1 | inner | 390.2.y.g | ✓ | 8 |
15.d | odd | 2 | 1 | 1170.2.bp.g | 8 | ||
39.i | odd | 6 | 1 | 1170.2.bp.g | 8 | ||
65.n | even | 6 | 1 | inner | 390.2.y.g | ✓ | 8 |
65.q | odd | 12 | 1 | 1950.2.i.bb | 4 | ||
65.q | odd | 12 | 1 | 1950.2.i.be | 4 | ||
195.x | odd | 6 | 1 | 1170.2.bp.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
390.2.y.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
390.2.y.g | ✓ | 8 | 5.b | even | 2 | 1 | inner |
390.2.y.g | ✓ | 8 | 13.c | even | 3 | 1 | inner |
390.2.y.g | ✓ | 8 | 65.n | even | 6 | 1 | inner |
1170.2.bp.g | 8 | 3.b | odd | 2 | 1 | ||
1170.2.bp.g | 8 | 15.d | odd | 2 | 1 | ||
1170.2.bp.g | 8 | 39.i | odd | 6 | 1 | ||
1170.2.bp.g | 8 | 195.x | odd | 6 | 1 | ||
1950.2.i.bb | 4 | 5.c | odd | 4 | 1 | ||
1950.2.i.bb | 4 | 65.q | odd | 12 | 1 | ||
1950.2.i.be | 4 | 5.c | odd | 4 | 1 | ||
1950.2.i.be | 4 | 65.q | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 24T_{7}^{6} + 476T_{7}^{4} - 2400T_{7}^{2} + 10000 \)
acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{2} \)
$3$
\( (T^{4} - T^{2} + 1)^{2} \)
$5$
\( (T^{2} + 3 T + 5)^{4} \)
$7$
\( T^{8} - 24 T^{6} + 476 T^{4} + \cdots + 10000 \)
$11$
\( (T^{4} - 6 T^{3} + 38 T^{2} + 12 T + 4)^{2} \)
$13$
\( (T^{4} + 23 T^{2} + 169)^{2} \)
$17$
\( T^{8} - 90 T^{6} + 6251 T^{4} + \cdots + 3418801 \)
$19$
\( (T^{4} - 2 T^{3} + 14 T^{2} + 20 T + 100)^{2} \)
$23$
\( T^{8} - 72 T^{6} + 4988 T^{4} + \cdots + 38416 \)
$29$
\( (T^{2} - 3 T + 9)^{4} \)
$31$
\( (T + 4)^{8} \)
$37$
\( T^{8} - 90 T^{6} + 6251 T^{4} + \cdots + 3418801 \)
$41$
\( (T^{4} - 8 T^{3} + 59 T^{2} - 40 T + 25)^{2} \)
$43$
\( T^{8} - 24 T^{6} + 476 T^{4} + \cdots + 10000 \)
$47$
\( (T^{4} + 72 T^{2} + 196)^{2} \)
$53$
\( (T^{4} + 150 T^{2} + 2809)^{2} \)
$59$
\( (T^{4} - 4 T^{3} + 56 T^{2} + 160 T + 1600)^{2} \)
$61$
\( (T^{4} + 8 T^{3} + 59 T^{2} + 40 T + 25)^{2} \)
$67$
\( T^{8} - 216 T^{6} + \cdots + 65610000 \)
$71$
\( (T^{4} + 10 T^{3} + 86 T^{2} + 140 T + 196)^{2} \)
$73$
\( (T^{4} + 54 T^{2} + 25)^{2} \)
$79$
\( (T - 4)^{8} \)
$83$
\( (T^{4} + 72 T^{2} + 196)^{2} \)
$89$
\( (T^{4} - 4 T^{3} + 188 T^{2} + 688 T + 29584)^{2} \)
$97$
\( (T^{4} - 36 T^{2} + 1296)^{2} \)
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