Newspace parameters
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.40960000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 7x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} + 7x^{4} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{6} + 8\nu^{2} ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} + \nu^{5} + 8\nu^{3} + 8\nu ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( ( 2\nu^{4} + 7 ) / 3 \) |
\(\beta_{4}\) | \(=\) | \( ( 2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu ) / 3 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) |
\(\beta_{6}\) | \(=\) | \( \nu^{6} + 6\nu^{2} \) |
\(\beta_{7}\) | \(=\) | \( ( 2\nu^{7} - \nu^{5} + 13\nu^{3} - 5\nu ) / 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{6} + 3\beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{7} - 2\beta_{5} - \beta_{4} + 2\beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 3\beta_{3} - 7 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -8\beta_{7} - 5\beta_{5} + 8\beta_{4} - 5\beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( 4\beta_{6} - 9\beta_1 \) |
\(\nu^{7}\) | \(=\) | \( ( 8\beta_{7} + 13\beta_{5} + 8\beta_{4} - 13\beta_{2} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).
\(n\) | \(92\) | \(379\) | \(703\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
818.1 |
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−2.23607 | 0 | 3.00000 | − | 2.00000i | 0 | −2.12132 | + | 1.58114i | −2.23607 | 0 | 4.47214i | |||||||||||||||||||||||||||||||||||||||
818.2 | −2.23607 | 0 | 3.00000 | − | 2.00000i | 0 | 2.12132 | − | 1.58114i | −2.23607 | 0 | 4.47214i | ||||||||||||||||||||||||||||||||||||||||
818.3 | −2.23607 | 0 | 3.00000 | 2.00000i | 0 | −2.12132 | − | 1.58114i | −2.23607 | 0 | − | 4.47214i | ||||||||||||||||||||||||||||||||||||||||
818.4 | −2.23607 | 0 | 3.00000 | 2.00000i | 0 | 2.12132 | + | 1.58114i | −2.23607 | 0 | − | 4.47214i | ||||||||||||||||||||||||||||||||||||||||
818.5 | 2.23607 | 0 | 3.00000 | − | 2.00000i | 0 | −2.12132 | − | 1.58114i | 2.23607 | 0 | − | 4.47214i | |||||||||||||||||||||||||||||||||||||||
818.6 | 2.23607 | 0 | 3.00000 | − | 2.00000i | 0 | 2.12132 | + | 1.58114i | 2.23607 | 0 | − | 4.47214i | |||||||||||||||||||||||||||||||||||||||
818.7 | 2.23607 | 0 | 3.00000 | 2.00000i | 0 | −2.12132 | + | 1.58114i | 2.23607 | 0 | 4.47214i | |||||||||||||||||||||||||||||||||||||||||
818.8 | 2.23607 | 0 | 3.00000 | 2.00000i | 0 | 2.12132 | − | 1.58114i | 2.23607 | 0 | 4.47214i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
39.d | odd | 2 | 1 | inner |
91.b | odd | 2 | 1 | inner |
273.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.g.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
13.b | even | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
21.c | even | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
39.d | odd | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
91.b | odd | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
273.g | even | 2 | 1 | inner | 819.2.g.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.g.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
819.2.g.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
819.2.g.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
819.2.g.b | ✓ | 8 | 13.b | even | 2 | 1 | inner |
819.2.g.b | ✓ | 8 | 21.c | even | 2 | 1 | inner |
819.2.g.b | ✓ | 8 | 39.d | odd | 2 | 1 | inner |
819.2.g.b | ✓ | 8 | 91.b | odd | 2 | 1 | inner |
819.2.g.b | ✓ | 8 | 273.g | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 5)^{4} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 4)^{4} \)
$7$
\( (T^{4} - 4 T^{2} + 49)^{2} \)
$11$
\( (T^{2} - 20)^{4} \)
$13$
\( (T^{4} - 6 T^{2} + 169)^{2} \)
$17$
\( (T^{2} - 10)^{4} \)
$19$
\( (T^{2} - 18)^{4} \)
$23$
\( T^{8} \)
$29$
\( (T^{2} + 18)^{4} \)
$31$
\( (T^{2} - 2)^{4} \)
$37$
\( (T^{2} + 40)^{4} \)
$41$
\( (T^{2} + 100)^{4} \)
$43$
\( (T + 4)^{8} \)
$47$
\( (T^{2} + 64)^{4} \)
$53$
\( (T^{2} + 2)^{4} \)
$59$
\( (T^{2} + 36)^{4} \)
$61$
\( (T^{2} + 180)^{4} \)
$67$
\( (T^{2} + 250)^{4} \)
$71$
\( (T^{2} - 80)^{4} \)
$73$
\( (T^{2} - 200)^{4} \)
$79$
\( (T + 4)^{8} \)
$83$
\( (T^{2} + 36)^{4} \)
$89$
\( (T^{2} + 36)^{4} \)
$97$
\( (T^{2} - 200)^{4} \)
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