Newspace parameters
Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 429.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.42558224671\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.151613669376.21 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 5x^{4} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \( x^{8} + 5x^{4} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{5} + 3\nu ) / 2 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} + \nu^{2} ) / 4 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} + 5\nu^{3} + 8\nu ) / 8 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{7} - 5\nu^{3} + 8\nu ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{6} + 9\nu^{2} ) / 4 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 3\nu^{3} ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( \nu^{4} + \nu^{2} + 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{4} + \beta_{3} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} - \beta_{2} ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 2\beta_{6} - \beta_{4} + \beta_{3} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 2\beta_{7} - \beta_{5} + \beta_{2} - 6 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -3\beta_{4} - 3\beta_{3} + 4\beta_1 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( -\beta_{5} + 9\beta_{2} ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( -10\beta_{6} - 3\beta_{4} + 3\beta_{3} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).
\(n\) | \(67\) | \(79\) | \(287\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
428.1 |
|
− | 2.39417i | −1.73205 | −3.73205 | −4.11439 | 4.14682i | 0 | 4.14682i | 3.00000 | 9.85055i | |||||||||||||||||||||||||||||||||||||||||
428.2 | − | 2.39417i | −1.73205 | −3.73205 | 4.11439 | 4.14682i | 0 | 4.14682i | 3.00000 | − | 9.85055i | |||||||||||||||||||||||||||||||||||||||||
428.3 | − | 1.50597i | 1.73205 | −0.267949 | −1.75265 | − | 2.60842i | 0 | − | 2.60842i | 3.00000 | 2.63945i | ||||||||||||||||||||||||||||||||||||||||
428.4 | − | 1.50597i | 1.73205 | −0.267949 | 1.75265 | − | 2.60842i | 0 | − | 2.60842i | 3.00000 | − | 2.63945i | |||||||||||||||||||||||||||||||||||||||
428.5 | 1.50597i | 1.73205 | −0.267949 | −1.75265 | 2.60842i | 0 | 2.60842i | 3.00000 | − | 2.63945i | ||||||||||||||||||||||||||||||||||||||||||
428.6 | 1.50597i | 1.73205 | −0.267949 | 1.75265 | 2.60842i | 0 | 2.60842i | 3.00000 | 2.63945i | |||||||||||||||||||||||||||||||||||||||||||
428.7 | 2.39417i | −1.73205 | −3.73205 | −4.11439 | − | 4.14682i | 0 | − | 4.14682i | 3.00000 | − | 9.85055i | ||||||||||||||||||||||||||||||||||||||||
428.8 | 2.39417i | −1.73205 | −3.73205 | 4.11439 | − | 4.14682i | 0 | − | 4.14682i | 3.00000 | 9.85055i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
39.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-39}) \) |
3.b | odd | 2 | 1 | inner |
11.b | odd | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
33.d | even | 2 | 1 | inner |
143.d | odd | 2 | 1 | inner |
429.e | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 429.2.e.a | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 429.2.e.a | ✓ | 8 |
11.b | odd | 2 | 1 | inner | 429.2.e.a | ✓ | 8 |
13.b | even | 2 | 1 | inner | 429.2.e.a | ✓ | 8 |
33.d | even | 2 | 1 | inner | 429.2.e.a | ✓ | 8 |
39.d | odd | 2 | 1 | CM | 429.2.e.a | ✓ | 8 |
143.d | odd | 2 | 1 | inner | 429.2.e.a | ✓ | 8 |
429.e | even | 2 | 1 | inner | 429.2.e.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
429.2.e.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
429.2.e.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
429.2.e.a | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
429.2.e.a | ✓ | 8 | 13.b | even | 2 | 1 | inner |
429.2.e.a | ✓ | 8 | 33.d | even | 2 | 1 | inner |
429.2.e.a | ✓ | 8 | 39.d | odd | 2 | 1 | CM |
429.2.e.a | ✓ | 8 | 143.d | odd | 2 | 1 | inner |
429.2.e.a | ✓ | 8 | 429.e | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 8T_{2}^{2} + 13 \)
acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 8 T^{2} + 13)^{2} \)
$3$
\( (T^{2} - 3)^{4} \)
$5$
\( (T^{4} - 20 T^{2} + 52)^{2} \)
$7$
\( T^{8} \)
$11$
\( T^{8} - 190 T^{4} + 14641 \)
$13$
\( (T^{2} + 13)^{4} \)
$17$
\( T^{8} \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( (T^{4} + 164 T^{2} + 6292)^{2} \)
$43$
\( (T^{2} + 156)^{4} \)
$47$
\( (T^{4} - 188 T^{2} + 8788)^{2} \)
$53$
\( T^{8} \)
$59$
\( (T^{4} - 236 T^{2} + 52)^{2} \)
$61$
\( (T^{2} + 52)^{4} \)
$67$
\( T^{8} \)
$71$
\( (T^{4} - 284 T^{2} + 6292)^{2} \)
$73$
\( T^{8} \)
$79$
\( (T^{2} + 208)^{4} \)
$83$
\( (T^{4} + 332 T^{2} + 27508)^{2} \)
$89$
\( (T^{4} - 356 T^{2} + 6292)^{2} \)
$97$
\( T^{8} \)
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