Newspace parameters
| Level: | \( N \) | \(=\) | \( 3969 = 3^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3969.n (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.98078903514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 441) |
| Projective image: | \(D_{4}\) |
| Projective field: | Galois closure of 4.0.189.1 |
| Artin image: | $C_3\times \SD_{16}$ |
| Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\) |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times\).
| \(n\) | \(2108\) | \(3727\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2321.1 |
|
−1.22474 | − | 0.707107i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 2321.2 | 1.22474 | + | 0.707107i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 3509.1 | −1.22474 | + | 0.707107i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 3509.2 | 1.22474 | − | 0.707107i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 3.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 63.g | even | 3 | 1 | inner |
| 63.k | odd | 6 | 1 | inner |
| 63.n | odd | 6 | 1 | inner |
| 63.s | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{2} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(3969, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 2T^{2} + 4 \)
$3$
\( T^{4} \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( (T^{2} + 2)^{2} \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( (T^{2} + 2)^{2} \)
$29$
\( T^{4} - 2T^{2} + 4 \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( T^{4} - 2T^{2} + 4 \)
$59$
\( T^{4} \)
$61$
\( T^{4} \)
$67$
\( (T^{2} - 2 T + 4)^{2} \)
$71$
\( (T^{2} + 2)^{2} \)
$73$
\( T^{4} \)
$79$
\( (T^{2} + 2 T + 4)^{2} \)
$83$
\( T^{4} \)
$89$
\( T^{4} \)
$97$
\( T^{4} \)
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