Newspace parameters
Level: | \( N \) | \(=\) | \( 1568 = 2^{5} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1568.o (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.782533939809\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 2x^{2} + 4 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 392) |
Projective image: | \(D_{4}\) |
Projective field: | Galois closure of 4.0.2744.1 |
Artin image: | $C_6\times D_8$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).
\(n\) | \(197\) | \(1471\) | \(1473\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
79.1 |
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0 | −0.707107 | − | 1.22474i | 0 | 0 | 0 | 0 | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||
79.2 | 0 | 0.707107 | + | 1.22474i | 0 | 0 | 0 | 0 | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||
655.1 | 0 | −0.707107 | + | 1.22474i | 0 | 0 | 0 | 0 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||
655.2 | 0 | 0.707107 | − | 1.22474i | 0 | 0 | 0 | 0 | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
7.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
56.e | even | 2 | 1 | inner |
56.k | odd | 6 | 1 | inner |
56.m | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{2} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(1568, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + 2T^{2} + 4 \)
$5$
\( T^{4} \)
$7$
\( T^{4} \)
$11$
\( T^{4} \)
$13$
\( T^{4} \)
$17$
\( T^{4} + 2T^{2} + 4 \)
$19$
\( T^{4} + 2T^{2} + 4 \)
$23$
\( T^{4} \)
$29$
\( T^{4} \)
$31$
\( T^{4} \)
$37$
\( T^{4} \)
$41$
\( (T^{2} - 2)^{2} \)
$43$
\( T^{4} \)
$47$
\( T^{4} \)
$53$
\( T^{4} \)
$59$
\( T^{4} + 2T^{2} + 4 \)
$61$
\( T^{4} \)
$67$
\( (T^{2} + 2 T + 4)^{2} \)
$71$
\( T^{4} \)
$73$
\( T^{4} + 2T^{2} + 4 \)
$79$
\( T^{4} \)
$83$
\( (T^{2} - 2)^{2} \)
$89$
\( T^{4} + 2T^{2} + 4 \)
$97$
\( (T^{2} - 2)^{2} \)
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