Newspace parameters
Level: | \( N \) | \(=\) | \( 1444 = 2^{2} \cdot 19^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 1444.j (of order \(18\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.720649878242\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Coefficient field: | \(\Q(\zeta_{18})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
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Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 76) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.76.1 |
Artin image: | $C_9\times S_3$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).
\(n\) | \(723\) | \(1085\) |
\(\chi(n)\) | \(1\) | \(\zeta_{18}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
333.1 |
|
0 | 0 | 0 | 0.939693 | + | 0.342020i | 0 | 0.500000 | − | 0.866025i | 0 | 0.173648 | − | 0.984808i | 0 | ||||||||||||||||||||||||||||||
477.1 | 0 | 0 | 0 | 0.939693 | − | 0.342020i | 0 | 0.500000 | + | 0.866025i | 0 | 0.173648 | + | 0.984808i | 0 | |||||||||||||||||||||||||||||||
849.1 | 0 | 0 | 0 | −0.173648 | − | 0.984808i | 0 | 0.500000 | − | 0.866025i | 0 | 0.766044 | + | 0.642788i | 0 | |||||||||||||||||||||||||||||||
1021.1 | 0 | 0 | 0 | −0.766044 | − | 0.642788i | 0 | 0.500000 | + | 0.866025i | 0 | −0.939693 | − | 0.342020i | 0 | |||||||||||||||||||||||||||||||
1029.1 | 0 | 0 | 0 | −0.173648 | + | 0.984808i | 0 | 0.500000 | + | 0.866025i | 0 | 0.766044 | − | 0.642788i | 0 | |||||||||||||||||||||||||||||||
1345.1 | 0 | 0 | 0 | −0.766044 | + | 0.642788i | 0 | 0.500000 | − | 0.866025i | 0 | −0.939693 | + | 0.342020i | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
19.c | even | 3 | 2 | inner |
19.d | odd | 6 | 2 | inner |
19.e | even | 9 | 3 | inner |
19.f | odd | 18 | 3 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} \)
$5$
\( T^{6} - T^{3} + 1 \)
$7$
\( (T^{2} - T + 1)^{3} \)
$11$
\( (T^{2} - T + 1)^{3} \)
$13$
\( T^{6} \)
$17$
\( T^{6} - T^{3} + 1 \)
$19$
\( T^{6} \)
$23$
\( T^{6} + 8T^{3} + 64 \)
$29$
\( T^{6} \)
$31$
\( T^{6} \)
$37$
\( T^{6} \)
$41$
\( T^{6} \)
$43$
\( T^{6} - T^{3} + 1 \)
$47$
\( T^{6} - T^{3} + 1 \)
$53$
\( T^{6} \)
$59$
\( T^{6} \)
$61$
\( T^{6} - T^{3} + 1 \)
$67$
\( T^{6} \)
$71$
\( T^{6} \)
$73$
\( T^{6} - T^{3} + 1 \)
$79$
\( T^{6} \)
$83$
\( (T^{2} + 2 T + 4)^{3} \)
$89$
\( T^{6} \)
$97$
\( T^{6} \)
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