Newspace parameters
Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 304.u (of order \(9\), degree \(6\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.42745222145\) |
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: | \( x^{6} - x^{3} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 19) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).
\(n\) | \(97\) | \(191\) | \(229\) |
\(\chi(n)\) | \(-\zeta_{18}^{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 |
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0 | 2.20574 | + | 1.85083i | 0 | −0.826352 | + | 0.300767i | 0 | 0.173648 | + | 0.300767i | 0 | 0.918748 | + | 5.21048i | 0 | ||||||||||||||||||||||||||||
81.1 | 0 | −0.0923963 | + | 0.524005i | 0 | −1.93969 | − | 1.62760i | 0 | −0.939693 | − | 1.62760i | 0 | 2.55303 | + | 0.929228i | 0 | |||||||||||||||||||||||||||||
161.1 | 0 | 2.20574 | − | 1.85083i | 0 | −0.826352 | − | 0.300767i | 0 | 0.173648 | − | 0.300767i | 0 | 0.918748 | − | 5.21048i | 0 | |||||||||||||||||||||||||||||
177.1 | 0 | −0.613341 | − | 0.223238i | 0 | −0.233956 | − | 1.32683i | 0 | 0.766044 | − | 1.32683i | 0 | −1.97178 | − | 1.65452i | 0 | |||||||||||||||||||||||||||||
225.1 | 0 | −0.613341 | + | 0.223238i | 0 | −0.233956 | + | 1.32683i | 0 | 0.766044 | + | 1.32683i | 0 | −1.97178 | + | 1.65452i | 0 | |||||||||||||||||||||||||||||
289.1 | 0 | −0.0923963 | − | 0.524005i | 0 | −1.93969 | + | 1.62760i | 0 | −0.939693 | + | 1.62760i | 0 | 2.55303 | − | 0.929228i | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 3T_{3}^{5} + 3T_{3}^{4} + 8T_{3}^{3} + 6T_{3}^{2} + 3T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} - 3 T^{5} + 3 T^{4} + 8 T^{3} + \cdots + 1 \)
$5$
\( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \)
$7$
\( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \)
$11$
\( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \)
$13$
\( T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1369 \)
$17$
\( T^{6} - 3 T^{5} + 30 T^{3} + 36 T^{2} + \cdots + 9 \)
$19$
\( T^{6} - 12 T^{5} + 78 T^{4} + \cdots + 6859 \)
$23$
\( T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576 \)
$29$
\( T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 12321 \)
$31$
\( T^{6} + 9 T^{5} + 75 T^{4} + \cdots + 2809 \)
$37$
\( (T^{3} - 21 T - 17)^{2} \)
$41$
\( T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 12321 \)
$43$
\( T^{6} - 3 T^{5} + 60 T^{4} + \cdots + 26569 \)
$47$
\( T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9 \)
$53$
\( T^{6} + 3 T^{5} - 84 T^{3} + \cdots + 2601 \)
$59$
\( T^{6} + 12 T^{5} + 18 T^{4} + \cdots + 71289 \)
$61$
\( T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761 \)
$67$
\( T^{6} - 30 T^{5} + 348 T^{4} + \cdots + 179776 \)
$71$
\( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 788544 \)
$73$
\( T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096 \)
$79$
\( T^{6} - 39 T^{5} + 708 T^{4} + \cdots + 654481 \)
$83$
\( T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681 \)
$89$
\( T^{6} + 12 T^{5} + 54 T^{4} + \cdots + 3249 \)
$97$
\( T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 16129 \)
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