Newspace parameters
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.e (of order \(6\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.69894358832\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | − | 3.00000i | 0.866025 | + | 0.500000i | −0.866025 | − | 0.500000i | 1.00000i | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | |||||||||||||||
| 23.2 | 0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 3.00000i | −0.866025 | − | 0.500000i | 0.866025 | + | 0.500000i | − | 1.00000i | 1.00000 | − | 1.73205i | 1.50000 | + | 2.59808i | ||||||||||||||||
| 147.1 | −0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 3.00000i | 0.866025 | − | 0.500000i | −0.866025 | + | 0.500000i | − | 1.00000i | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | ||||||||||||||||
| 147.2 | 0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | − | 3.00000i | −0.866025 | + | 0.500000i | 0.866025 | − | 0.500000i | 1.00000i | 1.00000 | + | 1.73205i | 1.50000 | − | 2.59808i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( (T^{2} + T + 1)^{2} \)
$5$
\( (T^{2} + 9)^{2} \)
$7$
\( T^{4} - T^{2} + 1 \)
$11$
\( T^{4} - 36T^{2} + 1296 \)
$13$
\( T^{4} \)
$17$
\( (T^{2} + 3 T + 9)^{2} \)
$19$
\( T^{4} - 4T^{2} + 16 \)
$23$
\( T^{4} \)
$29$
\( (T^{2} + 6 T + 36)^{2} \)
$31$
\( (T^{2} + 16)^{2} \)
$37$
\( T^{4} - 49T^{2} + 2401 \)
$41$
\( T^{4} \)
$43$
\( (T^{2} + T + 1)^{2} \)
$47$
\( (T^{2} + 9)^{2} \)
$53$
\( T^{4} \)
$59$
\( T^{4} - 36T^{2} + 1296 \)
$61$
\( (T^{2} + 8 T + 64)^{2} \)
$67$
\( T^{4} - 196 T^{2} + 38416 \)
$71$
\( T^{4} - 9T^{2} + 81 \)
$73$
\( (T^{2} + 4)^{2} \)
$79$
\( (T - 8)^{4} \)
$83$
\( (T^{2} + 144)^{2} \)
$89$
\( T^{4} - 36T^{2} + 1296 \)
$97$
\( T^{4} - 100 T^{2} + 10000 \)
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