Newspace parameters
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.q (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(53.8889634572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 42) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | 4.50000 | − | 7.79423i | 0 | 3.00000 | + | 5.19615i | 0 | −59.5000 | + | 115.181i | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||
| 289.1 | 0 | 4.50000 | + | 7.79423i | 0 | 3.00000 | − | 5.19615i | 0 | −59.5000 | − | 115.181i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.6.q.c | 2 | |
| 4.b | odd | 2 | 1 | 42.6.e.a | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 336.6.q.c | 2 | |
| 12.b | even | 2 | 1 | 126.6.g.c | 2 | ||
| 28.d | even | 2 | 1 | 294.6.e.e | 2 | ||
| 28.f | even | 6 | 1 | 294.6.a.j | 1 | ||
| 28.f | even | 6 | 1 | 294.6.e.e | 2 | ||
| 28.g | odd | 6 | 1 | 42.6.e.a | ✓ | 2 | |
| 28.g | odd | 6 | 1 | 294.6.a.l | 1 | ||
| 84.j | odd | 6 | 1 | 882.6.a.e | 1 | ||
| 84.n | even | 6 | 1 | 126.6.g.c | 2 | ||
| 84.n | even | 6 | 1 | 882.6.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.6.e.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 42.6.e.a | ✓ | 2 | 28.g | odd | 6 | 1 | |
| 126.6.g.c | 2 | 12.b | even | 2 | 1 | ||
| 126.6.g.c | 2 | 84.n | even | 6 | 1 | ||
| 294.6.a.j | 1 | 28.f | even | 6 | 1 | ||
| 294.6.a.l | 1 | 28.g | odd | 6 | 1 | ||
| 294.6.e.e | 2 | 28.d | even | 2 | 1 | ||
| 294.6.e.e | 2 | 28.f | even | 6 | 1 | ||
| 336.6.q.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 336.6.q.c | 2 | 7.c | even | 3 | 1 | inner | |
| 882.6.a.e | 1 | 84.j | odd | 6 | 1 | ||
| 882.6.a.f | 1 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 6T_{5} + 36 \)
acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 9T + 81 \)
$5$
\( T^{2} - 6T + 36 \)
$7$
\( T^{2} + 119T + 16807 \)
$11$
\( T^{2} + 666T + 443556 \)
$13$
\( (T + 559)^{2} \)
$17$
\( T^{2} - 1740 T + 3027600 \)
$19$
\( T^{2} - 1157 T + 1338649 \)
$23$
\( T^{2} + 3468 T + 12027024 \)
$29$
\( (T - 3372)^{2} \)
$31$
\( T^{2} - 6293 T + 39601849 \)
$37$
\( T^{2} + 3131 T + 9803161 \)
$41$
\( (T + 4866)^{2} \)
$43$
\( (T - 11407)^{2} \)
$47$
\( T^{2} - 2310 T + 5336100 \)
$53$
\( T^{2} - 28296 T + 800663616 \)
$59$
\( T^{2} - 20544 T + 422055936 \)
$61$
\( T^{2} - 4630 T + 21436900 \)
$67$
\( T^{2} + 18745 T + 351375025 \)
$71$
\( (T - 38226)^{2} \)
$73$
\( T^{2} + 70589 T + 4982806921 \)
$79$
\( T^{2} + 62293 T + 3880417849 \)
$83$
\( (T + 79818)^{2} \)
$89$
\( T^{2} - 18120 T + 328334400 \)
$97$
\( (T - 124754)^{2} \)
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