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: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{9601})\) |
| Defining polynomial: |
\( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \)
|
| 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 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
\( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 2401\nu^{2} - 2401\nu + 5760000 ) / 5762400 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 4801 ) / 2401 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2400\beta_{2} + \beta _1 - 2401 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2401\beta_{3} - 4801 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −4.50000 | + | 7.79423i | 0 | −11.2462 | − | 19.4789i | 0 | 96.4847 | + | 86.5893i | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||||||||
| 193.2 | 0 | −4.50000 | + | 7.79423i | 0 | 37.7462 | + | 65.3783i | 0 | −99.4847 | − | 83.1252i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||
| 289.1 | 0 | −4.50000 | − | 7.79423i | 0 | −11.2462 | + | 19.4789i | 0 | 96.4847 | − | 86.5893i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||
| 289.2 | 0 | −4.50000 | − | 7.79423i | 0 | 37.7462 | − | 65.3783i | 0 | −99.4847 | + | 83.1252i | 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.f | 4 | |
| 4.b | odd | 2 | 1 | 42.6.e.c | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 336.6.q.f | 4 | |
| 12.b | even | 2 | 1 | 126.6.g.h | 4 | ||
| 28.d | even | 2 | 1 | 294.6.e.s | 4 | ||
| 28.f | even | 6 | 1 | 294.6.a.w | 2 | ||
| 28.f | even | 6 | 1 | 294.6.e.s | 4 | ||
| 28.g | odd | 6 | 1 | 42.6.e.c | ✓ | 4 | |
| 28.g | odd | 6 | 1 | 294.6.a.r | 2 | ||
| 84.j | odd | 6 | 1 | 882.6.a.bb | 2 | ||
| 84.n | even | 6 | 1 | 126.6.g.h | 4 | ||
| 84.n | even | 6 | 1 | 882.6.a.bh | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.6.e.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 42.6.e.c | ✓ | 4 | 28.g | odd | 6 | 1 | |
| 126.6.g.h | 4 | 12.b | even | 2 | 1 | ||
| 126.6.g.h | 4 | 84.n | even | 6 | 1 | ||
| 294.6.a.r | 2 | 28.g | odd | 6 | 1 | ||
| 294.6.a.w | 2 | 28.f | even | 6 | 1 | ||
| 294.6.e.s | 4 | 28.d | even | 2 | 1 | ||
| 294.6.e.s | 4 | 28.f | even | 6 | 1 | ||
| 336.6.q.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 336.6.q.f | 4 | 7.c | even | 3 | 1 | inner | |
| 882.6.a.bb | 2 | 84.j | odd | 6 | 1 | ||
| 882.6.a.bh | 2 | 84.n | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 53T_{5}^{3} + 4507T_{5}^{2} + 89994T_{5} + 2883204 \)
acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} + 9 T + 81)^{2} \)
$5$
\( T^{4} - 53 T^{3} + 4507 T^{2} + \cdots + 2883204 \)
$7$
\( T^{4} + 6 T^{3} - 4781 T^{2} + \cdots + 282475249 \)
$11$
\( T^{4} - 191 T^{3} + \cdots + 2589384996 \)
$13$
\( (T^{2} + 379 T - 254520)^{2} \)
$17$
\( T^{4} - 340 T^{3} + \cdots + 867133440000 \)
$19$
\( T^{4} + 1769 T^{3} + \cdots + 8227948033600 \)
$23$
\( T^{4} - 3236 T^{3} + \cdots + 6653923430400 \)
$29$
\( (T^{2} - 4459 T - 3960660)^{2} \)
$31$
\( T^{4} - 1994 T^{3} + \cdots + 40\!\cdots\!25 \)
$37$
\( T^{4} + 20587 T^{3} + \cdots + 99\!\cdots\!64 \)
$41$
\( (T^{2} - 8814 T + 8966160)^{2} \)
$43$
\( (T^{2} + 15853 T + 44661910)^{2} \)
$47$
\( T^{4} - 33912 T^{3} + \cdots + 52\!\cdots\!04 \)
$53$
\( T^{4} + 49239 T^{3} + \cdots + 23\!\cdots\!76 \)
$59$
\( T^{4} + 56735 T^{3} + \cdots + 64\!\cdots\!96 \)
$61$
\( T^{4} + 67508 T^{3} + \cdots + 12\!\cdots\!00 \)
$67$
\( T^{4} - 75723 T^{3} + \cdots + 13\!\cdots\!00 \)
$71$
\( (T^{2} - 8992 T - 4289674884)^{2} \)
$73$
\( T^{4} - 3201 T^{3} + \cdots + 33\!\cdots\!36 \)
$79$
\( T^{4} - 26612 T^{3} + \cdots + 46\!\cdots\!69 \)
$83$
\( (T^{2} - 949 T - 7511023590)^{2} \)
$89$
\( T^{4} + 176562 T^{3} + \cdots + 59\!\cdots\!96 \)
$97$
\( (T^{2} + 129423 T - 5231869258)^{2} \)
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