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: | \(\Q(\sqrt{-3}, \sqrt{505})\) |
| Defining polynomial: |
\( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 7 \) |
| 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
\( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 125\nu^{3} + 127\nu^{2} - 32131\nu + 47754 ) / 16002 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 379\nu^{3} - 127\nu^{2} + 16129\nu + 63756 ) / 16002 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 2\beta_{2} + 887\beta _1 - 886 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 254\beta_{3} + 127\beta_{2} + 127\beta _1 - 1517 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | 4.50000 | − | 7.79423i | 0 | −43.5764 | − | 75.4765i | 0 | 113.236 | − | 63.1236i | 0 | −40.5000 | − | 70.1481i | 0 | ||||||||||||||||||||||
| 193.2 | 0 | 4.50000 | − | 7.79423i | 0 | 35.0764 | + | 60.7540i | 0 | 90.7639 | + | 92.5684i | 0 | −40.5000 | − | 70.1481i | 0 | |||||||||||||||||||||||
| 289.1 | 0 | 4.50000 | + | 7.79423i | 0 | −43.5764 | + | 75.4765i | 0 | 113.236 | + | 63.1236i | 0 | −40.5000 | + | 70.1481i | 0 | |||||||||||||||||||||||
| 289.2 | 0 | 4.50000 | + | 7.79423i | 0 | 35.0764 | − | 60.7540i | 0 | 90.7639 | − | 92.5684i | 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.h | 4 | |
| 4.b | odd | 2 | 1 | 42.6.e.d | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 336.6.q.h | 4 | |
| 12.b | even | 2 | 1 | 126.6.g.g | 4 | ||
| 28.d | even | 2 | 1 | 294.6.e.y | 4 | ||
| 28.f | even | 6 | 1 | 294.6.a.o | 2 | ||
| 28.f | even | 6 | 1 | 294.6.e.y | 4 | ||
| 28.g | odd | 6 | 1 | 42.6.e.d | ✓ | 4 | |
| 28.g | odd | 6 | 1 | 294.6.a.p | 2 | ||
| 84.j | odd | 6 | 1 | 882.6.a.bs | 2 | ||
| 84.n | even | 6 | 1 | 126.6.g.g | 4 | ||
| 84.n | even | 6 | 1 | 882.6.a.bm | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.6.e.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 42.6.e.d | ✓ | 4 | 28.g | odd | 6 | 1 | |
| 126.6.g.g | 4 | 12.b | even | 2 | 1 | ||
| 126.6.g.g | 4 | 84.n | even | 6 | 1 | ||
| 294.6.a.o | 2 | 28.f | even | 6 | 1 | ||
| 294.6.a.p | 2 | 28.g | odd | 6 | 1 | ||
| 294.6.e.y | 4 | 28.d | even | 2 | 1 | ||
| 294.6.e.y | 4 | 28.f | even | 6 | 1 | ||
| 336.6.q.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 336.6.q.h | 4 | 7.c | even | 3 | 1 | inner | |
| 882.6.a.bm | 2 | 84.n | even | 6 | 1 | ||
| 882.6.a.bs | 2 | 84.j | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 17T_{5}^{3} + 6403T_{5}^{2} - 103938T_{5} + 37380996 \)
acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - 9 T + 81)^{2} \)
$5$
\( T^{4} + 17 T^{3} + 6403 T^{2} + \cdots + 37380996 \)
$7$
\( T^{4} - 408 T^{3} + \cdots + 282475249 \)
$11$
\( T^{4} - 145 T^{3} + \cdots + 88726536900 \)
$13$
\( (T^{2} + 715 T + 121620)^{2} \)
$17$
\( T^{4} + 1372 T^{3} + \cdots + 4015631241216 \)
$19$
\( T^{4} - 1081 T^{3} + \cdots + 17788453428496 \)
$23$
\( T^{4} + 4508 T^{3} + \cdots + 24815700919296 \)
$29$
\( (T^{2} - 7865 T - 6069780)^{2} \)
$31$
\( T^{4} + \cdots + 331894030125681 \)
$37$
\( T^{4} + \cdots + 156377425969216 \)
$41$
\( (T^{2} - 7350 T - 13441680)^{2} \)
$43$
\( (T^{2} - 5921 T - 15788666)^{2} \)
$47$
\( T^{4} - 44808 T^{3} + \cdots + 24\!\cdots\!96 \)
$53$
\( T^{4} + 9417 T^{3} + \cdots + 92983900409856 \)
$59$
\( T^{4} + 5077 T^{3} + \cdots + 76\!\cdots\!36 \)
$61$
\( T^{4} + 42368 T^{3} + \cdots + 15\!\cdots\!96 \)
$67$
\( T^{4} - 30501 T^{3} + \cdots + 15\!\cdots\!76 \)
$71$
\( (T^{2} + 91744 T + 1804825884)^{2} \)
$73$
\( T^{4} - 85665 T^{3} + \cdots + 66\!\cdots\!00 \)
$79$
\( T^{4} - 94646 T^{3} + \cdots + 47\!\cdots\!81 \)
$83$
\( (T^{2} + 33841 T + 280358334)^{2} \)
$89$
\( T^{4} + 27558 T^{3} + \cdots + 6584027556096 \)
$97$
\( (T^{2} + 46671 T - 4062781666)^{2} \)
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