Newspace parameters
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.k (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(19.8246417619\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2\cdot 3 \) |
Twist minimal: | no (minimal twist has level 21) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
\(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
209.1 |
|
0 | − | 5.19615i | 0 | 0 | 0 | 10.0000 | + | 15.5885i | 0 | −27.0000 | 0 | |||||||||||||||||||||
209.2 | 0 | 5.19615i | 0 | 0 | 0 | 10.0000 | − | 15.5885i | 0 | −27.0000 | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 336.4.k.a | 2 | |
3.b | odd | 2 | 1 | CM | 336.4.k.a | 2 | |
4.b | odd | 2 | 1 | 21.4.c.a | ✓ | 2 | |
7.b | odd | 2 | 1 | inner | 336.4.k.a | 2 | |
12.b | even | 2 | 1 | 21.4.c.a | ✓ | 2 | |
21.c | even | 2 | 1 | inner | 336.4.k.a | 2 | |
28.d | even | 2 | 1 | 21.4.c.a | ✓ | 2 | |
28.f | even | 6 | 1 | 147.4.g.a | 2 | ||
28.f | even | 6 | 1 | 147.4.g.b | 2 | ||
28.g | odd | 6 | 1 | 147.4.g.a | 2 | ||
28.g | odd | 6 | 1 | 147.4.g.b | 2 | ||
84.h | odd | 2 | 1 | 21.4.c.a | ✓ | 2 | |
84.j | odd | 6 | 1 | 147.4.g.a | 2 | ||
84.j | odd | 6 | 1 | 147.4.g.b | 2 | ||
84.n | even | 6 | 1 | 147.4.g.a | 2 | ||
84.n | even | 6 | 1 | 147.4.g.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.4.c.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
21.4.c.a | ✓ | 2 | 12.b | even | 2 | 1 | |
21.4.c.a | ✓ | 2 | 28.d | even | 2 | 1 | |
21.4.c.a | ✓ | 2 | 84.h | odd | 2 | 1 | |
147.4.g.a | 2 | 28.f | even | 6 | 1 | ||
147.4.g.a | 2 | 28.g | odd | 6 | 1 | ||
147.4.g.a | 2 | 84.j | odd | 6 | 1 | ||
147.4.g.a | 2 | 84.n | even | 6 | 1 | ||
147.4.g.b | 2 | 28.f | even | 6 | 1 | ||
147.4.g.b | 2 | 28.g | odd | 6 | 1 | ||
147.4.g.b | 2 | 84.j | odd | 6 | 1 | ||
147.4.g.b | 2 | 84.n | even | 6 | 1 | ||
336.4.k.a | 2 | 1.a | even | 1 | 1 | trivial | |
336.4.k.a | 2 | 3.b | odd | 2 | 1 | CM | |
336.4.k.a | 2 | 7.b | odd | 2 | 1 | inner | |
336.4.k.a | 2 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 27 \)
$5$
\( T^{2} \)
$7$
\( T^{2} - 20T + 343 \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 3888 \)
$17$
\( T^{2} \)
$19$
\( T^{2} + 24300 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} + 24300 \)
$37$
\( (T + 110)^{2} \)
$41$
\( T^{2} \)
$43$
\( (T + 520)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 874800 \)
$67$
\( (T - 880)^{2} \)
$71$
\( T^{2} \)
$73$
\( T^{2} + 139968 \)
$79$
\( (T + 884)^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 1881792 \)
show more
show less