Newspace parameters
Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 336.o (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.167685844245\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(D_{2}\) |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\) |
Artin image: | $D_4$ |
Artin field: | Galois closure of 4.0.1008.2 |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
335.1 |
|
0 | −1.00000 | 0 | 0 | 0 | 1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
28.d | even | 2 | 1 | RM by \(\Q(\sqrt{7}) \) |
84.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-21}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 336.1.o.a | ✓ | 1 |
3.b | odd | 2 | 1 | CM | 336.1.o.a | ✓ | 1 |
4.b | odd | 2 | 1 | 336.1.o.b | yes | 1 | |
7.b | odd | 2 | 1 | 336.1.o.b | yes | 1 | |
7.c | even | 3 | 2 | 2352.1.z.c | 2 | ||
7.d | odd | 6 | 2 | 2352.1.z.b | 2 | ||
8.b | even | 2 | 1 | 1344.1.o.b | 1 | ||
8.d | odd | 2 | 1 | 1344.1.o.a | 1 | ||
12.b | even | 2 | 1 | 336.1.o.b | yes | 1 | |
21.c | even | 2 | 1 | 336.1.o.b | yes | 1 | |
21.g | even | 6 | 2 | 2352.1.z.b | 2 | ||
21.h | odd | 6 | 2 | 2352.1.z.c | 2 | ||
24.f | even | 2 | 1 | 1344.1.o.a | 1 | ||
24.h | odd | 2 | 1 | 1344.1.o.b | 1 | ||
28.d | even | 2 | 1 | RM | 336.1.o.a | ✓ | 1 |
28.f | even | 6 | 2 | 2352.1.z.c | 2 | ||
28.g | odd | 6 | 2 | 2352.1.z.b | 2 | ||
56.e | even | 2 | 1 | 1344.1.o.b | 1 | ||
56.h | odd | 2 | 1 | 1344.1.o.a | 1 | ||
84.h | odd | 2 | 1 | CM | 336.1.o.a | ✓ | 1 |
84.j | odd | 6 | 2 | 2352.1.z.c | 2 | ||
84.n | even | 6 | 2 | 2352.1.z.b | 2 | ||
168.e | odd | 2 | 1 | 1344.1.o.b | 1 | ||
168.i | even | 2 | 1 | 1344.1.o.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.1.o.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
336.1.o.a | ✓ | 1 | 3.b | odd | 2 | 1 | CM |
336.1.o.a | ✓ | 1 | 28.d | even | 2 | 1 | RM |
336.1.o.a | ✓ | 1 | 84.h | odd | 2 | 1 | CM |
336.1.o.b | yes | 1 | 4.b | odd | 2 | 1 | |
336.1.o.b | yes | 1 | 7.b | odd | 2 | 1 | |
336.1.o.b | yes | 1 | 12.b | even | 2 | 1 | |
336.1.o.b | yes | 1 | 21.c | even | 2 | 1 | |
1344.1.o.a | 1 | 8.d | odd | 2 | 1 | ||
1344.1.o.a | 1 | 24.f | even | 2 | 1 | ||
1344.1.o.a | 1 | 56.h | odd | 2 | 1 | ||
1344.1.o.a | 1 | 168.i | even | 2 | 1 | ||
1344.1.o.b | 1 | 8.b | even | 2 | 1 | ||
1344.1.o.b | 1 | 24.h | odd | 2 | 1 | ||
1344.1.o.b | 1 | 56.e | even | 2 | 1 | ||
1344.1.o.b | 1 | 168.e | odd | 2 | 1 | ||
2352.1.z.b | 2 | 7.d | odd | 6 | 2 | ||
2352.1.z.b | 2 | 21.g | even | 6 | 2 | ||
2352.1.z.b | 2 | 28.g | odd | 6 | 2 | ||
2352.1.z.b | 2 | 84.n | even | 6 | 2 | ||
2352.1.z.c | 2 | 7.c | even | 3 | 2 | ||
2352.1.z.c | 2 | 21.h | odd | 6 | 2 | ||
2352.1.z.c | 2 | 28.f | even | 6 | 2 | ||
2352.1.z.c | 2 | 84.j | odd | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{19} - 2 \)
acting on \(S_{1}^{\mathrm{new}}(336, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 1 \)
$5$
\( T \)
$7$
\( T - 1 \)
$11$
\( T \)
$13$
\( T \)
$17$
\( T \)
$19$
\( T - 2 \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T + 2 \)
$37$
\( T + 2 \)
$41$
\( T \)
$43$
\( T \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T \)
$67$
\( T \)
$71$
\( T \)
$73$
\( T \)
$79$
\( T \)
$83$
\( T \)
$89$
\( T \)
$97$
\( T \)
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