Newspace parameters
Level: | \( N \) | \(=\) | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2352.o (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.17380090971\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | no (minimal twist has level 336) |
Projective image: | \(D_{6}\) |
Projective field: | Galois closure of 6.2.38723328.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(1471\) | \(1765\) | \(2257\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2351.1 |
|
0 | 1.00000 | 0 | 0 | 0 | 0 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||
2351.2 | 0 | 1.00000 | 0 | 0 | 0 | 0 | 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 | inner |
84.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2352.1.o.b | 2 | |
3.b | odd | 2 | 1 | CM | 2352.1.o.b | 2 | |
4.b | odd | 2 | 1 | 2352.1.o.a | 2 | ||
7.b | odd | 2 | 1 | 2352.1.o.a | 2 | ||
7.c | even | 3 | 1 | 336.1.z.a | ✓ | 2 | |
7.c | even | 3 | 1 | 2352.1.z.a | 2 | ||
7.d | odd | 6 | 1 | 336.1.z.b | yes | 2 | |
7.d | odd | 6 | 1 | 2352.1.z.d | 2 | ||
12.b | even | 2 | 1 | 2352.1.o.a | 2 | ||
21.c | even | 2 | 1 | 2352.1.o.a | 2 | ||
21.g | even | 6 | 1 | 336.1.z.b | yes | 2 | |
21.g | even | 6 | 1 | 2352.1.z.d | 2 | ||
21.h | odd | 6 | 1 | 336.1.z.a | ✓ | 2 | |
21.h | odd | 6 | 1 | 2352.1.z.a | 2 | ||
28.d | even | 2 | 1 | inner | 2352.1.o.b | 2 | |
28.f | even | 6 | 1 | 336.1.z.a | ✓ | 2 | |
28.f | even | 6 | 1 | 2352.1.z.a | 2 | ||
28.g | odd | 6 | 1 | 336.1.z.b | yes | 2 | |
28.g | odd | 6 | 1 | 2352.1.z.d | 2 | ||
56.j | odd | 6 | 1 | 1344.1.z.a | 2 | ||
56.k | odd | 6 | 1 | 1344.1.z.a | 2 | ||
56.m | even | 6 | 1 | 1344.1.z.b | 2 | ||
56.p | even | 6 | 1 | 1344.1.z.b | 2 | ||
84.h | odd | 2 | 1 | inner | 2352.1.o.b | 2 | |
84.j | odd | 6 | 1 | 336.1.z.a | ✓ | 2 | |
84.j | odd | 6 | 1 | 2352.1.z.a | 2 | ||
84.n | even | 6 | 1 | 336.1.z.b | yes | 2 | |
84.n | even | 6 | 1 | 2352.1.z.d | 2 | ||
168.s | odd | 6 | 1 | 1344.1.z.b | 2 | ||
168.v | even | 6 | 1 | 1344.1.z.a | 2 | ||
168.ba | even | 6 | 1 | 1344.1.z.a | 2 | ||
168.be | odd | 6 | 1 | 1344.1.z.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.1.z.a | ✓ | 2 | 7.c | even | 3 | 1 | |
336.1.z.a | ✓ | 2 | 21.h | odd | 6 | 1 | |
336.1.z.a | ✓ | 2 | 28.f | even | 6 | 1 | |
336.1.z.a | ✓ | 2 | 84.j | odd | 6 | 1 | |
336.1.z.b | yes | 2 | 7.d | odd | 6 | 1 | |
336.1.z.b | yes | 2 | 21.g | even | 6 | 1 | |
336.1.z.b | yes | 2 | 28.g | odd | 6 | 1 | |
336.1.z.b | yes | 2 | 84.n | even | 6 | 1 | |
1344.1.z.a | 2 | 56.j | odd | 6 | 1 | ||
1344.1.z.a | 2 | 56.k | odd | 6 | 1 | ||
1344.1.z.a | 2 | 168.v | even | 6 | 1 | ||
1344.1.z.a | 2 | 168.ba | even | 6 | 1 | ||
1344.1.z.b | 2 | 56.m | even | 6 | 1 | ||
1344.1.z.b | 2 | 56.p | even | 6 | 1 | ||
1344.1.z.b | 2 | 168.s | odd | 6 | 1 | ||
1344.1.z.b | 2 | 168.be | odd | 6 | 1 | ||
2352.1.o.a | 2 | 4.b | odd | 2 | 1 | ||
2352.1.o.a | 2 | 7.b | odd | 2 | 1 | ||
2352.1.o.a | 2 | 12.b | even | 2 | 1 | ||
2352.1.o.a | 2 | 21.c | even | 2 | 1 | ||
2352.1.o.b | 2 | 1.a | even | 1 | 1 | trivial | |
2352.1.o.b | 2 | 3.b | odd | 2 | 1 | CM | |
2352.1.o.b | 2 | 28.d | even | 2 | 1 | inner | |
2352.1.o.b | 2 | 84.h | odd | 2 | 1 | inner | |
2352.1.z.a | 2 | 7.c | even | 3 | 1 | ||
2352.1.z.a | 2 | 21.h | odd | 6 | 1 | ||
2352.1.z.a | 2 | 28.f | even | 6 | 1 | ||
2352.1.z.a | 2 | 84.j | odd | 6 | 1 | ||
2352.1.z.d | 2 | 7.d | odd | 6 | 1 | ||
2352.1.z.d | 2 | 21.g | even | 6 | 1 | ||
2352.1.z.d | 2 | 28.g | odd | 6 | 1 | ||
2352.1.z.d | 2 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{19} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(2352, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 1)^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 3 \)
$17$
\( T^{2} \)
$19$
\( (T - 1)^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( (T + 1)^{2} \)
$37$
\( (T - 1)^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 3 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} \)
$67$
\( T^{2} + 3 \)
$71$
\( T^{2} \)
$73$
\( T^{2} + 3 \)
$79$
\( T^{2} + 3 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} \)
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