Properties

Label 2352.1.z.b
Level $2352$
Weight $1$
Character orbit 2352.z
Analytic conductor $1.174$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -84, 28
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2352.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{7})\)
Artin image $C_3\times D_4$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{3} -\zeta_{6} q^{9} + 2 \zeta_{6} q^{19} -\zeta_{6}^{2} q^{25} + q^{27} + 2 \zeta_{6}^{2} q^{31} + 2 \zeta_{6} q^{37} -2 q^{57} + \zeta_{6} q^{75} + \zeta_{6}^{2} q^{81} -2 \zeta_{6} q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{9} + 2q^{19} + q^{25} + 2q^{27} - 2q^{31} + 2q^{37} - 4q^{57} + q^{75} - q^{81} - 2q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
815.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0 0 0 0 −0.500000 0.866025i 0
1391.1 0 −0.500000 0.866025i 0 0 0 0 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 \(\Q(\sqrt{-21}) \)
7.c even 3 1 inner
21.h odd 6 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.1.z.b 2
3.b odd 2 1 CM 2352.1.z.b 2
4.b odd 2 1 2352.1.z.c 2
7.b odd 2 1 2352.1.z.c 2
7.c even 3 1 336.1.o.b yes 1
7.c even 3 1 inner 2352.1.z.b 2
7.d odd 6 1 336.1.o.a 1
7.d odd 6 1 2352.1.z.c 2
12.b even 2 1 2352.1.z.c 2
21.c even 2 1 2352.1.z.c 2
21.g even 6 1 336.1.o.a 1
21.g even 6 1 2352.1.z.c 2
21.h odd 6 1 336.1.o.b yes 1
21.h odd 6 1 inner 2352.1.z.b 2
28.d even 2 1 RM 2352.1.z.b 2
28.f even 6 1 336.1.o.b yes 1
28.f even 6 1 inner 2352.1.z.b 2
28.g odd 6 1 336.1.o.a 1
28.g odd 6 1 2352.1.z.c 2
56.j odd 6 1 1344.1.o.b 1
56.k odd 6 1 1344.1.o.b 1
56.m even 6 1 1344.1.o.a 1
56.p even 6 1 1344.1.o.a 1
84.h odd 2 1 CM 2352.1.z.b 2
84.j odd 6 1 336.1.o.b yes 1
84.j odd 6 1 inner 2352.1.z.b 2
84.n even 6 1 336.1.o.a 1
84.n even 6 1 2352.1.z.c 2
168.s odd 6 1 1344.1.o.a 1
168.v even 6 1 1344.1.o.b 1
168.ba even 6 1 1344.1.o.b 1
168.be odd 6 1 1344.1.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.1.o.a 1 7.d odd 6 1
336.1.o.a 1 21.g even 6 1
336.1.o.a 1 28.g odd 6 1
336.1.o.a 1 84.n even 6 1
336.1.o.b yes 1 7.c even 3 1
336.1.o.b yes 1 21.h odd 6 1
336.1.o.b yes 1 28.f even 6 1
336.1.o.b yes 1 84.j odd 6 1
1344.1.o.a 1 56.m even 6 1
1344.1.o.a 1 56.p even 6 1
1344.1.o.a 1 168.s odd 6 1
1344.1.o.a 1 168.be odd 6 1
1344.1.o.b 1 56.j odd 6 1
1344.1.o.b 1 56.k odd 6 1
1344.1.o.b 1 168.v even 6 1
1344.1.o.b 1 168.ba even 6 1
2352.1.z.b 2 1.a even 1 1 trivial
2352.1.z.b 2 3.b odd 2 1 CM
2352.1.z.b 2 7.c even 3 1 inner
2352.1.z.b 2 21.h odd 6 1 inner
2352.1.z.b 2 28.d even 2 1 RM
2352.1.z.b 2 28.f even 6 1 inner
2352.1.z.b 2 84.h odd 2 1 CM
2352.1.z.b 2 84.j odd 6 1 inner
2352.1.z.c 2 4.b odd 2 1
2352.1.z.c 2 7.b odd 2 1
2352.1.z.c 2 7.d odd 6 1
2352.1.z.c 2 12.b even 2 1
2352.1.z.c 2 21.c even 2 1
2352.1.z.c 2 21.g even 6 1
2352.1.z.c 2 28.g odd 6 1
2352.1.z.c 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2352, [\chi])\):

\( T_{13} \)
\( T_{19}^{2} - 2 T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( 4 - 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 4 + 2 T + T^{2} \)
$37$ \( 4 - 2 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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