Properties

Label 2352.1.o.a
Level $2352$
Weight $1$
Character orbit 2352.o
Analytic conductor $1.174$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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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: \(\Q(\zeta_{6})\)
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.

\(f(q)\) \(=\) \( q - q^{3} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{9} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{13} - q^{19} - q^{25} - q^{27} + q^{31} + q^{37} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{39} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{43} + q^{57} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{67} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{73} + q^{75} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{79} + q^{81} - q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} - 2q^{19} - 2q^{25} - 2q^{27} + 2q^{31} + 2q^{37} + 2q^{57} + 2q^{75} + 2q^{81} - 2q^{93} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0 0 0 0 0 1.00000 0
2351.2 0 −1.00000 0 0 0 0 0 1.00000 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 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.a 2
3.b odd 2 1 CM 2352.1.o.a 2
4.b odd 2 1 2352.1.o.b 2
7.b odd 2 1 2352.1.o.b 2
7.c even 3 1 336.1.z.b yes 2
7.c even 3 1 2352.1.z.d 2
7.d odd 6 1 336.1.z.a 2
7.d odd 6 1 2352.1.z.a 2
12.b even 2 1 2352.1.o.b 2
21.c even 2 1 2352.1.o.b 2
21.g even 6 1 336.1.z.a 2
21.g even 6 1 2352.1.z.a 2
21.h odd 6 1 336.1.z.b yes 2
21.h odd 6 1 2352.1.z.d 2
28.d even 2 1 inner 2352.1.o.a 2
28.f even 6 1 336.1.z.b yes 2
28.f even 6 1 2352.1.z.d 2
28.g odd 6 1 336.1.z.a 2
28.g odd 6 1 2352.1.z.a 2
56.j odd 6 1 1344.1.z.b 2
56.k odd 6 1 1344.1.z.b 2
56.m even 6 1 1344.1.z.a 2
56.p even 6 1 1344.1.z.a 2
84.h odd 2 1 inner 2352.1.o.a 2
84.j odd 6 1 336.1.z.b yes 2
84.j odd 6 1 2352.1.z.d 2
84.n even 6 1 336.1.z.a 2
84.n even 6 1 2352.1.z.a 2
168.s odd 6 1 1344.1.z.a 2
168.v even 6 1 1344.1.z.b 2
168.ba even 6 1 1344.1.z.b 2
168.be odd 6 1 1344.1.z.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.1.z.a 2 7.d odd 6 1
336.1.z.a 2 21.g even 6 1
336.1.z.a 2 28.g odd 6 1
336.1.z.a 2 84.n even 6 1
336.1.z.b yes 2 7.c even 3 1
336.1.z.b yes 2 21.h odd 6 1
336.1.z.b yes 2 28.f even 6 1
336.1.z.b yes 2 84.j odd 6 1
1344.1.z.a 2 56.m even 6 1
1344.1.z.a 2 56.p even 6 1
1344.1.z.a 2 168.s odd 6 1
1344.1.z.a 2 168.be odd 6 1
1344.1.z.b 2 56.j odd 6 1
1344.1.z.b 2 56.k odd 6 1
1344.1.z.b 2 168.v even 6 1
1344.1.z.b 2 168.ba even 6 1
2352.1.o.a 2 1.a even 1 1 trivial
2352.1.o.a 2 3.b odd 2 1 CM
2352.1.o.a 2 28.d even 2 1 inner
2352.1.o.a 2 84.h odd 2 1 inner
2352.1.o.b 2 4.b odd 2 1
2352.1.o.b 2 7.b odd 2 1
2352.1.o.b 2 12.b even 2 1
2352.1.o.b 2 21.c even 2 1
2352.1.z.a 2 7.d odd 6 1
2352.1.z.a 2 21.g even 6 1
2352.1.z.a 2 28.g odd 6 1
2352.1.z.a 2 84.n even 6 1
2352.1.z.d 2 7.c even 3 1
2352.1.z.d 2 21.h odd 6 1
2352.1.z.d 2 28.f even 6 1
2352.1.z.d 2 84.j odd 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

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