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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,1,Mod(815,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.815");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{1}^{\mathrm{new}}(2352, [\chi])\):

\( T_{13} \) Copy content Toggle raw display
\( T_{19}^{2} - 2T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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