Properties

Label 2352.1.o.b
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,1,Mod(2351,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.2351");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

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: \(\Z[a_1, \ldots, a_{13}]\)
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

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.

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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.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 \(S_{1}^{\mathrm{new}}(2352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) 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
$13$ \( T^{2} + 3 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3 \) 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
$67$ \( T^{2} + 3 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( T^{2} + 3 \) 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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