Properties

Label 1960.1.br.a
Level $1960$
Weight $1$
Character orbit 1960.br
Analytic conductor $0.978$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1960,1,Mod(177,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.177");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1960.br (of order \(12\), degree \(4\), 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: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.14000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{3} - \zeta_{24}^{11} q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{3} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{4} q^{11} + \zeta_{24}^{3} q^{13} + \zeta_{24}^{6} q^{15} - \zeta_{24} q^{17} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{19} - \zeta_{24}^{10} q^{25} + \zeta_{24}^{9} q^{27} - \zeta_{24}^{6} q^{29} + (\zeta_{24}^{7} + \zeta_{24}) q^{31} - \zeta_{24}^{11} q^{33} + \zeta_{24}^{10} q^{39} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{41} + ( - \zeta_{24}^{6} - 1) q^{43} - \zeta_{24}^{5} q^{47} - \zeta_{24}^{8} q^{51} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{53} - \zeta_{24}^{3} q^{55} + (\zeta_{24}^{6} - 1) q^{57} + (\zeta_{24}^{7} - \zeta_{24}) q^{59} + \zeta_{24}^{2} q^{65} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{67} + \zeta_{24}^{5} q^{75} - \zeta_{24}^{2} q^{79} - \zeta_{24}^{4} q^{81} - q^{85} + \zeta_{24} q^{87} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{93} + ( - \zeta_{24}^{10} + \zeta_{24}^{4}) q^{95} + \zeta_{24}^{9} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{11} - 8 q^{43} + 4 q^{51} + 4 q^{53} - 8 q^{57} + 4 q^{67} - 4 q^{81} - 8 q^{85} - 4 q^{93} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(\zeta_{24}^{8}\) \(-\zeta_{24}^{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
0 −0.965926 0.258819i 0 0.258819 + 0.965926i 0 0 0 0 0
177.2 0 0.965926 + 0.258819i 0 −0.258819 0.965926i 0 0 0 0 0
753.1 0 −0.965926 + 0.258819i 0 0.258819 0.965926i 0 0 0 0 0
753.2 0 0.965926 0.258819i 0 −0.258819 + 0.965926i 0 0 0 0 0
1353.1 0 −0.258819 + 0.965926i 0 0.965926 0.258819i 0 0 0 0 0
1353.2 0 0.258819 0.965926i 0 −0.965926 + 0.258819i 0 0 0 0 0
1537.1 0 −0.258819 0.965926i 0 0.965926 + 0.258819i 0 0 0 0 0
1537.2 0 0.258819 + 0.965926i 0 −0.965926 0.258819i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.1.br.a 8
4.b odd 2 1 3920.1.cl.b 8
5.c odd 4 1 inner 1960.1.br.a 8
7.b odd 2 1 inner 1960.1.br.a 8
7.c even 3 1 1960.1.v.a 4
7.c even 3 1 inner 1960.1.br.a 8
7.d odd 6 1 1960.1.v.a 4
7.d odd 6 1 inner 1960.1.br.a 8
20.e even 4 1 3920.1.cl.b 8
28.d even 2 1 3920.1.cl.b 8
28.f even 6 1 3920.1.bh.a 4
28.f even 6 1 3920.1.cl.b 8
28.g odd 6 1 3920.1.bh.a 4
28.g odd 6 1 3920.1.cl.b 8
35.f even 4 1 inner 1960.1.br.a 8
35.k even 12 1 1960.1.v.a 4
35.k even 12 1 inner 1960.1.br.a 8
35.l odd 12 1 1960.1.v.a 4
35.l odd 12 1 inner 1960.1.br.a 8
140.j odd 4 1 3920.1.cl.b 8
140.w even 12 1 3920.1.bh.a 4
140.w even 12 1 3920.1.cl.b 8
140.x odd 12 1 3920.1.bh.a 4
140.x odd 12 1 3920.1.cl.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.1.v.a 4 7.c even 3 1
1960.1.v.a 4 7.d odd 6 1
1960.1.v.a 4 35.k even 12 1
1960.1.v.a 4 35.l odd 12 1
1960.1.br.a 8 1.a even 1 1 trivial
1960.1.br.a 8 5.c odd 4 1 inner
1960.1.br.a 8 7.b odd 2 1 inner
1960.1.br.a 8 7.c even 3 1 inner
1960.1.br.a 8 7.d odd 6 1 inner
1960.1.br.a 8 35.f even 4 1 inner
1960.1.br.a 8 35.k even 12 1 inner
1960.1.br.a 8 35.l odd 12 1 inner
3920.1.bh.a 4 28.f even 6 1
3920.1.bh.a 4 28.g odd 6 1
3920.1.bh.a 4 140.w even 12 1
3920.1.bh.a 4 140.x odd 12 1
3920.1.cl.b 8 4.b odd 2 1
3920.1.cl.b 8 20.e even 4 1
3920.1.cl.b 8 28.d even 2 1
3920.1.cl.b 8 28.f even 6 1
3920.1.cl.b 8 28.g odd 6 1
3920.1.cl.b 8 140.j odd 4 1
3920.1.cl.b 8 140.w even 12 1
3920.1.cl.b 8 140.x odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
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