Properties

Label 3920.1.cl.b
Level $3920$
Weight $1$
Character orbit 3920.cl
Analytic conductor $1.956$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.cl (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1960)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.14000.1

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{24}^{8}\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
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
3313.1 0 −0.258819 + 0.965926i 0 −0.965926 + 0.258819i 0 0 0 0 0
3313.2 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 3313.2
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 3920.1.cl.b 8
4.b odd 2 1 1960.1.br.a 8
5.c odd 4 1 inner 3920.1.cl.b 8
7.b odd 2 1 inner 3920.1.cl.b 8
7.c even 3 1 3920.1.bh.a 4
7.c even 3 1 inner 3920.1.cl.b 8
7.d odd 6 1 3920.1.bh.a 4
7.d odd 6 1 inner 3920.1.cl.b 8
20.e even 4 1 1960.1.br.a 8
28.d even 2 1 1960.1.br.a 8
28.f even 6 1 1960.1.v.a 4
28.f even 6 1 1960.1.br.a 8
28.g odd 6 1 1960.1.v.a 4
28.g odd 6 1 1960.1.br.a 8
35.f even 4 1 inner 3920.1.cl.b 8
35.k even 12 1 3920.1.bh.a 4
35.k even 12 1 inner 3920.1.cl.b 8
35.l odd 12 1 3920.1.bh.a 4
35.l odd 12 1 inner 3920.1.cl.b 8
140.j odd 4 1 1960.1.br.a 8
140.w even 12 1 1960.1.v.a 4
140.w even 12 1 1960.1.br.a 8
140.x odd 12 1 1960.1.v.a 4
140.x odd 12 1 1960.1.br.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.1.v.a 4 28.f even 6 1
1960.1.v.a 4 28.g odd 6 1
1960.1.v.a 4 140.w even 12 1
1960.1.v.a 4 140.x odd 12 1
1960.1.br.a 8 4.b odd 2 1
1960.1.br.a 8 20.e even 4 1
1960.1.br.a 8 28.d even 2 1
1960.1.br.a 8 28.f even 6 1
1960.1.br.a 8 28.g odd 6 1
1960.1.br.a 8 140.j odd 4 1
1960.1.br.a 8 140.w even 12 1
1960.1.br.a 8 140.x odd 12 1
3920.1.bh.a 4 7.c even 3 1
3920.1.bh.a 4 7.d odd 6 1
3920.1.bh.a 4 35.k even 12 1
3920.1.bh.a 4 35.l odd 12 1
3920.1.cl.b 8 1.a even 1 1 trivial
3920.1.cl.b 8 5.c odd 4 1 inner
3920.1.cl.b 8 7.b odd 2 1 inner
3920.1.cl.b 8 7.c even 3 1 inner
3920.1.cl.b 8 7.d odd 6 1 inner
3920.1.cl.b 8 35.f even 4 1 inner
3920.1.cl.b 8 35.k even 12 1 inner
3920.1.cl.b 8 35.l odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

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