Properties

Label 3920.1.br.b
Level $3920$
Weight $1$
Character orbit 3920.br
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.br (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.140.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} -\zeta_{6}^{2} q^{5} +O(q^{10})\) \( q + \zeta_{6} q^{3} -\zeta_{6}^{2} q^{5} -\zeta_{6} q^{11} + q^{13} + q^{15} -\zeta_{6} q^{17} -\zeta_{6} q^{25} + q^{27} - q^{29} -\zeta_{6}^{2} q^{33} + \zeta_{6} q^{39} -\zeta_{6}^{2} q^{47} -\zeta_{6}^{2} q^{51} - q^{55} -\zeta_{6}^{2} q^{65} -2 q^{71} + 2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{75} + \zeta_{6}^{2} q^{79} + \zeta_{6} q^{81} + 2 q^{83} - q^{85} -\zeta_{6} q^{87} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + q^{5} + O(q^{10}) \) \( 2q + q^{3} + q^{5} - q^{11} + 2q^{13} + 2q^{15} - q^{17} - q^{25} + 2q^{27} - 2q^{29} + q^{33} + q^{39} + q^{47} + q^{51} - 2q^{55} + q^{65} - 4q^{71} + 2q^{73} + q^{75} - q^{79} + q^{81} + 4q^{83} - 2q^{85} - q^{87} + 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0 0 0
1489.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0 0 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
7.c even 3 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.br.b 2
4.b odd 2 1 980.1.n.a 2
5.b even 2 1 3920.1.br.a 2
7.b odd 2 1 3920.1.br.a 2
7.c even 3 1 560.1.p.a 1
7.c even 3 1 inner 3920.1.br.b 2
7.d odd 6 1 560.1.p.b 1
7.d odd 6 1 3920.1.br.a 2
20.d odd 2 1 980.1.n.b 2
28.d even 2 1 980.1.n.b 2
28.f even 6 1 140.1.h.a 1
28.f even 6 1 980.1.n.b 2
28.g odd 6 1 140.1.h.b yes 1
28.g odd 6 1 980.1.n.a 2
35.c odd 2 1 CM 3920.1.br.b 2
35.i odd 6 1 560.1.p.a 1
35.i odd 6 1 inner 3920.1.br.b 2
35.j even 6 1 560.1.p.b 1
35.j even 6 1 3920.1.br.a 2
35.k even 12 2 2800.1.f.c 2
35.l odd 12 2 2800.1.f.c 2
56.j odd 6 1 2240.1.p.a 1
56.k odd 6 1 2240.1.p.b 1
56.m even 6 1 2240.1.p.c 1
56.p even 6 1 2240.1.p.d 1
84.j odd 6 1 1260.1.p.a 1
84.n even 6 1 1260.1.p.b 1
140.c even 2 1 980.1.n.a 2
140.p odd 6 1 140.1.h.a 1
140.p odd 6 1 980.1.n.b 2
140.s even 6 1 140.1.h.b yes 1
140.s even 6 1 980.1.n.a 2
140.w even 12 2 700.1.d.a 2
140.x odd 12 2 700.1.d.a 2
280.ba even 6 1 2240.1.p.b 1
280.bf even 6 1 2240.1.p.a 1
280.bi odd 6 1 2240.1.p.c 1
280.bk odd 6 1 2240.1.p.d 1
420.ba even 6 1 1260.1.p.a 1
420.be odd 6 1 1260.1.p.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 28.f even 6 1
140.1.h.a 1 140.p odd 6 1
140.1.h.b yes 1 28.g odd 6 1
140.1.h.b yes 1 140.s even 6 1
560.1.p.a 1 7.c even 3 1
560.1.p.a 1 35.i odd 6 1
560.1.p.b 1 7.d odd 6 1
560.1.p.b 1 35.j even 6 1
700.1.d.a 2 140.w even 12 2
700.1.d.a 2 140.x odd 12 2
980.1.n.a 2 4.b odd 2 1
980.1.n.a 2 28.g odd 6 1
980.1.n.a 2 140.c even 2 1
980.1.n.a 2 140.s even 6 1
980.1.n.b 2 20.d odd 2 1
980.1.n.b 2 28.d even 2 1
980.1.n.b 2 28.f even 6 1
980.1.n.b 2 140.p odd 6 1
1260.1.p.a 1 84.j odd 6 1
1260.1.p.a 1 420.ba even 6 1
1260.1.p.b 1 84.n even 6 1
1260.1.p.b 1 420.be odd 6 1
2240.1.p.a 1 56.j odd 6 1
2240.1.p.a 1 280.bf even 6 1
2240.1.p.b 1 56.k odd 6 1
2240.1.p.b 1 280.ba even 6 1
2240.1.p.c 1 56.m even 6 1
2240.1.p.c 1 280.bi odd 6 1
2240.1.p.d 1 56.p even 6 1
2240.1.p.d 1 280.bk odd 6 1
2800.1.f.c 2 35.k even 12 2
2800.1.f.c 2 35.l odd 12 2
3920.1.br.a 2 5.b even 2 1
3920.1.br.a 2 7.b odd 2 1
3920.1.br.a 2 7.d odd 6 1
3920.1.br.a 2 35.j even 6 1
3920.1.br.b 2 1.a even 1 1 trivial
3920.1.br.b 2 7.c even 3 1 inner
3920.1.br.b 2 35.c odd 2 1 CM
3920.1.br.b 2 35.i odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

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