Properties

Label 3920.1.j.a
Level $3920$
Weight $1$
Character orbit 3920.j
Self dual yes
Analytic conductor $1.956$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
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.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.15680.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} - q^{9} + O(q^{10}) \) \( q + q^{5} - q^{9} + q^{25} + 2q^{29} + 2q^{41} - q^{45} + 2q^{61} + q^{81} - 2q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3039.1
0
0 0 0 1.00000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 RM by \(\Q(\sqrt{5}) \)
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.j.a 1
4.b odd 2 1 CM 3920.1.j.a 1
5.b even 2 1 RM 3920.1.j.a 1
7.b odd 2 1 80.1.h.a 1
7.c even 3 2 3920.1.bt.a 2
7.d odd 6 2 3920.1.bt.b 2
20.d odd 2 1 CM 3920.1.j.a 1
21.c even 2 1 720.1.j.a 1
28.d even 2 1 80.1.h.a 1
28.f even 6 2 3920.1.bt.b 2
28.g odd 6 2 3920.1.bt.a 2
35.c odd 2 1 80.1.h.a 1
35.f even 4 2 400.1.b.a 1
35.i odd 6 2 3920.1.bt.b 2
35.j even 6 2 3920.1.bt.a 2
56.e even 2 1 320.1.h.a 1
56.h odd 2 1 320.1.h.a 1
84.h odd 2 1 720.1.j.a 1
105.g even 2 1 720.1.j.a 1
105.k odd 4 2 3600.1.e.a 1
112.j even 4 2 1280.1.e.a 2
112.l odd 4 2 1280.1.e.a 2
140.c even 2 1 80.1.h.a 1
140.j odd 4 2 400.1.b.a 1
140.p odd 6 2 3920.1.bt.a 2
140.s even 6 2 3920.1.bt.b 2
168.e odd 2 1 2880.1.j.a 1
168.i even 2 1 2880.1.j.a 1
280.c odd 2 1 320.1.h.a 1
280.n even 2 1 320.1.h.a 1
280.s even 4 2 1600.1.b.a 1
280.y odd 4 2 1600.1.b.a 1
420.o odd 2 1 720.1.j.a 1
420.w even 4 2 3600.1.e.a 1
560.be even 4 2 1280.1.e.a 2
560.bf odd 4 2 1280.1.e.a 2
840.b odd 2 1 2880.1.j.a 1
840.u even 2 1 2880.1.j.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 7.b odd 2 1
80.1.h.a 1 28.d even 2 1
80.1.h.a 1 35.c odd 2 1
80.1.h.a 1 140.c even 2 1
320.1.h.a 1 56.e even 2 1
320.1.h.a 1 56.h odd 2 1
320.1.h.a 1 280.c odd 2 1
320.1.h.a 1 280.n even 2 1
400.1.b.a 1 35.f even 4 2
400.1.b.a 1 140.j odd 4 2
720.1.j.a 1 21.c even 2 1
720.1.j.a 1 84.h odd 2 1
720.1.j.a 1 105.g even 2 1
720.1.j.a 1 420.o odd 2 1
1280.1.e.a 2 112.j even 4 2
1280.1.e.a 2 112.l odd 4 2
1280.1.e.a 2 560.be even 4 2
1280.1.e.a 2 560.bf odd 4 2
1600.1.b.a 1 280.s even 4 2
1600.1.b.a 1 280.y odd 4 2
2880.1.j.a 1 168.e odd 2 1
2880.1.j.a 1 168.i even 2 1
2880.1.j.a 1 840.b odd 2 1
2880.1.j.a 1 840.u even 2 1
3600.1.e.a 1 105.k odd 4 2
3600.1.e.a 1 420.w even 4 2
3920.1.j.a 1 1.a even 1 1 trivial
3920.1.j.a 1 4.b odd 2 1 CM
3920.1.j.a 1 5.b even 2 1 RM
3920.1.j.a 1 20.d odd 2 1 CM
3920.1.bt.a 2 7.c even 3 2
3920.1.bt.a 2 28.g odd 6 2
3920.1.bt.a 2 35.j even 6 2
3920.1.bt.a 2 140.p odd 6 2
3920.1.bt.b 2 7.d odd 6 2
3920.1.bt.b 2 28.f even 6 2
3920.1.bt.b 2 35.i odd 6 2
3920.1.bt.b 2 140.s even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3920, [\chi])\):

\( T_{3} \)
\( T_{11} \)
\( T_{13} \)
\( T_{41} - 2 \)

Hecke characteristic polynomials

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