Properties

Label 3920.1.j.b
Level $3920$
Weight $1$
Character orbit 3920.j
Self dual yes
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -20
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 560)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.3841600.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{3} - q^{5} + 2 q^{9} +O(q^{10})\) \( q -\beta q^{3} - q^{5} + 2 q^{9} + \beta q^{15} -\beta q^{23} + q^{25} -\beta q^{27} - q^{29} + q^{41} -\beta q^{43} -2 q^{45} + q^{61} + \beta q^{67} + 3 q^{69} -\beta q^{75} + q^{81} + \beta q^{83} + \beta q^{87} - q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 4q^{9} + O(q^{10}) \) \( 2q - 2q^{5} + 4q^{9} + 2q^{25} - 2q^{29} + 2q^{41} - 4q^{45} + 2q^{61} + 6q^{69} + 2q^{81} - 2q^{89} + O(q^{100}) \)

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
1.73205
−1.73205
0 −1.73205 0 −1.00000 0 0 0 2.00000 0
3039.2 0 1.73205 0 −1.00000 0 0 0 2.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.j.b 2
4.b odd 2 1 inner 3920.1.j.b 2
5.b even 2 1 inner 3920.1.j.b 2
7.b odd 2 1 3920.1.j.d 2
7.c even 3 2 560.1.bt.a 4
7.d odd 6 2 3920.1.bt.c 4
20.d odd 2 1 CM 3920.1.j.b 2
28.d even 2 1 3920.1.j.d 2
28.f even 6 2 3920.1.bt.c 4
28.g odd 6 2 560.1.bt.a 4
35.c odd 2 1 3920.1.j.d 2
35.i odd 6 2 3920.1.bt.c 4
35.j even 6 2 560.1.bt.a 4
35.l odd 12 2 2800.1.ce.a 2
35.l odd 12 2 2800.1.ce.b 2
56.k odd 6 2 2240.1.bt.c 4
56.p even 6 2 2240.1.bt.c 4
140.c even 2 1 3920.1.j.d 2
140.p odd 6 2 560.1.bt.a 4
140.s even 6 2 3920.1.bt.c 4
140.w even 12 2 2800.1.ce.a 2
140.w even 12 2 2800.1.ce.b 2
280.bf even 6 2 2240.1.bt.c 4
280.bi odd 6 2 2240.1.bt.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.1.bt.a 4 7.c even 3 2
560.1.bt.a 4 28.g odd 6 2
560.1.bt.a 4 35.j even 6 2
560.1.bt.a 4 140.p odd 6 2
2240.1.bt.c 4 56.k odd 6 2
2240.1.bt.c 4 56.p even 6 2
2240.1.bt.c 4 280.bf even 6 2
2240.1.bt.c 4 280.bi odd 6 2
2800.1.ce.a 2 35.l odd 12 2
2800.1.ce.a 2 140.w even 12 2
2800.1.ce.b 2 35.l odd 12 2
2800.1.ce.b 2 140.w even 12 2
3920.1.j.b 2 1.a even 1 1 trivial
3920.1.j.b 2 4.b odd 2 1 inner
3920.1.j.b 2 5.b even 2 1 inner
3920.1.j.b 2 20.d odd 2 1 CM
3920.1.j.d 2 7.b odd 2 1
3920.1.j.d 2 28.d even 2 1
3920.1.j.d 2 35.c odd 2 1
3920.1.j.d 2 140.c even 2 1
3920.1.bt.c 4 7.d odd 6 2
3920.1.bt.c 4 28.f even 6 2
3920.1.bt.c 4 35.i odd 6 2
3920.1.bt.c 4 140.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 \(S_{1}^{\mathrm{new}}(3920, [\chi])\):

\( T_{3}^{2} - 3 \)
\( T_{11} \)
\( T_{13} \)
\( T_{41} - 1 \)

Hecke characteristic polynomials

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