Properties

Label 3920.1.bt.a
Level $3920$
Weight $1$
Character orbit 3920.bt
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.bt (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: \(\Z[a_1, \ldots, a_{5}]\)
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: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{5} + \zeta_{6} q^{9} + \zeta_{6}^{2} q^{25} + 2 q^{29} + 2 q^{41} -\zeta_{6}^{2} q^{45} -2 \zeta_{6} q^{61} + \zeta_{6}^{2} q^{81} + 2 \zeta_{6} q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} + q^{9} + O(q^{10}) \) \( 2q - q^{5} + q^{9} - q^{25} + 4q^{29} + 4q^{41} + q^{45} - 2q^{61} - q^{81} + 2q^{89} + O(q^{100}) \)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 0 0 0.500000 0.866025i 0
1439.1 0 0 0 −0.500000 0.866025i 0 0 0 0.500000 + 0.866025i 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 \(\Q(\sqrt{-5}) \)
7.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p 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.bt.a 2
4.b odd 2 1 CM 3920.1.bt.a 2
5.b even 2 1 RM 3920.1.bt.a 2
7.b odd 2 1 3920.1.bt.b 2
7.c even 3 1 3920.1.j.a 1
7.c even 3 1 inner 3920.1.bt.a 2
7.d odd 6 1 80.1.h.a 1
7.d odd 6 1 3920.1.bt.b 2
20.d odd 2 1 CM 3920.1.bt.a 2
21.g even 6 1 720.1.j.a 1
28.d even 2 1 3920.1.bt.b 2
28.f even 6 1 80.1.h.a 1
28.f even 6 1 3920.1.bt.b 2
28.g odd 6 1 3920.1.j.a 1
28.g odd 6 1 inner 3920.1.bt.a 2
35.c odd 2 1 3920.1.bt.b 2
35.i odd 6 1 80.1.h.a 1
35.i odd 6 1 3920.1.bt.b 2
35.j even 6 1 3920.1.j.a 1
35.j even 6 1 inner 3920.1.bt.a 2
35.k even 12 2 400.1.b.a 1
56.j odd 6 1 320.1.h.a 1
56.m even 6 1 320.1.h.a 1
84.j odd 6 1 720.1.j.a 1
105.p even 6 1 720.1.j.a 1
105.w odd 12 2 3600.1.e.a 1
112.v even 12 2 1280.1.e.a 2
112.x odd 12 2 1280.1.e.a 2
140.c even 2 1 3920.1.bt.b 2
140.p odd 6 1 3920.1.j.a 1
140.p odd 6 1 inner 3920.1.bt.a 2
140.s even 6 1 80.1.h.a 1
140.s even 6 1 3920.1.bt.b 2
140.x odd 12 2 400.1.b.a 1
168.ba even 6 1 2880.1.j.a 1
168.be odd 6 1 2880.1.j.a 1
280.ba even 6 1 320.1.h.a 1
280.bk odd 6 1 320.1.h.a 1
280.bp odd 12 2 1600.1.b.a 1
280.bv even 12 2 1600.1.b.a 1
420.be odd 6 1 720.1.j.a 1
420.br even 12 2 3600.1.e.a 1
560.cn odd 12 2 1280.1.e.a 2
560.co even 12 2 1280.1.e.a 2
840.cb even 6 1 2880.1.j.a 1
840.ct odd 6 1 2880.1.j.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 7.d odd 6 1
80.1.h.a 1 28.f even 6 1
80.1.h.a 1 35.i odd 6 1
80.1.h.a 1 140.s even 6 1
320.1.h.a 1 56.j odd 6 1
320.1.h.a 1 56.m even 6 1
320.1.h.a 1 280.ba even 6 1
320.1.h.a 1 280.bk odd 6 1
400.1.b.a 1 35.k even 12 2
400.1.b.a 1 140.x odd 12 2
720.1.j.a 1 21.g even 6 1
720.1.j.a 1 84.j odd 6 1
720.1.j.a 1 105.p even 6 1
720.1.j.a 1 420.be odd 6 1
1280.1.e.a 2 112.v even 12 2
1280.1.e.a 2 112.x odd 12 2
1280.1.e.a 2 560.cn odd 12 2
1280.1.e.a 2 560.co even 12 2
1600.1.b.a 1 280.bp odd 12 2
1600.1.b.a 1 280.bv even 12 2
2880.1.j.a 1 168.ba even 6 1
2880.1.j.a 1 168.be odd 6 1
2880.1.j.a 1 840.cb even 6 1
2880.1.j.a 1 840.ct odd 6 1
3600.1.e.a 1 105.w odd 12 2
3600.1.e.a 1 420.br even 12 2
3920.1.j.a 1 7.c even 3 1
3920.1.j.a 1 28.g odd 6 1
3920.1.j.a 1 35.j even 6 1
3920.1.j.a 1 140.p odd 6 1
3920.1.bt.a 2 1.a even 1 1 trivial
3920.1.bt.a 2 4.b odd 2 1 CM
3920.1.bt.a 2 5.b even 2 1 RM
3920.1.bt.a 2 7.c even 3 1 inner
3920.1.bt.a 2 20.d odd 2 1 CM
3920.1.bt.a 2 28.g odd 6 1 inner
3920.1.bt.a 2 35.j even 6 1 inner
3920.1.bt.a 2 140.p odd 6 1 inner
3920.1.bt.b 2 7.b odd 2 1
3920.1.bt.b 2 7.d odd 6 1
3920.1.bt.b 2 28.d even 2 1
3920.1.bt.b 2 28.f even 6 1
3920.1.bt.b 2 35.c odd 2 1
3920.1.bt.b 2 35.i odd 6 1
3920.1.bt.b 2 140.c even 2 1
3920.1.bt.b 2 140.s even 6 1

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

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