Properties

Label 3920.1.bt.f
Level $3920$
Weight $1$
Character orbit 3920.bt
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -4, -35, 140
Inner twists $16$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(i, \sqrt{35})\)
Artin image $C_3\times D_4:C_2$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{5} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{12} q^{5} -\zeta_{12}^{4} q^{9} -2 \zeta_{12}^{3} q^{13} + 2 \zeta_{12}^{5} q^{17} + \zeta_{12}^{2} q^{25} -2 q^{29} + \zeta_{12}^{5} q^{45} + 2 \zeta_{12}^{4} q^{65} + 2 \zeta_{12}^{5} q^{73} -\zeta_{12}^{2} q^{81} + 2 q^{85} -2 \zeta_{12}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} + 2q^{25} - 8q^{29} - 4q^{65} - 2q^{81} + 8q^{85} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

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
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 −0.866025 0.500000i 0 0 0 0.500000 0.866025i 0
79.2 0 0 0 0.866025 + 0.500000i 0 0 0 0.500000 0.866025i 0
1439.1 0 0 0 −0.866025 + 0.500000i 0 0 0 0.500000 + 0.866025i 0
1439.2 0 0 0 0.866025 0.500000i 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 \(\Q(\sqrt{-1}) \)
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
140.c even 2 1 RM by \(\Q(\sqrt{35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
140.p odd 6 1 inner
140.s even 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.f 4
4.b odd 2 1 CM 3920.1.bt.f 4
5.b even 2 1 inner 3920.1.bt.f 4
7.b odd 2 1 inner 3920.1.bt.f 4
7.c even 3 1 3920.1.j.c 2
7.c even 3 1 inner 3920.1.bt.f 4
7.d odd 6 1 3920.1.j.c 2
7.d odd 6 1 inner 3920.1.bt.f 4
20.d odd 2 1 inner 3920.1.bt.f 4
28.d even 2 1 inner 3920.1.bt.f 4
28.f even 6 1 3920.1.j.c 2
28.f even 6 1 inner 3920.1.bt.f 4
28.g odd 6 1 3920.1.j.c 2
28.g odd 6 1 inner 3920.1.bt.f 4
35.c odd 2 1 CM 3920.1.bt.f 4
35.i odd 6 1 3920.1.j.c 2
35.i odd 6 1 inner 3920.1.bt.f 4
35.j even 6 1 3920.1.j.c 2
35.j even 6 1 inner 3920.1.bt.f 4
140.c even 2 1 RM 3920.1.bt.f 4
140.p odd 6 1 3920.1.j.c 2
140.p odd 6 1 inner 3920.1.bt.f 4
140.s even 6 1 3920.1.j.c 2
140.s even 6 1 inner 3920.1.bt.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.j.c 2 7.c even 3 1
3920.1.j.c 2 7.d odd 6 1
3920.1.j.c 2 28.f even 6 1
3920.1.j.c 2 28.g odd 6 1
3920.1.j.c 2 35.i odd 6 1
3920.1.j.c 2 35.j even 6 1
3920.1.j.c 2 140.p odd 6 1
3920.1.j.c 2 140.s even 6 1
3920.1.bt.f 4 1.a even 1 1 trivial
3920.1.bt.f 4 4.b odd 2 1 CM
3920.1.bt.f 4 5.b even 2 1 inner
3920.1.bt.f 4 7.b odd 2 1 inner
3920.1.bt.f 4 7.c even 3 1 inner
3920.1.bt.f 4 7.d odd 6 1 inner
3920.1.bt.f 4 20.d odd 2 1 inner
3920.1.bt.f 4 28.d even 2 1 inner
3920.1.bt.f 4 28.f even 6 1 inner
3920.1.bt.f 4 28.g odd 6 1 inner
3920.1.bt.f 4 35.c odd 2 1 CM
3920.1.bt.f 4 35.i odd 6 1 inner
3920.1.bt.f 4 35.j even 6 1 inner
3920.1.bt.f 4 140.c even 2 1 RM
3920.1.bt.f 4 140.p odd 6 1 inner
3920.1.bt.f 4 140.s even 6 1 inner

Hecke kernels

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

\( T_{3} \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)
\( T_{41} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( 16 - 4 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( ( 4 + T^{2} )^{2} \)
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