Properties

Label 3920.1.cx.a
Level $3920$
Weight $1$
Character orbit 3920.cx
Analytic conductor $1.956$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -4
Inner twists $16$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3920.cx (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{8}\)
Projective field Galois closure of 8.2.3294172000000.3

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{48}^{5} q^{5} -\zeta_{48}^{20} q^{9} +O(q^{10})\) \( q + \zeta_{48}^{5} q^{5} -\zeta_{48}^{20} q^{9} + ( -\zeta_{48}^{15} + \zeta_{48}^{21} ) q^{13} + ( \zeta_{48}^{7} + \zeta_{48}^{13} ) q^{17} + \zeta_{48}^{10} q^{25} + ( \zeta_{48}^{6} + \zeta_{48}^{18} ) q^{29} -2 \zeta_{48}^{14} q^{37} + ( \zeta_{48}^{3} + \zeta_{48}^{21} ) q^{41} + \zeta_{48} q^{45} + ( -\zeta_{48}^{4} - \zeta_{48}^{16} ) q^{53} + ( -\zeta_{48}^{17} - \zeta_{48}^{23} ) q^{61} + ( -\zeta_{48}^{2} - \zeta_{48}^{20} ) q^{65} + ( -\zeta_{48} + \zeta_{48}^{19} ) q^{73} -\zeta_{48}^{16} q^{81} + ( \zeta_{48}^{12} + \zeta_{48}^{18} ) q^{85} + ( \zeta_{48}^{5} + \zeta_{48}^{11} ) q^{89} + ( -\zeta_{48}^{3} + \zeta_{48}^{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q + 8q^{53} + 8q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{48}^{8}\) \(\zeta_{48}^{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
0.793353 + 0.608761i
0.608761 0.793353i
−0.608761 + 0.793353i
−0.793353 0.608761i
0.793353 0.608761i
0.608761 + 0.793353i
−0.608761 0.793353i
−0.793353 + 0.608761i
−0.991445 0.130526i
−0.130526 + 0.991445i
0.130526 0.991445i
0.991445 + 0.130526i
−0.991445 + 0.130526i
−0.130526 0.991445i
0.130526 + 0.991445i
0.991445 0.130526i
0 0 0 −0.991445 0.130526i 0 0 0 −0.866025 0.500000i 0
607.2 0 0 0 −0.130526 + 0.991445i 0 0 0 −0.866025 0.500000i 0
607.3 0 0 0 0.130526 0.991445i 0 0 0 −0.866025 0.500000i 0
607.4 0 0 0 0.991445 + 0.130526i 0 0 0 −0.866025 0.500000i 0
2383.1 0 0 0 −0.991445 + 0.130526i 0 0 0 −0.866025 + 0.500000i 0
2383.2 0 0 0 −0.130526 0.991445i 0 0 0 −0.866025 + 0.500000i 0
2383.3 0 0 0 0.130526 + 0.991445i 0 0 0 −0.866025 + 0.500000i 0
2383.4 0 0 0 0.991445 0.130526i 0 0 0 −0.866025 + 0.500000i 0
3167.1 0 0 0 −0.793353 0.608761i 0 0 0 0.866025 0.500000i 0
3167.2 0 0 0 −0.608761 + 0.793353i 0 0 0 0.866025 0.500000i 0
3167.3 0 0 0 0.608761 0.793353i 0 0 0 0.866025 0.500000i 0
3167.4 0 0 0 0.793353 + 0.608761i 0 0 0 0.866025 0.500000i 0
3743.1 0 0 0 −0.793353 + 0.608761i 0 0 0 0.866025 + 0.500000i 0
3743.2 0 0 0 −0.608761 0.793353i 0 0 0 0.866025 + 0.500000i 0
3743.3 0 0 0 0.608761 + 0.793353i 0 0 0 0.866025 + 0.500000i 0
3743.4 0 0 0 0.793353 0.608761i 0 0 0 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3743.4
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.c odd 4 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.e even 4 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.f even 4 1 inner
35.k even 12 1 inner
35.l odd 12 1 inner
140.j odd 4 1 inner
140.w even 12 1 inner
140.x odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.cx.a 16
4.b odd 2 1 CM 3920.1.cx.a 16
5.c odd 4 1 inner 3920.1.cx.a 16
7.b odd 2 1 inner 3920.1.cx.a 16
7.c even 3 1 3920.1.v.a 8
7.c even 3 1 inner 3920.1.cx.a 16
7.d odd 6 1 3920.1.v.a 8
7.d odd 6 1 inner 3920.1.cx.a 16
20.e even 4 1 inner 3920.1.cx.a 16
28.d even 2 1 inner 3920.1.cx.a 16
28.f even 6 1 3920.1.v.a 8
28.f even 6 1 inner 3920.1.cx.a 16
28.g odd 6 1 3920.1.v.a 8
28.g odd 6 1 inner 3920.1.cx.a 16
35.f even 4 1 inner 3920.1.cx.a 16
35.k even 12 1 3920.1.v.a 8
35.k even 12 1 inner 3920.1.cx.a 16
35.l odd 12 1 3920.1.v.a 8
35.l odd 12 1 inner 3920.1.cx.a 16
140.j odd 4 1 inner 3920.1.cx.a 16
140.w even 12 1 3920.1.v.a 8
140.w even 12 1 inner 3920.1.cx.a 16
140.x odd 12 1 3920.1.v.a 8
140.x odd 12 1 inner 3920.1.cx.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.v.a 8 7.c even 3 1
3920.1.v.a 8 7.d odd 6 1
3920.1.v.a 8 28.f even 6 1
3920.1.v.a 8 28.g odd 6 1
3920.1.v.a 8 35.k even 12 1
3920.1.v.a 8 35.l odd 12 1
3920.1.v.a 8 140.w even 12 1
3920.1.v.a 8 140.x odd 12 1
3920.1.cx.a 16 1.a even 1 1 trivial
3920.1.cx.a 16 4.b odd 2 1 CM
3920.1.cx.a 16 5.c odd 4 1 inner
3920.1.cx.a 16 7.b odd 2 1 inner
3920.1.cx.a 16 7.c even 3 1 inner
3920.1.cx.a 16 7.d odd 6 1 inner
3920.1.cx.a 16 20.e even 4 1 inner
3920.1.cx.a 16 28.d even 2 1 inner
3920.1.cx.a 16 28.f even 6 1 inner
3920.1.cx.a 16 28.g odd 6 1 inner
3920.1.cx.a 16 35.f even 4 1 inner
3920.1.cx.a 16 35.k even 12 1 inner
3920.1.cx.a 16 35.l odd 12 1 inner
3920.1.cx.a 16 140.j odd 4 1 inner
3920.1.cx.a 16 140.w even 12 1 inner
3920.1.cx.a 16 140.x odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 1 - T^{8} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 4 + 12 T^{4} + T^{8} )^{2} \)
$17$ \( 16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( ( 2 + T^{2} )^{8} \)
$31$ \( T^{16} \)
$37$ \( ( 256 - 16 T^{4} + T^{8} )^{2} \)
$41$ \( ( 2 + 4 T^{2} + T^{4} )^{4} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( ( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{4} \)
$59$ \( T^{16} \)
$61$ \( ( 4 - 8 T^{2} + 14 T^{4} - 4 T^{6} + T^{8} )^{2} \)
$67$ \( T^{16} \)
$71$ \( T^{16} \)
$73$ \( 16 - 48 T^{4} + 140 T^{8} - 12 T^{12} + T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( ( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2} \)
$97$ \( ( 4 + 12 T^{4} + T^{8} )^{2} \)
show more
show less