Properties

Label 3920.1.v.a
Level $3920$
Weight $1$
Character orbit 3920.v
Analytic conductor $1.956$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -4
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.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{5} q^{5} -\zeta_{16}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{16}^{5} q^{5} -\zeta_{16}^{4} q^{9} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{13} + ( -\zeta_{16}^{5} + \zeta_{16}^{7} ) q^{17} -\zeta_{16}^{2} q^{25} + ( \zeta_{16}^{2} + \zeta_{16}^{6} ) q^{29} + 2 \zeta_{16}^{6} q^{37} + ( \zeta_{16}^{3} + \zeta_{16}^{5} ) q^{41} + \zeta_{16} q^{45} + ( -1 - \zeta_{16}^{4} ) q^{53} + ( -\zeta_{16} - \zeta_{16}^{7} ) q^{61} + ( -\zeta_{16}^{2} - \zeta_{16}^{4} ) q^{65} + ( -\zeta_{16} + \zeta_{16}^{3} ) q^{73} - q^{81} + ( \zeta_{16}^{2} - \zeta_{16}^{4} ) q^{85} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{89} + ( -\zeta_{16} - \zeta_{16}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{53} - 8q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{16}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
0 0 0 −0.923880 0.382683i 0 0 0 1.00000i 0
783.2 0 0 0 −0.382683 + 0.923880i 0 0 0 1.00000i 0
783.3 0 0 0 0.382683 0.923880i 0 0 0 1.00000i 0
783.4 0 0 0 0.923880 + 0.382683i 0 0 0 1.00000i 0
1567.1 0 0 0 −0.923880 + 0.382683i 0 0 0 1.00000i 0
1567.2 0 0 0 −0.382683 0.923880i 0 0 0 1.00000i 0
1567.3 0 0 0 0.382683 + 0.923880i 0 0 0 1.00000i 0
1567.4 0 0 0 0.923880 0.382683i 0 0 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.4
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
20.e even 4 1 inner
28.d even 2 1 inner
35.f even 4 1 inner
140.j odd 4 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

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