Properties

Label 3920.1.bh.b
Level $3920$
Weight $1$
Character orbit 3920.bh
Analytic conductor $1.956$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
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.bh (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 980)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.171500.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{8} q^{3} -\zeta_{8}^{3} q^{5} +O(q^{10})\) \( q -\zeta_{8} q^{3} -\zeta_{8}^{3} q^{5} + q^{11} + \zeta_{8} q^{13} - q^{15} -\zeta_{8}^{3} q^{17} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{19} + ( 1 + \zeta_{8}^{2} ) q^{23} -\zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + \zeta_{8}^{2} q^{29} -\zeta_{8} q^{33} + ( -1 + \zeta_{8}^{2} ) q^{37} -\zeta_{8}^{2} q^{39} + ( 1 + \zeta_{8}^{2} ) q^{43} -\zeta_{8}^{3} q^{47} - q^{51} -\zeta_{8}^{3} q^{55} + ( -1 + \zeta_{8}^{2} ) q^{57} + q^{65} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{69} + \zeta_{8}^{3} q^{75} -\zeta_{8}^{2} q^{79} + q^{81} -\zeta_{8}^{2} q^{85} -\zeta_{8}^{3} q^{87} + ( -1 - \zeta_{8}^{2} ) q^{95} + \zeta_{8}^{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{11} - 4q^{15} + 4q^{23} - 4q^{37} + 4q^{43} - 4q^{51} - 4q^{57} + 4q^{65} + 4q^{81} - 4q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(981\) \(1471\) \(3041\) \(3137\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2353.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0.707107 0.707107i 0 0 0 0 0
2353.2 0 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 0 0 0 0 0
3137.1 0 −0.707107 + 0.707107i 0 0.707107 + 0.707107i 0 0 0 0 0
3137.2 0 0.707107 0.707107i 0 −0.707107 0.707107i 0 0 0 0 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
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.bh.b 4
4.b odd 2 1 980.1.l.a 4
5.c odd 4 1 inner 3920.1.bh.b 4
7.b odd 2 1 inner 3920.1.bh.b 4
7.c even 3 2 3920.1.cl.a 8
7.d odd 6 2 3920.1.cl.a 8
20.e even 4 1 980.1.l.a 4
28.d even 2 1 980.1.l.a 4
28.f even 6 2 980.1.w.a 8
28.g odd 6 2 980.1.w.a 8
35.f even 4 1 inner 3920.1.bh.b 4
35.k even 12 2 3920.1.cl.a 8
35.l odd 12 2 3920.1.cl.a 8
140.j odd 4 1 980.1.l.a 4
140.w even 12 2 980.1.w.a 8
140.x odd 12 2 980.1.w.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.l.a 4 4.b odd 2 1
980.1.l.a 4 20.e even 4 1
980.1.l.a 4 28.d even 2 1
980.1.l.a 4 140.j odd 4 1
980.1.w.a 8 28.f even 6 2
980.1.w.a 8 28.g odd 6 2
980.1.w.a 8 140.w even 12 2
980.1.w.a 8 140.x odd 12 2
3920.1.bh.b 4 1.a even 1 1 trivial
3920.1.bh.b 4 5.c odd 4 1 inner
3920.1.bh.b 4 7.b odd 2 1 inner
3920.1.bh.b 4 35.f even 4 1 inner
3920.1.cl.a 8 7.c even 3 2
3920.1.cl.a 8 7.d odd 6 2
3920.1.cl.a 8 35.k even 12 2
3920.1.cl.a 8 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 1 \) acting on \(S_{1}^{\mathrm{new}}(3920, [\chi])\).

Hecke characteristic polynomials

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