Properties

Label 160.1.p.a
Level 160
Weight 1
Character orbit 160.p
Analytic conductor 0.080
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 160.p (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0798504020213\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.2000.1
Artin image $C_4\wr C_2$
Artin field Galois closure of 8.0.8192000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{5} + i q^{9} +O(q^{10})\) \( q -i q^{5} + i q^{9} + ( -1 + i ) q^{13} + ( -1 - i ) q^{17} - q^{25} + ( 1 + i ) q^{37} + q^{45} -i q^{49} + ( 1 - i ) q^{53} + ( 1 + i ) q^{65} + ( 1 - i ) q^{73} - q^{81} + ( -1 + i ) q^{85} + ( 1 + i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{13} - 2q^{17} - 2q^{25} + 2q^{37} + 2q^{45} + 2q^{53} + 2q^{65} + 2q^{73} - 2q^{81} - 2q^{85} + 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(i\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.00000i
1.00000i
0 0 0 1.00000i 0 0 0 1.00000i 0
97.1 0 0 0 1.00000i 0 0 0 1.00000i 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.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.1.p.a 2
3.b odd 2 1 1440.1.bh.b 2
4.b odd 2 1 CM 160.1.p.a 2
5.b even 2 1 800.1.p.b 2
5.c odd 4 1 inner 160.1.p.a 2
5.c odd 4 1 800.1.p.b 2
8.b even 2 1 320.1.p.a 2
8.d odd 2 1 320.1.p.a 2
12.b even 2 1 1440.1.bh.b 2
15.e even 4 1 1440.1.bh.b 2
16.e even 4 1 1280.1.m.a 2
16.e even 4 1 1280.1.m.b 2
16.f odd 4 1 1280.1.m.a 2
16.f odd 4 1 1280.1.m.b 2
20.d odd 2 1 800.1.p.b 2
20.e even 4 1 inner 160.1.p.a 2
20.e even 4 1 800.1.p.b 2
24.f even 2 1 2880.1.bh.b 2
24.h odd 2 1 2880.1.bh.b 2
40.e odd 2 1 1600.1.p.b 2
40.f even 2 1 1600.1.p.b 2
40.i odd 4 1 320.1.p.a 2
40.i odd 4 1 1600.1.p.b 2
40.k even 4 1 320.1.p.a 2
40.k even 4 1 1600.1.p.b 2
60.l odd 4 1 1440.1.bh.b 2
80.i odd 4 1 1280.1.m.b 2
80.j even 4 1 1280.1.m.a 2
80.s even 4 1 1280.1.m.b 2
80.t odd 4 1 1280.1.m.a 2
120.q odd 4 1 2880.1.bh.b 2
120.w even 4 1 2880.1.bh.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.1.p.a 2 1.a even 1 1 trivial
160.1.p.a 2 4.b odd 2 1 CM
160.1.p.a 2 5.c odd 4 1 inner
160.1.p.a 2 20.e even 4 1 inner
320.1.p.a 2 8.b even 2 1
320.1.p.a 2 8.d odd 2 1
320.1.p.a 2 40.i odd 4 1
320.1.p.a 2 40.k even 4 1
800.1.p.b 2 5.b even 2 1
800.1.p.b 2 5.c odd 4 1
800.1.p.b 2 20.d odd 2 1
800.1.p.b 2 20.e even 4 1
1280.1.m.a 2 16.e even 4 1
1280.1.m.a 2 16.f odd 4 1
1280.1.m.a 2 80.j even 4 1
1280.1.m.a 2 80.t odd 4 1
1280.1.m.b 2 16.e even 4 1
1280.1.m.b 2 16.f odd 4 1
1280.1.m.b 2 80.i odd 4 1
1280.1.m.b 2 80.s even 4 1
1440.1.bh.b 2 3.b odd 2 1
1440.1.bh.b 2 12.b even 2 1
1440.1.bh.b 2 15.e even 4 1
1440.1.bh.b 2 60.l odd 4 1
1600.1.p.b 2 40.e odd 2 1
1600.1.p.b 2 40.f even 2 1
1600.1.p.b 2 40.i odd 4 1
1600.1.p.b 2 40.k even 4 1
2880.1.bh.b 2 24.f even 2 1
2880.1.bh.b 2 24.h odd 2 1
2880.1.bh.b 2 120.q odd 4 1
2880.1.bh.b 2 120.w even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(160, [\chi])\).