Properties

Label 80.1.h.a
Level $80$
Weight $1$
Character orbit 80.h
Self dual yes
Analytic conductor $0.040$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -4, -20, 5
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 80.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0399252010106\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{5})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.320.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - q^{9} + O(q^{10}) \) \( q - q^{5} - q^{9} + q^{25} + 2q^{29} - 2q^{41} + q^{45} - q^{49} - 2q^{61} + q^{81} + 2q^{89} + O(q^{100}) \)

Expression as an eta quotient

\(f(z) = \eta(4z)\eta(20z)=q\prod_{n=1}^\infty(1 - q^{4n})^{}(1 - q^{20n})^{}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
0 0 0 −1.00000 0 0 0 −1.00000 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.b even 2 1 RM by \(\Q(\sqrt{5}) \)
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.1.h.a 1
3.b odd 2 1 720.1.j.a 1
4.b odd 2 1 CM 80.1.h.a 1
5.b even 2 1 RM 80.1.h.a 1
5.c odd 4 2 400.1.b.a 1
7.b odd 2 1 3920.1.j.a 1
7.c even 3 2 3920.1.bt.b 2
7.d odd 6 2 3920.1.bt.a 2
8.b even 2 1 320.1.h.a 1
8.d odd 2 1 320.1.h.a 1
12.b even 2 1 720.1.j.a 1
15.d odd 2 1 720.1.j.a 1
15.e even 4 2 3600.1.e.a 1
16.e even 4 2 1280.1.e.a 2
16.f odd 4 2 1280.1.e.a 2
20.d odd 2 1 CM 80.1.h.a 1
20.e even 4 2 400.1.b.a 1
24.f even 2 1 2880.1.j.a 1
24.h odd 2 1 2880.1.j.a 1
28.d even 2 1 3920.1.j.a 1
28.f even 6 2 3920.1.bt.a 2
28.g odd 6 2 3920.1.bt.b 2
35.c odd 2 1 3920.1.j.a 1
35.i odd 6 2 3920.1.bt.a 2
35.j even 6 2 3920.1.bt.b 2
40.e odd 2 1 320.1.h.a 1
40.f even 2 1 320.1.h.a 1
40.i odd 4 2 1600.1.b.a 1
40.k even 4 2 1600.1.b.a 1
60.h even 2 1 720.1.j.a 1
60.l odd 4 2 3600.1.e.a 1
80.k odd 4 2 1280.1.e.a 2
80.q even 4 2 1280.1.e.a 2
120.i odd 2 1 2880.1.j.a 1
120.m even 2 1 2880.1.j.a 1
140.c even 2 1 3920.1.j.a 1
140.p odd 6 2 3920.1.bt.b 2
140.s even 6 2 3920.1.bt.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 1.a even 1 1 trivial
80.1.h.a 1 4.b odd 2 1 CM
80.1.h.a 1 5.b even 2 1 RM
80.1.h.a 1 20.d odd 2 1 CM
320.1.h.a 1 8.b even 2 1
320.1.h.a 1 8.d odd 2 1
320.1.h.a 1 40.e odd 2 1
320.1.h.a 1 40.f even 2 1
400.1.b.a 1 5.c odd 4 2
400.1.b.a 1 20.e even 4 2
720.1.j.a 1 3.b odd 2 1
720.1.j.a 1 12.b even 2 1
720.1.j.a 1 15.d odd 2 1
720.1.j.a 1 60.h even 2 1
1280.1.e.a 2 16.e even 4 2
1280.1.e.a 2 16.f odd 4 2
1280.1.e.a 2 80.k odd 4 2
1280.1.e.a 2 80.q even 4 2
1600.1.b.a 1 40.i odd 4 2
1600.1.b.a 1 40.k even 4 2
2880.1.j.a 1 24.f even 2 1
2880.1.j.a 1 24.h odd 2 1
2880.1.j.a 1 120.i odd 2 1
2880.1.j.a 1 120.m even 2 1
3600.1.e.a 1 15.e even 4 2
3600.1.e.a 1 60.l odd 4 2
3920.1.j.a 1 7.b odd 2 1
3920.1.j.a 1 28.d even 2 1
3920.1.j.a 1 35.c odd 2 1
3920.1.j.a 1 140.c even 2 1
3920.1.bt.a 2 7.d odd 6 2
3920.1.bt.a 2 28.f even 6 2
3920.1.bt.a 2 35.i odd 6 2
3920.1.bt.a 2 140.s even 6 2
3920.1.bt.b 2 7.c even 3 2
3920.1.bt.b 2 28.g odd 6 2
3920.1.bt.b 2 35.j even 6 2
3920.1.bt.b 2 140.p odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

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