Properties

Label 176.1.h.a
Level $176$
Weight $1$
Character orbit 176.h
Self dual yes
Analytic conductor $0.088$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,1,Mod(65,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.65");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 176.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.0878354422234\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.44.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.30976.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{3} - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} - q^{11} - q^{15} + q^{23} - q^{27} + q^{31} - q^{33} - q^{37} - 2 q^{47} + q^{49} + 2 q^{53} + q^{55} + q^{59} + q^{67} + q^{69} + q^{71} - q^{81} - q^{89} + q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Expression as an eta quotient

\(f(z) = \dfrac{\eta(4z)^{3}\eta(44z)^{3}}{\eta(2z)\eta(8z)\eta(22z)\eta(88z)}=q\prod_{n=1}^\infty(1 - q^{2n})^{-1}(1 - q^{4n})^{3}(1 - q^{8n})^{-1}(1 - q^{22n})^{-1}(1 - q^{44n})^{3}(1 - q^{88n})^{-1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0
0 1.00000 0 −1.00000 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.1.h.a 1
3.b odd 2 1 1584.1.j.a 1
4.b odd 2 1 44.1.d.a 1
8.b even 2 1 704.1.h.a 1
8.d odd 2 1 704.1.h.b 1
11.b odd 2 1 CM 176.1.h.a 1
11.c even 5 4 1936.1.n.a 4
11.d odd 10 4 1936.1.n.a 4
12.b even 2 1 396.1.f.a 1
16.e even 4 2 2816.1.b.a 2
16.f odd 4 2 2816.1.b.b 2
20.d odd 2 1 1100.1.f.a 1
20.e even 4 2 1100.1.e.a 2
28.d even 2 1 2156.1.h.a 1
28.f even 6 2 2156.1.k.a 2
28.g odd 6 2 2156.1.k.b 2
33.d even 2 1 1584.1.j.a 1
36.f odd 6 2 3564.1.m.b 2
36.h even 6 2 3564.1.m.a 2
44.c even 2 1 44.1.d.a 1
44.g even 10 4 484.1.f.a 4
44.h odd 10 4 484.1.f.a 4
88.b odd 2 1 704.1.h.a 1
88.g even 2 1 704.1.h.b 1
132.d odd 2 1 396.1.f.a 1
176.i even 4 2 2816.1.b.b 2
176.l odd 4 2 2816.1.b.a 2
220.g even 2 1 1100.1.f.a 1
220.i odd 4 2 1100.1.e.a 2
308.g odd 2 1 2156.1.h.a 1
308.m odd 6 2 2156.1.k.a 2
308.n even 6 2 2156.1.k.b 2
396.k even 6 2 3564.1.m.b 2
396.o odd 6 2 3564.1.m.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.1.d.a 1 4.b odd 2 1
44.1.d.a 1 44.c even 2 1
176.1.h.a 1 1.a even 1 1 trivial
176.1.h.a 1 11.b odd 2 1 CM
396.1.f.a 1 12.b even 2 1
396.1.f.a 1 132.d odd 2 1
484.1.f.a 4 44.g even 10 4
484.1.f.a 4 44.h odd 10 4
704.1.h.a 1 8.b even 2 1
704.1.h.a 1 88.b odd 2 1
704.1.h.b 1 8.d odd 2 1
704.1.h.b 1 88.g even 2 1
1100.1.e.a 2 20.e even 4 2
1100.1.e.a 2 220.i odd 4 2
1100.1.f.a 1 20.d odd 2 1
1100.1.f.a 1 220.g even 2 1
1584.1.j.a 1 3.b odd 2 1
1584.1.j.a 1 33.d even 2 1
1936.1.n.a 4 11.c even 5 4
1936.1.n.a 4 11.d odd 10 4
2156.1.h.a 1 28.d even 2 1
2156.1.h.a 1 308.g odd 2 1
2156.1.k.a 2 28.f even 6 2
2156.1.k.a 2 308.m odd 6 2
2156.1.k.b 2 28.g odd 6 2
2156.1.k.b 2 308.n even 6 2
2816.1.b.a 2 16.e even 4 2
2816.1.b.a 2 176.l odd 4 2
2816.1.b.b 2 16.f odd 4 2
2816.1.b.b 2 176.i even 4 2
3564.1.m.a 2 36.h even 6 2
3564.1.m.a 2 396.o odd 6 2
3564.1.m.b 2 36.f odd 6 2
3564.1.m.b 2 396.k even 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(176, [\chi])\).

Hecke characteristic polynomials

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