Properties

Label 76.1.c.a
Level $76$
Weight $1$
Character orbit 76.c
Self dual yes
Analytic conductor $0.038$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -19
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,1,Mod(37,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("76.37");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), 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: \(0.0379289409601\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.76.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.76.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} - q^{7} + q^{9} - q^{11} - q^{17} + q^{19} + 2 q^{23} + q^{35} - q^{43} - q^{45} - q^{47} + q^{55} - q^{61} - q^{63} - q^{73} + q^{77} + q^{81} + 2 q^{83} + q^{85} - q^{95} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(1\) \(0\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0
0 0 0 −1.00000 0 −1.00000 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.1.c.a 1
3.b odd 2 1 684.1.h.a 1
4.b odd 2 1 304.1.e.a 1
5.b even 2 1 1900.1.e.a 1
5.c odd 4 2 1900.1.g.a 2
7.b odd 2 1 3724.1.e.c 1
7.c even 3 2 3724.1.bc.c 2
7.d odd 6 2 3724.1.bc.b 2
8.b even 2 1 1216.1.e.a 1
8.d odd 2 1 1216.1.e.b 1
12.b even 2 1 2736.1.o.b 1
19.b odd 2 1 CM 76.1.c.a 1
19.c even 3 2 1444.1.h.a 2
19.d odd 6 2 1444.1.h.a 2
19.e even 9 6 1444.1.j.a 6
19.f odd 18 6 1444.1.j.a 6
57.d even 2 1 684.1.h.a 1
76.d even 2 1 304.1.e.a 1
95.d odd 2 1 1900.1.e.a 1
95.g even 4 2 1900.1.g.a 2
133.c even 2 1 3724.1.e.c 1
133.o even 6 2 3724.1.bc.b 2
133.r odd 6 2 3724.1.bc.c 2
152.b even 2 1 1216.1.e.b 1
152.g odd 2 1 1216.1.e.a 1
228.b odd 2 1 2736.1.o.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 1.a even 1 1 trivial
76.1.c.a 1 19.b odd 2 1 CM
304.1.e.a 1 4.b odd 2 1
304.1.e.a 1 76.d even 2 1
684.1.h.a 1 3.b odd 2 1
684.1.h.a 1 57.d even 2 1
1216.1.e.a 1 8.b even 2 1
1216.1.e.a 1 152.g odd 2 1
1216.1.e.b 1 8.d odd 2 1
1216.1.e.b 1 152.b even 2 1
1444.1.h.a 2 19.c even 3 2
1444.1.h.a 2 19.d odd 6 2
1444.1.j.a 6 19.e even 9 6
1444.1.j.a 6 19.f odd 18 6
1900.1.e.a 1 5.b even 2 1
1900.1.e.a 1 95.d odd 2 1
1900.1.g.a 2 5.c odd 4 2
1900.1.g.a 2 95.g even 4 2
2736.1.o.b 1 12.b even 2 1
2736.1.o.b 1 228.b odd 2 1
3724.1.e.c 1 7.b odd 2 1
3724.1.e.c 1 133.c even 2 1
3724.1.bc.b 2 7.d odd 6 2
3724.1.bc.b 2 133.o even 6 2
3724.1.bc.c 2 7.c even 3 2
3724.1.bc.c 2 133.r odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(76, [\chi])\).

Hecke characteristic polynomials

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