Properties

Label 204.1.g.b
Level $204$
Weight $1$
Character orbit 204.g
Self dual yes
Analytic conductor $0.102$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -51
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [204,1,Mod(101,204)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(204, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("204.101");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 204 = 2^{2} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 204.g (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: \(0.101809262577\)
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 3.1.204.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.204.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{3} - q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + q^{9} - q^{11} - q^{13} - q^{15} + q^{17} - q^{19} - q^{23} + q^{27} + 2 q^{29} - q^{33} - q^{39} - q^{41} - q^{43} - q^{45} + q^{49} + q^{51} + q^{55} - q^{57} + q^{65} + 2 q^{67} - q^{69} + 2 q^{71} + q^{81} - q^{85} + 2 q^{87} + q^{95} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/204\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(103\) \(137\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0
0 1.00000 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
51.c odd 2 1 CM by \(\Q(\sqrt{-51}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 204.1.g.b yes 1
3.b odd 2 1 204.1.g.a 1
4.b odd 2 1 816.1.m.a 1
8.b even 2 1 3264.1.m.b 1
8.d odd 2 1 3264.1.m.d 1
12.b even 2 1 816.1.m.b 1
17.b even 2 1 204.1.g.a 1
17.c even 4 2 3468.1.d.a 2
17.d even 8 4 3468.1.i.a 4
17.e odd 16 8 3468.1.m.a 8
24.f even 2 1 3264.1.m.a 1
24.h odd 2 1 3264.1.m.c 1
51.c odd 2 1 CM 204.1.g.b yes 1
51.f odd 4 2 3468.1.d.a 2
51.g odd 8 4 3468.1.i.a 4
51.i even 16 8 3468.1.m.a 8
68.d odd 2 1 816.1.m.b 1
136.e odd 2 1 3264.1.m.a 1
136.h even 2 1 3264.1.m.c 1
204.h even 2 1 816.1.m.a 1
408.b odd 2 1 3264.1.m.b 1
408.h even 2 1 3264.1.m.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
204.1.g.a 1 3.b odd 2 1
204.1.g.a 1 17.b even 2 1
204.1.g.b yes 1 1.a even 1 1 trivial
204.1.g.b yes 1 51.c odd 2 1 CM
816.1.m.a 1 4.b odd 2 1
816.1.m.a 1 204.h even 2 1
816.1.m.b 1 12.b even 2 1
816.1.m.b 1 68.d odd 2 1
3264.1.m.a 1 24.f even 2 1
3264.1.m.a 1 136.e odd 2 1
3264.1.m.b 1 8.b even 2 1
3264.1.m.b 1 408.b odd 2 1
3264.1.m.c 1 24.h odd 2 1
3264.1.m.c 1 136.h even 2 1
3264.1.m.d 1 8.d odd 2 1
3264.1.m.d 1 408.h even 2 1
3468.1.d.a 2 17.c even 4 2
3468.1.d.a 2 51.f odd 4 2
3468.1.i.a 4 17.d even 8 4
3468.1.i.a 4 51.g odd 8 4
3468.1.m.a 8 17.e odd 16 8
3468.1.m.a 8 51.i even 16 8

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(204, [\chi])\). Copy content Toggle raw display

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
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 1 \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 2 \) Copy content Toggle raw display
$71$ \( T - 2 \) 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
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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