Properties

Label 608.1.g.b
Level $608$
Weight $1$
Character orbit 608.g
Self dual yes
Analytic conductor $0.303$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -152
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,1,Mod(113,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, 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("608.113");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 608.g (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: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.369664.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{3} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{7} - q^{13} - q^{17} - q^{19} + q^{21} + q^{23} + q^{25} - q^{27} - q^{29} + 2 q^{37} - q^{39} - 2 q^{47} - q^{51} - q^{53} - q^{57} + q^{59} + q^{67} + q^{69} - q^{73} + q^{75} - q^{81} - q^{87} - q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0
0 1.00000 0 0 0 1.00000 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
152.g odd 2 1 CM by \(\Q(\sqrt{-38}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.g.b 1
4.b odd 2 1 152.1.g.b yes 1
8.b even 2 1 608.1.g.a 1
8.d odd 2 1 152.1.g.a 1
12.b even 2 1 1368.1.i.a 1
19.b odd 2 1 608.1.g.a 1
20.d odd 2 1 3800.1.o.a 1
20.e even 4 2 3800.1.b.b 2
24.f even 2 1 1368.1.i.b 1
40.e odd 2 1 3800.1.o.b 1
40.k even 4 2 3800.1.b.a 2
76.d even 2 1 152.1.g.a 1
76.f even 6 2 2888.1.l.b 2
76.g odd 6 2 2888.1.l.a 2
76.k even 18 6 2888.1.s.b 6
76.l odd 18 6 2888.1.s.a 6
152.b even 2 1 152.1.g.b yes 1
152.g odd 2 1 CM 608.1.g.b 1
152.k odd 6 2 2888.1.l.b 2
152.o even 6 2 2888.1.l.a 2
152.u odd 18 6 2888.1.s.b 6
152.v even 18 6 2888.1.s.a 6
228.b odd 2 1 1368.1.i.b 1
380.d even 2 1 3800.1.o.b 1
380.j odd 4 2 3800.1.b.a 2
456.l odd 2 1 1368.1.i.a 1
760.p even 2 1 3800.1.o.a 1
760.y odd 4 2 3800.1.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 8.d odd 2 1
152.1.g.a 1 76.d even 2 1
152.1.g.b yes 1 4.b odd 2 1
152.1.g.b yes 1 152.b even 2 1
608.1.g.a 1 8.b even 2 1
608.1.g.a 1 19.b odd 2 1
608.1.g.b 1 1.a even 1 1 trivial
608.1.g.b 1 152.g odd 2 1 CM
1368.1.i.a 1 12.b even 2 1
1368.1.i.a 1 456.l odd 2 1
1368.1.i.b 1 24.f even 2 1
1368.1.i.b 1 228.b odd 2 1
2888.1.l.a 2 76.g odd 6 2
2888.1.l.a 2 152.o even 6 2
2888.1.l.b 2 76.f even 6 2
2888.1.l.b 2 152.k odd 6 2
2888.1.s.a 6 76.l odd 18 6
2888.1.s.a 6 152.v even 18 6
2888.1.s.b 6 76.k even 18 6
2888.1.s.b 6 152.u odd 18 6
3800.1.b.a 2 40.k even 4 2
3800.1.b.a 2 380.j odd 4 2
3800.1.b.b 2 20.e even 4 2
3800.1.b.b 2 760.y odd 4 2
3800.1.o.a 1 20.d odd 2 1
3800.1.o.a 1 760.p even 2 1
3800.1.o.b 1 40.e odd 2 1
3800.1.o.b 1 380.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(608, [\chi])\). 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
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) 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
$29$ \( T + 1 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 2 \) 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
$53$ \( T + 1 \) 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
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1 \) 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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