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Properties

Label 608.1.o.a
Level $608$
Weight $1$
Character orbit 608.o
Analytic conductor $0.303$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,1,Mod(239,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.239");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 608.o (of order \(6\), degree \(2\), 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: no
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2888.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.2186423566336.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{3} + q^{11} + \zeta_{6}^{2} q^{17} + \zeta_{6} q^{19} - \zeta_{6} q^{25} - q^{27} + \zeta_{6}^{2} q^{33} - \zeta_{6}^{2} q^{41} - \zeta_{6}^{2} q^{43} + q^{49} - 2 \zeta_{6} q^{51} - q^{57} + \zeta_{6}^{2} q^{59} - \zeta_{6} q^{67} - \zeta_{6}^{2} q^{73} + q^{75} - \zeta_{6}^{2} q^{81} + q^{83} - \zeta_{6} q^{89} - \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{11} - 2 q^{17} + q^{19} - q^{25} - 2 q^{27} - q^{33} + q^{41} + 2 q^{43} + 2 q^{49} - 2 q^{51} - 2 q^{57} - q^{59} - q^{67} + q^{73} + 2 q^{75} + q^{81} + 2 q^{83} - 2 q^{89} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-\zeta_{6}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −0.500000 + 0.866025i 0 0 0 0 0 0 0
463.1 0 −0.500000 − 0.866025i 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.c even 3 1 inner
152.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.1.o.a 2
4.b odd 2 1 152.1.k.a ✓ 2
8.b even 2 1 152.1.k.a ✓ 2
8.d odd 2 1 CM 608.1.o.a 2
12.b even 2 1 1368.1.bz.a 2
19.c even 3 1 inner 608.1.o.a 2
20.d odd 2 1 3800.1.bd.c 2
20.e even 4 2 3800.1.bn.b 4
24.h odd 2 1 1368.1.bz.a 2
40.f even 2 1 3800.1.bd.c 2
40.i odd 4 2 3800.1.bn.b 4
76.d even 2 1 2888.1.k.a 2
76.f even 6 1 2888.1.f.a 1
76.f even 6 1 2888.1.k.a 2
76.g odd 6 1 152.1.k.a ✓ 2
76.g odd 6 1 2888.1.f.b 1
76.k even 18 6 2888.1.u.d 6
76.l odd 18 6 2888.1.u.c 6
152.g odd 2 1 2888.1.k.a 2
152.k odd 6 1 inner 608.1.o.a 2
152.l odd 6 1 2888.1.f.a 1
152.l odd 6 1 2888.1.k.a 2
152.p even 6 1 152.1.k.a ✓ 2
152.p even 6 1 2888.1.f.b 1
152.s odd 18 6 2888.1.u.d 6
152.t even 18 6 2888.1.u.c 6
228.m even 6 1 1368.1.bz.a 2
380.p odd 6 1 3800.1.bd.c 2
380.v even 12 2 3800.1.bn.b 4
456.x odd 6 1 1368.1.bz.a 2
760.z even 6 1 3800.1.bd.c 2
760.br odd 12 2 3800.1.bn.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a ✓ 2 4.b odd 2 1
152.1.k.a ✓ 2 8.b even 2 1
152.1.k.a ✓ 2 76.g odd 6 1
152.1.k.a ✓ 2 152.p even 6 1
608.1.o.a 2 1.a even 1 1 trivial
608.1.o.a 2 8.d odd 2 1 CM
608.1.o.a 2 19.c even 3 1 inner
608.1.o.a 2 152.k odd 6 1 inner
1368.1.bz.a 2 12.b even 2 1
1368.1.bz.a 2 24.h odd 2 1
1368.1.bz.a 2 228.m even 6 1
1368.1.bz.a 2 456.x odd 6 1
2888.1.f.a 1 76.f even 6 1
2888.1.f.a 1 152.l odd 6 1
2888.1.f.b 1 76.g odd 6 1
2888.1.f.b 1 152.p even 6 1
2888.1.k.a 2 76.d even 2 1
2888.1.k.a 2 76.f even 6 1
2888.1.k.a 2 152.g odd 2 1
2888.1.k.a 2 152.l odd 6 1
2888.1.u.c 6 76.l odd 18 6
2888.1.u.c 6 152.t even 18 6
2888.1.u.d 6 76.k even 18 6
2888.1.u.d 6 152.s odd 18 6
3800.1.bd.c 2 20.d odd 2 1
3800.1.bd.c 2 40.f even 2 1
3800.1.bd.c 2 380.p odd 6 1
3800.1.bd.c 2 760.z even 6 1
3800.1.bn.b 4 20.e even 4 2
3800.1.bn.b 4 40.i odd 4 2
3800.1.bn.b 4 380.v even 12 2
3800.1.bn.b 4 760.br odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(608, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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