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Properties

Label 1620.1.p.c
Level $1620$
Weight $1$
Character orbit 1620.p
Analytic conductor $0.808$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,1,Mod(379,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.379");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1620.p (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: \(0.808485320465\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.157464000.3

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12} q^{5} - \zeta_{12}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} - \zeta_{12} q^{5} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{10} + (\zeta_{12}^{4} - 1) q^{13} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{3} q^{17} - \zeta_{12}^{3} q^{20} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{26} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{29} - \zeta_{12}^{5} q^{32} - \zeta_{12}^{4} q^{34} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{37} + \zeta_{12}^{4} q^{40} + \zeta_{12}^{2} q^{49} - \zeta_{12}^{3} q^{50} + ( - \zeta_{12}^{2} - 1) q^{52} + \zeta_{12}^{3} q^{53} + ( - \zeta_{12}^{4} + 1) q^{58} + \zeta_{12}^{4} q^{61} - q^{64} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{65} + \zeta_{12}^{5} q^{68} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{73} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{74} - \zeta_{12}^{5} q^{80} - \zeta_{12}^{4} q^{85} + (\zeta_{12}^{5} - \zeta_{12}) q^{89} - \zeta_{12}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{10} - 6 q^{13} - 2 q^{16} + 2 q^{25} + 2 q^{34} - 2 q^{40} + 2 q^{49} - 6 q^{52} + 6 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.866025 − 0.500000i 0 0 − 1.00000i 0 0.500000 + 0.866025i
379.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866025 + 0.500000i 0 0 1.00000i 0 0.500000 + 0.866025i
919.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.866025 + 0.500000i 0 0 1.00000i 0 0.500000 − 0.866025i
919.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.866025 − 0.500000i 0 0 − 1.00000i 0 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner
180.n even 6 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.p.c 4
3.b odd 2 1 inner 1620.1.p.c 4
4.b odd 2 1 CM 1620.1.p.c 4
5.b even 2 1 1620.1.p.e 4
9.c even 3 1 1620.1.f.e ✓ 4
9.c even 3 1 1620.1.p.e 4
9.d odd 6 1 1620.1.f.e ✓ 4
9.d odd 6 1 1620.1.p.e 4
12.b even 2 1 inner 1620.1.p.c 4
15.d odd 2 1 1620.1.p.e 4
20.d odd 2 1 1620.1.p.e 4
36.f odd 6 1 1620.1.f.e ✓ 4
36.f odd 6 1 1620.1.p.e 4
36.h even 6 1 1620.1.f.e ✓ 4
36.h even 6 1 1620.1.p.e 4
45.h odd 6 1 1620.1.f.e ✓ 4
45.h odd 6 1 inner 1620.1.p.c 4
45.j even 6 1 1620.1.f.e ✓ 4
45.j even 6 1 inner 1620.1.p.c 4
60.h even 2 1 1620.1.p.e 4
180.n even 6 1 1620.1.f.e ✓ 4
180.n even 6 1 inner 1620.1.p.c 4
180.p odd 6 1 1620.1.f.e ✓ 4
180.p odd 6 1 inner 1620.1.p.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.1.f.e ✓ 4 9.c even 3 1
1620.1.f.e ✓ 4 9.d odd 6 1
1620.1.f.e ✓ 4 36.f odd 6 1
1620.1.f.e ✓ 4 36.h even 6 1
1620.1.f.e ✓ 4 45.h odd 6 1
1620.1.f.e ✓ 4 45.j even 6 1
1620.1.f.e ✓ 4 180.n even 6 1
1620.1.f.e ✓ 4 180.p odd 6 1
1620.1.p.c 4 1.a even 1 1 trivial
1620.1.p.c 4 3.b odd 2 1 inner
1620.1.p.c 4 4.b odd 2 1 CM
1620.1.p.c 4 12.b even 2 1 inner
1620.1.p.c 4 45.h odd 6 1 inner
1620.1.p.c 4 45.j even 6 1 inner
1620.1.p.c 4 180.n even 6 1 inner
1620.1.p.c 4 180.p odd 6 1 inner
1620.1.p.e 4 5.b even 2 1
1620.1.p.e 4 9.c even 3 1
1620.1.p.e 4 9.d odd 6 1
1620.1.p.e 4 15.d odd 2 1
1620.1.p.e 4 20.d odd 2 1
1620.1.p.e 4 36.f odd 6 1
1620.1.p.e 4 36.h even 6 1
1620.1.p.e 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1620, [\chi])\):

\( T_{13}^{2} + 3T_{13} + 3 \) Copy content Toggle raw display
\( T_{17}^{2} + 1 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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