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Properties

Label 1620.2.r.f
Level $1620$
Weight $2$
Character orbit 1620.r
Analytic conductor $12.936$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(109,1620)]
 
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([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.r (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: \(12.9357651274\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + 2 \zeta_{12} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + 2 \zeta_{12} q^{7} + (2 \zeta_{12}^{2} - 2) q^{11} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{13} + \zeta_{12}^{3} q^{17} + 3 q^{19} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{23} + (3 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{25} + (6 \zeta_{12}^{2} - 6) q^{29} - \zeta_{12}^{2} q^{31} + (4 \zeta_{12}^{3} + 2) q^{35} - 8 \zeta_{12}^{3} q^{37} + 10 \zeta_{12}^{2} q^{41} - 2 \zeta_{12} q^{43} - 8 \zeta_{12} q^{47} - 3 \zeta_{12}^{2} q^{49} - 9 \zeta_{12}^{3} q^{53} + (2 \zeta_{12}^{3} - 4) q^{55} + 10 \zeta_{12}^{2} q^{59} + (5 \zeta_{12}^{2} - 5) q^{61} + (6 \zeta_{12}^{2} - 12 \zeta_{12} - 6) q^{65} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{67} + 6 q^{71} + 2 \zeta_{12}^{3} q^{73} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{77} + ( - 15 \zeta_{12}^{2} + 15) q^{79} + 13 \zeta_{12} q^{83} + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{85} + \cdots - 8 \zeta_{12} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} - 4 q^{11} + 12 q^{19} - 6 q^{25} - 12 q^{29} - 2 q^{31} + 8 q^{35} + 20 q^{41} - 6 q^{49} - 16 q^{55} + 20 q^{59} - 10 q^{61} - 12 q^{65} + 24 q^{71} + 30 q^{79} + 2 q^{85} - 8 q^{89} - 48 q^{91} + 12 q^{95}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
0 0 0 0.133975 − 2.23205i 0 −1.73205 + 1.00000i 0 0 0
109.2 0 0 0 1.86603 − 1.23205i 0 1.73205 − 1.00000i 0 0 0
1189.1 0 0 0 0.133975 + 2.23205i 0 −1.73205 − 1.00000i 0 0 0
1189.2 0 0 0 1.86603 + 1.23205i 0 1.73205 + 1.00000i 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
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.r.f 4
3.b odd 2 1 1620.2.r.a 4
5.b even 2 1 inner 1620.2.r.f 4
9.c even 3 1 540.2.d.a ✓ 2
9.c even 3 1 inner 1620.2.r.f 4
9.d odd 6 1 540.2.d.b yes 2
9.d odd 6 1 1620.2.r.a 4
15.d odd 2 1 1620.2.r.a 4
36.f odd 6 1 2160.2.f.a 2
36.h even 6 1 2160.2.f.h 2
45.h odd 6 1 540.2.d.b yes 2
45.h odd 6 1 1620.2.r.a 4
45.j even 6 1 540.2.d.a ✓ 2
45.j even 6 1 inner 1620.2.r.f 4
45.k odd 12 1 2700.2.a.h 1
45.k odd 12 1 2700.2.a.p 1
45.l even 12 1 2700.2.a.e 1
45.l even 12 1 2700.2.a.o 1
180.n even 6 1 2160.2.f.h 2
180.p odd 6 1 2160.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.a ✓ 2 9.c even 3 1
540.2.d.a ✓ 2 45.j even 6 1
540.2.d.b yes 2 9.d odd 6 1
540.2.d.b yes 2 45.h odd 6 1
1620.2.r.a 4 3.b odd 2 1
1620.2.r.a 4 9.d odd 6 1
1620.2.r.a 4 15.d odd 2 1
1620.2.r.a 4 45.h odd 6 1
1620.2.r.f 4 1.a even 1 1 trivial
1620.2.r.f 4 5.b even 2 1 inner
1620.2.r.f 4 9.c even 3 1 inner
1620.2.r.f 4 45.j even 6 1 inner
2160.2.f.a 2 36.f odd 6 1
2160.2.f.a 2 180.p odd 6 1
2160.2.f.h 2 36.h even 6 1
2160.2.f.h 2 180.n even 6 1
2700.2.a.e 1 45.l even 12 1
2700.2.a.h 1 45.k odd 12 1
2700.2.a.o 1 45.l even 12 1
2700.2.a.p 1 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T - 3)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 225)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
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