Properties

Label 1620.2.r.d
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - \beta_1) q^{5} + 2 \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - \beta_1) q^{5} + 2 \beta_1 q^{7} + (4 \beta_{2} - 4) q^{11} + 2 \beta_{3} q^{17} + (2 \beta_{3} - 2 \beta_1) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{25} + ( - 6 \beta_{2} + 6) q^{29} - 4 \beta_{2} q^{31} + (2 \beta_{3} - 8) q^{35} + 4 \beta_{3} q^{37} - 10 \beta_{2} q^{41} - 2 \beta_1 q^{43} + 2 \beta_1 q^{47} + 9 \beta_{2} q^{49} + 6 \beta_{3} q^{53} + ( - 4 \beta_{3} - 4) q^{55} - 4 \beta_{2} q^{59} + (2 \beta_{2} - 2) q^{61} + ( - 2 \beta_{3} + 2 \beta_1) q^{67} - 4 \beta_{3} q^{73} + (8 \beta_{3} - 8 \beta_1) q^{77} + (12 \beta_{2} - 12) q^{79} + 2 \beta_1 q^{83} + (2 \beta_{3} - 8 \beta_{2} - 2 \beta_1) q^{85} - 10 q^{89} + 4 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 8 q^{11} + 6 q^{25} + 12 q^{29} - 8 q^{31} - 32 q^{35} - 20 q^{41} + 18 q^{49} - 16 q^{55} - 8 q^{59} - 4 q^{61} - 24 q^{79} - 16 q^{85} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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 −1.23205 1.86603i 0 3.46410 2.00000i 0 0 0
109.2 0 0 0 2.23205 + 0.133975i 0 −3.46410 + 2.00000i 0 0 0
1189.1 0 0 0 −1.23205 + 1.86603i 0 3.46410 + 2.00000i 0 0 0
1189.2 0 0 0 2.23205 0.133975i 0 −3.46410 2.00000i 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.d 4
3.b odd 2 1 1620.2.r.c 4
5.b even 2 1 inner 1620.2.r.d 4
9.c even 3 1 180.2.d.a 2
9.c even 3 1 inner 1620.2.r.d 4
9.d odd 6 1 60.2.d.a 2
9.d odd 6 1 1620.2.r.c 4
15.d odd 2 1 1620.2.r.c 4
36.f odd 6 1 720.2.f.c 2
36.h even 6 1 240.2.f.b 2
45.h odd 6 1 60.2.d.a 2
45.h odd 6 1 1620.2.r.c 4
45.j even 6 1 180.2.d.a 2
45.j even 6 1 inner 1620.2.r.d 4
45.k odd 12 1 900.2.a.a 1
45.k odd 12 1 900.2.a.h 1
45.l even 12 1 300.2.a.a 1
45.l even 12 1 300.2.a.d 1
63.i even 6 1 2940.2.bb.e 4
63.j odd 6 1 2940.2.bb.d 4
63.n odd 6 1 2940.2.bb.d 4
63.o even 6 1 2940.2.k.c 2
63.s even 6 1 2940.2.bb.e 4
72.j odd 6 1 960.2.f.f 2
72.l even 6 1 960.2.f.c 2
72.n even 6 1 2880.2.f.l 2
72.p odd 6 1 2880.2.f.p 2
144.u even 12 1 3840.2.d.b 2
144.u even 12 1 3840.2.d.be 2
144.w odd 12 1 3840.2.d.o 2
144.w odd 12 1 3840.2.d.r 2
180.n even 6 1 240.2.f.b 2
180.p odd 6 1 720.2.f.c 2
180.v odd 12 1 1200.2.a.a 1
180.v odd 12 1 1200.2.a.s 1
180.x even 12 1 3600.2.a.d 1
180.x even 12 1 3600.2.a.bm 1
315.u even 6 1 2940.2.bb.e 4
315.v odd 6 1 2940.2.bb.d 4
315.z even 6 1 2940.2.k.c 2
315.bq even 6 1 2940.2.bb.e 4
315.br odd 6 1 2940.2.bb.d 4
360.z odd 6 1 2880.2.f.p 2
360.bd even 6 1 960.2.f.c 2
360.bh odd 6 1 960.2.f.f 2
360.bk even 6 1 2880.2.f.l 2
360.br even 12 1 4800.2.a.bj 1
360.br even 12 1 4800.2.a.bn 1
360.bt odd 12 1 4800.2.a.bf 1
360.bt odd 12 1 4800.2.a.bk 1
720.ch odd 12 1 3840.2.d.o 2
720.ch odd 12 1 3840.2.d.r 2
720.da even 12 1 3840.2.d.b 2
720.da even 12 1 3840.2.d.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 9.d odd 6 1
60.2.d.a 2 45.h odd 6 1
180.2.d.a 2 9.c even 3 1
180.2.d.a 2 45.j even 6 1
240.2.f.b 2 36.h even 6 1
240.2.f.b 2 180.n even 6 1
300.2.a.a 1 45.l even 12 1
300.2.a.d 1 45.l even 12 1
720.2.f.c 2 36.f odd 6 1
720.2.f.c 2 180.p odd 6 1
900.2.a.a 1 45.k odd 12 1
900.2.a.h 1 45.k odd 12 1
960.2.f.c 2 72.l even 6 1
960.2.f.c 2 360.bd even 6 1
960.2.f.f 2 72.j odd 6 1
960.2.f.f 2 360.bh odd 6 1
1200.2.a.a 1 180.v odd 12 1
1200.2.a.s 1 180.v odd 12 1
1620.2.r.c 4 3.b odd 2 1
1620.2.r.c 4 9.d odd 6 1
1620.2.r.c 4 15.d odd 2 1
1620.2.r.c 4 45.h odd 6 1
1620.2.r.d 4 1.a even 1 1 trivial
1620.2.r.d 4 5.b even 2 1 inner
1620.2.r.d 4 9.c even 3 1 inner
1620.2.r.d 4 45.j even 6 1 inner
2880.2.f.l 2 72.n even 6 1
2880.2.f.l 2 360.bk even 6 1
2880.2.f.p 2 72.p odd 6 1
2880.2.f.p 2 360.z odd 6 1
2940.2.k.c 2 63.o even 6 1
2940.2.k.c 2 315.z even 6 1
2940.2.bb.d 4 63.j odd 6 1
2940.2.bb.d 4 63.n odd 6 1
2940.2.bb.d 4 315.v odd 6 1
2940.2.bb.d 4 315.br odd 6 1
2940.2.bb.e 4 63.i even 6 1
2940.2.bb.e 4 63.s even 6 1
2940.2.bb.e 4 315.u even 6 1
2940.2.bb.e 4 315.bq even 6 1
3600.2.a.d 1 180.x even 12 1
3600.2.a.bm 1 180.x even 12 1
3840.2.d.b 2 144.u even 12 1
3840.2.d.b 2 720.da even 12 1
3840.2.d.o 2 144.w odd 12 1
3840.2.d.o 2 720.ch odd 12 1
3840.2.d.r 2 144.w odd 12 1
3840.2.d.r 2 720.ch odd 12 1
3840.2.d.be 2 144.u even 12 1
3840.2.d.be 2 720.da even 12 1
4800.2.a.bf 1 360.bt odd 12 1
4800.2.a.bj 1 360.br even 12 1
4800.2.a.bk 1 360.bt odd 12 1
4800.2.a.bn 1 360.br even 12 1

Hecke kernels

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

\( T_{7}^{4} - 16T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) 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
$5$ \( T^{4} - 2 T^{3} - T^{2} - 10 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) 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
$43$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T + 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
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