Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(541,1620)]
 
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, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + ( - 6 \zeta_{6} + 6) q^{11} + \zeta_{6} q^{13} - q^{19} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 6 \zeta_{6} + 6) q^{29} - 8 \zeta_{6} q^{31} + q^{35} - 7 q^{37} - 6 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + ( - 12 \zeta_{6} + 12) q^{47} + 6 \zeta_{6} q^{49} + 6 q^{53} + 6 q^{55} + (11 \zeta_{6} - 11) q^{61} + (\zeta_{6} - 1) q^{65} + 7 \zeta_{6} q^{67} + 6 q^{71} + 11 q^{73} - 6 \zeta_{6} q^{77} + ( - \zeta_{6} + 1) q^{79} + ( - 6 \zeta_{6} + 6) q^{83} + 12 q^{89} + q^{91} - \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + q^{7} + 6 q^{11} + q^{13} - 2 q^{19} + 6 q^{23} - q^{25} + 6 q^{29} - 8 q^{31} + 2 q^{35} - 14 q^{37} - 6 q^{41} + 4 q^{43} + 12 q^{47} + 6 q^{49} + 12 q^{53} + 12 q^{55} - 11 q^{61} - q^{65} + 7 q^{67} + 12 q^{71} + 22 q^{73} - 6 q^{77} + q^{79} + 6 q^{83} + 24 q^{89} + 2 q^{91} - q^{95} + 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.j 2
3.b odd 2 1 1620.2.i.d 2
9.c even 3 1 540.2.a.b 1
9.c even 3 1 inner 1620.2.i.j 2
9.d odd 6 1 540.2.a.e yes 1
9.d odd 6 1 1620.2.i.d 2
36.f odd 6 1 2160.2.a.h 1
36.h even 6 1 2160.2.a.s 1
45.h odd 6 1 2700.2.a.m 1
45.j even 6 1 2700.2.a.k 1
45.k odd 12 2 2700.2.d.a 2
45.l even 12 2 2700.2.d.k 2
72.j odd 6 1 8640.2.a.l 1
72.l even 6 1 8640.2.a.s 1
72.n even 6 1 8640.2.a.br 1
72.p odd 6 1 8640.2.a.bu 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 9.c even 3 1
540.2.a.e yes 1 9.d odd 6 1
1620.2.i.d 2 3.b odd 2 1
1620.2.i.d 2 9.d odd 6 1
1620.2.i.j 2 1.a even 1 1 trivial
1620.2.i.j 2 9.c even 3 1 inner
2160.2.a.h 1 36.f odd 6 1
2160.2.a.s 1 36.h even 6 1
2700.2.a.k 1 45.j even 6 1
2700.2.a.m 1 45.h odd 6 1
2700.2.d.a 2 45.k odd 12 2
2700.2.d.k 2 45.l even 12 2
8640.2.a.l 1 72.j odd 6 1
8640.2.a.s 1 72.l even 6 1
8640.2.a.br 1 72.n even 6 1
8640.2.a.bu 1 72.p odd 6 1

Hecke kernels

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

\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( (T + 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
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