Properties

Label 2160.2.q.j
Level $2160$
Weight $2$
Character orbit 2160.q
Analytic conductor $17.248$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,2,Mod(721,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("2160.721");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.q (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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 360)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{7} + (\beta_{3} - \beta_1) q^{11} + (2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{13} + ( - 2 \beta_{4} + \beta_{2} - 1) q^{17} + ( - \beta_{2} + 1) q^{19} + (3 \beta_{5} - 3 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{23} - \beta_{3} q^{25} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3}) q^{29} + (5 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{31} + (\beta_{4} + \beta_{2} - 1) q^{35} + (\beta_{2} + 1) q^{37} + (\beta_{2} + \beta_1) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1) q^{43} + ( - \beta_{5} + \beta_{4} + 6 \beta_{3} - 4 \beta_1) q^{47} + (2 \beta_{5} + \beta_{3} + 5 \beta_{2} + 5 \beta_1 - 1) q^{49} + (2 \beta_{4} + 4) q^{53} + (\beta_{2} + 1) q^{55} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{59} + (4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_1) q^{61} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1) q^{65} + (5 \beta_{5} - 2 \beta_{3} + 2) q^{67} + (2 \beta_{4} + 3 \beta_{2} - 3) q^{71} + ( - 4 \beta_{4} + 4) q^{73} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{77} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 2 \beta_1) q^{79} + (\beta_{5} - \beta_{4} + 7 \beta_{3} - \beta_1) q^{83} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{85} + 11 q^{89} + ( - 3 \beta_{2} + 9) q^{91} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{95} + (2 \beta_{5} - 2 \beta_{4} - 12 \beta_{3} + 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} - 5 q^{7} + 2 q^{11} - 4 q^{17} + 8 q^{19} + 7 q^{23} - 3 q^{25} - 7 q^{29} - 16 q^{31} - 10 q^{35} + 4 q^{37} - q^{41} + 2 q^{43} + 13 q^{47} - 10 q^{49} + 20 q^{53} + 4 q^{55} + 6 q^{59} + 11 q^{61} + q^{67} - 28 q^{71} + 32 q^{73} - 12 q^{77} - 6 q^{79} + 21 q^{83} - 2 q^{85} + 66 q^{89} + 60 q^{91} + 4 q^{95} - 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
−1.62241 + 0.606458i
0.403374 1.68443i
1.71903 + 0.211943i
−1.62241 0.606458i
0.403374 + 1.68443i
1.71903 0.211943i
0 0 0 0.500000 + 0.866025i 0 −2.62241 + 4.54214i 0 0 0
721.2 0 0 0 0.500000 + 0.866025i 0 −0.596626 + 1.03339i 0 0 0
721.3 0 0 0 0.500000 + 0.866025i 0 0.719035 1.24540i 0 0 0
1441.1 0 0 0 0.500000 0.866025i 0 −2.62241 4.54214i 0 0 0
1441.2 0 0 0 0.500000 0.866025i 0 −0.596626 1.03339i 0 0 0
1441.3 0 0 0 0.500000 0.866025i 0 0.719035 + 1.24540i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.3
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 2160.2.q.j 6
3.b odd 2 1 720.2.q.j 6
4.b odd 2 1 1080.2.q.d 6
9.c even 3 1 inner 2160.2.q.j 6
9.c even 3 1 6480.2.a.bu 3
9.d odd 6 1 720.2.q.j 6
9.d odd 6 1 6480.2.a.bx 3
12.b even 2 1 360.2.q.d 6
36.f odd 6 1 1080.2.q.d 6
36.f odd 6 1 3240.2.a.q 3
36.h even 6 1 360.2.q.d 6
36.h even 6 1 3240.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.d 6 12.b even 2 1
360.2.q.d 6 36.h even 6 1
720.2.q.j 6 3.b odd 2 1
720.2.q.j 6 9.d odd 6 1
1080.2.q.d 6 4.b odd 2 1
1080.2.q.d 6 36.f odd 6 1
2160.2.q.j 6 1.a even 1 1 trivial
2160.2.q.j 6 9.c even 3 1 inner
3240.2.a.q 3 36.f odd 6 1
3240.2.a.r 3 36.h even 6 1
6480.2.a.bu 3 9.c even 3 1
6480.2.a.bx 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{6} + 5T_{7}^{5} + 28T_{7}^{4} + 3T_{7}^{3} + 54T_{7}^{2} + 27T_{7} + 81 \) Copy content Toggle raw display
\( T_{11}^{6} - 2T_{11}^{5} + 12T_{11}^{4} - 8T_{11}^{3} + 88T_{11}^{2} - 96T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 5 T^{5} + 28 T^{4} + 3 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{6} + 24 T^{4} + 72 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 36 T - 108)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 7 T^{5} + 100 T^{4} + \cdots + 91809 \) Copy content Toggle raw display
$29$ \( T^{6} + 7 T^{5} + 54 T^{4} + 19 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( T^{6} + 16 T^{5} + 180 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + T^{5} + 10 T^{4} - 15 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{6} - 2 T^{5} + 40 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$47$ \( T^{6} - 13 T^{5} + 254 T^{4} + \cdots + 1261129 \) Copy content Toggle raw display
$53$ \( (T^{3} - 10 T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + 96 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{6} - 11 T^{5} + 162 T^{4} + \cdots + 269361 \) Copy content Toggle raw display
$67$ \( T^{6} - T^{5} + 134 T^{4} + \cdots + 12769 \) Copy content Toggle raw display
$71$ \( (T^{3} + 14 T^{2} - 20 T - 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 16 T^{2} + 384)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + 120 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$83$ \( T^{6} - 21 T^{5} + 312 T^{4} + \cdots + 59049 \) Copy content Toggle raw display
$89$ \( (T - 11)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 30 T^{5} + 744 T^{4} + \cdots + 1201216 \) Copy content Toggle raw display
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