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Properties

Label 720.2.q.k
Level $720$
Weight $2$
Character orbit 720.q
Analytic conductor $5.749$
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] = [720,2,Mod(241,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, 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("720.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.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: \(5.74922894553\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 180)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{2} + 1) q^{5} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{5} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{2} + 1) q^{5} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{5} - 6 \beta_{4} + \beta_{3} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 3 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 3 q^{5} + 3 q^{7} + 5 q^{9} - 6 q^{13} - q^{15} - 12 q^{19} - 20 q^{21} - 3 q^{23} - 3 q^{25} - 2 q^{27} + 3 q^{29} + 6 q^{31} + 12 q^{33} + 6 q^{35} + 24 q^{37} + 20 q^{39} - 3 q^{41} + 6 q^{43} - 2 q^{45} + 15 q^{47} - 18 q^{49} - 30 q^{51} - 12 q^{53} - 32 q^{57} + 6 q^{59} - 21 q^{61} + 29 q^{63} + 6 q^{65} + 9 q^{67} + 15 q^{69} - 48 q^{71} + 48 q^{73} - 2 q^{75} - 6 q^{77} + 6 q^{79} + 29 q^{81} - 21 q^{83} - 42 q^{87} - 18 q^{89} + 16 q^{93} - 6 q^{95} - 18 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 5\nu^{4} - \nu^{3} - 9\nu^{2} + 6\nu + 45 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{5} + \nu^{4} + 2\nu^{3} - 12\nu - 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - \nu^{4} + \nu^{3} + 6\nu^{2} + 3\nu - 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - \nu^{4} - 2\nu^{3} + 3\nu^{2} - 6\nu - 9 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} + 7\nu^{4} - 13\nu^{3} - 9\nu^{2} - 3\nu - 90 ) / 27 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 2\beta_{4} + \beta_{3} - 4\beta_{2} - 4\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} - 5\beta_{4} + 2\beta_{3} + 10\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - \beta_{2} - 10\beta _1 + 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 5\beta_{4} + 11\beta_{3} + 19\beta_{2} + 10\beta _1 + 38 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
−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 −1.62241 − 0.606458i 0 0.500000 + 0.866025i 0 2.05042 − 3.55142i 0 2.26442 + 1.96784i 0
241.2 0 0.403374 + 1.68443i 0 0.500000 + 0.866025i 0 −1.91751 + 3.32123i 0 −2.67458 + 1.35891i 0
241.3 0 1.71903 − 0.211943i 0 0.500000 + 0.866025i 0 1.36710 − 2.36788i 0 2.91016 − 0.728674i 0
481.1 0 −1.62241 + 0.606458i 0 0.500000 − 0.866025i 0 2.05042 + 3.55142i 0 2.26442 − 1.96784i 0
481.2 0 0.403374 − 1.68443i 0 0.500000 − 0.866025i 0 −1.91751 − 3.32123i 0 −2.67458 − 1.35891i 0
481.3 0 1.71903 + 0.211943i 0 0.500000 − 0.866025i 0 1.36710 + 2.36788i 0 2.91016 + 0.728674i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.3
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 720.2.q.k 6
3.b odd 2 1 2160.2.q.i 6
4.b odd 2 1 180.2.i.b ✓ 6
9.c even 3 1 inner 720.2.q.k 6
9.c even 3 1 6480.2.a.bt 3
9.d odd 6 1 2160.2.q.i 6
9.d odd 6 1 6480.2.a.bw 3
12.b even 2 1 540.2.i.b 6
20.d odd 2 1 900.2.i.c 6
20.e even 4 2 900.2.s.c 12
36.f odd 6 1 180.2.i.b ✓ 6
36.f odd 6 1 1620.2.a.i 3
36.h even 6 1 540.2.i.b 6
36.h even 6 1 1620.2.a.j 3
60.h even 2 1 2700.2.i.c 6
60.l odd 4 2 2700.2.s.c 12
180.n even 6 1 2700.2.i.c 6
180.n even 6 1 8100.2.a.u 3
180.p odd 6 1 900.2.i.c 6
180.p odd 6 1 8100.2.a.v 3
180.v odd 12 2 2700.2.s.c 12
180.v odd 12 2 8100.2.d.o 6
180.x even 12 2 900.2.s.c 12
180.x even 12 2 8100.2.d.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.b ✓ 6 4.b odd 2 1
180.2.i.b ✓ 6 36.f odd 6 1
540.2.i.b 6 12.b even 2 1
540.2.i.b 6 36.h even 6 1
720.2.q.k 6 1.a even 1 1 trivial
720.2.q.k 6 9.c even 3 1 inner
900.2.i.c 6 20.d odd 2 1
900.2.i.c 6 180.p odd 6 1
900.2.s.c 12 20.e even 4 2
900.2.s.c 12 180.x even 12 2
1620.2.a.i 3 36.f odd 6 1
1620.2.a.j 3 36.h even 6 1
2160.2.q.i 6 3.b odd 2 1
2160.2.q.i 6 9.d odd 6 1
2700.2.i.c 6 60.h even 2 1
2700.2.i.c 6 180.n even 6 1
2700.2.s.c 12 60.l odd 4 2
2700.2.s.c 12 180.v odd 12 2
6480.2.a.bt 3 9.c even 3 1
6480.2.a.bw 3 9.d odd 6 1
8100.2.a.u 3 180.n even 6 1
8100.2.a.v 3 180.p odd 6 1
8100.2.d.o 6 180.v odd 12 2
8100.2.d.p 6 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{6} - 3T_{7}^{5} + 24T_{7}^{4} - 41T_{7}^{3} + 354T_{7}^{2} - 645T_{7} + 1849 \) Copy content Toggle raw display
\( T_{11}^{6} + 24T_{11}^{4} + 72T_{11}^{3} + 576T_{11}^{2} + 864T_{11} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{4} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{6} + 6 T^{5} + \cdots + 5776 \) Copy content Toggle raw display
$17$ \( (T^{3} - 24 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 12 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{6} - 3 T^{5} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( (T^{3} - 12 T^{2} + \cdots + 436)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 3 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{6} - 6 T^{5} + \cdots + 5776 \) Copy content Toggle raw display
$47$ \( T^{6} - 15 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 60 T + 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$61$ \( T^{6} + 21 T^{5} + \cdots + 167281 \) Copy content Toggle raw display
$67$ \( T^{6} - 9 T^{5} + \cdots + 22801 \) Copy content Toggle raw display
$71$ \( (T^{3} + 24 T^{2} + \cdots - 324)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{6} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$83$ \( T^{6} + 21 T^{5} + \cdots + 59049 \) Copy content Toggle raw display
$89$ \( (T + 3)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} + \cdots + 179776 \) Copy content Toggle raw display
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