Properties

Label 567.2.g.i
Level $567$
Weight $2$
Character orbit 567.g
Analytic conductor $4.528$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(109,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.g (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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 189)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4} + 1) q^{2} + (\beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 1) q^{7} + ( - 2 \beta_{3} - \beta_1 - 4) q^{8}+ \cdots + ( - 5 \beta_{3} - \beta_{2} + \cdots - 10) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 2 q^{7} - 18 q^{8} + q^{10} - 14 q^{11} - 2 q^{13} + 4 q^{14} - 10 q^{16} - 5 q^{19} - 13 q^{20} + 4 q^{22} - 12 q^{23} - 4 q^{25} + 17 q^{26} - 30 q^{28} + 13 q^{29}+ \cdots - 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-\beta_{4}\) \(-1 + \beta_{4}\)

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
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
−0.730252 + 1.26483i 0 −0.0665372 0.115246i 0.593579 0 −2.25729 1.38008i −2.72665 0 −0.433463 + 0.750780i
109.2 0.380438 0.658939i 0 0.710533 + 1.23068i −3.18194 0 1.85185 + 1.88962i 2.60301 0 −1.21053 + 2.09671i
109.3 1.34981 2.33795i 0 −2.64400 4.57954i 1.58836 0 1.40545 2.24159i −8.87636 0 2.14400 3.71351i
541.1 −0.730252 1.26483i 0 −0.0665372 + 0.115246i 0.593579 0 −2.25729 + 1.38008i −2.72665 0 −0.433463 0.750780i
541.2 0.380438 + 0.658939i 0 0.710533 1.23068i −3.18194 0 1.85185 1.88962i 2.60301 0 −1.21053 2.09671i
541.3 1.34981 + 2.33795i 0 −2.64400 + 4.57954i 1.58836 0 1.40545 + 2.24159i −8.87636 0 2.14400 + 3.71351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.g.i 6
3.b odd 2 1 567.2.g.h 6
7.c even 3 1 567.2.h.h 6
9.c even 3 1 189.2.e.f yes 6
9.c even 3 1 567.2.h.h 6
9.d odd 6 1 189.2.e.e 6
9.d odd 6 1 567.2.h.i 6
21.h odd 6 1 567.2.h.i 6
63.g even 3 1 inner 567.2.g.i 6
63.g even 3 1 1323.2.a.x 3
63.h even 3 1 189.2.e.f yes 6
63.j odd 6 1 189.2.e.e 6
63.k odd 6 1 1323.2.a.y 3
63.n odd 6 1 567.2.g.h 6
63.n odd 6 1 1323.2.a.ba 3
63.s even 6 1 1323.2.a.z 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.e 6 9.d odd 6 1
189.2.e.e 6 63.j odd 6 1
189.2.e.f yes 6 9.c even 3 1
189.2.e.f yes 6 63.h even 3 1
567.2.g.h 6 3.b odd 2 1
567.2.g.h 6 63.n odd 6 1
567.2.g.i 6 1.a even 1 1 trivial
567.2.g.i 6 63.g even 3 1 inner
567.2.h.h 6 7.c even 3 1
567.2.h.h 6 9.c even 3 1
567.2.h.i 6 9.d odd 6 1
567.2.h.i 6 21.h odd 6 1
1323.2.a.x 3 63.g even 3 1
1323.2.a.y 3 63.k odd 6 1
1323.2.a.z 3 63.s even 6 1
1323.2.a.ba 3 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{6} - 2T_{2}^{5} + 7T_{2}^{4} + 15T_{2}^{2} - 9T_{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{6} + 2T_{13}^{5} + 23T_{13}^{4} + 56T_{13}^{3} + 455T_{13}^{2} + 893T_{13} + 2209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} + T^{2} - 6 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( (T^{3} + 7 T^{2} + 12 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 2209 \) Copy content Toggle raw display
$17$ \( T^{6} + 33 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{6} + 5 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( (T^{3} + 6 T^{2} + 3 T - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} - 13 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + \cdots + 4761 \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + \cdots + 8649 \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + \cdots + 149769 \) Copy content Toggle raw display
$43$ \( T^{6} - 9 T^{5} + \cdots + 10201 \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$53$ \( T^{6} - 24 T^{5} + \cdots + 59049 \) Copy content Toggle raw display
$59$ \( T^{6} + 15 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$61$ \( T^{6} - T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$67$ \( T^{6} + 14 T^{5} + \cdots + 961 \) Copy content Toggle raw display
$71$ \( (T^{3} - 3 T^{2} + \cdots - 243)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 7 T^{5} + \cdots + 962361 \) Copy content Toggle raw display
$79$ \( T^{6} + 6 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$83$ \( T^{6} - 3 T^{5} + \cdots + 531441 \) Copy content Toggle raw display
$89$ \( T^{6} + 5 T^{5} + \cdots + 239121 \) Copy content Toggle raw display
$97$ \( T^{6} - 14 T^{5} + \cdots + 576 \) Copy content Toggle raw display
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