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Properties

Label 729.2.e.f
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM discriminant -3
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [729,2,Mod(82,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), 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: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{U}(1)[D_{9}]$

$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{18}^{5} q^{4} - \zeta_{18}^{4} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{18}^{5} q^{4} - \zeta_{18}^{4} q^{7} + 5 \zeta_{18}^{2} q^{13} - 4 \zeta_{18} q^{16} + 7 \zeta_{18}^{3} q^{19} + (5 \zeta_{18}^{4} - 5 \zeta_{18}) q^{25} + 2 q^{28} + 4 \zeta_{18}^{5} q^{31} + (11 \zeta_{18}^{3} - 11) q^{37} - 8 \zeta_{18} q^{43} + ( - 6 \zeta_{18}^{5} + 6 \zeta_{18}^{2}) q^{49} + (10 \zeta_{18}^{4} - 10 \zeta_{18}) q^{52} - \zeta_{18}^{4} q^{61} + ( - 8 \zeta_{18}^{3} + 8) q^{64} + 5 \zeta_{18}^{2} q^{67} + 7 \zeta_{18}^{3} q^{73} + (14 \zeta_{18}^{5} - 14 \zeta_{18}^{2}) q^{76} + ( - 17 \zeta_{18}^{4} + 17 \zeta_{18}) q^{79} + ( - 5 \zeta_{18}^{3} + 5) q^{91} + 19 \zeta_{18} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 21 q^{19} + 12 q^{28} - 33 q^{37} + 24 q^{64} + 21 q^{73} + 15 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{18}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−0.766044 + 0.642788i
−0.173648 + 0.984808i
0.939693 + 0.342020i
0.939693 − 0.342020i
−0.173648 − 0.984808i
−0.766044 − 0.642788i
0 0 1.87939 − 0.684040i 0 0 0.939693 + 0.342020i 0 0 0
163.1 0 0 −1.53209 + 1.28558i 0 0 −0.766044 − 0.642788i 0 0 0
325.1 0 0 −0.347296 + 1.96962i 0 0 −0.173648 − 0.984808i 0 0 0
406.1 0 0 −0.347296 − 1.96962i 0 0 −0.173648 + 0.984808i 0 0 0
568.1 0 0 −1.53209 − 1.28558i 0 0 −0.766044 + 0.642788i 0 0 0
649.1 0 0 1.87939 + 0.684040i 0 0 0.939693 − 0.342020i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
9.c even 3 2 inner
9.d odd 6 2 inner
27.e even 9 3 inner
27.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.f 6
3.b odd 2 1 CM 729.2.e.f 6
9.c even 3 2 inner 729.2.e.f 6
9.d odd 6 2 inner 729.2.e.f 6
27.e even 9 1 27.2.a.a ✓ 1
27.e even 9 2 81.2.c.a 2
27.e even 9 3 inner 729.2.e.f 6
27.f odd 18 1 27.2.a.a ✓ 1
27.f odd 18 2 81.2.c.a 2
27.f odd 18 3 inner 729.2.e.f 6
108.j odd 18 1 432.2.a.e 1
108.j odd 18 2 1296.2.i.i 2
108.l even 18 1 432.2.a.e 1
108.l even 18 2 1296.2.i.i 2
135.n odd 18 1 675.2.a.e 1
135.p even 18 1 675.2.a.e 1
135.q even 36 2 675.2.b.f 2
135.r odd 36 2 675.2.b.f 2
189.y odd 18 1 1323.2.a.i 1
189.be even 18 1 1323.2.a.i 1
216.r odd 18 1 1728.2.a.o 1
216.t even 18 1 1728.2.a.n 1
216.v even 18 1 1728.2.a.o 1
216.x odd 18 1 1728.2.a.n 1
297.o even 18 1 3267.2.a.f 1
297.q odd 18 1 3267.2.a.f 1
351.bi odd 18 1 4563.2.a.e 1
351.bl even 18 1 4563.2.a.e 1
459.r even 18 1 7803.2.a.k 1
459.t odd 18 1 7803.2.a.k 1
513.bw even 18 1 9747.2.a.f 1
513.ca odd 18 1 9747.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a ✓ 1 27.e even 9 1
27.2.a.a ✓ 1 27.f odd 18 1
81.2.c.a 2 27.e even 9 2
81.2.c.a 2 27.f odd 18 2
432.2.a.e 1 108.j odd 18 1
432.2.a.e 1 108.l even 18 1
675.2.a.e 1 135.n odd 18 1
675.2.a.e 1 135.p even 18 1
675.2.b.f 2 135.q even 36 2
675.2.b.f 2 135.r odd 36 2
729.2.e.f 6 1.a even 1 1 trivial
729.2.e.f 6 3.b odd 2 1 CM
729.2.e.f 6 9.c even 3 2 inner
729.2.e.f 6 9.d odd 6 2 inner
729.2.e.f 6 27.e even 9 3 inner
729.2.e.f 6 27.f odd 18 3 inner
1296.2.i.i 2 108.j odd 18 2
1296.2.i.i 2 108.l even 18 2
1323.2.a.i 1 189.y odd 18 1
1323.2.a.i 1 189.be even 18 1
1728.2.a.n 1 216.t even 18 1
1728.2.a.n 1 216.x odd 18 1
1728.2.a.o 1 216.r odd 18 1
1728.2.a.o 1 216.v even 18 1
3267.2.a.f 1 297.o even 18 1
3267.2.a.f 1 297.q odd 18 1
4563.2.a.e 1 351.bi odd 18 1
4563.2.a.e 1 351.bl even 18 1
7803.2.a.k 1 459.r even 18 1
7803.2.a.k 1 459.t odd 18 1
9747.2.a.f 1 513.bw even 18 1
9747.2.a.f 1 513.ca odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{6} - T_{7}^{3} + 1 \) 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
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 125 T^{3} + 15625 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 49)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 64T^{3} + 4096 \) Copy content Toggle raw display
$37$ \( (T^{2} + 11 T + 121)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 512 T^{3} + 262144 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$67$ \( T^{6} + 125 T^{3} + 15625 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T^{2} - 7 T + 49)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} + 4913 T^{3} + 24137569 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 6859 T^{3} + 47045881 \) Copy content Toggle raw display
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