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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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
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

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.

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

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.

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 649.1
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 \(S_{2}^{\mathrm{new}}(729, [\chi])\):

\( T_{2} \)
\( T_{5} \)
\( T_{7}^{6} - T_{7}^{3} + 1 \)

Hecke characteristic polynomials

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