Properties

Label 729.2.e.o
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
CM no
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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 81)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + (\beta_{7} - \beta_{2}) q^{4} + \beta_{4} q^{5} - 2 \beta_1 q^{7} - \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + (\beta_{7} - \beta_{2}) q^{4} + \beta_{4} q^{5} - 2 \beta_1 q^{7} - \beta_{6} q^{8} + ( - 3 \beta_{3} + 3) q^{10} + (2 \beta_{11} - 2 \beta_{8}) q^{11} + \beta_{7} q^{13} + ( - 2 \beta_{9} + 2 \beta_{4}) q^{14} + (5 \beta_{5} - 5 \beta_1) q^{16} + (3 \beta_{10} - 3 \beta_{6}) q^{17} - 2 \beta_{3} q^{19} - \beta_{11} q^{20} - 6 \beta_{2} q^{22} + 2 \beta_{9} q^{23} - 2 \beta_{5} q^{25} + \beta_{10} q^{26} + 2 q^{28} - \beta_{8} q^{29} + (8 \beta_{7} - 8 \beta_{2}) q^{31} + 3 \beta_{4} q^{32} - 9 \beta_1 q^{34} - 2 \beta_{6} q^{35} + ( - 7 \beta_{3} + 7) q^{37} + ( - 2 \beta_{11} + 2 \beta_{8}) q^{38} - 3 \beta_{7} q^{40} + (4 \beta_{9} - 4 \beta_{4}) q^{41} + ( - 2 \beta_{5} + 2 \beta_1) q^{43} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{44} - 6 \beta_{3} q^{46} + 4 \beta_{11} q^{47} - 3 \beta_{2} q^{49} - 2 \beta_{9} q^{50} - \beta_{5} q^{52} + 6 q^{55} + 2 \beta_{8} q^{56} + (3 \beta_{7} - 3 \beta_{2}) q^{58} - 8 \beta_{4} q^{59} + 7 \beta_1 q^{61} + 8 \beta_{6} q^{62} + (\beta_{3} - 1) q^{64} + ( - \beta_{11} + \beta_{8}) q^{65} + 10 \beta_{7} q^{67} + ( - 3 \beta_{9} + 3 \beta_{4}) q^{68} + (6 \beta_{5} - 6 \beta_1) q^{70} + ( - 6 \beta_{10} + 6 \beta_{6}) q^{71} + 7 \beta_{3} q^{73} - 7 \beta_{11} q^{74} + 2 \beta_{2} q^{76} - 4 \beta_{9} q^{77} + 2 \beta_{5} q^{79} - 5 \beta_{10} q^{80} - 12 q^{82} + 8 \beta_{8} q^{83} + ( - 9 \beta_{7} + 9 \beta_{2}) q^{85} - 2 \beta_{4} q^{86} - 6 \beta_1 q^{88} - 3 \beta_{6} q^{89} + ( - 2 \beta_{3} + 2) q^{91} + ( - 2 \beta_{11} + 2 \beta_{8}) q^{92} - 12 \beta_{7} q^{94} + (2 \beta_{9} - 2 \beta_{4}) q^{95} + ( - 2 \beta_{5} + 2 \beta_1) q^{97} + ( - 3 \beta_{10} + 3 \beta_{6}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{10} - 12 q^{19} + 24 q^{28} + 42 q^{37} - 36 q^{46} + 72 q^{55} - 6 q^{64} + 42 q^{73} - 144 q^{82} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{36}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{36}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{36}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{36}^{7} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{36}^{8} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{36}^{9} + \zeta_{36}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{36}^{10} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \zeta_{36}^{11} + \zeta_{36}^{5} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{36}^{7} + 2\zeta_{36} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{36}^{9} + 2\zeta_{36}^{3} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{36}^{11} + 2\zeta_{36}^{5} \) Copy content Toggle raw display
\(\zeta_{36}\)\(=\) \( ( \beta_{9} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{36}^{3}\)\(=\) \( ( \beta_{10} + \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{36}^{5}\)\(=\) \( ( \beta_{11} + \beta_{8} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{36}^{7}\)\(=\) \( ( -\beta_{9} + 2\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{8}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{36}^{9}\)\(=\) \( ( -\beta_{10} + 2\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{10}\)\(=\) \( \beta_{7} \) Copy content Toggle raw display
\(\zeta_{36}^{11}\)\(=\) \( ( -\beta_{11} + 2\beta_{8} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{2} + \beta_{7}\)

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
0.984808 + 0.173648i
−0.984808 0.173648i
0.342020 0.939693i
−0.342020 + 0.939693i
−0.642788 + 0.766044i
0.642788 0.766044i
−0.642788 0.766044i
0.642788 + 0.766044i
0.342020 + 0.939693i
−0.342020 0.939693i
0.984808 0.173648i
−0.984808 + 0.173648i
−0.300767 1.70574i 0 −0.939693 + 0.342020i 1.32683 + 1.11334i 0 −1.87939 0.684040i −0.866025 1.50000i 0 1.50000 2.59808i
82.2 0.300767 + 1.70574i 0 −0.939693 + 0.342020i −1.32683 1.11334i 0 −1.87939 0.684040i 0.866025 + 1.50000i 0 1.50000 2.59808i
163.1 −1.62760 + 0.592396i 0 0.766044 0.642788i −0.300767 1.70574i 0 1.53209 + 1.28558i 0.866025 1.50000i 0 1.50000 + 2.59808i
163.2 1.62760 0.592396i 0 0.766044 0.642788i 0.300767 + 1.70574i 0 1.53209 + 1.28558i −0.866025 + 1.50000i 0 1.50000 + 2.59808i
325.1 −1.32683 + 1.11334i 0 0.173648 0.984808i −1.62760 + 0.592396i 0 0.347296 + 1.96962i −0.866025 1.50000i 0 1.50000 2.59808i
325.2 1.32683 1.11334i 0 0.173648 0.984808i 1.62760 0.592396i 0 0.347296 + 1.96962i 0.866025 + 1.50000i 0 1.50000 2.59808i
406.1 −1.32683 1.11334i 0 0.173648 + 0.984808i −1.62760 0.592396i 0 0.347296 1.96962i −0.866025 + 1.50000i 0 1.50000 + 2.59808i
406.2 1.32683 + 1.11334i 0 0.173648 + 0.984808i 1.62760 + 0.592396i 0 0.347296 1.96962i 0.866025 1.50000i 0 1.50000 + 2.59808i
568.1 −1.62760 0.592396i 0 0.766044 + 0.642788i −0.300767 + 1.70574i 0 1.53209 1.28558i 0.866025 + 1.50000i 0 1.50000 2.59808i
568.2 1.62760 + 0.592396i 0 0.766044 + 0.642788i 0.300767 1.70574i 0 1.53209 1.28558i −0.866025 1.50000i 0 1.50000 2.59808i
649.1 −0.300767 + 1.70574i 0 −0.939693 0.342020i 1.32683 1.11334i 0 −1.87939 + 0.684040i −0.866025 + 1.50000i 0 1.50000 + 2.59808i
649.2 0.300767 1.70574i 0 −0.939693 0.342020i −1.32683 + 1.11334i 0 −1.87939 + 0.684040i 0.866025 1.50000i 0 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
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.o 12
3.b odd 2 1 inner 729.2.e.o 12
9.c even 3 2 inner 729.2.e.o 12
9.d odd 6 2 inner 729.2.e.o 12
27.e even 9 1 81.2.a.a 2
27.e even 9 2 81.2.c.b 4
27.e even 9 3 inner 729.2.e.o 12
27.f odd 18 1 81.2.a.a 2
27.f odd 18 2 81.2.c.b 4
27.f odd 18 3 inner 729.2.e.o 12
108.j odd 18 1 1296.2.a.o 2
108.j odd 18 2 1296.2.i.s 4
108.l even 18 1 1296.2.a.o 2
108.l even 18 2 1296.2.i.s 4
135.n odd 18 1 2025.2.a.j 2
135.p even 18 1 2025.2.a.j 2
135.q even 36 2 2025.2.b.k 4
135.r odd 36 2 2025.2.b.k 4
189.y odd 18 1 3969.2.a.i 2
189.be even 18 1 3969.2.a.i 2
216.r odd 18 1 5184.2.a.bq 2
216.t even 18 1 5184.2.a.br 2
216.v even 18 1 5184.2.a.bq 2
216.x odd 18 1 5184.2.a.br 2
297.o even 18 1 9801.2.a.v 2
297.q odd 18 1 9801.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 27.e even 9 1
81.2.a.a 2 27.f odd 18 1
81.2.c.b 4 27.e even 9 2
81.2.c.b 4 27.f odd 18 2
729.2.e.o 12 1.a even 1 1 trivial
729.2.e.o 12 3.b odd 2 1 inner
729.2.e.o 12 9.c even 3 2 inner
729.2.e.o 12 9.d odd 6 2 inner
729.2.e.o 12 27.e even 9 3 inner
729.2.e.o 12 27.f odd 18 3 inner
1296.2.a.o 2 108.j odd 18 1
1296.2.a.o 2 108.l even 18 1
1296.2.i.s 4 108.j odd 18 2
1296.2.i.s 4 108.l even 18 2
2025.2.a.j 2 135.n odd 18 1
2025.2.a.j 2 135.p even 18 1
2025.2.b.k 4 135.q even 36 2
2025.2.b.k 4 135.r odd 36 2
3969.2.a.i 2 189.y odd 18 1
3969.2.a.i 2 189.be even 18 1
5184.2.a.bq 2 216.r odd 18 1
5184.2.a.bq 2 216.v even 18 1
5184.2.a.br 2 216.t even 18 1
5184.2.a.br 2 216.x odd 18 1
9801.2.a.v 2 297.o even 18 1
9801.2.a.v 2 297.q odd 18 1

Hecke kernels

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

\( T_{2}^{12} + 27T_{2}^{6} + 729 \) Copy content Toggle raw display
\( T_{5}^{12} + 27T_{5}^{6} + 729 \) Copy content Toggle raw display
\( T_{7}^{6} + 8T_{7}^{3} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$7$ \( (T^{6} + 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 1728 T^{6} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 27 T^{2} + 729)^{3} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 1728 T^{6} + \cdots + 2985984 \) Copy content Toggle raw display
$29$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$31$ \( (T^{6} + 512 T^{3} + 262144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 7 T + 49)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} + 110592 T^{6} + \cdots + 12230590464 \) Copy content Toggle raw display
$43$ \( (T^{6} + 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 110592 T^{6} + \cdots + 12230590464 \) Copy content Toggle raw display
$53$ \( T^{12} \) Copy content Toggle raw display
$59$ \( T^{12} + 7077888 T^{6} + \cdots + 50096498540544 \) Copy content Toggle raw display
$61$ \( (T^{6} - 343 T^{3} + 117649)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 1000 T^{3} + 1000000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 108 T^{2} + 11664)^{3} \) Copy content Toggle raw display
$73$ \( (T^{2} - 7 T + 49)^{6} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 7077888 T^{6} + \cdots + 50096498540544 \) Copy content Toggle raw display
$89$ \( (T^{4} + 27 T^{2} + 729)^{3} \) Copy content Toggle raw display
$97$ \( (T^{6} + 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
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