Properties

Label 729.2.e.e
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: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 243)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{18}^{5} q^{4} - 4 \zeta_{18}^{4} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{18}^{5} q^{4} - 4 \zeta_{18}^{4} q^{7} - 7 \zeta_{18}^{2} q^{13} - 4 \zeta_{18} q^{16} + \zeta_{18}^{3} q^{19} + (5 \zeta_{18}^{4} - 5 \zeta_{18}) q^{25} + 8 q^{28} - 11 \zeta_{18}^{5} q^{31} + ( - 10 \zeta_{18}^{3} + 10) q^{37} - 5 \zeta_{18} q^{43} + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{49} + ( - 14 \zeta_{18}^{4} + 14 \zeta_{18}) q^{52} - \zeta_{18}^{4} q^{61} + ( - 8 \zeta_{18}^{3} + 8) q^{64} + 5 \zeta_{18}^{2} q^{67} + 7 \zeta_{18}^{3} q^{73} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{2}) q^{76} + (13 \zeta_{18}^{4} - 13 \zeta_{18}) q^{79} + (28 \zeta_{18}^{3} - 28) q^{91} - 5 \zeta_{18} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{19} + 48 q^{28} + 30 q^{37} + 24 q^{64} + 21 q^{73} - 84 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 3.75877 + 1.36808i 0 0 0
163.1 0 0 −1.53209 + 1.28558i 0 0 −3.06418 2.57115i 0 0 0
325.1 0 0 −0.347296 + 1.96962i 0 0 −0.694593 3.93923i 0 0 0
406.1 0 0 −0.347296 1.96962i 0 0 −0.694593 + 3.93923i 0 0 0
568.1 0 0 −1.53209 1.28558i 0 0 −3.06418 + 2.57115i 0 0 0
649.1 0 0 1.87939 + 0.684040i 0 0 3.75877 1.36808i 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.e 6
3.b odd 2 1 CM 729.2.e.e 6
9.c even 3 2 inner 729.2.e.e 6
9.d odd 6 2 inner 729.2.e.e 6
27.e even 9 1 243.2.a.a 1
27.e even 9 2 243.2.c.b 2
27.e even 9 3 inner 729.2.e.e 6
27.f odd 18 1 243.2.a.a 1
27.f odd 18 2 243.2.c.b 2
27.f odd 18 3 inner 729.2.e.e 6
108.j odd 18 1 3888.2.a.n 1
108.l even 18 1 3888.2.a.n 1
135.n odd 18 1 6075.2.a.w 1
135.p even 18 1 6075.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.a 1 27.e even 9 1
243.2.a.a 1 27.f odd 18 1
243.2.c.b 2 27.e even 9 2
243.2.c.b 2 27.f odd 18 2
729.2.e.e 6 1.a even 1 1 trivial
729.2.e.e 6 3.b odd 2 1 CM
729.2.e.e 6 9.c even 3 2 inner
729.2.e.e 6 9.d odd 6 2 inner
729.2.e.e 6 27.e even 9 3 inner
729.2.e.e 6 27.f odd 18 3 inner
3888.2.a.n 1 108.j odd 18 1
3888.2.a.n 1 108.l even 18 1
6075.2.a.w 1 135.n odd 18 1
6075.2.a.w 1 135.p even 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
\( T_{7}^{6} - 64T_{7}^{3} + 4096 \) 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
$7$ \( T^{6} - 64T^{3} + 4096 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 343 T^{3} + 117649 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 1331 T^{3} + \cdots + 1771561 \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 100)^{3} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 125 T^{3} + 15625 \) 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
$79$ \( T^{6} - 2197 T^{3} + \cdots + 4826809 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 125 T^{3} + 15625 \) Copy content Toggle raw display
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