Properties

Label 729.2.e.p
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: 12.0.101559956668416.1
Defining polynomial: \( x^{12} - 8x^{6} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 243)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + 4 \beta_{5} q^{4} + \beta_{11} q^{5} - 2 \beta_{7} q^{7} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + 4 \beta_{5} q^{4} + \beta_{11} q^{5} - 2 \beta_{7} q^{7} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{8} + 6 \beta_{3} q^{10} - \beta_{9} q^{11} + (\beta_{5} - \beta_1) q^{13} + (2 \beta_{11} - 2 \beta_{8}) q^{14} + (4 \beta_{7} - 4 \beta_{2}) q^{16} + 3 \beta_{6} q^{17} + ( - \beta_{3} + 1) q^{19} + ( - 4 \beta_{9} + 4 \beta_{4}) q^{20} - 6 \beta_1 q^{22} - \beta_{8} q^{23} + \beta_{2} q^{25} - \beta_{10} q^{26} + 8 q^{28} - 2 \beta_{4} q^{29} - \beta_{5} q^{31} + 18 \beta_{7} q^{34} + (2 \beta_{10} - 2 \beta_{6}) q^{35} - 8 \beta_{3} q^{37} + \beta_{9} q^{38} + (12 \beta_{5} - 12 \beta_1) q^{40} + (2 \beta_{11} - 2 \beta_{8}) q^{41} + (11 \beta_{7} - 11 \beta_{2}) q^{43} - 4 \beta_{6} q^{44} + ( - 6 \beta_{3} + 6) q^{46} + (4 \beta_{9} - 4 \beta_{4}) q^{47} + 3 \beta_1 q^{49} + \beta_{8} q^{50} - 4 \beta_{2} q^{52} + 3 \beta_{10} q^{53} - 6 q^{55} + 4 \beta_{4} q^{56} - 12 \beta_{5} q^{58} + \beta_{11} q^{59} - 5 \beta_{7} q^{61} + (\beta_{10} - \beta_{6}) q^{62} + 8 \beta_{3} q^{64} - \beta_{9} q^{65} + (7 \beta_{5} - 7 \beta_1) q^{67} + ( - 12 \beta_{11} + 12 \beta_{8}) q^{68} + ( - 12 \beta_{7} + 12 \beta_{2}) q^{70} - 3 \beta_{6} q^{71} + (11 \beta_{3} - 11) q^{73} + (8 \beta_{9} - 8 \beta_{4}) q^{74} + 4 \beta_1 q^{76} + 2 \beta_{8} q^{77} - 7 \beta_{2} q^{79} - 4 \beta_{10} q^{80} + 12 q^{82} - 5 \beta_{4} q^{83} + 18 \beta_{5} q^{85} - 11 \beta_{11} q^{86} - 12 \beta_{7} q^{88} + 2 \beta_{3} q^{91} + 4 \beta_{9} q^{92} + ( - 24 \beta_{5} + 24 \beta_1) q^{94} + (\beta_{11} - \beta_{8}) q^{95} + ( - 7 \beta_{7} + 7 \beta_{2}) q^{97} + 3 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 36 q^{10} + 6 q^{19} + 96 q^{28} - 48 q^{37} + 36 q^{46} - 72 q^{55} + 48 q^{64} - 66 q^{73} + 144 q^{82} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 8x^{6} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{8} ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 8\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 8\nu^{5} ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{7} + 16\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{9} + 16\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} + 16\nu^{5} ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{10} + 2\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{11} + 4\beta_{8} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{9} + 16\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16\beta_{5} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -16\beta_{10} + 32\beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{7} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -32\beta_{11} + 64\beta_{8} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{5}\)

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
0.483690 1.32893i
−0.483690 + 1.32893i
−0.909039 + 1.08335i
0.909039 1.08335i
−1.39273 + 0.245576i
1.39273 0.245576i
−1.39273 0.245576i
1.39273 + 0.245576i
−0.909039 1.08335i
0.909039 + 1.08335i
0.483690 + 1.32893i
−0.483690 1.32893i
−0.425349 2.41228i 0 −3.75877 + 1.36808i 1.87642 + 1.57450i 0 −1.87939 0.684040i 2.44949 + 4.24264i 0 3.00000 5.19615i
82.2 0.425349 + 2.41228i 0 −3.75877 + 1.36808i −1.87642 1.57450i 0 −1.87939 0.684040i −2.44949 4.24264i 0 3.00000 5.19615i
163.1 −2.30177 + 0.837775i 0 3.06418 2.57115i −0.425349 2.41228i 0 1.53209 + 1.28558i −2.44949 + 4.24264i 0 3.00000 + 5.19615i
163.2 2.30177 0.837775i 0 3.06418 2.57115i 0.425349 + 2.41228i 0 1.53209 + 1.28558i 2.44949 4.24264i 0 3.00000 + 5.19615i
325.1 −1.87642 + 1.57450i 0 0.694593 3.93923i −2.30177 + 0.837775i 0 0.347296 + 1.96962i 2.44949 + 4.24264i 0 3.00000 5.19615i
325.2 1.87642 1.57450i 0 0.694593 3.93923i 2.30177 0.837775i 0 0.347296 + 1.96962i −2.44949 4.24264i 0 3.00000 5.19615i
406.1 −1.87642 1.57450i 0 0.694593 + 3.93923i −2.30177 0.837775i 0 0.347296 1.96962i 2.44949 4.24264i 0 3.00000 + 5.19615i
406.2 1.87642 + 1.57450i 0 0.694593 + 3.93923i 2.30177 + 0.837775i 0 0.347296 1.96962i −2.44949 + 4.24264i 0 3.00000 + 5.19615i
568.1 −2.30177 0.837775i 0 3.06418 + 2.57115i −0.425349 + 2.41228i 0 1.53209 1.28558i −2.44949 4.24264i 0 3.00000 5.19615i
568.2 2.30177 + 0.837775i 0 3.06418 + 2.57115i 0.425349 2.41228i 0 1.53209 1.28558i 2.44949 + 4.24264i 0 3.00000 5.19615i
649.1 −0.425349 + 2.41228i 0 −3.75877 1.36808i 1.87642 1.57450i 0 −1.87939 + 0.684040i 2.44949 4.24264i 0 3.00000 + 5.19615i
649.2 0.425349 2.41228i 0 −3.75877 1.36808i −1.87642 + 1.57450i 0 −1.87939 + 0.684040i −2.44949 + 4.24264i 0 3.00000 + 5.19615i
\(n\): 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.p 12
3.b odd 2 1 inner 729.2.e.p 12
9.c even 3 2 inner 729.2.e.p 12
9.d odd 6 2 inner 729.2.e.p 12
27.e even 9 1 243.2.a.d 2
27.e even 9 2 243.2.c.c 4
27.e even 9 3 inner 729.2.e.p 12
27.f odd 18 1 243.2.a.d 2
27.f odd 18 2 243.2.c.c 4
27.f odd 18 3 inner 729.2.e.p 12
108.j odd 18 1 3888.2.a.z 2
108.l even 18 1 3888.2.a.z 2
135.n odd 18 1 6075.2.a.bn 2
135.p even 18 1 6075.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.2.a.d 2 27.e even 9 1
243.2.a.d 2 27.f odd 18 1
243.2.c.c 4 27.e even 9 2
243.2.c.c 4 27.f odd 18 2
729.2.e.p 12 1.a even 1 1 trivial
729.2.e.p 12 3.b odd 2 1 inner
729.2.e.p 12 9.c even 3 2 inner
729.2.e.p 12 9.d odd 6 2 inner
729.2.e.p 12 27.e even 9 3 inner
729.2.e.p 12 27.f odd 18 3 inner
3888.2.a.z 2 108.j odd 18 1
3888.2.a.z 2 108.l even 18 1
6075.2.a.bn 2 135.n odd 18 1
6075.2.a.bn 2 135.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 \(S_{2}^{\mathrm{new}}(729, [\chi])\):

\( T_{2}^{12} + 216T_{2}^{6} + 46656 \) Copy content Toggle raw display
\( T_{5}^{12} + 216T_{5}^{6} + 46656 \) Copy content Toggle raw display
\( T_{7}^{6} + 8T_{7}^{3} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 216 T^{6} + 46656 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 216 T^{6} + 46656 \) Copy content Toggle raw display
$7$ \( (T^{6} + 8 T^{3} + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 216 T^{6} + 46656 \) Copy content Toggle raw display
$13$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 54 T^{2} + 2916)^{3} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 216 T^{6} + 46656 \) Copy content Toggle raw display
$29$ \( T^{12} + 13824 T^{6} + \cdots + 191102976 \) Copy content Toggle raw display
$31$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 64)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} + 13824 T^{6} + \cdots + 191102976 \) Copy content Toggle raw display
$43$ \( (T^{6} + 1331 T^{3} + 1771561)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 884736 T^{6} + \cdots + 782757789696 \) Copy content Toggle raw display
$53$ \( (T^{2} - 54)^{6} \) Copy content Toggle raw display
$59$ \( T^{12} + 216 T^{6} + 46656 \) Copy content Toggle raw display
$61$ \( (T^{6} + 125 T^{3} + 15625)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 343 T^{3} + 117649)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 54 T^{2} + 2916)^{3} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11 T + 121)^{6} \) Copy content Toggle raw display
$79$ \( (T^{6} - 343 T^{3} + 117649)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 3375000 T^{6} + \cdots + 11390625000000 \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} - 343 T^{3} + 117649)^{2} \) Copy content Toggle raw display
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