Properties

Label 729.2.a.e
Level $729$
Weight $2$
Character orbit 729.a
Self dual yes
Analytic conductor $5.821$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.7459857.1
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{4} + 3 \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{4} + \beta_{3}) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_1 + 1) q^{7} + (\beta_{4} + 3 \beta_{3} + \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{10} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{11} + (\beta_{5} + \beta_1) q^{13} + ( - \beta_{5} - \beta_{2} + \beta_1 + 4) q^{14} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2) q^{16} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{17} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{19} + (\beta_{5} - \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + \beta_1 - 4) q^{20} + (\beta_{5} - 5 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{22} + ( - \beta_{5} - \beta_{2} + \beta_1 - 2) q^{23} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{25} + (\beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 3) q^{26} + (5 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 1) q^{28} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{29} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 4) q^{31} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 2) q^{32} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - 1) q^{34} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{35} + (\beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 1) q^{37} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 1) q^{38} + ( - \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{40} + (2 \beta_{5} - \beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{41} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_1 + 2) q^{43} + (3 \beta_{5} - \beta_{4} - 3 \beta_{2} - \beta_1 - 6) q^{44} + ( - 2 \beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{46} + (2 \beta_{4} + 3 \beta_{3} - \beta_1 - 3) q^{47} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{49} + (\beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4) q^{50} + (\beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{52} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{2} - 1) q^{53} + (\beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{55} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 3) q^{56} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{58} + ( - \beta_{4} + \beta_{3} - 3 \beta_1 + 3) q^{59} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 5) q^{61} + (2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + \beta_1 - 5) q^{62} + (\beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{64} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{65} + (2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{67} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 6) q^{68} + ( - \beta_{5} - 2 \beta_{4} + 9 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 3) q^{70} + ( - 2 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{71} + ( - 3 \beta_{5} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{73} + ( - 4 \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} - \beta_1 + 7) q^{74} + ( - \beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + 5 \beta_1 - 2) q^{76} + (3 \beta_{5} - 7 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 3) q^{77} + ( - 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_1 + 3) q^{79} + (4 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 1) q^{80} + (\beta_{5} - 4 \beta_{4} - 9 \beta_{3} - \beta_{2} - 4 \beta_1) q^{82} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{83} + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 6 \beta_{2} - 5) q^{85} + (4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{86} + ( - 2 \beta_{5} - \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{88} + (4 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{89} + (\beta_{5} - 6 \beta_{3} + 3 \beta_{2} + \beta_1 + 2) q^{91} + (\beta_{5} - 5 \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 5) q^{92} + ( - 3 \beta_{5} + 5 \beta_{4} - \beta_{3} + 7 \beta_{2} - 2 \beta_1 - 1) q^{94} + ( - 2 \beta_{5} - \beta_{4} - 4 \beta_{3} + 4 \beta_{2} - 4) q^{95} + (2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 4) q^{97} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} + 9 q^{4} - 3 q^{5} + 6 q^{7} + 6 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} + 24 q^{14} + 15 q^{16} - 9 q^{17} + 12 q^{19} - 21 q^{20} + 3 q^{22} - 12 q^{23} + 9 q^{25} + 24 q^{26} + 3 q^{28} + 21 q^{29} + 15 q^{31} + 30 q^{35} + 3 q^{37} + 15 q^{38} + 3 q^{40} - 12 q^{41} + 6 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 12 q^{49} - 24 q^{50} + 3 q^{52} - 9 q^{53} + 15 q^{55} + 12 q^{56} - 15 q^{58} + 6 q^{59} + 24 q^{61} - 30 q^{62} + 6 q^{64} - 15 q^{65} + 15 q^{67} + 36 q^{68} - 15 q^{70} + 12 q^{73} + 24 q^{74} + 9 q^{76} + 15 q^{77} + 24 q^{79} - 21 q^{80} - 21 q^{82} - 6 q^{83} - 18 q^{85} - 30 q^{86} - 21 q^{88} - 9 q^{89} + 18 q^{91} + 6 q^{92} - 6 q^{94} - 33 q^{95} - 21 q^{97} + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 7\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 6\nu^{3} + 13\nu^{2} + 8\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 7\nu^{2} + 4\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - 9\nu^{4} - 14\nu^{3} + 31\nu^{2} + 12\nu - 16 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{5} - 2\beta_{4} - 5\beta_{3} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13\beta_{5} - 10\beta_{4} - 19\beta_{3} + 2\beta_{2} + 12\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 44\beta_{5} - 29\beta_{4} - 70\beta_{3} + 6\beta_{2} + 45\beta _1 + 53 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.578404
−1.12503
−1.70506
3.45779
1.77773
1.17298
−2.45779 0 4.04073 −3.08026 0 −2.65867 −5.01568 0 7.57064
1.2 −0.777732 0 −1.39513 −2.37635 0 −2.50138 2.64050 0 1.84816
1.3 −0.172976 0 −1.97008 3.73656 0 3.03150 0.686728 0 −0.646335
1.4 1.57840 0 0.491360 −1.67851 0 2.77928 −2.38124 0 −2.64936
1.5 2.12503 0 2.51575 2.07094 0 4.84867 1.09598 0 4.40081
1.6 2.70506 0 5.31738 −1.67238 0 0.500591 8.97372 0 −4.52391
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.a.e yes 6
3.b odd 2 1 729.2.a.b 6
9.c even 3 2 729.2.c.a 12
9.d odd 6 2 729.2.c.d 12
27.e even 9 2 729.2.e.k 12
27.e even 9 2 729.2.e.l 12
27.e even 9 2 729.2.e.u 12
27.f odd 18 2 729.2.e.j 12
27.f odd 18 2 729.2.e.s 12
27.f odd 18 2 729.2.e.t 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.b 6 3.b odd 2 1
729.2.a.e yes 6 1.a even 1 1 trivial
729.2.c.a 12 9.c even 3 2
729.2.c.d 12 9.d odd 6 2
729.2.e.j 12 27.f odd 18 2
729.2.e.k 12 27.e even 9 2
729.2.e.l 12 27.e even 9 2
729.2.e.s 12 27.f odd 18 2
729.2.e.t 12 27.f odd 18 2
729.2.e.u 12 27.e even 9 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3T_{2}^{5} - 6T_{2}^{4} + 21T_{2}^{3} - 18T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 3 T^{5} - 6 T^{4} + 21 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} - 15 T^{4} - 57 T^{3} + \cdots + 159 \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} - 9 T^{4} + 83 T^{3} + \cdots + 136 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} - 15 T^{4} - 141 T^{3} + \cdots + 888 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} - 18 T^{4} + 74 T^{3} + \cdots - 89 \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{5} - 135 T^{3} + \cdots + 459 \) Copy content Toggle raw display
$19$ \( T^{6} - 12 T^{5} + 15 T^{4} + \cdots - 296 \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{5} + 21 T^{4} + \cdots + 456 \) Copy content Toggle raw display
$29$ \( T^{6} - 21 T^{5} + 111 T^{4} + \cdots + 9879 \) Copy content Toggle raw display
$31$ \( T^{6} - 15 T^{5} + 45 T^{4} + \cdots - 1016 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} - 129 T^{4} + \cdots - 16109 \) Copy content Toggle raw display
$41$ \( T^{6} + 12 T^{5} - 60 T^{4} + \cdots + 23109 \) Copy content Toggle raw display
$43$ \( T^{6} - 6 T^{5} - 108 T^{4} + \cdots + 17344 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} - 42 T^{4} + \cdots + 17736 \) Copy content Toggle raw display
$53$ \( T^{6} + 9 T^{5} - 81 T^{4} + \cdots + 1944 \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} - 105 T^{4} + \cdots - 408 \) Copy content Toggle raw display
$61$ \( T^{6} - 24 T^{5} + 117 T^{4} + \cdots + 1576 \) Copy content Toggle raw display
$67$ \( T^{6} - 15 T^{5} - 54 T^{4} + \cdots - 17288 \) Copy content Toggle raw display
$71$ \( T^{6} - 180 T^{4} - 864 T^{3} + \cdots + 29376 \) Copy content Toggle raw display
$73$ \( T^{6} - 12 T^{5} - 156 T^{4} + \cdots + 57601 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + 27 T^{4} + \cdots - 93176 \) Copy content Toggle raw display
$83$ \( T^{6} + 6 T^{5} - 87 T^{4} + \cdots + 4344 \) Copy content Toggle raw display
$89$ \( T^{6} + 9 T^{5} - 225 T^{4} + \cdots - 123957 \) Copy content Toggle raw display
$97$ \( T^{6} + 21 T^{5} + 72 T^{4} + \cdots - 18251 \) Copy content Toggle raw display
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