Properties

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

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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: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{4} - 2) q^{7} + (\beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{4} - 2) q^{7} + (\beta_{3} - \beta_1) q^{8} + ( - \beta_{4} - 3 \beta_{2} - 2) q^{10} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{11} + ( - 3 \beta_{4} - 2 \beta_{2} - 2) q^{13} + (\beta_{5} - 2 \beta_1) q^{14} + (\beta_{4} - \beta_{2} - 2) q^{16} + (4 \beta_{5} + \beta_{3} + 2 \beta_1) q^{17} + (\beta_{4} - \beta_{2} - 4) q^{19} + ( - \beta_{5} - \beta_{3} - 3 \beta_1) q^{20} + (\beta_{4} + 3 \beta_{2} - 1) q^{22} + ( - 5 \beta_{5} - 2 \beta_1) q^{23} + (2 \beta_{4} + 5 \beta_{2}) q^{25} + ( - 3 \beta_{5} - 2 \beta_{3} - 4 \beta_1) q^{26} + ( - \beta_{4} - 2 \beta_{2} - 1) q^{28} + (\beta_{5} - \beta_{3} + 3 \beta_1) q^{29} + ( - 2 \beta_{4} - 5) q^{31} + (\beta_{5} - 3 \beta_{3} - \beta_1) q^{32} + (5 \beta_{4} + 4 \beta_{2}) q^{34} + (\beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{35} + (\beta_{4} + 2 \beta_{2} - 1) q^{37} + (\beta_{5} - \beta_{3} - 5 \beta_1) q^{38} + (\beta_{2} - 1) q^{40} + (4 \beta_{5} - \beta_{3} + \beta_1) q^{41} + (2 \beta_{4} + 2 \beta_{2} - 2) q^{43} + (3 \beta_{5} - \beta_{3} + 4 \beta_1) q^{44} + ( - 5 \beta_{4} - 2 \beta_{2} + 1) q^{46} + ( - \beta_{3} + \beta_1) q^{47} + ( - 4 \beta_{4} - \beta_{2} - 1) q^{49} + (2 \beta_{5} + 5 \beta_{3} + 5 \beta_1) q^{50} + (\beta_{4} - 4 \beta_{2} - 1) q^{52} + ( - 4 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{53} + ( - 2 \beta_{4} - 2 \beta_{2} - 5) q^{55} + ( - 3 \beta_{5} - 2 \beta_{3} + \beta_1) q^{56} + (\beta_{2} + 5) q^{58} + (2 \beta_{3} + 5 \beta_1) q^{59} + ( - 6 \beta_{4} - 2 \beta_{2} - 2) q^{61} + ( - 2 \beta_{5} - 5 \beta_1) q^{62} + ( - 4 \beta_{4} - 5 \beta_{2} + 1) q^{64} + ( - \beta_{5} + 4 \beta_{3} + 5 \beta_1) q^{65} + (5 \beta_{4} + 8 \beta_{2} + 1) q^{67} + ( - 3 \beta_{5} + 2 \beta_{3}) q^{68} + (3 \beta_{4} + 7 \beta_{2} + 5) q^{70} + ( - 4 \beta_{5} - 2 \beta_1) q^{71} + (3 \beta_{4} - 3 \beta_{2} + 2) q^{73} + (\beta_{5} + 2 \beta_{3} + \beta_1) q^{74} + ( - 2 \beta_{4} - 5 \beta_{2} - 3) q^{76} + ( - 3 \beta_{5} - 3 \beta_{3} - \beta_1) q^{77} + (3 \beta_{4} - 2 \beta_{2} - 8) q^{79} + (2 \beta_{5} + 3 \beta_{3} + 6 \beta_1) q^{80} + (3 \beta_{4} - \beta_{2} - 2) q^{82} + (8 \beta_{5} + 4 \beta_{3} + 5 \beta_1) q^{83} + (\beta_{4} - 4 \beta_{2} - 3) q^{85} + (2 \beta_{5} + 2 \beta_{3}) q^{86} + ( - 4 \beta_{2} + 7) q^{88} + ( - 3 \beta_{5} - 5 \beta_{3} + 3 \beta_1) q^{89} + (6 \beta_{4} + 7 \beta_{2}) q^{91} + (5 \beta_{5} - 2 \beta_{3} + 3 \beta_1) q^{92} + ( - \beta_{4} - \beta_{2} + 2) q^{94} + (2 \beta_{5} + 5 \beta_{3} + 8 \beta_1) q^{95} + ( - 7 \beta_{4} - 4 \beta_{2} - 2) q^{97} + ( - 4 \beta_{5} - \beta_{3} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{36} + \zeta_{36}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 5\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 5\beta_{3} + 11\beta_1 \) 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
−1.96962
−1.28558
−0.684040
0.684040
1.28558
1.96962
−1.96962 0 1.87939 3.70167 0 −2.34730 0.237565 0 −7.29086
1.2 −1.28558 0 −0.347296 −0.446476 0 −3.53209 3.01763 0 0.573978
1.3 −0.684040 0 −1.53209 −1.04801 0 −0.120615 2.41609 0 0.716881
1.4 0.684040 0 −1.53209 1.04801 0 −0.120615 −2.41609 0 0.716881
1.5 1.28558 0 −0.347296 0.446476 0 −3.53209 −3.01763 0 0.573978
1.6 1.96962 0 1.87939 −3.70167 0 −2.34730 −0.237565 0 −7.29086
\(n\): 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.a.c 6
3.b odd 2 1 inner 729.2.a.c 6
9.c even 3 2 729.2.c.c 12
9.d odd 6 2 729.2.c.c 12
27.e even 9 2 729.2.e.m 12
27.e even 9 2 729.2.e.q 12
27.e even 9 2 729.2.e.r 12
27.f odd 18 2 729.2.e.m 12
27.f odd 18 2 729.2.e.q 12
27.f odd 18 2 729.2.e.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.c 6 1.a even 1 1 trivial
729.2.a.c 6 3.b odd 2 1 inner
729.2.c.c 12 9.c even 3 2
729.2.c.c 12 9.d odd 6 2
729.2.e.m 12 27.e even 9 2
729.2.e.m 12 27.f odd 18 2
729.2.e.q 12 27.e even 9 2
729.2.e.q 12 27.f odd 18 2
729.2.e.r 12 27.e even 9 2
729.2.e.r 12 27.f odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6T_{2}^{4} + 9T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(729))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 6 T^{4} + 9 T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 15 T^{4} + 18 T^{2} - 3 \) Copy content Toggle raw display
$7$ \( (T^{3} + 6 T^{2} + 9 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} - 42 T^{4} + 405 T^{2} + \cdots - 1083 \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} - 9 T - 71)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 81 T^{4} + 1755 T^{2} + \cdots - 9747 \) Copy content Toggle raw display
$19$ \( (T^{3} + 12 T^{2} + 39 T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 114 T^{4} + 3249 T^{2} + \cdots - 867 \) Copy content Toggle raw display
$29$ \( T^{6} - 51 T^{4} + 810 T^{2} + \cdots - 4107 \) Copy content Toggle raw display
$31$ \( (T^{3} + 15 T^{2} + 63 T + 73)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 6 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 87 T^{4} + 1854 T^{2} + \cdots - 8427 \) Copy content Toggle raw display
$43$ \( (T^{3} + 6 T^{2} - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 15 T^{4} + 54 T^{2} - 3 \) Copy content Toggle raw display
$53$ \( T^{6} - 108 T^{4} + 2592 T^{2} + \cdots - 15552 \) Copy content Toggle raw display
$59$ \( T^{6} - 186 T^{4} + 1557 T^{2} + \cdots - 3 \) Copy content Toggle raw display
$61$ \( (T^{3} + 6 T^{2} - 72 T - 296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 3 T^{2} - 144 T - 251)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} - 72 T^{4} + 1296 T^{2} + \cdots - 1728 \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} - 69 T - 89)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 24 T^{2} + 135 T - 107)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 438 T^{4} + 51777 T^{2} + \cdots - 1550883 \) Copy content Toggle raw display
$89$ \( T^{6} - 387 T^{4} + 16146 T^{2} + \cdots - 7803 \) Copy content Toggle raw display
$97$ \( (T^{3} + 6 T^{2} - 99 T - 647)^{2} \) Copy content Toggle raw display
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