# 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 $2$

# Related objects

## 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: $$x^{6} - 6x^{4} + 9x^{2} - 3$$ x^6 - 6*x^4 + 9*x^2 - 3 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{4} - 2) q^{7} + (\beta_{3} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + b2 * q^4 + (-b3 - b1) * q^5 + (b4 - 2) * q^7 + (b3 - b1) * q^8 $$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})$$ q + b1 * q^2 + b2 * q^4 + (-b3 - b1) * q^5 + (b4 - 2) * q^7 + (b3 - b1) * q^8 + (-b4 - 3*b2 - 2) * q^10 + (-b5 + 2*b3 - b1) * q^11 + (-3*b4 - 2*b2 - 2) * q^13 + (b5 - 2*b1) * q^14 + (b4 - b2 - 2) * q^16 + (4*b5 + b3 + 2*b1) * q^17 + (b4 - b2 - 4) * q^19 + (-b5 - b3 - 3*b1) * q^20 + (b4 + 3*b2 - 1) * q^22 + (-5*b5 - 2*b1) * q^23 + (2*b4 + 5*b2) * q^25 + (-3*b5 - 2*b3 - 4*b1) * q^26 + (-b4 - 2*b2 - 1) * q^28 + (b5 - b3 + 3*b1) * q^29 + (-2*b4 - 5) * q^31 + (b5 - 3*b3 - b1) * q^32 + (5*b4 + 4*b2) * q^34 + (b5 + 2*b3 + 3*b1) * q^35 + (b4 + 2*b2 - 1) * q^37 + (b5 - b3 - 5*b1) * q^38 + (b2 - 1) * q^40 + (4*b5 - b3 + b1) * q^41 + (2*b4 + 2*b2 - 2) * q^43 + (3*b5 - b3 + 4*b1) * q^44 + (-5*b4 - 2*b2 + 1) * q^46 + (-b3 + b1) * q^47 + (-4*b4 - b2 - 1) * q^49 + (2*b5 + 5*b3 + 5*b1) * q^50 + (b4 - 4*b2 - 1) * q^52 + (-4*b5 - 2*b3 - 2*b1) * q^53 + (-2*b4 - 2*b2 - 5) * q^55 + (-3*b5 - 2*b3 + b1) * q^56 + (b2 + 5) * q^58 + (2*b3 + 5*b1) * q^59 + (-6*b4 - 2*b2 - 2) * q^61 + (-2*b5 - 5*b1) * q^62 + (-4*b4 - 5*b2 + 1) * q^64 + (-b5 + 4*b3 + 5*b1) * q^65 + (5*b4 + 8*b2 + 1) * q^67 + (-3*b5 + 2*b3) * q^68 + (3*b4 + 7*b2 + 5) * q^70 + (-4*b5 - 2*b1) * q^71 + (3*b4 - 3*b2 + 2) * q^73 + (b5 + 2*b3 + b1) * q^74 + (-2*b4 - 5*b2 - 3) * q^76 + (-3*b5 - 3*b3 - b1) * q^77 + (3*b4 - 2*b2 - 8) * q^79 + (2*b5 + 3*b3 + 6*b1) * q^80 + (3*b4 - b2 - 2) * q^82 + (8*b5 + 4*b3 + 5*b1) * q^83 + (b4 - 4*b2 - 3) * q^85 + (2*b5 + 2*b3) * q^86 + (-4*b2 + 7) * q^88 + (-3*b5 - 5*b3 + 3*b1) * q^89 + (6*b4 + 7*b2) * q^91 + (5*b5 - 2*b3 + 3*b1) * q^92 + (-b4 - b2 + 2) * q^94 + (2*b5 + 5*b3 + 8*b1) * q^95 + (-7*b4 - 4*b2 - 2) * q^97 + (-4*b5 - b3 - 2*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{7}+O(q^{10})$$ 6 * q - 12 * q^7 $$6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5\nu^{2} + 4$$ v^4 - 5*v^2 + 4 $$\beta_{5}$$ $$=$$ $$\nu^{5} - 5\nu^{3} + 4\nu$$ v^5 - 5*v^3 + 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 6$$ b4 + 5*b2 + 6 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 5\beta_{3} + 11\beta_1$$ b5 + 5*b3 + 11*b1

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(729))$$.

## Hecke characteristic polynomials

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