# 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 $12$

# Related objects

## 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: $$x^{12} - 8x^{6} + 64$$ x^12 - 8*x^6 + 64 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + 4 \beta_{5} q^{4} + \beta_{11} q^{5} - 2 \beta_{7} q^{7} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{8}+O(q^{10})$$ q + b4 * q^2 + 4*b5 * q^4 + b11 * q^5 - 2*b7 * q^7 + (-2*b10 + 2*b6) * q^8 $$q + \beta_{4} q^{2} + 4 \beta_{5} q^{4} + \beta_{11} q^{5} - 2 \beta_{7} q^{7} + ( - 2 \beta_{10} + 2 \beta_{6}) q^{8} + 6 \beta_{3} q^{10} - \beta_{9} q^{11} + (\beta_{5} - \beta_1) q^{13} + (2 \beta_{11} - 2 \beta_{8}) q^{14} + (4 \beta_{7} - 4 \beta_{2}) q^{16} + 3 \beta_{6} q^{17} + ( - \beta_{3} + 1) q^{19} + ( - 4 \beta_{9} + 4 \beta_{4}) q^{20} - 6 \beta_1 q^{22} - \beta_{8} q^{23} + \beta_{2} q^{25} - \beta_{10} q^{26} + 8 q^{28} - 2 \beta_{4} q^{29} - \beta_{5} q^{31} + 18 \beta_{7} q^{34} + (2 \beta_{10} - 2 \beta_{6}) q^{35} - 8 \beta_{3} q^{37} + \beta_{9} q^{38} + (12 \beta_{5} - 12 \beta_1) q^{40} + (2 \beta_{11} - 2 \beta_{8}) q^{41} + (11 \beta_{7} - 11 \beta_{2}) q^{43} - 4 \beta_{6} q^{44} + ( - 6 \beta_{3} + 6) q^{46} + (4 \beta_{9} - 4 \beta_{4}) q^{47} + 3 \beta_1 q^{49} + \beta_{8} q^{50} - 4 \beta_{2} q^{52} + 3 \beta_{10} q^{53} - 6 q^{55} + 4 \beta_{4} q^{56} - 12 \beta_{5} q^{58} + \beta_{11} q^{59} - 5 \beta_{7} q^{61} + (\beta_{10} - \beta_{6}) q^{62} + 8 \beta_{3} q^{64} - \beta_{9} q^{65} + (7 \beta_{5} - 7 \beta_1) q^{67} + ( - 12 \beta_{11} + 12 \beta_{8}) q^{68} + ( - 12 \beta_{7} + 12 \beta_{2}) q^{70} - 3 \beta_{6} q^{71} + (11 \beta_{3} - 11) q^{73} + (8 \beta_{9} - 8 \beta_{4}) q^{74} + 4 \beta_1 q^{76} + 2 \beta_{8} q^{77} - 7 \beta_{2} q^{79} - 4 \beta_{10} q^{80} + 12 q^{82} - 5 \beta_{4} q^{83} + 18 \beta_{5} q^{85} - 11 \beta_{11} q^{86} - 12 \beta_{7} q^{88} + 2 \beta_{3} q^{91} + 4 \beta_{9} q^{92} + ( - 24 \beta_{5} + 24 \beta_1) q^{94} + (\beta_{11} - \beta_{8}) q^{95} + ( - 7 \beta_{7} + 7 \beta_{2}) q^{97} + 3 \beta_{6} q^{98}+O(q^{100})$$ q + b4 * q^2 + 4*b5 * q^4 + b11 * q^5 - 2*b7 * q^7 + (-2*b10 + 2*b6) * q^8 + 6*b3 * q^10 - b9 * q^11 + (b5 - b1) * q^13 + (2*b11 - 2*b8) * q^14 + (4*b7 - 4*b2) * q^16 + 3*b6 * q^17 + (-b3 + 1) * q^19 + (-4*b9 + 4*b4) * q^20 - 6*b1 * q^22 - b8 * q^23 + b2 * q^25 - b10 * q^26 + 8 * q^28 - 2*b4 * q^29 - b5 * q^31 + 18*b7 * q^34 + (2*b10 - 2*b6) * q^35 - 8*b3 * q^37 + b9 * q^38 + (12*b5 - 12*b1) * q^40 + (2*b11 - 2*b8) * q^41 + (11*b7 - 11*b2) * q^43 - 4*b6 * q^44 + (-6*b3 + 6) * q^46 + (4*b9 - 4*b4) * q^47 + 3*b1 * q^49 + b8 * q^50 - 4*b2 * q^52 + 3*b10 * q^53 - 6 * q^55 + 4*b4 * q^56 - 12*b5 * q^58 + b11 * q^59 - 5*b7 * q^61 + (b10 - b6) * q^62 + 8*b3 * q^64 - b9 * q^65 + (7*b5 - 7*b1) * q^67 + (-12*b11 + 12*b8) * q^68 + (-12*b7 + 12*b2) * q^70 - 3*b6 * q^71 + (11*b3 - 11) * q^73 + (8*b9 - 8*b4) * q^74 + 4*b1 * q^76 + 2*b8 * q^77 - 7*b2 * q^79 - 4*b10 * q^80 + 12 * q^82 - 5*b4 * q^83 + 18*b5 * q^85 - 11*b11 * q^86 - 12*b7 * q^88 + 2*b3 * q^91 + 4*b9 * q^92 + (-24*b5 + 24*b1) * q^94 + (b11 - b8) * q^95 + (-7*b7 + 7*b2) * q^97 + 3*b6 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 36 q^{10} + 6 q^{19} + 96 q^{28} - 48 q^{37} + 36 q^{46} - 72 q^{55} + 48 q^{64} - 66 q^{73} + 144 q^{82} + 12 q^{91}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 8x^{6} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{4} ) / 4$$ (v^4) / 4 $$\beta_{3}$$ $$=$$ $$( \nu^{6} ) / 8$$ (v^6) / 8 $$\beta_{4}$$ $$=$$ $$( \nu^{7} + 8\nu ) / 8$$ (v^7 + 8*v) / 8 $$\beta_{5}$$ $$=$$ $$( \nu^{8} ) / 16$$ (v^8) / 16 $$\beta_{6}$$ $$=$$ $$( \nu^{9} + 8\nu^{3} ) / 16$$ (v^9 + 8*v^3) / 16 $$\beta_{7}$$ $$=$$ $$( \nu^{10} ) / 32$$ (v^10) / 32 $$\beta_{8}$$ $$=$$ $$( \nu^{11} + 8\nu^{5} ) / 32$$ (v^11 + 8*v^5) / 32 $$\beta_{9}$$ $$=$$ $$( -\nu^{7} + 16\nu ) / 8$$ (-v^7 + 16*v) / 8 $$\beta_{10}$$ $$=$$ $$( -\nu^{9} + 16\nu^{3} ) / 16$$ (-v^9 + 16*v^3) / 16 $$\beta_{11}$$ $$=$$ $$( -\nu^{11} + 16\nu^{5} ) / 32$$ (-v^11 + 16*v^5) / 32
 $$\nu$$ $$=$$ $$( \beta_{9} + \beta_{4} ) / 3$$ (b9 + b4) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( 2\beta_{10} + 2\beta_{6} ) / 3$$ (2*b10 + 2*b6) / 3 $$\nu^{4}$$ $$=$$ $$4\beta_{2}$$ 4*b2 $$\nu^{5}$$ $$=$$ $$( 4\beta_{11} + 4\beta_{8} ) / 3$$ (4*b11 + 4*b8) / 3 $$\nu^{6}$$ $$=$$ $$8\beta_{3}$$ 8*b3 $$\nu^{7}$$ $$=$$ $$( -8\beta_{9} + 16\beta_{4} ) / 3$$ (-8*b9 + 16*b4) / 3 $$\nu^{8}$$ $$=$$ $$16\beta_{5}$$ 16*b5 $$\nu^{9}$$ $$=$$ $$( -16\beta_{10} + 32\beta_{6} ) / 3$$ (-16*b10 + 32*b6) / 3 $$\nu^{10}$$ $$=$$ $$32\beta_{7}$$ 32*b7 $$\nu^{11}$$ $$=$$ $$( -32\beta_{11} + 64\beta_{8} ) / 3$$ (-32*b11 + 64*b8) / 3

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(729, [\chi])$$:

 $$T_{2}^{12} + 216T_{2}^{6} + 46656$$ T2^12 + 216*T2^6 + 46656 $$T_{5}^{12} + 216T_{5}^{6} + 46656$$ T5^12 + 216*T5^6 + 46656 $$T_{7}^{6} + 8T_{7}^{3} + 64$$ T7^6 + 8*T7^3 + 64

## Hecke characteristic polynomials

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