# Properties

 Label 1728.3.g.k Level $1728$ Weight $3$ Character orbit 1728.g Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.22581504.2 Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 864) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{5} + \beta_{7} q^{7}+O(q^{10})$$ q + (b2 - 1) * q^5 + b7 * q^7 $$q + (\beta_{2} - 1) q^{5} + \beta_{7} q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{11} + (\beta_{4} + 2) q^{13} + (\beta_{4} + 6) q^{17} + (\beta_{7} - 3 \beta_{5} + \beta_{3}) q^{19} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3}) q^{23} + ( - 6 \beta_{2} - \beta_1 + 6) q^{25} + (\beta_1 + 4) q^{29} + ( - 2 \beta_{7} + \beta_{6} + 5 \beta_{5} + \beta_{3}) q^{31} + ( - 4 \beta_{6} - \beta_{5}) q^{35} + (\beta_{4} - 4 \beta_{2} + 2 \beta_1 + 12) q^{37} + ( - \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 16) q^{41} + ( - 2 \beta_{7} - 4 \beta_{5} - 4 \beta_{3}) q^{43} + (4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3}) q^{47} + ( - 2 \beta_{2} - 3 \beta_1) q^{49} + (4 \beta_{4} + \beta_{2} + \beta_1 - 21) q^{53} + ( - 3 \beta_{7} - 6 \beta_{6} + 12 \beta_{5} - 4 \beta_{3}) q^{55} + ( - 2 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} - 6 \beta_{3}) q^{59} + (4 \beta_{4} + 12 \beta_{2} + 2 \beta_1 - 4) q^{61} + (3 \beta_{4} + 8 \beta_{2} + 2 \beta_1 - 14) q^{65} + ( - 2 \beta_{7} + 2 \beta_{6} - 12 \beta_{5} + 4 \beta_{3}) q^{67} + (3 \beta_{7} + 14 \beta_{6} - 3 \beta_{5} + \beta_{3}) q^{71} + ( - 6 \beta_{4} - 6 \beta_{2} - \beta_1 + 3) q^{73} + (2 \beta_{4} + 5 \beta_{2} + 4 \beta_1 + 55) q^{77} + (3 \beta_{7} + 13 \beta_{6} + 9 \beta_{5} + 5 \beta_{3}) q^{79} + (13 \beta_{7} + 3 \beta_{6} - 10 \beta_{5} + 3 \beta_{3}) q^{83} + (3 \beta_{4} + 12 \beta_{2} + 2 \beta_1 - 18) q^{85} + (4 \beta_{4} + 10 \beta_{2} + 4 \beta_1 + 78) q^{89} + (\beta_{7} - 12 \beta_{6} - 13 \beta_{5} - \beta_{3}) q^{91} + ( - \beta_{7} - 22 \beta_{6} + 11 \beta_{5} - 11 \beta_{3}) q^{95} + ( - 6 \beta_{4} + 4 \beta_{2} - 2 \beta_1 - 17) q^{97}+O(q^{100})$$ q + (b2 - 1) * q^5 + b7 * q^7 + (-b7 + b6 + b3) * q^11 + (b4 + 2) * q^13 + (b4 + 6) * q^17 + (b7 - 3*b5 + b3) * q^19 + (-3*b7 - 2*b6 + b5 - b3) * q^23 + (-6*b2 - b1 + 6) * q^25 + (b1 + 4) * q^29 + (-2*b7 + b6 + 5*b5 + b3) * q^31 + (-4*b6 - b5) * q^35 + (b4 - 4*b2 + 2*b1 + 12) * q^37 + (-b4 + 2*b2 + 2*b1 - 16) * q^41 + (-2*b7 - 4*b5 - 4*b3) * q^43 + (4*b7 + 2*b6 - 2*b5 + 4*b3) * q^47 + (-2*b2 - 3*b1) * q^49 + (4*b4 + b2 + b1 - 21) * q^53 + (-3*b7 - 6*b6 + 12*b5 - 4*b3) * q^55 + (-2*b7 + 4*b6 + 6*b5 - 6*b3) * q^59 + (4*b4 + 12*b2 + 2*b1 - 4) * q^61 + (3*b4 + 8*b2 + 2*b1 - 14) * q^65 + (-2*b7 + 2*b6 - 12*b5 + 4*b3) * q^67 + (3*b7 + 14*b6 - 3*b5 + b3) * q^71 + (-6*b4 - 6*b2 - b1 + 3) * q^73 + (2*b4 + 5*b2 + 4*b1 + 55) * q^77 + (3*b7 + 13*b6 + 9*b5 + 5*b3) * q^79 + (13*b7 + 3*b6 - 10*b5 + 3*b3) * q^83 + (3*b4 + 12*b2 + 2*b1 - 18) * q^85 + (4*b4 + 10*b2 + 4*b1 + 78) * q^89 + (b7 - 12*b6 - 13*b5 - b3) * q^91 + (-b7 - 22*b6 + 11*b5 - 11*b3) * q^95 + (-6*b4 + 4*b2 - 2*b1 - 17) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{5}+O(q^{10})$$ 8 * q - 8 * q^5 $$8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100})$$ 8 * q - 8 * q^5 + 16 * q^13 + 48 * q^17 + 48 * q^25 + 32 * q^29 + 96 * q^37 - 128 * q^41 - 168 * q^53 - 32 * q^61 - 112 * q^65 + 24 * q^73 + 440 * q^77 - 144 * q^85 + 624 * q^89 - 136 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$-\nu^{7} + 3\nu^{6} - \nu^{5} - 7\nu^{4} + 9\nu^{3} + 11\nu^{2} - 32\nu + 14$$ -v^7 + 3*v^6 - v^5 - 7*v^4 + 9*v^3 + 11*v^2 - 32*v + 14 $$\beta_{2}$$ $$=$$ $$( -11\nu^{7} + 30\nu^{6} - 11\nu^{5} - 44\nu^{4} + 57\nu^{3} + 46\nu^{2} - 160\nu + 112 ) / 4$$ (-11*v^7 + 30*v^6 - 11*v^5 - 44*v^4 + 57*v^3 + 46*v^2 - 160*v + 112) / 4 $$\beta_{3}$$ $$=$$ $$( 11\nu^{7} - 34\nu^{6} + 27\nu^{5} + 40\nu^{4} - 81\nu^{3} - 18\nu^{2} + 208\nu - 192 ) / 4$$ (11*v^7 - 34*v^6 + 27*v^5 + 40*v^4 - 81*v^3 - 18*v^2 + 208*v - 192) / 4 $$\beta_{4}$$ $$=$$ $$( -5\nu^{7} + 18\nu^{6} - 17\nu^{5} - 20\nu^{4} + 51\nu^{3} + 10\nu^{2} - 100\nu + 100 ) / 2$$ (-5*v^7 + 18*v^6 - 17*v^5 - 20*v^4 + 51*v^3 + 10*v^2 - 100*v + 100) / 2 $$\beta_{5}$$ $$=$$ $$( 9\nu^{7} - 21\nu^{6} + 9\nu^{5} + 33\nu^{4} - 45\nu^{3} - 33\nu^{2} + 120\nu - 90 ) / 2$$ (9*v^7 - 21*v^6 + 9*v^5 + 33*v^4 - 45*v^3 - 33*v^2 + 120*v - 90) / 2 $$\beta_{6}$$ $$=$$ $$4\nu^{7} - 11\nu^{6} + 6\nu^{5} + 17\nu^{4} - 24\nu^{3} - 15\nu^{2} + 62\nu - 48$$ 4*v^7 - 11*v^6 + 6*v^5 + 17*v^4 - 24*v^3 - 15*v^2 + 62*v - 48 $$\beta_{7}$$ $$=$$ $$( -25\nu^{7} + 64\nu^{6} - 33\nu^{5} - 94\nu^{4} + 127\nu^{3} + 104\nu^{2} - 360\nu + 268 ) / 4$$ (-25*v^7 + 64*v^6 - 33*v^5 - 94*v^4 + 127*v^3 + 104*v^2 - 360*v + 268) / 4
 $$\nu$$ $$=$$ $$( -\beta_{6} + \beta_{4} + 2\beta_{3} - \beta _1 + 12 ) / 24$$ (-b6 + b4 + 2*b3 - b1 + 12) / 24 $$\nu^{2}$$ $$=$$ $$( 3\beta_{7} + 3\beta_{6} + \beta_{5} + 2\beta_{4} + 3\beta_{3} + \beta _1 + 18 ) / 24$$ (3*b7 + 3*b6 + b5 + 2*b4 + 3*b3 + b1 + 18) / 24 $$\nu^{3}$$ $$=$$ $$( -3\beta_{7} + 3\beta_{6} - 5\beta_{5} + 4\beta_{4} + 3\beta_{3} + 2\beta_{2} + \beta _1 - 6 ) / 24$$ (-3*b7 + 3*b6 - 5*b5 + 4*b4 + 3*b3 + 2*b2 + b1 - 6) / 24 $$\nu^{4}$$ $$=$$ $$( 3\beta_{7} + 17\beta_{6} - 3\beta_{5} + 4\beta_{4} - \beta_{3} + 8\beta_{2} + \beta _1 - 6 ) / 24$$ (3*b7 + 17*b6 - 3*b5 + 4*b4 - b3 + 8*b2 + b1 - 6) / 24 $$\nu^{5}$$ $$=$$ $$( -9\beta_{7} + 8\beta_{6} - 7\beta_{5} + \beta_{4} - \beta_{3} + 18\beta_{2} + 2\beta _1 + 42 ) / 24$$ (-9*b7 + 8*b6 - 7*b5 + b4 - b3 + 18*b2 + 2*b1 + 42) / 24 $$\nu^{6}$$ $$=$$ $$( 3\beta_{6} + 6\beta_{5} + \beta_{4} + 14\beta_{2} - 2\beta_1 ) / 12$$ (3*b6 + 6*b5 + b4 + 14*b2 - 2*b1) / 12 $$\nu^{7}$$ $$=$$ $$( -6\beta_{7} - 17\beta_{6} + 30\beta_{5} + 3\beta_{4} + 4\beta_{3} + 28\beta_{2} + 7\beta _1 + 96 ) / 24$$ (-6*b7 - 17*b6 + 30*b5 + 3*b4 + 4*b3 + 28*b2 + 7*b1 + 96) / 24

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 1.20036 − 0.747754i 1.20036 + 0.747754i −1.27597 + 0.609843i −1.27597 − 0.609843i 1.40994 + 0.109843i 1.40994 − 0.109843i 0.665665 − 1.24775i 0.665665 + 1.24775i
0 0 0 −9.64469 0 1.12019i 0 0 0
703.2 0 0 0 −9.64469 0 1.12019i 0 0 0
703.3 0 0 0 −1.76102 0 12.0363i 0 0 0
703.4 0 0 0 −1.76102 0 12.0363i 0 0 0
703.5 0 0 0 3.22512 0 6.57221i 0 0 0
703.6 0 0 0 3.22512 0 6.57221i 0 0 0
703.7 0 0 0 4.18059 0 2.58429i 0 0 0
703.8 0 0 0 4.18059 0 2.58429i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 703.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.g.k 8
3.b odd 2 1 1728.3.g.n 8
4.b odd 2 1 inner 1728.3.g.k 8
8.b even 2 1 864.3.g.c yes 8
8.d odd 2 1 864.3.g.c yes 8
12.b even 2 1 1728.3.g.n 8
24.f even 2 1 864.3.g.a 8
24.h odd 2 1 864.3.g.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.g.a 8 24.f even 2 1
864.3.g.a 8 24.h odd 2 1
864.3.g.c yes 8 8.b even 2 1
864.3.g.c yes 8 8.d odd 2 1
1728.3.g.k 8 1.a even 1 1 trivial
1728.3.g.k 8 4.b odd 2 1 inner
1728.3.g.n 8 3.b odd 2 1
1728.3.g.n 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} + 4T_{5}^{3} - 54T_{5}^{2} + 28T_{5} + 229$$ T5^4 + 4*T5^3 - 54*T5^2 + 28*T5 + 229 $$T_{7}^{8} + 196T_{7}^{6} + 7758T_{7}^{4} + 51220T_{7}^{2} + 52441$$ T7^8 + 196*T7^6 + 7758*T7^4 + 51220*T7^2 + 52441

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 4 T^{3} - 54 T^{2} + 28 T + 229)^{2}$$
$7$ $$T^{8} + 196 T^{6} + 7758 T^{4} + \cdots + 52441$$
$11$ $$T^{8} + 604 T^{6} + \cdots + 269517889$$
$13$ $$(T^{4} - 8 T^{3} - 288 T^{2} + 640 T + 15616)^{2}$$
$17$ $$(T^{4} - 24 T^{3} - 96 T^{2} + 2304 T + 9216)^{2}$$
$19$ $$T^{8} + 1200 T^{6} + 360288 T^{4} + \cdots + 20736$$
$23$ $$T^{8} + 2128 T^{6} + \cdots + 8962787584$$
$29$ $$(T^{4} - 16 T^{3} - 648 T^{2} - 3520 T - 2288)^{2}$$
$31$ $$T^{8} + 3220 T^{6} + \cdots + 116920969$$
$37$ $$(T^{4} - 48 T^{3} - 3480 T^{2} + \cdots - 354096)^{2}$$
$41$ $$(T^{4} + 64 T^{3} - 2376 T^{2} + \cdots - 3217904)^{2}$$
$43$ $$T^{8} + 7696 T^{6} + \cdots + 947180525824$$
$47$ $$T^{8} + 7216 T^{6} + \cdots + 1218851328256$$
$53$ $$(T^{4} + 84 T^{3} - 1950 T^{2} + \cdots - 1618803)^{2}$$
$59$ $$T^{8} + \cdots + 157628225880064$$
$61$ $$(T^{4} + 16 T^{3} - 11136 T^{2} + \cdots + 3739648)^{2}$$
$67$ $$T^{8} + 23824 T^{6} + \cdots + 54344260884736$$
$71$ $$T^{8} + 15616 T^{6} + \cdots + 55832696848384$$
$73$ $$(T^{4} - 12 T^{3} - 10626 T^{2} + \cdots - 7709463)^{2}$$
$79$ $$T^{8} + 29440 T^{6} + \cdots + 19\!\cdots\!84$$
$83$ $$T^{8} + \cdots + 122448233690689$$
$89$ $$(T^{4} - 312 T^{3} + 21288 T^{2} + \cdots - 3170736)^{2}$$
$97$ $$(T^{4} + 68 T^{3} - 12090 T^{2} + \cdots - 19173311)^{2}$$