# Properties

 Label 1728.3.g.m 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: 8.0.56070144.2 Defining polynomial: $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ x^8 - 4*x^7 + 16*x^6 - 34*x^5 + 63*x^4 - 74*x^3 + 70*x^2 - 38*x + 13 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{5} + (\beta_{6} + \beta_{2}) q^{7}+O(q^{10})$$ q + (b3 + 1) * q^5 + (b6 + b2) * q^7 $$q + (\beta_{3} + 1) q^{5} + (\beta_{6} + \beta_{2}) q^{7} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2}) q^{11} + (\beta_{3} - \beta_1 - 1) q^{13} + ( - \beta_{3} - 2 \beta_1 - 3) q^{17} + (4 \beta_{7} - \beta_{4} + \beta_{2}) q^{19} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + \beta_{2}) q^{23} + (\beta_{5} + \beta_1 + 3) q^{25} + ( - 2 \beta_{5} - 2 \beta_1 - 16) q^{29} + (4 \beta_{7} + 3 \beta_{2}) q^{31} + (3 \beta_{7} + 3 \beta_{6} + 5 \beta_{4} + 5 \beta_{2}) q^{35} + (\beta_{5} - 7 \beta_{3} - 3) q^{37} + (2 \beta_{5} - 2 \beta_1 - 20) q^{41} + ( - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \beta_{2}) q^{43} + ( - 7 \beta_{7} + 3 \beta_{6} - 7 \beta_{4} - 5 \beta_{2}) q^{47} + ( - 10 \beta_{3} - 6 \beta_1 - 18) q^{49} + ( - 2 \beta_{5} + 6 \beta_{3} + 2 \beta_1 - 6) q^{53} + ( - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 9 \beta_{2}) q^{55} + (3 \beta_{7} - \beta_{6} + 9 \beta_{4} - 9 \beta_{2}) q^{59} + (4 \beta_{5} + \beta_{3} + 3 \beta_1 + 17) q^{61} + (2 \beta_{5} - 7 \beta_{3} - 4 \beta_1 + 35) q^{65} + ( - 8 \beta_{7} - \beta_{4} + 2 \beta_{2}) q^{67} + (14 \beta_{7} - 2 \beta_{6} - 14 \beta_{4} + 2 \beta_{2}) q^{71} + (4 \beta_{5} + 6 \beta_{3} - 2 \beta_1 + 9) q^{73} + ( - 2 \beta_{5} + 9 \beta_{3} + 6 \beta_1 + 65) q^{77} + ( - 4 \beta_{7} - 5 \beta_{6} + 4 \beta_{4} - 4 \beta_{2}) q^{79} + ( - 18 \beta_{7} + 2 \beta_{6} + 14 \beta_{4} + 8 \beta_{2}) q^{83} + (\beta_{5} - 12 \beta_{3} - 11 \beta_1 - 12) q^{85} + (4 \beta_{5} - 9 \beta_{3} + 2 \beta_1 + 21) q^{89} + ( - 4 \beta_{7} - 4 \beta_{6} + 5 \beta_{4} - 17 \beta_{2}) q^{91} + ( - \beta_{7} + 5 \beta_{6} - 9 \beta_{4} + 27 \beta_{2}) q^{95} + (5 \beta_{5} - 10 \beta_{3} - \beta_1 + 13) q^{97}+O(q^{100})$$ q + (b3 + 1) * q^5 + (b6 + b2) * q^7 + (-b7 - b6 + b4 - b2) * q^11 + (b3 - b1 - 1) * q^13 + (-b3 - 2*b1 - 3) * q^17 + (4*b7 - b4 + b2) * q^19 + (b7 - b6 - 3*b4 + b2) * q^23 + (b5 + b1 + 3) * q^25 + (-2*b5 - 2*b1 - 16) * q^29 + (4*b7 + 3*b2) * q^31 + (3*b7 + 3*b6 + 5*b4 + 5*b2) * q^35 + (b5 - 7*b3 - 3) * q^37 + (2*b5 - 2*b1 - 20) * q^41 + (-4*b7 + 2*b6 + 2*b4 + b2) * q^43 + (-7*b7 + 3*b6 - 7*b4 - 5*b2) * q^47 + (-10*b3 - 6*b1 - 18) * q^49 + (-2*b5 + 6*b3 + 2*b1 - 6) * q^53 + (-4*b7 - 2*b6 - 2*b4 - 9*b2) * q^55 + (3*b7 - b6 + 9*b4 - 9*b2) * q^59 + (4*b5 + b3 + 3*b1 + 17) * q^61 + (2*b5 - 7*b3 - 4*b1 + 35) * q^65 + (-8*b7 - b4 + 2*b2) * q^67 + (14*b7 - 2*b6 - 14*b4 + 2*b2) * q^71 + (4*b5 + 6*b3 - 2*b1 + 9) * q^73 + (-2*b5 + 9*b3 + 6*b1 + 65) * q^77 + (-4*b7 - 5*b6 + 4*b4 - 4*b2) * q^79 + (-18*b7 + 2*b6 + 14*b4 + 8*b2) * q^83 + (b5 - 12*b3 - 11*b1 - 12) * q^85 + (4*b5 - 9*b3 + 2*b1 + 21) * q^89 + (-4*b7 - 4*b6 + 5*b4 - 17*b2) * q^91 + (-b7 + 5*b6 - 9*b4 + 27*b2) * q^95 + (5*b5 - 10*b3 - b1 + 13) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{5}+O(q^{10})$$ 8 * q + 8 * q^5 $$8 q + 8 q^{5} - 8 q^{13} - 24 q^{17} + 24 q^{25} - 128 q^{29} - 24 q^{37} - 160 q^{41} - 144 q^{49} - 48 q^{53} + 136 q^{61} + 280 q^{65} + 72 q^{73} + 520 q^{77} - 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100})$$ 8 * q + 8 * q^5 - 8 * q^13 - 24 * q^17 + 24 * q^25 - 128 * q^29 - 24 * q^37 - 160 * q^41 - 144 * q^49 - 48 * q^53 + 136 * q^61 + 280 * q^65 + 72 * q^73 + 520 * q^77 - 96 * q^85 + 168 * q^89 + 104 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ :

 $$\beta_{1}$$ $$=$$ $$6\nu^{2} - 6\nu + 15$$ 6*v^2 - 6*v + 15 $$\beta_{2}$$ $$=$$ $$( 24\nu^{7} - 84\nu^{6} + 268\nu^{5} - 460\nu^{4} + 468\nu^{3} - 284\nu^{2} - 164\nu + 116 ) / 37$$ (24*v^7 - 84*v^6 + 268*v^5 - 460*v^4 + 468*v^3 - 284*v^2 - 164*v + 116) / 37 $$\beta_{3}$$ $$=$$ $$2\nu^{6} - 6\nu^{5} + 24\nu^{4} - 38\nu^{3} + 62\nu^{2} - 44\nu + 25$$ 2*v^6 - 6*v^5 + 24*v^4 - 38*v^3 + 62*v^2 - 44*v + 25 $$\beta_{4}$$ $$=$$ $$( -48\nu^{7} + 168\nu^{6} - 684\nu^{5} + 1290\nu^{4} - 2268\nu^{3} + 2196\nu^{2} - 1596\nu + 471 ) / 37$$ (-48*v^7 + 168*v^6 - 684*v^5 + 1290*v^4 - 2268*v^3 + 2196*v^2 - 1596*v + 471) / 37 $$\beta_{5}$$ $$=$$ $$-8\nu^{6} + 24\nu^{5} - 84\nu^{4} + 128\nu^{3} - 170\nu^{2} + 110\nu - 43$$ -8*v^6 + 24*v^5 - 84*v^4 + 128*v^3 - 170*v^2 + 110*v - 43 $$\beta_{6}$$ $$=$$ $$( -100\nu^{7} + 350\nu^{6} - 1314\nu^{5} + 2410\nu^{4} - 3874\nu^{3} + 3576\nu^{2} - 2622\nu + 787 ) / 37$$ (-100*v^7 + 350*v^6 - 1314*v^5 + 2410*v^4 - 3874*v^3 + 3576*v^2 - 2622*v + 787) / 37 $$\beta_{7}$$ $$=$$ $$( -112\nu^{7} + 392\nu^{6} - 1448\nu^{5} + 2640\nu^{4} - 4108\nu^{3} + 3718\nu^{2} - 2318\nu + 618 ) / 37$$ (-112*v^7 + 392*v^6 - 1448*v^5 + 2640*v^4 - 4108*v^3 + 3718*v^2 - 2318*v + 618) / 37
 $$\nu$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + \beta_{2} + 6 ) / 12$$ (2*b7 - 2*b6 + b2 + 6) / 12 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + \beta_{2} + 2\beta _1 - 24 ) / 12$$ (2*b7 - 2*b6 + b2 + 2*b1 - 24) / 12 $$\nu^{3}$$ $$=$$ $$( -7\beta_{7} + 4\beta_{6} + 4\beta_{4} - 8\beta_{2} + 3\beta _1 - 39 ) / 12$$ (-7*b7 + 4*b6 + 4*b4 - 8*b2 + 3*b1 - 39) / 12 $$\nu^{4}$$ $$=$$ $$( -16\beta_{7} + 10\beta_{6} + \beta_{5} + 8\beta_{4} + 4\beta_{3} - 17\beta_{2} - 7\beta _1 + 54 ) / 12$$ (-16*b7 + 10*b6 + b5 + 8*b4 + 4*b3 - 17*b2 - 7*b1 + 54) / 12 $$\nu^{5}$$ $$=$$ $$( 38\beta_{7} - 14\beta_{6} + 5\beta_{5} - 38\beta_{4} + 20\beta_{3} + 43\beta_{2} - 45\beta _1 + 402 ) / 24$$ (38*b7 - 14*b6 + 5*b5 - 38*b4 + 20*b3 + 43*b2 - 45*b1 + 402) / 24 $$\nu^{6}$$ $$=$$ $$( 196\beta_{7} - 94\beta_{6} - 9\beta_{5} - 154\beta_{4} - 24\beta_{3} + 215\beta_{2} + 23\beta _1 - 120 ) / 24$$ (196*b7 - 94*b6 - 9*b5 - 154*b4 - 24*b3 + 215*b2 + 23*b1 - 120) / 24 $$\nu^{7}$$ $$=$$ $$( -4\beta_{7} - 20\beta_{6} - 49\beta_{5} + 36\beta_{4} - 154\beta_{3} + 7\beta_{2} + 245\beta _1 - 1920 ) / 24$$ (-4*b7 - 20*b6 - 49*b5 + 36*b4 - 154*b3 + 7*b2 + 245*b1 - 1920) / 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
 0.5 + 1.19293i 0.5 − 1.19293i 0.5 + 1.56488i 0.5 − 1.56488i 0.5 + 2.19293i 0.5 − 2.19293i 0.5 + 0.564882i 0.5 − 0.564882i
0 0 0 −6.59655 0 4.56106i 0 0 0
703.2 0 0 0 −6.59655 0 4.56106i 0 0 0
703.3 0 0 0 −0.956810 0 6.34610i 0 0 0
703.4 0 0 0 −0.956810 0 6.34610i 0 0 0
703.5 0 0 0 5.13244 0 4.02516i 0 0 0
703.6 0 0 0 5.13244 0 4.02516i 0 0 0
703.7 0 0 0 6.42091 0 13.8102i 0 0 0
703.8 0 0 0 6.42091 0 13.8102i 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.m 8
3.b odd 2 1 1728.3.g.j 8
4.b odd 2 1 inner 1728.3.g.m 8
8.b even 2 1 864.3.g.b 8
8.d odd 2 1 864.3.g.b 8
12.b even 2 1 1728.3.g.j 8
24.f even 2 1 864.3.g.d yes 8
24.h odd 2 1 864.3.g.d yes 8

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

## Hecke kernels

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

 $$T_{5}^{4} - 4T_{5}^{3} - 48T_{5}^{2} + 176T_{5} + 208$$ T5^4 - 4*T5^3 - 48*T5^2 + 176*T5 + 208 $$T_{7}^{8} + 268T_{7}^{6} + 16566T_{7}^{4} + 362092T_{7}^{2} + 2588881$$ T7^8 + 268*T7^6 + 16566*T7^4 + 362092*T7^2 + 2588881

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 4 T^{3} - 48 T^{2} + 176 T + 208)^{2}$$
$7$ $$T^{8} + 268 T^{6} + 16566 T^{4} + \cdots + 2588881$$
$11$ $$T^{8} + 400 T^{6} + 43296 T^{4} + \cdots + 891136$$
$13$ $$(T^{4} + 4 T^{3} - 282 T^{2} - 2300 T - 3167)^{2}$$
$17$ $$(T^{4} + 12 T^{3} - 720 T^{2} - 3024 T + 5328)^{2}$$
$19$ $$T^{8} + 1548 T^{6} + \cdots + 13809305169$$
$23$ $$T^{8} + 1360 T^{6} + \cdots + 522762496$$
$29$ $$(T^{4} + 64 T^{3} - 768 T^{2} + \cdots - 966656)^{2}$$
$31$ $$T^{8} + 1984 T^{6} + \cdots + 7929856$$
$37$ $$(T^{4} + 12 T^{3} - 2826 T^{2} + \cdots - 661167)^{2}$$
$41$ $$(T^{4} + 80 T^{3} - 768 T^{2} + \cdots - 929792)^{2}$$
$43$ $$T^{8} + 2368 T^{6} + \cdots + 8446345216$$
$47$ $$T^{8} + 14800 T^{6} + \cdots + 80138017536256$$
$53$ $$(T^{4} + 24 T^{3} - 4032 T^{2} + \cdots + 3624192)^{2}$$
$59$ $$T^{8} + 16528 T^{6} + \cdots + 1539862591744$$
$61$ $$(T^{4} - 68 T^{3} - 6618 T^{2} + \cdots - 11656223)^{2}$$
$67$ $$T^{8} + 8620 T^{6} + \cdots + 701937325489$$
$71$ $$T^{8} + \cdots + 121012672331776$$
$73$ $$(T^{4} - 36 T^{3} - 12186 T^{2} + \cdots + 1325601)^{2}$$
$79$ $$T^{8} + 8620 T^{6} + \cdots + 40981548721$$
$83$ $$T^{8} + 46912 T^{6} + \cdots + 64\!\cdots\!44$$
$89$ $$(T^{4} - 84 T^{3} - 8784 T^{2} + \cdots - 146736)^{2}$$
$97$ $$(T^{4} - 52 T^{3} - 15114 T^{2} + \cdots + 32388241)^{2}$$