# Properties

 Label 1728.3.g.l Level $1728$ Weight $3$ Character orbit 1728.g Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## 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: 8.0.207360000.1 Defining polynomial: $$x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16$$ x^8 + 6*x^6 + 32*x^4 + 24*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{16}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 108) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + \beta_{7} q^{7}+O(q^{10})$$ q - b1 * q^5 + b7 * q^7 $$q - \beta_1 q^{5} + \beta_{7} q^{7} + \beta_{3} q^{11} + ( - \beta_{6} + 1) q^{13} + (\beta_{4} + 2 \beta_1) q^{17} + \beta_{5} q^{19} + ( - \beta_{3} + \beta_{2}) q^{23} + ( - 2 \beta_{6} + 3) q^{25} - 4 \beta_1 q^{29} + ( - 4 \beta_{7} + 4 \beta_{5}) q^{31} + (\beta_{3} + 2 \beta_{2}) q^{35} + (\beta_{6} + 7) q^{37} + ( - 2 \beta_{4} + 2 \beta_1) q^{41} + (2 \beta_{7} + 6 \beta_{5}) q^{43} + (3 \beta_{3} + \beta_{2}) q^{47} + (2 \beta_{6} - 14) q^{49} + ( - 2 \beta_{4} + 12 \beta_1) q^{53} + (2 \beta_{7} + 6 \beta_{5}) q^{55} + ( - 3 \beta_{3} - 2 \beta_{2}) q^{59} + ( - \beta_{6} - 5) q^{61} + ( - \beta_{4} - 14 \beta_1) q^{65} + ( - 8 \beta_{7} + 9 \beta_{5}) q^{67} + (2 \beta_{3} - 4 \beta_{2}) q^{71} + (2 \beta_{6} + 29) q^{73} + (4 \beta_{4} + 7 \beta_1) q^{77} + (7 \beta_{7} + 16 \beta_{5}) q^{79} - 2 \beta_{3} q^{83} + (6 \beta_{6} - 52) q^{85} + (\beta_{4} + 2 \beta_1) q^{89} + (4 \beta_{7} + 19 \beta_{5}) q^{91} + (\beta_{3} + \beta_{2}) q^{95} + (8 \beta_{6} - 31) q^{97}+O(q^{100})$$ q - b1 * q^5 + b7 * q^7 + b3 * q^11 + (-b6 + 1) * q^13 + (b4 + 2*b1) * q^17 + b5 * q^19 + (-b3 + b2) * q^23 + (-2*b6 + 3) * q^25 - 4*b1 * q^29 + (-4*b7 + 4*b5) * q^31 + (b3 + 2*b2) * q^35 + (b6 + 7) * q^37 + (-2*b4 + 2*b1) * q^41 + (2*b7 + 6*b5) * q^43 + (3*b3 + b2) * q^47 + (2*b6 - 14) * q^49 + (-2*b4 + 12*b1) * q^53 + (2*b7 + 6*b5) * q^55 + (-3*b3 - 2*b2) * q^59 + (-b6 - 5) * q^61 + (-b4 - 14*b1) * q^65 + (-8*b7 + 9*b5) * q^67 + (2*b3 - 4*b2) * q^71 + (2*b6 + 29) * q^73 + (4*b4 + 7*b1) * q^77 + (7*b7 + 16*b5) * q^79 - 2*b3 * q^83 + (6*b6 - 52) * q^85 + (b4 + 2*b1) * q^89 + (4*b7 + 19*b5) * q^91 + (b3 + b2) * q^95 + (8*b6 - 31) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 8 q^{13} + 24 q^{25} + 56 q^{37} - 112 q^{49} - 40 q^{61} + 232 q^{73} - 416 q^{85} - 248 q^{97}+O(q^{100})$$ 8 * q + 8 * q^13 + 24 * q^25 + 56 * q^37 - 112 * q^49 - 40 * q^61 + 232 * q^73 - 416 * q^85 - 248 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{7} + 16\nu^{5} + 64\nu^{3} + 8\nu ) / 32$$ (3*v^7 + 16*v^5 + 64*v^3 + 8*v) / 32 $$\beta_{2}$$ $$=$$ $$( 3\nu^{7} + 16\nu^{5} + 96\nu^{3} + 136\nu ) / 8$$ (3*v^7 + 16*v^5 + 96*v^3 + 136*v) / 8 $$\beta_{3}$$ $$=$$ $$( -15\nu^{7} - 112\nu^{5} - 576\nu^{3} - 808\nu ) / 32$$ (-15*v^7 - 112*v^5 - 576*v^3 - 808*v) / 32 $$\beta_{4}$$ $$=$$ $$( -57\nu^{7} - 304\nu^{5} - 1600\nu^{3} - 152\nu ) / 32$$ (-57*v^7 - 304*v^5 - 1600*v^3 - 152*v) / 32 $$\beta_{5}$$ $$=$$ $$( -9\nu^{6} - 48\nu^{4} - 288\nu^{2} - 120 ) / 32$$ (-9*v^6 - 48*v^4 - 288*v^2 - 120) / 32 $$\beta_{6}$$ $$=$$ $$( -3\nu^{6} + 216 ) / 16$$ (-3*v^6 + 216) / 16 $$\beta_{7}$$ $$=$$ $$( 13\nu^{6} + 80\nu^{4} + 352\nu^{2} + 152 ) / 32$$ (13*v^6 + 80*v^4 + 352*v^2 + 152) / 32
 $$\nu$$ $$=$$ $$( \beta_{4} + 3\beta_{2} + 7\beta_1 ) / 48$$ (b4 + 3*b2 + 7*b1) / 48 $$\nu^{2}$$ $$=$$ $$( -3\beta_{7} + \beta_{6} - 5\beta_{5} - 18 ) / 12$$ (-3*b7 + b6 - 5*b5 - 18) / 12 $$\nu^{3}$$ $$=$$ $$( -\beta_{4} - 19\beta_1 ) / 12$$ (-b4 - 19*b1) / 12 $$\nu^{4}$$ $$=$$ $$( 9\beta_{7} + 3\beta_{6} + 11\beta_{5} - 42 ) / 6$$ (9*b7 + 3*b6 + 11*b5 - 42) / 6 $$\nu^{5}$$ $$=$$ $$( \beta_{4} - 6\beta_{3} - 9\beta_{2} + 25\beta_1 ) / 6$$ (b4 - 6*b3 - 9*b2 + 25*b1) / 6 $$\nu^{6}$$ $$=$$ $$( -16\beta_{6} + 216 ) / 3$$ (-16*b6 + 216) / 3 $$\nu^{7}$$ $$=$$ $$( 5\beta_{4} + 32\beta_{3} + 47\beta_{2} + 131\beta_1 ) / 6$$ (5*b4 + 32*b3 + 47*b2 + 131*b1) / 6

## 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
 1.14412 + 1.98168i 1.14412 − 1.98168i −0.437016 + 0.756934i −0.437016 − 0.756934i 0.437016 − 0.756934i 0.437016 + 0.756934i −1.14412 − 1.98168i −1.14412 + 1.98168i
0 0 0 −7.40492 0 9.47802i 0 0 0
703.2 0 0 0 −7.40492 0 9.47802i 0 0 0
703.3 0 0 0 −1.08036 0 6.01392i 0 0 0
703.4 0 0 0 −1.08036 0 6.01392i 0 0 0
703.5 0 0 0 1.08036 0 6.01392i 0 0 0
703.6 0 0 0 1.08036 0 6.01392i 0 0 0
703.7 0 0 0 7.40492 0 9.47802i 0 0 0
703.8 0 0 0 7.40492 0 9.47802i 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 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.l 8
3.b odd 2 1 inner 1728.3.g.l 8
4.b odd 2 1 inner 1728.3.g.l 8
8.b even 2 1 108.3.d.d 8
8.d odd 2 1 108.3.d.d 8
12.b even 2 1 inner 1728.3.g.l 8
24.f even 2 1 108.3.d.d 8
24.h odd 2 1 108.3.d.d 8
72.j odd 6 1 324.3.f.o 8
72.j odd 6 1 324.3.f.p 8
72.l even 6 1 324.3.f.o 8
72.l even 6 1 324.3.f.p 8
72.n even 6 1 324.3.f.o 8
72.n even 6 1 324.3.f.p 8
72.p odd 6 1 324.3.f.o 8
72.p odd 6 1 324.3.f.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.3.d.d 8 8.b even 2 1
108.3.d.d 8 8.d odd 2 1
108.3.d.d 8 24.f even 2 1
108.3.d.d 8 24.h odd 2 1
324.3.f.o 8 72.j odd 6 1
324.3.f.o 8 72.l even 6 1
324.3.f.o 8 72.n even 6 1
324.3.f.o 8 72.p odd 6 1
324.3.f.p 8 72.j odd 6 1
324.3.f.p 8 72.l even 6 1
324.3.f.p 8 72.n even 6 1
324.3.f.p 8 72.p odd 6 1
1728.3.g.l 8 1.a even 1 1 trivial
1728.3.g.l 8 3.b odd 2 1 inner
1728.3.g.l 8 4.b odd 2 1 inner
1728.3.g.l 8 12.b even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 56T_{5}^{2} + 64$$ T5^4 - 56*T5^2 + 64 $$T_{7}^{4} + 126T_{7}^{2} + 3249$$ T7^4 + 126*T7^2 + 3249

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 56 T^{2} + 64)^{2}$$
$7$ $$(T^{4} + 126 T^{2} + 3249)^{2}$$
$11$ $$(T^{4} + 360 T^{2} + 14400)^{2}$$
$13$ $$(T^{2} - 2 T - 179)^{4}$$
$17$ $$(T^{4} - 1016 T^{2} + 222784)^{2}$$
$19$ $$(T^{2} + 27)^{4}$$
$23$ $$(T^{4} + 1512 T^{2} + 553536)^{2}$$
$29$ $$(T^{4} - 896 T^{2} + 16384)^{2}$$
$31$ $$(T^{4} + 2304 T^{2} + 589824)^{2}$$
$37$ $$(T^{2} - 14 T - 131)^{4}$$
$41$ $$(T^{4} - 3584 T^{2} + 262144)^{2}$$
$43$ $$(T^{4} + 2880 T^{2} + 921600)^{2}$$
$47$ $$(T^{4} + 4392 T^{2} + 4562496)^{2}$$
$53$ $$(T^{4} - 11744 T^{2} + 33547264)^{2}$$
$59$ $$(T^{4} + 7848 T^{2} + 5875776)^{2}$$
$61$ $$(T^{2} + 10 T - 155)^{4}$$
$67$ $$(T^{4} + 9846 T^{2} + 7601049)^{2}$$
$71$ $$(T^{4} + 19872 T^{2} + 90326016)^{2}$$
$73$ $$(T^{2} - 58 T + 121)^{4}$$
$79$ $$(T^{4} + 24030 T^{2} + 37638225)^{2}$$
$83$ $$(T^{4} + 1440 T^{2} + 230400)^{2}$$
$89$ $$(T^{4} - 1016 T^{2} + 222784)^{2}$$
$97$ $$(T^{2} + 62 T - 10559)^{4}$$
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