# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} + \beta_{4} q^{7}+O(q^{10})$$ q + b2 * q^5 + b4 * q^7 $$q + \beta_{2} q^{5} + \beta_{4} q^{7} + \beta_{6} q^{11} + ( - \beta_{5} - 6) q^{13} + (2 \beta_{2} + \beta_1) q^{17} + (\beta_{7} + 2 \beta_{4}) q^{19} + ( - \beta_{6} + \beta_{3}) q^{23} + ( - \beta_{5} - 4) q^{25} + ( - 2 \beta_{2} + 5 \beta_1) q^{29} + ( - \beta_{7} + 5 \beta_{4}) q^{31} + ( - 2 \beta_{6} + 3 \beta_{3}) q^{35} + (3 \beta_{5} + 20) q^{37} + 7 \beta_1 q^{41} + (2 \beta_{7} - 6 \beta_{4}) q^{43} + (2 \beta_{6} + 4 \beta_{3}) q^{47} + (3 \beta_{5} - 4) q^{49} + (9 \beta_{2} - 5 \beta_1) q^{53} + ( - 2 \beta_{7} + 11 \beta_{4}) q^{55} + ( - 2 \beta_{6} + 4 \beta_{3}) q^{59} + 4 q^{61} + ( - 10 \beta_{2} - 25 \beta_1) q^{65} + (4 \beta_{7} + 2 \beta_{4}) q^{67} + (\beta_{6} + 5 \beta_{3}) q^{71} + ( - \beta_{5} + 21) q^{73} + ( - 15 \beta_{2} - 24 \beta_1) q^{77} + ( - 5 \beta_{7} - 8 \beta_{4}) q^{79} + (\beta_{6} + 10 \beta_{3}) q^{83} + ( - 3 \beta_{5} - 62) q^{85} + (14 \beta_{2} + 12 \beta_1) q^{89} + (\beta_{7} - 20 \beta_{4}) q^{91} + (\beta_{6} + 11 \beta_{3}) q^{95} + 27 q^{97}+O(q^{100})$$ q + b2 * q^5 + b4 * q^7 + b6 * q^11 + (-b5 - 6) * q^13 + (2*b2 + b1) * q^17 + (b7 + 2*b4) * q^19 + (-b6 + b3) * q^23 + (-b5 - 4) * q^25 + (-2*b2 + 5*b1) * q^29 + (-b7 + 5*b4) * q^31 + (-2*b6 + 3*b3) * q^35 + (3*b5 + 20) * q^37 + 7*b1 * q^41 + (2*b7 - 6*b4) * q^43 + (2*b6 + 4*b3) * q^47 + (3*b5 - 4) * q^49 + (9*b2 - 5*b1) * q^53 + (-2*b7 + 11*b4) * q^55 + (-2*b6 + 4*b3) * q^59 + 4 * q^61 + (-10*b2 - 25*b1) * q^65 + (4*b7 + 2*b4) * q^67 + (b6 + 5*b3) * q^71 + (-b5 + 21) * q^73 + (-15*b2 - 24*b1) * q^77 + (-5*b7 - 8*b4) * q^79 + (b6 + 10*b3) * q^83 + (-3*b5 - 62) * q^85 + (14*b2 + 12*b1) * q^89 + (b7 - 20*b4) * q^91 + (b6 + 11*b3) * q^95 + 27 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100})$$ 8 * q - 48 * q^13 - 32 * q^25 + 160 * q^37 - 32 * q^49 + 32 * q^61 + 168 * q^73 - 496 * q^85 + 216 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24}$$ -2*v^5 - 2*v^3 + 2*v $$\beta_{2}$$ $$=$$ $$-\zeta_{24}^{5} + 6\zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} - 3$$ -v^5 + 6*v^4 - v^3 + v - 3 $$\beta_{3}$$ $$=$$ $$9\zeta_{24}^{6}$$ 9*v^6 $$\beta_{4}$$ $$=$$ $$-3\zeta_{24}^{6} + 3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 6\zeta_{24}^{2} - 3\zeta_{24}$$ -3*v^6 + 3*v^5 - 3*v^3 + 6*v^2 - 3*v $$\beta_{5}$$ $$=$$ $$12\zeta_{24}^{7} - 6\zeta_{24}^{5} - 6\zeta_{24}^{3} - 6\zeta_{24}$$ 12*v^7 - 6*v^5 - 6*v^3 - 6*v $$\beta_{6}$$ $$=$$ $$12\zeta_{24}^{7} - 3\zeta_{24}^{6} + 6\zeta_{24}^{5} - 6\zeta_{24}^{3} + 6\zeta_{24}$$ 12*v^7 - 3*v^6 + 6*v^5 - 6*v^3 + 6*v $$\beta_{7}$$ $$=$$ $$-6\zeta_{24}^{6} - 12\zeta_{24}^{5} + 12\zeta_{24}^{3} + 12\zeta_{24}^{2} + 12\zeta_{24}$$ -6*v^6 - 12*v^5 + 12*v^3 + 12*v^2 + 12*v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + 3\beta_{6} - 3\beta_{5} - 2\beta_{4} + \beta_{3} + 9\beta_1 ) / 72$$ (b7 + 3*b6 - 3*b5 - 2*b4 + b3 + 9*b1) / 72 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{7} + 4\beta_{4} + 2\beta_{3} ) / 36$$ (b7 + 4*b4 + 2*b3) / 36 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} - 2\beta_{4} - 9\beta_1 ) / 36$$ (b7 - 2*b4 - 9*b1) / 36 $$\zeta_{24}^{4}$$ $$=$$ $$( 2\beta_{2} - \beta _1 + 6 ) / 12$$ (2*b2 - b1 + 6) / 12 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} - 3\beta_{5} + 2\beta_{4} + \beta_{3} - 9\beta_1 ) / 72$$ (-b7 + 3*b6 - 3*b5 + 2*b4 + b3 - 9*b1) / 72 $$\zeta_{24}^{6}$$ $$=$$ $$( \beta_{3} ) / 9$$ (b3) / 9 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + 3\beta_{6} + 3\beta_{5} - 2\beta_{4} + \beta_{3} - 9\beta_1 ) / 72$$ (b7 + 3*b6 + 3*b5 - 2*b4 + b3 - 9*b1) / 72

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 −0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i 0.965926 − 0.258819i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 + 0.965926i −0.965926 − 0.258819i
0 0 0 6.61037i 0 −9.43879 0 0 0
1025.2 0 0 0 6.61037i 0 9.43879 0 0 0
1025.3 0 0 0 3.78194i 0 −0.953512 0 0 0
1025.4 0 0 0 3.78194i 0 0.953512 0 0 0
1025.5 0 0 0 3.78194i 0 −0.953512 0 0 0
1025.6 0 0 0 3.78194i 0 0.953512 0 0 0
1025.7 0 0 0 6.61037i 0 −9.43879 0 0 0
1025.8 0 0 0 6.61037i 0 9.43879 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1025.8 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.e.u 8
3.b odd 2 1 inner 1728.3.e.u 8
4.b odd 2 1 inner 1728.3.e.u 8
8.b even 2 1 864.3.e.f 8
8.d odd 2 1 864.3.e.f 8
12.b even 2 1 inner 1728.3.e.u 8
24.f even 2 1 864.3.e.f 8
24.h odd 2 1 864.3.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.e.f 8 8.b even 2 1
864.3.e.f 8 8.d odd 2 1
864.3.e.f 8 24.f even 2 1
864.3.e.f 8 24.h odd 2 1
1728.3.e.u 8 1.a even 1 1 trivial
1728.3.e.u 8 3.b odd 2 1 inner
1728.3.e.u 8 4.b odd 2 1 inner
1728.3.e.u 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 $$S_{3}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} + 58T_{5}^{2} + 625$$ T5^4 + 58*T5^2 + 625 $$T_{7}^{4} - 90T_{7}^{2} + 81$$ T7^4 - 90*T7^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 58 T^{2} + 625)^{2}$$
$7$ $$(T^{4} - 90 T^{2} + 81)^{2}$$
$11$ $$(T^{4} + 450 T^{2} + 42849)^{2}$$
$13$ $$(T^{2} + 12 T - 180)^{4}$$
$17$ $$(T^{4} + 280 T^{2} + 5776)^{2}$$
$19$ $$(T^{4} - 1008 T^{2} + 129600)^{2}$$
$23$ $$(T^{4} + 720 T^{2} + 5184)^{2}$$
$29$ $$(T^{4} + 472 T^{2} + 400)^{2}$$
$31$ $$(T^{4} - 3402 T^{2} + 1476225)^{2}$$
$37$ $$(T^{2} - 40 T - 1544)^{4}$$
$41$ $$(T^{2} + 392)^{4}$$
$43$ $$(T^{4} - 7272 T^{2} + 11696400)^{2}$$
$47$ $$(T^{4} + 3528 T^{2} + 1296)^{2}$$
$53$ $$(T^{4} + 4378 T^{2} + 4774225)^{2}$$
$59$ $$(T^{4} + 5256 T^{2} + 810000)^{2}$$
$61$ $$(T - 4)^{8}$$
$67$ $$(T^{4} - 12456 T^{2} + 685584)^{2}$$
$71$ $$(T^{4} + 3960 T^{2} + 2396304)^{2}$$
$73$ $$(T^{2} - 42 T + 225)^{4}$$
$79$ $$(T^{4} - 22680 T^{2} + 37896336)^{2}$$
$83$ $$(T^{4} + 15570 T^{2} + 54066609)^{2}$$
$89$ $$(T^{4} + 16360 T^{2} + 5779216)^{2}$$
$97$ $$(T - 27)^{8}$$