# Properties

 Label 1728.2.r.c Level $1728$ Weight $2$ Character orbit 1728.r Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM discriminant -8 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + ( - \beta_{3} - \beta_1) q^{11} + (\beta_{7} + \beta_{5} - 3) q^{17} + ( - 3 \beta_{6} - 2 \beta_{4}) q^{19} + (5 \beta_{2} - 5) q^{25} + (2 \beta_{7} + 3 \beta_{2}) q^{41} + (3 \beta_{3} - 4 \beta_1) q^{43} + 7 \beta_{2} q^{49} + ( - 5 \beta_{6} - \beta_{4} - 5 \beta_{3} + 4 \beta_1) q^{59} + ( - 3 \beta_{6} - 5 \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{67} + (3 \beta_{7} + 3 \beta_{5} - 1) q^{73} - 9 \beta_1 q^{83} - 18 q^{89} + (3 \beta_{5} - 5 \beta_{2} + 5) q^{97}+O(q^{100})$$ q + (-b3 - b1) * q^11 + (b7 + b5 - 3) * q^17 + (-3*b6 - 2*b4) * q^19 + (5*b2 - 5) * q^25 + (2*b7 + 3*b2) * q^41 + (3*b3 - 4*b1) * q^43 + 7*b2 * q^49 + (-5*b6 - b4 - 5*b3 + 4*b1) * q^59 + (-3*b6 - 5*b4 - 3*b3 - 2*b1) * q^67 + (3*b7 + 3*b5 - 1) * q^73 - 9*b1 * q^83 - 18 * q^89 + (3*b5 - 5*b2 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 24 q^{17} - 20 q^{25} + 12 q^{41} + 28 q^{49} - 8 q^{73} - 144 q^{89} + 20 q^{97}+O(q^{100})$$ 8 * q - 24 * q^17 - 20 * q^25 + 12 * q^41 + 28 * q^49 - 8 * q^73 - 144 * q^89 + 20 * q^97

Basis of coefficient ring

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

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
0 0 0 0 0 0 0 0 0
289.2 0 0 0 0 0 0 0 0 0
289.3 0 0 0 0 0 0 0 0 0
289.4 0 0 0 0 0 0 0 0 0
1441.1 0 0 0 0 0 0 0 0 0
1441.2 0 0 0 0 0 0 0 0 0
1441.3 0 0 0 0 0 0 0 0 0
1441.4 0 0 0 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1441.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.r.c 8
3.b odd 2 1 576.2.r.c 8
4.b odd 2 1 inner 1728.2.r.c 8
8.b even 2 1 inner 1728.2.r.c 8
8.d odd 2 1 CM 1728.2.r.c 8
9.c even 3 1 inner 1728.2.r.c 8
9.c even 3 1 5184.2.d.e 4
9.d odd 6 1 576.2.r.c 8
9.d odd 6 1 5184.2.d.l 4
12.b even 2 1 576.2.r.c 8
24.f even 2 1 576.2.r.c 8
24.h odd 2 1 576.2.r.c 8
36.f odd 6 1 inner 1728.2.r.c 8
36.f odd 6 1 5184.2.d.e 4
36.h even 6 1 576.2.r.c 8
36.h even 6 1 5184.2.d.l 4
72.j odd 6 1 576.2.r.c 8
72.j odd 6 1 5184.2.d.l 4
72.l even 6 1 576.2.r.c 8
72.l even 6 1 5184.2.d.l 4
72.n even 6 1 inner 1728.2.r.c 8
72.n even 6 1 5184.2.d.e 4
72.p odd 6 1 inner 1728.2.r.c 8
72.p odd 6 1 5184.2.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.c 8 3.b odd 2 1
576.2.r.c 8 9.d odd 6 1
576.2.r.c 8 12.b even 2 1
576.2.r.c 8 24.f even 2 1
576.2.r.c 8 24.h odd 2 1
576.2.r.c 8 36.h even 6 1
576.2.r.c 8 72.j odd 6 1
576.2.r.c 8 72.l even 6 1
1728.2.r.c 8 1.a even 1 1 trivial
1728.2.r.c 8 4.b odd 2 1 inner
1728.2.r.c 8 8.b even 2 1 inner
1728.2.r.c 8 8.d odd 2 1 CM
1728.2.r.c 8 9.c even 3 1 inner
1728.2.r.c 8 36.f odd 6 1 inner
1728.2.r.c 8 72.n even 6 1 inner
1728.2.r.c 8 72.p odd 6 1 inner
5184.2.d.e 4 9.c even 3 1
5184.2.d.e 4 36.f odd 6 1
5184.2.d.e 4 72.n even 6 1
5184.2.d.e 4 72.p odd 6 1
5184.2.d.l 4 9.d odd 6 1
5184.2.d.l 4 36.h even 6 1
5184.2.d.l 4 72.j odd 6 1
5184.2.d.l 4 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81$$
$13$ $$T^{8}$$
$17$ $$(T^{2} + 6 T - 15)^{4}$$
$19$ $$(T^{4} + 110 T^{2} + 2809)^{2}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$(T^{4} - 6 T^{3} + 123 T^{2} + 522 T + 7569)^{2}$$
$43$ $$T^{8} - 158 T^{6} + 24123 T^{4} + \cdots + 707281$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8} - 318 T^{6} + \cdots + 395254161$$
$61$ $$T^{8}$$
$67$ $$T^{8} - 206 T^{6} + 42411 T^{4} + \cdots + 625$$
$71$ $$T^{8}$$
$73$ $$(T^{2} + 2 T - 215)^{4}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} - 324 T^{2} + 104976)^{2}$$
$89$ $$(T + 18)^{8}$$
$97$ $$(T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481)^{2}$$