# Properties

 Label 1728.2.c.g Level $1728$ Weight $2$ Character orbit 1728.c Analytic conductor $13.798$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ 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 primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} +O(q^{10})$$ $$q + ( 1 + \zeta_{24} - \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{11} + ( 2 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{13} + ( 2 - 4 \zeta_{24}^{4} ) q^{17} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{23} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{25} + ( 2 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{29} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} - 6 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{35} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 4 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{41} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{43} + ( -4 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{47} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + ( 5 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 10 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{53} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 7 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{55} + ( -4 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{59} -4 q^{61} + ( 6 + 8 \zeta_{24} - 8 \zeta_{24}^{3} - 12 \zeta_{24}^{4} - 8 \zeta_{24}^{5} ) q^{65} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 10 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{67} + ( -2 \zeta_{24} - 12 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{71} + ( -3 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 5 + 5 \zeta_{24} - 5 \zeta_{24}^{3} - 10 \zeta_{24}^{4} - 5 \zeta_{24}^{5} ) q^{77} -2 \zeta_{24}^{6} q^{79} + ( 10 \zeta_{24} - 2 \zeta_{24}^{2} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{83} + ( -6 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( 2 - 6 \zeta_{24} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{89} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 14 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{91} + ( 6 \zeta_{24} + 8 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{95} -5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 16 q^{13} - 32 q^{61} - 24 q^{73} - 48 q^{85} - 40 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1727.1
 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 0.965926 − 0.258819i
0 0 0 3.14626i 0 3.44949i 0 0 0
1727.2 0 0 0 3.14626i 0 3.44949i 0 0 0
1727.3 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.4 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.5 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.6 0 0 0 0.317837i 0 1.44949i 0 0 0
1727.7 0 0 0 3.14626i 0 3.44949i 0 0 0
1727.8 0 0 0 3.14626i 0 3.44949i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.8 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 1728.2.c.g 8
3.b odd 2 1 inner 1728.2.c.g 8
4.b odd 2 1 inner 1728.2.c.g 8
8.b even 2 1 864.2.c.a 8
8.d odd 2 1 864.2.c.a 8
12.b even 2 1 inner 1728.2.c.g 8
24.f even 2 1 864.2.c.a 8
24.h odd 2 1 864.2.c.a 8
72.j odd 6 1 2592.2.s.a 8
72.j odd 6 1 2592.2.s.h 8
72.l even 6 1 2592.2.s.a 8
72.l even 6 1 2592.2.s.h 8
72.n even 6 1 2592.2.s.a 8
72.n even 6 1 2592.2.s.h 8
72.p odd 6 1 2592.2.s.a 8
72.p odd 6 1 2592.2.s.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.a 8 8.b even 2 1
864.2.c.a 8 8.d odd 2 1
864.2.c.a 8 24.f even 2 1
864.2.c.a 8 24.h odd 2 1
1728.2.c.g 8 1.a even 1 1 trivial
1728.2.c.g 8 3.b odd 2 1 inner
1728.2.c.g 8 4.b odd 2 1 inner
1728.2.c.g 8 12.b even 2 1 inner
2592.2.s.a 8 72.j odd 6 1
2592.2.s.a 8 72.l even 6 1
2592.2.s.a 8 72.n even 6 1
2592.2.s.a 8 72.p odd 6 1
2592.2.s.h 8 72.j odd 6 1
2592.2.s.h 8 72.l even 6 1
2592.2.s.h 8 72.n even 6 1
2592.2.s.h 8 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{4} + 10 T_{5}^{2} + 1$$ $$T_{7}^{4} + 14 T_{7}^{2} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 1 + 10 T^{2} + T^{4} )^{2}$$
$7$ $$( 25 + 14 T^{2} + T^{4} )^{2}$$
$11$ $$( 25 - 22 T^{2} + T^{4} )^{2}$$
$13$ $$( -20 - 4 T + T^{2} )^{4}$$
$17$ $$( 12 + T^{2} )^{4}$$
$19$ $$( 24 + T^{2} )^{4}$$
$23$ $$( -8 + T^{2} )^{4}$$
$29$ $$( 400 + 88 T^{2} + T^{4} )^{2}$$
$31$ $$( 361 + 62 T^{2} + T^{4} )^{2}$$
$37$ $$( -24 + T^{2} )^{4}$$
$41$ $$( 1600 + 112 T^{2} + T^{4} )^{2}$$
$43$ $$( 400 + 56 T^{2} + T^{4} )^{2}$$
$47$ $$( 400 - 88 T^{2} + T^{4} )^{2}$$
$53$ $$( 3249 + 186 T^{2} + T^{4} )^{2}$$
$59$ $$( 400 - 88 T^{2} + T^{4} )^{2}$$
$61$ $$( 4 + T )^{8}$$
$67$ $$( 5776 + 248 T^{2} + T^{4} )^{2}$$
$71$ $$( 10000 - 232 T^{2} + T^{4} )^{2}$$
$73$ $$( -15 + 6 T + T^{2} )^{4}$$
$79$ $$( 4 + T^{2} )^{4}$$
$83$ $$( 38809 - 406 T^{2} + T^{4} )^{2}$$
$89$ $$( 3600 + 168 T^{2} + T^{4} )^{2}$$
$97$ $$( 5 + T )^{8}$$