# Properties

 Label 1728.3.g.a Level $1728$ Weight $3$ Character orbit 1728.g Analytic conductor $47.085$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -7 q^{5} + ( 5 - 10 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -7 q^{5} + ( 5 - 10 \zeta_{6} ) q^{7} + ( 5 - 10 \zeta_{6} ) q^{11} -20 q^{13} -8 q^{17} + ( 6 - 12 \zeta_{6} ) q^{19} + ( 2 - 4 \zeta_{6} ) q^{23} + 24 q^{25} -10 q^{29} + ( 31 - 62 \zeta_{6} ) q^{31} + ( -35 + 70 \zeta_{6} ) q^{35} + 10 q^{37} -50 q^{41} + ( 10 - 20 \zeta_{6} ) q^{43} + ( -50 + 100 \zeta_{6} ) q^{47} -26 q^{49} + 47 q^{53} + ( -35 + 70 \zeta_{6} ) q^{55} + ( -20 + 40 \zeta_{6} ) q^{59} + 64 q^{61} + 140 q^{65} + ( 50 - 100 \zeta_{6} ) q^{67} -55 q^{73} -75 q^{77} + ( -4 + 8 \zeta_{6} ) q^{79} + ( 17 - 34 \zeta_{6} ) q^{83} + 56 q^{85} + 10 q^{89} + ( -100 + 200 \zeta_{6} ) q^{91} + ( -42 + 84 \zeta_{6} ) q^{95} -25 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{5} + O(q^{10})$$ $$2 q - 14 q^{5} - 40 q^{13} - 16 q^{17} + 48 q^{25} - 20 q^{29} + 20 q^{37} - 100 q^{41} - 52 q^{49} + 94 q^{53} + 128 q^{61} + 280 q^{65} - 110 q^{73} - 150 q^{77} + 112 q^{85} + 20 q^{89} - 50 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −7.00000 0 8.66025i 0 0 0
703.2 0 0 0 −7.00000 0 8.66025i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.a 2
3.b odd 2 1 1728.3.g.f 2
4.b odd 2 1 inner 1728.3.g.a 2
8.b even 2 1 108.3.d.a 2
8.d odd 2 1 108.3.d.a 2
12.b even 2 1 1728.3.g.f 2
24.f even 2 1 108.3.d.b yes 2
24.h odd 2 1 108.3.d.b yes 2
72.j odd 6 1 324.3.f.b 2
72.j odd 6 1 324.3.f.h 2
72.l even 6 1 324.3.f.b 2
72.l even 6 1 324.3.f.h 2
72.n even 6 1 324.3.f.c 2
72.n even 6 1 324.3.f.i 2
72.p odd 6 1 324.3.f.c 2
72.p odd 6 1 324.3.f.i 2

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

## Hecke kernels

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

 $$T_{5} + 7$$ $$T_{7}^{2} + 75$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 7 + T )^{2}$$
$7$ $$75 + T^{2}$$
$11$ $$75 + T^{2}$$
$13$ $$( 20 + T )^{2}$$
$17$ $$( 8 + T )^{2}$$
$19$ $$108 + T^{2}$$
$23$ $$12 + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$2883 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$( 50 + T )^{2}$$
$43$ $$300 + T^{2}$$
$47$ $$7500 + T^{2}$$
$53$ $$( -47 + T )^{2}$$
$59$ $$1200 + T^{2}$$
$61$ $$( -64 + T )^{2}$$
$67$ $$7500 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 55 + T )^{2}$$
$79$ $$48 + T^{2}$$
$83$ $$867 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( 25 + T )^{2}$$