# Properties

 Label 1728.3.g.f 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: $$x^{2} - x + 1$$ 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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.f 2
3.b odd 2 1 1728.3.g.a 2
4.b odd 2 1 inner 1728.3.g.f 2
8.b even 2 1 108.3.d.b yes 2
8.d odd 2 1 108.3.d.b yes 2
12.b even 2 1 1728.3.g.a 2
24.f even 2 1 108.3.d.a 2
24.h odd 2 1 108.3.d.a 2
72.j odd 6 1 324.3.f.c 2
72.j odd 6 1 324.3.f.i 2
72.l even 6 1 324.3.f.c 2
72.l even 6 1 324.3.f.i 2
72.n even 6 1 324.3.f.b 2
72.n even 6 1 324.3.f.h 2
72.p odd 6 1 324.3.f.b 2
72.p odd 6 1 324.3.f.h 2

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

## Hecke kernels

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

 $$T_{5} - 7$$ T5 - 7 $$T_{7}^{2} + 75$$ T7^2 + 75

## Hecke characteristic polynomials

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