# Properties

 Label 1728.2.i.d Level $1728$ Weight $2$ Character orbit 1728.i Analytic conductor $13.798$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 36) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -3 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} -\zeta_{6} q^{13} -6 q^{17} + 4 q^{19} -3 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -3 + 3 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -3 q^{35} -2 q^{37} + 3 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( -9 + 9 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} -6 q^{53} + 9 q^{55} + 3 \zeta_{6} q^{59} + ( -13 + 13 \zeta_{6} ) q^{61} + ( -3 + 3 \zeta_{6} ) q^{65} -7 \zeta_{6} q^{67} + 12 q^{71} -10 q^{73} + 3 \zeta_{6} q^{77} + ( -11 + 11 \zeta_{6} ) q^{79} + ( 9 - 9 \zeta_{6} ) q^{83} + 18 \zeta_{6} q^{85} -6 q^{89} - q^{91} -12 \zeta_{6} q^{95} + ( -11 + 11 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{5} + q^{7} + O(q^{10})$$ $$2q - 3q^{5} + q^{7} - 3q^{11} - q^{13} - 12q^{17} + 8q^{19} - 3q^{23} - 4q^{25} - 3q^{29} - 5q^{31} - 6q^{35} - 4q^{37} + 3q^{41} - q^{43} - 9q^{47} + 6q^{49} - 12q^{53} + 18q^{55} + 3q^{59} - 13q^{61} - 3q^{65} - 7q^{67} + 24q^{71} - 20q^{73} + 3q^{77} - 11q^{79} + 9q^{83} + 18q^{85} - 12q^{89} - 2q^{91} - 12q^{95} - 11q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 0 0
1153.1 0 0 0 −1.50000 2.59808i 0 0.500000 0.866025i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.d 2
3.b odd 2 1 576.2.i.f 2
4.b odd 2 1 1728.2.i.c 2
8.b even 2 1 108.2.e.a 2
8.d odd 2 1 432.2.i.c 2
9.c even 3 1 inner 1728.2.i.d 2
9.c even 3 1 5184.2.a.ba 1
9.d odd 6 1 576.2.i.f 2
9.d odd 6 1 5184.2.a.e 1
12.b even 2 1 576.2.i.e 2
24.f even 2 1 144.2.i.a 2
24.h odd 2 1 36.2.e.a 2
36.f odd 6 1 1728.2.i.c 2
36.f odd 6 1 5184.2.a.bb 1
36.h even 6 1 576.2.i.e 2
36.h even 6 1 5184.2.a.f 1
40.f even 2 1 2700.2.i.b 2
40.i odd 4 2 2700.2.s.b 4
56.h odd 2 1 5292.2.j.a 2
56.j odd 6 1 5292.2.i.a 2
56.j odd 6 1 5292.2.l.c 2
56.p even 6 1 5292.2.i.c 2
56.p even 6 1 5292.2.l.a 2
72.j odd 6 1 36.2.e.a 2
72.j odd 6 1 324.2.a.c 1
72.l even 6 1 144.2.i.a 2
72.l even 6 1 1296.2.a.k 1
72.n even 6 1 108.2.e.a 2
72.n even 6 1 324.2.a.a 1
72.p odd 6 1 432.2.i.c 2
72.p odd 6 1 1296.2.a.b 1
120.i odd 2 1 900.2.i.b 2
120.w even 4 2 900.2.s.b 4
168.i even 2 1 1764.2.j.b 2
168.s odd 6 1 1764.2.i.a 2
168.s odd 6 1 1764.2.l.c 2
168.ba even 6 1 1764.2.i.c 2
168.ba even 6 1 1764.2.l.a 2
360.bh odd 6 1 900.2.i.b 2
360.bh odd 6 1 8100.2.a.j 1
360.bk even 6 1 2700.2.i.b 2
360.bk even 6 1 8100.2.a.g 1
360.br even 12 2 900.2.s.b 4
360.br even 12 2 8100.2.d.h 2
360.bu odd 12 2 2700.2.s.b 4
360.bu odd 12 2 8100.2.d.c 2
504.w even 6 1 5292.2.i.c 2
504.y even 6 1 1764.2.i.c 2
504.bi odd 6 1 1764.2.l.c 2
504.bn odd 6 1 5292.2.j.a 2
504.bp odd 6 1 5292.2.l.c 2
504.ca even 6 1 1764.2.l.a 2
504.cc even 6 1 1764.2.j.b 2
504.cq even 6 1 5292.2.l.a 2
504.cw odd 6 1 5292.2.i.a 2
504.db odd 6 1 1764.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 24.h odd 2 1
36.2.e.a 2 72.j odd 6 1
108.2.e.a 2 8.b even 2 1
108.2.e.a 2 72.n even 6 1
144.2.i.a 2 24.f even 2 1
144.2.i.a 2 72.l even 6 1
324.2.a.a 1 72.n even 6 1
324.2.a.c 1 72.j odd 6 1
432.2.i.c 2 8.d odd 2 1
432.2.i.c 2 72.p odd 6 1
576.2.i.e 2 12.b even 2 1
576.2.i.e 2 36.h even 6 1
576.2.i.f 2 3.b odd 2 1
576.2.i.f 2 9.d odd 6 1
900.2.i.b 2 120.i odd 2 1
900.2.i.b 2 360.bh odd 6 1
900.2.s.b 4 120.w even 4 2
900.2.s.b 4 360.br even 12 2
1296.2.a.b 1 72.p odd 6 1
1296.2.a.k 1 72.l even 6 1
1728.2.i.c 2 4.b odd 2 1
1728.2.i.c 2 36.f odd 6 1
1728.2.i.d 2 1.a even 1 1 trivial
1728.2.i.d 2 9.c even 3 1 inner
1764.2.i.a 2 168.s odd 6 1
1764.2.i.a 2 504.db odd 6 1
1764.2.i.c 2 168.ba even 6 1
1764.2.i.c 2 504.y even 6 1
1764.2.j.b 2 168.i even 2 1
1764.2.j.b 2 504.cc even 6 1
1764.2.l.a 2 168.ba even 6 1
1764.2.l.a 2 504.ca even 6 1
1764.2.l.c 2 168.s odd 6 1
1764.2.l.c 2 504.bi odd 6 1
2700.2.i.b 2 40.f even 2 1
2700.2.i.b 2 360.bk even 6 1
2700.2.s.b 4 40.i odd 4 2
2700.2.s.b 4 360.bu odd 12 2
5184.2.a.e 1 9.d odd 6 1
5184.2.a.f 1 36.h even 6 1
5184.2.a.ba 1 9.c even 3 1
5184.2.a.bb 1 36.f odd 6 1
5292.2.i.a 2 56.j odd 6 1
5292.2.i.a 2 504.cw odd 6 1
5292.2.i.c 2 56.p even 6 1
5292.2.i.c 2 504.w even 6 1
5292.2.j.a 2 56.h odd 2 1
5292.2.j.a 2 504.bn odd 6 1
5292.2.l.a 2 56.p even 6 1
5292.2.l.a 2 504.cq even 6 1
5292.2.l.c 2 56.j odd 6 1
5292.2.l.c 2 504.bp odd 6 1
8100.2.a.g 1 360.bk even 6 1
8100.2.a.j 1 360.bh odd 6 1
8100.2.d.c 2 360.bu odd 12 2
8100.2.d.h 2 360.br even 12 2

## Hecke kernels

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

 $$T_{5}^{2} + 3 T_{5} + 9$$ $$T_{7}^{2} - T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$1 - T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$25 + 5 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$169 + 13 T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( 10 + T )^{2}$$
$79$ $$121 + 11 T + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$121 + 11 T + T^{2}$$