# Properties

 Label 432.2.i.c Level $432$ Weight $2$ Character orbit 432.i Analytic conductor $3.450$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ 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 $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 1.50000 2.59808i 0 −0.500000 0.866025i 0 0 0
289.1 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 432.2.i.c 2
3.b odd 2 1 144.2.i.a 2
4.b odd 2 1 108.2.e.a 2
8.b even 2 1 1728.2.i.c 2
8.d odd 2 1 1728.2.i.d 2
9.c even 3 1 inner 432.2.i.c 2
9.c even 3 1 1296.2.a.b 1
9.d odd 6 1 144.2.i.a 2
9.d odd 6 1 1296.2.a.k 1
12.b even 2 1 36.2.e.a 2
20.d odd 2 1 2700.2.i.b 2
20.e even 4 2 2700.2.s.b 4
24.f even 2 1 576.2.i.f 2
24.h odd 2 1 576.2.i.e 2
28.d even 2 1 5292.2.j.a 2
28.f even 6 1 5292.2.i.a 2
28.f even 6 1 5292.2.l.c 2
28.g odd 6 1 5292.2.i.c 2
28.g odd 6 1 5292.2.l.a 2
36.f odd 6 1 108.2.e.a 2
36.f odd 6 1 324.2.a.a 1
36.h even 6 1 36.2.e.a 2
36.h even 6 1 324.2.a.c 1
60.h even 2 1 900.2.i.b 2
60.l odd 4 2 900.2.s.b 4
72.j odd 6 1 576.2.i.e 2
72.j odd 6 1 5184.2.a.f 1
72.l even 6 1 576.2.i.f 2
72.l even 6 1 5184.2.a.e 1
72.n even 6 1 1728.2.i.c 2
72.n even 6 1 5184.2.a.bb 1
72.p odd 6 1 1728.2.i.d 2
72.p odd 6 1 5184.2.a.ba 1
84.h odd 2 1 1764.2.j.b 2
84.j odd 6 1 1764.2.i.c 2
84.j odd 6 1 1764.2.l.a 2
84.n even 6 1 1764.2.i.a 2
84.n even 6 1 1764.2.l.c 2
180.n even 6 1 900.2.i.b 2
180.n even 6 1 8100.2.a.j 1
180.p odd 6 1 2700.2.i.b 2
180.p odd 6 1 8100.2.a.g 1
180.v odd 12 2 900.2.s.b 4
180.v odd 12 2 8100.2.d.h 2
180.x even 12 2 2700.2.s.b 4
180.x even 12 2 8100.2.d.c 2
252.n even 6 1 5292.2.i.a 2
252.o even 6 1 1764.2.i.a 2
252.r odd 6 1 1764.2.l.a 2
252.s odd 6 1 1764.2.j.b 2
252.u odd 6 1 5292.2.l.a 2
252.bb even 6 1 1764.2.l.c 2
252.bi even 6 1 5292.2.j.a 2
252.bj even 6 1 5292.2.l.c 2
252.bl odd 6 1 5292.2.i.c 2
252.bn odd 6 1 1764.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 12.b even 2 1
36.2.e.a 2 36.h even 6 1
108.2.e.a 2 4.b odd 2 1
108.2.e.a 2 36.f odd 6 1
144.2.i.a 2 3.b odd 2 1
144.2.i.a 2 9.d odd 6 1
324.2.a.a 1 36.f odd 6 1
324.2.a.c 1 36.h even 6 1
432.2.i.c 2 1.a even 1 1 trivial
432.2.i.c 2 9.c even 3 1 inner
576.2.i.e 2 24.h odd 2 1
576.2.i.e 2 72.j odd 6 1
576.2.i.f 2 24.f even 2 1
576.2.i.f 2 72.l even 6 1
900.2.i.b 2 60.h even 2 1
900.2.i.b 2 180.n even 6 1
900.2.s.b 4 60.l odd 4 2
900.2.s.b 4 180.v odd 12 2
1296.2.a.b 1 9.c even 3 1
1296.2.a.k 1 9.d odd 6 1
1728.2.i.c 2 8.b even 2 1
1728.2.i.c 2 72.n even 6 1
1728.2.i.d 2 8.d odd 2 1
1728.2.i.d 2 72.p odd 6 1
1764.2.i.a 2 84.n even 6 1
1764.2.i.a 2 252.o even 6 1
1764.2.i.c 2 84.j odd 6 1
1764.2.i.c 2 252.bn odd 6 1
1764.2.j.b 2 84.h odd 2 1
1764.2.j.b 2 252.s odd 6 1
1764.2.l.a 2 84.j odd 6 1
1764.2.l.a 2 252.r odd 6 1
1764.2.l.c 2 84.n even 6 1
1764.2.l.c 2 252.bb even 6 1
2700.2.i.b 2 20.d odd 2 1
2700.2.i.b 2 180.p odd 6 1
2700.2.s.b 4 20.e even 4 2
2700.2.s.b 4 180.x even 12 2
5184.2.a.e 1 72.l even 6 1
5184.2.a.f 1 72.j odd 6 1
5184.2.a.ba 1 72.p odd 6 1
5184.2.a.bb 1 72.n even 6 1
5292.2.i.a 2 28.f even 6 1
5292.2.i.a 2 252.n even 6 1
5292.2.i.c 2 28.g odd 6 1
5292.2.i.c 2 252.bl odd 6 1
5292.2.j.a 2 28.d even 2 1
5292.2.j.a 2 252.bi even 6 1
5292.2.l.a 2 28.g odd 6 1
5292.2.l.a 2 252.u odd 6 1
5292.2.l.c 2 28.f even 6 1
5292.2.l.c 2 252.bj even 6 1
8100.2.a.g 1 180.p odd 6 1
8100.2.a.j 1 180.n even 6 1
8100.2.d.c 2 180.x even 12 2
8100.2.d.h 2 180.v odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3 T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$.

## Hecke characteristic polynomials

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