# Properties

 Label 432.3.e.c Level $432$ Weight $3$ Character orbit 432.e Analytic conductor $11.771$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 432.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.7711474204$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{5} -5 q^{7} +O(q^{10})$$ $$q + 3 i q^{5} -5 q^{7} -15 i q^{11} -10 q^{13} -18 i q^{17} + 16 q^{19} -12 i q^{23} + 16 q^{25} -30 i q^{29} + q^{31} -15 i q^{35} + 20 q^{37} -60 i q^{41} -50 q^{43} -6 i q^{47} -24 q^{49} + 27 i q^{53} + 45 q^{55} -30 i q^{59} -76 q^{61} -30 i q^{65} + 10 q^{67} -90 i q^{71} + 65 q^{73} + 75 i q^{77} -14 q^{79} + 3 i q^{83} + 54 q^{85} -90 i q^{89} + 50 q^{91} + 48 i q^{95} -85 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{7} + O(q^{10})$$ $$2q - 10q^{7} - 20q^{13} + 32q^{19} + 32q^{25} + 2q^{31} + 40q^{37} - 100q^{43} - 48q^{49} + 90q^{55} - 152q^{61} + 20q^{67} + 130q^{73} - 28q^{79} + 108q^{85} + 100q^{91} - 170q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.00000i 1.00000i
0 0 0 3.00000i 0 −5.00000 0 0 0
161.2 0 0 0 3.00000i 0 −5.00000 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.3.e.c 2
3.b odd 2 1 inner 432.3.e.c 2
4.b odd 2 1 27.3.b.b 2
8.b even 2 1 1728.3.e.g 2
8.d odd 2 1 1728.3.e.m 2
9.c even 3 2 1296.3.q.j 4
9.d odd 6 2 1296.3.q.j 4
12.b even 2 1 27.3.b.b 2
20.d odd 2 1 675.3.c.h 2
20.e even 4 1 675.3.d.a 2
20.e even 4 1 675.3.d.d 2
24.f even 2 1 1728.3.e.m 2
24.h odd 2 1 1728.3.e.g 2
36.f odd 6 2 81.3.d.b 4
36.h even 6 2 81.3.d.b 4
60.h even 2 1 675.3.c.h 2
60.l odd 4 1 675.3.d.a 2
60.l odd 4 1 675.3.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.3.b.b 2 4.b odd 2 1
27.3.b.b 2 12.b even 2 1
81.3.d.b 4 36.f odd 6 2
81.3.d.b 4 36.h even 6 2
432.3.e.c 2 1.a even 1 1 trivial
432.3.e.c 2 3.b odd 2 1 inner
675.3.c.h 2 20.d odd 2 1
675.3.c.h 2 60.h even 2 1
675.3.d.a 2 20.e even 4 1
675.3.d.a 2 60.l odd 4 1
675.3.d.d 2 20.e even 4 1
675.3.d.d 2 60.l odd 4 1
1296.3.q.j 4 9.c even 3 2
1296.3.q.j 4 9.d odd 6 2
1728.3.e.g 2 8.b even 2 1
1728.3.e.g 2 24.h odd 2 1
1728.3.e.m 2 8.d odd 2 1
1728.3.e.m 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}^{2} + 9$$ $$T_{7} + 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + T^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$225 + T^{2}$$
$13$ $$( 10 + T )^{2}$$
$17$ $$324 + T^{2}$$
$19$ $$( -16 + T )^{2}$$
$23$ $$144 + T^{2}$$
$29$ $$900 + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$( -20 + T )^{2}$$
$41$ $$3600 + T^{2}$$
$43$ $$( 50 + T )^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$729 + T^{2}$$
$59$ $$900 + T^{2}$$
$61$ $$( 76 + T )^{2}$$
$67$ $$( -10 + T )^{2}$$
$71$ $$8100 + T^{2}$$
$73$ $$( -65 + T )^{2}$$
$79$ $$( 14 + T )^{2}$$
$83$ $$9 + T^{2}$$
$89$ $$8100 + T^{2}$$
$97$ $$( 85 + T )^{2}$$