# Properties

 Label 432.2.c.b Level $432$ Weight $2$ Character orbit 432.c Analytic conductor $3.450$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.c (of order $$2$$, degree $$1$$, 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{7} + 7 q^{13} + ( 5 - 10 \zeta_{6} ) q^{19} + 5 q^{25} + ( -6 + 12 \zeta_{6} ) q^{31} - q^{37} + ( 6 - 12 \zeta_{6} ) q^{43} + 4 q^{49} -13 q^{61} + ( -7 + 14 \zeta_{6} ) q^{67} -17 q^{73} + ( -7 + 14 \zeta_{6} ) q^{79} + ( 7 - 14 \zeta_{6} ) q^{91} -5 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 14 q^{13} + 10 q^{25} - 2 q^{37} + 8 q^{49} - 26 q^{61} - 34 q^{73} - 10 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
431.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 1.73205i 0 0 0
431.2 0 0 0 0 0 1.73205i 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 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.c.b 2
3.b odd 2 1 CM 432.2.c.b 2
4.b odd 2 1 inner 432.2.c.b 2
8.b even 2 1 1728.2.c.a 2
8.d odd 2 1 1728.2.c.a 2
9.c even 3 1 1296.2.s.c 2
9.c even 3 1 1296.2.s.d 2
9.d odd 6 1 1296.2.s.c 2
9.d odd 6 1 1296.2.s.d 2
12.b even 2 1 inner 432.2.c.b 2
24.f even 2 1 1728.2.c.a 2
24.h odd 2 1 1728.2.c.a 2
36.f odd 6 1 1296.2.s.c 2
36.f odd 6 1 1296.2.s.d 2
36.h even 6 1 1296.2.s.c 2
36.h even 6 1 1296.2.s.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.c.b 2 1.a even 1 1 trivial
432.2.c.b 2 3.b odd 2 1 CM
432.2.c.b 2 4.b odd 2 1 inner
432.2.c.b 2 12.b even 2 1 inner
1296.2.s.c 2 9.c even 3 1
1296.2.s.c 2 9.d odd 6 1
1296.2.s.c 2 36.f odd 6 1
1296.2.s.c 2 36.h even 6 1
1296.2.s.d 2 9.c even 3 1
1296.2.s.d 2 9.d odd 6 1
1296.2.s.d 2 36.f odd 6 1
1296.2.s.d 2 36.h even 6 1
1728.2.c.a 2 8.b even 2 1
1728.2.c.a 2 8.d odd 2 1
1728.2.c.a 2 24.f even 2 1
1728.2.c.a 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(432, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{2} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$3 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -7 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$75 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$108 + T^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$108 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 13 + T )^{2}$$
$67$ $$147 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 17 + T )^{2}$$
$79$ $$147 + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 5 + T )^{2}$$