# Properties

 Label 432.2.k.a Level $432$ Weight $2$ Character orbit 432.k Analytic conductor $3.450$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} -\zeta_{8} q^{5} + 3 \zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} -2 q^{4} -\zeta_{8} q^{5} + 3 \zeta_{8}^{2} q^{7} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 1 - \zeta_{8}^{2} ) q^{10} -\zeta_{8} q^{11} + ( -5 + 5 \zeta_{8}^{2} ) q^{13} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + ( -2 + 2 \zeta_{8}^{2} ) q^{19} + 2 \zeta_{8} q^{20} + ( 1 - \zeta_{8}^{2} ) q^{22} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{23} -4 \zeta_{8}^{2} q^{25} -10 \zeta_{8} q^{26} -6 \zeta_{8}^{2} q^{28} + 8 \zeta_{8}^{3} q^{29} - q^{31} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} -6 \zeta_{8}^{2} q^{34} -3 \zeta_{8}^{3} q^{35} + ( 4 + 4 \zeta_{8}^{2} ) q^{37} -4 \zeta_{8} q^{38} + ( -2 + 2 \zeta_{8}^{2} ) q^{40} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{41} + ( 1 + \zeta_{8}^{2} ) q^{43} + 2 \zeta_{8} q^{44} + 8 q^{46} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{47} -2 q^{49} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{50} + ( 10 - 10 \zeta_{8}^{2} ) q^{52} + 11 \zeta_{8} q^{53} + \zeta_{8}^{2} q^{55} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{56} + ( -8 - 8 \zeta_{8}^{2} ) q^{58} -10 \zeta_{8} q^{59} + ( -2 + 2 \zeta_{8}^{2} ) q^{61} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{62} -8 q^{64} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{65} + ( -5 + 5 \zeta_{8}^{2} ) q^{67} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{68} + ( 3 + 3 \zeta_{8}^{2} ) q^{70} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{71} + 3 \zeta_{8}^{2} q^{73} + 8 \zeta_{8}^{3} q^{74} + ( 4 - 4 \zeta_{8}^{2} ) q^{76} -3 \zeta_{8}^{3} q^{77} + 14 q^{79} -4 \zeta_{8} q^{80} -4 q^{82} -7 \zeta_{8}^{3} q^{83} + ( 3 + 3 \zeta_{8}^{2} ) q^{85} + 2 \zeta_{8}^{3} q^{86} + ( -2 + 2 \zeta_{8}^{2} ) q^{88} + ( 11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{89} + ( -15 - 15 \zeta_{8}^{2} ) q^{91} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{92} + 6 \zeta_{8}^{2} q^{94} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{95} + 5 q^{97} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} + 4q^{10} - 20q^{13} + 16q^{16} - 8q^{19} + 4q^{22} - 4q^{31} + 16q^{37} - 8q^{40} + 4q^{43} + 32q^{46} - 8q^{49} + 40q^{52} - 32q^{58} - 8q^{61} - 32q^{64} - 20q^{67} + 12q^{70} + 16q^{76} + 56q^{79} - 16q^{82} + 12q^{85} - 8q^{88} - 60q^{91} + 20q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$1$$ $$\zeta_{8}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.41421i 0 −2.00000 −0.707107 + 0.707107i 0 3.00000i 2.82843i 0 1.00000 + 1.00000i
109.2 1.41421i 0 −2.00000 0.707107 0.707107i 0 3.00000i 2.82843i 0 1.00000 + 1.00000i
325.1 1.41421i 0 −2.00000 0.707107 + 0.707107i 0 3.00000i 2.82843i 0 1.00000 1.00000i
325.2 1.41421i 0 −2.00000 −0.707107 0.707107i 0 3.00000i 2.82843i 0 1.00000 1.00000i
 $$n$$: 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
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.2.k.a 4
3.b odd 2 1 inner 432.2.k.a 4
4.b odd 2 1 1728.2.k.a 4
12.b even 2 1 1728.2.k.a 4
16.e even 4 1 inner 432.2.k.a 4
16.f odd 4 1 1728.2.k.a 4
48.i odd 4 1 inner 432.2.k.a 4
48.k even 4 1 1728.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.2.k.a 4 1.a even 1 1 trivial
432.2.k.a 4 3.b odd 2 1 inner
432.2.k.a 4 16.e even 4 1 inner
432.2.k.a 4 48.i odd 4 1 inner
1728.2.k.a 4 4.b odd 2 1
1728.2.k.a 4 12.b even 2 1
1728.2.k.a 4 16.f odd 4 1
1728.2.k.a 4 48.k even 4 1

## Hecke kernels

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

 $$T_{5}^{4} + 1$$ $$T_{11}^{4} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$1 + T^{4}$$
$7$ $$( 9 + T^{2} )^{2}$$
$11$ $$1 + T^{4}$$
$13$ $$( 50 + 10 T + T^{2} )^{2}$$
$17$ $$( -18 + T^{2} )^{2}$$
$19$ $$( 8 + 4 T + T^{2} )^{2}$$
$23$ $$( 32 + T^{2} )^{2}$$
$29$ $$4096 + T^{4}$$
$31$ $$( 1 + T )^{4}$$
$37$ $$( 32 - 8 T + T^{2} )^{2}$$
$41$ $$( 8 + T^{2} )^{2}$$
$43$ $$( 2 - 2 T + T^{2} )^{2}$$
$47$ $$( -18 + T^{2} )^{2}$$
$53$ $$14641 + T^{4}$$
$59$ $$10000 + T^{4}$$
$61$ $$( 8 + 4 T + T^{2} )^{2}$$
$67$ $$( 50 + 10 T + T^{2} )^{2}$$
$71$ $$( 2 + T^{2} )^{2}$$
$73$ $$( 9 + T^{2} )^{2}$$
$79$ $$( -14 + T )^{4}$$
$83$ $$2401 + T^{4}$$
$89$ $$( 242 + T^{2} )^{2}$$
$97$ $$( -5 + T )^{4}$$