# Properties

 Label 162.2.c.d Level $162$ Weight $2$ Character orbit 162.c Analytic conductor $1.294$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.c (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} - q^{8} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{7} - q^{8} + 3 q^{10} + \zeta_{6} q^{13} -4 \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} -3 q^{17} -4 q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} + ( -4 + 4 \zeta_{6} ) q^{25} + q^{26} -4 q^{28} + ( -9 + 9 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( -3 + 3 \zeta_{6} ) q^{34} + 12 q^{35} - q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} -6 \zeta_{6} q^{41} + ( -8 + 8 \zeta_{6} ) q^{43} + ( 12 - 12 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + ( 1 - \zeta_{6} ) q^{52} -6 q^{53} + ( -4 + 4 \zeta_{6} ) q^{56} + 9 \zeta_{6} q^{58} + ( 1 - \zeta_{6} ) q^{61} + 4 q^{62} + q^{64} + ( -3 + 3 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + ( 12 - 12 \zeta_{6} ) q^{70} -12 q^{71} + 11 q^{73} + ( -1 + \zeta_{6} ) q^{74} + 4 \zeta_{6} q^{76} + ( 16 - 16 \zeta_{6} ) q^{79} -3 q^{80} -6 q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} + 8 \zeta_{6} q^{86} -3 q^{89} + 4 q^{91} -12 \zeta_{6} q^{94} -12 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} -9 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} - q^{4} + 3q^{5} + 4q^{7} - 2q^{8} + O(q^{10})$$ $$2q + q^{2} - q^{4} + 3q^{5} + 4q^{7} - 2q^{8} + 6q^{10} + q^{13} - 4q^{14} - q^{16} - 6q^{17} - 8q^{19} + 3q^{20} - 4q^{25} + 2q^{26} - 8q^{28} - 9q^{29} + 4q^{31} + q^{32} - 3q^{34} + 24q^{35} - 2q^{37} - 4q^{38} - 3q^{40} - 6q^{41} - 8q^{43} + 12q^{47} - 9q^{49} + 4q^{50} + q^{52} - 12q^{53} - 4q^{56} + 9q^{58} + q^{61} + 8q^{62} + 2q^{64} - 3q^{65} + 4q^{67} + 3q^{68} + 12q^{70} - 24q^{71} + 22q^{73} - q^{74} + 4q^{76} + 16q^{79} - 6q^{80} - 12q^{82} + 12q^{83} - 9q^{85} + 8q^{86} - 6q^{89} + 8q^{91} - 12q^{94} - 12q^{95} - 2q^{97} - 18q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$$.

 $$n$$ $$83$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 2.00000 3.46410i −1.00000 0 3.00000
109.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 2.00000 + 3.46410i −1.00000 0 3.00000
 $$n$$: 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 162.2.c.d 2
3.b odd 2 1 162.2.c.a 2
4.b odd 2 1 1296.2.i.n 2
9.c even 3 1 162.2.a.a 1
9.c even 3 1 inner 162.2.c.d 2
9.d odd 6 1 162.2.a.d yes 1
9.d odd 6 1 162.2.c.a 2
12.b even 2 1 1296.2.i.b 2
36.f odd 6 1 1296.2.a.c 1
36.f odd 6 1 1296.2.i.n 2
36.h even 6 1 1296.2.a.l 1
36.h even 6 1 1296.2.i.b 2
45.h odd 6 1 4050.2.a.r 1
45.j even 6 1 4050.2.a.bh 1
45.k odd 12 2 4050.2.c.g 2
45.l even 12 2 4050.2.c.n 2
63.l odd 6 1 7938.2.a.n 1
63.o even 6 1 7938.2.a.s 1
72.j odd 6 1 5184.2.a.c 1
72.l even 6 1 5184.2.a.h 1
72.n even 6 1 5184.2.a.y 1
72.p odd 6 1 5184.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.2.a.a 1 9.c even 3 1
162.2.a.d yes 1 9.d odd 6 1
162.2.c.a 2 3.b odd 2 1
162.2.c.a 2 9.d odd 6 1
162.2.c.d 2 1.a even 1 1 trivial
162.2.c.d 2 9.c even 3 1 inner
1296.2.a.c 1 36.f odd 6 1
1296.2.a.l 1 36.h even 6 1
1296.2.i.b 2 12.b even 2 1
1296.2.i.b 2 36.h even 6 1
1296.2.i.n 2 4.b odd 2 1
1296.2.i.n 2 36.f odd 6 1
4050.2.a.r 1 45.h odd 6 1
4050.2.a.bh 1 45.j even 6 1
4050.2.c.g 2 45.k odd 12 2
4050.2.c.n 2 45.l even 12 2
5184.2.a.c 1 72.j odd 6 1
5184.2.a.h 1 72.l even 6 1
5184.2.a.y 1 72.n even 6 1
5184.2.a.bd 1 72.p odd 6 1
7938.2.a.n 1 63.l odd 6 1
7938.2.a.s 1 63.o even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(162, [\chi])$$:

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{7}^{2} - 4 T_{7} + 16$$