# Properties

 Label 162.2.c.b 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: no (minimal twist has level 54) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( 1 - \zeta_{6} ) q^{7} + q^{8} -3 q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + \zeta_{6} q^{14} + ( -1 + \zeta_{6} ) q^{16} + 2 q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} -3 \zeta_{6} q^{22} -6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -4 q^{26} - q^{28} + ( 6 - 6 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -\zeta_{6} q^{32} + 3 q^{35} + 2 q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + 3 \zeta_{6} q^{40} -6 \zeta_{6} q^{41} + ( 10 - 10 \zeta_{6} ) q^{43} + 3 q^{44} + 6 q^{46} + ( 6 - 6 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} -4 \zeta_{6} q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} -9 q^{53} -9 q^{55} + ( 1 - \zeta_{6} ) q^{56} + 6 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} + ( -8 + 8 \zeta_{6} ) q^{61} + 5 q^{62} + q^{64} + ( -12 + 12 \zeta_{6} ) q^{65} -14 \zeta_{6} q^{67} + ( -3 + 3 \zeta_{6} ) q^{70} -7 q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} -2 \zeta_{6} q^{76} + 3 \zeta_{6} q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} -3 q^{80} + 6 q^{82} + ( -3 + 3 \zeta_{6} ) q^{83} + 10 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + 18 q^{89} + 4 q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + 6 \zeta_{6} q^{94} + 6 \zeta_{6} q^{95} + ( 1 - \zeta_{6} ) q^{97} -6 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 3q^{5} + q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 3q^{5} + q^{7} + 2q^{8} - 6q^{10} - 3q^{11} + 4q^{13} + q^{14} - q^{16} + 4q^{19} + 3q^{20} - 3q^{22} - 6q^{23} - 4q^{25} - 8q^{26} - 2q^{28} + 6q^{29} - 5q^{31} - q^{32} + 6q^{35} + 4q^{37} - 2q^{38} + 3q^{40} - 6q^{41} + 10q^{43} + 6q^{44} + 12q^{46} + 6q^{47} + 6q^{49} - 4q^{50} + 4q^{52} - 18q^{53} - 18q^{55} + q^{56} + 6q^{58} + 12q^{59} - 8q^{61} + 10q^{62} + 2q^{64} - 12q^{65} - 14q^{67} - 3q^{70} - 14q^{73} - 2q^{74} - 2q^{76} + 3q^{77} - 8q^{79} - 6q^{80} + 12q^{82} - 3q^{83} + 10q^{86} - 3q^{88} + 36q^{89} + 8q^{91} - 6q^{92} + 6q^{94} + 6q^{95} + q^{97} - 12q^{98} + O(q^{100})$$

## 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 0.500000 0.866025i 1.00000 0 −3.00000
109.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 0.500000 + 0.866025i 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.b 2
3.b odd 2 1 162.2.c.c 2
4.b odd 2 1 1296.2.i.o 2
9.c even 3 1 54.2.a.b yes 1
9.c even 3 1 inner 162.2.c.b 2
9.d odd 6 1 54.2.a.a 1
9.d odd 6 1 162.2.c.c 2
12.b even 2 1 1296.2.i.c 2
36.f odd 6 1 432.2.a.b 1
36.f odd 6 1 1296.2.i.o 2
36.h even 6 1 432.2.a.g 1
36.h even 6 1 1296.2.i.c 2
45.h odd 6 1 1350.2.a.r 1
45.j even 6 1 1350.2.a.h 1
45.k odd 12 2 1350.2.c.k 2
45.l even 12 2 1350.2.c.b 2
63.l odd 6 1 2646.2.a.bd 1
63.o even 6 1 2646.2.a.a 1
72.j odd 6 1 1728.2.a.c 1
72.l even 6 1 1728.2.a.d 1
72.n even 6 1 1728.2.a.y 1
72.p odd 6 1 1728.2.a.z 1
99.g even 6 1 6534.2.a.bc 1
99.h odd 6 1 6534.2.a.b 1
117.n odd 6 1 9126.2.a.u 1
117.t even 6 1 9126.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 9.d odd 6 1
54.2.a.b yes 1 9.c even 3 1
162.2.c.b 2 1.a even 1 1 trivial
162.2.c.b 2 9.c even 3 1 inner
162.2.c.c 2 3.b odd 2 1
162.2.c.c 2 9.d odd 6 1
432.2.a.b 1 36.f odd 6 1
432.2.a.g 1 36.h even 6 1
1296.2.i.c 2 12.b even 2 1
1296.2.i.c 2 36.h even 6 1
1296.2.i.o 2 4.b odd 2 1
1296.2.i.o 2 36.f odd 6 1
1350.2.a.h 1 45.j even 6 1
1350.2.a.r 1 45.h odd 6 1
1350.2.c.b 2 45.l even 12 2
1350.2.c.k 2 45.k odd 12 2
1728.2.a.c 1 72.j odd 6 1
1728.2.a.d 1 72.l even 6 1
1728.2.a.y 1 72.n even 6 1
1728.2.a.z 1 72.p odd 6 1
2646.2.a.a 1 63.o even 6 1
2646.2.a.bd 1 63.l odd 6 1
6534.2.a.b 1 99.h odd 6 1
6534.2.a.bc 1 99.g even 6 1
9126.2.a.r 1 117.t even 6 1
9126.2.a.u 1 117.n odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{7}^{2} - T_{7} + 1$$