# Properties

 Label 162.2.c.a 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.a 2
3.b odd 2 1 162.2.c.d 2
4.b odd 2 1 1296.2.i.b 2
9.c even 3 1 162.2.a.d yes 1
9.c even 3 1 inner 162.2.c.a 2
9.d odd 6 1 162.2.a.a 1
9.d odd 6 1 162.2.c.d 2
12.b even 2 1 1296.2.i.n 2
36.f odd 6 1 1296.2.a.l 1
36.f odd 6 1 1296.2.i.b 2
36.h even 6 1 1296.2.a.c 1
36.h even 6 1 1296.2.i.n 2
45.h odd 6 1 4050.2.a.bh 1
45.j even 6 1 4050.2.a.r 1
45.k odd 12 2 4050.2.c.n 2
45.l even 12 2 4050.2.c.g 2
63.l odd 6 1 7938.2.a.s 1
63.o even 6 1 7938.2.a.n 1
72.j odd 6 1 5184.2.a.y 1
72.l even 6 1 5184.2.a.bd 1
72.n even 6 1 5184.2.a.c 1
72.p odd 6 1 5184.2.a.h 1

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 - 5 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$
$11$ $$1 - 11 T^{2} + 121 T^{4}$$
$13$ $$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 3 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} )$$
$37$ $$( 1 + T + 37 T^{2} )^{2}$$
$41$ $$1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 5 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 + 12 T + 97 T^{2} + 564 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 6 T + 53 T^{2} )^{2}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 11 T + 73 T^{2} )^{2}$$
$79$ $$1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4}$$
$83$ $$1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 3 T + 89 T^{2} )^{2}$$
$97$ $$1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4}$$