# Properties

 Label 162.4.c.i Level $162$ Weight $4$ Character orbit 162.c Analytic conductor $9.558$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.55830942093$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{12}^{2} ) q^{2} -4 \zeta_{12}^{2} q^{4} + ( 3 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{5} + ( -8 + 6 \zeta_{12} + 8 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{7} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{12}^{2} ) q^{2} -4 \zeta_{12}^{2} q^{4} + ( 3 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{5} + ( -8 + 6 \zeta_{12} + 8 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{7} + 8 q^{8} + ( 12 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{10} + ( -18 - 24 \zeta_{12} + 18 \zeta_{12}^{2} + 48 \zeta_{12}^{3} ) q^{11} + ( -42 \zeta_{12} - 5 \zeta_{12}^{2} - 42 \zeta_{12}^{3} ) q^{13} + ( 12 \zeta_{12} - 16 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{14} + ( -16 + 16 \zeta_{12}^{2} ) q^{16} + ( 60 - 66 \zeta_{12} + 33 \zeta_{12}^{3} ) q^{17} + ( -4 - 132 \zeta_{12} + 66 \zeta_{12}^{3} ) q^{19} + ( -24 + 12 \zeta_{12} + 24 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{20} + ( -48 \zeta_{12} - 36 \zeta_{12}^{2} - 48 \zeta_{12}^{3} ) q^{22} + ( -12 \zeta_{12} - 90 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{23} + ( 62 + 36 \zeta_{12} - 62 \zeta_{12}^{2} - 72 \zeta_{12}^{3} ) q^{25} + ( 10 + 168 \zeta_{12} - 84 \zeta_{12}^{3} ) q^{26} + ( 32 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{28} + ( -162 + 21 \zeta_{12} + 162 \zeta_{12}^{2} - 42 \zeta_{12}^{3} ) q^{29} + ( -108 \zeta_{12} + 124 \zeta_{12}^{2} - 108 \zeta_{12}^{3} ) q^{31} -32 \zeta_{12}^{2} q^{32} + ( -120 + 66 \zeta_{12} + 120 \zeta_{12}^{2} - 132 \zeta_{12}^{3} ) q^{34} + ( 102 - 120 \zeta_{12} + 60 \zeta_{12}^{3} ) q^{35} + ( -217 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{37} + ( 8 + 132 \zeta_{12} - 8 \zeta_{12}^{2} - 264 \zeta_{12}^{3} ) q^{38} + ( 24 \zeta_{12} - 48 \zeta_{12}^{2} + 24 \zeta_{12}^{3} ) q^{40} + ( -132 \zeta_{12} - 96 \zeta_{12}^{2} - 132 \zeta_{12}^{3} ) q^{41} + ( 304 - 18 \zeta_{12} - 304 \zeta_{12}^{2} + 36 \zeta_{12}^{3} ) q^{43} + ( 72 + 192 \zeta_{12} - 96 \zeta_{12}^{3} ) q^{44} + ( 180 + 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{46} + ( 192 - 108 \zeta_{12} - 192 \zeta_{12}^{2} + 216 \zeta_{12}^{3} ) q^{47} + ( 96 \zeta_{12} + 171 \zeta_{12}^{2} + 96 \zeta_{12}^{3} ) q^{49} + ( 72 \zeta_{12} + 124 \zeta_{12}^{2} + 72 \zeta_{12}^{3} ) q^{50} + ( -20 - 168 \zeta_{12} + 20 \zeta_{12}^{2} + 336 \zeta_{12}^{3} ) q^{52} + ( -204 - 456 \zeta_{12} + 228 \zeta_{12}^{3} ) q^{53} + ( -108 + 180 \zeta_{12} - 90 \zeta_{12}^{3} ) q^{55} + ( -64 + 48 \zeta_{12} + 64 \zeta_{12}^{2} - 96 \zeta_{12}^{3} ) q^{56} + ( 42 \zeta_{12} - 324 \zeta_{12}^{2} + 42 \zeta_{12}^{3} ) q^{58} + ( 96 \zeta_{12} + 504 \zeta_{12}^{2} + 96 \zeta_{12}^{3} ) q^{59} + ( -371 - 54 \zeta_{12} + 371 \zeta_{12}^{2} + 108 \zeta_{12}^{3} ) q^{61} + ( -248 + 432 \zeta_{12} - 216 \zeta_{12}^{3} ) q^{62} + 64 q^{64} + ( 348 - 237 \zeta_{12} - 348 \zeta_{12}^{2} + 474 \zeta_{12}^{3} ) q^{65} + ( 414 \zeta_{12} + 52 \zeta_{12}^{2} + 414 \zeta_{12}^{3} ) q^{67} + ( 132 \zeta_{12} - 240 \zeta_{12}^{2} + 132 \zeta_{12}^{3} ) q^{68} + ( -204 + 120 \zeta_{12} + 204 \zeta_{12}^{2} - 240 \zeta_{12}^{3} ) q^{70} + ( -570 - 48 \zeta_{12} + 24 \zeta_{12}^{3} ) q^{71} + ( 425 + 576 \zeta_{12} - 288 \zeta_{12}^{3} ) q^{73} + ( 434 + 12 \zeta_{12} - 434 \zeta_{12}^{2} - 24 \zeta_{12}^{3} ) q^{74} + ( 264 \zeta_{12} + 16 \zeta_{12}^{2} + 264 \zeta_{12}^{3} ) q^{76} + ( -84 \zeta_{12} + 288 \zeta_{12}^{2} - 84 \zeta_{12}^{3} ) q^{77} + ( 220 - 150 \zeta_{12} - 220 \zeta_{12}^{2} + 300 \zeta_{12}^{3} ) q^{79} + ( 96 - 96 \zeta_{12} + 48 \zeta_{12}^{3} ) q^{80} + ( 192 + 528 \zeta_{12} - 264 \zeta_{12}^{3} ) q^{82} + ( -132 - 180 \zeta_{12} + 132 \zeta_{12}^{2} + 360 \zeta_{12}^{3} ) q^{83} + ( 378 \zeta_{12} - 657 \zeta_{12}^{2} + 378 \zeta_{12}^{3} ) q^{85} + ( -36 \zeta_{12} + 608 \zeta_{12}^{2} - 36 \zeta_{12}^{3} ) q^{86} + ( -144 - 192 \zeta_{12} + 144 \zeta_{12}^{2} + 384 \zeta_{12}^{3} ) q^{88} + ( 384 - 534 \zeta_{12} + 267 \zeta_{12}^{3} ) q^{89} + ( -716 + 612 \zeta_{12} - 306 \zeta_{12}^{3} ) q^{91} + ( -360 - 48 \zeta_{12} + 360 \zeta_{12}^{2} + 96 \zeta_{12}^{3} ) q^{92} + ( -216 \zeta_{12} + 384 \zeta_{12}^{2} - 216 \zeta_{12}^{3} ) q^{94} + ( 384 \zeta_{12} - 570 \zeta_{12}^{2} + 384 \zeta_{12}^{3} ) q^{95} + ( 382 - 168 \zeta_{12} - 382 \zeta_{12}^{2} + 336 \zeta_{12}^{3} ) q^{97} + ( -342 - 384 \zeta_{12} + 192 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} - 12q^{5} - 16q^{7} + 32q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} - 12q^{5} - 16q^{7} + 32q^{8} + 48q^{10} - 36q^{11} - 10q^{13} - 32q^{14} - 32q^{16} + 240q^{17} - 16q^{19} - 48q^{20} - 72q^{22} - 180q^{23} + 124q^{25} + 40q^{26} + 128q^{28} - 324q^{29} + 248q^{31} - 64q^{32} - 240q^{34} + 408q^{35} - 868q^{37} + 16q^{38} - 96q^{40} - 192q^{41} + 608q^{43} + 288q^{44} + 720q^{46} + 384q^{47} + 342q^{49} + 248q^{50} - 40q^{52} - 816q^{53} - 432q^{55} - 128q^{56} - 648q^{58} + 1008q^{59} - 742q^{61} - 992q^{62} + 256q^{64} + 696q^{65} + 104q^{67} - 480q^{68} - 408q^{70} - 2280q^{71} + 1700q^{73} + 868q^{74} + 32q^{76} + 576q^{77} + 440q^{79} + 384q^{80} + 768q^{82} - 264q^{83} - 1314q^{85} + 1216q^{86} - 288q^{88} + 1536q^{89} - 2864q^{91} - 720q^{92} + 768q^{94} - 1140q^{95} + 764q^{97} - 1368q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$83$$ $$\chi(n)$$ $$-\zeta_{12}$$

## 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
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −5.59808 9.69615i 0 −9.19615 + 15.9282i 8.00000 0 22.3923
55.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i −0.401924 0.696152i 0 1.19615 2.07180i 8.00000 0 1.60770
109.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −5.59808 + 9.69615i 0 −9.19615 15.9282i 8.00000 0 22.3923
109.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i −0.401924 + 0.696152i 0 1.19615 + 2.07180i 8.00000 0 1.60770
 $$n$$: 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 162.4.c.i 4
3.b odd 2 1 162.4.c.j 4
9.c even 3 1 162.4.a.h yes 2
9.c even 3 1 inner 162.4.c.i 4
9.d odd 6 1 162.4.a.e 2
9.d odd 6 1 162.4.c.j 4
36.f odd 6 1 1296.4.a.s 2
36.h even 6 1 1296.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.4.a.e 2 9.d odd 6 1
162.4.a.h yes 2 9.c even 3 1
162.4.c.i 4 1.a even 1 1 trivial
162.4.c.i 4 9.c even 3 1 inner
162.4.c.j 4 3.b odd 2 1
162.4.c.j 4 9.d odd 6 1
1296.4.a.j 2 36.h even 6 1
1296.4.a.s 2 36.f odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 12 T_{5}^{3} + 135 T_{5}^{2} + 108 T_{5} + 81$$ acting on $$S_{4}^{\mathrm{new}}(162, [\chi])$$.