# Properties

 Label 160.3.v.a Level $160$ Weight $3$ Character orbit 160.v Analytic conductor $4.360$ Analytic rank $0$ Dimension $184$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 160.v (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35968422976$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 4q^{2} - 4q^{3} - 4q^{5} - 8q^{6} + 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 4q^{2} - 4q^{3} - 4q^{5} - 8q^{6} + 8q^{8} + 32q^{10} - 8q^{11} + 44q^{12} - 4q^{13} + 32q^{14} - 8q^{16} - 20q^{18} - 64q^{19} + 80q^{20} - 8q^{21} - 116q^{22} + 32q^{24} - 4q^{25} - 8q^{26} + 32q^{27} + 60q^{28} + 152q^{30} - 16q^{31} - 144q^{32} - 8q^{33} - 88q^{34} - 8q^{36} - 4q^{37} + 220q^{38} + 176q^{40} - 8q^{41} + 20q^{42} - 132q^{43} + 176q^{44} - 4q^{45} - 8q^{46} - 344q^{48} - 952q^{49} + 12q^{50} - 200q^{51} + 96q^{52} - 4q^{53} - 56q^{54} + 252q^{55} - 344q^{56} - 356q^{58} - 68q^{60} + 56q^{61} - 272q^{62} - 8q^{63} - 432q^{64} - 8q^{65} - 280q^{66} + 284q^{67} - 376q^{68} + 72q^{69} - 132q^{70} + 248q^{71} - 168q^{72} - 40q^{75} + 312q^{76} + 192q^{77} - 496q^{78} - 272q^{80} - 420q^{82} + 156q^{83} + 392q^{84} - 4q^{85} + 216q^{86} + 888q^{87} + 328q^{88} + 1284q^{90} - 8q^{91} + 300q^{92} - 40q^{93} - 288q^{94} - 8q^{95} + 536q^{96} - 8q^{97} + 888q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.99562 0.132347i −4.49249 + 1.86085i 3.96497 + 0.528226i 4.34634 + 2.47171i 9.21157 3.11898i 9.26439i −7.84265 1.57889i 10.3558 10.3558i −8.34650 5.50781i
13.2 −1.98630 + 0.233717i 0.152407 0.0631292i 3.89075 0.928464i 1.13682 4.86905i −0.287972 + 0.161014i 5.77129i −7.51119 + 2.75354i −6.34472 + 6.34472i −1.12009 + 9.93707i
13.3 −1.98201 + 0.267624i 1.91566 0.793494i 3.85675 1.06087i −0.189284 + 4.99642i −3.58451 + 2.08539i 3.33475i −7.36023 + 3.13482i −3.32383 + 3.32383i −0.961999 9.95362i
13.4 −1.97439 0.319035i −3.58463 + 1.48480i 3.79643 + 1.25980i −4.99639 + 0.190010i 7.55117 1.78796i 8.87581i −7.09372 3.69853i 4.28100 4.28100i 9.92544 + 1.21887i
13.5 −1.96655 0.364277i 4.81555 1.99467i 3.73461 + 1.43273i −4.99995 + 0.0217470i −10.1966 + 2.16841i 5.72496i −6.82236 4.17796i 12.8469 12.8469i 9.84056 + 1.77860i
13.6 −1.86795 0.714680i −0.137258 + 0.0568543i 2.97847 + 2.66997i 4.82878 1.29725i 0.297024 0.00810515i 10.6903i −3.65545 7.11602i −6.34835 + 6.34835i −9.94704 1.02785i
13.7 −1.70391 + 1.04723i 5.07893 2.10376i 1.80660 3.56878i 2.76666 4.16480i −6.45089 + 8.90345i 13.0758i 0.659077 + 7.97280i 15.0058 15.0058i −0.352614 + 9.99378i
13.8 −1.66520 1.10776i 3.97729 1.64745i 1.54576 + 3.68926i 4.80144 + 1.39504i −8.44794 1.66254i 4.01011i 1.51281 7.85566i 6.74079 6.74079i −6.44998 7.64184i
13.9 −1.66483 + 1.10831i 1.14849 0.475719i 1.54331 3.69028i −4.93972 0.774033i −1.38479 + 2.06487i 0.575911i 1.52062 + 7.85415i −5.27125 + 5.27125i 9.08166 4.18610i
13.10 −1.60071 + 1.19905i −2.59734 + 1.07585i 1.12457 3.83867i 0.271832 + 4.99261i 2.86760 4.83647i 5.11582i 2.80264 + 7.49301i −0.775245 + 0.775245i −6.42150 7.66579i
13.11 −1.53272 1.28482i −0.518083 + 0.214597i 0.698458 + 3.93855i −1.24748 + 4.84188i 1.06980 + 0.336728i 7.03161i 3.98980 6.93408i −6.14160 + 6.14160i 8.13300 5.81845i
13.12 −1.50904 + 1.31255i −4.42876 + 1.83445i 0.554433 3.96139i −0.125524 4.99842i 4.27539 8.58123i 2.19283i 4.36285 + 6.70563i 9.88475 9.88475i 6.75010 + 7.37809i
13.13 −1.44356 1.38425i −1.90642 + 0.789667i 0.167724 + 3.99648i −3.37146 3.69233i 3.84513 + 1.49903i 6.30545i 5.29000 6.00133i −3.35308 + 3.35308i −0.244192 + 9.99702i
13.14 −1.42228 + 1.40610i 2.71037 1.12267i 0.0457753 3.99974i 4.68901 + 1.73586i −2.27632 + 5.40780i 12.7706i 5.55892 + 5.75312i −0.278249 + 0.278249i −9.10988 + 4.12433i
13.15 −1.01850 1.72124i −3.74118 + 1.54965i −1.92531 + 3.50616i 4.07056 2.90354i 6.47770 + 4.86113i 2.79047i 7.99587 0.257117i 5.23103 5.23103i −9.14355 4.04914i
13.16 −1.01554 1.72298i 2.65551 1.09995i −1.93734 + 3.49953i −4.54176 + 2.09103i −4.59197 3.45835i 12.6857i 7.99709 0.215915i −0.522125 + 0.522125i 8.21517 + 5.70185i
13.17 −0.756816 + 1.85128i −3.28804 + 1.36195i −2.85446 2.80215i −3.29788 + 3.75819i −0.0329077 7.11781i 11.5572i 7.34786 3.16368i 2.59232 2.59232i −4.46157 8.94955i
13.18 −0.710424 + 1.86957i 1.17810 0.487984i −2.99060 2.65638i −4.10083 2.86063i 0.0753728 + 2.54922i 3.86601i 7.09088 3.70398i −5.21417 + 5.21417i 8.26148 5.63454i
13.19 −0.702991 1.87238i 3.43142 1.42134i −3.01161 + 2.63253i 1.08056 4.88184i −5.07355 5.42573i 2.20595i 7.04623 + 3.78822i 3.39049 3.39049i −9.90028 + 1.40868i
13.20 −0.669438 + 1.88464i −0.764258 + 0.316566i −3.10371 2.52329i 4.97587 0.490617i −0.0849887 1.65227i 4.84631i 6.83323 4.16017i −5.88009 + 5.88009i −2.40640 + 9.70614i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 117.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.v odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.v.a 184
5.c odd 4 1 160.3.bb.a yes 184
32.g even 8 1 160.3.bb.a yes 184
160.v odd 8 1 inner 160.3.v.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.v.a 184 1.a even 1 1 trivial
160.3.v.a 184 160.v odd 8 1 inner
160.3.bb.a yes 184 5.c odd 4 1
160.3.bb.a yes 184 32.g even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(160, [\chi])$$.