# Properties

 Label 5070.2.b.m.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.m.1351.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{18} -2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} +6.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -2.00000i q^{28} +1.00000 q^{30} -8.00000i q^{31} +1.00000i q^{32} +2.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} +2.00000 q^{38} -1.00000 q^{40} -6.00000i q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000i q^{45} +6.00000i q^{46} +1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} -6.00000 q^{53} +1.00000i q^{54} +2.00000 q^{56} -2.00000i q^{57} +1.00000i q^{60} +14.0000 q^{61} +8.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} +6.00000 q^{69} +2.00000i q^{70} -1.00000i q^{72} -4.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} +2.00000i q^{76} -16.0000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000i q^{83} -2.00000i q^{84} +4.00000i q^{86} -6.00000i q^{89} +1.00000 q^{90} -6.00000 q^{92} -8.00000i q^{93} -2.00000 q^{95} +1.00000i q^{96} +4.00000i q^{97} +3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 12 q^{23} - 2 q^{25} + 2 q^{27} + 2 q^{30} + 4 q^{35} - 2 q^{36} + 4 q^{38} - 2 q^{40} - 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 12 q^{53} + 4 q^{56} + 28 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{69} - 4 q^{74} - 2 q^{75} - 32 q^{79} + 2 q^{81} + 12 q^{82} + 2 q^{90} - 12 q^{92} - 4 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$$.

 $$n$$ $$1691$$ $$1861$$ $$4057$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ 1.00000i 0.408248i
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −2.00000 −0.534522
$$15$$ − 1.00000i − 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ − 2.00000i − 0.458831i −0.973329 0.229416i $$-0.926318\pi$$
0.973329 0.229416i $$-0.0736815\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 2.00000i 0.436436i
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ − 2.00000i − 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 1.00000 0.182574
$$31$$ − 8.00000i − 1.43684i −0.695608 0.718421i $$-0.744865\pi$$
0.695608 0.718421i $$-0.255135\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ − 1.00000i − 0.149071i
$$46$$ 6.00000i 0.884652i
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 3.00000 0.428571
$$50$$ − 1.00000i − 0.141421i
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 0 0
$$56$$ 2.00000 0.267261
$$57$$ − 2.00000i − 0.264906i
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 1.00000i 0.129099i
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000i 0.488678i 0.969690 + 0.244339i $$0.0785709\pi$$
−0.969690 + 0.244339i $$0.921429\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 2.00000i 0.239046i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ − 4.00000i − 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ −1.00000 −0.115470
$$76$$ 2.00000i 0.229416i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ 0 0
$$86$$ 4.00000i 0.431331i
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 6.00000i − 0.635999i −0.948091 0.317999i $$-0.896989\pi$$
0.948091 0.317999i $$-0.103011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −6.00000 −0.625543
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ −2.00000 −0.205196
$$96$$ 1.00000i 0.102062i
$$97$$ 4.00000i 0.406138i 0.979164 + 0.203069i $$0.0650917\pi$$
−0.979164 + 0.203069i $$0.934908\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ − 6.00000i − 0.582772i
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.00000i 0.383131i 0.981480 + 0.191565i $$0.0613564\pi$$
−0.981480 + 0.191565i $$0.938644\pi$$
$$110$$ 0 0
$$111$$ 2.00000i 0.189832i
$$112$$ 2.00000i 0.188982i
$$113$$ 12.0000 1.12887 0.564433 0.825479i $$-0.309095\pi$$
0.564433 + 0.825479i $$0.309095\pi$$
$$114$$ 2.00000 0.187317
$$115$$ − 6.00000i − 0.559503i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ −1.00000 −0.0912871
$$121$$ 11.0000 1.00000
$$122$$ 14.0000i 1.26750i
$$123$$ − 6.00000i − 0.541002i
$$124$$ 8.00000i 0.718421i
$$125$$ 1.00000i 0.0894427i
$$126$$ −2.00000 −0.178174
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ −4.00000 −0.345547
$$135$$ − 1.00000i − 0.0860663i
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 6.00000i 0.510754i
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ 3.00000 0.247436
$$148$$ − 2.00000i − 0.164399i
$$149$$ − 18.0000i − 1.47462i −0.675556 0.737309i $$-0.736096\pi$$
0.675556 0.737309i $$-0.263904\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ 8.00000i 0.651031i 0.945537 + 0.325515i $$0.105538\pi$$
−0.945537 + 0.325515i $$0.894462\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ − 16.0000i − 1.27289i
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 12.0000i 0.945732i
$$162$$ 1.00000i 0.0785674i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 6.00000i 0.468521i
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 2.00000i − 0.152944i
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ − 2.00000i − 0.151186i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 14.0000 1.03491
$$184$$ − 6.00000i − 0.442326i
$$185$$ 2.00000 0.147043
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 2.00000i 0.145479i
$$190$$ − 2.00000i − 0.145095i
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 8.00000i 0.575853i 0.957653 + 0.287926i $$0.0929658\pi$$
−0.957653 + 0.287926i $$0.907034\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ 4.00000i 0.282138i
$$202$$ 12.0000i 0.844317i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −6.00000 −0.419058
$$206$$ 16.0000i 1.11477i
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 2.00000i 0.138013i
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ − 12.0000i − 0.820303i
$$215$$ − 4.00000i − 0.272798i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ 16.0000 1.08615
$$218$$ −4.00000 −0.270914
$$219$$ − 4.00000i − 0.270295i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −2.00000 −0.134231
$$223$$ 10.0000i 0.669650i 0.942280 + 0.334825i $$0.108677\pi$$
−0.942280 + 0.334825i $$0.891323\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ 12.0000i 0.798228i
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 2.00000i 0.132453i
$$229$$ − 4.00000i − 0.264327i −0.991228 0.132164i $$-0.957808\pi$$
0.991228 0.132164i $$-0.0421925\pi$$
$$230$$ 6.00000 0.395628
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −16.0000 −1.03931
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ − 1.00000i − 0.0645497i
$$241$$ 26.0000i 1.67481i 0.546585 + 0.837404i $$0.315928\pi$$
−0.546585 + 0.837404i $$0.684072\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ 1.00000 0.0641500
$$244$$ −14.0000 −0.896258
$$245$$ − 3.00000i − 0.191663i
$$246$$ 6.00000 0.382546
$$247$$ 0 0
$$248$$ −8.00000 −0.508001
$$249$$ 12.0000i 0.760469i
$$250$$ −1.00000 −0.0632456
$$251$$ 30.0000 1.89358 0.946792 0.321847i $$-0.104304\pi$$
0.946792 + 0.321847i $$0.104304\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 0 0
$$254$$ 16.0000i 1.00393i
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 6.00000i 0.370681i
$$263$$ 30.0000 1.84988 0.924940 0.380114i $$-0.124115\pi$$
0.924940 + 0.380114i $$0.124115\pi$$
$$264$$ 0 0
$$265$$ 6.00000i 0.368577i
$$266$$ 4.00000i 0.245256i
$$267$$ − 6.00000i − 0.367194i
$$268$$ − 4.00000i − 0.244339i
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 8.00000i 0.485965i 0.970031 + 0.242983i $$0.0781258\pi$$
−0.970031 + 0.242983i $$0.921874\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ − 8.00000i − 0.478947i
$$280$$ − 2.00000i − 0.119523i
$$281$$ − 30.0000i − 1.78965i −0.446417 0.894825i $$-0.647300\pi$$
0.446417 0.894825i $$-0.352700\pi$$
$$282$$ 0 0
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ 1.00000i 0.0589256i
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 4.00000i 0.234484i
$$292$$ 4.00000i 0.234082i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 3.00000i 0.174964i
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 18.0000 1.04271
$$299$$ 0 0
$$300$$ 1.00000 0.0577350
$$301$$ 8.00000i 0.461112i
$$302$$ −8.00000 −0.460348
$$303$$ 12.0000 0.689382
$$304$$ − 2.00000i − 0.114708i
$$305$$ − 14.0000i − 0.801638i
$$306$$ 0 0
$$307$$ − 4.00000i − 0.228292i −0.993464 0.114146i $$-0.963587\pi$$
0.993464 0.114146i $$-0.0364132\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ − 8.00000i − 0.454369i
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 2.00000i 0.112867i
$$315$$ 2.00000 0.112687
$$316$$ 16.0000 0.900070
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 0 0
$$320$$ 1.00000i 0.0559017i
$$321$$ −12.0000 −0.669775
$$322$$ −12.0000 −0.668734
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ 4.00000i 0.221201i
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 2.00000i − 0.109930i −0.998488 0.0549650i $$-0.982495\pi$$
0.998488 0.0549650i $$-0.0175047\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 2.00000i 0.109599i
$$334$$ −12.0000 −0.656611
$$335$$ 4.00000 0.218543
$$336$$ 2.00000i 0.109109i
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 2.00000 0.108148
$$343$$ 20.0000i 1.07990i
$$344$$ − 4.00000i − 0.215666i
$$345$$ − 6.00000i − 0.323029i
$$346$$ − 18.0000i − 0.967686i
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ 20.0000i 1.07058i 0.844670 + 0.535288i $$0.179797\pi$$
−0.844670 + 0.535288i $$0.820203\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000i 0.317999i
$$357$$ 0 0
$$358$$ 6.00000i 0.317110i
$$359$$ − 24.0000i − 1.26667i −0.773877 0.633336i $$-0.781685\pi$$
0.773877 0.633336i $$-0.218315\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 15.0000 0.789474
$$362$$ − 2.00000i − 0.105118i
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ −4.00000 −0.209370
$$366$$ 14.0000i 0.731792i
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 6.00000 0.312772
$$369$$ − 6.00000i − 0.312348i
$$370$$ 2.00000i 0.103975i
$$371$$ − 12.0000i − 0.623009i
$$372$$ 8.00000i 0.414781i
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ 1.00000i 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 34.0000i 1.74646i 0.487306 + 0.873231i $$0.337980\pi$$
−0.487306 + 0.873231i $$0.662020\pi$$
$$380$$ 2.00000 0.102598
$$381$$ 16.0000 0.819705
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 4.00000 0.203331
$$388$$ − 4.00000i − 0.203069i
$$389$$ −36.0000 −1.82527 −0.912636 0.408773i $$-0.865957\pi$$
−0.912636 + 0.408773i $$0.865957\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ − 3.00000i − 0.151523i
$$393$$ 6.00000 0.302660
$$394$$ 18.0000 0.906827
$$395$$ 16.0000i 0.805047i
$$396$$ 0 0
$$397$$ 26.0000i 1.30490i 0.757831 + 0.652451i $$0.226259\pi$$
−0.757831 + 0.652451i $$0.773741\pi$$
$$398$$ 16.0000i 0.802008i
$$399$$ 4.00000 0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ − 18.0000i − 0.898877i −0.893311 0.449439i $$-0.851624\pi$$
0.893311 0.449439i $$-0.148376\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 0 0
$$404$$ −12.0000 −0.597022
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ − 14.0000i − 0.692255i −0.938187 0.346128i $$-0.887496\pi$$
0.938187 0.346128i $$-0.112504\pi$$
$$410$$ − 6.00000i − 0.296319i
$$411$$ 6.00000i 0.295958i
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ 6.00000i 0.294884i
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ 6.00000 0.293119 0.146560 0.989202i $$-0.453180\pi$$
0.146560 + 0.989202i $$0.453180\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ − 32.0000i − 1.55958i −0.626038 0.779792i $$-0.715325\pi$$
0.626038 0.779792i $$-0.284675\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 0 0
$$424$$ 6.00000i 0.291386i
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 28.0000i 1.35501i
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ − 24.0000i − 1.15604i −0.816023 0.578020i $$-0.803826\pi$$
0.816023 0.578020i $$-0.196174\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 16.0000i 0.768025i
$$435$$ 0 0
$$436$$ − 4.00000i − 0.191565i
$$437$$ − 12.0000i − 0.574038i
$$438$$ 4.00000 0.191127
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ − 2.00000i − 0.0949158i
$$445$$ −6.00000 −0.284427
$$446$$ −10.0000 −0.473514
$$447$$ − 18.0000i − 0.851371i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ − 30.0000i − 1.41579i −0.706319 0.707894i $$-0.749646\pi$$
0.706319 0.707894i $$-0.250354\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 0 0
$$452$$ −12.0000 −0.564433
$$453$$ 8.00000i 0.375873i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 28.0000i 1.30978i 0.755722 + 0.654892i $$0.227286\pi$$
−0.755722 + 0.654892i $$0.772714\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 0 0
$$460$$ 6.00000i 0.279751i
$$461$$ 42.0000i 1.95614i 0.208288 + 0.978068i $$0.433211\pi$$
−0.208288 + 0.978068i $$0.566789\pi$$
$$462$$ 0 0
$$463$$ − 10.0000i − 0.464739i −0.972628 0.232370i $$-0.925352\pi$$
0.972628 0.232370i $$-0.0746479\pi$$
$$464$$ 0 0
$$465$$ −8.00000 −0.370991
$$466$$ 0 0
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ − 16.0000i − 0.734904i
$$475$$ 2.00000i 0.0917663i
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −26.0000 −1.18427
$$483$$ 12.0000i 0.546019i
$$484$$ −11.0000 −0.500000
$$485$$ 4.00000 0.181631
$$486$$ 1.00000i 0.0453609i
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ − 14.0000i − 0.633750i
$$489$$ − 16.0000i − 0.723545i
$$490$$ 3.00000 0.135526
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ − 8.00000i − 0.359211i
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ − 14.0000i − 0.626726i −0.949633 0.313363i $$-0.898544\pi$$
0.949633 0.313363i $$-0.101456\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 12.0000i 0.536120i
$$502$$ 30.0000i 1.33897i
$$503$$ −18.0000 −0.802580 −0.401290 0.915951i $$-0.631438\pi$$
−0.401290 + 0.915951i $$0.631438\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ − 12.0000i − 0.533993i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ − 6.00000i − 0.265945i −0.991120 0.132973i $$-0.957548\pi$$
0.991120 0.132973i $$-0.0424523\pi$$
$$510$$ 0 0
$$511$$ 8.00000 0.353899
$$512$$ 1.00000i 0.0441942i
$$513$$ − 2.00000i − 0.0883022i
$$514$$ 12.0000i 0.529297i
$$515$$ − 16.0000i − 0.705044i
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ − 4.00000i − 0.175750i
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ − 2.00000i − 0.0872872i
$$526$$ 30.0000i 1.30806i
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ −6.00000 −0.260623
$$531$$ 0 0
$$532$$ −4.00000 −0.173422
$$533$$ 0 0
$$534$$ 6.00000 0.259645
$$535$$ 12.0000i 0.518805i
$$536$$ 4.00000 0.172774
$$537$$ 6.00000 0.258919
$$538$$ 12.0000i 0.517357i
$$539$$ 0 0
$$540$$ 1.00000i 0.0430331i
$$541$$ 8.00000i 0.343947i 0.985102 + 0.171973i $$0.0550143\pi$$
−0.985102 + 0.171973i $$0.944986\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ −2.00000 −0.0858282
$$544$$ 0 0
$$545$$ 4.00000 0.171341
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ − 6.00000i − 0.256307i
$$549$$ 14.0000 0.597505
$$550$$ 0 0
$$551$$ 0 0
$$552$$ − 6.00000i − 0.255377i
$$553$$ − 32.0000i − 1.36078i
$$554$$ − 26.0000i − 1.10463i
$$555$$ 2.00000 0.0848953
$$556$$ −8.00000 −0.339276
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 8.00000 0.338667
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 30.0000 1.26547
$$563$$ −24.0000 −1.01148 −0.505740 0.862686i $$-0.668780\pi$$
−0.505740 + 0.862686i $$0.668780\pi$$
$$564$$ 0 0
$$565$$ − 12.0000i − 0.504844i
$$566$$ 28.0000i 1.17693i
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ − 2.00000i − 0.0837708i
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 12.0000i 0.500870i
$$575$$ −6.00000 −0.250217
$$576$$ −1.00000 −0.0416667
$$577$$ 16.0000i 0.666089i 0.942911 + 0.333044i $$0.108076\pi$$
−0.942911 + 0.333044i $$0.891924\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ 8.00000i 0.332469i
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ −4.00000 −0.165805
$$583$$ 0 0
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ − 18.0000i − 0.740421i
$$592$$ 2.00000i 0.0821995i
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000i 0.737309i
$$597$$ 16.0000 0.654836
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 4.00000i 0.162893i
$$604$$ − 8.00000i − 0.325515i
$$605$$ − 11.0000i − 0.447214i
$$606$$ 12.0000i 0.487467i
$$607$$ −40.0000 −1.62355 −0.811775 0.583970i $$-0.801498\pi$$
−0.811775 + 0.583970i $$0.801498\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 14.0000 0.566843
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 22.0000i 0.888572i 0.895885 + 0.444286i $$0.146543\pi$$
−0.895885 + 0.444286i $$0.853457\pi$$
$$614$$ 4.00000 0.161427
$$615$$ −6.00000 −0.241943
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ 38.0000i 1.52735i 0.645601 + 0.763674i $$0.276607\pi$$
−0.645601 + 0.763674i $$0.723393\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 6.00000 0.240772
$$622$$ − 24.0000i − 0.962312i
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 10.0000i − 0.399680i
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ 0 0
$$630$$ 2.00000i 0.0796819i
$$631$$ 20.0000i 0.796187i 0.917345 + 0.398094i $$0.130328\pi$$
−0.917345 + 0.398094i $$0.869672\pi$$
$$632$$ 16.0000i 0.636446i
$$633$$ −4.00000 −0.158986
$$634$$ −18.0000 −0.714871
$$635$$ − 16.0000i − 0.634941i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ − 12.0000i − 0.473602i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ − 12.0000i − 0.472866i
$$645$$ − 4.00000i − 0.157500i
$$646$$ 0 0
$$647$$ −6.00000 −0.235884 −0.117942 0.993020i $$-0.537630\pi$$
−0.117942 + 0.993020i $$0.537630\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ 16.0000i 0.626608i
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ − 6.00000i − 0.234439i
$$656$$ − 6.00000i − 0.234261i
$$657$$ − 4.00000i − 0.156055i
$$658$$ 0 0
$$659$$ −18.0000 −0.701180 −0.350590 0.936529i $$-0.614019\pi$$
−0.350590 + 0.936529i $$0.614019\pi$$
$$660$$ 0 0
$$661$$ − 16.0000i − 0.622328i −0.950356 0.311164i $$-0.899281\pi$$
0.950356 0.311164i $$-0.100719\pi$$
$$662$$ 2.00000 0.0777322
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ − 4.00000i − 0.155113i
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ − 12.0000i − 0.464294i
$$669$$ 10.0000i 0.386622i
$$670$$ 4.00000i 0.154533i
$$671$$ 0 0
$$672$$ −2.00000 −0.0771517
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ − 14.0000i − 0.539260i
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ − 12.0000i − 0.459841i
$$682$$ 0 0
$$683$$ − 36.0000i − 1.37750i −0.724998 0.688751i $$-0.758159\pi$$
0.724998 0.688751i $$-0.241841\pi$$
$$684$$ 2.00000i 0.0764719i
$$685$$ 6.00000 0.229248
$$686$$ −20.0000 −0.763604
$$687$$ − 4.00000i − 0.152610i
$$688$$ 4.00000 0.152499
$$689$$ 0 0
$$690$$ 6.00000 0.228416
$$691$$ − 26.0000i − 0.989087i −0.869153 0.494543i $$-0.835335\pi$$
0.869153 0.494543i $$-0.164665\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 24.0000i 0.911028i
$$695$$ − 8.00000i − 0.303457i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −20.0000 −0.757011
$$699$$ 0 0
$$700$$ 2.00000i 0.0755929i
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 24.0000i 0.902613i
$$708$$ 0 0
$$709$$ 20.0000i 0.751116i 0.926799 + 0.375558i $$0.122549\pi$$
−0.926799 + 0.375558i $$0.877451\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ −6.00000 −0.224860
$$713$$ − 48.0000i − 1.79761i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6.00000 −0.224231
$$717$$ 0 0
$$718$$ 24.0000 0.895672
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ 32.0000i 1.19174i
$$722$$ 15.0000i 0.558242i
$$723$$ 26.0000i 0.966950i
$$724$$ 2.00000 0.0743294
$$725$$ 0 0
$$726$$ 11.0000i 0.408248i
$$727$$ 4.00000 0.148352 0.0741759 0.997245i $$-0.476367\pi$$
0.0741759 + 0.997245i $$0.476367\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ − 4.00000i − 0.148047i
$$731$$ 0 0
$$732$$ −14.0000 −0.517455
$$733$$ − 50.0000i − 1.84679i −0.383849 0.923396i $$-0.625402\pi$$
0.383849 0.923396i $$-0.374598\pi$$
$$734$$ 8.00000i 0.295285i
$$735$$ − 3.00000i − 0.110657i
$$736$$ 6.00000i 0.221163i
$$737$$ 0 0
$$738$$ 6.00000 0.220863
$$739$$ 38.0000i 1.39785i 0.715194 + 0.698926i $$0.246338\pi$$
−0.715194 + 0.698926i $$0.753662\pi$$
$$740$$ −2.00000 −0.0735215
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ −18.0000 −0.659469
$$746$$ 26.0000i 0.951928i
$$747$$ 12.0000i 0.439057i
$$748$$ 0 0
$$749$$ − 24.0000i − 0.876941i
$$750$$ −1.00000 −0.0365148
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 30.0000 1.09326
$$754$$ 0 0
$$755$$ 8.00000 0.291150
$$756$$ − 2.00000i − 0.0727393i
$$757$$ −34.0000 −1.23575 −0.617876 0.786276i $$-0.712006\pi$$
−0.617876 + 0.786276i $$0.712006\pi$$
$$758$$ −34.0000 −1.23494
$$759$$ 0 0
$$760$$ 2.00000i 0.0725476i
$$761$$ 6.00000i 0.217500i 0.994069 + 0.108750i $$0.0346848\pi$$
−0.994069 + 0.108750i $$0.965315\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ −8.00000 −0.289619
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ 22.0000i 0.793340i 0.917961 + 0.396670i $$0.129834\pi$$
−0.917961 + 0.396670i $$0.870166\pi$$
$$770$$ 0 0
$$771$$ 12.0000 0.432169
$$772$$ − 8.00000i − 0.287926i
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 4.00000i 0.143777i
$$775$$ 8.00000i 0.287368i
$$776$$ 4.00000 0.143592
$$777$$ −4.00000 −0.143499
$$778$$ − 36.0000i − 1.29066i
$$779$$ −12.0000 −0.429945
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ − 2.00000i − 0.0713831i
$$786$$ 6.00000i 0.214013i
$$787$$ − 16.0000i − 0.570338i −0.958477 0.285169i $$-0.907950\pi$$
0.958477 0.285169i $$-0.0920498\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 30.0000 1.06803
$$790$$ −16.0000 −0.569254
$$791$$ 24.0000i 0.853342i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −26.0000 −0.922705
$$795$$ 6.00000i 0.212798i
$$796$$ −16.0000 −0.567105
$$797$$ 30.0000 1.06265 0.531327 0.847167i $$-0.321693\pi$$
0.531327 + 0.847167i $$0.321693\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 6.00000i − 0.212000i
$$802$$ 18.0000 0.635602
$$803$$ 0 0
$$804$$ − 4.00000i − 0.141069i
$$805$$ 12.0000 0.422944
$$806$$ 0 0
$$807$$ 12.0000 0.422420
$$808$$ − 12.0000i − 0.422159i
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ − 26.0000i − 0.912983i −0.889728 0.456492i $$-0.849106\pi$$
0.889728 0.456492i $$-0.150894\pi$$
$$812$$ 0 0
$$813$$ 8.00000i 0.280572i
$$814$$ 0 0
$$815$$ −16.0000 −0.560456
$$816$$ 0 0
$$817$$ − 8.00000i − 0.279885i
$$818$$ 14.0000 0.489499
$$819$$ 0 0
$$820$$ 6.00000 0.209529
$$821$$ 6.00000i 0.209401i 0.994504 + 0.104701i $$0.0333885\pi$$
−0.994504 + 0.104701i $$0.966612\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ − 16.0000i − 0.557386i
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 12.0000i 0.416526i
$$831$$ −26.0000 −0.901930
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 8.00000i 0.277017i
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ − 8.00000i − 0.276520i
$$838$$ 6.00000i 0.207267i
$$839$$ − 48.0000i − 1.65714i −0.559883 0.828572i $$-0.689154\pi$$
0.559883 0.828572i $$-0.310846\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −29.0000 −1.00000
$$842$$ 32.0000 1.10279
$$843$$ − 30.0000i − 1.03325i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 22.0000i 0.755929i
$$848$$ −6.00000 −0.206041
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ 12.0000i 0.411355i
$$852$$ 0 0
$$853$$ 26.0000i 0.890223i 0.895475 + 0.445112i $$0.146836\pi$$
−0.895475 + 0.445112i $$0.853164\pi$$
$$854$$ −28.0000 −0.958140
$$855$$ −2.00000 −0.0683986
$$856$$ 12.0000i 0.410152i
$$857$$ 12.0000 0.409912 0.204956 0.978771i $$-0.434295\pi$$
0.204956 + 0.978771i $$0.434295\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 4.00000i 0.136399i
$$861$$ 12.0000 0.408959
$$862$$ 24.0000 0.817443
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ 18.0000i 0.612018i
$$866$$ − 14.0000i − 0.475739i
$$867$$ −17.0000 −0.577350
$$868$$ −16.0000 −0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 4.00000 0.135457
$$873$$ 4.00000i 0.135379i
$$874$$ 12.0000 0.405906
$$875$$ −2.00000 −0.0676123
$$876$$ 4.00000i 0.135147i
$$877$$ 58.0000i 1.95852i 0.202606 + 0.979260i $$0.435059\pi$$
−0.202606 + 0.979260i $$0.564941\pi$$
$$878$$ − 32.0000i − 1.07995i
$$879$$ − 6.00000i − 0.202375i
$$880$$ 0 0
$$881$$ −6.00000 −0.202145 −0.101073 0.994879i $$-0.532227\pi$$
−0.101073 + 0.994879i $$0.532227\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −30.0000 −1.00730 −0.503651 0.863907i $$-0.668010\pi$$
−0.503651 + 0.863907i $$0.668010\pi$$
$$888$$ 2.00000 0.0671156
$$889$$ 32.0000i 1.07325i
$$890$$ − 6.00000i − 0.201120i
$$891$$ 0 0
$$892$$ − 10.0000i − 0.334825i
$$893$$ 0 0
$$894$$ 18.0000 0.602010
$$895$$ − 6.00000i − 0.200558i
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ − 12.0000i − 0.399114i
$$905$$ 2.00000i 0.0664822i
$$906$$ −8.00000 −0.265782
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 12.0000 0.398015
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ − 2.00000i − 0.0662266i
$$913$$ 0 0
$$914$$ −28.0000 −0.926158
$$915$$ − 14.0000i − 0.462826i
$$916$$ 4.00000i 0.132164i
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ −6.00000 −0.197814
$$921$$ − 4.00000i − 0.131804i
$$922$$ −42.0000 −1.38320
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 2.00000i − 0.0657596i
$$926$$ 10.0000 0.328620
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ 54.0000i 1.77168i 0.463988 + 0.885841i $$0.346418\pi$$
−0.463988 + 0.885841i $$0.653582\pi$$
$$930$$ − 8.00000i − 0.262330i
$$931$$ − 6.00000i − 0.196642i
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ − 36.0000i − 1.17796i
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ − 42.0000i − 1.36916i −0.728937 0.684580i $$-0.759985\pi$$
0.728937 0.684580i $$-0.240015\pi$$
$$942$$ 2.00000i 0.0651635i
$$943$$ − 36.0000i − 1.17232i
$$944$$ 0 0
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 0 0
$$950$$ −2.00000 −0.0648886
$$951$$ 18.0000i 0.583690i
$$952$$ 0 0
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ − 6.00000i − 0.194257i
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 1.00000i 0.0322749i
$$961$$ −33.0000 −1.06452
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ − 26.0000i − 0.837404i
$$965$$ 8.00000 0.257529
$$966$$ −12.0000 −0.386094
$$967$$ − 50.0000i − 1.60789i −0.594703 0.803946i $$-0.702730\pi$$
0.594703 0.803946i $$-0.297270\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 0 0
$$970$$ 4.00000i 0.128432i
$$971$$ −30.0000 −0.962746 −0.481373 0.876516i $$-0.659862\pi$$
−0.481373 + 0.876516i $$0.659862\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 16.0000i 0.512936i
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ 14.0000 0.448129
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 16.0000 0.511624
$$979$$ 0 0
$$980$$ 3.00000i 0.0958315i
$$981$$ 4.00000i 0.127710i
$$982$$ − 6.00000i − 0.191468i
$$983$$ 36.0000i 1.14822i 0.818778 + 0.574111i $$0.194652\pi$$
−0.818778 + 0.574111i $$0.805348\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 8.00000 0.254000
$$993$$ − 2.00000i − 0.0634681i
$$994$$ 0 0
$$995$$ − 16.0000i − 0.507234i
$$996$$ − 12.0000i − 0.380235i
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ 14.0000 0.443162
$$999$$ 2.00000i 0.0632772i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5070.2.b.m.1351.2 2
13.5 odd 4 5070.2.a.t.1.1 1
13.8 odd 4 390.2.a.d.1.1 1
13.12 even 2 inner 5070.2.b.m.1351.1 2
39.8 even 4 1170.2.a.k.1.1 1
52.47 even 4 3120.2.a.j.1.1 1
65.8 even 4 1950.2.e.d.1249.2 2
65.34 odd 4 1950.2.a.o.1.1 1
65.47 even 4 1950.2.e.d.1249.1 2
156.47 odd 4 9360.2.a.g.1.1 1
195.8 odd 4 5850.2.e.o.5149.1 2
195.47 odd 4 5850.2.e.o.5149.2 2
195.164 even 4 5850.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.d.1.1 1 13.8 odd 4
1170.2.a.k.1.1 1 39.8 even 4
1950.2.a.o.1.1 1 65.34 odd 4
1950.2.e.d.1249.1 2 65.47 even 4
1950.2.e.d.1249.2 2 65.8 even 4
3120.2.a.j.1.1 1 52.47 even 4
5070.2.a.t.1.1 1 13.5 odd 4
5070.2.b.m.1351.1 2 13.12 even 2 inner
5070.2.b.m.1351.2 2 1.1 even 1 trivial
5850.2.a.g.1.1 1 195.164 even 4
5850.2.e.o.5149.1 2 195.8 odd 4
5850.2.e.o.5149.2 2 195.47 odd 4
9360.2.a.g.1.1 1 156.47 odd 4