# Properties

 Label 1950.2.e.d.1249.1 Level $1950$ Weight $2$ Character 1950.1249 Analytic conductor $15.571$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.5708283941$$ 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 1249.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1950.1249 Dual form 1950.2.e.d.1249.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.00000i q^{12} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +1.00000i q^{18} -2.00000 q^{19} +2.00000 q^{21} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -2.00000i q^{28} +8.00000 q^{31} -1.00000i q^{32} +1.00000 q^{36} +2.00000i q^{37} +2.00000i q^{38} -1.00000 q^{39} +6.00000 q^{41} -2.00000i q^{42} +4.00000i q^{43} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -2.00000 q^{56} +2.00000i q^{57} +14.0000 q^{61} -8.00000i q^{62} -2.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} +1.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} +12.0000i q^{83} -2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} -4.00000i q^{97} -3.00000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 4 q^{14} + 2 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{24} - 2 q^{26} + 16 q^{31} + 2 q^{36} - 2 q^{39} + 12 q^{41} + 12 q^{46} + 6 q^{49} + 2 q^{54} - 4 q^{56} + 28 q^{61} - 2 q^{64} + 12 q^{69} + 4 q^{74} + 4 q^{76} + 32 q^{79} + 2 q^{81} - 4 q^{84} + 8 q^{86} + 12 q^{89} + 4 q^{91} - 2 q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ − 1.00000i − 0.577350i
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 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
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 1.00000i − 0.277350i
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000i 0.192450i
$$28$$ − 2.00000i − 0.377964i
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 2.00000i 0.324443i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 6.00000 0.884652
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000i 0.138675i
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 2.00000i 0.264906i
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 14.0000 1.79252 0.896258 0.443533i $$-0.146275\pi$$
0.896258 + 0.443533i $$0.146275\pi$$
$$62$$ − 8.00000i − 1.01600i
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ 0 0
$$69$$ 6.00000 0.722315
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ 0 0
$$78$$ 1.00000i 0.113228i
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 6.00000i − 0.662589i
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ − 6.00000i − 0.625543i
$$93$$ − 8.00000i − 0.829561i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ − 4.00000i − 0.406138i −0.979164 0.203069i $$-0.934908\pi$$
0.979164 0.203069i $$-0.0650917\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ 16.0000i 1.57653i 0.615338 + 0.788263i $$0.289020\pi$$
−0.615338 + 0.788263i $$0.710980\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ − 12.0000i − 1.16008i −0.814587 0.580042i $$-0.803036\pi$$
0.814587 0.580042i $$-0.196964\pi$$
$$108$$ − 1.00000i − 0.0962250i
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 2.00000i 0.188982i
$$113$$ − 12.0000i − 1.12887i −0.825479 0.564433i $$-0.809095\pi$$
0.825479 0.564433i $$-0.190905\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000i 0.0924500i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ − 14.0000i − 1.26750i
$$123$$ − 6.00000i − 0.541002i
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ 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
$$133$$ − 4.00000i − 0.346844i
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 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 $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 4.00000 0.331042
$$147$$ − 3.00000i − 0.247436i
$$148$$ − 2.00000i − 0.164399i
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ − 2.00000i − 0.162221i
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ − 16.0000i − 1.27289i
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ − 1.00000i − 0.0785674i
$$163$$ 16.0000i 1.25322i 0.779334 + 0.626608i $$0.215557\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ − 6.00000i − 0.449719i
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ − 2.00000i − 0.148250i
$$183$$ − 14.0000i − 1.03491i
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −2.00000 −0.145479
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 8.00000i − 0.575853i −0.957653 0.287926i $$-0.907034\pi$$
0.957653 0.287926i $$-0.0929658\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 12.0000i 0.844317i
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ − 6.00000i − 0.412082i
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 16.0000i 1.08615i
$$218$$ − 4.00000i − 0.270914i
$$219$$ 4.00000 0.270295
$$220$$ 0 0
$$221$$ 0 0
$$222$$ − 2.00000i − 0.134231i
$$223$$ 10.0000i 0.669650i 0.942280 + 0.334825i $$0.108677\pi$$
−0.942280 + 0.334825i $$0.891323\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ 12.0000i 0.796468i 0.917284 + 0.398234i $$0.130377\pi$$
−0.917284 + 0.398234i $$0.869623\pi$$
$$228$$ − 2.00000i − 0.132453i
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 16.0000i − 1.03931i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 11.0000i 0.707107i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 2.00000i 0.127257i
$$248$$ 8.00000i 0.508001i
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −30.0000 −1.89358 −0.946792 0.321847i $$-0.895696\pi$$
−0.946792 + 0.321847i $$0.895696\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 12.0000i − 0.748539i −0.927320 0.374270i $$-0.877893\pi$$
0.927320 0.374270i $$-0.122107\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ −4.00000 −0.248548
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 6.00000i − 0.370681i
$$263$$ − 30.0000i − 1.84988i −0.380114 0.924940i $$-0.624115\pi$$
0.380114 0.924940i $$-0.375885\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ − 6.00000i − 0.367194i
$$268$$ 4.00000i 0.244339i
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ − 2.00000i − 0.121046i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ −6.00000 −0.361158
$$277$$ 26.0000i 1.56219i 0.624413 + 0.781094i $$0.285338\pi$$
−0.624413 + 0.781094i $$0.714662\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 28.0000i 1.66443i 0.554455 + 0.832214i $$0.312927\pi$$
−0.554455 + 0.832214i $$0.687073\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000i 0.708338i
$$288$$ 1.00000i 0.0589256i
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ −4.00000 −0.234484
$$292$$ − 4.00000i − 0.234082i
$$293$$ 6.00000i 0.350524i 0.984522 + 0.175262i $$0.0560772\pi$$
−0.984522 + 0.175262i $$0.943923\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ 0 0
$$298$$ 18.0000i 1.04271i
$$299$$ 6.00000 0.346989
$$300$$ 0 0
$$301$$ −8.00000 −0.461112
$$302$$ − 8.00000i − 0.460348i
$$303$$ 12.0000i 0.689382i
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 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
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 10.0000i 0.565233i 0.959233 + 0.282617i $$0.0912024\pi$$
−0.959233 + 0.282617i $$0.908798\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 12.0000i 0.668734i
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ 16.0000 0.886158
$$327$$ − 4.00000i − 0.221201i
$$328$$ 6.00000i 0.331295i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ − 2.00000i − 0.109599i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ 1.00000i 0.0543928i
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ − 2.00000i − 0.108148i
$$343$$ 20.0000i 1.07990i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −18.0000 −0.967686
$$347$$ 24.0000i 1.28839i 0.764862 + 0.644194i $$0.222807\pi$$
−0.764862 + 0.644194i $$0.777193\pi$$
$$348$$ 0 0
$$349$$ −20.0000 −1.07058 −0.535288 0.844670i $$-0.679797\pi$$
−0.535288 + 0.844670i $$0.679797\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 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
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ − 6.00000i − 0.317110i
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 2.00000i − 0.105118i
$$363$$ 11.0000i 0.577350i
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 8.00000i 0.414781i
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 2.00000i 0.102869i
$$379$$ 34.0000 1.74646 0.873231 0.487306i $$-0.162020\pi$$
0.873231 + 0.487306i $$0.162020\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ − 12.0000i − 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ − 4.00000i − 0.203331i
$$388$$ 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
$$393$$ − 6.00000i − 0.302660i
$$394$$ 18.0000 0.906827
$$395$$ 0 0
$$396$$ 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
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 4.00000i 0.199502i
$$403$$ − 8.00000i − 0.398508i
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 6.00000 0.295958
$$412$$ − 16.0000i − 0.788263i
$$413$$ 0 0
$$414$$ −6.00000 −0.294884
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 8.00000i 0.391762i
$$418$$ 0 0
$$419$$ −6.00000 −0.293119 −0.146560 0.989202i $$-0.546820\pi$$
−0.146560 + 0.989202i $$0.546820\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 4.00000i 0.194717i
$$423$$ 0 0
$$424$$ −6.00000 −0.291386
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 28.0000i 1.35501i
$$428$$ 12.0000i 0.580042i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 1.00000i 0.0481125i
$$433$$ − 14.0000i − 0.672797i −0.941720 0.336399i $$-0.890791\pi$$
0.941720 0.336399i $$-0.109209\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ − 12.0000i − 0.574038i
$$438$$ − 4.00000i − 0.191127i
$$439$$ −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 1.00000 $$0$$
−1.00000 $$\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 10.0000 0.473514
$$447$$ 18.0000i 0.851371i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 12.0000i 0.564433i
$$453$$ − 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 $$-0.772714\pi$$
0.755722 0.654892i $$-0.227286\pi$$
$$458$$ − 4.00000i − 0.186908i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 0 0
$$463$$ 10.0000i 0.464739i 0.972628 + 0.232370i $$0.0746479\pi$$
−0.972628 + 0.232370i $$0.925352\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.0000i 1.66588i 0.553362 + 0.832941i $$0.313345\pi$$
−0.553362 + 0.832941i $$0.686655\pi$$
$$468$$ − 1.00000i − 0.0462250i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ 0 0
$$473$$ 0 0
$$474$$ −16.0000 −0.734904
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 6.00000i − 0.274721i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ − 26.0000i − 1.18427i
$$483$$ 12.0000i 0.546019i
$$484$$ 11.0000 0.500000
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 14.0000i 0.633750i
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 6.00000i 0.270501i
$$493$$ 0 0
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ − 12.0000i − 0.537733i
$$499$$ −14.0000 −0.626726 −0.313363 0.949633i $$-0.601456\pi$$
−0.313363 + 0.949633i $$0.601456\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 30.0000i 1.33897i
$$503$$ 18.0000i 0.802580i 0.915951 + 0.401290i $$0.131438\pi$$
−0.915951 + 0.401290i $$0.868562\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000i 0.0444116i
$$508$$ 16.0000i 0.709885i
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 2.00000i − 0.0883022i
$$514$$ −12.0000 −0.529297
$$515$$ 0 0
$$516$$ −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
$$523$$ − 20.0000i − 0.874539i −0.899331 0.437269i $$-0.855946\pi$$
0.899331 0.437269i $$-0.144054\pi$$
$$524$$ −6.00000 −0.262111
$$525$$ 0 0
$$526$$ −30.0000 −1.30806
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000i 0.173422i
$$533$$ − 6.00000i − 0.259889i
$$534$$ −6.00000 −0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ − 6.00000i − 0.258919i
$$538$$ 12.0000i 0.517357i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ − 8.00000i − 0.343629i
$$543$$ − 2.00000i − 0.0858282i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ −2.00000 −0.0855921
$$547$$ 20.0000i 0.855138i 0.903983 + 0.427569i $$0.140630\pi$$
−0.903983 + 0.427569i $$0.859370\pi$$
$$548$$ − 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
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ 8.00000 0.339276
$$557$$ 30.0000i 1.27114i 0.772043 + 0.635570i $$0.219235\pi$$
−0.772043 + 0.635570i $$0.780765\pi$$
$$558$$ 8.00000i 0.338667i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 30.0000i 1.26547i
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 28.0000 1.17693
$$567$$ 2.00000i 0.0839921i
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 16.0000i − 0.666089i −0.942911 0.333044i $$-0.891924\pi$$
0.942911 0.333044i $$-0.108076\pi$$
$$578$$ − 17.0000i − 0.707107i
$$579$$ −8.00000 −0.332469
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 4.00000i 0.165805i
$$583$$ 0 0
$$584$$ −4.00000 −0.165521
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ 3.00000i 0.123718i
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ 18.0000 0.740421
$$592$$ 2.00000i 0.0821995i
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 18.0000 0.737309
$$597$$ − 16.0000i − 0.654836i
$$598$$ − 6.00000i − 0.245358i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 8.00000i 0.326056i
$$603$$ 4.00000i 0.162893i
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 12.0000 0.487467
$$607$$ − 40.0000i − 1.62355i −0.583970 0.811775i $$-0.698502\pi$$
0.583970 0.811775i $$-0.301498\pi$$
$$608$$ 2.00000i 0.0811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 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
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ − 16.0000i − 0.643614i
$$619$$ −38.0000 −1.52735 −0.763674 0.645601i $$-0.776607\pi$$
−0.763674 + 0.645601i $$0.776607\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ − 24.0000i − 0.962312i
$$623$$ 12.0000i 0.480770i
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 0 0
$$628$$ − 2.00000i − 0.0798087i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 16.0000i 0.636446i
$$633$$ 4.00000i 0.158986i
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ − 3.00000i − 0.118864i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −6.00000 −0.236986 −0.118493 0.992955i $$-0.537806\pi$$
−0.118493 + 0.992955i $$0.537806\pi$$
$$642$$ 12.0000i 0.473602i
$$643$$ 4.00000i 0.157745i 0.996885 + 0.0788723i $$0.0251319\pi$$
−0.996885 + 0.0788723i $$0.974868\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 6.00000i 0.235884i 0.993020 + 0.117942i $$0.0376297\pi$$
−0.993020 + 0.117942i $$0.962370\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 16.0000 0.627089
$$652$$ − 16.0000i − 0.626608i
$$653$$ − 30.0000i − 1.17399i −0.809590 0.586995i $$-0.800311\pi$$
0.809590 0.586995i $$-0.199689\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ − 4.00000i − 0.156055i
$$658$$ 0 0
$$659$$ 18.0000 0.701180 0.350590 0.936529i $$-0.385981\pi$$
0.350590 + 0.936529i $$0.385981\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ − 2.00000i − 0.0777322i
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ − 12.0000i − 0.464294i
$$669$$ 10.0000 0.386622
$$670$$ 0 0
$$671$$ 0 0
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ − 42.0000i − 1.61419i −0.590421 0.807096i $$-0.701038\pi$$
0.590421 0.807096i $$-0.298962\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ − 4.00000i − 0.152610i
$$688$$ 4.00000i 0.152499i
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ 26.0000 0.989087 0.494543 0.869153i $$-0.335335\pi$$
0.494543 + 0.869153i $$0.335335\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 20.0000i 0.757011i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ − 1.00000i − 0.0377426i
$$703$$ − 4.00000i − 0.150863i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ − 24.0000i − 0.902613i
$$708$$ 0 0
$$709$$ −20.0000 −0.751116 −0.375558 0.926799i $$-0.622549\pi$$
−0.375558 + 0.926799i $$0.622549\pi$$
$$710$$ 0 0
$$711$$ −16.0000 −0.600047
$$712$$ 6.00000i 0.224860i
$$713$$ 48.0000i 1.79761i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6.00000 −0.224231
$$717$$ 0 0
$$718$$ − 24.0000i − 0.895672i
$$719$$ −12.0000 −0.447524 −0.223762 0.974644i $$-0.571834\pi$$
−0.223762 + 0.974644i $$0.571834\pi$$
$$720$$ 0 0
$$721$$ −32.0000 −1.19174
$$722$$ 15.0000i 0.558242i
$$723$$ − 26.0000i − 0.966950i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ 11.0000 0.408248
$$727$$ − 4.00000i − 0.148352i −0.997245 0.0741759i $$-0.976367\pi$$
0.997245 0.0741759i $$-0.0236326\pi$$
$$728$$ 2.00000i 0.0741249i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 14.0000i 0.517455i
$$733$$ − 50.0000i − 1.84679i −0.383849 0.923396i $$-0.625402\pi$$
0.383849 0.923396i $$-0.374598\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 6.00000 0.221163
$$737$$ 0 0
$$738$$ 6.00000i 0.220863i
$$739$$ −38.0000 −1.39785 −0.698926 0.715194i $$-0.746338\pi$$
−0.698926 + 0.715194i $$0.746338\pi$$
$$740$$ 0 0
$$741$$ 2.00000 0.0734718
$$742$$ 12.0000i 0.440534i
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ −26.0000 −0.951928
$$747$$ − 12.0000i − 0.439057i
$$748$$ 0 0
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ 30.0000i 1.09326i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 2.00000 0.0727393
$$757$$ − 34.0000i − 1.23575i −0.786276 0.617876i $$-0.787994\pi$$
0.786276 0.617876i $$-0.212006\pi$$
$$758$$ − 34.0000i − 1.23494i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 8.00000i 0.289619i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 0 0
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 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 $$0.965587\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 4.00000 0.143592
$$777$$ 4.00000i 0.143499i
$$778$$ 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
$$785$$ 0 0
$$786$$ −6.00000 −0.214013
$$787$$ − 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
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ − 14.0000i − 0.497155i
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000i 0.635602i
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 12.0000i 0.422420i
$$808$$ − 12.0000i − 0.422159i
$$809$$ 6.00000 0.210949 0.105474 0.994422i $$-0.466364\pi$$
0.105474 + 0.994422i $$0.466364\pi$$
$$810$$ 0 0
$$811$$ 26.0000 0.912983 0.456492 0.889728i $$-0.349106\pi$$
0.456492 + 0.889728i $$0.349106\pi$$
$$812$$ 0 0
$$813$$ − 8.00000i − 0.280572i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 8.00000i − 0.279885i
$$818$$ 14.0000i 0.489499i
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ − 6.00000i − 0.209274i
$$823$$ − 32.0000i − 1.11545i −0.830026 0.557725i $$-0.811674\pi$$
0.830026 0.557725i $$-0.188326\pi$$
$$824$$ −16.0000 −0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 12.0000i − 0.417281i −0.977992 0.208640i $$-0.933096\pi$$
0.977992 0.208640i $$-0.0669038\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 26.0000 0.901930
$$832$$ 1.00000i 0.0346688i
$$833$$ 0 0
$$834$$ 8.00000 0.277017
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000i 0.276520i
$$838$$ 6.00000i 0.207267i
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ − 32.0000i − 1.10279i
$$843$$ 30.0000i 1.03325i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 22.0000i − 0.755929i
$$848$$ 6.00000i 0.206041i
$$849$$ 28.0000 0.960958
$$850$$ 0 0
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ − 26.0000i − 0.890223i −0.895475 0.445112i $$-0.853164\pi$$
0.895475 0.445112i $$-0.146836\pi$$
$$854$$ 28.0000 0.958140
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ − 12.0000i − 0.409912i −0.978771 0.204956i $$-0.934295\pi$$
0.978771 0.204956i $$-0.0657052\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 12.0000 0.408959
$$862$$ − 24.0000i − 0.817443i
$$863$$ 12.0000i 0.408485i 0.978920 + 0.204242i $$0.0654731\pi$$
−0.978920 + 0.204242i $$0.934527\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −14.0000 −0.475739
$$867$$ − 17.0000i − 0.577350i
$$868$$ − 16.0000i − 0.543075i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −4.00000 −0.135535
$$872$$ 4.00000i 0.135457i
$$873$$ 4.00000i 0.135379i
$$874$$ −12.0000 −0.405906
$$875$$ 0 0
$$876$$ −4.00000 −0.135147
$$877$$ − 58.0000i − 1.95852i −0.202606 0.979260i $$-0.564941\pi$$
0.202606 0.979260i $$-0.435059\pi$$
$$878$$ 32.0000i 1.07995i
$$879$$ 6.00000 0.202375
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ 28.0000i 0.942275i 0.882060 + 0.471138i $$0.156156\pi$$
−0.882060 + 0.471138i $$0.843844\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 30.0000i − 1.00730i −0.863907 0.503651i $$-0.831990\pi$$
0.863907 0.503651i $$-0.168010\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ 32.0000 1.07325
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 10.0000i − 0.334825i
$$893$$ 0 0
$$894$$ 18.0000 0.602010
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ − 6.00000i − 0.200334i
$$898$$ − 30.0000i − 1.00111i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ 12.0000 0.399114
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ − 28.0000i − 0.929725i −0.885383 0.464862i $$-0.846104\pi$$
0.885383 0.464862i $$-0.153896\pi$$
$$908$$ − 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
$$915$$ 0 0
$$916$$ −4.00000 −0.132164
$$917$$ 12.0000i 0.396275i
$$918$$ 0 0
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 0 0
$$921$$ −4.00000 −0.131804
$$922$$ 42.0000i 1.38320i
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 10.0000 0.328620
$$927$$ − 16.0000i − 0.525509i
$$928$$ 0 0
$$929$$ 54.0000 1.77168 0.885841 0.463988i $$-0.153582\pi$$
0.885841 + 0.463988i $$0.153582\pi$$
$$930$$ 0 0
$$931$$ −6.00000 −0.196642
$$932$$ 0 0
$$933$$ − 24.0000i − 0.785725i
$$934$$ 36.0000 1.17796
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ − 8.00000i − 0.261209i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 36.0000i 1.17232i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 36.0000i 1.16984i 0.811090 + 0.584921i $$0.198875\pi$$
−0.811090 + 0.584921i $$0.801125\pi$$
$$948$$ 16.0000i 0.519656i
$$949$$ 4.00000 0.129845
$$950$$ 0 0
$$951$$ −18.0000 −0.583690
$$952$$ 0 0
$$953$$ − 24.0000i − 0.777436i −0.921357 0.388718i $$-0.872918\pi$$
0.921357 0.388718i $$-0.127082\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 12.0000i 0.386695i
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ 50.0000i 1.60789i 0.594703 + 0.803946i $$0.297270\pi$$
−0.594703 + 0.803946i $$0.702730\pi$$
$$968$$ − 11.0000i − 0.353553i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −30.0000 −0.962746 −0.481373 0.876516i $$-0.659862\pi$$
−0.481373 + 0.876516i $$0.659862\pi$$
$$972$$ 1.00000i 0.0320750i
$$973$$ − 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 $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 16.0000i − 0.511624i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ − 6.00000i − 0.191468i
$$983$$ − 36.0000i − 1.14822i −0.818778 0.574111i $$-0.805348\pi$$
0.818778 0.574111i $$-0.194652\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 2.00000i − 0.0636285i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ − 8.00000i − 0.254000i
$$993$$ − 2.00000i − 0.0634681i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −12.0000 −0.380235
$$997$$ − 46.0000i − 1.45683i −0.685134 0.728417i $$-0.740256\pi$$
0.685134 0.728417i $$-0.259744\pi$$
$$998$$ 14.0000i 0.443162i
$$999$$ −2.00000 −0.0632772
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 1950.2.e.d.1249.1 2
3.2 odd 2 5850.2.e.o.5149.2 2
5.2 odd 4 1950.2.a.o.1.1 1
5.3 odd 4 390.2.a.d.1.1 1
5.4 even 2 inner 1950.2.e.d.1249.2 2
15.2 even 4 5850.2.a.g.1.1 1
15.8 even 4 1170.2.a.k.1.1 1
15.14 odd 2 5850.2.e.o.5149.1 2
20.3 even 4 3120.2.a.j.1.1 1
60.23 odd 4 9360.2.a.g.1.1 1
65.8 even 4 5070.2.b.m.1351.1 2
65.18 even 4 5070.2.b.m.1351.2 2
65.38 odd 4 5070.2.a.t.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.d.1.1 1 5.3 odd 4
1170.2.a.k.1.1 1 15.8 even 4
1950.2.a.o.1.1 1 5.2 odd 4
1950.2.e.d.1249.1 2 1.1 even 1 trivial
1950.2.e.d.1249.2 2 5.4 even 2 inner
3120.2.a.j.1.1 1 20.3 even 4
5070.2.a.t.1.1 1 65.38 odd 4
5070.2.b.m.1351.1 2 65.8 even 4
5070.2.b.m.1351.2 2 65.18 even 4
5850.2.a.g.1.1 1 15.2 even 4
5850.2.e.o.5149.1 2 15.14 odd 2
5850.2.e.o.5149.2 2 3.2 odd 2
9360.2.a.g.1.1 1 60.23 odd 4