# Properties

 Label 950.2.b.c.799.1 Level $950$ Weight $2$ Character 950.799 Analytic conductor $7.586$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ 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 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 950.799 Dual form 950.2.b.c.799.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +2.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} -3.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} -3.00000 q^{21} -2.00000i q^{22} -1.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} -5.00000i q^{27} +3.00000i q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -2.00000i q^{33} -3.00000 q^{34} -2.00000 q^{36} +2.00000i q^{37} -1.00000i q^{38} -1.00000 q^{39} -8.00000 q^{41} +3.00000i q^{42} +4.00000i q^{43} -2.00000 q^{44} -1.00000 q^{46} -8.00000i q^{47} -1.00000i q^{48} -2.00000 q^{49} -3.00000 q^{51} +1.00000i q^{52} -1.00000i q^{53} -5.00000 q^{54} +3.00000 q^{56} -1.00000i q^{57} -5.00000i q^{58} -15.0000 q^{59} +2.00000 q^{61} +8.00000i q^{62} -6.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -3.00000i q^{67} +3.00000i q^{68} -1.00000 q^{69} +2.00000 q^{71} +2.00000i q^{72} +9.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -6.00000i q^{77} +1.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} +8.00000i q^{82} -6.00000i q^{83} +3.00000 q^{84} +4.00000 q^{86} -5.00000i q^{87} +2.00000i q^{88} -3.00000 q^{91} +1.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -1.00000 q^{96} +2.00000i q^{97} +2.00000i q^{98} +4.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} + 4q^{9} + 4q^{11} - 6q^{14} + 2q^{16} + 2q^{19} - 6q^{21} + 2q^{24} - 2q^{26} + 10q^{29} - 16q^{31} - 6q^{34} - 4q^{36} - 2q^{39} - 16q^{41} - 4q^{44} - 2q^{46} - 4q^{49} - 6q^{51} - 10q^{54} + 6q^{56} - 30q^{59} + 4q^{61} - 2q^{64} - 4q^{66} - 2q^{69} + 4q^{71} + 4q^{74} - 2q^{76} + 20q^{79} + 2q^{81} + 6q^{84} + 8q^{86} - 6q^{91} - 16q^{94} - 2q^{96} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 1.00000i 0.288675i
$$13$$ − 1.00000i − 0.277350i −0.990338 0.138675i $$-0.955716\pi$$
0.990338 0.138675i $$-0.0442844\pi$$
$$14$$ −3.00000 −0.801784
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ − 2.00000i − 0.426401i
$$23$$ − 1.00000i − 0.208514i −0.994550 0.104257i $$-0.966753\pi$$
0.994550 0.104257i $$-0.0332465\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ − 5.00000i − 0.962250i
$$28$$ 3.00000i 0.566947i
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ − 1.00000i − 0.162221i
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 3.00000i 0.462910i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ − 8.00000i − 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ − 1.00000i − 0.144338i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 1.00000i 0.138675i
$$53$$ − 1.00000i − 0.137361i −0.997639 0.0686803i $$-0.978121\pi$$
0.997639 0.0686803i $$-0.0218788\pi$$
$$54$$ −5.00000 −0.680414
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 5.00000i − 0.656532i
$$59$$ −15.0000 −1.95283 −0.976417 0.215894i $$-0.930733\pi$$
−0.976417 + 0.215894i $$0.930733\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ − 6.00000i − 0.755929i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −2.00000 −0.246183
$$67$$ − 3.00000i − 0.366508i −0.983066 0.183254i $$-0.941337\pi$$
0.983066 0.183254i $$-0.0586631\pi$$
$$68$$ 3.00000i 0.363803i
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 2.00000i 0.235702i
$$73$$ 9.00000i 1.05337i 0.850060 + 0.526685i $$0.176565\pi$$
−0.850060 + 0.526685i $$0.823435\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ − 6.00000i − 0.683763i
$$78$$ 1.00000i 0.113228i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 8.00000i 0.883452i
$$83$$ − 6.00000i − 0.658586i −0.944228 0.329293i $$-0.893190\pi$$
0.944228 0.329293i $$-0.106810\pi$$
$$84$$ 3.00000 0.327327
$$85$$ 0 0
$$86$$ 4.00000 0.431331
$$87$$ − 5.00000i − 0.536056i
$$88$$ 2.00000i 0.213201i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −3.00000 −0.314485
$$92$$ 1.00000i 0.104257i
$$93$$ 8.00000i 0.829561i
$$94$$ −8.00000 −0.825137
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ 2.00000i 0.202031i
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 3.00000i 0.297044i
$$103$$ − 6.00000i − 0.591198i −0.955312 0.295599i $$-0.904481\pi$$
0.955312 0.295599i $$-0.0955191\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −1.00000 −0.0971286
$$107$$ 7.00000i 0.676716i 0.941018 + 0.338358i $$0.109871\pi$$
−0.941018 + 0.338358i $$0.890129\pi$$
$$108$$ 5.00000i 0.481125i
$$109$$ 15.0000 1.43674 0.718370 0.695662i $$-0.244889\pi$$
0.718370 + 0.695662i $$0.244889\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ − 3.00000i − 0.283473i
$$113$$ 14.0000i 1.31701i 0.752577 + 0.658505i $$0.228811\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ 0 0
$$116$$ −5.00000 −0.464238
$$117$$ − 2.00000i − 0.184900i
$$118$$ 15.0000i 1.38086i
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ − 2.00000i − 0.181071i
$$123$$ 8.00000i 0.721336i
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ − 18.0000i − 1.59724i −0.601834 0.798621i $$-0.705563\pi$$
0.601834 0.798621i $$-0.294437\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ − 3.00000i − 0.260133i
$$134$$ −3.00000 −0.259161
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ 17.0000i 1.45241i 0.687479 + 0.726204i $$0.258717\pi$$
−0.687479 + 0.726204i $$0.741283\pi$$
$$138$$ 1.00000i 0.0851257i
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ − 2.00000i − 0.167836i
$$143$$ − 2.00000i − 0.167248i
$$144$$ 2.00000 0.166667
$$145$$ 0 0
$$146$$ 9.00000 0.744845
$$147$$ 2.00000i 0.164957i
$$148$$ − 2.00000i − 0.164399i
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 1.00000i 0.0811107i
$$153$$ − 6.00000i − 0.485071i
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 1.00000 0.0800641
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ −1.00000 −0.0793052
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 8.00000 0.624695
$$165$$ 0 0
$$166$$ −6.00000 −0.465690
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ − 3.00000i − 0.231455i
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 6.00000i − 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ −5.00000 −0.379049
$$175$$ 0 0
$$176$$ 2.00000 0.150756
$$177$$ 15.0000i 1.12747i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 3.00000i 0.222375i
$$183$$ − 2.00000i − 0.147844i
$$184$$ 1.00000 0.0737210
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ − 6.00000i − 0.438763i
$$188$$ 8.00000i 0.583460i
$$189$$ −15.0000 −1.09109
$$190$$ 0 0
$$191$$ 7.00000 0.506502 0.253251 0.967401i $$-0.418500\pi$$
0.253251 + 0.967401i $$0.418500\pi$$
$$192$$ 1.00000i 0.0721688i
$$193$$ − 6.00000i − 0.431889i −0.976406 0.215945i $$-0.930717\pi$$
0.976406 0.215945i $$-0.0692831\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ − 8.00000i − 0.569976i −0.958531 0.284988i $$-0.908010\pi$$
0.958531 0.284988i $$-0.0919897\pi$$
$$198$$ − 4.00000i − 0.284268i
$$199$$ 25.0000 1.77220 0.886102 0.463491i $$-0.153403\pi$$
0.886102 + 0.463491i $$0.153403\pi$$
$$200$$ 0 0
$$201$$ −3.00000 −0.211604
$$202$$ − 2.00000i − 0.140720i
$$203$$ − 15.0000i − 1.05279i
$$204$$ 3.00000 0.210042
$$205$$ 0 0
$$206$$ −6.00000 −0.418040
$$207$$ − 2.00000i − 0.139010i
$$208$$ − 1.00000i − 0.0693375i
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 27.0000 1.85876 0.929378 0.369129i $$-0.120344\pi$$
0.929378 + 0.369129i $$0.120344\pi$$
$$212$$ 1.00000i 0.0686803i
$$213$$ − 2.00000i − 0.137038i
$$214$$ 7.00000 0.478510
$$215$$ 0 0
$$216$$ 5.00000 0.340207
$$217$$ 24.0000i 1.62923i
$$218$$ − 15.0000i − 1.01593i
$$219$$ 9.00000 0.608164
$$220$$ 0 0
$$221$$ −3.00000 −0.201802
$$222$$ − 2.00000i − 0.134231i
$$223$$ 14.0000i 0.937509i 0.883328 + 0.468755i $$0.155297\pi$$
−0.883328 + 0.468755i $$0.844703\pi$$
$$224$$ −3.00000 −0.200446
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 17.0000i 1.12833i 0.825662 + 0.564165i $$0.190802\pi$$
−0.825662 + 0.564165i $$0.809198\pi$$
$$228$$ 1.00000i 0.0662266i
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 5.00000i 0.328266i
$$233$$ − 6.00000i − 0.393073i −0.980497 0.196537i $$-0.937031\pi$$
0.980497 0.196537i $$-0.0629694\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 15.0000 0.976417
$$237$$ − 10.0000i − 0.649570i
$$238$$ 9.00000i 0.583383i
$$239$$ −15.0000 −0.970269 −0.485135 0.874439i $$-0.661229\pi$$
−0.485135 + 0.874439i $$0.661229\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 7.00000i 0.449977i
$$243$$ − 16.0000i − 1.02640i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 8.00000 0.510061
$$247$$ − 1.00000i − 0.0636285i
$$248$$ − 8.00000i − 0.508001i
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 6.00000i 0.377964i
$$253$$ − 2.00000i − 0.125739i
$$254$$ −18.0000 −1.12942
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 8.00000i − 0.499026i −0.968371 0.249513i $$-0.919729\pi$$
0.968371 0.249513i $$-0.0802706\pi$$
$$258$$ − 4.00000i − 0.249029i
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 10.0000 0.618984
$$262$$ − 12.0000i − 0.741362i
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 2.00000 0.123091
$$265$$ 0 0
$$266$$ −3.00000 −0.183942
$$267$$ 0 0
$$268$$ 3.00000i 0.183254i
$$269$$ −30.0000 −1.82913 −0.914566 0.404436i $$-0.867468\pi$$
−0.914566 + 0.404436i $$0.867468\pi$$
$$270$$ 0 0
$$271$$ 7.00000 0.425220 0.212610 0.977137i $$-0.431804\pi$$
0.212610 + 0.977137i $$0.431804\pi$$
$$272$$ − 3.00000i − 0.181902i
$$273$$ 3.00000i 0.181568i
$$274$$ 17.0000 1.02701
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ − 28.0000i − 1.68236i −0.540758 0.841178i $$-0.681862\pi$$
0.540758 0.841178i $$-0.318138\pi$$
$$278$$ 0 0
$$279$$ −16.0000 −0.957895
$$280$$ 0 0
$$281$$ −8.00000 −0.477240 −0.238620 0.971113i $$-0.576695\pi$$
−0.238620 + 0.971113i $$0.576695\pi$$
$$282$$ 8.00000i 0.476393i
$$283$$ − 6.00000i − 0.356663i −0.983970 0.178331i $$-0.942930\pi$$
0.983970 0.178331i $$-0.0570699\pi$$
$$284$$ −2.00000 −0.118678
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 24.0000i 1.41668i
$$288$$ − 2.00000i − 0.117851i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ − 9.00000i − 0.526685i
$$293$$ 9.00000i 0.525786i 0.964825 + 0.262893i $$0.0846766\pi$$
−0.964825 + 0.262893i $$0.915323\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −2.00000 −0.116248
$$297$$ − 10.0000i − 0.580259i
$$298$$ 0 0
$$299$$ −1.00000 −0.0578315
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ − 2.00000i − 0.115087i
$$303$$ − 2.00000i − 0.114897i
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ 6.00000i 0.341882i
$$309$$ −6.00000 −0.341328
$$310$$ 0 0
$$311$$ 7.00000 0.396934 0.198467 0.980108i $$-0.436404\pi$$
0.198467 + 0.980108i $$0.436404\pi$$
$$312$$ − 1.00000i − 0.0566139i
$$313$$ 29.0000i 1.63918i 0.572953 + 0.819588i $$0.305798\pi$$
−0.572953 + 0.819588i $$0.694202\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 27.0000i 1.51647i 0.651981 + 0.758236i $$0.273938\pi$$
−0.651981 + 0.758236i $$0.726062\pi$$
$$318$$ 1.00000i 0.0560772i
$$319$$ 10.0000 0.559893
$$320$$ 0 0
$$321$$ 7.00000 0.390702
$$322$$ 3.00000i 0.167183i
$$323$$ − 3.00000i − 0.166924i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −16.0000 −0.886158
$$327$$ − 15.0000i − 0.829502i
$$328$$ − 8.00000i − 0.441726i
$$329$$ −24.0000 −1.32316
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 6.00000i 0.329293i
$$333$$ 4.00000i 0.219199i
$$334$$ 12.0000 0.656611
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ 32.0000i 1.74315i 0.490261 + 0.871576i $$0.336901\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ − 12.0000i − 0.652714i
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ − 2.00000i − 0.108148i
$$343$$ − 15.0000i − 0.809924i
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 2.00000i 0.107366i 0.998558 + 0.0536828i $$0.0170960\pi$$
−0.998558 + 0.0536828i $$0.982904\pi$$
$$348$$ 5.00000i 0.268028i
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ − 2.00000i − 0.106600i
$$353$$ 9.00000i 0.479022i 0.970894 + 0.239511i $$0.0769871\pi$$
−0.970894 + 0.239511i $$0.923013\pi$$
$$354$$ 15.0000 0.797241
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 9.00000i 0.476331i
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 22.0000i − 1.15629i
$$363$$ 7.00000i 0.367405i
$$364$$ 3.00000 0.157243
$$365$$ 0 0
$$366$$ −2.00000 −0.104542
$$367$$ − 28.0000i − 1.46159i −0.682598 0.730794i $$-0.739150\pi$$
0.682598 0.730794i $$-0.260850\pi$$
$$368$$ − 1.00000i − 0.0521286i
$$369$$ −16.0000 −0.832927
$$370$$ 0 0
$$371$$ −3.00000 −0.155752
$$372$$ − 8.00000i − 0.414781i
$$373$$ 29.0000i 1.50156i 0.660551 + 0.750782i $$0.270323\pi$$
−0.660551 + 0.750782i $$0.729677\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ − 5.00000i − 0.257513i
$$378$$ 15.0000i 0.771517i
$$379$$ −15.0000 −0.770498 −0.385249 0.922813i $$-0.625884\pi$$
−0.385249 + 0.922813i $$0.625884\pi$$
$$380$$ 0 0
$$381$$ −18.0000 −0.922168
$$382$$ − 7.00000i − 0.358151i
$$383$$ − 26.0000i − 1.32854i −0.747494 0.664269i $$-0.768743\pi$$
0.747494 0.664269i $$-0.231257\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −6.00000 −0.305392
$$387$$ 8.00000i 0.406663i
$$388$$ − 2.00000i − 0.101535i
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ − 2.00000i − 0.101015i
$$393$$ − 12.0000i − 0.605320i
$$394$$ −8.00000 −0.403034
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ − 8.00000i − 0.401508i −0.979642 0.200754i $$-0.935661\pi$$
0.979642 0.200754i $$-0.0643393\pi$$
$$398$$ − 25.0000i − 1.25314i
$$399$$ −3.00000 −0.150188
$$400$$ 0 0
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 3.00000i 0.149626i
$$403$$ 8.00000i 0.398508i
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ −15.0000 −0.744438
$$407$$ 4.00000i 0.198273i
$$408$$ − 3.00000i − 0.148522i
$$409$$ 20.0000 0.988936 0.494468 0.869196i $$-0.335363\pi$$
0.494468 + 0.869196i $$0.335363\pi$$
$$410$$ 0 0
$$411$$ 17.0000 0.838548
$$412$$ 6.00000i 0.295599i
$$413$$ 45.0000i 2.21431i
$$414$$ −2.00000 −0.0982946
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ − 2.00000i − 0.0978232i
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ − 27.0000i − 1.31434i
$$423$$ − 16.0000i − 0.777947i
$$424$$ 1.00000 0.0485643
$$425$$ 0 0
$$426$$ −2.00000 −0.0969003
$$427$$ − 6.00000i − 0.290360i
$$428$$ − 7.00000i − 0.338358i
$$429$$ −2.00000 −0.0965609
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ − 5.00000i − 0.240563i
$$433$$ 14.0000i 0.672797i 0.941720 + 0.336399i $$0.109209\pi$$
−0.941720 + 0.336399i $$0.890791\pi$$
$$434$$ 24.0000 1.15204
$$435$$ 0 0
$$436$$ −15.0000 −0.718370
$$437$$ − 1.00000i − 0.0478365i
$$438$$ − 9.00000i − 0.430037i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ −4.00000 −0.190476
$$442$$ 3.00000i 0.142695i
$$443$$ − 26.0000i − 1.23530i −0.786454 0.617649i $$-0.788085\pi$$
0.786454 0.617649i $$-0.211915\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ 0 0
$$446$$ 14.0000 0.662919
$$447$$ 0 0
$$448$$ 3.00000i 0.141737i
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −16.0000 −0.753411
$$452$$ − 14.0000i − 0.658505i
$$453$$ − 2.00000i − 0.0939682i
$$454$$ 17.0000 0.797850
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 7.00000i 0.327446i 0.986506 + 0.163723i $$0.0523504\pi$$
−0.986506 + 0.163723i $$0.947650\pi$$
$$458$$ − 10.0000i − 0.467269i
$$459$$ −15.0000 −0.700140
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 6.00000i 0.279145i
$$463$$ 4.00000i 0.185896i 0.995671 + 0.0929479i $$0.0296290\pi$$
−0.995671 + 0.0929479i $$0.970371\pi$$
$$464$$ 5.00000 0.232119
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 2.00000i 0.0925490i 0.998929 + 0.0462745i $$0.0147349\pi$$
−0.998929 + 0.0462745i $$0.985265\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ −9.00000 −0.415581
$$470$$ 0 0
$$471$$ 2.00000 0.0921551
$$472$$ − 15.0000i − 0.690431i
$$473$$ 8.00000i 0.367840i
$$474$$ −10.0000 −0.459315
$$475$$ 0 0
$$476$$ 9.00000 0.412514
$$477$$ − 2.00000i − 0.0915737i
$$478$$ 15.0000i 0.686084i
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 8.00000i 0.364390i
$$483$$ 3.00000i 0.136505i
$$484$$ 7.00000 0.318182
$$485$$ 0 0
$$486$$ −16.0000 −0.725775
$$487$$ 2.00000i 0.0906287i 0.998973 + 0.0453143i $$0.0144289\pi$$
−0.998973 + 0.0453143i $$0.985571\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ −16.0000 −0.723545
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ − 8.00000i − 0.360668i
$$493$$ − 15.0000i − 0.675566i
$$494$$ −1.00000 −0.0449921
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 6.00000i − 0.269137i
$$498$$ 6.00000i 0.268866i
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ − 2.00000i − 0.0892644i
$$503$$ 39.0000i 1.73892i 0.494000 + 0.869462i $$0.335534\pi$$
−0.494000 + 0.869462i $$0.664466\pi$$
$$504$$ 6.00000 0.267261
$$505$$ 0 0
$$506$$ −2.00000 −0.0889108
$$507$$ − 12.0000i − 0.532939i
$$508$$ 18.0000i 0.798621i
$$509$$ 30.0000 1.32973 0.664863 0.746965i $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 0 0
$$511$$ 27.0000 1.19441
$$512$$ − 1.00000i − 0.0441942i
$$513$$ − 5.00000i − 0.220755i
$$514$$ −8.00000 −0.352865
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ − 16.0000i − 0.703679i
$$518$$ − 6.00000i − 0.263625i
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ − 10.0000i − 0.437688i
$$523$$ 29.0000i 1.26808i 0.773300 + 0.634041i $$0.218605\pi$$
−0.773300 + 0.634041i $$0.781395\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ 24.0000i 1.04546i
$$528$$ − 2.00000i − 0.0870388i
$$529$$ 22.0000 0.956522
$$530$$ 0 0
$$531$$ −30.0000 −1.30189
$$532$$ 3.00000i 0.130066i
$$533$$ 8.00000i 0.346518i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 3.00000 0.129580
$$537$$ 0 0
$$538$$ 30.0000i 1.29339i
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ − 7.00000i − 0.300676i
$$543$$ − 22.0000i − 0.944110i
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 3.00000 0.128388
$$547$$ − 28.0000i − 1.19719i −0.801050 0.598597i $$-0.795725\pi$$
0.801050 0.598597i $$-0.204275\pi$$
$$548$$ − 17.0000i − 0.726204i
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ 5.00000 0.213007
$$552$$ − 1.00000i − 0.0425628i
$$553$$ − 30.0000i − 1.27573i
$$554$$ −28.0000 −1.18961
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 28.0000i − 1.18640i −0.805056 0.593199i $$-0.797865\pi$$
0.805056 0.593199i $$-0.202135\pi$$
$$558$$ 16.0000i 0.677334i
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −6.00000 −0.253320
$$562$$ 8.00000i 0.337460i
$$563$$ − 36.0000i − 1.51722i −0.651546 0.758610i $$-0.725879\pi$$
0.651546 0.758610i $$-0.274121\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ −6.00000 −0.252199
$$567$$ − 3.00000i − 0.125988i
$$568$$ 2.00000i 0.0839181i
$$569$$ −40.0000 −1.67689 −0.838444 0.544988i $$-0.816534\pi$$
−0.838444 + 0.544988i $$0.816534\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ − 7.00000i − 0.292429i
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ −2.00000 −0.0833333
$$577$$ 37.0000i 1.54033i 0.637845 + 0.770165i $$0.279826\pi$$
−0.637845 + 0.770165i $$0.720174\pi$$
$$578$$ − 8.00000i − 0.332756i
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ −18.0000 −0.746766
$$582$$ − 2.00000i − 0.0829027i
$$583$$ − 2.00000i − 0.0828315i
$$584$$ −9.00000 −0.372423
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 12.0000i 0.495293i 0.968850 + 0.247647i $$0.0796572\pi$$
−0.968850 + 0.247647i $$0.920343\pi$$
$$588$$ − 2.00000i − 0.0824786i
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ −8.00000 −0.329076
$$592$$ 2.00000i 0.0821995i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ −10.0000 −0.410305
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 25.0000i − 1.02318i
$$598$$ 1.00000i 0.0408930i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ − 12.0000i − 0.489083i
$$603$$ − 6.00000i − 0.244339i
$$604$$ −2.00000 −0.0813788
$$605$$ 0 0
$$606$$ −2.00000 −0.0812444
$$607$$ 22.0000i 0.892952i 0.894795 + 0.446476i $$0.147321\pi$$
−0.894795 + 0.446476i $$0.852679\pi$$
$$608$$ − 1.00000i − 0.0405554i
$$609$$ −15.0000 −0.607831
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ 6.00000i 0.242536i
$$613$$ 34.0000i 1.37325i 0.727013 + 0.686624i $$0.240908\pi$$
−0.727013 + 0.686624i $$0.759092\pi$$
$$614$$ 12.0000 0.484281
$$615$$ 0 0
$$616$$ 6.00000 0.241747
$$617$$ − 18.0000i − 0.724653i −0.932051 0.362326i $$-0.881983\pi$$
0.932051 0.362326i $$-0.118017\pi$$
$$618$$ 6.00000i 0.241355i
$$619$$ −10.0000 −0.401934 −0.200967 0.979598i $$-0.564408\pi$$
−0.200967 + 0.979598i $$0.564408\pi$$
$$620$$ 0 0
$$621$$ −5.00000 −0.200643
$$622$$ − 7.00000i − 0.280674i
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 0 0
$$626$$ 29.0000 1.15907
$$627$$ − 2.00000i − 0.0798723i
$$628$$ − 2.00000i − 0.0798087i
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 32.0000 1.27390 0.636950 0.770905i $$-0.280196\pi$$
0.636950 + 0.770905i $$0.280196\pi$$
$$632$$ 10.0000i 0.397779i
$$633$$ − 27.0000i − 1.07315i
$$634$$ 27.0000 1.07231
$$635$$ 0 0
$$636$$ 1.00000 0.0396526
$$637$$ 2.00000i 0.0792429i
$$638$$ − 10.0000i − 0.395904i
$$639$$ 4.00000 0.158238
$$640$$ 0 0
$$641$$ 42.0000 1.65890 0.829450 0.558581i $$-0.188654\pi$$
0.829450 + 0.558581i $$0.188654\pi$$
$$642$$ − 7.00000i − 0.276268i
$$643$$ − 26.0000i − 1.02534i −0.858586 0.512670i $$-0.828656\pi$$
0.858586 0.512670i $$-0.171344\pi$$
$$644$$ 3.00000 0.118217
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ − 23.0000i − 0.904223i −0.891961 0.452112i $$-0.850671\pi$$
0.891961 0.452112i $$-0.149329\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ −30.0000 −1.17760
$$650$$ 0 0
$$651$$ 24.0000 0.940634
$$652$$ 16.0000i 0.626608i
$$653$$ − 36.0000i − 1.40879i −0.709809 0.704394i $$-0.751219\pi$$
0.709809 0.704394i $$-0.248781\pi$$
$$654$$ −15.0000 −0.586546
$$655$$ 0 0
$$656$$ −8.00000 −0.312348
$$657$$ 18.0000i 0.702247i
$$658$$ 24.0000i 0.935617i
$$659$$ −5.00000 −0.194772 −0.0973862 0.995247i $$-0.531048\pi$$
−0.0973862 + 0.995247i $$0.531048\pi$$
$$660$$ 0 0
$$661$$ −23.0000 −0.894596 −0.447298 0.894385i $$-0.647614\pi$$
−0.447298 + 0.894385i $$0.647614\pi$$
$$662$$ − 17.0000i − 0.660724i
$$663$$ 3.00000i 0.116510i
$$664$$ 6.00000 0.232845
$$665$$ 0 0
$$666$$ 4.00000 0.154997
$$667$$ − 5.00000i − 0.193601i
$$668$$ − 12.0000i − 0.464294i
$$669$$ 14.0000 0.541271
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 3.00000i 0.115728i
$$673$$ 44.0000i 1.69608i 0.529936 + 0.848038i $$0.322216\pi$$
−0.529936 + 0.848038i $$0.677784\pi$$
$$674$$ 32.0000 1.23259
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ − 13.0000i − 0.499631i −0.968294 0.249815i $$-0.919630\pi$$
0.968294 0.249815i $$-0.0803699\pi$$
$$678$$ − 14.0000i − 0.537667i
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ 17.0000 0.651441
$$682$$ 16.0000i 0.612672i
$$683$$ 4.00000i 0.153056i 0.997067 + 0.0765279i $$0.0243834\pi$$
−0.997067 + 0.0765279i $$0.975617\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ −15.0000 −0.572703
$$687$$ − 10.0000i − 0.381524i
$$688$$ 4.00000i 0.152499i
$$689$$ −1.00000 −0.0380970
$$690$$ 0 0
$$691$$ 42.0000 1.59776 0.798878 0.601494i $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ − 12.0000i − 0.455842i
$$694$$ 2.00000 0.0759190
$$695$$ 0 0
$$696$$ 5.00000 0.189525
$$697$$ 24.0000i 0.909065i
$$698$$ 10.0000i 0.378506i
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −28.0000 −1.05755 −0.528773 0.848763i $$-0.677348\pi$$
−0.528773 + 0.848763i $$0.677348\pi$$
$$702$$ 5.00000i 0.188713i
$$703$$ 2.00000i 0.0754314i
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 9.00000 0.338719
$$707$$ − 6.00000i − 0.225653i
$$708$$ − 15.0000i − 0.563735i
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ 20.0000 0.750059
$$712$$ 0 0
$$713$$ 8.00000i 0.299602i
$$714$$ 9.00000 0.336817
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 15.0000i 0.560185i
$$718$$ − 15.0000i − 0.559795i
$$719$$ 5.00000 0.186469 0.0932343 0.995644i $$-0.470279\pi$$
0.0932343 + 0.995644i $$0.470279\pi$$
$$720$$ 0 0
$$721$$ −18.0000 −0.670355
$$722$$ − 1.00000i − 0.0372161i
$$723$$ 8.00000i 0.297523i
$$724$$ −22.0000 −0.817624
$$725$$ 0 0
$$726$$ 7.00000 0.259794
$$727$$ 17.0000i 0.630495i 0.949009 + 0.315248i $$0.102088\pi$$
−0.949009 + 0.315248i $$0.897912\pi$$
$$728$$ − 3.00000i − 0.111187i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 2.00000i 0.0739221i
$$733$$ − 36.0000i − 1.32969i −0.746981 0.664845i $$-0.768498\pi$$
0.746981 0.664845i $$-0.231502\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ − 6.00000i − 0.221013i
$$738$$ 16.0000i 0.588968i
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 0 0
$$741$$ −1.00000 −0.0367359
$$742$$ 3.00000i 0.110133i
$$743$$ − 16.0000i − 0.586983i −0.955962 0.293492i $$-0.905183\pi$$
0.955962 0.293492i $$-0.0948173\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 29.0000 1.06177
$$747$$ − 12.0000i − 0.439057i
$$748$$ 6.00000i 0.219382i
$$749$$ 21.0000 0.767323
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ − 8.00000i − 0.291730i
$$753$$ − 2.00000i − 0.0728841i
$$754$$ −5.00000 −0.182089
$$755$$ 0 0
$$756$$ 15.0000 0.545545
$$757$$ 2.00000i 0.0726912i 0.999339 + 0.0363456i $$0.0115717\pi$$
−0.999339 + 0.0363456i $$0.988428\pi$$
$$758$$ 15.0000i 0.544825i
$$759$$ −2.00000 −0.0725954
$$760$$ 0 0
$$761$$ 27.0000 0.978749 0.489375 0.872074i $$-0.337225\pi$$
0.489375 + 0.872074i $$0.337225\pi$$
$$762$$ 18.0000i 0.652071i
$$763$$ − 45.0000i − 1.62911i
$$764$$ −7.00000 −0.253251
$$765$$ 0 0
$$766$$ −26.0000 −0.939418
$$767$$ 15.0000i 0.541619i
$$768$$ − 1.00000i − 0.0360844i
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ 0 0
$$771$$ −8.00000 −0.288113
$$772$$ 6.00000i 0.215945i
$$773$$ 9.00000i 0.323708i 0.986815 + 0.161854i $$0.0517473\pi$$
−0.986815 + 0.161854i $$0.948253\pi$$
$$774$$ 8.00000 0.287554
$$775$$ 0 0
$$776$$ −2.00000 −0.0717958
$$777$$ − 6.00000i − 0.215249i
$$778$$ − 30.0000i − 1.07555i
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 3.00000i 0.107280i
$$783$$ − 25.0000i − 0.893427i
$$784$$ −2.00000 −0.0714286
$$785$$ 0 0
$$786$$ −12.0000 −0.428026
$$787$$ 17.0000i 0.605985i 0.952993 + 0.302992i $$0.0979856\pi$$
−0.952993 + 0.302992i $$0.902014\pi$$
$$788$$ 8.00000i 0.284988i
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 42.0000 1.49335
$$792$$ 4.00000i 0.142134i
$$793$$ − 2.00000i − 0.0710221i
$$794$$ −8.00000 −0.283909
$$795$$ 0 0
$$796$$ −25.0000 −0.886102
$$797$$ − 3.00000i − 0.106265i −0.998587 0.0531327i $$-0.983079\pi$$
0.998587 0.0531327i $$-0.0169206\pi$$
$$798$$ 3.00000i 0.106199i
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 8.00000i 0.282490i
$$803$$ 18.0000i 0.635206i
$$804$$ 3.00000 0.105802
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 30.0000i 1.05605i
$$808$$ 2.00000i 0.0703598i
$$809$$ 15.0000 0.527372 0.263686 0.964609i $$-0.415062\pi$$
0.263686 + 0.964609i $$0.415062\pi$$
$$810$$ 0 0
$$811$$ −3.00000 −0.105344 −0.0526721 0.998612i $$-0.516774\pi$$
−0.0526721 + 0.998612i $$0.516774\pi$$
$$812$$ 15.0000i 0.526397i
$$813$$ − 7.00000i − 0.245501i
$$814$$ 4.00000 0.140200
$$815$$ 0 0
$$816$$ −3.00000 −0.105021
$$817$$ 4.00000i 0.139942i
$$818$$ − 20.0000i − 0.699284i
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 12.0000 0.418803 0.209401 0.977830i $$-0.432848\pi$$
0.209401 + 0.977830i $$0.432848\pi$$
$$822$$ − 17.0000i − 0.592943i
$$823$$ 29.0000i 1.01088i 0.862863 + 0.505438i $$0.168669\pi$$
−0.862863 + 0.505438i $$0.831331\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 0 0
$$826$$ 45.0000 1.56575
$$827$$ − 23.0000i − 0.799788i −0.916561 0.399894i $$-0.869047\pi$$
0.916561 0.399894i $$-0.130953\pi$$
$$828$$ 2.00000i 0.0695048i
$$829$$ 15.0000 0.520972 0.260486 0.965478i $$-0.416117\pi$$
0.260486 + 0.965478i $$0.416117\pi$$
$$830$$ 0 0
$$831$$ −28.0000 −0.971309
$$832$$ 1.00000i 0.0346688i
$$833$$ 6.00000i 0.207888i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 40.0000i 1.38260i
$$838$$ 0 0
$$839$$ −20.0000 −0.690477 −0.345238 0.938515i $$-0.612202\pi$$
−0.345238 + 0.938515i $$0.612202\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 13.0000i 0.448010i
$$843$$ 8.00000i 0.275535i
$$844$$ −27.0000 −0.929378
$$845$$ 0 0
$$846$$ −16.0000 −0.550091
$$847$$ 21.0000i 0.721569i
$$848$$ − 1.00000i − 0.0343401i
$$849$$ −6.00000 −0.205919
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ 2.00000i 0.0685189i
$$853$$ − 6.00000i − 0.205436i −0.994711 0.102718i $$-0.967246\pi$$
0.994711 0.102718i $$-0.0327539\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ −7.00000 −0.239255
$$857$$ 12.0000i 0.409912i 0.978771 + 0.204956i $$0.0657052\pi$$
−0.978771 + 0.204956i $$0.934295\pi$$
$$858$$ 2.00000i 0.0682789i
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 18.0000i 0.613082i
$$863$$ 54.0000i 1.83818i 0.394046 + 0.919091i $$0.371075\pi$$
−0.394046 + 0.919091i $$0.628925\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 0 0
$$866$$ 14.0000 0.475739
$$867$$ − 8.00000i − 0.271694i
$$868$$ − 24.0000i − 0.814613i
$$869$$ 20.0000 0.678454
$$870$$ 0 0
$$871$$ −3.00000 −0.101651
$$872$$ 15.0000i 0.507964i
$$873$$ 4.00000i 0.135379i
$$874$$ −1.00000 −0.0338255
$$875$$ 0 0
$$876$$ −9.00000 −0.304082
$$877$$ − 13.0000i − 0.438979i −0.975615 0.219489i $$-0.929561\pi$$
0.975615 0.219489i $$-0.0704391\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ 9.00000 0.303562
$$880$$ 0 0
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 4.00000i 0.134687i
$$883$$ 34.0000i 1.14419i 0.820187 + 0.572096i $$0.193869\pi$$
−0.820187 + 0.572096i $$0.806131\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ −26.0000 −0.873487
$$887$$ 2.00000i 0.0671534i 0.999436 + 0.0335767i $$0.0106898\pi$$
−0.999436 + 0.0335767i $$0.989310\pi$$
$$888$$ 2.00000i 0.0671156i
$$889$$ −54.0000 −1.81110
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ − 14.0000i − 0.468755i
$$893$$ − 8.00000i − 0.267710i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 3.00000 0.100223
$$897$$ 1.00000i 0.0333890i
$$898$$ 10.0000i 0.333704i
$$899$$ −40.0000 −1.33407
$$900$$ 0 0
$$901$$ −3.00000 −0.0999445
$$902$$ 16.0000i 0.532742i
$$903$$ − 12.0000i − 0.399335i
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −2.00000 −0.0664455
$$907$$ − 53.0000i − 1.75984i −0.475125 0.879918i $$-0.657597\pi$$
0.475125 0.879918i $$-0.342403\pi$$
$$908$$ − 17.0000i − 0.564165i
$$909$$ 4.00000 0.132672
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ − 1.00000i − 0.0331133i
$$913$$ − 12.0000i − 0.397142i
$$914$$ 7.00000 0.231539
$$915$$ 0 0
$$916$$ −10.0000 −0.330409
$$917$$ − 36.0000i − 1.18882i
$$918$$ 15.0000i 0.495074i
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 28.0000i 0.922131i
$$923$$ − 2.00000i − 0.0658308i
$$924$$ 6.00000 0.197386
$$925$$ 0 0
$$926$$ 4.00000 0.131448
$$927$$ − 12.0000i − 0.394132i
$$928$$ − 5.00000i − 0.164133i
$$929$$ 55.0000 1.80449 0.902246 0.431222i $$-0.141918\pi$$
0.902246 + 0.431222i $$0.141918\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 6.00000i 0.196537i
$$933$$ − 7.00000i − 0.229170i
$$934$$ 2.00000 0.0654420
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 7.00000i 0.228680i 0.993442 + 0.114340i $$0.0364753\pi$$
−0.993442 + 0.114340i $$0.963525\pi$$
$$938$$ 9.00000i 0.293860i
$$939$$ 29.0000 0.946379
$$940$$ 0 0
$$941$$ 7.00000 0.228193 0.114097 0.993470i $$-0.463603\pi$$
0.114097 + 0.993470i $$0.463603\pi$$
$$942$$ − 2.00000i − 0.0651635i
$$943$$ 8.00000i 0.260516i
$$944$$ −15.0000 −0.488208
$$945$$ 0 0
$$946$$ 8.00000 0.260102
$$947$$ 12.0000i 0.389948i 0.980808 + 0.194974i $$0.0624622\pi$$
−0.980808 + 0.194974i $$0.937538\pi$$
$$948$$ 10.0000i 0.324785i
$$949$$ 9.00000 0.292152
$$950$$ 0 0
$$951$$ 27.0000 0.875535
$$952$$ − 9.00000i − 0.291692i
$$953$$ − 46.0000i − 1.49009i −0.667016 0.745043i $$-0.732429\pi$$
0.667016 0.745043i $$-0.267571\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ 15.0000 0.485135
$$957$$ − 10.0000i − 0.323254i
$$958$$ − 20.0000i − 0.646171i
$$959$$ 51.0000 1.64688
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ − 2.00000i − 0.0644826i
$$963$$ 14.0000i 0.451144i
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ 3.00000 0.0965234
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ − 7.00000i − 0.224989i
$$969$$ −3.00000 −0.0963739
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 16.0000i 0.513200i
$$973$$ 0 0
$$974$$ 2.00000 0.0640841
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ − 8.00000i − 0.255943i −0.991778 0.127971i $$-0.959153\pi$$
0.991778 0.127971i $$-0.0408466\pi$$
$$978$$ 16.0000i 0.511624i
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 30.0000 0.957826
$$982$$ 28.0000i 0.893516i
$$983$$ − 6.00000i − 0.191370i −0.995412 0.0956851i $$-0.969496\pi$$
0.995412 0.0956851i $$-0.0305042\pi$$
$$984$$ −8.00000 −0.255031
$$985$$ 0 0
$$986$$ −15.0000 −0.477697
$$987$$ 24.0000i 0.763928i
$$988$$ 1.00000i 0.0318142i
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ − 17.0000i − 0.539479i
$$994$$ −6.00000 −0.190308
$$995$$ 0 0
$$996$$ 6.00000 0.190117
$$997$$ − 28.0000i − 0.886769i −0.896332 0.443384i $$-0.853778\pi$$
0.896332 0.443384i $$-0.146222\pi$$
$$998$$ 40.0000i 1.26618i
$$999$$ 10.0000 0.316386
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 950.2.b.c.799.1 2
5.2 odd 4 38.2.a.b.1.1 1
5.3 odd 4 950.2.a.b.1.1 1
5.4 even 2 inner 950.2.b.c.799.2 2
15.2 even 4 342.2.a.d.1.1 1
15.8 even 4 8550.2.a.u.1.1 1
20.3 even 4 7600.2.a.h.1.1 1
20.7 even 4 304.2.a.d.1.1 1
35.27 even 4 1862.2.a.f.1.1 1
40.27 even 4 1216.2.a.g.1.1 1
40.37 odd 4 1216.2.a.n.1.1 1
55.32 even 4 4598.2.a.a.1.1 1
60.47 odd 4 2736.2.a.w.1.1 1
65.12 odd 4 6422.2.a.b.1.1 1
95.2 even 36 722.2.e.d.99.1 6
95.7 odd 12 722.2.c.d.429.1 2
95.12 even 12 722.2.c.f.429.1 2
95.17 odd 36 722.2.e.c.99.1 6
95.22 even 36 722.2.e.d.389.1 6
95.27 even 12 722.2.c.f.653.1 2
95.32 even 36 722.2.e.d.245.1 6
95.37 even 4 722.2.a.b.1.1 1
95.42 odd 36 722.2.e.c.415.1 6
95.47 odd 36 722.2.e.c.423.1 6
95.52 even 36 722.2.e.d.595.1 6
95.62 odd 36 722.2.e.c.595.1 6
95.67 even 36 722.2.e.d.423.1 6
95.72 even 36 722.2.e.d.415.1 6
95.82 odd 36 722.2.e.c.245.1 6
95.87 odd 12 722.2.c.d.653.1 2
95.92 odd 36 722.2.e.c.389.1 6
285.227 odd 4 6498.2.a.y.1.1 1
380.227 odd 4 5776.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.b.1.1 1 5.2 odd 4
304.2.a.d.1.1 1 20.7 even 4
342.2.a.d.1.1 1 15.2 even 4
722.2.a.b.1.1 1 95.37 even 4
722.2.c.d.429.1 2 95.7 odd 12
722.2.c.d.653.1 2 95.87 odd 12
722.2.c.f.429.1 2 95.12 even 12
722.2.c.f.653.1 2 95.27 even 12
722.2.e.c.99.1 6 95.17 odd 36
722.2.e.c.245.1 6 95.82 odd 36
722.2.e.c.389.1 6 95.92 odd 36
722.2.e.c.415.1 6 95.42 odd 36
722.2.e.c.423.1 6 95.47 odd 36
722.2.e.c.595.1 6 95.62 odd 36
722.2.e.d.99.1 6 95.2 even 36
722.2.e.d.245.1 6 95.32 even 36
722.2.e.d.389.1 6 95.22 even 36
722.2.e.d.415.1 6 95.72 even 36
722.2.e.d.423.1 6 95.67 even 36
722.2.e.d.595.1 6 95.52 even 36
950.2.a.b.1.1 1 5.3 odd 4
950.2.b.c.799.1 2 1.1 even 1 trivial
950.2.b.c.799.2 2 5.4 even 2 inner
1216.2.a.g.1.1 1 40.27 even 4
1216.2.a.n.1.1 1 40.37 odd 4
1862.2.a.f.1.1 1 35.27 even 4
2736.2.a.w.1.1 1 60.47 odd 4
4598.2.a.a.1.1 1 55.32 even 4
5776.2.a.d.1.1 1 380.227 odd 4
6422.2.a.b.1.1 1 65.12 odd 4
6498.2.a.y.1.1 1 285.227 odd 4
7600.2.a.h.1.1 1 20.3 even 4
8550.2.a.u.1.1 1 15.8 even 4