# Properties

 Label 3850.2.c.a.1849.2 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ 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 770) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1849.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3850.1849 Dual form 3850.2.c.a.1849.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -1.00000i q^{18} -2.00000 q^{19} -2.00000 q^{21} -1.00000i q^{22} +6.00000i q^{23} +2.00000 q^{24} +2.00000 q^{26} +4.00000i q^{27} -1.00000i q^{28} +8.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +8.00000i q^{37} -2.00000i q^{38} +4.00000 q^{39} -2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} +2.00000i q^{48} -1.00000 q^{49} -12.0000 q^{51} +2.00000i q^{52} -12.0000i q^{53} -4.00000 q^{54} +1.00000 q^{56} -4.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} +8.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} -4.00000i q^{67} -6.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -14.0000i q^{73} -8.00000 q^{74} +2.00000 q^{76} -1.00000i q^{77} +4.00000i q^{78} +10.0000 q^{79} -11.0000 q^{81} +2.00000 q^{84} -4.00000 q^{86} +1.00000i q^{88} +18.0000 q^{89} +2.00000 q^{91} -6.00000i q^{92} +16.0000i q^{93} -2.00000 q^{96} +8.00000i q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 4 q^{19} - 4 q^{21} + 4 q^{24} + 4 q^{26} + 16 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} - 24 q^{51} - 8 q^{54} + 2 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} - 16 q^{74} + 4 q^{76} + 20 q^{79} - 22 q^{81} + 4 q^{84} - 8 q^{86} + 36 q^{89} + 4 q^{91} - 4 q^{96} + 2 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ 1.00000i 0.377964i
$$8$$ − 1.00000i − 0.353553i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 2.00000i − 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000i 1.45521i 0.685994 + 0.727607i $$0.259367\pi$$
−0.685994 + 0.727607i $$0.740633\pi$$
$$18$$ − 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
$$22$$ − 1.00000i − 0.213201i
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 2.00000 0.408248
$$25$$ 0 0
$$26$$ 2.00000 0.392232
$$27$$ 4.00000i 0.769800i
$$28$$ − 1.00000i − 0.188982i
$$29$$ 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
$$33$$ − 2.00000i − 0.348155i
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ − 2.00000i − 0.324443i
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ − 2.00000i − 0.308607i
$$43$$ 4.00000i 0.609994i 0.952353 + 0.304997i $$0.0986555\pi$$
−0.952353 + 0.304997i $$0.901344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ −12.0000 −1.68034
$$52$$ 2.00000i 0.277350i
$$53$$ − 12.0000i − 1.64833i −0.566352 0.824163i $$-0.691646\pi$$
0.566352 0.824163i $$-0.308354\pi$$
$$54$$ −4.00000 −0.544331
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ − 4.00000i − 0.529813i
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 8.00000i 1.01600i
$$63$$ − 1.00000i − 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ − 4.00000i − 0.488678i −0.969690 0.244339i $$-0.921429\pi$$
0.969690 0.244339i $$-0.0785709\pi$$
$$68$$ − 6.00000i − 0.727607i
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 14.0000i − 1.63858i −0.573382 0.819288i $$-0.694369\pi$$
0.573382 0.819288i $$-0.305631\pi$$
$$74$$ −8.00000 −0.929981
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ − 1.00000i − 0.113961i
$$78$$ 4.00000i 0.452911i
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 1.00000i 0.106600i
$$89$$ 18.0000 1.90800 0.953998 0.299813i $$-0.0969242\pi$$
0.953998 + 0.299813i $$0.0969242\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ − 6.00000i − 0.625543i
$$93$$ 16.0000i 1.65912i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ 8.00000i 0.812277i 0.913812 + 0.406138i $$0.133125\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ − 12.0000i − 1.18818i
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ 12.0000i 1.16008i 0.814587 + 0.580042i $$0.196964\pi$$
−0.814587 + 0.580042i $$0.803036\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ −8.00000 −0.766261 −0.383131 0.923694i $$-0.625154\pi$$
−0.383131 + 0.923694i $$0.625154\pi$$
$$110$$ 0 0
$$111$$ −16.0000 −1.51865
$$112$$ 1.00000i 0.0944911i
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 2.00000i 0.184900i
$$118$$ − 12.0000i − 1.10469i
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ − 10.0000i − 0.905357i
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 1.00000 0.0890871
$$127$$ − 16.0000i − 1.41977i −0.704317 0.709885i $$-0.748747\pi$$
0.704317 0.709885i $$-0.251253\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ − 2.00000i − 0.173422i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ − 6.00000i − 0.512615i −0.966595 0.256307i $$-0.917494\pi$$
0.966595 0.256307i $$-0.0825059\pi$$
$$138$$ − 12.0000i − 1.02151i
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 12.0000i − 1.00702i
$$143$$ 2.00000i 0.167248i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ − 2.00000i − 0.164957i
$$148$$ − 8.00000i − 0.657596i
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 2.00000i 0.162221i
$$153$$ − 6.00000i − 0.485071i
$$154$$ 1.00000 0.0805823
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 10.0000i 0.795557i
$$159$$ 24.0000 1.90332
$$160$$ 0 0
$$161$$ −6.00000 −0.472866
$$162$$ − 11.0000i − 0.864242i
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 2.00000i 0.154303i
$$169$$ 9.00000 0.692308
$$170$$ 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 $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ − 24.0000i − 1.80395i
$$178$$ 18.0000i 1.34916i
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 2.00000i 0.148250i
$$183$$ − 20.0000i − 1.47844i
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ −16.0000 −1.17318
$$187$$ − 6.00000i − 0.438763i
$$188$$ 0 0
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ − 14.0000i − 1.00774i −0.863779 0.503871i $$-0.831909\pi$$
0.863779 0.503871i $$-0.168091\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 1.00000i 0.0710669i
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ − 18.0000i − 1.26648i
$$203$$ 0 0
$$204$$ 12.0000 0.840168
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ − 6.00000i − 0.417029i
$$208$$ − 2.00000i − 0.138675i
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 12.0000i 0.824163i
$$213$$ − 24.0000i − 1.64445i
$$214$$ −12.0000 −0.820303
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ 8.00000i 0.543075i
$$218$$ − 8.00000i − 0.541828i
$$219$$ 28.0000 1.89206
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ − 16.0000i − 1.07385i
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −6.00000 −0.399114
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 4.00000i 0.264906i
$$229$$ −26.0000 −1.71813 −0.859064 0.511868i $$-0.828954\pi$$
−0.859064 + 0.511868i $$0.828954\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 20.0000i 1.29914i
$$238$$ − 6.00000i − 0.388922i
$$239$$ −18.0000 −1.16432 −0.582162 0.813073i $$-0.697793\pi$$
−0.582162 + 0.813073i $$0.697793\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ − 10.0000i − 0.641500i
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ − 8.00000i − 0.508001i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 1.00000i 0.0629941i
$$253$$ − 6.00000i − 0.377217i
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ − 24.0000i − 1.49708i −0.663090 0.748539i $$-0.730755\pi$$
0.663090 0.748539i $$-0.269245\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 18.0000i − 1.11204i
$$263$$ 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
$$266$$ 2.00000 0.122628
$$267$$ 36.0000i 2.20316i
$$268$$ 4.00000i 0.244339i
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 6.00000i 0.363803i
$$273$$ 4.00000i 0.242091i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ − 10.0000i − 0.600842i −0.953807 0.300421i $$-0.902873\pi$$
0.953807 0.300421i $$-0.0971271\pi$$
$$278$$ 10.0000i 0.599760i
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 4.00000i 0.237775i 0.992908 + 0.118888i $$0.0379328\pi$$
−0.992908 + 0.118888i $$0.962067\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ −16.0000 −0.937937
$$292$$ 14.0000i 0.819288i
$$293$$ − 18.0000i − 1.05157i −0.850617 0.525786i $$-0.823771\pi$$
0.850617 0.525786i $$-0.176229\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ − 4.00000i − 0.232104i
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ − 10.0000i − 0.575435i
$$303$$ − 36.0000i − 2.06815i
$$304$$ −2.00000 −0.114708
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ 32.0000i 1.82634i 0.407583 + 0.913168i $$0.366372\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$308$$ 1.00000i 0.0569803i
$$309$$ −8.00000 −0.455104
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ − 4.00000i − 0.226455i
$$313$$ 16.0000i 0.904373i 0.891923 + 0.452187i $$0.149356\pi$$
−0.891923 + 0.452187i $$0.850644\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 24.0000i 1.34585i
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −24.0000 −1.33955
$$322$$ − 6.00000i − 0.334367i
$$323$$ − 12.0000i − 0.667698i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ − 16.0000i − 0.884802i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ − 8.00000i − 0.438397i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ −2.00000 −0.109109
$$337$$ − 22.0000i − 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ −12.0000 −0.651751
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 2.00000i 0.108148i
$$343$$ − 1.00000i − 0.0539949i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ − 12.0000i − 0.644194i −0.946707 0.322097i $$-0.895612\pi$$
0.946707 0.322097i $$-0.104388\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 8.00000 0.427008
$$352$$ − 1.00000i − 0.0533002i
$$353$$ 24.0000i 1.27739i 0.769460 + 0.638696i $$0.220526\pi$$
−0.769460 + 0.638696i $$0.779474\pi$$
$$354$$ 24.0000 1.27559
$$355$$ 0 0
$$356$$ −18.0000 −0.953998
$$357$$ − 12.0000i − 0.635107i
$$358$$ 12.0000i 0.634220i
$$359$$ 18.0000 0.950004 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ − 10.0000i − 0.525588i
$$363$$ 2.00000i 0.104973i
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ 20.0000 1.04542
$$367$$ 20.0000i 1.04399i 0.852948 + 0.521996i $$0.174812\pi$$
−0.852948 + 0.521996i $$0.825188\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ − 16.0000i − 0.829561i
$$373$$ 22.0000i 1.13912i 0.821951 + 0.569558i $$0.192886\pi$$
−0.821951 + 0.569558i $$0.807114\pi$$
$$374$$ 6.00000 0.310253
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ − 4.00000i − 0.205738i
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 32.0000 1.63941
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i 0.951843 + 0.306586i $$0.0991866\pi$$
−0.951843 + 0.306586i $$0.900813\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ − 4.00000i − 0.203331i
$$388$$ − 8.00000i − 0.406138i
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ −36.0000 −1.82060
$$392$$ 1.00000i 0.0505076i
$$393$$ − 36.0000i − 1.81596i
$$394$$ −18.0000 −0.906827
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ 2.00000i 0.100377i 0.998740 + 0.0501886i $$0.0159822\pi$$
−0.998740 + 0.0501886i $$0.984018\pi$$
$$398$$ − 8.00000i − 0.401004i
$$399$$ 4.00000 0.200250
$$400$$ 0 0
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 8.00000i 0.399004i
$$403$$ − 16.0000i − 0.797017i
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 8.00000i − 0.396545i
$$408$$ 12.0000i 0.594089i
$$409$$ −8.00000 −0.395575 −0.197787 0.980245i $$-0.563376\pi$$
−0.197787 + 0.980245i $$0.563376\pi$$
$$410$$ 0 0
$$411$$ 12.0000 0.591916
$$412$$ − 4.00000i − 0.197066i
$$413$$ − 12.0000i − 0.590481i
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ 2.00000 0.0980581
$$417$$ 20.0000i 0.979404i
$$418$$ 2.00000i 0.0978232i
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ 20.0000i 0.973585i
$$423$$ 0 0
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 24.0000 1.16280
$$427$$ − 10.0000i − 0.483934i
$$428$$ − 12.0000i − 0.580042i
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 4.00000i 0.192228i 0.995370 + 0.0961139i $$0.0306413\pi$$
−0.995370 + 0.0961139i $$0.969359\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ 0 0
$$436$$ 8.00000 0.383131
$$437$$ − 12.0000i − 0.574038i
$$438$$ 28.0000i 1.33789i
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 12.0000i 0.570782i
$$443$$ − 12.0000i − 0.570137i −0.958507 0.285069i $$-0.907984\pi$$
0.958507 0.285069i $$-0.0920164\pi$$
$$444$$ 16.0000 0.759326
$$445$$ 0 0
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ − 1.00000i − 0.0472456i
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ − 6.00000i − 0.282216i
$$453$$ − 20.0000i − 0.939682i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 26.0000i 1.21623i 0.793849 + 0.608114i $$0.208074\pi$$
−0.793849 + 0.608114i $$0.791926\pi$$
$$458$$ − 26.0000i − 1.21490i
$$459$$ −24.0000 −1.12022
$$460$$ 0 0
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 2.00000i 0.0930484i
$$463$$ − 2.00000i − 0.0929479i −0.998920 0.0464739i $$-0.985202\pi$$
0.998920 0.0464739i $$-0.0147984\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ −6.00000 −0.277945
$$467$$ 18.0000i 0.832941i 0.909149 + 0.416470i $$0.136733\pi$$
−0.909149 + 0.416470i $$0.863267\pi$$
$$468$$ − 2.00000i − 0.0924500i
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ −4.00000 −0.184310
$$472$$ 12.0000i 0.552345i
$$473$$ − 4.00000i − 0.183920i
$$474$$ −20.0000 −0.918630
$$475$$ 0 0
$$476$$ 6.00000 0.275010
$$477$$ 12.0000i 0.549442i
$$478$$ − 18.0000i − 0.823301i
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 16.0000 0.729537
$$482$$ 8.00000i 0.364390i
$$483$$ − 12.0000i − 0.546019i
$$484$$ −1.00000 −0.0454545
$$485$$ 0 0
$$486$$ 10.0000 0.453609
$$487$$ − 22.0000i − 0.996915i −0.866914 0.498458i $$-0.833900\pi$$
0.866914 0.498458i $$-0.166100\pi$$
$$488$$ 10.0000i 0.452679i
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ − 12.0000i − 0.538274i
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 12.0000i − 0.535586i
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ 18.0000i 0.799408i
$$508$$ 16.0000i 0.709885i
$$509$$ 42.0000 1.86162 0.930809 0.365507i $$-0.119104\pi$$
0.930809 + 0.365507i $$0.119104\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 1.00000i 0.0441942i
$$513$$ − 8.00000i − 0.353209i
$$514$$ 24.0000 1.05859
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ − 8.00000i − 0.351500i
$$519$$ 12.0000 0.526742
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 16.0000i 0.699631i 0.936819 + 0.349816i $$0.113756\pi$$
−0.936819 + 0.349816i $$0.886244\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 48.0000i 2.09091i
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 2.00000i 0.0867110i
$$533$$ 0 0
$$534$$ −36.0000 −1.55787
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 24.0000i 1.03568i
$$538$$ − 18.0000i − 0.776035i
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ 20.0000 0.859867 0.429934 0.902861i $$-0.358537\pi$$
0.429934 + 0.902861i $$0.358537\pi$$
$$542$$ 20.0000i 0.859074i
$$543$$ − 20.0000i − 0.858282i
$$544$$ −6.00000 −0.257248
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ 44.0000i 1.88130i 0.339372 + 0.940652i $$0.389785\pi$$
−0.339372 + 0.940652i $$0.610215\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 12.0000i 0.510754i
$$553$$ 10.0000i 0.425243i
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ − 6.00000i − 0.254228i −0.991888 0.127114i $$-0.959429\pi$$
0.991888 0.127114i $$-0.0405714\pi$$
$$558$$ − 8.00000i − 0.338667i
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 6.00000i 0.253095i
$$563$$ 36.0000i 1.51722i 0.651546 + 0.758610i $$0.274121\pi$$
−0.651546 + 0.758610i $$0.725879\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ − 11.0000i − 0.461957i
$$568$$ 12.0000i 0.503509i
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ − 2.00000i − 0.0836242i
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 16.0000i − 0.666089i −0.942911 0.333044i $$-0.891924\pi$$
0.942911 0.333044i $$-0.108076\pi$$
$$578$$ − 19.0000i − 0.790296i
$$579$$ 28.0000 1.16364
$$580$$ 0 0
$$581$$ 0 0
$$582$$ − 16.0000i − 0.663221i
$$583$$ 12.0000i 0.496989i
$$584$$ −14.0000 −0.579324
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ −16.0000 −0.659269
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ 8.00000i 0.328798i
$$593$$ − 30.0000i − 1.23195i −0.787765 0.615976i $$-0.788762\pi$$
0.787765 0.615976i $$-0.211238\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 16.0000i − 0.654836i
$$598$$ 12.0000i 0.490716i
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −28.0000 −1.14214 −0.571072 0.820900i $$-0.693472\pi$$
−0.571072 + 0.820900i $$0.693472\pi$$
$$602$$ − 4.00000i − 0.163028i
$$603$$ 4.00000i 0.162893i
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 36.0000 1.46240
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ − 2.00000i − 0.0811107i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 6.00000i 0.242536i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −32.0000 −1.29141
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ 30.0000i 1.20775i 0.797077 + 0.603877i $$0.206378\pi$$
−0.797077 + 0.603877i $$0.793622\pi$$
$$618$$ − 8.00000i − 0.321807i
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ −24.0000 −0.963087
$$622$$ 24.0000i 0.962312i
$$623$$ 18.0000i 0.721155i
$$624$$ 4.00000 0.160128
$$625$$ 0 0
$$626$$ −16.0000 −0.639489
$$627$$ 4.00000i 0.159745i
$$628$$ − 2.00000i − 0.0798087i
$$629$$ −48.0000 −1.91389
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ − 10.0000i − 0.397779i
$$633$$ 40.0000i 1.58986i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ −24.0000 −0.951662
$$637$$ 2.00000i 0.0792429i
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ − 24.0000i − 0.947204i
$$643$$ − 14.0000i − 0.552106i −0.961142 0.276053i $$-0.910973\pi$$
0.961142 0.276053i $$-0.0890266\pi$$
$$644$$ 6.00000 0.236433
$$645$$ 0 0
$$646$$ 12.0000 0.472134
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ 12.0000 0.471041
$$650$$ 0 0
$$651$$ −16.0000 −0.627089
$$652$$ − 4.00000i − 0.156652i
$$653$$ 36.0000i 1.40879i 0.709809 + 0.704394i $$0.248781\pi$$
−0.709809 + 0.704394i $$0.751219\pi$$
$$654$$ 16.0000 0.625650
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 14.0000i 0.546192i
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 20.0000i 0.777322i
$$663$$ 24.0000i 0.932083i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 0 0
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ − 2.00000i − 0.0771517i
$$673$$ − 2.00000i − 0.0770943i −0.999257 0.0385472i $$-0.987727\pi$$
0.999257 0.0385472i $$-0.0122730\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ 6.00000i 0.230599i 0.993331 + 0.115299i $$0.0367827\pi$$
−0.993331 + 0.115299i $$0.963217\pi$$
$$678$$ − 12.0000i − 0.460857i
$$679$$ −8.00000 −0.307012
$$680$$ 0 0
$$681$$ 24.0000 0.919682
$$682$$ − 8.00000i − 0.306336i
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ − 52.0000i − 1.98392i
$$688$$ 4.00000i 0.152499i
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 6.00000i 0.228086i
$$693$$ 1.00000i 0.0379869i
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ − 2.00000i − 0.0757011i
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ −36.0000 −1.35970 −0.679851 0.733351i $$-0.737955\pi$$
−0.679851 + 0.733351i $$0.737955\pi$$
$$702$$ 8.00000i 0.301941i
$$703$$ − 16.0000i − 0.603451i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −24.0000 −0.903252
$$707$$ − 18.0000i − 0.676960i
$$708$$ 24.0000i 0.901975i
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ − 18.0000i − 0.674579i
$$713$$ 48.0000i 1.79761i
$$714$$ 12.0000 0.449089
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ − 36.0000i − 1.34444i
$$718$$ 18.0000i 0.671754i
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −4.00000 −0.148968
$$722$$ − 15.0000i − 0.558242i
$$723$$ 16.0000i 0.595046i
$$724$$ 10.0000 0.371647
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ − 16.0000i − 0.593407i −0.954970 0.296704i $$-0.904113\pi$$
0.954970 0.296704i $$-0.0958873\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −24.0000 −0.887672
$$732$$ 20.0000i 0.739221i
$$733$$ − 14.0000i − 0.517102i −0.965998 0.258551i $$-0.916755\pi$$
0.965998 0.258551i $$-0.0832450\pi$$
$$734$$ −20.0000 −0.738213
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 4.00000i 0.147342i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 12.0000i 0.440534i
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 16.0000 0.586588
$$745$$ 0 0
$$746$$ −22.0000 −0.805477
$$747$$ 0 0
$$748$$ 6.00000i 0.219382i
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 44.0000 1.60558 0.802791 0.596260i $$-0.203347\pi$$
0.802791 + 0.596260i $$0.203347\pi$$
$$752$$ 0 0
$$753$$ − 24.0000i − 0.874609i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 4.00000 0.145479
$$757$$ − 52.0000i − 1.88997i −0.327111 0.944986i $$-0.606075\pi$$
0.327111 0.944986i $$-0.393925\pi$$
$$758$$ 28.0000i 1.01701i
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ 32.0000i 1.15924i
$$763$$ − 8.00000i − 0.289619i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −12.0000 −0.433578
$$767$$ 24.0000i 0.866590i
$$768$$ 2.00000i 0.0721688i
$$769$$ −20.0000 −0.721218 −0.360609 0.932717i $$-0.617431\pi$$
−0.360609 + 0.932717i $$0.617431\pi$$
$$770$$ 0 0
$$771$$ 48.0000 1.72868
$$772$$ 14.0000i 0.503871i
$$773$$ 42.0000i 1.51064i 0.655359 + 0.755318i $$0.272517\pi$$
−0.655359 + 0.755318i $$0.727483\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ 8.00000 0.287183
$$777$$ − 16.0000i − 0.573997i
$$778$$ 6.00000i 0.215110i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ − 36.0000i − 1.28736i
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 36.0000 1.28408
$$787$$ − 4.00000i − 0.142585i −0.997455 0.0712923i $$-0.977288\pi$$
0.997455 0.0712923i $$-0.0227123\pi$$
$$788$$ − 18.0000i − 0.641223i
$$789$$ −48.0000 −1.70885
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ − 1.00000i − 0.0355335i
$$793$$ 20.0000i 0.710221i
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 42.0000i 1.48772i 0.668338 + 0.743858i $$0.267006\pi$$
−0.668338 + 0.743858i $$0.732994\pi$$
$$798$$ 4.00000i 0.141598i
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ − 18.0000i − 0.635602i
$$803$$ 14.0000i 0.494049i
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 16.0000 0.563576
$$807$$ − 36.0000i − 1.26726i
$$808$$ 18.0000i 0.633238i
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 14.0000 0.491606 0.245803 0.969320i $$-0.420948\pi$$
0.245803 + 0.969320i $$0.420948\pi$$
$$812$$ 0 0
$$813$$ 40.0000i 1.40286i
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ − 8.00000i − 0.279885i
$$818$$ − 8.00000i − 0.279713i
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ 36.0000 1.25641 0.628204 0.778048i $$-0.283790\pi$$
0.628204 + 0.778048i $$0.283790\pi$$
$$822$$ 12.0000i 0.418548i
$$823$$ − 2.00000i − 0.0697156i −0.999392 0.0348578i $$-0.988902\pi$$
0.999392 0.0348578i $$-0.0110978\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 36.0000i 1.25184i 0.779886 + 0.625921i $$0.215277\pi$$
−0.779886 + 0.625921i $$0.784723\pi$$
$$828$$ 6.00000i 0.208514i
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 20.0000 0.693792
$$832$$ 2.00000i 0.0693375i
$$833$$ − 6.00000i − 0.207888i
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ −2.00000 −0.0691714
$$837$$ 32.0000i 1.10608i
$$838$$ 12.0000i 0.414533i
$$839$$ 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
$$842$$ 14.0000i 0.482472i
$$843$$ 12.0000i 0.413302i
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 1.00000i 0.0343604i
$$848$$ − 12.0000i − 0.412082i
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ −48.0000 −1.64542
$$852$$ 24.0000i 0.822226i
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 30.0000i 1.02478i 0.858753 + 0.512390i $$0.171240\pi$$
−0.858753 + 0.512390i $$0.828760\pi$$
$$858$$ − 4.00000i − 0.136558i
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 18.0000i − 0.613082i
$$863$$ 30.0000i 1.02121i 0.859815 + 0.510606i $$0.170579\pi$$
−0.859815 + 0.510606i $$0.829421\pi$$
$$864$$ −4.00000 −0.136083
$$865$$ 0 0
$$866$$ −4.00000 −0.135926
$$867$$ − 38.0000i − 1.29055i
$$868$$ − 8.00000i − 0.271538i
$$869$$ −10.0000 −0.339227
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 8.00000i 0.270914i
$$873$$ − 8.00000i − 0.270759i
$$874$$ 12.0000 0.405906
$$875$$ 0 0
$$876$$ −28.0000 −0.946032
$$877$$ − 22.0000i − 0.742887i −0.928456 0.371444i $$-0.878863\pi$$
0.928456 0.371444i $$-0.121137\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ 36.0000 1.21425
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ 16.0000i 0.538443i 0.963078 + 0.269221i $$0.0867663\pi$$
−0.963078 + 0.269221i $$0.913234\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 16.0000i 0.536925i
$$889$$ 16.0000 0.536623
$$890$$ 0 0
$$891$$ 11.0000 0.368514
$$892$$ − 4.00000i − 0.133930i
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 24.0000i 0.801337i
$$898$$ 18.0000i 0.600668i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 72.0000 2.39867
$$902$$ 0 0
$$903$$ − 8.00000i − 0.266223i
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 20.0000 0.664455
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ 0 0
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ 26.0000 0.859064
$$917$$ − 18.0000i − 0.594412i
$$918$$ − 24.0000i − 0.792118i
$$919$$ 34.0000 1.12156 0.560778 0.827966i $$-0.310502\pi$$
0.560778 + 0.827966i $$0.310502\pi$$
$$920$$ 0 0
$$921$$ −64.0000 −2.10887
$$922$$ − 30.0000i − 0.987997i
$$923$$ 24.0000i 0.789970i
$$924$$ −2.00000 −0.0657952
$$925$$ 0 0
$$926$$ 2.00000 0.0657241
$$927$$ − 4.00000i − 0.131377i
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ − 6.00000i − 0.196537i
$$933$$ 48.0000i 1.57145i
$$934$$ −18.0000 −0.588978
$$935$$ 0 0
$$936$$ 2.00000 0.0653720
$$937$$ 2.00000i 0.0653372i 0.999466 + 0.0326686i $$0.0104006\pi$$
−0.999466 + 0.0326686i $$0.989599\pi$$
$$938$$ 4.00000i 0.130605i
$$939$$ −32.0000 −1.04428
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ − 4.00000i − 0.130327i
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ − 48.0000i − 1.55979i −0.625910 0.779895i $$-0.715272\pi$$
0.625910 0.779895i $$-0.284728\pi$$
$$948$$ − 20.0000i − 0.649570i
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 6.00000i 0.194461i
$$953$$ − 6.00000i − 0.194359i −0.995267 0.0971795i $$-0.969018\pi$$
0.995267 0.0971795i $$-0.0309821\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 0 0
$$956$$ 18.0000 0.582162
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 16.0000i 0.515861i
$$963$$ − 12.0000i − 0.386695i
$$964$$ −8.00000 −0.257663
$$965$$ 0 0
$$966$$ 12.0000 0.386094
$$967$$ − 16.0000i − 0.514525i −0.966342 0.257263i $$-0.917179\pi$$
0.966342 0.257263i $$-0.0828206\pi$$
$$968$$ − 1.00000i − 0.0321412i
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ 60.0000 1.92549 0.962746 0.270408i $$-0.0871586\pi$$
0.962746 + 0.270408i $$0.0871586\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ 10.0000i 0.320585i
$$974$$ 22.0000 0.704925
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ − 8.00000i − 0.255812i
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ 8.00000 0.255420
$$982$$ 12.0000i 0.382935i
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ − 4.00000i − 0.127257i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 8.00000i 0.254000i
$$993$$ 40.0000i 1.26936i
$$994$$ 12.0000 0.380617
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 10.0000i − 0.316703i −0.987383 0.158352i $$-0.949382\pi$$
0.987383 0.158352i $$-0.0506179\pi$$
$$998$$ − 20.0000i − 0.633089i
$$999$$ −32.0000 −1.01244
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 3850.2.c.a.1849.2 2
5.2 odd 4 3850.2.a.j.1.1 1
5.3 odd 4 770.2.a.g.1.1 1
5.4 even 2 inner 3850.2.c.a.1849.1 2
15.8 even 4 6930.2.a.f.1.1 1
20.3 even 4 6160.2.a.n.1.1 1
35.13 even 4 5390.2.a.bh.1.1 1
55.43 even 4 8470.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.g.1.1 1 5.3 odd 4
3850.2.a.j.1.1 1 5.2 odd 4
3850.2.c.a.1849.1 2 5.4 even 2 inner
3850.2.c.a.1849.2 2 1.1 even 1 trivial
5390.2.a.bh.1.1 1 35.13 even 4
6160.2.a.n.1.1 1 20.3 even 4
6930.2.a.f.1.1 1 15.8 even 4
8470.2.a.e.1.1 1 55.43 even 4