# Properties

 Label 3850.2.c.c.1849.2 Level $3850$ Weight $2$ Character 3850.1849 Analytic conductor $30.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## 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: $$1$$ 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.c.1849.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} -1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{12} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} -6.00000 q^{19} +2.00000 q^{21} -1.00000i q^{22} +6.00000i q^{23} +2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} +1.00000i q^{28} -4.00000 q^{29} +1.00000i q^{32} -2.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} -6.00000i q^{38} -4.00000 q^{39} +2.00000i q^{42} +4.00000i q^{43} +1.00000 q^{44} -6.00000 q^{46} +4.00000i q^{47} +2.00000i q^{48} -1.00000 q^{49} +4.00000 q^{51} -2.00000i q^{52} -12.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} -12.0000i q^{57} -4.00000i q^{58} +2.00000 q^{61} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} +2.00000i q^{68} -12.0000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +8.00000 q^{74} +6.00000 q^{76} +1.00000i q^{77} -4.00000i q^{78} -10.0000 q^{79} -11.0000 q^{81} -12.0000i q^{83} -2.00000 q^{84} -4.00000 q^{86} -8.00000i q^{87} +1.00000i q^{88} -14.0000 q^{89} +2.00000 q^{91} -6.00000i q^{92} -4.00000 q^{94} -2.00000 q^{96} -4.00000i q^{97} -1.00000i q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 4 q^{21} + 4 q^{24} - 4 q^{26} - 8 q^{29} + 4 q^{34} + 2 q^{36} - 8 q^{39} + 2 q^{44} - 12 q^{46} - 2 q^{49} + 8 q^{51} - 8 q^{54} - 2 q^{56} + 4 q^{61} - 2 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} + 16 q^{74} + 12 q^{76} - 20 q^{79} - 22 q^{81} - 4 q^{84} - 8 q^{86} - 28 q^{89} + 4 q^{91} - 8 q^{94} - 4 q^{96} + 2 q^{99} + O(q^{100})$$

## 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 $$0.0894562\pi$$
−0.960769 + 0.277350i $$0.910544\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ − 2.00000i − 0.485071i −0.970143 0.242536i $$-0.922021\pi$$
0.970143 0.242536i $$-0.0779791\pi$$
$$18$$ − 1.00000i − 0.235702i
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 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
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 2.00000i − 0.348155i
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ − 6.00000i − 0.973329i
$$39$$ −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
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 2.00000i 0.288675i
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ − 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
$$57$$ − 12.0000i − 1.58944i
$$58$$ − 4.00000i − 0.525226i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 1.00000i 0.125988i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 2.00000 0.246183
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 2.00000i 0.242536i
$$69$$ −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
$$73$$ − 6.00000i − 0.702247i −0.936329 0.351123i $$-0.885800\pi$$
0.936329 0.351123i $$-0.114200\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 6.00000 0.688247
$$77$$ 1.00000i 0.113961i
$$78$$ − 4.00000i − 0.452911i
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ − 12.0000i − 1.31717i −0.752506 0.658586i $$-0.771155\pi$$
0.752506 0.658586i $$-0.228845\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ − 8.00000i − 0.857690i
$$88$$ 1.00000i 0.106600i
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ − 6.00000i − 0.625543i
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 0 0
$$96$$ −2.00000 −0.204124
$$97$$ − 4.00000i − 0.406138i −0.979164 0.203069i $$-0.934908\pi$$
0.979164 0.203069i $$-0.0650917\pi$$
$$98$$ − 1.00000i − 0.101015i
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ − 8.00000i − 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 2.00000 0.196116
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ − 20.0000i − 1.93347i −0.255774 0.966736i $$-0.582330\pi$$
0.255774 0.966736i $$-0.417670\pi$$
$$108$$ − 4.00000i − 0.384900i
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 16.0000 1.51865
$$112$$ − 1.00000i − 0.0944911i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ 12.0000 1.12390
$$115$$ 0 0
$$116$$ 4.00000 0.371391
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 2.00000i 0.181071i
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ − 8.00000i − 0.709885i −0.934888 0.354943i $$-0.884500\pi$$
0.934888 0.354943i $$-0.115500\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ −8.00000 −0.704361
$$130$$ 0 0
$$131$$ −6.00000 −0.524222 −0.262111 0.965038i $$-0.584419\pi$$
−0.262111 + 0.965038i $$0.584419\pi$$
$$132$$ 2.00000i 0.174078i
$$133$$ 6.00000i 0.520266i
$$134$$ −8.00000 −0.691095
$$135$$ 0 0
$$136$$ −2.00000 −0.171499
$$137$$ 22.0000i 1.87959i 0.341743 + 0.939793i $$0.388983\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ − 12.0000i − 1.02151i
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ − 12.0000i − 1.00702i
$$143$$ − 2.00000i − 0.167248i
$$144$$ −1.00000 −0.0833333
$$145$$ 0 0
$$146$$ 6.00000 0.496564
$$147$$ − 2.00000i − 0.164957i
$$148$$ 8.00000i 0.657596i
$$149$$ −4.00000 −0.327693 −0.163846 0.986486i $$-0.552390\pi$$
−0.163846 + 0.986486i $$0.552390\pi$$
$$150$$ 0 0
$$151$$ −6.00000 −0.488273 −0.244137 0.969741i $$-0.578505\pi$$
−0.244137 + 0.969741i $$0.578505\pi$$
$$152$$ 6.00000i 0.486664i
$$153$$ 2.00000i 0.161690i
$$154$$ −1.00000 −0.0805823
$$155$$ 0 0
$$156$$ 4.00000 0.320256
$$157$$ 10.0000i 0.798087i 0.916932 + 0.399043i $$0.130658\pi$$
−0.916932 + 0.399043i $$0.869342\pi$$
$$158$$ − 10.0000i − 0.795557i
$$159$$ 24.0000 1.90332
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ − 11.0000i − 0.864242i
$$163$$ − 24.0000i − 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ − 2.00000i − 0.154303i
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 6.00000 0.458831
$$172$$ − 4.00000i − 0.304997i
$$173$$ − 18.0000i − 1.36851i −0.729241 0.684257i $$-0.760127\pi$$
0.729241 0.684257i $$-0.239873\pi$$
$$174$$ 8.00000 0.606478
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ − 14.0000i − 1.04934i
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 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
$$183$$ 4.00000i 0.295689i
$$184$$ 6.00000 0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.00000i 0.146254i
$$188$$ − 4.00000i − 0.291730i
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ − 2.00000i − 0.144338i
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ 4.00000 0.287183
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 1.00000i 0.0710669i
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −16.0000 −1.12855
$$202$$ 10.0000i 0.703598i
$$203$$ 4.00000i 0.280745i
$$204$$ −4.00000 −0.280056
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ − 6.00000i − 0.417029i
$$208$$ 2.00000i 0.138675i
$$209$$ 6.00000 0.415029
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 12.0000i 0.824163i
$$213$$ − 24.0000i − 1.64445i
$$214$$ 20.0000 1.36717
$$215$$ 0 0
$$216$$ 4.00000 0.272166
$$217$$ 0 0
$$218$$ − 4.00000i − 0.270914i
$$219$$ 12.0000 0.810885
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 16.0000i 1.07385i
$$223$$ 16.0000i 1.07144i 0.844396 + 0.535720i $$0.179960\pi$$
−0.844396 + 0.535720i $$0.820040\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 24.0000i 1.59294i 0.604681 + 0.796468i $$0.293301\pi$$
−0.604681 + 0.796468i $$0.706699\pi$$
$$228$$ 12.0000i 0.794719i
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ −2.00000 −0.131590
$$232$$ 4.00000i 0.262613i
$$233$$ − 26.0000i − 1.70332i −0.524097 0.851658i $$-0.675597\pi$$
0.524097 0.851658i $$-0.324403\pi$$
$$234$$ 2.00000 0.130744
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 20.0000i − 1.29914i
$$238$$ − 2.00000i − 0.129641i
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 1.00000i 0.0642824i
$$243$$ − 10.0000i − 0.641500i
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 12.0000i − 0.763542i
$$248$$ 0 0
$$249$$ 24.0000 1.52094
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ − 1.00000i − 0.0629941i
$$253$$ − 6.00000i − 0.377217i
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 28.0000i 1.74659i 0.487190 + 0.873296i $$0.338022\pi$$
−0.487190 + 0.873296i $$0.661978\pi$$
$$258$$ − 8.00000i − 0.498058i
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 4.00000 0.247594
$$262$$ − 6.00000i − 0.370681i
$$263$$ 8.00000i 0.493301i 0.969104 + 0.246651i $$0.0793300\pi$$
−0.969104 + 0.246651i $$0.920670\pi$$
$$264$$ −2.00000 −0.123091
$$265$$ 0 0
$$266$$ −6.00000 −0.367884
$$267$$ − 28.0000i − 1.71357i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ 28.0000 1.70088 0.850439 0.526073i $$-0.176336\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ − 2.00000i − 0.121268i
$$273$$ 4.00000i 0.242091i
$$274$$ −22.0000 −1.32907
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ 22.0000i 1.32185i 0.750451 + 0.660926i $$0.229836\pi$$
−0.750451 + 0.660926i $$0.770164\pi$$
$$278$$ 6.00000i 0.359856i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ − 8.00000i − 0.476393i
$$283$$ 32.0000i 1.90220i 0.308879 + 0.951101i $$0.400046\pi$$
−0.308879 + 0.951101i $$0.599954\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 0 0
$$288$$ − 1.00000i − 0.0589256i
$$289$$ 13.0000 0.764706
$$290$$ 0 0
$$291$$ 8.00000 0.468968
$$292$$ 6.00000i 0.351123i
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 2.00000 0.116642
$$295$$ 0 0
$$296$$ −8.00000 −0.464991
$$297$$ − 4.00000i − 0.232104i
$$298$$ − 4.00000i − 0.231714i
$$299$$ −12.0000 −0.693978
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ − 6.00000i − 0.345261i
$$303$$ 20.0000i 1.14897i
$$304$$ −6.00000 −0.344124
$$305$$ 0 0
$$306$$ −2.00000 −0.114332
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ − 1.00000i − 0.0569803i
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 4.00000i 0.226455i
$$313$$ − 12.0000i − 0.678280i −0.940736 0.339140i $$-0.889864\pi$$
0.940736 0.339140i $$-0.110136\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 24.0000i 1.34585i
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ 40.0000 2.23258
$$322$$ 6.00000i 0.334367i
$$323$$ 12.0000i 0.667698i
$$324$$ 11.0000 0.611111
$$325$$ 0 0
$$326$$ 24.0000 1.32924
$$327$$ − 8.00000i − 0.442401i
$$328$$ 0 0
$$329$$ 4.00000 0.220527
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 12.0000i 0.658586i
$$333$$ 8.00000i 0.438397i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 2.00000 0.109109
$$337$$ 2.00000i 0.108947i 0.998515 + 0.0544735i $$0.0173480\pi$$
−0.998515 + 0.0544735i $$0.982652\pi$$
$$338$$ 9.00000i 0.489535i
$$339$$ 12.0000 0.651751
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 6.00000i 0.324443i
$$343$$ 1.00000i 0.0539949i
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ − 20.0000i − 1.07366i −0.843692 0.536828i $$-0.819622\pi$$
0.843692 0.536828i $$-0.180378\pi$$
$$348$$ 8.00000i 0.428845i
$$349$$ −6.00000 −0.321173 −0.160586 0.987022i $$-0.551338\pi$$
−0.160586 + 0.987022i $$0.551338\pi$$
$$350$$ 0 0
$$351$$ −8.00000 −0.427008
$$352$$ − 1.00000i − 0.0533002i
$$353$$ 20.0000i 1.06449i 0.846590 + 0.532246i $$0.178652\pi$$
−0.846590 + 0.532246i $$0.821348\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 14.0000 0.741999
$$357$$ − 4.00000i − 0.211702i
$$358$$ 4.00000i 0.211407i
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 2.00000i 0.105118i
$$363$$ 2.00000i 0.104973i
$$364$$ −2.00000 −0.104828
$$365$$ 0 0
$$366$$ −4.00000 −0.209083
$$367$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$368$$ 6.00000i 0.312772i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ − 26.0000i − 1.34623i −0.739538 0.673114i $$-0.764956\pi$$
0.739538 0.673114i $$-0.235044\pi$$
$$374$$ −2.00000 −0.103418
$$375$$ 0 0
$$376$$ 4.00000 0.206284
$$377$$ − 8.00000i − 0.412021i
$$378$$ 4.00000i 0.205738i
$$379$$ −4.00000 −0.205466 −0.102733 0.994709i $$-0.532759\pi$$
−0.102733 + 0.994709i $$0.532759\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 8.00000i 0.409316i
$$383$$ 16.0000i 0.817562i 0.912633 + 0.408781i $$0.134046\pi$$
−0.912633 + 0.408781i $$0.865954\pi$$
$$384$$ 2.00000 0.102062
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ − 4.00000i − 0.203331i
$$388$$ 4.00000i 0.203069i
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 1.00000i 0.0505076i
$$393$$ − 12.0000i − 0.605320i
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ −1.00000 −0.0502519
$$397$$ − 14.0000i − 0.702640i −0.936255 0.351320i $$-0.885733\pi$$
0.936255 0.351320i $$-0.114267\pi$$
$$398$$ − 16.0000i − 0.802008i
$$399$$ −12.0000 −0.600751
$$400$$ 0 0
$$401$$ −34.0000 −1.69788 −0.848939 0.528490i $$-0.822758\pi$$
−0.848939 + 0.528490i $$0.822758\pi$$
$$402$$ − 16.0000i − 0.798007i
$$403$$ 0 0
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ −4.00000 −0.198517
$$407$$ 8.00000i 0.396545i
$$408$$ − 4.00000i − 0.198030i
$$409$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$410$$ 0 0
$$411$$ −44.0000 −2.17036
$$412$$ 8.00000i 0.394132i
$$413$$ 0 0
$$414$$ 6.00000 0.294884
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 12.0000i 0.587643i
$$418$$ 6.00000i 0.293470i
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ − 20.0000i − 0.973585i
$$423$$ − 4.00000i − 0.194487i
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 24.0000 1.16280
$$427$$ − 2.00000i − 0.0967868i
$$428$$ 20.0000i 0.966736i
$$429$$ 4.00000 0.193122
$$430$$ 0 0
$$431$$ −22.0000 −1.05970 −0.529851 0.848091i $$-0.677752\pi$$
−0.529851 + 0.848091i $$0.677752\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 4.00000 0.191565
$$437$$ − 36.0000i − 1.72211i
$$438$$ 12.0000i 0.573382i
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 4.00000i 0.190261i
$$443$$ 40.0000i 1.90046i 0.311553 + 0.950229i $$0.399151\pi$$
−0.311553 + 0.950229i $$0.600849\pi$$
$$444$$ −16.0000 −0.759326
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ − 8.00000i − 0.378387i
$$448$$ 1.00000i 0.0472456i
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 6.00000i 0.282216i
$$453$$ − 12.0000i − 0.563809i
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ −12.0000 −0.561951
$$457$$ − 22.0000i − 1.02912i −0.857455 0.514558i $$-0.827956\pi$$
0.857455 0.514558i $$-0.172044\pi$$
$$458$$ − 6.00000i − 0.280362i
$$459$$ 8.00000 0.373408
$$460$$ 0 0
$$461$$ −26.0000 −1.21094 −0.605470 0.795868i $$-0.707015\pi$$
−0.605470 + 0.795868i $$0.707015\pi$$
$$462$$ − 2.00000i − 0.0930484i
$$463$$ − 10.0000i − 0.464739i −0.972628 0.232370i $$-0.925352\pi$$
0.972628 0.232370i $$-0.0746479\pi$$
$$464$$ −4.00000 −0.185695
$$465$$ 0 0
$$466$$ 26.0000 1.20443
$$467$$ − 38.0000i − 1.75843i −0.476425 0.879215i $$-0.658068\pi$$
0.476425 0.879215i $$-0.341932\pi$$
$$468$$ 2.00000i 0.0924500i
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ −20.0000 −0.921551
$$472$$ 0 0
$$473$$ − 4.00000i − 0.183920i
$$474$$ 20.0000 0.918630
$$475$$ 0 0
$$476$$ 2.00000 0.0916698
$$477$$ 12.0000i 0.549442i
$$478$$ − 6.00000i − 0.274434i
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 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
$$487$$ 18.0000i 0.815658i 0.913058 + 0.407829i $$0.133714\pi$$
−0.913058 + 0.407829i $$0.866286\pi$$
$$488$$ − 2.00000i − 0.0905357i
$$489$$ 48.0000 2.17064
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 8.00000i 0.360302i
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000i 0.538274i
$$498$$ 24.0000i 1.07547i
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 16.0000i 0.713405i 0.934218 + 0.356702i $$0.116099\pi$$
−0.934218 + 0.356702i $$0.883901\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 0 0
$$506$$ 6.00000 0.266733
$$507$$ 18.0000i 0.799408i
$$508$$ 8.00000i 0.354943i
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ −6.00000 −0.265424
$$512$$ 1.00000i 0.0441942i
$$513$$ − 24.0000i − 1.05963i
$$514$$ −28.0000 −1.23503
$$515$$ 0 0
$$516$$ 8.00000 0.352180
$$517$$ − 4.00000i − 0.175920i
$$518$$ − 8.00000i − 0.351500i
$$519$$ 36.0000 1.58022
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 4.00000i 0.175075i
$$523$$ 4.00000i 0.174908i 0.996169 + 0.0874539i $$0.0278730\pi$$
−0.996169 + 0.0874539i $$0.972127\pi$$
$$524$$ 6.00000 0.262111
$$525$$ 0 0
$$526$$ −8.00000 −0.348817
$$527$$ 0 0
$$528$$ − 2.00000i − 0.0870388i
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 6.00000i − 0.260133i
$$533$$ 0 0
$$534$$ 28.0000 1.21168
$$535$$ 0 0
$$536$$ 8.00000 0.345547
$$537$$ 8.00000i 0.345225i
$$538$$ − 14.0000i − 0.603583i
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ −16.0000 −0.687894 −0.343947 0.938989i $$-0.611764\pi$$
−0.343947 + 0.938989i $$0.611764\pi$$
$$542$$ 28.0000i 1.20270i
$$543$$ 4.00000i 0.171656i
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ −4.00000 −0.171184
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ − 22.0000i − 0.939793i
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ 24.0000 1.02243
$$552$$ 12.0000i 0.510754i
$$553$$ 10.0000i 0.425243i
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −6.00000 −0.254457
$$557$$ 18.0000i 0.762684i 0.924434 + 0.381342i $$0.124538\pi$$
−0.924434 + 0.381342i $$0.875462\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ − 2.00000i − 0.0843649i
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 8.00000 0.336861
$$565$$ 0 0
$$566$$ −32.0000 −1.34506
$$567$$ 11.0000i 0.461957i
$$568$$ 12.0000i 0.503509i
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 2.00000i 0.0836242i
$$573$$ 16.0000i 0.668410i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ − 4.00000i − 0.166522i −0.996528 0.0832611i $$-0.973466\pi$$
0.996528 0.0832611i $$-0.0265335\pi$$
$$578$$ 13.0000i 0.540729i
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 8.00000i 0.331611i
$$583$$ 12.0000i 0.496989i
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ 2.00000i 0.0825488i 0.999148 + 0.0412744i $$0.0131418\pi$$
−0.999148 + 0.0412744i $$0.986858\pi$$
$$588$$ 2.00000i 0.0824786i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −4.00000 −0.164538
$$592$$ − 8.00000i − 0.328798i
$$593$$ 26.0000i 1.06769i 0.845582 + 0.533846i $$0.179254\pi$$
−0.845582 + 0.533846i $$0.820746\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ − 32.0000i − 1.30967i
$$598$$ − 12.0000i − 0.490716i
$$599$$ −44.0000 −1.79779 −0.898896 0.438163i $$-0.855629\pi$$
−0.898896 + 0.438163i $$0.855629\pi$$
$$600$$ 0 0
$$601$$ 44.0000 1.79480 0.897399 0.441221i $$-0.145454\pi$$
0.897399 + 0.441221i $$0.145454\pi$$
$$602$$ 4.00000i 0.163028i
$$603$$ − 8.00000i − 0.325785i
$$604$$ 6.00000 0.244137
$$605$$ 0 0
$$606$$ −20.0000 −0.812444
$$607$$ − 16.0000i − 0.649420i −0.945814 0.324710i $$-0.894733\pi$$
0.945814 0.324710i $$-0.105267\pi$$
$$608$$ − 6.00000i − 0.243332i
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ −8.00000 −0.323645
$$612$$ − 2.00000i − 0.0808452i
$$613$$ − 26.0000i − 1.05013i −0.851062 0.525065i $$-0.824041\pi$$
0.851062 0.525065i $$-0.175959\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 1.00000 0.0402911
$$617$$ 42.0000i 1.69086i 0.534089 + 0.845428i $$0.320655\pi$$
−0.534089 + 0.845428i $$0.679345\pi$$
$$618$$ 16.0000i 0.643614i
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 0 0
$$621$$ −24.0000 −0.963087
$$622$$ 16.0000i 0.641542i
$$623$$ 14.0000i 0.560898i
$$624$$ −4.00000 −0.160128
$$625$$ 0 0
$$626$$ 12.0000 0.479616
$$627$$ 12.0000i 0.479234i
$$628$$ − 10.0000i − 0.399043i
$$629$$ −16.0000 −0.637962
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 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
$$638$$ 4.00000i 0.158362i
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 40.0000i 1.57867i
$$643$$ 34.0000i 1.34083i 0.741987 + 0.670415i $$0.233884\pi$$
−0.741987 + 0.670415i $$0.766116\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 11.0000i 0.432121i
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 24.0000i 0.939913i
$$653$$ − 36.0000i − 1.40879i −0.709809 0.704394i $$-0.751219\pi$$
0.709809 0.704394i $$-0.248781\pi$$
$$654$$ 8.00000 0.312825
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.00000i 0.234082i
$$658$$ 4.00000i 0.155936i
$$659$$ −8.00000 −0.311636 −0.155818 0.987786i $$-0.549801\pi$$
−0.155818 + 0.987786i $$0.549801\pi$$
$$660$$ 0 0
$$661$$ 42.0000 1.63361 0.816805 0.576913i $$-0.195743\pi$$
0.816805 + 0.576913i $$0.195743\pi$$
$$662$$ 12.0000i 0.466393i
$$663$$ 8.00000i 0.310694i
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −8.00000 −0.309994
$$667$$ − 24.0000i − 0.929284i
$$668$$ 0 0
$$669$$ −32.0000 −1.23719
$$670$$ 0 0
$$671$$ −2.00000 −0.0772091
$$672$$ 2.00000i 0.0771517i
$$673$$ − 42.0000i − 1.61898i −0.587133 0.809491i $$-0.699743\pi$$
0.587133 0.809491i $$-0.300257\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ −4.00000 −0.153506
$$680$$ 0 0
$$681$$ −48.0000 −1.83936
$$682$$ 0 0
$$683$$ − 4.00000i − 0.153056i −0.997067 0.0765279i $$-0.975617\pi$$
0.997067 0.0765279i $$-0.0243834\pi$$
$$684$$ −6.00000 −0.229416
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ − 12.0000i − 0.457829i
$$688$$ 4.00000i 0.152499i
$$689$$ 24.0000 0.914327
$$690$$ 0 0
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ 18.0000i 0.684257i
$$693$$ − 1.00000i − 0.0379869i
$$694$$ 20.0000 0.759190
$$695$$ 0 0
$$696$$ −8.00000 −0.303239
$$697$$ 0 0
$$698$$ − 6.00000i − 0.227103i
$$699$$ 52.0000 1.96682
$$700$$ 0 0
$$701$$ −8.00000 −0.302156 −0.151078 0.988522i $$-0.548274\pi$$
−0.151078 + 0.988522i $$0.548274\pi$$
$$702$$ − 8.00000i − 0.301941i
$$703$$ 48.0000i 1.81035i
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ −20.0000 −0.752710
$$707$$ − 10.0000i − 0.376089i
$$708$$ 0 0
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 14.0000i 0.524672i
$$713$$ 0 0
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ −4.00000 −0.149487
$$717$$ − 12.0000i − 0.448148i
$$718$$ 6.00000i 0.223918i
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 17.0000i 0.632674i
$$723$$ − 16.0000i − 0.595046i
$$724$$ −2.00000 −0.0743294
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ 44.0000i 1.63187i 0.578144 + 0.815935i $$0.303777\pi$$
−0.578144 + 0.815935i $$0.696223\pi$$
$$728$$ − 2.00000i − 0.0741249i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ − 4.00000i − 0.147844i
$$733$$ − 2.00000i − 0.0738717i −0.999318 0.0369358i $$-0.988240\pi$$
0.999318 0.0369358i $$-0.0117597\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ − 8.00000i − 0.294684i
$$738$$ 0 0
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ − 12.0000i − 0.440534i
$$743$$ 48.0000i 1.76095i 0.474093 + 0.880475i $$0.342776\pi$$
−0.474093 + 0.880475i $$0.657224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 26.0000 0.951928
$$747$$ 12.0000i 0.439057i
$$748$$ − 2.00000i − 0.0731272i
$$749$$ −20.0000 −0.730784
$$750$$ 0 0
$$751$$ 4.00000 0.145962 0.0729810 0.997333i $$-0.476749\pi$$
0.0729810 + 0.997333i $$0.476749\pi$$
$$752$$ 4.00000i 0.145865i
$$753$$ 0 0
$$754$$ 8.00000 0.291343
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ − 52.0000i − 1.88997i −0.327111 0.944986i $$-0.606075\pi$$
0.327111 0.944986i $$-0.393925\pi$$
$$758$$ − 4.00000i − 0.145287i
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −24.0000 −0.869999 −0.435000 0.900431i $$-0.643252\pi$$
−0.435000 + 0.900431i $$0.643252\pi$$
$$762$$ 16.0000i 0.579619i
$$763$$ 4.00000i 0.144810i
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 0 0
$$768$$ 2.00000i 0.0721688i
$$769$$ 28.0000 1.00971 0.504853 0.863205i $$-0.331547\pi$$
0.504853 + 0.863205i $$0.331547\pi$$
$$770$$ 0 0
$$771$$ −56.0000 −2.01679
$$772$$ − 2.00000i − 0.0719816i
$$773$$ − 6.00000i − 0.215805i −0.994161 0.107903i $$-0.965587\pi$$
0.994161 0.107903i $$-0.0344134\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 0 0
$$776$$ −4.00000 −0.143592
$$777$$ − 16.0000i − 0.573997i
$$778$$ − 18.0000i − 0.645331i
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 12.0000 0.429394
$$782$$ 12.0000i 0.429119i
$$783$$ − 16.0000i − 0.571793i
$$784$$ −1.00000 −0.0357143
$$785$$ 0 0
$$786$$ 12.0000 0.428026
$$787$$ 48.0000i 1.71102i 0.517790 + 0.855508i $$0.326755\pi$$
−0.517790 + 0.855508i $$0.673245\pi$$
$$788$$ − 2.00000i − 0.0712470i
$$789$$ −16.0000 −0.569615
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ − 1.00000i − 0.0355335i
$$793$$ 4.00000i 0.142044i
$$794$$ 14.0000 0.496841
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ − 30.0000i − 1.06265i −0.847167 0.531327i $$-0.821693\pi$$
0.847167 0.531327i $$-0.178307\pi$$
$$798$$ − 12.0000i − 0.424795i
$$799$$ 8.00000 0.283020
$$800$$ 0 0
$$801$$ 14.0000 0.494666
$$802$$ − 34.0000i − 1.20058i
$$803$$ 6.00000i 0.211735i
$$804$$ 16.0000 0.564276
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 28.0000i − 0.985647i
$$808$$ − 10.0000i − 0.351799i
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ −30.0000 −1.05344 −0.526721 0.850038i $$-0.676579\pi$$
−0.526721 + 0.850038i $$0.676579\pi$$
$$812$$ − 4.00000i − 0.140372i
$$813$$ 56.0000i 1.96401i
$$814$$ −8.00000 −0.280400
$$815$$ 0 0
$$816$$ 4.00000 0.140028
$$817$$ − 24.0000i − 0.839654i
$$818$$ 0 0
$$819$$ −2.00000 −0.0698857
$$820$$ 0 0
$$821$$ −8.00000 −0.279202 −0.139601 0.990208i $$-0.544582\pi$$
−0.139601 + 0.990208i $$0.544582\pi$$
$$822$$ − 44.0000i − 1.53468i
$$823$$ 14.0000i 0.488009i 0.969774 + 0.244005i $$0.0784612\pi$$
−0.969774 + 0.244005i $$0.921539\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 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
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ −44.0000 −1.52634
$$832$$ − 2.00000i − 0.0693375i
$$833$$ 2.00000i 0.0692959i
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ −6.00000 −0.207514
$$837$$ 0 0
$$838$$ 16.0000i 0.552711i
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ − 26.0000i − 0.896019i
$$843$$ − 4.00000i − 0.137767i
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 4.00000 0.137523
$$847$$ − 1.00000i − 0.0343604i
$$848$$ − 12.0000i − 0.412082i
$$849$$ −64.0000 −2.19647
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 24.0000i 0.822226i
$$853$$ 22.0000i 0.753266i 0.926363 + 0.376633i $$0.122918\pi$$
−0.926363 + 0.376633i $$0.877082\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ −20.0000 −0.683586
$$857$$ 30.0000i 1.02478i 0.858753 + 0.512390i $$0.171240\pi$$
−0.858753 + 0.512390i $$0.828760\pi$$
$$858$$ 4.00000i 0.136558i
$$859$$ −48.0000 −1.63774 −0.818869 0.573980i $$-0.805399\pi$$
−0.818869 + 0.573980i $$0.805399\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 22.0000i − 0.749323i
$$863$$ 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
$$866$$ −16.0000 −0.543702
$$867$$ 26.0000i 0.883006i
$$868$$ 0 0
$$869$$ 10.0000 0.339227
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 4.00000i 0.135457i
$$873$$ 4.00000i 0.135379i
$$874$$ 36.0000 1.21772
$$875$$ 0 0
$$876$$ −12.0000 −0.405442
$$877$$ − 14.0000i − 0.472746i −0.971662 0.236373i $$-0.924041\pi$$
0.971662 0.236373i $$-0.0759588\pi$$
$$878$$ − 12.0000i − 0.404980i
$$879$$ 12.0000 0.404750
$$880$$ 0 0
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ 1.00000i 0.0336718i
$$883$$ 28.0000i 0.942275i 0.882060 + 0.471138i $$0.156156\pi$$
−0.882060 + 0.471138i $$0.843844\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ −40.0000 −1.34383
$$887$$ − 48.0000i − 1.61168i −0.592132 0.805841i $$-0.701714\pi$$
0.592132 0.805841i $$-0.298286\pi$$
$$888$$ − 16.0000i − 0.536925i
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 11.0000 0.368514
$$892$$ − 16.0000i − 0.535720i
$$893$$ − 24.0000i − 0.803129i
$$894$$ 8.00000 0.267560
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ − 24.0000i − 0.801337i
$$898$$ − 14.0000i − 0.467186i
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −24.0000 −0.799556
$$902$$ 0 0
$$903$$ 8.00000i 0.266223i
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 12.0000 0.398673
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ − 24.0000i − 0.796468i
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ − 12.0000i − 0.397360i
$$913$$ 12.0000i 0.397142i
$$914$$ 22.0000 0.727695
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 6.00000i 0.198137i
$$918$$ 8.00000i 0.264039i
$$919$$ −26.0000 −0.857661 −0.428830 0.903385i $$-0.641074\pi$$
−0.428830 + 0.903385i $$0.641074\pi$$
$$920$$ 0 0
$$921$$ −24.0000 −0.790827
$$922$$ − 26.0000i − 0.856264i
$$923$$ − 24.0000i − 0.789970i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ 10.0000 0.328620
$$927$$ 8.00000i 0.262754i
$$928$$ − 4.00000i − 0.131306i
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 6.00000 0.196642
$$932$$ 26.0000i 0.851658i
$$933$$ 32.0000i 1.04763i
$$934$$ 38.0000 1.24340
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ − 6.00000i − 0.196011i −0.995186 0.0980057i $$-0.968754\pi$$
0.995186 0.0980057i $$-0.0312463\pi$$
$$938$$ 8.00000i 0.261209i
$$939$$ 24.0000 0.783210
$$940$$ 0 0
$$941$$ 18.0000 0.586783 0.293392 0.955992i $$-0.405216\pi$$
0.293392 + 0.955992i $$0.405216\pi$$
$$942$$ − 20.0000i − 0.651635i
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 52.0000i 1.68977i 0.534946 + 0.844886i $$0.320332\pi$$
−0.534946 + 0.844886i $$0.679668\pi$$
$$948$$ 20.0000i 0.649570i
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 2.00000i 0.0648204i
$$953$$ 18.0000i 0.583077i 0.956559 + 0.291539i $$0.0941672\pi$$
−0.956559 + 0.291539i $$0.905833\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 0 0
$$956$$ 6.00000 0.194054
$$957$$ 8.00000i 0.258603i
$$958$$ − 8.00000i − 0.258468i
$$959$$ 22.0000 0.710417
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 16.0000i 0.515861i
$$963$$ 20.0000i 0.644491i
$$964$$ 8.00000 0.257663
$$965$$ 0 0
$$966$$ −12.0000 −0.386094
$$967$$ 56.0000i 1.80084i 0.435023 + 0.900419i $$0.356740\pi$$
−0.435023 + 0.900419i $$0.643260\pi$$
$$968$$ − 1.00000i − 0.0321412i
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 10.0000i 0.320750i
$$973$$ − 6.00000i − 0.192351i
$$974$$ −18.0000 −0.576757
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 48.0000i 1.53487i
$$979$$ 14.0000 0.447442
$$980$$ 0 0
$$981$$ 4.00000 0.127710
$$982$$ 12.0000i 0.382935i
$$983$$ − 56.0000i − 1.78612i −0.449935 0.893061i $$-0.648553\pi$$
0.449935 0.893061i $$-0.351447\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −8.00000 −0.254772
$$987$$ 8.00000i 0.254643i
$$988$$ 12.0000i 0.381771i
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 24.0000i 0.761617i
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ −24.0000 −0.760469
$$997$$ 42.0000i 1.33015i 0.746775 + 0.665077i $$0.231601\pi$$
−0.746775 + 0.665077i $$0.768399\pi$$
$$998$$ 28.0000i 0.886325i
$$999$$ 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.c.1849.2 2
5.2 odd 4 770.2.a.e.1.1 1
5.3 odd 4 3850.2.a.m.1.1 1
5.4 even 2 inner 3850.2.c.c.1849.1 2
15.2 even 4 6930.2.a.bk.1.1 1
20.7 even 4 6160.2.a.a.1.1 1
35.27 even 4 5390.2.a.c.1.1 1
55.32 even 4 8470.2.a.bg.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.e.1.1 1 5.2 odd 4
3850.2.a.m.1.1 1 5.3 odd 4
3850.2.c.c.1849.1 2 5.4 even 2 inner
3850.2.c.c.1849.2 2 1.1 even 1 trivial
5390.2.a.c.1.1 1 35.27 even 4
6160.2.a.a.1.1 1 20.7 even 4
6930.2.a.bk.1.1 1 15.2 even 4
8470.2.a.bg.1.1 1 55.32 even 4