# Properties

 Label 5070.2.b.f.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$40.4841538248$$ Analytic rank: $$1$$ 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 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} +2.00000i q^{21} -4.00000 q^{22} -2.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} +2.00000i q^{28} +8.00000 q^{29} -1.00000 q^{30} -4.00000i q^{31} +1.00000i q^{32} -4.00000i q^{33} -4.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} +6.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} -10.0000i q^{41} -2.00000 q^{42} -4.00000 q^{43} -4.00000i q^{44} -1.00000i q^{45} -2.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +4.00000 q^{51} +6.00000 q^{53} -1.00000i q^{54} +4.00000 q^{55} -2.00000 q^{56} -2.00000i q^{57} +8.00000i q^{58} -12.0000i q^{59} -1.00000i q^{60} -2.00000 q^{61} +4.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +8.00000i q^{67} +4.00000 q^{68} +2.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -6.00000 q^{74} +1.00000 q^{75} -2.00000i q^{76} +8.00000 q^{77} -8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} +12.0000i q^{83} -2.00000i q^{84} +4.00000i q^{85} -4.00000i q^{86} -8.00000 q^{87} +4.00000 q^{88} -10.0000i q^{89} +1.00000 q^{90} +2.00000 q^{92} +4.00000i q^{93} +2.00000 q^{95} -1.00000i q^{96} +8.00000i q^{97} +3.00000i q^{98} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} - 8q^{17} - 8q^{22} - 4q^{23} - 2q^{25} - 2q^{27} + 16q^{29} - 2q^{30} - 4q^{35} - 2q^{36} - 4q^{38} - 2q^{40} - 4q^{42} - 8q^{43} - 2q^{48} + 6q^{49} + 8q^{51} + 12q^{53} + 8q^{55} - 4q^{56} - 4q^{61} + 8q^{62} - 2q^{64} + 8q^{66} + 8q^{68} + 4q^{69} - 12q^{74} + 2q^{75} + 16q^{77} - 16q^{79} + 2q^{81} + 20q^{82} - 16q^{87} + 8q^{88} + 2q^{90} + 4q^{92} + 4q^{95} + O(q^{100})$$

## Character values

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

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

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.707107i
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ − 1.00000i − 0.447214i
$$6$$ − 1.00000i − 0.408248i
$$7$$ − 2.00000i − 0.755929i −0.925820 0.377964i $$-0.876624\pi$$
0.925820 0.377964i $$-0.123376\pi$$
$$8$$ − 1.00000i − 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 4.00000i 1.20605i 0.797724 + 0.603023i $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000i 0.258199i
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000i 0.235702i
$$19$$ 2.00000i 0.458831i 0.973329 + 0.229416i $$0.0736815\pi$$
−0.973329 + 0.229416i $$0.926318\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ 2.00000i 0.436436i
$$22$$ −4.00000 −0.852803
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 1.00000i 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000i 0.377964i
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ − 4.00000i − 0.718421i −0.933257 0.359211i $$-0.883046\pi$$
0.933257 0.359211i $$-0.116954\pi$$
$$32$$ 1.00000i 0.176777i
$$33$$ − 4.00000i − 0.696311i
$$34$$ − 4.00000i − 0.685994i
$$35$$ −2.00000 −0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 6.00000i 0.986394i 0.869918 + 0.493197i $$0.164172\pi$$
−0.869918 + 0.493197i $$0.835828\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 0 0
$$40$$ −1.00000 −0.158114
$$41$$ − 10.0000i − 1.56174i −0.624695 0.780869i $$-0.714777\pi$$
0.624695 0.780869i $$-0.285223\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ − 4.00000i − 0.603023i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 2.00000i − 0.294884i
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 3.00000 0.428571
$$50$$ − 1.00000i − 0.141421i
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ − 1.00000i − 0.136083i
$$55$$ 4.00000 0.539360
$$56$$ −2.00000 −0.267261
$$57$$ − 2.00000i − 0.264906i
$$58$$ 8.00000i 1.05045i
$$59$$ − 12.0000i − 1.56227i −0.624364 0.781133i $$-0.714642\pi$$
0.624364 0.781133i $$-0.285358\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 4.00000 0.508001
$$63$$ − 2.00000i − 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 2.00000 0.240772
$$70$$ − 2.00000i − 0.239046i
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ − 1.00000i − 0.117851i
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 1.00000 0.115470
$$76$$ − 2.00000i − 0.229416i
$$77$$ 8.00000 0.911685
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 12.0000i 1.31717i 0.752506 + 0.658586i $$0.228845\pi$$
−0.752506 + 0.658586i $$0.771155\pi$$
$$84$$ − 2.00000i − 0.218218i
$$85$$ 4.00000i 0.433861i
$$86$$ − 4.00000i − 0.431331i
$$87$$ −8.00000 −0.857690
$$88$$ 4.00000 0.426401
$$89$$ − 10.0000i − 1.06000i −0.847998 0.529999i $$-0.822192\pi$$
0.847998 0.529999i $$-0.177808\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 2.00000 0.208514
$$93$$ 4.00000i 0.414781i
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ − 1.00000i − 0.102062i
$$97$$ 8.00000i 0.812277i 0.913812 + 0.406138i $$0.133125\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 3.00000i 0.303046i
$$99$$ 4.00000i 0.402015i
$$100$$ 1.00000 0.100000
$$101$$ −20.0000 −1.99007 −0.995037 0.0995037i $$-0.968274\pi$$
−0.995037 + 0.0995037i $$0.968274\pi$$
$$102$$ 4.00000i 0.396059i
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 6.00000i 0.582772i
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ − 4.00000i − 0.383131i −0.981480 0.191565i $$-0.938644\pi$$
0.981480 0.191565i $$-0.0613564\pi$$
$$110$$ 4.00000i 0.381385i
$$111$$ − 6.00000i − 0.569495i
$$112$$ − 2.00000i − 0.188982i
$$113$$ −16.0000 −1.50515 −0.752577 0.658505i $$-0.771189\pi$$
−0.752577 + 0.658505i $$0.771189\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 2.00000i 0.186501i
$$116$$ −8.00000 −0.742781
$$117$$ 0 0
$$118$$ 12.0000 1.10469
$$119$$ 8.00000i 0.733359i
$$120$$ 1.00000 0.0912871
$$121$$ −5.00000 −0.454545
$$122$$ − 2.00000i − 0.181071i
$$123$$ 10.0000i 0.901670i
$$124$$ 4.00000i 0.359211i
$$125$$ 1.00000i 0.0894427i
$$126$$ 2.00000 0.178174
$$127$$ −20.0000 −1.77471 −0.887357 0.461084i $$-0.847461\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ − 1.00000i − 0.0883883i
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 4.00000i 0.348155i
$$133$$ 4.00000 0.346844
$$134$$ −8.00000 −0.691095
$$135$$ 1.00000i 0.0860663i
$$136$$ 4.00000i 0.342997i
$$137$$ − 2.00000i − 0.170872i −0.996344 0.0854358i $$-0.972772\pi$$
0.996344 0.0854358i $$-0.0272282\pi$$
$$138$$ 2.00000i 0.170251i
$$139$$ −16.0000 −1.35710 −0.678551 0.734553i $$-0.737392\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 8.00000i − 0.664364i
$$146$$ 0 0
$$147$$ −3.00000 −0.247436
$$148$$ − 6.00000i − 0.493197i
$$149$$ 22.0000i 1.80231i 0.433497 + 0.901155i $$0.357280\pi$$
−0.433497 + 0.901155i $$0.642720\pi$$
$$150$$ 1.00000i 0.0816497i
$$151$$ − 4.00000i − 0.325515i −0.986666 0.162758i $$-0.947961\pi$$
0.986666 0.162758i $$-0.0520389\pi$$
$$152$$ 2.00000 0.162221
$$153$$ −4.00000 −0.323381
$$154$$ 8.00000i 0.644658i
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ − 8.00000i − 0.636446i
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 4.00000i 0.315244i
$$162$$ 1.00000i 0.0785674i
$$163$$ 12.0000i 0.939913i 0.882690 + 0.469956i $$0.155730\pi$$
−0.882690 + 0.469956i $$0.844270\pi$$
$$164$$ 10.0000i 0.780869i
$$165$$ −4.00000 −0.311400
$$166$$ −12.0000 −0.931381
$$167$$ − 12.0000i − 0.928588i −0.885681 0.464294i $$-0.846308\pi$$
0.885681 0.464294i $$-0.153692\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ −4.00000 −0.306786
$$171$$ 2.00000i 0.152944i
$$172$$ 4.00000 0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ − 8.00000i − 0.606478i
$$175$$ 2.00000i 0.151186i
$$176$$ 4.00000i 0.301511i
$$177$$ 12.0000i 0.901975i
$$178$$ 10.0000 0.749532
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ 2.00000 0.147844
$$184$$ 2.00000i 0.147442i
$$185$$ 6.00000 0.441129
$$186$$ −4.00000 −0.293294
$$187$$ − 16.0000i − 1.17004i
$$188$$ 0 0
$$189$$ 2.00000i 0.145479i
$$190$$ 2.00000i 0.145095i
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 12.0000i 0.863779i 0.901927 + 0.431889i $$0.142153\pi$$
−0.901927 + 0.431889i $$0.857847\pi$$
$$194$$ −8.00000 −0.574367
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ − 10.0000i − 0.712470i −0.934396 0.356235i $$-0.884060\pi$$
0.934396 0.356235i $$-0.115940\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 1.00000i 0.0707107i
$$201$$ − 8.00000i − 0.564276i
$$202$$ − 20.0000i − 1.40720i
$$203$$ − 16.0000i − 1.12298i
$$204$$ −4.00000 −0.280056
$$205$$ −10.0000 −0.698430
$$206$$ − 4.00000i − 0.278693i
$$207$$ −2.00000 −0.139010
$$208$$ 0 0
$$209$$ −8.00000 −0.553372
$$210$$ 2.00000i 0.138013i
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 4.00000i 0.273434i
$$215$$ 4.00000i 0.272798i
$$216$$ 1.00000i 0.0680414i
$$217$$ −8.00000 −0.543075
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ 0 0
$$222$$ 6.00000 0.402694
$$223$$ 22.0000i 1.47323i 0.676313 + 0.736614i $$0.263577\pi$$
−0.676313 + 0.736614i $$0.736423\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ − 16.0000i − 1.06430i
$$227$$ − 12.0000i − 0.796468i −0.917284 0.398234i $$-0.869623\pi$$
0.917284 0.398234i $$-0.130377\pi$$
$$228$$ 2.00000i 0.132453i
$$229$$ 28.0000i 1.85029i 0.379611 + 0.925146i $$0.376058\pi$$
−0.379611 + 0.925146i $$0.623942\pi$$
$$230$$ −2.00000 −0.131876
$$231$$ −8.00000 −0.526361
$$232$$ − 8.00000i − 0.525226i
$$233$$ −28.0000 −1.83434 −0.917170 0.398495i $$-0.869533\pi$$
−0.917170 + 0.398495i $$0.869533\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000i 0.781133i
$$237$$ 8.00000 0.519656
$$238$$ −8.00000 −0.518563
$$239$$ 8.00000i 0.517477i 0.965947 + 0.258738i $$0.0833068\pi$$
−0.965947 + 0.258738i $$0.916693\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ − 22.0000i − 1.41714i −0.705638 0.708572i $$-0.749340\pi$$
0.705638 0.708572i $$-0.250660\pi$$
$$242$$ − 5.00000i − 0.321412i
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ − 3.00000i − 0.191663i
$$246$$ −10.0000 −0.637577
$$247$$ 0 0
$$248$$ −4.00000 −0.254000
$$249$$ − 12.0000i − 0.760469i
$$250$$ −1.00000 −0.0632456
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 2.00000i 0.125988i
$$253$$ − 8.00000i − 0.502956i
$$254$$ − 20.0000i − 1.25491i
$$255$$ − 4.00000i − 0.250490i
$$256$$ 1.00000 0.0625000
$$257$$ 8.00000 0.499026 0.249513 0.968371i $$-0.419729\pi$$
0.249513 + 0.968371i $$0.419729\pi$$
$$258$$ 4.00000i 0.249029i
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 8.00000 0.495188
$$262$$ 14.0000i 0.864923i
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ − 6.00000i − 0.368577i
$$266$$ 4.00000i 0.245256i
$$267$$ 10.0000i 0.611990i
$$268$$ − 8.00000i − 0.488678i
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ − 28.0000i − 1.70088i −0.526073 0.850439i $$-0.676336\pi$$
0.526073 0.850439i $$-0.323664\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ 2.00000 0.120824
$$275$$ − 4.00000i − 0.241209i
$$276$$ −2.00000 −0.120386
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ − 16.0000i − 0.959616i
$$279$$ − 4.00000i − 0.239474i
$$280$$ 2.00000i 0.119523i
$$281$$ − 10.0000i − 0.596550i −0.954480 0.298275i $$-0.903589\pi$$
0.954480 0.298275i $$-0.0964112\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ −2.00000 −0.118470
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 1.00000i 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ 8.00000 0.469776
$$291$$ − 8.00000i − 0.468968i
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ − 3.00000i − 0.174964i
$$295$$ −12.0000 −0.698667
$$296$$ 6.00000 0.348743
$$297$$ − 4.00000i − 0.232104i
$$298$$ −22.0000 −1.27443
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 8.00000i 0.461112i
$$302$$ 4.00000 0.230174
$$303$$ 20.0000 1.14897
$$304$$ 2.00000i 0.114708i
$$305$$ 2.00000i 0.114520i
$$306$$ − 4.00000i − 0.228665i
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ 4.00000 0.227552
$$310$$ − 4.00000i − 0.227185i
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ − 10.0000i − 0.564333i
$$315$$ −2.00000 −0.112687
$$316$$ 8.00000 0.450035
$$317$$ 2.00000i 0.112331i 0.998421 + 0.0561656i $$0.0178875\pi$$
−0.998421 + 0.0561656i $$0.982113\pi$$
$$318$$ − 6.00000i − 0.336463i
$$319$$ 32.0000i 1.79166i
$$320$$ 1.00000i 0.0559017i
$$321$$ −4.00000 −0.223258
$$322$$ −4.00000 −0.222911
$$323$$ − 8.00000i − 0.445132i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −12.0000 −0.664619
$$327$$ 4.00000i 0.221201i
$$328$$ −10.0000 −0.552158
$$329$$ 0 0
$$330$$ − 4.00000i − 0.220193i
$$331$$ 10.0000i 0.549650i 0.961494 + 0.274825i $$0.0886199\pi$$
−0.961494 + 0.274825i $$0.911380\pi$$
$$332$$ − 12.0000i − 0.658586i
$$333$$ 6.00000i 0.328798i
$$334$$ 12.0000 0.656611
$$335$$ 8.00000 0.437087
$$336$$ 2.00000i 0.109109i
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 0 0
$$339$$ 16.0000 0.869001
$$340$$ − 4.00000i − 0.216930i
$$341$$ 16.0000 0.866449
$$342$$ −2.00000 −0.108148
$$343$$ − 20.0000i − 1.07990i
$$344$$ 4.00000i 0.215666i
$$345$$ − 2.00000i − 0.107676i
$$346$$ − 14.0000i − 0.752645i
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ 8.00000 0.428845
$$349$$ − 28.0000i − 1.49881i −0.662114 0.749403i $$-0.730341\pi$$
0.662114 0.749403i $$-0.269659\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ − 6.00000i − 0.319348i −0.987170 0.159674i $$-0.948956\pi$$
0.987170 0.159674i $$-0.0510443\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 10.0000i 0.529999i
$$357$$ − 8.00000i − 0.423405i
$$358$$ − 2.00000i − 0.105703i
$$359$$ 24.0000i 1.26667i 0.773877 + 0.633336i $$0.218315\pi$$
−0.773877 + 0.633336i $$0.781685\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 15.0000 0.789474
$$362$$ 22.0000i 1.15629i
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000i 0.104542i
$$367$$ −4.00000 −0.208798 −0.104399 0.994535i $$-0.533292\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ − 10.0000i − 0.520579i
$$370$$ 6.00000i 0.311925i
$$371$$ − 12.0000i − 0.623009i
$$372$$ − 4.00000i − 0.207390i
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 16.0000 0.827340
$$375$$ − 1.00000i − 0.0516398i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 14.0000i 0.719132i 0.933120 + 0.359566i $$0.117075\pi$$
−0.933120 + 0.359566i $$0.882925\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ 20.0000 1.02463
$$382$$ − 8.00000i − 0.409316i
$$383$$ 20.0000i 1.02195i 0.859595 + 0.510976i $$0.170716\pi$$
−0.859595 + 0.510976i $$0.829284\pi$$
$$384$$ 1.00000i 0.0510310i
$$385$$ − 8.00000i − 0.407718i
$$386$$ −12.0000 −0.610784
$$387$$ −4.00000 −0.203331
$$388$$ − 8.00000i − 0.406138i
$$389$$ −20.0000 −1.01404 −0.507020 0.861934i $$-0.669253\pi$$
−0.507020 + 0.861934i $$0.669253\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ − 3.00000i − 0.151523i
$$393$$ −14.0000 −0.706207
$$394$$ 10.0000 0.503793
$$395$$ 8.00000i 0.402524i
$$396$$ − 4.00000i − 0.201008i
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ − 24.0000i − 1.20301i
$$399$$ −4.00000 −0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ − 30.0000i − 1.49813i −0.662497 0.749064i $$-0.730503\pi$$
0.662497 0.749064i $$-0.269497\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 0 0
$$404$$ 20.0000 0.995037
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 16.0000 0.794067
$$407$$ −24.0000 −1.18964
$$408$$ − 4.00000i − 0.198030i
$$409$$ 18.0000i 0.890043i 0.895520 + 0.445021i $$0.146804\pi$$
−0.895520 + 0.445021i $$0.853196\pi$$
$$410$$ − 10.0000i − 0.493865i
$$411$$ 2.00000i 0.0986527i
$$412$$ 4.00000 0.197066
$$413$$ −24.0000 −1.18096
$$414$$ − 2.00000i − 0.0982946i
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 16.0000 0.783523
$$418$$ − 8.00000i − 0.391293i
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 16.0000i 0.779792i 0.920859 + 0.389896i $$0.127489\pi$$
−0.920859 + 0.389896i $$0.872511\pi$$
$$422$$ − 4.00000i − 0.194717i
$$423$$ 0 0
$$424$$ − 6.00000i − 0.291386i
$$425$$ 4.00000 0.194029
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ −4.00000 −0.192897
$$431$$ − 16.0000i − 0.770693i −0.922772 0.385346i $$-0.874082\pi$$
0.922772 0.385346i $$-0.125918\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ − 8.00000i − 0.384012i
$$435$$ 8.00000i 0.383571i
$$436$$ 4.00000i 0.191565i
$$437$$ − 4.00000i − 0.191346i
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ − 4.00000i − 0.190693i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 6.00000i 0.284747i
$$445$$ −10.0000 −0.474045
$$446$$ −22.0000 −1.04173
$$447$$ − 22.0000i − 1.04056i
$$448$$ 2.00000i 0.0944911i
$$449$$ − 18.0000i − 0.849473i −0.905317 0.424736i $$-0.860367\pi$$
0.905317 0.424736i $$-0.139633\pi$$
$$450$$ − 1.00000i − 0.0471405i
$$451$$ 40.0000 1.88353
$$452$$ 16.0000 0.752577
$$453$$ 4.00000i 0.187936i
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −2.00000 −0.0936586
$$457$$ 16.0000i 0.748448i 0.927338 + 0.374224i $$0.122091\pi$$
−0.927338 + 0.374224i $$0.877909\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 4.00000 0.186704
$$460$$ − 2.00000i − 0.0932505i
$$461$$ − 30.0000i − 1.39724i −0.715493 0.698620i $$-0.753798\pi$$
0.715493 0.698620i $$-0.246202\pi$$
$$462$$ − 8.00000i − 0.372194i
$$463$$ 2.00000i 0.0929479i 0.998920 + 0.0464739i $$0.0147984\pi$$
−0.998920 + 0.0464739i $$0.985202\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 4.00000 0.185496
$$466$$ − 28.0000i − 1.29707i
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −12.0000 −0.552345
$$473$$ − 16.0000i − 0.735681i
$$474$$ 8.00000i 0.367452i
$$475$$ − 2.00000i − 0.0917663i
$$476$$ − 8.00000i − 0.366679i
$$477$$ 6.00000 0.274721
$$478$$ −8.00000 −0.365911
$$479$$ 24.0000i 1.09659i 0.836286 + 0.548294i $$0.184723\pi$$
−0.836286 + 0.548294i $$0.815277\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ 22.0000 1.00207
$$483$$ − 4.00000i − 0.182006i
$$484$$ 5.00000 0.227273
$$485$$ 8.00000 0.363261
$$486$$ − 1.00000i − 0.0453609i
$$487$$ − 38.0000i − 1.72194i −0.508652 0.860972i $$-0.669856\pi$$
0.508652 0.860972i $$-0.330144\pi$$
$$488$$ 2.00000i 0.0905357i
$$489$$ − 12.0000i − 0.542659i
$$490$$ 3.00000 0.135526
$$491$$ −38.0000 −1.71492 −0.857458 0.514554i $$-0.827958\pi$$
−0.857458 + 0.514554i $$0.827958\pi$$
$$492$$ − 10.0000i − 0.450835i
$$493$$ −32.0000 −1.44121
$$494$$ 0 0
$$495$$ 4.00000 0.179787
$$496$$ − 4.00000i − 0.179605i
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 14.0000i 0.626726i 0.949633 + 0.313363i $$0.101456\pi$$
−0.949633 + 0.313363i $$0.898544\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ 12.0000i 0.536120i
$$502$$ − 10.0000i − 0.446322i
$$503$$ −42.0000 −1.87269 −0.936344 0.351085i $$-0.885813\pi$$
−0.936344 + 0.351085i $$0.885813\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ 20.0000i 0.889988i
$$506$$ 8.00000 0.355643
$$507$$ 0 0
$$508$$ 20.0000 0.887357
$$509$$ − 38.0000i − 1.68432i −0.539227 0.842160i $$-0.681284\pi$$
0.539227 0.842160i $$-0.318716\pi$$
$$510$$ 4.00000 0.177123
$$511$$ 0 0
$$512$$ 1.00000i 0.0441942i
$$513$$ − 2.00000i − 0.0883022i
$$514$$ 8.00000i 0.352865i
$$515$$ 4.00000i 0.176261i
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 12.0000i 0.527250i
$$519$$ 14.0000 0.614532
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 8.00000i 0.350150i
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ − 2.00000i − 0.0872872i
$$526$$ − 18.0000i − 0.784837i
$$527$$ 16.0000i 0.696971i
$$528$$ − 4.00000i − 0.174078i
$$529$$ −19.0000 −0.826087
$$530$$ 6.00000 0.260623
$$531$$ − 12.0000i − 0.520756i
$$532$$ −4.00000 −0.173422
$$533$$ 0 0
$$534$$ −10.0000 −0.432742
$$535$$ − 4.00000i − 0.172935i
$$536$$ 8.00000 0.345547
$$537$$ 2.00000 0.0863064
$$538$$ − 4.00000i − 0.172452i
$$539$$ 12.0000i 0.516877i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 28.0000 1.20270
$$543$$ −22.0000 −0.944110
$$544$$ − 4.00000i − 0.171499i
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 2.00000i 0.0854358i
$$549$$ −2.00000 −0.0853579
$$550$$ 4.00000 0.170561
$$551$$ 16.0000i 0.681623i
$$552$$ − 2.00000i − 0.0851257i
$$553$$ 16.0000i 0.680389i
$$554$$ − 22.0000i − 0.934690i
$$555$$ −6.00000 −0.254686
$$556$$ 16.0000 0.678551
$$557$$ − 10.0000i − 0.423714i −0.977301 0.211857i $$-0.932049\pi$$
0.977301 0.211857i $$-0.0679510\pi$$
$$558$$ 4.00000 0.169334
$$559$$ 0 0
$$560$$ −2.00000 −0.0845154
$$561$$ 16.0000i 0.675521i
$$562$$ 10.0000 0.421825
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ 16.0000i 0.673125i
$$566$$ − 28.0000i − 1.17693i
$$567$$ − 2.00000i − 0.0839921i
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ − 2.00000i − 0.0837708i
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ 8.00000 0.334205
$$574$$ − 20.0000i − 0.834784i
$$575$$ 2.00000 0.0834058
$$576$$ −1.00000 −0.0416667
$$577$$ 4.00000i 0.166522i 0.996528 + 0.0832611i $$0.0265335\pi$$
−0.996528 + 0.0832611i $$0.973466\pi$$
$$578$$ − 1.00000i − 0.0415945i
$$579$$ − 12.0000i − 0.498703i
$$580$$ 8.00000i 0.332182i
$$581$$ 24.0000 0.995688
$$582$$ 8.00000 0.331611
$$583$$ 24.0000i 0.993978i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ − 12.0000i − 0.495293i −0.968850 0.247647i $$-0.920343\pi$$
0.968850 0.247647i $$-0.0796572\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 8.00000 0.329634
$$590$$ − 12.0000i − 0.494032i
$$591$$ 10.0000i 0.411345i
$$592$$ 6.00000i 0.246598i
$$593$$ 34.0000i 1.39621i 0.715994 + 0.698106i $$0.245974\pi$$
−0.715994 + 0.698106i $$0.754026\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 8.00000 0.327968
$$596$$ − 22.0000i − 0.901155i
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ − 1.00000i − 0.0408248i
$$601$$ −42.0000 −1.71322 −0.856608 0.515968i $$-0.827432\pi$$
−0.856608 + 0.515968i $$0.827432\pi$$
$$602$$ −8.00000 −0.326056
$$603$$ 8.00000i 0.325785i
$$604$$ 4.00000i 0.162758i
$$605$$ 5.00000i 0.203279i
$$606$$ 20.0000i 0.812444i
$$607$$ −12.0000 −0.487065 −0.243532 0.969893i $$-0.578306\pi$$
−0.243532 + 0.969893i $$0.578306\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 16.0000i 0.648353i
$$610$$ −2.00000 −0.0809776
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ 2.00000i 0.0807792i 0.999184 + 0.0403896i $$0.0128599\pi$$
−0.999184 + 0.0403896i $$0.987140\pi$$
$$614$$ 0 0
$$615$$ 10.0000 0.403239
$$616$$ − 8.00000i − 0.322329i
$$617$$ − 38.0000i − 1.52982i −0.644136 0.764911i $$-0.722783\pi$$
0.644136 0.764911i $$-0.277217\pi$$
$$618$$ 4.00000i 0.160904i
$$619$$ 26.0000i 1.04503i 0.852631 + 0.522514i $$0.175006\pi$$
−0.852631 + 0.522514i $$0.824994\pi$$
$$620$$ 4.00000 0.160644
$$621$$ 2.00000 0.0802572
$$622$$ 16.0000i 0.641542i
$$623$$ −20.0000 −0.801283
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ − 10.0000i − 0.399680i
$$627$$ 8.00000 0.319489
$$628$$ 10.0000 0.399043
$$629$$ − 24.0000i − 0.956943i
$$630$$ − 2.00000i − 0.0796819i
$$631$$ 40.0000i 1.59237i 0.605050 + 0.796187i $$0.293153\pi$$
−0.605050 + 0.796187i $$0.706847\pi$$
$$632$$ 8.00000i 0.318223i
$$633$$ 4.00000 0.158986
$$634$$ −2.00000 −0.0794301
$$635$$ 20.0000i 0.793676i
$$636$$ 6.00000 0.237915
$$637$$ 0 0
$$638$$ −32.0000 −1.26689
$$639$$ 0 0
$$640$$ −1.00000 −0.0395285
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ 40.0000i 1.57745i 0.614749 + 0.788723i $$0.289257\pi$$
−0.614749 + 0.788723i $$0.710743\pi$$
$$644$$ − 4.00000i − 0.157622i
$$645$$ − 4.00000i − 0.157500i
$$646$$ 8.00000 0.314756
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ − 1.00000i − 0.0392837i
$$649$$ 48.0000 1.88416
$$650$$ 0 0
$$651$$ 8.00000 0.313545
$$652$$ − 12.0000i − 0.469956i
$$653$$ −38.0000 −1.48705 −0.743527 0.668705i $$-0.766849\pi$$
−0.743527 + 0.668705i $$0.766849\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ − 14.0000i − 0.547025i
$$656$$ − 10.0000i − 0.390434i
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 30.0000 1.16863 0.584317 0.811525i $$-0.301362\pi$$
0.584317 + 0.811525i $$0.301362\pi$$
$$660$$ 4.00000 0.155700
$$661$$ − 16.0000i − 0.622328i −0.950356 0.311164i $$-0.899281\pi$$
0.950356 0.311164i $$-0.100719\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ − 4.00000i − 0.155113i
$$666$$ −6.00000 −0.232495
$$667$$ −16.0000 −0.619522
$$668$$ 12.0000i 0.464294i
$$669$$ − 22.0000i − 0.850569i
$$670$$ 8.00000i 0.309067i
$$671$$ − 8.00000i − 0.308837i
$$672$$ −2.00000 −0.0771517
$$673$$ 46.0000 1.77317 0.886585 0.462566i $$-0.153071\pi$$
0.886585 + 0.462566i $$0.153071\pi$$
$$674$$ 26.0000i 1.00148i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 16.0000i 0.614476i
$$679$$ 16.0000 0.614024
$$680$$ 4.00000 0.153393
$$681$$ 12.0000i 0.459841i
$$682$$ 16.0000i 0.612672i
$$683$$ 36.0000i 1.37750i 0.724998 + 0.688751i $$0.241841\pi$$
−0.724998 + 0.688751i $$0.758159\pi$$
$$684$$ − 2.00000i − 0.0764719i
$$685$$ −2.00000 −0.0764161
$$686$$ 20.0000 0.763604
$$687$$ − 28.0000i − 1.06827i
$$688$$ −4.00000 −0.152499
$$689$$ 0 0
$$690$$ 2.00000 0.0761387
$$691$$ 18.0000i 0.684752i 0.939563 + 0.342376i $$0.111232\pi$$
−0.939563 + 0.342376i $$0.888768\pi$$
$$692$$ 14.0000 0.532200
$$693$$ 8.00000 0.303895
$$694$$ 16.0000i 0.607352i
$$695$$ 16.0000i 0.606915i
$$696$$ 8.00000i 0.303239i
$$697$$ 40.0000i 1.51511i
$$698$$ 28.0000 1.05982
$$699$$ 28.0000 1.05906
$$700$$ − 2.00000i − 0.0755929i
$$701$$ 20.0000 0.755390 0.377695 0.925930i $$-0.376717\pi$$
0.377695 + 0.925930i $$0.376717\pi$$
$$702$$ 0 0
$$703$$ −12.0000 −0.452589
$$704$$ − 4.00000i − 0.150756i
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 40.0000i 1.50435i
$$708$$ − 12.0000i − 0.450988i
$$709$$ 4.00000i 0.150223i 0.997175 + 0.0751116i $$0.0239313\pi$$
−0.997175 + 0.0751116i $$0.976069\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −10.0000 −0.374766
$$713$$ 8.00000i 0.299602i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ 2.00000 0.0747435
$$717$$ − 8.00000i − 0.298765i
$$718$$ −24.0000 −0.895672
$$719$$ −36.0000 −1.34257 −0.671287 0.741198i $$-0.734258\pi$$
−0.671287 + 0.741198i $$0.734258\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ 8.00000i 0.297936i
$$722$$ 15.0000i 0.558242i
$$723$$ 22.0000i 0.818189i
$$724$$ −22.0000 −0.817624
$$725$$ −8.00000 −0.297113
$$726$$ 5.00000i 0.185567i
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.0000 0.591781
$$732$$ −2.00000 −0.0739221
$$733$$ − 6.00000i − 0.221615i −0.993842 0.110808i $$-0.964656\pi$$
0.993842 0.110808i $$-0.0353437\pi$$
$$734$$ − 4.00000i − 0.147643i
$$735$$ 3.00000i 0.110657i
$$736$$ − 2.00000i − 0.0737210i
$$737$$ −32.0000 −1.17874
$$738$$ 10.0000 0.368105
$$739$$ 50.0000i 1.83928i 0.392763 + 0.919640i $$0.371519\pi$$
−0.392763 + 0.919640i $$0.628481\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ 0 0
$$742$$ 12.0000 0.440534
$$743$$ − 36.0000i − 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 4.00000 0.146647
$$745$$ 22.0000 0.806018
$$746$$ 14.0000i 0.512576i
$$747$$ 12.0000i 0.439057i
$$748$$ 16.0000i 0.585018i
$$749$$ − 8.00000i − 0.292314i
$$750$$ 1.00000 0.0365148
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ 10.0000 0.364420
$$754$$ 0 0
$$755$$ −4.00000 −0.145575
$$756$$ − 2.00000i − 0.0727393i
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ −14.0000 −0.508503
$$759$$ 8.00000i 0.290382i
$$760$$ − 2.00000i − 0.0725476i
$$761$$ − 46.0000i − 1.66750i −0.552143 0.833749i $$-0.686190\pi$$
0.552143 0.833749i $$-0.313810\pi$$
$$762$$ 20.0000i 0.724524i
$$763$$ −8.00000 −0.289619
$$764$$ 8.00000 0.289430
$$765$$ 4.00000i 0.144620i
$$766$$ −20.0000 −0.722629
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ − 10.0000i − 0.360609i −0.983611 0.180305i $$-0.942292\pi$$
0.983611 0.180305i $$-0.0577084\pi$$
$$770$$ 8.00000 0.288300
$$771$$ −8.00000 −0.288113
$$772$$ − 12.0000i − 0.431889i
$$773$$ − 26.0000i − 0.935155i −0.883952 0.467578i $$-0.845127\pi$$
0.883952 0.467578i $$-0.154873\pi$$
$$774$$ − 4.00000i − 0.143777i
$$775$$ 4.00000i 0.143684i
$$776$$ 8.00000 0.287183
$$777$$ −12.0000 −0.430498
$$778$$ − 20.0000i − 0.717035i
$$779$$ 20.0000 0.716574
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000i 0.286079i
$$783$$ −8.00000 −0.285897
$$784$$ 3.00000 0.107143
$$785$$ 10.0000i 0.356915i
$$786$$ − 14.0000i − 0.499363i
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 10.0000i 0.356235i
$$789$$ 18.0000 0.640817
$$790$$ −8.00000 −0.284627
$$791$$ 32.0000i 1.13779i
$$792$$ 4.00000 0.142134
$$793$$ 0 0
$$794$$ −6.00000 −0.212932
$$795$$ 6.00000i 0.212798i
$$796$$ 24.0000 0.850657
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ − 4.00000i − 0.141598i
$$799$$ 0 0
$$800$$ − 1.00000i − 0.0353553i
$$801$$ − 10.0000i − 0.353333i
$$802$$ 30.0000 1.05934
$$803$$ 0 0
$$804$$ 8.00000i 0.282138i
$$805$$ 4.00000 0.140981
$$806$$ 0 0
$$807$$ 4.00000 0.140807
$$808$$ 20.0000i 0.703598i
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 34.0000i 1.19390i 0.802278 + 0.596951i $$0.203621\pi$$
−0.802278 + 0.596951i $$0.796379\pi$$
$$812$$ 16.0000i 0.561490i
$$813$$ 28.0000i 0.982003i
$$814$$ − 24.0000i − 0.841200i
$$815$$ 12.0000 0.420342
$$816$$ 4.00000 0.140028
$$817$$ − 8.00000i − 0.279885i
$$818$$ −18.0000 −0.629355
$$819$$ 0 0
$$820$$ 10.0000 0.349215
$$821$$ − 42.0000i − 1.46581i −0.680331 0.732905i $$-0.738164\pi$$
0.680331 0.732905i $$-0.261836\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 4.00000i 0.139347i
$$825$$ 4.00000i 0.139262i
$$826$$ − 24.0000i − 0.835067i
$$827$$ − 44.0000i − 1.53003i −0.644013 0.765015i $$-0.722732\pi$$
0.644013 0.765015i $$-0.277268\pi$$
$$828$$ 2.00000 0.0695048
$$829$$ 10.0000 0.347314 0.173657 0.984806i $$-0.444442\pi$$
0.173657 + 0.984806i $$0.444442\pi$$
$$830$$ 12.0000i 0.416526i
$$831$$ 22.0000 0.763172
$$832$$ 0 0
$$833$$ −12.0000 −0.415775
$$834$$ 16.0000i 0.554035i
$$835$$ −12.0000 −0.415277
$$836$$ 8.00000 0.276686
$$837$$ 4.00000i 0.138260i
$$838$$ 14.0000i 0.483622i
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ 35.0000 1.20690
$$842$$ −16.0000 −0.551396
$$843$$ 10.0000i 0.344418i
$$844$$ 4.00000 0.137686
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.0000i 0.343604i
$$848$$ 6.00000 0.206041
$$849$$ 28.0000 0.960958
$$850$$ 4.00000i 0.137199i
$$851$$ − 12.0000i − 0.411355i
$$852$$ 0 0
$$853$$ − 10.0000i − 0.342393i −0.985237 0.171197i $$-0.945237\pi$$
0.985237 0.171197i $$-0.0547634\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 2.00000 0.0683986
$$856$$ − 4.00000i − 0.136717i
$$857$$ 16.0000 0.546550 0.273275 0.961936i $$-0.411893\pi$$
0.273275 + 0.961936i $$0.411893\pi$$
$$858$$ 0 0
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ − 4.00000i − 0.136399i
$$861$$ 20.0000 0.681598
$$862$$ 16.0000 0.544962
$$863$$ 36.0000i 1.22545i 0.790295 + 0.612727i $$0.209928\pi$$
−0.790295 + 0.612727i $$0.790072\pi$$
$$864$$ − 1.00000i − 0.0340207i
$$865$$ 14.0000i 0.476014i
$$866$$ 26.0000i 0.883516i
$$867$$ 1.00000 0.0339618
$$868$$ 8.00000 0.271538
$$869$$ − 32.0000i − 1.08553i
$$870$$ −8.00000 −0.271225
$$871$$ 0 0
$$872$$ −4.00000 −0.135457
$$873$$ 8.00000i 0.270759i
$$874$$ 4.00000 0.135302
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ − 8.00000i − 0.269987i
$$879$$ 6.00000i 0.202375i
$$880$$ 4.00000 0.134840
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 3.00000i 0.101015i
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 0 0
$$885$$ 12.0000 0.403376
$$886$$ − 16.0000i − 0.537531i
$$887$$ −6.00000 −0.201460 −0.100730 0.994914i $$-0.532118\pi$$
−0.100730 + 0.994914i $$0.532118\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 40.0000i 1.34156i
$$890$$ − 10.0000i − 0.335201i
$$891$$ 4.00000i 0.134005i
$$892$$ − 22.0000i − 0.736614i
$$893$$ 0 0
$$894$$ 22.0000 0.735790
$$895$$ 2.00000i 0.0668526i
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ − 32.0000i − 1.06726i
$$900$$ 1.00000 0.0333333
$$901$$ −24.0000 −0.799556
$$902$$ 40.0000i 1.33185i
$$903$$ − 8.00000i − 0.266223i
$$904$$ 16.0000i 0.532152i
$$905$$ − 22.0000i − 0.731305i
$$906$$ −4.00000 −0.132891
$$907$$ 44.0000 1.46100 0.730498 0.682915i $$-0.239288\pi$$
0.730498 + 0.682915i $$0.239288\pi$$
$$908$$ 12.0000i 0.398234i
$$909$$ −20.0000 −0.663358
$$910$$ 0 0
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ − 2.00000i − 0.0662266i
$$913$$ −48.0000 −1.58857
$$914$$ −16.0000 −0.529233
$$915$$ − 2.00000i − 0.0661180i
$$916$$ − 28.0000i − 0.925146i
$$917$$ − 28.0000i − 0.924641i
$$918$$ 4.00000i 0.132020i
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 2.00000 0.0659380
$$921$$ 0 0
$$922$$ 30.0000 0.987997
$$923$$ 0 0
$$924$$ 8.00000 0.263181
$$925$$ − 6.00000i − 0.197279i
$$926$$ −2.00000 −0.0657241
$$927$$ −4.00000 −0.131377
$$928$$ 8.00000i 0.262613i
$$929$$ 34.0000i 1.11550i 0.830008 + 0.557752i $$0.188336\pi$$
−0.830008 + 0.557752i $$0.811664\pi$$
$$930$$ 4.00000i 0.131165i
$$931$$ 6.00000i 0.196642i
$$932$$ 28.0000 0.917170
$$933$$ −16.0000 −0.523816
$$934$$ − 12.0000i − 0.392652i
$$935$$ −16.0000 −0.523256
$$936$$ 0 0
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ 38.0000i 1.23876i 0.785090 + 0.619382i $$0.212617\pi$$
−0.785090 + 0.619382i $$0.787383\pi$$
$$942$$ 10.0000i 0.325818i
$$943$$ 20.0000i 0.651290i
$$944$$ − 12.0000i − 0.390567i
$$945$$ 2.00000 0.0650600
$$946$$ 16.0000 0.520205
$$947$$ − 4.00000i − 0.129983i −0.997886 0.0649913i $$-0.979298\pi$$
0.997886 0.0649913i $$-0.0207020\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 0 0
$$950$$ 2.00000 0.0648886
$$951$$ − 2.00000i − 0.0648544i
$$952$$ 8.00000 0.259281
$$953$$ 36.0000 1.16615 0.583077 0.812417i $$-0.301849\pi$$
0.583077 + 0.812417i $$0.301849\pi$$
$$954$$ 6.00000i 0.194257i
$$955$$ 8.00000i 0.258874i
$$956$$ − 8.00000i − 0.258738i
$$957$$ − 32.0000i − 1.03441i
$$958$$ −24.0000 −0.775405
$$959$$ −4.00000 −0.129167
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 15.0000 0.483871
$$962$$ 0 0
$$963$$ 4.00000 0.128898
$$964$$ 22.0000i 0.708572i
$$965$$ 12.0000 0.386294
$$966$$ 4.00000 0.128698
$$967$$ − 14.0000i − 0.450210i −0.974335 0.225105i $$-0.927728\pi$$
0.974335 0.225105i $$-0.0722725\pi$$
$$968$$ 5.00000i 0.160706i
$$969$$ 8.00000i 0.256997i
$$970$$ 8.00000i 0.256865i
$$971$$ 42.0000 1.34784 0.673922 0.738802i $$-0.264608\pi$$
0.673922 + 0.738802i $$0.264608\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 32.0000i 1.02587i
$$974$$ 38.0000 1.21760
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ − 42.0000i − 1.34370i −0.740688 0.671850i $$-0.765500\pi$$
0.740688 0.671850i $$-0.234500\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 40.0000 1.27841
$$980$$ 3.00000i 0.0958315i
$$981$$ − 4.00000i − 0.127710i
$$982$$ − 38.0000i − 1.21263i
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 10.0000 0.318788
$$985$$ −10.0000 −0.318626
$$986$$ − 32.0000i − 1.01909i
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 8.00000 0.254385
$$990$$ 4.00000i 0.127128i
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 4.00000 0.127000
$$993$$ − 10.0000i − 0.317340i
$$994$$ 0 0
$$995$$ 24.0000i 0.760851i
$$996$$ 12.0000i 0.380235i
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ −14.0000 −0.443162
$$999$$ − 6.00000i − 0.189832i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

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

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