# Properties

 Label 5070.2.b.o.1351.2 Level $5070$ Weight $2$ Character 5070.1351 Analytic conductor $40.484$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 390) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1351.2 Root $$-0.866025 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1351 Dual form 5070.2.b.o.1351.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.46410i q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -7.46410i q^{19} +1.00000i q^{20} -2.00000i q^{21} +6.46410 q^{22} +3.73205 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} +0.267949 q^{29} +1.00000 q^{30} +1.73205i q^{31} -1.00000i q^{32} -6.46410i q^{33} +4.00000i q^{34} +2.00000 q^{35} -1.00000 q^{36} +9.19615i q^{37} -7.46410 q^{38} +1.00000 q^{40} +2.00000i q^{41} -2.00000 q^{42} +11.9282 q^{43} -6.46410i q^{44} -1.00000i q^{45} -3.73205i q^{46} +3.53590i q^{47} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +0.928203 q^{53} +1.00000i q^{54} +6.46410 q^{55} -2.00000 q^{56} +7.46410i q^{57} -0.267949i q^{58} -8.46410i q^{59} -1.00000i q^{60} -10.3923 q^{61} +1.73205 q^{62} +2.00000i q^{63} -1.00000 q^{64} -6.46410 q^{66} -11.4641i q^{67} +4.00000 q^{68} -3.73205 q^{69} -2.00000i q^{70} +12.3923i q^{71} +1.00000i q^{72} -2.00000i q^{73} +9.19615 q^{74} +1.00000 q^{75} +7.46410i q^{76} -12.9282 q^{77} -13.9282 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} -8.92820i q^{83} +2.00000i q^{84} +4.00000i q^{85} -11.9282i q^{86} -0.267949 q^{87} -6.46410 q^{88} -0.535898i q^{89} -1.00000 q^{90} -3.73205 q^{92} -1.73205i q^{93} +3.53590 q^{94} -7.46410 q^{95} +1.00000i q^{96} -0.535898i q^{97} -3.00000i q^{98} +6.46410i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} - 4q^{4} + 4q^{9} - 4q^{10} + 4q^{12} + 8q^{14} + 4q^{16} - 16q^{17} + 12q^{22} + 8q^{23} - 4q^{25} - 4q^{27} + 8q^{29} + 4q^{30} + 8q^{35} - 4q^{36} - 16q^{38} + 4q^{40} - 8q^{42} + 20q^{43} - 4q^{48} + 12q^{49} + 16q^{51} - 24q^{53} + 12q^{55} - 8q^{56} - 4q^{64} - 12q^{66} + 16q^{68} - 8q^{69} + 16q^{74} + 4q^{75} - 24q^{77} - 28q^{79} + 4q^{81} + 8q^{82} - 8q^{87} - 12q^{88} - 4q^{90} - 8q^{92} + 28q^{94} - 16q^{95} + O(q^{100})$$

## 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 $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 1.00000i 0.353553i
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ 6.46410i 1.94900i 0.224388 + 0.974500i $$0.427962\pi$$
−0.224388 + 0.974500i $$0.572038\pi$$
$$12$$ 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
$$19$$ − 7.46410i − 1.71238i −0.516659 0.856191i $$-0.672825\pi$$
0.516659 0.856191i $$-0.327175\pi$$
$$20$$ 1.00000i 0.223607i
$$21$$ − 2.00000i − 0.436436i
$$22$$ 6.46410 1.37815
$$23$$ 3.73205 0.778186 0.389093 0.921198i $$-0.372788\pi$$
0.389093 + 0.921198i $$0.372788\pi$$
$$24$$ − 1.00000i − 0.204124i
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ − 2.00000i − 0.377964i
$$29$$ 0.267949 0.0497569 0.0248785 0.999690i $$-0.492080\pi$$
0.0248785 + 0.999690i $$0.492080\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 1.73205i 0.311086i 0.987829 + 0.155543i $$0.0497126\pi$$
−0.987829 + 0.155543i $$0.950287\pi$$
$$32$$ − 1.00000i − 0.176777i
$$33$$ − 6.46410i − 1.12526i
$$34$$ 4.00000i 0.685994i
$$35$$ 2.00000 0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 9.19615i 1.51184i 0.654665 + 0.755919i $$0.272810\pi$$
−0.654665 + 0.755919i $$0.727190\pi$$
$$38$$ −7.46410 −1.21084
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 2.00000i 0.312348i 0.987730 + 0.156174i $$0.0499160\pi$$
−0.987730 + 0.156174i $$0.950084\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 11.9282 1.81903 0.909517 0.415667i $$-0.136452\pi$$
0.909517 + 0.415667i $$0.136452\pi$$
$$44$$ − 6.46410i − 0.974500i
$$45$$ − 1.00000i − 0.149071i
$$46$$ − 3.73205i − 0.550261i
$$47$$ 3.53590i 0.515764i 0.966176 + 0.257882i $$0.0830245\pi$$
−0.966176 + 0.257882i $$0.916975\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 3.00000 0.428571
$$50$$ 1.00000i 0.141421i
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 0.928203 0.127499 0.0637493 0.997966i $$-0.479694\pi$$
0.0637493 + 0.997966i $$0.479694\pi$$
$$54$$ 1.00000i 0.136083i
$$55$$ 6.46410 0.871619
$$56$$ −2.00000 −0.267261
$$57$$ 7.46410i 0.988644i
$$58$$ − 0.267949i − 0.0351835i
$$59$$ − 8.46410i − 1.10193i −0.834528 0.550966i $$-0.814259\pi$$
0.834528 0.550966i $$-0.185741\pi$$
$$60$$ − 1.00000i − 0.129099i
$$61$$ −10.3923 −1.33060 −0.665299 0.746577i $$-0.731696\pi$$
−0.665299 + 0.746577i $$0.731696\pi$$
$$62$$ 1.73205 0.219971
$$63$$ 2.00000i 0.251976i
$$64$$ −1.00000 −0.125000
$$65$$ 0 0
$$66$$ −6.46410 −0.795676
$$67$$ − 11.4641i − 1.40056i −0.713867 0.700281i $$-0.753058\pi$$
0.713867 0.700281i $$-0.246942\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −3.73205 −0.449286
$$70$$ − 2.00000i − 0.239046i
$$71$$ 12.3923i 1.47070i 0.677690 + 0.735348i $$0.262981\pi$$
−0.677690 + 0.735348i $$0.737019\pi$$
$$72$$ 1.00000i 0.117851i
$$73$$ − 2.00000i − 0.234082i −0.993127 0.117041i $$-0.962659\pi$$
0.993127 0.117041i $$-0.0373409\pi$$
$$74$$ 9.19615 1.06903
$$75$$ 1.00000 0.115470
$$76$$ 7.46410i 0.856191i
$$77$$ −12.9282 −1.47331
$$78$$ 0 0
$$79$$ −13.9282 −1.56705 −0.783523 0.621363i $$-0.786579\pi$$
−0.783523 + 0.621363i $$0.786579\pi$$
$$80$$ − 1.00000i − 0.111803i
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ − 8.92820i − 0.979998i −0.871723 0.489999i $$-0.836997\pi$$
0.871723 0.489999i $$-0.163003\pi$$
$$84$$ 2.00000i 0.218218i
$$85$$ 4.00000i 0.433861i
$$86$$ − 11.9282i − 1.28625i
$$87$$ −0.267949 −0.0287272
$$88$$ −6.46410 −0.689076
$$89$$ − 0.535898i − 0.0568051i −0.999597 0.0284026i $$-0.990958\pi$$
0.999597 0.0284026i $$-0.00904203\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 0 0
$$92$$ −3.73205 −0.389093
$$93$$ − 1.73205i − 0.179605i
$$94$$ 3.53590 0.364700
$$95$$ −7.46410 −0.765801
$$96$$ 1.00000i 0.102062i
$$97$$ − 0.535898i − 0.0544122i −0.999630 0.0272061i $$-0.991339\pi$$
0.999630 0.0272061i $$-0.00866105\pi$$
$$98$$ − 3.00000i − 0.303046i
$$99$$ 6.46410i 0.649667i
$$100$$ 1.00000 0.100000
$$101$$ 2.92820 0.291367 0.145684 0.989331i $$-0.453462\pi$$
0.145684 + 0.989331i $$0.453462\pi$$
$$102$$ − 4.00000i − 0.396059i
$$103$$ −11.8564 −1.16825 −0.584123 0.811665i $$-0.698562\pi$$
−0.584123 + 0.811665i $$0.698562\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ − 0.928203i − 0.0901551i
$$107$$ −7.85641 −0.759507 −0.379754 0.925088i $$-0.623991\pi$$
−0.379754 + 0.925088i $$0.623991\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 15.8564i 1.51877i 0.650643 + 0.759384i $$0.274500\pi$$
−0.650643 + 0.759384i $$0.725500\pi$$
$$110$$ − 6.46410i − 0.616328i
$$111$$ − 9.19615i − 0.872860i
$$112$$ 2.00000i 0.188982i
$$113$$ 0.803848 0.0756196 0.0378098 0.999285i $$-0.487962\pi$$
0.0378098 + 0.999285i $$0.487962\pi$$
$$114$$ 7.46410 0.699077
$$115$$ − 3.73205i − 0.348016i
$$116$$ −0.267949 −0.0248785
$$117$$ 0 0
$$118$$ −8.46410 −0.779184
$$119$$ − 8.00000i − 0.733359i
$$120$$ −1.00000 −0.0912871
$$121$$ −30.7846 −2.79860
$$122$$ 10.3923i 0.940875i
$$123$$ − 2.00000i − 0.180334i
$$124$$ − 1.73205i − 0.155543i
$$125$$ 1.00000i 0.0894427i
$$126$$ 2.00000 0.178174
$$127$$ −4.92820 −0.437307 −0.218654 0.975803i $$-0.570166\pi$$
−0.218654 + 0.975803i $$0.570166\pi$$
$$128$$ 1.00000i 0.0883883i
$$129$$ −11.9282 −1.05022
$$130$$ 0 0
$$131$$ −18.6603 −1.63035 −0.815177 0.579212i $$-0.803360\pi$$
−0.815177 + 0.579212i $$0.803360\pi$$
$$132$$ 6.46410i 0.562628i
$$133$$ 14.9282 1.29444
$$134$$ −11.4641 −0.990348
$$135$$ 1.00000i 0.0860663i
$$136$$ − 4.00000i − 0.342997i
$$137$$ 2.46410i 0.210522i 0.994445 + 0.105261i $$0.0335679\pi$$
−0.994445 + 0.105261i $$0.966432\pi$$
$$138$$ 3.73205i 0.317693i
$$139$$ −12.9282 −1.09656 −0.548278 0.836296i $$-0.684716\pi$$
−0.548278 + 0.836296i $$0.684716\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ − 3.53590i − 0.297776i
$$142$$ 12.3923 1.03994
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ − 0.267949i − 0.0222520i
$$146$$ −2.00000 −0.165521
$$147$$ −3.00000 −0.247436
$$148$$ − 9.19615i − 0.755919i
$$149$$ 13.5359i 1.10890i 0.832216 + 0.554452i $$0.187072\pi$$
−0.832216 + 0.554452i $$0.812928\pi$$
$$150$$ − 1.00000i − 0.0816497i
$$151$$ 10.3923i 0.845714i 0.906196 + 0.422857i $$0.138973\pi$$
−0.906196 + 0.422857i $$0.861027\pi$$
$$152$$ 7.46410 0.605419
$$153$$ −4.00000 −0.323381
$$154$$ 12.9282i 1.04178i
$$155$$ 1.73205 0.139122
$$156$$ 0 0
$$157$$ −5.00000 −0.399043 −0.199522 0.979893i $$-0.563939\pi$$
−0.199522 + 0.979893i $$0.563939\pi$$
$$158$$ 13.9282i 1.10807i
$$159$$ −0.928203 −0.0736113
$$160$$ −1.00000 −0.0790569
$$161$$ 7.46410i 0.588254i
$$162$$ − 1.00000i − 0.0785674i
$$163$$ − 15.0526i − 1.17901i −0.807766 0.589504i $$-0.799323\pi$$
0.807766 0.589504i $$-0.200677\pi$$
$$164$$ − 2.00000i − 0.156174i
$$165$$ −6.46410 −0.503230
$$166$$ −8.92820 −0.692963
$$167$$ 16.3205i 1.26292i 0.775409 + 0.631459i $$0.217544\pi$$
−0.775409 + 0.631459i $$0.782456\pi$$
$$168$$ 2.00000 0.154303
$$169$$ 0 0
$$170$$ 4.00000 0.306786
$$171$$ − 7.46410i − 0.570794i
$$172$$ −11.9282 −0.909517
$$173$$ 10.9282 0.830856 0.415428 0.909626i $$-0.363632\pi$$
0.415428 + 0.909626i $$0.363632\pi$$
$$174$$ 0.267949i 0.0203132i
$$175$$ − 2.00000i − 0.151186i
$$176$$ 6.46410i 0.487250i
$$177$$ 8.46410i 0.636201i
$$178$$ −0.535898 −0.0401673
$$179$$ −19.7321 −1.47484 −0.737421 0.675433i $$-0.763957\pi$$
−0.737421 + 0.675433i $$0.763957\pi$$
$$180$$ 1.00000i 0.0745356i
$$181$$ −2.92820 −0.217652 −0.108826 0.994061i $$-0.534709\pi$$
−0.108826 + 0.994061i $$0.534709\pi$$
$$182$$ 0 0
$$183$$ 10.3923 0.768221
$$184$$ 3.73205i 0.275130i
$$185$$ 9.19615 0.676115
$$186$$ −1.73205 −0.127000
$$187$$ − 25.8564i − 1.89081i
$$188$$ − 3.53590i − 0.257882i
$$189$$ − 2.00000i − 0.145479i
$$190$$ 7.46410i 0.541503i
$$191$$ 21.4641 1.55309 0.776544 0.630063i $$-0.216971\pi$$
0.776544 + 0.630063i $$0.216971\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 11.3205i 0.814868i 0.913235 + 0.407434i $$0.133576\pi$$
−0.913235 + 0.407434i $$0.866424\pi$$
$$194$$ −0.535898 −0.0384753
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 4.39230i 0.312939i 0.987683 + 0.156469i $$0.0500113\pi$$
−0.987683 + 0.156469i $$0.949989\pi$$
$$198$$ 6.46410 0.459384
$$199$$ −5.07180 −0.359530 −0.179765 0.983710i $$-0.557534\pi$$
−0.179765 + 0.983710i $$0.557534\pi$$
$$200$$ − 1.00000i − 0.0707107i
$$201$$ 11.4641i 0.808615i
$$202$$ − 2.92820i − 0.206028i
$$203$$ 0.535898i 0.0376127i
$$204$$ −4.00000 −0.280056
$$205$$ 2.00000 0.139686
$$206$$ 11.8564i 0.826075i
$$207$$ 3.73205 0.259395
$$208$$ 0 0
$$209$$ 48.2487 3.33743
$$210$$ 2.00000i 0.138013i
$$211$$ −11.3205 −0.779336 −0.389668 0.920955i $$-0.627410\pi$$
−0.389668 + 0.920955i $$0.627410\pi$$
$$212$$ −0.928203 −0.0637493
$$213$$ − 12.3923i − 0.849107i
$$214$$ 7.85641i 0.537053i
$$215$$ − 11.9282i − 0.813497i
$$216$$ − 1.00000i − 0.0680414i
$$217$$ −3.46410 −0.235159
$$218$$ 15.8564 1.07393
$$219$$ 2.00000i 0.135147i
$$220$$ −6.46410 −0.435810
$$221$$ 0 0
$$222$$ −9.19615 −0.617205
$$223$$ − 20.5359i − 1.37519i −0.726097 0.687593i $$-0.758667\pi$$
0.726097 0.687593i $$-0.241333\pi$$
$$224$$ 2.00000 0.133631
$$225$$ −1.00000 −0.0666667
$$226$$ − 0.803848i − 0.0534711i
$$227$$ 16.3923i 1.08800i 0.839087 + 0.543998i $$0.183090\pi$$
−0.839087 + 0.543998i $$0.816910\pi$$
$$228$$ − 7.46410i − 0.494322i
$$229$$ 7.85641i 0.519166i 0.965721 + 0.259583i $$0.0835851\pi$$
−0.965721 + 0.259583i $$0.916415\pi$$
$$230$$ −3.73205 −0.246084
$$231$$ 12.9282 0.850613
$$232$$ 0.267949i 0.0175917i
$$233$$ 6.12436 0.401220 0.200610 0.979671i $$-0.435708\pi$$
0.200610 + 0.979671i $$0.435708\pi$$
$$234$$ 0 0
$$235$$ 3.53590 0.230657
$$236$$ 8.46410i 0.550966i
$$237$$ 13.9282 0.904734
$$238$$ −8.00000 −0.518563
$$239$$ 16.3923i 1.06033i 0.847894 + 0.530165i $$0.177870\pi$$
−0.847894 + 0.530165i $$0.822130\pi$$
$$240$$ 1.00000i 0.0645497i
$$241$$ − 17.7321i − 1.14222i −0.820873 0.571111i $$-0.806513\pi$$
0.820873 0.571111i $$-0.193487\pi$$
$$242$$ 30.7846i 1.97891i
$$243$$ −1.00000 −0.0641500
$$244$$ 10.3923 0.665299
$$245$$ − 3.00000i − 0.191663i
$$246$$ −2.00000 −0.127515
$$247$$ 0 0
$$248$$ −1.73205 −0.109985
$$249$$ 8.92820i 0.565802i
$$250$$ 1.00000 0.0632456
$$251$$ 15.7321 0.992998 0.496499 0.868037i $$-0.334619\pi$$
0.496499 + 0.868037i $$0.334619\pi$$
$$252$$ − 2.00000i − 0.125988i
$$253$$ 24.1244i 1.51669i
$$254$$ 4.92820i 0.309223i
$$255$$ − 4.00000i − 0.250490i
$$256$$ 1.00000 0.0625000
$$257$$ −5.33975 −0.333084 −0.166542 0.986034i $$-0.553260\pi$$
−0.166542 + 0.986034i $$0.553260\pi$$
$$258$$ 11.9282i 0.742617i
$$259$$ −18.3923 −1.14284
$$260$$ 0 0
$$261$$ 0.267949 0.0165856
$$262$$ 18.6603i 1.15283i
$$263$$ 6.12436 0.377644 0.188822 0.982011i $$-0.439533\pi$$
0.188822 + 0.982011i $$0.439533\pi$$
$$264$$ 6.46410 0.397838
$$265$$ − 0.928203i − 0.0570191i
$$266$$ − 14.9282i − 0.915307i
$$267$$ 0.535898i 0.0327964i
$$268$$ 11.4641i 0.700281i
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 1.19615i 0.0726611i 0.999340 + 0.0363305i $$0.0115669\pi$$
−0.999340 + 0.0363305i $$0.988433\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ 2.46410 0.148862
$$275$$ − 6.46410i − 0.389800i
$$276$$ 3.73205 0.224643
$$277$$ −3.92820 −0.236023 −0.118011 0.993012i $$-0.537652\pi$$
−0.118011 + 0.993012i $$0.537652\pi$$
$$278$$ 12.9282i 0.775382i
$$279$$ 1.73205i 0.103695i
$$280$$ 2.00000i 0.119523i
$$281$$ 8.92820i 0.532612i 0.963889 + 0.266306i $$0.0858032\pi$$
−0.963889 + 0.266306i $$0.914197\pi$$
$$282$$ −3.53590 −0.210560
$$283$$ −9.92820 −0.590170 −0.295085 0.955471i $$-0.595348\pi$$
−0.295085 + 0.955471i $$0.595348\pi$$
$$284$$ − 12.3923i − 0.735348i
$$285$$ 7.46410 0.442135
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ − 1.00000i − 0.0589256i
$$289$$ −1.00000 −0.0588235
$$290$$ −0.267949 −0.0157345
$$291$$ 0.535898i 0.0314149i
$$292$$ 2.00000i 0.117041i
$$293$$ 31.8564i 1.86107i 0.366202 + 0.930536i $$0.380658\pi$$
−0.366202 + 0.930536i $$0.619342\pi$$
$$294$$ 3.00000i 0.174964i
$$295$$ −8.46410 −0.492799
$$296$$ −9.19615 −0.534516
$$297$$ − 6.46410i − 0.375085i
$$298$$ 13.5359 0.784114
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 23.8564i 1.37506i
$$302$$ 10.3923 0.598010
$$303$$ −2.92820 −0.168221
$$304$$ − 7.46410i − 0.428096i
$$305$$ 10.3923i 0.595062i
$$306$$ 4.00000i 0.228665i
$$307$$ − 19.4641i − 1.11087i −0.831558 0.555437i $$-0.812551\pi$$
0.831558 0.555437i $$-0.187449\pi$$
$$308$$ 12.9282 0.736653
$$309$$ 11.8564 0.674487
$$310$$ − 1.73205i − 0.0983739i
$$311$$ −28.3923 −1.60998 −0.804990 0.593288i $$-0.797829\pi$$
−0.804990 + 0.593288i $$0.797829\pi$$
$$312$$ 0 0
$$313$$ −28.0000 −1.58265 −0.791327 0.611393i $$-0.790609\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$314$$ 5.00000i 0.282166i
$$315$$ 2.00000 0.112687
$$316$$ 13.9282 0.783523
$$317$$ − 14.5359i − 0.816417i −0.912889 0.408209i $$-0.866154\pi$$
0.912889 0.408209i $$-0.133846\pi$$
$$318$$ 0.928203i 0.0520511i
$$319$$ 1.73205i 0.0969762i
$$320$$ 1.00000i 0.0559017i
$$321$$ 7.85641 0.438502
$$322$$ 7.46410 0.415958
$$323$$ 29.8564i 1.66125i
$$324$$ −1.00000 −0.0555556
$$325$$ 0 0
$$326$$ −15.0526 −0.833684
$$327$$ − 15.8564i − 0.876861i
$$328$$ −2.00000 −0.110432
$$329$$ −7.07180 −0.389881
$$330$$ 6.46410i 0.355837i
$$331$$ 16.7846i 0.922566i 0.887253 + 0.461283i $$0.152611\pi$$
−0.887253 + 0.461283i $$0.847389\pi$$
$$332$$ 8.92820i 0.489999i
$$333$$ 9.19615i 0.503946i
$$334$$ 16.3205 0.893018
$$335$$ −11.4641 −0.626351
$$336$$ − 2.00000i − 0.109109i
$$337$$ 9.32051 0.507720 0.253860 0.967241i $$-0.418300\pi$$
0.253860 + 0.967241i $$0.418300\pi$$
$$338$$ 0 0
$$339$$ −0.803848 −0.0436590
$$340$$ − 4.00000i − 0.216930i
$$341$$ −11.1962 −0.606306
$$342$$ −7.46410 −0.403612
$$343$$ 20.0000i 1.07990i
$$344$$ 11.9282i 0.643126i
$$345$$ 3.73205i 0.200927i
$$346$$ − 10.9282i − 0.587504i
$$347$$ 1.60770 0.0863056 0.0431528 0.999068i $$-0.486260\pi$$
0.0431528 + 0.999068i $$0.486260\pi$$
$$348$$ 0.267949 0.0143636
$$349$$ 21.4641i 1.14895i 0.818523 + 0.574474i $$0.194793\pi$$
−0.818523 + 0.574474i $$0.805207\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 0 0
$$352$$ 6.46410 0.344538
$$353$$ 2.00000i 0.106449i 0.998583 + 0.0532246i $$0.0169499\pi$$
−0.998583 + 0.0532246i $$0.983050\pi$$
$$354$$ 8.46410 0.449862
$$355$$ 12.3923 0.657715
$$356$$ 0.535898i 0.0284026i
$$357$$ 8.00000i 0.423405i
$$358$$ 19.7321i 1.04287i
$$359$$ − 5.07180i − 0.267679i −0.991003 0.133840i $$-0.957269\pi$$
0.991003 0.133840i $$-0.0427307\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −36.7128 −1.93225
$$362$$ 2.92820i 0.153903i
$$363$$ 30.7846 1.61577
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ − 10.3923i − 0.543214i
$$367$$ 15.6077 0.814715 0.407358 0.913269i $$-0.366450\pi$$
0.407358 + 0.913269i $$0.366450\pi$$
$$368$$ 3.73205 0.194547
$$369$$ 2.00000i 0.104116i
$$370$$ − 9.19615i − 0.478085i
$$371$$ 1.85641i 0.0963798i
$$372$$ 1.73205i 0.0898027i
$$373$$ −15.7846 −0.817296 −0.408648 0.912692i $$-0.634000\pi$$
−0.408648 + 0.912692i $$0.634000\pi$$
$$374$$ −25.8564 −1.33700
$$375$$ − 1.00000i − 0.0516398i
$$376$$ −3.53590 −0.182350
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 27.8564i 1.43089i 0.698670 + 0.715444i $$0.253775\pi$$
−0.698670 + 0.715444i $$0.746225\pi$$
$$380$$ 7.46410 0.382900
$$381$$ 4.92820 0.252479
$$382$$ − 21.4641i − 1.09820i
$$383$$ − 25.3923i − 1.29749i −0.761007 0.648743i $$-0.775295\pi$$
0.761007 0.648743i $$-0.224705\pi$$
$$384$$ − 1.00000i − 0.0510310i
$$385$$ 12.9282i 0.658882i
$$386$$ 11.3205 0.576199
$$387$$ 11.9282 0.606345
$$388$$ 0.535898i 0.0272061i
$$389$$ −23.7321 −1.20326 −0.601631 0.798774i $$-0.705482\pi$$
−0.601631 + 0.798774i $$0.705482\pi$$
$$390$$ 0 0
$$391$$ −14.9282 −0.754952
$$392$$ 3.00000i 0.151523i
$$393$$ 18.6603 0.941285
$$394$$ 4.39230 0.221281
$$395$$ 13.9282i 0.700804i
$$396$$ − 6.46410i − 0.324833i
$$397$$ − 12.1244i − 0.608504i −0.952592 0.304252i $$-0.901594\pi$$
0.952592 0.304252i $$-0.0984065\pi$$
$$398$$ 5.07180i 0.254226i
$$399$$ −14.9282 −0.747345
$$400$$ −1.00000 −0.0500000
$$401$$ − 32.0000i − 1.59800i −0.601329 0.799002i $$-0.705362\pi$$
0.601329 0.799002i $$-0.294638\pi$$
$$402$$ 11.4641 0.571777
$$403$$ 0 0
$$404$$ −2.92820 −0.145684
$$405$$ − 1.00000i − 0.0496904i
$$406$$ 0.535898 0.0265962
$$407$$ −59.4449 −2.94657
$$408$$ 4.00000i 0.198030i
$$409$$ − 4.00000i − 0.197787i −0.995098 0.0988936i $$-0.968470\pi$$
0.995098 0.0988936i $$-0.0315304\pi$$
$$410$$ − 2.00000i − 0.0987730i
$$411$$ − 2.46410i − 0.121545i
$$412$$ 11.8564 0.584123
$$413$$ 16.9282 0.832982
$$414$$ − 3.73205i − 0.183420i
$$415$$ −8.92820 −0.438268
$$416$$ 0 0
$$417$$ 12.9282 0.633097
$$418$$ − 48.2487i − 2.35992i
$$419$$ 22.3923 1.09394 0.546968 0.837154i $$-0.315782\pi$$
0.546968 + 0.837154i $$0.315782\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ − 4.39230i − 0.214068i −0.994255 0.107034i $$-0.965865\pi$$
0.994255 0.107034i $$-0.0341353\pi$$
$$422$$ 11.3205i 0.551074i
$$423$$ 3.53590i 0.171921i
$$424$$ 0.928203i 0.0450775i
$$425$$ 4.00000 0.194029
$$426$$ −12.3923 −0.600409
$$427$$ − 20.7846i − 1.00584i
$$428$$ 7.85641 0.379754
$$429$$ 0 0
$$430$$ −11.9282 −0.575229
$$431$$ 28.3923i 1.36761i 0.729665 + 0.683805i $$0.239676\pi$$
−0.729665 + 0.683805i $$0.760324\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 19.3205 0.928484 0.464242 0.885708i $$-0.346327\pi$$
0.464242 + 0.885708i $$0.346327\pi$$
$$434$$ 3.46410i 0.166282i
$$435$$ 0.267949i 0.0128472i
$$436$$ − 15.8564i − 0.759384i
$$437$$ − 27.8564i − 1.33255i
$$438$$ 2.00000 0.0955637
$$439$$ −17.8564 −0.852240 −0.426120 0.904667i $$-0.640120\pi$$
−0.426120 + 0.904667i $$0.640120\pi$$
$$440$$ 6.46410i 0.308164i
$$441$$ 3.00000 0.142857
$$442$$ 0 0
$$443$$ 16.3923 0.778822 0.389411 0.921064i $$-0.372679\pi$$
0.389411 + 0.921064i $$0.372679\pi$$
$$444$$ 9.19615i 0.436430i
$$445$$ −0.535898 −0.0254040
$$446$$ −20.5359 −0.972403
$$447$$ − 13.5359i − 0.640226i
$$448$$ − 2.00000i − 0.0944911i
$$449$$ 15.7128i 0.741533i 0.928726 + 0.370767i $$0.120905\pi$$
−0.928726 + 0.370767i $$0.879095\pi$$
$$450$$ 1.00000i 0.0471405i
$$451$$ −12.9282 −0.608765
$$452$$ −0.803848 −0.0378098
$$453$$ − 10.3923i − 0.488273i
$$454$$ 16.3923 0.769329
$$455$$ 0 0
$$456$$ −7.46410 −0.349539
$$457$$ 24.5359i 1.14774i 0.818946 + 0.573870i $$0.194559\pi$$
−0.818946 + 0.573870i $$0.805441\pi$$
$$458$$ 7.85641 0.367106
$$459$$ 4.00000 0.186704
$$460$$ 3.73205i 0.174008i
$$461$$ − 0.464102i − 0.0216154i −0.999942 0.0108077i $$-0.996560\pi$$
0.999942 0.0108077i $$-0.00344026\pi$$
$$462$$ − 12.9282i − 0.601474i
$$463$$ − 7.07180i − 0.328654i −0.986406 0.164327i $$-0.947455\pi$$
0.986406 0.164327i $$-0.0525453\pi$$
$$464$$ 0.267949 0.0124392
$$465$$ −1.73205 −0.0803219
$$466$$ − 6.12436i − 0.283705i
$$467$$ −15.8564 −0.733747 −0.366873 0.930271i $$-0.619572\pi$$
−0.366873 + 0.930271i $$0.619572\pi$$
$$468$$ 0 0
$$469$$ 22.9282 1.05873
$$470$$ − 3.53590i − 0.163099i
$$471$$ 5.00000 0.230388
$$472$$ 8.46410 0.389592
$$473$$ 77.1051i 3.54530i
$$474$$ − 13.9282i − 0.639744i
$$475$$ 7.46410i 0.342476i
$$476$$ 8.00000i 0.366679i
$$477$$ 0.928203 0.0424995
$$478$$ 16.3923 0.749767
$$479$$ 5.46410i 0.249661i 0.992178 + 0.124831i $$0.0398387\pi$$
−0.992178 + 0.124831i $$0.960161\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ 0 0
$$482$$ −17.7321 −0.807673
$$483$$ − 7.46410i − 0.339628i
$$484$$ 30.7846 1.39930
$$485$$ −0.535898 −0.0243339
$$486$$ 1.00000i 0.0453609i
$$487$$ − 23.1769i − 1.05025i −0.851026 0.525123i $$-0.824019\pi$$
0.851026 0.525123i $$-0.175981\pi$$
$$488$$ − 10.3923i − 0.470438i
$$489$$ 15.0526i 0.680700i
$$490$$ −3.00000 −0.135526
$$491$$ −17.3205 −0.781664 −0.390832 0.920462i $$-0.627813\pi$$
−0.390832 + 0.920462i $$0.627813\pi$$
$$492$$ 2.00000i 0.0901670i
$$493$$ −1.07180 −0.0482713
$$494$$ 0 0
$$495$$ 6.46410 0.290540
$$496$$ 1.73205i 0.0777714i
$$497$$ −24.7846 −1.11174
$$498$$ 8.92820 0.400082
$$499$$ − 6.53590i − 0.292587i −0.989241 0.146293i $$-0.953266\pi$$
0.989241 0.146293i $$-0.0467344\pi$$
$$500$$ − 1.00000i − 0.0447214i
$$501$$ − 16.3205i − 0.729147i
$$502$$ − 15.7321i − 0.702156i
$$503$$ −31.1769 −1.39011 −0.695055 0.718957i $$-0.744620\pi$$
−0.695055 + 0.718957i $$0.744620\pi$$
$$504$$ −2.00000 −0.0890871
$$505$$ − 2.92820i − 0.130303i
$$506$$ 24.1244 1.07246
$$507$$ 0 0
$$508$$ 4.92820 0.218654
$$509$$ 1.39230i 0.0617128i 0.999524 + 0.0308564i $$0.00982346\pi$$
−0.999524 + 0.0308564i $$0.990177\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ 4.00000 0.176950
$$512$$ − 1.00000i − 0.0441942i
$$513$$ 7.46410i 0.329548i
$$514$$ 5.33975i 0.235526i
$$515$$ 11.8564i 0.522456i
$$516$$ 11.9282 0.525110
$$517$$ −22.8564 −1.00522
$$518$$ 18.3923i 0.808111i
$$519$$ −10.9282 −0.479695
$$520$$ 0 0
$$521$$ −17.3205 −0.758825 −0.379413 0.925228i $$-0.623874\pi$$
−0.379413 + 0.925228i $$0.623874\pi$$
$$522$$ − 0.267949i − 0.0117278i
$$523$$ 11.7846 0.515305 0.257653 0.966238i $$-0.417051\pi$$
0.257653 + 0.966238i $$0.417051\pi$$
$$524$$ 18.6603 0.815177
$$525$$ 2.00000i 0.0872872i
$$526$$ − 6.12436i − 0.267035i
$$527$$ − 6.92820i − 0.301797i
$$528$$ − 6.46410i − 0.281314i
$$529$$ −9.07180 −0.394426
$$530$$ −0.928203 −0.0403186
$$531$$ − 8.46410i − 0.367311i
$$532$$ −14.9282 −0.647220
$$533$$ 0 0
$$534$$ 0.535898 0.0231906
$$535$$ 7.85641i 0.339662i
$$536$$ 11.4641 0.495174
$$537$$ 19.7321 0.851501
$$538$$ − 12.0000i − 0.517357i
$$539$$ 19.3923i 0.835286i
$$540$$ − 1.00000i − 0.0430331i
$$541$$ − 26.9282i − 1.15773i −0.815422 0.578867i $$-0.803495\pi$$
0.815422 0.578867i $$-0.196505\pi$$
$$542$$ 1.19615 0.0513791
$$543$$ 2.92820 0.125661
$$544$$ 4.00000i 0.171499i
$$545$$ 15.8564 0.679214
$$546$$ 0 0
$$547$$ 22.9282 0.980339 0.490170 0.871627i $$-0.336935\pi$$
0.490170 + 0.871627i $$0.336935\pi$$
$$548$$ − 2.46410i − 0.105261i
$$549$$ −10.3923 −0.443533
$$550$$ −6.46410 −0.275630
$$551$$ − 2.00000i − 0.0852029i
$$552$$ − 3.73205i − 0.158847i
$$553$$ − 27.8564i − 1.18457i
$$554$$ 3.92820i 0.166893i
$$555$$ −9.19615 −0.390355
$$556$$ 12.9282 0.548278
$$557$$ 17.7128i 0.750516i 0.926920 + 0.375258i $$0.122446\pi$$
−0.926920 + 0.375258i $$0.877554\pi$$
$$558$$ 1.73205 0.0733236
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ 25.8564i 1.09166i
$$562$$ 8.92820 0.376614
$$563$$ −4.67949 −0.197217 −0.0986085 0.995126i $$-0.531439\pi$$
−0.0986085 + 0.995126i $$0.531439\pi$$
$$564$$ 3.53590i 0.148888i
$$565$$ − 0.803848i − 0.0338181i
$$566$$ 9.92820i 0.417314i
$$567$$ 2.00000i 0.0839921i
$$568$$ −12.3923 −0.519970
$$569$$ 29.3205 1.22918 0.614590 0.788847i $$-0.289322\pi$$
0.614590 + 0.788847i $$0.289322\pi$$
$$570$$ − 7.46410i − 0.312637i
$$571$$ −17.1769 −0.718832 −0.359416 0.933178i $$-0.617024\pi$$
−0.359416 + 0.933178i $$0.617024\pi$$
$$572$$ 0 0
$$573$$ −21.4641 −0.896676
$$574$$ 4.00000i 0.166957i
$$575$$ −3.73205 −0.155637
$$576$$ −1.00000 −0.0416667
$$577$$ 10.0000i 0.416305i 0.978096 + 0.208153i $$0.0667451\pi$$
−0.978096 + 0.208153i $$0.933255\pi$$
$$578$$ 1.00000i 0.0415945i
$$579$$ − 11.3205i − 0.470464i
$$580$$ 0.267949i 0.0111260i
$$581$$ 17.8564 0.740809
$$582$$ 0.535898 0.0222137
$$583$$ 6.00000i 0.248495i
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ 31.8564 1.31598
$$587$$ − 2.39230i − 0.0987410i −0.998781 0.0493705i $$-0.984278\pi$$
0.998781 0.0493705i $$-0.0157215\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 12.9282 0.532697
$$590$$ 8.46410i 0.348462i
$$591$$ − 4.39230i − 0.180675i
$$592$$ 9.19615i 0.377960i
$$593$$ − 45.1051i − 1.85225i −0.377223 0.926123i $$-0.623121\pi$$
0.377223 0.926123i $$-0.376879\pi$$
$$594$$ −6.46410 −0.265225
$$595$$ −8.00000 −0.327968
$$596$$ − 13.5359i − 0.554452i
$$597$$ 5.07180 0.207575
$$598$$ 0 0
$$599$$ −10.3923 −0.424618 −0.212309 0.977203i $$-0.568098\pi$$
−0.212309 + 0.977203i $$0.568098\pi$$
$$600$$ 1.00000i 0.0408248i
$$601$$ −19.7846 −0.807031 −0.403516 0.914973i $$-0.632212\pi$$
−0.403516 + 0.914973i $$0.632212\pi$$
$$602$$ 23.8564 0.972315
$$603$$ − 11.4641i − 0.466854i
$$604$$ − 10.3923i − 0.422857i
$$605$$ 30.7846i 1.25157i
$$606$$ 2.92820i 0.118950i
$$607$$ −19.1769 −0.778367 −0.389183 0.921160i $$-0.627243\pi$$
−0.389183 + 0.921160i $$0.627243\pi$$
$$608$$ −7.46410 −0.302709
$$609$$ − 0.535898i − 0.0217157i
$$610$$ 10.3923 0.420772
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ − 39.0526i − 1.57732i −0.614831 0.788659i $$-0.710776\pi$$
0.614831 0.788659i $$-0.289224\pi$$
$$614$$ −19.4641 −0.785507
$$615$$ −2.00000 −0.0806478
$$616$$ − 12.9282i − 0.520892i
$$617$$ 22.4641i 0.904371i 0.891924 + 0.452185i $$0.149355\pi$$
−0.891924 + 0.452185i $$0.850645\pi$$
$$618$$ − 11.8564i − 0.476935i
$$619$$ 24.2487i 0.974638i 0.873224 + 0.487319i $$0.162025\pi$$
−0.873224 + 0.487319i $$0.837975\pi$$
$$620$$ −1.73205 −0.0695608
$$621$$ −3.73205 −0.149762
$$622$$ 28.3923i 1.13843i
$$623$$ 1.07180 0.0429406
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 28.0000i 1.11911i
$$627$$ −48.2487 −1.92687
$$628$$ 5.00000 0.199522
$$629$$ − 36.7846i − 1.46670i
$$630$$ − 2.00000i − 0.0796819i
$$631$$ − 31.4641i − 1.25257i −0.779596 0.626283i $$-0.784575\pi$$
0.779596 0.626283i $$-0.215425\pi$$
$$632$$ − 13.9282i − 0.554034i
$$633$$ 11.3205 0.449950
$$634$$ −14.5359 −0.577294
$$635$$ 4.92820i 0.195570i
$$636$$ 0.928203 0.0368057
$$637$$ 0 0
$$638$$ 1.73205 0.0685725
$$639$$ 12.3923i 0.490232i
$$640$$ 1.00000 0.0395285
$$641$$ 0.143594 0.00567160 0.00283580 0.999996i $$-0.499097\pi$$
0.00283580 + 0.999996i $$0.499097\pi$$
$$642$$ − 7.85641i − 0.310068i
$$643$$ 20.5359i 0.809857i 0.914348 + 0.404928i $$0.132704\pi$$
−0.914348 + 0.404928i $$0.867296\pi$$
$$644$$ − 7.46410i − 0.294127i
$$645$$ 11.9282i 0.469673i
$$646$$ 29.8564 1.17468
$$647$$ −13.3205 −0.523683 −0.261842 0.965111i $$-0.584330\pi$$
−0.261842 + 0.965111i $$0.584330\pi$$
$$648$$ 1.00000i 0.0392837i
$$649$$ 54.7128 2.14767
$$650$$ 0 0
$$651$$ 3.46410 0.135769
$$652$$ 15.0526i 0.589504i
$$653$$ 4.24871 0.166265 0.0831325 0.996539i $$-0.473508\pi$$
0.0831325 + 0.996539i $$0.473508\pi$$
$$654$$ −15.8564 −0.620035
$$655$$ 18.6603i 0.729116i
$$656$$ 2.00000i 0.0780869i
$$657$$ − 2.00000i − 0.0780274i
$$658$$ 7.07180i 0.275687i
$$659$$ −0.267949 −0.0104378 −0.00521891 0.999986i $$-0.501661\pi$$
−0.00521891 + 0.999986i $$0.501661\pi$$
$$660$$ 6.46410 0.251615
$$661$$ − 8.67949i − 0.337593i −0.985651 0.168797i $$-0.946012\pi$$
0.985651 0.168797i $$-0.0539881\pi$$
$$662$$ 16.7846 0.652352
$$663$$ 0 0
$$664$$ 8.92820 0.346481
$$665$$ − 14.9282i − 0.578891i
$$666$$ 9.19615 0.356344
$$667$$ 1.00000 0.0387202
$$668$$ − 16.3205i − 0.631459i
$$669$$ 20.5359i 0.793964i
$$670$$ 11.4641i 0.442897i
$$671$$ − 67.1769i − 2.59334i
$$672$$ −2.00000 −0.0771517
$$673$$ 32.0000 1.23351 0.616755 0.787155i $$-0.288447\pi$$
0.616755 + 0.787155i $$0.288447\pi$$
$$674$$ − 9.32051i − 0.359013i
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 32.3923 1.24494 0.622469 0.782645i $$-0.286130\pi$$
0.622469 + 0.782645i $$0.286130\pi$$
$$678$$ 0.803848i 0.0308716i
$$679$$ 1.07180 0.0411318
$$680$$ −4.00000 −0.153393
$$681$$ − 16.3923i − 0.628154i
$$682$$ 11.1962i 0.428723i
$$683$$ 40.7846i 1.56058i 0.625418 + 0.780290i $$0.284928\pi$$
−0.625418 + 0.780290i $$0.715072\pi$$
$$684$$ 7.46410i 0.285397i
$$685$$ 2.46410 0.0941485
$$686$$ 20.0000 0.763604
$$687$$ − 7.85641i − 0.299741i
$$688$$ 11.9282 0.454758
$$689$$ 0 0
$$690$$ 3.73205 0.142077
$$691$$ − 27.1769i − 1.03386i −0.856028 0.516929i $$-0.827075\pi$$
0.856028 0.516929i $$-0.172925\pi$$
$$692$$ −10.9282 −0.415428
$$693$$ −12.9282 −0.491102
$$694$$ − 1.60770i − 0.0610273i
$$695$$ 12.9282i 0.490395i
$$696$$ − 0.267949i − 0.0101566i
$$697$$ − 8.00000i − 0.303022i
$$698$$ 21.4641 0.812428
$$699$$ −6.12436 −0.231644
$$700$$ 2.00000i 0.0755929i
$$701$$ −0.267949 −0.0101203 −0.00506015 0.999987i $$-0.501611\pi$$
−0.00506015 + 0.999987i $$0.501611\pi$$
$$702$$ 0 0
$$703$$ 68.6410 2.58884
$$704$$ − 6.46410i − 0.243625i
$$705$$ −3.53590 −0.133170
$$706$$ 2.00000 0.0752710
$$707$$ 5.85641i 0.220253i
$$708$$ − 8.46410i − 0.318100i
$$709$$ 22.9282i 0.861087i 0.902570 + 0.430543i $$0.141678\pi$$
−0.902570 + 0.430543i $$0.858322\pi$$
$$710$$ − 12.3923i − 0.465075i
$$711$$ −13.9282 −0.522348
$$712$$ 0.535898 0.0200836
$$713$$ 6.46410i 0.242083i
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ 19.7321 0.737421
$$717$$ − 16.3923i − 0.612182i
$$718$$ −5.07180 −0.189278
$$719$$ −34.6410 −1.29189 −0.645946 0.763383i $$-0.723537\pi$$
−0.645946 + 0.763383i $$0.723537\pi$$
$$720$$ − 1.00000i − 0.0372678i
$$721$$ − 23.7128i − 0.883111i
$$722$$ 36.7128i 1.36631i
$$723$$ 17.7321i 0.659462i
$$724$$ 2.92820 0.108826
$$725$$ −0.267949 −0.00995138
$$726$$ − 30.7846i − 1.14252i
$$727$$ −31.7128 −1.17616 −0.588082 0.808802i $$-0.700117\pi$$
−0.588082 + 0.808802i $$0.700117\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000i 0.0740233i
$$731$$ −47.7128 −1.76472
$$732$$ −10.3923 −0.384111
$$733$$ 50.9282i 1.88108i 0.339688 + 0.940538i $$0.389678\pi$$
−0.339688 + 0.940538i $$0.610322\pi$$
$$734$$ − 15.6077i − 0.576091i
$$735$$ 3.00000i 0.110657i
$$736$$ − 3.73205i − 0.137565i
$$737$$ 74.1051 2.72970
$$738$$ 2.00000 0.0736210
$$739$$ − 19.3205i − 0.710716i −0.934730 0.355358i $$-0.884359\pi$$
0.934730 0.355358i $$-0.115641\pi$$
$$740$$ −9.19615 −0.338057
$$741$$ 0 0
$$742$$ 1.85641 0.0681508
$$743$$ − 40.4641i − 1.48448i −0.670132 0.742242i $$-0.733763\pi$$
0.670132 0.742242i $$-0.266237\pi$$
$$744$$ 1.73205 0.0635001
$$745$$ 13.5359 0.495917
$$746$$ 15.7846i 0.577916i
$$747$$ − 8.92820i − 0.326666i
$$748$$ 25.8564i 0.945404i
$$749$$ − 15.7128i − 0.574134i
$$750$$ −1.00000 −0.0365148
$$751$$ 14.0718 0.513487 0.256744 0.966480i $$-0.417350\pi$$
0.256744 + 0.966480i $$0.417350\pi$$
$$752$$ 3.53590i 0.128941i
$$753$$ −15.7321 −0.573308
$$754$$ 0 0
$$755$$ 10.3923 0.378215
$$756$$ 2.00000i 0.0727393i
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ 27.8564 1.01179
$$759$$ − 24.1244i − 0.875659i
$$760$$ − 7.46410i − 0.270751i
$$761$$ − 5.07180i − 0.183852i −0.995766 0.0919262i $$-0.970698\pi$$
0.995766 0.0919262i $$-0.0293024\pi$$
$$762$$ − 4.92820i − 0.178530i
$$763$$ −31.7128 −1.14808
$$764$$ −21.4641 −0.776544
$$765$$ 4.00000i 0.144620i
$$766$$ −25.3923 −0.917461
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ − 11.5885i − 0.417890i −0.977927 0.208945i $$-0.932997\pi$$
0.977927 0.208945i $$-0.0670030\pi$$
$$770$$ 12.9282 0.465900
$$771$$ 5.33975 0.192306
$$772$$ − 11.3205i − 0.407434i
$$773$$ − 27.7128i − 0.996761i −0.866959 0.498380i $$-0.833928\pi$$
0.866959 0.498380i $$-0.166072\pi$$
$$774$$ − 11.9282i − 0.428750i
$$775$$ − 1.73205i − 0.0622171i
$$776$$ 0.535898 0.0192376
$$777$$ 18.3923 0.659820
$$778$$ 23.7321i 0.850835i
$$779$$ 14.9282 0.534858
$$780$$ 0 0
$$781$$ −80.1051 −2.86639
$$782$$ 14.9282i 0.533831i
$$783$$ −0.267949 −0.00957572
$$784$$ 3.00000 0.107143
$$785$$ 5.00000i 0.178458i
$$786$$ − 18.6603i − 0.665589i
$$787$$ − 1.73205i − 0.0617409i −0.999523 0.0308705i $$-0.990172\pi$$
0.999523 0.0308705i $$-0.00982794\pi$$
$$788$$ − 4.39230i − 0.156469i
$$789$$ −6.12436 −0.218033
$$790$$ 13.9282 0.495543
$$791$$ 1.60770i 0.0571631i
$$792$$ −6.46410 −0.229692
$$793$$ 0 0
$$794$$ −12.1244 −0.430277
$$795$$ 0.928203i 0.0329200i
$$796$$ 5.07180 0.179765
$$797$$ −10.1436 −0.359305 −0.179652 0.983730i $$-0.557497\pi$$
−0.179652 + 0.983730i $$0.557497\pi$$
$$798$$ 14.9282i 0.528453i
$$799$$ − 14.1436i − 0.500364i
$$800$$ 1.00000i 0.0353553i
$$801$$ − 0.535898i − 0.0189350i
$$802$$ −32.0000 −1.12996
$$803$$ 12.9282 0.456226
$$804$$ − 11.4641i − 0.404308i
$$805$$ 7.46410 0.263075
$$806$$ 0 0
$$807$$ −12.0000 −0.422420
$$808$$ 2.92820i 0.103014i
$$809$$ −42.7846 −1.50423 −0.752113 0.659034i $$-0.770965\pi$$
−0.752113 + 0.659034i $$0.770965\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ − 40.7846i − 1.43214i −0.698028 0.716071i $$-0.745939\pi$$
0.698028 0.716071i $$-0.254061\pi$$
$$812$$ − 0.535898i − 0.0188063i
$$813$$ − 1.19615i − 0.0419509i
$$814$$ 59.4449i 2.08354i
$$815$$ −15.0526 −0.527268
$$816$$ 4.00000 0.140028
$$817$$ − 89.0333i − 3.11488i
$$818$$ −4.00000 −0.139857
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ 14.6077i 0.509812i 0.966966 + 0.254906i $$0.0820445\pi$$
−0.966966 + 0.254906i $$0.917955\pi$$
$$822$$ −2.46410 −0.0859454
$$823$$ 39.1769 1.36562 0.682811 0.730595i $$-0.260757\pi$$
0.682811 + 0.730595i $$0.260757\pi$$
$$824$$ − 11.8564i − 0.413037i
$$825$$ 6.46410i 0.225051i
$$826$$ − 16.9282i − 0.589008i
$$827$$ − 17.3205i − 0.602293i −0.953578 0.301147i $$-0.902631\pi$$
0.953578 0.301147i $$-0.0973693\pi$$
$$828$$ −3.73205 −0.129698
$$829$$ 16.5359 0.574315 0.287158 0.957883i $$-0.407290\pi$$
0.287158 + 0.957883i $$0.407290\pi$$
$$830$$ 8.92820i 0.309902i
$$831$$ 3.92820 0.136268
$$832$$ 0 0
$$833$$ −12.0000 −0.415775
$$834$$ − 12.9282i − 0.447667i
$$835$$ 16.3205 0.564794
$$836$$ −48.2487 −1.66872
$$837$$ − 1.73205i − 0.0598684i
$$838$$ − 22.3923i − 0.773529i
$$839$$ 43.5692i 1.50418i 0.659062 + 0.752088i $$0.270953\pi$$
−0.659062 + 0.752088i $$0.729047\pi$$
$$840$$ − 2.00000i − 0.0690066i
$$841$$ −28.9282 −0.997524
$$842$$ −4.39230 −0.151369
$$843$$ − 8.92820i − 0.307504i
$$844$$ 11.3205 0.389668
$$845$$ 0 0
$$846$$ 3.53590 0.121567
$$847$$ − 61.5692i − 2.11554i
$$848$$ 0.928203 0.0318746
$$849$$ 9.92820 0.340735
$$850$$ − 4.00000i − 0.137199i
$$851$$ 34.3205i 1.17649i
$$852$$ 12.3923i 0.424553i
$$853$$ − 43.8372i − 1.50096i −0.660895 0.750478i $$-0.729823\pi$$
0.660895 0.750478i $$-0.270177\pi$$
$$854$$ −20.7846 −0.711235
$$855$$ −7.46410 −0.255267
$$856$$ − 7.85641i − 0.268526i
$$857$$ 24.5167 0.837473 0.418737 0.908108i $$-0.362473\pi$$
0.418737 + 0.908108i $$0.362473\pi$$
$$858$$ 0 0
$$859$$ −35.1769 −1.20022 −0.600110 0.799917i $$-0.704877\pi$$
−0.600110 + 0.799917i $$0.704877\pi$$
$$860$$ 11.9282i 0.406748i
$$861$$ 4.00000 0.136320
$$862$$ 28.3923 0.967046
$$863$$ − 35.5359i − 1.20966i −0.796356 0.604828i $$-0.793242\pi$$
0.796356 0.604828i $$-0.206758\pi$$
$$864$$ 1.00000i 0.0340207i
$$865$$ − 10.9282i − 0.371570i
$$866$$ − 19.3205i − 0.656538i
$$867$$ 1.00000 0.0339618
$$868$$ 3.46410 0.117579
$$869$$ − 90.0333i − 3.05417i
$$870$$ 0.267949 0.00908433
$$871$$ 0 0
$$872$$ −15.8564 −0.536966
$$873$$ − 0.535898i − 0.0181374i
$$874$$ −27.8564 −0.942257
$$875$$ −2.00000 −0.0676123
$$876$$ − 2.00000i − 0.0675737i
$$877$$ 3.87564i 0.130871i 0.997857 + 0.0654356i $$0.0208437\pi$$
−0.997857 + 0.0654356i $$0.979156\pi$$
$$878$$ 17.8564i 0.602625i
$$879$$ − 31.8564i − 1.07449i
$$880$$ 6.46410 0.217905
$$881$$ −14.0000 −0.471672 −0.235836 0.971793i $$-0.575783\pi$$
−0.235836 + 0.971793i $$0.575783\pi$$
$$882$$ − 3.00000i − 0.101015i
$$883$$ 29.9282 1.00716 0.503582 0.863947i $$-0.332015\pi$$
0.503582 + 0.863947i $$0.332015\pi$$
$$884$$ 0 0
$$885$$ 8.46410 0.284518
$$886$$ − 16.3923i − 0.550710i
$$887$$ 14.1244 0.474249 0.237125 0.971479i $$-0.423795\pi$$
0.237125 + 0.971479i $$0.423795\pi$$
$$888$$ 9.19615 0.308603
$$889$$ − 9.85641i − 0.330573i
$$890$$ 0.535898i 0.0179634i
$$891$$ 6.46410i 0.216556i
$$892$$ 20.5359i 0.687593i
$$893$$ 26.3923 0.883185
$$894$$ −13.5359 −0.452708
$$895$$ 19.7321i 0.659570i
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ 15.7128 0.524343
$$899$$ 0.464102i 0.0154787i
$$900$$ 1.00000 0.0333333
$$901$$ −3.71281 −0.123692
$$902$$ 12.9282i 0.430462i
$$903$$ − 23.8564i − 0.793891i
$$904$$ 0.803848i 0.0267356i
$$905$$ 2.92820i 0.0973368i
$$906$$ −10.3923 −0.345261
$$907$$ −20.8564 −0.692526 −0.346263 0.938138i $$-0.612549\pi$$
−0.346263 + 0.938138i $$0.612549\pi$$
$$908$$ − 16.3923i − 0.543998i
$$909$$ 2.92820 0.0971224
$$910$$ 0 0
$$911$$ 24.2487 0.803396 0.401698 0.915772i $$-0.368420\pi$$
0.401698 + 0.915772i $$0.368420\pi$$
$$912$$ 7.46410i 0.247161i
$$913$$ 57.7128 1.91002
$$914$$ 24.5359 0.811575
$$915$$ − 10.3923i − 0.343559i
$$916$$ − 7.85641i − 0.259583i
$$917$$ − 37.3205i − 1.23243i
$$918$$ − 4.00000i − 0.132020i
$$919$$ 14.6410 0.482963 0.241481 0.970405i $$-0.422367\pi$$
0.241481 + 0.970405i $$0.422367\pi$$
$$920$$ 3.73205 0.123042
$$921$$ 19.4641i 0.641364i
$$922$$ −0.464102 −0.0152844
$$923$$ 0 0
$$924$$ −12.9282 −0.425307
$$925$$ − 9.19615i − 0.302368i
$$926$$ −7.07180 −0.232394
$$927$$ −11.8564 −0.389415
$$928$$ − 0.267949i − 0.00879586i
$$929$$ 6.14359i 0.201565i 0.994908 + 0.100782i $$0.0321346\pi$$
−0.994908 + 0.100782i $$0.967865\pi$$
$$930$$ 1.73205i 0.0567962i
$$931$$ − 22.3923i − 0.733878i
$$932$$ −6.12436 −0.200610
$$933$$ 28.3923 0.929522
$$934$$ 15.8564i 0.518837i
$$935$$ −25.8564 −0.845595
$$936$$ 0 0
$$937$$ −24.6410 −0.804987 −0.402493 0.915423i $$-0.631856\pi$$
−0.402493 + 0.915423i $$0.631856\pi$$
$$938$$ − 22.9282i − 0.748632i
$$939$$ 28.0000 0.913745
$$940$$ −3.53590 −0.115328
$$941$$ 14.7846i 0.481965i 0.970530 + 0.240982i $$0.0774696\pi$$
−0.970530 + 0.240982i $$0.922530\pi$$
$$942$$ − 5.00000i − 0.162909i
$$943$$ 7.46410i 0.243065i
$$944$$ − 8.46410i − 0.275483i
$$945$$ −2.00000 −0.0650600
$$946$$ 77.1051 2.50690
$$947$$ 9.46410i 0.307542i 0.988107 + 0.153771i $$0.0491418\pi$$
−0.988107 + 0.153771i $$0.950858\pi$$
$$948$$ −13.9282 −0.452367
$$949$$ 0 0
$$950$$ 7.46410 0.242167
$$951$$ 14.5359i 0.471359i
$$952$$ 8.00000 0.259281
$$953$$ −12.2679 −0.397398 −0.198699 0.980061i $$-0.563672\pi$$
−0.198699 + 0.980061i $$0.563672\pi$$
$$954$$ − 0.928203i − 0.0300517i
$$955$$ − 21.4641i − 0.694562i
$$956$$ − 16.3923i − 0.530165i
$$957$$ − 1.73205i − 0.0559893i
$$958$$ 5.46410 0.176537
$$959$$ −4.92820 −0.159140
$$960$$ − 1.00000i − 0.0322749i
$$961$$ 28.0000 0.903226
$$962$$ 0 0
$$963$$ −7.85641 −0.253169
$$964$$ 17.7321i 0.571111i
$$965$$ 11.3205 0.364420
$$966$$ −7.46410 −0.240154
$$967$$ 41.4641i 1.33340i 0.745328 + 0.666698i $$0.232293\pi$$
−0.745328 + 0.666698i $$0.767707\pi$$
$$968$$ − 30.7846i − 0.989455i
$$969$$ − 29.8564i − 0.959126i
$$970$$ 0.535898i 0.0172067i
$$971$$ 4.53590 0.145564 0.0727820 0.997348i $$-0.476812\pi$$
0.0727820 + 0.997348i $$0.476812\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ − 25.8564i − 0.828918i
$$974$$ −23.1769 −0.742636
$$975$$ 0 0
$$976$$ −10.3923 −0.332650
$$977$$ 12.6077i 0.403356i 0.979452 + 0.201678i $$0.0646394\pi$$
−0.979452 + 0.201678i $$0.935361\pi$$
$$978$$ 15.0526 0.481328
$$979$$ 3.46410 0.110713
$$980$$ 3.00000i 0.0958315i
$$981$$ 15.8564i 0.506256i
$$982$$ 17.3205i 0.552720i
$$983$$ 36.6077i 1.16760i 0.811896 + 0.583802i $$0.198436\pi$$
−0.811896 + 0.583802i $$0.801564\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 4.39230 0.139950
$$986$$ 1.07180i 0.0341330i
$$987$$ 7.07180 0.225098
$$988$$ 0 0
$$989$$ 44.5167 1.41555
$$990$$ − 6.46410i − 0.205443i
$$991$$ −52.8564 −1.67904 −0.839520 0.543329i $$-0.817163\pi$$
−0.839520 + 0.543329i $$0.817163\pi$$
$$992$$ 1.73205 0.0549927
$$993$$ − 16.7846i − 0.532643i
$$994$$ 24.7846i 0.786120i
$$995$$ 5.07180i 0.160787i
$$996$$ − 8.92820i − 0.282901i
$$997$$ 35.5692 1.12649 0.563244 0.826290i $$-0.309553\pi$$
0.563244 + 0.826290i $$0.309553\pi$$
$$998$$ −6.53590 −0.206890
$$999$$ − 9.19615i − 0.290953i
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.o.1351.2 4
13.5 odd 4 5070.2.a.y.1.1 2
13.8 odd 4 5070.2.a.bg.1.2 2
13.9 even 3 390.2.bb.b.361.1 yes 4
13.10 even 6 390.2.bb.b.121.1 4
13.12 even 2 inner 5070.2.b.o.1351.3 4
39.23 odd 6 1170.2.bs.e.901.2 4
39.35 odd 6 1170.2.bs.e.361.2 4
65.9 even 6 1950.2.bc.b.751.2 4
65.22 odd 12 1950.2.y.c.49.2 4
65.23 odd 12 1950.2.y.c.199.2 4
65.48 odd 12 1950.2.y.f.49.1 4
65.49 even 6 1950.2.bc.b.901.2 4
65.62 odd 12 1950.2.y.f.199.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.1 4 13.10 even 6
390.2.bb.b.361.1 yes 4 13.9 even 3
1170.2.bs.e.361.2 4 39.35 odd 6
1170.2.bs.e.901.2 4 39.23 odd 6
1950.2.y.c.49.2 4 65.22 odd 12
1950.2.y.c.199.2 4 65.23 odd 12
1950.2.y.f.49.1 4 65.48 odd 12
1950.2.y.f.199.1 4 65.62 odd 12
1950.2.bc.b.751.2 4 65.9 even 6
1950.2.bc.b.901.2 4 65.49 even 6
5070.2.a.y.1.1 2 13.5 odd 4
5070.2.a.bg.1.2 2 13.8 odd 4
5070.2.b.o.1351.2 4 1.1 even 1 trivial
5070.2.b.o.1351.3 4 13.12 even 2 inner