# Properties

 Label 7098.2.a.bv.1.1 Level $7098$ Weight $2$ Character 7098.1 Self dual yes Analytic conductor $56.678$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7098.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.6778153547$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 7098.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -1.43845 q^{11} -1.00000 q^{12} +1.00000 q^{14} +0.561553 q^{15} +1.00000 q^{16} -5.68466 q^{17} +1.00000 q^{18} +2.56155 q^{19} -0.561553 q^{20} -1.00000 q^{21} -1.43845 q^{22} -5.68466 q^{23} -1.00000 q^{24} -4.68466 q^{25} -1.00000 q^{27} +1.00000 q^{28} -2.56155 q^{29} +0.561553 q^{30} +10.2462 q^{31} +1.00000 q^{32} +1.43845 q^{33} -5.68466 q^{34} -0.561553 q^{35} +1.00000 q^{36} +1.68466 q^{37} +2.56155 q^{38} -0.561553 q^{40} +4.00000 q^{41} -1.00000 q^{42} +10.5616 q^{43} -1.43845 q^{44} -0.561553 q^{45} -5.68466 q^{46} -6.24621 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.68466 q^{50} +5.68466 q^{51} +13.1231 q^{53} -1.00000 q^{54} +0.807764 q^{55} +1.00000 q^{56} -2.56155 q^{57} -2.56155 q^{58} +12.2462 q^{59} +0.561553 q^{60} +2.56155 q^{61} +10.2462 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.43845 q^{66} +7.12311 q^{67} -5.68466 q^{68} +5.68466 q^{69} -0.561553 q^{70} -15.3693 q^{71} +1.00000 q^{72} -7.43845 q^{73} +1.68466 q^{74} +4.68466 q^{75} +2.56155 q^{76} -1.43845 q^{77} +16.0000 q^{79} -0.561553 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +3.19224 q^{85} +10.5616 q^{86} +2.56155 q^{87} -1.43845 q^{88} +8.00000 q^{89} -0.561553 q^{90} -5.68466 q^{92} -10.2462 q^{93} -6.24621 q^{94} -1.43845 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -1.43845 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 3q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 3q^{5} - 2q^{6} + 2q^{7} + 2q^{8} + 2q^{9} + 3q^{10} - 7q^{11} - 2q^{12} + 2q^{14} - 3q^{15} + 2q^{16} + q^{17} + 2q^{18} + q^{19} + 3q^{20} - 2q^{21} - 7q^{22} + q^{23} - 2q^{24} + 3q^{25} - 2q^{27} + 2q^{28} - q^{29} - 3q^{30} + 4q^{31} + 2q^{32} + 7q^{33} + q^{34} + 3q^{35} + 2q^{36} - 9q^{37} + q^{38} + 3q^{40} + 8q^{41} - 2q^{42} + 17q^{43} - 7q^{44} + 3q^{45} + q^{46} + 4q^{47} - 2q^{48} + 2q^{49} + 3q^{50} - q^{51} + 18q^{53} - 2q^{54} - 19q^{55} + 2q^{56} - q^{57} - q^{58} + 8q^{59} - 3q^{60} + q^{61} + 4q^{62} + 2q^{63} + 2q^{64} + 7q^{66} + 6q^{67} + q^{68} - q^{69} + 3q^{70} - 6q^{71} + 2q^{72} - 19q^{73} - 9q^{74} - 3q^{75} + q^{76} - 7q^{77} + 32q^{79} + 3q^{80} + 2q^{81} + 8q^{82} - 4q^{83} - 2q^{84} + 27q^{85} + 17q^{86} + q^{87} - 7q^{88} + 16q^{89} + 3q^{90} + q^{92} - 4q^{93} + 4q^{94} - 7q^{95} - 2q^{96} + 20q^{97} + 2q^{98} - 7q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −0.561553 −0.251134 −0.125567 0.992085i $$-0.540075\pi$$
−0.125567 + 0.992085i $$0.540075\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −0.561553 −0.177579
$$11$$ −1.43845 −0.433708 −0.216854 0.976204i $$-0.569580\pi$$
−0.216854 + 0.976204i $$0.569580\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 1.00000 0.267261
$$15$$ 0.561553 0.144992
$$16$$ 1.00000 0.250000
$$17$$ −5.68466 −1.37873 −0.689366 0.724413i $$-0.742111\pi$$
−0.689366 + 0.724413i $$0.742111\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.56155 0.587661 0.293830 0.955858i $$-0.405070\pi$$
0.293830 + 0.955858i $$0.405070\pi$$
$$20$$ −0.561553 −0.125567
$$21$$ −1.00000 −0.218218
$$22$$ −1.43845 −0.306678
$$23$$ −5.68466 −1.18533 −0.592667 0.805448i $$-0.701925\pi$$
−0.592667 + 0.805448i $$0.701925\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ −4.68466 −0.936932
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ −2.56155 −0.475668 −0.237834 0.971306i $$-0.576437\pi$$
−0.237834 + 0.971306i $$0.576437\pi$$
$$30$$ 0.561553 0.102525
$$31$$ 10.2462 1.84027 0.920137 0.391597i $$-0.128077\pi$$
0.920137 + 0.391597i $$0.128077\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.43845 0.250402
$$34$$ −5.68466 −0.974911
$$35$$ −0.561553 −0.0949197
$$36$$ 1.00000 0.166667
$$37$$ 1.68466 0.276956 0.138478 0.990366i $$-0.455779\pi$$
0.138478 + 0.990366i $$0.455779\pi$$
$$38$$ 2.56155 0.415539
$$39$$ 0 0
$$40$$ −0.561553 −0.0887893
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 10.5616 1.61062 0.805311 0.592853i $$-0.201998\pi$$
0.805311 + 0.592853i $$0.201998\pi$$
$$44$$ −1.43845 −0.216854
$$45$$ −0.561553 −0.0837114
$$46$$ −5.68466 −0.838157
$$47$$ −6.24621 −0.911104 −0.455552 0.890209i $$-0.650558\pi$$
−0.455552 + 0.890209i $$0.650558\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ −4.68466 −0.662511
$$51$$ 5.68466 0.796011
$$52$$ 0 0
$$53$$ 13.1231 1.80260 0.901299 0.433198i $$-0.142615\pi$$
0.901299 + 0.433198i $$0.142615\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0.807764 0.108919
$$56$$ 1.00000 0.133631
$$57$$ −2.56155 −0.339286
$$58$$ −2.56155 −0.336348
$$59$$ 12.2462 1.59432 0.797160 0.603768i $$-0.206334\pi$$
0.797160 + 0.603768i $$0.206334\pi$$
$$60$$ 0.561553 0.0724962
$$61$$ 2.56155 0.327973 0.163987 0.986463i $$-0.447565\pi$$
0.163987 + 0.986463i $$0.447565\pi$$
$$62$$ 10.2462 1.30127
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.43845 0.177061
$$67$$ 7.12311 0.870226 0.435113 0.900376i $$-0.356708\pi$$
0.435113 + 0.900376i $$0.356708\pi$$
$$68$$ −5.68466 −0.689366
$$69$$ 5.68466 0.684352
$$70$$ −0.561553 −0.0671184
$$71$$ −15.3693 −1.82400 −0.912001 0.410188i $$-0.865463\pi$$
−0.912001 + 0.410188i $$0.865463\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −7.43845 −0.870604 −0.435302 0.900284i $$-0.643358\pi$$
−0.435302 + 0.900284i $$0.643358\pi$$
$$74$$ 1.68466 0.195838
$$75$$ 4.68466 0.540938
$$76$$ 2.56155 0.293830
$$77$$ −1.43845 −0.163926
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ −0.561553 −0.0627835
$$81$$ 1.00000 0.111111
$$82$$ 4.00000 0.441726
$$83$$ −2.00000 −0.219529 −0.109764 0.993958i $$-0.535010\pi$$
−0.109764 + 0.993958i $$0.535010\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 3.19224 0.346247
$$86$$ 10.5616 1.13888
$$87$$ 2.56155 0.274627
$$88$$ −1.43845 −0.153339
$$89$$ 8.00000 0.847998 0.423999 0.905663i $$-0.360626\pi$$
0.423999 + 0.905663i $$0.360626\pi$$
$$90$$ −0.561553 −0.0591929
$$91$$ 0 0
$$92$$ −5.68466 −0.592667
$$93$$ −10.2462 −1.06248
$$94$$ −6.24621 −0.644247
$$95$$ −1.43845 −0.147582
$$96$$ −1.00000 −0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −1.43845 −0.144569
$$100$$ −4.68466 −0.468466
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 5.68466 0.562865
$$103$$ 18.8078 1.85318 0.926592 0.376068i $$-0.122724\pi$$
0.926592 + 0.376068i $$0.122724\pi$$
$$104$$ 0 0
$$105$$ 0.561553 0.0548019
$$106$$ 13.1231 1.27463
$$107$$ −13.1231 −1.26866 −0.634329 0.773063i $$-0.718724\pi$$
−0.634329 + 0.773063i $$0.718724\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −15.9309 −1.52590 −0.762950 0.646457i $$-0.776250\pi$$
−0.762950 + 0.646457i $$0.776250\pi$$
$$110$$ 0.807764 0.0770173
$$111$$ −1.68466 −0.159901
$$112$$ 1.00000 0.0944911
$$113$$ 3.75379 0.353127 0.176563 0.984289i $$-0.443502\pi$$
0.176563 + 0.984289i $$0.443502\pi$$
$$114$$ −2.56155 −0.239911
$$115$$ 3.19224 0.297678
$$116$$ −2.56155 −0.237834
$$117$$ 0 0
$$118$$ 12.2462 1.12736
$$119$$ −5.68466 −0.521112
$$120$$ 0.561553 0.0512625
$$121$$ −8.93087 −0.811897
$$122$$ 2.56155 0.231912
$$123$$ −4.00000 −0.360668
$$124$$ 10.2462 0.920137
$$125$$ 5.43845 0.486430
$$126$$ 1.00000 0.0890871
$$127$$ 6.24621 0.554262 0.277131 0.960832i $$-0.410616\pi$$
0.277131 + 0.960832i $$0.410616\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −10.5616 −0.929893
$$130$$ 0 0
$$131$$ 2.56155 0.223804 0.111902 0.993719i $$-0.464306\pi$$
0.111902 + 0.993719i $$0.464306\pi$$
$$132$$ 1.43845 0.125201
$$133$$ 2.56155 0.222115
$$134$$ 7.12311 0.615343
$$135$$ 0.561553 0.0483308
$$136$$ −5.68466 −0.487455
$$137$$ 5.68466 0.485673 0.242837 0.970067i $$-0.421922\pi$$
0.242837 + 0.970067i $$0.421922\pi$$
$$138$$ 5.68466 0.483910
$$139$$ −6.24621 −0.529797 −0.264898 0.964276i $$-0.585338\pi$$
−0.264898 + 0.964276i $$0.585338\pi$$
$$140$$ −0.561553 −0.0474599
$$141$$ 6.24621 0.526026
$$142$$ −15.3693 −1.28976
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 1.43845 0.119457
$$146$$ −7.43845 −0.615610
$$147$$ −1.00000 −0.0824786
$$148$$ 1.68466 0.138478
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 4.68466 0.382501
$$151$$ −4.31534 −0.351178 −0.175589 0.984464i $$-0.556183\pi$$
−0.175589 + 0.984464i $$0.556183\pi$$
$$152$$ 2.56155 0.207769
$$153$$ −5.68466 −0.459577
$$154$$ −1.43845 −0.115913
$$155$$ −5.75379 −0.462155
$$156$$ 0 0
$$157$$ 8.80776 0.702936 0.351468 0.936200i $$-0.385683\pi$$
0.351468 + 0.936200i $$0.385683\pi$$
$$158$$ 16.0000 1.27289
$$159$$ −13.1231 −1.04073
$$160$$ −0.561553 −0.0443946
$$161$$ −5.68466 −0.448014
$$162$$ 1.00000 0.0785674
$$163$$ −0.876894 −0.0686837 −0.0343418 0.999410i $$-0.510933\pi$$
−0.0343418 + 0.999410i $$0.510933\pi$$
$$164$$ 4.00000 0.312348
$$165$$ −0.807764 −0.0628843
$$166$$ −2.00000 −0.155230
$$167$$ −8.80776 −0.681565 −0.340783 0.940142i $$-0.610692\pi$$
−0.340783 + 0.940142i $$0.610692\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 0 0
$$170$$ 3.19224 0.244833
$$171$$ 2.56155 0.195887
$$172$$ 10.5616 0.805311
$$173$$ 20.2462 1.53929 0.769645 0.638471i $$-0.220433\pi$$
0.769645 + 0.638471i $$0.220433\pi$$
$$174$$ 2.56155 0.194191
$$175$$ −4.68466 −0.354127
$$176$$ −1.43845 −0.108427
$$177$$ −12.2462 −0.920482
$$178$$ 8.00000 0.599625
$$179$$ −2.87689 −0.215029 −0.107515 0.994204i $$-0.534289\pi$$
−0.107515 + 0.994204i $$0.534289\pi$$
$$180$$ −0.561553 −0.0418557
$$181$$ −11.3693 −0.845075 −0.422537 0.906346i $$-0.638860\pi$$
−0.422537 + 0.906346i $$0.638860\pi$$
$$182$$ 0 0
$$183$$ −2.56155 −0.189355
$$184$$ −5.68466 −0.419079
$$185$$ −0.946025 −0.0695531
$$186$$ −10.2462 −0.751289
$$187$$ 8.17708 0.597967
$$188$$ −6.24621 −0.455552
$$189$$ −1.00000 −0.0727393
$$190$$ −1.43845 −0.104356
$$191$$ 13.0540 0.944553 0.472276 0.881451i $$-0.343432\pi$$
0.472276 + 0.881451i $$0.343432\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 12.0000 0.863779 0.431889 0.901927i $$-0.357847\pi$$
0.431889 + 0.901927i $$0.357847\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −1.43845 −0.102226
$$199$$ 23.9309 1.69641 0.848207 0.529665i $$-0.177682\pi$$
0.848207 + 0.529665i $$0.177682\pi$$
$$200$$ −4.68466 −0.331255
$$201$$ −7.12311 −0.502425
$$202$$ −6.00000 −0.422159
$$203$$ −2.56155 −0.179786
$$204$$ 5.68466 0.398006
$$205$$ −2.24621 −0.156882
$$206$$ 18.8078 1.31040
$$207$$ −5.68466 −0.395111
$$208$$ 0 0
$$209$$ −3.68466 −0.254873
$$210$$ 0.561553 0.0387508
$$211$$ −12.8078 −0.881723 −0.440861 0.897575i $$-0.645327\pi$$
−0.440861 + 0.897575i $$0.645327\pi$$
$$212$$ 13.1231 0.901299
$$213$$ 15.3693 1.05309
$$214$$ −13.1231 −0.897077
$$215$$ −5.93087 −0.404482
$$216$$ −1.00000 −0.0680414
$$217$$ 10.2462 0.695558
$$218$$ −15.9309 −1.07897
$$219$$ 7.43845 0.502644
$$220$$ 0.807764 0.0544594
$$221$$ 0 0
$$222$$ −1.68466 −0.113067
$$223$$ −18.2462 −1.22186 −0.610928 0.791686i $$-0.709204\pi$$
−0.610928 + 0.791686i $$0.709204\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −4.68466 −0.312311
$$226$$ 3.75379 0.249698
$$227$$ 23.6155 1.56742 0.783709 0.621128i $$-0.213325\pi$$
0.783709 + 0.621128i $$0.213325\pi$$
$$228$$ −2.56155 −0.169643
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 3.19224 0.210490
$$231$$ 1.43845 0.0946429
$$232$$ −2.56155 −0.168174
$$233$$ −2.00000 −0.131024 −0.0655122 0.997852i $$-0.520868\pi$$
−0.0655122 + 0.997852i $$0.520868\pi$$
$$234$$ 0 0
$$235$$ 3.50758 0.228809
$$236$$ 12.2462 0.797160
$$237$$ −16.0000 −1.03931
$$238$$ −5.68466 −0.368482
$$239$$ −2.24621 −0.145295 −0.0726477 0.997358i $$-0.523145\pi$$
−0.0726477 + 0.997358i $$0.523145\pi$$
$$240$$ 0.561553 0.0362481
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ −8.93087 −0.574098
$$243$$ −1.00000 −0.0641500
$$244$$ 2.56155 0.163987
$$245$$ −0.561553 −0.0358763
$$246$$ −4.00000 −0.255031
$$247$$ 0 0
$$248$$ 10.2462 0.650635
$$249$$ 2.00000 0.126745
$$250$$ 5.43845 0.343958
$$251$$ 15.0540 0.950198 0.475099 0.879932i $$-0.342412\pi$$
0.475099 + 0.879932i $$0.342412\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 8.17708 0.514089
$$254$$ 6.24621 0.391922
$$255$$ −3.19224 −0.199906
$$256$$ 1.00000 0.0625000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ −10.5616 −0.657534
$$259$$ 1.68466 0.104680
$$260$$ 0 0
$$261$$ −2.56155 −0.158556
$$262$$ 2.56155 0.158253
$$263$$ 1.36932 0.0844357 0.0422178 0.999108i $$-0.486558\pi$$
0.0422178 + 0.999108i $$0.486558\pi$$
$$264$$ 1.43845 0.0885303
$$265$$ −7.36932 −0.452694
$$266$$ 2.56155 0.157059
$$267$$ −8.00000 −0.489592
$$268$$ 7.12311 0.435113
$$269$$ 6.63068 0.404280 0.202140 0.979357i $$-0.435210\pi$$
0.202140 + 0.979357i $$0.435210\pi$$
$$270$$ 0.561553 0.0341750
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ −5.68466 −0.344683
$$273$$ 0 0
$$274$$ 5.68466 0.343423
$$275$$ 6.73863 0.406355
$$276$$ 5.68466 0.342176
$$277$$ 19.1231 1.14900 0.574498 0.818506i $$-0.305197\pi$$
0.574498 + 0.818506i $$0.305197\pi$$
$$278$$ −6.24621 −0.374623
$$279$$ 10.2462 0.613425
$$280$$ −0.561553 −0.0335592
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 6.24621 0.371956
$$283$$ 13.1231 0.780088 0.390044 0.920796i $$-0.372460\pi$$
0.390044 + 0.920796i $$0.372460\pi$$
$$284$$ −15.3693 −0.912001
$$285$$ 1.43845 0.0852063
$$286$$ 0 0
$$287$$ 4.00000 0.236113
$$288$$ 1.00000 0.0589256
$$289$$ 15.3153 0.900902
$$290$$ 1.43845 0.0844685
$$291$$ −10.0000 −0.586210
$$292$$ −7.43845 −0.435302
$$293$$ 24.2462 1.41648 0.708239 0.705972i $$-0.249490\pi$$
0.708239 + 0.705972i $$0.249490\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ −6.87689 −0.400388
$$296$$ 1.68466 0.0979188
$$297$$ 1.43845 0.0834672
$$298$$ 10.0000 0.579284
$$299$$ 0 0
$$300$$ 4.68466 0.270469
$$301$$ 10.5616 0.608758
$$302$$ −4.31534 −0.248320
$$303$$ 6.00000 0.344691
$$304$$ 2.56155 0.146915
$$305$$ −1.43845 −0.0823652
$$306$$ −5.68466 −0.324970
$$307$$ −9.75379 −0.556678 −0.278339 0.960483i $$-0.589784\pi$$
−0.278339 + 0.960483i $$0.589784\pi$$
$$308$$ −1.43845 −0.0819631
$$309$$ −18.8078 −1.06994
$$310$$ −5.75379 −0.326793
$$311$$ −32.4924 −1.84248 −0.921238 0.388999i $$-0.872821\pi$$
−0.921238 + 0.388999i $$0.872821\pi$$
$$312$$ 0 0
$$313$$ 15.7538 0.890457 0.445228 0.895417i $$-0.353122\pi$$
0.445228 + 0.895417i $$0.353122\pi$$
$$314$$ 8.80776 0.497051
$$315$$ −0.561553 −0.0316399
$$316$$ 16.0000 0.900070
$$317$$ 21.3693 1.20022 0.600110 0.799917i $$-0.295123\pi$$
0.600110 + 0.799917i $$0.295123\pi$$
$$318$$ −13.1231 −0.735907
$$319$$ 3.68466 0.206301
$$320$$ −0.561553 −0.0313918
$$321$$ 13.1231 0.732460
$$322$$ −5.68466 −0.316794
$$323$$ −14.5616 −0.810226
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −0.876894 −0.0485667
$$327$$ 15.9309 0.880979
$$328$$ 4.00000 0.220863
$$329$$ −6.24621 −0.344365
$$330$$ −0.807764 −0.0444659
$$331$$ 7.12311 0.391521 0.195761 0.980652i $$-0.437282\pi$$
0.195761 + 0.980652i $$0.437282\pi$$
$$332$$ −2.00000 −0.109764
$$333$$ 1.68466 0.0923187
$$334$$ −8.80776 −0.481939
$$335$$ −4.00000 −0.218543
$$336$$ −1.00000 −0.0545545
$$337$$ 17.0540 0.928989 0.464495 0.885576i $$-0.346236\pi$$
0.464495 + 0.885576i $$0.346236\pi$$
$$338$$ 0 0
$$339$$ −3.75379 −0.203878
$$340$$ 3.19224 0.173123
$$341$$ −14.7386 −0.798142
$$342$$ 2.56155 0.138513
$$343$$ 1.00000 0.0539949
$$344$$ 10.5616 0.569441
$$345$$ −3.19224 −0.171864
$$346$$ 20.2462 1.08844
$$347$$ 8.49242 0.455897 0.227949 0.973673i $$-0.426798\pi$$
0.227949 + 0.973673i $$0.426798\pi$$
$$348$$ 2.56155 0.137314
$$349$$ 21.3693 1.14387 0.571937 0.820298i $$-0.306192\pi$$
0.571937 + 0.820298i $$0.306192\pi$$
$$350$$ −4.68466 −0.250406
$$351$$ 0 0
$$352$$ −1.43845 −0.0766695
$$353$$ −1.75379 −0.0933448 −0.0466724 0.998910i $$-0.514862\pi$$
−0.0466724 + 0.998910i $$0.514862\pi$$
$$354$$ −12.2462 −0.650879
$$355$$ 8.63068 0.458069
$$356$$ 8.00000 0.423999
$$357$$ 5.68466 0.300864
$$358$$ −2.87689 −0.152049
$$359$$ 17.6155 0.929712 0.464856 0.885386i $$-0.346106\pi$$
0.464856 + 0.885386i $$0.346106\pi$$
$$360$$ −0.561553 −0.0295964
$$361$$ −12.4384 −0.654655
$$362$$ −11.3693 −0.597558
$$363$$ 8.93087 0.468749
$$364$$ 0 0
$$365$$ 4.17708 0.218638
$$366$$ −2.56155 −0.133895
$$367$$ 25.3693 1.32427 0.662134 0.749386i $$-0.269651\pi$$
0.662134 + 0.749386i $$0.269651\pi$$
$$368$$ −5.68466 −0.296333
$$369$$ 4.00000 0.208232
$$370$$ −0.946025 −0.0491815
$$371$$ 13.1231 0.681318
$$372$$ −10.2462 −0.531241
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 8.17708 0.422827
$$375$$ −5.43845 −0.280840
$$376$$ −6.24621 −0.322124
$$377$$ 0 0
$$378$$ −1.00000 −0.0514344
$$379$$ −27.1231 −1.39322 −0.696610 0.717450i $$-0.745309\pi$$
−0.696610 + 0.717450i $$0.745309\pi$$
$$380$$ −1.43845 −0.0737908
$$381$$ −6.24621 −0.320003
$$382$$ 13.0540 0.667899
$$383$$ 13.4384 0.686673 0.343336 0.939213i $$-0.388443\pi$$
0.343336 + 0.939213i $$0.388443\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0.807764 0.0411675
$$386$$ 12.0000 0.610784
$$387$$ 10.5616 0.536874
$$388$$ 10.0000 0.507673
$$389$$ −18.8769 −0.957097 −0.478548 0.878061i $$-0.658837\pi$$
−0.478548 + 0.878061i $$0.658837\pi$$
$$390$$ 0 0
$$391$$ 32.3153 1.63426
$$392$$ 1.00000 0.0505076
$$393$$ −2.56155 −0.129213
$$394$$ −6.00000 −0.302276
$$395$$ −8.98485 −0.452077
$$396$$ −1.43845 −0.0722847
$$397$$ 9.36932 0.470233 0.235116 0.971967i $$-0.424453\pi$$
0.235116 + 0.971967i $$0.424453\pi$$
$$398$$ 23.9309 1.19955
$$399$$ −2.56155 −0.128238
$$400$$ −4.68466 −0.234233
$$401$$ −34.9848 −1.74706 −0.873530 0.486771i $$-0.838175\pi$$
−0.873530 + 0.486771i $$0.838175\pi$$
$$402$$ −7.12311 −0.355268
$$403$$ 0 0
$$404$$ −6.00000 −0.298511
$$405$$ −0.561553 −0.0279038
$$406$$ −2.56155 −0.127128
$$407$$ −2.42329 −0.120118
$$408$$ 5.68466 0.281433
$$409$$ −6.80776 −0.336622 −0.168311 0.985734i $$-0.553831\pi$$
−0.168311 + 0.985734i $$0.553831\pi$$
$$410$$ −2.24621 −0.110932
$$411$$ −5.68466 −0.280404
$$412$$ 18.8078 0.926592
$$413$$ 12.2462 0.602597
$$414$$ −5.68466 −0.279386
$$415$$ 1.12311 0.0551311
$$416$$ 0 0
$$417$$ 6.24621 0.305878
$$418$$ −3.68466 −0.180223
$$419$$ −8.31534 −0.406231 −0.203116 0.979155i $$-0.565107\pi$$
−0.203116 + 0.979155i $$0.565107\pi$$
$$420$$ 0.561553 0.0274010
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −12.8078 −0.623472
$$423$$ −6.24621 −0.303701
$$424$$ 13.1231 0.637314
$$425$$ 26.6307 1.29178
$$426$$ 15.3693 0.744646
$$427$$ 2.56155 0.123962
$$428$$ −13.1231 −0.634329
$$429$$ 0 0
$$430$$ −5.93087 −0.286012
$$431$$ −27.8617 −1.34205 −0.671026 0.741433i $$-0.734146\pi$$
−0.671026 + 0.741433i $$0.734146\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 1.36932 0.0658052 0.0329026 0.999459i $$-0.489525\pi$$
0.0329026 + 0.999459i $$0.489525\pi$$
$$434$$ 10.2462 0.491834
$$435$$ −1.43845 −0.0689683
$$436$$ −15.9309 −0.762950
$$437$$ −14.5616 −0.696574
$$438$$ 7.43845 0.355423
$$439$$ −19.9309 −0.951249 −0.475624 0.879649i $$-0.657778\pi$$
−0.475624 + 0.879649i $$0.657778\pi$$
$$440$$ 0.807764 0.0385086
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 31.3693 1.49040 0.745201 0.666840i $$-0.232354\pi$$
0.745201 + 0.666840i $$0.232354\pi$$
$$444$$ −1.68466 −0.0799504
$$445$$ −4.49242 −0.212961
$$446$$ −18.2462 −0.863983
$$447$$ −10.0000 −0.472984
$$448$$ 1.00000 0.0472456
$$449$$ 21.6847 1.02336 0.511681 0.859175i $$-0.329023\pi$$
0.511681 + 0.859175i $$0.329023\pi$$
$$450$$ −4.68466 −0.220837
$$451$$ −5.75379 −0.270935
$$452$$ 3.75379 0.176563
$$453$$ 4.31534 0.202752
$$454$$ 23.6155 1.10833
$$455$$ 0 0
$$456$$ −2.56155 −0.119956
$$457$$ −16.0000 −0.748448 −0.374224 0.927338i $$-0.622091\pi$$
−0.374224 + 0.927338i $$0.622091\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ 5.68466 0.265337
$$460$$ 3.19224 0.148839
$$461$$ 8.06913 0.375817 0.187908 0.982187i $$-0.439829\pi$$
0.187908 + 0.982187i $$0.439829\pi$$
$$462$$ 1.43845 0.0669226
$$463$$ −39.5464 −1.83788 −0.918938 0.394401i $$-0.870952\pi$$
−0.918938 + 0.394401i $$0.870952\pi$$
$$464$$ −2.56155 −0.118917
$$465$$ 5.75379 0.266826
$$466$$ −2.00000 −0.0926482
$$467$$ 2.56155 0.118535 0.0592673 0.998242i $$-0.481124\pi$$
0.0592673 + 0.998242i $$0.481124\pi$$
$$468$$ 0 0
$$469$$ 7.12311 0.328914
$$470$$ 3.50758 0.161792
$$471$$ −8.80776 −0.405840
$$472$$ 12.2462 0.563678
$$473$$ −15.1922 −0.698540
$$474$$ −16.0000 −0.734904
$$475$$ −12.0000 −0.550598
$$476$$ −5.68466 −0.260556
$$477$$ 13.1231 0.600866
$$478$$ −2.24621 −0.102739
$$479$$ 41.3002 1.88705 0.943527 0.331296i $$-0.107486\pi$$
0.943527 + 0.331296i $$0.107486\pi$$
$$480$$ 0.561553 0.0256313
$$481$$ 0 0
$$482$$ −6.00000 −0.273293
$$483$$ 5.68466 0.258661
$$484$$ −8.93087 −0.405949
$$485$$ −5.61553 −0.254988
$$486$$ −1.00000 −0.0453609
$$487$$ −30.2462 −1.37059 −0.685293 0.728267i $$-0.740326\pi$$
−0.685293 + 0.728267i $$0.740326\pi$$
$$488$$ 2.56155 0.115956
$$489$$ 0.876894 0.0396545
$$490$$ −0.561553 −0.0253684
$$491$$ −6.24621 −0.281888 −0.140944 0.990018i $$-0.545014\pi$$
−0.140944 + 0.990018i $$0.545014\pi$$
$$492$$ −4.00000 −0.180334
$$493$$ 14.5616 0.655819
$$494$$ 0 0
$$495$$ 0.807764 0.0363063
$$496$$ 10.2462 0.460068
$$497$$ −15.3693 −0.689408
$$498$$ 2.00000 0.0896221
$$499$$ −34.4924 −1.54409 −0.772046 0.635566i $$-0.780767\pi$$
−0.772046 + 0.635566i $$0.780767\pi$$
$$500$$ 5.43845 0.243215
$$501$$ 8.80776 0.393502
$$502$$ 15.0540 0.671892
$$503$$ −5.61553 −0.250384 −0.125192 0.992133i $$-0.539955\pi$$
−0.125192 + 0.992133i $$0.539955\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 3.36932 0.149933
$$506$$ 8.17708 0.363516
$$507$$ 0 0
$$508$$ 6.24621 0.277131
$$509$$ −9.68466 −0.429265 −0.214632 0.976695i $$-0.568855\pi$$
−0.214632 + 0.976695i $$0.568855\pi$$
$$510$$ −3.19224 −0.141355
$$511$$ −7.43845 −0.329058
$$512$$ 1.00000 0.0441942
$$513$$ −2.56155 −0.113095
$$514$$ 26.0000 1.14681
$$515$$ −10.5616 −0.465398
$$516$$ −10.5616 −0.464946
$$517$$ 8.98485 0.395153
$$518$$ 1.68466 0.0740196
$$519$$ −20.2462 −0.888710
$$520$$ 0 0
$$521$$ −29.0540 −1.27288 −0.636439 0.771327i $$-0.719593\pi$$
−0.636439 + 0.771327i $$0.719593\pi$$
$$522$$ −2.56155 −0.112116
$$523$$ 24.4924 1.07098 0.535489 0.844542i $$-0.320127\pi$$
0.535489 + 0.844542i $$0.320127\pi$$
$$524$$ 2.56155 0.111902
$$525$$ 4.68466 0.204455
$$526$$ 1.36932 0.0597051
$$527$$ −58.2462 −2.53724
$$528$$ 1.43845 0.0626004
$$529$$ 9.31534 0.405015
$$530$$ −7.36932 −0.320103
$$531$$ 12.2462 0.531440
$$532$$ 2.56155 0.111057
$$533$$ 0 0
$$534$$ −8.00000 −0.346194
$$535$$ 7.36932 0.318603
$$536$$ 7.12311 0.307671
$$537$$ 2.87689 0.124147
$$538$$ 6.63068 0.285869
$$539$$ −1.43845 −0.0619583
$$540$$ 0.561553 0.0241654
$$541$$ 26.8078 1.15256 0.576278 0.817254i $$-0.304505\pi$$
0.576278 + 0.817254i $$0.304505\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 11.3693 0.487904
$$544$$ −5.68466 −0.243728
$$545$$ 8.94602 0.383206
$$546$$ 0 0
$$547$$ −36.9848 −1.58136 −0.790679 0.612231i $$-0.790272\pi$$
−0.790679 + 0.612231i $$0.790272\pi$$
$$548$$ 5.68466 0.242837
$$549$$ 2.56155 0.109324
$$550$$ 6.73863 0.287336
$$551$$ −6.56155 −0.279532
$$552$$ 5.68466 0.241955
$$553$$ 16.0000 0.680389
$$554$$ 19.1231 0.812463
$$555$$ 0.946025 0.0401565
$$556$$ −6.24621 −0.264898
$$557$$ 0.246211 0.0104323 0.00521615 0.999986i $$-0.498340\pi$$
0.00521615 + 0.999986i $$0.498340\pi$$
$$558$$ 10.2462 0.433757
$$559$$ 0 0
$$560$$ −0.561553 −0.0237299
$$561$$ −8.17708 −0.345237
$$562$$ −6.00000 −0.253095
$$563$$ −0.946025 −0.0398702 −0.0199351 0.999801i $$-0.506346\pi$$
−0.0199351 + 0.999801i $$0.506346\pi$$
$$564$$ 6.24621 0.263013
$$565$$ −2.10795 −0.0886821
$$566$$ 13.1231 0.551605
$$567$$ 1.00000 0.0419961
$$568$$ −15.3693 −0.644882
$$569$$ 18.0000 0.754599 0.377300 0.926091i $$-0.376853\pi$$
0.377300 + 0.926091i $$0.376853\pi$$
$$570$$ 1.43845 0.0602499
$$571$$ 1.75379 0.0733938 0.0366969 0.999326i $$-0.488316\pi$$
0.0366969 + 0.999326i $$0.488316\pi$$
$$572$$ 0 0
$$573$$ −13.0540 −0.545338
$$574$$ 4.00000 0.166957
$$575$$ 26.6307 1.11058
$$576$$ 1.00000 0.0416667
$$577$$ 24.2462 1.00938 0.504691 0.863300i $$-0.331606\pi$$
0.504691 + 0.863300i $$0.331606\pi$$
$$578$$ 15.3153 0.637034
$$579$$ −12.0000 −0.498703
$$580$$ 1.43845 0.0597283
$$581$$ −2.00000 −0.0829740
$$582$$ −10.0000 −0.414513
$$583$$ −18.8769 −0.781801
$$584$$ −7.43845 −0.307805
$$585$$ 0 0
$$586$$ 24.2462 1.00160
$$587$$ −28.8769 −1.19188 −0.595938 0.803030i $$-0.703220\pi$$
−0.595938 + 0.803030i $$0.703220\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 26.2462 1.08146
$$590$$ −6.87689 −0.283117
$$591$$ 6.00000 0.246807
$$592$$ 1.68466 0.0692390
$$593$$ 28.0000 1.14982 0.574911 0.818216i $$-0.305037\pi$$
0.574911 + 0.818216i $$0.305037\pi$$
$$594$$ 1.43845 0.0590202
$$595$$ 3.19224 0.130869
$$596$$ 10.0000 0.409616
$$597$$ −23.9309 −0.979425
$$598$$ 0 0
$$599$$ −23.3002 −0.952020 −0.476010 0.879440i $$-0.657917\pi$$
−0.476010 + 0.879440i $$0.657917\pi$$
$$600$$ 4.68466 0.191250
$$601$$ 25.8617 1.05492 0.527461 0.849579i $$-0.323144\pi$$
0.527461 + 0.849579i $$0.323144\pi$$
$$602$$ 10.5616 0.430457
$$603$$ 7.12311 0.290075
$$604$$ −4.31534 −0.175589
$$605$$ 5.01515 0.203895
$$606$$ 6.00000 0.243733
$$607$$ 41.5464 1.68632 0.843158 0.537666i $$-0.180694\pi$$
0.843158 + 0.537666i $$0.180694\pi$$
$$608$$ 2.56155 0.103885
$$609$$ 2.56155 0.103799
$$610$$ −1.43845 −0.0582410
$$611$$ 0 0
$$612$$ −5.68466 −0.229789
$$613$$ 18.8078 0.759638 0.379819 0.925061i $$-0.375986\pi$$
0.379819 + 0.925061i $$0.375986\pi$$
$$614$$ −9.75379 −0.393631
$$615$$ 2.24621 0.0905760
$$616$$ −1.43845 −0.0579567
$$617$$ 26.8078 1.07924 0.539620 0.841909i $$-0.318568\pi$$
0.539620 + 0.841909i $$0.318568\pi$$
$$618$$ −18.8078 −0.756559
$$619$$ 2.06913 0.0831654 0.0415827 0.999135i $$-0.486760\pi$$
0.0415827 + 0.999135i $$0.486760\pi$$
$$620$$ −5.75379 −0.231078
$$621$$ 5.68466 0.228117
$$622$$ −32.4924 −1.30283
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ 20.3693 0.814773
$$626$$ 15.7538 0.629648
$$627$$ 3.68466 0.147151
$$628$$ 8.80776 0.351468
$$629$$ −9.57671 −0.381848
$$630$$ −0.561553 −0.0223728
$$631$$ −0.315342 −0.0125535 −0.00627677 0.999980i $$-0.501998\pi$$
−0.00627677 + 0.999980i $$0.501998\pi$$
$$632$$ 16.0000 0.636446
$$633$$ 12.8078 0.509063
$$634$$ 21.3693 0.848684
$$635$$ −3.50758 −0.139194
$$636$$ −13.1231 −0.520365
$$637$$ 0 0
$$638$$ 3.68466 0.145877
$$639$$ −15.3693 −0.608001
$$640$$ −0.561553 −0.0221973
$$641$$ 44.1080 1.74216 0.871080 0.491142i $$-0.163420\pi$$
0.871080 + 0.491142i $$0.163420\pi$$
$$642$$ 13.1231 0.517928
$$643$$ −20.8078 −0.820578 −0.410289 0.911956i $$-0.634572\pi$$
−0.410289 + 0.911956i $$0.634572\pi$$
$$644$$ −5.68466 −0.224007
$$645$$ 5.93087 0.233528
$$646$$ −14.5616 −0.572917
$$647$$ −35.3693 −1.39051 −0.695256 0.718763i $$-0.744709\pi$$
−0.695256 + 0.718763i $$0.744709\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −17.6155 −0.691470
$$650$$ 0 0
$$651$$ −10.2462 −0.401581
$$652$$ −0.876894 −0.0343418
$$653$$ −25.4384 −0.995483 −0.497742 0.867325i $$-0.665837\pi$$
−0.497742 + 0.867325i $$0.665837\pi$$
$$654$$ 15.9309 0.622946
$$655$$ −1.43845 −0.0562048
$$656$$ 4.00000 0.156174
$$657$$ −7.43845 −0.290201
$$658$$ −6.24621 −0.243503
$$659$$ 21.1231 0.822839 0.411420 0.911446i $$-0.365033\pi$$
0.411420 + 0.911446i $$0.365033\pi$$
$$660$$ −0.807764 −0.0314422
$$661$$ −4.73863 −0.184311 −0.0921557 0.995745i $$-0.529376\pi$$
−0.0921557 + 0.995745i $$0.529376\pi$$
$$662$$ 7.12311 0.276847
$$663$$ 0 0
$$664$$ −2.00000 −0.0776151
$$665$$ −1.43845 −0.0557806
$$666$$ 1.68466 0.0652792
$$667$$ 14.5616 0.563826
$$668$$ −8.80776 −0.340783
$$669$$ 18.2462 0.705439
$$670$$ −4.00000 −0.154533
$$671$$ −3.68466 −0.142245
$$672$$ −1.00000 −0.0385758
$$673$$ −17.1922 −0.662712 −0.331356 0.943506i $$-0.607506\pi$$
−0.331356 + 0.943506i $$0.607506\pi$$
$$674$$ 17.0540 0.656895
$$675$$ 4.68466 0.180313
$$676$$ 0 0
$$677$$ 6.63068 0.254838 0.127419 0.991849i $$-0.459331\pi$$
0.127419 + 0.991849i $$0.459331\pi$$
$$678$$ −3.75379 −0.144163
$$679$$ 10.0000 0.383765
$$680$$ 3.19224 0.122417
$$681$$ −23.6155 −0.904949
$$682$$ −14.7386 −0.564371
$$683$$ 11.6847 0.447101 0.223551 0.974692i $$-0.428235\pi$$
0.223551 + 0.974692i $$0.428235\pi$$
$$684$$ 2.56155 0.0979434
$$685$$ −3.19224 −0.121969
$$686$$ 1.00000 0.0381802
$$687$$ 2.00000 0.0763048
$$688$$ 10.5616 0.402655
$$689$$ 0 0
$$690$$ −3.19224 −0.121526
$$691$$ 3.50758 0.133435 0.0667173 0.997772i $$-0.478747\pi$$
0.0667173 + 0.997772i $$0.478747\pi$$
$$692$$ 20.2462 0.769645
$$693$$ −1.43845 −0.0546421
$$694$$ 8.49242 0.322368
$$695$$ 3.50758 0.133050
$$696$$ 2.56155 0.0970954
$$697$$ −22.7386 −0.861287
$$698$$ 21.3693 0.808841
$$699$$ 2.00000 0.0756469
$$700$$ −4.68466 −0.177063
$$701$$ −37.6155 −1.42072 −0.710359 0.703839i $$-0.751468\pi$$
−0.710359 + 0.703839i $$0.751468\pi$$
$$702$$ 0 0
$$703$$ 4.31534 0.162756
$$704$$ −1.43845 −0.0542135
$$705$$ −3.50758 −0.132103
$$706$$ −1.75379 −0.0660047
$$707$$ −6.00000 −0.225653
$$708$$ −12.2462 −0.460241
$$709$$ −48.2462 −1.81192 −0.905962 0.423358i $$-0.860851\pi$$
−0.905962 + 0.423358i $$0.860851\pi$$
$$710$$ 8.63068 0.323904
$$711$$ 16.0000 0.600047
$$712$$ 8.00000 0.299813
$$713$$ −58.2462 −2.18134
$$714$$ 5.68466 0.212743
$$715$$ 0 0
$$716$$ −2.87689 −0.107515
$$717$$ 2.24621 0.0838863
$$718$$ 17.6155 0.657406
$$719$$ −30.2462 −1.12799 −0.563997 0.825777i $$-0.690737\pi$$
−0.563997 + 0.825777i $$0.690737\pi$$
$$720$$ −0.561553 −0.0209278
$$721$$ 18.8078 0.700438
$$722$$ −12.4384 −0.462911
$$723$$ 6.00000 0.223142
$$724$$ −11.3693 −0.422537
$$725$$ 12.0000 0.445669
$$726$$ 8.93087 0.331456
$$727$$ 20.0691 0.744323 0.372161 0.928168i $$-0.378617\pi$$
0.372161 + 0.928168i $$0.378617\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 4.17708 0.154601
$$731$$ −60.0388 −2.22062
$$732$$ −2.56155 −0.0946777
$$733$$ −31.1231 −1.14956 −0.574779 0.818309i $$-0.694912\pi$$
−0.574779 + 0.818309i $$0.694912\pi$$
$$734$$ 25.3693 0.936399
$$735$$ 0.561553 0.0207132
$$736$$ −5.68466 −0.209539
$$737$$ −10.2462 −0.377424
$$738$$ 4.00000 0.147242
$$739$$ −23.6155 −0.868711 −0.434356 0.900741i $$-0.643024\pi$$
−0.434356 + 0.900741i $$0.643024\pi$$
$$740$$ −0.946025 −0.0347766
$$741$$ 0 0
$$742$$ 13.1231 0.481764
$$743$$ −16.0000 −0.586983 −0.293492 0.955962i $$-0.594817\pi$$
−0.293492 + 0.955962i $$0.594817\pi$$
$$744$$ −10.2462 −0.375644
$$745$$ −5.61553 −0.205737
$$746$$ −6.00000 −0.219676
$$747$$ −2.00000 −0.0731762
$$748$$ 8.17708 0.298984
$$749$$ −13.1231 −0.479508
$$750$$ −5.43845 −0.198584
$$751$$ 14.2462 0.519852 0.259926 0.965629i $$-0.416302\pi$$
0.259926 + 0.965629i $$0.416302\pi$$
$$752$$ −6.24621 −0.227776
$$753$$ −15.0540 −0.548597
$$754$$ 0 0
$$755$$ 2.42329 0.0881926
$$756$$ −1.00000 −0.0363696
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −27.1231 −0.985156
$$759$$ −8.17708 −0.296809
$$760$$ −1.43845 −0.0521780
$$761$$ 30.8769 1.11929 0.559643 0.828734i $$-0.310938\pi$$
0.559643 + 0.828734i $$0.310938\pi$$
$$762$$ −6.24621 −0.226276
$$763$$ −15.9309 −0.576736
$$764$$ 13.0540 0.472276
$$765$$ 3.19224 0.115416
$$766$$ 13.4384 0.485551
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −12.5616 −0.452981 −0.226491 0.974013i $$-0.572725\pi$$
−0.226491 + 0.974013i $$0.572725\pi$$
$$770$$ 0.807764 0.0291098
$$771$$ −26.0000 −0.936367
$$772$$ 12.0000 0.431889
$$773$$ −15.9309 −0.572994 −0.286497 0.958081i $$-0.592491\pi$$
−0.286497 + 0.958081i $$0.592491\pi$$
$$774$$ 10.5616 0.379627
$$775$$ −48.0000 −1.72421
$$776$$ 10.0000 0.358979
$$777$$ −1.68466 −0.0604368
$$778$$ −18.8769 −0.676769
$$779$$ 10.2462 0.367109
$$780$$ 0 0
$$781$$ 22.1080 0.791085
$$782$$ 32.3153 1.15559
$$783$$ 2.56155 0.0915424
$$784$$ 1.00000 0.0357143
$$785$$ −4.94602 −0.176531
$$786$$ −2.56155 −0.0913676
$$787$$ −45.9309 −1.63726 −0.818629 0.574322i $$-0.805266\pi$$
−0.818629 + 0.574322i $$0.805266\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ −1.36932 −0.0487490
$$790$$ −8.98485 −0.319666
$$791$$ 3.75379 0.133469
$$792$$ −1.43845 −0.0511130
$$793$$ 0 0
$$794$$ 9.36932 0.332505
$$795$$ 7.36932 0.261363
$$796$$ 23.9309 0.848207
$$797$$ −20.2462 −0.717158 −0.358579 0.933499i $$-0.616739\pi$$
−0.358579 + 0.933499i $$0.616739\pi$$
$$798$$ −2.56155 −0.0906780
$$799$$ 35.5076 1.25617
$$800$$ −4.68466 −0.165628
$$801$$ 8.00000 0.282666
$$802$$ −34.9848 −1.23536
$$803$$ 10.6998 0.377588
$$804$$ −7.12311 −0.251213
$$805$$ 3.19224 0.112512
$$806$$ 0 0
$$807$$ −6.63068 −0.233411
$$808$$ −6.00000 −0.211079
$$809$$ −30.6307 −1.07692 −0.538459 0.842652i $$-0.680993\pi$$
−0.538459 + 0.842652i $$0.680993\pi$$
$$810$$ −0.561553 −0.0197310
$$811$$ 35.0540 1.23091 0.615456 0.788171i $$-0.288972\pi$$
0.615456 + 0.788171i $$0.288972\pi$$
$$812$$ −2.56155 −0.0898929
$$813$$ −8.00000 −0.280572
$$814$$ −2.42329 −0.0849363
$$815$$ 0.492423 0.0172488
$$816$$ 5.68466 0.199003
$$817$$ 27.0540 0.946499
$$818$$ −6.80776 −0.238028
$$819$$ 0 0
$$820$$ −2.24621 −0.0784411
$$821$$ −39.1231 −1.36541 −0.682703 0.730696i $$-0.739196\pi$$
−0.682703 + 0.730696i $$0.739196\pi$$
$$822$$ −5.68466 −0.198275
$$823$$ −18.7386 −0.653188 −0.326594 0.945165i $$-0.605901\pi$$
−0.326594 + 0.945165i $$0.605901\pi$$
$$824$$ 18.8078 0.655200
$$825$$ −6.73863 −0.234609
$$826$$ 12.2462 0.426100
$$827$$ 15.0540 0.523478 0.261739 0.965139i $$-0.415704\pi$$
0.261739 + 0.965139i $$0.415704\pi$$
$$828$$ −5.68466 −0.197556
$$829$$ −4.94602 −0.171783 −0.0858913 0.996305i $$-0.527374\pi$$
−0.0858913 + 0.996305i $$0.527374\pi$$
$$830$$ 1.12311 0.0389836
$$831$$ −19.1231 −0.663373
$$832$$ 0 0
$$833$$ −5.68466 −0.196962
$$834$$ 6.24621 0.216289
$$835$$ 4.94602 0.171164
$$836$$ −3.68466 −0.127437
$$837$$ −10.2462 −0.354161
$$838$$ −8.31534 −0.287249
$$839$$ −42.2462 −1.45850 −0.729251 0.684247i $$-0.760131\pi$$
−0.729251 + 0.684247i $$0.760131\pi$$
$$840$$ 0.561553 0.0193754
$$841$$ −22.4384 −0.773740
$$842$$ −10.0000 −0.344623
$$843$$ 6.00000 0.206651
$$844$$ −12.8078 −0.440861
$$845$$ 0 0
$$846$$ −6.24621 −0.214749
$$847$$ −8.93087 −0.306868
$$848$$ 13.1231 0.450649
$$849$$ −13.1231 −0.450384
$$850$$ 26.6307 0.913425
$$851$$ −9.57671 −0.328285
$$852$$ 15.3693 0.526544
$$853$$ −19.1231 −0.654763 −0.327381 0.944892i $$-0.606166\pi$$
−0.327381 + 0.944892i $$0.606166\pi$$
$$854$$ 2.56155 0.0876545
$$855$$ −1.43845 −0.0491939
$$856$$ −13.1231 −0.448539
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ 0 0
$$859$$ 41.6155 1.41990 0.709952 0.704250i $$-0.248717\pi$$
0.709952 + 0.704250i $$0.248717\pi$$
$$860$$ −5.93087 −0.202241
$$861$$ −4.00000 −0.136320
$$862$$ −27.8617 −0.948975
$$863$$ 6.73863 0.229386 0.114693 0.993401i $$-0.463412\pi$$
0.114693 + 0.993401i $$0.463412\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −11.3693 −0.386568
$$866$$ 1.36932 0.0465313
$$867$$ −15.3153 −0.520136
$$868$$ 10.2462 0.347779
$$869$$ −23.0152 −0.780736
$$870$$ −1.43845 −0.0487679
$$871$$ 0 0
$$872$$ −15.9309 −0.539487
$$873$$ 10.0000 0.338449
$$874$$ −14.5616 −0.492552
$$875$$ 5.43845 0.183853
$$876$$ 7.43845 0.251322
$$877$$ 42.9848 1.45150 0.725748 0.687961i $$-0.241494\pi$$
0.725748 + 0.687961i $$0.241494\pi$$
$$878$$ −19.9309 −0.672634
$$879$$ −24.2462 −0.817804
$$880$$ 0.807764 0.0272297
$$881$$ 39.3002 1.32406 0.662028 0.749479i $$-0.269696\pi$$
0.662028 + 0.749479i $$0.269696\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 18.5616 0.624646 0.312323 0.949976i $$-0.398893\pi$$
0.312323 + 0.949976i $$0.398893\pi$$
$$884$$ 0 0
$$885$$ 6.87689 0.231164
$$886$$ 31.3693 1.05387
$$887$$ 19.8617 0.666892 0.333446 0.942769i $$-0.391789\pi$$
0.333446 + 0.942769i $$0.391789\pi$$
$$888$$ −1.68466 −0.0565334
$$889$$ 6.24621 0.209491
$$890$$ −4.49242 −0.150586
$$891$$ −1.43845 −0.0481898
$$892$$ −18.2462 −0.610928
$$893$$ −16.0000 −0.535420
$$894$$ −10.0000 −0.334450
$$895$$ 1.61553 0.0540011
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 21.6847 0.723626
$$899$$ −26.2462 −0.875360
$$900$$ −4.68466 −0.156155
$$901$$ −74.6004 −2.48530
$$902$$ −5.75379 −0.191580
$$903$$ −10.5616 −0.351466
$$904$$ 3.75379 0.124849
$$905$$ 6.38447 0.212227
$$906$$ 4.31534 0.143368
$$907$$ −8.49242 −0.281986 −0.140993 0.990011i $$-0.545030\pi$$
−0.140993 + 0.990011i $$0.545030\pi$$
$$908$$ 23.6155 0.783709
$$909$$ −6.00000 −0.199007
$$910$$ 0 0
$$911$$ −8.56155 −0.283657 −0.141828 0.989891i $$-0.545298\pi$$
−0.141828 + 0.989891i $$0.545298\pi$$
$$912$$ −2.56155 −0.0848215
$$913$$ 2.87689 0.0952113
$$914$$ −16.0000 −0.529233
$$915$$ 1.43845 0.0475536
$$916$$ −2.00000 −0.0660819
$$917$$ 2.56155 0.0845899
$$918$$ 5.68466 0.187622
$$919$$ 39.8617 1.31492 0.657459 0.753491i $$-0.271631\pi$$
0.657459 + 0.753491i $$0.271631\pi$$
$$920$$ 3.19224 0.105245
$$921$$ 9.75379 0.321398
$$922$$ 8.06913 0.265743
$$923$$ 0 0
$$924$$ 1.43845 0.0473214
$$925$$ −7.89205 −0.259489
$$926$$ −39.5464 −1.29958
$$927$$ 18.8078 0.617728
$$928$$ −2.56155 −0.0840871
$$929$$ −49.1231 −1.61168 −0.805838 0.592136i $$-0.798285\pi$$
−0.805838 + 0.592136i $$0.798285\pi$$
$$930$$ 5.75379 0.188674
$$931$$ 2.56155 0.0839515
$$932$$ −2.00000 −0.0655122
$$933$$ 32.4924 1.06375
$$934$$ 2.56155 0.0838166
$$935$$ −4.59186 −0.150170
$$936$$ 0 0
$$937$$ 37.8617 1.23689 0.618445 0.785828i $$-0.287763\pi$$
0.618445 + 0.785828i $$0.287763\pi$$
$$938$$ 7.12311 0.232578
$$939$$ −15.7538 −0.514105
$$940$$ 3.50758 0.114405
$$941$$ 6.49242 0.211647 0.105823 0.994385i $$-0.466252\pi$$
0.105823 + 0.994385i $$0.466252\pi$$
$$942$$ −8.80776 −0.286972
$$943$$ −22.7386 −0.740472
$$944$$ 12.2462 0.398580
$$945$$ 0.561553 0.0182673
$$946$$ −15.1922 −0.493942
$$947$$ −7.19224 −0.233716 −0.116858 0.993149i $$-0.537282\pi$$
−0.116858 + 0.993149i $$0.537282\pi$$
$$948$$ −16.0000 −0.519656
$$949$$ 0 0
$$950$$ −12.0000 −0.389331
$$951$$ −21.3693 −0.692948
$$952$$ −5.68466 −0.184241
$$953$$ 48.7386 1.57880 0.789400 0.613880i $$-0.210392\pi$$
0.789400 + 0.613880i $$0.210392\pi$$
$$954$$ 13.1231 0.424876
$$955$$ −7.33050 −0.237209
$$956$$ −2.24621 −0.0726477
$$957$$ −3.68466 −0.119108
$$958$$ 41.3002 1.33435
$$959$$ 5.68466 0.183567
$$960$$ 0.561553 0.0181240
$$961$$ 73.9848 2.38661
$$962$$ 0 0
$$963$$ −13.1231 −0.422886
$$964$$ −6.00000 −0.193247
$$965$$ −6.73863 −0.216924
$$966$$ 5.68466 0.182901
$$967$$ −25.9309 −0.833881 −0.416940 0.908934i $$-0.636898\pi$$
−0.416940 + 0.908934i $$0.636898\pi$$
$$968$$ −8.93087 −0.287049
$$969$$ 14.5616 0.467784
$$970$$ −5.61553 −0.180304
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −6.24621 −0.200244
$$974$$ −30.2462 −0.969151
$$975$$ 0 0
$$976$$ 2.56155 0.0819933
$$977$$ 39.7926 1.27308 0.636539 0.771244i $$-0.280365\pi$$
0.636539 + 0.771244i $$0.280365\pi$$
$$978$$ 0.876894 0.0280400
$$979$$ −11.5076 −0.367784
$$980$$ −0.561553 −0.0179381
$$981$$ −15.9309 −0.508634
$$982$$ −6.24621 −0.199325
$$983$$ 6.56155 0.209281 0.104641 0.994510i $$-0.466631\pi$$
0.104641 + 0.994510i $$0.466631\pi$$
$$984$$ −4.00000 −0.127515
$$985$$ 3.36932 0.107355
$$986$$ 14.5616 0.463734
$$987$$ 6.24621 0.198819
$$988$$ 0 0
$$989$$ −60.0388 −1.90912
$$990$$ 0.807764 0.0256724
$$991$$ −13.7538 −0.436903 −0.218452 0.975848i $$-0.570101\pi$$
−0.218452 + 0.975848i $$0.570101\pi$$
$$992$$ 10.2462 0.325318
$$993$$ −7.12311 −0.226045
$$994$$ −15.3693 −0.487485
$$995$$ −13.4384 −0.426027
$$996$$ 2.00000 0.0633724
$$997$$ −59.3693 −1.88025 −0.940123 0.340837i $$-0.889290\pi$$
−0.940123 + 0.340837i $$0.889290\pi$$
$$998$$ −34.4924 −1.09184
$$999$$ −1.68466 −0.0533002
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 7098.2.a.bv.1.1 2
13.5 odd 4 546.2.c.e.337.2 4
13.8 odd 4 546.2.c.e.337.3 yes 4
13.12 even 2 7098.2.a.bg.1.2 2
39.5 even 4 1638.2.c.h.883.3 4
39.8 even 4 1638.2.c.h.883.2 4
52.31 even 4 4368.2.h.n.337.3 4
52.47 even 4 4368.2.h.n.337.2 4
91.34 even 4 3822.2.c.h.883.4 4
91.83 even 4 3822.2.c.h.883.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.c.e.337.2 4 13.5 odd 4
546.2.c.e.337.3 yes 4 13.8 odd 4
1638.2.c.h.883.2 4 39.8 even 4
1638.2.c.h.883.3 4 39.5 even 4
3822.2.c.h.883.1 4 91.83 even 4
3822.2.c.h.883.4 4 91.34 even 4
4368.2.h.n.337.2 4 52.47 even 4
4368.2.h.n.337.3 4 52.31 even 4
7098.2.a.bg.1.2 2 13.12 even 2
7098.2.a.bv.1.1 2 1.1 even 1 trivial