# Properties

 Label 4598.2.a.br.1.4 Level $4598$ Weight $2$ Character 4598.1 Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4598 = 2 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4598.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7152148494$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.33452.1 Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + 7x - 1$$ x^4 - x^3 - 8*x^2 + 7*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$2.94749$$ of defining polynomial Character $$\chi$$ $$=$$ 4598.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.94749 q^{3} +1.00000 q^{4} +2.36624 q^{5} -2.94749 q^{6} +1.66073 q^{7} -1.00000 q^{8} +5.68769 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.94749 q^{3} +1.00000 q^{4} +2.36624 q^{5} -2.94749 q^{6} +1.66073 q^{7} -1.00000 q^{8} +5.68769 q^{9} -2.36624 q^{10} +2.94749 q^{12} +0.633765 q^{13} -1.66073 q^{14} +6.97445 q^{15} +1.00000 q^{16} -0.581254 q^{17} -5.68769 q^{18} +1.00000 q^{19} +2.36624 q^{20} +4.89498 q^{21} +3.60822 q^{23} -2.94749 q^{24} +0.599069 q^{25} -0.633765 q^{26} +7.92194 q^{27} +1.66073 q^{28} -9.13696 q^{29} -6.97445 q^{30} +1.02555 q^{31} -1.00000 q^{32} +0.581254 q^{34} +3.92967 q^{35} +5.68769 q^{36} +2.15895 q^{37} -1.00000 q^{38} +1.86801 q^{39} -2.36624 q^{40} +4.40093 q^{41} -4.89498 q^{42} +9.58267 q^{43} +13.4584 q^{45} -3.60822 q^{46} +8.73247 q^{47} +2.94749 q^{48} -4.24198 q^{49} -0.599069 q^{50} -1.71324 q^{51} +0.633765 q^{52} -12.5302 q^{53} -7.92194 q^{54} -1.66073 q^{56} +2.94749 q^{57} +9.13696 q^{58} +6.47623 q^{59} +6.97445 q^{60} +1.32146 q^{61} -1.02555 q^{62} +9.44571 q^{63} +1.00000 q^{64} +1.49964 q^{65} -0.393199 q^{67} -0.581254 q^{68} +10.6352 q^{69} -3.92967 q^{70} -0.312308 q^{71} -5.68769 q^{72} -9.31372 q^{73} -2.15895 q^{74} +1.76575 q^{75} +1.00000 q^{76} -1.86801 q^{78} +11.2704 q^{79} +2.36624 q^{80} +6.28676 q^{81} -4.40093 q^{82} -6.90413 q^{83} +4.89498 q^{84} -1.37538 q^{85} -9.58267 q^{86} -26.9311 q^{87} +0.519595 q^{89} -13.4584 q^{90} +1.05251 q^{91} +3.60822 q^{92} +3.02279 q^{93} -8.73247 q^{94} +2.36624 q^{95} -2.94749 q^{96} -8.37894 q^{97} +4.24198 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} - q^{6} + q^{7} - 4 q^{8} + 5 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + q^3 + 4 * q^4 + 3 * q^5 - q^6 + q^7 - 4 * q^8 + 5 * q^9 $$4 q - 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} - q^{6} + q^{7} - 4 q^{8} + 5 q^{9} - 3 q^{10} + q^{12} + 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} + 2 q^{17} - 5 q^{18} + 4 q^{19} + 3 q^{20} - 2 q^{21} - 2 q^{23} - q^{24} + 15 q^{25} - 9 q^{26} - 2 q^{27} + q^{28} - 5 q^{29} - 5 q^{30} + 27 q^{31} - 4 q^{32} - 2 q^{34} - 12 q^{35} + 5 q^{36} + 6 q^{37} - 4 q^{38} - 2 q^{39} - 3 q^{40} + 5 q^{41} + 2 q^{42} - q^{43} + 11 q^{45} + 2 q^{46} + 22 q^{47} + q^{48} - 7 q^{49} - 15 q^{50} - 12 q^{51} + 9 q^{52} + 2 q^{54} - q^{56} + q^{57} + 5 q^{58} + 5 q^{60} - 6 q^{61} - 27 q^{62} + 30 q^{63} + 4 q^{64} - 26 q^{65} + 17 q^{67} + 2 q^{68} + 14 q^{69} + 12 q^{70} - 19 q^{71} - 5 q^{72} - 20 q^{73} - 6 q^{74} + 23 q^{75} + 4 q^{76} + 2 q^{78} - 12 q^{79} + 3 q^{80} + 20 q^{81} - 5 q^{82} + 23 q^{83} - 2 q^{84} + 30 q^{85} + q^{86} - 45 q^{87} + 16 q^{89} - 11 q^{90} + 15 q^{91} - 2 q^{92} - 12 q^{93} - 22 q^{94} + 3 q^{95} - q^{96} + 8 q^{97} + 7 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + q^3 + 4 * q^4 + 3 * q^5 - q^6 + q^7 - 4 * q^8 + 5 * q^9 - 3 * q^10 + q^12 + 9 * q^13 - q^14 + 5 * q^15 + 4 * q^16 + 2 * q^17 - 5 * q^18 + 4 * q^19 + 3 * q^20 - 2 * q^21 - 2 * q^23 - q^24 + 15 * q^25 - 9 * q^26 - 2 * q^27 + q^28 - 5 * q^29 - 5 * q^30 + 27 * q^31 - 4 * q^32 - 2 * q^34 - 12 * q^35 + 5 * q^36 + 6 * q^37 - 4 * q^38 - 2 * q^39 - 3 * q^40 + 5 * q^41 + 2 * q^42 - q^43 + 11 * q^45 + 2 * q^46 + 22 * q^47 + q^48 - 7 * q^49 - 15 * q^50 - 12 * q^51 + 9 * q^52 + 2 * q^54 - q^56 + q^57 + 5 * q^58 + 5 * q^60 - 6 * q^61 - 27 * q^62 + 30 * q^63 + 4 * q^64 - 26 * q^65 + 17 * q^67 + 2 * q^68 + 14 * q^69 + 12 * q^70 - 19 * q^71 - 5 * q^72 - 20 * q^73 - 6 * q^74 + 23 * q^75 + 4 * q^76 + 2 * q^78 - 12 * q^79 + 3 * q^80 + 20 * q^81 - 5 * q^82 + 23 * q^83 - 2 * q^84 + 30 * q^85 + q^86 - 45 * q^87 + 16 * q^89 - 11 * q^90 + 15 * q^91 - 2 * q^92 - 12 * q^93 - 22 * q^94 + 3 * q^95 - q^96 + 8 * q^97 + 7 * q^98

## 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
$$3$$ 2.94749 1.70173 0.850867 0.525381i $$-0.176077\pi$$
0.850867 + 0.525381i $$0.176077\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 2.36624 1.05821 0.529106 0.848556i $$-0.322527\pi$$
0.529106 + 0.848556i $$0.322527\pi$$
$$6$$ −2.94749 −1.20331
$$7$$ 1.66073 0.627696 0.313848 0.949473i $$-0.398382\pi$$
0.313848 + 0.949473i $$0.398382\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 5.68769 1.89590
$$10$$ −2.36624 −0.748269
$$11$$ 0 0
$$12$$ 2.94749 0.850867
$$13$$ 0.633765 0.175775 0.0878874 0.996130i $$-0.471988\pi$$
0.0878874 + 0.996130i $$0.471988\pi$$
$$14$$ −1.66073 −0.443848
$$15$$ 6.97445 1.80080
$$16$$ 1.00000 0.250000
$$17$$ −0.581254 −0.140975 −0.0704874 0.997513i $$-0.522455\pi$$
−0.0704874 + 0.997513i $$0.522455\pi$$
$$18$$ −5.68769 −1.34060
$$19$$ 1.00000 0.229416
$$20$$ 2.36624 0.529106
$$21$$ 4.89498 1.06817
$$22$$ 0 0
$$23$$ 3.60822 0.752365 0.376183 0.926546i $$-0.377237\pi$$
0.376183 + 0.926546i $$0.377237\pi$$
$$24$$ −2.94749 −0.601654
$$25$$ 0.599069 0.119814
$$26$$ −0.633765 −0.124291
$$27$$ 7.92194 1.52458
$$28$$ 1.66073 0.313848
$$29$$ −9.13696 −1.69669 −0.848345 0.529443i $$-0.822401\pi$$
−0.848345 + 0.529443i $$0.822401\pi$$
$$30$$ −6.97445 −1.27335
$$31$$ 1.02555 0.184194 0.0920969 0.995750i $$-0.470643\pi$$
0.0920969 + 0.995750i $$0.470643\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 0.581254 0.0996842
$$35$$ 3.92967 0.664236
$$36$$ 5.68769 0.947949
$$37$$ 2.15895 0.354929 0.177464 0.984127i $$-0.443211\pi$$
0.177464 + 0.984127i $$0.443211\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 1.86801 0.299122
$$40$$ −2.36624 −0.374135
$$41$$ 4.40093 0.687310 0.343655 0.939096i $$-0.388335\pi$$
0.343655 + 0.939096i $$0.388335\pi$$
$$42$$ −4.89498 −0.755312
$$43$$ 9.58267 1.46134 0.730672 0.682729i $$-0.239207\pi$$
0.730672 + 0.682729i $$0.239207\pi$$
$$44$$ 0 0
$$45$$ 13.4584 2.00626
$$46$$ −3.60822 −0.532003
$$47$$ 8.73247 1.27376 0.636881 0.770962i $$-0.280224\pi$$
0.636881 + 0.770962i $$0.280224\pi$$
$$48$$ 2.94749 0.425433
$$49$$ −4.24198 −0.605997
$$50$$ −0.599069 −0.0847212
$$51$$ −1.71324 −0.239901
$$52$$ 0.633765 0.0878874
$$53$$ −12.5302 −1.72115 −0.860575 0.509324i $$-0.829895\pi$$
−0.860575 + 0.509324i $$0.829895\pi$$
$$54$$ −7.92194 −1.07804
$$55$$ 0 0
$$56$$ −1.66073 −0.221924
$$57$$ 2.94749 0.390404
$$58$$ 9.13696 1.19974
$$59$$ 6.47623 0.843134 0.421567 0.906797i $$-0.361480\pi$$
0.421567 + 0.906797i $$0.361480\pi$$
$$60$$ 6.97445 0.900398
$$61$$ 1.32146 0.169195 0.0845976 0.996415i $$-0.473040\pi$$
0.0845976 + 0.996415i $$0.473040\pi$$
$$62$$ −1.02555 −0.130245
$$63$$ 9.44571 1.19005
$$64$$ 1.00000 0.125000
$$65$$ 1.49964 0.186007
$$66$$ 0 0
$$67$$ −0.393199 −0.0480369 −0.0240184 0.999712i $$-0.507646\pi$$
−0.0240184 + 0.999712i $$0.507646\pi$$
$$68$$ −0.581254 −0.0704874
$$69$$ 10.6352 1.28033
$$70$$ −3.92967 −0.469686
$$71$$ −0.312308 −0.0370642 −0.0185321 0.999828i $$-0.505899\pi$$
−0.0185321 + 0.999828i $$0.505899\pi$$
$$72$$ −5.68769 −0.670301
$$73$$ −9.31372 −1.09009 −0.545044 0.838407i $$-0.683487\pi$$
−0.545044 + 0.838407i $$0.683487\pi$$
$$74$$ −2.15895 −0.250973
$$75$$ 1.76575 0.203891
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ −1.86801 −0.211511
$$79$$ 11.2704 1.26801 0.634007 0.773327i $$-0.281409\pi$$
0.634007 + 0.773327i $$0.281409\pi$$
$$80$$ 2.36624 0.264553
$$81$$ 6.28676 0.698529
$$82$$ −4.40093 −0.486001
$$83$$ −6.90413 −0.757826 −0.378913 0.925432i $$-0.623702\pi$$
−0.378913 + 0.925432i $$0.623702\pi$$
$$84$$ 4.89498 0.534086
$$85$$ −1.37538 −0.149181
$$86$$ −9.58267 −1.03333
$$87$$ −26.9311 −2.88732
$$88$$ 0 0
$$89$$ 0.519595 0.0550769 0.0275385 0.999621i $$-0.491233\pi$$
0.0275385 + 0.999621i $$0.491233\pi$$
$$90$$ −13.4584 −1.41864
$$91$$ 1.05251 0.110333
$$92$$ 3.60822 0.376183
$$93$$ 3.02279 0.313449
$$94$$ −8.73247 −0.900686
$$95$$ 2.36624 0.242771
$$96$$ −2.94749 −0.300827
$$97$$ −8.37894 −0.850753 −0.425376 0.905017i $$-0.639858\pi$$
−0.425376 + 0.905017i $$0.639858\pi$$
$$98$$ 4.24198 0.428505
$$99$$ 0 0
$$100$$ 0.599069 0.0599069
$$101$$ −16.5379 −1.64558 −0.822791 0.568344i $$-0.807584\pi$$
−0.822791 + 0.568344i $$0.807584\pi$$
$$102$$ 1.71324 0.169636
$$103$$ 2.79627 0.275525 0.137762 0.990465i $$-0.456009\pi$$
0.137762 + 0.990465i $$0.456009\pi$$
$$104$$ −0.633765 −0.0621457
$$105$$ 11.5827 1.13035
$$106$$ 12.5302 1.21704
$$107$$ 18.9681 1.83372 0.916859 0.399210i $$-0.130716\pi$$
0.916859 + 0.399210i $$0.130716\pi$$
$$108$$ 7.92194 0.762289
$$109$$ 17.0732 1.63531 0.817656 0.575707i $$-0.195273\pi$$
0.817656 + 0.575707i $$0.195273\pi$$
$$110$$ 0 0
$$111$$ 6.36348 0.603995
$$112$$ 1.66073 0.156924
$$113$$ 21.1114 1.98599 0.992997 0.118137i $$-0.0376922\pi$$
0.992997 + 0.118137i $$0.0376922\pi$$
$$114$$ −2.94749 −0.276058
$$115$$ 8.53789 0.796162
$$116$$ −9.13696 −0.848345
$$117$$ 3.60466 0.333251
$$118$$ −6.47623 −0.596185
$$119$$ −0.965305 −0.0884893
$$120$$ −6.97445 −0.636677
$$121$$ 0 0
$$122$$ −1.32146 −0.119639
$$123$$ 12.9717 1.16962
$$124$$ 1.02555 0.0920969
$$125$$ −10.4136 −0.931424
$$126$$ −9.44571 −0.841491
$$127$$ 4.62745 0.410620 0.205310 0.978697i $$-0.434180\pi$$
0.205310 + 0.978697i $$0.434180\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 28.2448 2.48682
$$130$$ −1.49964 −0.131527
$$131$$ −15.7763 −1.37838 −0.689191 0.724579i $$-0.742034\pi$$
−0.689191 + 0.724579i $$0.742034\pi$$
$$132$$ 0 0
$$133$$ 1.66073 0.144003
$$134$$ 0.393199 0.0339672
$$135$$ 18.7452 1.61333
$$136$$ 0.581254 0.0498421
$$137$$ −20.6366 −1.76310 −0.881552 0.472087i $$-0.843501\pi$$
−0.881552 + 0.472087i $$0.843501\pi$$
$$138$$ −10.6352 −0.905327
$$139$$ −9.56721 −0.811480 −0.405740 0.913989i $$-0.632986\pi$$
−0.405740 + 0.913989i $$0.632986\pi$$
$$140$$ 3.92967 0.332118
$$141$$ 25.7389 2.16760
$$142$$ 0.312308 0.0262083
$$143$$ 0 0
$$144$$ 5.68769 0.473974
$$145$$ −21.6202 −1.79546
$$146$$ 9.31372 0.770809
$$147$$ −12.5032 −1.03125
$$148$$ 2.15895 0.177464
$$149$$ 18.3096 1.49998 0.749988 0.661451i $$-0.230059\pi$$
0.749988 + 0.661451i $$0.230059\pi$$
$$150$$ −1.76575 −0.144173
$$151$$ 6.25562 0.509075 0.254538 0.967063i $$-0.418077\pi$$
0.254538 + 0.967063i $$0.418077\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −3.30599 −0.267274
$$154$$ 0 0
$$155$$ 2.42669 0.194916
$$156$$ 1.86801 0.149561
$$157$$ −6.15619 −0.491318 −0.245659 0.969356i $$-0.579004\pi$$
−0.245659 + 0.969356i $$0.579004\pi$$
$$158$$ −11.2704 −0.896622
$$159$$ −36.9325 −2.92894
$$160$$ −2.36624 −0.187067
$$161$$ 5.99227 0.472257
$$162$$ −6.28676 −0.493935
$$163$$ −11.8439 −0.927685 −0.463842 0.885918i $$-0.653530\pi$$
−0.463842 + 0.885918i $$0.653530\pi$$
$$164$$ 4.40093 0.343655
$$165$$ 0 0
$$166$$ 6.90413 0.535864
$$167$$ −4.89142 −0.378509 −0.189255 0.981928i $$-0.560607\pi$$
−0.189255 + 0.981928i $$0.560607\pi$$
$$168$$ −4.89498 −0.377656
$$169$$ −12.5983 −0.969103
$$170$$ 1.37538 0.105487
$$171$$ 5.68769 0.434949
$$172$$ 9.58267 0.730672
$$173$$ −18.1206 −1.37768 −0.688840 0.724913i $$-0.741880\pi$$
−0.688840 + 0.724913i $$0.741880\pi$$
$$174$$ 26.9311 2.04164
$$175$$ 0.994891 0.0752067
$$176$$ 0 0
$$177$$ 19.0886 1.43479
$$178$$ −0.519595 −0.0389453
$$179$$ 3.07947 0.230171 0.115085 0.993356i $$-0.463286\pi$$
0.115085 + 0.993356i $$0.463286\pi$$
$$180$$ 13.4584 1.00313
$$181$$ 0.0693910 0.00515779 0.00257889 0.999997i $$-0.499179\pi$$
0.00257889 + 0.999997i $$0.499179\pi$$
$$182$$ −1.05251 −0.0780173
$$183$$ 3.89498 0.287925
$$184$$ −3.60822 −0.266001
$$185$$ 5.10858 0.375590
$$186$$ −3.02279 −0.221642
$$187$$ 0 0
$$188$$ 8.73247 0.636881
$$189$$ 13.1562 0.956972
$$190$$ −2.36624 −0.171665
$$191$$ 13.6776 0.989677 0.494838 0.868985i $$-0.335227\pi$$
0.494838 + 0.868985i $$0.335227\pi$$
$$192$$ 2.94749 0.212717
$$193$$ 5.61830 0.404414 0.202207 0.979343i $$-0.435189\pi$$
0.202207 + 0.979343i $$0.435189\pi$$
$$194$$ 8.37894 0.601573
$$195$$ 4.42016 0.316534
$$196$$ −4.24198 −0.302999
$$197$$ 5.46494 0.389361 0.194680 0.980867i $$-0.437633\pi$$
0.194680 + 0.980867i $$0.437633\pi$$
$$198$$ 0 0
$$199$$ −13.5571 −0.961039 −0.480519 0.876984i $$-0.659552\pi$$
−0.480519 + 0.876984i $$0.659552\pi$$
$$200$$ −0.599069 −0.0423606
$$201$$ −1.15895 −0.0817459
$$202$$ 16.5379 1.16360
$$203$$ −15.1740 −1.06501
$$204$$ −1.71324 −0.119951
$$205$$ 10.4136 0.727320
$$206$$ −2.79627 −0.194826
$$207$$ 20.5224 1.42641
$$208$$ 0.633765 0.0439437
$$209$$ 0 0
$$210$$ −11.5827 −0.799280
$$211$$ 14.3562 0.988318 0.494159 0.869371i $$-0.335476\pi$$
0.494159 + 0.869371i $$0.335476\pi$$
$$212$$ −12.5302 −0.860575
$$213$$ −0.920526 −0.0630734
$$214$$ −18.9681 −1.29664
$$215$$ 22.6749 1.54641
$$216$$ −7.92194 −0.539020
$$217$$ 1.70316 0.115618
$$218$$ −17.0732 −1.15634
$$219$$ −27.4521 −1.85504
$$220$$ 0 0
$$221$$ −0.368378 −0.0247798
$$222$$ −6.36348 −0.427089
$$223$$ −5.16251 −0.345707 −0.172854 0.984948i $$-0.555299\pi$$
−0.172854 + 0.984948i $$0.555299\pi$$
$$224$$ −1.66073 −0.110962
$$225$$ 3.40732 0.227155
$$226$$ −21.1114 −1.40431
$$227$$ −18.3435 −1.21750 −0.608751 0.793361i $$-0.708329\pi$$
−0.608751 + 0.793361i $$0.708329\pi$$
$$228$$ 2.94749 0.195202
$$229$$ 18.3344 1.21157 0.605785 0.795629i $$-0.292859\pi$$
0.605785 + 0.795629i $$0.292859\pi$$
$$230$$ −8.53789 −0.562972
$$231$$ 0 0
$$232$$ 9.13696 0.599871
$$233$$ −14.6429 −0.959289 −0.479645 0.877463i $$-0.659234\pi$$
−0.479645 + 0.877463i $$0.659234\pi$$
$$234$$ −3.60466 −0.235644
$$235$$ 20.6631 1.34791
$$236$$ 6.47623 0.421567
$$237$$ 33.2193 2.15782
$$238$$ 0.965305 0.0625714
$$239$$ −12.5868 −0.814175 −0.407088 0.913389i $$-0.633456\pi$$
−0.407088 + 0.913389i $$0.633456\pi$$
$$240$$ 6.97445 0.450199
$$241$$ −2.79627 −0.180124 −0.0900619 0.995936i $$-0.528706\pi$$
−0.0900619 + 0.995936i $$0.528706\pi$$
$$242$$ 0 0
$$243$$ −5.23567 −0.335868
$$244$$ 1.32146 0.0845976
$$245$$ −10.0375 −0.641274
$$246$$ −12.9717 −0.827045
$$247$$ 0.633765 0.0403255
$$248$$ −1.02555 −0.0651223
$$249$$ −20.3498 −1.28962
$$250$$ 10.4136 0.658616
$$251$$ −1.72891 −0.109128 −0.0545640 0.998510i $$-0.517377\pi$$
−0.0545640 + 0.998510i $$0.517377\pi$$
$$252$$ 9.44571 0.595024
$$253$$ 0 0
$$254$$ −4.62745 −0.290352
$$255$$ −4.05393 −0.253867
$$256$$ 1.00000 0.0625000
$$257$$ 21.5414 1.34372 0.671859 0.740679i $$-0.265496\pi$$
0.671859 + 0.740679i $$0.265496\pi$$
$$258$$ −28.2448 −1.75845
$$259$$ 3.58543 0.222788
$$260$$ 1.49964 0.0930035
$$261$$ −51.9682 −3.21675
$$262$$ 15.7763 0.974664
$$263$$ −22.7920 −1.40541 −0.702707 0.711479i $$-0.748026\pi$$
−0.702707 + 0.711479i $$0.748026\pi$$
$$264$$ 0 0
$$265$$ −29.6493 −1.82134
$$266$$ −1.66073 −0.101826
$$267$$ 1.53150 0.0937263
$$268$$ −0.393199 −0.0240184
$$269$$ −18.6429 −1.13668 −0.568339 0.822794i $$-0.692414\pi$$
−0.568339 + 0.822794i $$0.692414\pi$$
$$270$$ −18.7452 −1.14080
$$271$$ −3.33571 −0.202630 −0.101315 0.994854i $$-0.532305\pi$$
−0.101315 + 0.994854i $$0.532305\pi$$
$$272$$ −0.581254 −0.0352437
$$273$$ 3.10226 0.187758
$$274$$ 20.6366 1.24670
$$275$$ 0 0
$$276$$ 10.6352 0.640163
$$277$$ 23.5883 1.41728 0.708641 0.705570i $$-0.249309\pi$$
0.708641 + 0.705570i $$0.249309\pi$$
$$278$$ 9.56721 0.573803
$$279$$ 5.83300 0.349212
$$280$$ −3.92967 −0.234843
$$281$$ 31.6366 1.88728 0.943641 0.330972i $$-0.107377\pi$$
0.943641 + 0.330972i $$0.107377\pi$$
$$282$$ −25.7389 −1.53273
$$283$$ 6.06757 0.360680 0.180340 0.983604i $$-0.442280\pi$$
0.180340 + 0.983604i $$0.442280\pi$$
$$284$$ −0.312308 −0.0185321
$$285$$ 6.97445 0.413131
$$286$$ 0 0
$$287$$ 7.30875 0.431422
$$288$$ −5.68769 −0.335150
$$289$$ −16.6621 −0.980126
$$290$$ 21.6202 1.26958
$$291$$ −24.6968 −1.44775
$$292$$ −9.31372 −0.545044
$$293$$ 16.8312 0.983288 0.491644 0.870796i $$-0.336396\pi$$
0.491644 + 0.870796i $$0.336396\pi$$
$$294$$ 12.5032 0.729201
$$295$$ 15.3243 0.892215
$$296$$ −2.15895 −0.125486
$$297$$ 0 0
$$298$$ −18.3096 −1.06064
$$299$$ 2.28676 0.132247
$$300$$ 1.76575 0.101946
$$301$$ 15.9142 0.917280
$$302$$ −6.25562 −0.359971
$$303$$ −48.7453 −2.80034
$$304$$ 1.00000 0.0573539
$$305$$ 3.12688 0.179044
$$306$$ 3.30599 0.188991
$$307$$ −5.10858 −0.291562 −0.145781 0.989317i $$-0.546570\pi$$
−0.145781 + 0.989317i $$0.546570\pi$$
$$308$$ 0 0
$$309$$ 8.24198 0.468870
$$310$$ −2.42669 −0.137827
$$311$$ −24.5635 −1.39287 −0.696435 0.717620i $$-0.745231\pi$$
−0.696435 + 0.717620i $$0.745231\pi$$
$$312$$ −1.86801 −0.105756
$$313$$ −24.2868 −1.37277 −0.686387 0.727237i $$-0.740804\pi$$
−0.686387 + 0.727237i $$0.740804\pi$$
$$314$$ 6.15619 0.347414
$$315$$ 22.3508 1.25932
$$316$$ 11.2704 0.634007
$$317$$ −28.6225 −1.60760 −0.803801 0.594898i $$-0.797192\pi$$
−0.803801 + 0.594898i $$0.797192\pi$$
$$318$$ 36.9325 2.07107
$$319$$ 0 0
$$320$$ 2.36624 0.132277
$$321$$ 55.9084 3.12050
$$322$$ −5.99227 −0.333936
$$323$$ −0.581254 −0.0323418
$$324$$ 6.28676 0.349264
$$325$$ 0.379669 0.0210602
$$326$$ 11.8439 0.655972
$$327$$ 50.3229 2.78287
$$328$$ −4.40093 −0.243001
$$329$$ 14.5023 0.799535
$$330$$ 0 0
$$331$$ −2.40960 −0.132444 −0.0662218 0.997805i $$-0.521094\pi$$
−0.0662218 + 0.997805i $$0.521094\pi$$
$$332$$ −6.90413 −0.378913
$$333$$ 12.2794 0.672909
$$334$$ 4.89142 0.267647
$$335$$ −0.930401 −0.0508332
$$336$$ 4.89498 0.267043
$$337$$ −17.5195 −0.954346 −0.477173 0.878809i $$-0.658338\pi$$
−0.477173 + 0.878809i $$0.658338\pi$$
$$338$$ 12.5983 0.685259
$$339$$ 62.2257 3.37963
$$340$$ −1.37538 −0.0745906
$$341$$ 0 0
$$342$$ −5.68769 −0.307555
$$343$$ −18.6699 −1.00808
$$344$$ −9.58267 −0.516663
$$345$$ 25.1653 1.35486
$$346$$ 18.1206 0.974167
$$347$$ −32.1534 −1.72609 −0.863043 0.505130i $$-0.831445\pi$$
−0.863043 + 0.505130i $$0.831445\pi$$
$$348$$ −26.9311 −1.44366
$$349$$ −0.407455 −0.0218106 −0.0109053 0.999941i $$-0.503471\pi$$
−0.0109053 + 0.999941i $$0.503471\pi$$
$$350$$ −0.994891 −0.0531792
$$351$$ 5.02065 0.267982
$$352$$ 0 0
$$353$$ 3.94158 0.209789 0.104895 0.994483i $$-0.466549\pi$$
0.104895 + 0.994483i $$0.466549\pi$$
$$354$$ −19.0886 −1.01455
$$355$$ −0.738995 −0.0392218
$$356$$ 0.519595 0.0275385
$$357$$ −2.84522 −0.150585
$$358$$ −3.07947 −0.162755
$$359$$ 20.0781 1.05968 0.529842 0.848097i $$-0.322251\pi$$
0.529842 + 0.848097i $$0.322251\pi$$
$$360$$ −13.4584 −0.709321
$$361$$ 1.00000 0.0526316
$$362$$ −0.0693910 −0.00364711
$$363$$ 0 0
$$364$$ 1.05251 0.0551666
$$365$$ −22.0385 −1.15355
$$366$$ −3.89498 −0.203594
$$367$$ −20.5918 −1.07488 −0.537442 0.843301i $$-0.680609\pi$$
−0.537442 + 0.843301i $$0.680609\pi$$
$$368$$ 3.60822 0.188091
$$369$$ 25.0311 1.30307
$$370$$ −5.10858 −0.265582
$$371$$ −20.8092 −1.08036
$$372$$ 3.02279 0.156724
$$373$$ 20.8813 1.08119 0.540597 0.841282i $$-0.318198\pi$$
0.540597 + 0.841282i $$0.318198\pi$$
$$374$$ 0 0
$$375$$ −30.6941 −1.58504
$$376$$ −8.73247 −0.450343
$$377$$ −5.79068 −0.298235
$$378$$ −13.1562 −0.676681
$$379$$ −0.753845 −0.0387224 −0.0193612 0.999813i $$-0.506163\pi$$
−0.0193612 + 0.999813i $$0.506163\pi$$
$$380$$ 2.36624 0.121385
$$381$$ 13.6394 0.698765
$$382$$ −13.6776 −0.699807
$$383$$ −18.3142 −0.935812 −0.467906 0.883778i $$-0.654991\pi$$
−0.467906 + 0.883778i $$0.654991\pi$$
$$384$$ −2.94749 −0.150413
$$385$$ 0 0
$$386$$ −5.61830 −0.285964
$$387$$ 54.5033 2.77056
$$388$$ −8.37894 −0.425376
$$389$$ −26.1206 −1.32436 −0.662182 0.749343i $$-0.730370\pi$$
−0.662182 + 0.749343i $$0.730370\pi$$
$$390$$ −4.42016 −0.223824
$$391$$ −2.09729 −0.106065
$$392$$ 4.24198 0.214252
$$393$$ −46.5005 −2.34564
$$394$$ −5.46494 −0.275320
$$395$$ 26.6683 1.34183
$$396$$ 0 0
$$397$$ −34.9079 −1.75198 −0.875988 0.482332i $$-0.839790\pi$$
−0.875988 + 0.482332i $$0.839790\pi$$
$$398$$ 13.5571 0.679557
$$399$$ 4.89498 0.245055
$$400$$ 0.599069 0.0299535
$$401$$ 6.35353 0.317280 0.158640 0.987336i $$-0.449289\pi$$
0.158640 + 0.987336i $$0.449289\pi$$
$$402$$ 1.15895 0.0578031
$$403$$ 0.649956 0.0323766
$$404$$ −16.5379 −0.822791
$$405$$ 14.8760 0.739192
$$406$$ 15.1740 0.753073
$$407$$ 0 0
$$408$$ 1.71324 0.0848180
$$409$$ −26.1816 −1.29460 −0.647299 0.762237i $$-0.724101\pi$$
−0.647299 + 0.762237i $$0.724101\pi$$
$$410$$ −10.4136 −0.514293
$$411$$ −60.8261 −3.00033
$$412$$ 2.79627 0.137762
$$413$$ 10.7553 0.529232
$$414$$ −20.5224 −1.00862
$$415$$ −16.3368 −0.801941
$$416$$ −0.633765 −0.0310729
$$417$$ −28.1992 −1.38092
$$418$$ 0 0
$$419$$ 8.15060 0.398183 0.199091 0.979981i $$-0.436201\pi$$
0.199091 + 0.979981i $$0.436201\pi$$
$$420$$ 11.5827 0.565176
$$421$$ 10.9531 0.533820 0.266910 0.963721i $$-0.413997\pi$$
0.266910 + 0.963721i $$0.413997\pi$$
$$422$$ −14.3562 −0.698847
$$423$$ 49.6676 2.41492
$$424$$ 12.5302 0.608518
$$425$$ −0.348211 −0.0168907
$$426$$ 0.920526 0.0445996
$$427$$ 2.19458 0.106203
$$428$$ 18.9681 0.916859
$$429$$ 0 0
$$430$$ −22.6749 −1.09348
$$431$$ −17.8510 −0.859852 −0.429926 0.902864i $$-0.641460\pi$$
−0.429926 + 0.902864i $$0.641460\pi$$
$$432$$ 7.92194 0.381145
$$433$$ 39.4138 1.89411 0.947054 0.321074i $$-0.104044\pi$$
0.947054 + 0.321074i $$0.104044\pi$$
$$434$$ −1.70316 −0.0817541
$$435$$ −63.7253 −3.05539
$$436$$ 17.0732 0.817656
$$437$$ 3.60822 0.172604
$$438$$ 27.4521 1.31171
$$439$$ −16.9993 −0.811331 −0.405666 0.914022i $$-0.632960\pi$$
−0.405666 + 0.914022i $$0.632960\pi$$
$$440$$ 0 0
$$441$$ −24.1271 −1.14891
$$442$$ 0.368378 0.0175220
$$443$$ 16.2668 0.772859 0.386430 0.922319i $$-0.373708\pi$$
0.386430 + 0.922319i $$0.373708\pi$$
$$444$$ 6.36348 0.301997
$$445$$ 1.22948 0.0582831
$$446$$ 5.16251 0.244452
$$447$$ 53.9672 2.55256
$$448$$ 1.66073 0.0784620
$$449$$ 16.5534 0.781201 0.390601 0.920560i $$-0.372267\pi$$
0.390601 + 0.920560i $$0.372267\pi$$
$$450$$ −3.40732 −0.160623
$$451$$ 0 0
$$452$$ 21.1114 0.992997
$$453$$ 18.4384 0.866311
$$454$$ 18.3435 0.860904
$$455$$ 2.49049 0.116756
$$456$$ −2.94749 −0.138029
$$457$$ 11.5805 0.541714 0.270857 0.962620i $$-0.412693\pi$$
0.270857 + 0.962620i $$0.412693\pi$$
$$458$$ −18.3344 −0.856709
$$459$$ −4.60466 −0.214927
$$460$$ 8.53789 0.398081
$$461$$ 5.42648 0.252736 0.126368 0.991983i $$-0.459668\pi$$
0.126368 + 0.991983i $$0.459668\pi$$
$$462$$ 0 0
$$463$$ 26.9168 1.25093 0.625466 0.780252i $$-0.284909\pi$$
0.625466 + 0.780252i $$0.284909\pi$$
$$464$$ −9.13696 −0.424173
$$465$$ 7.15263 0.331695
$$466$$ 14.6429 0.678320
$$467$$ 12.5581 0.581118 0.290559 0.956857i $$-0.406159\pi$$
0.290559 + 0.956857i $$0.406159\pi$$
$$468$$ 3.60466 0.166625
$$469$$ −0.652996 −0.0301526
$$470$$ −20.6631 −0.953117
$$471$$ −18.1453 −0.836092
$$472$$ −6.47623 −0.298093
$$473$$ 0 0
$$474$$ −33.2193 −1.52581
$$475$$ 0.599069 0.0274872
$$476$$ −0.965305 −0.0442447
$$477$$ −71.2677 −3.26312
$$478$$ 12.5868 0.575709
$$479$$ −40.2322 −1.83826 −0.919128 0.393960i $$-0.871105\pi$$
−0.919128 + 0.393960i $$0.871105\pi$$
$$480$$ −6.97445 −0.318339
$$481$$ 1.36827 0.0623875
$$482$$ 2.79627 0.127367
$$483$$ 17.6621 0.803655
$$484$$ 0 0
$$485$$ −19.8265 −0.900277
$$486$$ 5.23567 0.237495
$$487$$ −14.2795 −0.647066 −0.323533 0.946217i $$-0.604871\pi$$
−0.323533 + 0.946217i $$0.604871\pi$$
$$488$$ −1.32146 −0.0598195
$$489$$ −34.9097 −1.57867
$$490$$ 10.0375 0.453449
$$491$$ −14.0094 −0.632233 −0.316117 0.948720i $$-0.602379\pi$$
−0.316117 + 0.948720i $$0.602379\pi$$
$$492$$ 12.9717 0.584809
$$493$$ 5.31089 0.239191
$$494$$ −0.633765 −0.0285144
$$495$$ 0 0
$$496$$ 1.02555 0.0460484
$$497$$ −0.518659 −0.0232651
$$498$$ 20.3498 0.911898
$$499$$ 18.1506 0.812533 0.406266 0.913755i $$-0.366831\pi$$
0.406266 + 0.913755i $$0.366831\pi$$
$$500$$ −10.4136 −0.465712
$$501$$ −14.4174 −0.644122
$$502$$ 1.72891 0.0771651
$$503$$ 29.8672 1.33171 0.665855 0.746081i $$-0.268067\pi$$
0.665855 + 0.746081i $$0.268067\pi$$
$$504$$ −9.44571 −0.420745
$$505$$ −39.1325 −1.74138
$$506$$ 0 0
$$507$$ −37.1335 −1.64916
$$508$$ 4.62745 0.205310
$$509$$ 24.2045 1.07285 0.536423 0.843949i $$-0.319775\pi$$
0.536423 + 0.843949i $$0.319775\pi$$
$$510$$ 4.05393 0.179511
$$511$$ −15.4676 −0.684245
$$512$$ −1.00000 −0.0441942
$$513$$ 7.92194 0.349762
$$514$$ −21.5414 −0.950153
$$515$$ 6.61664 0.291564
$$516$$ 28.2448 1.24341
$$517$$ 0 0
$$518$$ −3.58543 −0.157535
$$519$$ −53.4102 −2.34445
$$520$$ −1.49964 −0.0657634
$$521$$ −6.36899 −0.279031 −0.139515 0.990220i $$-0.544554\pi$$
−0.139515 + 0.990220i $$0.544554\pi$$
$$522$$ 51.9682 2.27459
$$523$$ 21.2749 0.930284 0.465142 0.885236i $$-0.346003\pi$$
0.465142 + 0.885236i $$0.346003\pi$$
$$524$$ −15.7763 −0.689191
$$525$$ 2.93243 0.127982
$$526$$ 22.7920 0.993778
$$527$$ −0.596103 −0.0259667
$$528$$ 0 0
$$529$$ −9.98077 −0.433946
$$530$$ 29.6493 1.28788
$$531$$ 36.8348 1.59849
$$532$$ 1.66073 0.0720017
$$533$$ 2.78915 0.120812
$$534$$ −1.53150 −0.0662745
$$535$$ 44.8831 1.94046
$$536$$ 0.393199 0.0169836
$$537$$ 9.07672 0.391689
$$538$$ 18.6429 0.803753
$$539$$ 0 0
$$540$$ 18.7452 0.806664
$$541$$ 34.2668 1.47324 0.736622 0.676304i $$-0.236420\pi$$
0.736622 + 0.676304i $$0.236420\pi$$
$$542$$ 3.33571 0.143281
$$543$$ 0.204529 0.00877718
$$544$$ 0.581254 0.0249211
$$545$$ 40.3991 1.73051
$$546$$ −3.10226 −0.132765
$$547$$ 31.2732 1.33715 0.668573 0.743647i $$-0.266906\pi$$
0.668573 + 0.743647i $$0.266906\pi$$
$$548$$ −20.6366 −0.881552
$$549$$ 7.51604 0.320777
$$550$$ 0 0
$$551$$ −9.13696 −0.389248
$$552$$ −10.6352 −0.452663
$$553$$ 18.7170 0.795928
$$554$$ −23.5883 −1.00217
$$555$$ 15.0575 0.639155
$$556$$ −9.56721 −0.405740
$$557$$ −24.7198 −1.04741 −0.523707 0.851899i $$-0.675451\pi$$
−0.523707 + 0.851899i $$0.675451\pi$$
$$558$$ −5.83300 −0.246931
$$559$$ 6.07316 0.256867
$$560$$ 3.92967 0.166059
$$561$$ 0 0
$$562$$ −31.6366 −1.33451
$$563$$ −8.94251 −0.376882 −0.188441 0.982085i $$-0.560343\pi$$
−0.188441 + 0.982085i $$0.560343\pi$$
$$564$$ 25.7389 1.08380
$$565$$ 49.9546 2.10160
$$566$$ −6.06757 −0.255039
$$567$$ 10.4406 0.438464
$$568$$ 0.312308 0.0131042
$$569$$ 0.764409 0.0320457 0.0160228 0.999872i $$-0.494900\pi$$
0.0160228 + 0.999872i $$0.494900\pi$$
$$570$$ −6.97445 −0.292128
$$571$$ −4.11866 −0.172361 −0.0861804 0.996280i $$-0.527466\pi$$
−0.0861804 + 0.996280i $$0.527466\pi$$
$$572$$ 0 0
$$573$$ 40.3146 1.68417
$$574$$ −7.30875 −0.305061
$$575$$ 2.16157 0.0901438
$$576$$ 5.68769 0.236987
$$577$$ 26.8538 1.11794 0.558968 0.829189i $$-0.311197\pi$$
0.558968 + 0.829189i $$0.311197\pi$$
$$578$$ 16.6621 0.693054
$$579$$ 16.5599 0.688205
$$580$$ −21.6202 −0.897730
$$581$$ −11.4659 −0.475685
$$582$$ 24.6968 1.02372
$$583$$ 0 0
$$584$$ 9.31372 0.385405
$$585$$ 8.52947 0.352650
$$586$$ −16.8312 −0.695289
$$587$$ 11.4832 0.473964 0.236982 0.971514i $$-0.423842\pi$$
0.236982 + 0.971514i $$0.423842\pi$$
$$588$$ −12.5032 −0.515623
$$589$$ 1.02555 0.0422570
$$590$$ −15.3243 −0.630891
$$591$$ 16.1079 0.662589
$$592$$ 2.15895 0.0887322
$$593$$ −32.3508 −1.32849 −0.664245 0.747515i $$-0.731247\pi$$
−0.664245 + 0.747515i $$0.731247\pi$$
$$594$$ 0 0
$$595$$ −2.28414 −0.0936405
$$596$$ 18.3096 0.749988
$$597$$ −39.9595 −1.63543
$$598$$ −2.28676 −0.0935126
$$599$$ 6.06380 0.247760 0.123880 0.992297i $$-0.460466\pi$$
0.123880 + 0.992297i $$0.460466\pi$$
$$600$$ −1.76575 −0.0720864
$$601$$ 23.0923 0.941953 0.470976 0.882146i $$-0.343902\pi$$
0.470976 + 0.882146i $$0.343902\pi$$
$$602$$ −15.9142 −0.648615
$$603$$ −2.23639 −0.0910730
$$604$$ 6.25562 0.254538
$$605$$ 0 0
$$606$$ 48.7453 1.98014
$$607$$ 11.1808 0.453815 0.226907 0.973916i $$-0.427139\pi$$
0.226907 + 0.973916i $$0.427139\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ −44.7252 −1.81236
$$610$$ −3.12688 −0.126604
$$611$$ 5.53433 0.223895
$$612$$ −3.30599 −0.133637
$$613$$ 33.8355 1.36660 0.683302 0.730136i $$-0.260543\pi$$
0.683302 + 0.730136i $$0.260543\pi$$
$$614$$ 5.10858 0.206166
$$615$$ 30.6941 1.23770
$$616$$ 0 0
$$617$$ 24.9927 1.00617 0.503085 0.864237i $$-0.332198\pi$$
0.503085 + 0.864237i $$0.332198\pi$$
$$618$$ −8.24198 −0.331541
$$619$$ 7.68493 0.308884 0.154442 0.988002i $$-0.450642\pi$$
0.154442 + 0.988002i $$0.450642\pi$$
$$620$$ 2.42669 0.0974581
$$621$$ 28.5841 1.14704
$$622$$ 24.5635 0.984907
$$623$$ 0.862906 0.0345716
$$624$$ 1.86801 0.0747804
$$625$$ −27.6365 −1.10546
$$626$$ 24.2868 0.970697
$$627$$ 0 0
$$628$$ −6.15619 −0.245659
$$629$$ −1.25490 −0.0500360
$$630$$ −22.3508 −0.890476
$$631$$ 25.7004 1.02312 0.511558 0.859249i $$-0.329068\pi$$
0.511558 + 0.859249i $$0.329068\pi$$
$$632$$ −11.2704 −0.448311
$$633$$ 42.3146 1.68185
$$634$$ 28.6225 1.13675
$$635$$ 10.9496 0.434523
$$636$$ −36.9325 −1.46447
$$637$$ −2.68842 −0.106519
$$638$$ 0 0
$$639$$ −1.77631 −0.0702699
$$640$$ −2.36624 −0.0935337
$$641$$ 4.03846 0.159510 0.0797548 0.996815i $$-0.474586\pi$$
0.0797548 + 0.996815i $$0.474586\pi$$
$$642$$ −55.9084 −2.20653
$$643$$ 5.44194 0.214609 0.107305 0.994226i $$-0.465778\pi$$
0.107305 + 0.994226i $$0.465778\pi$$
$$644$$ 5.99227 0.236128
$$645$$ 66.8339 2.63158
$$646$$ 0.581254 0.0228691
$$647$$ −32.4046 −1.27395 −0.636977 0.770882i $$-0.719816\pi$$
−0.636977 + 0.770882i $$0.719816\pi$$
$$648$$ −6.28676 −0.246967
$$649$$ 0 0
$$650$$ −0.379669 −0.0148918
$$651$$ 5.02003 0.196751
$$652$$ −11.8439 −0.463842
$$653$$ 24.7672 0.969217 0.484609 0.874731i $$-0.338962\pi$$
0.484609 + 0.874731i $$0.338962\pi$$
$$654$$ −50.3229 −1.96778
$$655$$ −37.3305 −1.45862
$$656$$ 4.40093 0.171827
$$657$$ −52.9736 −2.06670
$$658$$ −14.5023 −0.565357
$$659$$ −27.5600 −1.07358 −0.536792 0.843715i $$-0.680364\pi$$
−0.536792 + 0.843715i $$0.680364\pi$$
$$660$$ 0 0
$$661$$ −7.53372 −0.293028 −0.146514 0.989209i $$-0.546805\pi$$
−0.146514 + 0.989209i $$0.546805\pi$$
$$662$$ 2.40960 0.0936517
$$663$$ −1.08579 −0.0421686
$$664$$ 6.90413 0.267932
$$665$$ 3.92967 0.152386
$$666$$ −12.2794 −0.475818
$$667$$ −32.9681 −1.27653
$$668$$ −4.89142 −0.189255
$$669$$ −15.2164 −0.588301
$$670$$ 0.930401 0.0359445
$$671$$ 0 0
$$672$$ −4.89498 −0.188828
$$673$$ −8.79534 −0.339035 −0.169518 0.985527i $$-0.554221\pi$$
−0.169518 + 0.985527i $$0.554221\pi$$
$$674$$ 17.5195 0.674824
$$675$$ 4.74579 0.182666
$$676$$ −12.5983 −0.484552
$$677$$ 15.8310 0.608434 0.304217 0.952603i $$-0.401605\pi$$
0.304217 + 0.952603i $$0.401605\pi$$
$$678$$ −62.2257 −2.38976
$$679$$ −13.9151 −0.534014
$$680$$ 1.37538 0.0527435
$$681$$ −54.0673 −2.07186
$$682$$ 0 0
$$683$$ 38.6640 1.47944 0.739719 0.672916i $$-0.234958\pi$$
0.739719 + 0.672916i $$0.234958\pi$$
$$684$$ 5.68769 0.217474
$$685$$ −48.8310 −1.86574
$$686$$ 18.6699 0.712819
$$687$$ 54.0404 2.06177
$$688$$ 9.58267 0.365336
$$689$$ −7.94117 −0.302535
$$690$$ −25.1653 −0.958028
$$691$$ −27.4678 −1.04492 −0.522462 0.852663i $$-0.674986\pi$$
−0.522462 + 0.852663i $$0.674986\pi$$
$$692$$ −18.1206 −0.688840
$$693$$ 0 0
$$694$$ 32.1534 1.22053
$$695$$ −22.6383 −0.858718
$$696$$ 26.9311 1.02082
$$697$$ −2.55806 −0.0968933
$$698$$ 0.407455 0.0154224
$$699$$ −43.1598 −1.63245
$$700$$ 0.994891 0.0376034
$$701$$ −1.50892 −0.0569911 −0.0284955 0.999594i $$-0.509072\pi$$
−0.0284955 + 0.999594i $$0.509072\pi$$
$$702$$ −5.02065 −0.189492
$$703$$ 2.15895 0.0814263
$$704$$ 0 0
$$705$$ 60.9042 2.29378
$$706$$ −3.94158 −0.148343
$$707$$ −27.4649 −1.03293
$$708$$ 19.0886 0.717394
$$709$$ 16.2959 0.612006 0.306003 0.952031i $$-0.401008\pi$$
0.306003 + 0.952031i $$0.401008\pi$$
$$710$$ 0.738995 0.0277340
$$711$$ 64.1023 2.40403
$$712$$ −0.519595 −0.0194726
$$713$$ 3.70040 0.138581
$$714$$ 2.84522 0.106480
$$715$$ 0 0
$$716$$ 3.07947 0.115085
$$717$$ −37.0996 −1.38551
$$718$$ −20.0781 −0.749309
$$719$$ 40.6785 1.51705 0.758526 0.651643i $$-0.225920\pi$$
0.758526 + 0.651643i $$0.225920\pi$$
$$720$$ 13.4584 0.501566
$$721$$ 4.64385 0.172946
$$722$$ −1.00000 −0.0372161
$$723$$ −8.24198 −0.306523
$$724$$ 0.0693910 0.00257889
$$725$$ −5.47367 −0.203287
$$726$$ 0 0
$$727$$ 27.7545 1.02936 0.514679 0.857383i $$-0.327911\pi$$
0.514679 + 0.857383i $$0.327911\pi$$
$$728$$ −1.05251 −0.0390087
$$729$$ −34.2924 −1.27009
$$730$$ 22.0385 0.815680
$$731$$ −5.56996 −0.206013
$$732$$ 3.89498 0.143963
$$733$$ −6.89854 −0.254803 −0.127402 0.991851i $$-0.540664\pi$$
−0.127402 + 0.991851i $$0.540664\pi$$
$$734$$ 20.5918 0.760058
$$735$$ −29.5855 −1.09128
$$736$$ −3.60822 −0.133001
$$737$$ 0 0
$$738$$ −25.0311 −0.921409
$$739$$ 49.2039 1.80999 0.904997 0.425418i $$-0.139873\pi$$
0.904997 + 0.425418i $$0.139873\pi$$
$$740$$ 5.10858 0.187795
$$741$$ 1.86801 0.0686232
$$742$$ 20.8092 0.763929
$$743$$ 10.4046 0.381709 0.190854 0.981618i $$-0.438874\pi$$
0.190854 + 0.981618i $$0.438874\pi$$
$$744$$ −3.02279 −0.110821
$$745$$ 43.3247 1.58729
$$746$$ −20.8813 −0.764520
$$747$$ −39.2685 −1.43676
$$748$$ 0 0
$$749$$ 31.5009 1.15102
$$750$$ 30.6941 1.12079
$$751$$ 37.7103 1.37607 0.688035 0.725677i $$-0.258474\pi$$
0.688035 + 0.725677i $$0.258474\pi$$
$$752$$ 8.73247 0.318440
$$753$$ −5.09595 −0.185707
$$754$$ 5.79068 0.210884
$$755$$ 14.8023 0.538710
$$756$$ 13.1562 0.478486
$$757$$ 14.3972 0.523274 0.261637 0.965166i $$-0.415738\pi$$
0.261637 + 0.965166i $$0.415738\pi$$
$$758$$ 0.753845 0.0273809
$$759$$ 0 0
$$760$$ −2.36624 −0.0858324
$$761$$ −39.7311 −1.44025 −0.720126 0.693843i $$-0.755916\pi$$
−0.720126 + 0.693843i $$0.755916\pi$$
$$762$$ −13.6394 −0.494102
$$763$$ 28.3539 1.02648
$$764$$ 13.6776 0.494838
$$765$$ −7.82276 −0.282832
$$766$$ 18.3142 0.661719
$$767$$ 4.10441 0.148202
$$768$$ 2.94749 0.106358
$$769$$ −2.27170 −0.0819197 −0.0409598 0.999161i $$-0.513042\pi$$
−0.0409598 + 0.999161i $$0.513042\pi$$
$$770$$ 0 0
$$771$$ 63.4932 2.28665
$$772$$ 5.61830 0.202207
$$773$$ 3.64157 0.130978 0.0654891 0.997853i $$-0.479139\pi$$
0.0654891 + 0.997853i $$0.479139\pi$$
$$774$$ −54.5033 −1.95908
$$775$$ 0.614374 0.0220690
$$776$$ 8.37894 0.300786
$$777$$ 10.5680 0.379125
$$778$$ 26.1206 0.936467
$$779$$ 4.40093 0.157680
$$780$$ 4.42016 0.158267
$$781$$ 0 0
$$782$$ 2.09729 0.0749989
$$783$$ −72.3825 −2.58674
$$784$$ −4.24198 −0.151499
$$785$$ −14.5670 −0.519918
$$786$$ 46.5005 1.65862
$$787$$ 0.279643 0.00996821 0.00498410 0.999988i $$-0.498414\pi$$
0.00498410 + 0.999988i $$0.498414\pi$$
$$788$$ 5.46494 0.194680
$$789$$ −67.1791 −2.39164
$$790$$ −26.6683 −0.948816
$$791$$ 35.0603 1.24660
$$792$$ 0 0
$$793$$ 0.837492 0.0297402
$$794$$ 34.9079 1.23883
$$795$$ −87.3910 −3.09944
$$796$$ −13.5571 −0.480519
$$797$$ −3.66014 −0.129649 −0.0648243 0.997897i $$-0.520649\pi$$
−0.0648243 + 0.997897i $$0.520649\pi$$
$$798$$ −4.89498 −0.173280
$$799$$ −5.07578 −0.179568
$$800$$ −0.599069 −0.0211803
$$801$$ 2.95530 0.104420
$$802$$ −6.35353 −0.224351
$$803$$ 0 0
$$804$$ −1.15895 −0.0408730
$$805$$ 14.1791 0.499748
$$806$$ −0.649956 −0.0228937
$$807$$ −54.9498 −1.93432
$$808$$ 16.5379 0.581801
$$809$$ −27.0990 −0.952749 −0.476375 0.879242i $$-0.658049\pi$$
−0.476375 + 0.879242i $$0.658049\pi$$
$$810$$ −14.8760 −0.522688
$$811$$ −31.4212 −1.10335 −0.551673 0.834060i $$-0.686010\pi$$
−0.551673 + 0.834060i $$0.686010\pi$$
$$812$$ −15.1740 −0.532503
$$813$$ −9.83198 −0.344823
$$814$$ 0 0
$$815$$ −28.0254 −0.981687
$$816$$ −1.71324 −0.0599754
$$817$$ 9.58267 0.335255
$$818$$ 26.1816 0.915418
$$819$$ 5.98636 0.209180
$$820$$ 10.4136 0.363660
$$821$$ 20.1304 0.702557 0.351279 0.936271i $$-0.385747\pi$$
0.351279 + 0.936271i $$0.385747\pi$$
$$822$$ 60.8261 2.12156
$$823$$ −7.23922 −0.252344 −0.126172 0.992008i $$-0.540269\pi$$
−0.126172 + 0.992008i $$0.540269\pi$$
$$824$$ −2.79627 −0.0974128
$$825$$ 0 0
$$826$$ −10.7553 −0.374223
$$827$$ −23.1088 −0.803571 −0.401786 0.915734i $$-0.631610\pi$$
−0.401786 + 0.915734i $$0.631610\pi$$
$$828$$ 20.5224 0.713204
$$829$$ 21.7081 0.753955 0.376977 0.926223i $$-0.376963\pi$$
0.376977 + 0.926223i $$0.376963\pi$$
$$830$$ 16.3368 0.567058
$$831$$ 69.5261 2.41183
$$832$$ 0.633765 0.0219718
$$833$$ 2.46567 0.0854303
$$834$$ 28.1992 0.976460
$$835$$ −11.5742 −0.400543
$$836$$ 0 0
$$837$$ 8.12433 0.280818
$$838$$ −8.15060 −0.281558
$$839$$ −21.1935 −0.731681 −0.365841 0.930678i $$-0.619218\pi$$
−0.365841 + 0.930678i $$0.619218\pi$$
$$840$$ −11.5827 −0.399640
$$841$$ 54.4840 1.87876
$$842$$ −10.9531 −0.377468
$$843$$ 93.2485 3.21165
$$844$$ 14.3562 0.494159
$$845$$ −29.8106 −1.02552
$$846$$ −49.6676 −1.70761
$$847$$ 0 0
$$848$$ −12.5302 −0.430287
$$849$$ 17.8841 0.613780
$$850$$ 0.348211 0.0119435
$$851$$ 7.78996 0.267036
$$852$$ −0.920526 −0.0315367
$$853$$ −30.5835 −1.04716 −0.523579 0.851977i $$-0.675404\pi$$
−0.523579 + 0.851977i $$0.675404\pi$$
$$854$$ −2.19458 −0.0750970
$$855$$ 13.4584 0.460268
$$856$$ −18.9681 −0.648318
$$857$$ −48.3757 −1.65248 −0.826241 0.563317i $$-0.809525\pi$$
−0.826241 + 0.563317i $$0.809525\pi$$
$$858$$ 0 0
$$859$$ −57.0941 −1.94802 −0.974012 0.226495i $$-0.927273\pi$$
−0.974012 + 0.226495i $$0.927273\pi$$
$$860$$ 22.6749 0.773206
$$861$$ 21.5425 0.734165
$$862$$ 17.8510 0.608007
$$863$$ 3.31963 0.113002 0.0565008 0.998403i $$-0.482006\pi$$
0.0565008 + 0.998403i $$0.482006\pi$$
$$864$$ −7.92194 −0.269510
$$865$$ −42.8775 −1.45788
$$866$$ −39.4138 −1.33934
$$867$$ −49.1115 −1.66791
$$868$$ 1.70316 0.0578089
$$869$$ 0 0
$$870$$ 63.7253 2.16049
$$871$$ −0.249195 −0.00844367
$$872$$ −17.0732 −0.578170
$$873$$ −47.6568 −1.61294
$$874$$ −3.60822 −0.122050
$$875$$ −17.2942 −0.584651
$$876$$ −27.4521 −0.927521
$$877$$ −4.46676 −0.150832 −0.0754159 0.997152i $$-0.524028\pi$$
−0.0754159 + 0.997152i $$0.524028\pi$$
$$878$$ 16.9993 0.573698
$$879$$ 49.6097 1.67329
$$880$$ 0 0
$$881$$ 6.10320 0.205622 0.102811 0.994701i $$-0.467216\pi$$
0.102811 + 0.994701i $$0.467216\pi$$
$$882$$ 24.1271 0.812401
$$883$$ 6.98526 0.235073 0.117536 0.993069i $$-0.462500\pi$$
0.117536 + 0.993069i $$0.462500\pi$$
$$884$$ −0.368378 −0.0123899
$$885$$ 45.1682 1.51831
$$886$$ −16.2668 −0.546494
$$887$$ −39.3670 −1.32182 −0.660908 0.750467i $$-0.729829\pi$$
−0.660908 + 0.750467i $$0.729829\pi$$
$$888$$ −6.36348 −0.213544
$$889$$ 7.68493 0.257744
$$890$$ −1.22948 −0.0412124
$$891$$ 0 0
$$892$$ −5.16251 −0.172854
$$893$$ 8.73247 0.292221
$$894$$ −53.9672 −1.80493
$$895$$ 7.28676 0.243570
$$896$$ −1.66073 −0.0554810
$$897$$ 6.74020 0.225049
$$898$$ −16.5534 −0.552393
$$899$$ −9.37039 −0.312520
$$900$$ 3.40732 0.113577
$$901$$ 7.28320 0.242639
$$902$$ 0 0
$$903$$ 46.9070 1.56097
$$904$$ −21.1114 −0.702155
$$905$$ 0.164195 0.00545804
$$906$$ −18.4384 −0.612574
$$907$$ 56.7300 1.88369 0.941844 0.336050i $$-0.109091\pi$$
0.941844 + 0.336050i $$0.109091\pi$$
$$908$$ −18.3435 −0.608751
$$909$$ −94.0624 −3.11985
$$910$$ −2.49049 −0.0825589
$$911$$ −47.8355 −1.58486 −0.792431 0.609962i $$-0.791185\pi$$
−0.792431 + 0.609962i $$0.791185\pi$$
$$912$$ 2.94749 0.0976011
$$913$$ 0 0
$$914$$ −11.5805 −0.383050
$$915$$ 9.21643 0.304686
$$916$$ 18.3344 0.605785
$$917$$ −26.2002 −0.865206
$$918$$ 4.60466 0.151976
$$919$$ 0.295588 0.00975054 0.00487527 0.999988i $$-0.498448\pi$$
0.00487527 + 0.999988i $$0.498448\pi$$
$$920$$ −8.53789 −0.281486
$$921$$ −15.0575 −0.496161
$$922$$ −5.42648 −0.178712
$$923$$ −0.197930 −0.00651495
$$924$$ 0 0
$$925$$ 1.29336 0.0425254
$$926$$ −26.9168 −0.884542
$$927$$ 15.9043 0.522367
$$928$$ 9.13696 0.299935
$$929$$ −18.6254 −0.611080 −0.305540 0.952179i $$-0.598837\pi$$
−0.305540 + 0.952179i $$0.598837\pi$$
$$930$$ −7.15263 −0.234544
$$931$$ −4.24198 −0.139025
$$932$$ −14.6429 −0.479645
$$933$$ −72.4007 −2.37029
$$934$$ −12.5581 −0.410912
$$935$$ 0 0
$$936$$ −3.60466 −0.117822
$$937$$ −50.5786 −1.65233 −0.826165 0.563428i $$-0.809482\pi$$
−0.826165 + 0.563428i $$0.809482\pi$$
$$938$$ 0.652996 0.0213211
$$939$$ −71.5852 −2.33609
$$940$$ 20.6631 0.673955
$$941$$ 47.1169 1.53597 0.767983 0.640470i $$-0.221261\pi$$
0.767983 + 0.640470i $$0.221261\pi$$
$$942$$ 18.1453 0.591206
$$943$$ 15.8795 0.517108
$$944$$ 6.47623 0.210783
$$945$$ 31.1306 1.01268
$$946$$ 0 0
$$947$$ 8.28227 0.269138 0.134569 0.990904i $$-0.457035\pi$$
0.134569 + 0.990904i $$0.457035\pi$$
$$948$$ 33.2193 1.07891
$$949$$ −5.90271 −0.191610
$$950$$ −0.599069 −0.0194364
$$951$$ −84.3646 −2.73571
$$952$$ 0.965305 0.0312857
$$953$$ −10.0765 −0.326410 −0.163205 0.986592i $$-0.552183\pi$$
−0.163205 + 0.986592i $$0.552183\pi$$
$$954$$ 71.2677 2.30738
$$955$$ 32.3644 1.04729
$$956$$ −12.5868 −0.407088
$$957$$ 0 0
$$958$$ 40.2322 1.29984
$$959$$ −34.2718 −1.10669
$$960$$ 6.97445 0.225099
$$961$$ −29.9483 −0.966073
$$962$$ −1.36827 −0.0441147
$$963$$ 107.885 3.47654
$$964$$ −2.79627 −0.0900619
$$965$$ 13.2942 0.427956
$$966$$ −17.6621 −0.568270
$$967$$ 48.6402 1.56416 0.782082 0.623175i $$-0.214158\pi$$
0.782082 + 0.623175i $$0.214158\pi$$
$$968$$ 0 0
$$969$$ −1.71324 −0.0550372
$$970$$ 19.8265 0.636592
$$971$$ −4.89578 −0.157113 −0.0785565 0.996910i $$-0.525031\pi$$
−0.0785565 + 0.996910i $$0.525031\pi$$
$$972$$ −5.23567 −0.167934
$$973$$ −15.8885 −0.509363
$$974$$ 14.2795 0.457545
$$975$$ 1.11907 0.0358389
$$976$$ 1.32146 0.0422988
$$977$$ −36.1052 −1.15511 −0.577553 0.816353i $$-0.695992\pi$$
−0.577553 + 0.816353i $$0.695992\pi$$
$$978$$ 34.9097 1.11629
$$979$$ 0 0
$$980$$ −10.0375 −0.320637
$$981$$ 97.1069 3.10038
$$982$$ 14.0094 0.447057
$$983$$ −18.7371 −0.597621 −0.298811 0.954312i $$-0.596590\pi$$
−0.298811 + 0.954312i $$0.596590\pi$$
$$984$$ −12.9717 −0.413523
$$985$$ 12.9313 0.412027
$$986$$ −5.31089 −0.169133
$$987$$ 42.7453 1.36060
$$988$$ 0.633765 0.0201627
$$989$$ 34.5764 1.09946
$$990$$ 0 0
$$991$$ −8.60669 −0.273400 −0.136700 0.990612i $$-0.543650\pi$$
−0.136700 + 0.990612i $$0.543650\pi$$
$$992$$ −1.02555 −0.0325612
$$993$$ −7.10226 −0.225384
$$994$$ 0.518659 0.0164509
$$995$$ −32.0793 −1.01698
$$996$$ −20.3498 −0.644809
$$997$$ −13.3500 −0.422798 −0.211399 0.977400i $$-0.567802\pi$$
−0.211399 + 0.977400i $$0.567802\pi$$
$$998$$ −18.1506 −0.574547
$$999$$ 17.1031 0.541117
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 4598.2.a.br.1.4 4
11.10 odd 2 4598.2.a.bu.1.4 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.br.1.4 4 1.1 even 1 trivial
4598.2.a.bu.1.4 yes 4 11.10 odd 2