# Properties

 Label 5445.2.a.bv.1.1 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.09529$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} +O(q^{10})$$ $$q-1.39026 q^{2} -0.0671858 q^{4} +1.00000 q^{5} -1.27759 q^{7} +2.87392 q^{8} -1.39026 q^{10} +1.41837 q^{13} +1.77618 q^{14} -3.86111 q^{16} +5.00000 q^{17} -0.158146 q^{19} -0.0671858 q^{20} +5.00829 q^{23} +1.00000 q^{25} -1.97189 q^{26} +0.0858360 q^{28} +6.27515 q^{29} +3.04981 q^{31} -0.379898 q^{32} -6.95128 q^{34} -1.27759 q^{35} -4.69991 q^{37} +0.219863 q^{38} +2.87392 q^{40} +7.58597 q^{41} -5.41324 q^{43} -6.96281 q^{46} +8.23211 q^{47} -5.36776 q^{49} -1.39026 q^{50} -0.0952940 q^{52} -9.36648 q^{53} -3.67169 q^{56} -8.72406 q^{58} -8.09846 q^{59} +14.2917 q^{61} -4.24002 q^{62} +8.25038 q^{64} +1.41837 q^{65} -7.38362 q^{67} -0.335929 q^{68} +1.77618 q^{70} +6.77618 q^{71} -8.66107 q^{73} +6.53409 q^{74} +0.0106252 q^{76} +2.54572 q^{79} -3.86111 q^{80} -10.5464 q^{82} +10.1221 q^{83} +5.00000 q^{85} +7.52580 q^{86} -11.0447 q^{89} -1.81209 q^{91} -0.336486 q^{92} -11.4447 q^{94} -0.158146 q^{95} +6.41196 q^{97} +7.46257 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8}+O(q^{10})$$ 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8 $$4 q + 5 q^{2} + 9 q^{4} + 4 q^{5} + 2 q^{7} + 15 q^{8} + 5 q^{10} - 3 q^{13} + 5 q^{14} + 15 q^{16} + 20 q^{17} - 3 q^{19} + 9 q^{20} + 5 q^{23} + 4 q^{25} - 6 q^{26} - 3 q^{28} + 5 q^{29} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} - 7 q^{37} - q^{38} + 15 q^{40} + 20 q^{41} + 2 q^{43} - 7 q^{46} + 20 q^{47} + 8 q^{49} + 5 q^{50} + 7 q^{52} - 6 q^{53} - 10 q^{56} - 21 q^{58} + 5 q^{59} + 7 q^{61} - 12 q^{62} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} + 5 q^{70} + 25 q^{71} - 23 q^{73} - 7 q^{74} + 7 q^{76} + 15 q^{80} + 11 q^{82} + 33 q^{83} + 20 q^{85} + 12 q^{86} - 16 q^{89} - 24 q^{91} + 17 q^{94} - 3 q^{95} + 25 q^{98}+O(q^{100})$$ 4 * q + 5 * q^2 + 9 * q^4 + 4 * q^5 + 2 * q^7 + 15 * q^8 + 5 * q^10 - 3 * q^13 + 5 * q^14 + 15 * q^16 + 20 * q^17 - 3 * q^19 + 9 * q^20 + 5 * q^23 + 4 * q^25 - 6 * q^26 - 3 * q^28 + 5 * q^29 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 - 7 * q^37 - q^38 + 15 * q^40 + 20 * q^41 + 2 * q^43 - 7 * q^46 + 20 * q^47 + 8 * q^49 + 5 * q^50 + 7 * q^52 - 6 * q^53 - 10 * q^56 - 21 * q^58 + 5 * q^59 + 7 * q^61 - 12 * q^62 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 + 5 * q^70 + 25 * q^71 - 23 * q^73 - 7 * q^74 + 7 * q^76 + 15 * q^80 + 11 * q^82 + 33 * q^83 + 20 * q^85 + 12 * q^86 - 16 * q^89 - 24 * q^91 + 17 * q^94 - 3 * q^95 + 25 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.39026 −0.983060 −0.491530 0.870861i $$-0.663562\pi$$
−0.491530 + 0.870861i $$0.663562\pi$$
$$3$$ 0 0
$$4$$ −0.0671858 −0.0335929
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.27759 −0.482884 −0.241442 0.970415i $$-0.577620\pi$$
−0.241442 + 0.970415i $$0.577620\pi$$
$$8$$ 2.87392 1.01608
$$9$$ 0 0
$$10$$ −1.39026 −0.439638
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 1.41837 0.393384 0.196692 0.980465i $$-0.436980\pi$$
0.196692 + 0.980465i $$0.436980\pi$$
$$14$$ 1.77618 0.474704
$$15$$ 0 0
$$16$$ −3.86111 −0.965279
$$17$$ 5.00000 1.21268 0.606339 0.795206i $$-0.292637\pi$$
0.606339 + 0.795206i $$0.292637\pi$$
$$18$$ 0 0
$$19$$ −0.158146 −0.0362811 −0.0181406 0.999835i $$-0.505775\pi$$
−0.0181406 + 0.999835i $$0.505775\pi$$
$$20$$ −0.0671858 −0.0150232
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.00829 1.04430 0.522150 0.852853i $$-0.325130\pi$$
0.522150 + 0.852853i $$0.325130\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −1.97189 −0.386720
$$27$$ 0 0
$$28$$ 0.0858360 0.0162215
$$29$$ 6.27515 1.16527 0.582633 0.812736i $$-0.302023\pi$$
0.582633 + 0.812736i $$0.302023\pi$$
$$30$$ 0 0
$$31$$ 3.04981 0.547763 0.273881 0.961763i $$-0.411692\pi$$
0.273881 + 0.961763i $$0.411692\pi$$
$$32$$ −0.379898 −0.0671570
$$33$$ 0 0
$$34$$ −6.95128 −1.19214
$$35$$ −1.27759 −0.215952
$$36$$ 0 0
$$37$$ −4.69991 −0.772661 −0.386330 0.922360i $$-0.626258\pi$$
−0.386330 + 0.922360i $$0.626258\pi$$
$$38$$ 0.219863 0.0356665
$$39$$ 0 0
$$40$$ 2.87392 0.454407
$$41$$ 7.58597 1.18473 0.592365 0.805670i $$-0.298194\pi$$
0.592365 + 0.805670i $$0.298194\pi$$
$$42$$ 0 0
$$43$$ −5.41324 −0.825512 −0.412756 0.910842i $$-0.635434\pi$$
−0.412756 + 0.910842i $$0.635434\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −6.96281 −1.02661
$$47$$ 8.23211 1.20078 0.600388 0.799709i $$-0.295013\pi$$
0.600388 + 0.799709i $$0.295013\pi$$
$$48$$ 0 0
$$49$$ −5.36776 −0.766823
$$50$$ −1.39026 −0.196612
$$51$$ 0 0
$$52$$ −0.0952940 −0.0132149
$$53$$ −9.36648 −1.28659 −0.643293 0.765620i $$-0.722432\pi$$
−0.643293 + 0.765620i $$0.722432\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.67169 −0.490651
$$57$$ 0 0
$$58$$ −8.72406 −1.14553
$$59$$ −8.09846 −1.05433 −0.527165 0.849763i $$-0.676745\pi$$
−0.527165 + 0.849763i $$0.676745\pi$$
$$60$$ 0 0
$$61$$ 14.2917 1.82987 0.914934 0.403603i $$-0.132242\pi$$
0.914934 + 0.403603i $$0.132242\pi$$
$$62$$ −4.24002 −0.538484
$$63$$ 0 0
$$64$$ 8.25038 1.03130
$$65$$ 1.41837 0.175927
$$66$$ 0 0
$$67$$ −7.38362 −0.902053 −0.451026 0.892511i $$-0.648942\pi$$
−0.451026 + 0.892511i $$0.648942\pi$$
$$68$$ −0.335929 −0.0407374
$$69$$ 0 0
$$70$$ 1.77618 0.212294
$$71$$ 6.77618 0.804185 0.402092 0.915599i $$-0.368283\pi$$
0.402092 + 0.915599i $$0.368283\pi$$
$$72$$ 0 0
$$73$$ −8.66107 −1.01370 −0.506851 0.862034i $$-0.669190\pi$$
−0.506851 + 0.862034i $$0.669190\pi$$
$$74$$ 6.53409 0.759572
$$75$$ 0 0
$$76$$ 0.0106252 0.00121879
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 2.54572 0.286416 0.143208 0.989693i $$-0.454258\pi$$
0.143208 + 0.989693i $$0.454258\pi$$
$$80$$ −3.86111 −0.431686
$$81$$ 0 0
$$82$$ −10.5464 −1.16466
$$83$$ 10.1221 1.11105 0.555524 0.831501i $$-0.312518\pi$$
0.555524 + 0.831501i $$0.312518\pi$$
$$84$$ 0 0
$$85$$ 5.00000 0.542326
$$86$$ 7.52580 0.811527
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −11.0447 −1.17073 −0.585367 0.810768i $$-0.699050\pi$$
−0.585367 + 0.810768i $$0.699050\pi$$
$$90$$ 0 0
$$91$$ −1.81209 −0.189959
$$92$$ −0.336486 −0.0350811
$$93$$ 0 0
$$94$$ −11.4447 −1.18044
$$95$$ −0.158146 −0.0162254
$$96$$ 0 0
$$97$$ 6.41196 0.651036 0.325518 0.945536i $$-0.394461\pi$$
0.325518 + 0.945536i $$0.394461\pi$$
$$98$$ 7.46257 0.753833
$$99$$ 0 0
$$100$$ −0.0671858 −0.00671858
$$101$$ 8.77143 0.872790 0.436395 0.899755i $$-0.356255\pi$$
0.436395 + 0.899755i $$0.356255\pi$$
$$102$$ 0 0
$$103$$ 11.5666 1.13969 0.569847 0.821751i $$-0.307002\pi$$
0.569847 + 0.821751i $$0.307002\pi$$
$$104$$ 4.07627 0.399711
$$105$$ 0 0
$$106$$ 13.0218 1.26479
$$107$$ 15.3467 1.48362 0.741809 0.670611i $$-0.233968\pi$$
0.741809 + 0.670611i $$0.233968\pi$$
$$108$$ 0 0
$$109$$ −14.4004 −1.37931 −0.689656 0.724137i $$-0.742238\pi$$
−0.689656 + 0.724137i $$0.742238\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 4.93293 0.466118
$$113$$ −19.3645 −1.82166 −0.910831 0.412780i $$-0.864558\pi$$
−0.910831 + 0.412780i $$0.864558\pi$$
$$114$$ 0 0
$$115$$ 5.00829 0.467026
$$116$$ −0.421601 −0.0391446
$$117$$ 0 0
$$118$$ 11.2589 1.03647
$$119$$ −6.38796 −0.585583
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −19.8692 −1.79887
$$123$$ 0 0
$$124$$ −0.204904 −0.0184009
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 6.02415 0.534557 0.267278 0.963619i $$-0.413876\pi$$
0.267278 + 0.963619i $$0.413876\pi$$
$$128$$ −10.7104 −0.946671
$$129$$ 0 0
$$130$$ −1.97189 −0.172946
$$131$$ −18.7278 −1.63626 −0.818130 0.575034i $$-0.804989\pi$$
−0.818130 + 0.575034i $$0.804989\pi$$
$$132$$ 0 0
$$133$$ 0.202046 0.0175196
$$134$$ 10.2651 0.886772
$$135$$ 0 0
$$136$$ 14.3696 1.23218
$$137$$ −3.03719 −0.259485 −0.129742 0.991548i $$-0.541415\pi$$
−0.129742 + 0.991548i $$0.541415\pi$$
$$138$$ 0 0
$$139$$ 2.21192 0.187612 0.0938062 0.995590i $$-0.470097\pi$$
0.0938062 + 0.995590i $$0.470097\pi$$
$$140$$ 0.0858360 0.00725446
$$141$$ 0 0
$$142$$ −9.42063 −0.790562
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.27515 0.521122
$$146$$ 12.0411 0.996529
$$147$$ 0 0
$$148$$ 0.315767 0.0259559
$$149$$ 1.48122 0.121346 0.0606730 0.998158i $$-0.480675\pi$$
0.0606730 + 0.998158i $$0.480675\pi$$
$$150$$ 0 0
$$151$$ 9.03395 0.735173 0.367586 0.929989i $$-0.380184\pi$$
0.367586 + 0.929989i $$0.380184\pi$$
$$152$$ −0.454498 −0.0368647
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3.04981 0.244967
$$156$$ 0 0
$$157$$ −17.6377 −1.40764 −0.703820 0.710379i $$-0.748524\pi$$
−0.703820 + 0.710379i $$0.748524\pi$$
$$158$$ −3.53921 −0.281564
$$159$$ 0 0
$$160$$ −0.379898 −0.0300335
$$161$$ −6.39855 −0.504276
$$162$$ 0 0
$$163$$ 23.5040 1.84098 0.920489 0.390770i $$-0.127791\pi$$
0.920489 + 0.390770i $$0.127791\pi$$
$$164$$ −0.509669 −0.0397985
$$165$$ 0 0
$$166$$ −14.0724 −1.09223
$$167$$ 0.602731 0.0466407 0.0233203 0.999728i $$-0.492576\pi$$
0.0233203 + 0.999728i $$0.492576\pi$$
$$168$$ 0 0
$$169$$ −10.9882 −0.845249
$$170$$ −6.95128 −0.533139
$$171$$ 0 0
$$172$$ 0.363693 0.0277313
$$173$$ 9.66389 0.734732 0.367366 0.930076i $$-0.380260\pi$$
0.367366 + 0.930076i $$0.380260\pi$$
$$174$$ 0 0
$$175$$ −1.27759 −0.0965768
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 15.3550 1.15090
$$179$$ 1.19571 0.0893717 0.0446859 0.999001i $$-0.485771\pi$$
0.0446859 + 0.999001i $$0.485771\pi$$
$$180$$ 0 0
$$181$$ 15.5741 1.15761 0.578806 0.815466i $$-0.303519\pi$$
0.578806 + 0.815466i $$0.303519\pi$$
$$182$$ 2.51927 0.186741
$$183$$ 0 0
$$184$$ 14.3934 1.06110
$$185$$ −4.69991 −0.345544
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −0.553081 −0.0403376
$$189$$ 0 0
$$190$$ 0.219863 0.0159506
$$191$$ −18.4016 −1.33149 −0.665747 0.746178i $$-0.731887\pi$$
−0.665747 + 0.746178i $$0.731887\pi$$
$$192$$ 0 0
$$193$$ −1.56799 −0.112866 −0.0564331 0.998406i $$-0.517973\pi$$
−0.0564331 + 0.998406i $$0.517973\pi$$
$$194$$ −8.91428 −0.640008
$$195$$ 0 0
$$196$$ 0.360637 0.0257598
$$197$$ 7.97000 0.567839 0.283920 0.958848i $$-0.408365\pi$$
0.283920 + 0.958848i $$0.408365\pi$$
$$198$$ 0 0
$$199$$ 3.53141 0.250335 0.125167 0.992136i $$-0.460053\pi$$
0.125167 + 0.992136i $$0.460053\pi$$
$$200$$ 2.87392 0.203217
$$201$$ 0 0
$$202$$ −12.1945 −0.858005
$$203$$ −8.01707 −0.562688
$$204$$ 0 0
$$205$$ 7.58597 0.529827
$$206$$ −16.0806 −1.12039
$$207$$ 0 0
$$208$$ −5.47647 −0.379725
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −20.2550 −1.39441 −0.697207 0.716870i $$-0.745574\pi$$
−0.697207 + 0.716870i $$0.745574\pi$$
$$212$$ 0.629295 0.0432201
$$213$$ 0 0
$$214$$ −21.3358 −1.45849
$$215$$ −5.41324 −0.369180
$$216$$ 0 0
$$217$$ −3.89642 −0.264506
$$218$$ 20.0203 1.35595
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 7.09183 0.477048
$$222$$ 0 0
$$223$$ 22.5500 1.51006 0.755029 0.655691i $$-0.227623\pi$$
0.755029 + 0.655691i $$0.227623\pi$$
$$224$$ 0.485354 0.0324291
$$225$$ 0 0
$$226$$ 26.9217 1.79080
$$227$$ 18.8062 1.24821 0.624105 0.781341i $$-0.285464\pi$$
0.624105 + 0.781341i $$0.285464\pi$$
$$228$$ 0 0
$$229$$ −23.8320 −1.57486 −0.787431 0.616403i $$-0.788589\pi$$
−0.787431 + 0.616403i $$0.788589\pi$$
$$230$$ −6.96281 −0.459114
$$231$$ 0 0
$$232$$ 18.0343 1.18401
$$233$$ −10.0918 −0.661137 −0.330569 0.943782i $$-0.607241\pi$$
−0.330569 + 0.943782i $$0.607241\pi$$
$$234$$ 0 0
$$235$$ 8.23211 0.537004
$$236$$ 0.544102 0.0354180
$$237$$ 0 0
$$238$$ 8.88090 0.575663
$$239$$ 0.167227 0.0108170 0.00540850 0.999985i $$-0.498278\pi$$
0.00540850 + 0.999985i $$0.498278\pi$$
$$240$$ 0 0
$$241$$ −0.965256 −0.0621776 −0.0310888 0.999517i $$-0.509897\pi$$
−0.0310888 + 0.999517i $$0.509897\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −0.960201 −0.0614706
$$245$$ −5.36776 −0.342934
$$246$$ 0 0
$$247$$ −0.224308 −0.0142724
$$248$$ 8.76492 0.556573
$$249$$ 0 0
$$250$$ −1.39026 −0.0879276
$$251$$ 23.5102 1.48395 0.741975 0.670428i $$-0.233889\pi$$
0.741975 + 0.670428i $$0.233889\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.37512 −0.525502
$$255$$ 0 0
$$256$$ −1.61062 −0.100664
$$257$$ 6.58273 0.410620 0.205310 0.978697i $$-0.434180\pi$$
0.205310 + 0.978697i $$0.434180\pi$$
$$258$$ 0 0
$$259$$ 6.00457 0.373106
$$260$$ −0.0952940 −0.00590988
$$261$$ 0 0
$$262$$ 26.0365 1.60854
$$263$$ 12.4538 0.767936 0.383968 0.923346i $$-0.374557\pi$$
0.383968 + 0.923346i $$0.374557\pi$$
$$264$$ 0 0
$$265$$ −9.36648 −0.575378
$$266$$ −0.280895 −0.0172228
$$267$$ 0 0
$$268$$ 0.496074 0.0303026
$$269$$ −20.7729 −1.26655 −0.633274 0.773927i $$-0.718290\pi$$
−0.633274 + 0.773927i $$0.718290\pi$$
$$270$$ 0 0
$$271$$ −14.3903 −0.874146 −0.437073 0.899426i $$-0.643985\pi$$
−0.437073 + 0.899426i $$0.643985\pi$$
$$272$$ −19.3056 −1.17057
$$273$$ 0 0
$$274$$ 4.22247 0.255089
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 17.4872 1.05070 0.525352 0.850885i $$-0.323934\pi$$
0.525352 + 0.850885i $$0.323934\pi$$
$$278$$ −3.07513 −0.184434
$$279$$ 0 0
$$280$$ −3.67169 −0.219426
$$281$$ −18.8860 −1.12664 −0.563322 0.826238i $$-0.690477\pi$$
−0.563322 + 0.826238i $$0.690477\pi$$
$$282$$ 0 0
$$283$$ 13.5813 0.807326 0.403663 0.914908i $$-0.367737\pi$$
0.403663 + 0.914908i $$0.367737\pi$$
$$284$$ −0.455263 −0.0270149
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −9.69177 −0.572087
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ −8.72406 −0.512295
$$291$$ 0 0
$$292$$ 0.581901 0.0340532
$$293$$ 19.9023 1.16271 0.581353 0.813651i $$-0.302523\pi$$
0.581353 + 0.813651i $$0.302523\pi$$
$$294$$ 0 0
$$295$$ −8.09846 −0.471511
$$296$$ −13.5072 −0.785088
$$297$$ 0 0
$$298$$ −2.05927 −0.119290
$$299$$ 7.10358 0.410811
$$300$$ 0 0
$$301$$ 6.91591 0.398626
$$302$$ −12.5595 −0.722719
$$303$$ 0 0
$$304$$ 0.610619 0.0350214
$$305$$ 14.2917 0.818342
$$306$$ 0 0
$$307$$ 14.9354 0.852410 0.426205 0.904627i $$-0.359850\pi$$
0.426205 + 0.904627i $$0.359850\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.24002 −0.240817
$$311$$ 4.99700 0.283354 0.141677 0.989913i $$-0.454751\pi$$
0.141677 + 0.989913i $$0.454751\pi$$
$$312$$ 0 0
$$313$$ 9.39648 0.531120 0.265560 0.964094i $$-0.414443\pi$$
0.265560 + 0.964094i $$0.414443\pi$$
$$314$$ 24.5209 1.38379
$$315$$ 0 0
$$316$$ −0.171037 −0.00962155
$$317$$ −2.15253 −0.120898 −0.0604492 0.998171i $$-0.519253\pi$$
−0.0604492 + 0.998171i $$0.519253\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 8.25038 0.461210
$$321$$ 0 0
$$322$$ 8.89563 0.495734
$$323$$ −0.790729 −0.0439973
$$324$$ 0 0
$$325$$ 1.41837 0.0786767
$$326$$ −32.6766 −1.80979
$$327$$ 0 0
$$328$$ 21.8015 1.20378
$$329$$ −10.5173 −0.579836
$$330$$ 0 0
$$331$$ −19.5116 −1.07245 −0.536227 0.844074i $$-0.680151\pi$$
−0.536227 + 0.844074i $$0.680151\pi$$
$$332$$ −0.680063 −0.0373233
$$333$$ 0 0
$$334$$ −0.837950 −0.0458506
$$335$$ −7.38362 −0.403410
$$336$$ 0 0
$$337$$ −31.6047 −1.72162 −0.860809 0.508929i $$-0.830042\pi$$
−0.860809 + 0.508929i $$0.830042\pi$$
$$338$$ 15.2765 0.830931
$$339$$ 0 0
$$340$$ −0.335929 −0.0182183
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 15.8009 0.853171
$$344$$ −15.5572 −0.838789
$$345$$ 0 0
$$346$$ −13.4353 −0.722286
$$347$$ 9.38313 0.503713 0.251856 0.967765i $$-0.418959\pi$$
0.251856 + 0.967765i $$0.418959\pi$$
$$348$$ 0 0
$$349$$ 24.8131 1.32822 0.664108 0.747637i $$-0.268812\pi$$
0.664108 + 0.747637i $$0.268812\pi$$
$$350$$ 1.77618 0.0949408
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 19.4788 1.03675 0.518375 0.855153i $$-0.326537\pi$$
0.518375 + 0.855153i $$0.326537\pi$$
$$354$$ 0 0
$$355$$ 6.77618 0.359642
$$356$$ 0.742046 0.0393284
$$357$$ 0 0
$$358$$ −1.66235 −0.0878578
$$359$$ 29.7701 1.57121 0.785603 0.618732i $$-0.212353\pi$$
0.785603 + 0.618732i $$0.212353\pi$$
$$360$$ 0 0
$$361$$ −18.9750 −0.998684
$$362$$ −21.6520 −1.13800
$$363$$ 0 0
$$364$$ 0.121747 0.00638126
$$365$$ −8.66107 −0.453341
$$366$$ 0 0
$$367$$ −6.43597 −0.335955 −0.167977 0.985791i $$-0.553724\pi$$
−0.167977 + 0.985791i $$0.553724\pi$$
$$368$$ −19.3376 −1.00804
$$369$$ 0 0
$$370$$ 6.53409 0.339691
$$371$$ 11.9665 0.621272
$$372$$ 0 0
$$373$$ 20.8924 1.08177 0.540883 0.841098i $$-0.318090\pi$$
0.540883 + 0.841098i $$0.318090\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 23.6584 1.22009
$$377$$ 8.90045 0.458396
$$378$$ 0 0
$$379$$ −14.1991 −0.729359 −0.364680 0.931133i $$-0.618822\pi$$
−0.364680 + 0.931133i $$0.618822\pi$$
$$380$$ 0.0106252 0.000545059 0
$$381$$ 0 0
$$382$$ 25.5830 1.30894
$$383$$ 26.4783 1.35298 0.676489 0.736453i $$-0.263501\pi$$
0.676489 + 0.736453i $$0.263501\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.17990 0.110954
$$387$$ 0 0
$$388$$ −0.430793 −0.0218702
$$389$$ 36.4965 1.85044 0.925222 0.379427i $$-0.123879\pi$$
0.925222 + 0.379427i $$0.123879\pi$$
$$390$$ 0 0
$$391$$ 25.0415 1.26640
$$392$$ −15.4265 −0.779157
$$393$$ 0 0
$$394$$ −11.0804 −0.558220
$$395$$ 2.54572 0.128089
$$396$$ 0 0
$$397$$ −8.03969 −0.403500 −0.201750 0.979437i $$-0.564663\pi$$
−0.201750 + 0.979437i $$0.564663\pi$$
$$398$$ −4.90956 −0.246094
$$399$$ 0 0
$$400$$ −3.86111 −0.193056
$$401$$ 28.0902 1.40276 0.701379 0.712788i $$-0.252568\pi$$
0.701379 + 0.712788i $$0.252568\pi$$
$$402$$ 0 0
$$403$$ 4.32575 0.215481
$$404$$ −0.589316 −0.0293196
$$405$$ 0 0
$$406$$ 11.1458 0.553156
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 5.97950 0.295667 0.147834 0.989012i $$-0.452770\pi$$
0.147834 + 0.989012i $$0.452770\pi$$
$$410$$ −10.5464 −0.520852
$$411$$ 0 0
$$412$$ −0.777114 −0.0382857
$$413$$ 10.3465 0.509119
$$414$$ 0 0
$$415$$ 10.1221 0.496876
$$416$$ −0.538833 −0.0264185
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 15.4707 0.755795 0.377897 0.925847i $$-0.376647\pi$$
0.377897 + 0.925847i $$0.376647\pi$$
$$420$$ 0 0
$$421$$ 34.5746 1.68506 0.842532 0.538647i $$-0.181064\pi$$
0.842532 + 0.538647i $$0.181064\pi$$
$$422$$ 28.1597 1.37079
$$423$$ 0 0
$$424$$ −26.9185 −1.30728
$$425$$ 5.00000 0.242536
$$426$$ 0 0
$$427$$ −18.2590 −0.883614
$$428$$ −1.03108 −0.0498390
$$429$$ 0 0
$$430$$ 7.52580 0.362926
$$431$$ 3.36052 0.161870 0.0809352 0.996719i $$-0.474209\pi$$
0.0809352 + 0.996719i $$0.474209\pi$$
$$432$$ 0 0
$$433$$ 3.67779 0.176744 0.0883718 0.996088i $$-0.471834\pi$$
0.0883718 + 0.996088i $$0.471834\pi$$
$$434$$ 5.41702 0.260025
$$435$$ 0 0
$$436$$ 0.967505 0.0463351
$$437$$ −0.792040 −0.0378884
$$438$$ 0 0
$$439$$ 1.55432 0.0741835 0.0370917 0.999312i $$-0.488191\pi$$
0.0370917 + 0.999312i $$0.488191\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −9.85946 −0.468967
$$443$$ −21.2059 −1.00752 −0.503761 0.863843i $$-0.668051\pi$$
−0.503761 + 0.863843i $$0.668051\pi$$
$$444$$ 0 0
$$445$$ −11.0447 −0.523569
$$446$$ −31.3503 −1.48448
$$447$$ 0 0
$$448$$ −10.5406 −0.497997
$$449$$ −15.1106 −0.713113 −0.356557 0.934274i $$-0.616049\pi$$
−0.356557 + 0.934274i $$0.616049\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 1.30102 0.0611949
$$453$$ 0 0
$$454$$ −26.1454 −1.22707
$$455$$ −1.81209 −0.0849521
$$456$$ 0 0
$$457$$ −41.3042 −1.93213 −0.966064 0.258303i $$-0.916837\pi$$
−0.966064 + 0.258303i $$0.916837\pi$$
$$458$$ 33.1326 1.54818
$$459$$ 0 0
$$460$$ −0.336486 −0.0156887
$$461$$ −16.3158 −0.759901 −0.379951 0.925007i $$-0.624059\pi$$
−0.379951 + 0.925007i $$0.624059\pi$$
$$462$$ 0 0
$$463$$ 8.47904 0.394054 0.197027 0.980398i $$-0.436871\pi$$
0.197027 + 0.980398i $$0.436871\pi$$
$$464$$ −24.2291 −1.12481
$$465$$ 0 0
$$466$$ 14.0302 0.649938
$$467$$ 36.5669 1.69212 0.846058 0.533090i $$-0.178969\pi$$
0.846058 + 0.533090i $$0.178969\pi$$
$$468$$ 0 0
$$469$$ 9.43325 0.435587
$$470$$ −11.4447 −0.527907
$$471$$ 0 0
$$472$$ −23.2743 −1.07129
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −0.158146 −0.00725623
$$476$$ 0.429180 0.0196714
$$477$$ 0 0
$$478$$ −0.232488 −0.0106338
$$479$$ −38.9513 −1.77973 −0.889866 0.456222i $$-0.849202\pi$$
−0.889866 + 0.456222i $$0.849202\pi$$
$$480$$ 0 0
$$481$$ −6.66619 −0.303952
$$482$$ 1.34195 0.0611243
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 6.41196 0.291152
$$486$$ 0 0
$$487$$ 8.27209 0.374844 0.187422 0.982279i $$-0.439987\pi$$
0.187422 + 0.982279i $$0.439987\pi$$
$$488$$ 41.0733 1.85930
$$489$$ 0 0
$$490$$ 7.46257 0.337124
$$491$$ −12.1322 −0.547518 −0.273759 0.961798i $$-0.588267\pi$$
−0.273759 + 0.961798i $$0.588267\pi$$
$$492$$ 0 0
$$493$$ 31.3757 1.41309
$$494$$ 0.311846 0.0140306
$$495$$ 0 0
$$496$$ −11.7757 −0.528744
$$497$$ −8.65719 −0.388328
$$498$$ 0 0
$$499$$ −28.3581 −1.26948 −0.634740 0.772725i $$-0.718893\pi$$
−0.634740 + 0.772725i $$0.718893\pi$$
$$500$$ −0.0671858 −0.00300464
$$501$$ 0 0
$$502$$ −32.6852 −1.45881
$$503$$ −15.2800 −0.681300 −0.340650 0.940190i $$-0.610647\pi$$
−0.340650 + 0.940190i $$0.610647\pi$$
$$504$$ 0 0
$$505$$ 8.77143 0.390324
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −0.404737 −0.0179573
$$509$$ 22.0494 0.977321 0.488661 0.872474i $$-0.337486\pi$$
0.488661 + 0.872474i $$0.337486\pi$$
$$510$$ 0 0
$$511$$ 11.0653 0.489500
$$512$$ 23.6599 1.04563
$$513$$ 0 0
$$514$$ −9.15169 −0.403664
$$515$$ 11.5666 0.509687
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −8.34789 −0.366785
$$519$$ 0 0
$$520$$ 4.07627 0.178756
$$521$$ 23.9380 1.04874 0.524371 0.851490i $$-0.324300\pi$$
0.524371 + 0.851490i $$0.324300\pi$$
$$522$$ 0 0
$$523$$ 25.3421 1.10813 0.554066 0.832473i $$-0.313075\pi$$
0.554066 + 0.832473i $$0.313075\pi$$
$$524$$ 1.25824 0.0549667
$$525$$ 0 0
$$526$$ −17.3140 −0.754927
$$527$$ 15.2491 0.664260
$$528$$ 0 0
$$529$$ 2.08298 0.0905642
$$530$$ 13.0218 0.565632
$$531$$ 0 0
$$532$$ −0.0135746 −0.000588534 0
$$533$$ 10.7597 0.466053
$$534$$ 0 0
$$535$$ 15.3467 0.663494
$$536$$ −21.2199 −0.916561
$$537$$ 0 0
$$538$$ 28.8797 1.24509
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 35.0634 1.50749 0.753747 0.657165i $$-0.228245\pi$$
0.753747 + 0.657165i $$0.228245\pi$$
$$542$$ 20.0062 0.859338
$$543$$ 0 0
$$544$$ −1.89949 −0.0814399
$$545$$ −14.4004 −0.616847
$$546$$ 0 0
$$547$$ 1.10787 0.0473689 0.0236845 0.999719i $$-0.492460\pi$$
0.0236845 + 0.999719i $$0.492460\pi$$
$$548$$ 0.204056 0.00871684
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −0.992388 −0.0422771
$$552$$ 0 0
$$553$$ −3.25239 −0.138306
$$554$$ −24.3117 −1.03291
$$555$$ 0 0
$$556$$ −0.148609 −0.00630244
$$557$$ −5.41988 −0.229648 −0.114824 0.993386i $$-0.536630\pi$$
−0.114824 + 0.993386i $$0.536630\pi$$
$$558$$ 0 0
$$559$$ −7.67795 −0.324743
$$560$$ 4.93293 0.208454
$$561$$ 0 0
$$562$$ 26.2564 1.10756
$$563$$ 32.7448 1.38003 0.690015 0.723795i $$-0.257604\pi$$
0.690015 + 0.723795i $$0.257604\pi$$
$$564$$ 0 0
$$565$$ −19.3645 −0.814672
$$566$$ −18.8815 −0.793650
$$567$$ 0 0
$$568$$ 19.4742 0.817119
$$569$$ −3.51736 −0.147455 −0.0737277 0.997278i $$-0.523490\pi$$
−0.0737277 + 0.997278i $$0.523490\pi$$
$$570$$ 0 0
$$571$$ 36.0252 1.50761 0.753804 0.657099i $$-0.228217\pi$$
0.753804 + 0.657099i $$0.228217\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 13.4740 0.562396
$$575$$ 5.00829 0.208860
$$576$$ 0 0
$$577$$ 15.5501 0.647357 0.323679 0.946167i $$-0.395080\pi$$
0.323679 + 0.946167i $$0.395080\pi$$
$$578$$ −11.1221 −0.462617
$$579$$ 0 0
$$580$$ −0.421601 −0.0175060
$$581$$ −12.9319 −0.536507
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −24.8912 −1.03001
$$585$$ 0 0
$$586$$ −27.6694 −1.14301
$$587$$ 24.2604 1.00133 0.500667 0.865640i $$-0.333088\pi$$
0.500667 + 0.865640i $$0.333088\pi$$
$$588$$ 0 0
$$589$$ −0.482315 −0.0198735
$$590$$ 11.2589 0.463523
$$591$$ 0 0
$$592$$ 18.1469 0.745833
$$593$$ 27.4019 1.12526 0.562630 0.826709i $$-0.309790\pi$$
0.562630 + 0.826709i $$0.309790\pi$$
$$594$$ 0 0
$$595$$ −6.38796 −0.261881
$$596$$ −0.0995167 −0.00407636
$$597$$ 0 0
$$598$$ −9.87581 −0.403852
$$599$$ −10.6773 −0.436262 −0.218131 0.975920i $$-0.569996\pi$$
−0.218131 + 0.975920i $$0.569996\pi$$
$$600$$ 0 0
$$601$$ −11.0663 −0.451403 −0.225701 0.974197i $$-0.572467\pi$$
−0.225701 + 0.974197i $$0.572467\pi$$
$$602$$ −9.61489 −0.391874
$$603$$ 0 0
$$604$$ −0.606953 −0.0246966
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0.514518 0.0208836 0.0104418 0.999945i $$-0.496676\pi$$
0.0104418 + 0.999945i $$0.496676\pi$$
$$608$$ 0.0600792 0.00243653
$$609$$ 0 0
$$610$$ −19.8692 −0.804479
$$611$$ 11.6761 0.472366
$$612$$ 0 0
$$613$$ −19.0977 −0.771348 −0.385674 0.922635i $$-0.626031\pi$$
−0.385674 + 0.922635i $$0.626031\pi$$
$$614$$ −20.7641 −0.837970
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 4.70745 0.189515 0.0947573 0.995500i $$-0.469792\pi$$
0.0947573 + 0.995500i $$0.469792\pi$$
$$618$$ 0 0
$$619$$ 37.3691 1.50199 0.750996 0.660307i $$-0.229574\pi$$
0.750996 + 0.660307i $$0.229574\pi$$
$$620$$ −0.204904 −0.00822915
$$621$$ 0 0
$$622$$ −6.94711 −0.278554
$$623$$ 14.1106 0.565329
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −13.0635 −0.522123
$$627$$ 0 0
$$628$$ 1.18500 0.0472867
$$629$$ −23.4996 −0.936989
$$630$$ 0 0
$$631$$ 35.5669 1.41590 0.707949 0.706264i $$-0.249621\pi$$
0.707949 + 0.706264i $$0.249621\pi$$
$$632$$ 7.31621 0.291023
$$633$$ 0 0
$$634$$ 2.99257 0.118850
$$635$$ 6.02415 0.239061
$$636$$ 0 0
$$637$$ −7.61344 −0.301656
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −10.7104 −0.423364
$$641$$ −12.2098 −0.482260 −0.241130 0.970493i $$-0.577518\pi$$
−0.241130 + 0.970493i $$0.577518\pi$$
$$642$$ 0 0
$$643$$ −20.9596 −0.826565 −0.413282 0.910603i $$-0.635618\pi$$
−0.413282 + 0.910603i $$0.635618\pi$$
$$644$$ 0.429892 0.0169401
$$645$$ 0 0
$$646$$ 1.09932 0.0432520
$$647$$ −14.8766 −0.584859 −0.292430 0.956287i $$-0.594464\pi$$
−0.292430 + 0.956287i $$0.594464\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −1.97189 −0.0773440
$$651$$ 0 0
$$652$$ −1.57914 −0.0618438
$$653$$ 12.1308 0.474713 0.237357 0.971423i $$-0.423719\pi$$
0.237357 + 0.971423i $$0.423719\pi$$
$$654$$ 0 0
$$655$$ −18.7278 −0.731757
$$656$$ −29.2903 −1.14359
$$657$$ 0 0
$$658$$ 14.6217 0.570014
$$659$$ 7.49994 0.292156 0.146078 0.989273i $$-0.453335\pi$$
0.146078 + 0.989273i $$0.453335\pi$$
$$660$$ 0 0
$$661$$ 9.65248 0.375438 0.187719 0.982223i $$-0.439891\pi$$
0.187719 + 0.982223i $$0.439891\pi$$
$$662$$ 27.1261 1.05429
$$663$$ 0 0
$$664$$ 29.0902 1.12892
$$665$$ 0.202046 0.00783500
$$666$$ 0 0
$$667$$ 31.4278 1.21689
$$668$$ −0.0404949 −0.00156680
$$669$$ 0 0
$$670$$ 10.2651 0.396577
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −24.4496 −0.942463 −0.471231 0.882010i $$-0.656190\pi$$
−0.471231 + 0.882010i $$0.656190\pi$$
$$674$$ 43.9386 1.69245
$$675$$ 0 0
$$676$$ 0.738254 0.0283944
$$677$$ 46.7394 1.79634 0.898171 0.439646i $$-0.144896\pi$$
0.898171 + 0.439646i $$0.144896\pi$$
$$678$$ 0 0
$$679$$ −8.19187 −0.314375
$$680$$ 14.3696 0.551049
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 28.1941 1.07882 0.539408 0.842045i $$-0.318648\pi$$
0.539408 + 0.842045i $$0.318648\pi$$
$$684$$ 0 0
$$685$$ −3.03719 −0.116045
$$686$$ −21.9674 −0.838718
$$687$$ 0 0
$$688$$ 20.9011 0.796849
$$689$$ −13.2851 −0.506122
$$690$$ 0 0
$$691$$ 11.5971 0.441175 0.220587 0.975367i $$-0.429203\pi$$
0.220587 + 0.975367i $$0.429203\pi$$
$$692$$ −0.649276 −0.0246818
$$693$$ 0 0
$$694$$ −13.0450 −0.495180
$$695$$ 2.21192 0.0839028
$$696$$ 0 0
$$697$$ 37.9298 1.43670
$$698$$ −34.4966 −1.30572
$$699$$ 0 0
$$700$$ 0.0858360 0.00324429
$$701$$ −36.7019 −1.38621 −0.693107 0.720835i $$-0.743759\pi$$
−0.693107 + 0.720835i $$0.743759\pi$$
$$702$$ 0 0
$$703$$ 0.743272 0.0280330
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −27.0805 −1.01919
$$707$$ −11.2063 −0.421456
$$708$$ 0 0
$$709$$ 36.4905 1.37043 0.685214 0.728341i $$-0.259708\pi$$
0.685214 + 0.728341i $$0.259708\pi$$
$$710$$ −9.42063 −0.353550
$$711$$ 0 0
$$712$$ −31.7415 −1.18956
$$713$$ 15.2744 0.572029
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −0.0803349 −0.00300225
$$717$$ 0 0
$$718$$ −41.3881 −1.54459
$$719$$ 9.32888 0.347909 0.173954 0.984754i $$-0.444346\pi$$
0.173954 + 0.984754i $$0.444346\pi$$
$$720$$ 0 0
$$721$$ −14.7774 −0.550340
$$722$$ 26.3801 0.981766
$$723$$ 0 0
$$724$$ −1.04636 −0.0388875
$$725$$ 6.27515 0.233053
$$726$$ 0 0
$$727$$ −8.46883 −0.314091 −0.157046 0.987591i $$-0.550197\pi$$
−0.157046 + 0.987591i $$0.550197\pi$$
$$728$$ −5.20780 −0.193014
$$729$$ 0 0
$$730$$ 12.0411 0.445661
$$731$$ −27.0662 −1.00108
$$732$$ 0 0
$$733$$ −29.0470 −1.07288 −0.536438 0.843940i $$-0.680230\pi$$
−0.536438 + 0.843940i $$0.680230\pi$$
$$734$$ 8.94765 0.330264
$$735$$ 0 0
$$736$$ −1.90264 −0.0701322
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 13.7551 0.505990 0.252995 0.967468i $$-0.418584\pi$$
0.252995 + 0.967468i $$0.418584\pi$$
$$740$$ 0.315767 0.0116078
$$741$$ 0 0
$$742$$ −16.6366 −0.610747
$$743$$ −34.9135 −1.28085 −0.640427 0.768019i $$-0.721242\pi$$
−0.640427 + 0.768019i $$0.721242\pi$$
$$744$$ 0 0
$$745$$ 1.48122 0.0542676
$$746$$ −29.0458 −1.06344
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −19.6068 −0.716416
$$750$$ 0 0
$$751$$ 22.0992 0.806412 0.403206 0.915109i $$-0.367896\pi$$
0.403206 + 0.915109i $$0.367896\pi$$
$$752$$ −31.7851 −1.15908
$$753$$ 0 0
$$754$$ −12.3739 −0.450631
$$755$$ 9.03395 0.328779
$$756$$ 0 0
$$757$$ −45.0300 −1.63664 −0.818321 0.574761i $$-0.805095\pi$$
−0.818321 + 0.574761i $$0.805095\pi$$
$$758$$ 19.7404 0.717004
$$759$$ 0 0
$$760$$ −0.454498 −0.0164864
$$761$$ 33.6001 1.21800 0.609001 0.793169i $$-0.291570\pi$$
0.609001 + 0.793169i $$0.291570\pi$$
$$762$$ 0 0
$$763$$ 18.3979 0.666048
$$764$$ 1.23633 0.0447287
$$765$$ 0 0
$$766$$ −36.8116 −1.33006
$$767$$ −11.4866 −0.414756
$$768$$ 0 0
$$769$$ −24.6086 −0.887408 −0.443704 0.896173i $$-0.646336\pi$$
−0.443704 + 0.896173i $$0.646336\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0.105346 0.00379150
$$773$$ 23.6816 0.851768 0.425884 0.904778i $$-0.359963\pi$$
0.425884 + 0.904778i $$0.359963\pi$$
$$774$$ 0 0
$$775$$ 3.04981 0.109553
$$776$$ 18.4275 0.661507
$$777$$ 0 0
$$778$$ −50.7394 −1.81910
$$779$$ −1.19969 −0.0429833
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −34.8141 −1.24495
$$783$$ 0 0
$$784$$ 20.7255 0.740198
$$785$$ −17.6377 −0.629515
$$786$$ 0 0
$$787$$ 18.4210 0.656639 0.328320 0.944567i $$-0.393518\pi$$
0.328320 + 0.944567i $$0.393518\pi$$
$$788$$ −0.535471 −0.0190754
$$789$$ 0 0
$$790$$ −3.53921 −0.125919
$$791$$ 24.7399 0.879651
$$792$$ 0 0
$$793$$ 20.2709 0.719840
$$794$$ 11.1772 0.396665
$$795$$ 0 0
$$796$$ −0.237260 −0.00840947
$$797$$ 6.50665 0.230477 0.115239 0.993338i $$-0.463237\pi$$
0.115239 + 0.993338i $$0.463237\pi$$
$$798$$ 0 0
$$799$$ 41.1606 1.45616
$$800$$ −0.379898 −0.0134314
$$801$$ 0 0
$$802$$ −39.0526 −1.37900
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −6.39855 −0.225519
$$806$$ −6.01390 −0.211831
$$807$$ 0 0
$$808$$ 25.2084 0.886828
$$809$$ 11.8505 0.416642 0.208321 0.978060i $$-0.433200\pi$$
0.208321 + 0.978060i $$0.433200\pi$$
$$810$$ 0 0
$$811$$ −7.51391 −0.263849 −0.131925 0.991260i $$-0.542116\pi$$
−0.131925 + 0.991260i $$0.542116\pi$$
$$812$$ 0.538633 0.0189023
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 23.5040 0.823310
$$816$$ 0 0
$$817$$ 0.856081 0.0299505
$$818$$ −8.31305 −0.290659
$$819$$ 0 0
$$820$$ −0.509669 −0.0177984
$$821$$ −37.3365 −1.30305 −0.651527 0.758625i $$-0.725871\pi$$
−0.651527 + 0.758625i $$0.725871\pi$$
$$822$$ 0 0
$$823$$ −43.0498 −1.50062 −0.750311 0.661085i $$-0.770096\pi$$
−0.750311 + 0.661085i $$0.770096\pi$$
$$824$$ 33.2416 1.15803
$$825$$ 0 0
$$826$$ −14.3843 −0.500495
$$827$$ −20.0304 −0.696524 −0.348262 0.937397i $$-0.613228\pi$$
−0.348262 + 0.937397i $$0.613228\pi$$
$$828$$ 0 0
$$829$$ −47.9207 −1.66435 −0.832177 0.554510i $$-0.812906\pi$$
−0.832177 + 0.554510i $$0.812906\pi$$
$$830$$ −14.0724 −0.488458
$$831$$ 0 0
$$832$$ 11.7021 0.405696
$$833$$ −26.8388 −0.929909
$$834$$ 0 0
$$835$$ 0.602731 0.0208584
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −21.5083 −0.742992
$$839$$ −2.22093 −0.0766750 −0.0383375 0.999265i $$-0.512206\pi$$
−0.0383375 + 0.999265i $$0.512206\pi$$
$$840$$ 0 0
$$841$$ 10.3775 0.357843
$$842$$ −48.0676 −1.65652
$$843$$ 0 0
$$844$$ 1.36085 0.0468424
$$845$$ −10.9882 −0.378007
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 36.1651 1.24191
$$849$$ 0 0
$$850$$ −6.95128 −0.238427
$$851$$ −23.5385 −0.806890
$$852$$ 0 0
$$853$$ 6.37161 0.218160 0.109080 0.994033i $$-0.465210\pi$$
0.109080 + 0.994033i $$0.465210\pi$$
$$854$$ 25.3847 0.868646
$$855$$ 0 0
$$856$$ 44.1051 1.50748
$$857$$ 35.9060 1.22653 0.613263 0.789878i $$-0.289856\pi$$
0.613263 + 0.789878i $$0.289856\pi$$
$$858$$ 0 0
$$859$$ 56.4697 1.92672 0.963360 0.268212i $$-0.0864328\pi$$
0.963360 + 0.268212i $$0.0864328\pi$$
$$860$$ 0.363693 0.0124018
$$861$$ 0 0
$$862$$ −4.67198 −0.159128
$$863$$ 31.0742 1.05778 0.528888 0.848691i $$-0.322609\pi$$
0.528888 + 0.848691i $$0.322609\pi$$
$$864$$ 0 0
$$865$$ 9.66389 0.328582
$$866$$ −5.11308 −0.173749
$$867$$ 0 0
$$868$$ 0.261784 0.00888552
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −10.4727 −0.354853
$$872$$ −41.3857 −1.40150
$$873$$ 0 0
$$874$$ 1.10114 0.0372466
$$875$$ −1.27759 −0.0431905
$$876$$ 0 0
$$877$$ −16.8224 −0.568053 −0.284026 0.958816i $$-0.591670\pi$$
−0.284026 + 0.958816i $$0.591670\pi$$
$$878$$ −2.16090 −0.0729268
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −26.5633 −0.894940 −0.447470 0.894299i $$-0.647675\pi$$
−0.447470 + 0.894299i $$0.647675\pi$$
$$882$$ 0 0
$$883$$ −54.3742 −1.82984 −0.914919 0.403638i $$-0.867746\pi$$
−0.914919 + 0.403638i $$0.867746\pi$$
$$884$$ −0.476470 −0.0160254
$$885$$ 0 0
$$886$$ 29.4816 0.990455
$$887$$ −41.2262 −1.38424 −0.692120 0.721783i $$-0.743323\pi$$
−0.692120 + 0.721783i $$0.743323\pi$$
$$888$$ 0 0
$$889$$ −7.69640 −0.258129
$$890$$ 15.3550 0.514699
$$891$$ 0 0
$$892$$ −1.51504 −0.0507273
$$893$$ −1.30187 −0.0435655
$$894$$ 0 0
$$895$$ 1.19571 0.0399682
$$896$$ 13.6835 0.457132
$$897$$ 0 0
$$898$$ 21.0076 0.701033
$$899$$ 19.1380 0.638289
$$900$$ 0 0
$$901$$ −46.8324 −1.56021
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −55.6521 −1.85096
$$905$$ 15.5741 0.517699
$$906$$ 0 0
$$907$$ 42.4046 1.40802 0.704010 0.710190i $$-0.251391\pi$$
0.704010 + 0.710190i $$0.251391\pi$$
$$908$$ −1.26351 −0.0419310
$$909$$ 0 0
$$910$$ 2.51927 0.0835130
$$911$$ 8.01199 0.265449 0.132725 0.991153i $$-0.457627\pi$$
0.132725 + 0.991153i $$0.457627\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 57.4234 1.89940
$$915$$ 0 0
$$916$$ 1.60117 0.0529042
$$917$$ 23.9265 0.790123
$$918$$ 0 0
$$919$$ −41.2336 −1.36017 −0.680086 0.733132i $$-0.738058\pi$$
−0.680086 + 0.733132i $$0.738058\pi$$
$$920$$ 14.3934 0.474537
$$921$$ 0 0
$$922$$ 22.6831 0.747028
$$923$$ 9.61110 0.316353
$$924$$ 0 0
$$925$$ −4.69991 −0.154532
$$926$$ −11.7880 −0.387379
$$927$$ 0 0
$$928$$ −2.38391 −0.0782558
$$929$$ 15.1939 0.498495 0.249247 0.968440i $$-0.419817\pi$$
0.249247 + 0.968440i $$0.419817\pi$$
$$930$$ 0 0
$$931$$ 0.848889 0.0278212
$$932$$ 0.678027 0.0222095
$$933$$ 0 0
$$934$$ −50.8375 −1.66345
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −25.5213 −0.833746 −0.416873 0.908965i $$-0.636874\pi$$
−0.416873 + 0.908965i $$0.636874\pi$$
$$938$$ −13.1146 −0.428208
$$939$$ 0 0
$$940$$ −0.553081 −0.0180395
$$941$$ 32.3673 1.05514 0.527572 0.849511i $$-0.323103\pi$$
0.527572 + 0.849511i $$0.323103\pi$$
$$942$$ 0 0
$$943$$ 37.9927 1.23721
$$944$$ 31.2691 1.01772
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 40.1516 1.30475 0.652375 0.757896i $$-0.273773\pi$$
0.652375 + 0.757896i $$0.273773\pi$$
$$948$$ 0 0
$$949$$ −12.2846 −0.398774
$$950$$ 0.219863 0.00713331
$$951$$ 0 0
$$952$$ −18.3585 −0.595001
$$953$$ 35.6513 1.15486 0.577429 0.816441i $$-0.304056\pi$$
0.577429 + 0.816441i $$0.304056\pi$$
$$954$$ 0 0
$$955$$ −18.4016 −0.595462
$$956$$ −0.0112353 −0.000363374 0
$$957$$ 0 0
$$958$$ 54.1524 1.74958
$$959$$ 3.88029 0.125301
$$960$$ 0 0
$$961$$ −21.6986 −0.699956
$$962$$ 9.26772 0.298803
$$963$$ 0 0
$$964$$ 0.0648515 0.00208873
$$965$$ −1.56799 −0.0504753
$$966$$ 0 0
$$967$$ 20.0622 0.645156 0.322578 0.946543i $$-0.395451\pi$$
0.322578 + 0.946543i $$0.395451\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −8.91428 −0.286220
$$971$$ 41.6307 1.33599 0.667996 0.744165i $$-0.267152\pi$$
0.667996 + 0.744165i $$0.267152\pi$$
$$972$$ 0 0
$$973$$ −2.82593 −0.0905950
$$974$$ −11.5003 −0.368494
$$975$$ 0 0
$$976$$ −55.1820 −1.76633
$$977$$ −19.0722 −0.610173 −0.305087 0.952325i $$-0.598685\pi$$
−0.305087 + 0.952325i $$0.598685\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0.360637 0.0115201
$$981$$ 0 0
$$982$$ 16.8668 0.538243
$$983$$ −26.5563 −0.847015 −0.423507 0.905893i $$-0.639201\pi$$
−0.423507 + 0.905893i $$0.639201\pi$$
$$984$$ 0 0
$$985$$ 7.97000 0.253945
$$986$$ −43.6203 −1.38915
$$987$$ 0 0
$$988$$ 0.0150703 0.000479452 0
$$989$$ −27.1111 −0.862082
$$990$$ 0 0
$$991$$ −24.2494 −0.770307 −0.385153 0.922853i $$-0.625851\pi$$
−0.385153 + 0.922853i $$0.625851\pi$$
$$992$$ −1.15862 −0.0367861
$$993$$ 0 0
$$994$$ 12.0357 0.381750
$$995$$ 3.53141 0.111953
$$996$$ 0 0
$$997$$ −16.0616 −0.508675 −0.254338 0.967116i $$-0.581857\pi$$
−0.254338 + 0.967116i $$0.581857\pi$$
$$998$$ 39.4250 1.24798
$$999$$ 0 0
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 5445.2.a.bv.1.1 4
3.2 odd 2 1815.2.a.o.1.4 4
11.2 odd 10 495.2.n.d.136.1 8
11.6 odd 10 495.2.n.d.91.1 8
11.10 odd 2 5445.2.a.be.1.4 4
15.14 odd 2 9075.2.a.dj.1.1 4
33.2 even 10 165.2.m.a.136.2 yes 8
33.17 even 10 165.2.m.a.91.2 8
33.32 even 2 1815.2.a.x.1.1 4
165.2 odd 20 825.2.bx.h.499.2 16
165.17 odd 20 825.2.bx.h.124.3 16
165.68 odd 20 825.2.bx.h.499.3 16
165.83 odd 20 825.2.bx.h.124.2 16
165.134 even 10 825.2.n.k.301.1 8
165.149 even 10 825.2.n.k.751.1 8
165.164 even 2 9075.2.a.cl.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.91.2 8 33.17 even 10
165.2.m.a.136.2 yes 8 33.2 even 10
495.2.n.d.91.1 8 11.6 odd 10
495.2.n.d.136.1 8 11.2 odd 10
825.2.n.k.301.1 8 165.134 even 10
825.2.n.k.751.1 8 165.149 even 10
825.2.bx.h.124.2 16 165.83 odd 20
825.2.bx.h.124.3 16 165.17 odd 20
825.2.bx.h.499.2 16 165.2 odd 20
825.2.bx.h.499.3 16 165.68 odd 20
1815.2.a.o.1.4 4 3.2 odd 2
1815.2.a.x.1.1 4 33.32 even 2
5445.2.a.be.1.4 4 11.10 odd 2
5445.2.a.bv.1.1 4 1.1 even 1 trivial
9075.2.a.cl.1.4 4 165.164 even 2
9075.2.a.dj.1.1 4 15.14 odd 2