# Properties

 Label 2842.2.a.p.1.4 Level $2842$ Weight $2$ Character 2842.1 Self dual yes Analytic conductor $22.693$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2842 = 2 \cdot 7^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2842.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$22.6934842544$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.9248.1 Defining polynomial: $$x^{4} - 5x^{2} + 2$$ x^4 - 5*x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$0.662153$$ of defining polynomial Character $$\chi$$ $$=$$ 2842.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.35829 q^{3} +1.00000 q^{4} -0.662153 q^{5} -2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.35829 q^{3} +1.00000 q^{4} -0.662153 q^{5} -2.35829 q^{6} -1.00000 q^{8} +2.56155 q^{9} +0.662153 q^{10} -0.438447 q^{11} +2.35829 q^{12} -4.05444 q^{13} -1.56155 q^{15} +1.00000 q^{16} -4.34475 q^{17} -2.56155 q^{18} +7.36520 q^{19} -0.662153 q^{20} +0.438447 q^{22} -8.24621 q^{23} -2.35829 q^{24} -4.56155 q^{25} +4.05444 q^{26} -1.03399 q^{27} +1.00000 q^{29} +1.56155 q^{30} -4.05444 q^{31} -1.00000 q^{32} -1.03399 q^{33} +4.34475 q^{34} +2.56155 q^{36} +7.12311 q^{37} -7.36520 q^{38} -9.56155 q^{39} +0.662153 q^{40} -4.34475 q^{41} -4.68466 q^{43} -0.438447 q^{44} -1.69614 q^{45} +8.24621 q^{46} -0.662153 q^{47} +2.35829 q^{48} +4.56155 q^{50} -10.2462 q^{51} -4.05444 q^{52} -11.5616 q^{53} +1.03399 q^{54} +0.290319 q^{55} +17.3693 q^{57} -1.00000 q^{58} +4.34475 q^{59} -1.56155 q^{60} +1.32431 q^{61} +4.05444 q^{62} +1.00000 q^{64} +2.68466 q^{65} +1.03399 q^{66} -4.87689 q^{67} -4.34475 q^{68} -19.4470 q^{69} +11.3693 q^{71} -2.56155 q^{72} -8.48071 q^{73} -7.12311 q^{74} -10.7575 q^{75} +7.36520 q^{76} +9.56155 q^{78} +7.80776 q^{79} -0.662153 q^{80} -10.1231 q^{81} +4.34475 q^{82} -5.08842 q^{83} +2.87689 q^{85} +4.68466 q^{86} +2.35829 q^{87} +0.438447 q^{88} -9.06134 q^{89} +1.69614 q^{90} -8.24621 q^{92} -9.56155 q^{93} +0.662153 q^{94} -4.87689 q^{95} -2.35829 q^{96} -6.99337 q^{97} -1.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9 $$4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{15} + 4 q^{16} - 2 q^{18} + 10 q^{22} - 10 q^{25} + 4 q^{29} - 2 q^{30} - 4 q^{32} + 2 q^{36} + 12 q^{37} - 30 q^{39} + 6 q^{43} - 10 q^{44} + 10 q^{50} - 8 q^{51} - 38 q^{53} + 20 q^{57} - 4 q^{58} + 2 q^{60} + 4 q^{64} - 14 q^{65} - 36 q^{67} - 4 q^{71} - 2 q^{72} - 12 q^{74} + 30 q^{78} - 10 q^{79} - 24 q^{81} + 28 q^{85} - 6 q^{86} + 10 q^{88} - 30 q^{93} - 36 q^{95} + 12 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 2 * q^9 - 10 * q^11 + 2 * q^15 + 4 * q^16 - 2 * q^18 + 10 * q^22 - 10 * q^25 + 4 * q^29 - 2 * q^30 - 4 * q^32 + 2 * q^36 + 12 * q^37 - 30 * q^39 + 6 * q^43 - 10 * q^44 + 10 * q^50 - 8 * q^51 - 38 * q^53 + 20 * q^57 - 4 * q^58 + 2 * q^60 + 4 * q^64 - 14 * q^65 - 36 * q^67 - 4 * q^71 - 2 * q^72 - 12 * q^74 + 30 * q^78 - 10 * q^79 - 24 * q^81 + 28 * q^85 - 6 * q^86 + 10 * q^88 - 30 * q^93 - 36 * q^95 + 12 * q^99

## 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
$$3$$ 2.35829 1.36156 0.680781 0.732487i $$-0.261641\pi$$
0.680781 + 0.732487i $$0.261641\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −0.662153 −0.296124 −0.148062 0.988978i $$-0.547304\pi$$
−0.148062 + 0.988978i $$0.547304\pi$$
$$6$$ −2.35829 −0.962770
$$7$$ 0 0
$$8$$ −1.00000 −0.353553
$$9$$ 2.56155 0.853851
$$10$$ 0.662153 0.209391
$$11$$ −0.438447 −0.132197 −0.0660984 0.997813i $$-0.521055\pi$$
−0.0660984 + 0.997813i $$0.521055\pi$$
$$12$$ 2.35829 0.680781
$$13$$ −4.05444 −1.12450 −0.562249 0.826968i $$-0.690064\pi$$
−0.562249 + 0.826968i $$0.690064\pi$$
$$14$$ 0 0
$$15$$ −1.56155 −0.403191
$$16$$ 1.00000 0.250000
$$17$$ −4.34475 −1.05376 −0.526879 0.849940i $$-0.676638\pi$$
−0.526879 + 0.849940i $$0.676638\pi$$
$$18$$ −2.56155 −0.603764
$$19$$ 7.36520 1.68969 0.844847 0.535008i $$-0.179692\pi$$
0.844847 + 0.535008i $$0.179692\pi$$
$$20$$ −0.662153 −0.148062
$$21$$ 0 0
$$22$$ 0.438447 0.0934773
$$23$$ −8.24621 −1.71945 −0.859727 0.510754i $$-0.829366\pi$$
−0.859727 + 0.510754i $$0.829366\pi$$
$$24$$ −2.35829 −0.481385
$$25$$ −4.56155 −0.912311
$$26$$ 4.05444 0.795140
$$27$$ −1.03399 −0.198991
$$28$$ 0 0
$$29$$ 1.00000 0.185695
$$30$$ 1.56155 0.285099
$$31$$ −4.05444 −0.728198 −0.364099 0.931360i $$-0.618623\pi$$
−0.364099 + 0.931360i $$0.618623\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.03399 −0.179994
$$34$$ 4.34475 0.745119
$$35$$ 0 0
$$36$$ 2.56155 0.426925
$$37$$ 7.12311 1.17103 0.585516 0.810661i $$-0.300892\pi$$
0.585516 + 0.810661i $$0.300892\pi$$
$$38$$ −7.36520 −1.19479
$$39$$ −9.56155 −1.53107
$$40$$ 0.662153 0.104696
$$41$$ −4.34475 −0.678537 −0.339268 0.940690i $$-0.610179\pi$$
−0.339268 + 0.940690i $$0.610179\pi$$
$$42$$ 0 0
$$43$$ −4.68466 −0.714404 −0.357202 0.934027i $$-0.616269\pi$$
−0.357202 + 0.934027i $$0.616269\pi$$
$$44$$ −0.438447 −0.0660984
$$45$$ −1.69614 −0.252846
$$46$$ 8.24621 1.21584
$$47$$ −0.662153 −0.0965850 −0.0482925 0.998833i $$-0.515378\pi$$
−0.0482925 + 0.998833i $$0.515378\pi$$
$$48$$ 2.35829 0.340390
$$49$$ 0 0
$$50$$ 4.56155 0.645101
$$51$$ −10.2462 −1.43476
$$52$$ −4.05444 −0.562249
$$53$$ −11.5616 −1.58810 −0.794051 0.607852i $$-0.792032\pi$$
−0.794051 + 0.607852i $$0.792032\pi$$
$$54$$ 1.03399 0.140708
$$55$$ 0.290319 0.0391466
$$56$$ 0 0
$$57$$ 17.3693 2.30062
$$58$$ −1.00000 −0.131306
$$59$$ 4.34475 0.565639 0.282819 0.959173i $$-0.408730\pi$$
0.282819 + 0.959173i $$0.408730\pi$$
$$60$$ −1.56155 −0.201596
$$61$$ 1.32431 0.169560 0.0847801 0.996400i $$-0.472981\pi$$
0.0847801 + 0.996400i $$0.472981\pi$$
$$62$$ 4.05444 0.514914
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 2.68466 0.332991
$$66$$ 1.03399 0.127275
$$67$$ −4.87689 −0.595807 −0.297904 0.954596i $$-0.596287\pi$$
−0.297904 + 0.954596i $$0.596287\pi$$
$$68$$ −4.34475 −0.526879
$$69$$ −19.4470 −2.34114
$$70$$ 0 0
$$71$$ 11.3693 1.34929 0.674645 0.738142i $$-0.264297\pi$$
0.674645 + 0.738142i $$0.264297\pi$$
$$72$$ −2.56155 −0.301882
$$73$$ −8.48071 −0.992591 −0.496296 0.868154i $$-0.665307\pi$$
−0.496296 + 0.868154i $$0.665307\pi$$
$$74$$ −7.12311 −0.828044
$$75$$ −10.7575 −1.24217
$$76$$ 7.36520 0.844847
$$77$$ 0 0
$$78$$ 9.56155 1.08263
$$79$$ 7.80776 0.878442 0.439221 0.898379i $$-0.355254\pi$$
0.439221 + 0.898379i $$0.355254\pi$$
$$80$$ −0.662153 −0.0740310
$$81$$ −10.1231 −1.12479
$$82$$ 4.34475 0.479798
$$83$$ −5.08842 −0.558527 −0.279263 0.960215i $$-0.590090\pi$$
−0.279263 + 0.960215i $$0.590090\pi$$
$$84$$ 0 0
$$85$$ 2.87689 0.312043
$$86$$ 4.68466 0.505160
$$87$$ 2.35829 0.252836
$$88$$ 0.438447 0.0467386
$$89$$ −9.06134 −0.960501 −0.480250 0.877132i $$-0.659454\pi$$
−0.480250 + 0.877132i $$0.659454\pi$$
$$90$$ 1.69614 0.178789
$$91$$ 0 0
$$92$$ −8.24621 −0.859727
$$93$$ −9.56155 −0.991487
$$94$$ 0.662153 0.0682959
$$95$$ −4.87689 −0.500359
$$96$$ −2.35829 −0.240692
$$97$$ −6.99337 −0.710069 −0.355034 0.934853i $$-0.615531\pi$$
−0.355034 + 0.934853i $$0.615531\pi$$
$$98$$ 0 0
$$99$$ −1.12311 −0.112876
$$100$$ −4.56155 −0.456155
$$101$$ 11.3381 1.12819 0.564093 0.825711i $$-0.309226\pi$$
0.564093 + 0.825711i $$0.309226\pi$$
$$102$$ 10.2462 1.01453
$$103$$ 12.8255 1.26373 0.631865 0.775078i $$-0.282290\pi$$
0.631865 + 0.775078i $$0.282290\pi$$
$$104$$ 4.05444 0.397570
$$105$$ 0 0
$$106$$ 11.5616 1.12296
$$107$$ −0.876894 −0.0847726 −0.0423863 0.999101i $$-0.513496\pi$$
−0.0423863 + 0.999101i $$0.513496\pi$$
$$108$$ −1.03399 −0.0994955
$$109$$ 10.6847 1.02340 0.511702 0.859163i $$-0.329015\pi$$
0.511702 + 0.859163i $$0.329015\pi$$
$$110$$ −0.290319 −0.0276809
$$111$$ 16.7984 1.59443
$$112$$ 0 0
$$113$$ −19.3693 −1.82211 −0.911056 0.412283i $$-0.864732\pi$$
−0.911056 + 0.412283i $$0.864732\pi$$
$$114$$ −17.3693 −1.62679
$$115$$ 5.46026 0.509172
$$116$$ 1.00000 0.0928477
$$117$$ −10.3857 −0.960154
$$118$$ −4.34475 −0.399967
$$119$$ 0 0
$$120$$ 1.56155 0.142550
$$121$$ −10.8078 −0.982524
$$122$$ −1.32431 −0.119897
$$123$$ −10.2462 −0.923870
$$124$$ −4.05444 −0.364099
$$125$$ 6.33122 0.566281
$$126$$ 0 0
$$127$$ −2.24621 −0.199319 −0.0996595 0.995022i $$-0.531775\pi$$
−0.0996595 + 0.995022i $$0.531775\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −11.0478 −0.972705
$$130$$ −2.68466 −0.235460
$$131$$ −2.06798 −0.180680 −0.0903399 0.995911i $$-0.528795\pi$$
−0.0903399 + 0.995911i $$0.528795\pi$$
$$132$$ −1.03399 −0.0899971
$$133$$ 0 0
$$134$$ 4.87689 0.421300
$$135$$ 0.684658 0.0589260
$$136$$ 4.34475 0.372560
$$137$$ −8.24621 −0.704521 −0.352261 0.935902i $$-0.614587\pi$$
−0.352261 + 0.935902i $$0.614587\pi$$
$$138$$ 19.4470 1.65544
$$139$$ −8.31768 −0.705496 −0.352748 0.935718i $$-0.614753\pi$$
−0.352748 + 0.935718i $$0.614753\pi$$
$$140$$ 0 0
$$141$$ −1.56155 −0.131506
$$142$$ −11.3693 −0.954092
$$143$$ 1.77766 0.148655
$$144$$ 2.56155 0.213463
$$145$$ −0.662153 −0.0549889
$$146$$ 8.48071 0.701868
$$147$$ 0 0
$$148$$ 7.12311 0.585516
$$149$$ 2.68466 0.219936 0.109968 0.993935i $$-0.464925\pi$$
0.109968 + 0.993935i $$0.464925\pi$$
$$150$$ 10.7575 0.878345
$$151$$ 4.24621 0.345552 0.172776 0.984961i $$-0.444726\pi$$
0.172776 + 0.984961i $$0.444726\pi$$
$$152$$ −7.36520 −0.597397
$$153$$ −11.1293 −0.899752
$$154$$ 0 0
$$155$$ 2.68466 0.215637
$$156$$ −9.56155 −0.765537
$$157$$ −17.3790 −1.38700 −0.693498 0.720458i $$-0.743932\pi$$
−0.693498 + 0.720458i $$0.743932\pi$$
$$158$$ −7.80776 −0.621152
$$159$$ −27.2655 −2.16230
$$160$$ 0.662153 0.0523478
$$161$$ 0 0
$$162$$ 10.1231 0.795346
$$163$$ −4.43845 −0.347646 −0.173823 0.984777i $$-0.555612\pi$$
−0.173823 + 0.984777i $$0.555612\pi$$
$$164$$ −4.34475 −0.339268
$$165$$ 0.684658 0.0533006
$$166$$ 5.08842 0.394938
$$167$$ 9.43318 0.729961 0.364981 0.931015i $$-0.381076\pi$$
0.364981 + 0.931015i $$0.381076\pi$$
$$168$$ 0 0
$$169$$ 3.43845 0.264496
$$170$$ −2.87689 −0.220648
$$171$$ 18.8664 1.44275
$$172$$ −4.68466 −0.357202
$$173$$ 11.1293 0.846146 0.423073 0.906095i $$-0.360951\pi$$
0.423073 + 0.906095i $$0.360951\pi$$
$$174$$ −2.35829 −0.178782
$$175$$ 0 0
$$176$$ −0.438447 −0.0330492
$$177$$ 10.2462 0.770152
$$178$$ 9.06134 0.679176
$$179$$ −1.75379 −0.131084 −0.0655422 0.997850i $$-0.520878\pi$$
−0.0655422 + 0.997850i $$0.520878\pi$$
$$180$$ −1.69614 −0.126423
$$181$$ 22.1771 1.64841 0.824206 0.566290i $$-0.191622\pi$$
0.824206 + 0.566290i $$0.191622\pi$$
$$182$$ 0 0
$$183$$ 3.12311 0.230867
$$184$$ 8.24621 0.607919
$$185$$ −4.71659 −0.346771
$$186$$ 9.56155 0.701087
$$187$$ 1.90495 0.139303
$$188$$ −0.662153 −0.0482925
$$189$$ 0 0
$$190$$ 4.87689 0.353807
$$191$$ −10.2462 −0.741390 −0.370695 0.928755i $$-0.620880\pi$$
−0.370695 + 0.928755i $$0.620880\pi$$
$$192$$ 2.35829 0.170195
$$193$$ −3.36932 −0.242529 −0.121264 0.992620i $$-0.538695\pi$$
−0.121264 + 0.992620i $$0.538695\pi$$
$$194$$ 6.99337 0.502095
$$195$$ 6.33122 0.453388
$$196$$ 0 0
$$197$$ −3.75379 −0.267446 −0.133723 0.991019i $$-0.542693\pi$$
−0.133723 + 0.991019i $$0.542693\pi$$
$$198$$ 1.12311 0.0798156
$$199$$ 20.7713 1.47244 0.736219 0.676743i $$-0.236609\pi$$
0.736219 + 0.676743i $$0.236609\pi$$
$$200$$ 4.56155 0.322550
$$201$$ −11.5012 −0.811229
$$202$$ −11.3381 −0.797748
$$203$$ 0 0
$$204$$ −10.2462 −0.717378
$$205$$ 2.87689 0.200931
$$206$$ −12.8255 −0.893592
$$207$$ −21.1231 −1.46816
$$208$$ −4.05444 −0.281125
$$209$$ −3.22925 −0.223372
$$210$$ 0 0
$$211$$ 4.68466 0.322505 0.161253 0.986913i $$-0.448447\pi$$
0.161253 + 0.986913i $$0.448447\pi$$
$$212$$ −11.5616 −0.794051
$$213$$ 26.8122 1.83714
$$214$$ 0.876894 0.0599433
$$215$$ 3.10196 0.211552
$$216$$ 1.03399 0.0703539
$$217$$ 0 0
$$218$$ −10.6847 −0.723656
$$219$$ −20.0000 −1.35147
$$220$$ 0.290319 0.0195733
$$221$$ 17.6155 1.18495
$$222$$ −16.7984 −1.12743
$$223$$ 3.39228 0.227164 0.113582 0.993529i $$-0.463768\pi$$
0.113582 + 0.993529i $$0.463768\pi$$
$$224$$ 0 0
$$225$$ −11.6847 −0.778977
$$226$$ 19.3693 1.28843
$$227$$ 12.4536 0.826576 0.413288 0.910600i $$-0.364380\pi$$
0.413288 + 0.910600i $$0.364380\pi$$
$$228$$ 17.3693 1.15031
$$229$$ 23.5829 1.55840 0.779202 0.626772i $$-0.215624\pi$$
0.779202 + 0.626772i $$0.215624\pi$$
$$230$$ −5.46026 −0.360039
$$231$$ 0 0
$$232$$ −1.00000 −0.0656532
$$233$$ −12.9309 −0.847129 −0.423565 0.905866i $$-0.639221\pi$$
−0.423565 + 0.905866i $$0.639221\pi$$
$$234$$ 10.3857 0.678931
$$235$$ 0.438447 0.0286011
$$236$$ 4.34475 0.282819
$$237$$ 18.4130 1.19605
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −1.56155 −0.100798
$$241$$ −24.6169 −1.58572 −0.792858 0.609406i $$-0.791408\pi$$
−0.792858 + 0.609406i $$0.791408\pi$$
$$242$$ 10.8078 0.694749
$$243$$ −20.7713 −1.33248
$$244$$ 1.32431 0.0847801
$$245$$ 0 0
$$246$$ 10.2462 0.653275
$$247$$ −29.8617 −1.90006
$$248$$ 4.05444 0.257457
$$249$$ −12.0000 −0.760469
$$250$$ −6.33122 −0.400421
$$251$$ 22.5490 1.42328 0.711639 0.702546i $$-0.247953\pi$$
0.711639 + 0.702546i $$0.247953\pi$$
$$252$$ 0 0
$$253$$ 3.61553 0.227306
$$254$$ 2.24621 0.140940
$$255$$ 6.78456 0.424866
$$256$$ 1.00000 0.0625000
$$257$$ 13.8594 0.864529 0.432264 0.901747i $$-0.357715\pi$$
0.432264 + 0.901747i $$0.357715\pi$$
$$258$$ 11.0478 0.687806
$$259$$ 0 0
$$260$$ 2.68466 0.166495
$$261$$ 2.56155 0.158556
$$262$$ 2.06798 0.127760
$$263$$ −15.8078 −0.974748 −0.487374 0.873193i $$-0.662045\pi$$
−0.487374 + 0.873193i $$0.662045\pi$$
$$264$$ 1.03399 0.0636375
$$265$$ 7.65552 0.470275
$$266$$ 0 0
$$267$$ −21.3693 −1.30778
$$268$$ −4.87689 −0.297904
$$269$$ 24.9073 1.51862 0.759311 0.650728i $$-0.225536\pi$$
0.759311 + 0.650728i $$0.225536\pi$$
$$270$$ −0.684658 −0.0416670
$$271$$ −12.0003 −0.728965 −0.364482 0.931210i $$-0.618754\pi$$
−0.364482 + 0.931210i $$0.618754\pi$$
$$272$$ −4.34475 −0.263439
$$273$$ 0 0
$$274$$ 8.24621 0.498172
$$275$$ 2.00000 0.120605
$$276$$ −19.4470 −1.17057
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 8.31768 0.498861
$$279$$ −10.3857 −0.621773
$$280$$ 0 0
$$281$$ −29.1771 −1.74056 −0.870279 0.492558i $$-0.836062\pi$$
−0.870279 + 0.492558i $$0.836062\pi$$
$$282$$ 1.56155 0.0929891
$$283$$ 10.5487 0.627054 0.313527 0.949579i $$-0.398489\pi$$
0.313527 + 0.949579i $$0.398489\pi$$
$$284$$ 11.3693 0.674645
$$285$$ −11.5012 −0.681270
$$286$$ −1.77766 −0.105115
$$287$$ 0 0
$$288$$ −2.56155 −0.150941
$$289$$ 1.87689 0.110406
$$290$$ 0.662153 0.0388830
$$291$$ −16.4924 −0.966803
$$292$$ −8.48071 −0.496296
$$293$$ −11.5012 −0.671905 −0.335952 0.941879i $$-0.609058\pi$$
−0.335952 + 0.941879i $$0.609058\pi$$
$$294$$ 0 0
$$295$$ −2.87689 −0.167499
$$296$$ −7.12311 −0.414022
$$297$$ 0.453349 0.0263060
$$298$$ −2.68466 −0.155518
$$299$$ 33.4337 1.93352
$$300$$ −10.7575 −0.621084
$$301$$ 0 0
$$302$$ −4.24621 −0.244342
$$303$$ 26.7386 1.53609
$$304$$ 7.36520 0.422423
$$305$$ −0.876894 −0.0502108
$$306$$ 11.1293 0.636221
$$307$$ 11.2108 0.639836 0.319918 0.947445i $$-0.396345\pi$$
0.319918 + 0.947445i $$0.396345\pi$$
$$308$$ 0 0
$$309$$ 30.2462 1.72065
$$310$$ −2.68466 −0.152478
$$311$$ −9.80501 −0.555991 −0.277996 0.960582i $$-0.589670\pi$$
−0.277996 + 0.960582i $$0.589670\pi$$
$$312$$ 9.56155 0.541316
$$313$$ 27.2655 1.54114 0.770570 0.637356i $$-0.219972\pi$$
0.770570 + 0.637356i $$0.219972\pi$$
$$314$$ 17.3790 0.980755
$$315$$ 0 0
$$316$$ 7.80776 0.439221
$$317$$ 14.4924 0.813976 0.406988 0.913434i $$-0.366579\pi$$
0.406988 + 0.913434i $$0.366579\pi$$
$$318$$ 27.2655 1.52898
$$319$$ −0.438447 −0.0245483
$$320$$ −0.662153 −0.0370155
$$321$$ −2.06798 −0.115423
$$322$$ 0 0
$$323$$ −32.0000 −1.78053
$$324$$ −10.1231 −0.562395
$$325$$ 18.4945 1.02589
$$326$$ 4.43845 0.245823
$$327$$ 25.1976 1.39343
$$328$$ 4.34475 0.239899
$$329$$ 0 0
$$330$$ −0.684658 −0.0376892
$$331$$ 36.3002 1.99524 0.997619 0.0689611i $$-0.0219684\pi$$
0.997619 + 0.0689611i $$0.0219684\pi$$
$$332$$ −5.08842 −0.279263
$$333$$ 18.2462 0.999886
$$334$$ −9.43318 −0.516161
$$335$$ 3.22925 0.176433
$$336$$ 0 0
$$337$$ −24.2462 −1.32078 −0.660388 0.750925i $$-0.729608\pi$$
−0.660388 + 0.750925i $$0.729608\pi$$
$$338$$ −3.43845 −0.187027
$$339$$ −45.6786 −2.48092
$$340$$ 2.87689 0.156022
$$341$$ 1.77766 0.0962655
$$342$$ −18.8664 −1.02018
$$343$$ 0 0
$$344$$ 4.68466 0.252580
$$345$$ 12.8769 0.693269
$$346$$ −11.1293 −0.598316
$$347$$ −31.1231 −1.67078 −0.835388 0.549661i $$-0.814757\pi$$
−0.835388 + 0.549661i $$0.814757\pi$$
$$348$$ 2.35829 0.126418
$$349$$ −22.9208 −1.22692 −0.613461 0.789725i $$-0.710223\pi$$
−0.613461 + 0.789725i $$0.710223\pi$$
$$350$$ 0 0
$$351$$ 4.19224 0.223765
$$352$$ 0.438447 0.0233693
$$353$$ −9.43318 −0.502077 −0.251039 0.967977i $$-0.580772\pi$$
−0.251039 + 0.967977i $$0.580772\pi$$
$$354$$ −10.2462 −0.544580
$$355$$ −7.52823 −0.399557
$$356$$ −9.06134 −0.480250
$$357$$ 0 0
$$358$$ 1.75379 0.0926906
$$359$$ −28.6847 −1.51392 −0.756959 0.653462i $$-0.773316\pi$$
−0.756959 + 0.653462i $$0.773316\pi$$
$$360$$ 1.69614 0.0893945
$$361$$ 35.2462 1.85506
$$362$$ −22.1771 −1.16560
$$363$$ −25.4879 −1.33777
$$364$$ 0 0
$$365$$ 5.61553 0.293930
$$366$$ −3.12311 −0.163247
$$367$$ −0.371834 −0.0194096 −0.00970479 0.999953i $$-0.503089\pi$$
−0.00970479 + 0.999953i $$0.503089\pi$$
$$368$$ −8.24621 −0.429863
$$369$$ −11.1293 −0.579369
$$370$$ 4.71659 0.245204
$$371$$ 0 0
$$372$$ −9.56155 −0.495743
$$373$$ −21.8078 −1.12916 −0.564582 0.825377i $$-0.690962\pi$$
−0.564582 + 0.825377i $$0.690962\pi$$
$$374$$ −1.90495 −0.0985024
$$375$$ 14.9309 0.771027
$$376$$ 0.662153 0.0341480
$$377$$ −4.05444 −0.208814
$$378$$ 0 0
$$379$$ 32.4924 1.66902 0.834512 0.550990i $$-0.185750\pi$$
0.834512 + 0.550990i $$0.185750\pi$$
$$380$$ −4.87689 −0.250179
$$381$$ −5.29723 −0.271385
$$382$$ 10.2462 0.524242
$$383$$ −7.36520 −0.376344 −0.188172 0.982136i $$-0.560256\pi$$
−0.188172 + 0.982136i $$0.560256\pi$$
$$384$$ −2.35829 −0.120346
$$385$$ 0 0
$$386$$ 3.36932 0.171494
$$387$$ −12.0000 −0.609994
$$388$$ −6.99337 −0.355034
$$389$$ −32.9848 −1.67240 −0.836199 0.548426i $$-0.815227\pi$$
−0.836199 + 0.548426i $$0.815227\pi$$
$$390$$ −6.33122 −0.320594
$$391$$ 35.8278 1.81189
$$392$$ 0 0
$$393$$ −4.87689 −0.246007
$$394$$ 3.75379 0.189113
$$395$$ −5.16994 −0.260128
$$396$$ −1.12311 −0.0564382
$$397$$ 10.2584 0.514852 0.257426 0.966298i $$-0.417126\pi$$
0.257426 + 0.966298i $$0.417126\pi$$
$$398$$ −20.7713 −1.04117
$$399$$ 0 0
$$400$$ −4.56155 −0.228078
$$401$$ 14.1922 0.708726 0.354363 0.935108i $$-0.384698\pi$$
0.354363 + 0.935108i $$0.384698\pi$$
$$402$$ 11.5012 0.573625
$$403$$ 16.4384 0.818857
$$404$$ 11.3381 0.564093
$$405$$ 6.70305 0.333077
$$406$$ 0 0
$$407$$ −3.12311 −0.154807
$$408$$ 10.2462 0.507263
$$409$$ −29.9957 −1.48319 −0.741595 0.670848i $$-0.765931\pi$$
−0.741595 + 0.670848i $$0.765931\pi$$
$$410$$ −2.87689 −0.142080
$$411$$ −19.4470 −0.959249
$$412$$ 12.8255 0.631865
$$413$$ 0 0
$$414$$ 21.1231 1.03814
$$415$$ 3.36932 0.165393
$$416$$ 4.05444 0.198785
$$417$$ −19.6155 −0.960577
$$418$$ 3.22925 0.157948
$$419$$ −5.83209 −0.284916 −0.142458 0.989801i $$-0.545501\pi$$
−0.142458 + 0.989801i $$0.545501\pi$$
$$420$$ 0 0
$$421$$ −31.1231 −1.51685 −0.758424 0.651762i $$-0.774030\pi$$
−0.758424 + 0.651762i $$0.774030\pi$$
$$422$$ −4.68466 −0.228046
$$423$$ −1.69614 −0.0824692
$$424$$ 11.5616 0.561479
$$425$$ 19.8188 0.961354
$$426$$ −26.8122 −1.29906
$$427$$ 0 0
$$428$$ −0.876894 −0.0423863
$$429$$ 4.19224 0.202403
$$430$$ −3.10196 −0.149590
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ −1.03399 −0.0497478
$$433$$ 16.5896 0.797244 0.398622 0.917115i $$-0.369489\pi$$
0.398622 + 0.917115i $$0.369489\pi$$
$$434$$ 0 0
$$435$$ −1.56155 −0.0748707
$$436$$ 10.6847 0.511702
$$437$$ −60.7350 −2.90535
$$438$$ 20.0000 0.955637
$$439$$ −35.6647 −1.70218 −0.851092 0.525016i $$-0.824059\pi$$
−0.851092 + 0.525016i $$0.824059\pi$$
$$440$$ −0.290319 −0.0138404
$$441$$ 0 0
$$442$$ −17.6155 −0.837885
$$443$$ 33.1231 1.57373 0.786863 0.617128i $$-0.211704\pi$$
0.786863 + 0.617128i $$0.211704\pi$$
$$444$$ 16.7984 0.797216
$$445$$ 6.00000 0.284427
$$446$$ −3.39228 −0.160629
$$447$$ 6.33122 0.299456
$$448$$ 0 0
$$449$$ −19.3693 −0.914095 −0.457047 0.889442i $$-0.651093\pi$$
−0.457047 + 0.889442i $$0.651093\pi$$
$$450$$ 11.6847 0.550820
$$451$$ 1.90495 0.0897004
$$452$$ −19.3693 −0.911056
$$453$$ 10.0138 0.470490
$$454$$ −12.4536 −0.584478
$$455$$ 0 0
$$456$$ −17.3693 −0.813393
$$457$$ 29.3693 1.37384 0.686919 0.726734i $$-0.258963\pi$$
0.686919 + 0.726734i $$0.258963\pi$$
$$458$$ −23.5829 −1.10196
$$459$$ 4.49242 0.209688
$$460$$ 5.46026 0.254586
$$461$$ −23.4199 −1.09077 −0.545387 0.838184i $$-0.683617\pi$$
−0.545387 + 0.838184i $$0.683617\pi$$
$$462$$ 0 0
$$463$$ −23.3693 −1.08606 −0.543032 0.839712i $$-0.682724\pi$$
−0.543032 + 0.839712i $$0.682724\pi$$
$$464$$ 1.00000 0.0464238
$$465$$ 6.33122 0.293603
$$466$$ 12.9309 0.599011
$$467$$ −6.33122 −0.292974 −0.146487 0.989213i $$-0.546797\pi$$
−0.146487 + 0.989213i $$0.546797\pi$$
$$468$$ −10.3857 −0.480077
$$469$$ 0 0
$$470$$ −0.438447 −0.0202241
$$471$$ −40.9848 −1.88848
$$472$$ −4.34475 −0.199984
$$473$$ 2.05398 0.0944419
$$474$$ −18.4130 −0.845737
$$475$$ −33.5968 −1.54153
$$476$$ 0 0
$$477$$ −29.6155 −1.35600
$$478$$ 0 0
$$479$$ −40.2998 −1.84135 −0.920673 0.390336i $$-0.872359\pi$$
−0.920673 + 0.390336i $$0.872359\pi$$
$$480$$ 1.56155 0.0712748
$$481$$ −28.8802 −1.31682
$$482$$ 24.6169 1.12127
$$483$$ 0 0
$$484$$ −10.8078 −0.491262
$$485$$ 4.63068 0.210268
$$486$$ 20.7713 0.942205
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −1.32431 −0.0599486
$$489$$ −10.4672 −0.473342
$$490$$ 0 0
$$491$$ −25.1771 −1.13623 −0.568113 0.822951i $$-0.692326\pi$$
−0.568113 + 0.822951i $$0.692326\pi$$
$$492$$ −10.2462 −0.461935
$$493$$ −4.34475 −0.195678
$$494$$ 29.8617 1.34354
$$495$$ 0.743668 0.0334254
$$496$$ −4.05444 −0.182050
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 3.12311 0.139809 0.0699047 0.997554i $$-0.477730\pi$$
0.0699047 + 0.997554i $$0.477730\pi$$
$$500$$ 6.33122 0.283141
$$501$$ 22.2462 0.993887
$$502$$ −22.5490 −1.00641
$$503$$ 33.6783 1.50164 0.750820 0.660507i $$-0.229659\pi$$
0.750820 + 0.660507i $$0.229659\pi$$
$$504$$ 0 0
$$505$$ −7.50758 −0.334083
$$506$$ −3.61553 −0.160730
$$507$$ 8.10887 0.360128
$$508$$ −2.24621 −0.0996595
$$509$$ 34.4219 1.52573 0.762863 0.646560i $$-0.223793\pi$$
0.762863 + 0.646560i $$0.223793\pi$$
$$510$$ −6.78456 −0.300426
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ −7.61553 −0.336234
$$514$$ −13.8594 −0.611314
$$515$$ −8.49242 −0.374221
$$516$$ −11.0478 −0.486352
$$517$$ 0.290319 0.0127682
$$518$$ 0 0
$$519$$ 26.2462 1.15208
$$520$$ −2.68466 −0.117730
$$521$$ −24.4539 −1.07134 −0.535672 0.844426i $$-0.679942\pi$$
−0.535672 + 0.844426i $$0.679942\pi$$
$$522$$ −2.56155 −0.112116
$$523$$ 43.8194 1.91609 0.958044 0.286621i $$-0.0925322\pi$$
0.958044 + 0.286621i $$0.0925322\pi$$
$$524$$ −2.06798 −0.0903399
$$525$$ 0 0
$$526$$ 15.8078 0.689251
$$527$$ 17.6155 0.767344
$$528$$ −1.03399 −0.0449985
$$529$$ 45.0000 1.95652
$$530$$ −7.65552 −0.332535
$$531$$ 11.1293 0.482971
$$532$$ 0 0
$$533$$ 17.6155 0.763013
$$534$$ 21.3693 0.924741
$$535$$ 0.580639 0.0251032
$$536$$ 4.87689 0.210650
$$537$$ −4.13595 −0.178479
$$538$$ −24.9073 −1.07383
$$539$$ 0 0
$$540$$ 0.684658 0.0294630
$$541$$ 9.12311 0.392233 0.196116 0.980581i $$-0.437167\pi$$
0.196116 + 0.980581i $$0.437167\pi$$
$$542$$ 12.0003 0.515456
$$543$$ 52.3002 2.24442
$$544$$ 4.34475 0.186280
$$545$$ −7.07488 −0.303055
$$546$$ 0 0
$$547$$ 7.61553 0.325616 0.162808 0.986658i $$-0.447945\pi$$
0.162808 + 0.986658i $$0.447945\pi$$
$$548$$ −8.24621 −0.352261
$$549$$ 3.39228 0.144779
$$550$$ −2.00000 −0.0852803
$$551$$ 7.36520 0.313768
$$552$$ 19.4470 0.827719
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ −11.1231 −0.472150
$$556$$ −8.31768 −0.352748
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 10.3857 0.439660
$$559$$ 18.9936 0.803346
$$560$$ 0 0
$$561$$ 4.49242 0.189670
$$562$$ 29.1771 1.23076
$$563$$ 18.4130 0.776016 0.388008 0.921656i $$-0.373163\pi$$
0.388008 + 0.921656i $$0.373163\pi$$
$$564$$ −1.56155 −0.0657532
$$565$$ 12.8255 0.539571
$$566$$ −10.5487 −0.443394
$$567$$ 0 0
$$568$$ −11.3693 −0.477046
$$569$$ 30.0000 1.25767 0.628833 0.777541i $$-0.283533\pi$$
0.628833 + 0.777541i $$0.283533\pi$$
$$570$$ 11.5012 0.481730
$$571$$ 16.8769 0.706276 0.353138 0.935571i $$-0.385115\pi$$
0.353138 + 0.935571i $$0.385115\pi$$
$$572$$ 1.77766 0.0743275
$$573$$ −24.1636 −1.00945
$$574$$ 0 0
$$575$$ 37.6155 1.56868
$$576$$ 2.56155 0.106731
$$577$$ 4.18173 0.174087 0.0870437 0.996204i $$-0.472258\pi$$
0.0870437 + 0.996204i $$0.472258\pi$$
$$578$$ −1.87689 −0.0780685
$$579$$ −7.94584 −0.330218
$$580$$ −0.662153 −0.0274944
$$581$$ 0 0
$$582$$ 16.4924 0.683633
$$583$$ 5.06913 0.209942
$$584$$ 8.48071 0.350934
$$585$$ 6.87689 0.284325
$$586$$ 11.5012 0.475108
$$587$$ 35.8735 1.48066 0.740330 0.672244i $$-0.234669\pi$$
0.740330 + 0.672244i $$0.234669\pi$$
$$588$$ 0 0
$$589$$ −29.8617 −1.23043
$$590$$ 2.87689 0.118440
$$591$$ −8.85254 −0.364145
$$592$$ 7.12311 0.292758
$$593$$ −11.6284 −0.477523 −0.238761 0.971078i $$-0.576741\pi$$
−0.238761 + 0.971078i $$0.576741\pi$$
$$594$$ −0.453349 −0.0186011
$$595$$ 0 0
$$596$$ 2.68466 0.109968
$$597$$ 48.9848 2.00482
$$598$$ −33.4337 −1.36721
$$599$$ −24.6847 −1.00859 −0.504294 0.863532i $$-0.668247\pi$$
−0.504294 + 0.863532i $$0.668247\pi$$
$$600$$ 10.7575 0.439172
$$601$$ 5.83209 0.237896 0.118948 0.992900i $$-0.462048\pi$$
0.118948 + 0.992900i $$0.462048\pi$$
$$602$$ 0 0
$$603$$ −12.4924 −0.508731
$$604$$ 4.24621 0.172776
$$605$$ 7.15640 0.290949
$$606$$ −26.7386 −1.08618
$$607$$ 32.9346 1.33677 0.668387 0.743813i $$-0.266985\pi$$
0.668387 + 0.743813i $$0.266985\pi$$
$$608$$ −7.36520 −0.298698
$$609$$ 0 0
$$610$$ 0.876894 0.0355044
$$611$$ 2.68466 0.108610
$$612$$ −11.1293 −0.449876
$$613$$ 20.0540 0.809972 0.404986 0.914323i $$-0.367276\pi$$
0.404986 + 0.914323i $$0.367276\pi$$
$$614$$ −11.2108 −0.452432
$$615$$ 6.78456 0.273580
$$616$$ 0 0
$$617$$ 7.36932 0.296678 0.148339 0.988937i $$-0.452607\pi$$
0.148339 + 0.988937i $$0.452607\pi$$
$$618$$ −30.2462 −1.21668
$$619$$ 31.4015 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$620$$ 2.68466 0.107818
$$621$$ 8.52648 0.342156
$$622$$ 9.80501 0.393145
$$623$$ 0 0
$$624$$ −9.56155 −0.382768
$$625$$ 18.6155 0.744621
$$626$$ −27.2655 −1.08975
$$627$$ −7.61553 −0.304135
$$628$$ −17.3790 −0.693498
$$629$$ −30.9481 −1.23398
$$630$$ 0 0
$$631$$ 22.2462 0.885608 0.442804 0.896619i $$-0.353984\pi$$
0.442804 + 0.896619i $$0.353984\pi$$
$$632$$ −7.80776 −0.310576
$$633$$ 11.0478 0.439111
$$634$$ −14.4924 −0.575568
$$635$$ 1.48734 0.0590231
$$636$$ −27.2655 −1.08115
$$637$$ 0 0
$$638$$ 0.438447 0.0173583
$$639$$ 29.1231 1.15209
$$640$$ 0.662153 0.0261739
$$641$$ 7.36932 0.291071 0.145535 0.989353i $$-0.453510\pi$$
0.145535 + 0.989353i $$0.453510\pi$$
$$642$$ 2.06798 0.0816165
$$643$$ −13.0343 −0.514021 −0.257011 0.966409i $$-0.582738\pi$$
−0.257011 + 0.966409i $$0.582738\pi$$
$$644$$ 0 0
$$645$$ 7.31534 0.288041
$$646$$ 32.0000 1.25902
$$647$$ 0.743668 0.0292366 0.0146183 0.999893i $$-0.495347\pi$$
0.0146183 + 0.999893i $$0.495347\pi$$
$$648$$ 10.1231 0.397673
$$649$$ −1.90495 −0.0747757
$$650$$ −18.4945 −0.725415
$$651$$ 0 0
$$652$$ −4.43845 −0.173823
$$653$$ 47.8617 1.87297 0.936487 0.350701i $$-0.114057\pi$$
0.936487 + 0.350701i $$0.114057\pi$$
$$654$$ −25.1976 −0.985303
$$655$$ 1.36932 0.0535036
$$656$$ −4.34475 −0.169634
$$657$$ −21.7238 −0.847525
$$658$$ 0 0
$$659$$ −34.3002 −1.33615 −0.668073 0.744096i $$-0.732881\pi$$
−0.668073 + 0.744096i $$0.732881\pi$$
$$660$$ 0.684658 0.0266503
$$661$$ −25.8597 −1.00583 −0.502913 0.864337i $$-0.667739\pi$$
−0.502913 + 0.864337i $$0.667739\pi$$
$$662$$ −36.3002 −1.41085
$$663$$ 41.5426 1.61338
$$664$$ 5.08842 0.197469
$$665$$ 0 0
$$666$$ −18.2462 −0.707026
$$667$$ −8.24621 −0.319295
$$668$$ 9.43318 0.364981
$$669$$ 8.00000 0.309298
$$670$$ −3.22925 −0.124757
$$671$$ −0.580639 −0.0224153
$$672$$ 0 0
$$673$$ 41.6695 1.60624 0.803121 0.595816i $$-0.203171\pi$$
0.803121 + 0.595816i $$0.203171\pi$$
$$674$$ 24.2462 0.933929
$$675$$ 4.71659 0.181542
$$676$$ 3.43845 0.132248
$$677$$ 1.16128 0.0446315 0.0223158 0.999751i $$-0.492896\pi$$
0.0223158 + 0.999751i $$0.492896\pi$$
$$678$$ 45.6786 1.75427
$$679$$ 0 0
$$680$$ −2.87689 −0.110324
$$681$$ 29.3693 1.12543
$$682$$ −1.77766 −0.0680700
$$683$$ −22.2462 −0.851228 −0.425614 0.904905i $$-0.639942\pi$$
−0.425614 + 0.904905i $$0.639942\pi$$
$$684$$ 18.8664 0.721373
$$685$$ 5.46026 0.208626
$$686$$ 0 0
$$687$$ 55.6155 2.12186
$$688$$ −4.68466 −0.178601
$$689$$ 46.8756 1.78582
$$690$$ −12.8769 −0.490215
$$691$$ −29.9957 −1.14109 −0.570545 0.821267i $$-0.693268\pi$$
−0.570545 + 0.821267i $$0.693268\pi$$
$$692$$ 11.1293 0.423073
$$693$$ 0 0
$$694$$ 31.1231 1.18142
$$695$$ 5.50758 0.208914
$$696$$ −2.35829 −0.0893909
$$697$$ 18.8769 0.715013
$$698$$ 22.9208 0.867565
$$699$$ −30.4948 −1.15342
$$700$$ 0 0
$$701$$ −12.0540 −0.455272 −0.227636 0.973746i $$-0.573100\pi$$
−0.227636 + 0.973746i $$0.573100\pi$$
$$702$$ −4.19224 −0.158226
$$703$$ 52.4631 1.97868
$$704$$ −0.438447 −0.0165246
$$705$$ 1.03399 0.0389422
$$706$$ 9.43318 0.355022
$$707$$ 0 0
$$708$$ 10.2462 0.385076
$$709$$ 3.06913 0.115264 0.0576318 0.998338i $$-0.481645\pi$$
0.0576318 + 0.998338i $$0.481645\pi$$
$$710$$ 7.52823 0.282530
$$711$$ 20.0000 0.750059
$$712$$ 9.06134 0.339588
$$713$$ 33.4337 1.25210
$$714$$ 0 0
$$715$$ −1.17708 −0.0440203
$$716$$ −1.75379 −0.0655422
$$717$$ 0 0
$$718$$ 28.6847 1.07050
$$719$$ −29.6238 −1.10478 −0.552391 0.833585i $$-0.686285\pi$$
−0.552391 + 0.833585i $$0.686285\pi$$
$$720$$ −1.69614 −0.0632114
$$721$$ 0 0
$$722$$ −35.2462 −1.31173
$$723$$ −58.0540 −2.15905
$$724$$ 22.1771 0.824206
$$725$$ −4.56155 −0.169412
$$726$$ 25.4879 0.945944
$$727$$ −34.7123 −1.28741 −0.643703 0.765275i $$-0.722603\pi$$
−0.643703 + 0.765275i $$0.722603\pi$$
$$728$$ 0 0
$$729$$ −18.6155 −0.689464
$$730$$ −5.61553 −0.207840
$$731$$ 20.3537 0.752809
$$732$$ 3.12311 0.115433
$$733$$ 24.3266 0.898524 0.449262 0.893400i $$-0.351687\pi$$
0.449262 + 0.893400i $$0.351687\pi$$
$$734$$ 0.371834 0.0137246
$$735$$ 0 0
$$736$$ 8.24621 0.303959
$$737$$ 2.13826 0.0787638
$$738$$ 11.1293 0.409676
$$739$$ −22.0540 −0.811269 −0.405634 0.914035i $$-0.632949\pi$$
−0.405634 + 0.914035i $$0.632949\pi$$
$$740$$ −4.71659 −0.173385
$$741$$ −70.4228 −2.58705
$$742$$ 0 0
$$743$$ 44.9848 1.65033 0.825167 0.564889i $$-0.191081\pi$$
0.825167 + 0.564889i $$0.191081\pi$$
$$744$$ 9.56155 0.350544
$$745$$ −1.77766 −0.0651283
$$746$$ 21.8078 0.798439
$$747$$ −13.0343 −0.476899
$$748$$ 1.90495 0.0696517
$$749$$ 0 0
$$750$$ −14.9309 −0.545198
$$751$$ 30.7386 1.12167 0.560834 0.827928i $$-0.310480\pi$$
0.560834 + 0.827928i $$0.310480\pi$$
$$752$$ −0.662153 −0.0241463
$$753$$ 53.1771 1.93788
$$754$$ 4.05444 0.147654
$$755$$ −2.81164 −0.102326
$$756$$ 0 0
$$757$$ −0.492423 −0.0178974 −0.00894870 0.999960i $$-0.502848\pi$$
−0.00894870 + 0.999960i $$0.502848\pi$$
$$758$$ −32.4924 −1.18018
$$759$$ 8.52648 0.309492
$$760$$ 4.87689 0.176904
$$761$$ −27.1383 −0.983761 −0.491881 0.870663i $$-0.663690\pi$$
−0.491881 + 0.870663i $$0.663690\pi$$
$$762$$ 5.29723 0.191898
$$763$$ 0 0
$$764$$ −10.2462 −0.370695
$$765$$ 7.36932 0.266438
$$766$$ 7.36520 0.266116
$$767$$ −17.6155 −0.636060
$$768$$ 2.35829 0.0850976
$$769$$ −24.6984 −0.890649 −0.445324 0.895369i $$-0.646912\pi$$
−0.445324 + 0.895369i $$0.646912\pi$$
$$770$$ 0 0
$$771$$ 32.6847 1.17711
$$772$$ −3.36932 −0.121264
$$773$$ 43.0299 1.54768 0.773840 0.633382i $$-0.218334\pi$$
0.773840 + 0.633382i $$0.218334\pi$$
$$774$$ 12.0000 0.431331
$$775$$ 18.4945 0.664343
$$776$$ 6.99337 0.251047
$$777$$ 0 0
$$778$$ 32.9848 1.18256
$$779$$ −32.0000 −1.14652
$$780$$ 6.33122 0.226694
$$781$$ −4.98485 −0.178372
$$782$$ −35.8278 −1.28120
$$783$$ −1.03399 −0.0369517
$$784$$ 0 0
$$785$$ 11.5076 0.410723
$$786$$ 4.87689 0.173953
$$787$$ 29.9957 1.06923 0.534615 0.845096i $$-0.320457\pi$$
0.534615 + 0.845096i $$0.320457\pi$$
$$788$$ −3.75379 −0.133723
$$789$$ −37.2794 −1.32718
$$790$$ 5.16994 0.183938
$$791$$ 0 0
$$792$$ 1.12311 0.0399078
$$793$$ −5.36932 −0.190670
$$794$$ −10.2584 −0.364056
$$795$$ 18.0540 0.640309
$$796$$ 20.7713 0.736219
$$797$$ 47.1659 1.67070 0.835351 0.549717i $$-0.185265\pi$$
0.835351 + 0.549717i $$0.185265\pi$$
$$798$$ 0 0
$$799$$ 2.87689 0.101777
$$800$$ 4.56155 0.161275
$$801$$ −23.2111 −0.820124
$$802$$ −14.1922 −0.501145
$$803$$ 3.71834 0.131217
$$804$$ −11.5012 −0.405614
$$805$$ 0 0
$$806$$ −16.4384 −0.579020
$$807$$ 58.7386 2.06770
$$808$$ −11.3381 −0.398874
$$809$$ 42.9848 1.51127 0.755633 0.654995i $$-0.227329\pi$$
0.755633 + 0.654995i $$0.227329\pi$$
$$810$$ −6.70305 −0.235521
$$811$$ 54.7399 1.92218 0.961089 0.276239i $$-0.0890882\pi$$
0.961089 + 0.276239i $$0.0890882\pi$$
$$812$$ 0 0
$$813$$ −28.3002 −0.992531
$$814$$ 3.12311 0.109465
$$815$$ 2.93893 0.102946
$$816$$ −10.2462 −0.358689
$$817$$ −34.5035 −1.20712
$$818$$ 29.9957 1.04877
$$819$$ 0 0
$$820$$ 2.87689 0.100466
$$821$$ −16.9309 −0.590891 −0.295446 0.955360i $$-0.595468\pi$$
−0.295446 + 0.955360i $$0.595468\pi$$
$$822$$ 19.4470 0.678292
$$823$$ 26.2462 0.914885 0.457443 0.889239i $$-0.348765\pi$$
0.457443 + 0.889239i $$0.348765\pi$$
$$824$$ −12.8255 −0.446796
$$825$$ 4.71659 0.164211
$$826$$ 0 0
$$827$$ −25.3153 −0.880301 −0.440150 0.897924i $$-0.645075\pi$$
−0.440150 + 0.897924i $$0.645075\pi$$
$$828$$ −21.1231 −0.734079
$$829$$ 22.2586 0.773074 0.386537 0.922274i $$-0.373671\pi$$
0.386537 + 0.922274i $$0.373671\pi$$
$$830$$ −3.36932 −0.116951
$$831$$ −23.5829 −0.818083
$$832$$ −4.05444 −0.140562
$$833$$ 0 0
$$834$$ 19.6155 0.679230
$$835$$ −6.24621 −0.216159
$$836$$ −3.22925 −0.111686
$$837$$ 4.19224 0.144905
$$838$$ 5.83209 0.201466
$$839$$ 21.4335 0.739965 0.369983 0.929039i $$-0.379364\pi$$
0.369983 + 0.929039i $$0.379364\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 31.1231 1.07257
$$843$$ −68.8081 −2.36988
$$844$$ 4.68466 0.161253
$$845$$ −2.27678 −0.0783236
$$846$$ 1.69614 0.0583145
$$847$$ 0 0
$$848$$ −11.5616 −0.397025
$$849$$ 24.8769 0.853773
$$850$$ −19.8188 −0.679780
$$851$$ −58.7386 −2.01353
$$852$$ 26.8122 0.918571
$$853$$ 36.0823 1.23544 0.617718 0.786400i $$-0.288057\pi$$
0.617718 + 0.786400i $$0.288057\pi$$
$$854$$ 0 0
$$855$$ −12.4924 −0.427232
$$856$$ 0.876894 0.0299716
$$857$$ −32.5628 −1.11232 −0.556162 0.831074i $$-0.687726\pi$$
−0.556162 + 0.831074i $$0.687726\pi$$
$$858$$ −4.19224 −0.143121
$$859$$ −33.8871 −1.15621 −0.578106 0.815962i $$-0.696208\pi$$
−0.578106 + 0.815962i $$0.696208\pi$$
$$860$$ 3.10196 0.105776
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ −14.0000 −0.476566 −0.238283 0.971196i $$-0.576585\pi$$
−0.238283 + 0.971196i $$0.576585\pi$$
$$864$$ 1.03399 0.0351770
$$865$$ −7.36932 −0.250564
$$866$$ −16.5896 −0.563737
$$867$$ 4.42627 0.150324
$$868$$ 0 0
$$869$$ −3.42329 −0.116127
$$870$$ 1.56155 0.0529416
$$871$$ 19.7731 0.669984
$$872$$ −10.6847 −0.361828
$$873$$ −17.9139 −0.606293
$$874$$ 60.7350 2.05439
$$875$$ 0 0
$$876$$ −20.0000 −0.675737
$$877$$ −30.6847 −1.03615 −0.518074 0.855336i $$-0.673351\pi$$
−0.518074 + 0.855336i $$0.673351\pi$$
$$878$$ 35.6647 1.20363
$$879$$ −27.1231 −0.914840
$$880$$ 0.290319 0.00978666
$$881$$ 32.4813 1.09432 0.547161 0.837028i $$-0.315709\pi$$
0.547161 + 0.837028i $$0.315709\pi$$
$$882$$ 0 0
$$883$$ −35.1231 −1.18199 −0.590993 0.806676i $$-0.701264\pi$$
−0.590993 + 0.806676i $$0.701264\pi$$
$$884$$ 17.6155 0.592474
$$885$$ −6.78456 −0.228061
$$886$$ −33.1231 −1.11279
$$887$$ −6.28544 −0.211044 −0.105522 0.994417i $$-0.533651\pi$$
−0.105522 + 0.994417i $$0.533651\pi$$
$$888$$ −16.7984 −0.563717
$$889$$ 0 0
$$890$$ −6.00000 −0.201120
$$891$$ 4.43845 0.148694
$$892$$ 3.39228 0.113582
$$893$$ −4.87689 −0.163199
$$894$$ −6.33122 −0.211748
$$895$$ 1.16128 0.0388172
$$896$$ 0 0
$$897$$ 78.8466 2.63261
$$898$$ 19.3693 0.646362
$$899$$ −4.05444 −0.135223
$$900$$ −11.6847 −0.389489
$$901$$ 50.2321 1.67347
$$902$$ −1.90495 −0.0634277
$$903$$ 0 0
$$904$$ 19.3693 0.644214
$$905$$ −14.6847 −0.488135
$$906$$ −10.0138 −0.332687
$$907$$ 19.3693 0.643148 0.321574 0.946885i $$-0.395788\pi$$
0.321574 + 0.946885i $$0.395788\pi$$
$$908$$ 12.4536 0.413288
$$909$$ 29.0432 0.963302
$$910$$ 0 0
$$911$$ 0.300187 0.00994562 0.00497281 0.999988i $$-0.498417\pi$$
0.00497281 + 0.999988i $$0.498417\pi$$
$$912$$ 17.3693 0.575156
$$913$$ 2.23100 0.0738355
$$914$$ −29.3693 −0.971451
$$915$$ −2.06798 −0.0683651
$$916$$ 23.5829 0.779202
$$917$$ 0 0
$$918$$ −4.49242 −0.148272
$$919$$ 18.0000 0.593765 0.296883 0.954914i $$-0.404053\pi$$
0.296883 + 0.954914i $$0.404053\pi$$
$$920$$ −5.46026 −0.180019
$$921$$ 26.4384 0.871176
$$922$$ 23.4199 0.771294
$$923$$ −46.0962 −1.51727
$$924$$ 0 0
$$925$$ −32.4924 −1.06834
$$926$$ 23.3693 0.767963
$$927$$ 32.8531 1.07904
$$928$$ −1.00000 −0.0328266
$$929$$ −24.1636 −0.792781 −0.396391 0.918082i $$-0.629737\pi$$
−0.396391 + 0.918082i $$0.629737\pi$$
$$930$$ −6.33122 −0.207609
$$931$$ 0 0
$$932$$ −12.9309 −0.423565
$$933$$ −23.1231 −0.757016
$$934$$ 6.33122 0.207164
$$935$$ −1.26137 −0.0412511
$$936$$ 10.3857 0.339466
$$937$$ −15.0565 −0.491873 −0.245937 0.969286i $$-0.579096\pi$$
−0.245937 + 0.969286i $$0.579096\pi$$
$$938$$ 0 0
$$939$$ 64.3002 2.09836
$$940$$ 0.438447 0.0143006
$$941$$ 10.0953 0.329098 0.164549 0.986369i $$-0.447383\pi$$
0.164549 + 0.986369i $$0.447383\pi$$
$$942$$ 40.9848 1.33536
$$943$$ 35.8278 1.16671
$$944$$ 4.34475 0.141410
$$945$$ 0 0
$$946$$ −2.05398 −0.0667805
$$947$$ −52.6847 −1.71202 −0.856011 0.516958i $$-0.827064\pi$$
−0.856011 + 0.516958i $$0.827064\pi$$
$$948$$ 18.4130 0.598027
$$949$$ 34.3845 1.11617
$$950$$ 33.5968 1.09002
$$951$$ 34.1774 1.10828
$$952$$ 0 0
$$953$$ −38.0540 −1.23269 −0.616345 0.787477i $$-0.711387\pi$$
−0.616345 + 0.787477i $$0.711387\pi$$
$$954$$ 29.6155 0.958838
$$955$$ 6.78456 0.219543
$$956$$ 0 0
$$957$$ −1.03399 −0.0334241
$$958$$ 40.2998 1.30203
$$959$$ 0 0
$$960$$ −1.56155 −0.0503989
$$961$$ −14.5616 −0.469728
$$962$$ 28.8802 0.931134
$$963$$ −2.24621 −0.0723831
$$964$$ −24.6169 −0.792858
$$965$$ 2.23100 0.0718186
$$966$$ 0 0
$$967$$ −22.9309 −0.737407 −0.368704 0.929547i $$-0.620198\pi$$
−0.368704 + 0.929547i $$0.620198\pi$$
$$968$$ 10.8078 0.347375
$$969$$ −75.4654 −2.42430
$$970$$ −4.63068 −0.148682
$$971$$ −11.5012 −0.369090 −0.184545 0.982824i $$-0.559081\pi$$
−0.184545 + 0.982824i $$0.559081\pi$$
$$972$$ −20.7713 −0.666240
$$973$$ 0 0
$$974$$ 8.00000 0.256337
$$975$$ 43.6155 1.39681
$$976$$ 1.32431 0.0423900
$$977$$ 17.6695 0.565297 0.282649 0.959223i $$-0.408787\pi$$
0.282649 + 0.959223i $$0.408787\pi$$
$$978$$ 10.4672 0.334703
$$979$$ 3.97292 0.126975
$$980$$ 0 0
$$981$$ 27.3693 0.873835
$$982$$ 25.1771 0.803433
$$983$$ 39.1385 1.24833 0.624163 0.781294i $$-0.285440\pi$$
0.624163 + 0.781294i $$0.285440\pi$$
$$984$$ 10.2462 0.326637
$$985$$ 2.48558 0.0791973
$$986$$ 4.34475 0.138365
$$987$$ 0 0
$$988$$ −29.8617 −0.950028
$$989$$ 38.6307 1.22838
$$990$$ −0.743668 −0.0236353
$$991$$ −39.6155 −1.25843 −0.629214 0.777232i $$-0.716623\pi$$
−0.629214 + 0.777232i $$0.716623\pi$$
$$992$$ 4.05444 0.128728
$$993$$ 85.6065 2.71664
$$994$$ 0 0
$$995$$ −13.7538 −0.436024
$$996$$ −12.0000 −0.380235
$$997$$ −25.4879 −0.807210 −0.403605 0.914933i $$-0.632243\pi$$
−0.403605 + 0.914933i $$0.632243\pi$$
$$998$$ −3.12311 −0.0988602
$$999$$ −7.36520 −0.233025
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 2842.2.a.p.1.4 yes 4
7.6 odd 2 inner 2842.2.a.p.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
2842.2.a.p.1.1 4 7.6 odd 2 inner
2842.2.a.p.1.4 yes 4 1.1 even 1 trivial