# Properties

 Label 3822.2.a.bu.1.1 Level $3822$ Weight $2$ Character 3822.1 Self dual yes Analytic conductor $30.519$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 3822.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.585786 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.585786 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +0.585786 q^{10} -0.414214 q^{11} +1.00000 q^{12} -1.00000 q^{13} +0.585786 q^{15} +1.00000 q^{16} +2.41421 q^{17} +1.00000 q^{18} +2.17157 q^{19} +0.585786 q^{20} -0.414214 q^{22} +1.41421 q^{23} +1.00000 q^{24} -4.65685 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.82843 q^{29} +0.585786 q^{30} +8.48528 q^{31} +1.00000 q^{32} -0.414214 q^{33} +2.41421 q^{34} +1.00000 q^{36} +1.41421 q^{37} +2.17157 q^{38} -1.00000 q^{39} +0.585786 q^{40} +9.89949 q^{41} +6.48528 q^{43} -0.414214 q^{44} +0.585786 q^{45} +1.41421 q^{46} -1.00000 q^{47} +1.00000 q^{48} -4.65685 q^{50} +2.41421 q^{51} -1.00000 q^{52} +9.48528 q^{53} +1.00000 q^{54} -0.242641 q^{55} +2.17157 q^{57} -1.82843 q^{58} -2.07107 q^{59} +0.585786 q^{60} +4.41421 q^{61} +8.48528 q^{62} +1.00000 q^{64} -0.585786 q^{65} -0.414214 q^{66} +1.82843 q^{67} +2.41421 q^{68} +1.41421 q^{69} -5.00000 q^{71} +1.00000 q^{72} -1.41421 q^{73} +1.41421 q^{74} -4.65685 q^{75} +2.17157 q^{76} -1.00000 q^{78} -11.6569 q^{79} +0.585786 q^{80} +1.00000 q^{81} +9.89949 q^{82} -7.65685 q^{83} +1.41421 q^{85} +6.48528 q^{86} -1.82843 q^{87} -0.414214 q^{88} -2.58579 q^{89} +0.585786 q^{90} +1.41421 q^{92} +8.48528 q^{93} -1.00000 q^{94} +1.27208 q^{95} +1.00000 q^{96} +0.928932 q^{97} -0.414214 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27} + 2 q^{29} + 4 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 10 q^{38} - 2 q^{39} + 4 q^{40} - 4 q^{43} + 2 q^{44} + 4 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 8 q^{55} + 10 q^{57} + 2 q^{58} + 10 q^{59} + 4 q^{60} + 6 q^{61} + 2 q^{64} - 4 q^{65} + 2 q^{66} - 2 q^{67} + 2 q^{68} - 10 q^{71} + 2 q^{72} + 2 q^{75} + 10 q^{76} - 2 q^{78} - 12 q^{79} + 4 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{86} + 2 q^{87} + 2 q^{88} - 8 q^{89} + 4 q^{90} - 2 q^{94} + 28 q^{95} + 2 q^{96} + 16 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 4 * q^5 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 4 * q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 10 * q^19 + 4 * q^20 + 2 * q^22 + 2 * q^24 + 2 * q^25 - 2 * q^26 + 2 * q^27 + 2 * q^29 + 4 * q^30 + 2 * q^32 + 2 * q^33 + 2 * q^34 + 2 * q^36 + 10 * q^38 - 2 * q^39 + 4 * q^40 - 4 * q^43 + 2 * q^44 + 4 * q^45 - 2 * q^47 + 2 * q^48 + 2 * q^50 + 2 * q^51 - 2 * q^52 + 2 * q^53 + 2 * q^54 + 8 * q^55 + 10 * q^57 + 2 * q^58 + 10 * q^59 + 4 * q^60 + 6 * q^61 + 2 * q^64 - 4 * q^65 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 10 * q^71 + 2 * q^72 + 2 * q^75 + 10 * q^76 - 2 * q^78 - 12 * q^79 + 4 * q^80 + 2 * q^81 - 4 * q^83 - 4 * q^86 + 2 * q^87 + 2 * q^88 - 8 * q^89 + 4 * q^90 - 2 * q^94 + 28 * q^95 + 2 * q^96 + 16 * q^97 + 2 * 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
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0.585786 0.261972 0.130986 0.991384i $$-0.458186\pi$$
0.130986 + 0.991384i $$0.458186\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0.585786 0.185242
$$11$$ −0.414214 −0.124890 −0.0624450 0.998048i $$-0.519890\pi$$
−0.0624450 + 0.998048i $$0.519890\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0.585786 0.151249
$$16$$ 1.00000 0.250000
$$17$$ 2.41421 0.585533 0.292766 0.956184i $$-0.405424\pi$$
0.292766 + 0.956184i $$0.405424\pi$$
$$18$$ 1.00000 0.235702
$$19$$ 2.17157 0.498193 0.249096 0.968479i $$-0.419866\pi$$
0.249096 + 0.968479i $$0.419866\pi$$
$$20$$ 0.585786 0.130986
$$21$$ 0 0
$$22$$ −0.414214 −0.0883106
$$23$$ 1.41421 0.294884 0.147442 0.989071i $$-0.452896\pi$$
0.147442 + 0.989071i $$0.452896\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −4.65685 −0.931371
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −1.82843 −0.339530 −0.169765 0.985485i $$-0.554301\pi$$
−0.169765 + 0.985485i $$0.554301\pi$$
$$30$$ 0.585786 0.106949
$$31$$ 8.48528 1.52400 0.762001 0.647576i $$-0.224217\pi$$
0.762001 + 0.647576i $$0.224217\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −0.414214 −0.0721053
$$34$$ 2.41421 0.414034
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.41421 0.232495 0.116248 0.993220i $$-0.462913\pi$$
0.116248 + 0.993220i $$0.462913\pi$$
$$38$$ 2.17157 0.352276
$$39$$ −1.00000 −0.160128
$$40$$ 0.585786 0.0926210
$$41$$ 9.89949 1.54604 0.773021 0.634381i $$-0.218745\pi$$
0.773021 + 0.634381i $$0.218745\pi$$
$$42$$ 0 0
$$43$$ 6.48528 0.988996 0.494498 0.869179i $$-0.335352\pi$$
0.494498 + 0.869179i $$0.335352\pi$$
$$44$$ −0.414214 −0.0624450
$$45$$ 0.585786 0.0873239
$$46$$ 1.41421 0.208514
$$47$$ −1.00000 −0.145865 −0.0729325 0.997337i $$-0.523236\pi$$
−0.0729325 + 0.997337i $$0.523236\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 0 0
$$50$$ −4.65685 −0.658579
$$51$$ 2.41421 0.338058
$$52$$ −1.00000 −0.138675
$$53$$ 9.48528 1.30290 0.651452 0.758690i $$-0.274160\pi$$
0.651452 + 0.758690i $$0.274160\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −0.242641 −0.0327177
$$56$$ 0 0
$$57$$ 2.17157 0.287632
$$58$$ −1.82843 −0.240084
$$59$$ −2.07107 −0.269630 −0.134815 0.990871i $$-0.543044\pi$$
−0.134815 + 0.990871i $$0.543044\pi$$
$$60$$ 0.585786 0.0756247
$$61$$ 4.41421 0.565182 0.282591 0.959240i $$-0.408806\pi$$
0.282591 + 0.959240i $$0.408806\pi$$
$$62$$ 8.48528 1.07763
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −0.585786 −0.0726579
$$66$$ −0.414214 −0.0509862
$$67$$ 1.82843 0.223378 0.111689 0.993743i $$-0.464374\pi$$
0.111689 + 0.993743i $$0.464374\pi$$
$$68$$ 2.41421 0.292766
$$69$$ 1.41421 0.170251
$$70$$ 0 0
$$71$$ −5.00000 −0.593391 −0.296695 0.954972i $$-0.595885\pi$$
−0.296695 + 0.954972i $$0.595885\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −1.41421 −0.165521 −0.0827606 0.996569i $$-0.526374\pi$$
−0.0827606 + 0.996569i $$0.526374\pi$$
$$74$$ 1.41421 0.164399
$$75$$ −4.65685 −0.537727
$$76$$ 2.17157 0.249096
$$77$$ 0 0
$$78$$ −1.00000 −0.113228
$$79$$ −11.6569 −1.31150 −0.655749 0.754979i $$-0.727647\pi$$
−0.655749 + 0.754979i $$0.727647\pi$$
$$80$$ 0.585786 0.0654929
$$81$$ 1.00000 0.111111
$$82$$ 9.89949 1.09322
$$83$$ −7.65685 −0.840449 −0.420224 0.907420i $$-0.638049\pi$$
−0.420224 + 0.907420i $$0.638049\pi$$
$$84$$ 0 0
$$85$$ 1.41421 0.153393
$$86$$ 6.48528 0.699326
$$87$$ −1.82843 −0.196028
$$88$$ −0.414214 −0.0441553
$$89$$ −2.58579 −0.274093 −0.137046 0.990565i $$-0.543761\pi$$
−0.137046 + 0.990565i $$0.543761\pi$$
$$90$$ 0.585786 0.0617473
$$91$$ 0 0
$$92$$ 1.41421 0.147442
$$93$$ 8.48528 0.879883
$$94$$ −1.00000 −0.103142
$$95$$ 1.27208 0.130512
$$96$$ 1.00000 0.102062
$$97$$ 0.928932 0.0943188 0.0471594 0.998887i $$-0.484983\pi$$
0.0471594 + 0.998887i $$0.484983\pi$$
$$98$$ 0 0
$$99$$ −0.414214 −0.0416300
$$100$$ −4.65685 −0.465685
$$101$$ 14.8284 1.47548 0.737742 0.675083i $$-0.235892\pi$$
0.737742 + 0.675083i $$0.235892\pi$$
$$102$$ 2.41421 0.239043
$$103$$ 16.8284 1.65815 0.829077 0.559134i $$-0.188866\pi$$
0.829077 + 0.559134i $$0.188866\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 9.48528 0.921292
$$107$$ −19.3137 −1.86713 −0.933563 0.358412i $$-0.883318\pi$$
−0.933563 + 0.358412i $$0.883318\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −9.65685 −0.924959 −0.462479 0.886630i $$-0.653040\pi$$
−0.462479 + 0.886630i $$0.653040\pi$$
$$110$$ −0.242641 −0.0231349
$$111$$ 1.41421 0.134231
$$112$$ 0 0
$$113$$ 4.89949 0.460906 0.230453 0.973083i $$-0.425979\pi$$
0.230453 + 0.973083i $$0.425979\pi$$
$$114$$ 2.17157 0.203386
$$115$$ 0.828427 0.0772512
$$116$$ −1.82843 −0.169765
$$117$$ −1.00000 −0.0924500
$$118$$ −2.07107 −0.190657
$$119$$ 0 0
$$120$$ 0.585786 0.0534747
$$121$$ −10.8284 −0.984402
$$122$$ 4.41421 0.399644
$$123$$ 9.89949 0.892607
$$124$$ 8.48528 0.762001
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ −7.89949 −0.700967 −0.350483 0.936569i $$-0.613983\pi$$
−0.350483 + 0.936569i $$0.613983\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 6.48528 0.570997
$$130$$ −0.585786 −0.0513769
$$131$$ 9.55635 0.834942 0.417471 0.908690i $$-0.362917\pi$$
0.417471 + 0.908690i $$0.362917\pi$$
$$132$$ −0.414214 −0.0360527
$$133$$ 0 0
$$134$$ 1.82843 0.157952
$$135$$ 0.585786 0.0504165
$$136$$ 2.41421 0.207017
$$137$$ 11.0711 0.945865 0.472933 0.881099i $$-0.343195\pi$$
0.472933 + 0.881099i $$0.343195\pi$$
$$138$$ 1.41421 0.120386
$$139$$ 13.0711 1.10867 0.554337 0.832292i $$-0.312972\pi$$
0.554337 + 0.832292i $$0.312972\pi$$
$$140$$ 0 0
$$141$$ −1.00000 −0.0842152
$$142$$ −5.00000 −0.419591
$$143$$ 0.414214 0.0346383
$$144$$ 1.00000 0.0833333
$$145$$ −1.07107 −0.0889473
$$146$$ −1.41421 −0.117041
$$147$$ 0 0
$$148$$ 1.41421 0.116248
$$149$$ −4.92893 −0.403794 −0.201897 0.979407i $$-0.564711\pi$$
−0.201897 + 0.979407i $$0.564711\pi$$
$$150$$ −4.65685 −0.380231
$$151$$ 4.07107 0.331299 0.165649 0.986185i $$-0.447028\pi$$
0.165649 + 0.986185i $$0.447028\pi$$
$$152$$ 2.17157 0.176138
$$153$$ 2.41421 0.195178
$$154$$ 0 0
$$155$$ 4.97056 0.399245
$$156$$ −1.00000 −0.0800641
$$157$$ −9.72792 −0.776373 −0.388186 0.921581i $$-0.626898\pi$$
−0.388186 + 0.921581i $$0.626898\pi$$
$$158$$ −11.6569 −0.927370
$$159$$ 9.48528 0.752232
$$160$$ 0.585786 0.0463105
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −18.6569 −1.46132 −0.730659 0.682743i $$-0.760787\pi$$
−0.730659 + 0.682743i $$0.760787\pi$$
$$164$$ 9.89949 0.773021
$$165$$ −0.242641 −0.0188896
$$166$$ −7.65685 −0.594287
$$167$$ 0.656854 0.0508289 0.0254145 0.999677i $$-0.491909\pi$$
0.0254145 + 0.999677i $$0.491909\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 1.41421 0.108465
$$171$$ 2.17157 0.166064
$$172$$ 6.48528 0.494498
$$173$$ −5.48528 −0.417038 −0.208519 0.978018i $$-0.566864\pi$$
−0.208519 + 0.978018i $$0.566864\pi$$
$$174$$ −1.82843 −0.138613
$$175$$ 0 0
$$176$$ −0.414214 −0.0312225
$$177$$ −2.07107 −0.155671
$$178$$ −2.58579 −0.193813
$$179$$ 19.6569 1.46922 0.734611 0.678488i $$-0.237365\pi$$
0.734611 + 0.678488i $$0.237365\pi$$
$$180$$ 0.585786 0.0436619
$$181$$ −2.89949 −0.215518 −0.107759 0.994177i $$-0.534367\pi$$
−0.107759 + 0.994177i $$0.534367\pi$$
$$182$$ 0 0
$$183$$ 4.41421 0.326308
$$184$$ 1.41421 0.104257
$$185$$ 0.828427 0.0609072
$$186$$ 8.48528 0.622171
$$187$$ −1.00000 −0.0731272
$$188$$ −1.00000 −0.0729325
$$189$$ 0 0
$$190$$ 1.27208 0.0922862
$$191$$ −10.3431 −0.748404 −0.374202 0.927347i $$-0.622083\pi$$
−0.374202 + 0.927347i $$0.622083\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −2.92893 −0.210829 −0.105415 0.994428i $$-0.533617\pi$$
−0.105415 + 0.994428i $$0.533617\pi$$
$$194$$ 0.928932 0.0666934
$$195$$ −0.585786 −0.0419490
$$196$$ 0 0
$$197$$ −15.5563 −1.10834 −0.554172 0.832402i $$-0.686965\pi$$
−0.554172 + 0.832402i $$0.686965\pi$$
$$198$$ −0.414214 −0.0294369
$$199$$ −2.72792 −0.193377 −0.0966886 0.995315i $$-0.530825\pi$$
−0.0966886 + 0.995315i $$0.530825\pi$$
$$200$$ −4.65685 −0.329289
$$201$$ 1.82843 0.128967
$$202$$ 14.8284 1.04332
$$203$$ 0 0
$$204$$ 2.41421 0.169029
$$205$$ 5.79899 0.405019
$$206$$ 16.8284 1.17249
$$207$$ 1.41421 0.0982946
$$208$$ −1.00000 −0.0693375
$$209$$ −0.899495 −0.0622194
$$210$$ 0 0
$$211$$ −12.9289 −0.890064 −0.445032 0.895515i $$-0.646808\pi$$
−0.445032 + 0.895515i $$0.646808\pi$$
$$212$$ 9.48528 0.651452
$$213$$ −5.00000 −0.342594
$$214$$ −19.3137 −1.32026
$$215$$ 3.79899 0.259089
$$216$$ 1.00000 0.0680414
$$217$$ 0 0
$$218$$ −9.65685 −0.654045
$$219$$ −1.41421 −0.0955637
$$220$$ −0.242641 −0.0163588
$$221$$ −2.41421 −0.162398
$$222$$ 1.41421 0.0949158
$$223$$ 7.92893 0.530961 0.265480 0.964116i $$-0.414469\pi$$
0.265480 + 0.964116i $$0.414469\pi$$
$$224$$ 0 0
$$225$$ −4.65685 −0.310457
$$226$$ 4.89949 0.325910
$$227$$ 21.1716 1.40521 0.702603 0.711582i $$-0.252021\pi$$
0.702603 + 0.711582i $$0.252021\pi$$
$$228$$ 2.17157 0.143816
$$229$$ −26.4853 −1.75020 −0.875098 0.483945i $$-0.839203\pi$$
−0.875098 + 0.483945i $$0.839203\pi$$
$$230$$ 0.828427 0.0546249
$$231$$ 0 0
$$232$$ −1.82843 −0.120042
$$233$$ −17.7279 −1.16139 −0.580697 0.814119i $$-0.697220\pi$$
−0.580697 + 0.814119i $$0.697220\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ −0.585786 −0.0382125
$$236$$ −2.07107 −0.134815
$$237$$ −11.6569 −0.757194
$$238$$ 0 0
$$239$$ 4.51472 0.292033 0.146016 0.989282i $$-0.453355\pi$$
0.146016 + 0.989282i $$0.453355\pi$$
$$240$$ 0.585786 0.0378124
$$241$$ 2.14214 0.137987 0.0689935 0.997617i $$-0.478021\pi$$
0.0689935 + 0.997617i $$0.478021\pi$$
$$242$$ −10.8284 −0.696078
$$243$$ 1.00000 0.0641500
$$244$$ 4.41421 0.282591
$$245$$ 0 0
$$246$$ 9.89949 0.631169
$$247$$ −2.17157 −0.138174
$$248$$ 8.48528 0.538816
$$249$$ −7.65685 −0.485233
$$250$$ −5.65685 −0.357771
$$251$$ −24.8284 −1.56716 −0.783578 0.621293i $$-0.786608\pi$$
−0.783578 + 0.621293i $$0.786608\pi$$
$$252$$ 0 0
$$253$$ −0.585786 −0.0368281
$$254$$ −7.89949 −0.495658
$$255$$ 1.41421 0.0885615
$$256$$ 1.00000 0.0625000
$$257$$ −26.4853 −1.65211 −0.826053 0.563592i $$-0.809419\pi$$
−0.826053 + 0.563592i $$0.809419\pi$$
$$258$$ 6.48528 0.403756
$$259$$ 0 0
$$260$$ −0.585786 −0.0363289
$$261$$ −1.82843 −0.113177
$$262$$ 9.55635 0.590393
$$263$$ 20.5858 1.26937 0.634687 0.772769i $$-0.281129\pi$$
0.634687 + 0.772769i $$0.281129\pi$$
$$264$$ −0.414214 −0.0254931
$$265$$ 5.55635 0.341324
$$266$$ 0 0
$$267$$ −2.58579 −0.158248
$$268$$ 1.82843 0.111689
$$269$$ 9.14214 0.557406 0.278703 0.960377i $$-0.410096\pi$$
0.278703 + 0.960377i $$0.410096\pi$$
$$270$$ 0.585786 0.0356498
$$271$$ −26.2132 −1.59234 −0.796169 0.605074i $$-0.793144\pi$$
−0.796169 + 0.605074i $$0.793144\pi$$
$$272$$ 2.41421 0.146383
$$273$$ 0 0
$$274$$ 11.0711 0.668828
$$275$$ 1.92893 0.116319
$$276$$ 1.41421 0.0851257
$$277$$ 18.0711 1.08579 0.542893 0.839802i $$-0.317329\pi$$
0.542893 + 0.839802i $$0.317329\pi$$
$$278$$ 13.0711 0.783951
$$279$$ 8.48528 0.508001
$$280$$ 0 0
$$281$$ 0.686292 0.0409407 0.0204704 0.999790i $$-0.493484\pi$$
0.0204704 + 0.999790i $$0.493484\pi$$
$$282$$ −1.00000 −0.0595491
$$283$$ 15.7574 0.936678 0.468339 0.883549i $$-0.344853\pi$$
0.468339 + 0.883549i $$0.344853\pi$$
$$284$$ −5.00000 −0.296695
$$285$$ 1.27208 0.0753514
$$286$$ 0.414214 0.0244930
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ −11.1716 −0.657151
$$290$$ −1.07107 −0.0628953
$$291$$ 0.928932 0.0544550
$$292$$ −1.41421 −0.0827606
$$293$$ 9.89949 0.578335 0.289167 0.957279i $$-0.406622\pi$$
0.289167 + 0.957279i $$0.406622\pi$$
$$294$$ 0 0
$$295$$ −1.21320 −0.0706354
$$296$$ 1.41421 0.0821995
$$297$$ −0.414214 −0.0240351
$$298$$ −4.92893 −0.285525
$$299$$ −1.41421 −0.0817861
$$300$$ −4.65685 −0.268864
$$301$$ 0 0
$$302$$ 4.07107 0.234264
$$303$$ 14.8284 0.851871
$$304$$ 2.17157 0.124548
$$305$$ 2.58579 0.148062
$$306$$ 2.41421 0.138011
$$307$$ 28.1127 1.60448 0.802238 0.597004i $$-0.203642\pi$$
0.802238 + 0.597004i $$0.203642\pi$$
$$308$$ 0 0
$$309$$ 16.8284 0.957336
$$310$$ 4.97056 0.282309
$$311$$ 8.10051 0.459338 0.229669 0.973269i $$-0.426236\pi$$
0.229669 + 0.973269i $$0.426236\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −3.51472 −0.198664 −0.0993318 0.995054i $$-0.531671\pi$$
−0.0993318 + 0.995054i $$0.531671\pi$$
$$314$$ −9.72792 −0.548978
$$315$$ 0 0
$$316$$ −11.6569 −0.655749
$$317$$ 29.7990 1.67368 0.836839 0.547449i $$-0.184401\pi$$
0.836839 + 0.547449i $$0.184401\pi$$
$$318$$ 9.48528 0.531908
$$319$$ 0.757359 0.0424040
$$320$$ 0.585786 0.0327465
$$321$$ −19.3137 −1.07799
$$322$$ 0 0
$$323$$ 5.24264 0.291708
$$324$$ 1.00000 0.0555556
$$325$$ 4.65685 0.258316
$$326$$ −18.6569 −1.03331
$$327$$ −9.65685 −0.534025
$$328$$ 9.89949 0.546608
$$329$$ 0 0
$$330$$ −0.242641 −0.0133569
$$331$$ −22.4853 −1.23590 −0.617951 0.786216i $$-0.712037\pi$$
−0.617951 + 0.786216i $$0.712037\pi$$
$$332$$ −7.65685 −0.420224
$$333$$ 1.41421 0.0774984
$$334$$ 0.656854 0.0359415
$$335$$ 1.07107 0.0585187
$$336$$ 0 0
$$337$$ −20.3137 −1.10656 −0.553279 0.832996i $$-0.686624\pi$$
−0.553279 + 0.832996i $$0.686624\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 4.89949 0.266104
$$340$$ 1.41421 0.0766965
$$341$$ −3.51472 −0.190333
$$342$$ 2.17157 0.117425
$$343$$ 0 0
$$344$$ 6.48528 0.349663
$$345$$ 0.828427 0.0446010
$$346$$ −5.48528 −0.294891
$$347$$ −8.72792 −0.468539 −0.234270 0.972172i $$-0.575270\pi$$
−0.234270 + 0.972172i $$0.575270\pi$$
$$348$$ −1.82843 −0.0980140
$$349$$ 6.72792 0.360137 0.180069 0.983654i $$-0.442368\pi$$
0.180069 + 0.983654i $$0.442368\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ −0.414214 −0.0220777
$$353$$ −12.6274 −0.672090 −0.336045 0.941846i $$-0.609089\pi$$
−0.336045 + 0.941846i $$0.609089\pi$$
$$354$$ −2.07107 −0.110076
$$355$$ −2.92893 −0.155452
$$356$$ −2.58579 −0.137046
$$357$$ 0 0
$$358$$ 19.6569 1.03890
$$359$$ −0.828427 −0.0437227 −0.0218614 0.999761i $$-0.506959\pi$$
−0.0218614 + 0.999761i $$0.506959\pi$$
$$360$$ 0.585786 0.0308737
$$361$$ −14.2843 −0.751804
$$362$$ −2.89949 −0.152394
$$363$$ −10.8284 −0.568345
$$364$$ 0 0
$$365$$ −0.828427 −0.0433619
$$366$$ 4.41421 0.230735
$$367$$ −12.6274 −0.659146 −0.329573 0.944130i $$-0.606905\pi$$
−0.329573 + 0.944130i $$0.606905\pi$$
$$368$$ 1.41421 0.0737210
$$369$$ 9.89949 0.515347
$$370$$ 0.828427 0.0430679
$$371$$ 0 0
$$372$$ 8.48528 0.439941
$$373$$ −5.58579 −0.289221 −0.144611 0.989489i $$-0.546193\pi$$
−0.144611 + 0.989489i $$0.546193\pi$$
$$374$$ −1.00000 −0.0517088
$$375$$ −5.65685 −0.292119
$$376$$ −1.00000 −0.0515711
$$377$$ 1.82843 0.0941688
$$378$$ 0 0
$$379$$ 18.6274 0.956826 0.478413 0.878135i $$-0.341212\pi$$
0.478413 + 0.878135i $$0.341212\pi$$
$$380$$ 1.27208 0.0652562
$$381$$ −7.89949 −0.404703
$$382$$ −10.3431 −0.529201
$$383$$ −11.1716 −0.570841 −0.285420 0.958402i $$-0.592133\pi$$
−0.285420 + 0.958402i $$0.592133\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −2.92893 −0.149079
$$387$$ 6.48528 0.329665
$$388$$ 0.928932 0.0471594
$$389$$ −4.51472 −0.228905 −0.114453 0.993429i $$-0.536511\pi$$
−0.114453 + 0.993429i $$0.536511\pi$$
$$390$$ −0.585786 −0.0296624
$$391$$ 3.41421 0.172664
$$392$$ 0 0
$$393$$ 9.55635 0.482054
$$394$$ −15.5563 −0.783718
$$395$$ −6.82843 −0.343575
$$396$$ −0.414214 −0.0208150
$$397$$ 9.55635 0.479619 0.239810 0.970820i $$-0.422915\pi$$
0.239810 + 0.970820i $$0.422915\pi$$
$$398$$ −2.72792 −0.136738
$$399$$ 0 0
$$400$$ −4.65685 −0.232843
$$401$$ 19.8995 0.993733 0.496867 0.867827i $$-0.334484\pi$$
0.496867 + 0.867827i $$0.334484\pi$$
$$402$$ 1.82843 0.0911937
$$403$$ −8.48528 −0.422682
$$404$$ 14.8284 0.737742
$$405$$ 0.585786 0.0291080
$$406$$ 0 0
$$407$$ −0.585786 −0.0290364
$$408$$ 2.41421 0.119521
$$409$$ −32.5858 −1.61126 −0.805632 0.592417i $$-0.798174\pi$$
−0.805632 + 0.592417i $$0.798174\pi$$
$$410$$ 5.79899 0.286392
$$411$$ 11.0711 0.546096
$$412$$ 16.8284 0.829077
$$413$$ 0 0
$$414$$ 1.41421 0.0695048
$$415$$ −4.48528 −0.220174
$$416$$ −1.00000 −0.0490290
$$417$$ 13.0711 0.640093
$$418$$ −0.899495 −0.0439957
$$419$$ −27.4558 −1.34131 −0.670653 0.741771i $$-0.733986\pi$$
−0.670653 + 0.741771i $$0.733986\pi$$
$$420$$ 0 0
$$421$$ 1.07107 0.0522007 0.0261003 0.999659i $$-0.491691\pi$$
0.0261003 + 0.999659i $$0.491691\pi$$
$$422$$ −12.9289 −0.629371
$$423$$ −1.00000 −0.0486217
$$424$$ 9.48528 0.460646
$$425$$ −11.2426 −0.545348
$$426$$ −5.00000 −0.242251
$$427$$ 0 0
$$428$$ −19.3137 −0.933563
$$429$$ 0.414214 0.0199984
$$430$$ 3.79899 0.183204
$$431$$ 30.4853 1.46842 0.734212 0.678920i $$-0.237552\pi$$
0.734212 + 0.678920i $$0.237552\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −29.9706 −1.44029 −0.720147 0.693822i $$-0.755926\pi$$
−0.720147 + 0.693822i $$0.755926\pi$$
$$434$$ 0 0
$$435$$ −1.07107 −0.0513538
$$436$$ −9.65685 −0.462479
$$437$$ 3.07107 0.146909
$$438$$ −1.41421 −0.0675737
$$439$$ −16.5858 −0.791596 −0.395798 0.918338i $$-0.629532\pi$$
−0.395798 + 0.918338i $$0.629532\pi$$
$$440$$ −0.242641 −0.0115674
$$441$$ 0 0
$$442$$ −2.41421 −0.114832
$$443$$ −33.6985 −1.60106 −0.800532 0.599290i $$-0.795449\pi$$
−0.800532 + 0.599290i $$0.795449\pi$$
$$444$$ 1.41421 0.0671156
$$445$$ −1.51472 −0.0718045
$$446$$ 7.92893 0.375446
$$447$$ −4.92893 −0.233130
$$448$$ 0 0
$$449$$ 24.9706 1.17843 0.589217 0.807975i $$-0.299436\pi$$
0.589217 + 0.807975i $$0.299436\pi$$
$$450$$ −4.65685 −0.219526
$$451$$ −4.10051 −0.193085
$$452$$ 4.89949 0.230453
$$453$$ 4.07107 0.191275
$$454$$ 21.1716 0.993631
$$455$$ 0 0
$$456$$ 2.17157 0.101693
$$457$$ −3.07107 −0.143658 −0.0718292 0.997417i $$-0.522884\pi$$
−0.0718292 + 0.997417i $$0.522884\pi$$
$$458$$ −26.4853 −1.23758
$$459$$ 2.41421 0.112686
$$460$$ 0.828427 0.0386256
$$461$$ 4.48528 0.208900 0.104450 0.994530i $$-0.466692\pi$$
0.104450 + 0.994530i $$0.466692\pi$$
$$462$$ 0 0
$$463$$ −20.9706 −0.974585 −0.487292 0.873239i $$-0.662015\pi$$
−0.487292 + 0.873239i $$0.662015\pi$$
$$464$$ −1.82843 −0.0848826
$$465$$ 4.97056 0.230504
$$466$$ −17.7279 −0.821230
$$467$$ 11.4142 0.528187 0.264093 0.964497i $$-0.414927\pi$$
0.264093 + 0.964497i $$0.414927\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 0 0
$$470$$ −0.585786 −0.0270203
$$471$$ −9.72792 −0.448239
$$472$$ −2.07107 −0.0953286
$$473$$ −2.68629 −0.123516
$$474$$ −11.6569 −0.535417
$$475$$ −10.1127 −0.464002
$$476$$ 0 0
$$477$$ 9.48528 0.434301
$$478$$ 4.51472 0.206498
$$479$$ 32.1127 1.46727 0.733633 0.679546i $$-0.237823\pi$$
0.733633 + 0.679546i $$0.237823\pi$$
$$480$$ 0.585786 0.0267374
$$481$$ −1.41421 −0.0644826
$$482$$ 2.14214 0.0975716
$$483$$ 0 0
$$484$$ −10.8284 −0.492201
$$485$$ 0.544156 0.0247088
$$486$$ 1.00000 0.0453609
$$487$$ 16.2132 0.734690 0.367345 0.930085i $$-0.380267\pi$$
0.367345 + 0.930085i $$0.380267\pi$$
$$488$$ 4.41421 0.199822
$$489$$ −18.6569 −0.843692
$$490$$ 0 0
$$491$$ 23.6569 1.06762 0.533809 0.845605i $$-0.320760\pi$$
0.533809 + 0.845605i $$0.320760\pi$$
$$492$$ 9.89949 0.446304
$$493$$ −4.41421 −0.198806
$$494$$ −2.17157 −0.0977037
$$495$$ −0.242641 −0.0109059
$$496$$ 8.48528 0.381000
$$497$$ 0 0
$$498$$ −7.65685 −0.343112
$$499$$ 3.65685 0.163703 0.0818516 0.996645i $$-0.473917\pi$$
0.0818516 + 0.996645i $$0.473917\pi$$
$$500$$ −5.65685 −0.252982
$$501$$ 0.656854 0.0293461
$$502$$ −24.8284 −1.10815
$$503$$ −26.1421 −1.16562 −0.582810 0.812608i $$-0.698047\pi$$
−0.582810 + 0.812608i $$0.698047\pi$$
$$504$$ 0 0
$$505$$ 8.68629 0.386535
$$506$$ −0.585786 −0.0260414
$$507$$ 1.00000 0.0444116
$$508$$ −7.89949 −0.350483
$$509$$ −10.6274 −0.471052 −0.235526 0.971868i $$-0.575681\pi$$
−0.235526 + 0.971868i $$0.575681\pi$$
$$510$$ 1.41421 0.0626224
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 2.17157 0.0958773
$$514$$ −26.4853 −1.16822
$$515$$ 9.85786 0.434389
$$516$$ 6.48528 0.285499
$$517$$ 0.414214 0.0182171
$$518$$ 0 0
$$519$$ −5.48528 −0.240777
$$520$$ −0.585786 −0.0256884
$$521$$ −40.6274 −1.77992 −0.889960 0.456039i $$-0.849268\pi$$
−0.889960 + 0.456039i $$0.849268\pi$$
$$522$$ −1.82843 −0.0800281
$$523$$ 16.8284 0.735856 0.367928 0.929854i $$-0.380067\pi$$
0.367928 + 0.929854i $$0.380067\pi$$
$$524$$ 9.55635 0.417471
$$525$$ 0 0
$$526$$ 20.5858 0.897583
$$527$$ 20.4853 0.892353
$$528$$ −0.414214 −0.0180263
$$529$$ −21.0000 −0.913043
$$530$$ 5.55635 0.241352
$$531$$ −2.07107 −0.0898767
$$532$$ 0 0
$$533$$ −9.89949 −0.428795
$$534$$ −2.58579 −0.111898
$$535$$ −11.3137 −0.489134
$$536$$ 1.82843 0.0789760
$$537$$ 19.6569 0.848256
$$538$$ 9.14214 0.394145
$$539$$ 0 0
$$540$$ 0.585786 0.0252082
$$541$$ 23.0711 0.991903 0.495951 0.868350i $$-0.334819\pi$$
0.495951 + 0.868350i $$0.334819\pi$$
$$542$$ −26.2132 −1.12595
$$543$$ −2.89949 −0.124429
$$544$$ 2.41421 0.103509
$$545$$ −5.65685 −0.242313
$$546$$ 0 0
$$547$$ −30.8701 −1.31991 −0.659954 0.751306i $$-0.729424\pi$$
−0.659954 + 0.751306i $$0.729424\pi$$
$$548$$ 11.0711 0.472933
$$549$$ 4.41421 0.188394
$$550$$ 1.92893 0.0822499
$$551$$ −3.97056 −0.169152
$$552$$ 1.41421 0.0601929
$$553$$ 0 0
$$554$$ 18.0711 0.767766
$$555$$ 0.828427 0.0351648
$$556$$ 13.0711 0.554337
$$557$$ −32.6274 −1.38247 −0.691234 0.722631i $$-0.742933\pi$$
−0.691234 + 0.722631i $$0.742933\pi$$
$$558$$ 8.48528 0.359211
$$559$$ −6.48528 −0.274298
$$560$$ 0 0
$$561$$ −1.00000 −0.0422200
$$562$$ 0.686292 0.0289495
$$563$$ 32.7696 1.38107 0.690536 0.723298i $$-0.257375\pi$$
0.690536 + 0.723298i $$0.257375\pi$$
$$564$$ −1.00000 −0.0421076
$$565$$ 2.87006 0.120744
$$566$$ 15.7574 0.662331
$$567$$ 0 0
$$568$$ −5.00000 −0.209795
$$569$$ 23.7279 0.994726 0.497363 0.867542i $$-0.334302\pi$$
0.497363 + 0.867542i $$0.334302\pi$$
$$570$$ 1.27208 0.0532815
$$571$$ −31.3137 −1.31044 −0.655219 0.755439i $$-0.727424\pi$$
−0.655219 + 0.755439i $$0.727424\pi$$
$$572$$ 0.414214 0.0173191
$$573$$ −10.3431 −0.432091
$$574$$ 0 0
$$575$$ −6.58579 −0.274646
$$576$$ 1.00000 0.0416667
$$577$$ −16.3431 −0.680374 −0.340187 0.940358i $$-0.610490\pi$$
−0.340187 + 0.940358i $$0.610490\pi$$
$$578$$ −11.1716 −0.464676
$$579$$ −2.92893 −0.121722
$$580$$ −1.07107 −0.0444737
$$581$$ 0 0
$$582$$ 0.928932 0.0385055
$$583$$ −3.92893 −0.162720
$$584$$ −1.41421 −0.0585206
$$585$$ −0.585786 −0.0242193
$$586$$ 9.89949 0.408944
$$587$$ 1.58579 0.0654524 0.0327262 0.999464i $$-0.489581\pi$$
0.0327262 + 0.999464i $$0.489581\pi$$
$$588$$ 0 0
$$589$$ 18.4264 0.759247
$$590$$ −1.21320 −0.0499468
$$591$$ −15.5563 −0.639903
$$592$$ 1.41421 0.0581238
$$593$$ 34.3848 1.41201 0.706007 0.708205i $$-0.250495\pi$$
0.706007 + 0.708205i $$0.250495\pi$$
$$594$$ −0.414214 −0.0169954
$$595$$ 0 0
$$596$$ −4.92893 −0.201897
$$597$$ −2.72792 −0.111646
$$598$$ −1.41421 −0.0578315
$$599$$ 1.55635 0.0635907 0.0317954 0.999494i $$-0.489878\pi$$
0.0317954 + 0.999494i $$0.489878\pi$$
$$600$$ −4.65685 −0.190115
$$601$$ −1.20101 −0.0489902 −0.0244951 0.999700i $$-0.507798\pi$$
−0.0244951 + 0.999700i $$0.507798\pi$$
$$602$$ 0 0
$$603$$ 1.82843 0.0744593
$$604$$ 4.07107 0.165649
$$605$$ −6.34315 −0.257886
$$606$$ 14.8284 0.602364
$$607$$ −34.5269 −1.40140 −0.700702 0.713454i $$-0.747130\pi$$
−0.700702 + 0.713454i $$0.747130\pi$$
$$608$$ 2.17157 0.0880689
$$609$$ 0 0
$$610$$ 2.58579 0.104695
$$611$$ 1.00000 0.0404557
$$612$$ 2.41421 0.0975888
$$613$$ −7.79899 −0.314998 −0.157499 0.987519i $$-0.550343\pi$$
−0.157499 + 0.987519i $$0.550343\pi$$
$$614$$ 28.1127 1.13454
$$615$$ 5.79899 0.233838
$$616$$ 0 0
$$617$$ −24.1421 −0.971926 −0.485963 0.873979i $$-0.661531\pi$$
−0.485963 + 0.873979i $$0.661531\pi$$
$$618$$ 16.8284 0.676939
$$619$$ −20.8284 −0.837165 −0.418583 0.908179i $$-0.637473\pi$$
−0.418583 + 0.908179i $$0.637473\pi$$
$$620$$ 4.97056 0.199623
$$621$$ 1.41421 0.0567504
$$622$$ 8.10051 0.324801
$$623$$ 0 0
$$624$$ −1.00000 −0.0400320
$$625$$ 19.9706 0.798823
$$626$$ −3.51472 −0.140476
$$627$$ −0.899495 −0.0359224
$$628$$ −9.72792 −0.388186
$$629$$ 3.41421 0.136134
$$630$$ 0 0
$$631$$ 32.6274 1.29888 0.649438 0.760414i $$-0.275004\pi$$
0.649438 + 0.760414i $$0.275004\pi$$
$$632$$ −11.6569 −0.463685
$$633$$ −12.9289 −0.513879
$$634$$ 29.7990 1.18347
$$635$$ −4.62742 −0.183633
$$636$$ 9.48528 0.376116
$$637$$ 0 0
$$638$$ 0.757359 0.0299841
$$639$$ −5.00000 −0.197797
$$640$$ 0.585786 0.0231552
$$641$$ 31.7990 1.25598 0.627992 0.778220i $$-0.283877\pi$$
0.627992 + 0.778220i $$0.283877\pi$$
$$642$$ −19.3137 −0.762251
$$643$$ 2.51472 0.0991708 0.0495854 0.998770i $$-0.484210\pi$$
0.0495854 + 0.998770i $$0.484210\pi$$
$$644$$ 0 0
$$645$$ 3.79899 0.149585
$$646$$ 5.24264 0.206269
$$647$$ 9.02944 0.354984 0.177492 0.984122i $$-0.443202\pi$$
0.177492 + 0.984122i $$0.443202\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0.857864 0.0336741
$$650$$ 4.65685 0.182657
$$651$$ 0 0
$$652$$ −18.6569 −0.730659
$$653$$ 22.1421 0.866489 0.433244 0.901276i $$-0.357369\pi$$
0.433244 + 0.901276i $$0.357369\pi$$
$$654$$ −9.65685 −0.377613
$$655$$ 5.59798 0.218731
$$656$$ 9.89949 0.386510
$$657$$ −1.41421 −0.0551737
$$658$$ 0 0
$$659$$ 10.5858 0.412364 0.206182 0.978514i $$-0.433896\pi$$
0.206182 + 0.978514i $$0.433896\pi$$
$$660$$ −0.242641 −0.00944478
$$661$$ −40.2426 −1.56526 −0.782629 0.622489i $$-0.786122\pi$$
−0.782629 + 0.622489i $$0.786122\pi$$
$$662$$ −22.4853 −0.873915
$$663$$ −2.41421 −0.0937603
$$664$$ −7.65685 −0.297144
$$665$$ 0 0
$$666$$ 1.41421 0.0547997
$$667$$ −2.58579 −0.100122
$$668$$ 0.656854 0.0254145
$$669$$ 7.92893 0.306550
$$670$$ 1.07107 0.0413790
$$671$$ −1.82843 −0.0705856
$$672$$ 0 0
$$673$$ 42.1421 1.62446 0.812230 0.583337i $$-0.198253\pi$$
0.812230 + 0.583337i $$0.198253\pi$$
$$674$$ −20.3137 −0.782455
$$675$$ −4.65685 −0.179242
$$676$$ 1.00000 0.0384615
$$677$$ −15.0000 −0.576497 −0.288248 0.957556i $$-0.593073\pi$$
−0.288248 + 0.957556i $$0.593073\pi$$
$$678$$ 4.89949 0.188164
$$679$$ 0 0
$$680$$ 1.41421 0.0542326
$$681$$ 21.1716 0.811296
$$682$$ −3.51472 −0.134586
$$683$$ −14.1421 −0.541134 −0.270567 0.962701i $$-0.587211\pi$$
−0.270567 + 0.962701i $$0.587211\pi$$
$$684$$ 2.17157 0.0830322
$$685$$ 6.48528 0.247790
$$686$$ 0 0
$$687$$ −26.4853 −1.01048
$$688$$ 6.48528 0.247249
$$689$$ −9.48528 −0.361360
$$690$$ 0.828427 0.0315377
$$691$$ 20.6569 0.785824 0.392912 0.919576i $$-0.371468\pi$$
0.392912 + 0.919576i $$0.371468\pi$$
$$692$$ −5.48528 −0.208519
$$693$$ 0 0
$$694$$ −8.72792 −0.331307
$$695$$ 7.65685 0.290441
$$696$$ −1.82843 −0.0693064
$$697$$ 23.8995 0.905258
$$698$$ 6.72792 0.254656
$$699$$ −17.7279 −0.670532
$$700$$ 0 0
$$701$$ −29.1716 −1.10180 −0.550898 0.834573i $$-0.685715\pi$$
−0.550898 + 0.834573i $$0.685715\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 3.07107 0.115828
$$704$$ −0.414214 −0.0156113
$$705$$ −0.585786 −0.0220620
$$706$$ −12.6274 −0.475239
$$707$$ 0 0
$$708$$ −2.07107 −0.0778355
$$709$$ −36.8701 −1.38468 −0.692342 0.721569i $$-0.743421\pi$$
−0.692342 + 0.721569i $$0.743421\pi$$
$$710$$ −2.92893 −0.109921
$$711$$ −11.6569 −0.437166
$$712$$ −2.58579 −0.0969064
$$713$$ 12.0000 0.449404
$$714$$ 0 0
$$715$$ 0.242641 0.00907425
$$716$$ 19.6569 0.734611
$$717$$ 4.51472 0.168605
$$718$$ −0.828427 −0.0309166
$$719$$ −9.31371 −0.347343 −0.173671 0.984804i $$-0.555563\pi$$
−0.173671 + 0.984804i $$0.555563\pi$$
$$720$$ 0.585786 0.0218310
$$721$$ 0 0
$$722$$ −14.2843 −0.531606
$$723$$ 2.14214 0.0796669
$$724$$ −2.89949 −0.107759
$$725$$ 8.51472 0.316229
$$726$$ −10.8284 −0.401881
$$727$$ 50.4264 1.87021 0.935106 0.354368i $$-0.115304\pi$$
0.935106 + 0.354368i $$0.115304\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −0.828427 −0.0306615
$$731$$ 15.6569 0.579090
$$732$$ 4.41421 0.163154
$$733$$ 28.5858 1.05584 0.527920 0.849294i $$-0.322972\pi$$
0.527920 + 0.849294i $$0.322972\pi$$
$$734$$ −12.6274 −0.466086
$$735$$ 0 0
$$736$$ 1.41421 0.0521286
$$737$$ −0.757359 −0.0278977
$$738$$ 9.89949 0.364405
$$739$$ −30.0000 −1.10357 −0.551784 0.833987i $$-0.686053\pi$$
−0.551784 + 0.833987i $$0.686053\pi$$
$$740$$ 0.828427 0.0304536
$$741$$ −2.17157 −0.0797747
$$742$$ 0 0
$$743$$ 12.1716 0.446532 0.223266 0.974758i $$-0.428328\pi$$
0.223266 + 0.974758i $$0.428328\pi$$
$$744$$ 8.48528 0.311086
$$745$$ −2.88730 −0.105783
$$746$$ −5.58579 −0.204510
$$747$$ −7.65685 −0.280150
$$748$$ −1.00000 −0.0365636
$$749$$ 0 0
$$750$$ −5.65685 −0.206559
$$751$$ −19.3137 −0.704767 −0.352384 0.935856i $$-0.614629\pi$$
−0.352384 + 0.935856i $$0.614629\pi$$
$$752$$ −1.00000 −0.0364662
$$753$$ −24.8284 −0.904798
$$754$$ 1.82843 0.0665874
$$755$$ 2.38478 0.0867909
$$756$$ 0 0
$$757$$ −30.7574 −1.11790 −0.558948 0.829203i $$-0.688795\pi$$
−0.558948 + 0.829203i $$0.688795\pi$$
$$758$$ 18.6274 0.676578
$$759$$ −0.585786 −0.0212627
$$760$$ 1.27208 0.0461431
$$761$$ −43.1127 −1.56283 −0.781417 0.624009i $$-0.785503\pi$$
−0.781417 + 0.624009i $$0.785503\pi$$
$$762$$ −7.89949 −0.286169
$$763$$ 0 0
$$764$$ −10.3431 −0.374202
$$765$$ 1.41421 0.0511310
$$766$$ −11.1716 −0.403645
$$767$$ 2.07107 0.0747819
$$768$$ 1.00000 0.0360844
$$769$$ −9.89949 −0.356985 −0.178492 0.983941i $$-0.557122\pi$$
−0.178492 + 0.983941i $$0.557122\pi$$
$$770$$ 0 0
$$771$$ −26.4853 −0.953844
$$772$$ −2.92893 −0.105415
$$773$$ −18.9706 −0.682324 −0.341162 0.940005i $$-0.610820\pi$$
−0.341162 + 0.940005i $$0.610820\pi$$
$$774$$ 6.48528 0.233109
$$775$$ −39.5147 −1.41941
$$776$$ 0.928932 0.0333467
$$777$$ 0 0
$$778$$ −4.51472 −0.161861
$$779$$ 21.4975 0.770227
$$780$$ −0.585786 −0.0209745
$$781$$ 2.07107 0.0741086
$$782$$ 3.41421 0.122092
$$783$$ −1.82843 −0.0653427
$$784$$ 0 0
$$785$$ −5.69848 −0.203388
$$786$$ 9.55635 0.340864
$$787$$ 26.3137 0.937982 0.468991 0.883203i $$-0.344618\pi$$
0.468991 + 0.883203i $$0.344618\pi$$
$$788$$ −15.5563 −0.554172
$$789$$ 20.5858 0.732873
$$790$$ −6.82843 −0.242945
$$791$$ 0 0
$$792$$ −0.414214 −0.0147184
$$793$$ −4.41421 −0.156753
$$794$$ 9.55635 0.339142
$$795$$ 5.55635 0.197063
$$796$$ −2.72792 −0.0966886
$$797$$ −34.3431 −1.21650 −0.608248 0.793747i $$-0.708128\pi$$
−0.608248 + 0.793747i $$0.708128\pi$$
$$798$$ 0 0
$$799$$ −2.41421 −0.0854087
$$800$$ −4.65685 −0.164645
$$801$$ −2.58579 −0.0913643
$$802$$ 19.8995 0.702676
$$803$$ 0.585786 0.0206720
$$804$$ 1.82843 0.0644837
$$805$$ 0 0
$$806$$ −8.48528 −0.298881
$$807$$ 9.14214 0.321818
$$808$$ 14.8284 0.521662
$$809$$ −17.5858 −0.618283 −0.309142 0.951016i $$-0.600042\pi$$
−0.309142 + 0.951016i $$0.600042\pi$$
$$810$$ 0.585786 0.0205824
$$811$$ −55.4558 −1.94732 −0.973659 0.228009i $$-0.926778\pi$$
−0.973659 + 0.228009i $$0.926778\pi$$
$$812$$ 0 0
$$813$$ −26.2132 −0.919337
$$814$$ −0.585786 −0.0205318
$$815$$ −10.9289 −0.382824
$$816$$ 2.41421 0.0845144
$$817$$ 14.0833 0.492711
$$818$$ −32.5858 −1.13934
$$819$$ 0 0
$$820$$ 5.79899 0.202510
$$821$$ 47.8995 1.67170 0.835852 0.548955i $$-0.184974\pi$$
0.835852 + 0.548955i $$0.184974\pi$$
$$822$$ 11.0711 0.386148
$$823$$ −43.8406 −1.52819 −0.764094 0.645105i $$-0.776814\pi$$
−0.764094 + 0.645105i $$0.776814\pi$$
$$824$$ 16.8284 0.586246
$$825$$ 1.92893 0.0671568
$$826$$ 0 0
$$827$$ 46.3553 1.61193 0.805967 0.591961i $$-0.201646\pi$$
0.805967 + 0.591961i $$0.201646\pi$$
$$828$$ 1.41421 0.0491473
$$829$$ 26.6985 0.927277 0.463638 0.886025i $$-0.346544\pi$$
0.463638 + 0.886025i $$0.346544\pi$$
$$830$$ −4.48528 −0.155686
$$831$$ 18.0711 0.626878
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ 13.0711 0.452614
$$835$$ 0.384776 0.0133157
$$836$$ −0.899495 −0.0311097
$$837$$ 8.48528 0.293294
$$838$$ −27.4558 −0.948446
$$839$$ 31.8284 1.09884 0.549420 0.835547i $$-0.314849\pi$$
0.549420 + 0.835547i $$0.314849\pi$$
$$840$$ 0 0
$$841$$ −25.6569 −0.884719
$$842$$ 1.07107 0.0369114
$$843$$ 0.686292 0.0236371
$$844$$ −12.9289 −0.445032
$$845$$ 0.585786 0.0201517
$$846$$ −1.00000 −0.0343807
$$847$$ 0 0
$$848$$ 9.48528 0.325726
$$849$$ 15.7574 0.540791
$$850$$ −11.2426 −0.385619
$$851$$ 2.00000 0.0685591
$$852$$ −5.00000 −0.171297
$$853$$ −6.72792 −0.230360 −0.115180 0.993345i $$-0.536744\pi$$
−0.115180 + 0.993345i $$0.536744\pi$$
$$854$$ 0 0
$$855$$ 1.27208 0.0435041
$$856$$ −19.3137 −0.660129
$$857$$ −20.5563 −0.702192 −0.351096 0.936340i $$-0.614191\pi$$
−0.351096 + 0.936340i $$0.614191\pi$$
$$858$$ 0.414214 0.0141410
$$859$$ 11.7990 0.402576 0.201288 0.979532i $$-0.435487\pi$$
0.201288 + 0.979532i $$0.435487\pi$$
$$860$$ 3.79899 0.129544
$$861$$ 0 0
$$862$$ 30.4853 1.03833
$$863$$ 39.3137 1.33825 0.669127 0.743148i $$-0.266668\pi$$
0.669127 + 0.743148i $$0.266668\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −3.21320 −0.109252
$$866$$ −29.9706 −1.01844
$$867$$ −11.1716 −0.379407
$$868$$ 0 0
$$869$$ 4.82843 0.163793
$$870$$ −1.07107 −0.0363126
$$871$$ −1.82843 −0.0619539
$$872$$ −9.65685 −0.327022
$$873$$ 0.928932 0.0314396
$$874$$ 3.07107 0.103880
$$875$$ 0 0
$$876$$ −1.41421 −0.0477818
$$877$$ −40.0416 −1.35211 −0.676055 0.736851i $$-0.736312\pi$$
−0.676055 + 0.736851i $$0.736312\pi$$
$$878$$ −16.5858 −0.559743
$$879$$ 9.89949 0.333902
$$880$$ −0.242641 −0.00817942
$$881$$ −18.4853 −0.622785 −0.311392 0.950281i $$-0.600795\pi$$
−0.311392 + 0.950281i $$0.600795\pi$$
$$882$$ 0 0
$$883$$ −9.75736 −0.328361 −0.164181 0.986430i $$-0.552498\pi$$
−0.164181 + 0.986430i $$0.552498\pi$$
$$884$$ −2.41421 −0.0811988
$$885$$ −1.21320 −0.0407814
$$886$$ −33.6985 −1.13212
$$887$$ −48.9117 −1.64229 −0.821147 0.570717i $$-0.806665\pi$$
−0.821147 + 0.570717i $$0.806665\pi$$
$$888$$ 1.41421 0.0474579
$$889$$ 0 0
$$890$$ −1.51472 −0.0507735
$$891$$ −0.414214 −0.0138767
$$892$$ 7.92893 0.265480
$$893$$ −2.17157 −0.0726689
$$894$$ −4.92893 −0.164848
$$895$$ 11.5147 0.384895
$$896$$ 0 0
$$897$$ −1.41421 −0.0472192
$$898$$ 24.9706 0.833278
$$899$$ −15.5147 −0.517445
$$900$$ −4.65685 −0.155228
$$901$$ 22.8995 0.762893
$$902$$ −4.10051 −0.136532
$$903$$ 0 0
$$904$$ 4.89949 0.162955
$$905$$ −1.69848 −0.0564595
$$906$$ 4.07107 0.135252
$$907$$ 54.3848 1.80582 0.902908 0.429833i $$-0.141428\pi$$
0.902908 + 0.429833i $$0.141428\pi$$
$$908$$ 21.1716 0.702603
$$909$$ 14.8284 0.491828
$$910$$ 0 0
$$911$$ −4.34315 −0.143895 −0.0719474 0.997408i $$-0.522921\pi$$
−0.0719474 + 0.997408i $$0.522921\pi$$
$$912$$ 2.17157 0.0719080
$$913$$ 3.17157 0.104964
$$914$$ −3.07107 −0.101582
$$915$$ 2.58579 0.0854835
$$916$$ −26.4853 −0.875098
$$917$$ 0 0
$$918$$ 2.41421 0.0796809
$$919$$ 1.79899 0.0593432 0.0296716 0.999560i $$-0.490554\pi$$
0.0296716 + 0.999560i $$0.490554\pi$$
$$920$$ 0.828427 0.0273124
$$921$$ 28.1127 0.926345
$$922$$ 4.48528 0.147715
$$923$$ 5.00000 0.164577
$$924$$ 0 0
$$925$$ −6.58579 −0.216539
$$926$$ −20.9706 −0.689135
$$927$$ 16.8284 0.552718
$$928$$ −1.82843 −0.0600211
$$929$$ −34.6863 −1.13802 −0.569010 0.822330i $$-0.692673\pi$$
−0.569010 + 0.822330i $$0.692673\pi$$
$$930$$ 4.97056 0.162991
$$931$$ 0 0
$$932$$ −17.7279 −0.580697
$$933$$ 8.10051 0.265199
$$934$$ 11.4142 0.373484
$$935$$ −0.585786 −0.0191573
$$936$$ −1.00000 −0.0326860
$$937$$ −42.6569 −1.39354 −0.696769 0.717295i $$-0.745380\pi$$
−0.696769 + 0.717295i $$0.745380\pi$$
$$938$$ 0 0
$$939$$ −3.51472 −0.114699
$$940$$ −0.585786 −0.0191062
$$941$$ 12.9706 0.422828 0.211414 0.977397i $$-0.432193\pi$$
0.211414 + 0.977397i $$0.432193\pi$$
$$942$$ −9.72792 −0.316953
$$943$$ 14.0000 0.455903
$$944$$ −2.07107 −0.0674075
$$945$$ 0 0
$$946$$ −2.68629 −0.0873389
$$947$$ 8.41421 0.273425 0.136713 0.990611i $$-0.456346\pi$$
0.136713 + 0.990611i $$0.456346\pi$$
$$948$$ −11.6569 −0.378597
$$949$$ 1.41421 0.0459073
$$950$$ −10.1127 −0.328099
$$951$$ 29.7990 0.966298
$$952$$ 0 0
$$953$$ −41.5858 −1.34710 −0.673548 0.739144i $$-0.735230\pi$$
−0.673548 + 0.739144i $$0.735230\pi$$
$$954$$ 9.48528 0.307097
$$955$$ −6.05887 −0.196061
$$956$$ 4.51472 0.146016
$$957$$ 0.757359 0.0244819
$$958$$ 32.1127 1.03751
$$959$$ 0 0
$$960$$ 0.585786 0.0189062
$$961$$ 41.0000 1.32258
$$962$$ −1.41421 −0.0455961
$$963$$ −19.3137 −0.622376
$$964$$ 2.14214 0.0689935
$$965$$ −1.71573 −0.0552313
$$966$$ 0 0
$$967$$ −13.5858 −0.436889 −0.218445 0.975849i $$-0.570098\pi$$
−0.218445 + 0.975849i $$0.570098\pi$$
$$968$$ −10.8284 −0.348039
$$969$$ 5.24264 0.168418
$$970$$ 0.544156 0.0174718
$$971$$ 32.5269 1.04384 0.521919 0.852995i $$-0.325216\pi$$
0.521919 + 0.852995i $$0.325216\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 16.2132 0.519505
$$975$$ 4.65685 0.149139
$$976$$ 4.41421 0.141296
$$977$$ 43.1716 1.38118 0.690590 0.723246i $$-0.257351\pi$$
0.690590 + 0.723246i $$0.257351\pi$$
$$978$$ −18.6569 −0.596580
$$979$$ 1.07107 0.0342315
$$980$$ 0 0
$$981$$ −9.65685 −0.308320
$$982$$ 23.6569 0.754921
$$983$$ 29.4853 0.940434 0.470217 0.882551i $$-0.344176\pi$$
0.470217 + 0.882551i $$0.344176\pi$$
$$984$$ 9.89949 0.315584
$$985$$ −9.11270 −0.290355
$$986$$ −4.41421 −0.140577
$$987$$ 0 0
$$988$$ −2.17157 −0.0690869
$$989$$ 9.17157 0.291639
$$990$$ −0.242641 −0.00771163
$$991$$ −0.686292 −0.0218008 −0.0109004 0.999941i $$-0.503470\pi$$
−0.0109004 + 0.999941i $$0.503470\pi$$
$$992$$ 8.48528 0.269408
$$993$$ −22.4853 −0.713549
$$994$$ 0 0
$$995$$ −1.59798 −0.0506594
$$996$$ −7.65685 −0.242617
$$997$$ −52.4975 −1.66261 −0.831306 0.555815i $$-0.812406\pi$$
−0.831306 + 0.555815i $$0.812406\pi$$
$$998$$ 3.65685 0.115756
$$999$$ 1.41421 0.0447437
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 3822.2.a.bu.1.1 2
7.3 odd 6 546.2.i.i.79.1 4
7.5 odd 6 546.2.i.i.235.1 yes 4
7.6 odd 2 3822.2.a.bn.1.2 2
21.5 even 6 1638.2.j.m.235.2 4
21.17 even 6 1638.2.j.m.1171.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.i.79.1 4 7.3 odd 6
546.2.i.i.235.1 yes 4 7.5 odd 6
1638.2.j.m.235.2 4 21.5 even 6
1638.2.j.m.1171.2 4 21.17 even 6
3822.2.a.bn.1.2 2 7.6 odd 2
3822.2.a.bu.1.1 2 1.1 even 1 trivial