Properties

 Label 5070.2.a.y.1.2 Level $5070$ Weight $2$ Character 5070.1 Self dual yes Analytic conductor $40.484$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Newspace parameters

 Level: $$N$$ $$=$$ $$5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5070.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$40.4841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +0.464102 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -0.535898 q^{19} -1.00000 q^{20} +2.00000 q^{21} -0.464102 q^{22} -0.267949 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{28} +3.73205 q^{29} -1.00000 q^{30} -1.73205 q^{31} -1.00000 q^{32} -0.464102 q^{33} -4.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +1.19615 q^{37} +0.535898 q^{38} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} +1.92820 q^{43} +0.464102 q^{44} -1.00000 q^{45} +0.267949 q^{46} -10.4641 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} -12.9282 q^{53} +1.00000 q^{54} -0.464102 q^{55} +2.00000 q^{56} +0.535898 q^{57} -3.73205 q^{58} +1.53590 q^{59} +1.00000 q^{60} +10.3923 q^{61} +1.73205 q^{62} -2.00000 q^{63} +1.00000 q^{64} +0.464102 q^{66} -4.53590 q^{67} +4.00000 q^{68} +0.267949 q^{69} -2.00000 q^{70} -8.39230 q^{71} -1.00000 q^{72} +2.00000 q^{73} -1.19615 q^{74} -1.00000 q^{75} -0.535898 q^{76} -0.928203 q^{77} -0.0717968 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.92820 q^{83} +2.00000 q^{84} -4.00000 q^{85} -1.92820 q^{86} -3.73205 q^{87} -0.464102 q^{88} +7.46410 q^{89} +1.00000 q^{90} -0.267949 q^{92} +1.73205 q^{93} +10.4641 q^{94} +0.535898 q^{95} +1.00000 q^{96} -7.46410 q^{97} +3.00000 q^{98} +0.464102 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 4q^{7} - 2q^{8} + 2q^{9} + 2q^{10} - 6q^{11} - 2q^{12} + 4q^{14} + 2q^{15} + 2q^{16} + 8q^{17} - 2q^{18} - 8q^{19} - 2q^{20} + 4q^{21} + 6q^{22} - 4q^{23} + 2q^{24} + 2q^{25} - 2q^{27} - 4q^{28} + 4q^{29} - 2q^{30} - 2q^{32} + 6q^{33} - 8q^{34} + 4q^{35} + 2q^{36} - 8q^{37} + 8q^{38} + 2q^{40} + 4q^{41} - 4q^{42} - 10q^{43} - 6q^{44} - 2q^{45} + 4q^{46} - 14q^{47} - 2q^{48} - 6q^{49} - 2q^{50} - 8q^{51} - 12q^{53} + 2q^{54} + 6q^{55} + 4q^{56} + 8q^{57} - 4q^{58} + 10q^{59} + 2q^{60} - 4q^{63} + 2q^{64} - 6q^{66} - 16q^{67} + 8q^{68} + 4q^{69} - 4q^{70} + 4q^{71} - 2q^{72} + 4q^{73} + 8q^{74} - 2q^{75} - 8q^{76} + 12q^{77} - 14q^{79} - 2q^{80} + 2q^{81} - 4q^{82} - 4q^{83} + 4q^{84} - 8q^{85} + 10q^{86} - 4q^{87} + 6q^{88} + 8q^{89} + 2q^{90} - 4q^{92} + 14q^{94} + 8q^{95} + 2q^{96} - 8q^{97} + 6q^{98} - 6q^{99} + O(q^{100})$$

Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ 1.00000 0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0.464102 0.139932 0.0699660 0.997549i $$-0.477711\pi$$
0.0699660 + 0.997549i $$0.477711\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ 2.00000 0.534522
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −0.535898 −0.122944 −0.0614718 0.998109i $$-0.519579\pi$$
−0.0614718 + 0.998109i $$0.519579\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 2.00000 0.436436
$$22$$ −0.464102 −0.0989468
$$23$$ −0.267949 −0.0558713 −0.0279356 0.999610i $$-0.508893\pi$$
−0.0279356 + 0.999610i $$0.508893\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −2.00000 −0.377964
$$29$$ 3.73205 0.693024 0.346512 0.938045i $$-0.387366\pi$$
0.346512 + 0.938045i $$0.387366\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −1.73205 −0.311086 −0.155543 0.987829i $$-0.549713\pi$$
−0.155543 + 0.987829i $$0.549713\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −0.464102 −0.0807897
$$34$$ −4.00000 −0.685994
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ 1.19615 0.196646 0.0983231 0.995155i $$-0.468652\pi$$
0.0983231 + 0.995155i $$0.468652\pi$$
$$38$$ 0.535898 0.0869342
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ 1.92820 0.294048 0.147024 0.989133i $$-0.453031\pi$$
0.147024 + 0.989133i $$0.453031\pi$$
$$44$$ 0.464102 0.0699660
$$45$$ −1.00000 −0.149071
$$46$$ 0.267949 0.0395070
$$47$$ −10.4641 −1.52635 −0.763173 0.646194i $$-0.776360\pi$$
−0.763173 + 0.646194i $$0.776360\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ −12.9282 −1.77583 −0.887913 0.460012i $$-0.847845\pi$$
−0.887913 + 0.460012i $$0.847845\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −0.464102 −0.0625794
$$56$$ 2.00000 0.267261
$$57$$ 0.535898 0.0709815
$$58$$ −3.73205 −0.490042
$$59$$ 1.53590 0.199957 0.0999785 0.994990i $$-0.468123\pi$$
0.0999785 + 0.994990i $$0.468123\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 10.3923 1.33060 0.665299 0.746577i $$-0.268304\pi$$
0.665299 + 0.746577i $$0.268304\pi$$
$$62$$ 1.73205 0.219971
$$63$$ −2.00000 −0.251976
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.464102 0.0571270
$$67$$ −4.53590 −0.554148 −0.277074 0.960849i $$-0.589365\pi$$
−0.277074 + 0.960849i $$0.589365\pi$$
$$68$$ 4.00000 0.485071
$$69$$ 0.267949 0.0322573
$$70$$ −2.00000 −0.239046
$$71$$ −8.39230 −0.995983 −0.497992 0.867182i $$-0.665929\pi$$
−0.497992 + 0.867182i $$0.665929\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −1.19615 −0.139050
$$75$$ −1.00000 −0.115470
$$76$$ −0.535898 −0.0614718
$$77$$ −0.928203 −0.105779
$$78$$ 0 0
$$79$$ −0.0717968 −0.00807777 −0.00403888 0.999992i $$-0.501286\pi$$
−0.00403888 + 0.999992i $$0.501286\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 4.92820 0.540941 0.270470 0.962728i $$-0.412821\pi$$
0.270470 + 0.962728i $$0.412821\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −4.00000 −0.433861
$$86$$ −1.92820 −0.207924
$$87$$ −3.73205 −0.400118
$$88$$ −0.464102 −0.0494734
$$89$$ 7.46410 0.791193 0.395597 0.918424i $$-0.370538\pi$$
0.395597 + 0.918424i $$0.370538\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ −0.267949 −0.0279356
$$93$$ 1.73205 0.179605
$$94$$ 10.4641 1.07929
$$95$$ 0.535898 0.0549820
$$96$$ 1.00000 0.102062
$$97$$ −7.46410 −0.757865 −0.378932 0.925424i $$-0.623709\pi$$
−0.378932 + 0.925424i $$0.623709\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0.464102 0.0466440
$$100$$ 1.00000 0.100000
$$101$$ 10.9282 1.08740 0.543698 0.839281i $$-0.317024\pi$$
0.543698 + 0.839281i $$0.317024\pi$$
$$102$$ 4.00000 0.396059
$$103$$ −15.8564 −1.56238 −0.781189 0.624295i $$-0.785387\pi$$
−0.781189 + 0.624295i $$0.785387\pi$$
$$104$$ 0 0
$$105$$ −2.00000 −0.195180
$$106$$ 12.9282 1.25570
$$107$$ 19.8564 1.91959 0.959796 0.280700i $$-0.0905665\pi$$
0.959796 + 0.280700i $$0.0905665\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −11.8564 −1.13564 −0.567819 0.823154i $$-0.692213\pi$$
−0.567819 + 0.823154i $$0.692213\pi$$
$$110$$ 0.464102 0.0442504
$$111$$ −1.19615 −0.113534
$$112$$ −2.00000 −0.188982
$$113$$ 11.1962 1.05325 0.526623 0.850099i $$-0.323458\pi$$
0.526623 + 0.850099i $$0.323458\pi$$
$$114$$ −0.535898 −0.0501915
$$115$$ 0.267949 0.0249864
$$116$$ 3.73205 0.346512
$$117$$ 0 0
$$118$$ −1.53590 −0.141391
$$119$$ −8.00000 −0.733359
$$120$$ −1.00000 −0.0912871
$$121$$ −10.7846 −0.980419
$$122$$ −10.3923 −0.940875
$$123$$ −2.00000 −0.180334
$$124$$ −1.73205 −0.155543
$$125$$ −1.00000 −0.0894427
$$126$$ 2.00000 0.178174
$$127$$ −8.92820 −0.792250 −0.396125 0.918197i $$-0.629645\pi$$
−0.396125 + 0.918197i $$0.629645\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −1.92820 −0.169769
$$130$$ 0 0
$$131$$ −1.33975 −0.117054 −0.0585271 0.998286i $$-0.518640\pi$$
−0.0585271 + 0.998286i $$0.518640\pi$$
$$132$$ −0.464102 −0.0403949
$$133$$ 1.07180 0.0929366
$$134$$ 4.53590 0.391842
$$135$$ 1.00000 0.0860663
$$136$$ −4.00000 −0.342997
$$137$$ 4.46410 0.381394 0.190697 0.981649i $$-0.438925\pi$$
0.190697 + 0.981649i $$0.438925\pi$$
$$138$$ −0.267949 −0.0228093
$$139$$ 0.928203 0.0787292 0.0393646 0.999225i $$-0.487467\pi$$
0.0393646 + 0.999225i $$0.487467\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 10.4641 0.881236
$$142$$ 8.39230 0.704267
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −3.73205 −0.309930
$$146$$ −2.00000 −0.165521
$$147$$ 3.00000 0.247436
$$148$$ 1.19615 0.0983231
$$149$$ 20.4641 1.67648 0.838242 0.545298i $$-0.183584\pi$$
0.838242 + 0.545298i $$0.183584\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 10.3923 0.845714 0.422857 0.906196i $$-0.361027\pi$$
0.422857 + 0.906196i $$0.361027\pi$$
$$152$$ 0.535898 0.0434671
$$153$$ 4.00000 0.323381
$$154$$ 0.928203 0.0747967
$$155$$ 1.73205 0.139122
$$156$$ 0 0
$$157$$ −5.00000 −0.399043 −0.199522 0.979893i $$-0.563939\pi$$
−0.199522 + 0.979893i $$0.563939\pi$$
$$158$$ 0.0717968 0.00571184
$$159$$ 12.9282 1.02527
$$160$$ 1.00000 0.0790569
$$161$$ 0.535898 0.0422347
$$162$$ −1.00000 −0.0785674
$$163$$ −23.0526 −1.80562 −0.902808 0.430044i $$-0.858498\pi$$
−0.902808 + 0.430044i $$0.858498\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0.464102 0.0361303
$$166$$ −4.92820 −0.382503
$$167$$ 18.3205 1.41768 0.708842 0.705368i $$-0.249218\pi$$
0.708842 + 0.705368i $$0.249218\pi$$
$$168$$ −2.00000 −0.154303
$$169$$ 0 0
$$170$$ 4.00000 0.306786
$$171$$ −0.535898 −0.0409812
$$172$$ 1.92820 0.147024
$$173$$ 2.92820 0.222627 0.111314 0.993785i $$-0.464494\pi$$
0.111314 + 0.993785i $$0.464494\pi$$
$$174$$ 3.73205 0.282926
$$175$$ −2.00000 −0.151186
$$176$$ 0.464102 0.0349830
$$177$$ −1.53590 −0.115445
$$178$$ −7.46410 −0.559458
$$179$$ 16.2679 1.21592 0.607962 0.793966i $$-0.291987\pi$$
0.607962 + 0.793966i $$0.291987\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −10.9282 −0.812287 −0.406143 0.913809i $$-0.633127\pi$$
−0.406143 + 0.913809i $$0.633127\pi$$
$$182$$ 0 0
$$183$$ −10.3923 −0.768221
$$184$$ 0.267949 0.0197535
$$185$$ −1.19615 −0.0879429
$$186$$ −1.73205 −0.127000
$$187$$ 1.85641 0.135754
$$188$$ −10.4641 −0.763173
$$189$$ 2.00000 0.145479
$$190$$ −0.535898 −0.0388782
$$191$$ 14.5359 1.05178 0.525890 0.850552i $$-0.323732\pi$$
0.525890 + 0.850552i $$0.323732\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 23.3205 1.67865 0.839323 0.543632i $$-0.182951\pi$$
0.839323 + 0.543632i $$0.182951\pi$$
$$194$$ 7.46410 0.535891
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −16.3923 −1.16790 −0.583952 0.811788i $$-0.698494\pi$$
−0.583952 + 0.811788i $$0.698494\pi$$
$$198$$ −0.464102 −0.0329823
$$199$$ 18.9282 1.34178 0.670892 0.741555i $$-0.265911\pi$$
0.670892 + 0.741555i $$0.265911\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 4.53590 0.319938
$$202$$ −10.9282 −0.768906
$$203$$ −7.46410 −0.523877
$$204$$ −4.00000 −0.280056
$$205$$ −2.00000 −0.139686
$$206$$ 15.8564 1.10477
$$207$$ −0.267949 −0.0186238
$$208$$ 0 0
$$209$$ −0.248711 −0.0172037
$$210$$ 2.00000 0.138013
$$211$$ 23.3205 1.60545 0.802725 0.596349i $$-0.203383\pi$$
0.802725 + 0.596349i $$0.203383\pi$$
$$212$$ −12.9282 −0.887913
$$213$$ 8.39230 0.575031
$$214$$ −19.8564 −1.35736
$$215$$ −1.92820 −0.131502
$$216$$ 1.00000 0.0680414
$$217$$ 3.46410 0.235159
$$218$$ 11.8564 0.803017
$$219$$ −2.00000 −0.135147
$$220$$ −0.464102 −0.0312897
$$221$$ 0 0
$$222$$ 1.19615 0.0802805
$$223$$ −27.4641 −1.83913 −0.919566 0.392935i $$-0.871460\pi$$
−0.919566 + 0.392935i $$0.871460\pi$$
$$224$$ 2.00000 0.133631
$$225$$ 1.00000 0.0666667
$$226$$ −11.1962 −0.744757
$$227$$ −4.39230 −0.291528 −0.145764 0.989319i $$-0.546564\pi$$
−0.145764 + 0.989319i $$0.546564\pi$$
$$228$$ 0.535898 0.0354907
$$229$$ 19.8564 1.31215 0.656074 0.754696i $$-0.272216\pi$$
0.656074 + 0.754696i $$0.272216\pi$$
$$230$$ −0.267949 −0.0176680
$$231$$ 0.928203 0.0610713
$$232$$ −3.73205 −0.245021
$$233$$ 18.1244 1.18737 0.593683 0.804699i $$-0.297673\pi$$
0.593683 + 0.804699i $$0.297673\pi$$
$$234$$ 0 0
$$235$$ 10.4641 0.682603
$$236$$ 1.53590 0.0999785
$$237$$ 0.0717968 0.00466370
$$238$$ 8.00000 0.518563
$$239$$ −4.39230 −0.284115 −0.142057 0.989858i $$-0.545372\pi$$
−0.142057 + 0.989858i $$0.545372\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 14.2679 0.919079 0.459540 0.888157i $$-0.348014\pi$$
0.459540 + 0.888157i $$0.348014\pi$$
$$242$$ 10.7846 0.693261
$$243$$ −1.00000 −0.0641500
$$244$$ 10.3923 0.665299
$$245$$ 3.00000 0.191663
$$246$$ 2.00000 0.127515
$$247$$ 0 0
$$248$$ 1.73205 0.109985
$$249$$ −4.92820 −0.312312
$$250$$ 1.00000 0.0632456
$$251$$ −12.2679 −0.774346 −0.387173 0.922007i $$-0.626548\pi$$
−0.387173 + 0.922007i $$0.626548\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ −0.124356 −0.00781817
$$254$$ 8.92820 0.560205
$$255$$ 4.00000 0.250490
$$256$$ 1.00000 0.0625000
$$257$$ 22.6603 1.41351 0.706754 0.707459i $$-0.250159\pi$$
0.706754 + 0.707459i $$0.250159\pi$$
$$258$$ 1.92820 0.120045
$$259$$ −2.39230 −0.148651
$$260$$ 0 0
$$261$$ 3.73205 0.231008
$$262$$ 1.33975 0.0827698
$$263$$ −18.1244 −1.11760 −0.558798 0.829304i $$-0.688737\pi$$
−0.558798 + 0.829304i $$0.688737\pi$$
$$264$$ 0.464102 0.0285635
$$265$$ 12.9282 0.794173
$$266$$ −1.07180 −0.0657161
$$267$$ −7.46410 −0.456796
$$268$$ −4.53590 −0.277074
$$269$$ 12.0000 0.731653 0.365826 0.930683i $$-0.380786\pi$$
0.365826 + 0.930683i $$0.380786\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 9.19615 0.558626 0.279313 0.960200i $$-0.409893\pi$$
0.279313 + 0.960200i $$0.409893\pi$$
$$272$$ 4.00000 0.242536
$$273$$ 0 0
$$274$$ −4.46410 −0.269686
$$275$$ 0.464102 0.0279864
$$276$$ 0.267949 0.0161286
$$277$$ −9.92820 −0.596528 −0.298264 0.954483i $$-0.596408\pi$$
−0.298264 + 0.954483i $$0.596408\pi$$
$$278$$ −0.928203 −0.0556699
$$279$$ −1.73205 −0.103695
$$280$$ −2.00000 −0.119523
$$281$$ 4.92820 0.293992 0.146996 0.989137i $$-0.453040\pi$$
0.146996 + 0.989137i $$0.453040\pi$$
$$282$$ −10.4641 −0.623128
$$283$$ −3.92820 −0.233507 −0.116754 0.993161i $$-0.537249\pi$$
−0.116754 + 0.993161i $$0.537249\pi$$
$$284$$ −8.39230 −0.497992
$$285$$ −0.535898 −0.0317439
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ −1.00000 −0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 3.73205 0.219154
$$291$$ 7.46410 0.437553
$$292$$ 2.00000 0.117041
$$293$$ −4.14359 −0.242071 −0.121036 0.992648i $$-0.538622\pi$$
−0.121036 + 0.992648i $$0.538622\pi$$
$$294$$ −3.00000 −0.174964
$$295$$ −1.53590 −0.0894235
$$296$$ −1.19615 −0.0695249
$$297$$ −0.464102 −0.0269299
$$298$$ −20.4641 −1.18545
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ −3.85641 −0.222280
$$302$$ −10.3923 −0.598010
$$303$$ −10.9282 −0.627809
$$304$$ −0.535898 −0.0307359
$$305$$ −10.3923 −0.595062
$$306$$ −4.00000 −0.228665
$$307$$ 12.5359 0.715462 0.357731 0.933825i $$-0.383551\pi$$
0.357731 + 0.933825i $$0.383551\pi$$
$$308$$ −0.928203 −0.0528893
$$309$$ 15.8564 0.902039
$$310$$ −1.73205 −0.0983739
$$311$$ 7.60770 0.431393 0.215696 0.976460i $$-0.430798\pi$$
0.215696 + 0.976460i $$0.430798\pi$$
$$312$$ 0 0
$$313$$ −28.0000 −1.58265 −0.791327 0.611393i $$-0.790609\pi$$
−0.791327 + 0.611393i $$0.790609\pi$$
$$314$$ 5.00000 0.282166
$$315$$ 2.00000 0.112687
$$316$$ −0.0717968 −0.00403888
$$317$$ −21.4641 −1.20554 −0.602772 0.797913i $$-0.705937\pi$$
−0.602772 + 0.797913i $$0.705937\pi$$
$$318$$ −12.9282 −0.724978
$$319$$ 1.73205 0.0969762
$$320$$ −1.00000 −0.0559017
$$321$$ −19.8564 −1.10828
$$322$$ −0.535898 −0.0298644
$$323$$ −2.14359 −0.119273
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 23.0526 1.27676
$$327$$ 11.8564 0.655661
$$328$$ −2.00000 −0.110432
$$329$$ 20.9282 1.15381
$$330$$ −0.464102 −0.0255480
$$331$$ −24.7846 −1.36229 −0.681143 0.732151i $$-0.738517\pi$$
−0.681143 + 0.732151i $$0.738517\pi$$
$$332$$ 4.92820 0.270470
$$333$$ 1.19615 0.0655487
$$334$$ −18.3205 −1.00245
$$335$$ 4.53590 0.247823
$$336$$ 2.00000 0.109109
$$337$$ 25.3205 1.37930 0.689648 0.724145i $$-0.257765\pi$$
0.689648 + 0.724145i $$0.257765\pi$$
$$338$$ 0 0
$$339$$ −11.1962 −0.608092
$$340$$ −4.00000 −0.216930
$$341$$ −0.803848 −0.0435308
$$342$$ 0.535898 0.0289781
$$343$$ 20.0000 1.07990
$$344$$ −1.92820 −0.103962
$$345$$ −0.267949 −0.0144259
$$346$$ −2.92820 −0.157421
$$347$$ 22.3923 1.20208 0.601041 0.799218i $$-0.294753\pi$$
0.601041 + 0.799218i $$0.294753\pi$$
$$348$$ −3.73205 −0.200059
$$349$$ −14.5359 −0.778089 −0.389044 0.921219i $$-0.627195\pi$$
−0.389044 + 0.921219i $$0.627195\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ −0.464102 −0.0247367
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 1.53590 0.0816321
$$355$$ 8.39230 0.445417
$$356$$ 7.46410 0.395597
$$357$$ 8.00000 0.423405
$$358$$ −16.2679 −0.859788
$$359$$ 18.9282 0.998992 0.499496 0.866316i $$-0.333518\pi$$
0.499496 + 0.866316i $$0.333518\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −18.7128 −0.984885
$$362$$ 10.9282 0.574374
$$363$$ 10.7846 0.566045
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ 10.3923 0.543214
$$367$$ 36.3923 1.89966 0.949831 0.312762i $$-0.101254\pi$$
0.949831 + 0.312762i $$0.101254\pi$$
$$368$$ −0.267949 −0.0139678
$$369$$ 2.00000 0.104116
$$370$$ 1.19615 0.0621850
$$371$$ 25.8564 1.34240
$$372$$ 1.73205 0.0898027
$$373$$ 25.7846 1.33508 0.667538 0.744576i $$-0.267348\pi$$
0.667538 + 0.744576i $$0.267348\pi$$
$$374$$ −1.85641 −0.0959925
$$375$$ 1.00000 0.0516398
$$376$$ 10.4641 0.539645
$$377$$ 0 0
$$378$$ −2.00000 −0.102869
$$379$$ 0.143594 0.00737590 0.00368795 0.999993i $$-0.498826\pi$$
0.00368795 + 0.999993i $$0.498826\pi$$
$$380$$ 0.535898 0.0274910
$$381$$ 8.92820 0.457406
$$382$$ −14.5359 −0.743721
$$383$$ −4.60770 −0.235442 −0.117721 0.993047i $$-0.537559\pi$$
−0.117721 + 0.993047i $$0.537559\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0.928203 0.0473056
$$386$$ −23.3205 −1.18698
$$387$$ 1.92820 0.0980161
$$388$$ −7.46410 −0.378932
$$389$$ 20.2679 1.02763 0.513813 0.857902i $$-0.328233\pi$$
0.513813 + 0.857902i $$0.328233\pi$$
$$390$$ 0 0
$$391$$ −1.07180 −0.0542031
$$392$$ 3.00000 0.151523
$$393$$ 1.33975 0.0675812
$$394$$ 16.3923 0.825832
$$395$$ 0.0717968 0.00361249
$$396$$ 0.464102 0.0233220
$$397$$ −12.1244 −0.608504 −0.304252 0.952592i $$-0.598406\pi$$
−0.304252 + 0.952592i $$0.598406\pi$$
$$398$$ −18.9282 −0.948785
$$399$$ −1.07180 −0.0536570
$$400$$ 1.00000 0.0500000
$$401$$ 32.0000 1.59800 0.799002 0.601329i $$-0.205362\pi$$
0.799002 + 0.601329i $$0.205362\pi$$
$$402$$ −4.53590 −0.226230
$$403$$ 0 0
$$404$$ 10.9282 0.543698
$$405$$ −1.00000 −0.0496904
$$406$$ 7.46410 0.370437
$$407$$ 0.555136 0.0275171
$$408$$ 4.00000 0.198030
$$409$$ −4.00000 −0.197787 −0.0988936 0.995098i $$-0.531530\pi$$
−0.0988936 + 0.995098i $$0.531530\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ −4.46410 −0.220198
$$412$$ −15.8564 −0.781189
$$413$$ −3.07180 −0.151153
$$414$$ 0.267949 0.0131690
$$415$$ −4.92820 −0.241916
$$416$$ 0 0
$$417$$ −0.928203 −0.0454543
$$418$$ 0.248711 0.0121649
$$419$$ 1.60770 0.0785410 0.0392705 0.999229i $$-0.487497\pi$$
0.0392705 + 0.999229i $$0.487497\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 16.3923 0.798912 0.399456 0.916752i $$-0.369199\pi$$
0.399456 + 0.916752i $$0.369199\pi$$
$$422$$ −23.3205 −1.13522
$$423$$ −10.4641 −0.508782
$$424$$ 12.9282 0.627849
$$425$$ 4.00000 0.194029
$$426$$ −8.39230 −0.406608
$$427$$ −20.7846 −1.00584
$$428$$ 19.8564 0.959796
$$429$$ 0 0
$$430$$ 1.92820 0.0929862
$$431$$ 7.60770 0.366450 0.183225 0.983071i $$-0.441346\pi$$
0.183225 + 0.983071i $$0.441346\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 15.3205 0.736257 0.368128 0.929775i $$-0.379999\pi$$
0.368128 + 0.929775i $$0.379999\pi$$
$$434$$ −3.46410 −0.166282
$$435$$ 3.73205 0.178938
$$436$$ −11.8564 −0.567819
$$437$$ 0.143594 0.00686901
$$438$$ 2.00000 0.0955637
$$439$$ −9.85641 −0.470421 −0.235210 0.971945i $$-0.575578\pi$$
−0.235210 + 0.971945i $$0.575578\pi$$
$$440$$ 0.464102 0.0221252
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −4.39230 −0.208685 −0.104342 0.994541i $$-0.533274\pi$$
−0.104342 + 0.994541i $$0.533274\pi$$
$$444$$ −1.19615 −0.0567669
$$445$$ −7.46410 −0.353832
$$446$$ 27.4641 1.30046
$$447$$ −20.4641 −0.967919
$$448$$ −2.00000 −0.0944911
$$449$$ 39.7128 1.87416 0.937082 0.349110i $$-0.113516\pi$$
0.937082 + 0.349110i $$0.113516\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0.928203 0.0437074
$$452$$ 11.1962 0.526623
$$453$$ −10.3923 −0.488273
$$454$$ 4.39230 0.206141
$$455$$ 0 0
$$456$$ −0.535898 −0.0250957
$$457$$ 31.4641 1.47183 0.735914 0.677075i $$-0.236753\pi$$
0.735914 + 0.677075i $$0.236753\pi$$
$$458$$ −19.8564 −0.927829
$$459$$ −4.00000 −0.186704
$$460$$ 0.267949 0.0124932
$$461$$ 6.46410 0.301063 0.150532 0.988605i $$-0.451901\pi$$
0.150532 + 0.988605i $$0.451901\pi$$
$$462$$ −0.928203 −0.0431839
$$463$$ 20.9282 0.972616 0.486308 0.873787i $$-0.338343\pi$$
0.486308 + 0.873787i $$0.338343\pi$$
$$464$$ 3.73205 0.173256
$$465$$ −1.73205 −0.0803219
$$466$$ −18.1244 −0.839595
$$467$$ −11.8564 −0.548649 −0.274325 0.961637i $$-0.588454\pi$$
−0.274325 + 0.961637i $$0.588454\pi$$
$$468$$ 0 0
$$469$$ 9.07180 0.418897
$$470$$ −10.4641 −0.482673
$$471$$ 5.00000 0.230388
$$472$$ −1.53590 −0.0706955
$$473$$ 0.894882 0.0411467
$$474$$ −0.0717968 −0.00329773
$$475$$ −0.535898 −0.0245887
$$476$$ −8.00000 −0.366679
$$477$$ −12.9282 −0.591942
$$478$$ 4.39230 0.200899
$$479$$ 1.46410 0.0668965 0.0334483 0.999440i $$-0.489351\pi$$
0.0334483 + 0.999440i $$0.489351\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −14.2679 −0.649887
$$483$$ −0.535898 −0.0243842
$$484$$ −10.7846 −0.490210
$$485$$ 7.46410 0.338927
$$486$$ 1.00000 0.0453609
$$487$$ 39.1769 1.77528 0.887638 0.460542i $$-0.152345\pi$$
0.887638 + 0.460542i $$0.152345\pi$$
$$488$$ −10.3923 −0.470438
$$489$$ 23.0526 1.04247
$$490$$ −3.00000 −0.135526
$$491$$ −17.3205 −0.781664 −0.390832 0.920462i $$-0.627813\pi$$
−0.390832 + 0.920462i $$0.627813\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 14.9282 0.672332
$$494$$ 0 0
$$495$$ −0.464102 −0.0208598
$$496$$ −1.73205 −0.0777714
$$497$$ 16.7846 0.752893
$$498$$ 4.92820 0.220838
$$499$$ −13.4641 −0.602736 −0.301368 0.953508i $$-0.597443\pi$$
−0.301368 + 0.953508i $$0.597443\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −18.3205 −0.818500
$$502$$ 12.2679 0.547545
$$503$$ 31.1769 1.39011 0.695055 0.718957i $$-0.255380\pi$$
0.695055 + 0.718957i $$0.255380\pi$$
$$504$$ 2.00000 0.0890871
$$505$$ −10.9282 −0.486299
$$506$$ 0.124356 0.00552828
$$507$$ 0 0
$$508$$ −8.92820 −0.396125
$$509$$ −19.3923 −0.859549 −0.429774 0.902936i $$-0.641407\pi$$
−0.429774 + 0.902936i $$0.641407\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ −4.00000 −0.176950
$$512$$ −1.00000 −0.0441942
$$513$$ 0.535898 0.0236605
$$514$$ −22.6603 −0.999501
$$515$$ 15.8564 0.698717
$$516$$ −1.92820 −0.0848844
$$517$$ −4.85641 −0.213585
$$518$$ 2.39230 0.105112
$$519$$ −2.92820 −0.128534
$$520$$ 0 0
$$521$$ 17.3205 0.758825 0.379413 0.925228i $$-0.376126\pi$$
0.379413 + 0.925228i $$0.376126\pi$$
$$522$$ −3.73205 −0.163347
$$523$$ −29.7846 −1.30239 −0.651195 0.758910i $$-0.725732\pi$$
−0.651195 + 0.758910i $$0.725732\pi$$
$$524$$ −1.33975 −0.0585271
$$525$$ 2.00000 0.0872872
$$526$$ 18.1244 0.790259
$$527$$ −6.92820 −0.301797
$$528$$ −0.464102 −0.0201974
$$529$$ −22.9282 −0.996878
$$530$$ −12.9282 −0.561565
$$531$$ 1.53590 0.0666523
$$532$$ 1.07180 0.0464683
$$533$$ 0 0
$$534$$ 7.46410 0.323003
$$535$$ −19.8564 −0.858467
$$536$$ 4.53590 0.195921
$$537$$ −16.2679 −0.702014
$$538$$ −12.0000 −0.517357
$$539$$ −1.39230 −0.0599708
$$540$$ 1.00000 0.0430331
$$541$$ 13.0718 0.562000 0.281000 0.959708i $$-0.409334\pi$$
0.281000 + 0.959708i $$0.409334\pi$$
$$542$$ −9.19615 −0.395009
$$543$$ 10.9282 0.468974
$$544$$ −4.00000 −0.171499
$$545$$ 11.8564 0.507873
$$546$$ 0 0
$$547$$ 9.07180 0.387882 0.193941 0.981013i $$-0.437873\pi$$
0.193941 + 0.981013i $$0.437873\pi$$
$$548$$ 4.46410 0.190697
$$549$$ 10.3923 0.443533
$$550$$ −0.464102 −0.0197894
$$551$$ −2.00000 −0.0852029
$$552$$ −0.267949 −0.0114047
$$553$$ 0.143594 0.00610622
$$554$$ 9.92820 0.421809
$$555$$ 1.19615 0.0507738
$$556$$ 0.928203 0.0393646
$$557$$ 37.7128 1.59794 0.798972 0.601369i $$-0.205378\pi$$
0.798972 + 0.601369i $$0.205378\pi$$
$$558$$ 1.73205 0.0733236
$$559$$ 0 0
$$560$$ 2.00000 0.0845154
$$561$$ −1.85641 −0.0783775
$$562$$ −4.92820 −0.207884
$$563$$ 39.3205 1.65716 0.828581 0.559869i $$-0.189149\pi$$
0.828581 + 0.559869i $$0.189149\pi$$
$$564$$ 10.4641 0.440618
$$565$$ −11.1962 −0.471026
$$566$$ 3.92820 0.165115
$$567$$ −2.00000 −0.0839921
$$568$$ 8.39230 0.352133
$$569$$ 5.32051 0.223047 0.111524 0.993762i $$-0.464427\pi$$
0.111524 + 0.993762i $$0.464427\pi$$
$$570$$ 0.535898 0.0224463
$$571$$ −45.1769 −1.89060 −0.945298 0.326209i $$-0.894229\pi$$
−0.945298 + 0.326209i $$0.894229\pi$$
$$572$$ 0 0
$$573$$ −14.5359 −0.607246
$$574$$ 4.00000 0.166957
$$575$$ −0.267949 −0.0111743
$$576$$ 1.00000 0.0416667
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ −23.3205 −0.969167
$$580$$ −3.73205 −0.154965
$$581$$ −9.85641 −0.408913
$$582$$ −7.46410 −0.309397
$$583$$ −6.00000 −0.248495
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 4.14359 0.171170
$$587$$ 18.3923 0.759132 0.379566 0.925165i $$-0.376073\pi$$
0.379566 + 0.925165i $$0.376073\pi$$
$$588$$ 3.00000 0.123718
$$589$$ 0.928203 0.0382459
$$590$$ 1.53590 0.0632319
$$591$$ 16.3923 0.674289
$$592$$ 1.19615 0.0491616
$$593$$ −31.1051 −1.27733 −0.638667 0.769483i $$-0.720514\pi$$
−0.638667 + 0.769483i $$0.720514\pi$$
$$594$$ 0.464102 0.0190423
$$595$$ 8.00000 0.327968
$$596$$ 20.4641 0.838242
$$597$$ −18.9282 −0.774680
$$598$$ 0 0
$$599$$ 10.3923 0.424618 0.212309 0.977203i $$-0.431902\pi$$
0.212309 + 0.977203i $$0.431902\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 21.7846 0.888613 0.444306 0.895875i $$-0.353450\pi$$
0.444306 + 0.895875i $$0.353450\pi$$
$$602$$ 3.85641 0.157175
$$603$$ −4.53590 −0.184716
$$604$$ 10.3923 0.422857
$$605$$ 10.7846 0.438457
$$606$$ 10.9282 0.443928
$$607$$ 43.1769 1.75250 0.876248 0.481860i $$-0.160038\pi$$
0.876248 + 0.481860i $$0.160038\pi$$
$$608$$ 0.535898 0.0217335
$$609$$ 7.46410 0.302461
$$610$$ 10.3923 0.420772
$$611$$ 0 0
$$612$$ 4.00000 0.161690
$$613$$ −0.947441 −0.0382668 −0.0191334 0.999817i $$-0.506091\pi$$
−0.0191334 + 0.999817i $$0.506091\pi$$
$$614$$ −12.5359 −0.505908
$$615$$ 2.00000 0.0806478
$$616$$ 0.928203 0.0373984
$$617$$ 15.5359 0.625452 0.312726 0.949843i $$-0.398758\pi$$
0.312726 + 0.949843i $$0.398758\pi$$
$$618$$ −15.8564 −0.637838
$$619$$ 24.2487 0.974638 0.487319 0.873224i $$-0.337975\pi$$
0.487319 + 0.873224i $$0.337975\pi$$
$$620$$ 1.73205 0.0695608
$$621$$ 0.267949 0.0107524
$$622$$ −7.60770 −0.305041
$$623$$ −14.9282 −0.598086
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 28.0000 1.11911
$$627$$ 0.248711 0.00993257
$$628$$ −5.00000 −0.199522
$$629$$ 4.78461 0.190775
$$630$$ −2.00000 −0.0796819
$$631$$ 24.5359 0.976759 0.488379 0.872631i $$-0.337588\pi$$
0.488379 + 0.872631i $$0.337588\pi$$
$$632$$ 0.0717968 0.00285592
$$633$$ −23.3205 −0.926907
$$634$$ 21.4641 0.852448
$$635$$ 8.92820 0.354305
$$636$$ 12.9282 0.512637
$$637$$ 0 0
$$638$$ −1.73205 −0.0685725
$$639$$ −8.39230 −0.331994
$$640$$ 1.00000 0.0395285
$$641$$ −27.8564 −1.10026 −0.550131 0.835078i $$-0.685422\pi$$
−0.550131 + 0.835078i $$0.685422\pi$$
$$642$$ 19.8564 0.783670
$$643$$ 27.4641 1.08308 0.541539 0.840675i $$-0.317842\pi$$
0.541539 + 0.840675i $$0.317842\pi$$
$$644$$ 0.535898 0.0211174
$$645$$ 1.92820 0.0759229
$$646$$ 2.14359 0.0843386
$$647$$ −21.3205 −0.838196 −0.419098 0.907941i $$-0.637654\pi$$
−0.419098 + 0.907941i $$0.637654\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0.712813 0.0279804
$$650$$ 0 0
$$651$$ −3.46410 −0.135769
$$652$$ −23.0526 −0.902808
$$653$$ −44.2487 −1.73159 −0.865793 0.500402i $$-0.833185\pi$$
−0.865793 + 0.500402i $$0.833185\pi$$
$$654$$ −11.8564 −0.463622
$$655$$ 1.33975 0.0523482
$$656$$ 2.00000 0.0780869
$$657$$ 2.00000 0.0780274
$$658$$ −20.9282 −0.815866
$$659$$ −3.73205 −0.145380 −0.0726900 0.997355i $$-0.523158\pi$$
−0.0726900 + 0.997355i $$0.523158\pi$$
$$660$$ 0.464102 0.0180651
$$661$$ 43.3205 1.68497 0.842486 0.538718i $$-0.181091\pi$$
0.842486 + 0.538718i $$0.181091\pi$$
$$662$$ 24.7846 0.963281
$$663$$ 0 0
$$664$$ −4.92820 −0.191251
$$665$$ −1.07180 −0.0415625
$$666$$ −1.19615 −0.0463500
$$667$$ −1.00000 −0.0387202
$$668$$ 18.3205 0.708842
$$669$$ 27.4641 1.06182
$$670$$ −4.53590 −0.175237
$$671$$ 4.82309 0.186193
$$672$$ −2.00000 −0.0771517
$$673$$ −32.0000 −1.23351 −0.616755 0.787155i $$-0.711553\pi$$
−0.616755 + 0.787155i $$0.711553\pi$$
$$674$$ −25.3205 −0.975310
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 11.6077 0.446120 0.223060 0.974805i $$-0.428395\pi$$
0.223060 + 0.974805i $$0.428395\pi$$
$$678$$ 11.1962 0.429986
$$679$$ 14.9282 0.572892
$$680$$ 4.00000 0.153393
$$681$$ 4.39230 0.168313
$$682$$ 0.803848 0.0307809
$$683$$ 0.784610 0.0300223 0.0150111 0.999887i $$-0.495222\pi$$
0.0150111 + 0.999887i $$0.495222\pi$$
$$684$$ −0.535898 −0.0204906
$$685$$ −4.46410 −0.170565
$$686$$ −20.0000 −0.763604
$$687$$ −19.8564 −0.757569
$$688$$ 1.92820 0.0735121
$$689$$ 0 0
$$690$$ 0.267949 0.0102007
$$691$$ 35.1769 1.33819 0.669096 0.743176i $$-0.266681\pi$$
0.669096 + 0.743176i $$0.266681\pi$$
$$692$$ 2.92820 0.111314
$$693$$ −0.928203 −0.0352595
$$694$$ −22.3923 −0.850000
$$695$$ −0.928203 −0.0352088
$$696$$ 3.73205 0.141463
$$697$$ 8.00000 0.303022
$$698$$ 14.5359 0.550192
$$699$$ −18.1244 −0.685526
$$700$$ −2.00000 −0.0755929
$$701$$ 3.73205 0.140958 0.0704788 0.997513i $$-0.477547\pi$$
0.0704788 + 0.997513i $$0.477547\pi$$
$$702$$ 0 0
$$703$$ −0.641016 −0.0241764
$$704$$ 0.464102 0.0174915
$$705$$ −10.4641 −0.394101
$$706$$ −2.00000 −0.0752710
$$707$$ −21.8564 −0.821995
$$708$$ −1.53590 −0.0577226
$$709$$ −9.07180 −0.340698 −0.170349 0.985384i $$-0.554490\pi$$
−0.170349 + 0.985384i $$0.554490\pi$$
$$710$$ −8.39230 −0.314958
$$711$$ −0.0717968 −0.00269259
$$712$$ −7.46410 −0.279729
$$713$$ 0.464102 0.0173807
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ 16.2679 0.607962
$$717$$ 4.39230 0.164034
$$718$$ −18.9282 −0.706394
$$719$$ −34.6410 −1.29189 −0.645946 0.763383i $$-0.723537\pi$$
−0.645946 + 0.763383i $$0.723537\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 31.7128 1.18105
$$722$$ 18.7128 0.696419
$$723$$ −14.2679 −0.530631
$$724$$ −10.9282 −0.406143
$$725$$ 3.73205 0.138605
$$726$$ −10.7846 −0.400254
$$727$$ −23.7128 −0.879460 −0.439730 0.898130i $$-0.644926\pi$$
−0.439730 + 0.898130i $$0.644926\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000 0.0740233
$$731$$ 7.71281 0.285269
$$732$$ −10.3923 −0.384111
$$733$$ 37.0718 1.36928 0.684639 0.728882i $$-0.259960\pi$$
0.684639 + 0.728882i $$0.259960\pi$$
$$734$$ −36.3923 −1.34326
$$735$$ −3.00000 −0.110657
$$736$$ 0.267949 0.00987674
$$737$$ −2.10512 −0.0775430
$$738$$ −2.00000 −0.0736210
$$739$$ −15.3205 −0.563574 −0.281787 0.959477i $$-0.590927\pi$$
−0.281787 + 0.959477i $$0.590927\pi$$
$$740$$ −1.19615 −0.0439714
$$741$$ 0 0
$$742$$ −25.8564 −0.949219
$$743$$ −33.5359 −1.23031 −0.615156 0.788405i $$-0.710907\pi$$
−0.615156 + 0.788405i $$0.710907\pi$$
$$744$$ −1.73205 −0.0635001
$$745$$ −20.4641 −0.749747
$$746$$ −25.7846 −0.944042
$$747$$ 4.92820 0.180314
$$748$$ 1.85641 0.0678769
$$749$$ −39.7128 −1.45107
$$750$$ −1.00000 −0.0365148
$$751$$ −27.9282 −1.01911 −0.509557 0.860437i $$-0.670191\pi$$
−0.509557 + 0.860437i $$0.670191\pi$$
$$752$$ −10.4641 −0.381587
$$753$$ 12.2679 0.447069
$$754$$ 0 0
$$755$$ −10.3923 −0.378215
$$756$$ 2.00000 0.0727393
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ −0.143594 −0.00521555
$$759$$ 0.124356 0.00451382
$$760$$ −0.535898 −0.0194391
$$761$$ 18.9282 0.686147 0.343073 0.939309i $$-0.388532\pi$$
0.343073 + 0.939309i $$0.388532\pi$$
$$762$$ −8.92820 −0.323435
$$763$$ 23.7128 0.858461
$$764$$ 14.5359 0.525890
$$765$$ −4.00000 −0.144620
$$766$$ 4.60770 0.166483
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 19.5885 0.706378 0.353189 0.935552i $$-0.385097\pi$$
0.353189 + 0.935552i $$0.385097\pi$$
$$770$$ −0.928203 −0.0334501
$$771$$ −22.6603 −0.816089
$$772$$ 23.3205 0.839323
$$773$$ 27.7128 0.996761 0.498380 0.866959i $$-0.333928\pi$$
0.498380 + 0.866959i $$0.333928\pi$$
$$774$$ −1.92820 −0.0693078
$$775$$ −1.73205 −0.0622171
$$776$$ 7.46410 0.267946
$$777$$ 2.39230 0.0858235
$$778$$ −20.2679 −0.726641
$$779$$ −1.07180 −0.0384011
$$780$$ 0 0
$$781$$ −3.89488 −0.139370
$$782$$ 1.07180 0.0383274
$$783$$ −3.73205 −0.133373
$$784$$ −3.00000 −0.107143
$$785$$ 5.00000 0.178458
$$786$$ −1.33975 −0.0477872
$$787$$ −1.73205 −0.0617409 −0.0308705 0.999523i $$-0.509828\pi$$
−0.0308705 + 0.999523i $$0.509828\pi$$
$$788$$ −16.3923 −0.583952
$$789$$ 18.1244 0.645244
$$790$$ −0.0717968 −0.00255441
$$791$$ −22.3923 −0.796179
$$792$$ −0.464102 −0.0164911
$$793$$ 0 0
$$794$$ 12.1244 0.430277
$$795$$ −12.9282 −0.458516
$$796$$ 18.9282 0.670892
$$797$$ 37.8564 1.34094 0.670471 0.741935i $$-0.266092\pi$$
0.670471 + 0.741935i $$0.266092\pi$$
$$798$$ 1.07180 0.0379412
$$799$$ −41.8564 −1.48077
$$800$$ −1.00000 −0.0353553
$$801$$ 7.46410 0.263731
$$802$$ −32.0000 −1.12996
$$803$$ 0.928203 0.0327556
$$804$$ 4.53590 0.159969
$$805$$ −0.535898 −0.0188879
$$806$$ 0 0
$$807$$ −12.0000 −0.422420
$$808$$ −10.9282 −0.384453
$$809$$ −1.21539 −0.0427308 −0.0213654 0.999772i $$-0.506801\pi$$
−0.0213654 + 0.999772i $$0.506801\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 0.784610 0.0275514 0.0137757 0.999905i $$-0.495615\pi$$
0.0137757 + 0.999905i $$0.495615\pi$$
$$812$$ −7.46410 −0.261939
$$813$$ −9.19615 −0.322523
$$814$$ −0.555136 −0.0194575
$$815$$ 23.0526 0.807496
$$816$$ −4.00000 −0.140028
$$817$$ −1.03332 −0.0361513
$$818$$ 4.00000 0.139857
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ 35.3923 1.23520 0.617600 0.786492i $$-0.288105\pi$$
0.617600 + 0.786492i $$0.288105\pi$$
$$822$$ 4.46410 0.155703
$$823$$ 23.1769 0.807896 0.403948 0.914782i $$-0.367638\pi$$
0.403948 + 0.914782i $$0.367638\pi$$
$$824$$ 15.8564 0.552384
$$825$$ −0.464102 −0.0161579
$$826$$ 3.07180 0.106881
$$827$$ −17.3205 −0.602293 −0.301147 0.953578i $$-0.597369\pi$$
−0.301147 + 0.953578i $$0.597369\pi$$
$$828$$ −0.267949 −0.00931188
$$829$$ −23.4641 −0.814942 −0.407471 0.913218i $$-0.633589\pi$$
−0.407471 + 0.913218i $$0.633589\pi$$
$$830$$ 4.92820 0.171060
$$831$$ 9.92820 0.344406
$$832$$ 0 0
$$833$$ −12.0000 −0.415775
$$834$$ 0.928203 0.0321410
$$835$$ −18.3205 −0.634007
$$836$$ −0.248711 −0.00860186
$$837$$ 1.73205 0.0598684
$$838$$ −1.60770 −0.0555369
$$839$$ 39.5692 1.36608 0.683041 0.730380i $$-0.260657\pi$$
0.683041 + 0.730380i $$0.260657\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ −15.0718 −0.519717
$$842$$ −16.3923 −0.564916
$$843$$ −4.92820 −0.169736
$$844$$ 23.3205 0.802725
$$845$$ 0 0
$$846$$ 10.4641 0.359763
$$847$$ 21.5692 0.741127
$$848$$ −12.9282 −0.443956
$$849$$ 3.92820 0.134816
$$850$$ −4.00000 −0.137199
$$851$$ −0.320508 −0.0109869
$$852$$ 8.39230 0.287516
$$853$$ −35.8372 −1.22704 −0.613521 0.789679i $$-0.710247\pi$$
−0.613521 + 0.789679i $$0.710247\pi$$
$$854$$ 20.7846 0.711235
$$855$$ 0.535898 0.0183273
$$856$$ −19.8564 −0.678678
$$857$$ 20.5167 0.700836 0.350418 0.936593i $$-0.386040\pi$$
0.350418 + 0.936593i $$0.386040\pi$$
$$858$$ 0 0
$$859$$ 27.1769 0.927264 0.463632 0.886028i $$-0.346546\pi$$
0.463632 + 0.886028i $$0.346546\pi$$
$$860$$ −1.92820 −0.0657512
$$861$$ 4.00000 0.136320
$$862$$ −7.60770 −0.259119
$$863$$ −42.4641 −1.44549 −0.722747 0.691112i $$-0.757121\pi$$
−0.722747 + 0.691112i $$0.757121\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −2.92820 −0.0995619
$$866$$ −15.3205 −0.520612
$$867$$ 1.00000 0.0339618
$$868$$ 3.46410 0.117579
$$869$$ −0.0333210 −0.00113034
$$870$$ −3.73205 −0.126528
$$871$$ 0 0
$$872$$ 11.8564 0.401509
$$873$$ −7.46410 −0.252622
$$874$$ −0.143594 −0.00485712
$$875$$ 2.00000 0.0676123
$$876$$ −2.00000 −0.0675737
$$877$$ 28.1244 0.949692 0.474846 0.880069i $$-0.342504\pi$$
0.474846 + 0.880069i $$0.342504\pi$$
$$878$$ 9.85641 0.332638
$$879$$ 4.14359 0.139760
$$880$$ −0.464102 −0.0156449
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 3.00000 0.101015
$$883$$ −16.0718 −0.540859 −0.270430 0.962740i $$-0.587166\pi$$
−0.270430 + 0.962740i $$0.587166\pi$$
$$884$$ 0 0
$$885$$ 1.53590 0.0516287
$$886$$ 4.39230 0.147562
$$887$$ −10.1244 −0.339943 −0.169971 0.985449i $$-0.554368\pi$$
−0.169971 + 0.985449i $$0.554368\pi$$
$$888$$ 1.19615 0.0401402
$$889$$ 17.8564 0.598885
$$890$$ 7.46410 0.250197
$$891$$ 0.464102 0.0155480
$$892$$ −27.4641 −0.919566
$$893$$ 5.60770 0.187654
$$894$$ 20.4641 0.684422
$$895$$ −16.2679 −0.543778
$$896$$ 2.00000 0.0668153
$$897$$ 0 0
$$898$$ −39.7128 −1.32523
$$899$$ −6.46410 −0.215590
$$900$$ 1.00000 0.0333333
$$901$$ −51.7128 −1.72280
$$902$$ −0.928203 −0.0309058
$$903$$ 3.85641 0.128333
$$904$$ −11.1962 −0.372378
$$905$$ 10.9282 0.363266
$$906$$ 10.3923 0.345261
$$907$$ −6.85641 −0.227663 −0.113832 0.993500i $$-0.536312\pi$$
−0.113832 + 0.993500i $$0.536312\pi$$
$$908$$ −4.39230 −0.145764
$$909$$ 10.9282 0.362466
$$910$$ 0 0
$$911$$ −24.2487 −0.803396 −0.401698 0.915772i $$-0.631580\pi$$
−0.401698 + 0.915772i $$0.631580\pi$$
$$912$$ 0.535898 0.0177454
$$913$$ 2.28719 0.0756948
$$914$$ −31.4641 −1.04074
$$915$$ 10.3923 0.343559
$$916$$ 19.8564 0.656074
$$917$$ 2.67949 0.0884846
$$918$$ 4.00000 0.132020
$$919$$ −54.6410 −1.80244 −0.901220 0.433361i $$-0.857328\pi$$
−0.901220 + 0.433361i $$0.857328\pi$$
$$920$$ −0.267949 −0.00883402
$$921$$ −12.5359 −0.413072
$$922$$ −6.46410 −0.212884
$$923$$ 0 0
$$924$$ 0.928203 0.0305356
$$925$$ 1.19615 0.0393292
$$926$$ −20.9282 −0.687743
$$927$$ −15.8564 −0.520793
$$928$$ −3.73205 −0.122511
$$929$$ 33.8564 1.11079 0.555396 0.831586i $$-0.312567\pi$$
0.555396 + 0.831586i $$0.312567\pi$$
$$930$$ 1.73205 0.0567962
$$931$$ 1.60770 0.0526901
$$932$$ 18.1244 0.593683
$$933$$ −7.60770 −0.249065
$$934$$ 11.8564 0.387953
$$935$$ −1.85641 −0.0607110
$$936$$ 0 0
$$937$$ 44.6410 1.45836 0.729179 0.684323i $$-0.239902\pi$$
0.729179 + 0.684323i $$0.239902\pi$$
$$938$$ −9.07180 −0.296205
$$939$$ 28.0000 0.913745
$$940$$ 10.4641 0.341301
$$941$$ −26.7846 −0.873153 −0.436577 0.899667i $$-0.643809\pi$$
−0.436577 + 0.899667i $$0.643809\pi$$
$$942$$ −5.00000 −0.162909
$$943$$ −0.535898 −0.0174513
$$944$$ 1.53590 0.0499892
$$945$$ −2.00000 −0.0650600
$$946$$ −0.894882 −0.0290951
$$947$$ −2.53590 −0.0824056 −0.0412028 0.999151i $$-0.513119\pi$$
−0.0412028 + 0.999151i $$0.513119\pi$$
$$948$$ 0.0717968 0.00233185
$$949$$ 0 0
$$950$$ 0.535898 0.0173868
$$951$$ 21.4641 0.696021
$$952$$ 8.00000 0.259281
$$953$$ 15.7321 0.509611 0.254806 0.966992i $$-0.417989\pi$$
0.254806 + 0.966992i $$0.417989\pi$$
$$954$$ 12.9282 0.418566
$$955$$ −14.5359 −0.470371
$$956$$ −4.39230 −0.142057
$$957$$ −1.73205 −0.0559893
$$958$$ −1.46410 −0.0473030
$$959$$ −8.92820 −0.288307
$$960$$ 1.00000 0.0322749
$$961$$ −28.0000 −0.903226
$$962$$ 0 0
$$963$$ 19.8564 0.639864
$$964$$ 14.2679 0.459540
$$965$$ −23.3205 −0.750714
$$966$$ 0.535898 0.0172422
$$967$$ 34.5359 1.11060 0.555300 0.831650i $$-0.312604\pi$$
0.555300 + 0.831650i $$0.312604\pi$$
$$968$$ 10.7846 0.346630
$$969$$ 2.14359 0.0688621
$$970$$ −7.46410 −0.239658
$$971$$ 11.4641 0.367901 0.183950 0.982936i $$-0.441111\pi$$
0.183950 + 0.982936i $$0.441111\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −1.85641 −0.0595137
$$974$$ −39.1769 −1.25531
$$975$$ 0 0
$$976$$ 10.3923 0.332650
$$977$$ 33.3923 1.06831 0.534157 0.845385i $$-0.320629\pi$$
0.534157 + 0.845385i $$0.320629\pi$$
$$978$$ −23.0526 −0.737140
$$979$$ 3.46410 0.110713
$$980$$ 3.00000 0.0958315
$$981$$ −11.8564 −0.378546
$$982$$ 17.3205 0.552720
$$983$$ −57.3923 −1.83053 −0.915265 0.402852i $$-0.868019\pi$$
−0.915265 + 0.402852i $$0.868019\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 16.3923 0.522302
$$986$$ −14.9282 −0.475411
$$987$$ −20.9282 −0.666152
$$988$$ 0 0
$$989$$ −0.516660 −0.0164288
$$990$$ 0.464102 0.0147501
$$991$$ −25.1436 −0.798713 −0.399356 0.916796i $$-0.630766\pi$$
−0.399356 + 0.916796i $$0.630766\pi$$
$$992$$ 1.73205 0.0549927
$$993$$ 24.7846 0.786516
$$994$$ −16.7846 −0.532375
$$995$$ −18.9282 −0.600064
$$996$$ −4.92820 −0.156156
$$997$$ −47.5692 −1.50653 −0.753266 0.657716i $$-0.771523\pi$$
−0.753266 + 0.657716i $$0.771523\pi$$
$$998$$ 13.4641 0.426199
$$999$$ −1.19615 −0.0378446
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 5070.2.a.y.1.2 2
13.5 odd 4 5070.2.b.o.1351.4 4
13.6 odd 12 390.2.bb.b.361.2 yes 4
13.8 odd 4 5070.2.b.o.1351.1 4
13.11 odd 12 390.2.bb.b.121.2 4
13.12 even 2 5070.2.a.bg.1.1 2
39.11 even 12 1170.2.bs.e.901.1 4
39.32 even 12 1170.2.bs.e.361.1 4
65.19 odd 12 1950.2.bc.b.751.1 4
65.24 odd 12 1950.2.bc.b.901.1 4
65.32 even 12 1950.2.y.f.49.2 4
65.37 even 12 1950.2.y.c.199.1 4
65.58 even 12 1950.2.y.c.49.1 4
65.63 even 12 1950.2.y.f.199.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.b.121.2 4 13.11 odd 12
390.2.bb.b.361.2 yes 4 13.6 odd 12
1170.2.bs.e.361.1 4 39.32 even 12
1170.2.bs.e.901.1 4 39.11 even 12
1950.2.y.c.49.1 4 65.58 even 12
1950.2.y.c.199.1 4 65.37 even 12
1950.2.y.f.49.2 4 65.32 even 12
1950.2.y.f.199.2 4 65.63 even 12
1950.2.bc.b.751.1 4 65.19 odd 12
1950.2.bc.b.901.1 4 65.24 odd 12
5070.2.a.y.1.2 2 1.1 even 1 trivial
5070.2.a.bg.1.1 2 13.12 even 2
5070.2.b.o.1351.1 4 13.8 odd 4
5070.2.b.o.1351.4 4 13.5 odd 4