# Properties

 Label 1110.2.a.p.1.1 Level $1110$ Weight $2$ Character 1110.1 Self dual yes Analytic conductor $8.863$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 1110.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.37228 q^{11} -1.00000 q^{12} +1.37228 q^{13} +3.37228 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.37228 q^{17} -1.00000 q^{18} -1.37228 q^{19} +1.00000 q^{20} +3.37228 q^{21} +1.37228 q^{22} +3.37228 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.37228 q^{26} -1.00000 q^{27} -3.37228 q^{28} +6.00000 q^{29} +1.00000 q^{30} +2.74456 q^{31} -1.00000 q^{32} +1.37228 q^{33} +1.37228 q^{34} -3.37228 q^{35} +1.00000 q^{36} -1.00000 q^{37} +1.37228 q^{38} -1.37228 q^{39} -1.00000 q^{40} +8.74456 q^{41} -3.37228 q^{42} -4.00000 q^{43} -1.37228 q^{44} +1.00000 q^{45} -3.37228 q^{46} -4.74456 q^{47} -1.00000 q^{48} +4.37228 q^{49} -1.00000 q^{50} +1.37228 q^{51} +1.37228 q^{52} +5.37228 q^{53} +1.00000 q^{54} -1.37228 q^{55} +3.37228 q^{56} +1.37228 q^{57} -6.00000 q^{58} +14.7446 q^{59} -1.00000 q^{60} -2.74456 q^{61} -2.74456 q^{62} -3.37228 q^{63} +1.00000 q^{64} +1.37228 q^{65} -1.37228 q^{66} +2.74456 q^{67} -1.37228 q^{68} -3.37228 q^{69} +3.37228 q^{70} +1.25544 q^{71} -1.00000 q^{72} -4.11684 q^{73} +1.00000 q^{74} -1.00000 q^{75} -1.37228 q^{76} +4.62772 q^{77} +1.37228 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -8.74456 q^{82} +0.627719 q^{83} +3.37228 q^{84} -1.37228 q^{85} +4.00000 q^{86} -6.00000 q^{87} +1.37228 q^{88} +13.3723 q^{89} -1.00000 q^{90} -4.62772 q^{91} +3.37228 q^{92} -2.74456 q^{93} +4.74456 q^{94} -1.37228 q^{95} +1.00000 q^{96} +13.4891 q^{97} -4.37228 q^{98} -1.37228 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 3q^{11} - 2q^{12} - 3q^{13} + q^{14} - 2q^{15} + 2q^{16} + 3q^{17} - 2q^{18} + 3q^{19} + 2q^{20} + q^{21} - 3q^{22} + q^{23} + 2q^{24} + 2q^{25} + 3q^{26} - 2q^{27} - q^{28} + 12q^{29} + 2q^{30} - 6q^{31} - 2q^{32} - 3q^{33} - 3q^{34} - q^{35} + 2q^{36} - 2q^{37} - 3q^{38} + 3q^{39} - 2q^{40} + 6q^{41} - q^{42} - 8q^{43} + 3q^{44} + 2q^{45} - q^{46} + 2q^{47} - 2q^{48} + 3q^{49} - 2q^{50} - 3q^{51} - 3q^{52} + 5q^{53} + 2q^{54} + 3q^{55} + q^{56} - 3q^{57} - 12q^{58} + 18q^{59} - 2q^{60} + 6q^{61} + 6q^{62} - q^{63} + 2q^{64} - 3q^{65} + 3q^{66} - 6q^{67} + 3q^{68} - q^{69} + q^{70} + 14q^{71} - 2q^{72} + 9q^{73} + 2q^{74} - 2q^{75} + 3q^{76} + 15q^{77} - 3q^{78} + 8q^{79} + 2q^{80} + 2q^{81} - 6q^{82} + 7q^{83} + q^{84} + 3q^{85} + 8q^{86} - 12q^{87} - 3q^{88} + 21q^{89} - 2q^{90} - 15q^{91} + q^{92} + 6q^{93} - 2q^{94} + 3q^{95} + 2q^{96} + 4q^{97} - 3q^{98} + 3q^{99} + O(q^{100})$$

## 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
$$7$$ −3.37228 −1.27460 −0.637301 0.770615i $$-0.719949\pi$$
−0.637301 + 0.770615i $$0.719949\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −1.37228 −0.413758 −0.206879 0.978366i $$-0.566331\pi$$
−0.206879 + 0.978366i $$0.566331\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 1.37228 0.380602 0.190301 0.981726i $$-0.439054\pi$$
0.190301 + 0.981726i $$0.439054\pi$$
$$14$$ 3.37228 0.901280
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −1.37228 −0.332827 −0.166414 0.986056i $$-0.553219\pi$$
−0.166414 + 0.986056i $$0.553219\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −1.37228 −0.314823 −0.157411 0.987533i $$-0.550315\pi$$
−0.157411 + 0.987533i $$0.550315\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 3.37228 0.735892
$$22$$ 1.37228 0.292571
$$23$$ 3.37228 0.703169 0.351585 0.936156i $$-0.385643\pi$$
0.351585 + 0.936156i $$0.385643\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ −1.37228 −0.269127
$$27$$ −1.00000 −0.192450
$$28$$ −3.37228 −0.637301
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 2.74456 0.492938 0.246469 0.969151i $$-0.420730\pi$$
0.246469 + 0.969151i $$0.420730\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 1.37228 0.238884
$$34$$ 1.37228 0.235344
$$35$$ −3.37228 −0.570020
$$36$$ 1.00000 0.166667
$$37$$ −1.00000 −0.164399
$$38$$ 1.37228 0.222613
$$39$$ −1.37228 −0.219741
$$40$$ −1.00000 −0.158114
$$41$$ 8.74456 1.36567 0.682836 0.730572i $$-0.260747\pi$$
0.682836 + 0.730572i $$0.260747\pi$$
$$42$$ −3.37228 −0.520354
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −1.37228 −0.206879
$$45$$ 1.00000 0.149071
$$46$$ −3.37228 −0.497216
$$47$$ −4.74456 −0.692066 −0.346033 0.938222i $$-0.612471\pi$$
−0.346033 + 0.938222i $$0.612471\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 4.37228 0.624612
$$50$$ −1.00000 −0.141421
$$51$$ 1.37228 0.192158
$$52$$ 1.37228 0.190301
$$53$$ 5.37228 0.737940 0.368970 0.929441i $$-0.379711\pi$$
0.368970 + 0.929441i $$0.379711\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −1.37228 −0.185038
$$56$$ 3.37228 0.450640
$$57$$ 1.37228 0.181763
$$58$$ −6.00000 −0.787839
$$59$$ 14.7446 1.91958 0.959789 0.280721i $$-0.0905737\pi$$
0.959789 + 0.280721i $$0.0905737\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −2.74456 −0.351405 −0.175703 0.984443i $$-0.556220\pi$$
−0.175703 + 0.984443i $$0.556220\pi$$
$$62$$ −2.74456 −0.348560
$$63$$ −3.37228 −0.424868
$$64$$ 1.00000 0.125000
$$65$$ 1.37228 0.170211
$$66$$ −1.37228 −0.168916
$$67$$ 2.74456 0.335302 0.167651 0.985846i $$-0.446382\pi$$
0.167651 + 0.985846i $$0.446382\pi$$
$$68$$ −1.37228 −0.166414
$$69$$ −3.37228 −0.405975
$$70$$ 3.37228 0.403065
$$71$$ 1.25544 0.148993 0.0744965 0.997221i $$-0.476265\pi$$
0.0744965 + 0.997221i $$0.476265\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −4.11684 −0.481840 −0.240920 0.970545i $$-0.577449\pi$$
−0.240920 + 0.970545i $$0.577449\pi$$
$$74$$ 1.00000 0.116248
$$75$$ −1.00000 −0.115470
$$76$$ −1.37228 −0.157411
$$77$$ 4.62772 0.527377
$$78$$ 1.37228 0.155380
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ −8.74456 −0.965675
$$83$$ 0.627719 0.0689011 0.0344505 0.999406i $$-0.489032\pi$$
0.0344505 + 0.999406i $$0.489032\pi$$
$$84$$ 3.37228 0.367946
$$85$$ −1.37228 −0.148845
$$86$$ 4.00000 0.431331
$$87$$ −6.00000 −0.643268
$$88$$ 1.37228 0.146286
$$89$$ 13.3723 1.41746 0.708729 0.705480i $$-0.249269\pi$$
0.708729 + 0.705480i $$0.249269\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ −4.62772 −0.485117
$$92$$ 3.37228 0.351585
$$93$$ −2.74456 −0.284598
$$94$$ 4.74456 0.489364
$$95$$ −1.37228 −0.140793
$$96$$ 1.00000 0.102062
$$97$$ 13.4891 1.36961 0.684807 0.728725i $$-0.259887\pi$$
0.684807 + 0.728725i $$0.259887\pi$$
$$98$$ −4.37228 −0.441667
$$99$$ −1.37228 −0.137919
$$100$$ 1.00000 0.100000
$$101$$ 10.7446 1.06912 0.534562 0.845129i $$-0.320477\pi$$
0.534562 + 0.845129i $$0.320477\pi$$
$$102$$ −1.37228 −0.135876
$$103$$ −0.744563 −0.0733639 −0.0366820 0.999327i $$-0.511679\pi$$
−0.0366820 + 0.999327i $$0.511679\pi$$
$$104$$ −1.37228 −0.134563
$$105$$ 3.37228 0.329101
$$106$$ −5.37228 −0.521802
$$107$$ 3.37228 0.326011 0.163005 0.986625i $$-0.447881\pi$$
0.163005 + 0.986625i $$0.447881\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 4.62772 0.443255 0.221628 0.975131i $$-0.428863\pi$$
0.221628 + 0.975131i $$0.428863\pi$$
$$110$$ 1.37228 0.130842
$$111$$ 1.00000 0.0949158
$$112$$ −3.37228 −0.318651
$$113$$ 19.4891 1.83338 0.916691 0.399596i $$-0.130850\pi$$
0.916691 + 0.399596i $$0.130850\pi$$
$$114$$ −1.37228 −0.128526
$$115$$ 3.37228 0.314467
$$116$$ 6.00000 0.557086
$$117$$ 1.37228 0.126867
$$118$$ −14.7446 −1.35735
$$119$$ 4.62772 0.424222
$$120$$ 1.00000 0.0912871
$$121$$ −9.11684 −0.828804
$$122$$ 2.74456 0.248481
$$123$$ −8.74456 −0.788471
$$124$$ 2.74456 0.246469
$$125$$ 1.00000 0.0894427
$$126$$ 3.37228 0.300427
$$127$$ 3.37228 0.299242 0.149621 0.988743i $$-0.452195\pi$$
0.149621 + 0.988743i $$0.452195\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ −1.37228 −0.120357
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 1.37228 0.119442
$$133$$ 4.62772 0.401274
$$134$$ −2.74456 −0.237094
$$135$$ −1.00000 −0.0860663
$$136$$ 1.37228 0.117672
$$137$$ 10.7446 0.917970 0.458985 0.888444i $$-0.348213\pi$$
0.458985 + 0.888444i $$0.348213\pi$$
$$138$$ 3.37228 0.287068
$$139$$ 2.74456 0.232791 0.116395 0.993203i $$-0.462866\pi$$
0.116395 + 0.993203i $$0.462866\pi$$
$$140$$ −3.37228 −0.285010
$$141$$ 4.74456 0.399564
$$142$$ −1.25544 −0.105354
$$143$$ −1.88316 −0.157477
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ 4.11684 0.340712
$$147$$ −4.37228 −0.360620
$$148$$ −1.00000 −0.0821995
$$149$$ 9.25544 0.758235 0.379117 0.925349i $$-0.376228\pi$$
0.379117 + 0.925349i $$0.376228\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −3.37228 −0.274432 −0.137216 0.990541i $$-0.543816\pi$$
−0.137216 + 0.990541i $$0.543816\pi$$
$$152$$ 1.37228 0.111307
$$153$$ −1.37228 −0.110942
$$154$$ −4.62772 −0.372912
$$155$$ 2.74456 0.220449
$$156$$ −1.37228 −0.109870
$$157$$ 3.25544 0.259812 0.129906 0.991526i $$-0.458532\pi$$
0.129906 + 0.991526i $$0.458532\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ −5.37228 −0.426050
$$160$$ −1.00000 −0.0790569
$$161$$ −11.3723 −0.896261
$$162$$ −1.00000 −0.0785674
$$163$$ 4.86141 0.380775 0.190387 0.981709i $$-0.439026\pi$$
0.190387 + 0.981709i $$0.439026\pi$$
$$164$$ 8.74456 0.682836
$$165$$ 1.37228 0.106832
$$166$$ −0.627719 −0.0487204
$$167$$ −1.88316 −0.145723 −0.0728615 0.997342i $$-0.523213\pi$$
−0.0728615 + 0.997342i $$0.523213\pi$$
$$168$$ −3.37228 −0.260177
$$169$$ −11.1168 −0.855142
$$170$$ 1.37228 0.105249
$$171$$ −1.37228 −0.104941
$$172$$ −4.00000 −0.304997
$$173$$ −6.86141 −0.521663 −0.260832 0.965384i $$-0.583997\pi$$
−0.260832 + 0.965384i $$0.583997\pi$$
$$174$$ 6.00000 0.454859
$$175$$ −3.37228 −0.254921
$$176$$ −1.37228 −0.103440
$$177$$ −14.7446 −1.10827
$$178$$ −13.3723 −1.00229
$$179$$ 5.48913 0.410276 0.205138 0.978733i $$-0.434236\pi$$
0.205138 + 0.978733i $$0.434236\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −20.9783 −1.55930 −0.779651 0.626215i $$-0.784603\pi$$
−0.779651 + 0.626215i $$0.784603\pi$$
$$182$$ 4.62772 0.343029
$$183$$ 2.74456 0.202884
$$184$$ −3.37228 −0.248608
$$185$$ −1.00000 −0.0735215
$$186$$ 2.74456 0.201241
$$187$$ 1.88316 0.137710
$$188$$ −4.74456 −0.346033
$$189$$ 3.37228 0.245297
$$190$$ 1.37228 0.0995558
$$191$$ −19.3723 −1.40173 −0.700865 0.713294i $$-0.747202\pi$$
−0.700865 + 0.713294i $$0.747202\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −1.25544 −0.0903684 −0.0451842 0.998979i $$-0.514387\pi$$
−0.0451842 + 0.998979i $$0.514387\pi$$
$$194$$ −13.4891 −0.968463
$$195$$ −1.37228 −0.0982711
$$196$$ 4.37228 0.312306
$$197$$ −18.8614 −1.34382 −0.671910 0.740633i $$-0.734526\pi$$
−0.671910 + 0.740633i $$0.734526\pi$$
$$198$$ 1.37228 0.0975238
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −2.74456 −0.193587
$$202$$ −10.7446 −0.755985
$$203$$ −20.2337 −1.42013
$$204$$ 1.37228 0.0960789
$$205$$ 8.74456 0.610747
$$206$$ 0.744563 0.0518761
$$207$$ 3.37228 0.234390
$$208$$ 1.37228 0.0951506
$$209$$ 1.88316 0.130261
$$210$$ −3.37228 −0.232710
$$211$$ −9.25544 −0.637171 −0.318585 0.947894i $$-0.603208\pi$$
−0.318585 + 0.947894i $$0.603208\pi$$
$$212$$ 5.37228 0.368970
$$213$$ −1.25544 −0.0860212
$$214$$ −3.37228 −0.230524
$$215$$ −4.00000 −0.272798
$$216$$ 1.00000 0.0680414
$$217$$ −9.25544 −0.628300
$$218$$ −4.62772 −0.313429
$$219$$ 4.11684 0.278191
$$220$$ −1.37228 −0.0925192
$$221$$ −1.88316 −0.126675
$$222$$ −1.00000 −0.0671156
$$223$$ 18.9783 1.27088 0.635439 0.772151i $$-0.280819\pi$$
0.635439 + 0.772151i $$0.280819\pi$$
$$224$$ 3.37228 0.225320
$$225$$ 1.00000 0.0666667
$$226$$ −19.4891 −1.29640
$$227$$ 4.00000 0.265489 0.132745 0.991150i $$-0.457621\pi$$
0.132745 + 0.991150i $$0.457621\pi$$
$$228$$ 1.37228 0.0908816
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −3.37228 −0.222362
$$231$$ −4.62772 −0.304482
$$232$$ −6.00000 −0.393919
$$233$$ −22.7446 −1.49005 −0.745023 0.667039i $$-0.767561\pi$$
−0.745023 + 0.667039i $$0.767561\pi$$
$$234$$ −1.37228 −0.0897088
$$235$$ −4.74456 −0.309501
$$236$$ 14.7446 0.959789
$$237$$ −4.00000 −0.259828
$$238$$ −4.62772 −0.299970
$$239$$ −5.48913 −0.355062 −0.177531 0.984115i $$-0.556811\pi$$
−0.177531 + 0.984115i $$0.556811\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ −0.510875 −0.0329083 −0.0164542 0.999865i $$-0.505238\pi$$
−0.0164542 + 0.999865i $$0.505238\pi$$
$$242$$ 9.11684 0.586053
$$243$$ −1.00000 −0.0641500
$$244$$ −2.74456 −0.175703
$$245$$ 4.37228 0.279335
$$246$$ 8.74456 0.557533
$$247$$ −1.88316 −0.119822
$$248$$ −2.74456 −0.174280
$$249$$ −0.627719 −0.0397801
$$250$$ −1.00000 −0.0632456
$$251$$ −26.7446 −1.68810 −0.844051 0.536263i $$-0.819836\pi$$
−0.844051 + 0.536263i $$0.819836\pi$$
$$252$$ −3.37228 −0.212434
$$253$$ −4.62772 −0.290942
$$254$$ −3.37228 −0.211596
$$255$$ 1.37228 0.0859356
$$256$$ 1.00000 0.0625000
$$257$$ −5.37228 −0.335114 −0.167557 0.985862i $$-0.553588\pi$$
−0.167557 + 0.985862i $$0.553588\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 3.37228 0.209543
$$260$$ 1.37228 0.0851053
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 10.2337 0.631036 0.315518 0.948920i $$-0.397822\pi$$
0.315518 + 0.948920i $$0.397822\pi$$
$$264$$ −1.37228 −0.0844581
$$265$$ 5.37228 0.330017
$$266$$ −4.62772 −0.283744
$$267$$ −13.3723 −0.818370
$$268$$ 2.74456 0.167651
$$269$$ 0.627719 0.0382727 0.0191363 0.999817i $$-0.493908\pi$$
0.0191363 + 0.999817i $$0.493908\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 13.4891 0.819406 0.409703 0.912219i $$-0.365632\pi$$
0.409703 + 0.912219i $$0.365632\pi$$
$$272$$ −1.37228 −0.0832068
$$273$$ 4.62772 0.280082
$$274$$ −10.7446 −0.649103
$$275$$ −1.37228 −0.0827517
$$276$$ −3.37228 −0.202987
$$277$$ 25.6060 1.53851 0.769257 0.638940i $$-0.220627\pi$$
0.769257 + 0.638940i $$0.220627\pi$$
$$278$$ −2.74456 −0.164608
$$279$$ 2.74456 0.164313
$$280$$ 3.37228 0.201532
$$281$$ 1.37228 0.0818634 0.0409317 0.999162i $$-0.486967\pi$$
0.0409317 + 0.999162i $$0.486967\pi$$
$$282$$ −4.74456 −0.282535
$$283$$ −11.6060 −0.689903 −0.344952 0.938620i $$-0.612105\pi$$
−0.344952 + 0.938620i $$0.612105\pi$$
$$284$$ 1.25544 0.0744965
$$285$$ 1.37228 0.0812869
$$286$$ 1.88316 0.111353
$$287$$ −29.4891 −1.74069
$$288$$ −1.00000 −0.0589256
$$289$$ −15.1168 −0.889226
$$290$$ −6.00000 −0.352332
$$291$$ −13.4891 −0.790747
$$292$$ −4.11684 −0.240920
$$293$$ −4.11684 −0.240509 −0.120254 0.992743i $$-0.538371\pi$$
−0.120254 + 0.992743i $$0.538371\pi$$
$$294$$ 4.37228 0.254997
$$295$$ 14.7446 0.858462
$$296$$ 1.00000 0.0581238
$$297$$ 1.37228 0.0796278
$$298$$ −9.25544 −0.536153
$$299$$ 4.62772 0.267628
$$300$$ −1.00000 −0.0577350
$$301$$ 13.4891 0.777500
$$302$$ 3.37228 0.194053
$$303$$ −10.7446 −0.617259
$$304$$ −1.37228 −0.0787057
$$305$$ −2.74456 −0.157153
$$306$$ 1.37228 0.0784481
$$307$$ −10.7446 −0.613225 −0.306612 0.951834i $$-0.599195\pi$$
−0.306612 + 0.951834i $$0.599195\pi$$
$$308$$ 4.62772 0.263689
$$309$$ 0.744563 0.0423567
$$310$$ −2.74456 −0.155881
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ 1.37228 0.0776901
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ −3.25544 −0.183715
$$315$$ −3.37228 −0.190007
$$316$$ 4.00000 0.225018
$$317$$ −16.9783 −0.953594 −0.476797 0.879014i $$-0.658202\pi$$
−0.476797 + 0.879014i $$0.658202\pi$$
$$318$$ 5.37228 0.301263
$$319$$ −8.23369 −0.460998
$$320$$ 1.00000 0.0559017
$$321$$ −3.37228 −0.188222
$$322$$ 11.3723 0.633752
$$323$$ 1.88316 0.104782
$$324$$ 1.00000 0.0555556
$$325$$ 1.37228 0.0761205
$$326$$ −4.86141 −0.269248
$$327$$ −4.62772 −0.255913
$$328$$ −8.74456 −0.482838
$$329$$ 16.0000 0.882109
$$330$$ −1.37228 −0.0755416
$$331$$ −8.74456 −0.480645 −0.240322 0.970693i $$-0.577253\pi$$
−0.240322 + 0.970693i $$0.577253\pi$$
$$332$$ 0.627719 0.0344505
$$333$$ −1.00000 −0.0547997
$$334$$ 1.88316 0.103042
$$335$$ 2.74456 0.149951
$$336$$ 3.37228 0.183973
$$337$$ −4.11684 −0.224259 −0.112129 0.993694i $$-0.535767\pi$$
−0.112129 + 0.993694i $$0.535767\pi$$
$$338$$ 11.1168 0.604677
$$339$$ −19.4891 −1.05850
$$340$$ −1.37228 −0.0744224
$$341$$ −3.76631 −0.203957
$$342$$ 1.37228 0.0742045
$$343$$ 8.86141 0.478471
$$344$$ 4.00000 0.215666
$$345$$ −3.37228 −0.181558
$$346$$ 6.86141 0.368872
$$347$$ 1.48913 0.0799404 0.0399702 0.999201i $$-0.487274\pi$$
0.0399702 + 0.999201i $$0.487274\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 16.7446 0.896316 0.448158 0.893954i $$-0.352080\pi$$
0.448158 + 0.893954i $$0.352080\pi$$
$$350$$ 3.37228 0.180256
$$351$$ −1.37228 −0.0732470
$$352$$ 1.37228 0.0731428
$$353$$ 3.48913 0.185707 0.0928537 0.995680i $$-0.470401\pi$$
0.0928537 + 0.995680i $$0.470401\pi$$
$$354$$ 14.7446 0.783665
$$355$$ 1.25544 0.0666317
$$356$$ 13.3723 0.708729
$$357$$ −4.62772 −0.244925
$$358$$ −5.48913 −0.290109
$$359$$ 32.4674 1.71356 0.856781 0.515680i $$-0.172461\pi$$
0.856781 + 0.515680i $$0.172461\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ −17.1168 −0.900887
$$362$$ 20.9783 1.10259
$$363$$ 9.11684 0.478510
$$364$$ −4.62772 −0.242558
$$365$$ −4.11684 −0.215485
$$366$$ −2.74456 −0.143461
$$367$$ 20.6277 1.07676 0.538379 0.842703i $$-0.319037\pi$$
0.538379 + 0.842703i $$0.319037\pi$$
$$368$$ 3.37228 0.175792
$$369$$ 8.74456 0.455224
$$370$$ 1.00000 0.0519875
$$371$$ −18.1168 −0.940580
$$372$$ −2.74456 −0.142299
$$373$$ 3.48913 0.180660 0.0903300 0.995912i $$-0.471208\pi$$
0.0903300 + 0.995912i $$0.471208\pi$$
$$374$$ −1.88316 −0.0973757
$$375$$ −1.00000 −0.0516398
$$376$$ 4.74456 0.244682
$$377$$ 8.23369 0.424057
$$378$$ −3.37228 −0.173451
$$379$$ 24.2337 1.24480 0.622400 0.782699i $$-0.286158\pi$$
0.622400 + 0.782699i $$0.286158\pi$$
$$380$$ −1.37228 −0.0703965
$$381$$ −3.37228 −0.172767
$$382$$ 19.3723 0.991172
$$383$$ 27.6060 1.41060 0.705300 0.708909i $$-0.250812\pi$$
0.705300 + 0.708909i $$0.250812\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 4.62772 0.235850
$$386$$ 1.25544 0.0639001
$$387$$ −4.00000 −0.203331
$$388$$ 13.4891 0.684807
$$389$$ 23.4891 1.19095 0.595473 0.803375i $$-0.296965\pi$$
0.595473 + 0.803375i $$0.296965\pi$$
$$390$$ 1.37228 0.0694882
$$391$$ −4.62772 −0.234034
$$392$$ −4.37228 −0.220834
$$393$$ 0 0
$$394$$ 18.8614 0.950224
$$395$$ 4.00000 0.201262
$$396$$ −1.37228 −0.0689597
$$397$$ 20.7446 1.04114 0.520570 0.853819i $$-0.325720\pi$$
0.520570 + 0.853819i $$0.325720\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ −4.62772 −0.231676
$$400$$ 1.00000 0.0500000
$$401$$ −36.1168 −1.80359 −0.901795 0.432165i $$-0.857750\pi$$
−0.901795 + 0.432165i $$0.857750\pi$$
$$402$$ 2.74456 0.136886
$$403$$ 3.76631 0.187613
$$404$$ 10.7446 0.534562
$$405$$ 1.00000 0.0496904
$$406$$ 20.2337 1.00418
$$407$$ 1.37228 0.0680215
$$408$$ −1.37228 −0.0679380
$$409$$ −8.74456 −0.432391 −0.216195 0.976350i $$-0.569365\pi$$
−0.216195 + 0.976350i $$0.569365\pi$$
$$410$$ −8.74456 −0.431863
$$411$$ −10.7446 −0.529990
$$412$$ −0.744563 −0.0366820
$$413$$ −49.7228 −2.44670
$$414$$ −3.37228 −0.165739
$$415$$ 0.627719 0.0308135
$$416$$ −1.37228 −0.0672816
$$417$$ −2.74456 −0.134402
$$418$$ −1.88316 −0.0921082
$$419$$ 7.88316 0.385117 0.192559 0.981285i $$-0.438321\pi$$
0.192559 + 0.981285i $$0.438321\pi$$
$$420$$ 3.37228 0.164550
$$421$$ 24.2337 1.18108 0.590539 0.807009i $$-0.298915\pi$$
0.590539 + 0.807009i $$0.298915\pi$$
$$422$$ 9.25544 0.450548
$$423$$ −4.74456 −0.230689
$$424$$ −5.37228 −0.260901
$$425$$ −1.37228 −0.0665654
$$426$$ 1.25544 0.0608261
$$427$$ 9.25544 0.447902
$$428$$ 3.37228 0.163005
$$429$$ 1.88316 0.0909196
$$430$$ 4.00000 0.192897
$$431$$ −31.6060 −1.52241 −0.761203 0.648514i $$-0.775391\pi$$
−0.761203 + 0.648514i $$0.775391\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −10.8614 −0.521966 −0.260983 0.965343i $$-0.584047\pi$$
−0.260983 + 0.965343i $$0.584047\pi$$
$$434$$ 9.25544 0.444275
$$435$$ −6.00000 −0.287678
$$436$$ 4.62772 0.221628
$$437$$ −4.62772 −0.221374
$$438$$ −4.11684 −0.196710
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 1.37228 0.0654209
$$441$$ 4.37228 0.208204
$$442$$ 1.88316 0.0895726
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 13.3723 0.633907
$$446$$ −18.9783 −0.898646
$$447$$ −9.25544 −0.437767
$$448$$ −3.37228 −0.159325
$$449$$ 36.9783 1.74511 0.872556 0.488515i $$-0.162461\pi$$
0.872556 + 0.488515i $$0.162461\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −12.0000 −0.565058
$$452$$ 19.4891 0.916691
$$453$$ 3.37228 0.158444
$$454$$ −4.00000 −0.187729
$$455$$ −4.62772 −0.216951
$$456$$ −1.37228 −0.0642630
$$457$$ −40.4674 −1.89298 −0.946492 0.322727i $$-0.895400\pi$$
−0.946492 + 0.322727i $$0.895400\pi$$
$$458$$ 6.00000 0.280362
$$459$$ 1.37228 0.0640526
$$460$$ 3.37228 0.157233
$$461$$ 7.48913 0.348803 0.174402 0.984675i $$-0.444201\pi$$
0.174402 + 0.984675i $$0.444201\pi$$
$$462$$ 4.62772 0.215301
$$463$$ −19.7228 −0.916597 −0.458298 0.888798i $$-0.651541\pi$$
−0.458298 + 0.888798i $$0.651541\pi$$
$$464$$ 6.00000 0.278543
$$465$$ −2.74456 −0.127276
$$466$$ 22.7446 1.05362
$$467$$ 1.48913 0.0689085 0.0344543 0.999406i $$-0.489031\pi$$
0.0344543 + 0.999406i $$0.489031\pi$$
$$468$$ 1.37228 0.0634337
$$469$$ −9.25544 −0.427376
$$470$$ 4.74456 0.218850
$$471$$ −3.25544 −0.150003
$$472$$ −14.7446 −0.678674
$$473$$ 5.48913 0.252390
$$474$$ 4.00000 0.183726
$$475$$ −1.37228 −0.0629646
$$476$$ 4.62772 0.212111
$$477$$ 5.37228 0.245980
$$478$$ 5.48913 0.251067
$$479$$ −18.1168 −0.827780 −0.413890 0.910327i $$-0.635830\pi$$
−0.413890 + 0.910327i $$0.635830\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −1.37228 −0.0625706
$$482$$ 0.510875 0.0232697
$$483$$ 11.3723 0.517457
$$484$$ −9.11684 −0.414402
$$485$$ 13.4891 0.612510
$$486$$ 1.00000 0.0453609
$$487$$ −30.4674 −1.38061 −0.690304 0.723519i $$-0.742523\pi$$
−0.690304 + 0.723519i $$0.742523\pi$$
$$488$$ 2.74456 0.124241
$$489$$ −4.86141 −0.219840
$$490$$ −4.37228 −0.197520
$$491$$ 2.62772 0.118587 0.0592936 0.998241i $$-0.481115\pi$$
0.0592936 + 0.998241i $$0.481115\pi$$
$$492$$ −8.74456 −0.394235
$$493$$ −8.23369 −0.370827
$$494$$ 1.88316 0.0847272
$$495$$ −1.37228 −0.0616795
$$496$$ 2.74456 0.123235
$$497$$ −4.23369 −0.189907
$$498$$ 0.627719 0.0281287
$$499$$ 20.3505 0.911015 0.455507 0.890232i $$-0.349458\pi$$
0.455507 + 0.890232i $$0.349458\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 1.88316 0.0841332
$$502$$ 26.7446 1.19367
$$503$$ 5.48913 0.244748 0.122374 0.992484i $$-0.460949\pi$$
0.122374 + 0.992484i $$0.460949\pi$$
$$504$$ 3.37228 0.150213
$$505$$ 10.7446 0.478127
$$506$$ 4.62772 0.205727
$$507$$ 11.1168 0.493716
$$508$$ 3.37228 0.149621
$$509$$ 18.3505 0.813373 0.406687 0.913568i $$-0.366684\pi$$
0.406687 + 0.913568i $$0.366684\pi$$
$$510$$ −1.37228 −0.0607656
$$511$$ 13.8832 0.614155
$$512$$ −1.00000 −0.0441942
$$513$$ 1.37228 0.0605877
$$514$$ 5.37228 0.236961
$$515$$ −0.744563 −0.0328094
$$516$$ 4.00000 0.176090
$$517$$ 6.51087 0.286348
$$518$$ −3.37228 −0.148170
$$519$$ 6.86141 0.301182
$$520$$ −1.37228 −0.0601785
$$521$$ −5.76631 −0.252627 −0.126313 0.991990i $$-0.540314\pi$$
−0.126313 + 0.991990i $$0.540314\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −6.97825 −0.305138 −0.152569 0.988293i $$-0.548755\pi$$
−0.152569 + 0.988293i $$0.548755\pi$$
$$524$$ 0 0
$$525$$ 3.37228 0.147178
$$526$$ −10.2337 −0.446210
$$527$$ −3.76631 −0.164063
$$528$$ 1.37228 0.0597209
$$529$$ −11.6277 −0.505553
$$530$$ −5.37228 −0.233357
$$531$$ 14.7446 0.639860
$$532$$ 4.62772 0.200637
$$533$$ 12.0000 0.519778
$$534$$ 13.3723 0.578675
$$535$$ 3.37228 0.145796
$$536$$ −2.74456 −0.118547
$$537$$ −5.48913 −0.236873
$$538$$ −0.627719 −0.0270629
$$539$$ −6.00000 −0.258438
$$540$$ −1.00000 −0.0430331
$$541$$ 4.86141 0.209008 0.104504 0.994524i $$-0.466674\pi$$
0.104504 + 0.994524i $$0.466674\pi$$
$$542$$ −13.4891 −0.579408
$$543$$ 20.9783 0.900263
$$544$$ 1.37228 0.0588361
$$545$$ 4.62772 0.198230
$$546$$ −4.62772 −0.198048
$$547$$ −2.11684 −0.0905097 −0.0452549 0.998975i $$-0.514410\pi$$
−0.0452549 + 0.998975i $$0.514410\pi$$
$$548$$ 10.7446 0.458985
$$549$$ −2.74456 −0.117135
$$550$$ 1.37228 0.0585143
$$551$$ −8.23369 −0.350767
$$552$$ 3.37228 0.143534
$$553$$ −13.4891 −0.573616
$$554$$ −25.6060 −1.08789
$$555$$ 1.00000 0.0424476
$$556$$ 2.74456 0.116395
$$557$$ 40.7446 1.72640 0.863201 0.504860i $$-0.168456\pi$$
0.863201 + 0.504860i $$0.168456\pi$$
$$558$$ −2.74456 −0.116187
$$559$$ −5.48913 −0.232165
$$560$$ −3.37228 −0.142505
$$561$$ −1.88316 −0.0795069
$$562$$ −1.37228 −0.0578862
$$563$$ 13.2554 0.558650 0.279325 0.960197i $$-0.409889\pi$$
0.279325 + 0.960197i $$0.409889\pi$$
$$564$$ 4.74456 0.199782
$$565$$ 19.4891 0.819914
$$566$$ 11.6060 0.487835
$$567$$ −3.37228 −0.141623
$$568$$ −1.25544 −0.0526770
$$569$$ −32.3505 −1.35620 −0.678102 0.734967i $$-0.737197\pi$$
−0.678102 + 0.734967i $$0.737197\pi$$
$$570$$ −1.37228 −0.0574785
$$571$$ −1.25544 −0.0525384 −0.0262692 0.999655i $$-0.508363\pi$$
−0.0262692 + 0.999655i $$0.508363\pi$$
$$572$$ −1.88316 −0.0787387
$$573$$ 19.3723 0.809289
$$574$$ 29.4891 1.23085
$$575$$ 3.37228 0.140634
$$576$$ 1.00000 0.0416667
$$577$$ 26.7446 1.11339 0.556695 0.830717i $$-0.312069\pi$$
0.556695 + 0.830717i $$0.312069\pi$$
$$578$$ 15.1168 0.628778
$$579$$ 1.25544 0.0521742
$$580$$ 6.00000 0.249136
$$581$$ −2.11684 −0.0878215
$$582$$ 13.4891 0.559142
$$583$$ −7.37228 −0.305329
$$584$$ 4.11684 0.170356
$$585$$ 1.37228 0.0567368
$$586$$ 4.11684 0.170065
$$587$$ 1.48913 0.0614628 0.0307314 0.999528i $$-0.490216\pi$$
0.0307314 + 0.999528i $$0.490216\pi$$
$$588$$ −4.37228 −0.180310
$$589$$ −3.76631 −0.155188
$$590$$ −14.7446 −0.607024
$$591$$ 18.8614 0.775855
$$592$$ −1.00000 −0.0410997
$$593$$ −22.9783 −0.943604 −0.471802 0.881705i $$-0.656396\pi$$
−0.471802 + 0.881705i $$0.656396\pi$$
$$594$$ −1.37228 −0.0563054
$$595$$ 4.62772 0.189718
$$596$$ 9.25544 0.379117
$$597$$ −8.00000 −0.327418
$$598$$ −4.62772 −0.189241
$$599$$ −22.9783 −0.938866 −0.469433 0.882968i $$-0.655542\pi$$
−0.469433 + 0.882968i $$0.655542\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ 30.8614 1.25886 0.629432 0.777056i $$-0.283288\pi$$
0.629432 + 0.777056i $$0.283288\pi$$
$$602$$ −13.4891 −0.549776
$$603$$ 2.74456 0.111767
$$604$$ −3.37228 −0.137216
$$605$$ −9.11684 −0.370652
$$606$$ 10.7446 0.436468
$$607$$ 38.4674 1.56134 0.780671 0.624942i $$-0.214877\pi$$
0.780671 + 0.624942i $$0.214877\pi$$
$$608$$ 1.37228 0.0556534
$$609$$ 20.2337 0.819910
$$610$$ 2.74456 0.111124
$$611$$ −6.51087 −0.263402
$$612$$ −1.37228 −0.0554712
$$613$$ 30.4674 1.23057 0.615283 0.788306i $$-0.289042\pi$$
0.615283 + 0.788306i $$0.289042\pi$$
$$614$$ 10.7446 0.433615
$$615$$ −8.74456 −0.352615
$$616$$ −4.62772 −0.186456
$$617$$ 32.2337 1.29768 0.648840 0.760925i $$-0.275255\pi$$
0.648840 + 0.760925i $$0.275255\pi$$
$$618$$ −0.744563 −0.0299507
$$619$$ −26.9783 −1.08435 −0.542174 0.840266i $$-0.682399\pi$$
−0.542174 + 0.840266i $$0.682399\pi$$
$$620$$ 2.74456 0.110224
$$621$$ −3.37228 −0.135325
$$622$$ 8.00000 0.320771
$$623$$ −45.0951 −1.80670
$$624$$ −1.37228 −0.0549352
$$625$$ 1.00000 0.0400000
$$626$$ −8.00000 −0.319744
$$627$$ −1.88316 −0.0752060
$$628$$ 3.25544 0.129906
$$629$$ 1.37228 0.0547164
$$630$$ 3.37228 0.134355
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 9.25544 0.367871
$$634$$ 16.9783 0.674292
$$635$$ 3.37228 0.133825
$$636$$ −5.37228 −0.213025
$$637$$ 6.00000 0.237729
$$638$$ 8.23369 0.325975
$$639$$ 1.25544 0.0496643
$$640$$ −1.00000 −0.0395285
$$641$$ 43.7228 1.72695 0.863474 0.504394i $$-0.168284\pi$$
0.863474 + 0.504394i $$0.168284\pi$$
$$642$$ 3.37228 0.133093
$$643$$ 23.6060 0.930929 0.465464 0.885067i $$-0.345887\pi$$
0.465464 + 0.885067i $$0.345887\pi$$
$$644$$ −11.3723 −0.448131
$$645$$ 4.00000 0.157500
$$646$$ −1.88316 −0.0740918
$$647$$ −39.6060 −1.55707 −0.778536 0.627600i $$-0.784037\pi$$
−0.778536 + 0.627600i $$0.784037\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −20.2337 −0.794242
$$650$$ −1.37228 −0.0538253
$$651$$ 9.25544 0.362749
$$652$$ 4.86141 0.190387
$$653$$ 19.7228 0.771813 0.385907 0.922538i $$-0.373889\pi$$
0.385907 + 0.922538i $$0.373889\pi$$
$$654$$ 4.62772 0.180958
$$655$$ 0 0
$$656$$ 8.74456 0.341418
$$657$$ −4.11684 −0.160613
$$658$$ −16.0000 −0.623745
$$659$$ −2.23369 −0.0870121 −0.0435061 0.999053i $$-0.513853\pi$$
−0.0435061 + 0.999053i $$0.513853\pi$$
$$660$$ 1.37228 0.0534160
$$661$$ −26.1168 −1.01583 −0.507914 0.861408i $$-0.669583\pi$$
−0.507914 + 0.861408i $$0.669583\pi$$
$$662$$ 8.74456 0.339867
$$663$$ 1.88316 0.0731357
$$664$$ −0.627719 −0.0243602
$$665$$ 4.62772 0.179455
$$666$$ 1.00000 0.0387492
$$667$$ 20.2337 0.783452
$$668$$ −1.88316 −0.0728615
$$669$$ −18.9783 −0.733742
$$670$$ −2.74456 −0.106032
$$671$$ 3.76631 0.145397
$$672$$ −3.37228 −0.130089
$$673$$ 2.86141 0.110299 0.0551496 0.998478i $$-0.482436\pi$$
0.0551496 + 0.998478i $$0.482436\pi$$
$$674$$ 4.11684 0.158575
$$675$$ −1.00000 −0.0384900
$$676$$ −11.1168 −0.427571
$$677$$ −2.86141 −0.109973 −0.0549864 0.998487i $$-0.517512\pi$$
−0.0549864 + 0.998487i $$0.517512\pi$$
$$678$$ 19.4891 0.748475
$$679$$ −45.4891 −1.74571
$$680$$ 1.37228 0.0526246
$$681$$ −4.00000 −0.153280
$$682$$ 3.76631 0.144220
$$683$$ 30.9783 1.18535 0.592675 0.805442i $$-0.298072\pi$$
0.592675 + 0.805442i $$0.298072\pi$$
$$684$$ −1.37228 −0.0524705
$$685$$ 10.7446 0.410529
$$686$$ −8.86141 −0.338330
$$687$$ 6.00000 0.228914
$$688$$ −4.00000 −0.152499
$$689$$ 7.37228 0.280862
$$690$$ 3.37228 0.128381
$$691$$ 22.7446 0.865244 0.432622 0.901575i $$-0.357588\pi$$
0.432622 + 0.901575i $$0.357588\pi$$
$$692$$ −6.86141 −0.260832
$$693$$ 4.62772 0.175792
$$694$$ −1.48913 −0.0565264
$$695$$ 2.74456 0.104107
$$696$$ 6.00000 0.227429
$$697$$ −12.0000 −0.454532
$$698$$ −16.7446 −0.633791
$$699$$ 22.7446 0.860278
$$700$$ −3.37228 −0.127460
$$701$$ −47.7228 −1.80247 −0.901233 0.433335i $$-0.857337\pi$$
−0.901233 + 0.433335i $$0.857337\pi$$
$$702$$ 1.37228 0.0517934
$$703$$ 1.37228 0.0517566
$$704$$ −1.37228 −0.0517198
$$705$$ 4.74456 0.178691
$$706$$ −3.48913 −0.131315
$$707$$ −36.2337 −1.36271
$$708$$ −14.7446 −0.554135
$$709$$ −20.6277 −0.774690 −0.387345 0.921935i $$-0.626608\pi$$
−0.387345 + 0.921935i $$0.626608\pi$$
$$710$$ −1.25544 −0.0471157
$$711$$ 4.00000 0.150012
$$712$$ −13.3723 −0.501147
$$713$$ 9.25544 0.346619
$$714$$ 4.62772 0.173188
$$715$$ −1.88316 −0.0704260
$$716$$ 5.48913 0.205138
$$717$$ 5.48913 0.204995
$$718$$ −32.4674 −1.21167
$$719$$ −13.7228 −0.511775 −0.255887 0.966707i $$-0.582368\pi$$
−0.255887 + 0.966707i $$0.582368\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 2.51087 0.0935099
$$722$$ 17.1168 0.637023
$$723$$ 0.510875 0.0189996
$$724$$ −20.9783 −0.779651
$$725$$ 6.00000 0.222834
$$726$$ −9.11684 −0.338358
$$727$$ 8.97825 0.332985 0.166492 0.986043i $$-0.446756\pi$$
0.166492 + 0.986043i $$0.446756\pi$$
$$728$$ 4.62772 0.171515
$$729$$ 1.00000 0.0370370
$$730$$ 4.11684 0.152371
$$731$$ 5.48913 0.203023
$$732$$ 2.74456 0.101442
$$733$$ 3.48913 0.128874 0.0644369 0.997922i $$-0.479475\pi$$
0.0644369 + 0.997922i $$0.479475\pi$$
$$734$$ −20.6277 −0.761383
$$735$$ −4.37228 −0.161274
$$736$$ −3.37228 −0.124304
$$737$$ −3.76631 −0.138734
$$738$$ −8.74456 −0.321892
$$739$$ 25.7228 0.946229 0.473114 0.881001i $$-0.343130\pi$$
0.473114 + 0.881001i $$0.343130\pi$$
$$740$$ −1.00000 −0.0367607
$$741$$ 1.88316 0.0691795
$$742$$ 18.1168 0.665090
$$743$$ 22.4674 0.824248 0.412124 0.911128i $$-0.364787\pi$$
0.412124 + 0.911128i $$0.364787\pi$$
$$744$$ 2.74456 0.100621
$$745$$ 9.25544 0.339093
$$746$$ −3.48913 −0.127746
$$747$$ 0.627719 0.0229670
$$748$$ 1.88316 0.0688550
$$749$$ −11.3723 −0.415534
$$750$$ 1.00000 0.0365148
$$751$$ −10.5109 −0.383547 −0.191774 0.981439i $$-0.561424\pi$$
−0.191774 + 0.981439i $$0.561424\pi$$
$$752$$ −4.74456 −0.173016
$$753$$ 26.7446 0.974626
$$754$$ −8.23369 −0.299853
$$755$$ −3.37228 −0.122730
$$756$$ 3.37228 0.122649
$$757$$ 44.1168 1.60345 0.801727 0.597690i $$-0.203915\pi$$
0.801727 + 0.597690i $$0.203915\pi$$
$$758$$ −24.2337 −0.880207
$$759$$ 4.62772 0.167976
$$760$$ 1.37228 0.0497779
$$761$$ −12.5109 −0.453519 −0.226759 0.973951i $$-0.572813\pi$$
−0.226759 + 0.973951i $$0.572813\pi$$
$$762$$ 3.37228 0.122165
$$763$$ −15.6060 −0.564974
$$764$$ −19.3723 −0.700865
$$765$$ −1.37228 −0.0496149
$$766$$ −27.6060 −0.997444
$$767$$ 20.2337 0.730596
$$768$$ −1.00000 −0.0360844
$$769$$ −51.4891 −1.85675 −0.928373 0.371651i $$-0.878792\pi$$
−0.928373 + 0.371651i $$0.878792\pi$$
$$770$$ −4.62772 −0.166771
$$771$$ 5.37228 0.193478
$$772$$ −1.25544 −0.0451842
$$773$$ 20.1168 0.723553 0.361776 0.932265i $$-0.382170\pi$$
0.361776 + 0.932265i $$0.382170\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 2.74456 0.0985876
$$776$$ −13.4891 −0.484231
$$777$$ −3.37228 −0.120980
$$778$$ −23.4891 −0.842126
$$779$$ −12.0000 −0.429945
$$780$$ −1.37228 −0.0491356
$$781$$ −1.72281 −0.0616471
$$782$$ 4.62772 0.165487
$$783$$ −6.00000 −0.214423
$$784$$ 4.37228 0.156153
$$785$$ 3.25544 0.116192
$$786$$ 0 0
$$787$$ 10.7446 0.383002 0.191501 0.981492i $$-0.438664\pi$$
0.191501 + 0.981492i $$0.438664\pi$$
$$788$$ −18.8614 −0.671910
$$789$$ −10.2337 −0.364329
$$790$$ −4.00000 −0.142314
$$791$$ −65.7228 −2.33683
$$792$$ 1.37228 0.0487619
$$793$$ −3.76631 −0.133746
$$794$$ −20.7446 −0.736197
$$795$$ −5.37228 −0.190535
$$796$$ 8.00000 0.283552
$$797$$ 4.51087 0.159783 0.0798917 0.996804i $$-0.474543\pi$$
0.0798917 + 0.996804i $$0.474543\pi$$
$$798$$ 4.62772 0.163819
$$799$$ 6.51087 0.230338
$$800$$ −1.00000 −0.0353553
$$801$$ 13.3723 0.472486
$$802$$ 36.1168 1.27533
$$803$$ 5.64947 0.199365
$$804$$ −2.74456 −0.0967933
$$805$$ −11.3723 −0.400820
$$806$$ −3.76631 −0.132663
$$807$$ −0.627719 −0.0220967
$$808$$ −10.7446 −0.377992
$$809$$ −24.1168 −0.847903 −0.423952 0.905685i $$-0.639357\pi$$
−0.423952 + 0.905685i $$0.639357\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −20.2337 −0.710063
$$813$$ −13.4891 −0.473084
$$814$$ −1.37228 −0.0480984
$$815$$ 4.86141 0.170288
$$816$$ 1.37228 0.0480395
$$817$$ 5.48913 0.192040
$$818$$ 8.74456 0.305746
$$819$$ −4.62772 −0.161706
$$820$$ 8.74456 0.305373
$$821$$ 26.3505 0.919640 0.459820 0.888012i $$-0.347914\pi$$
0.459820 + 0.888012i $$0.347914\pi$$
$$822$$ 10.7446 0.374760
$$823$$ 25.8832 0.902230 0.451115 0.892466i $$-0.351026\pi$$
0.451115 + 0.892466i $$0.351026\pi$$
$$824$$ 0.744563 0.0259381
$$825$$ 1.37228 0.0477767
$$826$$ 49.7228 1.73008
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ 3.37228 0.117195
$$829$$ −35.8397 −1.24476 −0.622381 0.782714i $$-0.713835\pi$$
−0.622381 + 0.782714i $$0.713835\pi$$
$$830$$ −0.627719 −0.0217884
$$831$$ −25.6060 −0.888261
$$832$$ 1.37228 0.0475753
$$833$$ −6.00000 −0.207888
$$834$$ 2.74456 0.0950364
$$835$$ −1.88316 −0.0651693
$$836$$ 1.88316 0.0651303
$$837$$ −2.74456 −0.0948660
$$838$$ −7.88316 −0.272319
$$839$$ −9.25544 −0.319533 −0.159767 0.987155i $$-0.551074\pi$$
−0.159767 + 0.987155i $$0.551074\pi$$
$$840$$ −3.37228 −0.116355
$$841$$ 7.00000 0.241379
$$842$$ −24.2337 −0.835148
$$843$$ −1.37228 −0.0472639
$$844$$ −9.25544 −0.318585
$$845$$ −11.1168 −0.382431
$$846$$ 4.74456 0.163121
$$847$$ 30.7446 1.05640
$$848$$ 5.37228 0.184485
$$849$$ 11.6060 0.398316
$$850$$ 1.37228 0.0470689
$$851$$ −3.37228 −0.115600
$$852$$ −1.25544 −0.0430106
$$853$$ 40.3505 1.38158 0.690788 0.723057i $$-0.257264\pi$$
0.690788 + 0.723057i $$0.257264\pi$$
$$854$$ −9.25544 −0.316715
$$855$$ −1.37228 −0.0469310
$$856$$ −3.37228 −0.115262
$$857$$ 17.8397 0.609391 0.304696 0.952450i $$-0.401445\pi$$
0.304696 + 0.952450i $$0.401445\pi$$
$$858$$ −1.88316 −0.0642899
$$859$$ 17.8397 0.608681 0.304341 0.952563i $$-0.401564\pi$$
0.304341 + 0.952563i $$0.401564\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 29.4891 1.00499
$$862$$ 31.6060 1.07650
$$863$$ 3.25544 0.110816 0.0554082 0.998464i $$-0.482354\pi$$
0.0554082 + 0.998464i $$0.482354\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ −6.86141 −0.233295
$$866$$ 10.8614 0.369086
$$867$$ 15.1168 0.513395
$$868$$ −9.25544 −0.314150
$$869$$ −5.48913 −0.186206
$$870$$ 6.00000 0.203419
$$871$$ 3.76631 0.127617
$$872$$ −4.62772 −0.156714
$$873$$ 13.4891 0.456538
$$874$$ 4.62772 0.156535
$$875$$ −3.37228 −0.114004
$$876$$ 4.11684 0.139095
$$877$$ −54.2337 −1.83134 −0.915671 0.401929i $$-0.868340\pi$$
−0.915671 + 0.401929i $$0.868340\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 4.11684 0.138858
$$880$$ −1.37228 −0.0462596
$$881$$ −14.2337 −0.479545 −0.239773 0.970829i $$-0.577073\pi$$
−0.239773 + 0.970829i $$0.577073\pi$$
$$882$$ −4.37228 −0.147222
$$883$$ −50.1168 −1.68657 −0.843283 0.537470i $$-0.819380\pi$$
−0.843283 + 0.537470i $$0.819380\pi$$
$$884$$ −1.88316 −0.0633374
$$885$$ −14.7446 −0.495633
$$886$$ 4.00000 0.134383
$$887$$ −42.0000 −1.41022 −0.705111 0.709097i $$-0.749103\pi$$
−0.705111 + 0.709097i $$0.749103\pi$$
$$888$$ −1.00000 −0.0335578
$$889$$ −11.3723 −0.381414
$$890$$ −13.3723 −0.448240
$$891$$ −1.37228 −0.0459732
$$892$$ 18.9783 0.635439
$$893$$ 6.51087 0.217878
$$894$$ 9.25544 0.309548
$$895$$ 5.48913 0.183481
$$896$$ 3.37228 0.112660
$$897$$ −4.62772 −0.154515
$$898$$ −36.9783 −1.23398
$$899$$ 16.4674 0.549218
$$900$$ 1.00000 0.0333333
$$901$$ −7.37228 −0.245606
$$902$$ 12.0000 0.399556
$$903$$ −13.4891 −0.448890
$$904$$ −19.4891 −0.648199
$$905$$ −20.9783 −0.697341
$$906$$ −3.37228 −0.112037
$$907$$ −3.60597 −0.119734 −0.0598671 0.998206i $$-0.519068\pi$$
−0.0598671 + 0.998206i $$0.519068\pi$$
$$908$$ 4.00000 0.132745
$$909$$ 10.7446 0.356375
$$910$$ 4.62772 0.153407
$$911$$ 34.9783 1.15888 0.579441 0.815014i $$-0.303271\pi$$
0.579441 + 0.815014i $$0.303271\pi$$
$$912$$ 1.37228 0.0454408
$$913$$ −0.861407 −0.0285084
$$914$$ 40.4674 1.33854
$$915$$ 2.74456 0.0907324
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ −1.37228 −0.0452920
$$919$$ −34.5109 −1.13841 −0.569204 0.822196i $$-0.692749\pi$$
−0.569204 + 0.822196i $$0.692749\pi$$
$$920$$ −3.37228 −0.111181
$$921$$ 10.7446 0.354045
$$922$$ −7.48913 −0.246641
$$923$$ 1.72281 0.0567071
$$924$$ −4.62772 −0.152241
$$925$$ −1.00000 −0.0328798
$$926$$ 19.7228 0.648132
$$927$$ −0.744563 −0.0244546
$$928$$ −6.00000 −0.196960
$$929$$ −12.5109 −0.410468 −0.205234 0.978713i $$-0.565796\pi$$
−0.205234 + 0.978713i $$0.565796\pi$$
$$930$$ 2.74456 0.0899978
$$931$$ −6.00000 −0.196642
$$932$$ −22.7446 −0.745023
$$933$$ 8.00000 0.261908
$$934$$ −1.48913 −0.0487257
$$935$$ 1.88316 0.0615858
$$936$$ −1.37228 −0.0448544
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 9.25544 0.302201
$$939$$ −8.00000 −0.261070
$$940$$ −4.74456 −0.154751
$$941$$ 15.7663 0.513967 0.256984 0.966416i $$-0.417271\pi$$
0.256984 + 0.966416i $$0.417271\pi$$
$$942$$ 3.25544 0.106068
$$943$$ 29.4891 0.960298
$$944$$ 14.7446 0.479895
$$945$$ 3.37228 0.109700
$$946$$ −5.48913 −0.178467
$$947$$ −28.4674 −0.925065 −0.462533 0.886602i $$-0.653059\pi$$
−0.462533 + 0.886602i $$0.653059\pi$$
$$948$$ −4.00000 −0.129914
$$949$$ −5.64947 −0.183389
$$950$$ 1.37228 0.0445227
$$951$$ 16.9783 0.550557
$$952$$ −4.62772 −0.149985
$$953$$ −50.7446 −1.64378 −0.821889 0.569648i $$-0.807080\pi$$
−0.821889 + 0.569648i $$0.807080\pi$$
$$954$$ −5.37228 −0.173934
$$955$$ −19.3723 −0.626872
$$956$$ −5.48913 −0.177531
$$957$$ 8.23369 0.266157
$$958$$ 18.1168 0.585329
$$959$$ −36.2337 −1.17005
$$960$$ −1.00000 −0.0322749
$$961$$ −23.4674 −0.757012
$$962$$ 1.37228 0.0442441
$$963$$ 3.37228 0.108670
$$964$$ −0.510875 −0.0164542
$$965$$ −1.25544 −0.0404140
$$966$$ −11.3723 −0.365897
$$967$$ −16.9783 −0.545984 −0.272992 0.962016i $$-0.588013\pi$$
−0.272992 + 0.962016i $$0.588013\pi$$
$$968$$ 9.11684 0.293026
$$969$$ −1.88316 −0.0604957
$$970$$ −13.4891 −0.433110
$$971$$ 50.2337 1.61208 0.806038 0.591864i $$-0.201608\pi$$
0.806038 + 0.591864i $$0.201608\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −9.25544 −0.296716
$$974$$ 30.4674 0.976238
$$975$$ −1.37228 −0.0439482
$$976$$ −2.74456 −0.0878513
$$977$$ 25.1386 0.804255 0.402127 0.915584i $$-0.368271\pi$$
0.402127 + 0.915584i $$0.368271\pi$$
$$978$$ 4.86141 0.155451
$$979$$ −18.3505 −0.586486
$$980$$ 4.37228 0.139667
$$981$$ 4.62772 0.147752
$$982$$ −2.62772 −0.0838539
$$983$$ −12.5109 −0.399035 −0.199517 0.979894i $$-0.563937\pi$$
−0.199517 + 0.979894i $$0.563937\pi$$
$$984$$ 8.74456 0.278766
$$985$$ −18.8614 −0.600974
$$986$$ 8.23369 0.262214
$$987$$ −16.0000 −0.509286
$$988$$ −1.88316 −0.0599112
$$989$$ −13.4891 −0.428929
$$990$$ 1.37228 0.0436140
$$991$$ 22.9783 0.729928 0.364964 0.931022i $$-0.381081\pi$$
0.364964 + 0.931022i $$0.381081\pi$$
$$992$$ −2.74456 −0.0871400
$$993$$ 8.74456 0.277500
$$994$$ 4.23369 0.134284
$$995$$ 8.00000 0.253617
$$996$$ −0.627719 −0.0198900
$$997$$ 16.1168 0.510426 0.255213 0.966885i $$-0.417854\pi$$
0.255213 + 0.966885i $$0.417854\pi$$
$$998$$ −20.3505 −0.644185
$$999$$ 1.00000 0.0316386
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 1110.2.a.p.1.1 2
3.2 odd 2 3330.2.a.be.1.1 2
4.3 odd 2 8880.2.a.br.1.2 2
5.4 even 2 5550.2.a.ca.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.p.1.1 2 1.1 even 1 trivial
3330.2.a.be.1.1 2 3.2 odd 2
5550.2.a.ca.1.2 2 5.4 even 2
8880.2.a.br.1.2 2 4.3 odd 2