# Properties

 Label 165.2.a.b.1.1 Level $165$ Weight $2$ Character 165.1 Self dual yes Analytic conductor $1.318$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 165.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.31753163335$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 165.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +2.00000 q^{7} +1.73205 q^{8} +1.00000 q^{9} +1.73205 q^{10} -1.00000 q^{11} +1.00000 q^{12} +5.46410 q^{13} -3.46410 q^{14} -1.00000 q^{15} -5.00000 q^{16} -1.73205 q^{18} +5.46410 q^{19} -1.00000 q^{20} +2.00000 q^{21} +1.73205 q^{22} +6.92820 q^{23} +1.73205 q^{24} +1.00000 q^{25} -9.46410 q^{26} +1.00000 q^{27} +2.00000 q^{28} -3.46410 q^{29} +1.73205 q^{30} -10.9282 q^{31} +5.19615 q^{32} -1.00000 q^{33} -2.00000 q^{35} +1.00000 q^{36} -4.92820 q^{37} -9.46410 q^{38} +5.46410 q^{39} -1.73205 q^{40} +3.46410 q^{41} -3.46410 q^{42} -4.92820 q^{43} -1.00000 q^{44} -1.00000 q^{45} -12.0000 q^{46} -6.92820 q^{47} -5.00000 q^{48} -3.00000 q^{49} -1.73205 q^{50} +5.46410 q^{52} +0.928203 q^{53} -1.73205 q^{54} +1.00000 q^{55} +3.46410 q^{56} +5.46410 q^{57} +6.00000 q^{58} -6.92820 q^{59} -1.00000 q^{60} +2.00000 q^{61} +18.9282 q^{62} +2.00000 q^{63} +1.00000 q^{64} -5.46410 q^{65} +1.73205 q^{66} +8.00000 q^{67} +6.92820 q^{69} +3.46410 q^{70} +13.8564 q^{71} +1.73205 q^{72} -8.39230 q^{73} +8.53590 q^{74} +1.00000 q^{75} +5.46410 q^{76} -2.00000 q^{77} -9.46410 q^{78} -6.53590 q^{79} +5.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.53590 q^{83} +2.00000 q^{84} +8.53590 q^{86} -3.46410 q^{87} -1.73205 q^{88} +0.928203 q^{89} +1.73205 q^{90} +10.9282 q^{91} +6.92820 q^{92} -10.9282 q^{93} +12.0000 q^{94} -5.46410 q^{95} +5.19615 q^{96} -10.0000 q^{97} +5.19615 q^{98} -1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{4} - 2q^{5} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{4} - 2q^{5} + 4q^{7} + 2q^{9} - 2q^{11} + 2q^{12} + 4q^{13} - 2q^{15} - 10q^{16} + 4q^{19} - 2q^{20} + 4q^{21} + 2q^{25} - 12q^{26} + 2q^{27} + 4q^{28} - 8q^{31} - 2q^{33} - 4q^{35} + 2q^{36} + 4q^{37} - 12q^{38} + 4q^{39} + 4q^{43} - 2q^{44} - 2q^{45} - 24q^{46} - 10q^{48} - 6q^{49} + 4q^{52} - 12q^{53} + 2q^{55} + 4q^{57} + 12q^{58} - 2q^{60} + 4q^{61} + 24q^{62} + 4q^{63} + 2q^{64} - 4q^{65} + 16q^{67} + 4q^{73} + 24q^{74} + 2q^{75} + 4q^{76} - 4q^{77} - 12q^{78} - 20q^{79} + 10q^{80} + 2q^{81} - 12q^{82} + 24q^{83} + 4q^{84} + 24q^{86} - 12q^{89} + 8q^{91} - 8q^{93} + 24q^{94} - 4q^{95} - 20q^{97} - 2q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.73205 −0.707107
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 1.73205 0.612372
$$9$$ 1.00000 0.333333
$$10$$ 1.73205 0.547723
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000 0.288675
$$13$$ 5.46410 1.51547 0.757735 0.652563i $$-0.226306\pi$$
0.757735 + 0.652563i $$0.226306\pi$$
$$14$$ −3.46410 −0.925820
$$15$$ −1.00000 −0.258199
$$16$$ −5.00000 −1.25000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ −1.73205 −0.408248
$$19$$ 5.46410 1.25355 0.626775 0.779200i $$-0.284374\pi$$
0.626775 + 0.779200i $$0.284374\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 2.00000 0.436436
$$22$$ 1.73205 0.369274
$$23$$ 6.92820 1.44463 0.722315 0.691564i $$-0.243078\pi$$
0.722315 + 0.691564i $$0.243078\pi$$
$$24$$ 1.73205 0.353553
$$25$$ 1.00000 0.200000
$$26$$ −9.46410 −1.85606
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ −3.46410 −0.643268 −0.321634 0.946864i $$-0.604232\pi$$
−0.321634 + 0.946864i $$0.604232\pi$$
$$30$$ 1.73205 0.316228
$$31$$ −10.9282 −1.96276 −0.981382 0.192068i $$-0.938481\pi$$
−0.981382 + 0.192068i $$0.938481\pi$$
$$32$$ 5.19615 0.918559
$$33$$ −1.00000 −0.174078
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −4.92820 −0.810192 −0.405096 0.914274i $$-0.632762\pi$$
−0.405096 + 0.914274i $$0.632762\pi$$
$$38$$ −9.46410 −1.53528
$$39$$ 5.46410 0.874957
$$40$$ −1.73205 −0.273861
$$41$$ 3.46410 0.541002 0.270501 0.962720i $$-0.412811\pi$$
0.270501 + 0.962720i $$0.412811\pi$$
$$42$$ −3.46410 −0.534522
$$43$$ −4.92820 −0.751544 −0.375772 0.926712i $$-0.622622\pi$$
−0.375772 + 0.926712i $$0.622622\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −1.00000 −0.149071
$$46$$ −12.0000 −1.76930
$$47$$ −6.92820 −1.01058 −0.505291 0.862949i $$-0.668615\pi$$
−0.505291 + 0.862949i $$0.668615\pi$$
$$48$$ −5.00000 −0.721688
$$49$$ −3.00000 −0.428571
$$50$$ −1.73205 −0.244949
$$51$$ 0 0
$$52$$ 5.46410 0.757735
$$53$$ 0.928203 0.127499 0.0637493 0.997966i $$-0.479694\pi$$
0.0637493 + 0.997966i $$0.479694\pi$$
$$54$$ −1.73205 −0.235702
$$55$$ 1.00000 0.134840
$$56$$ 3.46410 0.462910
$$57$$ 5.46410 0.723738
$$58$$ 6.00000 0.787839
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 18.9282 2.40388
$$63$$ 2.00000 0.251976
$$64$$ 1.00000 0.125000
$$65$$ −5.46410 −0.677738
$$66$$ 1.73205 0.213201
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ 6.92820 0.834058
$$70$$ 3.46410 0.414039
$$71$$ 13.8564 1.64445 0.822226 0.569160i $$-0.192732\pi$$
0.822226 + 0.569160i $$0.192732\pi$$
$$72$$ 1.73205 0.204124
$$73$$ −8.39230 −0.982245 −0.491122 0.871091i $$-0.663413\pi$$
−0.491122 + 0.871091i $$0.663413\pi$$
$$74$$ 8.53590 0.992278
$$75$$ 1.00000 0.115470
$$76$$ 5.46410 0.626775
$$77$$ −2.00000 −0.227921
$$78$$ −9.46410 −1.07160
$$79$$ −6.53590 −0.735346 −0.367673 0.929955i $$-0.619845\pi$$
−0.367673 + 0.929955i $$0.619845\pi$$
$$80$$ 5.00000 0.559017
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 8.53590 0.936937 0.468468 0.883480i $$-0.344806\pi$$
0.468468 + 0.883480i $$0.344806\pi$$
$$84$$ 2.00000 0.218218
$$85$$ 0 0
$$86$$ 8.53590 0.920450
$$87$$ −3.46410 −0.371391
$$88$$ −1.73205 −0.184637
$$89$$ 0.928203 0.0983893 0.0491947 0.998789i $$-0.484335\pi$$
0.0491947 + 0.998789i $$0.484335\pi$$
$$90$$ 1.73205 0.182574
$$91$$ 10.9282 1.14559
$$92$$ 6.92820 0.722315
$$93$$ −10.9282 −1.13320
$$94$$ 12.0000 1.23771
$$95$$ −5.46410 −0.560605
$$96$$ 5.19615 0.530330
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 5.19615 0.524891
$$99$$ −1.00000 −0.100504
$$100$$ 1.00000 0.100000
$$101$$ −10.3923 −1.03407 −0.517036 0.855963i $$-0.672965\pi$$
−0.517036 + 0.855963i $$0.672965\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 9.46410 0.928032
$$105$$ −2.00000 −0.195180
$$106$$ −1.60770 −0.156153
$$107$$ 8.53590 0.825196 0.412598 0.910913i $$-0.364621\pi$$
0.412598 + 0.910913i $$0.364621\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ −1.73205 −0.165145
$$111$$ −4.92820 −0.467764
$$112$$ −10.0000 −0.944911
$$113$$ −12.9282 −1.21618 −0.608092 0.793867i $$-0.708065\pi$$
−0.608092 + 0.793867i $$0.708065\pi$$
$$114$$ −9.46410 −0.886394
$$115$$ −6.92820 −0.646058
$$116$$ −3.46410 −0.321634
$$117$$ 5.46410 0.505156
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ −1.73205 −0.158114
$$121$$ 1.00000 0.0909091
$$122$$ −3.46410 −0.313625
$$123$$ 3.46410 0.312348
$$124$$ −10.9282 −0.981382
$$125$$ −1.00000 −0.0894427
$$126$$ −3.46410 −0.308607
$$127$$ 8.92820 0.792250 0.396125 0.918197i $$-0.370355\pi$$
0.396125 + 0.918197i $$0.370355\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ −4.92820 −0.433904
$$130$$ 9.46410 0.830057
$$131$$ 18.9282 1.65376 0.826882 0.562375i $$-0.190112\pi$$
0.826882 + 0.562375i $$0.190112\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 10.9282 0.947595
$$134$$ −13.8564 −1.19701
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ −12.0000 −1.02151
$$139$$ 12.3923 1.05110 0.525551 0.850762i $$-0.323859\pi$$
0.525551 + 0.850762i $$0.323859\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −6.92820 −0.583460
$$142$$ −24.0000 −2.01404
$$143$$ −5.46410 −0.456931
$$144$$ −5.00000 −0.416667
$$145$$ 3.46410 0.287678
$$146$$ 14.5359 1.20300
$$147$$ −3.00000 −0.247436
$$148$$ −4.92820 −0.405096
$$149$$ −15.4641 −1.26687 −0.633434 0.773796i $$-0.718355\pi$$
−0.633434 + 0.773796i $$0.718355\pi$$
$$150$$ −1.73205 −0.141421
$$151$$ −20.3923 −1.65950 −0.829751 0.558134i $$-0.811518\pi$$
−0.829751 + 0.558134i $$0.811518\pi$$
$$152$$ 9.46410 0.767640
$$153$$ 0 0
$$154$$ 3.46410 0.279145
$$155$$ 10.9282 0.877774
$$156$$ 5.46410 0.437478
$$157$$ −3.07180 −0.245156 −0.122578 0.992459i $$-0.539116\pi$$
−0.122578 + 0.992459i $$0.539116\pi$$
$$158$$ 11.3205 0.900611
$$159$$ 0.928203 0.0736113
$$160$$ −5.19615 −0.410792
$$161$$ 13.8564 1.09204
$$162$$ −1.73205 −0.136083
$$163$$ 9.85641 0.772013 0.386007 0.922496i $$-0.373854\pi$$
0.386007 + 0.922496i $$0.373854\pi$$
$$164$$ 3.46410 0.270501
$$165$$ 1.00000 0.0778499
$$166$$ −14.7846 −1.14751
$$167$$ −10.3923 −0.804181 −0.402090 0.915600i $$-0.631716\pi$$
−0.402090 + 0.915600i $$0.631716\pi$$
$$168$$ 3.46410 0.267261
$$169$$ 16.8564 1.29665
$$170$$ 0 0
$$171$$ 5.46410 0.417850
$$172$$ −4.92820 −0.375772
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 2.00000 0.151186
$$176$$ 5.00000 0.376889
$$177$$ −6.92820 −0.520756
$$178$$ −1.60770 −0.120502
$$179$$ 6.92820 0.517838 0.258919 0.965899i $$-0.416634\pi$$
0.258919 + 0.965899i $$0.416634\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 15.8564 1.17860 0.589299 0.807915i $$-0.299404\pi$$
0.589299 + 0.807915i $$0.299404\pi$$
$$182$$ −18.9282 −1.40305
$$183$$ 2.00000 0.147844
$$184$$ 12.0000 0.884652
$$185$$ 4.92820 0.362329
$$186$$ 18.9282 1.38788
$$187$$ 0 0
$$188$$ −6.92820 −0.505291
$$189$$ 2.00000 0.145479
$$190$$ 9.46410 0.686598
$$191$$ −18.9282 −1.36960 −0.684798 0.728733i $$-0.740110\pi$$
−0.684798 + 0.728733i $$0.740110\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 24.3923 1.75580 0.877898 0.478847i $$-0.158945\pi$$
0.877898 + 0.478847i $$0.158945\pi$$
$$194$$ 17.3205 1.24354
$$195$$ −5.46410 −0.391292
$$196$$ −3.00000 −0.214286
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 1.73205 0.123091
$$199$$ −24.7846 −1.75693 −0.878467 0.477803i $$-0.841433\pi$$
−0.878467 + 0.477803i $$0.841433\pi$$
$$200$$ 1.73205 0.122474
$$201$$ 8.00000 0.564276
$$202$$ 18.0000 1.26648
$$203$$ −6.92820 −0.486265
$$204$$ 0 0
$$205$$ −3.46410 −0.241943
$$206$$ −13.8564 −0.965422
$$207$$ 6.92820 0.481543
$$208$$ −27.3205 −1.89434
$$209$$ −5.46410 −0.377960
$$210$$ 3.46410 0.239046
$$211$$ −8.39230 −0.577750 −0.288875 0.957367i $$-0.593281\pi$$
−0.288875 + 0.957367i $$0.593281\pi$$
$$212$$ 0.928203 0.0637493
$$213$$ 13.8564 0.949425
$$214$$ −14.7846 −1.01066
$$215$$ 4.92820 0.336101
$$216$$ 1.73205 0.117851
$$217$$ −21.8564 −1.48371
$$218$$ 17.3205 1.17309
$$219$$ −8.39230 −0.567099
$$220$$ 1.00000 0.0674200
$$221$$ 0 0
$$222$$ 8.53590 0.572892
$$223$$ 9.85641 0.660034 0.330017 0.943975i $$-0.392946\pi$$
0.330017 + 0.943975i $$0.392946\pi$$
$$224$$ 10.3923 0.694365
$$225$$ 1.00000 0.0666667
$$226$$ 22.3923 1.48951
$$227$$ 15.4641 1.02639 0.513194 0.858272i $$-0.328462\pi$$
0.513194 + 0.858272i $$0.328462\pi$$
$$228$$ 5.46410 0.361869
$$229$$ −23.8564 −1.57648 −0.788238 0.615371i $$-0.789006\pi$$
−0.788238 + 0.615371i $$0.789006\pi$$
$$230$$ 12.0000 0.791257
$$231$$ −2.00000 −0.131590
$$232$$ −6.00000 −0.393919
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ −9.46410 −0.618688
$$235$$ 6.92820 0.451946
$$236$$ −6.92820 −0.450988
$$237$$ −6.53590 −0.424552
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 5.00000 0.322749
$$241$$ 0.143594 0.00924967 0.00462484 0.999989i $$-0.498528\pi$$
0.00462484 + 0.999989i $$0.498528\pi$$
$$242$$ −1.73205 −0.111340
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 3.00000 0.191663
$$246$$ −6.00000 −0.382546
$$247$$ 29.8564 1.89972
$$248$$ −18.9282 −1.20194
$$249$$ 8.53590 0.540941
$$250$$ 1.73205 0.109545
$$251$$ −1.85641 −0.117175 −0.0585877 0.998282i $$-0.518660\pi$$
−0.0585877 + 0.998282i $$0.518660\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −6.92820 −0.435572
$$254$$ −15.4641 −0.970304
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ −19.8564 −1.23861 −0.619304 0.785151i $$-0.712585\pi$$
−0.619304 + 0.785151i $$0.712585\pi$$
$$258$$ 8.53590 0.531422
$$259$$ −9.85641 −0.612447
$$260$$ −5.46410 −0.338869
$$261$$ −3.46410 −0.214423
$$262$$ −32.7846 −2.02544
$$263$$ 20.5359 1.26630 0.633149 0.774030i $$-0.281762\pi$$
0.633149 + 0.774030i $$0.281762\pi$$
$$264$$ −1.73205 −0.106600
$$265$$ −0.928203 −0.0570191
$$266$$ −18.9282 −1.16056
$$267$$ 0.928203 0.0568051
$$268$$ 8.00000 0.488678
$$269$$ 19.8564 1.21067 0.605333 0.795972i $$-0.293040\pi$$
0.605333 + 0.795972i $$0.293040\pi$$
$$270$$ 1.73205 0.105409
$$271$$ −11.6077 −0.705117 −0.352559 0.935790i $$-0.614688\pi$$
−0.352559 + 0.935790i $$0.614688\pi$$
$$272$$ 0 0
$$273$$ 10.9282 0.661405
$$274$$ 31.1769 1.88347
$$275$$ −1.00000 −0.0603023
$$276$$ 6.92820 0.417029
$$277$$ 29.4641 1.77033 0.885163 0.465281i $$-0.154047\pi$$
0.885163 + 0.465281i $$0.154047\pi$$
$$278$$ −21.4641 −1.28733
$$279$$ −10.9282 −0.654254
$$280$$ −3.46410 −0.207020
$$281$$ 3.46410 0.206651 0.103325 0.994648i $$-0.467052\pi$$
0.103325 + 0.994648i $$0.467052\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −4.92820 −0.292951 −0.146476 0.989214i $$-0.546793\pi$$
−0.146476 + 0.989214i $$0.546793\pi$$
$$284$$ 13.8564 0.822226
$$285$$ −5.46410 −0.323665
$$286$$ 9.46410 0.559624
$$287$$ 6.92820 0.408959
$$288$$ 5.19615 0.306186
$$289$$ −17.0000 −1.00000
$$290$$ −6.00000 −0.352332
$$291$$ −10.0000 −0.586210
$$292$$ −8.39230 −0.491122
$$293$$ −13.8564 −0.809500 −0.404750 0.914427i $$-0.632641\pi$$
−0.404750 + 0.914427i $$0.632641\pi$$
$$294$$ 5.19615 0.303046
$$295$$ 6.92820 0.403376
$$296$$ −8.53590 −0.496139
$$297$$ −1.00000 −0.0580259
$$298$$ 26.7846 1.55159
$$299$$ 37.8564 2.18929
$$300$$ 1.00000 0.0577350
$$301$$ −9.85641 −0.568114
$$302$$ 35.3205 2.03247
$$303$$ −10.3923 −0.597022
$$304$$ −27.3205 −1.56694
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ 14.0000 0.799022 0.399511 0.916728i $$-0.369180\pi$$
0.399511 + 0.916728i $$0.369180\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ 8.00000 0.455104
$$310$$ −18.9282 −1.07505
$$311$$ 5.07180 0.287595 0.143798 0.989607i $$-0.454069\pi$$
0.143798 + 0.989607i $$0.454069\pi$$
$$312$$ 9.46410 0.535799
$$313$$ 20.9282 1.18293 0.591466 0.806330i $$-0.298549\pi$$
0.591466 + 0.806330i $$0.298549\pi$$
$$314$$ 5.32051 0.300254
$$315$$ −2.00000 −0.112687
$$316$$ −6.53590 −0.367673
$$317$$ −24.9282 −1.40011 −0.700054 0.714090i $$-0.746841\pi$$
−0.700054 + 0.714090i $$0.746841\pi$$
$$318$$ −1.60770 −0.0901551
$$319$$ 3.46410 0.193952
$$320$$ −1.00000 −0.0559017
$$321$$ 8.53590 0.476427
$$322$$ −24.0000 −1.33747
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ 5.46410 0.303094
$$326$$ −17.0718 −0.945519
$$327$$ −10.0000 −0.553001
$$328$$ 6.00000 0.331295
$$329$$ −13.8564 −0.763928
$$330$$ −1.73205 −0.0953463
$$331$$ 9.85641 0.541757 0.270879 0.962614i $$-0.412686\pi$$
0.270879 + 0.962614i $$0.412686\pi$$
$$332$$ 8.53590 0.468468
$$333$$ −4.92820 −0.270064
$$334$$ 18.0000 0.984916
$$335$$ −8.00000 −0.437087
$$336$$ −10.0000 −0.545545
$$337$$ 33.1769 1.80726 0.903631 0.428312i $$-0.140892\pi$$
0.903631 + 0.428312i $$0.140892\pi$$
$$338$$ −29.1962 −1.58806
$$339$$ −12.9282 −0.702164
$$340$$ 0 0
$$341$$ 10.9282 0.591795
$$342$$ −9.46410 −0.511760
$$343$$ −20.0000 −1.07990
$$344$$ −8.53590 −0.460225
$$345$$ −6.92820 −0.373002
$$346$$ 20.7846 1.11739
$$347$$ 22.3923 1.20208 0.601041 0.799218i $$-0.294753\pi$$
0.601041 + 0.799218i $$0.294753\pi$$
$$348$$ −3.46410 −0.185695
$$349$$ −8.14359 −0.435917 −0.217958 0.975958i $$-0.569940\pi$$
−0.217958 + 0.975958i $$0.569940\pi$$
$$350$$ −3.46410 −0.185164
$$351$$ 5.46410 0.291652
$$352$$ −5.19615 −0.276956
$$353$$ 12.9282 0.688099 0.344049 0.938952i $$-0.388201\pi$$
0.344049 + 0.938952i $$0.388201\pi$$
$$354$$ 12.0000 0.637793
$$355$$ −13.8564 −0.735422
$$356$$ 0.928203 0.0491947
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −20.7846 −1.09697 −0.548485 0.836160i $$-0.684795\pi$$
−0.548485 + 0.836160i $$0.684795\pi$$
$$360$$ −1.73205 −0.0912871
$$361$$ 10.8564 0.571390
$$362$$ −27.4641 −1.44348
$$363$$ 1.00000 0.0524864
$$364$$ 10.9282 0.572793
$$365$$ 8.39230 0.439273
$$366$$ −3.46410 −0.181071
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ −34.6410 −1.80579
$$369$$ 3.46410 0.180334
$$370$$ −8.53590 −0.443760
$$371$$ 1.85641 0.0963798
$$372$$ −10.9282 −0.566601
$$373$$ −20.3923 −1.05587 −0.527937 0.849284i $$-0.677034\pi$$
−0.527937 + 0.849284i $$0.677034\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ −12.0000 −0.618853
$$377$$ −18.9282 −0.974852
$$378$$ −3.46410 −0.178174
$$379$$ −17.8564 −0.917222 −0.458611 0.888637i $$-0.651653\pi$$
−0.458611 + 0.888637i $$0.651653\pi$$
$$380$$ −5.46410 −0.280302
$$381$$ 8.92820 0.457406
$$382$$ 32.7846 1.67741
$$383$$ −13.8564 −0.708029 −0.354015 0.935240i $$-0.615184\pi$$
−0.354015 + 0.935240i $$0.615184\pi$$
$$384$$ −12.1244 −0.618718
$$385$$ 2.00000 0.101929
$$386$$ −42.2487 −2.15040
$$387$$ −4.92820 −0.250515
$$388$$ −10.0000 −0.507673
$$389$$ 11.0718 0.561362 0.280681 0.959801i $$-0.409440\pi$$
0.280681 + 0.959801i $$0.409440\pi$$
$$390$$ 9.46410 0.479233
$$391$$ 0 0
$$392$$ −5.19615 −0.262445
$$393$$ 18.9282 0.954802
$$394$$ 20.7846 1.04711
$$395$$ 6.53590 0.328857
$$396$$ −1.00000 −0.0502519
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 42.9282 2.15180
$$399$$ 10.9282 0.547094
$$400$$ −5.00000 −0.250000
$$401$$ −7.85641 −0.392330 −0.196165 0.980571i $$-0.562849\pi$$
−0.196165 + 0.980571i $$0.562849\pi$$
$$402$$ −13.8564 −0.691095
$$403$$ −59.7128 −2.97451
$$404$$ −10.3923 −0.517036
$$405$$ −1.00000 −0.0496904
$$406$$ 12.0000 0.595550
$$407$$ 4.92820 0.244282
$$408$$ 0 0
$$409$$ −6.78461 −0.335477 −0.167739 0.985831i $$-0.553646\pi$$
−0.167739 + 0.985831i $$0.553646\pi$$
$$410$$ 6.00000 0.296319
$$411$$ −18.0000 −0.887875
$$412$$ 8.00000 0.394132
$$413$$ −13.8564 −0.681829
$$414$$ −12.0000 −0.589768
$$415$$ −8.53590 −0.419011
$$416$$ 28.3923 1.39205
$$417$$ 12.3923 0.606854
$$418$$ 9.46410 0.462904
$$419$$ 30.9282 1.51094 0.755471 0.655182i $$-0.227408\pi$$
0.755471 + 0.655182i $$0.227408\pi$$
$$420$$ −2.00000 −0.0975900
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 14.5359 0.707596
$$423$$ −6.92820 −0.336861
$$424$$ 1.60770 0.0780766
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ 4.00000 0.193574
$$428$$ 8.53590 0.412598
$$429$$ −5.46410 −0.263809
$$430$$ −8.53590 −0.411638
$$431$$ 8.78461 0.423140 0.211570 0.977363i $$-0.432142\pi$$
0.211570 + 0.977363i $$0.432142\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 0.143594 0.00690067 0.00345033 0.999994i $$-0.498902\pi$$
0.00345033 + 0.999994i $$0.498902\pi$$
$$434$$ 37.8564 1.81717
$$435$$ 3.46410 0.166091
$$436$$ −10.0000 −0.478913
$$437$$ 37.8564 1.81092
$$438$$ 14.5359 0.694552
$$439$$ 33.1769 1.58345 0.791724 0.610879i $$-0.209184\pi$$
0.791724 + 0.610879i $$0.209184\pi$$
$$440$$ 1.73205 0.0825723
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −4.92820 −0.233882
$$445$$ −0.928203 −0.0440011
$$446$$ −17.0718 −0.808373
$$447$$ −15.4641 −0.731427
$$448$$ 2.00000 0.0944911
$$449$$ −26.7846 −1.26404 −0.632022 0.774950i $$-0.717775\pi$$
−0.632022 + 0.774950i $$0.717775\pi$$
$$450$$ −1.73205 −0.0816497
$$451$$ −3.46410 −0.163118
$$452$$ −12.9282 −0.608092
$$453$$ −20.3923 −0.958114
$$454$$ −26.7846 −1.25706
$$455$$ −10.9282 −0.512322
$$456$$ 9.46410 0.443197
$$457$$ 12.3923 0.579688 0.289844 0.957074i $$-0.406397\pi$$
0.289844 + 0.957074i $$0.406397\pi$$
$$458$$ 41.3205 1.93078
$$459$$ 0 0
$$460$$ −6.92820 −0.323029
$$461$$ 36.2487 1.68827 0.844135 0.536130i $$-0.180114\pi$$
0.844135 + 0.536130i $$0.180114\pi$$
$$462$$ 3.46410 0.161165
$$463$$ −28.0000 −1.30127 −0.650635 0.759390i $$-0.725497\pi$$
−0.650635 + 0.759390i $$0.725497\pi$$
$$464$$ 17.3205 0.804084
$$465$$ 10.9282 0.506783
$$466$$ −20.7846 −0.962828
$$467$$ 5.07180 0.234695 0.117347 0.993091i $$-0.462561\pi$$
0.117347 + 0.993091i $$0.462561\pi$$
$$468$$ 5.46410 0.252578
$$469$$ 16.0000 0.738811
$$470$$ −12.0000 −0.553519
$$471$$ −3.07180 −0.141541
$$472$$ −12.0000 −0.552345
$$473$$ 4.92820 0.226599
$$474$$ 11.3205 0.519968
$$475$$ 5.46410 0.250710
$$476$$ 0 0
$$477$$ 0.928203 0.0424995
$$478$$ 20.7846 0.950666
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ −5.19615 −0.237171
$$481$$ −26.9282 −1.22782
$$482$$ −0.248711 −0.0113285
$$483$$ 13.8564 0.630488
$$484$$ 1.00000 0.0454545
$$485$$ 10.0000 0.454077
$$486$$ −1.73205 −0.0785674
$$487$$ −31.7128 −1.43704 −0.718522 0.695504i $$-0.755181\pi$$
−0.718522 + 0.695504i $$0.755181\pi$$
$$488$$ 3.46410 0.156813
$$489$$ 9.85641 0.445722
$$490$$ −5.19615 −0.234738
$$491$$ −30.9282 −1.39577 −0.697885 0.716210i $$-0.745875\pi$$
−0.697885 + 0.716210i $$0.745875\pi$$
$$492$$ 3.46410 0.156174
$$493$$ 0 0
$$494$$ −51.7128 −2.32667
$$495$$ 1.00000 0.0449467
$$496$$ 54.6410 2.45345
$$497$$ 27.7128 1.24309
$$498$$ −14.7846 −0.662514
$$499$$ 28.7846 1.28858 0.644288 0.764783i $$-0.277154\pi$$
0.644288 + 0.764783i $$0.277154\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −10.3923 −0.464294
$$502$$ 3.21539 0.143510
$$503$$ 31.1769 1.39011 0.695055 0.718957i $$-0.255380\pi$$
0.695055 + 0.718957i $$0.255380\pi$$
$$504$$ 3.46410 0.154303
$$505$$ 10.3923 0.462451
$$506$$ 12.0000 0.533465
$$507$$ 16.8564 0.748619
$$508$$ 8.92820 0.396125
$$509$$ 19.8564 0.880120 0.440060 0.897968i $$-0.354957\pi$$
0.440060 + 0.897968i $$0.354957\pi$$
$$510$$ 0 0
$$511$$ −16.7846 −0.742507
$$512$$ −8.66025 −0.382733
$$513$$ 5.46410 0.241246
$$514$$ 34.3923 1.51698
$$515$$ −8.00000 −0.352522
$$516$$ −4.92820 −0.216952
$$517$$ 6.92820 0.304702
$$518$$ 17.0718 0.750092
$$519$$ −12.0000 −0.526742
$$520$$ −9.46410 −0.415028
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ 18.9282 0.826882
$$525$$ 2.00000 0.0872872
$$526$$ −35.5692 −1.55089
$$527$$ 0 0
$$528$$ 5.00000 0.217597
$$529$$ 25.0000 1.08696
$$530$$ 1.60770 0.0698338
$$531$$ −6.92820 −0.300658
$$532$$ 10.9282 0.473798
$$533$$ 18.9282 0.819871
$$534$$ −1.60770 −0.0695718
$$535$$ −8.53590 −0.369039
$$536$$ 13.8564 0.598506
$$537$$ 6.92820 0.298974
$$538$$ −34.3923 −1.48276
$$539$$ 3.00000 0.129219
$$540$$ −1.00000 −0.0430331
$$541$$ 27.8564 1.19764 0.598820 0.800883i $$-0.295636\pi$$
0.598820 + 0.800883i $$0.295636\pi$$
$$542$$ 20.1051 0.863589
$$543$$ 15.8564 0.680464
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ −18.9282 −0.810052
$$547$$ 2.00000 0.0855138 0.0427569 0.999086i $$-0.486386\pi$$
0.0427569 + 0.999086i $$0.486386\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 2.00000 0.0853579
$$550$$ 1.73205 0.0738549
$$551$$ −18.9282 −0.806369
$$552$$ 12.0000 0.510754
$$553$$ −13.0718 −0.555869
$$554$$ −51.0333 −2.16820
$$555$$ 4.92820 0.209191
$$556$$ 12.3923 0.525551
$$557$$ 3.21539 0.136240 0.0681202 0.997677i $$-0.478300\pi$$
0.0681202 + 0.997677i $$0.478300\pi$$
$$558$$ 18.9282 0.801295
$$559$$ −26.9282 −1.13894
$$560$$ 10.0000 0.422577
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ 10.3923 0.437983 0.218992 0.975727i $$-0.429723\pi$$
0.218992 + 0.975727i $$0.429723\pi$$
$$564$$ −6.92820 −0.291730
$$565$$ 12.9282 0.543894
$$566$$ 8.53590 0.358791
$$567$$ 2.00000 0.0839921
$$568$$ 24.0000 1.00702
$$569$$ 5.32051 0.223047 0.111524 0.993762i $$-0.464427\pi$$
0.111524 + 0.993762i $$0.464427\pi$$
$$570$$ 9.46410 0.396408
$$571$$ 3.60770 0.150977 0.0754887 0.997147i $$-0.475948\pi$$
0.0754887 + 0.997147i $$0.475948\pi$$
$$572$$ −5.46410 −0.228466
$$573$$ −18.9282 −0.790737
$$574$$ −12.0000 −0.500870
$$575$$ 6.92820 0.288926
$$576$$ 1.00000 0.0416667
$$577$$ −18.7846 −0.782014 −0.391007 0.920388i $$-0.627873\pi$$
−0.391007 + 0.920388i $$0.627873\pi$$
$$578$$ 29.4449 1.22474
$$579$$ 24.3923 1.01371
$$580$$ 3.46410 0.143839
$$581$$ 17.0718 0.708257
$$582$$ 17.3205 0.717958
$$583$$ −0.928203 −0.0384422
$$584$$ −14.5359 −0.601500
$$585$$ −5.46410 −0.225913
$$586$$ 24.0000 0.991431
$$587$$ −18.9282 −0.781251 −0.390625 0.920550i $$-0.627741\pi$$
−0.390625 + 0.920550i $$0.627741\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ −59.7128 −2.46042
$$590$$ −12.0000 −0.494032
$$591$$ −12.0000 −0.493614
$$592$$ 24.6410 1.01274
$$593$$ 8.78461 0.360741 0.180370 0.983599i $$-0.442270\pi$$
0.180370 + 0.983599i $$0.442270\pi$$
$$594$$ 1.73205 0.0710669
$$595$$ 0 0
$$596$$ −15.4641 −0.633434
$$597$$ −24.7846 −1.01437
$$598$$ −65.5692 −2.68132
$$599$$ −37.8564 −1.54677 −0.773385 0.633936i $$-0.781438\pi$$
−0.773385 + 0.633936i $$0.781438\pi$$
$$600$$ 1.73205 0.0707107
$$601$$ −32.6410 −1.33145 −0.665727 0.746195i $$-0.731879\pi$$
−0.665727 + 0.746195i $$0.731879\pi$$
$$602$$ 17.0718 0.695794
$$603$$ 8.00000 0.325785
$$604$$ −20.3923 −0.829751
$$605$$ −1.00000 −0.0406558
$$606$$ 18.0000 0.731200
$$607$$ −18.7846 −0.762444 −0.381222 0.924484i $$-0.624497\pi$$
−0.381222 + 0.924484i $$0.624497\pi$$
$$608$$ 28.3923 1.15146
$$609$$ −6.92820 −0.280745
$$610$$ 3.46410 0.140257
$$611$$ −37.8564 −1.53151
$$612$$ 0 0
$$613$$ −20.3923 −0.823637 −0.411819 0.911266i $$-0.635106\pi$$
−0.411819 + 0.911266i $$0.635106\pi$$
$$614$$ −24.2487 −0.978598
$$615$$ −3.46410 −0.139686
$$616$$ −3.46410 −0.139573
$$617$$ −36.9282 −1.48667 −0.743337 0.668917i $$-0.766758\pi$$
−0.743337 + 0.668917i $$0.766758\pi$$
$$618$$ −13.8564 −0.557386
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 10.9282 0.438887
$$621$$ 6.92820 0.278019
$$622$$ −8.78461 −0.352231
$$623$$ 1.85641 0.0743754
$$624$$ −27.3205 −1.09370
$$625$$ 1.00000 0.0400000
$$626$$ −36.2487 −1.44879
$$627$$ −5.46410 −0.218215
$$628$$ −3.07180 −0.122578
$$629$$ 0 0
$$630$$ 3.46410 0.138013
$$631$$ −21.0718 −0.838855 −0.419427 0.907789i $$-0.637769\pi$$
−0.419427 + 0.907789i $$0.637769\pi$$
$$632$$ −11.3205 −0.450306
$$633$$ −8.39230 −0.333564
$$634$$ 43.1769 1.71477
$$635$$ −8.92820 −0.354305
$$636$$ 0.928203 0.0368057
$$637$$ −16.3923 −0.649487
$$638$$ −6.00000 −0.237542
$$639$$ 13.8564 0.548151
$$640$$ 12.1244 0.479257
$$641$$ −12.9282 −0.510633 −0.255317 0.966857i $$-0.582180\pi$$
−0.255317 + 0.966857i $$0.582180\pi$$
$$642$$ −14.7846 −0.583502
$$643$$ 37.5692 1.48159 0.740793 0.671734i $$-0.234450\pi$$
0.740793 + 0.671734i $$0.234450\pi$$
$$644$$ 13.8564 0.546019
$$645$$ 4.92820 0.194048
$$646$$ 0 0
$$647$$ 27.7128 1.08950 0.544752 0.838597i $$-0.316624\pi$$
0.544752 + 0.838597i $$0.316624\pi$$
$$648$$ 1.73205 0.0680414
$$649$$ 6.92820 0.271956
$$650$$ −9.46410 −0.371213
$$651$$ −21.8564 −0.856620
$$652$$ 9.85641 0.386007
$$653$$ 19.8564 0.777041 0.388521 0.921440i $$-0.372986\pi$$
0.388521 + 0.921440i $$0.372986\pi$$
$$654$$ 17.3205 0.677285
$$655$$ −18.9282 −0.739586
$$656$$ −17.3205 −0.676252
$$657$$ −8.39230 −0.327415
$$658$$ 24.0000 0.935617
$$659$$ −15.7128 −0.612084 −0.306042 0.952018i $$-0.599005\pi$$
−0.306042 + 0.952018i $$0.599005\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −17.0718 −0.663514
$$663$$ 0 0
$$664$$ 14.7846 0.573754
$$665$$ −10.9282 −0.423778
$$666$$ 8.53590 0.330759
$$667$$ −24.0000 −0.929284
$$668$$ −10.3923 −0.402090
$$669$$ 9.85641 0.381071
$$670$$ 13.8564 0.535320
$$671$$ −2.00000 −0.0772091
$$672$$ 10.3923 0.400892
$$673$$ −3.32051 −0.127996 −0.0639981 0.997950i $$-0.520385\pi$$
−0.0639981 + 0.997950i $$0.520385\pi$$
$$674$$ −57.4641 −2.21343
$$675$$ 1.00000 0.0384900
$$676$$ 16.8564 0.648323
$$677$$ 8.78461 0.337620 0.168810 0.985649i $$-0.446008\pi$$
0.168810 + 0.985649i $$0.446008\pi$$
$$678$$ 22.3923 0.859971
$$679$$ −20.0000 −0.767530
$$680$$ 0 0
$$681$$ 15.4641 0.592586
$$682$$ −18.9282 −0.724798
$$683$$ −32.7846 −1.25447 −0.627234 0.778831i $$-0.715813\pi$$
−0.627234 + 0.778831i $$0.715813\pi$$
$$684$$ 5.46410 0.208925
$$685$$ 18.0000 0.687745
$$686$$ 34.6410 1.32260
$$687$$ −23.8564 −0.910179
$$688$$ 24.6410 0.939430
$$689$$ 5.07180 0.193220
$$690$$ 12.0000 0.456832
$$691$$ 47.7128 1.81508 0.907540 0.419965i $$-0.137958\pi$$
0.907540 + 0.419965i $$0.137958\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ −2.00000 −0.0759737
$$694$$ −38.7846 −1.47224
$$695$$ −12.3923 −0.470067
$$696$$ −6.00000 −0.227429
$$697$$ 0 0
$$698$$ 14.1051 0.533887
$$699$$ 12.0000 0.453882
$$700$$ 2.00000 0.0755929
$$701$$ 39.4641 1.49054 0.745269 0.666764i $$-0.232321\pi$$
0.745269 + 0.666764i $$0.232321\pi$$
$$702$$ −9.46410 −0.357199
$$703$$ −26.9282 −1.01562
$$704$$ −1.00000 −0.0376889
$$705$$ 6.92820 0.260931
$$706$$ −22.3923 −0.842746
$$707$$ −20.7846 −0.781686
$$708$$ −6.92820 −0.260378
$$709$$ −11.8564 −0.445277 −0.222638 0.974901i $$-0.571467\pi$$
−0.222638 + 0.974901i $$0.571467\pi$$
$$710$$ 24.0000 0.900704
$$711$$ −6.53590 −0.245115
$$712$$ 1.60770 0.0602509
$$713$$ −75.7128 −2.83547
$$714$$ 0 0
$$715$$ 5.46410 0.204346
$$716$$ 6.92820 0.258919
$$717$$ −12.0000 −0.448148
$$718$$ 36.0000 1.34351
$$719$$ −5.07180 −0.189146 −0.0945731 0.995518i $$-0.530149\pi$$
−0.0945731 + 0.995518i $$0.530149\pi$$
$$720$$ 5.00000 0.186339
$$721$$ 16.0000 0.595871
$$722$$ −18.8038 −0.699807
$$723$$ 0.143594 0.00534030
$$724$$ 15.8564 0.589299
$$725$$ −3.46410 −0.128654
$$726$$ −1.73205 −0.0642824
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 18.9282 0.701526
$$729$$ 1.00000 0.0370370
$$730$$ −14.5359 −0.537998
$$731$$ 0 0
$$732$$ 2.00000 0.0739221
$$733$$ 53.9615 1.99311 0.996557 0.0829082i $$-0.0264208\pi$$
0.996557 + 0.0829082i $$0.0264208\pi$$
$$734$$ −34.6410 −1.27862
$$735$$ 3.00000 0.110657
$$736$$ 36.0000 1.32698
$$737$$ −8.00000 −0.294684
$$738$$ −6.00000 −0.220863
$$739$$ 17.4641 0.642427 0.321214 0.947007i $$-0.395909\pi$$
0.321214 + 0.947007i $$0.395909\pi$$
$$740$$ 4.92820 0.181164
$$741$$ 29.8564 1.09680
$$742$$ −3.21539 −0.118041
$$743$$ 25.6077 0.939455 0.469728 0.882811i $$-0.344352\pi$$
0.469728 + 0.882811i $$0.344352\pi$$
$$744$$ −18.9282 −0.693942
$$745$$ 15.4641 0.566561
$$746$$ 35.3205 1.29318
$$747$$ 8.53590 0.312312
$$748$$ 0 0
$$749$$ 17.0718 0.623790
$$750$$ 1.73205 0.0632456
$$751$$ 26.9282 0.982624 0.491312 0.870984i $$-0.336517\pi$$
0.491312 + 0.870984i $$0.336517\pi$$
$$752$$ 34.6410 1.26323
$$753$$ −1.85641 −0.0676512
$$754$$ 32.7846 1.19395
$$755$$ 20.3923 0.742152
$$756$$ 2.00000 0.0727393
$$757$$ 34.7846 1.26427 0.632134 0.774859i $$-0.282179\pi$$
0.632134 + 0.774859i $$0.282179\pi$$
$$758$$ 30.9282 1.12336
$$759$$ −6.92820 −0.251478
$$760$$ −9.46410 −0.343299
$$761$$ −32.5359 −1.17943 −0.589713 0.807613i $$-0.700759\pi$$
−0.589713 + 0.807613i $$0.700759\pi$$
$$762$$ −15.4641 −0.560205
$$763$$ −20.0000 −0.724049
$$764$$ −18.9282 −0.684798
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −37.8564 −1.36692
$$768$$ 19.0000 0.685603
$$769$$ 50.4974 1.82098 0.910492 0.413527i $$-0.135703\pi$$
0.910492 + 0.413527i $$0.135703\pi$$
$$770$$ −3.46410 −0.124838
$$771$$ −19.8564 −0.715111
$$772$$ 24.3923 0.877898
$$773$$ −4.14359 −0.149035 −0.0745174 0.997220i $$-0.523742\pi$$
−0.0745174 + 0.997220i $$0.523742\pi$$
$$774$$ 8.53590 0.306817
$$775$$ −10.9282 −0.392553
$$776$$ −17.3205 −0.621770
$$777$$ −9.85641 −0.353597
$$778$$ −19.1769 −0.687526
$$779$$ 18.9282 0.678173
$$780$$ −5.46410 −0.195646
$$781$$ −13.8564 −0.495821
$$782$$ 0 0
$$783$$ −3.46410 −0.123797
$$784$$ 15.0000 0.535714
$$785$$ 3.07180 0.109637
$$786$$ −32.7846 −1.16939
$$787$$ 22.7846 0.812184 0.406092 0.913832i $$-0.366891\pi$$
0.406092 + 0.913832i $$0.366891\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 20.5359 0.731097
$$790$$ −11.3205 −0.402766
$$791$$ −25.8564 −0.919348
$$792$$ −1.73205 −0.0615457
$$793$$ 10.9282 0.388072
$$794$$ −3.46410 −0.122936
$$795$$ −0.928203 −0.0329200
$$796$$ −24.7846 −0.878467
$$797$$ −52.6410 −1.86464 −0.932320 0.361634i $$-0.882219\pi$$
−0.932320 + 0.361634i $$0.882219\pi$$
$$798$$ −18.9282 −0.670051
$$799$$ 0 0
$$800$$ 5.19615 0.183712
$$801$$ 0.928203 0.0327964
$$802$$ 13.6077 0.480504
$$803$$ 8.39230 0.296158
$$804$$ 8.00000 0.282138
$$805$$ −13.8564 −0.488374
$$806$$ 103.426 3.64301
$$807$$ 19.8564 0.698979
$$808$$ −18.0000 −0.633238
$$809$$ −15.4641 −0.543689 −0.271844 0.962341i $$-0.587634\pi$$
−0.271844 + 0.962341i $$0.587634\pi$$
$$810$$ 1.73205 0.0608581
$$811$$ 12.3923 0.435153 0.217576 0.976043i $$-0.430185\pi$$
0.217576 + 0.976043i $$0.430185\pi$$
$$812$$ −6.92820 −0.243132
$$813$$ −11.6077 −0.407100
$$814$$ −8.53590 −0.299183
$$815$$ −9.85641 −0.345255
$$816$$ 0 0
$$817$$ −26.9282 −0.942099
$$818$$ 11.7513 0.410874
$$819$$ 10.9282 0.381862
$$820$$ −3.46410 −0.120972
$$821$$ 20.5359 0.716708 0.358354 0.933586i $$-0.383338\pi$$
0.358354 + 0.933586i $$0.383338\pi$$
$$822$$ 31.1769 1.08742
$$823$$ −33.5692 −1.17015 −0.585075 0.810979i $$-0.698935\pi$$
−0.585075 + 0.810979i $$0.698935\pi$$
$$824$$ 13.8564 0.482711
$$825$$ −1.00000 −0.0348155
$$826$$ 24.0000 0.835067
$$827$$ 22.3923 0.778657 0.389328 0.921099i $$-0.372707\pi$$
0.389328 + 0.921099i $$0.372707\pi$$
$$828$$ 6.92820 0.240772
$$829$$ 29.7128 1.03197 0.515984 0.856598i $$-0.327426\pi$$
0.515984 + 0.856598i $$0.327426\pi$$
$$830$$ 14.7846 0.513181
$$831$$ 29.4641 1.02210
$$832$$ 5.46410 0.189434
$$833$$ 0 0
$$834$$ −21.4641 −0.743241
$$835$$ 10.3923 0.359641
$$836$$ −5.46410 −0.188980
$$837$$ −10.9282 −0.377734
$$838$$ −53.5692 −1.85052
$$839$$ 56.7846 1.96042 0.980211 0.197954i $$-0.0634298\pi$$
0.980211 + 0.197954i $$0.0634298\pi$$
$$840$$ −3.46410 −0.119523
$$841$$ −17.0000 −0.586207
$$842$$ −3.46410 −0.119381
$$843$$ 3.46410 0.119310
$$844$$ −8.39230 −0.288875
$$845$$ −16.8564 −0.579878
$$846$$ 12.0000 0.412568
$$847$$ 2.00000 0.0687208
$$848$$ −4.64102 −0.159373
$$849$$ −4.92820 −0.169135
$$850$$ 0 0
$$851$$ −34.1436 −1.17043
$$852$$ 13.8564 0.474713
$$853$$ 3.60770 0.123525 0.0617626 0.998091i $$-0.480328\pi$$
0.0617626 + 0.998091i $$0.480328\pi$$
$$854$$ −6.92820 −0.237078
$$855$$ −5.46410 −0.186868
$$856$$ 14.7846 0.505328
$$857$$ −37.8564 −1.29315 −0.646575 0.762850i $$-0.723799\pi$$
−0.646575 + 0.762850i $$0.723799\pi$$
$$858$$ 9.46410 0.323099
$$859$$ −7.71281 −0.263158 −0.131579 0.991306i $$-0.542005\pi$$
−0.131579 + 0.991306i $$0.542005\pi$$
$$860$$ 4.92820 0.168050
$$861$$ 6.92820 0.236113
$$862$$ −15.2154 −0.518238
$$863$$ −37.8564 −1.28865 −0.644324 0.764753i $$-0.722861\pi$$
−0.644324 + 0.764753i $$0.722861\pi$$
$$864$$ 5.19615 0.176777
$$865$$ 12.0000 0.408012
$$866$$ −0.248711 −0.00845155
$$867$$ −17.0000 −0.577350
$$868$$ −21.8564 −0.741855
$$869$$ 6.53590 0.221715
$$870$$ −6.00000 −0.203419
$$871$$ 43.7128 1.48115
$$872$$ −17.3205 −0.586546
$$873$$ −10.0000 −0.338449
$$874$$ −65.5692 −2.21791
$$875$$ −2.00000 −0.0676123
$$876$$ −8.39230 −0.283550
$$877$$ −34.2487 −1.15650 −0.578248 0.815861i $$-0.696264\pi$$
−0.578248 + 0.815861i $$0.696264\pi$$
$$878$$ −57.4641 −1.93932
$$879$$ −13.8564 −0.467365
$$880$$ −5.00000 −0.168550
$$881$$ −0.928203 −0.0312720 −0.0156360 0.999878i $$-0.504977\pi$$
−0.0156360 + 0.999878i $$0.504977\pi$$
$$882$$ 5.19615 0.174964
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 0 0
$$885$$ 6.92820 0.232889
$$886$$ −20.7846 −0.698273
$$887$$ 12.2487 0.411271 0.205636 0.978629i $$-0.434074\pi$$
0.205636 + 0.978629i $$0.434074\pi$$
$$888$$ −8.53590 −0.286446
$$889$$ 17.8564 0.598885
$$890$$ 1.60770 0.0538901
$$891$$ −1.00000 −0.0335013
$$892$$ 9.85641 0.330017
$$893$$ −37.8564 −1.26682
$$894$$ 26.7846 0.895811
$$895$$ −6.92820 −0.231584
$$896$$ −24.2487 −0.810093
$$897$$ 37.8564 1.26399
$$898$$ 46.3923 1.54813
$$899$$ 37.8564 1.26258
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 6.00000 0.199778
$$903$$ −9.85641 −0.328001
$$904$$ −22.3923 −0.744757
$$905$$ −15.8564 −0.527085
$$906$$ 35.3205 1.17345
$$907$$ 18.1436 0.602448 0.301224 0.953553i $$-0.402605\pi$$
0.301224 + 0.953553i $$0.402605\pi$$
$$908$$ 15.4641 0.513194
$$909$$ −10.3923 −0.344691
$$910$$ 18.9282 0.627464
$$911$$ 18.9282 0.627119 0.313560 0.949568i $$-0.398478\pi$$
0.313560 + 0.949568i $$0.398478\pi$$
$$912$$ −27.3205 −0.904672
$$913$$ −8.53590 −0.282497
$$914$$ −21.4641 −0.709969
$$915$$ −2.00000 −0.0661180
$$916$$ −23.8564 −0.788238
$$917$$ 37.8564 1.25013
$$918$$ 0 0
$$919$$ −32.3923 −1.06852 −0.534262 0.845319i $$-0.679410\pi$$
−0.534262 + 0.845319i $$0.679410\pi$$
$$920$$ −12.0000 −0.395628
$$921$$ 14.0000 0.461316
$$922$$ −62.7846 −2.06770
$$923$$ 75.7128 2.49212
$$924$$ −2.00000 −0.0657952
$$925$$ −4.92820 −0.162038
$$926$$ 48.4974 1.59372
$$927$$ 8.00000 0.262754
$$928$$ −18.0000 −0.590879
$$929$$ 2.78461 0.0913601 0.0456800 0.998956i $$-0.485455\pi$$
0.0456800 + 0.998956i $$0.485455\pi$$
$$930$$ −18.9282 −0.620680
$$931$$ −16.3923 −0.537236
$$932$$ 12.0000 0.393073
$$933$$ 5.07180 0.166043
$$934$$ −8.78461 −0.287441
$$935$$ 0 0
$$936$$ 9.46410 0.309344
$$937$$ −20.3923 −0.666188 −0.333094 0.942894i $$-0.608093\pi$$
−0.333094 + 0.942894i $$0.608093\pi$$
$$938$$ −27.7128 −0.904855
$$939$$ 20.9282 0.682966
$$940$$ 6.92820 0.225973
$$941$$ 27.4641 0.895304 0.447652 0.894208i $$-0.352260\pi$$
0.447652 + 0.894208i $$0.352260\pi$$
$$942$$ 5.32051 0.173352
$$943$$ 24.0000 0.781548
$$944$$ 34.6410 1.12747
$$945$$ −2.00000 −0.0650600
$$946$$ −8.53590 −0.277526
$$947$$ −18.9282 −0.615084 −0.307542 0.951535i $$-0.599506\pi$$
−0.307542 + 0.951535i $$0.599506\pi$$
$$948$$ −6.53590 −0.212276
$$949$$ −45.8564 −1.48856
$$950$$ −9.46410 −0.307056
$$951$$ −24.9282 −0.808352
$$952$$ 0 0
$$953$$ −3.21539 −0.104157 −0.0520784 0.998643i $$-0.516585\pi$$
−0.0520784 + 0.998643i $$0.516585\pi$$
$$954$$ −1.60770 −0.0520511
$$955$$ 18.9282 0.612502
$$956$$ −12.0000 −0.388108
$$957$$ 3.46410 0.111979
$$958$$ −20.7846 −0.671520
$$959$$ −36.0000 −1.16250
$$960$$ −1.00000 −0.0322749
$$961$$ 88.4256 2.85244
$$962$$ 46.6410 1.50377
$$963$$ 8.53590 0.275065
$$964$$ 0.143594 0.00462484
$$965$$ −24.3923 −0.785216
$$966$$ −24.0000 −0.772187
$$967$$ 22.7846 0.732704 0.366352 0.930476i $$-0.380607\pi$$
0.366352 + 0.930476i $$0.380607\pi$$
$$968$$ 1.73205 0.0556702
$$969$$ 0 0
$$970$$ −17.3205 −0.556128
$$971$$ 25.8564 0.829772 0.414886 0.909873i $$-0.363822\pi$$
0.414886 + 0.909873i $$0.363822\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 24.7846 0.794558
$$974$$ 54.9282 1.76001
$$975$$ 5.46410 0.174991
$$976$$ −10.0000 −0.320092
$$977$$ 47.5692 1.52187 0.760937 0.648826i $$-0.224740\pi$$
0.760937 + 0.648826i $$0.224740\pi$$
$$978$$ −17.0718 −0.545896
$$979$$ −0.928203 −0.0296655
$$980$$ 3.00000 0.0958315
$$981$$ −10.0000 −0.319275
$$982$$ 53.5692 1.70946
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ −13.8564 −0.441054
$$988$$ 29.8564 0.949859
$$989$$ −34.1436 −1.08570
$$990$$ −1.73205 −0.0550482
$$991$$ −7.21539 −0.229204 −0.114602 0.993411i $$-0.536559\pi$$
−0.114602 + 0.993411i $$0.536559\pi$$
$$992$$ −56.7846 −1.80291
$$993$$ 9.85641 0.312784
$$994$$ −48.0000 −1.52247
$$995$$ 24.7846 0.785725
$$996$$ 8.53590 0.270470
$$997$$ 27.6077 0.874344 0.437172 0.899378i $$-0.355980\pi$$
0.437172 + 0.899378i $$0.355980\pi$$
$$998$$ −49.8564 −1.57818
$$999$$ −4.92820 −0.155921
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 165.2.a.b.1.1 2
3.2 odd 2 495.2.a.c.1.2 2
4.3 odd 2 2640.2.a.x.1.2 2
5.2 odd 4 825.2.c.c.199.1 4
5.3 odd 4 825.2.c.c.199.4 4
5.4 even 2 825.2.a.e.1.2 2
7.6 odd 2 8085.2.a.bd.1.1 2
11.10 odd 2 1815.2.a.i.1.2 2
12.11 even 2 7920.2.a.bz.1.2 2
15.2 even 4 2475.2.c.n.199.4 4
15.8 even 4 2475.2.c.n.199.1 4
15.14 odd 2 2475.2.a.r.1.1 2
33.32 even 2 5445.2.a.s.1.1 2
55.54 odd 2 9075.2.a.bh.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 1.1 even 1 trivial
495.2.a.c.1.2 2 3.2 odd 2
825.2.a.e.1.2 2 5.4 even 2
825.2.c.c.199.1 4 5.2 odd 4
825.2.c.c.199.4 4 5.3 odd 4
1815.2.a.i.1.2 2 11.10 odd 2
2475.2.a.r.1.1 2 15.14 odd 2
2475.2.c.n.199.1 4 15.8 even 4
2475.2.c.n.199.4 4 15.2 even 4
2640.2.a.x.1.2 2 4.3 odd 2
5445.2.a.s.1.1 2 33.32 even 2
7920.2.a.bz.1.2 2 12.11 even 2
8085.2.a.bd.1.1 2 7.6 odd 2
9075.2.a.bh.1.1 2 55.54 odd 2