# Properties

 Label 5070.2.a.z.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 $1$

# Related objects

## 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 390) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 5070.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.60555 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -2.60555 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.60555 q^{17} -1.00000 q^{18} -2.60555 q^{19} -1.00000 q^{20} -2.60555 q^{21} +8.60555 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.60555 q^{28} -2.60555 q^{29} -1.00000 q^{30} -6.00000 q^{31} -1.00000 q^{32} -2.60555 q^{34} -2.60555 q^{35} +1.00000 q^{36} +5.21110 q^{37} +2.60555 q^{38} +1.00000 q^{40} +11.2111 q^{41} +2.60555 q^{42} -8.00000 q^{43} -1.00000 q^{45} -8.60555 q^{46} +5.21110 q^{47} -1.00000 q^{48} -0.211103 q^{49} -1.00000 q^{50} -2.60555 q^{51} +6.00000 q^{53} +1.00000 q^{54} -2.60555 q^{56} +2.60555 q^{57} +2.60555 q^{58} +5.21110 q^{59} +1.00000 q^{60} +3.21110 q^{61} +6.00000 q^{62} +2.60555 q^{63} +1.00000 q^{64} -11.2111 q^{67} +2.60555 q^{68} -8.60555 q^{69} +2.60555 q^{70} +5.21110 q^{71} -1.00000 q^{72} -8.60555 q^{73} -5.21110 q^{74} -1.00000 q^{75} -2.60555 q^{76} +14.4222 q^{79} -1.00000 q^{80} +1.00000 q^{81} -11.2111 q^{82} -17.2111 q^{83} -2.60555 q^{84} -2.60555 q^{85} +8.00000 q^{86} +2.60555 q^{87} +0.788897 q^{89} +1.00000 q^{90} +8.60555 q^{92} +6.00000 q^{93} -5.21110 q^{94} +2.60555 q^{95} +1.00000 q^{96} -8.60555 q^{97} +0.211103 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} + 2q^{14} + 2q^{15} + 2q^{16} - 2q^{17} - 2q^{18} + 2q^{19} - 2q^{20} + 2q^{21} + 10q^{23} + 2q^{24} + 2q^{25} - 2q^{27} - 2q^{28} + 2q^{29} - 2q^{30} - 12q^{31} - 2q^{32} + 2q^{34} + 2q^{35} + 2q^{36} - 4q^{37} - 2q^{38} + 2q^{40} + 8q^{41} - 2q^{42} - 16q^{43} - 2q^{45} - 10q^{46} - 4q^{47} - 2q^{48} + 14q^{49} - 2q^{50} + 2q^{51} + 12q^{53} + 2q^{54} + 2q^{56} - 2q^{57} - 2q^{58} - 4q^{59} + 2q^{60} - 8q^{61} + 12q^{62} - 2q^{63} + 2q^{64} - 8q^{67} - 2q^{68} - 10q^{69} - 2q^{70} - 4q^{71} - 2q^{72} - 10q^{73} + 4q^{74} - 2q^{75} + 2q^{76} - 2q^{80} + 2q^{81} - 8q^{82} - 20q^{83} + 2q^{84} + 2q^{85} + 16q^{86} - 2q^{87} + 16q^{89} + 2q^{90} + 10q^{92} + 12q^{93} + 4q^{94} - 2q^{95} + 2q^{96} - 10q^{97} - 14q^{98} + O(q^{100})$$

## 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
$$7$$ 2.60555 0.984806 0.492403 0.870367i $$-0.336119\pi$$
0.492403 + 0.870367i $$0.336119\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0
$$14$$ −2.60555 −0.696363
$$15$$ 1.00000 0.258199
$$16$$ 1.00000 0.250000
$$17$$ 2.60555 0.631939 0.315970 0.948769i $$-0.397670\pi$$
0.315970 + 0.948769i $$0.397670\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −2.60555 −0.597754 −0.298877 0.954292i $$-0.596612\pi$$
−0.298877 + 0.954292i $$0.596612\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ −2.60555 −0.568578
$$22$$ 0 0
$$23$$ 8.60555 1.79438 0.897191 0.441643i $$-0.145604\pi$$
0.897191 + 0.441643i $$0.145604\pi$$
$$24$$ 1.00000 0.204124
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 2.60555 0.492403
$$29$$ −2.60555 −0.483839 −0.241919 0.970296i $$-0.577777\pi$$
−0.241919 + 0.970296i $$0.577777\pi$$
$$30$$ −1.00000 −0.182574
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ −2.60555 −0.446848
$$35$$ −2.60555 −0.440419
$$36$$ 1.00000 0.166667
$$37$$ 5.21110 0.856700 0.428350 0.903613i $$-0.359095\pi$$
0.428350 + 0.903613i $$0.359095\pi$$
$$38$$ 2.60555 0.422676
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 11.2111 1.75088 0.875440 0.483327i $$-0.160572\pi$$
0.875440 + 0.483327i $$0.160572\pi$$
$$42$$ 2.60555 0.402045
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ −8.60555 −1.26882
$$47$$ 5.21110 0.760117 0.380059 0.924962i $$-0.375904\pi$$
0.380059 + 0.924962i $$0.375904\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −0.211103 −0.0301575
$$50$$ −1.00000 −0.141421
$$51$$ −2.60555 −0.364850
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −2.60555 −0.348181
$$57$$ 2.60555 0.345114
$$58$$ 2.60555 0.342126
$$59$$ 5.21110 0.678428 0.339214 0.940709i $$-0.389839\pi$$
0.339214 + 0.940709i $$0.389839\pi$$
$$60$$ 1.00000 0.129099
$$61$$ 3.21110 0.411140 0.205570 0.978642i $$-0.434095\pi$$
0.205570 + 0.978642i $$0.434095\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 2.60555 0.328269
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.2111 −1.36965 −0.684827 0.728706i $$-0.740122\pi$$
−0.684827 + 0.728706i $$0.740122\pi$$
$$68$$ 2.60555 0.315970
$$69$$ −8.60555 −1.03599
$$70$$ 2.60555 0.311423
$$71$$ 5.21110 0.618444 0.309222 0.950990i $$-0.399931\pi$$
0.309222 + 0.950990i $$0.399931\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −8.60555 −1.00720 −0.503602 0.863936i $$-0.667992\pi$$
−0.503602 + 0.863936i $$0.667992\pi$$
$$74$$ −5.21110 −0.605778
$$75$$ −1.00000 −0.115470
$$76$$ −2.60555 −0.298877
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −11.2111 −1.23806
$$83$$ −17.2111 −1.88916 −0.944582 0.328276i $$-0.893533\pi$$
−0.944582 + 0.328276i $$0.893533\pi$$
$$84$$ −2.60555 −0.284289
$$85$$ −2.60555 −0.282612
$$86$$ 8.00000 0.862662
$$87$$ 2.60555 0.279344
$$88$$ 0 0
$$89$$ 0.788897 0.0836230 0.0418115 0.999126i $$-0.486687\pi$$
0.0418115 + 0.999126i $$0.486687\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 8.60555 0.897191
$$93$$ 6.00000 0.622171
$$94$$ −5.21110 −0.537484
$$95$$ 2.60555 0.267324
$$96$$ 1.00000 0.102062
$$97$$ −8.60555 −0.873761 −0.436881 0.899519i $$-0.643917\pi$$
−0.436881 + 0.899519i $$0.643917\pi$$
$$98$$ 0.211103 0.0213246
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 14.6056 1.45331 0.726653 0.687004i $$-0.241075\pi$$
0.726653 + 0.687004i $$0.241075\pi$$
$$102$$ 2.60555 0.257988
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 2.60555 0.254276
$$106$$ −6.00000 −0.582772
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −8.60555 −0.824262 −0.412131 0.911125i $$-0.635215\pi$$
−0.412131 + 0.911125i $$0.635215\pi$$
$$110$$ 0 0
$$111$$ −5.21110 −0.494616
$$112$$ 2.60555 0.246201
$$113$$ 7.81665 0.735329 0.367664 0.929959i $$-0.380157\pi$$
0.367664 + 0.929959i $$0.380157\pi$$
$$114$$ −2.60555 −0.244032
$$115$$ −8.60555 −0.802472
$$116$$ −2.60555 −0.241919
$$117$$ 0 0
$$118$$ −5.21110 −0.479721
$$119$$ 6.78890 0.622337
$$120$$ −1.00000 −0.0912871
$$121$$ −11.0000 −1.00000
$$122$$ −3.21110 −0.290720
$$123$$ −11.2111 −1.01087
$$124$$ −6.00000 −0.538816
$$125$$ −1.00000 −0.0894427
$$126$$ −2.60555 −0.232121
$$127$$ 13.2111 1.17230 0.586148 0.810204i $$-0.300644\pi$$
0.586148 + 0.810204i $$0.300644\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ 15.3944 1.34502 0.672510 0.740088i $$-0.265216\pi$$
0.672510 + 0.740088i $$0.265216\pi$$
$$132$$ 0 0
$$133$$ −6.78890 −0.588672
$$134$$ 11.2111 0.968492
$$135$$ 1.00000 0.0860663
$$136$$ −2.60555 −0.223424
$$137$$ −11.2111 −0.957829 −0.478915 0.877862i $$-0.658970\pi$$
−0.478915 + 0.877862i $$0.658970\pi$$
$$138$$ 8.60555 0.732553
$$139$$ 2.78890 0.236551 0.118276 0.992981i $$-0.462263\pi$$
0.118276 + 0.992981i $$0.462263\pi$$
$$140$$ −2.60555 −0.220209
$$141$$ −5.21110 −0.438854
$$142$$ −5.21110 −0.437306
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 2.60555 0.216379
$$146$$ 8.60555 0.712200
$$147$$ 0.211103 0.0174114
$$148$$ 5.21110 0.428350
$$149$$ −0.788897 −0.0646290 −0.0323145 0.999478i $$-0.510288\pi$$
−0.0323145 + 0.999478i $$0.510288\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 6.00000 0.488273 0.244137 0.969741i $$-0.421495\pi$$
0.244137 + 0.969741i $$0.421495\pi$$
$$152$$ 2.60555 0.211338
$$153$$ 2.60555 0.210646
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −8.42221 −0.672165 −0.336083 0.941833i $$-0.609102\pi$$
−0.336083 + 0.941833i $$0.609102\pi$$
$$158$$ −14.4222 −1.14737
$$159$$ −6.00000 −0.475831
$$160$$ 1.00000 0.0790569
$$161$$ 22.4222 1.76712
$$162$$ −1.00000 −0.0785674
$$163$$ −4.42221 −0.346374 −0.173187 0.984889i $$-0.555406\pi$$
−0.173187 + 0.984889i $$0.555406\pi$$
$$164$$ 11.2111 0.875440
$$165$$ 0 0
$$166$$ 17.2111 1.33584
$$167$$ −5.21110 −0.403247 −0.201624 0.979463i $$-0.564622\pi$$
−0.201624 + 0.979463i $$0.564622\pi$$
$$168$$ 2.60555 0.201023
$$169$$ 0 0
$$170$$ 2.60555 0.199837
$$171$$ −2.60555 −0.199251
$$172$$ −8.00000 −0.609994
$$173$$ −16.4222 −1.24856 −0.624279 0.781202i $$-0.714607\pi$$
−0.624279 + 0.781202i $$0.714607\pi$$
$$174$$ −2.60555 −0.197526
$$175$$ 2.60555 0.196961
$$176$$ 0 0
$$177$$ −5.21110 −0.391690
$$178$$ −0.788897 −0.0591304
$$179$$ −1.81665 −0.135783 −0.0678915 0.997693i $$-0.521627\pi$$
−0.0678915 + 0.997693i $$0.521627\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 20.4222 1.51797 0.758985 0.651108i $$-0.225695\pi$$
0.758985 + 0.651108i $$0.225695\pi$$
$$182$$ 0 0
$$183$$ −3.21110 −0.237372
$$184$$ −8.60555 −0.634410
$$185$$ −5.21110 −0.383128
$$186$$ −6.00000 −0.439941
$$187$$ 0 0
$$188$$ 5.21110 0.380059
$$189$$ −2.60555 −0.189526
$$190$$ −2.60555 −0.189027
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −13.8167 −0.994545 −0.497272 0.867595i $$-0.665665\pi$$
−0.497272 + 0.867595i $$0.665665\pi$$
$$194$$ 8.60555 0.617843
$$195$$ 0 0
$$196$$ −0.211103 −0.0150788
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 6.42221 0.455258 0.227629 0.973748i $$-0.426903\pi$$
0.227629 + 0.973748i $$0.426903\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 11.2111 0.790770
$$202$$ −14.6056 −1.02764
$$203$$ −6.78890 −0.476487
$$204$$ −2.60555 −0.182425
$$205$$ −11.2111 −0.783017
$$206$$ −4.00000 −0.278693
$$207$$ 8.60555 0.598127
$$208$$ 0 0
$$209$$ 0 0
$$210$$ −2.60555 −0.179800
$$211$$ −2.78890 −0.191996 −0.0959978 0.995382i $$-0.530604\pi$$
−0.0959978 + 0.995382i $$0.530604\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −5.21110 −0.357059
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 1.00000 0.0680414
$$217$$ −15.6333 −1.06126
$$218$$ 8.60555 0.582841
$$219$$ 8.60555 0.581509
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 5.21110 0.349746
$$223$$ 19.8167 1.32702 0.663511 0.748167i $$-0.269066\pi$$
0.663511 + 0.748167i $$0.269066\pi$$
$$224$$ −2.60555 −0.174091
$$225$$ 1.00000 0.0666667
$$226$$ −7.81665 −0.519956
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ 2.60555 0.172557
$$229$$ 1.81665 0.120048 0.0600239 0.998197i $$-0.480882\pi$$
0.0600239 + 0.998197i $$0.480882\pi$$
$$230$$ 8.60555 0.567433
$$231$$ 0 0
$$232$$ 2.60555 0.171063
$$233$$ −19.8167 −1.29823 −0.649116 0.760689i $$-0.724861\pi$$
−0.649116 + 0.760689i $$0.724861\pi$$
$$234$$ 0 0
$$235$$ −5.21110 −0.339935
$$236$$ 5.21110 0.339214
$$237$$ −14.4222 −0.936823
$$238$$ −6.78890 −0.440059
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 1.00000 0.0645497
$$241$$ 22.4222 1.44434 0.722171 0.691715i $$-0.243145\pi$$
0.722171 + 0.691715i $$0.243145\pi$$
$$242$$ 11.0000 0.707107
$$243$$ −1.00000 −0.0641500
$$244$$ 3.21110 0.205570
$$245$$ 0.211103 0.0134868
$$246$$ 11.2111 0.714794
$$247$$ 0 0
$$248$$ 6.00000 0.381000
$$249$$ 17.2111 1.09071
$$250$$ 1.00000 0.0632456
$$251$$ 20.6056 1.30061 0.650305 0.759673i $$-0.274641\pi$$
0.650305 + 0.759673i $$0.274641\pi$$
$$252$$ 2.60555 0.164134
$$253$$ 0 0
$$254$$ −13.2111 −0.828938
$$255$$ 2.60555 0.163166
$$256$$ 1.00000 0.0625000
$$257$$ 21.3944 1.33455 0.667275 0.744812i $$-0.267461\pi$$
0.667275 + 0.744812i $$0.267461\pi$$
$$258$$ −8.00000 −0.498058
$$259$$ 13.5778 0.843683
$$260$$ 0 0
$$261$$ −2.60555 −0.161280
$$262$$ −15.3944 −0.951072
$$263$$ 13.8167 0.851971 0.425986 0.904730i $$-0.359927\pi$$
0.425986 + 0.904730i $$0.359927\pi$$
$$264$$ 0 0
$$265$$ −6.00000 −0.368577
$$266$$ 6.78890 0.416254
$$267$$ −0.788897 −0.0482797
$$268$$ −11.2111 −0.684827
$$269$$ 4.18335 0.255063 0.127532 0.991835i $$-0.459295\pi$$
0.127532 + 0.991835i $$0.459295\pi$$
$$270$$ −1.00000 −0.0608581
$$271$$ 28.4222 1.72653 0.863263 0.504754i $$-0.168417\pi$$
0.863263 + 0.504754i $$0.168417\pi$$
$$272$$ 2.60555 0.157985
$$273$$ 0 0
$$274$$ 11.2111 0.677287
$$275$$ 0 0
$$276$$ −8.60555 −0.517993
$$277$$ −12.4222 −0.746378 −0.373189 0.927755i $$-0.621736\pi$$
−0.373189 + 0.927755i $$0.621736\pi$$
$$278$$ −2.78890 −0.167267
$$279$$ −6.00000 −0.359211
$$280$$ 2.60555 0.155711
$$281$$ 12.7889 0.762922 0.381461 0.924385i $$-0.375421\pi$$
0.381461 + 0.924385i $$0.375421\pi$$
$$282$$ 5.21110 0.310317
$$283$$ 18.4222 1.09509 0.547543 0.836777i $$-0.315563\pi$$
0.547543 + 0.836777i $$0.315563\pi$$
$$284$$ 5.21110 0.309222
$$285$$ −2.60555 −0.154340
$$286$$ 0 0
$$287$$ 29.2111 1.72428
$$288$$ −1.00000 −0.0589256
$$289$$ −10.2111 −0.600653
$$290$$ −2.60555 −0.153003
$$291$$ 8.60555 0.504466
$$292$$ −8.60555 −0.503602
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ −0.211103 −0.0123118
$$295$$ −5.21110 −0.303402
$$296$$ −5.21110 −0.302889
$$297$$ 0 0
$$298$$ 0.788897 0.0456996
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ −20.8444 −1.20145
$$302$$ −6.00000 −0.345261
$$303$$ −14.6056 −0.839067
$$304$$ −2.60555 −0.149439
$$305$$ −3.21110 −0.183867
$$306$$ −2.60555 −0.148949
$$307$$ −23.2111 −1.32473 −0.662364 0.749182i $$-0.730447\pi$$
−0.662364 + 0.749182i $$0.730447\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ −6.00000 −0.340777
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 32.4222 1.83261 0.916306 0.400480i $$-0.131156\pi$$
0.916306 + 0.400480i $$0.131156\pi$$
$$314$$ 8.42221 0.475293
$$315$$ −2.60555 −0.146806
$$316$$ 14.4222 0.811312
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ −22.4222 −1.24954
$$323$$ −6.78890 −0.377744
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 4.42221 0.244923
$$327$$ 8.60555 0.475888
$$328$$ −11.2111 −0.619030
$$329$$ 13.5778 0.748568
$$330$$ 0 0
$$331$$ 9.39445 0.516366 0.258183 0.966096i $$-0.416876\pi$$
0.258183 + 0.966096i $$0.416876\pi$$
$$332$$ −17.2111 −0.944582
$$333$$ 5.21110 0.285567
$$334$$ 5.21110 0.285139
$$335$$ 11.2111 0.612528
$$336$$ −2.60555 −0.142144
$$337$$ 29.6333 1.61423 0.807115 0.590395i $$-0.201028\pi$$
0.807115 + 0.590395i $$0.201028\pi$$
$$338$$ 0 0
$$339$$ −7.81665 −0.424542
$$340$$ −2.60555 −0.141306
$$341$$ 0 0
$$342$$ 2.60555 0.140892
$$343$$ −18.7889 −1.01451
$$344$$ 8.00000 0.431331
$$345$$ 8.60555 0.463307
$$346$$ 16.4222 0.882863
$$347$$ 15.6333 0.839240 0.419620 0.907700i $$-0.362163\pi$$
0.419620 + 0.907700i $$0.362163\pi$$
$$348$$ 2.60555 0.139672
$$349$$ 13.8167 0.739589 0.369794 0.929114i $$-0.379428\pi$$
0.369794 + 0.929114i $$0.379428\pi$$
$$350$$ −2.60555 −0.139273
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 23.2111 1.23540 0.617701 0.786413i $$-0.288064\pi$$
0.617701 + 0.786413i $$0.288064\pi$$
$$354$$ 5.21110 0.276967
$$355$$ −5.21110 −0.276577
$$356$$ 0.788897 0.0418115
$$357$$ −6.78890 −0.359307
$$358$$ 1.81665 0.0960131
$$359$$ −27.6333 −1.45843 −0.729215 0.684285i $$-0.760115\pi$$
−0.729215 + 0.684285i $$0.760115\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −12.2111 −0.642690
$$362$$ −20.4222 −1.07337
$$363$$ 11.0000 0.577350
$$364$$ 0 0
$$365$$ 8.60555 0.450435
$$366$$ 3.21110 0.167847
$$367$$ 23.6333 1.23365 0.616824 0.787101i $$-0.288419\pi$$
0.616824 + 0.787101i $$0.288419\pi$$
$$368$$ 8.60555 0.448595
$$369$$ 11.2111 0.583627
$$370$$ 5.21110 0.270912
$$371$$ 15.6333 0.811641
$$372$$ 6.00000 0.311086
$$373$$ 8.42221 0.436085 0.218043 0.975939i $$-0.430033\pi$$
0.218043 + 0.975939i $$0.430033\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ −5.21110 −0.268742
$$377$$ 0 0
$$378$$ 2.60555 0.134015
$$379$$ 1.02776 0.0527923 0.0263961 0.999652i $$-0.491597\pi$$
0.0263961 + 0.999652i $$0.491597\pi$$
$$380$$ 2.60555 0.133662
$$381$$ −13.2111 −0.676825
$$382$$ 12.0000 0.613973
$$383$$ −15.6333 −0.798825 −0.399412 0.916771i $$-0.630786\pi$$
−0.399412 + 0.916771i $$0.630786\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 13.8167 0.703249
$$387$$ −8.00000 −0.406663
$$388$$ −8.60555 −0.436881
$$389$$ 2.60555 0.132107 0.0660533 0.997816i $$-0.478959\pi$$
0.0660533 + 0.997816i $$0.478959\pi$$
$$390$$ 0 0
$$391$$ 22.4222 1.13394
$$392$$ 0.211103 0.0106623
$$393$$ −15.3944 −0.776547
$$394$$ −6.00000 −0.302276
$$395$$ −14.4222 −0.725660
$$396$$ 0 0
$$397$$ 39.6333 1.98914 0.994569 0.104076i $$-0.0331885\pi$$
0.994569 + 0.104076i $$0.0331885\pi$$
$$398$$ −6.42221 −0.321916
$$399$$ 6.78890 0.339870
$$400$$ 1.00000 0.0500000
$$401$$ 23.2111 1.15911 0.579554 0.814934i $$-0.303227\pi$$
0.579554 + 0.814934i $$0.303227\pi$$
$$402$$ −11.2111 −0.559159
$$403$$ 0 0
$$404$$ 14.6056 0.726653
$$405$$ −1.00000 −0.0496904
$$406$$ 6.78890 0.336927
$$407$$ 0 0
$$408$$ 2.60555 0.128994
$$409$$ 29.2111 1.44440 0.722198 0.691686i $$-0.243132\pi$$
0.722198 + 0.691686i $$0.243132\pi$$
$$410$$ 11.2111 0.553677
$$411$$ 11.2111 0.553003
$$412$$ 4.00000 0.197066
$$413$$ 13.5778 0.668120
$$414$$ −8.60555 −0.422940
$$415$$ 17.2111 0.844860
$$416$$ 0 0
$$417$$ −2.78890 −0.136573
$$418$$ 0 0
$$419$$ 25.8167 1.26123 0.630613 0.776097i $$-0.282804\pi$$
0.630613 + 0.776097i $$0.282804\pi$$
$$420$$ 2.60555 0.127138
$$421$$ −1.81665 −0.0885383 −0.0442691 0.999020i $$-0.514096\pi$$
−0.0442691 + 0.999020i $$0.514096\pi$$
$$422$$ 2.78890 0.135761
$$423$$ 5.21110 0.253372
$$424$$ −6.00000 −0.291386
$$425$$ 2.60555 0.126388
$$426$$ 5.21110 0.252479
$$427$$ 8.36669 0.404893
$$428$$ 0 0
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −4.78890 −0.230140 −0.115070 0.993357i $$-0.536709\pi$$
−0.115070 + 0.993357i $$0.536709\pi$$
$$434$$ 15.6333 0.750423
$$435$$ −2.60555 −0.124927
$$436$$ −8.60555 −0.412131
$$437$$ −22.4222 −1.07260
$$438$$ −8.60555 −0.411189
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −0.211103 −0.0100525
$$442$$ 0 0
$$443$$ −27.6333 −1.31290 −0.656449 0.754370i $$-0.727942\pi$$
−0.656449 + 0.754370i $$0.727942\pi$$
$$444$$ −5.21110 −0.247308
$$445$$ −0.788897 −0.0373973
$$446$$ −19.8167 −0.938346
$$447$$ 0.788897 0.0373136
$$448$$ 2.60555 0.123101
$$449$$ −9.63331 −0.454624 −0.227312 0.973822i $$-0.572994\pi$$
−0.227312 + 0.973822i $$0.572994\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ 7.81665 0.367664
$$453$$ −6.00000 −0.281905
$$454$$ 24.0000 1.12638
$$455$$ 0 0
$$456$$ −2.60555 −0.122016
$$457$$ 12.2389 0.572510 0.286255 0.958154i $$-0.407590\pi$$
0.286255 + 0.958154i $$0.407590\pi$$
$$458$$ −1.81665 −0.0848867
$$459$$ −2.60555 −0.121617
$$460$$ −8.60555 −0.401236
$$461$$ −9.63331 −0.448668 −0.224334 0.974512i $$-0.572021\pi$$
−0.224334 + 0.974512i $$0.572021\pi$$
$$462$$ 0 0
$$463$$ −38.6056 −1.79415 −0.897076 0.441876i $$-0.854313\pi$$
−0.897076 + 0.441876i $$0.854313\pi$$
$$464$$ −2.60555 −0.120960
$$465$$ −6.00000 −0.278243
$$466$$ 19.8167 0.917989
$$467$$ 1.57779 0.0730116 0.0365058 0.999333i $$-0.488377\pi$$
0.0365058 + 0.999333i $$0.488377\pi$$
$$468$$ 0 0
$$469$$ −29.2111 −1.34884
$$470$$ 5.21110 0.240370
$$471$$ 8.42221 0.388075
$$472$$ −5.21110 −0.239860
$$473$$ 0 0
$$474$$ 14.4222 0.662434
$$475$$ −2.60555 −0.119551
$$476$$ 6.78890 0.311169
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 34.4222 1.57279 0.786395 0.617724i $$-0.211945\pi$$
0.786395 + 0.617724i $$0.211945\pi$$
$$480$$ −1.00000 −0.0456435
$$481$$ 0 0
$$482$$ −22.4222 −1.02130
$$483$$ −22.4222 −1.02025
$$484$$ −11.0000 −0.500000
$$485$$ 8.60555 0.390758
$$486$$ 1.00000 0.0453609
$$487$$ 37.0278 1.67789 0.838944 0.544218i $$-0.183174\pi$$
0.838944 + 0.544218i $$0.183174\pi$$
$$488$$ −3.21110 −0.145360
$$489$$ 4.42221 0.199979
$$490$$ −0.211103 −0.00953664
$$491$$ 13.8167 0.623537 0.311768 0.950158i $$-0.399079\pi$$
0.311768 + 0.950158i $$0.399079\pi$$
$$492$$ −11.2111 −0.505436
$$493$$ −6.78890 −0.305757
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −6.00000 −0.269408
$$497$$ 13.5778 0.609047
$$498$$ −17.2111 −0.771248
$$499$$ 13.0278 0.583202 0.291601 0.956540i $$-0.405812\pi$$
0.291601 + 0.956540i $$0.405812\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 5.21110 0.232815
$$502$$ −20.6056 −0.919671
$$503$$ −41.4500 −1.84816 −0.924081 0.382196i $$-0.875168\pi$$
−0.924081 + 0.382196i $$0.875168\pi$$
$$504$$ −2.60555 −0.116060
$$505$$ −14.6056 −0.649939
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 13.2111 0.586148
$$509$$ −9.63331 −0.426989 −0.213494 0.976944i $$-0.568485\pi$$
−0.213494 + 0.976944i $$0.568485\pi$$
$$510$$ −2.60555 −0.115376
$$511$$ −22.4222 −0.991900
$$512$$ −1.00000 −0.0441942
$$513$$ 2.60555 0.115038
$$514$$ −21.3944 −0.943669
$$515$$ −4.00000 −0.176261
$$516$$ 8.00000 0.352180
$$517$$ 0 0
$$518$$ −13.5778 −0.596574
$$519$$ 16.4222 0.720855
$$520$$ 0 0
$$521$$ −21.6333 −0.947772 −0.473886 0.880586i $$-0.657149\pi$$
−0.473886 + 0.880586i $$0.657149\pi$$
$$522$$ 2.60555 0.114042
$$523$$ −24.8444 −1.08637 −0.543185 0.839613i $$-0.682782\pi$$
−0.543185 + 0.839613i $$0.682782\pi$$
$$524$$ 15.3944 0.672510
$$525$$ −2.60555 −0.113716
$$526$$ −13.8167 −0.602435
$$527$$ −15.6333 −0.680998
$$528$$ 0 0
$$529$$ 51.0555 2.21980
$$530$$ 6.00000 0.260623
$$531$$ 5.21110 0.226143
$$532$$ −6.78890 −0.294336
$$533$$ 0 0
$$534$$ 0.788897 0.0341389
$$535$$ 0 0
$$536$$ 11.2111 0.484246
$$537$$ 1.81665 0.0783944
$$538$$ −4.18335 −0.180357
$$539$$ 0 0
$$540$$ 1.00000 0.0430331
$$541$$ 43.0278 1.84991 0.924954 0.380079i $$-0.124103\pi$$
0.924954 + 0.380079i $$0.124103\pi$$
$$542$$ −28.4222 −1.22084
$$543$$ −20.4222 −0.876401
$$544$$ −2.60555 −0.111712
$$545$$ 8.60555 0.368621
$$546$$ 0 0
$$547$$ −14.4222 −0.616649 −0.308324 0.951281i $$-0.599768\pi$$
−0.308324 + 0.951281i $$0.599768\pi$$
$$548$$ −11.2111 −0.478915
$$549$$ 3.21110 0.137047
$$550$$ 0 0
$$551$$ 6.78890 0.289217
$$552$$ 8.60555 0.366277
$$553$$ 37.5778 1.59797
$$554$$ 12.4222 0.527769
$$555$$ 5.21110 0.221199
$$556$$ 2.78890 0.118276
$$557$$ 40.4222 1.71274 0.856372 0.516360i $$-0.172713\pi$$
0.856372 + 0.516360i $$0.172713\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 0 0
$$560$$ −2.60555 −0.110105
$$561$$ 0 0
$$562$$ −12.7889 −0.539467
$$563$$ 38.0555 1.60385 0.801924 0.597426i $$-0.203810\pi$$
0.801924 + 0.597426i $$0.203810\pi$$
$$564$$ −5.21110 −0.219427
$$565$$ −7.81665 −0.328849
$$566$$ −18.4222 −0.774343
$$567$$ 2.60555 0.109423
$$568$$ −5.21110 −0.218653
$$569$$ 9.63331 0.403849 0.201925 0.979401i $$-0.435280\pi$$
0.201925 + 0.979401i $$0.435280\pi$$
$$570$$ 2.60555 0.109135
$$571$$ −42.0555 −1.75997 −0.879984 0.475003i $$-0.842447\pi$$
−0.879984 + 0.475003i $$0.842447\pi$$
$$572$$ 0 0
$$573$$ 12.0000 0.501307
$$574$$ −29.2111 −1.21925
$$575$$ 8.60555 0.358876
$$576$$ 1.00000 0.0416667
$$577$$ 44.6056 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$578$$ 10.2111 0.424726
$$579$$ 13.8167 0.574201
$$580$$ 2.60555 0.108190
$$581$$ −44.8444 −1.86046
$$582$$ −8.60555 −0.356712
$$583$$ 0 0
$$584$$ 8.60555 0.356100
$$585$$ 0 0
$$586$$ −18.0000 −0.743573
$$587$$ −22.4222 −0.925463 −0.462732 0.886498i $$-0.653131\pi$$
−0.462732 + 0.886498i $$0.653131\pi$$
$$588$$ 0.211103 0.00870572
$$589$$ 15.6333 0.644159
$$590$$ 5.21110 0.214538
$$591$$ −6.00000 −0.246807
$$592$$ 5.21110 0.214175
$$593$$ 4.42221 0.181598 0.0907991 0.995869i $$-0.471058\pi$$
0.0907991 + 0.995869i $$0.471058\pi$$
$$594$$ 0 0
$$595$$ −6.78890 −0.278318
$$596$$ −0.788897 −0.0323145
$$597$$ −6.42221 −0.262843
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 1.00000 0.0408248
$$601$$ −41.6333 −1.69826 −0.849129 0.528185i $$-0.822873\pi$$
−0.849129 + 0.528185i $$0.822873\pi$$
$$602$$ 20.8444 0.849555
$$603$$ −11.2111 −0.456551
$$604$$ 6.00000 0.244137
$$605$$ 11.0000 0.447214
$$606$$ 14.6056 0.593310
$$607$$ 2.78890 0.113198 0.0565989 0.998397i $$-0.481974\pi$$
0.0565989 + 0.998397i $$0.481974\pi$$
$$608$$ 2.60555 0.105669
$$609$$ 6.78890 0.275100
$$610$$ 3.21110 0.130014
$$611$$ 0 0
$$612$$ 2.60555 0.105323
$$613$$ 18.7889 0.758876 0.379438 0.925217i $$-0.376117\pi$$
0.379438 + 0.925217i $$0.376117\pi$$
$$614$$ 23.2111 0.936724
$$615$$ 11.2111 0.452075
$$616$$ 0 0
$$617$$ 16.4222 0.661133 0.330567 0.943783i $$-0.392760\pi$$
0.330567 + 0.943783i $$0.392760\pi$$
$$618$$ 4.00000 0.160904
$$619$$ 4.18335 0.168143 0.0840714 0.996460i $$-0.473208\pi$$
0.0840714 + 0.996460i $$0.473208\pi$$
$$620$$ 6.00000 0.240966
$$621$$ −8.60555 −0.345329
$$622$$ 12.0000 0.481156
$$623$$ 2.05551 0.0823524
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −32.4222 −1.29585
$$627$$ 0 0
$$628$$ −8.42221 −0.336083
$$629$$ 13.5778 0.541382
$$630$$ 2.60555 0.103808
$$631$$ −11.2111 −0.446307 −0.223153 0.974783i $$-0.571635\pi$$
−0.223153 + 0.974783i $$0.571635\pi$$
$$632$$ −14.4222 −0.573685
$$633$$ 2.78890 0.110849
$$634$$ −18.0000 −0.714871
$$635$$ −13.2111 −0.524267
$$636$$ −6.00000 −0.237915
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 5.21110 0.206148
$$640$$ 1.00000 0.0395285
$$641$$ 28.4222 1.12261 0.561305 0.827609i $$-0.310300\pi$$
0.561305 + 0.827609i $$0.310300\pi$$
$$642$$ 0 0
$$643$$ 33.6333 1.32637 0.663184 0.748456i $$-0.269205\pi$$
0.663184 + 0.748456i $$0.269205\pi$$
$$644$$ 22.4222 0.883559
$$645$$ −8.00000 −0.315000
$$646$$ 6.78890 0.267106
$$647$$ 27.3944 1.07699 0.538493 0.842630i $$-0.318994\pi$$
0.538493 + 0.842630i $$0.318994\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 15.6333 0.612718
$$652$$ −4.42221 −0.173187
$$653$$ 24.7889 0.970065 0.485032 0.874496i $$-0.338808\pi$$
0.485032 + 0.874496i $$0.338808\pi$$
$$654$$ −8.60555 −0.336504
$$655$$ −15.3944 −0.601511
$$656$$ 11.2111 0.437720
$$657$$ −8.60555 −0.335735
$$658$$ −13.5778 −0.529318
$$659$$ 24.2389 0.944212 0.472106 0.881542i $$-0.343494\pi$$
0.472106 + 0.881542i $$0.343494\pi$$
$$660$$ 0 0
$$661$$ −0.238859 −0.00929054 −0.00464527 0.999989i $$-0.501479\pi$$
−0.00464527 + 0.999989i $$0.501479\pi$$
$$662$$ −9.39445 −0.365126
$$663$$ 0 0
$$664$$ 17.2111 0.667920
$$665$$ 6.78890 0.263262
$$666$$ −5.21110 −0.201926
$$667$$ −22.4222 −0.868191
$$668$$ −5.21110 −0.201624
$$669$$ −19.8167 −0.766156
$$670$$ −11.2111 −0.433123
$$671$$ 0 0
$$672$$ 2.60555 0.100511
$$673$$ −5.63331 −0.217148 −0.108574 0.994088i $$-0.534628\pi$$
−0.108574 + 0.994088i $$0.534628\pi$$
$$674$$ −29.6333 −1.14143
$$675$$ −1.00000 −0.0384900
$$676$$ 0 0
$$677$$ 44.0555 1.69319 0.846595 0.532237i $$-0.178648\pi$$
0.846595 + 0.532237i $$0.178648\pi$$
$$678$$ 7.81665 0.300197
$$679$$ −22.4222 −0.860485
$$680$$ 2.60555 0.0999183
$$681$$ 24.0000 0.919682
$$682$$ 0 0
$$683$$ −5.21110 −0.199397 −0.0996986 0.995018i $$-0.531788\pi$$
−0.0996986 + 0.995018i $$0.531788\pi$$
$$684$$ −2.60555 −0.0996257
$$685$$ 11.2111 0.428354
$$686$$ 18.7889 0.717363
$$687$$ −1.81665 −0.0693097
$$688$$ −8.00000 −0.304997
$$689$$ 0 0
$$690$$ −8.60555 −0.327608
$$691$$ 30.2389 1.15034 0.575170 0.818034i $$-0.304936\pi$$
0.575170 + 0.818034i $$0.304936\pi$$
$$692$$ −16.4222 −0.624279
$$693$$ 0 0
$$694$$ −15.6333 −0.593432
$$695$$ −2.78890 −0.105789
$$696$$ −2.60555 −0.0987632
$$697$$ 29.2111 1.10645
$$698$$ −13.8167 −0.522968
$$699$$ 19.8167 0.749535
$$700$$ 2.60555 0.0984806
$$701$$ −10.9722 −0.414416 −0.207208 0.978297i $$-0.566438\pi$$
−0.207208 + 0.978297i $$0.566438\pi$$
$$702$$ 0 0
$$703$$ −13.5778 −0.512096
$$704$$ 0 0
$$705$$ 5.21110 0.196261
$$706$$ −23.2111 −0.873561
$$707$$ 38.0555 1.43122
$$708$$ −5.21110 −0.195845
$$709$$ −8.60555 −0.323188 −0.161594 0.986857i $$-0.551664\pi$$
−0.161594 + 0.986857i $$0.551664\pi$$
$$710$$ 5.21110 0.195569
$$711$$ 14.4222 0.540875
$$712$$ −0.788897 −0.0295652
$$713$$ −51.6333 −1.93368
$$714$$ 6.78890 0.254068
$$715$$ 0 0
$$716$$ −1.81665 −0.0678915
$$717$$ 0 0
$$718$$ 27.6333 1.03127
$$719$$ 8.36669 0.312025 0.156012 0.987755i $$-0.450136\pi$$
0.156012 + 0.987755i $$0.450136\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 10.4222 0.388143
$$722$$ 12.2111 0.454450
$$723$$ −22.4222 −0.833891
$$724$$ 20.4222 0.758985
$$725$$ −2.60555 −0.0967677
$$726$$ −11.0000 −0.408248
$$727$$ −14.4222 −0.534890 −0.267445 0.963573i $$-0.586179\pi$$
−0.267445 + 0.963573i $$0.586179\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −8.60555 −0.318506
$$731$$ −20.8444 −0.770958
$$732$$ −3.21110 −0.118686
$$733$$ −38.0555 −1.40561 −0.702806 0.711381i $$-0.748070\pi$$
−0.702806 + 0.711381i $$0.748070\pi$$
$$734$$ −23.6333 −0.872321
$$735$$ −0.211103 −0.00778663
$$736$$ −8.60555 −0.317205
$$737$$ 0 0
$$738$$ −11.2111 −0.412686
$$739$$ −30.2389 −1.11235 −0.556177 0.831064i $$-0.687732\pi$$
−0.556177 + 0.831064i $$0.687732\pi$$
$$740$$ −5.21110 −0.191564
$$741$$ 0 0
$$742$$ −15.6333 −0.573917
$$743$$ 20.8444 0.764707 0.382354 0.924016i $$-0.375114\pi$$
0.382354 + 0.924016i $$0.375114\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 0.788897 0.0289030
$$746$$ −8.42221 −0.308359
$$747$$ −17.2111 −0.629721
$$748$$ 0 0
$$749$$ 0 0
$$750$$ −1.00000 −0.0365148
$$751$$ −18.4222 −0.672236 −0.336118 0.941820i $$-0.609114\pi$$
−0.336118 + 0.941820i $$0.609114\pi$$
$$752$$ 5.21110 0.190029
$$753$$ −20.6056 −0.750908
$$754$$ 0 0
$$755$$ −6.00000 −0.218362
$$756$$ −2.60555 −0.0947630
$$757$$ 27.2111 0.989004 0.494502 0.869176i $$-0.335350\pi$$
0.494502 + 0.869176i $$0.335350\pi$$
$$758$$ −1.02776 −0.0373298
$$759$$ 0 0
$$760$$ −2.60555 −0.0945133
$$761$$ −9.63331 −0.349207 −0.174604 0.984639i $$-0.555864\pi$$
−0.174604 + 0.984639i $$0.555864\pi$$
$$762$$ 13.2111 0.478588
$$763$$ −22.4222 −0.811738
$$764$$ −12.0000 −0.434145
$$765$$ −2.60555 −0.0942039
$$766$$ 15.6333 0.564854
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −44.8444 −1.61713 −0.808565 0.588406i $$-0.799756\pi$$
−0.808565 + 0.588406i $$0.799756\pi$$
$$770$$ 0 0
$$771$$ −21.3944 −0.770502
$$772$$ −13.8167 −0.497272
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ 8.00000 0.287554
$$775$$ −6.00000 −0.215526
$$776$$ 8.60555 0.308921
$$777$$ −13.5778 −0.487101
$$778$$ −2.60555 −0.0934135
$$779$$ −29.2111 −1.04660
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −22.4222 −0.801816
$$783$$ 2.60555 0.0931148
$$784$$ −0.211103 −0.00753938
$$785$$ 8.42221 0.300601
$$786$$ 15.3944 0.549102
$$787$$ −37.2666 −1.32841 −0.664206 0.747550i $$-0.731230\pi$$
−0.664206 + 0.747550i $$0.731230\pi$$
$$788$$ 6.00000 0.213741
$$789$$ −13.8167 −0.491886
$$790$$ 14.4222 0.513119
$$791$$ 20.3667 0.724156
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −39.6333 −1.40653
$$795$$ 6.00000 0.212798
$$796$$ 6.42221 0.227629
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ −6.78890 −0.240324
$$799$$ 13.5778 0.480348
$$800$$ −1.00000 −0.0353553
$$801$$ 0.788897 0.0278743
$$802$$ −23.2111 −0.819613
$$803$$ 0 0
$$804$$ 11.2111 0.395385
$$805$$ −22.4222 −0.790279
$$806$$ 0 0
$$807$$ −4.18335 −0.147261
$$808$$ −14.6056 −0.513822
$$809$$ 50.8444 1.78759 0.893797 0.448471i $$-0.148031\pi$$
0.893797 + 0.448471i $$0.148031\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 18.2389 0.640453 0.320226 0.947341i $$-0.396241\pi$$
0.320226 + 0.947341i $$0.396241\pi$$
$$812$$ −6.78890 −0.238244
$$813$$ −28.4222 −0.996810
$$814$$ 0 0
$$815$$ 4.42221 0.154903
$$816$$ −2.60555 −0.0912125
$$817$$ 20.8444 0.729254
$$818$$ −29.2111 −1.02134
$$819$$ 0 0
$$820$$ −11.2111 −0.391509
$$821$$ −11.2111 −0.391270 −0.195635 0.980677i $$-0.562677\pi$$
−0.195635 + 0.980677i $$0.562677\pi$$
$$822$$ −11.2111 −0.391032
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ −13.5778 −0.472432
$$827$$ 15.6333 0.543623 0.271812 0.962350i $$-0.412377\pi$$
0.271812 + 0.962350i $$0.412377\pi$$
$$828$$ 8.60555 0.299064
$$829$$ 10.8444 0.376642 0.188321 0.982108i $$-0.439695\pi$$
0.188321 + 0.982108i $$0.439695\pi$$
$$830$$ −17.2111 −0.597406
$$831$$ 12.4222 0.430922
$$832$$ 0 0
$$833$$ −0.550039 −0.0190577
$$834$$ 2.78890 0.0965716
$$835$$ 5.21110 0.180338
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ −25.8167 −0.891822
$$839$$ 10.4222 0.359814 0.179907 0.983684i $$-0.442420\pi$$
0.179907 + 0.983684i $$0.442420\pi$$
$$840$$ −2.60555 −0.0899001
$$841$$ −22.2111 −0.765900
$$842$$ 1.81665 0.0626060
$$843$$ −12.7889 −0.440473
$$844$$ −2.78890 −0.0959978
$$845$$ 0 0
$$846$$ −5.21110 −0.179161
$$847$$ −28.6611 −0.984806
$$848$$ 6.00000 0.206041
$$849$$ −18.4222 −0.632248
$$850$$ −2.60555 −0.0893697
$$851$$ 44.8444 1.53725
$$852$$ −5.21110 −0.178529
$$853$$ −29.2111 −1.00017 −0.500085 0.865977i $$-0.666698\pi$$
−0.500085 + 0.865977i $$0.666698\pi$$
$$854$$ −8.36669 −0.286302
$$855$$ 2.60555 0.0891080
$$856$$ 0 0
$$857$$ −13.0278 −0.445020 −0.222510 0.974930i $$-0.571425\pi$$
−0.222510 + 0.974930i $$0.571425\pi$$
$$858$$ 0 0
$$859$$ 10.7889 0.368112 0.184056 0.982916i $$-0.441077\pi$$
0.184056 + 0.982916i $$0.441077\pi$$
$$860$$ 8.00000 0.272798
$$861$$ −29.2111 −0.995512
$$862$$ 12.0000 0.408722
$$863$$ −8.36669 −0.284806 −0.142403 0.989809i $$-0.545483\pi$$
−0.142403 + 0.989809i $$0.545483\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 16.4222 0.558372
$$866$$ 4.78890 0.162733
$$867$$ 10.2111 0.346787
$$868$$ −15.6333 −0.530629
$$869$$ 0 0
$$870$$ 2.60555 0.0883365
$$871$$ 0 0
$$872$$ 8.60555 0.291421
$$873$$ −8.60555 −0.291254
$$874$$ 22.4222 0.758442
$$875$$ −2.60555 −0.0880837
$$876$$ 8.60555 0.290755
$$877$$ −32.8444 −1.10908 −0.554538 0.832158i $$-0.687105\pi$$
−0.554538 + 0.832158i $$0.687105\pi$$
$$878$$ 8.00000 0.269987
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ −24.7889 −0.835159 −0.417580 0.908640i $$-0.637122\pi$$
−0.417580 + 0.908640i $$0.637122\pi$$
$$882$$ 0.211103 0.00710819
$$883$$ −38.4222 −1.29301 −0.646505 0.762910i $$-0.723770\pi$$
−0.646505 + 0.762910i $$0.723770\pi$$
$$884$$ 0 0
$$885$$ 5.21110 0.175169
$$886$$ 27.6333 0.928359
$$887$$ 43.0278 1.44473 0.722365 0.691512i $$-0.243055\pi$$
0.722365 + 0.691512i $$0.243055\pi$$
$$888$$ 5.21110 0.174873
$$889$$ 34.4222 1.15448
$$890$$ 0.788897 0.0264439
$$891$$ 0 0
$$892$$ 19.8167 0.663511
$$893$$ −13.5778 −0.454364
$$894$$ −0.788897 −0.0263847
$$895$$ 1.81665 0.0607240
$$896$$ −2.60555 −0.0870454
$$897$$ 0 0
$$898$$ 9.63331 0.321468
$$899$$ 15.6333 0.521400
$$900$$ 1.00000 0.0333333
$$901$$ 15.6333 0.520821
$$902$$ 0 0
$$903$$ 20.8444 0.693659
$$904$$ −7.81665 −0.259978
$$905$$ −20.4222 −0.678857
$$906$$ 6.00000 0.199337
$$907$$ −50.4222 −1.67424 −0.837121 0.547018i $$-0.815763\pi$$
−0.837121 + 0.547018i $$0.815763\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 14.6056 0.484436
$$910$$ 0 0
$$911$$ 15.6333 0.517955 0.258977 0.965883i $$-0.416615\pi$$
0.258977 + 0.965883i $$0.416615\pi$$
$$912$$ 2.60555 0.0862784
$$913$$ 0 0
$$914$$ −12.2389 −0.404825
$$915$$ 3.21110 0.106156
$$916$$ 1.81665 0.0600239
$$917$$ 40.1110 1.32458
$$918$$ 2.60555 0.0859960
$$919$$ 16.0000 0.527791 0.263896 0.964551i $$-0.414993\pi$$
0.263896 + 0.964551i $$0.414993\pi$$
$$920$$ 8.60555 0.283717
$$921$$ 23.2111 0.764832
$$922$$ 9.63331 0.317256
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 5.21110 0.171340
$$926$$ 38.6056 1.26866
$$927$$ 4.00000 0.131377
$$928$$ 2.60555 0.0855314
$$929$$ −24.7889 −0.813297 −0.406649 0.913585i $$-0.633303\pi$$
−0.406649 + 0.913585i $$0.633303\pi$$
$$930$$ 6.00000 0.196748
$$931$$ 0.550039 0.0180268
$$932$$ −19.8167 −0.649116
$$933$$ 12.0000 0.392862
$$934$$ −1.57779 −0.0516270
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −53.6333 −1.75212 −0.876062 0.482199i $$-0.839838\pi$$
−0.876062 + 0.482199i $$0.839838\pi$$
$$938$$ 29.2111 0.953776
$$939$$ −32.4222 −1.05806
$$940$$ −5.21110 −0.169967
$$941$$ 54.0000 1.76035 0.880175 0.474650i $$-0.157425\pi$$
0.880175 + 0.474650i $$0.157425\pi$$
$$942$$ −8.42221 −0.274410
$$943$$ 96.4777 3.14175
$$944$$ 5.21110 0.169607
$$945$$ 2.60555 0.0847586
$$946$$ 0 0
$$947$$ 27.6333 0.897962 0.448981 0.893541i $$-0.351787\pi$$
0.448981 + 0.893541i $$0.351787\pi$$
$$948$$ −14.4222 −0.468411
$$949$$ 0 0
$$950$$ 2.60555 0.0845352
$$951$$ −18.0000 −0.583690
$$952$$ −6.78890 −0.220029
$$953$$ −30.2389 −0.979533 −0.489766 0.871854i $$-0.662918\pi$$
−0.489766 + 0.871854i $$0.662918\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 12.0000 0.388311
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −34.4222 −1.11213
$$959$$ −29.2111 −0.943276
$$960$$ 1.00000 0.0322749
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 22.4222 0.722171
$$965$$ 13.8167 0.444774
$$966$$ 22.4222 0.721423
$$967$$ −42.2389 −1.35831 −0.679155 0.733995i $$-0.737654\pi$$
−0.679155 + 0.733995i $$0.737654\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 6.78890 0.218091
$$970$$ −8.60555 −0.276308
$$971$$ −16.9722 −0.544665 −0.272333 0.962203i $$-0.587795\pi$$
−0.272333 + 0.962203i $$0.587795\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 7.26662 0.232957
$$974$$ −37.0278 −1.18645
$$975$$ 0 0
$$976$$ 3.21110 0.102785
$$977$$ 38.8444 1.24274 0.621371 0.783516i $$-0.286576\pi$$
0.621371 + 0.783516i $$0.286576\pi$$
$$978$$ −4.42221 −0.141407
$$979$$ 0 0
$$980$$ 0.211103 0.00674342
$$981$$ −8.60555 −0.274754
$$982$$ −13.8167 −0.440907
$$983$$ −13.5778 −0.433064 −0.216532 0.976275i $$-0.569475\pi$$
−0.216532 + 0.976275i $$0.569475\pi$$
$$984$$ 11.2111 0.357397
$$985$$ −6.00000 −0.191176
$$986$$ 6.78890 0.216203
$$987$$ −13.5778 −0.432186
$$988$$ 0 0
$$989$$ −68.8444 −2.18912
$$990$$ 0 0
$$991$$ 6.42221 0.204008 0.102004 0.994784i $$-0.467475\pi$$
0.102004 + 0.994784i $$0.467475\pi$$
$$992$$ 6.00000 0.190500
$$993$$ −9.39445 −0.298124
$$994$$ −13.5778 −0.430662
$$995$$ −6.42221 −0.203598
$$996$$ 17.2111 0.545355
$$997$$ 12.4222 0.393415 0.196708 0.980462i $$-0.436975\pi$$
0.196708 + 0.980462i $$0.436975\pi$$
$$998$$ −13.0278 −0.412386
$$999$$ −5.21110 −0.164872
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.z.1.2 2
13.5 odd 4 390.2.b.c.181.4 yes 4
13.8 odd 4 390.2.b.c.181.1 4
13.12 even 2 5070.2.a.bf.1.1 2
39.5 even 4 1170.2.b.d.181.2 4
39.8 even 4 1170.2.b.d.181.3 4
52.31 even 4 3120.2.g.q.961.3 4
52.47 even 4 3120.2.g.q.961.2 4
65.8 even 4 1950.2.f.m.649.1 4
65.18 even 4 1950.2.f.n.649.2 4
65.34 odd 4 1950.2.b.k.1351.4 4
65.44 odd 4 1950.2.b.k.1351.1 4
65.47 even 4 1950.2.f.n.649.4 4
65.57 even 4 1950.2.f.m.649.3 4

By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.1 4 13.8 odd 4
390.2.b.c.181.4 yes 4 13.5 odd 4
1170.2.b.d.181.2 4 39.5 even 4
1170.2.b.d.181.3 4 39.8 even 4
1950.2.b.k.1351.1 4 65.44 odd 4
1950.2.b.k.1351.4 4 65.34 odd 4
1950.2.f.m.649.1 4 65.8 even 4
1950.2.f.m.649.3 4 65.57 even 4
1950.2.f.n.649.2 4 65.18 even 4
1950.2.f.n.649.4 4 65.47 even 4
3120.2.g.q.961.2 4 52.47 even 4
3120.2.g.q.961.3 4 52.31 even 4
5070.2.a.z.1.2 2 1.1 even 1 trivial
5070.2.a.bf.1.1 2 13.12 even 2