# Properties

 Label 1815.2.a.m.1.2 Level $1815$ Weight $2$ Character 1815.1 Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 1815.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} +1.00000 q^{5} +0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} +1.00000 q^{5} +0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +0.193937 q^{10} -1.96239 q^{12} -2.96239 q^{13} -0.649738 q^{14} +1.00000 q^{15} +3.77575 q^{16} +4.57452 q^{17} +0.193937 q^{18} +4.31265 q^{19} -1.96239 q^{20} -3.35026 q^{21} -6.70052 q^{23} -0.768452 q^{24} +1.00000 q^{25} -0.574515 q^{26} +1.00000 q^{27} +6.57452 q^{28} +3.61213 q^{29} +0.193937 q^{30} +9.92478 q^{31} +2.26916 q^{32} +0.887166 q^{34} -3.35026 q^{35} -1.96239 q^{36} -2.00000 q^{37} +0.836381 q^{38} -2.96239 q^{39} -0.768452 q^{40} +4.38787 q^{41} -0.649738 q^{42} +9.27504 q^{43} +1.00000 q^{45} -1.29948 q^{46} -9.92478 q^{47} +3.77575 q^{48} +4.22425 q^{49} +0.193937 q^{50} +4.57452 q^{51} +5.81336 q^{52} +4.70052 q^{53} +0.193937 q^{54} +2.57452 q^{56} +4.31265 q^{57} +0.700523 q^{58} +10.7005 q^{59} -1.96239 q^{60} +8.70052 q^{61} +1.92478 q^{62} -3.35026 q^{63} -7.11142 q^{64} -2.96239 q^{65} +5.92478 q^{67} -8.97698 q^{68} -6.70052 q^{69} -0.649738 q^{70} +9.92478 q^{71} -0.768452 q^{72} +7.73813 q^{73} -0.387873 q^{74} +1.00000 q^{75} -8.46310 q^{76} -0.574515 q^{78} -11.5369 q^{79} +3.77575 q^{80} +1.00000 q^{81} +0.850969 q^{82} -10.8872 q^{83} +6.57452 q^{84} +4.57452 q^{85} +1.79877 q^{86} +3.61213 q^{87} -2.77575 q^{89} +0.193937 q^{90} +9.92478 q^{91} +13.1490 q^{92} +9.92478 q^{93} -1.92478 q^{94} +4.31265 q^{95} +2.26916 q^{96} +0.0752228 q^{97} +0.819237 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} + q^{6} + 9 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q + q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} + q^{6} + 9 q^{8} + 3 q^{9} + q^{10} + 5 q^{12} + 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} + 2 q^{17} + q^{18} - 8 q^{19} + 5 q^{20} + 9 q^{24} + 3 q^{25} + 10 q^{26} + 3 q^{27} + 8 q^{28} + 10 q^{29} + q^{30} + 8 q^{31} + 29 q^{32} - 30 q^{34} + 5 q^{36} - 6 q^{37} + 2 q^{39} + 9 q^{40} + 14 q^{41} - 12 q^{42} - 4 q^{43} + 3 q^{45} - 24 q^{46} - 8 q^{47} + 13 q^{48} + 11 q^{49} + q^{50} + 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 4 q^{56} - 8 q^{57} - 18 q^{58} + 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} + 2 q^{65} - 4 q^{67} - 42 q^{68} - 12 q^{70} + 8 q^{71} + 9 q^{72} + 14 q^{73} - 2 q^{74} + 3 q^{75} - 48 q^{76} + 10 q^{78} - 12 q^{79} + 13 q^{80} + 3 q^{81} + 26 q^{82} + 8 q^{84} + 2 q^{85} - 8 q^{86} + 10 q^{87} - 10 q^{89} + q^{90} + 8 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} - 8 q^{95} + 29 q^{96} + 22 q^{97} - 39 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.193937 0.137134 0.0685669 0.997647i $$-0.478157\pi$$
0.0685669 + 0.997647i $$0.478157\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.96239 −0.981194
$$5$$ 1.00000 0.447214
$$6$$ 0.193937 0.0791743
$$7$$ −3.35026 −1.26628 −0.633140 0.774037i $$-0.718234\pi$$
−0.633140 + 0.774037i $$0.718234\pi$$
$$8$$ −0.768452 −0.271689
$$9$$ 1.00000 0.333333
$$10$$ 0.193937 0.0613281
$$11$$ 0 0
$$12$$ −1.96239 −0.566493
$$13$$ −2.96239 −0.821619 −0.410809 0.911721i $$-0.634754\pi$$
−0.410809 + 0.911721i $$0.634754\pi$$
$$14$$ −0.649738 −0.173650
$$15$$ 1.00000 0.258199
$$16$$ 3.77575 0.943937
$$17$$ 4.57452 1.10948 0.554741 0.832023i $$-0.312817\pi$$
0.554741 + 0.832023i $$0.312817\pi$$
$$18$$ 0.193937 0.0457113
$$19$$ 4.31265 0.989390 0.494695 0.869067i $$-0.335280\pi$$
0.494695 + 0.869067i $$0.335280\pi$$
$$20$$ −1.96239 −0.438803
$$21$$ −3.35026 −0.731087
$$22$$ 0 0
$$23$$ −6.70052 −1.39716 −0.698578 0.715534i $$-0.746183\pi$$
−0.698578 + 0.715534i $$0.746183\pi$$
$$24$$ −0.768452 −0.156860
$$25$$ 1.00000 0.200000
$$26$$ −0.574515 −0.112672
$$27$$ 1.00000 0.192450
$$28$$ 6.57452 1.24247
$$29$$ 3.61213 0.670755 0.335378 0.942084i $$-0.391136\pi$$
0.335378 + 0.942084i $$0.391136\pi$$
$$30$$ 0.193937 0.0354078
$$31$$ 9.92478 1.78254 0.891271 0.453470i $$-0.149814\pi$$
0.891271 + 0.453470i $$0.149814\pi$$
$$32$$ 2.26916 0.401134
$$33$$ 0 0
$$34$$ 0.887166 0.152148
$$35$$ −3.35026 −0.566298
$$36$$ −1.96239 −0.327065
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0.836381 0.135679
$$39$$ −2.96239 −0.474362
$$40$$ −0.768452 −0.121503
$$41$$ 4.38787 0.685271 0.342635 0.939468i $$-0.388680\pi$$
0.342635 + 0.939468i $$0.388680\pi$$
$$42$$ −0.649738 −0.100257
$$43$$ 9.27504 1.41443 0.707215 0.706998i $$-0.249951\pi$$
0.707215 + 0.706998i $$0.249951\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ −1.29948 −0.191597
$$47$$ −9.92478 −1.44768 −0.723839 0.689969i $$-0.757624\pi$$
−0.723839 + 0.689969i $$0.757624\pi$$
$$48$$ 3.77575 0.544982
$$49$$ 4.22425 0.603465
$$50$$ 0.193937 0.0274268
$$51$$ 4.57452 0.640560
$$52$$ 5.81336 0.806168
$$53$$ 4.70052 0.645667 0.322833 0.946456i $$-0.395365\pi$$
0.322833 + 0.946456i $$0.395365\pi$$
$$54$$ 0.193937 0.0263914
$$55$$ 0 0
$$56$$ 2.57452 0.344034
$$57$$ 4.31265 0.571224
$$58$$ 0.700523 0.0919832
$$59$$ 10.7005 1.39309 0.696545 0.717513i $$-0.254720\pi$$
0.696545 + 0.717513i $$0.254720\pi$$
$$60$$ −1.96239 −0.253343
$$61$$ 8.70052 1.11399 0.556994 0.830517i $$-0.311955\pi$$
0.556994 + 0.830517i $$0.311955\pi$$
$$62$$ 1.92478 0.244447
$$63$$ −3.35026 −0.422093
$$64$$ −7.11142 −0.888927
$$65$$ −2.96239 −0.367439
$$66$$ 0 0
$$67$$ 5.92478 0.723827 0.361913 0.932212i $$-0.382124\pi$$
0.361913 + 0.932212i $$0.382124\pi$$
$$68$$ −8.97698 −1.08862
$$69$$ −6.70052 −0.806648
$$70$$ −0.649738 −0.0776586
$$71$$ 9.92478 1.17785 0.588927 0.808186i $$-0.299550\pi$$
0.588927 + 0.808186i $$0.299550\pi$$
$$72$$ −0.768452 −0.0905629
$$73$$ 7.73813 0.905680 0.452840 0.891592i $$-0.350411\pi$$
0.452840 + 0.891592i $$0.350411\pi$$
$$74$$ −0.387873 −0.0450893
$$75$$ 1.00000 0.115470
$$76$$ −8.46310 −0.970784
$$77$$ 0 0
$$78$$ −0.574515 −0.0650511
$$79$$ −11.5369 −1.29800 −0.649002 0.760787i $$-0.724813\pi$$
−0.649002 + 0.760787i $$0.724813\pi$$
$$80$$ 3.77575 0.422141
$$81$$ 1.00000 0.111111
$$82$$ 0.850969 0.0939738
$$83$$ −10.8872 −1.19502 −0.597511 0.801861i $$-0.703844\pi$$
−0.597511 + 0.801861i $$0.703844\pi$$
$$84$$ 6.57452 0.717338
$$85$$ 4.57452 0.496176
$$86$$ 1.79877 0.193966
$$87$$ 3.61213 0.387261
$$88$$ 0 0
$$89$$ −2.77575 −0.294229 −0.147114 0.989120i $$-0.546999\pi$$
−0.147114 + 0.989120i $$0.546999\pi$$
$$90$$ 0.193937 0.0204427
$$91$$ 9.92478 1.04040
$$92$$ 13.1490 1.37088
$$93$$ 9.92478 1.02915
$$94$$ −1.92478 −0.198526
$$95$$ 4.31265 0.442469
$$96$$ 2.26916 0.231595
$$97$$ 0.0752228 0.00763772 0.00381886 0.999993i $$-0.498784\pi$$
0.00381886 + 0.999993i $$0.498784\pi$$
$$98$$ 0.819237 0.0827555
$$99$$ 0 0
$$100$$ −1.96239 −0.196239
$$101$$ 15.0884 1.50135 0.750676 0.660671i $$-0.229728\pi$$
0.750676 + 0.660671i $$0.229728\pi$$
$$102$$ 0.887166 0.0878425
$$103$$ −3.22425 −0.317695 −0.158848 0.987303i $$-0.550778\pi$$
−0.158848 + 0.987303i $$0.550778\pi$$
$$104$$ 2.27645 0.223225
$$105$$ −3.35026 −0.326952
$$106$$ 0.911603 0.0885427
$$107$$ 0.962389 0.0930376 0.0465188 0.998917i $$-0.485187\pi$$
0.0465188 + 0.998917i $$0.485187\pi$$
$$108$$ −1.96239 −0.188831
$$109$$ −11.4010 −1.09202 −0.546011 0.837778i $$-0.683854\pi$$
−0.546011 + 0.837778i $$0.683854\pi$$
$$110$$ 0 0
$$111$$ −2.00000 −0.189832
$$112$$ −12.6497 −1.19529
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0.836381 0.0783342
$$115$$ −6.70052 −0.624827
$$116$$ −7.08840 −0.658141
$$117$$ −2.96239 −0.273873
$$118$$ 2.07522 0.191040
$$119$$ −15.3258 −1.40492
$$120$$ −0.768452 −0.0701498
$$121$$ 0 0
$$122$$ 1.68735 0.152765
$$123$$ 4.38787 0.395641
$$124$$ −19.4763 −1.74902
$$125$$ 1.00000 0.0894427
$$126$$ −0.649738 −0.0578833
$$127$$ 14.5745 1.29328 0.646640 0.762796i $$-0.276174\pi$$
0.646640 + 0.762796i $$0.276174\pi$$
$$128$$ −5.91748 −0.523037
$$129$$ 9.27504 0.816622
$$130$$ −0.574515 −0.0503883
$$131$$ 5.92478 0.517650 0.258825 0.965924i $$-0.416665\pi$$
0.258825 + 0.965924i $$0.416665\pi$$
$$132$$ 0 0
$$133$$ −14.4485 −1.25284
$$134$$ 1.14903 0.0992612
$$135$$ 1.00000 0.0860663
$$136$$ −3.51530 −0.301434
$$137$$ 13.8496 1.18325 0.591624 0.806214i $$-0.298487\pi$$
0.591624 + 0.806214i $$0.298487\pi$$
$$138$$ −1.29948 −0.110619
$$139$$ −13.6121 −1.15457 −0.577283 0.816544i $$-0.695887\pi$$
−0.577283 + 0.816544i $$0.695887\pi$$
$$140$$ 6.57452 0.555648
$$141$$ −9.92478 −0.835817
$$142$$ 1.92478 0.161524
$$143$$ 0 0
$$144$$ 3.77575 0.314646
$$145$$ 3.61213 0.299971
$$146$$ 1.50071 0.124199
$$147$$ 4.22425 0.348411
$$148$$ 3.92478 0.322615
$$149$$ −1.53690 −0.125908 −0.0629540 0.998016i $$-0.520052\pi$$
−0.0629540 + 0.998016i $$0.520052\pi$$
$$150$$ 0.193937 0.0158349
$$151$$ 6.76116 0.550215 0.275108 0.961413i $$-0.411287\pi$$
0.275108 + 0.961413i $$0.411287\pi$$
$$152$$ −3.31406 −0.268806
$$153$$ 4.57452 0.369828
$$154$$ 0 0
$$155$$ 9.92478 0.797177
$$156$$ 5.81336 0.465441
$$157$$ −5.47627 −0.437054 −0.218527 0.975831i $$-0.570125\pi$$
−0.218527 + 0.975831i $$0.570125\pi$$
$$158$$ −2.23743 −0.178000
$$159$$ 4.70052 0.372776
$$160$$ 2.26916 0.179393
$$161$$ 22.4485 1.76919
$$162$$ 0.193937 0.0152371
$$163$$ 12.6253 0.988890 0.494445 0.869209i $$-0.335371\pi$$
0.494445 + 0.869209i $$0.335371\pi$$
$$164$$ −8.61071 −0.672384
$$165$$ 0 0
$$166$$ −2.11142 −0.163878
$$167$$ −18.3634 −1.42101 −0.710503 0.703695i $$-0.751532\pi$$
−0.710503 + 0.703695i $$0.751532\pi$$
$$168$$ 2.57452 0.198628
$$169$$ −4.22425 −0.324943
$$170$$ 0.887166 0.0680425
$$171$$ 4.31265 0.329797
$$172$$ −18.2012 −1.38783
$$173$$ 8.57452 0.651908 0.325954 0.945386i $$-0.394314\pi$$
0.325954 + 0.945386i $$0.394314\pi$$
$$174$$ 0.700523 0.0531065
$$175$$ −3.35026 −0.253256
$$176$$ 0 0
$$177$$ 10.7005 0.804301
$$178$$ −0.538319 −0.0403487
$$179$$ 14.1768 1.05962 0.529812 0.848115i $$-0.322263\pi$$
0.529812 + 0.848115i $$0.322263\pi$$
$$180$$ −1.96239 −0.146268
$$181$$ −5.22425 −0.388316 −0.194158 0.980970i $$-0.562197\pi$$
−0.194158 + 0.980970i $$0.562197\pi$$
$$182$$ 1.92478 0.142674
$$183$$ 8.70052 0.643161
$$184$$ 5.14903 0.379592
$$185$$ −2.00000 −0.147043
$$186$$ 1.92478 0.141132
$$187$$ 0 0
$$188$$ 19.4763 1.42045
$$189$$ −3.35026 −0.243696
$$190$$ 0.836381 0.0606774
$$191$$ −16.6253 −1.20296 −0.601482 0.798886i $$-0.705423\pi$$
−0.601482 + 0.798886i $$0.705423\pi$$
$$192$$ −7.11142 −0.513222
$$193$$ 16.3634 1.17787 0.588933 0.808182i $$-0.299548\pi$$
0.588933 + 0.808182i $$0.299548\pi$$
$$194$$ 0.0145884 0.00104739
$$195$$ −2.96239 −0.212141
$$196$$ −8.28963 −0.592116
$$197$$ 20.4241 1.45515 0.727577 0.686026i $$-0.240646\pi$$
0.727577 + 0.686026i $$0.240646\pi$$
$$198$$ 0 0
$$199$$ −8.62530 −0.611431 −0.305716 0.952123i $$-0.598896\pi$$
−0.305716 + 0.952123i $$0.598896\pi$$
$$200$$ −0.768452 −0.0543378
$$201$$ 5.92478 0.417902
$$202$$ 2.92619 0.205886
$$203$$ −12.1016 −0.849364
$$204$$ −8.97698 −0.628514
$$205$$ 4.38787 0.306462
$$206$$ −0.625301 −0.0435668
$$207$$ −6.70052 −0.465719
$$208$$ −11.1852 −0.775556
$$209$$ 0 0
$$210$$ −0.649738 −0.0448362
$$211$$ −9.08840 −0.625671 −0.312836 0.949807i $$-0.601279\pi$$
−0.312836 + 0.949807i $$0.601279\pi$$
$$212$$ −9.22425 −0.633524
$$213$$ 9.92478 0.680035
$$214$$ 0.186642 0.0127586
$$215$$ 9.27504 0.632552
$$216$$ −0.768452 −0.0522865
$$217$$ −33.2506 −2.25720
$$218$$ −2.21108 −0.149753
$$219$$ 7.73813 0.522895
$$220$$ 0 0
$$221$$ −13.5515 −0.911572
$$222$$ −0.387873 −0.0260323
$$223$$ −6.70052 −0.448700 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$224$$ −7.60228 −0.507949
$$225$$ 1.00000 0.0666667
$$226$$ −1.16362 −0.0774028
$$227$$ −16.9624 −1.12583 −0.562917 0.826514i $$-0.690321\pi$$
−0.562917 + 0.826514i $$0.690321\pi$$
$$228$$ −8.46310 −0.560482
$$229$$ 25.8496 1.70819 0.854093 0.520120i $$-0.174113\pi$$
0.854093 + 0.520120i $$0.174113\pi$$
$$230$$ −1.29948 −0.0856849
$$231$$ 0 0
$$232$$ −2.77575 −0.182237
$$233$$ 19.2750 1.26275 0.631375 0.775478i $$-0.282491\pi$$
0.631375 + 0.775478i $$0.282491\pi$$
$$234$$ −0.574515 −0.0375573
$$235$$ −9.92478 −0.647421
$$236$$ −20.9986 −1.36689
$$237$$ −11.5369 −0.749402
$$238$$ −2.97224 −0.192662
$$239$$ −26.5501 −1.71738 −0.858691 0.512494i $$-0.828722\pi$$
−0.858691 + 0.512494i $$0.828722\pi$$
$$240$$ 3.77575 0.243723
$$241$$ −28.5501 −1.83907 −0.919536 0.393006i $$-0.871435\pi$$
−0.919536 + 0.393006i $$0.871435\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ −17.0738 −1.09304
$$245$$ 4.22425 0.269878
$$246$$ 0.850969 0.0542558
$$247$$ −12.7757 −0.812901
$$248$$ −7.62672 −0.484297
$$249$$ −10.8872 −0.689946
$$250$$ 0.193937 0.0122656
$$251$$ 29.9248 1.88884 0.944418 0.328748i $$-0.106627\pi$$
0.944418 + 0.328748i $$0.106627\pi$$
$$252$$ 6.57452 0.414156
$$253$$ 0 0
$$254$$ 2.82653 0.177352
$$255$$ 4.57452 0.286467
$$256$$ 13.0752 0.817201
$$257$$ 8.70052 0.542724 0.271362 0.962477i $$-0.412526\pi$$
0.271362 + 0.962477i $$0.412526\pi$$
$$258$$ 1.79877 0.111986
$$259$$ 6.70052 0.416350
$$260$$ 5.81336 0.360529
$$261$$ 3.61213 0.223585
$$262$$ 1.14903 0.0709874
$$263$$ −12.2882 −0.757724 −0.378862 0.925453i $$-0.623684\pi$$
−0.378862 + 0.925453i $$0.623684\pi$$
$$264$$ 0 0
$$265$$ 4.70052 0.288751
$$266$$ −2.80209 −0.171807
$$267$$ −2.77575 −0.169873
$$268$$ −11.6267 −0.710215
$$269$$ −5.84955 −0.356654 −0.178327 0.983971i $$-0.557068\pi$$
−0.178327 + 0.983971i $$0.557068\pi$$
$$270$$ 0.193937 0.0118026
$$271$$ 5.08840 0.309098 0.154549 0.987985i $$-0.450608\pi$$
0.154549 + 0.987985i $$0.450608\pi$$
$$272$$ 17.2722 1.04728
$$273$$ 9.92478 0.600675
$$274$$ 2.68594 0.162263
$$275$$ 0 0
$$276$$ 13.1490 0.791479
$$277$$ −1.41090 −0.0847725 −0.0423863 0.999101i $$-0.513496\pi$$
−0.0423863 + 0.999101i $$0.513496\pi$$
$$278$$ −2.63989 −0.158330
$$279$$ 9.92478 0.594181
$$280$$ 2.57452 0.153857
$$281$$ 4.38787 0.261759 0.130879 0.991398i $$-0.458220\pi$$
0.130879 + 0.991398i $$0.458220\pi$$
$$282$$ −1.92478 −0.114619
$$283$$ −26.5745 −1.57969 −0.789845 0.613306i $$-0.789839\pi$$
−0.789845 + 0.613306i $$0.789839\pi$$
$$284$$ −19.4763 −1.15570
$$285$$ 4.31265 0.255459
$$286$$ 0 0
$$287$$ −14.7005 −0.867744
$$288$$ 2.26916 0.133711
$$289$$ 3.92619 0.230952
$$290$$ 0.700523 0.0411362
$$291$$ 0.0752228 0.00440964
$$292$$ −15.1852 −0.888648
$$293$$ 3.42548 0.200119 0.100059 0.994981i $$-0.468097\pi$$
0.100059 + 0.994981i $$0.468097\pi$$
$$294$$ 0.819237 0.0477789
$$295$$ 10.7005 0.623009
$$296$$ 1.53690 0.0893307
$$297$$ 0 0
$$298$$ −0.298062 −0.0172663
$$299$$ 19.8496 1.14793
$$300$$ −1.96239 −0.113299
$$301$$ −31.0738 −1.79106
$$302$$ 1.31124 0.0754531
$$303$$ 15.0884 0.866806
$$304$$ 16.2835 0.933921
$$305$$ 8.70052 0.498191
$$306$$ 0.887166 0.0507159
$$307$$ 16.6497 0.950251 0.475125 0.879918i $$-0.342403\pi$$
0.475125 + 0.879918i $$0.342403\pi$$
$$308$$ 0 0
$$309$$ −3.22425 −0.183421
$$310$$ 1.92478 0.109320
$$311$$ 32.9986 1.87118 0.935589 0.353091i $$-0.114869\pi$$
0.935589 + 0.353091i $$0.114869\pi$$
$$312$$ 2.27645 0.128879
$$313$$ 15.4010 0.870519 0.435259 0.900305i $$-0.356657\pi$$
0.435259 + 0.900305i $$0.356657\pi$$
$$314$$ −1.06205 −0.0599349
$$315$$ −3.35026 −0.188766
$$316$$ 22.6399 1.27359
$$317$$ 2.15045 0.120781 0.0603905 0.998175i $$-0.480765\pi$$
0.0603905 + 0.998175i $$0.480765\pi$$
$$318$$ 0.911603 0.0511202
$$319$$ 0 0
$$320$$ −7.11142 −0.397540
$$321$$ 0.962389 0.0537153
$$322$$ 4.35359 0.242616
$$323$$ 19.7283 1.09771
$$324$$ −1.96239 −0.109022
$$325$$ −2.96239 −0.164324
$$326$$ 2.44851 0.135610
$$327$$ −11.4010 −0.630479
$$328$$ −3.37187 −0.186180
$$329$$ 33.2506 1.83316
$$330$$ 0 0
$$331$$ −14.5501 −0.799745 −0.399872 0.916571i $$-0.630946\pi$$
−0.399872 + 0.916571i $$0.630946\pi$$
$$332$$ 21.3649 1.17255
$$333$$ −2.00000 −0.109599
$$334$$ −3.56134 −0.194868
$$335$$ 5.92478 0.323705
$$336$$ −12.6497 −0.690100
$$337$$ −16.2619 −0.885840 −0.442920 0.896561i $$-0.646057\pi$$
−0.442920 + 0.896561i $$0.646057\pi$$
$$338$$ −0.819237 −0.0445606
$$339$$ −6.00000 −0.325875
$$340$$ −8.97698 −0.486845
$$341$$ 0 0
$$342$$ 0.836381 0.0452263
$$343$$ 9.29948 0.502125
$$344$$ −7.12742 −0.384285
$$345$$ −6.70052 −0.360744
$$346$$ 1.66291 0.0893987
$$347$$ 0.962389 0.0516637 0.0258319 0.999666i $$-0.491777\pi$$
0.0258319 + 0.999666i $$0.491777\pi$$
$$348$$ −7.08840 −0.379978
$$349$$ −20.7005 −1.10807 −0.554037 0.832492i $$-0.686913\pi$$
−0.554037 + 0.832492i $$0.686913\pi$$
$$350$$ −0.649738 −0.0347300
$$351$$ −2.96239 −0.158121
$$352$$ 0 0
$$353$$ 20.5501 1.09377 0.546885 0.837208i $$-0.315813\pi$$
0.546885 + 0.837208i $$0.315813\pi$$
$$354$$ 2.07522 0.110297
$$355$$ 9.92478 0.526752
$$356$$ 5.44709 0.288695
$$357$$ −15.3258 −0.811129
$$358$$ 2.74940 0.145310
$$359$$ −17.9248 −0.946034 −0.473017 0.881053i $$-0.656835\pi$$
−0.473017 + 0.881053i $$0.656835\pi$$
$$360$$ −0.768452 −0.0405010
$$361$$ −0.401047 −0.0211077
$$362$$ −1.01317 −0.0532512
$$363$$ 0 0
$$364$$ −19.4763 −1.02083
$$365$$ 7.73813 0.405032
$$366$$ 1.68735 0.0881992
$$367$$ −29.6531 −1.54788 −0.773939 0.633261i $$-0.781716\pi$$
−0.773939 + 0.633261i $$0.781716\pi$$
$$368$$ −25.2995 −1.31883
$$369$$ 4.38787 0.228424
$$370$$ −0.387873 −0.0201646
$$371$$ −15.7480 −0.817595
$$372$$ −19.4763 −1.00980
$$373$$ 9.13918 0.473209 0.236604 0.971606i $$-0.423965\pi$$
0.236604 + 0.971606i $$0.423965\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 7.62672 0.393318
$$377$$ −10.7005 −0.551105
$$378$$ −0.649738 −0.0334189
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −8.46310 −0.434148
$$381$$ 14.5745 0.746675
$$382$$ −3.22425 −0.164967
$$383$$ −34.9234 −1.78450 −0.892250 0.451541i $$-0.850874\pi$$
−0.892250 + 0.451541i $$0.850874\pi$$
$$384$$ −5.91748 −0.301975
$$385$$ 0 0
$$386$$ 3.17347 0.161525
$$387$$ 9.27504 0.471477
$$388$$ −0.147616 −0.00749408
$$389$$ 2.77575 0.140736 0.0703680 0.997521i $$-0.477583\pi$$
0.0703680 + 0.997521i $$0.477583\pi$$
$$390$$ −0.574515 −0.0290917
$$391$$ −30.6516 −1.55012
$$392$$ −3.24614 −0.163955
$$393$$ 5.92478 0.298865
$$394$$ 3.96097 0.199551
$$395$$ −11.5369 −0.580485
$$396$$ 0 0
$$397$$ −19.9248 −0.999996 −0.499998 0.866027i $$-0.666666\pi$$
−0.499998 + 0.866027i $$0.666666\pi$$
$$398$$ −1.67276 −0.0838479
$$399$$ −14.4485 −0.723330
$$400$$ 3.77575 0.188787
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 1.14903 0.0573085
$$403$$ −29.4010 −1.46457
$$404$$ −29.6093 −1.47312
$$405$$ 1.00000 0.0496904
$$406$$ −2.34694 −0.116477
$$407$$ 0 0
$$408$$ −3.51530 −0.174033
$$409$$ 13.0738 0.646458 0.323229 0.946321i $$-0.395232\pi$$
0.323229 + 0.946321i $$0.395232\pi$$
$$410$$ 0.850969 0.0420264
$$411$$ 13.8496 0.683148
$$412$$ 6.32724 0.311721
$$413$$ −35.8496 −1.76404
$$414$$ −1.29948 −0.0638658
$$415$$ −10.8872 −0.534430
$$416$$ −6.72213 −0.329580
$$417$$ −13.6121 −0.666589
$$418$$ 0 0
$$419$$ 7.22425 0.352928 0.176464 0.984307i $$-0.443534\pi$$
0.176464 + 0.984307i $$0.443534\pi$$
$$420$$ 6.57452 0.320804
$$421$$ 30.6253 1.49259 0.746293 0.665618i $$-0.231832\pi$$
0.746293 + 0.665618i $$0.231832\pi$$
$$422$$ −1.76257 −0.0858007
$$423$$ −9.92478 −0.482559
$$424$$ −3.61213 −0.175420
$$425$$ 4.57452 0.221897
$$426$$ 1.92478 0.0932558
$$427$$ −29.1490 −1.41062
$$428$$ −1.88858 −0.0912880
$$429$$ 0 0
$$430$$ 1.79877 0.0867444
$$431$$ 33.8759 1.63174 0.815872 0.578232i $$-0.196257\pi$$
0.815872 + 0.578232i $$0.196257\pi$$
$$432$$ 3.77575 0.181661
$$433$$ −9.47627 −0.455400 −0.227700 0.973731i $$-0.573121\pi$$
−0.227700 + 0.973731i $$0.573121\pi$$
$$434$$ −6.44851 −0.309538
$$435$$ 3.61213 0.173188
$$436$$ 22.3733 1.07149
$$437$$ −28.8970 −1.38233
$$438$$ 1.50071 0.0717066
$$439$$ 29.4617 1.40613 0.703065 0.711126i $$-0.251814\pi$$
0.703065 + 0.711126i $$0.251814\pi$$
$$440$$ 0 0
$$441$$ 4.22425 0.201155
$$442$$ −2.62813 −0.125007
$$443$$ −19.0738 −0.906224 −0.453112 0.891454i $$-0.649686\pi$$
−0.453112 + 0.891454i $$0.649686\pi$$
$$444$$ 3.92478 0.186262
$$445$$ −2.77575 −0.131583
$$446$$ −1.29948 −0.0615320
$$447$$ −1.53690 −0.0726931
$$448$$ 23.8251 1.12563
$$449$$ 35.8759 1.69309 0.846544 0.532318i $$-0.178679\pi$$
0.846544 + 0.532318i $$0.178679\pi$$
$$450$$ 0.193937 0.00914226
$$451$$ 0 0
$$452$$ 11.7743 0.553818
$$453$$ 6.76116 0.317667
$$454$$ −3.28963 −0.154390
$$455$$ 9.92478 0.465281
$$456$$ −3.31406 −0.155195
$$457$$ −5.28963 −0.247438 −0.123719 0.992317i $$-0.539482\pi$$
−0.123719 + 0.992317i $$0.539482\pi$$
$$458$$ 5.01317 0.234250
$$459$$ 4.57452 0.213520
$$460$$ 13.1490 0.613077
$$461$$ −36.3390 −1.69248 −0.846238 0.532805i $$-0.821138\pi$$
−0.846238 + 0.532805i $$0.821138\pi$$
$$462$$ 0 0
$$463$$ 10.5501 0.490304 0.245152 0.969485i $$-0.421162\pi$$
0.245152 + 0.969485i $$0.421162\pi$$
$$464$$ 13.6385 0.633150
$$465$$ 9.92478 0.460251
$$466$$ 3.73813 0.173166
$$467$$ 18.7005 0.865357 0.432679 0.901548i $$-0.357569\pi$$
0.432679 + 0.901548i $$0.357569\pi$$
$$468$$ 5.81336 0.268723
$$469$$ −19.8496 −0.916567
$$470$$ −1.92478 −0.0887834
$$471$$ −5.47627 −0.252333
$$472$$ −8.22284 −0.378487
$$473$$ 0 0
$$474$$ −2.23743 −0.102768
$$475$$ 4.31265 0.197878
$$476$$ 30.0752 1.37850
$$477$$ 4.70052 0.215222
$$478$$ −5.14903 −0.235511
$$479$$ 9.29948 0.424904 0.212452 0.977172i $$-0.431855\pi$$
0.212452 + 0.977172i $$0.431855\pi$$
$$480$$ 2.26916 0.103572
$$481$$ 5.92478 0.270147
$$482$$ −5.53690 −0.252199
$$483$$ 22.4485 1.02144
$$484$$ 0 0
$$485$$ 0.0752228 0.00341569
$$486$$ 0.193937 0.00879714
$$487$$ −35.4763 −1.60758 −0.803792 0.594911i $$-0.797187\pi$$
−0.803792 + 0.594911i $$0.797187\pi$$
$$488$$ −6.68594 −0.302658
$$489$$ 12.6253 0.570936
$$490$$ 0.819237 0.0370094
$$491$$ −24.7757 −1.11811 −0.559057 0.829129i $$-0.688837\pi$$
−0.559057 + 0.829129i $$0.688837\pi$$
$$492$$ −8.61071 −0.388201
$$493$$ 16.5237 0.744191
$$494$$ −2.47768 −0.111476
$$495$$ 0 0
$$496$$ 37.4734 1.68261
$$497$$ −33.2506 −1.49149
$$498$$ −2.11142 −0.0946150
$$499$$ 14.1768 0.634640 0.317320 0.948318i $$-0.397217\pi$$
0.317320 + 0.948318i $$0.397217\pi$$
$$500$$ −1.96239 −0.0877607
$$501$$ −18.3634 −0.820418
$$502$$ 5.80351 0.259023
$$503$$ 8.43866 0.376261 0.188131 0.982144i $$-0.439757\pi$$
0.188131 + 0.982144i $$0.439757\pi$$
$$504$$ 2.57452 0.114678
$$505$$ 15.0884 0.671425
$$506$$ 0 0
$$507$$ −4.22425 −0.187606
$$508$$ −28.6009 −1.26896
$$509$$ 1.10299 0.0488890 0.0244445 0.999701i $$-0.492218\pi$$
0.0244445 + 0.999701i $$0.492218\pi$$
$$510$$ 0.887166 0.0392844
$$511$$ −25.9248 −1.14684
$$512$$ 14.3707 0.635103
$$513$$ 4.31265 0.190408
$$514$$ 1.68735 0.0744258
$$515$$ −3.22425 −0.142078
$$516$$ −18.2012 −0.801265
$$517$$ 0 0
$$518$$ 1.29948 0.0570957
$$519$$ 8.57452 0.376379
$$520$$ 2.27645 0.0998291
$$521$$ −12.4485 −0.545379 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$522$$ 0.700523 0.0306611
$$523$$ −30.0508 −1.31403 −0.657015 0.753878i $$-0.728181\pi$$
−0.657015 + 0.753878i $$0.728181\pi$$
$$524$$ −11.6267 −0.507915
$$525$$ −3.35026 −0.146217
$$526$$ −2.38313 −0.103910
$$527$$ 45.4010 1.97770
$$528$$ 0 0
$$529$$ 21.8970 0.952044
$$530$$ 0.911603 0.0395975
$$531$$ 10.7005 0.464363
$$532$$ 28.3536 1.22928
$$533$$ −12.9986 −0.563031
$$534$$ −0.538319 −0.0232953
$$535$$ 0.962389 0.0416077
$$536$$ −4.55291 −0.196656
$$537$$ 14.1768 0.611774
$$538$$ −1.13444 −0.0489093
$$539$$ 0 0
$$540$$ −1.96239 −0.0844478
$$541$$ 18.0000 0.773880 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$542$$ 0.986826 0.0423878
$$543$$ −5.22425 −0.224194
$$544$$ 10.3803 0.445052
$$545$$ −11.4010 −0.488367
$$546$$ 1.92478 0.0823729
$$547$$ 14.3028 0.611544 0.305772 0.952105i $$-0.401086\pi$$
0.305772 + 0.952105i $$0.401086\pi$$
$$548$$ −27.1782 −1.16100
$$549$$ 8.70052 0.371329
$$550$$ 0 0
$$551$$ 15.5778 0.663638
$$552$$ 5.14903 0.219157
$$553$$ 38.6516 1.64364
$$554$$ −0.273624 −0.0116252
$$555$$ −2.00000 −0.0848953
$$556$$ 26.7123 1.13285
$$557$$ 11.7988 0.499930 0.249965 0.968255i $$-0.419581\pi$$
0.249965 + 0.968255i $$0.419581\pi$$
$$558$$ 1.92478 0.0814823
$$559$$ −27.4763 −1.16212
$$560$$ −12.6497 −0.534549
$$561$$ 0 0
$$562$$ 0.850969 0.0358960
$$563$$ −30.4847 −1.28478 −0.642389 0.766379i $$-0.722056\pi$$
−0.642389 + 0.766379i $$0.722056\pi$$
$$564$$ 19.4763 0.820099
$$565$$ −6.00000 −0.252422
$$566$$ −5.15377 −0.216629
$$567$$ −3.35026 −0.140698
$$568$$ −7.62672 −0.320010
$$569$$ 27.0884 1.13560 0.567802 0.823165i $$-0.307794\pi$$
0.567802 + 0.823165i $$0.307794\pi$$
$$570$$ 0.836381 0.0350321
$$571$$ −7.28489 −0.304863 −0.152432 0.988314i $$-0.548710\pi$$
−0.152432 + 0.988314i $$0.548710\pi$$
$$572$$ 0 0
$$573$$ −16.6253 −0.694532
$$574$$ −2.85097 −0.118997
$$575$$ −6.70052 −0.279431
$$576$$ −7.11142 −0.296309
$$577$$ −31.6239 −1.31652 −0.658260 0.752791i $$-0.728707\pi$$
−0.658260 + 0.752791i $$0.728707\pi$$
$$578$$ 0.761432 0.0316714
$$579$$ 16.3634 0.680041
$$580$$ −7.08840 −0.294330
$$581$$ 36.4749 1.51323
$$582$$ 0.0145884 0.000604711 0
$$583$$ 0 0
$$584$$ −5.94639 −0.246063
$$585$$ −2.96239 −0.122480
$$586$$ 0.664327 0.0274431
$$587$$ 33.1490 1.36821 0.684103 0.729385i $$-0.260194\pi$$
0.684103 + 0.729385i $$0.260194\pi$$
$$588$$ −8.28963 −0.341858
$$589$$ 42.8021 1.76363
$$590$$ 2.07522 0.0854356
$$591$$ 20.4241 0.840134
$$592$$ −7.55149 −0.310364
$$593$$ −34.4993 −1.41672 −0.708358 0.705853i $$-0.750564\pi$$
−0.708358 + 0.705853i $$0.750564\pi$$
$$594$$ 0 0
$$595$$ −15.3258 −0.628298
$$596$$ 3.01600 0.123540
$$597$$ −8.62530 −0.353010
$$598$$ 3.84955 0.157420
$$599$$ −14.4485 −0.590350 −0.295175 0.955443i $$-0.595378\pi$$
−0.295175 + 0.955443i $$0.595378\pi$$
$$600$$ −0.768452 −0.0313719
$$601$$ 15.9248 0.649585 0.324793 0.945785i $$-0.394705\pi$$
0.324793 + 0.945785i $$0.394705\pi$$
$$602$$ −6.02635 −0.245616
$$603$$ 5.92478 0.241276
$$604$$ −13.2680 −0.539868
$$605$$ 0 0
$$606$$ 2.92619 0.118868
$$607$$ 14.5745 0.591561 0.295781 0.955256i $$-0.404420\pi$$
0.295781 + 0.955256i $$0.404420\pi$$
$$608$$ 9.78609 0.396878
$$609$$ −12.1016 −0.490380
$$610$$ 1.68735 0.0683188
$$611$$ 29.4010 1.18944
$$612$$ −8.97698 −0.362873
$$613$$ −16.4123 −0.662887 −0.331443 0.943475i $$-0.607536\pi$$
−0.331443 + 0.943475i $$0.607536\pi$$
$$614$$ 3.22899 0.130312
$$615$$ 4.38787 0.176936
$$616$$ 0 0
$$617$$ −17.8496 −0.718596 −0.359298 0.933223i $$-0.616984\pi$$
−0.359298 + 0.933223i $$0.616984\pi$$
$$618$$ −0.625301 −0.0251533
$$619$$ −0.402462 −0.0161763 −0.00808815 0.999967i $$-0.502575\pi$$
−0.00808815 + 0.999967i $$0.502575\pi$$
$$620$$ −19.4763 −0.782186
$$621$$ −6.70052 −0.268883
$$622$$ 6.39963 0.256602
$$623$$ 9.29948 0.372576
$$624$$ −11.1852 −0.447767
$$625$$ 1.00000 0.0400000
$$626$$ 2.98683 0.119378
$$627$$ 0 0
$$628$$ 10.7466 0.428835
$$629$$ −9.14903 −0.364796
$$630$$ −0.649738 −0.0258862
$$631$$ −38.0263 −1.51380 −0.756902 0.653528i $$-0.773288\pi$$
−0.756902 + 0.653528i $$0.773288\pi$$
$$632$$ 8.86556 0.352653
$$633$$ −9.08840 −0.361231
$$634$$ 0.417050 0.0165632
$$635$$ 14.5745 0.578372
$$636$$ −9.22425 −0.365765
$$637$$ −12.5139 −0.495818
$$638$$ 0 0
$$639$$ 9.92478 0.392618
$$640$$ −5.91748 −0.233909
$$641$$ −28.0263 −1.10697 −0.553487 0.832858i $$-0.686703\pi$$
−0.553487 + 0.832858i $$0.686703\pi$$
$$642$$ 0.186642 0.00736619
$$643$$ −4.62530 −0.182404 −0.0912020 0.995832i $$-0.529071\pi$$
−0.0912020 + 0.995832i $$0.529071\pi$$
$$644$$ −44.0527 −1.73592
$$645$$ 9.27504 0.365204
$$646$$ 3.82604 0.150533
$$647$$ 23.5778 0.926941 0.463470 0.886112i $$-0.346604\pi$$
0.463470 + 0.886112i $$0.346604\pi$$
$$648$$ −0.768452 −0.0301876
$$649$$ 0 0
$$650$$ −0.574515 −0.0225344
$$651$$ −33.2506 −1.30319
$$652$$ −24.7757 −0.970293
$$653$$ −2.25202 −0.0881282 −0.0440641 0.999029i $$-0.514031\pi$$
−0.0440641 + 0.999029i $$0.514031\pi$$
$$654$$ −2.21108 −0.0864601
$$655$$ 5.92478 0.231500
$$656$$ 16.5675 0.646852
$$657$$ 7.73813 0.301893
$$658$$ 6.44851 0.251389
$$659$$ 41.4010 1.61276 0.806378 0.591401i $$-0.201425\pi$$
0.806378 + 0.591401i $$0.201425\pi$$
$$660$$ 0 0
$$661$$ 3.40105 0.132285 0.0661427 0.997810i $$-0.478931\pi$$
0.0661427 + 0.997810i $$0.478931\pi$$
$$662$$ −2.82179 −0.109672
$$663$$ −13.5515 −0.526296
$$664$$ 8.36626 0.324674
$$665$$ −14.4485 −0.560289
$$666$$ −0.387873 −0.0150298
$$667$$ −24.2031 −0.937149
$$668$$ 36.0362 1.39428
$$669$$ −6.70052 −0.259057
$$670$$ 1.14903 0.0443909
$$671$$ 0 0
$$672$$ −7.60228 −0.293264
$$673$$ −0.887166 −0.0341977 −0.0170989 0.999854i $$-0.505443\pi$$
−0.0170989 + 0.999854i $$0.505443\pi$$
$$674$$ −3.15377 −0.121479
$$675$$ 1.00000 0.0384900
$$676$$ 8.28963 0.318832
$$677$$ −18.9018 −0.726453 −0.363227 0.931701i $$-0.618325\pi$$
−0.363227 + 0.931701i $$0.618325\pi$$
$$678$$ −1.16362 −0.0446885
$$679$$ −0.252016 −0.00967149
$$680$$ −3.51530 −0.134805
$$681$$ −16.9624 −0.650000
$$682$$ 0 0
$$683$$ −20.8773 −0.798848 −0.399424 0.916766i $$-0.630790\pi$$
−0.399424 + 0.916766i $$0.630790\pi$$
$$684$$ −8.46310 −0.323595
$$685$$ 13.8496 0.529164
$$686$$ 1.80351 0.0688583
$$687$$ 25.8496 0.986222
$$688$$ 35.0202 1.33513
$$689$$ −13.9248 −0.530492
$$690$$ −1.29948 −0.0494702
$$691$$ −2.44851 −0.0931456 −0.0465728 0.998915i $$-0.514830\pi$$
−0.0465728 + 0.998915i $$0.514830\pi$$
$$692$$ −16.8265 −0.639649
$$693$$ 0 0
$$694$$ 0.186642 0.00708485
$$695$$ −13.6121 −0.516337
$$696$$ −2.77575 −0.105214
$$697$$ 20.0724 0.760296
$$698$$ −4.01459 −0.151954
$$699$$ 19.2750 0.729049
$$700$$ 6.57452 0.248493
$$701$$ 2.98683 0.112811 0.0564054 0.998408i $$-0.482036\pi$$
0.0564054 + 0.998408i $$0.482036\pi$$
$$702$$ −0.574515 −0.0216837
$$703$$ −8.62530 −0.325309
$$704$$ 0 0
$$705$$ −9.92478 −0.373789
$$706$$ 3.98541 0.149993
$$707$$ −50.5501 −1.90113
$$708$$ −20.9986 −0.789175
$$709$$ 24.1768 0.907979 0.453989 0.891007i $$-0.350000\pi$$
0.453989 + 0.891007i $$0.350000\pi$$
$$710$$ 1.92478 0.0722356
$$711$$ −11.5369 −0.432668
$$712$$ 2.13303 0.0799386
$$713$$ −66.5012 −2.49049
$$714$$ −2.97224 −0.111233
$$715$$ 0 0
$$716$$ −27.8204 −1.03970
$$717$$ −26.5501 −0.991531
$$718$$ −3.47627 −0.129733
$$719$$ −30.0263 −1.11979 −0.559897 0.828562i $$-0.689159\pi$$
−0.559897 + 0.828562i $$0.689159\pi$$
$$720$$ 3.77575 0.140714
$$721$$ 10.8021 0.402291
$$722$$ −0.0777777 −0.00289459
$$723$$ −28.5501 −1.06179
$$724$$ 10.2520 0.381013
$$725$$ 3.61213 0.134151
$$726$$ 0 0
$$727$$ 14.9525 0.554559 0.277279 0.960789i $$-0.410567\pi$$
0.277279 + 0.960789i $$0.410567\pi$$
$$728$$ −7.62672 −0.282665
$$729$$ 1.00000 0.0370370
$$730$$ 1.50071 0.0555437
$$731$$ 42.4288 1.56929
$$732$$ −17.0738 −0.631066
$$733$$ 19.1128 0.705949 0.352974 0.935633i $$-0.385170\pi$$
0.352974 + 0.935633i $$0.385170\pi$$
$$734$$ −5.75081 −0.212266
$$735$$ 4.22425 0.155814
$$736$$ −15.2046 −0.560447
$$737$$ 0 0
$$738$$ 0.850969 0.0313246
$$739$$ 3.31406 0.121910 0.0609549 0.998141i $$-0.480585\pi$$
0.0609549 + 0.998141i $$0.480585\pi$$
$$740$$ 3.92478 0.144278
$$741$$ −12.7757 −0.469329
$$742$$ −3.05411 −0.112120
$$743$$ 34.9887 1.28361 0.641806 0.766867i $$-0.278185\pi$$
0.641806 + 0.766867i $$0.278185\pi$$
$$744$$ −7.62672 −0.279609
$$745$$ −1.53690 −0.0563078
$$746$$ 1.77242 0.0648930
$$747$$ −10.8872 −0.398341
$$748$$ 0 0
$$749$$ −3.22425 −0.117812
$$750$$ 0.193937 0.00708156
$$751$$ −26.9234 −0.982447 −0.491224 0.871033i $$-0.663450\pi$$
−0.491224 + 0.871033i $$0.663450\pi$$
$$752$$ −37.4734 −1.36652
$$753$$ 29.9248 1.09052
$$754$$ −2.07522 −0.0755752
$$755$$ 6.76116 0.246064
$$756$$ 6.57452 0.239113
$$757$$ 15.9248 0.578796 0.289398 0.957209i $$-0.406545\pi$$
0.289398 + 0.957209i $$0.406545\pi$$
$$758$$ −3.87873 −0.140882
$$759$$ 0 0
$$760$$ −3.31406 −0.120214
$$761$$ 30.9380 1.12150 0.560750 0.827985i $$-0.310513\pi$$
0.560750 + 0.827985i $$0.310513\pi$$
$$762$$ 2.82653 0.102394
$$763$$ 38.1965 1.38281
$$764$$ 32.6253 1.18034
$$765$$ 4.57452 0.165392
$$766$$ −6.77292 −0.244715
$$767$$ −31.6991 −1.14459
$$768$$ 13.0752 0.471811
$$769$$ −9.32582 −0.336298 −0.168149 0.985762i $$-0.553779\pi$$
−0.168149 + 0.985762i $$0.553779\pi$$
$$770$$ 0 0
$$771$$ 8.70052 0.313342
$$772$$ −32.1114 −1.15572
$$773$$ 44.7005 1.60777 0.803883 0.594787i $$-0.202764\pi$$
0.803883 + 0.594787i $$0.202764\pi$$
$$774$$ 1.79877 0.0646554
$$775$$ 9.92478 0.356509
$$776$$ −0.0578051 −0.00207508
$$777$$ 6.70052 0.240380
$$778$$ 0.538319 0.0192997
$$779$$ 18.9234 0.678000
$$780$$ 5.81336 0.208152
$$781$$ 0 0
$$782$$ −5.94448 −0.212574
$$783$$ 3.61213 0.129087
$$784$$ 15.9497 0.569633
$$785$$ −5.47627 −0.195456
$$786$$ 1.14903 0.0409846
$$787$$ −21.6775 −0.772719 −0.386360 0.922348i $$-0.626268\pi$$
−0.386360 + 0.922348i $$0.626268\pi$$
$$788$$ −40.0800 −1.42779
$$789$$ −12.2882 −0.437472
$$790$$ −2.23743 −0.0796041
$$791$$ 20.1016 0.714730
$$792$$ 0 0
$$793$$ −25.7743 −0.915273
$$794$$ −3.86414 −0.137133
$$795$$ 4.70052 0.166710
$$796$$ 16.9262 0.599933
$$797$$ 22.7466 0.805725 0.402862 0.915261i $$-0.368015\pi$$
0.402862 + 0.915261i $$0.368015\pi$$
$$798$$ −2.80209 −0.0991930
$$799$$ −45.4010 −1.60617
$$800$$ 2.26916 0.0802269
$$801$$ −2.77575 −0.0980762
$$802$$ 0.387873 0.0136963
$$803$$ 0 0
$$804$$ −11.6267 −0.410043
$$805$$ 22.4485 0.791206
$$806$$ −5.70194 −0.200842
$$807$$ −5.84955 −0.205914
$$808$$ −11.5947 −0.407900
$$809$$ 23.6121 0.830158 0.415079 0.909785i $$-0.363754\pi$$
0.415079 + 0.909785i $$0.363754\pi$$
$$810$$ 0.193937 0.00681424
$$811$$ 26.0870 0.916038 0.458019 0.888942i $$-0.348559\pi$$
0.458019 + 0.888942i $$0.348559\pi$$
$$812$$ 23.7480 0.833391
$$813$$ 5.08840 0.178458
$$814$$ 0 0
$$815$$ 12.6253 0.442245
$$816$$ 17.2722 0.604648
$$817$$ 40.0000 1.39942
$$818$$ 2.53549 0.0886513
$$819$$ 9.92478 0.346800
$$820$$ −8.61071 −0.300699
$$821$$ 54.4142 1.89907 0.949535 0.313662i $$-0.101556\pi$$
0.949535 + 0.313662i $$0.101556\pi$$
$$822$$ 2.68594 0.0936827
$$823$$ 0.121269 0.00422716 0.00211358 0.999998i $$-0.499327\pi$$
0.00211358 + 0.999998i $$0.499327\pi$$
$$824$$ 2.47768 0.0863142
$$825$$ 0 0
$$826$$ −6.95254 −0.241910
$$827$$ 18.2130 0.633328 0.316664 0.948538i $$-0.397437\pi$$
0.316664 + 0.948538i $$0.397437\pi$$
$$828$$ 13.1490 0.456960
$$829$$ 13.0738 0.454072 0.227036 0.973886i $$-0.427096\pi$$
0.227036 + 0.973886i $$0.427096\pi$$
$$830$$ −2.11142 −0.0732884
$$831$$ −1.41090 −0.0489434
$$832$$ 21.0668 0.730359
$$833$$ 19.3239 0.669534
$$834$$ −2.63989 −0.0914119
$$835$$ −18.3634 −0.635493
$$836$$ 0 0
$$837$$ 9.92478 0.343050
$$838$$ 1.40105 0.0483984
$$839$$ −26.5501 −0.916610 −0.458305 0.888795i $$-0.651543\pi$$
−0.458305 + 0.888795i $$0.651543\pi$$
$$840$$ 2.57452 0.0888292
$$841$$ −15.9525 −0.550088
$$842$$ 5.93937 0.204684
$$843$$ 4.38787 0.151126
$$844$$ 17.8350 0.613905
$$845$$ −4.22425 −0.145319
$$846$$ −1.92478 −0.0661752
$$847$$ 0 0
$$848$$ 17.7480 0.609468
$$849$$ −26.5745 −0.912035
$$850$$ 0.887166 0.0304295
$$851$$ 13.4010 0.459382
$$852$$ −19.4763 −0.667246
$$853$$ −40.6155 −1.39065 −0.695323 0.718697i $$-0.744739\pi$$
−0.695323 + 0.718697i $$0.744739\pi$$
$$854$$ −5.65306 −0.193444
$$855$$ 4.31265 0.147490
$$856$$ −0.739549 −0.0252773
$$857$$ −20.1721 −0.689064 −0.344532 0.938775i $$-0.611962\pi$$
−0.344532 + 0.938775i $$0.611962\pi$$
$$858$$ 0 0
$$859$$ 21.8035 0.743926 0.371963 0.928248i $$-0.378685\pi$$
0.371963 + 0.928248i $$0.378685\pi$$
$$860$$ −18.2012 −0.620657
$$861$$ −14.7005 −0.500993
$$862$$ 6.56978 0.223767
$$863$$ 35.4274 1.20596 0.602981 0.797755i $$-0.293979\pi$$
0.602981 + 0.797755i $$0.293979\pi$$
$$864$$ 2.26916 0.0771984
$$865$$ 8.57452 0.291542
$$866$$ −1.83780 −0.0624508
$$867$$ 3.92619 0.133340
$$868$$ 65.2506 2.21475
$$869$$ 0 0
$$870$$ 0.700523 0.0237500
$$871$$ −17.5515 −0.594710
$$872$$ 8.76116 0.296690
$$873$$ 0.0752228 0.00254591
$$874$$ −5.60419 −0.189564
$$875$$ −3.35026 −0.113260
$$876$$ −15.1852 −0.513061
$$877$$ 14.0362 0.473969 0.236984 0.971513i $$-0.423841\pi$$
0.236984 + 0.971513i $$0.423841\pi$$
$$878$$ 5.71370 0.192828
$$879$$ 3.42548 0.115539
$$880$$ 0 0
$$881$$ −21.0738 −0.709995 −0.354997 0.934867i $$-0.615518\pi$$
−0.354997 + 0.934867i $$0.615518\pi$$
$$882$$ 0.819237 0.0275852
$$883$$ −42.1476 −1.41838 −0.709190 0.705017i $$-0.750939\pi$$
−0.709190 + 0.705017i $$0.750939\pi$$
$$884$$ 26.5933 0.894429
$$885$$ 10.7005 0.359694
$$886$$ −3.69911 −0.124274
$$887$$ −6.93604 −0.232889 −0.116445 0.993197i $$-0.537150\pi$$
−0.116445 + 0.993197i $$0.537150\pi$$
$$888$$ 1.53690 0.0515751
$$889$$ −48.8284 −1.63765
$$890$$ −0.538319 −0.0180445
$$891$$ 0 0
$$892$$ 13.1490 0.440262
$$893$$ −42.8021 −1.43232
$$894$$ −0.298062 −0.00996868
$$895$$ 14.1768 0.473878
$$896$$ 19.8251 0.662311
$$897$$ 19.8496 0.662757
$$898$$ 6.95765 0.232180
$$899$$ 35.8496 1.19565
$$900$$ −1.96239 −0.0654130
$$901$$ 21.5026 0.716356
$$902$$ 0 0
$$903$$ −31.0738 −1.03407
$$904$$ 4.61071 0.153350
$$905$$ −5.22425 −0.173660
$$906$$ 1.31124 0.0435629
$$907$$ 53.2017 1.76653 0.883267 0.468870i $$-0.155339\pi$$
0.883267 + 0.468870i $$0.155339\pi$$
$$908$$ 33.2868 1.10466
$$909$$ 15.0884 0.500451
$$910$$ 1.92478 0.0638057
$$911$$ 36.4749 1.20847 0.604233 0.796808i $$-0.293480\pi$$
0.604233 + 0.796808i $$0.293480\pi$$
$$912$$ 16.2835 0.539200
$$913$$ 0 0
$$914$$ −1.02585 −0.0339322
$$915$$ 8.70052 0.287630
$$916$$ −50.7269 −1.67606
$$917$$ −19.8496 −0.655490
$$918$$ 0.887166 0.0292808
$$919$$ −9.73340 −0.321075 −0.160538 0.987030i $$-0.551323\pi$$
−0.160538 + 0.987030i $$0.551323\pi$$
$$920$$ 5.14903 0.169759
$$921$$ 16.6497 0.548628
$$922$$ −7.04746 −0.232096
$$923$$ −29.4010 −0.967747
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ 2.04605 0.0672372
$$927$$ −3.22425 −0.105898
$$928$$ 8.19649 0.269063
$$929$$ −24.1768 −0.793215 −0.396607 0.917988i $$-0.629813\pi$$
−0.396607 + 0.917988i $$0.629813\pi$$
$$930$$ 1.92478 0.0631159
$$931$$ 18.2177 0.597062
$$932$$ −37.8251 −1.23900
$$933$$ 32.9986 1.08033
$$934$$ 3.62672 0.118670
$$935$$ 0 0
$$936$$ 2.27645 0.0744082
$$937$$ 7.48612 0.244561 0.122280 0.992496i $$-0.460979\pi$$
0.122280 + 0.992496i $$0.460979\pi$$
$$938$$ −3.84955 −0.125692
$$939$$ 15.4010 0.502594
$$940$$ 19.4763 0.635246
$$941$$ 21.2360 0.692274 0.346137 0.938184i $$-0.387493\pi$$
0.346137 + 0.938184i $$0.387493\pi$$
$$942$$ −1.06205 −0.0346034
$$943$$ −29.4010 −0.957430
$$944$$ 40.4025 1.31499
$$945$$ −3.35026 −0.108984
$$946$$ 0 0
$$947$$ −15.4763 −0.502911 −0.251456 0.967869i $$-0.580909\pi$$
−0.251456 + 0.967869i $$0.580909\pi$$
$$948$$ 22.6399 0.735309
$$949$$ −22.9234 −0.744124
$$950$$ 0.836381 0.0271358
$$951$$ 2.15045 0.0697330
$$952$$ 11.7772 0.381700
$$953$$ 32.0508 1.03823 0.519113 0.854705i $$-0.326262\pi$$
0.519113 + 0.854705i $$0.326262\pi$$
$$954$$ 0.911603 0.0295142
$$955$$ −16.6253 −0.537982
$$956$$ 52.1016 1.68509
$$957$$ 0 0
$$958$$ 1.80351 0.0582687
$$959$$ −46.3996 −1.49832
$$960$$ −7.11142 −0.229520
$$961$$ 67.5012 2.17746
$$962$$ 1.14903 0.0370462
$$963$$ 0.962389 0.0310125
$$964$$ 56.0263 1.80449
$$965$$ 16.3634 0.526758
$$966$$ 4.35359 0.140074
$$967$$ 17.3766 0.558794 0.279397 0.960176i $$-0.409865\pi$$
0.279397 + 0.960176i $$0.409865\pi$$
$$968$$ 0 0
$$969$$ 19.7283 0.633764
$$970$$ 0.0145884 0.000468407 0
$$971$$ 36.2031 1.16181 0.580907 0.813970i $$-0.302698\pi$$
0.580907 + 0.813970i $$0.302698\pi$$
$$972$$ −1.96239 −0.0629436
$$973$$ 45.6042 1.46200
$$974$$ −6.88015 −0.220454
$$975$$ −2.96239 −0.0948724
$$976$$ 32.8510 1.05153
$$977$$ −28.1476 −0.900522 −0.450261 0.892897i $$-0.648669\pi$$
−0.450261 + 0.892897i $$0.648669\pi$$
$$978$$ 2.44851 0.0782946
$$979$$ 0 0
$$980$$ −8.28963 −0.264802
$$981$$ −11.4010 −0.364007
$$982$$ −4.80492 −0.153331
$$983$$ 7.07381 0.225619 0.112810 0.993617i $$-0.464015\pi$$
0.112810 + 0.993617i $$0.464015\pi$$
$$984$$ −3.37187 −0.107491
$$985$$ 20.4241 0.650765
$$986$$ 3.20456 0.102054
$$987$$ 33.2506 1.05838
$$988$$ 25.0710 0.797614
$$989$$ −62.1476 −1.97618
$$990$$ 0 0
$$991$$ 44.4260 1.41124 0.705619 0.708592i $$-0.250669\pi$$
0.705619 + 0.708592i $$0.250669\pi$$
$$992$$ 22.5209 0.715039
$$993$$ −14.5501 −0.461733
$$994$$ −6.44851 −0.204534
$$995$$ −8.62530 −0.273440
$$996$$ 21.3649 0.676971
$$997$$ 28.4847 0.902120 0.451060 0.892494i $$-0.351046\pi$$
0.451060 + 0.892494i $$0.351046\pi$$
$$998$$ 2.74940 0.0870307
$$999$$ −2.00000 −0.0632772
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 1815.2.a.m.1.2 3
3.2 odd 2 5445.2.a.z.1.2 3
5.4 even 2 9075.2.a.cf.1.2 3
11.10 odd 2 165.2.a.c.1.2 3
33.32 even 2 495.2.a.e.1.2 3
44.43 even 2 2640.2.a.be.1.1 3
55.32 even 4 825.2.c.g.199.3 6
55.43 even 4 825.2.c.g.199.4 6
55.54 odd 2 825.2.a.k.1.2 3
77.76 even 2 8085.2.a.bk.1.2 3
132.131 odd 2 7920.2.a.cj.1.1 3
165.32 odd 4 2475.2.c.r.199.4 6
165.98 odd 4 2475.2.c.r.199.3 6
165.164 even 2 2475.2.a.bb.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 11.10 odd 2
495.2.a.e.1.2 3 33.32 even 2
825.2.a.k.1.2 3 55.54 odd 2
825.2.c.g.199.3 6 55.32 even 4
825.2.c.g.199.4 6 55.43 even 4
1815.2.a.m.1.2 3 1.1 even 1 trivial
2475.2.a.bb.1.2 3 165.164 even 2
2475.2.c.r.199.3 6 165.98 odd 4
2475.2.c.r.199.4 6 165.32 odd 4
2640.2.a.be.1.1 3 44.43 even 2
5445.2.a.z.1.2 3 3.2 odd 2
7920.2.a.cj.1.1 3 132.131 odd 2
8085.2.a.bk.1.2 3 77.76 even 2
9075.2.a.cf.1.2 3 5.4 even 2