# Properties

 Label 5445.2.a.o.1.2 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1815) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} +3.23607 q^{7} -2.23607 q^{8} +O(q^{10})$$ $$q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} +3.23607 q^{7} -2.23607 q^{8} +0.618034 q^{10} +5.23607 q^{13} +2.00000 q^{14} +1.85410 q^{16} +5.47214 q^{17} +6.47214 q^{19} -1.61803 q^{20} +4.70820 q^{23} +1.00000 q^{25} +3.23607 q^{26} -5.23607 q^{28} +1.23607 q^{29} -6.70820 q^{31} +5.61803 q^{32} +3.38197 q^{34} +3.23607 q^{35} +0.763932 q^{37} +4.00000 q^{38} -2.23607 q^{40} -3.52786 q^{41} -5.23607 q^{43} +2.90983 q^{46} -8.70820 q^{47} +3.47214 q^{49} +0.618034 q^{50} -8.47214 q^{52} -9.94427 q^{53} -7.23607 q^{56} +0.763932 q^{58} -11.7082 q^{59} -1.47214 q^{61} -4.14590 q^{62} -0.236068 q^{64} +5.23607 q^{65} +11.2361 q^{67} -8.85410 q^{68} +2.00000 q^{70} +14.4721 q^{71} -10.4721 q^{73} +0.472136 q^{74} -10.4721 q^{76} +12.7082 q^{79} +1.85410 q^{80} -2.18034 q^{82} +4.00000 q^{83} +5.47214 q^{85} -3.23607 q^{86} -4.76393 q^{89} +16.9443 q^{91} -7.61803 q^{92} -5.38197 q^{94} +6.47214 q^{95} +12.7639 q^{97} +2.14590 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 2q^{5} + 2q^{7} - q^{10} + 6q^{13} + 4q^{14} - 3q^{16} + 2q^{17} + 4q^{19} - q^{20} - 4q^{23} + 2q^{25} + 2q^{26} - 6q^{28} - 2q^{29} + 9q^{32} + 9q^{34} + 2q^{35} + 6q^{37} + 8q^{38} - 16q^{41} - 6q^{43} + 17q^{46} - 4q^{47} - 2q^{49} - q^{50} - 8q^{52} - 2q^{53} - 10q^{56} + 6q^{58} - 10q^{59} + 6q^{61} - 15q^{62} + 4q^{64} + 6q^{65} + 18q^{67} - 11q^{68} + 4q^{70} + 20q^{71} - 12q^{73} - 8q^{74} - 12q^{76} + 12q^{79} - 3q^{80} + 18q^{82} + 8q^{83} + 2q^{85} - 2q^{86} - 14q^{89} + 16q^{91} - 13q^{92} - 13q^{94} + 4q^{95} + 30q^{97} + 11q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.618034 0.437016 0.218508 0.975835i $$-0.429881\pi$$
0.218508 + 0.975835i $$0.429881\pi$$
$$3$$ 0 0
$$4$$ −1.61803 −0.809017
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.23607 1.22312 0.611559 0.791199i $$-0.290543\pi$$
0.611559 + 0.791199i $$0.290543\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ 0 0
$$10$$ 0.618034 0.195440
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 5.23607 1.45222 0.726112 0.687576i $$-0.241325\pi$$
0.726112 + 0.687576i $$0.241325\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 5.47214 1.32719 0.663594 0.748093i $$-0.269030\pi$$
0.663594 + 0.748093i $$0.269030\pi$$
$$18$$ 0 0
$$19$$ 6.47214 1.48481 0.742405 0.669951i $$-0.233685\pi$$
0.742405 + 0.669951i $$0.233685\pi$$
$$20$$ −1.61803 −0.361803
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.70820 0.981728 0.490864 0.871236i $$-0.336681\pi$$
0.490864 + 0.871236i $$0.336681\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 3.23607 0.634645
$$27$$ 0 0
$$28$$ −5.23607 −0.989524
$$29$$ 1.23607 0.229532 0.114766 0.993393i $$-0.463388\pi$$
0.114766 + 0.993393i $$0.463388\pi$$
$$30$$ 0 0
$$31$$ −6.70820 −1.20483 −0.602414 0.798183i $$-0.705795\pi$$
−0.602414 + 0.798183i $$0.705795\pi$$
$$32$$ 5.61803 0.993137
$$33$$ 0 0
$$34$$ 3.38197 0.580002
$$35$$ 3.23607 0.546995
$$36$$ 0 0
$$37$$ 0.763932 0.125590 0.0627948 0.998026i $$-0.479999\pi$$
0.0627948 + 0.998026i $$0.479999\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ −2.23607 −0.353553
$$41$$ −3.52786 −0.550960 −0.275480 0.961307i $$-0.588837\pi$$
−0.275480 + 0.961307i $$0.588837\pi$$
$$42$$ 0 0
$$43$$ −5.23607 −0.798493 −0.399246 0.916844i $$-0.630728\pi$$
−0.399246 + 0.916844i $$0.630728\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 2.90983 0.429031
$$47$$ −8.70820 −1.27022 −0.635111 0.772421i $$-0.719046\pi$$
−0.635111 + 0.772421i $$0.719046\pi$$
$$48$$ 0 0
$$49$$ 3.47214 0.496019
$$50$$ 0.618034 0.0874032
$$51$$ 0 0
$$52$$ −8.47214 −1.17487
$$53$$ −9.94427 −1.36595 −0.682975 0.730441i $$-0.739314\pi$$
−0.682975 + 0.730441i $$0.739314\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −7.23607 −0.966960
$$57$$ 0 0
$$58$$ 0.763932 0.100309
$$59$$ −11.7082 −1.52428 −0.762139 0.647413i $$-0.775851\pi$$
−0.762139 + 0.647413i $$0.775851\pi$$
$$60$$ 0 0
$$61$$ −1.47214 −0.188488 −0.0942438 0.995549i $$-0.530043\pi$$
−0.0942438 + 0.995549i $$0.530043\pi$$
$$62$$ −4.14590 −0.526530
$$63$$ 0 0
$$64$$ −0.236068 −0.0295085
$$65$$ 5.23607 0.649454
$$66$$ 0 0
$$67$$ 11.2361 1.37270 0.686352 0.727269i $$-0.259211\pi$$
0.686352 + 0.727269i $$0.259211\pi$$
$$68$$ −8.85410 −1.07372
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ 14.4721 1.71753 0.858763 0.512373i $$-0.171233\pi$$
0.858763 + 0.512373i $$0.171233\pi$$
$$72$$ 0 0
$$73$$ −10.4721 −1.22567 −0.612835 0.790211i $$-0.709971\pi$$
−0.612835 + 0.790211i $$0.709971\pi$$
$$74$$ 0.472136 0.0548847
$$75$$ 0 0
$$76$$ −10.4721 −1.20124
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.7082 1.42978 0.714892 0.699235i $$-0.246476\pi$$
0.714892 + 0.699235i $$0.246476\pi$$
$$80$$ 1.85410 0.207295
$$81$$ 0 0
$$82$$ −2.18034 −0.240778
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 5.47214 0.593536
$$86$$ −3.23607 −0.348954
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −4.76393 −0.504976 −0.252488 0.967600i $$-0.581249\pi$$
−0.252488 + 0.967600i $$0.581249\pi$$
$$90$$ 0 0
$$91$$ 16.9443 1.77624
$$92$$ −7.61803 −0.794235
$$93$$ 0 0
$$94$$ −5.38197 −0.555107
$$95$$ 6.47214 0.664027
$$96$$ 0 0
$$97$$ 12.7639 1.29598 0.647990 0.761648i $$-0.275610\pi$$
0.647990 + 0.761648i $$0.275610\pi$$
$$98$$ 2.14590 0.216768
$$99$$ 0 0
$$100$$ −1.61803 −0.161803
$$101$$ 5.52786 0.550043 0.275022 0.961438i $$-0.411315\pi$$
0.275022 + 0.961438i $$0.411315\pi$$
$$102$$ 0 0
$$103$$ −0.944272 −0.0930419 −0.0465209 0.998917i $$-0.514813\pi$$
−0.0465209 + 0.998917i $$0.514813\pi$$
$$104$$ −11.7082 −1.14808
$$105$$ 0 0
$$106$$ −6.14590 −0.596942
$$107$$ −14.2361 −1.37625 −0.688126 0.725591i $$-0.741567\pi$$
−0.688126 + 0.725591i $$0.741567\pi$$
$$108$$ 0 0
$$109$$ 14.9443 1.43140 0.715701 0.698407i $$-0.246107\pi$$
0.715701 + 0.698407i $$0.246107\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 6.00000 0.566947
$$113$$ 12.4164 1.16804 0.584019 0.811740i $$-0.301479\pi$$
0.584019 + 0.811740i $$0.301479\pi$$
$$114$$ 0 0
$$115$$ 4.70820 0.439042
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ −7.23607 −0.666134
$$119$$ 17.7082 1.62331
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −0.909830 −0.0823721
$$123$$ 0 0
$$124$$ 10.8541 0.974727
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 3.70820 0.329050 0.164525 0.986373i $$-0.447391\pi$$
0.164525 + 0.986373i $$0.447391\pi$$
$$128$$ −11.3820 −1.00603
$$129$$ 0 0
$$130$$ 3.23607 0.283822
$$131$$ 0.472136 0.0412507 0.0206254 0.999787i $$-0.493434\pi$$
0.0206254 + 0.999787i $$0.493434\pi$$
$$132$$ 0 0
$$133$$ 20.9443 1.81610
$$134$$ 6.94427 0.599894
$$135$$ 0 0
$$136$$ −12.2361 −1.04923
$$137$$ −17.4721 −1.49275 −0.746373 0.665528i $$-0.768206\pi$$
−0.746373 + 0.665528i $$0.768206\pi$$
$$138$$ 0 0
$$139$$ −7.29180 −0.618482 −0.309241 0.950984i $$-0.600075\pi$$
−0.309241 + 0.950984i $$0.600075\pi$$
$$140$$ −5.23607 −0.442529
$$141$$ 0 0
$$142$$ 8.94427 0.750587
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.23607 0.102650
$$146$$ −6.47214 −0.535638
$$147$$ 0 0
$$148$$ −1.23607 −0.101604
$$149$$ 4.29180 0.351598 0.175799 0.984426i $$-0.443749\pi$$
0.175799 + 0.984426i $$0.443749\pi$$
$$150$$ 0 0
$$151$$ 7.29180 0.593398 0.296699 0.954971i $$-0.404114\pi$$
0.296699 + 0.954971i $$0.404114\pi$$
$$152$$ −14.4721 −1.17385
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.70820 −0.538816
$$156$$ 0 0
$$157$$ 2.29180 0.182905 0.0914526 0.995809i $$-0.470849\pi$$
0.0914526 + 0.995809i $$0.470849\pi$$
$$158$$ 7.85410 0.624839
$$159$$ 0 0
$$160$$ 5.61803 0.444145
$$161$$ 15.2361 1.20077
$$162$$ 0 0
$$163$$ 6.18034 0.484082 0.242041 0.970266i $$-0.422183\pi$$
0.242041 + 0.970266i $$0.422183\pi$$
$$164$$ 5.70820 0.445736
$$165$$ 0 0
$$166$$ 2.47214 0.191875
$$167$$ −3.18034 −0.246102 −0.123051 0.992400i $$-0.539268\pi$$
−0.123051 + 0.992400i $$0.539268\pi$$
$$168$$ 0 0
$$169$$ 14.4164 1.10895
$$170$$ 3.38197 0.259385
$$171$$ 0 0
$$172$$ 8.47214 0.645994
$$173$$ −2.94427 −0.223849 −0.111924 0.993717i $$-0.535701\pi$$
−0.111924 + 0.993717i $$0.535701\pi$$
$$174$$ 0 0
$$175$$ 3.23607 0.244624
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −2.94427 −0.220683
$$179$$ 20.1803 1.50835 0.754175 0.656674i $$-0.228037\pi$$
0.754175 + 0.656674i $$0.228037\pi$$
$$180$$ 0 0
$$181$$ −21.4164 −1.59187 −0.795935 0.605383i $$-0.793020\pi$$
−0.795935 + 0.605383i $$0.793020\pi$$
$$182$$ 10.4721 0.776246
$$183$$ 0 0
$$184$$ −10.5279 −0.776124
$$185$$ 0.763932 0.0561654
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 14.0902 1.02763
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ −27.5967 −1.99683 −0.998415 0.0562752i $$-0.982078\pi$$
−0.998415 + 0.0562752i $$0.982078\pi$$
$$192$$ 0 0
$$193$$ 16.1803 1.16469 0.582343 0.812943i $$-0.302136\pi$$
0.582343 + 0.812943i $$0.302136\pi$$
$$194$$ 7.88854 0.566364
$$195$$ 0 0
$$196$$ −5.61803 −0.401288
$$197$$ 1.41641 0.100915 0.0504574 0.998726i $$-0.483932\pi$$
0.0504574 + 0.998726i $$0.483932\pi$$
$$198$$ 0 0
$$199$$ 16.2361 1.15094 0.575472 0.817821i $$-0.304818\pi$$
0.575472 + 0.817821i $$0.304818\pi$$
$$200$$ −2.23607 −0.158114
$$201$$ 0 0
$$202$$ 3.41641 0.240378
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ −3.52786 −0.246397
$$206$$ −0.583592 −0.0406608
$$207$$ 0 0
$$208$$ 9.70820 0.673143
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 1.76393 0.121434 0.0607170 0.998155i $$-0.480661\pi$$
0.0607170 + 0.998155i $$0.480661\pi$$
$$212$$ 16.0902 1.10508
$$213$$ 0 0
$$214$$ −8.79837 −0.601444
$$215$$ −5.23607 −0.357097
$$216$$ 0 0
$$217$$ −21.7082 −1.47365
$$218$$ 9.23607 0.625545
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 28.6525 1.92737
$$222$$ 0 0
$$223$$ 9.05573 0.606416 0.303208 0.952924i $$-0.401942\pi$$
0.303208 + 0.952924i $$0.401942\pi$$
$$224$$ 18.1803 1.21473
$$225$$ 0 0
$$226$$ 7.67376 0.510451
$$227$$ −16.7082 −1.10896 −0.554481 0.832196i $$-0.687083\pi$$
−0.554481 + 0.832196i $$0.687083\pi$$
$$228$$ 0 0
$$229$$ 7.00000 0.462573 0.231287 0.972886i $$-0.425707\pi$$
0.231287 + 0.972886i $$0.425707\pi$$
$$230$$ 2.90983 0.191869
$$231$$ 0 0
$$232$$ −2.76393 −0.181461
$$233$$ −10.8885 −0.713332 −0.356666 0.934232i $$-0.616087\pi$$
−0.356666 + 0.934232i $$0.616087\pi$$
$$234$$ 0 0
$$235$$ −8.70820 −0.568061
$$236$$ 18.9443 1.23317
$$237$$ 0 0
$$238$$ 10.9443 0.709412
$$239$$ 24.6525 1.59464 0.797318 0.603559i $$-0.206251\pi$$
0.797318 + 0.603559i $$0.206251\pi$$
$$240$$ 0 0
$$241$$ −17.9443 −1.15589 −0.577946 0.816075i $$-0.696146\pi$$
−0.577946 + 0.816075i $$0.696146\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 2.38197 0.152490
$$245$$ 3.47214 0.221827
$$246$$ 0 0
$$247$$ 33.8885 2.15628
$$248$$ 15.0000 0.952501
$$249$$ 0 0
$$250$$ 0.618034 0.0390879
$$251$$ 23.7082 1.49645 0.748224 0.663446i $$-0.230907\pi$$
0.748224 + 0.663446i $$0.230907\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 2.29180 0.143800
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ −28.8885 −1.80202 −0.901009 0.433801i $$-0.857172\pi$$
−0.901009 + 0.433801i $$0.857172\pi$$
$$258$$ 0 0
$$259$$ 2.47214 0.153611
$$260$$ −8.47214 −0.525420
$$261$$ 0 0
$$262$$ 0.291796 0.0180272
$$263$$ −20.1246 −1.24094 −0.620468 0.784231i $$-0.713057\pi$$
−0.620468 + 0.784231i $$0.713057\pi$$
$$264$$ 0 0
$$265$$ −9.94427 −0.610872
$$266$$ 12.9443 0.793664
$$267$$ 0 0
$$268$$ −18.1803 −1.11054
$$269$$ −7.05573 −0.430195 −0.215098 0.976593i $$-0.569007\pi$$
−0.215098 + 0.976593i $$0.569007\pi$$
$$270$$ 0 0
$$271$$ −24.2361 −1.47224 −0.736118 0.676853i $$-0.763343\pi$$
−0.736118 + 0.676853i $$0.763343\pi$$
$$272$$ 10.1459 0.615185
$$273$$ 0 0
$$274$$ −10.7984 −0.652354
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −4.94427 −0.297073 −0.148536 0.988907i $$-0.547456\pi$$
−0.148536 + 0.988907i $$0.547456\pi$$
$$278$$ −4.50658 −0.270287
$$279$$ 0 0
$$280$$ −7.23607 −0.432438
$$281$$ −15.2361 −0.908908 −0.454454 0.890770i $$-0.650166\pi$$
−0.454454 + 0.890770i $$0.650166\pi$$
$$282$$ 0 0
$$283$$ 11.8885 0.706701 0.353350 0.935491i $$-0.385042\pi$$
0.353350 + 0.935491i $$0.385042\pi$$
$$284$$ −23.4164 −1.38951
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −11.4164 −0.673889
$$288$$ 0 0
$$289$$ 12.9443 0.761428
$$290$$ 0.763932 0.0448596
$$291$$ 0 0
$$292$$ 16.9443 0.991589
$$293$$ −21.4721 −1.25442 −0.627208 0.778852i $$-0.715802\pi$$
−0.627208 + 0.778852i $$0.715802\pi$$
$$294$$ 0 0
$$295$$ −11.7082 −0.681678
$$296$$ −1.70820 −0.0992873
$$297$$ 0 0
$$298$$ 2.65248 0.153654
$$299$$ 24.6525 1.42569
$$300$$ 0 0
$$301$$ −16.9443 −0.976652
$$302$$ 4.50658 0.259324
$$303$$ 0 0
$$304$$ 12.0000 0.688247
$$305$$ −1.47214 −0.0842943
$$306$$ 0 0
$$307$$ −11.8885 −0.678515 −0.339258 0.940694i $$-0.610176\pi$$
−0.339258 + 0.940694i $$0.610176\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −4.14590 −0.235471
$$311$$ 3.23607 0.183501 0.0917503 0.995782i $$-0.470754\pi$$
0.0917503 + 0.995782i $$0.470754\pi$$
$$312$$ 0 0
$$313$$ 22.7639 1.28669 0.643347 0.765575i $$-0.277545\pi$$
0.643347 + 0.765575i $$0.277545\pi$$
$$314$$ 1.41641 0.0799325
$$315$$ 0 0
$$316$$ −20.5623 −1.15672
$$317$$ 3.94427 0.221532 0.110766 0.993846i $$-0.464670\pi$$
0.110766 + 0.993846i $$0.464670\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −0.236068 −0.0131966
$$321$$ 0 0
$$322$$ 9.41641 0.524756
$$323$$ 35.4164 1.97062
$$324$$ 0 0
$$325$$ 5.23607 0.290445
$$326$$ 3.81966 0.211551
$$327$$ 0 0
$$328$$ 7.88854 0.435572
$$329$$ −28.1803 −1.55363
$$330$$ 0 0
$$331$$ −17.1803 −0.944317 −0.472158 0.881514i $$-0.656525\pi$$
−0.472158 + 0.881514i $$0.656525\pi$$
$$332$$ −6.47214 −0.355205
$$333$$ 0 0
$$334$$ −1.96556 −0.107551
$$335$$ 11.2361 0.613892
$$336$$ 0 0
$$337$$ −7.88854 −0.429716 −0.214858 0.976645i $$-0.568929\pi$$
−0.214858 + 0.976645i $$0.568929\pi$$
$$338$$ 8.90983 0.484631
$$339$$ 0 0
$$340$$ −8.85410 −0.480181
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −11.4164 −0.616428
$$344$$ 11.7082 0.631264
$$345$$ 0 0
$$346$$ −1.81966 −0.0978255
$$347$$ 14.2361 0.764232 0.382116 0.924114i $$-0.375195\pi$$
0.382116 + 0.924114i $$0.375195\pi$$
$$348$$ 0 0
$$349$$ −15.0000 −0.802932 −0.401466 0.915874i $$-0.631499\pi$$
−0.401466 + 0.915874i $$0.631499\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7.00000 0.372572 0.186286 0.982496i $$-0.440355\pi$$
0.186286 + 0.982496i $$0.440355\pi$$
$$354$$ 0 0
$$355$$ 14.4721 0.768101
$$356$$ 7.70820 0.408534
$$357$$ 0 0
$$358$$ 12.4721 0.659173
$$359$$ −7.41641 −0.391423 −0.195712 0.980662i $$-0.562702\pi$$
−0.195712 + 0.980662i $$0.562702\pi$$
$$360$$ 0 0
$$361$$ 22.8885 1.20466
$$362$$ −13.2361 −0.695672
$$363$$ 0 0
$$364$$ −27.4164 −1.43701
$$365$$ −10.4721 −0.548137
$$366$$ 0 0
$$367$$ 4.76393 0.248675 0.124338 0.992240i $$-0.460319\pi$$
0.124338 + 0.992240i $$0.460319\pi$$
$$368$$ 8.72949 0.455056
$$369$$ 0 0
$$370$$ 0.472136 0.0245452
$$371$$ −32.1803 −1.67072
$$372$$ 0 0
$$373$$ 29.8885 1.54757 0.773785 0.633448i $$-0.218361\pi$$
0.773785 + 0.633448i $$0.218361\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 19.4721 1.00420
$$377$$ 6.47214 0.333332
$$378$$ 0 0
$$379$$ 30.5967 1.57165 0.785825 0.618449i $$-0.212239\pi$$
0.785825 + 0.618449i $$0.212239\pi$$
$$380$$ −10.4721 −0.537209
$$381$$ 0 0
$$382$$ −17.0557 −0.872647
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10.0000 0.508987
$$387$$ 0 0
$$388$$ −20.6525 −1.04847
$$389$$ 21.5967 1.09500 0.547499 0.836806i $$-0.315580\pi$$
0.547499 + 0.836806i $$0.315580\pi$$
$$390$$ 0 0
$$391$$ 25.7639 1.30294
$$392$$ −7.76393 −0.392138
$$393$$ 0 0
$$394$$ 0.875388 0.0441014
$$395$$ 12.7082 0.639419
$$396$$ 0 0
$$397$$ −17.4164 −0.874104 −0.437052 0.899436i $$-0.643978\pi$$
−0.437052 + 0.899436i $$0.643978\pi$$
$$398$$ 10.0344 0.502981
$$399$$ 0 0
$$400$$ 1.85410 0.0927051
$$401$$ −20.9443 −1.04591 −0.522954 0.852361i $$-0.675170\pi$$
−0.522954 + 0.852361i $$0.675170\pi$$
$$402$$ 0 0
$$403$$ −35.1246 −1.74968
$$404$$ −8.94427 −0.444994
$$405$$ 0 0
$$406$$ 2.47214 0.122690
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 36.7771 1.81851 0.909255 0.416240i $$-0.136652\pi$$
0.909255 + 0.416240i $$0.136652\pi$$
$$410$$ −2.18034 −0.107679
$$411$$ 0 0
$$412$$ 1.52786 0.0752725
$$413$$ −37.8885 −1.86437
$$414$$ 0 0
$$415$$ 4.00000 0.196352
$$416$$ 29.4164 1.44226
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 14.0000 0.683945 0.341972 0.939710i $$-0.388905\pi$$
0.341972 + 0.939710i $$0.388905\pi$$
$$420$$ 0 0
$$421$$ 29.8328 1.45396 0.726981 0.686657i $$-0.240923\pi$$
0.726981 + 0.686657i $$0.240923\pi$$
$$422$$ 1.09017 0.0530686
$$423$$ 0 0
$$424$$ 22.2361 1.07988
$$425$$ 5.47214 0.265438
$$426$$ 0 0
$$427$$ −4.76393 −0.230543
$$428$$ 23.0344 1.11341
$$429$$ 0 0
$$430$$ −3.23607 −0.156057
$$431$$ 5.81966 0.280323 0.140162 0.990129i $$-0.455238\pi$$
0.140162 + 0.990129i $$0.455238\pi$$
$$432$$ 0 0
$$433$$ 25.2361 1.21277 0.606384 0.795172i $$-0.292619\pi$$
0.606384 + 0.795172i $$0.292619\pi$$
$$434$$ −13.4164 −0.644008
$$435$$ 0 0
$$436$$ −24.1803 −1.15803
$$437$$ 30.4721 1.45768
$$438$$ 0 0
$$439$$ −8.12461 −0.387767 −0.193883 0.981025i $$-0.562108\pi$$
−0.193883 + 0.981025i $$0.562108\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 17.7082 0.842293
$$443$$ −7.41641 −0.352364 −0.176182 0.984358i $$-0.556375\pi$$
−0.176182 + 0.984358i $$0.556375\pi$$
$$444$$ 0 0
$$445$$ −4.76393 −0.225832
$$446$$ 5.59675 0.265014
$$447$$ 0 0
$$448$$ −0.763932 −0.0360924
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −20.0902 −0.944962
$$453$$ 0 0
$$454$$ −10.3262 −0.484634
$$455$$ 16.9443 0.794360
$$456$$ 0 0
$$457$$ −34.3607 −1.60732 −0.803662 0.595085i $$-0.797118\pi$$
−0.803662 + 0.595085i $$0.797118\pi$$
$$458$$ 4.32624 0.202152
$$459$$ 0 0
$$460$$ −7.61803 −0.355193
$$461$$ −30.1803 −1.40564 −0.702819 0.711368i $$-0.748076\pi$$
−0.702819 + 0.711368i $$0.748076\pi$$
$$462$$ 0 0
$$463$$ 6.00000 0.278844 0.139422 0.990233i $$-0.455476\pi$$
0.139422 + 0.990233i $$0.455476\pi$$
$$464$$ 2.29180 0.106394
$$465$$ 0 0
$$466$$ −6.72949 −0.311738
$$467$$ 3.76393 0.174174 0.0870870 0.996201i $$-0.472244\pi$$
0.0870870 + 0.996201i $$0.472244\pi$$
$$468$$ 0 0
$$469$$ 36.3607 1.67898
$$470$$ −5.38197 −0.248252
$$471$$ 0 0
$$472$$ 26.1803 1.20505
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 6.47214 0.296962
$$476$$ −28.6525 −1.31328
$$477$$ 0 0
$$478$$ 15.2361 0.696882
$$479$$ 0.291796 0.0133325 0.00666625 0.999978i $$-0.497878\pi$$
0.00666625 + 0.999978i $$0.497878\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ −11.0902 −0.505143
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.7639 0.579580
$$486$$ 0 0
$$487$$ −11.7082 −0.530549 −0.265275 0.964173i $$-0.585463\pi$$
−0.265275 + 0.964173i $$0.585463\pi$$
$$488$$ 3.29180 0.149013
$$489$$ 0 0
$$490$$ 2.14590 0.0969418
$$491$$ 38.4721 1.73622 0.868112 0.496369i $$-0.165334\pi$$
0.868112 + 0.496369i $$0.165334\pi$$
$$492$$ 0 0
$$493$$ 6.76393 0.304632
$$494$$ 20.9443 0.942327
$$495$$ 0 0
$$496$$ −12.4377 −0.558469
$$497$$ 46.8328 2.10074
$$498$$ 0 0
$$499$$ −0.944272 −0.0422714 −0.0211357 0.999777i $$-0.506728\pi$$
−0.0211357 + 0.999777i $$0.506728\pi$$
$$500$$ −1.61803 −0.0723607
$$501$$ 0 0
$$502$$ 14.6525 0.653972
$$503$$ 23.1803 1.03356 0.516780 0.856118i $$-0.327130\pi$$
0.516780 + 0.856118i $$0.327130\pi$$
$$504$$ 0 0
$$505$$ 5.52786 0.245987
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −6.00000 −0.266207
$$509$$ −15.5967 −0.691314 −0.345657 0.938361i $$-0.612344\pi$$
−0.345657 + 0.938361i $$0.612344\pi$$
$$510$$ 0 0
$$511$$ −33.8885 −1.49914
$$512$$ 18.7082 0.826794
$$513$$ 0 0
$$514$$ −17.8541 −0.787511
$$515$$ −0.944272 −0.0416096
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 1.52786 0.0671305
$$519$$ 0 0
$$520$$ −11.7082 −0.513439
$$521$$ −17.8197 −0.780693 −0.390347 0.920668i $$-0.627645\pi$$
−0.390347 + 0.920668i $$0.627645\pi$$
$$522$$ 0 0
$$523$$ −24.5410 −1.07310 −0.536552 0.843867i $$-0.680273\pi$$
−0.536552 + 0.843867i $$0.680273\pi$$
$$524$$ −0.763932 −0.0333725
$$525$$ 0 0
$$526$$ −12.4377 −0.542309
$$527$$ −36.7082 −1.59903
$$528$$ 0 0
$$529$$ −0.832816 −0.0362094
$$530$$ −6.14590 −0.266961
$$531$$ 0 0
$$532$$ −33.8885 −1.46925
$$533$$ −18.4721 −0.800117
$$534$$ 0 0
$$535$$ −14.2361 −0.615479
$$536$$ −25.1246 −1.08522
$$537$$ 0 0
$$538$$ −4.36068 −0.188002
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −43.3050 −1.86183 −0.930913 0.365242i $$-0.880986\pi$$
−0.930913 + 0.365242i $$0.880986\pi$$
$$542$$ −14.9787 −0.643391
$$543$$ 0 0
$$544$$ 30.7426 1.31808
$$545$$ 14.9443 0.640142
$$546$$ 0 0
$$547$$ −9.59675 −0.410327 −0.205164 0.978728i $$-0.565773\pi$$
−0.205164 + 0.978728i $$0.565773\pi$$
$$548$$ 28.2705 1.20766
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 41.1246 1.74880
$$554$$ −3.05573 −0.129825
$$555$$ 0 0
$$556$$ 11.7984 0.500363
$$557$$ 8.52786 0.361337 0.180669 0.983544i $$-0.442174\pi$$
0.180669 + 0.983544i $$0.442174\pi$$
$$558$$ 0 0
$$559$$ −27.4164 −1.15959
$$560$$ 6.00000 0.253546
$$561$$ 0 0
$$562$$ −9.41641 −0.397207
$$563$$ 0.944272 0.0397963 0.0198982 0.999802i $$-0.493666\pi$$
0.0198982 + 0.999802i $$0.493666\pi$$
$$564$$ 0 0
$$565$$ 12.4164 0.522362
$$566$$ 7.34752 0.308839
$$567$$ 0 0
$$568$$ −32.3607 −1.35782
$$569$$ 19.7082 0.826211 0.413105 0.910683i $$-0.364444\pi$$
0.413105 + 0.910683i $$0.364444\pi$$
$$570$$ 0 0
$$571$$ 0.124612 0.00521484 0.00260742 0.999997i $$-0.499170\pi$$
0.00260742 + 0.999997i $$0.499170\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −7.05573 −0.294500
$$575$$ 4.70820 0.196346
$$576$$ 0 0
$$577$$ 20.0000 0.832611 0.416305 0.909225i $$-0.363325\pi$$
0.416305 + 0.909225i $$0.363325\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ −2.00000 −0.0830455
$$581$$ 12.9443 0.537019
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 23.4164 0.968978
$$585$$ 0 0
$$586$$ −13.2705 −0.548200
$$587$$ 11.6525 0.480949 0.240475 0.970655i $$-0.422697\pi$$
0.240475 + 0.970655i $$0.422697\pi$$
$$588$$ 0 0
$$589$$ −43.4164 −1.78894
$$590$$ −7.23607 −0.297904
$$591$$ 0 0
$$592$$ 1.41641 0.0582140
$$593$$ 34.9443 1.43499 0.717495 0.696564i $$-0.245289\pi$$
0.717495 + 0.696564i $$0.245289\pi$$
$$594$$ 0 0
$$595$$ 17.7082 0.725966
$$596$$ −6.94427 −0.284448
$$597$$ 0 0
$$598$$ 15.2361 0.623049
$$599$$ 31.0132 1.26716 0.633582 0.773676i $$-0.281584\pi$$
0.633582 + 0.773676i $$0.281584\pi$$
$$600$$ 0 0
$$601$$ 28.4721 1.16140 0.580701 0.814117i $$-0.302778\pi$$
0.580701 + 0.814117i $$0.302778\pi$$
$$602$$ −10.4721 −0.426812
$$603$$ 0 0
$$604$$ −11.7984 −0.480069
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 6.29180 0.255376 0.127688 0.991814i $$-0.459244\pi$$
0.127688 + 0.991814i $$0.459244\pi$$
$$608$$ 36.3607 1.47462
$$609$$ 0 0
$$610$$ −0.909830 −0.0368379
$$611$$ −45.5967 −1.84465
$$612$$ 0 0
$$613$$ −14.8328 −0.599092 −0.299546 0.954082i $$-0.596835\pi$$
−0.299546 + 0.954082i $$0.596835\pi$$
$$614$$ −7.34752 −0.296522
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −41.7771 −1.68188 −0.840941 0.541127i $$-0.817998\pi$$
−0.840941 + 0.541127i $$0.817998\pi$$
$$618$$ 0 0
$$619$$ −9.52786 −0.382957 −0.191479 0.981497i $$-0.561328\pi$$
−0.191479 + 0.981497i $$0.561328\pi$$
$$620$$ 10.8541 0.435911
$$621$$ 0 0
$$622$$ 2.00000 0.0801927
$$623$$ −15.4164 −0.617645
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 14.0689 0.562306
$$627$$ 0 0
$$628$$ −3.70820 −0.147973
$$629$$ 4.18034 0.166681
$$630$$ 0 0
$$631$$ 9.18034 0.365464 0.182732 0.983163i $$-0.441506\pi$$
0.182732 + 0.983163i $$0.441506\pi$$
$$632$$ −28.4164 −1.13034
$$633$$ 0 0
$$634$$ 2.43769 0.0968132
$$635$$ 3.70820 0.147156
$$636$$ 0 0
$$637$$ 18.1803 0.720331
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −11.3820 −0.449912
$$641$$ 9.12461 0.360400 0.180200 0.983630i $$-0.442325\pi$$
0.180200 + 0.983630i $$0.442325\pi$$
$$642$$ 0 0
$$643$$ −28.8328 −1.13706 −0.568528 0.822664i $$-0.692487\pi$$
−0.568528 + 0.822664i $$0.692487\pi$$
$$644$$ −24.6525 −0.971444
$$645$$ 0 0
$$646$$ 21.8885 0.861193
$$647$$ 16.2361 0.638306 0.319153 0.947703i $$-0.396602\pi$$
0.319153 + 0.947703i $$0.396602\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 3.23607 0.126929
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ −16.8328 −0.658719 −0.329359 0.944205i $$-0.606833\pi$$
−0.329359 + 0.944205i $$0.606833\pi$$
$$654$$ 0 0
$$655$$ 0.472136 0.0184479
$$656$$ −6.54102 −0.255384
$$657$$ 0 0
$$658$$ −17.4164 −0.678962
$$659$$ −35.5967 −1.38665 −0.693326 0.720624i $$-0.743855\pi$$
−0.693326 + 0.720624i $$0.743855\pi$$
$$660$$ 0 0
$$661$$ 45.7771 1.78052 0.890261 0.455450i $$-0.150522\pi$$
0.890261 + 0.455450i $$0.150522\pi$$
$$662$$ −10.6180 −0.412682
$$663$$ 0 0
$$664$$ −8.94427 −0.347105
$$665$$ 20.9443 0.812184
$$666$$ 0 0
$$667$$ 5.81966 0.225338
$$668$$ 5.14590 0.199101
$$669$$ 0 0
$$670$$ 6.94427 0.268281
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −27.5967 −1.06378 −0.531888 0.846815i $$-0.678517\pi$$
−0.531888 + 0.846815i $$0.678517\pi$$
$$674$$ −4.87539 −0.187793
$$675$$ 0 0
$$676$$ −23.3262 −0.897163
$$677$$ −5.41641 −0.208169 −0.104085 0.994568i $$-0.533191\pi$$
−0.104085 + 0.994568i $$0.533191\pi$$
$$678$$ 0 0
$$679$$ 41.3050 1.58514
$$680$$ −12.2361 −0.469232
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ −17.4721 −0.667576
$$686$$ −7.05573 −0.269389
$$687$$ 0 0
$$688$$ −9.70820 −0.370122
$$689$$ −52.0689 −1.98367
$$690$$ 0 0
$$691$$ −37.5410 −1.42813 −0.714064 0.700081i $$-0.753147\pi$$
−0.714064 + 0.700081i $$0.753147\pi$$
$$692$$ 4.76393 0.181098
$$693$$ 0 0
$$694$$ 8.79837 0.333982
$$695$$ −7.29180 −0.276594
$$696$$ 0 0
$$697$$ −19.3050 −0.731227
$$698$$ −9.27051 −0.350894
$$699$$ 0 0
$$700$$ −5.23607 −0.197905
$$701$$ −36.1803 −1.36651 −0.683256 0.730179i $$-0.739437\pi$$
−0.683256 + 0.730179i $$0.739437\pi$$
$$702$$ 0 0
$$703$$ 4.94427 0.186477
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 4.32624 0.162820
$$707$$ 17.8885 0.672768
$$708$$ 0 0
$$709$$ −17.9443 −0.673911 −0.336956 0.941521i $$-0.609397\pi$$
−0.336956 + 0.941521i $$0.609397\pi$$
$$710$$ 8.94427 0.335673
$$711$$ 0 0
$$712$$ 10.6525 0.399218
$$713$$ −31.5836 −1.18281
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −32.6525 −1.22028
$$717$$ 0 0
$$718$$ −4.58359 −0.171058
$$719$$ −26.3607 −0.983087 −0.491544 0.870853i $$-0.663567\pi$$
−0.491544 + 0.870853i $$0.663567\pi$$
$$720$$ 0 0
$$721$$ −3.05573 −0.113801
$$722$$ 14.1459 0.526456
$$723$$ 0 0
$$724$$ 34.6525 1.28785
$$725$$ 1.23607 0.0459064
$$726$$ 0 0
$$727$$ 44.8328 1.66276 0.831379 0.555706i $$-0.187552\pi$$
0.831379 + 0.555706i $$0.187552\pi$$
$$728$$ −37.8885 −1.40424
$$729$$ 0 0
$$730$$ −6.47214 −0.239544
$$731$$ −28.6525 −1.05975
$$732$$ 0 0
$$733$$ 47.8885 1.76880 0.884402 0.466726i $$-0.154567\pi$$
0.884402 + 0.466726i $$0.154567\pi$$
$$734$$ 2.94427 0.108675
$$735$$ 0 0
$$736$$ 26.4508 0.974991
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 1.65248 0.0607873 0.0303937 0.999538i $$-0.490324\pi$$
0.0303937 + 0.999538i $$0.490324\pi$$
$$740$$ −1.23607 −0.0454388
$$741$$ 0 0
$$742$$ −19.8885 −0.730131
$$743$$ −2.23607 −0.0820334 −0.0410167 0.999158i $$-0.513060\pi$$
−0.0410167 + 0.999158i $$0.513060\pi$$
$$744$$ 0 0
$$745$$ 4.29180 0.157239
$$746$$ 18.4721 0.676313
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −46.0689 −1.68332
$$750$$ 0 0
$$751$$ −34.0132 −1.24116 −0.620579 0.784144i $$-0.713102\pi$$
−0.620579 + 0.784144i $$0.713102\pi$$
$$752$$ −16.1459 −0.588780
$$753$$ 0 0
$$754$$ 4.00000 0.145671
$$755$$ 7.29180 0.265376
$$756$$ 0 0
$$757$$ 18.9443 0.688541 0.344271 0.938870i $$-0.388126\pi$$
0.344271 + 0.938870i $$0.388126\pi$$
$$758$$ 18.9098 0.686836
$$759$$ 0 0
$$760$$ −14.4721 −0.524960
$$761$$ 44.6525 1.61865 0.809325 0.587360i $$-0.199833\pi$$
0.809325 + 0.587360i $$0.199833\pi$$
$$762$$ 0 0
$$763$$ 48.3607 1.75077
$$764$$ 44.6525 1.61547
$$765$$ 0 0
$$766$$ −9.88854 −0.357288
$$767$$ −61.3050 −2.21359
$$768$$ 0 0
$$769$$ −24.5279 −0.884497 −0.442249 0.896892i $$-0.645819\pi$$
−0.442249 + 0.896892i $$0.645819\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −26.1803 −0.942251
$$773$$ 7.94427 0.285736 0.142868 0.989742i $$-0.454368\pi$$
0.142868 + 0.989742i $$0.454368\pi$$
$$774$$ 0 0
$$775$$ −6.70820 −0.240966
$$776$$ −28.5410 −1.02456
$$777$$ 0 0
$$778$$ 13.3475 0.478532
$$779$$ −22.8328 −0.818071
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 15.9230 0.569405
$$783$$ 0 0
$$784$$ 6.43769 0.229918
$$785$$ 2.29180 0.0817977
$$786$$ 0 0
$$787$$ −27.0557 −0.964433 −0.482216 0.876052i $$-0.660168\pi$$
−0.482216 + 0.876052i $$0.660168\pi$$
$$788$$ −2.29180 −0.0816419
$$789$$ 0 0
$$790$$ 7.85410 0.279436
$$791$$ 40.1803 1.42865
$$792$$ 0 0
$$793$$ −7.70820 −0.273726
$$794$$ −10.7639 −0.381998
$$795$$ 0 0
$$796$$ −26.2705 −0.931134
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ 0 0
$$799$$ −47.6525 −1.68582
$$800$$ 5.61803 0.198627
$$801$$ 0 0
$$802$$ −12.9443 −0.457078
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 15.2361 0.537001
$$806$$ −21.7082 −0.764639
$$807$$ 0 0
$$808$$ −12.3607 −0.434847
$$809$$ 8.00000 0.281265 0.140633 0.990062i $$-0.455086\pi$$
0.140633 + 0.990062i $$0.455086\pi$$
$$810$$ 0 0
$$811$$ 7.87539 0.276542 0.138271 0.990394i $$-0.455845\pi$$
0.138271 + 0.990394i $$0.455845\pi$$
$$812$$ −6.47214 −0.227127
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 6.18034 0.216488
$$816$$ 0 0
$$817$$ −33.8885 −1.18561
$$818$$ 22.7295 0.794718
$$819$$ 0 0
$$820$$ 5.70820 0.199339
$$821$$ 33.7771 1.17883 0.589414 0.807831i $$-0.299359\pi$$
0.589414 + 0.807831i $$0.299359\pi$$
$$822$$ 0 0
$$823$$ 38.2492 1.33328 0.666642 0.745378i $$-0.267731\pi$$
0.666642 + 0.745378i $$0.267731\pi$$
$$824$$ 2.11146 0.0735561
$$825$$ 0 0
$$826$$ −23.4164 −0.814761
$$827$$ −8.94427 −0.311023 −0.155511 0.987834i $$-0.549703\pi$$
−0.155511 + 0.987834i $$0.549703\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 2.47214 0.0858091
$$831$$ 0 0
$$832$$ −1.23607 −0.0428529
$$833$$ 19.0000 0.658311
$$834$$ 0 0
$$835$$ −3.18034 −0.110060
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 8.65248 0.298895
$$839$$ −28.3607 −0.979119 −0.489560 0.871970i $$-0.662842\pi$$
−0.489560 + 0.871970i $$0.662842\pi$$
$$840$$ 0 0
$$841$$ −27.4721 −0.947315
$$842$$ 18.4377 0.635405
$$843$$ 0 0
$$844$$ −2.85410 −0.0982422
$$845$$ 14.4164 0.495940
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −18.4377 −0.633153
$$849$$ 0 0
$$850$$ 3.38197 0.116000
$$851$$ 3.59675 0.123295
$$852$$ 0 0
$$853$$ −31.8885 −1.09184 −0.545921 0.837836i $$-0.683820\pi$$
−0.545921 + 0.837836i $$0.683820\pi$$
$$854$$ −2.94427 −0.100751
$$855$$ 0 0
$$856$$ 31.8328 1.08802
$$857$$ −1.00000 −0.0341593 −0.0170797 0.999854i $$-0.505437\pi$$
−0.0170797 + 0.999854i $$0.505437\pi$$
$$858$$ 0 0
$$859$$ −26.8328 −0.915524 −0.457762 0.889075i $$-0.651349\pi$$
−0.457762 + 0.889075i $$0.651349\pi$$
$$860$$ 8.47214 0.288897
$$861$$ 0 0
$$862$$ 3.59675 0.122506
$$863$$ 37.8885 1.28974 0.644871 0.764292i $$-0.276911\pi$$
0.644871 + 0.764292i $$0.276911\pi$$
$$864$$ 0 0
$$865$$ −2.94427 −0.100108
$$866$$ 15.5967 0.529999
$$867$$ 0 0
$$868$$ 35.1246 1.19221
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 58.8328 1.99347
$$872$$ −33.4164 −1.13162
$$873$$ 0 0
$$874$$ 18.8328 0.637029
$$875$$ 3.23607 0.109399
$$876$$ 0 0
$$877$$ 11.3050 0.381741 0.190871 0.981615i $$-0.438869\pi$$
0.190871 + 0.981615i $$0.438869\pi$$
$$878$$ −5.02129 −0.169460
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −38.8328 −1.30831 −0.654155 0.756360i $$-0.726976\pi$$
−0.654155 + 0.756360i $$0.726976\pi$$
$$882$$ 0 0
$$883$$ 10.1803 0.342596 0.171298 0.985219i $$-0.445204\pi$$
0.171298 + 0.985219i $$0.445204\pi$$
$$884$$ −46.3607 −1.55928
$$885$$ 0 0
$$886$$ −4.58359 −0.153989
$$887$$ −16.9443 −0.568933 −0.284466 0.958686i $$-0.591816\pi$$
−0.284466 + 0.958686i $$0.591816\pi$$
$$888$$ 0 0
$$889$$ 12.0000 0.402467
$$890$$ −2.94427 −0.0986922
$$891$$ 0 0
$$892$$ −14.6525 −0.490601
$$893$$ −56.3607 −1.88604
$$894$$ 0 0
$$895$$ 20.1803 0.674554
$$896$$ −36.8328 −1.23050
$$897$$ 0 0
$$898$$ 6.18034 0.206241
$$899$$ −8.29180 −0.276547
$$900$$ 0 0
$$901$$ −54.4164 −1.81287
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −27.7639 −0.923415
$$905$$ −21.4164 −0.711905
$$906$$ 0 0
$$907$$ −42.0000 −1.39459 −0.697294 0.716786i $$-0.745613\pi$$
−0.697294 + 0.716786i $$0.745613\pi$$
$$908$$ 27.0344 0.897169
$$909$$ 0 0
$$910$$ 10.4721 0.347148
$$911$$ −47.9574 −1.58890 −0.794450 0.607329i $$-0.792241\pi$$
−0.794450 + 0.607329i $$0.792241\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −21.2361 −0.702427
$$915$$ 0 0
$$916$$ −11.3262 −0.374229
$$917$$ 1.52786 0.0504545
$$918$$ 0 0
$$919$$ 3.41641 0.112697 0.0563484 0.998411i $$-0.482054\pi$$
0.0563484 + 0.998411i $$0.482054\pi$$
$$920$$ −10.5279 −0.347093
$$921$$ 0 0
$$922$$ −18.6525 −0.614287
$$923$$ 75.7771 2.49423
$$924$$ 0 0
$$925$$ 0.763932 0.0251179
$$926$$ 3.70820 0.121859
$$927$$ 0 0
$$928$$ 6.94427 0.227957
$$929$$ −50.9443 −1.67143 −0.835714 0.549165i $$-0.814946\pi$$
−0.835714 + 0.549165i $$0.814946\pi$$
$$930$$ 0 0
$$931$$ 22.4721 0.736495
$$932$$ 17.6180 0.577098
$$933$$ 0 0
$$934$$ 2.32624 0.0761168
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 33.1246 1.08213 0.541067 0.840980i $$-0.318021\pi$$
0.541067 + 0.840980i $$0.318021\pi$$
$$938$$ 22.4721 0.733741
$$939$$ 0 0
$$940$$ 14.0902 0.459571
$$941$$ 45.2361 1.47465 0.737327 0.675536i $$-0.236088\pi$$
0.737327 + 0.675536i $$0.236088\pi$$
$$942$$ 0 0
$$943$$ −16.6099 −0.540893
$$944$$ −21.7082 −0.706542
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 18.2361 0.592593 0.296296 0.955096i $$-0.404248\pi$$
0.296296 + 0.955096i $$0.404248\pi$$
$$948$$ 0 0
$$949$$ −54.8328 −1.77995
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ −39.5967 −1.28334
$$953$$ −31.8885 −1.03297 −0.516486 0.856296i $$-0.672760\pi$$
−0.516486 + 0.856296i $$0.672760\pi$$
$$954$$ 0 0
$$955$$ −27.5967 −0.893010
$$956$$ −39.8885 −1.29009
$$957$$ 0 0
$$958$$ 0.180340 0.00582652
$$959$$ −56.5410 −1.82580
$$960$$ 0 0
$$961$$ 14.0000 0.451613
$$962$$ 2.47214 0.0797049
$$963$$ 0 0
$$964$$ 29.0344 0.935136
$$965$$ 16.1803 0.520864
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 7.88854 0.253286
$$971$$ −29.2361 −0.938230 −0.469115 0.883137i $$-0.655427\pi$$
−0.469115 + 0.883137i $$0.655427\pi$$
$$972$$ 0 0
$$973$$ −23.5967 −0.756477
$$974$$ −7.23607 −0.231859
$$975$$ 0 0
$$976$$ −2.72949 −0.0873689
$$977$$ 15.9443 0.510102 0.255051 0.966928i $$-0.417908\pi$$
0.255051 + 0.966928i $$0.417908\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −5.61803 −0.179462
$$981$$ 0 0
$$982$$ 23.7771 0.758757
$$983$$ 55.1803 1.75998 0.879990 0.474993i $$-0.157549\pi$$
0.879990 + 0.474993i $$0.157549\pi$$
$$984$$ 0 0
$$985$$ 1.41641 0.0451305
$$986$$ 4.18034 0.133129
$$987$$ 0 0
$$988$$ −54.8328 −1.74446
$$989$$ −24.6525 −0.783903
$$990$$ 0 0
$$991$$ 6.23607 0.198095 0.0990476 0.995083i $$-0.468420\pi$$
0.0990476 + 0.995083i $$0.468420\pi$$
$$992$$ −37.6869 −1.19656
$$993$$ 0 0
$$994$$ 28.9443 0.918057
$$995$$ 16.2361 0.514718
$$996$$ 0 0
$$997$$ 38.7639 1.22767 0.613833 0.789436i $$-0.289627\pi$$
0.613833 + 0.789436i $$0.289627\pi$$
$$998$$ −0.583592 −0.0184733
$$999$$ 0 0
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 5445.2.a.o.1.2 2
3.2 odd 2 1815.2.a.j.1.1 yes 2
11.10 odd 2 5445.2.a.x.1.1 2
15.14 odd 2 9075.2.a.bd.1.2 2
33.32 even 2 1815.2.a.f.1.2 2
165.164 even 2 9075.2.a.bx.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.2 2 33.32 even 2
1815.2.a.j.1.1 yes 2 3.2 odd 2
5445.2.a.o.1.2 2 1.1 even 1 trivial
5445.2.a.x.1.1 2 11.10 odd 2
9075.2.a.bd.1.2 2 15.14 odd 2
9075.2.a.bx.1.1 2 165.164 even 2