# Properties

 Label 9075.2.a.bx.1.1 Level $9075$ Weight $2$ Character 9075.1 Self dual yes Analytic conductor $72.464$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$9075 = 3 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.4642398343$$ Analytic rank: $$1$$ 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 1.1 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 9075.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +3.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +3.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{12} +5.23607 q^{13} -2.00000 q^{14} +1.85410 q^{16} -5.47214 q^{17} -0.618034 q^{18} -6.47214 q^{19} +3.23607 q^{21} +4.70820 q^{23} +2.23607 q^{24} -3.23607 q^{26} +1.00000 q^{27} -5.23607 q^{28} +1.23607 q^{29} -6.70820 q^{31} -5.61803 q^{32} +3.38197 q^{34} -1.61803 q^{36} -0.763932 q^{37} +4.00000 q^{38} +5.23607 q^{39} -3.52786 q^{41} -2.00000 q^{42} -5.23607 q^{43} -2.90983 q^{46} -8.70820 q^{47} +1.85410 q^{48} +3.47214 q^{49} -5.47214 q^{51} -8.47214 q^{52} -9.94427 q^{53} -0.618034 q^{54} +7.23607 q^{56} -6.47214 q^{57} -0.763932 q^{58} +11.7082 q^{59} +1.47214 q^{61} +4.14590 q^{62} +3.23607 q^{63} -0.236068 q^{64} -11.2361 q^{67} +8.85410 q^{68} +4.70820 q^{69} -14.4721 q^{71} +2.23607 q^{72} -10.4721 q^{73} +0.472136 q^{74} +10.4721 q^{76} -3.23607 q^{78} -12.7082 q^{79} +1.00000 q^{81} +2.18034 q^{82} -4.00000 q^{83} -5.23607 q^{84} +3.23607 q^{86} +1.23607 q^{87} +4.76393 q^{89} +16.9443 q^{91} -7.61803 q^{92} -6.70820 q^{93} +5.38197 q^{94} -5.61803 q^{96} -12.7639 q^{97} -2.14590 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + q^{2} + 2q^{3} - q^{4} + q^{6} + 2q^{7} + 2q^{9} - q^{12} + 6q^{13} - 4q^{14} - 3q^{16} - 2q^{17} + q^{18} - 4q^{19} + 2q^{21} - 4q^{23} - 2q^{26} + 2q^{27} - 6q^{28} - 2q^{29} - 9q^{32} + 9q^{34} - q^{36} - 6q^{37} + 8q^{38} + 6q^{39} - 16q^{41} - 4q^{42} - 6q^{43} - 17q^{46} - 4q^{47} - 3q^{48} - 2q^{49} - 2q^{51} - 8q^{52} - 2q^{53} + q^{54} + 10q^{56} - 4q^{57} - 6q^{58} + 10q^{59} - 6q^{61} + 15q^{62} + 2q^{63} + 4q^{64} - 18q^{67} + 11q^{68} - 4q^{69} - 20q^{71} - 12q^{73} - 8q^{74} + 12q^{76} - 2q^{78} - 12q^{79} + 2q^{81} - 18q^{82} - 8q^{83} - 6q^{84} + 2q^{86} - 2q^{87} + 14q^{89} + 16q^{91} - 13q^{92} + 13q^{94} - 9q^{96} - 30q^{97} - 11q^{98} + O(q^{100})$$

## 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 $$-0.570119\pi$$
−0.218508 + 0.975835i $$0.570119\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.61803 −0.809017
$$5$$ 0 0
$$6$$ −0.618034 −0.252311
$$7$$ 3.23607 1.22312 0.611559 0.791199i $$-0.290543\pi$$
0.611559 + 0.791199i $$0.290543\pi$$
$$8$$ 2.23607 0.790569
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ −1.61803 −0.467086
$$13$$ 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 $$-0.730970\pi$$
−0.663594 + 0.748093i $$0.730970\pi$$
$$18$$ −0.618034 −0.145672
$$19$$ −6.47214 −1.48481 −0.742405 0.669951i $$-0.766315\pi$$
−0.742405 + 0.669951i $$0.766315\pi$$
$$20$$ 0 0
$$21$$ 3.23607 0.706168
$$22$$ 0 0
$$23$$ 4.70820 0.981728 0.490864 0.871236i $$-0.336681\pi$$
0.490864 + 0.871236i $$0.336681\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −3.23607 −0.634645
$$27$$ 1.00000 0.192450
$$28$$ −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
$$35$$ 0 0
$$36$$ −1.61803 −0.269672
$$37$$ −0.763932 −0.125590 −0.0627948 0.998026i $$-0.520001\pi$$
−0.0627948 + 0.998026i $$0.520001\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 5.23607 0.838442
$$40$$ 0 0
$$41$$ −3.52786 −0.550960 −0.275480 0.961307i $$-0.588837\pi$$
−0.275480 + 0.961307i $$0.588837\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −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 $$0.719046\pi$$
$$48$$ 1.85410 0.267617
$$49$$ 3.47214 0.496019
$$50$$ 0 0
$$51$$ −5.47214 −0.766252
$$52$$ −8.47214 −1.17487
$$53$$ −9.94427 −1.36595 −0.682975 0.730441i $$-0.739314\pi$$
−0.682975 + 0.730441i $$0.739314\pi$$
$$54$$ −0.618034 −0.0841038
$$55$$ 0 0
$$56$$ 7.23607 0.966960
$$57$$ −6.47214 −0.857255
$$58$$ −0.763932 −0.100309
$$59$$ 11.7082 1.52428 0.762139 0.647413i $$-0.224149\pi$$
0.762139 + 0.647413i $$0.224149\pi$$
$$60$$ 0 0
$$61$$ 1.47214 0.188488 0.0942438 0.995549i $$-0.469957\pi$$
0.0942438 + 0.995549i $$0.469957\pi$$
$$62$$ 4.14590 0.526530
$$63$$ 3.23607 0.407706
$$64$$ −0.236068 −0.0295085
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −11.2361 −1.37270 −0.686352 0.727269i $$-0.740789\pi$$
−0.686352 + 0.727269i $$0.740789\pi$$
$$68$$ 8.85410 1.07372
$$69$$ 4.70820 0.566801
$$70$$ 0 0
$$71$$ −14.4721 −1.71753 −0.858763 0.512373i $$-0.828767\pi$$
−0.858763 + 0.512373i $$0.828767\pi$$
$$72$$ 2.23607 0.263523
$$73$$ −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
$$78$$ −3.23607 −0.366413
$$79$$ −12.7082 −1.42978 −0.714892 0.699235i $$-0.753524\pi$$
−0.714892 + 0.699235i $$0.753524\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 2.18034 0.240778
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ −5.23607 −0.571302
$$85$$ 0 0
$$86$$ 3.23607 0.348954
$$87$$ 1.23607 0.132520
$$88$$ 0 0
$$89$$ 4.76393 0.504976 0.252488 0.967600i $$-0.418751\pi$$
0.252488 + 0.967600i $$0.418751\pi$$
$$90$$ 0 0
$$91$$ 16.9443 1.77624
$$92$$ −7.61803 −0.794235
$$93$$ −6.70820 −0.695608
$$94$$ 5.38197 0.555107
$$95$$ 0 0
$$96$$ −5.61803 −0.573388
$$97$$ −12.7639 −1.29598 −0.647990 0.761648i $$-0.724390\pi$$
−0.647990 + 0.761648i $$0.724390\pi$$
$$98$$ −2.14590 −0.216768
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 5.52786 0.550043 0.275022 0.961438i $$-0.411315\pi$$
0.275022 + 0.961438i $$0.411315\pi$$
$$102$$ 3.38197 0.334865
$$103$$ 0.944272 0.0930419 0.0465209 0.998917i $$-0.485187\pi$$
0.0465209 + 0.998917i $$0.485187\pi$$
$$104$$ 11.7082 1.14808
$$105$$ 0 0
$$106$$ 6.14590 0.596942
$$107$$ 14.2361 1.37625 0.688126 0.725591i $$-0.258433\pi$$
0.688126 + 0.725591i $$0.258433\pi$$
$$108$$ −1.61803 −0.155695
$$109$$ −14.9443 −1.43140 −0.715701 0.698407i $$-0.753893\pi$$
−0.715701 + 0.698407i $$0.753893\pi$$
$$110$$ 0 0
$$111$$ −0.763932 −0.0725092
$$112$$ 6.00000 0.566947
$$113$$ 12.4164 1.16804 0.584019 0.811740i $$-0.301479\pi$$
0.584019 + 0.811740i $$0.301479\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 5.23607 0.484075
$$118$$ −7.23607 −0.666134
$$119$$ −17.7082 −1.62331
$$120$$ 0 0
$$121$$ 0 0
$$122$$ −0.909830 −0.0823721
$$123$$ −3.52786 −0.318097
$$124$$ 10.8541 0.974727
$$125$$ 0 0
$$126$$ −2.00000 −0.178174
$$127$$ 3.70820 0.329050 0.164525 0.986373i $$-0.447391\pi$$
0.164525 + 0.986373i $$0.447391\pi$$
$$128$$ 11.3820 1.00603
$$129$$ −5.23607 −0.461010
$$130$$ 0 0
$$131$$ 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 $$0.768206\pi$$
$$138$$ −2.90983 −0.247701
$$139$$ 7.29180 0.618482 0.309241 0.950984i $$-0.399925\pi$$
0.309241 + 0.950984i $$0.399925\pi$$
$$140$$ 0 0
$$141$$ −8.70820 −0.733363
$$142$$ 8.94427 0.750587
$$143$$ 0 0
$$144$$ 1.85410 0.154508
$$145$$ 0 0
$$146$$ 6.47214 0.535638
$$147$$ 3.47214 0.286377
$$148$$ 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 $$-0.595886\pi$$
−0.296699 + 0.954971i $$0.595886\pi$$
$$152$$ −14.4721 −1.17385
$$153$$ −5.47214 −0.442396
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −8.47214 −0.678314
$$157$$ −2.29180 −0.182905 −0.0914526 0.995809i $$-0.529151\pi$$
−0.0914526 + 0.995809i $$0.529151\pi$$
$$158$$ 7.85410 0.624839
$$159$$ −9.94427 −0.788632
$$160$$ 0 0
$$161$$ 15.2361 1.20077
$$162$$ −0.618034 −0.0485573
$$163$$ −6.18034 −0.484082 −0.242041 0.970266i $$-0.577817\pi$$
−0.242041 + 0.970266i $$0.577817\pi$$
$$164$$ 5.70820 0.445736
$$165$$ 0 0
$$166$$ 2.47214 0.191875
$$167$$ 3.18034 0.246102 0.123051 0.992400i $$-0.460732\pi$$
0.123051 + 0.992400i $$0.460732\pi$$
$$168$$ 7.23607 0.558275
$$169$$ 14.4164 1.10895
$$170$$ 0 0
$$171$$ −6.47214 −0.494937
$$172$$ 8.47214 0.645994
$$173$$ 2.94427 0.223849 0.111924 0.993717i $$-0.464299\pi$$
0.111924 + 0.993717i $$0.464299\pi$$
$$174$$ −0.763932 −0.0579135
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 11.7082 0.880042
$$178$$ −2.94427 −0.220683
$$179$$ −20.1803 −1.50835 −0.754175 0.656674i $$-0.771963\pi$$
−0.754175 + 0.656674i $$0.771963\pi$$
$$180$$ 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
$$183$$ 1.47214 0.108823
$$184$$ 10.5279 0.776124
$$185$$ 0 0
$$186$$ 4.14590 0.303992
$$187$$ 0 0
$$188$$ 14.0902 1.02763
$$189$$ 3.23607 0.235389
$$190$$ 0 0
$$191$$ 27.5967 1.99683 0.998415 0.0562752i $$-0.0179224\pi$$
0.998415 + 0.0562752i $$0.0179224\pi$$
$$192$$ −0.236068 −0.0170367
$$193$$ 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 $$-0.516068\pi$$
−0.0504574 + 0.998726i $$0.516068\pi$$
$$198$$ 0 0
$$199$$ 16.2361 1.15094 0.575472 0.817821i $$-0.304818\pi$$
0.575472 + 0.817821i $$0.304818\pi$$
$$200$$ 0 0
$$201$$ −11.2361 −0.792531
$$202$$ −3.41641 −0.240378
$$203$$ 4.00000 0.280745
$$204$$ 8.85410 0.619911
$$205$$ 0 0
$$206$$ −0.583592 −0.0406608
$$207$$ 4.70820 0.327243
$$208$$ 9.70820 0.673143
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −1.76393 −0.121434 −0.0607170 0.998155i $$-0.519339\pi$$
−0.0607170 + 0.998155i $$0.519339\pi$$
$$212$$ 16.0902 1.10508
$$213$$ −14.4721 −0.991614
$$214$$ −8.79837 −0.601444
$$215$$ 0 0
$$216$$ 2.23607 0.152145
$$217$$ −21.7082 −1.47365
$$218$$ 9.23607 0.625545
$$219$$ −10.4721 −0.707641
$$220$$ 0 0
$$221$$ −28.6525 −1.92737
$$222$$ 0.472136 0.0316877
$$223$$ −9.05573 −0.606416 −0.303208 0.952924i $$-0.598058\pi$$
−0.303208 + 0.952924i $$0.598058\pi$$
$$224$$ −18.1803 −1.21473
$$225$$ 0 0
$$226$$ −7.67376 −0.510451
$$227$$ 16.7082 1.10896 0.554481 0.832196i $$-0.312917\pi$$
0.554481 + 0.832196i $$0.312917\pi$$
$$228$$ 10.4721 0.693534
$$229$$ 7.00000 0.462573 0.231287 0.972886i $$-0.425707\pi$$
0.231287 + 0.972886i $$0.425707\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.76393 0.181461
$$233$$ 10.8885 0.713332 0.356666 0.934232i $$-0.383913\pi$$
0.356666 + 0.934232i $$0.383913\pi$$
$$234$$ −3.23607 −0.211548
$$235$$ 0 0
$$236$$ −18.9443 −1.23317
$$237$$ −12.7082 −0.825487
$$238$$ 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 $$-0.303854\pi$$
0.577946 + 0.816075i $$0.303854\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ −2.38197 −0.152490
$$245$$ 0 0
$$246$$ 2.18034 0.139013
$$247$$ −33.8885 −2.15628
$$248$$ −15.0000 −0.952501
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ −23.7082 −1.49645 −0.748224 0.663446i $$-0.769093\pi$$
−0.748224 + 0.663446i $$0.769093\pi$$
$$252$$ −5.23607 −0.329841
$$253$$ 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 $$0.857172\pi$$
$$258$$ 3.23607 0.201469
$$259$$ −2.47214 −0.153611
$$260$$ 0 0
$$261$$ 1.23607 0.0765107
$$262$$ −0.291796 −0.0180272
$$263$$ 20.1246 1.24094 0.620468 0.784231i $$-0.286943\pi$$
0.620468 + 0.784231i $$0.286943\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 12.9443 0.793664
$$267$$ 4.76393 0.291548
$$268$$ 18.1803 1.11054
$$269$$ 7.05573 0.430195 0.215098 0.976593i $$-0.430993\pi$$
0.215098 + 0.976593i $$0.430993\pi$$
$$270$$ 0 0
$$271$$ 24.2361 1.47224 0.736118 0.676853i $$-0.236657\pi$$
0.736118 + 0.676853i $$0.236657\pi$$
$$272$$ −10.1459 −0.615185
$$273$$ 16.9443 1.02551
$$274$$ 10.7984 0.652354
$$275$$ 0 0
$$276$$ −7.61803 −0.458552
$$277$$ −4.94427 −0.297073 −0.148536 0.988907i $$-0.547456\pi$$
−0.148536 + 0.988907i $$0.547456\pi$$
$$278$$ −4.50658 −0.270287
$$279$$ −6.70820 −0.401610
$$280$$ 0 0
$$281$$ −15.2361 −0.908908 −0.454454 0.890770i $$-0.650166\pi$$
−0.454454 + 0.890770i $$0.650166\pi$$
$$282$$ 5.38197 0.320491
$$283$$ 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
$$288$$ −5.61803 −0.331046
$$289$$ 12.9443 0.761428
$$290$$ 0 0
$$291$$ −12.7639 −0.748235
$$292$$ 16.9443 0.991589
$$293$$ 21.4721 1.25442 0.627208 0.778852i $$-0.284198\pi$$
0.627208 + 0.778852i $$0.284198\pi$$
$$294$$ −2.14590 −0.125151
$$295$$ 0 0
$$296$$ −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
$$303$$ 5.52786 0.317567
$$304$$ −12.0000 −0.688247
$$305$$ 0 0
$$306$$ 3.38197 0.193334
$$307$$ −11.8885 −0.678515 −0.339258 0.940694i $$-0.610176\pi$$
−0.339258 + 0.940694i $$0.610176\pi$$
$$308$$ 0 0
$$309$$ 0.944272 0.0537178
$$310$$ 0 0
$$311$$ −3.23607 −0.183501 −0.0917503 0.995782i $$-0.529246\pi$$
−0.0917503 + 0.995782i $$0.529246\pi$$
$$312$$ 11.7082 0.662847
$$313$$ −22.7639 −1.28669 −0.643347 0.765575i $$-0.722455\pi$$
−0.643347 + 0.765575i $$0.722455\pi$$
$$314$$ 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 $$0.464670\pi$$
$$318$$ 6.14590 0.344645
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 14.2361 0.794580
$$322$$ −9.41641 −0.524756
$$323$$ 35.4164 1.97062
$$324$$ −1.61803 −0.0898908
$$325$$ 0 0
$$326$$ 3.81966 0.211551
$$327$$ −14.9443 −0.826420
$$328$$ −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
$$333$$ −0.763932 −0.0418632
$$334$$ −1.96556 −0.107551
$$335$$ 0 0
$$336$$ 6.00000 0.327327
$$337$$ −7.88854 −0.429716 −0.214858 0.976645i $$-0.568929\pi$$
−0.214858 + 0.976645i $$0.568929\pi$$
$$338$$ −8.90983 −0.484631
$$339$$ 12.4164 0.674367
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ −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 $$-0.624805\pi$$
−0.382116 + 0.924114i $$0.624805\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ 0 0
$$351$$ 5.23607 0.279481
$$352$$ 0 0
$$353$$ 7.00000 0.372572 0.186286 0.982496i $$-0.440355\pi$$
0.186286 + 0.982496i $$0.440355\pi$$
$$354$$ −7.23607 −0.384593
$$355$$ 0 0
$$356$$ −7.70820 −0.408534
$$357$$ −17.7082 −0.937218
$$358$$ 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
$$365$$ 0 0
$$366$$ −0.909830 −0.0475576
$$367$$ −4.76393 −0.248675 −0.124338 0.992240i $$-0.539681\pi$$
−0.124338 + 0.992240i $$0.539681\pi$$
$$368$$ 8.72949 0.455056
$$369$$ −3.52786 −0.183653
$$370$$ 0 0
$$371$$ −32.1803 −1.67072
$$372$$ 10.8541 0.562759
$$373$$ 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
$$378$$ −2.00000 −0.102869
$$379$$ 30.5967 1.57165 0.785825 0.618449i $$-0.212239\pi$$
0.785825 + 0.618449i $$0.212239\pi$$
$$380$$ 0 0
$$381$$ 3.70820 0.189977
$$382$$ −17.0557 −0.872647
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 11.3820 0.580834
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ −5.23607 −0.266164
$$388$$ 20.6525 1.04847
$$389$$ −21.5967 −1.09500 −0.547499 0.836806i $$-0.684420\pi$$
−0.547499 + 0.836806i $$0.684420\pi$$
$$390$$ 0 0
$$391$$ −25.7639 −1.30294
$$392$$ 7.76393 0.392138
$$393$$ 0.472136 0.0238161
$$394$$ 0.875388 0.0441014
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 17.4164 0.874104 0.437052 0.899436i $$-0.356022\pi$$
0.437052 + 0.899436i $$0.356022\pi$$
$$398$$ −10.0344 −0.502981
$$399$$ −20.9443 −1.04853
$$400$$ 0 0
$$401$$ 20.9443 1.04591 0.522954 0.852361i $$-0.324830\pi$$
0.522954 + 0.852361i $$0.324830\pi$$
$$402$$ 6.94427 0.346349
$$403$$ −35.1246 −1.74968
$$404$$ −8.94427 −0.444994
$$405$$ 0 0
$$406$$ −2.47214 −0.122690
$$407$$ 0 0
$$408$$ −12.2361 −0.605776
$$409$$ −36.7771 −1.81851 −0.909255 0.416240i $$-0.863348\pi$$
−0.909255 + 0.416240i $$0.863348\pi$$
$$410$$ 0 0
$$411$$ −17.4721 −0.861837
$$412$$ −1.52786 −0.0752725
$$413$$ 37.8885 1.86437
$$414$$ −2.90983 −0.143010
$$415$$ 0 0
$$416$$ −29.4164 −1.44226
$$417$$ 7.29180 0.357081
$$418$$ 0 0
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ 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
$$423$$ −8.70820 −0.423407
$$424$$ −22.2361 −1.07988
$$425$$ 0 0
$$426$$ 8.94427 0.433351
$$427$$ 4.76393 0.230543
$$428$$ −23.0344 −1.11341
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.81966 0.280323 0.140162 0.990129i $$-0.455238\pi$$
0.140162 + 0.990129i $$0.455238\pi$$
$$432$$ 1.85410 0.0892055
$$433$$ −25.2361 −1.21277 −0.606384 0.795172i $$-0.707381\pi$$
−0.606384 + 0.795172i $$0.707381\pi$$
$$434$$ 13.4164 0.644008
$$435$$ 0 0
$$436$$ 24.1803 1.15803
$$437$$ −30.4721 −1.45768
$$438$$ 6.47214 0.309251
$$439$$ 8.12461 0.387767 0.193883 0.981025i $$-0.437892\pi$$
0.193883 + 0.981025i $$0.437892\pi$$
$$440$$ 0 0
$$441$$ 3.47214 0.165340
$$442$$ 17.7082 0.842293
$$443$$ −7.41641 −0.352364 −0.176182 0.984358i $$-0.556375\pi$$
−0.176182 + 0.984358i $$0.556375\pi$$
$$444$$ 1.23607 0.0586612
$$445$$ 0 0
$$446$$ 5.59675 0.265014
$$447$$ 4.29180 0.202995
$$448$$ −0.763932 −0.0360924
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −20.0902 −0.944962
$$453$$ −7.29180 −0.342598
$$454$$ −10.3262 −0.484634
$$455$$ 0 0
$$456$$ −14.4721 −0.677720
$$457$$ −34.3607 −1.60732 −0.803662 0.595085i $$-0.797118\pi$$
−0.803662 + 0.595085i $$0.797118\pi$$
$$458$$ −4.32624 −0.202152
$$459$$ −5.47214 −0.255417
$$460$$ 0 0
$$461$$ −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 $$-0.544524\pi$$
−0.139422 + 0.990233i $$0.544524\pi$$
$$464$$ 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 $$0.472244\pi$$
$$468$$ −8.47214 −0.391625
$$469$$ −36.3607 −1.67898
$$470$$ 0 0
$$471$$ −2.29180 −0.105600
$$472$$ 26.1803 1.20505
$$473$$ 0 0
$$474$$ 7.85410 0.360751
$$475$$ 0 0
$$476$$ 28.6525 1.31328
$$477$$ −9.94427 −0.455317
$$478$$ −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
$$483$$ 15.2361 0.693265
$$484$$ 0 0
$$485$$ 0 0
$$486$$ −0.618034 −0.0280346
$$487$$ 11.7082 0.530549 0.265275 0.964173i $$-0.414537\pi$$
0.265275 + 0.964173i $$0.414537\pi$$
$$488$$ 3.29180 0.149013
$$489$$ −6.18034 −0.279485
$$490$$ 0 0
$$491$$ 38.4721 1.73622 0.868112 0.496369i $$-0.165334\pi$$
0.868112 + 0.496369i $$0.165334\pi$$
$$492$$ 5.70820 0.257346
$$493$$ −6.76393 −0.304632
$$494$$ 20.9443 0.942327
$$495$$ 0 0
$$496$$ −12.4377 −0.558469
$$497$$ −46.8328 −2.10074
$$498$$ 2.47214 0.110779
$$499$$ −0.944272 −0.0422714 −0.0211357 0.999777i $$-0.506728\pi$$
−0.0211357 + 0.999777i $$0.506728\pi$$
$$500$$ 0 0
$$501$$ 3.18034 0.142087
$$502$$ 14.6525 0.653972
$$503$$ −23.1803 −1.03356 −0.516780 0.856118i $$-0.672870\pi$$
−0.516780 + 0.856118i $$0.672870\pi$$
$$504$$ 7.23607 0.322320
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 14.4164 0.640255
$$508$$ −6.00000 −0.266207
$$509$$ 15.5967 0.691314 0.345657 0.938361i $$-0.387656\pi$$
0.345657 + 0.938361i $$0.387656\pi$$
$$510$$ 0 0
$$511$$ −33.8885 −1.49914
$$512$$ −18.7082 −0.826794
$$513$$ −6.47214 −0.285752
$$514$$ 17.8541 0.787511
$$515$$ 0 0
$$516$$ 8.47214 0.372965
$$517$$ 0 0
$$518$$ 1.52786 0.0671305
$$519$$ 2.94427 0.129239
$$520$$ 0 0
$$521$$ 17.8197 0.780693 0.390347 0.920668i $$-0.372355\pi$$
0.390347 + 0.920668i $$0.372355\pi$$
$$522$$ −0.763932 −0.0334364
$$523$$ −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
$$530$$ 0 0
$$531$$ 11.7082 0.508093
$$532$$ 33.8885 1.46925
$$533$$ −18.4721 −0.800117
$$534$$ −2.94427 −0.127411
$$535$$ 0 0
$$536$$ −25.1246 −1.08522
$$537$$ −20.1803 −0.870846
$$538$$ −4.36068 −0.188002
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 43.3050 1.86183 0.930913 0.365242i $$-0.119014\pi$$
0.930913 + 0.365242i $$0.119014\pi$$
$$542$$ −14.9787 −0.643391
$$543$$ −21.4164 −0.919066
$$544$$ 30.7426 1.31808
$$545$$ 0 0
$$546$$ −10.4721 −0.448166
$$547$$ −9.59675 −0.410327 −0.205164 0.978728i $$-0.565773\pi$$
−0.205164 + 0.978728i $$0.565773\pi$$
$$548$$ 28.2705 1.20766
$$549$$ 1.47214 0.0628292
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 10.5279 0.448096
$$553$$ −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 $$-0.557826\pi$$
−0.180669 + 0.983544i $$0.557826\pi$$
$$558$$ 4.14590 0.175510
$$559$$ −27.4164 −1.15959
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 9.41641 0.397207
$$563$$ −0.944272 −0.0397963 −0.0198982 0.999802i $$-0.506334\pi$$
−0.0198982 + 0.999802i $$0.506334\pi$$
$$564$$ 14.0902 0.593303
$$565$$ 0 0
$$566$$ −7.34752 −0.308839
$$567$$ 3.23607 0.135902
$$568$$ −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 $$-0.500830\pi$$
−0.00260742 + 0.999997i $$0.500830\pi$$
$$572$$ 0 0
$$573$$ 27.5967 1.15287
$$574$$ 7.05573 0.294500
$$575$$ 0 0
$$576$$ −0.236068 −0.00983617
$$577$$ −20.0000 −0.832611 −0.416305 0.909225i $$-0.636675\pi$$
−0.416305 + 0.909225i $$0.636675\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 16.1803 0.672432
$$580$$ 0 0
$$581$$ −12.9443 −0.537019
$$582$$ 7.88854 0.326991
$$583$$ 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 $$0.422697\pi$$
$$588$$ −5.61803 −0.231684
$$589$$ 43.4164 1.78894
$$590$$ 0 0
$$591$$ −1.41641 −0.0582632
$$592$$ −1.41641 −0.0582140
$$593$$ −34.9443 −1.43499 −0.717495 0.696564i $$-0.754711\pi$$
−0.717495 + 0.696564i $$0.754711\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.94427 −0.284448
$$597$$ 16.2361 0.664498
$$598$$ −15.2361 −0.623049
$$599$$ −31.0132 −1.26716 −0.633582 0.773676i $$-0.718416\pi$$
−0.633582 + 0.773676i $$0.718416\pi$$
$$600$$ 0 0
$$601$$ −28.4721 −1.16140 −0.580701 0.814117i $$-0.697222\pi$$
−0.580701 + 0.814117i $$0.697222\pi$$
$$602$$ 10.4721 0.426812
$$603$$ −11.2361 −0.457568
$$604$$ 11.7984 0.480069
$$605$$ 0 0
$$606$$ −3.41641 −0.138782
$$607$$ 6.29180 0.255376 0.127688 0.991814i $$-0.459244\pi$$
0.127688 + 0.991814i $$0.459244\pi$$
$$608$$ 36.3607 1.47462
$$609$$ 4.00000 0.162088
$$610$$ 0 0
$$611$$ −45.5967 −1.84465
$$612$$ 8.85410 0.357906
$$613$$ −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 $$0.817998\pi$$
$$618$$ −0.583592 −0.0234755
$$619$$ −9.52786 −0.382957 −0.191479 0.981497i $$-0.561328\pi$$
−0.191479 + 0.981497i $$0.561328\pi$$
$$620$$ 0 0
$$621$$ 4.70820 0.188934
$$622$$ 2.00000 0.0801927
$$623$$ 15.4164 0.617645
$$624$$ 9.70820 0.388639
$$625$$ 0 0
$$626$$ 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
$$633$$ −1.76393 −0.0701100
$$634$$ −2.43769 −0.0968132
$$635$$ 0 0
$$636$$ 16.0902 0.638017
$$637$$ 18.1803 0.720331
$$638$$ 0 0
$$639$$ −14.4721 −0.572509
$$640$$ 0 0
$$641$$ −9.12461 −0.360400 −0.180200 0.983630i $$-0.557675\pi$$
−0.180200 + 0.983630i $$0.557675\pi$$
$$642$$ −8.79837 −0.347244
$$643$$ 28.8328 1.13706 0.568528 0.822664i $$-0.307513\pi$$
0.568528 + 0.822664i $$0.307513\pi$$
$$644$$ −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 $$0.396602\pi$$
$$648$$ 2.23607 0.0878410
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −21.7082 −0.850812
$$652$$ 10.0000 0.391630
$$653$$ −16.8328 −0.658719 −0.329359 0.944205i $$-0.606833\pi$$
−0.329359 + 0.944205i $$0.606833\pi$$
$$654$$ 9.23607 0.361159
$$655$$ 0 0
$$656$$ −6.54102 −0.255384
$$657$$ −10.4721 −0.408557
$$658$$ 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
$$663$$ −28.6525 −1.11277
$$664$$ −8.94427 −0.347105
$$665$$ 0 0
$$666$$ 0.472136 0.0182949
$$667$$ 5.81966 0.225338
$$668$$ −5.14590 −0.199101
$$669$$ −9.05573 −0.350115
$$670$$ 0 0
$$671$$ 0 0
$$672$$ −18.1803 −0.701322
$$673$$ −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 $$-0.466809\pi$$
0.104085 + 0.994568i $$0.466809\pi$$
$$678$$ −7.67376 −0.294709
$$679$$ −41.3050 −1.58514
$$680$$ 0 0
$$681$$ 16.7082 0.640260
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 10.4721 0.400412
$$685$$ 0 0
$$686$$ 7.05573 0.269389
$$687$$ 7.00000 0.267067
$$688$$ −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
$$695$$ 0 0
$$696$$ 2.76393 0.104767
$$697$$ 19.3050 0.731227
$$698$$ −9.27051 −0.350894
$$699$$ 10.8885 0.411843
$$700$$ 0 0
$$701$$ −36.1803 −1.36651 −0.683256 0.730179i $$-0.739437\pi$$
−0.683256 + 0.730179i $$0.739437\pi$$
$$702$$ −3.23607 −0.122138
$$703$$ 4.94427 0.186477
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −4.32624 −0.162820
$$707$$ 17.8885 0.672768
$$708$$ −18.9443 −0.711969
$$709$$ −17.9443 −0.673911 −0.336956 0.941521i $$-0.609397\pi$$
−0.336956 + 0.941521i $$0.609397\pi$$
$$710$$ 0 0
$$711$$ −12.7082 −0.476595
$$712$$ 10.6525 0.399218
$$713$$ −31.5836 −1.18281
$$714$$ 10.9443 0.409579
$$715$$ 0 0
$$716$$ 32.6525 1.22028
$$717$$ 24.6525 0.920664
$$718$$ 4.58359 0.171058
$$719$$ 26.3607 0.983087 0.491544 0.870853i $$-0.336433\pi$$
0.491544 + 0.870853i $$0.336433\pi$$
$$720$$ 0 0
$$721$$ 3.05573 0.113801
$$722$$ −14.1459 −0.526456
$$723$$ 17.9443 0.667355
$$724$$ 34.6525 1.28785
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −44.8328 −1.66276 −0.831379 0.555706i $$-0.812448\pi$$
−0.831379 + 0.555706i $$0.812448\pi$$
$$728$$ 37.8885 1.40424
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 28.6525 1.05975
$$732$$ −2.38197 −0.0880400
$$733$$ 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
$$738$$ 2.18034 0.0802594
$$739$$ −1.65248 −0.0607873 −0.0303937 0.999538i $$-0.509676\pi$$
−0.0303937 + 0.999538i $$0.509676\pi$$
$$740$$ 0 0
$$741$$ −33.8885 −1.24493
$$742$$ 19.8885 0.730131
$$743$$ 2.23607 0.0820334 0.0410167 0.999158i $$-0.486940\pi$$
0.0410167 + 0.999158i $$0.486940\pi$$
$$744$$ −15.0000 −0.549927
$$745$$ 0 0
$$746$$ −18.4721 −0.676313
$$747$$ −4.00000 −0.146352
$$748$$ 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
$$753$$ −23.7082 −0.863975
$$754$$ −4.00000 −0.145671
$$755$$ 0 0
$$756$$ −5.23607 −0.190434
$$757$$ −18.9443 −0.688541 −0.344271 0.938870i $$-0.611874\pi$$
−0.344271 + 0.938870i $$0.611874\pi$$
$$758$$ −18.9098 −0.686836
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 44.6525 1.61865 0.809325 0.587360i $$-0.199833\pi$$
0.809325 + 0.587360i $$0.199833\pi$$
$$762$$ −2.29180 −0.0830230
$$763$$ −48.3607 −1.75077
$$764$$ −44.6525 −1.61547
$$765$$ 0 0
$$766$$ 9.88854 0.357288
$$767$$ 61.3050 2.21359
$$768$$ −6.56231 −0.236797
$$769$$ 24.5279 0.884497 0.442249 0.896892i $$-0.354181\pi$$
0.442249 + 0.896892i $$0.354181\pi$$
$$770$$ 0 0
$$771$$ −28.8885 −1.04040
$$772$$ −26.1803 −0.942251
$$773$$ 7.94427 0.285736 0.142868 0.989742i $$-0.454368\pi$$
0.142868 + 0.989742i $$0.454368\pi$$
$$774$$ 3.23607 0.116318
$$775$$ 0 0
$$776$$ −28.5410 −1.02456
$$777$$ −2.47214 −0.0886874
$$778$$ 13.3475 0.478532
$$779$$ 22.8328 0.818071
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 15.9230 0.569405
$$783$$ 1.23607 0.0441735
$$784$$ 6.43769 0.229918
$$785$$ 0 0
$$786$$ −0.291796 −0.0104080
$$787$$ −27.0557 −0.964433 −0.482216 0.876052i $$-0.660168\pi$$
−0.482216 + 0.876052i $$0.660168\pi$$
$$788$$ 2.29180 0.0816419
$$789$$ 20.1246 0.716455
$$790$$ 0 0
$$791$$ 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 $$0.678307\pi$$
$$798$$ 12.9443 0.458222
$$799$$ 47.6525 1.68582
$$800$$ 0 0
$$801$$ 4.76393 0.168325
$$802$$ −12.9443 −0.457078
$$803$$ 0 0
$$804$$ 18.1803 0.641171
$$805$$ 0 0
$$806$$ 21.7082 0.764639
$$807$$ 7.05573 0.248373
$$808$$ 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 $$-0.544155\pi$$
−0.138271 + 0.990394i $$0.544155\pi$$
$$812$$ −6.47214 −0.227127
$$813$$ 24.2361 0.849996
$$814$$ 0 0
$$815$$ 0 0
$$816$$ −10.1459 −0.355177
$$817$$ 33.8885 1.18561
$$818$$ 22.7295 0.794718
$$819$$ 16.9443 0.592081
$$820$$ 0 0
$$821$$ 33.7771 1.17883 0.589414 0.807831i $$-0.299359\pi$$
0.589414 + 0.807831i $$0.299359\pi$$
$$822$$ 10.7984 0.376637
$$823$$ −38.2492 −1.33328 −0.666642 0.745378i $$-0.732269\pi$$
−0.666642 + 0.745378i $$0.732269\pi$$
$$824$$ 2.11146 0.0735561
$$825$$ 0 0
$$826$$ −23.4164 −0.814761
$$827$$ 8.94427 0.311023 0.155511 0.987834i $$-0.450297\pi$$
0.155511 + 0.987834i $$0.450297\pi$$
$$828$$ −7.61803 −0.264745
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ −4.94427 −0.171515
$$832$$ −1.23607 −0.0428529
$$833$$ −19.0000 −0.658311
$$834$$ −4.50658 −0.156050
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −6.70820 −0.231869
$$838$$ 8.65248 0.298895
$$839$$ 28.3607 0.979119 0.489560 0.871970i $$-0.337158\pi$$
0.489560 + 0.871970i $$0.337158\pi$$
$$840$$ 0 0
$$841$$ −27.4721 −0.947315
$$842$$ −18.4377 −0.635405
$$843$$ −15.2361 −0.524758
$$844$$ 2.85410 0.0982422
$$845$$ 0 0
$$846$$ 5.38197 0.185036
$$847$$ 0 0
$$848$$ −18.4377 −0.633153
$$849$$ 11.8885 0.408014
$$850$$ 0 0
$$851$$ −3.59675 −0.123295
$$852$$ 23.4164 0.802233
$$853$$ −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 $$-0.494563\pi$$
0.0170797 + 0.999854i $$0.494563\pi$$
$$858$$ 0 0
$$859$$ −26.8328 −0.915524 −0.457762 0.889075i $$-0.651349\pi$$
−0.457762 + 0.889075i $$0.651349\pi$$
$$860$$ 0 0
$$861$$ −11.4164 −0.389070
$$862$$ −3.59675 −0.122506
$$863$$ 37.8885 1.28974 0.644871 0.764292i $$-0.276911\pi$$
0.644871 + 0.764292i $$0.276911\pi$$
$$864$$ −5.61803 −0.191129
$$865$$ 0 0
$$866$$ 15.5967 0.529999
$$867$$ 12.9443 0.439611
$$868$$ 35.1246 1.19221
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −58.8328 −1.99347
$$872$$ −33.4164 −1.13162
$$873$$ −12.7639 −0.431994
$$874$$ 18.8328 0.637029
$$875$$ 0 0
$$876$$ 16.9443 0.572494
$$877$$ 11.3050 0.381741 0.190871 0.981615i $$-0.438869\pi$$
0.190871 + 0.981615i $$0.438869\pi$$
$$878$$ −5.02129 −0.169460
$$879$$ 21.4721 0.724237
$$880$$ 0 0
$$881$$ 38.8328 1.30831 0.654155 0.756360i $$-0.273024\pi$$
0.654155 + 0.756360i $$0.273024\pi$$
$$882$$ −2.14590 −0.0722561
$$883$$ −10.1803 −0.342596 −0.171298 0.985219i $$-0.554796\pi$$
−0.171298 + 0.985219i $$0.554796\pi$$
$$884$$ 46.3607 1.55928
$$885$$ 0 0
$$886$$ 4.58359 0.153989
$$887$$ 16.9443 0.568933 0.284466 0.958686i $$-0.408184\pi$$
0.284466 + 0.958686i $$0.408184\pi$$
$$888$$ −1.70820 −0.0573236
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 14.6525 0.490601
$$893$$ 56.3607 1.88604
$$894$$ −2.65248 −0.0887121
$$895$$ 0 0
$$896$$ 36.8328 1.23050
$$897$$ 24.6525 0.823122
$$898$$ 6.18034 0.206241
$$899$$ −8.29180 −0.276547
$$900$$ 0 0
$$901$$ 54.4164 1.81287
$$902$$ 0 0
$$903$$ −16.9443 −0.563870
$$904$$ 27.7639 0.923415
$$905$$ 0 0
$$906$$ 4.50658 0.149721
$$907$$ 42.0000 1.39459 0.697294 0.716786i $$-0.254387\pi$$
0.697294 + 0.716786i $$0.254387\pi$$
$$908$$ −27.0344 −0.897169
$$909$$ 5.52786 0.183348
$$910$$ 0 0
$$911$$ 47.9574 1.58890 0.794450 0.607329i $$-0.207759\pi$$
0.794450 + 0.607329i $$0.207759\pi$$
$$912$$ −12.0000 −0.397360
$$913$$ 0 0
$$914$$ 21.2361 0.702427
$$915$$ 0 0
$$916$$ −11.3262 −0.374229
$$917$$ 1.52786 0.0504545
$$918$$ 3.38197 0.111622
$$919$$ −3.41641 −0.112697 −0.0563484 0.998411i $$-0.517946\pi$$
−0.0563484 + 0.998411i $$0.517946\pi$$
$$920$$ 0 0
$$921$$ −11.8885 −0.391741
$$922$$ 18.6525 0.614287
$$923$$ −75.7771 −2.49423
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 3.70820 0.121859
$$927$$ 0.944272 0.0310140
$$928$$ −6.94427 −0.227957
$$929$$ 50.9443 1.67143 0.835714 0.549165i $$-0.185054\pi$$
0.835714 + 0.549165i $$0.185054\pi$$
$$930$$ 0 0
$$931$$ −22.4721 −0.736495
$$932$$ −17.6180 −0.577098
$$933$$ −3.23607 −0.105944
$$934$$ −2.32624 −0.0761168
$$935$$ 0 0
$$936$$ 11.7082 0.382695
$$937$$ 33.1246 1.08213 0.541067 0.840980i $$-0.318021\pi$$
0.541067 + 0.840980i $$0.318021\pi$$
$$938$$ 22.4721 0.733741
$$939$$ −22.7639 −0.742873
$$940$$ 0 0
$$941$$ 45.2361 1.47465 0.737327 0.675536i $$-0.236088\pi$$
0.737327 + 0.675536i $$0.236088\pi$$
$$942$$ 1.41641 0.0461491
$$943$$ −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 $$0.404248\pi$$
$$948$$ 20.5623 0.667833
$$949$$ −54.8328 −1.77995
$$950$$ 0 0
$$951$$ 3.94427 0.127902
$$952$$ −39.5967 −1.28334
$$953$$ 31.8885 1.03297 0.516486 0.856296i $$-0.327240\pi$$
0.516486 + 0.856296i $$0.327240\pi$$
$$954$$ 6.14590 0.198981
$$955$$ 0 0
$$956$$ −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
$$963$$ 14.2361 0.458751
$$964$$ −29.0344 −0.935136
$$965$$ 0 0
$$966$$ −9.41641 −0.302968
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 0 0
$$969$$ 35.4164 1.13774
$$970$$ 0 0
$$971$$ 29.2361 0.938230 0.469115 0.883137i $$-0.344573\pi$$
0.469115 + 0.883137i $$0.344573\pi$$
$$972$$ −1.61803 −0.0518985
$$973$$ 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 $$0.417908\pi$$
$$978$$ 3.81966 0.122139
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −14.9443 −0.477134
$$982$$ −23.7771 −0.758757
$$983$$ 55.1803 1.75998 0.879990 0.474993i $$-0.157549\pi$$
0.879990 + 0.474993i $$0.157549\pi$$
$$984$$ −7.88854 −0.251478
$$985$$ 0 0
$$986$$ 4.18034 0.133129
$$987$$ −28.1803 −0.896990
$$988$$ 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
$$993$$ −17.1803 −0.545202
$$994$$ 28.9443 0.918057
$$995$$ 0 0
$$996$$ 6.47214 0.205077
$$997$$ 38.7639 1.22767 0.613833 0.789436i $$-0.289627\pi$$
0.613833 + 0.789436i $$0.289627\pi$$
$$998$$ 0.583592 0.0184733
$$999$$ −0.763932 −0.0241697
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 9075.2.a.bx.1.1 2
5.4 even 2 1815.2.a.f.1.2 2
11.10 odd 2 9075.2.a.bd.1.2 2
15.14 odd 2 5445.2.a.x.1.1 2
55.54 odd 2 1815.2.a.j.1.1 yes 2
165.164 even 2 5445.2.a.o.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.f.1.2 2 5.4 even 2
1815.2.a.j.1.1 yes 2 55.54 odd 2
5445.2.a.o.1.2 2 165.164 even 2
5445.2.a.x.1.1 2 15.14 odd 2
9075.2.a.bd.1.2 2 11.10 odd 2
9075.2.a.bx.1.1 2 1.1 even 1 trivial