# Properties

 Label 4560.2.a.bv.1.3 Level $4560$ Weight $2$ Character 4560.1 Self dual yes Analytic conductor $36.412$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 2280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.470683$$ of defining polynomial Character $$\chi$$ $$=$$ 4560.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} +3.30777 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} +3.30777 q^{7} +1.00000 q^{9} -2.24914 q^{11} -3.19051 q^{13} +1.00000 q^{15} +3.55691 q^{17} +1.00000 q^{19} +3.30777 q^{21} -1.55691 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.24914 q^{29} +7.43965 q^{31} -2.24914 q^{33} +3.30777 q^{35} -5.30777 q^{37} -3.19051 q^{39} +2.36641 q^{41} -3.80605 q^{43} +1.00000 q^{45} +9.55691 q^{47} +3.94137 q^{49} +3.55691 q^{51} -3.55691 q^{53} -2.24914 q^{55} +1.00000 q^{57} -0.498281 q^{59} -1.17590 q^{61} +3.30777 q^{63} -3.19051 q^{65} +12.9966 q^{67} -1.55691 q^{69} +10.3810 q^{71} -0.117266 q^{73} +1.00000 q^{75} -7.43965 q^{77} +1.00000 q^{81} +12.1725 q^{83} +3.55691 q^{85} +4.24914 q^{87} +10.6302 q^{89} -10.5535 q^{91} +7.43965 q^{93} +1.00000 q^{95} +7.68879 q^{97} -2.24914 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 $$3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{15} - 6 q^{17} + 3 q^{19} + 2 q^{21} + 12 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 14 q^{43} + 3 q^{45} + 12 q^{47} + 11 q^{49} - 6 q^{51} + 6 q^{53} + 2 q^{55} + 3 q^{57} + 16 q^{59} - 6 q^{61} + 2 q^{63} + 4 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} + 3 q^{75} - 4 q^{77} + 3 q^{81} + 4 q^{83} - 6 q^{85} + 4 q^{87} + 4 q^{89} + 20 q^{91} + 4 q^{93} + 3 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 2 * q^11 + 3 * q^15 - 6 * q^17 + 3 * q^19 + 2 * q^21 + 12 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 14 * q^43 + 3 * q^45 + 12 * q^47 + 11 * q^49 - 6 * q^51 + 6 * q^53 + 2 * q^55 + 3 * q^57 + 16 * q^59 - 6 * q^61 + 2 * q^63 + 4 * q^67 + 12 * q^69 + 12 * q^71 - 2 * q^73 + 3 * q^75 - 4 * q^77 + 3 * q^81 + 4 * q^83 - 6 * q^85 + 4 * q^87 + 4 * q^89 + 20 * q^91 + 4 * q^93 + 3 * q^95 - 4 * q^97 + 2 * q^99

## 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 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 3.30777 1.25022 0.625110 0.780536i $$-0.285054\pi$$
0.625110 + 0.780536i $$0.285054\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −2.24914 −0.678141 −0.339071 0.940761i $$-0.610113\pi$$
−0.339071 + 0.940761i $$0.610113\pi$$
$$12$$ 0 0
$$13$$ −3.19051 −0.884888 −0.442444 0.896796i $$-0.645888\pi$$
−0.442444 + 0.896796i $$0.645888\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 3.55691 0.862678 0.431339 0.902190i $$-0.358041\pi$$
0.431339 + 0.902190i $$0.358041\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 3.30777 0.721815
$$22$$ 0 0
$$23$$ −1.55691 −0.324639 −0.162320 0.986738i $$-0.551898\pi$$
−0.162320 + 0.986738i $$0.551898\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 4.24914 0.789046 0.394523 0.918886i $$-0.370910\pi$$
0.394523 + 0.918886i $$0.370910\pi$$
$$30$$ 0 0
$$31$$ 7.43965 1.33620 0.668100 0.744071i $$-0.267108\pi$$
0.668100 + 0.744071i $$0.267108\pi$$
$$32$$ 0 0
$$33$$ −2.24914 −0.391525
$$34$$ 0 0
$$35$$ 3.30777 0.559116
$$36$$ 0 0
$$37$$ −5.30777 −0.872593 −0.436296 0.899803i $$-0.643710\pi$$
−0.436296 + 0.899803i $$0.643710\pi$$
$$38$$ 0 0
$$39$$ −3.19051 −0.510890
$$40$$ 0 0
$$41$$ 2.36641 0.369571 0.184785 0.982779i $$-0.440841\pi$$
0.184785 + 0.982779i $$0.440841\pi$$
$$42$$ 0 0
$$43$$ −3.80605 −0.580418 −0.290209 0.956963i $$-0.593725\pi$$
−0.290209 + 0.956963i $$0.593725\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 9.55691 1.39402 0.697010 0.717062i $$-0.254513\pi$$
0.697010 + 0.717062i $$0.254513\pi$$
$$48$$ 0 0
$$49$$ 3.94137 0.563052
$$50$$ 0 0
$$51$$ 3.55691 0.498068
$$52$$ 0 0
$$53$$ −3.55691 −0.488580 −0.244290 0.969702i $$-0.578555\pi$$
−0.244290 + 0.969702i $$0.578555\pi$$
$$54$$ 0 0
$$55$$ −2.24914 −0.303274
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 0 0
$$59$$ −0.498281 −0.0648707 −0.0324353 0.999474i $$-0.510326\pi$$
−0.0324353 + 0.999474i $$0.510326\pi$$
$$60$$ 0 0
$$61$$ −1.17590 −0.150559 −0.0752793 0.997162i $$-0.523985\pi$$
−0.0752793 + 0.997162i $$0.523985\pi$$
$$62$$ 0 0
$$63$$ 3.30777 0.416740
$$64$$ 0 0
$$65$$ −3.19051 −0.395734
$$66$$ 0 0
$$67$$ 12.9966 1.58778 0.793891 0.608060i $$-0.208052\pi$$
0.793891 + 0.608060i $$0.208052\pi$$
$$68$$ 0 0
$$69$$ −1.55691 −0.187430
$$70$$ 0 0
$$71$$ 10.3810 1.23200 0.616000 0.787746i $$-0.288752\pi$$
0.616000 + 0.787746i $$0.288752\pi$$
$$72$$ 0 0
$$73$$ −0.117266 −0.0137250 −0.00686249 0.999976i $$-0.502184\pi$$
−0.00686249 + 0.999976i $$0.502184\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −7.43965 −0.847827
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.1725 1.33610 0.668051 0.744116i $$-0.267129\pi$$
0.668051 + 0.744116i $$0.267129\pi$$
$$84$$ 0 0
$$85$$ 3.55691 0.385802
$$86$$ 0 0
$$87$$ 4.24914 0.455556
$$88$$ 0 0
$$89$$ 10.6302 1.12679 0.563397 0.826186i $$-0.309494\pi$$
0.563397 + 0.826186i $$0.309494\pi$$
$$90$$ 0 0
$$91$$ −10.5535 −1.10630
$$92$$ 0 0
$$93$$ 7.43965 0.771456
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 7.68879 0.780678 0.390339 0.920671i $$-0.372358\pi$$
0.390339 + 0.920671i $$0.372358\pi$$
$$98$$ 0 0
$$99$$ −2.24914 −0.226047
$$100$$ 0 0
$$101$$ 8.38101 0.833942 0.416971 0.908920i $$-0.363092\pi$$
0.416971 + 0.908920i $$0.363092\pi$$
$$102$$ 0 0
$$103$$ 3.76547 0.371023 0.185511 0.982642i $$-0.440606\pi$$
0.185511 + 0.982642i $$0.440606\pi$$
$$104$$ 0 0
$$105$$ 3.30777 0.322806
$$106$$ 0 0
$$107$$ 6.11727 0.591378 0.295689 0.955284i $$-0.404451\pi$$
0.295689 + 0.955284i $$0.404451\pi$$
$$108$$ 0 0
$$109$$ −15.4948 −1.48414 −0.742068 0.670324i $$-0.766155\pi$$
−0.742068 + 0.670324i $$0.766155\pi$$
$$110$$ 0 0
$$111$$ −5.30777 −0.503792
$$112$$ 0 0
$$113$$ −8.94137 −0.841133 −0.420567 0.907262i $$-0.638169\pi$$
−0.420567 + 0.907262i $$0.638169\pi$$
$$114$$ 0 0
$$115$$ −1.55691 −0.145183
$$116$$ 0 0
$$117$$ −3.19051 −0.294963
$$118$$ 0 0
$$119$$ 11.7655 1.07854
$$120$$ 0 0
$$121$$ −5.94137 −0.540124
$$122$$ 0 0
$$123$$ 2.36641 0.213372
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 10.1173 0.897762 0.448881 0.893591i $$-0.351823\pi$$
0.448881 + 0.893591i $$0.351823\pi$$
$$128$$ 0 0
$$129$$ −3.80605 −0.335104
$$130$$ 0 0
$$131$$ 1.51633 0.132482 0.0662410 0.997804i $$-0.478899\pi$$
0.0662410 + 0.997804i $$0.478899\pi$$
$$132$$ 0 0
$$133$$ 3.30777 0.286820
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −11.8827 −1.01521 −0.507605 0.861590i $$-0.669469\pi$$
−0.507605 + 0.861590i $$0.669469\pi$$
$$138$$ 0 0
$$139$$ −16.4983 −1.39937 −0.699683 0.714453i $$-0.746675\pi$$
−0.699683 + 0.714453i $$0.746675\pi$$
$$140$$ 0 0
$$141$$ 9.55691 0.804837
$$142$$ 0 0
$$143$$ 7.17590 0.600079
$$144$$ 0 0
$$145$$ 4.24914 0.352872
$$146$$ 0 0
$$147$$ 3.94137 0.325078
$$148$$ 0 0
$$149$$ −7.23109 −0.592394 −0.296197 0.955127i $$-0.595719\pi$$
−0.296197 + 0.955127i $$0.595719\pi$$
$$150$$ 0 0
$$151$$ 3.67418 0.299001 0.149500 0.988762i $$-0.452234\pi$$
0.149500 + 0.988762i $$0.452234\pi$$
$$152$$ 0 0
$$153$$ 3.55691 0.287559
$$154$$ 0 0
$$155$$ 7.43965 0.597567
$$156$$ 0 0
$$157$$ 0.381015 0.0304083 0.0152041 0.999884i $$-0.495160\pi$$
0.0152041 + 0.999884i $$0.495160\pi$$
$$158$$ 0 0
$$159$$ −3.55691 −0.282082
$$160$$ 0 0
$$161$$ −5.14992 −0.405871
$$162$$ 0 0
$$163$$ −2.80949 −0.220056 −0.110028 0.993928i $$-0.535094\pi$$
−0.110028 + 0.993928i $$0.535094\pi$$
$$164$$ 0 0
$$165$$ −2.24914 −0.175095
$$166$$ 0 0
$$167$$ 7.17590 0.555288 0.277644 0.960684i $$-0.410446\pi$$
0.277644 + 0.960684i $$0.410446\pi$$
$$168$$ 0 0
$$169$$ −2.82066 −0.216974
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ 0 0
$$173$$ −4.94137 −0.375685 −0.187843 0.982199i $$-0.560149\pi$$
−0.187843 + 0.982199i $$0.560149\pi$$
$$174$$ 0 0
$$175$$ 3.30777 0.250044
$$176$$ 0 0
$$177$$ −0.498281 −0.0374531
$$178$$ 0 0
$$179$$ −2.61555 −0.195495 −0.0977476 0.995211i $$-0.531164\pi$$
−0.0977476 + 0.995211i $$0.531164\pi$$
$$180$$ 0 0
$$181$$ −17.6121 −1.30910 −0.654549 0.756020i $$-0.727141\pi$$
−0.654549 + 0.756020i $$0.727141\pi$$
$$182$$ 0 0
$$183$$ −1.17590 −0.0869250
$$184$$ 0 0
$$185$$ −5.30777 −0.390235
$$186$$ 0 0
$$187$$ −8.00000 −0.585018
$$188$$ 0 0
$$189$$ 3.30777 0.240605
$$190$$ 0 0
$$191$$ −11.7440 −0.849765 −0.424882 0.905249i $$-0.639685\pi$$
−0.424882 + 0.905249i $$0.639685\pi$$
$$192$$ 0 0
$$193$$ −0.574960 −0.0413865 −0.0206933 0.999786i $$-0.506587\pi$$
−0.0206933 + 0.999786i $$0.506587\pi$$
$$194$$ 0 0
$$195$$ −3.19051 −0.228477
$$196$$ 0 0
$$197$$ −11.5569 −0.823396 −0.411698 0.911320i $$-0.635064\pi$$
−0.411698 + 0.911320i $$0.635064\pi$$
$$198$$ 0 0
$$199$$ −3.50172 −0.248230 −0.124115 0.992268i $$-0.539609\pi$$
−0.124115 + 0.992268i $$0.539609\pi$$
$$200$$ 0 0
$$201$$ 12.9966 0.916707
$$202$$ 0 0
$$203$$ 14.0552 0.986481
$$204$$ 0 0
$$205$$ 2.36641 0.165277
$$206$$ 0 0
$$207$$ −1.55691 −0.108213
$$208$$ 0 0
$$209$$ −2.24914 −0.155576
$$210$$ 0 0
$$211$$ −9.23109 −0.635495 −0.317747 0.948175i $$-0.602926\pi$$
−0.317747 + 0.948175i $$0.602926\pi$$
$$212$$ 0 0
$$213$$ 10.3810 0.711295
$$214$$ 0 0
$$215$$ −3.80605 −0.259571
$$216$$ 0 0
$$217$$ 24.6087 1.67055
$$218$$ 0 0
$$219$$ −0.117266 −0.00792412
$$220$$ 0 0
$$221$$ −11.3484 −0.763373
$$222$$ 0 0
$$223$$ −17.9931 −1.20491 −0.602454 0.798153i $$-0.705810\pi$$
−0.602454 + 0.798153i $$0.705810\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 5.94480 0.394571 0.197285 0.980346i $$-0.436787\pi$$
0.197285 + 0.980346i $$0.436787\pi$$
$$228$$ 0 0
$$229$$ 14.1725 0.936543 0.468271 0.883585i $$-0.344877\pi$$
0.468271 + 0.883585i $$0.344877\pi$$
$$230$$ 0 0
$$231$$ −7.43965 −0.489493
$$232$$ 0 0
$$233$$ −13.3484 −0.874480 −0.437240 0.899345i $$-0.644044\pi$$
−0.437240 + 0.899345i $$0.644044\pi$$
$$234$$ 0 0
$$235$$ 9.55691 0.623424
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 24.2423 1.56810 0.784051 0.620697i $$-0.213150\pi$$
0.784051 + 0.620697i $$0.213150\pi$$
$$240$$ 0 0
$$241$$ −11.2311 −0.723458 −0.361729 0.932283i $$-0.617814\pi$$
−0.361729 + 0.932283i $$0.617814\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 3.94137 0.251805
$$246$$ 0 0
$$247$$ −3.19051 −0.203007
$$248$$ 0 0
$$249$$ 12.1725 0.771398
$$250$$ 0 0
$$251$$ 7.48024 0.472148 0.236074 0.971735i $$-0.424139\pi$$
0.236074 + 0.971735i $$0.424139\pi$$
$$252$$ 0 0
$$253$$ 3.50172 0.220151
$$254$$ 0 0
$$255$$ 3.55691 0.222743
$$256$$ 0 0
$$257$$ 0.0551953 0.00344299 0.00172149 0.999999i $$-0.499452\pi$$
0.00172149 + 0.999999i $$0.499452\pi$$
$$258$$ 0 0
$$259$$ −17.5569 −1.09093
$$260$$ 0 0
$$261$$ 4.24914 0.263015
$$262$$ 0 0
$$263$$ −3.93793 −0.242823 −0.121412 0.992602i $$-0.538742\pi$$
−0.121412 + 0.992602i $$0.538742\pi$$
$$264$$ 0 0
$$265$$ −3.55691 −0.218500
$$266$$ 0 0
$$267$$ 10.6302 0.650555
$$268$$ 0 0
$$269$$ 17.8681 1.08944 0.544719 0.838618i $$-0.316636\pi$$
0.544719 + 0.838618i $$0.316636\pi$$
$$270$$ 0 0
$$271$$ 27.8466 1.69156 0.845782 0.533529i $$-0.179135\pi$$
0.845782 + 0.533529i $$0.179135\pi$$
$$272$$ 0 0
$$273$$ −10.5535 −0.638725
$$274$$ 0 0
$$275$$ −2.24914 −0.135628
$$276$$ 0 0
$$277$$ −11.9931 −0.720597 −0.360299 0.932837i $$-0.617325\pi$$
−0.360299 + 0.932837i $$0.617325\pi$$
$$278$$ 0 0
$$279$$ 7.43965 0.445400
$$280$$ 0 0
$$281$$ −10.8647 −0.648133 −0.324066 0.946034i $$-0.605050\pi$$
−0.324066 + 0.946034i $$0.605050\pi$$
$$282$$ 0 0
$$283$$ −16.6922 −0.992250 −0.496125 0.868251i $$-0.665244\pi$$
−0.496125 + 0.868251i $$0.665244\pi$$
$$284$$ 0 0
$$285$$ 1.00000 0.0592349
$$286$$ 0 0
$$287$$ 7.82754 0.462045
$$288$$ 0 0
$$289$$ −4.34836 −0.255786
$$290$$ 0 0
$$291$$ 7.68879 0.450725
$$292$$ 0 0
$$293$$ 31.1690 1.82091 0.910457 0.413604i $$-0.135730\pi$$
0.910457 + 0.413604i $$0.135730\pi$$
$$294$$ 0 0
$$295$$ −0.498281 −0.0290110
$$296$$ 0 0
$$297$$ −2.24914 −0.130508
$$298$$ 0 0
$$299$$ 4.96735 0.287269
$$300$$ 0 0
$$301$$ −12.5896 −0.725651
$$302$$ 0 0
$$303$$ 8.38101 0.481477
$$304$$ 0 0
$$305$$ −1.17590 −0.0673318
$$306$$ 0 0
$$307$$ 0.498281 0.0284384 0.0142192 0.999899i $$-0.495474\pi$$
0.0142192 + 0.999899i $$0.495474\pi$$
$$308$$ 0 0
$$309$$ 3.76547 0.214210
$$310$$ 0 0
$$311$$ −1.01805 −0.0577281 −0.0288640 0.999583i $$-0.509189\pi$$
−0.0288640 + 0.999583i $$0.509189\pi$$
$$312$$ 0 0
$$313$$ 12.3810 0.699816 0.349908 0.936784i $$-0.386213\pi$$
0.349908 + 0.936784i $$0.386213\pi$$
$$314$$ 0 0
$$315$$ 3.30777 0.186372
$$316$$ 0 0
$$317$$ 4.78801 0.268921 0.134461 0.990919i $$-0.457070\pi$$
0.134461 + 0.990919i $$0.457070\pi$$
$$318$$ 0 0
$$319$$ −9.55691 −0.535084
$$320$$ 0 0
$$321$$ 6.11727 0.341433
$$322$$ 0 0
$$323$$ 3.55691 0.197912
$$324$$ 0 0
$$325$$ −3.19051 −0.176978
$$326$$ 0 0
$$327$$ −15.4948 −0.856867
$$328$$ 0 0
$$329$$ 31.6121 1.74283
$$330$$ 0 0
$$331$$ 12.4362 0.683556 0.341778 0.939781i $$-0.388971\pi$$
0.341778 + 0.939781i $$0.388971\pi$$
$$332$$ 0 0
$$333$$ −5.30777 −0.290864
$$334$$ 0 0
$$335$$ 12.9966 0.710078
$$336$$ 0 0
$$337$$ −15.4543 −0.841847 −0.420923 0.907096i $$-0.638294\pi$$
−0.420923 + 0.907096i $$0.638294\pi$$
$$338$$ 0 0
$$339$$ −8.94137 −0.485628
$$340$$ 0 0
$$341$$ −16.7328 −0.906133
$$342$$ 0 0
$$343$$ −10.1173 −0.546281
$$344$$ 0 0
$$345$$ −1.55691 −0.0838214
$$346$$ 0 0
$$347$$ 12.8241 0.688434 0.344217 0.938890i $$-0.388144\pi$$
0.344217 + 0.938890i $$0.388144\pi$$
$$348$$ 0 0
$$349$$ 28.8793 1.54587 0.772937 0.634483i $$-0.218787\pi$$
0.772937 + 0.634483i $$0.218787\pi$$
$$350$$ 0 0
$$351$$ −3.19051 −0.170297
$$352$$ 0 0
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 0 0
$$355$$ 10.3810 0.550967
$$356$$ 0 0
$$357$$ 11.7655 0.622695
$$358$$ 0 0
$$359$$ −19.7440 −1.04205 −0.521024 0.853542i $$-0.674450\pi$$
−0.521024 + 0.853542i $$0.674450\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −5.94137 −0.311841
$$364$$ 0 0
$$365$$ −0.117266 −0.00613800
$$366$$ 0 0
$$367$$ −13.6888 −0.714549 −0.357274 0.933999i $$-0.616294\pi$$
−0.357274 + 0.933999i $$0.616294\pi$$
$$368$$ 0 0
$$369$$ 2.36641 0.123190
$$370$$ 0 0
$$371$$ −11.7655 −0.610833
$$372$$ 0 0
$$373$$ 30.1510 1.56116 0.780579 0.625057i $$-0.214924\pi$$
0.780579 + 0.625057i $$0.214924\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ −13.5569 −0.698217
$$378$$ 0 0
$$379$$ 25.6673 1.31844 0.659220 0.751950i $$-0.270886\pi$$
0.659220 + 0.751950i $$0.270886\pi$$
$$380$$ 0 0
$$381$$ 10.1173 0.518323
$$382$$ 0 0
$$383$$ 23.8759 1.22000 0.610000 0.792402i $$-0.291170\pi$$
0.610000 + 0.792402i $$0.291170\pi$$
$$384$$ 0 0
$$385$$ −7.43965 −0.379160
$$386$$ 0 0
$$387$$ −3.80605 −0.193473
$$388$$ 0 0
$$389$$ 29.3484 1.48802 0.744010 0.668168i $$-0.232921\pi$$
0.744010 + 0.668168i $$0.232921\pi$$
$$390$$ 0 0
$$391$$ −5.53781 −0.280059
$$392$$ 0 0
$$393$$ 1.51633 0.0764886
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −12.1173 −0.608148 −0.304074 0.952648i $$-0.598347\pi$$
−0.304074 + 0.952648i $$0.598347\pi$$
$$398$$ 0 0
$$399$$ 3.30777 0.165596
$$400$$ 0 0
$$401$$ −27.4734 −1.37195 −0.685977 0.727623i $$-0.740625\pi$$
−0.685977 + 0.727623i $$0.740625\pi$$
$$402$$ 0 0
$$403$$ −23.7363 −1.18239
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 11.9379 0.591741
$$408$$ 0 0
$$409$$ 32.9605 1.62979 0.814895 0.579608i $$-0.196794\pi$$
0.814895 + 0.579608i $$0.196794\pi$$
$$410$$ 0 0
$$411$$ −11.8827 −0.586132
$$412$$ 0 0
$$413$$ −1.64820 −0.0811027
$$414$$ 0 0
$$415$$ 12.1725 0.597523
$$416$$ 0 0
$$417$$ −16.4983 −0.807924
$$418$$ 0 0
$$419$$ 14.7474 0.720459 0.360229 0.932864i $$-0.382698\pi$$
0.360229 + 0.932864i $$0.382698\pi$$
$$420$$ 0 0
$$421$$ 24.9897 1.21792 0.608961 0.793200i $$-0.291586\pi$$
0.608961 + 0.793200i $$0.291586\pi$$
$$422$$ 0 0
$$423$$ 9.55691 0.464673
$$424$$ 0 0
$$425$$ 3.55691 0.172536
$$426$$ 0 0
$$427$$ −3.88961 −0.188231
$$428$$ 0 0
$$429$$ 7.17590 0.346456
$$430$$ 0 0
$$431$$ −10.8501 −0.522630 −0.261315 0.965254i $$-0.584156\pi$$
−0.261315 + 0.965254i $$0.584156\pi$$
$$432$$ 0 0
$$433$$ −11.0371 −0.530412 −0.265206 0.964192i $$-0.585440\pi$$
−0.265206 + 0.964192i $$0.585440\pi$$
$$434$$ 0 0
$$435$$ 4.24914 0.203731
$$436$$ 0 0
$$437$$ −1.55691 −0.0744773
$$438$$ 0 0
$$439$$ −30.2277 −1.44269 −0.721344 0.692577i $$-0.756475\pi$$
−0.721344 + 0.692577i $$0.756475\pi$$
$$440$$ 0 0
$$441$$ 3.94137 0.187684
$$442$$ 0 0
$$443$$ −21.9018 −1.04059 −0.520294 0.853987i $$-0.674178\pi$$
−0.520294 + 0.853987i $$0.674178\pi$$
$$444$$ 0 0
$$445$$ 10.6302 0.503918
$$446$$ 0 0
$$447$$ −7.23109 −0.342019
$$448$$ 0 0
$$449$$ 3.01805 0.142430 0.0712152 0.997461i $$-0.477312\pi$$
0.0712152 + 0.997461i $$0.477312\pi$$
$$450$$ 0 0
$$451$$ −5.32238 −0.250621
$$452$$ 0 0
$$453$$ 3.67418 0.172628
$$454$$ 0 0
$$455$$ −10.5535 −0.494755
$$456$$ 0 0
$$457$$ 13.5017 0.631584 0.315792 0.948828i $$-0.397730\pi$$
0.315792 + 0.948828i $$0.397730\pi$$
$$458$$ 0 0
$$459$$ 3.55691 0.166023
$$460$$ 0 0
$$461$$ −3.64820 −0.169914 −0.0849568 0.996385i $$-0.527075\pi$$
−0.0849568 + 0.996385i $$0.527075\pi$$
$$462$$ 0 0
$$463$$ 8.53887 0.396835 0.198417 0.980118i $$-0.436420\pi$$
0.198417 + 0.980118i $$0.436420\pi$$
$$464$$ 0 0
$$465$$ 7.43965 0.345005
$$466$$ 0 0
$$467$$ −22.5535 −1.04365 −0.521825 0.853052i $$-0.674749\pi$$
−0.521825 + 0.853052i $$0.674749\pi$$
$$468$$ 0 0
$$469$$ 42.9897 1.98508
$$470$$ 0 0
$$471$$ 0.381015 0.0175562
$$472$$ 0 0
$$473$$ 8.56035 0.393605
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −3.55691 −0.162860
$$478$$ 0 0
$$479$$ 33.6267 1.53644 0.768222 0.640184i $$-0.221142\pi$$
0.768222 + 0.640184i $$0.221142\pi$$
$$480$$ 0 0
$$481$$ 16.9345 0.772146
$$482$$ 0 0
$$483$$ −5.14992 −0.234329
$$484$$ 0 0
$$485$$ 7.68879 0.349130
$$486$$ 0 0
$$487$$ −11.8466 −0.536823 −0.268411 0.963304i $$-0.586499\pi$$
−0.268411 + 0.963304i $$0.586499\pi$$
$$488$$ 0 0
$$489$$ −2.80949 −0.127050
$$490$$ 0 0
$$491$$ −35.3269 −1.59428 −0.797140 0.603795i $$-0.793655\pi$$
−0.797140 + 0.603795i $$0.793655\pi$$
$$492$$ 0 0
$$493$$ 15.1138 0.680693
$$494$$ 0 0
$$495$$ −2.24914 −0.101091
$$496$$ 0 0
$$497$$ 34.3380 1.54027
$$498$$ 0 0
$$499$$ −30.7259 −1.37548 −0.687741 0.725956i $$-0.741398\pi$$
−0.687741 + 0.725956i $$0.741398\pi$$
$$500$$ 0 0
$$501$$ 7.17590 0.320596
$$502$$ 0 0
$$503$$ −42.1656 −1.88007 −0.940035 0.341077i $$-0.889208\pi$$
−0.940035 + 0.341077i $$0.889208\pi$$
$$504$$ 0 0
$$505$$ 8.38101 0.372950
$$506$$ 0 0
$$507$$ −2.82066 −0.125270
$$508$$ 0 0
$$509$$ 41.7440 1.85027 0.925135 0.379639i $$-0.123952\pi$$
0.925135 + 0.379639i $$0.123952\pi$$
$$510$$ 0 0
$$511$$ −0.387890 −0.0171593
$$512$$ 0 0
$$513$$ 1.00000 0.0441511
$$514$$ 0 0
$$515$$ 3.76547 0.165926
$$516$$ 0 0
$$517$$ −21.4948 −0.945342
$$518$$ 0 0
$$519$$ −4.94137 −0.216902
$$520$$ 0 0
$$521$$ −18.7405 −0.821038 −0.410519 0.911852i $$-0.634652\pi$$
−0.410519 + 0.911852i $$0.634652\pi$$
$$522$$ 0 0
$$523$$ 14.7259 0.643920 0.321960 0.946753i $$-0.395658\pi$$
0.321960 + 0.946753i $$0.395658\pi$$
$$524$$ 0 0
$$525$$ 3.30777 0.144363
$$526$$ 0 0
$$527$$ 26.4622 1.15271
$$528$$ 0 0
$$529$$ −20.5760 −0.894609
$$530$$ 0 0
$$531$$ −0.498281 −0.0216236
$$532$$ 0 0
$$533$$ −7.55004 −0.327028
$$534$$ 0 0
$$535$$ 6.11727 0.264472
$$536$$ 0 0
$$537$$ −2.61555 −0.112869
$$538$$ 0 0
$$539$$ −8.86469 −0.381829
$$540$$ 0 0
$$541$$ −40.2277 −1.72952 −0.864761 0.502184i $$-0.832530\pi$$
−0.864761 + 0.502184i $$0.832530\pi$$
$$542$$ 0 0
$$543$$ −17.6121 −0.755808
$$544$$ 0 0
$$545$$ −15.4948 −0.663726
$$546$$ 0 0
$$547$$ −37.7294 −1.61319 −0.806596 0.591103i $$-0.798693\pi$$
−0.806596 + 0.591103i $$0.798693\pi$$
$$548$$ 0 0
$$549$$ −1.17590 −0.0501862
$$550$$ 0 0
$$551$$ 4.24914 0.181019
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −5.30777 −0.225302
$$556$$ 0 0
$$557$$ −19.9931 −0.847136 −0.423568 0.905864i $$-0.639223\pi$$
−0.423568 + 0.905864i $$0.639223\pi$$
$$558$$ 0 0
$$559$$ 12.1432 0.513605
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ −44.0483 −1.85642 −0.928208 0.372063i $$-0.878651\pi$$
−0.928208 + 0.372063i $$0.878651\pi$$
$$564$$ 0 0
$$565$$ −8.94137 −0.376166
$$566$$ 0 0
$$567$$ 3.30777 0.138913
$$568$$ 0 0
$$569$$ −16.7474 −0.702088 −0.351044 0.936359i $$-0.614173\pi$$
−0.351044 + 0.936359i $$0.614173\pi$$
$$570$$ 0 0
$$571$$ −36.1396 −1.51240 −0.756198 0.654343i $$-0.772945\pi$$
−0.756198 + 0.654343i $$0.772945\pi$$
$$572$$ 0 0
$$573$$ −11.7440 −0.490612
$$574$$ 0 0
$$575$$ −1.55691 −0.0649278
$$576$$ 0 0
$$577$$ −16.3810 −0.681951 −0.340975 0.940072i $$-0.610757\pi$$
−0.340975 + 0.940072i $$0.610757\pi$$
$$578$$ 0 0
$$579$$ −0.574960 −0.0238945
$$580$$ 0 0
$$581$$ 40.2637 1.67042
$$582$$ 0 0
$$583$$ 8.00000 0.331326
$$584$$ 0 0
$$585$$ −3.19051 −0.131911
$$586$$ 0 0
$$587$$ −12.5604 −0.518421 −0.259211 0.965821i $$-0.583462\pi$$
−0.259211 + 0.965821i $$0.583462\pi$$
$$588$$ 0 0
$$589$$ 7.43965 0.306545
$$590$$ 0 0
$$591$$ −11.5569 −0.475388
$$592$$ 0 0
$$593$$ −38.9966 −1.60140 −0.800698 0.599068i $$-0.795538\pi$$
−0.800698 + 0.599068i $$0.795538\pi$$
$$594$$ 0 0
$$595$$ 11.7655 0.482337
$$596$$ 0 0
$$597$$ −3.50172 −0.143316
$$598$$ 0 0
$$599$$ −17.3415 −0.708554 −0.354277 0.935141i $$-0.615273\pi$$
−0.354277 + 0.935141i $$0.615273\pi$$
$$600$$ 0 0
$$601$$ −15.9119 −0.649062 −0.324531 0.945875i $$-0.605206\pi$$
−0.324531 + 0.945875i $$0.605206\pi$$
$$602$$ 0 0
$$603$$ 12.9966 0.529261
$$604$$ 0 0
$$605$$ −5.94137 −0.241551
$$606$$ 0 0
$$607$$ 24.2637 0.984835 0.492418 0.870359i $$-0.336113\pi$$
0.492418 + 0.870359i $$0.336113\pi$$
$$608$$ 0 0
$$609$$ 14.0552 0.569545
$$610$$ 0 0
$$611$$ −30.4914 −1.23355
$$612$$ 0 0
$$613$$ −44.7259 −1.80646 −0.903232 0.429153i $$-0.858812\pi$$
−0.903232 + 0.429153i $$0.858812\pi$$
$$614$$ 0 0
$$615$$ 2.36641 0.0954227
$$616$$ 0 0
$$617$$ −10.6707 −0.429588 −0.214794 0.976659i $$-0.568908\pi$$
−0.214794 + 0.976659i $$0.568908\pi$$
$$618$$ 0 0
$$619$$ −47.3776 −1.90427 −0.952133 0.305685i $$-0.901115\pi$$
−0.952133 + 0.305685i $$0.901115\pi$$
$$620$$ 0 0
$$621$$ −1.55691 −0.0624768
$$622$$ 0 0
$$623$$ 35.1621 1.40874
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −2.24914 −0.0898220
$$628$$ 0 0
$$629$$ −18.8793 −0.752767
$$630$$ 0 0
$$631$$ −10.4622 −0.416493 −0.208247 0.978076i $$-0.566776\pi$$
−0.208247 + 0.978076i $$0.566776\pi$$
$$632$$ 0 0
$$633$$ −9.23109 −0.366903
$$634$$ 0 0
$$635$$ 10.1173 0.401491
$$636$$ 0 0
$$637$$ −12.5750 −0.498238
$$638$$ 0 0
$$639$$ 10.3810 0.410667
$$640$$ 0 0
$$641$$ 20.6233 0.814571 0.407285 0.913301i $$-0.366475\pi$$
0.407285 + 0.913301i $$0.366475\pi$$
$$642$$ 0 0
$$643$$ −34.4216 −1.35746 −0.678728 0.734390i $$-0.737468\pi$$
−0.678728 + 0.734390i $$0.737468\pi$$
$$644$$ 0 0
$$645$$ −3.80605 −0.149863
$$646$$ 0 0
$$647$$ 25.8207 1.01511 0.507557 0.861618i $$-0.330548\pi$$
0.507557 + 0.861618i $$0.330548\pi$$
$$648$$ 0 0
$$649$$ 1.12070 0.0439915
$$650$$ 0 0
$$651$$ 24.6087 0.964490
$$652$$ 0 0
$$653$$ 45.3155 1.77333 0.886666 0.462410i $$-0.153015\pi$$
0.886666 + 0.462410i $$0.153015\pi$$
$$654$$ 0 0
$$655$$ 1.51633 0.0592478
$$656$$ 0 0
$$657$$ −0.117266 −0.00457499
$$658$$ 0 0
$$659$$ 45.3415 1.76625 0.883127 0.469134i $$-0.155434\pi$$
0.883127 + 0.469134i $$0.155434\pi$$
$$660$$ 0 0
$$661$$ −17.6121 −0.685032 −0.342516 0.939512i $$-0.611279\pi$$
−0.342516 + 0.939512i $$0.611279\pi$$
$$662$$ 0 0
$$663$$ −11.3484 −0.440734
$$664$$ 0 0
$$665$$ 3.30777 0.128270
$$666$$ 0 0
$$667$$ −6.61555 −0.256155
$$668$$ 0 0
$$669$$ −17.9931 −0.695654
$$670$$ 0 0
$$671$$ 2.64476 0.102100
$$672$$ 0 0
$$673$$ −13.8061 −0.532184 −0.266092 0.963948i $$-0.585733\pi$$
−0.266092 + 0.963948i $$0.585733\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −23.7914 −0.914380 −0.457190 0.889369i $$-0.651144\pi$$
−0.457190 + 0.889369i $$0.651144\pi$$
$$678$$ 0 0
$$679$$ 25.4328 0.976020
$$680$$ 0 0
$$681$$ 5.94480 0.227805
$$682$$ 0 0
$$683$$ −7.11383 −0.272203 −0.136102 0.990695i $$-0.543457\pi$$
−0.136102 + 0.990695i $$0.543457\pi$$
$$684$$ 0 0
$$685$$ −11.8827 −0.454016
$$686$$ 0 0
$$687$$ 14.1725 0.540713
$$688$$ 0 0
$$689$$ 11.3484 0.432338
$$690$$ 0 0
$$691$$ 18.1465 0.690325 0.345162 0.938543i $$-0.387824\pi$$
0.345162 + 0.938543i $$0.387824\pi$$
$$692$$ 0 0
$$693$$ −7.43965 −0.282609
$$694$$ 0 0
$$695$$ −16.4983 −0.625815
$$696$$ 0 0
$$697$$ 8.41711 0.318821
$$698$$ 0 0
$$699$$ −13.3484 −0.504881
$$700$$ 0 0
$$701$$ 24.9897 0.943847 0.471924 0.881639i $$-0.343560\pi$$
0.471924 + 0.881639i $$0.343560\pi$$
$$702$$ 0 0
$$703$$ −5.30777 −0.200186
$$704$$ 0 0
$$705$$ 9.55691 0.359934
$$706$$ 0 0
$$707$$ 27.7225 1.04261
$$708$$ 0 0
$$709$$ −31.9311 −1.19920 −0.599598 0.800301i $$-0.704673\pi$$
−0.599598 + 0.800301i $$0.704673\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −11.5829 −0.433783
$$714$$ 0 0
$$715$$ 7.17590 0.268363
$$716$$ 0 0
$$717$$ 24.2423 0.905344
$$718$$ 0 0
$$719$$ 38.5941 1.43932 0.719658 0.694329i $$-0.244299\pi$$
0.719658 + 0.694329i $$0.244299\pi$$
$$720$$ 0 0
$$721$$ 12.4553 0.463860
$$722$$ 0 0
$$723$$ −11.2311 −0.417689
$$724$$ 0 0
$$725$$ 4.24914 0.157809
$$726$$ 0 0
$$727$$ 38.0337 1.41059 0.705296 0.708913i $$-0.250814\pi$$
0.705296 + 0.708913i $$0.250814\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −13.5378 −0.500714
$$732$$ 0 0
$$733$$ 14.5275 0.536585 0.268293 0.963337i $$-0.413541\pi$$
0.268293 + 0.963337i $$0.413541\pi$$
$$734$$ 0 0
$$735$$ 3.94137 0.145380
$$736$$ 0 0
$$737$$ −29.2311 −1.07674
$$738$$ 0 0
$$739$$ 5.46563 0.201056 0.100528 0.994934i $$-0.467947\pi$$
0.100528 + 0.994934i $$0.467947\pi$$
$$740$$ 0 0
$$741$$ −3.19051 −0.117206
$$742$$ 0 0
$$743$$ 44.5795 1.63546 0.817731 0.575601i $$-0.195232\pi$$
0.817731 + 0.575601i $$0.195232\pi$$
$$744$$ 0 0
$$745$$ −7.23109 −0.264927
$$746$$ 0 0
$$747$$ 12.1725 0.445367
$$748$$ 0 0
$$749$$ 20.2345 0.739354
$$750$$ 0 0
$$751$$ −7.43965 −0.271477 −0.135738 0.990745i $$-0.543341\pi$$
−0.135738 + 0.990745i $$0.543341\pi$$
$$752$$ 0 0
$$753$$ 7.48024 0.272595
$$754$$ 0 0
$$755$$ 3.67418 0.133717
$$756$$ 0 0
$$757$$ −23.6190 −0.858447 −0.429223 0.903198i $$-0.641213\pi$$
−0.429223 + 0.903198i $$0.641213\pi$$
$$758$$ 0 0
$$759$$ 3.50172 0.127104
$$760$$ 0 0
$$761$$ 18.9966 0.688625 0.344312 0.938855i $$-0.388112\pi$$
0.344312 + 0.938855i $$0.388112\pi$$
$$762$$ 0 0
$$763$$ −51.2534 −1.85550
$$764$$ 0 0
$$765$$ 3.55691 0.128601
$$766$$ 0 0
$$767$$ 1.58977 0.0574032
$$768$$ 0 0
$$769$$ −11.2311 −0.405004 −0.202502 0.979282i $$-0.564907\pi$$
−0.202502 + 0.979282i $$0.564907\pi$$
$$770$$ 0 0
$$771$$ 0.0551953 0.00198781
$$772$$ 0 0
$$773$$ 26.0191 0.935842 0.467921 0.883770i $$-0.345003\pi$$
0.467921 + 0.883770i $$0.345003\pi$$
$$774$$ 0 0
$$775$$ 7.43965 0.267240
$$776$$ 0 0
$$777$$ −17.5569 −0.629851
$$778$$ 0 0
$$779$$ 2.36641 0.0847853
$$780$$ 0 0
$$781$$ −23.3484 −0.835470
$$782$$ 0 0
$$783$$ 4.24914 0.151852
$$784$$ 0 0
$$785$$ 0.381015 0.0135990
$$786$$ 0 0
$$787$$ −14.1984 −0.506120 −0.253060 0.967451i $$-0.581437\pi$$
−0.253060 + 0.967451i $$0.581437\pi$$
$$788$$ 0 0
$$789$$ −3.93793 −0.140194
$$790$$ 0 0
$$791$$ −29.5760 −1.05160
$$792$$ 0 0
$$793$$ 3.75172 0.133227
$$794$$ 0 0
$$795$$ −3.55691 −0.126151
$$796$$ 0 0
$$797$$ −34.1725 −1.21045 −0.605225 0.796054i $$-0.706917\pi$$
−0.605225 + 0.796054i $$0.706917\pi$$
$$798$$ 0 0
$$799$$ 33.9931 1.20259
$$800$$ 0 0
$$801$$ 10.6302 0.375598
$$802$$ 0 0
$$803$$ 0.263748 0.00930748
$$804$$ 0 0
$$805$$ −5.14992 −0.181511
$$806$$ 0 0
$$807$$ 17.8681 0.628988
$$808$$ 0 0
$$809$$ 20.1173 0.707285 0.353643 0.935381i $$-0.384943\pi$$
0.353643 + 0.935381i $$0.384943\pi$$
$$810$$ 0 0
$$811$$ 0.762030 0.0267585 0.0133792 0.999910i $$-0.495741\pi$$
0.0133792 + 0.999910i $$0.495741\pi$$
$$812$$ 0 0
$$813$$ 27.8466 0.976624
$$814$$ 0 0
$$815$$ −2.80949 −0.0984122
$$816$$ 0 0
$$817$$ −3.80605 −0.133157
$$818$$ 0 0
$$819$$ −10.5535 −0.368768
$$820$$ 0 0
$$821$$ −4.50516 −0.157231 −0.0786155 0.996905i $$-0.525050\pi$$
−0.0786155 + 0.996905i $$0.525050\pi$$
$$822$$ 0 0
$$823$$ 11.1836 0.389837 0.194918 0.980819i $$-0.437556\pi$$
0.194918 + 0.980819i $$0.437556\pi$$
$$824$$ 0 0
$$825$$ −2.24914 −0.0783050
$$826$$ 0 0
$$827$$ 16.2829 0.566210 0.283105 0.959089i $$-0.408635\pi$$
0.283105 + 0.959089i $$0.408635\pi$$
$$828$$ 0 0
$$829$$ 15.3845 0.534324 0.267162 0.963652i $$-0.413914\pi$$
0.267162 + 0.963652i $$0.413914\pi$$
$$830$$ 0 0
$$831$$ −11.9931 −0.416037
$$832$$ 0 0
$$833$$ 14.0191 0.485733
$$834$$ 0 0
$$835$$ 7.17590 0.248332
$$836$$ 0 0
$$837$$ 7.43965 0.257152
$$838$$ 0 0
$$839$$ 16.0812 0.555184 0.277592 0.960699i $$-0.410464\pi$$
0.277592 + 0.960699i $$0.410464\pi$$
$$840$$ 0 0
$$841$$ −10.9448 −0.377407
$$842$$ 0 0
$$843$$ −10.8647 −0.374200
$$844$$ 0 0
$$845$$ −2.82066 −0.0970337
$$846$$ 0 0
$$847$$ −19.6527 −0.675275
$$848$$ 0 0
$$849$$ −16.6922 −0.572876
$$850$$ 0 0
$$851$$ 8.26375 0.283278
$$852$$ 0 0
$$853$$ −19.9119 −0.681772 −0.340886 0.940105i $$-0.610727\pi$$
−0.340886 + 0.940105i $$0.610727\pi$$
$$854$$ 0 0
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ −28.1364 −0.961120 −0.480560 0.876962i $$-0.659567\pi$$
−0.480560 + 0.876962i $$0.659567\pi$$
$$858$$ 0 0
$$859$$ −33.5760 −1.14560 −0.572799 0.819696i $$-0.694143\pi$$
−0.572799 + 0.819696i $$0.694143\pi$$
$$860$$ 0 0
$$861$$ 7.82754 0.266762
$$862$$ 0 0
$$863$$ −32.8724 −1.11899 −0.559495 0.828834i $$-0.689005\pi$$
−0.559495 + 0.828834i $$0.689005\pi$$
$$864$$ 0 0
$$865$$ −4.94137 −0.168012
$$866$$ 0 0
$$867$$ −4.34836 −0.147678
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −41.4656 −1.40501
$$872$$ 0 0
$$873$$ 7.68879 0.260226
$$874$$ 0 0
$$875$$ 3.30777 0.111823
$$876$$ 0 0
$$877$$ −24.1579 −0.815753 −0.407876 0.913037i $$-0.633731\pi$$
−0.407876 + 0.913037i $$0.633731\pi$$
$$878$$ 0 0
$$879$$ 31.1690 1.05131
$$880$$ 0 0
$$881$$ 32.4914 1.09466 0.547332 0.836916i $$-0.315644\pi$$
0.547332 + 0.836916i $$0.315644\pi$$
$$882$$ 0 0
$$883$$ −26.3404 −0.886426 −0.443213 0.896416i $$-0.646161\pi$$
−0.443213 + 0.896416i $$0.646161\pi$$
$$884$$ 0 0
$$885$$ −0.498281 −0.0167495
$$886$$ 0 0
$$887$$ −0.996562 −0.0334613 −0.0167306 0.999860i $$-0.505326\pi$$
−0.0167306 + 0.999860i $$0.505326\pi$$
$$888$$ 0 0
$$889$$ 33.4656 1.12240
$$890$$ 0 0
$$891$$ −2.24914 −0.0753490
$$892$$ 0 0
$$893$$ 9.55691 0.319810
$$894$$ 0 0
$$895$$ −2.61555 −0.0874281
$$896$$ 0 0
$$897$$ 4.96735 0.165855
$$898$$ 0 0
$$899$$ 31.6121 1.05432
$$900$$ 0 0
$$901$$ −12.6516 −0.421487
$$902$$ 0 0
$$903$$ −12.5896 −0.418955
$$904$$ 0 0
$$905$$ −17.6121 −0.585446
$$906$$ 0 0
$$907$$ −43.1430 −1.43254 −0.716271 0.697823i $$-0.754152\pi$$
−0.716271 + 0.697823i $$0.754152\pi$$
$$908$$ 0 0
$$909$$ 8.38101 0.277981
$$910$$ 0 0
$$911$$ −19.6413 −0.650746 −0.325373 0.945586i $$-0.605490\pi$$
−0.325373 + 0.945586i $$0.605490\pi$$
$$912$$ 0 0
$$913$$ −27.3776 −0.906066
$$914$$ 0 0
$$915$$ −1.17590 −0.0388740
$$916$$ 0 0
$$917$$ 5.01567 0.165632
$$918$$ 0 0
$$919$$ 37.8827 1.24964 0.624818 0.780770i $$-0.285173\pi$$
0.624818 + 0.780770i $$0.285173\pi$$
$$920$$ 0 0
$$921$$ 0.498281 0.0164189
$$922$$ 0 0
$$923$$ −33.1207 −1.09018
$$924$$ 0 0
$$925$$ −5.30777 −0.174519
$$926$$ 0 0
$$927$$ 3.76547 0.123674
$$928$$ 0 0
$$929$$ −5.39133 −0.176884 −0.0884419 0.996081i $$-0.528189\pi$$
−0.0884419 + 0.996081i $$0.528189\pi$$
$$930$$ 0 0
$$931$$ 3.94137 0.129173
$$932$$ 0 0
$$933$$ −1.01805 −0.0333293
$$934$$ 0 0
$$935$$ −8.00000 −0.261628
$$936$$ 0 0
$$937$$ 26.7328 0.873323 0.436661 0.899626i $$-0.356161\pi$$
0.436661 + 0.899626i $$0.356161\pi$$
$$938$$ 0 0
$$939$$ 12.3810 0.404039
$$940$$ 0 0
$$941$$ −4.35953 −0.142117 −0.0710583 0.997472i $$-0.522638\pi$$
−0.0710583 + 0.997472i $$0.522638\pi$$
$$942$$ 0 0
$$943$$ −3.68429 −0.119977
$$944$$ 0 0
$$945$$ 3.30777 0.107602
$$946$$ 0 0
$$947$$ −31.9379 −1.03784 −0.518922 0.854822i $$-0.673666\pi$$
−0.518922 + 0.854822i $$0.673666\pi$$
$$948$$ 0 0
$$949$$ 0.374139 0.0121451
$$950$$ 0 0
$$951$$ 4.78801 0.155262
$$952$$ 0 0
$$953$$ 1.82754 0.0591998 0.0295999 0.999562i $$-0.490577\pi$$
0.0295999 + 0.999562i $$0.490577\pi$$
$$954$$ 0 0
$$955$$ −11.7440 −0.380026
$$956$$ 0 0
$$957$$ −9.55691 −0.308931
$$958$$ 0 0
$$959$$ −39.3054 −1.26924
$$960$$ 0 0
$$961$$ 24.3484 0.785431
$$962$$ 0 0
$$963$$ 6.11727 0.197126
$$964$$ 0 0
$$965$$ −0.574960 −0.0185086
$$966$$ 0 0
$$967$$ 0.926759 0.0298026 0.0149013 0.999889i $$-0.495257\pi$$
0.0149013 + 0.999889i $$0.495257\pi$$
$$968$$ 0 0
$$969$$ 3.55691 0.114265
$$970$$ 0 0
$$971$$ 58.1035 1.86463 0.932315 0.361647i $$-0.117785\pi$$
0.932315 + 0.361647i $$0.117785\pi$$
$$972$$ 0 0
$$973$$ −54.5726 −1.74952
$$974$$ 0 0
$$975$$ −3.19051 −0.102178
$$976$$ 0 0
$$977$$ −8.02598 −0.256774 −0.128387 0.991724i $$-0.540980\pi$$
−0.128387 + 0.991724i $$0.540980\pi$$
$$978$$ 0 0
$$979$$ −23.9087 −0.764126
$$980$$ 0 0
$$981$$ −15.4948 −0.494712
$$982$$ 0 0
$$983$$ 44.8103 1.42923 0.714614 0.699519i $$-0.246602\pi$$
0.714614 + 0.699519i $$0.246602\pi$$
$$984$$ 0 0
$$985$$ −11.5569 −0.368234
$$986$$ 0 0
$$987$$ 31.6121 1.00622
$$988$$ 0 0
$$989$$ 5.92570 0.188426
$$990$$ 0 0
$$991$$ 36.7620 1.16778 0.583892 0.811831i $$-0.301529\pi$$
0.583892 + 0.811831i $$0.301529\pi$$
$$992$$ 0 0
$$993$$ 12.4362 0.394651
$$994$$ 0 0
$$995$$ −3.50172 −0.111012
$$996$$ 0 0
$$997$$ 26.3741 0.835277 0.417639 0.908613i $$-0.362858\pi$$
0.417639 + 0.908613i $$0.362858\pi$$
$$998$$ 0 0
$$999$$ −5.30777 −0.167931
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 4560.2.a.bv.1.3 3
4.3 odd 2 2280.2.a.s.1.1 3
12.11 even 2 6840.2.a.bf.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.s.1.1 3 4.3 odd 2
4560.2.a.bv.1.3 3 1.1 even 1 trivial
6840.2.a.bf.1.1 3 12.11 even 2