# Properties

 Label 1380.2.a.j.1.3 Level $1380$ Weight $2$ Character 1380.1 Self dual yes Analytic conductor $11.019$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.0193554789$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-3.20905$$ of defining polynomial Character $$\chi$$ $$=$$ 1380.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +4.20905 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +4.20905 q^{7} +1.00000 q^{9} +2.75353 q^{11} -0.753525 q^{13} -1.00000 q^{15} -4.96257 q^{17} +4.75353 q^{19} +4.20905 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.96257 q^{29} -0.209050 q^{31} +2.75353 q^{33} -4.20905 q^{35} -5.71610 q^{37} -0.753525 q^{39} -9.38067 q^{41} +12.4181 q^{43} -1.00000 q^{45} +7.17162 q^{47} +10.7161 q^{49} -4.96257 q^{51} -9.38067 q^{53} -2.75353 q^{55} +4.75353 q^{57} -4.96257 q^{59} -5.17162 q^{61} +4.20905 q^{63} +0.753525 q^{65} -0.209050 q^{67} +1.00000 q^{69} +9.38067 q^{71} +10.2606 q^{73} +1.00000 q^{75} +11.5897 q^{77} -2.41810 q^{79} +1.00000 q^{81} +5.45552 q^{83} +4.96257 q^{85} +4.96257 q^{87} +4.91105 q^{89} -3.17162 q^{91} -0.209050 q^{93} -4.75353 q^{95} +10.3432 q^{97} +2.75353 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 $$3 q + 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} - 3 q^{15} + 10 q^{19} + 2 q^{21} + 3 q^{23} + 3 q^{25} + 3 q^{27} + 10 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} + 8 q^{41} + 16 q^{43} - 3 q^{45} - 4 q^{47} + 13 q^{49} + 8 q^{53} - 4 q^{55} + 10 q^{57} + 10 q^{61} + 2 q^{63} - 2 q^{65} + 10 q^{67} + 3 q^{69} - 8 q^{71} + 18 q^{73} + 3 q^{75} - 12 q^{77} + 14 q^{79} + 3 q^{81} + 10 q^{83} + 2 q^{89} + 16 q^{91} + 10 q^{93} - 10 q^{95} - 20 q^{97} + 4 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 - 3 * q^15 + 10 * q^19 + 2 * q^21 + 3 * q^23 + 3 * q^25 + 3 * q^27 + 10 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 + 8 * q^41 + 16 * q^43 - 3 * q^45 - 4 * q^47 + 13 * q^49 + 8 * q^53 - 4 * q^55 + 10 * q^57 + 10 * q^61 + 2 * q^63 - 2 * q^65 + 10 * q^67 + 3 * q^69 - 8 * q^71 + 18 * q^73 + 3 * q^75 - 12 * q^77 + 14 * q^79 + 3 * q^81 + 10 * q^83 + 2 * q^89 + 16 * q^91 + 10 * q^93 - 10 * q^95 - 20 * q^97 + 4 * 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
$$7$$ 4.20905 1.59087 0.795436 0.606038i $$-0.207242\pi$$
0.795436 + 0.606038i $$0.207242\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.75353 0.830219 0.415110 0.909771i $$-0.363743\pi$$
0.415110 + 0.909771i $$0.363743\pi$$
$$12$$ 0 0
$$13$$ −0.753525 −0.208990 −0.104495 0.994525i $$-0.533323\pi$$
−0.104495 + 0.994525i $$0.533323\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −4.96257 −1.20360 −0.601801 0.798646i $$-0.705550\pi$$
−0.601801 + 0.798646i $$0.705550\pi$$
$$18$$ 0 0
$$19$$ 4.75353 1.09053 0.545267 0.838263i $$-0.316428\pi$$
0.545267 + 0.838263i $$0.316428\pi$$
$$20$$ 0 0
$$21$$ 4.20905 0.918490
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 4.96257 0.921527 0.460764 0.887523i $$-0.347576\pi$$
0.460764 + 0.887523i $$0.347576\pi$$
$$30$$ 0 0
$$31$$ −0.209050 −0.0375464 −0.0187732 0.999824i $$-0.505976\pi$$
−0.0187732 + 0.999824i $$0.505976\pi$$
$$32$$ 0 0
$$33$$ 2.75353 0.479327
$$34$$ 0 0
$$35$$ −4.20905 −0.711459
$$36$$ 0 0
$$37$$ −5.71610 −0.939721 −0.469861 0.882741i $$-0.655696\pi$$
−0.469861 + 0.882741i $$0.655696\pi$$
$$38$$ 0 0
$$39$$ −0.753525 −0.120661
$$40$$ 0 0
$$41$$ −9.38067 −1.46502 −0.732508 0.680759i $$-0.761650\pi$$
−0.732508 + 0.680759i $$0.761650\pi$$
$$42$$ 0 0
$$43$$ 12.4181 1.89374 0.946871 0.321613i $$-0.104225\pi$$
0.946871 + 0.321613i $$0.104225\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 7.17162 1.04609 0.523044 0.852305i $$-0.324796\pi$$
0.523044 + 0.852305i $$0.324796\pi$$
$$48$$ 0 0
$$49$$ 10.7161 1.53087
$$50$$ 0 0
$$51$$ −4.96257 −0.694899
$$52$$ 0 0
$$53$$ −9.38067 −1.28853 −0.644267 0.764800i $$-0.722838\pi$$
−0.644267 + 0.764800i $$0.722838\pi$$
$$54$$ 0 0
$$55$$ −2.75353 −0.371285
$$56$$ 0 0
$$57$$ 4.75353 0.629620
$$58$$ 0 0
$$59$$ −4.96257 −0.646072 −0.323036 0.946387i $$-0.604704\pi$$
−0.323036 + 0.946387i $$0.604704\pi$$
$$60$$ 0 0
$$61$$ −5.17162 −0.662159 −0.331079 0.943603i $$-0.607413\pi$$
−0.331079 + 0.943603i $$0.607413\pi$$
$$62$$ 0 0
$$63$$ 4.20905 0.530290
$$64$$ 0 0
$$65$$ 0.753525 0.0934633
$$66$$ 0 0
$$67$$ −0.209050 −0.0255395 −0.0127697 0.999918i $$-0.504065\pi$$
−0.0127697 + 0.999918i $$0.504065\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 9.38067 1.11328 0.556641 0.830753i $$-0.312090\pi$$
0.556641 + 0.830753i $$0.312090\pi$$
$$72$$ 0 0
$$73$$ 10.2606 1.20091 0.600455 0.799659i $$-0.294986\pi$$
0.600455 + 0.799659i $$0.294986\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 11.5897 1.32077
$$78$$ 0 0
$$79$$ −2.41810 −0.272057 −0.136029 0.990705i $$-0.543434\pi$$
−0.136029 + 0.990705i $$0.543434\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 5.45552 0.598822 0.299411 0.954124i $$-0.403210\pi$$
0.299411 + 0.954124i $$0.403210\pi$$
$$84$$ 0 0
$$85$$ 4.96257 0.538267
$$86$$ 0 0
$$87$$ 4.96257 0.532044
$$88$$ 0 0
$$89$$ 4.91105 0.520570 0.260285 0.965532i $$-0.416183\pi$$
0.260285 + 0.965532i $$0.416183\pi$$
$$90$$ 0 0
$$91$$ −3.17162 −0.332477
$$92$$ 0 0
$$93$$ −0.209050 −0.0216775
$$94$$ 0 0
$$95$$ −4.75353 −0.487701
$$96$$ 0 0
$$97$$ 10.3432 1.05020 0.525099 0.851041i $$-0.324028\pi$$
0.525099 + 0.851041i $$0.324028\pi$$
$$98$$ 0 0
$$99$$ 2.75353 0.276740
$$100$$ 0 0
$$101$$ 14.8877 1.48138 0.740692 0.671845i $$-0.234498\pi$$
0.740692 + 0.671845i $$0.234498\pi$$
$$102$$ 0 0
$$103$$ 17.9251 1.76622 0.883109 0.469168i $$-0.155446\pi$$
0.883109 + 0.469168i $$0.155446\pi$$
$$104$$ 0 0
$$105$$ −4.20905 −0.410761
$$106$$ 0 0
$$107$$ −16.4696 −1.59218 −0.796089 0.605179i $$-0.793102\pi$$
−0.796089 + 0.605179i $$0.793102\pi$$
$$108$$ 0 0
$$109$$ −6.26058 −0.599654 −0.299827 0.953994i $$-0.596929\pi$$
−0.299827 + 0.953994i $$0.596929\pi$$
$$110$$ 0 0
$$111$$ −5.71610 −0.542548
$$112$$ 0 0
$$113$$ −9.38067 −0.882460 −0.441230 0.897394i $$-0.645458\pi$$
−0.441230 + 0.897394i $$0.645458\pi$$
$$114$$ 0 0
$$115$$ −1.00000 −0.0932505
$$116$$ 0 0
$$117$$ −0.753525 −0.0696634
$$118$$ 0 0
$$119$$ −20.8877 −1.91477
$$120$$ 0 0
$$121$$ −3.41810 −0.310736
$$122$$ 0 0
$$123$$ −9.38067 −0.845827
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −22.0827 −1.95952 −0.979760 0.200175i $$-0.935849\pi$$
−0.979760 + 0.200175i $$0.935849\pi$$
$$128$$ 0 0
$$129$$ 12.4181 1.09335
$$130$$ 0 0
$$131$$ −1.58190 −0.138211 −0.0691056 0.997609i $$-0.522015\pi$$
−0.0691056 + 0.997609i $$0.522015\pi$$
$$132$$ 0 0
$$133$$ 20.0078 1.73490
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −4.91105 −0.419579 −0.209790 0.977747i $$-0.567278\pi$$
−0.209790 + 0.977747i $$0.567278\pi$$
$$138$$ 0 0
$$139$$ 14.1342 1.19885 0.599424 0.800432i $$-0.295397\pi$$
0.599424 + 0.800432i $$0.295397\pi$$
$$140$$ 0 0
$$141$$ 7.17162 0.603960
$$142$$ 0 0
$$143$$ −2.07485 −0.173508
$$144$$ 0 0
$$145$$ −4.96257 −0.412119
$$146$$ 0 0
$$147$$ 10.7161 0.883849
$$148$$ 0 0
$$149$$ 3.24647 0.265962 0.132981 0.991119i $$-0.457545\pi$$
0.132981 + 0.991119i $$0.457545\pi$$
$$150$$ 0 0
$$151$$ 0.418100 0.0340245 0.0170122 0.999855i $$-0.494585\pi$$
0.0170122 + 0.999855i $$0.494585\pi$$
$$152$$ 0 0
$$153$$ −4.96257 −0.401200
$$154$$ 0 0
$$155$$ 0.209050 0.0167913
$$156$$ 0 0
$$157$$ −21.0452 −1.67959 −0.839797 0.542901i $$-0.817326\pi$$
−0.839797 + 0.542901i $$0.817326\pi$$
$$158$$ 0 0
$$159$$ −9.38067 −0.743936
$$160$$ 0 0
$$161$$ 4.20905 0.331720
$$162$$ 0 0
$$163$$ −7.43220 −0.582135 −0.291067 0.956703i $$-0.594010\pi$$
−0.291067 + 0.956703i $$0.594010\pi$$
$$164$$ 0 0
$$165$$ −2.75353 −0.214362
$$166$$ 0 0
$$167$$ −19.1716 −1.48354 −0.741772 0.670652i $$-0.766015\pi$$
−0.741772 + 0.670652i $$0.766015\pi$$
$$168$$ 0 0
$$169$$ −12.4322 −0.956323
$$170$$ 0 0
$$171$$ 4.75353 0.363511
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 4.20905 0.318174
$$176$$ 0 0
$$177$$ −4.96257 −0.373010
$$178$$ 0 0
$$179$$ 13.8503 1.03522 0.517610 0.855617i $$-0.326822\pi$$
0.517610 + 0.855617i $$0.326822\pi$$
$$180$$ 0 0
$$181$$ 12.9110 0.959671 0.479835 0.877359i $$-0.340696\pi$$
0.479835 + 0.877359i $$0.340696\pi$$
$$182$$ 0 0
$$183$$ −5.17162 −0.382297
$$184$$ 0 0
$$185$$ 5.71610 0.420256
$$186$$ 0 0
$$187$$ −13.6646 −0.999253
$$188$$ 0 0
$$189$$ 4.20905 0.306163
$$190$$ 0 0
$$191$$ −8.15752 −0.590258 −0.295129 0.955457i $$-0.595363\pi$$
−0.295129 + 0.955457i $$0.595363\pi$$
$$192$$ 0 0
$$193$$ −16.7613 −1.20651 −0.603254 0.797549i $$-0.706130\pi$$
−0.603254 + 0.797549i $$0.706130\pi$$
$$194$$ 0 0
$$195$$ 0.753525 0.0539611
$$196$$ 0 0
$$197$$ −10.9110 −0.777380 −0.388690 0.921369i $$-0.627072\pi$$
−0.388690 + 0.921369i $$0.627072\pi$$
$$198$$ 0 0
$$199$$ −4.49295 −0.318497 −0.159248 0.987239i $$-0.550907\pi$$
−0.159248 + 0.987239i $$0.550907\pi$$
$$200$$ 0 0
$$201$$ −0.209050 −0.0147452
$$202$$ 0 0
$$203$$ 20.8877 1.46603
$$204$$ 0 0
$$205$$ 9.38067 0.655175
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ 13.0890 0.905382
$$210$$ 0 0
$$211$$ 9.71610 0.668884 0.334442 0.942416i $$-0.391452\pi$$
0.334442 + 0.942416i $$0.391452\pi$$
$$212$$ 0 0
$$213$$ 9.38067 0.642753
$$214$$ 0 0
$$215$$ −12.4181 −0.846907
$$216$$ 0 0
$$217$$ −0.879901 −0.0597316
$$218$$ 0 0
$$219$$ 10.2606 0.693345
$$220$$ 0 0
$$221$$ 3.73942 0.251541
$$222$$ 0 0
$$223$$ −19.4322 −1.30128 −0.650638 0.759388i $$-0.725499\pi$$
−0.650638 + 0.759388i $$0.725499\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −0.985900 −0.0654365 −0.0327182 0.999465i $$-0.510416\pi$$
−0.0327182 + 0.999465i $$0.510416\pi$$
$$228$$ 0 0
$$229$$ 3.08895 0.204124 0.102062 0.994778i $$-0.467456\pi$$
0.102062 + 0.994778i $$0.467456\pi$$
$$230$$ 0 0
$$231$$ 11.5897 0.762548
$$232$$ 0 0
$$233$$ 8.83620 0.578879 0.289439 0.957196i $$-0.406531\pi$$
0.289439 + 0.957196i $$0.406531\pi$$
$$234$$ 0 0
$$235$$ −7.17162 −0.467825
$$236$$ 0 0
$$237$$ −2.41810 −0.157072
$$238$$ 0 0
$$239$$ −21.3807 −1.38300 −0.691500 0.722376i $$-0.743050\pi$$
−0.691500 + 0.722376i $$0.743050\pi$$
$$240$$ 0 0
$$241$$ 22.2606 1.43393 0.716965 0.697109i $$-0.245531\pi$$
0.716965 + 0.697109i $$0.245531\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −10.7161 −0.684627
$$246$$ 0 0
$$247$$ −3.58190 −0.227911
$$248$$ 0 0
$$249$$ 5.45552 0.345730
$$250$$ 0 0
$$251$$ −6.59600 −0.416336 −0.208168 0.978093i $$-0.566750\pi$$
−0.208168 + 0.978093i $$0.566750\pi$$
$$252$$ 0 0
$$253$$ 2.75353 0.173113
$$254$$ 0 0
$$255$$ 4.96257 0.310768
$$256$$ 0 0
$$257$$ −18.1857 −1.13439 −0.567197 0.823582i $$-0.691972\pi$$
−0.567197 + 0.823582i $$0.691972\pi$$
$$258$$ 0 0
$$259$$ −24.0593 −1.49498
$$260$$ 0 0
$$261$$ 4.96257 0.307176
$$262$$ 0 0
$$263$$ −15.3807 −0.948413 −0.474207 0.880414i $$-0.657265\pi$$
−0.474207 + 0.880414i $$0.657265\pi$$
$$264$$ 0 0
$$265$$ 9.38067 0.576250
$$266$$ 0 0
$$267$$ 4.91105 0.300551
$$268$$ 0 0
$$269$$ 0.544475 0.0331972 0.0165986 0.999862i $$-0.494716\pi$$
0.0165986 + 0.999862i $$0.494716\pi$$
$$270$$ 0 0
$$271$$ −12.2090 −0.741647 −0.370823 0.928703i $$-0.620925\pi$$
−0.370823 + 0.928703i $$0.620925\pi$$
$$272$$ 0 0
$$273$$ −3.17162 −0.191955
$$274$$ 0 0
$$275$$ 2.75353 0.166044
$$276$$ 0 0
$$277$$ −21.0141 −1.26261 −0.631307 0.775533i $$-0.717481\pi$$
−0.631307 + 0.775533i $$0.717481\pi$$
$$278$$ 0 0
$$279$$ −0.209050 −0.0125155
$$280$$ 0 0
$$281$$ −24.1857 −1.44280 −0.721400 0.692519i $$-0.756501\pi$$
−0.721400 + 0.692519i $$0.756501\pi$$
$$282$$ 0 0
$$283$$ 15.1201 0.898797 0.449398 0.893331i $$-0.351638\pi$$
0.449398 + 0.893331i $$0.351638\pi$$
$$284$$ 0 0
$$285$$ −4.75353 −0.281575
$$286$$ 0 0
$$287$$ −39.4837 −2.33065
$$288$$ 0 0
$$289$$ 7.62715 0.448656
$$290$$ 0 0
$$291$$ 10.3432 0.606332
$$292$$ 0 0
$$293$$ 10.4696 0.611642 0.305821 0.952089i $$-0.401069\pi$$
0.305821 + 0.952089i $$0.401069\pi$$
$$294$$ 0 0
$$295$$ 4.96257 0.288932
$$296$$ 0 0
$$297$$ 2.75353 0.159776
$$298$$ 0 0
$$299$$ −0.753525 −0.0435775
$$300$$ 0 0
$$301$$ 52.2684 3.01270
$$302$$ 0 0
$$303$$ 14.8877 0.855277
$$304$$ 0 0
$$305$$ 5.17162 0.296126
$$306$$ 0 0
$$307$$ −12.1575 −0.693867 −0.346933 0.937890i $$-0.612777\pi$$
−0.346933 + 0.937890i $$0.612777\pi$$
$$308$$ 0 0
$$309$$ 17.9251 1.01973
$$310$$ 0 0
$$311$$ −34.4181 −1.95167 −0.975836 0.218506i $$-0.929882\pi$$
−0.975836 + 0.218506i $$0.929882\pi$$
$$312$$ 0 0
$$313$$ 29.5664 1.67119 0.835596 0.549345i $$-0.185123\pi$$
0.835596 + 0.549345i $$0.185123\pi$$
$$314$$ 0 0
$$315$$ −4.20905 −0.237153
$$316$$ 0 0
$$317$$ 1.66457 0.0934918 0.0467459 0.998907i $$-0.485115\pi$$
0.0467459 + 0.998907i $$0.485115\pi$$
$$318$$ 0 0
$$319$$ 13.6646 0.765069
$$320$$ 0 0
$$321$$ −16.4696 −0.919245
$$322$$ 0 0
$$323$$ −23.5897 −1.31257
$$324$$ 0 0
$$325$$ −0.753525 −0.0417981
$$326$$ 0 0
$$327$$ −6.26058 −0.346211
$$328$$ 0 0
$$329$$ 30.1857 1.66419
$$330$$ 0 0
$$331$$ 3.12010 0.171496 0.0857481 0.996317i $$-0.472672\pi$$
0.0857481 + 0.996317i $$0.472672\pi$$
$$332$$ 0 0
$$333$$ −5.71610 −0.313240
$$334$$ 0 0
$$335$$ 0.209050 0.0114216
$$336$$ 0 0
$$337$$ −11.5819 −0.630906 −0.315453 0.948941i $$-0.602157\pi$$
−0.315453 + 0.948941i $$0.602157\pi$$
$$338$$ 0 0
$$339$$ −9.38067 −0.509488
$$340$$ 0 0
$$341$$ −0.575624 −0.0311718
$$342$$ 0 0
$$343$$ 15.6412 0.844548
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 0 0
$$347$$ 26.3432 1.41418 0.707090 0.707124i $$-0.250008\pi$$
0.707090 + 0.707124i $$0.250008\pi$$
$$348$$ 0 0
$$349$$ 9.22315 0.493704 0.246852 0.969053i $$-0.420604\pi$$
0.246852 + 0.969053i $$0.420604\pi$$
$$350$$ 0 0
$$351$$ −0.753525 −0.0402202
$$352$$ 0 0
$$353$$ −18.1857 −0.967928 −0.483964 0.875088i $$-0.660804\pi$$
−0.483964 + 0.875088i $$0.660804\pi$$
$$354$$ 0 0
$$355$$ −9.38067 −0.497875
$$356$$ 0 0
$$357$$ −20.8877 −1.10550
$$358$$ 0 0
$$359$$ 7.17162 0.378504 0.189252 0.981929i $$-0.439394\pi$$
0.189252 + 0.981929i $$0.439394\pi$$
$$360$$ 0 0
$$361$$ 3.59600 0.189263
$$362$$ 0 0
$$363$$ −3.41810 −0.179404
$$364$$ 0 0
$$365$$ −10.2606 −0.537063
$$366$$ 0 0
$$367$$ 17.5664 0.916959 0.458479 0.888705i $$-0.348394\pi$$
0.458479 + 0.888705i $$0.348394\pi$$
$$368$$ 0 0
$$369$$ −9.38067 −0.488338
$$370$$ 0 0
$$371$$ −39.4837 −2.04989
$$372$$ 0 0
$$373$$ −33.6724 −1.74349 −0.871745 0.489959i $$-0.837012\pi$$
−0.871745 + 0.489959i $$0.837012\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ −3.73942 −0.192590
$$378$$ 0 0
$$379$$ 18.4181 0.946074 0.473037 0.881043i $$-0.343158\pi$$
0.473037 + 0.881043i $$0.343158\pi$$
$$380$$ 0 0
$$381$$ −22.0827 −1.13133
$$382$$ 0 0
$$383$$ −20.7847 −1.06205 −0.531024 0.847357i $$-0.678192\pi$$
−0.531024 + 0.847357i $$0.678192\pi$$
$$384$$ 0 0
$$385$$ −11.5897 −0.590667
$$386$$ 0 0
$$387$$ 12.4181 0.631247
$$388$$ 0 0
$$389$$ 22.5212 1.14187 0.570934 0.820996i $$-0.306581\pi$$
0.570934 + 0.820996i $$0.306581\pi$$
$$390$$ 0 0
$$391$$ −4.96257 −0.250968
$$392$$ 0 0
$$393$$ −1.58190 −0.0797963
$$394$$ 0 0
$$395$$ 2.41810 0.121668
$$396$$ 0 0
$$397$$ −15.4040 −0.773105 −0.386552 0.922267i $$-0.626334\pi$$
−0.386552 + 0.922267i $$0.626334\pi$$
$$398$$ 0 0
$$399$$ 20.0078 1.00164
$$400$$ 0 0
$$401$$ −10.5212 −0.525401 −0.262701 0.964877i $$-0.584613\pi$$
−0.262701 + 0.964877i $$0.584613\pi$$
$$402$$ 0 0
$$403$$ 0.157524 0.00784684
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −15.7394 −0.780174
$$408$$ 0 0
$$409$$ −26.0593 −1.28855 −0.644276 0.764793i $$-0.722841\pi$$
−0.644276 + 0.764793i $$0.722841\pi$$
$$410$$ 0 0
$$411$$ −4.91105 −0.242244
$$412$$ 0 0
$$413$$ −20.8877 −1.02782
$$414$$ 0 0
$$415$$ −5.45552 −0.267801
$$416$$ 0 0
$$417$$ 14.1342 0.692155
$$418$$ 0 0
$$419$$ −3.73942 −0.182683 −0.0913414 0.995820i $$-0.529115\pi$$
−0.0913414 + 0.995820i $$0.529115\pi$$
$$420$$ 0 0
$$421$$ −3.09677 −0.150928 −0.0754638 0.997149i $$-0.524044\pi$$
−0.0754638 + 0.997149i $$0.524044\pi$$
$$422$$ 0 0
$$423$$ 7.17162 0.348696
$$424$$ 0 0
$$425$$ −4.96257 −0.240720
$$426$$ 0 0
$$427$$ −21.7676 −1.05341
$$428$$ 0 0
$$429$$ −2.07485 −0.100175
$$430$$ 0 0
$$431$$ −36.1653 −1.74202 −0.871012 0.491262i $$-0.836536\pi$$
−0.871012 + 0.491262i $$0.836536\pi$$
$$432$$ 0 0
$$433$$ −16.6271 −0.799050 −0.399525 0.916722i $$-0.630825\pi$$
−0.399525 + 0.916722i $$0.630825\pi$$
$$434$$ 0 0
$$435$$ −4.96257 −0.237937
$$436$$ 0 0
$$437$$ 4.75353 0.227392
$$438$$ 0 0
$$439$$ 38.7613 1.84998 0.924989 0.379994i $$-0.124074\pi$$
0.924989 + 0.379994i $$0.124074\pi$$
$$440$$ 0 0
$$441$$ 10.7161 0.510290
$$442$$ 0 0
$$443$$ 33.5149 1.59234 0.796170 0.605073i $$-0.206856\pi$$
0.796170 + 0.605073i $$0.206856\pi$$
$$444$$ 0 0
$$445$$ −4.91105 −0.232806
$$446$$ 0 0
$$447$$ 3.24647 0.153553
$$448$$ 0 0
$$449$$ −13.9018 −0.656068 −0.328034 0.944666i $$-0.606386\pi$$
−0.328034 + 0.944666i $$0.606386\pi$$
$$450$$ 0 0
$$451$$ −25.8299 −1.21628
$$452$$ 0 0
$$453$$ 0.418100 0.0196440
$$454$$ 0 0
$$455$$ 3.17162 0.148688
$$456$$ 0 0
$$457$$ −2.28390 −0.106836 −0.0534182 0.998572i $$-0.517012\pi$$
−0.0534182 + 0.998572i $$0.517012\pi$$
$$458$$ 0 0
$$459$$ −4.96257 −0.231633
$$460$$ 0 0
$$461$$ 8.34325 0.388584 0.194292 0.980944i $$-0.437759\pi$$
0.194292 + 0.980944i $$0.437759\pi$$
$$462$$ 0 0
$$463$$ −27.6928 −1.28699 −0.643496 0.765449i $$-0.722517\pi$$
−0.643496 + 0.765449i $$0.722517\pi$$
$$464$$ 0 0
$$465$$ 0.209050 0.00969445
$$466$$ 0 0
$$467$$ 7.53037 0.348464 0.174232 0.984705i $$-0.444256\pi$$
0.174232 + 0.984705i $$0.444256\pi$$
$$468$$ 0 0
$$469$$ −0.879901 −0.0406300
$$470$$ 0 0
$$471$$ −21.0452 −0.969714
$$472$$ 0 0
$$473$$ 34.1935 1.57222
$$474$$ 0 0
$$475$$ 4.75353 0.218107
$$476$$ 0 0
$$477$$ −9.38067 −0.429512
$$478$$ 0 0
$$479$$ −18.1857 −0.830927 −0.415463 0.909610i $$-0.636381\pi$$
−0.415463 + 0.909610i $$0.636381\pi$$
$$480$$ 0 0
$$481$$ 4.30722 0.196393
$$482$$ 0 0
$$483$$ 4.20905 0.191518
$$484$$ 0 0
$$485$$ −10.3432 −0.469663
$$486$$ 0 0
$$487$$ −24.2606 −1.09935 −0.549676 0.835378i $$-0.685249\pi$$
−0.549676 + 0.835378i $$0.685249\pi$$
$$488$$ 0 0
$$489$$ −7.43220 −0.336096
$$490$$ 0 0
$$491$$ 14.8877 0.671874 0.335937 0.941885i $$-0.390947\pi$$
0.335937 + 0.941885i $$0.390947\pi$$
$$492$$ 0 0
$$493$$ −24.6271 −1.10915
$$494$$ 0 0
$$495$$ −2.75353 −0.123762
$$496$$ 0 0
$$497$$ 39.4837 1.77109
$$498$$ 0 0
$$499$$ −0.312101 −0.0139716 −0.00698578 0.999976i $$-0.502224\pi$$
−0.00698578 + 0.999976i $$0.502224\pi$$
$$500$$ 0 0
$$501$$ −19.1716 −0.856525
$$502$$ 0 0
$$503$$ 20.8877 0.931338 0.465669 0.884959i $$-0.345814\pi$$
0.465669 + 0.884959i $$0.345814\pi$$
$$504$$ 0 0
$$505$$ −14.8877 −0.662495
$$506$$ 0 0
$$507$$ −12.4322 −0.552133
$$508$$ 0 0
$$509$$ 27.9251 1.23776 0.618880 0.785485i $$-0.287587\pi$$
0.618880 + 0.785485i $$0.287587\pi$$
$$510$$ 0 0
$$511$$ 43.1873 1.91049
$$512$$ 0 0
$$513$$ 4.75353 0.209873
$$514$$ 0 0
$$515$$ −17.9251 −0.789876
$$516$$ 0 0
$$517$$ 19.7472 0.868483
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 32.0360 1.40352 0.701762 0.712412i $$-0.252397\pi$$
0.701762 + 0.712412i $$0.252397\pi$$
$$522$$ 0 0
$$523$$ 38.8644 1.69942 0.849711 0.527249i $$-0.176777\pi$$
0.849711 + 0.527249i $$0.176777\pi$$
$$524$$ 0 0
$$525$$ 4.20905 0.183698
$$526$$ 0 0
$$527$$ 1.03743 0.0451909
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −4.96257 −0.215357
$$532$$ 0 0
$$533$$ 7.06857 0.306174
$$534$$ 0 0
$$535$$ 16.4696 0.712044
$$536$$ 0 0
$$537$$ 13.8503 0.597685
$$538$$ 0 0
$$539$$ 29.5071 1.27096
$$540$$ 0 0
$$541$$ 4.07485 0.175191 0.0875957 0.996156i $$-0.472082\pi$$
0.0875957 + 0.996156i $$0.472082\pi$$
$$542$$ 0 0
$$543$$ 12.9110 0.554066
$$544$$ 0 0
$$545$$ 6.26058 0.268174
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ −5.17162 −0.220720
$$550$$ 0 0
$$551$$ 23.5897 1.00496
$$552$$ 0 0
$$553$$ −10.1779 −0.432808
$$554$$ 0 0
$$555$$ 5.71610 0.242635
$$556$$ 0 0
$$557$$ 1.63343 0.0692105 0.0346052 0.999401i $$-0.488983\pi$$
0.0346052 + 0.999401i $$0.488983\pi$$
$$558$$ 0 0
$$559$$ −9.35735 −0.395774
$$560$$ 0 0
$$561$$ −13.6646 −0.576919
$$562$$ 0 0
$$563$$ −23.2310 −0.979069 −0.489534 0.871984i $$-0.662833\pi$$
−0.489534 + 0.871984i $$0.662833\pi$$
$$564$$ 0 0
$$565$$ 9.38067 0.394648
$$566$$ 0 0
$$567$$ 4.20905 0.176763
$$568$$ 0 0
$$569$$ 21.3291 0.894164 0.447082 0.894493i $$-0.352463\pi$$
0.447082 + 0.894493i $$0.352463\pi$$
$$570$$ 0 0
$$571$$ −4.18573 −0.175167 −0.0875836 0.996157i $$-0.527914\pi$$
−0.0875836 + 0.996157i $$0.527914\pi$$
$$572$$ 0 0
$$573$$ −8.15752 −0.340785
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 10.5678 0.439943 0.219972 0.975506i $$-0.429404\pi$$
0.219972 + 0.975506i $$0.429404\pi$$
$$578$$ 0 0
$$579$$ −16.7613 −0.696578
$$580$$ 0 0
$$581$$ 22.9626 0.952648
$$582$$ 0 0
$$583$$ −25.8299 −1.06977
$$584$$ 0 0
$$585$$ 0.753525 0.0311544
$$586$$ 0 0
$$587$$ 28.4181 1.17294 0.586470 0.809971i $$-0.300517\pi$$
0.586470 + 0.809971i $$0.300517\pi$$
$$588$$ 0 0
$$589$$ −0.993723 −0.0409457
$$590$$ 0 0
$$591$$ −10.9110 −0.448821
$$592$$ 0 0
$$593$$ 28.9937 1.19063 0.595315 0.803493i $$-0.297027\pi$$
0.595315 + 0.803493i $$0.297027\pi$$
$$594$$ 0 0
$$595$$ 20.8877 0.856313
$$596$$ 0 0
$$597$$ −4.49295 −0.183884
$$598$$ 0 0
$$599$$ −6.10305 −0.249364 −0.124682 0.992197i $$-0.539791\pi$$
−0.124682 + 0.992197i $$0.539791\pi$$
$$600$$ 0 0
$$601$$ −29.3885 −1.19878 −0.599391 0.800456i $$-0.704590\pi$$
−0.599391 + 0.800456i $$0.704590\pi$$
$$602$$ 0 0
$$603$$ −0.209050 −0.00851316
$$604$$ 0 0
$$605$$ 3.41810 0.138966
$$606$$ 0 0
$$607$$ −3.58972 −0.145702 −0.0728512 0.997343i $$-0.523210\pi$$
−0.0728512 + 0.997343i $$0.523210\pi$$
$$608$$ 0 0
$$609$$ 20.8877 0.846413
$$610$$ 0 0
$$611$$ −5.40400 −0.218622
$$612$$ 0 0
$$613$$ −21.4040 −0.864499 −0.432250 0.901754i $$-0.642280\pi$$
−0.432250 + 0.901754i $$0.642280\pi$$
$$614$$ 0 0
$$615$$ 9.38067 0.378265
$$616$$ 0 0
$$617$$ 32.2917 1.30002 0.650008 0.759927i $$-0.274766\pi$$
0.650008 + 0.759927i $$0.274766\pi$$
$$618$$ 0 0
$$619$$ 47.1046 1.89329 0.946647 0.322273i $$-0.104447\pi$$
0.946647 + 0.322273i $$0.104447\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 20.6709 0.828160
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 13.0890 0.522722
$$628$$ 0 0
$$629$$ 28.3666 1.13105
$$630$$ 0 0
$$631$$ 23.5149 0.936112 0.468056 0.883699i $$-0.344954\pi$$
0.468056 + 0.883699i $$0.344954\pi$$
$$632$$ 0 0
$$633$$ 9.71610 0.386180
$$634$$ 0 0
$$635$$ 22.0827 0.876324
$$636$$ 0 0
$$637$$ −8.07485 −0.319937
$$638$$ 0 0
$$639$$ 9.38067 0.371094
$$640$$ 0 0
$$641$$ −6.67867 −0.263792 −0.131896 0.991264i $$-0.542106\pi$$
−0.131896 + 0.991264i $$0.542106\pi$$
$$642$$ 0 0
$$643$$ 27.1201 1.06951 0.534756 0.845006i $$-0.320403\pi$$
0.534756 + 0.845006i $$0.320403\pi$$
$$644$$ 0 0
$$645$$ −12.4181 −0.488962
$$646$$ 0 0
$$647$$ −27.7394 −1.09055 −0.545275 0.838257i $$-0.683575\pi$$
−0.545275 + 0.838257i $$0.683575\pi$$
$$648$$ 0 0
$$649$$ −13.6646 −0.536381
$$650$$ 0 0
$$651$$ −0.879901 −0.0344860
$$652$$ 0 0
$$653$$ −2.75353 −0.107754 −0.0538769 0.998548i $$-0.517158\pi$$
−0.0538769 + 0.998548i $$0.517158\pi$$
$$654$$ 0 0
$$655$$ 1.58190 0.0618100
$$656$$ 0 0
$$657$$ 10.2606 0.400303
$$658$$ 0 0
$$659$$ 11.5897 0.451472 0.225736 0.974189i $$-0.427521\pi$$
0.225736 + 0.974189i $$0.427521\pi$$
$$660$$ 0 0
$$661$$ 28.3432 1.10242 0.551212 0.834365i $$-0.314165\pi$$
0.551212 + 0.834365i $$0.314165\pi$$
$$662$$ 0 0
$$663$$ 3.73942 0.145227
$$664$$ 0 0
$$665$$ −20.0078 −0.775870
$$666$$ 0 0
$$667$$ 4.96257 0.192152
$$668$$ 0 0
$$669$$ −19.4322 −0.751292
$$670$$ 0 0
$$671$$ −14.2402 −0.549737
$$672$$ 0 0
$$673$$ −48.0360 −1.85165 −0.925826 0.377949i $$-0.876629\pi$$
−0.925826 + 0.377949i $$0.876629\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 2.51627 0.0967083 0.0483541 0.998830i $$-0.484602\pi$$
0.0483541 + 0.998830i $$0.484602\pi$$
$$678$$ 0 0
$$679$$ 43.5353 1.67073
$$680$$ 0 0
$$681$$ −0.985900 −0.0377798
$$682$$ 0 0
$$683$$ −3.84248 −0.147028 −0.0735141 0.997294i $$-0.523421\pi$$
−0.0735141 + 0.997294i $$0.523421\pi$$
$$684$$ 0 0
$$685$$ 4.91105 0.187642
$$686$$ 0 0
$$687$$ 3.08895 0.117851
$$688$$ 0 0
$$689$$ 7.06857 0.269291
$$690$$ 0 0
$$691$$ 2.59600 0.0987565 0.0493783 0.998780i $$-0.484276\pi$$
0.0493783 + 0.998780i $$0.484276\pi$$
$$692$$ 0 0
$$693$$ 11.5897 0.440257
$$694$$ 0 0
$$695$$ −14.1342 −0.536141
$$696$$ 0 0
$$697$$ 46.5523 1.76329
$$698$$ 0 0
$$699$$ 8.83620 0.334216
$$700$$ 0 0
$$701$$ 41.5897 1.57082 0.785411 0.618974i $$-0.212452\pi$$
0.785411 + 0.618974i $$0.212452\pi$$
$$702$$ 0 0
$$703$$ −27.1716 −1.02480
$$704$$ 0 0
$$705$$ −7.17162 −0.270099
$$706$$ 0 0
$$707$$ 62.6632 2.35669
$$708$$ 0 0
$$709$$ −53.1716 −1.99690 −0.998451 0.0556356i $$-0.982281\pi$$
−0.998451 + 0.0556356i $$0.982281\pi$$
$$710$$ 0 0
$$711$$ −2.41810 −0.0906858
$$712$$ 0 0
$$713$$ −0.209050 −0.00782898
$$714$$ 0 0
$$715$$ 2.07485 0.0775950
$$716$$ 0 0
$$717$$ −21.3807 −0.798476
$$718$$ 0 0
$$719$$ 45.3807 1.69241 0.846207 0.532855i $$-0.178881\pi$$
0.846207 + 0.532855i $$0.178881\pi$$
$$720$$ 0 0
$$721$$ 75.4478 2.80982
$$722$$ 0 0
$$723$$ 22.2606 0.827880
$$724$$ 0 0
$$725$$ 4.96257 0.184305
$$726$$ 0 0
$$727$$ −46.4026 −1.72098 −0.860489 0.509470i $$-0.829842\pi$$
−0.860489 + 0.509470i $$0.829842\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −61.6257 −2.27931
$$732$$ 0 0
$$733$$ 22.8051 0.842324 0.421162 0.906985i $$-0.361622\pi$$
0.421162 + 0.906985i $$0.361622\pi$$
$$734$$ 0 0
$$735$$ −10.7161 −0.395269
$$736$$ 0 0
$$737$$ −0.575624 −0.0212034
$$738$$ 0 0
$$739$$ −16.5241 −0.607849 −0.303924 0.952696i $$-0.598297\pi$$
−0.303924 + 0.952696i $$0.598297\pi$$
$$740$$ 0 0
$$741$$ −3.58190 −0.131584
$$742$$ 0 0
$$743$$ 43.8503 1.60871 0.804356 0.594148i $$-0.202511\pi$$
0.804356 + 0.594148i $$0.202511\pi$$
$$744$$ 0 0
$$745$$ −3.24647 −0.118942
$$746$$ 0 0
$$747$$ 5.45552 0.199607
$$748$$ 0 0
$$749$$ −69.3215 −2.53295
$$750$$ 0 0
$$751$$ 5.73942 0.209435 0.104717 0.994502i $$-0.466606\pi$$
0.104717 + 0.994502i $$0.466606\pi$$
$$752$$ 0 0
$$753$$ −6.59600 −0.240372
$$754$$ 0 0
$$755$$ −0.418100 −0.0152162
$$756$$ 0 0
$$757$$ −9.86580 −0.358579 −0.179289 0.983796i $$-0.557380\pi$$
−0.179289 + 0.983796i $$0.557380\pi$$
$$758$$ 0 0
$$759$$ 2.75353 0.0999466
$$760$$ 0 0
$$761$$ 4.69418 0.170164 0.0850819 0.996374i $$-0.472885\pi$$
0.0850819 + 0.996374i $$0.472885\pi$$
$$762$$ 0 0
$$763$$ −26.3511 −0.953973
$$764$$ 0 0
$$765$$ 4.96257 0.179422
$$766$$ 0 0
$$767$$ 3.73942 0.135023
$$768$$ 0 0
$$769$$ 45.5431 1.64233 0.821163 0.570694i $$-0.193326\pi$$
0.821163 + 0.570694i $$0.193326\pi$$
$$770$$ 0 0
$$771$$ −18.1857 −0.654943
$$772$$ 0 0
$$773$$ 12.4929 0.449340 0.224670 0.974435i $$-0.427870\pi$$
0.224670 + 0.974435i $$0.427870\pi$$
$$774$$ 0 0
$$775$$ −0.209050 −0.00750929
$$776$$ 0 0
$$777$$ −24.0593 −0.863124
$$778$$ 0 0
$$779$$ −44.5913 −1.59765
$$780$$ 0 0
$$781$$ 25.8299 0.924267
$$782$$ 0 0
$$783$$ 4.96257 0.177348
$$784$$ 0 0
$$785$$ 21.0452 0.751137
$$786$$ 0 0
$$787$$ −5.71610 −0.203757 −0.101878 0.994797i $$-0.532485\pi$$
−0.101878 + 0.994797i $$0.532485\pi$$
$$788$$ 0 0
$$789$$ −15.3807 −0.547567
$$790$$ 0 0
$$791$$ −39.4837 −1.40388
$$792$$ 0 0
$$793$$ 3.89695 0.138385
$$794$$ 0 0
$$795$$ 9.38067 0.332698
$$796$$ 0 0
$$797$$ −9.48373 −0.335931 −0.167965 0.985793i $$-0.553720\pi$$
−0.167965 + 0.985793i $$0.553720\pi$$
$$798$$ 0 0
$$799$$ −35.5897 −1.25907
$$800$$ 0 0
$$801$$ 4.91105 0.173523
$$802$$ 0 0
$$803$$ 28.2528 0.997018
$$804$$ 0 0
$$805$$ −4.20905 −0.148350
$$806$$ 0 0
$$807$$ 0.544475 0.0191664
$$808$$ 0 0
$$809$$ 50.0672 1.76027 0.880134 0.474725i $$-0.157453\pi$$
0.880134 + 0.474725i $$0.157453\pi$$
$$810$$ 0 0
$$811$$ −9.14830 −0.321240 −0.160620 0.987016i $$-0.551349\pi$$
−0.160620 + 0.987016i $$0.551349\pi$$
$$812$$ 0 0
$$813$$ −12.2090 −0.428190
$$814$$ 0 0
$$815$$ 7.43220 0.260339
$$816$$ 0 0
$$817$$ 59.0297 2.06519
$$818$$ 0 0
$$819$$ −3.17162 −0.110826
$$820$$ 0 0
$$821$$ −4.91105 −0.171397 −0.0856984 0.996321i $$-0.527312\pi$$
−0.0856984 + 0.996321i $$0.527312\pi$$
$$822$$ 0 0
$$823$$ 23.7006 0.826151 0.413075 0.910697i $$-0.364455\pi$$
0.413075 + 0.910697i $$0.364455\pi$$
$$824$$ 0 0
$$825$$ 2.75353 0.0958654
$$826$$ 0 0
$$827$$ 13.1405 0.456939 0.228470 0.973551i $$-0.426628\pi$$
0.228470 + 0.973551i $$0.426628\pi$$
$$828$$ 0 0
$$829$$ −45.6412 −1.58519 −0.792593 0.609751i $$-0.791269\pi$$
−0.792593 + 0.609751i $$0.791269\pi$$
$$830$$ 0 0
$$831$$ −21.0141 −0.728971
$$832$$ 0 0
$$833$$ −53.1794 −1.84256
$$834$$ 0 0
$$835$$ 19.1716 0.663461
$$836$$ 0 0
$$837$$ −0.209050 −0.00722582
$$838$$ 0 0
$$839$$ −5.40400 −0.186567 −0.0932834 0.995640i $$-0.529736\pi$$
−0.0932834 + 0.995640i $$0.529736\pi$$
$$840$$ 0 0
$$841$$ −4.37285 −0.150788
$$842$$ 0 0
$$843$$ −24.1857 −0.833001
$$844$$ 0 0
$$845$$ 12.4322 0.427681
$$846$$ 0 0
$$847$$ −14.3870 −0.494341
$$848$$ 0 0
$$849$$ 15.1201 0.518920
$$850$$ 0 0
$$851$$ −5.71610 −0.195945
$$852$$ 0 0
$$853$$ −53.0297 −1.81570 −0.907852 0.419291i $$-0.862279\pi$$
−0.907852 + 0.419291i $$0.862279\pi$$
$$854$$ 0 0
$$855$$ −4.75353 −0.162567
$$856$$ 0 0
$$857$$ −14.4463 −0.493476 −0.246738 0.969082i $$-0.579359\pi$$
−0.246738 + 0.969082i $$0.579359\pi$$
$$858$$ 0 0
$$859$$ 13.8658 0.473095 0.236548 0.971620i $$-0.423984\pi$$
0.236548 + 0.971620i $$0.423984\pi$$
$$860$$ 0 0
$$861$$ −39.4837 −1.34560
$$862$$ 0 0
$$863$$ 52.5834 1.78996 0.894981 0.446105i $$-0.147189\pi$$
0.894981 + 0.446105i $$0.147189\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 7.62715 0.259032
$$868$$ 0 0
$$869$$ −6.65830 −0.225867
$$870$$ 0 0
$$871$$ 0.157524 0.00533751
$$872$$ 0 0
$$873$$ 10.3432 0.350066
$$874$$ 0 0
$$875$$ −4.20905 −0.142292
$$876$$ 0 0
$$877$$ −25.3291 −0.855305 −0.427652 0.903943i $$-0.640659\pi$$
−0.427652 + 0.903943i $$0.640659\pi$$
$$878$$ 0 0
$$879$$ 10.4696 0.353132
$$880$$ 0 0
$$881$$ 14.2606 0.480451 0.240225 0.970717i $$-0.422779\pi$$
0.240225 + 0.970717i $$0.422779\pi$$
$$882$$ 0 0
$$883$$ 40.6320 1.36738 0.683688 0.729774i $$-0.260375\pi$$
0.683688 + 0.729774i $$0.260375\pi$$
$$884$$ 0 0
$$885$$ 4.96257 0.166815
$$886$$ 0 0
$$887$$ −19.9534 −0.669968 −0.334984 0.942224i $$-0.608731\pi$$
−0.334984 + 0.942224i $$0.608731\pi$$
$$888$$ 0 0
$$889$$ −92.9471 −3.11734
$$890$$ 0 0
$$891$$ 2.75353 0.0922466
$$892$$ 0 0
$$893$$ 34.0905 1.14080
$$894$$ 0 0
$$895$$ −13.8503 −0.462964
$$896$$ 0 0
$$897$$ −0.753525 −0.0251595
$$898$$ 0 0
$$899$$ −1.03743 −0.0346001
$$900$$ 0 0
$$901$$ 46.5523 1.55088
$$902$$ 0 0
$$903$$ 52.2684 1.73938
$$904$$ 0 0
$$905$$ −12.9110 −0.429178
$$906$$ 0 0
$$907$$ −29.8814 −0.992197 −0.496099 0.868266i $$-0.665235\pi$$
−0.496099 + 0.868266i $$0.665235\pi$$
$$908$$ 0 0
$$909$$ 14.8877 0.493795
$$910$$ 0 0
$$911$$ 18.8644 0.625005 0.312503 0.949917i $$-0.398833\pi$$
0.312503 + 0.949917i $$0.398833\pi$$
$$912$$ 0 0
$$913$$ 15.0219 0.497153
$$914$$ 0 0
$$915$$ 5.17162 0.170969
$$916$$ 0 0
$$917$$ −6.65830 −0.219876
$$918$$ 0 0
$$919$$ 16.2402 0.535715 0.267857 0.963459i $$-0.413684\pi$$
0.267857 + 0.963459i $$0.413684\pi$$
$$920$$ 0 0
$$921$$ −12.1575 −0.400604
$$922$$ 0 0
$$923$$ −7.06857 −0.232665
$$924$$ 0 0
$$925$$ −5.71610 −0.187944
$$926$$ 0 0
$$927$$ 17.9251 0.588739
$$928$$ 0 0
$$929$$ 29.3340 0.962418 0.481209 0.876606i $$-0.340198\pi$$
0.481209 + 0.876606i $$0.340198\pi$$
$$930$$ 0 0
$$931$$ 50.9393 1.66947
$$932$$ 0 0
$$933$$ −34.4181 −1.12680
$$934$$ 0 0
$$935$$ 13.6646 0.446879
$$936$$ 0 0
$$937$$ −41.3573 −1.35109 −0.675543 0.737321i $$-0.736091\pi$$
−0.675543 + 0.737321i $$0.736091\pi$$
$$938$$ 0 0
$$939$$ 29.5664 0.964863
$$940$$ 0 0
$$941$$ 50.4259 1.64384 0.821919 0.569604i $$-0.192904\pi$$
0.821919 + 0.569604i $$0.192904\pi$$
$$942$$ 0 0
$$943$$ −9.38067 −0.305477
$$944$$ 0 0
$$945$$ −4.20905 −0.136920
$$946$$ 0 0
$$947$$ 5.50705 0.178955 0.0894775 0.995989i $$-0.471480\pi$$
0.0894775 + 0.995989i $$0.471480\pi$$
$$948$$ 0 0
$$949$$ −7.73160 −0.250978
$$950$$ 0 0
$$951$$ 1.66457 0.0539775
$$952$$ 0 0
$$953$$ 0.864400 0.0280007 0.0140003 0.999902i $$-0.495543\pi$$
0.0140003 + 0.999902i $$0.495543\pi$$
$$954$$ 0 0
$$955$$ 8.15752 0.263971
$$956$$ 0 0
$$957$$ 13.6646 0.441713
$$958$$ 0 0
$$959$$ −20.6709 −0.667497
$$960$$ 0 0
$$961$$ −30.9563 −0.998590
$$962$$ 0 0
$$963$$ −16.4696 −0.530726
$$964$$ 0 0
$$965$$ 16.7613 0.539567
$$966$$ 0 0
$$967$$ 10.7535 0.345810 0.172905 0.984939i $$-0.444685\pi$$
0.172905 + 0.984939i $$0.444685\pi$$
$$968$$ 0 0
$$969$$ −23.5897 −0.757811
$$970$$ 0 0
$$971$$ 35.1794 1.12896 0.564481 0.825446i $$-0.309076\pi$$
0.564481 + 0.825446i $$0.309076\pi$$
$$972$$ 0 0
$$973$$ 59.4915 1.90721
$$974$$ 0 0
$$975$$ −0.753525 −0.0241321
$$976$$ 0 0
$$977$$ 15.9767 0.511139 0.255570 0.966791i $$-0.417737\pi$$
0.255570 + 0.966791i $$0.417737\pi$$
$$978$$ 0 0
$$979$$ 13.5227 0.432187
$$980$$ 0 0
$$981$$ −6.26058 −0.199885
$$982$$ 0 0
$$983$$ −57.2592 −1.82628 −0.913142 0.407642i $$-0.866351\pi$$
−0.913142 + 0.407642i $$0.866351\pi$$
$$984$$ 0 0
$$985$$ 10.9110 0.347655
$$986$$ 0 0
$$987$$ 30.1857 0.960822
$$988$$ 0 0
$$989$$ 12.4181 0.394873
$$990$$ 0 0
$$991$$ 12.8799 0.409144 0.204572 0.978852i $$-0.434420\pi$$
0.204572 + 0.978852i $$0.434420\pi$$
$$992$$ 0 0
$$993$$ 3.12010 0.0990134
$$994$$ 0 0
$$995$$ 4.49295 0.142436
$$996$$ 0 0
$$997$$ −39.7754 −1.25970 −0.629851 0.776716i $$-0.716884\pi$$
−0.629851 + 0.776716i $$0.716884\pi$$
$$998$$ 0 0
$$999$$ −5.71610 −0.180849
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 1380.2.a.j.1.3 3
3.2 odd 2 4140.2.a.s.1.3 3
4.3 odd 2 5520.2.a.bv.1.1 3
5.2 odd 4 6900.2.f.r.6349.3 6
5.3 odd 4 6900.2.f.r.6349.4 6
5.4 even 2 6900.2.a.x.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.j.1.3 3 1.1 even 1 trivial
4140.2.a.s.1.3 3 3.2 odd 2
5520.2.a.bv.1.1 3 4.3 odd 2
6900.2.a.x.1.1 3 5.4 even 2
6900.2.f.r.6349.3 6 5.2 odd 4
6900.2.f.r.6349.4 6 5.3 odd 4