# Properties

 Label 3120.2.a.bj.1.3 Level $3120$ Weight $2$ Character 3120.1 Self dual yes Analytic conductor $24.913$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.9133254306$$ 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.470683$$ of defining polynomial Character $$\chi$$ $$=$$ 3120.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +2.71982 q^{7} +1.00000 q^{9} +2.71982 q^{11} +1.00000 q^{13} -1.00000 q^{15} -2.83709 q^{17} +3.55691 q^{19} +2.71982 q^{21} +4.83709 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -7.55691 q^{31} +2.71982 q^{33} -2.71982 q^{35} -4.27674 q^{37} +1.00000 q^{39} +2.83709 q^{41} -11.1138 q^{43} -1.00000 q^{45} +11.5569 q^{47} +0.397442 q^{49} -2.83709 q^{51} +1.16291 q^{53} -2.71982 q^{55} +3.55691 q^{57} +2.11727 q^{59} +6.60256 q^{61} +2.71982 q^{63} -1.00000 q^{65} -1.88273 q^{67} +4.83709 q^{69} +6.71982 q^{71} +9.11383 q^{73} +1.00000 q^{75} +7.39744 q^{77} -10.2767 q^{79} +1.00000 q^{81} -2.11727 q^{83} +2.83709 q^{85} +6.00000 q^{87} +1.16291 q^{89} +2.71982 q^{91} -7.55691 q^{93} -3.55691 q^{95} -10.8371 q^{97} +2.71982 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 $$3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - 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
$$7$$ 2.71982 1.02800 0.513998 0.857791i $$-0.328164\pi$$
0.513998 + 0.857791i $$0.328164\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.71982 0.820058 0.410029 0.912073i $$-0.365519\pi$$
0.410029 + 0.912073i $$0.365519\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −2.83709 −0.688095 −0.344048 0.938952i $$-0.611798\pi$$
−0.344048 + 0.938952i $$0.611798\pi$$
$$18$$ 0 0
$$19$$ 3.55691 0.816012 0.408006 0.912979i $$-0.366224\pi$$
0.408006 + 0.912979i $$0.366224\pi$$
$$20$$ 0 0
$$21$$ 2.71982 0.593514
$$22$$ 0 0
$$23$$ 4.83709 1.00860 0.504302 0.863528i $$-0.331750\pi$$
0.504302 + 0.863528i $$0.331750\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −7.55691 −1.35726 −0.678631 0.734479i $$-0.737426\pi$$
−0.678631 + 0.734479i $$0.737426\pi$$
$$32$$ 0 0
$$33$$ 2.71982 0.473461
$$34$$ 0 0
$$35$$ −2.71982 −0.459734
$$36$$ 0 0
$$37$$ −4.27674 −0.703091 −0.351546 0.936171i $$-0.614344\pi$$
−0.351546 + 0.936171i $$0.614344\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 2.83709 0.443079 0.221540 0.975151i $$-0.428892\pi$$
0.221540 + 0.975151i $$0.428892\pi$$
$$42$$ 0 0
$$43$$ −11.1138 −1.69484 −0.847421 0.530921i $$-0.821846\pi$$
−0.847421 + 0.530921i $$0.821846\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 11.5569 1.68575 0.842875 0.538110i $$-0.180862\pi$$
0.842875 + 0.538110i $$0.180862\pi$$
$$48$$ 0 0
$$49$$ 0.397442 0.0567775
$$50$$ 0 0
$$51$$ −2.83709 −0.397272
$$52$$ 0 0
$$53$$ 1.16291 0.159738 0.0798690 0.996805i $$-0.474550\pi$$
0.0798690 + 0.996805i $$0.474550\pi$$
$$54$$ 0 0
$$55$$ −2.71982 −0.366741
$$56$$ 0 0
$$57$$ 3.55691 0.471125
$$58$$ 0 0
$$59$$ 2.11727 0.275645 0.137822 0.990457i $$-0.455990\pi$$
0.137822 + 0.990457i $$0.455990\pi$$
$$60$$ 0 0
$$61$$ 6.60256 0.845371 0.422685 0.906276i $$-0.361088\pi$$
0.422685 + 0.906276i $$0.361088\pi$$
$$62$$ 0 0
$$63$$ 2.71982 0.342666
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −1.88273 −0.230013 −0.115006 0.993365i $$-0.536689\pi$$
−0.115006 + 0.993365i $$0.536689\pi$$
$$68$$ 0 0
$$69$$ 4.83709 0.582317
$$70$$ 0 0
$$71$$ 6.71982 0.797496 0.398748 0.917060i $$-0.369445\pi$$
0.398748 + 0.917060i $$0.369445\pi$$
$$72$$ 0 0
$$73$$ 9.11383 1.06669 0.533346 0.845897i $$-0.320934\pi$$
0.533346 + 0.845897i $$0.320934\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 7.39744 0.843017
$$78$$ 0 0
$$79$$ −10.2767 −1.15622 −0.578112 0.815958i $$-0.696210\pi$$
−0.578112 + 0.815958i $$0.696210\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −2.11727 −0.232400 −0.116200 0.993226i $$-0.537071\pi$$
−0.116200 + 0.993226i $$0.537071\pi$$
$$84$$ 0 0
$$85$$ 2.83709 0.307726
$$86$$ 0 0
$$87$$ 6.00000 0.643268
$$88$$ 0 0
$$89$$ 1.16291 0.123268 0.0616341 0.998099i $$-0.480369\pi$$
0.0616341 + 0.998099i $$0.480369\pi$$
$$90$$ 0 0
$$91$$ 2.71982 0.285115
$$92$$ 0 0
$$93$$ −7.55691 −0.783616
$$94$$ 0 0
$$95$$ −3.55691 −0.364932
$$96$$ 0 0
$$97$$ −10.8371 −1.10034 −0.550170 0.835053i $$-0.685437\pi$$
−0.550170 + 0.835053i $$0.685437\pi$$
$$98$$ 0 0
$$99$$ 2.71982 0.273353
$$100$$ 0 0
$$101$$ 7.67418 0.763610 0.381805 0.924243i $$-0.375303\pi$$
0.381805 + 0.924243i $$0.375303\pi$$
$$102$$ 0 0
$$103$$ −3.76547 −0.371023 −0.185511 0.982642i $$-0.559394\pi$$
−0.185511 + 0.982642i $$0.559394\pi$$
$$104$$ 0 0
$$105$$ −2.71982 −0.265428
$$106$$ 0 0
$$107$$ 12.6026 1.21834 0.609168 0.793041i $$-0.291504\pi$$
0.609168 + 0.793041i $$0.291504\pi$$
$$108$$ 0 0
$$109$$ 11.4396 1.09572 0.547860 0.836570i $$-0.315443\pi$$
0.547860 + 0.836570i $$0.315443\pi$$
$$110$$ 0 0
$$111$$ −4.27674 −0.405930
$$112$$ 0 0
$$113$$ −13.1138 −1.23365 −0.616823 0.787102i $$-0.711580\pi$$
−0.616823 + 0.787102i $$0.711580\pi$$
$$114$$ 0 0
$$115$$ −4.83709 −0.451061
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −7.71639 −0.707360
$$120$$ 0 0
$$121$$ −3.60256 −0.327505
$$122$$ 0 0
$$123$$ 2.83709 0.255812
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 13.4396 1.19258 0.596288 0.802771i $$-0.296642\pi$$
0.596288 + 0.802771i $$0.296642\pi$$
$$128$$ 0 0
$$129$$ −11.1138 −0.978518
$$130$$ 0 0
$$131$$ 9.43965 0.824746 0.412373 0.911015i $$-0.364700\pi$$
0.412373 + 0.911015i $$0.364700\pi$$
$$132$$ 0 0
$$133$$ 9.67418 0.838858
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ 1.76547 0.150834 0.0754170 0.997152i $$-0.475971\pi$$
0.0754170 + 0.997152i $$0.475971\pi$$
$$138$$ 0 0
$$139$$ 6.27674 0.532386 0.266193 0.963920i $$-0.414234\pi$$
0.266193 + 0.963920i $$0.414234\pi$$
$$140$$ 0 0
$$141$$ 11.5569 0.973268
$$142$$ 0 0
$$143$$ 2.71982 0.227443
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 0.397442 0.0327805
$$148$$ 0 0
$$149$$ −20.8302 −1.70648 −0.853239 0.521520i $$-0.825365\pi$$
−0.853239 + 0.521520i $$0.825365\pi$$
$$150$$ 0 0
$$151$$ −4.99656 −0.406614 −0.203307 0.979115i $$-0.565169\pi$$
−0.203307 + 0.979115i $$0.565169\pi$$
$$152$$ 0 0
$$153$$ −2.83709 −0.229365
$$154$$ 0 0
$$155$$ 7.55691 0.606986
$$156$$ 0 0
$$157$$ 8.87930 0.708645 0.354322 0.935123i $$-0.384712\pi$$
0.354322 + 0.935123i $$0.384712\pi$$
$$158$$ 0 0
$$159$$ 1.16291 0.0922247
$$160$$ 0 0
$$161$$ 13.1560 1.03684
$$162$$ 0 0
$$163$$ 13.8337 1.08354 0.541768 0.840528i $$-0.317755\pi$$
0.541768 + 0.840528i $$0.317755\pi$$
$$164$$ 0 0
$$165$$ −2.71982 −0.211738
$$166$$ 0 0
$$167$$ 9.88273 0.764749 0.382374 0.924007i $$-0.375106\pi$$
0.382374 + 0.924007i $$0.375106\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 3.55691 0.272004
$$172$$ 0 0
$$173$$ 13.1138 0.997026 0.498513 0.866882i $$-0.333880\pi$$
0.498513 + 0.866882i $$0.333880\pi$$
$$174$$ 0 0
$$175$$ 2.71982 0.205599
$$176$$ 0 0
$$177$$ 2.11727 0.159143
$$178$$ 0 0
$$179$$ −8.55348 −0.639317 −0.319658 0.947533i $$-0.603568\pi$$
−0.319658 + 0.947533i $$0.603568\pi$$
$$180$$ 0 0
$$181$$ 3.72326 0.276748 0.138374 0.990380i $$-0.455812\pi$$
0.138374 + 0.990380i $$0.455812\pi$$
$$182$$ 0 0
$$183$$ 6.60256 0.488075
$$184$$ 0 0
$$185$$ 4.27674 0.314432
$$186$$ 0 0
$$187$$ −7.71639 −0.564278
$$188$$ 0 0
$$189$$ 2.71982 0.197838
$$190$$ 0 0
$$191$$ 4.23453 0.306400 0.153200 0.988195i $$-0.451042\pi$$
0.153200 + 0.988195i $$0.451042\pi$$
$$192$$ 0 0
$$193$$ −23.3906 −1.68369 −0.841845 0.539719i $$-0.818530\pi$$
−0.841845 + 0.539719i $$0.818530\pi$$
$$194$$ 0 0
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ 14.5535 1.03689 0.518446 0.855110i $$-0.326511\pi$$
0.518446 + 0.855110i $$0.326511\pi$$
$$198$$ 0 0
$$199$$ 15.1138 1.07139 0.535695 0.844411i $$-0.320050\pi$$
0.535695 + 0.844411i $$0.320050\pi$$
$$200$$ 0 0
$$201$$ −1.88273 −0.132798
$$202$$ 0 0
$$203$$ 16.3189 1.14537
$$204$$ 0 0
$$205$$ −2.83709 −0.198151
$$206$$ 0 0
$$207$$ 4.83709 0.336201
$$208$$ 0 0
$$209$$ 9.67418 0.669177
$$210$$ 0 0
$$211$$ 18.2277 1.25484 0.627422 0.778680i $$-0.284110\pi$$
0.627422 + 0.778680i $$0.284110\pi$$
$$212$$ 0 0
$$213$$ 6.71982 0.460435
$$214$$ 0 0
$$215$$ 11.1138 0.757957
$$216$$ 0 0
$$217$$ −20.5535 −1.39526
$$218$$ 0 0
$$219$$ 9.11383 0.615855
$$220$$ 0 0
$$221$$ −2.83709 −0.190843
$$222$$ 0 0
$$223$$ −10.1173 −0.677502 −0.338751 0.940876i $$-0.610004\pi$$
−0.338751 + 0.940876i $$0.610004\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −11.3224 −0.751493 −0.375746 0.926723i $$-0.622614\pi$$
−0.375746 + 0.926723i $$0.622614\pi$$
$$228$$ 0 0
$$229$$ −6.23453 −0.411990 −0.205995 0.978553i $$-0.566043\pi$$
−0.205995 + 0.978553i $$0.566043\pi$$
$$230$$ 0 0
$$231$$ 7.39744 0.486716
$$232$$ 0 0
$$233$$ 6.83709 0.447913 0.223956 0.974599i $$-0.428103\pi$$
0.223956 + 0.974599i $$0.428103\pi$$
$$234$$ 0 0
$$235$$ −11.5569 −0.753890
$$236$$ 0 0
$$237$$ −10.2767 −0.667546
$$238$$ 0 0
$$239$$ −1.28018 −0.0828077 −0.0414039 0.999142i $$-0.513183\pi$$
−0.0414039 + 0.999142i $$0.513183\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −0.397442 −0.0253917
$$246$$ 0 0
$$247$$ 3.55691 0.226321
$$248$$ 0 0
$$249$$ −2.11727 −0.134176
$$250$$ 0 0
$$251$$ −18.2277 −1.15052 −0.575260 0.817971i $$-0.695099\pi$$
−0.575260 + 0.817971i $$0.695099\pi$$
$$252$$ 0 0
$$253$$ 13.1560 0.827113
$$254$$ 0 0
$$255$$ 2.83709 0.177665
$$256$$ 0 0
$$257$$ 1.11383 0.0694787 0.0347394 0.999396i $$-0.488940\pi$$
0.0347394 + 0.999396i $$0.488940\pi$$
$$258$$ 0 0
$$259$$ −11.6320 −0.722776
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ −1.16291 −0.0714370
$$266$$ 0 0
$$267$$ 1.16291 0.0711689
$$268$$ 0 0
$$269$$ 15.6742 0.955672 0.477836 0.878449i $$-0.341421\pi$$
0.477836 + 0.878449i $$0.341421\pi$$
$$270$$ 0 0
$$271$$ −0.443086 −0.0269155 −0.0134578 0.999909i $$-0.504284\pi$$
−0.0134578 + 0.999909i $$0.504284\pi$$
$$272$$ 0 0
$$273$$ 2.71982 0.164611
$$274$$ 0 0
$$275$$ 2.71982 0.164012
$$276$$ 0 0
$$277$$ −4.87930 −0.293168 −0.146584 0.989198i $$-0.546828\pi$$
−0.146584 + 0.989198i $$0.546828\pi$$
$$278$$ 0 0
$$279$$ −7.55691 −0.452421
$$280$$ 0 0
$$281$$ −9.11383 −0.543685 −0.271843 0.962342i $$-0.587633\pi$$
−0.271843 + 0.962342i $$0.587633\pi$$
$$282$$ 0 0
$$283$$ −33.3415 −1.98195 −0.990973 0.134063i $$-0.957198\pi$$
−0.990973 + 0.134063i $$0.957198\pi$$
$$284$$ 0 0
$$285$$ −3.55691 −0.210693
$$286$$ 0 0
$$287$$ 7.71639 0.455484
$$288$$ 0 0
$$289$$ −8.95092 −0.526525
$$290$$ 0 0
$$291$$ −10.8371 −0.635281
$$292$$ 0 0
$$293$$ −29.4328 −1.71948 −0.859740 0.510731i $$-0.829375\pi$$
−0.859740 + 0.510731i $$0.829375\pi$$
$$294$$ 0 0
$$295$$ −2.11727 −0.123272
$$296$$ 0 0
$$297$$ 2.71982 0.157820
$$298$$ 0 0
$$299$$ 4.83709 0.279736
$$300$$ 0 0
$$301$$ −30.2277 −1.74229
$$302$$ 0 0
$$303$$ 7.67418 0.440870
$$304$$ 0 0
$$305$$ −6.60256 −0.378061
$$306$$ 0 0
$$307$$ 21.8337 1.24611 0.623056 0.782177i $$-0.285891\pi$$
0.623056 + 0.782177i $$0.285891\pi$$
$$308$$ 0 0
$$309$$ −3.76547 −0.214210
$$310$$ 0 0
$$311$$ −25.1070 −1.42368 −0.711842 0.702339i $$-0.752139\pi$$
−0.711842 + 0.702339i $$0.752139\pi$$
$$312$$ 0 0
$$313$$ 8.22766 0.465055 0.232527 0.972590i $$-0.425300\pi$$
0.232527 + 0.972590i $$0.425300\pi$$
$$314$$ 0 0
$$315$$ −2.71982 −0.153245
$$316$$ 0 0
$$317$$ 27.6742 1.55434 0.777168 0.629293i $$-0.216655\pi$$
0.777168 + 0.629293i $$0.216655\pi$$
$$318$$ 0 0
$$319$$ 16.3189 0.913685
$$320$$ 0 0
$$321$$ 12.6026 0.703406
$$322$$ 0 0
$$323$$ −10.0913 −0.561494
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 0 0
$$327$$ 11.4396 0.632614
$$328$$ 0 0
$$329$$ 31.4328 1.73294
$$330$$ 0 0
$$331$$ 13.2311 0.727247 0.363623 0.931546i $$-0.381540\pi$$
0.363623 + 0.931546i $$0.381540\pi$$
$$332$$ 0 0
$$333$$ −4.27674 −0.234364
$$334$$ 0 0
$$335$$ 1.88273 0.102865
$$336$$ 0 0
$$337$$ −4.32582 −0.235642 −0.117821 0.993035i $$-0.537591\pi$$
−0.117821 + 0.993035i $$0.537591\pi$$
$$338$$ 0 0
$$339$$ −13.1138 −0.712245
$$340$$ 0 0
$$341$$ −20.5535 −1.11303
$$342$$ 0 0
$$343$$ −17.9578 −0.969630
$$344$$ 0 0
$$345$$ −4.83709 −0.260420
$$346$$ 0 0
$$347$$ −6.27674 −0.336953 −0.168476 0.985706i $$-0.553885\pi$$
−0.168476 + 0.985706i $$0.553885\pi$$
$$348$$ 0 0
$$349$$ 17.6673 0.945709 0.472855 0.881140i $$-0.343224\pi$$
0.472855 + 0.881140i $$0.343224\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 0 0
$$353$$ −13.7655 −0.732662 −0.366331 0.930485i $$-0.619386\pi$$
−0.366331 + 0.930485i $$0.619386\pi$$
$$354$$ 0 0
$$355$$ −6.71982 −0.356651
$$356$$ 0 0
$$357$$ −7.71639 −0.408394
$$358$$ 0 0
$$359$$ −0.996562 −0.0525965 −0.0262983 0.999654i $$-0.508372\pi$$
−0.0262983 + 0.999654i $$0.508372\pi$$
$$360$$ 0 0
$$361$$ −6.34836 −0.334124
$$362$$ 0 0
$$363$$ −3.60256 −0.189085
$$364$$ 0 0
$$365$$ −9.11383 −0.477040
$$366$$ 0 0
$$367$$ −14.2277 −0.742678 −0.371339 0.928497i $$-0.621101\pi$$
−0.371339 + 0.928497i $$0.621101\pi$$
$$368$$ 0 0
$$369$$ 2.83709 0.147693
$$370$$ 0 0
$$371$$ 3.16291 0.164210
$$372$$ 0 0
$$373$$ 15.6742 0.811578 0.405789 0.913967i $$-0.366997\pi$$
0.405789 + 0.913967i $$0.366997\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 6.00000 0.309016
$$378$$ 0 0
$$379$$ −26.2017 −1.34589 −0.672945 0.739693i $$-0.734971\pi$$
−0.672945 + 0.739693i $$0.734971\pi$$
$$380$$ 0 0
$$381$$ 13.4396 0.688534
$$382$$ 0 0
$$383$$ −22.4362 −1.14644 −0.573218 0.819403i $$-0.694305\pi$$
−0.573218 + 0.819403i $$0.694305\pi$$
$$384$$ 0 0
$$385$$ −7.39744 −0.377009
$$386$$ 0 0
$$387$$ −11.1138 −0.564948
$$388$$ 0 0
$$389$$ 31.6742 1.60594 0.802972 0.596016i $$-0.203251\pi$$
0.802972 + 0.596016i $$0.203251\pi$$
$$390$$ 0 0
$$391$$ −13.7233 −0.694015
$$392$$ 0 0
$$393$$ 9.43965 0.476167
$$394$$ 0 0
$$395$$ 10.2767 0.517079
$$396$$ 0 0
$$397$$ 17.9509 0.900931 0.450465 0.892794i $$-0.351258\pi$$
0.450465 + 0.892794i $$0.351258\pi$$
$$398$$ 0 0
$$399$$ 9.67418 0.484315
$$400$$ 0 0
$$401$$ 13.5829 0.678297 0.339149 0.940733i $$-0.389861\pi$$
0.339149 + 0.940733i $$0.389861\pi$$
$$402$$ 0 0
$$403$$ −7.55691 −0.376437
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −11.6320 −0.576576
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 1.76547 0.0870841
$$412$$ 0 0
$$413$$ 5.75859 0.283362
$$414$$ 0 0
$$415$$ 2.11727 0.103933
$$416$$ 0 0
$$417$$ 6.27674 0.307373
$$418$$ 0 0
$$419$$ −12.3189 −0.601820 −0.300910 0.953653i $$-0.597290\pi$$
−0.300910 + 0.953653i $$0.597290\pi$$
$$420$$ 0 0
$$421$$ 22.7880 1.11062 0.555310 0.831644i $$-0.312600\pi$$
0.555310 + 0.831644i $$0.312600\pi$$
$$422$$ 0 0
$$423$$ 11.5569 0.561916
$$424$$ 0 0
$$425$$ −2.83709 −0.137619
$$426$$ 0 0
$$427$$ 17.9578 0.869039
$$428$$ 0 0
$$429$$ 2.71982 0.131314
$$430$$ 0 0
$$431$$ 8.99656 0.433349 0.216675 0.976244i $$-0.430479\pi$$
0.216675 + 0.976244i $$0.430479\pi$$
$$432$$ 0 0
$$433$$ −20.3258 −0.976797 −0.488398 0.872621i $$-0.662419\pi$$
−0.488398 + 0.872621i $$0.662419\pi$$
$$434$$ 0 0
$$435$$ −6.00000 −0.287678
$$436$$ 0 0
$$437$$ 17.2051 0.823032
$$438$$ 0 0
$$439$$ 25.3906 1.21183 0.605913 0.795531i $$-0.292808\pi$$
0.605913 + 0.795531i $$0.292808\pi$$
$$440$$ 0 0
$$441$$ 0.397442 0.0189258
$$442$$ 0 0
$$443$$ −10.9284 −0.519223 −0.259611 0.965713i $$-0.583594\pi$$
−0.259611 + 0.965713i $$0.583594\pi$$
$$444$$ 0 0
$$445$$ −1.16291 −0.0551272
$$446$$ 0 0
$$447$$ −20.8302 −0.985235
$$448$$ 0 0
$$449$$ 2.83709 0.133891 0.0669453 0.997757i $$-0.478675\pi$$
0.0669453 + 0.997757i $$0.478675\pi$$
$$450$$ 0 0
$$451$$ 7.71639 0.363350
$$452$$ 0 0
$$453$$ −4.99656 −0.234759
$$454$$ 0 0
$$455$$ −2.71982 −0.127507
$$456$$ 0 0
$$457$$ −13.7164 −0.641625 −0.320813 0.947143i $$-0.603956\pi$$
−0.320813 + 0.947143i $$0.603956\pi$$
$$458$$ 0 0
$$459$$ −2.83709 −0.132424
$$460$$ 0 0
$$461$$ −19.6251 −0.914032 −0.457016 0.889458i $$-0.651082\pi$$
−0.457016 + 0.889458i $$0.651082\pi$$
$$462$$ 0 0
$$463$$ −27.0388 −1.25660 −0.628299 0.777972i $$-0.716249\pi$$
−0.628299 + 0.777972i $$0.716249\pi$$
$$464$$ 0 0
$$465$$ 7.55691 0.350444
$$466$$ 0 0
$$467$$ −28.9215 −1.33833 −0.669164 0.743115i $$-0.733348\pi$$
−0.669164 + 0.743115i $$0.733348\pi$$
$$468$$ 0 0
$$469$$ −5.12070 −0.236452
$$470$$ 0 0
$$471$$ 8.87930 0.409136
$$472$$ 0 0
$$473$$ −30.2277 −1.38987
$$474$$ 0 0
$$475$$ 3.55691 0.163202
$$476$$ 0 0
$$477$$ 1.16291 0.0532460
$$478$$ 0 0
$$479$$ 12.1595 0.555580 0.277790 0.960642i $$-0.410398\pi$$
0.277790 + 0.960642i $$0.410398\pi$$
$$480$$ 0 0
$$481$$ −4.27674 −0.195002
$$482$$ 0 0
$$483$$ 13.1560 0.598620
$$484$$ 0 0
$$485$$ 10.8371 0.492087
$$486$$ 0 0
$$487$$ 0.159472 0.00722636 0.00361318 0.999993i $$-0.498850\pi$$
0.00361318 + 0.999993i $$0.498850\pi$$
$$488$$ 0 0
$$489$$ 13.8337 0.625579
$$490$$ 0 0
$$491$$ 42.2277 1.90571 0.952854 0.303430i $$-0.0981318\pi$$
0.952854 + 0.303430i $$0.0981318\pi$$
$$492$$ 0 0
$$493$$ −17.0225 −0.766657
$$494$$ 0 0
$$495$$ −2.71982 −0.122247
$$496$$ 0 0
$$497$$ 18.2767 0.819824
$$498$$ 0 0
$$499$$ −7.79145 −0.348793 −0.174397 0.984676i $$-0.555797\pi$$
−0.174397 + 0.984676i $$0.555797\pi$$
$$500$$ 0 0
$$501$$ 9.88273 0.441528
$$502$$ 0 0
$$503$$ −27.3484 −1.21940 −0.609702 0.792631i $$-0.708711\pi$$
−0.609702 + 0.792631i $$0.708711\pi$$
$$504$$ 0 0
$$505$$ −7.67418 −0.341497
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −33.4819 −1.48406 −0.742029 0.670368i $$-0.766136\pi$$
−0.742029 + 0.670368i $$0.766136\pi$$
$$510$$ 0 0
$$511$$ 24.7880 1.09656
$$512$$ 0 0
$$513$$ 3.55691 0.157042
$$514$$ 0 0
$$515$$ 3.76547 0.165926
$$516$$ 0 0
$$517$$ 31.4328 1.38241
$$518$$ 0 0
$$519$$ 13.1138 0.575633
$$520$$ 0 0
$$521$$ −17.3484 −0.760045 −0.380023 0.924977i $$-0.624084\pi$$
−0.380023 + 0.924977i $$0.624084\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ 2.71982 0.118703
$$526$$ 0 0
$$527$$ 21.4396 0.933926
$$528$$ 0 0
$$529$$ 0.397442 0.0172801
$$530$$ 0 0
$$531$$ 2.11727 0.0918815
$$532$$ 0 0
$$533$$ 2.83709 0.122888
$$534$$ 0 0
$$535$$ −12.6026 −0.544856
$$536$$ 0 0
$$537$$ −8.55348 −0.369110
$$538$$ 0 0
$$539$$ 1.08097 0.0465608
$$540$$ 0 0
$$541$$ −32.6448 −1.40351 −0.701754 0.712419i $$-0.747599\pi$$
−0.701754 + 0.712419i $$0.747599\pi$$
$$542$$ 0 0
$$543$$ 3.72326 0.159780
$$544$$ 0 0
$$545$$ −11.4396 −0.490021
$$546$$ 0 0
$$547$$ −34.2277 −1.46347 −0.731734 0.681590i $$-0.761289\pi$$
−0.731734 + 0.681590i $$0.761289\pi$$
$$548$$ 0 0
$$549$$ 6.60256 0.281790
$$550$$ 0 0
$$551$$ 21.3415 0.909178
$$552$$ 0 0
$$553$$ −27.9509 −1.18859
$$554$$ 0 0
$$555$$ 4.27674 0.181537
$$556$$ 0 0
$$557$$ 6.65164 0.281839 0.140919 0.990021i $$-0.454994\pi$$
0.140919 + 0.990021i $$0.454994\pi$$
$$558$$ 0 0
$$559$$ −11.1138 −0.470065
$$560$$ 0 0
$$561$$ −7.71639 −0.325786
$$562$$ 0 0
$$563$$ −40.2699 −1.69717 −0.848586 0.529057i $$-0.822546\pi$$
−0.848586 + 0.529057i $$0.822546\pi$$
$$564$$ 0 0
$$565$$ 13.1138 0.551703
$$566$$ 0 0
$$567$$ 2.71982 0.114222
$$568$$ 0 0
$$569$$ −13.4328 −0.563131 −0.281566 0.959542i $$-0.590854\pi$$
−0.281566 + 0.959542i $$0.590854\pi$$
$$570$$ 0 0
$$571$$ 35.7164 1.49468 0.747342 0.664439i $$-0.231330\pi$$
0.747342 + 0.664439i $$0.231330\pi$$
$$572$$ 0 0
$$573$$ 4.23453 0.176900
$$574$$ 0 0
$$575$$ 4.83709 0.201721
$$576$$ 0 0
$$577$$ −13.7164 −0.571021 −0.285510 0.958376i $$-0.592163\pi$$
−0.285510 + 0.958376i $$0.592163\pi$$
$$578$$ 0 0
$$579$$ −23.3906 −0.972079
$$580$$ 0 0
$$581$$ −5.75859 −0.238907
$$582$$ 0 0
$$583$$ 3.16291 0.130994
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 0 0
$$587$$ −30.6707 −1.26592 −0.632959 0.774186i $$-0.718160\pi$$
−0.632959 + 0.774186i $$0.718160\pi$$
$$588$$ 0 0
$$589$$ −26.8793 −1.10754
$$590$$ 0 0
$$591$$ 14.5535 0.598650
$$592$$ 0 0
$$593$$ −45.6673 −1.87533 −0.937666 0.347538i $$-0.887018\pi$$
−0.937666 + 0.347538i $$0.887018\pi$$
$$594$$ 0 0
$$595$$ 7.71639 0.316341
$$596$$ 0 0
$$597$$ 15.1138 0.618568
$$598$$ 0 0
$$599$$ 40.2208 1.64338 0.821688 0.569937i $$-0.193032\pi$$
0.821688 + 0.569937i $$0.193032\pi$$
$$600$$ 0 0
$$601$$ 17.3974 0.709656 0.354828 0.934932i $$-0.384539\pi$$
0.354828 + 0.934932i $$0.384539\pi$$
$$602$$ 0 0
$$603$$ −1.88273 −0.0766708
$$604$$ 0 0
$$605$$ 3.60256 0.146465
$$606$$ 0 0
$$607$$ 14.2277 0.577483 0.288741 0.957407i $$-0.406763\pi$$
0.288741 + 0.957407i $$0.406763\pi$$
$$608$$ 0 0
$$609$$ 16.3189 0.661277
$$610$$ 0 0
$$611$$ 11.5569 0.467543
$$612$$ 0 0
$$613$$ −40.8302 −1.64912 −0.824558 0.565777i $$-0.808576\pi$$
−0.824558 + 0.565777i $$0.808576\pi$$
$$614$$ 0 0
$$615$$ −2.83709 −0.114403
$$616$$ 0 0
$$617$$ 5.11383 0.205875 0.102937 0.994688i $$-0.467176\pi$$
0.102937 + 0.994688i $$0.467176\pi$$
$$618$$ 0 0
$$619$$ −11.5569 −0.464512 −0.232256 0.972655i $$-0.574611\pi$$
−0.232256 + 0.972655i $$0.574611\pi$$
$$620$$ 0 0
$$621$$ 4.83709 0.194106
$$622$$ 0 0
$$623$$ 3.16291 0.126719
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 9.67418 0.386350
$$628$$ 0 0
$$629$$ 12.1335 0.483794
$$630$$ 0 0
$$631$$ −35.2242 −1.40225 −0.701127 0.713036i $$-0.747319\pi$$
−0.701127 + 0.713036i $$0.747319\pi$$
$$632$$ 0 0
$$633$$ 18.2277 0.724484
$$634$$ 0 0
$$635$$ −13.4396 −0.533336
$$636$$ 0 0
$$637$$ 0.397442 0.0157472
$$638$$ 0 0
$$639$$ 6.71982 0.265832
$$640$$ 0 0
$$641$$ −21.9018 −0.865071 −0.432535 0.901617i $$-0.642381\pi$$
−0.432535 + 0.901617i $$0.642381\pi$$
$$642$$ 0 0
$$643$$ −7.50783 −0.296080 −0.148040 0.988981i $$-0.547296\pi$$
−0.148040 + 0.988981i $$0.547296\pi$$
$$644$$ 0 0
$$645$$ 11.1138 0.437607
$$646$$ 0 0
$$647$$ 14.0422 0.552056 0.276028 0.961150i $$-0.410982\pi$$
0.276028 + 0.961150i $$0.410982\pi$$
$$648$$ 0 0
$$649$$ 5.75859 0.226044
$$650$$ 0 0
$$651$$ −20.5535 −0.805554
$$652$$ 0 0
$$653$$ 7.99312 0.312795 0.156398 0.987694i $$-0.450012\pi$$
0.156398 + 0.987694i $$0.450012\pi$$
$$654$$ 0 0
$$655$$ −9.43965 −0.368838
$$656$$ 0 0
$$657$$ 9.11383 0.355564
$$658$$ 0 0
$$659$$ 25.3415 0.987164 0.493582 0.869699i $$-0.335687\pi$$
0.493582 + 0.869699i $$0.335687\pi$$
$$660$$ 0 0
$$661$$ 27.4396 1.06728 0.533639 0.845712i $$-0.320824\pi$$
0.533639 + 0.845712i $$0.320824\pi$$
$$662$$ 0 0
$$663$$ −2.83709 −0.110183
$$664$$ 0 0
$$665$$ −9.67418 −0.375149
$$666$$ 0 0
$$667$$ 29.0225 1.12376
$$668$$ 0 0
$$669$$ −10.1173 −0.391156
$$670$$ 0 0
$$671$$ 17.9578 0.693253
$$672$$ 0 0
$$673$$ 27.1070 1.04490 0.522448 0.852671i $$-0.325019\pi$$
0.522448 + 0.852671i $$0.325019\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −36.5957 −1.40649 −0.703243 0.710949i $$-0.748265\pi$$
−0.703243 + 0.710949i $$0.748265\pi$$
$$678$$ 0 0
$$679$$ −29.4750 −1.13115
$$680$$ 0 0
$$681$$ −11.3224 −0.433875
$$682$$ 0 0
$$683$$ 13.4656 0.515248 0.257624 0.966245i $$-0.417060\pi$$
0.257624 + 0.966245i $$0.417060\pi$$
$$684$$ 0 0
$$685$$ −1.76547 −0.0674550
$$686$$ 0 0
$$687$$ −6.23453 −0.237862
$$688$$ 0 0
$$689$$ 1.16291 0.0443033
$$690$$ 0 0
$$691$$ −29.5500 −1.12414 −0.562068 0.827091i $$-0.689994\pi$$
−0.562068 + 0.827091i $$0.689994\pi$$
$$692$$ 0 0
$$693$$ 7.39744 0.281006
$$694$$ 0 0
$$695$$ −6.27674 −0.238090
$$696$$ 0 0
$$697$$ −8.04908 −0.304881
$$698$$ 0 0
$$699$$ 6.83709 0.258603
$$700$$ 0 0
$$701$$ 43.6604 1.64903 0.824516 0.565839i $$-0.191448\pi$$
0.824516 + 0.565839i $$0.191448\pi$$
$$702$$ 0 0
$$703$$ −15.2120 −0.573731
$$704$$ 0 0
$$705$$ −11.5569 −0.435259
$$706$$ 0 0
$$707$$ 20.8724 0.784988
$$708$$ 0 0
$$709$$ −26.7880 −1.00604 −0.503022 0.864273i $$-0.667779\pi$$
−0.503022 + 0.864273i $$0.667779\pi$$
$$710$$ 0 0
$$711$$ −10.2767 −0.385408
$$712$$ 0 0
$$713$$ −36.5535 −1.36894
$$714$$ 0 0
$$715$$ −2.71982 −0.101716
$$716$$ 0 0
$$717$$ −1.28018 −0.0478091
$$718$$ 0 0
$$719$$ 34.8793 1.30078 0.650389 0.759601i $$-0.274606\pi$$
0.650389 + 0.759601i $$0.274606\pi$$
$$720$$ 0 0
$$721$$ −10.2414 −0.381410
$$722$$ 0 0
$$723$$ −6.00000 −0.223142
$$724$$ 0 0
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ −37.4396 −1.38856 −0.694280 0.719705i $$-0.744277\pi$$
−0.694280 + 0.719705i $$0.744277\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 31.5309 1.16621
$$732$$ 0 0
$$733$$ −47.1560 −1.74175 −0.870874 0.491506i $$-0.836446\pi$$
−0.870874 + 0.491506i $$0.836446\pi$$
$$734$$ 0 0
$$735$$ −0.397442 −0.0146599
$$736$$ 0 0
$$737$$ −5.12070 −0.188624
$$738$$ 0 0
$$739$$ 31.7914 1.16947 0.584734 0.811225i $$-0.301199\pi$$
0.584734 + 0.811225i $$0.301199\pi$$
$$740$$ 0 0
$$741$$ 3.55691 0.130667
$$742$$ 0 0
$$743$$ −16.6776 −0.611842 −0.305921 0.952057i $$-0.598964\pi$$
−0.305921 + 0.952057i $$0.598964\pi$$
$$744$$ 0 0
$$745$$ 20.8302 0.763160
$$746$$ 0 0
$$747$$ −2.11727 −0.0774667
$$748$$ 0 0
$$749$$ 34.2767 1.25244
$$750$$ 0 0
$$751$$ −16.1855 −0.590616 −0.295308 0.955402i $$-0.595422\pi$$
−0.295308 + 0.955402i $$0.595422\pi$$
$$752$$ 0 0
$$753$$ −18.2277 −0.664253
$$754$$ 0 0
$$755$$ 4.99656 0.181844
$$756$$ 0 0
$$757$$ 12.3258 0.447990 0.223995 0.974590i $$-0.428090\pi$$
0.223995 + 0.974590i $$0.428090\pi$$
$$758$$ 0 0
$$759$$ 13.1560 0.477534
$$760$$ 0 0
$$761$$ −0.00687569 −0.000249244 0 −0.000124622 1.00000i $$-0.500040\pi$$
−0.000124622 1.00000i $$0.500040\pi$$
$$762$$ 0 0
$$763$$ 31.1138 1.12640
$$764$$ 0 0
$$765$$ 2.83709 0.102575
$$766$$ 0 0
$$767$$ 2.11727 0.0764501
$$768$$ 0 0
$$769$$ −20.3258 −0.732968 −0.366484 0.930424i $$-0.619439\pi$$
−0.366484 + 0.930424i $$0.619439\pi$$
$$770$$ 0 0
$$771$$ 1.11383 0.0401136
$$772$$ 0 0
$$773$$ 9.90184 0.356144 0.178072 0.984017i $$-0.443014\pi$$
0.178072 + 0.984017i $$0.443014\pi$$
$$774$$ 0 0
$$775$$ −7.55691 −0.271452
$$776$$ 0 0
$$777$$ −11.6320 −0.417295
$$778$$ 0 0
$$779$$ 10.0913 0.361558
$$780$$ 0 0
$$781$$ 18.2767 0.653993
$$782$$ 0 0
$$783$$ 6.00000 0.214423
$$784$$ 0 0
$$785$$ −8.87930 −0.316916
$$786$$ 0 0
$$787$$ −36.3449 −1.29556 −0.647778 0.761829i $$-0.724302\pi$$
−0.647778 + 0.761829i $$0.724302\pi$$
$$788$$ 0 0
$$789$$ −8.00000 −0.284808
$$790$$ 0 0
$$791$$ −35.6673 −1.26818
$$792$$ 0 0
$$793$$ 6.60256 0.234464
$$794$$ 0 0
$$795$$ −1.16291 −0.0412442
$$796$$ 0 0
$$797$$ 18.8371 0.667244 0.333622 0.942707i $$-0.391729\pi$$
0.333622 + 0.942707i $$0.391729\pi$$
$$798$$ 0 0
$$799$$ −32.7880 −1.15996
$$800$$ 0 0
$$801$$ 1.16291 0.0410894
$$802$$ 0 0
$$803$$ 24.7880 0.874750
$$804$$ 0 0
$$805$$ −13.1560 −0.463689
$$806$$ 0 0
$$807$$ 15.6742 0.551757
$$808$$ 0 0
$$809$$ 32.2277 1.13306 0.566532 0.824040i $$-0.308285\pi$$
0.566532 + 0.824040i $$0.308285\pi$$
$$810$$ 0 0
$$811$$ 23.0034 0.807760 0.403880 0.914812i $$-0.367661\pi$$
0.403880 + 0.914812i $$0.367661\pi$$
$$812$$ 0 0
$$813$$ −0.443086 −0.0155397
$$814$$ 0 0
$$815$$ −13.8337 −0.484572
$$816$$ 0 0
$$817$$ −39.5309 −1.38301
$$818$$ 0 0
$$819$$ 2.71982 0.0950383
$$820$$ 0 0
$$821$$ 49.9372 1.74282 0.871410 0.490556i $$-0.163206\pi$$
0.871410 + 0.490556i $$0.163206\pi$$
$$822$$ 0 0
$$823$$ −28.2345 −0.984194 −0.492097 0.870540i $$-0.663769\pi$$
−0.492097 + 0.870540i $$0.663769\pi$$
$$824$$ 0 0
$$825$$ 2.71982 0.0946921
$$826$$ 0 0
$$827$$ 9.55004 0.332087 0.166044 0.986118i $$-0.446901\pi$$
0.166044 + 0.986118i $$0.446901\pi$$
$$828$$ 0 0
$$829$$ −37.9862 −1.31932 −0.659658 0.751565i $$-0.729299\pi$$
−0.659658 + 0.751565i $$0.729299\pi$$
$$830$$ 0 0
$$831$$ −4.87930 −0.169261
$$832$$ 0 0
$$833$$ −1.12758 −0.0390683
$$834$$ 0 0
$$835$$ −9.88273 −0.342006
$$836$$ 0 0
$$837$$ −7.55691 −0.261205
$$838$$ 0 0
$$839$$ 4.72670 0.163184 0.0815919 0.996666i $$-0.474000\pi$$
0.0815919 + 0.996666i $$0.474000\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ −9.11383 −0.313897
$$844$$ 0 0
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ −9.79832 −0.336674
$$848$$ 0 0
$$849$$ −33.3415 −1.14428
$$850$$ 0 0
$$851$$ −20.6870 −0.709140
$$852$$ 0 0
$$853$$ −48.3611 −1.65585 −0.827927 0.560836i $$-0.810480\pi$$
−0.827927 + 0.560836i $$0.810480\pi$$
$$854$$ 0 0
$$855$$ −3.55691 −0.121644
$$856$$ 0 0
$$857$$ 6.83709 0.233551 0.116775 0.993158i $$-0.462744\pi$$
0.116775 + 0.993158i $$0.462744\pi$$
$$858$$ 0 0
$$859$$ −12.6026 −0.429994 −0.214997 0.976615i $$-0.568974\pi$$
−0.214997 + 0.976615i $$0.568974\pi$$
$$860$$ 0 0
$$861$$ 7.71639 0.262974
$$862$$ 0 0
$$863$$ −8.20855 −0.279422 −0.139711 0.990192i $$-0.544617\pi$$
−0.139711 + 0.990192i $$0.544617\pi$$
$$864$$ 0 0
$$865$$ −13.1138 −0.445884
$$866$$ 0 0
$$867$$ −8.95092 −0.303989
$$868$$ 0 0
$$869$$ −27.9509 −0.948170
$$870$$ 0 0
$$871$$ −1.88273 −0.0637940
$$872$$ 0 0
$$873$$ −10.8371 −0.366780
$$874$$ 0 0
$$875$$ −2.71982 −0.0919468
$$876$$ 0 0
$$877$$ 13.5309 0.456907 0.228454 0.973555i $$-0.426633\pi$$
0.228454 + 0.973555i $$0.426633\pi$$
$$878$$ 0 0
$$879$$ −29.4328 −0.992743
$$880$$ 0 0
$$881$$ −9.34836 −0.314954 −0.157477 0.987523i $$-0.550336\pi$$
−0.157477 + 0.987523i $$0.550336\pi$$
$$882$$ 0 0
$$883$$ 55.1001 1.85427 0.927133 0.374733i $$-0.122266\pi$$
0.927133 + 0.374733i $$0.122266\pi$$
$$884$$ 0 0
$$885$$ −2.11727 −0.0711711
$$886$$ 0 0
$$887$$ −0.133492 −0.00448223 −0.00224112 0.999997i $$-0.500713\pi$$
−0.00224112 + 0.999997i $$0.500713\pi$$
$$888$$ 0 0
$$889$$ 36.5535 1.22596
$$890$$ 0 0
$$891$$ 2.71982 0.0911175
$$892$$ 0 0
$$893$$ 41.1070 1.37559
$$894$$ 0 0
$$895$$ 8.55348 0.285911
$$896$$ 0 0
$$897$$ 4.83709 0.161506
$$898$$ 0 0
$$899$$ −45.3415 −1.51222
$$900$$ 0 0
$$901$$ −3.29928 −0.109915
$$902$$ 0 0
$$903$$ −30.2277 −1.00591
$$904$$ 0 0
$$905$$ −3.72326 −0.123765
$$906$$ 0 0
$$907$$ 58.5466 1.94401 0.972004 0.234964i $$-0.0754973\pi$$
0.972004 + 0.234964i $$0.0754973\pi$$
$$908$$ 0 0
$$909$$ 7.67418 0.254537
$$910$$ 0 0
$$911$$ −50.4622 −1.67189 −0.835943 0.548816i $$-0.815079\pi$$
−0.835943 + 0.548816i $$0.815079\pi$$
$$912$$ 0 0
$$913$$ −5.75859 −0.190582
$$914$$ 0 0
$$915$$ −6.60256 −0.218274
$$916$$ 0 0
$$917$$ 25.6742 0.847836
$$918$$ 0 0
$$919$$ −56.9735 −1.87938 −0.939691 0.342026i $$-0.888887\pi$$
−0.939691 + 0.342026i $$0.888887\pi$$
$$920$$ 0 0
$$921$$ 21.8337 0.719443
$$922$$ 0 0
$$923$$ 6.71982 0.221186
$$924$$ 0 0
$$925$$ −4.27674 −0.140618
$$926$$ 0 0
$$927$$ −3.76547 −0.123674
$$928$$ 0 0
$$929$$ −36.5957 −1.20067 −0.600333 0.799750i $$-0.704965\pi$$
−0.600333 + 0.799750i $$0.704965\pi$$
$$930$$ 0 0
$$931$$ 1.41367 0.0463311
$$932$$ 0 0
$$933$$ −25.1070 −0.821965
$$934$$ 0 0
$$935$$ 7.71639 0.252353
$$936$$ 0 0
$$937$$ −47.1070 −1.53892 −0.769459 0.638697i $$-0.779474\pi$$
−0.769459 + 0.638697i $$0.779474\pi$$
$$938$$ 0 0
$$939$$ 8.22766 0.268499
$$940$$ 0 0
$$941$$ −36.3611 −1.18534 −0.592670 0.805446i $$-0.701926\pi$$
−0.592670 + 0.805446i $$0.701926\pi$$
$$942$$ 0 0
$$943$$ 13.7233 0.446891
$$944$$ 0 0
$$945$$ −2.71982 −0.0884759
$$946$$ 0 0
$$947$$ −44.5795 −1.44864 −0.724319 0.689465i $$-0.757846\pi$$
−0.724319 + 0.689465i $$0.757846\pi$$
$$948$$ 0 0
$$949$$ 9.11383 0.295847
$$950$$ 0 0
$$951$$ 27.6742 0.897397
$$952$$ 0 0
$$953$$ 19.8596 0.643317 0.321658 0.946856i $$-0.395760\pi$$
0.321658 + 0.946856i $$0.395760\pi$$
$$954$$ 0 0
$$955$$ −4.23453 −0.137026
$$956$$ 0 0
$$957$$ 16.3189 0.527517
$$958$$ 0 0
$$959$$ 4.80176 0.155057
$$960$$ 0 0
$$961$$ 26.1070 0.842160
$$962$$ 0 0
$$963$$ 12.6026 0.406112
$$964$$ 0 0
$$965$$ 23.3906 0.752969
$$966$$ 0 0
$$967$$ −47.4068 −1.52450 −0.762250 0.647283i $$-0.775905\pi$$
−0.762250 + 0.647283i $$0.775905\pi$$
$$968$$ 0 0
$$969$$ −10.0913 −0.324179
$$970$$ 0 0
$$971$$ 10.6448 0.341607 0.170803 0.985305i $$-0.445364\pi$$
0.170803 + 0.985305i $$0.445364\pi$$
$$972$$ 0 0
$$973$$ 17.0716 0.547291
$$974$$ 0 0
$$975$$ 1.00000 0.0320256
$$976$$ 0 0
$$977$$ −1.21199 −0.0387750 −0.0193875 0.999812i $$-0.506172\pi$$
−0.0193875 + 0.999812i $$0.506172\pi$$
$$978$$ 0 0
$$979$$ 3.16291 0.101087
$$980$$ 0 0
$$981$$ 11.4396 0.365240
$$982$$ 0 0
$$983$$ −51.8759 −1.65458 −0.827291 0.561773i $$-0.810119\pi$$
−0.827291 + 0.561773i $$0.810119\pi$$
$$984$$ 0 0
$$985$$ −14.5535 −0.463712
$$986$$ 0 0
$$987$$ 31.4328 1.00052
$$988$$ 0 0
$$989$$ −53.7586 −1.70942
$$990$$ 0 0
$$991$$ 21.6251 0.686944 0.343472 0.939163i $$-0.388397\pi$$
0.343472 + 0.939163i $$0.388397\pi$$
$$992$$ 0 0
$$993$$ 13.2311 0.419876
$$994$$ 0 0
$$995$$ −15.1138 −0.479141
$$996$$ 0 0
$$997$$ 23.2051 0.734913 0.367457 0.930041i $$-0.380229\pi$$
0.367457 + 0.930041i $$0.380229\pi$$
$$998$$ 0 0
$$999$$ −4.27674 −0.135310
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 3120.2.a.bj.1.3 3
3.2 odd 2 9360.2.a.dd.1.3 3
4.3 odd 2 195.2.a.e.1.3 3
12.11 even 2 585.2.a.n.1.1 3
20.3 even 4 975.2.c.i.274.1 6
20.7 even 4 975.2.c.i.274.6 6
20.19 odd 2 975.2.a.o.1.1 3
28.27 even 2 9555.2.a.bq.1.3 3
52.51 odd 2 2535.2.a.bc.1.1 3
60.23 odd 4 2925.2.c.w.2224.6 6
60.47 odd 4 2925.2.c.w.2224.1 6
60.59 even 2 2925.2.a.bh.1.3 3
156.155 even 2 7605.2.a.bx.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.3 3 4.3 odd 2
585.2.a.n.1.1 3 12.11 even 2
975.2.a.o.1.1 3 20.19 odd 2
975.2.c.i.274.1 6 20.3 even 4
975.2.c.i.274.6 6 20.7 even 4
2535.2.a.bc.1.1 3 52.51 odd 2
2925.2.a.bh.1.3 3 60.59 even 2
2925.2.c.w.2224.1 6 60.47 odd 4
2925.2.c.w.2224.6 6 60.23 odd 4
3120.2.a.bj.1.3 3 1.1 even 1 trivial
7605.2.a.bx.1.3 3 156.155 even 2
9360.2.a.dd.1.3 3 3.2 odd 2
9555.2.a.bq.1.3 3 28.27 even 2