# Properties

 Label 8550.2.a.cf.1.3 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 570) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +3.35026 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +3.35026 q^{7} -1.00000 q^{8} +1.61213 q^{11} -1.35026 q^{13} -3.35026 q^{14} +1.00000 q^{16} -6.96239 q^{17} -1.00000 q^{19} -1.61213 q^{22} +1.35026 q^{23} +1.35026 q^{26} +3.35026 q^{28} -3.61213 q^{29} -2.31265 q^{31} -1.00000 q^{32} +6.96239 q^{34} +11.2750 q^{37} +1.00000 q^{38} +3.35026 q^{41} -10.3127 q^{43} +1.61213 q^{44} -1.35026 q^{46} +4.57452 q^{47} +4.22425 q^{49} -1.35026 q^{52} -11.9248 q^{53} -3.35026 q^{56} +3.61213 q^{58} -1.03761 q^{59} +2.00000 q^{61} +2.31265 q^{62} +1.00000 q^{64} -9.92478 q^{67} -6.96239 q^{68} +0.775746 q^{71} -3.22425 q^{73} -11.2750 q^{74} -1.00000 q^{76} +5.40105 q^{77} +14.3127 q^{79} -3.35026 q^{82} -10.8872 q^{83} +10.3127 q^{86} -1.61213 q^{88} +2.57452 q^{89} -4.52373 q^{91} +1.35026 q^{92} -4.57452 q^{94} -1.16362 q^{97} -4.22425 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 4 q^{11} + 6 q^{13} + 3 q^{16} - 10 q^{17} - 3 q^{19} - 4 q^{22} - 6 q^{23} - 6 q^{26} - 10 q^{29} + 14 q^{31} - 3 q^{32} + 10 q^{34} + 2 q^{37} + 3 q^{38} - 10 q^{43} + 4 q^{44} + 6 q^{46} + 2 q^{47} + 11 q^{49} + 6 q^{52} - 14 q^{53} + 10 q^{58} - 14 q^{59} + 6 q^{61} - 14 q^{62} + 3 q^{64} - 8 q^{67} - 10 q^{68} + 4 q^{71} - 8 q^{73} - 2 q^{74} - 3 q^{76} - 24 q^{77} + 22 q^{79} + 10 q^{86} - 4 q^{88} - 4 q^{89} - 32 q^{91} - 6 q^{92} - 2 q^{94} - 6 q^{97} - 11 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 - 3 * q^8 + 4 * q^11 + 6 * q^13 + 3 * q^16 - 10 * q^17 - 3 * q^19 - 4 * q^22 - 6 * q^23 - 6 * q^26 - 10 * q^29 + 14 * q^31 - 3 * q^32 + 10 * q^34 + 2 * q^37 + 3 * q^38 - 10 * q^43 + 4 * q^44 + 6 * q^46 + 2 * q^47 + 11 * q^49 + 6 * q^52 - 14 * q^53 + 10 * q^58 - 14 * q^59 + 6 * q^61 - 14 * q^62 + 3 * q^64 - 8 * q^67 - 10 * q^68 + 4 * q^71 - 8 * q^73 - 2 * q^74 - 3 * q^76 - 24 * q^77 + 22 * q^79 + 10 * q^86 - 4 * q^88 - 4 * q^89 - 32 * q^91 - 6 * q^92 - 2 * q^94 - 6 * q^97 - 11 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.35026 1.26628 0.633140 0.774037i $$-0.281766\pi$$
0.633140 + 0.774037i $$0.281766\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.61213 0.486075 0.243037 0.970017i $$-0.421856\pi$$
0.243037 + 0.970017i $$0.421856\pi$$
$$12$$ 0 0
$$13$$ −1.35026 −0.374495 −0.187248 0.982313i $$-0.559957\pi$$
−0.187248 + 0.982313i $$0.559957\pi$$
$$14$$ −3.35026 −0.895395
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.96239 −1.68863 −0.844314 0.535849i $$-0.819992\pi$$
−0.844314 + 0.535849i $$0.819992\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −1.61213 −0.343707
$$23$$ 1.35026 0.281549 0.140775 0.990042i $$-0.455041\pi$$
0.140775 + 0.990042i $$0.455041\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.35026 0.264808
$$27$$ 0 0
$$28$$ 3.35026 0.633140
$$29$$ −3.61213 −0.670755 −0.335378 0.942084i $$-0.608864\pi$$
−0.335378 + 0.942084i $$0.608864\pi$$
$$30$$ 0 0
$$31$$ −2.31265 −0.415364 −0.207682 0.978196i $$-0.566592\pi$$
−0.207682 + 0.978196i $$0.566592\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 6.96239 1.19404
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 11.2750 1.85360 0.926802 0.375549i $$-0.122546\pi$$
0.926802 + 0.375549i $$0.122546\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.35026 0.523223 0.261611 0.965173i $$-0.415746\pi$$
0.261611 + 0.965173i $$0.415746\pi$$
$$42$$ 0 0
$$43$$ −10.3127 −1.57266 −0.786332 0.617804i $$-0.788023\pi$$
−0.786332 + 0.617804i $$0.788023\pi$$
$$44$$ 1.61213 0.243037
$$45$$ 0 0
$$46$$ −1.35026 −0.199085
$$47$$ 4.57452 0.667262 0.333631 0.942704i $$-0.391726\pi$$
0.333631 + 0.942704i $$0.391726\pi$$
$$48$$ 0 0
$$49$$ 4.22425 0.603465
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.35026 −0.187248
$$53$$ −11.9248 −1.63799 −0.818997 0.573798i $$-0.805470\pi$$
−0.818997 + 0.573798i $$0.805470\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.35026 −0.447698
$$57$$ 0 0
$$58$$ 3.61213 0.474295
$$59$$ −1.03761 −0.135085 −0.0675427 0.997716i $$-0.521516\pi$$
−0.0675427 + 0.997716i $$0.521516\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 2.31265 0.293707
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.92478 −1.21250 −0.606252 0.795272i $$-0.707328\pi$$
−0.606252 + 0.795272i $$0.707328\pi$$
$$68$$ −6.96239 −0.844314
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0.775746 0.0920641 0.0460321 0.998940i $$-0.485342\pi$$
0.0460321 + 0.998940i $$0.485342\pi$$
$$72$$ 0 0
$$73$$ −3.22425 −0.377370 −0.188685 0.982038i $$-0.560423\pi$$
−0.188685 + 0.982038i $$0.560423\pi$$
$$74$$ −11.2750 −1.31070
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 5.40105 0.615506
$$78$$ 0 0
$$79$$ 14.3127 1.61030 0.805149 0.593072i $$-0.202085\pi$$
0.805149 + 0.593072i $$0.202085\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −3.35026 −0.369975
$$83$$ −10.8872 −1.19502 −0.597511 0.801861i $$-0.703844\pi$$
−0.597511 + 0.801861i $$0.703844\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.3127 1.11204
$$87$$ 0 0
$$88$$ −1.61213 −0.171853
$$89$$ 2.57452 0.272898 0.136449 0.990647i $$-0.456431\pi$$
0.136449 + 0.990647i $$0.456431\pi$$
$$90$$ 0 0
$$91$$ −4.52373 −0.474216
$$92$$ 1.35026 0.140775
$$93$$ 0 0
$$94$$ −4.57452 −0.471825
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1.16362 −0.118148 −0.0590738 0.998254i $$-0.518815\pi$$
−0.0590738 + 0.998254i $$0.518815\pi$$
$$98$$ −4.22425 −0.426714
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.88717 −0.884306 −0.442153 0.896940i $$-0.645785\pi$$
−0.442153 + 0.896940i $$0.645785\pi$$
$$102$$ 0 0
$$103$$ −7.03761 −0.693436 −0.346718 0.937969i $$-0.612704\pi$$
−0.346718 + 0.937969i $$0.612704\pi$$
$$104$$ 1.35026 0.132404
$$105$$ 0 0
$$106$$ 11.9248 1.15824
$$107$$ −0.775746 −0.0749942 −0.0374971 0.999297i $$-0.511938\pi$$
−0.0374971 + 0.999297i $$0.511938\pi$$
$$108$$ 0 0
$$109$$ −20.1622 −1.93119 −0.965594 0.260052i $$-0.916260\pi$$
−0.965594 + 0.260052i $$0.916260\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 3.35026 0.316570
$$113$$ 11.1490 1.04881 0.524406 0.851468i $$-0.324287\pi$$
0.524406 + 0.851468i $$0.324287\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −3.61213 −0.335378
$$117$$ 0 0
$$118$$ 1.03761 0.0955199
$$119$$ −23.3258 −2.13827
$$120$$ 0 0
$$121$$ −8.40105 −0.763732
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ −2.31265 −0.207682
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −13.7381 −1.21906 −0.609531 0.792762i $$-0.708642\pi$$
−0.609531 + 0.792762i $$0.708642\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.61213 −0.490334 −0.245167 0.969481i $$-0.578843\pi$$
−0.245167 + 0.969481i $$0.578843\pi$$
$$132$$ 0 0
$$133$$ −3.35026 −0.290505
$$134$$ 9.92478 0.857370
$$135$$ 0 0
$$136$$ 6.96239 0.597020
$$137$$ −17.6629 −1.50904 −0.754522 0.656275i $$-0.772131\pi$$
−0.754522 + 0.656275i $$0.772131\pi$$
$$138$$ 0 0
$$139$$ 10.7005 0.907607 0.453803 0.891102i $$-0.350067\pi$$
0.453803 + 0.891102i $$0.350067\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −0.775746 −0.0650992
$$143$$ −2.17679 −0.182033
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3.22425 0.266841
$$147$$ 0 0
$$148$$ 11.2750 0.926802
$$149$$ −1.03761 −0.0850044 −0.0425022 0.999096i $$-0.513533\pi$$
−0.0425022 + 0.999096i $$0.513533\pi$$
$$150$$ 0 0
$$151$$ 1.16362 0.0946940 0.0473470 0.998879i $$-0.484923\pi$$
0.0473470 + 0.998879i $$0.484923\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ −5.40105 −0.435229
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −10.9624 −0.874894 −0.437447 0.899244i $$-0.644117\pi$$
−0.437447 + 0.899244i $$0.644117\pi$$
$$158$$ −14.3127 −1.13865
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4.52373 0.356520
$$162$$ 0 0
$$163$$ −21.0132 −1.64588 −0.822939 0.568129i $$-0.807667\pi$$
−0.822939 + 0.568129i $$0.807667\pi$$
$$164$$ 3.35026 0.261611
$$165$$ 0 0
$$166$$ 10.8872 0.845008
$$167$$ 9.92478 0.768002 0.384001 0.923333i $$-0.374546\pi$$
0.384001 + 0.923333i $$0.374546\pi$$
$$168$$ 0 0
$$169$$ −11.1768 −0.859753
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −10.3127 −0.786332
$$173$$ 14.6253 1.11194 0.555971 0.831202i $$-0.312347\pi$$
0.555971 + 0.831202i $$0.312347\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.61213 0.121519
$$177$$ 0 0
$$178$$ −2.57452 −0.192968
$$179$$ −11.7381 −0.877349 −0.438675 0.898646i $$-0.644552\pi$$
−0.438675 + 0.898646i $$0.644552\pi$$
$$180$$ 0 0
$$181$$ 21.4617 1.59523 0.797617 0.603164i $$-0.206094\pi$$
0.797617 + 0.603164i $$0.206094\pi$$
$$182$$ 4.52373 0.335321
$$183$$ 0 0
$$184$$ −1.35026 −0.0995426
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −11.2243 −0.820799
$$188$$ 4.57452 0.333631
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 21.2750 1.53941 0.769704 0.638401i $$-0.220404\pi$$
0.769704 + 0.638401i $$0.220404\pi$$
$$192$$ 0 0
$$193$$ 14.1622 1.01942 0.509709 0.860347i $$-0.329753\pi$$
0.509709 + 0.860347i $$0.329753\pi$$
$$194$$ 1.16362 0.0835430
$$195$$ 0 0
$$196$$ 4.22425 0.301732
$$197$$ 3.87399 0.276011 0.138005 0.990431i $$-0.455931\pi$$
0.138005 + 0.990431i $$0.455931\pi$$
$$198$$ 0 0
$$199$$ 3.47627 0.246426 0.123213 0.992380i $$-0.460680\pi$$
0.123213 + 0.992380i $$0.460680\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 8.88717 0.625299
$$203$$ −12.1016 −0.849364
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 7.03761 0.490334
$$207$$ 0 0
$$208$$ −1.35026 −0.0936238
$$209$$ −1.61213 −0.111513
$$210$$ 0 0
$$211$$ 9.92478 0.683250 0.341625 0.939836i $$-0.389023\pi$$
0.341625 + 0.939836i $$0.389023\pi$$
$$212$$ −11.9248 −0.818997
$$213$$ 0 0
$$214$$ 0.775746 0.0530289
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −7.74798 −0.525967
$$218$$ 20.1622 1.36556
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.40105 0.632383
$$222$$ 0 0
$$223$$ −7.03761 −0.471273 −0.235637 0.971841i $$-0.575718\pi$$
−0.235637 + 0.971841i $$0.575718\pi$$
$$224$$ −3.35026 −0.223849
$$225$$ 0 0
$$226$$ −11.1490 −0.741623
$$227$$ −14.5501 −0.965723 −0.482861 0.875697i $$-0.660402\pi$$
−0.482861 + 0.875697i $$0.660402\pi$$
$$228$$ 0 0
$$229$$ 11.4010 0.753402 0.376701 0.926335i $$-0.377058\pi$$
0.376701 + 0.926335i $$0.377058\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.61213 0.237148
$$233$$ 21.9149 1.43569 0.717847 0.696201i $$-0.245128\pi$$
0.717847 + 0.696201i $$0.245128\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1.03761 −0.0675427
$$237$$ 0 0
$$238$$ 23.3258 1.51199
$$239$$ 13.2750 0.858691 0.429345 0.903140i $$-0.358744\pi$$
0.429345 + 0.903140i $$0.358744\pi$$
$$240$$ 0 0
$$241$$ 21.3258 1.37372 0.686859 0.726791i $$-0.258989\pi$$
0.686859 + 0.726791i $$0.258989\pi$$
$$242$$ 8.40105 0.540040
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.35026 0.0859151
$$248$$ 2.31265 0.146853
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −16.3127 −1.02965 −0.514823 0.857297i $$-0.672142\pi$$
−0.514823 + 0.857297i $$0.672142\pi$$
$$252$$ 0 0
$$253$$ 2.17679 0.136854
$$254$$ 13.7381 0.862007
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −24.5501 −1.53139 −0.765696 0.643203i $$-0.777605\pi$$
−0.765696 + 0.643203i $$0.777605\pi$$
$$258$$ 0 0
$$259$$ 37.7743 2.34718
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 5.61213 0.346718
$$263$$ −30.3488 −1.87139 −0.935695 0.352810i $$-0.885226\pi$$
−0.935695 + 0.352810i $$0.885226\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 3.35026 0.205418
$$267$$ 0 0
$$268$$ −9.92478 −0.606252
$$269$$ −14.3127 −0.872658 −0.436329 0.899787i $$-0.643722\pi$$
−0.436329 + 0.899787i $$0.643722\pi$$
$$270$$ 0 0
$$271$$ −7.32582 −0.445012 −0.222506 0.974931i $$-0.571424\pi$$
−0.222506 + 0.974931i $$0.571424\pi$$
$$272$$ −6.96239 −0.422157
$$273$$ 0 0
$$274$$ 17.6629 1.06706
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −14.4387 −0.867535 −0.433767 0.901025i $$-0.642816\pi$$
−0.433767 + 0.901025i $$0.642816\pi$$
$$278$$ −10.7005 −0.641775
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 11.9756 0.714402 0.357201 0.934028i $$-0.383731\pi$$
0.357201 + 0.934028i $$0.383731\pi$$
$$282$$ 0 0
$$283$$ −24.4894 −1.45575 −0.727873 0.685712i $$-0.759491\pi$$
−0.727873 + 0.685712i $$0.759491\pi$$
$$284$$ 0.775746 0.0460321
$$285$$ 0 0
$$286$$ 2.17679 0.128716
$$287$$ 11.2243 0.662547
$$288$$ 0 0
$$289$$ 31.4749 1.85146
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −3.22425 −0.188685
$$293$$ 18.1016 1.05751 0.528753 0.848776i $$-0.322660\pi$$
0.528753 + 0.848776i $$0.322660\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −11.2750 −0.655348
$$297$$ 0 0
$$298$$ 1.03761 0.0601072
$$299$$ −1.82321 −0.105439
$$300$$ 0 0
$$301$$ −34.5501 −1.99143
$$302$$ −1.16362 −0.0669588
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −29.9248 −1.70790 −0.853949 0.520357i $$-0.825799\pi$$
−0.853949 + 0.520357i $$0.825799\pi$$
$$308$$ 5.40105 0.307753
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 21.2750 1.20640 0.603198 0.797591i $$-0.293893\pi$$
0.603198 + 0.797591i $$0.293893\pi$$
$$312$$ 0 0
$$313$$ −17.4010 −0.983565 −0.491783 0.870718i $$-0.663655\pi$$
−0.491783 + 0.870718i $$0.663655\pi$$
$$314$$ 10.9624 0.618643
$$315$$ 0 0
$$316$$ 14.3127 0.805149
$$317$$ 14.1016 0.792023 0.396012 0.918246i $$-0.370394\pi$$
0.396012 + 0.918246i $$0.370394\pi$$
$$318$$ 0 0
$$319$$ −5.82321 −0.326037
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −4.52373 −0.252098
$$323$$ 6.96239 0.387398
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 21.0132 1.16381
$$327$$ 0 0
$$328$$ −3.35026 −0.184987
$$329$$ 15.3258 0.844940
$$330$$ 0 0
$$331$$ 6.85097 0.376563 0.188282 0.982115i $$-0.439708\pi$$
0.188282 + 0.982115i $$0.439708\pi$$
$$332$$ −10.8872 −0.597511
$$333$$ 0 0
$$334$$ −9.92478 −0.543060
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −24.2374 −1.32030 −0.660148 0.751135i $$-0.729507\pi$$
−0.660148 + 0.751135i $$0.729507\pi$$
$$338$$ 11.1768 0.607937
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −3.72829 −0.201898
$$342$$ 0 0
$$343$$ −9.29948 −0.502125
$$344$$ 10.3127 0.556021
$$345$$ 0 0
$$346$$ −14.6253 −0.786261
$$347$$ 0.962389 0.0516637 0.0258319 0.999666i $$-0.491777\pi$$
0.0258319 + 0.999666i $$0.491777\pi$$
$$348$$ 0 0
$$349$$ −1.37470 −0.0735860 −0.0367930 0.999323i $$-0.511714\pi$$
−0.0367930 + 0.999323i $$0.511714\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1.61213 −0.0859267
$$353$$ 28.7367 1.52950 0.764751 0.644326i $$-0.222862\pi$$
0.764751 + 0.644326i $$0.222862\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 2.57452 0.136449
$$357$$ 0 0
$$358$$ 11.7381 0.620380
$$359$$ −31.1998 −1.64666 −0.823332 0.567561i $$-0.807887\pi$$
−0.823332 + 0.567561i $$0.807887\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −21.4617 −1.12800
$$363$$ 0 0
$$364$$ −4.52373 −0.237108
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −20.1260 −1.05057 −0.525285 0.850927i $$-0.676041\pi$$
−0.525285 + 0.850927i $$0.676041\pi$$
$$368$$ 1.35026 0.0703873
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −39.9511 −2.07416
$$372$$ 0 0
$$373$$ 17.9756 0.930739 0.465370 0.885116i $$-0.345921\pi$$
0.465370 + 0.885116i $$0.345921\pi$$
$$374$$ 11.2243 0.580392
$$375$$ 0 0
$$376$$ −4.57452 −0.235913
$$377$$ 4.87732 0.251195
$$378$$ 0 0
$$379$$ 1.67276 0.0859240 0.0429620 0.999077i $$-0.486321\pi$$
0.0429620 + 0.999077i $$0.486321\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −21.2750 −1.08853
$$383$$ 31.8496 1.62744 0.813718 0.581260i $$-0.197440\pi$$
0.813718 + 0.581260i $$0.197440\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −14.1622 −0.720837
$$387$$ 0 0
$$388$$ −1.16362 −0.0590738
$$389$$ 10.3371 0.524111 0.262056 0.965053i $$-0.415600\pi$$
0.262056 + 0.965053i $$0.415600\pi$$
$$390$$ 0 0
$$391$$ −9.40105 −0.475431
$$392$$ −4.22425 −0.213357
$$393$$ 0 0
$$394$$ −3.87399 −0.195169
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 31.4372 1.57779 0.788895 0.614528i $$-0.210654\pi$$
0.788895 + 0.614528i $$0.210654\pi$$
$$398$$ −3.47627 −0.174250
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.94921 −0.297090 −0.148545 0.988906i $$-0.547459\pi$$
−0.148545 + 0.988906i $$0.547459\pi$$
$$402$$ 0 0
$$403$$ 3.12268 0.155552
$$404$$ −8.88717 −0.442153
$$405$$ 0 0
$$406$$ 12.1016 0.600591
$$407$$ 18.1768 0.900990
$$408$$ 0 0
$$409$$ −30.9986 −1.53278 −0.766391 0.642375i $$-0.777949\pi$$
−0.766391 + 0.642375i $$0.777949\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −7.03761 −0.346718
$$413$$ −3.47627 −0.171056
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.35026 0.0662020
$$417$$ 0 0
$$418$$ 1.61213 0.0788517
$$419$$ 34.3390 1.67757 0.838785 0.544463i $$-0.183266\pi$$
0.838785 + 0.544463i $$0.183266\pi$$
$$420$$ 0 0
$$421$$ 22.8627 1.11426 0.557131 0.830425i $$-0.311902\pi$$
0.557131 + 0.830425i $$0.311902\pi$$
$$422$$ −9.92478 −0.483131
$$423$$ 0 0
$$424$$ 11.9248 0.579118
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6.70052 0.324261
$$428$$ −0.775746 −0.0374971
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −11.5975 −0.558634 −0.279317 0.960199i $$-0.590108\pi$$
−0.279317 + 0.960199i $$0.590108\pi$$
$$432$$ 0 0
$$433$$ −27.8350 −1.33766 −0.668832 0.743414i $$-0.733205\pi$$
−0.668832 + 0.743414i $$0.733205\pi$$
$$434$$ 7.74798 0.371915
$$435$$ 0 0
$$436$$ −20.1622 −0.965594
$$437$$ −1.35026 −0.0645918
$$438$$ 0 0
$$439$$ −38.7875 −1.85123 −0.925613 0.378471i $$-0.876450\pi$$
−0.925613 + 0.378471i $$0.876450\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −9.40105 −0.447162
$$443$$ −29.2144 −1.38802 −0.694009 0.719966i $$-0.744157\pi$$
−0.694009 + 0.719966i $$0.744157\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 7.03761 0.333241
$$447$$ 0 0
$$448$$ 3.35026 0.158285
$$449$$ 20.4993 0.967421 0.483711 0.875228i $$-0.339289\pi$$
0.483711 + 0.875228i $$0.339289\pi$$
$$450$$ 0 0
$$451$$ 5.40105 0.254325
$$452$$ 11.1490 0.524406
$$453$$ 0 0
$$454$$ 14.5501 0.682869
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −6.44851 −0.301648 −0.150824 0.988561i $$-0.548193\pi$$
−0.150824 + 0.988561i $$0.548193\pi$$
$$458$$ −11.4010 −0.532736
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 13.5125 0.629338 0.314669 0.949201i $$-0.398106\pi$$
0.314669 + 0.949201i $$0.398106\pi$$
$$462$$ 0 0
$$463$$ −6.20123 −0.288196 −0.144098 0.989563i $$-0.546028\pi$$
−0.144098 + 0.989563i $$0.546028\pi$$
$$464$$ −3.61213 −0.167689
$$465$$ 0 0
$$466$$ −21.9149 −1.01519
$$467$$ −16.5599 −0.766302 −0.383151 0.923686i $$-0.625161\pi$$
−0.383151 + 0.923686i $$0.625161\pi$$
$$468$$ 0 0
$$469$$ −33.2506 −1.53537
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 1.03761 0.0477599
$$473$$ −16.6253 −0.764432
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −23.3258 −1.06914
$$477$$ 0 0
$$478$$ −13.2750 −0.607186
$$479$$ 30.5256 1.39475 0.697376 0.716705i $$-0.254351\pi$$
0.697376 + 0.716705i $$0.254351\pi$$
$$480$$ 0 0
$$481$$ −15.2243 −0.694166
$$482$$ −21.3258 −0.971365
$$483$$ 0 0
$$484$$ −8.40105 −0.381866
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 2.51388 0.113915 0.0569574 0.998377i $$-0.481860\pi$$
0.0569574 + 0.998377i $$0.481860\pi$$
$$488$$ −2.00000 −0.0905357
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −1.46168 −0.0659648 −0.0329824 0.999456i $$-0.510501\pi$$
−0.0329824 + 0.999456i $$0.510501\pi$$
$$492$$ 0 0
$$493$$ 25.1490 1.13266
$$494$$ −1.35026 −0.0607511
$$495$$ 0 0
$$496$$ −2.31265 −0.103841
$$497$$ 2.59895 0.116579
$$498$$ 0 0
$$499$$ −5.55149 −0.248519 −0.124259 0.992250i $$-0.539656\pi$$
−0.124259 + 0.992250i $$0.539656\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 16.3127 0.728069
$$503$$ 6.90175 0.307734 0.153867 0.988092i $$-0.450827\pi$$
0.153867 + 0.988092i $$0.450827\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −2.17679 −0.0967703
$$507$$ 0 0
$$508$$ −13.7381 −0.609531
$$509$$ −8.76116 −0.388331 −0.194166 0.980969i $$-0.562200\pi$$
−0.194166 + 0.980969i $$0.562200\pi$$
$$510$$ 0 0
$$511$$ −10.8021 −0.477857
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 24.5501 1.08286
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 7.37470 0.324339
$$518$$ −37.7743 −1.65971
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −39.4518 −1.72842 −0.864208 0.503135i $$-0.832180\pi$$
−0.864208 + 0.503135i $$0.832180\pi$$
$$522$$ 0 0
$$523$$ 37.5026 1.63987 0.819937 0.572453i $$-0.194008\pi$$
0.819937 + 0.572453i $$0.194008\pi$$
$$524$$ −5.61213 −0.245167
$$525$$ 0 0
$$526$$ 30.3488 1.32327
$$527$$ 16.1016 0.701395
$$528$$ 0 0
$$529$$ −21.1768 −0.920730
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −3.35026 −0.145252
$$533$$ −4.52373 −0.195945
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 9.92478 0.428685
$$537$$ 0 0
$$538$$ 14.3127 0.617062
$$539$$ 6.81003 0.293329
$$540$$ 0 0
$$541$$ −30.7757 −1.32315 −0.661576 0.749878i $$-0.730112\pi$$
−0.661576 + 0.749878i $$0.730112\pi$$
$$542$$ 7.32582 0.314671
$$543$$ 0 0
$$544$$ 6.96239 0.298510
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1.77433 0.0758649 0.0379325 0.999280i $$-0.487923\pi$$
0.0379325 + 0.999280i $$0.487923\pi$$
$$548$$ −17.6629 −0.754522
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.61213 0.153882
$$552$$ 0 0
$$553$$ 47.9511 2.03909
$$554$$ 14.4387 0.613440
$$555$$ 0 0
$$556$$ 10.7005 0.453803
$$557$$ −8.90175 −0.377179 −0.188590 0.982056i $$-0.560392\pi$$
−0.188590 + 0.982056i $$0.560392\pi$$
$$558$$ 0 0
$$559$$ 13.9248 0.588955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −11.9756 −0.505159
$$563$$ −10.9525 −0.461595 −0.230797 0.973002i $$-0.574133\pi$$
−0.230797 + 0.973002i $$0.574133\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 24.4894 1.02937
$$567$$ 0 0
$$568$$ −0.775746 −0.0325496
$$569$$ 10.2012 0.427658 0.213829 0.976871i $$-0.431406\pi$$
0.213829 + 0.976871i $$0.431406\pi$$
$$570$$ 0 0
$$571$$ −25.6531 −1.07355 −0.536774 0.843726i $$-0.680357\pi$$
−0.536774 + 0.843726i $$0.680357\pi$$
$$572$$ −2.17679 −0.0910163
$$573$$ 0 0
$$574$$ −11.2243 −0.468491
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −6.44851 −0.268455 −0.134227 0.990951i $$-0.542855\pi$$
−0.134227 + 0.990951i $$0.542855\pi$$
$$578$$ −31.4749 −1.30918
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −36.4749 −1.51323
$$582$$ 0 0
$$583$$ −19.2243 −0.796187
$$584$$ 3.22425 0.133421
$$585$$ 0 0
$$586$$ −18.1016 −0.747769
$$587$$ −41.8397 −1.72691 −0.863455 0.504426i $$-0.831704\pi$$
−0.863455 + 0.504426i $$0.831704\pi$$
$$588$$ 0 0
$$589$$ 2.31265 0.0952911
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 11.2750 0.463401
$$593$$ 8.73672 0.358774 0.179387 0.983779i $$-0.442589\pi$$
0.179387 + 0.983779i $$0.442589\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −1.03761 −0.0425022
$$597$$ 0 0
$$598$$ 1.82321 0.0745565
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 30.6253 1.24923 0.624616 0.780932i $$-0.285255\pi$$
0.624616 + 0.780932i $$0.285255\pi$$
$$602$$ 34.5501 1.40816
$$603$$ 0 0
$$604$$ 1.16362 0.0473470
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2.51388 0.102035 0.0510176 0.998698i $$-0.483754\pi$$
0.0510176 + 0.998698i $$0.483754\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.17679 −0.249886
$$612$$ 0 0
$$613$$ 2.96239 0.119650 0.0598249 0.998209i $$-0.480946\pi$$
0.0598249 + 0.998209i $$0.480946\pi$$
$$614$$ 29.9248 1.20767
$$615$$ 0 0
$$616$$ −5.40105 −0.217614
$$617$$ −18.3371 −0.738223 −0.369112 0.929385i $$-0.620338\pi$$
−0.369112 + 0.929385i $$0.620338\pi$$
$$618$$ 0 0
$$619$$ −40.7269 −1.63695 −0.818476 0.574541i $$-0.805180\pi$$
−0.818476 + 0.574541i $$0.805180\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −21.2750 −0.853051
$$623$$ 8.62530 0.345565
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 17.4010 0.695486
$$627$$ 0 0
$$628$$ −10.9624 −0.437447
$$629$$ −78.5012 −3.13005
$$630$$ 0 0
$$631$$ −5.92478 −0.235862 −0.117931 0.993022i $$-0.537626\pi$$
−0.117931 + 0.993022i $$0.537626\pi$$
$$632$$ −14.3127 −0.569327
$$633$$ 0 0
$$634$$ −14.1016 −0.560045
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −5.70385 −0.225995
$$638$$ 5.82321 0.230543
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −19.0494 −0.752405 −0.376202 0.926537i $$-0.622770\pi$$
−0.376202 + 0.926537i $$0.622770\pi$$
$$642$$ 0 0
$$643$$ 1.06205 0.0418831 0.0209416 0.999781i $$-0.493334\pi$$
0.0209416 + 0.999781i $$0.493334\pi$$
$$644$$ 4.52373 0.178260
$$645$$ 0 0
$$646$$ −6.96239 −0.273932
$$647$$ 26.6497 1.04771 0.523855 0.851808i $$-0.324493\pi$$
0.523855 + 0.851808i $$0.324493\pi$$
$$648$$ 0 0
$$649$$ −1.67276 −0.0656616
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −21.0132 −0.822939
$$653$$ −9.64832 −0.377568 −0.188784 0.982019i $$-0.560455\pi$$
−0.188784 + 0.982019i $$0.560455\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 3.35026 0.130806
$$657$$ 0 0
$$658$$ −15.3258 −0.597463
$$659$$ −10.0654 −0.392091 −0.196046 0.980595i $$-0.562810\pi$$
−0.196046 + 0.980595i $$0.562810\pi$$
$$660$$ 0 0
$$661$$ 13.5633 0.527549 0.263775 0.964584i $$-0.415032\pi$$
0.263775 + 0.964584i $$0.415032\pi$$
$$662$$ −6.85097 −0.266270
$$663$$ 0 0
$$664$$ 10.8872 0.422504
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −4.87732 −0.188850
$$668$$ 9.92478 0.384001
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 3.22425 0.124471
$$672$$ 0 0
$$673$$ 5.93937 0.228946 0.114473 0.993426i $$-0.463482\pi$$
0.114473 + 0.993426i $$0.463482\pi$$
$$674$$ 24.2374 0.933591
$$675$$ 0 0
$$676$$ −11.1768 −0.429877
$$677$$ −24.9525 −0.959004 −0.479502 0.877541i $$-0.659183\pi$$
−0.479502 + 0.877541i $$0.659183\pi$$
$$678$$ 0 0
$$679$$ −3.89843 −0.149608
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 3.72829 0.142763
$$683$$ 45.6239 1.74575 0.872875 0.487944i $$-0.162253\pi$$
0.872875 + 0.487944i $$0.162253\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 9.29948 0.355056
$$687$$ 0 0
$$688$$ −10.3127 −0.393166
$$689$$ 16.1016 0.613421
$$690$$ 0 0
$$691$$ −11.7480 −0.446914 −0.223457 0.974714i $$-0.571734\pi$$
−0.223457 + 0.974714i $$0.571734\pi$$
$$692$$ 14.6253 0.555971
$$693$$ 0 0
$$694$$ −0.962389 −0.0365318
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −23.3258 −0.883529
$$698$$ 1.37470 0.0520331
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 43.8105 1.65470 0.827350 0.561686i $$-0.189847\pi$$
0.827350 + 0.561686i $$0.189847\pi$$
$$702$$ 0 0
$$703$$ −11.2750 −0.425246
$$704$$ 1.61213 0.0607593
$$705$$ 0 0
$$706$$ −28.7367 −1.08152
$$707$$ −29.7743 −1.11978
$$708$$ 0 0
$$709$$ −29.5975 −1.11156 −0.555779 0.831330i $$-0.687580\pi$$
−0.555779 + 0.831330i $$0.687580\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −2.57452 −0.0964840
$$713$$ −3.12268 −0.116945
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −11.7381 −0.438675
$$717$$ 0 0
$$718$$ 31.1998 1.16437
$$719$$ −7.45183 −0.277906 −0.138953 0.990299i $$-0.544374\pi$$
−0.138953 + 0.990299i $$0.544374\pi$$
$$720$$ 0 0
$$721$$ −23.5778 −0.878085
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ 21.4617 0.797617
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0.600863 0.0222848 0.0111424 0.999938i $$-0.496453\pi$$
0.0111424 + 0.999938i $$0.496453\pi$$
$$728$$ 4.52373 0.167661
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 71.8007 2.65564
$$732$$ 0 0
$$733$$ 36.0625 1.33200 0.666000 0.745952i $$-0.268005\pi$$
0.666000 + 0.745952i $$0.268005\pi$$
$$734$$ 20.1260 0.742865
$$735$$ 0 0
$$736$$ −1.35026 −0.0497713
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ −14.1768 −0.521502 −0.260751 0.965406i $$-0.583970\pi$$
−0.260751 + 0.965406i $$0.583970\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 39.9511 1.46665
$$743$$ −24.9986 −0.917109 −0.458555 0.888666i $$-0.651633\pi$$
−0.458555 + 0.888666i $$0.651633\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −17.9756 −0.658132
$$747$$ 0 0
$$748$$ −11.2243 −0.410399
$$749$$ −2.59895 −0.0949637
$$750$$ 0 0
$$751$$ −10.2111 −0.372608 −0.186304 0.982492i $$-0.559651\pi$$
−0.186304 + 0.982492i $$0.559651\pi$$
$$752$$ 4.57452 0.166815
$$753$$ 0 0
$$754$$ −4.87732 −0.177621
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 44.4847 1.61682 0.808412 0.588617i $$-0.200327\pi$$
0.808412 + 0.588617i $$0.200327\pi$$
$$758$$ −1.67276 −0.0607574
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.1490 −0.984152 −0.492076 0.870552i $$-0.663762\pi$$
−0.492076 + 0.870552i $$0.663762\pi$$
$$762$$ 0 0
$$763$$ −67.5487 −2.44543
$$764$$ 21.2750 0.769704
$$765$$ 0 0
$$766$$ −31.8496 −1.15077
$$767$$ 1.40105 0.0505889
$$768$$ 0 0
$$769$$ −31.4010 −1.13235 −0.566175 0.824285i $$-0.691578\pi$$
−0.566175 + 0.824285i $$0.691578\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 14.1622 0.509709
$$773$$ −4.02635 −0.144818 −0.0724088 0.997375i $$-0.523069\pi$$
−0.0724088 + 0.997375i $$0.523069\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 1.16362 0.0417715
$$777$$ 0 0
$$778$$ −10.3371 −0.370603
$$779$$ −3.35026 −0.120036
$$780$$ 0 0
$$781$$ 1.25060 0.0447500
$$782$$ 9.40105 0.336181
$$783$$ 0 0
$$784$$ 4.22425 0.150866
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 18.2981 0.652255 0.326128 0.945326i $$-0.394256\pi$$
0.326128 + 0.945326i $$0.394256\pi$$
$$788$$ 3.87399 0.138005
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 37.3522 1.32809
$$792$$ 0 0
$$793$$ −2.70052 −0.0958984
$$794$$ −31.4372 −1.11567
$$795$$ 0 0
$$796$$ 3.47627 0.123213
$$797$$ 8.55008 0.302859 0.151430 0.988468i $$-0.451612\pi$$
0.151430 + 0.988468i $$0.451612\pi$$
$$798$$ 0 0
$$799$$ −31.8496 −1.12676
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 5.94921 0.210074
$$803$$ −5.19791 −0.183430
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3.12268 −0.109992
$$807$$ 0 0
$$808$$ 8.88717 0.312649
$$809$$ −50.7269 −1.78346 −0.891731 0.452566i $$-0.850509\pi$$
−0.891731 + 0.452566i $$0.850509\pi$$
$$810$$ 0 0
$$811$$ 10.3272 0.362638 0.181319 0.983424i $$-0.441963\pi$$
0.181319 + 0.983424i $$0.441963\pi$$
$$812$$ −12.1016 −0.424682
$$813$$ 0 0
$$814$$ −18.1768 −0.637096
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 10.3127 0.360794
$$818$$ 30.9986 1.08384
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −36.5139 −1.27434 −0.637172 0.770722i $$-0.719896\pi$$
−0.637172 + 0.770722i $$0.719896\pi$$
$$822$$ 0 0
$$823$$ 23.0982 0.805154 0.402577 0.915386i $$-0.368115\pi$$
0.402577 + 0.915386i $$0.368115\pi$$
$$824$$ 7.03761 0.245167
$$825$$ 0 0
$$826$$ 3.47627 0.120955
$$827$$ 3.37470 0.117350 0.0586749 0.998277i $$-0.481312\pi$$
0.0586749 + 0.998277i $$0.481312\pi$$
$$828$$ 0 0
$$829$$ 22.6399 0.786316 0.393158 0.919471i $$-0.371383\pi$$
0.393158 + 0.919471i $$0.371383\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.35026 −0.0468119
$$833$$ −29.4109 −1.01903
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −1.61213 −0.0557566
$$837$$ 0 0
$$838$$ −34.3390 −1.18622
$$839$$ 9.02776 0.311673 0.155836 0.987783i $$-0.450193\pi$$
0.155836 + 0.987783i $$0.450193\pi$$
$$840$$ 0 0
$$841$$ −15.9525 −0.550088
$$842$$ −22.8627 −0.787902
$$843$$ 0 0
$$844$$ 9.92478 0.341625
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −28.1457 −0.967098
$$848$$ −11.9248 −0.409499
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 15.2243 0.521881
$$852$$ 0 0
$$853$$ −43.1852 −1.47863 −0.739317 0.673358i $$-0.764851\pi$$
−0.739317 + 0.673358i $$0.764851\pi$$
$$854$$ −6.70052 −0.229287
$$855$$ 0 0
$$856$$ 0.775746 0.0265145
$$857$$ 33.0249 1.12811 0.564055 0.825737i $$-0.309241\pi$$
0.564055 + 0.825737i $$0.309241\pi$$
$$858$$ 0 0
$$859$$ 15.1754 0.517777 0.258889 0.965907i $$-0.416644\pi$$
0.258889 + 0.965907i $$0.416644\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 11.5975 0.395014
$$863$$ 39.4763 1.34379 0.671894 0.740647i $$-0.265481\pi$$
0.671894 + 0.740647i $$0.265481\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 27.8350 0.945871
$$867$$ 0 0
$$868$$ −7.74798 −0.262984
$$869$$ 23.0738 0.782725
$$870$$ 0 0
$$871$$ 13.4010 0.454077
$$872$$ 20.1622 0.682778
$$873$$ 0 0
$$874$$ 1.35026 0.0456733
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −52.5256 −1.77366 −0.886832 0.462091i $$-0.847099\pi$$
−0.886832 + 0.462091i $$0.847099\pi$$
$$878$$ 38.7875 1.30901
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 41.8007 1.40830 0.704150 0.710051i $$-0.251328\pi$$
0.704150 + 0.710051i $$0.251328\pi$$
$$882$$ 0 0
$$883$$ 36.9643 1.24395 0.621974 0.783038i $$-0.286331\pi$$
0.621974 + 0.783038i $$0.286331\pi$$
$$884$$ 9.40105 0.316191
$$885$$ 0 0
$$886$$ 29.2144 0.981477
$$887$$ 36.0724 1.21119 0.605596 0.795772i $$-0.292935\pi$$
0.605596 + 0.795772i $$0.292935\pi$$
$$888$$ 0 0
$$889$$ −46.0263 −1.54367
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −7.03761 −0.235637
$$893$$ −4.57452 −0.153080
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −3.35026 −0.111924
$$897$$ 0 0
$$898$$ −20.4993 −0.684070
$$899$$ 8.35359 0.278608
$$900$$ 0 0
$$901$$ 83.0249 2.76596
$$902$$ −5.40105 −0.179835
$$903$$ 0 0
$$904$$ −11.1490 −0.370811
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 57.6239 1.91337 0.956685 0.291126i $$-0.0940297\pi$$
0.956685 + 0.291126i $$0.0940297\pi$$
$$908$$ −14.5501 −0.482861
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.9234 −1.02454 −0.512268 0.858825i $$-0.671195\pi$$
−0.512268 + 0.858825i $$0.671195\pi$$
$$912$$ 0 0
$$913$$ −17.5515 −0.580870
$$914$$ 6.44851 0.213298
$$915$$ 0 0
$$916$$ 11.4010 0.376701
$$917$$ −18.8021 −0.620900
$$918$$ 0 0
$$919$$ 33.7743 1.11411 0.557056 0.830475i $$-0.311931\pi$$
0.557056 + 0.830475i $$0.311931\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −13.5125 −0.445009
$$923$$ −1.04746 −0.0344776
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 6.20123 0.203785
$$927$$ 0 0
$$928$$ 3.61213 0.118574
$$929$$ −48.6516 −1.59621 −0.798104 0.602519i $$-0.794164\pi$$
−0.798104 + 0.602519i $$0.794164\pi$$
$$930$$ 0 0
$$931$$ −4.22425 −0.138444
$$932$$ 21.9149 0.717847
$$933$$ 0 0
$$934$$ 16.5599 0.541857
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 18.7005 0.610919 0.305460 0.952205i $$-0.401190\pi$$
0.305460 + 0.952205i $$0.401190\pi$$
$$938$$ 33.2506 1.08567
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −38.4142 −1.25227 −0.626134 0.779716i $$-0.715364\pi$$
−0.626134 + 0.779716i $$0.715364\pi$$
$$942$$ 0 0
$$943$$ 4.52373 0.147313
$$944$$ −1.03761 −0.0337714
$$945$$ 0 0
$$946$$ 16.6253 0.540535
$$947$$ 44.0362 1.43098 0.715492 0.698621i $$-0.246203\pi$$
0.715492 + 0.698621i $$0.246203\pi$$
$$948$$ 0 0
$$949$$ 4.35359 0.141323
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 23.3258 0.755994
$$953$$ −38.0724 −1.23329 −0.616643 0.787243i $$-0.711508\pi$$
−0.616643 + 0.787243i $$0.711508\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 13.2750 0.429345
$$957$$ 0 0
$$958$$ −30.5256 −0.986239
$$959$$ −59.1754 −1.91087
$$960$$ 0 0
$$961$$ −25.6516 −0.827473
$$962$$ 15.2243 0.490850
$$963$$ 0 0
$$964$$ 21.3258 0.686859
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −42.5256 −1.36753 −0.683766 0.729701i $$-0.739659\pi$$
−0.683766 + 0.729701i $$0.739659\pi$$
$$968$$ 8.40105 0.270020
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 22.1378 0.710435 0.355217 0.934784i $$-0.384407\pi$$
0.355217 + 0.934784i $$0.384407\pi$$
$$972$$ 0 0
$$973$$ 35.8496 1.14928
$$974$$ −2.51388 −0.0805499
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ −21.7480 −0.695780 −0.347890 0.937535i $$-0.613102\pi$$
−0.347890 + 0.937535i $$0.613102\pi$$
$$978$$ 0 0
$$979$$ 4.15045 0.132649
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 1.46168 0.0466441
$$983$$ 9.04746 0.288569 0.144285 0.989536i $$-0.453912\pi$$
0.144285 + 0.989536i $$0.453912\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −25.1490 −0.800908
$$987$$ 0 0
$$988$$ 1.35026 0.0429575
$$989$$ −13.9248 −0.442782
$$990$$ 0 0
$$991$$ −26.0606 −0.827843 −0.413922 0.910313i $$-0.635841\pi$$
−0.413922 + 0.910313i $$0.635841\pi$$
$$992$$ 2.31265 0.0734267
$$993$$ 0 0
$$994$$ −2.59895 −0.0824338
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −28.2130 −0.893514 −0.446757 0.894655i $$-0.647421\pi$$
−0.446757 + 0.894655i $$0.647421\pi$$
$$998$$ 5.55149 0.175729
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8550.2.a.cf.1.3 3
3.2 odd 2 2850.2.a.bn.1.3 3
5.2 odd 4 1710.2.d.e.1369.3 6
5.3 odd 4 1710.2.d.e.1369.6 6
5.4 even 2 8550.2.a.cr.1.1 3
15.2 even 4 570.2.d.d.229.4 yes 6
15.8 even 4 570.2.d.d.229.1 6
15.14 odd 2 2850.2.a.bk.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.d.d.229.1 6 15.8 even 4
570.2.d.d.229.4 yes 6 15.2 even 4
1710.2.d.e.1369.3 6 5.2 odd 4
1710.2.d.e.1369.6 6 5.3 odd 4
2850.2.a.bk.1.1 3 15.14 odd 2
2850.2.a.bn.1.3 3 3.2 odd 2
8550.2.a.cf.1.3 3 1.1 even 1 trivial
8550.2.a.cr.1.1 3 5.4 even 2