# Properties

 Label 3969.2.a.y.1.4 Level $3969$ Weight $2$ Character 3969.1 Self dual yes Analytic conductor $31.693$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3969 = 3^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3969.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.6926245622$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6x^{2} + 4$$ x^4 - 6*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-0.874032$$ of defining polynomial Character $$\chi$$ $$=$$ 3969.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} +0.618034 q^{4} +0.874032 q^{5} -2.23607 q^{8} +O(q^{10})$$ $$q+1.61803 q^{2} +0.618034 q^{4} +0.874032 q^{5} -2.23607 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.74806 q^{13} -4.85410 q^{16} -3.16228 q^{17} -5.45052 q^{19} +0.540182 q^{20} +1.61803 q^{22} -2.76393 q^{23} -4.23607 q^{25} +2.82843 q^{26} -3.23607 q^{29} -0.333851 q^{31} -3.38197 q^{32} -5.11667 q^{34} +2.70820 q^{37} -8.81913 q^{38} -1.95440 q^{40} -10.0270 q^{41} -11.9443 q^{43} +0.618034 q^{44} -4.47214 q^{46} +12.5216 q^{47} -6.85410 q^{50} +1.08036 q^{52} -2.70820 q^{53} +0.874032 q^{55} -5.23607 q^{58} +8.15143 q^{59} +13.1893 q^{61} -0.540182 q^{62} +4.23607 q^{64} +1.52786 q^{65} -7.47214 q^{67} -1.95440 q^{68} -3.47214 q^{71} +13.0618 q^{73} +4.38197 q^{74} -3.36861 q^{76} -12.2361 q^{79} -4.24264 q^{80} -16.2241 q^{82} -7.19859 q^{83} -2.76393 q^{85} -19.3262 q^{86} -2.23607 q^{88} +6.32456 q^{89} -1.70820 q^{92} +20.2604 q^{94} -4.76393 q^{95} -9.35931 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 $$4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^11 - 6 * q^16 + 2 * q^22 - 20 * q^23 - 8 * q^25 - 4 * q^29 - 18 * q^32 - 16 * q^37 - 12 * q^43 - 2 * q^44 - 14 * q^50 + 16 * q^53 - 12 * q^58 + 8 * q^64 + 24 * q^65 - 12 * q^67 + 4 * q^71 + 22 * q^74 - 40 * q^79 - 20 * q^85 - 46 * q^86 + 20 * q^92 - 28 * q^95

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ 0 0
$$4$$ 0.618034 0.309017
$$5$$ 0.874032 0.390879 0.195440 0.980716i $$-0.437387\pi$$
0.195440 + 0.980716i $$0.437387\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −2.23607 −0.790569
$$9$$ 0 0
$$10$$ 1.41421 0.447214
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ 1.74806 0.484826 0.242413 0.970173i $$-0.422061\pi$$
0.242413 + 0.970173i $$0.422061\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ −3.16228 −0.766965 −0.383482 0.923548i $$-0.625275\pi$$
−0.383482 + 0.923548i $$0.625275\pi$$
$$18$$ 0 0
$$19$$ −5.45052 −1.25044 −0.625218 0.780450i $$-0.714990\pi$$
−0.625218 + 0.780450i $$0.714990\pi$$
$$20$$ 0.540182 0.120788
$$21$$ 0 0
$$22$$ 1.61803 0.344966
$$23$$ −2.76393 −0.576320 −0.288160 0.957582i $$-0.593043\pi$$
−0.288160 + 0.957582i $$0.593043\pi$$
$$24$$ 0 0
$$25$$ −4.23607 −0.847214
$$26$$ 2.82843 0.554700
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −3.23607 −0.600923 −0.300461 0.953794i $$-0.597141\pi$$
−0.300461 + 0.953794i $$0.597141\pi$$
$$30$$ 0 0
$$31$$ −0.333851 −0.0599613 −0.0299807 0.999550i $$-0.509545\pi$$
−0.0299807 + 0.999550i $$0.509545\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 0 0
$$34$$ −5.11667 −0.877502
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.70820 0.445226 0.222613 0.974907i $$-0.428541\pi$$
0.222613 + 0.974907i $$0.428541\pi$$
$$38$$ −8.81913 −1.43065
$$39$$ 0 0
$$40$$ −1.95440 −0.309017
$$41$$ −10.0270 −1.56596 −0.782978 0.622049i $$-0.786300\pi$$
−0.782978 + 0.622049i $$0.786300\pi$$
$$42$$ 0 0
$$43$$ −11.9443 −1.82148 −0.910742 0.412975i $$-0.864490\pi$$
−0.910742 + 0.412975i $$0.864490\pi$$
$$44$$ 0.618034 0.0931721
$$45$$ 0 0
$$46$$ −4.47214 −0.659380
$$47$$ 12.5216 1.82646 0.913231 0.407442i $$-0.133579\pi$$
0.913231 + 0.407442i $$0.133579\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ −6.85410 −0.969316
$$51$$ 0 0
$$52$$ 1.08036 0.149819
$$53$$ −2.70820 −0.372000 −0.186000 0.982550i $$-0.559553\pi$$
−0.186000 + 0.982550i $$0.559553\pi$$
$$54$$ 0 0
$$55$$ 0.874032 0.117854
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −5.23607 −0.687529
$$59$$ 8.15143 1.06123 0.530613 0.847614i $$-0.321962\pi$$
0.530613 + 0.847614i $$0.321962\pi$$
$$60$$ 0 0
$$61$$ 13.1893 1.68872 0.844358 0.535780i $$-0.179982\pi$$
0.844358 + 0.535780i $$0.179982\pi$$
$$62$$ −0.540182 −0.0686031
$$63$$ 0 0
$$64$$ 4.23607 0.529508
$$65$$ 1.52786 0.189508
$$66$$ 0 0
$$67$$ −7.47214 −0.912867 −0.456433 0.889758i $$-0.650873\pi$$
−0.456433 + 0.889758i $$0.650873\pi$$
$$68$$ −1.95440 −0.237005
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −3.47214 −0.412067 −0.206033 0.978545i $$-0.566056\pi$$
−0.206033 + 0.978545i $$0.566056\pi$$
$$72$$ 0 0
$$73$$ 13.0618 1.52876 0.764382 0.644763i $$-0.223044\pi$$
0.764382 + 0.644763i $$0.223044\pi$$
$$74$$ 4.38197 0.509393
$$75$$ 0 0
$$76$$ −3.36861 −0.386406
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −12.2361 −1.37667 −0.688333 0.725395i $$-0.741657\pi$$
−0.688333 + 0.725395i $$0.741657\pi$$
$$80$$ −4.24264 −0.474342
$$81$$ 0 0
$$82$$ −16.2241 −1.79165
$$83$$ −7.19859 −0.790148 −0.395074 0.918649i $$-0.629281\pi$$
−0.395074 + 0.918649i $$0.629281\pi$$
$$84$$ 0 0
$$85$$ −2.76393 −0.299791
$$86$$ −19.3262 −2.08400
$$87$$ 0 0
$$88$$ −2.23607 −0.238366
$$89$$ 6.32456 0.670402 0.335201 0.942147i $$-0.391196\pi$$
0.335201 + 0.942147i $$0.391196\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1.70820 −0.178093
$$93$$ 0 0
$$94$$ 20.2604 2.08970
$$95$$ −4.76393 −0.488769
$$96$$ 0 0
$$97$$ −9.35931 −0.950294 −0.475147 0.879906i $$-0.657605\pi$$
−0.475147 + 0.879906i $$0.657605\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.61803 −0.261803
$$101$$ −12.3941 −1.23326 −0.616628 0.787255i $$-0.711502\pi$$
−0.616628 + 0.787255i $$0.711502\pi$$
$$102$$ 0 0
$$103$$ −10.0270 −0.987991 −0.493996 0.869464i $$-0.664464\pi$$
−0.493996 + 0.869464i $$0.664464\pi$$
$$104$$ −3.90879 −0.383288
$$105$$ 0 0
$$106$$ −4.38197 −0.425614
$$107$$ −5.94427 −0.574654 −0.287327 0.957832i $$-0.592767\pi$$
−0.287327 + 0.957832i $$0.592767\pi$$
$$108$$ 0 0
$$109$$ 2.94427 0.282010 0.141005 0.990009i $$-0.454967\pi$$
0.141005 + 0.990009i $$0.454967\pi$$
$$110$$ 1.41421 0.134840
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −8.23607 −0.774784 −0.387392 0.921915i $$-0.626624\pi$$
−0.387392 + 0.921915i $$0.626624\pi$$
$$114$$ 0 0
$$115$$ −2.41577 −0.225271
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ 13.1893 1.21417
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 21.3407 1.93210
$$123$$ 0 0
$$124$$ −0.206331 −0.0185291
$$125$$ −8.07262 −0.722037
$$126$$ 0 0
$$127$$ 13.6525 1.21146 0.605731 0.795670i $$-0.292881\pi$$
0.605731 + 0.795670i $$0.292881\pi$$
$$128$$ 13.6180 1.20368
$$129$$ 0 0
$$130$$ 2.47214 0.216821
$$131$$ 1.87558 0.163871 0.0819353 0.996638i $$-0.473890\pi$$
0.0819353 + 0.996638i $$0.473890\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −12.0902 −1.04443
$$135$$ 0 0
$$136$$ 7.07107 0.606339
$$137$$ 3.76393 0.321574 0.160787 0.986989i $$-0.448597\pi$$
0.160787 + 0.986989i $$0.448597\pi$$
$$138$$ 0 0
$$139$$ 0.206331 0.0175008 0.00875038 0.999962i $$-0.497215\pi$$
0.00875038 + 0.999962i $$0.497215\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −5.61803 −0.471455
$$143$$ 1.74806 0.146180
$$144$$ 0 0
$$145$$ −2.82843 −0.234888
$$146$$ 21.1344 1.74909
$$147$$ 0 0
$$148$$ 1.67376 0.137582
$$149$$ −19.4721 −1.59522 −0.797610 0.603174i $$-0.793903\pi$$
−0.797610 + 0.603174i $$0.793903\pi$$
$$150$$ 0 0
$$151$$ −5.29180 −0.430640 −0.215320 0.976544i $$-0.569079\pi$$
−0.215320 + 0.976544i $$0.569079\pi$$
$$152$$ 12.1877 0.988556
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −0.291796 −0.0234376
$$156$$ 0 0
$$157$$ 2.16073 0.172445 0.0862224 0.996276i $$-0.472520\pi$$
0.0862224 + 0.996276i $$0.472520\pi$$
$$158$$ −19.7984 −1.57507
$$159$$ 0 0
$$160$$ −2.95595 −0.233688
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −8.23607 −0.645099 −0.322549 0.946553i $$-0.604540\pi$$
−0.322549 + 0.946553i $$0.604540\pi$$
$$164$$ −6.19704 −0.483907
$$165$$ 0 0
$$166$$ −11.6476 −0.904026
$$167$$ −8.69161 −0.672577 −0.336289 0.941759i $$-0.609172\pi$$
−0.336289 + 0.941759i $$0.609172\pi$$
$$168$$ 0 0
$$169$$ −9.94427 −0.764944
$$170$$ −4.47214 −0.342997
$$171$$ 0 0
$$172$$ −7.38197 −0.562870
$$173$$ 19.2588 1.46422 0.732110 0.681186i $$-0.238536\pi$$
0.732110 + 0.681186i $$0.238536\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −4.85410 −0.365892
$$177$$ 0 0
$$178$$ 10.2333 0.767022
$$179$$ 20.3607 1.52183 0.760914 0.648852i $$-0.224751\pi$$
0.760914 + 0.648852i $$0.224751\pi$$
$$180$$ 0 0
$$181$$ −22.9613 −1.70670 −0.853349 0.521340i $$-0.825432\pi$$
−0.853349 + 0.521340i $$0.825432\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 6.18034 0.455621
$$185$$ 2.36706 0.174029
$$186$$ 0 0
$$187$$ −3.16228 −0.231249
$$188$$ 7.73877 0.564408
$$189$$ 0 0
$$190$$ −7.70820 −0.559212
$$191$$ −4.41641 −0.319560 −0.159780 0.987153i $$-0.551078\pi$$
−0.159780 + 0.987153i $$0.551078\pi$$
$$192$$ 0 0
$$193$$ −16.4721 −1.18569 −0.592845 0.805316i $$-0.701995\pi$$
−0.592845 + 0.805316i $$0.701995\pi$$
$$194$$ −15.1437 −1.08725
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.4164 1.31211 0.656057 0.754711i $$-0.272223\pi$$
0.656057 + 0.754711i $$0.272223\pi$$
$$198$$ 0 0
$$199$$ 11.4412 0.811047 0.405524 0.914085i $$-0.367089\pi$$
0.405524 + 0.914085i $$0.367089\pi$$
$$200$$ 9.47214 0.669781
$$201$$ 0 0
$$202$$ −20.0540 −1.41100
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −8.76393 −0.612100
$$206$$ −16.2241 −1.13038
$$207$$ 0 0
$$208$$ −8.48528 −0.588348
$$209$$ −5.45052 −0.377021
$$210$$ 0 0
$$211$$ −3.29180 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$212$$ −1.67376 −0.114954
$$213$$ 0 0
$$214$$ −9.61803 −0.657475
$$215$$ −10.4397 −0.711980
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 4.76393 0.322654
$$219$$ 0 0
$$220$$ 0.540182 0.0364190
$$221$$ −5.52786 −0.371844
$$222$$ 0 0
$$223$$ −11.6476 −0.779978 −0.389989 0.920819i $$-0.627521\pi$$
−0.389989 + 0.920819i $$0.627521\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −13.3262 −0.886448
$$227$$ −23.0888 −1.53246 −0.766228 0.642568i $$-0.777869\pi$$
−0.766228 + 0.642568i $$0.777869\pi$$
$$228$$ 0 0
$$229$$ 10.0270 0.662604 0.331302 0.943525i $$-0.392512\pi$$
0.331302 + 0.943525i $$0.392512\pi$$
$$230$$ −3.90879 −0.257738
$$231$$ 0 0
$$232$$ 7.23607 0.475071
$$233$$ 22.9443 1.50313 0.751565 0.659659i $$-0.229299\pi$$
0.751565 + 0.659659i $$0.229299\pi$$
$$234$$ 0 0
$$235$$ 10.9443 0.713926
$$236$$ 5.03786 0.327937
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 13.9443 0.901980 0.450990 0.892529i $$-0.351071\pi$$
0.450990 + 0.892529i $$0.351071\pi$$
$$240$$ 0 0
$$241$$ 12.1089 0.780005 0.390002 0.920814i $$-0.372474\pi$$
0.390002 + 0.920814i $$0.372474\pi$$
$$242$$ −16.1803 −1.04011
$$243$$ 0 0
$$244$$ 8.15143 0.521842
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9.52786 −0.606243
$$248$$ 0.746512 0.0474036
$$249$$ 0 0
$$250$$ −13.0618 −0.826099
$$251$$ 2.70091 0.170480 0.0852399 0.996360i $$-0.472834\pi$$
0.0852399 + 0.996360i $$0.472834\pi$$
$$252$$ 0 0
$$253$$ −2.76393 −0.173767
$$254$$ 22.0902 1.38606
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −15.8902 −0.991203 −0.495602 0.868550i $$-0.665052\pi$$
−0.495602 + 0.868550i $$0.665052\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0.944272 0.0585613
$$261$$ 0 0
$$262$$ 3.03476 0.187488
$$263$$ 14.2361 0.877834 0.438917 0.898528i $$-0.355362\pi$$
0.438917 + 0.898528i $$0.355362\pi$$
$$264$$ 0 0
$$265$$ −2.36706 −0.145407
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −4.61803 −0.282091
$$269$$ 21.1344 1.28859 0.644293 0.764778i $$-0.277152\pi$$
0.644293 + 0.764778i $$0.277152\pi$$
$$270$$ 0 0
$$271$$ 27.5865 1.67576 0.837879 0.545856i $$-0.183795\pi$$
0.837879 + 0.545856i $$0.183795\pi$$
$$272$$ 15.3500 0.930732
$$273$$ 0 0
$$274$$ 6.09017 0.367921
$$275$$ −4.23607 −0.255445
$$276$$ 0 0
$$277$$ 5.76393 0.346321 0.173161 0.984894i $$-0.444602\pi$$
0.173161 + 0.984894i $$0.444602\pi$$
$$278$$ 0.333851 0.0200230
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −5.00000 −0.298275 −0.149137 0.988816i $$-0.547650\pi$$
−0.149137 + 0.988816i $$0.547650\pi$$
$$282$$ 0 0
$$283$$ −12.1089 −0.719801 −0.359901 0.932991i $$-0.617189\pi$$
−0.359901 + 0.932991i $$0.617189\pi$$
$$284$$ −2.14590 −0.127336
$$285$$ 0 0
$$286$$ 2.82843 0.167248
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −7.00000 −0.411765
$$290$$ −4.57649 −0.268741
$$291$$ 0 0
$$292$$ 8.07262 0.472414
$$293$$ 14.9374 0.872650 0.436325 0.899789i $$-0.356280\pi$$
0.436325 + 0.899789i $$0.356280\pi$$
$$294$$ 0 0
$$295$$ 7.12461 0.414811
$$296$$ −6.05573 −0.351982
$$297$$ 0 0
$$298$$ −31.5066 −1.82513
$$299$$ −4.83153 −0.279415
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −8.56231 −0.492705
$$303$$ 0 0
$$304$$ 26.4574 1.51744
$$305$$ 11.5279 0.660084
$$306$$ 0 0
$$307$$ −18.6398 −1.06383 −0.531915 0.846798i $$-0.678528\pi$$
−0.531915 + 0.846798i $$0.678528\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −0.472136 −0.0268155
$$311$$ 28.4118 1.61108 0.805542 0.592538i $$-0.201874\pi$$
0.805542 + 0.592538i $$0.201874\pi$$
$$312$$ 0 0
$$313$$ −13.3956 −0.757165 −0.378583 0.925567i $$-0.623588\pi$$
−0.378583 + 0.925567i $$0.623588\pi$$
$$314$$ 3.49613 0.197298
$$315$$ 0 0
$$316$$ −7.56231 −0.425413
$$317$$ 18.8328 1.05776 0.528878 0.848698i $$-0.322613\pi$$
0.528878 + 0.848698i $$0.322613\pi$$
$$318$$ 0 0
$$319$$ −3.23607 −0.181185
$$320$$ 3.70246 0.206974
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 17.2361 0.959040
$$324$$ 0 0
$$325$$ −7.40492 −0.410751
$$326$$ −13.3262 −0.738072
$$327$$ 0 0
$$328$$ 22.4211 1.23800
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2.00000 0.109930 0.0549650 0.998488i $$-0.482495\pi$$
0.0549650 + 0.998488i $$0.482495\pi$$
$$332$$ −4.44897 −0.244169
$$333$$ 0 0
$$334$$ −14.0633 −0.769511
$$335$$ −6.53089 −0.356820
$$336$$ 0 0
$$337$$ −17.1803 −0.935873 −0.467936 0.883762i $$-0.655002\pi$$
−0.467936 + 0.883762i $$0.655002\pi$$
$$338$$ −16.0902 −0.875190
$$339$$ 0 0
$$340$$ −1.70820 −0.0926404
$$341$$ −0.333851 −0.0180790
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 26.7082 1.44001
$$345$$ 0 0
$$346$$ 31.1614 1.67525
$$347$$ 21.9443 1.17803 0.589015 0.808122i $$-0.299516\pi$$
0.589015 + 0.808122i $$0.299516\pi$$
$$348$$ 0 0
$$349$$ 28.6668 1.53450 0.767250 0.641348i $$-0.221625\pi$$
0.767250 + 0.641348i $$0.221625\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.38197 −0.180259
$$353$$ 20.5154 1.09192 0.545962 0.837810i $$-0.316164\pi$$
0.545962 + 0.837810i $$0.316164\pi$$
$$354$$ 0 0
$$355$$ −3.03476 −0.161068
$$356$$ 3.90879 0.207165
$$357$$ 0 0
$$358$$ 32.9443 1.74116
$$359$$ 17.1803 0.906744 0.453372 0.891321i $$-0.350221\pi$$
0.453372 + 0.891321i $$0.350221\pi$$
$$360$$ 0 0
$$361$$ 10.7082 0.563590
$$362$$ −37.1521 −1.95267
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 11.4164 0.597562
$$366$$ 0 0
$$367$$ −1.74806 −0.0912482 −0.0456241 0.998959i $$-0.514528\pi$$
−0.0456241 + 0.998959i $$0.514528\pi$$
$$368$$ 13.4164 0.699379
$$369$$ 0 0
$$370$$ 3.82998 0.199111
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −9.18034 −0.475340 −0.237670 0.971346i $$-0.576384\pi$$
−0.237670 + 0.971346i $$0.576384\pi$$
$$374$$ −5.11667 −0.264577
$$375$$ 0 0
$$376$$ −27.9991 −1.44394
$$377$$ −5.65685 −0.291343
$$378$$ 0 0
$$379$$ −31.6525 −1.62588 −0.812939 0.582349i $$-0.802134\pi$$
−0.812939 + 0.582349i $$0.802134\pi$$
$$380$$ −2.94427 −0.151038
$$381$$ 0 0
$$382$$ −7.14590 −0.365616
$$383$$ −13.3168 −0.680457 −0.340229 0.940343i $$-0.610504\pi$$
−0.340229 + 0.940343i $$0.610504\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.6525 −1.35658
$$387$$ 0 0
$$388$$ −5.78437 −0.293657
$$389$$ 10.9443 0.554897 0.277448 0.960741i $$-0.410511\pi$$
0.277448 + 0.960741i $$0.410511\pi$$
$$390$$ 0 0
$$391$$ 8.74032 0.442017
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 29.7984 1.50122
$$395$$ −10.6947 −0.538110
$$396$$ 0 0
$$397$$ 28.8245 1.44666 0.723329 0.690504i $$-0.242611\pi$$
0.723329 + 0.690504i $$0.242611\pi$$
$$398$$ 18.5123 0.927938
$$399$$ 0 0
$$400$$ 20.5623 1.02812
$$401$$ 37.0689 1.85113 0.925566 0.378587i $$-0.123590\pi$$
0.925566 + 0.378587i $$0.123590\pi$$
$$402$$ 0 0
$$403$$ −0.583592 −0.0290708
$$404$$ −7.65996 −0.381097
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2.70820 0.134241
$$408$$ 0 0
$$409$$ −4.11512 −0.203480 −0.101740 0.994811i $$-0.532441\pi$$
−0.101740 + 0.994811i $$0.532441\pi$$
$$410$$ −14.1803 −0.700317
$$411$$ 0 0
$$412$$ −6.19704 −0.305306
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −6.29180 −0.308852
$$416$$ −5.91189 −0.289854
$$417$$ 0 0
$$418$$ −8.81913 −0.431358
$$419$$ −24.4242 −1.19320 −0.596600 0.802539i $$-0.703482\pi$$
−0.596600 + 0.802539i $$0.703482\pi$$
$$420$$ 0 0
$$421$$ 14.8885 0.725623 0.362812 0.931863i $$-0.381817\pi$$
0.362812 + 0.931863i $$0.381817\pi$$
$$422$$ −5.32624 −0.259277
$$423$$ 0 0
$$424$$ 6.05573 0.294092
$$425$$ 13.3956 0.649783
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −3.67376 −0.177578
$$429$$ 0 0
$$430$$ −16.8918 −0.814593
$$431$$ −23.4164 −1.12793 −0.563964 0.825799i $$-0.690724\pi$$
−0.563964 + 0.825799i $$0.690724\pi$$
$$432$$ 0 0
$$433$$ −20.5154 −0.985907 −0.492954 0.870056i $$-0.664083\pi$$
−0.492954 + 0.870056i $$0.664083\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 1.81966 0.0871459
$$437$$ 15.0649 0.720651
$$438$$ 0 0
$$439$$ −11.6963 −0.558232 −0.279116 0.960257i $$-0.590041\pi$$
−0.279116 + 0.960257i $$0.590041\pi$$
$$440$$ −1.95440 −0.0931721
$$441$$ 0 0
$$442$$ −8.94427 −0.425436
$$443$$ −2.29180 −0.108887 −0.0544433 0.998517i $$-0.517338\pi$$
−0.0544433 + 0.998517i $$0.517338\pi$$
$$444$$ 0 0
$$445$$ 5.52786 0.262046
$$446$$ −18.8461 −0.892391
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 31.4721 1.48526 0.742631 0.669701i $$-0.233578\pi$$
0.742631 + 0.669701i $$0.233578\pi$$
$$450$$ 0 0
$$451$$ −10.0270 −0.472154
$$452$$ −5.09017 −0.239421
$$453$$ 0 0
$$454$$ −37.3584 −1.75332
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 23.8328 1.11485 0.557426 0.830227i $$-0.311789\pi$$
0.557426 + 0.830227i $$0.311789\pi$$
$$458$$ 16.2241 0.758100
$$459$$ 0 0
$$460$$ −1.49302 −0.0696126
$$461$$ −18.1784 −0.846655 −0.423327 0.905977i $$-0.639138\pi$$
−0.423327 + 0.905977i $$0.639138\pi$$
$$462$$ 0 0
$$463$$ −36.8885 −1.71436 −0.857178 0.515020i $$-0.827784\pi$$
−0.857178 + 0.515020i $$0.827784\pi$$
$$464$$ 15.7082 0.729235
$$465$$ 0 0
$$466$$ 37.1246 1.71976
$$467$$ −40.3144 −1.86553 −0.932764 0.360488i $$-0.882610\pi$$
−0.932764 + 0.360488i $$0.882610\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 17.7082 0.816819
$$471$$ 0 0
$$472$$ −18.2272 −0.838973
$$473$$ −11.9443 −0.549198
$$474$$ 0 0
$$475$$ 23.0888 1.05939
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 22.5623 1.03198
$$479$$ 11.2349 0.513336 0.256668 0.966500i $$-0.417375\pi$$
0.256668 + 0.966500i $$0.417375\pi$$
$$480$$ 0 0
$$481$$ 4.73411 0.215857
$$482$$ 19.5927 0.892421
$$483$$ 0 0
$$484$$ −6.18034 −0.280925
$$485$$ −8.18034 −0.371450
$$486$$ 0 0
$$487$$ −13.9443 −0.631875 −0.315938 0.948780i $$-0.602319\pi$$
−0.315938 + 0.948780i $$0.602319\pi$$
$$488$$ −29.4922 −1.33505
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 22.0000 0.992846 0.496423 0.868081i $$-0.334646\pi$$
0.496423 + 0.868081i $$0.334646\pi$$
$$492$$ 0 0
$$493$$ 10.2333 0.460887
$$494$$ −15.4164 −0.693617
$$495$$ 0 0
$$496$$ 1.62054 0.0727646
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −13.8885 −0.621737 −0.310868 0.950453i $$-0.600620\pi$$
−0.310868 + 0.950453i $$0.600620\pi$$
$$500$$ −4.98915 −0.223122
$$501$$ 0 0
$$502$$ 4.37016 0.195050
$$503$$ 44.4295 1.98101 0.990507 0.137463i $$-0.0438947\pi$$
0.990507 + 0.137463i $$0.0438947\pi$$
$$504$$ 0 0
$$505$$ −10.8328 −0.482054
$$506$$ −4.47214 −0.198811
$$507$$ 0 0
$$508$$ 8.43769 0.374362
$$509$$ 10.1545 0.450092 0.225046 0.974348i $$-0.427747\pi$$
0.225046 + 0.974348i $$0.427747\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −5.29180 −0.233867
$$513$$ 0 0
$$514$$ −25.7109 −1.13406
$$515$$ −8.76393 −0.386185
$$516$$ 0 0
$$517$$ 12.5216 0.550699
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −3.41641 −0.149819
$$521$$ 21.8809 0.958620 0.479310 0.877646i $$-0.340887\pi$$
0.479310 + 0.877646i $$0.340887\pi$$
$$522$$ 0 0
$$523$$ 17.0981 0.747647 0.373823 0.927500i $$-0.378047\pi$$
0.373823 + 0.927500i $$0.378047\pi$$
$$524$$ 1.15917 0.0506388
$$525$$ 0 0
$$526$$ 23.0344 1.00435
$$527$$ 1.05573 0.0459882
$$528$$ 0 0
$$529$$ −15.3607 −0.667856
$$530$$ −3.82998 −0.166364
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −17.5279 −0.759216
$$534$$ 0 0
$$535$$ −5.19548 −0.224620
$$536$$ 16.7082 0.721684
$$537$$ 0 0
$$538$$ 34.1962 1.47430
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.5967 −0.799537 −0.399768 0.916616i $$-0.630909\pi$$
−0.399768 + 0.916616i $$0.630909\pi$$
$$542$$ 44.6358 1.91727
$$543$$ 0 0
$$544$$ 10.6947 0.458532
$$545$$ 2.57339 0.110232
$$546$$ 0 0
$$547$$ −2.70820 −0.115794 −0.0578972 0.998323i $$-0.518440\pi$$
−0.0578972 + 0.998323i $$0.518440\pi$$
$$548$$ 2.32624 0.0993720
$$549$$ 0 0
$$550$$ −6.85410 −0.292260
$$551$$ 17.6383 0.751415
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 9.32624 0.396234
$$555$$ 0 0
$$556$$ 0.127520 0.00540803
$$557$$ 20.7082 0.877435 0.438717 0.898625i $$-0.355433\pi$$
0.438717 + 0.898625i $$0.355433\pi$$
$$558$$ 0 0
$$559$$ −20.8794 −0.883103
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −8.09017 −0.341263
$$563$$ −34.2750 −1.44452 −0.722259 0.691623i $$-0.756896\pi$$
−0.722259 + 0.691623i $$0.756896\pi$$
$$564$$ 0 0
$$565$$ −7.19859 −0.302847
$$566$$ −19.5927 −0.823541
$$567$$ 0 0
$$568$$ 7.76393 0.325767
$$569$$ −42.4164 −1.77819 −0.889094 0.457724i $$-0.848665\pi$$
−0.889094 + 0.457724i $$0.848665\pi$$
$$570$$ 0 0
$$571$$ −40.1803 −1.68149 −0.840747 0.541427i $$-0.817884\pi$$
−0.840747 + 0.541427i $$0.817884\pi$$
$$572$$ 1.08036 0.0451722
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 11.7082 0.488266
$$576$$ 0 0
$$577$$ −12.5216 −0.521281 −0.260640 0.965436i $$-0.583934\pi$$
−0.260640 + 0.965436i $$0.583934\pi$$
$$578$$ −11.3262 −0.471109
$$579$$ 0 0
$$580$$ −1.74806 −0.0725844
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −2.70820 −0.112162
$$584$$ −29.2070 −1.20859
$$585$$ 0 0
$$586$$ 24.1692 0.998418
$$587$$ −2.90724 −0.119995 −0.0599973 0.998199i $$-0.519109\pi$$
−0.0599973 + 0.998199i $$0.519109\pi$$
$$588$$ 0 0
$$589$$ 1.81966 0.0749778
$$590$$ 11.5279 0.474595
$$591$$ 0 0
$$592$$ −13.1459 −0.540293
$$593$$ −27.9504 −1.14779 −0.573893 0.818930i $$-0.694568\pi$$
−0.573893 + 0.818930i $$0.694568\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −12.0344 −0.492950
$$597$$ 0 0
$$598$$ −7.81758 −0.319685
$$599$$ 39.8328 1.62752 0.813762 0.581198i $$-0.197416\pi$$
0.813762 + 0.581198i $$0.197416\pi$$
$$600$$ 0 0
$$601$$ −31.9867 −1.30477 −0.652383 0.757889i $$-0.726231\pi$$
−0.652383 + 0.757889i $$0.726231\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −3.27051 −0.133075
$$605$$ −8.74032 −0.355345
$$606$$ 0 0
$$607$$ 26.2511 1.06550 0.532749 0.846273i $$-0.321159\pi$$
0.532749 + 0.846273i $$0.321159\pi$$
$$608$$ 18.4335 0.747577
$$609$$ 0 0
$$610$$ 18.6525 0.755217
$$611$$ 21.8885 0.885516
$$612$$ 0 0
$$613$$ 39.1803 1.58248 0.791240 0.611506i $$-0.209436\pi$$
0.791240 + 0.611506i $$0.209436\pi$$
$$614$$ −30.1599 −1.21715
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 21.4164 0.862192 0.431096 0.902306i $$-0.358127\pi$$
0.431096 + 0.902306i $$0.358127\pi$$
$$618$$ 0 0
$$619$$ −35.5617 −1.42934 −0.714672 0.699460i $$-0.753424\pi$$
−0.714672 + 0.699460i $$0.753424\pi$$
$$620$$ −0.180340 −0.00724262
$$621$$ 0 0
$$622$$ 45.9712 1.84328
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 14.1246 0.564984
$$626$$ −21.6746 −0.866290
$$627$$ 0 0
$$628$$ 1.33540 0.0532883
$$629$$ −8.56409 −0.341473
$$630$$ 0 0
$$631$$ −11.3475 −0.451738 −0.225869 0.974158i $$-0.572522\pi$$
−0.225869 + 0.974158i $$0.572522\pi$$
$$632$$ 27.3607 1.08835
$$633$$ 0 0
$$634$$ 30.4721 1.21020
$$635$$ 11.9327 0.473535
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −5.23607 −0.207298
$$639$$ 0 0
$$640$$ 11.9026 0.470492
$$641$$ −12.5279 −0.494821 −0.247410 0.968911i $$-0.579580\pi$$
−0.247410 + 0.968911i $$0.579580\pi$$
$$642$$ 0 0
$$643$$ 28.5393 1.12548 0.562740 0.826634i $$-0.309747\pi$$
0.562740 + 0.826634i $$0.309747\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 27.8885 1.09726
$$647$$ 29.6684 1.16638 0.583192 0.812334i $$-0.301803\pi$$
0.583192 + 0.812334i $$0.301803\pi$$
$$648$$ 0 0
$$649$$ 8.15143 0.319972
$$650$$ −11.9814 −0.469950
$$651$$ 0 0
$$652$$ −5.09017 −0.199346
$$653$$ 3.29180 0.128818 0.0644090 0.997924i $$-0.479484\pi$$
0.0644090 + 0.997924i $$0.479484\pi$$
$$654$$ 0 0
$$655$$ 1.63932 0.0640535
$$656$$ 48.6722 1.90033
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −11.7639 −0.458258 −0.229129 0.973396i $$-0.573588\pi$$
−0.229129 + 0.973396i $$0.573588\pi$$
$$660$$ 0 0
$$661$$ −17.6383 −0.686049 −0.343024 0.939326i $$-0.611451\pi$$
−0.343024 + 0.939326i $$0.611451\pi$$
$$662$$ 3.23607 0.125773
$$663$$ 0 0
$$664$$ 16.0965 0.624667
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.94427 0.346324
$$668$$ −5.37171 −0.207838
$$669$$ 0 0
$$670$$ −10.5672 −0.408246
$$671$$ 13.1893 0.509167
$$672$$ 0 0
$$673$$ 9.65248 0.372076 0.186038 0.982543i $$-0.440435\pi$$
0.186038 + 0.982543i $$0.440435\pi$$
$$674$$ −27.7984 −1.07075
$$675$$ 0 0
$$676$$ −6.14590 −0.236381
$$677$$ −7.07107 −0.271763 −0.135882 0.990725i $$-0.543387\pi$$
−0.135882 + 0.990725i $$0.543387\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 6.18034 0.237005
$$681$$ 0 0
$$682$$ −0.540182 −0.0206846
$$683$$ −17.7639 −0.679718 −0.339859 0.940476i $$-0.610379\pi$$
−0.339859 + 0.940476i $$0.610379\pi$$
$$684$$ 0 0
$$685$$ 3.28980 0.125697
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 57.9787 2.21042
$$689$$ −4.73411 −0.180355
$$690$$ 0 0
$$691$$ 18.7974 0.715088 0.357544 0.933896i $$-0.383614\pi$$
0.357544 + 0.933896i $$0.383614\pi$$
$$692$$ 11.9026 0.452469
$$693$$ 0 0
$$694$$ 35.5066 1.34781
$$695$$ 0.180340 0.00684068
$$696$$ 0 0
$$697$$ 31.7082 1.20103
$$698$$ 46.3839 1.75566
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −13.1803 −0.497815 −0.248907 0.968527i $$-0.580071\pi$$
−0.248907 + 0.968527i $$0.580071\pi$$
$$702$$ 0 0
$$703$$ −14.7611 −0.556727
$$704$$ 4.23607 0.159653
$$705$$ 0 0
$$706$$ 33.1946 1.24930
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −5.65248 −0.212283 −0.106142 0.994351i $$-0.533850\pi$$
−0.106142 + 0.994351i $$0.533850\pi$$
$$710$$ −4.91034 −0.184282
$$711$$ 0 0
$$712$$ −14.1421 −0.529999
$$713$$ 0.922740 0.0345569
$$714$$ 0 0
$$715$$ 1.52786 0.0571389
$$716$$ 12.5836 0.470271
$$717$$ 0 0
$$718$$ 27.7984 1.03743
$$719$$ 17.8145 0.664368 0.332184 0.943215i $$-0.392214\pi$$
0.332184 + 0.943215i $$0.392214\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 17.3262 0.644816
$$723$$ 0 0
$$724$$ −14.1908 −0.527399
$$725$$ 13.7082 0.509110
$$726$$ 0 0
$$727$$ 7.02236 0.260445 0.130222 0.991485i $$-0.458431\pi$$
0.130222 + 0.991485i $$0.458431\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 18.4721 0.683684
$$731$$ 37.7711 1.39701
$$732$$ 0 0
$$733$$ −10.0270 −0.370356 −0.185178 0.982705i $$-0.559286\pi$$
−0.185178 + 0.982705i $$0.559286\pi$$
$$734$$ −2.82843 −0.104399
$$735$$ 0 0
$$736$$ 9.34752 0.344554
$$737$$ −7.47214 −0.275240
$$738$$ 0 0
$$739$$ −33.3607 −1.22719 −0.613596 0.789620i $$-0.710278\pi$$
−0.613596 + 0.789620i $$0.710278\pi$$
$$740$$ 1.46292 0.0537781
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −41.2492 −1.51329 −0.756644 0.653828i $$-0.773162\pi$$
−0.756644 + 0.653828i $$0.773162\pi$$
$$744$$ 0 0
$$745$$ −17.0193 −0.623538
$$746$$ −14.8541 −0.543847
$$747$$ 0 0
$$748$$ −1.95440 −0.0714598
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 30.3050 1.10584 0.552922 0.833233i $$-0.313513\pi$$
0.552922 + 0.833233i $$0.313513\pi$$
$$752$$ −60.7811 −2.21646
$$753$$ 0 0
$$754$$ −9.15298 −0.333332
$$755$$ −4.62520 −0.168328
$$756$$ 0 0
$$757$$ 16.0689 0.584034 0.292017 0.956413i $$-0.405674\pi$$
0.292017 + 0.956413i $$0.405674\pi$$
$$758$$ −51.2148 −1.86020
$$759$$ 0 0
$$760$$ 10.6525 0.386406
$$761$$ 1.87558 0.0679899 0.0339949 0.999422i $$-0.489177\pi$$
0.0339949 + 0.999422i $$0.489177\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −2.72949 −0.0987495
$$765$$ 0 0
$$766$$ −21.5471 −0.778527
$$767$$ 14.2492 0.514510
$$768$$ 0 0
$$769$$ −52.2958 −1.88583 −0.942917 0.333027i $$-0.891930\pi$$
−0.942917 + 0.333027i $$0.891930\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.1803 −0.366398
$$773$$ −14.3184 −0.514996 −0.257498 0.966279i $$-0.582898\pi$$
−0.257498 + 0.966279i $$0.582898\pi$$
$$774$$ 0 0
$$775$$ 1.41421 0.0508001
$$776$$ 20.9281 0.751274
$$777$$ 0 0
$$778$$ 17.7082 0.634870
$$779$$ 54.6525 1.95813
$$780$$ 0 0
$$781$$ −3.47214 −0.124243
$$782$$ 14.1421 0.505722
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.88854 0.0674050
$$786$$ 0 0
$$787$$ −34.1475 −1.21723 −0.608613 0.793467i $$-0.708274\pi$$
−0.608613 + 0.793467i $$0.708274\pi$$
$$788$$ 11.3820 0.405466
$$789$$ 0 0
$$790$$ −17.3044 −0.615663
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 23.0557 0.818733
$$794$$ 46.6389 1.65515
$$795$$ 0 0
$$796$$ 7.07107 0.250627
$$797$$ 27.2039 0.963612 0.481806 0.876278i $$-0.339981\pi$$
0.481806 + 0.876278i $$0.339981\pi$$
$$798$$ 0 0
$$799$$ −39.5967 −1.40083
$$800$$ 14.3262 0.506509
$$801$$ 0 0
$$802$$ 59.9787 2.11792
$$803$$ 13.0618 0.460940
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −0.944272 −0.0332606
$$807$$ 0 0
$$808$$ 27.7140 0.974975
$$809$$ 43.5410 1.53082 0.765410 0.643543i $$-0.222536\pi$$
0.765410 + 0.643543i $$0.222536\pi$$
$$810$$ 0 0
$$811$$ 31.5254 1.10701 0.553503 0.832847i $$-0.313291\pi$$
0.553503 + 0.832847i $$0.313291\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 4.38197 0.153588
$$815$$ −7.19859 −0.252156
$$816$$ 0 0
$$817$$ 65.1025 2.27765
$$818$$ −6.65841 −0.232806
$$819$$ 0 0
$$820$$ −5.41641 −0.189149
$$821$$ −33.1803 −1.15800 −0.579001 0.815327i $$-0.696557\pi$$
−0.579001 + 0.815327i $$0.696557\pi$$
$$822$$ 0 0
$$823$$ −9.41641 −0.328235 −0.164118 0.986441i $$-0.552478\pi$$
−0.164118 + 0.986441i $$0.552478\pi$$
$$824$$ 22.4211 0.781076
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −21.6525 −0.752930 −0.376465 0.926431i $$-0.622861\pi$$
−0.376465 + 0.926431i $$0.622861\pi$$
$$828$$ 0 0
$$829$$ 22.2148 0.771550 0.385775 0.922593i $$-0.373934\pi$$
0.385775 + 0.922593i $$0.373934\pi$$
$$830$$ −10.1803 −0.353365
$$831$$ 0 0
$$832$$ 7.40492 0.256719
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −7.59675 −0.262896
$$836$$ −3.36861 −0.116506
$$837$$ 0 0
$$838$$ −39.5192 −1.36517
$$839$$ −47.4643 −1.63865 −0.819324 0.573330i $$-0.805651\pi$$
−0.819324 + 0.573330i $$0.805651\pi$$
$$840$$ 0 0
$$841$$ −18.5279 −0.638892
$$842$$ 24.0902 0.830202
$$843$$ 0 0
$$844$$ −2.03444 −0.0700284
$$845$$ −8.69161 −0.299001
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 13.1459 0.451432
$$849$$ 0 0
$$850$$ 21.6746 0.743432
$$851$$ −7.48529 −0.256592
$$852$$ 0 0
$$853$$ −47.6405 −1.63118 −0.815590 0.578631i $$-0.803587\pi$$
−0.815590 + 0.578631i $$0.803587\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 13.2918 0.454304
$$857$$ −26.9675 −0.921191 −0.460596 0.887610i $$-0.652364\pi$$
−0.460596 + 0.887610i $$0.652364\pi$$
$$858$$ 0 0
$$859$$ 35.0215 1.19492 0.597459 0.801900i $$-0.296177\pi$$
0.597459 + 0.801900i $$0.296177\pi$$
$$860$$ −6.45207 −0.220014
$$861$$ 0 0
$$862$$ −37.8885 −1.29049
$$863$$ −22.3050 −0.759269 −0.379635 0.925136i $$-0.623950\pi$$
−0.379635 + 0.925136i $$0.623950\pi$$
$$864$$ 0 0
$$865$$ 16.8328 0.572333
$$866$$ −33.1946 −1.12800
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −12.2361 −0.415080
$$870$$ 0 0
$$871$$ −13.0618 −0.442581
$$872$$ −6.58359 −0.222949
$$873$$ 0 0
$$874$$ 24.3755 0.824513
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 9.76393 0.329705 0.164852 0.986318i $$-0.447285\pi$$
0.164852 + 0.986318i $$0.447285\pi$$
$$878$$ −18.9250 −0.638686
$$879$$ 0 0
$$880$$ −4.24264 −0.143019
$$881$$ −21.7534 −0.732890 −0.366445 0.930440i $$-0.619425\pi$$
−0.366445 + 0.930440i $$0.619425\pi$$
$$882$$ 0 0
$$883$$ 41.6525 1.40172 0.700859 0.713300i $$-0.252800\pi$$
0.700859 + 0.713300i $$0.252800\pi$$
$$884$$ −3.41641 −0.114906
$$885$$ 0 0
$$886$$ −3.70820 −0.124580
$$887$$ 14.1421 0.474846 0.237423 0.971406i $$-0.423697\pi$$
0.237423 + 0.971406i $$0.423697\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 8.94427 0.299813
$$891$$ 0 0
$$892$$ −7.19859 −0.241027
$$893$$ −68.2492 −2.28387
$$894$$ 0 0
$$895$$ 17.7959 0.594851
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 50.9230 1.69932
$$899$$ 1.08036 0.0360321
$$900$$ 0 0
$$901$$ 8.56409 0.285311
$$902$$ −16.2241 −0.540202
$$903$$ 0 0
$$904$$ 18.4164 0.612521
$$905$$ −20.0689 −0.667112
$$906$$ 0 0
$$907$$ 11.5410 0.383213 0.191607 0.981472i $$-0.438630\pi$$
0.191607 + 0.981472i $$0.438630\pi$$
$$908$$ −14.2697 −0.473555
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −40.2492 −1.33352 −0.666758 0.745274i $$-0.732319\pi$$
−0.666758 + 0.745274i $$0.732319\pi$$
$$912$$ 0 0
$$913$$ −7.19859 −0.238238
$$914$$ 38.5623 1.27553
$$915$$ 0 0
$$916$$ 6.19704 0.204756
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 19.1803 0.632701 0.316351 0.948642i $$-0.397542\pi$$
0.316351 + 0.948642i $$0.397542\pi$$
$$920$$ 5.40182 0.178093
$$921$$ 0 0
$$922$$ −29.4133 −0.968677
$$923$$ −6.06952 −0.199781
$$924$$ 0 0
$$925$$ −11.4721 −0.377202
$$926$$ −59.6869 −1.96143
$$927$$ 0 0
$$928$$ 10.9443 0.359263
$$929$$ 53.2974 1.74863 0.874315 0.485360i $$-0.161311\pi$$
0.874315 + 0.485360i $$0.161311\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 14.1803 0.464492
$$933$$ 0 0
$$934$$ −65.2301 −2.13439
$$935$$ −2.76393 −0.0903902
$$936$$ 0 0
$$937$$ −19.5440 −0.638473 −0.319237 0.947675i $$-0.603426\pi$$
−0.319237 + 0.947675i $$0.603426\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 6.76393 0.220615
$$941$$ −46.5902 −1.51880 −0.759399 0.650625i $$-0.774507\pi$$
−0.759399 + 0.650625i $$0.774507\pi$$
$$942$$ 0 0
$$943$$ 27.7140 0.902492
$$944$$ −39.5679 −1.28782
$$945$$ 0 0
$$946$$ −19.3262 −0.628350
$$947$$ −47.2361 −1.53497 −0.767483 0.641069i $$-0.778491\pi$$
−0.767483 + 0.641069i $$0.778491\pi$$
$$948$$ 0 0
$$949$$ 22.8328 0.741185
$$950$$ 37.3584 1.21207
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 30.5836 0.990700 0.495350 0.868694i $$-0.335040\pi$$
0.495350 + 0.868694i $$0.335040\pi$$
$$954$$ 0 0
$$955$$ −3.86008 −0.124909
$$956$$ 8.61803 0.278727
$$957$$ 0 0
$$958$$ 18.1784 0.587319
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −30.8885 −0.996405
$$962$$ 7.65996 0.246967
$$963$$ 0 0
$$964$$ 7.48373 0.241035
$$965$$ −14.3972 −0.463461
$$966$$ 0 0
$$967$$ 8.47214 0.272446 0.136223 0.990678i $$-0.456504\pi$$
0.136223 + 0.990678i $$0.456504\pi$$
$$968$$ 22.3607 0.718699
$$969$$ 0 0
$$970$$ −13.2361 −0.424985
$$971$$ −0.333851 −0.0107138 −0.00535689 0.999986i $$-0.501705\pi$$
−0.00535689 + 0.999986i $$0.501705\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −22.5623 −0.722943
$$975$$ 0 0
$$976$$ −64.0222 −2.04930
$$977$$ −31.3050 −1.00153 −0.500767 0.865582i $$-0.666949\pi$$
−0.500767 + 0.865582i $$0.666949\pi$$
$$978$$ 0 0
$$979$$ 6.32456 0.202134
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 35.5967 1.13594
$$983$$ −24.5030 −0.781524 −0.390762 0.920492i $$-0.627789\pi$$
−0.390762 + 0.920492i $$0.627789\pi$$
$$984$$ 0 0
$$985$$ 16.0965 0.512878
$$986$$ 16.5579 0.527311
$$987$$ 0 0
$$988$$ −5.88854 −0.187340
$$989$$ 33.0132 1.04976
$$990$$ 0 0
$$991$$ −6.41641 −0.203824 −0.101912 0.994793i $$-0.532496\pi$$
−0.101912 + 0.994793i $$0.532496\pi$$
$$992$$ 1.12907 0.0358480
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 10.0000 0.317021
$$996$$ 0 0
$$997$$ −45.1760 −1.43074 −0.715369 0.698746i $$-0.753742\pi$$
−0.715369 + 0.698746i $$0.753742\pi$$
$$998$$ −22.4721 −0.711343
$$999$$ 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 3969.2.a.y.1.4 yes 4
3.2 odd 2 3969.2.a.r.1.1 4
7.6 odd 2 inner 3969.2.a.y.1.3 yes 4
21.20 even 2 3969.2.a.r.1.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
3969.2.a.r.1.1 4 3.2 odd 2
3969.2.a.r.1.2 yes 4 21.20 even 2
3969.2.a.y.1.3 yes 4 7.6 odd 2 inner
3969.2.a.y.1.4 yes 4 1.1 even 1 trivial