# Properties

 Label 585.2.a.m.1.2 Level $585$ Weight $2$ Character 585.1 Self dual yes Analytic conductor $4.671$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$585 = 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 585.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.67124851824$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 585.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} +O(q^{10})$$ $$q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +4.82843 q^{7} +4.41421 q^{8} -2.41421 q^{10} -3.41421 q^{11} -1.00000 q^{13} +11.6569 q^{14} +3.00000 q^{16} -0.828427 q^{17} +0.585786 q^{19} -3.82843 q^{20} -8.24264 q^{22} -1.41421 q^{23} +1.00000 q^{25} -2.41421 q^{26} +18.4853 q^{28} +5.65685 q^{29} +1.75736 q^{31} -1.58579 q^{32} -2.00000 q^{34} -4.82843 q^{35} -8.48528 q^{37} +1.41421 q^{38} -4.41421 q^{40} +3.17157 q^{41} -11.0711 q^{43} -13.0711 q^{44} -3.41421 q^{46} +4.82843 q^{47} +16.3137 q^{49} +2.41421 q^{50} -3.82843 q^{52} -2.48528 q^{53} +3.41421 q^{55} +21.3137 q^{56} +13.6569 q^{58} -1.75736 q^{59} -8.00000 q^{61} +4.24264 q^{62} -9.82843 q^{64} +1.00000 q^{65} -2.00000 q^{67} -3.17157 q^{68} -11.6569 q^{70} -11.8995 q^{71} +8.48528 q^{73} -20.4853 q^{74} +2.24264 q^{76} -16.4853 q^{77} -8.48528 q^{79} -3.00000 q^{80} +7.65685 q^{82} +3.17157 q^{83} +0.828427 q^{85} -26.7279 q^{86} -15.0711 q^{88} -6.00000 q^{89} -4.82843 q^{91} -5.41421 q^{92} +11.6569 q^{94} -0.585786 q^{95} -7.65685 q^{97} +39.3848 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} - 2 q^{13} + 12 q^{14} + 6 q^{16} + 4 q^{17} + 4 q^{19} - 2 q^{20} - 8 q^{22} + 2 q^{25} - 2 q^{26} + 20 q^{28} + 12 q^{31} - 6 q^{32} - 4 q^{34} - 4 q^{35} - 6 q^{40} + 12 q^{41} - 8 q^{43} - 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{55} + 20 q^{56} + 16 q^{58} - 12 q^{59} - 16 q^{61} - 14 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 12 q^{70} - 4 q^{71} - 24 q^{74} - 4 q^{76} - 16 q^{77} - 6 q^{80} + 4 q^{82} + 12 q^{83} - 4 q^{85} - 28 q^{86} - 16 q^{88} - 12 q^{89} - 4 q^{91} - 8 q^{92} + 12 q^{94} - 4 q^{95} - 4 q^{97} + 42 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 + 6 * q^8 - 2 * q^10 - 4 * q^11 - 2 * q^13 + 12 * q^14 + 6 * q^16 + 4 * q^17 + 4 * q^19 - 2 * q^20 - 8 * q^22 + 2 * q^25 - 2 * q^26 + 20 * q^28 + 12 * q^31 - 6 * q^32 - 4 * q^34 - 4 * q^35 - 6 * q^40 + 12 * q^41 - 8 * q^43 - 12 * q^44 - 4 * q^46 + 4 * q^47 + 10 * q^49 + 2 * q^50 - 2 * q^52 + 12 * q^53 + 4 * q^55 + 20 * q^56 + 16 * q^58 - 12 * q^59 - 16 * q^61 - 14 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 12 * q^70 - 4 * q^71 - 24 * q^74 - 4 * q^76 - 16 * q^77 - 6 * q^80 + 4 * q^82 + 12 * q^83 - 4 * q^85 - 28 * q^86 - 16 * q^88 - 12 * q^89 - 4 * q^91 - 8 * q^92 + 12 * q^94 - 4 * q^95 - 4 * q^97 + 42 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41421 1.70711 0.853553 0.521005i $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 0 0
$$4$$ 3.82843 1.91421
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.82843 1.82497 0.912487 0.409106i $$-0.134159\pi$$
0.912487 + 0.409106i $$0.134159\pi$$
$$8$$ 4.41421 1.56066
$$9$$ 0 0
$$10$$ −2.41421 −0.763441
$$11$$ −3.41421 −1.02942 −0.514712 0.857363i $$-0.672101\pi$$
−0.514712 + 0.857363i $$0.672101\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 11.6569 3.11543
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ −0.828427 −0.200923 −0.100462 0.994941i $$-0.532032\pi$$
−0.100462 + 0.994941i $$0.532032\pi$$
$$18$$ 0 0
$$19$$ 0.585786 0.134389 0.0671943 0.997740i $$-0.478595\pi$$
0.0671943 + 0.997740i $$0.478595\pi$$
$$20$$ −3.82843 −0.856062
$$21$$ 0 0
$$22$$ −8.24264 −1.75734
$$23$$ −1.41421 −0.294884 −0.147442 0.989071i $$-0.547104\pi$$
−0.147442 + 0.989071i $$0.547104\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.41421 −0.473466
$$27$$ 0 0
$$28$$ 18.4853 3.49339
$$29$$ 5.65685 1.05045 0.525226 0.850963i $$-0.323981\pi$$
0.525226 + 0.850963i $$0.323981\pi$$
$$30$$ 0 0
$$31$$ 1.75736 0.315631 0.157816 0.987469i $$-0.449555\pi$$
0.157816 + 0.987469i $$0.449555\pi$$
$$32$$ −1.58579 −0.280330
$$33$$ 0 0
$$34$$ −2.00000 −0.342997
$$35$$ −4.82843 −0.816153
$$36$$ 0 0
$$37$$ −8.48528 −1.39497 −0.697486 0.716599i $$-0.745698\pi$$
−0.697486 + 0.716599i $$0.745698\pi$$
$$38$$ 1.41421 0.229416
$$39$$ 0 0
$$40$$ −4.41421 −0.697948
$$41$$ 3.17157 0.495316 0.247658 0.968847i $$-0.420339\pi$$
0.247658 + 0.968847i $$0.420339\pi$$
$$42$$ 0 0
$$43$$ −11.0711 −1.68832 −0.844161 0.536090i $$-0.819901\pi$$
−0.844161 + 0.536090i $$0.819901\pi$$
$$44$$ −13.0711 −1.97054
$$45$$ 0 0
$$46$$ −3.41421 −0.503398
$$47$$ 4.82843 0.704298 0.352149 0.935944i $$-0.385451\pi$$
0.352149 + 0.935944i $$0.385451\pi$$
$$48$$ 0 0
$$49$$ 16.3137 2.33053
$$50$$ 2.41421 0.341421
$$51$$ 0 0
$$52$$ −3.82843 −0.530907
$$53$$ −2.48528 −0.341380 −0.170690 0.985325i $$-0.554600\pi$$
−0.170690 + 0.985325i $$0.554600\pi$$
$$54$$ 0 0
$$55$$ 3.41421 0.460372
$$56$$ 21.3137 2.84816
$$57$$ 0 0
$$58$$ 13.6569 1.79323
$$59$$ −1.75736 −0.228789 −0.114394 0.993435i $$-0.536493\pi$$
−0.114394 + 0.993435i $$0.536493\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ 4.24264 0.538816
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ −3.17157 −0.384610
$$69$$ 0 0
$$70$$ −11.6569 −1.39326
$$71$$ −11.8995 −1.41221 −0.706105 0.708107i $$-0.749549\pi$$
−0.706105 + 0.708107i $$0.749549\pi$$
$$72$$ 0 0
$$73$$ 8.48528 0.993127 0.496564 0.868000i $$-0.334595\pi$$
0.496564 + 0.868000i $$0.334595\pi$$
$$74$$ −20.4853 −2.38137
$$75$$ 0 0
$$76$$ 2.24264 0.257249
$$77$$ −16.4853 −1.87867
$$78$$ 0 0
$$79$$ −8.48528 −0.954669 −0.477334 0.878722i $$-0.658397\pi$$
−0.477334 + 0.878722i $$0.658397\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 0 0
$$82$$ 7.65685 0.845558
$$83$$ 3.17157 0.348125 0.174063 0.984735i $$-0.444310\pi$$
0.174063 + 0.984735i $$0.444310\pi$$
$$84$$ 0 0
$$85$$ 0.828427 0.0898555
$$86$$ −26.7279 −2.88215
$$87$$ 0 0
$$88$$ −15.0711 −1.60658
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −4.82843 −0.506157
$$92$$ −5.41421 −0.564471
$$93$$ 0 0
$$94$$ 11.6569 1.20231
$$95$$ −0.585786 −0.0601004
$$96$$ 0 0
$$97$$ −7.65685 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$98$$ 39.3848 3.97846
$$99$$ 0 0
$$100$$ 3.82843 0.382843
$$101$$ 3.65685 0.363871 0.181935 0.983311i $$-0.441764\pi$$
0.181935 + 0.983311i $$0.441764\pi$$
$$102$$ 0 0
$$103$$ 14.5858 1.43718 0.718590 0.695434i $$-0.244788\pi$$
0.718590 + 0.695434i $$0.244788\pi$$
$$104$$ −4.41421 −0.432849
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 9.41421 0.910106 0.455053 0.890464i $$-0.349620\pi$$
0.455053 + 0.890464i $$0.349620\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 8.24264 0.785905
$$111$$ 0 0
$$112$$ 14.4853 1.36873
$$113$$ 8.82843 0.830509 0.415254 0.909705i $$-0.363693\pi$$
0.415254 + 0.909705i $$0.363693\pi$$
$$114$$ 0 0
$$115$$ 1.41421 0.131876
$$116$$ 21.6569 2.01079
$$117$$ 0 0
$$118$$ −4.24264 −0.390567
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 0.656854 0.0597140
$$122$$ −19.3137 −1.74858
$$123$$ 0 0
$$124$$ 6.72792 0.604185
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −6.58579 −0.584394 −0.292197 0.956358i $$-0.594386\pi$$
−0.292197 + 0.956358i $$0.594386\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ 0 0
$$130$$ 2.41421 0.211741
$$131$$ 16.9706 1.48272 0.741362 0.671105i $$-0.234180\pi$$
0.741362 + 0.671105i $$0.234180\pi$$
$$132$$ 0 0
$$133$$ 2.82843 0.245256
$$134$$ −4.82843 −0.417113
$$135$$ 0 0
$$136$$ −3.65685 −0.313573
$$137$$ 17.3137 1.47921 0.739605 0.673041i $$-0.235012\pi$$
0.739605 + 0.673041i $$0.235012\pi$$
$$138$$ 0 0
$$139$$ 4.48528 0.380437 0.190218 0.981742i $$-0.439080\pi$$
0.190218 + 0.981742i $$0.439080\pi$$
$$140$$ −18.4853 −1.56229
$$141$$ 0 0
$$142$$ −28.7279 −2.41079
$$143$$ 3.41421 0.285511
$$144$$ 0 0
$$145$$ −5.65685 −0.469776
$$146$$ 20.4853 1.69537
$$147$$ 0 0
$$148$$ −32.4853 −2.67027
$$149$$ 11.6569 0.954967 0.477483 0.878641i $$-0.341549\pi$$
0.477483 + 0.878641i $$0.341549\pi$$
$$150$$ 0 0
$$151$$ 9.75736 0.794043 0.397021 0.917809i $$-0.370044\pi$$
0.397021 + 0.917809i $$0.370044\pi$$
$$152$$ 2.58579 0.209735
$$153$$ 0 0
$$154$$ −39.7990 −3.20709
$$155$$ −1.75736 −0.141154
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −20.4853 −1.62972
$$159$$ 0 0
$$160$$ 1.58579 0.125367
$$161$$ −6.82843 −0.538155
$$162$$ 0 0
$$163$$ 18.9706 1.48589 0.742945 0.669353i $$-0.233429\pi$$
0.742945 + 0.669353i $$0.233429\pi$$
$$164$$ 12.1421 0.948141
$$165$$ 0 0
$$166$$ 7.65685 0.594287
$$167$$ 3.17157 0.245424 0.122712 0.992442i $$-0.460841\pi$$
0.122712 + 0.992442i $$0.460841\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ −42.3848 −3.23181
$$173$$ −16.8284 −1.27944 −0.639721 0.768607i $$-0.720950\pi$$
−0.639721 + 0.768607i $$0.720950\pi$$
$$174$$ 0 0
$$175$$ 4.82843 0.364995
$$176$$ −10.2426 −0.772068
$$177$$ 0 0
$$178$$ −14.4853 −1.08572
$$179$$ −5.65685 −0.422813 −0.211407 0.977398i $$-0.567804\pi$$
−0.211407 + 0.977398i $$0.567804\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ −11.6569 −0.864064
$$183$$ 0 0
$$184$$ −6.24264 −0.460214
$$185$$ 8.48528 0.623850
$$186$$ 0 0
$$187$$ 2.82843 0.206835
$$188$$ 18.4853 1.34818
$$189$$ 0 0
$$190$$ −1.41421 −0.102598
$$191$$ −2.34315 −0.169544 −0.0847720 0.996400i $$-0.527016\pi$$
−0.0847720 + 0.996400i $$0.527016\pi$$
$$192$$ 0 0
$$193$$ 4.34315 0.312626 0.156313 0.987708i $$-0.450039\pi$$
0.156313 + 0.987708i $$0.450039\pi$$
$$194$$ −18.4853 −1.32717
$$195$$ 0 0
$$196$$ 62.4558 4.46113
$$197$$ −10.9706 −0.781620 −0.390810 0.920471i $$-0.627805\pi$$
−0.390810 + 0.920471i $$0.627805\pi$$
$$198$$ 0 0
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 4.41421 0.312132
$$201$$ 0 0
$$202$$ 8.82843 0.621166
$$203$$ 27.3137 1.91705
$$204$$ 0 0
$$205$$ −3.17157 −0.221512
$$206$$ 35.2132 2.45342
$$207$$ 0 0
$$208$$ −3.00000 −0.208013
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ 3.31371 0.228125 0.114063 0.993474i $$-0.463614\pi$$
0.114063 + 0.993474i $$0.463614\pi$$
$$212$$ −9.51472 −0.653474
$$213$$ 0 0
$$214$$ 22.7279 1.55365
$$215$$ 11.0711 0.755041
$$216$$ 0 0
$$217$$ 8.48528 0.576018
$$218$$ −4.82843 −0.327022
$$219$$ 0 0
$$220$$ 13.0711 0.881251
$$221$$ 0.828427 0.0557260
$$222$$ 0 0
$$223$$ 9.51472 0.637153 0.318576 0.947897i $$-0.396795\pi$$
0.318576 + 0.947897i $$0.396795\pi$$
$$224$$ −7.65685 −0.511595
$$225$$ 0 0
$$226$$ 21.3137 1.41777
$$227$$ 16.3431 1.08473 0.542366 0.840142i $$-0.317528\pi$$
0.542366 + 0.840142i $$0.317528\pi$$
$$228$$ 0 0
$$229$$ −4.82843 −0.319071 −0.159536 0.987192i $$-0.551000\pi$$
−0.159536 + 0.987192i $$0.551000\pi$$
$$230$$ 3.41421 0.225127
$$231$$ 0 0
$$232$$ 24.9706 1.63940
$$233$$ 20.6274 1.35135 0.675674 0.737201i $$-0.263853\pi$$
0.675674 + 0.737201i $$0.263853\pi$$
$$234$$ 0 0
$$235$$ −4.82843 −0.314972
$$236$$ −6.72792 −0.437950
$$237$$ 0 0
$$238$$ −9.65685 −0.625961
$$239$$ 3.41421 0.220847 0.110424 0.993885i $$-0.464779\pi$$
0.110424 + 0.993885i $$0.464779\pi$$
$$240$$ 0 0
$$241$$ −14.4853 −0.933079 −0.466539 0.884500i $$-0.654499\pi$$
−0.466539 + 0.884500i $$0.654499\pi$$
$$242$$ 1.58579 0.101938
$$243$$ 0 0
$$244$$ −30.6274 −1.96072
$$245$$ −16.3137 −1.04224
$$246$$ 0 0
$$247$$ −0.585786 −0.0372727
$$248$$ 7.75736 0.492593
$$249$$ 0 0
$$250$$ −2.41421 −0.152688
$$251$$ −19.7990 −1.24970 −0.624851 0.780744i $$-0.714840\pi$$
−0.624851 + 0.780744i $$0.714840\pi$$
$$252$$ 0 0
$$253$$ 4.82843 0.303561
$$254$$ −15.8995 −0.997623
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ −27.6569 −1.72519 −0.862594 0.505898i $$-0.831161\pi$$
−0.862594 + 0.505898i $$0.831161\pi$$
$$258$$ 0 0
$$259$$ −40.9706 −2.54579
$$260$$ 3.82843 0.237429
$$261$$ 0 0
$$262$$ 40.9706 2.53117
$$263$$ 10.5858 0.652748 0.326374 0.945241i $$-0.394173\pi$$
0.326374 + 0.945241i $$0.394173\pi$$
$$264$$ 0 0
$$265$$ 2.48528 0.152670
$$266$$ 6.82843 0.418678
$$267$$ 0 0
$$268$$ −7.65685 −0.467717
$$269$$ 25.3137 1.54340 0.771702 0.635984i $$-0.219406\pi$$
0.771702 + 0.635984i $$0.219406\pi$$
$$270$$ 0 0
$$271$$ 26.7279 1.62361 0.811803 0.583932i $$-0.198486\pi$$
0.811803 + 0.583932i $$0.198486\pi$$
$$272$$ −2.48528 −0.150692
$$273$$ 0 0
$$274$$ 41.7990 2.52517
$$275$$ −3.41421 −0.205885
$$276$$ 0 0
$$277$$ −12.8284 −0.770785 −0.385393 0.922753i $$-0.625934\pi$$
−0.385393 + 0.922753i $$0.625934\pi$$
$$278$$ 10.8284 0.649446
$$279$$ 0 0
$$280$$ −21.3137 −1.27374
$$281$$ −21.7990 −1.30042 −0.650209 0.759755i $$-0.725319\pi$$
−0.650209 + 0.759755i $$0.725319\pi$$
$$282$$ 0 0
$$283$$ −16.7279 −0.994372 −0.497186 0.867644i $$-0.665633\pi$$
−0.497186 + 0.867644i $$0.665633\pi$$
$$284$$ −45.5563 −2.70327
$$285$$ 0 0
$$286$$ 8.24264 0.487398
$$287$$ 15.3137 0.903940
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ −13.6569 −0.801958
$$291$$ 0 0
$$292$$ 32.4853 1.90106
$$293$$ −26.1421 −1.52724 −0.763620 0.645666i $$-0.776580\pi$$
−0.763620 + 0.645666i $$0.776580\pi$$
$$294$$ 0 0
$$295$$ 1.75736 0.102317
$$296$$ −37.4558 −2.17708
$$297$$ 0 0
$$298$$ 28.1421 1.63023
$$299$$ 1.41421 0.0817861
$$300$$ 0 0
$$301$$ −53.4558 −3.08114
$$302$$ 23.5563 1.35552
$$303$$ 0 0
$$304$$ 1.75736 0.100791
$$305$$ 8.00000 0.458079
$$306$$ 0 0
$$307$$ 24.8284 1.41703 0.708517 0.705694i $$-0.249365\pi$$
0.708517 + 0.705694i $$0.249365\pi$$
$$308$$ −63.1127 −3.59618
$$309$$ 0 0
$$310$$ −4.24264 −0.240966
$$311$$ −8.48528 −0.481156 −0.240578 0.970630i $$-0.577337\pi$$
−0.240578 + 0.970630i $$0.577337\pi$$
$$312$$ 0 0
$$313$$ −4.82843 −0.272919 −0.136459 0.990646i $$-0.543572\pi$$
−0.136459 + 0.990646i $$0.543572\pi$$
$$314$$ 43.4558 2.45236
$$315$$ 0 0
$$316$$ −32.4853 −1.82744
$$317$$ 2.14214 0.120314 0.0601572 0.998189i $$-0.480840\pi$$
0.0601572 + 0.998189i $$0.480840\pi$$
$$318$$ 0 0
$$319$$ −19.3137 −1.08136
$$320$$ 9.82843 0.549426
$$321$$ 0 0
$$322$$ −16.4853 −0.918689
$$323$$ −0.485281 −0.0270018
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 45.7990 2.53657
$$327$$ 0 0
$$328$$ 14.0000 0.773021
$$329$$ 23.3137 1.28533
$$330$$ 0 0
$$331$$ −26.0416 −1.43138 −0.715689 0.698419i $$-0.753887\pi$$
−0.715689 + 0.698419i $$0.753887\pi$$
$$332$$ 12.1421 0.666386
$$333$$ 0 0
$$334$$ 7.65685 0.418964
$$335$$ 2.00000 0.109272
$$336$$ 0 0
$$337$$ 12.8284 0.698809 0.349404 0.936972i $$-0.386384\pi$$
0.349404 + 0.936972i $$0.386384\pi$$
$$338$$ 2.41421 0.131316
$$339$$ 0 0
$$340$$ 3.17157 0.172003
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 44.9706 2.42818
$$344$$ −48.8701 −2.63490
$$345$$ 0 0
$$346$$ −40.6274 −2.18414
$$347$$ 4.24264 0.227757 0.113878 0.993495i $$-0.463673\pi$$
0.113878 + 0.993495i $$0.463673\pi$$
$$348$$ 0 0
$$349$$ 18.4853 0.989494 0.494747 0.869037i $$-0.335261\pi$$
0.494747 + 0.869037i $$0.335261\pi$$
$$350$$ 11.6569 0.623085
$$351$$ 0 0
$$352$$ 5.41421 0.288579
$$353$$ −14.8284 −0.789238 −0.394619 0.918845i $$-0.629123\pi$$
−0.394619 + 0.918845i $$0.629123\pi$$
$$354$$ 0 0
$$355$$ 11.8995 0.631560
$$356$$ −22.9706 −1.21744
$$357$$ 0 0
$$358$$ −13.6569 −0.721787
$$359$$ 8.10051 0.427528 0.213764 0.976885i $$-0.431428\pi$$
0.213764 + 0.976885i $$0.431428\pi$$
$$360$$ 0 0
$$361$$ −18.6569 −0.981940
$$362$$ 0 0
$$363$$ 0 0
$$364$$ −18.4853 −0.968892
$$365$$ −8.48528 −0.444140
$$366$$ 0 0
$$367$$ 35.5563 1.85603 0.928013 0.372547i $$-0.121516\pi$$
0.928013 + 0.372547i $$0.121516\pi$$
$$368$$ −4.24264 −0.221163
$$369$$ 0 0
$$370$$ 20.4853 1.06498
$$371$$ −12.0000 −0.623009
$$372$$ 0 0
$$373$$ −2.68629 −0.139091 −0.0695455 0.997579i $$-0.522155\pi$$
−0.0695455 + 0.997579i $$0.522155\pi$$
$$374$$ 6.82843 0.353090
$$375$$ 0 0
$$376$$ 21.3137 1.09917
$$377$$ −5.65685 −0.291343
$$378$$ 0 0
$$379$$ 29.0711 1.49328 0.746640 0.665228i $$-0.231666\pi$$
0.746640 + 0.665228i $$0.231666\pi$$
$$380$$ −2.24264 −0.115045
$$381$$ 0 0
$$382$$ −5.65685 −0.289430
$$383$$ 29.1127 1.48759 0.743795 0.668408i $$-0.233024\pi$$
0.743795 + 0.668408i $$0.233024\pi$$
$$384$$ 0 0
$$385$$ 16.4853 0.840168
$$386$$ 10.4853 0.533687
$$387$$ 0 0
$$388$$ −29.3137 −1.48818
$$389$$ −28.6274 −1.45147 −0.725734 0.687976i $$-0.758500\pi$$
−0.725734 + 0.687976i $$0.758500\pi$$
$$390$$ 0 0
$$391$$ 1.17157 0.0592490
$$392$$ 72.0122 3.63717
$$393$$ 0 0
$$394$$ −26.4853 −1.33431
$$395$$ 8.48528 0.426941
$$396$$ 0 0
$$397$$ 11.7990 0.592174 0.296087 0.955161i $$-0.404318\pi$$
0.296087 + 0.955161i $$0.404318\pi$$
$$398$$ 9.65685 0.484054
$$399$$ 0 0
$$400$$ 3.00000 0.150000
$$401$$ 5.31371 0.265354 0.132677 0.991159i $$-0.457643\pi$$
0.132677 + 0.991159i $$0.457643\pi$$
$$402$$ 0 0
$$403$$ −1.75736 −0.0875403
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ 65.9411 3.27260
$$407$$ 28.9706 1.43602
$$408$$ 0 0
$$409$$ 7.17157 0.354611 0.177306 0.984156i $$-0.443262\pi$$
0.177306 + 0.984156i $$0.443262\pi$$
$$410$$ −7.65685 −0.378145
$$411$$ 0 0
$$412$$ 55.8406 2.75107
$$413$$ −8.48528 −0.417533
$$414$$ 0 0
$$415$$ −3.17157 −0.155686
$$416$$ 1.58579 0.0777496
$$417$$ 0 0
$$418$$ −4.82843 −0.236166
$$419$$ −10.8284 −0.529003 −0.264502 0.964385i $$-0.585207\pi$$
−0.264502 + 0.964385i $$0.585207\pi$$
$$420$$ 0 0
$$421$$ −34.9706 −1.70436 −0.852180 0.523248i $$-0.824720\pi$$
−0.852180 + 0.523248i $$0.824720\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 0 0
$$424$$ −10.9706 −0.532778
$$425$$ −0.828427 −0.0401846
$$426$$ 0 0
$$427$$ −38.6274 −1.86931
$$428$$ 36.0416 1.74214
$$429$$ 0 0
$$430$$ 26.7279 1.28893
$$431$$ −40.3848 −1.94527 −0.972633 0.232346i $$-0.925360\pi$$
−0.972633 + 0.232346i $$0.925360\pi$$
$$432$$ 0 0
$$433$$ 7.65685 0.367965 0.183982 0.982930i $$-0.441101\pi$$
0.183982 + 0.982930i $$0.441101\pi$$
$$434$$ 20.4853 0.983325
$$435$$ 0 0
$$436$$ −7.65685 −0.366697
$$437$$ −0.828427 −0.0396290
$$438$$ 0 0
$$439$$ 0.970563 0.0463224 0.0231612 0.999732i $$-0.492627\pi$$
0.0231612 + 0.999732i $$0.492627\pi$$
$$440$$ 15.0711 0.718485
$$441$$ 0 0
$$442$$ 2.00000 0.0951303
$$443$$ 9.41421 0.447283 0.223641 0.974671i $$-0.428206\pi$$
0.223641 + 0.974671i $$0.428206\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 22.9706 1.08769
$$447$$ 0 0
$$448$$ −47.4558 −2.24208
$$449$$ 33.1127 1.56268 0.781342 0.624103i $$-0.214535\pi$$
0.781342 + 0.624103i $$0.214535\pi$$
$$450$$ 0 0
$$451$$ −10.8284 −0.509891
$$452$$ 33.7990 1.58977
$$453$$ 0 0
$$454$$ 39.4558 1.85175
$$455$$ 4.82843 0.226360
$$456$$ 0 0
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ −11.6569 −0.544689
$$459$$ 0 0
$$460$$ 5.41421 0.252439
$$461$$ −9.51472 −0.443145 −0.221572 0.975144i $$-0.571119\pi$$
−0.221572 + 0.975144i $$0.571119\pi$$
$$462$$ 0 0
$$463$$ −4.34315 −0.201843 −0.100922 0.994894i $$-0.532179\pi$$
−0.100922 + 0.994894i $$0.532179\pi$$
$$464$$ 16.9706 0.787839
$$465$$ 0 0
$$466$$ 49.7990 2.30689
$$467$$ 13.4142 0.620736 0.310368 0.950617i $$-0.399548\pi$$
0.310368 + 0.950617i $$0.399548\pi$$
$$468$$ 0 0
$$469$$ −9.65685 −0.445912
$$470$$ −11.6569 −0.537691
$$471$$ 0 0
$$472$$ −7.75736 −0.357061
$$473$$ 37.7990 1.73800
$$474$$ 0 0
$$475$$ 0.585786 0.0268777
$$476$$ −15.3137 −0.701903
$$477$$ 0 0
$$478$$ 8.24264 0.377010
$$479$$ 30.7279 1.40399 0.701997 0.712180i $$-0.252292\pi$$
0.701997 + 0.712180i $$0.252292\pi$$
$$480$$ 0 0
$$481$$ 8.48528 0.386896
$$482$$ −34.9706 −1.59287
$$483$$ 0 0
$$484$$ 2.51472 0.114305
$$485$$ 7.65685 0.347680
$$486$$ 0 0
$$487$$ −10.9706 −0.497124 −0.248562 0.968616i $$-0.579958\pi$$
−0.248562 + 0.968616i $$0.579958\pi$$
$$488$$ −35.3137 −1.59858
$$489$$ 0 0
$$490$$ −39.3848 −1.77922
$$491$$ −5.17157 −0.233390 −0.116695 0.993168i $$-0.537230\pi$$
−0.116695 + 0.993168i $$0.537230\pi$$
$$492$$ 0 0
$$493$$ −4.68629 −0.211060
$$494$$ −1.41421 −0.0636285
$$495$$ 0 0
$$496$$ 5.27208 0.236723
$$497$$ −57.4558 −2.57725
$$498$$ 0 0
$$499$$ 41.5563 1.86032 0.930159 0.367157i $$-0.119669\pi$$
0.930159 + 0.367157i $$0.119669\pi$$
$$500$$ −3.82843 −0.171212
$$501$$ 0 0
$$502$$ −47.7990 −2.13337
$$503$$ −37.8995 −1.68985 −0.844927 0.534881i $$-0.820356\pi$$
−0.844927 + 0.534881i $$0.820356\pi$$
$$504$$ 0 0
$$505$$ −3.65685 −0.162728
$$506$$ 11.6569 0.518210
$$507$$ 0 0
$$508$$ −25.2132 −1.11866
$$509$$ −41.1127 −1.82229 −0.911144 0.412088i $$-0.864800\pi$$
−0.911144 + 0.412088i $$0.864800\pi$$
$$510$$ 0 0
$$511$$ 40.9706 1.81243
$$512$$ −31.2426 −1.38074
$$513$$ 0 0
$$514$$ −66.7696 −2.94508
$$515$$ −14.5858 −0.642727
$$516$$ 0 0
$$517$$ −16.4853 −0.725022
$$518$$ −98.9117 −4.34593
$$519$$ 0 0
$$520$$ 4.41421 0.193576
$$521$$ 17.6569 0.773561 0.386780 0.922172i $$-0.373587\pi$$
0.386780 + 0.922172i $$0.373587\pi$$
$$522$$ 0 0
$$523$$ −19.7574 −0.863929 −0.431965 0.901891i $$-0.642179\pi$$
−0.431965 + 0.901891i $$0.642179\pi$$
$$524$$ 64.9706 2.83825
$$525$$ 0 0
$$526$$ 25.5563 1.11431
$$527$$ −1.45584 −0.0634176
$$528$$ 0 0
$$529$$ −21.0000 −0.913043
$$530$$ 6.00000 0.260623
$$531$$ 0 0
$$532$$ 10.8284 0.469472
$$533$$ −3.17157 −0.137376
$$534$$ 0 0
$$535$$ −9.41421 −0.407012
$$536$$ −8.82843 −0.381330
$$537$$ 0 0
$$538$$ 61.1127 2.63476
$$539$$ −55.6985 −2.39910
$$540$$ 0 0
$$541$$ −7.17157 −0.308330 −0.154165 0.988045i $$-0.549269\pi$$
−0.154165 + 0.988045i $$0.549269\pi$$
$$542$$ 64.5269 2.77167
$$543$$ 0 0
$$544$$ 1.31371 0.0563248
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 13.2132 0.564956 0.282478 0.959274i $$-0.408844\pi$$
0.282478 + 0.959274i $$0.408844\pi$$
$$548$$ 66.2843 2.83152
$$549$$ 0 0
$$550$$ −8.24264 −0.351467
$$551$$ 3.31371 0.141169
$$552$$ 0 0
$$553$$ −40.9706 −1.74225
$$554$$ −30.9706 −1.31581
$$555$$ 0 0
$$556$$ 17.1716 0.728237
$$557$$ 35.7990 1.51685 0.758426 0.651759i $$-0.225969\pi$$
0.758426 + 0.651759i $$0.225969\pi$$
$$558$$ 0 0
$$559$$ 11.0711 0.468256
$$560$$ −14.4853 −0.612115
$$561$$ 0 0
$$562$$ −52.6274 −2.21995
$$563$$ 7.75736 0.326934 0.163467 0.986549i $$-0.447732\pi$$
0.163467 + 0.986549i $$0.447732\pi$$
$$564$$ 0 0
$$565$$ −8.82843 −0.371415
$$566$$ −40.3848 −1.69750
$$567$$ 0 0
$$568$$ −52.5269 −2.20398
$$569$$ 10.3431 0.433607 0.216804 0.976215i $$-0.430437\pi$$
0.216804 + 0.976215i $$0.430437\pi$$
$$570$$ 0 0
$$571$$ −11.5147 −0.481876 −0.240938 0.970541i $$-0.577455\pi$$
−0.240938 + 0.970541i $$0.577455\pi$$
$$572$$ 13.0711 0.546529
$$573$$ 0 0
$$574$$ 36.9706 1.54312
$$575$$ −1.41421 −0.0589768
$$576$$ 0 0
$$577$$ −34.8284 −1.44993 −0.724963 0.688788i $$-0.758143\pi$$
−0.724963 + 0.688788i $$0.758143\pi$$
$$578$$ −39.3848 −1.63819
$$579$$ 0 0
$$580$$ −21.6569 −0.899252
$$581$$ 15.3137 0.635320
$$582$$ 0 0
$$583$$ 8.48528 0.351424
$$584$$ 37.4558 1.54993
$$585$$ 0 0
$$586$$ −63.1127 −2.60716
$$587$$ −20.3431 −0.839651 −0.419826 0.907605i $$-0.637909\pi$$
−0.419826 + 0.907605i $$0.637909\pi$$
$$588$$ 0 0
$$589$$ 1.02944 0.0424172
$$590$$ 4.24264 0.174667
$$591$$ 0 0
$$592$$ −25.4558 −1.04623
$$593$$ 24.6274 1.01133 0.505663 0.862731i $$-0.331248\pi$$
0.505663 + 0.862731i $$0.331248\pi$$
$$594$$ 0 0
$$595$$ 4.00000 0.163984
$$596$$ 44.6274 1.82801
$$597$$ 0 0
$$598$$ 3.41421 0.139618
$$599$$ −25.4558 −1.04010 −0.520049 0.854137i $$-0.674086\pi$$
−0.520049 + 0.854137i $$0.674086\pi$$
$$600$$ 0 0
$$601$$ −44.6274 −1.82039 −0.910195 0.414180i $$-0.864069\pi$$
−0.910195 + 0.414180i $$0.864069\pi$$
$$602$$ −129.054 −5.25984
$$603$$ 0 0
$$604$$ 37.3553 1.51997
$$605$$ −0.656854 −0.0267049
$$606$$ 0 0
$$607$$ 31.7574 1.28899 0.644496 0.764608i $$-0.277067\pi$$
0.644496 + 0.764608i $$0.277067\pi$$
$$608$$ −0.928932 −0.0376732
$$609$$ 0 0
$$610$$ 19.3137 0.781989
$$611$$ −4.82843 −0.195337
$$612$$ 0 0
$$613$$ −14.6863 −0.593174 −0.296587 0.955006i $$-0.595848\pi$$
−0.296587 + 0.955006i $$0.595848\pi$$
$$614$$ 59.9411 2.41903
$$615$$ 0 0
$$616$$ −72.7696 −2.93197
$$617$$ −10.9706 −0.441658 −0.220829 0.975313i $$-0.570876\pi$$
−0.220829 + 0.975313i $$0.570876\pi$$
$$618$$ 0 0
$$619$$ 1.75736 0.0706342 0.0353171 0.999376i $$-0.488756\pi$$
0.0353171 + 0.999376i $$0.488756\pi$$
$$620$$ −6.72792 −0.270200
$$621$$ 0 0
$$622$$ −20.4853 −0.821385
$$623$$ −28.9706 −1.16068
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −11.6569 −0.465902
$$627$$ 0 0
$$628$$ 68.9117 2.74988
$$629$$ 7.02944 0.280282
$$630$$ 0 0
$$631$$ −9.75736 −0.388434 −0.194217 0.980959i $$-0.562217\pi$$
−0.194217 + 0.980959i $$0.562217\pi$$
$$632$$ −37.4558 −1.48991
$$633$$ 0 0
$$634$$ 5.17157 0.205389
$$635$$ 6.58579 0.261349
$$636$$ 0 0
$$637$$ −16.3137 −0.646373
$$638$$ −46.6274 −1.84600
$$639$$ 0 0
$$640$$ 20.5563 0.812561
$$641$$ −47.6569 −1.88233 −0.941166 0.337944i $$-0.890269\pi$$
−0.941166 + 0.337944i $$0.890269\pi$$
$$642$$ 0 0
$$643$$ 9.51472 0.375224 0.187612 0.982243i $$-0.439925\pi$$
0.187612 + 0.982243i $$0.439925\pi$$
$$644$$ −26.1421 −1.03014
$$645$$ 0 0
$$646$$ −1.17157 −0.0460949
$$647$$ −9.41421 −0.370111 −0.185055 0.982728i $$-0.559246\pi$$
−0.185055 + 0.982728i $$0.559246\pi$$
$$648$$ 0 0
$$649$$ 6.00000 0.235521
$$650$$ −2.41421 −0.0946932
$$651$$ 0 0
$$652$$ 72.6274 2.84431
$$653$$ −46.9706 −1.83810 −0.919050 0.394141i $$-0.871042\pi$$
−0.919050 + 0.394141i $$0.871042\pi$$
$$654$$ 0 0
$$655$$ −16.9706 −0.663095
$$656$$ 9.51472 0.371487
$$657$$ 0 0
$$658$$ 56.2843 2.19419
$$659$$ −17.8579 −0.695644 −0.347822 0.937561i $$-0.613079\pi$$
−0.347822 + 0.937561i $$0.613079\pi$$
$$660$$ 0 0
$$661$$ 29.5980 1.15123 0.575614 0.817722i $$-0.304763\pi$$
0.575614 + 0.817722i $$0.304763\pi$$
$$662$$ −62.8701 −2.44351
$$663$$ 0 0
$$664$$ 14.0000 0.543305
$$665$$ −2.82843 −0.109682
$$666$$ 0 0
$$667$$ −8.00000 −0.309761
$$668$$ 12.1421 0.469793
$$669$$ 0 0
$$670$$ 4.82843 0.186538
$$671$$ 27.3137 1.05443
$$672$$ 0 0
$$673$$ 6.48528 0.249989 0.124995 0.992157i $$-0.460109\pi$$
0.124995 + 0.992157i $$0.460109\pi$$
$$674$$ 30.9706 1.19294
$$675$$ 0 0
$$676$$ 3.82843 0.147247
$$677$$ 20.1421 0.774125 0.387063 0.922053i $$-0.373490\pi$$
0.387063 + 0.922053i $$0.373490\pi$$
$$678$$ 0 0
$$679$$ −36.9706 −1.41880
$$680$$ 3.65685 0.140234
$$681$$ 0 0
$$682$$ −14.4853 −0.554670
$$683$$ −10.6863 −0.408900 −0.204450 0.978877i $$-0.565541\pi$$
−0.204450 + 0.978877i $$0.565541\pi$$
$$684$$ 0 0
$$685$$ −17.3137 −0.661523
$$686$$ 108.569 4.14517
$$687$$ 0 0
$$688$$ −33.2132 −1.26624
$$689$$ 2.48528 0.0946817
$$690$$ 0 0
$$691$$ 6.92893 0.263589 0.131795 0.991277i $$-0.457926\pi$$
0.131795 + 0.991277i $$0.457926\pi$$
$$692$$ −64.4264 −2.44912
$$693$$ 0 0
$$694$$ 10.2426 0.388805
$$695$$ −4.48528 −0.170136
$$696$$ 0 0
$$697$$ −2.62742 −0.0995205
$$698$$ 44.6274 1.68917
$$699$$ 0 0
$$700$$ 18.4853 0.698678
$$701$$ −14.6863 −0.554694 −0.277347 0.960770i $$-0.589455\pi$$
−0.277347 + 0.960770i $$0.589455\pi$$
$$702$$ 0 0
$$703$$ −4.97056 −0.187468
$$704$$ 33.5563 1.26470
$$705$$ 0 0
$$706$$ −35.7990 −1.34731
$$707$$ 17.6569 0.664054
$$708$$ 0 0
$$709$$ 45.1127 1.69424 0.847121 0.531399i $$-0.178334\pi$$
0.847121 + 0.531399i $$0.178334\pi$$
$$710$$ 28.7279 1.07814
$$711$$ 0 0
$$712$$ −26.4853 −0.992578
$$713$$ −2.48528 −0.0930745
$$714$$ 0 0
$$715$$ −3.41421 −0.127684
$$716$$ −21.6569 −0.809355
$$717$$ 0 0
$$718$$ 19.5563 0.729836
$$719$$ −28.9706 −1.08042 −0.540210 0.841530i $$-0.681655\pi$$
−0.540210 + 0.841530i $$0.681655\pi$$
$$720$$ 0 0
$$721$$ 70.4264 2.62282
$$722$$ −45.0416 −1.67628
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 5.65685 0.210090
$$726$$ 0 0
$$727$$ −51.3553 −1.90466 −0.952332 0.305063i $$-0.901322\pi$$
−0.952332 + 0.305063i $$0.901322\pi$$
$$728$$ −21.3137 −0.789939
$$729$$ 0 0
$$730$$ −20.4853 −0.758194
$$731$$ 9.17157 0.339223
$$732$$ 0 0
$$733$$ 21.3137 0.787240 0.393620 0.919273i $$-0.371223\pi$$
0.393620 + 0.919273i $$0.371223\pi$$
$$734$$ 85.8406 3.16844
$$735$$ 0 0
$$736$$ 2.24264 0.0826648
$$737$$ 6.82843 0.251528
$$738$$ 0 0
$$739$$ 5.27208 0.193937 0.0969683 0.995287i $$-0.469085\pi$$
0.0969683 + 0.995287i $$0.469085\pi$$
$$740$$ 32.4853 1.19418
$$741$$ 0 0
$$742$$ −28.9706 −1.06354
$$743$$ 21.5147 0.789298 0.394649 0.918832i $$-0.370866\pi$$
0.394649 + 0.918832i $$0.370866\pi$$
$$744$$ 0 0
$$745$$ −11.6569 −0.427074
$$746$$ −6.48528 −0.237443
$$747$$ 0 0
$$748$$ 10.8284 0.395927
$$749$$ 45.4558 1.66092
$$750$$ 0 0
$$751$$ −27.5147 −1.00403 −0.502013 0.864860i $$-0.667407\pi$$
−0.502013 + 0.864860i $$0.667407\pi$$
$$752$$ 14.4853 0.528224
$$753$$ 0 0
$$754$$ −13.6569 −0.497353
$$755$$ −9.75736 −0.355107
$$756$$ 0 0
$$757$$ −24.1421 −0.877461 −0.438730 0.898619i $$-0.644572\pi$$
−0.438730 + 0.898619i $$0.644572\pi$$
$$758$$ 70.1838 2.54919
$$759$$ 0 0
$$760$$ −2.58579 −0.0937963
$$761$$ −8.62742 −0.312744 −0.156372 0.987698i $$-0.549980\pi$$
−0.156372 + 0.987698i $$0.549980\pi$$
$$762$$ 0 0
$$763$$ −9.65685 −0.349602
$$764$$ −8.97056 −0.324544
$$765$$ 0 0
$$766$$ 70.2843 2.53947
$$767$$ 1.75736 0.0634546
$$768$$ 0 0
$$769$$ −22.9706 −0.828340 −0.414170 0.910200i $$-0.635928\pi$$
−0.414170 + 0.910200i $$0.635928\pi$$
$$770$$ 39.7990 1.43426
$$771$$ 0 0
$$772$$ 16.6274 0.598434
$$773$$ 22.1421 0.796397 0.398199 0.917299i $$-0.369635\pi$$
0.398199 + 0.917299i $$0.369635\pi$$
$$774$$ 0 0
$$775$$ 1.75736 0.0631262
$$776$$ −33.7990 −1.21331
$$777$$ 0 0
$$778$$ −69.1127 −2.47781
$$779$$ 1.85786 0.0665649
$$780$$ 0 0
$$781$$ 40.6274 1.45376
$$782$$ 2.82843 0.101144
$$783$$ 0 0
$$784$$ 48.9411 1.74790
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ 22.4853 0.801514 0.400757 0.916184i $$-0.368747\pi$$
0.400757 + 0.916184i $$0.368747\pi$$
$$788$$ −42.0000 −1.49619
$$789$$ 0 0
$$790$$ 20.4853 0.728834
$$791$$ 42.6274 1.51566
$$792$$ 0 0
$$793$$ 8.00000 0.284088
$$794$$ 28.4853 1.01090
$$795$$ 0 0
$$796$$ 15.3137 0.542780
$$797$$ 22.9706 0.813659 0.406830 0.913504i $$-0.366634\pi$$
0.406830 + 0.913504i $$0.366634\pi$$
$$798$$ 0 0
$$799$$ −4.00000 −0.141510
$$800$$ −1.58579 −0.0560660
$$801$$ 0 0
$$802$$ 12.8284 0.452988
$$803$$ −28.9706 −1.02235
$$804$$ 0 0
$$805$$ 6.82843 0.240670
$$806$$ −4.24264 −0.149441
$$807$$ 0 0
$$808$$ 16.1421 0.567878
$$809$$ −45.2548 −1.59108 −0.795538 0.605904i $$-0.792811\pi$$
−0.795538 + 0.605904i $$0.792811\pi$$
$$810$$ 0 0
$$811$$ −28.3848 −0.996724 −0.498362 0.866969i $$-0.666065\pi$$
−0.498362 + 0.866969i $$0.666065\pi$$
$$812$$ 104.569 3.66964
$$813$$ 0 0
$$814$$ 69.9411 2.45144
$$815$$ −18.9706 −0.664510
$$816$$ 0 0
$$817$$ −6.48528 −0.226891
$$818$$ 17.3137 0.605360
$$819$$ 0 0
$$820$$ −12.1421 −0.424022
$$821$$ 51.2548 1.78881 0.894403 0.447262i $$-0.147601\pi$$
0.894403 + 0.447262i $$0.147601\pi$$
$$822$$ 0 0
$$823$$ −2.38478 −0.0831281 −0.0415640 0.999136i $$-0.513234\pi$$
−0.0415640 + 0.999136i $$0.513234\pi$$
$$824$$ 64.3848 2.24295
$$825$$ 0 0
$$826$$ −20.4853 −0.712774
$$827$$ −56.1421 −1.95225 −0.976127 0.217202i $$-0.930307\pi$$
−0.976127 + 0.217202i $$0.930307\pi$$
$$828$$ 0 0
$$829$$ 40.9706 1.42297 0.711483 0.702703i $$-0.248024\pi$$
0.711483 + 0.702703i $$0.248024\pi$$
$$830$$ −7.65685 −0.265773
$$831$$ 0 0
$$832$$ 9.82843 0.340739
$$833$$ −13.5147 −0.468257
$$834$$ 0 0
$$835$$ −3.17157 −0.109757
$$836$$ −7.65685 −0.264818
$$837$$ 0 0
$$838$$ −26.1421 −0.903065
$$839$$ −6.72792 −0.232274 −0.116137 0.993233i $$-0.537051\pi$$
−0.116137 + 0.993233i $$0.537051\pi$$
$$840$$ 0 0
$$841$$ 3.00000 0.103448
$$842$$ −84.4264 −2.90953
$$843$$ 0 0
$$844$$ 12.6863 0.436680
$$845$$ −1.00000 −0.0344010
$$846$$ 0 0
$$847$$ 3.17157 0.108977
$$848$$ −7.45584 −0.256035
$$849$$ 0 0
$$850$$ −2.00000 −0.0685994
$$851$$ 12.0000 0.411355
$$852$$ 0 0
$$853$$ −13.4558 −0.460719 −0.230360 0.973106i $$-0.573990\pi$$
−0.230360 + 0.973106i $$0.573990\pi$$
$$854$$ −93.2548 −3.19111
$$855$$ 0 0
$$856$$ 41.5563 1.42037
$$857$$ −11.6569 −0.398191 −0.199095 0.979980i $$-0.563800\pi$$
−0.199095 + 0.979980i $$0.563800\pi$$
$$858$$ 0 0
$$859$$ −27.7990 −0.948489 −0.474245 0.880393i $$-0.657279\pi$$
−0.474245 + 0.880393i $$0.657279\pi$$
$$860$$ 42.3848 1.44531
$$861$$ 0 0
$$862$$ −97.4975 −3.32078
$$863$$ 31.4558 1.07077 0.535385 0.844608i $$-0.320167\pi$$
0.535385 + 0.844608i $$0.320167\pi$$
$$864$$ 0 0
$$865$$ 16.8284 0.572184
$$866$$ 18.4853 0.628155
$$867$$ 0 0
$$868$$ 32.4853 1.10262
$$869$$ 28.9706 0.982759
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ −8.82843 −0.298968
$$873$$ 0 0
$$874$$ −2.00000 −0.0676510
$$875$$ −4.82843 −0.163231
$$876$$ 0 0
$$877$$ 25.3137 0.854783 0.427392 0.904067i $$-0.359433\pi$$
0.427392 + 0.904067i $$0.359433\pi$$
$$878$$ 2.34315 0.0790773
$$879$$ 0 0
$$880$$ 10.2426 0.345279
$$881$$ −19.0294 −0.641118 −0.320559 0.947229i $$-0.603871\pi$$
−0.320559 + 0.947229i $$0.603871\pi$$
$$882$$ 0 0
$$883$$ −23.7574 −0.799499 −0.399749 0.916624i $$-0.630903\pi$$
−0.399749 + 0.916624i $$0.630903\pi$$
$$884$$ 3.17157 0.106672
$$885$$ 0 0
$$886$$ 22.7279 0.763559
$$887$$ 22.3848 0.751607 0.375804 0.926699i $$-0.377367\pi$$
0.375804 + 0.926699i $$0.377367\pi$$
$$888$$ 0 0
$$889$$ −31.7990 −1.06650
$$890$$ 14.4853 0.485548
$$891$$ 0 0
$$892$$ 36.4264 1.21965
$$893$$ 2.82843 0.0946497
$$894$$ 0 0
$$895$$ 5.65685 0.189088
$$896$$ −99.2548 −3.31587
$$897$$ 0 0
$$898$$ 79.9411 2.66767
$$899$$ 9.94113 0.331555
$$900$$ 0 0
$$901$$ 2.05887 0.0685911
$$902$$ −26.1421 −0.870438
$$903$$ 0 0
$$904$$ 38.9706 1.29614
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 9.21320 0.305919 0.152960 0.988232i $$-0.451120\pi$$
0.152960 + 0.988232i $$0.451120\pi$$
$$908$$ 62.5685 2.07641
$$909$$ 0 0
$$910$$ 11.6569 0.386421
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ −10.8284 −0.358369
$$914$$ −43.4558 −1.43739
$$915$$ 0 0
$$916$$ −18.4853 −0.610771
$$917$$ 81.9411 2.70593
$$918$$ 0 0
$$919$$ 0.485281 0.0160080 0.00800398 0.999968i $$-0.497452\pi$$
0.00800398 + 0.999968i $$0.497452\pi$$
$$920$$ 6.24264 0.205814
$$921$$ 0 0
$$922$$ −22.9706 −0.756495
$$923$$ 11.8995 0.391677
$$924$$ 0 0
$$925$$ −8.48528 −0.278994
$$926$$ −10.4853 −0.344568
$$927$$ 0 0
$$928$$ −8.97056 −0.294473
$$929$$ −16.8284 −0.552123 −0.276061 0.961140i $$-0.589029\pi$$
−0.276061 + 0.961140i $$0.589029\pi$$
$$930$$ 0 0
$$931$$ 9.55635 0.313197
$$932$$ 78.9706 2.58677
$$933$$ 0 0
$$934$$ 32.3848 1.05966
$$935$$ −2.82843 −0.0924995
$$936$$ 0 0
$$937$$ −22.9706 −0.750416 −0.375208 0.926941i $$-0.622429\pi$$
−0.375208 + 0.926941i $$0.622429\pi$$
$$938$$ −23.3137 −0.761220
$$939$$ 0 0
$$940$$ −18.4853 −0.602923
$$941$$ −18.7696 −0.611870 −0.305935 0.952052i $$-0.598969\pi$$
−0.305935 + 0.952052i $$0.598969\pi$$
$$942$$ 0 0
$$943$$ −4.48528 −0.146061
$$944$$ −5.27208 −0.171592
$$945$$ 0 0
$$946$$ 91.2548 2.96695
$$947$$ −17.1127 −0.556088 −0.278044 0.960568i $$-0.589686\pi$$
−0.278044 + 0.960568i $$0.589686\pi$$
$$948$$ 0 0
$$949$$ −8.48528 −0.275444
$$950$$ 1.41421 0.0458831
$$951$$ 0 0
$$952$$ −17.6569 −0.572262
$$953$$ −35.2548 −1.14202 −0.571008 0.820944i $$-0.693447\pi$$
−0.571008 + 0.820944i $$0.693447\pi$$
$$954$$ 0 0
$$955$$ 2.34315 0.0758224
$$956$$ 13.0711 0.422749
$$957$$ 0 0
$$958$$ 74.1838 2.39677
$$959$$ 83.5980 2.69952
$$960$$ 0 0
$$961$$ −27.9117 −0.900377
$$962$$ 20.4853 0.660472
$$963$$ 0 0
$$964$$ −55.4558 −1.78611
$$965$$ −4.34315 −0.139811
$$966$$ 0 0
$$967$$ 47.9411 1.54168 0.770841 0.637027i $$-0.219836\pi$$
0.770841 + 0.637027i $$0.219836\pi$$
$$968$$ 2.89949 0.0931933
$$969$$ 0 0
$$970$$ 18.4853 0.593527
$$971$$ −44.2843 −1.42115 −0.710575 0.703622i $$-0.751565\pi$$
−0.710575 + 0.703622i $$0.751565\pi$$
$$972$$ 0 0
$$973$$ 21.6569 0.694287
$$974$$ −26.4853 −0.848643
$$975$$ 0 0
$$976$$ −24.0000 −0.768221
$$977$$ 39.5147 1.26419 0.632094 0.774892i $$-0.282196\pi$$
0.632094 + 0.774892i $$0.282196\pi$$
$$978$$ 0 0
$$979$$ 20.4853 0.654712
$$980$$ −62.4558 −1.99508
$$981$$ 0 0
$$982$$ −12.4853 −0.398421
$$983$$ 1.02944 0.0328339 0.0164170 0.999865i $$-0.494774\pi$$
0.0164170 + 0.999865i $$0.494774\pi$$
$$984$$ 0 0
$$985$$ 10.9706 0.349551
$$986$$ −11.3137 −0.360302
$$987$$ 0 0
$$988$$ −2.24264 −0.0713479
$$989$$ 15.6569 0.497859
$$990$$ 0 0
$$991$$ 48.9706 1.55560 0.777801 0.628511i $$-0.216335\pi$$
0.777801 + 0.628511i $$0.216335\pi$$
$$992$$ −2.78680 −0.0884809
$$993$$ 0 0
$$994$$ −138.711 −4.39964
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ 28.8284 0.913005 0.456503 0.889722i $$-0.349102\pi$$
0.456503 + 0.889722i $$0.349102\pi$$
$$998$$ 100.326 3.17576
$$999$$ 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 585.2.a.m.1.2 2
3.2 odd 2 65.2.a.b.1.1 2
4.3 odd 2 9360.2.a.cd.1.1 2
5.2 odd 4 2925.2.c.r.2224.4 4
5.3 odd 4 2925.2.c.r.2224.1 4
5.4 even 2 2925.2.a.u.1.1 2
12.11 even 2 1040.2.a.j.1.2 2
13.12 even 2 7605.2.a.x.1.1 2
15.2 even 4 325.2.b.f.274.1 4
15.8 even 4 325.2.b.f.274.4 4
15.14 odd 2 325.2.a.i.1.2 2
21.20 even 2 3185.2.a.j.1.1 2
24.5 odd 2 4160.2.a.bf.1.2 2
24.11 even 2 4160.2.a.z.1.1 2
33.32 even 2 7865.2.a.j.1.2 2
39.2 even 12 845.2.m.f.316.1 8
39.5 even 4 845.2.c.b.506.4 4
39.8 even 4 845.2.c.b.506.1 4
39.11 even 12 845.2.m.f.316.4 8
39.17 odd 6 845.2.e.c.146.1 4
39.20 even 12 845.2.m.f.361.1 8
39.23 odd 6 845.2.e.c.191.1 4
39.29 odd 6 845.2.e.h.191.2 4
39.32 even 12 845.2.m.f.361.4 8
39.35 odd 6 845.2.e.h.146.2 4
39.38 odd 2 845.2.a.g.1.2 2
60.59 even 2 5200.2.a.bu.1.1 2
195.194 odd 2 4225.2.a.r.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.b.1.1 2 3.2 odd 2
325.2.a.i.1.2 2 15.14 odd 2
325.2.b.f.274.1 4 15.2 even 4
325.2.b.f.274.4 4 15.8 even 4
585.2.a.m.1.2 2 1.1 even 1 trivial
845.2.a.g.1.2 2 39.38 odd 2
845.2.c.b.506.1 4 39.8 even 4
845.2.c.b.506.4 4 39.5 even 4
845.2.e.c.146.1 4 39.17 odd 6
845.2.e.c.191.1 4 39.23 odd 6
845.2.e.h.146.2 4 39.35 odd 6
845.2.e.h.191.2 4 39.29 odd 6
845.2.m.f.316.1 8 39.2 even 12
845.2.m.f.316.4 8 39.11 even 12
845.2.m.f.361.1 8 39.20 even 12
845.2.m.f.361.4 8 39.32 even 12
1040.2.a.j.1.2 2 12.11 even 2
2925.2.a.u.1.1 2 5.4 even 2
2925.2.c.r.2224.1 4 5.3 odd 4
2925.2.c.r.2224.4 4 5.2 odd 4
3185.2.a.j.1.1 2 21.20 even 2
4160.2.a.z.1.1 2 24.11 even 2
4160.2.a.bf.1.2 2 24.5 odd 2
4225.2.a.r.1.1 2 195.194 odd 2
5200.2.a.bu.1.1 2 60.59 even 2
7605.2.a.x.1.1 2 13.12 even 2
7865.2.a.j.1.2 2 33.32 even 2
9360.2.a.cd.1.1 2 4.3 odd 2