# Properties

 Label 585.2.a.l.1.1 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 585.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} +O(q^{10})$$ $$q-1.56155 q^{2} +0.438447 q^{4} +1.00000 q^{5} -4.56155 q^{7} +2.43845 q^{8} -1.56155 q^{10} +2.56155 q^{11} -1.00000 q^{13} +7.12311 q^{14} -4.68466 q^{16} -2.56155 q^{17} +3.12311 q^{19} +0.438447 q^{20} -4.00000 q^{22} +6.56155 q^{23} +1.00000 q^{25} +1.56155 q^{26} -2.00000 q^{28} +1.12311 q^{29} +6.00000 q^{31} +2.43845 q^{32} +4.00000 q^{34} -4.56155 q^{35} +1.68466 q^{37} -4.87689 q^{38} +2.43845 q^{40} -0.561553 q^{41} -5.12311 q^{43} +1.12311 q^{44} -10.2462 q^{46} +2.87689 q^{47} +13.8078 q^{49} -1.56155 q^{50} -0.438447 q^{52} +7.68466 q^{53} +2.56155 q^{55} -11.1231 q^{56} -1.75379 q^{58} +12.0000 q^{59} +5.68466 q^{61} -9.36932 q^{62} +5.56155 q^{64} -1.00000 q^{65} +13.3693 q^{67} -1.12311 q^{68} +7.12311 q^{70} +14.5616 q^{71} -6.00000 q^{73} -2.63068 q^{74} +1.36932 q^{76} -11.6847 q^{77} -7.68466 q^{79} -4.68466 q^{80} +0.876894 q^{82} +16.4924 q^{83} -2.56155 q^{85} +8.00000 q^{86} +6.24621 q^{88} -1.68466 q^{89} +4.56155 q^{91} +2.87689 q^{92} -4.49242 q^{94} +3.12311 q^{95} -11.9309 q^{97} -21.5616 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8 $$2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29} + 12 q^{31} + 9 q^{32} + 8 q^{34} - 5 q^{35} - 9 q^{37} - 18 q^{38} + 9 q^{40} + 3 q^{41} - 2 q^{43} - 6 q^{44} - 4 q^{46} + 14 q^{47} + 7 q^{49} + q^{50} - 5 q^{52} + 3 q^{53} + q^{55} - 14 q^{56} - 20 q^{58} + 24 q^{59} - q^{61} + 6 q^{62} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 6 q^{68} + 6 q^{70} + 25 q^{71} - 12 q^{73} - 30 q^{74} - 22 q^{76} - 11 q^{77} - 3 q^{79} + 3 q^{80} + 10 q^{82} - q^{85} + 16 q^{86} - 4 q^{88} + 9 q^{89} + 5 q^{91} + 14 q^{92} + 24 q^{94} - 2 q^{95} + 5 q^{97} - 39 q^{98}+O(q^{100})$$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8 + q^10 + q^11 - 2 * q^13 + 6 * q^14 + 3 * q^16 - q^17 - 2 * q^19 + 5 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 - q^26 - 4 * q^28 - 6 * q^29 + 12 * q^31 + 9 * q^32 + 8 * q^34 - 5 * q^35 - 9 * q^37 - 18 * q^38 + 9 * q^40 + 3 * q^41 - 2 * q^43 - 6 * q^44 - 4 * q^46 + 14 * q^47 + 7 * q^49 + q^50 - 5 * q^52 + 3 * q^53 + q^55 - 14 * q^56 - 20 * q^58 + 24 * q^59 - q^61 + 6 * q^62 + 7 * q^64 - 2 * q^65 + 2 * q^67 + 6 * q^68 + 6 * q^70 + 25 * q^71 - 12 * q^73 - 30 * q^74 - 22 * q^76 - 11 * q^77 - 3 * q^79 + 3 * q^80 + 10 * q^82 - q^85 + 16 * q^86 - 4 * q^88 + 9 * q^89 + 5 * q^91 + 14 * q^92 + 24 * q^94 - 2 * q^95 + 5 * q^97 - 39 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.56155 −1.10418 −0.552092 0.833783i $$-0.686170\pi$$
−0.552092 + 0.833783i $$0.686170\pi$$
$$3$$ 0 0
$$4$$ 0.438447 0.219224
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.56155 −1.72410 −0.862052 0.506819i $$-0.830821\pi$$
−0.862052 + 0.506819i $$0.830821\pi$$
$$8$$ 2.43845 0.862121
$$9$$ 0 0
$$10$$ −1.56155 −0.493806
$$11$$ 2.56155 0.772337 0.386169 0.922428i $$-0.373798\pi$$
0.386169 + 0.922428i $$0.373798\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 7.12311 1.90373
$$15$$ 0 0
$$16$$ −4.68466 −1.17116
$$17$$ −2.56155 −0.621268 −0.310634 0.950530i $$-0.600541\pi$$
−0.310634 + 0.950530i $$0.600541\pi$$
$$18$$ 0 0
$$19$$ 3.12311 0.716490 0.358245 0.933628i $$-0.383375\pi$$
0.358245 + 0.933628i $$0.383375\pi$$
$$20$$ 0.438447 0.0980398
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 6.56155 1.36818 0.684089 0.729398i $$-0.260200\pi$$
0.684089 + 0.729398i $$0.260200\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.56155 0.306246
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ 1.12311 0.208555 0.104278 0.994548i $$-0.466747\pi$$
0.104278 + 0.994548i $$0.466747\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 2.43845 0.431061
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ −4.56155 −0.771043
$$36$$ 0 0
$$37$$ 1.68466 0.276956 0.138478 0.990366i $$-0.455779\pi$$
0.138478 + 0.990366i $$0.455779\pi$$
$$38$$ −4.87689 −0.791137
$$39$$ 0 0
$$40$$ 2.43845 0.385552
$$41$$ −0.561553 −0.0876998 −0.0438499 0.999038i $$-0.513962\pi$$
−0.0438499 + 0.999038i $$0.513962\pi$$
$$42$$ 0 0
$$43$$ −5.12311 −0.781266 −0.390633 0.920546i $$-0.627744\pi$$
−0.390633 + 0.920546i $$0.627744\pi$$
$$44$$ 1.12311 0.169315
$$45$$ 0 0
$$46$$ −10.2462 −1.51072
$$47$$ 2.87689 0.419638 0.209819 0.977740i $$-0.432712\pi$$
0.209819 + 0.977740i $$0.432712\pi$$
$$48$$ 0 0
$$49$$ 13.8078 1.97254
$$50$$ −1.56155 −0.220837
$$51$$ 0 0
$$52$$ −0.438447 −0.0608017
$$53$$ 7.68466 1.05557 0.527785 0.849378i $$-0.323023\pi$$
0.527785 + 0.849378i $$0.323023\pi$$
$$54$$ 0 0
$$55$$ 2.56155 0.345400
$$56$$ −11.1231 −1.48639
$$57$$ 0 0
$$58$$ −1.75379 −0.230284
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 5.68466 0.727846 0.363923 0.931429i $$-0.381437\pi$$
0.363923 + 0.931429i $$0.381437\pi$$
$$62$$ −9.36932 −1.18990
$$63$$ 0 0
$$64$$ 5.56155 0.695194
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ 13.3693 1.63332 0.816661 0.577118i $$-0.195823\pi$$
0.816661 + 0.577118i $$0.195823\pi$$
$$68$$ −1.12311 −0.136197
$$69$$ 0 0
$$70$$ 7.12311 0.851374
$$71$$ 14.5616 1.72814 0.864069 0.503373i $$-0.167908\pi$$
0.864069 + 0.503373i $$0.167908\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ −2.63068 −0.305811
$$75$$ 0 0
$$76$$ 1.36932 0.157071
$$77$$ −11.6847 −1.33159
$$78$$ 0 0
$$79$$ −7.68466 −0.864592 −0.432296 0.901732i $$-0.642296\pi$$
−0.432296 + 0.901732i $$0.642296\pi$$
$$80$$ −4.68466 −0.523761
$$81$$ 0 0
$$82$$ 0.876894 0.0968368
$$83$$ 16.4924 1.81028 0.905139 0.425115i $$-0.139766\pi$$
0.905139 + 0.425115i $$0.139766\pi$$
$$84$$ 0 0
$$85$$ −2.56155 −0.277839
$$86$$ 8.00000 0.862662
$$87$$ 0 0
$$88$$ 6.24621 0.665848
$$89$$ −1.68466 −0.178573 −0.0892867 0.996006i $$-0.528459\pi$$
−0.0892867 + 0.996006i $$0.528459\pi$$
$$90$$ 0 0
$$91$$ 4.56155 0.478181
$$92$$ 2.87689 0.299937
$$93$$ 0 0
$$94$$ −4.49242 −0.463358
$$95$$ 3.12311 0.320424
$$96$$ 0 0
$$97$$ −11.9309 −1.21140 −0.605698 0.795695i $$-0.707106\pi$$
−0.605698 + 0.795695i $$0.707106\pi$$
$$98$$ −21.5616 −2.17805
$$99$$ 0 0
$$100$$ 0.438447 0.0438447
$$101$$ −6.24621 −0.621521 −0.310761 0.950488i $$-0.600584\pi$$
−0.310761 + 0.950488i $$0.600584\pi$$
$$102$$ 0 0
$$103$$ −6.87689 −0.677601 −0.338800 0.940858i $$-0.610021\pi$$
−0.338800 + 0.940858i $$0.610021\pi$$
$$104$$ −2.43845 −0.239109
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ −17.9309 −1.73344 −0.866721 0.498793i $$-0.833777\pi$$
−0.866721 + 0.498793i $$0.833777\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ 21.3693 2.01921
$$113$$ −13.1231 −1.23452 −0.617259 0.786760i $$-0.711757\pi$$
−0.617259 + 0.786760i $$0.711757\pi$$
$$114$$ 0 0
$$115$$ 6.56155 0.611868
$$116$$ 0.492423 0.0457203
$$117$$ 0 0
$$118$$ −18.7386 −1.72503
$$119$$ 11.6847 1.07113
$$120$$ 0 0
$$121$$ −4.43845 −0.403495
$$122$$ −8.87689 −0.803676
$$123$$ 0 0
$$124$$ 2.63068 0.236242
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 18.2462 1.61909 0.809545 0.587058i $$-0.199714\pi$$
0.809545 + 0.587058i $$0.199714\pi$$
$$128$$ −13.5616 −1.19868
$$129$$ 0 0
$$130$$ 1.56155 0.136957
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −14.2462 −1.23530
$$134$$ −20.8769 −1.80349
$$135$$ 0 0
$$136$$ −6.24621 −0.535608
$$137$$ −7.12311 −0.608568 −0.304284 0.952581i $$-0.598417\pi$$
−0.304284 + 0.952581i $$0.598417\pi$$
$$138$$ 0 0
$$139$$ 15.6847 1.33036 0.665178 0.746685i $$-0.268356\pi$$
0.665178 + 0.746685i $$0.268356\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ −22.7386 −1.90818
$$143$$ −2.56155 −0.214208
$$144$$ 0 0
$$145$$ 1.12311 0.0932688
$$146$$ 9.36932 0.775410
$$147$$ 0 0
$$148$$ 0.738634 0.0607153
$$149$$ 2.80776 0.230021 0.115010 0.993364i $$-0.463310\pi$$
0.115010 + 0.993364i $$0.463310\pi$$
$$150$$ 0 0
$$151$$ −13.3693 −1.08798 −0.543990 0.839092i $$-0.683087\pi$$
−0.543990 + 0.839092i $$0.683087\pi$$
$$152$$ 7.61553 0.617701
$$153$$ 0 0
$$154$$ 18.2462 1.47032
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −21.3693 −1.70546 −0.852729 0.522354i $$-0.825054\pi$$
−0.852729 + 0.522354i $$0.825054\pi$$
$$158$$ 12.0000 0.954669
$$159$$ 0 0
$$160$$ 2.43845 0.192776
$$161$$ −29.9309 −2.35888
$$162$$ 0 0
$$163$$ −21.0540 −1.64907 −0.824537 0.565808i $$-0.808564\pi$$
−0.824537 + 0.565808i $$0.808564\pi$$
$$164$$ −0.246211 −0.0192259
$$165$$ 0 0
$$166$$ −25.7538 −1.99888
$$167$$ 13.1231 1.01550 0.507748 0.861506i $$-0.330478\pi$$
0.507748 + 0.861506i $$0.330478\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ −2.24621 −0.171272
$$173$$ 21.1231 1.60596 0.802980 0.596006i $$-0.203247\pi$$
0.802980 + 0.596006i $$0.203247\pi$$
$$174$$ 0 0
$$175$$ −4.56155 −0.344821
$$176$$ −12.0000 −0.904534
$$177$$ 0 0
$$178$$ 2.63068 0.197178
$$179$$ −13.1231 −0.980867 −0.490433 0.871479i $$-0.663162\pi$$
−0.490433 + 0.871479i $$0.663162\pi$$
$$180$$ 0 0
$$181$$ 25.6847 1.90913 0.954563 0.298010i $$-0.0963228\pi$$
0.954563 + 0.298010i $$0.0963228\pi$$
$$182$$ −7.12311 −0.528000
$$183$$ 0 0
$$184$$ 16.0000 1.17954
$$185$$ 1.68466 0.123859
$$186$$ 0 0
$$187$$ −6.56155 −0.479828
$$188$$ 1.26137 0.0919946
$$189$$ 0 0
$$190$$ −4.87689 −0.353807
$$191$$ −21.6155 −1.56404 −0.782022 0.623250i $$-0.785812\pi$$
−0.782022 + 0.623250i $$0.785812\pi$$
$$192$$ 0 0
$$193$$ 15.4384 1.11128 0.555642 0.831422i $$-0.312473\pi$$
0.555642 + 0.831422i $$0.312473\pi$$
$$194$$ 18.6307 1.33761
$$195$$ 0 0
$$196$$ 6.05398 0.432427
$$197$$ 21.3693 1.52250 0.761250 0.648458i $$-0.224586\pi$$
0.761250 + 0.648458i $$0.224586\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 2.43845 0.172424
$$201$$ 0 0
$$202$$ 9.75379 0.686274
$$203$$ −5.12311 −0.359572
$$204$$ 0 0
$$205$$ −0.561553 −0.0392205
$$206$$ 10.7386 0.748196
$$207$$ 0 0
$$208$$ 4.68466 0.324823
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −10.2462 −0.705378 −0.352689 0.935741i $$-0.614733\pi$$
−0.352689 + 0.935741i $$0.614733\pi$$
$$212$$ 3.36932 0.231406
$$213$$ 0 0
$$214$$ 28.0000 1.91404
$$215$$ −5.12311 −0.349393
$$216$$ 0 0
$$217$$ −27.3693 −1.85795
$$218$$ 3.12311 0.211523
$$219$$ 0 0
$$220$$ 1.12311 0.0757198
$$221$$ 2.56155 0.172309
$$222$$ 0 0
$$223$$ 9.36932 0.627416 0.313708 0.949520i $$-0.398429\pi$$
0.313708 + 0.949520i $$0.398429\pi$$
$$224$$ −11.1231 −0.743194
$$225$$ 0 0
$$226$$ 20.4924 1.36314
$$227$$ 22.2462 1.47653 0.738266 0.674509i $$-0.235645\pi$$
0.738266 + 0.674509i $$0.235645\pi$$
$$228$$ 0 0
$$229$$ 8.87689 0.586602 0.293301 0.956020i $$-0.405246\pi$$
0.293301 + 0.956020i $$0.405246\pi$$
$$230$$ −10.2462 −0.675615
$$231$$ 0 0
$$232$$ 2.73863 0.179800
$$233$$ −7.19224 −0.471179 −0.235590 0.971853i $$-0.575702\pi$$
−0.235590 + 0.971853i $$0.575702\pi$$
$$234$$ 0 0
$$235$$ 2.87689 0.187668
$$236$$ 5.26137 0.342486
$$237$$ 0 0
$$238$$ −18.2462 −1.18273
$$239$$ −5.93087 −0.383636 −0.191818 0.981431i $$-0.561438\pi$$
−0.191818 + 0.981431i $$0.561438\pi$$
$$240$$ 0 0
$$241$$ −24.7386 −1.59356 −0.796778 0.604272i $$-0.793464\pi$$
−0.796778 + 0.604272i $$0.793464\pi$$
$$242$$ 6.93087 0.445533
$$243$$ 0 0
$$244$$ 2.49242 0.159561
$$245$$ 13.8078 0.882146
$$246$$ 0 0
$$247$$ −3.12311 −0.198718
$$248$$ 14.6307 0.929049
$$249$$ 0 0
$$250$$ −1.56155 −0.0987613
$$251$$ 9.75379 0.615654 0.307827 0.951442i $$-0.400398\pi$$
0.307827 + 0.951442i $$0.400398\pi$$
$$252$$ 0 0
$$253$$ 16.8078 1.05670
$$254$$ −28.4924 −1.78777
$$255$$ 0 0
$$256$$ 10.0540 0.628373
$$257$$ −21.1231 −1.31762 −0.658812 0.752308i $$-0.728941\pi$$
−0.658812 + 0.752308i $$0.728941\pi$$
$$258$$ 0 0
$$259$$ −7.68466 −0.477501
$$260$$ −0.438447 −0.0271913
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 14.2462 0.878459 0.439230 0.898375i $$-0.355251\pi$$
0.439230 + 0.898375i $$0.355251\pi$$
$$264$$ 0 0
$$265$$ 7.68466 0.472065
$$266$$ 22.2462 1.36400
$$267$$ 0 0
$$268$$ 5.86174 0.358063
$$269$$ −9.12311 −0.556246 −0.278123 0.960546i $$-0.589712\pi$$
−0.278123 + 0.960546i $$0.589712\pi$$
$$270$$ 0 0
$$271$$ 5.36932 0.326163 0.163081 0.986613i $$-0.447857\pi$$
0.163081 + 0.986613i $$0.447857\pi$$
$$272$$ 12.0000 0.727607
$$273$$ 0 0
$$274$$ 11.1231 0.671971
$$275$$ 2.56155 0.154467
$$276$$ 0 0
$$277$$ 4.24621 0.255130 0.127565 0.991830i $$-0.459284\pi$$
0.127565 + 0.991830i $$0.459284\pi$$
$$278$$ −24.4924 −1.46896
$$279$$ 0 0
$$280$$ −11.1231 −0.664733
$$281$$ −12.2462 −0.730548 −0.365274 0.930900i $$-0.619025\pi$$
−0.365274 + 0.930900i $$0.619025\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 6.38447 0.378849
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 2.56155 0.151204
$$288$$ 0 0
$$289$$ −10.4384 −0.614026
$$290$$ −1.75379 −0.102986
$$291$$ 0 0
$$292$$ −2.63068 −0.153949
$$293$$ −3.75379 −0.219299 −0.109649 0.993970i $$-0.534973\pi$$
−0.109649 + 0.993970i $$0.534973\pi$$
$$294$$ 0 0
$$295$$ 12.0000 0.698667
$$296$$ 4.10795 0.238770
$$297$$ 0 0
$$298$$ −4.38447 −0.253986
$$299$$ −6.56155 −0.379464
$$300$$ 0 0
$$301$$ 23.3693 1.34699
$$302$$ 20.8769 1.20133
$$303$$ 0 0
$$304$$ −14.6307 −0.839127
$$305$$ 5.68466 0.325503
$$306$$ 0 0
$$307$$ 3.43845 0.196243 0.0981213 0.995174i $$-0.468717\pi$$
0.0981213 + 0.995174i $$0.468717\pi$$
$$308$$ −5.12311 −0.291916
$$309$$ 0 0
$$310$$ −9.36932 −0.532141
$$311$$ 27.3693 1.55197 0.775986 0.630750i $$-0.217253\pi$$
0.775986 + 0.630750i $$0.217253\pi$$
$$312$$ 0 0
$$313$$ 20.8769 1.18003 0.590016 0.807392i $$-0.299121\pi$$
0.590016 + 0.807392i $$0.299121\pi$$
$$314$$ 33.3693 1.88314
$$315$$ 0 0
$$316$$ −3.36932 −0.189539
$$317$$ 34.4924 1.93729 0.968644 0.248454i $$-0.0799224\pi$$
0.968644 + 0.248454i $$0.0799224\pi$$
$$318$$ 0 0
$$319$$ 2.87689 0.161075
$$320$$ 5.56155 0.310900
$$321$$ 0 0
$$322$$ 46.7386 2.60464
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 32.8769 1.82088
$$327$$ 0 0
$$328$$ −1.36932 −0.0756079
$$329$$ −13.1231 −0.723500
$$330$$ 0 0
$$331$$ 20.8769 1.14750 0.573749 0.819031i $$-0.305489\pi$$
0.573749 + 0.819031i $$0.305489\pi$$
$$332$$ 7.23106 0.396856
$$333$$ 0 0
$$334$$ −20.4924 −1.12130
$$335$$ 13.3693 0.730444
$$336$$ 0 0
$$337$$ 2.49242 0.135771 0.0678855 0.997693i $$-0.478375\pi$$
0.0678855 + 0.997693i $$0.478375\pi$$
$$338$$ −1.56155 −0.0849373
$$339$$ 0 0
$$340$$ −1.12311 −0.0609090
$$341$$ 15.3693 0.832295
$$342$$ 0 0
$$343$$ −31.0540 −1.67676
$$344$$ −12.4924 −0.673546
$$345$$ 0 0
$$346$$ −32.9848 −1.77328
$$347$$ 11.0540 0.593408 0.296704 0.954969i $$-0.404112\pi$$
0.296704 + 0.954969i $$0.404112\pi$$
$$348$$ 0 0
$$349$$ 1.36932 0.0732979 0.0366489 0.999328i $$-0.488332\pi$$
0.0366489 + 0.999328i $$0.488332\pi$$
$$350$$ 7.12311 0.380746
$$351$$ 0 0
$$352$$ 6.24621 0.332924
$$353$$ 10.4924 0.558455 0.279228 0.960225i $$-0.409922\pi$$
0.279228 + 0.960225i $$0.409922\pi$$
$$354$$ 0 0
$$355$$ 14.5616 0.772847
$$356$$ −0.738634 −0.0391475
$$357$$ 0 0
$$358$$ 20.4924 1.08306
$$359$$ 2.24621 0.118550 0.0592752 0.998242i $$-0.481121\pi$$
0.0592752 + 0.998242i $$0.481121\pi$$
$$360$$ 0 0
$$361$$ −9.24621 −0.486643
$$362$$ −40.1080 −2.10803
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ −6.00000 −0.314054
$$366$$ 0 0
$$367$$ −18.2462 −0.952444 −0.476222 0.879325i $$-0.657994\pi$$
−0.476222 + 0.879325i $$0.657994\pi$$
$$368$$ −30.7386 −1.60236
$$369$$ 0 0
$$370$$ −2.63068 −0.136763
$$371$$ −35.0540 −1.81991
$$372$$ 0 0
$$373$$ −4.24621 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$374$$ 10.2462 0.529819
$$375$$ 0 0
$$376$$ 7.01515 0.361779
$$377$$ −1.12311 −0.0578429
$$378$$ 0 0
$$379$$ 2.49242 0.128027 0.0640136 0.997949i $$-0.479610\pi$$
0.0640136 + 0.997949i $$0.479610\pi$$
$$380$$ 1.36932 0.0702445
$$381$$ 0 0
$$382$$ 33.7538 1.72699
$$383$$ 36.4924 1.86468 0.932338 0.361588i $$-0.117765\pi$$
0.932338 + 0.361588i $$0.117765\pi$$
$$384$$ 0 0
$$385$$ −11.6847 −0.595505
$$386$$ −24.1080 −1.22706
$$387$$ 0 0
$$388$$ −5.23106 −0.265567
$$389$$ −19.8617 −1.00703 −0.503515 0.863986i $$-0.667960\pi$$
−0.503515 + 0.863986i $$0.667960\pi$$
$$390$$ 0 0
$$391$$ −16.8078 −0.850005
$$392$$ 33.6695 1.70057
$$393$$ 0 0
$$394$$ −33.3693 −1.68112
$$395$$ −7.68466 −0.386657
$$396$$ 0 0
$$397$$ −8.56155 −0.429692 −0.214846 0.976648i $$-0.568925\pi$$
−0.214846 + 0.976648i $$0.568925\pi$$
$$398$$ 12.4924 0.626189
$$399$$ 0 0
$$400$$ −4.68466 −0.234233
$$401$$ 20.2462 1.01105 0.505524 0.862813i $$-0.331299\pi$$
0.505524 + 0.862813i $$0.331299\pi$$
$$402$$ 0 0
$$403$$ −6.00000 −0.298881
$$404$$ −2.73863 −0.136252
$$405$$ 0 0
$$406$$ 8.00000 0.397033
$$407$$ 4.31534 0.213904
$$408$$ 0 0
$$409$$ −15.1231 −0.747789 −0.373895 0.927471i $$-0.621978\pi$$
−0.373895 + 0.927471i $$0.621978\pi$$
$$410$$ 0.876894 0.0433067
$$411$$ 0 0
$$412$$ −3.01515 −0.148546
$$413$$ −54.7386 −2.69351
$$414$$ 0 0
$$415$$ 16.4924 0.809581
$$416$$ −2.43845 −0.119555
$$417$$ 0 0
$$418$$ −12.4924 −0.611024
$$419$$ −33.6155 −1.64223 −0.821113 0.570766i $$-0.806646\pi$$
−0.821113 + 0.570766i $$0.806646\pi$$
$$420$$ 0 0
$$421$$ 14.6307 0.713056 0.356528 0.934285i $$-0.383960\pi$$
0.356528 + 0.934285i $$0.383960\pi$$
$$422$$ 16.0000 0.778868
$$423$$ 0 0
$$424$$ 18.7386 0.910029
$$425$$ −2.56155 −0.124254
$$426$$ 0 0
$$427$$ −25.9309 −1.25488
$$428$$ −7.86174 −0.380012
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ −36.4924 −1.75778 −0.878889 0.477026i $$-0.841715\pi$$
−0.878889 + 0.477026i $$0.841715\pi$$
$$432$$ 0 0
$$433$$ 31.6155 1.51935 0.759673 0.650306i $$-0.225359\pi$$
0.759673 + 0.650306i $$0.225359\pi$$
$$434$$ 42.7386 2.05152
$$435$$ 0 0
$$436$$ −0.876894 −0.0419956
$$437$$ 20.4924 0.980286
$$438$$ 0 0
$$439$$ −15.0540 −0.718487 −0.359244 0.933244i $$-0.616965\pi$$
−0.359244 + 0.933244i $$0.616965\pi$$
$$440$$ 6.24621 0.297776
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ 12.8078 0.608515 0.304258 0.952590i $$-0.401592\pi$$
0.304258 + 0.952590i $$0.401592\pi$$
$$444$$ 0 0
$$445$$ −1.68466 −0.0798605
$$446$$ −14.6307 −0.692783
$$447$$ 0 0
$$448$$ −25.3693 −1.19859
$$449$$ 10.1771 0.480286 0.240143 0.970738i $$-0.422806\pi$$
0.240143 + 0.970738i $$0.422806\pi$$
$$450$$ 0 0
$$451$$ −1.43845 −0.0677338
$$452$$ −5.75379 −0.270635
$$453$$ 0 0
$$454$$ −34.7386 −1.63036
$$455$$ 4.56155 0.213849
$$456$$ 0 0
$$457$$ −25.6847 −1.20148 −0.600739 0.799445i $$-0.705127\pi$$
−0.600739 + 0.799445i $$0.705127\pi$$
$$458$$ −13.8617 −0.647717
$$459$$ 0 0
$$460$$ 2.87689 0.134136
$$461$$ −29.0540 −1.35318 −0.676589 0.736361i $$-0.736543\pi$$
−0.676589 + 0.736361i $$0.736543\pi$$
$$462$$ 0 0
$$463$$ −0.0691303 −0.00321276 −0.00160638 0.999999i $$-0.500511\pi$$
−0.00160638 + 0.999999i $$0.500511\pi$$
$$464$$ −5.26137 −0.244253
$$465$$ 0 0
$$466$$ 11.2311 0.520269
$$467$$ −21.9309 −1.01484 −0.507420 0.861699i $$-0.669401\pi$$
−0.507420 + 0.861699i $$0.669401\pi$$
$$468$$ 0 0
$$469$$ −60.9848 −2.81602
$$470$$ −4.49242 −0.207220
$$471$$ 0 0
$$472$$ 29.2614 1.34686
$$473$$ −13.1231 −0.603401
$$474$$ 0 0
$$475$$ 3.12311 0.143298
$$476$$ 5.12311 0.234817
$$477$$ 0 0
$$478$$ 9.26137 0.423605
$$479$$ −11.0540 −0.505069 −0.252535 0.967588i $$-0.581264\pi$$
−0.252535 + 0.967588i $$0.581264\pi$$
$$480$$ 0 0
$$481$$ −1.68466 −0.0768138
$$482$$ 38.6307 1.75958
$$483$$ 0 0
$$484$$ −1.94602 −0.0884557
$$485$$ −11.9309 −0.541753
$$486$$ 0 0
$$487$$ 5.05398 0.229017 0.114509 0.993422i $$-0.463471\pi$$
0.114509 + 0.993422i $$0.463471\pi$$
$$488$$ 13.8617 0.627491
$$489$$ 0 0
$$490$$ −21.5616 −0.974052
$$491$$ 13.6155 0.614460 0.307230 0.951635i $$-0.400598\pi$$
0.307230 + 0.951635i $$0.400598\pi$$
$$492$$ 0 0
$$493$$ −2.87689 −0.129569
$$494$$ 4.87689 0.219422
$$495$$ 0 0
$$496$$ −28.1080 −1.26208
$$497$$ −66.4233 −2.97949
$$498$$ 0 0
$$499$$ −30.9848 −1.38707 −0.693536 0.720422i $$-0.743948\pi$$
−0.693536 + 0.720422i $$0.743948\pi$$
$$500$$ 0.438447 0.0196080
$$501$$ 0 0
$$502$$ −15.2311 −0.679795
$$503$$ 30.2462 1.34861 0.674306 0.738452i $$-0.264443\pi$$
0.674306 + 0.738452i $$0.264443\pi$$
$$504$$ 0 0
$$505$$ −6.24621 −0.277953
$$506$$ −26.2462 −1.16679
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −19.4384 −0.861594 −0.430797 0.902449i $$-0.641768\pi$$
−0.430797 + 0.902449i $$0.641768\pi$$
$$510$$ 0 0
$$511$$ 27.3693 1.21075
$$512$$ 11.4233 0.504843
$$513$$ 0 0
$$514$$ 32.9848 1.45490
$$515$$ −6.87689 −0.303032
$$516$$ 0 0
$$517$$ 7.36932 0.324102
$$518$$ 12.0000 0.527250
$$519$$ 0 0
$$520$$ −2.43845 −0.106933
$$521$$ 31.8617 1.39589 0.697944 0.716152i $$-0.254098\pi$$
0.697944 + 0.716152i $$0.254098\pi$$
$$522$$ 0 0
$$523$$ 18.7386 0.819383 0.409692 0.912224i $$-0.365636\pi$$
0.409692 + 0.912224i $$0.365636\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −22.2462 −0.969981
$$527$$ −15.3693 −0.669498
$$528$$ 0 0
$$529$$ 20.0540 0.871912
$$530$$ −12.0000 −0.521247
$$531$$ 0 0
$$532$$ −6.24621 −0.270808
$$533$$ 0.561553 0.0243236
$$534$$ 0 0
$$535$$ −17.9309 −0.775219
$$536$$ 32.6004 1.40812
$$537$$ 0 0
$$538$$ 14.2462 0.614198
$$539$$ 35.3693 1.52346
$$540$$ 0 0
$$541$$ 27.1231 1.16611 0.583057 0.812431i $$-0.301857\pi$$
0.583057 + 0.812431i $$0.301857\pi$$
$$542$$ −8.38447 −0.360144
$$543$$ 0 0
$$544$$ −6.24621 −0.267804
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ 19.3693 0.828172 0.414086 0.910238i $$-0.364101\pi$$
0.414086 + 0.910238i $$0.364101\pi$$
$$548$$ −3.12311 −0.133412
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ 3.50758 0.149428
$$552$$ 0 0
$$553$$ 35.0540 1.49065
$$554$$ −6.63068 −0.281711
$$555$$ 0 0
$$556$$ 6.87689 0.291645
$$557$$ −26.4924 −1.12252 −0.561260 0.827640i $$-0.689683\pi$$
−0.561260 + 0.827640i $$0.689683\pi$$
$$558$$ 0 0
$$559$$ 5.12311 0.216684
$$560$$ 21.3693 0.903018
$$561$$ 0 0
$$562$$ 19.1231 0.806660
$$563$$ −31.6847 −1.33535 −0.667675 0.744453i $$-0.732710\pi$$
−0.667675 + 0.744453i $$0.732710\pi$$
$$564$$ 0 0
$$565$$ −13.1231 −0.552093
$$566$$ 6.24621 0.262548
$$567$$ 0 0
$$568$$ 35.5076 1.48986
$$569$$ −41.1231 −1.72397 −0.861985 0.506934i $$-0.830779\pi$$
−0.861985 + 0.506934i $$0.830779\pi$$
$$570$$ 0 0
$$571$$ 15.0540 0.629989 0.314995 0.949093i $$-0.397997\pi$$
0.314995 + 0.949093i $$0.397997\pi$$
$$572$$ −1.12311 −0.0469594
$$573$$ 0 0
$$574$$ −4.00000 −0.166957
$$575$$ 6.56155 0.273636
$$576$$ 0 0
$$577$$ 28.5616 1.18903 0.594517 0.804083i $$-0.297343\pi$$
0.594517 + 0.804083i $$0.297343\pi$$
$$578$$ 16.3002 0.677998
$$579$$ 0 0
$$580$$ 0.492423 0.0204467
$$581$$ −75.2311 −3.12111
$$582$$ 0 0
$$583$$ 19.6847 0.815255
$$584$$ −14.6307 −0.605422
$$585$$ 0 0
$$586$$ 5.86174 0.242146
$$587$$ 0.492423 0.0203245 0.0101622 0.999948i $$-0.496765\pi$$
0.0101622 + 0.999948i $$0.496765\pi$$
$$588$$ 0 0
$$589$$ 18.7386 0.772112
$$590$$ −18.7386 −0.771457
$$591$$ 0 0
$$592$$ −7.89205 −0.324361
$$593$$ 7.75379 0.318410 0.159205 0.987246i $$-0.449107\pi$$
0.159205 + 0.987246i $$0.449107\pi$$
$$594$$ 0 0
$$595$$ 11.6847 0.479024
$$596$$ 1.23106 0.0504260
$$597$$ 0 0
$$598$$ 10.2462 0.418999
$$599$$ 15.3693 0.627973 0.313987 0.949427i $$-0.398335\pi$$
0.313987 + 0.949427i $$0.398335\pi$$
$$600$$ 0 0
$$601$$ 41.5464 1.69471 0.847356 0.531025i $$-0.178193\pi$$
0.847356 + 0.531025i $$0.178193\pi$$
$$602$$ −36.4924 −1.48732
$$603$$ 0 0
$$604$$ −5.86174 −0.238511
$$605$$ −4.43845 −0.180449
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 7.61553 0.308850
$$609$$ 0 0
$$610$$ −8.87689 −0.359415
$$611$$ −2.87689 −0.116387
$$612$$ 0 0
$$613$$ 37.5464 1.51648 0.758242 0.651973i $$-0.226058\pi$$
0.758242 + 0.651973i $$0.226058\pi$$
$$614$$ −5.36932 −0.216688
$$615$$ 0 0
$$616$$ −28.4924 −1.14799
$$617$$ −48.7386 −1.96214 −0.981072 0.193645i $$-0.937969\pi$$
−0.981072 + 0.193645i $$0.937969\pi$$
$$618$$ 0 0
$$619$$ −33.3693 −1.34123 −0.670613 0.741807i $$-0.733969\pi$$
−0.670613 + 0.741807i $$0.733969\pi$$
$$620$$ 2.63068 0.105651
$$621$$ 0 0
$$622$$ −42.7386 −1.71366
$$623$$ 7.68466 0.307879
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −32.6004 −1.30297
$$627$$ 0 0
$$628$$ −9.36932 −0.373876
$$629$$ −4.31534 −0.172064
$$630$$ 0 0
$$631$$ 16.7386 0.666354 0.333177 0.942864i $$-0.391879\pi$$
0.333177 + 0.942864i $$0.391879\pi$$
$$632$$ −18.7386 −0.745383
$$633$$ 0 0
$$634$$ −53.8617 −2.13912
$$635$$ 18.2462 0.724079
$$636$$ 0 0
$$637$$ −13.8078 −0.547084
$$638$$ −4.49242 −0.177857
$$639$$ 0 0
$$640$$ −13.5616 −0.536067
$$641$$ 32.9848 1.30282 0.651412 0.758725i $$-0.274177\pi$$
0.651412 + 0.758725i $$0.274177\pi$$
$$642$$ 0 0
$$643$$ 1.68466 0.0664364 0.0332182 0.999448i $$-0.489424\pi$$
0.0332182 + 0.999448i $$0.489424\pi$$
$$644$$ −13.1231 −0.517123
$$645$$ 0 0
$$646$$ 12.4924 0.491508
$$647$$ 11.1922 0.440012 0.220006 0.975498i $$-0.429392\pi$$
0.220006 + 0.975498i $$0.429392\pi$$
$$648$$ 0 0
$$649$$ 30.7386 1.20660
$$650$$ 1.56155 0.0612491
$$651$$ 0 0
$$652$$ −9.23106 −0.361516
$$653$$ −8.63068 −0.337745 −0.168872 0.985638i $$-0.554013\pi$$
−0.168872 + 0.985638i $$0.554013\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 2.63068 0.102711
$$657$$ 0 0
$$658$$ 20.4924 0.798878
$$659$$ 9.12311 0.355386 0.177693 0.984086i $$-0.443137\pi$$
0.177693 + 0.984086i $$0.443137\pi$$
$$660$$ 0 0
$$661$$ −26.4924 −1.03044 −0.515218 0.857059i $$-0.672289\pi$$
−0.515218 + 0.857059i $$0.672289\pi$$
$$662$$ −32.6004 −1.26705
$$663$$ 0 0
$$664$$ 40.2159 1.56068
$$665$$ −14.2462 −0.552444
$$666$$ 0 0
$$667$$ 7.36932 0.285341
$$668$$ 5.75379 0.222621
$$669$$ 0 0
$$670$$ −20.8769 −0.806545
$$671$$ 14.5616 0.562143
$$672$$ 0 0
$$673$$ −32.7386 −1.26198 −0.630991 0.775790i $$-0.717351\pi$$
−0.630991 + 0.775790i $$0.717351\pi$$
$$674$$ −3.89205 −0.149916
$$675$$ 0 0
$$676$$ 0.438447 0.0168634
$$677$$ −18.5616 −0.713378 −0.356689 0.934223i $$-0.616094\pi$$
−0.356689 + 0.934223i $$0.616094\pi$$
$$678$$ 0 0
$$679$$ 54.4233 2.08857
$$680$$ −6.24621 −0.239531
$$681$$ 0 0
$$682$$ −24.0000 −0.919007
$$683$$ −0.492423 −0.0188420 −0.00942101 0.999956i $$-0.502999\pi$$
−0.00942101 + 0.999956i $$0.502999\pi$$
$$684$$ 0 0
$$685$$ −7.12311 −0.272160
$$686$$ 48.4924 1.85145
$$687$$ 0 0
$$688$$ 24.0000 0.914991
$$689$$ −7.68466 −0.292762
$$690$$ 0 0
$$691$$ 19.6155 0.746210 0.373105 0.927789i $$-0.378293\pi$$
0.373105 + 0.927789i $$0.378293\pi$$
$$692$$ 9.26137 0.352064
$$693$$ 0 0
$$694$$ −17.2614 −0.655233
$$695$$ 15.6847 0.594953
$$696$$ 0 0
$$697$$ 1.43845 0.0544851
$$698$$ −2.13826 −0.0809344
$$699$$ 0 0
$$700$$ −2.00000 −0.0755929
$$701$$ 16.9848 0.641509 0.320754 0.947162i $$-0.396064\pi$$
0.320754 + 0.947162i $$0.396064\pi$$
$$702$$ 0 0
$$703$$ 5.26137 0.198436
$$704$$ 14.2462 0.536924
$$705$$ 0 0
$$706$$ −16.3845 −0.616638
$$707$$ 28.4924 1.07157
$$708$$ 0 0
$$709$$ 0.876894 0.0329325 0.0164662 0.999864i $$-0.494758\pi$$
0.0164662 + 0.999864i $$0.494758\pi$$
$$710$$ −22.7386 −0.853366
$$711$$ 0 0
$$712$$ −4.10795 −0.153952
$$713$$ 39.3693 1.47439
$$714$$ 0 0
$$715$$ −2.56155 −0.0957966
$$716$$ −5.75379 −0.215029
$$717$$ 0 0
$$718$$ −3.50758 −0.130902
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 31.3693 1.16825
$$722$$ 14.4384 0.537343
$$723$$ 0 0
$$724$$ 11.2614 0.418525
$$725$$ 1.12311 0.0417111
$$726$$ 0 0
$$727$$ 21.1231 0.783413 0.391706 0.920090i $$-0.371885\pi$$
0.391706 + 0.920090i $$0.371885\pi$$
$$728$$ 11.1231 0.412250
$$729$$ 0 0
$$730$$ 9.36932 0.346774
$$731$$ 13.1231 0.485376
$$732$$ 0 0
$$733$$ −20.5616 −0.759458 −0.379729 0.925098i $$-0.623983\pi$$
−0.379729 + 0.925098i $$0.623983\pi$$
$$734$$ 28.4924 1.05167
$$735$$ 0 0
$$736$$ 16.0000 0.589768
$$737$$ 34.2462 1.26148
$$738$$ 0 0
$$739$$ −0.738634 −0.0271711 −0.0135855 0.999908i $$-0.504325\pi$$
−0.0135855 + 0.999908i $$0.504325\pi$$
$$740$$ 0.738634 0.0271527
$$741$$ 0 0
$$742$$ 54.7386 2.00952
$$743$$ −30.7386 −1.12769 −0.563846 0.825880i $$-0.690679\pi$$
−0.563846 + 0.825880i $$0.690679\pi$$
$$744$$ 0 0
$$745$$ 2.80776 0.102869
$$746$$ 6.63068 0.242767
$$747$$ 0 0
$$748$$ −2.87689 −0.105190
$$749$$ 81.7926 2.98864
$$750$$ 0 0
$$751$$ −23.0540 −0.841252 −0.420626 0.907234i $$-0.638189\pi$$
−0.420626 + 0.907234i $$0.638189\pi$$
$$752$$ −13.4773 −0.491465
$$753$$ 0 0
$$754$$ 1.75379 0.0638692
$$755$$ −13.3693 −0.486559
$$756$$ 0 0
$$757$$ 18.4924 0.672119 0.336059 0.941841i $$-0.390906\pi$$
0.336059 + 0.941841i $$0.390906\pi$$
$$758$$ −3.89205 −0.141366
$$759$$ 0 0
$$760$$ 7.61553 0.276244
$$761$$ −37.2311 −1.34962 −0.674812 0.737989i $$-0.735775\pi$$
−0.674812 + 0.737989i $$0.735775\pi$$
$$762$$ 0 0
$$763$$ 9.12311 0.330279
$$764$$ −9.47727 −0.342876
$$765$$ 0 0
$$766$$ −56.9848 −2.05895
$$767$$ −12.0000 −0.433295
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 18.2462 0.657548
$$771$$ 0 0
$$772$$ 6.76894 0.243620
$$773$$ −0.876894 −0.0315397 −0.0157698 0.999876i $$-0.505020\pi$$
−0.0157698 + 0.999876i $$0.505020\pi$$
$$774$$ 0 0
$$775$$ 6.00000 0.215526
$$776$$ −29.0928 −1.04437
$$777$$ 0 0
$$778$$ 31.0152 1.11195
$$779$$ −1.75379 −0.0628360
$$780$$ 0 0
$$781$$ 37.3002 1.33471
$$782$$ 26.2462 0.938563
$$783$$ 0 0
$$784$$ −64.6847 −2.31017
$$785$$ −21.3693 −0.762704
$$786$$ 0 0
$$787$$ −13.3693 −0.476565 −0.238282 0.971196i $$-0.576584\pi$$
−0.238282 + 0.971196i $$0.576584\pi$$
$$788$$ 9.36932 0.333768
$$789$$ 0 0
$$790$$ 12.0000 0.426941
$$791$$ 59.8617 2.12844
$$792$$ 0 0
$$793$$ −5.68466 −0.201868
$$794$$ 13.3693 0.474459
$$795$$ 0 0
$$796$$ −3.50758 −0.124323
$$797$$ −7.68466 −0.272205 −0.136102 0.990695i $$-0.543458\pi$$
−0.136102 + 0.990695i $$0.543458\pi$$
$$798$$ 0 0
$$799$$ −7.36932 −0.260708
$$800$$ 2.43845 0.0862121
$$801$$ 0 0
$$802$$ −31.6155 −1.11638
$$803$$ −15.3693 −0.542371
$$804$$ 0 0
$$805$$ −29.9309 −1.05492
$$806$$ 9.36932 0.330020
$$807$$ 0 0
$$808$$ −15.2311 −0.535827
$$809$$ 40.4924 1.42364 0.711819 0.702363i $$-0.247872\pi$$
0.711819 + 0.702363i $$0.247872\pi$$
$$810$$ 0 0
$$811$$ −12.2462 −0.430023 −0.215011 0.976612i $$-0.568979\pi$$
−0.215011 + 0.976612i $$0.568979\pi$$
$$812$$ −2.24621 −0.0788266
$$813$$ 0 0
$$814$$ −6.73863 −0.236189
$$815$$ −21.0540 −0.737489
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 23.6155 0.825698
$$819$$ 0 0
$$820$$ −0.246211 −0.00859807
$$821$$ −21.5464 −0.751974 −0.375987 0.926625i $$-0.622696\pi$$
−0.375987 + 0.926625i $$0.622696\pi$$
$$822$$ 0 0
$$823$$ −46.2462 −1.61204 −0.806021 0.591887i $$-0.798383\pi$$
−0.806021 + 0.591887i $$0.798383\pi$$
$$824$$ −16.7689 −0.584174
$$825$$ 0 0
$$826$$ 85.4773 2.97413
$$827$$ −1.12311 −0.0390542 −0.0195271 0.999809i $$-0.506216\pi$$
−0.0195271 + 0.999809i $$0.506216\pi$$
$$828$$ 0 0
$$829$$ −48.7386 −1.69276 −0.846381 0.532577i $$-0.821224\pi$$
−0.846381 + 0.532577i $$0.821224\pi$$
$$830$$ −25.7538 −0.893927
$$831$$ 0 0
$$832$$ −5.56155 −0.192812
$$833$$ −35.3693 −1.22547
$$834$$ 0 0
$$835$$ 13.1231 0.454144
$$836$$ 3.50758 0.121312
$$837$$ 0 0
$$838$$ 52.4924 1.81332
$$839$$ 41.7926 1.44284 0.721421 0.692497i $$-0.243490\pi$$
0.721421 + 0.692497i $$0.243490\pi$$
$$840$$ 0 0
$$841$$ −27.7386 −0.956505
$$842$$ −22.8466 −0.787345
$$843$$ 0 0
$$844$$ −4.49242 −0.154636
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 20.2462 0.695668
$$848$$ −36.0000 −1.23625
$$849$$ 0 0
$$850$$ 4.00000 0.137199
$$851$$ 11.0540 0.378925
$$852$$ 0 0
$$853$$ −20.4233 −0.699280 −0.349640 0.936884i $$-0.613696\pi$$
−0.349640 + 0.936884i $$0.613696\pi$$
$$854$$ 40.4924 1.38562
$$855$$ 0 0
$$856$$ −43.7235 −1.49444
$$857$$ 6.56155 0.224138 0.112069 0.993700i $$-0.464252\pi$$
0.112069 + 0.993700i $$0.464252\pi$$
$$858$$ 0 0
$$859$$ −16.8078 −0.573474 −0.286737 0.958009i $$-0.592571\pi$$
−0.286737 + 0.958009i $$0.592571\pi$$
$$860$$ −2.24621 −0.0765952
$$861$$ 0 0
$$862$$ 56.9848 1.94091
$$863$$ 15.3693 0.523178 0.261589 0.965179i $$-0.415754\pi$$
0.261589 + 0.965179i $$0.415754\pi$$
$$864$$ 0 0
$$865$$ 21.1231 0.718207
$$866$$ −49.3693 −1.67764
$$867$$ 0 0
$$868$$ −12.0000 −0.407307
$$869$$ −19.6847 −0.667756
$$870$$ 0 0
$$871$$ −13.3693 −0.453002
$$872$$ −4.87689 −0.165152
$$873$$ 0 0
$$874$$ −32.0000 −1.08242
$$875$$ −4.56155 −0.154209
$$876$$ 0 0
$$877$$ −18.9848 −0.641073 −0.320536 0.947236i $$-0.603863\pi$$
−0.320536 + 0.947236i $$0.603863\pi$$
$$878$$ 23.5076 0.793342
$$879$$ 0 0
$$880$$ −12.0000 −0.404520
$$881$$ 3.36932 0.113515 0.0567576 0.998388i $$-0.481924\pi$$
0.0567576 + 0.998388i $$0.481924\pi$$
$$882$$ 0 0
$$883$$ 18.1080 0.609381 0.304691 0.952451i $$-0.401447\pi$$
0.304691 + 0.952451i $$0.401447\pi$$
$$884$$ 1.12311 0.0377741
$$885$$ 0 0
$$886$$ −20.0000 −0.671913
$$887$$ 1.43845 0.0482983 0.0241492 0.999708i $$-0.492312\pi$$
0.0241492 + 0.999708i $$0.492312\pi$$
$$888$$ 0 0
$$889$$ −83.2311 −2.79148
$$890$$ 2.63068 0.0881807
$$891$$ 0 0
$$892$$ 4.10795 0.137544
$$893$$ 8.98485 0.300666
$$894$$ 0 0
$$895$$ −13.1231 −0.438657
$$896$$ 61.8617 2.06666
$$897$$ 0 0
$$898$$ −15.8920 −0.530325
$$899$$ 6.73863 0.224746
$$900$$ 0 0
$$901$$ −19.6847 −0.655791
$$902$$ 2.24621 0.0747907
$$903$$ 0 0
$$904$$ −32.0000 −1.06430
$$905$$ 25.6847 0.853787
$$906$$ 0 0
$$907$$ −27.3693 −0.908783 −0.454392 0.890802i $$-0.650143\pi$$
−0.454392 + 0.890802i $$0.650143\pi$$
$$908$$ 9.75379 0.323691
$$909$$ 0 0
$$910$$ −7.12311 −0.236129
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 42.2462 1.39815
$$914$$ 40.1080 1.32665
$$915$$ 0 0
$$916$$ 3.89205 0.128597
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 39.0540 1.28827 0.644136 0.764911i $$-0.277217\pi$$
0.644136 + 0.764911i $$0.277217\pi$$
$$920$$ 16.0000 0.527504
$$921$$ 0 0
$$922$$ 45.3693 1.49416
$$923$$ −14.5616 −0.479299
$$924$$ 0 0
$$925$$ 1.68466 0.0553912
$$926$$ 0.107951 0.00354748
$$927$$ 0 0
$$928$$ 2.73863 0.0899001
$$929$$ −38.6695 −1.26871 −0.634353 0.773044i $$-0.718733\pi$$
−0.634353 + 0.773044i $$0.718733\pi$$
$$930$$ 0 0
$$931$$ 43.1231 1.41330
$$932$$ −3.15342 −0.103294
$$933$$ 0 0
$$934$$ 34.2462 1.12057
$$935$$ −6.56155 −0.214586
$$936$$ 0 0
$$937$$ −2.63068 −0.0859407 −0.0429703 0.999076i $$-0.513682\pi$$
−0.0429703 + 0.999076i $$0.513682\pi$$
$$938$$ 95.2311 3.10940
$$939$$ 0 0
$$940$$ 1.26137 0.0411412
$$941$$ −21.1922 −0.690847 −0.345424 0.938447i $$-0.612265\pi$$
−0.345424 + 0.938447i $$0.612265\pi$$
$$942$$ 0 0
$$943$$ −3.68466 −0.119989
$$944$$ −56.2159 −1.82967
$$945$$ 0 0
$$946$$ 20.4924 0.666266
$$947$$ −46.6004 −1.51431 −0.757154 0.653236i $$-0.773411\pi$$
−0.757154 + 0.653236i $$0.773411\pi$$
$$948$$ 0 0
$$949$$ 6.00000 0.194768
$$950$$ −4.87689 −0.158227
$$951$$ 0 0
$$952$$ 28.4924 0.923445
$$953$$ 25.4384 0.824032 0.412016 0.911177i $$-0.364825\pi$$
0.412016 + 0.911177i $$0.364825\pi$$
$$954$$ 0 0
$$955$$ −21.6155 −0.699462
$$956$$ −2.60037 −0.0841021
$$957$$ 0 0
$$958$$ 17.2614 0.557689
$$959$$ 32.4924 1.04924
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 2.63068 0.0848166
$$963$$ 0 0
$$964$$ −10.8466 −0.349345
$$965$$ 15.4384 0.496981
$$966$$ 0 0
$$967$$ 17.3693 0.558560 0.279280 0.960210i $$-0.409904\pi$$
0.279280 + 0.960210i $$0.409904\pi$$
$$968$$ −10.8229 −0.347862
$$969$$ 0 0
$$970$$ 18.6307 0.598195
$$971$$ 8.49242 0.272535 0.136267 0.990672i $$-0.456489\pi$$
0.136267 + 0.990672i $$0.456489\pi$$
$$972$$ 0 0
$$973$$ −71.5464 −2.29367
$$974$$ −7.89205 −0.252878
$$975$$ 0 0
$$976$$ −26.6307 −0.852427
$$977$$ 55.4773 1.77488 0.887438 0.460928i $$-0.152483\pi$$
0.887438 + 0.460928i $$0.152483\pi$$
$$978$$ 0 0
$$979$$ −4.31534 −0.137919
$$980$$ 6.05398 0.193387
$$981$$ 0 0
$$982$$ −21.2614 −0.678477
$$983$$ −6.73863 −0.214929 −0.107465 0.994209i $$-0.534273\pi$$
−0.107465 + 0.994209i $$0.534273\pi$$
$$984$$ 0 0
$$985$$ 21.3693 0.680883
$$986$$ 4.49242 0.143068
$$987$$ 0 0
$$988$$ −1.36932 −0.0435638
$$989$$ −33.6155 −1.06891
$$990$$ 0 0
$$991$$ −27.0540 −0.859398 −0.429699 0.902972i $$-0.641380\pi$$
−0.429699 + 0.902972i $$0.641380\pi$$
$$992$$ 14.6307 0.464525
$$993$$ 0 0
$$994$$ 103.723 3.28991
$$995$$ −8.00000 −0.253617
$$996$$ 0 0
$$997$$ 11.7538 0.372246 0.186123 0.982526i $$-0.440408\pi$$
0.186123 + 0.982526i $$0.440408\pi$$
$$998$$ 48.3845 1.53158
$$999$$ 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.l.1.1 yes 2
3.2 odd 2 585.2.a.j.1.2 2
4.3 odd 2 9360.2.a.cw.1.2 2
5.2 odd 4 2925.2.c.p.2224.2 4
5.3 odd 4 2925.2.c.p.2224.3 4
5.4 even 2 2925.2.a.x.1.2 2
12.11 even 2 9360.2.a.cl.1.2 2
13.12 even 2 7605.2.a.bd.1.2 2
15.2 even 4 2925.2.c.o.2224.3 4
15.8 even 4 2925.2.c.o.2224.2 4
15.14 odd 2 2925.2.a.bc.1.1 2
39.38 odd 2 7605.2.a.bi.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.a.j.1.2 2 3.2 odd 2
585.2.a.l.1.1 yes 2 1.1 even 1 trivial
2925.2.a.x.1.2 2 5.4 even 2
2925.2.a.bc.1.1 2 15.14 odd 2
2925.2.c.o.2224.2 4 15.8 even 4
2925.2.c.o.2224.3 4 15.2 even 4
2925.2.c.p.2224.2 4 5.2 odd 4
2925.2.c.p.2224.3 4 5.3 odd 4
7605.2.a.bd.1.2 2 13.12 even 2
7605.2.a.bi.1.1 2 39.38 odd 2
9360.2.a.cl.1.2 2 12.11 even 2
9360.2.a.cw.1.2 2 4.3 odd 2