# Properties

 Label 3549.2.a.bf.1.8 Level $3549$ Weight $2$ Character 3549.1 Self dual yes Analytic conductor $28.339$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$3549 = 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3549.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.3389076774$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - x^{8} - 11 x^{7} + 8 x^{6} + 37 x^{5} - 18 x^{4} - 41 x^{3} + 12 x^{2} + 6 x - 1$$ 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 1.8 Root $$1.90042$$ of defining polynomial Character $$\chi$$ $$=$$ 3549.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.90042 q^{2} -1.00000 q^{3} +1.61158 q^{4} -1.76582 q^{5} -1.90042 q^{6} -1.00000 q^{7} -0.738153 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.90042 q^{2} -1.00000 q^{3} +1.61158 q^{4} -1.76582 q^{5} -1.90042 q^{6} -1.00000 q^{7} -0.738153 q^{8} +1.00000 q^{9} -3.35579 q^{10} +3.13241 q^{11} -1.61158 q^{12} -1.90042 q^{14} +1.76582 q^{15} -4.62597 q^{16} -0.934195 q^{17} +1.90042 q^{18} -2.13289 q^{19} -2.84577 q^{20} +1.00000 q^{21} +5.95289 q^{22} +5.87145 q^{23} +0.738153 q^{24} -1.88188 q^{25} -1.00000 q^{27} -1.61158 q^{28} -2.37261 q^{29} +3.35579 q^{30} +6.06765 q^{31} -7.31496 q^{32} -3.13241 q^{33} -1.77536 q^{34} +1.76582 q^{35} +1.61158 q^{36} +2.02644 q^{37} -4.05338 q^{38} +1.30344 q^{40} +6.16399 q^{41} +1.90042 q^{42} +5.68247 q^{43} +5.04815 q^{44} -1.76582 q^{45} +11.1582 q^{46} -5.26708 q^{47} +4.62597 q^{48} +1.00000 q^{49} -3.57636 q^{50} +0.934195 q^{51} +10.5891 q^{53} -1.90042 q^{54} -5.53127 q^{55} +0.738153 q^{56} +2.13289 q^{57} -4.50895 q^{58} -4.83882 q^{59} +2.84577 q^{60} +12.6018 q^{61} +11.5311 q^{62} -1.00000 q^{63} -4.64954 q^{64} -5.95289 q^{66} +6.65056 q^{67} -1.50553 q^{68} -5.87145 q^{69} +3.35579 q^{70} +7.17531 q^{71} -0.738153 q^{72} -4.85580 q^{73} +3.85108 q^{74} +1.88188 q^{75} -3.43733 q^{76} -3.13241 q^{77} +12.0565 q^{79} +8.16862 q^{80} +1.00000 q^{81} +11.7142 q^{82} -11.7189 q^{83} +1.61158 q^{84} +1.64962 q^{85} +10.7991 q^{86} +2.37261 q^{87} -2.31220 q^{88} -8.30211 q^{89} -3.35579 q^{90} +9.46233 q^{92} -6.06765 q^{93} -10.0096 q^{94} +3.76630 q^{95} +7.31496 q^{96} +18.2921 q^{97} +1.90042 q^{98} +3.13241 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q + q^{2} - 9q^{3} + 5q^{4} - 9q^{5} - q^{6} - 9q^{7} + 6q^{8} + 9q^{9} + O(q^{10})$$ $$9q + q^{2} - 9q^{3} + 5q^{4} - 9q^{5} - q^{6} - 9q^{7} + 6q^{8} + 9q^{9} + q^{10} + q^{11} - 5q^{12} - q^{14} + 9q^{15} + 5q^{16} + 11q^{17} + q^{18} - 7q^{19} - 23q^{20} + 9q^{21} - 3q^{22} + 22q^{23} - 6q^{24} - 8q^{25} - 9q^{27} - 5q^{28} + 11q^{29} - q^{30} - 7q^{31} + 18q^{32} - q^{33} + 6q^{34} + 9q^{35} + 5q^{36} + q^{37} - 6q^{38} - 14q^{40} - 16q^{41} + q^{42} + 32q^{43} - 18q^{44} - 9q^{45} + 9q^{46} + 12q^{47} - 5q^{48} + 9q^{49} - 10q^{50} - 11q^{51} + 13q^{53} - q^{54} + 9q^{55} - 6q^{56} + 7q^{57} - 4q^{58} - 29q^{59} + 23q^{60} - 12q^{61} + 30q^{62} - 9q^{63} + 6q^{64} + 3q^{66} + 20q^{67} + 34q^{68} - 22q^{69} - q^{70} + 2q^{71} + 6q^{72} - q^{73} + 43q^{74} + 8q^{75} - 13q^{76} - q^{77} + 3q^{79} + 39q^{80} + 9q^{81} - 19q^{82} - 24q^{83} + 5q^{84} - 15q^{85} + 28q^{86} - 11q^{87} - 19q^{88} - 11q^{89} + q^{90} + 73q^{92} + 7q^{93} + 15q^{94} + 39q^{95} - 18q^{96} - 20q^{97} + q^{98} + q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.90042 1.34380 0.671899 0.740643i $$-0.265479\pi$$
0.671899 + 0.740643i $$0.265479\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 1.61158 0.805792
$$5$$ −1.76582 −0.789698 −0.394849 0.918746i $$-0.629203\pi$$
−0.394849 + 0.918746i $$0.629203\pi$$
$$6$$ −1.90042 −0.775842
$$7$$ −1.00000 −0.377964
$$8$$ −0.738153 −0.260976
$$9$$ 1.00000 0.333333
$$10$$ −3.35579 −1.06119
$$11$$ 3.13241 0.944458 0.472229 0.881476i $$-0.343450\pi$$
0.472229 + 0.881476i $$0.343450\pi$$
$$12$$ −1.61158 −0.465224
$$13$$ 0 0
$$14$$ −1.90042 −0.507908
$$15$$ 1.76582 0.455932
$$16$$ −4.62597 −1.15649
$$17$$ −0.934195 −0.226576 −0.113288 0.993562i $$-0.536138\pi$$
−0.113288 + 0.993562i $$0.536138\pi$$
$$18$$ 1.90042 0.447933
$$19$$ −2.13289 −0.489319 −0.244659 0.969609i $$-0.578676\pi$$
−0.244659 + 0.969609i $$0.578676\pi$$
$$20$$ −2.84577 −0.636332
$$21$$ 1.00000 0.218218
$$22$$ 5.95289 1.26916
$$23$$ 5.87145 1.22428 0.612141 0.790749i $$-0.290309\pi$$
0.612141 + 0.790749i $$0.290309\pi$$
$$24$$ 0.738153 0.150675
$$25$$ −1.88188 −0.376377
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ −1.61158 −0.304561
$$29$$ −2.37261 −0.440583 −0.220292 0.975434i $$-0.570701\pi$$
−0.220292 + 0.975434i $$0.570701\pi$$
$$30$$ 3.35579 0.612681
$$31$$ 6.06765 1.08978 0.544891 0.838507i $$-0.316571\pi$$
0.544891 + 0.838507i $$0.316571\pi$$
$$32$$ −7.31496 −1.29311
$$33$$ −3.13241 −0.545283
$$34$$ −1.77536 −0.304472
$$35$$ 1.76582 0.298478
$$36$$ 1.61158 0.268597
$$37$$ 2.02644 0.333145 0.166572 0.986029i $$-0.446730\pi$$
0.166572 + 0.986029i $$0.446730\pi$$
$$38$$ −4.05338 −0.657546
$$39$$ 0 0
$$40$$ 1.30344 0.206093
$$41$$ 6.16399 0.962654 0.481327 0.876541i $$-0.340155\pi$$
0.481327 + 0.876541i $$0.340155\pi$$
$$42$$ 1.90042 0.293241
$$43$$ 5.68247 0.866569 0.433284 0.901257i $$-0.357355\pi$$
0.433284 + 0.901257i $$0.357355\pi$$
$$44$$ 5.04815 0.761037
$$45$$ −1.76582 −0.263233
$$46$$ 11.1582 1.64519
$$47$$ −5.26708 −0.768283 −0.384141 0.923274i $$-0.625502\pi$$
−0.384141 + 0.923274i $$0.625502\pi$$
$$48$$ 4.62597 0.667701
$$49$$ 1.00000 0.142857
$$50$$ −3.57636 −0.505774
$$51$$ 0.934195 0.130813
$$52$$ 0 0
$$53$$ 10.5891 1.45452 0.727260 0.686362i $$-0.240794\pi$$
0.727260 + 0.686362i $$0.240794\pi$$
$$54$$ −1.90042 −0.258614
$$55$$ −5.53127 −0.745837
$$56$$ 0.738153 0.0986398
$$57$$ 2.13289 0.282508
$$58$$ −4.50895 −0.592055
$$59$$ −4.83882 −0.629960 −0.314980 0.949098i $$-0.601998\pi$$
−0.314980 + 0.949098i $$0.601998\pi$$
$$60$$ 2.84577 0.367387
$$61$$ 12.6018 1.61350 0.806749 0.590894i $$-0.201225\pi$$
0.806749 + 0.590894i $$0.201225\pi$$
$$62$$ 11.5311 1.46445
$$63$$ −1.00000 −0.125988
$$64$$ −4.64954 −0.581192
$$65$$ 0 0
$$66$$ −5.95289 −0.732750
$$67$$ 6.65056 0.812495 0.406248 0.913763i $$-0.366837\pi$$
0.406248 + 0.913763i $$0.366837\pi$$
$$68$$ −1.50553 −0.182573
$$69$$ −5.87145 −0.706839
$$70$$ 3.35579 0.401094
$$71$$ 7.17531 0.851552 0.425776 0.904829i $$-0.360001\pi$$
0.425776 + 0.904829i $$0.360001\pi$$
$$72$$ −0.738153 −0.0869921
$$73$$ −4.85580 −0.568328 −0.284164 0.958776i $$-0.591716\pi$$
−0.284164 + 0.958776i $$0.591716\pi$$
$$74$$ 3.85108 0.447679
$$75$$ 1.88188 0.217301
$$76$$ −3.43733 −0.394289
$$77$$ −3.13241 −0.356972
$$78$$ 0 0
$$79$$ 12.0565 1.35646 0.678231 0.734848i $$-0.262747\pi$$
0.678231 + 0.734848i $$0.262747\pi$$
$$80$$ 8.16862 0.913279
$$81$$ 1.00000 0.111111
$$82$$ 11.7142 1.29361
$$83$$ −11.7189 −1.28632 −0.643160 0.765732i $$-0.722377\pi$$
−0.643160 + 0.765732i $$0.722377\pi$$
$$84$$ 1.61158 0.175838
$$85$$ 1.64962 0.178926
$$86$$ 10.7991 1.16449
$$87$$ 2.37261 0.254371
$$88$$ −2.31220 −0.246481
$$89$$ −8.30211 −0.880021 −0.440011 0.897993i $$-0.645025\pi$$
−0.440011 + 0.897993i $$0.645025\pi$$
$$90$$ −3.35579 −0.353731
$$91$$ 0 0
$$92$$ 9.46233 0.986516
$$93$$ −6.06765 −0.629186
$$94$$ −10.0096 −1.03242
$$95$$ 3.76630 0.386414
$$96$$ 7.31496 0.746580
$$97$$ 18.2921 1.85728 0.928642 0.370977i $$-0.120977\pi$$
0.928642 + 0.370977i $$0.120977\pi$$
$$98$$ 1.90042 0.191971
$$99$$ 3.13241 0.314819
$$100$$ −3.03281 −0.303281
$$101$$ 12.5967 1.25342 0.626710 0.779252i $$-0.284401\pi$$
0.626710 + 0.779252i $$0.284401\pi$$
$$102$$ 1.77536 0.175787
$$103$$ −4.92445 −0.485220 −0.242610 0.970124i $$-0.578004\pi$$
−0.242610 + 0.970124i $$0.578004\pi$$
$$104$$ 0 0
$$105$$ −1.76582 −0.172326
$$106$$ 20.1236 1.95458
$$107$$ −3.69523 −0.357231 −0.178616 0.983919i $$-0.557162\pi$$
−0.178616 + 0.983919i $$0.557162\pi$$
$$108$$ −1.61158 −0.155075
$$109$$ 14.6664 1.40479 0.702393 0.711789i $$-0.252115\pi$$
0.702393 + 0.711789i $$0.252115\pi$$
$$110$$ −10.5117 −1.00225
$$111$$ −2.02644 −0.192341
$$112$$ 4.62597 0.437113
$$113$$ 14.4311 1.35756 0.678780 0.734342i $$-0.262509\pi$$
0.678780 + 0.734342i $$0.262509\pi$$
$$114$$ 4.05338 0.379634
$$115$$ −10.3679 −0.966813
$$116$$ −3.82366 −0.355018
$$117$$ 0 0
$$118$$ −9.19577 −0.846539
$$119$$ 0.934195 0.0856375
$$120$$ −1.30344 −0.118988
$$121$$ −1.18799 −0.107999
$$122$$ 23.9487 2.16821
$$123$$ −6.16399 −0.555789
$$124$$ 9.77853 0.878138
$$125$$ 12.1522 1.08692
$$126$$ −1.90042 −0.169303
$$127$$ −4.80045 −0.425971 −0.212986 0.977055i $$-0.568319\pi$$
−0.212986 + 0.977055i $$0.568319\pi$$
$$128$$ 5.79386 0.512110
$$129$$ −5.68247 −0.500314
$$130$$ 0 0
$$131$$ −7.33712 −0.641047 −0.320524 0.947241i $$-0.603859\pi$$
−0.320524 + 0.947241i $$0.603859\pi$$
$$132$$ −5.04815 −0.439385
$$133$$ 2.13289 0.184945
$$134$$ 12.6388 1.09183
$$135$$ 1.76582 0.151977
$$136$$ 0.689579 0.0591309
$$137$$ −0.951272 −0.0812726 −0.0406363 0.999174i $$-0.512939\pi$$
−0.0406363 + 0.999174i $$0.512939\pi$$
$$138$$ −11.1582 −0.949849
$$139$$ −9.89366 −0.839169 −0.419584 0.907716i $$-0.637824\pi$$
−0.419584 + 0.907716i $$0.637824\pi$$
$$140$$ 2.84577 0.240511
$$141$$ 5.26708 0.443568
$$142$$ 13.6361 1.14431
$$143$$ 0 0
$$144$$ −4.62597 −0.385497
$$145$$ 4.18960 0.347928
$$146$$ −9.22804 −0.763718
$$147$$ −1.00000 −0.0824786
$$148$$ 3.26578 0.268445
$$149$$ 19.7207 1.61558 0.807792 0.589467i $$-0.200662\pi$$
0.807792 + 0.589467i $$0.200662\pi$$
$$150$$ 3.57636 0.292009
$$151$$ −12.6508 −1.02951 −0.514753 0.857338i $$-0.672116\pi$$
−0.514753 + 0.857338i $$0.672116\pi$$
$$152$$ 1.57440 0.127701
$$153$$ −0.934195 −0.0755252
$$154$$ −5.95289 −0.479698
$$155$$ −10.7144 −0.860599
$$156$$ 0 0
$$157$$ −15.9485 −1.27283 −0.636413 0.771348i $$-0.719583\pi$$
−0.636413 + 0.771348i $$0.719583\pi$$
$$158$$ 22.9124 1.82281
$$159$$ −10.5891 −0.839767
$$160$$ 12.9169 1.02117
$$161$$ −5.87145 −0.462735
$$162$$ 1.90042 0.149311
$$163$$ 9.44073 0.739455 0.369727 0.929140i $$-0.379451\pi$$
0.369727 + 0.929140i $$0.379451\pi$$
$$164$$ 9.93379 0.775699
$$165$$ 5.53127 0.430609
$$166$$ −22.2708 −1.72855
$$167$$ −4.14823 −0.320999 −0.160500 0.987036i $$-0.551311\pi$$
−0.160500 + 0.987036i $$0.551311\pi$$
$$168$$ −0.738153 −0.0569497
$$169$$ 0 0
$$170$$ 3.13496 0.240441
$$171$$ −2.13289 −0.163106
$$172$$ 9.15778 0.698274
$$173$$ 3.23618 0.246042 0.123021 0.992404i $$-0.460742\pi$$
0.123021 + 0.992404i $$0.460742\pi$$
$$174$$ 4.50895 0.341823
$$175$$ 1.88188 0.142257
$$176$$ −14.4904 −1.09226
$$177$$ 4.83882 0.363708
$$178$$ −15.7775 −1.18257
$$179$$ 19.8786 1.48579 0.742896 0.669406i $$-0.233452\pi$$
0.742896 + 0.669406i $$0.233452\pi$$
$$180$$ −2.84577 −0.212111
$$181$$ 8.91114 0.662360 0.331180 0.943568i $$-0.392553\pi$$
0.331180 + 0.943568i $$0.392553\pi$$
$$182$$ 0 0
$$183$$ −12.6018 −0.931553
$$184$$ −4.33403 −0.319509
$$185$$ −3.57833 −0.263084
$$186$$ −11.5311 −0.845499
$$187$$ −2.92629 −0.213991
$$188$$ −8.48834 −0.619076
$$189$$ 1.00000 0.0727393
$$190$$ 7.15754 0.519262
$$191$$ 17.4590 1.26329 0.631646 0.775257i $$-0.282380\pi$$
0.631646 + 0.775257i $$0.282380\pi$$
$$192$$ 4.64954 0.335551
$$193$$ −19.3101 −1.38997 −0.694984 0.719025i $$-0.744589\pi$$
−0.694984 + 0.719025i $$0.744589\pi$$
$$194$$ 34.7627 2.49581
$$195$$ 0 0
$$196$$ 1.61158 0.115113
$$197$$ −25.9848 −1.85134 −0.925669 0.378334i $$-0.876497\pi$$
−0.925669 + 0.378334i $$0.876497\pi$$
$$198$$ 5.95289 0.423054
$$199$$ 20.1033 1.42508 0.712542 0.701629i $$-0.247544\pi$$
0.712542 + 0.701629i $$0.247544\pi$$
$$200$$ 1.38912 0.0982254
$$201$$ −6.65056 −0.469094
$$202$$ 23.9390 1.68434
$$203$$ 2.37261 0.166525
$$204$$ 1.50553 0.105408
$$205$$ −10.8845 −0.760206
$$206$$ −9.35850 −0.652038
$$207$$ 5.87145 0.408094
$$208$$ 0 0
$$209$$ −6.68110 −0.462141
$$210$$ −3.35579 −0.231572
$$211$$ −7.25390 −0.499379 −0.249689 0.968326i $$-0.580329\pi$$
−0.249689 + 0.968326i $$0.580329\pi$$
$$212$$ 17.0652 1.17204
$$213$$ −7.17531 −0.491644
$$214$$ −7.02247 −0.480046
$$215$$ −10.0342 −0.684328
$$216$$ 0.738153 0.0502249
$$217$$ −6.06765 −0.411899
$$218$$ 27.8723 1.88775
$$219$$ 4.85580 0.328124
$$220$$ −8.91411 −0.600989
$$221$$ 0 0
$$222$$ −3.85108 −0.258468
$$223$$ 8.41784 0.563701 0.281850 0.959458i $$-0.409052\pi$$
0.281850 + 0.959458i $$0.409052\pi$$
$$224$$ 7.31496 0.488751
$$225$$ −1.88188 −0.125459
$$226$$ 27.4250 1.82429
$$227$$ 10.4603 0.694275 0.347138 0.937814i $$-0.387154\pi$$
0.347138 + 0.937814i $$0.387154\pi$$
$$228$$ 3.43733 0.227643
$$229$$ −10.6287 −0.702362 −0.351181 0.936308i $$-0.614220\pi$$
−0.351181 + 0.936308i $$0.614220\pi$$
$$230$$ −19.7034 −1.29920
$$231$$ 3.13241 0.206098
$$232$$ 1.75135 0.114982
$$233$$ 22.9731 1.50502 0.752510 0.658580i $$-0.228843\pi$$
0.752510 + 0.658580i $$0.228843\pi$$
$$234$$ 0 0
$$235$$ 9.30071 0.606711
$$236$$ −7.79816 −0.507617
$$237$$ −12.0565 −0.783154
$$238$$ 1.77536 0.115079
$$239$$ −15.7349 −1.01781 −0.508904 0.860823i $$-0.669949\pi$$
−0.508904 + 0.860823i $$0.669949\pi$$
$$240$$ −8.16862 −0.527282
$$241$$ −9.04536 −0.582663 −0.291331 0.956622i $$-0.594098\pi$$
−0.291331 + 0.956622i $$0.594098\pi$$
$$242$$ −2.25767 −0.145128
$$243$$ −1.00000 −0.0641500
$$244$$ 20.3089 1.30014
$$245$$ −1.76582 −0.112814
$$246$$ −11.7142 −0.746867
$$247$$ 0 0
$$248$$ −4.47885 −0.284407
$$249$$ 11.7189 0.742657
$$250$$ 23.0942 1.46060
$$251$$ 17.0593 1.07677 0.538386 0.842698i $$-0.319034\pi$$
0.538386 + 0.842698i $$0.319034\pi$$
$$252$$ −1.61158 −0.101520
$$253$$ 18.3918 1.15628
$$254$$ −9.12286 −0.572419
$$255$$ −1.64962 −0.103303
$$256$$ 20.3098 1.26936
$$257$$ −7.93028 −0.494677 −0.247339 0.968929i $$-0.579556\pi$$
−0.247339 + 0.968929i $$0.579556\pi$$
$$258$$ −10.7991 −0.672320
$$259$$ −2.02644 −0.125917
$$260$$ 0 0
$$261$$ −2.37261 −0.146861
$$262$$ −13.9436 −0.861438
$$263$$ 18.4617 1.13840 0.569199 0.822200i $$-0.307253\pi$$
0.569199 + 0.822200i $$0.307253\pi$$
$$264$$ 2.31220 0.142306
$$265$$ −18.6984 −1.14863
$$266$$ 4.05338 0.248529
$$267$$ 8.30211 0.508081
$$268$$ 10.7179 0.654702
$$269$$ −10.2133 −0.622716 −0.311358 0.950293i $$-0.600784\pi$$
−0.311358 + 0.950293i $$0.600784\pi$$
$$270$$ 3.35579 0.204227
$$271$$ −24.1804 −1.46886 −0.734428 0.678687i $$-0.762549\pi$$
−0.734428 + 0.678687i $$0.762549\pi$$
$$272$$ 4.32155 0.262033
$$273$$ 0 0
$$274$$ −1.80781 −0.109214
$$275$$ −5.89484 −0.355472
$$276$$ −9.46233 −0.569565
$$277$$ −18.3142 −1.10039 −0.550195 0.835036i $$-0.685447\pi$$
−0.550195 + 0.835036i $$0.685447\pi$$
$$278$$ −18.8021 −1.12767
$$279$$ 6.06765 0.363261
$$280$$ −1.30344 −0.0778957
$$281$$ −9.11661 −0.543852 −0.271926 0.962318i $$-0.587661\pi$$
−0.271926 + 0.962318i $$0.587661\pi$$
$$282$$ 10.0096 0.596066
$$283$$ −9.22502 −0.548371 −0.274185 0.961677i $$-0.588408\pi$$
−0.274185 + 0.961677i $$0.588408\pi$$
$$284$$ 11.5636 0.686174
$$285$$ −3.76630 −0.223096
$$286$$ 0 0
$$287$$ −6.16399 −0.363849
$$288$$ −7.31496 −0.431038
$$289$$ −16.1273 −0.948664
$$290$$ 7.96199 0.467544
$$291$$ −18.2921 −1.07230
$$292$$ −7.82553 −0.457954
$$293$$ 18.0748 1.05594 0.527970 0.849263i $$-0.322953\pi$$
0.527970 + 0.849263i $$0.322953\pi$$
$$294$$ −1.90042 −0.110835
$$295$$ 8.54448 0.497479
$$296$$ −1.49582 −0.0869429
$$297$$ −3.13241 −0.181761
$$298$$ 37.4776 2.17102
$$299$$ 0 0
$$300$$ 3.03281 0.175100
$$301$$ −5.68247 −0.327532
$$302$$ −24.0418 −1.38345
$$303$$ −12.5967 −0.723663
$$304$$ 9.86668 0.565893
$$305$$ −22.2525 −1.27418
$$306$$ −1.77536 −0.101491
$$307$$ −15.5831 −0.889373 −0.444686 0.895686i $$-0.646685\pi$$
−0.444686 + 0.895686i $$0.646685\pi$$
$$308$$ −5.04815 −0.287645
$$309$$ 4.92445 0.280142
$$310$$ −20.3618 −1.15647
$$311$$ −2.20637 −0.125112 −0.0625558 0.998041i $$-0.519925\pi$$
−0.0625558 + 0.998041i $$0.519925\pi$$
$$312$$ 0 0
$$313$$ 26.6495 1.50632 0.753160 0.657837i $$-0.228528\pi$$
0.753160 + 0.657837i $$0.228528\pi$$
$$314$$ −30.3087 −1.71042
$$315$$ 1.76582 0.0994926
$$316$$ 19.4301 1.09303
$$317$$ −22.8415 −1.28291 −0.641455 0.767161i $$-0.721669\pi$$
−0.641455 + 0.767161i $$0.721669\pi$$
$$318$$ −20.1236 −1.12848
$$319$$ −7.43200 −0.416112
$$320$$ 8.21024 0.458966
$$321$$ 3.69523 0.206247
$$322$$ −11.1582 −0.621822
$$323$$ 1.99254 0.110868
$$324$$ 1.61158 0.0895324
$$325$$ 0 0
$$326$$ 17.9413 0.993678
$$327$$ −14.6664 −0.811054
$$328$$ −4.54997 −0.251230
$$329$$ 5.26708 0.290383
$$330$$ 10.5117 0.578652
$$331$$ −3.66559 −0.201479 −0.100739 0.994913i $$-0.532121\pi$$
−0.100739 + 0.994913i $$0.532121\pi$$
$$332$$ −18.8860 −1.03651
$$333$$ 2.02644 0.111048
$$334$$ −7.88336 −0.431358
$$335$$ −11.7437 −0.641626
$$336$$ −4.62597 −0.252367
$$337$$ −32.6505 −1.77859 −0.889293 0.457339i $$-0.848803\pi$$
−0.889293 + 0.457339i $$0.848803\pi$$
$$338$$ 0 0
$$339$$ −14.4311 −0.783788
$$340$$ 2.65850 0.144177
$$341$$ 19.0064 1.02925
$$342$$ −4.05338 −0.219182
$$343$$ −1.00000 −0.0539949
$$344$$ −4.19453 −0.226154
$$345$$ 10.3679 0.558190
$$346$$ 6.15009 0.330631
$$347$$ 21.9592 1.17883 0.589417 0.807829i $$-0.299358\pi$$
0.589417 + 0.807829i $$0.299358\pi$$
$$348$$ 3.82366 0.204970
$$349$$ −35.8913 −1.92121 −0.960607 0.277909i $$-0.910359\pi$$
−0.960607 + 0.277909i $$0.910359\pi$$
$$350$$ 3.57636 0.191165
$$351$$ 0 0
$$352$$ −22.9135 −1.22129
$$353$$ 5.10983 0.271969 0.135984 0.990711i $$-0.456580\pi$$
0.135984 + 0.990711i $$0.456580\pi$$
$$354$$ 9.19577 0.488750
$$355$$ −12.6703 −0.672469
$$356$$ −13.3795 −0.709114
$$357$$ −0.934195 −0.0494428
$$358$$ 37.7775 1.99660
$$359$$ 2.15127 0.113540 0.0567698 0.998387i $$-0.481920\pi$$
0.0567698 + 0.998387i $$0.481920\pi$$
$$360$$ 1.30344 0.0686975
$$361$$ −14.4508 −0.760567
$$362$$ 16.9349 0.890077
$$363$$ 1.18799 0.0623531
$$364$$ 0 0
$$365$$ 8.57446 0.448808
$$366$$ −23.9487 −1.25182
$$367$$ 19.3500 1.01006 0.505031 0.863101i $$-0.331481\pi$$
0.505031 + 0.863101i $$0.331481\pi$$
$$368$$ −27.1611 −1.41587
$$369$$ 6.16399 0.320885
$$370$$ −6.80031 −0.353532
$$371$$ −10.5891 −0.549757
$$372$$ −9.77853 −0.506993
$$373$$ −16.3147 −0.844741 −0.422371 0.906423i $$-0.638802\pi$$
−0.422371 + 0.906423i $$0.638802\pi$$
$$374$$ −5.56116 −0.287561
$$375$$ −12.1522 −0.627535
$$376$$ 3.88791 0.200504
$$377$$ 0 0
$$378$$ 1.90042 0.0977469
$$379$$ 8.99682 0.462135 0.231068 0.972938i $$-0.425778\pi$$
0.231068 + 0.972938i $$0.425778\pi$$
$$380$$ 6.06971 0.311369
$$381$$ 4.80045 0.245935
$$382$$ 33.1794 1.69761
$$383$$ −35.9745 −1.83821 −0.919105 0.394012i $$-0.871087\pi$$
−0.919105 + 0.394012i $$0.871087\pi$$
$$384$$ −5.79386 −0.295667
$$385$$ 5.53127 0.281900
$$386$$ −36.6972 −1.86784
$$387$$ 5.68247 0.288856
$$388$$ 29.4793 1.49658
$$389$$ 21.7976 1.10518 0.552591 0.833453i $$-0.313639\pi$$
0.552591 + 0.833453i $$0.313639\pi$$
$$390$$ 0 0
$$391$$ −5.48508 −0.277392
$$392$$ −0.738153 −0.0372823
$$393$$ 7.33712 0.370109
$$394$$ −49.3819 −2.48782
$$395$$ −21.2896 −1.07120
$$396$$ 5.04815 0.253679
$$397$$ 25.9786 1.30383 0.651913 0.758293i $$-0.273967\pi$$
0.651913 + 0.758293i $$0.273967\pi$$
$$398$$ 38.2046 1.91503
$$399$$ −2.13289 −0.106778
$$400$$ 8.70553 0.435276
$$401$$ 24.2479 1.21088 0.605442 0.795890i $$-0.292996\pi$$
0.605442 + 0.795890i $$0.292996\pi$$
$$402$$ −12.6388 −0.630368
$$403$$ 0 0
$$404$$ 20.3007 1.01000
$$405$$ −1.76582 −0.0877442
$$406$$ 4.50895 0.223776
$$407$$ 6.34765 0.314641
$$408$$ −0.689579 −0.0341392
$$409$$ 21.5639 1.06627 0.533134 0.846031i $$-0.321014\pi$$
0.533134 + 0.846031i $$0.321014\pi$$
$$410$$ −20.6851 −1.02156
$$411$$ 0.951272 0.0469228
$$412$$ −7.93616 −0.390987
$$413$$ 4.83882 0.238103
$$414$$ 11.1582 0.548396
$$415$$ 20.6935 1.01580
$$416$$ 0 0
$$417$$ 9.89366 0.484494
$$418$$ −12.6969 −0.621024
$$419$$ 0.199171 0.00973014 0.00486507 0.999988i $$-0.498451\pi$$
0.00486507 + 0.999988i $$0.498451\pi$$
$$420$$ −2.84577 −0.138859
$$421$$ −20.8702 −1.01715 −0.508576 0.861017i $$-0.669828\pi$$
−0.508576 + 0.861017i $$0.669828\pi$$
$$422$$ −13.7854 −0.671064
$$423$$ −5.26708 −0.256094
$$424$$ −7.81634 −0.379595
$$425$$ 1.75805 0.0852778
$$426$$ −13.6361 −0.660670
$$427$$ −12.6018 −0.609845
$$428$$ −5.95517 −0.287854
$$429$$ 0 0
$$430$$ −19.0692 −0.919598
$$431$$ 13.2296 0.637246 0.318623 0.947882i $$-0.396780\pi$$
0.318623 + 0.947882i $$0.396780\pi$$
$$432$$ 4.62597 0.222567
$$433$$ −31.5518 −1.51628 −0.758142 0.652090i $$-0.773892\pi$$
−0.758142 + 0.652090i $$0.773892\pi$$
$$434$$ −11.5311 −0.553509
$$435$$ −4.18960 −0.200876
$$436$$ 23.6361 1.13197
$$437$$ −12.5232 −0.599064
$$438$$ 9.22804 0.440933
$$439$$ 27.1399 1.29532 0.647658 0.761931i $$-0.275748\pi$$
0.647658 + 0.761931i $$0.275748\pi$$
$$440$$ 4.08292 0.194646
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ −0.825961 −0.0392426 −0.0196213 0.999807i $$-0.506246\pi$$
−0.0196213 + 0.999807i $$0.506246\pi$$
$$444$$ −3.26578 −0.154987
$$445$$ 14.6600 0.694951
$$446$$ 15.9974 0.757499
$$447$$ −19.7207 −0.932758
$$448$$ 4.64954 0.219670
$$449$$ −27.5595 −1.30061 −0.650306 0.759673i $$-0.725359\pi$$
−0.650306 + 0.759673i $$0.725359\pi$$
$$450$$ −3.57636 −0.168591
$$451$$ 19.3082 0.909187
$$452$$ 23.2569 1.09391
$$453$$ 12.6508 0.594386
$$454$$ 19.8789 0.932965
$$455$$ 0 0
$$456$$ −1.57440 −0.0737280
$$457$$ 0.0445936 0.00208600 0.00104300 0.999999i $$-0.499668\pi$$
0.00104300 + 0.999999i $$0.499668\pi$$
$$458$$ −20.1989 −0.943833
$$459$$ 0.934195 0.0436045
$$460$$ −16.7088 −0.779050
$$461$$ −2.86714 −0.133536 −0.0667680 0.997769i $$-0.521269\pi$$
−0.0667680 + 0.997769i $$0.521269\pi$$
$$462$$ 5.95289 0.276954
$$463$$ 11.7612 0.546590 0.273295 0.961930i $$-0.411886\pi$$
0.273295 + 0.961930i $$0.411886\pi$$
$$464$$ 10.9756 0.509531
$$465$$ 10.7144 0.496867
$$466$$ 43.6586 2.02244
$$467$$ 39.3716 1.82190 0.910949 0.412518i $$-0.135351\pi$$
0.910949 + 0.412518i $$0.135351\pi$$
$$468$$ 0 0
$$469$$ −6.65056 −0.307094
$$470$$ 17.6752 0.815297
$$471$$ 15.9485 0.734867
$$472$$ 3.57179 0.164405
$$473$$ 17.7999 0.818438
$$474$$ −22.9124 −1.05240
$$475$$ 4.01385 0.184168
$$476$$ 1.50553 0.0690060
$$477$$ 10.5891 0.484840
$$478$$ −29.9029 −1.36773
$$479$$ −15.5574 −0.710835 −0.355417 0.934708i $$-0.615661\pi$$
−0.355417 + 0.934708i $$0.615661\pi$$
$$480$$ −12.9169 −0.589573
$$481$$ 0 0
$$482$$ −17.1900 −0.782981
$$483$$ 5.87145 0.267160
$$484$$ −1.91454 −0.0870245
$$485$$ −32.3006 −1.46669
$$486$$ −1.90042 −0.0862047
$$487$$ −23.7460 −1.07604 −0.538018 0.842933i $$-0.680827\pi$$
−0.538018 + 0.842933i $$0.680827\pi$$
$$488$$ −9.30207 −0.421085
$$489$$ −9.44073 −0.426924
$$490$$ −3.35579 −0.151599
$$491$$ −5.86527 −0.264696 −0.132348 0.991203i $$-0.542252\pi$$
−0.132348 + 0.991203i $$0.542252\pi$$
$$492$$ −9.93379 −0.447850
$$493$$ 2.21648 0.0998254
$$494$$ 0 0
$$495$$ −5.53127 −0.248612
$$496$$ −28.0687 −1.26032
$$497$$ −7.17531 −0.321856
$$498$$ 22.2708 0.997981
$$499$$ −7.79729 −0.349054 −0.174527 0.984652i $$-0.555840\pi$$
−0.174527 + 0.984652i $$0.555840\pi$$
$$500$$ 19.5842 0.875833
$$501$$ 4.14823 0.185329
$$502$$ 32.4197 1.44696
$$503$$ −34.3648 −1.53225 −0.766126 0.642690i $$-0.777818\pi$$
−0.766126 + 0.642690i $$0.777818\pi$$
$$504$$ 0.738153 0.0328799
$$505$$ −22.2435 −0.989824
$$506$$ 34.9521 1.55381
$$507$$ 0 0
$$508$$ −7.73633 −0.343244
$$509$$ 10.4900 0.464960 0.232480 0.972601i $$-0.425316\pi$$
0.232480 + 0.972601i $$0.425316\pi$$
$$510$$ −3.13496 −0.138819
$$511$$ 4.85580 0.214808
$$512$$ 27.0094 1.19366
$$513$$ 2.13289 0.0941695
$$514$$ −15.0708 −0.664746
$$515$$ 8.69568 0.383178
$$516$$ −9.15778 −0.403149
$$517$$ −16.4987 −0.725611
$$518$$ −3.85108 −0.169207
$$519$$ −3.23618 −0.142053
$$520$$ 0 0
$$521$$ −2.34072 −0.102549 −0.0512743 0.998685i $$-0.516328\pi$$
−0.0512743 + 0.998685i $$0.516328\pi$$
$$522$$ −4.50895 −0.197352
$$523$$ 32.9438 1.44053 0.720266 0.693698i $$-0.244020\pi$$
0.720266 + 0.693698i $$0.244020\pi$$
$$524$$ −11.8244 −0.516551
$$525$$ −1.88188 −0.0821321
$$526$$ 35.0849 1.52978
$$527$$ −5.66837 −0.246918
$$528$$ 14.4904 0.630615
$$529$$ 11.4739 0.498866
$$530$$ −35.5347 −1.54353
$$531$$ −4.83882 −0.209987
$$532$$ 3.43733 0.149027
$$533$$ 0 0
$$534$$ 15.7775 0.682758
$$535$$ 6.52510 0.282105
$$536$$ −4.90913 −0.212042
$$537$$ −19.8786 −0.857823
$$538$$ −19.4095 −0.836805
$$539$$ 3.13241 0.134923
$$540$$ 2.84577 0.122462
$$541$$ −7.71023 −0.331489 −0.165744 0.986169i $$-0.553003\pi$$
−0.165744 + 0.986169i $$0.553003\pi$$
$$542$$ −45.9529 −1.97384
$$543$$ −8.91114 −0.382414
$$544$$ 6.83360 0.292988
$$545$$ −25.8982 −1.10936
$$546$$ 0 0
$$547$$ 23.6877 1.01281 0.506407 0.862294i $$-0.330973\pi$$
0.506407 + 0.862294i $$0.330973\pi$$
$$548$$ −1.53305 −0.0654888
$$549$$ 12.6018 0.537833
$$550$$ −11.2027 −0.477683
$$551$$ 5.06053 0.215586
$$552$$ 4.33403 0.184468
$$553$$ −12.0565 −0.512695
$$554$$ −34.8045 −1.47870
$$555$$ 3.57833 0.151892
$$556$$ −15.9445 −0.676195
$$557$$ 11.8899 0.503792 0.251896 0.967754i $$-0.418946\pi$$
0.251896 + 0.967754i $$0.418946\pi$$
$$558$$ 11.5311 0.488149
$$559$$ 0 0
$$560$$ −8.16862 −0.345187
$$561$$ 2.92629 0.123548
$$562$$ −17.3254 −0.730826
$$563$$ 45.2378 1.90655 0.953273 0.302109i $$-0.0976907\pi$$
0.953273 + 0.302109i $$0.0976907\pi$$
$$564$$ 8.48834 0.357424
$$565$$ −25.4826 −1.07206
$$566$$ −17.5314 −0.736899
$$567$$ −1.00000 −0.0419961
$$568$$ −5.29647 −0.222235
$$569$$ −17.7145 −0.742629 −0.371314 0.928507i $$-0.621093\pi$$
−0.371314 + 0.928507i $$0.621093\pi$$
$$570$$ −7.15754 −0.299796
$$571$$ 20.5380 0.859487 0.429743 0.902951i $$-0.358604\pi$$
0.429743 + 0.902951i $$0.358604\pi$$
$$572$$ 0 0
$$573$$ −17.4590 −0.729361
$$574$$ −11.7142 −0.488940
$$575$$ −11.0494 −0.460791
$$576$$ −4.64954 −0.193731
$$577$$ 11.9204 0.496253 0.248127 0.968728i $$-0.420185\pi$$
0.248127 + 0.968728i $$0.420185\pi$$
$$578$$ −30.6486 −1.27481
$$579$$ 19.3101 0.802499
$$580$$ 6.75190 0.280357
$$581$$ 11.7189 0.486183
$$582$$ −34.7627 −1.44096
$$583$$ 33.1693 1.37373
$$584$$ 3.58432 0.148320
$$585$$ 0 0
$$586$$ 34.3496 1.41897
$$587$$ −18.9993 −0.784185 −0.392093 0.919926i $$-0.628249\pi$$
−0.392093 + 0.919926i $$0.628249\pi$$
$$588$$ −1.61158 −0.0664606
$$589$$ −12.9416 −0.533251
$$590$$ 16.2381 0.668511
$$591$$ 25.9848 1.06887
$$592$$ −9.37425 −0.385279
$$593$$ 14.9494 0.613899 0.306949 0.951726i $$-0.400692\pi$$
0.306949 + 0.951726i $$0.400692\pi$$
$$594$$ −5.95289 −0.244250
$$595$$ −1.64962 −0.0676278
$$596$$ 31.7816 1.30183
$$597$$ −20.1033 −0.822773
$$598$$ 0 0
$$599$$ 45.5130 1.85961 0.929806 0.368051i $$-0.119975\pi$$
0.929806 + 0.368051i $$0.119975\pi$$
$$600$$ −1.38912 −0.0567105
$$601$$ 31.0256 1.26556 0.632781 0.774331i $$-0.281913\pi$$
0.632781 + 0.774331i $$0.281913\pi$$
$$602$$ −10.7991 −0.440137
$$603$$ 6.65056 0.270832
$$604$$ −20.3878 −0.829568
$$605$$ 2.09777 0.0852864
$$606$$ −23.9390 −0.972456
$$607$$ −1.55815 −0.0632435 −0.0316218 0.999500i $$-0.510067\pi$$
−0.0316218 + 0.999500i $$0.510067\pi$$
$$608$$ 15.6020 0.632745
$$609$$ −2.37261 −0.0961431
$$610$$ −42.2891 −1.71224
$$611$$ 0 0
$$612$$ −1.50553 −0.0608576
$$613$$ 17.4821 0.706096 0.353048 0.935605i $$-0.385145\pi$$
0.353048 + 0.935605i $$0.385145\pi$$
$$614$$ −29.6143 −1.19514
$$615$$ 10.8845 0.438905
$$616$$ 2.31220 0.0931612
$$617$$ 31.7640 1.27877 0.639385 0.768887i $$-0.279189\pi$$
0.639385 + 0.768887i $$0.279189\pi$$
$$618$$ 9.35850 0.376454
$$619$$ −1.58220 −0.0635941 −0.0317970 0.999494i $$-0.510123\pi$$
−0.0317970 + 0.999494i $$0.510123\pi$$
$$620$$ −17.2671 −0.693464
$$621$$ −5.87145 −0.235613
$$622$$ −4.19301 −0.168125
$$623$$ 8.30211 0.332617
$$624$$ 0 0
$$625$$ −12.0491 −0.481964
$$626$$ 50.6452 2.02419
$$627$$ 6.68110 0.266817
$$628$$ −25.7023 −1.02563
$$629$$ −1.89309 −0.0754825
$$630$$ 3.35579 0.133698
$$631$$ 3.86214 0.153749 0.0768746 0.997041i $$-0.475506\pi$$
0.0768746 + 0.997041i $$0.475506\pi$$
$$632$$ −8.89954 −0.354005
$$633$$ 7.25390 0.288317
$$634$$ −43.4085 −1.72397
$$635$$ 8.47673 0.336389
$$636$$ −17.0652 −0.676678
$$637$$ 0 0
$$638$$ −14.1239 −0.559171
$$639$$ 7.17531 0.283851
$$640$$ −10.2309 −0.404412
$$641$$ −17.1008 −0.675442 −0.337721 0.941246i $$-0.609656\pi$$
−0.337721 + 0.941246i $$0.609656\pi$$
$$642$$ 7.02247 0.277155
$$643$$ 49.0673 1.93503 0.967513 0.252821i $$-0.0813583\pi$$
0.967513 + 0.252821i $$0.0813583\pi$$
$$644$$ −9.46233 −0.372868
$$645$$ 10.0342 0.395097
$$646$$ 3.78665 0.148984
$$647$$ −18.3702 −0.722209 −0.361104 0.932525i $$-0.617600\pi$$
−0.361104 + 0.932525i $$0.617600\pi$$
$$648$$ −0.738153 −0.0289974
$$649$$ −15.1572 −0.594971
$$650$$ 0 0
$$651$$ 6.06765 0.237810
$$652$$ 15.2145 0.595847
$$653$$ 18.7518 0.733814 0.366907 0.930258i $$-0.380417\pi$$
0.366907 + 0.930258i $$0.380417\pi$$
$$654$$ −27.8723 −1.08989
$$655$$ 12.9560 0.506234
$$656$$ −28.5144 −1.11330
$$657$$ −4.85580 −0.189443
$$658$$ 10.0096 0.390217
$$659$$ 27.0220 1.05263 0.526315 0.850290i $$-0.323573\pi$$
0.526315 + 0.850290i $$0.323573\pi$$
$$660$$ 8.91411 0.346981
$$661$$ −17.3755 −0.675830 −0.337915 0.941177i $$-0.609722\pi$$
−0.337915 + 0.941177i $$0.609722\pi$$
$$662$$ −6.96614 −0.270747
$$663$$ 0 0
$$664$$ 8.65036 0.335699
$$665$$ −3.76630 −0.146051
$$666$$ 3.85108 0.149226
$$667$$ −13.9307 −0.539398
$$668$$ −6.68522 −0.258659
$$669$$ −8.41784 −0.325453
$$670$$ −22.3179 −0.862216
$$671$$ 39.4741 1.52388
$$672$$ −7.31496 −0.282181
$$673$$ 8.26424 0.318563 0.159281 0.987233i $$-0.449082\pi$$
0.159281 + 0.987233i $$0.449082\pi$$
$$674$$ −62.0495 −2.39006
$$675$$ 1.88188 0.0724337
$$676$$ 0 0
$$677$$ −33.9430 −1.30454 −0.652268 0.757988i $$-0.726182\pi$$
−0.652268 + 0.757988i $$0.726182\pi$$
$$678$$ −27.4250 −1.05325
$$679$$ −18.2921 −0.701988
$$680$$ −1.21767 −0.0466955
$$681$$ −10.4603 −0.400840
$$682$$ 36.1201 1.38311
$$683$$ 3.32745 0.127321 0.0636606 0.997972i $$-0.479722\pi$$
0.0636606 + 0.997972i $$0.479722\pi$$
$$684$$ −3.43733 −0.131430
$$685$$ 1.67977 0.0641809
$$686$$ −1.90042 −0.0725582
$$687$$ 10.6287 0.405509
$$688$$ −26.2869 −1.00218
$$689$$ 0 0
$$690$$ 19.7034 0.750094
$$691$$ −13.6990 −0.521133 −0.260567 0.965456i $$-0.583909\pi$$
−0.260567 + 0.965456i $$0.583909\pi$$
$$692$$ 5.21538 0.198259
$$693$$ −3.13241 −0.118991
$$694$$ 41.7317 1.58411
$$695$$ 17.4704 0.662690
$$696$$ −1.75135 −0.0663848
$$697$$ −5.75837 −0.218114
$$698$$ −68.2083 −2.58172
$$699$$ −22.9731 −0.868924
$$700$$ 3.03281 0.114630
$$701$$ −25.4300 −0.960478 −0.480239 0.877138i $$-0.659450\pi$$
−0.480239 + 0.877138i $$0.659450\pi$$
$$702$$ 0 0
$$703$$ −4.32218 −0.163014
$$704$$ −14.5643 −0.548911
$$705$$ −9.30071 −0.350285
$$706$$ 9.71081 0.365471
$$707$$ −12.5967 −0.473749
$$708$$ 7.79816 0.293073
$$709$$ −28.2448 −1.06076 −0.530378 0.847761i $$-0.677950\pi$$
−0.530378 + 0.847761i $$0.677950\pi$$
$$710$$ −24.0788 −0.903662
$$711$$ 12.0565 0.452154
$$712$$ 6.12822 0.229665
$$713$$ 35.6259 1.33420
$$714$$ −1.77536 −0.0664412
$$715$$ 0 0
$$716$$ 32.0360 1.19724
$$717$$ 15.7349 0.587631
$$718$$ 4.08831 0.152574
$$719$$ 16.4857 0.614812 0.307406 0.951578i $$-0.400539\pi$$
0.307406 + 0.951578i $$0.400539\pi$$
$$720$$ 8.16862 0.304426
$$721$$ 4.92445 0.183396
$$722$$ −27.4625 −1.02205
$$723$$ 9.04536 0.336401
$$724$$ 14.3610 0.533724
$$725$$ 4.46498 0.165825
$$726$$ 2.25767 0.0837899
$$727$$ −34.4487 −1.27763 −0.638816 0.769359i $$-0.720576\pi$$
−0.638816 + 0.769359i $$0.720576\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 16.2950 0.603107
$$731$$ −5.30854 −0.196343
$$732$$ −20.3089 −0.750638
$$733$$ −51.0217 −1.88453 −0.942265 0.334869i $$-0.891308\pi$$
−0.942265 + 0.334869i $$0.891308\pi$$
$$734$$ 36.7731 1.35732
$$735$$ 1.76582 0.0651332
$$736$$ −42.9494 −1.58314
$$737$$ 20.8323 0.767368
$$738$$ 11.7142 0.431204
$$739$$ 33.2683 1.22379 0.611897 0.790938i $$-0.290407\pi$$
0.611897 + 0.790938i $$0.290407\pi$$
$$740$$ −5.76678 −0.211991
$$741$$ 0 0
$$742$$ −20.1236 −0.738762
$$743$$ 48.4232 1.77648 0.888238 0.459384i $$-0.151930\pi$$
0.888238 + 0.459384i $$0.151930\pi$$
$$744$$ 4.47885 0.164203
$$745$$ −34.8232 −1.27582
$$746$$ −31.0047 −1.13516
$$747$$ −11.7189 −0.428773
$$748$$ −4.71595 −0.172432
$$749$$ 3.69523 0.135021
$$750$$ −23.0942 −0.843280
$$751$$ 51.7999 1.89021 0.945103 0.326771i $$-0.105961\pi$$
0.945103 + 0.326771i $$0.105961\pi$$
$$752$$ 24.3653 0.888512
$$753$$ −17.0593 −0.621675
$$754$$ 0 0
$$755$$ 22.3390 0.812999
$$756$$ 1.61158 0.0586127
$$757$$ −2.92984 −0.106487 −0.0532434 0.998582i $$-0.516956\pi$$
−0.0532434 + 0.998582i $$0.516956\pi$$
$$758$$ 17.0977 0.621016
$$759$$ −18.3918 −0.667580
$$760$$ −2.78010 −0.100845
$$761$$ 18.2514 0.661613 0.330806 0.943699i $$-0.392679\pi$$
0.330806 + 0.943699i $$0.392679\pi$$
$$762$$ 9.12286 0.330486
$$763$$ −14.6664 −0.530959
$$764$$ 28.1367 1.01795
$$765$$ 1.64962 0.0596421
$$766$$ −68.3665 −2.47018
$$767$$ 0 0
$$768$$ −20.3098 −0.732867
$$769$$ −10.3062 −0.371652 −0.185826 0.982583i $$-0.559496\pi$$
−0.185826 + 0.982583i $$0.559496\pi$$
$$770$$ 10.5117 0.378816
$$771$$ 7.93028 0.285602
$$772$$ −31.1198 −1.12003
$$773$$ 41.7219 1.50063 0.750316 0.661079i $$-0.229901\pi$$
0.750316 + 0.661079i $$0.229901\pi$$
$$774$$ 10.7991 0.388164
$$775$$ −11.4186 −0.410169
$$776$$ −13.5024 −0.484707
$$777$$ 2.02644 0.0726982
$$778$$ 41.4245 1.48514
$$779$$ −13.1471 −0.471045
$$780$$ 0 0
$$781$$ 22.4760 0.804255
$$782$$ −10.4239 −0.372759
$$783$$ 2.37261 0.0847903
$$784$$ −4.62597 −0.165213
$$785$$ 28.1621 1.00515
$$786$$ 13.9436 0.497351
$$787$$ −22.8437 −0.814289 −0.407144 0.913364i $$-0.633475\pi$$
−0.407144 + 0.913364i $$0.633475\pi$$
$$788$$ −41.8766 −1.49179
$$789$$ −18.4617 −0.657254
$$790$$ −40.4591 −1.43947
$$791$$ −14.4311 −0.513110
$$792$$ −2.31220 −0.0821604
$$793$$ 0 0
$$794$$ 49.3701 1.75208
$$795$$ 18.6984 0.663163
$$796$$ 32.3981 1.14832
$$797$$ 30.3269 1.07423 0.537116 0.843508i $$-0.319514\pi$$
0.537116 + 0.843508i $$0.319514\pi$$
$$798$$ −4.05338 −0.143488
$$799$$ 4.92048 0.174074
$$800$$ 13.7659 0.486698
$$801$$ −8.30211 −0.293340
$$802$$ 46.0812 1.62718
$$803$$ −15.2104 −0.536762
$$804$$ −10.7179 −0.377992
$$805$$ 10.3679 0.365421
$$806$$ 0 0
$$807$$ 10.2133 0.359525
$$808$$ −9.29830 −0.327113
$$809$$ 27.4074 0.963594 0.481797 0.876283i $$-0.339984\pi$$
0.481797 + 0.876283i $$0.339984\pi$$
$$810$$ −3.35579 −0.117910
$$811$$ −10.9233 −0.383568 −0.191784 0.981437i $$-0.561427\pi$$
−0.191784 + 0.981437i $$0.561427\pi$$
$$812$$ 3.82366 0.134184
$$813$$ 24.1804 0.848044
$$814$$ 12.0632 0.422814
$$815$$ −16.6706 −0.583946
$$816$$ −4.32155 −0.151285
$$817$$ −12.1201 −0.424028
$$818$$ 40.9804 1.43285
$$819$$ 0 0
$$820$$ −17.5413 −0.612568
$$821$$ 3.84174 0.134078 0.0670388 0.997750i $$-0.478645\pi$$
0.0670388 + 0.997750i $$0.478645\pi$$
$$822$$ 1.80781 0.0630547
$$823$$ 46.5183 1.62153 0.810764 0.585373i $$-0.199052\pi$$
0.810764 + 0.585373i $$0.199052\pi$$
$$824$$ 3.63499 0.126631
$$825$$ 5.89484 0.205232
$$826$$ 9.19577 0.319962
$$827$$ −1.60867 −0.0559390 −0.0279695 0.999609i $$-0.508904\pi$$
−0.0279695 + 0.999609i $$0.508904\pi$$
$$828$$ 9.46233 0.328839
$$829$$ 40.8933 1.42028 0.710142 0.704058i $$-0.248631\pi$$
0.710142 + 0.704058i $$0.248631\pi$$
$$830$$ 39.3263 1.36504
$$831$$ 18.3142 0.635311
$$832$$ 0 0
$$833$$ −0.934195 −0.0323679
$$834$$ 18.8021 0.651062
$$835$$ 7.32502 0.253493
$$836$$ −10.7671 −0.372390
$$837$$ −6.06765 −0.209729
$$838$$ 0.378508 0.0130753
$$839$$ −45.4468 −1.56900 −0.784499 0.620130i $$-0.787080\pi$$
−0.784499 + 0.620130i $$0.787080\pi$$
$$840$$ 1.30344 0.0449731
$$841$$ −23.3707 −0.805886
$$842$$ −39.6621 −1.36685
$$843$$ 9.11661 0.313993
$$844$$ −11.6903 −0.402396
$$845$$ 0 0
$$846$$ −10.0096 −0.344139
$$847$$ 1.18799 0.0408197
$$848$$ −48.9846 −1.68214
$$849$$ 9.22502 0.316602
$$850$$ 3.34102 0.114596
$$851$$ 11.8981 0.407863
$$852$$ −11.5636 −0.396163
$$853$$ −34.0573 −1.16610 −0.583050 0.812436i $$-0.698141\pi$$
−0.583050 + 0.812436i $$0.698141\pi$$
$$854$$ −23.9487 −0.819508
$$855$$ 3.76630 0.128805
$$856$$ 2.72764 0.0932289
$$857$$ −30.3420 −1.03646 −0.518231 0.855240i $$-0.673409\pi$$
−0.518231 + 0.855240i $$0.673409\pi$$
$$858$$ 0 0
$$859$$ 54.1644 1.84806 0.924032 0.382315i $$-0.124873\pi$$
0.924032 + 0.382315i $$0.124873\pi$$
$$860$$ −16.1710 −0.551426
$$861$$ 6.16399 0.210068
$$862$$ 25.1417 0.856329
$$863$$ −53.2784 −1.81362 −0.906808 0.421543i $$-0.861489\pi$$
−0.906808 + 0.421543i $$0.861489\pi$$
$$864$$ 7.31496 0.248860
$$865$$ −5.71451 −0.194299
$$866$$ −59.9616 −2.03758
$$867$$ 16.1273 0.547711
$$868$$ −9.77853 −0.331905
$$869$$ 37.7660 1.28112
$$870$$ −7.96199 −0.269937
$$871$$ 0 0
$$872$$ −10.8260 −0.366616
$$873$$ 18.2921 0.619095
$$874$$ −23.7992 −0.805021
$$875$$ −12.1522 −0.410818
$$876$$ 7.82553 0.264400
$$877$$ −32.5325 −1.09854 −0.549272 0.835643i $$-0.685095\pi$$
−0.549272 + 0.835643i $$0.685095\pi$$
$$878$$ 51.5771 1.74064
$$879$$ −18.0748 −0.609647
$$880$$ 25.5875 0.862554
$$881$$ −12.1091 −0.407965 −0.203983 0.978975i $$-0.565389\pi$$
−0.203983 + 0.978975i $$0.565389\pi$$
$$882$$ 1.90042 0.0639904
$$883$$ −39.1408 −1.31719 −0.658596 0.752497i $$-0.728849\pi$$
−0.658596 + 0.752497i $$0.728849\pi$$
$$884$$ 0 0
$$885$$ −8.54448 −0.287219
$$886$$ −1.56967 −0.0527341
$$887$$ −18.8647 −0.633414 −0.316707 0.948523i $$-0.602577\pi$$
−0.316707 + 0.948523i $$0.602577\pi$$
$$888$$ 1.49582 0.0501965
$$889$$ 4.80045 0.161002
$$890$$ 27.8601 0.933874
$$891$$ 3.13241 0.104940
$$892$$ 13.5661 0.454225
$$893$$ 11.2341 0.375935
$$894$$ −37.4776 −1.25344
$$895$$ −35.1019 −1.17333
$$896$$ −5.79386 −0.193559
$$897$$ 0 0
$$898$$ −52.3744 −1.74776
$$899$$ −14.3962 −0.480140
$$900$$ −3.03281 −0.101094
$$901$$ −9.89225 −0.329559
$$902$$ 36.6936 1.22176
$$903$$ 5.68247 0.189101
$$904$$ −10.6523 −0.354291
$$905$$ −15.7355 −0.523064
$$906$$ 24.0418 0.798734
$$907$$ −7.69608 −0.255544 −0.127772 0.991804i $$-0.540783\pi$$
−0.127772 + 0.991804i $$0.540783\pi$$
$$908$$ 16.8577 0.559441
$$909$$ 12.5967 0.417807
$$910$$ 0 0
$$911$$ −37.7524 −1.25079 −0.625396 0.780307i $$-0.715063\pi$$
−0.625396 + 0.780307i $$0.715063\pi$$
$$912$$ −9.86668 −0.326718
$$913$$ −36.7085 −1.21488
$$914$$ 0.0847464 0.00280316
$$915$$ 22.2525 0.735646
$$916$$ −17.1290 −0.565958
$$917$$ 7.33712 0.242293
$$918$$ 1.77536 0.0585956
$$919$$ −55.4581 −1.82939 −0.914696 0.404142i $$-0.867570\pi$$
−0.914696 + 0.404142i $$0.867570\pi$$
$$920$$ 7.65310 0.252315
$$921$$ 15.5831 0.513480
$$922$$ −5.44876 −0.179445
$$923$$ 0 0
$$924$$ 5.04815 0.166072
$$925$$ −3.81353 −0.125388
$$926$$ 22.3512 0.734507
$$927$$ −4.92445 −0.161740
$$928$$ 17.3556 0.569724
$$929$$ 25.9014 0.849799 0.424899 0.905241i $$-0.360309\pi$$
0.424899 + 0.905241i $$0.360309\pi$$
$$930$$ 20.3618 0.667689
$$931$$ −2.13289 −0.0699027
$$932$$ 37.0232 1.21273
$$933$$ 2.20637 0.0722332
$$934$$ 74.8224 2.44826
$$935$$ 5.16729 0.168988
$$936$$ 0 0
$$937$$ −52.0461 −1.70027 −0.850136 0.526562i $$-0.823481\pi$$
−0.850136 + 0.526562i $$0.823481\pi$$
$$938$$ −12.6388 −0.412673
$$939$$ −26.6495 −0.869675
$$940$$ 14.9889 0.488883
$$941$$ 16.0103 0.521921 0.260960 0.965349i $$-0.415961\pi$$
0.260960 + 0.965349i $$0.415961\pi$$
$$942$$ 30.3087 0.987512
$$943$$ 36.1916 1.17856
$$944$$ 22.3842 0.728544
$$945$$ −1.76582 −0.0574421
$$946$$ 33.8271 1.09982
$$947$$ 12.0064 0.390155 0.195078 0.980788i $$-0.437504\pi$$
0.195078 + 0.980788i $$0.437504\pi$$
$$948$$ −19.4301 −0.631059
$$949$$ 0 0
$$950$$ 7.62800 0.247485
$$951$$ 22.8415 0.740688
$$952$$ −0.689579 −0.0223494
$$953$$ −0.168086 −0.00544483 −0.00272241 0.999996i $$-0.500867\pi$$
−0.00272241 + 0.999996i $$0.500867\pi$$
$$954$$ 20.1236 0.651527
$$955$$ −30.8295 −0.997619
$$956$$ −25.3581 −0.820141
$$957$$ 7.43200 0.240243
$$958$$ −29.5655 −0.955218
$$959$$ 0.951272 0.0307182
$$960$$ −8.21024 −0.264984
$$961$$ 5.81638 0.187625
$$962$$ 0 0
$$963$$ −3.69523 −0.119077
$$964$$ −14.5774 −0.469505
$$965$$ 34.0981 1.09766
$$966$$ 11.1582 0.359009
$$967$$ −13.9167 −0.447531 −0.223766 0.974643i $$-0.571835\pi$$
−0.223766 + 0.974643i $$0.571835\pi$$
$$968$$ 0.876915 0.0281851
$$969$$ −1.99254 −0.0640095
$$970$$ −61.3846 −1.97094
$$971$$ 46.8956 1.50495 0.752476 0.658619i $$-0.228859\pi$$
0.752476 + 0.658619i $$0.228859\pi$$
$$972$$ −1.61158 −0.0516916
$$973$$ 9.89366 0.317176
$$974$$ −45.1274 −1.44597
$$975$$ 0 0
$$976$$ −58.2956 −1.86600
$$977$$ 5.97584 0.191184 0.0955920 0.995421i $$-0.469526\pi$$
0.0955920 + 0.995421i $$0.469526\pi$$
$$978$$ −17.9413 −0.573700
$$979$$ −26.0056 −0.831144
$$980$$ −2.84577 −0.0909046
$$981$$ 14.6664 0.468262
$$982$$ −11.1465 −0.355698
$$983$$ 36.0862 1.15097 0.575485 0.817812i $$-0.304813\pi$$
0.575485 + 0.817812i $$0.304813\pi$$
$$984$$ 4.54997 0.145048
$$985$$ 45.8844 1.46200
$$986$$ 4.21224 0.134145
$$987$$ −5.26708 −0.167653
$$988$$ 0 0
$$989$$ 33.3643 1.06092
$$990$$ −10.5117 −0.334085
$$991$$ −45.0778 −1.43194 −0.715972 0.698129i $$-0.754016\pi$$
−0.715972 + 0.698129i $$0.754016\pi$$
$$992$$ −44.3846 −1.40921
$$993$$ 3.66559 0.116324
$$994$$ −13.6361 −0.432510
$$995$$ −35.4988 −1.12539
$$996$$ 18.8860 0.598427
$$997$$ 15.1512 0.479842 0.239921 0.970792i $$-0.422878\pi$$
0.239921 + 0.970792i $$0.422878\pi$$
$$998$$ −14.8181 −0.469059
$$999$$ −2.02644 −0.0641138
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 3549.2.a.bf.1.8 yes 9
13.12 even 2 3549.2.a.be.1.2 9

By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.be.1.2 9 13.12 even 2
3549.2.a.bf.1.8 yes 9 1.1 even 1 trivial