# Properties

 Label 5577.2.a.y.1.7 Level $5577$ Weight $2$ Character 5577.1 Self dual yes Analytic conductor $44.533$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5577 = 3 \cdot 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5577.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.5325692073$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3 x^{6} - 7 x^{5} + 21 x^{4} + 13 x^{3} - 33 x^{2} - 7 x + 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 429) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.7 Root $$2.73878$$ of defining polynomial Character $$\chi$$ $$=$$ 5577.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.73878 q^{2} +1.00000 q^{3} +5.50093 q^{4} -2.84154 q^{5} +2.73878 q^{6} +3.93129 q^{7} +9.58828 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+2.73878 q^{2} +1.00000 q^{3} +5.50093 q^{4} -2.84154 q^{5} +2.73878 q^{6} +3.93129 q^{7} +9.58828 q^{8} +1.00000 q^{9} -7.78236 q^{10} +1.00000 q^{11} +5.50093 q^{12} +10.7670 q^{14} -2.84154 q^{15} +15.2584 q^{16} +3.81818 q^{17} +2.73878 q^{18} -2.94082 q^{19} -15.6311 q^{20} +3.93129 q^{21} +2.73878 q^{22} -1.89484 q^{23} +9.58828 q^{24} +3.07435 q^{25} +1.00000 q^{27} +21.6258 q^{28} -2.09928 q^{29} -7.78236 q^{30} -6.16032 q^{31} +22.6127 q^{32} +1.00000 q^{33} +10.4572 q^{34} -11.1709 q^{35} +5.50093 q^{36} +8.34404 q^{37} -8.05426 q^{38} -27.2455 q^{40} +6.35787 q^{41} +10.7670 q^{42} -11.7275 q^{43} +5.50093 q^{44} -2.84154 q^{45} -5.18956 q^{46} +5.31137 q^{47} +15.2584 q^{48} +8.45506 q^{49} +8.41997 q^{50} +3.81818 q^{51} -2.37985 q^{53} +2.73878 q^{54} -2.84154 q^{55} +37.6943 q^{56} -2.94082 q^{57} -5.74947 q^{58} -5.38624 q^{59} -15.6311 q^{60} -2.34857 q^{61} -16.8718 q^{62} +3.93129 q^{63} +31.4147 q^{64} +2.73878 q^{66} +10.4731 q^{67} +21.0035 q^{68} -1.89484 q^{69} -30.5947 q^{70} +10.0939 q^{71} +9.58828 q^{72} -15.1079 q^{73} +22.8525 q^{74} +3.07435 q^{75} -16.1772 q^{76} +3.93129 q^{77} +1.57120 q^{79} -43.3572 q^{80} +1.00000 q^{81} +17.4128 q^{82} +10.6736 q^{83} +21.6258 q^{84} -10.8495 q^{85} -32.1190 q^{86} -2.09928 q^{87} +9.58828 q^{88} +3.23647 q^{89} -7.78236 q^{90} -10.4234 q^{92} -6.16032 q^{93} +14.5467 q^{94} +8.35645 q^{95} +22.6127 q^{96} -17.7914 q^{97} +23.1566 q^{98} +1.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 3q^{2} + 7q^{3} + 9q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 15q^{8} + 7q^{9} + O(q^{10})$$ $$7q + 3q^{2} + 7q^{3} + 9q^{4} + 6q^{5} + 3q^{6} + 6q^{7} + 15q^{8} + 7q^{9} + 7q^{11} + 9q^{12} + 8q^{14} + 6q^{15} + 17q^{16} - 2q^{17} + 3q^{18} + 8q^{19} - 2q^{20} + 6q^{21} + 3q^{22} + 4q^{23} + 15q^{24} + 13q^{25} + 7q^{27} + 12q^{28} - 12q^{29} - 10q^{31} + 33q^{32} + 7q^{33} + 28q^{34} - 4q^{35} + 9q^{36} + 6q^{37} + 16q^{38} - 10q^{40} + 2q^{41} + 8q^{42} - 16q^{43} + 9q^{44} + 6q^{45} - 26q^{46} + 18q^{47} + 17q^{48} + 23q^{49} + 39q^{50} - 2q^{51} + 10q^{53} + 3q^{54} + 6q^{55} + 16q^{56} + 8q^{57} + 10q^{58} + 2q^{59} - 2q^{60} - 10q^{61} - 36q^{62} + 6q^{63} + 29q^{64} + 3q^{66} + 8q^{67} - 10q^{68} + 4q^{69} - 20q^{70} + 36q^{71} + 15q^{72} + 20q^{73} + 13q^{75} + 10q^{76} + 6q^{77} + 6q^{79} - 20q^{80} + 7q^{81} - 10q^{82} + 30q^{83} + 12q^{84} - 40q^{85} + 6q^{86} - 12q^{87} + 15q^{88} + 34q^{89} - 12q^{92} - 10q^{93} + 32q^{94} + 18q^{95} + 33q^{96} + 16q^{97} + q^{98} + 7q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.73878 1.93661 0.968306 0.249768i $$-0.0803543\pi$$
0.968306 + 0.249768i $$0.0803543\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 5.50093 2.75046
$$5$$ −2.84154 −1.27078 −0.635388 0.772193i $$-0.719160\pi$$
−0.635388 + 0.772193i $$0.719160\pi$$
$$6$$ 2.73878 1.11810
$$7$$ 3.93129 1.48589 0.742944 0.669353i $$-0.233429\pi$$
0.742944 + 0.669353i $$0.233429\pi$$
$$8$$ 9.58828 3.38997
$$9$$ 1.00000 0.333333
$$10$$ −7.78236 −2.46100
$$11$$ 1.00000 0.301511
$$12$$ 5.50093 1.58798
$$13$$ 0 0
$$14$$ 10.7670 2.87759
$$15$$ −2.84154 −0.733682
$$16$$ 15.2584 3.81459
$$17$$ 3.81818 0.926044 0.463022 0.886347i $$-0.346765\pi$$
0.463022 + 0.886347i $$0.346765\pi$$
$$18$$ 2.73878 0.645537
$$19$$ −2.94082 −0.674670 −0.337335 0.941385i $$-0.609526\pi$$
−0.337335 + 0.941385i $$0.609526\pi$$
$$20$$ −15.6311 −3.49522
$$21$$ 3.93129 0.857878
$$22$$ 2.73878 0.583910
$$23$$ −1.89484 −0.395102 −0.197551 0.980293i $$-0.563299\pi$$
−0.197551 + 0.980293i $$0.563299\pi$$
$$24$$ 9.58828 1.95720
$$25$$ 3.07435 0.614869
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 21.6258 4.08688
$$29$$ −2.09928 −0.389826 −0.194913 0.980821i $$-0.562443\pi$$
−0.194913 + 0.980821i $$0.562443\pi$$
$$30$$ −7.78236 −1.42086
$$31$$ −6.16032 −1.10643 −0.553213 0.833040i $$-0.686598\pi$$
−0.553213 + 0.833040i $$0.686598\pi$$
$$32$$ 22.6127 3.99741
$$33$$ 1.00000 0.174078
$$34$$ 10.4572 1.79339
$$35$$ −11.1709 −1.88823
$$36$$ 5.50093 0.916821
$$37$$ 8.34404 1.37175 0.685876 0.727719i $$-0.259419\pi$$
0.685876 + 0.727719i $$0.259419\pi$$
$$38$$ −8.05426 −1.30657
$$39$$ 0 0
$$40$$ −27.2455 −4.30789
$$41$$ 6.35787 0.992933 0.496466 0.868056i $$-0.334631\pi$$
0.496466 + 0.868056i $$0.334631\pi$$
$$42$$ 10.7670 1.66138
$$43$$ −11.7275 −1.78842 −0.894212 0.447643i $$-0.852264\pi$$
−0.894212 + 0.447643i $$0.852264\pi$$
$$44$$ 5.50093 0.829296
$$45$$ −2.84154 −0.423592
$$46$$ −5.18956 −0.765159
$$47$$ 5.31137 0.774743 0.387371 0.921924i $$-0.373383\pi$$
0.387371 + 0.921924i $$0.373383\pi$$
$$48$$ 15.2584 2.20235
$$49$$ 8.45506 1.20787
$$50$$ 8.41997 1.19076
$$51$$ 3.81818 0.534652
$$52$$ 0 0
$$53$$ −2.37985 −0.326898 −0.163449 0.986552i $$-0.552262\pi$$
−0.163449 + 0.986552i $$0.552262\pi$$
$$54$$ 2.73878 0.372701
$$55$$ −2.84154 −0.383153
$$56$$ 37.6943 5.03712
$$57$$ −2.94082 −0.389521
$$58$$ −5.74947 −0.754942
$$59$$ −5.38624 −0.701229 −0.350615 0.936520i $$-0.614027\pi$$
−0.350615 + 0.936520i $$0.614027\pi$$
$$60$$ −15.6311 −2.01797
$$61$$ −2.34857 −0.300703 −0.150352 0.988633i $$-0.548041\pi$$
−0.150352 + 0.988633i $$0.548041\pi$$
$$62$$ −16.8718 −2.14272
$$63$$ 3.93129 0.495296
$$64$$ 31.4147 3.92684
$$65$$ 0 0
$$66$$ 2.73878 0.337121
$$67$$ 10.4731 1.27949 0.639744 0.768588i $$-0.279040\pi$$
0.639744 + 0.768588i $$0.279040\pi$$
$$68$$ 21.0035 2.54705
$$69$$ −1.89484 −0.228112
$$70$$ −30.5947 −3.65677
$$71$$ 10.0939 1.19793 0.598964 0.800776i $$-0.295579\pi$$
0.598964 + 0.800776i $$0.295579\pi$$
$$72$$ 9.58828 1.12999
$$73$$ −15.1079 −1.76824 −0.884120 0.467260i $$-0.845241\pi$$
−0.884120 + 0.467260i $$0.845241\pi$$
$$74$$ 22.8525 2.65655
$$75$$ 3.07435 0.354995
$$76$$ −16.1772 −1.85566
$$77$$ 3.93129 0.448012
$$78$$ 0 0
$$79$$ 1.57120 0.176774 0.0883872 0.996086i $$-0.471829\pi$$
0.0883872 + 0.996086i $$0.471829\pi$$
$$80$$ −43.3572 −4.84748
$$81$$ 1.00000 0.111111
$$82$$ 17.4128 1.92293
$$83$$ 10.6736 1.17157 0.585787 0.810465i $$-0.300785\pi$$
0.585787 + 0.810465i $$0.300785\pi$$
$$84$$ 21.6258 2.35956
$$85$$ −10.8495 −1.17679
$$86$$ −32.1190 −3.46348
$$87$$ −2.09928 −0.225066
$$88$$ 9.58828 1.02211
$$89$$ 3.23647 0.343065 0.171533 0.985178i $$-0.445128\pi$$
0.171533 + 0.985178i $$0.445128\pi$$
$$90$$ −7.78236 −0.820333
$$91$$ 0 0
$$92$$ −10.4234 −1.08671
$$93$$ −6.16032 −0.638795
$$94$$ 14.5467 1.50038
$$95$$ 8.35645 0.857354
$$96$$ 22.6127 2.30790
$$97$$ −17.7914 −1.80645 −0.903223 0.429172i $$-0.858805\pi$$
−0.903223 + 0.429172i $$0.858805\pi$$
$$98$$ 23.1566 2.33917
$$99$$ 1.00000 0.100504
$$100$$ 16.9118 1.69118
$$101$$ −12.1229 −1.20627 −0.603136 0.797638i $$-0.706083\pi$$
−0.603136 + 0.797638i $$0.706083\pi$$
$$102$$ 10.4572 1.03541
$$103$$ −3.40983 −0.335981 −0.167990 0.985789i $$-0.553728\pi$$
−0.167990 + 0.985789i $$0.553728\pi$$
$$104$$ 0 0
$$105$$ −11.1709 −1.09017
$$106$$ −6.51790 −0.633075
$$107$$ −12.2352 −1.18282 −0.591411 0.806370i $$-0.701429\pi$$
−0.591411 + 0.806370i $$0.701429\pi$$
$$108$$ 5.50093 0.529327
$$109$$ −8.72763 −0.835956 −0.417978 0.908457i $$-0.637261\pi$$
−0.417978 + 0.908457i $$0.637261\pi$$
$$110$$ −7.78236 −0.742019
$$111$$ 8.34404 0.791981
$$112$$ 59.9850 5.66805
$$113$$ 0.0155525 0.00146305 0.000731527 1.00000i $$-0.499767\pi$$
0.000731527 1.00000i $$0.499767\pi$$
$$114$$ −8.05426 −0.754351
$$115$$ 5.38427 0.502086
$$116$$ −11.5480 −1.07220
$$117$$ 0 0
$$118$$ −14.7517 −1.35801
$$119$$ 15.0104 1.37600
$$120$$ −27.2455 −2.48716
$$121$$ 1.00000 0.0909091
$$122$$ −6.43222 −0.582346
$$123$$ 6.35787 0.573270
$$124$$ −33.8875 −3.04318
$$125$$ 5.47182 0.489414
$$126$$ 10.7670 0.959196
$$127$$ −5.91514 −0.524883 −0.262442 0.964948i $$-0.584528\pi$$
−0.262442 + 0.964948i $$0.584528\pi$$
$$128$$ 40.8125 3.60735
$$129$$ −11.7275 −1.03255
$$130$$ 0 0
$$131$$ −1.32191 −0.115496 −0.0577481 0.998331i $$-0.518392\pi$$
−0.0577481 + 0.998331i $$0.518392\pi$$
$$132$$ 5.50093 0.478794
$$133$$ −11.5612 −1.00248
$$134$$ 28.6835 2.47787
$$135$$ −2.84154 −0.244561
$$136$$ 36.6097 3.13926
$$137$$ 4.82127 0.411909 0.205955 0.978562i $$-0.433970\pi$$
0.205955 + 0.978562i $$0.433970\pi$$
$$138$$ −5.18956 −0.441765
$$139$$ 17.1877 1.45784 0.728919 0.684600i $$-0.240023\pi$$
0.728919 + 0.684600i $$0.240023\pi$$
$$140$$ −61.4504 −5.19351
$$141$$ 5.31137 0.447298
$$142$$ 27.6451 2.31992
$$143$$ 0 0
$$144$$ 15.2584 1.27153
$$145$$ 5.96518 0.495382
$$146$$ −41.3771 −3.42439
$$147$$ 8.45506 0.697361
$$148$$ 45.8999 3.77295
$$149$$ −5.31754 −0.435629 −0.217815 0.975990i $$-0.569893\pi$$
−0.217815 + 0.975990i $$0.569893\pi$$
$$150$$ 8.41997 0.687487
$$151$$ −6.77573 −0.551401 −0.275700 0.961244i $$-0.588910\pi$$
−0.275700 + 0.961244i $$0.588910\pi$$
$$152$$ −28.1974 −2.28711
$$153$$ 3.81818 0.308681
$$154$$ 10.7670 0.867626
$$155$$ 17.5048 1.40602
$$156$$ 0 0
$$157$$ 16.2884 1.29995 0.649977 0.759954i $$-0.274779\pi$$
0.649977 + 0.759954i $$0.274779\pi$$
$$158$$ 4.30319 0.342343
$$159$$ −2.37985 −0.188735
$$160$$ −64.2550 −5.07981
$$161$$ −7.44918 −0.587077
$$162$$ 2.73878 0.215179
$$163$$ 11.4032 0.893170 0.446585 0.894741i $$-0.352640\pi$$
0.446585 + 0.894741i $$0.352640\pi$$
$$164$$ 34.9742 2.73103
$$165$$ −2.84154 −0.221214
$$166$$ 29.2325 2.26888
$$167$$ −14.7302 −1.13985 −0.569927 0.821695i $$-0.693029\pi$$
−0.569927 + 0.821695i $$0.693029\pi$$
$$168$$ 37.6943 2.90818
$$169$$ 0 0
$$170$$ −29.7144 −2.27899
$$171$$ −2.94082 −0.224890
$$172$$ −64.5121 −4.91900
$$173$$ 0.186647 0.0141905 0.00709527 0.999975i $$-0.497741\pi$$
0.00709527 + 0.999975i $$0.497741\pi$$
$$174$$ −5.74947 −0.435866
$$175$$ 12.0862 0.913627
$$176$$ 15.2584 1.15014
$$177$$ −5.38624 −0.404855
$$178$$ 8.86399 0.664384
$$179$$ 10.9555 0.818856 0.409428 0.912342i $$-0.365728\pi$$
0.409428 + 0.912342i $$0.365728\pi$$
$$180$$ −15.6311 −1.16507
$$181$$ −8.52034 −0.633312 −0.316656 0.948540i $$-0.602560\pi$$
−0.316656 + 0.948540i $$0.602560\pi$$
$$182$$ 0 0
$$183$$ −2.34857 −0.173611
$$184$$ −18.1683 −1.33938
$$185$$ −23.7099 −1.74319
$$186$$ −16.8718 −1.23710
$$187$$ 3.81818 0.279213
$$188$$ 29.2175 2.13090
$$189$$ 3.93129 0.285959
$$190$$ 22.8865 1.66036
$$191$$ −11.8003 −0.853840 −0.426920 0.904289i $$-0.640401\pi$$
−0.426920 + 0.904289i $$0.640401\pi$$
$$192$$ 31.4147 2.26716
$$193$$ 10.9405 0.787518 0.393759 0.919214i $$-0.371174\pi$$
0.393759 + 0.919214i $$0.371174\pi$$
$$194$$ −48.7268 −3.49838
$$195$$ 0 0
$$196$$ 46.5107 3.32219
$$197$$ 9.43307 0.672079 0.336039 0.941848i $$-0.390912\pi$$
0.336039 + 0.941848i $$0.390912\pi$$
$$198$$ 2.73878 0.194637
$$199$$ −15.3198 −1.08599 −0.542996 0.839735i $$-0.682710\pi$$
−0.542996 + 0.839735i $$0.682710\pi$$
$$200$$ 29.4777 2.08439
$$201$$ 10.4731 0.738713
$$202$$ −33.2020 −2.33608
$$203$$ −8.25288 −0.579238
$$204$$ 21.0035 1.47054
$$205$$ −18.0661 −1.26179
$$206$$ −9.33878 −0.650664
$$207$$ −1.89484 −0.131701
$$208$$ 0 0
$$209$$ −2.94082 −0.203421
$$210$$ −30.5947 −2.11124
$$211$$ 2.91243 0.200500 0.100250 0.994962i $$-0.468036\pi$$
0.100250 + 0.994962i $$0.468036\pi$$
$$212$$ −13.0914 −0.899122
$$213$$ 10.0939 0.691625
$$214$$ −33.5096 −2.29067
$$215$$ 33.3241 2.27269
$$216$$ 9.58828 0.652400
$$217$$ −24.2180 −1.64403
$$218$$ −23.9031 −1.61892
$$219$$ −15.1079 −1.02089
$$220$$ −15.6311 −1.05385
$$221$$ 0 0
$$222$$ 22.8525 1.53376
$$223$$ 10.8511 0.726641 0.363320 0.931664i $$-0.381643\pi$$
0.363320 + 0.931664i $$0.381643\pi$$
$$224$$ 88.8973 5.93970
$$225$$ 3.07435 0.204956
$$226$$ 0.0425948 0.00283337
$$227$$ 0.322636 0.0214141 0.0107070 0.999943i $$-0.496592\pi$$
0.0107070 + 0.999943i $$0.496592\pi$$
$$228$$ −16.1772 −1.07136
$$229$$ −5.94060 −0.392566 −0.196283 0.980547i $$-0.562887\pi$$
−0.196283 + 0.980547i $$0.562887\pi$$
$$230$$ 14.7463 0.972345
$$231$$ 3.93129 0.258660
$$232$$ −20.1285 −1.32150
$$233$$ −12.6202 −0.826774 −0.413387 0.910555i $$-0.635654\pi$$
−0.413387 + 0.910555i $$0.635654\pi$$
$$234$$ 0 0
$$235$$ −15.0925 −0.984524
$$236$$ −29.6293 −1.92871
$$237$$ 1.57120 0.102061
$$238$$ 41.1101 2.66477
$$239$$ 20.7706 1.34354 0.671768 0.740762i $$-0.265535\pi$$
0.671768 + 0.740762i $$0.265535\pi$$
$$240$$ −43.3572 −2.79870
$$241$$ −18.2163 −1.17341 −0.586706 0.809800i $$-0.699576\pi$$
−0.586706 + 0.809800i $$0.699576\pi$$
$$242$$ 2.73878 0.176056
$$243$$ 1.00000 0.0641500
$$244$$ −12.9193 −0.827074
$$245$$ −24.0254 −1.53492
$$246$$ 17.4128 1.11020
$$247$$ 0 0
$$248$$ −59.0668 −3.75075
$$249$$ 10.6736 0.676409
$$250$$ 14.9861 0.947806
$$251$$ −22.6662 −1.43068 −0.715340 0.698776i $$-0.753728\pi$$
−0.715340 + 0.698776i $$0.753728\pi$$
$$252$$ 21.6258 1.36229
$$253$$ −1.89484 −0.119128
$$254$$ −16.2003 −1.01650
$$255$$ −10.8495 −0.679422
$$256$$ 48.9471 3.05920
$$257$$ 17.7333 1.10617 0.553087 0.833124i $$-0.313450\pi$$
0.553087 + 0.833124i $$0.313450\pi$$
$$258$$ −32.1190 −1.99964
$$259$$ 32.8028 2.03827
$$260$$ 0 0
$$261$$ −2.09928 −0.129942
$$262$$ −3.62044 −0.223671
$$263$$ 29.5107 1.81971 0.909853 0.414930i $$-0.136194\pi$$
0.909853 + 0.414930i $$0.136194\pi$$
$$264$$ 9.58828 0.590118
$$265$$ 6.76245 0.415414
$$266$$ −31.6637 −1.94142
$$267$$ 3.23647 0.198069
$$268$$ 57.6116 3.51919
$$269$$ −9.02091 −0.550015 −0.275007 0.961442i $$-0.588680\pi$$
−0.275007 + 0.961442i $$0.588680\pi$$
$$270$$ −7.78236 −0.473619
$$271$$ 4.58320 0.278410 0.139205 0.990264i $$-0.455545\pi$$
0.139205 + 0.990264i $$0.455545\pi$$
$$272$$ 58.2591 3.53248
$$273$$ 0 0
$$274$$ 13.2044 0.797708
$$275$$ 3.07435 0.185390
$$276$$ −10.4234 −0.627414
$$277$$ 20.9859 1.26092 0.630461 0.776221i $$-0.282866\pi$$
0.630461 + 0.776221i $$0.282866\pi$$
$$278$$ 47.0732 2.82327
$$279$$ −6.16032 −0.368809
$$280$$ −107.110 −6.40104
$$281$$ 10.5360 0.628524 0.314262 0.949336i $$-0.398243\pi$$
0.314262 + 0.949336i $$0.398243\pi$$
$$282$$ 14.5467 0.866242
$$283$$ −2.57169 −0.152871 −0.0764355 0.997075i $$-0.524354\pi$$
−0.0764355 + 0.997075i $$0.524354\pi$$
$$284$$ 55.5260 3.29486
$$285$$ 8.35645 0.494993
$$286$$ 0 0
$$287$$ 24.9947 1.47539
$$288$$ 22.6127 1.33247
$$289$$ −2.42153 −0.142443
$$290$$ 16.3373 0.959362
$$291$$ −17.7914 −1.04295
$$292$$ −83.1072 −4.86348
$$293$$ 5.16162 0.301545 0.150772 0.988568i $$-0.451824\pi$$
0.150772 + 0.988568i $$0.451824\pi$$
$$294$$ 23.1566 1.35052
$$295$$ 15.3052 0.891105
$$296$$ 80.0050 4.65019
$$297$$ 1.00000 0.0580259
$$298$$ −14.5636 −0.843645
$$299$$ 0 0
$$300$$ 16.9118 0.976401
$$301$$ −46.1042 −2.65740
$$302$$ −18.5572 −1.06785
$$303$$ −12.1229 −0.696442
$$304$$ −44.8720 −2.57359
$$305$$ 6.67355 0.382126
$$306$$ 10.4572 0.597796
$$307$$ −30.0561 −1.71539 −0.857697 0.514156i $$-0.828105\pi$$
−0.857697 + 0.514156i $$0.828105\pi$$
$$308$$ 21.6258 1.23224
$$309$$ −3.40983 −0.193978
$$310$$ 47.9418 2.72291
$$311$$ −21.7462 −1.23311 −0.616557 0.787310i $$-0.711473\pi$$
−0.616557 + 0.787310i $$0.711473\pi$$
$$312$$ 0 0
$$313$$ −24.4361 −1.38121 −0.690606 0.723231i $$-0.742656\pi$$
−0.690606 + 0.723231i $$0.742656\pi$$
$$314$$ 44.6103 2.51751
$$315$$ −11.1709 −0.629410
$$316$$ 8.64309 0.486212
$$317$$ 21.1040 1.18532 0.592658 0.805454i $$-0.298079\pi$$
0.592658 + 0.805454i $$0.298079\pi$$
$$318$$ −6.51790 −0.365506
$$319$$ −2.09928 −0.117537
$$320$$ −89.2661 −4.99013
$$321$$ −12.2352 −0.682903
$$322$$ −20.4017 −1.13694
$$323$$ −11.2286 −0.624774
$$324$$ 5.50093 0.305607
$$325$$ 0 0
$$326$$ 31.2310 1.72972
$$327$$ −8.72763 −0.482639
$$328$$ 60.9611 3.36601
$$329$$ 20.8805 1.15118
$$330$$ −7.78236 −0.428405
$$331$$ 0.560518 0.0308089 0.0154044 0.999881i $$-0.495096\pi$$
0.0154044 + 0.999881i $$0.495096\pi$$
$$332$$ 58.7144 3.22237
$$333$$ 8.34404 0.457250
$$334$$ −40.3427 −2.20745
$$335$$ −29.7596 −1.62594
$$336$$ 59.9850 3.27245
$$337$$ −7.76850 −0.423177 −0.211589 0.977359i $$-0.567864\pi$$
−0.211589 + 0.977359i $$0.567864\pi$$
$$338$$ 0 0
$$339$$ 0.0155525 0.000844695 0
$$340$$ −59.6823 −3.23673
$$341$$ −6.16032 −0.333600
$$342$$ −8.05426 −0.435525
$$343$$ 5.72025 0.308864
$$344$$ −112.446 −6.06270
$$345$$ 5.38427 0.289879
$$346$$ 0.511186 0.0274816
$$347$$ −10.1357 −0.544110 −0.272055 0.962282i $$-0.587703\pi$$
−0.272055 + 0.962282i $$0.587703\pi$$
$$348$$ −11.5480 −0.619037
$$349$$ −20.0030 −1.07074 −0.535369 0.844619i $$-0.679827\pi$$
−0.535369 + 0.844619i $$0.679827\pi$$
$$350$$ 33.1013 1.76934
$$351$$ 0 0
$$352$$ 22.6127 1.20526
$$353$$ −26.7905 −1.42591 −0.712957 0.701208i $$-0.752645\pi$$
−0.712957 + 0.701208i $$0.752645\pi$$
$$354$$ −14.7517 −0.784047
$$355$$ −28.6823 −1.52230
$$356$$ 17.8036 0.943589
$$357$$ 15.0104 0.794433
$$358$$ 30.0049 1.58581
$$359$$ 26.7347 1.41101 0.705503 0.708707i $$-0.250721\pi$$
0.705503 + 0.708707i $$0.250721\pi$$
$$360$$ −27.2455 −1.43596
$$361$$ −10.3516 −0.544820
$$362$$ −23.3354 −1.22648
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ 42.9296 2.24704
$$366$$ −6.43222 −0.336218
$$367$$ 1.89563 0.0989512 0.0494756 0.998775i $$-0.484245\pi$$
0.0494756 + 0.998775i $$0.484245\pi$$
$$368$$ −28.9122 −1.50715
$$369$$ 6.35787 0.330978
$$370$$ −64.9363 −3.37588
$$371$$ −9.35590 −0.485734
$$372$$ −33.8875 −1.75698
$$373$$ −36.7517 −1.90293 −0.951464 0.307759i $$-0.900421\pi$$
−0.951464 + 0.307759i $$0.900421\pi$$
$$374$$ 10.4572 0.540726
$$375$$ 5.47182 0.282564
$$376$$ 50.9269 2.62635
$$377$$ 0 0
$$378$$ 10.7670 0.553792
$$379$$ −26.1328 −1.34235 −0.671176 0.741298i $$-0.734211\pi$$
−0.671176 + 0.741298i $$0.734211\pi$$
$$380$$ 45.9682 2.35812
$$381$$ −5.91514 −0.303042
$$382$$ −32.3185 −1.65356
$$383$$ 16.9211 0.864629 0.432315 0.901723i $$-0.357697\pi$$
0.432315 + 0.901723i $$0.357697\pi$$
$$384$$ 40.8125 2.08270
$$385$$ −11.1709 −0.569323
$$386$$ 29.9638 1.52512
$$387$$ −11.7275 −0.596142
$$388$$ −97.8693 −4.96856
$$389$$ −5.05607 −0.256353 −0.128177 0.991751i $$-0.540912\pi$$
−0.128177 + 0.991751i $$0.540912\pi$$
$$390$$ 0 0
$$391$$ −7.23484 −0.365882
$$392$$ 81.0694 4.09462
$$393$$ −1.32191 −0.0666818
$$394$$ 25.8351 1.30156
$$395$$ −4.46464 −0.224640
$$396$$ 5.50093 0.276432
$$397$$ 15.7731 0.791631 0.395815 0.918330i $$-0.370462\pi$$
0.395815 + 0.918330i $$0.370462\pi$$
$$398$$ −41.9576 −2.10314
$$399$$ −11.5612 −0.578785
$$400$$ 46.9095 2.34547
$$401$$ 8.42218 0.420584 0.210292 0.977639i $$-0.432559\pi$$
0.210292 + 0.977639i $$0.432559\pi$$
$$402$$ 28.6835 1.43060
$$403$$ 0 0
$$404$$ −66.6872 −3.31781
$$405$$ −2.84154 −0.141197
$$406$$ −22.6028 −1.12176
$$407$$ 8.34404 0.413599
$$408$$ 36.6097 1.81245
$$409$$ 31.3861 1.55194 0.775971 0.630768i $$-0.217260\pi$$
0.775971 + 0.630768i $$0.217260\pi$$
$$410$$ −49.4792 −2.44361
$$411$$ 4.82127 0.237816
$$412$$ −18.7572 −0.924102
$$413$$ −21.1749 −1.04195
$$414$$ −5.18956 −0.255053
$$415$$ −30.3293 −1.48881
$$416$$ 0 0
$$417$$ 17.1877 0.841683
$$418$$ −8.05426 −0.393947
$$419$$ −6.62928 −0.323861 −0.161931 0.986802i $$-0.551772\pi$$
−0.161931 + 0.986802i $$0.551772\pi$$
$$420$$ −61.4504 −2.99847
$$421$$ −2.61327 −0.127363 −0.0636816 0.997970i $$-0.520284\pi$$
−0.0636816 + 0.997970i $$0.520284\pi$$
$$422$$ 7.97651 0.388290
$$423$$ 5.31137 0.258248
$$424$$ −22.8187 −1.10817
$$425$$ 11.7384 0.569396
$$426$$ 27.6451 1.33941
$$427$$ −9.23291 −0.446812
$$428$$ −67.3050 −3.25331
$$429$$ 0 0
$$430$$ 91.2675 4.40131
$$431$$ −24.8745 −1.19816 −0.599081 0.800688i $$-0.704467\pi$$
−0.599081 + 0.800688i $$0.704467\pi$$
$$432$$ 15.2584 0.734118
$$433$$ 6.55556 0.315040 0.157520 0.987516i $$-0.449650\pi$$
0.157520 + 0.987516i $$0.449650\pi$$
$$434$$ −66.3278 −3.18384
$$435$$ 5.96518 0.286009
$$436$$ −48.0101 −2.29927
$$437$$ 5.57239 0.266563
$$438$$ −41.3771 −1.97707
$$439$$ −38.2180 −1.82405 −0.912023 0.410138i $$-0.865480\pi$$
−0.912023 + 0.410138i $$0.865480\pi$$
$$440$$ −27.2455 −1.29888
$$441$$ 8.45506 0.402622
$$442$$ 0 0
$$443$$ −12.7407 −0.605330 −0.302665 0.953097i $$-0.597876\pi$$
−0.302665 + 0.953097i $$0.597876\pi$$
$$444$$ 45.8999 2.17831
$$445$$ −9.19656 −0.435959
$$446$$ 29.7187 1.40722
$$447$$ −5.31754 −0.251511
$$448$$ 123.500 5.83484
$$449$$ 25.9513 1.22472 0.612358 0.790581i $$-0.290221\pi$$
0.612358 + 0.790581i $$0.290221\pi$$
$$450$$ 8.41997 0.396921
$$451$$ 6.35787 0.299381
$$452$$ 0.0855531 0.00402408
$$453$$ −6.77573 −0.318351
$$454$$ 0.883629 0.0414708
$$455$$ 0 0
$$456$$ −28.1974 −1.32046
$$457$$ 16.0372 0.750190 0.375095 0.926986i $$-0.377610\pi$$
0.375095 + 0.926986i $$0.377610\pi$$
$$458$$ −16.2700 −0.760247
$$459$$ 3.81818 0.178217
$$460$$ 29.6185 1.38097
$$461$$ 3.48566 0.162343 0.0811716 0.996700i $$-0.474134\pi$$
0.0811716 + 0.996700i $$0.474134\pi$$
$$462$$ 10.7670 0.500924
$$463$$ −27.0784 −1.25844 −0.629220 0.777227i $$-0.716626\pi$$
−0.629220 + 0.777227i $$0.716626\pi$$
$$464$$ −32.0315 −1.48703
$$465$$ 17.5048 0.811765
$$466$$ −34.5639 −1.60114
$$467$$ −15.6899 −0.726041 −0.363020 0.931781i $$-0.618254\pi$$
−0.363020 + 0.931781i $$0.618254\pi$$
$$468$$ 0 0
$$469$$ 41.1727 1.90118
$$470$$ −41.3350 −1.90664
$$471$$ 16.2884 0.750529
$$472$$ −51.6448 −2.37715
$$473$$ −11.7275 −0.539230
$$474$$ 4.30319 0.197652
$$475$$ −9.04109 −0.414834
$$476$$ 82.5709 3.78463
$$477$$ −2.37985 −0.108966
$$478$$ 56.8860 2.60191
$$479$$ 33.0061 1.50809 0.754043 0.656825i $$-0.228101\pi$$
0.754043 + 0.656825i $$0.228101\pi$$
$$480$$ −64.2550 −2.93283
$$481$$ 0 0
$$482$$ −49.8904 −2.27244
$$483$$ −7.44918 −0.338949
$$484$$ 5.50093 0.250042
$$485$$ 50.5550 2.29559
$$486$$ 2.73878 0.124234
$$487$$ 18.6684 0.845945 0.422973 0.906142i $$-0.360987\pi$$
0.422973 + 0.906142i $$0.360987\pi$$
$$488$$ −22.5187 −1.01938
$$489$$ 11.4032 0.515672
$$490$$ −65.8003 −2.97255
$$491$$ −6.78238 −0.306084 −0.153042 0.988220i $$-0.548907\pi$$
−0.153042 + 0.988220i $$0.548907\pi$$
$$492$$ 34.9742 1.57676
$$493$$ −8.01542 −0.360996
$$494$$ 0 0
$$495$$ −2.84154 −0.127718
$$496$$ −93.9963 −4.22056
$$497$$ 39.6822 1.77999
$$498$$ 29.2325 1.30994
$$499$$ −0.212900 −0.00953070 −0.00476535 0.999989i $$-0.501517\pi$$
−0.00476535 + 0.999989i $$0.501517\pi$$
$$500$$ 30.1001 1.34612
$$501$$ −14.7302 −0.658095
$$502$$ −62.0779 −2.77067
$$503$$ −10.5282 −0.469430 −0.234715 0.972064i $$-0.575416\pi$$
−0.234715 + 0.972064i $$0.575416\pi$$
$$504$$ 37.6943 1.67904
$$505$$ 34.4477 1.53290
$$506$$ −5.18956 −0.230704
$$507$$ 0 0
$$508$$ −32.5387 −1.44367
$$509$$ −26.1335 −1.15835 −0.579175 0.815204i $$-0.696625\pi$$
−0.579175 + 0.815204i $$0.696625\pi$$
$$510$$ −29.7144 −1.31578
$$511$$ −59.3934 −2.62741
$$512$$ 52.4306 2.31713
$$513$$ −2.94082 −0.129840
$$514$$ 48.5677 2.14223
$$515$$ 9.68917 0.426956
$$516$$ −64.5121 −2.83999
$$517$$ 5.31137 0.233594
$$518$$ 89.8398 3.94734
$$519$$ 0.186647 0.00819291
$$520$$ 0 0
$$521$$ 44.5983 1.95389 0.976944 0.213495i $$-0.0684847\pi$$
0.976944 + 0.213495i $$0.0684847\pi$$
$$522$$ −5.74947 −0.251647
$$523$$ −35.9788 −1.57324 −0.786622 0.617435i $$-0.788172\pi$$
−0.786622 + 0.617435i $$0.788172\pi$$
$$524$$ −7.27176 −0.317668
$$525$$ 12.0862 0.527483
$$526$$ 80.8233 3.52406
$$527$$ −23.5212 −1.02460
$$528$$ 15.2584 0.664035
$$529$$ −19.4096 −0.843895
$$530$$ 18.5209 0.804495
$$531$$ −5.38624 −0.233743
$$532$$ −63.5974 −2.75730
$$533$$ 0 0
$$534$$ 8.86399 0.383583
$$535$$ 34.7668 1.50310
$$536$$ 100.419 4.33743
$$537$$ 10.9555 0.472767
$$538$$ −24.7063 −1.06516
$$539$$ 8.45506 0.364185
$$540$$ −15.6311 −0.672656
$$541$$ 19.2806 0.828937 0.414469 0.910064i $$-0.363967\pi$$
0.414469 + 0.910064i $$0.363967\pi$$
$$542$$ 12.5524 0.539172
$$543$$ −8.52034 −0.365643
$$544$$ 86.3395 3.70177
$$545$$ 24.7999 1.06231
$$546$$ 0 0
$$547$$ −18.3680 −0.785359 −0.392679 0.919675i $$-0.628452\pi$$
−0.392679 + 0.919675i $$0.628452\pi$$
$$548$$ 26.5215 1.13294
$$549$$ −2.34857 −0.100234
$$550$$ 8.41997 0.359029
$$551$$ 6.17360 0.263004
$$552$$ −18.1683 −0.773293
$$553$$ 6.17686 0.262667
$$554$$ 57.4759 2.44192
$$555$$ −23.7099 −1.00643
$$556$$ 94.5481 4.00973
$$557$$ −20.9920 −0.889459 −0.444730 0.895665i $$-0.646700\pi$$
−0.444730 + 0.895665i $$0.646700\pi$$
$$558$$ −16.8718 −0.714239
$$559$$ 0 0
$$560$$ −170.450 −7.20282
$$561$$ 3.81818 0.161204
$$562$$ 28.8558 1.21721
$$563$$ 3.18924 0.134411 0.0672053 0.997739i $$-0.478592\pi$$
0.0672053 + 0.997739i $$0.478592\pi$$
$$564$$ 29.2175 1.23028
$$565$$ −0.0441930 −0.00185921
$$566$$ −7.04329 −0.296052
$$567$$ 3.93129 0.165099
$$568$$ 96.7834 4.06094
$$569$$ 19.0570 0.798909 0.399455 0.916753i $$-0.369200\pi$$
0.399455 + 0.916753i $$0.369200\pi$$
$$570$$ 22.8865 0.958610
$$571$$ 13.1418 0.549967 0.274984 0.961449i $$-0.411328\pi$$
0.274984 + 0.961449i $$0.411328\pi$$
$$572$$ 0 0
$$573$$ −11.8003 −0.492965
$$574$$ 68.4549 2.85725
$$575$$ −5.82540 −0.242936
$$576$$ 31.4147 1.30895
$$577$$ 6.38825 0.265946 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$578$$ −6.63205 −0.275857
$$579$$ 10.9405 0.454674
$$580$$ 32.8140 1.36253
$$581$$ 41.9608 1.74083
$$582$$ −48.7268 −2.01979
$$583$$ −2.37985 −0.0985635
$$584$$ −144.858 −5.99428
$$585$$ 0 0
$$586$$ 14.1365 0.583975
$$587$$ −8.66483 −0.357636 −0.178818 0.983882i $$-0.557227\pi$$
−0.178818 + 0.983882i $$0.557227\pi$$
$$588$$ 46.5107 1.91807
$$589$$ 18.1164 0.746472
$$590$$ 41.9177 1.72572
$$591$$ 9.43307 0.388025
$$592$$ 127.316 5.23267
$$593$$ 5.41552 0.222389 0.111194 0.993799i $$-0.464532\pi$$
0.111194 + 0.993799i $$0.464532\pi$$
$$594$$ 2.73878 0.112374
$$595$$ −42.6525 −1.74858
$$596$$ −29.2514 −1.19818
$$597$$ −15.3198 −0.626997
$$598$$ 0 0
$$599$$ 17.1821 0.702041 0.351021 0.936368i $$-0.385835\pi$$
0.351021 + 0.936368i $$0.385835\pi$$
$$600$$ 29.4777 1.20342
$$601$$ 4.44618 0.181364 0.0906818 0.995880i $$-0.471095\pi$$
0.0906818 + 0.995880i $$0.471095\pi$$
$$602$$ −126.269 −5.14635
$$603$$ 10.4731 0.426496
$$604$$ −37.2728 −1.51661
$$605$$ −2.84154 −0.115525
$$606$$ −33.2020 −1.34874
$$607$$ 12.1499 0.493148 0.246574 0.969124i $$-0.420695\pi$$
0.246574 + 0.969124i $$0.420695\pi$$
$$608$$ −66.5000 −2.69693
$$609$$ −8.25288 −0.334423
$$610$$ 18.2774 0.740031
$$611$$ 0 0
$$612$$ 21.0035 0.849017
$$613$$ −30.6073 −1.23622 −0.618108 0.786093i $$-0.712101\pi$$
−0.618108 + 0.786093i $$0.712101\pi$$
$$614$$ −82.3172 −3.32205
$$615$$ −18.0661 −0.728497
$$616$$ 37.6943 1.51875
$$617$$ 15.3414 0.617622 0.308811 0.951123i $$-0.400069\pi$$
0.308811 + 0.951123i $$0.400069\pi$$
$$618$$ −9.33878 −0.375661
$$619$$ −10.0391 −0.403505 −0.201752 0.979437i $$-0.564664\pi$$
−0.201752 + 0.979437i $$0.564664\pi$$
$$620$$ 96.2926 3.86720
$$621$$ −1.89484 −0.0760374
$$622$$ −59.5581 −2.38806
$$623$$ 12.7235 0.509757
$$624$$ 0 0
$$625$$ −30.9201 −1.23681
$$626$$ −66.9253 −2.67487
$$627$$ −2.94082 −0.117445
$$628$$ 89.6012 3.57548
$$629$$ 31.8590 1.27030
$$630$$ −30.5947 −1.21892
$$631$$ 20.8523 0.830119 0.415059 0.909794i $$-0.363761\pi$$
0.415059 + 0.909794i $$0.363761\pi$$
$$632$$ 15.0652 0.599260
$$633$$ 2.91243 0.115759
$$634$$ 57.7991 2.29550
$$635$$ 16.8081 0.667009
$$636$$ −13.0914 −0.519108
$$637$$ 0 0
$$638$$ −5.74947 −0.227624
$$639$$ 10.0939 0.399310
$$640$$ −115.970 −4.58413
$$641$$ 29.3934 1.16097 0.580484 0.814272i $$-0.302863\pi$$
0.580484 + 0.814272i $$0.302863\pi$$
$$642$$ −33.5096 −1.32252
$$643$$ −7.23812 −0.285444 −0.142722 0.989763i $$-0.545585\pi$$
−0.142722 + 0.989763i $$0.545585\pi$$
$$644$$ −40.9774 −1.61474
$$645$$ 33.3241 1.31214
$$646$$ −30.7526 −1.20994
$$647$$ −25.4069 −0.998848 −0.499424 0.866358i $$-0.666455\pi$$
−0.499424 + 0.866358i $$0.666455\pi$$
$$648$$ 9.58828 0.376663
$$649$$ −5.38624 −0.211429
$$650$$ 0 0
$$651$$ −24.2180 −0.949178
$$652$$ 62.7284 2.45663
$$653$$ −10.1486 −0.397146 −0.198573 0.980086i $$-0.563631\pi$$
−0.198573 + 0.980086i $$0.563631\pi$$
$$654$$ −23.9031 −0.934685
$$655$$ 3.75627 0.146770
$$656$$ 97.0107 3.78763
$$657$$ −15.1079 −0.589413
$$658$$ 57.1873 2.22939
$$659$$ −18.8648 −0.734869 −0.367434 0.930049i $$-0.619764\pi$$
−0.367434 + 0.930049i $$0.619764\pi$$
$$660$$ −15.6311 −0.608440
$$661$$ −7.39059 −0.287461 −0.143730 0.989617i $$-0.545910\pi$$
−0.143730 + 0.989617i $$0.545910\pi$$
$$662$$ 1.53514 0.0596648
$$663$$ 0 0
$$664$$ 102.341 3.97160
$$665$$ 32.8516 1.27393
$$666$$ 22.8525 0.885516
$$667$$ 3.97780 0.154021
$$668$$ −81.0296 −3.13513
$$669$$ 10.8511 0.419526
$$670$$ −81.5052 −3.14882
$$671$$ −2.34857 −0.0906655
$$672$$ 88.8973 3.42929
$$673$$ −13.8241 −0.532879 −0.266439 0.963852i $$-0.585847\pi$$
−0.266439 + 0.963852i $$0.585847\pi$$
$$674$$ −21.2762 −0.819530
$$675$$ 3.07435 0.118332
$$676$$ 0 0
$$677$$ −28.1714 −1.08272 −0.541358 0.840792i $$-0.682090\pi$$
−0.541358 + 0.840792i $$0.682090\pi$$
$$678$$ 0.0425948 0.00163585
$$679$$ −69.9433 −2.68418
$$680$$ −104.028 −3.98929
$$681$$ 0.322636 0.0123634
$$682$$ −16.8718 −0.646053
$$683$$ −20.1458 −0.770856 −0.385428 0.922738i $$-0.625946\pi$$
−0.385428 + 0.922738i $$0.625946\pi$$
$$684$$ −16.1772 −0.618552
$$685$$ −13.6998 −0.523444
$$686$$ 15.6665 0.598150
$$687$$ −5.94060 −0.226648
$$688$$ −178.942 −6.82211
$$689$$ 0 0
$$690$$ 14.7463 0.561384
$$691$$ −4.60090 −0.175026 −0.0875132 0.996163i $$-0.527892\pi$$
−0.0875132 + 0.996163i $$0.527892\pi$$
$$692$$ 1.02673 0.0390306
$$693$$ 3.93129 0.149337
$$694$$ −27.7593 −1.05373
$$695$$ −48.8394 −1.85258
$$696$$ −20.1285 −0.762968
$$697$$ 24.2755 0.919499
$$698$$ −54.7839 −2.07360
$$699$$ −12.6202 −0.477338
$$700$$ 66.4851 2.51290
$$701$$ −2.14477 −0.0810067 −0.0405033 0.999179i $$-0.512896\pi$$
−0.0405033 + 0.999179i $$0.512896\pi$$
$$702$$ 0 0
$$703$$ −24.5383 −0.925479
$$704$$ 31.4147 1.18399
$$705$$ −15.0925 −0.568415
$$706$$ −73.3733 −2.76144
$$707$$ −47.6586 −1.79239
$$708$$ −29.6293 −1.11354
$$709$$ −18.0751 −0.678826 −0.339413 0.940637i $$-0.610228\pi$$
−0.339413 + 0.940637i $$0.610228\pi$$
$$710$$ −78.5546 −2.94810
$$711$$ 1.57120 0.0589248
$$712$$ 31.0322 1.16298
$$713$$ 11.6728 0.437151
$$714$$ 41.1101 1.53851
$$715$$ 0 0
$$716$$ 60.2657 2.25223
$$717$$ 20.7706 0.775691
$$718$$ 73.2206 2.73257
$$719$$ −5.67258 −0.211552 −0.105776 0.994390i $$-0.533733\pi$$
−0.105776 + 0.994390i $$0.533733\pi$$
$$720$$ −43.3572 −1.61583
$$721$$ −13.4050 −0.499230
$$722$$ −28.3507 −1.05511
$$723$$ −18.2163 −0.677470
$$724$$ −46.8698 −1.74190
$$725$$ −6.45391 −0.239692
$$726$$ 2.73878 0.101646
$$727$$ 20.7279 0.768756 0.384378 0.923176i $$-0.374416\pi$$
0.384378 + 0.923176i $$0.374416\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 117.575 4.35163
$$731$$ −44.7776 −1.65616
$$732$$ −12.9193 −0.477511
$$733$$ 13.8875 0.512945 0.256473 0.966552i $$-0.417440\pi$$
0.256473 + 0.966552i $$0.417440\pi$$
$$734$$ 5.19172 0.191630
$$735$$ −24.0254 −0.886189
$$736$$ −42.8476 −1.57938
$$737$$ 10.4731 0.385780
$$738$$ 17.4128 0.640975
$$739$$ −2.10062 −0.0772725 −0.0386362 0.999253i $$-0.512301\pi$$
−0.0386362 + 0.999253i $$0.512301\pi$$
$$740$$ −130.427 −4.79457
$$741$$ 0 0
$$742$$ −25.6238 −0.940678
$$743$$ 39.5255 1.45005 0.725025 0.688722i $$-0.241828\pi$$
0.725025 + 0.688722i $$0.241828\pi$$
$$744$$ −59.0668 −2.16550
$$745$$ 15.1100 0.553587
$$746$$ −100.655 −3.68523
$$747$$ 10.6736 0.390525
$$748$$ 21.0035 0.767964
$$749$$ −48.1002 −1.75754
$$750$$ 14.9861 0.547216
$$751$$ −7.96076 −0.290492 −0.145246 0.989396i $$-0.546397\pi$$
−0.145246 + 0.989396i $$0.546397\pi$$
$$752$$ 81.0427 2.95532
$$753$$ −22.6662 −0.826004
$$754$$ 0 0
$$755$$ 19.2535 0.700707
$$756$$ 21.6258 0.786521
$$757$$ −21.7720 −0.791316 −0.395658 0.918398i $$-0.629484\pi$$
−0.395658 + 0.918398i $$0.629484\pi$$
$$758$$ −71.5721 −2.59962
$$759$$ −1.89484 −0.0687784
$$760$$ 80.1240 2.90640
$$761$$ −25.3978 −0.920668 −0.460334 0.887746i $$-0.652270\pi$$
−0.460334 + 0.887746i $$0.652270\pi$$
$$762$$ −16.2003 −0.586874
$$763$$ −34.3109 −1.24214
$$764$$ −64.9126 −2.34846
$$765$$ −10.8495 −0.392264
$$766$$ 46.3433 1.67445
$$767$$ 0 0
$$768$$ 48.9471 1.76623
$$769$$ −17.5471 −0.632763 −0.316382 0.948632i $$-0.602468\pi$$
−0.316382 + 0.948632i $$0.602468\pi$$
$$770$$ −30.5947 −1.10256
$$771$$ 17.7333 0.638649
$$772$$ 60.1832 2.16604
$$773$$ 16.9181 0.608503 0.304251 0.952592i $$-0.401594\pi$$
0.304251 + 0.952592i $$0.401594\pi$$
$$774$$ −32.1190 −1.15449
$$775$$ −18.9389 −0.680307
$$776$$ −170.589 −6.12379
$$777$$ 32.8028 1.17680
$$778$$ −13.8475 −0.496456
$$779$$ −18.6974 −0.669902
$$780$$ 0 0
$$781$$ 10.0939 0.361189
$$782$$ −19.8147 −0.708570
$$783$$ −2.09928 −0.0750221
$$784$$ 129.010 4.60751
$$785$$ −46.2841 −1.65195
$$786$$ −3.62044 −0.129137
$$787$$ 27.1659 0.968361 0.484181 0.874968i $$-0.339118\pi$$
0.484181 + 0.874968i $$0.339118\pi$$
$$788$$ 51.8907 1.84853
$$789$$ 29.5107 1.05061
$$790$$ −12.2277 −0.435041
$$791$$ 0.0611413 0.00217394
$$792$$ 9.58828 0.340705
$$793$$ 0 0
$$794$$ 43.1992 1.53308
$$795$$ 6.76245 0.239839
$$796$$ −84.2731 −2.98698
$$797$$ 12.3253 0.436583 0.218292 0.975884i $$-0.429952\pi$$
0.218292 + 0.975884i $$0.429952\pi$$
$$798$$ −31.6637 −1.12088
$$799$$ 20.2797 0.717446
$$800$$ 69.5194 2.45788
$$801$$ 3.23647 0.114355
$$802$$ 23.0665 0.814507
$$803$$ −15.1079 −0.533144
$$804$$ 57.6116 2.03180
$$805$$ 21.1671 0.746043
$$806$$ 0 0
$$807$$ −9.02091 −0.317551
$$808$$ −116.238 −4.08923
$$809$$ 51.6306 1.81523 0.907617 0.419798i $$-0.137899\pi$$
0.907617 + 0.419798i $$0.137899\pi$$
$$810$$ −7.78236 −0.273444
$$811$$ 22.5443 0.791637 0.395818 0.918329i $$-0.370461\pi$$
0.395818 + 0.918329i $$0.370461\pi$$
$$812$$ −45.3985 −1.59317
$$813$$ 4.58320 0.160740
$$814$$ 22.8525 0.800980
$$815$$ −32.4027 −1.13502
$$816$$ 58.2591 2.03948
$$817$$ 34.4884 1.20660
$$818$$ 85.9597 3.00551
$$819$$ 0 0
$$820$$ −99.3806 −3.47052
$$821$$ −18.1907 −0.634861 −0.317430 0.948282i $$-0.602820\pi$$
−0.317430 + 0.948282i $$0.602820\pi$$
$$822$$ 13.2044 0.460557
$$823$$ 44.4631 1.54989 0.774944 0.632030i $$-0.217778\pi$$
0.774944 + 0.632030i $$0.217778\pi$$
$$824$$ −32.6944 −1.13896
$$825$$ 3.07435 0.107035
$$826$$ −57.9934 −2.01785
$$827$$ −6.33341 −0.220234 −0.110117 0.993919i $$-0.535123\pi$$
−0.110117 + 0.993919i $$0.535123\pi$$
$$828$$ −10.4234 −0.362238
$$829$$ 7.54455 0.262033 0.131017 0.991380i $$-0.458176\pi$$
0.131017 + 0.991380i $$0.458176\pi$$
$$830$$ −83.0654 −2.88324
$$831$$ 20.9859 0.727994
$$832$$ 0 0
$$833$$ 32.2829 1.11854
$$834$$ 47.0732 1.63001
$$835$$ 41.8563 1.44850
$$836$$ −16.1772 −0.559501
$$837$$ −6.16032 −0.212932
$$838$$ −18.1561 −0.627194
$$839$$ 30.1703 1.04159 0.520797 0.853681i $$-0.325635\pi$$
0.520797 + 0.853681i $$0.325635\pi$$
$$840$$ −107.110 −3.69564
$$841$$ −24.5930 −0.848035
$$842$$ −7.15719 −0.246653
$$843$$ 10.5360 0.362878
$$844$$ 16.0211 0.551468
$$845$$ 0 0
$$846$$ 14.5467 0.500125
$$847$$ 3.93129 0.135081
$$848$$ −36.3127 −1.24698
$$849$$ −2.57169 −0.0882601
$$850$$ 32.1489 1.10270
$$851$$ −15.8106 −0.541981
$$852$$ 55.5260 1.90229
$$853$$ −14.6801 −0.502637 −0.251319 0.967904i $$-0.580864\pi$$
−0.251319 + 0.967904i $$0.580864\pi$$
$$854$$ −25.2869 −0.865301
$$855$$ 8.35645 0.285785
$$856$$ −117.315 −4.00973
$$857$$ −21.1257 −0.721641 −0.360821 0.932635i $$-0.617503\pi$$
−0.360821 + 0.932635i $$0.617503\pi$$
$$858$$ 0 0
$$859$$ 29.4462 1.00469 0.502346 0.864667i $$-0.332470\pi$$
0.502346 + 0.864667i $$0.332470\pi$$
$$860$$ 183.314 6.25094
$$861$$ 24.9947 0.851816
$$862$$ −68.1258 −2.32038
$$863$$ 33.8929 1.15373 0.576865 0.816840i $$-0.304276\pi$$
0.576865 + 0.816840i $$0.304276\pi$$
$$864$$ 22.6127 0.769301
$$865$$ −0.530366 −0.0180330
$$866$$ 17.9543 0.610111
$$867$$ −2.42153 −0.0822396
$$868$$ −133.221 −4.52183
$$869$$ 1.57120 0.0532995
$$870$$ 16.3373 0.553888
$$871$$ 0 0
$$872$$ −83.6830 −2.83386
$$873$$ −17.7914 −0.602148
$$874$$ 15.2616 0.516230
$$875$$ 21.5113 0.727215
$$876$$ −83.1072 −2.80793
$$877$$ −6.62356 −0.223662 −0.111831 0.993727i $$-0.535671\pi$$
−0.111831 + 0.993727i $$0.535671\pi$$
$$878$$ −104.671 −3.53247
$$879$$ 5.16162 0.174097
$$880$$ −43.3572 −1.46157
$$881$$ 18.7993 0.633366 0.316683 0.948531i $$-0.397431\pi$$
0.316683 + 0.948531i $$0.397431\pi$$
$$882$$ 23.1566 0.779722
$$883$$ −35.0323 −1.17893 −0.589465 0.807794i $$-0.700661\pi$$
−0.589465 + 0.807794i $$0.700661\pi$$
$$884$$ 0 0
$$885$$ 15.3052 0.514479
$$886$$ −34.8941 −1.17229
$$887$$ 12.8619 0.431860 0.215930 0.976409i $$-0.430722\pi$$
0.215930 + 0.976409i $$0.430722\pi$$
$$888$$ 80.0050 2.68479
$$889$$ −23.2541 −0.779918
$$890$$ −25.1874 −0.844283
$$891$$ 1.00000 0.0335013
$$892$$ 59.6909 1.99860
$$893$$ −15.6198 −0.522696
$$894$$ −14.5636 −0.487079
$$895$$ −31.1306 −1.04058
$$896$$ 160.446 5.36012
$$897$$ 0 0
$$898$$ 71.0749 2.37180
$$899$$ 12.9322 0.431314
$$900$$ 16.9118 0.563725
$$901$$ −9.08670 −0.302722
$$902$$ 17.4128 0.579784
$$903$$ −46.1042 −1.53425
$$904$$ 0.149121 0.00495971
$$905$$ 24.2109 0.804797
$$906$$ −18.5572 −0.616523
$$907$$ 47.0378 1.56187 0.780933 0.624615i $$-0.214744\pi$$
0.780933 + 0.624615i $$0.214744\pi$$
$$908$$ 1.77480 0.0588987
$$909$$ −12.1229 −0.402091
$$910$$ 0 0
$$911$$ 40.3206 1.33588 0.667940 0.744215i $$-0.267176\pi$$
0.667940 + 0.744215i $$0.267176\pi$$
$$912$$ −44.8720 −1.48586
$$913$$ 10.6736 0.353243
$$914$$ 43.9225 1.45283
$$915$$ 6.67355 0.220621
$$916$$ −32.6788 −1.07974
$$917$$ −5.19683 −0.171614
$$918$$ 10.4572 0.345137
$$919$$ −1.80179 −0.0594356 −0.0297178 0.999558i $$-0.509461\pi$$
−0.0297178 + 0.999558i $$0.509461\pi$$
$$920$$ 51.6259 1.70205
$$921$$ −30.0561 −0.990383
$$922$$ 9.54646 0.314396
$$923$$ 0 0
$$924$$ 21.6258 0.711435
$$925$$ 25.6525 0.843448
$$926$$ −74.1619 −2.43711
$$927$$ −3.40983 −0.111994
$$928$$ −47.4705 −1.55829
$$929$$ 26.3015 0.862924 0.431462 0.902131i $$-0.357998\pi$$
0.431462 + 0.902131i $$0.357998\pi$$
$$930$$ 47.9418 1.57207
$$931$$ −24.8648 −0.814910
$$932$$ −69.4226 −2.27401
$$933$$ −21.7462 −0.711939
$$934$$ −42.9711 −1.40606
$$935$$ −10.8495 −0.354817
$$936$$ 0 0
$$937$$ 8.45280 0.276141 0.138070 0.990422i $$-0.455910\pi$$
0.138070 + 0.990422i $$0.455910\pi$$
$$938$$ 112.763 3.68184
$$939$$ −24.4361 −0.797443
$$940$$ −83.0226 −2.70790
$$941$$ −8.34188 −0.271937 −0.135969 0.990713i $$-0.543415\pi$$
−0.135969 + 0.990713i $$0.543415\pi$$
$$942$$ 44.6103 1.45348
$$943$$ −12.0472 −0.392310
$$944$$ −82.1852 −2.67490
$$945$$ −11.1709 −0.363390
$$946$$ −32.1190 −1.04428
$$947$$ −21.8593 −0.710330 −0.355165 0.934804i $$-0.615575\pi$$
−0.355165 + 0.934804i $$0.615575\pi$$
$$948$$ 8.64309 0.280714
$$949$$ 0 0
$$950$$ −24.7616 −0.803372
$$951$$ 21.1040 0.684343
$$952$$ 143.924 4.66459
$$953$$ 57.6509 1.86750 0.933748 0.357932i $$-0.116518\pi$$
0.933748 + 0.357932i $$0.116518\pi$$
$$954$$ −6.51790 −0.211025
$$955$$ 33.5310 1.08504
$$956$$ 114.257 3.69535
$$957$$ −2.09928 −0.0678600
$$958$$ 90.3964 2.92058
$$959$$ 18.9538 0.612051
$$960$$ −89.2661 −2.88105
$$961$$ 6.94950 0.224177
$$962$$ 0 0
$$963$$ −12.2352 −0.394274
$$964$$ −100.206 −3.22743
$$965$$ −31.0880 −1.00076
$$966$$ −20.4017 −0.656413
$$967$$ 41.2637 1.32695 0.663476 0.748197i $$-0.269080\pi$$
0.663476 + 0.748197i $$0.269080\pi$$
$$968$$ 9.58828 0.308179
$$969$$ −11.2286 −0.360713
$$970$$ 138.459 4.44566
$$971$$ 10.2876 0.330146 0.165073 0.986281i $$-0.447214\pi$$
0.165073 + 0.986281i $$0.447214\pi$$
$$972$$ 5.50093 0.176442
$$973$$ 67.5697 2.16618
$$974$$ 51.1286 1.63827
$$975$$ 0 0
$$976$$ −35.8353 −1.14706
$$977$$ −54.0502 −1.72922 −0.864609 0.502445i $$-0.832434\pi$$
−0.864609 + 0.502445i $$0.832434\pi$$
$$978$$ 31.2310 0.998657
$$979$$ 3.23647 0.103438
$$980$$ −132.162 −4.22176
$$981$$ −8.72763 −0.278652
$$982$$ −18.5755 −0.592766
$$983$$ −33.3316 −1.06311 −0.531557 0.847022i $$-0.678393\pi$$
−0.531557 + 0.847022i $$0.678393\pi$$
$$984$$ 60.9611 1.94337
$$985$$ −26.8045 −0.854061
$$986$$ −21.9525 −0.699109
$$987$$ 20.8805 0.664635
$$988$$ 0 0
$$989$$ 22.2217 0.706610
$$990$$ −7.78236 −0.247340
$$991$$ 15.1783 0.482155 0.241077 0.970506i $$-0.422499\pi$$
0.241077 + 0.970506i $$0.422499\pi$$
$$992$$ −139.302 −4.42283
$$993$$ 0.560518 0.0177875
$$994$$ 108.681 3.44715
$$995$$ 43.5318 1.38005
$$996$$ 58.7144 1.86044
$$997$$ −23.6131 −0.747834 −0.373917 0.927462i $$-0.621986\pi$$
−0.373917 + 0.927462i $$0.621986\pi$$
$$998$$ −0.583086 −0.0184573
$$999$$ 8.34404 0.263994
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 5577.2.a.y.1.7 7
13.5 odd 4 429.2.b.b.298.1 14
13.8 odd 4 429.2.b.b.298.14 yes 14
13.12 even 2 5577.2.a.x.1.1 7
39.5 even 4 1287.2.b.c.298.14 14
39.8 even 4 1287.2.b.c.298.1 14

By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.b.b.298.1 14 13.5 odd 4
429.2.b.b.298.14 yes 14 13.8 odd 4
1287.2.b.c.298.1 14 39.8 even 4
1287.2.b.c.298.14 14 39.5 even 4
5577.2.a.x.1.1 7 13.12 even 2
5577.2.a.y.1.7 7 1.1 even 1 trivial