Properties

 Label 625.2.a.c.1.2 Level $625$ Weight $2$ Character 625.1 Self dual yes Analytic conductor $4.991$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.99065012633$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 625.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -0.618034 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})$$ $$q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -0.618034 q^{7} -2.23607 q^{8} -2.00000 q^{9} -5.23607 q^{11} -0.618034 q^{12} -1.85410 q^{13} -1.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} -3.23607 q^{18} +0.854102 q^{19} +0.618034 q^{21} -8.47214 q^{22} -3.76393 q^{23} +2.23607 q^{24} -3.00000 q^{26} +5.00000 q^{27} -0.381966 q^{28} -3.61803 q^{29} -3.00000 q^{31} -3.38197 q^{32} +5.23607 q^{33} +8.47214 q^{34} -1.23607 q^{36} +0.236068 q^{37} +1.38197 q^{38} +1.85410 q^{39} -0.763932 q^{41} +1.00000 q^{42} +4.85410 q^{43} -3.23607 q^{44} -6.09017 q^{46} -0.618034 q^{47} +4.85410 q^{48} -6.61803 q^{49} -5.23607 q^{51} -1.14590 q^{52} +3.47214 q^{53} +8.09017 q^{54} +1.38197 q^{56} -0.854102 q^{57} -5.85410 q^{58} -10.8541 q^{59} +8.70820 q^{61} -4.85410 q^{62} +1.23607 q^{63} +4.23607 q^{64} +8.47214 q^{66} -4.76393 q^{67} +3.23607 q^{68} +3.76393 q^{69} -6.61803 q^{71} +4.47214 q^{72} +9.00000 q^{73} +0.381966 q^{74} +0.527864 q^{76} +3.23607 q^{77} +3.00000 q^{78} -8.09017 q^{79} +1.00000 q^{81} -1.23607 q^{82} +6.23607 q^{83} +0.381966 q^{84} +7.85410 q^{86} +3.61803 q^{87} +11.7082 q^{88} -8.94427 q^{89} +1.14590 q^{91} -2.32624 q^{92} +3.00000 q^{93} -1.00000 q^{94} +3.38197 q^{96} +3.85410 q^{97} -10.7082 q^{98} +10.4721 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} + O(q^{10})$$ $$2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} - 6 q^{11} + q^{12} + 3 q^{13} - 2 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 5 q^{19} - q^{21} - 8 q^{22} - 12 q^{23} - 6 q^{26} + 10 q^{27} - 3 q^{28} - 5 q^{29} - 6 q^{31} - 9 q^{32} + 6 q^{33} + 8 q^{34} + 2 q^{36} - 4 q^{37} + 5 q^{38} - 3 q^{39} - 6 q^{41} + 2 q^{42} + 3 q^{43} - 2 q^{44} - q^{46} + q^{47} + 3 q^{48} - 11 q^{49} - 6 q^{51} - 9 q^{52} - 2 q^{53} + 5 q^{54} + 5 q^{56} + 5 q^{57} - 5 q^{58} - 15 q^{59} + 4 q^{61} - 3 q^{62} - 2 q^{63} + 4 q^{64} + 8 q^{66} - 14 q^{67} + 2 q^{68} + 12 q^{69} - 11 q^{71} + 18 q^{73} + 3 q^{74} + 10 q^{76} + 2 q^{77} + 6 q^{78} - 5 q^{79} + 2 q^{81} + 2 q^{82} + 8 q^{83} + 3 q^{84} + 9 q^{86} + 5 q^{87} + 10 q^{88} + 9 q^{91} + 11 q^{92} + 6 q^{93} - 2 q^{94} + 9 q^{96} + q^{97} - 8 q^{98} + 12 q^{99} + O(q^{100})$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.61803 1.14412 0.572061 0.820211i $$-0.306144\pi$$
0.572061 + 0.820211i $$0.306144\pi$$
$$3$$ −1.00000 −0.577350 −0.288675 0.957427i $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0.618034 0.309017
$$5$$ 0 0
$$6$$ −1.61803 −0.660560
$$7$$ −0.618034 −0.233595 −0.116797 0.993156i $$-0.537263\pi$$
−0.116797 + 0.993156i $$0.537263\pi$$
$$8$$ −2.23607 −0.790569
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −5.23607 −1.57873 −0.789367 0.613922i $$-0.789591\pi$$
−0.789367 + 0.613922i $$0.789591\pi$$
$$12$$ −0.618034 −0.178411
$$13$$ −1.85410 −0.514235 −0.257118 0.966380i $$-0.582773\pi$$
−0.257118 + 0.966380i $$0.582773\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ 5.23607 1.26993 0.634967 0.772540i $$-0.281014\pi$$
0.634967 + 0.772540i $$0.281014\pi$$
$$18$$ −3.23607 −0.762749
$$19$$ 0.854102 0.195944 0.0979722 0.995189i $$-0.468764\pi$$
0.0979722 + 0.995189i $$0.468764\pi$$
$$20$$ 0 0
$$21$$ 0.618034 0.134866
$$22$$ −8.47214 −1.80627
$$23$$ −3.76393 −0.784834 −0.392417 0.919787i $$-0.628361\pi$$
−0.392417 + 0.919787i $$0.628361\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −3.00000 −0.588348
$$27$$ 5.00000 0.962250
$$28$$ −0.381966 −0.0721848
$$29$$ −3.61803 −0.671852 −0.335926 0.941888i $$-0.609049\pi$$
−0.335926 + 0.941888i $$0.609049\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ −3.38197 −0.597853
$$33$$ 5.23607 0.911482
$$34$$ 8.47214 1.45296
$$35$$ 0 0
$$36$$ −1.23607 −0.206011
$$37$$ 0.236068 0.0388093 0.0194047 0.999812i $$-0.493823\pi$$
0.0194047 + 0.999812i $$0.493823\pi$$
$$38$$ 1.38197 0.224184
$$39$$ 1.85410 0.296894
$$40$$ 0 0
$$41$$ −0.763932 −0.119306 −0.0596531 0.998219i $$-0.518999\pi$$
−0.0596531 + 0.998219i $$0.518999\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 4.85410 0.740244 0.370122 0.928983i $$-0.379316\pi$$
0.370122 + 0.928983i $$0.379316\pi$$
$$44$$ −3.23607 −0.487856
$$45$$ 0 0
$$46$$ −6.09017 −0.897947
$$47$$ −0.618034 −0.0901495 −0.0450748 0.998984i $$-0.514353\pi$$
−0.0450748 + 0.998984i $$0.514353\pi$$
$$48$$ 4.85410 0.700629
$$49$$ −6.61803 −0.945433
$$50$$ 0 0
$$51$$ −5.23607 −0.733196
$$52$$ −1.14590 −0.158907
$$53$$ 3.47214 0.476935 0.238467 0.971151i $$-0.423355\pi$$
0.238467 + 0.971151i $$0.423355\pi$$
$$54$$ 8.09017 1.10093
$$55$$ 0 0
$$56$$ 1.38197 0.184673
$$57$$ −0.854102 −0.113129
$$58$$ −5.85410 −0.768681
$$59$$ −10.8541 −1.41308 −0.706542 0.707671i $$-0.749746\pi$$
−0.706542 + 0.707671i $$0.749746\pi$$
$$60$$ 0 0
$$61$$ 8.70820 1.11497 0.557486 0.830187i $$-0.311766\pi$$
0.557486 + 0.830187i $$0.311766\pi$$
$$62$$ −4.85410 −0.616472
$$63$$ 1.23607 0.155730
$$64$$ 4.23607 0.529508
$$65$$ 0 0
$$66$$ 8.47214 1.04285
$$67$$ −4.76393 −0.582007 −0.291003 0.956722i $$-0.593989\pi$$
−0.291003 + 0.956722i $$0.593989\pi$$
$$68$$ 3.23607 0.392431
$$69$$ 3.76393 0.453124
$$70$$ 0 0
$$71$$ −6.61803 −0.785416 −0.392708 0.919663i $$-0.628462\pi$$
−0.392708 + 0.919663i $$0.628462\pi$$
$$72$$ 4.47214 0.527046
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 0.381966 0.0444026
$$75$$ 0 0
$$76$$ 0.527864 0.0605502
$$77$$ 3.23607 0.368784
$$78$$ 3.00000 0.339683
$$79$$ −8.09017 −0.910215 −0.455108 0.890436i $$-0.650399\pi$$
−0.455108 + 0.890436i $$0.650399\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −1.23607 −0.136501
$$83$$ 6.23607 0.684497 0.342249 0.939609i $$-0.388811\pi$$
0.342249 + 0.939609i $$0.388811\pi$$
$$84$$ 0.381966 0.0416759
$$85$$ 0 0
$$86$$ 7.85410 0.846930
$$87$$ 3.61803 0.387894
$$88$$ 11.7082 1.24810
$$89$$ −8.94427 −0.948091 −0.474045 0.880500i $$-0.657207\pi$$
−0.474045 + 0.880500i $$0.657207\pi$$
$$90$$ 0 0
$$91$$ 1.14590 0.120123
$$92$$ −2.32624 −0.242527
$$93$$ 3.00000 0.311086
$$94$$ −1.00000 −0.103142
$$95$$ 0 0
$$96$$ 3.38197 0.345170
$$97$$ 3.85410 0.391325 0.195662 0.980671i $$-0.437314\pi$$
0.195662 + 0.980671i $$0.437314\pi$$
$$98$$ −10.7082 −1.08169
$$99$$ 10.4721 1.05249
$$100$$ 0 0
$$101$$ 1.47214 0.146483 0.0732415 0.997314i $$-0.476666\pi$$
0.0732415 + 0.997314i $$0.476666\pi$$
$$102$$ −8.47214 −0.838866
$$103$$ −8.56231 −0.843669 −0.421835 0.906673i $$-0.638614\pi$$
−0.421835 + 0.906673i $$0.638614\pi$$
$$104$$ 4.14590 0.406539
$$105$$ 0 0
$$106$$ 5.61803 0.545672
$$107$$ 16.4164 1.58703 0.793517 0.608548i $$-0.208248\pi$$
0.793517 + 0.608548i $$0.208248\pi$$
$$108$$ 3.09017 0.297352
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −0.236068 −0.0224066
$$112$$ 3.00000 0.283473
$$113$$ −16.8541 −1.58550 −0.792750 0.609547i $$-0.791352\pi$$
−0.792750 + 0.609547i $$0.791352\pi$$
$$114$$ −1.38197 −0.129433
$$115$$ 0 0
$$116$$ −2.23607 −0.207614
$$117$$ 3.70820 0.342824
$$118$$ −17.5623 −1.61674
$$119$$ −3.23607 −0.296650
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ 14.0902 1.27566
$$123$$ 0.763932 0.0688814
$$124$$ −1.85410 −0.166503
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ −19.8885 −1.76482 −0.882411 0.470479i $$-0.844081\pi$$
−0.882411 + 0.470479i $$0.844081\pi$$
$$128$$ 13.6180 1.20368
$$129$$ −4.85410 −0.427380
$$130$$ 0 0
$$131$$ 6.79837 0.593977 0.296988 0.954881i $$-0.404018\pi$$
0.296988 + 0.954881i $$0.404018\pi$$
$$132$$ 3.23607 0.281664
$$133$$ −0.527864 −0.0457716
$$134$$ −7.70820 −0.665887
$$135$$ 0 0
$$136$$ −11.7082 −1.00397
$$137$$ 11.9443 1.02047 0.510234 0.860036i $$-0.329559\pi$$
0.510234 + 0.860036i $$0.329559\pi$$
$$138$$ 6.09017 0.518430
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ 0.618034 0.0520479
$$142$$ −10.7082 −0.898613
$$143$$ 9.70820 0.811841
$$144$$ 9.70820 0.809017
$$145$$ 0 0
$$146$$ 14.5623 1.20519
$$147$$ 6.61803 0.545846
$$148$$ 0.145898 0.0119927
$$149$$ −3.94427 −0.323127 −0.161564 0.986862i $$-0.551654\pi$$
−0.161564 + 0.986862i $$0.551654\pi$$
$$150$$ 0 0
$$151$$ 14.5623 1.18506 0.592532 0.805547i $$-0.298128\pi$$
0.592532 + 0.805547i $$0.298128\pi$$
$$152$$ −1.90983 −0.154908
$$153$$ −10.4721 −0.846622
$$154$$ 5.23607 0.421934
$$155$$ 0 0
$$156$$ 1.14590 0.0917453
$$157$$ −13.1803 −1.05191 −0.525953 0.850514i $$-0.676291\pi$$
−0.525953 + 0.850514i $$0.676291\pi$$
$$158$$ −13.0902 −1.04140
$$159$$ −3.47214 −0.275358
$$160$$ 0 0
$$161$$ 2.32624 0.183333
$$162$$ 1.61803 0.127125
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ −0.472136 −0.0368676
$$165$$ 0 0
$$166$$ 10.0902 0.783149
$$167$$ −14.5623 −1.12687 −0.563433 0.826162i $$-0.690520\pi$$
−0.563433 + 0.826162i $$0.690520\pi$$
$$168$$ −1.38197 −0.106621
$$169$$ −9.56231 −0.735562
$$170$$ 0 0
$$171$$ −1.70820 −0.130630
$$172$$ 3.00000 0.228748
$$173$$ −18.8885 −1.43607 −0.718035 0.696007i $$-0.754958\pi$$
−0.718035 + 0.696007i $$0.754958\pi$$
$$174$$ 5.85410 0.443798
$$175$$ 0 0
$$176$$ 25.4164 1.91583
$$177$$ 10.8541 0.815844
$$178$$ −14.4721 −1.08473
$$179$$ −0.527864 −0.0394544 −0.0197272 0.999805i $$-0.506280\pi$$
−0.0197272 + 0.999805i $$0.506280\pi$$
$$180$$ 0 0
$$181$$ 0.291796 0.0216890 0.0108445 0.999941i $$-0.496548\pi$$
0.0108445 + 0.999941i $$0.496548\pi$$
$$182$$ 1.85410 0.137435
$$183$$ −8.70820 −0.643729
$$184$$ 8.41641 0.620466
$$185$$ 0 0
$$186$$ 4.85410 0.355920
$$187$$ −27.4164 −2.00489
$$188$$ −0.381966 −0.0278577
$$189$$ −3.09017 −0.224777
$$190$$ 0 0
$$191$$ −1.81966 −0.131666 −0.0658330 0.997831i $$-0.520970\pi$$
−0.0658330 + 0.997831i $$0.520970\pi$$
$$192$$ −4.23607 −0.305712
$$193$$ −7.70820 −0.554849 −0.277424 0.960747i $$-0.589481\pi$$
−0.277424 + 0.960747i $$0.589481\pi$$
$$194$$ 6.23607 0.447724
$$195$$ 0 0
$$196$$ −4.09017 −0.292155
$$197$$ −3.70820 −0.264199 −0.132099 0.991236i $$-0.542172\pi$$
−0.132099 + 0.991236i $$0.542172\pi$$
$$198$$ 16.9443 1.20418
$$199$$ −17.5623 −1.24496 −0.622479 0.782636i $$-0.713875\pi$$
−0.622479 + 0.782636i $$0.713875\pi$$
$$200$$ 0 0
$$201$$ 4.76393 0.336022
$$202$$ 2.38197 0.167595
$$203$$ 2.23607 0.156941
$$204$$ −3.23607 −0.226570
$$205$$ 0 0
$$206$$ −13.8541 −0.965261
$$207$$ 7.52786 0.523223
$$208$$ 9.00000 0.624038
$$209$$ −4.47214 −0.309344
$$210$$ 0 0
$$211$$ −9.18034 −0.632001 −0.316000 0.948759i $$-0.602340\pi$$
−0.316000 + 0.948759i $$0.602340\pi$$
$$212$$ 2.14590 0.147381
$$213$$ 6.61803 0.453460
$$214$$ 26.5623 1.81576
$$215$$ 0 0
$$216$$ −11.1803 −0.760726
$$217$$ 1.85410 0.125865
$$218$$ 16.1803 1.09587
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ −9.70820 −0.653044
$$222$$ −0.381966 −0.0256359
$$223$$ 0.180340 0.0120765 0.00603823 0.999982i $$-0.498078\pi$$
0.00603823 + 0.999982i $$0.498078\pi$$
$$224$$ 2.09017 0.139655
$$225$$ 0 0
$$226$$ −27.2705 −1.81401
$$227$$ −14.7639 −0.979917 −0.489958 0.871746i $$-0.662988\pi$$
−0.489958 + 0.871746i $$0.662988\pi$$
$$228$$ −0.527864 −0.0349587
$$229$$ 21.7082 1.43452 0.717259 0.696806i $$-0.245396\pi$$
0.717259 + 0.696806i $$0.245396\pi$$
$$230$$ 0 0
$$231$$ −3.23607 −0.212918
$$232$$ 8.09017 0.531146
$$233$$ 2.94427 0.192886 0.0964428 0.995339i $$-0.469254\pi$$
0.0964428 + 0.995339i $$0.469254\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ −6.70820 −0.436667
$$237$$ 8.09017 0.525513
$$238$$ −5.23607 −0.339404
$$239$$ −20.5279 −1.32784 −0.663919 0.747805i $$-0.731108\pi$$
−0.663919 + 0.747805i $$0.731108\pi$$
$$240$$ 0 0
$$241$$ 2.52786 0.162834 0.0814170 0.996680i $$-0.474055\pi$$
0.0814170 + 0.996680i $$0.474055\pi$$
$$242$$ 26.5623 1.70749
$$243$$ −16.0000 −1.02640
$$244$$ 5.38197 0.344545
$$245$$ 0 0
$$246$$ 1.23607 0.0788088
$$247$$ −1.58359 −0.100762
$$248$$ 6.70820 0.425971
$$249$$ −6.23607 −0.395195
$$250$$ 0 0
$$251$$ −29.1803 −1.84185 −0.920923 0.389744i $$-0.872564\pi$$
−0.920923 + 0.389744i $$0.872564\pi$$
$$252$$ 0.763932 0.0481232
$$253$$ 19.7082 1.23904
$$254$$ −32.1803 −2.01917
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ −22.8541 −1.42560 −0.712800 0.701367i $$-0.752573\pi$$
−0.712800 + 0.701367i $$0.752573\pi$$
$$258$$ −7.85410 −0.488975
$$259$$ −0.145898 −0.00906566
$$260$$ 0 0
$$261$$ 7.23607 0.447901
$$262$$ 11.0000 0.679582
$$263$$ 10.9098 0.672729 0.336364 0.941732i $$-0.390803\pi$$
0.336364 + 0.941732i $$0.390803\pi$$
$$264$$ −11.7082 −0.720590
$$265$$ 0 0
$$266$$ −0.854102 −0.0523684
$$267$$ 8.94427 0.547381
$$268$$ −2.94427 −0.179850
$$269$$ −12.7639 −0.778231 −0.389115 0.921189i $$-0.627219\pi$$
−0.389115 + 0.921189i $$0.627219\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −25.4164 −1.54110
$$273$$ −1.14590 −0.0693529
$$274$$ 19.3262 1.16754
$$275$$ 0 0
$$276$$ 2.32624 0.140023
$$277$$ 24.7082 1.48457 0.742286 0.670083i $$-0.233742\pi$$
0.742286 + 0.670083i $$0.233742\pi$$
$$278$$ 8.09017 0.485216
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 10.0902 0.601929 0.300965 0.953635i $$-0.402691\pi$$
0.300965 + 0.953635i $$0.402691\pi$$
$$282$$ 1.00000 0.0595491
$$283$$ 29.8541 1.77464 0.887321 0.461152i $$-0.152564\pi$$
0.887321 + 0.461152i $$0.152564\pi$$
$$284$$ −4.09017 −0.242707
$$285$$ 0 0
$$286$$ 15.7082 0.928846
$$287$$ 0.472136 0.0278693
$$288$$ 6.76393 0.398569
$$289$$ 10.4164 0.612730
$$290$$ 0 0
$$291$$ −3.85410 −0.225931
$$292$$ 5.56231 0.325509
$$293$$ 19.5279 1.14083 0.570415 0.821357i $$-0.306782\pi$$
0.570415 + 0.821357i $$0.306782\pi$$
$$294$$ 10.7082 0.624515
$$295$$ 0 0
$$296$$ −0.527864 −0.0306815
$$297$$ −26.1803 −1.51914
$$298$$ −6.38197 −0.369697
$$299$$ 6.97871 0.403589
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 23.5623 1.35586
$$303$$ −1.47214 −0.0845720
$$304$$ −4.14590 −0.237784
$$305$$ 0 0
$$306$$ −16.9443 −0.968640
$$307$$ −9.23607 −0.527130 −0.263565 0.964642i $$-0.584898\pi$$
−0.263565 + 0.964642i $$0.584898\pi$$
$$308$$ 2.00000 0.113961
$$309$$ 8.56231 0.487093
$$310$$ 0 0
$$311$$ 8.50658 0.482364 0.241182 0.970480i $$-0.422465\pi$$
0.241182 + 0.970480i $$0.422465\pi$$
$$312$$ −4.14590 −0.234715
$$313$$ 16.7639 0.947553 0.473777 0.880645i $$-0.342890\pi$$
0.473777 + 0.880645i $$0.342890\pi$$
$$314$$ −21.3262 −1.20351
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ −7.65248 −0.429806 −0.214903 0.976635i $$-0.568944\pi$$
−0.214903 + 0.976635i $$0.568944\pi$$
$$318$$ −5.61803 −0.315044
$$319$$ 18.9443 1.06068
$$320$$ 0 0
$$321$$ −16.4164 −0.916275
$$322$$ 3.76393 0.209756
$$323$$ 4.47214 0.248836
$$324$$ 0.618034 0.0343352
$$325$$ 0 0
$$326$$ −17.7984 −0.985761
$$327$$ −10.0000 −0.553001
$$328$$ 1.70820 0.0943198
$$329$$ 0.381966 0.0210585
$$330$$ 0 0
$$331$$ −23.1246 −1.27104 −0.635522 0.772083i $$-0.719215\pi$$
−0.635522 + 0.772083i $$0.719215\pi$$
$$332$$ 3.85410 0.211521
$$333$$ −0.472136 −0.0258729
$$334$$ −23.5623 −1.28927
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ −7.85410 −0.427840 −0.213920 0.976851i $$-0.568623\pi$$
−0.213920 + 0.976851i $$0.568623\pi$$
$$338$$ −15.4721 −0.841573
$$339$$ 16.8541 0.915389
$$340$$ 0 0
$$341$$ 15.7082 0.850647
$$342$$ −2.76393 −0.149456
$$343$$ 8.41641 0.454443
$$344$$ −10.8541 −0.585214
$$345$$ 0 0
$$346$$ −30.5623 −1.64304
$$347$$ 19.9098 1.06882 0.534408 0.845227i $$-0.320535\pi$$
0.534408 + 0.845227i $$0.320535\pi$$
$$348$$ 2.23607 0.119866
$$349$$ 21.7082 1.16201 0.581007 0.813899i $$-0.302659\pi$$
0.581007 + 0.813899i $$0.302659\pi$$
$$350$$ 0 0
$$351$$ −9.27051 −0.494823
$$352$$ 17.7082 0.943850
$$353$$ −12.9098 −0.687121 −0.343560 0.939131i $$-0.611633\pi$$
−0.343560 + 0.939131i $$0.611633\pi$$
$$354$$ 17.5623 0.933426
$$355$$ 0 0
$$356$$ −5.52786 −0.292976
$$357$$ 3.23607 0.171271
$$358$$ −0.854102 −0.0451407
$$359$$ 13.7426 0.725309 0.362655 0.931924i $$-0.381870\pi$$
0.362655 + 0.931924i $$0.381870\pi$$
$$360$$ 0 0
$$361$$ −18.2705 −0.961606
$$362$$ 0.472136 0.0248149
$$363$$ −16.4164 −0.861638
$$364$$ 0.708204 0.0371200
$$365$$ 0 0
$$366$$ −14.0902 −0.736505
$$367$$ 25.5623 1.33434 0.667171 0.744905i $$-0.267505\pi$$
0.667171 + 0.744905i $$0.267505\pi$$
$$368$$ 18.2705 0.952416
$$369$$ 1.52786 0.0795374
$$370$$ 0 0
$$371$$ −2.14590 −0.111409
$$372$$ 1.85410 0.0961307
$$373$$ 28.2705 1.46379 0.731896 0.681417i $$-0.238636\pi$$
0.731896 + 0.681417i $$0.238636\pi$$
$$374$$ −44.3607 −2.29384
$$375$$ 0 0
$$376$$ 1.38197 0.0712695
$$377$$ 6.70820 0.345490
$$378$$ −5.00000 −0.257172
$$379$$ 14.5967 0.749785 0.374892 0.927068i $$-0.377680\pi$$
0.374892 + 0.927068i $$0.377680\pi$$
$$380$$ 0 0
$$381$$ 19.8885 1.01892
$$382$$ −2.94427 −0.150642
$$383$$ −33.3607 −1.70465 −0.852326 0.523012i $$-0.824808\pi$$
−0.852326 + 0.523012i $$0.824808\pi$$
$$384$$ −13.6180 −0.694942
$$385$$ 0 0
$$386$$ −12.4721 −0.634815
$$387$$ −9.70820 −0.493496
$$388$$ 2.38197 0.120926
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ −19.7082 −0.996687
$$392$$ 14.7984 0.747431
$$393$$ −6.79837 −0.342933
$$394$$ −6.00000 −0.302276
$$395$$ 0 0
$$396$$ 6.47214 0.325237
$$397$$ −29.0344 −1.45720 −0.728598 0.684941i $$-0.759828\pi$$
−0.728598 + 0.684941i $$0.759828\pi$$
$$398$$ −28.4164 −1.42439
$$399$$ 0.527864 0.0264263
$$400$$ 0 0
$$401$$ 26.5967 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$402$$ 7.70820 0.384450
$$403$$ 5.56231 0.277078
$$404$$ 0.909830 0.0452657
$$405$$ 0 0
$$406$$ 3.61803 0.179560
$$407$$ −1.23607 −0.0612696
$$408$$ 11.7082 0.579642
$$409$$ 1.58359 0.0783036 0.0391518 0.999233i $$-0.487534\pi$$
0.0391518 + 0.999233i $$0.487534\pi$$
$$410$$ 0 0
$$411$$ −11.9443 −0.589167
$$412$$ −5.29180 −0.260708
$$413$$ 6.70820 0.330089
$$414$$ 12.1803 0.598631
$$415$$ 0 0
$$416$$ 6.27051 0.307437
$$417$$ −5.00000 −0.244851
$$418$$ −7.23607 −0.353928
$$419$$ −9.47214 −0.462744 −0.231372 0.972865i $$-0.574321\pi$$
−0.231372 + 0.972865i $$0.574321\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ −14.8541 −0.723086
$$423$$ 1.23607 0.0600997
$$424$$ −7.76393 −0.377050
$$425$$ 0 0
$$426$$ 10.7082 0.518814
$$427$$ −5.38197 −0.260452
$$428$$ 10.1459 0.490420
$$429$$ −9.70820 −0.468717
$$430$$ 0 0
$$431$$ −29.8328 −1.43700 −0.718498 0.695529i $$-0.755170\pi$$
−0.718498 + 0.695529i $$0.755170\pi$$
$$432$$ −24.2705 −1.16772
$$433$$ −26.8541 −1.29053 −0.645263 0.763961i $$-0.723252\pi$$
−0.645263 + 0.763961i $$0.723252\pi$$
$$434$$ 3.00000 0.144005
$$435$$ 0 0
$$436$$ 6.18034 0.295985
$$437$$ −3.21478 −0.153784
$$438$$ −14.5623 −0.695814
$$439$$ −40.9787 −1.95581 −0.977904 0.209056i $$-0.932961\pi$$
−0.977904 + 0.209056i $$0.932961\pi$$
$$440$$ 0 0
$$441$$ 13.2361 0.630289
$$442$$ −15.7082 −0.747163
$$443$$ −29.9443 −1.42270 −0.711348 0.702840i $$-0.751915\pi$$
−0.711348 + 0.702840i $$0.751915\pi$$
$$444$$ −0.145898 −0.00692401
$$445$$ 0 0
$$446$$ 0.291796 0.0138169
$$447$$ 3.94427 0.186558
$$448$$ −2.61803 −0.123690
$$449$$ 4.67376 0.220568 0.110284 0.993900i $$-0.464824\pi$$
0.110284 + 0.993900i $$0.464824\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ −10.4164 −0.489947
$$453$$ −14.5623 −0.684197
$$454$$ −23.8885 −1.12114
$$455$$ 0 0
$$456$$ 1.90983 0.0894360
$$457$$ 21.4164 1.00182 0.500909 0.865500i $$-0.332999\pi$$
0.500909 + 0.865500i $$0.332999\pi$$
$$458$$ 35.1246 1.64127
$$459$$ 26.1803 1.22199
$$460$$ 0 0
$$461$$ 0.819660 0.0381754 0.0190877 0.999818i $$-0.493924\pi$$
0.0190877 + 0.999818i $$0.493924\pi$$
$$462$$ −5.23607 −0.243604
$$463$$ 24.1246 1.12117 0.560583 0.828098i $$-0.310577\pi$$
0.560583 + 0.828098i $$0.310577\pi$$
$$464$$ 17.5623 0.815310
$$465$$ 0 0
$$466$$ 4.76393 0.220685
$$467$$ −27.4508 −1.27027 −0.635137 0.772400i $$-0.719056\pi$$
−0.635137 + 0.772400i $$0.719056\pi$$
$$468$$ 2.29180 0.105938
$$469$$ 2.94427 0.135954
$$470$$ 0 0
$$471$$ 13.1803 0.607318
$$472$$ 24.2705 1.11714
$$473$$ −25.4164 −1.16865
$$474$$ 13.0902 0.601251
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ −6.94427 −0.317956
$$478$$ −33.2148 −1.51921
$$479$$ −10.8541 −0.495937 −0.247968 0.968768i $$-0.579763\pi$$
−0.247968 + 0.968768i $$0.579763\pi$$
$$480$$ 0 0
$$481$$ −0.437694 −0.0199571
$$482$$ 4.09017 0.186302
$$483$$ −2.32624 −0.105847
$$484$$ 10.1459 0.461177
$$485$$ 0 0
$$486$$ −25.8885 −1.17433
$$487$$ 36.4164 1.65018 0.825092 0.564998i $$-0.191123\pi$$
0.825092 + 0.564998i $$0.191123\pi$$
$$488$$ −19.4721 −0.881462
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ −43.2492 −1.95181 −0.975905 0.218196i $$-0.929983\pi$$
−0.975905 + 0.218196i $$0.929983\pi$$
$$492$$ 0.472136 0.0212855
$$493$$ −18.9443 −0.853207
$$494$$ −2.56231 −0.115284
$$495$$ 0 0
$$496$$ 14.5623 0.653867
$$497$$ 4.09017 0.183469
$$498$$ −10.0902 −0.452151
$$499$$ 7.56231 0.338535 0.169268 0.985570i $$-0.445860\pi$$
0.169268 + 0.985570i $$0.445860\pi$$
$$500$$ 0 0
$$501$$ 14.5623 0.650596
$$502$$ −47.2148 −2.10730
$$503$$ 37.4164 1.66832 0.834158 0.551526i $$-0.185954\pi$$
0.834158 + 0.551526i $$0.185954\pi$$
$$504$$ −2.76393 −0.123115
$$505$$ 0 0
$$506$$ 31.8885 1.41762
$$507$$ 9.56231 0.424677
$$508$$ −12.2918 −0.545360
$$509$$ −20.3262 −0.900945 −0.450472 0.892790i $$-0.648744\pi$$
−0.450472 + 0.892790i $$0.648744\pi$$
$$510$$ 0 0
$$511$$ −5.56231 −0.246062
$$512$$ −5.29180 −0.233867
$$513$$ 4.27051 0.188548
$$514$$ −36.9787 −1.63106
$$515$$ 0 0
$$516$$ −3.00000 −0.132068
$$517$$ 3.23607 0.142322
$$518$$ −0.236068 −0.0103722
$$519$$ 18.8885 0.829115
$$520$$ 0 0
$$521$$ 29.3607 1.28631 0.643157 0.765734i $$-0.277624\pi$$
0.643157 + 0.765734i $$0.277624\pi$$
$$522$$ 11.7082 0.512454
$$523$$ 13.1459 0.574830 0.287415 0.957806i $$-0.407204\pi$$
0.287415 + 0.957806i $$0.407204\pi$$
$$524$$ 4.20163 0.183549
$$525$$ 0 0
$$526$$ 17.6525 0.769685
$$527$$ −15.7082 −0.684260
$$528$$ −25.4164 −1.10611
$$529$$ −8.83282 −0.384035
$$530$$ 0 0
$$531$$ 21.7082 0.942056
$$532$$ −0.326238 −0.0141442
$$533$$ 1.41641 0.0613514
$$534$$ 14.4721 0.626271
$$535$$ 0 0
$$536$$ 10.6525 0.460117
$$537$$ 0.527864 0.0227790
$$538$$ −20.6525 −0.890391
$$539$$ 34.6525 1.49259
$$540$$ 0 0
$$541$$ 27.1246 1.16618 0.583089 0.812408i $$-0.301844\pi$$
0.583089 + 0.812408i $$0.301844\pi$$
$$542$$ −12.9443 −0.556004
$$543$$ −0.291796 −0.0125222
$$544$$ −17.7082 −0.759233
$$545$$ 0 0
$$546$$ −1.85410 −0.0793482
$$547$$ 21.2918 0.910371 0.455186 0.890397i $$-0.349573\pi$$
0.455186 + 0.890397i $$0.349573\pi$$
$$548$$ 7.38197 0.315342
$$549$$ −17.4164 −0.743314
$$550$$ 0 0
$$551$$ −3.09017 −0.131646
$$552$$ −8.41641 −0.358226
$$553$$ 5.00000 0.212622
$$554$$ 39.9787 1.69853
$$555$$ 0 0
$$556$$ 3.09017 0.131052
$$557$$ −4.76393 −0.201854 −0.100927 0.994894i $$-0.532181\pi$$
−0.100927 + 0.994894i $$0.532181\pi$$
$$558$$ 9.70820 0.410981
$$559$$ −9.00000 −0.380659
$$560$$ 0 0
$$561$$ 27.4164 1.15752
$$562$$ 16.3262 0.688681
$$563$$ −7.38197 −0.311113 −0.155556 0.987827i $$-0.549717\pi$$
−0.155556 + 0.987827i $$0.549717\pi$$
$$564$$ 0.381966 0.0160837
$$565$$ 0 0
$$566$$ 48.3050 2.03041
$$567$$ −0.618034 −0.0259550
$$568$$ 14.7984 0.620926
$$569$$ 20.5279 0.860573 0.430286 0.902692i $$-0.358413\pi$$
0.430286 + 0.902692i $$0.358413\pi$$
$$570$$ 0 0
$$571$$ −8.12461 −0.340004 −0.170002 0.985444i $$-0.554377\pi$$
−0.170002 + 0.985444i $$0.554377\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 1.81966 0.0760174
$$574$$ 0.763932 0.0318859
$$575$$ 0 0
$$576$$ −8.47214 −0.353006
$$577$$ 33.7771 1.40616 0.703079 0.711111i $$-0.251808\pi$$
0.703079 + 0.711111i $$0.251808\pi$$
$$578$$ 16.8541 0.701038
$$579$$ 7.70820 0.320342
$$580$$ 0 0
$$581$$ −3.85410 −0.159895
$$582$$ −6.23607 −0.258493
$$583$$ −18.1803 −0.752953
$$584$$ −20.1246 −0.832762
$$585$$ 0 0
$$586$$ 31.5967 1.30525
$$587$$ −5.29180 −0.218416 −0.109208 0.994019i $$-0.534831\pi$$
−0.109208 + 0.994019i $$0.534831\pi$$
$$588$$ 4.09017 0.168676
$$589$$ −2.56231 −0.105578
$$590$$ 0 0
$$591$$ 3.70820 0.152535
$$592$$ −1.14590 −0.0470961
$$593$$ 10.9098 0.448013 0.224007 0.974588i $$-0.428086\pi$$
0.224007 + 0.974588i $$0.428086\pi$$
$$594$$ −42.3607 −1.73808
$$595$$ 0 0
$$596$$ −2.43769 −0.0998518
$$597$$ 17.5623 0.718777
$$598$$ 11.2918 0.461756
$$599$$ −9.47214 −0.387021 −0.193510 0.981098i $$-0.561987\pi$$
−0.193510 + 0.981098i $$0.561987\pi$$
$$600$$ 0 0
$$601$$ 2.72949 0.111338 0.0556691 0.998449i $$-0.482271\pi$$
0.0556691 + 0.998449i $$0.482271\pi$$
$$602$$ −4.85410 −0.197838
$$603$$ 9.52786 0.388005
$$604$$ 9.00000 0.366205
$$605$$ 0 0
$$606$$ −2.38197 −0.0967608
$$607$$ 35.5623 1.44343 0.721715 0.692191i $$-0.243354\pi$$
0.721715 + 0.692191i $$0.243354\pi$$
$$608$$ −2.88854 −0.117146
$$609$$ −2.23607 −0.0906100
$$610$$ 0 0
$$611$$ 1.14590 0.0463581
$$612$$ −6.47214 −0.261621
$$613$$ 14.9787 0.604985 0.302492 0.953152i $$-0.402181\pi$$
0.302492 + 0.953152i $$0.402181\pi$$
$$614$$ −14.9443 −0.603102
$$615$$ 0 0
$$616$$ −7.23607 −0.291549
$$617$$ −14.2361 −0.573123 −0.286561 0.958062i $$-0.592512\pi$$
−0.286561 + 0.958062i $$0.592512\pi$$
$$618$$ 13.8541 0.557294
$$619$$ 30.5279 1.22702 0.613509 0.789688i $$-0.289757\pi$$
0.613509 + 0.789688i $$0.289757\pi$$
$$620$$ 0 0
$$621$$ −18.8197 −0.755207
$$622$$ 13.7639 0.551883
$$623$$ 5.52786 0.221469
$$624$$ −9.00000 −0.360288
$$625$$ 0 0
$$626$$ 27.1246 1.08412
$$627$$ 4.47214 0.178600
$$628$$ −8.14590 −0.325057
$$629$$ 1.23607 0.0492853
$$630$$ 0 0
$$631$$ −10.2361 −0.407491 −0.203746 0.979024i $$-0.565312\pi$$
−0.203746 + 0.979024i $$0.565312\pi$$
$$632$$ 18.0902 0.719588
$$633$$ 9.18034 0.364886
$$634$$ −12.3820 −0.491751
$$635$$ 0 0
$$636$$ −2.14590 −0.0850904
$$637$$ 12.2705 0.486175
$$638$$ 30.6525 1.21354
$$639$$ 13.2361 0.523611
$$640$$ 0 0
$$641$$ −1.09017 −0.0430591 −0.0215296 0.999768i $$-0.506854\pi$$
−0.0215296 + 0.999768i $$0.506854\pi$$
$$642$$ −26.5623 −1.04833
$$643$$ 30.8328 1.21593 0.607964 0.793965i $$-0.291987\pi$$
0.607964 + 0.793965i $$0.291987\pi$$
$$644$$ 1.43769 0.0566531
$$645$$ 0 0
$$646$$ 7.23607 0.284699
$$647$$ 36.5410 1.43658 0.718288 0.695746i $$-0.244926\pi$$
0.718288 + 0.695746i $$0.244926\pi$$
$$648$$ −2.23607 −0.0878410
$$649$$ 56.8328 2.23088
$$650$$ 0 0
$$651$$ −1.85410 −0.0726680
$$652$$ −6.79837 −0.266245
$$653$$ −19.0902 −0.747056 −0.373528 0.927619i $$-0.621852\pi$$
−0.373528 + 0.927619i $$0.621852\pi$$
$$654$$ −16.1803 −0.632701
$$655$$ 0 0
$$656$$ 3.70820 0.144781
$$657$$ −18.0000 −0.702247
$$658$$ 0.618034 0.0240935
$$659$$ 15.5279 0.604880 0.302440 0.953168i $$-0.402199\pi$$
0.302440 + 0.953168i $$0.402199\pi$$
$$660$$ 0 0
$$661$$ 19.6869 0.765732 0.382866 0.923804i $$-0.374937\pi$$
0.382866 + 0.923804i $$0.374937\pi$$
$$662$$ −37.4164 −1.45423
$$663$$ 9.70820 0.377035
$$664$$ −13.9443 −0.541143
$$665$$ 0 0
$$666$$ −0.763932 −0.0296018
$$667$$ 13.6180 0.527292
$$668$$ −9.00000 −0.348220
$$669$$ −0.180340 −0.00697234
$$670$$ 0 0
$$671$$ −45.5967 −1.76024
$$672$$ −2.09017 −0.0806301
$$673$$ −12.1803 −0.469518 −0.234759 0.972054i $$-0.575430\pi$$
−0.234759 + 0.972054i $$0.575430\pi$$
$$674$$ −12.7082 −0.489502
$$675$$ 0 0
$$676$$ −5.90983 −0.227301
$$677$$ −10.6180 −0.408084 −0.204042 0.978962i $$-0.565408\pi$$
−0.204042 + 0.978962i $$0.565408\pi$$
$$678$$ 27.2705 1.04732
$$679$$ −2.38197 −0.0914115
$$680$$ 0 0
$$681$$ 14.7639 0.565755
$$682$$ 25.4164 0.973245
$$683$$ 13.4721 0.515497 0.257748 0.966212i $$-0.417019\pi$$
0.257748 + 0.966212i $$0.417019\pi$$
$$684$$ −1.05573 −0.0403668
$$685$$ 0 0
$$686$$ 13.6180 0.519939
$$687$$ −21.7082 −0.828220
$$688$$ −23.5623 −0.898304
$$689$$ −6.43769 −0.245257
$$690$$ 0 0
$$691$$ 36.2705 1.37980 0.689898 0.723907i $$-0.257656\pi$$
0.689898 + 0.723907i $$0.257656\pi$$
$$692$$ −11.6738 −0.443770
$$693$$ −6.47214 −0.245856
$$694$$ 32.2148 1.22286
$$695$$ 0 0
$$696$$ −8.09017 −0.306657
$$697$$ −4.00000 −0.151511
$$698$$ 35.1246 1.32949
$$699$$ −2.94427 −0.111363
$$700$$ 0 0
$$701$$ −41.0132 −1.54905 −0.774523 0.632546i $$-0.782010\pi$$
−0.774523 + 0.632546i $$0.782010\pi$$
$$702$$ −15.0000 −0.566139
$$703$$ 0.201626 0.00760447
$$704$$ −22.1803 −0.835953
$$705$$ 0 0
$$706$$ −20.8885 −0.786151
$$707$$ −0.909830 −0.0342177
$$708$$ 6.70820 0.252110
$$709$$ −33.5410 −1.25966 −0.629830 0.776733i $$-0.716875\pi$$
−0.629830 + 0.776733i $$0.716875\pi$$
$$710$$ 0 0
$$711$$ 16.1803 0.606810
$$712$$ 20.0000 0.749532
$$713$$ 11.2918 0.422881
$$714$$ 5.23607 0.195955
$$715$$ 0 0
$$716$$ −0.326238 −0.0121921
$$717$$ 20.5279 0.766627
$$718$$ 22.2361 0.829843
$$719$$ −23.2918 −0.868637 −0.434319 0.900759i $$-0.643011\pi$$
−0.434319 + 0.900759i $$0.643011\pi$$
$$720$$ 0 0
$$721$$ 5.29180 0.197077
$$722$$ −29.5623 −1.10020
$$723$$ −2.52786 −0.0940123
$$724$$ 0.180340 0.00670228
$$725$$ 0 0
$$726$$ −26.5623 −0.985820
$$727$$ −24.5623 −0.910966 −0.455483 0.890245i $$-0.650533\pi$$
−0.455483 + 0.890245i $$0.650533\pi$$
$$728$$ −2.56231 −0.0949654
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 25.4164 0.940060
$$732$$ −5.38197 −0.198923
$$733$$ 19.9787 0.737931 0.368965 0.929443i $$-0.379712\pi$$
0.368965 + 0.929443i $$0.379712\pi$$
$$734$$ 41.3607 1.52665
$$735$$ 0 0
$$736$$ 12.7295 0.469215
$$737$$ 24.9443 0.918834
$$738$$ 2.47214 0.0910006
$$739$$ 15.9787 0.587786 0.293893 0.955838i $$-0.405049\pi$$
0.293893 + 0.955838i $$0.405049\pi$$
$$740$$ 0 0
$$741$$ 1.58359 0.0581747
$$742$$ −3.47214 −0.127466
$$743$$ −28.3607 −1.04045 −0.520226 0.854029i $$-0.674152\pi$$
−0.520226 + 0.854029i $$0.674152\pi$$
$$744$$ −6.70820 −0.245935
$$745$$ 0 0
$$746$$ 45.7426 1.67476
$$747$$ −12.4721 −0.456332
$$748$$ −16.9443 −0.619544
$$749$$ −10.1459 −0.370723
$$750$$ 0 0
$$751$$ −5.11146 −0.186520 −0.0932598 0.995642i $$-0.529729\pi$$
−0.0932598 + 0.995642i $$0.529729\pi$$
$$752$$ 3.00000 0.109399
$$753$$ 29.1803 1.06339
$$754$$ 10.8541 0.395283
$$755$$ 0 0
$$756$$ −1.90983 −0.0694598
$$757$$ −30.4164 −1.10550 −0.552752 0.833346i $$-0.686422\pi$$
−0.552752 + 0.833346i $$0.686422\pi$$
$$758$$ 23.6180 0.857846
$$759$$ −19.7082 −0.715362
$$760$$ 0 0
$$761$$ −18.4508 −0.668843 −0.334421 0.942424i $$-0.608541\pi$$
−0.334421 + 0.942424i $$0.608541\pi$$
$$762$$ 32.1803 1.16577
$$763$$ −6.18034 −0.223743
$$764$$ −1.12461 −0.0406870
$$765$$ 0 0
$$766$$ −53.9787 −1.95033
$$767$$ 20.1246 0.726658
$$768$$ −13.5623 −0.489388
$$769$$ 13.4164 0.483808 0.241904 0.970300i $$-0.422228\pi$$
0.241904 + 0.970300i $$0.422228\pi$$
$$770$$ 0 0
$$771$$ 22.8541 0.823070
$$772$$ −4.76393 −0.171458
$$773$$ 36.1591 1.30055 0.650275 0.759699i $$-0.274654\pi$$
0.650275 + 0.759699i $$0.274654\pi$$
$$774$$ −15.7082 −0.564620
$$775$$ 0 0
$$776$$ −8.61803 −0.309369
$$777$$ 0.145898 0.00523406
$$778$$ 24.2705 0.870140
$$779$$ −0.652476 −0.0233774
$$780$$ 0 0
$$781$$ 34.6525 1.23996
$$782$$ −31.8885 −1.14033
$$783$$ −18.0902 −0.646490
$$784$$ 32.1246 1.14731
$$785$$ 0 0
$$786$$ −11.0000 −0.392357
$$787$$ 11.8197 0.421325 0.210663 0.977559i $$-0.432438\pi$$
0.210663 + 0.977559i $$0.432438\pi$$
$$788$$ −2.29180 −0.0816419
$$789$$ −10.9098 −0.388400
$$790$$ 0 0
$$791$$ 10.4164 0.370365
$$792$$ −23.4164 −0.832066
$$793$$ −16.1459 −0.573358
$$794$$ −46.9787 −1.66721
$$795$$ 0 0
$$796$$ −10.8541 −0.384713
$$797$$ −9.76393 −0.345856 −0.172928 0.984934i $$-0.555323\pi$$
−0.172928 + 0.984934i $$0.555323\pi$$
$$798$$ 0.854102 0.0302349
$$799$$ −3.23607 −0.114484
$$800$$ 0 0
$$801$$ 17.8885 0.632061
$$802$$ 43.0344 1.51960
$$803$$ −47.1246 −1.66299
$$804$$ 2.94427 0.103836
$$805$$ 0 0
$$806$$ 9.00000 0.317011
$$807$$ 12.7639 0.449312
$$808$$ −3.29180 −0.115805
$$809$$ 30.9787 1.08915 0.544577 0.838711i $$-0.316690\pi$$
0.544577 + 0.838711i $$0.316690\pi$$
$$810$$ 0 0
$$811$$ −14.7082 −0.516475 −0.258237 0.966081i $$-0.583142\pi$$
−0.258237 + 0.966081i $$0.583142\pi$$
$$812$$ 1.38197 0.0484975
$$813$$ 8.00000 0.280572
$$814$$ −2.00000 −0.0701000
$$815$$ 0 0
$$816$$ 25.4164 0.889752
$$817$$ 4.14590 0.145047
$$818$$ 2.56231 0.0895889
$$819$$ −2.29180 −0.0800818
$$820$$ 0 0
$$821$$ −40.6869 −1.41998 −0.709992 0.704210i $$-0.751301\pi$$
−0.709992 + 0.704210i $$0.751301\pi$$
$$822$$ −19.3262 −0.674080
$$823$$ −47.7082 −1.66300 −0.831502 0.555522i $$-0.812518\pi$$
−0.831502 + 0.555522i $$0.812518\pi$$
$$824$$ 19.1459 0.666979
$$825$$ 0 0
$$826$$ 10.8541 0.377663
$$827$$ 0.965558 0.0335757 0.0167879 0.999859i $$-0.494656\pi$$
0.0167879 + 0.999859i $$0.494656\pi$$
$$828$$ 4.65248 0.161685
$$829$$ −35.8541 −1.24526 −0.622632 0.782515i $$-0.713937\pi$$
−0.622632 + 0.782515i $$0.713937\pi$$
$$830$$ 0 0
$$831$$ −24.7082 −0.857118
$$832$$ −7.85410 −0.272292
$$833$$ −34.6525 −1.20064
$$834$$ −8.09017 −0.280140
$$835$$ 0 0
$$836$$ −2.76393 −0.0955926
$$837$$ −15.0000 −0.518476
$$838$$ −15.3262 −0.529436
$$839$$ 10.8541 0.374725 0.187363 0.982291i $$-0.440006\pi$$
0.187363 + 0.982291i $$0.440006\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ 51.7771 1.78436
$$843$$ −10.0902 −0.347524
$$844$$ −5.67376 −0.195299
$$845$$ 0 0
$$846$$ 2.00000 0.0687614
$$847$$ −10.1459 −0.348617
$$848$$ −16.8541 −0.578772
$$849$$ −29.8541 −1.02459
$$850$$ 0 0
$$851$$ −0.888544 −0.0304589
$$852$$ 4.09017 0.140127
$$853$$ 15.3050 0.524032 0.262016 0.965064i $$-0.415613\pi$$
0.262016 + 0.965064i $$0.415613\pi$$
$$854$$ −8.70820 −0.297989
$$855$$ 0 0
$$856$$ −36.7082 −1.25466
$$857$$ −19.6869 −0.672492 −0.336246 0.941774i $$-0.609157\pi$$
−0.336246 + 0.941774i $$0.609157\pi$$
$$858$$ −15.7082 −0.536269
$$859$$ −1.58359 −0.0540315 −0.0270157 0.999635i $$-0.508600\pi$$
−0.0270157 + 0.999635i $$0.508600\pi$$
$$860$$ 0 0
$$861$$ −0.472136 −0.0160904
$$862$$ −48.2705 −1.64410
$$863$$ 21.4377 0.729748 0.364874 0.931057i $$-0.381112\pi$$
0.364874 + 0.931057i $$0.381112\pi$$
$$864$$ −16.9098 −0.575284
$$865$$ 0 0
$$866$$ −43.4508 −1.47652
$$867$$ −10.4164 −0.353760
$$868$$ 1.14590 0.0388943
$$869$$ 42.3607 1.43699
$$870$$ 0 0
$$871$$ 8.83282 0.299289
$$872$$ −22.3607 −0.757228
$$873$$ −7.70820 −0.260883
$$874$$ −5.20163 −0.175948
$$875$$ 0 0
$$876$$ −5.56231 −0.187933
$$877$$ 36.5410 1.23390 0.616951 0.787001i $$-0.288368\pi$$
0.616951 + 0.787001i $$0.288368\pi$$
$$878$$ −66.3050 −2.23768
$$879$$ −19.5279 −0.658659
$$880$$ 0 0
$$881$$ −40.3607 −1.35979 −0.679893 0.733311i $$-0.737974\pi$$
−0.679893 + 0.733311i $$0.737974\pi$$
$$882$$ 21.4164 0.721128
$$883$$ 20.5836 0.692693 0.346347 0.938107i $$-0.387422\pi$$
0.346347 + 0.938107i $$0.387422\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ −48.4508 −1.62774
$$887$$ −29.8885 −1.00356 −0.501780 0.864996i $$-0.667321\pi$$
−0.501780 + 0.864996i $$0.667321\pi$$
$$888$$ 0.527864 0.0177140
$$889$$ 12.2918 0.412254
$$890$$ 0 0
$$891$$ −5.23607 −0.175415
$$892$$ 0.111456 0.00373183
$$893$$ −0.527864 −0.0176643
$$894$$ 6.38197 0.213445
$$895$$ 0 0
$$896$$ −8.41641 −0.281172
$$897$$ −6.97871 −0.233012
$$898$$ 7.56231 0.252357
$$899$$ 10.8541 0.362005
$$900$$ 0 0
$$901$$ 18.1803 0.605675
$$902$$ 6.47214 0.215499
$$903$$ 3.00000 0.0998337
$$904$$ 37.6869 1.25345
$$905$$ 0 0
$$906$$ −23.5623 −0.782805
$$907$$ 33.2492 1.10402 0.552011 0.833837i $$-0.313861\pi$$
0.552011 + 0.833837i $$0.313861\pi$$
$$908$$ −9.12461 −0.302811
$$909$$ −2.94427 −0.0976553
$$910$$ 0 0
$$911$$ −40.2361 −1.33308 −0.666540 0.745469i $$-0.732226\pi$$
−0.666540 + 0.745469i $$0.732226\pi$$
$$912$$ 4.14590 0.137284
$$913$$ −32.6525 −1.08064
$$914$$ 34.6525 1.14620
$$915$$ 0 0
$$916$$ 13.4164 0.443291
$$917$$ −4.20163 −0.138750
$$918$$ 42.3607 1.39811
$$919$$ −53.2148 −1.75539 −0.877697 0.479216i $$-0.840921\pi$$
−0.877697 + 0.479216i $$0.840921\pi$$
$$920$$ 0 0
$$921$$ 9.23607 0.304339
$$922$$ 1.32624 0.0436773
$$923$$ 12.2705 0.403889
$$924$$ −2.00000 −0.0657952
$$925$$ 0 0
$$926$$ 39.0344 1.28275
$$927$$ 17.1246 0.562446
$$928$$ 12.2361 0.401669
$$929$$ 41.6312 1.36588 0.682938 0.730477i $$-0.260702\pi$$
0.682938 + 0.730477i $$0.260702\pi$$
$$930$$ 0 0
$$931$$ −5.65248 −0.185252
$$932$$ 1.81966 0.0596049
$$933$$ −8.50658 −0.278493
$$934$$ −44.4164 −1.45335
$$935$$ 0 0
$$936$$ −8.29180 −0.271026
$$937$$ −17.7295 −0.579197 −0.289599 0.957148i $$-0.593522\pi$$
−0.289599 + 0.957148i $$0.593522\pi$$
$$938$$ 4.76393 0.155548
$$939$$ −16.7639 −0.547070
$$940$$ 0 0
$$941$$ −46.4164 −1.51313 −0.756566 0.653918i $$-0.773124\pi$$
−0.756566 + 0.653918i $$0.773124\pi$$
$$942$$ 21.3262 0.694846
$$943$$ 2.87539 0.0936355
$$944$$ 52.6869 1.71481
$$945$$ 0 0
$$946$$ −41.1246 −1.33708
$$947$$ −2.65248 −0.0861939 −0.0430969 0.999071i $$-0.513722\pi$$
−0.0430969 + 0.999071i $$0.513722\pi$$
$$948$$ 5.00000 0.162392
$$949$$ −16.6869 −0.541680
$$950$$ 0 0
$$951$$ 7.65248 0.248149
$$952$$ 7.23607 0.234522
$$953$$ 7.74265 0.250809 0.125404 0.992106i $$-0.459977\pi$$
0.125404 + 0.992106i $$0.459977\pi$$
$$954$$ −11.2361 −0.363781
$$955$$ 0 0
$$956$$ −12.6869 −0.410324
$$957$$ −18.9443 −0.612381
$$958$$ −17.5623 −0.567412
$$959$$ −7.38197 −0.238376
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −0.708204 −0.0228334
$$963$$ −32.8328 −1.05802
$$964$$ 1.56231 0.0503185
$$965$$ 0 0
$$966$$ −3.76393 −0.121103
$$967$$ −4.11146 −0.132216 −0.0661078 0.997812i $$-0.521058\pi$$
−0.0661078 + 0.997812i $$0.521058\pi$$
$$968$$ −36.7082 −1.17985
$$969$$ −4.47214 −0.143666
$$970$$ 0 0
$$971$$ 5.61803 0.180291 0.0901456 0.995929i $$-0.471267\pi$$
0.0901456 + 0.995929i $$0.471267\pi$$
$$972$$ −9.88854 −0.317175
$$973$$ −3.09017 −0.0990663
$$974$$ 58.9230 1.88801
$$975$$ 0 0
$$976$$ −42.2705 −1.35305
$$977$$ 2.34752 0.0751040 0.0375520 0.999295i $$-0.488044\pi$$
0.0375520 + 0.999295i $$0.488044\pi$$
$$978$$ 17.7984 0.569129
$$979$$ 46.8328 1.49678
$$980$$ 0 0
$$981$$ −20.0000 −0.638551
$$982$$ −69.9787 −2.23311
$$983$$ −9.61803 −0.306768 −0.153384 0.988167i $$-0.549017\pi$$
−0.153384 + 0.988167i $$0.549017\pi$$
$$984$$ −1.70820 −0.0544556
$$985$$ 0 0
$$986$$ −30.6525 −0.976174
$$987$$ −0.381966 −0.0121581
$$988$$ −0.978714 −0.0311370
$$989$$ −18.2705 −0.580968
$$990$$ 0 0
$$991$$ −15.3607 −0.487948 −0.243974 0.969782i $$-0.578451\pi$$
−0.243974 + 0.969782i $$0.578451\pi$$
$$992$$ 10.1459 0.322133
$$993$$ 23.1246 0.733837
$$994$$ 6.61803 0.209911
$$995$$ 0 0
$$996$$ −3.85410 −0.122122
$$997$$ −24.8885 −0.788228 −0.394114 0.919062i $$-0.628949\pi$$
−0.394114 + 0.919062i $$0.628949\pi$$
$$998$$ 12.2361 0.387326
$$999$$ 1.18034 0.0373443
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 625.2.a.c.1.2 2
3.2 odd 2 5625.2.a.d.1.1 2
4.3 odd 2 10000.2.a.l.1.2 2
5.2 odd 4 625.2.b.a.624.4 4
5.3 odd 4 625.2.b.a.624.1 4
5.4 even 2 625.2.a.b.1.1 2
15.14 odd 2 5625.2.a.f.1.2 2
20.19 odd 2 10000.2.a.c.1.1 2
25.2 odd 20 125.2.e.a.24.2 8
25.3 odd 20 625.2.e.c.249.1 8
25.4 even 10 625.2.d.h.376.1 4
25.6 even 5 625.2.d.b.251.1 4
25.8 odd 20 625.2.e.c.374.2 8
25.9 even 10 25.2.d.a.6.1 4
25.11 even 5 125.2.d.a.101.1 4
25.12 odd 20 125.2.e.a.99.1 8
25.13 odd 20 125.2.e.a.99.2 8
25.14 even 10 25.2.d.a.21.1 yes 4
25.16 even 5 125.2.d.a.26.1 4
25.17 odd 20 625.2.e.c.374.1 8
25.19 even 10 625.2.d.h.251.1 4
25.21 even 5 625.2.d.b.376.1 4
25.22 odd 20 625.2.e.c.249.2 8
25.23 odd 20 125.2.e.a.24.1 8
75.14 odd 10 225.2.h.b.46.1 4
75.59 odd 10 225.2.h.b.181.1 4
100.39 odd 10 400.2.u.b.321.1 4
100.59 odd 10 400.2.u.b.81.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 25.9 even 10
25.2.d.a.21.1 yes 4 25.14 even 10
125.2.d.a.26.1 4 25.16 even 5
125.2.d.a.101.1 4 25.11 even 5
125.2.e.a.24.1 8 25.23 odd 20
125.2.e.a.24.2 8 25.2 odd 20
125.2.e.a.99.1 8 25.12 odd 20
125.2.e.a.99.2 8 25.13 odd 20
225.2.h.b.46.1 4 75.14 odd 10
225.2.h.b.181.1 4 75.59 odd 10
400.2.u.b.81.1 4 100.59 odd 10
400.2.u.b.321.1 4 100.39 odd 10
625.2.a.b.1.1 2 5.4 even 2
625.2.a.c.1.2 2 1.1 even 1 trivial
625.2.b.a.624.1 4 5.3 odd 4
625.2.b.a.624.4 4 5.2 odd 4
625.2.d.b.251.1 4 25.6 even 5
625.2.d.b.376.1 4 25.21 even 5
625.2.d.h.251.1 4 25.19 even 10
625.2.d.h.376.1 4 25.4 even 10
625.2.e.c.249.1 8 25.3 odd 20
625.2.e.c.249.2 8 25.22 odd 20
625.2.e.c.374.1 8 25.17 odd 20
625.2.e.c.374.2 8 25.8 odd 20
5625.2.a.d.1.1 2 3.2 odd 2
5625.2.a.f.1.2 2 15.14 odd 2
10000.2.a.c.1.1 2 20.19 odd 2
10000.2.a.l.1.2 2 4.3 odd 2