# Properties

 Label 5054.2.a.g.1.2 Level $5054$ Weight $2$ Character 5054.1 Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5054 = 2 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5054.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.3563931816$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 5054.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +2.61803 q^{10} -4.85410 q^{11} +1.61803 q^{12} -4.47214 q^{13} -1.00000 q^{14} -4.23607 q^{15} +1.00000 q^{16} +0.763932 q^{17} +0.381966 q^{18} -2.61803 q^{20} +1.61803 q^{21} +4.85410 q^{22} +8.94427 q^{23} -1.61803 q^{24} +1.85410 q^{25} +4.47214 q^{26} -5.47214 q^{27} +1.00000 q^{28} +0.145898 q^{29} +4.23607 q^{30} +6.00000 q^{31} -1.00000 q^{32} -7.85410 q^{33} -0.763932 q^{34} -2.61803 q^{35} -0.381966 q^{36} +7.85410 q^{37} -7.23607 q^{39} +2.61803 q^{40} -9.56231 q^{41} -1.61803 q^{42} +3.85410 q^{43} -4.85410 q^{44} +1.00000 q^{45} -8.94427 q^{46} -10.8541 q^{47} +1.61803 q^{48} +1.00000 q^{49} -1.85410 q^{50} +1.23607 q^{51} -4.47214 q^{52} -5.14590 q^{53} +5.47214 q^{54} +12.7082 q^{55} -1.00000 q^{56} -0.145898 q^{58} -11.5623 q^{59} -4.23607 q^{60} -8.56231 q^{61} -6.00000 q^{62} -0.381966 q^{63} +1.00000 q^{64} +11.7082 q^{65} +7.85410 q^{66} +5.23607 q^{67} +0.763932 q^{68} +14.4721 q^{69} +2.61803 q^{70} +8.56231 q^{71} +0.381966 q^{72} -7.85410 q^{74} +3.00000 q^{75} -4.85410 q^{77} +7.23607 q^{78} +0.326238 q^{79} -2.61803 q^{80} -7.70820 q^{81} +9.56231 q^{82} +11.2361 q^{83} +1.61803 q^{84} -2.00000 q^{85} -3.85410 q^{86} +0.236068 q^{87} +4.85410 q^{88} -3.14590 q^{89} -1.00000 q^{90} -4.47214 q^{91} +8.94427 q^{92} +9.70820 q^{93} +10.8541 q^{94} -1.61803 q^{96} +7.14590 q^{97} -1.00000 q^{98} +1.85410 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + O(q^{10})$$ $$2 q - 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 3 q^{10} - 3 q^{11} + q^{12} - 2 q^{14} - 4 q^{15} + 2 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{20} + q^{21} + 3 q^{22} - q^{24} - 3 q^{25} - 2 q^{27} + 2 q^{28} + 7 q^{29} + 4 q^{30} + 12 q^{31} - 2 q^{32} - 9 q^{33} - 6 q^{34} - 3 q^{35} - 3 q^{36} + 9 q^{37} - 10 q^{39} + 3 q^{40} + q^{41} - q^{42} + q^{43} - 3 q^{44} + 2 q^{45} - 15 q^{47} + q^{48} + 2 q^{49} + 3 q^{50} - 2 q^{51} - 17 q^{53} + 2 q^{54} + 12 q^{55} - 2 q^{56} - 7 q^{58} - 3 q^{59} - 4 q^{60} + 3 q^{61} - 12 q^{62} - 3 q^{63} + 2 q^{64} + 10 q^{65} + 9 q^{66} + 6 q^{67} + 6 q^{68} + 20 q^{69} + 3 q^{70} - 3 q^{71} + 3 q^{72} - 9 q^{74} + 6 q^{75} - 3 q^{77} + 10 q^{78} - 15 q^{79} - 3 q^{80} - 2 q^{81} - q^{82} + 18 q^{83} + q^{84} - 4 q^{85} - q^{86} - 4 q^{87} + 3 q^{88} - 13 q^{89} - 2 q^{90} + 6 q^{93} + 15 q^{94} - q^{96} + 21 q^{97} - 2 q^{98} - 3 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.61803 0.934172 0.467086 0.884212i $$-0.345304\pi$$
0.467086 + 0.884212i $$0.345304\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −2.61803 −1.17082 −0.585410 0.810737i $$-0.699067\pi$$
−0.585410 + 0.810737i $$0.699067\pi$$
$$6$$ −1.61803 −0.660560
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ −0.381966 −0.127322
$$10$$ 2.61803 0.827895
$$11$$ −4.85410 −1.46357 −0.731783 0.681537i $$-0.761312\pi$$
−0.731783 + 0.681537i $$0.761312\pi$$
$$12$$ 1.61803 0.467086
$$13$$ −4.47214 −1.24035 −0.620174 0.784465i $$-0.712938\pi$$
−0.620174 + 0.784465i $$0.712938\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −4.23607 −1.09375
$$16$$ 1.00000 0.250000
$$17$$ 0.763932 0.185281 0.0926404 0.995700i $$-0.470469\pi$$
0.0926404 + 0.995700i $$0.470469\pi$$
$$18$$ 0.381966 0.0900303
$$19$$ 0 0
$$20$$ −2.61803 −0.585410
$$21$$ 1.61803 0.353084
$$22$$ 4.85410 1.03490
$$23$$ 8.94427 1.86501 0.932505 0.361158i $$-0.117618\pi$$
0.932505 + 0.361158i $$0.117618\pi$$
$$24$$ −1.61803 −0.330280
$$25$$ 1.85410 0.370820
$$26$$ 4.47214 0.877058
$$27$$ −5.47214 −1.05311
$$28$$ 1.00000 0.188982
$$29$$ 0.145898 0.0270926 0.0135463 0.999908i $$-0.495688\pi$$
0.0135463 + 0.999908i $$0.495688\pi$$
$$30$$ 4.23607 0.773397
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −7.85410 −1.36722
$$34$$ −0.763932 −0.131013
$$35$$ −2.61803 −0.442529
$$36$$ −0.381966 −0.0636610
$$37$$ 7.85410 1.29121 0.645603 0.763673i $$-0.276606\pi$$
0.645603 + 0.763673i $$0.276606\pi$$
$$38$$ 0 0
$$39$$ −7.23607 −1.15870
$$40$$ 2.61803 0.413948
$$41$$ −9.56231 −1.49338 −0.746691 0.665171i $$-0.768358\pi$$
−0.746691 + 0.665171i $$0.768358\pi$$
$$42$$ −1.61803 −0.249668
$$43$$ 3.85410 0.587745 0.293873 0.955845i $$-0.405056\pi$$
0.293873 + 0.955845i $$0.405056\pi$$
$$44$$ −4.85410 −0.731783
$$45$$ 1.00000 0.149071
$$46$$ −8.94427 −1.31876
$$47$$ −10.8541 −1.58323 −0.791617 0.611018i $$-0.790760\pi$$
−0.791617 + 0.611018i $$0.790760\pi$$
$$48$$ 1.61803 0.233543
$$49$$ 1.00000 0.142857
$$50$$ −1.85410 −0.262210
$$51$$ 1.23607 0.173084
$$52$$ −4.47214 −0.620174
$$53$$ −5.14590 −0.706843 −0.353422 0.935464i $$-0.614982\pi$$
−0.353422 + 0.935464i $$0.614982\pi$$
$$54$$ 5.47214 0.744663
$$55$$ 12.7082 1.71357
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ −0.145898 −0.0191574
$$59$$ −11.5623 −1.50528 −0.752642 0.658430i $$-0.771221\pi$$
−0.752642 + 0.658430i $$0.771221\pi$$
$$60$$ −4.23607 −0.546874
$$61$$ −8.56231 −1.09629 −0.548145 0.836383i $$-0.684666\pi$$
−0.548145 + 0.836383i $$0.684666\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ −0.381966 −0.0481232
$$64$$ 1.00000 0.125000
$$65$$ 11.7082 1.45222
$$66$$ 7.85410 0.966773
$$67$$ 5.23607 0.639688 0.319844 0.947470i $$-0.396370\pi$$
0.319844 + 0.947470i $$0.396370\pi$$
$$68$$ 0.763932 0.0926404
$$69$$ 14.4721 1.74224
$$70$$ 2.61803 0.312915
$$71$$ 8.56231 1.01616 0.508079 0.861310i $$-0.330356\pi$$
0.508079 + 0.861310i $$0.330356\pi$$
$$72$$ 0.381966 0.0450151
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −7.85410 −0.913021
$$75$$ 3.00000 0.346410
$$76$$ 0 0
$$77$$ −4.85410 −0.553176
$$78$$ 7.23607 0.819323
$$79$$ 0.326238 0.0367046 0.0183523 0.999832i $$-0.494158\pi$$
0.0183523 + 0.999832i $$0.494158\pi$$
$$80$$ −2.61803 −0.292705
$$81$$ −7.70820 −0.856467
$$82$$ 9.56231 1.05598
$$83$$ 11.2361 1.23332 0.616659 0.787230i $$-0.288486\pi$$
0.616659 + 0.787230i $$0.288486\pi$$
$$84$$ 1.61803 0.176542
$$85$$ −2.00000 −0.216930
$$86$$ −3.85410 −0.415599
$$87$$ 0.236068 0.0253091
$$88$$ 4.85410 0.517449
$$89$$ −3.14590 −0.333465 −0.166732 0.986002i $$-0.553322\pi$$
−0.166732 + 0.986002i $$0.553322\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ −4.47214 −0.468807
$$92$$ 8.94427 0.932505
$$93$$ 9.70820 1.00669
$$94$$ 10.8541 1.11952
$$95$$ 0 0
$$96$$ −1.61803 −0.165140
$$97$$ 7.14590 0.725556 0.362778 0.931876i $$-0.381828\pi$$
0.362778 + 0.931876i $$0.381828\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 1.85410 0.186344
$$100$$ 1.85410 0.185410
$$101$$ 2.94427 0.292966 0.146483 0.989213i $$-0.453205\pi$$
0.146483 + 0.989213i $$0.453205\pi$$
$$102$$ −1.23607 −0.122389
$$103$$ 8.94427 0.881305 0.440653 0.897678i $$-0.354747\pi$$
0.440653 + 0.897678i $$0.354747\pi$$
$$104$$ 4.47214 0.438529
$$105$$ −4.23607 −0.413398
$$106$$ 5.14590 0.499814
$$107$$ 19.7082 1.90526 0.952632 0.304125i $$-0.0983642\pi$$
0.952632 + 0.304125i $$0.0983642\pi$$
$$108$$ −5.47214 −0.526557
$$109$$ 10.8541 1.03963 0.519817 0.854278i $$-0.326000\pi$$
0.519817 + 0.854278i $$0.326000\pi$$
$$110$$ −12.7082 −1.21168
$$111$$ 12.7082 1.20621
$$112$$ 1.00000 0.0944911
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 0 0
$$115$$ −23.4164 −2.18359
$$116$$ 0.145898 0.0135463
$$117$$ 1.70820 0.157924
$$118$$ 11.5623 1.06440
$$119$$ 0.763932 0.0700295
$$120$$ 4.23607 0.386698
$$121$$ 12.5623 1.14203
$$122$$ 8.56231 0.775195
$$123$$ −15.4721 −1.39508
$$124$$ 6.00000 0.538816
$$125$$ 8.23607 0.736656
$$126$$ 0.381966 0.0340282
$$127$$ 22.0902 1.96019 0.980093 0.198540i $$-0.0636199\pi$$
0.980093 + 0.198540i $$0.0636199\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 6.23607 0.549055
$$130$$ −11.7082 −1.02688
$$131$$ −4.47214 −0.390732 −0.195366 0.980730i $$-0.562590\pi$$
−0.195366 + 0.980730i $$0.562590\pi$$
$$132$$ −7.85410 −0.683612
$$133$$ 0 0
$$134$$ −5.23607 −0.452327
$$135$$ 14.3262 1.23301
$$136$$ −0.763932 −0.0655066
$$137$$ 3.38197 0.288941 0.144470 0.989509i $$-0.453852\pi$$
0.144470 + 0.989509i $$0.453852\pi$$
$$138$$ −14.4721 −1.23195
$$139$$ 6.00000 0.508913 0.254457 0.967084i $$-0.418103\pi$$
0.254457 + 0.967084i $$0.418103\pi$$
$$140$$ −2.61803 −0.221264
$$141$$ −17.5623 −1.47901
$$142$$ −8.56231 −0.718533
$$143$$ 21.7082 1.81533
$$144$$ −0.381966 −0.0318305
$$145$$ −0.381966 −0.0317206
$$146$$ 0 0
$$147$$ 1.61803 0.133453
$$148$$ 7.85410 0.645603
$$149$$ −17.2361 −1.41203 −0.706017 0.708195i $$-0.749510\pi$$
−0.706017 + 0.708195i $$0.749510\pi$$
$$150$$ −3.00000 −0.244949
$$151$$ 3.05573 0.248672 0.124336 0.992240i $$-0.460320\pi$$
0.124336 + 0.992240i $$0.460320\pi$$
$$152$$ 0 0
$$153$$ −0.291796 −0.0235903
$$154$$ 4.85410 0.391155
$$155$$ −15.7082 −1.26171
$$156$$ −7.23607 −0.579349
$$157$$ 2.85410 0.227782 0.113891 0.993493i $$-0.463669\pi$$
0.113891 + 0.993493i $$0.463669\pi$$
$$158$$ −0.326238 −0.0259541
$$159$$ −8.32624 −0.660314
$$160$$ 2.61803 0.206974
$$161$$ 8.94427 0.704907
$$162$$ 7.70820 0.605614
$$163$$ −20.2705 −1.58771 −0.793854 0.608108i $$-0.791929\pi$$
−0.793854 + 0.608108i $$0.791929\pi$$
$$164$$ −9.56231 −0.746691
$$165$$ 20.5623 1.60077
$$166$$ −11.2361 −0.872088
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ −1.61803 −0.124834
$$169$$ 7.00000 0.538462
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ 3.85410 0.293873
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −0.236068 −0.0178963
$$175$$ 1.85410 0.140157
$$176$$ −4.85410 −0.365892
$$177$$ −18.7082 −1.40619
$$178$$ 3.14590 0.235795
$$179$$ 17.7082 1.32357 0.661787 0.749692i $$-0.269798\pi$$
0.661787 + 0.749692i $$0.269798\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ −7.41641 −0.551257 −0.275629 0.961264i $$-0.588886\pi$$
−0.275629 + 0.961264i $$0.588886\pi$$
$$182$$ 4.47214 0.331497
$$183$$ −13.8541 −1.02412
$$184$$ −8.94427 −0.659380
$$185$$ −20.5623 −1.51177
$$186$$ −9.70820 −0.711840
$$187$$ −3.70820 −0.271171
$$188$$ −10.8541 −0.791617
$$189$$ −5.47214 −0.398039
$$190$$ 0 0
$$191$$ 14.9443 1.08133 0.540665 0.841238i $$-0.318173\pi$$
0.540665 + 0.841238i $$0.318173\pi$$
$$192$$ 1.61803 0.116772
$$193$$ −20.9443 −1.50760 −0.753801 0.657103i $$-0.771782\pi$$
−0.753801 + 0.657103i $$0.771782\pi$$
$$194$$ −7.14590 −0.513046
$$195$$ 18.9443 1.35663
$$196$$ 1.00000 0.0714286
$$197$$ −5.23607 −0.373054 −0.186527 0.982450i $$-0.559723\pi$$
−0.186527 + 0.982450i $$0.559723\pi$$
$$198$$ −1.85410 −0.131765
$$199$$ 1.56231 0.110749 0.0553745 0.998466i $$-0.482365\pi$$
0.0553745 + 0.998466i $$0.482365\pi$$
$$200$$ −1.85410 −0.131105
$$201$$ 8.47214 0.597578
$$202$$ −2.94427 −0.207158
$$203$$ 0.145898 0.0102400
$$204$$ 1.23607 0.0865421
$$205$$ 25.0344 1.74848
$$206$$ −8.94427 −0.623177
$$207$$ −3.41641 −0.237457
$$208$$ −4.47214 −0.310087
$$209$$ 0 0
$$210$$ 4.23607 0.292316
$$211$$ 17.2361 1.18658 0.593290 0.804989i $$-0.297829\pi$$
0.593290 + 0.804989i $$0.297829\pi$$
$$212$$ −5.14590 −0.353422
$$213$$ 13.8541 0.949267
$$214$$ −19.7082 −1.34723
$$215$$ −10.0902 −0.688144
$$216$$ 5.47214 0.372332
$$217$$ 6.00000 0.407307
$$218$$ −10.8541 −0.735133
$$219$$ 0 0
$$220$$ 12.7082 0.856787
$$221$$ −3.41641 −0.229812
$$222$$ −12.7082 −0.852919
$$223$$ −5.23607 −0.350633 −0.175317 0.984512i $$-0.556095\pi$$
−0.175317 + 0.984512i $$0.556095\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −0.708204 −0.0472136
$$226$$ −10.0000 −0.665190
$$227$$ 19.4164 1.28871 0.644356 0.764726i $$-0.277125\pi$$
0.644356 + 0.764726i $$0.277125\pi$$
$$228$$ 0 0
$$229$$ 23.8541 1.57632 0.788162 0.615468i $$-0.211033\pi$$
0.788162 + 0.615468i $$0.211033\pi$$
$$230$$ 23.4164 1.54403
$$231$$ −7.85410 −0.516762
$$232$$ −0.145898 −0.00957868
$$233$$ 4.09017 0.267956 0.133978 0.990984i $$-0.457225\pi$$
0.133978 + 0.990984i $$0.457225\pi$$
$$234$$ −1.70820 −0.111669
$$235$$ 28.4164 1.85368
$$236$$ −11.5623 −0.752642
$$237$$ 0.527864 0.0342885
$$238$$ −0.763932 −0.0495184
$$239$$ 14.1803 0.917250 0.458625 0.888630i $$-0.348342\pi$$
0.458625 + 0.888630i $$0.348342\pi$$
$$240$$ −4.23607 −0.273437
$$241$$ 17.5623 1.13129 0.565644 0.824650i $$-0.308628\pi$$
0.565644 + 0.824650i $$0.308628\pi$$
$$242$$ −12.5623 −0.807536
$$243$$ 3.94427 0.253025
$$244$$ −8.56231 −0.548145
$$245$$ −2.61803 −0.167260
$$246$$ 15.4721 0.986467
$$247$$ 0 0
$$248$$ −6.00000 −0.381000
$$249$$ 18.1803 1.15213
$$250$$ −8.23607 −0.520895
$$251$$ 9.05573 0.571592 0.285796 0.958290i $$-0.407742\pi$$
0.285796 + 0.958290i $$0.407742\pi$$
$$252$$ −0.381966 −0.0240616
$$253$$ −43.4164 −2.72957
$$254$$ −22.0902 −1.38606
$$255$$ −3.23607 −0.202650
$$256$$ 1.00000 0.0625000
$$257$$ 9.85410 0.614682 0.307341 0.951599i $$-0.400561\pi$$
0.307341 + 0.951599i $$0.400561\pi$$
$$258$$ −6.23607 −0.388241
$$259$$ 7.85410 0.488030
$$260$$ 11.7082 0.726112
$$261$$ −0.0557281 −0.00344948
$$262$$ 4.47214 0.276289
$$263$$ 12.6525 0.780185 0.390093 0.920776i $$-0.372443\pi$$
0.390093 + 0.920776i $$0.372443\pi$$
$$264$$ 7.85410 0.483387
$$265$$ 13.4721 0.827587
$$266$$ 0 0
$$267$$ −5.09017 −0.311513
$$268$$ 5.23607 0.319844
$$269$$ −4.58359 −0.279467 −0.139733 0.990189i $$-0.544625\pi$$
−0.139733 + 0.990189i $$0.544625\pi$$
$$270$$ −14.3262 −0.871867
$$271$$ 2.43769 0.148079 0.0740397 0.997255i $$-0.476411\pi$$
0.0740397 + 0.997255i $$0.476411\pi$$
$$272$$ 0.763932 0.0463202
$$273$$ −7.23607 −0.437947
$$274$$ −3.38197 −0.204312
$$275$$ −9.00000 −0.542720
$$276$$ 14.4721 0.871120
$$277$$ −25.4164 −1.52712 −0.763562 0.645735i $$-0.776551\pi$$
−0.763562 + 0.645735i $$0.776551\pi$$
$$278$$ −6.00000 −0.359856
$$279$$ −2.29180 −0.137206
$$280$$ 2.61803 0.156457
$$281$$ −23.4164 −1.39691 −0.698453 0.715656i $$-0.746128\pi$$
−0.698453 + 0.715656i $$0.746128\pi$$
$$282$$ 17.5623 1.04582
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 8.56231 0.508079
$$285$$ 0 0
$$286$$ −21.7082 −1.28363
$$287$$ −9.56231 −0.564445
$$288$$ 0.381966 0.0225076
$$289$$ −16.4164 −0.965671
$$290$$ 0.381966 0.0224298
$$291$$ 11.5623 0.677794
$$292$$ 0 0
$$293$$ 31.4164 1.83537 0.917683 0.397313i $$-0.130057\pi$$
0.917683 + 0.397313i $$0.130057\pi$$
$$294$$ −1.61803 −0.0943657
$$295$$ 30.2705 1.76242
$$296$$ −7.85410 −0.456510
$$297$$ 26.5623 1.54130
$$298$$ 17.2361 0.998459
$$299$$ −40.0000 −2.31326
$$300$$ 3.00000 0.173205
$$301$$ 3.85410 0.222147
$$302$$ −3.05573 −0.175837
$$303$$ 4.76393 0.273681
$$304$$ 0 0
$$305$$ 22.4164 1.28356
$$306$$ 0.291796 0.0166809
$$307$$ 18.9787 1.08317 0.541586 0.840645i $$-0.317824\pi$$
0.541586 + 0.840645i $$0.317824\pi$$
$$308$$ −4.85410 −0.276588
$$309$$ 14.4721 0.823291
$$310$$ 15.7082 0.892166
$$311$$ −21.3262 −1.20930 −0.604650 0.796491i $$-0.706687\pi$$
−0.604650 + 0.796491i $$0.706687\pi$$
$$312$$ 7.23607 0.409662
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ −2.85410 −0.161066
$$315$$ 1.00000 0.0563436
$$316$$ 0.326238 0.0183523
$$317$$ −25.6869 −1.44272 −0.721361 0.692560i $$-0.756483\pi$$
−0.721361 + 0.692560i $$0.756483\pi$$
$$318$$ 8.32624 0.466912
$$319$$ −0.708204 −0.0396518
$$320$$ −2.61803 −0.146353
$$321$$ 31.8885 1.77984
$$322$$ −8.94427 −0.498445
$$323$$ 0 0
$$324$$ −7.70820 −0.428234
$$325$$ −8.29180 −0.459946
$$326$$ 20.2705 1.12268
$$327$$ 17.5623 0.971198
$$328$$ 9.56231 0.527990
$$329$$ −10.8541 −0.598406
$$330$$ −20.5623 −1.13192
$$331$$ −3.70820 −0.203821 −0.101911 0.994794i $$-0.532496\pi$$
−0.101911 + 0.994794i $$0.532496\pi$$
$$332$$ 11.2361 0.616659
$$333$$ −3.00000 −0.164399
$$334$$ 10.0000 0.547176
$$335$$ −13.7082 −0.748959
$$336$$ 1.61803 0.0882710
$$337$$ 9.70820 0.528840 0.264420 0.964408i $$-0.414820\pi$$
0.264420 + 0.964408i $$0.414820\pi$$
$$338$$ −7.00000 −0.380750
$$339$$ 16.1803 0.878795
$$340$$ −2.00000 −0.108465
$$341$$ −29.1246 −1.57719
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −3.85410 −0.207799
$$345$$ −37.8885 −2.03985
$$346$$ −18.0000 −0.967686
$$347$$ 8.94427 0.480154 0.240077 0.970754i $$-0.422827\pi$$
0.240077 + 0.970754i $$0.422827\pi$$
$$348$$ 0.236068 0.0126546
$$349$$ −16.8328 −0.901040 −0.450520 0.892766i $$-0.648761\pi$$
−0.450520 + 0.892766i $$0.648761\pi$$
$$350$$ −1.85410 −0.0991059
$$351$$ 24.4721 1.30623
$$352$$ 4.85410 0.258725
$$353$$ 13.4164 0.714083 0.357042 0.934088i $$-0.383785\pi$$
0.357042 + 0.934088i $$0.383785\pi$$
$$354$$ 18.7082 0.994330
$$355$$ −22.4164 −1.18974
$$356$$ −3.14590 −0.166732
$$357$$ 1.23607 0.0654197
$$358$$ −17.7082 −0.935908
$$359$$ 2.29180 0.120956 0.0604782 0.998170i $$-0.480737\pi$$
0.0604782 + 0.998170i $$0.480737\pi$$
$$360$$ −1.00000 −0.0527046
$$361$$ 0 0
$$362$$ 7.41641 0.389798
$$363$$ 20.3262 1.06685
$$364$$ −4.47214 −0.234404
$$365$$ 0 0
$$366$$ 13.8541 0.724166
$$367$$ −6.27051 −0.327318 −0.163659 0.986517i $$-0.552330\pi$$
−0.163659 + 0.986517i $$0.552330\pi$$
$$368$$ 8.94427 0.466252
$$369$$ 3.65248 0.190140
$$370$$ 20.5623 1.06898
$$371$$ −5.14590 −0.267162
$$372$$ 9.70820 0.503347
$$373$$ 14.6180 0.756893 0.378447 0.925623i $$-0.376458\pi$$
0.378447 + 0.925623i $$0.376458\pi$$
$$374$$ 3.70820 0.191747
$$375$$ 13.3262 0.688164
$$376$$ 10.8541 0.559758
$$377$$ −0.652476 −0.0336042
$$378$$ 5.47214 0.281456
$$379$$ 32.1803 1.65299 0.826497 0.562942i $$-0.190331\pi$$
0.826497 + 0.562942i $$0.190331\pi$$
$$380$$ 0 0
$$381$$ 35.7426 1.83115
$$382$$ −14.9443 −0.764615
$$383$$ −13.1246 −0.670636 −0.335318 0.942105i $$-0.608844\pi$$
−0.335318 + 0.942105i $$0.608844\pi$$
$$384$$ −1.61803 −0.0825700
$$385$$ 12.7082 0.647670
$$386$$ 20.9443 1.06604
$$387$$ −1.47214 −0.0748329
$$388$$ 7.14590 0.362778
$$389$$ −37.3050 −1.89144 −0.945718 0.324988i $$-0.894640\pi$$
−0.945718 + 0.324988i $$0.894640\pi$$
$$390$$ −18.9443 −0.959280
$$391$$ 6.83282 0.345550
$$392$$ −1.00000 −0.0505076
$$393$$ −7.23607 −0.365011
$$394$$ 5.23607 0.263789
$$395$$ −0.854102 −0.0429745
$$396$$ 1.85410 0.0931721
$$397$$ −28.5623 −1.43350 −0.716751 0.697330i $$-0.754371\pi$$
−0.716751 + 0.697330i $$0.754371\pi$$
$$398$$ −1.56231 −0.0783113
$$399$$ 0 0
$$400$$ 1.85410 0.0927051
$$401$$ −25.4164 −1.26923 −0.634617 0.772826i $$-0.718842\pi$$
−0.634617 + 0.772826i $$0.718842\pi$$
$$402$$ −8.47214 −0.422552
$$403$$ −26.8328 −1.33664
$$404$$ 2.94427 0.146483
$$405$$ 20.1803 1.00277
$$406$$ −0.145898 −0.00724080
$$407$$ −38.1246 −1.88977
$$408$$ −1.23607 −0.0611945
$$409$$ −1.79837 −0.0889239 −0.0444619 0.999011i $$-0.514157\pi$$
−0.0444619 + 0.999011i $$0.514157\pi$$
$$410$$ −25.0344 −1.23636
$$411$$ 5.47214 0.269921
$$412$$ 8.94427 0.440653
$$413$$ −11.5623 −0.568944
$$414$$ 3.41641 0.167907
$$415$$ −29.4164 −1.44399
$$416$$ 4.47214 0.219265
$$417$$ 9.70820 0.475413
$$418$$ 0 0
$$419$$ −12.6525 −0.618114 −0.309057 0.951044i $$-0.600013\pi$$
−0.309057 + 0.951044i $$0.600013\pi$$
$$420$$ −4.23607 −0.206699
$$421$$ 19.5279 0.951730 0.475865 0.879518i $$-0.342135\pi$$
0.475865 + 0.879518i $$0.342135\pi$$
$$422$$ −17.2361 −0.839039
$$423$$ 4.14590 0.201580
$$424$$ 5.14590 0.249907
$$425$$ 1.41641 0.0687059
$$426$$ −13.8541 −0.671233
$$427$$ −8.56231 −0.414359
$$428$$ 19.7082 0.952632
$$429$$ 35.1246 1.69583
$$430$$ 10.0902 0.486591
$$431$$ 22.8541 1.10084 0.550422 0.834887i $$-0.314467\pi$$
0.550422 + 0.834887i $$0.314467\pi$$
$$432$$ −5.47214 −0.263278
$$433$$ −3.38197 −0.162527 −0.0812635 0.996693i $$-0.525896\pi$$
−0.0812635 + 0.996693i $$0.525896\pi$$
$$434$$ −6.00000 −0.288009
$$435$$ −0.618034 −0.0296325
$$436$$ 10.8541 0.519817
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −2.18034 −0.104062 −0.0520310 0.998645i $$-0.516569\pi$$
−0.0520310 + 0.998645i $$0.516569\pi$$
$$440$$ −12.7082 −0.605840
$$441$$ −0.381966 −0.0181889
$$442$$ 3.41641 0.162502
$$443$$ 1.09017 0.0517955 0.0258978 0.999665i $$-0.491756\pi$$
0.0258978 + 0.999665i $$0.491756\pi$$
$$444$$ 12.7082 0.603105
$$445$$ 8.23607 0.390427
$$446$$ 5.23607 0.247935
$$447$$ −27.8885 −1.31908
$$448$$ 1.00000 0.0472456
$$449$$ −0.291796 −0.0137707 −0.00688535 0.999976i $$-0.502192\pi$$
−0.00688535 + 0.999976i $$0.502192\pi$$
$$450$$ 0.708204 0.0333851
$$451$$ 46.4164 2.18566
$$452$$ 10.0000 0.470360
$$453$$ 4.94427 0.232302
$$454$$ −19.4164 −0.911257
$$455$$ 11.7082 0.548889
$$456$$ 0 0
$$457$$ 31.1459 1.45694 0.728472 0.685076i $$-0.240231\pi$$
0.728472 + 0.685076i $$0.240231\pi$$
$$458$$ −23.8541 −1.11463
$$459$$ −4.18034 −0.195122
$$460$$ −23.4164 −1.09180
$$461$$ −41.5066 −1.93315 −0.966577 0.256376i $$-0.917471\pi$$
−0.966577 + 0.256376i $$0.917471\pi$$
$$462$$ 7.85410 0.365406
$$463$$ −12.0000 −0.557687 −0.278844 0.960337i $$-0.589951\pi$$
−0.278844 + 0.960337i $$0.589951\pi$$
$$464$$ 0.145898 0.00677315
$$465$$ −25.4164 −1.17866
$$466$$ −4.09017 −0.189473
$$467$$ −40.3607 −1.86767 −0.933835 0.357705i $$-0.883559\pi$$
−0.933835 + 0.357705i $$0.883559\pi$$
$$468$$ 1.70820 0.0789618
$$469$$ 5.23607 0.241779
$$470$$ −28.4164 −1.31075
$$471$$ 4.61803 0.212788
$$472$$ 11.5623 0.532198
$$473$$ −18.7082 −0.860204
$$474$$ −0.527864 −0.0242456
$$475$$ 0 0
$$476$$ 0.763932 0.0350148
$$477$$ 1.96556 0.0899967
$$478$$ −14.1803 −0.648594
$$479$$ 36.3262 1.65979 0.829894 0.557921i $$-0.188401\pi$$
0.829894 + 0.557921i $$0.188401\pi$$
$$480$$ 4.23607 0.193349
$$481$$ −35.1246 −1.60154
$$482$$ −17.5623 −0.799941
$$483$$ 14.4721 0.658505
$$484$$ 12.5623 0.571014
$$485$$ −18.7082 −0.849496
$$486$$ −3.94427 −0.178916
$$487$$ 16.9098 0.766258 0.383129 0.923695i $$-0.374847\pi$$
0.383129 + 0.923695i $$0.374847\pi$$
$$488$$ 8.56231 0.387597
$$489$$ −32.7984 −1.48319
$$490$$ 2.61803 0.118271
$$491$$ −40.3607 −1.82145 −0.910726 0.413011i $$-0.864477\pi$$
−0.910726 + 0.413011i $$0.864477\pi$$
$$492$$ −15.4721 −0.697538
$$493$$ 0.111456 0.00501973
$$494$$ 0 0
$$495$$ −4.85410 −0.218176
$$496$$ 6.00000 0.269408
$$497$$ 8.56231 0.384072
$$498$$ −18.1803 −0.814681
$$499$$ −18.6869 −0.836541 −0.418271 0.908322i $$-0.637364\pi$$
−0.418271 + 0.908322i $$0.637364\pi$$
$$500$$ 8.23607 0.368328
$$501$$ −16.1803 −0.722884
$$502$$ −9.05573 −0.404177
$$503$$ 30.2148 1.34721 0.673605 0.739091i $$-0.264745\pi$$
0.673605 + 0.739091i $$0.264745\pi$$
$$504$$ 0.381966 0.0170141
$$505$$ −7.70820 −0.343011
$$506$$ 43.4164 1.93009
$$507$$ 11.3262 0.503016
$$508$$ 22.0902 0.980093
$$509$$ −35.1246 −1.55687 −0.778436 0.627725i $$-0.783986\pi$$
−0.778436 + 0.627725i $$0.783986\pi$$
$$510$$ 3.23607 0.143295
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −9.85410 −0.434646
$$515$$ −23.4164 −1.03185
$$516$$ 6.23607 0.274528
$$517$$ 52.6869 2.31717
$$518$$ −7.85410 −0.345089
$$519$$ 29.1246 1.27843
$$520$$ −11.7082 −0.513439
$$521$$ −21.4164 −0.938270 −0.469135 0.883127i $$-0.655434\pi$$
−0.469135 + 0.883127i $$0.655434\pi$$
$$522$$ 0.0557281 0.00243915
$$523$$ 24.0000 1.04945 0.524723 0.851273i $$-0.324169\pi$$
0.524723 + 0.851273i $$0.324169\pi$$
$$524$$ −4.47214 −0.195366
$$525$$ 3.00000 0.130931
$$526$$ −12.6525 −0.551674
$$527$$ 4.58359 0.199664
$$528$$ −7.85410 −0.341806
$$529$$ 57.0000 2.47826
$$530$$ −13.4721 −0.585192
$$531$$ 4.41641 0.191656
$$532$$ 0 0
$$533$$ 42.7639 1.85231
$$534$$ 5.09017 0.220273
$$535$$ −51.5967 −2.23072
$$536$$ −5.23607 −0.226164
$$537$$ 28.6525 1.23645
$$538$$ 4.58359 0.197613
$$539$$ −4.85410 −0.209081
$$540$$ 14.3262 0.616503
$$541$$ 19.1246 0.822231 0.411116 0.911583i $$-0.365139\pi$$
0.411116 + 0.911583i $$0.365139\pi$$
$$542$$ −2.43769 −0.104708
$$543$$ −12.0000 −0.514969
$$544$$ −0.763932 −0.0327533
$$545$$ −28.4164 −1.21723
$$546$$ 7.23607 0.309675
$$547$$ 12.6525 0.540981 0.270490 0.962723i $$-0.412814\pi$$
0.270490 + 0.962723i $$0.412814\pi$$
$$548$$ 3.38197 0.144470
$$549$$ 3.27051 0.139582
$$550$$ 9.00000 0.383761
$$551$$ 0 0
$$552$$ −14.4721 −0.615975
$$553$$ 0.326238 0.0138730
$$554$$ 25.4164 1.07984
$$555$$ −33.2705 −1.41225
$$556$$ 6.00000 0.254457
$$557$$ −32.0689 −1.35880 −0.679401 0.733767i $$-0.737760\pi$$
−0.679401 + 0.733767i $$0.737760\pi$$
$$558$$ 2.29180 0.0970195
$$559$$ −17.2361 −0.729008
$$560$$ −2.61803 −0.110632
$$561$$ −6.00000 −0.253320
$$562$$ 23.4164 0.987762
$$563$$ −6.56231 −0.276568 −0.138284 0.990393i $$-0.544159\pi$$
−0.138284 + 0.990393i $$0.544159\pi$$
$$564$$ −17.5623 −0.739506
$$565$$ −26.1803 −1.10142
$$566$$ −8.00000 −0.336265
$$567$$ −7.70820 −0.323714
$$568$$ −8.56231 −0.359266
$$569$$ −25.7082 −1.07774 −0.538872 0.842388i $$-0.681149\pi$$
−0.538872 + 0.842388i $$0.681149\pi$$
$$570$$ 0 0
$$571$$ 15.6869 0.656477 0.328239 0.944595i $$-0.393545\pi$$
0.328239 + 0.944595i $$0.393545\pi$$
$$572$$ 21.7082 0.907666
$$573$$ 24.1803 1.01015
$$574$$ 9.56231 0.399123
$$575$$ 16.5836 0.691584
$$576$$ −0.381966 −0.0159153
$$577$$ −13.1246 −0.546385 −0.273192 0.961959i $$-0.588080\pi$$
−0.273192 + 0.961959i $$0.588080\pi$$
$$578$$ 16.4164 0.682833
$$579$$ −33.8885 −1.40836
$$580$$ −0.381966 −0.0158603
$$581$$ 11.2361 0.466151
$$582$$ −11.5623 −0.479273
$$583$$ 24.9787 1.03451
$$584$$ 0 0
$$585$$ −4.47214 −0.184900
$$586$$ −31.4164 −1.29780
$$587$$ 19.5279 0.806001 0.403001 0.915200i $$-0.367967\pi$$
0.403001 + 0.915200i $$0.367967\pi$$
$$588$$ 1.61803 0.0667266
$$589$$ 0 0
$$590$$ −30.2705 −1.24622
$$591$$ −8.47214 −0.348497
$$592$$ 7.85410 0.322802
$$593$$ 16.4721 0.676430 0.338215 0.941069i $$-0.390177\pi$$
0.338215 + 0.941069i $$0.390177\pi$$
$$594$$ −26.5623 −1.08986
$$595$$ −2.00000 −0.0819920
$$596$$ −17.2361 −0.706017
$$597$$ 2.52786 0.103459
$$598$$ 40.0000 1.63572
$$599$$ 11.9787 0.489437 0.244718 0.969594i $$-0.421304\pi$$
0.244718 + 0.969594i $$0.421304\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ 10.3607 0.422621 0.211310 0.977419i $$-0.432227\pi$$
0.211310 + 0.977419i $$0.432227\pi$$
$$602$$ −3.85410 −0.157081
$$603$$ −2.00000 −0.0814463
$$604$$ 3.05573 0.124336
$$605$$ −32.8885 −1.33711
$$606$$ −4.76393 −0.193522
$$607$$ 16.3607 0.664060 0.332030 0.943269i $$-0.392267\pi$$
0.332030 + 0.943269i $$0.392267\pi$$
$$608$$ 0 0
$$609$$ 0.236068 0.00956596
$$610$$ −22.4164 −0.907614
$$611$$ 48.5410 1.96376
$$612$$ −0.291796 −0.0117952
$$613$$ 12.2918 0.496461 0.248230 0.968701i $$-0.420151\pi$$
0.248230 + 0.968701i $$0.420151\pi$$
$$614$$ −18.9787 −0.765919
$$615$$ 40.5066 1.63338
$$616$$ 4.85410 0.195577
$$617$$ 33.9230 1.36569 0.682844 0.730564i $$-0.260743\pi$$
0.682844 + 0.730564i $$0.260743\pi$$
$$618$$ −14.4721 −0.582155
$$619$$ −35.4164 −1.42351 −0.711753 0.702430i $$-0.752098\pi$$
−0.711753 + 0.702430i $$0.752098\pi$$
$$620$$ −15.7082 −0.630857
$$621$$ −48.9443 −1.96407
$$622$$ 21.3262 0.855104
$$623$$ −3.14590 −0.126038
$$624$$ −7.23607 −0.289675
$$625$$ −30.8328 −1.23331
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ 2.85410 0.113891
$$629$$ 6.00000 0.239236
$$630$$ −1.00000 −0.0398410
$$631$$ 22.8328 0.908960 0.454480 0.890757i $$-0.349825\pi$$
0.454480 + 0.890757i $$0.349825\pi$$
$$632$$ −0.326238 −0.0129770
$$633$$ 27.8885 1.10847
$$634$$ 25.6869 1.02016
$$635$$ −57.8328 −2.29503
$$636$$ −8.32624 −0.330157
$$637$$ −4.47214 −0.177192
$$638$$ 0.708204 0.0280381
$$639$$ −3.27051 −0.129379
$$640$$ 2.61803 0.103487
$$641$$ −7.41641 −0.292930 −0.146465 0.989216i $$-0.546790\pi$$
−0.146465 + 0.989216i $$0.546790\pi$$
$$642$$ −31.8885 −1.25854
$$643$$ 21.4164 0.844581 0.422290 0.906461i $$-0.361226\pi$$
0.422290 + 0.906461i $$0.361226\pi$$
$$644$$ 8.94427 0.352454
$$645$$ −16.3262 −0.642845
$$646$$ 0 0
$$647$$ −0.326238 −0.0128257 −0.00641287 0.999979i $$-0.502041\pi$$
−0.00641287 + 0.999979i $$0.502041\pi$$
$$648$$ 7.70820 0.302807
$$649$$ 56.1246 2.20308
$$650$$ 8.29180 0.325231
$$651$$ 9.70820 0.380495
$$652$$ −20.2705 −0.793854
$$653$$ −13.5279 −0.529386 −0.264693 0.964333i $$-0.585271\pi$$
−0.264693 + 0.964333i $$0.585271\pi$$
$$654$$ −17.5623 −0.686741
$$655$$ 11.7082 0.457477
$$656$$ −9.56231 −0.373345
$$657$$ 0 0
$$658$$ 10.8541 0.423137
$$659$$ −0.875388 −0.0341003 −0.0170501 0.999855i $$-0.505427\pi$$
−0.0170501 + 0.999855i $$0.505427\pi$$
$$660$$ 20.5623 0.800387
$$661$$ −26.8328 −1.04368 −0.521838 0.853045i $$-0.674753\pi$$
−0.521838 + 0.853045i $$0.674753\pi$$
$$662$$ 3.70820 0.144123
$$663$$ −5.52786 −0.214684
$$664$$ −11.2361 −0.436044
$$665$$ 0 0
$$666$$ 3.00000 0.116248
$$667$$ 1.30495 0.0505279
$$668$$ −10.0000 −0.386912
$$669$$ −8.47214 −0.327552
$$670$$ 13.7082 0.529594
$$671$$ 41.5623 1.60450
$$672$$ −1.61803 −0.0624170
$$673$$ −3.05573 −0.117790 −0.0588948 0.998264i $$-0.518758\pi$$
−0.0588948 + 0.998264i $$0.518758\pi$$
$$674$$ −9.70820 −0.373946
$$675$$ −10.1459 −0.390516
$$676$$ 7.00000 0.269231
$$677$$ −41.7082 −1.60298 −0.801488 0.598011i $$-0.795958\pi$$
−0.801488 + 0.598011i $$0.795958\pi$$
$$678$$ −16.1803 −0.621402
$$679$$ 7.14590 0.274234
$$680$$ 2.00000 0.0766965
$$681$$ 31.4164 1.20388
$$682$$ 29.1246 1.11524
$$683$$ −2.29180 −0.0876931 −0.0438466 0.999038i $$-0.513961\pi$$
−0.0438466 + 0.999038i $$0.513961\pi$$
$$684$$ 0 0
$$685$$ −8.85410 −0.338298
$$686$$ −1.00000 −0.0381802
$$687$$ 38.5967 1.47256
$$688$$ 3.85410 0.146936
$$689$$ 23.0132 0.876731
$$690$$ 37.8885 1.44239
$$691$$ 20.5410 0.781417 0.390709 0.920514i $$-0.372230\pi$$
0.390709 + 0.920514i $$0.372230\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 1.85410 0.0704315
$$694$$ −8.94427 −0.339520
$$695$$ −15.7082 −0.595846
$$696$$ −0.236068 −0.00894813
$$697$$ −7.30495 −0.276695
$$698$$ 16.8328 0.637131
$$699$$ 6.61803 0.250317
$$700$$ 1.85410 0.0700785
$$701$$ −5.23607 −0.197764 −0.0988818 0.995099i $$-0.531527\pi$$
−0.0988818 + 0.995099i $$0.531527\pi$$
$$702$$ −24.4721 −0.923641
$$703$$ 0 0
$$704$$ −4.85410 −0.182946
$$705$$ 45.9787 1.73166
$$706$$ −13.4164 −0.504933
$$707$$ 2.94427 0.110731
$$708$$ −18.7082 −0.703097
$$709$$ 9.12461 0.342682 0.171341 0.985212i $$-0.445190\pi$$
0.171341 + 0.985212i $$0.445190\pi$$
$$710$$ 22.4164 0.841273
$$711$$ −0.124612 −0.00467331
$$712$$ 3.14590 0.117898
$$713$$ 53.6656 2.00979
$$714$$ −1.23607 −0.0462587
$$715$$ −56.8328 −2.12543
$$716$$ 17.7082 0.661787
$$717$$ 22.9443 0.856870
$$718$$ −2.29180 −0.0855291
$$719$$ 16.3607 0.610150 0.305075 0.952328i $$-0.401318\pi$$
0.305075 + 0.952328i $$0.401318\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 8.94427 0.333102
$$722$$ 0 0
$$723$$ 28.4164 1.05682
$$724$$ −7.41641 −0.275629
$$725$$ 0.270510 0.0100465
$$726$$ −20.3262 −0.754377
$$727$$ 28.5623 1.05932 0.529659 0.848211i $$-0.322320\pi$$
0.529659 + 0.848211i $$0.322320\pi$$
$$728$$ 4.47214 0.165748
$$729$$ 29.5066 1.09284
$$730$$ 0 0
$$731$$ 2.94427 0.108898
$$732$$ −13.8541 −0.512062
$$733$$ −19.5623 −0.722550 −0.361275 0.932459i $$-0.617658\pi$$
−0.361275 + 0.932459i $$0.617658\pi$$
$$734$$ 6.27051 0.231449
$$735$$ −4.23607 −0.156250
$$736$$ −8.94427 −0.329690
$$737$$ −25.4164 −0.936225
$$738$$ −3.65248 −0.134449
$$739$$ 35.2705 1.29745 0.648723 0.761024i $$-0.275303\pi$$
0.648723 + 0.761024i $$0.275303\pi$$
$$740$$ −20.5623 −0.755885
$$741$$ 0 0
$$742$$ 5.14590 0.188912
$$743$$ 24.9787 0.916380 0.458190 0.888854i $$-0.348498\pi$$
0.458190 + 0.888854i $$0.348498\pi$$
$$744$$ −9.70820 −0.355920
$$745$$ 45.1246 1.65324
$$746$$ −14.6180 −0.535204
$$747$$ −4.29180 −0.157029
$$748$$ −3.70820 −0.135585
$$749$$ 19.7082 0.720122
$$750$$ −13.3262 −0.486605
$$751$$ −0.326238 −0.0119046 −0.00595230 0.999982i $$-0.501895\pi$$
−0.00595230 + 0.999982i $$0.501895\pi$$
$$752$$ −10.8541 −0.395808
$$753$$ 14.6525 0.533966
$$754$$ 0.652476 0.0237618
$$755$$ −8.00000 −0.291150
$$756$$ −5.47214 −0.199020
$$757$$ 46.2492 1.68096 0.840478 0.541845i $$-0.182274\pi$$
0.840478 + 0.541845i $$0.182274\pi$$
$$758$$ −32.1803 −1.16884
$$759$$ −70.2492 −2.54989
$$760$$ 0 0
$$761$$ −17.7771 −0.644419 −0.322209 0.946668i $$-0.604426\pi$$
−0.322209 + 0.946668i $$0.604426\pi$$
$$762$$ −35.7426 −1.29482
$$763$$ 10.8541 0.392945
$$764$$ 14.9443 0.540665
$$765$$ 0.763932 0.0276200
$$766$$ 13.1246 0.474212
$$767$$ 51.7082 1.86708
$$768$$ 1.61803 0.0583858
$$769$$ 12.2918 0.443254 0.221627 0.975132i $$-0.428863\pi$$
0.221627 + 0.975132i $$0.428863\pi$$
$$770$$ −12.7082 −0.457972
$$771$$ 15.9443 0.574219
$$772$$ −20.9443 −0.753801
$$773$$ 1.70820 0.0614398 0.0307199 0.999528i $$-0.490220\pi$$
0.0307199 + 0.999528i $$0.490220\pi$$
$$774$$ 1.47214 0.0529148
$$775$$ 11.1246 0.399608
$$776$$ −7.14590 −0.256523
$$777$$ 12.7082 0.455904
$$778$$ 37.3050 1.33745
$$779$$ 0 0
$$780$$ 18.9443 0.678314
$$781$$ −41.5623 −1.48722
$$782$$ −6.83282 −0.244341
$$783$$ −0.798374 −0.0285316
$$784$$ 1.00000 0.0357143
$$785$$ −7.47214 −0.266692
$$786$$ 7.23607 0.258102
$$787$$ −27.2705 −0.972089 −0.486044 0.873934i $$-0.661561\pi$$
−0.486044 + 0.873934i $$0.661561\pi$$
$$788$$ −5.23607 −0.186527
$$789$$ 20.4721 0.728827
$$790$$ 0.854102 0.0303876
$$791$$ 10.0000 0.355559
$$792$$ −1.85410 −0.0658826
$$793$$ 38.2918 1.35978
$$794$$ 28.5623 1.01364
$$795$$ 21.7984 0.773109
$$796$$ 1.56231 0.0553745
$$797$$ −44.8328 −1.58806 −0.794030 0.607879i $$-0.792021\pi$$
−0.794030 + 0.607879i $$0.792021\pi$$
$$798$$ 0 0
$$799$$ −8.29180 −0.293343
$$800$$ −1.85410 −0.0655524
$$801$$ 1.20163 0.0424574
$$802$$ 25.4164 0.897485
$$803$$ 0 0
$$804$$ 8.47214 0.298789
$$805$$ −23.4164 −0.825320
$$806$$ 26.8328 0.945146
$$807$$ −7.41641 −0.261070
$$808$$ −2.94427 −0.103579
$$809$$ −16.0344 −0.563741 −0.281870 0.959452i $$-0.590955\pi$$
−0.281870 + 0.959452i $$0.590955\pi$$
$$810$$ −20.1803 −0.709065
$$811$$ 2.02129 0.0709770 0.0354885 0.999370i $$-0.488701\pi$$
0.0354885 + 0.999370i $$0.488701\pi$$
$$812$$ 0.145898 0.00512002
$$813$$ 3.94427 0.138332
$$814$$ 38.1246 1.33627
$$815$$ 53.0689 1.85892
$$816$$ 1.23607 0.0432710
$$817$$ 0 0
$$818$$ 1.79837 0.0628787
$$819$$ 1.70820 0.0596895
$$820$$ 25.0344 0.874241
$$821$$ 6.87539 0.239953 0.119976 0.992777i $$-0.461718\pi$$
0.119976 + 0.992777i $$0.461718\pi$$
$$822$$ −5.47214 −0.190863
$$823$$ −2.29180 −0.0798870 −0.0399435 0.999202i $$-0.512718\pi$$
−0.0399435 + 0.999202i $$0.512718\pi$$
$$824$$ −8.94427 −0.311588
$$825$$ −14.5623 −0.506994
$$826$$ 11.5623 0.402304
$$827$$ −43.9574 −1.52855 −0.764275 0.644891i $$-0.776903\pi$$
−0.764275 + 0.644891i $$0.776903\pi$$
$$828$$ −3.41641 −0.118728
$$829$$ −35.1246 −1.21993 −0.609964 0.792429i $$-0.708816\pi$$
−0.609964 + 0.792429i $$0.708816\pi$$
$$830$$ 29.4164 1.02106
$$831$$ −41.1246 −1.42660
$$832$$ −4.47214 −0.155043
$$833$$ 0.763932 0.0264687
$$834$$ −9.70820 −0.336168
$$835$$ 26.1803 0.906008
$$836$$ 0 0
$$837$$ −32.8328 −1.13487
$$838$$ 12.6525 0.437073
$$839$$ −4.83282 −0.166847 −0.0834237 0.996514i $$-0.526585\pi$$
−0.0834237 + 0.996514i $$0.526585\pi$$
$$840$$ 4.23607 0.146158
$$841$$ −28.9787 −0.999266
$$842$$ −19.5279 −0.672975
$$843$$ −37.8885 −1.30495
$$844$$ 17.2361 0.593290
$$845$$ −18.3262 −0.630442
$$846$$ −4.14590 −0.142539
$$847$$ 12.5623 0.431646
$$848$$ −5.14590 −0.176711
$$849$$ 12.9443 0.444246
$$850$$ −1.41641 −0.0485824
$$851$$ 70.2492 2.40811
$$852$$ 13.8541 0.474634
$$853$$ −23.2705 −0.796767 −0.398384 0.917219i $$-0.630429\pi$$
−0.398384 + 0.917219i $$0.630429\pi$$
$$854$$ 8.56231 0.292996
$$855$$ 0 0
$$856$$ −19.7082 −0.673613
$$857$$ 32.2492 1.10161 0.550806 0.834633i $$-0.314320\pi$$
0.550806 + 0.834633i $$0.314320\pi$$
$$858$$ −35.1246 −1.19913
$$859$$ −34.0000 −1.16007 −0.580033 0.814593i $$-0.696960\pi$$
−0.580033 + 0.814593i $$0.696960\pi$$
$$860$$ −10.0902 −0.344072
$$861$$ −15.4721 −0.527289
$$862$$ −22.8541 −0.778414
$$863$$ 48.9787 1.66725 0.833627 0.552327i $$-0.186260\pi$$
0.833627 + 0.552327i $$0.186260\pi$$
$$864$$ 5.47214 0.186166
$$865$$ −47.1246 −1.60228
$$866$$ 3.38197 0.114924
$$867$$ −26.5623 −0.902103
$$868$$ 6.00000 0.203653
$$869$$ −1.58359 −0.0537197
$$870$$ 0.618034 0.0209533
$$871$$ −23.4164 −0.793435
$$872$$ −10.8541 −0.367566
$$873$$ −2.72949 −0.0923792
$$874$$ 0 0
$$875$$ 8.23607 0.278430
$$876$$ 0 0
$$877$$ −47.3951 −1.60042 −0.800210 0.599720i $$-0.795279\pi$$
−0.800210 + 0.599720i $$0.795279\pi$$
$$878$$ 2.18034 0.0735829
$$879$$ 50.8328 1.71455
$$880$$ 12.7082 0.428393
$$881$$ 9.81966 0.330833 0.165416 0.986224i $$-0.447103\pi$$
0.165416 + 0.986224i $$0.447103\pi$$
$$882$$ 0.381966 0.0128615
$$883$$ −27.6869 −0.931739 −0.465869 0.884853i $$-0.654258\pi$$
−0.465869 + 0.884853i $$0.654258\pi$$
$$884$$ −3.41641 −0.114906
$$885$$ 48.9787 1.64640
$$886$$ −1.09017 −0.0366250
$$887$$ 2.83282 0.0951166 0.0475583 0.998868i $$-0.484856\pi$$
0.0475583 + 0.998868i $$0.484856\pi$$
$$888$$ −12.7082 −0.426459
$$889$$ 22.0902 0.740881
$$890$$ −8.23607 −0.276074
$$891$$ 37.4164 1.25350
$$892$$ −5.23607 −0.175317
$$893$$ 0 0
$$894$$ 27.8885 0.932732
$$895$$ −46.3607 −1.54967
$$896$$ −1.00000 −0.0334077
$$897$$ −64.7214 −2.16098
$$898$$ 0.291796 0.00973736
$$899$$ 0.875388 0.0291958
$$900$$ −0.708204 −0.0236068
$$901$$ −3.93112 −0.130964
$$902$$ −46.4164 −1.54550
$$903$$ 6.23607 0.207523
$$904$$ −10.0000 −0.332595
$$905$$ 19.4164 0.645423
$$906$$ −4.94427 −0.164262
$$907$$ 48.0000 1.59381 0.796907 0.604102i $$-0.206468\pi$$
0.796907 + 0.604102i $$0.206468\pi$$
$$908$$ 19.4164 0.644356
$$909$$ −1.12461 −0.0373010
$$910$$ −11.7082 −0.388123
$$911$$ −49.8541 −1.65174 −0.825870 0.563861i $$-0.809316\pi$$
−0.825870 + 0.563861i $$0.809316\pi$$
$$912$$ 0 0
$$913$$ −54.5410 −1.80504
$$914$$ −31.1459 −1.03021
$$915$$ 36.2705 1.19907
$$916$$ 23.8541 0.788162
$$917$$ −4.47214 −0.147683
$$918$$ 4.18034 0.137972
$$919$$ −12.2918 −0.405469 −0.202734 0.979234i $$-0.564983\pi$$
−0.202734 + 0.979234i $$0.564983\pi$$
$$920$$ 23.4164 0.772016
$$921$$ 30.7082 1.01187
$$922$$ 41.5066 1.36695
$$923$$ −38.2918 −1.26039
$$924$$ −7.85410 −0.258381
$$925$$ 14.5623 0.478806
$$926$$ 12.0000 0.394344
$$927$$ −3.41641 −0.112210
$$928$$ −0.145898 −0.00478934
$$929$$ 42.7639 1.40304 0.701520 0.712650i $$-0.252505\pi$$
0.701520 + 0.712650i $$0.252505\pi$$
$$930$$ 25.4164 0.833437
$$931$$ 0 0
$$932$$ 4.09017 0.133978
$$933$$ −34.5066 −1.12969
$$934$$ 40.3607 1.32064
$$935$$ 9.70820 0.317492
$$936$$ −1.70820 −0.0558344
$$937$$ 13.7082 0.447828 0.223914 0.974609i $$-0.428117\pi$$
0.223914 + 0.974609i $$0.428117\pi$$
$$938$$ −5.23607 −0.170964
$$939$$ 16.1803 0.528025
$$940$$ 28.4164 0.926841
$$941$$ −33.7082 −1.09886 −0.549428 0.835541i $$-0.685154\pi$$
−0.549428 + 0.835541i $$0.685154\pi$$
$$942$$ −4.61803 −0.150464
$$943$$ −85.5279 −2.78517
$$944$$ −11.5623 −0.376321
$$945$$ 14.3262 0.466033
$$946$$ 18.7082 0.608256
$$947$$ −43.1459 −1.40205 −0.701027 0.713135i $$-0.747275\pi$$
−0.701027 + 0.713135i $$0.747275\pi$$
$$948$$ 0.527864 0.0171442
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −41.5623 −1.34775
$$952$$ −0.763932 −0.0247592
$$953$$ 2.29180 0.0742386 0.0371193 0.999311i $$-0.488182\pi$$
0.0371193 + 0.999311i $$0.488182\pi$$
$$954$$ −1.96556 −0.0636373
$$955$$ −39.1246 −1.26604
$$956$$ 14.1803 0.458625
$$957$$ −1.14590 −0.0370416
$$958$$ −36.3262 −1.17365
$$959$$ 3.38197 0.109209
$$960$$ −4.23607 −0.136719
$$961$$ 5.00000 0.161290
$$962$$ 35.1246 1.13246
$$963$$ −7.52786 −0.242582
$$964$$ 17.5623 0.565644
$$965$$ 54.8328 1.76513
$$966$$ −14.4721 −0.465633
$$967$$ 54.2492 1.74454 0.872269 0.489027i $$-0.162648\pi$$
0.872269 + 0.489027i $$0.162648\pi$$
$$968$$ −12.5623 −0.403768
$$969$$ 0 0
$$970$$ 18.7082 0.600684
$$971$$ 11.7295 0.376417 0.188209 0.982129i $$-0.439732\pi$$
0.188209 + 0.982129i $$0.439732\pi$$
$$972$$ 3.94427 0.126513
$$973$$ 6.00000 0.192351
$$974$$ −16.9098 −0.541826
$$975$$ −13.4164 −0.429669
$$976$$ −8.56231 −0.274073
$$977$$ 35.7082 1.14241 0.571203 0.820809i $$-0.306477\pi$$
0.571203 + 0.820809i $$0.306477\pi$$
$$978$$ 32.7984 1.04878
$$979$$ 15.2705 0.488048
$$980$$ −2.61803 −0.0836300
$$981$$ −4.14590 −0.132368
$$982$$ 40.3607 1.28796
$$983$$ 19.1246 0.609980 0.304990 0.952355i $$-0.401347\pi$$
0.304990 + 0.952355i $$0.401347\pi$$
$$984$$ 15.4721 0.493234
$$985$$ 13.7082 0.436780
$$986$$ −0.111456 −0.00354949
$$987$$ −17.5623 −0.559014
$$988$$ 0 0
$$989$$ 34.4721 1.09615
$$990$$ 4.85410 0.154273
$$991$$ 47.3951 1.50556 0.752778 0.658275i $$-0.228713\pi$$
0.752778 + 0.658275i $$0.228713\pi$$
$$992$$ −6.00000 −0.190500
$$993$$ −6.00000 −0.190404
$$994$$ −8.56231 −0.271580
$$995$$ −4.09017 −0.129667
$$996$$ 18.1803 0.576066
$$997$$ −51.8541 −1.64224 −0.821118 0.570759i $$-0.806649\pi$$
−0.821118 + 0.570759i $$0.806649\pi$$
$$998$$ 18.6869 0.591524
$$999$$ −42.9787 −1.35979
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 5054.2.a.g.1.2 2
19.18 odd 2 5054.2.a.l.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.g.1.2 2 1.1 even 1 trivial
5054.2.a.l.1.1 yes 2 19.18 odd 2