# Properties

 Label 5054.2.a.l.1.1 Level $5054$ Weight $2$ Character 5054.1 Self dual yes Analytic conductor $40.356$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ 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 1.1 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 $$2 q + 2 q^{2} - q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} - 3 q^{10} - 3 q^{11} - q^{12} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 6 q^{17} - 3 q^{18} - 3 q^{20} - q^{21} - 3 q^{22} - q^{24} - 3 q^{25} + 2 q^{27} + 2 q^{28} - 7 q^{29} + 4 q^{30} - 12 q^{31} + 2 q^{32} + 9 q^{33} + 6 q^{34} - 3 q^{35} - 3 q^{36} - 9 q^{37} - 10 q^{39} - 3 q^{40} - q^{41} - q^{42} + q^{43} - 3 q^{44} + 2 q^{45} - 15 q^{47} - q^{48} + 2 q^{49} - 3 q^{50} + 2 q^{51} + 17 q^{53} + 2 q^{54} + 12 q^{55} + 2 q^{56} - 7 q^{58} + 3 q^{59} + 4 q^{60} + 3 q^{61} - 12 q^{62} - 3 q^{63} + 2 q^{64} - 10 q^{65} + 9 q^{66} - 6 q^{67} + 6 q^{68} - 20 q^{69} - 3 q^{70} + 3 q^{71} - 3 q^{72} - 9 q^{74} - 6 q^{75} - 3 q^{77} - 10 q^{78} + 15 q^{79} - 3 q^{80} - 2 q^{81} - q^{82} + 18 q^{83} - q^{84} - 4 q^{85} + q^{86} - 4 q^{87} - 3 q^{88} + 13 q^{89} + 2 q^{90} + 6 q^{93} - 15 q^{94} - q^{96} - 21 q^{97} + 2 q^{98} - 3 q^{99}+O(q^{100})$$ 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

## 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 $$-0.654696\pi$$
−0.467086 + 0.884212i $$0.654696\pi$$
$$4$$ 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 $$-0.287062\pi$$
0.620174 + 0.784465i $$0.287062\pi$$
$$14$$ 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 $$-0.504312\pi$$
−0.0135463 + 0.999908i $$0.504312\pi$$
$$30$$ 4.23607 0.773397
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ 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 $$-0.723394\pi$$
−0.645603 + 0.763673i $$0.723394\pi$$
$$38$$ 0 0
$$39$$ −7.23607 −1.15870
$$40$$ −2.61803 −0.413948
$$41$$ 9.56231 1.49338 0.746691 0.665171i $$-0.231642\pi$$
0.746691 + 0.665171i $$0.231642\pi$$
$$42$$ −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 $$-0.385018\pi$$
0.353422 + 0.935464i $$0.385018\pi$$
$$54$$ 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 $$-0.228779\pi$$
0.752642 + 0.658430i $$0.228779\pi$$
$$60$$ 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 $$-0.603630\pi$$
−0.319844 + 0.947470i $$0.603630\pi$$
$$68$$ 0.763932 0.0926404
$$69$$ −14.4721 −1.74224
$$70$$ −2.61803 −0.312915
$$71$$ −8.56231 −1.01616 −0.508079 0.861310i $$-0.669644\pi$$
−0.508079 + 0.861310i $$0.669644\pi$$
$$72$$ −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 $$-0.505842\pi$$
−0.0183523 + 0.999832i $$0.505842\pi$$
$$80$$ −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 $$-0.446678\pi$$
0.166732 + 0.986002i $$0.446678\pi$$
$$90$$ 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 $$-0.618172\pi$$
−0.362778 + 0.931876i $$0.618172\pi$$
$$98$$ 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 $$-0.645253\pi$$
−0.440653 + 0.897678i $$0.645253\pi$$
$$104$$ 4.47214 0.438529
$$105$$ 4.23607 0.413398
$$106$$ 5.14590 0.499814
$$107$$ −19.7082 −1.90526 −0.952632 0.304125i $$-0.901636\pi$$
−0.952632 + 0.304125i $$0.901636\pi$$
$$108$$ 5.47214 0.526557
$$109$$ −10.8541 −1.03963 −0.519817 0.854278i $$-0.674000\pi$$
−0.519817 + 0.854278i $$0.674000\pi$$
$$110$$ 12.7082 1.21168
$$111$$ 12.7082 1.20621
$$112$$ 1.00000 0.0944911
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 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 $$-0.936380\pi$$
−0.980093 + 0.198540i $$0.936380\pi$$
$$128$$ 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 $$-0.539680\pi$$
−0.124336 + 0.992240i $$0.539680\pi$$
$$152$$ 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 $$-0.373542\pi$$
0.386912 + 0.922117i $$0.373542\pi$$
$$168$$ −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 $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 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 $$-0.730202\pi$$
−0.661787 + 0.749692i $$0.730202\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 7.41641 0.551257 0.275629 0.961264i $$-0.411114\pi$$
0.275629 + 0.961264i $$0.411114\pi$$
$$182$$ 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 $$-0.228218\pi$$
0.753801 + 0.657103i $$0.228218\pi$$
$$194$$ −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 $$-0.702171\pi$$
−0.593290 + 0.804989i $$0.702171\pi$$
$$212$$ 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 $$-0.443905\pi$$
0.175317 + 0.984512i $$0.443905\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −0.708204 −0.0472136
$$226$$ −10.0000 −0.665190
$$227$$ −19.4164 −1.28871 −0.644356 0.764726i $$-0.722875\pi$$
−0.644356 + 0.764726i $$0.722875\pi$$
$$228$$ 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 $$-0.691372\pi$$
−0.565644 + 0.824650i $$0.691372\pi$$
$$242$$ 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 $$-0.599439\pi$$
−0.307341 + 0.951599i $$0.599439\pi$$
$$258$$ −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 $$-0.455375\pi$$
0.139733 + 0.990189i $$0.455375\pi$$
$$270$$ −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 $$-0.253872\pi$$
0.698453 + 0.715656i $$0.253872\pi$$
$$282$$ 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 $$-0.869943\pi$$
−0.917683 + 0.397313i $$0.869943\pi$$
$$294$$ −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 $$-0.682176\pi$$
−0.541586 + 0.840645i $$0.682176\pi$$
$$308$$ −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 $$-0.243517\pi$$
0.721361 + 0.692560i $$0.243517\pi$$
$$318$$ −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 $$-0.467504\pi$$
0.101911 + 0.994794i $$0.467504\pi$$
$$332$$ 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 $$-0.585180\pi$$
−0.264420 + 0.964408i $$0.585180\pi$$
$$338$$ 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 $$-0.623542\pi$$
−0.378447 + 0.925623i $$0.623542\pi$$
$$374$$ −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 $$-0.809669\pi$$
−0.826497 + 0.562942i $$0.809669\pi$$
$$380$$ 0 0
$$381$$ 35.7426 1.83115
$$382$$ 14.9443 0.764615
$$383$$ 13.1246 0.670636 0.335318 0.942105i $$-0.391156\pi$$
0.335318 + 0.942105i $$0.391156\pi$$
$$384$$ −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 $$-0.281158\pi$$
0.634617 + 0.772826i $$0.281158\pi$$
$$402$$ 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 $$-0.485843\pi$$
0.0444619 + 0.999011i $$0.485843\pi$$
$$410$$ −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 $$-0.657865\pi$$
−0.475865 + 0.879518i $$0.657865\pi$$
$$422$$ −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 $$-0.685533\pi$$
−0.550422 + 0.834887i $$0.685533\pi$$
$$432$$ 5.47214 0.263278
$$433$$ 3.38197 0.162527 0.0812635 0.996693i $$-0.474104\pi$$
0.0812635 + 0.996693i $$0.474104\pi$$
$$434$$ −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 $$-0.483431\pi$$
0.0520310 + 0.998645i $$0.483431\pi$$
$$440$$ 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 $$-0.497808\pi$$
0.00688535 + 0.999976i $$0.497808\pi$$
$$450$$ −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 $$-0.625153\pi$$
−0.383129 + 0.923695i $$0.625153\pi$$
$$488$$ −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 $$-0.216014\pi$$
0.778436 + 0.627725i $$0.216014\pi$$
$$510$$ 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 $$-0.344566\pi$$
0.469135 + 0.883127i $$0.344566\pi$$
$$522$$ 0.0557281 0.00243915
$$523$$ −24.0000 −1.04945 −0.524723 0.851273i $$-0.675831\pi$$
−0.524723 + 0.851273i $$0.675831\pi$$
$$524$$ −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 $$-0.587186\pi$$
−0.270490 + 0.962723i $$0.587186\pi$$
$$548$$ 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 $$-0.455841\pi$$
0.138284 + 0.990393i $$0.455841\pi$$
$$564$$ 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 $$-0.318851\pi$$
0.538872 + 0.842388i $$0.318851\pi$$
$$570$$ 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 $$-0.578696\pi$$
−0.244718 + 0.969594i $$0.578696\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −10.3607 −0.422621 −0.211310 0.977419i $$-0.567773\pi$$
−0.211310 + 0.977419i $$0.567773\pi$$
$$602$$ 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 $$-0.607733\pi$$
−0.332030 + 0.943269i $$0.607733\pi$$
$$608$$ 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 $$-0.453210\pi$$
0.146465 + 0.989216i $$0.453210\pi$$
$$642$$ 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 $$-0.494573\pi$$
0.0170501 + 0.999855i $$0.494573\pi$$
$$660$$ −20.5623 −0.800387
$$661$$ 26.8328 1.04368 0.521838 0.853045i $$-0.325247\pi$$
0.521838 + 0.853045i $$0.325247\pi$$
$$662$$ 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 $$-0.481242\pi$$
0.0588948 + 0.998264i $$0.481242\pi$$
$$674$$ −9.70820 −0.373946
$$675$$ 10.1459 0.390516
$$676$$ 7.00000 0.269231
$$677$$ 41.7082 1.60298 0.801488 0.598011i $$-0.204042\pi$$
0.801488 + 0.598011i $$0.204042\pi$$
$$678$$ 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 $$-0.486039\pi$$
0.0438466 + 0.999038i $$0.486039\pi$$
$$684$$ 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 $$-0.651502\pi$$
−0.458190 + 0.888854i $$0.651502\pi$$
$$744$$ 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 $$-0.498105\pi$$
0.00595230 + 0.999982i $$0.498105\pi$$
$$752$$ −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 $$-0.509780\pi$$
−0.0307199 + 0.999528i $$0.509780\pi$$
$$774$$ −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 $$-0.338439\pi$$
0.486044 + 0.873934i $$0.338439\pi$$
$$788$$ −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 $$-0.207979\pi$$
0.794030 + 0.607879i $$0.207979\pi$$
$$798$$ 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 $$-0.511299\pi$$
−0.0354885 + 0.999370i $$0.511299\pi$$
$$812$$ −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 $$-0.223097\pi$$
0.764275 + 0.644891i $$0.223097\pi$$
$$828$$ −3.41641 −0.118728
$$829$$ 35.1246 1.21993 0.609964 0.792429i $$-0.291184\pi$$
0.609964 + 0.792429i $$0.291184\pi$$
$$830$$ −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 $$-0.473415\pi$$
0.0834237 + 0.996514i $$0.473415\pi$$
$$840$$ 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 $$-0.685680\pi$$
−0.550806 + 0.834633i $$0.685680\pi$$
$$858$$ 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 $$-0.813740\pi$$
−0.833627 + 0.552327i $$0.813740\pi$$
$$864$$ 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 $$-0.204721\pi$$
0.800210 + 0.599720i $$0.204721\pi$$
$$878$$ 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 $$-0.515144\pi$$
−0.0475583 + 0.998868i $$0.515144\pi$$
$$888$$ 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 $$-0.793532\pi$$
−0.796907 + 0.604102i $$0.793532\pi$$
$$908$$ −19.4164 −0.644356
$$909$$ −1.12461 −0.0373010
$$910$$ −11.7082 −0.388123
$$911$$ 49.8541 1.65174 0.825870 0.563861i $$-0.190684\pi$$
0.825870 + 0.563861i $$0.190684\pi$$
$$912$$ 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 $$-0.314846\pi$$
0.549428 + 0.835541i $$0.314846\pi$$
$$942$$ −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 $$-0.511818\pi$$
−0.0371193 + 0.999311i $$0.511818\pi$$
$$954$$ −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 $$-0.560268\pi$$
−0.188209 + 0.982129i $$0.560268\pi$$
$$972$$ −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 $$-0.693523\pi$$
−0.571203 + 0.820809i $$0.693523\pi$$
$$978$$ 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 $$-0.598653\pi$$
−0.304990 + 0.952355i $$0.598653\pi$$
$$984$$ −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 $$-0.771287\pi$$
−0.752778 + 0.658275i $$0.771287\pi$$
$$992$$ −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.l.1.1 yes 2
19.18 odd 2 5054.2.a.g.1.2 2

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