Properties

 Label 525.2.a.j.1.2 Level $525$ Weight $2$ Character 525.1 Self dual yes Analytic conductor $4.192$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 525.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-1.48119$$ of defining polynomial Character $$\chi$$ $$=$$ 525.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} +1.00000 q^{7} +0.768452 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} -0.193937 q^{6} +1.00000 q^{7} +0.768452 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.96239 q^{12} -1.35026 q^{13} -0.193937 q^{14} +3.77575 q^{16} +3.35026 q^{17} -0.193937 q^{18} +5.35026 q^{19} +1.00000 q^{21} -0.387873 q^{22} -4.96239 q^{23} +0.768452 q^{24} +0.261865 q^{26} +1.00000 q^{27} -1.96239 q^{28} +7.92478 q^{29} +4.57452 q^{31} -2.26916 q^{32} +2.00000 q^{33} -0.649738 q^{34} -1.96239 q^{36} +0.775746 q^{37} -1.03761 q^{38} -1.35026 q^{39} +3.73813 q^{41} -0.193937 q^{42} +12.6253 q^{43} -3.92478 q^{44} +0.962389 q^{46} -9.92478 q^{47} +3.77575 q^{48} +1.00000 q^{49} +3.35026 q^{51} +2.64974 q^{52} -8.57452 q^{53} -0.193937 q^{54} +0.768452 q^{56} +5.35026 q^{57} -1.53690 q^{58} -8.62530 q^{59} -8.70052 q^{61} -0.887166 q^{62} +1.00000 q^{63} -7.11142 q^{64} -0.387873 q^{66} -9.92478 q^{67} -6.57452 q^{68} -4.96239 q^{69} +2.00000 q^{71} +0.768452 q^{72} +9.35026 q^{73} -0.150446 q^{74} -10.4993 q^{76} +2.00000 q^{77} +0.261865 q^{78} +10.7005 q^{79} +1.00000 q^{81} -0.724961 q^{82} -3.22425 q^{83} -1.96239 q^{84} -2.44851 q^{86} +7.92478 q^{87} +1.53690 q^{88} +1.03761 q^{89} -1.35026 q^{91} +9.73813 q^{92} +4.57452 q^{93} +1.92478 q^{94} -2.26916 q^{96} +18.4993 q^{97} -0.193937 q^{98} +2.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 3 * q^7 - 9 * q^8 + 3 * q^9 $$3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 3 q^{7} - 9 q^{8} + 3 q^{9} + 6 q^{11} + 5 q^{12} + 6 q^{13} - q^{14} + 13 q^{16} - q^{18} + 6 q^{19} + 3 q^{21} - 2 q^{22} - 4 q^{23} - 9 q^{24} + 10 q^{26} + 3 q^{27} + 5 q^{28} + 2 q^{29} + 2 q^{31} - 29 q^{32} + 6 q^{33} - 12 q^{34} + 5 q^{36} + 4 q^{37} - 14 q^{38} + 6 q^{39} + 2 q^{41} - q^{42} - 4 q^{43} + 10 q^{44} - 8 q^{46} - 8 q^{47} + 13 q^{48} + 3 q^{49} + 18 q^{52} - 14 q^{53} - q^{54} - 9 q^{56} + 6 q^{57} + 18 q^{58} + 16 q^{59} - 6 q^{61} + 30 q^{62} + 3 q^{63} + 13 q^{64} - 2 q^{66} - 8 q^{67} - 8 q^{68} - 4 q^{69} + 6 q^{71} - 9 q^{72} + 18 q^{73} - 44 q^{74} + 2 q^{76} + 6 q^{77} + 10 q^{78} + 12 q^{79} + 3 q^{81} - 34 q^{82} - 8 q^{83} + 5 q^{84} - 4 q^{86} + 2 q^{87} - 18 q^{88} + 14 q^{89} + 6 q^{91} + 20 q^{92} + 2 q^{93} - 16 q^{94} - 29 q^{96} + 22 q^{97} - q^{98} + 6 q^{99}+O(q^{100})$$ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 3 * q^7 - 9 * q^8 + 3 * q^9 + 6 * q^11 + 5 * q^12 + 6 * q^13 - q^14 + 13 * q^16 - q^18 + 6 * q^19 + 3 * q^21 - 2 * q^22 - 4 * q^23 - 9 * q^24 + 10 * q^26 + 3 * q^27 + 5 * q^28 + 2 * q^29 + 2 * q^31 - 29 * q^32 + 6 * q^33 - 12 * q^34 + 5 * q^36 + 4 * q^37 - 14 * q^38 + 6 * q^39 + 2 * q^41 - q^42 - 4 * q^43 + 10 * q^44 - 8 * q^46 - 8 * q^47 + 13 * q^48 + 3 * q^49 + 18 * q^52 - 14 * q^53 - q^54 - 9 * q^56 + 6 * q^57 + 18 * q^58 + 16 * q^59 - 6 * q^61 + 30 * q^62 + 3 * q^63 + 13 * q^64 - 2 * q^66 - 8 * q^67 - 8 * q^68 - 4 * q^69 + 6 * q^71 - 9 * q^72 + 18 * q^73 - 44 * q^74 + 2 * q^76 + 6 * q^77 + 10 * q^78 + 12 * q^79 + 3 * q^81 - 34 * q^82 - 8 * q^83 + 5 * q^84 - 4 * q^86 + 2 * q^87 - 18 * q^88 + 14 * q^89 + 6 * q^91 + 20 * q^92 + 2 * q^93 - 16 * q^94 - 29 * q^96 + 22 * q^97 - q^98 + 6 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.193937 −0.137134 −0.0685669 0.997647i $$-0.521843\pi$$
−0.0685669 + 0.997647i $$0.521843\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.96239 −0.981194
$$5$$ 0 0
$$6$$ −0.193937 −0.0791743
$$7$$ 1.00000 0.377964
$$8$$ 0.768452 0.271689
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ −1.96239 −0.566493
$$13$$ −1.35026 −0.374495 −0.187248 0.982313i $$-0.559957\pi$$
−0.187248 + 0.982313i $$0.559957\pi$$
$$14$$ −0.193937 −0.0518317
$$15$$ 0 0
$$16$$ 3.77575 0.943937
$$17$$ 3.35026 0.812558 0.406279 0.913749i $$-0.366826\pi$$
0.406279 + 0.913749i $$0.366826\pi$$
$$18$$ −0.193937 −0.0457113
$$19$$ 5.35026 1.22743 0.613717 0.789526i $$-0.289674\pi$$
0.613717 + 0.789526i $$0.289674\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ −0.387873 −0.0826948
$$23$$ −4.96239 −1.03473 −0.517365 0.855765i $$-0.673087\pi$$
−0.517365 + 0.855765i $$0.673087\pi$$
$$24$$ 0.768452 0.156860
$$25$$ 0 0
$$26$$ 0.261865 0.0513560
$$27$$ 1.00000 0.192450
$$28$$ −1.96239 −0.370857
$$29$$ 7.92478 1.47159 0.735797 0.677202i $$-0.236808\pi$$
0.735797 + 0.677202i $$0.236808\pi$$
$$30$$ 0 0
$$31$$ 4.57452 0.821607 0.410804 0.911724i $$-0.365248\pi$$
0.410804 + 0.911724i $$0.365248\pi$$
$$32$$ −2.26916 −0.401134
$$33$$ 2.00000 0.348155
$$34$$ −0.649738 −0.111429
$$35$$ 0 0
$$36$$ −1.96239 −0.327065
$$37$$ 0.775746 0.127532 0.0637660 0.997965i $$-0.479689\pi$$
0.0637660 + 0.997965i $$0.479689\pi$$
$$38$$ −1.03761 −0.168323
$$39$$ −1.35026 −0.216215
$$40$$ 0 0
$$41$$ 3.73813 0.583799 0.291899 0.956449i $$-0.405713\pi$$
0.291899 + 0.956449i $$0.405713\pi$$
$$42$$ −0.193937 −0.0299251
$$43$$ 12.6253 1.92534 0.962670 0.270677i $$-0.0872476\pi$$
0.962670 + 0.270677i $$0.0872476\pi$$
$$44$$ −3.92478 −0.591682
$$45$$ 0 0
$$46$$ 0.962389 0.141896
$$47$$ −9.92478 −1.44768 −0.723839 0.689969i $$-0.757624\pi$$
−0.723839 + 0.689969i $$0.757624\pi$$
$$48$$ 3.77575 0.544982
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 3.35026 0.469130
$$52$$ 2.64974 0.367453
$$53$$ −8.57452 −1.17780 −0.588900 0.808206i $$-0.700439\pi$$
−0.588900 + 0.808206i $$0.700439\pi$$
$$54$$ −0.193937 −0.0263914
$$55$$ 0 0
$$56$$ 0.768452 0.102689
$$57$$ 5.35026 0.708659
$$58$$ −1.53690 −0.201805
$$59$$ −8.62530 −1.12292 −0.561459 0.827504i $$-0.689760\pi$$
−0.561459 + 0.827504i $$0.689760\pi$$
$$60$$ 0 0
$$61$$ −8.70052 −1.11399 −0.556994 0.830517i $$-0.688045\pi$$
−0.556994 + 0.830517i $$0.688045\pi$$
$$62$$ −0.887166 −0.112670
$$63$$ 1.00000 0.125988
$$64$$ −7.11142 −0.888927
$$65$$ 0 0
$$66$$ −0.387873 −0.0477439
$$67$$ −9.92478 −1.21250 −0.606252 0.795272i $$-0.707328\pi$$
−0.606252 + 0.795272i $$0.707328\pi$$
$$68$$ −6.57452 −0.797277
$$69$$ −4.96239 −0.597401
$$70$$ 0 0
$$71$$ 2.00000 0.237356 0.118678 0.992933i $$-0.462134\pi$$
0.118678 + 0.992933i $$0.462134\pi$$
$$72$$ 0.768452 0.0905629
$$73$$ 9.35026 1.09437 0.547183 0.837013i $$-0.315700\pi$$
0.547183 + 0.837013i $$0.315700\pi$$
$$74$$ −0.150446 −0.0174889
$$75$$ 0 0
$$76$$ −10.4993 −1.20435
$$77$$ 2.00000 0.227921
$$78$$ 0.261865 0.0296504
$$79$$ 10.7005 1.20390 0.601951 0.798533i $$-0.294390\pi$$
0.601951 + 0.798533i $$0.294390\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −0.724961 −0.0800586
$$83$$ −3.22425 −0.353908 −0.176954 0.984219i $$-0.556624\pi$$
−0.176954 + 0.984219i $$0.556624\pi$$
$$84$$ −1.96239 −0.214114
$$85$$ 0 0
$$86$$ −2.44851 −0.264029
$$87$$ 7.92478 0.849625
$$88$$ 1.53690 0.163835
$$89$$ 1.03761 0.109987 0.0549933 0.998487i $$-0.482486\pi$$
0.0549933 + 0.998487i $$0.482486\pi$$
$$90$$ 0 0
$$91$$ −1.35026 −0.141546
$$92$$ 9.73813 1.01527
$$93$$ 4.57452 0.474355
$$94$$ 1.92478 0.198526
$$95$$ 0 0
$$96$$ −2.26916 −0.231595
$$97$$ 18.4993 1.87832 0.939159 0.343482i $$-0.111606\pi$$
0.939159 + 0.343482i $$0.111606\pi$$
$$98$$ −0.193937 −0.0195906
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ −17.6629 −1.75753 −0.878763 0.477259i $$-0.841630\pi$$
−0.878763 + 0.477259i $$0.841630\pi$$
$$102$$ −0.649738 −0.0643337
$$103$$ −6.70052 −0.660222 −0.330111 0.943942i $$-0.607086\pi$$
−0.330111 + 0.943942i $$0.607086\pi$$
$$104$$ −1.03761 −0.101746
$$105$$ 0 0
$$106$$ 1.66291 0.161516
$$107$$ −13.7381 −1.32812 −0.664058 0.747681i $$-0.731167\pi$$
−0.664058 + 0.747681i $$0.731167\pi$$
$$108$$ −1.96239 −0.188831
$$109$$ −2.77575 −0.265868 −0.132934 0.991125i $$-0.542440\pi$$
−0.132934 + 0.991125i $$0.542440\pi$$
$$110$$ 0 0
$$111$$ 0.775746 0.0736306
$$112$$ 3.77575 0.356774
$$113$$ −12.0508 −1.13364 −0.566821 0.823841i $$-0.691827\pi$$
−0.566821 + 0.823841i $$0.691827\pi$$
$$114$$ −1.03761 −0.0971812
$$115$$ 0 0
$$116$$ −15.5515 −1.44392
$$117$$ −1.35026 −0.124832
$$118$$ 1.67276 0.153990
$$119$$ 3.35026 0.307118
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 1.68735 0.152765
$$123$$ 3.73813 0.337056
$$124$$ −8.97698 −0.806156
$$125$$ 0 0
$$126$$ −0.193937 −0.0172772
$$127$$ −2.70052 −0.239633 −0.119816 0.992796i $$-0.538231\pi$$
−0.119816 + 0.992796i $$0.538231\pi$$
$$128$$ 5.91748 0.523037
$$129$$ 12.6253 1.11160
$$130$$ 0 0
$$131$$ 20.6253 1.80204 0.901020 0.433777i $$-0.142819\pi$$
0.901020 + 0.433777i $$0.142819\pi$$
$$132$$ −3.92478 −0.341608
$$133$$ 5.35026 0.463927
$$134$$ 1.92478 0.166275
$$135$$ 0 0
$$136$$ 2.57452 0.220763
$$137$$ −22.4993 −1.92224 −0.961122 0.276124i $$-0.910950\pi$$
−0.961122 + 0.276124i $$0.910950\pi$$
$$138$$ 0.962389 0.0819240
$$139$$ −3.27504 −0.277785 −0.138893 0.990307i $$-0.544354\pi$$
−0.138893 + 0.990307i $$0.544354\pi$$
$$140$$ 0 0
$$141$$ −9.92478 −0.835817
$$142$$ −0.387873 −0.0325496
$$143$$ −2.70052 −0.225829
$$144$$ 3.77575 0.314646
$$145$$ 0 0
$$146$$ −1.81336 −0.150075
$$147$$ 1.00000 0.0824786
$$148$$ −1.52232 −0.125134
$$149$$ −4.44851 −0.364436 −0.182218 0.983258i $$-0.558328\pi$$
−0.182218 + 0.983258i $$0.558328\pi$$
$$150$$ 0 0
$$151$$ 1.29948 0.105750 0.0528749 0.998601i $$-0.483162\pi$$
0.0528749 + 0.998601i $$0.483162\pi$$
$$152$$ 4.11142 0.333480
$$153$$ 3.35026 0.270853
$$154$$ −0.387873 −0.0312557
$$155$$ 0 0
$$156$$ 2.64974 0.212149
$$157$$ 2.64974 0.211472 0.105736 0.994394i $$-0.466280\pi$$
0.105736 + 0.994394i $$0.466280\pi$$
$$158$$ −2.07522 −0.165096
$$159$$ −8.57452 −0.680003
$$160$$ 0 0
$$161$$ −4.96239 −0.391091
$$162$$ −0.193937 −0.0152371
$$163$$ −5.29948 −0.415087 −0.207544 0.978226i $$-0.566547\pi$$
−0.207544 + 0.978226i $$0.566547\pi$$
$$164$$ −7.33567 −0.572820
$$165$$ 0 0
$$166$$ 0.625301 0.0485327
$$167$$ 14.5501 1.12592 0.562959 0.826485i $$-0.309663\pi$$
0.562959 + 0.826485i $$0.309663\pi$$
$$168$$ 0.768452 0.0592874
$$169$$ −11.1768 −0.859753
$$170$$ 0 0
$$171$$ 5.35026 0.409145
$$172$$ −24.7757 −1.88913
$$173$$ 4.49929 0.342075 0.171037 0.985265i $$-0.445288\pi$$
0.171037 + 0.985265i $$0.445288\pi$$
$$174$$ −1.53690 −0.116512
$$175$$ 0 0
$$176$$ 7.55149 0.569215
$$177$$ −8.62530 −0.648317
$$178$$ −0.201231 −0.0150829
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 10.6253 0.789772 0.394886 0.918730i $$-0.370784\pi$$
0.394886 + 0.918730i $$0.370784\pi$$
$$182$$ 0.261865 0.0194107
$$183$$ −8.70052 −0.643161
$$184$$ −3.81336 −0.281124
$$185$$ 0 0
$$186$$ −0.887166 −0.0650502
$$187$$ 6.70052 0.489991
$$188$$ 19.4763 1.42045
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ −13.8496 −1.00212 −0.501059 0.865413i $$-0.667056\pi$$
−0.501059 + 0.865413i $$0.667056\pi$$
$$192$$ −7.11142 −0.513222
$$193$$ −15.3258 −1.10318 −0.551588 0.834116i $$-0.685978\pi$$
−0.551588 + 0.834116i $$0.685978\pi$$
$$194$$ −3.58769 −0.257581
$$195$$ 0 0
$$196$$ −1.96239 −0.140171
$$197$$ 0.574515 0.0409325 0.0204663 0.999791i $$-0.493485\pi$$
0.0204663 + 0.999791i $$0.493485\pi$$
$$198$$ −0.387873 −0.0275649
$$199$$ −0.201231 −0.0142649 −0.00713244 0.999975i $$-0.502270\pi$$
−0.00713244 + 0.999975i $$0.502270\pi$$
$$200$$ 0 0
$$201$$ −9.92478 −0.700040
$$202$$ 3.42548 0.241016
$$203$$ 7.92478 0.556210
$$204$$ −6.57452 −0.460308
$$205$$ 0 0
$$206$$ 1.29948 0.0905388
$$207$$ −4.96239 −0.344910
$$208$$ −5.09825 −0.353500
$$209$$ 10.7005 0.740171
$$210$$ 0 0
$$211$$ 6.44851 0.443934 0.221967 0.975054i $$-0.428752\pi$$
0.221967 + 0.975054i $$0.428752\pi$$
$$212$$ 16.8265 1.15565
$$213$$ 2.00000 0.137038
$$214$$ 2.66433 0.182130
$$215$$ 0 0
$$216$$ 0.768452 0.0522865
$$217$$ 4.57452 0.310538
$$218$$ 0.538319 0.0364595
$$219$$ 9.35026 0.631832
$$220$$ 0 0
$$221$$ −4.52373 −0.304299
$$222$$ −0.150446 −0.0100972
$$223$$ −1.55149 −0.103896 −0.0519478 0.998650i $$-0.516543\pi$$
−0.0519478 + 0.998650i $$0.516543\pi$$
$$224$$ −2.26916 −0.151615
$$225$$ 0 0
$$226$$ 2.33709 0.155461
$$227$$ 13.1490 0.872732 0.436366 0.899769i $$-0.356265\pi$$
0.436366 + 0.899769i $$0.356265\pi$$
$$228$$ −10.4993 −0.695333
$$229$$ 2.77575 0.183426 0.0917132 0.995785i $$-0.470766\pi$$
0.0917132 + 0.995785i $$0.470766\pi$$
$$230$$ 0 0
$$231$$ 2.00000 0.131590
$$232$$ 6.08981 0.399816
$$233$$ 0.0507852 0.00332705 0.00166353 0.999999i $$-0.499470\pi$$
0.00166353 + 0.999999i $$0.499470\pi$$
$$234$$ 0.261865 0.0171187
$$235$$ 0 0
$$236$$ 16.9262 1.10180
$$237$$ 10.7005 0.695074
$$238$$ −0.649738 −0.0421163
$$239$$ 5.84955 0.378376 0.189188 0.981941i $$-0.439414\pi$$
0.189188 + 0.981941i $$0.439414\pi$$
$$240$$ 0 0
$$241$$ −0.0752228 −0.00484553 −0.00242276 0.999997i $$-0.500771\pi$$
−0.00242276 + 0.999997i $$0.500771\pi$$
$$242$$ 1.35756 0.0872670
$$243$$ 1.00000 0.0641500
$$244$$ 17.0738 1.09304
$$245$$ 0 0
$$246$$ −0.724961 −0.0462218
$$247$$ −7.22425 −0.459668
$$248$$ 3.51530 0.223222
$$249$$ −3.22425 −0.204329
$$250$$ 0 0
$$251$$ 19.2243 1.21342 0.606712 0.794922i $$-0.292488\pi$$
0.606712 + 0.794922i $$0.292488\pi$$
$$252$$ −1.96239 −0.123619
$$253$$ −9.92478 −0.623965
$$254$$ 0.523730 0.0328618
$$255$$ 0 0
$$256$$ 13.0752 0.817201
$$257$$ −7.35026 −0.458497 −0.229248 0.973368i $$-0.573627\pi$$
−0.229248 + 0.973368i $$0.573627\pi$$
$$258$$ −2.44851 −0.152437
$$259$$ 0.775746 0.0482025
$$260$$ 0 0
$$261$$ 7.92478 0.490531
$$262$$ −4.00000 −0.247121
$$263$$ 12.9624 0.799295 0.399648 0.916669i $$-0.369133\pi$$
0.399648 + 0.916669i $$0.369133\pi$$
$$264$$ 1.53690 0.0945899
$$265$$ 0 0
$$266$$ −1.03761 −0.0636200
$$267$$ 1.03761 0.0635008
$$268$$ 19.4763 1.18970
$$269$$ 4.11142 0.250678 0.125339 0.992114i $$-0.459998\pi$$
0.125339 + 0.992114i $$0.459998\pi$$
$$270$$ 0 0
$$271$$ −16.4241 −0.997691 −0.498846 0.866691i $$-0.666243\pi$$
−0.498846 + 0.866691i $$0.666243\pi$$
$$272$$ 12.6497 0.767003
$$273$$ −1.35026 −0.0817216
$$274$$ 4.36344 0.263605
$$275$$ 0 0
$$276$$ 9.73813 0.586167
$$277$$ 11.0738 0.665361 0.332680 0.943040i $$-0.392047\pi$$
0.332680 + 0.943040i $$0.392047\pi$$
$$278$$ 0.635150 0.0380938
$$279$$ 4.57452 0.273869
$$280$$ 0 0
$$281$$ 14.3733 0.857438 0.428719 0.903438i $$-0.358965\pi$$
0.428719 + 0.903438i $$0.358965\pi$$
$$282$$ 1.92478 0.114619
$$283$$ −1.14903 −0.0683028 −0.0341514 0.999417i $$-0.510873\pi$$
−0.0341514 + 0.999417i $$0.510873\pi$$
$$284$$ −3.92478 −0.232893
$$285$$ 0 0
$$286$$ 0.523730 0.0309688
$$287$$ 3.73813 0.220655
$$288$$ −2.26916 −0.133711
$$289$$ −5.77575 −0.339750
$$290$$ 0 0
$$291$$ 18.4993 1.08445
$$292$$ −18.3488 −1.07379
$$293$$ −0.649738 −0.0379581 −0.0189791 0.999820i $$-0.506042\pi$$
−0.0189791 + 0.999820i $$0.506042\pi$$
$$294$$ −0.193937 −0.0113106
$$295$$ 0 0
$$296$$ 0.596124 0.0346490
$$297$$ 2.00000 0.116052
$$298$$ 0.862728 0.0499765
$$299$$ 6.70052 0.387501
$$300$$ 0 0
$$301$$ 12.6253 0.727710
$$302$$ −0.252016 −0.0145019
$$303$$ −17.6629 −1.01471
$$304$$ 20.2012 1.15862
$$305$$ 0 0
$$306$$ −0.649738 −0.0371431
$$307$$ −24.1016 −1.37555 −0.687775 0.725924i $$-0.741412\pi$$
−0.687775 + 0.725924i $$0.741412\pi$$
$$308$$ −3.92478 −0.223635
$$309$$ −6.70052 −0.381179
$$310$$ 0 0
$$311$$ 8.25202 0.467929 0.233964 0.972245i $$-0.424830\pi$$
0.233964 + 0.972245i $$0.424830\pi$$
$$312$$ −1.03761 −0.0587432
$$313$$ 14.9018 0.842297 0.421148 0.906992i $$-0.361627\pi$$
0.421148 + 0.906992i $$0.361627\pi$$
$$314$$ −0.513881 −0.0290000
$$315$$ 0 0
$$316$$ −20.9986 −1.18126
$$317$$ −10.1260 −0.568733 −0.284367 0.958716i $$-0.591783\pi$$
−0.284367 + 0.958716i $$0.591783\pi$$
$$318$$ 1.66291 0.0932515
$$319$$ 15.8496 0.887405
$$320$$ 0 0
$$321$$ −13.7381 −0.766788
$$322$$ 0.962389 0.0536318
$$323$$ 17.9248 0.997361
$$324$$ −1.96239 −0.109022
$$325$$ 0 0
$$326$$ 1.02776 0.0569225
$$327$$ −2.77575 −0.153499
$$328$$ 2.87258 0.158612
$$329$$ −9.92478 −0.547171
$$330$$ 0 0
$$331$$ 27.8496 1.53075 0.765375 0.643585i $$-0.222554\pi$$
0.765375 + 0.643585i $$0.222554\pi$$
$$332$$ 6.32724 0.347252
$$333$$ 0.775746 0.0425106
$$334$$ −2.82179 −0.154402
$$335$$ 0 0
$$336$$ 3.77575 0.205984
$$337$$ 3.84955 0.209699 0.104849 0.994488i $$-0.466564\pi$$
0.104849 + 0.994488i $$0.466564\pi$$
$$338$$ 2.16759 0.117901
$$339$$ −12.0508 −0.654509
$$340$$ 0 0
$$341$$ 9.14903 0.495448
$$342$$ −1.03761 −0.0561076
$$343$$ 1.00000 0.0539949
$$344$$ 9.70194 0.523093
$$345$$ 0 0
$$346$$ −0.872577 −0.0469101
$$347$$ −9.58769 −0.514694 −0.257347 0.966319i $$-0.582848\pi$$
−0.257347 + 0.966319i $$0.582848\pi$$
$$348$$ −15.5515 −0.833648
$$349$$ −15.1490 −0.810909 −0.405455 0.914115i $$-0.632887\pi$$
−0.405455 + 0.914115i $$0.632887\pi$$
$$350$$ 0 0
$$351$$ −1.35026 −0.0720716
$$352$$ −4.53832 −0.241893
$$353$$ 20.3488 1.08306 0.541530 0.840681i $$-0.317845\pi$$
0.541530 + 0.840681i $$0.317845\pi$$
$$354$$ 1.67276 0.0889063
$$355$$ 0 0
$$356$$ −2.03620 −0.107918
$$357$$ 3.35026 0.177315
$$358$$ −1.93937 −0.102499
$$359$$ −31.4010 −1.65728 −0.828642 0.559779i $$-0.810886\pi$$
−0.828642 + 0.559779i $$0.810886\pi$$
$$360$$ 0 0
$$361$$ 9.62530 0.506595
$$362$$ −2.06063 −0.108305
$$363$$ −7.00000 −0.367405
$$364$$ 2.64974 0.138884
$$365$$ 0 0
$$366$$ 1.68735 0.0881992
$$367$$ 29.4010 1.53472 0.767361 0.641215i $$-0.221569\pi$$
0.767361 + 0.641215i $$0.221569\pi$$
$$368$$ −18.7367 −0.976719
$$369$$ 3.73813 0.194600
$$370$$ 0 0
$$371$$ −8.57452 −0.445167
$$372$$ −8.97698 −0.465435
$$373$$ −16.0000 −0.828449 −0.414224 0.910175i $$-0.635947\pi$$
−0.414224 + 0.910175i $$0.635947\pi$$
$$374$$ −1.29948 −0.0671943
$$375$$ 0 0
$$376$$ −7.62672 −0.393318
$$377$$ −10.7005 −0.551105
$$378$$ −0.193937 −0.00997502
$$379$$ 10.7005 0.549649 0.274824 0.961494i $$-0.411380\pi$$
0.274824 + 0.961494i $$0.411380\pi$$
$$380$$ 0 0
$$381$$ −2.70052 −0.138352
$$382$$ 2.68594 0.137424
$$383$$ 16.7757 0.857201 0.428600 0.903494i $$-0.359007\pi$$
0.428600 + 0.903494i $$0.359007\pi$$
$$384$$ 5.91748 0.301975
$$385$$ 0 0
$$386$$ 2.97224 0.151283
$$387$$ 12.6253 0.641780
$$388$$ −36.3028 −1.84300
$$389$$ −29.3258 −1.48688 −0.743439 0.668804i $$-0.766807\pi$$
−0.743439 + 0.668804i $$0.766807\pi$$
$$390$$ 0 0
$$391$$ −16.6253 −0.840778
$$392$$ 0.768452 0.0388127
$$393$$ 20.6253 1.04041
$$394$$ −0.111420 −0.00561324
$$395$$ 0 0
$$396$$ −3.92478 −0.197227
$$397$$ −18.3488 −0.920902 −0.460451 0.887685i $$-0.652312\pi$$
−0.460451 + 0.887685i $$0.652312\pi$$
$$398$$ 0.0390260 0.00195620
$$399$$ 5.35026 0.267848
$$400$$ 0 0
$$401$$ −37.3258 −1.86396 −0.931981 0.362506i $$-0.881921\pi$$
−0.931981 + 0.362506i $$0.881921\pi$$
$$402$$ 1.92478 0.0959992
$$403$$ −6.17679 −0.307688
$$404$$ 34.6615 1.72447
$$405$$ 0 0
$$406$$ −1.53690 −0.0762753
$$407$$ 1.55149 0.0769046
$$408$$ 2.57452 0.127458
$$409$$ −22.3733 −1.10629 −0.553144 0.833086i $$-0.686572\pi$$
−0.553144 + 0.833086i $$0.686572\pi$$
$$410$$ 0 0
$$411$$ −22.4993 −1.10981
$$412$$ 13.1490 0.647806
$$413$$ −8.62530 −0.424423
$$414$$ 0.962389 0.0472988
$$415$$ 0 0
$$416$$ 3.06396 0.150223
$$417$$ −3.27504 −0.160379
$$418$$ −2.07522 −0.101502
$$419$$ 23.4763 1.14689 0.573445 0.819244i $$-0.305606\pi$$
0.573445 + 0.819244i $$0.305606\pi$$
$$420$$ 0 0
$$421$$ −25.2243 −1.22935 −0.614677 0.788779i $$-0.710714\pi$$
−0.614677 + 0.788779i $$0.710714\pi$$
$$422$$ −1.25060 −0.0608783
$$423$$ −9.92478 −0.482559
$$424$$ −6.58910 −0.319995
$$425$$ 0 0
$$426$$ −0.387873 −0.0187925
$$427$$ −8.70052 −0.421048
$$428$$ 26.9596 1.30314
$$429$$ −2.70052 −0.130383
$$430$$ 0 0
$$431$$ −19.4010 −0.934516 −0.467258 0.884121i $$-0.654758\pi$$
−0.467258 + 0.884121i $$0.654758\pi$$
$$432$$ 3.77575 0.181661
$$433$$ −6.49929 −0.312336 −0.156168 0.987731i $$-0.549914\pi$$
−0.156168 + 0.987731i $$0.549914\pi$$
$$434$$ −0.887166 −0.0425853
$$435$$ 0 0
$$436$$ 5.44709 0.260868
$$437$$ −26.5501 −1.27006
$$438$$ −1.81336 −0.0866456
$$439$$ −14.6497 −0.699194 −0.349597 0.936900i $$-0.613681\pi$$
−0.349597 + 0.936900i $$0.613681\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0.877317 0.0417297
$$443$$ −19.1392 −0.909330 −0.454665 0.890663i $$-0.650241\pi$$
−0.454665 + 0.890663i $$0.650241\pi$$
$$444$$ −1.52232 −0.0722459
$$445$$ 0 0
$$446$$ 0.300891 0.0142476
$$447$$ −4.44851 −0.210407
$$448$$ −7.11142 −0.335983
$$449$$ −32.8021 −1.54803 −0.774013 0.633169i $$-0.781754\pi$$
−0.774013 + 0.633169i $$0.781754\pi$$
$$450$$ 0 0
$$451$$ 7.47627 0.352044
$$452$$ 23.6483 1.11232
$$453$$ 1.29948 0.0610547
$$454$$ −2.55008 −0.119681
$$455$$ 0 0
$$456$$ 4.11142 0.192535
$$457$$ 18.7005 0.874774 0.437387 0.899273i $$-0.355904\pi$$
0.437387 + 0.899273i $$0.355904\pi$$
$$458$$ −0.538319 −0.0251540
$$459$$ 3.35026 0.156377
$$460$$ 0 0
$$461$$ −6.96239 −0.324271 −0.162135 0.986769i $$-0.551838\pi$$
−0.162135 + 0.986769i $$0.551838\pi$$
$$462$$ −0.387873 −0.0180455
$$463$$ −5.29948 −0.246288 −0.123144 0.992389i $$-0.539298\pi$$
−0.123144 + 0.992389i $$0.539298\pi$$
$$464$$ 29.9219 1.38909
$$465$$ 0 0
$$466$$ −0.00984911 −0.000456251 0
$$467$$ 13.1490 0.608465 0.304232 0.952598i $$-0.401600\pi$$
0.304232 + 0.952598i $$0.401600\pi$$
$$468$$ 2.64974 0.122484
$$469$$ −9.92478 −0.458284
$$470$$ 0 0
$$471$$ 2.64974 0.122093
$$472$$ −6.62813 −0.305084
$$473$$ 25.2506 1.16102
$$474$$ −2.07522 −0.0953181
$$475$$ 0 0
$$476$$ −6.57452 −0.301342
$$477$$ −8.57452 −0.392600
$$478$$ −1.13444 −0.0518882
$$479$$ 5.14903 0.235265 0.117633 0.993057i $$-0.462469\pi$$
0.117633 + 0.993057i $$0.462469\pi$$
$$480$$ 0 0
$$481$$ −1.04746 −0.0477601
$$482$$ 0.0145884 0.000664486 0
$$483$$ −4.96239 −0.225797
$$484$$ 13.7367 0.624396
$$485$$ 0 0
$$486$$ −0.193937 −0.00879714
$$487$$ 22.1768 1.00493 0.502463 0.864599i $$-0.332427\pi$$
0.502463 + 0.864599i $$0.332427\pi$$
$$488$$ −6.68594 −0.302658
$$489$$ −5.29948 −0.239651
$$490$$ 0 0
$$491$$ 2.00000 0.0902587 0.0451294 0.998981i $$-0.485630\pi$$
0.0451294 + 0.998981i $$0.485630\pi$$
$$492$$ −7.33567 −0.330718
$$493$$ 26.5501 1.19576
$$494$$ 1.40105 0.0630361
$$495$$ 0 0
$$496$$ 17.2722 0.775545
$$497$$ 2.00000 0.0897123
$$498$$ 0.625301 0.0280204
$$499$$ −6.55008 −0.293222 −0.146611 0.989194i $$-0.546837\pi$$
−0.146611 + 0.989194i $$0.546837\pi$$
$$500$$ 0 0
$$501$$ 14.5501 0.650050
$$502$$ −3.72829 −0.166402
$$503$$ −8.77575 −0.391291 −0.195646 0.980675i $$-0.562680\pi$$
−0.195646 + 0.980675i $$0.562680\pi$$
$$504$$ 0.768452 0.0342296
$$505$$ 0 0
$$506$$ 1.92478 0.0855668
$$507$$ −11.1768 −0.496379
$$508$$ 5.29948 0.235126
$$509$$ 13.1392 0.582384 0.291192 0.956665i $$-0.405948\pi$$
0.291192 + 0.956665i $$0.405948\pi$$
$$510$$ 0 0
$$511$$ 9.35026 0.413631
$$512$$ −14.3707 −0.635103
$$513$$ 5.35026 0.236220
$$514$$ 1.42548 0.0628754
$$515$$ 0 0
$$516$$ −24.7757 −1.09069
$$517$$ −19.8496 −0.872982
$$518$$ −0.150446 −0.00661020
$$519$$ 4.49929 0.197497
$$520$$ 0 0
$$521$$ −37.6629 −1.65004 −0.825021 0.565102i $$-0.808837\pi$$
−0.825021 + 0.565102i $$0.808837\pi$$
$$522$$ −1.53690 −0.0672685
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −40.4749 −1.76815
$$525$$ 0 0
$$526$$ −2.51388 −0.109610
$$527$$ 15.3258 0.667603
$$528$$ 7.55149 0.328637
$$529$$ 1.62530 0.0706652
$$530$$ 0 0
$$531$$ −8.62530 −0.374306
$$532$$ −10.4993 −0.455202
$$533$$ −5.04746 −0.218630
$$534$$ −0.201231 −0.00870811
$$535$$ 0 0
$$536$$ −7.62672 −0.329424
$$537$$ 10.0000 0.431532
$$538$$ −0.797355 −0.0343764
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ −22.4749 −0.966269 −0.483135 0.875546i $$-0.660502\pi$$
−0.483135 + 0.875546i $$0.660502\pi$$
$$542$$ 3.18523 0.136817
$$543$$ 10.6253 0.455975
$$544$$ −7.60228 −0.325945
$$545$$ 0 0
$$546$$ 0.261865 0.0112068
$$547$$ 25.9248 1.10846 0.554232 0.832362i $$-0.313012\pi$$
0.554232 + 0.832362i $$0.313012\pi$$
$$548$$ 44.1524 1.88610
$$549$$ −8.70052 −0.371329
$$550$$ 0 0
$$551$$ 42.3996 1.80629
$$552$$ −3.81336 −0.162307
$$553$$ 10.7005 0.455033
$$554$$ −2.14762 −0.0912435
$$555$$ 0 0
$$556$$ 6.42690 0.272561
$$557$$ 28.5256 1.20867 0.604335 0.796730i $$-0.293439\pi$$
0.604335 + 0.796730i $$0.293439\pi$$
$$558$$ −0.887166 −0.0375567
$$559$$ −17.0475 −0.721031
$$560$$ 0 0
$$561$$ 6.70052 0.282896
$$562$$ −2.78751 −0.117584
$$563$$ 11.6267 0.490008 0.245004 0.969522i $$-0.421211\pi$$
0.245004 + 0.969522i $$0.421211\pi$$
$$564$$ 19.4763 0.820099
$$565$$ 0 0
$$566$$ 0.222839 0.00936663
$$567$$ 1.00000 0.0419961
$$568$$ 1.53690 0.0644871
$$569$$ 9.32582 0.390959 0.195479 0.980708i $$-0.437374\pi$$
0.195479 + 0.980708i $$0.437374\pi$$
$$570$$ 0 0
$$571$$ −19.6991 −0.824382 −0.412191 0.911097i $$-0.635236\pi$$
−0.412191 + 0.911097i $$0.635236\pi$$
$$572$$ 5.29948 0.221582
$$573$$ −13.8496 −0.578573
$$574$$ −0.724961 −0.0302593
$$575$$ 0 0
$$576$$ −7.11142 −0.296309
$$577$$ −32.7974 −1.36537 −0.682686 0.730712i $$-0.739188\pi$$
−0.682686 + 0.730712i $$0.739188\pi$$
$$578$$ 1.12013 0.0465912
$$579$$ −15.3258 −0.636920
$$580$$ 0 0
$$581$$ −3.22425 −0.133765
$$582$$ −3.58769 −0.148715
$$583$$ −17.1490 −0.710240
$$584$$ 7.18523 0.297327
$$585$$ 0 0
$$586$$ 0.126008 0.00520534
$$587$$ −18.8218 −0.776859 −0.388429 0.921479i $$-0.626982\pi$$
−0.388429 + 0.921479i $$0.626982\pi$$
$$588$$ −1.96239 −0.0809275
$$589$$ 24.4749 1.00847
$$590$$ 0 0
$$591$$ 0.574515 0.0236324
$$592$$ 2.92902 0.120382
$$593$$ −33.7499 −1.38594 −0.692971 0.720965i $$-0.743699\pi$$
−0.692971 + 0.720965i $$0.743699\pi$$
$$594$$ −0.387873 −0.0159146
$$595$$ 0 0
$$596$$ 8.72970 0.357582
$$597$$ −0.201231 −0.00823583
$$598$$ −1.29948 −0.0531395
$$599$$ −20.2981 −0.829356 −0.414678 0.909968i $$-0.636106\pi$$
−0.414678 + 0.909968i $$0.636106\pi$$
$$600$$ 0 0
$$601$$ −13.8496 −0.564935 −0.282468 0.959277i $$-0.591153\pi$$
−0.282468 + 0.959277i $$0.591153\pi$$
$$602$$ −2.44851 −0.0997937
$$603$$ −9.92478 −0.404168
$$604$$ −2.55008 −0.103761
$$605$$ 0 0
$$606$$ 3.42548 0.139151
$$607$$ 25.2506 1.02489 0.512445 0.858720i $$-0.328740\pi$$
0.512445 + 0.858720i $$0.328740\pi$$
$$608$$ −12.1406 −0.492366
$$609$$ 7.92478 0.321128
$$610$$ 0 0
$$611$$ 13.4010 0.542148
$$612$$ −6.57452 −0.265759
$$613$$ 9.14903 0.369526 0.184763 0.982783i $$-0.440848\pi$$
0.184763 + 0.982783i $$0.440848\pi$$
$$614$$ 4.67418 0.188634
$$615$$ 0 0
$$616$$ 1.53690 0.0619236
$$617$$ −15.9492 −0.642091 −0.321046 0.947064i $$-0.604034\pi$$
−0.321046 + 0.947064i $$0.604034\pi$$
$$618$$ 1.29948 0.0522726
$$619$$ 11.1735 0.449100 0.224550 0.974463i $$-0.427909\pi$$
0.224550 + 0.974463i $$0.427909\pi$$
$$620$$ 0 0
$$621$$ −4.96239 −0.199134
$$622$$ −1.60037 −0.0641689
$$623$$ 1.03761 0.0415710
$$624$$ −5.09825 −0.204093
$$625$$ 0 0
$$626$$ −2.89000 −0.115507
$$627$$ 10.7005 0.427338
$$628$$ −5.19982 −0.207495
$$629$$ 2.59895 0.103627
$$630$$ 0 0
$$631$$ −14.5501 −0.579229 −0.289615 0.957143i $$-0.593527\pi$$
−0.289615 + 0.957143i $$0.593527\pi$$
$$632$$ 8.22284 0.327087
$$633$$ 6.44851 0.256305
$$634$$ 1.96380 0.0779926
$$635$$ 0 0
$$636$$ 16.8265 0.667215
$$637$$ −1.35026 −0.0534993
$$638$$ −3.07381 −0.121693
$$639$$ 2.00000 0.0791188
$$640$$ 0 0
$$641$$ −38.7269 −1.52962 −0.764810 0.644256i $$-0.777167\pi$$
−0.764810 + 0.644256i $$0.777167\pi$$
$$642$$ 2.66433 0.105153
$$643$$ 11.9511 0.471306 0.235653 0.971837i $$-0.424277\pi$$
0.235653 + 0.971837i $$0.424277\pi$$
$$644$$ 9.73813 0.383736
$$645$$ 0 0
$$646$$ −3.47627 −0.136772
$$647$$ 14.5501 0.572023 0.286011 0.958226i $$-0.407671\pi$$
0.286011 + 0.958226i $$0.407671\pi$$
$$648$$ 0.768452 0.0301876
$$649$$ −17.2506 −0.677145
$$650$$ 0 0
$$651$$ 4.57452 0.179289
$$652$$ 10.3996 0.407281
$$653$$ −49.9756 −1.95569 −0.977847 0.209319i $$-0.932875\pi$$
−0.977847 + 0.209319i $$0.932875\pi$$
$$654$$ 0.538319 0.0210499
$$655$$ 0 0
$$656$$ 14.1142 0.551069
$$657$$ 9.35026 0.364788
$$658$$ 1.92478 0.0750356
$$659$$ −16.9525 −0.660377 −0.330189 0.943915i $$-0.607112\pi$$
−0.330189 + 0.943915i $$0.607112\pi$$
$$660$$ 0 0
$$661$$ −15.6531 −0.608834 −0.304417 0.952539i $$-0.598462\pi$$
−0.304417 + 0.952539i $$0.598462\pi$$
$$662$$ −5.40105 −0.209918
$$663$$ −4.52373 −0.175687
$$664$$ −2.47768 −0.0961528
$$665$$ 0 0
$$666$$ −0.150446 −0.00582965
$$667$$ −39.3258 −1.52270
$$668$$ −28.5529 −1.10475
$$669$$ −1.55149 −0.0599842
$$670$$ 0 0
$$671$$ −17.4010 −0.671760
$$672$$ −2.26916 −0.0875347
$$673$$ −26.0263 −1.00324 −0.501621 0.865088i $$-0.667263\pi$$
−0.501621 + 0.865088i $$0.667263\pi$$
$$674$$ −0.746569 −0.0287568
$$675$$ 0 0
$$676$$ 21.9332 0.843585
$$677$$ 35.4518 1.36252 0.681262 0.732039i $$-0.261431\pi$$
0.681262 + 0.732039i $$0.261431\pi$$
$$678$$ 2.33709 0.0897553
$$679$$ 18.4993 0.709938
$$680$$ 0 0
$$681$$ 13.1490 0.503872
$$682$$ −1.77433 −0.0679427
$$683$$ 23.6629 0.905436 0.452718 0.891654i $$-0.350454\pi$$
0.452718 + 0.891654i $$0.350454\pi$$
$$684$$ −10.4993 −0.401450
$$685$$ 0 0
$$686$$ −0.193937 −0.00740453
$$687$$ 2.77575 0.105901
$$688$$ 47.6699 1.81740
$$689$$ 11.5778 0.441081
$$690$$ 0 0
$$691$$ −0.574515 −0.0218556 −0.0109278 0.999940i $$-0.503478\pi$$
−0.0109278 + 0.999940i $$0.503478\pi$$
$$692$$ −8.82936 −0.335642
$$693$$ 2.00000 0.0759737
$$694$$ 1.85940 0.0705820
$$695$$ 0 0
$$696$$ 6.08981 0.230834
$$697$$ 12.5237 0.474370
$$698$$ 2.93795 0.111203
$$699$$ 0.0507852 0.00192087
$$700$$ 0 0
$$701$$ 42.7269 1.61377 0.806886 0.590707i $$-0.201151\pi$$
0.806886 + 0.590707i $$0.201151\pi$$
$$702$$ 0.261865 0.00988346
$$703$$ 4.15045 0.156537
$$704$$ −14.2228 −0.536043
$$705$$ 0 0
$$706$$ −3.94639 −0.148524
$$707$$ −17.6629 −0.664282
$$708$$ 16.9262 0.636125
$$709$$ 27.2506 1.02342 0.511709 0.859159i $$-0.329013\pi$$
0.511709 + 0.859159i $$0.329013\pi$$
$$710$$ 0 0
$$711$$ 10.7005 0.401301
$$712$$ 0.797355 0.0298821
$$713$$ −22.7005 −0.850141
$$714$$ −0.649738 −0.0243158
$$715$$ 0 0
$$716$$ −19.6239 −0.733379
$$717$$ 5.84955 0.218456
$$718$$ 6.08981 0.227270
$$719$$ −10.7005 −0.399062 −0.199531 0.979891i $$-0.563942\pi$$
−0.199531 + 0.979891i $$0.563942\pi$$
$$720$$ 0 0
$$721$$ −6.70052 −0.249541
$$722$$ −1.86670 −0.0694713
$$723$$ −0.0752228 −0.00279757
$$724$$ −20.8510 −0.774920
$$725$$ 0 0
$$726$$ 1.35756 0.0503836
$$727$$ −39.9511 −1.48171 −0.740853 0.671668i $$-0.765578\pi$$
−0.740853 + 0.671668i $$0.765578\pi$$
$$728$$ −1.03761 −0.0384564
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 42.2981 1.56445
$$732$$ 17.0738 0.631066
$$733$$ 30.3488 1.12096 0.560480 0.828168i $$-0.310617\pi$$
0.560480 + 0.828168i $$0.310617\pi$$
$$734$$ −5.70194 −0.210462
$$735$$ 0 0
$$736$$ 11.2605 0.415066
$$737$$ −19.8496 −0.731168
$$738$$ −0.724961 −0.0266862
$$739$$ 37.2506 1.37029 0.685143 0.728409i $$-0.259740\pi$$
0.685143 + 0.728409i $$0.259740\pi$$
$$740$$ 0 0
$$741$$ −7.22425 −0.265390
$$742$$ 1.66291 0.0610474
$$743$$ −26.3634 −0.967181 −0.483590 0.875294i $$-0.660668\pi$$
−0.483590 + 0.875294i $$0.660668\pi$$
$$744$$ 3.51530 0.128877
$$745$$ 0 0
$$746$$ 3.10299 0.113608
$$747$$ −3.22425 −0.117969
$$748$$ −13.1490 −0.480776
$$749$$ −13.7381 −0.501981
$$750$$ 0 0
$$751$$ 50.6516 1.84830 0.924152 0.382024i $$-0.124773\pi$$
0.924152 + 0.382024i $$0.124773\pi$$
$$752$$ −37.4734 −1.36652
$$753$$ 19.2243 0.700571
$$754$$ 2.07522 0.0755752
$$755$$ 0 0
$$756$$ −1.96239 −0.0713714
$$757$$ −38.9525 −1.41575 −0.707877 0.706336i $$-0.750347\pi$$
−0.707877 + 0.706336i $$0.750347\pi$$
$$758$$ −2.07522 −0.0753755
$$759$$ −9.92478 −0.360247
$$760$$ 0 0
$$761$$ 48.2130 1.74772 0.873860 0.486178i $$-0.161609\pi$$
0.873860 + 0.486178i $$0.161609\pi$$
$$762$$ 0.523730 0.0189727
$$763$$ −2.77575 −0.100489
$$764$$ 27.1782 0.983273
$$765$$ 0 0
$$766$$ −3.25343 −0.117551
$$767$$ 11.6464 0.420528
$$768$$ 13.0752 0.471811
$$769$$ 4.44851 0.160417 0.0802086 0.996778i $$-0.474441\pi$$
0.0802086 + 0.996778i $$0.474441\pi$$
$$770$$ 0 0
$$771$$ −7.35026 −0.264713
$$772$$ 30.0752 1.08243
$$773$$ −39.3014 −1.41357 −0.706786 0.707427i $$-0.749856\pi$$
−0.706786 + 0.707427i $$0.749856\pi$$
$$774$$ −2.44851 −0.0880098
$$775$$ 0 0
$$776$$ 14.2158 0.510318
$$777$$ 0.775746 0.0278297
$$778$$ 5.68735 0.203901
$$779$$ 20.0000 0.716574
$$780$$ 0 0
$$781$$ 4.00000 0.143131
$$782$$ 3.22425 0.115299
$$783$$ 7.92478 0.283208
$$784$$ 3.77575 0.134848
$$785$$ 0 0
$$786$$ −4.00000 −0.142675
$$787$$ −0.897015 −0.0319751 −0.0159876 0.999872i $$-0.505089\pi$$
−0.0159876 + 0.999872i $$0.505089\pi$$
$$788$$ −1.12742 −0.0401628
$$789$$ 12.9624 0.461473
$$790$$ 0 0
$$791$$ −12.0508 −0.428477
$$792$$ 1.53690 0.0546115
$$793$$ 11.7480 0.417183
$$794$$ 3.55851 0.126287
$$795$$ 0 0
$$796$$ 0.394893 0.0139966
$$797$$ −3.19982 −0.113343 −0.0566717 0.998393i $$-0.518049\pi$$
−0.0566717 + 0.998393i $$0.518049\pi$$
$$798$$ −1.03761 −0.0367310
$$799$$ −33.2506 −1.17632
$$800$$ 0 0
$$801$$ 1.03761 0.0366622
$$802$$ 7.23884 0.255612
$$803$$ 18.7005 0.659927
$$804$$ 19.4763 0.686875
$$805$$ 0 0
$$806$$ 1.19791 0.0421944
$$807$$ 4.11142 0.144729
$$808$$ −13.5731 −0.477500
$$809$$ 4.44851 0.156401 0.0782006 0.996938i $$-0.475083\pi$$
0.0782006 + 0.996938i $$0.475083\pi$$
$$810$$ 0 0
$$811$$ 37.6747 1.32294 0.661468 0.749973i $$-0.269934\pi$$
0.661468 + 0.749973i $$0.269934\pi$$
$$812$$ −15.5515 −0.545750
$$813$$ −16.4241 −0.576017
$$814$$ −0.300891 −0.0105462
$$815$$ 0 0
$$816$$ 12.6497 0.442829
$$817$$ 67.5487 2.36323
$$818$$ 4.33900 0.151710
$$819$$ −1.35026 −0.0471820
$$820$$ 0 0
$$821$$ −0.749399 −0.0261542 −0.0130771 0.999914i $$-0.504163\pi$$
−0.0130771 + 0.999914i $$0.504163\pi$$
$$822$$ 4.36344 0.152192
$$823$$ 26.3996 0.920233 0.460117 0.887858i $$-0.347808\pi$$
0.460117 + 0.887858i $$0.347808\pi$$
$$824$$ −5.14903 −0.179375
$$825$$ 0 0
$$826$$ 1.67276 0.0582028
$$827$$ −5.43724 −0.189071 −0.0945357 0.995521i $$-0.530137\pi$$
−0.0945357 + 0.995521i $$0.530137\pi$$
$$828$$ 9.73813 0.338424
$$829$$ 22.7757 0.791034 0.395517 0.918459i $$-0.370565\pi$$
0.395517 + 0.918459i $$0.370565\pi$$
$$830$$ 0 0
$$831$$ 11.0738 0.384146
$$832$$ 9.60228 0.332899
$$833$$ 3.35026 0.116080
$$834$$ 0.635150 0.0219934
$$835$$ 0 0
$$836$$ −20.9986 −0.726251
$$837$$ 4.57452 0.158118
$$838$$ −4.55291 −0.157278
$$839$$ 15.8496 0.547187 0.273594 0.961845i $$-0.411788\pi$$
0.273594 + 0.961845i $$0.411788\pi$$
$$840$$ 0 0
$$841$$ 33.8021 1.16559
$$842$$ 4.89191 0.168586
$$843$$ 14.3733 0.495042
$$844$$ −12.6545 −0.435585
$$845$$ 0 0
$$846$$ 1.92478 0.0661752
$$847$$ −7.00000 −0.240523
$$848$$ −32.3752 −1.11177
$$849$$ −1.14903 −0.0394346
$$850$$ 0 0
$$851$$ −3.84955 −0.131961
$$852$$ −3.92478 −0.134461
$$853$$ 21.0494 0.720717 0.360358 0.932814i $$-0.382654\pi$$
0.360358 + 0.932814i $$0.382654\pi$$
$$854$$ 1.68735 0.0577399
$$855$$ 0 0
$$856$$ −10.5571 −0.360834
$$857$$ −50.1524 −1.71317 −0.856586 0.516004i $$-0.827419\pi$$
−0.856586 + 0.516004i $$0.827419\pi$$
$$858$$ 0.523730 0.0178799
$$859$$ −5.35026 −0.182549 −0.0912743 0.995826i $$-0.529094\pi$$
−0.0912743 + 0.995826i $$0.529094\pi$$
$$860$$ 0 0
$$861$$ 3.73813 0.127395
$$862$$ 3.76257 0.128154
$$863$$ 33.6893 1.14680 0.573398 0.819277i $$-0.305625\pi$$
0.573398 + 0.819277i $$0.305625\pi$$
$$864$$ −2.26916 −0.0771984
$$865$$ 0 0
$$866$$ 1.26045 0.0428319
$$867$$ −5.77575 −0.196155
$$868$$ −8.97698 −0.304698
$$869$$ 21.4010 0.725981
$$870$$ 0 0
$$871$$ 13.4010 0.454077
$$872$$ −2.13303 −0.0722334
$$873$$ 18.4993 0.626106
$$874$$ 5.14903 0.174169
$$875$$ 0 0
$$876$$ −18.3488 −0.619950
$$877$$ 21.5026 0.726092 0.363046 0.931771i $$-0.381737\pi$$
0.363046 + 0.931771i $$0.381737\pi$$
$$878$$ 2.84112 0.0958832
$$879$$ −0.649738 −0.0219151
$$880$$ 0 0
$$881$$ 32.3634 1.09035 0.545176 0.838322i $$-0.316463\pi$$
0.545176 + 0.838322i $$0.316463\pi$$
$$882$$ −0.193937 −0.00653018
$$883$$ 2.59895 0.0874617 0.0437309 0.999043i $$-0.486076\pi$$
0.0437309 + 0.999043i $$0.486076\pi$$
$$884$$ 8.87732 0.298576
$$885$$ 0 0
$$886$$ 3.71179 0.124700
$$887$$ −38.2784 −1.28526 −0.642631 0.766176i $$-0.722157\pi$$
−0.642631 + 0.766176i $$0.722157\pi$$
$$888$$ 0.596124 0.0200046
$$889$$ −2.70052 −0.0905727
$$890$$ 0 0
$$891$$ 2.00000 0.0670025
$$892$$ 3.04463 0.101942
$$893$$ −53.1002 −1.77693
$$894$$ 0.862728 0.0288539
$$895$$ 0 0
$$896$$ 5.91748 0.197689
$$897$$ 6.70052 0.223724
$$898$$ 6.36153 0.212287
$$899$$ 36.2520 1.20907
$$900$$ 0 0
$$901$$ −28.7269 −0.957031
$$902$$ −1.44992 −0.0482771
$$903$$ 12.6253 0.420144
$$904$$ −9.26045 −0.307998
$$905$$ 0 0
$$906$$ −0.252016 −0.00837267
$$907$$ 49.9972 1.66013 0.830064 0.557668i $$-0.188304\pi$$
0.830064 + 0.557668i $$0.188304\pi$$
$$908$$ −25.8035 −0.856320
$$909$$ −17.6629 −0.585842
$$910$$ 0 0
$$911$$ −24.9525 −0.826715 −0.413357 0.910569i $$-0.635644\pi$$
−0.413357 + 0.910569i $$0.635644\pi$$
$$912$$ 20.2012 0.668930
$$913$$ −6.44851 −0.213414
$$914$$ −3.62672 −0.119961
$$915$$ 0 0
$$916$$ −5.44709 −0.179977
$$917$$ 20.6253 0.681107
$$918$$ −0.649738 −0.0214446
$$919$$ −11.6991 −0.385918 −0.192959 0.981207i $$-0.561808\pi$$
−0.192959 + 0.981207i $$0.561808\pi$$
$$920$$ 0 0
$$921$$ −24.1016 −0.794174
$$922$$ 1.35026 0.0444685
$$923$$ −2.70052 −0.0888888
$$924$$ −3.92478 −0.129116
$$925$$ 0 0
$$926$$ 1.02776 0.0337744
$$927$$ −6.70052 −0.220074
$$928$$ −17.9826 −0.590307
$$929$$ 23.7090 0.777866 0.388933 0.921266i $$-0.372844\pi$$
0.388933 + 0.921266i $$0.372844\pi$$
$$930$$ 0 0
$$931$$ 5.35026 0.175348
$$932$$ −0.0996603 −0.00326448
$$933$$ 8.25202 0.270159
$$934$$ −2.55008 −0.0834411
$$935$$ 0 0
$$936$$ −1.03761 −0.0339154
$$937$$ 19.9003 0.650116 0.325058 0.945694i $$-0.394616\pi$$
0.325058 + 0.945694i $$0.394616\pi$$
$$938$$ 1.92478 0.0628462
$$939$$ 14.9018 0.486300
$$940$$ 0 0
$$941$$ −6.28821 −0.204990 −0.102495 0.994734i $$-0.532683\pi$$
−0.102495 + 0.994734i $$0.532683\pi$$
$$942$$ −0.513881 −0.0167432
$$943$$ −18.5501 −0.604074
$$944$$ −32.5669 −1.05996
$$945$$ 0 0
$$946$$ −4.89701 −0.159216
$$947$$ 40.0362 1.30100 0.650501 0.759506i $$-0.274559\pi$$
0.650501 + 0.759506i $$0.274559\pi$$
$$948$$ −20.9986 −0.682002
$$949$$ −12.6253 −0.409835
$$950$$ 0 0
$$951$$ −10.1260 −0.328358
$$952$$ 2.57452 0.0834405
$$953$$ −40.9478 −1.32643 −0.663215 0.748429i $$-0.730808\pi$$
−0.663215 + 0.748429i $$0.730808\pi$$
$$954$$ 1.66291 0.0538388
$$955$$ 0 0
$$956$$ −11.4791 −0.371261
$$957$$ 15.8496 0.512343
$$958$$ −0.998585 −0.0322628
$$959$$ −22.4993 −0.726540
$$960$$ 0 0
$$961$$ −10.0738 −0.324962
$$962$$ 0.203141 0.00654953
$$963$$ −13.7381 −0.442705
$$964$$ 0.147616 0.00475440
$$965$$ 0 0
$$966$$ 0.962389 0.0309643
$$967$$ −38.2784 −1.23095 −0.615475 0.788157i $$-0.711036\pi$$
−0.615475 + 0.788157i $$0.711036\pi$$
$$968$$ −5.37916 −0.172893
$$969$$ 17.9248 0.575827
$$970$$ 0 0
$$971$$ −28.7269 −0.921889 −0.460945 0.887429i $$-0.652489\pi$$
−0.460945 + 0.887429i $$0.652489\pi$$
$$972$$ −1.96239 −0.0629436
$$973$$ −3.27504 −0.104993
$$974$$ −4.30089 −0.137809
$$975$$ 0 0
$$976$$ −32.8510 −1.05153
$$977$$ 41.3014 1.32135 0.660674 0.750673i $$-0.270270\pi$$
0.660674 + 0.750673i $$0.270270\pi$$
$$978$$ 1.02776 0.0328642
$$979$$ 2.07522 0.0663244
$$980$$ 0 0
$$981$$ −2.77575 −0.0886228
$$982$$ −0.387873 −0.0123775
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 2.87258 0.0915744
$$985$$ 0 0
$$986$$ −5.14903 −0.163979
$$987$$ −9.92478 −0.315909
$$988$$ 14.1768 0.451024
$$989$$ −62.6516 −1.99221
$$990$$ 0 0
$$991$$ 25.1002 0.797333 0.398666 0.917096i $$-0.369473\pi$$
0.398666 + 0.917096i $$0.369473\pi$$
$$992$$ −10.3803 −0.329575
$$993$$ 27.8496 0.883779
$$994$$ −0.387873 −0.0123026
$$995$$ 0 0
$$996$$ 6.32724 0.200486
$$997$$ −32.7974 −1.03870 −0.519351 0.854561i $$-0.673826\pi$$
−0.519351 + 0.854561i $$0.673826\pi$$
$$998$$ 1.27030 0.0402106
$$999$$ 0.775746 0.0245435
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 525.2.a.j.1.2 3
3.2 odd 2 1575.2.a.x.1.2 3
4.3 odd 2 8400.2.a.dg.1.1 3
5.2 odd 4 105.2.d.b.64.3 6
5.3 odd 4 105.2.d.b.64.4 yes 6
5.4 even 2 525.2.a.k.1.2 3
7.6 odd 2 3675.2.a.bi.1.2 3
15.2 even 4 315.2.d.e.64.4 6
15.8 even 4 315.2.d.e.64.3 6
15.14 odd 2 1575.2.a.w.1.2 3
20.3 even 4 1680.2.t.k.1009.1 6
20.7 even 4 1680.2.t.k.1009.4 6
20.19 odd 2 8400.2.a.dj.1.3 3
35.2 odd 12 735.2.q.e.214.3 12
35.3 even 12 735.2.q.f.79.3 12
35.12 even 12 735.2.q.f.214.3 12
35.13 even 4 735.2.d.b.589.4 6
35.17 even 12 735.2.q.f.79.4 12
35.18 odd 12 735.2.q.e.79.3 12
35.23 odd 12 735.2.q.e.214.4 12
35.27 even 4 735.2.d.b.589.3 6
35.32 odd 12 735.2.q.e.79.4 12
35.33 even 12 735.2.q.f.214.4 12
35.34 odd 2 3675.2.a.bj.1.2 3
60.23 odd 4 5040.2.t.v.1009.5 6
60.47 odd 4 5040.2.t.v.1009.6 6
105.62 odd 4 2205.2.d.l.1324.4 6
105.83 odd 4 2205.2.d.l.1324.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.3 6 5.2 odd 4
105.2.d.b.64.4 yes 6 5.3 odd 4
315.2.d.e.64.3 6 15.8 even 4
315.2.d.e.64.4 6 15.2 even 4
525.2.a.j.1.2 3 1.1 even 1 trivial
525.2.a.k.1.2 3 5.4 even 2
735.2.d.b.589.3 6 35.27 even 4
735.2.d.b.589.4 6 35.13 even 4
735.2.q.e.79.3 12 35.18 odd 12
735.2.q.e.79.4 12 35.32 odd 12
735.2.q.e.214.3 12 35.2 odd 12
735.2.q.e.214.4 12 35.23 odd 12
735.2.q.f.79.3 12 35.3 even 12
735.2.q.f.79.4 12 35.17 even 12
735.2.q.f.214.3 12 35.12 even 12
735.2.q.f.214.4 12 35.33 even 12
1575.2.a.w.1.2 3 15.14 odd 2
1575.2.a.x.1.2 3 3.2 odd 2
1680.2.t.k.1009.1 6 20.3 even 4
1680.2.t.k.1009.4 6 20.7 even 4
2205.2.d.l.1324.3 6 105.83 odd 4
2205.2.d.l.1324.4 6 105.62 odd 4
3675.2.a.bi.1.2 3 7.6 odd 2
3675.2.a.bj.1.2 3 35.34 odd 2
5040.2.t.v.1009.5 6 60.23 odd 4
5040.2.t.v.1009.6 6 60.47 odd 4
8400.2.a.dg.1.1 3 4.3 odd 2
8400.2.a.dj.1.3 3 20.19 odd 2