Properties

 Label 147.4.a.e.1.1 Level $147$ Weight $4$ Character 147.1 Self dual yes Analytic conductor $8.673$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$8.67328077084$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 147.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -12.0000 q^{5} -3.00000 q^{6} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} -12.0000 q^{5} -3.00000 q^{6} +15.0000 q^{8} +9.00000 q^{9} +12.0000 q^{10} +20.0000 q^{11} -21.0000 q^{12} +84.0000 q^{13} -36.0000 q^{15} +41.0000 q^{16} +96.0000 q^{17} -9.00000 q^{18} -12.0000 q^{19} +84.0000 q^{20} -20.0000 q^{22} -176.000 q^{23} +45.0000 q^{24} +19.0000 q^{25} -84.0000 q^{26} +27.0000 q^{27} +58.0000 q^{29} +36.0000 q^{30} +264.000 q^{31} -161.000 q^{32} +60.0000 q^{33} -96.0000 q^{34} -63.0000 q^{36} +258.000 q^{37} +12.0000 q^{38} +252.000 q^{39} -180.000 q^{40} +156.000 q^{43} -140.000 q^{44} -108.000 q^{45} +176.000 q^{46} +408.000 q^{47} +123.000 q^{48} -19.0000 q^{50} +288.000 q^{51} -588.000 q^{52} -722.000 q^{53} -27.0000 q^{54} -240.000 q^{55} -36.0000 q^{57} -58.0000 q^{58} -492.000 q^{59} +252.000 q^{60} +492.000 q^{61} -264.000 q^{62} -167.000 q^{64} -1008.00 q^{65} -60.0000 q^{66} +412.000 q^{67} -672.000 q^{68} -528.000 q^{69} +296.000 q^{71} +135.000 q^{72} -240.000 q^{73} -258.000 q^{74} +57.0000 q^{75} +84.0000 q^{76} -252.000 q^{78} +776.000 q^{79} -492.000 q^{80} +81.0000 q^{81} -924.000 q^{83} -1152.00 q^{85} -156.000 q^{86} +174.000 q^{87} +300.000 q^{88} +744.000 q^{89} +108.000 q^{90} +1232.00 q^{92} +792.000 q^{93} -408.000 q^{94} +144.000 q^{95} -483.000 q^{96} +168.000 q^{97} +180.000 q^{99} +O(q^{100})$$

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 −0.353553 −0.176777 0.984251i $$-0.556567\pi$$
−0.176777 + 0.984251i $$0.556567\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −7.00000 −0.875000
$$5$$ −12.0000 −1.07331 −0.536656 0.843801i $$-0.680313\pi$$
−0.536656 + 0.843801i $$0.680313\pi$$
$$6$$ −3.00000 −0.204124
$$7$$ 0 0
$$8$$ 15.0000 0.662913
$$9$$ 9.00000 0.333333
$$10$$ 12.0000 0.379473
$$11$$ 20.0000 0.548202 0.274101 0.961701i $$-0.411620\pi$$
0.274101 + 0.961701i $$0.411620\pi$$
$$12$$ −21.0000 −0.505181
$$13$$ 84.0000 1.79211 0.896054 0.443945i $$-0.146421\pi$$
0.896054 + 0.443945i $$0.146421\pi$$
$$14$$ 0 0
$$15$$ −36.0000 −0.619677
$$16$$ 41.0000 0.640625
$$17$$ 96.0000 1.36961 0.684806 0.728725i $$-0.259887\pi$$
0.684806 + 0.728725i $$0.259887\pi$$
$$18$$ −9.00000 −0.117851
$$19$$ −12.0000 −0.144894 −0.0724471 0.997372i $$-0.523081\pi$$
−0.0724471 + 0.997372i $$0.523081\pi$$
$$20$$ 84.0000 0.939149
$$21$$ 0 0
$$22$$ −20.0000 −0.193819
$$23$$ −176.000 −1.59559 −0.797794 0.602930i $$-0.794000\pi$$
−0.797794 + 0.602930i $$0.794000\pi$$
$$24$$ 45.0000 0.382733
$$25$$ 19.0000 0.152000
$$26$$ −84.0000 −0.633606
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 58.0000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 36.0000 0.219089
$$31$$ 264.000 1.52954 0.764771 0.644302i $$-0.222852\pi$$
0.764771 + 0.644302i $$0.222852\pi$$
$$32$$ −161.000 −0.889408
$$33$$ 60.0000 0.316505
$$34$$ −96.0000 −0.484231
$$35$$ 0 0
$$36$$ −63.0000 −0.291667
$$37$$ 258.000 1.14635 0.573175 0.819433i $$-0.305712\pi$$
0.573175 + 0.819433i $$0.305712\pi$$
$$38$$ 12.0000 0.0512278
$$39$$ 252.000 1.03467
$$40$$ −180.000 −0.711512
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 156.000 0.553251 0.276625 0.960978i $$-0.410784\pi$$
0.276625 + 0.960978i $$0.410784\pi$$
$$44$$ −140.000 −0.479677
$$45$$ −108.000 −0.357771
$$46$$ 176.000 0.564126
$$47$$ 408.000 1.26623 0.633116 0.774057i $$-0.281776\pi$$
0.633116 + 0.774057i $$0.281776\pi$$
$$48$$ 123.000 0.369865
$$49$$ 0 0
$$50$$ −19.0000 −0.0537401
$$51$$ 288.000 0.790746
$$52$$ −588.000 −1.56809
$$53$$ −722.000 −1.87121 −0.935607 0.353044i $$-0.885147\pi$$
−0.935607 + 0.353044i $$0.885147\pi$$
$$54$$ −27.0000 −0.0680414
$$55$$ −240.000 −0.588393
$$56$$ 0 0
$$57$$ −36.0000 −0.0836547
$$58$$ −58.0000 −0.131306
$$59$$ −492.000 −1.08564 −0.542822 0.839848i $$-0.682644\pi$$
−0.542822 + 0.839848i $$0.682644\pi$$
$$60$$ 252.000 0.542218
$$61$$ 492.000 1.03269 0.516345 0.856380i $$-0.327292\pi$$
0.516345 + 0.856380i $$0.327292\pi$$
$$62$$ −264.000 −0.540775
$$63$$ 0 0
$$64$$ −167.000 −0.326172
$$65$$ −1008.00 −1.92349
$$66$$ −60.0000 −0.111901
$$67$$ 412.000 0.751251 0.375625 0.926772i $$-0.377428\pi$$
0.375625 + 0.926772i $$0.377428\pi$$
$$68$$ −672.000 −1.19841
$$69$$ −528.000 −0.921213
$$70$$ 0 0
$$71$$ 296.000 0.494771 0.247385 0.968917i $$-0.420429\pi$$
0.247385 + 0.968917i $$0.420429\pi$$
$$72$$ 135.000 0.220971
$$73$$ −240.000 −0.384793 −0.192396 0.981317i $$-0.561626\pi$$
−0.192396 + 0.981317i $$0.561626\pi$$
$$74$$ −258.000 −0.405296
$$75$$ 57.0000 0.0877572
$$76$$ 84.0000 0.126782
$$77$$ 0 0
$$78$$ −252.000 −0.365813
$$79$$ 776.000 1.10515 0.552575 0.833463i $$-0.313645\pi$$
0.552575 + 0.833463i $$0.313645\pi$$
$$80$$ −492.000 −0.687591
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −924.000 −1.22195 −0.610977 0.791648i $$-0.709223\pi$$
−0.610977 + 0.791648i $$0.709223\pi$$
$$84$$ 0 0
$$85$$ −1152.00 −1.47002
$$86$$ −156.000 −0.195604
$$87$$ 174.000 0.214423
$$88$$ 300.000 0.363410
$$89$$ 744.000 0.886111 0.443055 0.896494i $$-0.353895\pi$$
0.443055 + 0.896494i $$0.353895\pi$$
$$90$$ 108.000 0.126491
$$91$$ 0 0
$$92$$ 1232.00 1.39614
$$93$$ 792.000 0.883081
$$94$$ −408.000 −0.447681
$$95$$ 144.000 0.155517
$$96$$ −483.000 −0.513500
$$97$$ 168.000 0.175854 0.0879269 0.996127i $$-0.471976\pi$$
0.0879269 + 0.996127i $$0.471976\pi$$
$$98$$ 0 0
$$99$$ 180.000 0.182734
$$100$$ −133.000 −0.133000
$$101$$ 1524.00 1.50142 0.750711 0.660630i $$-0.229711\pi$$
0.750711 + 0.660630i $$0.229711\pi$$
$$102$$ −288.000 −0.279571
$$103$$ 408.000 0.390305 0.195153 0.980773i $$-0.437480\pi$$
0.195153 + 0.980773i $$0.437480\pi$$
$$104$$ 1260.00 1.18801
$$105$$ 0 0
$$106$$ 722.000 0.661574
$$107$$ −820.000 −0.740863 −0.370432 0.928860i $$-0.620790\pi$$
−0.370432 + 0.928860i $$0.620790\pi$$
$$108$$ −189.000 −0.168394
$$109$$ −918.000 −0.806683 −0.403342 0.915050i $$-0.632151\pi$$
−0.403342 + 0.915050i $$0.632151\pi$$
$$110$$ 240.000 0.208028
$$111$$ 774.000 0.661845
$$112$$ 0 0
$$113$$ −110.000 −0.0915746 −0.0457873 0.998951i $$-0.514580\pi$$
−0.0457873 + 0.998951i $$0.514580\pi$$
$$114$$ 36.0000 0.0295764
$$115$$ 2112.00 1.71257
$$116$$ −406.000 −0.324967
$$117$$ 756.000 0.597369
$$118$$ 492.000 0.383833
$$119$$ 0 0
$$120$$ −540.000 −0.410792
$$121$$ −931.000 −0.699474
$$122$$ −492.000 −0.365111
$$123$$ 0 0
$$124$$ −1848.00 −1.33835
$$125$$ 1272.00 0.910169
$$126$$ 0 0
$$127$$ 16.0000 0.0111793 0.00558965 0.999984i $$-0.498221\pi$$
0.00558965 + 0.999984i $$0.498221\pi$$
$$128$$ 1455.00 1.00473
$$129$$ 468.000 0.319419
$$130$$ 1008.00 0.680057
$$131$$ −1692.00 −1.12848 −0.564239 0.825611i $$-0.690831\pi$$
−0.564239 + 0.825611i $$0.690831\pi$$
$$132$$ −420.000 −0.276942
$$133$$ 0 0
$$134$$ −412.000 −0.265607
$$135$$ −324.000 −0.206559
$$136$$ 1440.00 0.907934
$$137$$ 1126.00 0.702195 0.351097 0.936339i $$-0.385809\pi$$
0.351097 + 0.936339i $$0.385809\pi$$
$$138$$ 528.000 0.325698
$$139$$ −1092.00 −0.666347 −0.333173 0.942866i $$-0.608119\pi$$
−0.333173 + 0.942866i $$0.608119\pi$$
$$140$$ 0 0
$$141$$ 1224.00 0.731060
$$142$$ −296.000 −0.174928
$$143$$ 1680.00 0.982438
$$144$$ 369.000 0.213542
$$145$$ −696.000 −0.398618
$$146$$ 240.000 0.136045
$$147$$ 0 0
$$148$$ −1806.00 −1.00306
$$149$$ 1070.00 0.588307 0.294154 0.955758i $$-0.404962\pi$$
0.294154 + 0.955758i $$0.404962\pi$$
$$150$$ −57.0000 −0.0310269
$$151$$ −120.000 −0.0646719 −0.0323360 0.999477i $$-0.510295\pi$$
−0.0323360 + 0.999477i $$0.510295\pi$$
$$152$$ −180.000 −0.0960522
$$153$$ 864.000 0.456538
$$154$$ 0 0
$$155$$ −3168.00 −1.64168
$$156$$ −1764.00 −0.905340
$$157$$ −1836.00 −0.933304 −0.466652 0.884441i $$-0.654540\pi$$
−0.466652 + 0.884441i $$0.654540\pi$$
$$158$$ −776.000 −0.390729
$$159$$ −2166.00 −1.08035
$$160$$ 1932.00 0.954613
$$161$$ 0 0
$$162$$ −81.0000 −0.0392837
$$163$$ 916.000 0.440164 0.220082 0.975481i $$-0.429368\pi$$
0.220082 + 0.975481i $$0.429368\pi$$
$$164$$ 0 0
$$165$$ −720.000 −0.339709
$$166$$ 924.000 0.432026
$$167$$ −504.000 −0.233537 −0.116769 0.993159i $$-0.537254\pi$$
−0.116769 + 0.993159i $$0.537254\pi$$
$$168$$ 0 0
$$169$$ 4859.00 2.21165
$$170$$ 1152.00 0.519732
$$171$$ −108.000 −0.0482980
$$172$$ −1092.00 −0.484094
$$173$$ 1836.00 0.806870 0.403435 0.915008i $$-0.367816\pi$$
0.403435 + 0.915008i $$0.367816\pi$$
$$174$$ −174.000 −0.0758098
$$175$$ 0 0
$$176$$ 820.000 0.351192
$$177$$ −1476.00 −0.626796
$$178$$ −744.000 −0.313287
$$179$$ 2372.00 0.990456 0.495228 0.868763i $$-0.335085\pi$$
0.495228 + 0.868763i $$0.335085\pi$$
$$180$$ 756.000 0.313050
$$181$$ 1092.00 0.448440 0.224220 0.974539i $$-0.428017\pi$$
0.224220 + 0.974539i $$0.428017\pi$$
$$182$$ 0 0
$$183$$ 1476.00 0.596224
$$184$$ −2640.00 −1.05774
$$185$$ −3096.00 −1.23039
$$186$$ −792.000 −0.312216
$$187$$ 1920.00 0.750825
$$188$$ −2856.00 −1.10795
$$189$$ 0 0
$$190$$ −144.000 −0.0549835
$$191$$ 2512.00 0.951633 0.475817 0.879545i $$-0.342153\pi$$
0.475817 + 0.879545i $$0.342153\pi$$
$$192$$ −501.000 −0.188315
$$193$$ −2430.00 −0.906297 −0.453148 0.891435i $$-0.649699\pi$$
−0.453148 + 0.891435i $$0.649699\pi$$
$$194$$ −168.000 −0.0621737
$$195$$ −3024.00 −1.11053
$$196$$ 0 0
$$197$$ −1762.00 −0.637245 −0.318623 0.947882i $$-0.603220\pi$$
−0.318623 + 0.947882i $$0.603220\pi$$
$$198$$ −180.000 −0.0646063
$$199$$ −3096.00 −1.10286 −0.551431 0.834220i $$-0.685918\pi$$
−0.551431 + 0.834220i $$0.685918\pi$$
$$200$$ 285.000 0.100763
$$201$$ 1236.00 0.433735
$$202$$ −1524.00 −0.530833
$$203$$ 0 0
$$204$$ −2016.00 −0.691903
$$205$$ 0 0
$$206$$ −408.000 −0.137994
$$207$$ −1584.00 −0.531863
$$208$$ 3444.00 1.14807
$$209$$ −240.000 −0.0794313
$$210$$ 0 0
$$211$$ 156.000 0.0508980 0.0254490 0.999676i $$-0.491898\pi$$
0.0254490 + 0.999676i $$0.491898\pi$$
$$212$$ 5054.00 1.63731
$$213$$ 888.000 0.285656
$$214$$ 820.000 0.261935
$$215$$ −1872.00 −0.593811
$$216$$ 405.000 0.127578
$$217$$ 0 0
$$218$$ 918.000 0.285206
$$219$$ −720.000 −0.222160
$$220$$ 1680.00 0.514844
$$221$$ 8064.00 2.45449
$$222$$ −774.000 −0.233998
$$223$$ −5040.00 −1.51347 −0.756734 0.653723i $$-0.773206\pi$$
−0.756734 + 0.653723i $$0.773206\pi$$
$$224$$ 0 0
$$225$$ 171.000 0.0506667
$$226$$ 110.000 0.0323765
$$227$$ −2172.00 −0.635069 −0.317535 0.948247i $$-0.602855\pi$$
−0.317535 + 0.948247i $$0.602855\pi$$
$$228$$ 252.000 0.0731978
$$229$$ −2700.00 −0.779131 −0.389566 0.920999i $$-0.627375\pi$$
−0.389566 + 0.920999i $$0.627375\pi$$
$$230$$ −2112.00 −0.605483
$$231$$ 0 0
$$232$$ 870.000 0.246200
$$233$$ −3802.00 −1.06900 −0.534501 0.845168i $$-0.679500\pi$$
−0.534501 + 0.845168i $$0.679500\pi$$
$$234$$ −756.000 −0.211202
$$235$$ −4896.00 −1.35906
$$236$$ 3444.00 0.949938
$$237$$ 2328.00 0.638058
$$238$$ 0 0
$$239$$ −4408.00 −1.19301 −0.596506 0.802609i $$-0.703445\pi$$
−0.596506 + 0.802609i $$0.703445\pi$$
$$240$$ −1476.00 −0.396981
$$241$$ −3096.00 −0.827514 −0.413757 0.910387i $$-0.635784\pi$$
−0.413757 + 0.910387i $$0.635784\pi$$
$$242$$ 931.000 0.247301
$$243$$ 243.000 0.0641500
$$244$$ −3444.00 −0.903605
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1008.00 −0.259666
$$248$$ 3960.00 1.01395
$$249$$ −2772.00 −0.705495
$$250$$ −1272.00 −0.321793
$$251$$ 924.000 0.232360 0.116180 0.993228i $$-0.462935\pi$$
0.116180 + 0.993228i $$0.462935\pi$$
$$252$$ 0 0
$$253$$ −3520.00 −0.874706
$$254$$ −16.0000 −0.00395248
$$255$$ −3456.00 −0.848718
$$256$$ −119.000 −0.0290527
$$257$$ 2760.00 0.669899 0.334950 0.942236i $$-0.391281\pi$$
0.334950 + 0.942236i $$0.391281\pi$$
$$258$$ −468.000 −0.112932
$$259$$ 0 0
$$260$$ 7056.00 1.68306
$$261$$ 522.000 0.123797
$$262$$ 1692.00 0.398978
$$263$$ −2360.00 −0.553323 −0.276661 0.960967i $$-0.589228\pi$$
−0.276661 + 0.960967i $$0.589228\pi$$
$$264$$ 900.000 0.209815
$$265$$ 8664.00 2.00840
$$266$$ 0 0
$$267$$ 2232.00 0.511596
$$268$$ −2884.00 −0.657345
$$269$$ −4020.00 −0.911166 −0.455583 0.890193i $$-0.650569\pi$$
−0.455583 + 0.890193i $$0.650569\pi$$
$$270$$ 324.000 0.0730297
$$271$$ −4800.00 −1.07594 −0.537969 0.842965i $$-0.680808\pi$$
−0.537969 + 0.842965i $$0.680808\pi$$
$$272$$ 3936.00 0.877408
$$273$$ 0 0
$$274$$ −1126.00 −0.248263
$$275$$ 380.000 0.0833268
$$276$$ 3696.00 0.806062
$$277$$ 6446.00 1.39820 0.699102 0.715022i $$-0.253583\pi$$
0.699102 + 0.715022i $$0.253583\pi$$
$$278$$ 1092.00 0.235589
$$279$$ 2376.00 0.509847
$$280$$ 0 0
$$281$$ −2602.00 −0.552393 −0.276196 0.961101i $$-0.589074\pi$$
−0.276196 + 0.961101i $$0.589074\pi$$
$$282$$ −1224.00 −0.258469
$$283$$ 6900.00 1.44934 0.724669 0.689098i $$-0.241993\pi$$
0.724669 + 0.689098i $$0.241993\pi$$
$$284$$ −2072.00 −0.432925
$$285$$ 432.000 0.0897876
$$286$$ −1680.00 −0.347344
$$287$$ 0 0
$$288$$ −1449.00 −0.296469
$$289$$ 4303.00 0.875840
$$290$$ 696.000 0.140933
$$291$$ 504.000 0.101529
$$292$$ 1680.00 0.336694
$$293$$ 4452.00 0.887674 0.443837 0.896107i $$-0.353617\pi$$
0.443837 + 0.896107i $$0.353617\pi$$
$$294$$ 0 0
$$295$$ 5904.00 1.16523
$$296$$ 3870.00 0.759930
$$297$$ 540.000 0.105502
$$298$$ −1070.00 −0.207998
$$299$$ −14784.0 −2.85947
$$300$$ −399.000 −0.0767876
$$301$$ 0 0
$$302$$ 120.000 0.0228650
$$303$$ 4572.00 0.866847
$$304$$ −492.000 −0.0928228
$$305$$ −5904.00 −1.10840
$$306$$ −864.000 −0.161410
$$307$$ 2436.00 0.452866 0.226433 0.974027i $$-0.427294\pi$$
0.226433 + 0.974027i $$0.427294\pi$$
$$308$$ 0 0
$$309$$ 1224.00 0.225343
$$310$$ 3168.00 0.580420
$$311$$ 7488.00 1.36529 0.682646 0.730750i $$-0.260829\pi$$
0.682646 + 0.730750i $$0.260829\pi$$
$$312$$ 3780.00 0.685899
$$313$$ 1752.00 0.316386 0.158193 0.987408i $$-0.449433\pi$$
0.158193 + 0.987408i $$0.449433\pi$$
$$314$$ 1836.00 0.329973
$$315$$ 0 0
$$316$$ −5432.00 −0.967006
$$317$$ −1562.00 −0.276753 −0.138376 0.990380i $$-0.544188\pi$$
−0.138376 + 0.990380i $$0.544188\pi$$
$$318$$ 2166.00 0.381960
$$319$$ 1160.00 0.203597
$$320$$ 2004.00 0.350084
$$321$$ −2460.00 −0.427738
$$322$$ 0 0
$$323$$ −1152.00 −0.198449
$$324$$ −567.000 −0.0972222
$$325$$ 1596.00 0.272400
$$326$$ −916.000 −0.155621
$$327$$ −2754.00 −0.465739
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 720.000 0.120105
$$331$$ −7092.00 −1.17768 −0.588839 0.808250i $$-0.700415\pi$$
−0.588839 + 0.808250i $$0.700415\pi$$
$$332$$ 6468.00 1.06921
$$333$$ 2322.00 0.382117
$$334$$ 504.000 0.0825678
$$335$$ −4944.00 −0.806327
$$336$$ 0 0
$$337$$ 366.000 0.0591611 0.0295805 0.999562i $$-0.490583\pi$$
0.0295805 + 0.999562i $$0.490583\pi$$
$$338$$ −4859.00 −0.781937
$$339$$ −330.000 −0.0528706
$$340$$ 8064.00 1.28627
$$341$$ 5280.00 0.838499
$$342$$ 108.000 0.0170759
$$343$$ 0 0
$$344$$ 2340.00 0.366757
$$345$$ 6336.00 0.988750
$$346$$ −1836.00 −0.285272
$$347$$ −6364.00 −0.984546 −0.492273 0.870441i $$-0.663834\pi$$
−0.492273 + 0.870441i $$0.663834\pi$$
$$348$$ −1218.00 −0.187620
$$349$$ 10500.0 1.61046 0.805232 0.592960i $$-0.202041\pi$$
0.805232 + 0.592960i $$0.202041\pi$$
$$350$$ 0 0
$$351$$ 2268.00 0.344891
$$352$$ −3220.00 −0.487576
$$353$$ −408.000 −0.0615174 −0.0307587 0.999527i $$-0.509792\pi$$
−0.0307587 + 0.999527i $$0.509792\pi$$
$$354$$ 1476.00 0.221606
$$355$$ −3552.00 −0.531044
$$356$$ −5208.00 −0.775347
$$357$$ 0 0
$$358$$ −2372.00 −0.350179
$$359$$ −11936.0 −1.75476 −0.877379 0.479798i $$-0.840710\pi$$
−0.877379 + 0.479798i $$0.840710\pi$$
$$360$$ −1620.00 −0.237171
$$361$$ −6715.00 −0.979006
$$362$$ −1092.00 −0.158548
$$363$$ −2793.00 −0.403842
$$364$$ 0 0
$$365$$ 2880.00 0.413003
$$366$$ −1476.00 −0.210797
$$367$$ 2448.00 0.348187 0.174093 0.984729i $$-0.444301\pi$$
0.174093 + 0.984729i $$0.444301\pi$$
$$368$$ −7216.00 −1.02217
$$369$$ 0 0
$$370$$ 3096.00 0.435009
$$371$$ 0 0
$$372$$ −5544.00 −0.772696
$$373$$ 11374.0 1.57888 0.789442 0.613826i $$-0.210370\pi$$
0.789442 + 0.613826i $$0.210370\pi$$
$$374$$ −1920.00 −0.265457
$$375$$ 3816.00 0.525486
$$376$$ 6120.00 0.839401
$$377$$ 4872.00 0.665572
$$378$$ 0 0
$$379$$ −5892.00 −0.798553 −0.399277 0.916830i $$-0.630739\pi$$
−0.399277 + 0.916830i $$0.630739\pi$$
$$380$$ −1008.00 −0.136077
$$381$$ 48.0000 0.00645437
$$382$$ −2512.00 −0.336453
$$383$$ 10488.0 1.39925 0.699624 0.714511i $$-0.253351\pi$$
0.699624 + 0.714511i $$0.253351\pi$$
$$384$$ 4365.00 0.580079
$$385$$ 0 0
$$386$$ 2430.00 0.320424
$$387$$ 1404.00 0.184417
$$388$$ −1176.00 −0.153872
$$389$$ 4514.00 0.588352 0.294176 0.955751i $$-0.404955\pi$$
0.294176 + 0.955751i $$0.404955\pi$$
$$390$$ 3024.00 0.392631
$$391$$ −16896.0 −2.18534
$$392$$ 0 0
$$393$$ −5076.00 −0.651528
$$394$$ 1762.00 0.225300
$$395$$ −9312.00 −1.18617
$$396$$ −1260.00 −0.159892
$$397$$ 6036.00 0.763068 0.381534 0.924355i $$-0.375396\pi$$
0.381534 + 0.924355i $$0.375396\pi$$
$$398$$ 3096.00 0.389921
$$399$$ 0 0
$$400$$ 779.000 0.0973750
$$401$$ −6770.00 −0.843086 −0.421543 0.906808i $$-0.638511\pi$$
−0.421543 + 0.906808i $$0.638511\pi$$
$$402$$ −1236.00 −0.153348
$$403$$ 22176.0 2.74110
$$404$$ −10668.0 −1.31374
$$405$$ −972.000 −0.119257
$$406$$ 0 0
$$407$$ 5160.00 0.628432
$$408$$ 4320.00 0.524196
$$409$$ −12504.0 −1.51169 −0.755847 0.654748i $$-0.772775\pi$$
−0.755847 + 0.654748i $$0.772775\pi$$
$$410$$ 0 0
$$411$$ 3378.00 0.405412
$$412$$ −2856.00 −0.341517
$$413$$ 0 0
$$414$$ 1584.00 0.188042
$$415$$ 11088.0 1.31154
$$416$$ −13524.0 −1.59392
$$417$$ −3276.00 −0.384716
$$418$$ 240.000 0.0280832
$$419$$ −9492.00 −1.10672 −0.553359 0.832943i $$-0.686654\pi$$
−0.553359 + 0.832943i $$0.686654\pi$$
$$420$$ 0 0
$$421$$ 5182.00 0.599894 0.299947 0.953956i $$-0.403031\pi$$
0.299947 + 0.953956i $$0.403031\pi$$
$$422$$ −156.000 −0.0179952
$$423$$ 3672.00 0.422077
$$424$$ −10830.0 −1.24045
$$425$$ 1824.00 0.208181
$$426$$ −888.000 −0.100995
$$427$$ 0 0
$$428$$ 5740.00 0.648256
$$429$$ 5040.00 0.567211
$$430$$ 1872.00 0.209944
$$431$$ −5720.00 −0.639264 −0.319632 0.947542i $$-0.603559\pi$$
−0.319632 + 0.947542i $$0.603559\pi$$
$$432$$ 1107.00 0.123288
$$433$$ −13608.0 −1.51030 −0.755149 0.655554i $$-0.772435\pi$$
−0.755149 + 0.655554i $$0.772435\pi$$
$$434$$ 0 0
$$435$$ −2088.00 −0.230142
$$436$$ 6426.00 0.705848
$$437$$ 2112.00 0.231191
$$438$$ 720.000 0.0785455
$$439$$ −12864.0 −1.39855 −0.699277 0.714851i $$-0.746495\pi$$
−0.699277 + 0.714851i $$0.746495\pi$$
$$440$$ −3600.00 −0.390053
$$441$$ 0 0
$$442$$ −8064.00 −0.867795
$$443$$ −13252.0 −1.42127 −0.710634 0.703562i $$-0.751592\pi$$
−0.710634 + 0.703562i $$0.751592\pi$$
$$444$$ −5418.00 −0.579115
$$445$$ −8928.00 −0.951074
$$446$$ 5040.00 0.535092
$$447$$ 3210.00 0.339659
$$448$$ 0 0
$$449$$ 226.000 0.0237541 0.0118771 0.999929i $$-0.496219\pi$$
0.0118771 + 0.999929i $$0.496219\pi$$
$$450$$ −171.000 −0.0179134
$$451$$ 0 0
$$452$$ 770.000 0.0801278
$$453$$ −360.000 −0.0373384
$$454$$ 2172.00 0.224531
$$455$$ 0 0
$$456$$ −540.000 −0.0554557
$$457$$ −11334.0 −1.16014 −0.580068 0.814568i $$-0.696974\pi$$
−0.580068 + 0.814568i $$0.696974\pi$$
$$458$$ 2700.00 0.275464
$$459$$ 2592.00 0.263582
$$460$$ −14784.0 −1.49849
$$461$$ 1596.00 0.161243 0.0806216 0.996745i $$-0.474309\pi$$
0.0806216 + 0.996745i $$0.474309\pi$$
$$462$$ 0 0
$$463$$ 12728.0 1.27758 0.638791 0.769380i $$-0.279435\pi$$
0.638791 + 0.769380i $$0.279435\pi$$
$$464$$ 2378.00 0.237922
$$465$$ −9504.00 −0.947822
$$466$$ 3802.00 0.377949
$$467$$ 3012.00 0.298456 0.149228 0.988803i $$-0.452321\pi$$
0.149228 + 0.988803i $$0.452321\pi$$
$$468$$ −5292.00 −0.522698
$$469$$ 0 0
$$470$$ 4896.00 0.480501
$$471$$ −5508.00 −0.538843
$$472$$ −7380.00 −0.719687
$$473$$ 3120.00 0.303293
$$474$$ −2328.00 −0.225588
$$475$$ −228.000 −0.0220239
$$476$$ 0 0
$$477$$ −6498.00 −0.623738
$$478$$ 4408.00 0.421793
$$479$$ 4296.00 0.409790 0.204895 0.978784i $$-0.434315\pi$$
0.204895 + 0.978784i $$0.434315\pi$$
$$480$$ 5796.00 0.551146
$$481$$ 21672.0 2.05438
$$482$$ 3096.00 0.292570
$$483$$ 0 0
$$484$$ 6517.00 0.612040
$$485$$ −2016.00 −0.188746
$$486$$ −243.000 −0.0226805
$$487$$ −8184.00 −0.761504 −0.380752 0.924677i $$-0.624335\pi$$
−0.380752 + 0.924677i $$0.624335\pi$$
$$488$$ 7380.00 0.684584
$$489$$ 2748.00 0.254129
$$490$$ 0 0
$$491$$ −12164.0 −1.11803 −0.559016 0.829157i $$-0.688821\pi$$
−0.559016 + 0.829157i $$0.688821\pi$$
$$492$$ 0 0
$$493$$ 5568.00 0.508661
$$494$$ 1008.00 0.0918058
$$495$$ −2160.00 −0.196131
$$496$$ 10824.0 0.979863
$$497$$ 0 0
$$498$$ 2772.00 0.249430
$$499$$ 972.000 0.0871998 0.0435999 0.999049i $$-0.486117\pi$$
0.0435999 + 0.999049i $$0.486117\pi$$
$$500$$ −8904.00 −0.796398
$$501$$ −1512.00 −0.134833
$$502$$ −924.000 −0.0821517
$$503$$ −7728.00 −0.685039 −0.342519 0.939511i $$-0.611280\pi$$
−0.342519 + 0.939511i $$0.611280\pi$$
$$504$$ 0 0
$$505$$ −18288.0 −1.61150
$$506$$ 3520.00 0.309255
$$507$$ 14577.0 1.27690
$$508$$ −112.000 −0.00978188
$$509$$ −11604.0 −1.01049 −0.505244 0.862977i $$-0.668597\pi$$
−0.505244 + 0.862977i $$0.668597\pi$$
$$510$$ 3456.00 0.300067
$$511$$ 0 0
$$512$$ −11521.0 −0.994455
$$513$$ −324.000 −0.0278849
$$514$$ −2760.00 −0.236845
$$515$$ −4896.00 −0.418919
$$516$$ −3276.00 −0.279492
$$517$$ 8160.00 0.694152
$$518$$ 0 0
$$519$$ 5508.00 0.465847
$$520$$ −15120.0 −1.27511
$$521$$ 10848.0 0.912206 0.456103 0.889927i $$-0.349245\pi$$
0.456103 + 0.889927i $$0.349245\pi$$
$$522$$ −522.000 −0.0437688
$$523$$ 18132.0 1.51598 0.757989 0.652267i $$-0.226182\pi$$
0.757989 + 0.652267i $$0.226182\pi$$
$$524$$ 11844.0 0.987419
$$525$$ 0 0
$$526$$ 2360.00 0.195629
$$527$$ 25344.0 2.09488
$$528$$ 2460.00 0.202761
$$529$$ 18809.0 1.54590
$$530$$ −8664.00 −0.710076
$$531$$ −4428.00 −0.361881
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −2232.00 −0.180877
$$535$$ 9840.00 0.795178
$$536$$ 6180.00 0.498014
$$537$$ 7116.00 0.571840
$$538$$ 4020.00 0.322146
$$539$$ 0 0
$$540$$ 2268.00 0.180739
$$541$$ 6950.00 0.552318 0.276159 0.961112i $$-0.410938\pi$$
0.276159 + 0.961112i $$0.410938\pi$$
$$542$$ 4800.00 0.380402
$$543$$ 3276.00 0.258907
$$544$$ −15456.0 −1.21814
$$545$$ 11016.0 0.865823
$$546$$ 0 0
$$547$$ 17012.0 1.32976 0.664882 0.746949i $$-0.268482\pi$$
0.664882 + 0.746949i $$0.268482\pi$$
$$548$$ −7882.00 −0.614420
$$549$$ 4428.00 0.344230
$$550$$ −380.000 −0.0294605
$$551$$ −696.000 −0.0538123
$$552$$ −7920.00 −0.610684
$$553$$ 0 0
$$554$$ −6446.00 −0.494340
$$555$$ −9288.00 −0.710367
$$556$$ 7644.00 0.583054
$$557$$ 3926.00 0.298653 0.149327 0.988788i $$-0.452289\pi$$
0.149327 + 0.988788i $$0.452289\pi$$
$$558$$ −2376.00 −0.180258
$$559$$ 13104.0 0.991485
$$560$$ 0 0
$$561$$ 5760.00 0.433489
$$562$$ 2602.00 0.195300
$$563$$ 18828.0 1.40942 0.704712 0.709494i $$-0.251076\pi$$
0.704712 + 0.709494i $$0.251076\pi$$
$$564$$ −8568.00 −0.639677
$$565$$ 1320.00 0.0982882
$$566$$ −6900.00 −0.512418
$$567$$ 0 0
$$568$$ 4440.00 0.327990
$$569$$ 11990.0 0.883387 0.441693 0.897166i $$-0.354378\pi$$
0.441693 + 0.897166i $$0.354378\pi$$
$$570$$ −432.000 −0.0317447
$$571$$ −15716.0 −1.15183 −0.575914 0.817510i $$-0.695354\pi$$
−0.575914 + 0.817510i $$0.695354\pi$$
$$572$$ −11760.0 −0.859633
$$573$$ 7536.00 0.549426
$$574$$ 0 0
$$575$$ −3344.00 −0.242529
$$576$$ −1503.00 −0.108724
$$577$$ 13872.0 1.00086 0.500432 0.865776i $$-0.333174\pi$$
0.500432 + 0.865776i $$0.333174\pi$$
$$578$$ −4303.00 −0.309656
$$579$$ −7290.00 −0.523251
$$580$$ 4872.00 0.348791
$$581$$ 0 0
$$582$$ −504.000 −0.0358960
$$583$$ −14440.0 −1.02580
$$584$$ −3600.00 −0.255084
$$585$$ −9072.00 −0.641164
$$586$$ −4452.00 −0.313840
$$587$$ −8820.00 −0.620171 −0.310085 0.950709i $$-0.600358\pi$$
−0.310085 + 0.950709i $$0.600358\pi$$
$$588$$ 0 0
$$589$$ −3168.00 −0.221622
$$590$$ −5904.00 −0.411973
$$591$$ −5286.00 −0.367914
$$592$$ 10578.0 0.734380
$$593$$ 16872.0 1.16838 0.584191 0.811617i $$-0.301412\pi$$
0.584191 + 0.811617i $$0.301412\pi$$
$$594$$ −540.000 −0.0373005
$$595$$ 0 0
$$596$$ −7490.00 −0.514769
$$597$$ −9288.00 −0.636738
$$598$$ 14784.0 1.01097
$$599$$ −6056.00 −0.413091 −0.206545 0.978437i $$-0.566222\pi$$
−0.206545 + 0.978437i $$0.566222\pi$$
$$600$$ 855.000 0.0581754
$$601$$ −10752.0 −0.729756 −0.364878 0.931055i $$-0.618889\pi$$
−0.364878 + 0.931055i $$0.618889\pi$$
$$602$$ 0 0
$$603$$ 3708.00 0.250417
$$604$$ 840.000 0.0565879
$$605$$ 11172.0 0.750754
$$606$$ −4572.00 −0.306477
$$607$$ −20256.0 −1.35447 −0.677237 0.735765i $$-0.736823\pi$$
−0.677237 + 0.735765i $$0.736823\pi$$
$$608$$ 1932.00 0.128870
$$609$$ 0 0
$$610$$ 5904.00 0.391879
$$611$$ 34272.0 2.26923
$$612$$ −6048.00 −0.399470
$$613$$ −28190.0 −1.85740 −0.928698 0.370838i $$-0.879071\pi$$
−0.928698 + 0.370838i $$0.879071\pi$$
$$614$$ −2436.00 −0.160112
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 29318.0 1.91296 0.956482 0.291793i $$-0.0942518\pi$$
0.956482 + 0.291793i $$0.0942518\pi$$
$$618$$ −1224.00 −0.0796707
$$619$$ −24348.0 −1.58098 −0.790492 0.612473i $$-0.790175\pi$$
−0.790492 + 0.612473i $$0.790175\pi$$
$$620$$ 22176.0 1.43647
$$621$$ −4752.00 −0.307071
$$622$$ −7488.00 −0.482703
$$623$$ 0 0
$$624$$ 10332.0 0.662838
$$625$$ −17639.0 −1.12890
$$626$$ −1752.00 −0.111859
$$627$$ −720.000 −0.0458597
$$628$$ 12852.0 0.816641
$$629$$ 24768.0 1.57006
$$630$$ 0 0
$$631$$ −25184.0 −1.58884 −0.794421 0.607368i $$-0.792226\pi$$
−0.794421 + 0.607368i $$0.792226\pi$$
$$632$$ 11640.0 0.732618
$$633$$ 468.000 0.0293860
$$634$$ 1562.00 0.0978469
$$635$$ −192.000 −0.0119989
$$636$$ 15162.0 0.945303
$$637$$ 0 0
$$638$$ −1160.00 −0.0719825
$$639$$ 2664.00 0.164924
$$640$$ −17460.0 −1.07839
$$641$$ 32318.0 1.99140 0.995698 0.0926628i $$-0.0295379\pi$$
0.995698 + 0.0926628i $$0.0295379\pi$$
$$642$$ 2460.00 0.151228
$$643$$ 3948.00 0.242137 0.121068 0.992644i $$-0.461368\pi$$
0.121068 + 0.992644i $$0.461368\pi$$
$$644$$ 0 0
$$645$$ −5616.00 −0.342837
$$646$$ 1152.00 0.0701623
$$647$$ −13848.0 −0.841454 −0.420727 0.907187i $$-0.638225\pi$$
−0.420727 + 0.907187i $$0.638225\pi$$
$$648$$ 1215.00 0.0736570
$$649$$ −9840.00 −0.595152
$$650$$ −1596.00 −0.0963081
$$651$$ 0 0
$$652$$ −6412.00 −0.385143
$$653$$ −3158.00 −0.189253 −0.0946264 0.995513i $$-0.530166\pi$$
−0.0946264 + 0.995513i $$0.530166\pi$$
$$654$$ 2754.00 0.164663
$$655$$ 20304.0 1.21121
$$656$$ 0 0
$$657$$ −2160.00 −0.128264
$$658$$ 0 0
$$659$$ −24596.0 −1.45391 −0.726953 0.686687i $$-0.759064\pi$$
−0.726953 + 0.686687i $$0.759064\pi$$
$$660$$ 5040.00 0.297245
$$661$$ 15468.0 0.910190 0.455095 0.890443i $$-0.349605\pi$$
0.455095 + 0.890443i $$0.349605\pi$$
$$662$$ 7092.00 0.416372
$$663$$ 24192.0 1.41710
$$664$$ −13860.0 −0.810049
$$665$$ 0 0
$$666$$ −2322.00 −0.135099
$$667$$ −10208.0 −0.592587
$$668$$ 3528.00 0.204345
$$669$$ −15120.0 −0.873801
$$670$$ 4944.00 0.285080
$$671$$ 9840.00 0.566124
$$672$$ 0 0
$$673$$ 13470.0 0.771516 0.385758 0.922600i $$-0.373940\pi$$
0.385758 + 0.922600i $$0.373940\pi$$
$$674$$ −366.000 −0.0209166
$$675$$ 513.000 0.0292524
$$676$$ −34013.0 −1.93520
$$677$$ 9564.00 0.542946 0.271473 0.962446i $$-0.412489\pi$$
0.271473 + 0.962446i $$0.412489\pi$$
$$678$$ 330.000 0.0186926
$$679$$ 0 0
$$680$$ −17280.0 −0.974497
$$681$$ −6516.00 −0.366657
$$682$$ −5280.00 −0.296454
$$683$$ 13852.0 0.776035 0.388018 0.921652i $$-0.373160\pi$$
0.388018 + 0.921652i $$0.373160\pi$$
$$684$$ 756.000 0.0422608
$$685$$ −13512.0 −0.753674
$$686$$ 0 0
$$687$$ −8100.00 −0.449832
$$688$$ 6396.00 0.354426
$$689$$ −60648.0 −3.35342
$$690$$ −6336.00 −0.349576
$$691$$ 324.000 0.0178373 0.00891863 0.999960i $$-0.497161\pi$$
0.00891863 + 0.999960i $$0.497161\pi$$
$$692$$ −12852.0 −0.706011
$$693$$ 0 0
$$694$$ 6364.00 0.348090
$$695$$ 13104.0 0.715199
$$696$$ 2610.00 0.142143
$$697$$ 0 0
$$698$$ −10500.0 −0.569385
$$699$$ −11406.0 −0.617188
$$700$$ 0 0
$$701$$ 24922.0 1.34278 0.671392 0.741103i $$-0.265697\pi$$
0.671392 + 0.741103i $$0.265697\pi$$
$$702$$ −2268.00 −0.121938
$$703$$ −3096.00 −0.166099
$$704$$ −3340.00 −0.178808
$$705$$ −14688.0 −0.784655
$$706$$ 408.000 0.0217497
$$707$$ 0 0
$$708$$ 10332.0 0.548447
$$709$$ −17886.0 −0.947423 −0.473711 0.880680i $$-0.657086\pi$$
−0.473711 + 0.880680i $$0.657086\pi$$
$$710$$ 3552.00 0.187752
$$711$$ 6984.00 0.368383
$$712$$ 11160.0 0.587414
$$713$$ −46464.0 −2.44052
$$714$$ 0 0
$$715$$ −20160.0 −1.05446
$$716$$ −16604.0 −0.866649
$$717$$ −13224.0 −0.688786
$$718$$ 11936.0 0.620401
$$719$$ 6792.00 0.352293 0.176147 0.984364i $$-0.443637\pi$$
0.176147 + 0.984364i $$0.443637\pi$$
$$720$$ −4428.00 −0.229197
$$721$$ 0 0
$$722$$ 6715.00 0.346131
$$723$$ −9288.00 −0.477765
$$724$$ −7644.00 −0.392385
$$725$$ 1102.00 0.0564514
$$726$$ 2793.00 0.142780
$$727$$ −1512.00 −0.0771348 −0.0385674 0.999256i $$-0.512279\pi$$
−0.0385674 + 0.999256i $$0.512279\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ −2880.00 −0.146019
$$731$$ 14976.0 0.757739
$$732$$ −10332.0 −0.521696
$$733$$ 11244.0 0.566585 0.283292 0.959034i $$-0.408573\pi$$
0.283292 + 0.959034i $$0.408573\pi$$
$$734$$ −2448.00 −0.123103
$$735$$ 0 0
$$736$$ 28336.0 1.41913
$$737$$ 8240.00 0.411838
$$738$$ 0 0
$$739$$ −1996.00 −0.0993559 −0.0496780 0.998765i $$-0.515820\pi$$
−0.0496780 + 0.998765i $$0.515820\pi$$
$$740$$ 21672.0 1.07659
$$741$$ −3024.00 −0.149918
$$742$$ 0 0
$$743$$ −656.000 −0.0323907 −0.0161954 0.999869i $$-0.505155\pi$$
−0.0161954 + 0.999869i $$0.505155\pi$$
$$744$$ 11880.0 0.585406
$$745$$ −12840.0 −0.631438
$$746$$ −11374.0 −0.558219
$$747$$ −8316.00 −0.407318
$$748$$ −13440.0 −0.656972
$$749$$ 0 0
$$750$$ −3816.00 −0.185787
$$751$$ 1056.00 0.0513102 0.0256551 0.999671i $$-0.491833\pi$$
0.0256551 + 0.999671i $$0.491833\pi$$
$$752$$ 16728.0 0.811180
$$753$$ 2772.00 0.134153
$$754$$ −4872.00 −0.235315
$$755$$ 1440.00 0.0694132
$$756$$ 0 0
$$757$$ −18702.0 −0.897934 −0.448967 0.893548i $$-0.648208\pi$$
−0.448967 + 0.893548i $$0.648208\pi$$
$$758$$ 5892.00 0.282331
$$759$$ −10560.0 −0.505011
$$760$$ 2160.00 0.103094
$$761$$ −17904.0 −0.852851 −0.426425 0.904523i $$-0.640227\pi$$
−0.426425 + 0.904523i $$0.640227\pi$$
$$762$$ −48.0000 −0.00228196
$$763$$ 0 0
$$764$$ −17584.0 −0.832679
$$765$$ −10368.0 −0.490008
$$766$$ −10488.0 −0.494709
$$767$$ −41328.0 −1.94559
$$768$$ −357.000 −0.0167736
$$769$$ 7560.00 0.354513 0.177257 0.984165i $$-0.443278\pi$$
0.177257 + 0.984165i $$0.443278\pi$$
$$770$$ 0 0
$$771$$ 8280.00 0.386766
$$772$$ 17010.0 0.793009
$$773$$ 14292.0 0.665003 0.332502 0.943103i $$-0.392107\pi$$
0.332502 + 0.943103i $$0.392107\pi$$
$$774$$ −1404.00 −0.0652012
$$775$$ 5016.00 0.232490
$$776$$ 2520.00 0.116576
$$777$$ 0 0
$$778$$ −4514.00 −0.208014
$$779$$ 0 0
$$780$$ 21168.0 0.971713
$$781$$ 5920.00 0.271235
$$782$$ 16896.0 0.772634
$$783$$ 1566.00 0.0714742
$$784$$ 0 0
$$785$$ 22032.0 1.00173
$$786$$ 5076.00 0.230350
$$787$$ −26364.0 −1.19412 −0.597062 0.802195i $$-0.703665\pi$$
−0.597062 + 0.802195i $$0.703665\pi$$
$$788$$ 12334.0 0.557590
$$789$$ −7080.00 −0.319461
$$790$$ 9312.00 0.419375
$$791$$ 0 0
$$792$$ 2700.00 0.121137
$$793$$ 41328.0 1.85069
$$794$$ −6036.00 −0.269785
$$795$$ 25992.0 1.15955
$$796$$ 21672.0 0.965005
$$797$$ −17220.0 −0.765325 −0.382662 0.923888i $$-0.624993\pi$$
−0.382662 + 0.923888i $$0.624993\pi$$
$$798$$ 0 0
$$799$$ 39168.0 1.73425
$$800$$ −3059.00 −0.135190
$$801$$ 6696.00 0.295370
$$802$$ 6770.00 0.298076
$$803$$ −4800.00 −0.210944
$$804$$ −8652.00 −0.379518
$$805$$ 0 0
$$806$$ −22176.0 −0.969127
$$807$$ −12060.0 −0.526062
$$808$$ 22860.0 0.995312
$$809$$ 16442.0 0.714549 0.357274 0.933999i $$-0.383706\pi$$
0.357274 + 0.933999i $$0.383706\pi$$
$$810$$ 972.000 0.0421637
$$811$$ 31332.0 1.35662 0.678308 0.734778i $$-0.262714\pi$$
0.678308 + 0.734778i $$0.262714\pi$$
$$812$$ 0 0
$$813$$ −14400.0 −0.621193
$$814$$ −5160.00 −0.222184
$$815$$ −10992.0 −0.472433
$$816$$ 11808.0 0.506572
$$817$$ −1872.00 −0.0801628
$$818$$ 12504.0 0.534465
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −25810.0 −1.09717 −0.548584 0.836095i $$-0.684833\pi$$
−0.548584 + 0.836095i $$0.684833\pi$$
$$822$$ −3378.00 −0.143335
$$823$$ 12368.0 0.523841 0.261921 0.965089i $$-0.415644\pi$$
0.261921 + 0.965089i $$0.415644\pi$$
$$824$$ 6120.00 0.258738
$$825$$ 1140.00 0.0481087
$$826$$ 0 0
$$827$$ 6316.00 0.265573 0.132786 0.991145i $$-0.457608\pi$$
0.132786 + 0.991145i $$0.457608\pi$$
$$828$$ 11088.0 0.465380
$$829$$ 23868.0 0.999964 0.499982 0.866036i $$-0.333340\pi$$
0.499982 + 0.866036i $$0.333340\pi$$
$$830$$ −11088.0 −0.463699
$$831$$ 19338.0 0.807254
$$832$$ −14028.0 −0.584535
$$833$$ 0 0
$$834$$ 3276.00 0.136018
$$835$$ 6048.00 0.250658
$$836$$ 1680.00 0.0695024
$$837$$ 7128.00 0.294360
$$838$$ 9492.00 0.391284
$$839$$ 48216.0 1.98403 0.992015 0.126120i $$-0.0402524\pi$$
0.992015 + 0.126120i $$0.0402524\pi$$
$$840$$ 0 0
$$841$$ −21025.0 −0.862069
$$842$$ −5182.00 −0.212094
$$843$$ −7806.00 −0.318924
$$844$$ −1092.00 −0.0445358
$$845$$ −58308.0 −2.37379
$$846$$ −3672.00 −0.149227
$$847$$ 0 0
$$848$$ −29602.0 −1.19875
$$849$$ 20700.0 0.836775
$$850$$ −1824.00 −0.0736032
$$851$$ −45408.0 −1.82910
$$852$$ −6216.00 −0.249949
$$853$$ 27300.0 1.09582 0.547910 0.836537i $$-0.315424\pi$$
0.547910 + 0.836537i $$0.315424\pi$$
$$854$$ 0 0
$$855$$ 1296.00 0.0518389
$$856$$ −12300.0 −0.491128
$$857$$ −8640.00 −0.344384 −0.172192 0.985063i $$-0.555085\pi$$
−0.172192 + 0.985063i $$0.555085\pi$$
$$858$$ −5040.00 −0.200539
$$859$$ −24372.0 −0.968058 −0.484029 0.875052i $$-0.660827\pi$$
−0.484029 + 0.875052i $$0.660827\pi$$
$$860$$ 13104.0 0.519585
$$861$$ 0 0
$$862$$ 5720.00 0.226014
$$863$$ 2176.00 0.0858307 0.0429154 0.999079i $$-0.486335\pi$$
0.0429154 + 0.999079i $$0.486335\pi$$
$$864$$ −4347.00 −0.171167
$$865$$ −22032.0 −0.866024
$$866$$ 13608.0 0.533971
$$867$$ 12909.0 0.505666
$$868$$ 0 0
$$869$$ 15520.0 0.605846
$$870$$ 2088.00 0.0813676
$$871$$ 34608.0 1.34632
$$872$$ −13770.0 −0.534760
$$873$$ 1512.00 0.0586179
$$874$$ −2112.00 −0.0817385
$$875$$ 0 0
$$876$$ 5040.00 0.194390
$$877$$ −27574.0 −1.06170 −0.530848 0.847467i $$-0.678127\pi$$
−0.530848 + 0.847467i $$0.678127\pi$$
$$878$$ 12864.0 0.494464
$$879$$ 13356.0 0.512499
$$880$$ −9840.00 −0.376939
$$881$$ −16968.0 −0.648884 −0.324442 0.945906i $$-0.605176\pi$$
−0.324442 + 0.945906i $$0.605176\pi$$
$$882$$ 0 0
$$883$$ −1860.00 −0.0708879 −0.0354439 0.999372i $$-0.511285\pi$$
−0.0354439 + 0.999372i $$0.511285\pi$$
$$884$$ −56448.0 −2.14768
$$885$$ 17712.0 0.672748
$$886$$ 13252.0 0.502494
$$887$$ −2280.00 −0.0863077 −0.0431538 0.999068i $$-0.513741\pi$$
−0.0431538 + 0.999068i $$0.513741\pi$$
$$888$$ 11610.0 0.438746
$$889$$ 0 0
$$890$$ 8928.00 0.336255
$$891$$ 1620.00 0.0609114
$$892$$ 35280.0 1.32428
$$893$$ −4896.00 −0.183470
$$894$$ −3210.00 −0.120088
$$895$$ −28464.0 −1.06307
$$896$$ 0 0
$$897$$ −44352.0 −1.65091
$$898$$ −226.000 −0.00839835
$$899$$ 15312.0 0.568058
$$900$$ −1197.00 −0.0443333
$$901$$ −69312.0 −2.56284
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −1650.00 −0.0607060
$$905$$ −13104.0 −0.481317
$$906$$ 360.000 0.0132011
$$907$$ 36084.0 1.32100 0.660501 0.750825i $$-0.270344\pi$$
0.660501 + 0.750825i $$0.270344\pi$$
$$908$$ 15204.0 0.555686
$$909$$ 13716.0 0.500474
$$910$$ 0 0
$$911$$ 24152.0 0.878366 0.439183 0.898398i $$-0.355268\pi$$
0.439183 + 0.898398i $$0.355268\pi$$
$$912$$ −1476.00 −0.0535913
$$913$$ −18480.0 −0.669878
$$914$$ 11334.0 0.410170
$$915$$ −17712.0 −0.639935
$$916$$ 18900.0 0.681740
$$917$$ 0 0
$$918$$ −2592.00 −0.0931904
$$919$$ 36336.0 1.30426 0.652130 0.758108i $$-0.273876\pi$$
0.652130 + 0.758108i $$0.273876\pi$$
$$920$$ 31680.0 1.13528
$$921$$ 7308.00 0.261462
$$922$$ −1596.00 −0.0570081
$$923$$ 24864.0 0.886683
$$924$$ 0 0
$$925$$ 4902.00 0.174245
$$926$$ −12728.0 −0.451693
$$927$$ 3672.00 0.130102
$$928$$ −9338.00 −0.330318
$$929$$ −432.000 −0.0152567 −0.00762834 0.999971i $$-0.502428\pi$$
−0.00762834 + 0.999971i $$0.502428\pi$$
$$930$$ 9504.00 0.335106
$$931$$ 0 0
$$932$$ 26614.0 0.935376
$$933$$ 22464.0 0.788251
$$934$$ −3012.00 −0.105520
$$935$$ −23040.0 −0.805870
$$936$$ 11340.0 0.396004
$$937$$ −22176.0 −0.773168 −0.386584 0.922254i $$-0.626345\pi$$
−0.386584 + 0.922254i $$0.626345\pi$$
$$938$$ 0 0
$$939$$ 5256.00 0.182666
$$940$$ 34272.0 1.18918
$$941$$ 43524.0 1.50780 0.753901 0.656988i $$-0.228170\pi$$
0.753901 + 0.656988i $$0.228170\pi$$
$$942$$ 5508.00 0.190510
$$943$$ 0 0
$$944$$ −20172.0 −0.695490
$$945$$ 0 0
$$946$$ −3120.00 −0.107230
$$947$$ 1868.00 0.0640991 0.0320495 0.999486i $$-0.489797\pi$$
0.0320495 + 0.999486i $$0.489797\pi$$
$$948$$ −16296.0 −0.558301
$$949$$ −20160.0 −0.689590
$$950$$ 228.000 0.00778663
$$951$$ −4686.00 −0.159783
$$952$$ 0 0
$$953$$ −9238.00 −0.314006 −0.157003 0.987598i $$-0.550183\pi$$
−0.157003 + 0.987598i $$0.550183\pi$$
$$954$$ 6498.00 0.220525
$$955$$ −30144.0 −1.02140
$$956$$ 30856.0 1.04389
$$957$$ 3480.00 0.117547
$$958$$ −4296.00 −0.144883
$$959$$ 0 0
$$960$$ 6012.00 0.202121
$$961$$ 39905.0 1.33950
$$962$$ −21672.0 −0.726334
$$963$$ −7380.00 −0.246954
$$964$$ 21672.0 0.724075
$$965$$ 29160.0 0.972739
$$966$$ 0 0
$$967$$ −30616.0 −1.01814 −0.509071 0.860724i $$-0.670011\pi$$
−0.509071 + 0.860724i $$0.670011\pi$$
$$968$$ −13965.0 −0.463690
$$969$$ −3456.00 −0.114575
$$970$$ 2016.00 0.0667318
$$971$$ 27540.0 0.910196 0.455098 0.890441i $$-0.349604\pi$$
0.455098 + 0.890441i $$0.349604\pi$$
$$972$$ −1701.00 −0.0561313
$$973$$ 0 0
$$974$$ 8184.00 0.269232
$$975$$ 4788.00 0.157270
$$976$$ 20172.0 0.661568
$$977$$ −16402.0 −0.537100 −0.268550 0.963266i $$-0.586544\pi$$
−0.268550 + 0.963266i $$0.586544\pi$$
$$978$$ −2748.00 −0.0898480
$$979$$ 14880.0 0.485768
$$980$$ 0 0
$$981$$ −8262.00 −0.268894
$$982$$ 12164.0 0.395284
$$983$$ −55176.0 −1.79028 −0.895138 0.445789i $$-0.852923\pi$$
−0.895138 + 0.445789i $$0.852923\pi$$
$$984$$ 0 0
$$985$$ 21144.0 0.683963
$$986$$ −5568.00 −0.179839
$$987$$ 0 0
$$988$$ 7056.00 0.227208
$$989$$ −27456.0 −0.882760
$$990$$ 2160.00 0.0693427
$$991$$ 27096.0 0.868550 0.434275 0.900780i $$-0.357005\pi$$
0.434275 + 0.900780i $$0.357005\pi$$
$$992$$ −42504.0 −1.36039
$$993$$ −21276.0 −0.679933
$$994$$ 0 0
$$995$$ 37152.0 1.18372
$$996$$ 19404.0 0.617309
$$997$$ 16812.0 0.534044 0.267022 0.963691i $$-0.413960\pi$$
0.267022 + 0.963691i $$0.413960\pi$$
$$998$$ −972.000 −0.0308298
$$999$$ 6966.00 0.220615
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 147.4.a.e.1.1 yes 1
3.2 odd 2 441.4.a.h.1.1 1
4.3 odd 2 2352.4.a.b.1.1 1
7.2 even 3 147.4.e.e.67.1 2
7.3 odd 6 147.4.e.f.79.1 2
7.4 even 3 147.4.e.e.79.1 2
7.5 odd 6 147.4.e.f.67.1 2
7.6 odd 2 147.4.a.d.1.1 1
21.2 odd 6 441.4.e.f.361.1 2
21.5 even 6 441.4.e.g.361.1 2
21.11 odd 6 441.4.e.f.226.1 2
21.17 even 6 441.4.e.g.226.1 2
21.20 even 2 441.4.a.g.1.1 1
28.27 even 2 2352.4.a.bi.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.d.1.1 1 7.6 odd 2
147.4.a.e.1.1 yes 1 1.1 even 1 trivial
147.4.e.e.67.1 2 7.2 even 3
147.4.e.e.79.1 2 7.4 even 3
147.4.e.f.67.1 2 7.5 odd 6
147.4.e.f.79.1 2 7.3 odd 6
441.4.a.g.1.1 1 21.20 even 2
441.4.a.h.1.1 1 3.2 odd 2
441.4.e.f.226.1 2 21.11 odd 6
441.4.e.f.361.1 2 21.2 odd 6
441.4.e.g.226.1 2 21.17 even 6
441.4.e.g.361.1 2 21.5 even 6
2352.4.a.b.1.1 1 4.3 odd 2
2352.4.a.bi.1.1 1 28.27 even 2