Properties

Label 3381.2.a.bj.1.9
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(2.51520\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.51520 q^{2} +1.00000 q^{3} +4.32625 q^{4} -4.31735 q^{5} +2.51520 q^{6} +5.85100 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.51520 q^{2} +1.00000 q^{3} +4.32625 q^{4} -4.31735 q^{5} +2.51520 q^{6} +5.85100 q^{8} +1.00000 q^{9} -10.8590 q^{10} +1.72915 q^{11} +4.32625 q^{12} +2.88941 q^{13} -4.31735 q^{15} +6.06395 q^{16} +4.93124 q^{17} +2.51520 q^{18} -2.44156 q^{19} -18.6780 q^{20} +4.34917 q^{22} +1.00000 q^{23} +5.85100 q^{24} +13.6396 q^{25} +7.26746 q^{26} +1.00000 q^{27} +8.05817 q^{29} -10.8590 q^{30} -0.915859 q^{31} +3.55007 q^{32} +1.72915 q^{33} +12.4031 q^{34} +4.32625 q^{36} -9.15581 q^{37} -6.14102 q^{38} +2.88941 q^{39} -25.2608 q^{40} +6.60783 q^{41} +10.7440 q^{43} +7.48074 q^{44} -4.31735 q^{45} +2.51520 q^{46} +0.399649 q^{47} +6.06395 q^{48} +34.3063 q^{50} +4.93124 q^{51} +12.5003 q^{52} +6.54512 q^{53} +2.51520 q^{54} -7.46536 q^{55} -2.44156 q^{57} +20.2679 q^{58} +1.83508 q^{59} -18.6780 q^{60} +1.30126 q^{61} -2.30357 q^{62} -3.19874 q^{64} -12.4746 q^{65} +4.34917 q^{66} -10.2140 q^{67} +21.3338 q^{68} +1.00000 q^{69} -9.00190 q^{71} +5.85100 q^{72} +2.61716 q^{73} -23.0287 q^{74} +13.6396 q^{75} -10.5628 q^{76} +7.26746 q^{78} +10.1870 q^{79} -26.1802 q^{80} +1.00000 q^{81} +16.6200 q^{82} +10.8838 q^{83} -21.2899 q^{85} +27.0234 q^{86} +8.05817 q^{87} +10.1173 q^{88} -14.9862 q^{89} -10.8590 q^{90} +4.32625 q^{92} -0.915859 q^{93} +1.00520 q^{94} +10.5411 q^{95} +3.55007 q^{96} +3.07116 q^{97} +1.72915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} + 10q^{3} + 15q^{4} + 5q^{5} + 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} + 10q^{3} + 15q^{4} + 5q^{5} + 3q^{6} + 9q^{8} + 10q^{9} - 11q^{10} + 8q^{11} + 15q^{12} + 5q^{15} + 37q^{16} + 11q^{17} + 3q^{18} - q^{19} + 15q^{20} + 6q^{22} + 10q^{23} + 9q^{24} + 21q^{25} - q^{26} + 10q^{27} + 22q^{29} - 11q^{30} + 3q^{31} + 11q^{32} + 8q^{33} + 3q^{34} + 15q^{36} - 3q^{37} - 16q^{38} - 39q^{40} + 26q^{41} + 27q^{43} + 16q^{44} + 5q^{45} + 3q^{46} - 11q^{47} + 37q^{48} + 2q^{50} + 11q^{51} - 29q^{52} + 5q^{53} + 3q^{54} + 18q^{55} - q^{57} + 16q^{58} + 10q^{59} + 15q^{60} - 22q^{61} + 32q^{62} + 69q^{64} - 11q^{65} + 6q^{66} - 2q^{67} + 21q^{68} + 10q^{69} + 27q^{71} + 9q^{72} + 8q^{73} + 14q^{74} + 21q^{75} + 22q^{76} - q^{78} + 21q^{79} + 53q^{80} + 10q^{81} - 36q^{82} + 12q^{83} + 23q^{85} + 18q^{86} + 22q^{87} - 10q^{88} - 6q^{89} - 11q^{90} + 15q^{92} + 3q^{93} - 35q^{94} + 44q^{95} + 11q^{96} - 6q^{97} + 8q^{99} + O(q^{100}) \)

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 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51520 1.77852 0.889259 0.457404i \(-0.151221\pi\)
0.889259 + 0.457404i \(0.151221\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.32625 2.16313
\(5\) −4.31735 −1.93078 −0.965390 0.260811i \(-0.916010\pi\)
−0.965390 + 0.260811i \(0.916010\pi\)
\(6\) 2.51520 1.02683
\(7\) 0 0
\(8\) 5.85100 2.06864
\(9\) 1.00000 0.333333
\(10\) −10.8590 −3.43393
\(11\) 1.72915 0.521359 0.260679 0.965425i \(-0.416053\pi\)
0.260679 + 0.965425i \(0.416053\pi\)
\(12\) 4.32625 1.24888
\(13\) 2.88941 0.801378 0.400689 0.916214i \(-0.368771\pi\)
0.400689 + 0.916214i \(0.368771\pi\)
\(14\) 0 0
\(15\) −4.31735 −1.11474
\(16\) 6.06395 1.51599
\(17\) 4.93124 1.19600 0.598001 0.801495i \(-0.295962\pi\)
0.598001 + 0.801495i \(0.295962\pi\)
\(18\) 2.51520 0.592839
\(19\) −2.44156 −0.560132 −0.280066 0.959981i \(-0.590356\pi\)
−0.280066 + 0.959981i \(0.590356\pi\)
\(20\) −18.6780 −4.17652
\(21\) 0 0
\(22\) 4.34917 0.927245
\(23\) 1.00000 0.208514
\(24\) 5.85100 1.19433
\(25\) 13.6396 2.72791
\(26\) 7.26746 1.42527
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.05817 1.49636 0.748182 0.663493i \(-0.230927\pi\)
0.748182 + 0.663493i \(0.230927\pi\)
\(30\) −10.8590 −1.98258
\(31\) −0.915859 −0.164493 −0.0822466 0.996612i \(-0.526210\pi\)
−0.0822466 + 0.996612i \(0.526210\pi\)
\(32\) 3.55007 0.627570
\(33\) 1.72915 0.301007
\(34\) 12.4031 2.12711
\(35\) 0 0
\(36\) 4.32625 0.721042
\(37\) −9.15581 −1.50521 −0.752603 0.658475i \(-0.771202\pi\)
−0.752603 + 0.658475i \(0.771202\pi\)
\(38\) −6.14102 −0.996205
\(39\) 2.88941 0.462676
\(40\) −25.2608 −3.99409
\(41\) 6.60783 1.03197 0.515985 0.856598i \(-0.327426\pi\)
0.515985 + 0.856598i \(0.327426\pi\)
\(42\) 0 0
\(43\) 10.7440 1.63845 0.819223 0.573475i \(-0.194405\pi\)
0.819223 + 0.573475i \(0.194405\pi\)
\(44\) 7.48074 1.12776
\(45\) −4.31735 −0.643593
\(46\) 2.51520 0.370847
\(47\) 0.399649 0.0582948 0.0291474 0.999575i \(-0.490721\pi\)
0.0291474 + 0.999575i \(0.490721\pi\)
\(48\) 6.06395 0.875255
\(49\) 0 0
\(50\) 34.3063 4.85164
\(51\) 4.93124 0.690512
\(52\) 12.5003 1.73348
\(53\) 6.54512 0.899042 0.449521 0.893270i \(-0.351595\pi\)
0.449521 + 0.893270i \(0.351595\pi\)
\(54\) 2.51520 0.342276
\(55\) −7.46536 −1.00663
\(56\) 0 0
\(57\) −2.44156 −0.323393
\(58\) 20.2679 2.66131
\(59\) 1.83508 0.238907 0.119454 0.992840i \(-0.461886\pi\)
0.119454 + 0.992840i \(0.461886\pi\)
\(60\) −18.6780 −2.41131
\(61\) 1.30126 0.166609 0.0833045 0.996524i \(-0.473453\pi\)
0.0833045 + 0.996524i \(0.473453\pi\)
\(62\) −2.30357 −0.292554
\(63\) 0 0
\(64\) −3.19874 −0.399843
\(65\) −12.4746 −1.54728
\(66\) 4.34917 0.535345
\(67\) −10.2140 −1.24784 −0.623918 0.781490i \(-0.714460\pi\)
−0.623918 + 0.781490i \(0.714460\pi\)
\(68\) 21.3338 2.58710
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −9.00190 −1.06833 −0.534164 0.845381i \(-0.679374\pi\)
−0.534164 + 0.845381i \(0.679374\pi\)
\(72\) 5.85100 0.689547
\(73\) 2.61716 0.306315 0.153158 0.988202i \(-0.451056\pi\)
0.153158 + 0.988202i \(0.451056\pi\)
\(74\) −23.0287 −2.67703
\(75\) 13.6396 1.57496
\(76\) −10.5628 −1.21164
\(77\) 0 0
\(78\) 7.26746 0.822877
\(79\) 10.1870 1.14613 0.573064 0.819510i \(-0.305755\pi\)
0.573064 + 0.819510i \(0.305755\pi\)
\(80\) −26.1802 −2.92704
\(81\) 1.00000 0.111111
\(82\) 16.6200 1.83538
\(83\) 10.8838 1.19465 0.597327 0.801998i \(-0.296229\pi\)
0.597327 + 0.801998i \(0.296229\pi\)
\(84\) 0 0
\(85\) −21.2899 −2.30922
\(86\) 27.0234 2.91400
\(87\) 8.05817 0.863926
\(88\) 10.1173 1.07850
\(89\) −14.9862 −1.58854 −0.794269 0.607566i \(-0.792146\pi\)
−0.794269 + 0.607566i \(0.792146\pi\)
\(90\) −10.8590 −1.14464
\(91\) 0 0
\(92\) 4.32625 0.451043
\(93\) −0.915859 −0.0949702
\(94\) 1.00520 0.103678
\(95\) 10.5411 1.08149
\(96\) 3.55007 0.362328
\(97\) 3.07116 0.311829 0.155914 0.987771i \(-0.450168\pi\)
0.155914 + 0.987771i \(0.450168\pi\)
\(98\) 0 0
\(99\) 1.72915 0.173786
\(100\) 59.0081 5.90081
\(101\) −1.08810 −0.108270 −0.0541349 0.998534i \(-0.517240\pi\)
−0.0541349 + 0.998534i \(0.517240\pi\)
\(102\) 12.4031 1.22809
\(103\) 3.23110 0.318369 0.159185 0.987249i \(-0.449113\pi\)
0.159185 + 0.987249i \(0.449113\pi\)
\(104\) 16.9059 1.65776
\(105\) 0 0
\(106\) 16.4623 1.59896
\(107\) −0.936899 −0.0905734 −0.0452867 0.998974i \(-0.514420\pi\)
−0.0452867 + 0.998974i \(0.514420\pi\)
\(108\) 4.32625 0.416294
\(109\) −14.9782 −1.43465 −0.717324 0.696739i \(-0.754633\pi\)
−0.717324 + 0.696739i \(0.754633\pi\)
\(110\) −18.7769 −1.79031
\(111\) −9.15581 −0.869031
\(112\) 0 0
\(113\) −14.1424 −1.33040 −0.665201 0.746665i \(-0.731654\pi\)
−0.665201 + 0.746665i \(0.731654\pi\)
\(114\) −6.14102 −0.575159
\(115\) −4.31735 −0.402595
\(116\) 34.8617 3.23682
\(117\) 2.88941 0.267126
\(118\) 4.61560 0.424901
\(119\) 0 0
\(120\) −25.2608 −2.30599
\(121\) −8.01004 −0.728185
\(122\) 3.27293 0.296317
\(123\) 6.60783 0.595808
\(124\) −3.96224 −0.355819
\(125\) −37.3000 −3.33621
\(126\) 0 0
\(127\) −1.77968 −0.157921 −0.0789604 0.996878i \(-0.525160\pi\)
−0.0789604 + 0.996878i \(0.525160\pi\)
\(128\) −15.1456 −1.33870
\(129\) 10.7440 0.945957
\(130\) −31.3762 −2.75187
\(131\) −8.89264 −0.776954 −0.388477 0.921459i \(-0.626999\pi\)
−0.388477 + 0.921459i \(0.626999\pi\)
\(132\) 7.48074 0.651115
\(133\) 0 0
\(134\) −25.6902 −2.21930
\(135\) −4.31735 −0.371579
\(136\) 28.8527 2.47410
\(137\) 0.247404 0.0211371 0.0105686 0.999944i \(-0.496636\pi\)
0.0105686 + 0.999944i \(0.496636\pi\)
\(138\) 2.51520 0.214108
\(139\) 8.59949 0.729399 0.364699 0.931125i \(-0.381172\pi\)
0.364699 + 0.931125i \(0.381172\pi\)
\(140\) 0 0
\(141\) 0.399649 0.0336565
\(142\) −22.6416 −1.90004
\(143\) 4.99623 0.417805
\(144\) 6.06395 0.505329
\(145\) −34.7900 −2.88915
\(146\) 6.58268 0.544787
\(147\) 0 0
\(148\) −39.6103 −3.25595
\(149\) 4.50669 0.369202 0.184601 0.982814i \(-0.440901\pi\)
0.184601 + 0.982814i \(0.440901\pi\)
\(150\) 34.3063 2.80109
\(151\) −4.52766 −0.368455 −0.184228 0.982884i \(-0.558978\pi\)
−0.184228 + 0.982884i \(0.558978\pi\)
\(152\) −14.2856 −1.15871
\(153\) 4.93124 0.398667
\(154\) 0 0
\(155\) 3.95409 0.317600
\(156\) 12.5003 1.00083
\(157\) 3.26727 0.260757 0.130378 0.991464i \(-0.458381\pi\)
0.130378 + 0.991464i \(0.458381\pi\)
\(158\) 25.6224 2.03841
\(159\) 6.54512 0.519062
\(160\) −15.3269 −1.21170
\(161\) 0 0
\(162\) 2.51520 0.197613
\(163\) 10.6420 0.833549 0.416774 0.909010i \(-0.363160\pi\)
0.416774 + 0.909010i \(0.363160\pi\)
\(164\) 28.5871 2.23228
\(165\) −7.46536 −0.581177
\(166\) 27.3750 2.12471
\(167\) −2.99812 −0.232001 −0.116001 0.993249i \(-0.537007\pi\)
−0.116001 + 0.993249i \(0.537007\pi\)
\(168\) 0 0
\(169\) −4.65131 −0.357793
\(170\) −53.5485 −4.10698
\(171\) −2.44156 −0.186711
\(172\) 46.4813 3.54416
\(173\) 13.2996 1.01115 0.505576 0.862782i \(-0.331280\pi\)
0.505576 + 0.862782i \(0.331280\pi\)
\(174\) 20.2679 1.53651
\(175\) 0 0
\(176\) 10.4855 0.790373
\(177\) 1.83508 0.137933
\(178\) −37.6934 −2.82524
\(179\) 19.6297 1.46719 0.733596 0.679585i \(-0.237840\pi\)
0.733596 + 0.679585i \(0.237840\pi\)
\(180\) −18.6780 −1.39217
\(181\) −19.2095 −1.42783 −0.713915 0.700232i \(-0.753080\pi\)
−0.713915 + 0.700232i \(0.753080\pi\)
\(182\) 0 0
\(183\) 1.30126 0.0961917
\(184\) 5.85100 0.431341
\(185\) 39.5289 2.90622
\(186\) −2.30357 −0.168906
\(187\) 8.52686 0.623546
\(188\) 1.72898 0.126099
\(189\) 0 0
\(190\) 26.5130 1.92345
\(191\) −7.15375 −0.517627 −0.258814 0.965927i \(-0.583332\pi\)
−0.258814 + 0.965927i \(0.583332\pi\)
\(192\) −3.19874 −0.230849
\(193\) −21.9573 −1.58052 −0.790259 0.612772i \(-0.790054\pi\)
−0.790259 + 0.612772i \(0.790054\pi\)
\(194\) 7.72458 0.554593
\(195\) −12.4746 −0.893325
\(196\) 0 0
\(197\) 3.21028 0.228723 0.114361 0.993439i \(-0.463518\pi\)
0.114361 + 0.993439i \(0.463518\pi\)
\(198\) 4.34917 0.309082
\(199\) −12.7170 −0.901483 −0.450741 0.892655i \(-0.648840\pi\)
−0.450741 + 0.892655i \(0.648840\pi\)
\(200\) 79.8050 5.64306
\(201\) −10.2140 −0.720438
\(202\) −2.73679 −0.192560
\(203\) 0 0
\(204\) 21.3338 1.49366
\(205\) −28.5283 −1.99251
\(206\) 8.12686 0.566225
\(207\) 1.00000 0.0695048
\(208\) 17.5212 1.21488
\(209\) −4.22183 −0.292030
\(210\) 0 0
\(211\) −17.0500 −1.17377 −0.586884 0.809671i \(-0.699646\pi\)
−0.586884 + 0.809671i \(0.699646\pi\)
\(212\) 28.3159 1.94474
\(213\) −9.00190 −0.616800
\(214\) −2.35649 −0.161086
\(215\) −46.3857 −3.16348
\(216\) 5.85100 0.398110
\(217\) 0 0
\(218\) −37.6732 −2.55155
\(219\) 2.61716 0.176851
\(220\) −32.2970 −2.17746
\(221\) 14.2484 0.958450
\(222\) −23.0287 −1.54559
\(223\) −23.3278 −1.56215 −0.781073 0.624440i \(-0.785327\pi\)
−0.781073 + 0.624440i \(0.785327\pi\)
\(224\) 0 0
\(225\) 13.6396 0.909303
\(226\) −35.5709 −2.36614
\(227\) 7.67594 0.509470 0.254735 0.967011i \(-0.418012\pi\)
0.254735 + 0.967011i \(0.418012\pi\)
\(228\) −10.5628 −0.699539
\(229\) 22.8736 1.51153 0.755766 0.654842i \(-0.227265\pi\)
0.755766 + 0.654842i \(0.227265\pi\)
\(230\) −10.8590 −0.716023
\(231\) 0 0
\(232\) 47.1483 3.09544
\(233\) 23.5559 1.54320 0.771599 0.636110i \(-0.219457\pi\)
0.771599 + 0.636110i \(0.219457\pi\)
\(234\) 7.26746 0.475089
\(235\) −1.72543 −0.112554
\(236\) 7.93902 0.516786
\(237\) 10.1870 0.661718
\(238\) 0 0
\(239\) 13.1829 0.852732 0.426366 0.904551i \(-0.359794\pi\)
0.426366 + 0.904551i \(0.359794\pi\)
\(240\) −26.1802 −1.68993
\(241\) 17.2670 1.11227 0.556134 0.831093i \(-0.312284\pi\)
0.556134 + 0.831093i \(0.312284\pi\)
\(242\) −20.1469 −1.29509
\(243\) 1.00000 0.0641500
\(244\) 5.62957 0.360396
\(245\) 0 0
\(246\) 16.6200 1.05966
\(247\) −7.05467 −0.448878
\(248\) −5.35869 −0.340277
\(249\) 10.8838 0.689734
\(250\) −93.8171 −5.93352
\(251\) −4.74956 −0.299790 −0.149895 0.988702i \(-0.547894\pi\)
−0.149895 + 0.988702i \(0.547894\pi\)
\(252\) 0 0
\(253\) 1.72915 0.108711
\(254\) −4.47625 −0.280865
\(255\) −21.2899 −1.33323
\(256\) −31.6969 −1.98105
\(257\) −1.42884 −0.0891283 −0.0445641 0.999007i \(-0.514190\pi\)
−0.0445641 + 0.999007i \(0.514190\pi\)
\(258\) 27.0234 1.68240
\(259\) 0 0
\(260\) −53.9683 −3.34697
\(261\) 8.05817 0.498788
\(262\) −22.3668 −1.38183
\(263\) 8.81746 0.543708 0.271854 0.962339i \(-0.412363\pi\)
0.271854 + 0.962339i \(0.412363\pi\)
\(264\) 10.1173 0.622674
\(265\) −28.2576 −1.73585
\(266\) 0 0
\(267\) −14.9862 −0.917143
\(268\) −44.1882 −2.69922
\(269\) 12.9185 0.787658 0.393829 0.919184i \(-0.371150\pi\)
0.393829 + 0.919184i \(0.371150\pi\)
\(270\) −10.8590 −0.660859
\(271\) −13.3485 −0.810865 −0.405433 0.914125i \(-0.632879\pi\)
−0.405433 + 0.914125i \(0.632879\pi\)
\(272\) 29.9028 1.81312
\(273\) 0 0
\(274\) 0.622271 0.0375927
\(275\) 23.5848 1.42222
\(276\) 4.32625 0.260410
\(277\) −15.4986 −0.931221 −0.465610 0.884990i \(-0.654165\pi\)
−0.465610 + 0.884990i \(0.654165\pi\)
\(278\) 21.6295 1.29725
\(279\) −0.915859 −0.0548311
\(280\) 0 0
\(281\) 23.6088 1.40838 0.704192 0.710009i \(-0.251309\pi\)
0.704192 + 0.710009i \(0.251309\pi\)
\(282\) 1.00520 0.0598587
\(283\) −23.6814 −1.40772 −0.703858 0.710341i \(-0.748541\pi\)
−0.703858 + 0.710341i \(0.748541\pi\)
\(284\) −38.9445 −2.31093
\(285\) 10.5411 0.624400
\(286\) 12.5665 0.743074
\(287\) 0 0
\(288\) 3.55007 0.209190
\(289\) 7.31715 0.430421
\(290\) −87.5039 −5.13840
\(291\) 3.07116 0.180034
\(292\) 11.3225 0.662598
\(293\) −21.1268 −1.23424 −0.617121 0.786868i \(-0.711701\pi\)
−0.617121 + 0.786868i \(0.711701\pi\)
\(294\) 0 0
\(295\) −7.92270 −0.461277
\(296\) −53.5706 −3.11373
\(297\) 1.72915 0.100336
\(298\) 11.3352 0.656633
\(299\) 2.88941 0.167099
\(300\) 59.0081 3.40684
\(301\) 0 0
\(302\) −11.3880 −0.655305
\(303\) −1.08810 −0.0625096
\(304\) −14.8055 −0.849153
\(305\) −5.61799 −0.321685
\(306\) 12.4031 0.709037
\(307\) 1.97923 0.112961 0.0564804 0.998404i \(-0.482012\pi\)
0.0564804 + 0.998404i \(0.482012\pi\)
\(308\) 0 0
\(309\) 3.23110 0.183811
\(310\) 9.94534 0.564857
\(311\) 3.89679 0.220967 0.110483 0.993878i \(-0.464760\pi\)
0.110483 + 0.993878i \(0.464760\pi\)
\(312\) 16.9059 0.957110
\(313\) −17.6584 −0.998111 −0.499056 0.866570i \(-0.666320\pi\)
−0.499056 + 0.866570i \(0.666320\pi\)
\(314\) 8.21786 0.463761
\(315\) 0 0
\(316\) 44.0716 2.47922
\(317\) −12.3323 −0.692652 −0.346326 0.938114i \(-0.612571\pi\)
−0.346326 + 0.938114i \(0.612571\pi\)
\(318\) 16.4623 0.923161
\(319\) 13.9338 0.780142
\(320\) 13.8101 0.772008
\(321\) −0.936899 −0.0522926
\(322\) 0 0
\(323\) −12.0399 −0.669919
\(324\) 4.32625 0.240347
\(325\) 39.4103 2.18609
\(326\) 26.7669 1.48248
\(327\) −14.9782 −0.828295
\(328\) 38.6624 2.13477
\(329\) 0 0
\(330\) −18.7769 −1.03363
\(331\) 5.80565 0.319107 0.159554 0.987189i \(-0.448995\pi\)
0.159554 + 0.987189i \(0.448995\pi\)
\(332\) 47.0861 2.58419
\(333\) −9.15581 −0.501735
\(334\) −7.54087 −0.412618
\(335\) 44.0973 2.40930
\(336\) 0 0
\(337\) −22.5441 −1.22805 −0.614027 0.789285i \(-0.710452\pi\)
−0.614027 + 0.789285i \(0.710452\pi\)
\(338\) −11.6990 −0.636341
\(339\) −14.1424 −0.768108
\(340\) −92.1056 −4.99512
\(341\) −1.58366 −0.0857599
\(342\) −6.14102 −0.332068
\(343\) 0 0
\(344\) 62.8631 3.38935
\(345\) −4.31735 −0.232439
\(346\) 33.4513 1.79835
\(347\) −7.49730 −0.402476 −0.201238 0.979542i \(-0.564497\pi\)
−0.201238 + 0.979542i \(0.564497\pi\)
\(348\) 34.8617 1.86878
\(349\) 7.97347 0.426810 0.213405 0.976964i \(-0.431545\pi\)
0.213405 + 0.976964i \(0.431545\pi\)
\(350\) 0 0
\(351\) 2.88941 0.154225
\(352\) 6.13861 0.327189
\(353\) 29.5289 1.57167 0.785833 0.618439i \(-0.212235\pi\)
0.785833 + 0.618439i \(0.212235\pi\)
\(354\) 4.61560 0.245317
\(355\) 38.8644 2.06271
\(356\) −64.8342 −3.43621
\(357\) 0 0
\(358\) 49.3727 2.60943
\(359\) −13.1206 −0.692481 −0.346241 0.938146i \(-0.612542\pi\)
−0.346241 + 0.938146i \(0.612542\pi\)
\(360\) −25.2608 −1.33136
\(361\) −13.0388 −0.686252
\(362\) −48.3158 −2.53942
\(363\) −8.01004 −0.420418
\(364\) 0 0
\(365\) −11.2992 −0.591427
\(366\) 3.27293 0.171079
\(367\) −5.26368 −0.274762 −0.137381 0.990518i \(-0.543868\pi\)
−0.137381 + 0.990518i \(0.543868\pi\)
\(368\) 6.06395 0.316105
\(369\) 6.60783 0.343990
\(370\) 99.4232 5.16876
\(371\) 0 0
\(372\) −3.96224 −0.205432
\(373\) −12.6495 −0.654968 −0.327484 0.944857i \(-0.606201\pi\)
−0.327484 + 0.944857i \(0.606201\pi\)
\(374\) 21.4468 1.10899
\(375\) −37.3000 −1.92616
\(376\) 2.33834 0.120591
\(377\) 23.2834 1.19915
\(378\) 0 0
\(379\) 12.7987 0.657426 0.328713 0.944430i \(-0.393385\pi\)
0.328713 + 0.944430i \(0.393385\pi\)
\(380\) 45.6034 2.33940
\(381\) −1.77968 −0.0911756
\(382\) −17.9931 −0.920609
\(383\) 16.6269 0.849595 0.424797 0.905288i \(-0.360345\pi\)
0.424797 + 0.905288i \(0.360345\pi\)
\(384\) −15.1456 −0.772897
\(385\) 0 0
\(386\) −55.2270 −2.81098
\(387\) 10.7440 0.546149
\(388\) 13.2866 0.674525
\(389\) −29.8703 −1.51448 −0.757242 0.653134i \(-0.773454\pi\)
−0.757242 + 0.653134i \(0.773454\pi\)
\(390\) −31.3762 −1.58880
\(391\) 4.93124 0.249384
\(392\) 0 0
\(393\) −8.89264 −0.448574
\(394\) 8.07450 0.406787
\(395\) −43.9810 −2.21292
\(396\) 7.48074 0.375921
\(397\) 5.98437 0.300347 0.150173 0.988660i \(-0.452017\pi\)
0.150173 + 0.988660i \(0.452017\pi\)
\(398\) −31.9858 −1.60330
\(399\) 0 0
\(400\) 82.7095 4.13548
\(401\) −15.8635 −0.792185 −0.396092 0.918211i \(-0.629634\pi\)
−0.396092 + 0.918211i \(0.629634\pi\)
\(402\) −25.6902 −1.28131
\(403\) −2.64629 −0.131821
\(404\) −4.70738 −0.234201
\(405\) −4.31735 −0.214531
\(406\) 0 0
\(407\) −15.8318 −0.784752
\(408\) 28.8527 1.42842
\(409\) −24.9384 −1.23313 −0.616563 0.787306i \(-0.711475\pi\)
−0.616563 + 0.787306i \(0.711475\pi\)
\(410\) −71.7546 −3.54371
\(411\) 0.247404 0.0122035
\(412\) 13.9785 0.688673
\(413\) 0 0
\(414\) 2.51520 0.123616
\(415\) −46.9893 −2.30661
\(416\) 10.2576 0.502921
\(417\) 8.59949 0.421119
\(418\) −10.6188 −0.519380
\(419\) 13.2262 0.646140 0.323070 0.946375i \(-0.395285\pi\)
0.323070 + 0.946375i \(0.395285\pi\)
\(420\) 0 0
\(421\) −5.82168 −0.283731 −0.141866 0.989886i \(-0.545310\pi\)
−0.141866 + 0.989886i \(0.545310\pi\)
\(422\) −42.8841 −2.08757
\(423\) 0.399649 0.0194316
\(424\) 38.2955 1.85979
\(425\) 67.2599 3.26259
\(426\) −22.6416 −1.09699
\(427\) 0 0
\(428\) −4.05326 −0.195922
\(429\) 4.99623 0.241220
\(430\) −116.669 −5.62630
\(431\) 6.21912 0.299565 0.149782 0.988719i \(-0.452143\pi\)
0.149782 + 0.988719i \(0.452143\pi\)
\(432\) 6.06395 0.291752
\(433\) 2.16316 0.103955 0.0519774 0.998648i \(-0.483448\pi\)
0.0519774 + 0.998648i \(0.483448\pi\)
\(434\) 0 0
\(435\) −34.7900 −1.66805
\(436\) −64.7993 −3.10333
\(437\) −2.44156 −0.116796
\(438\) 6.58268 0.314533
\(439\) −11.5256 −0.550085 −0.275042 0.961432i \(-0.588692\pi\)
−0.275042 + 0.961432i \(0.588692\pi\)
\(440\) −43.6798 −2.08235
\(441\) 0 0
\(442\) 35.8376 1.70462
\(443\) 9.13040 0.433799 0.216899 0.976194i \(-0.430406\pi\)
0.216899 + 0.976194i \(0.430406\pi\)
\(444\) −39.6103 −1.87982
\(445\) 64.7009 3.06712
\(446\) −58.6742 −2.77830
\(447\) 4.50669 0.213159
\(448\) 0 0
\(449\) −20.6827 −0.976079 −0.488039 0.872822i \(-0.662288\pi\)
−0.488039 + 0.872822i \(0.662288\pi\)
\(450\) 34.3063 1.61721
\(451\) 11.4259 0.538026
\(452\) −61.1834 −2.87783
\(453\) −4.52766 −0.212728
\(454\) 19.3066 0.906102
\(455\) 0 0
\(456\) −14.2856 −0.668983
\(457\) 24.5958 1.15054 0.575271 0.817963i \(-0.304897\pi\)
0.575271 + 0.817963i \(0.304897\pi\)
\(458\) 57.5318 2.68829
\(459\) 4.93124 0.230171
\(460\) −18.6780 −0.870864
\(461\) 3.35047 0.156047 0.0780235 0.996952i \(-0.475139\pi\)
0.0780235 + 0.996952i \(0.475139\pi\)
\(462\) 0 0
\(463\) −30.2309 −1.40495 −0.702476 0.711708i \(-0.747922\pi\)
−0.702476 + 0.711708i \(0.747922\pi\)
\(464\) 48.8643 2.26847
\(465\) 3.95409 0.183366
\(466\) 59.2479 2.74460
\(467\) 30.1416 1.39479 0.697394 0.716688i \(-0.254343\pi\)
0.697394 + 0.716688i \(0.254343\pi\)
\(468\) 12.5003 0.577827
\(469\) 0 0
\(470\) −4.33980 −0.200180
\(471\) 3.26727 0.150548
\(472\) 10.7371 0.494213
\(473\) 18.5780 0.854218
\(474\) 25.6224 1.17688
\(475\) −33.3018 −1.52799
\(476\) 0 0
\(477\) 6.54512 0.299681
\(478\) 33.1577 1.51660
\(479\) −29.0344 −1.32662 −0.663308 0.748347i \(-0.730848\pi\)
−0.663308 + 0.748347i \(0.730848\pi\)
\(480\) −15.3269 −0.699575
\(481\) −26.4549 −1.20624
\(482\) 43.4301 1.97819
\(483\) 0 0
\(484\) −34.6534 −1.57516
\(485\) −13.2593 −0.602072
\(486\) 2.51520 0.114092
\(487\) −17.3057 −0.784195 −0.392097 0.919924i \(-0.628250\pi\)
−0.392097 + 0.919924i \(0.628250\pi\)
\(488\) 7.61365 0.344654
\(489\) 10.6420 0.481250
\(490\) 0 0
\(491\) 14.2287 0.642133 0.321067 0.947057i \(-0.395959\pi\)
0.321067 + 0.947057i \(0.395959\pi\)
\(492\) 28.5871 1.28881
\(493\) 39.7368 1.78965
\(494\) −17.7439 −0.798337
\(495\) −7.46536 −0.335543
\(496\) −5.55372 −0.249369
\(497\) 0 0
\(498\) 27.3750 1.22670
\(499\) 11.1172 0.497672 0.248836 0.968546i \(-0.419952\pi\)
0.248836 + 0.968546i \(0.419952\pi\)
\(500\) −161.369 −7.21665
\(501\) −2.99812 −0.133946
\(502\) −11.9461 −0.533181
\(503\) −25.5401 −1.13878 −0.569389 0.822068i \(-0.692820\pi\)
−0.569389 + 0.822068i \(0.692820\pi\)
\(504\) 0 0
\(505\) 4.69770 0.209045
\(506\) 4.34917 0.193344
\(507\) −4.65131 −0.206572
\(508\) −7.69933 −0.341602
\(509\) −30.3594 −1.34566 −0.672829 0.739798i \(-0.734921\pi\)
−0.672829 + 0.739798i \(0.734921\pi\)
\(510\) −53.5485 −2.37117
\(511\) 0 0
\(512\) −49.4328 −2.18464
\(513\) −2.44156 −0.107798
\(514\) −3.59381 −0.158516
\(515\) −13.9498 −0.614701
\(516\) 46.4813 2.04622
\(517\) 0.691053 0.0303925
\(518\) 0 0
\(519\) 13.2996 0.583789
\(520\) −72.9889 −3.20077
\(521\) 24.7007 1.08216 0.541079 0.840972i \(-0.318016\pi\)
0.541079 + 0.840972i \(0.318016\pi\)
\(522\) 20.2679 0.887104
\(523\) 28.5353 1.24776 0.623881 0.781519i \(-0.285555\pi\)
0.623881 + 0.781519i \(0.285555\pi\)
\(524\) −38.4718 −1.68065
\(525\) 0 0
\(526\) 22.1777 0.966994
\(527\) −4.51632 −0.196734
\(528\) 10.4855 0.456322
\(529\) 1.00000 0.0434783
\(530\) −71.0737 −3.08724
\(531\) 1.83508 0.0796357
\(532\) 0 0
\(533\) 19.0927 0.826998
\(534\) −37.6934 −1.63115
\(535\) 4.04493 0.174877
\(536\) −59.7619 −2.58132
\(537\) 19.6297 0.847084
\(538\) 32.4928 1.40086
\(539\) 0 0
\(540\) −18.6780 −0.803771
\(541\) −7.74392 −0.332937 −0.166469 0.986047i \(-0.553236\pi\)
−0.166469 + 0.986047i \(0.553236\pi\)
\(542\) −33.5743 −1.44214
\(543\) −19.2095 −0.824358
\(544\) 17.5063 0.750575
\(545\) 64.6661 2.76999
\(546\) 0 0
\(547\) 36.8996 1.57771 0.788856 0.614578i \(-0.210674\pi\)
0.788856 + 0.614578i \(0.210674\pi\)
\(548\) 1.07033 0.0457222
\(549\) 1.30126 0.0555363
\(550\) 59.3207 2.52944
\(551\) −19.6745 −0.838162
\(552\) 5.85100 0.249035
\(553\) 0 0
\(554\) −38.9821 −1.65619
\(555\) 39.5289 1.67791
\(556\) 37.2035 1.57778
\(557\) 28.2645 1.19760 0.598802 0.800897i \(-0.295644\pi\)
0.598802 + 0.800897i \(0.295644\pi\)
\(558\) −2.30357 −0.0975180
\(559\) 31.0438 1.31301
\(560\) 0 0
\(561\) 8.52686 0.360004
\(562\) 59.3810 2.50484
\(563\) 9.62212 0.405524 0.202762 0.979228i \(-0.435008\pi\)
0.202762 + 0.979228i \(0.435008\pi\)
\(564\) 1.72898 0.0728032
\(565\) 61.0576 2.56871
\(566\) −59.5637 −2.50365
\(567\) 0 0
\(568\) −52.6701 −2.20999
\(569\) −15.7146 −0.658791 −0.329396 0.944192i \(-0.606845\pi\)
−0.329396 + 0.944192i \(0.606845\pi\)
\(570\) 26.5130 1.11051
\(571\) 11.3442 0.474739 0.237369 0.971419i \(-0.423715\pi\)
0.237369 + 0.971419i \(0.423715\pi\)
\(572\) 21.6149 0.903766
\(573\) −7.15375 −0.298852
\(574\) 0 0
\(575\) 13.6396 0.568809
\(576\) −3.19874 −0.133281
\(577\) 7.95051 0.330984 0.165492 0.986211i \(-0.447079\pi\)
0.165492 + 0.986211i \(0.447079\pi\)
\(578\) 18.4041 0.765511
\(579\) −21.9573 −0.912513
\(580\) −150.510 −6.24959
\(581\) 0 0
\(582\) 7.72458 0.320194
\(583\) 11.3175 0.468723
\(584\) 15.3130 0.633655
\(585\) −12.4746 −0.515762
\(586\) −53.1383 −2.19512
\(587\) −21.2897 −0.878718 −0.439359 0.898311i \(-0.644794\pi\)
−0.439359 + 0.898311i \(0.644794\pi\)
\(588\) 0 0
\(589\) 2.23613 0.0921379
\(590\) −19.9272 −0.820390
\(591\) 3.21028 0.132053
\(592\) −55.5203 −2.28187
\(593\) 9.67715 0.397393 0.198696 0.980061i \(-0.436329\pi\)
0.198696 + 0.980061i \(0.436329\pi\)
\(594\) 4.34917 0.178448
\(595\) 0 0
\(596\) 19.4971 0.798631
\(597\) −12.7170 −0.520471
\(598\) 7.26746 0.297188
\(599\) 25.9792 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(600\) 79.8050 3.25802
\(601\) −31.2857 −1.27617 −0.638084 0.769967i \(-0.720273\pi\)
−0.638084 + 0.769967i \(0.720273\pi\)
\(602\) 0 0
\(603\) −10.2140 −0.415945
\(604\) −19.5878 −0.797015
\(605\) 34.5822 1.40597
\(606\) −2.73679 −0.111174
\(607\) −28.9163 −1.17368 −0.586838 0.809704i \(-0.699628\pi\)
−0.586838 + 0.809704i \(0.699628\pi\)
\(608\) −8.66771 −0.351522
\(609\) 0 0
\(610\) −14.1304 −0.572123
\(611\) 1.15475 0.0467162
\(612\) 21.3338 0.862367
\(613\) −47.9848 −1.93809 −0.969043 0.246892i \(-0.920591\pi\)
−0.969043 + 0.246892i \(0.920591\pi\)
\(614\) 4.97818 0.200903
\(615\) −28.5283 −1.15037
\(616\) 0 0
\(617\) 41.9964 1.69071 0.845355 0.534205i \(-0.179389\pi\)
0.845355 + 0.534205i \(0.179389\pi\)
\(618\) 8.12686 0.326910
\(619\) 9.42381 0.378775 0.189388 0.981902i \(-0.439350\pi\)
0.189388 + 0.981902i \(0.439350\pi\)
\(620\) 17.1064 0.687009
\(621\) 1.00000 0.0401286
\(622\) 9.80123 0.392994
\(623\) 0 0
\(624\) 17.5212 0.701411
\(625\) 92.8396 3.71358
\(626\) −44.4144 −1.77516
\(627\) −4.22183 −0.168603
\(628\) 14.1350 0.564050
\(629\) −45.1495 −1.80023
\(630\) 0 0
\(631\) 6.84273 0.272405 0.136202 0.990681i \(-0.456510\pi\)
0.136202 + 0.990681i \(0.456510\pi\)
\(632\) 59.6042 2.37093
\(633\) −17.0500 −0.677675
\(634\) −31.0183 −1.23189
\(635\) 7.68350 0.304910
\(636\) 28.3159 1.12280
\(637\) 0 0
\(638\) 35.0463 1.38750
\(639\) −9.00190 −0.356110
\(640\) 65.3891 2.58473
\(641\) 7.02021 0.277282 0.138641 0.990343i \(-0.455727\pi\)
0.138641 + 0.990343i \(0.455727\pi\)
\(642\) −2.35649 −0.0930033
\(643\) 47.1743 1.86037 0.930185 0.367090i \(-0.119646\pi\)
0.930185 + 0.367090i \(0.119646\pi\)
\(644\) 0 0
\(645\) −46.3857 −1.82643
\(646\) −30.2829 −1.19146
\(647\) −23.9670 −0.942239 −0.471119 0.882070i \(-0.656150\pi\)
−0.471119 + 0.882070i \(0.656150\pi\)
\(648\) 5.85100 0.229849
\(649\) 3.17313 0.124556
\(650\) 99.1249 3.88800
\(651\) 0 0
\(652\) 46.0401 1.80307
\(653\) −18.3313 −0.717359 −0.358679 0.933461i \(-0.616773\pi\)
−0.358679 + 0.933461i \(0.616773\pi\)
\(654\) −37.6732 −1.47314
\(655\) 38.3927 1.50013
\(656\) 40.0695 1.56445
\(657\) 2.61716 0.102105
\(658\) 0 0
\(659\) 16.4341 0.640183 0.320092 0.947387i \(-0.396286\pi\)
0.320092 + 0.947387i \(0.396286\pi\)
\(660\) −32.2970 −1.25716
\(661\) 30.0350 1.16823 0.584113 0.811672i \(-0.301442\pi\)
0.584113 + 0.811672i \(0.301442\pi\)
\(662\) 14.6024 0.567538
\(663\) 14.2484 0.553361
\(664\) 63.6812 2.47131
\(665\) 0 0
\(666\) −23.0287 −0.892345
\(667\) 8.05817 0.312014
\(668\) −12.9706 −0.501848
\(669\) −23.3278 −0.901906
\(670\) 110.914 4.28497
\(671\) 2.25007 0.0868630
\(672\) 0 0
\(673\) 17.0642 0.657776 0.328888 0.944369i \(-0.393326\pi\)
0.328888 + 0.944369i \(0.393326\pi\)
\(674\) −56.7030 −2.18412
\(675\) 13.6396 0.524987
\(676\) −20.1227 −0.773951
\(677\) 37.6555 1.44722 0.723610 0.690209i \(-0.242482\pi\)
0.723610 + 0.690209i \(0.242482\pi\)
\(678\) −35.5709 −1.36609
\(679\) 0 0
\(680\) −124.567 −4.77694
\(681\) 7.67594 0.294143
\(682\) −3.98322 −0.152526
\(683\) −40.7749 −1.56021 −0.780104 0.625649i \(-0.784834\pi\)
−0.780104 + 0.625649i \(0.784834\pi\)
\(684\) −10.5628 −0.403879
\(685\) −1.06813 −0.0408111
\(686\) 0 0
\(687\) 22.8736 0.872683
\(688\) 65.1511 2.48386
\(689\) 18.9116 0.720473
\(690\) −10.8590 −0.413396
\(691\) −3.98906 −0.151751 −0.0758756 0.997117i \(-0.524175\pi\)
−0.0758756 + 0.997117i \(0.524175\pi\)
\(692\) 57.5376 2.18725
\(693\) 0 0
\(694\) −18.8572 −0.715811
\(695\) −37.1270 −1.40831
\(696\) 47.1483 1.78715
\(697\) 32.5848 1.23424
\(698\) 20.0549 0.759089
\(699\) 23.5559 0.890965
\(700\) 0 0
\(701\) 3.27653 0.123753 0.0618763 0.998084i \(-0.480292\pi\)
0.0618763 + 0.998084i \(0.480292\pi\)
\(702\) 7.26746 0.274292
\(703\) 22.3545 0.843114
\(704\) −5.53111 −0.208461
\(705\) −1.72543 −0.0649833
\(706\) 74.2712 2.79523
\(707\) 0 0
\(708\) 7.93902 0.298367
\(709\) −7.16279 −0.269004 −0.134502 0.990913i \(-0.542944\pi\)
−0.134502 + 0.990913i \(0.542944\pi\)
\(710\) 97.7519 3.66856
\(711\) 10.1870 0.382043
\(712\) −87.6844 −3.28611
\(713\) −0.915859 −0.0342992
\(714\) 0 0
\(715\) −21.5705 −0.806690
\(716\) 84.9230 3.17372
\(717\) 13.1829 0.492325
\(718\) −33.0011 −1.23159
\(719\) −49.3742 −1.84135 −0.920673 0.390335i \(-0.872359\pi\)
−0.920673 + 0.390335i \(0.872359\pi\)
\(720\) −26.1802 −0.975679
\(721\) 0 0
\(722\) −32.7952 −1.22051
\(723\) 17.2670 0.642168
\(724\) −83.1051 −3.08858
\(725\) 109.910 4.08195
\(726\) −20.1469 −0.747721
\(727\) 15.4034 0.571281 0.285640 0.958337i \(-0.407794\pi\)
0.285640 + 0.958337i \(0.407794\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −28.4198 −1.05186
\(731\) 52.9813 1.95958
\(732\) 5.62957 0.208075
\(733\) −12.2110 −0.451025 −0.225512 0.974240i \(-0.572406\pi\)
−0.225512 + 0.974240i \(0.572406\pi\)
\(734\) −13.2392 −0.488669
\(735\) 0 0
\(736\) 3.55007 0.130857
\(737\) −17.6615 −0.650570
\(738\) 16.6200 0.611792
\(739\) 32.4803 1.19481 0.597404 0.801940i \(-0.296199\pi\)
0.597404 + 0.801940i \(0.296199\pi\)
\(740\) 171.012 6.28652
\(741\) −7.05467 −0.259160
\(742\) 0 0
\(743\) −40.1671 −1.47359 −0.736795 0.676117i \(-0.763662\pi\)
−0.736795 + 0.676117i \(0.763662\pi\)
\(744\) −5.35869 −0.196459
\(745\) −19.4570 −0.712848
\(746\) −31.8162 −1.16487
\(747\) 10.8838 0.398218
\(748\) 36.8893 1.34881
\(749\) 0 0
\(750\) −93.8171 −3.42572
\(751\) 44.3036 1.61666 0.808330 0.588729i \(-0.200372\pi\)
0.808330 + 0.588729i \(0.200372\pi\)
\(752\) 2.42345 0.0883741
\(753\) −4.74956 −0.173084
\(754\) 58.5624 2.13272
\(755\) 19.5475 0.711406
\(756\) 0 0
\(757\) 41.4696 1.50724 0.753620 0.657311i \(-0.228306\pi\)
0.753620 + 0.657311i \(0.228306\pi\)
\(758\) 32.1914 1.16924
\(759\) 1.72915 0.0627642
\(760\) 61.6758 2.23722
\(761\) 3.17300 0.115021 0.0575107 0.998345i \(-0.481684\pi\)
0.0575107 + 0.998345i \(0.481684\pi\)
\(762\) −4.47625 −0.162157
\(763\) 0 0
\(764\) −30.9489 −1.11969
\(765\) −21.2899 −0.769739
\(766\) 41.8200 1.51102
\(767\) 5.30230 0.191455
\(768\) −31.6969 −1.14376
\(769\) 13.6487 0.492184 0.246092 0.969246i \(-0.420853\pi\)
0.246092 + 0.969246i \(0.420853\pi\)
\(770\) 0 0
\(771\) −1.42884 −0.0514582
\(772\) −94.9927 −3.41886
\(773\) 28.4748 1.02417 0.512083 0.858936i \(-0.328874\pi\)
0.512083 + 0.858936i \(0.328874\pi\)
\(774\) 27.0234 0.971335
\(775\) −12.4919 −0.448723
\(776\) 17.9693 0.645061
\(777\) 0 0
\(778\) −75.1299 −2.69354
\(779\) −16.1334 −0.578040
\(780\) −53.9683 −1.93237
\(781\) −15.5656 −0.556982
\(782\) 12.4031 0.443533
\(783\) 8.05817 0.287975
\(784\) 0 0
\(785\) −14.1060 −0.503464
\(786\) −22.3668 −0.797797
\(787\) −48.0679 −1.71343 −0.856717 0.515786i \(-0.827500\pi\)
−0.856717 + 0.515786i \(0.827500\pi\)
\(788\) 13.8885 0.494756
\(789\) 8.81746 0.313910
\(790\) −110.621 −3.93572
\(791\) 0 0
\(792\) 10.1173 0.359501
\(793\) 3.75987 0.133517
\(794\) 15.0519 0.534172
\(795\) −28.2576 −1.00219
\(796\) −55.0168 −1.95002
\(797\) 28.0705 0.994308 0.497154 0.867662i \(-0.334378\pi\)
0.497154 + 0.867662i \(0.334378\pi\)
\(798\) 0 0
\(799\) 1.97077 0.0697207
\(800\) 48.4214 1.71195
\(801\) −14.9862 −0.529513
\(802\) −39.8999 −1.40892
\(803\) 4.52546 0.159700
\(804\) −44.1882 −1.55840
\(805\) 0 0
\(806\) −6.65597 −0.234446
\(807\) 12.9185 0.454754
\(808\) −6.36645 −0.223971
\(809\) 42.7600 1.50336 0.751681 0.659527i \(-0.229243\pi\)
0.751681 + 0.659527i \(0.229243\pi\)
\(810\) −10.8590 −0.381547
\(811\) −50.7214 −1.78107 −0.890534 0.454916i \(-0.849669\pi\)
−0.890534 + 0.454916i \(0.849669\pi\)
\(812\) 0 0
\(813\) −13.3485 −0.468153
\(814\) −39.8201 −1.39569
\(815\) −45.9455 −1.60940
\(816\) 29.9028 1.04681
\(817\) −26.2321 −0.917746
\(818\) −62.7252 −2.19314
\(819\) 0 0
\(820\) −123.421 −4.31004
\(821\) 42.2837 1.47571 0.737856 0.674959i \(-0.235839\pi\)
0.737856 + 0.674959i \(0.235839\pi\)
\(822\) 0.622271 0.0217042
\(823\) −16.4856 −0.574650 −0.287325 0.957833i \(-0.592766\pi\)
−0.287325 + 0.957833i \(0.592766\pi\)
\(824\) 18.9051 0.658591
\(825\) 23.5848 0.821119
\(826\) 0 0
\(827\) 21.3447 0.742229 0.371115 0.928587i \(-0.378976\pi\)
0.371115 + 0.928587i \(0.378976\pi\)
\(828\) 4.32625 0.150348
\(829\) −8.82638 −0.306553 −0.153276 0.988183i \(-0.548982\pi\)
−0.153276 + 0.988183i \(0.548982\pi\)
\(830\) −118.188 −4.10235
\(831\) −15.4986 −0.537640
\(832\) −9.24248 −0.320425
\(833\) 0 0
\(834\) 21.6295 0.748967
\(835\) 12.9439 0.447943
\(836\) −18.2647 −0.631697
\(837\) −0.915859 −0.0316567
\(838\) 33.2665 1.14917
\(839\) 32.7270 1.12986 0.564932 0.825138i \(-0.308903\pi\)
0.564932 + 0.825138i \(0.308903\pi\)
\(840\) 0 0
\(841\) 35.9341 1.23911
\(842\) −14.6427 −0.504621
\(843\) 23.6088 0.813131
\(844\) −73.7624 −2.53901
\(845\) 20.0813 0.690819
\(846\) 1.00520 0.0345594
\(847\) 0 0
\(848\) 39.6893 1.36294
\(849\) −23.6814 −0.812745
\(850\) 169.172 5.80257
\(851\) −9.15581 −0.313857
\(852\) −38.9445 −1.33422
\(853\) −8.42097 −0.288329 −0.144164 0.989554i \(-0.546049\pi\)
−0.144164 + 0.989554i \(0.546049\pi\)
\(854\) 0 0
\(855\) 10.5411 0.360497
\(856\) −5.48179 −0.187364
\(857\) 5.60723 0.191539 0.0957697 0.995404i \(-0.469469\pi\)
0.0957697 + 0.995404i \(0.469469\pi\)
\(858\) 12.5665 0.429014
\(859\) −6.20674 −0.211771 −0.105886 0.994378i \(-0.533768\pi\)
−0.105886 + 0.994378i \(0.533768\pi\)
\(860\) −200.676 −6.84300
\(861\) 0 0
\(862\) 15.6424 0.532781
\(863\) −7.30992 −0.248833 −0.124416 0.992230i \(-0.539706\pi\)
−0.124416 + 0.992230i \(0.539706\pi\)
\(864\) 3.55007 0.120776
\(865\) −57.4192 −1.95231
\(866\) 5.44079 0.184886
\(867\) 7.31715 0.248504
\(868\) 0 0
\(869\) 17.6149 0.597544
\(870\) −87.5039 −2.96666
\(871\) −29.5124 −0.999988
\(872\) −87.6372 −2.96777
\(873\) 3.07116 0.103943
\(874\) −6.14102 −0.207723
\(875\) 0 0
\(876\) 11.3225 0.382551
\(877\) −32.7872 −1.10715 −0.553573 0.832801i \(-0.686736\pi\)
−0.553573 + 0.832801i \(0.686736\pi\)
\(878\) −28.9891 −0.978335
\(879\) −21.1268 −0.712590
\(880\) −45.2695 −1.52604
\(881\) −17.0828 −0.575533 −0.287767 0.957701i \(-0.592913\pi\)
−0.287767 + 0.957701i \(0.592913\pi\)
\(882\) 0 0
\(883\) 26.9824 0.908030 0.454015 0.890994i \(-0.349991\pi\)
0.454015 + 0.890994i \(0.349991\pi\)
\(884\) 61.6421 2.07325
\(885\) −7.92270 −0.266319
\(886\) 22.9648 0.771518
\(887\) −6.80466 −0.228478 −0.114239 0.993453i \(-0.536443\pi\)
−0.114239 + 0.993453i \(0.536443\pi\)
\(888\) −53.5706 −1.79771
\(889\) 0 0
\(890\) 162.736 5.45492
\(891\) 1.72915 0.0579287
\(892\) −100.922 −3.37912
\(893\) −0.975766 −0.0326528
\(894\) 11.3352 0.379107
\(895\) −84.7484 −2.83283
\(896\) 0 0
\(897\) 2.88941 0.0964746
\(898\) −52.0213 −1.73597
\(899\) −7.38015 −0.246142
\(900\) 59.0081 1.96694
\(901\) 32.2756 1.07526
\(902\) 28.7386 0.956889
\(903\) 0 0
\(904\) −82.7469 −2.75212
\(905\) 82.9342 2.75683
\(906\) −11.3880 −0.378340
\(907\) −5.42913 −0.180271 −0.0901357 0.995929i \(-0.528730\pi\)
−0.0901357 + 0.995929i \(0.528730\pi\)
\(908\) 33.2081 1.10205
\(909\) −1.08810 −0.0360899
\(910\) 0 0
\(911\) −15.8106 −0.523829 −0.261914 0.965091i \(-0.584354\pi\)
−0.261914 + 0.965091i \(0.584354\pi\)
\(912\) −14.8055 −0.490259
\(913\) 18.8198 0.622843
\(914\) 61.8634 2.04626
\(915\) −5.61799 −0.185725
\(916\) 98.9571 3.26963
\(917\) 0 0
\(918\) 12.4031 0.409363
\(919\) 37.3581 1.23233 0.616165 0.787617i \(-0.288686\pi\)
0.616165 + 0.787617i \(0.288686\pi\)
\(920\) −25.2608 −0.832825
\(921\) 1.97923 0.0652180
\(922\) 8.42712 0.277532
\(923\) −26.0102 −0.856135
\(924\) 0 0
\(925\) −124.881 −4.10607
\(926\) −76.0370 −2.49873
\(927\) 3.23110 0.106123
\(928\) 28.6071 0.939073
\(929\) −32.7069 −1.07308 −0.536539 0.843875i \(-0.680269\pi\)
−0.536539 + 0.843875i \(0.680269\pi\)
\(930\) 9.94534 0.326121
\(931\) 0 0
\(932\) 101.909 3.33813
\(933\) 3.89679 0.127575
\(934\) 75.8123 2.48065
\(935\) −36.8135 −1.20393
\(936\) 16.9059 0.552588
\(937\) −49.2657 −1.60944 −0.804721 0.593654i \(-0.797685\pi\)
−0.804721 + 0.593654i \(0.797685\pi\)
\(938\) 0 0
\(939\) −17.6584 −0.576260
\(940\) −7.46462 −0.243469
\(941\) 10.8684 0.354299 0.177149 0.984184i \(-0.443312\pi\)
0.177149 + 0.984184i \(0.443312\pi\)
\(942\) 8.21786 0.267752
\(943\) 6.60783 0.215181
\(944\) 11.1278 0.362180
\(945\) 0 0
\(946\) 46.7275 1.51924
\(947\) 31.5907 1.02656 0.513279 0.858222i \(-0.328431\pi\)
0.513279 + 0.858222i \(0.328431\pi\)
\(948\) 44.0716 1.43138
\(949\) 7.56204 0.245474
\(950\) −83.7608 −2.71756
\(951\) −12.3323 −0.399903
\(952\) 0 0
\(953\) −7.60480 −0.246344 −0.123172 0.992385i \(-0.539307\pi\)
−0.123172 + 0.992385i \(0.539307\pi\)
\(954\) 16.4623 0.532987
\(955\) 30.8853 0.999424
\(956\) 57.0326 1.84457
\(957\) 13.9338 0.450415
\(958\) −73.0274 −2.35941
\(959\) 0 0
\(960\) 13.8101 0.445719
\(961\) −30.1612 −0.972942
\(962\) −66.5394 −2.14532
\(963\) −0.936899 −0.0301911
\(964\) 74.7015 2.40597
\(965\) 94.7973 3.05163
\(966\) 0 0
\(967\) 10.3379 0.332446 0.166223 0.986088i \(-0.446843\pi\)
0.166223 + 0.986088i \(0.446843\pi\)
\(968\) −46.8667 −1.50635
\(969\) −12.0399 −0.386778
\(970\) −33.3498 −1.07080
\(971\) −19.7589 −0.634095 −0.317047 0.948410i \(-0.602691\pi\)
−0.317047 + 0.948410i \(0.602691\pi\)
\(972\) 4.32625 0.138765
\(973\) 0 0
\(974\) −43.5273 −1.39470
\(975\) 39.4103 1.26214
\(976\) 7.89075 0.252577
\(977\) −19.6177 −0.627626 −0.313813 0.949485i \(-0.601607\pi\)
−0.313813 + 0.949485i \(0.601607\pi\)
\(978\) 26.7669 0.855911
\(979\) −25.9135 −0.828198
\(980\) 0 0
\(981\) −14.9782 −0.478216
\(982\) 35.7881 1.14205
\(983\) 15.1056 0.481794 0.240897 0.970551i \(-0.422558\pi\)
0.240897 + 0.970551i \(0.422558\pi\)
\(984\) 38.6624 1.23251
\(985\) −13.8599 −0.441613
\(986\) 99.9461 3.18293
\(987\) 0 0
\(988\) −30.5203 −0.970979
\(989\) 10.7440 0.341640
\(990\) −18.7769 −0.596769
\(991\) 41.2825 1.31138 0.655690 0.755030i \(-0.272378\pi\)
0.655690 + 0.755030i \(0.272378\pi\)
\(992\) −3.25136 −0.103231
\(993\) 5.80565 0.184237
\(994\) 0 0
\(995\) 54.9037 1.74056
\(996\) 47.0861 1.49198
\(997\) −18.5381 −0.587107 −0.293554 0.955943i \(-0.594838\pi\)
−0.293554 + 0.955943i \(0.594838\pi\)
\(998\) 27.9619 0.885119
\(999\) −9.15581 −0.289677
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3381.2.a.bj.1.9 10
7.3 odd 6 483.2.i.h.415.2 yes 20
7.5 odd 6 483.2.i.h.277.2 20
7.6 odd 2 3381.2.a.bi.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.2 20 7.5 odd 6
483.2.i.h.415.2 yes 20 7.3 odd 6
3381.2.a.bi.1.9 10 7.6 odd 2
3381.2.a.bj.1.9 10 1.1 even 1 trivial