Properties

Label 3381.2.a.bi.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 \(-0.0839900\pi\)
0.965390 + 0.260811i \(0.0839900\pi\)
\(6\) −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 \(-0.631229\pi\)
−0.400689 + 0.916214i \(0.631229\pi\)
\(14\) 0 0
\(15\) −4.31735 −1.11474
\(16\) 6.06395 1.51599
\(17\) −4.93124 −1.19600 −0.598001 0.801495i \(-0.704038\pi\)
−0.598001 + 0.801495i \(0.704038\pi\)
\(18\) 2.51520 0.592839
\(19\) 2.44156 0.560132 0.280066 0.959981i \(-0.409644\pi\)
0.280066 + 0.959981i \(0.409644\pi\)
\(20\) 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 \(-0.473790\pi\)
0.0822466 + 0.996612i \(0.473790\pi\)
\(32\) 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 \(-0.672574\pi\)
−0.515985 + 0.856598i \(0.672574\pi\)
\(42\) 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 \(-0.509279\pi\)
−0.0291474 + 0.999575i \(0.509279\pi\)
\(48\) −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 \(-0.538114\pi\)
−0.119454 + 0.992840i \(0.538114\pi\)
\(60\) −18.6780 −2.41131
\(61\) −1.30126 −0.166609 −0.0833045 0.996524i \(-0.526547\pi\)
−0.0833045 + 0.996524i \(0.526547\pi\)
\(62\) 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 \(-0.548944\pi\)
−0.153158 + 0.988202i \(0.548944\pi\)
\(74\) −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 \(-0.703771\pi\)
−0.597327 + 0.801998i \(0.703771\pi\)
\(84\) 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 \(-0.207854\pi\)
0.794269 + 0.607566i \(0.207854\pi\)
\(90\) 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 \(-0.549832\pi\)
−0.155914 + 0.987771i \(0.549832\pi\)
\(98\) 0 0
\(99\) 1.72915 0.173786
\(100\) 59.0081 5.90081
\(101\) 1.08810 0.108270 0.0541349 0.998534i \(-0.482760\pi\)
0.0541349 + 0.998534i \(0.482760\pi\)
\(102\) 12.4031 1.22809
\(103\) −3.23110 −0.318369 −0.159185 0.987249i \(-0.550887\pi\)
−0.159185 + 0.987249i \(0.550887\pi\)
\(104\) −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 \(-0.373001\pi\)
0.388477 + 0.921459i \(0.373001\pi\)
\(132\) −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 \(-0.618828\pi\)
−0.364699 + 0.931125i \(0.618828\pi\)
\(140\) 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 \(-0.541619\pi\)
−0.130378 + 0.991464i \(0.541619\pi\)
\(158\) 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 \(-0.462993\pi\)
0.116001 + 0.993249i \(0.462993\pi\)
\(168\) 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 \(-0.668720\pi\)
−0.505576 + 0.862782i \(0.668720\pi\)
\(174\) −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 \(-0.246920\pi\)
0.713915 + 0.700232i \(0.246920\pi\)
\(182\) 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 \(-0.351160\pi\)
0.450741 + 0.892655i \(0.351160\pi\)
\(200\) 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 \(-0.214673\pi\)
0.781073 + 0.624440i \(0.214673\pi\)
\(224\) 0 0
\(225\) 13.6396 0.909303
\(226\) −35.5709 −2.36614
\(227\) −7.67594 −0.509470 −0.254735 0.967011i \(-0.581988\pi\)
−0.254735 + 0.967011i \(0.581988\pi\)
\(228\) −10.5628 −0.699539
\(229\) −22.8736 −1.51153 −0.755766 0.654842i \(-0.772735\pi\)
−0.755766 + 0.654842i \(0.772735\pi\)
\(230\) 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 \(-0.687716\pi\)
−0.556134 + 0.831093i \(0.687716\pi\)
\(242\) −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 \(-0.452106\pi\)
0.149895 + 0.988702i \(0.452106\pi\)
\(252\) 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 \(-0.485810\pi\)
0.0445641 + 0.999007i \(0.485810\pi\)
\(258\) −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 \(-0.628850\pi\)
−0.393829 + 0.919184i \(0.628850\pi\)
\(270\) −10.8590 −0.660859
\(271\) 13.3485 0.810865 0.405433 0.914125i \(-0.367121\pi\)
0.405433 + 0.914125i \(0.367121\pi\)
\(272\) −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 \(-0.251459\pi\)
0.703858 + 0.710341i \(0.251459\pi\)
\(284\) −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 \(-0.288299\pi\)
0.617121 + 0.786868i \(0.288299\pi\)
\(294\) 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 \(-0.517988\pi\)
−0.0564804 + 0.998404i \(0.517988\pi\)
\(308\) 0 0
\(309\) 3.23110 0.183811
\(310\) 9.94534 0.564857
\(311\) −3.89679 −0.220967 −0.110483 0.993878i \(-0.535240\pi\)
−0.110483 + 0.993878i \(0.535240\pi\)
\(312\) 16.9059 0.957110
\(313\) 17.6584 0.998111 0.499056 0.866570i \(-0.333680\pi\)
0.499056 + 0.866570i \(0.333680\pi\)
\(314\) −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 \(-0.568455\pi\)
−0.213405 + 0.976964i \(0.568455\pi\)
\(350\) 0 0
\(351\) 2.88941 0.154225
\(352\) 6.13861 0.327189
\(353\) −29.5289 −1.57167 −0.785833 0.618439i \(-0.787765\pi\)
−0.785833 + 0.618439i \(0.787765\pi\)
\(354\) 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 \(-0.456132\pi\)
0.137381 + 0.990518i \(0.456132\pi\)
\(368\) 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 \(-0.639655\pi\)
−0.424797 + 0.905288i \(0.639655\pi\)
\(384\) 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 \(-0.547983\pi\)
−0.150173 + 0.988660i \(0.547983\pi\)
\(398\) 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 \(-0.288525\pi\)
0.616563 + 0.787306i \(0.288525\pi\)
\(410\) −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 \(-0.604715\pi\)
−0.323070 + 0.946375i \(0.604715\pi\)
\(420\) 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 \(-0.516552\pi\)
−0.0519774 + 0.998648i \(0.516552\pi\)
\(434\) 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 \(-0.411308\pi\)
0.275042 + 0.961432i \(0.411308\pi\)
\(440\) 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 \(-0.524861\pi\)
−0.0780235 + 0.996952i \(0.524861\pi\)
\(462\) 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 \(-0.745657\pi\)
−0.697394 + 0.716688i \(0.745657\pi\)
\(468\) −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 \(-0.269152\pi\)
0.663308 + 0.748347i \(0.269152\pi\)
\(480\) −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 \(-0.307180\pi\)
0.569389 + 0.822068i \(0.307180\pi\)
\(504\) 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 \(-0.265079\pi\)
0.672829 + 0.739798i \(0.265079\pi\)
\(510\) 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 \(-0.681984\pi\)
−0.541079 + 0.840972i \(0.681984\pi\)
\(522\) 20.2679 0.887104
\(523\) −28.5353 −1.24776 −0.623881 0.781519i \(-0.714445\pi\)
−0.623881 + 0.781519i \(0.714445\pi\)
\(524\) 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 \(-0.564992\pi\)
−0.202762 + 0.979228i \(0.564992\pi\)
\(564\) 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 \(-0.552921\pi\)
−0.165492 + 0.986211i \(0.552921\pi\)
\(578\) 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 \(-0.355206\pi\)
0.439359 + 0.898311i \(0.355206\pi\)
\(588\) 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 \(-0.563671\pi\)
−0.198696 + 0.980061i \(0.563671\pi\)
\(594\) −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 \(-0.279727\pi\)
0.638084 + 0.769967i \(0.279727\pi\)
\(602\) 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 \(-0.300372\pi\)
0.586838 + 0.809704i \(0.300372\pi\)
\(608\) 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 \(-0.560650\pi\)
−0.189388 + 0.981902i \(0.560650\pi\)
\(620\) 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 \(-0.880354\pi\)
−0.930185 + 0.367090i \(0.880354\pi\)
\(644\) 0 0
\(645\) −46.3857 −1.82643
\(646\) −30.2829 −1.19146
\(647\) 23.9670 0.942239 0.471119 0.882070i \(-0.343850\pi\)
0.471119 + 0.882070i \(0.343850\pi\)
\(648\) 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 \(-0.698558\pi\)
−0.584113 + 0.811672i \(0.698558\pi\)
\(662\) 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 \(-0.757518\pi\)
−0.723610 + 0.690209i \(0.757518\pi\)
\(678\) 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 \(-0.475825\pi\)
0.0758756 + 0.997117i \(0.475825\pi\)
\(692\) −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 \(-0.127641\pi\)
0.920673 + 0.390335i \(0.127641\pi\)
\(720\) 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 \(-0.592206\pi\)
−0.285640 + 0.958337i \(0.592206\pi\)
\(728\) 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 \(-0.427594\pi\)
0.225512 + 0.974240i \(0.427594\pi\)
\(734\) 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 \(-0.518316\pi\)
−0.0575107 + 0.998345i \(0.518316\pi\)
\(762\) 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 \(-0.579147\pi\)
−0.246092 + 0.969246i \(0.579147\pi\)
\(770\) 0 0
\(771\) −1.42884 −0.0514582
\(772\) −94.9927 −3.41886
\(773\) −28.4748 −1.02417 −0.512083 0.858936i \(-0.671126\pi\)
−0.512083 + 0.858936i \(0.671126\pi\)
\(774\) 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 \(-0.172500\pi\)
0.856717 + 0.515786i \(0.172500\pi\)
\(788\) 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 \(-0.665622\pi\)
−0.497154 + 0.867662i \(0.665622\pi\)
\(798\) 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 \(-0.150331\pi\)
0.890534 + 0.454916i \(0.150331\pi\)
\(812\) 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 \(-0.451018\pi\)
0.153276 + 0.988183i \(0.451018\pi\)
\(830\) −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 \(-0.691097\pi\)
−0.564932 + 0.825138i \(0.691097\pi\)
\(840\) 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 \(-0.453951\pi\)
0.144164 + 0.989554i \(0.453951\pi\)
\(854\) 0 0
\(855\) 10.5411 0.360497
\(856\) −5.48179 −0.187364
\(857\) −5.60723 −0.191539 −0.0957697 0.995404i \(-0.530531\pi\)
−0.0957697 + 0.995404i \(0.530531\pi\)
\(858\) 12.5665 0.429014
\(859\) 6.20674 0.211771 0.105886 0.994378i \(-0.466232\pi\)
0.105886 + 0.994378i \(0.466232\pi\)
\(860\) 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 \(-0.407087\pi\)
0.287767 + 0.957701i \(0.407087\pi\)
\(882\) 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 \(-0.463557\pi\)
0.114239 + 0.993453i \(0.463557\pi\)
\(888\) 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 \(-0.319731\pi\)
0.536539 + 0.843875i \(0.319731\pi\)
\(930\) −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 \(-0.202315\pi\)
0.804721 + 0.593654i \(0.202315\pi\)
\(938\) 0 0
\(939\) −17.6584 −0.576260
\(940\) −7.46462 −0.243469
\(941\) −10.8684 −0.354299 −0.177149 0.984184i \(-0.556688\pi\)
−0.177149 + 0.984184i \(0.556688\pi\)
\(942\) 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 \(-0.397309\pi\)
0.317047 + 0.948410i \(0.397309\pi\)
\(972\) −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 \(-0.577442\pi\)
−0.240897 + 0.970551i \(0.577442\pi\)
\(984\) 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 \(-0.405162\pi\)
0.293554 + 0.955943i \(0.405162\pi\)
\(998\) 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.bi.1.9 10
7.2 even 3 483.2.i.h.277.2 20
7.4 even 3 483.2.i.h.415.2 yes 20
7.6 odd 2 3381.2.a.bj.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.2 20 7.2 even 3
483.2.i.h.415.2 yes 20 7.4 even 3
3381.2.a.bi.1.9 10 1.1 even 1 trivial
3381.2.a.bj.1.9 10 7.6 odd 2