Properties

Label 3381.2.a.bj.1.7
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 1.7
Root \(1.19920\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.19920 q^{2} +1.00000 q^{3} -0.561916 q^{4} +0.572653 q^{5} +1.19920 q^{6} -3.07225 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.19920 q^{2} +1.00000 q^{3} -0.561916 q^{4} +0.572653 q^{5} +1.19920 q^{6} -3.07225 q^{8} +1.00000 q^{9} +0.686727 q^{10} +2.52510 q^{11} -0.561916 q^{12} +5.05314 q^{13} +0.572653 q^{15} -2.56042 q^{16} +2.25223 q^{17} +1.19920 q^{18} -2.39935 q^{19} -0.321783 q^{20} +3.02811 q^{22} +1.00000 q^{23} -3.07225 q^{24} -4.67207 q^{25} +6.05973 q^{26} +1.00000 q^{27} +0.796522 q^{29} +0.686727 q^{30} +3.45731 q^{31} +3.07405 q^{32} +2.52510 q^{33} +2.70088 q^{34} -0.561916 q^{36} +8.80485 q^{37} -2.87730 q^{38} +5.05314 q^{39} -1.75934 q^{40} -10.9790 q^{41} -1.90984 q^{43} -1.41890 q^{44} +0.572653 q^{45} +1.19920 q^{46} +1.93652 q^{47} -2.56042 q^{48} -5.60275 q^{50} +2.25223 q^{51} -2.83944 q^{52} +9.57060 q^{53} +1.19920 q^{54} +1.44601 q^{55} -2.39935 q^{57} +0.955190 q^{58} +4.40927 q^{59} -0.321783 q^{60} -2.12670 q^{61} +4.14602 q^{62} +8.80724 q^{64} +2.89370 q^{65} +3.02811 q^{66} +4.01556 q^{67} -1.26557 q^{68} +1.00000 q^{69} +0.424132 q^{71} -3.07225 q^{72} +14.0009 q^{73} +10.5588 q^{74} -4.67207 q^{75} +1.34823 q^{76} +6.05973 q^{78} +4.97332 q^{79} -1.46623 q^{80} +1.00000 q^{81} -13.1661 q^{82} +10.1555 q^{83} +1.28975 q^{85} -2.29028 q^{86} +0.796522 q^{87} -7.75776 q^{88} -3.82713 q^{89} +0.686727 q^{90} -0.561916 q^{92} +3.45731 q^{93} +2.32227 q^{94} -1.37399 q^{95} +3.07405 q^{96} -15.4364 q^{97} +2.52510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} + 10q^{3} + 15q^{4} + 5q^{5} + 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} + 10q^{3} + 15q^{4} + 5q^{5} + 3q^{6} + 9q^{8} + 10q^{9} - 11q^{10} + 8q^{11} + 15q^{12} + 5q^{15} + 37q^{16} + 11q^{17} + 3q^{18} - q^{19} + 15q^{20} + 6q^{22} + 10q^{23} + 9q^{24} + 21q^{25} - q^{26} + 10q^{27} + 22q^{29} - 11q^{30} + 3q^{31} + 11q^{32} + 8q^{33} + 3q^{34} + 15q^{36} - 3q^{37} - 16q^{38} - 39q^{40} + 26q^{41} + 27q^{43} + 16q^{44} + 5q^{45} + 3q^{46} - 11q^{47} + 37q^{48} + 2q^{50} + 11q^{51} - 29q^{52} + 5q^{53} + 3q^{54} + 18q^{55} - q^{57} + 16q^{58} + 10q^{59} + 15q^{60} - 22q^{61} + 32q^{62} + 69q^{64} - 11q^{65} + 6q^{66} - 2q^{67} + 21q^{68} + 10q^{69} + 27q^{71} + 9q^{72} + 8q^{73} + 14q^{74} + 21q^{75} + 22q^{76} - q^{78} + 21q^{79} + 53q^{80} + 10q^{81} - 36q^{82} + 12q^{83} + 23q^{85} + 18q^{86} + 22q^{87} - 10q^{88} - 6q^{89} - 11q^{90} + 15q^{92} + 3q^{93} - 35q^{94} + 44q^{95} + 11q^{96} - 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19920 0.847964 0.423982 0.905671i \(-0.360632\pi\)
0.423982 + 0.905671i \(0.360632\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.561916 −0.280958
\(5\) 0.572653 0.256098 0.128049 0.991768i \(-0.459128\pi\)
0.128049 + 0.991768i \(0.459128\pi\)
\(6\) 1.19920 0.489572
\(7\) 0 0
\(8\) −3.07225 −1.08621
\(9\) 1.00000 0.333333
\(10\) 0.686727 0.217162
\(11\) 2.52510 0.761347 0.380674 0.924709i \(-0.375692\pi\)
0.380674 + 0.924709i \(0.375692\pi\)
\(12\) −0.561916 −0.162211
\(13\) 5.05314 1.40149 0.700744 0.713413i \(-0.252851\pi\)
0.700744 + 0.713413i \(0.252851\pi\)
\(14\) 0 0
\(15\) 0.572653 0.147858
\(16\) −2.56042 −0.640105
\(17\) 2.25223 0.546247 0.273123 0.961979i \(-0.411943\pi\)
0.273123 + 0.961979i \(0.411943\pi\)
\(18\) 1.19920 0.282655
\(19\) −2.39935 −0.550448 −0.275224 0.961380i \(-0.588752\pi\)
−0.275224 + 0.961380i \(0.588752\pi\)
\(20\) −0.321783 −0.0719528
\(21\) 0 0
\(22\) 3.02811 0.645595
\(23\) 1.00000 0.208514
\(24\) −3.07225 −0.627121
\(25\) −4.67207 −0.934414
\(26\) 6.05973 1.18841
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.796522 0.147910 0.0739552 0.997262i \(-0.476438\pi\)
0.0739552 + 0.997262i \(0.476438\pi\)
\(30\) 0.686727 0.125379
\(31\) 3.45731 0.620952 0.310476 0.950581i \(-0.399512\pi\)
0.310476 + 0.950581i \(0.399512\pi\)
\(32\) 3.07405 0.543420
\(33\) 2.52510 0.439564
\(34\) 2.70088 0.463197
\(35\) 0 0
\(36\) −0.561916 −0.0936526
\(37\) 8.80485 1.44751 0.723754 0.690058i \(-0.242415\pi\)
0.723754 + 0.690058i \(0.242415\pi\)
\(38\) −2.87730 −0.466760
\(39\) 5.05314 0.809150
\(40\) −1.75934 −0.278175
\(41\) −10.9790 −1.71464 −0.857319 0.514785i \(-0.827872\pi\)
−0.857319 + 0.514785i \(0.827872\pi\)
\(42\) 0 0
\(43\) −1.90984 −0.291248 −0.145624 0.989340i \(-0.546519\pi\)
−0.145624 + 0.989340i \(0.546519\pi\)
\(44\) −1.41890 −0.213906
\(45\) 0.572653 0.0853661
\(46\) 1.19920 0.176813
\(47\) 1.93652 0.282470 0.141235 0.989976i \(-0.454893\pi\)
0.141235 + 0.989976i \(0.454893\pi\)
\(48\) −2.56042 −0.369565
\(49\) 0 0
\(50\) −5.60275 −0.792349
\(51\) 2.25223 0.315376
\(52\) −2.83944 −0.393759
\(53\) 9.57060 1.31462 0.657312 0.753619i \(-0.271693\pi\)
0.657312 + 0.753619i \(0.271693\pi\)
\(54\) 1.19920 0.163191
\(55\) 1.44601 0.194980
\(56\) 0 0
\(57\) −2.39935 −0.317801
\(58\) 0.955190 0.125423
\(59\) 4.40927 0.574038 0.287019 0.957925i \(-0.407336\pi\)
0.287019 + 0.957925i \(0.407336\pi\)
\(60\) −0.321783 −0.0415420
\(61\) −2.12670 −0.272296 −0.136148 0.990689i \(-0.543472\pi\)
−0.136148 + 0.990689i \(0.543472\pi\)
\(62\) 4.14602 0.526545
\(63\) 0 0
\(64\) 8.80724 1.10091
\(65\) 2.89370 0.358919
\(66\) 3.02811 0.372734
\(67\) 4.01556 0.490579 0.245289 0.969450i \(-0.421117\pi\)
0.245289 + 0.969450i \(0.421117\pi\)
\(68\) −1.26557 −0.153472
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0.424132 0.0503353 0.0251676 0.999683i \(-0.491988\pi\)
0.0251676 + 0.999683i \(0.491988\pi\)
\(72\) −3.07225 −0.362069
\(73\) 14.0009 1.63868 0.819341 0.573306i \(-0.194339\pi\)
0.819341 + 0.573306i \(0.194339\pi\)
\(74\) 10.5588 1.22743
\(75\) −4.67207 −0.539484
\(76\) 1.34823 0.154653
\(77\) 0 0
\(78\) 6.05973 0.686129
\(79\) 4.97332 0.559542 0.279771 0.960067i \(-0.409741\pi\)
0.279771 + 0.960067i \(0.409741\pi\)
\(80\) −1.46623 −0.163930
\(81\) 1.00000 0.111111
\(82\) −13.1661 −1.45395
\(83\) 10.1555 1.11471 0.557355 0.830275i \(-0.311816\pi\)
0.557355 + 0.830275i \(0.311816\pi\)
\(84\) 0 0
\(85\) 1.28975 0.139893
\(86\) −2.29028 −0.246967
\(87\) 0.796522 0.0853961
\(88\) −7.75776 −0.826980
\(89\) −3.82713 −0.405675 −0.202838 0.979212i \(-0.565016\pi\)
−0.202838 + 0.979212i \(0.565016\pi\)
\(90\) 0.686727 0.0723874
\(91\) 0 0
\(92\) −0.561916 −0.0585838
\(93\) 3.45731 0.358507
\(94\) 2.32227 0.239524
\(95\) −1.37399 −0.140969
\(96\) 3.07405 0.313744
\(97\) −15.4364 −1.56733 −0.783665 0.621184i \(-0.786652\pi\)
−0.783665 + 0.621184i \(0.786652\pi\)
\(98\) 0 0
\(99\) 2.52510 0.253782
\(100\) 2.62531 0.262531
\(101\) 6.89184 0.685763 0.342882 0.939379i \(-0.388597\pi\)
0.342882 + 0.939379i \(0.388597\pi\)
\(102\) 2.70088 0.267427
\(103\) −10.7593 −1.06014 −0.530071 0.847953i \(-0.677835\pi\)
−0.530071 + 0.847953i \(0.677835\pi\)
\(104\) −15.5245 −1.52230
\(105\) 0 0
\(106\) 11.4771 1.11475
\(107\) 8.73468 0.844414 0.422207 0.906500i \(-0.361256\pi\)
0.422207 + 0.906500i \(0.361256\pi\)
\(108\) −0.561916 −0.0540704
\(109\) −2.99347 −0.286722 −0.143361 0.989670i \(-0.545791\pi\)
−0.143361 + 0.989670i \(0.545791\pi\)
\(110\) 1.73406 0.165336
\(111\) 8.80485 0.835720
\(112\) 0 0
\(113\) 16.3312 1.53631 0.768156 0.640263i \(-0.221175\pi\)
0.768156 + 0.640263i \(0.221175\pi\)
\(114\) −2.87730 −0.269484
\(115\) 0.572653 0.0534002
\(116\) −0.447578 −0.0415566
\(117\) 5.05314 0.467163
\(118\) 5.28761 0.486764
\(119\) 0 0
\(120\) −1.75934 −0.160605
\(121\) −4.62385 −0.420350
\(122\) −2.55034 −0.230897
\(123\) −10.9790 −0.989947
\(124\) −1.94272 −0.174461
\(125\) −5.53874 −0.495400
\(126\) 0 0
\(127\) 18.5959 1.65012 0.825061 0.565044i \(-0.191141\pi\)
0.825061 + 0.565044i \(0.191141\pi\)
\(128\) 4.41356 0.390107
\(129\) −1.90984 −0.168152
\(130\) 3.47013 0.304350
\(131\) −17.9120 −1.56498 −0.782490 0.622663i \(-0.786051\pi\)
−0.782490 + 0.622663i \(0.786051\pi\)
\(132\) −1.41890 −0.123499
\(133\) 0 0
\(134\) 4.81547 0.415993
\(135\) 0.572653 0.0492862
\(136\) −6.91943 −0.593336
\(137\) 16.1522 1.37998 0.689989 0.723820i \(-0.257615\pi\)
0.689989 + 0.723820i \(0.257615\pi\)
\(138\) 1.19920 0.102083
\(139\) 1.14087 0.0967677 0.0483839 0.998829i \(-0.484593\pi\)
0.0483839 + 0.998829i \(0.484593\pi\)
\(140\) 0 0
\(141\) 1.93652 0.163084
\(142\) 0.508620 0.0426825
\(143\) 12.7597 1.06702
\(144\) −2.56042 −0.213368
\(145\) 0.456131 0.0378796
\(146\) 16.7899 1.38954
\(147\) 0 0
\(148\) −4.94758 −0.406689
\(149\) 8.02342 0.657304 0.328652 0.944451i \(-0.393406\pi\)
0.328652 + 0.944451i \(0.393406\pi\)
\(150\) −5.60275 −0.457463
\(151\) 20.2695 1.64951 0.824753 0.565494i \(-0.191314\pi\)
0.824753 + 0.565494i \(0.191314\pi\)
\(152\) 7.37140 0.597899
\(153\) 2.25223 0.182082
\(154\) 0 0
\(155\) 1.97984 0.159025
\(156\) −2.83944 −0.227337
\(157\) 6.64919 0.530663 0.265331 0.964157i \(-0.414519\pi\)
0.265331 + 0.964157i \(0.414519\pi\)
\(158\) 5.96401 0.474471
\(159\) 9.57060 0.758998
\(160\) 1.76036 0.139169
\(161\) 0 0
\(162\) 1.19920 0.0942182
\(163\) 3.74710 0.293496 0.146748 0.989174i \(-0.453119\pi\)
0.146748 + 0.989174i \(0.453119\pi\)
\(164\) 6.16930 0.481741
\(165\) 1.44601 0.112572
\(166\) 12.1785 0.945233
\(167\) 2.97131 0.229927 0.114964 0.993370i \(-0.463325\pi\)
0.114964 + 0.993370i \(0.463325\pi\)
\(168\) 0 0
\(169\) 12.5342 0.964170
\(170\) 1.54667 0.118624
\(171\) −2.39935 −0.183483
\(172\) 1.07317 0.0818283
\(173\) −11.5181 −0.875701 −0.437851 0.899048i \(-0.644260\pi\)
−0.437851 + 0.899048i \(0.644260\pi\)
\(174\) 0.955190 0.0724128
\(175\) 0 0
\(176\) −6.46532 −0.487342
\(177\) 4.40927 0.331421
\(178\) −4.58950 −0.343998
\(179\) −11.9135 −0.890455 −0.445228 0.895417i \(-0.646877\pi\)
−0.445228 + 0.895417i \(0.646877\pi\)
\(180\) −0.321783 −0.0239843
\(181\) −25.9847 −1.93143 −0.965713 0.259612i \(-0.916405\pi\)
−0.965713 + 0.259612i \(0.916405\pi\)
\(182\) 0 0
\(183\) −2.12670 −0.157210
\(184\) −3.07225 −0.226490
\(185\) 5.04213 0.370705
\(186\) 4.14602 0.304001
\(187\) 5.68712 0.415884
\(188\) −1.08816 −0.0793621
\(189\) 0 0
\(190\) −1.64770 −0.119536
\(191\) −18.0234 −1.30412 −0.652062 0.758165i \(-0.726096\pi\)
−0.652062 + 0.758165i \(0.726096\pi\)
\(192\) 8.80724 0.635608
\(193\) −4.68289 −0.337082 −0.168541 0.985695i \(-0.553906\pi\)
−0.168541 + 0.985695i \(0.553906\pi\)
\(194\) −18.5114 −1.32904
\(195\) 2.89370 0.207222
\(196\) 0 0
\(197\) −22.4797 −1.60161 −0.800804 0.598926i \(-0.795594\pi\)
−0.800804 + 0.598926i \(0.795594\pi\)
\(198\) 3.02811 0.215198
\(199\) −2.73372 −0.193788 −0.0968941 0.995295i \(-0.530891\pi\)
−0.0968941 + 0.995295i \(0.530891\pi\)
\(200\) 14.3538 1.01497
\(201\) 4.01556 0.283236
\(202\) 8.26470 0.581502
\(203\) 0 0
\(204\) −1.26557 −0.0886073
\(205\) −6.28719 −0.439116
\(206\) −12.9025 −0.898961
\(207\) 1.00000 0.0695048
\(208\) −12.9382 −0.897100
\(209\) −6.05860 −0.419082
\(210\) 0 0
\(211\) −3.94428 −0.271536 −0.135768 0.990741i \(-0.543350\pi\)
−0.135768 + 0.990741i \(0.543350\pi\)
\(212\) −5.37787 −0.369354
\(213\) 0.424132 0.0290611
\(214\) 10.4746 0.716032
\(215\) −1.09368 −0.0745881
\(216\) −3.07225 −0.209040
\(217\) 0 0
\(218\) −3.58977 −0.243130
\(219\) 14.0009 0.946094
\(220\) −0.812535 −0.0547811
\(221\) 11.3808 0.765559
\(222\) 10.5588 0.708660
\(223\) 17.8955 1.19837 0.599187 0.800609i \(-0.295491\pi\)
0.599187 + 0.800609i \(0.295491\pi\)
\(224\) 0 0
\(225\) −4.67207 −0.311471
\(226\) 19.5844 1.30274
\(227\) −18.1777 −1.20649 −0.603247 0.797554i \(-0.706127\pi\)
−0.603247 + 0.797554i \(0.706127\pi\)
\(228\) 1.34823 0.0892887
\(229\) −7.17469 −0.474117 −0.237059 0.971495i \(-0.576183\pi\)
−0.237059 + 0.971495i \(0.576183\pi\)
\(230\) 0.686727 0.0452814
\(231\) 0 0
\(232\) −2.44712 −0.160661
\(233\) −14.3870 −0.942525 −0.471262 0.881993i \(-0.656201\pi\)
−0.471262 + 0.881993i \(0.656201\pi\)
\(234\) 6.05973 0.396137
\(235\) 1.10895 0.0723401
\(236\) −2.47764 −0.161281
\(237\) 4.97332 0.323052
\(238\) 0 0
\(239\) −1.51078 −0.0977241 −0.0488620 0.998806i \(-0.515559\pi\)
−0.0488620 + 0.998806i \(0.515559\pi\)
\(240\) −1.46623 −0.0946449
\(241\) −30.7830 −1.98291 −0.991453 0.130464i \(-0.958353\pi\)
−0.991453 + 0.130464i \(0.958353\pi\)
\(242\) −5.54493 −0.356442
\(243\) 1.00000 0.0641500
\(244\) 1.19503 0.0765037
\(245\) 0 0
\(246\) −13.1661 −0.839439
\(247\) −12.1242 −0.771446
\(248\) −10.6217 −0.674481
\(249\) 10.1555 0.643578
\(250\) −6.64207 −0.420081
\(251\) −12.8572 −0.811540 −0.405770 0.913975i \(-0.632997\pi\)
−0.405770 + 0.913975i \(0.632997\pi\)
\(252\) 0 0
\(253\) 2.52510 0.158752
\(254\) 22.3003 1.39924
\(255\) 1.28975 0.0807672
\(256\) −12.3217 −0.770108
\(257\) −4.16813 −0.260001 −0.130000 0.991514i \(-0.541498\pi\)
−0.130000 + 0.991514i \(0.541498\pi\)
\(258\) −2.29028 −0.142587
\(259\) 0 0
\(260\) −1.62601 −0.100841
\(261\) 0.796522 0.0493035
\(262\) −21.4801 −1.32705
\(263\) 9.13504 0.563291 0.281645 0.959519i \(-0.409120\pi\)
0.281645 + 0.959519i \(0.409120\pi\)
\(264\) −7.75776 −0.477457
\(265\) 5.48064 0.336673
\(266\) 0 0
\(267\) −3.82713 −0.234217
\(268\) −2.25641 −0.137832
\(269\) 9.03201 0.550691 0.275346 0.961345i \(-0.411208\pi\)
0.275346 + 0.961345i \(0.411208\pi\)
\(270\) 0.686727 0.0417929
\(271\) −7.05301 −0.428440 −0.214220 0.976785i \(-0.568721\pi\)
−0.214220 + 0.976785i \(0.568721\pi\)
\(272\) −5.76666 −0.349655
\(273\) 0 0
\(274\) 19.3698 1.17017
\(275\) −11.7975 −0.711413
\(276\) −0.561916 −0.0338233
\(277\) 11.3120 0.679671 0.339835 0.940485i \(-0.389629\pi\)
0.339835 + 0.940485i \(0.389629\pi\)
\(278\) 1.36814 0.0820555
\(279\) 3.45731 0.206984
\(280\) 0 0
\(281\) −22.3499 −1.33329 −0.666643 0.745377i \(-0.732269\pi\)
−0.666643 + 0.745377i \(0.732269\pi\)
\(282\) 2.32227 0.138289
\(283\) 18.4769 1.09834 0.549168 0.835712i \(-0.314945\pi\)
0.549168 + 0.835712i \(0.314945\pi\)
\(284\) −0.238327 −0.0141421
\(285\) −1.37399 −0.0813884
\(286\) 15.3014 0.904794
\(287\) 0 0
\(288\) 3.07405 0.181140
\(289\) −11.9274 −0.701614
\(290\) 0.546993 0.0321205
\(291\) −15.4364 −0.904898
\(292\) −7.86733 −0.460401
\(293\) −0.717401 −0.0419110 −0.0209555 0.999780i \(-0.506671\pi\)
−0.0209555 + 0.999780i \(0.506671\pi\)
\(294\) 0 0
\(295\) 2.52498 0.147010
\(296\) −27.0507 −1.57229
\(297\) 2.52510 0.146521
\(298\) 9.62170 0.557370
\(299\) 5.05314 0.292231
\(300\) 2.62531 0.151572
\(301\) 0 0
\(302\) 24.3072 1.39872
\(303\) 6.89184 0.395926
\(304\) 6.14333 0.352344
\(305\) −1.21786 −0.0697346
\(306\) 2.70088 0.154399
\(307\) 8.27982 0.472555 0.236277 0.971686i \(-0.424073\pi\)
0.236277 + 0.971686i \(0.424073\pi\)
\(308\) 0 0
\(309\) −10.7593 −0.612073
\(310\) 2.37423 0.134847
\(311\) 8.94613 0.507288 0.253644 0.967298i \(-0.418371\pi\)
0.253644 + 0.967298i \(0.418371\pi\)
\(312\) −15.5245 −0.878903
\(313\) 4.93844 0.279137 0.139569 0.990212i \(-0.455428\pi\)
0.139569 + 0.990212i \(0.455428\pi\)
\(314\) 7.97371 0.449983
\(315\) 0 0
\(316\) −2.79458 −0.157208
\(317\) −15.7414 −0.884124 −0.442062 0.896985i \(-0.645753\pi\)
−0.442062 + 0.896985i \(0.645753\pi\)
\(318\) 11.4771 0.643603
\(319\) 2.01130 0.112611
\(320\) 5.04350 0.281940
\(321\) 8.73468 0.487522
\(322\) 0 0
\(323\) −5.40389 −0.300680
\(324\) −0.561916 −0.0312175
\(325\) −23.6086 −1.30957
\(326\) 4.49353 0.248874
\(327\) −2.99347 −0.165539
\(328\) 33.7304 1.86245
\(329\) 0 0
\(330\) 1.73406 0.0954567
\(331\) −27.3344 −1.50244 −0.751218 0.660054i \(-0.770533\pi\)
−0.751218 + 0.660054i \(0.770533\pi\)
\(332\) −5.70653 −0.313186
\(333\) 8.80485 0.482503
\(334\) 3.56320 0.194970
\(335\) 2.29952 0.125636
\(336\) 0 0
\(337\) 9.73803 0.530465 0.265232 0.964185i \(-0.414551\pi\)
0.265232 + 0.964185i \(0.414551\pi\)
\(338\) 15.0310 0.817581
\(339\) 16.3312 0.886990
\(340\) −0.724730 −0.0393040
\(341\) 8.73008 0.472760
\(342\) −2.87730 −0.155587
\(343\) 0 0
\(344\) 5.86751 0.316355
\(345\) 0.572653 0.0308306
\(346\) −13.8125 −0.742563
\(347\) −28.9292 −1.55300 −0.776502 0.630115i \(-0.783008\pi\)
−0.776502 + 0.630115i \(0.783008\pi\)
\(348\) −0.447578 −0.0239927
\(349\) −19.9972 −1.07042 −0.535212 0.844718i \(-0.679768\pi\)
−0.535212 + 0.844718i \(0.679768\pi\)
\(350\) 0 0
\(351\) 5.05314 0.269717
\(352\) 7.76229 0.413731
\(353\) 24.5968 1.30915 0.654577 0.755995i \(-0.272847\pi\)
0.654577 + 0.755995i \(0.272847\pi\)
\(354\) 5.28761 0.281033
\(355\) 0.242881 0.0128908
\(356\) 2.15052 0.113978
\(357\) 0 0
\(358\) −14.2867 −0.755074
\(359\) 20.3150 1.07218 0.536091 0.844160i \(-0.319900\pi\)
0.536091 + 0.844160i \(0.319900\pi\)
\(360\) −1.75934 −0.0927252
\(361\) −13.2431 −0.697007
\(362\) −31.1609 −1.63778
\(363\) −4.62385 −0.242689
\(364\) 0 0
\(365\) 8.01767 0.419664
\(366\) −2.55034 −0.133309
\(367\) 3.48022 0.181666 0.0908330 0.995866i \(-0.471047\pi\)
0.0908330 + 0.995866i \(0.471047\pi\)
\(368\) −2.56042 −0.133471
\(369\) −10.9790 −0.571546
\(370\) 6.04653 0.314344
\(371\) 0 0
\(372\) −1.94272 −0.100725
\(373\) −30.1089 −1.55898 −0.779490 0.626415i \(-0.784521\pi\)
−0.779490 + 0.626415i \(0.784521\pi\)
\(374\) 6.82000 0.352654
\(375\) −5.53874 −0.286019
\(376\) −5.94947 −0.306820
\(377\) 4.02494 0.207295
\(378\) 0 0
\(379\) −26.6106 −1.36689 −0.683446 0.730001i \(-0.739520\pi\)
−0.683446 + 0.730001i \(0.739520\pi\)
\(380\) 0.772069 0.0396063
\(381\) 18.5959 0.952698
\(382\) −21.6136 −1.10585
\(383\) 22.0176 1.12505 0.562524 0.826781i \(-0.309831\pi\)
0.562524 + 0.826781i \(0.309831\pi\)
\(384\) 4.41356 0.225229
\(385\) 0 0
\(386\) −5.61573 −0.285833
\(387\) −1.90984 −0.0970826
\(388\) 8.67396 0.440354
\(389\) 18.3864 0.932227 0.466113 0.884725i \(-0.345654\pi\)
0.466113 + 0.884725i \(0.345654\pi\)
\(390\) 3.47013 0.175717
\(391\) 2.25223 0.113900
\(392\) 0 0
\(393\) −17.9120 −0.903542
\(394\) −26.9576 −1.35811
\(395\) 2.84799 0.143298
\(396\) −1.41890 −0.0713022
\(397\) −10.0731 −0.505557 −0.252778 0.967524i \(-0.581344\pi\)
−0.252778 + 0.967524i \(0.581344\pi\)
\(398\) −3.27828 −0.164325
\(399\) 0 0
\(400\) 11.9625 0.598123
\(401\) −10.8915 −0.543896 −0.271948 0.962312i \(-0.587668\pi\)
−0.271948 + 0.962312i \(0.587668\pi\)
\(402\) 4.81547 0.240174
\(403\) 17.4703 0.870257
\(404\) −3.87263 −0.192671
\(405\) 0.572653 0.0284554
\(406\) 0 0
\(407\) 22.2332 1.10206
\(408\) −6.91943 −0.342563
\(409\) 12.5106 0.618608 0.309304 0.950963i \(-0.399904\pi\)
0.309304 + 0.950963i \(0.399904\pi\)
\(410\) −7.53960 −0.372354
\(411\) 16.1522 0.796731
\(412\) 6.04580 0.297855
\(413\) 0 0
\(414\) 1.19920 0.0589375
\(415\) 5.81557 0.285475
\(416\) 15.5336 0.761597
\(417\) 1.14087 0.0558689
\(418\) −7.26548 −0.355366
\(419\) −32.3891 −1.58231 −0.791157 0.611613i \(-0.790521\pi\)
−0.791157 + 0.611613i \(0.790521\pi\)
\(420\) 0 0
\(421\) −2.76745 −0.134877 −0.0674387 0.997723i \(-0.521483\pi\)
−0.0674387 + 0.997723i \(0.521483\pi\)
\(422\) −4.72999 −0.230252
\(423\) 1.93652 0.0941566
\(424\) −29.4033 −1.42795
\(425\) −10.5226 −0.510420
\(426\) 0.508620 0.0246427
\(427\) 0 0
\(428\) −4.90815 −0.237245
\(429\) 12.7597 0.616044
\(430\) −1.31154 −0.0632480
\(431\) 37.9249 1.82678 0.913389 0.407088i \(-0.133456\pi\)
0.913389 + 0.407088i \(0.133456\pi\)
\(432\) −2.56042 −0.123188
\(433\) −17.8080 −0.855798 −0.427899 0.903827i \(-0.640746\pi\)
−0.427899 + 0.903827i \(0.640746\pi\)
\(434\) 0 0
\(435\) 0.456131 0.0218698
\(436\) 1.68208 0.0805568
\(437\) −2.39935 −0.114776
\(438\) 16.7899 0.802253
\(439\) −39.0320 −1.86290 −0.931448 0.363874i \(-0.881454\pi\)
−0.931448 + 0.363874i \(0.881454\pi\)
\(440\) −4.44251 −0.211788
\(441\) 0 0
\(442\) 13.6479 0.649166
\(443\) 19.9008 0.945513 0.472757 0.881193i \(-0.343259\pi\)
0.472757 + 0.881193i \(0.343259\pi\)
\(444\) −4.94758 −0.234802
\(445\) −2.19162 −0.103893
\(446\) 21.4603 1.01618
\(447\) 8.02342 0.379495
\(448\) 0 0
\(449\) 31.7323 1.49754 0.748769 0.662831i \(-0.230645\pi\)
0.748769 + 0.662831i \(0.230645\pi\)
\(450\) −5.60275 −0.264116
\(451\) −27.7232 −1.30544
\(452\) −9.17677 −0.431639
\(453\) 20.2695 0.952342
\(454\) −21.7987 −1.02306
\(455\) 0 0
\(456\) 7.37140 0.345197
\(457\) −6.27845 −0.293694 −0.146847 0.989159i \(-0.546912\pi\)
−0.146847 + 0.989159i \(0.546912\pi\)
\(458\) −8.60390 −0.402034
\(459\) 2.25223 0.105125
\(460\) −0.321783 −0.0150032
\(461\) 20.4402 0.951996 0.475998 0.879446i \(-0.342087\pi\)
0.475998 + 0.879446i \(0.342087\pi\)
\(462\) 0 0
\(463\) −8.31436 −0.386401 −0.193201 0.981159i \(-0.561887\pi\)
−0.193201 + 0.981159i \(0.561887\pi\)
\(464\) −2.03943 −0.0946782
\(465\) 1.97984 0.0918130
\(466\) −17.2529 −0.799226
\(467\) −3.47840 −0.160961 −0.0804807 0.996756i \(-0.525646\pi\)
−0.0804807 + 0.996756i \(0.525646\pi\)
\(468\) −2.83944 −0.131253
\(469\) 0 0
\(470\) 1.32986 0.0613417
\(471\) 6.64919 0.306378
\(472\) −13.5464 −0.623524
\(473\) −4.82254 −0.221741
\(474\) 5.96401 0.273936
\(475\) 11.2099 0.514346
\(476\) 0 0
\(477\) 9.57060 0.438208
\(478\) −1.81173 −0.0828665
\(479\) −20.7534 −0.948249 −0.474124 0.880458i \(-0.657235\pi\)
−0.474124 + 0.880458i \(0.657235\pi\)
\(480\) 1.76036 0.0803492
\(481\) 44.4921 2.02867
\(482\) −36.9150 −1.68143
\(483\) 0 0
\(484\) 2.59822 0.118101
\(485\) −8.83971 −0.401391
\(486\) 1.19920 0.0543969
\(487\) −26.2962 −1.19159 −0.595796 0.803135i \(-0.703163\pi\)
−0.595796 + 0.803135i \(0.703163\pi\)
\(488\) 6.53376 0.295769
\(489\) 3.74710 0.169450
\(490\) 0 0
\(491\) −5.35237 −0.241549 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(492\) 6.16930 0.278133
\(493\) 1.79395 0.0807956
\(494\) −14.5394 −0.654158
\(495\) 1.44601 0.0649933
\(496\) −8.85217 −0.397474
\(497\) 0 0
\(498\) 12.1785 0.545730
\(499\) 38.8594 1.73959 0.869794 0.493416i \(-0.164252\pi\)
0.869794 + 0.493416i \(0.164252\pi\)
\(500\) 3.11231 0.139187
\(501\) 2.97131 0.132749
\(502\) −15.4184 −0.688156
\(503\) 22.5980 1.00760 0.503798 0.863822i \(-0.331936\pi\)
0.503798 + 0.863822i \(0.331936\pi\)
\(504\) 0 0
\(505\) 3.94663 0.175623
\(506\) 3.02811 0.134616
\(507\) 12.5342 0.556664
\(508\) −10.4493 −0.463615
\(509\) 41.6437 1.84583 0.922913 0.385010i \(-0.125802\pi\)
0.922913 + 0.385010i \(0.125802\pi\)
\(510\) 1.54667 0.0684877
\(511\) 0 0
\(512\) −23.6034 −1.04313
\(513\) −2.39935 −0.105934
\(514\) −4.99843 −0.220471
\(515\) −6.16133 −0.271500
\(516\) 1.07317 0.0472436
\(517\) 4.88990 0.215058
\(518\) 0 0
\(519\) −11.5181 −0.505586
\(520\) −8.89017 −0.389860
\(521\) −17.4077 −0.762646 −0.381323 0.924442i \(-0.624531\pi\)
−0.381323 + 0.924442i \(0.624531\pi\)
\(522\) 0.955190 0.0418076
\(523\) −18.4315 −0.805955 −0.402977 0.915210i \(-0.632025\pi\)
−0.402977 + 0.915210i \(0.632025\pi\)
\(524\) 10.0650 0.439694
\(525\) 0 0
\(526\) 10.9548 0.477650
\(527\) 7.78668 0.339193
\(528\) −6.46532 −0.281367
\(529\) 1.00000 0.0434783
\(530\) 6.57239 0.285486
\(531\) 4.40927 0.191346
\(532\) 0 0
\(533\) −55.4786 −2.40305
\(534\) −4.58950 −0.198607
\(535\) 5.00195 0.216253
\(536\) −12.3368 −0.532869
\(537\) −11.9135 −0.514105
\(538\) 10.8312 0.466966
\(539\) 0 0
\(540\) −0.321783 −0.0138473
\(541\) −16.8540 −0.724611 −0.362305 0.932059i \(-0.618010\pi\)
−0.362305 + 0.932059i \(0.618010\pi\)
\(542\) −8.45798 −0.363302
\(543\) −25.9847 −1.11511
\(544\) 6.92347 0.296841
\(545\) −1.71422 −0.0734291
\(546\) 0 0
\(547\) 34.9994 1.49647 0.748233 0.663436i \(-0.230903\pi\)
0.748233 + 0.663436i \(0.230903\pi\)
\(548\) −9.07619 −0.387716
\(549\) −2.12670 −0.0907654
\(550\) −14.1475 −0.603253
\(551\) −1.91113 −0.0814170
\(552\) −3.07225 −0.130764
\(553\) 0 0
\(554\) 13.5653 0.576336
\(555\) 5.04213 0.214026
\(556\) −0.641075 −0.0271876
\(557\) −25.4013 −1.07629 −0.538143 0.842853i \(-0.680874\pi\)
−0.538143 + 0.842853i \(0.680874\pi\)
\(558\) 4.14602 0.175515
\(559\) −9.65068 −0.408180
\(560\) 0 0
\(561\) 5.68712 0.240110
\(562\) −26.8021 −1.13058
\(563\) 10.4735 0.441404 0.220702 0.975341i \(-0.429165\pi\)
0.220702 + 0.975341i \(0.429165\pi\)
\(564\) −1.08816 −0.0458197
\(565\) 9.35213 0.393447
\(566\) 22.1575 0.931350
\(567\) 0 0
\(568\) −1.30304 −0.0546744
\(569\) 24.0842 1.00966 0.504830 0.863219i \(-0.331555\pi\)
0.504830 + 0.863219i \(0.331555\pi\)
\(570\) −1.64770 −0.0690144
\(571\) 25.7969 1.07957 0.539783 0.841804i \(-0.318506\pi\)
0.539783 + 0.841804i \(0.318506\pi\)
\(572\) −7.16987 −0.299787
\(573\) −18.0234 −0.752937
\(574\) 0 0
\(575\) −4.67207 −0.194839
\(576\) 8.80724 0.366968
\(577\) −11.5859 −0.482326 −0.241163 0.970485i \(-0.577529\pi\)
−0.241163 + 0.970485i \(0.577529\pi\)
\(578\) −14.3034 −0.594943
\(579\) −4.68289 −0.194614
\(580\) −0.256307 −0.0106426
\(581\) 0 0
\(582\) −18.5114 −0.767321
\(583\) 24.1668 1.00089
\(584\) −43.0143 −1.77995
\(585\) 2.89370 0.119640
\(586\) −0.860308 −0.0355390
\(587\) 8.06567 0.332906 0.166453 0.986049i \(-0.446769\pi\)
0.166453 + 0.986049i \(0.446769\pi\)
\(588\) 0 0
\(589\) −8.29529 −0.341802
\(590\) 3.02797 0.124659
\(591\) −22.4797 −0.924689
\(592\) −22.5441 −0.926558
\(593\) 24.5938 1.00995 0.504973 0.863135i \(-0.331502\pi\)
0.504973 + 0.863135i \(0.331502\pi\)
\(594\) 3.02811 0.124245
\(595\) 0 0
\(596\) −4.50849 −0.184675
\(597\) −2.73372 −0.111884
\(598\) 6.05973 0.247801
\(599\) −17.4274 −0.712064 −0.356032 0.934474i \(-0.615871\pi\)
−0.356032 + 0.934474i \(0.615871\pi\)
\(600\) 14.3538 0.585990
\(601\) −46.7989 −1.90897 −0.954484 0.298263i \(-0.903593\pi\)
−0.954484 + 0.298263i \(0.903593\pi\)
\(602\) 0 0
\(603\) 4.01556 0.163526
\(604\) −11.3897 −0.463441
\(605\) −2.64787 −0.107651
\(606\) 8.26470 0.335731
\(607\) 17.9031 0.726665 0.363333 0.931659i \(-0.381639\pi\)
0.363333 + 0.931659i \(0.381639\pi\)
\(608\) −7.37570 −0.299124
\(609\) 0 0
\(610\) −1.46046 −0.0591324
\(611\) 9.78548 0.395878
\(612\) −1.26557 −0.0511574
\(613\) −34.8986 −1.40954 −0.704770 0.709436i \(-0.748950\pi\)
−0.704770 + 0.709436i \(0.748950\pi\)
\(614\) 9.92918 0.400709
\(615\) −6.28719 −0.253524
\(616\) 0 0
\(617\) −47.6661 −1.91896 −0.959482 0.281771i \(-0.909078\pi\)
−0.959482 + 0.281771i \(0.909078\pi\)
\(618\) −12.9025 −0.519015
\(619\) −39.4321 −1.58491 −0.792455 0.609930i \(-0.791197\pi\)
−0.792455 + 0.609930i \(0.791197\pi\)
\(620\) −1.11250 −0.0446793
\(621\) 1.00000 0.0401286
\(622\) 10.7282 0.430162
\(623\) 0 0
\(624\) −12.9382 −0.517941
\(625\) 20.1886 0.807542
\(626\) 5.92219 0.236698
\(627\) −6.05860 −0.241957
\(628\) −3.73628 −0.149094
\(629\) 19.8306 0.790697
\(630\) 0 0
\(631\) −6.93619 −0.276125 −0.138063 0.990423i \(-0.544088\pi\)
−0.138063 + 0.990423i \(0.544088\pi\)
\(632\) −15.2793 −0.607777
\(633\) −3.94428 −0.156771
\(634\) −18.8771 −0.749705
\(635\) 10.6490 0.422593
\(636\) −5.37787 −0.213247
\(637\) 0 0
\(638\) 2.41195 0.0954902
\(639\) 0.424132 0.0167784
\(640\) 2.52744 0.0999059
\(641\) 25.6360 1.01256 0.506281 0.862368i \(-0.331020\pi\)
0.506281 + 0.862368i \(0.331020\pi\)
\(642\) 10.4746 0.413401
\(643\) −14.2975 −0.563840 −0.281920 0.959438i \(-0.590971\pi\)
−0.281920 + 0.959438i \(0.590971\pi\)
\(644\) 0 0
\(645\) −1.09368 −0.0430635
\(646\) −6.48035 −0.254966
\(647\) 33.3271 1.31022 0.655111 0.755533i \(-0.272622\pi\)
0.655111 + 0.755533i \(0.272622\pi\)
\(648\) −3.07225 −0.120690
\(649\) 11.1339 0.437043
\(650\) −28.3115 −1.11047
\(651\) 0 0
\(652\) −2.10556 −0.0824599
\(653\) 0.716990 0.0280580 0.0140290 0.999902i \(-0.495534\pi\)
0.0140290 + 0.999902i \(0.495534\pi\)
\(654\) −3.58977 −0.140371
\(655\) −10.2574 −0.400789
\(656\) 28.1110 1.09755
\(657\) 14.0009 0.546228
\(658\) 0 0
\(659\) −22.0825 −0.860214 −0.430107 0.902778i \(-0.641524\pi\)
−0.430107 + 0.902778i \(0.641524\pi\)
\(660\) −0.812535 −0.0316279
\(661\) 20.7710 0.807898 0.403949 0.914782i \(-0.367637\pi\)
0.403949 + 0.914782i \(0.367637\pi\)
\(662\) −32.7795 −1.27401
\(663\) 11.3808 0.441995
\(664\) −31.2002 −1.21080
\(665\) 0 0
\(666\) 10.5588 0.409145
\(667\) 0.796522 0.0308415
\(668\) −1.66963 −0.0645998
\(669\) 17.8955 0.691881
\(670\) 2.75759 0.106535
\(671\) −5.37014 −0.207312
\(672\) 0 0
\(673\) 10.5462 0.406525 0.203262 0.979124i \(-0.434846\pi\)
0.203262 + 0.979124i \(0.434846\pi\)
\(674\) 11.6779 0.449815
\(675\) −4.67207 −0.179828
\(676\) −7.04317 −0.270891
\(677\) 14.0346 0.539393 0.269696 0.962945i \(-0.413077\pi\)
0.269696 + 0.962945i \(0.413077\pi\)
\(678\) 19.5844 0.752135
\(679\) 0 0
\(680\) −3.96244 −0.151952
\(681\) −18.1777 −0.696570
\(682\) 10.4691 0.400883
\(683\) 43.3231 1.65771 0.828856 0.559462i \(-0.188992\pi\)
0.828856 + 0.559462i \(0.188992\pi\)
\(684\) 1.34823 0.0515509
\(685\) 9.24963 0.353410
\(686\) 0 0
\(687\) −7.17469 −0.273732
\(688\) 4.88999 0.186429
\(689\) 48.3616 1.84243
\(690\) 0.686727 0.0261432
\(691\) −37.9027 −1.44189 −0.720944 0.692993i \(-0.756292\pi\)
−0.720944 + 0.692993i \(0.756292\pi\)
\(692\) 6.47217 0.246035
\(693\) 0 0
\(694\) −34.6920 −1.31689
\(695\) 0.653326 0.0247821
\(696\) −2.44712 −0.0927577
\(697\) −24.7274 −0.936616
\(698\) −23.9806 −0.907681
\(699\) −14.3870 −0.544167
\(700\) 0 0
\(701\) −17.9525 −0.678058 −0.339029 0.940776i \(-0.610099\pi\)
−0.339029 + 0.940776i \(0.610099\pi\)
\(702\) 6.05973 0.228710
\(703\) −21.1259 −0.796778
\(704\) 22.2392 0.838171
\(705\) 1.10895 0.0417656
\(706\) 29.4965 1.11011
\(707\) 0 0
\(708\) −2.47764 −0.0931154
\(709\) 31.9105 1.19843 0.599213 0.800590i \(-0.295480\pi\)
0.599213 + 0.800590i \(0.295480\pi\)
\(710\) 0.291263 0.0109309
\(711\) 4.97332 0.186514
\(712\) 11.7579 0.440646
\(713\) 3.45731 0.129477
\(714\) 0 0
\(715\) 7.30688 0.273262
\(716\) 6.69437 0.250180
\(717\) −1.51078 −0.0564210
\(718\) 24.3617 0.909172
\(719\) −10.3283 −0.385182 −0.192591 0.981279i \(-0.561689\pi\)
−0.192591 + 0.981279i \(0.561689\pi\)
\(720\) −1.46623 −0.0546433
\(721\) 0 0
\(722\) −15.8812 −0.591037
\(723\) −30.7830 −1.14483
\(724\) 14.6012 0.542649
\(725\) −3.72141 −0.138210
\(726\) −5.54493 −0.205792
\(727\) −32.5415 −1.20690 −0.603448 0.797402i \(-0.706207\pi\)
−0.603448 + 0.797402i \(0.706207\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 9.61480 0.355860
\(731\) −4.30140 −0.159093
\(732\) 1.19503 0.0441694
\(733\) −30.9326 −1.14252 −0.571262 0.820768i \(-0.693546\pi\)
−0.571262 + 0.820768i \(0.693546\pi\)
\(734\) 4.17349 0.154046
\(735\) 0 0
\(736\) 3.07405 0.113311
\(737\) 10.1397 0.373501
\(738\) −13.1661 −0.484650
\(739\) 31.6017 1.16249 0.581244 0.813730i \(-0.302566\pi\)
0.581244 + 0.813730i \(0.302566\pi\)
\(740\) −2.83325 −0.104152
\(741\) −12.1242 −0.445395
\(742\) 0 0
\(743\) −10.6217 −0.389674 −0.194837 0.980836i \(-0.562418\pi\)
−0.194837 + 0.980836i \(0.562418\pi\)
\(744\) −10.6217 −0.389412
\(745\) 4.59464 0.168335
\(746\) −36.1066 −1.32196
\(747\) 10.1555 0.371570
\(748\) −3.19568 −0.116846
\(749\) 0 0
\(750\) −6.64207 −0.242534
\(751\) −0.0261115 −0.000952821 0 −0.000476410 1.00000i \(-0.500152\pi\)
−0.000476410 1.00000i \(0.500152\pi\)
\(752\) −4.95829 −0.180810
\(753\) −12.8572 −0.468543
\(754\) 4.82671 0.175778
\(755\) 11.6074 0.422436
\(756\) 0 0
\(757\) 1.50079 0.0545471 0.0272735 0.999628i \(-0.491317\pi\)
0.0272735 + 0.999628i \(0.491317\pi\)
\(758\) −31.9114 −1.15907
\(759\) 2.52510 0.0916554
\(760\) 4.22126 0.153121
\(761\) 37.4931 1.35912 0.679562 0.733618i \(-0.262170\pi\)
0.679562 + 0.733618i \(0.262170\pi\)
\(762\) 22.3003 0.807853
\(763\) 0 0
\(764\) 10.1276 0.366404
\(765\) 1.28975 0.0466310
\(766\) 26.4036 0.953999
\(767\) 22.2807 0.804508
\(768\) −12.3217 −0.444622
\(769\) −32.9794 −1.18927 −0.594633 0.803997i \(-0.702703\pi\)
−0.594633 + 0.803997i \(0.702703\pi\)
\(770\) 0 0
\(771\) −4.16813 −0.150112
\(772\) 2.63139 0.0947057
\(773\) 42.7241 1.53668 0.768340 0.640042i \(-0.221083\pi\)
0.768340 + 0.640042i \(0.221083\pi\)
\(774\) −2.29028 −0.0823225
\(775\) −16.1528 −0.580226
\(776\) 47.4245 1.70244
\(777\) 0 0
\(778\) 22.0490 0.790494
\(779\) 26.3425 0.943819
\(780\) −1.62601 −0.0582206
\(781\) 1.07098 0.0383226
\(782\) 2.70088 0.0965833
\(783\) 0.796522 0.0284654
\(784\) 0 0
\(785\) 3.80768 0.135902
\(786\) −21.4801 −0.766171
\(787\) −11.7935 −0.420394 −0.210197 0.977659i \(-0.567411\pi\)
−0.210197 + 0.977659i \(0.567411\pi\)
\(788\) 12.6317 0.449985
\(789\) 9.13504 0.325216
\(790\) 3.41531 0.121511
\(791\) 0 0
\(792\) −7.75776 −0.275660
\(793\) −10.7465 −0.381620
\(794\) −12.0797 −0.428694
\(795\) 5.48064 0.194378
\(796\) 1.53612 0.0544463
\(797\) 30.7118 1.08787 0.543934 0.839128i \(-0.316934\pi\)
0.543934 + 0.839128i \(0.316934\pi\)
\(798\) 0 0
\(799\) 4.36148 0.154298
\(800\) −14.3622 −0.507779
\(801\) −3.82713 −0.135225
\(802\) −13.0611 −0.461204
\(803\) 35.3537 1.24761
\(804\) −2.25641 −0.0795773
\(805\) 0 0
\(806\) 20.9504 0.737946
\(807\) 9.03201 0.317942
\(808\) −21.1735 −0.744880
\(809\) −41.2129 −1.44897 −0.724484 0.689291i \(-0.757922\pi\)
−0.724484 + 0.689291i \(0.757922\pi\)
\(810\) 0.686727 0.0241291
\(811\) −37.4927 −1.31655 −0.658273 0.752779i \(-0.728713\pi\)
−0.658273 + 0.752779i \(0.728713\pi\)
\(812\) 0 0
\(813\) −7.05301 −0.247360
\(814\) 26.6620 0.934504
\(815\) 2.14579 0.0751638
\(816\) −5.76666 −0.201874
\(817\) 4.58237 0.160317
\(818\) 15.0027 0.524557
\(819\) 0 0
\(820\) 3.53287 0.123373
\(821\) 30.8579 1.07695 0.538473 0.842643i \(-0.319001\pi\)
0.538473 + 0.842643i \(0.319001\pi\)
\(822\) 19.3698 0.675599
\(823\) −7.20897 −0.251289 −0.125644 0.992075i \(-0.540100\pi\)
−0.125644 + 0.992075i \(0.540100\pi\)
\(824\) 33.0552 1.15153
\(825\) −11.7975 −0.410735
\(826\) 0 0
\(827\) 49.7245 1.72909 0.864546 0.502554i \(-0.167606\pi\)
0.864546 + 0.502554i \(0.167606\pi\)
\(828\) −0.561916 −0.0195279
\(829\) −22.3658 −0.776796 −0.388398 0.921492i \(-0.626971\pi\)
−0.388398 + 0.921492i \(0.626971\pi\)
\(830\) 6.97404 0.242073
\(831\) 11.3120 0.392408
\(832\) 44.5042 1.54291
\(833\) 0 0
\(834\) 1.36814 0.0473748
\(835\) 1.70153 0.0588840
\(836\) 3.40442 0.117744
\(837\) 3.45731 0.119502
\(838\) −38.8411 −1.34174
\(839\) −25.3478 −0.875102 −0.437551 0.899194i \(-0.644154\pi\)
−0.437551 + 0.899194i \(0.644154\pi\)
\(840\) 0 0
\(841\) −28.3656 −0.978123
\(842\) −3.31873 −0.114371
\(843\) −22.3499 −0.769773
\(844\) 2.21635 0.0762901
\(845\) 7.17776 0.246922
\(846\) 2.32227 0.0798414
\(847\) 0 0
\(848\) −24.5048 −0.841497
\(849\) 18.4769 0.634125
\(850\) −12.6187 −0.432818
\(851\) 8.80485 0.301826
\(852\) −0.238327 −0.00816494
\(853\) −7.98030 −0.273240 −0.136620 0.990624i \(-0.543624\pi\)
−0.136620 + 0.990624i \(0.543624\pi\)
\(854\) 0 0
\(855\) −1.37399 −0.0469896
\(856\) −26.8352 −0.917207
\(857\) −18.6632 −0.637522 −0.318761 0.947835i \(-0.603267\pi\)
−0.318761 + 0.947835i \(0.603267\pi\)
\(858\) 15.3014 0.522383
\(859\) −6.59371 −0.224975 −0.112487 0.993653i \(-0.535882\pi\)
−0.112487 + 0.993653i \(0.535882\pi\)
\(860\) 0.614554 0.0209561
\(861\) 0 0
\(862\) 45.4796 1.54904
\(863\) −6.50683 −0.221495 −0.110748 0.993849i \(-0.535325\pi\)
−0.110748 + 0.993849i \(0.535325\pi\)
\(864\) 3.07405 0.104581
\(865\) −6.59585 −0.224266
\(866\) −21.3554 −0.725685
\(867\) −11.9274 −0.405077
\(868\) 0 0
\(869\) 12.5581 0.426006
\(870\) 0.546993 0.0185448
\(871\) 20.2912 0.687541
\(872\) 9.19669 0.311439
\(873\) −15.4364 −0.522443
\(874\) −2.87730 −0.0973261
\(875\) 0 0
\(876\) −7.86733 −0.265812
\(877\) 28.9978 0.979186 0.489593 0.871951i \(-0.337145\pi\)
0.489593 + 0.871951i \(0.337145\pi\)
\(878\) −46.8072 −1.57967
\(879\) −0.717401 −0.0241973
\(880\) −3.70239 −0.124808
\(881\) 0.393718 0.0132647 0.00663235 0.999978i \(-0.497889\pi\)
0.00663235 + 0.999978i \(0.497889\pi\)
\(882\) 0 0
\(883\) 56.9604 1.91687 0.958435 0.285310i \(-0.0920964\pi\)
0.958435 + 0.285310i \(0.0920964\pi\)
\(884\) −6.39508 −0.215090
\(885\) 2.52498 0.0848764
\(886\) 23.8650 0.801761
\(887\) −9.84441 −0.330543 −0.165271 0.986248i \(-0.552850\pi\)
−0.165271 + 0.986248i \(0.552850\pi\)
\(888\) −27.0507 −0.907763
\(889\) 0 0
\(890\) −2.62819 −0.0880972
\(891\) 2.52510 0.0845941
\(892\) −10.0558 −0.336692
\(893\) −4.64637 −0.155485
\(894\) 9.62170 0.321798
\(895\) −6.82229 −0.228044
\(896\) 0 0
\(897\) 5.05314 0.168719
\(898\) 38.0534 1.26986
\(899\) 2.75383 0.0918453
\(900\) 2.62531 0.0875103
\(901\) 21.5552 0.718109
\(902\) −33.2457 −1.10696
\(903\) 0 0
\(904\) −50.1736 −1.66875
\(905\) −14.8802 −0.494635
\(906\) 24.3072 0.807551
\(907\) −22.2794 −0.739774 −0.369887 0.929077i \(-0.620604\pi\)
−0.369887 + 0.929077i \(0.620604\pi\)
\(908\) 10.2143 0.338974
\(909\) 6.89184 0.228588
\(910\) 0 0
\(911\) 5.74501 0.190341 0.0951703 0.995461i \(-0.469660\pi\)
0.0951703 + 0.995461i \(0.469660\pi\)
\(912\) 6.14333 0.203426
\(913\) 25.6436 0.848681
\(914\) −7.52913 −0.249041
\(915\) −1.21786 −0.0402613
\(916\) 4.03157 0.133207
\(917\) 0 0
\(918\) 2.70088 0.0891424
\(919\) 20.0743 0.662192 0.331096 0.943597i \(-0.392582\pi\)
0.331096 + 0.943597i \(0.392582\pi\)
\(920\) −1.75934 −0.0580036
\(921\) 8.27982 0.272829
\(922\) 24.5120 0.807258
\(923\) 2.14320 0.0705443
\(924\) 0 0
\(925\) −41.1369 −1.35257
\(926\) −9.97059 −0.327654
\(927\) −10.7593 −0.353380
\(928\) 2.44855 0.0803775
\(929\) 37.1362 1.21840 0.609199 0.793017i \(-0.291491\pi\)
0.609199 + 0.793017i \(0.291491\pi\)
\(930\) 2.37423 0.0778541
\(931\) 0 0
\(932\) 8.08429 0.264810
\(933\) 8.94613 0.292883
\(934\) −4.17131 −0.136489
\(935\) 3.25675 0.106507
\(936\) −15.5245 −0.507435
\(937\) 7.45112 0.243418 0.121709 0.992566i \(-0.461163\pi\)
0.121709 + 0.992566i \(0.461163\pi\)
\(938\) 0 0
\(939\) 4.93844 0.161160
\(940\) −0.623138 −0.0203245
\(941\) −21.3759 −0.696835 −0.348417 0.937339i \(-0.613281\pi\)
−0.348417 + 0.937339i \(0.613281\pi\)
\(942\) 7.97371 0.259798
\(943\) −10.9790 −0.357527
\(944\) −11.2896 −0.367445
\(945\) 0 0
\(946\) −5.78320 −0.188028
\(947\) −8.80385 −0.286087 −0.143043 0.989716i \(-0.545689\pi\)
−0.143043 + 0.989716i \(0.545689\pi\)
\(948\) −2.79458 −0.0907639
\(949\) 70.7485 2.29659
\(950\) 13.4429 0.436146
\(951\) −15.7414 −0.510449
\(952\) 0 0
\(953\) 45.2400 1.46547 0.732734 0.680515i \(-0.238244\pi\)
0.732734 + 0.680515i \(0.238244\pi\)
\(954\) 11.4771 0.371584
\(955\) −10.3211 −0.333984
\(956\) 0.848930 0.0274563
\(957\) 2.01130 0.0650161
\(958\) −24.8876 −0.804081
\(959\) 0 0
\(960\) 5.04350 0.162778
\(961\) −19.0470 −0.614419
\(962\) 53.3550 1.72024
\(963\) 8.73468 0.281471
\(964\) 17.2974 0.557113
\(965\) −2.68167 −0.0863261
\(966\) 0 0
\(967\) −5.60142 −0.180130 −0.0900648 0.995936i \(-0.528707\pi\)
−0.0900648 + 0.995936i \(0.528707\pi\)
\(968\) 14.2056 0.456587
\(969\) −5.40389 −0.173598
\(970\) −10.6006 −0.340365
\(971\) −12.4528 −0.399628 −0.199814 0.979834i \(-0.564034\pi\)
−0.199814 + 0.979834i \(0.564034\pi\)
\(972\) −0.561916 −0.0180235
\(973\) 0 0
\(974\) −31.5344 −1.01043
\(975\) −23.6086 −0.756081
\(976\) 5.44524 0.174298
\(977\) −40.1908 −1.28582 −0.642909 0.765942i \(-0.722273\pi\)
−0.642909 + 0.765942i \(0.722273\pi\)
\(978\) 4.49353 0.143687
\(979\) −9.66390 −0.308860
\(980\) 0 0
\(981\) −2.99347 −0.0955740
\(982\) −6.41857 −0.204825
\(983\) −46.5198 −1.48375 −0.741875 0.670538i \(-0.766063\pi\)
−0.741875 + 0.670538i \(0.766063\pi\)
\(984\) 33.7304 1.07529
\(985\) −12.8731 −0.410169
\(986\) 2.15131 0.0685117
\(987\) 0 0
\(988\) 6.81279 0.216744
\(989\) −1.90984 −0.0607294
\(990\) 1.73406 0.0551119
\(991\) −14.5301 −0.461565 −0.230782 0.973005i \(-0.574129\pi\)
−0.230782 + 0.973005i \(0.574129\pi\)
\(992\) 10.6279 0.337438
\(993\) −27.3344 −0.867432
\(994\) 0 0
\(995\) −1.56547 −0.0496288
\(996\) −5.70653 −0.180818
\(997\) 4.08275 0.129302 0.0646510 0.997908i \(-0.479407\pi\)
0.0646510 + 0.997908i \(0.479407\pi\)
\(998\) 46.6003 1.47511
\(999\) 8.80485 0.278573
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.1.7 10
7.3 odd 6 483.2.i.h.415.4 yes 20
7.5 odd 6 483.2.i.h.277.4 20
7.6 odd 2 3381.2.a.bi.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.4 20 7.5 odd 6
483.2.i.h.415.4 yes 20 7.3 odd 6
3381.2.a.bi.1.7 10 7.6 odd 2
3381.2.a.bj.1.7 10 1.1 even 1 trivial