Properties

Label 3330.2.a.bl.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.23544108.1
Defining polynomial: \(x^{5} - x^{4} - 20 x^{3} + 39 x^{2} + 9 x - 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.43959\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.23800 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.23800 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.01784 q^{11} -3.86299 q^{13} -4.23800 q^{14} +1.00000 q^{16} +6.23800 q^{17} -7.49220 q^{19} +1.00000 q^{20} +2.01784 q^{22} +5.86299 q^{23} +1.00000 q^{25} -3.86299 q^{26} -4.23800 q^{28} +0.375011 q^{29} +9.47436 q^{31} +1.00000 q^{32} +6.23800 q^{34} -4.23800 q^{35} +1.00000 q^{37} -7.49220 q^{38} +1.00000 q^{40} +6.72433 q^{41} +8.72433 q^{43} +2.01784 q^{44} +5.86299 q^{46} +11.3195 q^{47} +10.9607 q^{49} +1.00000 q^{50} -3.86299 q^{52} -0.861339 q^{53} +2.01784 q^{55} -4.23800 q^{56} +0.375011 q^{58} -4.03569 q^{59} +4.12247 q^{61} +9.47436 q^{62} +1.00000 q^{64} -3.86299 q^{65} +12.5117 q^{67} +6.23800 q^{68} -4.23800 q^{70} -14.1987 q^{71} -10.3714 q^{73} +1.00000 q^{74} -7.49220 q^{76} -8.55164 q^{77} -12.1987 q^{79} +1.00000 q^{80} +6.72433 q^{82} +1.82730 q^{83} +6.23800 q^{85} +8.72433 q^{86} +2.01784 q^{88} +13.0890 q^{89} +16.3714 q^{91} +5.86299 q^{92} +11.3195 q^{94} -7.49220 q^{95} +6.59517 q^{97} +10.9607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + O(q^{10}) \) \( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 5 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 7 q^{17} - q^{19} + 5 q^{20} + 5 q^{22} + 5 q^{23} + 5 q^{25} + 5 q^{26} + 3 q^{28} + 2 q^{29} + 16 q^{31} + 5 q^{32} + 7 q^{34} + 3 q^{35} + 5 q^{37} - q^{38} + 5 q^{40} + 2 q^{41} + 12 q^{43} + 5 q^{44} + 5 q^{46} + 6 q^{47} + 16 q^{49} + 5 q^{50} + 5 q^{52} + 3 q^{53} + 5 q^{55} + 3 q^{56} + 2 q^{58} - 10 q^{59} + 16 q^{61} + 16 q^{62} + 5 q^{64} + 5 q^{65} + 4 q^{67} + 7 q^{68} + 3 q^{70} - 8 q^{71} - 3 q^{73} + 5 q^{74} - q^{76} + 3 q^{77} + 2 q^{79} + 5 q^{80} + 2 q^{82} - 5 q^{83} + 7 q^{85} + 12 q^{86} + 5 q^{88} - 7 q^{89} + 33 q^{91} + 5 q^{92} + 6 q^{94} - q^{95} + 14 q^{97} + 16 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.23800 −1.60182 −0.800908 0.598788i \(-0.795649\pi\)
−0.800908 + 0.598788i \(0.795649\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.01784 0.608403 0.304202 0.952608i \(-0.401610\pi\)
0.304202 + 0.952608i \(0.401610\pi\)
\(12\) 0 0
\(13\) −3.86299 −1.07140 −0.535701 0.844408i \(-0.679953\pi\)
−0.535701 + 0.844408i \(0.679953\pi\)
\(14\) −4.23800 −1.13265
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.23800 1.51294 0.756469 0.654029i \(-0.226923\pi\)
0.756469 + 0.654029i \(0.226923\pi\)
\(18\) 0 0
\(19\) −7.49220 −1.71883 −0.859414 0.511280i \(-0.829172\pi\)
−0.859414 + 0.511280i \(0.829172\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.01784 0.430206
\(23\) 5.86299 1.22252 0.611259 0.791430i \(-0.290663\pi\)
0.611259 + 0.791430i \(0.290663\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −3.86299 −0.757595
\(27\) 0 0
\(28\) −4.23800 −0.800908
\(29\) 0.375011 0.0696378 0.0348189 0.999394i \(-0.488915\pi\)
0.0348189 + 0.999394i \(0.488915\pi\)
\(30\) 0 0
\(31\) 9.47436 1.70164 0.850822 0.525454i \(-0.176104\pi\)
0.850822 + 0.525454i \(0.176104\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.23800 1.06981
\(35\) −4.23800 −0.716354
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −7.49220 −1.21540
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 6.72433 1.05016 0.525082 0.851052i \(-0.324035\pi\)
0.525082 + 0.851052i \(0.324035\pi\)
\(42\) 0 0
\(43\) 8.72433 1.33045 0.665224 0.746644i \(-0.268336\pi\)
0.665224 + 0.746644i \(0.268336\pi\)
\(44\) 2.01784 0.304202
\(45\) 0 0
\(46\) 5.86299 0.864451
\(47\) 11.3195 1.65112 0.825560 0.564315i \(-0.190860\pi\)
0.825560 + 0.564315i \(0.190860\pi\)
\(48\) 0 0
\(49\) 10.9607 1.56581
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −3.86299 −0.535701
\(53\) −0.861339 −0.118314 −0.0591570 0.998249i \(-0.518841\pi\)
−0.0591570 + 0.998249i \(0.518841\pi\)
\(54\) 0 0
\(55\) 2.01784 0.272086
\(56\) −4.23800 −0.566327
\(57\) 0 0
\(58\) 0.375011 0.0492414
\(59\) −4.03569 −0.525402 −0.262701 0.964877i \(-0.584613\pi\)
−0.262701 + 0.964877i \(0.584613\pi\)
\(60\) 0 0
\(61\) 4.12247 0.527828 0.263914 0.964546i \(-0.414986\pi\)
0.263914 + 0.964546i \(0.414986\pi\)
\(62\) 9.47436 1.20324
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.86299 −0.479145
\(66\) 0 0
\(67\) 12.5117 1.52855 0.764274 0.644892i \(-0.223098\pi\)
0.764274 + 0.644892i \(0.223098\pi\)
\(68\) 6.23800 0.756469
\(69\) 0 0
\(70\) −4.23800 −0.506538
\(71\) −14.1987 −1.68507 −0.842537 0.538638i \(-0.818939\pi\)
−0.842537 + 0.538638i \(0.818939\pi\)
\(72\) 0 0
\(73\) −10.3714 −1.21388 −0.606939 0.794748i \(-0.707603\pi\)
−0.606939 + 0.794748i \(0.707603\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −7.49220 −0.859414
\(77\) −8.55164 −0.974549
\(78\) 0 0
\(79\) −12.1987 −1.37246 −0.686230 0.727385i \(-0.740736\pi\)
−0.686230 + 0.727385i \(0.740736\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 6.72433 0.742578
\(83\) 1.82730 0.200573 0.100286 0.994959i \(-0.468024\pi\)
0.100286 + 0.994959i \(0.468024\pi\)
\(84\) 0 0
\(85\) 6.23800 0.676607
\(86\) 8.72433 0.940769
\(87\) 0 0
\(88\) 2.01784 0.215103
\(89\) 13.0890 1.38743 0.693717 0.720248i \(-0.255972\pi\)
0.693717 + 0.720248i \(0.255972\pi\)
\(90\) 0 0
\(91\) 16.3714 1.71619
\(92\) 5.86299 0.611259
\(93\) 0 0
\(94\) 11.3195 1.16752
\(95\) −7.49220 −0.768684
\(96\) 0 0
\(97\) 6.59517 0.669638 0.334819 0.942282i \(-0.391325\pi\)
0.334819 + 0.942282i \(0.391325\pi\)
\(98\) 10.9607 1.10720
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 13.0422 1.29775 0.648873 0.760897i \(-0.275241\pi\)
0.648873 + 0.760897i \(0.275241\pi\)
\(102\) 0 0
\(103\) −9.31950 −0.918278 −0.459139 0.888364i \(-0.651842\pi\)
−0.459139 + 0.888364i \(0.651842\pi\)
\(104\) −3.86299 −0.378798
\(105\) 0 0
\(106\) −0.861339 −0.0836607
\(107\) 5.11297 0.494290 0.247145 0.968979i \(-0.420508\pi\)
0.247145 + 0.968979i \(0.420508\pi\)
\(108\) 0 0
\(109\) 14.1808 1.35828 0.679139 0.734009i \(-0.262353\pi\)
0.679139 + 0.734009i \(0.262353\pi\)
\(110\) 2.01784 0.192394
\(111\) 0 0
\(112\) −4.23800 −0.400454
\(113\) 3.66068 0.344368 0.172184 0.985065i \(-0.444918\pi\)
0.172184 + 0.985065i \(0.444918\pi\)
\(114\) 0 0
\(115\) 5.86299 0.546727
\(116\) 0.375011 0.0348189
\(117\) 0 0
\(118\) −4.03569 −0.371515
\(119\) −26.4367 −2.42345
\(120\) 0 0
\(121\) −6.92830 −0.629846
\(122\) 4.12247 0.373231
\(123\) 0 0
\(124\) 9.47436 0.850822
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.609707 −0.0541028 −0.0270514 0.999634i \(-0.508612\pi\)
−0.0270514 + 0.999634i \(0.508612\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −3.86299 −0.338807
\(131\) −20.9163 −1.82747 −0.913734 0.406313i \(-0.866814\pi\)
−0.913734 + 0.406313i \(0.866814\pi\)
\(132\) 0 0
\(133\) 31.7520 2.75325
\(134\) 12.5117 1.08085
\(135\) 0 0
\(136\) 6.23800 0.534904
\(137\) −2.87918 −0.245985 −0.122993 0.992408i \(-0.539249\pi\)
−0.122993 + 0.992408i \(0.539249\pi\)
\(138\) 0 0
\(139\) −15.8594 −1.34517 −0.672587 0.740018i \(-0.734817\pi\)
−0.672587 + 0.740018i \(0.734817\pi\)
\(140\) −4.23800 −0.358177
\(141\) 0 0
\(142\) −14.1987 −1.19153
\(143\) −7.79492 −0.651844
\(144\) 0 0
\(145\) 0.375011 0.0311430
\(146\) −10.3714 −0.858342
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 5.35189 0.438444 0.219222 0.975675i \(-0.429648\pi\)
0.219222 + 0.975675i \(0.429648\pi\)
\(150\) 0 0
\(151\) 17.0857 1.39042 0.695208 0.718809i \(-0.255312\pi\)
0.695208 + 0.718809i \(0.255312\pi\)
\(152\) −7.49220 −0.607698
\(153\) 0 0
\(154\) −8.55164 −0.689110
\(155\) 9.47436 0.760999
\(156\) 0 0
\(157\) 1.16072 0.0926358 0.0463179 0.998927i \(-0.485251\pi\)
0.0463179 + 0.998927i \(0.485251\pi\)
\(158\) −12.1987 −0.970475
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −24.8474 −1.95825
\(162\) 0 0
\(163\) 9.36973 0.733894 0.366947 0.930242i \(-0.380403\pi\)
0.366947 + 0.930242i \(0.380403\pi\)
\(164\) 6.72433 0.525082
\(165\) 0 0
\(166\) 1.82730 0.141826
\(167\) −5.86299 −0.453692 −0.226846 0.973931i \(-0.572841\pi\)
−0.226846 + 0.973931i \(0.572841\pi\)
\(168\) 0 0
\(169\) 1.92272 0.147901
\(170\) 6.23800 0.478433
\(171\) 0 0
\(172\) 8.72433 0.665224
\(173\) 2.36138 0.179533 0.0897663 0.995963i \(-0.471388\pi\)
0.0897663 + 0.995963i \(0.471388\pi\)
\(174\) 0 0
\(175\) −4.23800 −0.320363
\(176\) 2.01784 0.152101
\(177\) 0 0
\(178\) 13.0890 0.981064
\(179\) 13.9247 1.04078 0.520390 0.853929i \(-0.325787\pi\)
0.520390 + 0.853929i \(0.325787\pi\)
\(180\) 0 0
\(181\) −2.51170 −0.186693 −0.0933466 0.995634i \(-0.529756\pi\)
−0.0933466 + 0.995634i \(0.529756\pi\)
\(182\) 16.3714 1.21353
\(183\) 0 0
\(184\) 5.86299 0.432226
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 12.5873 0.920476
\(188\) 11.3195 0.825560
\(189\) 0 0
\(190\) −7.49220 −0.543541
\(191\) −25.6867 −1.85862 −0.929311 0.369298i \(-0.879598\pi\)
−0.929311 + 0.369298i \(0.879598\pi\)
\(192\) 0 0
\(193\) −6.03384 −0.434325 −0.217163 0.976135i \(-0.569680\pi\)
−0.217163 + 0.976135i \(0.569680\pi\)
\(194\) 6.59517 0.473506
\(195\) 0 0
\(196\) 10.9607 0.782906
\(197\) 22.3747 1.59413 0.797065 0.603893i \(-0.206385\pi\)
0.797065 + 0.603893i \(0.206385\pi\)
\(198\) 0 0
\(199\) −18.1663 −1.28778 −0.643888 0.765120i \(-0.722680\pi\)
−0.643888 + 0.765120i \(0.722680\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 13.0422 0.917645
\(203\) −1.58930 −0.111547
\(204\) 0 0
\(205\) 6.72433 0.469648
\(206\) −9.31950 −0.649321
\(207\) 0 0
\(208\) −3.86299 −0.267850
\(209\) −15.1181 −1.04574
\(210\) 0 0
\(211\) −17.3594 −1.19507 −0.597536 0.801842i \(-0.703853\pi\)
−0.597536 + 0.801842i \(0.703853\pi\)
\(212\) −0.861339 −0.0591570
\(213\) 0 0
\(214\) 5.11297 0.349516
\(215\) 8.72433 0.594995
\(216\) 0 0
\(217\) −40.1524 −2.72572
\(218\) 14.1808 0.960448
\(219\) 0 0
\(220\) 2.01784 0.136043
\(221\) −24.0974 −1.62096
\(222\) 0 0
\(223\) 0.652333 0.0436834 0.0218417 0.999761i \(-0.493047\pi\)
0.0218417 + 0.999761i \(0.493047\pi\)
\(224\) −4.23800 −0.283164
\(225\) 0 0
\(226\) 3.66068 0.243505
\(227\) 10.1367 0.672796 0.336398 0.941720i \(-0.390791\pi\)
0.336398 + 0.941720i \(0.390791\pi\)
\(228\) 0 0
\(229\) −26.1747 −1.72968 −0.864838 0.502051i \(-0.832579\pi\)
−0.864838 + 0.502051i \(0.832579\pi\)
\(230\) 5.86299 0.386594
\(231\) 0 0
\(232\) 0.375011 0.0246207
\(233\) 27.0812 1.77415 0.887074 0.461628i \(-0.152735\pi\)
0.887074 + 0.461628i \(0.152735\pi\)
\(234\) 0 0
\(235\) 11.3195 0.738403
\(236\) −4.03569 −0.262701
\(237\) 0 0
\(238\) −26.4367 −1.71364
\(239\) −12.3426 −0.798378 −0.399189 0.916869i \(-0.630708\pi\)
−0.399189 + 0.916869i \(0.630708\pi\)
\(240\) 0 0
\(241\) 12.5000 0.805193 0.402596 0.915378i \(-0.368108\pi\)
0.402596 + 0.915378i \(0.368108\pi\)
\(242\) −6.92830 −0.445368
\(243\) 0 0
\(244\) 4.12247 0.263914
\(245\) 10.9607 0.700252
\(246\) 0 0
\(247\) 28.9423 1.84156
\(248\) 9.47436 0.601622
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 9.41298 0.594142 0.297071 0.954855i \(-0.403990\pi\)
0.297071 + 0.954855i \(0.403990\pi\)
\(252\) 0 0
\(253\) 11.8306 0.743784
\(254\) −0.609707 −0.0382564
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.26431 0.390757 0.195379 0.980728i \(-0.437406\pi\)
0.195379 + 0.980728i \(0.437406\pi\)
\(258\) 0 0
\(259\) −4.23800 −0.263337
\(260\) −3.86299 −0.239573
\(261\) 0 0
\(262\) −20.9163 −1.29222
\(263\) −25.6392 −1.58098 −0.790491 0.612474i \(-0.790174\pi\)
−0.790491 + 0.612474i \(0.790174\pi\)
\(264\) 0 0
\(265\) −0.861339 −0.0529117
\(266\) 31.7520 1.94684
\(267\) 0 0
\(268\) 12.5117 0.764274
\(269\) 6.74549 0.411280 0.205640 0.978628i \(-0.434072\pi\)
0.205640 + 0.978628i \(0.434072\pi\)
\(270\) 0 0
\(271\) −14.6390 −0.889256 −0.444628 0.895715i \(-0.646664\pi\)
−0.444628 + 0.895715i \(0.646664\pi\)
\(272\) 6.23800 0.378235
\(273\) 0 0
\(274\) −2.87918 −0.173938
\(275\) 2.01784 0.121681
\(276\) 0 0
\(277\) 17.0533 1.02464 0.512318 0.858796i \(-0.328787\pi\)
0.512318 + 0.858796i \(0.328787\pi\)
\(278\) −15.8594 −0.951182
\(279\) 0 0
\(280\) −4.23800 −0.253269
\(281\) −26.0617 −1.55471 −0.777355 0.629062i \(-0.783439\pi\)
−0.777355 + 0.629062i \(0.783439\pi\)
\(282\) 0 0
\(283\) 15.1214 0.898874 0.449437 0.893312i \(-0.351625\pi\)
0.449437 + 0.893312i \(0.351625\pi\)
\(284\) −14.1987 −0.842537
\(285\) 0 0
\(286\) −7.79492 −0.460923
\(287\) −28.4978 −1.68217
\(288\) 0 0
\(289\) 21.9127 1.28898
\(290\) 0.375011 0.0220214
\(291\) 0 0
\(292\) −10.3714 −0.606939
\(293\) 6.82896 0.398952 0.199476 0.979903i \(-0.436076\pi\)
0.199476 + 0.979903i \(0.436076\pi\)
\(294\) 0 0
\(295\) −4.03569 −0.234967
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 5.35189 0.310026
\(299\) −22.6487 −1.30981
\(300\) 0 0
\(301\) −36.9738 −2.13113
\(302\) 17.0857 0.983173
\(303\) 0 0
\(304\) −7.49220 −0.429707
\(305\) 4.12247 0.236052
\(306\) 0 0
\(307\) −8.03569 −0.458621 −0.229311 0.973353i \(-0.573647\pi\)
−0.229311 + 0.973353i \(0.573647\pi\)
\(308\) −8.55164 −0.487275
\(309\) 0 0
\(310\) 9.47436 0.538107
\(311\) 25.5614 1.44945 0.724727 0.689036i \(-0.241966\pi\)
0.724727 + 0.689036i \(0.241966\pi\)
\(312\) 0 0
\(313\) 20.4182 1.15411 0.577053 0.816707i \(-0.304203\pi\)
0.577053 + 0.816707i \(0.304203\pi\)
\(314\) 1.16072 0.0655034
\(315\) 0 0
\(316\) −12.1987 −0.686230
\(317\) 0.280705 0.0157660 0.00788298 0.999969i \(-0.497491\pi\)
0.00788298 + 0.999969i \(0.497491\pi\)
\(318\) 0 0
\(319\) 0.756714 0.0423679
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −24.8474 −1.38469
\(323\) −46.7364 −2.60048
\(324\) 0 0
\(325\) −3.86299 −0.214280
\(326\) 9.36973 0.518941
\(327\) 0 0
\(328\) 6.72433 0.371289
\(329\) −47.9721 −2.64479
\(330\) 0 0
\(331\) −1.85184 −0.101786 −0.0508932 0.998704i \(-0.516207\pi\)
−0.0508932 + 0.998704i \(0.516207\pi\)
\(332\) 1.82730 0.100286
\(333\) 0 0
\(334\) −5.86299 −0.320809
\(335\) 12.5117 0.683587
\(336\) 0 0
\(337\) −13.4344 −0.731819 −0.365910 0.930650i \(-0.619242\pi\)
−0.365910 + 0.930650i \(0.619242\pi\)
\(338\) 1.92272 0.104582
\(339\) 0 0
\(340\) 6.23800 0.338303
\(341\) 19.1178 1.03529
\(342\) 0 0
\(343\) −16.7854 −0.906326
\(344\) 8.72433 0.470384
\(345\) 0 0
\(346\) 2.36138 0.126949
\(347\) 13.2260 0.710011 0.355005 0.934864i \(-0.384479\pi\)
0.355005 + 0.934864i \(0.384479\pi\)
\(348\) 0 0
\(349\) 34.1591 1.82849 0.914246 0.405159i \(-0.132784\pi\)
0.914246 + 0.405159i \(0.132784\pi\)
\(350\) −4.23800 −0.226531
\(351\) 0 0
\(352\) 2.01784 0.107551
\(353\) −9.08934 −0.483777 −0.241888 0.970304i \(-0.577767\pi\)
−0.241888 + 0.970304i \(0.577767\pi\)
\(354\) 0 0
\(355\) −14.1987 −0.753588
\(356\) 13.0890 0.693717
\(357\) 0 0
\(358\) 13.9247 0.735942
\(359\) −19.3214 −1.01974 −0.509871 0.860251i \(-0.670307\pi\)
−0.509871 + 0.860251i \(0.670307\pi\)
\(360\) 0 0
\(361\) 37.1331 1.95437
\(362\) −2.51170 −0.132012
\(363\) 0 0
\(364\) 16.3714 0.858094
\(365\) −10.3714 −0.542863
\(366\) 0 0
\(367\) 3.77043 0.196815 0.0984074 0.995146i \(-0.468625\pi\)
0.0984074 + 0.995146i \(0.468625\pi\)
\(368\) 5.86299 0.305630
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) 3.65036 0.189517
\(372\) 0 0
\(373\) −12.0390 −0.623356 −0.311678 0.950188i \(-0.600891\pi\)
−0.311678 + 0.950188i \(0.600891\pi\)
\(374\) 12.5873 0.650875
\(375\) 0 0
\(376\) 11.3195 0.583759
\(377\) −1.44867 −0.0746101
\(378\) 0 0
\(379\) 38.1624 1.96027 0.980134 0.198334i \(-0.0635532\pi\)
0.980134 + 0.198334i \(0.0635532\pi\)
\(380\) −7.49220 −0.384342
\(381\) 0 0
\(382\) −25.6867 −1.31424
\(383\) −33.1280 −1.69276 −0.846381 0.532577i \(-0.821224\pi\)
−0.846381 + 0.532577i \(0.821224\pi\)
\(384\) 0 0
\(385\) −8.55164 −0.435832
\(386\) −6.03384 −0.307114
\(387\) 0 0
\(388\) 6.59517 0.334819
\(389\) 13.1178 0.665098 0.332549 0.943086i \(-0.392091\pi\)
0.332549 + 0.943086i \(0.392091\pi\)
\(390\) 0 0
\(391\) 36.5734 1.84960
\(392\) 10.9607 0.553598
\(393\) 0 0
\(394\) 22.3747 1.12722
\(395\) −12.1987 −0.613783
\(396\) 0 0
\(397\) −2.74159 −0.137596 −0.0687982 0.997631i \(-0.521916\pi\)
−0.0687982 + 0.997631i \(0.521916\pi\)
\(398\) −18.1663 −0.910595
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 10.3423 0.516470 0.258235 0.966082i \(-0.416859\pi\)
0.258235 + 0.966082i \(0.416859\pi\)
\(402\) 0 0
\(403\) −36.5994 −1.82314
\(404\) 13.0422 0.648873
\(405\) 0 0
\(406\) −1.58930 −0.0788756
\(407\) 2.01784 0.100021
\(408\) 0 0
\(409\) −7.68699 −0.380097 −0.190049 0.981775i \(-0.560865\pi\)
−0.190049 + 0.981775i \(0.560865\pi\)
\(410\) 6.72433 0.332091
\(411\) 0 0
\(412\) −9.31950 −0.459139
\(413\) 17.1033 0.841597
\(414\) 0 0
\(415\) 1.82730 0.0896988
\(416\) −3.86299 −0.189399
\(417\) 0 0
\(418\) −15.1181 −0.739450
\(419\) −6.70318 −0.327472 −0.163736 0.986504i \(-0.552354\pi\)
−0.163736 + 0.986504i \(0.552354\pi\)
\(420\) 0 0
\(421\) 11.1208 0.541995 0.270998 0.962580i \(-0.412646\pi\)
0.270998 + 0.962580i \(0.412646\pi\)
\(422\) −17.3594 −0.845043
\(423\) 0 0
\(424\) −0.861339 −0.0418303
\(425\) 6.23800 0.302588
\(426\) 0 0
\(427\) −17.4710 −0.845483
\(428\) 5.11297 0.247145
\(429\) 0 0
\(430\) 8.72433 0.420725
\(431\) 26.7035 1.28626 0.643130 0.765757i \(-0.277635\pi\)
0.643130 + 0.765757i \(0.277635\pi\)
\(432\) 0 0
\(433\) −3.38698 −0.162768 −0.0813840 0.996683i \(-0.525934\pi\)
−0.0813840 + 0.996683i \(0.525934\pi\)
\(434\) −40.1524 −1.92737
\(435\) 0 0
\(436\) 14.1808 0.679139
\(437\) −43.9267 −2.10130
\(438\) 0 0
\(439\) −27.5847 −1.31655 −0.658274 0.752779i \(-0.728713\pi\)
−0.658274 + 0.752779i \(0.728713\pi\)
\(440\) 2.01784 0.0961970
\(441\) 0 0
\(442\) −24.0974 −1.14620
\(443\) 15.6474 0.743428 0.371714 0.928347i \(-0.378770\pi\)
0.371714 + 0.928347i \(0.378770\pi\)
\(444\) 0 0
\(445\) 13.0890 0.620479
\(446\) 0.652333 0.0308888
\(447\) 0 0
\(448\) −4.23800 −0.200227
\(449\) −13.6903 −0.646085 −0.323043 0.946384i \(-0.604706\pi\)
−0.323043 + 0.946384i \(0.604706\pi\)
\(450\) 0 0
\(451\) 13.5687 0.638923
\(452\) 3.66068 0.172184
\(453\) 0 0
\(454\) 10.1367 0.475738
\(455\) 16.3714 0.767502
\(456\) 0 0
\(457\) 4.66324 0.218137 0.109069 0.994034i \(-0.465213\pi\)
0.109069 + 0.994034i \(0.465213\pi\)
\(458\) −26.1747 −1.22307
\(459\) 0 0
\(460\) 5.86299 0.273363
\(461\) −22.8048 −1.06212 −0.531062 0.847333i \(-0.678207\pi\)
−0.531062 + 0.847333i \(0.678207\pi\)
\(462\) 0 0
\(463\) 15.8551 0.736851 0.368426 0.929657i \(-0.379897\pi\)
0.368426 + 0.929657i \(0.379897\pi\)
\(464\) 0.375011 0.0174095
\(465\) 0 0
\(466\) 27.0812 1.25451
\(467\) −27.2841 −1.26256 −0.631278 0.775556i \(-0.717469\pi\)
−0.631278 + 0.775556i \(0.717469\pi\)
\(468\) 0 0
\(469\) −53.0246 −2.44845
\(470\) 11.3195 0.522130
\(471\) 0 0
\(472\) −4.03569 −0.185758
\(473\) 17.6044 0.809449
\(474\) 0 0
\(475\) −7.49220 −0.343766
\(476\) −26.4367 −1.21172
\(477\) 0 0
\(478\) −12.3426 −0.564539
\(479\) −38.3033 −1.75012 −0.875062 0.484012i \(-0.839179\pi\)
−0.875062 + 0.484012i \(0.839179\pi\)
\(480\) 0 0
\(481\) −3.86299 −0.176137
\(482\) 12.5000 0.569357
\(483\) 0 0
\(484\) −6.92830 −0.314923
\(485\) 6.59517 0.299471
\(486\) 0 0
\(487\) 7.66820 0.347480 0.173740 0.984792i \(-0.444415\pi\)
0.173740 + 0.984792i \(0.444415\pi\)
\(488\) 4.12247 0.186615
\(489\) 0 0
\(490\) 10.9607 0.495153
\(491\) 42.1786 1.90350 0.951748 0.306882i \(-0.0992856\pi\)
0.951748 + 0.306882i \(0.0992856\pi\)
\(492\) 0 0
\(493\) 2.33932 0.105358
\(494\) 28.9423 1.30218
\(495\) 0 0
\(496\) 9.47436 0.425411
\(497\) 60.1741 2.69918
\(498\) 0 0
\(499\) −4.70318 −0.210543 −0.105272 0.994444i \(-0.533571\pi\)
−0.105272 + 0.994444i \(0.533571\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 9.41298 0.420122
\(503\) −25.7650 −1.14880 −0.574402 0.818573i \(-0.694765\pi\)
−0.574402 + 0.818573i \(0.694765\pi\)
\(504\) 0 0
\(505\) 13.0422 0.580369
\(506\) 11.8306 0.525935
\(507\) 0 0
\(508\) −0.609707 −0.0270514
\(509\) −9.86114 −0.437087 −0.218544 0.975827i \(-0.570131\pi\)
−0.218544 + 0.975827i \(0.570131\pi\)
\(510\) 0 0
\(511\) 43.9540 1.94441
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 6.26431 0.276307
\(515\) −9.31950 −0.410666
\(516\) 0 0
\(517\) 22.8410 1.00455
\(518\) −4.23800 −0.186207
\(519\) 0 0
\(520\) −3.86299 −0.169403
\(521\) 19.1374 0.838425 0.419212 0.907888i \(-0.362306\pi\)
0.419212 + 0.907888i \(0.362306\pi\)
\(522\) 0 0
\(523\) −31.9313 −1.39626 −0.698129 0.715972i \(-0.745984\pi\)
−0.698129 + 0.715972i \(0.745984\pi\)
\(524\) −20.9163 −0.913734
\(525\) 0 0
\(526\) −25.6392 −1.11792
\(527\) 59.1011 2.57448
\(528\) 0 0
\(529\) 11.3747 0.494552
\(530\) −0.861339 −0.0374142
\(531\) 0 0
\(532\) 31.7520 1.37662
\(533\) −25.9761 −1.12515
\(534\) 0 0
\(535\) 5.11297 0.221053
\(536\) 12.5117 0.540423
\(537\) 0 0
\(538\) 6.74549 0.290819
\(539\) 22.1170 0.952645
\(540\) 0 0
\(541\) −16.6825 −0.717238 −0.358619 0.933484i \(-0.616752\pi\)
−0.358619 + 0.933484i \(0.616752\pi\)
\(542\) −14.6390 −0.628799
\(543\) 0 0
\(544\) 6.23800 0.267452
\(545\) 14.1808 0.607441
\(546\) 0 0
\(547\) 15.5600 0.665297 0.332648 0.943051i \(-0.392058\pi\)
0.332648 + 0.943051i \(0.392058\pi\)
\(548\) −2.87918 −0.122993
\(549\) 0 0
\(550\) 2.01784 0.0860412
\(551\) −2.80966 −0.119695
\(552\) 0 0
\(553\) 51.6981 2.19843
\(554\) 17.0533 0.724527
\(555\) 0 0
\(556\) −15.8594 −0.672587
\(557\) 30.6986 1.30074 0.650372 0.759616i \(-0.274613\pi\)
0.650372 + 0.759616i \(0.274613\pi\)
\(558\) 0 0
\(559\) −33.7020 −1.42544
\(560\) −4.23800 −0.179088
\(561\) 0 0
\(562\) −26.0617 −1.09935
\(563\) 15.0173 0.632905 0.316452 0.948608i \(-0.397508\pi\)
0.316452 + 0.948608i \(0.397508\pi\)
\(564\) 0 0
\(565\) 3.66068 0.154006
\(566\) 15.1214 0.635600
\(567\) 0 0
\(568\) −14.1987 −0.595764
\(569\) 16.8084 0.704645 0.352322 0.935879i \(-0.385392\pi\)
0.352322 + 0.935879i \(0.385392\pi\)
\(570\) 0 0
\(571\) 40.0050 1.67416 0.837079 0.547082i \(-0.184261\pi\)
0.837079 + 0.547082i \(0.184261\pi\)
\(572\) −7.79492 −0.325922
\(573\) 0 0
\(574\) −28.4978 −1.18947
\(575\) 5.86299 0.244504
\(576\) 0 0
\(577\) −36.7442 −1.52968 −0.764841 0.644219i \(-0.777182\pi\)
−0.764841 + 0.644219i \(0.777182\pi\)
\(578\) 21.9127 0.911448
\(579\) 0 0
\(580\) 0.375011 0.0155715
\(581\) −7.74412 −0.321280
\(582\) 0 0
\(583\) −1.73805 −0.0719826
\(584\) −10.3714 −0.429171
\(585\) 0 0
\(586\) 6.82896 0.282102
\(587\) 7.28472 0.300673 0.150336 0.988635i \(-0.451964\pi\)
0.150336 + 0.988635i \(0.451964\pi\)
\(588\) 0 0
\(589\) −70.9838 −2.92484
\(590\) −4.03569 −0.166147
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −31.0025 −1.27312 −0.636561 0.771226i \(-0.719644\pi\)
−0.636561 + 0.771226i \(0.719644\pi\)
\(594\) 0 0
\(595\) −26.4367 −1.08380
\(596\) 5.35189 0.219222
\(597\) 0 0
\(598\) −22.6487 −0.926174
\(599\) −16.5117 −0.674650 −0.337325 0.941388i \(-0.609522\pi\)
−0.337325 + 0.941388i \(0.609522\pi\)
\(600\) 0 0
\(601\) −13.9997 −0.571059 −0.285529 0.958370i \(-0.592169\pi\)
−0.285529 + 0.958370i \(0.592169\pi\)
\(602\) −36.9738 −1.50694
\(603\) 0 0
\(604\) 17.0857 0.695208
\(605\) −6.92830 −0.281676
\(606\) 0 0
\(607\) −36.5422 −1.48320 −0.741602 0.670841i \(-0.765933\pi\)
−0.741602 + 0.670841i \(0.765933\pi\)
\(608\) −7.49220 −0.303849
\(609\) 0 0
\(610\) 4.12247 0.166914
\(611\) −43.7272 −1.76901
\(612\) 0 0
\(613\) 5.90231 0.238392 0.119196 0.992871i \(-0.461968\pi\)
0.119196 + 0.992871i \(0.461968\pi\)
\(614\) −8.03569 −0.324294
\(615\) 0 0
\(616\) −8.55164 −0.344555
\(617\) 19.4115 0.781478 0.390739 0.920501i \(-0.372219\pi\)
0.390739 + 0.920501i \(0.372219\pi\)
\(618\) 0 0
\(619\) −28.6723 −1.15244 −0.576219 0.817295i \(-0.695473\pi\)
−0.576219 + 0.817295i \(0.695473\pi\)
\(620\) 9.47436 0.380499
\(621\) 0 0
\(622\) 25.5614 1.02492
\(623\) −55.4713 −2.22241
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 20.4182 0.816076
\(627\) 0 0
\(628\) 1.16072 0.0463179
\(629\) 6.23800 0.248726
\(630\) 0 0
\(631\) −17.5167 −0.697327 −0.348664 0.937248i \(-0.613364\pi\)
−0.348664 + 0.937248i \(0.613364\pi\)
\(632\) −12.1987 −0.485238
\(633\) 0 0
\(634\) 0.280705 0.0111482
\(635\) −0.609707 −0.0241955
\(636\) 0 0
\(637\) −42.3410 −1.67761
\(638\) 0.756714 0.0299586
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 29.7444 1.17483 0.587417 0.809284i \(-0.300145\pi\)
0.587417 + 0.809284i \(0.300145\pi\)
\(642\) 0 0
\(643\) 16.3541 0.644944 0.322472 0.946579i \(-0.395486\pi\)
0.322472 + 0.946579i \(0.395486\pi\)
\(644\) −24.8474 −0.979124
\(645\) 0 0
\(646\) −46.7364 −1.83882
\(647\) 9.67605 0.380405 0.190202 0.981745i \(-0.439086\pi\)
0.190202 + 0.981745i \(0.439086\pi\)
\(648\) 0 0
\(649\) −8.14340 −0.319656
\(650\) −3.86299 −0.151519
\(651\) 0 0
\(652\) 9.36973 0.366947
\(653\) −30.9006 −1.20924 −0.604618 0.796516i \(-0.706674\pi\)
−0.604618 + 0.796516i \(0.706674\pi\)
\(654\) 0 0
\(655\) −20.9163 −0.817269
\(656\) 6.72433 0.262541
\(657\) 0 0
\(658\) −47.9721 −1.87015
\(659\) −30.5779 −1.19115 −0.595573 0.803301i \(-0.703075\pi\)
−0.595573 + 0.803301i \(0.703075\pi\)
\(660\) 0 0
\(661\) 13.4749 0.524115 0.262057 0.965052i \(-0.415599\pi\)
0.262057 + 0.965052i \(0.415599\pi\)
\(662\) −1.85184 −0.0719738
\(663\) 0 0
\(664\) 1.82730 0.0709131
\(665\) 31.7520 1.23129
\(666\) 0 0
\(667\) 2.19869 0.0851335
\(668\) −5.86299 −0.226846
\(669\) 0 0
\(670\) 12.5117 0.483369
\(671\) 8.31851 0.321132
\(672\) 0 0
\(673\) −21.4937 −0.828519 −0.414260 0.910159i \(-0.635959\pi\)
−0.414260 + 0.910159i \(0.635959\pi\)
\(674\) −13.4344 −0.517474
\(675\) 0 0
\(676\) 1.92272 0.0739507
\(677\) −27.9981 −1.07605 −0.538027 0.842927i \(-0.680830\pi\)
−0.538027 + 0.842927i \(0.680830\pi\)
\(678\) 0 0
\(679\) −27.9504 −1.07264
\(680\) 6.23800 0.239217
\(681\) 0 0
\(682\) 19.1178 0.732058
\(683\) 20.6366 0.789639 0.394820 0.918759i \(-0.370807\pi\)
0.394820 + 0.918759i \(0.370807\pi\)
\(684\) 0 0
\(685\) −2.87918 −0.110008
\(686\) −16.7854 −0.640869
\(687\) 0 0
\(688\) 8.72433 0.332612
\(689\) 3.32735 0.126762
\(690\) 0 0
\(691\) 33.2874 1.26631 0.633156 0.774024i \(-0.281759\pi\)
0.633156 + 0.774024i \(0.281759\pi\)
\(692\) 2.36138 0.0897663
\(693\) 0 0
\(694\) 13.2260 0.502053
\(695\) −15.8594 −0.601580
\(696\) 0 0
\(697\) 41.9464 1.58883
\(698\) 34.1591 1.29294
\(699\) 0 0
\(700\) −4.23800 −0.160182
\(701\) −3.29741 −0.124542 −0.0622708 0.998059i \(-0.519834\pi\)
−0.0622708 + 0.998059i \(0.519834\pi\)
\(702\) 0 0
\(703\) −7.49220 −0.282574
\(704\) 2.01784 0.0760504
\(705\) 0 0
\(706\) −9.08934 −0.342082
\(707\) −55.2728 −2.07875
\(708\) 0 0
\(709\) −27.8744 −1.04685 −0.523423 0.852073i \(-0.675345\pi\)
−0.523423 + 0.852073i \(0.675345\pi\)
\(710\) −14.1987 −0.532867
\(711\) 0 0
\(712\) 13.0890 0.490532
\(713\) 55.5481 2.08029
\(714\) 0 0
\(715\) −7.79492 −0.291514
\(716\) 13.9247 0.520390
\(717\) 0 0
\(718\) −19.3214 −0.721067
\(719\) −34.0982 −1.27165 −0.635825 0.771833i \(-0.719340\pi\)
−0.635825 + 0.771833i \(0.719340\pi\)
\(720\) 0 0
\(721\) 39.4961 1.47091
\(722\) 37.1331 1.38195
\(723\) 0 0
\(724\) −2.51170 −0.0933466
\(725\) 0.375011 0.0139276
\(726\) 0 0
\(727\) −33.3122 −1.23548 −0.617741 0.786381i \(-0.711952\pi\)
−0.617741 + 0.786381i \(0.711952\pi\)
\(728\) 16.3714 0.606764
\(729\) 0 0
\(730\) −10.3714 −0.383862
\(731\) 54.4224 2.01289
\(732\) 0 0
\(733\) −17.5569 −0.648480 −0.324240 0.945975i \(-0.605108\pi\)
−0.324240 + 0.945975i \(0.605108\pi\)
\(734\) 3.77043 0.139169
\(735\) 0 0
\(736\) 5.86299 0.216113
\(737\) 25.2467 0.929973
\(738\) 0 0
\(739\) −18.4464 −0.678561 −0.339281 0.940685i \(-0.610184\pi\)
−0.339281 + 0.940685i \(0.610184\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 3.65036 0.134009
\(743\) −19.0679 −0.699532 −0.349766 0.936837i \(-0.613739\pi\)
−0.349766 + 0.936837i \(0.613739\pi\)
\(744\) 0 0
\(745\) 5.35189 0.196078
\(746\) −12.0390 −0.440779
\(747\) 0 0
\(748\) 12.5873 0.460238
\(749\) −21.6688 −0.791761
\(750\) 0 0
\(751\) −35.9571 −1.31209 −0.656046 0.754721i \(-0.727772\pi\)
−0.656046 + 0.754721i \(0.727772\pi\)
\(752\) 11.3195 0.412780
\(753\) 0 0
\(754\) −1.44867 −0.0527573
\(755\) 17.0857 0.621813
\(756\) 0 0
\(757\) −22.5940 −0.821194 −0.410597 0.911817i \(-0.634680\pi\)
−0.410597 + 0.911817i \(0.634680\pi\)
\(758\) 38.1624 1.38612
\(759\) 0 0
\(760\) −7.49220 −0.271771
\(761\) −25.4347 −0.922008 −0.461004 0.887398i \(-0.652511\pi\)
−0.461004 + 0.887398i \(0.652511\pi\)
\(762\) 0 0
\(763\) −60.0985 −2.17571
\(764\) −25.6867 −0.929311
\(765\) 0 0
\(766\) −33.1280 −1.19696
\(767\) 15.5898 0.562917
\(768\) 0 0
\(769\) −2.20263 −0.0794290 −0.0397145 0.999211i \(-0.512645\pi\)
−0.0397145 + 0.999211i \(0.512645\pi\)
\(770\) −8.55164 −0.308180
\(771\) 0 0
\(772\) −6.03384 −0.217163
\(773\) −20.5939 −0.740713 −0.370356 0.928890i \(-0.620764\pi\)
−0.370356 + 0.928890i \(0.620764\pi\)
\(774\) 0 0
\(775\) 9.47436 0.340329
\(776\) 6.59517 0.236753
\(777\) 0 0
\(778\) 13.1178 0.470295
\(779\) −50.3800 −1.80505
\(780\) 0 0
\(781\) −28.6508 −1.02520
\(782\) 36.5734 1.30786
\(783\) 0 0
\(784\) 10.9607 0.391453
\(785\) 1.16072 0.0414280
\(786\) 0 0
\(787\) 5.94036 0.211751 0.105876 0.994379i \(-0.466235\pi\)
0.105876 + 0.994379i \(0.466235\pi\)
\(788\) 22.3747 0.797065
\(789\) 0 0
\(790\) −12.1987 −0.434010
\(791\) −15.5140 −0.551613
\(792\) 0 0
\(793\) −15.9251 −0.565516
\(794\) −2.74159 −0.0972953
\(795\) 0 0
\(796\) −18.1663 −0.643888
\(797\) 13.3214 0.471867 0.235933 0.971769i \(-0.424185\pi\)
0.235933 + 0.971769i \(0.424185\pi\)
\(798\) 0 0
\(799\) 70.6111 2.49804
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 10.3423 0.365200
\(803\) −20.9278 −0.738528
\(804\) 0 0
\(805\) −24.8474 −0.875756
\(806\) −36.5994 −1.28916
\(807\) 0 0
\(808\) 13.0422 0.458822
\(809\) 3.52974 0.124099 0.0620496 0.998073i \(-0.480236\pi\)
0.0620496 + 0.998073i \(0.480236\pi\)
\(810\) 0 0
\(811\) −3.46105 −0.121534 −0.0607669 0.998152i \(-0.519355\pi\)
−0.0607669 + 0.998152i \(0.519355\pi\)
\(812\) −1.58930 −0.0557735
\(813\) 0 0
\(814\) 2.01784 0.0707254
\(815\) 9.36973 0.328207
\(816\) 0 0
\(817\) −65.3644 −2.28681
\(818\) −7.68699 −0.268769
\(819\) 0 0
\(820\) 6.72433 0.234824
\(821\) −6.13452 −0.214096 −0.107048 0.994254i \(-0.534140\pi\)
−0.107048 + 0.994254i \(0.534140\pi\)
\(822\) 0 0
\(823\) −7.18104 −0.250315 −0.125158 0.992137i \(-0.539944\pi\)
−0.125158 + 0.992137i \(0.539944\pi\)
\(824\) −9.31950 −0.324660
\(825\) 0 0
\(826\) 17.1033 0.595099
\(827\) 34.6127 1.20360 0.601801 0.798646i \(-0.294450\pi\)
0.601801 + 0.798646i \(0.294450\pi\)
\(828\) 0 0
\(829\) −21.3225 −0.740561 −0.370281 0.928920i \(-0.620738\pi\)
−0.370281 + 0.928920i \(0.620738\pi\)
\(830\) 1.82730 0.0634266
\(831\) 0 0
\(832\) −3.86299 −0.133925
\(833\) 68.3728 2.36898
\(834\) 0 0
\(835\) −5.86299 −0.202897
\(836\) −15.1181 −0.522870
\(837\) 0 0
\(838\) −6.70318 −0.231557
\(839\) 38.1267 1.31628 0.658139 0.752896i \(-0.271344\pi\)
0.658139 + 0.752896i \(0.271344\pi\)
\(840\) 0 0
\(841\) −28.8594 −0.995151
\(842\) 11.1208 0.383249
\(843\) 0 0
\(844\) −17.3594 −0.597536
\(845\) 1.92272 0.0661435
\(846\) 0 0
\(847\) 29.3622 1.00890
\(848\) −0.861339 −0.0295785
\(849\) 0 0
\(850\) 6.23800 0.213962
\(851\) 5.86299 0.200981
\(852\) 0 0
\(853\) 25.8306 0.884423 0.442212 0.896911i \(-0.354194\pi\)
0.442212 + 0.896911i \(0.354194\pi\)
\(854\) −17.4710 −0.597847
\(855\) 0 0
\(856\) 5.11297 0.174758
\(857\) 39.5354 1.35050 0.675252 0.737587i \(-0.264035\pi\)
0.675252 + 0.737587i \(0.264035\pi\)
\(858\) 0 0
\(859\) −48.9118 −1.66885 −0.834424 0.551122i \(-0.814200\pi\)
−0.834424 + 0.551122i \(0.814200\pi\)
\(860\) 8.72433 0.297497
\(861\) 0 0
\(862\) 26.7035 0.909523
\(863\) 33.7426 1.14861 0.574305 0.818641i \(-0.305272\pi\)
0.574305 + 0.818641i \(0.305272\pi\)
\(864\) 0 0
\(865\) 2.36138 0.0802895
\(866\) −3.38698 −0.115094
\(867\) 0 0
\(868\) −40.1524 −1.36286
\(869\) −24.6151 −0.835009
\(870\) 0 0
\(871\) −48.3326 −1.63769
\(872\) 14.1808 0.480224
\(873\) 0 0
\(874\) −43.9267 −1.48584
\(875\) −4.23800 −0.143271
\(876\) 0 0
\(877\) −41.2829 −1.39402 −0.697012 0.717059i \(-0.745488\pi\)
−0.697012 + 0.717059i \(0.745488\pi\)
\(878\) −27.5847 −0.930940
\(879\) 0 0
\(880\) 2.01784 0.0680215
\(881\) 53.0893 1.78862 0.894312 0.447443i \(-0.147665\pi\)
0.894312 + 0.447443i \(0.147665\pi\)
\(882\) 0 0
\(883\) 11.6180 0.390976 0.195488 0.980706i \(-0.437371\pi\)
0.195488 + 0.980706i \(0.437371\pi\)
\(884\) −24.0974 −0.810482
\(885\) 0 0
\(886\) 15.6474 0.525683
\(887\) −28.7458 −0.965189 −0.482595 0.875844i \(-0.660306\pi\)
−0.482595 + 0.875844i \(0.660306\pi\)
\(888\) 0 0
\(889\) 2.58394 0.0866627
\(890\) 13.0890 0.438745
\(891\) 0 0
\(892\) 0.652333 0.0218417
\(893\) −84.8080 −2.83799
\(894\) 0 0
\(895\) 13.9247 0.465451
\(896\) −4.23800 −0.141582
\(897\) 0 0
\(898\) −13.6903 −0.456851
\(899\) 3.55299 0.118499
\(900\) 0 0
\(901\) −5.37304 −0.179002
\(902\) 13.5687 0.451787
\(903\) 0 0
\(904\) 3.66068 0.121752
\(905\) −2.51170 −0.0834917
\(906\) 0 0
\(907\) 24.6046 0.816982 0.408491 0.912762i \(-0.366055\pi\)
0.408491 + 0.912762i \(0.366055\pi\)
\(908\) 10.1367 0.336398
\(909\) 0 0
\(910\) 16.3714 0.542706
\(911\) −41.3494 −1.36997 −0.684983 0.728559i \(-0.740191\pi\)
−0.684983 + 0.728559i \(0.740191\pi\)
\(912\) 0 0
\(913\) 3.68722 0.122029
\(914\) 4.66324 0.154246
\(915\) 0 0
\(916\) −26.1747 −0.864838
\(917\) 88.6435 2.92727
\(918\) 0 0
\(919\) 49.8444 1.64422 0.822108 0.569332i \(-0.192798\pi\)
0.822108 + 0.569332i \(0.192798\pi\)
\(920\) 5.86299 0.193297
\(921\) 0 0
\(922\) −22.8048 −0.751035
\(923\) 54.8494 1.80539
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 15.8551 0.521033
\(927\) 0 0
\(928\) 0.375011 0.0123103
\(929\) 15.1440 0.496859 0.248429 0.968650i \(-0.420086\pi\)
0.248429 + 0.968650i \(0.420086\pi\)
\(930\) 0 0
\(931\) −82.1196 −2.69136
\(932\) 27.0812 0.887074
\(933\) 0 0
\(934\) −27.2841 −0.892763
\(935\) 12.5873 0.411650
\(936\) 0 0
\(937\) −33.2735 −1.08700 −0.543498 0.839410i \(-0.682901\pi\)
−0.543498 + 0.839410i \(0.682901\pi\)
\(938\) −53.0246 −1.73132
\(939\) 0 0
\(940\) 11.3195 0.369201
\(941\) −47.6286 −1.55265 −0.776323 0.630335i \(-0.782918\pi\)
−0.776323 + 0.630335i \(0.782918\pi\)
\(942\) 0 0
\(943\) 39.4247 1.28385
\(944\) −4.03569 −0.131351
\(945\) 0 0
\(946\) 17.6044 0.572367
\(947\) 20.6366 0.670601 0.335300 0.942111i \(-0.391162\pi\)
0.335300 + 0.942111i \(0.391162\pi\)
\(948\) 0 0
\(949\) 40.0646 1.30055
\(950\) −7.49220 −0.243079
\(951\) 0 0
\(952\) −26.4367 −0.856818
\(953\) 22.8435 0.739973 0.369987 0.929037i \(-0.379362\pi\)
0.369987 + 0.929037i \(0.379362\pi\)
\(954\) 0 0
\(955\) −25.6867 −0.831201
\(956\) −12.3426 −0.399189
\(957\) 0 0
\(958\) −38.3033 −1.23752
\(959\) 12.2020 0.394023
\(960\) 0 0
\(961\) 58.7634 1.89559
\(962\) −3.86299 −0.124548
\(963\) 0 0
\(964\) 12.5000 0.402596
\(965\) −6.03384 −0.194236
\(966\) 0 0
\(967\) 0.0934721 0.00300586 0.00150293 0.999999i \(-0.499522\pi\)
0.00150293 + 0.999999i \(0.499522\pi\)
\(968\) −6.92830 −0.222684
\(969\) 0 0
\(970\) 6.59517 0.211758
\(971\) 34.5628 1.10917 0.554586 0.832126i \(-0.312877\pi\)
0.554586 + 0.832126i \(0.312877\pi\)
\(972\) 0 0
\(973\) 67.2121 2.15472
\(974\) 7.66820 0.245705
\(975\) 0 0
\(976\) 4.12247 0.131957
\(977\) −61.6264 −1.97160 −0.985802 0.167915i \(-0.946297\pi\)
−0.985802 + 0.167915i \(0.946297\pi\)
\(978\) 0 0
\(979\) 26.4116 0.844119
\(980\) 10.9607 0.350126
\(981\) 0 0
\(982\) 42.1786 1.34597
\(983\) −6.38987 −0.203805 −0.101903 0.994794i \(-0.532493\pi\)
−0.101903 + 0.994794i \(0.532493\pi\)
\(984\) 0 0
\(985\) 22.3747 0.712917
\(986\) 2.33932 0.0744992
\(987\) 0 0
\(988\) 28.9423 0.920778
\(989\) 51.1507 1.62650
\(990\) 0 0
\(991\) −24.0777 −0.764852 −0.382426 0.923986i \(-0.624911\pi\)
−0.382426 + 0.923986i \(0.624911\pi\)
\(992\) 9.47436 0.300811
\(993\) 0 0
\(994\) 60.1741 1.90861
\(995\) −18.1663 −0.575911
\(996\) 0 0
\(997\) 50.1397 1.58794 0.793970 0.607957i \(-0.208011\pi\)
0.793970 + 0.607957i \(0.208011\pi\)
\(998\) −4.70318 −0.148877
\(999\) 0 0
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 3330.2.a.bl.1.1 yes 5
3.2 odd 2 3330.2.a.bk.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.bk.1.1 5 3.2 odd 2
3330.2.a.bl.1.1 yes 5 1.1 even 1 trivial