Properties

Label 7728.2.a.by.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1229.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.75153\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.75153 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.75153 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.32246 q^{11} +1.57093 q^{13} +2.75153 q^{15} +0.819397 q^{17} +2.81940 q^{19} -1.00000 q^{21} -1.00000 q^{23} +2.57093 q^{25} +1.00000 q^{27} -0.819397 q^{29} -8.32246 q^{31} -4.32246 q^{33} -2.75153 q^{35} +8.68367 q^{37} +1.57093 q^{39} +10.3225 q^{41} +0.429071 q^{43} +2.75153 q^{45} +10.1480 q^{47} +1.00000 q^{49} +0.819397 q^{51} -7.39645 q^{53} -11.8934 q^{55} +2.81940 q^{57} +6.39033 q^{59} +13.4352 q^{61} -1.00000 q^{63} +4.32246 q^{65} +1.60967 q^{67} -1.00000 q^{69} +2.06786 q^{71} -13.4643 q^{73} +2.57093 q^{75} +4.32246 q^{77} +6.81940 q^{79} +1.00000 q^{81} +4.32246 q^{83} +2.25460 q^{85} -0.819397 q^{87} +7.93214 q^{89} -1.57093 q^{91} -8.32246 q^{93} +7.75766 q^{95} +1.67754 q^{97} -4.32246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} + q^{15} + 2 q^{17} + 8 q^{19} - 3 q^{21} - 3 q^{23} + 3 q^{27} - 2 q^{29} - 10 q^{31} + 2 q^{33} - q^{35} + 12 q^{37} - 3 q^{39} + 16 q^{41} + 9 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} + 2 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} + 19 q^{61} - 3 q^{63} - 2 q^{65} + 13 q^{67} - 3 q^{69} + 13 q^{71} - 10 q^{73} - 2 q^{77} + 20 q^{79} + 3 q^{81} - 2 q^{83} - 15 q^{85} - 2 q^{87} + 17 q^{89} + 3 q^{91} - 10 q^{93} - 13 q^{95} + 20 q^{97} + 2 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.75153 1.23052 0.615261 0.788323i \(-0.289051\pi\)
0.615261 + 0.788323i \(0.289051\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.32246 −1.30327 −0.651635 0.758532i \(-0.725917\pi\)
−0.651635 + 0.758532i \(0.725917\pi\)
\(12\) 0 0
\(13\) 1.57093 0.435697 0.217849 0.975983i \(-0.430096\pi\)
0.217849 + 0.975983i \(0.430096\pi\)
\(14\) 0 0
\(15\) 2.75153 0.710443
\(16\) 0 0
\(17\) 0.819397 0.198733 0.0993664 0.995051i \(-0.468318\pi\)
0.0993664 + 0.995051i \(0.468318\pi\)
\(18\) 0 0
\(19\) 2.81940 0.646814 0.323407 0.946260i \(-0.395172\pi\)
0.323407 + 0.946260i \(0.395172\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.57093 0.514186
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.819397 −0.152158 −0.0760791 0.997102i \(-0.524240\pi\)
−0.0760791 + 0.997102i \(0.524240\pi\)
\(30\) 0 0
\(31\) −8.32246 −1.49476 −0.747379 0.664398i \(-0.768688\pi\)
−0.747379 + 0.664398i \(0.768688\pi\)
\(32\) 0 0
\(33\) −4.32246 −0.752444
\(34\) 0 0
\(35\) −2.75153 −0.465094
\(36\) 0 0
\(37\) 8.68367 1.42759 0.713793 0.700357i \(-0.246976\pi\)
0.713793 + 0.700357i \(0.246976\pi\)
\(38\) 0 0
\(39\) 1.57093 0.251550
\(40\) 0 0
\(41\) 10.3225 1.61210 0.806049 0.591849i \(-0.201602\pi\)
0.806049 + 0.591849i \(0.201602\pi\)
\(42\) 0 0
\(43\) 0.429071 0.0654328 0.0327164 0.999465i \(-0.489584\pi\)
0.0327164 + 0.999465i \(0.489584\pi\)
\(44\) 0 0
\(45\) 2.75153 0.410174
\(46\) 0 0
\(47\) 10.1480 1.48024 0.740118 0.672477i \(-0.234770\pi\)
0.740118 + 0.672477i \(0.234770\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.819397 0.114738
\(52\) 0 0
\(53\) −7.39645 −1.01598 −0.507991 0.861363i \(-0.669612\pi\)
−0.507991 + 0.861363i \(0.669612\pi\)
\(54\) 0 0
\(55\) −11.8934 −1.60370
\(56\) 0 0
\(57\) 2.81940 0.373438
\(58\) 0 0
\(59\) 6.39033 0.831950 0.415975 0.909376i \(-0.363440\pi\)
0.415975 + 0.909376i \(0.363440\pi\)
\(60\) 0 0
\(61\) 13.4352 1.72020 0.860101 0.510125i \(-0.170401\pi\)
0.860101 + 0.510125i \(0.170401\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 4.32246 0.536135
\(66\) 0 0
\(67\) 1.60967 0.196653 0.0983265 0.995154i \(-0.468651\pi\)
0.0983265 + 0.995154i \(0.468651\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.06786 0.245410 0.122705 0.992443i \(-0.460843\pi\)
0.122705 + 0.992443i \(0.460843\pi\)
\(72\) 0 0
\(73\) −13.4643 −1.57588 −0.787940 0.615752i \(-0.788852\pi\)
−0.787940 + 0.615752i \(0.788852\pi\)
\(74\) 0 0
\(75\) 2.57093 0.296865
\(76\) 0 0
\(77\) 4.32246 0.492590
\(78\) 0 0
\(79\) 6.81940 0.767242 0.383621 0.923491i \(-0.374677\pi\)
0.383621 + 0.923491i \(0.374677\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.32246 0.474452 0.237226 0.971455i \(-0.423762\pi\)
0.237226 + 0.971455i \(0.423762\pi\)
\(84\) 0 0
\(85\) 2.25460 0.244545
\(86\) 0 0
\(87\) −0.819397 −0.0878485
\(88\) 0 0
\(89\) 7.93214 0.840805 0.420402 0.907338i \(-0.361889\pi\)
0.420402 + 0.907338i \(0.361889\pi\)
\(90\) 0 0
\(91\) −1.57093 −0.164678
\(92\) 0 0
\(93\) −8.32246 −0.862999
\(94\) 0 0
\(95\) 7.75766 0.795919
\(96\) 0 0
\(97\) 1.67754 0.170328 0.0851641 0.996367i \(-0.472859\pi\)
0.0851641 + 0.996367i \(0.472859\pi\)
\(98\) 0 0
\(99\) −4.32246 −0.434424
\(100\) 0 0
\(101\) 8.39033 0.834869 0.417434 0.908707i \(-0.362929\pi\)
0.417434 + 0.908707i \(0.362929\pi\)
\(102\) 0 0
\(103\) 11.0061 1.08447 0.542233 0.840228i \(-0.317579\pi\)
0.542233 + 0.840228i \(0.317579\pi\)
\(104\) 0 0
\(105\) −2.75153 −0.268522
\(106\) 0 0
\(107\) 4.29334 0.415053 0.207527 0.978229i \(-0.433459\pi\)
0.207527 + 0.978229i \(0.433459\pi\)
\(108\) 0 0
\(109\) −5.89339 −0.564484 −0.282242 0.959343i \(-0.591078\pi\)
−0.282242 + 0.959343i \(0.591078\pi\)
\(110\) 0 0
\(111\) 8.68367 0.824217
\(112\) 0 0
\(113\) 3.74540 0.352338 0.176169 0.984360i \(-0.443629\pi\)
0.176169 + 0.984360i \(0.443629\pi\)
\(114\) 0 0
\(115\) −2.75153 −0.256582
\(116\) 0 0
\(117\) 1.57093 0.145232
\(118\) 0 0
\(119\) −0.819397 −0.0751140
\(120\) 0 0
\(121\) 7.68367 0.698515
\(122\) 0 0
\(123\) 10.3225 0.930745
\(124\) 0 0
\(125\) −6.68367 −0.597805
\(126\) 0 0
\(127\) −5.07399 −0.450244 −0.225122 0.974331i \(-0.572278\pi\)
−0.225122 + 0.974331i \(0.572278\pi\)
\(128\) 0 0
\(129\) 0.429071 0.0377776
\(130\) 0 0
\(131\) −12.3225 −1.07662 −0.538309 0.842747i \(-0.680937\pi\)
−0.538309 + 0.842747i \(0.680937\pi\)
\(132\) 0 0
\(133\) −2.81940 −0.244473
\(134\) 0 0
\(135\) 2.75153 0.236814
\(136\) 0 0
\(137\) 1.67754 0.143322 0.0716609 0.997429i \(-0.477170\pi\)
0.0716609 + 0.997429i \(0.477170\pi\)
\(138\) 0 0
\(139\) 11.7577 0.997272 0.498636 0.866812i \(-0.333834\pi\)
0.498636 + 0.866812i \(0.333834\pi\)
\(140\) 0 0
\(141\) 10.1480 0.854615
\(142\) 0 0
\(143\) −6.79028 −0.567832
\(144\) 0 0
\(145\) −2.25460 −0.187234
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −0.496936 −0.0407106 −0.0203553 0.999793i \(-0.506480\pi\)
−0.0203553 + 0.999793i \(0.506480\pi\)
\(150\) 0 0
\(151\) 10.1480 0.825831 0.412916 0.910769i \(-0.364510\pi\)
0.412916 + 0.910769i \(0.364510\pi\)
\(152\) 0 0
\(153\) 0.819397 0.0662443
\(154\) 0 0
\(155\) −22.8995 −1.83933
\(156\) 0 0
\(157\) 19.4256 1.55033 0.775165 0.631759i \(-0.217667\pi\)
0.775165 + 0.631759i \(0.217667\pi\)
\(158\) 0 0
\(159\) −7.39645 −0.586577
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −2.25460 −0.176594 −0.0882968 0.996094i \(-0.528142\pi\)
−0.0882968 + 0.996094i \(0.528142\pi\)
\(164\) 0 0
\(165\) −11.8934 −0.925899
\(166\) 0 0
\(167\) 5.31633 0.411390 0.205695 0.978616i \(-0.434054\pi\)
0.205695 + 0.978616i \(0.434054\pi\)
\(168\) 0 0
\(169\) −10.5322 −0.810168
\(170\) 0 0
\(171\) 2.81940 0.215605
\(172\) 0 0
\(173\) 19.9613 1.51763 0.758813 0.651309i \(-0.225780\pi\)
0.758813 + 0.651309i \(0.225780\pi\)
\(174\) 0 0
\(175\) −2.57093 −0.194344
\(176\) 0 0
\(177\) 6.39033 0.480326
\(178\) 0 0
\(179\) −4.02912 −0.301150 −0.150575 0.988599i \(-0.548113\pi\)
−0.150575 + 0.988599i \(0.548113\pi\)
\(180\) 0 0
\(181\) 0.955126 0.0709939 0.0354970 0.999370i \(-0.488699\pi\)
0.0354970 + 0.999370i \(0.488699\pi\)
\(182\) 0 0
\(183\) 13.4352 0.993159
\(184\) 0 0
\(185\) 23.8934 1.75668
\(186\) 0 0
\(187\) −3.54181 −0.259003
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.85814 −0.640953 −0.320476 0.947257i \(-0.603843\pi\)
−0.320476 + 0.947257i \(0.603843\pi\)
\(192\) 0 0
\(193\) 9.14186 0.658045 0.329023 0.944322i \(-0.393281\pi\)
0.329023 + 0.944322i \(0.393281\pi\)
\(194\) 0 0
\(195\) 4.32246 0.309538
\(196\) 0 0
\(197\) −6.21585 −0.442861 −0.221431 0.975176i \(-0.571073\pi\)
−0.221431 + 0.975176i \(0.571073\pi\)
\(198\) 0 0
\(199\) −21.3965 −1.51675 −0.758377 0.651816i \(-0.774007\pi\)
−0.758377 + 0.651816i \(0.774007\pi\)
\(200\) 0 0
\(201\) 1.60967 0.113538
\(202\) 0 0
\(203\) 0.819397 0.0575104
\(204\) 0 0
\(205\) 28.4026 1.98372
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −12.1867 −0.842974
\(210\) 0 0
\(211\) −5.18060 −0.356647 −0.178324 0.983972i \(-0.557067\pi\)
−0.178324 + 0.983972i \(0.557067\pi\)
\(212\) 0 0
\(213\) 2.06786 0.141688
\(214\) 0 0
\(215\) 1.18060 0.0805165
\(216\) 0 0
\(217\) 8.32246 0.564965
\(218\) 0 0
\(219\) −13.4643 −0.909834
\(220\) 0 0
\(221\) 1.28721 0.0865874
\(222\) 0 0
\(223\) 16.0291 1.07339 0.536695 0.843777i \(-0.319673\pi\)
0.536695 + 0.843777i \(0.319673\pi\)
\(224\) 0 0
\(225\) 2.57093 0.171395
\(226\) 0 0
\(227\) 13.2607 0.880145 0.440073 0.897962i \(-0.354953\pi\)
0.440073 + 0.897962i \(0.354953\pi\)
\(228\) 0 0
\(229\) −0.254596 −0.0168242 −0.00841210 0.999965i \(-0.502678\pi\)
−0.00841210 + 0.999965i \(0.502678\pi\)
\(230\) 0 0
\(231\) 4.32246 0.284397
\(232\) 0 0
\(233\) 3.74540 0.245370 0.122685 0.992446i \(-0.460850\pi\)
0.122685 + 0.992446i \(0.460850\pi\)
\(234\) 0 0
\(235\) 27.9225 1.82146
\(236\) 0 0
\(237\) 6.81940 0.442967
\(238\) 0 0
\(239\) −8.61580 −0.557310 −0.278655 0.960391i \(-0.589889\pi\)
−0.278655 + 0.960391i \(0.589889\pi\)
\(240\) 0 0
\(241\) −0.683667 −0.0440389 −0.0220194 0.999758i \(-0.507010\pi\)
−0.0220194 + 0.999758i \(0.507010\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.75153 0.175789
\(246\) 0 0
\(247\) 4.42907 0.281815
\(248\) 0 0
\(249\) 4.32246 0.273925
\(250\) 0 0
\(251\) 24.5092 1.54701 0.773503 0.633792i \(-0.218503\pi\)
0.773503 + 0.633792i \(0.218503\pi\)
\(252\) 0 0
\(253\) 4.32246 0.271751
\(254\) 0 0
\(255\) 2.25460 0.141188
\(256\) 0 0
\(257\) 28.7929 1.79605 0.898026 0.439942i \(-0.145001\pi\)
0.898026 + 0.439942i \(0.145001\pi\)
\(258\) 0 0
\(259\) −8.68367 −0.539577
\(260\) 0 0
\(261\) −0.819397 −0.0507194
\(262\) 0 0
\(263\) −23.1929 −1.43013 −0.715067 0.699056i \(-0.753604\pi\)
−0.715067 + 0.699056i \(0.753604\pi\)
\(264\) 0 0
\(265\) −20.3516 −1.25019
\(266\) 0 0
\(267\) 7.93214 0.485439
\(268\) 0 0
\(269\) 9.57093 0.583550 0.291775 0.956487i \(-0.405754\pi\)
0.291775 + 0.956487i \(0.405754\pi\)
\(270\) 0 0
\(271\) 19.1154 1.16118 0.580588 0.814198i \(-0.302823\pi\)
0.580588 + 0.814198i \(0.302823\pi\)
\(272\) 0 0
\(273\) −1.57093 −0.0950769
\(274\) 0 0
\(275\) −11.1127 −0.670123
\(276\) 0 0
\(277\) 4.84852 0.291319 0.145660 0.989335i \(-0.453470\pi\)
0.145660 + 0.989335i \(0.453470\pi\)
\(278\) 0 0
\(279\) −8.32246 −0.498253
\(280\) 0 0
\(281\) −8.14799 −0.486068 −0.243034 0.970018i \(-0.578143\pi\)
−0.243034 + 0.970018i \(0.578143\pi\)
\(282\) 0 0
\(283\) −20.0291 −1.19061 −0.595304 0.803501i \(-0.702968\pi\)
−0.595304 + 0.803501i \(0.702968\pi\)
\(284\) 0 0
\(285\) 7.75766 0.459524
\(286\) 0 0
\(287\) −10.3225 −0.609316
\(288\) 0 0
\(289\) −16.3286 −0.960505
\(290\) 0 0
\(291\) 1.67754 0.0983391
\(292\) 0 0
\(293\) −5.92251 −0.345997 −0.172998 0.984922i \(-0.555346\pi\)
−0.172998 + 0.984922i \(0.555346\pi\)
\(294\) 0 0
\(295\) 17.5832 1.02373
\(296\) 0 0
\(297\) −4.32246 −0.250815
\(298\) 0 0
\(299\) −1.57093 −0.0908492
\(300\) 0 0
\(301\) −0.429071 −0.0247313
\(302\) 0 0
\(303\) 8.39033 0.482012
\(304\) 0 0
\(305\) 36.9674 2.11675
\(306\) 0 0
\(307\) −3.40608 −0.194395 −0.0971976 0.995265i \(-0.530988\pi\)
−0.0971976 + 0.995265i \(0.530988\pi\)
\(308\) 0 0
\(309\) 11.0061 0.626117
\(310\) 0 0
\(311\) −26.4413 −1.49935 −0.749675 0.661806i \(-0.769790\pi\)
−0.749675 + 0.661806i \(0.769790\pi\)
\(312\) 0 0
\(313\) 7.63879 0.431770 0.215885 0.976419i \(-0.430736\pi\)
0.215885 + 0.976419i \(0.430736\pi\)
\(314\) 0 0
\(315\) −2.75153 −0.155031
\(316\) 0 0
\(317\) −9.75766 −0.548045 −0.274022 0.961723i \(-0.588354\pi\)
−0.274022 + 0.961723i \(0.588354\pi\)
\(318\) 0 0
\(319\) 3.54181 0.198303
\(320\) 0 0
\(321\) 4.29334 0.239631
\(322\) 0 0
\(323\) 2.31020 0.128543
\(324\) 0 0
\(325\) 4.03875 0.224029
\(326\) 0 0
\(327\) −5.89339 −0.325905
\(328\) 0 0
\(329\) −10.1480 −0.559477
\(330\) 0 0
\(331\) −4.64492 −0.255308 −0.127654 0.991819i \(-0.540745\pi\)
−0.127654 + 0.991819i \(0.540745\pi\)
\(332\) 0 0
\(333\) 8.68367 0.475862
\(334\) 0 0
\(335\) 4.42907 0.241986
\(336\) 0 0
\(337\) 3.15148 0.171672 0.0858362 0.996309i \(-0.472644\pi\)
0.0858362 + 0.996309i \(0.472644\pi\)
\(338\) 0 0
\(339\) 3.74540 0.203422
\(340\) 0 0
\(341\) 35.9735 1.94807
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −2.75153 −0.148138
\(346\) 0 0
\(347\) −21.6123 −1.16021 −0.580105 0.814542i \(-0.696988\pi\)
−0.580105 + 0.814542i \(0.696988\pi\)
\(348\) 0 0
\(349\) 3.60967 0.193221 0.0966107 0.995322i \(-0.469200\pi\)
0.0966107 + 0.995322i \(0.469200\pi\)
\(350\) 0 0
\(351\) 1.57093 0.0838500
\(352\) 0 0
\(353\) −32.2572 −1.71688 −0.858439 0.512915i \(-0.828566\pi\)
−0.858439 + 0.512915i \(0.828566\pi\)
\(354\) 0 0
\(355\) 5.68980 0.301983
\(356\) 0 0
\(357\) −0.819397 −0.0433671
\(358\) 0 0
\(359\) −9.47395 −0.500016 −0.250008 0.968244i \(-0.580433\pi\)
−0.250008 + 0.968244i \(0.580433\pi\)
\(360\) 0 0
\(361\) −11.0510 −0.581632
\(362\) 0 0
\(363\) 7.68367 0.403288
\(364\) 0 0
\(365\) −37.0475 −1.93915
\(366\) 0 0
\(367\) −13.0740 −0.682457 −0.341228 0.939980i \(-0.610843\pi\)
−0.341228 + 0.939980i \(0.610843\pi\)
\(368\) 0 0
\(369\) 10.3225 0.537366
\(370\) 0 0
\(371\) 7.39645 0.384005
\(372\) 0 0
\(373\) −32.3347 −1.67423 −0.837114 0.547028i \(-0.815759\pi\)
−0.837114 + 0.547028i \(0.815759\pi\)
\(374\) 0 0
\(375\) −6.68367 −0.345143
\(376\) 0 0
\(377\) −1.28721 −0.0662949
\(378\) 0 0
\(379\) 11.7285 0.602455 0.301227 0.953552i \(-0.402604\pi\)
0.301227 + 0.953552i \(0.402604\pi\)
\(380\) 0 0
\(381\) −5.07399 −0.259949
\(382\) 0 0
\(383\) −19.3286 −0.987645 −0.493822 0.869563i \(-0.664401\pi\)
−0.493822 + 0.869563i \(0.664401\pi\)
\(384\) 0 0
\(385\) 11.8934 0.606143
\(386\) 0 0
\(387\) 0.429071 0.0218109
\(388\) 0 0
\(389\) −31.7480 −1.60969 −0.804845 0.593486i \(-0.797751\pi\)
−0.804845 + 0.593486i \(0.797751\pi\)
\(390\) 0 0
\(391\) −0.819397 −0.0414387
\(392\) 0 0
\(393\) −12.3225 −0.621586
\(394\) 0 0
\(395\) 18.7638 0.944109
\(396\) 0 0
\(397\) −25.7868 −1.29420 −0.647101 0.762405i \(-0.724019\pi\)
−0.647101 + 0.762405i \(0.724019\pi\)
\(398\) 0 0
\(399\) −2.81940 −0.141146
\(400\) 0 0
\(401\) 15.7480 0.786419 0.393210 0.919449i \(-0.371365\pi\)
0.393210 + 0.919449i \(0.371365\pi\)
\(402\) 0 0
\(403\) −13.0740 −0.651262
\(404\) 0 0
\(405\) 2.75153 0.136725
\(406\) 0 0
\(407\) −37.5348 −1.86053
\(408\) 0 0
\(409\) −3.96125 −0.195871 −0.0979357 0.995193i \(-0.531224\pi\)
−0.0979357 + 0.995193i \(0.531224\pi\)
\(410\) 0 0
\(411\) 1.67754 0.0827469
\(412\) 0 0
\(413\) −6.39033 −0.314447
\(414\) 0 0
\(415\) 11.8934 0.583824
\(416\) 0 0
\(417\) 11.7577 0.575775
\(418\) 0 0
\(419\) 16.7515 0.818366 0.409183 0.912452i \(-0.365814\pi\)
0.409183 + 0.912452i \(0.365814\pi\)
\(420\) 0 0
\(421\) −2.77802 −0.135392 −0.0676962 0.997706i \(-0.521565\pi\)
−0.0676962 + 0.997706i \(0.521565\pi\)
\(422\) 0 0
\(423\) 10.1480 0.493412
\(424\) 0 0
\(425\) 2.10661 0.102186
\(426\) 0 0
\(427\) −13.4352 −0.650175
\(428\) 0 0
\(429\) −6.79028 −0.327838
\(430\) 0 0
\(431\) −9.58319 −0.461606 −0.230803 0.973001i \(-0.574135\pi\)
−0.230803 + 0.973001i \(0.574135\pi\)
\(432\) 0 0
\(433\) 26.4317 1.27023 0.635113 0.772419i \(-0.280953\pi\)
0.635113 + 0.772419i \(0.280953\pi\)
\(434\) 0 0
\(435\) −2.25460 −0.108100
\(436\) 0 0
\(437\) −2.81940 −0.134870
\(438\) 0 0
\(439\) 21.6898 1.03520 0.517599 0.855623i \(-0.326826\pi\)
0.517599 + 0.855623i \(0.326826\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −7.00613 −0.332871 −0.166436 0.986052i \(-0.553226\pi\)
−0.166436 + 0.986052i \(0.553226\pi\)
\(444\) 0 0
\(445\) 21.8255 1.03463
\(446\) 0 0
\(447\) −0.496936 −0.0235043
\(448\) 0 0
\(449\) 19.5322 0.921781 0.460890 0.887457i \(-0.347530\pi\)
0.460890 + 0.887457i \(0.347530\pi\)
\(450\) 0 0
\(451\) −44.6184 −2.10100
\(452\) 0 0
\(453\) 10.1480 0.476794
\(454\) 0 0
\(455\) −4.32246 −0.202640
\(456\) 0 0
\(457\) −38.4026 −1.79640 −0.898199 0.439590i \(-0.855124\pi\)
−0.898199 + 0.439590i \(0.855124\pi\)
\(458\) 0 0
\(459\) 0.819397 0.0382462
\(460\) 0 0
\(461\) 30.6740 1.42863 0.714316 0.699823i \(-0.246738\pi\)
0.714316 + 0.699823i \(0.246738\pi\)
\(462\) 0 0
\(463\) −39.8960 −1.85413 −0.927063 0.374907i \(-0.877675\pi\)
−0.927063 + 0.374907i \(0.877675\pi\)
\(464\) 0 0
\(465\) −22.8995 −1.06194
\(466\) 0 0
\(467\) 0.671411 0.0310692 0.0155346 0.999879i \(-0.495055\pi\)
0.0155346 + 0.999879i \(0.495055\pi\)
\(468\) 0 0
\(469\) −1.60967 −0.0743279
\(470\) 0 0
\(471\) 19.4256 0.895083
\(472\) 0 0
\(473\) −1.85464 −0.0852766
\(474\) 0 0
\(475\) 7.24847 0.332583
\(476\) 0 0
\(477\) −7.39645 −0.338660
\(478\) 0 0
\(479\) −16.9091 −0.772599 −0.386299 0.922373i \(-0.626247\pi\)
−0.386299 + 0.922373i \(0.626247\pi\)
\(480\) 0 0
\(481\) 13.6414 0.621995
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 4.61580 0.209593
\(486\) 0 0
\(487\) 2.03875 0.0923844 0.0461922 0.998933i \(-0.485291\pi\)
0.0461922 + 0.998933i \(0.485291\pi\)
\(488\) 0 0
\(489\) −2.25460 −0.101956
\(490\) 0 0
\(491\) 12.7250 0.574273 0.287137 0.957890i \(-0.407297\pi\)
0.287137 + 0.957890i \(0.407297\pi\)
\(492\) 0 0
\(493\) −0.671411 −0.0302388
\(494\) 0 0
\(495\) −11.8934 −0.534568
\(496\) 0 0
\(497\) −2.06786 −0.0927564
\(498\) 0 0
\(499\) −21.9056 −0.980631 −0.490316 0.871545i \(-0.663119\pi\)
−0.490316 + 0.871545i \(0.663119\pi\)
\(500\) 0 0
\(501\) 5.31633 0.237516
\(502\) 0 0
\(503\) 18.9770 0.846143 0.423072 0.906096i \(-0.360952\pi\)
0.423072 + 0.906096i \(0.360952\pi\)
\(504\) 0 0
\(505\) 23.0862 1.02732
\(506\) 0 0
\(507\) −10.5322 −0.467751
\(508\) 0 0
\(509\) 40.5214 1.79608 0.898041 0.439912i \(-0.144990\pi\)
0.898041 + 0.439912i \(0.144990\pi\)
\(510\) 0 0
\(511\) 13.4643 0.595626
\(512\) 0 0
\(513\) 2.81940 0.124479
\(514\) 0 0
\(515\) 30.2837 1.33446
\(516\) 0 0
\(517\) −43.8643 −1.92915
\(518\) 0 0
\(519\) 19.9613 0.876202
\(520\) 0 0
\(521\) 3.71629 0.162813 0.0814067 0.996681i \(-0.474059\pi\)
0.0814067 + 0.996681i \(0.474059\pi\)
\(522\) 0 0
\(523\) −11.3551 −0.496523 −0.248261 0.968693i \(-0.579859\pi\)
−0.248261 + 0.968693i \(0.579859\pi\)
\(524\) 0 0
\(525\) −2.57093 −0.112205
\(526\) 0 0
\(527\) −6.81940 −0.297058
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.39033 0.277317
\(532\) 0 0
\(533\) 16.2159 0.702386
\(534\) 0 0
\(535\) 11.8133 0.510732
\(536\) 0 0
\(537\) −4.02912 −0.173869
\(538\) 0 0
\(539\) −4.32246 −0.186182
\(540\) 0 0
\(541\) −30.5092 −1.31169 −0.655846 0.754894i \(-0.727688\pi\)
−0.655846 + 0.754894i \(0.727688\pi\)
\(542\) 0 0
\(543\) 0.955126 0.0409884
\(544\) 0 0
\(545\) −16.2159 −0.694611
\(546\) 0 0
\(547\) 31.6802 1.35455 0.677273 0.735732i \(-0.263162\pi\)
0.677273 + 0.735732i \(0.263162\pi\)
\(548\) 0 0
\(549\) 13.4352 0.573400
\(550\) 0 0
\(551\) −2.31020 −0.0984180
\(552\) 0 0
\(553\) −6.81940 −0.289990
\(554\) 0 0
\(555\) 23.8934 1.01422
\(556\) 0 0
\(557\) 16.4194 0.695714 0.347857 0.937548i \(-0.386909\pi\)
0.347857 + 0.937548i \(0.386909\pi\)
\(558\) 0 0
\(559\) 0.674040 0.0285089
\(560\) 0 0
\(561\) −3.54181 −0.149535
\(562\) 0 0
\(563\) −12.5648 −0.529543 −0.264772 0.964311i \(-0.585297\pi\)
−0.264772 + 0.964311i \(0.585297\pi\)
\(564\) 0 0
\(565\) 10.3056 0.433560
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 32.9286 1.38044 0.690220 0.723599i \(-0.257514\pi\)
0.690220 + 0.723599i \(0.257514\pi\)
\(570\) 0 0
\(571\) 44.7542 1.87290 0.936452 0.350797i \(-0.114089\pi\)
0.936452 + 0.350797i \(0.114089\pi\)
\(572\) 0 0
\(573\) −8.85814 −0.370054
\(574\) 0 0
\(575\) −2.57093 −0.107215
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 9.14186 0.379923
\(580\) 0 0
\(581\) −4.32246 −0.179326
\(582\) 0 0
\(583\) 31.9709 1.32410
\(584\) 0 0
\(585\) 4.32246 0.178712
\(586\) 0 0
\(587\) 41.2342 1.70192 0.850960 0.525231i \(-0.176021\pi\)
0.850960 + 0.525231i \(0.176021\pi\)
\(588\) 0 0
\(589\) −23.4643 −0.966830
\(590\) 0 0
\(591\) −6.21585 −0.255686
\(592\) 0 0
\(593\) 39.4256 1.61901 0.809507 0.587110i \(-0.199734\pi\)
0.809507 + 0.587110i \(0.199734\pi\)
\(594\) 0 0
\(595\) −2.25460 −0.0924294
\(596\) 0 0
\(597\) −21.3965 −0.875699
\(598\) 0 0
\(599\) −0.293342 −0.0119856 −0.00599282 0.999982i \(-0.501908\pi\)
−0.00599282 + 0.999982i \(0.501908\pi\)
\(600\) 0 0
\(601\) 11.4739 0.468032 0.234016 0.972233i \(-0.424813\pi\)
0.234016 + 0.972233i \(0.424813\pi\)
\(602\) 0 0
\(603\) 1.60967 0.0655510
\(604\) 0 0
\(605\) 21.1419 0.859539
\(606\) 0 0
\(607\) −8.72504 −0.354139 −0.177069 0.984198i \(-0.556662\pi\)
−0.177069 + 0.984198i \(0.556662\pi\)
\(608\) 0 0
\(609\) 0.819397 0.0332036
\(610\) 0 0
\(611\) 15.9418 0.644935
\(612\) 0 0
\(613\) −37.1154 −1.49908 −0.749538 0.661962i \(-0.769724\pi\)
−0.749538 + 0.661962i \(0.769724\pi\)
\(614\) 0 0
\(615\) 28.4026 1.14530
\(616\) 0 0
\(617\) −22.3516 −0.899841 −0.449920 0.893069i \(-0.648548\pi\)
−0.449920 + 0.893069i \(0.648548\pi\)
\(618\) 0 0
\(619\) 33.2342 1.33580 0.667898 0.744253i \(-0.267194\pi\)
0.667898 + 0.744253i \(0.267194\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −7.93214 −0.317794
\(624\) 0 0
\(625\) −31.2450 −1.24980
\(626\) 0 0
\(627\) −12.1867 −0.486691
\(628\) 0 0
\(629\) 7.11537 0.283708
\(630\) 0 0
\(631\) 3.60005 0.143316 0.0716578 0.997429i \(-0.477171\pi\)
0.0716578 + 0.997429i \(0.477171\pi\)
\(632\) 0 0
\(633\) −5.18060 −0.205910
\(634\) 0 0
\(635\) −13.9613 −0.554035
\(636\) 0 0
\(637\) 1.57093 0.0622425
\(638\) 0 0
\(639\) 2.06786 0.0818035
\(640\) 0 0
\(641\) −22.7250 −0.897585 −0.448793 0.893636i \(-0.648146\pi\)
−0.448793 + 0.893636i \(0.648146\pi\)
\(642\) 0 0
\(643\) 42.9505 1.69380 0.846902 0.531750i \(-0.178465\pi\)
0.846902 + 0.531750i \(0.178465\pi\)
\(644\) 0 0
\(645\) 1.18060 0.0464862
\(646\) 0 0
\(647\) −17.2607 −0.678589 −0.339295 0.940680i \(-0.610188\pi\)
−0.339295 + 0.940680i \(0.610188\pi\)
\(648\) 0 0
\(649\) −27.6219 −1.08426
\(650\) 0 0
\(651\) 8.32246 0.326183
\(652\) 0 0
\(653\) 24.7638 0.969082 0.484541 0.874769i \(-0.338987\pi\)
0.484541 + 0.874769i \(0.338987\pi\)
\(654\) 0 0
\(655\) −33.9056 −1.32480
\(656\) 0 0
\(657\) −13.4643 −0.525293
\(658\) 0 0
\(659\) 13.2388 0.515712 0.257856 0.966183i \(-0.416984\pi\)
0.257856 + 0.966183i \(0.416984\pi\)
\(660\) 0 0
\(661\) 24.6837 0.960083 0.480042 0.877246i \(-0.340622\pi\)
0.480042 + 0.877246i \(0.340622\pi\)
\(662\) 0 0
\(663\) 1.28721 0.0499912
\(664\) 0 0
\(665\) −7.75766 −0.300829
\(666\) 0 0
\(667\) 0.819397 0.0317272
\(668\) 0 0
\(669\) 16.0291 0.619722
\(670\) 0 0
\(671\) −58.0731 −2.24189
\(672\) 0 0
\(673\) −18.4000 −0.709266 −0.354633 0.935006i \(-0.615394\pi\)
−0.354633 + 0.935006i \(0.615394\pi\)
\(674\) 0 0
\(675\) 2.57093 0.0989551
\(676\) 0 0
\(677\) −48.1506 −1.85058 −0.925289 0.379262i \(-0.876178\pi\)
−0.925289 + 0.379262i \(0.876178\pi\)
\(678\) 0 0
\(679\) −1.67754 −0.0643780
\(680\) 0 0
\(681\) 13.2607 0.508152
\(682\) 0 0
\(683\) 36.0510 1.37945 0.689727 0.724070i \(-0.257731\pi\)
0.689727 + 0.724070i \(0.257731\pi\)
\(684\) 0 0
\(685\) 4.61580 0.176361
\(686\) 0 0
\(687\) −0.254596 −0.00971345
\(688\) 0 0
\(689\) −11.6193 −0.442660
\(690\) 0 0
\(691\) 19.0088 0.723127 0.361564 0.932347i \(-0.382243\pi\)
0.361564 + 0.932347i \(0.382243\pi\)
\(692\) 0 0
\(693\) 4.32246 0.164197
\(694\) 0 0
\(695\) 32.3516 1.22717
\(696\) 0 0
\(697\) 8.45819 0.320377
\(698\) 0 0
\(699\) 3.74540 0.141664
\(700\) 0 0
\(701\) −7.60967 −0.287413 −0.143707 0.989620i \(-0.545902\pi\)
−0.143707 + 0.989620i \(0.545902\pi\)
\(702\) 0 0
\(703\) 24.4827 0.923383
\(704\) 0 0
\(705\) 27.9225 1.05162
\(706\) 0 0
\(707\) −8.39033 −0.315551
\(708\) 0 0
\(709\) −34.2933 −1.28791 −0.643957 0.765062i \(-0.722708\pi\)
−0.643957 + 0.765062i \(0.722708\pi\)
\(710\) 0 0
\(711\) 6.81940 0.255747
\(712\) 0 0
\(713\) 8.32246 0.311679
\(714\) 0 0
\(715\) −18.6837 −0.698730
\(716\) 0 0
\(717\) −8.61580 −0.321763
\(718\) 0 0
\(719\) −0.807140 −0.0301012 −0.0150506 0.999887i \(-0.504791\pi\)
−0.0150506 + 0.999887i \(0.504791\pi\)
\(720\) 0 0
\(721\) −11.0061 −0.409890
\(722\) 0 0
\(723\) −0.683667 −0.0254259
\(724\) 0 0
\(725\) −2.10661 −0.0782375
\(726\) 0 0
\(727\) −5.37457 −0.199332 −0.0996659 0.995021i \(-0.531777\pi\)
−0.0996659 + 0.995021i \(0.531777\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.351580 0.0130036
\(732\) 0 0
\(733\) −37.7093 −1.39282 −0.696412 0.717642i \(-0.745221\pi\)
−0.696412 + 0.717642i \(0.745221\pi\)
\(734\) 0 0
\(735\) 2.75153 0.101492
\(736\) 0 0
\(737\) −6.95776 −0.256292
\(738\) 0 0
\(739\) −5.18060 −0.190572 −0.0952858 0.995450i \(-0.530376\pi\)
−0.0952858 + 0.995450i \(0.530376\pi\)
\(740\) 0 0
\(741\) 4.42907 0.162706
\(742\) 0 0
\(743\) 32.8343 1.20457 0.602287 0.798280i \(-0.294256\pi\)
0.602287 + 0.798280i \(0.294256\pi\)
\(744\) 0 0
\(745\) −1.36733 −0.0500953
\(746\) 0 0
\(747\) 4.32246 0.158151
\(748\) 0 0
\(749\) −4.29334 −0.156875
\(750\) 0 0
\(751\) −21.8282 −0.796521 −0.398260 0.917272i \(-0.630386\pi\)
−0.398260 + 0.917272i \(0.630386\pi\)
\(752\) 0 0
\(753\) 24.5092 0.893165
\(754\) 0 0
\(755\) 27.9225 1.01620
\(756\) 0 0
\(757\) −39.9613 −1.45242 −0.726208 0.687475i \(-0.758719\pi\)
−0.726208 + 0.687475i \(0.758719\pi\)
\(758\) 0 0
\(759\) 4.32246 0.156895
\(760\) 0 0
\(761\) 18.3735 0.666038 0.333019 0.942920i \(-0.391933\pi\)
0.333019 + 0.942920i \(0.391933\pi\)
\(762\) 0 0
\(763\) 5.89339 0.213355
\(764\) 0 0
\(765\) 2.25460 0.0815151
\(766\) 0 0
\(767\) 10.0387 0.362478
\(768\) 0 0
\(769\) −52.7154 −1.90097 −0.950483 0.310776i \(-0.899411\pi\)
−0.950483 + 0.310776i \(0.899411\pi\)
\(770\) 0 0
\(771\) 28.7929 1.03695
\(772\) 0 0
\(773\) −31.9083 −1.14766 −0.573830 0.818974i \(-0.694543\pi\)
−0.573830 + 0.818974i \(0.694543\pi\)
\(774\) 0 0
\(775\) −21.3965 −0.768583
\(776\) 0 0
\(777\) −8.68367 −0.311525
\(778\) 0 0
\(779\) 29.1031 1.04273
\(780\) 0 0
\(781\) −8.93826 −0.319836
\(782\) 0 0
\(783\) −0.819397 −0.0292828
\(784\) 0 0
\(785\) 53.4501 1.90772
\(786\) 0 0
\(787\) −6.11887 −0.218114 −0.109057 0.994035i \(-0.534783\pi\)
−0.109057 + 0.994035i \(0.534783\pi\)
\(788\) 0 0
\(789\) −23.1929 −0.825688
\(790\) 0 0
\(791\) −3.74540 −0.133171
\(792\) 0 0
\(793\) 21.1057 0.749487
\(794\) 0 0
\(795\) −20.3516 −0.721796
\(796\) 0 0
\(797\) −41.7868 −1.48016 −0.740082 0.672517i \(-0.765213\pi\)
−0.740082 + 0.672517i \(0.765213\pi\)
\(798\) 0 0
\(799\) 8.31523 0.294172
\(800\) 0 0
\(801\) 7.93214 0.280268
\(802\) 0 0
\(803\) 58.1990 2.05380
\(804\) 0 0
\(805\) 2.75153 0.0969788
\(806\) 0 0
\(807\) 9.57093 0.336913
\(808\) 0 0
\(809\) 23.1322 0.813286 0.406643 0.913587i \(-0.366699\pi\)
0.406643 + 0.913587i \(0.366699\pi\)
\(810\) 0 0
\(811\) −29.8255 −1.04732 −0.523658 0.851929i \(-0.675433\pi\)
−0.523658 + 0.851929i \(0.675433\pi\)
\(812\) 0 0
\(813\) 19.1154 0.670405
\(814\) 0 0
\(815\) −6.20359 −0.217302
\(816\) 0 0
\(817\) 1.20972 0.0423228
\(818\) 0 0
\(819\) −1.57093 −0.0548927
\(820\) 0 0
\(821\) −28.7929 −1.00488 −0.502440 0.864612i \(-0.667564\pi\)
−0.502440 + 0.864612i \(0.667564\pi\)
\(822\) 0 0
\(823\) −8.67404 −0.302358 −0.151179 0.988506i \(-0.548307\pi\)
−0.151179 + 0.988506i \(0.548307\pi\)
\(824\) 0 0
\(825\) −11.1127 −0.386896
\(826\) 0 0
\(827\) 2.89952 0.100826 0.0504131 0.998728i \(-0.483946\pi\)
0.0504131 + 0.998728i \(0.483946\pi\)
\(828\) 0 0
\(829\) 41.7021 1.44837 0.724186 0.689605i \(-0.242216\pi\)
0.724186 + 0.689605i \(0.242216\pi\)
\(830\) 0 0
\(831\) 4.84852 0.168193
\(832\) 0 0
\(833\) 0.819397 0.0283904
\(834\) 0 0
\(835\) 14.6281 0.506225
\(836\) 0 0
\(837\) −8.32246 −0.287666
\(838\) 0 0
\(839\) 19.3577 0.668302 0.334151 0.942520i \(-0.391550\pi\)
0.334151 + 0.942520i \(0.391550\pi\)
\(840\) 0 0
\(841\) −28.3286 −0.976848
\(842\) 0 0
\(843\) −8.14799 −0.280632
\(844\) 0 0
\(845\) −28.9796 −0.996930
\(846\) 0 0
\(847\) −7.68367 −0.264014
\(848\) 0 0
\(849\) −20.0291 −0.687398
\(850\) 0 0
\(851\) −8.68367 −0.297672
\(852\) 0 0
\(853\) −1.56830 −0.0536975 −0.0268488 0.999640i \(-0.508547\pi\)
−0.0268488 + 0.999640i \(0.508547\pi\)
\(854\) 0 0
\(855\) 7.75766 0.265306
\(856\) 0 0
\(857\) 42.1602 1.44017 0.720083 0.693888i \(-0.244104\pi\)
0.720083 + 0.693888i \(0.244104\pi\)
\(858\) 0 0
\(859\) −8.94089 −0.305059 −0.152530 0.988299i \(-0.548742\pi\)
−0.152530 + 0.988299i \(0.548742\pi\)
\(860\) 0 0
\(861\) −10.3225 −0.351789
\(862\) 0 0
\(863\) 47.5153 1.61744 0.808720 0.588194i \(-0.200161\pi\)
0.808720 + 0.588194i \(0.200161\pi\)
\(864\) 0 0
\(865\) 54.9240 1.86747
\(866\) 0 0
\(867\) −16.3286 −0.554548
\(868\) 0 0
\(869\) −29.4766 −0.999924
\(870\) 0 0
\(871\) 2.52868 0.0856812
\(872\) 0 0
\(873\) 1.67754 0.0567761
\(874\) 0 0
\(875\) 6.68367 0.225949
\(876\) 0 0
\(877\) −40.2307 −1.35850 −0.679248 0.733909i \(-0.737694\pi\)
−0.679248 + 0.733909i \(0.737694\pi\)
\(878\) 0 0
\(879\) −5.92251 −0.199761
\(880\) 0 0
\(881\) −6.70316 −0.225835 −0.112918 0.993604i \(-0.536020\pi\)
−0.112918 + 0.993604i \(0.536020\pi\)
\(882\) 0 0
\(883\) 27.2802 0.918052 0.459026 0.888423i \(-0.348198\pi\)
0.459026 + 0.888423i \(0.348198\pi\)
\(884\) 0 0
\(885\) 17.5832 0.591052
\(886\) 0 0
\(887\) 16.2933 0.547077 0.273538 0.961861i \(-0.411806\pi\)
0.273538 + 0.961861i \(0.411806\pi\)
\(888\) 0 0
\(889\) 5.07399 0.170176
\(890\) 0 0
\(891\) −4.32246 −0.144808
\(892\) 0 0
\(893\) 28.6112 0.957437
\(894\) 0 0
\(895\) −11.0862 −0.370572
\(896\) 0 0
\(897\) −1.57093 −0.0524518
\(898\) 0 0
\(899\) 6.81940 0.227440
\(900\) 0 0
\(901\) −6.06063 −0.201909
\(902\) 0 0
\(903\) −0.429071 −0.0142786
\(904\) 0 0
\(905\) 2.62806 0.0873597
\(906\) 0 0
\(907\) −8.80977 −0.292524 −0.146262 0.989246i \(-0.546724\pi\)
−0.146262 + 0.989246i \(0.546724\pi\)
\(908\) 0 0
\(909\) 8.39033 0.278290
\(910\) 0 0
\(911\) 30.4440 1.00865 0.504327 0.863513i \(-0.331741\pi\)
0.504327 + 0.863513i \(0.331741\pi\)
\(912\) 0 0
\(913\) −18.6837 −0.618339
\(914\) 0 0
\(915\) 36.9674 1.22210
\(916\) 0 0
\(917\) 12.3225 0.406924
\(918\) 0 0
\(919\) 28.6184 0.944035 0.472017 0.881589i \(-0.343526\pi\)
0.472017 + 0.881589i \(0.343526\pi\)
\(920\) 0 0
\(921\) −3.40608 −0.112234
\(922\) 0 0
\(923\) 3.24847 0.106925
\(924\) 0 0
\(925\) 22.3251 0.734044
\(926\) 0 0
\(927\) 11.0061 0.361489
\(928\) 0 0
\(929\) 39.7117 1.30290 0.651449 0.758692i \(-0.274161\pi\)
0.651449 + 0.758692i \(0.274161\pi\)
\(930\) 0 0
\(931\) 2.81940 0.0924020
\(932\) 0 0
\(933\) −26.4413 −0.865650
\(934\) 0 0
\(935\) −9.74540 −0.318709
\(936\) 0 0
\(937\) 38.1867 1.24751 0.623753 0.781621i \(-0.285607\pi\)
0.623753 + 0.781621i \(0.285607\pi\)
\(938\) 0 0
\(939\) 7.63879 0.249283
\(940\) 0 0
\(941\) 33.2246 1.08309 0.541546 0.840671i \(-0.317839\pi\)
0.541546 + 0.840671i \(0.317839\pi\)
\(942\) 0 0
\(943\) −10.3225 −0.336146
\(944\) 0 0
\(945\) −2.75153 −0.0895073
\(946\) 0 0
\(947\) 8.19397 0.266268 0.133134 0.991098i \(-0.457496\pi\)
0.133134 + 0.991098i \(0.457496\pi\)
\(948\) 0 0
\(949\) −21.1515 −0.686606
\(950\) 0 0
\(951\) −9.75766 −0.316414
\(952\) 0 0
\(953\) 12.7710 0.413694 0.206847 0.978373i \(-0.433680\pi\)
0.206847 + 0.978373i \(0.433680\pi\)
\(954\) 0 0
\(955\) −24.3735 −0.788707
\(956\) 0 0
\(957\) 3.54181 0.114490
\(958\) 0 0
\(959\) −1.67754 −0.0541706
\(960\) 0 0
\(961\) 38.2634 1.23430
\(962\) 0 0
\(963\) 4.29334 0.138351
\(964\) 0 0
\(965\) 25.1541 0.809740
\(966\) 0 0
\(967\) 37.3673 1.20165 0.600826 0.799380i \(-0.294838\pi\)
0.600826 + 0.799380i \(0.294838\pi\)
\(968\) 0 0
\(969\) 2.31020 0.0742145
\(970\) 0 0
\(971\) −35.0088 −1.12348 −0.561742 0.827312i \(-0.689869\pi\)
−0.561742 + 0.827312i \(0.689869\pi\)
\(972\) 0 0
\(973\) −11.7577 −0.376933
\(974\) 0 0
\(975\) 4.03875 0.129343
\(976\) 0 0
\(977\) 21.2995 0.681430 0.340715 0.940167i \(-0.389331\pi\)
0.340715 + 0.940167i \(0.389331\pi\)
\(978\) 0 0
\(979\) −34.2863 −1.09580
\(980\) 0 0
\(981\) −5.89339 −0.188161
\(982\) 0 0
\(983\) 18.4122 0.587258 0.293629 0.955919i \(-0.405137\pi\)
0.293629 + 0.955919i \(0.405137\pi\)
\(984\) 0 0
\(985\) −17.1031 −0.544950
\(986\) 0 0
\(987\) −10.1480 −0.323014
\(988\) 0 0
\(989\) −0.429071 −0.0136437
\(990\) 0 0
\(991\) 57.3700 1.82242 0.911208 0.411945i \(-0.135151\pi\)
0.911208 + 0.411945i \(0.135151\pi\)
\(992\) 0 0
\(993\) −4.64492 −0.147402
\(994\) 0 0
\(995\) −58.8730 −1.86640
\(996\) 0 0
\(997\) −18.5939 −0.588875 −0.294438 0.955671i \(-0.595132\pi\)
−0.294438 + 0.955671i \(0.595132\pi\)
\(998\) 0 0
\(999\) 8.68367 0.274739
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 7728.2.a.by.1.3 3
4.3 odd 2 3864.2.a.n.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.n.1.3 3 4.3 odd 2
7728.2.a.by.1.3 3 1.1 even 1 trivial