Properties

Label 3969.2.a.be.1.1
Level $3969$
Weight $2$
Character 3969.1
Self dual yes
Analytic conductor $31.693$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3969 = 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3969.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.6926245622\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.59351616.1
Defining polynomial: \(x^{6} - 12 x^{4} + 21 x^{2} - 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 441)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.16032\) of defining polynomial
Character \(\chi\) \(=\) 3969.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.69963 q^{2} +0.888736 q^{4} -0.949271 q^{5} +1.88874 q^{8} +O(q^{10})\) \(q-1.69963 q^{2} +0.888736 q^{4} -0.949271 q^{5} +1.88874 q^{8} +1.61341 q^{10} -0.588364 q^{11} -5.01974 q^{13} -4.98762 q^{16} -7.58242 q^{17} -4.46122 q^{19} -0.843651 q^{20} +1.00000 q^{22} +2.47710 q^{23} -4.09888 q^{25} +8.53169 q^{26} +5.47710 q^{29} -6.07463 q^{31} +4.69963 q^{32} +12.8873 q^{34} -6.98762 q^{37} +7.58242 q^{38} -1.79292 q^{40} -1.05489 q^{41} +6.98762 q^{43} -0.522900 q^{44} -4.21015 q^{46} -7.47680 q^{47} +6.96658 q^{50} -4.46122 q^{52} +6.92216 q^{53} +0.558517 q^{55} -9.30903 q^{58} +10.4302 q^{59} +11.6529 q^{61} +10.3246 q^{62} +1.98762 q^{64} +4.76509 q^{65} -11.8640 q^{67} -6.73877 q^{68} +4.30037 q^{71} +4.46122 q^{73} +11.8764 q^{74} -3.96485 q^{76} -1.33379 q^{79} +4.73460 q^{80} +1.79292 q^{82} -5.68387 q^{83} +7.19777 q^{85} -11.8764 q^{86} -1.11126 q^{88} +0.843651 q^{89} +2.20149 q^{92} +12.7078 q^{94} +4.23491 q^{95} -3.40633 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 6q^{4} + 12q^{8} + 8q^{11} + 6q^{16} + 6q^{22} + 4q^{23} + 12q^{25} + 22q^{29} + 16q^{32} - 6q^{37} + 6q^{43} - 14q^{44} + 12q^{46} + 56q^{50} + 28q^{53} + 18q^{58} - 24q^{64} - 6q^{65} + 38q^{71} + 36q^{74} - 6q^{79} - 30q^{85} - 36q^{86} - 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.69963 −1.20182 −0.600909 0.799317i \(-0.705195\pi\)
−0.600909 + 0.799317i \(0.705195\pi\)
\(3\) 0 0
\(4\) 0.888736 0.444368
\(5\) −0.949271 −0.424527 −0.212263 0.977212i \(-0.568084\pi\)
−0.212263 + 0.977212i \(0.568084\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.88874 0.667769
\(9\) 0 0
\(10\) 1.61341 0.510204
\(11\) −0.588364 −0.177398 −0.0886992 0.996058i \(-0.528271\pi\)
−0.0886992 + 0.996058i \(0.528271\pi\)
\(12\) 0 0
\(13\) −5.01974 −1.39222 −0.696112 0.717933i \(-0.745088\pi\)
−0.696112 + 0.717933i \(0.745088\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.98762 −1.24691
\(17\) −7.58242 −1.83901 −0.919503 0.393083i \(-0.871409\pi\)
−0.919503 + 0.393083i \(0.871409\pi\)
\(18\) 0 0
\(19\) −4.46122 −1.02347 −0.511737 0.859142i \(-0.670998\pi\)
−0.511737 + 0.859142i \(0.670998\pi\)
\(20\) −0.843651 −0.188646
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.47710 0.516511 0.258256 0.966077i \(-0.416852\pi\)
0.258256 + 0.966077i \(0.416852\pi\)
\(24\) 0 0
\(25\) −4.09888 −0.819777
\(26\) 8.53169 1.67320
\(27\) 0 0
\(28\) 0 0
\(29\) 5.47710 1.01707 0.508536 0.861041i \(-0.330187\pi\)
0.508536 + 0.861041i \(0.330187\pi\)
\(30\) 0 0
\(31\) −6.07463 −1.09104 −0.545518 0.838099i \(-0.683667\pi\)
−0.545518 + 0.838099i \(0.683667\pi\)
\(32\) 4.69963 0.830785
\(33\) 0 0
\(34\) 12.8873 2.21015
\(35\) 0 0
\(36\) 0 0
\(37\) −6.98762 −1.14876 −0.574379 0.818590i \(-0.694756\pi\)
−0.574379 + 0.818590i \(0.694756\pi\)
\(38\) 7.58242 1.23003
\(39\) 0 0
\(40\) −1.79292 −0.283486
\(41\) −1.05489 −0.164746 −0.0823731 0.996602i \(-0.526250\pi\)
−0.0823731 + 0.996602i \(0.526250\pi\)
\(42\) 0 0
\(43\) 6.98762 1.06560 0.532801 0.846241i \(-0.321139\pi\)
0.532801 + 0.846241i \(0.321139\pi\)
\(44\) −0.522900 −0.0788302
\(45\) 0 0
\(46\) −4.21015 −0.620753
\(47\) −7.47680 −1.09060 −0.545301 0.838240i \(-0.683585\pi\)
−0.545301 + 0.838240i \(0.683585\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 6.96658 0.985223
\(51\) 0 0
\(52\) −4.46122 −0.618660
\(53\) 6.92216 0.950831 0.475416 0.879761i \(-0.342298\pi\)
0.475416 + 0.879761i \(0.342298\pi\)
\(54\) 0 0
\(55\) 0.558517 0.0753104
\(56\) 0 0
\(57\) 0 0
\(58\) −9.30903 −1.22234
\(59\) 10.4302 1.35790 0.678950 0.734184i \(-0.262435\pi\)
0.678950 + 0.734184i \(0.262435\pi\)
\(60\) 0 0
\(61\) 11.6529 1.49200 0.745999 0.665947i \(-0.231972\pi\)
0.745999 + 0.665947i \(0.231972\pi\)
\(62\) 10.3246 1.31123
\(63\) 0 0
\(64\) 1.98762 0.248453
\(65\) 4.76509 0.591037
\(66\) 0 0
\(67\) −11.8640 −1.44942 −0.724708 0.689056i \(-0.758025\pi\)
−0.724708 + 0.689056i \(0.758025\pi\)
\(68\) −6.73877 −0.817195
\(69\) 0 0
\(70\) 0 0
\(71\) 4.30037 0.510360 0.255180 0.966894i \(-0.417865\pi\)
0.255180 + 0.966894i \(0.417865\pi\)
\(72\) 0 0
\(73\) 4.46122 0.522146 0.261073 0.965319i \(-0.415924\pi\)
0.261073 + 0.965319i \(0.415924\pi\)
\(74\) 11.8764 1.38060
\(75\) 0 0
\(76\) −3.96485 −0.454799
\(77\) 0 0
\(78\) 0 0
\(79\) −1.33379 −0.150063 −0.0750317 0.997181i \(-0.523906\pi\)
−0.0750317 + 0.997181i \(0.523906\pi\)
\(80\) 4.73460 0.529345
\(81\) 0 0
\(82\) 1.79292 0.197995
\(83\) −5.68387 −0.623886 −0.311943 0.950101i \(-0.600980\pi\)
−0.311943 + 0.950101i \(0.600980\pi\)
\(84\) 0 0
\(85\) 7.19777 0.780708
\(86\) −11.8764 −1.28066
\(87\) 0 0
\(88\) −1.11126 −0.118461
\(89\) 0.843651 0.0894269 0.0447134 0.999000i \(-0.485763\pi\)
0.0447134 + 0.999000i \(0.485763\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.20149 0.229521
\(93\) 0 0
\(94\) 12.7078 1.31071
\(95\) 4.23491 0.434492
\(96\) 0 0
\(97\) −3.40633 −0.345860 −0.172930 0.984934i \(-0.555324\pi\)
−0.172930 + 0.984934i \(0.555324\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.64283 −0.364283
\(101\) −9.58658 −0.953900 −0.476950 0.878930i \(-0.658258\pi\)
−0.476950 + 0.878930i \(0.658258\pi\)
\(102\) 0 0
\(103\) −11.6529 −1.14819 −0.574096 0.818788i \(-0.694647\pi\)
−0.574096 + 0.818788i \(0.694647\pi\)
\(104\) −9.48096 −0.929685
\(105\) 0 0
\(106\) −11.7651 −1.14273
\(107\) −3.79851 −0.367216 −0.183608 0.983000i \(-0.558778\pi\)
−0.183608 + 0.983000i \(0.558778\pi\)
\(108\) 0 0
\(109\) −12.8640 −1.23215 −0.616073 0.787689i \(-0.711277\pi\)
−0.616073 + 0.787689i \(0.711277\pi\)
\(110\) −0.949271 −0.0905094
\(111\) 0 0
\(112\) 0 0
\(113\) 9.02104 0.848628 0.424314 0.905515i \(-0.360515\pi\)
0.424314 + 0.905515i \(0.360515\pi\)
\(114\) 0 0
\(115\) −2.35144 −0.219273
\(116\) 4.86769 0.451954
\(117\) 0 0
\(118\) −17.7275 −1.63195
\(119\) 0 0
\(120\) 0 0
\(121\) −10.6538 −0.968530
\(122\) −19.8056 −1.79311
\(123\) 0 0
\(124\) −5.39874 −0.484821
\(125\) 8.63731 0.772544
\(126\) 0 0
\(127\) 6.43268 0.570808 0.285404 0.958407i \(-0.407872\pi\)
0.285404 + 0.958407i \(0.407872\pi\)
\(128\) −12.7775 −1.12938
\(129\) 0 0
\(130\) −8.09888 −0.710319
\(131\) 6.63315 0.579541 0.289770 0.957096i \(-0.406421\pi\)
0.289770 + 0.957096i \(0.406421\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 20.1643 1.74193
\(135\) 0 0
\(136\) −14.3212 −1.22803
\(137\) 14.0334 1.19896 0.599478 0.800391i \(-0.295375\pi\)
0.599478 + 0.800391i \(0.295375\pi\)
\(138\) 0 0
\(139\) 8.80507 0.746836 0.373418 0.927663i \(-0.378186\pi\)
0.373418 + 0.927663i \(0.378186\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.30903 −0.613360
\(143\) 2.95343 0.246978
\(144\) 0 0
\(145\) −5.19925 −0.431774
\(146\) −7.58242 −0.627525
\(147\) 0 0
\(148\) −6.21015 −0.510471
\(149\) −4.36584 −0.357663 −0.178832 0.983880i \(-0.557232\pi\)
−0.178832 + 0.983880i \(0.557232\pi\)
\(150\) 0 0
\(151\) −12.6538 −1.02975 −0.514877 0.857264i \(-0.672162\pi\)
−0.514877 + 0.857264i \(0.672162\pi\)
\(152\) −8.42607 −0.683444
\(153\) 0 0
\(154\) 0 0
\(155\) 5.76647 0.463174
\(156\) 0 0
\(157\) 11.2739 0.899754 0.449877 0.893091i \(-0.351468\pi\)
0.449877 + 0.893091i \(0.351468\pi\)
\(158\) 2.26695 0.180349
\(159\) 0 0
\(160\) −4.46122 −0.352690
\(161\) 0 0
\(162\) 0 0
\(163\) −1.66621 −0.130507 −0.0652537 0.997869i \(-0.520786\pi\)
−0.0652537 + 0.997869i \(0.520786\pi\)
\(164\) −0.937519 −0.0732080
\(165\) 0 0
\(166\) 9.66047 0.749798
\(167\) 3.90270 0.302000 0.151000 0.988534i \(-0.451751\pi\)
0.151000 + 0.988534i \(0.451751\pi\)
\(168\) 0 0
\(169\) 12.1978 0.938290
\(170\) −12.2335 −0.938269
\(171\) 0 0
\(172\) 6.21015 0.473519
\(173\) 16.1141 1.22513 0.612566 0.790419i \(-0.290137\pi\)
0.612566 + 0.790419i \(0.290137\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.93454 0.221199
\(177\) 0 0
\(178\) −1.43389 −0.107475
\(179\) 14.2880 1.06793 0.533967 0.845505i \(-0.320701\pi\)
0.533967 + 0.845505i \(0.320701\pi\)
\(180\) 0 0
\(181\) 12.8873 0.957905 0.478952 0.877841i \(-0.341017\pi\)
0.478952 + 0.877841i \(0.341017\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.67859 0.344910
\(185\) 6.63315 0.487679
\(186\) 0 0
\(187\) 4.46122 0.326237
\(188\) −6.64490 −0.484629
\(189\) 0 0
\(190\) −7.19777 −0.522181
\(191\) −2.16435 −0.156607 −0.0783034 0.996930i \(-0.524950\pi\)
−0.0783034 + 0.996930i \(0.524950\pi\)
\(192\) 0 0
\(193\) 10.4313 0.750861 0.375431 0.926850i \(-0.377495\pi\)
0.375431 + 0.926850i \(0.377495\pi\)
\(194\) 5.78949 0.415661
\(195\) 0 0
\(196\) 0 0
\(197\) −18.7848 −1.33836 −0.669179 0.743101i \(-0.733354\pi\)
−0.669179 + 0.743101i \(0.733354\pi\)
\(198\) 0 0
\(199\) 8.42607 0.597308 0.298654 0.954361i \(-0.403462\pi\)
0.298654 + 0.954361i \(0.403462\pi\)
\(200\) −7.74171 −0.547422
\(201\) 0 0
\(202\) 16.2936 1.14642
\(203\) 0 0
\(204\) 0 0
\(205\) 1.00138 0.0699392
\(206\) 19.8056 1.37992
\(207\) 0 0
\(208\) 25.0365 1.73597
\(209\) 2.62482 0.181563
\(210\) 0 0
\(211\) 11.2225 0.772591 0.386295 0.922375i \(-0.373755\pi\)
0.386295 + 0.922375i \(0.373755\pi\)
\(212\) 6.15197 0.422519
\(213\) 0 0
\(214\) 6.45606 0.441327
\(215\) −6.63315 −0.452377
\(216\) 0 0
\(217\) 0 0
\(218\) 21.8640 1.48082
\(219\) 0 0
\(220\) 0.496374 0.0334655
\(221\) 38.0617 2.56031
\(222\) 0 0
\(223\) −20.7548 −1.38985 −0.694923 0.719084i \(-0.744562\pi\)
−0.694923 + 0.719084i \(0.744562\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −15.3324 −1.01990
\(227\) −10.4302 −0.692279 −0.346139 0.938183i \(-0.612508\pi\)
−0.346139 + 0.938183i \(0.612508\pi\)
\(228\) 0 0
\(229\) 15.0592 0.995141 0.497570 0.867424i \(-0.334226\pi\)
0.497570 + 0.867424i \(0.334226\pi\)
\(230\) 3.99657 0.263526
\(231\) 0 0
\(232\) 10.3448 0.679169
\(233\) 4.38688 0.287394 0.143697 0.989622i \(-0.454101\pi\)
0.143697 + 0.989622i \(0.454101\pi\)
\(234\) 0 0
\(235\) 7.09751 0.462990
\(236\) 9.26972 0.603407
\(237\) 0 0
\(238\) 0 0
\(239\) −9.55122 −0.617817 −0.308909 0.951092i \(-0.599964\pi\)
−0.308909 + 0.951092i \(0.599964\pi\)
\(240\) 0 0
\(241\) 10.5358 0.678674 0.339337 0.940665i \(-0.389797\pi\)
0.339337 + 0.940665i \(0.389797\pi\)
\(242\) 18.1075 1.16400
\(243\) 0 0
\(244\) 10.3563 0.662996
\(245\) 0 0
\(246\) 0 0
\(247\) 22.3942 1.42491
\(248\) −11.4734 −0.728560
\(249\) 0 0
\(250\) −14.6802 −0.928458
\(251\) −24.4346 −1.54230 −0.771148 0.636656i \(-0.780317\pi\)
−0.771148 + 0.636656i \(0.780317\pi\)
\(252\) 0 0
\(253\) −1.45744 −0.0916282
\(254\) −10.9332 −0.686007
\(255\) 0 0
\(256\) 17.7417 1.10886
\(257\) 4.00832 0.250032 0.125016 0.992155i \(-0.460102\pi\)
0.125016 + 0.992155i \(0.460102\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 4.23491 0.262638
\(261\) 0 0
\(262\) −11.2739 −0.696503
\(263\) 17.6872 1.09064 0.545321 0.838227i \(-0.316408\pi\)
0.545321 + 0.838227i \(0.316408\pi\)
\(264\) 0 0
\(265\) −6.57100 −0.403653
\(266\) 0 0
\(267\) 0 0
\(268\) −10.5439 −0.644074
\(269\) −14.2273 −0.867455 −0.433727 0.901044i \(-0.642802\pi\)
−0.433727 + 0.901044i \(0.642802\pi\)
\(270\) 0 0
\(271\) −5.39874 −0.327950 −0.163975 0.986464i \(-0.552432\pi\)
−0.163975 + 0.986464i \(0.552432\pi\)
\(272\) 37.8182 2.29307
\(273\) 0 0
\(274\) −23.8516 −1.44093
\(275\) 2.41164 0.145427
\(276\) 0 0
\(277\) 7.66621 0.460618 0.230309 0.973118i \(-0.426026\pi\)
0.230309 + 0.973118i \(0.426026\pi\)
\(278\) −14.9653 −0.897562
\(279\) 0 0
\(280\) 0 0
\(281\) −22.6625 −1.35193 −0.675965 0.736933i \(-0.736273\pi\)
−0.675965 + 0.736933i \(0.736273\pi\)
\(282\) 0 0
\(283\) −31.8492 −1.89324 −0.946619 0.322353i \(-0.895526\pi\)
−0.946619 + 0.322353i \(0.895526\pi\)
\(284\) 3.82189 0.226788
\(285\) 0 0
\(286\) −5.01974 −0.296823
\(287\) 0 0
\(288\) 0 0
\(289\) 40.4930 2.38194
\(290\) 8.83680 0.518915
\(291\) 0 0
\(292\) 3.96485 0.232025
\(293\) 27.4936 1.60619 0.803097 0.595849i \(-0.203184\pi\)
0.803097 + 0.595849i \(0.203184\pi\)
\(294\) 0 0
\(295\) −9.90112 −0.576465
\(296\) −13.1978 −0.767105
\(297\) 0 0
\(298\) 7.42030 0.429846
\(299\) −12.4344 −0.719099
\(300\) 0 0
\(301\) 0 0
\(302\) 21.5068 1.23758
\(303\) 0 0
\(304\) 22.2509 1.27618
\(305\) −11.0617 −0.633394
\(306\) 0 0
\(307\) 14.8176 0.845683 0.422841 0.906204i \(-0.361033\pi\)
0.422841 + 0.906204i \(0.361033\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −9.80085 −0.556651
\(311\) 29.0635 1.64804 0.824021 0.566559i \(-0.191726\pi\)
0.824021 + 0.566559i \(0.191726\pi\)
\(312\) 0 0
\(313\) −24.4780 −1.38358 −0.691790 0.722099i \(-0.743178\pi\)
−0.691790 + 0.722099i \(0.743178\pi\)
\(314\) −19.1614 −1.08134
\(315\) 0 0
\(316\) −1.18539 −0.0666834
\(317\) −7.38688 −0.414888 −0.207444 0.978247i \(-0.566515\pi\)
−0.207444 + 0.978247i \(0.566515\pi\)
\(318\) 0 0
\(319\) −3.22253 −0.180427
\(320\) −1.88679 −0.105475
\(321\) 0 0
\(322\) 0 0
\(323\) 33.8268 1.88218
\(324\) 0 0
\(325\) 20.5753 1.14131
\(326\) 2.83193 0.156846
\(327\) 0 0
\(328\) −1.99241 −0.110012
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0617 1.10269 0.551347 0.834276i \(-0.314114\pi\)
0.551347 + 0.834276i \(0.314114\pi\)
\(332\) −5.05146 −0.277235
\(333\) 0 0
\(334\) −6.63315 −0.362950
\(335\) 11.2621 0.615316
\(336\) 0 0
\(337\) 6.40654 0.348986 0.174493 0.984658i \(-0.444171\pi\)
0.174493 + 0.984658i \(0.444171\pi\)
\(338\) −20.7317 −1.12765
\(339\) 0 0
\(340\) 6.39692 0.346921
\(341\) 3.57409 0.193548
\(342\) 0 0
\(343\) 0 0
\(344\) 13.1978 0.711576
\(345\) 0 0
\(346\) −27.3880 −1.47239
\(347\) −29.1927 −1.56714 −0.783572 0.621300i \(-0.786605\pi\)
−0.783572 + 0.621300i \(0.786605\pi\)
\(348\) 0 0
\(349\) −4.34385 −0.232521 −0.116260 0.993219i \(-0.537091\pi\)
−0.116260 + 0.993219i \(0.537091\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.76509 −0.147380
\(353\) 25.7007 1.36791 0.683955 0.729525i \(-0.260259\pi\)
0.683955 + 0.729525i \(0.260259\pi\)
\(354\) 0 0
\(355\) −4.08222 −0.216662
\(356\) 0.749783 0.0397384
\(357\) 0 0
\(358\) −24.2843 −1.28346
\(359\) 20.6872 1.09183 0.545916 0.837840i \(-0.316182\pi\)
0.545916 + 0.837840i \(0.316182\pi\)
\(360\) 0 0
\(361\) 0.902493 0.0474996
\(362\) −21.9036 −1.15123
\(363\) 0 0
\(364\) 0 0
\(365\) −4.23491 −0.221665
\(366\) 0 0
\(367\) −2.84781 −0.148655 −0.0743273 0.997234i \(-0.523681\pi\)
−0.0743273 + 0.997234i \(0.523681\pi\)
\(368\) −12.3548 −0.644040
\(369\) 0 0
\(370\) −11.2739 −0.586101
\(371\) 0 0
\(372\) 0 0
\(373\) 21.4327 1.10974 0.554871 0.831936i \(-0.312768\pi\)
0.554871 + 0.831936i \(0.312768\pi\)
\(374\) −7.58242 −0.392077
\(375\) 0 0
\(376\) −14.1217 −0.728271
\(377\) −27.4936 −1.41599
\(378\) 0 0
\(379\) 27.0494 1.38943 0.694716 0.719284i \(-0.255530\pi\)
0.694716 + 0.719284i \(0.255530\pi\)
\(380\) 3.76371 0.193074
\(381\) 0 0
\(382\) 3.67859 0.188213
\(383\) −14.4268 −0.737175 −0.368588 0.929593i \(-0.620159\pi\)
−0.368588 + 0.929593i \(0.620159\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −17.7293 −0.902399
\(387\) 0 0
\(388\) −3.02733 −0.153689
\(389\) −6.10755 −0.309665 −0.154832 0.987941i \(-0.549484\pi\)
−0.154832 + 0.987941i \(0.549484\pi\)
\(390\) 0 0
\(391\) −18.7824 −0.949867
\(392\) 0 0
\(393\) 0 0
\(394\) 31.9271 1.60846
\(395\) 1.26613 0.0637059
\(396\) 0 0
\(397\) −12.8873 −0.646794 −0.323397 0.946263i \(-0.604825\pi\)
−0.323397 + 0.946263i \(0.604825\pi\)
\(398\) −14.3212 −0.717856
\(399\) 0 0
\(400\) 20.4437 1.02218
\(401\) 8.39060 0.419006 0.209503 0.977808i \(-0.432815\pi\)
0.209503 + 0.977808i \(0.432815\pi\)
\(402\) 0 0
\(403\) 30.4930 1.51897
\(404\) −8.51994 −0.423883
\(405\) 0 0
\(406\) 0 0
\(407\) 4.11126 0.203788
\(408\) 0 0
\(409\) 6.81266 0.336864 0.168432 0.985713i \(-0.446130\pi\)
0.168432 + 0.985713i \(0.446130\pi\)
\(410\) −1.70197 −0.0840543
\(411\) 0 0
\(412\) −10.3563 −0.510220
\(413\) 0 0
\(414\) 0 0
\(415\) 5.39554 0.264857
\(416\) −23.5909 −1.15664
\(417\) 0 0
\(418\) −4.46122 −0.218205
\(419\) −10.3246 −0.504390 −0.252195 0.967676i \(-0.581152\pi\)
−0.252195 + 0.967676i \(0.581152\pi\)
\(420\) 0 0
\(421\) 3.13602 0.152840 0.0764202 0.997076i \(-0.475651\pi\)
0.0764202 + 0.997076i \(0.475651\pi\)
\(422\) −19.0741 −0.928514
\(423\) 0 0
\(424\) 13.0741 0.634936
\(425\) 31.0795 1.50757
\(426\) 0 0
\(427\) 0 0
\(428\) −3.37587 −0.163179
\(429\) 0 0
\(430\) 11.2739 0.543675
\(431\) 31.8726 1.53525 0.767625 0.640899i \(-0.221438\pi\)
0.767625 + 0.640899i \(0.221438\pi\)
\(432\) 0 0
\(433\) 7.48855 0.359877 0.179938 0.983678i \(-0.442410\pi\)
0.179938 + 0.983678i \(0.442410\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −11.4327 −0.547526
\(437\) −11.0509 −0.528636
\(438\) 0 0
\(439\) 2.28930 0.109262 0.0546311 0.998507i \(-0.482602\pi\)
0.0546311 + 0.998507i \(0.482602\pi\)
\(440\) 1.05489 0.0502900
\(441\) 0 0
\(442\) −64.6908 −3.07703
\(443\) 37.3497 1.77454 0.887270 0.461251i \(-0.152599\pi\)
0.887270 + 0.461251i \(0.152599\pi\)
\(444\) 0 0
\(445\) −0.800854 −0.0379641
\(446\) 35.2755 1.67034
\(447\) 0 0
\(448\) 0 0
\(449\) −6.20286 −0.292731 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(450\) 0 0
\(451\) 0.620660 0.0292257
\(452\) 8.01732 0.377103
\(453\) 0 0
\(454\) 17.7275 0.831993
\(455\) 0 0
\(456\) 0 0
\(457\) 20.1716 0.943589 0.471795 0.881709i \(-0.343606\pi\)
0.471795 + 0.881709i \(0.343606\pi\)
\(458\) −25.5951 −1.19598
\(459\) 0 0
\(460\) −2.08981 −0.0974378
\(461\) −22.5360 −1.04961 −0.524803 0.851224i \(-0.675861\pi\)
−0.524803 + 0.851224i \(0.675861\pi\)
\(462\) 0 0
\(463\) −27.6291 −1.28403 −0.642016 0.766691i \(-0.721902\pi\)
−0.642016 + 0.766691i \(0.721902\pi\)
\(464\) −27.3177 −1.26819
\(465\) 0 0
\(466\) −7.45606 −0.345395
\(467\) 20.1224 0.931155 0.465577 0.885007i \(-0.345847\pi\)
0.465577 + 0.885007i \(0.345847\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −12.0631 −0.556430
\(471\) 0 0
\(472\) 19.6999 0.906764
\(473\) −4.11126 −0.189036
\(474\) 0 0
\(475\) 18.2860 0.839021
\(476\) 0 0
\(477\) 0 0
\(478\) 16.2335 0.742504
\(479\) −9.58658 −0.438022 −0.219011 0.975722i \(-0.570283\pi\)
−0.219011 + 0.975722i \(0.570283\pi\)
\(480\) 0 0
\(481\) 35.0760 1.59933
\(482\) −17.9070 −0.815643
\(483\) 0 0
\(484\) −9.46844 −0.430384
\(485\) 3.23353 0.146827
\(486\) 0 0
\(487\) 13.0741 0.592445 0.296223 0.955119i \(-0.404273\pi\)
0.296223 + 0.955119i \(0.404273\pi\)
\(488\) 22.0092 0.996311
\(489\) 0 0
\(490\) 0 0
\(491\) 15.3411 0.692333 0.346167 0.938173i \(-0.387483\pi\)
0.346167 + 0.938173i \(0.387483\pi\)
\(492\) 0 0
\(493\) −41.5297 −1.87040
\(494\) −38.0617 −1.71248
\(495\) 0 0
\(496\) 30.2979 1.36042
\(497\) 0 0
\(498\) 0 0
\(499\) 4.86535 0.217803 0.108902 0.994053i \(-0.465267\pi\)
0.108902 + 0.994053i \(0.465267\pi\)
\(500\) 7.67628 0.343294
\(501\) 0 0
\(502\) 41.5297 1.85356
\(503\) 16.0085 0.713783 0.356892 0.934146i \(-0.383837\pi\)
0.356892 + 0.934146i \(0.383837\pi\)
\(504\) 0 0
\(505\) 9.10026 0.404956
\(506\) 2.47710 0.110121
\(507\) 0 0
\(508\) 5.71695 0.253649
\(509\) −31.1851 −1.38225 −0.691127 0.722733i \(-0.742885\pi\)
−0.691127 + 0.722733i \(0.742885\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −4.59937 −0.203265
\(513\) 0 0
\(514\) −6.81266 −0.300494
\(515\) 11.0617 0.487439
\(516\) 0 0
\(517\) 4.39908 0.193471
\(518\) 0 0
\(519\) 0 0
\(520\) 9.00000 0.394676
\(521\) 20.9661 0.918541 0.459270 0.888297i \(-0.348111\pi\)
0.459270 + 0.888297i \(0.348111\pi\)
\(522\) 0 0
\(523\) 43.5642 1.90493 0.952465 0.304647i \(-0.0985383\pi\)
0.952465 + 0.304647i \(0.0985383\pi\)
\(524\) 5.89511 0.257529
\(525\) 0 0
\(526\) −30.0617 −1.31075
\(527\) 46.0604 2.00642
\(528\) 0 0
\(529\) −16.8640 −0.733216
\(530\) 11.1683 0.485118
\(531\) 0 0
\(532\) 0 0
\(533\) 5.29528 0.229364
\(534\) 0 0
\(535\) 3.60582 0.155893
\(536\) −22.4079 −0.967875
\(537\) 0 0
\(538\) 24.1811 1.04252
\(539\) 0 0
\(540\) 0 0
\(541\) 9.86535 0.424145 0.212072 0.977254i \(-0.431979\pi\)
0.212072 + 0.977254i \(0.431979\pi\)
\(542\) 9.17585 0.394137
\(543\) 0 0
\(544\) −35.6345 −1.52782
\(545\) 12.2114 0.523079
\(546\) 0 0
\(547\) 0.568701 0.0243159 0.0121579 0.999926i \(-0.496130\pi\)
0.0121579 + 0.999926i \(0.496130\pi\)
\(548\) 12.4720 0.532778
\(549\) 0 0
\(550\) −4.09888 −0.174777
\(551\) −24.4346 −1.04095
\(552\) 0 0
\(553\) 0 0
\(554\) −13.0297 −0.553579
\(555\) 0 0
\(556\) 7.82538 0.331870
\(557\) −2.58699 −0.109614 −0.0548071 0.998497i \(-0.517454\pi\)
−0.0548071 + 0.998497i \(0.517454\pi\)
\(558\) 0 0
\(559\) −35.0760 −1.48356
\(560\) 0 0
\(561\) 0 0
\(562\) 38.5178 1.62478
\(563\) −33.2831 −1.40272 −0.701358 0.712809i \(-0.747422\pi\)
−0.701358 + 0.712809i \(0.747422\pi\)
\(564\) 0 0
\(565\) −8.56341 −0.360265
\(566\) 54.1318 2.27533
\(567\) 0 0
\(568\) 8.12227 0.340803
\(569\) −5.35346 −0.224429 −0.112214 0.993684i \(-0.535794\pi\)
−0.112214 + 0.993684i \(0.535794\pi\)
\(570\) 0 0
\(571\) 4.90112 0.205105 0.102553 0.994728i \(-0.467299\pi\)
0.102553 + 0.994728i \(0.467299\pi\)
\(572\) 2.62482 0.109749
\(573\) 0 0
\(574\) 0 0
\(575\) −10.1533 −0.423424
\(576\) 0 0
\(577\) −36.0757 −1.50185 −0.750925 0.660387i \(-0.770392\pi\)
−0.750925 + 0.660387i \(0.770392\pi\)
\(578\) −68.8231 −2.86266
\(579\) 0 0
\(580\) −4.62076 −0.191867
\(581\) 0 0
\(582\) 0 0
\(583\) −4.07275 −0.168676
\(584\) 8.42607 0.348673
\(585\) 0 0
\(586\) −46.7289 −1.93035
\(587\) 1.05489 0.0435400 0.0217700 0.999763i \(-0.493070\pi\)
0.0217700 + 0.999763i \(0.493070\pi\)
\(588\) 0 0
\(589\) 27.1003 1.11665
\(590\) 16.8282 0.692807
\(591\) 0 0
\(592\) 34.8516 1.43239
\(593\) −15.0710 −0.618890 −0.309445 0.950917i \(-0.600143\pi\)
−0.309445 + 0.950917i \(0.600143\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.88007 −0.158934
\(597\) 0 0
\(598\) 21.1338 0.864227
\(599\) 42.0566 1.71839 0.859194 0.511650i \(-0.170965\pi\)
0.859194 + 0.511650i \(0.170965\pi\)
\(600\) 0 0
\(601\) 18.8998 0.770938 0.385469 0.922721i \(-0.374040\pi\)
0.385469 + 0.922721i \(0.374040\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −11.2459 −0.457590
\(605\) 10.1134 0.411167
\(606\) 0 0
\(607\) −29.4425 −1.19504 −0.597518 0.801856i \(-0.703846\pi\)
−0.597518 + 0.801856i \(0.703846\pi\)
\(608\) −20.9661 −0.850287
\(609\) 0 0
\(610\) 18.8009 0.761224
\(611\) 37.5316 1.51836
\(612\) 0 0
\(613\) −11.6676 −0.471249 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(614\) −25.1843 −1.01636
\(615\) 0 0
\(616\) 0 0
\(617\) −32.8109 −1.32092 −0.660458 0.750863i \(-0.729638\pi\)
−0.660458 + 0.750863i \(0.729638\pi\)
\(618\) 0 0
\(619\) −24.1612 −0.971119 −0.485560 0.874204i \(-0.661384\pi\)
−0.485560 + 0.874204i \(0.661384\pi\)
\(620\) 5.12487 0.205820
\(621\) 0 0
\(622\) −49.3972 −1.98065
\(623\) 0 0
\(624\) 0 0
\(625\) 12.2953 0.491811
\(626\) 41.6035 1.66281
\(627\) 0 0
\(628\) 10.0195 0.399822
\(629\) 52.9830 2.11257
\(630\) 0 0
\(631\) −11.1003 −0.441894 −0.220947 0.975286i \(-0.570915\pi\)
−0.220947 + 0.975286i \(0.570915\pi\)
\(632\) −2.51918 −0.100208
\(633\) 0 0
\(634\) 12.5549 0.498620
\(635\) −6.10635 −0.242323
\(636\) 0 0
\(637\) 0 0
\(638\) 5.47710 0.216840
\(639\) 0 0
\(640\) 12.1293 0.479452
\(641\) 7.30037 0.288347 0.144174 0.989552i \(-0.453948\pi\)
0.144174 + 0.989552i \(0.453948\pi\)
\(642\) 0 0
\(643\) 21.2512 0.838066 0.419033 0.907971i \(-0.362369\pi\)
0.419033 + 0.907971i \(0.362369\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −57.4930 −2.26203
\(647\) 16.9460 0.666216 0.333108 0.942889i \(-0.391903\pi\)
0.333108 + 0.942889i \(0.391903\pi\)
\(648\) 0 0
\(649\) −6.13677 −0.240889
\(650\) −34.9704 −1.37165
\(651\) 0 0
\(652\) −1.48082 −0.0579933
\(653\) −3.73305 −0.146085 −0.0730427 0.997329i \(-0.523271\pi\)
−0.0730427 + 0.997329i \(0.523271\pi\)
\(654\) 0 0
\(655\) −6.29665 −0.246031
\(656\) 5.26140 0.205423
\(657\) 0 0
\(658\) 0 0
\(659\) −23.5984 −0.919263 −0.459632 0.888110i \(-0.652019\pi\)
−0.459632 + 0.888110i \(0.652019\pi\)
\(660\) 0 0
\(661\) 34.5175 1.34258 0.671288 0.741197i \(-0.265742\pi\)
0.671288 + 0.741197i \(0.265742\pi\)
\(662\) −34.0975 −1.32524
\(663\) 0 0
\(664\) −10.7353 −0.416612
\(665\) 0 0
\(666\) 0 0
\(667\) 13.5673 0.525329
\(668\) 3.46847 0.134199
\(669\) 0 0
\(670\) −19.1414 −0.739498
\(671\) −6.85614 −0.264678
\(672\) 0 0
\(673\) −24.4574 −0.942765 −0.471382 0.881929i \(-0.656245\pi\)
−0.471382 + 0.881929i \(0.656245\pi\)
\(674\) −10.8887 −0.419418
\(675\) 0 0
\(676\) 10.8406 0.416946
\(677\) −8.32045 −0.319781 −0.159890 0.987135i \(-0.551114\pi\)
−0.159890 + 0.987135i \(0.551114\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 13.5947 0.521332
\(681\) 0 0
\(682\) −6.07463 −0.232610
\(683\) −42.4624 −1.62478 −0.812389 0.583116i \(-0.801833\pi\)
−0.812389 + 0.583116i \(0.801833\pi\)
\(684\) 0 0
\(685\) −13.3215 −0.508989
\(686\) 0 0
\(687\) 0 0
\(688\) −34.8516 −1.32870
\(689\) −34.7474 −1.32377
\(690\) 0 0
\(691\) 35.3929 1.34641 0.673204 0.739456i \(-0.264917\pi\)
0.673204 + 0.739456i \(0.264917\pi\)
\(692\) 14.3212 0.544410
\(693\) 0 0
\(694\) 49.6167 1.88342
\(695\) −8.35840 −0.317052
\(696\) 0 0
\(697\) 7.99862 0.302969
\(698\) 7.38293 0.279448
\(699\) 0 0
\(700\) 0 0
\(701\) −7.00372 −0.264527 −0.132263 0.991215i \(-0.542224\pi\)
−0.132263 + 0.991215i \(0.542224\pi\)
\(702\) 0 0
\(703\) 31.1733 1.17572
\(704\) −1.16944 −0.0440751
\(705\) 0 0
\(706\) −43.6816 −1.64398
\(707\) 0 0
\(708\) 0 0
\(709\) −2.22253 −0.0834688 −0.0417344 0.999129i \(-0.513288\pi\)
−0.0417344 + 0.999129i \(0.513288\pi\)
\(710\) 6.93825 0.260388
\(711\) 0 0
\(712\) 1.59343 0.0597165
\(713\) −15.0475 −0.563532
\(714\) 0 0
\(715\) −2.80361 −0.104849
\(716\) 12.6983 0.474556
\(717\) 0 0
\(718\) −35.1606 −1.31218
\(719\) 26.0175 0.970291 0.485145 0.874434i \(-0.338767\pi\)
0.485145 + 0.874434i \(0.338767\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.53390 −0.0570859
\(723\) 0 0
\(724\) 11.4534 0.425662
\(725\) −22.4500 −0.833772
\(726\) 0 0
\(727\) −1.37175 −0.0508754 −0.0254377 0.999676i \(-0.508098\pi\)
−0.0254377 + 0.999676i \(0.508098\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 7.19777 0.266401
\(731\) −52.9830 −1.95965
\(732\) 0 0
\(733\) −0.800174 −0.0295551 −0.0147776 0.999891i \(-0.504704\pi\)
−0.0147776 + 0.999891i \(0.504704\pi\)
\(734\) 4.84022 0.178656
\(735\) 0 0
\(736\) 11.6414 0.429109
\(737\) 6.98034 0.257124
\(738\) 0 0
\(739\) 5.37093 0.197573 0.0987865 0.995109i \(-0.468504\pi\)
0.0987865 + 0.995109i \(0.468504\pi\)
\(740\) 5.89511 0.216709
\(741\) 0 0
\(742\) 0 0
\(743\) −13.2632 −0.486581 −0.243290 0.969953i \(-0.578227\pi\)
−0.243290 + 0.969953i \(0.578227\pi\)
\(744\) 0 0
\(745\) 4.14436 0.151838
\(746\) −36.4276 −1.33371
\(747\) 0 0
\(748\) 3.96485 0.144969
\(749\) 0 0
\(750\) 0 0
\(751\) 5.55632 0.202753 0.101377 0.994848i \(-0.467675\pi\)
0.101377 + 0.994848i \(0.467675\pi\)
\(752\) 37.2914 1.35988
\(753\) 0 0
\(754\) 46.7289 1.70177
\(755\) 12.0119 0.437158
\(756\) 0 0
\(757\) −13.3942 −0.486819 −0.243410 0.969924i \(-0.578266\pi\)
−0.243410 + 0.969924i \(0.578266\pi\)
\(758\) −45.9739 −1.66985
\(759\) 0 0
\(760\) 7.99862 0.290141
\(761\) −12.8438 −0.465588 −0.232794 0.972526i \(-0.574787\pi\)
−0.232794 + 0.972526i \(0.574787\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.92353 −0.0695910
\(765\) 0 0
\(766\) 24.5202 0.885951
\(767\) −52.3570 −1.89050
\(768\) 0 0
\(769\) 2.96518 0.106927 0.0534636 0.998570i \(-0.482974\pi\)
0.0534636 + 0.998570i \(0.482974\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.27067 0.333659
\(773\) −19.2788 −0.693409 −0.346705 0.937974i \(-0.612699\pi\)
−0.346705 + 0.937974i \(0.612699\pi\)
\(774\) 0 0
\(775\) 24.8992 0.894406
\(776\) −6.43366 −0.230955
\(777\) 0 0
\(778\) 10.3806 0.372161
\(779\) 4.70610 0.168614
\(780\) 0 0
\(781\) −2.53018 −0.0905371
\(782\) 31.9231 1.14157
\(783\) 0 0
\(784\) 0 0
\(785\) −10.7020 −0.381970
\(786\) 0 0
\(787\) 13.6453 0.486402 0.243201 0.969976i \(-0.421803\pi\)
0.243201 + 0.969976i \(0.421803\pi\)
\(788\) −16.6947 −0.594724
\(789\) 0 0
\(790\) −2.15195 −0.0765630
\(791\) 0 0
\(792\) 0 0
\(793\) −58.4944 −2.07720
\(794\) 21.9036 0.777330
\(795\) 0 0
\(796\) 7.48855 0.265425
\(797\) −22.9585 −0.813231 −0.406616 0.913599i \(-0.633291\pi\)
−0.406616 + 0.913599i \(0.633291\pi\)
\(798\) 0 0
\(799\) 56.6922 2.00563
\(800\) −19.2632 −0.681058
\(801\) 0 0
\(802\) −14.2609 −0.503570
\(803\) −2.62482 −0.0926279
\(804\) 0 0
\(805\) 0 0
\(806\) −51.8268 −1.82552
\(807\) 0 0
\(808\) −18.1065 −0.636985
\(809\) 39.4582 1.38728 0.693639 0.720323i \(-0.256006\pi\)
0.693639 + 0.720323i \(0.256006\pi\)
\(810\) 0 0
\(811\) −0.496374 −0.0174300 −0.00871502 0.999962i \(-0.502774\pi\)
−0.00871502 + 0.999962i \(0.502774\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −6.98762 −0.244916
\(815\) 1.58168 0.0554039
\(816\) 0 0
\(817\) −31.1733 −1.09062
\(818\) −11.5790 −0.404850
\(819\) 0 0
\(820\) 0.889960 0.0310788
\(821\) 47.1038 1.64393 0.821967 0.569534i \(-0.192876\pi\)
0.821967 + 0.569534i \(0.192876\pi\)
\(822\) 0 0
\(823\) −2.19777 −0.0766094 −0.0383047 0.999266i \(-0.512196\pi\)
−0.0383047 + 0.999266i \(0.512196\pi\)
\(824\) −22.0092 −0.766727
\(825\) 0 0
\(826\) 0 0
\(827\) −55.3360 −1.92422 −0.962110 0.272661i \(-0.912096\pi\)
−0.962110 + 0.272661i \(0.912096\pi\)
\(828\) 0 0
\(829\) −20.3206 −0.705764 −0.352882 0.935668i \(-0.614798\pi\)
−0.352882 + 0.935668i \(0.614798\pi\)
\(830\) −9.17041 −0.318310
\(831\) 0 0
\(832\) −9.97733 −0.345902
\(833\) 0 0
\(834\) 0 0
\(835\) −3.70472 −0.128207
\(836\) 2.33277 0.0806806
\(837\) 0 0
\(838\) 17.5480 0.606186
\(839\) 24.5519 0.847627 0.423813 0.905750i \(-0.360691\pi\)
0.423813 + 0.905750i \(0.360691\pi\)
\(840\) 0 0
\(841\) 0.998623 0.0344353
\(842\) −5.33007 −0.183686
\(843\) 0 0
\(844\) 9.97386 0.343315
\(845\) −11.5790 −0.398329
\(846\) 0 0
\(847\) 0 0
\(848\) −34.5251 −1.18560
\(849\) 0 0
\(850\) −52.8235 −1.81183
\(851\) −17.3090 −0.593346
\(852\) 0 0
\(853\) 53.5416 1.83323 0.916614 0.399773i \(-0.130911\pi\)
0.916614 + 0.399773i \(0.130911\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.17439 −0.245215
\(857\) 54.1553 1.84991 0.924955 0.380076i \(-0.124102\pi\)
0.924955 + 0.380076i \(0.124102\pi\)
\(858\) 0 0
\(859\) −1.79292 −0.0611737 −0.0305869 0.999532i \(-0.509738\pi\)
−0.0305869 + 0.999532i \(0.509738\pi\)
\(860\) −5.89511 −0.201022
\(861\) 0 0
\(862\) −54.1716 −1.84509
\(863\) −32.5709 −1.10873 −0.554363 0.832275i \(-0.687038\pi\)
−0.554363 + 0.832275i \(0.687038\pi\)
\(864\) 0 0
\(865\) −15.2967 −0.520102
\(866\) −12.7277 −0.432506
\(867\) 0 0
\(868\) 0 0
\(869\) 0.784755 0.0266210
\(870\) 0 0
\(871\) 59.5541 2.01791
\(872\) −24.2967 −0.822789
\(873\) 0 0
\(874\) 18.7824 0.635324
\(875\) 0 0
\(876\) 0 0
\(877\) −36.7293 −1.24026 −0.620131 0.784499i \(-0.712920\pi\)
−0.620131 + 0.784499i \(0.712920\pi\)
\(878\) −3.89095 −0.131313
\(879\) 0 0
\(880\) −2.78567 −0.0939049
\(881\) −25.3721 −0.854807 −0.427403 0.904061i \(-0.640572\pi\)
−0.427403 + 0.904061i \(0.640572\pi\)
\(882\) 0 0
\(883\) −16.9381 −0.570012 −0.285006 0.958526i \(-0.591996\pi\)
−0.285006 + 0.958526i \(0.591996\pi\)
\(884\) 33.8268 1.13772
\(885\) 0 0
\(886\) −63.4807 −2.13267
\(887\) 48.0137 1.61214 0.806071 0.591819i \(-0.201590\pi\)
0.806071 + 0.591819i \(0.201590\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.36115 0.0456260
\(891\) 0 0
\(892\) −18.4456 −0.617603
\(893\) 33.3556 1.11620
\(894\) 0 0
\(895\) −13.5632 −0.453367
\(896\) 0 0
\(897\) 0 0
\(898\) 10.5426 0.351810
\(899\) −33.2713 −1.10966
\(900\) 0 0
\(901\) −52.4867 −1.74858
\(902\) −1.05489 −0.0351240
\(903\) 0 0
\(904\) 17.0384 0.566688
\(905\) −12.2335 −0.406656
\(906\) 0 0
\(907\) −24.7775 −0.822722 −0.411361 0.911472i \(-0.634947\pi\)
−0.411361 + 0.911472i \(0.634947\pi\)
\(908\) −9.26972 −0.307626
\(909\) 0 0
\(910\) 0 0
\(911\) 31.5833 1.04640 0.523200 0.852210i \(-0.324738\pi\)
0.523200 + 0.852210i \(0.324738\pi\)
\(912\) 0 0
\(913\) 3.34419 0.110676
\(914\) −34.2843 −1.13402
\(915\) 0 0
\(916\) 13.3837 0.442209
\(917\) 0 0
\(918\) 0 0
\(919\) 1.59208 0.0525180 0.0262590 0.999655i \(-0.491641\pi\)
0.0262590 + 0.999655i \(0.491641\pi\)
\(920\) −4.44125 −0.146424
\(921\) 0 0
\(922\) 38.3028 1.26144
\(923\) −21.5867 −0.710536
\(924\) 0 0
\(925\) 28.6414 0.941725
\(926\) 46.9591 1.54317
\(927\) 0 0
\(928\) 25.7403 0.844968
\(929\) 27.0711 0.888175 0.444087 0.895983i \(-0.353528\pi\)
0.444087 + 0.895983i \(0.353528\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 3.89877 0.127709
\(933\) 0 0
\(934\) −34.2006 −1.11908
\(935\) −4.23491 −0.138496
\(936\) 0 0
\(937\) 32.6624 1.06704 0.533518 0.845789i \(-0.320870\pi\)
0.533518 + 0.845789i \(0.320870\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.30781 0.205738
\(941\) 4.72285 0.153961 0.0769803 0.997033i \(-0.475472\pi\)
0.0769803 + 0.997033i \(0.475472\pi\)
\(942\) 0 0
\(943\) −2.61307 −0.0850933
\(944\) −52.0220 −1.69317
\(945\) 0 0
\(946\) 6.98762 0.227187
\(947\) 56.7810 1.84514 0.922568 0.385835i \(-0.126087\pi\)
0.922568 + 0.385835i \(0.126087\pi\)
\(948\) 0 0
\(949\) −22.3942 −0.726945
\(950\) −31.0795 −1.00835
\(951\) 0 0
\(952\) 0 0
\(953\) −47.1693 −1.52796 −0.763982 0.645238i \(-0.776758\pi\)
−0.763982 + 0.645238i \(0.776758\pi\)
\(954\) 0 0
\(955\) 2.05455 0.0664838
\(956\) −8.48852 −0.274538
\(957\) 0 0
\(958\) 16.2936 0.526423
\(959\) 0 0
\(960\) 0 0
\(961\) 5.90112 0.190359
\(962\) −59.6162 −1.92210
\(963\) 0 0
\(964\) 9.36359 0.301581
\(965\) −9.90213 −0.318761
\(966\) 0 0
\(967\) 47.3969 1.52418 0.762091 0.647470i \(-0.224173\pi\)
0.762091 + 0.647470i \(0.224173\pi\)
\(968\) −20.1223 −0.646754
\(969\) 0 0
\(970\) −5.49580 −0.176459
\(971\) 22.7473 0.729994 0.364997 0.931009i \(-0.381070\pi\)
0.364997 + 0.931009i \(0.381070\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −22.2212 −0.712012
\(975\) 0 0
\(976\) −58.1202 −1.86038
\(977\) 35.6849 1.14166 0.570831 0.821068i \(-0.306621\pi\)
0.570831 + 0.821068i \(0.306621\pi\)
\(978\) 0 0
\(979\) −0.496374 −0.0158642
\(980\) 0 0
\(981\) 0 0
\(982\) −26.0741 −0.832059
\(983\) 24.0134 0.765908 0.382954 0.923767i \(-0.374907\pi\)
0.382954 + 0.923767i \(0.374907\pi\)
\(984\) 0 0
\(985\) 17.8318 0.568169
\(986\) 70.5850 2.24788
\(987\) 0 0
\(988\) 19.9025 0.633183
\(989\) 17.3090 0.550395
\(990\) 0 0
\(991\) −44.4189 −1.41101 −0.705507 0.708703i \(-0.749281\pi\)
−0.705507 + 0.708703i \(0.749281\pi\)
\(992\) −28.5485 −0.906416
\(993\) 0 0
\(994\) 0 0
\(995\) −7.99862 −0.253573
\(996\) 0 0
\(997\) 9.04673 0.286513 0.143256 0.989686i \(-0.454243\pi\)
0.143256 + 0.989686i \(0.454243\pi\)
\(998\) −8.26929 −0.261760
\(999\) 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 3969.2.a.be.1.1 6
3.2 odd 2 3969.2.a.bd.1.6 6
7.6 odd 2 inner 3969.2.a.be.1.2 6
9.2 odd 6 1323.2.f.g.442.1 12
9.4 even 3 441.2.f.g.295.6 yes 12
9.5 odd 6 1323.2.f.g.883.1 12
9.7 even 3 441.2.f.g.148.6 yes 12
21.20 even 2 3969.2.a.bd.1.5 6
63.2 odd 6 1323.2.g.g.361.2 12
63.4 even 3 441.2.g.g.79.5 12
63.5 even 6 1323.2.h.g.802.6 12
63.11 odd 6 1323.2.h.g.226.5 12
63.13 odd 6 441.2.f.g.295.5 yes 12
63.16 even 3 441.2.g.g.67.5 12
63.20 even 6 1323.2.f.g.442.2 12
63.23 odd 6 1323.2.h.g.802.5 12
63.25 even 3 441.2.h.g.373.2 12
63.31 odd 6 441.2.g.g.79.6 12
63.32 odd 6 1323.2.g.g.667.2 12
63.34 odd 6 441.2.f.g.148.5 12
63.38 even 6 1323.2.h.g.226.6 12
63.40 odd 6 441.2.h.g.214.1 12
63.41 even 6 1323.2.f.g.883.2 12
63.47 even 6 1323.2.g.g.361.1 12
63.52 odd 6 441.2.h.g.373.1 12
63.58 even 3 441.2.h.g.214.2 12
63.59 even 6 1323.2.g.g.667.1 12
63.61 odd 6 441.2.g.g.67.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.g.148.5 12 63.34 odd 6
441.2.f.g.148.6 yes 12 9.7 even 3
441.2.f.g.295.5 yes 12 63.13 odd 6
441.2.f.g.295.6 yes 12 9.4 even 3
441.2.g.g.67.5 12 63.16 even 3
441.2.g.g.67.6 12 63.61 odd 6
441.2.g.g.79.5 12 63.4 even 3
441.2.g.g.79.6 12 63.31 odd 6
441.2.h.g.214.1 12 63.40 odd 6
441.2.h.g.214.2 12 63.58 even 3
441.2.h.g.373.1 12 63.52 odd 6
441.2.h.g.373.2 12 63.25 even 3
1323.2.f.g.442.1 12 9.2 odd 6
1323.2.f.g.442.2 12 63.20 even 6
1323.2.f.g.883.1 12 9.5 odd 6
1323.2.f.g.883.2 12 63.41 even 6
1323.2.g.g.361.1 12 63.47 even 6
1323.2.g.g.361.2 12 63.2 odd 6
1323.2.g.g.667.1 12 63.59 even 6
1323.2.g.g.667.2 12 63.32 odd 6
1323.2.h.g.226.5 12 63.11 odd 6
1323.2.h.g.226.6 12 63.38 even 6
1323.2.h.g.802.5 12 63.23 odd 6
1323.2.h.g.802.6 12 63.5 even 6
3969.2.a.bd.1.5 6 21.20 even 2
3969.2.a.bd.1.6 6 3.2 odd 2
3969.2.a.be.1.1 6 1.1 even 1 trivial
3969.2.a.be.1.2 6 7.6 odd 2 inner