Properties

Label 3969.2.a.be.1.2
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 1.2
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 \(-0.431916\pi\)
0.212263 + 0.977212i \(0.431916\pi\)
\(6\) 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 \(-0.254912\pi\)
0.696112 + 0.717933i \(0.254912\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.98762 −1.24691
\(17\) 7.58242 1.83901 0.919503 0.393083i \(-0.128591\pi\)
0.919503 + 0.393083i \(0.128591\pi\)
\(18\) 0 0
\(19\) 4.46122 1.02347 0.511737 0.859142i \(-0.329002\pi\)
0.511737 + 0.859142i \(0.329002\pi\)
\(20\) 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 \(-0.316333\pi\)
0.545518 + 0.838099i \(0.316333\pi\)
\(32\) 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 \(-0.473750\pi\)
0.0823731 + 0.996602i \(0.473750\pi\)
\(42\) 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 \(-0.316415\pi\)
0.545301 + 0.838240i \(0.316415\pi\)
\(48\) 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 \(-0.737565\pi\)
−0.678950 + 0.734184i \(0.737565\pi\)
\(60\) 0 0
\(61\) −11.6529 −1.49200 −0.745999 0.665947i \(-0.768028\pi\)
−0.745999 + 0.665947i \(0.768028\pi\)
\(62\) −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 \(-0.584076\pi\)
−0.261073 + 0.965319i \(0.584076\pi\)
\(74\) 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 \(-0.399020\pi\)
0.311943 + 0.950101i \(0.399020\pi\)
\(84\) 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 \(-0.514237\pi\)
−0.0447134 + 0.999000i \(0.514237\pi\)
\(90\) 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 \(-0.444676\pi\)
0.172930 + 0.984934i \(0.444676\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.64283 −0.364283
\(101\) 9.58658 0.953900 0.476950 0.878930i \(-0.341742\pi\)
0.476950 + 0.878930i \(0.341742\pi\)
\(102\) 0 0
\(103\) 11.6529 1.14819 0.574096 0.818788i \(-0.305353\pi\)
0.574096 + 0.818788i \(0.305353\pi\)
\(104\) 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 \(-0.593579\pi\)
−0.289770 + 0.957096i \(0.593579\pi\)
\(132\) 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 \(-0.621814\pi\)
−0.373418 + 0.927663i \(0.621814\pi\)
\(140\) 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 \(-0.648532\pi\)
−0.449877 + 0.893091i \(0.648532\pi\)
\(158\) 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 \(-0.548249\pi\)
−0.151000 + 0.988534i \(0.548249\pi\)
\(168\) 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 \(-0.709863\pi\)
−0.612566 + 0.790419i \(0.709863\pi\)
\(174\) 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 \(-0.658983\pi\)
−0.478952 + 0.877841i \(0.658983\pi\)
\(182\) 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 \(-0.596538\pi\)
−0.298654 + 0.954361i \(0.596538\pi\)
\(200\) −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 \(-0.255438\pi\)
0.694923 + 0.719084i \(0.255438\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −15.3324 −1.01990
\(227\) 10.4302 0.692279 0.346139 0.938183i \(-0.387492\pi\)
0.346139 + 0.938183i \(0.387492\pi\)
\(228\) 0 0
\(229\) −15.0592 −0.995141 −0.497570 0.867424i \(-0.665774\pi\)
−0.497570 + 0.867424i \(0.665774\pi\)
\(230\) −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 \(-0.610203\pi\)
−0.339337 + 0.940665i \(0.610203\pi\)
\(242\) 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 \(-0.219683\pi\)
0.771148 + 0.636656i \(0.219683\pi\)
\(252\) 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 \(-0.539898\pi\)
−0.125016 + 0.992155i \(0.539898\pi\)
\(258\) 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 \(-0.357198\pi\)
0.433727 + 0.901044i \(0.357198\pi\)
\(270\) 0 0
\(271\) 5.39874 0.327950 0.163975 0.986464i \(-0.447568\pi\)
0.163975 + 0.986464i \(0.447568\pi\)
\(272\) −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 \(-0.104474\pi\)
0.946619 + 0.322353i \(0.104474\pi\)
\(284\) 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 \(-0.796816\pi\)
−0.803097 + 0.595849i \(0.796816\pi\)
\(294\) 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 \(-0.638967\pi\)
−0.422841 + 0.906204i \(0.638967\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −9.80085 −0.556651
\(311\) −29.0635 −1.64804 −0.824021 0.566559i \(-0.808274\pi\)
−0.824021 + 0.566559i \(0.808274\pi\)
\(312\) 0 0
\(313\) 24.4780 1.38358 0.691790 0.722099i \(-0.256822\pi\)
0.691790 + 0.722099i \(0.256822\pi\)
\(314\) 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 \(-0.462909\pi\)
0.116260 + 0.993219i \(0.462909\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.76509 −0.147380
\(353\) −25.7007 −1.36791 −0.683955 0.729525i \(-0.739741\pi\)
−0.683955 + 0.729525i \(0.739741\pi\)
\(354\) 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 \(-0.476319\pi\)
0.0743273 + 0.997234i \(0.476319\pi\)
\(368\) −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 \(-0.379841\pi\)
0.368588 + 0.929593i \(0.379841\pi\)
\(384\) 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 \(-0.395175\pi\)
0.323397 + 0.946263i \(0.395175\pi\)
\(398\) 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 \(-0.553870\pi\)
−0.168432 + 0.985713i \(0.553870\pi\)
\(410\) −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 \(-0.418848\pi\)
0.252195 + 0.967676i \(0.418848\pi\)
\(420\) 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 \(-0.557590\pi\)
−0.179938 + 0.983678i \(0.557590\pi\)
\(434\) 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 \(-0.517398\pi\)
−0.0546311 + 0.998507i \(0.517398\pi\)
\(440\) −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 \(-0.324139\pi\)
0.524803 + 0.851224i \(0.324139\pi\)
\(462\) 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 \(-0.654153\pi\)
−0.465577 + 0.885007i \(0.654153\pi\)
\(468\) 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 \(-0.429717\pi\)
0.219011 + 0.975722i \(0.429717\pi\)
\(480\) 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 \(-0.616163\pi\)
−0.356892 + 0.934146i \(0.616163\pi\)
\(504\) 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 \(-0.257115\pi\)
0.691127 + 0.722733i \(0.257115\pi\)
\(510\) 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 \(-0.651889\pi\)
−0.459270 + 0.888297i \(0.651889\pi\)
\(522\) 0 0
\(523\) −43.5642 −1.90493 −0.952465 0.304647i \(-0.901462\pi\)
−0.952465 + 0.304647i \(0.901462\pi\)
\(524\) −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 \(-0.252578\pi\)
0.701358 + 0.712809i \(0.252578\pi\)
\(564\) 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 \(-0.229608\pi\)
0.750925 + 0.660387i \(0.229608\pi\)
\(578\) −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 \(-0.506930\pi\)
−0.0217700 + 0.999763i \(0.506930\pi\)
\(588\) 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 \(-0.399857\pi\)
0.309445 + 0.950917i \(0.399857\pi\)
\(594\) 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 \(-0.625960\pi\)
−0.385469 + 0.922721i \(0.625960\pi\)
\(602\) 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 \(-0.296154\pi\)
0.597518 + 0.801856i \(0.296154\pi\)
\(608\) 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 \(-0.338616\pi\)
0.485560 + 0.874204i \(0.338616\pi\)
\(620\) 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 \(-0.637631\pi\)
−0.419033 + 0.907971i \(0.637631\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −57.4930 −2.26203
\(647\) −16.9460 −0.666216 −0.333108 0.942889i \(-0.608097\pi\)
−0.333108 + 0.942889i \(0.608097\pi\)
\(648\) 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 \(-0.734258\pi\)
−0.671288 + 0.741197i \(0.734258\pi\)
\(662\) −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 \(-0.448886\pi\)
0.159890 + 0.987135i \(0.448886\pi\)
\(678\) 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 \(-0.735083\pi\)
−0.673204 + 0.739456i \(0.735083\pi\)
\(692\) −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 \(-0.661233\pi\)
−0.485145 + 0.874434i \(0.661233\pi\)
\(720\) 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 \(-0.491902\pi\)
0.0254377 + 0.999676i \(0.491902\pi\)
\(728\) 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 \(-0.495296\pi\)
0.0147776 + 0.999891i \(0.495296\pi\)
\(734\) −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 \(-0.425213\pi\)
0.232794 + 0.972526i \(0.425213\pi\)
\(762\) 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 \(-0.517026\pi\)
−0.0534636 + 0.998570i \(0.517026\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.27067 0.333659
\(773\) 19.2788 0.693409 0.346705 0.937974i \(-0.387301\pi\)
0.346705 + 0.937974i \(0.387301\pi\)
\(774\) 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 \(-0.578197\pi\)
−0.243201 + 0.969976i \(0.578197\pi\)
\(788\) −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 \(-0.366709\pi\)
0.406616 + 0.913599i \(0.366709\pi\)
\(798\) 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 \(-0.497226\pi\)
0.00871502 + 0.999962i \(0.497226\pi\)
\(812\) 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 \(-0.385202\pi\)
0.352882 + 0.935668i \(0.385202\pi\)
\(830\) −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 \(-0.639309\pi\)
−0.423813 + 0.905750i \(0.639309\pi\)
\(840\) 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 \(-0.869089\pi\)
−0.916614 + 0.399773i \(0.869089\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.17439 −0.245215
\(857\) −54.1553 −1.84991 −0.924955 0.380076i \(-0.875898\pi\)
−0.924955 + 0.380076i \(0.875898\pi\)
\(858\) 0 0
\(859\) 1.79292 0.0611737 0.0305869 0.999532i \(-0.490262\pi\)
0.0305869 + 0.999532i \(0.490262\pi\)
\(860\) 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 \(-0.359428\pi\)
0.427403 + 0.904061i \(0.359428\pi\)
\(882\) 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 \(-0.798410\pi\)
−0.806071 + 0.591819i \(0.798410\pi\)
\(888\) 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 \(-0.646472\pi\)
−0.444087 + 0.895983i \(0.646472\pi\)
\(930\) 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 \(-0.679130\pi\)
−0.533518 + 0.845789i \(0.679130\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.30781 0.205738
\(941\) −4.72285 −0.153961 −0.0769803 0.997033i \(-0.524528\pi\)
−0.0769803 + 0.997033i \(0.524528\pi\)
\(942\) 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 \(-0.618930\pi\)
−0.364997 + 0.931009i \(0.618930\pi\)
\(972\) 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 \(-0.625093\pi\)
−0.382954 + 0.923767i \(0.625093\pi\)
\(984\) 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 \(-0.545757\pi\)
−0.143256 + 0.989686i \(0.545757\pi\)
\(998\) −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.1.2 6
3.2 odd 2 3969.2.a.bd.1.5 6
7.6 odd 2 inner 3969.2.a.be.1.1 6
9.2 odd 6 1323.2.f.g.442.2 12
9.4 even 3 441.2.f.g.295.5 yes 12
9.5 odd 6 1323.2.f.g.883.2 12
9.7 even 3 441.2.f.g.148.5 12
21.20 even 2 3969.2.a.bd.1.6 6
63.2 odd 6 1323.2.g.g.361.1 12
63.4 even 3 441.2.g.g.79.6 12
63.5 even 6 1323.2.h.g.802.5 12
63.11 odd 6 1323.2.h.g.226.6 12
63.13 odd 6 441.2.f.g.295.6 yes 12
63.16 even 3 441.2.g.g.67.6 12
63.20 even 6 1323.2.f.g.442.1 12
63.23 odd 6 1323.2.h.g.802.6 12
63.25 even 3 441.2.h.g.373.1 12
63.31 odd 6 441.2.g.g.79.5 12
63.32 odd 6 1323.2.g.g.667.1 12
63.34 odd 6 441.2.f.g.148.6 yes 12
63.38 even 6 1323.2.h.g.226.5 12
63.40 odd 6 441.2.h.g.214.2 12
63.41 even 6 1323.2.f.g.883.1 12
63.47 even 6 1323.2.g.g.361.2 12
63.52 odd 6 441.2.h.g.373.2 12
63.58 even 3 441.2.h.g.214.1 12
63.59 even 6 1323.2.g.g.667.2 12
63.61 odd 6 441.2.g.g.67.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.2.f.g.148.5 12 9.7 even 3
441.2.f.g.148.6 yes 12 63.34 odd 6
441.2.f.g.295.5 yes 12 9.4 even 3
441.2.f.g.295.6 yes 12 63.13 odd 6
441.2.g.g.67.5 12 63.61 odd 6
441.2.g.g.67.6 12 63.16 even 3
441.2.g.g.79.5 12 63.31 odd 6
441.2.g.g.79.6 12 63.4 even 3
441.2.h.g.214.1 12 63.58 even 3
441.2.h.g.214.2 12 63.40 odd 6
441.2.h.g.373.1 12 63.25 even 3
441.2.h.g.373.2 12 63.52 odd 6
1323.2.f.g.442.1 12 63.20 even 6
1323.2.f.g.442.2 12 9.2 odd 6
1323.2.f.g.883.1 12 63.41 even 6
1323.2.f.g.883.2 12 9.5 odd 6
1323.2.g.g.361.1 12 63.2 odd 6
1323.2.g.g.361.2 12 63.47 even 6
1323.2.g.g.667.1 12 63.32 odd 6
1323.2.g.g.667.2 12 63.59 even 6
1323.2.h.g.226.5 12 63.38 even 6
1323.2.h.g.226.6 12 63.11 odd 6
1323.2.h.g.802.5 12 63.5 even 6
1323.2.h.g.802.6 12 63.23 odd 6
3969.2.a.bd.1.5 6 3.2 odd 2
3969.2.a.bd.1.6 6 21.20 even 2
3969.2.a.be.1.1 6 7.6 odd 2 inner
3969.2.a.be.1.2 6 1.1 even 1 trivial