Properties

Label 8004.2.a.i.1.5
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.83485\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.83485 q^{5} +1.71276 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.83485 q^{5} +1.71276 q^{7} +1.00000 q^{9} -0.324628 q^{11} +2.24187 q^{13} +1.83485 q^{15} +3.03434 q^{17} -6.93143 q^{19} -1.71276 q^{21} +1.00000 q^{23} -1.63333 q^{25} -1.00000 q^{27} +1.00000 q^{29} +0.153463 q^{31} +0.324628 q^{33} -3.14266 q^{35} +7.96250 q^{37} -2.24187 q^{39} +5.42941 q^{41} +4.81157 q^{43} -1.83485 q^{45} -4.18978 q^{47} -4.06644 q^{49} -3.03434 q^{51} +11.5727 q^{53} +0.595642 q^{55} +6.93143 q^{57} -2.10296 q^{59} -1.10334 q^{61} +1.71276 q^{63} -4.11350 q^{65} -3.92964 q^{67} -1.00000 q^{69} -11.2432 q^{71} -6.02817 q^{73} +1.63333 q^{75} -0.556010 q^{77} -9.02404 q^{79} +1.00000 q^{81} -0.162205 q^{83} -5.56755 q^{85} -1.00000 q^{87} +9.59572 q^{89} +3.83980 q^{91} -0.153463 q^{93} +12.7181 q^{95} -1.19093 q^{97} -0.324628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) −1.83485 −0.820569 −0.410284 0.911958i \(-0.634571\pi\)
−0.410284 + 0.911958i \(0.634571\pi\)
\(6\) 0 0
\(7\) 1.71276 0.647363 0.323682 0.946166i \(-0.395079\pi\)
0.323682 + 0.946166i \(0.395079\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.324628 −0.0978789 −0.0489395 0.998802i \(-0.515584\pi\)
−0.0489395 + 0.998802i \(0.515584\pi\)
\(12\) 0 0
\(13\) 2.24187 0.621784 0.310892 0.950445i \(-0.399372\pi\)
0.310892 + 0.950445i \(0.399372\pi\)
\(14\) 0 0
\(15\) 1.83485 0.473756
\(16\) 0 0
\(17\) 3.03434 0.735935 0.367967 0.929839i \(-0.380054\pi\)
0.367967 + 0.929839i \(0.380054\pi\)
\(18\) 0 0
\(19\) −6.93143 −1.59018 −0.795089 0.606492i \(-0.792576\pi\)
−0.795089 + 0.606492i \(0.792576\pi\)
\(20\) 0 0
\(21\) −1.71276 −0.373755
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −1.63333 −0.326667
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 0.153463 0.0275627 0.0137813 0.999905i \(-0.495613\pi\)
0.0137813 + 0.999905i \(0.495613\pi\)
\(32\) 0 0
\(33\) 0.324628 0.0565104
\(34\) 0 0
\(35\) −3.14266 −0.531206
\(36\) 0 0
\(37\) 7.96250 1.30903 0.654514 0.756050i \(-0.272873\pi\)
0.654514 + 0.756050i \(0.272873\pi\)
\(38\) 0 0
\(39\) −2.24187 −0.358987
\(40\) 0 0
\(41\) 5.42941 0.847931 0.423966 0.905678i \(-0.360638\pi\)
0.423966 + 0.905678i \(0.360638\pi\)
\(42\) 0 0
\(43\) 4.81157 0.733758 0.366879 0.930269i \(-0.380426\pi\)
0.366879 + 0.930269i \(0.380426\pi\)
\(44\) 0 0
\(45\) −1.83485 −0.273523
\(46\) 0 0
\(47\) −4.18978 −0.611142 −0.305571 0.952169i \(-0.598847\pi\)
−0.305571 + 0.952169i \(0.598847\pi\)
\(48\) 0 0
\(49\) −4.06644 −0.580921
\(50\) 0 0
\(51\) −3.03434 −0.424892
\(52\) 0 0
\(53\) 11.5727 1.58963 0.794817 0.606849i \(-0.207567\pi\)
0.794817 + 0.606849i \(0.207567\pi\)
\(54\) 0 0
\(55\) 0.595642 0.0803164
\(56\) 0 0
\(57\) 6.93143 0.918090
\(58\) 0 0
\(59\) −2.10296 −0.273782 −0.136891 0.990586i \(-0.543711\pi\)
−0.136891 + 0.990586i \(0.543711\pi\)
\(60\) 0 0
\(61\) −1.10334 −0.141268 −0.0706342 0.997502i \(-0.522502\pi\)
−0.0706342 + 0.997502i \(0.522502\pi\)
\(62\) 0 0
\(63\) 1.71276 0.215788
\(64\) 0 0
\(65\) −4.11350 −0.510217
\(66\) 0 0
\(67\) −3.92964 −0.480082 −0.240041 0.970763i \(-0.577161\pi\)
−0.240041 + 0.970763i \(0.577161\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −11.2432 −1.33432 −0.667161 0.744913i \(-0.732491\pi\)
−0.667161 + 0.744913i \(0.732491\pi\)
\(72\) 0 0
\(73\) −6.02817 −0.705544 −0.352772 0.935709i \(-0.614761\pi\)
−0.352772 + 0.935709i \(0.614761\pi\)
\(74\) 0 0
\(75\) 1.63333 0.188601
\(76\) 0 0
\(77\) −0.556010 −0.0633632
\(78\) 0 0
\(79\) −9.02404 −1.01528 −0.507642 0.861568i \(-0.669483\pi\)
−0.507642 + 0.861568i \(0.669483\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.162205 −0.0178043 −0.00890215 0.999960i \(-0.502834\pi\)
−0.00890215 + 0.999960i \(0.502834\pi\)
\(84\) 0 0
\(85\) −5.56755 −0.603885
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 9.59572 1.01714 0.508572 0.861019i \(-0.330174\pi\)
0.508572 + 0.861019i \(0.330174\pi\)
\(90\) 0 0
\(91\) 3.83980 0.402520
\(92\) 0 0
\(93\) −0.153463 −0.0159133
\(94\) 0 0
\(95\) 12.7181 1.30485
\(96\) 0 0
\(97\) −1.19093 −0.120920 −0.0604601 0.998171i \(-0.519257\pi\)
−0.0604601 + 0.998171i \(0.519257\pi\)
\(98\) 0 0
\(99\) −0.324628 −0.0326263
\(100\) 0 0
\(101\) 1.51959 0.151205 0.0756025 0.997138i \(-0.475912\pi\)
0.0756025 + 0.997138i \(0.475912\pi\)
\(102\) 0 0
\(103\) 19.7934 1.95030 0.975149 0.221548i \(-0.0711109\pi\)
0.975149 + 0.221548i \(0.0711109\pi\)
\(104\) 0 0
\(105\) 3.14266 0.306692
\(106\) 0 0
\(107\) 17.5955 1.70102 0.850511 0.525957i \(-0.176293\pi\)
0.850511 + 0.525957i \(0.176293\pi\)
\(108\) 0 0
\(109\) 15.6414 1.49817 0.749086 0.662473i \(-0.230493\pi\)
0.749086 + 0.662473i \(0.230493\pi\)
\(110\) 0 0
\(111\) −7.96250 −0.755767
\(112\) 0 0
\(113\) −8.49236 −0.798894 −0.399447 0.916756i \(-0.630798\pi\)
−0.399447 + 0.916756i \(0.630798\pi\)
\(114\) 0 0
\(115\) −1.83485 −0.171100
\(116\) 0 0
\(117\) 2.24187 0.207261
\(118\) 0 0
\(119\) 5.19710 0.476417
\(120\) 0 0
\(121\) −10.8946 −0.990420
\(122\) 0 0
\(123\) −5.42941 −0.489553
\(124\) 0 0
\(125\) 12.1712 1.08862
\(126\) 0 0
\(127\) 2.78311 0.246961 0.123480 0.992347i \(-0.460594\pi\)
0.123480 + 0.992347i \(0.460594\pi\)
\(128\) 0 0
\(129\) −4.81157 −0.423635
\(130\) 0 0
\(131\) −3.49691 −0.305526 −0.152763 0.988263i \(-0.548817\pi\)
−0.152763 + 0.988263i \(0.548817\pi\)
\(132\) 0 0
\(133\) −11.8719 −1.02942
\(134\) 0 0
\(135\) 1.83485 0.157919
\(136\) 0 0
\(137\) −17.9808 −1.53620 −0.768102 0.640328i \(-0.778799\pi\)
−0.768102 + 0.640328i \(0.778799\pi\)
\(138\) 0 0
\(139\) −8.22207 −0.697387 −0.348694 0.937237i \(-0.613375\pi\)
−0.348694 + 0.937237i \(0.613375\pi\)
\(140\) 0 0
\(141\) 4.18978 0.352843
\(142\) 0 0
\(143\) −0.727774 −0.0608596
\(144\) 0 0
\(145\) −1.83485 −0.152376
\(146\) 0 0
\(147\) 4.06644 0.335395
\(148\) 0 0
\(149\) −1.98612 −0.162709 −0.0813546 0.996685i \(-0.525925\pi\)
−0.0813546 + 0.996685i \(0.525925\pi\)
\(150\) 0 0
\(151\) −8.62021 −0.701503 −0.350752 0.936469i \(-0.614074\pi\)
−0.350752 + 0.936469i \(0.614074\pi\)
\(152\) 0 0
\(153\) 3.03434 0.245312
\(154\) 0 0
\(155\) −0.281580 −0.0226171
\(156\) 0 0
\(157\) −3.47315 −0.277188 −0.138594 0.990349i \(-0.544258\pi\)
−0.138594 + 0.990349i \(0.544258\pi\)
\(158\) 0 0
\(159\) −11.5727 −0.917776
\(160\) 0 0
\(161\) 1.71276 0.134985
\(162\) 0 0
\(163\) 14.9038 1.16735 0.583676 0.811987i \(-0.301614\pi\)
0.583676 + 0.811987i \(0.301614\pi\)
\(164\) 0 0
\(165\) −0.595642 −0.0463707
\(166\) 0 0
\(167\) 5.58054 0.431835 0.215917 0.976412i \(-0.430726\pi\)
0.215917 + 0.976412i \(0.430726\pi\)
\(168\) 0 0
\(169\) −7.97400 −0.613384
\(170\) 0 0
\(171\) −6.93143 −0.530060
\(172\) 0 0
\(173\) −17.3399 −1.31833 −0.659164 0.751999i \(-0.729090\pi\)
−0.659164 + 0.751999i \(0.729090\pi\)
\(174\) 0 0
\(175\) −2.79751 −0.211472
\(176\) 0 0
\(177\) 2.10296 0.158068
\(178\) 0 0
\(179\) −8.86053 −0.662267 −0.331134 0.943584i \(-0.607431\pi\)
−0.331134 + 0.943584i \(0.607431\pi\)
\(180\) 0 0
\(181\) −24.0277 −1.78596 −0.892981 0.450094i \(-0.851391\pi\)
−0.892981 + 0.450094i \(0.851391\pi\)
\(182\) 0 0
\(183\) 1.10334 0.0815613
\(184\) 0 0
\(185\) −14.6100 −1.07415
\(186\) 0 0
\(187\) −0.985030 −0.0720325
\(188\) 0 0
\(189\) −1.71276 −0.124585
\(190\) 0 0
\(191\) 20.0670 1.45200 0.725998 0.687696i \(-0.241378\pi\)
0.725998 + 0.687696i \(0.241378\pi\)
\(192\) 0 0
\(193\) 14.5004 1.04376 0.521881 0.853019i \(-0.325231\pi\)
0.521881 + 0.853019i \(0.325231\pi\)
\(194\) 0 0
\(195\) 4.11350 0.294574
\(196\) 0 0
\(197\) 25.3961 1.80940 0.904698 0.426054i \(-0.140097\pi\)
0.904698 + 0.426054i \(0.140097\pi\)
\(198\) 0 0
\(199\) −22.9534 −1.62713 −0.813563 0.581477i \(-0.802475\pi\)
−0.813563 + 0.581477i \(0.802475\pi\)
\(200\) 0 0
\(201\) 3.92964 0.277176
\(202\) 0 0
\(203\) 1.71276 0.120212
\(204\) 0 0
\(205\) −9.96214 −0.695786
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 2.25013 0.155645
\(210\) 0 0
\(211\) −14.9031 −1.02597 −0.512985 0.858398i \(-0.671460\pi\)
−0.512985 + 0.858398i \(0.671460\pi\)
\(212\) 0 0
\(213\) 11.2432 0.770372
\(214\) 0 0
\(215\) −8.82850 −0.602099
\(216\) 0 0
\(217\) 0.262845 0.0178431
\(218\) 0 0
\(219\) 6.02817 0.407346
\(220\) 0 0
\(221\) 6.80260 0.457593
\(222\) 0 0
\(223\) 3.21974 0.215610 0.107805 0.994172i \(-0.465618\pi\)
0.107805 + 0.994172i \(0.465618\pi\)
\(224\) 0 0
\(225\) −1.63333 −0.108889
\(226\) 0 0
\(227\) 22.1194 1.46811 0.734057 0.679088i \(-0.237625\pi\)
0.734057 + 0.679088i \(0.237625\pi\)
\(228\) 0 0
\(229\) 16.6341 1.09921 0.549605 0.835425i \(-0.314778\pi\)
0.549605 + 0.835425i \(0.314778\pi\)
\(230\) 0 0
\(231\) 0.556010 0.0365828
\(232\) 0 0
\(233\) 9.19416 0.602329 0.301165 0.953572i \(-0.402625\pi\)
0.301165 + 0.953572i \(0.402625\pi\)
\(234\) 0 0
\(235\) 7.68761 0.501484
\(236\) 0 0
\(237\) 9.02404 0.586175
\(238\) 0 0
\(239\) 14.3208 0.926334 0.463167 0.886271i \(-0.346713\pi\)
0.463167 + 0.886271i \(0.346713\pi\)
\(240\) 0 0
\(241\) 19.6949 1.26866 0.634329 0.773063i \(-0.281276\pi\)
0.634329 + 0.773063i \(0.281276\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 7.46131 0.476685
\(246\) 0 0
\(247\) −15.5394 −0.988748
\(248\) 0 0
\(249\) 0.162205 0.0102793
\(250\) 0 0
\(251\) 19.1162 1.20660 0.603301 0.797514i \(-0.293852\pi\)
0.603301 + 0.797514i \(0.293852\pi\)
\(252\) 0 0
\(253\) −0.324628 −0.0204092
\(254\) 0 0
\(255\) 5.56755 0.348653
\(256\) 0 0
\(257\) −10.5799 −0.659954 −0.329977 0.943989i \(-0.607041\pi\)
−0.329977 + 0.943989i \(0.607041\pi\)
\(258\) 0 0
\(259\) 13.6379 0.847416
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 3.00001 0.184988 0.0924942 0.995713i \(-0.470516\pi\)
0.0924942 + 0.995713i \(0.470516\pi\)
\(264\) 0 0
\(265\) −21.2342 −1.30440
\(266\) 0 0
\(267\) −9.59572 −0.587249
\(268\) 0 0
\(269\) 10.4320 0.636048 0.318024 0.948083i \(-0.396981\pi\)
0.318024 + 0.948083i \(0.396981\pi\)
\(270\) 0 0
\(271\) −16.1350 −0.980129 −0.490064 0.871686i \(-0.663027\pi\)
−0.490064 + 0.871686i \(0.663027\pi\)
\(272\) 0 0
\(273\) −3.83980 −0.232395
\(274\) 0 0
\(275\) 0.530225 0.0319738
\(276\) 0 0
\(277\) 21.6156 1.29876 0.649378 0.760466i \(-0.275029\pi\)
0.649378 + 0.760466i \(0.275029\pi\)
\(278\) 0 0
\(279\) 0.153463 0.00918757
\(280\) 0 0
\(281\) 26.5768 1.58544 0.792721 0.609585i \(-0.208664\pi\)
0.792721 + 0.609585i \(0.208664\pi\)
\(282\) 0 0
\(283\) 2.73513 0.162587 0.0812933 0.996690i \(-0.474095\pi\)
0.0812933 + 0.996690i \(0.474095\pi\)
\(284\) 0 0
\(285\) −12.7181 −0.753356
\(286\) 0 0
\(287\) 9.29929 0.548920
\(288\) 0 0
\(289\) −7.79280 −0.458400
\(290\) 0 0
\(291\) 1.19093 0.0698133
\(292\) 0 0
\(293\) 8.83703 0.516265 0.258132 0.966110i \(-0.416893\pi\)
0.258132 + 0.966110i \(0.416893\pi\)
\(294\) 0 0
\(295\) 3.85861 0.224657
\(296\) 0 0
\(297\) 0.324628 0.0188368
\(298\) 0 0
\(299\) 2.24187 0.129651
\(300\) 0 0
\(301\) 8.24108 0.475008
\(302\) 0 0
\(303\) −1.51959 −0.0872982
\(304\) 0 0
\(305\) 2.02446 0.115920
\(306\) 0 0
\(307\) 6.01045 0.343034 0.171517 0.985181i \(-0.445133\pi\)
0.171517 + 0.985181i \(0.445133\pi\)
\(308\) 0 0
\(309\) −19.7934 −1.12601
\(310\) 0 0
\(311\) 25.6664 1.45541 0.727703 0.685893i \(-0.240588\pi\)
0.727703 + 0.685893i \(0.240588\pi\)
\(312\) 0 0
\(313\) −17.3009 −0.977907 −0.488954 0.872310i \(-0.662621\pi\)
−0.488954 + 0.872310i \(0.662621\pi\)
\(314\) 0 0
\(315\) −3.14266 −0.177069
\(316\) 0 0
\(317\) 22.3014 1.25257 0.626285 0.779594i \(-0.284575\pi\)
0.626285 + 0.779594i \(0.284575\pi\)
\(318\) 0 0
\(319\) −0.324628 −0.0181757
\(320\) 0 0
\(321\) −17.5955 −0.982086
\(322\) 0 0
\(323\) −21.0323 −1.17027
\(324\) 0 0
\(325\) −3.66173 −0.203116
\(326\) 0 0
\(327\) −15.6414 −0.864969
\(328\) 0 0
\(329\) −7.17610 −0.395631
\(330\) 0 0
\(331\) −19.5061 −1.07215 −0.536076 0.844169i \(-0.680094\pi\)
−0.536076 + 0.844169i \(0.680094\pi\)
\(332\) 0 0
\(333\) 7.96250 0.436342
\(334\) 0 0
\(335\) 7.21030 0.393941
\(336\) 0 0
\(337\) 15.5982 0.849687 0.424844 0.905267i \(-0.360329\pi\)
0.424844 + 0.905267i \(0.360329\pi\)
\(338\) 0 0
\(339\) 8.49236 0.461242
\(340\) 0 0
\(341\) −0.0498182 −0.00269781
\(342\) 0 0
\(343\) −18.9542 −1.02343
\(344\) 0 0
\(345\) 1.83485 0.0987849
\(346\) 0 0
\(347\) 31.9578 1.71559 0.857793 0.513996i \(-0.171835\pi\)
0.857793 + 0.513996i \(0.171835\pi\)
\(348\) 0 0
\(349\) 30.7086 1.64380 0.821898 0.569635i \(-0.192915\pi\)
0.821898 + 0.569635i \(0.192915\pi\)
\(350\) 0 0
\(351\) −2.24187 −0.119662
\(352\) 0 0
\(353\) 4.64406 0.247178 0.123589 0.992333i \(-0.460560\pi\)
0.123589 + 0.992333i \(0.460560\pi\)
\(354\) 0 0
\(355\) 20.6296 1.09490
\(356\) 0 0
\(357\) −5.19710 −0.275060
\(358\) 0 0
\(359\) −12.3358 −0.651059 −0.325530 0.945532i \(-0.605543\pi\)
−0.325530 + 0.945532i \(0.605543\pi\)
\(360\) 0 0
\(361\) 29.0447 1.52867
\(362\) 0 0
\(363\) 10.8946 0.571819
\(364\) 0 0
\(365\) 11.0608 0.578948
\(366\) 0 0
\(367\) 11.7419 0.612923 0.306462 0.951883i \(-0.400855\pi\)
0.306462 + 0.951883i \(0.400855\pi\)
\(368\) 0 0
\(369\) 5.42941 0.282644
\(370\) 0 0
\(371\) 19.8213 1.02907
\(372\) 0 0
\(373\) 8.28462 0.428961 0.214481 0.976728i \(-0.431194\pi\)
0.214481 + 0.976728i \(0.431194\pi\)
\(374\) 0 0
\(375\) −12.1712 −0.628516
\(376\) 0 0
\(377\) 2.24187 0.115462
\(378\) 0 0
\(379\) −12.8329 −0.659182 −0.329591 0.944124i \(-0.606911\pi\)
−0.329591 + 0.944124i \(0.606911\pi\)
\(380\) 0 0
\(381\) −2.78311 −0.142583
\(382\) 0 0
\(383\) 36.8683 1.88388 0.941940 0.335781i \(-0.109000\pi\)
0.941940 + 0.335781i \(0.109000\pi\)
\(384\) 0 0
\(385\) 1.02019 0.0519939
\(386\) 0 0
\(387\) 4.81157 0.244586
\(388\) 0 0
\(389\) 19.2985 0.978470 0.489235 0.872152i \(-0.337276\pi\)
0.489235 + 0.872152i \(0.337276\pi\)
\(390\) 0 0
\(391\) 3.03434 0.153453
\(392\) 0 0
\(393\) 3.49691 0.176396
\(394\) 0 0
\(395\) 16.5577 0.833111
\(396\) 0 0
\(397\) 22.0507 1.10669 0.553346 0.832952i \(-0.313351\pi\)
0.553346 + 0.832952i \(0.313351\pi\)
\(398\) 0 0
\(399\) 11.8719 0.594338
\(400\) 0 0
\(401\) −8.46339 −0.422642 −0.211321 0.977417i \(-0.567777\pi\)
−0.211321 + 0.977417i \(0.567777\pi\)
\(402\) 0 0
\(403\) 0.344044 0.0171380
\(404\) 0 0
\(405\) −1.83485 −0.0911743
\(406\) 0 0
\(407\) −2.58485 −0.128126
\(408\) 0 0
\(409\) 1.84815 0.0913853 0.0456926 0.998956i \(-0.485451\pi\)
0.0456926 + 0.998956i \(0.485451\pi\)
\(410\) 0 0
\(411\) 17.9808 0.886928
\(412\) 0 0
\(413\) −3.60187 −0.177237
\(414\) 0 0
\(415\) 0.297621 0.0146097
\(416\) 0 0
\(417\) 8.22207 0.402637
\(418\) 0 0
\(419\) −2.32702 −0.113683 −0.0568413 0.998383i \(-0.518103\pi\)
−0.0568413 + 0.998383i \(0.518103\pi\)
\(420\) 0 0
\(421\) 14.8437 0.723435 0.361718 0.932288i \(-0.382190\pi\)
0.361718 + 0.932288i \(0.382190\pi\)
\(422\) 0 0
\(423\) −4.18978 −0.203714
\(424\) 0 0
\(425\) −4.95609 −0.240406
\(426\) 0 0
\(427\) −1.88976 −0.0914519
\(428\) 0 0
\(429\) 0.727774 0.0351373
\(430\) 0 0
\(431\) 6.17218 0.297303 0.148652 0.988890i \(-0.452507\pi\)
0.148652 + 0.988890i \(0.452507\pi\)
\(432\) 0 0
\(433\) −18.8245 −0.904649 −0.452325 0.891853i \(-0.649405\pi\)
−0.452325 + 0.891853i \(0.649405\pi\)
\(434\) 0 0
\(435\) 1.83485 0.0879742
\(436\) 0 0
\(437\) −6.93143 −0.331575
\(438\) 0 0
\(439\) −24.0226 −1.14654 −0.573268 0.819368i \(-0.694325\pi\)
−0.573268 + 0.819368i \(0.694325\pi\)
\(440\) 0 0
\(441\) −4.06644 −0.193640
\(442\) 0 0
\(443\) 10.3626 0.492340 0.246170 0.969227i \(-0.420828\pi\)
0.246170 + 0.969227i \(0.420828\pi\)
\(444\) 0 0
\(445\) −17.6067 −0.834637
\(446\) 0 0
\(447\) 1.98612 0.0939402
\(448\) 0 0
\(449\) 3.43394 0.162058 0.0810288 0.996712i \(-0.474179\pi\)
0.0810288 + 0.996712i \(0.474179\pi\)
\(450\) 0 0
\(451\) −1.76254 −0.0829946
\(452\) 0 0
\(453\) 8.62021 0.405013
\(454\) 0 0
\(455\) −7.04545 −0.330296
\(456\) 0 0
\(457\) 17.5875 0.822711 0.411355 0.911475i \(-0.365056\pi\)
0.411355 + 0.911475i \(0.365056\pi\)
\(458\) 0 0
\(459\) −3.03434 −0.141631
\(460\) 0 0
\(461\) 22.4078 1.04364 0.521818 0.853057i \(-0.325254\pi\)
0.521818 + 0.853057i \(0.325254\pi\)
\(462\) 0 0
\(463\) 27.2511 1.26647 0.633234 0.773960i \(-0.281727\pi\)
0.633234 + 0.773960i \(0.281727\pi\)
\(464\) 0 0
\(465\) 0.281580 0.0130580
\(466\) 0 0
\(467\) −11.7419 −0.543351 −0.271675 0.962389i \(-0.587578\pi\)
−0.271675 + 0.962389i \(0.587578\pi\)
\(468\) 0 0
\(469\) −6.73055 −0.310788
\(470\) 0 0
\(471\) 3.47315 0.160034
\(472\) 0 0
\(473\) −1.56197 −0.0718194
\(474\) 0 0
\(475\) 11.3213 0.519459
\(476\) 0 0
\(477\) 11.5727 0.529878
\(478\) 0 0
\(479\) −35.2411 −1.61020 −0.805102 0.593136i \(-0.797890\pi\)
−0.805102 + 0.593136i \(0.797890\pi\)
\(480\) 0 0
\(481\) 17.8509 0.813932
\(482\) 0 0
\(483\) −1.71276 −0.0779334
\(484\) 0 0
\(485\) 2.18517 0.0992234
\(486\) 0 0
\(487\) −30.4349 −1.37914 −0.689569 0.724220i \(-0.742200\pi\)
−0.689569 + 0.724220i \(0.742200\pi\)
\(488\) 0 0
\(489\) −14.9038 −0.673971
\(490\) 0 0
\(491\) 11.8872 0.536461 0.268231 0.963355i \(-0.413561\pi\)
0.268231 + 0.963355i \(0.413561\pi\)
\(492\) 0 0
\(493\) 3.03434 0.136660
\(494\) 0 0
\(495\) 0.595642 0.0267721
\(496\) 0 0
\(497\) −19.2569 −0.863792
\(498\) 0 0
\(499\) 10.5750 0.473401 0.236701 0.971583i \(-0.423934\pi\)
0.236701 + 0.971583i \(0.423934\pi\)
\(500\) 0 0
\(501\) −5.58054 −0.249320
\(502\) 0 0
\(503\) 19.6881 0.877851 0.438926 0.898523i \(-0.355359\pi\)
0.438926 + 0.898523i \(0.355359\pi\)
\(504\) 0 0
\(505\) −2.78822 −0.124074
\(506\) 0 0
\(507\) 7.97400 0.354138
\(508\) 0 0
\(509\) 31.6510 1.40291 0.701453 0.712716i \(-0.252535\pi\)
0.701453 + 0.712716i \(0.252535\pi\)
\(510\) 0 0
\(511\) −10.3248 −0.456743
\(512\) 0 0
\(513\) 6.93143 0.306030
\(514\) 0 0
\(515\) −36.3178 −1.60035
\(516\) 0 0
\(517\) 1.36012 0.0598180
\(518\) 0 0
\(519\) 17.3399 0.761137
\(520\) 0 0
\(521\) −18.5910 −0.814488 −0.407244 0.913319i \(-0.633510\pi\)
−0.407244 + 0.913319i \(0.633510\pi\)
\(522\) 0 0
\(523\) −28.7656 −1.25783 −0.628916 0.777473i \(-0.716501\pi\)
−0.628916 + 0.777473i \(0.716501\pi\)
\(524\) 0 0
\(525\) 2.79751 0.122093
\(526\) 0 0
\(527\) 0.465657 0.0202844
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.10296 −0.0912608
\(532\) 0 0
\(533\) 12.1721 0.527230
\(534\) 0 0
\(535\) −32.2851 −1.39581
\(536\) 0 0
\(537\) 8.86053 0.382360
\(538\) 0 0
\(539\) 1.32008 0.0568599
\(540\) 0 0
\(541\) −18.9433 −0.814436 −0.407218 0.913331i \(-0.633501\pi\)
−0.407218 + 0.913331i \(0.633501\pi\)
\(542\) 0 0
\(543\) 24.0277 1.03113
\(544\) 0 0
\(545\) −28.6995 −1.22935
\(546\) 0 0
\(547\) 43.4893 1.85947 0.929733 0.368233i \(-0.120037\pi\)
0.929733 + 0.368233i \(0.120037\pi\)
\(548\) 0 0
\(549\) −1.10334 −0.0470894
\(550\) 0 0
\(551\) −6.93143 −0.295289
\(552\) 0 0
\(553\) −15.4560 −0.657258
\(554\) 0 0
\(555\) 14.6100 0.620159
\(556\) 0 0
\(557\) −2.35638 −0.0998432 −0.0499216 0.998753i \(-0.515897\pi\)
−0.0499216 + 0.998753i \(0.515897\pi\)
\(558\) 0 0
\(559\) 10.7869 0.456239
\(560\) 0 0
\(561\) 0.985030 0.0415880
\(562\) 0 0
\(563\) 7.13063 0.300520 0.150260 0.988647i \(-0.451989\pi\)
0.150260 + 0.988647i \(0.451989\pi\)
\(564\) 0 0
\(565\) 15.5822 0.655547
\(566\) 0 0
\(567\) 1.71276 0.0719293
\(568\) 0 0
\(569\) −40.3302 −1.69073 −0.845365 0.534189i \(-0.820617\pi\)
−0.845365 + 0.534189i \(0.820617\pi\)
\(570\) 0 0
\(571\) 24.3196 1.01774 0.508872 0.860842i \(-0.330063\pi\)
0.508872 + 0.860842i \(0.330063\pi\)
\(572\) 0 0
\(573\) −20.0670 −0.838311
\(574\) 0 0
\(575\) −1.63333 −0.0681147
\(576\) 0 0
\(577\) 37.7001 1.56948 0.784738 0.619827i \(-0.212797\pi\)
0.784738 + 0.619827i \(0.212797\pi\)
\(578\) 0 0
\(579\) −14.5004 −0.602616
\(580\) 0 0
\(581\) −0.277818 −0.0115258
\(582\) 0 0
\(583\) −3.75682 −0.155592
\(584\) 0 0
\(585\) −4.11350 −0.170072
\(586\) 0 0
\(587\) 20.4809 0.845336 0.422668 0.906285i \(-0.361094\pi\)
0.422668 + 0.906285i \(0.361094\pi\)
\(588\) 0 0
\(589\) −1.06372 −0.0438296
\(590\) 0 0
\(591\) −25.3961 −1.04466
\(592\) 0 0
\(593\) −1.99892 −0.0820859 −0.0410430 0.999157i \(-0.513068\pi\)
−0.0410430 + 0.999157i \(0.513068\pi\)
\(594\) 0 0
\(595\) −9.53589 −0.390933
\(596\) 0 0
\(597\) 22.9534 0.939422
\(598\) 0 0
\(599\) 12.8750 0.526057 0.263029 0.964788i \(-0.415279\pi\)
0.263029 + 0.964788i \(0.415279\pi\)
\(600\) 0 0
\(601\) 15.6499 0.638373 0.319187 0.947692i \(-0.396590\pi\)
0.319187 + 0.947692i \(0.396590\pi\)
\(602\) 0 0
\(603\) −3.92964 −0.160027
\(604\) 0 0
\(605\) 19.9900 0.812708
\(606\) 0 0
\(607\) 45.3361 1.84013 0.920067 0.391762i \(-0.128134\pi\)
0.920067 + 0.391762i \(0.128134\pi\)
\(608\) 0 0
\(609\) −1.71276 −0.0694046
\(610\) 0 0
\(611\) −9.39297 −0.379999
\(612\) 0 0
\(613\) 8.87478 0.358449 0.179224 0.983808i \(-0.442641\pi\)
0.179224 + 0.983808i \(0.442641\pi\)
\(614\) 0 0
\(615\) 9.96214 0.401712
\(616\) 0 0
\(617\) −28.6912 −1.15506 −0.577532 0.816368i \(-0.695984\pi\)
−0.577532 + 0.816368i \(0.695984\pi\)
\(618\) 0 0
\(619\) 5.93342 0.238484 0.119242 0.992865i \(-0.461954\pi\)
0.119242 + 0.992865i \(0.461954\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 16.4352 0.658462
\(624\) 0 0
\(625\) −14.1655 −0.566622
\(626\) 0 0
\(627\) −2.25013 −0.0898617
\(628\) 0 0
\(629\) 24.1609 0.963359
\(630\) 0 0
\(631\) −0.435051 −0.0173191 −0.00865955 0.999963i \(-0.502756\pi\)
−0.00865955 + 0.999963i \(0.502756\pi\)
\(632\) 0 0
\(633\) 14.9031 0.592344
\(634\) 0 0
\(635\) −5.10658 −0.202648
\(636\) 0 0
\(637\) −9.11646 −0.361207
\(638\) 0 0
\(639\) −11.2432 −0.444774
\(640\) 0 0
\(641\) 3.55955 0.140594 0.0702970 0.997526i \(-0.477605\pi\)
0.0702970 + 0.997526i \(0.477605\pi\)
\(642\) 0 0
\(643\) −22.6722 −0.894105 −0.447052 0.894508i \(-0.647526\pi\)
−0.447052 + 0.894508i \(0.647526\pi\)
\(644\) 0 0
\(645\) 8.82850 0.347622
\(646\) 0 0
\(647\) −32.2678 −1.26858 −0.634289 0.773096i \(-0.718707\pi\)
−0.634289 + 0.773096i \(0.718707\pi\)
\(648\) 0 0
\(649\) 0.682679 0.0267975
\(650\) 0 0
\(651\) −0.262845 −0.0103017
\(652\) 0 0
\(653\) 17.4321 0.682172 0.341086 0.940032i \(-0.389205\pi\)
0.341086 + 0.940032i \(0.389205\pi\)
\(654\) 0 0
\(655\) 6.41630 0.250705
\(656\) 0 0
\(657\) −6.02817 −0.235181
\(658\) 0 0
\(659\) −44.7795 −1.74436 −0.872181 0.489183i \(-0.837295\pi\)
−0.872181 + 0.489183i \(0.837295\pi\)
\(660\) 0 0
\(661\) 11.7323 0.456333 0.228166 0.973622i \(-0.426727\pi\)
0.228166 + 0.973622i \(0.426727\pi\)
\(662\) 0 0
\(663\) −6.80260 −0.264191
\(664\) 0 0
\(665\) 21.7831 0.844713
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −3.21974 −0.124482
\(670\) 0 0
\(671\) 0.358175 0.0138272
\(672\) 0 0
\(673\) 28.5123 1.09907 0.549534 0.835471i \(-0.314805\pi\)
0.549534 + 0.835471i \(0.314805\pi\)
\(674\) 0 0
\(675\) 1.63333 0.0628671
\(676\) 0 0
\(677\) −45.2187 −1.73790 −0.868948 0.494904i \(-0.835203\pi\)
−0.868948 + 0.494904i \(0.835203\pi\)
\(678\) 0 0
\(679\) −2.03977 −0.0782793
\(680\) 0 0
\(681\) −22.1194 −0.847616
\(682\) 0 0
\(683\) 26.4359 1.01154 0.505771 0.862668i \(-0.331208\pi\)
0.505771 + 0.862668i \(0.331208\pi\)
\(684\) 0 0
\(685\) 32.9920 1.26056
\(686\) 0 0
\(687\) −16.6341 −0.634629
\(688\) 0 0
\(689\) 25.9446 0.988409
\(690\) 0 0
\(691\) −15.8905 −0.604503 −0.302252 0.953228i \(-0.597738\pi\)
−0.302252 + 0.953228i \(0.597738\pi\)
\(692\) 0 0
\(693\) −0.556010 −0.0211211
\(694\) 0 0
\(695\) 15.0862 0.572254
\(696\) 0 0
\(697\) 16.4747 0.624022
\(698\) 0 0
\(699\) −9.19416 −0.347755
\(700\) 0 0
\(701\) 47.6090 1.79817 0.899084 0.437776i \(-0.144234\pi\)
0.899084 + 0.437776i \(0.144234\pi\)
\(702\) 0 0
\(703\) −55.1915 −2.08159
\(704\) 0 0
\(705\) −7.68761 −0.289532
\(706\) 0 0
\(707\) 2.60270 0.0978845
\(708\) 0 0
\(709\) −4.09427 −0.153764 −0.0768818 0.997040i \(-0.524496\pi\)
−0.0768818 + 0.997040i \(0.524496\pi\)
\(710\) 0 0
\(711\) −9.02404 −0.338428
\(712\) 0 0
\(713\) 0.153463 0.00574722
\(714\) 0 0
\(715\) 1.33536 0.0499395
\(716\) 0 0
\(717\) −14.3208 −0.534819
\(718\) 0 0
\(719\) −10.7902 −0.402408 −0.201204 0.979549i \(-0.564485\pi\)
−0.201204 + 0.979549i \(0.564485\pi\)
\(720\) 0 0
\(721\) 33.9013 1.26255
\(722\) 0 0
\(723\) −19.6949 −0.732460
\(724\) 0 0
\(725\) −1.63333 −0.0606605
\(726\) 0 0
\(727\) 46.1386 1.71119 0.855593 0.517648i \(-0.173192\pi\)
0.855593 + 0.517648i \(0.173192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.5999 0.539998
\(732\) 0 0
\(733\) −29.2622 −1.08082 −0.540412 0.841401i \(-0.681732\pi\)
−0.540412 + 0.841401i \(0.681732\pi\)
\(734\) 0 0
\(735\) −7.46131 −0.275214
\(736\) 0 0
\(737\) 1.27567 0.0469899
\(738\) 0 0
\(739\) 36.8527 1.35565 0.677824 0.735224i \(-0.262923\pi\)
0.677824 + 0.735224i \(0.262923\pi\)
\(740\) 0 0
\(741\) 15.5394 0.570854
\(742\) 0 0
\(743\) 35.5920 1.30574 0.652872 0.757469i \(-0.273564\pi\)
0.652872 + 0.757469i \(0.273564\pi\)
\(744\) 0 0
\(745\) 3.64423 0.133514
\(746\) 0 0
\(747\) −0.162205 −0.00593477
\(748\) 0 0
\(749\) 30.1369 1.10118
\(750\) 0 0
\(751\) −41.8311 −1.52644 −0.763220 0.646139i \(-0.776383\pi\)
−0.763220 + 0.646139i \(0.776383\pi\)
\(752\) 0 0
\(753\) −19.1162 −0.696632
\(754\) 0 0
\(755\) 15.8168 0.575632
\(756\) 0 0
\(757\) 28.0872 1.02085 0.510424 0.859923i \(-0.329488\pi\)
0.510424 + 0.859923i \(0.329488\pi\)
\(758\) 0 0
\(759\) 0.324628 0.0117832
\(760\) 0 0
\(761\) 24.7936 0.898768 0.449384 0.893339i \(-0.351643\pi\)
0.449384 + 0.893339i \(0.351643\pi\)
\(762\) 0 0
\(763\) 26.7899 0.969861
\(764\) 0 0
\(765\) −5.56755 −0.201295
\(766\) 0 0
\(767\) −4.71458 −0.170233
\(768\) 0 0
\(769\) 11.8990 0.429087 0.214544 0.976714i \(-0.431174\pi\)
0.214544 + 0.976714i \(0.431174\pi\)
\(770\) 0 0
\(771\) 10.5799 0.381025
\(772\) 0 0
\(773\) −32.6895 −1.17576 −0.587881 0.808948i \(-0.700038\pi\)
−0.587881 + 0.808948i \(0.700038\pi\)
\(774\) 0 0
\(775\) −0.250656 −0.00900382
\(776\) 0 0
\(777\) −13.6379 −0.489256
\(778\) 0 0
\(779\) −37.6336 −1.34836
\(780\) 0 0
\(781\) 3.64985 0.130602
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 6.37270 0.227452
\(786\) 0 0
\(787\) −52.1372 −1.85849 −0.929246 0.369463i \(-0.879542\pi\)
−0.929246 + 0.369463i \(0.879542\pi\)
\(788\) 0 0
\(789\) −3.00001 −0.106803
\(790\) 0 0
\(791\) −14.5454 −0.517175
\(792\) 0 0
\(793\) −2.47355 −0.0878384
\(794\) 0 0
\(795\) 21.2342 0.753098
\(796\) 0 0
\(797\) 23.5361 0.833691 0.416846 0.908977i \(-0.363136\pi\)
0.416846 + 0.908977i \(0.363136\pi\)
\(798\) 0 0
\(799\) −12.7132 −0.449761
\(800\) 0 0
\(801\) 9.59572 0.339048
\(802\) 0 0
\(803\) 1.95691 0.0690579
\(804\) 0 0
\(805\) −3.14266 −0.110764
\(806\) 0 0
\(807\) −10.4320 −0.367222
\(808\) 0 0
\(809\) −26.4692 −0.930608 −0.465304 0.885151i \(-0.654055\pi\)
−0.465304 + 0.885151i \(0.654055\pi\)
\(810\) 0 0
\(811\) 18.2602 0.641202 0.320601 0.947214i \(-0.396115\pi\)
0.320601 + 0.947214i \(0.396115\pi\)
\(812\) 0 0
\(813\) 16.1350 0.565877
\(814\) 0 0
\(815\) −27.3461 −0.957893
\(816\) 0 0
\(817\) −33.3511 −1.16681
\(818\) 0 0
\(819\) 3.83980 0.134173
\(820\) 0 0
\(821\) −33.2407 −1.16011 −0.580055 0.814577i \(-0.696969\pi\)
−0.580055 + 0.814577i \(0.696969\pi\)
\(822\) 0 0
\(823\) 29.2349 1.01906 0.509531 0.860452i \(-0.329819\pi\)
0.509531 + 0.860452i \(0.329819\pi\)
\(824\) 0 0
\(825\) −0.530225 −0.0184601
\(826\) 0 0
\(827\) 1.88879 0.0656798 0.0328399 0.999461i \(-0.489545\pi\)
0.0328399 + 0.999461i \(0.489545\pi\)
\(828\) 0 0
\(829\) −27.2251 −0.945568 −0.472784 0.881178i \(-0.656751\pi\)
−0.472784 + 0.881178i \(0.656751\pi\)
\(830\) 0 0
\(831\) −21.6156 −0.749837
\(832\) 0 0
\(833\) −12.3390 −0.427520
\(834\) 0 0
\(835\) −10.2394 −0.354350
\(836\) 0 0
\(837\) −0.153463 −0.00530444
\(838\) 0 0
\(839\) −52.2304 −1.80320 −0.901598 0.432576i \(-0.857605\pi\)
−0.901598 + 0.432576i \(0.857605\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −26.5768 −0.915355
\(844\) 0 0
\(845\) 14.6311 0.503324
\(846\) 0 0
\(847\) −18.6599 −0.641161
\(848\) 0 0
\(849\) −2.73513 −0.0938694
\(850\) 0 0
\(851\) 7.96250 0.272951
\(852\) 0 0
\(853\) −3.03850 −0.104036 −0.0520182 0.998646i \(-0.516565\pi\)
−0.0520182 + 0.998646i \(0.516565\pi\)
\(854\) 0 0
\(855\) 12.7181 0.434950
\(856\) 0 0
\(857\) −17.0934 −0.583899 −0.291949 0.956434i \(-0.594304\pi\)
−0.291949 + 0.956434i \(0.594304\pi\)
\(858\) 0 0
\(859\) 8.49832 0.289959 0.144980 0.989435i \(-0.453688\pi\)
0.144980 + 0.989435i \(0.453688\pi\)
\(860\) 0 0
\(861\) −9.29929 −0.316919
\(862\) 0 0
\(863\) −4.51995 −0.153861 −0.0769304 0.997036i \(-0.524512\pi\)
−0.0769304 + 0.997036i \(0.524512\pi\)
\(864\) 0 0
\(865\) 31.8161 1.08178
\(866\) 0 0
\(867\) 7.79280 0.264657
\(868\) 0 0
\(869\) 2.92945 0.0993749
\(870\) 0 0
\(871\) −8.80977 −0.298508
\(872\) 0 0
\(873\) −1.19093 −0.0403067
\(874\) 0 0
\(875\) 20.8463 0.704734
\(876\) 0 0
\(877\) −3.65379 −0.123380 −0.0616899 0.998095i \(-0.519649\pi\)
−0.0616899 + 0.998095i \(0.519649\pi\)
\(878\) 0 0
\(879\) −8.83703 −0.298066
\(880\) 0 0
\(881\) 10.5490 0.355405 0.177703 0.984084i \(-0.443133\pi\)
0.177703 + 0.984084i \(0.443133\pi\)
\(882\) 0 0
\(883\) 28.5933 0.962240 0.481120 0.876655i \(-0.340230\pi\)
0.481120 + 0.876655i \(0.340230\pi\)
\(884\) 0 0
\(885\) −3.85861 −0.129706
\(886\) 0 0
\(887\) −11.9607 −0.401603 −0.200801 0.979632i \(-0.564355\pi\)
−0.200801 + 0.979632i \(0.564355\pi\)
\(888\) 0 0
\(889\) 4.76680 0.159873
\(890\) 0 0
\(891\) −0.324628 −0.0108754
\(892\) 0 0
\(893\) 29.0412 0.971826
\(894\) 0 0
\(895\) 16.2577 0.543436
\(896\) 0 0
\(897\) −2.24187 −0.0748540
\(898\) 0 0
\(899\) 0.153463 0.00511826
\(900\) 0 0
\(901\) 35.1155 1.16987
\(902\) 0 0
\(903\) −8.24108 −0.274246
\(904\) 0 0
\(905\) 44.0871 1.46551
\(906\) 0 0
\(907\) −52.7883 −1.75281 −0.876403 0.481578i \(-0.840064\pi\)
−0.876403 + 0.481578i \(0.840064\pi\)
\(908\) 0 0
\(909\) 1.51959 0.0504016
\(910\) 0 0
\(911\) 38.6152 1.27938 0.639690 0.768633i \(-0.279063\pi\)
0.639690 + 0.768633i \(0.279063\pi\)
\(912\) 0 0
\(913\) 0.0526562 0.00174267
\(914\) 0 0
\(915\) −2.02446 −0.0669267
\(916\) 0 0
\(917\) −5.98938 −0.197787
\(918\) 0 0
\(919\) −5.06764 −0.167166 −0.0835829 0.996501i \(-0.526636\pi\)
−0.0835829 + 0.996501i \(0.526636\pi\)
\(920\) 0 0
\(921\) −6.01045 −0.198051
\(922\) 0 0
\(923\) −25.2058 −0.829661
\(924\) 0 0
\(925\) −13.0054 −0.427616
\(926\) 0 0
\(927\) 19.7934 0.650100
\(928\) 0 0
\(929\) 32.9328 1.08049 0.540246 0.841507i \(-0.318331\pi\)
0.540246 + 0.841507i \(0.318331\pi\)
\(930\) 0 0
\(931\) 28.1863 0.923768
\(932\) 0 0
\(933\) −25.6664 −0.840279
\(934\) 0 0
\(935\) 1.80738 0.0591076
\(936\) 0 0
\(937\) 8.94072 0.292081 0.146040 0.989279i \(-0.453347\pi\)
0.146040 + 0.989279i \(0.453347\pi\)
\(938\) 0 0
\(939\) 17.3009 0.564595
\(940\) 0 0
\(941\) −20.7057 −0.674987 −0.337493 0.941328i \(-0.609579\pi\)
−0.337493 + 0.941328i \(0.609579\pi\)
\(942\) 0 0
\(943\) 5.42941 0.176806
\(944\) 0 0
\(945\) 3.14266 0.102231
\(946\) 0 0
\(947\) −18.3482 −0.596237 −0.298119 0.954529i \(-0.596359\pi\)
−0.298119 + 0.954529i \(0.596359\pi\)
\(948\) 0 0
\(949\) −13.5144 −0.438696
\(950\) 0 0
\(951\) −22.3014 −0.723171
\(952\) 0 0
\(953\) 34.3438 1.11250 0.556252 0.831013i \(-0.312239\pi\)
0.556252 + 0.831013i \(0.312239\pi\)
\(954\) 0 0
\(955\) −36.8199 −1.19146
\(956\) 0 0
\(957\) 0.324628 0.0104937
\(958\) 0 0
\(959\) −30.7968 −0.994482
\(960\) 0 0
\(961\) −30.9764 −0.999240
\(962\) 0 0
\(963\) 17.5955 0.567007
\(964\) 0 0
\(965\) −26.6060 −0.856478
\(966\) 0 0
\(967\) −55.1712 −1.77419 −0.887094 0.461589i \(-0.847279\pi\)
−0.887094 + 0.461589i \(0.847279\pi\)
\(968\) 0 0
\(969\) 21.0323 0.675655
\(970\) 0 0
\(971\) 46.7876 1.50149 0.750743 0.660594i \(-0.229696\pi\)
0.750743 + 0.660594i \(0.229696\pi\)
\(972\) 0 0
\(973\) −14.0825 −0.451463
\(974\) 0 0
\(975\) 3.66173 0.117269
\(976\) 0 0
\(977\) 31.2119 0.998558 0.499279 0.866441i \(-0.333598\pi\)
0.499279 + 0.866441i \(0.333598\pi\)
\(978\) 0 0
\(979\) −3.11504 −0.0995570
\(980\) 0 0
\(981\) 15.6414 0.499390
\(982\) 0 0
\(983\) −45.7017 −1.45766 −0.728830 0.684695i \(-0.759935\pi\)
−0.728830 + 0.684695i \(0.759935\pi\)
\(984\) 0 0
\(985\) −46.5979 −1.48473
\(986\) 0 0
\(987\) 7.17610 0.228418
\(988\) 0 0
\(989\) 4.81157 0.152999
\(990\) 0 0
\(991\) 20.4553 0.649783 0.324892 0.945751i \(-0.394672\pi\)
0.324892 + 0.945751i \(0.394672\pi\)
\(992\) 0 0
\(993\) 19.5061 0.619008
\(994\) 0 0
\(995\) 42.1161 1.33517
\(996\) 0 0
\(997\) −14.7908 −0.468428 −0.234214 0.972185i \(-0.575252\pi\)
−0.234214 + 0.972185i \(0.575252\pi\)
\(998\) 0 0
\(999\) −7.96250 −0.251922
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 8004.2.a.i.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.5 16 1.1 even 1 trivial