Properties

Label 7360.2.a.bu.1.1
Level $7360$
Weight $2$
Character 7360.1
Self dual yes
Analytic conductor $58.770$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 7360.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.302776 q^{3} -1.00000 q^{5} -3.30278 q^{7} -2.90833 q^{9} +O(q^{10})\) \(q-0.302776 q^{3} -1.00000 q^{5} -3.30278 q^{7} -2.90833 q^{9} -1.69722 q^{11} -3.30278 q^{13} +0.302776 q^{15} +6.90833 q^{17} +5.90833 q^{19} +1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.78890 q^{27} +2.60555 q^{29} +7.90833 q^{31} +0.513878 q^{33} +3.30278 q^{35} -8.00000 q^{37} +1.00000 q^{39} +0.908327 q^{41} -9.21110 q^{43} +2.90833 q^{45} +2.60555 q^{47} +3.90833 q^{49} -2.09167 q^{51} +11.2111 q^{53} +1.69722 q^{55} -1.78890 q^{57} -3.39445 q^{59} -11.5139 q^{61} +9.60555 q^{63} +3.30278 q^{65} -4.00000 q^{67} -0.302776 q^{69} +16.3028 q^{71} -5.81665 q^{73} -0.302776 q^{75} +5.60555 q^{77} +14.4222 q^{79} +8.18335 q^{81} +11.2111 q^{83} -6.90833 q^{85} -0.788897 q^{87} +10.9083 q^{91} -2.39445 q^{93} -5.90833 q^{95} +6.30278 q^{97} +4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{3} - 2q^{5} - 3q^{7} + 5q^{9} + O(q^{10}) \) \( 2q + 3q^{3} - 2q^{5} - 3q^{7} + 5q^{9} - 7q^{11} - 3q^{13} - 3q^{15} + 3q^{17} + q^{19} + 2q^{21} + 2q^{23} + 2q^{25} + 18q^{27} - 2q^{29} + 5q^{31} - 17q^{33} + 3q^{35} - 16q^{37} + 2q^{39} - 9q^{41} - 4q^{43} - 5q^{45} - 2q^{47} - 3q^{49} - 15q^{51} + 8q^{53} + 7q^{55} - 18q^{57} - 14q^{59} - 5q^{61} + 12q^{63} + 3q^{65} - 8q^{67} + 3q^{69} + 29q^{71} + 10q^{73} + 3q^{75} + 4q^{77} + 38q^{81} + 8q^{83} - 3q^{85} - 16q^{87} + 11q^{91} - 12q^{93} - q^{95} + 9q^{97} - 37q^{99} + O(q^{100}) \)

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
\(3\) −0.302776 −0.174808 −0.0874038 0.996173i \(-0.527857\pi\)
−0.0874038 + 0.996173i \(0.527857\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.30278 −1.24833 −0.624166 0.781292i \(-0.714561\pi\)
−0.624166 + 0.781292i \(0.714561\pi\)
\(8\) 0 0
\(9\) −2.90833 −0.969442
\(10\) 0 0
\(11\) −1.69722 −0.511732 −0.255866 0.966712i \(-0.582361\pi\)
−0.255866 + 0.966712i \(0.582361\pi\)
\(12\) 0 0
\(13\) −3.30278 −0.916025 −0.458013 0.888946i \(-0.651439\pi\)
−0.458013 + 0.888946i \(0.651439\pi\)
\(14\) 0 0
\(15\) 0.302776 0.0781763
\(16\) 0 0
\(17\) 6.90833 1.67552 0.837758 0.546042i \(-0.183866\pi\)
0.837758 + 0.546042i \(0.183866\pi\)
\(18\) 0 0
\(19\) 5.90833 1.35546 0.677732 0.735309i \(-0.262963\pi\)
0.677732 + 0.735309i \(0.262963\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.78890 0.344273
\(28\) 0 0
\(29\) 2.60555 0.483839 0.241919 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) 7.90833 1.42038 0.710189 0.704011i \(-0.248610\pi\)
0.710189 + 0.704011i \(0.248610\pi\)
\(32\) 0 0
\(33\) 0.513878 0.0894547
\(34\) 0 0
\(35\) 3.30278 0.558271
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0.908327 0.141857 0.0709284 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 0 0
\(43\) −9.21110 −1.40468 −0.702340 0.711842i \(-0.747861\pi\)
−0.702340 + 0.711842i \(0.747861\pi\)
\(44\) 0 0
\(45\) 2.90833 0.433548
\(46\) 0 0
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0 0
\(49\) 3.90833 0.558332
\(50\) 0 0
\(51\) −2.09167 −0.292893
\(52\) 0 0
\(53\) 11.2111 1.53996 0.769982 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) 0 0
\(55\) 1.69722 0.228854
\(56\) 0 0
\(57\) −1.78890 −0.236945
\(58\) 0 0
\(59\) −3.39445 −0.441920 −0.220960 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(60\) 0 0
\(61\) −11.5139 −1.47420 −0.737101 0.675783i \(-0.763806\pi\)
−0.737101 + 0.675783i \(0.763806\pi\)
\(62\) 0 0
\(63\) 9.60555 1.21019
\(64\) 0 0
\(65\) 3.30278 0.409659
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −0.302776 −0.0364499
\(70\) 0 0
\(71\) 16.3028 1.93478 0.967392 0.253285i \(-0.0815110\pi\)
0.967392 + 0.253285i \(0.0815110\pi\)
\(72\) 0 0
\(73\) −5.81665 −0.680788 −0.340394 0.940283i \(-0.610560\pi\)
−0.340394 + 0.940283i \(0.610560\pi\)
\(74\) 0 0
\(75\) −0.302776 −0.0349615
\(76\) 0 0
\(77\) 5.60555 0.638812
\(78\) 0 0
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0 0
\(83\) 11.2111 1.23058 0.615289 0.788301i \(-0.289039\pi\)
0.615289 + 0.788301i \(0.289039\pi\)
\(84\) 0 0
\(85\) −6.90833 −0.749313
\(86\) 0 0
\(87\) −0.788897 −0.0845787
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 10.9083 1.14350
\(92\) 0 0
\(93\) −2.39445 −0.248293
\(94\) 0 0
\(95\) −5.90833 −0.606182
\(96\) 0 0
\(97\) 6.30278 0.639950 0.319975 0.947426i \(-0.396325\pi\)
0.319975 + 0.947426i \(0.396325\pi\)
\(98\) 0 0
\(99\) 4.93608 0.496095
\(100\) 0 0
\(101\) −2.60555 −0.259262 −0.129631 0.991562i \(-0.541379\pi\)
−0.129631 + 0.991562i \(0.541379\pi\)
\(102\) 0 0
\(103\) −8.11943 −0.800031 −0.400016 0.916508i \(-0.630995\pi\)
−0.400016 + 0.916508i \(0.630995\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −2.60555 −0.251888 −0.125944 0.992037i \(-0.540196\pi\)
−0.125944 + 0.992037i \(0.540196\pi\)
\(108\) 0 0
\(109\) −1.48612 −0.142345 −0.0711723 0.997464i \(-0.522674\pi\)
−0.0711723 + 0.997464i \(0.522674\pi\)
\(110\) 0 0
\(111\) 2.42221 0.229906
\(112\) 0 0
\(113\) −16.4222 −1.54487 −0.772436 0.635093i \(-0.780962\pi\)
−0.772436 + 0.635093i \(0.780962\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 9.60555 0.888034
\(118\) 0 0
\(119\) −22.8167 −2.09160
\(120\) 0 0
\(121\) −8.11943 −0.738130
\(122\) 0 0
\(123\) −0.275019 −0.0247977
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −9.81665 −0.871087 −0.435544 0.900168i \(-0.643444\pi\)
−0.435544 + 0.900168i \(0.643444\pi\)
\(128\) 0 0
\(129\) 2.78890 0.245549
\(130\) 0 0
\(131\) −11.2111 −0.979519 −0.489759 0.871858i \(-0.662915\pi\)
−0.489759 + 0.871858i \(0.662915\pi\)
\(132\) 0 0
\(133\) −19.5139 −1.69207
\(134\) 0 0
\(135\) −1.78890 −0.153964
\(136\) 0 0
\(137\) 3.90833 0.333911 0.166955 0.985964i \(-0.446606\pi\)
0.166955 + 0.985964i \(0.446606\pi\)
\(138\) 0 0
\(139\) −12.6056 −1.06919 −0.534594 0.845109i \(-0.679536\pi\)
−0.534594 + 0.845109i \(0.679536\pi\)
\(140\) 0 0
\(141\) −0.788897 −0.0664372
\(142\) 0 0
\(143\) 5.60555 0.468760
\(144\) 0 0
\(145\) −2.60555 −0.216379
\(146\) 0 0
\(147\) −1.18335 −0.0976007
\(148\) 0 0
\(149\) −13.3028 −1.08981 −0.544903 0.838499i \(-0.683433\pi\)
−0.544903 + 0.838499i \(0.683433\pi\)
\(150\) 0 0
\(151\) −8.90833 −0.724949 −0.362475 0.931994i \(-0.618068\pi\)
−0.362475 + 0.931994i \(0.618068\pi\)
\(152\) 0 0
\(153\) −20.0917 −1.62432
\(154\) 0 0
\(155\) −7.90833 −0.635212
\(156\) 0 0
\(157\) 18.6056 1.48488 0.742442 0.669910i \(-0.233667\pi\)
0.742442 + 0.669910i \(0.233667\pi\)
\(158\) 0 0
\(159\) −3.39445 −0.269197
\(160\) 0 0
\(161\) −3.30278 −0.260295
\(162\) 0 0
\(163\) 9.30278 0.728650 0.364325 0.931272i \(-0.381300\pi\)
0.364325 + 0.931272i \(0.381300\pi\)
\(164\) 0 0
\(165\) −0.513878 −0.0400054
\(166\) 0 0
\(167\) −6.78890 −0.525341 −0.262670 0.964886i \(-0.584603\pi\)
−0.262670 + 0.964886i \(0.584603\pi\)
\(168\) 0 0
\(169\) −2.09167 −0.160898
\(170\) 0 0
\(171\) −17.1833 −1.31404
\(172\) 0 0
\(173\) −19.6972 −1.49755 −0.748776 0.662823i \(-0.769358\pi\)
−0.748776 + 0.662823i \(0.769358\pi\)
\(174\) 0 0
\(175\) −3.30278 −0.249666
\(176\) 0 0
\(177\) 1.02776 0.0772509
\(178\) 0 0
\(179\) 9.39445 0.702174 0.351087 0.936343i \(-0.385812\pi\)
0.351087 + 0.936343i \(0.385812\pi\)
\(180\) 0 0
\(181\) −17.1194 −1.27248 −0.636239 0.771492i \(-0.719511\pi\)
−0.636239 + 0.771492i \(0.719511\pi\)
\(182\) 0 0
\(183\) 3.48612 0.257702
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −11.7250 −0.857416
\(188\) 0 0
\(189\) −5.90833 −0.429768
\(190\) 0 0
\(191\) 8.60555 0.622676 0.311338 0.950299i \(-0.399223\pi\)
0.311338 + 0.950299i \(0.399223\pi\)
\(192\) 0 0
\(193\) −17.8167 −1.28247 −0.641235 0.767344i \(-0.721578\pi\)
−0.641235 + 0.767344i \(0.721578\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −4.30278 −0.306560 −0.153280 0.988183i \(-0.548984\pi\)
−0.153280 + 0.988183i \(0.548984\pi\)
\(198\) 0 0
\(199\) 20.4222 1.44769 0.723846 0.689962i \(-0.242373\pi\)
0.723846 + 0.689962i \(0.242373\pi\)
\(200\) 0 0
\(201\) 1.21110 0.0854246
\(202\) 0 0
\(203\) −8.60555 −0.603991
\(204\) 0 0
\(205\) −0.908327 −0.0634403
\(206\) 0 0
\(207\) −2.90833 −0.202143
\(208\) 0 0
\(209\) −10.0278 −0.693634
\(210\) 0 0
\(211\) 7.21110 0.496433 0.248216 0.968705i \(-0.420156\pi\)
0.248216 + 0.968705i \(0.420156\pi\)
\(212\) 0 0
\(213\) −4.93608 −0.338215
\(214\) 0 0
\(215\) 9.21110 0.628192
\(216\) 0 0
\(217\) −26.1194 −1.77310
\(218\) 0 0
\(219\) 1.76114 0.119007
\(220\) 0 0
\(221\) −22.8167 −1.53481
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −2.90833 −0.193888
\(226\) 0 0
\(227\) −14.6056 −0.969404 −0.484702 0.874679i \(-0.661072\pi\)
−0.484702 + 0.874679i \(0.661072\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) −1.69722 −0.111669
\(232\) 0 0
\(233\) 25.8167 1.69131 0.845653 0.533734i \(-0.179212\pi\)
0.845653 + 0.533734i \(0.179212\pi\)
\(234\) 0 0
\(235\) −2.60555 −0.169967
\(236\) 0 0
\(237\) −4.36669 −0.283647
\(238\) 0 0
\(239\) −5.21110 −0.337078 −0.168539 0.985695i \(-0.553905\pi\)
−0.168539 + 0.985695i \(0.553905\pi\)
\(240\) 0 0
\(241\) −14.4222 −0.929016 −0.464508 0.885569i \(-0.653769\pi\)
−0.464508 + 0.885569i \(0.653769\pi\)
\(242\) 0 0
\(243\) −7.84441 −0.503219
\(244\) 0 0
\(245\) −3.90833 −0.249694
\(246\) 0 0
\(247\) −19.5139 −1.24164
\(248\) 0 0
\(249\) −3.39445 −0.215114
\(250\) 0 0
\(251\) 12.5139 0.789869 0.394934 0.918709i \(-0.370767\pi\)
0.394934 + 0.918709i \(0.370767\pi\)
\(252\) 0 0
\(253\) −1.69722 −0.106704
\(254\) 0 0
\(255\) 2.09167 0.130986
\(256\) 0 0
\(257\) −1.81665 −0.113320 −0.0566599 0.998394i \(-0.518045\pi\)
−0.0566599 + 0.998394i \(0.518045\pi\)
\(258\) 0 0
\(259\) 26.4222 1.64180
\(260\) 0 0
\(261\) −7.57779 −0.469054
\(262\) 0 0
\(263\) −3.51388 −0.216675 −0.108338 0.994114i \(-0.534553\pi\)
−0.108338 + 0.994114i \(0.534553\pi\)
\(264\) 0 0
\(265\) −11.2111 −0.688693
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.18335 −0.255063 −0.127532 0.991835i \(-0.540705\pi\)
−0.127532 + 0.991835i \(0.540705\pi\)
\(270\) 0 0
\(271\) 2.69722 0.163845 0.0819224 0.996639i \(-0.473894\pi\)
0.0819224 + 0.996639i \(0.473894\pi\)
\(272\) 0 0
\(273\) −3.30278 −0.199893
\(274\) 0 0
\(275\) −1.69722 −0.102346
\(276\) 0 0
\(277\) 27.2111 1.63496 0.817478 0.575959i \(-0.195371\pi\)
0.817478 + 0.575959i \(0.195371\pi\)
\(278\) 0 0
\(279\) −23.0000 −1.37697
\(280\) 0 0
\(281\) −26.6056 −1.58715 −0.793577 0.608470i \(-0.791784\pi\)
−0.793577 + 0.608470i \(0.791784\pi\)
\(282\) 0 0
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 0 0
\(285\) 1.78890 0.105965
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) 30.7250 1.80735
\(290\) 0 0
\(291\) −1.90833 −0.111868
\(292\) 0 0
\(293\) −23.2111 −1.35601 −0.678004 0.735059i \(-0.737155\pi\)
−0.678004 + 0.735059i \(0.737155\pi\)
\(294\) 0 0
\(295\) 3.39445 0.197632
\(296\) 0 0
\(297\) −3.03616 −0.176176
\(298\) 0 0
\(299\) −3.30278 −0.191004
\(300\) 0 0
\(301\) 30.4222 1.75351
\(302\) 0 0
\(303\) 0.788897 0.0453210
\(304\) 0 0
\(305\) 11.5139 0.659283
\(306\) 0 0
\(307\) −11.6972 −0.667596 −0.333798 0.942645i \(-0.608330\pi\)
−0.333798 + 0.942645i \(0.608330\pi\)
\(308\) 0 0
\(309\) 2.45837 0.139852
\(310\) 0 0
\(311\) −22.4222 −1.27145 −0.635723 0.771917i \(-0.719298\pi\)
−0.635723 + 0.771917i \(0.719298\pi\)
\(312\) 0 0
\(313\) 19.7250 1.11492 0.557461 0.830203i \(-0.311776\pi\)
0.557461 + 0.830203i \(0.311776\pi\)
\(314\) 0 0
\(315\) −9.60555 −0.541212
\(316\) 0 0
\(317\) −17.7250 −0.995534 −0.497767 0.867311i \(-0.665847\pi\)
−0.497767 + 0.867311i \(0.665847\pi\)
\(318\) 0 0
\(319\) −4.42221 −0.247596
\(320\) 0 0
\(321\) 0.788897 0.0440320
\(322\) 0 0
\(323\) 40.8167 2.27110
\(324\) 0 0
\(325\) −3.30278 −0.183205
\(326\) 0 0
\(327\) 0.449961 0.0248829
\(328\) 0 0
\(329\) −8.60555 −0.474439
\(330\) 0 0
\(331\) 16.6056 0.912724 0.456362 0.889794i \(-0.349152\pi\)
0.456362 + 0.889794i \(0.349152\pi\)
\(332\) 0 0
\(333\) 23.2666 1.27500
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −22.5139 −1.22641 −0.613205 0.789924i \(-0.710120\pi\)
−0.613205 + 0.789924i \(0.710120\pi\)
\(338\) 0 0
\(339\) 4.97224 0.270055
\(340\) 0 0
\(341\) −13.4222 −0.726853
\(342\) 0 0
\(343\) 10.2111 0.551348
\(344\) 0 0
\(345\) 0.302776 0.0163009
\(346\) 0 0
\(347\) 28.5416 1.53220 0.766098 0.642724i \(-0.222196\pi\)
0.766098 + 0.642724i \(0.222196\pi\)
\(348\) 0 0
\(349\) 27.2111 1.45658 0.728288 0.685271i \(-0.240316\pi\)
0.728288 + 0.685271i \(0.240316\pi\)
\(350\) 0 0
\(351\) −5.90833 −0.315363
\(352\) 0 0
\(353\) −10.4222 −0.554718 −0.277359 0.960766i \(-0.589459\pi\)
−0.277359 + 0.960766i \(0.589459\pi\)
\(354\) 0 0
\(355\) −16.3028 −0.865261
\(356\) 0 0
\(357\) 6.90833 0.365627
\(358\) 0 0
\(359\) −11.2111 −0.591699 −0.295850 0.955235i \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) 0 0
\(361\) 15.9083 0.837280
\(362\) 0 0
\(363\) 2.45837 0.129031
\(364\) 0 0
\(365\) 5.81665 0.304458
\(366\) 0 0
\(367\) −14.7889 −0.771974 −0.385987 0.922504i \(-0.626139\pi\)
−0.385987 + 0.922504i \(0.626139\pi\)
\(368\) 0 0
\(369\) −2.64171 −0.137522
\(370\) 0 0
\(371\) −37.0278 −1.92239
\(372\) 0 0
\(373\) −4.60555 −0.238466 −0.119233 0.992866i \(-0.538044\pi\)
−0.119233 + 0.992866i \(0.538044\pi\)
\(374\) 0 0
\(375\) 0.302776 0.0156353
\(376\) 0 0
\(377\) −8.60555 −0.443208
\(378\) 0 0
\(379\) 14.9083 0.765789 0.382895 0.923792i \(-0.374927\pi\)
0.382895 + 0.923792i \(0.374927\pi\)
\(380\) 0 0
\(381\) 2.97224 0.152273
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −5.60555 −0.285685
\(386\) 0 0
\(387\) 26.7889 1.36176
\(388\) 0 0
\(389\) 25.9361 1.31501 0.657506 0.753449i \(-0.271612\pi\)
0.657506 + 0.753449i \(0.271612\pi\)
\(390\) 0 0
\(391\) 6.90833 0.349369
\(392\) 0 0
\(393\) 3.39445 0.171227
\(394\) 0 0
\(395\) −14.4222 −0.725660
\(396\) 0 0
\(397\) −10.7250 −0.538271 −0.269136 0.963102i \(-0.586738\pi\)
−0.269136 + 0.963102i \(0.586738\pi\)
\(398\) 0 0
\(399\) 5.90833 0.295786
\(400\) 0 0
\(401\) −8.60555 −0.429741 −0.214870 0.976643i \(-0.568933\pi\)
−0.214870 + 0.976643i \(0.568933\pi\)
\(402\) 0 0
\(403\) −26.1194 −1.30110
\(404\) 0 0
\(405\) −8.18335 −0.406634
\(406\) 0 0
\(407\) 13.5778 0.673026
\(408\) 0 0
\(409\) −25.9083 −1.28108 −0.640542 0.767923i \(-0.721290\pi\)
−0.640542 + 0.767923i \(0.721290\pi\)
\(410\) 0 0
\(411\) −1.18335 −0.0583702
\(412\) 0 0
\(413\) 11.2111 0.551662
\(414\) 0 0
\(415\) −11.2111 −0.550331
\(416\) 0 0
\(417\) 3.81665 0.186902
\(418\) 0 0
\(419\) 3.63331 0.177499 0.0887493 0.996054i \(-0.471713\pi\)
0.0887493 + 0.996054i \(0.471713\pi\)
\(420\) 0 0
\(421\) −30.6972 −1.49609 −0.748046 0.663647i \(-0.769008\pi\)
−0.748046 + 0.663647i \(0.769008\pi\)
\(422\) 0 0
\(423\) −7.57779 −0.368445
\(424\) 0 0
\(425\) 6.90833 0.335103
\(426\) 0 0
\(427\) 38.0278 1.84029
\(428\) 0 0
\(429\) −1.69722 −0.0819428
\(430\) 0 0
\(431\) 30.2389 1.45655 0.728277 0.685283i \(-0.240321\pi\)
0.728277 + 0.685283i \(0.240321\pi\)
\(432\) 0 0
\(433\) −24.0917 −1.15777 −0.578886 0.815409i \(-0.696512\pi\)
−0.578886 + 0.815409i \(0.696512\pi\)
\(434\) 0 0
\(435\) 0.788897 0.0378247
\(436\) 0 0
\(437\) 5.90833 0.282634
\(438\) 0 0
\(439\) 14.6972 0.701460 0.350730 0.936477i \(-0.385933\pi\)
0.350730 + 0.936477i \(0.385933\pi\)
\(440\) 0 0
\(441\) −11.3667 −0.541271
\(442\) 0 0
\(443\) 17.4861 0.830791 0.415395 0.909641i \(-0.363643\pi\)
0.415395 + 0.909641i \(0.363643\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.02776 0.190506
\(448\) 0 0
\(449\) −2.09167 −0.0987122 −0.0493561 0.998781i \(-0.515717\pi\)
−0.0493561 + 0.998781i \(0.515717\pi\)
\(450\) 0 0
\(451\) −1.54163 −0.0725927
\(452\) 0 0
\(453\) 2.69722 0.126727
\(454\) 0 0
\(455\) −10.9083 −0.511390
\(456\) 0 0
\(457\) −32.4222 −1.51665 −0.758323 0.651879i \(-0.773981\pi\)
−0.758323 + 0.651879i \(0.773981\pi\)
\(458\) 0 0
\(459\) 12.3583 0.576836
\(460\) 0 0
\(461\) −10.1833 −0.474286 −0.237143 0.971475i \(-0.576211\pi\)
−0.237143 + 0.971475i \(0.576211\pi\)
\(462\) 0 0
\(463\) −17.6333 −0.819489 −0.409745 0.912200i \(-0.634382\pi\)
−0.409745 + 0.912200i \(0.634382\pi\)
\(464\) 0 0
\(465\) 2.39445 0.111040
\(466\) 0 0
\(467\) −1.81665 −0.0840647 −0.0420324 0.999116i \(-0.513383\pi\)
−0.0420324 + 0.999116i \(0.513383\pi\)
\(468\) 0 0
\(469\) 13.2111 0.610032
\(470\) 0 0
\(471\) −5.63331 −0.259569
\(472\) 0 0
\(473\) 15.6333 0.718820
\(474\) 0 0
\(475\) 5.90833 0.271093
\(476\) 0 0
\(477\) −32.6056 −1.49291
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 26.4222 1.20475
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −6.30278 −0.286194
\(486\) 0 0
\(487\) −9.81665 −0.444835 −0.222418 0.974952i \(-0.571395\pi\)
−0.222418 + 0.974952i \(0.571395\pi\)
\(488\) 0 0
\(489\) −2.81665 −0.127373
\(490\) 0 0
\(491\) −4.18335 −0.188792 −0.0943959 0.995535i \(-0.530092\pi\)
−0.0943959 + 0.995535i \(0.530092\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 0 0
\(495\) −4.93608 −0.221860
\(496\) 0 0
\(497\) −53.8444 −2.41525
\(498\) 0 0
\(499\) −31.6333 −1.41610 −0.708051 0.706162i \(-0.750425\pi\)
−0.708051 + 0.706162i \(0.750425\pi\)
\(500\) 0 0
\(501\) 2.05551 0.0918335
\(502\) 0 0
\(503\) −29.7250 −1.32537 −0.662686 0.748898i \(-0.730583\pi\)
−0.662686 + 0.748898i \(0.730583\pi\)
\(504\) 0 0
\(505\) 2.60555 0.115946
\(506\) 0 0
\(507\) 0.633308 0.0281262
\(508\) 0 0
\(509\) 35.4500 1.57129 0.785646 0.618676i \(-0.212331\pi\)
0.785646 + 0.618676i \(0.212331\pi\)
\(510\) 0 0
\(511\) 19.2111 0.849849
\(512\) 0 0
\(513\) 10.5694 0.466650
\(514\) 0 0
\(515\) 8.11943 0.357785
\(516\) 0 0
\(517\) −4.42221 −0.194488
\(518\) 0 0
\(519\) 5.96384 0.261784
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −20.4222 −0.893001 −0.446500 0.894783i \(-0.647330\pi\)
−0.446500 + 0.894783i \(0.647330\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 54.6333 2.37986
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 9.87217 0.428416
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 0 0
\(535\) 2.60555 0.112648
\(536\) 0 0
\(537\) −2.84441 −0.122745
\(538\) 0 0
\(539\) −6.63331 −0.285717
\(540\) 0 0
\(541\) −28.8444 −1.24012 −0.620059 0.784555i \(-0.712891\pi\)
−0.620059 + 0.784555i \(0.712891\pi\)
\(542\) 0 0
\(543\) 5.18335 0.222439
\(544\) 0 0
\(545\) 1.48612 0.0636585
\(546\) 0 0
\(547\) −10.5139 −0.449541 −0.224770 0.974412i \(-0.572163\pi\)
−0.224770 + 0.974412i \(0.572163\pi\)
\(548\) 0 0
\(549\) 33.4861 1.42915
\(550\) 0 0
\(551\) 15.3944 0.655826
\(552\) 0 0
\(553\) −47.6333 −2.02557
\(554\) 0 0
\(555\) −2.42221 −0.102817
\(556\) 0 0
\(557\) −22.4222 −0.950059 −0.475030 0.879970i \(-0.657563\pi\)
−0.475030 + 0.879970i \(0.657563\pi\)
\(558\) 0 0
\(559\) 30.4222 1.28672
\(560\) 0 0
\(561\) 3.55004 0.149883
\(562\) 0 0
\(563\) −3.63331 −0.153126 −0.0765628 0.997065i \(-0.524395\pi\)
−0.0765628 + 0.997065i \(0.524395\pi\)
\(564\) 0 0
\(565\) 16.4222 0.690887
\(566\) 0 0
\(567\) −27.0278 −1.13506
\(568\) 0 0
\(569\) 28.4222 1.19152 0.595760 0.803162i \(-0.296851\pi\)
0.595760 + 0.803162i \(0.296851\pi\)
\(570\) 0 0
\(571\) −16.1194 −0.674577 −0.337289 0.941401i \(-0.609510\pi\)
−0.337289 + 0.941401i \(0.609510\pi\)
\(572\) 0 0
\(573\) −2.60555 −0.108848
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 5.39445 0.224186
\(580\) 0 0
\(581\) −37.0278 −1.53617
\(582\) 0 0
\(583\) −19.0278 −0.788049
\(584\) 0 0
\(585\) −9.60555 −0.397141
\(586\) 0 0
\(587\) 16.5416 0.682746 0.341373 0.939928i \(-0.389108\pi\)
0.341373 + 0.939928i \(0.389108\pi\)
\(588\) 0 0
\(589\) 46.7250 1.92527
\(590\) 0 0
\(591\) 1.30278 0.0535890
\(592\) 0 0
\(593\) −1.81665 −0.0746010 −0.0373005 0.999304i \(-0.511876\pi\)
−0.0373005 + 0.999304i \(0.511876\pi\)
\(594\) 0 0
\(595\) 22.8167 0.935392
\(596\) 0 0
\(597\) −6.18335 −0.253068
\(598\) 0 0
\(599\) 35.3305 1.44357 0.721783 0.692119i \(-0.243323\pi\)
0.721783 + 0.692119i \(0.243323\pi\)
\(600\) 0 0
\(601\) 42.9361 1.75140 0.875700 0.482856i \(-0.160401\pi\)
0.875700 + 0.482856i \(0.160401\pi\)
\(602\) 0 0
\(603\) 11.6333 0.473745
\(604\) 0 0
\(605\) 8.11943 0.330102
\(606\) 0 0
\(607\) −46.0555 −1.86934 −0.934668 0.355522i \(-0.884303\pi\)
−0.934668 + 0.355522i \(0.884303\pi\)
\(608\) 0 0
\(609\) 2.60555 0.105582
\(610\) 0 0
\(611\) −8.60555 −0.348143
\(612\) 0 0
\(613\) −3.57779 −0.144506 −0.0722529 0.997386i \(-0.523019\pi\)
−0.0722529 + 0.997386i \(0.523019\pi\)
\(614\) 0 0
\(615\) 0.275019 0.0110898
\(616\) 0 0
\(617\) 18.9083 0.761221 0.380610 0.924736i \(-0.375714\pi\)
0.380610 + 0.924736i \(0.375714\pi\)
\(618\) 0 0
\(619\) −12.3305 −0.495606 −0.247803 0.968810i \(-0.579709\pi\)
−0.247803 + 0.968810i \(0.579709\pi\)
\(620\) 0 0
\(621\) 1.78890 0.0717860
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.03616 0.121253
\(628\) 0 0
\(629\) −55.2666 −2.20362
\(630\) 0 0
\(631\) −23.3944 −0.931318 −0.465659 0.884964i \(-0.654183\pi\)
−0.465659 + 0.884964i \(0.654183\pi\)
\(632\) 0 0
\(633\) −2.18335 −0.0867802
\(634\) 0 0
\(635\) 9.81665 0.389562
\(636\) 0 0
\(637\) −12.9083 −0.511447
\(638\) 0 0
\(639\) −47.4138 −1.87566
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 0 0
\(643\) −34.2389 −1.35025 −0.675124 0.737704i \(-0.735910\pi\)
−0.675124 + 0.737704i \(0.735910\pi\)
\(644\) 0 0
\(645\) −2.78890 −0.109813
\(646\) 0 0
\(647\) −26.8444 −1.05536 −0.527681 0.849442i \(-0.676938\pi\)
−0.527681 + 0.849442i \(0.676938\pi\)
\(648\) 0 0
\(649\) 5.76114 0.226145
\(650\) 0 0
\(651\) 7.90833 0.309952
\(652\) 0 0
\(653\) −41.7250 −1.63282 −0.816412 0.577469i \(-0.804040\pi\)
−0.816412 + 0.577469i \(0.804040\pi\)
\(654\) 0 0
\(655\) 11.2111 0.438054
\(656\) 0 0
\(657\) 16.9167 0.659985
\(658\) 0 0
\(659\) 15.6333 0.608987 0.304494 0.952514i \(-0.401513\pi\)
0.304494 + 0.952514i \(0.401513\pi\)
\(660\) 0 0
\(661\) 34.9083 1.35778 0.678888 0.734242i \(-0.262462\pi\)
0.678888 + 0.734242i \(0.262462\pi\)
\(662\) 0 0
\(663\) 6.90833 0.268297
\(664\) 0 0
\(665\) 19.5139 0.756716
\(666\) 0 0
\(667\) 2.60555 0.100887
\(668\) 0 0
\(669\) −1.21110 −0.0468239
\(670\) 0 0
\(671\) 19.5416 0.754396
\(672\) 0 0
\(673\) −37.6333 −1.45066 −0.725329 0.688403i \(-0.758312\pi\)
−0.725329 + 0.688403i \(0.758312\pi\)
\(674\) 0 0
\(675\) 1.78890 0.0688547
\(676\) 0 0
\(677\) −16.4222 −0.631157 −0.315578 0.948900i \(-0.602198\pi\)
−0.315578 + 0.948900i \(0.602198\pi\)
\(678\) 0 0
\(679\) −20.8167 −0.798870
\(680\) 0 0
\(681\) 4.42221 0.169459
\(682\) 0 0
\(683\) −0.275019 −0.0105233 −0.00526166 0.999986i \(-0.501675\pi\)
−0.00526166 + 0.999986i \(0.501675\pi\)
\(684\) 0 0
\(685\) −3.90833 −0.149329
\(686\) 0 0
\(687\) 0.605551 0.0231032
\(688\) 0 0
\(689\) −37.0278 −1.41065
\(690\) 0 0
\(691\) 51.8167 1.97120 0.985599 0.169098i \(-0.0540855\pi\)
0.985599 + 0.169098i \(0.0540855\pi\)
\(692\) 0 0
\(693\) −16.3028 −0.619291
\(694\) 0 0
\(695\) 12.6056 0.478156
\(696\) 0 0
\(697\) 6.27502 0.237683
\(698\) 0 0
\(699\) −7.81665 −0.295653
\(700\) 0 0
\(701\) −32.0917 −1.21209 −0.606043 0.795432i \(-0.707244\pi\)
−0.606043 + 0.795432i \(0.707244\pi\)
\(702\) 0 0
\(703\) −47.2666 −1.78269
\(704\) 0 0
\(705\) 0.788897 0.0297116
\(706\) 0 0
\(707\) 8.60555 0.323645
\(708\) 0 0
\(709\) 15.8806 0.596407 0.298204 0.954502i \(-0.403613\pi\)
0.298204 + 0.954502i \(0.403613\pi\)
\(710\) 0 0
\(711\) −41.9445 −1.57304
\(712\) 0 0
\(713\) 7.90833 0.296169
\(714\) 0 0
\(715\) −5.60555 −0.209636
\(716\) 0 0
\(717\) 1.57779 0.0589238
\(718\) 0 0
\(719\) −10.6972 −0.398939 −0.199470 0.979904i \(-0.563922\pi\)
−0.199470 + 0.979904i \(0.563922\pi\)
\(720\) 0 0
\(721\) 26.8167 0.998704
\(722\) 0 0
\(723\) 4.36669 0.162399
\(724\) 0 0
\(725\) 2.60555 0.0967677
\(726\) 0 0
\(727\) −2.90833 −0.107864 −0.0539319 0.998545i \(-0.517175\pi\)
−0.0539319 + 0.998545i \(0.517175\pi\)
\(728\) 0 0
\(729\) −22.1749 −0.821294
\(730\) 0 0
\(731\) −63.6333 −2.35356
\(732\) 0 0
\(733\) −29.6333 −1.09453 −0.547266 0.836959i \(-0.684331\pi\)
−0.547266 + 0.836959i \(0.684331\pi\)
\(734\) 0 0
\(735\) 1.18335 0.0436484
\(736\) 0 0
\(737\) 6.78890 0.250072
\(738\) 0 0
\(739\) 35.6333 1.31079 0.655396 0.755285i \(-0.272502\pi\)
0.655396 + 0.755285i \(0.272502\pi\)
\(740\) 0 0
\(741\) 5.90833 0.217048
\(742\) 0 0
\(743\) 32.3305 1.18609 0.593046 0.805169i \(-0.297925\pi\)
0.593046 + 0.805169i \(0.297925\pi\)
\(744\) 0 0
\(745\) 13.3028 0.487376
\(746\) 0 0
\(747\) −32.6056 −1.19297
\(748\) 0 0
\(749\) 8.60555 0.314440
\(750\) 0 0
\(751\) −21.8167 −0.796101 −0.398051 0.917364i \(-0.630313\pi\)
−0.398051 + 0.917364i \(0.630313\pi\)
\(752\) 0 0
\(753\) −3.78890 −0.138075
\(754\) 0 0
\(755\) 8.90833 0.324207
\(756\) 0 0
\(757\) −13.2111 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(758\) 0 0
\(759\) 0.513878 0.0186526
\(760\) 0 0
\(761\) −49.5416 −1.79588 −0.897941 0.440115i \(-0.854938\pi\)
−0.897941 + 0.440115i \(0.854938\pi\)
\(762\) 0 0
\(763\) 4.90833 0.177693
\(764\) 0 0
\(765\) 20.0917 0.726416
\(766\) 0 0
\(767\) 11.2111 0.404809
\(768\) 0 0
\(769\) 45.2666 1.63236 0.816178 0.577801i \(-0.196089\pi\)
0.816178 + 0.577801i \(0.196089\pi\)
\(770\) 0 0
\(771\) 0.550039 0.0198092
\(772\) 0 0
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) 0 0
\(775\) 7.90833 0.284075
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) 5.36669 0.192282
\(780\) 0 0
\(781\) −27.6695 −0.990091
\(782\) 0 0
\(783\) 4.66106 0.166573
\(784\) 0 0
\(785\) −18.6056 −0.664061
\(786\) 0 0
\(787\) 37.4500 1.33495 0.667473 0.744634i \(-0.267376\pi\)
0.667473 + 0.744634i \(0.267376\pi\)
\(788\) 0 0
\(789\) 1.06392 0.0378764
\(790\) 0 0
\(791\) 54.2389 1.92851
\(792\) 0 0
\(793\) 38.0278 1.35041
\(794\) 0 0
\(795\) 3.39445 0.120389
\(796\) 0 0
\(797\) −1.81665 −0.0643492 −0.0321746 0.999482i \(-0.510243\pi\)
−0.0321746 + 0.999482i \(0.510243\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.87217 0.348381
\(804\) 0 0
\(805\) 3.30278 0.116408
\(806\) 0 0
\(807\) 1.26662 0.0445870
\(808\) 0 0
\(809\) 50.7250 1.78340 0.891698 0.452631i \(-0.149515\pi\)
0.891698 + 0.452631i \(0.149515\pi\)
\(810\) 0 0
\(811\) −41.0278 −1.44068 −0.720340 0.693621i \(-0.756014\pi\)
−0.720340 + 0.693621i \(0.756014\pi\)
\(812\) 0 0
\(813\) −0.816654 −0.0286413
\(814\) 0 0
\(815\) −9.30278 −0.325862
\(816\) 0 0
\(817\) −54.4222 −1.90399
\(818\) 0 0
\(819\) −31.7250 −1.10856
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 15.2111 0.530226 0.265113 0.964217i \(-0.414591\pi\)
0.265113 + 0.964217i \(0.414591\pi\)
\(824\) 0 0
\(825\) 0.513878 0.0178909
\(826\) 0 0
\(827\) −29.4500 −1.02408 −0.512038 0.858963i \(-0.671109\pi\)
−0.512038 + 0.858963i \(0.671109\pi\)
\(828\) 0 0
\(829\) −31.2111 −1.08401 −0.542003 0.840376i \(-0.682334\pi\)
−0.542003 + 0.840376i \(0.682334\pi\)
\(830\) 0 0
\(831\) −8.23886 −0.285803
\(832\) 0 0
\(833\) 27.0000 0.935495
\(834\) 0 0
\(835\) 6.78890 0.234939
\(836\) 0 0
\(837\) 14.1472 0.488998
\(838\) 0 0
\(839\) 43.8167 1.51272 0.756359 0.654156i \(-0.226976\pi\)
0.756359 + 0.654156i \(0.226976\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) 0 0
\(843\) 8.05551 0.277447
\(844\) 0 0
\(845\) 2.09167 0.0719557
\(846\) 0 0
\(847\) 26.8167 0.921431
\(848\) 0 0
\(849\) −0.605551 −0.0207825
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 21.7250 0.743849 0.371925 0.928263i \(-0.378698\pi\)
0.371925 + 0.928263i \(0.378698\pi\)
\(854\) 0 0
\(855\) 17.1833 0.587658
\(856\) 0 0
\(857\) 9.63331 0.329068 0.164534 0.986371i \(-0.447388\pi\)
0.164534 + 0.986371i \(0.447388\pi\)
\(858\) 0 0
\(859\) −35.8167 −1.22205 −0.611024 0.791612i \(-0.709242\pi\)
−0.611024 + 0.791612i \(0.709242\pi\)
\(860\) 0 0
\(861\) 0.908327 0.0309557
\(862\) 0 0
\(863\) 41.4500 1.41097 0.705487 0.708723i \(-0.250729\pi\)
0.705487 + 0.708723i \(0.250729\pi\)
\(864\) 0 0
\(865\) 19.6972 0.669726
\(866\) 0 0
\(867\) −9.30278 −0.315939
\(868\) 0 0
\(869\) −24.4777 −0.830350
\(870\) 0 0
\(871\) 13.2111 0.447641
\(872\) 0 0
\(873\) −18.3305 −0.620395
\(874\) 0 0
\(875\) 3.30278 0.111654
\(876\) 0 0
\(877\) 48.1749 1.62675 0.813376 0.581738i \(-0.197627\pi\)
0.813376 + 0.581738i \(0.197627\pi\)
\(878\) 0 0
\(879\) 7.02776 0.237040
\(880\) 0 0
\(881\) 55.2666 1.86198 0.930990 0.365045i \(-0.118946\pi\)
0.930990 + 0.365045i \(0.118946\pi\)
\(882\) 0 0
\(883\) 8.27502 0.278477 0.139238 0.990259i \(-0.455535\pi\)
0.139238 + 0.990259i \(0.455535\pi\)
\(884\) 0 0
\(885\) −1.02776 −0.0345477
\(886\) 0 0
\(887\) −27.6333 −0.927836 −0.463918 0.885878i \(-0.653557\pi\)
−0.463918 + 0.885878i \(0.653557\pi\)
\(888\) 0 0
\(889\) 32.4222 1.08741
\(890\) 0 0
\(891\) −13.8890 −0.465298
\(892\) 0 0
\(893\) 15.3944 0.515156
\(894\) 0 0
\(895\) −9.39445 −0.314022
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) 0 0
\(899\) 20.6056 0.687234
\(900\) 0 0
\(901\) 77.4500 2.58023
\(902\) 0 0
\(903\) −9.21110 −0.306526
\(904\) 0 0
\(905\) 17.1194 0.569069
\(906\) 0 0
\(907\) 48.6611 1.61576 0.807882 0.589344i \(-0.200614\pi\)
0.807882 + 0.589344i \(0.200614\pi\)
\(908\) 0 0
\(909\) 7.57779 0.251340
\(910\) 0 0
\(911\) 4.18335 0.138600 0.0693002 0.997596i \(-0.477923\pi\)
0.0693002 + 0.997596i \(0.477923\pi\)
\(912\) 0 0
\(913\) −19.0278 −0.629727
\(914\) 0 0
\(915\) −3.48612 −0.115248
\(916\) 0 0
\(917\) 37.0278 1.22276
\(918\) 0 0
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 3.54163 0.116701
\(922\) 0 0
\(923\) −53.8444 −1.77231
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) 23.6140 0.775584
\(928\) 0 0
\(929\) −14.3667 −0.471356 −0.235678 0.971831i \(-0.575731\pi\)
−0.235678 + 0.971831i \(0.575731\pi\)
\(930\) 0 0
\(931\) 23.0917 0.756799
\(932\) 0 0
\(933\) 6.78890 0.222259
\(934\) 0 0
\(935\) 11.7250 0.383448
\(936\) 0 0
\(937\) 37.9638 1.24022 0.620112 0.784513i \(-0.287087\pi\)
0.620112 + 0.784513i \(0.287087\pi\)
\(938\) 0 0
\(939\) −5.97224 −0.194897
\(940\) 0 0
\(941\) −25.9361 −0.845492 −0.422746 0.906248i \(-0.638934\pi\)
−0.422746 + 0.906248i \(0.638934\pi\)
\(942\) 0 0
\(943\) 0.908327 0.0295792
\(944\) 0 0
\(945\) 5.90833 0.192198
\(946\) 0 0
\(947\) −4.93608 −0.160401 −0.0802006 0.996779i \(-0.525556\pi\)
−0.0802006 + 0.996779i \(0.525556\pi\)
\(948\) 0 0
\(949\) 19.2111 0.623619
\(950\) 0 0
\(951\) 5.36669 0.174027
\(952\) 0 0
\(953\) −41.3305 −1.33883 −0.669414 0.742890i \(-0.733455\pi\)
−0.669414 + 0.742890i \(0.733455\pi\)
\(954\) 0 0
\(955\) −8.60555 −0.278469
\(956\) 0 0
\(957\) 1.33894 0.0432817
\(958\) 0 0
\(959\) −12.9083 −0.416832
\(960\) 0 0
\(961\) 31.5416 1.01747
\(962\) 0 0
\(963\) 7.57779 0.244191
\(964\) 0 0
\(965\) 17.8167 0.573538
\(966\) 0 0
\(967\) 12.6056 0.405367 0.202684 0.979244i \(-0.435034\pi\)
0.202684 + 0.979244i \(0.435034\pi\)
\(968\) 0 0
\(969\) −12.3583 −0.397005
\(970\) 0 0
\(971\) −17.0917 −0.548498 −0.274249 0.961659i \(-0.588429\pi\)
−0.274249 + 0.961659i \(0.588429\pi\)
\(972\) 0 0
\(973\) 41.6333 1.33470
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 6.51388 0.208397 0.104199 0.994556i \(-0.466772\pi\)
0.104199 + 0.994556i \(0.466772\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.32213 0.137995
\(982\) 0 0
\(983\) −34.5416 −1.10171 −0.550854 0.834602i \(-0.685698\pi\)
−0.550854 + 0.834602i \(0.685698\pi\)
\(984\) 0 0
\(985\) 4.30278 0.137098
\(986\) 0 0
\(987\) 2.60555 0.0829356
\(988\) 0 0
\(989\) −9.21110 −0.292896
\(990\) 0 0
\(991\) 15.3305 0.486990 0.243495 0.969902i \(-0.421706\pi\)
0.243495 + 0.969902i \(0.421706\pi\)
\(992\) 0 0
\(993\) −5.02776 −0.159551
\(994\) 0 0
\(995\) −20.4222 −0.647427
\(996\) 0 0
\(997\) 16.7889 0.531710 0.265855 0.964013i \(-0.414346\pi\)
0.265855 + 0.964013i \(0.414346\pi\)
\(998\) 0 0
\(999\) −14.3112 −0.452786
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 7360.2.a.bu.1.1 2
4.3 odd 2 7360.2.a.bc.1.2 2
8.3 odd 2 230.2.a.b.1.1 2
8.5 even 2 1840.2.a.j.1.2 2
24.11 even 2 2070.2.a.w.1.2 2
40.3 even 4 1150.2.b.f.599.3 4
40.19 odd 2 1150.2.a.m.1.2 2
40.27 even 4 1150.2.b.f.599.2 4
40.29 even 2 9200.2.a.ca.1.1 2
184.91 even 2 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 2 8.3 odd 2
1150.2.a.m.1.2 2 40.19 odd 2
1150.2.b.f.599.2 4 40.27 even 4
1150.2.b.f.599.3 4 40.3 even 4
1840.2.a.j.1.2 2 8.5 even 2
2070.2.a.w.1.2 2 24.11 even 2
5290.2.a.j.1.1 2 184.91 even 2
7360.2.a.bc.1.2 2 4.3 odd 2
7360.2.a.bu.1.1 2 1.1 even 1 trivial
9200.2.a.ca.1.1 2 40.29 even 2