Properties

Label 6930.2.a.g.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +8.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -1.00000 q^{35} +4.00000 q^{37} -8.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{46} +4.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -2.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +10.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +14.0000 q^{67} +4.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -10.0000 q^{73} -4.00000 q^{74} +8.00000 q^{76} +1.00000 q^{77} +4.00000 q^{79} -1.00000 q^{80} +2.00000 q^{82} +4.00000 q^{83} -4.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} -16.0000 q^{89} +4.00000 q^{91} -2.00000 q^{92} -4.00000 q^{94} -8.00000 q^{95} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 10.0000 0.827606
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −4.00000 −0.318223
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 16.0000 1.19925
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 16.0000 1.11477
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 4.00000 0.266076
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 32.0000 2.03611
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 20.0000 1.23325 0.616626 0.787256i \(-0.288499\pi\)
0.616626 + 0.787256i \(0.288499\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) 14.0000 0.855186
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −12.0000 −0.719712
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 4.00000 0.238620 0.119310 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 2.00000 0.111456
\(323\) 32.0000 1.78053
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −14.0000 −0.775388
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) −14.0000 −0.764902
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) −4.00000 −0.216930
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) −16.0000 −0.847998
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 8.00000 0.420471
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) −2.00000 −0.103835
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) 20.0000 1.02329
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −14.0000 −0.710742
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −6.00000 −0.292075
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) −4.00000 −0.188144
\(453\) 0 0
\(454\) −4.00000 −0.187729
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 12.0000 0.560723
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 4.00000 0.184506
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) −32.0000 −1.43975
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 0 0
\(523\) −42.0000 −1.83653 −0.918266 0.395964i \(-0.870410\pi\)
−0.918266 + 0.395964i \(0.870410\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −2.00000 −0.0868744
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −4.00000 −0.170872
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 48.0000 2.04487
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −4.00000 −0.168730
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) 14.0000 0.588464
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −40.0000 −1.67689 −0.838444 0.544988i \(-0.816534\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 2.00000 0.0834784
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 10.0000 0.413803
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) 32.0000 1.31408 0.657041 0.753855i \(-0.271808\pi\)
0.657041 + 0.753855i \(0.271808\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 4.00000 0.161034 0.0805170 0.996753i \(-0.474343\pi\)
0.0805170 + 0.996753i \(0.474343\pi\)
\(618\) 0 0
\(619\) 46.0000 1.84890 0.924448 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −30.0000 −1.20289
\(623\) −16.0000 −0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −4.00000 −0.159111
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) −6.00000 −0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) −32.0000 −1.25902
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 14.0000 0.548282
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) −4.00000 −0.155936
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) 14.0000 0.540867
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 4.00000 0.153393
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) −26.0000 −0.989087 −0.494543 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 16.0000 0.599625
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 14.0000 0.522475
\(719\) 42.0000 1.56634 0.783168 0.621810i \(-0.213603\pi\)
0.783168 + 0.621810i \(0.213603\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) −8.00000 −0.297318
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) −4.00000 −0.148250
\(729\) 0 0
\(730\) −10.0000 −0.370117
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −20.0000 −0.738717 −0.369358 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 14.0000 0.515697
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 2.00000 0.0734223
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −6.00000 −0.219676
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) −24.0000 −0.874028
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) −20.0000 −0.723575
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) −30.0000 −1.06938 −0.534692 0.845047i \(-0.679572\pi\)
−0.534692 + 0.845047i \(0.679572\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 4.00000 0.142314
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 12.0000 0.425329
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −6.00000 −0.211867
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 16.0000 0.563576
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) −48.0000 −1.68759 −0.843795 0.536666i \(-0.819684\pi\)
−0.843795 + 0.536666i \(0.819684\pi\)
\(810\) 0 0
\(811\) 56.0000 1.96643 0.983213 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) −14.0000 −0.490399
\(816\) 0 0
\(817\) 32.0000 1.11954
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 16.0000 0.557386
\(825\) 0 0
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 4.00000 0.138842
\(831\) 0 0
\(832\) 4.00000 0.138675
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 8.00000 0.276686
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) 54.0000 1.86429 0.932144 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 6.00000 0.206529
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −2.00000 −0.0686803
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −14.0000 −0.476842
\(863\) −34.0000 −1.15737 −0.578687 0.815550i \(-0.696435\pi\)
−0.578687 + 0.815550i \(0.696435\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 14.0000 0.475739
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 56.0000 1.89749
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 16.0000 0.541208
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −40.0000 −1.34993
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −40.0000 −1.34763 −0.673817 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 0 0
\(883\) −2.00000 −0.0673054 −0.0336527 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16.0000 −0.536321
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 4.00000 0.133038
\(905\) 8.00000 0.265929
\(906\) 0 0
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) 4.00000 0.132745
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) −12.0000 −0.396491
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 0 0
\(922\) 14.0000 0.461065
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −14.0000 −0.457116
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −16.0000 −0.519930 −0.259965 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) 0 0
\(949\) −40.0000 −1.29845
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) −4.00000 −0.129641
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 20.0000 0.647185
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −16.0000 −0.515861
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −32.0000 −1.02377 −0.511885 0.859054i \(-0.671053\pi\)
−0.511885 + 0.859054i \(0.671053\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −28.0000 −0.893516
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) 32.0000 1.01806
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) 8.00000 0.253236
\(999\) 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 6930.2.a.g.1.1 1
3.2 odd 2 2310.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.p.1.1 1 3.2 odd 2
6930.2.a.g.1.1 1 1.1 even 1 trivial