Properties

Label 7098.2.a.bd.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -7.00000 q^{17} +1.00000 q^{18} +5.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -3.00000 q^{22} +6.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +5.00000 q^{29} +2.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} -7.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +5.00000 q^{38} +2.00000 q^{40} +5.00000 q^{41} +1.00000 q^{42} +2.00000 q^{43} -3.00000 q^{44} +2.00000 q^{45} +6.00000 q^{46} +1.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -7.00000 q^{51} +3.00000 q^{53} +1.00000 q^{54} -6.00000 q^{55} +1.00000 q^{56} +5.00000 q^{57} +5.00000 q^{58} +6.00000 q^{59} +2.00000 q^{60} +7.00000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -3.00000 q^{66} -2.00000 q^{67} -7.00000 q^{68} +6.00000 q^{69} +2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -12.0000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +5.00000 q^{76} -3.00000 q^{77} +3.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +5.00000 q^{82} +8.00000 q^{83} +1.00000 q^{84} -14.0000 q^{85} +2.00000 q^{86} +5.00000 q^{87} -3.00000 q^{88} +11.0000 q^{89} +2.00000 q^{90} +6.00000 q^{92} +2.00000 q^{93} +1.00000 q^{94} +10.0000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) −3.00000 −0.639602
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 2.00000 0.365148
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) −7.00000 −1.20049
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 5.00000 0.811107
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 1.00000 0.154303
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −3.00000 −0.452267
\(45\) 2.00000 0.298142
\(46\) 6.00000 0.884652
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.00000 −0.809040
\(56\) 1.00000 0.133631
\(57\) 5.00000 0.662266
\(58\) 5.00000 0.656532
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 2.00000 0.258199
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −7.00000 −0.848875
\(69\) 6.00000 0.722315
\(70\) 2.00000 0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 5.00000 0.573539
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 5.00000 0.552158
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 1.00000 0.109109
\(85\) −14.0000 −1.51851
\(86\) 2.00000 0.215666
\(87\) 5.00000 0.536056
\(88\) −3.00000 −0.319801
\(89\) 11.0000 1.16600 0.582999 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 2.00000 0.207390
\(94\) 1.00000 0.103142
\(95\) 10.0000 1.02598
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) −3.00000 −0.301511
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −7.00000 −0.693103
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 3.00000 0.291386
\(107\) 15.0000 1.45010 0.725052 0.688694i \(-0.241816\pi\)
0.725052 + 0.688694i \(0.241816\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) −6.00000 −0.572078
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 5.00000 0.468293
\(115\) 12.0000 1.11901
\(116\) 5.00000 0.464238
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −7.00000 −0.641689
\(120\) 2.00000 0.182574
\(121\) −2.00000 −0.181818
\(122\) 7.00000 0.633750
\(123\) 5.00000 0.450835
\(124\) 2.00000 0.179605
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −3.00000 −0.261116
\(133\) 5.00000 0.433555
\(134\) −2.00000 −0.172774
\(135\) 2.00000 0.172133
\(136\) −7.00000 −0.600245
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 6.00000 0.510754
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 2.00000 0.169031
\(141\) 1.00000 0.0842152
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 10.0000 0.830455
\(146\) −12.0000 −0.993127
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) 5.00000 0.405554
\(153\) −7.00000 −0.565916
\(154\) −3.00000 −0.241747
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 3.00000 0.238667
\(159\) 3.00000 0.237915
\(160\) 2.00000 0.158114
\(161\) 6.00000 0.472866
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 5.00000 0.390434
\(165\) −6.00000 −0.467099
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −14.0000 −1.07375
\(171\) 5.00000 0.382360
\(172\) 2.00000 0.152499
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 5.00000 0.379049
\(175\) −1.00000 −0.0755929
\(176\) −3.00000 −0.226134
\(177\) 6.00000 0.450988
\(178\) 11.0000 0.824485
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 2.00000 0.149071
\(181\) −9.00000 −0.668965 −0.334482 0.942402i \(-0.608561\pi\)
−0.334482 + 0.942402i \(0.608561\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 6.00000 0.442326
\(185\) 4.00000 0.294086
\(186\) 2.00000 0.146647
\(187\) 21.0000 1.53567
\(188\) 1.00000 0.0729325
\(189\) 1.00000 0.0727393
\(190\) 10.0000 0.725476
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 1.00000 0.0721688
\(193\) −13.0000 −0.935760 −0.467880 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 9.00000 0.641223 0.320612 0.947211i \(-0.396112\pi\)
0.320612 + 0.947211i \(0.396112\pi\)
\(198\) −3.00000 −0.213201
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.00000 −0.141069
\(202\) 14.0000 0.985037
\(203\) 5.00000 0.350931
\(204\) −7.00000 −0.490098
\(205\) 10.0000 0.698430
\(206\) 10.0000 0.696733
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −15.0000 −1.03757
\(210\) 2.00000 0.138013
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 3.00000 0.206041
\(213\) −8.00000 −0.548151
\(214\) 15.0000 1.02538
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) 2.00000 0.135769
\(218\) 16.0000 1.08366
\(219\) −12.0000 −0.810885
\(220\) −6.00000 −0.404520
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 6.00000 0.399114
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 5.00000 0.331133
\(229\) −27.0000 −1.78421 −0.892105 0.451828i \(-0.850772\pi\)
−0.892105 + 0.451828i \(0.850772\pi\)
\(230\) 12.0000 0.791257
\(231\) −3.00000 −0.197386
\(232\) 5.00000 0.328266
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 6.00000 0.390567
\(237\) 3.00000 0.194871
\(238\) −7.00000 −0.453743
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 2.00000 0.129099
\(241\) −28.0000 −1.80364 −0.901819 0.432113i \(-0.857768\pi\)
−0.901819 + 0.432113i \(0.857768\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) 7.00000 0.448129
\(245\) 2.00000 0.127775
\(246\) 5.00000 0.318788
\(247\) 0 0
\(248\) 2.00000 0.127000
\(249\) 8.00000 0.506979
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) −18.0000 −1.13165
\(254\) −12.0000 −0.752947
\(255\) −14.0000 −0.876714
\(256\) 1.00000 0.0625000
\(257\) −15.0000 −0.935674 −0.467837 0.883815i \(-0.654967\pi\)
−0.467837 + 0.883815i \(0.654967\pi\)
\(258\) 2.00000 0.124515
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 8.00000 0.494242
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −3.00000 −0.184637
\(265\) 6.00000 0.368577
\(266\) 5.00000 0.306570
\(267\) 11.0000 0.673189
\(268\) −2.00000 −0.122169
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 2.00000 0.121716
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 3.00000 0.180907
\(276\) 6.00000 0.361158
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) −19.0000 −1.13954
\(279\) 2.00000 0.119737
\(280\) 2.00000 0.119523
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 1.00000 0.0595491
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) 10.0000 0.592349
\(286\) 0 0
\(287\) 5.00000 0.295141
\(288\) 1.00000 0.0589256
\(289\) 32.0000 1.88235
\(290\) 10.0000 0.587220
\(291\) 2.00000 0.117242
\(292\) −12.0000 −0.702247
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 1.00000 0.0583212
\(295\) 12.0000 0.698667
\(296\) 2.00000 0.116248
\(297\) −3.00000 −0.174078
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 2.00000 0.115278
\(302\) −13.0000 −0.748066
\(303\) 14.0000 0.804279
\(304\) 5.00000 0.286770
\(305\) 14.0000 0.801638
\(306\) −7.00000 −0.400163
\(307\) −9.00000 −0.513657 −0.256829 0.966457i \(-0.582678\pi\)
−0.256829 + 0.966457i \(0.582678\pi\)
\(308\) −3.00000 −0.170941
\(309\) 10.0000 0.568880
\(310\) 4.00000 0.227185
\(311\) 23.0000 1.30421 0.652105 0.758129i \(-0.273886\pi\)
0.652105 + 0.758129i \(0.273886\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) −18.0000 −1.01580
\(315\) 2.00000 0.112687
\(316\) 3.00000 0.168763
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 3.00000 0.168232
\(319\) −15.0000 −0.839839
\(320\) 2.00000 0.111803
\(321\) 15.0000 0.837218
\(322\) 6.00000 0.334367
\(323\) −35.0000 −1.94745
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 14.0000 0.775388
\(327\) 16.0000 0.884802
\(328\) 5.00000 0.276079
\(329\) 1.00000 0.0551318
\(330\) −6.00000 −0.330289
\(331\) −22.0000 −1.20923 −0.604615 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 8.00000 0.439057
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 1.00000 0.0545545
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) −14.0000 −0.759257
\(341\) −6.00000 −0.324918
\(342\) 5.00000 0.270369
\(343\) 1.00000 0.0539949
\(344\) 2.00000 0.107833
\(345\) 12.0000 0.646058
\(346\) 12.0000 0.645124
\(347\) 17.0000 0.912608 0.456304 0.889824i \(-0.349173\pi\)
0.456304 + 0.889824i \(0.349173\pi\)
\(348\) 5.00000 0.268028
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 6.00000 0.318896
\(355\) −16.0000 −0.849192
\(356\) 11.0000 0.582999
\(357\) −7.00000 −0.370479
\(358\) 20.0000 1.05703
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 2.00000 0.105409
\(361\) 6.00000 0.315789
\(362\) −9.00000 −0.473029
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −24.0000 −1.25622
\(366\) 7.00000 0.365896
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 6.00000 0.312772
\(369\) 5.00000 0.260290
\(370\) 4.00000 0.207950
\(371\) 3.00000 0.155752
\(372\) 2.00000 0.103695
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 21.0000 1.08588
\(375\) −12.0000 −0.619677
\(376\) 1.00000 0.0515711
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 10.0000 0.512989
\(381\) −12.0000 −0.614779
\(382\) −4.00000 −0.204658
\(383\) −31.0000 −1.58403 −0.792013 0.610504i \(-0.790967\pi\)
−0.792013 + 0.610504i \(0.790967\pi\)
\(384\) 1.00000 0.0510310
\(385\) −6.00000 −0.305788
\(386\) −13.0000 −0.661683
\(387\) 2.00000 0.101666
\(388\) 2.00000 0.101535
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −42.0000 −2.12403
\(392\) 1.00000 0.0505076
\(393\) 8.00000 0.403547
\(394\) 9.00000 0.453413
\(395\) 6.00000 0.301893
\(396\) −3.00000 −0.150756
\(397\) 15.0000 0.752828 0.376414 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(398\) −26.0000 −1.30326
\(399\) 5.00000 0.250313
\(400\) −1.00000 −0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 2.00000 0.0993808
\(406\) 5.00000 0.248146
\(407\) −6.00000 −0.297409
\(408\) −7.00000 −0.346552
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 10.0000 0.493865
\(411\) −18.0000 −0.887875
\(412\) 10.0000 0.492665
\(413\) 6.00000 0.295241
\(414\) 6.00000 0.294884
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) −19.0000 −0.930434
\(418\) −15.0000 −0.733674
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 2.00000 0.0975900
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −10.0000 −0.486792
\(423\) 1.00000 0.0486217
\(424\) 3.00000 0.145693
\(425\) 7.00000 0.339550
\(426\) −8.00000 −0.387601
\(427\) 7.00000 0.338754
\(428\) 15.0000 0.725052
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 2.00000 0.0960031
\(435\) 10.0000 0.479463
\(436\) 16.0000 0.766261
\(437\) 30.0000 1.43509
\(438\) −12.0000 −0.573382
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) −6.00000 −0.286039
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 15.0000 0.712672 0.356336 0.934358i \(-0.384026\pi\)
0.356336 + 0.934358i \(0.384026\pi\)
\(444\) 2.00000 0.0949158
\(445\) 22.0000 1.04290
\(446\) 4.00000 0.189405
\(447\) −14.0000 −0.662177
\(448\) 1.00000 0.0472456
\(449\) 38.0000 1.79333 0.896665 0.442709i \(-0.145982\pi\)
0.896665 + 0.442709i \(0.145982\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −15.0000 −0.706322
\(452\) 6.00000 0.282216
\(453\) −13.0000 −0.610793
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) −27.0000 −1.26163
\(459\) −7.00000 −0.326732
\(460\) 12.0000 0.559503
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) −3.00000 −0.139573
\(463\) 31.0000 1.44069 0.720346 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 5.00000 0.232119
\(465\) 4.00000 0.185496
\(466\) 26.0000 1.20443
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 2.00000 0.0922531
\(471\) −18.0000 −0.829396
\(472\) 6.00000 0.276172
\(473\) −6.00000 −0.275880
\(474\) 3.00000 0.137795
\(475\) −5.00000 −0.229416
\(476\) −7.00000 −0.320844
\(477\) 3.00000 0.137361
\(478\) −12.0000 −0.548867
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −28.0000 −1.27537
\(483\) 6.00000 0.273009
\(484\) −2.00000 −0.0909091
\(485\) 4.00000 0.181631
\(486\) 1.00000 0.0453609
\(487\) −35.0000 −1.58600 −0.793001 0.609221i \(-0.791482\pi\)
−0.793001 + 0.609221i \(0.791482\pi\)
\(488\) 7.00000 0.316875
\(489\) 14.0000 0.633102
\(490\) 2.00000 0.0903508
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 5.00000 0.225417
\(493\) −35.0000 −1.57632
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 2.00000 0.0898027
\(497\) −8.00000 −0.358849
\(498\) 8.00000 0.358489
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 1.00000 0.0445435
\(505\) 28.0000 1.24598
\(506\) −18.0000 −0.800198
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 40.0000 1.77297 0.886484 0.462758i \(-0.153140\pi\)
0.886484 + 0.462758i \(0.153140\pi\)
\(510\) −14.0000 −0.619930
\(511\) −12.0000 −0.530849
\(512\) 1.00000 0.0441942
\(513\) 5.00000 0.220755
\(514\) −15.0000 −0.661622
\(515\) 20.0000 0.881305
\(516\) 2.00000 0.0880451
\(517\) −3.00000 −0.131940
\(518\) 2.00000 0.0878750
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 21.0000 0.920027 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) 5.00000 0.218844
\(523\) 39.0000 1.70535 0.852675 0.522441i \(-0.174978\pi\)
0.852675 + 0.522441i \(0.174978\pi\)
\(524\) 8.00000 0.349482
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) −14.0000 −0.609850
\(528\) −3.00000 −0.130558
\(529\) 13.0000 0.565217
\(530\) 6.00000 0.260623
\(531\) 6.00000 0.260378
\(532\) 5.00000 0.216777
\(533\) 0 0
\(534\) 11.0000 0.476017
\(535\) 30.0000 1.29701
\(536\) −2.00000 −0.0863868
\(537\) 20.0000 0.863064
\(538\) −24.0000 −1.03471
\(539\) −3.00000 −0.129219
\(540\) 2.00000 0.0860663
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −9.00000 −0.386227
\(544\) −7.00000 −0.300123
\(545\) 32.0000 1.37073
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −18.0000 −0.768922
\(549\) 7.00000 0.298753
\(550\) 3.00000 0.127920
\(551\) 25.0000 1.06504
\(552\) 6.00000 0.255377
\(553\) 3.00000 0.127573
\(554\) 6.00000 0.254916
\(555\) 4.00000 0.169791
\(556\) −19.0000 −0.805779
\(557\) 15.0000 0.635570 0.317785 0.948163i \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 2.00000 0.0846668
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 21.0000 0.886621
\(562\) −16.0000 −0.674919
\(563\) 30.0000 1.26435 0.632175 0.774826i \(-0.282163\pi\)
0.632175 + 0.774826i \(0.282163\pi\)
\(564\) 1.00000 0.0421076
\(565\) 12.0000 0.504844
\(566\) 28.0000 1.17693
\(567\) 1.00000 0.0419961
\(568\) −8.00000 −0.335673
\(569\) −44.0000 −1.84458 −0.922288 0.386503i \(-0.873683\pi\)
−0.922288 + 0.386503i \(0.873683\pi\)
\(570\) 10.0000 0.418854
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 5.00000 0.208696
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 32.0000 1.33102
\(579\) −13.0000 −0.540262
\(580\) 10.0000 0.415227
\(581\) 8.00000 0.331896
\(582\) 2.00000 0.0829027
\(583\) −9.00000 −0.372742
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) −10.0000 −0.412744 −0.206372 0.978474i \(-0.566166\pi\)
−0.206372 + 0.978474i \(0.566166\pi\)
\(588\) 1.00000 0.0412393
\(589\) 10.0000 0.412043
\(590\) 12.0000 0.494032
\(591\) 9.00000 0.370211
\(592\) 2.00000 0.0821995
\(593\) 13.0000 0.533846 0.266923 0.963718i \(-0.413993\pi\)
0.266923 + 0.963718i \(0.413993\pi\)
\(594\) −3.00000 −0.123091
\(595\) −14.0000 −0.573944
\(596\) −14.0000 −0.573462
\(597\) −26.0000 −1.06411
\(598\) 0 0
\(599\) −26.0000 −1.06233 −0.531166 0.847268i \(-0.678246\pi\)
−0.531166 + 0.847268i \(0.678246\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 2.00000 0.0815139
\(603\) −2.00000 −0.0814463
\(604\) −13.0000 −0.528962
\(605\) −4.00000 −0.162623
\(606\) 14.0000 0.568711
\(607\) −30.0000 −1.21766 −0.608831 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) 5.00000 0.202777
\(609\) 5.00000 0.202610
\(610\) 14.0000 0.566843
\(611\) 0 0
\(612\) −7.00000 −0.282958
\(613\) 28.0000 1.13091 0.565455 0.824779i \(-0.308701\pi\)
0.565455 + 0.824779i \(0.308701\pi\)
\(614\) −9.00000 −0.363210
\(615\) 10.0000 0.403239
\(616\) −3.00000 −0.120873
\(617\) 20.0000 0.805170 0.402585 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(618\) 10.0000 0.402259
\(619\) −41.0000 −1.64793 −0.823965 0.566641i \(-0.808243\pi\)
−0.823965 + 0.566641i \(0.808243\pi\)
\(620\) 4.00000 0.160644
\(621\) 6.00000 0.240772
\(622\) 23.0000 0.922216
\(623\) 11.0000 0.440706
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 28.0000 1.11911
\(627\) −15.0000 −0.599042
\(628\) −18.0000 −0.718278
\(629\) −14.0000 −0.558217
\(630\) 2.00000 0.0796819
\(631\) −15.0000 −0.597141 −0.298570 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 3.00000 0.119334
\(633\) −10.0000 −0.397464
\(634\) −30.0000 −1.19145
\(635\) −24.0000 −0.952411
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) −15.0000 −0.593856
\(639\) −8.00000 −0.316475
\(640\) 2.00000 0.0790569
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 15.0000 0.592003
\(643\) 49.0000 1.93237 0.966186 0.257847i \(-0.0830131\pi\)
0.966186 + 0.257847i \(0.0830131\pi\)
\(644\) 6.00000 0.236433
\(645\) 4.00000 0.157500
\(646\) −35.0000 −1.37706
\(647\) 1.00000 0.0393141 0.0196570 0.999807i \(-0.493743\pi\)
0.0196570 + 0.999807i \(0.493743\pi\)
\(648\) 1.00000 0.0392837
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 14.0000 0.548282
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 16.0000 0.625650
\(655\) 16.0000 0.625172
\(656\) 5.00000 0.195217
\(657\) −12.0000 −0.468165
\(658\) 1.00000 0.0389841
\(659\) −27.0000 −1.05177 −0.525885 0.850555i \(-0.676266\pi\)
−0.525885 + 0.850555i \(0.676266\pi\)
\(660\) −6.00000 −0.233550
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −22.0000 −0.855054
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 10.0000 0.387783
\(666\) 2.00000 0.0774984
\(667\) 30.0000 1.16160
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) −4.00000 −0.154533
\(671\) −21.0000 −0.810696
\(672\) 1.00000 0.0385758
\(673\) 23.0000 0.886585 0.443292 0.896377i \(-0.353810\pi\)
0.443292 + 0.896377i \(0.353810\pi\)
\(674\) −17.0000 −0.654816
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 6.00000 0.230429
\(679\) 2.00000 0.0767530
\(680\) −14.0000 −0.536875
\(681\) −14.0000 −0.536481
\(682\) −6.00000 −0.229752
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 5.00000 0.191180
\(685\) −36.0000 −1.37549
\(686\) 1.00000 0.0381802
\(687\) −27.0000 −1.03011
\(688\) 2.00000 0.0762493
\(689\) 0 0
\(690\) 12.0000 0.456832
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 12.0000 0.456172
\(693\) −3.00000 −0.113961
\(694\) 17.0000 0.645311
\(695\) −38.0000 −1.44142
\(696\) 5.00000 0.189525
\(697\) −35.0000 −1.32572
\(698\) 18.0000 0.681310
\(699\) 26.0000 0.983410
\(700\) −1.00000 −0.0377964
\(701\) 13.0000 0.491003 0.245502 0.969396i \(-0.421047\pi\)
0.245502 + 0.969396i \(0.421047\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) −3.00000 −0.113067
\(705\) 2.00000 0.0753244
\(706\) −14.0000 −0.526897
\(707\) 14.0000 0.526524
\(708\) 6.00000 0.225494
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) −16.0000 −0.600469
\(711\) 3.00000 0.112509
\(712\) 11.0000 0.412242
\(713\) 12.0000 0.449404
\(714\) −7.00000 −0.261968
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) −12.0000 −0.448148
\(718\) −18.0000 −0.671754
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 2.00000 0.0745356
\(721\) 10.0000 0.372419
\(722\) 6.00000 0.223297
\(723\) −28.0000 −1.04133
\(724\) −9.00000 −0.334482
\(725\) −5.00000 −0.185695
\(726\) −2.00000 −0.0742270
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −24.0000 −0.888280
\(731\) −14.0000 −0.517809
\(732\) 7.00000 0.258727
\(733\) 7.00000 0.258551 0.129275 0.991609i \(-0.458735\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) −10.0000 −0.369107
\(735\) 2.00000 0.0737711
\(736\) 6.00000 0.221163
\(737\) 6.00000 0.221013
\(738\) 5.00000 0.184053
\(739\) 14.0000 0.514998 0.257499 0.966279i \(-0.417102\pi\)
0.257499 + 0.966279i \(0.417102\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) 6.00000 0.220119 0.110059 0.993925i \(-0.464896\pi\)
0.110059 + 0.993925i \(0.464896\pi\)
\(744\) 2.00000 0.0733236
\(745\) −28.0000 −1.02584
\(746\) 4.00000 0.146450
\(747\) 8.00000 0.292705
\(748\) 21.0000 0.767836
\(749\) 15.0000 0.548088
\(750\) −12.0000 −0.438178
\(751\) −33.0000 −1.20419 −0.602094 0.798426i \(-0.705667\pi\)
−0.602094 + 0.798426i \(0.705667\pi\)
\(752\) 1.00000 0.0364662
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −26.0000 −0.946237
\(756\) 1.00000 0.0363696
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 20.0000 0.726433
\(759\) −18.0000 −0.653359
\(760\) 10.0000 0.362738
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) −12.0000 −0.434714
\(763\) 16.0000 0.579239
\(764\) −4.00000 −0.144715
\(765\) −14.0000 −0.506171
\(766\) −31.0000 −1.12008
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) −6.00000 −0.216225
\(771\) −15.0000 −0.540212
\(772\) −13.0000 −0.467880
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 2.00000 0.0718885
\(775\) −2.00000 −0.0718421
\(776\) 2.00000 0.0717958
\(777\) 2.00000 0.0717496
\(778\) 6.00000 0.215110
\(779\) 25.0000 0.895718
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) −42.0000 −1.50192
\(783\) 5.00000 0.178685
\(784\) 1.00000 0.0357143
\(785\) −36.0000 −1.28490
\(786\) 8.00000 0.285351
\(787\) −25.0000 −0.891154 −0.445577 0.895244i \(-0.647001\pi\)
−0.445577 + 0.895244i \(0.647001\pi\)
\(788\) 9.00000 0.320612
\(789\) −24.0000 −0.854423
\(790\) 6.00000 0.213470
\(791\) 6.00000 0.213335
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) 15.0000 0.532330
\(795\) 6.00000 0.212798
\(796\) −26.0000 −0.921546
\(797\) 56.0000 1.98362 0.991811 0.127715i \(-0.0407643\pi\)
0.991811 + 0.127715i \(0.0407643\pi\)
\(798\) 5.00000 0.176998
\(799\) −7.00000 −0.247642
\(800\) −1.00000 −0.0353553
\(801\) 11.0000 0.388666
\(802\) −18.0000 −0.635602
\(803\) 36.0000 1.27041
\(804\) −2.00000 −0.0705346
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 14.0000 0.492518
\(809\) 52.0000 1.82822 0.914111 0.405463i \(-0.132890\pi\)
0.914111 + 0.405463i \(0.132890\pi\)
\(810\) 2.00000 0.0702728
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 5.00000 0.175466
\(813\) −2.00000 −0.0701431
\(814\) −6.00000 −0.210300
\(815\) 28.0000 0.980797
\(816\) −7.00000 −0.245049
\(817\) 10.0000 0.349856
\(818\) 0 0
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) −15.0000 −0.523504 −0.261752 0.965135i \(-0.584300\pi\)
−0.261752 + 0.965135i \(0.584300\pi\)
\(822\) −18.0000 −0.627822
\(823\) −48.0000 −1.67317 −0.836587 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) 10.0000 0.348367
\(825\) 3.00000 0.104447
\(826\) 6.00000 0.208767
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 6.00000 0.208514
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 16.0000 0.555368
\(831\) 6.00000 0.208138
\(832\) 0 0
\(833\) −7.00000 −0.242536
\(834\) −19.0000 −0.657916
\(835\) 0 0
\(836\) −15.0000 −0.518786
\(837\) 2.00000 0.0691301
\(838\) 4.00000 0.138178
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 2.00000 0.0690066
\(841\) −4.00000 −0.137931
\(842\) 22.0000 0.758170
\(843\) −16.0000 −0.551069
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 1.00000 0.0343807
\(847\) −2.00000 −0.0687208
\(848\) 3.00000 0.103020
\(849\) 28.0000 0.960958
\(850\) 7.00000 0.240098
\(851\) 12.0000 0.411355
\(852\) −8.00000 −0.274075
\(853\) −41.0000 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) 7.00000 0.239535
\(855\) 10.0000 0.341993
\(856\) 15.0000 0.512689
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 35.0000 1.19418 0.597092 0.802173i \(-0.296323\pi\)
0.597092 + 0.802173i \(0.296323\pi\)
\(860\) 4.00000 0.136399
\(861\) 5.00000 0.170400
\(862\) 8.00000 0.272481
\(863\) −34.0000 −1.15737 −0.578687 0.815550i \(-0.696435\pi\)
−0.578687 + 0.815550i \(0.696435\pi\)
\(864\) 1.00000 0.0340207
\(865\) 24.0000 0.816024
\(866\) 26.0000 0.883516
\(867\) 32.0000 1.08678
\(868\) 2.00000 0.0678844
\(869\) −9.00000 −0.305304
\(870\) 10.0000 0.339032
\(871\) 0 0
\(872\) 16.0000 0.541828
\(873\) 2.00000 0.0676897
\(874\) 30.0000 1.01477
\(875\) −12.0000 −0.405674
\(876\) −12.0000 −0.405442
\(877\) −40.0000 −1.35070 −0.675352 0.737496i \(-0.736008\pi\)
−0.675352 + 0.737496i \(0.736008\pi\)
\(878\) 4.00000 0.134993
\(879\) −12.0000 −0.404750
\(880\) −6.00000 −0.202260
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 1.00000 0.0336718
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 15.0000 0.503935
\(887\) 37.0000 1.24234 0.621169 0.783676i \(-0.286658\pi\)
0.621169 + 0.783676i \(0.286658\pi\)
\(888\) 2.00000 0.0671156
\(889\) −12.0000 −0.402467
\(890\) 22.0000 0.737442
\(891\) −3.00000 −0.100504
\(892\) 4.00000 0.133930
\(893\) 5.00000 0.167319
\(894\) −14.0000 −0.468230
\(895\) 40.0000 1.33705
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 38.0000 1.26808
\(899\) 10.0000 0.333519
\(900\) −1.00000 −0.0333333
\(901\) −21.0000 −0.699611
\(902\) −15.0000 −0.499445
\(903\) 2.00000 0.0665558
\(904\) 6.00000 0.199557
\(905\) −18.0000 −0.598340
\(906\) −13.0000 −0.431896
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) −14.0000 −0.464606
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −52.0000 −1.72284 −0.861418 0.507896i \(-0.830423\pi\)
−0.861418 + 0.507896i \(0.830423\pi\)
\(912\) 5.00000 0.165567
\(913\) −24.0000 −0.794284
\(914\) −26.0000 −0.860004
\(915\) 14.0000 0.462826
\(916\) −27.0000 −0.892105
\(917\) 8.00000 0.264183
\(918\) −7.00000 −0.231034
\(919\) −11.0000 −0.362857 −0.181428 0.983404i \(-0.558072\pi\)
−0.181428 + 0.983404i \(0.558072\pi\)
\(920\) 12.0000 0.395628
\(921\) −9.00000 −0.296560
\(922\) −8.00000 −0.263466
\(923\) 0 0
\(924\) −3.00000 −0.0986928
\(925\) −2.00000 −0.0657596
\(926\) 31.0000 1.01872
\(927\) 10.0000 0.328443
\(928\) 5.00000 0.164133
\(929\) 13.0000 0.426516 0.213258 0.976996i \(-0.431592\pi\)
0.213258 + 0.976996i \(0.431592\pi\)
\(930\) 4.00000 0.131165
\(931\) 5.00000 0.163868
\(932\) 26.0000 0.851658
\(933\) 23.0000 0.752986
\(934\) −12.0000 −0.392652
\(935\) 42.0000 1.37355
\(936\) 0 0
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) −2.00000 −0.0653023
\(939\) 28.0000 0.913745
\(940\) 2.00000 0.0652328
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −18.0000 −0.586472
\(943\) 30.0000 0.976934
\(944\) 6.00000 0.195283
\(945\) 2.00000 0.0650600
\(946\) −6.00000 −0.195077
\(947\) 31.0000 1.00736 0.503682 0.863889i \(-0.331978\pi\)
0.503682 + 0.863889i \(0.331978\pi\)
\(948\) 3.00000 0.0974355
\(949\) 0 0
\(950\) −5.00000 −0.162221
\(951\) −30.0000 −0.972817
\(952\) −7.00000 −0.226871
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 3.00000 0.0971286
\(955\) −8.00000 −0.258874
\(956\) −12.0000 −0.388108
\(957\) −15.0000 −0.484881
\(958\) −21.0000 −0.678479
\(959\) −18.0000 −0.581250
\(960\) 2.00000 0.0645497
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 15.0000 0.483368
\(964\) −28.0000 −0.901819
\(965\) −26.0000 −0.836970
\(966\) 6.00000 0.193047
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −35.0000 −1.12436
\(970\) 4.00000 0.128432
\(971\) 22.0000 0.706014 0.353007 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 1.00000 0.0320750
\(973\) −19.0000 −0.609112
\(974\) −35.0000 −1.12147
\(975\) 0 0
\(976\) 7.00000 0.224065
\(977\) −8.00000 −0.255943 −0.127971 0.991778i \(-0.540847\pi\)
−0.127971 + 0.991778i \(0.540847\pi\)
\(978\) 14.0000 0.447671
\(979\) −33.0000 −1.05468
\(980\) 2.00000 0.0638877
\(981\) 16.0000 0.510841
\(982\) −8.00000 −0.255290
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 5.00000 0.159394
\(985\) 18.0000 0.573528
\(986\) −35.0000 −1.11463
\(987\) 1.00000 0.0318304
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) −6.00000 −0.190693
\(991\) −3.00000 −0.0952981 −0.0476491 0.998864i \(-0.515173\pi\)
−0.0476491 + 0.998864i \(0.515173\pi\)
\(992\) 2.00000 0.0635001
\(993\) −22.0000 −0.698149
\(994\) −8.00000 −0.253745
\(995\) −52.0000 −1.64851
\(996\) 8.00000 0.253490
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) −14.0000 −0.443162
\(999\) 2.00000 0.0632772
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 7098.2.a.bd.1.1 1
13.4 even 6 546.2.l.c.211.1 2
13.10 even 6 546.2.l.c.295.1 yes 2
13.12 even 2 7098.2.a.i.1.1 1
39.17 odd 6 1638.2.r.l.757.1 2
39.23 odd 6 1638.2.r.l.1387.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.c.211.1 2 13.4 even 6
546.2.l.c.295.1 yes 2 13.10 even 6
1638.2.r.l.757.1 2 39.17 odd 6
1638.2.r.l.1387.1 2 39.23 odd 6
7098.2.a.i.1.1 1 13.12 even 2
7098.2.a.bd.1.1 1 1.1 even 1 trivial