Properties

Label 2310.2.a.p.1.1
Level $2310$
Weight $2$
Character 2310.1
Self dual yes
Analytic conductor $18.445$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2310.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.4454428669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2310.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +4.00000 q^{37} +8.00000 q^{38} -4.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} +2.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +4.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} -8.00000 q^{57} -6.00000 q^{58} -1.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} +14.0000 q^{67} -4.00000 q^{68} -2.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +4.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} -1.00000 q^{77} -4.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +16.0000 q^{89} +1.00000 q^{90} +4.00000 q^{91} +2.00000 q^{92} +4.00000 q^{93} -4.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −1.00000 −0.213201
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −1.00000 −0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −4.00000 −0.685994
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 8.00000 1.29777
\(39\) −4.00000 −0.640513
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.00000 0.149071
\(46\) 2.00000 0.294884
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 4.00000 0.554700
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) −8.00000 −1.05963
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 1.00000 0.123091
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) −4.00000 −0.485071
\(69\) −2.00000 −0.240772
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 4.00000 0.464991
\(75\) −1.00000 −0.115470
\(76\) 8.00000 0.917663
\(77\) −1.00000 −0.113961
\(78\) −4.00000 −0.452911
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −1.00000 −0.109109
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 6.00000 0.643268
\(88\) −1.00000 −0.106600
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.00000 0.419314
\(92\) 2.00000 0.208514
\(93\) 4.00000 0.414781
\(94\) −4.00000 −0.412568
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 4.00000 0.396059
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 4.00000 0.392232
\(105\) −1.00000 −0.0975900
\(106\) 2.00000 0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) −8.00000 −0.749269
\(115\) 2.00000 0.186501
\(116\) −6.00000 −0.557086
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) −4.00000 −0.359211
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 4.00000 0.350823
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 1.00000 0.0870388
\(133\) 8.00000 0.693688
\(134\) 14.0000 1.20942
\(135\) −1.00000 −0.0860663
\(136\) −4.00000 −0.342997
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) −2.00000 −0.170251
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 1.00000 0.0845154
\(141\) 4.00000 0.336861
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −10.0000 −0.827606
\(147\) −1.00000 −0.0824786
\(148\) 4.00000 0.328798
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 8.00000 0.648886
\(153\) −4.00000 −0.323381
\(154\) −1.00000 −0.0805823
\(155\) −4.00000 −0.321288
\(156\) −4.00000 −0.320256
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 4.00000 0.318223
\(159\) −2.00000 −0.158610
\(160\) 1.00000 0.0790569
\(161\) 2.00000 0.157622
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) 2.00000 0.156174
\(165\) 1.00000 0.0778499
\(166\) −4.00000 −0.310460
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 8.00000 0.611775
\(172\) 4.00000 0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.00000 0.454859
\(175\) 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 \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 1.00000 0.0745356
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) 4.00000 0.296500
\(183\) −10.0000 −0.739221
\(184\) 2.00000 0.147442
\(185\) 4.00000 0.294086
\(186\) 4.00000 0.293294
\(187\) 4.00000 0.292509
\(188\) −4.00000 −0.291730
\(189\) −1.00000 −0.0727393
\(190\) 8.00000 0.580381
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −14.0000 −1.00514
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 1.00000 0.0707107
\(201\) −14.0000 −0.987484
\(202\) −6.00000 −0.422159
\(203\) −6.00000 −0.421117
\(204\) 4.00000 0.280056
\(205\) 2.00000 0.139686
\(206\) −16.0000 −1.11477
\(207\) 2.00000 0.139010
\(208\) 4.00000 0.277350
\(209\) −8.00000 −0.553372
\(210\) −1.00000 −0.0690066
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 2.00000 0.137361
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) −4.00000 −0.270914
\(219\) 10.0000 0.675737
\(220\) −1.00000 −0.0674200
\(221\) −16.0000 −1.07628
\(222\) −4.00000 −0.268462
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 4.00000 0.266076
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −8.00000 −0.529813
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 2.00000 0.131876
\(231\) 1.00000 0.0657952
\(232\) −6.00000 −0.393919
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 4.00000 0.261488
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) −4.00000 −0.259281
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) 1.00000 0.0638877
\(246\) −2.00000 −0.127515
\(247\) 32.0000 2.03611
\(248\) −4.00000 −0.254000
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 1.00000 0.0629941
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) 4.00000 0.248548
\(260\) 4.00000 0.248069
\(261\) −6.00000 −0.371391
\(262\) −12.0000 −0.741362
\(263\) −20.0000 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(264\) 1.00000 0.0615457
\(265\) 2.00000 0.122859
\(266\) 8.00000 0.490511
\(267\) −16.0000 −0.979184
\(268\) 14.0000 0.855186
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −4.00000 −0.242536
\(273\) −4.00000 −0.242091
\(274\) 4.00000 0.241649
\(275\) −1.00000 −0.0603023
\(276\) −2.00000 −0.120386
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 12.0000 0.719712
\(279\) −4.00000 −0.239474
\(280\) 1.00000 0.0597614
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 4.00000 0.238197
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 8.00000 0.474713
\(285\) −8.00000 −0.473879
\(286\) −4.00000 −0.236525
\(287\) 2.00000 0.118056
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −6.00000 −0.352332
\(291\) 14.0000 0.820695
\(292\) −10.0000 −0.585206
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 1.00000 0.0580259
\(298\) −10.0000 −0.579284
\(299\) 8.00000 0.462652
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 12.0000 0.690522
\(303\) 6.00000 0.344691
\(304\) 8.00000 0.458831
\(305\) 10.0000 0.572598
\(306\) −4.00000 −0.228665
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 16.0000 0.910208
\(310\) −4.00000 −0.227185
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) −4.00000 −0.226455
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 6.00000 0.338600
\(315\) 1.00000 0.0563436
\(316\) 4.00000 0.225018
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −2.00000 −0.112154
\(319\) 6.00000 0.335936
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 2.00000 0.111456
\(323\) −32.0000 −1.78053
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 14.0000 0.775388
\(327\) 4.00000 0.221201
\(328\) 2.00000 0.110432
\(329\) −4.00000 −0.220527
\(330\) 1.00000 0.0550482
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) 4.00000 0.219199
\(334\) 18.0000 0.984916
\(335\) 14.0000 0.764902
\(336\) −1.00000 −0.0545545
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 3.00000 0.163178
\(339\) −4.00000 −0.217250
\(340\) −4.00000 −0.216930
\(341\) 4.00000 0.216612
\(342\) 8.00000 0.432590
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) −2.00000 −0.107676
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 6.00000 0.321634
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 1.00000 0.0534522
\(351\) −4.00000 −0.213504
\(352\) −1.00000 −0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 16.0000 0.847998
\(357\) 4.00000 0.211702
\(358\) −4.00000 −0.211407
\(359\) 14.0000 0.738892 0.369446 0.929252i \(-0.379548\pi\)
0.369446 + 0.929252i \(0.379548\pi\)
\(360\) 1.00000 0.0527046
\(361\) 45.0000 2.36842
\(362\) −8.00000 −0.420471
\(363\) −1.00000 −0.0524864
\(364\) 4.00000 0.209657
\(365\) −10.0000 −0.523424
\(366\) −10.0000 −0.522708
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 2.00000 0.104257
\(369\) 2.00000 0.104116
\(370\) 4.00000 0.207950
\(371\) 2.00000 0.103835
\(372\) 4.00000 0.207390
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 4.00000 0.206835
\(375\) −1.00000 −0.0516398
\(376\) −4.00000 −0.206284
\(377\) −24.0000 −1.23606
\(378\) −1.00000 −0.0514344
\(379\) −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 \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.00000 −0.0509647
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) −4.00000 −0.202548
\(391\) −8.00000 −0.404577
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 10.0000 0.503793
\(395\) 4.00000 0.201262
\(396\) −1.00000 −0.0502519
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 12.0000 0.601506
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −14.0000 −0.698257
\(403\) −16.0000 −0.797017
\(404\) −6.00000 −0.298511
\(405\) 1.00000 0.0496904
\(406\) −6.00000 −0.297775
\(407\) −4.00000 −0.198273
\(408\) 4.00000 0.198030
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 2.00000 0.0987730
\(411\) −4.00000 −0.197305
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) −4.00000 −0.196352
\(416\) 4.00000 0.196116
\(417\) −12.0000 −0.587643
\(418\) −8.00000 −0.391293
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) −1.00000 −0.0487950
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 6.00000 0.292075
\(423\) −4.00000 −0.194487
\(424\) 2.00000 0.0971286
\(425\) −4.00000 −0.194029
\(426\) −8.00000 −0.387601
\(427\) 10.0000 0.483934
\(428\) 12.0000 0.580042
\(429\) 4.00000 0.193122
\(430\) 4.00000 0.192897
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) −4.00000 −0.192006
\(435\) 6.00000 0.287678
\(436\) −4.00000 −0.191565
\(437\) 16.0000 0.765384
\(438\) 10.0000 0.477818
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) −16.0000 −0.761042
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.00000 −0.189832
\(445\) 16.0000 0.758473
\(446\) −24.0000 −1.13643
\(447\) 10.0000 0.472984
\(448\) 1.00000 0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) −2.00000 −0.0941763
\(452\) 4.00000 0.188144
\(453\) −12.0000 −0.563809
\(454\) −4.00000 −0.187729
\(455\) 4.00000 0.187523
\(456\) −8.00000 −0.374634
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −12.0000 −0.560723
\(459\) 4.00000 0.186704
\(460\) 2.00000 0.0932505
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 1.00000 0.0465242
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −6.00000 −0.278543
\(465\) 4.00000 0.185496
\(466\) −14.0000 −0.648537
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 4.00000 0.184900
\(469\) 14.0000 0.646460
\(470\) −4.00000 −0.184506
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) −4.00000 −0.183726
\(475\) 8.00000 0.367065
\(476\) −4.00000 −0.183340
\(477\) 2.00000 0.0915737
\(478\) −6.00000 −0.274434
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 16.0000 0.729537
\(482\) 2.00000 0.0910975
\(483\) −2.00000 −0.0910032
\(484\) 1.00000 0.0454545
\(485\) −14.0000 −0.635707
\(486\) −1.00000 −0.0453609
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 10.0000 0.452679
\(489\) −14.0000 −0.633102
\(490\) 1.00000 0.0451754
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 24.0000 1.08091
\(494\) 32.0000 1.43975
\(495\) −1.00000 −0.0449467
\(496\) −4.00000 −0.179605
\(497\) 8.00000 0.358849
\(498\) 4.00000 0.179244
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 1.00000 0.0447214
\(501\) −18.0000 −0.804181
\(502\) 24.0000 1.07117
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 1.00000 0.0445435
\(505\) −6.00000 −0.266996
\(506\) −2.00000 −0.0889108
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 4.00000 0.177123
\(511\) −10.0000 −0.442374
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −14.0000 −0.617514
\(515\) −16.0000 −0.705044
\(516\) −4.00000 −0.176090
\(517\) 4.00000 0.175920
\(518\) 4.00000 0.175750
\(519\) 18.0000 0.790112
\(520\) 4.00000 0.175412
\(521\) −32.0000 −1.40195 −0.700973 0.713188i \(-0.747251\pi\)
−0.700973 + 0.713188i \(0.747251\pi\)
\(522\) −6.00000 −0.262613
\(523\) −42.0000 −1.83653 −0.918266 0.395964i \(-0.870410\pi\)
−0.918266 + 0.395964i \(0.870410\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) −20.0000 −0.872041
\(527\) 16.0000 0.696971
\(528\) 1.00000 0.0435194
\(529\) −19.0000 −0.826087
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) 8.00000 0.346518
\(534\) −16.0000 −0.692388
\(535\) 12.0000 0.518805
\(536\) 14.0000 0.604708
\(537\) 4.00000 0.172613
\(538\) −14.0000 −0.603583
\(539\) −1.00000 −0.0430730
\(540\) −1.00000 −0.0430331
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 16.0000 0.687259
\(543\) 8.00000 0.343313
\(544\) −4.00000 −0.171499
\(545\) −4.00000 −0.171341
\(546\) −4.00000 −0.171184
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 4.00000 0.170872
\(549\) 10.0000 0.426790
\(550\) −1.00000 −0.0426401
\(551\) −48.0000 −2.04487
\(552\) −2.00000 −0.0851257
\(553\) 4.00000 0.170097
\(554\) 10.0000 0.424859
\(555\) −4.00000 −0.169791
\(556\) 12.0000 0.508913
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −4.00000 −0.169334
\(559\) 16.0000 0.676728
\(560\) 1.00000 0.0422577
\(561\) −4.00000 −0.168880
\(562\) −4.00000 −0.168730
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 4.00000 0.168430
\(565\) 4.00000 0.168281
\(566\) −14.0000 −0.588464
\(567\) 1.00000 0.0419961
\(568\) 8.00000 0.335673
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) −8.00000 −0.335083
\(571\) 18.0000 0.753277 0.376638 0.926360i \(-0.377080\pi\)
0.376638 + 0.926360i \(0.377080\pi\)
\(572\) −4.00000 −0.167248
\(573\) −20.0000 −0.835512
\(574\) 2.00000 0.0834784
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 14.0000 0.581820
\(580\) −6.00000 −0.249136
\(581\) −4.00000 −0.165948
\(582\) 14.0000 0.580319
\(583\) −2.00000 −0.0828315
\(584\) −10.0000 −0.413803
\(585\) 4.00000 0.165380
\(586\) −14.0000 −0.578335
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 4.00000 0.164399
\(593\) −32.0000 −1.31408 −0.657041 0.753855i \(-0.728192\pi\)
−0.657041 + 0.753855i \(0.728192\pi\)
\(594\) 1.00000 0.0410305
\(595\) −4.00000 −0.163984
\(596\) −10.0000 −0.409616
\(597\) −12.0000 −0.491127
\(598\) 8.00000 0.327144
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 4.00000 0.163028
\(603\) 14.0000 0.570124
\(604\) 12.0000 0.488273
\(605\) 1.00000 0.0406558
\(606\) 6.00000 0.243733
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 8.00000 0.324443
\(609\) 6.00000 0.243132
\(610\) 10.0000 0.404888
\(611\) −16.0000 −0.647291
\(612\) −4.00000 −0.161690
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −14.0000 −0.564994
\(615\) −2.00000 −0.0806478
\(616\) −1.00000 −0.0402911
\(617\) −4.00000 −0.161034 −0.0805170 0.996753i \(-0.525657\pi\)
−0.0805170 + 0.996753i \(0.525657\pi\)
\(618\) 16.0000 0.643614
\(619\) 46.0000 1.84890 0.924448 0.381308i \(-0.124526\pi\)
0.924448 + 0.381308i \(0.124526\pi\)
\(620\) −4.00000 −0.160644
\(621\) −2.00000 −0.0802572
\(622\) −30.0000 −1.20289
\(623\) 16.0000 0.641026
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) 8.00000 0.319489
\(628\) 6.00000 0.239426
\(629\) −16.0000 −0.637962
\(630\) 1.00000 0.0398410
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 4.00000 0.159111
\(633\) −6.00000 −0.238479
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 4.00000 0.158486
\(638\) 6.00000 0.237542
\(639\) 8.00000 0.316475
\(640\) 1.00000 0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 2.00000 0.0788110
\(645\) −4.00000 −0.157500
\(646\) −32.0000 −1.25902
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 4.00000 0.156772
\(652\) 14.0000 0.548282
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 4.00000 0.156412
\(655\) −12.0000 −0.468879
\(656\) 2.00000 0.0780869
\(657\) −10.0000 −0.390137
\(658\) −4.00000 −0.155936
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 1.00000 0.0389249
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −20.0000 −0.777322
\(663\) 16.0000 0.621389
\(664\) −4.00000 −0.155230
\(665\) 8.00000 0.310227
\(666\) 4.00000 0.154997
\(667\) −12.0000 −0.464642
\(668\) 18.0000 0.696441
\(669\) 24.0000 0.927894
\(670\) 14.0000 0.540867
\(671\) −10.0000 −0.386046
\(672\) −1.00000 −0.0385758
\(673\) −50.0000 −1.92736 −0.963679 0.267063i \(-0.913947\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) −34.0000 −1.30963
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −4.00000 −0.153619
\(679\) −14.0000 −0.537271
\(680\) −4.00000 −0.153393
\(681\) 4.00000 0.153280
\(682\) 4.00000 0.153168
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 8.00000 0.305888
\(685\) 4.00000 0.152832
\(686\) 1.00000 0.0381802
\(687\) 12.0000 0.457829
\(688\) 4.00000 0.152499
\(689\) 8.00000 0.304776
\(690\) −2.00000 −0.0761387
\(691\) −26.0000 −0.989087 −0.494543 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) −18.0000 −0.684257
\(693\) −1.00000 −0.0379869
\(694\) −12.0000 −0.455514
\(695\) 12.0000 0.455186
\(696\) 6.00000 0.227429
\(697\) −8.00000 −0.303022
\(698\) −14.0000 −0.529908
\(699\) 14.0000 0.529529
\(700\) 1.00000 0.0377964
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −4.00000 −0.150970
\(703\) 32.0000 1.20690
\(704\) −1.00000 −0.0376889
\(705\) 4.00000 0.150649
\(706\) −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
\(711\) 4.00000 0.150012
\(712\) 16.0000 0.599625
\(713\) −8.00000 −0.299602
\(714\) 4.00000 0.149696
\(715\) −4.00000 −0.149592
\(716\) −4.00000 −0.149487
\(717\) 6.00000 0.224074
\(718\) 14.0000 0.522475
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 1.00000 0.0372678
\(721\) −16.0000 −0.595871
\(722\) 45.0000 1.67473
\(723\) −2.00000 −0.0743808
\(724\) −8.00000 −0.297318
\(725\) −6.00000 −0.222834
\(726\) −1.00000 −0.0371135
\(727\) 48.0000 1.78022 0.890111 0.455744i \(-0.150627\pi\)
0.890111 + 0.455744i \(0.150627\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) −16.0000 −0.591781
\(732\) −10.0000 −0.369611
\(733\) −20.0000 −0.738717 −0.369358 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) −24.0000 −0.885856
\(735\) −1.00000 −0.0368856
\(736\) 2.00000 0.0737210
\(737\) −14.0000 −0.515697
\(738\) 2.00000 0.0736210
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 4.00000 0.147043
\(741\) −32.0000 −1.17555
\(742\) 2.00000 0.0734223
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 4.00000 0.146647
\(745\) −10.0000 −0.366372
\(746\) 6.00000 0.219676
\(747\) −4.00000 −0.146352
\(748\) 4.00000 0.146254
\(749\) 12.0000 0.438470
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −4.00000 −0.145865
\(753\) −24.0000 −0.874609
\(754\) −24.0000 −0.874028
\(755\) 12.0000 0.436725
\(756\) −1.00000 −0.0363696
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) −20.0000 −0.726433
\(759\) 2.00000 0.0725954
\(760\) 8.00000 0.290191
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 20.0000 0.723575
\(765\) −4.00000 −0.144620
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 14.0000 0.504198
\(772\) −14.0000 −0.503871
\(773\) 26.0000 0.935155 0.467578 0.883952i \(-0.345127\pi\)
0.467578 + 0.883952i \(0.345127\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) −14.0000 −0.502571
\(777\) −4.00000 −0.143499
\(778\) −14.0000 −0.501924
\(779\) 16.0000 0.573259
\(780\) −4.00000 −0.143223
\(781\) −8.00000 −0.286263
\(782\) −8.00000 −0.286079
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 6.00000 0.214149
\(786\) 12.0000 0.428026
\(787\) −30.0000 −1.06938 −0.534692 0.845047i \(-0.679572\pi\)
−0.534692 + 0.845047i \(0.679572\pi\)
\(788\) 10.0000 0.356235
\(789\) 20.0000 0.712019
\(790\) 4.00000 0.142314
\(791\) 4.00000 0.142224
\(792\) −1.00000 −0.0355335
\(793\) 40.0000 1.42044
\(794\) −30.0000 −1.06466
\(795\) −2.00000 −0.0709327
\(796\) 12.0000 0.425329
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) −8.00000 −0.283197
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) 16.0000 0.565332
\(802\) −6.00000 −0.211867
\(803\) 10.0000 0.352892
\(804\) −14.0000 −0.493742
\(805\) 2.00000 0.0704907
\(806\) −16.0000 −0.563576
\(807\) 14.0000 0.492823
\(808\) −6.00000 −0.211079
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 1.00000 0.0351364
\(811\) 56.0000 1.96643 0.983213 0.182462i \(-0.0584065\pi\)
0.983213 + 0.182462i \(0.0584065\pi\)
\(812\) −6.00000 −0.210559
\(813\) −16.0000 −0.561144
\(814\) −4.00000 −0.140200
\(815\) 14.0000 0.490399
\(816\) 4.00000 0.140028
\(817\) 32.0000 1.11954
\(818\) 14.0000 0.489499
\(819\) 4.00000 0.139771
\(820\) 2.00000 0.0698430
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −4.00000 −0.139516
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −16.0000 −0.557386
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) 2.00000 0.0695048
\(829\) 16.0000 0.555703 0.277851 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) −4.00000 −0.138842
\(831\) −10.0000 −0.346896
\(832\) 4.00000 0.138675
\(833\) −4.00000 −0.138592
\(834\) −12.0000 −0.415526
\(835\) 18.0000 0.622916
\(836\) −8.00000 −0.276686
\(837\) 4.00000 0.138260
\(838\) −28.0000 −0.967244
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) 4.00000 0.137767
\(844\) 6.00000 0.206529
\(845\) 3.00000 0.103203
\(846\) −4.00000 −0.137523
\(847\) 1.00000 0.0343604
\(848\) 2.00000 0.0686803
\(849\) 14.0000 0.480479
\(850\) −4.00000 −0.137199
\(851\) 8.00000 0.274236
\(852\) −8.00000 −0.274075
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 10.0000 0.342193
\(855\) 8.00000 0.273594
\(856\) 12.0000 0.410152
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 4.00000 0.136558
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 4.00000 0.136399
\(861\) −2.00000 −0.0681598
\(862\) −14.0000 −0.476842
\(863\) 34.0000 1.15737 0.578687 0.815550i \(-0.303565\pi\)
0.578687 + 0.815550i \(0.303565\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −18.0000 −0.612018
\(866\) −14.0000 −0.475739
\(867\) 1.00000 0.0339618
\(868\) −4.00000 −0.135769
\(869\) −4.00000 −0.135691
\(870\) 6.00000 0.203419
\(871\) 56.0000 1.89749
\(872\) −4.00000 −0.135457
\(873\) −14.0000 −0.473828
\(874\) 16.0000 0.541208
\(875\) 1.00000 0.0338062
\(876\) 10.0000 0.337869
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 40.0000 1.34993
\(879\) 14.0000 0.472208
\(880\) −1.00000 −0.0337100
\(881\) 40.0000 1.34763 0.673817 0.738898i \(-0.264654\pi\)
0.673817 + 0.738898i \(0.264654\pi\)
\(882\) 1.00000 0.0336718
\(883\) −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 \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) −4.00000 −0.134231
\(889\) 0 0
\(890\) 16.0000 0.536321
\(891\) −1.00000 −0.0335013
\(892\) −24.0000 −0.803579
\(893\) −32.0000 −1.07084
\(894\) 10.0000 0.334450
\(895\) −4.00000 −0.133705
\(896\) 1.00000 0.0334077
\(897\) −8.00000 −0.267112
\(898\) −6.00000 −0.200223
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) −8.00000 −0.266519
\(902\) −2.00000 −0.0665927
\(903\) −4.00000 −0.133112
\(904\) 4.00000 0.133038
\(905\) −8.00000 −0.265929
\(906\) −12.0000 −0.398673
\(907\) −38.0000 −1.26177 −0.630885 0.775877i \(-0.717308\pi\)
−0.630885 + 0.775877i \(0.717308\pi\)
\(908\) −4.00000 −0.132745
\(909\) −6.00000 −0.199007
\(910\) 4.00000 0.132599
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) −8.00000 −0.264906
\(913\) 4.00000 0.132381
\(914\) 10.0000 0.330771
\(915\) −10.0000 −0.330590
\(916\) −12.0000 −0.396491
\(917\) −12.0000 −0.396275
\(918\) 4.00000 0.132020
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 2.00000 0.0659380
\(921\) 14.0000 0.461316
\(922\) 14.0000 0.461065
\(923\) 32.0000 1.05329
\(924\) 1.00000 0.0328976
\(925\) 4.00000 0.131519
\(926\) 16.0000 0.525793
\(927\) −16.0000 −0.525509
\(928\) −6.00000 −0.196960
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 4.00000 0.131165
\(931\) 8.00000 0.262189
\(932\) −14.0000 −0.458585
\(933\) 30.0000 0.982156
\(934\) 36.0000 1.17796
\(935\) 4.00000 0.130814
\(936\) 4.00000 0.130744
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 14.0000 0.457116
\(939\) −6.00000 −0.195803
\(940\) −4.00000 −0.130466
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) −6.00000 −0.195491
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) −4.00000 −0.130051
\(947\) 16.0000 0.519930 0.259965 0.965618i \(-0.416289\pi\)
0.259965 + 0.965618i \(0.416289\pi\)
\(948\) −4.00000 −0.129914
\(949\) −40.0000 −1.29845
\(950\) 8.00000 0.259554
\(951\) 6.00000 0.194563
\(952\) −4.00000 −0.129641
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 2.00000 0.0647524
\(955\) 20.0000 0.647185
\(956\) −6.00000 −0.194054
\(957\) −6.00000 −0.193952
\(958\) 24.0000 0.775405
\(959\) 4.00000 0.129167
\(960\) −1.00000 −0.0322749
\(961\) −15.0000 −0.483871
\(962\) 16.0000 0.515861
\(963\) 12.0000 0.386695
\(964\) 2.00000 0.0644157
\(965\) −14.0000 −0.450676
\(966\) −2.00000 −0.0643489
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) 32.0000 1.02799
\(970\) −14.0000 −0.449513
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 12.0000 0.384702
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 10.0000 0.320092
\(977\) 32.0000 1.02377 0.511885 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) −14.0000 −0.447671
\(979\) −16.0000 −0.511362
\(980\) 1.00000 0.0319438
\(981\) −4.00000 −0.127710
\(982\) −28.0000 −0.893516
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 10.0000 0.318626
\(986\) 24.0000 0.764316
\(987\) 4.00000 0.127321
\(988\) 32.0000 1.01806
\(989\) 8.00000 0.254385
\(990\) −1.00000 −0.0317821
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −4.00000 −0.127000
\(993\) 20.0000 0.634681
\(994\) 8.00000 0.253745
\(995\) 12.0000 0.380426
\(996\) 4.00000 0.126745
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) −8.00000 −0.253236
\(999\) −4.00000 −0.126554
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 2310.2.a.p.1.1 1
3.2 odd 2 6930.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.p.1.1 1 1.1 even 1 trivial
6930.2.a.g.1.1 1 3.2 odd 2