Properties

Label 3630.2.a.bp.1.2
Level $3630$
Weight $2$
Character 3630.1
Self dual yes
Analytic conductor $28.986$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3630.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9856959337\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3630.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.73205 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +3.73205 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} +3.00000 q^{13} +3.73205 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.267949 q^{17} +1.00000 q^{18} -4.46410 q^{19} +1.00000 q^{20} +3.73205 q^{21} -4.73205 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.00000 q^{26} +1.00000 q^{27} +3.73205 q^{28} -3.00000 q^{29} +1.00000 q^{30} +8.19615 q^{31} +1.00000 q^{32} +0.267949 q^{34} +3.73205 q^{35} +1.00000 q^{36} -1.73205 q^{37} -4.46410 q^{38} +3.00000 q^{39} +1.00000 q^{40} +4.73205 q^{41} +3.73205 q^{42} +6.73205 q^{43} +1.00000 q^{45} -4.73205 q^{46} +8.92820 q^{47} +1.00000 q^{48} +6.92820 q^{49} +1.00000 q^{50} +0.267949 q^{51} +3.00000 q^{52} -5.26795 q^{53} +1.00000 q^{54} +3.73205 q^{56} -4.46410 q^{57} -3.00000 q^{58} -4.19615 q^{59} +1.00000 q^{60} -2.19615 q^{61} +8.19615 q^{62} +3.73205 q^{63} +1.00000 q^{64} +3.00000 q^{65} -11.4641 q^{67} +0.267949 q^{68} -4.73205 q^{69} +3.73205 q^{70} -9.92820 q^{71} +1.00000 q^{72} +6.19615 q^{73} -1.73205 q^{74} +1.00000 q^{75} -4.46410 q^{76} +3.00000 q^{78} +0.535898 q^{79} +1.00000 q^{80} +1.00000 q^{81} +4.73205 q^{82} -11.3923 q^{83} +3.73205 q^{84} +0.267949 q^{85} +6.73205 q^{86} -3.00000 q^{87} -16.3923 q^{89} +1.00000 q^{90} +11.1962 q^{91} -4.73205 q^{92} +8.19615 q^{93} +8.92820 q^{94} -4.46410 q^{95} +1.00000 q^{96} -18.7321 q^{97} +6.92820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 6q^{13} + 4q^{14} + 2q^{15} + 2q^{16} + 4q^{17} + 2q^{18} - 2q^{19} + 2q^{20} + 4q^{21} - 6q^{23} + 2q^{24} + 2q^{25} + 6q^{26} + 2q^{27} + 4q^{28} - 6q^{29} + 2q^{30} + 6q^{31} + 2q^{32} + 4q^{34} + 4q^{35} + 2q^{36} - 2q^{38} + 6q^{39} + 2q^{40} + 6q^{41} + 4q^{42} + 10q^{43} + 2q^{45} - 6q^{46} + 4q^{47} + 2q^{48} + 2q^{50} + 4q^{51} + 6q^{52} - 14q^{53} + 2q^{54} + 4q^{56} - 2q^{57} - 6q^{58} + 2q^{59} + 2q^{60} + 6q^{61} + 6q^{62} + 4q^{63} + 2q^{64} + 6q^{65} - 16q^{67} + 4q^{68} - 6q^{69} + 4q^{70} - 6q^{71} + 2q^{72} + 2q^{73} + 2q^{75} - 2q^{76} + 6q^{78} + 8q^{79} + 2q^{80} + 2q^{81} + 6q^{82} - 2q^{83} + 4q^{84} + 4q^{85} + 10q^{86} - 6q^{87} - 12q^{89} + 2q^{90} + 12q^{91} - 6q^{92} + 6q^{93} + 4q^{94} - 2q^{95} + 2q^{96} - 34q^{97} + O(q^{100}) \)

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
\(7\) 3.73205 1.41058 0.705291 0.708918i \(-0.250816\pi\)
0.705291 + 0.708918i \(0.250816\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 3.73205 0.997433
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0.267949 0.0649872 0.0324936 0.999472i \(-0.489655\pi\)
0.0324936 + 0.999472i \(0.489655\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.46410 −1.02414 −0.512068 0.858945i \(-0.671120\pi\)
−0.512068 + 0.858945i \(0.671120\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.73205 0.814400
\(22\) 0 0
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 3.00000 0.588348
\(27\) 1.00000 0.192450
\(28\) 3.73205 0.705291
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 1.00000 0.182574
\(31\) 8.19615 1.47207 0.736036 0.676942i \(-0.236695\pi\)
0.736036 + 0.676942i \(0.236695\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0.267949 0.0459529
\(35\) 3.73205 0.630832
\(36\) 1.00000 0.166667
\(37\) −1.73205 −0.284747 −0.142374 0.989813i \(-0.545473\pi\)
−0.142374 + 0.989813i \(0.545473\pi\)
\(38\) −4.46410 −0.724173
\(39\) 3.00000 0.480384
\(40\) 1.00000 0.158114
\(41\) 4.73205 0.739022 0.369511 0.929226i \(-0.379525\pi\)
0.369511 + 0.929226i \(0.379525\pi\)
\(42\) 3.73205 0.575868
\(43\) 6.73205 1.02663 0.513314 0.858201i \(-0.328418\pi\)
0.513314 + 0.858201i \(0.328418\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −4.73205 −0.697703
\(47\) 8.92820 1.30231 0.651156 0.758944i \(-0.274284\pi\)
0.651156 + 0.758944i \(0.274284\pi\)
\(48\) 1.00000 0.144338
\(49\) 6.92820 0.989743
\(50\) 1.00000 0.141421
\(51\) 0.267949 0.0375204
\(52\) 3.00000 0.416025
\(53\) −5.26795 −0.723608 −0.361804 0.932254i \(-0.617839\pi\)
−0.361804 + 0.932254i \(0.617839\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.73205 0.498716
\(57\) −4.46410 −0.591285
\(58\) −3.00000 −0.393919
\(59\) −4.19615 −0.546293 −0.273146 0.961973i \(-0.588064\pi\)
−0.273146 + 0.961973i \(0.588064\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.19615 −0.281189 −0.140594 0.990067i \(-0.544901\pi\)
−0.140594 + 0.990067i \(0.544901\pi\)
\(62\) 8.19615 1.04091
\(63\) 3.73205 0.470194
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −11.4641 −1.40056 −0.700281 0.713867i \(-0.746942\pi\)
−0.700281 + 0.713867i \(0.746942\pi\)
\(68\) 0.267949 0.0324936
\(69\) −4.73205 −0.569672
\(70\) 3.73205 0.446065
\(71\) −9.92820 −1.17826 −0.589130 0.808038i \(-0.700530\pi\)
−0.589130 + 0.808038i \(0.700530\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.19615 0.725205 0.362602 0.931944i \(-0.381888\pi\)
0.362602 + 0.931944i \(0.381888\pi\)
\(74\) −1.73205 −0.201347
\(75\) 1.00000 0.115470
\(76\) −4.46410 −0.512068
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) 0.535898 0.0602933 0.0301466 0.999545i \(-0.490403\pi\)
0.0301466 + 0.999545i \(0.490403\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 4.73205 0.522568
\(83\) −11.3923 −1.25047 −0.625234 0.780437i \(-0.714996\pi\)
−0.625234 + 0.780437i \(0.714996\pi\)
\(84\) 3.73205 0.407200
\(85\) 0.267949 0.0290632
\(86\) 6.73205 0.725936
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −16.3923 −1.73758 −0.868790 0.495180i \(-0.835102\pi\)
−0.868790 + 0.495180i \(0.835102\pi\)
\(90\) 1.00000 0.105409
\(91\) 11.1962 1.17368
\(92\) −4.73205 −0.493350
\(93\) 8.19615 0.849901
\(94\) 8.92820 0.920874
\(95\) −4.46410 −0.458007
\(96\) 1.00000 0.102062
\(97\) −18.7321 −1.90195 −0.950976 0.309265i \(-0.899917\pi\)
−0.950976 + 0.309265i \(0.899917\pi\)
\(98\) 6.92820 0.699854
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −5.53590 −0.550842 −0.275421 0.961324i \(-0.588817\pi\)
−0.275421 + 0.961324i \(0.588817\pi\)
\(102\) 0.267949 0.0265309
\(103\) −18.3205 −1.80517 −0.902587 0.430508i \(-0.858334\pi\)
−0.902587 + 0.430508i \(0.858334\pi\)
\(104\) 3.00000 0.294174
\(105\) 3.73205 0.364211
\(106\) −5.26795 −0.511668
\(107\) 11.4641 1.10828 0.554138 0.832425i \(-0.313048\pi\)
0.554138 + 0.832425i \(0.313048\pi\)
\(108\) 1.00000 0.0962250
\(109\) 13.4641 1.28963 0.644814 0.764340i \(-0.276935\pi\)
0.644814 + 0.764340i \(0.276935\pi\)
\(110\) 0 0
\(111\) −1.73205 −0.164399
\(112\) 3.73205 0.352646
\(113\) 13.8564 1.30350 0.651751 0.758433i \(-0.274035\pi\)
0.651751 + 0.758433i \(0.274035\pi\)
\(114\) −4.46410 −0.418101
\(115\) −4.73205 −0.441266
\(116\) −3.00000 −0.278543
\(117\) 3.00000 0.277350
\(118\) −4.19615 −0.386287
\(119\) 1.00000 0.0916698
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) −2.19615 −0.198830
\(123\) 4.73205 0.426675
\(124\) 8.19615 0.736036
\(125\) 1.00000 0.0894427
\(126\) 3.73205 0.332478
\(127\) −0.392305 −0.0348114 −0.0174057 0.999849i \(-0.505541\pi\)
−0.0174057 + 0.999849i \(0.505541\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.73205 0.592724
\(130\) 3.00000 0.263117
\(131\) 11.6603 1.01876 0.509381 0.860541i \(-0.329875\pi\)
0.509381 + 0.860541i \(0.329875\pi\)
\(132\) 0 0
\(133\) −16.6603 −1.44463
\(134\) −11.4641 −0.990348
\(135\) 1.00000 0.0860663
\(136\) 0.267949 0.0229765
\(137\) −19.7846 −1.69031 −0.845157 0.534519i \(-0.820493\pi\)
−0.845157 + 0.534519i \(0.820493\pi\)
\(138\) −4.73205 −0.402819
\(139\) 20.3205 1.72356 0.861781 0.507280i \(-0.169349\pi\)
0.861781 + 0.507280i \(0.169349\pi\)
\(140\) 3.73205 0.315416
\(141\) 8.92820 0.751890
\(142\) −9.92820 −0.833156
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) 6.19615 0.512797
\(147\) 6.92820 0.571429
\(148\) −1.73205 −0.142374
\(149\) −0.392305 −0.0321389 −0.0160694 0.999871i \(-0.505115\pi\)
−0.0160694 + 0.999871i \(0.505115\pi\)
\(150\) 1.00000 0.0816497
\(151\) 13.2679 1.07973 0.539865 0.841751i \(-0.318475\pi\)
0.539865 + 0.841751i \(0.318475\pi\)
\(152\) −4.46410 −0.362086
\(153\) 0.267949 0.0216624
\(154\) 0 0
\(155\) 8.19615 0.658331
\(156\) 3.00000 0.240192
\(157\) 13.1962 1.05317 0.526584 0.850123i \(-0.323473\pi\)
0.526584 + 0.850123i \(0.323473\pi\)
\(158\) 0.535898 0.0426338
\(159\) −5.26795 −0.417776
\(160\) 1.00000 0.0790569
\(161\) −17.6603 −1.39182
\(162\) 1.00000 0.0785674
\(163\) 18.0526 1.41399 0.706993 0.707221i \(-0.250051\pi\)
0.706993 + 0.707221i \(0.250051\pi\)
\(164\) 4.73205 0.369511
\(165\) 0 0
\(166\) −11.3923 −0.884214
\(167\) −21.4641 −1.66094 −0.830471 0.557062i \(-0.811929\pi\)
−0.830471 + 0.557062i \(0.811929\pi\)
\(168\) 3.73205 0.287934
\(169\) −4.00000 −0.307692
\(170\) 0.267949 0.0205508
\(171\) −4.46410 −0.341378
\(172\) 6.73205 0.513314
\(173\) 14.5885 1.10914 0.554570 0.832137i \(-0.312883\pi\)
0.554570 + 0.832137i \(0.312883\pi\)
\(174\) −3.00000 −0.227429
\(175\) 3.73205 0.282117
\(176\) 0 0
\(177\) −4.19615 −0.315402
\(178\) −16.3923 −1.22866
\(179\) 7.12436 0.532499 0.266250 0.963904i \(-0.414215\pi\)
0.266250 + 0.963904i \(0.414215\pi\)
\(180\) 1.00000 0.0745356
\(181\) 4.19615 0.311898 0.155949 0.987765i \(-0.450157\pi\)
0.155949 + 0.987765i \(0.450157\pi\)
\(182\) 11.1962 0.829914
\(183\) −2.19615 −0.162344
\(184\) −4.73205 −0.348851
\(185\) −1.73205 −0.127343
\(186\) 8.19615 0.600971
\(187\) 0 0
\(188\) 8.92820 0.651156
\(189\) 3.73205 0.271467
\(190\) −4.46410 −0.323860
\(191\) 18.4641 1.33602 0.668008 0.744154i \(-0.267147\pi\)
0.668008 + 0.744154i \(0.267147\pi\)
\(192\) 1.00000 0.0721688
\(193\) 9.46410 0.681241 0.340620 0.940201i \(-0.389363\pi\)
0.340620 + 0.940201i \(0.389363\pi\)
\(194\) −18.7321 −1.34488
\(195\) 3.00000 0.214834
\(196\) 6.92820 0.494872
\(197\) −13.4641 −0.959278 −0.479639 0.877466i \(-0.659232\pi\)
−0.479639 + 0.877466i \(0.659232\pi\)
\(198\) 0 0
\(199\) −16.3923 −1.16202 −0.581010 0.813897i \(-0.697342\pi\)
−0.581010 + 0.813897i \(0.697342\pi\)
\(200\) 1.00000 0.0707107
\(201\) −11.4641 −0.808615
\(202\) −5.53590 −0.389504
\(203\) −11.1962 −0.785816
\(204\) 0.267949 0.0187602
\(205\) 4.73205 0.330501
\(206\) −18.3205 −1.27645
\(207\) −4.73205 −0.328900
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 3.73205 0.257536
\(211\) 17.7846 1.22434 0.612172 0.790725i \(-0.290296\pi\)
0.612172 + 0.790725i \(0.290296\pi\)
\(212\) −5.26795 −0.361804
\(213\) −9.92820 −0.680269
\(214\) 11.4641 0.783670
\(215\) 6.73205 0.459122
\(216\) 1.00000 0.0680414
\(217\) 30.5885 2.07648
\(218\) 13.4641 0.911904
\(219\) 6.19615 0.418697
\(220\) 0 0
\(221\) 0.803848 0.0540726
\(222\) −1.73205 −0.116248
\(223\) 6.46410 0.432868 0.216434 0.976297i \(-0.430557\pi\)
0.216434 + 0.976297i \(0.430557\pi\)
\(224\) 3.73205 0.249358
\(225\) 1.00000 0.0666667
\(226\) 13.8564 0.921714
\(227\) −10.3923 −0.689761 −0.344881 0.938647i \(-0.612081\pi\)
−0.344881 + 0.938647i \(0.612081\pi\)
\(228\) −4.46410 −0.295642
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) −4.73205 −0.312022
\(231\) 0 0
\(232\) −3.00000 −0.196960
\(233\) 1.85641 0.121617 0.0608086 0.998149i \(-0.480632\pi\)
0.0608086 + 0.998149i \(0.480632\pi\)
\(234\) 3.00000 0.196116
\(235\) 8.92820 0.582412
\(236\) −4.19615 −0.273146
\(237\) 0.535898 0.0348103
\(238\) 1.00000 0.0648204
\(239\) −6.26795 −0.405440 −0.202720 0.979237i \(-0.564978\pi\)
−0.202720 + 0.979237i \(0.564978\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.6603 −1.45968 −0.729838 0.683621i \(-0.760404\pi\)
−0.729838 + 0.683621i \(0.760404\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.19615 −0.140594
\(245\) 6.92820 0.442627
\(246\) 4.73205 0.301705
\(247\) −13.3923 −0.852132
\(248\) 8.19615 0.520456
\(249\) −11.3923 −0.721958
\(250\) 1.00000 0.0632456
\(251\) 18.5359 1.16998 0.584988 0.811042i \(-0.301099\pi\)
0.584988 + 0.811042i \(0.301099\pi\)
\(252\) 3.73205 0.235097
\(253\) 0 0
\(254\) −0.392305 −0.0246154
\(255\) 0.267949 0.0167796
\(256\) 1.00000 0.0625000
\(257\) 6.32051 0.394262 0.197131 0.980377i \(-0.436837\pi\)
0.197131 + 0.980377i \(0.436837\pi\)
\(258\) 6.73205 0.419119
\(259\) −6.46410 −0.401660
\(260\) 3.00000 0.186052
\(261\) −3.00000 −0.185695
\(262\) 11.6603 0.720373
\(263\) −25.1244 −1.54923 −0.774617 0.632431i \(-0.782057\pi\)
−0.774617 + 0.632431i \(0.782057\pi\)
\(264\) 0 0
\(265\) −5.26795 −0.323608
\(266\) −16.6603 −1.02151
\(267\) −16.3923 −1.00319
\(268\) −11.4641 −0.700281
\(269\) 11.5885 0.706561 0.353280 0.935517i \(-0.385066\pi\)
0.353280 + 0.935517i \(0.385066\pi\)
\(270\) 1.00000 0.0608581
\(271\) −23.8564 −1.44917 −0.724587 0.689184i \(-0.757969\pi\)
−0.724587 + 0.689184i \(0.757969\pi\)
\(272\) 0.267949 0.0162468
\(273\) 11.1962 0.677622
\(274\) −19.7846 −1.19523
\(275\) 0 0
\(276\) −4.73205 −0.284836
\(277\) 18.9282 1.13729 0.568643 0.822585i \(-0.307469\pi\)
0.568643 + 0.822585i \(0.307469\pi\)
\(278\) 20.3205 1.21874
\(279\) 8.19615 0.490691
\(280\) 3.73205 0.223033
\(281\) 26.0526 1.55417 0.777083 0.629399i \(-0.216699\pi\)
0.777083 + 0.629399i \(0.216699\pi\)
\(282\) 8.92820 0.531667
\(283\) −30.2487 −1.79810 −0.899050 0.437847i \(-0.855741\pi\)
−0.899050 + 0.437847i \(0.855741\pi\)
\(284\) −9.92820 −0.589130
\(285\) −4.46410 −0.264431
\(286\) 0 0
\(287\) 17.6603 1.04245
\(288\) 1.00000 0.0589256
\(289\) −16.9282 −0.995777
\(290\) −3.00000 −0.176166
\(291\) −18.7321 −1.09809
\(292\) 6.19615 0.362602
\(293\) −18.9282 −1.10580 −0.552899 0.833248i \(-0.686478\pi\)
−0.552899 + 0.833248i \(0.686478\pi\)
\(294\) 6.92820 0.404061
\(295\) −4.19615 −0.244309
\(296\) −1.73205 −0.100673
\(297\) 0 0
\(298\) −0.392305 −0.0227256
\(299\) −14.1962 −0.820985
\(300\) 1.00000 0.0577350
\(301\) 25.1244 1.44814
\(302\) 13.2679 0.763485
\(303\) −5.53590 −0.318029
\(304\) −4.46410 −0.256034
\(305\) −2.19615 −0.125751
\(306\) 0.267949 0.0153176
\(307\) 6.73205 0.384218 0.192109 0.981374i \(-0.438467\pi\)
0.192109 + 0.981374i \(0.438467\pi\)
\(308\) 0 0
\(309\) −18.3205 −1.04222
\(310\) 8.19615 0.465510
\(311\) 13.8564 0.785725 0.392862 0.919597i \(-0.371485\pi\)
0.392862 + 0.919597i \(0.371485\pi\)
\(312\) 3.00000 0.169842
\(313\) −19.1244 −1.08097 −0.540486 0.841353i \(-0.681760\pi\)
−0.540486 + 0.841353i \(0.681760\pi\)
\(314\) 13.1962 0.744702
\(315\) 3.73205 0.210277
\(316\) 0.535898 0.0301466
\(317\) −15.1244 −0.849468 −0.424734 0.905318i \(-0.639632\pi\)
−0.424734 + 0.905318i \(0.639632\pi\)
\(318\) −5.26795 −0.295412
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 11.4641 0.639864
\(322\) −17.6603 −0.984167
\(323\) −1.19615 −0.0665557
\(324\) 1.00000 0.0555556
\(325\) 3.00000 0.166410
\(326\) 18.0526 0.999839
\(327\) 13.4641 0.744567
\(328\) 4.73205 0.261284
\(329\) 33.3205 1.83702
\(330\) 0 0
\(331\) −30.5167 −1.67735 −0.838674 0.544634i \(-0.816668\pi\)
−0.838674 + 0.544634i \(0.816668\pi\)
\(332\) −11.3923 −0.625234
\(333\) −1.73205 −0.0949158
\(334\) −21.4641 −1.17446
\(335\) −11.4641 −0.626351
\(336\) 3.73205 0.203600
\(337\) 23.2679 1.26749 0.633743 0.773544i \(-0.281518\pi\)
0.633743 + 0.773544i \(0.281518\pi\)
\(338\) −4.00000 −0.217571
\(339\) 13.8564 0.752577
\(340\) 0.267949 0.0145316
\(341\) 0 0
\(342\) −4.46410 −0.241391
\(343\) −0.267949 −0.0144679
\(344\) 6.73205 0.362968
\(345\) −4.73205 −0.254765
\(346\) 14.5885 0.784280
\(347\) 25.3923 1.36313 0.681565 0.731757i \(-0.261300\pi\)
0.681565 + 0.731757i \(0.261300\pi\)
\(348\) −3.00000 −0.160817
\(349\) 7.85641 0.420544 0.210272 0.977643i \(-0.432565\pi\)
0.210272 + 0.977643i \(0.432565\pi\)
\(350\) 3.73205 0.199487
\(351\) 3.00000 0.160128
\(352\) 0 0
\(353\) 6.92820 0.368751 0.184376 0.982856i \(-0.440974\pi\)
0.184376 + 0.982856i \(0.440974\pi\)
\(354\) −4.19615 −0.223023
\(355\) −9.92820 −0.526934
\(356\) −16.3923 −0.868790
\(357\) 1.00000 0.0529256
\(358\) 7.12436 0.376534
\(359\) −27.4641 −1.44950 −0.724750 0.689012i \(-0.758045\pi\)
−0.724750 + 0.689012i \(0.758045\pi\)
\(360\) 1.00000 0.0527046
\(361\) 0.928203 0.0488528
\(362\) 4.19615 0.220545
\(363\) 0 0
\(364\) 11.1962 0.586838
\(365\) 6.19615 0.324321
\(366\) −2.19615 −0.114795
\(367\) 14.8564 0.775498 0.387749 0.921765i \(-0.373253\pi\)
0.387749 + 0.921765i \(0.373253\pi\)
\(368\) −4.73205 −0.246675
\(369\) 4.73205 0.246341
\(370\) −1.73205 −0.0900450
\(371\) −19.6603 −1.02071
\(372\) 8.19615 0.424951
\(373\) 6.07180 0.314386 0.157193 0.987568i \(-0.449756\pi\)
0.157193 + 0.987568i \(0.449756\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 8.92820 0.460437
\(377\) −9.00000 −0.463524
\(378\) 3.73205 0.191956
\(379\) 21.4449 1.10155 0.550774 0.834654i \(-0.314332\pi\)
0.550774 + 0.834654i \(0.314332\pi\)
\(380\) −4.46410 −0.229004
\(381\) −0.392305 −0.0200984
\(382\) 18.4641 0.944706
\(383\) −16.9282 −0.864991 −0.432495 0.901636i \(-0.642367\pi\)
−0.432495 + 0.901636i \(0.642367\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 9.46410 0.481710
\(387\) 6.73205 0.342209
\(388\) −18.7321 −0.950976
\(389\) −37.8564 −1.91940 −0.959698 0.281033i \(-0.909323\pi\)
−0.959698 + 0.281033i \(0.909323\pi\)
\(390\) 3.00000 0.151911
\(391\) −1.26795 −0.0641229
\(392\) 6.92820 0.349927
\(393\) 11.6603 0.588182
\(394\) −13.4641 −0.678312
\(395\) 0.535898 0.0269640
\(396\) 0 0
\(397\) 25.9808 1.30394 0.651969 0.758246i \(-0.273943\pi\)
0.651969 + 0.758246i \(0.273943\pi\)
\(398\) −16.3923 −0.821672
\(399\) −16.6603 −0.834056
\(400\) 1.00000 0.0500000
\(401\) 17.6603 0.881911 0.440956 0.897529i \(-0.354640\pi\)
0.440956 + 0.897529i \(0.354640\pi\)
\(402\) −11.4641 −0.571777
\(403\) 24.5885 1.22484
\(404\) −5.53590 −0.275421
\(405\) 1.00000 0.0496904
\(406\) −11.1962 −0.555656
\(407\) 0 0
\(408\) 0.267949 0.0132655
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 4.73205 0.233699
\(411\) −19.7846 −0.975903
\(412\) −18.3205 −0.902587
\(413\) −15.6603 −0.770591
\(414\) −4.73205 −0.232568
\(415\) −11.3923 −0.559226
\(416\) 3.00000 0.147087
\(417\) 20.3205 0.995100
\(418\) 0 0
\(419\) −11.4641 −0.560058 −0.280029 0.959992i \(-0.590344\pi\)
−0.280029 + 0.959992i \(0.590344\pi\)
\(420\) 3.73205 0.182105
\(421\) −4.05256 −0.197510 −0.0987548 0.995112i \(-0.531486\pi\)
−0.0987548 + 0.995112i \(0.531486\pi\)
\(422\) 17.7846 0.865741
\(423\) 8.92820 0.434104
\(424\) −5.26795 −0.255834
\(425\) 0.267949 0.0129974
\(426\) −9.92820 −0.481023
\(427\) −8.19615 −0.396640
\(428\) 11.4641 0.554138
\(429\) 0 0
\(430\) 6.73205 0.324648
\(431\) 6.12436 0.295000 0.147500 0.989062i \(-0.452877\pi\)
0.147500 + 0.989062i \(0.452877\pi\)
\(432\) 1.00000 0.0481125
\(433\) −10.9282 −0.525176 −0.262588 0.964908i \(-0.584576\pi\)
−0.262588 + 0.964908i \(0.584576\pi\)
\(434\) 30.5885 1.46829
\(435\) −3.00000 −0.143839
\(436\) 13.4641 0.644814
\(437\) 21.1244 1.01051
\(438\) 6.19615 0.296064
\(439\) 4.92820 0.235210 0.117605 0.993060i \(-0.462478\pi\)
0.117605 + 0.993060i \(0.462478\pi\)
\(440\) 0 0
\(441\) 6.92820 0.329914
\(442\) 0.803848 0.0382351
\(443\) −5.19615 −0.246877 −0.123438 0.992352i \(-0.539392\pi\)
−0.123438 + 0.992352i \(0.539392\pi\)
\(444\) −1.73205 −0.0821995
\(445\) −16.3923 −0.777070
\(446\) 6.46410 0.306084
\(447\) −0.392305 −0.0185554
\(448\) 3.73205 0.176323
\(449\) −3.46410 −0.163481 −0.0817405 0.996654i \(-0.526048\pi\)
−0.0817405 + 0.996654i \(0.526048\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 13.8564 0.651751
\(453\) 13.2679 0.623383
\(454\) −10.3923 −0.487735
\(455\) 11.1962 0.524884
\(456\) −4.46410 −0.209051
\(457\) −30.4449 −1.42415 −0.712075 0.702103i \(-0.752245\pi\)
−0.712075 + 0.702103i \(0.752245\pi\)
\(458\) −12.0000 −0.560723
\(459\) 0.267949 0.0125068
\(460\) −4.73205 −0.220633
\(461\) −17.2487 −0.803353 −0.401676 0.915782i \(-0.631572\pi\)
−0.401676 + 0.915782i \(0.631572\pi\)
\(462\) 0 0
\(463\) −11.4641 −0.532782 −0.266391 0.963865i \(-0.585831\pi\)
−0.266391 + 0.963865i \(0.585831\pi\)
\(464\) −3.00000 −0.139272
\(465\) 8.19615 0.380087
\(466\) 1.85641 0.0859964
\(467\) −30.1244 −1.39399 −0.696994 0.717077i \(-0.745480\pi\)
−0.696994 + 0.717077i \(0.745480\pi\)
\(468\) 3.00000 0.138675
\(469\) −42.7846 −1.97561
\(470\) 8.92820 0.411827
\(471\) 13.1962 0.608047
\(472\) −4.19615 −0.193144
\(473\) 0 0
\(474\) 0.535898 0.0246146
\(475\) −4.46410 −0.204827
\(476\) 1.00000 0.0458349
\(477\) −5.26795 −0.241203
\(478\) −6.26795 −0.286689
\(479\) −8.26795 −0.377772 −0.188886 0.981999i \(-0.560488\pi\)
−0.188886 + 0.981999i \(0.560488\pi\)
\(480\) 1.00000 0.0456435
\(481\) −5.19615 −0.236924
\(482\) −22.6603 −1.03215
\(483\) −17.6603 −0.803569
\(484\) 0 0
\(485\) −18.7321 −0.850579
\(486\) 1.00000 0.0453609
\(487\) 6.46410 0.292916 0.146458 0.989217i \(-0.453213\pi\)
0.146458 + 0.989217i \(0.453213\pi\)
\(488\) −2.19615 −0.0994151
\(489\) 18.0526 0.816365
\(490\) 6.92820 0.312984
\(491\) 1.26795 0.0572217 0.0286109 0.999591i \(-0.490892\pi\)
0.0286109 + 0.999591i \(0.490892\pi\)
\(492\) 4.73205 0.213337
\(493\) −0.803848 −0.0362035
\(494\) −13.3923 −0.602548
\(495\) 0 0
\(496\) 8.19615 0.368018
\(497\) −37.0526 −1.66203
\(498\) −11.3923 −0.510501
\(499\) −21.4449 −0.960004 −0.480002 0.877267i \(-0.659364\pi\)
−0.480002 + 0.877267i \(0.659364\pi\)
\(500\) 1.00000 0.0447214
\(501\) −21.4641 −0.958945
\(502\) 18.5359 0.827298
\(503\) −17.2679 −0.769940 −0.384970 0.922929i \(-0.625788\pi\)
−0.384970 + 0.922929i \(0.625788\pi\)
\(504\) 3.73205 0.166239
\(505\) −5.53590 −0.246344
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) −0.392305 −0.0174057
\(509\) −8.14359 −0.360958 −0.180479 0.983579i \(-0.557765\pi\)
−0.180479 + 0.983579i \(0.557765\pi\)
\(510\) 0.267949 0.0118650
\(511\) 23.1244 1.02296
\(512\) 1.00000 0.0441942
\(513\) −4.46410 −0.197095
\(514\) 6.32051 0.278786
\(515\) −18.3205 −0.807298
\(516\) 6.73205 0.296362
\(517\) 0 0
\(518\) −6.46410 −0.284016
\(519\) 14.5885 0.640362
\(520\) 3.00000 0.131559
\(521\) −22.5885 −0.989618 −0.494809 0.869002i \(-0.664762\pi\)
−0.494809 + 0.869002i \(0.664762\pi\)
\(522\) −3.00000 −0.131306
\(523\) −19.2679 −0.842529 −0.421264 0.906938i \(-0.638414\pi\)
−0.421264 + 0.906938i \(0.638414\pi\)
\(524\) 11.6603 0.509381
\(525\) 3.73205 0.162880
\(526\) −25.1244 −1.09547
\(527\) 2.19615 0.0956659
\(528\) 0 0
\(529\) −0.607695 −0.0264215
\(530\) −5.26795 −0.228825
\(531\) −4.19615 −0.182098
\(532\) −16.6603 −0.722314
\(533\) 14.1962 0.614904
\(534\) −16.3923 −0.709364
\(535\) 11.4641 0.495636
\(536\) −11.4641 −0.495174
\(537\) 7.12436 0.307439
\(538\) 11.5885 0.499614
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 8.58846 0.369247 0.184623 0.982809i \(-0.440893\pi\)
0.184623 + 0.982809i \(0.440893\pi\)
\(542\) −23.8564 −1.02472
\(543\) 4.19615 0.180074
\(544\) 0.267949 0.0114882
\(545\) 13.4641 0.576739
\(546\) 11.1962 0.479151
\(547\) 15.0718 0.644423 0.322212 0.946668i \(-0.395574\pi\)
0.322212 + 0.946668i \(0.395574\pi\)
\(548\) −19.7846 −0.845157
\(549\) −2.19615 −0.0937295
\(550\) 0 0
\(551\) 13.3923 0.570531
\(552\) −4.73205 −0.201409
\(553\) 2.00000 0.0850487
\(554\) 18.9282 0.804182
\(555\) −1.73205 −0.0735215
\(556\) 20.3205 0.861781
\(557\) −10.3923 −0.440336 −0.220168 0.975462i \(-0.570661\pi\)
−0.220168 + 0.975462i \(0.570661\pi\)
\(558\) 8.19615 0.346971
\(559\) 20.1962 0.854206
\(560\) 3.73205 0.157708
\(561\) 0 0
\(562\) 26.0526 1.09896
\(563\) 41.1051 1.73237 0.866187 0.499720i \(-0.166564\pi\)
0.866187 + 0.499720i \(0.166564\pi\)
\(564\) 8.92820 0.375945
\(565\) 13.8564 0.582943
\(566\) −30.2487 −1.27145
\(567\) 3.73205 0.156731
\(568\) −9.92820 −0.416578
\(569\) −25.8038 −1.08175 −0.540877 0.841102i \(-0.681907\pi\)
−0.540877 + 0.841102i \(0.681907\pi\)
\(570\) −4.46410 −0.186981
\(571\) −45.3205 −1.89660 −0.948302 0.317369i \(-0.897201\pi\)
−0.948302 + 0.317369i \(0.897201\pi\)
\(572\) 0 0
\(573\) 18.4641 0.771349
\(574\) 17.6603 0.737125
\(575\) −4.73205 −0.197340
\(576\) 1.00000 0.0416667
\(577\) −17.3205 −0.721062 −0.360531 0.932747i \(-0.617405\pi\)
−0.360531 + 0.932747i \(0.617405\pi\)
\(578\) −16.9282 −0.704120
\(579\) 9.46410 0.393315
\(580\) −3.00000 −0.124568
\(581\) −42.5167 −1.76389
\(582\) −18.7321 −0.776468
\(583\) 0 0
\(584\) 6.19615 0.256399
\(585\) 3.00000 0.124035
\(586\) −18.9282 −0.781917
\(587\) 22.9090 0.945554 0.472777 0.881182i \(-0.343252\pi\)
0.472777 + 0.881182i \(0.343252\pi\)
\(588\) 6.92820 0.285714
\(589\) −36.5885 −1.50760
\(590\) −4.19615 −0.172753
\(591\) −13.4641 −0.553839
\(592\) −1.73205 −0.0711868
\(593\) −4.92820 −0.202377 −0.101189 0.994867i \(-0.532265\pi\)
−0.101189 + 0.994867i \(0.532265\pi\)
\(594\) 0 0
\(595\) 1.00000 0.0409960
\(596\) −0.392305 −0.0160694
\(597\) −16.3923 −0.670892
\(598\) −14.1962 −0.580524
\(599\) −34.3923 −1.40523 −0.702616 0.711569i \(-0.747985\pi\)
−0.702616 + 0.711569i \(0.747985\pi\)
\(600\) 1.00000 0.0408248
\(601\) 39.8564 1.62578 0.812888 0.582420i \(-0.197894\pi\)
0.812888 + 0.582420i \(0.197894\pi\)
\(602\) 25.1244 1.02399
\(603\) −11.4641 −0.466854
\(604\) 13.2679 0.539865
\(605\) 0 0
\(606\) −5.53590 −0.224880
\(607\) 8.41154 0.341414 0.170707 0.985322i \(-0.445395\pi\)
0.170707 + 0.985322i \(0.445395\pi\)
\(608\) −4.46410 −0.181043
\(609\) −11.1962 −0.453691
\(610\) −2.19615 −0.0889196
\(611\) 26.7846 1.08359
\(612\) 0.267949 0.0108312
\(613\) 33.6410 1.35875 0.679374 0.733792i \(-0.262251\pi\)
0.679374 + 0.733792i \(0.262251\pi\)
\(614\) 6.73205 0.271683
\(615\) 4.73205 0.190815
\(616\) 0 0
\(617\) 27.7846 1.11857 0.559283 0.828977i \(-0.311076\pi\)
0.559283 + 0.828977i \(0.311076\pi\)
\(618\) −18.3205 −0.736959
\(619\) −19.9808 −0.803095 −0.401547 0.915838i \(-0.631527\pi\)
−0.401547 + 0.915838i \(0.631527\pi\)
\(620\) 8.19615 0.329165
\(621\) −4.73205 −0.189891
\(622\) 13.8564 0.555591
\(623\) −61.1769 −2.45100
\(624\) 3.00000 0.120096
\(625\) 1.00000 0.0400000
\(626\) −19.1244 −0.764363
\(627\) 0 0
\(628\) 13.1962 0.526584
\(629\) −0.464102 −0.0185049
\(630\) 3.73205 0.148688
\(631\) 17.8038 0.708760 0.354380 0.935102i \(-0.384692\pi\)
0.354380 + 0.935102i \(0.384692\pi\)
\(632\) 0.535898 0.0213169
\(633\) 17.7846 0.706875
\(634\) −15.1244 −0.600665
\(635\) −0.392305 −0.0155681
\(636\) −5.26795 −0.208888
\(637\) 20.7846 0.823516
\(638\) 0 0
\(639\) −9.92820 −0.392754
\(640\) 1.00000 0.0395285
\(641\) 8.92820 0.352643 0.176321 0.984333i \(-0.443580\pi\)
0.176321 + 0.984333i \(0.443580\pi\)
\(642\) 11.4641 0.452452
\(643\) 37.8564 1.49291 0.746455 0.665435i \(-0.231754\pi\)
0.746455 + 0.665435i \(0.231754\pi\)
\(644\) −17.6603 −0.695911
\(645\) 6.73205 0.265074
\(646\) −1.19615 −0.0470620
\(647\) 8.67949 0.341226 0.170613 0.985338i \(-0.445425\pi\)
0.170613 + 0.985338i \(0.445425\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 3.00000 0.117670
\(651\) 30.5885 1.19886
\(652\) 18.0526 0.706993
\(653\) −26.1962 −1.02513 −0.512567 0.858647i \(-0.671306\pi\)
−0.512567 + 0.858647i \(0.671306\pi\)
\(654\) 13.4641 0.526488
\(655\) 11.6603 0.455604
\(656\) 4.73205 0.184756
\(657\) 6.19615 0.241735
\(658\) 33.3205 1.29897
\(659\) 18.3923 0.716462 0.358231 0.933633i \(-0.383380\pi\)
0.358231 + 0.933633i \(0.383380\pi\)
\(660\) 0 0
\(661\) −18.1962 −0.707748 −0.353874 0.935293i \(-0.615136\pi\)
−0.353874 + 0.935293i \(0.615136\pi\)
\(662\) −30.5167 −1.18606
\(663\) 0.803848 0.0312189
\(664\) −11.3923 −0.442107
\(665\) −16.6603 −0.646057
\(666\) −1.73205 −0.0671156
\(667\) 14.1962 0.549677
\(668\) −21.4641 −0.830471
\(669\) 6.46410 0.249917
\(670\) −11.4641 −0.442897
\(671\) 0 0
\(672\) 3.73205 0.143967
\(673\) −17.8564 −0.688314 −0.344157 0.938912i \(-0.611835\pi\)
−0.344157 + 0.938912i \(0.611835\pi\)
\(674\) 23.2679 0.896248
\(675\) 1.00000 0.0384900
\(676\) −4.00000 −0.153846
\(677\) −34.0526 −1.30875 −0.654373 0.756172i \(-0.727067\pi\)
−0.654373 + 0.756172i \(0.727067\pi\)
\(678\) 13.8564 0.532152
\(679\) −69.9090 −2.68286
\(680\) 0.267949 0.0102754
\(681\) −10.3923 −0.398234
\(682\) 0 0
\(683\) −5.87564 −0.224825 −0.112413 0.993662i \(-0.535858\pi\)
−0.112413 + 0.993662i \(0.535858\pi\)
\(684\) −4.46410 −0.170689
\(685\) −19.7846 −0.755931
\(686\) −0.267949 −0.0102303
\(687\) −12.0000 −0.457829
\(688\) 6.73205 0.256657
\(689\) −15.8038 −0.602079
\(690\) −4.73205 −0.180146
\(691\) 51.0526 1.94213 0.971065 0.238814i \(-0.0767585\pi\)
0.971065 + 0.238814i \(0.0767585\pi\)
\(692\) 14.5885 0.554570
\(693\) 0 0
\(694\) 25.3923 0.963879
\(695\) 20.3205 0.770801
\(696\) −3.00000 −0.113715
\(697\) 1.26795 0.0480270
\(698\) 7.85641 0.297369
\(699\) 1.85641 0.0702157
\(700\) 3.73205 0.141058
\(701\) −48.4641 −1.83046 −0.915232 0.402927i \(-0.867993\pi\)
−0.915232 + 0.402927i \(0.867993\pi\)
\(702\) 3.00000 0.113228
\(703\) 7.73205 0.291620
\(704\) 0 0
\(705\) 8.92820 0.336256
\(706\) 6.92820 0.260746
\(707\) −20.6603 −0.777009
\(708\) −4.19615 −0.157701
\(709\) −4.33975 −0.162983 −0.0814913 0.996674i \(-0.525968\pi\)
−0.0814913 + 0.996674i \(0.525968\pi\)
\(710\) −9.92820 −0.372599
\(711\) 0.535898 0.0200978
\(712\) −16.3923 −0.614328
\(713\) −38.7846 −1.45250
\(714\) 1.00000 0.0374241
\(715\) 0 0
\(716\) 7.12436 0.266250
\(717\) −6.26795 −0.234081
\(718\) −27.4641 −1.02495
\(719\) 24.5359 0.915035 0.457517 0.889201i \(-0.348739\pi\)
0.457517 + 0.889201i \(0.348739\pi\)
\(720\) 1.00000 0.0372678
\(721\) −68.3731 −2.54635
\(722\) 0.928203 0.0345441
\(723\) −22.6603 −0.842744
\(724\) 4.19615 0.155949
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 47.4974 1.76158 0.880791 0.473505i \(-0.157012\pi\)
0.880791 + 0.473505i \(0.157012\pi\)
\(728\) 11.1962 0.414957
\(729\) 1.00000 0.0370370
\(730\) 6.19615 0.229330
\(731\) 1.80385 0.0667177
\(732\) −2.19615 −0.0811721
\(733\) 17.8564 0.659541 0.329771 0.944061i \(-0.393029\pi\)
0.329771 + 0.944061i \(0.393029\pi\)
\(734\) 14.8564 0.548360
\(735\) 6.92820 0.255551
\(736\) −4.73205 −0.174426
\(737\) 0 0
\(738\) 4.73205 0.174189
\(739\) 16.4641 0.605642 0.302821 0.953047i \(-0.402072\pi\)
0.302821 + 0.953047i \(0.402072\pi\)
\(740\) −1.73205 −0.0636715
\(741\) −13.3923 −0.491979
\(742\) −19.6603 −0.721751
\(743\) −34.0526 −1.24927 −0.624634 0.780918i \(-0.714752\pi\)
−0.624634 + 0.780918i \(0.714752\pi\)
\(744\) 8.19615 0.300486
\(745\) −0.392305 −0.0143729
\(746\) 6.07180 0.222304
\(747\) −11.3923 −0.416823
\(748\) 0 0
\(749\) 42.7846 1.56332
\(750\) 1.00000 0.0365148
\(751\) 27.8038 1.01458 0.507288 0.861776i \(-0.330648\pi\)
0.507288 + 0.861776i \(0.330648\pi\)
\(752\) 8.92820 0.325578
\(753\) 18.5359 0.675486
\(754\) −9.00000 −0.327761
\(755\) 13.2679 0.482870
\(756\) 3.73205 0.135733
\(757\) −3.46410 −0.125905 −0.0629525 0.998017i \(-0.520052\pi\)
−0.0629525 + 0.998017i \(0.520052\pi\)
\(758\) 21.4449 0.778913
\(759\) 0 0
\(760\) −4.46410 −0.161930
\(761\) 12.6795 0.459631 0.229816 0.973234i \(-0.426188\pi\)
0.229816 + 0.973234i \(0.426188\pi\)
\(762\) −0.392305 −0.0142117
\(763\) 50.2487 1.81913
\(764\) 18.4641 0.668008
\(765\) 0.267949 0.00968772
\(766\) −16.9282 −0.611641
\(767\) −12.5885 −0.454543
\(768\) 1.00000 0.0360844
\(769\) 10.5167 0.379240 0.189620 0.981858i \(-0.439274\pi\)
0.189620 + 0.981858i \(0.439274\pi\)
\(770\) 0 0
\(771\) 6.32051 0.227628
\(772\) 9.46410 0.340620
\(773\) 37.6603 1.35455 0.677273 0.735732i \(-0.263162\pi\)
0.677273 + 0.735732i \(0.263162\pi\)
\(774\) 6.73205 0.241979
\(775\) 8.19615 0.294414
\(776\) −18.7321 −0.672441
\(777\) −6.46410 −0.231898
\(778\) −37.8564 −1.35722
\(779\) −21.1244 −0.756859
\(780\) 3.00000 0.107417
\(781\) 0 0
\(782\) −1.26795 −0.0453418
\(783\) −3.00000 −0.107211
\(784\) 6.92820 0.247436
\(785\) 13.1962 0.470991
\(786\) 11.6603 0.415907
\(787\) 30.9808 1.10434 0.552172 0.833730i \(-0.313799\pi\)
0.552172 + 0.833730i \(0.313799\pi\)
\(788\) −13.4641 −0.479639
\(789\) −25.1244 −0.894451
\(790\) 0.535898 0.0190664
\(791\) 51.7128 1.83870
\(792\) 0 0
\(793\) −6.58846 −0.233963
\(794\) 25.9808 0.922023
\(795\) −5.26795 −0.186835
\(796\) −16.3923 −0.581010
\(797\) −25.3731 −0.898760 −0.449380 0.893341i \(-0.648355\pi\)
−0.449380 + 0.893341i \(0.648355\pi\)
\(798\) −16.6603 −0.589767
\(799\) 2.39230 0.0846337
\(800\) 1.00000 0.0353553
\(801\) −16.3923 −0.579194
\(802\) 17.6603 0.623605
\(803\) 0 0
\(804\) −11.4641 −0.404308
\(805\) −17.6603 −0.622442
\(806\) 24.5885 0.866091
\(807\) 11.5885 0.407933
\(808\) −5.53590 −0.194752
\(809\) −19.2679 −0.677425 −0.338713 0.940890i \(-0.609991\pi\)
−0.338713 + 0.940890i \(0.609991\pi\)
\(810\) 1.00000 0.0351364
\(811\) −13.0000 −0.456492 −0.228246 0.973604i \(-0.573299\pi\)
−0.228246 + 0.973604i \(0.573299\pi\)
\(812\) −11.1962 −0.392908
\(813\) −23.8564 −0.836681
\(814\) 0 0
\(815\) 18.0526 0.632354
\(816\) 0.267949 0.00938010
\(817\) −30.0526 −1.05141
\(818\) 14.0000 0.489499
\(819\) 11.1962 0.391225
\(820\) 4.73205 0.165250
\(821\) −12.6795 −0.442517 −0.221259 0.975215i \(-0.571017\pi\)
−0.221259 + 0.975215i \(0.571017\pi\)
\(822\) −19.7846 −0.690068
\(823\) −19.3923 −0.675973 −0.337987 0.941151i \(-0.609746\pi\)
−0.337987 + 0.941151i \(0.609746\pi\)
\(824\) −18.3205 −0.638225
\(825\) 0 0
\(826\) −15.6603 −0.544890
\(827\) 54.5692 1.89756 0.948779 0.315941i \(-0.102320\pi\)
0.948779 + 0.315941i \(0.102320\pi\)
\(828\) −4.73205 −0.164450
\(829\) 12.8756 0.447190 0.223595 0.974682i \(-0.428221\pi\)
0.223595 + 0.974682i \(0.428221\pi\)
\(830\) −11.3923 −0.395433
\(831\) 18.9282 0.656612
\(832\) 3.00000 0.104006
\(833\) 1.85641 0.0643207
\(834\) 20.3205 0.703642
\(835\) −21.4641 −0.742796
\(836\) 0 0
\(837\) 8.19615 0.283300
\(838\) −11.4641 −0.396021
\(839\) 29.7846 1.02828 0.514139 0.857707i \(-0.328111\pi\)
0.514139 + 0.857707i \(0.328111\pi\)
\(840\) 3.73205 0.128768
\(841\) −20.0000 −0.689655
\(842\) −4.05256 −0.139660
\(843\) 26.0526 0.897298
\(844\) 17.7846 0.612172
\(845\) −4.00000 −0.137604
\(846\) 8.92820 0.306958
\(847\) 0 0
\(848\) −5.26795 −0.180902
\(849\) −30.2487 −1.03813
\(850\) 0.267949 0.00919058
\(851\) 8.19615 0.280960
\(852\) −9.92820 −0.340135
\(853\) 5.67949 0.194462 0.0972310 0.995262i \(-0.469001\pi\)
0.0972310 + 0.995262i \(0.469001\pi\)
\(854\) −8.19615 −0.280467
\(855\) −4.46410 −0.152669
\(856\) 11.4641 0.391835
\(857\) −44.6603 −1.52557 −0.762783 0.646655i \(-0.776167\pi\)
−0.762783 + 0.646655i \(0.776167\pi\)
\(858\) 0 0
\(859\) −1.60770 −0.0548539 −0.0274269 0.999624i \(-0.508731\pi\)
−0.0274269 + 0.999624i \(0.508731\pi\)
\(860\) 6.73205 0.229561
\(861\) 17.6603 0.601860
\(862\) 6.12436 0.208596
\(863\) 12.2487 0.416951 0.208475 0.978028i \(-0.433150\pi\)
0.208475 + 0.978028i \(0.433150\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.5885 0.496022
\(866\) −10.9282 −0.371355
\(867\) −16.9282 −0.574912
\(868\) 30.5885 1.03824
\(869\) 0 0
\(870\) −3.00000 −0.101710
\(871\) −34.3923 −1.16534
\(872\) 13.4641 0.455952
\(873\) −18.7321 −0.633984
\(874\) 21.1244 0.714542
\(875\) 3.73205 0.126166
\(876\) 6.19615 0.209349
\(877\) 33.6410 1.13598 0.567988 0.823037i \(-0.307722\pi\)
0.567988 + 0.823037i \(0.307722\pi\)
\(878\) 4.92820 0.166319
\(879\) −18.9282 −0.638432
\(880\) 0 0
\(881\) 50.4449 1.69953 0.849765 0.527161i \(-0.176744\pi\)
0.849765 + 0.527161i \(0.176744\pi\)
\(882\) 6.92820 0.233285
\(883\) 22.8756 0.769827 0.384913 0.922953i \(-0.374231\pi\)
0.384913 + 0.922953i \(0.374231\pi\)
\(884\) 0.803848 0.0270363
\(885\) −4.19615 −0.141052
\(886\) −5.19615 −0.174568
\(887\) 44.7846 1.50372 0.751860 0.659323i \(-0.229157\pi\)
0.751860 + 0.659323i \(0.229157\pi\)
\(888\) −1.73205 −0.0581238
\(889\) −1.46410 −0.0491044
\(890\) −16.3923 −0.549471
\(891\) 0 0
\(892\) 6.46410 0.216434
\(893\) −39.8564 −1.33374
\(894\) −0.392305 −0.0131206
\(895\) 7.12436 0.238141
\(896\) 3.73205 0.124679
\(897\) −14.1962 −0.473996
\(898\) −3.46410 −0.115599
\(899\) −24.5885 −0.820071
\(900\) 1.00000 0.0333333
\(901\) −1.41154 −0.0470253
\(902\) 0 0
\(903\) 25.1244 0.836086
\(904\) 13.8564 0.460857
\(905\) 4.19615 0.139485
\(906\) 13.2679 0.440798
\(907\) 13.2154 0.438810 0.219405 0.975634i \(-0.429588\pi\)
0.219405 + 0.975634i \(0.429588\pi\)
\(908\) −10.3923 −0.344881
\(909\) −5.53590 −0.183614
\(910\) 11.1962 0.371149
\(911\) −30.5692 −1.01280 −0.506402 0.862298i \(-0.669025\pi\)
−0.506402 + 0.862298i \(0.669025\pi\)
\(912\) −4.46410 −0.147821
\(913\) 0 0
\(914\) −30.4449 −1.00703
\(915\) −2.19615 −0.0726026
\(916\) −12.0000 −0.396491
\(917\) 43.5167 1.43705
\(918\) 0.267949 0.00884364
\(919\) −14.3397 −0.473025 −0.236512 0.971628i \(-0.576004\pi\)
−0.236512 + 0.971628i \(0.576004\pi\)
\(920\) −4.73205 −0.156011
\(921\) 6.73205 0.221829
\(922\) −17.2487 −0.568056
\(923\) −29.7846 −0.980372
\(924\) 0 0
\(925\) −1.73205 −0.0569495
\(926\) −11.4641 −0.376734
\(927\) −18.3205 −0.601724
\(928\) −3.00000 −0.0984798
\(929\) 9.12436 0.299360 0.149680 0.988734i \(-0.452176\pi\)
0.149680 + 0.988734i \(0.452176\pi\)
\(930\) 8.19615 0.268762
\(931\) −30.9282 −1.01363
\(932\) 1.85641 0.0608086
\(933\) 13.8564 0.453638
\(934\) −30.1244 −0.985699
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) −5.71281 −0.186629 −0.0933147 0.995637i \(-0.529746\pi\)
−0.0933147 + 0.995637i \(0.529746\pi\)
\(938\) −42.7846 −1.39697
\(939\) −19.1244 −0.624100
\(940\) 8.92820 0.291206
\(941\) −19.9282 −0.649641 −0.324820 0.945776i \(-0.605304\pi\)
−0.324820 + 0.945776i \(0.605304\pi\)
\(942\) 13.1962 0.429954
\(943\) −22.3923 −0.729194
\(944\) −4.19615 −0.136573
\(945\) 3.73205 0.121404
\(946\) 0 0
\(947\) −29.1962 −0.948747 −0.474374 0.880324i \(-0.657325\pi\)
−0.474374 + 0.880324i \(0.657325\pi\)
\(948\) 0.535898 0.0174052
\(949\) 18.5885 0.603407
\(950\) −4.46410 −0.144835
\(951\) −15.1244 −0.490441
\(952\) 1.00000 0.0324102
\(953\) −4.14359 −0.134224 −0.0671121 0.997745i \(-0.521379\pi\)
−0.0671121 + 0.997745i \(0.521379\pi\)
\(954\) −5.26795 −0.170556
\(955\) 18.4641 0.597484
\(956\) −6.26795 −0.202720
\(957\) 0 0
\(958\) −8.26795 −0.267125
\(959\) −73.8372 −2.38433
\(960\) 1.00000 0.0322749
\(961\) 36.1769 1.16700
\(962\) −5.19615 −0.167531
\(963\) 11.4641 0.369426
\(964\) −22.6603 −0.729838
\(965\) 9.46410 0.304660
\(966\) −17.6603 −0.568209
\(967\) 35.4641 1.14045 0.570224 0.821489i \(-0.306856\pi\)
0.570224 + 0.821489i \(0.306856\pi\)
\(968\) 0 0
\(969\) −1.19615 −0.0384260
\(970\) −18.7321 −0.601450
\(971\) −9.46410 −0.303717 −0.151859 0.988402i \(-0.548526\pi\)
−0.151859 + 0.988402i \(0.548526\pi\)
\(972\) 1.00000 0.0320750
\(973\) 75.8372 2.43123
\(974\) 6.46410 0.207123
\(975\) 3.00000 0.0960769
\(976\) −2.19615 −0.0702971
\(977\) 10.0000 0.319928 0.159964 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 18.0526 0.577257
\(979\) 0 0
\(980\) 6.92820 0.221313
\(981\) 13.4641 0.429876
\(982\) 1.26795 0.0404619
\(983\) 32.5885 1.03941 0.519705 0.854346i \(-0.326042\pi\)
0.519705 + 0.854346i \(0.326042\pi\)
\(984\) 4.73205 0.150852
\(985\) −13.4641 −0.429002
\(986\) −0.803848 −0.0255997
\(987\) 33.3205 1.06060
\(988\) −13.3923 −0.426066
\(989\) −31.8564 −1.01297
\(990\) 0 0
\(991\) 9.21539 0.292737 0.146368 0.989230i \(-0.453242\pi\)
0.146368 + 0.989230i \(0.453242\pi\)
\(992\) 8.19615 0.260228
\(993\) −30.5167 −0.968417
\(994\) −37.0526 −1.17524
\(995\) −16.3923 −0.519671
\(996\) −11.3923 −0.360979
\(997\) 59.3923 1.88097 0.940487 0.339831i \(-0.110370\pi\)
0.940487 + 0.339831i \(0.110370\pi\)
\(998\) −21.4449 −0.678825
\(999\) −1.73205 −0.0547997
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 3630.2.a.bp.1.2 yes 2
11.10 odd 2 3630.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3630.2.a.bh.1.1 2 11.10 odd 2
3630.2.a.bp.1.2 yes 2 1.1 even 1 trivial