Properties

Label 5070.2.a.bz.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.131472.2
Defining polynomial: \(x^{4} - 2 x^{3} - 19 x^{2} + 20 x + 52\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.64466\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.64466 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.64466 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.40013 q^{11} +1.00000 q^{12} +4.64466 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -8.04479 q^{19} +1.00000 q^{20} -4.64466 q^{21} -4.40013 q^{22} -0.976584 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -4.64466 q^{28} -4.31274 q^{29} -1.00000 q^{30} -6.44069 q^{31} -1.00000 q^{32} +4.40013 q^{33} -4.00000 q^{34} -4.64466 q^{35} +1.00000 q^{36} +3.79603 q^{37} +8.04479 q^{38} -1.00000 q^{40} +7.28932 q^{41} +4.64466 q^{42} +0.716456 q^{43} +4.40013 q^{44} +1.00000 q^{45} +0.976584 q^{46} +9.75342 q^{47} +1.00000 q^{48} +14.5729 q^{49} -1.00000 q^{50} +4.00000 q^{51} +13.5089 q^{53} -1.00000 q^{54} +4.40013 q^{55} +4.64466 q^{56} -8.04479 q^{57} +4.31274 q^{58} +2.18056 q^{59} +1.00000 q^{60} +7.46410 q^{61} +6.44069 q^{62} -4.64466 q^{63} +1.00000 q^{64} -4.40013 q^{66} +1.82522 q^{67} +4.00000 q^{68} -0.976584 q^{69} +4.64466 q^{70} -7.95317 q^{71} -1.00000 q^{72} -4.36112 q^{73} -3.79603 q^{74} +1.00000 q^{75} -8.04479 q^{76} -20.4371 q^{77} -14.9340 q^{79} +1.00000 q^{80} +1.00000 q^{81} -7.28932 q^{82} -3.51093 q^{83} -4.64466 q^{84} +4.00000 q^{85} -0.716456 q^{86} -4.31274 q^{87} -4.40013 q^{88} +8.17274 q^{89} -1.00000 q^{90} -0.976584 q^{92} -6.44069 q^{93} -9.75342 q^{94} -8.04479 q^{95} -1.00000 q^{96} +13.8252 q^{97} -14.5729 q^{98} +4.40013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 2q^{7} - 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 2q^{7} - 4q^{8} + 4q^{9} - 4q^{10} + 2q^{11} + 4q^{12} + 2q^{14} + 4q^{15} + 4q^{16} + 16q^{17} - 4q^{18} + 4q^{20} - 2q^{21} - 2q^{22} + 4q^{23} - 4q^{24} + 4q^{25} + 4q^{27} - 2q^{28} + 8q^{29} - 4q^{30} - 4q^{31} - 4q^{32} + 2q^{33} - 16q^{34} - 2q^{35} + 4q^{36} + 10q^{37} - 4q^{40} - 4q^{41} + 2q^{42} + 14q^{43} + 2q^{44} + 4q^{45} - 4q^{46} - 8q^{47} + 4q^{48} + 14q^{49} - 4q^{50} + 16q^{51} + 8q^{53} - 4q^{54} + 2q^{55} + 2q^{56} - 8q^{58} + 6q^{59} + 4q^{60} + 16q^{61} + 4q^{62} - 2q^{63} + 4q^{64} - 2q^{66} - 12q^{67} + 16q^{68} + 4q^{69} + 2q^{70} - 16q^{71} - 4q^{72} - 12q^{73} - 10q^{74} + 4q^{75} - 8q^{77} - 10q^{79} + 4q^{80} + 4q^{81} + 4q^{82} - 16q^{83} - 2q^{84} + 16q^{85} - 14q^{86} + 8q^{87} - 2q^{88} + 4q^{89} - 4q^{90} + 4q^{92} - 4q^{93} + 8q^{94} - 4q^{96} + 36q^{97} - 14q^{98} + 2q^{99} + O(q^{100}) \)

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
\(7\) −4.64466 −1.75552 −0.877758 0.479104i \(-0.840962\pi\)
−0.877758 + 0.479104i \(0.840962\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.40013 1.32669 0.663344 0.748315i \(-0.269137\pi\)
0.663344 + 0.748315i \(0.269137\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 4.64466 1.24134
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) −8.04479 −1.84560 −0.922800 0.385279i \(-0.874105\pi\)
−0.922800 + 0.385279i \(0.874105\pi\)
\(20\) 1.00000 0.223607
\(21\) −4.64466 −1.01355
\(22\) −4.40013 −0.938110
\(23\) −0.976584 −0.203632 −0.101816 0.994803i \(-0.532465\pi\)
−0.101816 + 0.994803i \(0.532465\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −4.64466 −0.877758
\(29\) −4.31274 −0.800855 −0.400427 0.916328i \(-0.631138\pi\)
−0.400427 + 0.916328i \(0.631138\pi\)
\(30\) −1.00000 −0.182574
\(31\) −6.44069 −1.15678 −0.578391 0.815760i \(-0.696319\pi\)
−0.578391 + 0.815760i \(0.696319\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.40013 0.765964
\(34\) −4.00000 −0.685994
\(35\) −4.64466 −0.785091
\(36\) 1.00000 0.166667
\(37\) 3.79603 0.624063 0.312031 0.950072i \(-0.398991\pi\)
0.312031 + 0.950072i \(0.398991\pi\)
\(38\) 8.04479 1.30504
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 7.28932 1.13840 0.569200 0.822199i \(-0.307253\pi\)
0.569200 + 0.822199i \(0.307253\pi\)
\(42\) 4.64466 0.716687
\(43\) 0.716456 0.109259 0.0546293 0.998507i \(-0.482602\pi\)
0.0546293 + 0.998507i \(0.482602\pi\)
\(44\) 4.40013 0.663344
\(45\) 1.00000 0.149071
\(46\) 0.976584 0.143989
\(47\) 9.75342 1.42268 0.711341 0.702847i \(-0.248088\pi\)
0.711341 + 0.702847i \(0.248088\pi\)
\(48\) 1.00000 0.144338
\(49\) 14.5729 2.08184
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 13.5089 1.85559 0.927794 0.373092i \(-0.121703\pi\)
0.927794 + 0.373092i \(0.121703\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.40013 0.593313
\(56\) 4.64466 0.620669
\(57\) −8.04479 −1.06556
\(58\) 4.31274 0.566290
\(59\) 2.18056 0.283884 0.141942 0.989875i \(-0.454665\pi\)
0.141942 + 0.989875i \(0.454665\pi\)
\(60\) 1.00000 0.129099
\(61\) 7.46410 0.955680 0.477840 0.878447i \(-0.341420\pi\)
0.477840 + 0.878447i \(0.341420\pi\)
\(62\) 6.44069 0.817968
\(63\) −4.64466 −0.585172
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.40013 −0.541618
\(67\) 1.82522 0.222986 0.111493 0.993765i \(-0.464437\pi\)
0.111493 + 0.993765i \(0.464437\pi\)
\(68\) 4.00000 0.485071
\(69\) −0.976584 −0.117567
\(70\) 4.64466 0.555143
\(71\) −7.95317 −0.943867 −0.471934 0.881634i \(-0.656444\pi\)
−0.471934 + 0.881634i \(0.656444\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.36112 −0.510430 −0.255215 0.966884i \(-0.582146\pi\)
−0.255215 + 0.966884i \(0.582146\pi\)
\(74\) −3.79603 −0.441279
\(75\) 1.00000 0.115470
\(76\) −8.04479 −0.922800
\(77\) −20.4371 −2.32902
\(78\) 0 0
\(79\) −14.9340 −1.68020 −0.840102 0.542429i \(-0.817505\pi\)
−0.840102 + 0.542429i \(0.817505\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −7.28932 −0.804971
\(83\) −3.51093 −0.385375 −0.192688 0.981260i \(-0.561720\pi\)
−0.192688 + 0.981260i \(0.561720\pi\)
\(84\) −4.64466 −0.506774
\(85\) 4.00000 0.433861
\(86\) −0.716456 −0.0772575
\(87\) −4.31274 −0.462374
\(88\) −4.40013 −0.469055
\(89\) 8.17274 0.866308 0.433154 0.901320i \(-0.357401\pi\)
0.433154 + 0.901320i \(0.357401\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.976584 −0.101816
\(93\) −6.44069 −0.667868
\(94\) −9.75342 −1.00599
\(95\) −8.04479 −0.825378
\(96\) −1.00000 −0.102062
\(97\) 13.8252 1.40374 0.701869 0.712306i \(-0.252349\pi\)
0.701869 + 0.712306i \(0.252349\pi\)
\(98\) −14.5729 −1.47208
\(99\) 4.40013 0.442229
\(100\) 1.00000 0.100000
\(101\) −6.80025 −0.676651 −0.338325 0.941029i \(-0.609860\pi\)
−0.338325 + 0.941029i \(0.609860\pi\)
\(102\) −4.00000 −0.396059
\(103\) 5.29137 0.521374 0.260687 0.965423i \(-0.416051\pi\)
0.260687 + 0.965423i \(0.416051\pi\)
\(104\) 0 0
\(105\) −4.64466 −0.453272
\(106\) −13.5089 −1.31210
\(107\) 16.9282 1.63651 0.818256 0.574855i \(-0.194941\pi\)
0.818256 + 0.574855i \(0.194941\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.8003 −1.22604 −0.613021 0.790067i \(-0.710046\pi\)
−0.613021 + 0.790067i \(0.710046\pi\)
\(110\) −4.40013 −0.419536
\(111\) 3.79603 0.360303
\(112\) −4.64466 −0.438879
\(113\) 13.0662 1.22916 0.614580 0.788854i \(-0.289325\pi\)
0.614580 + 0.788854i \(0.289325\pi\)
\(114\) 8.04479 0.753463
\(115\) −0.976584 −0.0910669
\(116\) −4.31274 −0.400427
\(117\) 0 0
\(118\) −2.18056 −0.200737
\(119\) −18.5786 −1.70310
\(120\) −1.00000 −0.0912871
\(121\) 8.36112 0.760101
\(122\) −7.46410 −0.675768
\(123\) 7.28932 0.657256
\(124\) −6.44069 −0.578391
\(125\) 1.00000 0.0894427
\(126\) 4.64466 0.413779
\(127\) 12.2196 1.08431 0.542156 0.840278i \(-0.317608\pi\)
0.542156 + 0.840278i \(0.317608\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.716456 0.0630805
\(130\) 0 0
\(131\) −0.378757 −0.0330921 −0.0165461 0.999863i \(-0.505267\pi\)
−0.0165461 + 0.999863i \(0.505267\pi\)
\(132\) 4.40013 0.382982
\(133\) 37.3653 3.23998
\(134\) −1.82522 −0.157675
\(135\) 1.00000 0.0860663
\(136\) −4.00000 −0.342997
\(137\) 1.25235 0.106996 0.0534979 0.998568i \(-0.482963\pi\)
0.0534979 + 0.998568i \(0.482963\pi\)
\(138\) 0.976584 0.0831323
\(139\) 3.35534 0.284596 0.142298 0.989824i \(-0.454551\pi\)
0.142298 + 0.989824i \(0.454551\pi\)
\(140\) −4.64466 −0.392545
\(141\) 9.75342 0.821386
\(142\) 7.95317 0.667415
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.31274 −0.358153
\(146\) 4.36112 0.360929
\(147\) 14.5729 1.20195
\(148\) 3.79603 0.312031
\(149\) −4.61970 −0.378460 −0.189230 0.981933i \(-0.560599\pi\)
−0.189230 + 0.981933i \(0.560599\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 11.2425 0.914901 0.457450 0.889235i \(-0.348763\pi\)
0.457450 + 0.889235i \(0.348763\pi\)
\(152\) 8.04479 0.652518
\(153\) 4.00000 0.323381
\(154\) 20.4371 1.64687
\(155\) −6.44069 −0.517328
\(156\) 0 0
\(157\) −1.50311 −0.119961 −0.0599807 0.998200i \(-0.519104\pi\)
−0.0599807 + 0.998200i \(0.519104\pi\)
\(158\) 14.9340 1.18808
\(159\) 13.5089 1.07132
\(160\) −1.00000 −0.0790569
\(161\) 4.53590 0.357479
\(162\) −1.00000 −0.0785674
\(163\) 0.176330 0.0138112 0.00690561 0.999976i \(-0.497802\pi\)
0.00690561 + 0.999976i \(0.497802\pi\)
\(164\) 7.28932 0.569200
\(165\) 4.40013 0.342549
\(166\) 3.51093 0.272501
\(167\) 8.68162 0.671804 0.335902 0.941897i \(-0.390959\pi\)
0.335902 + 0.941897i \(0.390959\pi\)
\(168\) 4.64466 0.358343
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) −8.04479 −0.615200
\(172\) 0.716456 0.0546293
\(173\) 1.78043 0.135364 0.0676818 0.997707i \(-0.478440\pi\)
0.0676818 + 0.997707i \(0.478440\pi\)
\(174\) 4.31274 0.326948
\(175\) −4.64466 −0.351103
\(176\) 4.40013 0.331672
\(177\) 2.18056 0.163901
\(178\) −8.17274 −0.612572
\(179\) −17.4157 −1.30171 −0.650856 0.759201i \(-0.725590\pi\)
−0.650856 + 0.759201i \(0.725590\pi\)
\(180\) 1.00000 0.0745356
\(181\) 10.3611 0.770136 0.385068 0.922888i \(-0.374178\pi\)
0.385068 + 0.922888i \(0.374178\pi\)
\(182\) 0 0
\(183\) 7.46410 0.551762
\(184\) 0.976584 0.0719947
\(185\) 3.79603 0.279089
\(186\) 6.44069 0.472254
\(187\) 17.6005 1.28708
\(188\) 9.75342 0.711341
\(189\) −4.64466 −0.337849
\(190\) 8.04479 0.583630
\(191\) −0.897014 −0.0649057 −0.0324528 0.999473i \(-0.510332\pi\)
−0.0324528 + 0.999473i \(0.510332\pi\)
\(192\) 1.00000 0.0721688
\(193\) 20.2644 1.45866 0.729330 0.684162i \(-0.239832\pi\)
0.729330 + 0.684162i \(0.239832\pi\)
\(194\) −13.8252 −0.992593
\(195\) 0 0
\(196\) 14.5729 1.04092
\(197\) −9.18056 −0.654088 −0.327044 0.945009i \(-0.606052\pi\)
−0.327044 + 0.945009i \(0.606052\pi\)
\(198\) −4.40013 −0.312703
\(199\) −16.2175 −1.14963 −0.574815 0.818284i \(-0.694926\pi\)
−0.574815 + 0.818284i \(0.694926\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 1.82522 0.128741
\(202\) 6.80025 0.478464
\(203\) 20.0312 1.40591
\(204\) 4.00000 0.280056
\(205\) 7.28932 0.509108
\(206\) −5.29137 −0.368667
\(207\) −0.976584 −0.0678773
\(208\) 0 0
\(209\) −35.3981 −2.44854
\(210\) 4.64466 0.320512
\(211\) −1.61970 −0.111504 −0.0557522 0.998445i \(-0.517756\pi\)
−0.0557522 + 0.998445i \(0.517756\pi\)
\(212\) 13.5089 0.927794
\(213\) −7.95317 −0.544942
\(214\) −16.9282 −1.15719
\(215\) 0.716456 0.0488619
\(216\) −1.00000 −0.0680414
\(217\) 29.9148 2.03075
\(218\) 12.8003 0.866943
\(219\) −4.36112 −0.294697
\(220\) 4.40013 0.296656
\(221\) 0 0
\(222\) −3.79603 −0.254773
\(223\) −2.23671 −0.149781 −0.0748906 0.997192i \(-0.523861\pi\)
−0.0748906 + 0.997192i \(0.523861\pi\)
\(224\) 4.64466 0.310334
\(225\) 1.00000 0.0666667
\(226\) −13.0662 −0.869148
\(227\) −15.3205 −1.01686 −0.508429 0.861104i \(-0.669774\pi\)
−0.508429 + 0.861104i \(0.669774\pi\)
\(228\) −8.04479 −0.532779
\(229\) −10.1279 −0.669274 −0.334637 0.942347i \(-0.608614\pi\)
−0.334637 + 0.942347i \(0.608614\pi\)
\(230\) 0.976584 0.0643940
\(231\) −20.4371 −1.34466
\(232\) 4.31274 0.283145
\(233\) −20.7519 −1.35950 −0.679750 0.733444i \(-0.737912\pi\)
−0.679750 + 0.733444i \(0.737912\pi\)
\(234\) 0 0
\(235\) 9.75342 0.636243
\(236\) 2.18056 0.141942
\(237\) −14.9340 −0.970066
\(238\) 18.5786 1.20427
\(239\) 24.3539 1.57532 0.787662 0.616107i \(-0.211291\pi\)
0.787662 + 0.616107i \(0.211291\pi\)
\(240\) 1.00000 0.0645497
\(241\) 19.8685 1.27984 0.639920 0.768442i \(-0.278968\pi\)
0.639920 + 0.768442i \(0.278968\pi\)
\(242\) −8.36112 −0.537473
\(243\) 1.00000 0.0641500
\(244\) 7.46410 0.477840
\(245\) 14.5729 0.931026
\(246\) −7.28932 −0.464750
\(247\) 0 0
\(248\) 6.44069 0.408984
\(249\) −3.51093 −0.222496
\(250\) −1.00000 −0.0632456
\(251\) 13.5713 0.856614 0.428307 0.903633i \(-0.359110\pi\)
0.428307 + 0.903633i \(0.359110\pi\)
\(252\) −4.64466 −0.292586
\(253\) −4.29709 −0.270156
\(254\) −12.2196 −0.766724
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 0.662300 0.0413132 0.0206566 0.999787i \(-0.493424\pi\)
0.0206566 + 0.999787i \(0.493424\pi\)
\(258\) −0.716456 −0.0446046
\(259\) −17.6312 −1.09555
\(260\) 0 0
\(261\) −4.31274 −0.266952
\(262\) 0.378757 0.0233997
\(263\) 30.7498 1.89612 0.948058 0.318098i \(-0.103044\pi\)
0.948058 + 0.318098i \(0.103044\pi\)
\(264\) −4.40013 −0.270809
\(265\) 13.5089 0.829844
\(266\) −37.3653 −2.29101
\(267\) 8.17274 0.500163
\(268\) 1.82522 0.111493
\(269\) −3.74410 −0.228282 −0.114141 0.993465i \(-0.536412\pi\)
−0.114141 + 0.993465i \(0.536412\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 27.7612 1.68637 0.843186 0.537622i \(-0.180677\pi\)
0.843186 + 0.537622i \(0.180677\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −1.25235 −0.0756575
\(275\) 4.40013 0.265338
\(276\) −0.976584 −0.0587834
\(277\) 11.9922 0.720540 0.360270 0.932848i \(-0.382685\pi\)
0.360270 + 0.932848i \(0.382685\pi\)
\(278\) −3.35534 −0.201240
\(279\) −6.44069 −0.385594
\(280\) 4.64466 0.277572
\(281\) 3.63888 0.217078 0.108539 0.994092i \(-0.465383\pi\)
0.108539 + 0.994092i \(0.465383\pi\)
\(282\) −9.75342 −0.580808
\(283\) 4.84441 0.287970 0.143985 0.989580i \(-0.454008\pi\)
0.143985 + 0.989580i \(0.454008\pi\)
\(284\) −7.95317 −0.471934
\(285\) −8.04479 −0.476532
\(286\) 0 0
\(287\) −33.8564 −1.99848
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 4.31274 0.253253
\(291\) 13.8252 0.810449
\(292\) −4.36112 −0.255215
\(293\) −3.70081 −0.216204 −0.108102 0.994140i \(-0.534477\pi\)
−0.108102 + 0.994140i \(0.534477\pi\)
\(294\) −14.5729 −0.849907
\(295\) 2.18056 0.126957
\(296\) −3.79603 −0.220640
\(297\) 4.40013 0.255321
\(298\) 4.61970 0.267612
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −3.32770 −0.191805
\(302\) −11.2425 −0.646932
\(303\) −6.80025 −0.390664
\(304\) −8.04479 −0.461400
\(305\) 7.46410 0.427393
\(306\) −4.00000 −0.228665
\(307\) −7.11454 −0.406048 −0.203024 0.979174i \(-0.565077\pi\)
−0.203024 + 0.979174i \(0.565077\pi\)
\(308\) −20.4371 −1.16451
\(309\) 5.29137 0.301015
\(310\) 6.44069 0.365806
\(311\) 19.9148 1.12926 0.564632 0.825343i \(-0.309018\pi\)
0.564632 + 0.825343i \(0.309018\pi\)
\(312\) 0 0
\(313\) 6.13950 0.347025 0.173513 0.984832i \(-0.444488\pi\)
0.173513 + 0.984832i \(0.444488\pi\)
\(314\) 1.50311 0.0848255
\(315\) −4.64466 −0.261697
\(316\) −14.9340 −0.840102
\(317\) −7.85286 −0.441061 −0.220530 0.975380i \(-0.570779\pi\)
−0.220530 + 0.975380i \(0.570779\pi\)
\(318\) −13.5089 −0.757541
\(319\) −18.9766 −1.06248
\(320\) 1.00000 0.0559017
\(321\) 16.9282 0.944840
\(322\) −4.53590 −0.252776
\(323\) −32.1791 −1.79050
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −0.176330 −0.00976601
\(327\) −12.8003 −0.707856
\(328\) −7.28932 −0.402485
\(329\) −45.3013 −2.49754
\(330\) −4.40013 −0.242219
\(331\) 31.9959 1.75865 0.879327 0.476218i \(-0.157993\pi\)
0.879327 + 0.476218i \(0.157993\pi\)
\(332\) −3.51093 −0.192688
\(333\) 3.79603 0.208021
\(334\) −8.68162 −0.475037
\(335\) 1.82522 0.0997223
\(336\) −4.64466 −0.253387
\(337\) 35.6432 1.94161 0.970806 0.239867i \(-0.0771039\pi\)
0.970806 + 0.239867i \(0.0771039\pi\)
\(338\) 0 0
\(339\) 13.0662 0.709656
\(340\) 4.00000 0.216930
\(341\) −28.3398 −1.53469
\(342\) 8.04479 0.435012
\(343\) −35.1734 −1.89918
\(344\) −0.716456 −0.0386287
\(345\) −0.976584 −0.0525775
\(346\) −1.78043 −0.0957166
\(347\) −6.89701 −0.370251 −0.185126 0.982715i \(-0.559269\pi\)
−0.185126 + 0.982715i \(0.559269\pi\)
\(348\) −4.31274 −0.231187
\(349\) 19.3205 1.03420 0.517102 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(350\) 4.64466 0.248267
\(351\) 0 0
\(352\) −4.40013 −0.234528
\(353\) 27.4173 1.45927 0.729637 0.683835i \(-0.239689\pi\)
0.729637 + 0.683835i \(0.239689\pi\)
\(354\) −2.18056 −0.115895
\(355\) −7.95317 −0.422110
\(356\) 8.17274 0.433154
\(357\) −18.5786 −0.983286
\(358\) 17.4157 0.920449
\(359\) 14.3611 0.757951 0.378975 0.925407i \(-0.376277\pi\)
0.378975 + 0.925407i \(0.376277\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 45.7186 2.40624
\(362\) −10.3611 −0.544568
\(363\) 8.36112 0.438845
\(364\) 0 0
\(365\) −4.36112 −0.228271
\(366\) −7.46410 −0.390155
\(367\) −13.7753 −0.719064 −0.359532 0.933133i \(-0.617064\pi\)
−0.359532 + 0.933133i \(0.617064\pi\)
\(368\) −0.976584 −0.0509079
\(369\) 7.28932 0.379467
\(370\) −3.79603 −0.197346
\(371\) −62.7442 −3.25752
\(372\) −6.44069 −0.333934
\(373\) −33.4627 −1.73263 −0.866316 0.499496i \(-0.833519\pi\)
−0.866316 + 0.499496i \(0.833519\pi\)
\(374\) −17.6005 −0.910101
\(375\) 1.00000 0.0516398
\(376\) −9.75342 −0.502994
\(377\) 0 0
\(378\) 4.64466 0.238896
\(379\) 33.3768 1.71445 0.857227 0.514939i \(-0.172186\pi\)
0.857227 + 0.514939i \(0.172186\pi\)
\(380\) −8.04479 −0.412689
\(381\) 12.2196 0.626027
\(382\) 0.897014 0.0458952
\(383\) 7.33038 0.374565 0.187282 0.982306i \(-0.440032\pi\)
0.187282 + 0.982306i \(0.440032\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −20.4371 −1.04157
\(386\) −20.2644 −1.03143
\(387\) 0.716456 0.0364195
\(388\) 13.8252 0.701869
\(389\) 32.6198 1.65389 0.826946 0.562282i \(-0.190076\pi\)
0.826946 + 0.562282i \(0.190076\pi\)
\(390\) 0 0
\(391\) −3.90633 −0.197552
\(392\) −14.5729 −0.736041
\(393\) −0.378757 −0.0191058
\(394\) 9.18056 0.462510
\(395\) −14.9340 −0.751410
\(396\) 4.40013 0.221115
\(397\) −14.4683 −0.726145 −0.363072 0.931761i \(-0.618272\pi\)
−0.363072 + 0.931761i \(0.618272\pi\)
\(398\) 16.2175 0.812911
\(399\) 37.3653 1.87060
\(400\) 1.00000 0.0500000
\(401\) −7.28727 −0.363909 −0.181955 0.983307i \(-0.558242\pi\)
−0.181955 + 0.983307i \(0.558242\pi\)
\(402\) −1.82522 −0.0910336
\(403\) 0 0
\(404\) −6.80025 −0.338325
\(405\) 1.00000 0.0496904
\(406\) −20.0312 −0.994131
\(407\) 16.7030 0.827937
\(408\) −4.00000 −0.198030
\(409\) −24.5766 −1.21523 −0.607617 0.794230i \(-0.707874\pi\)
−0.607617 + 0.794230i \(0.707874\pi\)
\(410\) −7.28932 −0.359994
\(411\) 1.25235 0.0617741
\(412\) 5.29137 0.260687
\(413\) −10.1279 −0.498364
\(414\) 0.976584 0.0479965
\(415\) −3.51093 −0.172345
\(416\) 0 0
\(417\) 3.35534 0.164312
\(418\) 35.3981 1.73138
\(419\) 15.5537 0.759847 0.379923 0.925018i \(-0.375950\pi\)
0.379923 + 0.925018i \(0.375950\pi\)
\(420\) −4.64466 −0.226636
\(421\) 22.3143 1.08753 0.543766 0.839237i \(-0.316998\pi\)
0.543766 + 0.839237i \(0.316998\pi\)
\(422\) 1.61970 0.0788456
\(423\) 9.75342 0.474228
\(424\) −13.5089 −0.656050
\(425\) 4.00000 0.194029
\(426\) 7.95317 0.385332
\(427\) −34.6682 −1.67771
\(428\) 16.9282 0.818256
\(429\) 0 0
\(430\) −0.716456 −0.0345506
\(431\) −24.2644 −1.16877 −0.584386 0.811476i \(-0.698665\pi\)
−0.584386 + 0.811476i \(0.698665\pi\)
\(432\) 1.00000 0.0481125
\(433\) −23.0989 −1.11006 −0.555031 0.831830i \(-0.687294\pi\)
−0.555031 + 0.831830i \(0.687294\pi\)
\(434\) −29.9148 −1.43596
\(435\) −4.31274 −0.206780
\(436\) −12.8003 −0.613021
\(437\) 7.85641 0.375823
\(438\) 4.36112 0.208382
\(439\) −39.9959 −1.90890 −0.954450 0.298370i \(-0.903557\pi\)
−0.954450 + 0.298370i \(0.903557\pi\)
\(440\) −4.40013 −0.209768
\(441\) 14.5729 0.693946
\(442\) 0 0
\(443\) −15.6036 −0.741350 −0.370675 0.928763i \(-0.620874\pi\)
−0.370675 + 0.928763i \(0.620874\pi\)
\(444\) 3.79603 0.180151
\(445\) 8.17274 0.387425
\(446\) 2.23671 0.105911
\(447\) −4.61970 −0.218504
\(448\) −4.64466 −0.219440
\(449\) −27.0042 −1.27441 −0.637203 0.770696i \(-0.719909\pi\)
−0.637203 + 0.770696i \(0.719909\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 32.0739 1.51030
\(452\) 13.0662 0.614580
\(453\) 11.2425 0.528218
\(454\) 15.3205 0.719027
\(455\) 0 0
\(456\) 8.04479 0.376732
\(457\) −17.7868 −0.832033 −0.416017 0.909357i \(-0.636574\pi\)
−0.416017 + 0.909357i \(0.636574\pi\)
\(458\) 10.1279 0.473248
\(459\) 4.00000 0.186704
\(460\) −0.976584 −0.0455334
\(461\) 24.9652 1.16274 0.581372 0.813638i \(-0.302516\pi\)
0.581372 + 0.813638i \(0.302516\pi\)
\(462\) 20.4371 0.950820
\(463\) 15.6389 0.726801 0.363400 0.931633i \(-0.381616\pi\)
0.363400 + 0.931633i \(0.381616\pi\)
\(464\) −4.31274 −0.200214
\(465\) −6.44069 −0.298680
\(466\) 20.7519 0.961312
\(467\) 2.43914 0.112870 0.0564349 0.998406i \(-0.482027\pi\)
0.0564349 + 0.998406i \(0.482027\pi\)
\(468\) 0 0
\(469\) −8.47751 −0.391455
\(470\) −9.75342 −0.449892
\(471\) −1.50311 −0.0692598
\(472\) −2.18056 −0.100368
\(473\) 3.15250 0.144952
\(474\) 14.9340 0.685940
\(475\) −8.04479 −0.369120
\(476\) −18.5786 −0.851550
\(477\) 13.5089 0.618529
\(478\) −24.3539 −1.11392
\(479\) 14.1863 0.648190 0.324095 0.946025i \(-0.394940\pi\)
0.324095 + 0.946025i \(0.394940\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −19.8685 −0.904983
\(483\) 4.53590 0.206391
\(484\) 8.36112 0.380051
\(485\) 13.8252 0.627771
\(486\) −1.00000 −0.0453609
\(487\) 5.76329 0.261160 0.130580 0.991438i \(-0.458316\pi\)
0.130580 + 0.991438i \(0.458316\pi\)
\(488\) −7.46410 −0.337884
\(489\) 0.176330 0.00797392
\(490\) −14.5729 −0.658335
\(491\) −1.07534 −0.0485295 −0.0242647 0.999706i \(-0.507724\pi\)
−0.0242647 + 0.999706i \(0.507724\pi\)
\(492\) 7.28932 0.328628
\(493\) −17.2509 −0.776943
\(494\) 0 0
\(495\) 4.40013 0.197771
\(496\) −6.44069 −0.289195
\(497\) 36.9398 1.65697
\(498\) 3.51093 0.157329
\(499\) −14.7534 −0.660454 −0.330227 0.943902i \(-0.607125\pi\)
−0.330227 + 0.943902i \(0.607125\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.68162 0.387866
\(502\) −13.5713 −0.605717
\(503\) −26.6618 −1.18879 −0.594395 0.804173i \(-0.702609\pi\)
−0.594395 + 0.804173i \(0.702609\pi\)
\(504\) 4.64466 0.206890
\(505\) −6.80025 −0.302607
\(506\) 4.29709 0.191029
\(507\) 0 0
\(508\) 12.2196 0.542156
\(509\) −14.1806 −0.628542 −0.314271 0.949333i \(-0.601760\pi\)
−0.314271 + 0.949333i \(0.601760\pi\)
\(510\) −4.00000 −0.177123
\(511\) 20.2559 0.896068
\(512\) −1.00000 −0.0441942
\(513\) −8.04479 −0.355186
\(514\) −0.662300 −0.0292128
\(515\) 5.29137 0.233165
\(516\) 0.716456 0.0315402
\(517\) 42.9163 1.88746
\(518\) 17.6312 0.774673
\(519\) 1.78043 0.0781523
\(520\) 0 0
\(521\) 23.7476 1.04040 0.520202 0.854043i \(-0.325857\pi\)
0.520202 + 0.854043i \(0.325857\pi\)
\(522\) 4.31274 0.188763
\(523\) 13.8506 0.605646 0.302823 0.953047i \(-0.402071\pi\)
0.302823 + 0.953047i \(0.402071\pi\)
\(524\) −0.378757 −0.0165461
\(525\) −4.64466 −0.202710
\(526\) −30.7498 −1.34076
\(527\) −25.7627 −1.12224
\(528\) 4.40013 0.191491
\(529\) −22.0463 −0.958534
\(530\) −13.5089 −0.586789
\(531\) 2.18056 0.0946282
\(532\) 37.3653 1.61999
\(533\) 0 0
\(534\) −8.17274 −0.353669
\(535\) 16.9282 0.731870
\(536\) −1.82522 −0.0788374
\(537\) −17.4157 −0.751544
\(538\) 3.74410 0.161420
\(539\) 64.1224 2.76195
\(540\) 1.00000 0.0430331
\(541\) 14.8898 0.640164 0.320082 0.947390i \(-0.396290\pi\)
0.320082 + 0.947390i \(0.396290\pi\)
\(542\) −27.7612 −1.19245
\(543\) 10.3611 0.444638
\(544\) −4.00000 −0.171499
\(545\) −12.8003 −0.548303
\(546\) 0 0
\(547\) −22.2019 −0.949284 −0.474642 0.880179i \(-0.657422\pi\)
−0.474642 + 0.880179i \(0.657422\pi\)
\(548\) 1.25235 0.0534979
\(549\) 7.46410 0.318560
\(550\) −4.40013 −0.187622
\(551\) 34.6950 1.47806
\(552\) 0.976584 0.0415662
\(553\) 69.3632 2.94963
\(554\) −11.9922 −0.509499
\(555\) 3.79603 0.161132
\(556\) 3.35534 0.142298
\(557\) 1.89969 0.0804927 0.0402463 0.999190i \(-0.487186\pi\)
0.0402463 + 0.999190i \(0.487186\pi\)
\(558\) 6.44069 0.272656
\(559\) 0 0
\(560\) −4.64466 −0.196273
\(561\) 17.6005 0.743094
\(562\) −3.63888 −0.153497
\(563\) −1.72000 −0.0724894 −0.0362447 0.999343i \(-0.511540\pi\)
−0.0362447 + 0.999343i \(0.511540\pi\)
\(564\) 9.75342 0.410693
\(565\) 13.0662 0.549697
\(566\) −4.84441 −0.203626
\(567\) −4.64466 −0.195057
\(568\) 7.95317 0.333707
\(569\) −0.601920 −0.0252338 −0.0126169 0.999920i \(-0.504016\pi\)
−0.0126169 + 0.999920i \(0.504016\pi\)
\(570\) 8.04479 0.336959
\(571\) −11.9808 −0.501381 −0.250691 0.968067i \(-0.580658\pi\)
−0.250691 + 0.968067i \(0.580658\pi\)
\(572\) 0 0
\(573\) −0.897014 −0.0374733
\(574\) 33.8564 1.41314
\(575\) −0.976584 −0.0407264
\(576\) 1.00000 0.0416667
\(577\) 43.0293 1.79133 0.895667 0.444725i \(-0.146699\pi\)
0.895667 + 0.444725i \(0.146699\pi\)
\(578\) 1.00000 0.0415945
\(579\) 20.2644 0.842158
\(580\) −4.31274 −0.179077
\(581\) 16.3071 0.676532
\(582\) −13.8252 −0.573074
\(583\) 59.4408 2.46179
\(584\) 4.36112 0.180464
\(585\) 0 0
\(586\) 3.70081 0.152879
\(587\) 17.9148 0.739423 0.369711 0.929147i \(-0.379457\pi\)
0.369711 + 0.929147i \(0.379457\pi\)
\(588\) 14.5729 0.600975
\(589\) 51.8139 2.13496
\(590\) −2.18056 −0.0897722
\(591\) −9.18056 −0.377638
\(592\) 3.79603 0.156016
\(593\) 0.669624 0.0274981 0.0137491 0.999905i \(-0.495623\pi\)
0.0137491 + 0.999905i \(0.495623\pi\)
\(594\) −4.40013 −0.180539
\(595\) −18.5786 −0.761650
\(596\) −4.61970 −0.189230
\(597\) −16.2175 −0.663739
\(598\) 0 0
\(599\) 1.29241 0.0528066 0.0264033 0.999651i \(-0.491595\pi\)
0.0264033 + 0.999651i \(0.491595\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −11.4734 −0.468011 −0.234005 0.972235i \(-0.575183\pi\)
−0.234005 + 0.972235i \(0.575183\pi\)
\(602\) 3.32770 0.135627
\(603\) 1.82522 0.0743286
\(604\) 11.2425 0.457450
\(605\) 8.36112 0.339928
\(606\) 6.80025 0.276241
\(607\) −24.1344 −0.979583 −0.489792 0.871839i \(-0.662927\pi\)
−0.489792 + 0.871839i \(0.662927\pi\)
\(608\) 8.04479 0.326259
\(609\) 20.0312 0.811705
\(610\) −7.46410 −0.302213
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) −28.9502 −1.16929 −0.584644 0.811290i \(-0.698766\pi\)
−0.584644 + 0.811290i \(0.698766\pi\)
\(614\) 7.11454 0.287119
\(615\) 7.28932 0.293934
\(616\) 20.4371 0.823434
\(617\) −0.841311 −0.0338699 −0.0169349 0.999857i \(-0.505391\pi\)
−0.0169349 + 0.999857i \(0.505391\pi\)
\(618\) −5.29137 −0.212850
\(619\) −6.25076 −0.251239 −0.125620 0.992078i \(-0.540092\pi\)
−0.125620 + 0.992078i \(0.540092\pi\)
\(620\) −6.44069 −0.258664
\(621\) −0.976584 −0.0391889
\(622\) −19.9148 −0.798510
\(623\) −37.9596 −1.52082
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.13950 −0.245384
\(627\) −35.3981 −1.41366
\(628\) −1.50311 −0.0599807
\(629\) 15.1841 0.605430
\(630\) 4.64466 0.185048
\(631\) −3.02496 −0.120422 −0.0602110 0.998186i \(-0.519177\pi\)
−0.0602110 + 0.998186i \(0.519177\pi\)
\(632\) 14.9340 0.594042
\(633\) −1.61970 −0.0643771
\(634\) 7.85286 0.311877
\(635\) 12.2196 0.484919
\(636\) 13.5089 0.535662
\(637\) 0 0
\(638\) 18.9766 0.751290
\(639\) −7.95317 −0.314622
\(640\) −1.00000 −0.0395285
\(641\) 0.772609 0.0305162 0.0152581 0.999884i \(-0.495143\pi\)
0.0152581 + 0.999884i \(0.495143\pi\)
\(642\) −16.9282 −0.668103
\(643\) 19.3745 0.764057 0.382028 0.924151i \(-0.375226\pi\)
0.382028 + 0.924151i \(0.375226\pi\)
\(644\) 4.53590 0.178739
\(645\) 0.716456 0.0282104
\(646\) 32.1791 1.26607
\(647\) −27.1905 −1.06897 −0.534485 0.845178i \(-0.679494\pi\)
−0.534485 + 0.845178i \(0.679494\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 9.59473 0.376626
\(650\) 0 0
\(651\) 29.9148 1.17245
\(652\) 0.176330 0.00690561
\(653\) 15.7960 0.618144 0.309072 0.951039i \(-0.399982\pi\)
0.309072 + 0.951039i \(0.399982\pi\)
\(654\) 12.8003 0.500530
\(655\) −0.378757 −0.0147993
\(656\) 7.28932 0.284600
\(657\) −4.36112 −0.170143
\(658\) 45.3013 1.76603
\(659\) 30.4762 1.18719 0.593593 0.804766i \(-0.297709\pi\)
0.593593 + 0.804766i \(0.297709\pi\)
\(660\) 4.40013 0.171275
\(661\) −27.8876 −1.08470 −0.542351 0.840152i \(-0.682466\pi\)
−0.542351 + 0.840152i \(0.682466\pi\)
\(662\) −31.9959 −1.24356
\(663\) 0 0
\(664\) 3.51093 0.136251
\(665\) 37.3653 1.44896
\(666\) −3.79603 −0.147093
\(667\) 4.21175 0.163079
\(668\) 8.68162 0.335902
\(669\) −2.23671 −0.0864762
\(670\) −1.82522 −0.0705143
\(671\) 32.8430 1.26789
\(672\) 4.64466 0.179172
\(673\) −0.978131 −0.0377042 −0.0188521 0.999822i \(-0.506001\pi\)
−0.0188521 + 0.999822i \(0.506001\pi\)
\(674\) −35.6432 −1.37293
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −19.1926 −0.737630 −0.368815 0.929503i \(-0.620236\pi\)
−0.368815 + 0.929503i \(0.620236\pi\)
\(678\) −13.0662 −0.501803
\(679\) −64.2134 −2.46429
\(680\) −4.00000 −0.153393
\(681\) −15.3205 −0.587083
\(682\) 28.3398 1.08519
\(683\) 1.51093 0.0578143 0.0289071 0.999582i \(-0.490797\pi\)
0.0289071 + 0.999582i \(0.490797\pi\)
\(684\) −8.04479 −0.307600
\(685\) 1.25235 0.0478500
\(686\) 35.1734 1.34293
\(687\) −10.1279 −0.386405
\(688\) 0.716456 0.0273146
\(689\) 0 0
\(690\) 0.976584 0.0371779
\(691\) −33.1166 −1.25981 −0.629907 0.776670i \(-0.716907\pi\)
−0.629907 + 0.776670i \(0.716907\pi\)
\(692\) 1.78043 0.0676818
\(693\) −20.4371 −0.776341
\(694\) 6.89701 0.261807
\(695\) 3.35534 0.127275
\(696\) 4.31274 0.163474
\(697\) 29.1573 1.10441
\(698\) −19.3205 −0.731292
\(699\) −20.7519 −0.784908
\(700\) −4.64466 −0.175552
\(701\) −27.8695 −1.05262 −0.526308 0.850294i \(-0.676424\pi\)
−0.526308 + 0.850294i \(0.676424\pi\)
\(702\) 0 0
\(703\) −30.5382 −1.15177
\(704\) 4.40013 0.165836
\(705\) 9.75342 0.367335
\(706\) −27.4173 −1.03186
\(707\) 31.5849 1.18787
\(708\) 2.18056 0.0819504
\(709\) 11.0562 0.415223 0.207611 0.978211i \(-0.433431\pi\)
0.207611 + 0.978211i \(0.433431\pi\)
\(710\) 7.95317 0.298477
\(711\) −14.9340 −0.560068
\(712\) −8.17274 −0.306286
\(713\) 6.28987 0.235557
\(714\) 18.5786 0.695288
\(715\) 0 0
\(716\) −17.4157 −0.650856
\(717\) 24.3539 0.909514
\(718\) −14.3611 −0.535952
\(719\) 11.7128 0.436814 0.218407 0.975858i \(-0.429914\pi\)
0.218407 + 0.975858i \(0.429914\pi\)
\(720\) 1.00000 0.0372678
\(721\) −24.5766 −0.915280
\(722\) −45.7186 −1.70147
\(723\) 19.8685 0.738916
\(724\) 10.3611 0.385068
\(725\) −4.31274 −0.160171
\(726\) −8.36112 −0.310310
\(727\) 19.4152 0.720071 0.360035 0.932939i \(-0.382765\pi\)
0.360035 + 0.932939i \(0.382765\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.36112 0.161412
\(731\) 2.86583 0.105996
\(732\) 7.46410 0.275881
\(733\) −11.9340 −0.440792 −0.220396 0.975411i \(-0.570735\pi\)
−0.220396 + 0.975411i \(0.570735\pi\)
\(734\) 13.7753 0.508455
\(735\) 14.5729 0.537528
\(736\) 0.976584 0.0359973
\(737\) 8.03119 0.295833
\(738\) −7.28932 −0.268324
\(739\) 19.8628 0.730666 0.365333 0.930877i \(-0.380955\pi\)
0.365333 + 0.930877i \(0.380955\pi\)
\(740\) 3.79603 0.139545
\(741\) 0 0
\(742\) 62.7442 2.30341
\(743\) −16.4877 −0.604873 −0.302437 0.953169i \(-0.597800\pi\)
−0.302437 + 0.953169i \(0.597800\pi\)
\(744\) 6.44069 0.236127
\(745\) −4.61970 −0.169253
\(746\) 33.4627 1.22516
\(747\) −3.51093 −0.128458
\(748\) 17.6005 0.643538
\(749\) −78.6257 −2.87292
\(750\) −1.00000 −0.0365148
\(751\) −46.6624 −1.70274 −0.851368 0.524569i \(-0.824227\pi\)
−0.851368 + 0.524569i \(0.824227\pi\)
\(752\) 9.75342 0.355671
\(753\) 13.5713 0.494566
\(754\) 0 0
\(755\) 11.2425 0.409156
\(756\) −4.64466 −0.168925
\(757\) 2.63597 0.0958061 0.0479030 0.998852i \(-0.484746\pi\)
0.0479030 + 0.998852i \(0.484746\pi\)
\(758\) −33.3768 −1.21230
\(759\) −4.29709 −0.155975
\(760\) 8.04479 0.291815
\(761\) −0.0800681 −0.00290246 −0.00145123 0.999999i \(-0.500462\pi\)
−0.00145123 + 0.999999i \(0.500462\pi\)
\(762\) −12.2196 −0.442668
\(763\) 59.4528 2.15234
\(764\) −0.897014 −0.0324528
\(765\) 4.00000 0.144620
\(766\) −7.33038 −0.264857
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −5.68317 −0.204940 −0.102470 0.994736i \(-0.532675\pi\)
−0.102470 + 0.994736i \(0.532675\pi\)
\(770\) 20.4371 0.736502
\(771\) 0.662300 0.0238522
\(772\) 20.2644 0.729330
\(773\) −7.58851 −0.272940 −0.136470 0.990644i \(-0.543576\pi\)
−0.136470 + 0.990644i \(0.543576\pi\)
\(774\) −0.716456 −0.0257525
\(775\) −6.44069 −0.231356
\(776\) −13.8252 −0.496296
\(777\) −17.6312 −0.632517
\(778\) −32.6198 −1.16948
\(779\) −58.6410 −2.10103
\(780\) 0 0
\(781\) −34.9949 −1.25222
\(782\) 3.90633 0.139690
\(783\) −4.31274 −0.154125
\(784\) 14.5729 0.520459
\(785\) −1.50311 −0.0536484
\(786\) 0.378757 0.0135098
\(787\) 4.73623 0.168828 0.0844142 0.996431i \(-0.473098\pi\)
0.0844142 + 0.996431i \(0.473098\pi\)
\(788\) −9.18056 −0.327044
\(789\) 30.7498 1.09472
\(790\) 14.9340 0.531327
\(791\) −60.6878 −2.15781
\(792\) −4.40013 −0.156352
\(793\) 0 0
\(794\) 14.4683 0.513462
\(795\) 13.5089 0.479111
\(796\) −16.2175 −0.574815
\(797\) −41.0636 −1.45455 −0.727274 0.686347i \(-0.759213\pi\)
−0.727274 + 0.686347i \(0.759213\pi\)
\(798\) −37.3653 −1.32272
\(799\) 39.0137 1.38020
\(800\) −1.00000 −0.0353553
\(801\) 8.17274 0.288769
\(802\) 7.28727 0.257323
\(803\) −19.1895 −0.677181
\(804\) 1.82522 0.0643705
\(805\) 4.53590 0.159869
\(806\) 0 0
\(807\) −3.74410 −0.131799
\(808\) 6.80025 0.239232
\(809\) −27.6005 −0.970382 −0.485191 0.874408i \(-0.661250\pi\)
−0.485191 + 0.874408i \(0.661250\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −10.6930 −0.375482 −0.187741 0.982219i \(-0.560117\pi\)
−0.187741 + 0.982219i \(0.560117\pi\)
\(812\) 20.0312 0.702957
\(813\) 27.7612 0.973628
\(814\) −16.7030 −0.585440
\(815\) 0.176330 0.00617657
\(816\) 4.00000 0.140028
\(817\) −5.76374 −0.201648
\(818\) 24.5766 0.859300
\(819\) 0 0
\(820\) 7.28932 0.254554
\(821\) −17.5635 −0.612972 −0.306486 0.951875i \(-0.599153\pi\)
−0.306486 + 0.951875i \(0.599153\pi\)
\(822\) −1.25235 −0.0436809
\(823\) −38.7130 −1.34945 −0.674725 0.738069i \(-0.735738\pi\)
−0.674725 + 0.738069i \(0.735738\pi\)
\(824\) −5.29137 −0.184333
\(825\) 4.40013 0.153193
\(826\) 10.1279 0.352396
\(827\) 28.4703 0.990010 0.495005 0.868890i \(-0.335166\pi\)
0.495005 + 0.868890i \(0.335166\pi\)
\(828\) −0.976584 −0.0339386
\(829\) 8.28709 0.287822 0.143911 0.989591i \(-0.454032\pi\)
0.143911 + 0.989591i \(0.454032\pi\)
\(830\) 3.51093 0.121866
\(831\) 11.9922 0.416004
\(832\) 0 0
\(833\) 58.2915 2.01968
\(834\) −3.35534 −0.116186
\(835\) 8.68162 0.300440
\(836\) −35.3981 −1.22427
\(837\) −6.44069 −0.222623
\(838\) −15.5537 −0.537293
\(839\) 42.5630 1.46944 0.734719 0.678372i \(-0.237314\pi\)
0.734719 + 0.678372i \(0.237314\pi\)
\(840\) 4.64466 0.160256
\(841\) −10.4003 −0.358631
\(842\) −22.3143 −0.769001
\(843\) 3.63888 0.125330
\(844\) −1.61970 −0.0557522
\(845\) 0 0
\(846\) −9.75342 −0.335330
\(847\) −38.8345 −1.33437
\(848\) 13.5089 0.463897
\(849\) 4.84441 0.166260
\(850\) −4.00000 −0.137199
\(851\) −3.70714 −0.127079
\(852\) −7.95317 −0.272471
\(853\) 28.1380 0.963425 0.481713 0.876329i \(-0.340015\pi\)
0.481713 + 0.876329i \(0.340015\pi\)
\(854\) 34.6682 1.18632
\(855\) −8.04479 −0.275126
\(856\) −16.9282 −0.578594
\(857\) 10.3355 0.353053 0.176526 0.984296i \(-0.443514\pi\)
0.176526 + 0.984296i \(0.443514\pi\)
\(858\) 0 0
\(859\) 0.380304 0.0129758 0.00648791 0.999979i \(-0.497935\pi\)
0.00648791 + 0.999979i \(0.497935\pi\)
\(860\) 0.716456 0.0244310
\(861\) −33.8564 −1.15382
\(862\) 24.2644 0.826447
\(863\) −47.1484 −1.60495 −0.802475 0.596685i \(-0.796484\pi\)
−0.802475 + 0.596685i \(0.796484\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 1.78043 0.0605365
\(866\) 23.0989 0.784932
\(867\) −1.00000 −0.0339618
\(868\) 29.9148 1.01537
\(869\) −65.7114 −2.22911
\(870\) 4.31274 0.146215
\(871\) 0 0
\(872\) 12.8003 0.433471
\(873\) 13.8252 0.467913
\(874\) −7.85641 −0.265747
\(875\) −4.64466 −0.157018
\(876\) −4.36112 −0.147348
\(877\) 40.0412 1.35209 0.676047 0.736858i \(-0.263691\pi\)
0.676047 + 0.736858i \(0.263691\pi\)
\(878\) 39.9959 1.34980
\(879\) −3.70081 −0.124825
\(880\) 4.40013 0.148328
\(881\) −15.4449 −0.520352 −0.260176 0.965561i \(-0.583781\pi\)
−0.260176 + 0.965561i \(0.583781\pi\)
\(882\) −14.5729 −0.490694
\(883\) 6.02142 0.202637 0.101318 0.994854i \(-0.467694\pi\)
0.101318 + 0.994854i \(0.467694\pi\)
\(884\) 0 0
\(885\) 2.18056 0.0732987
\(886\) 15.6036 0.524213
\(887\) −49.1380 −1.64989 −0.824947 0.565210i \(-0.808795\pi\)
−0.824947 + 0.565210i \(0.808795\pi\)
\(888\) −3.79603 −0.127386
\(889\) −56.7557 −1.90353
\(890\) −8.17274 −0.273951
\(891\) 4.40013 0.147410
\(892\) −2.23671 −0.0748906
\(893\) −78.4642 −2.62570
\(894\) 4.61970 0.154506
\(895\) −17.4157 −0.582143
\(896\) 4.64466 0.155167
\(897\) 0 0
\(898\) 27.0042 0.901141
\(899\) 27.7770 0.926414
\(900\) 1.00000 0.0333333
\(901\) 54.0356 1.80019
\(902\) −32.0739 −1.06795
\(903\) −3.32770 −0.110739
\(904\) −13.0662 −0.434574
\(905\) 10.3611 0.344415
\(906\) −11.2425 −0.373507
\(907\) 16.3669 0.543454 0.271727 0.962374i \(-0.412405\pi\)
0.271727 + 0.962374i \(0.412405\pi\)
\(908\) −15.3205 −0.508429
\(909\) −6.80025 −0.225550
\(910\) 0 0
\(911\) −10.4819 −0.347280 −0.173640 0.984809i \(-0.555553\pi\)
−0.173640 + 0.984809i \(0.555553\pi\)
\(912\) −8.04479 −0.266389
\(913\) −15.4486 −0.511273
\(914\) 17.7868 0.588336
\(915\) 7.46410 0.246756
\(916\) −10.1279 −0.334637
\(917\) 1.75920 0.0580938
\(918\) −4.00000 −0.132020
\(919\) 3.21130 0.105931 0.0529655 0.998596i \(-0.483133\pi\)
0.0529655 + 0.998596i \(0.483133\pi\)
\(920\) 0.976584 0.0321970
\(921\) −7.11454 −0.234432
\(922\) −24.9652 −0.822184
\(923\) 0 0
\(924\) −20.4371 −0.672331
\(925\) 3.79603 0.124813
\(926\) −15.6389 −0.513926
\(927\) 5.29137 0.173791
\(928\) 4.31274 0.141572
\(929\) 46.4788 1.52492 0.762460 0.647036i \(-0.223992\pi\)
0.762460 + 0.647036i \(0.223992\pi\)
\(930\) 6.44069 0.211198
\(931\) −117.236 −3.84224
\(932\) −20.7519 −0.679750
\(933\) 19.9148 0.651981
\(934\) −2.43914 −0.0798110
\(935\) 17.6005 0.575598
\(936\) 0 0
\(937\) −27.7627 −0.906969 −0.453485 0.891264i \(-0.649819\pi\)
−0.453485 + 0.891264i \(0.649819\pi\)
\(938\) 8.47751 0.276801
\(939\) 6.13950 0.200355
\(940\) 9.75342 0.318122
\(941\) 20.7962 0.677935 0.338968 0.940798i \(-0.389922\pi\)
0.338968 + 0.940798i \(0.389922\pi\)
\(942\) 1.50311 0.0489741
\(943\) −7.11863 −0.231814
\(944\) 2.18056 0.0709711
\(945\) −4.64466 −0.151091
\(946\) −3.15250 −0.102497
\(947\) 9.05925 0.294386 0.147193 0.989108i \(-0.452976\pi\)
0.147193 + 0.989108i \(0.452976\pi\)
\(948\) −14.9340 −0.485033
\(949\) 0 0
\(950\) 8.04479 0.261007
\(951\) −7.85286 −0.254646
\(952\) 18.5786 0.602137
\(953\) −15.2762 −0.494845 −0.247423 0.968908i \(-0.579584\pi\)
−0.247423 + 0.968908i \(0.579584\pi\)
\(954\) −13.5089 −0.437366
\(955\) −0.897014 −0.0290267
\(956\) 24.3539 0.787662
\(957\) −18.9766 −0.613426
\(958\) −14.1863 −0.458340
\(959\) −5.81676 −0.187833
\(960\) 1.00000 0.0322749
\(961\) 10.4824 0.338143
\(962\) 0 0
\(963\) 16.9282 0.545504
\(964\) 19.8685 0.639920
\(965\) 20.2644 0.652333
\(966\) −4.53590 −0.145940
\(967\) −46.3036 −1.48902 −0.744511 0.667610i \(-0.767317\pi\)
−0.744511 + 0.667610i \(0.767317\pi\)
\(968\) −8.36112 −0.268736
\(969\) −32.1791 −1.03374
\(970\) −13.8252 −0.443901
\(971\) 11.9652 0.383981 0.191990 0.981397i \(-0.438506\pi\)
0.191990 + 0.981397i \(0.438506\pi\)
\(972\) 1.00000 0.0320750
\(973\) −15.5844 −0.499613
\(974\) −5.76329 −0.184668
\(975\) 0 0
\(976\) 7.46410 0.238920
\(977\) −1.76415 −0.0564403 −0.0282201 0.999602i \(-0.508984\pi\)
−0.0282201 + 0.999602i \(0.508984\pi\)
\(978\) −0.176330 −0.00563841
\(979\) 35.9611 1.14932
\(980\) 14.5729 0.465513
\(981\) −12.8003 −0.408681
\(982\) 1.07534 0.0343155
\(983\) 54.2821 1.73133 0.865666 0.500623i \(-0.166896\pi\)
0.865666 + 0.500623i \(0.166896\pi\)
\(984\) −7.28932 −0.232375
\(985\) −9.18056 −0.292517
\(986\) 17.2509 0.549382
\(987\) −45.3013 −1.44196
\(988\) 0 0
\(989\) −0.699680 −0.0222485
\(990\) −4.40013 −0.139845
\(991\) −17.4783 −0.555218 −0.277609 0.960694i \(-0.589542\pi\)
−0.277609 + 0.960694i \(0.589542\pi\)
\(992\) 6.44069 0.204492
\(993\) 31.9959 1.01536
\(994\) −36.9398 −1.17166
\(995\) −16.2175 −0.514130
\(996\) −3.51093 −0.111248
\(997\) −21.2914 −0.674304 −0.337152 0.941450i \(-0.609464\pi\)
−0.337152 + 0.941450i \(0.609464\pi\)
\(998\) 14.7534 0.467011
\(999\) 3.79603 0.120101
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 5070.2.a.bz.1.1 4
13.5 odd 4 5070.2.b.ba.1351.5 8
13.6 odd 12 390.2.bb.c.361.4 yes 8
13.8 odd 4 5070.2.b.ba.1351.4 8
13.11 odd 12 390.2.bb.c.121.4 8
13.12 even 2 5070.2.a.ca.1.4 4
39.11 even 12 1170.2.bs.f.901.2 8
39.32 even 12 1170.2.bs.f.361.2 8
65.19 odd 12 1950.2.bc.g.751.1 8
65.24 odd 12 1950.2.bc.g.901.1 8
65.32 even 12 1950.2.y.k.49.1 8
65.37 even 12 1950.2.y.j.199.4 8
65.58 even 12 1950.2.y.j.49.4 8
65.63 even 12 1950.2.y.k.199.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.c.121.4 8 13.11 odd 12
390.2.bb.c.361.4 yes 8 13.6 odd 12
1170.2.bs.f.361.2 8 39.32 even 12
1170.2.bs.f.901.2 8 39.11 even 12
1950.2.y.j.49.4 8 65.58 even 12
1950.2.y.j.199.4 8 65.37 even 12
1950.2.y.k.49.1 8 65.32 even 12
1950.2.y.k.199.1 8 65.63 even 12
1950.2.bc.g.751.1 8 65.19 odd 12
1950.2.bc.g.901.1 8 65.24 odd 12
5070.2.a.bz.1.1 4 1.1 even 1 trivial
5070.2.a.ca.1.4 4 13.12 even 2
5070.2.b.ba.1351.4 8 13.8 odd 4
5070.2.b.ba.1351.5 8 13.5 odd 4