Properties

Label 5070.2.a.ba.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.73205 q^{11} -1.00000 q^{12} +3.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +2.26795 q^{19} +1.00000 q^{20} +3.00000 q^{21} -3.73205 q^{22} +3.46410 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -3.00000 q^{28} -5.46410 q^{29} +1.00000 q^{30} -8.92820 q^{31} -1.00000 q^{32} -3.73205 q^{33} +4.00000 q^{34} -3.00000 q^{35} +1.00000 q^{36} +7.92820 q^{37} -2.26795 q^{38} -1.00000 q^{40} +4.00000 q^{41} -3.00000 q^{42} -6.00000 q^{43} +3.73205 q^{44} +1.00000 q^{45} -3.46410 q^{46} -0.464102 q^{47} -1.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} +4.00000 q^{51} -3.73205 q^{53} +1.00000 q^{54} +3.73205 q^{55} +3.00000 q^{56} -2.26795 q^{57} +5.46410 q^{58} +4.53590 q^{59} -1.00000 q^{60} -7.46410 q^{61} +8.92820 q^{62} -3.00000 q^{63} +1.00000 q^{64} +3.73205 q^{66} +5.46410 q^{67} -4.00000 q^{68} -3.46410 q^{69} +3.00000 q^{70} +0.928203 q^{71} -1.00000 q^{72} -6.92820 q^{73} -7.92820 q^{74} -1.00000 q^{75} +2.26795 q^{76} -11.1962 q^{77} +16.9282 q^{79} +1.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +2.53590 q^{83} +3.00000 q^{84} -4.00000 q^{85} +6.00000 q^{86} +5.46410 q^{87} -3.73205 q^{88} +10.1244 q^{89} -1.00000 q^{90} +3.46410 q^{92} +8.92820 q^{93} +0.464102 q^{94} +2.26795 q^{95} +1.00000 q^{96} -12.3923 q^{97} -2.00000 q^{98} +3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 6q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 6q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 4q^{11} - 2q^{12} + 6q^{14} - 2q^{15} + 2q^{16} - 8q^{17} - 2q^{18} + 8q^{19} + 2q^{20} + 6q^{21} - 4q^{22} + 2q^{24} + 2q^{25} - 2q^{27} - 6q^{28} - 4q^{29} + 2q^{30} - 4q^{31} - 2q^{32} - 4q^{33} + 8q^{34} - 6q^{35} + 2q^{36} + 2q^{37} - 8q^{38} - 2q^{40} + 8q^{41} - 6q^{42} - 12q^{43} + 4q^{44} + 2q^{45} + 6q^{47} - 2q^{48} + 4q^{49} - 2q^{50} + 8q^{51} - 4q^{53} + 2q^{54} + 4q^{55} + 6q^{56} - 8q^{57} + 4q^{58} + 16q^{59} - 2q^{60} - 8q^{61} + 4q^{62} - 6q^{63} + 2q^{64} + 4q^{66} + 4q^{67} - 8q^{68} + 6q^{70} - 12q^{71} - 2q^{72} - 2q^{74} - 2q^{75} + 8q^{76} - 12q^{77} + 20q^{79} + 2q^{80} + 2q^{81} - 8q^{82} + 12q^{83} + 6q^{84} - 8q^{85} + 12q^{86} + 4q^{87} - 4q^{88} - 4q^{89} - 2q^{90} + 4q^{93} - 6q^{94} + 8q^{95} + 2q^{96} - 4q^{97} - 4q^{98} + 4q^{99} + O(q^{100}) \)

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
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.73205 1.12526 0.562628 0.826710i \(-0.309790\pi\)
0.562628 + 0.826710i \(0.309790\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 3.00000 0.801784
\(15\) −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
\(19\) 2.26795 0.520303 0.260152 0.965568i \(-0.416227\pi\)
0.260152 + 0.965568i \(0.416227\pi\)
\(20\) 1.00000 0.223607
\(21\) 3.00000 0.654654
\(22\) −3.73205 −0.795676
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −3.00000 −0.566947
\(29\) −5.46410 −1.01466 −0.507329 0.861752i \(-0.669367\pi\)
−0.507329 + 0.861752i \(0.669367\pi\)
\(30\) 1.00000 0.182574
\(31\) −8.92820 −1.60355 −0.801776 0.597624i \(-0.796111\pi\)
−0.801776 + 0.597624i \(0.796111\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.73205 −0.649667
\(34\) 4.00000 0.685994
\(35\) −3.00000 −0.507093
\(36\) 1.00000 0.166667
\(37\) 7.92820 1.30339 0.651694 0.758482i \(-0.274059\pi\)
0.651694 + 0.758482i \(0.274059\pi\)
\(38\) −2.26795 −0.367910
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) −3.00000 −0.462910
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 3.73205 0.562628
\(45\) 1.00000 0.149071
\(46\) −3.46410 −0.510754
\(47\) −0.464102 −0.0676962 −0.0338481 0.999427i \(-0.510776\pi\)
−0.0338481 + 0.999427i \(0.510776\pi\)
\(48\) −1.00000 −0.144338
\(49\) 2.00000 0.285714
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) −3.73205 −0.512637 −0.256318 0.966592i \(-0.582510\pi\)
−0.256318 + 0.966592i \(0.582510\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.73205 0.503230
\(56\) 3.00000 0.400892
\(57\) −2.26795 −0.300397
\(58\) 5.46410 0.717472
\(59\) 4.53590 0.590524 0.295262 0.955416i \(-0.404593\pi\)
0.295262 + 0.955416i \(0.404593\pi\)
\(60\) −1.00000 −0.129099
\(61\) −7.46410 −0.955680 −0.477840 0.878447i \(-0.658580\pi\)
−0.477840 + 0.878447i \(0.658580\pi\)
\(62\) 8.92820 1.13388
\(63\) −3.00000 −0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.73205 0.459384
\(67\) 5.46410 0.667546 0.333773 0.942653i \(-0.391678\pi\)
0.333773 + 0.942653i \(0.391678\pi\)
\(68\) −4.00000 −0.485071
\(69\) −3.46410 −0.417029
\(70\) 3.00000 0.358569
\(71\) 0.928203 0.110157 0.0550787 0.998482i \(-0.482459\pi\)
0.0550787 + 0.998482i \(0.482459\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) −7.92820 −0.921635
\(75\) −1.00000 −0.115470
\(76\) 2.26795 0.260152
\(77\) −11.1962 −1.27592
\(78\) 0 0
\(79\) 16.9282 1.90457 0.952286 0.305208i \(-0.0987259\pi\)
0.952286 + 0.305208i \(0.0987259\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) 2.53590 0.278351 0.139176 0.990268i \(-0.455555\pi\)
0.139176 + 0.990268i \(0.455555\pi\)
\(84\) 3.00000 0.327327
\(85\) −4.00000 −0.433861
\(86\) 6.00000 0.646997
\(87\) 5.46410 0.585813
\(88\) −3.73205 −0.397838
\(89\) 10.1244 1.07318 0.536590 0.843843i \(-0.319712\pi\)
0.536590 + 0.843843i \(0.319712\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 3.46410 0.361158
\(93\) 8.92820 0.925812
\(94\) 0.464102 0.0478684
\(95\) 2.26795 0.232687
\(96\) 1.00000 0.102062
\(97\) −12.3923 −1.25825 −0.629124 0.777305i \(-0.716586\pi\)
−0.629124 + 0.777305i \(0.716586\pi\)
\(98\) −2.00000 −0.202031
\(99\) 3.73205 0.375085
\(100\) 1.00000 0.100000
\(101\) 8.39230 0.835066 0.417533 0.908662i \(-0.362895\pi\)
0.417533 + 0.908662i \(0.362895\pi\)
\(102\) −4.00000 −0.396059
\(103\) −19.5885 −1.93011 −0.965054 0.262051i \(-0.915601\pi\)
−0.965054 + 0.262051i \(0.915601\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 3.73205 0.362489
\(107\) 0.928203 0.0897328 0.0448664 0.998993i \(-0.485714\pi\)
0.0448664 + 0.998993i \(0.485714\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.3923 0.995402 0.497701 0.867349i \(-0.334178\pi\)
0.497701 + 0.867349i \(0.334178\pi\)
\(110\) −3.73205 −0.355837
\(111\) −7.92820 −0.752512
\(112\) −3.00000 −0.283473
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 2.26795 0.212413
\(115\) 3.46410 0.323029
\(116\) −5.46410 −0.507329
\(117\) 0 0
\(118\) −4.53590 −0.417563
\(119\) 12.0000 1.10004
\(120\) 1.00000 0.0912871
\(121\) 2.92820 0.266200
\(122\) 7.46410 0.675768
\(123\) −4.00000 −0.360668
\(124\) −8.92820 −0.801776
\(125\) 1.00000 0.0894427
\(126\) 3.00000 0.267261
\(127\) −4.66025 −0.413531 −0.206765 0.978391i \(-0.566294\pi\)
−0.206765 + 0.978391i \(0.566294\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −20.3205 −1.77541 −0.887706 0.460412i \(-0.847702\pi\)
−0.887706 + 0.460412i \(0.847702\pi\)
\(132\) −3.73205 −0.324833
\(133\) −6.80385 −0.589968
\(134\) −5.46410 −0.472026
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) −6.53590 −0.558399 −0.279200 0.960233i \(-0.590069\pi\)
−0.279200 + 0.960233i \(0.590069\pi\)
\(138\) 3.46410 0.294884
\(139\) 21.7846 1.84775 0.923873 0.382699i \(-0.125005\pi\)
0.923873 + 0.382699i \(0.125005\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0.464102 0.0390844
\(142\) −0.928203 −0.0778931
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −5.46410 −0.453769
\(146\) 6.92820 0.573382
\(147\) −2.00000 −0.164957
\(148\) 7.92820 0.651694
\(149\) −22.7846 −1.86659 −0.933294 0.359113i \(-0.883079\pi\)
−0.933294 + 0.359113i \(0.883079\pi\)
\(150\) 1.00000 0.0816497
\(151\) 18.7846 1.52867 0.764335 0.644819i \(-0.223067\pi\)
0.764335 + 0.644819i \(0.223067\pi\)
\(152\) −2.26795 −0.183955
\(153\) −4.00000 −0.323381
\(154\) 11.1962 0.902212
\(155\) −8.92820 −0.717131
\(156\) 0 0
\(157\) 10.8038 0.862241 0.431120 0.902294i \(-0.358118\pi\)
0.431120 + 0.902294i \(0.358118\pi\)
\(158\) −16.9282 −1.34674
\(159\) 3.73205 0.295971
\(160\) −1.00000 −0.0790569
\(161\) −10.3923 −0.819028
\(162\) −1.00000 −0.0785674
\(163\) −10.9282 −0.855963 −0.427981 0.903788i \(-0.640775\pi\)
−0.427981 + 0.903788i \(0.640775\pi\)
\(164\) 4.00000 0.312348
\(165\) −3.73205 −0.290540
\(166\) −2.53590 −0.196824
\(167\) 6.46410 0.500207 0.250104 0.968219i \(-0.419535\pi\)
0.250104 + 0.968219i \(0.419535\pi\)
\(168\) −3.00000 −0.231455
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) 2.26795 0.173434
\(172\) −6.00000 −0.457496
\(173\) −22.1244 −1.68208 −0.841042 0.540970i \(-0.818057\pi\)
−0.841042 + 0.540970i \(0.818057\pi\)
\(174\) −5.46410 −0.414232
\(175\) −3.00000 −0.226779
\(176\) 3.73205 0.281314
\(177\) −4.53590 −0.340939
\(178\) −10.1244 −0.758853
\(179\) −22.9282 −1.71373 −0.856867 0.515537i \(-0.827592\pi\)
−0.856867 + 0.515537i \(0.827592\pi\)
\(180\) 1.00000 0.0745356
\(181\) −3.07180 −0.228325 −0.114162 0.993462i \(-0.536418\pi\)
−0.114162 + 0.993462i \(0.536418\pi\)
\(182\) 0 0
\(183\) 7.46410 0.551762
\(184\) −3.46410 −0.255377
\(185\) 7.92820 0.582893
\(186\) −8.92820 −0.654648
\(187\) −14.9282 −1.09166
\(188\) −0.464102 −0.0338481
\(189\) 3.00000 0.218218
\(190\) −2.26795 −0.164534
\(191\) 17.3205 1.25327 0.626634 0.779314i \(-0.284432\pi\)
0.626634 + 0.779314i \(0.284432\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 17.8564 1.28533 0.642666 0.766146i \(-0.277828\pi\)
0.642666 + 0.766146i \(0.277828\pi\)
\(194\) 12.3923 0.889716
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −9.39230 −0.669174 −0.334587 0.942365i \(-0.608597\pi\)
−0.334587 + 0.942365i \(0.608597\pi\)
\(198\) −3.73205 −0.265225
\(199\) 11.0718 0.784859 0.392429 0.919782i \(-0.371635\pi\)
0.392429 + 0.919782i \(0.371635\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −5.46410 −0.385408
\(202\) −8.39230 −0.590481
\(203\) 16.3923 1.15051
\(204\) 4.00000 0.280056
\(205\) 4.00000 0.279372
\(206\) 19.5885 1.36479
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) 8.46410 0.585474
\(210\) −3.00000 −0.207020
\(211\) −21.9282 −1.50960 −0.754800 0.655955i \(-0.772266\pi\)
−0.754800 + 0.655955i \(0.772266\pi\)
\(212\) −3.73205 −0.256318
\(213\) −0.928203 −0.0635994
\(214\) −0.928203 −0.0634507
\(215\) −6.00000 −0.409197
\(216\) 1.00000 0.0680414
\(217\) 26.7846 1.81826
\(218\) −10.3923 −0.703856
\(219\) 6.92820 0.468165
\(220\) 3.73205 0.251615
\(221\) 0 0
\(222\) 7.92820 0.532106
\(223\) −8.85641 −0.593069 −0.296534 0.955022i \(-0.595831\pi\)
−0.296534 + 0.955022i \(0.595831\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) 12.0000 0.798228
\(227\) 13.4641 0.893644 0.446822 0.894623i \(-0.352556\pi\)
0.446822 + 0.894623i \(0.352556\pi\)
\(228\) −2.26795 −0.150199
\(229\) −11.4641 −0.757569 −0.378785 0.925485i \(-0.623658\pi\)
−0.378785 + 0.925485i \(0.623658\pi\)
\(230\) −3.46410 −0.228416
\(231\) 11.1962 0.736653
\(232\) 5.46410 0.358736
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −0.464102 −0.0302747
\(236\) 4.53590 0.295262
\(237\) −16.9282 −1.09960
\(238\) −12.0000 −0.777844
\(239\) 3.46410 0.224074 0.112037 0.993704i \(-0.464262\pi\)
0.112037 + 0.993704i \(0.464262\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 14.8038 0.953600 0.476800 0.879012i \(-0.341797\pi\)
0.476800 + 0.879012i \(0.341797\pi\)
\(242\) −2.92820 −0.188232
\(243\) −1.00000 −0.0641500
\(244\) −7.46410 −0.477840
\(245\) 2.00000 0.127775
\(246\) 4.00000 0.255031
\(247\) 0 0
\(248\) 8.92820 0.566941
\(249\) −2.53590 −0.160706
\(250\) −1.00000 −0.0632456
\(251\) −26.4641 −1.67040 −0.835200 0.549947i \(-0.814648\pi\)
−0.835200 + 0.549947i \(0.814648\pi\)
\(252\) −3.00000 −0.188982
\(253\) 12.9282 0.812789
\(254\) 4.66025 0.292410
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −21.4641 −1.33889 −0.669447 0.742860i \(-0.733469\pi\)
−0.669447 + 0.742860i \(0.733469\pi\)
\(258\) −6.00000 −0.373544
\(259\) −23.7846 −1.47790
\(260\) 0 0
\(261\) −5.46410 −0.338219
\(262\) 20.3205 1.25541
\(263\) 20.5167 1.26511 0.632556 0.774515i \(-0.282006\pi\)
0.632556 + 0.774515i \(0.282006\pi\)
\(264\) 3.73205 0.229692
\(265\) −3.73205 −0.229258
\(266\) 6.80385 0.417171
\(267\) −10.1244 −0.619601
\(268\) 5.46410 0.333773
\(269\) −30.9282 −1.88573 −0.942863 0.333181i \(-0.891878\pi\)
−0.942863 + 0.333181i \(0.891878\pi\)
\(270\) 1.00000 0.0608581
\(271\) 3.60770 0.219152 0.109576 0.993978i \(-0.465051\pi\)
0.109576 + 0.993978i \(0.465051\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 6.53590 0.394848
\(275\) 3.73205 0.225051
\(276\) −3.46410 −0.208514
\(277\) 19.5885 1.17696 0.588478 0.808513i \(-0.299727\pi\)
0.588478 + 0.808513i \(0.299727\pi\)
\(278\) −21.7846 −1.30655
\(279\) −8.92820 −0.534518
\(280\) 3.00000 0.179284
\(281\) −6.92820 −0.413302 −0.206651 0.978415i \(-0.566256\pi\)
−0.206651 + 0.978415i \(0.566256\pi\)
\(282\) −0.464102 −0.0276368
\(283\) −14.3923 −0.855534 −0.427767 0.903889i \(-0.640700\pi\)
−0.427767 + 0.903889i \(0.640700\pi\)
\(284\) 0.928203 0.0550787
\(285\) −2.26795 −0.134342
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 5.46410 0.320863
\(291\) 12.3923 0.726450
\(292\) −6.92820 −0.405442
\(293\) 19.2487 1.12452 0.562261 0.826960i \(-0.309932\pi\)
0.562261 + 0.826960i \(0.309932\pi\)
\(294\) 2.00000 0.116642
\(295\) 4.53590 0.264090
\(296\) −7.92820 −0.460817
\(297\) −3.73205 −0.216556
\(298\) 22.7846 1.31988
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 18.0000 1.03750
\(302\) −18.7846 −1.08093
\(303\) −8.39230 −0.482125
\(304\) 2.26795 0.130076
\(305\) −7.46410 −0.427393
\(306\) 4.00000 0.228665
\(307\) 20.2487 1.15565 0.577827 0.816159i \(-0.303901\pi\)
0.577827 + 0.816159i \(0.303901\pi\)
\(308\) −11.1962 −0.637960
\(309\) 19.5885 1.11435
\(310\) 8.92820 0.507088
\(311\) −5.07180 −0.287595 −0.143798 0.989607i \(-0.545931\pi\)
−0.143798 + 0.989607i \(0.545931\pi\)
\(312\) 0 0
\(313\) 1.32051 0.0746395 0.0373198 0.999303i \(-0.488118\pi\)
0.0373198 + 0.999303i \(0.488118\pi\)
\(314\) −10.8038 −0.609696
\(315\) −3.00000 −0.169031
\(316\) 16.9282 0.952286
\(317\) −23.5359 −1.32191 −0.660954 0.750427i \(-0.729848\pi\)
−0.660954 + 0.750427i \(0.729848\pi\)
\(318\) −3.73205 −0.209283
\(319\) −20.3923 −1.14175
\(320\) 1.00000 0.0559017
\(321\) −0.928203 −0.0518073
\(322\) 10.3923 0.579141
\(323\) −9.07180 −0.504768
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 10.9282 0.605257
\(327\) −10.3923 −0.574696
\(328\) −4.00000 −0.220863
\(329\) 1.39230 0.0767603
\(330\) 3.73205 0.205443
\(331\) −6.39230 −0.351353 −0.175676 0.984448i \(-0.556211\pi\)
−0.175676 + 0.984448i \(0.556211\pi\)
\(332\) 2.53590 0.139176
\(333\) 7.92820 0.434463
\(334\) −6.46410 −0.353700
\(335\) 5.46410 0.298536
\(336\) 3.00000 0.163663
\(337\) 5.60770 0.305471 0.152735 0.988267i \(-0.451192\pi\)
0.152735 + 0.988267i \(0.451192\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) −4.00000 −0.216930
\(341\) −33.3205 −1.80441
\(342\) −2.26795 −0.122637
\(343\) 15.0000 0.809924
\(344\) 6.00000 0.323498
\(345\) −3.46410 −0.186501
\(346\) 22.1244 1.18941
\(347\) −16.3923 −0.879985 −0.439993 0.898001i \(-0.645019\pi\)
−0.439993 + 0.898001i \(0.645019\pi\)
\(348\) 5.46410 0.292907
\(349\) −17.4641 −0.934832 −0.467416 0.884038i \(-0.654815\pi\)
−0.467416 + 0.884038i \(0.654815\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −3.73205 −0.198919
\(353\) −28.3923 −1.51117 −0.755585 0.655051i \(-0.772647\pi\)
−0.755585 + 0.655051i \(0.772647\pi\)
\(354\) 4.53590 0.241080
\(355\) 0.928203 0.0492639
\(356\) 10.1244 0.536590
\(357\) −12.0000 −0.635107
\(358\) 22.9282 1.21179
\(359\) −12.9282 −0.682324 −0.341162 0.940004i \(-0.610821\pi\)
−0.341162 + 0.940004i \(0.610821\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −13.8564 −0.729285
\(362\) 3.07180 0.161450
\(363\) −2.92820 −0.153691
\(364\) 0 0
\(365\) −6.92820 −0.362639
\(366\) −7.46410 −0.390155
\(367\) −13.3205 −0.695325 −0.347662 0.937620i \(-0.613024\pi\)
−0.347662 + 0.937620i \(0.613024\pi\)
\(368\) 3.46410 0.180579
\(369\) 4.00000 0.208232
\(370\) −7.92820 −0.412168
\(371\) 11.1962 0.581275
\(372\) 8.92820 0.462906
\(373\) 22.9282 1.18718 0.593589 0.804769i \(-0.297711\pi\)
0.593589 + 0.804769i \(0.297711\pi\)
\(374\) 14.9282 0.771919
\(375\) −1.00000 −0.0516398
\(376\) 0.464102 0.0239342
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) 6.26795 0.321963 0.160981 0.986957i \(-0.448534\pi\)
0.160981 + 0.986957i \(0.448534\pi\)
\(380\) 2.26795 0.116343
\(381\) 4.66025 0.238752
\(382\) −17.3205 −0.886194
\(383\) −36.7846 −1.87961 −0.939803 0.341717i \(-0.888992\pi\)
−0.939803 + 0.341717i \(0.888992\pi\)
\(384\) 1.00000 0.0510310
\(385\) −11.1962 −0.570609
\(386\) −17.8564 −0.908867
\(387\) −6.00000 −0.304997
\(388\) −12.3923 −0.629124
\(389\) 9.85641 0.499740 0.249870 0.968279i \(-0.419612\pi\)
0.249870 + 0.968279i \(0.419612\pi\)
\(390\) 0 0
\(391\) −13.8564 −0.700749
\(392\) −2.00000 −0.101015
\(393\) 20.3205 1.02503
\(394\) 9.39230 0.473177
\(395\) 16.9282 0.851750
\(396\) 3.73205 0.187543
\(397\) −7.92820 −0.397905 −0.198953 0.980009i \(-0.563754\pi\)
−0.198953 + 0.980009i \(0.563754\pi\)
\(398\) −11.0718 −0.554979
\(399\) 6.80385 0.340618
\(400\) 1.00000 0.0500000
\(401\) 2.12436 0.106085 0.0530426 0.998592i \(-0.483108\pi\)
0.0530426 + 0.998592i \(0.483108\pi\)
\(402\) 5.46410 0.272525
\(403\) 0 0
\(404\) 8.39230 0.417533
\(405\) 1.00000 0.0496904
\(406\) −16.3923 −0.813536
\(407\) 29.5885 1.46665
\(408\) −4.00000 −0.198030
\(409\) −0.947441 −0.0468479 −0.0234240 0.999726i \(-0.507457\pi\)
−0.0234240 + 0.999726i \(0.507457\pi\)
\(410\) −4.00000 −0.197546
\(411\) 6.53590 0.322392
\(412\) −19.5885 −0.965054
\(413\) −13.6077 −0.669591
\(414\) −3.46410 −0.170251
\(415\) 2.53590 0.124482
\(416\) 0 0
\(417\) −21.7846 −1.06680
\(418\) −8.46410 −0.413993
\(419\) −17.8564 −0.872343 −0.436171 0.899864i \(-0.643666\pi\)
−0.436171 + 0.899864i \(0.643666\pi\)
\(420\) 3.00000 0.146385
\(421\) 5.85641 0.285424 0.142712 0.989764i \(-0.454418\pi\)
0.142712 + 0.989764i \(0.454418\pi\)
\(422\) 21.9282 1.06745
\(423\) −0.464102 −0.0225654
\(424\) 3.73205 0.181244
\(425\) −4.00000 −0.194029
\(426\) 0.928203 0.0449716
\(427\) 22.3923 1.08364
\(428\) 0.928203 0.0448664
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −13.8564 −0.667440 −0.333720 0.942672i \(-0.608304\pi\)
−0.333720 + 0.942672i \(0.608304\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −32.7846 −1.57553 −0.787764 0.615977i \(-0.788761\pi\)
−0.787764 + 0.615977i \(0.788761\pi\)
\(434\) −26.7846 −1.28570
\(435\) 5.46410 0.261984
\(436\) 10.3923 0.497701
\(437\) 7.85641 0.375823
\(438\) −6.92820 −0.331042
\(439\) −21.3205 −1.01757 −0.508786 0.860893i \(-0.669906\pi\)
−0.508786 + 0.860893i \(0.669906\pi\)
\(440\) −3.73205 −0.177919
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) 7.85641 0.373269 0.186635 0.982429i \(-0.440242\pi\)
0.186635 + 0.982429i \(0.440242\pi\)
\(444\) −7.92820 −0.376256
\(445\) 10.1244 0.479940
\(446\) 8.85641 0.419363
\(447\) 22.7846 1.07768
\(448\) −3.00000 −0.141737
\(449\) −18.1244 −0.855341 −0.427671 0.903935i \(-0.640666\pi\)
−0.427671 + 0.903935i \(0.640666\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 14.9282 0.702942
\(452\) −12.0000 −0.564433
\(453\) −18.7846 −0.882578
\(454\) −13.4641 −0.631902
\(455\) 0 0
\(456\) 2.26795 0.106206
\(457\) 0.535898 0.0250683 0.0125341 0.999921i \(-0.496010\pi\)
0.0125341 + 0.999921i \(0.496010\pi\)
\(458\) 11.4641 0.535682
\(459\) 4.00000 0.186704
\(460\) 3.46410 0.161515
\(461\) −32.3923 −1.50866 −0.754330 0.656495i \(-0.772038\pi\)
−0.754330 + 0.656495i \(0.772038\pi\)
\(462\) −11.1962 −0.520892
\(463\) 0.784610 0.0364639 0.0182320 0.999834i \(-0.494196\pi\)
0.0182320 + 0.999834i \(0.494196\pi\)
\(464\) −5.46410 −0.253665
\(465\) 8.92820 0.414036
\(466\) 18.0000 0.833834
\(467\) −3.60770 −0.166944 −0.0834721 0.996510i \(-0.526601\pi\)
−0.0834721 + 0.996510i \(0.526601\pi\)
\(468\) 0 0
\(469\) −16.3923 −0.756926
\(470\) 0.464102 0.0214074
\(471\) −10.8038 −0.497815
\(472\) −4.53590 −0.208782
\(473\) −22.3923 −1.02960
\(474\) 16.9282 0.777538
\(475\) 2.26795 0.104061
\(476\) 12.0000 0.550019
\(477\) −3.73205 −0.170879
\(478\) −3.46410 −0.158444
\(479\) 26.2487 1.19933 0.599667 0.800250i \(-0.295300\pi\)
0.599667 + 0.800250i \(0.295300\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −14.8038 −0.674297
\(483\) 10.3923 0.472866
\(484\) 2.92820 0.133100
\(485\) −12.3923 −0.562706
\(486\) 1.00000 0.0453609
\(487\) 21.0000 0.951601 0.475800 0.879553i \(-0.342158\pi\)
0.475800 + 0.879553i \(0.342158\pi\)
\(488\) 7.46410 0.337884
\(489\) 10.9282 0.494190
\(490\) −2.00000 −0.0903508
\(491\) 15.3923 0.694645 0.347322 0.937746i \(-0.387091\pi\)
0.347322 + 0.937746i \(0.387091\pi\)
\(492\) −4.00000 −0.180334
\(493\) 21.8564 0.984363
\(494\) 0 0
\(495\) 3.73205 0.167743
\(496\) −8.92820 −0.400888
\(497\) −2.78461 −0.124907
\(498\) 2.53590 0.113636
\(499\) 1.32051 0.0591141 0.0295570 0.999563i \(-0.490590\pi\)
0.0295570 + 0.999563i \(0.490590\pi\)
\(500\) 1.00000 0.0447214
\(501\) −6.46410 −0.288795
\(502\) 26.4641 1.18115
\(503\) 0.267949 0.0119473 0.00597363 0.999982i \(-0.498099\pi\)
0.00597363 + 0.999982i \(0.498099\pi\)
\(504\) 3.00000 0.133631
\(505\) 8.39230 0.373453
\(506\) −12.9282 −0.574729
\(507\) 0 0
\(508\) −4.66025 −0.206765
\(509\) 5.32051 0.235827 0.117914 0.993024i \(-0.462379\pi\)
0.117914 + 0.993024i \(0.462379\pi\)
\(510\) −4.00000 −0.177123
\(511\) 20.7846 0.919457
\(512\) −1.00000 −0.0441942
\(513\) −2.26795 −0.100132
\(514\) 21.4641 0.946741
\(515\) −19.5885 −0.863171
\(516\) 6.00000 0.264135
\(517\) −1.73205 −0.0761755
\(518\) 23.7846 1.04504
\(519\) 22.1244 0.971151
\(520\) 0 0
\(521\) −27.3923 −1.20008 −0.600039 0.799970i \(-0.704848\pi\)
−0.600039 + 0.799970i \(0.704848\pi\)
\(522\) 5.46410 0.239157
\(523\) 13.7128 0.599619 0.299810 0.953999i \(-0.403077\pi\)
0.299810 + 0.953999i \(0.403077\pi\)
\(524\) −20.3205 −0.887706
\(525\) 3.00000 0.130931
\(526\) −20.5167 −0.894569
\(527\) 35.7128 1.55567
\(528\) −3.73205 −0.162417
\(529\) −11.0000 −0.478261
\(530\) 3.73205 0.162110
\(531\) 4.53590 0.196841
\(532\) −6.80385 −0.294984
\(533\) 0 0
\(534\) 10.1244 0.438124
\(535\) 0.928203 0.0401297
\(536\) −5.46410 −0.236013
\(537\) 22.9282 0.989425
\(538\) 30.9282 1.33341
\(539\) 7.46410 0.321502
\(540\) −1.00000 −0.0430331
\(541\) −26.7846 −1.15156 −0.575780 0.817605i \(-0.695302\pi\)
−0.575780 + 0.817605i \(0.695302\pi\)
\(542\) −3.60770 −0.154964
\(543\) 3.07180 0.131823
\(544\) 4.00000 0.171499
\(545\) 10.3923 0.445157
\(546\) 0 0
\(547\) 29.3205 1.25365 0.626827 0.779158i \(-0.284353\pi\)
0.626827 + 0.779158i \(0.284353\pi\)
\(548\) −6.53590 −0.279200
\(549\) −7.46410 −0.318560
\(550\) −3.73205 −0.159135
\(551\) −12.3923 −0.527930
\(552\) 3.46410 0.147442
\(553\) −50.7846 −2.15958
\(554\) −19.5885 −0.832234
\(555\) −7.92820 −0.336533
\(556\) 21.7846 0.923873
\(557\) −43.3923 −1.83859 −0.919295 0.393568i \(-0.871241\pi\)
−0.919295 + 0.393568i \(0.871241\pi\)
\(558\) 8.92820 0.377961
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) 14.9282 0.630269
\(562\) 6.92820 0.292249
\(563\) −19.3205 −0.814262 −0.407131 0.913370i \(-0.633471\pi\)
−0.407131 + 0.913370i \(0.633471\pi\)
\(564\) 0.464102 0.0195422
\(565\) −12.0000 −0.504844
\(566\) 14.3923 0.604954
\(567\) −3.00000 −0.125988
\(568\) −0.928203 −0.0389465
\(569\) −18.3205 −0.768036 −0.384018 0.923326i \(-0.625460\pi\)
−0.384018 + 0.923326i \(0.625460\pi\)
\(570\) 2.26795 0.0949939
\(571\) 16.8564 0.705419 0.352709 0.935733i \(-0.385260\pi\)
0.352709 + 0.935733i \(0.385260\pi\)
\(572\) 0 0
\(573\) −17.3205 −0.723575
\(574\) 12.0000 0.500870
\(575\) 3.46410 0.144463
\(576\) 1.00000 0.0416667
\(577\) −25.3205 −1.05411 −0.527053 0.849832i \(-0.676703\pi\)
−0.527053 + 0.849832i \(0.676703\pi\)
\(578\) 1.00000 0.0415945
\(579\) −17.8564 −0.742087
\(580\) −5.46410 −0.226884
\(581\) −7.60770 −0.315620
\(582\) −12.3923 −0.513678
\(583\) −13.9282 −0.576847
\(584\) 6.92820 0.286691
\(585\) 0 0
\(586\) −19.2487 −0.795157
\(587\) −4.92820 −0.203409 −0.101704 0.994815i \(-0.532430\pi\)
−0.101704 + 0.994815i \(0.532430\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −20.2487 −0.834334
\(590\) −4.53590 −0.186740
\(591\) 9.39230 0.386348
\(592\) 7.92820 0.325847
\(593\) −3.21539 −0.132040 −0.0660201 0.997818i \(-0.521030\pi\)
−0.0660201 + 0.997818i \(0.521030\pi\)
\(594\) 3.73205 0.153128
\(595\) 12.0000 0.491952
\(596\) −22.7846 −0.933294
\(597\) −11.0718 −0.453138
\(598\) 0 0
\(599\) 15.0718 0.615817 0.307908 0.951416i \(-0.400371\pi\)
0.307908 + 0.951416i \(0.400371\pi\)
\(600\) 1.00000 0.0408248
\(601\) −24.7128 −1.00806 −0.504028 0.863687i \(-0.668149\pi\)
−0.504028 + 0.863687i \(0.668149\pi\)
\(602\) −18.0000 −0.733625
\(603\) 5.46410 0.222515
\(604\) 18.7846 0.764335
\(605\) 2.92820 0.119048
\(606\) 8.39230 0.340914
\(607\) 37.4449 1.51984 0.759920 0.650017i \(-0.225238\pi\)
0.759920 + 0.650017i \(0.225238\pi\)
\(608\) −2.26795 −0.0919775
\(609\) −16.3923 −0.664250
\(610\) 7.46410 0.302213
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) −45.0000 −1.81753 −0.908766 0.417305i \(-0.862975\pi\)
−0.908766 + 0.417305i \(0.862975\pi\)
\(614\) −20.2487 −0.817171
\(615\) −4.00000 −0.161296
\(616\) 11.1962 0.451106
\(617\) −28.5359 −1.14881 −0.574406 0.818571i \(-0.694767\pi\)
−0.574406 + 0.818571i \(0.694767\pi\)
\(618\) −19.5885 −0.787963
\(619\) −42.5167 −1.70889 −0.854444 0.519543i \(-0.826102\pi\)
−0.854444 + 0.519543i \(0.826102\pi\)
\(620\) −8.92820 −0.358565
\(621\) −3.46410 −0.139010
\(622\) 5.07180 0.203361
\(623\) −30.3731 −1.21687
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −1.32051 −0.0527781
\(627\) −8.46410 −0.338024
\(628\) 10.8038 0.431120
\(629\) −31.7128 −1.26447
\(630\) 3.00000 0.119523
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −16.9282 −0.673368
\(633\) 21.9282 0.871568
\(634\) 23.5359 0.934730
\(635\) −4.66025 −0.184937
\(636\) 3.73205 0.147985
\(637\) 0 0
\(638\) 20.3923 0.807339
\(639\) 0.928203 0.0367192
\(640\) −1.00000 −0.0395285
\(641\) −7.53590 −0.297650 −0.148825 0.988864i \(-0.547549\pi\)
−0.148825 + 0.988864i \(0.547549\pi\)
\(642\) 0.928203 0.0366333
\(643\) −28.7846 −1.13515 −0.567577 0.823320i \(-0.692119\pi\)
−0.567577 + 0.823320i \(0.692119\pi\)
\(644\) −10.3923 −0.409514
\(645\) 6.00000 0.236250
\(646\) 9.07180 0.356925
\(647\) 32.2679 1.26858 0.634292 0.773094i \(-0.281292\pi\)
0.634292 + 0.773094i \(0.281292\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 16.9282 0.664490
\(650\) 0 0
\(651\) −26.7846 −1.04977
\(652\) −10.9282 −0.427981
\(653\) 29.5885 1.15789 0.578943 0.815368i \(-0.303465\pi\)
0.578943 + 0.815368i \(0.303465\pi\)
\(654\) 10.3923 0.406371
\(655\) −20.3205 −0.793988
\(656\) 4.00000 0.156174
\(657\) −6.92820 −0.270295
\(658\) −1.39230 −0.0542777
\(659\) 15.7128 0.612084 0.306042 0.952018i \(-0.400995\pi\)
0.306042 + 0.952018i \(0.400995\pi\)
\(660\) −3.73205 −0.145270
\(661\) −42.3923 −1.64887 −0.824435 0.565957i \(-0.808507\pi\)
−0.824435 + 0.565957i \(0.808507\pi\)
\(662\) 6.39230 0.248444
\(663\) 0 0
\(664\) −2.53590 −0.0984119
\(665\) −6.80385 −0.263842
\(666\) −7.92820 −0.307212
\(667\) −18.9282 −0.732903
\(668\) 6.46410 0.250104
\(669\) 8.85641 0.342408
\(670\) −5.46410 −0.211097
\(671\) −27.8564 −1.07538
\(672\) −3.00000 −0.115728
\(673\) −24.7846 −0.955376 −0.477688 0.878529i \(-0.658525\pi\)
−0.477688 + 0.878529i \(0.658525\pi\)
\(674\) −5.60770 −0.216000
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −2.92820 −0.112540 −0.0562700 0.998416i \(-0.517921\pi\)
−0.0562700 + 0.998416i \(0.517921\pi\)
\(678\) −12.0000 −0.460857
\(679\) 37.1769 1.42672
\(680\) 4.00000 0.153393
\(681\) −13.4641 −0.515945
\(682\) 33.3205 1.27591
\(683\) −6.67949 −0.255584 −0.127792 0.991801i \(-0.540789\pi\)
−0.127792 + 0.991801i \(0.540789\pi\)
\(684\) 2.26795 0.0867172
\(685\) −6.53590 −0.249724
\(686\) −15.0000 −0.572703
\(687\) 11.4641 0.437383
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) 3.46410 0.131876
\(691\) 23.0526 0.876961 0.438480 0.898741i \(-0.355517\pi\)
0.438480 + 0.898741i \(0.355517\pi\)
\(692\) −22.1244 −0.841042
\(693\) −11.1962 −0.425307
\(694\) 16.3923 0.622243
\(695\) 21.7846 0.826337
\(696\) −5.46410 −0.207116
\(697\) −16.0000 −0.606043
\(698\) 17.4641 0.661026
\(699\) 18.0000 0.680823
\(700\) −3.00000 −0.113389
\(701\) −16.3923 −0.619129 −0.309564 0.950878i \(-0.600183\pi\)
−0.309564 + 0.950878i \(0.600183\pi\)
\(702\) 0 0
\(703\) 17.9808 0.678157
\(704\) 3.73205 0.140657
\(705\) 0.464102 0.0174791
\(706\) 28.3923 1.06856
\(707\) −25.1769 −0.946875
\(708\) −4.53590 −0.170470
\(709\) −7.32051 −0.274927 −0.137464 0.990507i \(-0.543895\pi\)
−0.137464 + 0.990507i \(0.543895\pi\)
\(710\) −0.928203 −0.0348348
\(711\) 16.9282 0.634857
\(712\) −10.1244 −0.379426
\(713\) −30.9282 −1.15827
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −22.9282 −0.856867
\(717\) −3.46410 −0.129369
\(718\) 12.9282 0.482476
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 1.00000 0.0372678
\(721\) 58.7654 2.18854
\(722\) 13.8564 0.515682
\(723\) −14.8038 −0.550561
\(724\) −3.07180 −0.114162
\(725\) −5.46410 −0.202932
\(726\) 2.92820 0.108676
\(727\) −12.6603 −0.469543 −0.234771 0.972051i \(-0.575434\pi\)
−0.234771 + 0.972051i \(0.575434\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.92820 0.256424
\(731\) 24.0000 0.887672
\(732\) 7.46410 0.275881
\(733\) 6.85641 0.253247 0.126624 0.991951i \(-0.459586\pi\)
0.126624 + 0.991951i \(0.459586\pi\)
\(734\) 13.3205 0.491669
\(735\) −2.00000 −0.0737711
\(736\) −3.46410 −0.127688
\(737\) 20.3923 0.751160
\(738\) −4.00000 −0.147242
\(739\) −43.8372 −1.61258 −0.806288 0.591523i \(-0.798527\pi\)
−0.806288 + 0.591523i \(0.798527\pi\)
\(740\) 7.92820 0.291447
\(741\) 0 0
\(742\) −11.1962 −0.411024
\(743\) −2.92820 −0.107425 −0.0537127 0.998556i \(-0.517106\pi\)
−0.0537127 + 0.998556i \(0.517106\pi\)
\(744\) −8.92820 −0.327324
\(745\) −22.7846 −0.834764
\(746\) −22.9282 −0.839461
\(747\) 2.53590 0.0927837
\(748\) −14.9282 −0.545829
\(749\) −2.78461 −0.101747
\(750\) 1.00000 0.0365148
\(751\) −41.1769 −1.50257 −0.751283 0.659980i \(-0.770565\pi\)
−0.751283 + 0.659980i \(0.770565\pi\)
\(752\) −0.464102 −0.0169240
\(753\) 26.4641 0.964405
\(754\) 0 0
\(755\) 18.7846 0.683642
\(756\) 3.00000 0.109109
\(757\) 18.2679 0.663960 0.331980 0.943286i \(-0.392283\pi\)
0.331980 + 0.943286i \(0.392283\pi\)
\(758\) −6.26795 −0.227662
\(759\) −12.9282 −0.469264
\(760\) −2.26795 −0.0822672
\(761\) 43.9808 1.59430 0.797151 0.603780i \(-0.206340\pi\)
0.797151 + 0.603780i \(0.206340\pi\)
\(762\) −4.66025 −0.168823
\(763\) −31.1769 −1.12868
\(764\) 17.3205 0.626634
\(765\) −4.00000 −0.144620
\(766\) 36.7846 1.32908
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 39.7128 1.43208 0.716040 0.698059i \(-0.245953\pi\)
0.716040 + 0.698059i \(0.245953\pi\)
\(770\) 11.1962 0.403481
\(771\) 21.4641 0.773011
\(772\) 17.8564 0.642666
\(773\) −53.3923 −1.92039 −0.960194 0.279334i \(-0.909886\pi\)
−0.960194 + 0.279334i \(0.909886\pi\)
\(774\) 6.00000 0.215666
\(775\) −8.92820 −0.320711
\(776\) 12.3923 0.444858
\(777\) 23.7846 0.853268
\(778\) −9.85641 −0.353369
\(779\) 9.07180 0.325031
\(780\) 0 0
\(781\) 3.46410 0.123955
\(782\) 13.8564 0.495504
\(783\) 5.46410 0.195271
\(784\) 2.00000 0.0714286
\(785\) 10.8038 0.385606
\(786\) −20.3205 −0.724809
\(787\) 5.46410 0.194774 0.0973871 0.995247i \(-0.468952\pi\)
0.0973871 + 0.995247i \(0.468952\pi\)
\(788\) −9.39230 −0.334587
\(789\) −20.5167 −0.730412
\(790\) −16.9282 −0.602278
\(791\) 36.0000 1.28001
\(792\) −3.73205 −0.132613
\(793\) 0 0
\(794\) 7.92820 0.281361
\(795\) 3.73205 0.132362
\(796\) 11.0718 0.392429
\(797\) 37.8564 1.34094 0.670471 0.741935i \(-0.266092\pi\)
0.670471 + 0.741935i \(0.266092\pi\)
\(798\) −6.80385 −0.240854
\(799\) 1.85641 0.0656749
\(800\) −1.00000 −0.0353553
\(801\) 10.1244 0.357727
\(802\) −2.12436 −0.0750136
\(803\) −25.8564 −0.912453
\(804\) −5.46410 −0.192704
\(805\) −10.3923 −0.366281
\(806\) 0 0
\(807\) 30.9282 1.08872
\(808\) −8.39230 −0.295240
\(809\) 34.7846 1.22296 0.611481 0.791259i \(-0.290574\pi\)
0.611481 + 0.791259i \(0.290574\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 7.58846 0.266467 0.133233 0.991085i \(-0.457464\pi\)
0.133233 + 0.991085i \(0.457464\pi\)
\(812\) 16.3923 0.575257
\(813\) −3.60770 −0.126527
\(814\) −29.5885 −1.03707
\(815\) −10.9282 −0.382798
\(816\) 4.00000 0.140028
\(817\) −13.6077 −0.476073
\(818\) 0.947441 0.0331265
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 49.3205 1.72130 0.860649 0.509199i \(-0.170058\pi\)
0.860649 + 0.509199i \(0.170058\pi\)
\(822\) −6.53590 −0.227966
\(823\) 39.5885 1.37997 0.689983 0.723825i \(-0.257618\pi\)
0.689983 + 0.723825i \(0.257618\pi\)
\(824\) 19.5885 0.682396
\(825\) −3.73205 −0.129933
\(826\) 13.6077 0.473472
\(827\) −50.4974 −1.75597 −0.877984 0.478690i \(-0.841112\pi\)
−0.877984 + 0.478690i \(0.841112\pi\)
\(828\) 3.46410 0.120386
\(829\) 39.3205 1.36566 0.682829 0.730578i \(-0.260749\pi\)
0.682829 + 0.730578i \(0.260749\pi\)
\(830\) −2.53590 −0.0880223
\(831\) −19.5885 −0.679516
\(832\) 0 0
\(833\) −8.00000 −0.277184
\(834\) 21.7846 0.754339
\(835\) 6.46410 0.223699
\(836\) 8.46410 0.292737
\(837\) 8.92820 0.308604
\(838\) 17.8564 0.616839
\(839\) 43.1769 1.49063 0.745316 0.666711i \(-0.232298\pi\)
0.745316 + 0.666711i \(0.232298\pi\)
\(840\) −3.00000 −0.103510
\(841\) 0.856406 0.0295313
\(842\) −5.85641 −0.201825
\(843\) 6.92820 0.238620
\(844\) −21.9282 −0.754800
\(845\) 0 0
\(846\) 0.464102 0.0159561
\(847\) −8.78461 −0.301843
\(848\) −3.73205 −0.128159
\(849\) 14.3923 0.493943
\(850\) 4.00000 0.137199
\(851\) 27.4641 0.941457
\(852\) −0.928203 −0.0317997
\(853\) 44.6410 1.52848 0.764240 0.644932i \(-0.223114\pi\)
0.764240 + 0.644932i \(0.223114\pi\)
\(854\) −22.3923 −0.766249
\(855\) 2.26795 0.0775622
\(856\) −0.928203 −0.0317253
\(857\) 15.0718 0.514843 0.257421 0.966299i \(-0.417127\pi\)
0.257421 + 0.966299i \(0.417127\pi\)
\(858\) 0 0
\(859\) −19.9282 −0.679942 −0.339971 0.940436i \(-0.610417\pi\)
−0.339971 + 0.940436i \(0.610417\pi\)
\(860\) −6.00000 −0.204598
\(861\) 12.0000 0.408959
\(862\) 13.8564 0.471951
\(863\) 10.9282 0.372000 0.186000 0.982550i \(-0.440447\pi\)
0.186000 + 0.982550i \(0.440447\pi\)
\(864\) 1.00000 0.0340207
\(865\) −22.1244 −0.752251
\(866\) 32.7846 1.11407
\(867\) 1.00000 0.0339618
\(868\) 26.7846 0.909129
\(869\) 63.1769 2.14313
\(870\) −5.46410 −0.185250
\(871\) 0 0
\(872\) −10.3923 −0.351928
\(873\) −12.3923 −0.419416
\(874\) −7.85641 −0.265747
\(875\) −3.00000 −0.101419
\(876\) 6.92820 0.234082
\(877\) −33.7128 −1.13840 −0.569200 0.822199i \(-0.692747\pi\)
−0.569200 + 0.822199i \(0.692747\pi\)
\(878\) 21.3205 0.719532
\(879\) −19.2487 −0.649243
\(880\) 3.73205 0.125807
\(881\) 37.1051 1.25010 0.625052 0.780583i \(-0.285078\pi\)
0.625052 + 0.780583i \(0.285078\pi\)
\(882\) −2.00000 −0.0673435
\(883\) −21.3205 −0.717492 −0.358746 0.933435i \(-0.616796\pi\)
−0.358746 + 0.933435i \(0.616796\pi\)
\(884\) 0 0
\(885\) −4.53590 −0.152473
\(886\) −7.85641 −0.263941
\(887\) 52.2679 1.75499 0.877493 0.479589i \(-0.159214\pi\)
0.877493 + 0.479589i \(0.159214\pi\)
\(888\) 7.92820 0.266053
\(889\) 13.9808 0.468900
\(890\) −10.1244 −0.339369
\(891\) 3.73205 0.125028
\(892\) −8.85641 −0.296534
\(893\) −1.05256 −0.0352225
\(894\) −22.7846 −0.762031
\(895\) −22.9282 −0.766405
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 18.1244 0.604818
\(899\) 48.7846 1.62706
\(900\) 1.00000 0.0333333
\(901\) 14.9282 0.497331
\(902\) −14.9282 −0.497055
\(903\) −18.0000 −0.599002
\(904\) 12.0000 0.399114
\(905\) −3.07180 −0.102110
\(906\) 18.7846 0.624077
\(907\) −7.07180 −0.234815 −0.117408 0.993084i \(-0.537458\pi\)
−0.117408 + 0.993084i \(0.537458\pi\)
\(908\) 13.4641 0.446822
\(909\) 8.39230 0.278355
\(910\) 0 0
\(911\) −28.7846 −0.953677 −0.476838 0.878991i \(-0.658217\pi\)
−0.476838 + 0.878991i \(0.658217\pi\)
\(912\) −2.26795 −0.0750993
\(913\) 9.46410 0.313216
\(914\) −0.535898 −0.0177259
\(915\) 7.46410 0.246756
\(916\) −11.4641 −0.378785
\(917\) 60.9615 2.01313
\(918\) −4.00000 −0.132020
\(919\) 55.9615 1.84600 0.923000 0.384800i \(-0.125729\pi\)
0.923000 + 0.384800i \(0.125729\pi\)
\(920\) −3.46410 −0.114208
\(921\) −20.2487 −0.667218
\(922\) 32.3923 1.06678
\(923\) 0 0
\(924\) 11.1962 0.368326
\(925\) 7.92820 0.260678
\(926\) −0.784610 −0.0257839
\(927\) −19.5885 −0.643369
\(928\) 5.46410 0.179368
\(929\) 30.1436 0.988979 0.494490 0.869184i \(-0.335355\pi\)
0.494490 + 0.869184i \(0.335355\pi\)
\(930\) −8.92820 −0.292767
\(931\) 4.53590 0.148658
\(932\) −18.0000 −0.589610
\(933\) 5.07180 0.166043
\(934\) 3.60770 0.118047
\(935\) −14.9282 −0.488204
\(936\) 0 0
\(937\) 33.7128 1.10135 0.550675 0.834720i \(-0.314370\pi\)
0.550675 + 0.834720i \(0.314370\pi\)
\(938\) 16.3923 0.535228
\(939\) −1.32051 −0.0430932
\(940\) −0.464102 −0.0151373
\(941\) −36.4974 −1.18978 −0.594891 0.803806i \(-0.702805\pi\)
−0.594891 + 0.803806i \(0.702805\pi\)
\(942\) 10.8038 0.352008
\(943\) 13.8564 0.451227
\(944\) 4.53590 0.147631
\(945\) 3.00000 0.0975900
\(946\) 22.3923 0.728037
\(947\) 13.6077 0.442191 0.221095 0.975252i \(-0.429037\pi\)
0.221095 + 0.975252i \(0.429037\pi\)
\(948\) −16.9282 −0.549802
\(949\) 0 0
\(950\) −2.26795 −0.0735820
\(951\) 23.5359 0.763204
\(952\) −12.0000 −0.388922
\(953\) 18.6795 0.605088 0.302544 0.953135i \(-0.402164\pi\)
0.302544 + 0.953135i \(0.402164\pi\)
\(954\) 3.73205 0.120830
\(955\) 17.3205 0.560478
\(956\) 3.46410 0.112037
\(957\) 20.3923 0.659190
\(958\) −26.2487 −0.848057
\(959\) 19.6077 0.633165
\(960\) −1.00000 −0.0322749
\(961\) 48.7128 1.57138
\(962\) 0 0
\(963\) 0.928203 0.0299109
\(964\) 14.8038 0.476800
\(965\) 17.8564 0.574818
\(966\) −10.3923 −0.334367
\(967\) 28.8564 0.927959 0.463980 0.885846i \(-0.346421\pi\)
0.463980 + 0.885846i \(0.346421\pi\)
\(968\) −2.92820 −0.0941160
\(969\) 9.07180 0.291428
\(970\) 12.3923 0.397893
\(971\) −20.6077 −0.661332 −0.330666 0.943748i \(-0.607273\pi\)
−0.330666 + 0.943748i \(0.607273\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −65.3538 −2.09515
\(974\) −21.0000 −0.672883
\(975\) 0 0
\(976\) −7.46410 −0.238920
\(977\) −1.46410 −0.0468408 −0.0234204 0.999726i \(-0.507456\pi\)
−0.0234204 + 0.999726i \(0.507456\pi\)
\(978\) −10.9282 −0.349445
\(979\) 37.7846 1.20760
\(980\) 2.00000 0.0638877
\(981\) 10.3923 0.331801
\(982\) −15.3923 −0.491188
\(983\) −16.1769 −0.515963 −0.257982 0.966150i \(-0.583057\pi\)
−0.257982 + 0.966150i \(0.583057\pi\)
\(984\) 4.00000 0.127515
\(985\) −9.39230 −0.299264
\(986\) −21.8564 −0.696050
\(987\) −1.39230 −0.0443176
\(988\) 0 0
\(989\) −20.7846 −0.660912
\(990\) −3.73205 −0.118612
\(991\) −23.1769 −0.736239 −0.368119 0.929778i \(-0.619998\pi\)
−0.368119 + 0.929778i \(0.619998\pi\)
\(992\) 8.92820 0.283471
\(993\) 6.39230 0.202854
\(994\) 2.78461 0.0883225
\(995\) 11.0718 0.351000
\(996\) −2.53590 −0.0803530
\(997\) 10.0192 0.317312 0.158656 0.987334i \(-0.449284\pi\)
0.158656 + 0.987334i \(0.449284\pi\)
\(998\) −1.32051 −0.0418000
\(999\) −7.92820 −0.250837
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.ba.1.2 2
13.2 odd 12 390.2.bb.a.121.1 4
13.5 odd 4 5070.2.b.p.1351.4 4
13.7 odd 12 390.2.bb.a.361.1 yes 4
13.8 odd 4 5070.2.b.p.1351.1 4
13.12 even 2 5070.2.a.be.1.1 2
39.2 even 12 1170.2.bs.d.901.2 4
39.20 even 12 1170.2.bs.d.361.2 4
65.2 even 12 1950.2.y.e.199.1 4
65.7 even 12 1950.2.y.d.49.2 4
65.28 even 12 1950.2.y.d.199.2 4
65.33 even 12 1950.2.y.e.49.1 4
65.54 odd 12 1950.2.bc.a.901.2 4
65.59 odd 12 1950.2.bc.a.751.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.1 4 13.2 odd 12
390.2.bb.a.361.1 yes 4 13.7 odd 12
1170.2.bs.d.361.2 4 39.20 even 12
1170.2.bs.d.901.2 4 39.2 even 12
1950.2.y.d.49.2 4 65.7 even 12
1950.2.y.d.199.2 4 65.28 even 12
1950.2.y.e.49.1 4 65.33 even 12
1950.2.y.e.199.1 4 65.2 even 12
1950.2.bc.a.751.2 4 65.59 odd 12
1950.2.bc.a.901.2 4 65.54 odd 12
5070.2.a.ba.1.2 2 1.1 even 1 trivial
5070.2.a.be.1.1 2 13.12 even 2
5070.2.b.p.1351.1 4 13.8 odd 4
5070.2.b.p.1351.4 4 13.5 odd 4