Properties

Label 5070.2.a.be.1.1
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 1.1
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 \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.73205 −1.12526 −0.562628 0.826710i \(-0.690210\pi\)
−0.562628 + 0.826710i \(0.690210\pi\)
\(12\) −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 \(-0.583773\pi\)
−0.260152 + 0.965568i \(0.583773\pi\)
\(20\) −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 \(-0.203889\pi\)
0.801776 + 0.597624i \(0.203889\pi\)
\(32\) 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 \(-0.725941\pi\)
−0.651694 + 0.758482i \(0.725941\pi\)
\(38\) −2.26795 −0.367910
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −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 \(-0.489224\pi\)
0.0338481 + 0.999427i \(0.489224\pi\)
\(48\) −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 \(-0.595407\pi\)
−0.295262 + 0.955416i \(0.595407\pi\)
\(60\) 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 \(-0.608322\pi\)
−0.333773 + 0.942653i \(0.608322\pi\)
\(68\) −4.00000 −0.485071
\(69\) −3.46410 −0.417029
\(70\) −3.00000 −0.358569
\(71\) −0.928203 −0.110157 −0.0550787 0.998482i \(-0.517541\pi\)
−0.0550787 + 0.998482i \(0.517541\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) −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 \(-0.544445\pi\)
−0.139176 + 0.990268i \(0.544445\pi\)
\(84\) −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 \(-0.680288\pi\)
−0.536590 + 0.843843i \(0.680288\pi\)
\(90\) −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 \(-0.283414\pi\)
0.629124 + 0.777305i \(0.283414\pi\)
\(98\) 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 \(-0.665822\pi\)
−0.497701 + 0.867349i \(0.665822\pi\)
\(110\) 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 \(-0.409931\pi\)
0.279200 + 0.960233i \(0.409931\pi\)
\(138\) −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 \(-0.116921\pi\)
0.933294 + 0.359113i \(0.116921\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −18.7846 −1.52867 −0.764335 0.644819i \(-0.776933\pi\)
−0.764335 + 0.644819i \(0.776933\pi\)
\(152\) −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 \(-0.359225\pi\)
0.427981 + 0.903788i \(0.359225\pi\)
\(164\) −4.00000 −0.312348
\(165\) −3.73205 −0.290540
\(166\) −2.53590 −0.196824
\(167\) −6.46410 −0.500207 −0.250104 0.968219i \(-0.580465\pi\)
−0.250104 + 0.968219i \(0.580465\pi\)
\(168\) −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 \(-0.722172\pi\)
−0.642666 + 0.766146i \(0.722172\pi\)
\(194\) 12.3923 0.889716
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 9.39230 0.669174 0.334587 0.942365i \(-0.391403\pi\)
0.334587 + 0.942365i \(0.391403\pi\)
\(198\) −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 \(-0.404169\pi\)
0.296534 + 0.955022i \(0.404169\pi\)
\(224\) 3.00000 0.200446
\(225\) 1.00000 0.0666667
\(226\) −12.0000 −0.798228
\(227\) −13.4641 −0.893644 −0.446822 0.894623i \(-0.647444\pi\)
−0.446822 + 0.894623i \(0.647444\pi\)
\(228\) 2.26795 0.150199
\(229\) 11.4641 0.757569 0.378785 0.925485i \(-0.376342\pi\)
0.378785 + 0.925485i \(0.376342\pi\)
\(230\) −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 \(-0.535738\pi\)
−0.112037 + 0.993704i \(0.535738\pi\)
\(240\) 1.00000 0.0645497
\(241\) −14.8038 −0.953600 −0.476800 0.879012i \(-0.658203\pi\)
−0.476800 + 0.879012i \(0.658203\pi\)
\(242\) 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 \(-0.534949\pi\)
−0.109576 + 0.993978i \(0.534949\pi\)
\(272\) −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 \(-0.433744\pi\)
0.206651 + 0.978415i \(0.433744\pi\)
\(282\) −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 \(-0.690068\pi\)
−0.562261 + 0.826960i \(0.690068\pi\)
\(294\) −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 \(-0.696099\pi\)
−0.577827 + 0.816159i \(0.696099\pi\)
\(308\) −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 \(-0.270152\pi\)
0.660954 + 0.750427i \(0.270152\pi\)
\(318\) 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 \(-0.443789\pi\)
0.175676 + 0.984448i \(0.443789\pi\)
\(332\) −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 \(-0.345185\pi\)
0.467416 + 0.884038i \(0.345185\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) −3.73205 −0.198919
\(353\) 28.3923 1.51117 0.755585 0.655051i \(-0.227353\pi\)
0.755585 + 0.655051i \(0.227353\pi\)
\(354\) 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 \(-0.389179\pi\)
0.341162 + 0.940004i \(0.389179\pi\)
\(360\) −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 \(-0.551466\pi\)
−0.160981 + 0.986957i \(0.551466\pi\)
\(380\) 2.26795 0.116343
\(381\) 4.66025 0.238752
\(382\) 17.3205 0.886194
\(383\) 36.7846 1.87961 0.939803 0.341717i \(-0.111008\pi\)
0.939803 + 0.341717i \(0.111008\pi\)
\(384\) −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 \(-0.436246\pi\)
0.198953 + 0.980009i \(0.436246\pi\)
\(398\) 11.0718 0.554979
\(399\) 6.80385 0.340618
\(400\) 1.00000 0.0500000
\(401\) −2.12436 −0.106085 −0.0530426 0.998592i \(-0.516892\pi\)
−0.0530426 + 0.998592i \(0.516892\pi\)
\(402\) 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 \(-0.492543\pi\)
0.0234240 + 0.999726i \(0.492543\pi\)
\(410\) 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 \(-0.545582\pi\)
−0.142712 + 0.989764i \(0.545582\pi\)
\(422\) −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 \(-0.391696\pi\)
0.333720 + 0.942672i \(0.391696\pi\)
\(432\) −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 \(-0.359334\pi\)
0.427671 + 0.903935i \(0.359334\pi\)
\(450\) 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 \(-0.503990\pi\)
−0.0125341 + 0.999921i \(0.503990\pi\)
\(458\) 11.4641 0.535682
\(459\) 4.00000 0.186704
\(460\) −3.46410 −0.161515
\(461\) 32.3923 1.50866 0.754330 0.656495i \(-0.227962\pi\)
0.754330 + 0.656495i \(0.227962\pi\)
\(462\) 11.1962 0.520892
\(463\) −0.784610 −0.0364639 −0.0182320 0.999834i \(-0.505804\pi\)
−0.0182320 + 0.999834i \(0.505804\pi\)
\(464\) −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 \(-0.704700\pi\)
−0.599667 + 0.800250i \(0.704700\pi\)
\(480\) 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 \(-0.657842\pi\)
−0.475800 + 0.879553i \(0.657842\pi\)
\(488\) −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 \(-0.509410\pi\)
−0.0295570 + 0.999563i \(0.509410\pi\)
\(500\) −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 \(-0.537621\pi\)
−0.117914 + 0.993024i \(0.537621\pi\)
\(510\) −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 \(-0.304698\pi\)
0.575780 + 0.817605i \(0.304698\pi\)
\(542\) −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 \(-0.128759\pi\)
0.919295 + 0.393568i \(0.128759\pi\)
\(558\) 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 \(-0.323297\pi\)
0.527053 + 0.849832i \(0.323297\pi\)
\(578\) −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 \(-0.467570\pi\)
0.101704 + 0.994815i \(0.467570\pi\)
\(588\) −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 \(-0.478970\pi\)
0.0660201 + 0.997818i \(0.478970\pi\)
\(594\) 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 \(-0.137025\pi\)
0.908766 + 0.417305i \(0.137025\pi\)
\(614\) −20.2487 −0.817171
\(615\) −4.00000 −0.161296
\(616\) −11.1962 −0.451106
\(617\) 28.5359 1.14881 0.574406 0.818571i \(-0.305233\pi\)
0.574406 + 0.818571i \(0.305233\pi\)
\(618\) 19.5885 0.787963
\(619\) 42.5167 1.70889 0.854444 0.519543i \(-0.173898\pi\)
0.854444 + 0.519543i \(0.173898\pi\)
\(620\) −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 \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 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 \(-0.307881\pi\)
0.567577 + 0.823320i \(0.307881\pi\)
\(644\) 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 \(-0.191493\pi\)
0.824435 + 0.565957i \(0.191493\pi\)
\(662\) 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 \(-0.459211\pi\)
0.127792 + 0.991801i \(0.459211\pi\)
\(684\) −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 \(-0.644483\pi\)
−0.438480 + 0.898741i \(0.644483\pi\)
\(692\) −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 \(-0.456105\pi\)
0.137464 + 0.990507i \(0.456105\pi\)
\(710\) 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 \(-0.540414\pi\)
−0.126624 + 0.991951i \(0.540414\pi\)
\(734\) −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 \(-0.201473\pi\)
0.806288 + 0.591523i \(0.201473\pi\)
\(740\) 7.92820 0.291447
\(741\) 0 0
\(742\) −11.1962 −0.411024
\(743\) 2.92820 0.107425 0.0537127 0.998556i \(-0.482894\pi\)
0.0537127 + 0.998556i \(0.482894\pi\)
\(744\) −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 \(-0.793660\pi\)
−0.797151 + 0.603780i \(0.793660\pi\)
\(762\) 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 \(-0.754047\pi\)
−0.716040 + 0.698059i \(0.754047\pi\)
\(770\) 11.1962 0.403481
\(771\) 21.4641 0.773011
\(772\) −17.8564 −0.642666
\(773\) 53.3923 1.92039 0.960194 0.279334i \(-0.0901138\pi\)
0.960194 + 0.279334i \(0.0901138\pi\)
\(774\) −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 \(-0.531048\pi\)
−0.0973871 + 0.995247i \(0.531048\pi\)
\(788\) 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 \(-0.542536\pi\)
−0.133233 + 0.991085i \(0.542536\pi\)
\(812\) −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 \(-0.829942\pi\)
−0.860649 + 0.509199i \(0.829942\pi\)
\(822\) −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 \(-0.158888\pi\)
0.877984 + 0.478690i \(0.158888\pi\)
\(828\) 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 \(-0.767702\pi\)
−0.745316 + 0.666711i \(0.767702\pi\)
\(840\) 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 \(-0.776886\pi\)
−0.764240 + 0.644932i \(0.776886\pi\)
\(854\) −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 \(-0.559553\pi\)
−0.186000 + 0.982550i \(0.559553\pi\)
\(864\) −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 \(-0.307253\pi\)
0.569200 + 0.822199i \(0.307253\pi\)
\(878\) −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 \(-0.664645\pi\)
−0.494490 + 0.869184i \(0.664645\pi\)
\(930\) 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 \(-0.297195\pi\)
0.594891 + 0.803806i \(0.297195\pi\)
\(942\) −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 \(-0.570963\pi\)
−0.221095 + 0.975252i \(0.570963\pi\)
\(948\) −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 \(-0.653579\pi\)
−0.463980 + 0.885846i \(0.653579\pi\)
\(968\) 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 \(-0.492544\pi\)
0.0234204 + 0.999726i \(0.492544\pi\)
\(978\) −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 \(-0.416943\pi\)
0.257982 + 0.966150i \(0.416943\pi\)
\(984\) 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.be.1.1 2
13.5 odd 4 5070.2.b.p.1351.1 4
13.6 odd 12 390.2.bb.a.361.1 yes 4
13.8 odd 4 5070.2.b.p.1351.4 4
13.11 odd 12 390.2.bb.a.121.1 4
13.12 even 2 5070.2.a.ba.1.2 2
39.11 even 12 1170.2.bs.d.901.2 4
39.32 even 12 1170.2.bs.d.361.2 4
65.19 odd 12 1950.2.bc.a.751.2 4
65.24 odd 12 1950.2.bc.a.901.2 4
65.32 even 12 1950.2.y.d.49.2 4
65.37 even 12 1950.2.y.e.199.1 4
65.58 even 12 1950.2.y.e.49.1 4
65.63 even 12 1950.2.y.d.199.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.1 4 13.11 odd 12
390.2.bb.a.361.1 yes 4 13.6 odd 12
1170.2.bs.d.361.2 4 39.32 even 12
1170.2.bs.d.901.2 4 39.11 even 12
1950.2.y.d.49.2 4 65.32 even 12
1950.2.y.d.199.2 4 65.63 even 12
1950.2.y.e.49.1 4 65.58 even 12
1950.2.y.e.199.1 4 65.37 even 12
1950.2.bc.a.751.2 4 65.19 odd 12
1950.2.bc.a.901.2 4 65.24 odd 12
5070.2.a.ba.1.2 2 13.12 even 2
5070.2.a.be.1.1 2 1.1 even 1 trivial
5070.2.b.p.1351.1 4 13.5 odd 4
5070.2.b.p.1351.4 4 13.8 odd 4