Properties

Label 5070.2.b.p.1351.4
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.4
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.p.1351.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 3.00000i − 1.13389i −0.823754 0.566947i \(-0.808125\pi\)
0.823754 0.566947i \(-0.191875\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 3.73205i 1.12526i 0.826710 + 0.562628i \(0.190210\pi\)
−0.826710 + 0.562628i \(0.809790\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.00000 0.801784
\(15\) 1.00000i 0.258199i
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 2.26795i − 0.520303i −0.965568 0.260152i \(-0.916227\pi\)
0.965568 0.260152i \(-0.0837725\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 3.00000i 0.654654i
\(22\) −3.73205 −0.795676
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.00000i 0.566947i
\(29\) −5.46410 −1.01466 −0.507329 0.861752i \(-0.669367\pi\)
−0.507329 + 0.861752i \(0.669367\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.92820i 1.60355i 0.597624 + 0.801776i \(0.296111\pi\)
−0.597624 + 0.801776i \(0.703889\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 3.73205i − 0.649667i
\(34\) 4.00000i 0.685994i
\(35\) −3.00000 −0.507093
\(36\) −1.00000 −0.166667
\(37\) 7.92820i 1.30339i 0.758482 + 0.651694i \(0.225941\pi\)
−0.758482 + 0.651694i \(0.774059\pi\)
\(38\) 2.26795 0.367910
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) − 4.00000i − 0.624695i −0.949968 0.312348i \(-0.898885\pi\)
0.949968 0.312348i \(-0.101115\pi\)
\(42\) −3.00000 −0.462910
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) − 3.73205i − 0.562628i
\(45\) − 1.00000i − 0.149071i
\(46\) − 3.46410i − 0.510754i
\(47\) − 0.464102i − 0.0676962i −0.999427 0.0338481i \(-0.989224\pi\)
0.999427 0.0338481i \(-0.0107762\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.00000 −0.285714
\(50\) − 1.00000i − 0.141421i
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −3.73205 −0.512637 −0.256318 0.966592i \(-0.582510\pi\)
−0.256318 + 0.966592i \(0.582510\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 3.73205 0.503230
\(56\) −3.00000 −0.400892
\(57\) 2.26795i 0.300397i
\(58\) − 5.46410i − 0.717472i
\(59\) 4.53590i 0.590524i 0.955416 + 0.295262i \(0.0954069\pi\)
−0.955416 + 0.295262i \(0.904593\pi\)
\(60\) − 1.00000i − 0.129099i
\(61\) −7.46410 −0.955680 −0.477840 0.878447i \(-0.658580\pi\)
−0.477840 + 0.878447i \(0.658580\pi\)
\(62\) −8.92820 −1.13388
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.73205 0.459384
\(67\) − 5.46410i − 0.667546i −0.942653 0.333773i \(-0.891678\pi\)
0.942653 0.333773i \(-0.108322\pi\)
\(68\) −4.00000 −0.485071
\(69\) 3.46410 0.417029
\(70\) − 3.00000i − 0.358569i
\(71\) − 0.928203i − 0.110157i −0.998482 0.0550787i \(-0.982459\pi\)
0.998482 0.0550787i \(-0.0175410\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 6.92820i − 0.810885i −0.914121 0.405442i \(-0.867117\pi\)
0.914121 0.405442i \(-0.132883\pi\)
\(74\) −7.92820 −0.921635
\(75\) 1.00000 0.115470
\(76\) 2.26795i 0.260152i
\(77\) 11.1962 1.27592
\(78\) 0 0
\(79\) 16.9282 1.90457 0.952286 0.305208i \(-0.0987259\pi\)
0.952286 + 0.305208i \(0.0987259\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) − 2.53590i − 0.278351i −0.990268 0.139176i \(-0.955555\pi\)
0.990268 0.139176i \(-0.0444452\pi\)
\(84\) − 3.00000i − 0.327327i
\(85\) − 4.00000i − 0.433861i
\(86\) 6.00000i 0.646997i
\(87\) 5.46410 0.585813
\(88\) 3.73205 0.397838
\(89\) 10.1244i 1.07318i 0.843843 + 0.536590i \(0.180288\pi\)
−0.843843 + 0.536590i \(0.819712\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 3.46410 0.361158
\(93\) − 8.92820i − 0.925812i
\(94\) 0.464102 0.0478684
\(95\) −2.26795 −0.232687
\(96\) − 1.00000i − 0.102062i
\(97\) 12.3923i 1.25825i 0.777305 + 0.629124i \(0.216586\pi\)
−0.777305 + 0.629124i \(0.783414\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 3.73205i 0.375085i
\(100\) 1.00000 0.100000
\(101\) −8.39230 −0.835066 −0.417533 0.908662i \(-0.637105\pi\)
−0.417533 + 0.908662i \(0.637105\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) 19.5885 1.93011 0.965054 0.262051i \(-0.0843989\pi\)
0.965054 + 0.262051i \(0.0843989\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) − 3.73205i − 0.362489i
\(107\) 0.928203 0.0897328 0.0448664 0.998993i \(-0.485714\pi\)
0.0448664 + 0.998993i \(0.485714\pi\)
\(108\) 1.00000 0.0962250
\(109\) − 10.3923i − 0.995402i −0.867349 0.497701i \(-0.834178\pi\)
0.867349 0.497701i \(-0.165822\pi\)
\(110\) 3.73205i 0.355837i
\(111\) − 7.92820i − 0.752512i
\(112\) − 3.00000i − 0.283473i
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) −2.26795 −0.212413
\(115\) 3.46410i 0.323029i
\(116\) 5.46410 0.507329
\(117\) 0 0
\(118\) −4.53590 −0.417563
\(119\) − 12.0000i − 1.10004i
\(120\) 1.00000 0.0912871
\(121\) −2.92820 −0.266200
\(122\) − 7.46410i − 0.675768i
\(123\) 4.00000i 0.360668i
\(124\) − 8.92820i − 0.801776i
\(125\) 1.00000i 0.0894427i
\(126\) 3.00000 0.267261
\(127\) 4.66025 0.413531 0.206765 0.978391i \(-0.433706\pi\)
0.206765 + 0.978391i \(0.433706\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −20.3205 −1.77541 −0.887706 0.460412i \(-0.847702\pi\)
−0.887706 + 0.460412i \(0.847702\pi\)
\(132\) 3.73205i 0.324833i
\(133\) −6.80385 −0.589968
\(134\) 5.46410 0.472026
\(135\) 1.00000i 0.0860663i
\(136\) − 4.00000i − 0.342997i
\(137\) − 6.53590i − 0.558399i −0.960233 0.279200i \(-0.909931\pi\)
0.960233 0.279200i \(-0.0900692\pi\)
\(138\) 3.46410i 0.294884i
\(139\) 21.7846 1.84775 0.923873 0.382699i \(-0.125005\pi\)
0.923873 + 0.382699i \(0.125005\pi\)
\(140\) 3.00000 0.253546
\(141\) 0.464102i 0.0390844i
\(142\) 0.928203 0.0778931
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 5.46410i 0.453769i
\(146\) 6.92820 0.573382
\(147\) 2.00000 0.164957
\(148\) − 7.92820i − 0.651694i
\(149\) 22.7846i 1.86659i 0.359113 + 0.933294i \(0.383079\pi\)
−0.359113 + 0.933294i \(0.616921\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 18.7846i 1.52867i 0.644819 + 0.764335i \(0.276933\pi\)
−0.644819 + 0.764335i \(0.723067\pi\)
\(152\) −2.26795 −0.183955
\(153\) 4.00000 0.323381
\(154\) 11.1962i 0.902212i
\(155\) 8.92820 0.717131
\(156\) 0 0
\(157\) 10.8038 0.862241 0.431120 0.902294i \(-0.358118\pi\)
0.431120 + 0.902294i \(0.358118\pi\)
\(158\) 16.9282i 1.34674i
\(159\) 3.73205 0.295971
\(160\) 1.00000 0.0790569
\(161\) 10.3923i 0.819028i
\(162\) 1.00000i 0.0785674i
\(163\) − 10.9282i − 0.855963i −0.903788 0.427981i \(-0.859225\pi\)
0.903788 0.427981i \(-0.140775\pi\)
\(164\) 4.00000i 0.312348i
\(165\) −3.73205 −0.290540
\(166\) 2.53590 0.196824
\(167\) 6.46410i 0.500207i 0.968219 + 0.250104i \(0.0804647\pi\)
−0.968219 + 0.250104i \(0.919535\pi\)
\(168\) 3.00000 0.231455
\(169\) 0 0
\(170\) 4.00000 0.306786
\(171\) − 2.26795i − 0.173434i
\(172\) −6.00000 −0.457496
\(173\) 22.1244 1.68208 0.841042 0.540970i \(-0.181943\pi\)
0.841042 + 0.540970i \(0.181943\pi\)
\(174\) 5.46410i 0.414232i
\(175\) 3.00000i 0.226779i
\(176\) 3.73205i 0.281314i
\(177\) − 4.53590i − 0.340939i
\(178\) −10.1244 −0.758853
\(179\) 22.9282 1.71373 0.856867 0.515537i \(-0.172408\pi\)
0.856867 + 0.515537i \(0.172408\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 3.07180 0.228325 0.114162 0.993462i \(-0.463582\pi\)
0.114162 + 0.993462i \(0.463582\pi\)
\(182\) 0 0
\(183\) 7.46410 0.551762
\(184\) 3.46410i 0.255377i
\(185\) 7.92820 0.582893
\(186\) 8.92820 0.654648
\(187\) 14.9282i 1.09166i
\(188\) 0.464102i 0.0338481i
\(189\) 3.00000i 0.218218i
\(190\) − 2.26795i − 0.164534i
\(191\) 17.3205 1.25327 0.626634 0.779314i \(-0.284432\pi\)
0.626634 + 0.779314i \(0.284432\pi\)
\(192\) 1.00000 0.0721688
\(193\) 17.8564i 1.28533i 0.766146 + 0.642666i \(0.222172\pi\)
−0.766146 + 0.642666i \(0.777828\pi\)
\(194\) −12.3923 −0.889716
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 9.39230i 0.669174i 0.942365 + 0.334587i \(0.108597\pi\)
−0.942365 + 0.334587i \(0.891403\pi\)
\(198\) −3.73205 −0.265225
\(199\) −11.0718 −0.784859 −0.392429 0.919782i \(-0.628365\pi\)
−0.392429 + 0.919782i \(0.628365\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 5.46410i 0.385408i
\(202\) − 8.39230i − 0.590481i
\(203\) 16.3923i 1.15051i
\(204\) 4.00000 0.280056
\(205\) −4.00000 −0.279372
\(206\) 19.5885i 1.36479i
\(207\) −3.46410 −0.240772
\(208\) 0 0
\(209\) 8.46410 0.585474
\(210\) 3.00000i 0.207020i
\(211\) −21.9282 −1.50960 −0.754800 0.655955i \(-0.772266\pi\)
−0.754800 + 0.655955i \(0.772266\pi\)
\(212\) 3.73205 0.256318
\(213\) 0.928203i 0.0635994i
\(214\) 0.928203i 0.0634507i
\(215\) − 6.00000i − 0.409197i
\(216\) 1.00000i 0.0680414i
\(217\) 26.7846 1.81826
\(218\) 10.3923 0.703856
\(219\) 6.92820i 0.468165i
\(220\) −3.73205 −0.251615
\(221\) 0 0
\(222\) 7.92820 0.532106
\(223\) 8.85641i 0.593069i 0.955022 + 0.296534i \(0.0958309\pi\)
−0.955022 + 0.296534i \(0.904169\pi\)
\(224\) 3.00000 0.200446
\(225\) −1.00000 −0.0666667
\(226\) − 12.0000i − 0.798228i
\(227\) − 13.4641i − 0.893644i −0.894623 0.446822i \(-0.852556\pi\)
0.894623 0.446822i \(-0.147444\pi\)
\(228\) − 2.26795i − 0.150199i
\(229\) − 11.4641i − 0.757569i −0.925485 0.378785i \(-0.876342\pi\)
0.925485 0.378785i \(-0.123658\pi\)
\(230\) −3.46410 −0.228416
\(231\) −11.1962 −0.736653
\(232\) 5.46410i 0.358736i
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −0.464102 −0.0302747
\(236\) − 4.53590i − 0.295262i
\(237\) −16.9282 −1.09960
\(238\) 12.0000 0.777844
\(239\) − 3.46410i − 0.224074i −0.993704 0.112037i \(-0.964262\pi\)
0.993704 0.112037i \(-0.0357375\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 14.8038i 0.953600i 0.879012 + 0.476800i \(0.158203\pi\)
−0.879012 + 0.476800i \(0.841797\pi\)
\(242\) − 2.92820i − 0.188232i
\(243\) −1.00000 −0.0641500
\(244\) 7.46410 0.477840
\(245\) 2.00000i 0.127775i
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) 8.92820 0.566941
\(249\) 2.53590i 0.160706i
\(250\) −1.00000 −0.0632456
\(251\) 26.4641 1.67040 0.835200 0.549947i \(-0.185352\pi\)
0.835200 + 0.549947i \(0.185352\pi\)
\(252\) 3.00000i 0.188982i
\(253\) − 12.9282i − 0.812789i
\(254\) 4.66025i 0.292410i
\(255\) 4.00000i 0.250490i
\(256\) 1.00000 0.0625000
\(257\) 21.4641 1.33889 0.669447 0.742860i \(-0.266531\pi\)
0.669447 + 0.742860i \(0.266531\pi\)
\(258\) − 6.00000i − 0.373544i
\(259\) 23.7846 1.47790
\(260\) 0 0
\(261\) −5.46410 −0.338219
\(262\) − 20.3205i − 1.25541i
\(263\) 20.5167 1.26511 0.632556 0.774515i \(-0.282006\pi\)
0.632556 + 0.774515i \(0.282006\pi\)
\(264\) −3.73205 −0.229692
\(265\) 3.73205i 0.229258i
\(266\) − 6.80385i − 0.417171i
\(267\) − 10.1244i − 0.619601i
\(268\) 5.46410i 0.333773i
\(269\) −30.9282 −1.88573 −0.942863 0.333181i \(-0.891878\pi\)
−0.942863 + 0.333181i \(0.891878\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 3.60770i 0.219152i 0.993978 + 0.109576i \(0.0349493\pi\)
−0.993978 + 0.109576i \(0.965051\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 6.53590 0.394848
\(275\) − 3.73205i − 0.225051i
\(276\) −3.46410 −0.208514
\(277\) −19.5885 −1.17696 −0.588478 0.808513i \(-0.700273\pi\)
−0.588478 + 0.808513i \(0.700273\pi\)
\(278\) 21.7846i 1.30655i
\(279\) 8.92820i 0.534518i
\(280\) 3.00000i 0.179284i
\(281\) − 6.92820i − 0.413302i −0.978415 0.206651i \(-0.933744\pi\)
0.978415 0.206651i \(-0.0662565\pi\)
\(282\) −0.464102 −0.0276368
\(283\) 14.3923 0.855534 0.427767 0.903889i \(-0.359300\pi\)
0.427767 + 0.903889i \(0.359300\pi\)
\(284\) 0.928203i 0.0550787i
\(285\) 2.26795 0.134342
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 1.00000i 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) −5.46410 −0.320863
\(291\) − 12.3923i − 0.726450i
\(292\) 6.92820i 0.405442i
\(293\) 19.2487i 1.12452i 0.826960 + 0.562261i \(0.190068\pi\)
−0.826960 + 0.562261i \(0.809932\pi\)
\(294\) 2.00000i 0.116642i
\(295\) 4.53590 0.264090
\(296\) 7.92820 0.460817
\(297\) − 3.73205i − 0.216556i
\(298\) −22.7846 −1.31988
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 18.0000i − 1.03750i
\(302\) −18.7846 −1.08093
\(303\) 8.39230 0.482125
\(304\) − 2.26795i − 0.130076i
\(305\) 7.46410i 0.427393i
\(306\) 4.00000i 0.228665i
\(307\) 20.2487i 1.15565i 0.816159 + 0.577827i \(0.196099\pi\)
−0.816159 + 0.577827i \(0.803901\pi\)
\(308\) −11.1962 −0.637960
\(309\) −19.5885 −1.11435
\(310\) 8.92820i 0.507088i
\(311\) 5.07180 0.287595 0.143798 0.989607i \(-0.454069\pi\)
0.143798 + 0.989607i \(0.454069\pi\)
\(312\) 0 0
\(313\) 1.32051 0.0746395 0.0373198 0.999303i \(-0.488118\pi\)
0.0373198 + 0.999303i \(0.488118\pi\)
\(314\) 10.8038i 0.609696i
\(315\) −3.00000 −0.169031
\(316\) −16.9282 −0.952286
\(317\) 23.5359i 1.32191i 0.750427 + 0.660954i \(0.229848\pi\)
−0.750427 + 0.660954i \(0.770152\pi\)
\(318\) 3.73205i 0.209283i
\(319\) − 20.3923i − 1.14175i
\(320\) 1.00000i 0.0559017i
\(321\) −0.928203 −0.0518073
\(322\) −10.3923 −0.579141
\(323\) − 9.07180i − 0.504768i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 10.9282 0.605257
\(327\) 10.3923i 0.574696i
\(328\) −4.00000 −0.220863
\(329\) −1.39230 −0.0767603
\(330\) − 3.73205i − 0.205443i
\(331\) 6.39230i 0.351353i 0.984448 + 0.175676i \(0.0562112\pi\)
−0.984448 + 0.175676i \(0.943789\pi\)
\(332\) 2.53590i 0.139176i
\(333\) 7.92820i 0.434463i
\(334\) −6.46410 −0.353700
\(335\) −5.46410 −0.298536
\(336\) 3.00000i 0.163663i
\(337\) −5.60770 −0.305471 −0.152735 0.988267i \(-0.548808\pi\)
−0.152735 + 0.988267i \(0.548808\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 4.00000i 0.216930i
\(341\) −33.3205 −1.80441
\(342\) 2.26795 0.122637
\(343\) − 15.0000i − 0.809924i
\(344\) − 6.00000i − 0.323498i
\(345\) − 3.46410i − 0.186501i
\(346\) 22.1244i 1.18941i
\(347\) −16.3923 −0.879985 −0.439993 0.898001i \(-0.645019\pi\)
−0.439993 + 0.898001i \(0.645019\pi\)
\(348\) −5.46410 −0.292907
\(349\) − 17.4641i − 0.934832i −0.884038 0.467416i \(-0.845185\pi\)
0.884038 0.467416i \(-0.154815\pi\)
\(350\) −3.00000 −0.160357
\(351\) 0 0
\(352\) −3.73205 −0.198919
\(353\) 28.3923i 1.51117i 0.655051 + 0.755585i \(0.272647\pi\)
−0.655051 + 0.755585i \(0.727353\pi\)
\(354\) 4.53590 0.241080
\(355\) −0.928203 −0.0492639
\(356\) − 10.1244i − 0.536590i
\(357\) 12.0000i 0.635107i
\(358\) 22.9282i 1.21179i
\(359\) − 12.9282i − 0.682324i −0.940004 0.341162i \(-0.889179\pi\)
0.940004 0.341162i \(-0.110821\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 13.8564 0.729285
\(362\) 3.07180i 0.161450i
\(363\) 2.92820 0.153691
\(364\) 0 0
\(365\) −6.92820 −0.362639
\(366\) 7.46410i 0.390155i
\(367\) −13.3205 −0.695325 −0.347662 0.937620i \(-0.613024\pi\)
−0.347662 + 0.937620i \(0.613024\pi\)
\(368\) −3.46410 −0.180579
\(369\) − 4.00000i − 0.208232i
\(370\) 7.92820i 0.412168i
\(371\) 11.1962i 0.581275i
\(372\) 8.92820i 0.462906i
\(373\) 22.9282 1.18718 0.593589 0.804769i \(-0.297711\pi\)
0.593589 + 0.804769i \(0.297711\pi\)
\(374\) −14.9282 −0.771919
\(375\) − 1.00000i − 0.0516398i
\(376\) −0.464102 −0.0239342
\(377\) 0 0
\(378\) −3.00000 −0.154303
\(379\) − 6.26795i − 0.321963i −0.986957 0.160981i \(-0.948534\pi\)
0.986957 0.160981i \(-0.0514659\pi\)
\(380\) 2.26795 0.116343
\(381\) −4.66025 −0.238752
\(382\) 17.3205i 0.886194i
\(383\) 36.7846i 1.87961i 0.341717 + 0.939803i \(0.388992\pi\)
−0.341717 + 0.939803i \(0.611008\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 11.1962i − 0.570609i
\(386\) −17.8564 −0.908867
\(387\) 6.00000 0.304997
\(388\) − 12.3923i − 0.629124i
\(389\) −9.85641 −0.499740 −0.249870 0.968279i \(-0.580388\pi\)
−0.249870 + 0.968279i \(0.580388\pi\)
\(390\) 0 0
\(391\) −13.8564 −0.700749
\(392\) 2.00000i 0.101015i
\(393\) 20.3205 1.02503
\(394\) −9.39230 −0.473177
\(395\) − 16.9282i − 0.851750i
\(396\) − 3.73205i − 0.187543i
\(397\) − 7.92820i − 0.397905i −0.980009 0.198953i \(-0.936246\pi\)
0.980009 0.198953i \(-0.0637540\pi\)
\(398\) − 11.0718i − 0.554979i
\(399\) 6.80385 0.340618
\(400\) −1.00000 −0.0500000
\(401\) 2.12436i 0.106085i 0.998592 + 0.0530426i \(0.0168919\pi\)
−0.998592 + 0.0530426i \(0.983108\pi\)
\(402\) −5.46410 −0.272525
\(403\) 0 0
\(404\) 8.39230 0.417533
\(405\) − 1.00000i − 0.0496904i
\(406\) −16.3923 −0.813536
\(407\) −29.5885 −1.46665
\(408\) 4.00000i 0.198030i
\(409\) 0.947441i 0.0468479i 0.999726 + 0.0234240i \(0.00745676\pi\)
−0.999726 + 0.0234240i \(0.992543\pi\)
\(410\) − 4.00000i − 0.197546i
\(411\) 6.53590i 0.322392i
\(412\) −19.5885 −0.965054
\(413\) 13.6077 0.669591
\(414\) − 3.46410i − 0.170251i
\(415\) −2.53590 −0.124482
\(416\) 0 0
\(417\) −21.7846 −1.06680
\(418\) 8.46410i 0.413993i
\(419\) −17.8564 −0.872343 −0.436171 0.899864i \(-0.643666\pi\)
−0.436171 + 0.899864i \(0.643666\pi\)
\(420\) −3.00000 −0.146385
\(421\) − 5.85641i − 0.285424i −0.989764 0.142712i \(-0.954418\pi\)
0.989764 0.142712i \(-0.0455822\pi\)
\(422\) − 21.9282i − 1.06745i
\(423\) − 0.464102i − 0.0225654i
\(424\) 3.73205i 0.181244i
\(425\) −4.00000 −0.194029
\(426\) −0.928203 −0.0449716
\(427\) 22.3923i 1.08364i
\(428\) −0.928203 −0.0448664
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) 13.8564i 0.667440i 0.942672 + 0.333720i \(0.108304\pi\)
−0.942672 + 0.333720i \(0.891696\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 32.7846 1.57553 0.787764 0.615977i \(-0.211239\pi\)
0.787764 + 0.615977i \(0.211239\pi\)
\(434\) 26.7846i 1.28570i
\(435\) − 5.46410i − 0.261984i
\(436\) 10.3923i 0.497701i
\(437\) 7.85641i 0.375823i
\(438\) −6.92820 −0.331042
\(439\) 21.3205 1.01757 0.508786 0.860893i \(-0.330094\pi\)
0.508786 + 0.860893i \(0.330094\pi\)
\(440\) − 3.73205i − 0.177919i
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 7.85641 0.373269 0.186635 0.982429i \(-0.440242\pi\)
0.186635 + 0.982429i \(0.440242\pi\)
\(444\) 7.92820i 0.376256i
\(445\) 10.1244 0.479940
\(446\) −8.85641 −0.419363
\(447\) − 22.7846i − 1.07768i
\(448\) 3.00000i 0.141737i
\(449\) − 18.1244i − 0.855341i −0.903935 0.427671i \(-0.859334\pi\)
0.903935 0.427671i \(-0.140666\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) 14.9282 0.702942
\(452\) 12.0000 0.564433
\(453\) − 18.7846i − 0.882578i
\(454\) 13.4641 0.631902
\(455\) 0 0
\(456\) 2.26795 0.106206
\(457\) − 0.535898i − 0.0250683i −0.999921 0.0125341i \(-0.996010\pi\)
0.999921 0.0125341i \(-0.00398984\pi\)
\(458\) 11.4641 0.535682
\(459\) −4.00000 −0.186704
\(460\) − 3.46410i − 0.161515i
\(461\) 32.3923i 1.50866i 0.656495 + 0.754330i \(0.272038\pi\)
−0.656495 + 0.754330i \(0.727962\pi\)
\(462\) − 11.1962i − 0.520892i
\(463\) 0.784610i 0.0364639i 0.999834 + 0.0182320i \(0.00580373\pi\)
−0.999834 + 0.0182320i \(0.994196\pi\)
\(464\) −5.46410 −0.253665
\(465\) −8.92820 −0.414036
\(466\) 18.0000i 0.833834i
\(467\) 3.60770 0.166944 0.0834721 0.996510i \(-0.473399\pi\)
0.0834721 + 0.996510i \(0.473399\pi\)
\(468\) 0 0
\(469\) −16.3923 −0.756926
\(470\) − 0.464102i − 0.0214074i
\(471\) −10.8038 −0.497815
\(472\) 4.53590 0.208782
\(473\) 22.3923i 1.02960i
\(474\) − 16.9282i − 0.777538i
\(475\) 2.26795i 0.104061i
\(476\) 12.0000i 0.550019i
\(477\) −3.73205 −0.170879
\(478\) 3.46410 0.158444
\(479\) 26.2487i 1.19933i 0.800250 + 0.599667i \(0.204700\pi\)
−0.800250 + 0.599667i \(0.795300\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) −14.8038 −0.674297
\(483\) − 10.3923i − 0.472866i
\(484\) 2.92820 0.133100
\(485\) 12.3923 0.562706
\(486\) − 1.00000i − 0.0453609i
\(487\) − 21.0000i − 0.951601i −0.879553 0.475800i \(-0.842158\pi\)
0.879553 0.475800i \(-0.157842\pi\)
\(488\) 7.46410i 0.337884i
\(489\) 10.9282i 0.494190i
\(490\) −2.00000 −0.0903508
\(491\) −15.3923 −0.694645 −0.347322 0.937746i \(-0.612909\pi\)
−0.347322 + 0.937746i \(0.612909\pi\)
\(492\) − 4.00000i − 0.180334i
\(493\) −21.8564 −0.984363
\(494\) 0 0
\(495\) 3.73205 0.167743
\(496\) 8.92820i 0.400888i
\(497\) −2.78461 −0.124907
\(498\) −2.53590 −0.113636
\(499\) − 1.32051i − 0.0591141i −0.999563 0.0295570i \(-0.990590\pi\)
0.999563 0.0295570i \(-0.00940967\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) − 6.46410i − 0.288795i
\(502\) 26.4641i 1.18115i
\(503\) 0.267949 0.0119473 0.00597363 0.999982i \(-0.498099\pi\)
0.00597363 + 0.999982i \(0.498099\pi\)
\(504\) −3.00000 −0.133631
\(505\) 8.39230i 0.373453i
\(506\) 12.9282 0.574729
\(507\) 0 0
\(508\) −4.66025 −0.206765
\(509\) − 5.32051i − 0.235827i −0.993024 0.117914i \(-0.962379\pi\)
0.993024 0.117914i \(-0.0376206\pi\)
\(510\) −4.00000 −0.177123
\(511\) −20.7846 −0.919457
\(512\) 1.00000i 0.0441942i
\(513\) 2.26795i 0.100132i
\(514\) 21.4641i 0.946741i
\(515\) − 19.5885i − 0.863171i
\(516\) 6.00000 0.264135
\(517\) 1.73205 0.0761755
\(518\) 23.7846i 1.04504i
\(519\) −22.1244 −0.971151
\(520\) 0 0
\(521\) −27.3923 −1.20008 −0.600039 0.799970i \(-0.704848\pi\)
−0.600039 + 0.799970i \(0.704848\pi\)
\(522\) − 5.46410i − 0.239157i
\(523\) 13.7128 0.599619 0.299810 0.953999i \(-0.403077\pi\)
0.299810 + 0.953999i \(0.403077\pi\)
\(524\) 20.3205 0.887706
\(525\) − 3.00000i − 0.130931i
\(526\) 20.5167i 0.894569i
\(527\) 35.7128i 1.55567i
\(528\) − 3.73205i − 0.162417i
\(529\) −11.0000 −0.478261
\(530\) −3.73205 −0.162110
\(531\) 4.53590i 0.196841i
\(532\) 6.80385 0.294984
\(533\) 0 0
\(534\) 10.1244 0.438124
\(535\) − 0.928203i − 0.0401297i
\(536\) −5.46410 −0.236013
\(537\) −22.9282 −0.989425
\(538\) − 30.9282i − 1.33341i
\(539\) − 7.46410i − 0.321502i
\(540\) − 1.00000i − 0.0430331i
\(541\) − 26.7846i − 1.15156i −0.817605 0.575780i \(-0.804698\pi\)
0.817605 0.575780i \(-0.195302\pi\)
\(542\) −3.60770 −0.154964
\(543\) −3.07180 −0.131823
\(544\) 4.00000i 0.171499i
\(545\) −10.3923 −0.445157
\(546\) 0 0
\(547\) 29.3205 1.25365 0.626827 0.779158i \(-0.284353\pi\)
0.626827 + 0.779158i \(0.284353\pi\)
\(548\) 6.53590i 0.279200i
\(549\) −7.46410 −0.318560
\(550\) 3.73205 0.159135
\(551\) 12.3923i 0.527930i
\(552\) − 3.46410i − 0.147442i
\(553\) − 50.7846i − 2.15958i
\(554\) − 19.5885i − 0.832234i
\(555\) −7.92820 −0.336533
\(556\) −21.7846 −0.923873
\(557\) − 43.3923i − 1.83859i −0.393568 0.919295i \(-0.628759\pi\)
0.393568 0.919295i \(-0.371241\pi\)
\(558\) −8.92820 −0.377961
\(559\) 0 0
\(560\) −3.00000 −0.126773
\(561\) − 14.9282i − 0.630269i
\(562\) 6.92820 0.292249
\(563\) 19.3205 0.814262 0.407131 0.913370i \(-0.366529\pi\)
0.407131 + 0.913370i \(0.366529\pi\)
\(564\) − 0.464102i − 0.0195422i
\(565\) 12.0000i 0.504844i
\(566\) 14.3923i 0.604954i
\(567\) − 3.00000i − 0.125988i
\(568\) −0.928203 −0.0389465
\(569\) 18.3205 0.768036 0.384018 0.923326i \(-0.374540\pi\)
0.384018 + 0.923326i \(0.374540\pi\)
\(570\) 2.26795i 0.0949939i
\(571\) −16.8564 −0.705419 −0.352709 0.935733i \(-0.614740\pi\)
−0.352709 + 0.935733i \(0.614740\pi\)
\(572\) 0 0
\(573\) −17.3205 −0.723575
\(574\) − 12.0000i − 0.500870i
\(575\) 3.46410 0.144463
\(576\) −1.00000 −0.0416667
\(577\) 25.3205i 1.05411i 0.849832 + 0.527053i \(0.176703\pi\)
−0.849832 + 0.527053i \(0.823297\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) − 17.8564i − 0.742087i
\(580\) − 5.46410i − 0.226884i
\(581\) −7.60770 −0.315620
\(582\) 12.3923 0.513678
\(583\) − 13.9282i − 0.576847i
\(584\) −6.92820 −0.286691
\(585\) 0 0
\(586\) −19.2487 −0.795157
\(587\) 4.92820i 0.203409i 0.994815 + 0.101704i \(0.0324296\pi\)
−0.994815 + 0.101704i \(0.967570\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 20.2487 0.834334
\(590\) 4.53590i 0.186740i
\(591\) − 9.39230i − 0.386348i
\(592\) 7.92820i 0.325847i
\(593\) − 3.21539i − 0.132040i −0.997818 0.0660201i \(-0.978970\pi\)
0.997818 0.0660201i \(-0.0210302\pi\)
\(594\) 3.73205 0.153128
\(595\) −12.0000 −0.491952
\(596\) − 22.7846i − 0.933294i
\(597\) 11.0718 0.453138
\(598\) 0 0
\(599\) 15.0718 0.615817 0.307908 0.951416i \(-0.400371\pi\)
0.307908 + 0.951416i \(0.400371\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −24.7128 −1.00806 −0.504028 0.863687i \(-0.668149\pi\)
−0.504028 + 0.863687i \(0.668149\pi\)
\(602\) 18.0000 0.733625
\(603\) − 5.46410i − 0.222515i
\(604\) − 18.7846i − 0.764335i
\(605\) 2.92820i 0.119048i
\(606\) 8.39230i 0.340914i
\(607\) 37.4449 1.51984 0.759920 0.650017i \(-0.225238\pi\)
0.759920 + 0.650017i \(0.225238\pi\)
\(608\) 2.26795 0.0919775
\(609\) − 16.3923i − 0.664250i
\(610\) −7.46410 −0.302213
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 45.0000i 1.81753i 0.417305 + 0.908766i \(0.362975\pi\)
−0.417305 + 0.908766i \(0.637025\pi\)
\(614\) −20.2487 −0.817171
\(615\) 4.00000 0.161296
\(616\) − 11.1962i − 0.451106i
\(617\) 28.5359i 1.14881i 0.818571 + 0.574406i \(0.194767\pi\)
−0.818571 + 0.574406i \(0.805233\pi\)
\(618\) − 19.5885i − 0.787963i
\(619\) − 42.5167i − 1.70889i −0.519543 0.854444i \(-0.673898\pi\)
0.519543 0.854444i \(-0.326102\pi\)
\(620\) −8.92820 −0.358565
\(621\) 3.46410 0.139010
\(622\) 5.07180i 0.203361i
\(623\) 30.3731 1.21687
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 1.32051i 0.0527781i
\(627\) −8.46410 −0.338024
\(628\) −10.8038 −0.431120
\(629\) 31.7128i 1.26447i
\(630\) − 3.00000i − 0.119523i
\(631\) 8.00000i 0.318475i 0.987240 + 0.159237i \(0.0509036\pi\)
−0.987240 + 0.159237i \(0.949096\pi\)
\(632\) − 16.9282i − 0.673368i
\(633\) 21.9282 0.871568
\(634\) −23.5359 −0.934730
\(635\) − 4.66025i − 0.184937i
\(636\) −3.73205 −0.147985
\(637\) 0 0
\(638\) 20.3923 0.807339
\(639\) − 0.928203i − 0.0367192i
\(640\) −1.00000 −0.0395285
\(641\) 7.53590 0.297650 0.148825 0.988864i \(-0.452451\pi\)
0.148825 + 0.988864i \(0.452451\pi\)
\(642\) − 0.928203i − 0.0366333i
\(643\) 28.7846i 1.13515i 0.823320 + 0.567577i \(0.192119\pi\)
−0.823320 + 0.567577i \(0.807881\pi\)
\(644\) − 10.3923i − 0.409514i
\(645\) 6.00000i 0.236250i
\(646\) 9.07180 0.356925
\(647\) −32.2679 −1.26858 −0.634292 0.773094i \(-0.718708\pi\)
−0.634292 + 0.773094i \(0.718708\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −16.9282 −0.664490
\(650\) 0 0
\(651\) −26.7846 −1.04977
\(652\) 10.9282i 0.427981i
\(653\) 29.5885 1.15789 0.578943 0.815368i \(-0.303465\pi\)
0.578943 + 0.815368i \(0.303465\pi\)
\(654\) −10.3923 −0.406371
\(655\) 20.3205i 0.793988i
\(656\) − 4.00000i − 0.156174i
\(657\) − 6.92820i − 0.270295i
\(658\) − 1.39230i − 0.0542777i
\(659\) 15.7128 0.612084 0.306042 0.952018i \(-0.400995\pi\)
0.306042 + 0.952018i \(0.400995\pi\)
\(660\) 3.73205 0.145270
\(661\) − 42.3923i − 1.64887i −0.565957 0.824435i \(-0.691493\pi\)
0.565957 0.824435i \(-0.308507\pi\)
\(662\) −6.39230 −0.248444
\(663\) 0 0
\(664\) −2.53590 −0.0984119
\(665\) 6.80385i 0.263842i
\(666\) −7.92820 −0.307212
\(667\) 18.9282 0.732903
\(668\) − 6.46410i − 0.250104i
\(669\) − 8.85641i − 0.342408i
\(670\) − 5.46410i − 0.211097i
\(671\) − 27.8564i − 1.07538i
\(672\) −3.00000 −0.115728
\(673\) 24.7846 0.955376 0.477688 0.878529i \(-0.341475\pi\)
0.477688 + 0.878529i \(0.341475\pi\)
\(674\) − 5.60770i − 0.216000i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −2.92820 −0.112540 −0.0562700 0.998416i \(-0.517921\pi\)
−0.0562700 + 0.998416i \(0.517921\pi\)
\(678\) 12.0000i 0.460857i
\(679\) 37.1769 1.42672
\(680\) −4.00000 −0.153393
\(681\) 13.4641i 0.515945i
\(682\) − 33.3205i − 1.27591i
\(683\) − 6.67949i − 0.255584i −0.991801 0.127792i \(-0.959211\pi\)
0.991801 0.127792i \(-0.0407889\pi\)
\(684\) 2.26795i 0.0867172i
\(685\) −6.53590 −0.249724
\(686\) 15.0000 0.572703
\(687\) 11.4641i 0.437383i
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) 3.46410 0.131876
\(691\) − 23.0526i − 0.876961i −0.898741 0.438480i \(-0.855517\pi\)
0.898741 0.438480i \(-0.144483\pi\)
\(692\) −22.1244 −0.841042
\(693\) 11.1962 0.425307
\(694\) − 16.3923i − 0.622243i
\(695\) − 21.7846i − 0.826337i
\(696\) − 5.46410i − 0.207116i
\(697\) − 16.0000i − 0.606043i
\(698\) 17.4641 0.661026
\(699\) −18.0000 −0.680823
\(700\) − 3.00000i − 0.113389i
\(701\) 16.3923 0.619129 0.309564 0.950878i \(-0.399817\pi\)
0.309564 + 0.950878i \(0.399817\pi\)
\(702\) 0 0
\(703\) 17.9808 0.678157
\(704\) − 3.73205i − 0.140657i
\(705\) 0.464102 0.0174791
\(706\) −28.3923 −1.06856
\(707\) 25.1769i 0.946875i
\(708\) 4.53590i 0.170470i
\(709\) − 7.32051i − 0.274927i −0.990507 0.137464i \(-0.956105\pi\)
0.990507 0.137464i \(-0.0438951\pi\)
\(710\) − 0.928203i − 0.0348348i
\(711\) 16.9282 0.634857
\(712\) 10.1244 0.379426
\(713\) − 30.9282i − 1.15827i
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −22.9282 −0.856867
\(717\) 3.46410i 0.129369i
\(718\) 12.9282 0.482476
\(719\) 16.0000 0.596699 0.298350 0.954457i \(-0.403564\pi\)
0.298350 + 0.954457i \(0.403564\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 58.7654i − 2.18854i
\(722\) 13.8564i 0.515682i
\(723\) − 14.8038i − 0.550561i
\(724\) −3.07180 −0.114162
\(725\) 5.46410 0.202932
\(726\) 2.92820i 0.108676i
\(727\) 12.6603 0.469543 0.234771 0.972051i \(-0.424566\pi\)
0.234771 + 0.972051i \(0.424566\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 6.92820i − 0.256424i
\(731\) 24.0000 0.887672
\(732\) −7.46410 −0.275881
\(733\) − 6.85641i − 0.253247i −0.991951 0.126624i \(-0.959586\pi\)
0.991951 0.126624i \(-0.0404140\pi\)
\(734\) − 13.3205i − 0.491669i
\(735\) − 2.00000i − 0.0737711i
\(736\) − 3.46410i − 0.127688i
\(737\) 20.3923 0.751160
\(738\) 4.00000 0.147242
\(739\) − 43.8372i − 1.61258i −0.591523 0.806288i \(-0.701473\pi\)
0.591523 0.806288i \(-0.298527\pi\)
\(740\) −7.92820 −0.291447
\(741\) 0 0
\(742\) −11.1962 −0.411024
\(743\) 2.92820i 0.107425i 0.998556 + 0.0537127i \(0.0171055\pi\)
−0.998556 + 0.0537127i \(0.982894\pi\)
\(744\) −8.92820 −0.327324
\(745\) 22.7846 0.834764
\(746\) 22.9282i 0.839461i
\(747\) − 2.53590i − 0.0927837i
\(748\) − 14.9282i − 0.545829i
\(749\) − 2.78461i − 0.101747i
\(750\) 1.00000 0.0365148
\(751\) 41.1769 1.50257 0.751283 0.659980i \(-0.229435\pi\)
0.751283 + 0.659980i \(0.229435\pi\)
\(752\) − 0.464102i − 0.0169240i
\(753\) −26.4641 −0.964405
\(754\) 0 0
\(755\) 18.7846 0.683642
\(756\) − 3.00000i − 0.109109i
\(757\) 18.2679 0.663960 0.331980 0.943286i \(-0.392283\pi\)
0.331980 + 0.943286i \(0.392283\pi\)
\(758\) 6.26795 0.227662
\(759\) 12.9282i 0.469264i
\(760\) 2.26795i 0.0822672i
\(761\) 43.9808i 1.59430i 0.603780 + 0.797151i \(0.293660\pi\)
−0.603780 + 0.797151i \(0.706340\pi\)
\(762\) − 4.66025i − 0.168823i
\(763\) −31.1769 −1.12868
\(764\) −17.3205 −0.626634
\(765\) − 4.00000i − 0.144620i
\(766\) −36.7846 −1.32908
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) − 39.7128i − 1.43208i −0.698059 0.716040i \(-0.745953\pi\)
0.698059 0.716040i \(-0.254047\pi\)
\(770\) 11.1962 0.403481
\(771\) −21.4641 −0.773011
\(772\) − 17.8564i − 0.642666i
\(773\) 53.3923i 1.92039i 0.279334 + 0.960194i \(0.409886\pi\)
−0.279334 + 0.960194i \(0.590114\pi\)
\(774\) 6.00000i 0.215666i
\(775\) − 8.92820i − 0.320711i
\(776\) 12.3923 0.444858
\(777\) −23.7846 −0.853268
\(778\) − 9.85641i − 0.353369i
\(779\) −9.07180 −0.325031
\(780\) 0 0
\(781\) 3.46410 0.123955
\(782\) − 13.8564i − 0.495504i
\(783\) 5.46410 0.195271
\(784\) −2.00000 −0.0714286
\(785\) − 10.8038i − 0.385606i
\(786\) 20.3205i 0.724809i
\(787\) 5.46410i 0.194774i 0.995247 + 0.0973871i \(0.0310485\pi\)
−0.995247 + 0.0973871i \(0.968952\pi\)
\(788\) − 9.39230i − 0.334587i
\(789\) −20.5167 −0.730412
\(790\) 16.9282 0.602278
\(791\) 36.0000i 1.28001i
\(792\) 3.73205 0.132613
\(793\) 0 0
\(794\) 7.92820 0.281361
\(795\) − 3.73205i − 0.132362i
\(796\) 11.0718 0.392429
\(797\) −37.8564 −1.34094 −0.670471 0.741935i \(-0.733908\pi\)
−0.670471 + 0.741935i \(0.733908\pi\)
\(798\) 6.80385i 0.240854i
\(799\) − 1.85641i − 0.0656749i
\(800\) − 1.00000i − 0.0353553i
\(801\) 10.1244i 0.357727i
\(802\) −2.12436 −0.0750136
\(803\) 25.8564 0.912453
\(804\) − 5.46410i − 0.192704i
\(805\) 10.3923 0.366281
\(806\) 0 0
\(807\) 30.9282 1.08872
\(808\) 8.39230i 0.295240i
\(809\) 34.7846 1.22296 0.611481 0.791259i \(-0.290574\pi\)
0.611481 + 0.791259i \(0.290574\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 7.58846i − 0.266467i −0.991085 0.133233i \(-0.957464\pi\)
0.991085 0.133233i \(-0.0425360\pi\)
\(812\) − 16.3923i − 0.575257i
\(813\) − 3.60770i − 0.126527i
\(814\) − 29.5885i − 1.03707i
\(815\) −10.9282 −0.382798
\(816\) −4.00000 −0.140028
\(817\) − 13.6077i − 0.476073i
\(818\) −0.947441 −0.0331265
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) − 49.3205i − 1.72130i −0.509199 0.860649i \(-0.670058\pi\)
0.509199 0.860649i \(-0.329942\pi\)
\(822\) −6.53590 −0.227966
\(823\) −39.5885 −1.37997 −0.689983 0.723825i \(-0.742382\pi\)
−0.689983 + 0.723825i \(0.742382\pi\)
\(824\) − 19.5885i − 0.682396i
\(825\) 3.73205i 0.129933i
\(826\) 13.6077i 0.473472i
\(827\) − 50.4974i − 1.75597i −0.478690 0.877984i \(-0.658888\pi\)
0.478690 0.877984i \(-0.341112\pi\)
\(828\) 3.46410 0.120386
\(829\) −39.3205 −1.36566 −0.682829 0.730578i \(-0.739251\pi\)
−0.682829 + 0.730578i \(0.739251\pi\)
\(830\) − 2.53590i − 0.0880223i
\(831\) 19.5885 0.679516
\(832\) 0 0
\(833\) −8.00000 −0.277184
\(834\) − 21.7846i − 0.754339i
\(835\) 6.46410 0.223699
\(836\) −8.46410 −0.292737
\(837\) − 8.92820i − 0.308604i
\(838\) − 17.8564i − 0.616839i
\(839\) 43.1769i 1.49063i 0.666711 + 0.745316i \(0.267702\pi\)
−0.666711 + 0.745316i \(0.732298\pi\)
\(840\) − 3.00000i − 0.103510i
\(841\) 0.856406 0.0295313
\(842\) 5.85641 0.201825
\(843\) 6.92820i 0.238620i
\(844\) 21.9282 0.754800
\(845\) 0 0
\(846\) 0.464102 0.0159561
\(847\) 8.78461i 0.301843i
\(848\) −3.73205 −0.128159
\(849\) −14.3923 −0.493943
\(850\) − 4.00000i − 0.137199i
\(851\) − 27.4641i − 0.941457i
\(852\) − 0.928203i − 0.0317997i
\(853\) 44.6410i 1.52848i 0.644932 + 0.764240i \(0.276886\pi\)
−0.644932 + 0.764240i \(0.723114\pi\)
\(854\) −22.3923 −0.766249
\(855\) −2.26795 −0.0775622
\(856\) − 0.928203i − 0.0317253i
\(857\) −15.0718 −0.514843 −0.257421 0.966299i \(-0.582873\pi\)
−0.257421 + 0.966299i \(0.582873\pi\)
\(858\) 0 0
\(859\) −19.9282 −0.679942 −0.339971 0.940436i \(-0.610417\pi\)
−0.339971 + 0.940436i \(0.610417\pi\)
\(860\) 6.00000i 0.204598i
\(861\) 12.0000 0.408959
\(862\) −13.8564 −0.471951
\(863\) − 10.9282i − 0.372000i −0.982550 0.186000i \(-0.940447\pi\)
0.982550 0.186000i \(-0.0595525\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) − 22.1244i − 0.752251i
\(866\) 32.7846i 1.11407i
\(867\) 1.00000 0.0339618
\(868\) −26.7846 −0.909129
\(869\) 63.1769i 2.14313i
\(870\) 5.46410 0.185250
\(871\) 0 0
\(872\) −10.3923 −0.351928
\(873\) 12.3923i 0.419416i
\(874\) −7.85641 −0.265747
\(875\) 3.00000 0.101419
\(876\) − 6.92820i − 0.234082i
\(877\) 33.7128i 1.13840i 0.822199 + 0.569200i \(0.192747\pi\)
−0.822199 + 0.569200i \(0.807253\pi\)
\(878\) 21.3205i 0.719532i
\(879\) − 19.2487i − 0.649243i
\(880\) 3.73205 0.125807
\(881\) −37.1051 −1.25010 −0.625052 0.780583i \(-0.714922\pi\)
−0.625052 + 0.780583i \(0.714922\pi\)
\(882\) − 2.00000i − 0.0673435i
\(883\) 21.3205 0.717492 0.358746 0.933435i \(-0.383204\pi\)
0.358746 + 0.933435i \(0.383204\pi\)
\(884\) 0 0
\(885\) −4.53590 −0.152473
\(886\) 7.85641i 0.263941i
\(887\) 52.2679 1.75499 0.877493 0.479589i \(-0.159214\pi\)
0.877493 + 0.479589i \(0.159214\pi\)
\(888\) −7.92820 −0.266053
\(889\) − 13.9808i − 0.468900i
\(890\) 10.1244i 0.339369i
\(891\) 3.73205i 0.125028i
\(892\) − 8.85641i − 0.296534i
\(893\) −1.05256 −0.0352225
\(894\) 22.7846 0.762031
\(895\) − 22.9282i − 0.766405i
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 18.1244 0.604818
\(899\) − 48.7846i − 1.62706i
\(900\) 1.00000 0.0333333
\(901\) −14.9282 −0.497331
\(902\) 14.9282i 0.497055i
\(903\) 18.0000i 0.599002i
\(904\) 12.0000i 0.399114i
\(905\) − 3.07180i − 0.102110i
\(906\) 18.7846 0.624077
\(907\) 7.07180 0.234815 0.117408 0.993084i \(-0.462542\pi\)
0.117408 + 0.993084i \(0.462542\pi\)
\(908\) 13.4641i 0.446822i
\(909\) −8.39230 −0.278355
\(910\) 0 0
\(911\) −28.7846 −0.953677 −0.476838 0.878991i \(-0.658217\pi\)
−0.476838 + 0.878991i \(0.658217\pi\)
\(912\) 2.26795i 0.0750993i
\(913\) 9.46410 0.313216
\(914\) 0.535898 0.0177259
\(915\) − 7.46410i − 0.246756i
\(916\) 11.4641i 0.378785i
\(917\) 60.9615i 2.01313i
\(918\) − 4.00000i − 0.132020i
\(919\) 55.9615 1.84600 0.923000 0.384800i \(-0.125729\pi\)
0.923000 + 0.384800i \(0.125729\pi\)
\(920\) 3.46410 0.114208
\(921\) − 20.2487i − 0.667218i
\(922\) −32.3923 −1.06678
\(923\) 0 0
\(924\) 11.1962 0.368326
\(925\) − 7.92820i − 0.260678i
\(926\) −0.784610 −0.0257839
\(927\) 19.5885 0.643369
\(928\) − 5.46410i − 0.179368i
\(929\) − 30.1436i − 0.988979i −0.869184 0.494490i \(-0.835355\pi\)
0.869184 0.494490i \(-0.164645\pi\)
\(930\) − 8.92820i − 0.292767i
\(931\) 4.53590i 0.148658i
\(932\) −18.0000 −0.589610
\(933\) −5.07180 −0.166043
\(934\) 3.60770i 0.118047i
\(935\) 14.9282 0.488204
\(936\) 0 0
\(937\) 33.7128 1.10135 0.550675 0.834720i \(-0.314370\pi\)
0.550675 + 0.834720i \(0.314370\pi\)
\(938\) − 16.3923i − 0.535228i
\(939\) −1.32051 −0.0430932
\(940\) 0.464102 0.0151373
\(941\) 36.4974i 1.18978i 0.803806 + 0.594891i \(0.202805\pi\)
−0.803806 + 0.594891i \(0.797195\pi\)
\(942\) − 10.8038i − 0.352008i
\(943\) 13.8564i 0.451227i
\(944\) 4.53590i 0.147631i
\(945\) 3.00000 0.0975900
\(946\) −22.3923 −0.728037
\(947\) 13.6077i 0.442191i 0.975252 + 0.221095i \(0.0709632\pi\)
−0.975252 + 0.221095i \(0.929037\pi\)
\(948\) 16.9282 0.549802
\(949\) 0 0
\(950\) −2.26795 −0.0735820
\(951\) − 23.5359i − 0.763204i
\(952\) −12.0000 −0.388922
\(953\) −18.6795 −0.605088 −0.302544 0.953135i \(-0.597836\pi\)
−0.302544 + 0.953135i \(0.597836\pi\)
\(954\) − 3.73205i − 0.120830i
\(955\) − 17.3205i − 0.560478i
\(956\) 3.46410i 0.112037i
\(957\) 20.3923i 0.659190i
\(958\) −26.2487 −0.848057
\(959\) −19.6077 −0.633165
\(960\) − 1.00000i − 0.0322749i
\(961\) −48.7128 −1.57138
\(962\) 0 0
\(963\) 0.928203 0.0299109
\(964\) − 14.8038i − 0.476800i
\(965\) 17.8564 0.574818
\(966\) 10.3923 0.334367
\(967\) − 28.8564i − 0.927959i −0.885846 0.463980i \(-0.846421\pi\)
0.885846 0.463980i \(-0.153579\pi\)
\(968\) 2.92820i 0.0941160i
\(969\) 9.07180i 0.291428i
\(970\) 12.3923i 0.397893i
\(971\) −20.6077 −0.661332 −0.330666 0.943748i \(-0.607273\pi\)
−0.330666 + 0.943748i \(0.607273\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 65.3538i − 2.09515i
\(974\) 21.0000 0.672883
\(975\) 0 0
\(976\) −7.46410 −0.238920
\(977\) 1.46410i 0.0468408i 0.999726 + 0.0234204i \(0.00745562\pi\)
−0.999726 + 0.0234204i \(0.992544\pi\)
\(978\) −10.9282 −0.349445
\(979\) −37.7846 −1.20760
\(980\) − 2.00000i − 0.0638877i
\(981\) − 10.3923i − 0.331801i
\(982\) − 15.3923i − 0.491188i
\(983\) − 16.1769i − 0.515963i −0.966150 0.257982i \(-0.916943\pi\)
0.966150 0.257982i \(-0.0830574\pi\)
\(984\) 4.00000 0.127515
\(985\) 9.39230 0.299264
\(986\) − 21.8564i − 0.696050i
\(987\) 1.39230 0.0443176
\(988\) 0 0
\(989\) −20.7846 −0.660912
\(990\) 3.73205i 0.118612i
\(991\) −23.1769 −0.736239 −0.368119 0.929778i \(-0.619998\pi\)
−0.368119 + 0.929778i \(0.619998\pi\)
\(992\) −8.92820 −0.283471
\(993\) − 6.39230i − 0.202854i
\(994\) − 2.78461i − 0.0883225i
\(995\) 11.0718i 0.351000i
\(996\) − 2.53590i − 0.0803530i
\(997\) 10.0192 0.317312 0.158656 0.987334i \(-0.449284\pi\)
0.158656 + 0.987334i \(0.449284\pi\)
\(998\) 1.32051 0.0418000
\(999\) − 7.92820i − 0.250837i
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.b.p.1351.4 4
13.3 even 3 390.2.bb.a.121.1 4
13.4 even 6 390.2.bb.a.361.1 yes 4
13.5 odd 4 5070.2.a.be.1.1 2
13.8 odd 4 5070.2.a.ba.1.2 2
13.12 even 2 inner 5070.2.b.p.1351.1 4
39.17 odd 6 1170.2.bs.d.361.2 4
39.29 odd 6 1170.2.bs.d.901.2 4
65.3 odd 12 1950.2.y.d.199.2 4
65.4 even 6 1950.2.bc.a.751.2 4
65.17 odd 12 1950.2.y.d.49.2 4
65.29 even 6 1950.2.bc.a.901.2 4
65.42 odd 12 1950.2.y.e.199.1 4
65.43 odd 12 1950.2.y.e.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.bb.a.121.1 4 13.3 even 3
390.2.bb.a.361.1 yes 4 13.4 even 6
1170.2.bs.d.361.2 4 39.17 odd 6
1170.2.bs.d.901.2 4 39.29 odd 6
1950.2.y.d.49.2 4 65.17 odd 12
1950.2.y.d.199.2 4 65.3 odd 12
1950.2.y.e.49.1 4 65.43 odd 12
1950.2.y.e.199.1 4 65.42 odd 12
1950.2.bc.a.751.2 4 65.4 even 6
1950.2.bc.a.901.2 4 65.29 even 6
5070.2.a.ba.1.2 2 13.8 odd 4
5070.2.a.be.1.1 2 13.5 odd 4
5070.2.b.p.1351.1 4 13.12 even 2 inner
5070.2.b.p.1351.4 4 1.1 even 1 trivial