Properties

Label 507.2.b.e.337.3
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.e.337.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214i q^{2} +1.00000 q^{3} +1.82843 q^{4} -2.82843i q^{5} +0.414214i q^{6} -2.82843i q^{7} +1.58579i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214i q^{2} +1.00000 q^{3} +1.82843 q^{4} -2.82843i q^{5} +0.414214i q^{6} -2.82843i q^{7} +1.58579i q^{8} +1.00000 q^{9} +1.17157 q^{10} +2.00000i q^{11} +1.82843 q^{12} +1.17157 q^{14} -2.82843i q^{15} +3.00000 q^{16} -7.65685 q^{17} +0.414214i q^{18} -2.82843i q^{19} -5.17157i q^{20} -2.82843i q^{21} -0.828427 q^{22} +4.00000 q^{23} +1.58579i q^{24} -3.00000 q^{25} +1.00000 q^{27} -5.17157i q^{28} +2.00000 q^{29} +1.17157 q^{30} -1.17157i q^{31} +4.41421i q^{32} +2.00000i q^{33} -3.17157i q^{34} -8.00000 q^{35} +1.82843 q^{36} +7.65685i q^{37} +1.17157 q^{38} +4.48528 q^{40} +5.17157i q^{41} +1.17157 q^{42} +1.65685 q^{43} +3.65685i q^{44} -2.82843i q^{45} +1.65685i q^{46} +11.6569i q^{47} +3.00000 q^{48} -1.00000 q^{49} -1.24264i q^{50} -7.65685 q^{51} -2.00000 q^{53} +0.414214i q^{54} +5.65685 q^{55} +4.48528 q^{56} -2.82843i q^{57} +0.828427i q^{58} -7.65685i q^{59} -5.17157i q^{60} +13.3137 q^{61} +0.485281 q^{62} -2.82843i q^{63} +4.17157 q^{64} -0.828427 q^{66} +6.82843i q^{67} -14.0000 q^{68} +4.00000 q^{69} -3.31371i q^{70} +2.00000i q^{71} +1.58579i q^{72} -0.343146i q^{73} -3.17157 q^{74} -3.00000 q^{75} -5.17157i q^{76} +5.65685 q^{77} -11.3137 q^{79} -8.48528i q^{80} +1.00000 q^{81} -2.14214 q^{82} +3.65685i q^{83} -5.17157i q^{84} +21.6569i q^{85} +0.686292i q^{86} +2.00000 q^{87} -3.17157 q^{88} -14.8284i q^{89} +1.17157 q^{90} +7.31371 q^{92} -1.17157i q^{93} -4.82843 q^{94} -8.00000 q^{95} +4.41421i q^{96} +3.65685i q^{97} -0.414214i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{4} + 4q^{9} + 16q^{10} - 4q^{12} + 16q^{14} + 12q^{16} - 8q^{17} + 8q^{22} + 16q^{23} - 12q^{25} + 4q^{27} + 8q^{29} + 16q^{30} - 32q^{35} - 4q^{36} + 16q^{38} - 16q^{40} + 16q^{42} - 16q^{43} + 12q^{48} - 4q^{49} - 8q^{51} - 8q^{53} - 16q^{56} + 8q^{61} - 32q^{62} + 28q^{64} + 8q^{66} - 56q^{68} + 16q^{69} - 24q^{74} - 12q^{75} + 4q^{81} + 48q^{82} + 8q^{87} - 24q^{88} + 16q^{90} - 16q^{92} - 8q^{94} - 32q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214i 0.292893i 0.989219 + 0.146447i \(0.0467837\pi\)
−0.989219 + 0.146447i \(0.953216\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.82843 0.914214
\(5\) − 2.82843i − 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0.414214i 0.169102i
\(7\) − 2.82843i − 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 1.58579i 0.560660i
\(9\) 1.00000 0.333333
\(10\) 1.17157 0.370484
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 1.82843 0.527821
\(13\) 0 0
\(14\) 1.17157 0.313116
\(15\) − 2.82843i − 0.730297i
\(16\) 3.00000 0.750000
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0.414214i 0.0976311i
\(19\) − 2.82843i − 0.648886i −0.945905 0.324443i \(-0.894823\pi\)
0.945905 0.324443i \(-0.105177\pi\)
\(20\) − 5.17157i − 1.15640i
\(21\) − 2.82843i − 0.617213i
\(22\) −0.828427 −0.176621
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.58579i 0.323697i
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) − 5.17157i − 0.977335i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 1.17157 0.213899
\(31\) − 1.17157i − 0.210421i −0.994450 0.105210i \(-0.966448\pi\)
0.994450 0.105210i \(-0.0335516\pi\)
\(32\) 4.41421i 0.780330i
\(33\) 2.00000i 0.348155i
\(34\) − 3.17157i − 0.543920i
\(35\) −8.00000 −1.35225
\(36\) 1.82843 0.304738
\(37\) 7.65685i 1.25878i 0.777090 + 0.629390i \(0.216695\pi\)
−0.777090 + 0.629390i \(0.783305\pi\)
\(38\) 1.17157 0.190054
\(39\) 0 0
\(40\) 4.48528 0.709185
\(41\) 5.17157i 0.807664i 0.914833 + 0.403832i \(0.132322\pi\)
−0.914833 + 0.403832i \(0.867678\pi\)
\(42\) 1.17157 0.180778
\(43\) 1.65685 0.252668 0.126334 0.991988i \(-0.459679\pi\)
0.126334 + 0.991988i \(0.459679\pi\)
\(44\) 3.65685i 0.551292i
\(45\) − 2.82843i − 0.421637i
\(46\) 1.65685i 0.244290i
\(47\) 11.6569i 1.70033i 0.526519 + 0.850163i \(0.323497\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(48\) 3.00000 0.433013
\(49\) −1.00000 −0.142857
\(50\) − 1.24264i − 0.175736i
\(51\) −7.65685 −1.07217
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0.414214i 0.0563673i
\(55\) 5.65685 0.762770
\(56\) 4.48528 0.599371
\(57\) − 2.82843i − 0.374634i
\(58\) 0.828427i 0.108778i
\(59\) − 7.65685i − 0.996838i −0.866936 0.498419i \(-0.833914\pi\)
0.866936 0.498419i \(-0.166086\pi\)
\(60\) − 5.17157i − 0.667647i
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0.485281 0.0616308
\(63\) − 2.82843i − 0.356348i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) −0.828427 −0.101972
\(67\) 6.82843i 0.834225i 0.908855 + 0.417113i \(0.136958\pi\)
−0.908855 + 0.417113i \(0.863042\pi\)
\(68\) −14.0000 −1.69775
\(69\) 4.00000 0.481543
\(70\) − 3.31371i − 0.396064i
\(71\) 2.00000i 0.237356i 0.992933 + 0.118678i \(0.0378657\pi\)
−0.992933 + 0.118678i \(0.962134\pi\)
\(72\) 1.58579i 0.186887i
\(73\) − 0.343146i − 0.0401622i −0.999798 0.0200811i \(-0.993608\pi\)
0.999798 0.0200811i \(-0.00639244\pi\)
\(74\) −3.17157 −0.368688
\(75\) −3.00000 −0.346410
\(76\) − 5.17157i − 0.593220i
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) − 8.48528i − 0.948683i
\(81\) 1.00000 0.111111
\(82\) −2.14214 −0.236559
\(83\) 3.65685i 0.401392i 0.979654 + 0.200696i \(0.0643203\pi\)
−0.979654 + 0.200696i \(0.935680\pi\)
\(84\) − 5.17157i − 0.564265i
\(85\) 21.6569i 2.34902i
\(86\) 0.686292i 0.0740047i
\(87\) 2.00000 0.214423
\(88\) −3.17157 −0.338091
\(89\) − 14.8284i − 1.57181i −0.618347 0.785905i \(-0.712197\pi\)
0.618347 0.785905i \(-0.287803\pi\)
\(90\) 1.17157 0.123495
\(91\) 0 0
\(92\) 7.31371 0.762507
\(93\) − 1.17157i − 0.121486i
\(94\) −4.82843 −0.498014
\(95\) −8.00000 −0.820783
\(96\) 4.41421i 0.450524i
\(97\) 3.65685i 0.371297i 0.982616 + 0.185649i \(0.0594386\pi\)
−0.982616 + 0.185649i \(0.940561\pi\)
\(98\) − 0.414214i − 0.0418419i
\(99\) 2.00000i 0.201008i
\(100\) −5.48528 −0.548528
\(101\) −7.65685 −0.761885 −0.380943 0.924599i \(-0.624401\pi\)
−0.380943 + 0.924599i \(0.624401\pi\)
\(102\) − 3.17157i − 0.314033i
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) − 0.828427i − 0.0804640i
\(107\) −11.3137 −1.09374 −0.546869 0.837218i \(-0.684180\pi\)
−0.546869 + 0.837218i \(0.684180\pi\)
\(108\) 1.82843 0.175940
\(109\) 5.31371i 0.508961i 0.967078 + 0.254480i \(0.0819045\pi\)
−0.967078 + 0.254480i \(0.918096\pi\)
\(110\) 2.34315i 0.223410i
\(111\) 7.65685i 0.726756i
\(112\) − 8.48528i − 0.801784i
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) 1.17157 0.109728
\(115\) − 11.3137i − 1.05501i
\(116\) 3.65685 0.339530
\(117\) 0 0
\(118\) 3.17157 0.291967
\(119\) 21.6569i 1.98528i
\(120\) 4.48528 0.409448
\(121\) 7.00000 0.636364
\(122\) 5.51472i 0.499279i
\(123\) 5.17157i 0.466305i
\(124\) − 2.14214i − 0.192369i
\(125\) − 5.65685i − 0.505964i
\(126\) 1.17157 0.104372
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 10.5563i 0.933058i
\(129\) 1.65685 0.145878
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 3.65685i 0.318288i
\(133\) −8.00000 −0.693688
\(134\) −2.82843 −0.244339
\(135\) − 2.82843i − 0.243432i
\(136\) − 12.1421i − 1.04118i
\(137\) 10.8284i 0.925135i 0.886584 + 0.462567i \(0.153072\pi\)
−0.886584 + 0.462567i \(0.846928\pi\)
\(138\) 1.65685i 0.141041i
\(139\) −7.31371 −0.620341 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(140\) −14.6274 −1.23624
\(141\) 11.6569i 0.981684i
\(142\) −0.828427 −0.0695201
\(143\) 0 0
\(144\) 3.00000 0.250000
\(145\) − 5.65685i − 0.469776i
\(146\) 0.142136 0.0117632
\(147\) −1.00000 −0.0824786
\(148\) 14.0000i 1.15079i
\(149\) − 9.17157i − 0.751365i −0.926749 0.375682i \(-0.877408\pi\)
0.926749 0.375682i \(-0.122592\pi\)
\(150\) − 1.24264i − 0.101461i
\(151\) 3.51472i 0.286024i 0.989721 + 0.143012i \(0.0456787\pi\)
−0.989721 + 0.143012i \(0.954321\pi\)
\(152\) 4.48528 0.363804
\(153\) −7.65685 −0.619020
\(154\) 2.34315i 0.188816i
\(155\) −3.31371 −0.266163
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) − 4.68629i − 0.372821i
\(159\) −2.00000 −0.158610
\(160\) 12.4853 0.987048
\(161\) − 11.3137i − 0.891645i
\(162\) 0.414214i 0.0325437i
\(163\) − 18.8284i − 1.47476i −0.675480 0.737378i \(-0.736064\pi\)
0.675480 0.737378i \(-0.263936\pi\)
\(164\) 9.45584i 0.738377i
\(165\) 5.65685 0.440386
\(166\) −1.51472 −0.117565
\(167\) 3.65685i 0.282976i 0.989940 + 0.141488i \(0.0451886\pi\)
−0.989940 + 0.141488i \(0.954811\pi\)
\(168\) 4.48528 0.346047
\(169\) 0 0
\(170\) −8.97056 −0.688011
\(171\) − 2.82843i − 0.216295i
\(172\) 3.02944 0.230992
\(173\) 11.6569 0.886254 0.443127 0.896459i \(-0.353869\pi\)
0.443127 + 0.896459i \(0.353869\pi\)
\(174\) 0.828427i 0.0628029i
\(175\) 8.48528i 0.641427i
\(176\) 6.00000i 0.452267i
\(177\) − 7.65685i − 0.575524i
\(178\) 6.14214 0.460373
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) − 5.17157i − 0.385466i
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 13.3137 0.984178
\(184\) 6.34315i 0.467623i
\(185\) 21.6569 1.59224
\(186\) 0.485281 0.0355826
\(187\) − 15.3137i − 1.11985i
\(188\) 21.3137i 1.55446i
\(189\) − 2.82843i − 0.205738i
\(190\) − 3.31371i − 0.240402i
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 4.17157 0.301057
\(193\) − 5.31371i − 0.382489i −0.981542 0.191245i \(-0.938748\pi\)
0.981542 0.191245i \(-0.0612524\pi\)
\(194\) −1.51472 −0.108750
\(195\) 0 0
\(196\) −1.82843 −0.130602
\(197\) 0.485281i 0.0345749i 0.999851 + 0.0172874i \(0.00550304\pi\)
−0.999851 + 0.0172874i \(0.994497\pi\)
\(198\) −0.828427 −0.0588738
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) − 4.75736i − 0.336396i
\(201\) 6.82843i 0.481640i
\(202\) − 3.17157i − 0.223151i
\(203\) − 5.65685i − 0.397033i
\(204\) −14.0000 −0.980196
\(205\) 14.6274 1.02162
\(206\) − 0.970563i − 0.0676223i
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 5.65685 0.391293
\(210\) − 3.31371i − 0.228668i
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −3.65685 −0.251154
\(213\) 2.00000i 0.137038i
\(214\) − 4.68629i − 0.320348i
\(215\) − 4.68629i − 0.319602i
\(216\) 1.58579i 0.107899i
\(217\) −3.31371 −0.224949
\(218\) −2.20101 −0.149071
\(219\) − 0.343146i − 0.0231876i
\(220\) 10.3431 0.697335
\(221\) 0 0
\(222\) −3.17157 −0.212862
\(223\) − 12.4853i − 0.836076i −0.908429 0.418038i \(-0.862718\pi\)
0.908429 0.418038i \(-0.137282\pi\)
\(224\) 12.4853 0.834208
\(225\) −3.00000 −0.200000
\(226\) − 2.20101i − 0.146409i
\(227\) − 17.3137i − 1.14915i −0.818452 0.574576i \(-0.805167\pi\)
0.818452 0.574576i \(-0.194833\pi\)
\(228\) − 5.17157i − 0.342496i
\(229\) 1.31371i 0.0868123i 0.999058 + 0.0434062i \(0.0138209\pi\)
−0.999058 + 0.0434062i \(0.986179\pi\)
\(230\) 4.68629 0.309005
\(231\) 5.65685 0.372194
\(232\) 3.17157i 0.208224i
\(233\) −6.97056 −0.456657 −0.228328 0.973584i \(-0.573326\pi\)
−0.228328 + 0.973584i \(0.573326\pi\)
\(234\) 0 0
\(235\) 32.9706 2.15076
\(236\) − 14.0000i − 0.911322i
\(237\) −11.3137 −0.734904
\(238\) −8.97056 −0.581475
\(239\) 2.00000i 0.129369i 0.997906 + 0.0646846i \(0.0206041\pi\)
−0.997906 + 0.0646846i \(0.979396\pi\)
\(240\) − 8.48528i − 0.547723i
\(241\) − 0.343146i − 0.0221040i −0.999939 0.0110520i \(-0.996482\pi\)
0.999939 0.0110520i \(-0.00351803\pi\)
\(242\) 2.89949i 0.186387i
\(243\) 1.00000 0.0641500
\(244\) 24.3431 1.55841
\(245\) 2.82843i 0.180702i
\(246\) −2.14214 −0.136578
\(247\) 0 0
\(248\) 1.85786 0.117975
\(249\) 3.65685i 0.231744i
\(250\) 2.34315 0.148194
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) − 5.17157i − 0.325778i
\(253\) 8.00000i 0.502956i
\(254\) − 2.34315i − 0.147022i
\(255\) 21.6569i 1.35620i
\(256\) 3.97056 0.248160
\(257\) 4.34315 0.270918 0.135459 0.990783i \(-0.456749\pi\)
0.135459 + 0.990783i \(0.456749\pi\)
\(258\) 0.686292i 0.0427266i
\(259\) 21.6569 1.34569
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 3.31371i − 0.204722i
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −3.17157 −0.195197
\(265\) 5.65685i 0.347498i
\(266\) − 3.31371i − 0.203177i
\(267\) − 14.8284i − 0.907485i
\(268\) 12.4853i 0.762660i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 1.17157 0.0712997
\(271\) 27.7990i 1.68867i 0.535817 + 0.844334i \(0.320004\pi\)
−0.535817 + 0.844334i \(0.679996\pi\)
\(272\) −22.9706 −1.39279
\(273\) 0 0
\(274\) −4.48528 −0.270966
\(275\) − 6.00000i − 0.361814i
\(276\) 7.31371 0.440234
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) − 3.02944i − 0.181694i
\(279\) − 1.17157i − 0.0701402i
\(280\) − 12.6863i − 0.758151i
\(281\) − 21.1716i − 1.26299i −0.775380 0.631495i \(-0.782442\pi\)
0.775380 0.631495i \(-0.217558\pi\)
\(282\) −4.82843 −0.287529
\(283\) −28.9706 −1.72212 −0.861061 0.508502i \(-0.830199\pi\)
−0.861061 + 0.508502i \(0.830199\pi\)
\(284\) 3.65685i 0.216994i
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 14.6274 0.863429
\(288\) 4.41421i 0.260110i
\(289\) 41.6274 2.44867
\(290\) 2.34315 0.137594
\(291\) 3.65685i 0.214369i
\(292\) − 0.627417i − 0.0367168i
\(293\) − 2.14214i − 0.125145i −0.998040 0.0625724i \(-0.980070\pi\)
0.998040 0.0625724i \(-0.0199304\pi\)
\(294\) − 0.414214i − 0.0241574i
\(295\) −21.6569 −1.26091
\(296\) −12.1421 −0.705747
\(297\) 2.00000i 0.116052i
\(298\) 3.79899 0.220070
\(299\) 0 0
\(300\) −5.48528 −0.316693
\(301\) − 4.68629i − 0.270113i
\(302\) −1.45584 −0.0837744
\(303\) −7.65685 −0.439875
\(304\) − 8.48528i − 0.486664i
\(305\) − 37.6569i − 2.15623i
\(306\) − 3.17157i − 0.181307i
\(307\) 22.8284i 1.30289i 0.758697 + 0.651444i \(0.225836\pi\)
−0.758697 + 0.651444i \(0.774164\pi\)
\(308\) 10.3431 0.589355
\(309\) −2.34315 −0.133297
\(310\) − 1.37258i − 0.0779575i
\(311\) 10.6274 0.602626 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) − 4.14214i − 0.233754i
\(315\) −8.00000 −0.450749
\(316\) −20.6863 −1.16369
\(317\) 8.48528i 0.476581i 0.971194 + 0.238290i \(0.0765870\pi\)
−0.971194 + 0.238290i \(0.923413\pi\)
\(318\) − 0.828427i − 0.0464559i
\(319\) 4.00000i 0.223957i
\(320\) − 11.7990i − 0.659584i
\(321\) −11.3137 −0.631470
\(322\) 4.68629 0.261157
\(323\) 21.6569i 1.20502i
\(324\) 1.82843 0.101579
\(325\) 0 0
\(326\) 7.79899 0.431946
\(327\) 5.31371i 0.293849i
\(328\) −8.20101 −0.452825
\(329\) 32.9706 1.81773
\(330\) 2.34315i 0.128986i
\(331\) 26.1421i 1.43690i 0.695578 + 0.718451i \(0.255148\pi\)
−0.695578 + 0.718451i \(0.744852\pi\)
\(332\) 6.68629i 0.366958i
\(333\) 7.65685i 0.419593i
\(334\) −1.51472 −0.0828817
\(335\) 19.3137 1.05522
\(336\) − 8.48528i − 0.462910i
\(337\) −9.31371 −0.507350 −0.253675 0.967290i \(-0.581639\pi\)
−0.253675 + 0.967290i \(0.581639\pi\)
\(338\) 0 0
\(339\) −5.31371 −0.288601
\(340\) 39.5980i 2.14750i
\(341\) 2.34315 0.126888
\(342\) 1.17157 0.0633514
\(343\) − 16.9706i − 0.916324i
\(344\) 2.62742i 0.141661i
\(345\) − 11.3137i − 0.609110i
\(346\) 4.82843i 0.259578i
\(347\) −8.68629 −0.466305 −0.233152 0.972440i \(-0.574904\pi\)
−0.233152 + 0.972440i \(0.574904\pi\)
\(348\) 3.65685 0.196028
\(349\) − 3.65685i − 0.195747i −0.995199 0.0978735i \(-0.968796\pi\)
0.995199 0.0978735i \(-0.0312040\pi\)
\(350\) −3.51472 −0.187870
\(351\) 0 0
\(352\) −8.82843 −0.470557
\(353\) − 33.4558i − 1.78067i −0.455301 0.890337i \(-0.650468\pi\)
0.455301 0.890337i \(-0.349532\pi\)
\(354\) 3.17157 0.168567
\(355\) 5.65685 0.300235
\(356\) − 27.1127i − 1.43697i
\(357\) 21.6569i 1.14620i
\(358\) 9.65685i 0.510381i
\(359\) − 34.9706i − 1.84568i −0.385189 0.922838i \(-0.625864\pi\)
0.385189 0.922838i \(-0.374136\pi\)
\(360\) 4.48528 0.236395
\(361\) 11.0000 0.578947
\(362\) − 5.79899i − 0.304788i
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −0.970563 −0.0508016
\(366\) 5.51472i 0.288259i
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 12.0000 0.625543
\(369\) 5.17157i 0.269221i
\(370\) 8.97056i 0.466357i
\(371\) 5.65685i 0.293689i
\(372\) − 2.14214i − 0.111065i
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 6.34315 0.327996
\(375\) − 5.65685i − 0.292119i
\(376\) −18.4853 −0.953306
\(377\) 0 0
\(378\) 1.17157 0.0602592
\(379\) 0.485281i 0.0249272i 0.999922 + 0.0124636i \(0.00396739\pi\)
−0.999922 + 0.0124636i \(0.996033\pi\)
\(380\) −14.6274 −0.750371
\(381\) −5.65685 −0.289809
\(382\) 1.37258i 0.0702275i
\(383\) 30.9706i 1.58252i 0.611479 + 0.791261i \(0.290575\pi\)
−0.611479 + 0.791261i \(0.709425\pi\)
\(384\) 10.5563i 0.538701i
\(385\) − 16.0000i − 0.815436i
\(386\) 2.20101 0.112028
\(387\) 1.65685 0.0842226
\(388\) 6.68629i 0.339445i
\(389\) 26.9706 1.36746 0.683731 0.729734i \(-0.260356\pi\)
0.683731 + 0.729734i \(0.260356\pi\)
\(390\) 0 0
\(391\) −30.6274 −1.54890
\(392\) − 1.58579i − 0.0800943i
\(393\) −8.00000 −0.403547
\(394\) −0.201010 −0.0101267
\(395\) 32.0000i 1.61009i
\(396\) 3.65685i 0.183764i
\(397\) − 30.9706i − 1.55437i −0.629273 0.777184i \(-0.716647\pi\)
0.629273 0.777184i \(-0.283353\pi\)
\(398\) − 8.97056i − 0.449654i
\(399\) −8.00000 −0.400501
\(400\) −9.00000 −0.450000
\(401\) − 26.1421i − 1.30548i −0.757584 0.652738i \(-0.773620\pi\)
0.757584 0.652738i \(-0.226380\pi\)
\(402\) −2.82843 −0.141069
\(403\) 0 0
\(404\) −14.0000 −0.696526
\(405\) − 2.82843i − 0.140546i
\(406\) 2.34315 0.116288
\(407\) −15.3137 −0.759072
\(408\) − 12.1421i − 0.601125i
\(409\) − 34.9706i − 1.72918i −0.502475 0.864592i \(-0.667577\pi\)
0.502475 0.864592i \(-0.332423\pi\)
\(410\) 6.05887i 0.299226i
\(411\) 10.8284i 0.534127i
\(412\) −4.28427 −0.211071
\(413\) −21.6569 −1.06566
\(414\) 1.65685i 0.0814299i
\(415\) 10.3431 0.507725
\(416\) 0 0
\(417\) −7.31371 −0.358154
\(418\) 2.34315i 0.114607i
\(419\) 14.6274 0.714596 0.357298 0.933990i \(-0.383698\pi\)
0.357298 + 0.933990i \(0.383698\pi\)
\(420\) −14.6274 −0.713745
\(421\) 37.3137i 1.81856i 0.416186 + 0.909279i \(0.363366\pi\)
−0.416186 + 0.909279i \(0.636634\pi\)
\(422\) − 4.97056i − 0.241963i
\(423\) 11.6569i 0.566776i
\(424\) − 3.17157i − 0.154025i
\(425\) 22.9706 1.11424
\(426\) −0.828427 −0.0401374
\(427\) − 37.6569i − 1.82234i
\(428\) −20.6863 −0.999910
\(429\) 0 0
\(430\) 1.94113 0.0936094
\(431\) 8.34315i 0.401875i 0.979604 + 0.200938i \(0.0643988\pi\)
−0.979604 + 0.200938i \(0.935601\pi\)
\(432\) 3.00000 0.144338
\(433\) 21.3137 1.02427 0.512136 0.858905i \(-0.328854\pi\)
0.512136 + 0.858905i \(0.328854\pi\)
\(434\) − 1.37258i − 0.0658861i
\(435\) − 5.65685i − 0.271225i
\(436\) 9.71573i 0.465299i
\(437\) − 11.3137i − 0.541208i
\(438\) 0.142136 0.00679150
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 8.97056i 0.427655i
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) −25.9411 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(444\) 14.0000i 0.664411i
\(445\) −41.9411 −1.98820
\(446\) 5.17157 0.244881
\(447\) − 9.17157i − 0.433801i
\(448\) − 11.7990i − 0.557450i
\(449\) 31.7990i 1.50069i 0.661048 + 0.750344i \(0.270112\pi\)
−0.661048 + 0.750344i \(0.729888\pi\)
\(450\) − 1.24264i − 0.0585786i
\(451\) −10.3431 −0.487040
\(452\) −9.71573 −0.456989
\(453\) 3.51472i 0.165136i
\(454\) 7.17157 0.336579
\(455\) 0 0
\(456\) 4.48528 0.210043
\(457\) − 7.65685i − 0.358173i −0.983833 0.179086i \(-0.942686\pi\)
0.983833 0.179086i \(-0.0573141\pi\)
\(458\) −0.544156 −0.0254267
\(459\) −7.65685 −0.357391
\(460\) − 20.6863i − 0.964503i
\(461\) 5.17157i 0.240864i 0.992722 + 0.120432i \(0.0384280\pi\)
−0.992722 + 0.120432i \(0.961572\pi\)
\(462\) 2.34315i 0.109013i
\(463\) 24.4853i 1.13793i 0.822363 + 0.568964i \(0.192656\pi\)
−0.822363 + 0.568964i \(0.807344\pi\)
\(464\) 6.00000 0.278543
\(465\) −3.31371 −0.153670
\(466\) − 2.88730i − 0.133752i
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 19.3137 0.891824
\(470\) 13.6569i 0.629944i
\(471\) −10.0000 −0.460776
\(472\) 12.1421 0.558887
\(473\) 3.31371i 0.152364i
\(474\) − 4.68629i − 0.215248i
\(475\) 8.48528i 0.389331i
\(476\) 39.5980i 1.81497i
\(477\) −2.00000 −0.0915737
\(478\) −0.828427 −0.0378914
\(479\) − 25.3137i − 1.15661i −0.815820 0.578306i \(-0.803714\pi\)
0.815820 0.578306i \(-0.196286\pi\)
\(480\) 12.4853 0.569873
\(481\) 0 0
\(482\) 0.142136 0.00647410
\(483\) − 11.3137i − 0.514792i
\(484\) 12.7990 0.581772
\(485\) 10.3431 0.469658
\(486\) 0.414214i 0.0187891i
\(487\) − 7.79899i − 0.353406i −0.984264 0.176703i \(-0.943457\pi\)
0.984264 0.176703i \(-0.0565432\pi\)
\(488\) 21.1127i 0.955727i
\(489\) − 18.8284i − 0.851451i
\(490\) −1.17157 −0.0529263
\(491\) −30.6274 −1.38220 −0.691098 0.722761i \(-0.742873\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(492\) 9.45584i 0.426302i
\(493\) −15.3137 −0.689695
\(494\) 0 0
\(495\) 5.65685 0.254257
\(496\) − 3.51472i − 0.157816i
\(497\) 5.65685 0.253745
\(498\) −1.51472 −0.0678762
\(499\) 26.1421i 1.17028i 0.810931 + 0.585141i \(0.198961\pi\)
−0.810931 + 0.585141i \(0.801039\pi\)
\(500\) − 10.3431i − 0.462560i
\(501\) 3.65685i 0.163376i
\(502\) 0 0
\(503\) −7.31371 −0.326102 −0.163051 0.986618i \(-0.552134\pi\)
−0.163051 + 0.986618i \(0.552134\pi\)
\(504\) 4.48528 0.199790
\(505\) 21.6569i 0.963717i
\(506\) −3.31371 −0.147312
\(507\) 0 0
\(508\) −10.3431 −0.458903
\(509\) 11.7990i 0.522981i 0.965206 + 0.261491i \(0.0842140\pi\)
−0.965206 + 0.261491i \(0.915786\pi\)
\(510\) −8.97056 −0.397223
\(511\) −0.970563 −0.0429352
\(512\) 22.7574i 1.00574i
\(513\) − 2.82843i − 0.124878i
\(514\) 1.79899i 0.0793500i
\(515\) 6.62742i 0.292039i
\(516\) 3.02944 0.133364
\(517\) −23.3137 −1.02534
\(518\) 8.97056i 0.394144i
\(519\) 11.6569 0.511679
\(520\) 0 0
\(521\) 25.3137 1.10901 0.554507 0.832179i \(-0.312907\pi\)
0.554507 + 0.832179i \(0.312907\pi\)
\(522\) 0.828427i 0.0362593i
\(523\) −15.3137 −0.669622 −0.334811 0.942285i \(-0.608672\pi\)
−0.334811 + 0.942285i \(0.608672\pi\)
\(524\) −14.6274 −0.639002
\(525\) 8.48528i 0.370328i
\(526\) 4.97056i 0.216727i
\(527\) 8.97056i 0.390764i
\(528\) 6.00000i 0.261116i
\(529\) −7.00000 −0.304348
\(530\) −2.34315 −0.101780
\(531\) − 7.65685i − 0.332279i
\(532\) −14.6274 −0.634179
\(533\) 0 0
\(534\) 6.14214 0.265796
\(535\) 32.0000i 1.38348i
\(536\) −10.8284 −0.467717
\(537\) 23.3137 1.00606
\(538\) 7.45584i 0.321444i
\(539\) − 2.00000i − 0.0861461i
\(540\) − 5.17157i − 0.222549i
\(541\) − 10.0000i − 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) −11.5147 −0.494600
\(543\) −14.0000 −0.600798
\(544\) − 33.7990i − 1.44912i
\(545\) 15.0294 0.643790
\(546\) 0 0
\(547\) 23.3137 0.996822 0.498411 0.866941i \(-0.333917\pi\)
0.498411 + 0.866941i \(0.333917\pi\)
\(548\) 19.7990i 0.845771i
\(549\) 13.3137 0.568215
\(550\) 2.48528 0.105973
\(551\) − 5.65685i − 0.240990i
\(552\) 6.34315i 0.269982i
\(553\) 32.0000i 1.36078i
\(554\) 0.828427i 0.0351965i
\(555\) 21.6569 0.919282
\(556\) −13.3726 −0.567124
\(557\) 7.79899i 0.330454i 0.986256 + 0.165227i \(0.0528356\pi\)
−0.986256 + 0.165227i \(0.947164\pi\)
\(558\) 0.485281 0.0205436
\(559\) 0 0
\(560\) −24.0000 −1.01419
\(561\) − 15.3137i − 0.646545i
\(562\) 8.76955 0.369921
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 21.3137i 0.897469i
\(565\) 15.0294i 0.632293i
\(566\) − 12.0000i − 0.504398i
\(567\) − 2.82843i − 0.118783i
\(568\) −3.17157 −0.133076
\(569\) 42.9706 1.80142 0.900710 0.434421i \(-0.143047\pi\)
0.900710 + 0.434421i \(0.143047\pi\)
\(570\) − 3.31371i − 0.138796i
\(571\) 12.9706 0.542801 0.271401 0.962466i \(-0.412513\pi\)
0.271401 + 0.962466i \(0.412513\pi\)
\(572\) 0 0
\(573\) 3.31371 0.138432
\(574\) 6.05887i 0.252893i
\(575\) −12.0000 −0.500435
\(576\) 4.17157 0.173816
\(577\) − 31.9411i − 1.32973i −0.746965 0.664863i \(-0.768490\pi\)
0.746965 0.664863i \(-0.231510\pi\)
\(578\) 17.2426i 0.717199i
\(579\) − 5.31371i − 0.220830i
\(580\) − 10.3431i − 0.429476i
\(581\) 10.3431 0.429106
\(582\) −1.51472 −0.0627871
\(583\) − 4.00000i − 0.165663i
\(584\) 0.544156 0.0225173
\(585\) 0 0
\(586\) 0.887302 0.0366541
\(587\) − 10.9706i − 0.452804i −0.974034 0.226402i \(-0.927304\pi\)
0.974034 0.226402i \(-0.0726962\pi\)
\(588\) −1.82843 −0.0754031
\(589\) −3.31371 −0.136539
\(590\) − 8.97056i − 0.369312i
\(591\) 0.485281i 0.0199618i
\(592\) 22.9706i 0.944084i
\(593\) 20.4853i 0.841230i 0.907239 + 0.420615i \(0.138186\pi\)
−0.907239 + 0.420615i \(0.861814\pi\)
\(594\) −0.828427 −0.0339908
\(595\) 61.2548 2.51120
\(596\) − 16.7696i − 0.686908i
\(597\) −21.6569 −0.886356
\(598\) 0 0
\(599\) −23.3137 −0.952572 −0.476286 0.879290i \(-0.658017\pi\)
−0.476286 + 0.879290i \(0.658017\pi\)
\(600\) − 4.75736i − 0.194218i
\(601\) −0.627417 −0.0255929 −0.0127964 0.999918i \(-0.504073\pi\)
−0.0127964 + 0.999918i \(0.504073\pi\)
\(602\) 1.94113 0.0791144
\(603\) 6.82843i 0.278075i
\(604\) 6.42641i 0.261487i
\(605\) − 19.7990i − 0.804943i
\(606\) − 3.17157i − 0.128836i
\(607\) 41.9411 1.70234 0.851169 0.524892i \(-0.175894\pi\)
0.851169 + 0.524892i \(0.175894\pi\)
\(608\) 12.4853 0.506345
\(609\) − 5.65685i − 0.229227i
\(610\) 15.5980 0.631544
\(611\) 0 0
\(612\) −14.0000 −0.565916
\(613\) − 47.6569i − 1.92484i −0.271561 0.962421i \(-0.587540\pi\)
0.271561 0.962421i \(-0.412460\pi\)
\(614\) −9.45584 −0.381607
\(615\) 14.6274 0.589834
\(616\) 8.97056i 0.361434i
\(617\) − 34.8284i − 1.40214i −0.713093 0.701070i \(-0.752706\pi\)
0.713093 0.701070i \(-0.247294\pi\)
\(618\) − 0.970563i − 0.0390418i
\(619\) − 23.7990i − 0.956562i −0.878207 0.478281i \(-0.841260\pi\)
0.878207 0.478281i \(-0.158740\pi\)
\(620\) −6.05887 −0.243330
\(621\) 4.00000 0.160514
\(622\) 4.40202i 0.176505i
\(623\) −41.9411 −1.68034
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 2.48528i 0.0993318i
\(627\) 5.65685 0.225913
\(628\) −18.2843 −0.729622
\(629\) − 58.6274i − 2.33763i
\(630\) − 3.31371i − 0.132021i
\(631\) − 43.1127i − 1.71629i −0.513408 0.858145i \(-0.671617\pi\)
0.513408 0.858145i \(-0.328383\pi\)
\(632\) − 17.9411i − 0.713660i
\(633\) −12.0000 −0.476957
\(634\) −3.51472 −0.139587
\(635\) 16.0000i 0.634941i
\(636\) −3.65685 −0.145004
\(637\) 0 0
\(638\) −1.65685 −0.0655955
\(639\) 2.00000i 0.0791188i
\(640\) 29.8579 1.18024
\(641\) −30.2843 −1.19616 −0.598078 0.801438i \(-0.704069\pi\)
−0.598078 + 0.801438i \(0.704069\pi\)
\(642\) − 4.68629i − 0.184953i
\(643\) 22.8284i 0.900265i 0.892962 + 0.450133i \(0.148623\pi\)
−0.892962 + 0.450133i \(0.851377\pi\)
\(644\) − 20.6863i − 0.815154i
\(645\) − 4.68629i − 0.184523i
\(646\) −8.97056 −0.352942
\(647\) −11.3137 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(648\) 1.58579i 0.0622956i
\(649\) 15.3137 0.601116
\(650\) 0 0
\(651\) −3.31371 −0.129874
\(652\) − 34.4264i − 1.34824i
\(653\) −25.3137 −0.990602 −0.495301 0.868721i \(-0.664942\pi\)
−0.495301 + 0.868721i \(0.664942\pi\)
\(654\) −2.20101 −0.0860663
\(655\) 22.6274i 0.884126i
\(656\) 15.5147i 0.605748i
\(657\) − 0.343146i − 0.0133874i
\(658\) 13.6569i 0.532400i
\(659\) −47.3137 −1.84308 −0.921540 0.388283i \(-0.873068\pi\)
−0.921540 + 0.388283i \(0.873068\pi\)
\(660\) 10.3431 0.402606
\(661\) 34.9706i 1.36020i 0.733121 + 0.680099i \(0.238063\pi\)
−0.733121 + 0.680099i \(0.761937\pi\)
\(662\) −10.8284 −0.420859
\(663\) 0 0
\(664\) −5.79899 −0.225044
\(665\) 22.6274i 0.877454i
\(666\) −3.17157 −0.122896
\(667\) 8.00000 0.309761
\(668\) 6.68629i 0.258700i
\(669\) − 12.4853i − 0.482709i
\(670\) 8.00000i 0.309067i
\(671\) 26.6274i 1.02794i
\(672\) 12.4853 0.481630
\(673\) −16.6274 −0.640940 −0.320470 0.947259i \(-0.603841\pi\)
−0.320470 + 0.947259i \(0.603841\pi\)
\(674\) − 3.85786i − 0.148599i
\(675\) −3.00000 −0.115470
\(676\) 0 0
\(677\) 26.6863 1.02564 0.512819 0.858497i \(-0.328601\pi\)
0.512819 + 0.858497i \(0.328601\pi\)
\(678\) − 2.20101i − 0.0845293i
\(679\) 10.3431 0.396934
\(680\) −34.3431 −1.31700
\(681\) − 17.3137i − 0.663463i
\(682\) 0.970563i 0.0371648i
\(683\) − 47.9411i − 1.83442i −0.398408 0.917208i \(-0.630437\pi\)
0.398408 0.917208i \(-0.369563\pi\)
\(684\) − 5.17157i − 0.197740i
\(685\) 30.6274 1.17021
\(686\) 7.02944 0.268385
\(687\) 1.31371i 0.0501211i
\(688\) 4.97056 0.189501
\(689\) 0 0
\(690\) 4.68629 0.178404
\(691\) − 5.85786i − 0.222844i −0.993773 0.111422i \(-0.964460\pi\)
0.993773 0.111422i \(-0.0355405\pi\)
\(692\) 21.3137 0.810226
\(693\) 5.65685 0.214886
\(694\) − 3.59798i − 0.136577i
\(695\) 20.6863i 0.784676i
\(696\) 3.17157i 0.120218i
\(697\) − 39.5980i − 1.49988i
\(698\) 1.51472 0.0573329
\(699\) −6.97056 −0.263651
\(700\) 15.5147i 0.586401i
\(701\) −5.02944 −0.189959 −0.0949796 0.995479i \(-0.530279\pi\)
−0.0949796 + 0.995479i \(0.530279\pi\)
\(702\) 0 0
\(703\) 21.6569 0.816804
\(704\) 8.34315i 0.314444i
\(705\) 32.9706 1.24174
\(706\) 13.8579 0.521548
\(707\) 21.6569i 0.814490i
\(708\) − 14.0000i − 0.526152i
\(709\) 4.62742i 0.173786i 0.996218 + 0.0868931i \(0.0276939\pi\)
−0.996218 + 0.0868931i \(0.972306\pi\)
\(710\) 2.34315i 0.0879367i
\(711\) −11.3137 −0.424297
\(712\) 23.5147 0.881251
\(713\) − 4.68629i − 0.175503i
\(714\) −8.97056 −0.335715
\(715\) 0 0
\(716\) 42.6274 1.59306
\(717\) 2.00000i 0.0746914i
\(718\) 14.4853 0.540586
\(719\) 29.9411 1.11662 0.558308 0.829634i \(-0.311451\pi\)
0.558308 + 0.829634i \(0.311451\pi\)
\(720\) − 8.48528i − 0.316228i
\(721\) 6.62742i 0.246818i
\(722\) 4.55635i 0.169570i
\(723\) − 0.343146i − 0.0127617i
\(724\) −25.5980 −0.951341
\(725\) −6.00000 −0.222834
\(726\) 2.89949i 0.107610i
\(727\) 10.3431 0.383606 0.191803 0.981433i \(-0.438567\pi\)
0.191803 + 0.981433i \(0.438567\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 0.402020i − 0.0148794i
\(731\) −12.6863 −0.469219
\(732\) 24.3431 0.899749
\(733\) − 36.6274i − 1.35286i −0.736505 0.676432i \(-0.763525\pi\)
0.736505 0.676432i \(-0.236475\pi\)
\(734\) − 9.94113i − 0.366934i
\(735\) 2.82843i 0.104328i
\(736\) 17.6569i 0.650840i
\(737\) −13.6569 −0.503057
\(738\) −2.14214 −0.0788531
\(739\) 18.1421i 0.667369i 0.942685 + 0.333685i \(0.108292\pi\)
−0.942685 + 0.333685i \(0.891708\pi\)
\(740\) 39.5980 1.45565
\(741\) 0 0
\(742\) −2.34315 −0.0860196
\(743\) 2.00000i 0.0733729i 0.999327 + 0.0366864i \(0.0116803\pi\)
−0.999327 + 0.0366864i \(0.988320\pi\)
\(744\) 1.85786 0.0681126
\(745\) −25.9411 −0.950409
\(746\) 4.14214i 0.151654i
\(747\) 3.65685i 0.133797i
\(748\) − 28.0000i − 1.02378i
\(749\) 32.0000i 1.16925i
\(750\) 2.34315 0.0855596
\(751\) 0.970563 0.0354163 0.0177082 0.999843i \(-0.494363\pi\)
0.0177082 + 0.999843i \(0.494363\pi\)
\(752\) 34.9706i 1.27525i
\(753\) 0 0
\(754\) 0 0
\(755\) 9.94113 0.361795
\(756\) − 5.17157i − 0.188088i
\(757\) 51.9411 1.88783 0.943916 0.330185i \(-0.107111\pi\)
0.943916 + 0.330185i \(0.107111\pi\)
\(758\) −0.201010 −0.00730102
\(759\) 8.00000i 0.290382i
\(760\) − 12.6863i − 0.460180i
\(761\) − 32.4853i − 1.17759i −0.808282 0.588795i \(-0.799602\pi\)
0.808282 0.588795i \(-0.200398\pi\)
\(762\) − 2.34315i − 0.0848832i
\(763\) 15.0294 0.544102
\(764\) 6.05887 0.219202
\(765\) 21.6569i 0.783005i
\(766\) −12.8284 −0.463510
\(767\) 0 0
\(768\) 3.97056 0.143275
\(769\) 42.0000i 1.51456i 0.653091 + 0.757279i \(0.273472\pi\)
−0.653091 + 0.757279i \(0.726528\pi\)
\(770\) 6.62742 0.238836
\(771\) 4.34315 0.156415
\(772\) − 9.71573i − 0.349677i
\(773\) 34.1421i 1.22801i 0.789303 + 0.614004i \(0.210442\pi\)
−0.789303 + 0.614004i \(0.789558\pi\)
\(774\) 0.686292i 0.0246682i
\(775\) 3.51472i 0.126252i
\(776\) −5.79899 −0.208172
\(777\) 21.6569 0.776935
\(778\) 11.1716i 0.400520i
\(779\) 14.6274 0.524082
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) − 12.6863i − 0.453661i
\(783\) 2.00000 0.0714742
\(784\) −3.00000 −0.107143
\(785\) 28.2843i 1.00951i
\(786\) − 3.31371i − 0.118196i
\(787\) 40.7696i 1.45328i 0.687020 + 0.726639i \(0.258919\pi\)
−0.687020 + 0.726639i \(0.741081\pi\)
\(788\) 0.887302i 0.0316088i
\(789\) 12.0000 0.427211
\(790\) −13.2548 −0.471586
\(791\) 15.0294i 0.534385i
\(792\) −3.17157 −0.112697
\(793\) 0 0
\(794\) 12.8284 0.455264
\(795\) 5.65685i 0.200628i
\(796\) −39.5980 −1.40351
\(797\) −24.3431 −0.862278 −0.431139 0.902285i \(-0.641888\pi\)
−0.431139 + 0.902285i \(0.641888\pi\)
\(798\) − 3.31371i − 0.117304i
\(799\) − 89.2548i − 3.15761i
\(800\) − 13.2426i − 0.468198i
\(801\) − 14.8284i − 0.523937i
\(802\) 10.8284 0.382365
\(803\) 0.686292 0.0242187
\(804\) 12.4853i 0.440322i
\(805\) −32.0000 −1.12785
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) − 12.1421i − 0.427159i
\(809\) 18.6863 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(810\) 1.17157 0.0411649
\(811\) − 30.1421i − 1.05843i −0.848487 0.529217i \(-0.822486\pi\)
0.848487 0.529217i \(-0.177514\pi\)
\(812\) − 10.3431i − 0.362973i
\(813\) 27.7990i 0.974953i
\(814\) − 6.34315i − 0.222327i
\(815\) −53.2548 −1.86544
\(816\) −22.9706 −0.804131
\(817\) − 4.68629i − 0.163953i
\(818\) 14.4853 0.506466
\(819\) 0 0
\(820\) 26.7452 0.933982
\(821\) − 23.7990i − 0.830590i −0.909687 0.415295i \(-0.863678\pi\)
0.909687 0.415295i \(-0.136322\pi\)
\(822\) −4.48528 −0.156442
\(823\) −15.0294 −0.523893 −0.261947 0.965082i \(-0.584364\pi\)
−0.261947 + 0.965082i \(0.584364\pi\)
\(824\) − 3.71573i − 0.129444i
\(825\) − 6.00000i − 0.208893i
\(826\) − 8.97056i − 0.312126i
\(827\) 26.0000i 0.904109i 0.891990 + 0.452054i \(0.149309\pi\)
−0.891990 + 0.452054i \(0.850691\pi\)
\(828\) 7.31371 0.254169
\(829\) 17.3137 0.601330 0.300665 0.953730i \(-0.402791\pi\)
0.300665 + 0.953730i \(0.402791\pi\)
\(830\) 4.28427i 0.148709i
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 7.65685 0.265294
\(834\) − 3.02944i − 0.104901i
\(835\) 10.3431 0.357939
\(836\) 10.3431 0.357725
\(837\) − 1.17157i − 0.0404955i
\(838\) 6.05887i 0.209300i
\(839\) − 43.2548i − 1.49332i −0.665204 0.746661i \(-0.731656\pi\)
0.665204 0.746661i \(-0.268344\pi\)
\(840\) − 12.6863i − 0.437719i
\(841\) −25.0000 −0.862069
\(842\) −15.4558 −0.532644
\(843\) − 21.1716i − 0.729188i
\(844\) −21.9411 −0.755245
\(845\) 0 0
\(846\) −4.82843 −0.166005
\(847\) − 19.7990i − 0.680301i
\(848\) −6.00000 −0.206041
\(849\) −28.9706 −0.994267
\(850\) 9.51472i 0.326352i
\(851\) 30.6274i 1.04989i
\(852\) 3.65685i 0.125282i
\(853\) − 3.65685i − 0.125208i −0.998038 0.0626042i \(-0.980059\pi\)
0.998038 0.0626042i \(-0.0199406\pi\)
\(854\) 15.5980 0.533752
\(855\) −8.00000 −0.273594
\(856\) − 17.9411i − 0.613215i
\(857\) 49.5980 1.69423 0.847117 0.531406i \(-0.178336\pi\)
0.847117 + 0.531406i \(0.178336\pi\)
\(858\) 0 0
\(859\) −0.686292 −0.0234160 −0.0117080 0.999931i \(-0.503727\pi\)
−0.0117080 + 0.999931i \(0.503727\pi\)
\(860\) − 8.56854i − 0.292185i
\(861\) 14.6274 0.498501
\(862\) −3.45584 −0.117707
\(863\) − 28.3431i − 0.964812i −0.875948 0.482406i \(-0.839763\pi\)
0.875948 0.482406i \(-0.160237\pi\)
\(864\) 4.41421i 0.150175i
\(865\) − 32.9706i − 1.12103i
\(866\) 8.82843i 0.300002i
\(867\) 41.6274 1.41374
\(868\) −6.05887 −0.205652
\(869\) − 22.6274i − 0.767583i
\(870\) 2.34315 0.0794401
\(871\) 0 0
\(872\) −8.42641 −0.285354
\(873\) 3.65685i 0.123766i
\(874\) 4.68629 0.158516
\(875\) −16.0000 −0.540899
\(876\) − 0.627417i − 0.0211985i
\(877\) 42.2843i 1.42784i 0.700228 + 0.713919i \(0.253082\pi\)
−0.700228 + 0.713919i \(0.746918\pi\)
\(878\) − 7.02944i − 0.237232i
\(879\) − 2.14214i − 0.0722524i
\(880\) 16.9706 0.572078
\(881\) 25.5980 0.862418 0.431209 0.902252i \(-0.358087\pi\)
0.431209 + 0.902252i \(0.358087\pi\)
\(882\) − 0.414214i − 0.0139473i
\(883\) −27.5980 −0.928746 −0.464373 0.885640i \(-0.653720\pi\)
−0.464373 + 0.885640i \(0.653720\pi\)
\(884\) 0 0
\(885\) −21.6569 −0.727987
\(886\) − 10.7452i − 0.360991i
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −12.1421 −0.407463
\(889\) 16.0000i 0.536623i
\(890\) − 17.3726i − 0.582330i
\(891\) 2.00000i 0.0670025i
\(892\) − 22.8284i − 0.764352i
\(893\) 32.9706 1.10332
\(894\) 3.79899 0.127057
\(895\) − 65.9411i − 2.20417i
\(896\) 29.8579 0.997481
\(897\) 0 0
\(898\) −13.1716 −0.439541
\(899\) − 2.34315i − 0.0781483i
\(900\) −5.48528 −0.182843
\(901\) 15.3137 0.510174
\(902\) − 4.28427i − 0.142651i
\(903\) − 4.68629i − 0.155950i
\(904\) − 8.42641i − 0.280258i
\(905\) 39.5980i 1.31628i
\(906\) −1.45584 −0.0483672
\(907\) 12.9706 0.430680 0.215340 0.976539i \(-0.430914\pi\)
0.215340 + 0.976539i \(0.430914\pi\)
\(908\) − 31.6569i − 1.05057i
\(909\) −7.65685 −0.253962
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) − 8.48528i − 0.280976i
\(913\) −7.31371 −0.242048
\(914\) 3.17157 0.104906
\(915\) − 37.6569i − 1.24490i
\(916\) 2.40202i 0.0793650i
\(917\) 22.6274i 0.747223i
\(918\) − 3.17157i − 0.104678i
\(919\) 3.31371 0.109309 0.0546546 0.998505i \(-0.482594\pi\)
0.0546546 + 0.998505i \(0.482594\pi\)
\(920\) 17.9411 0.591501
\(921\) 22.8284i 0.752222i
\(922\) −2.14214 −0.0705475
\(923\) 0 0
\(924\) 10.3431 0.340265
\(925\) − 22.9706i − 0.755267i
\(926\) −10.1421 −0.333291
\(927\) −2.34315 −0.0769590
\(928\) 8.82843i 0.289807i
\(929\) 11.7990i 0.387112i 0.981089 + 0.193556i \(0.0620022\pi\)
−0.981089 + 0.193556i \(0.937998\pi\)
\(930\) − 1.37258i − 0.0450088i
\(931\) 2.82843i 0.0926980i
\(932\) −12.7452 −0.417482
\(933\) 10.6274 0.347926
\(934\) 3.31371i 0.108428i
\(935\) −43.3137 −1.41651
\(936\) 0 0
\(937\) −21.3137 −0.696289 −0.348144 0.937441i \(-0.613188\pi\)
−0.348144 + 0.937441i \(0.613188\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 6.00000 0.195803
\(940\) 60.2843 1.96626
\(941\) 34.1421i 1.11300i 0.830847 + 0.556501i \(0.187856\pi\)
−0.830847 + 0.556501i \(0.812144\pi\)
\(942\) − 4.14214i − 0.134958i
\(943\) 20.6863i 0.673638i
\(944\) − 22.9706i − 0.747628i
\(945\) −8.00000 −0.260240
\(946\) −1.37258 −0.0446265
\(947\) 21.0294i 0.683365i 0.939815 + 0.341682i \(0.110997\pi\)
−0.939815 + 0.341682i \(0.889003\pi\)
\(948\) −20.6863 −0.671860
\(949\) 0 0
\(950\) −3.51472 −0.114033
\(951\) 8.48528i 0.275154i
\(952\) −34.3431 −1.11307
\(953\) −40.3431 −1.30684 −0.653421 0.756994i \(-0.726667\pi\)
−0.653421 + 0.756994i \(0.726667\pi\)
\(954\) − 0.828427i − 0.0268213i
\(955\) − 9.37258i − 0.303290i
\(956\) 3.65685i 0.118271i
\(957\) 4.00000i 0.129302i
\(958\) 10.4853 0.338764
\(959\) 30.6274 0.989011
\(960\) − 11.7990i − 0.380811i
\(961\) 29.6274 0.955723
\(962\) 0 0
\(963\) −11.3137 −0.364579
\(964\) − 0.627417i − 0.0202077i
\(965\) −15.0294 −0.483815
\(966\) 4.68629 0.150779
\(967\) 18.1421i 0.583412i 0.956508 + 0.291706i \(0.0942228\pi\)
−0.956508 + 0.291706i \(0.905777\pi\)
\(968\) 11.1005i 0.356784i
\(969\) 21.6569i 0.695718i
\(970\) 4.28427i 0.137560i
\(971\) −15.3137 −0.491440 −0.245720 0.969341i \(-0.579024\pi\)
−0.245720 + 0.969341i \(0.579024\pi\)
\(972\) 1.82843 0.0586468
\(973\) 20.6863i 0.663172i
\(974\) 3.23045 0.103510
\(975\) 0 0
\(976\) 39.9411 1.27848
\(977\) 42.1421i 1.34825i 0.738619 + 0.674123i \(0.235478\pi\)
−0.738619 + 0.674123i \(0.764522\pi\)
\(978\) 7.79899 0.249384
\(979\) 29.6569 0.947837
\(980\) 5.17157i 0.165200i
\(981\) 5.31371i 0.169654i
\(982\) − 12.6863i − 0.404836i
\(983\) − 25.3137i − 0.807382i −0.914895 0.403691i \(-0.867727\pi\)
0.914895 0.403691i \(-0.132273\pi\)
\(984\) −8.20101 −0.261439
\(985\) 1.37258 0.0437341
\(986\) − 6.34315i − 0.202007i
\(987\) 32.9706 1.04946
\(988\) 0 0
\(989\) 6.62742 0.210740
\(990\) 2.34315i 0.0744701i
\(991\) 4.68629 0.148865 0.0744325 0.997226i \(-0.476285\pi\)
0.0744325 + 0.997226i \(0.476285\pi\)
\(992\) 5.17157 0.164198
\(993\) 26.1421i 0.829596i
\(994\) 2.34315i 0.0743201i
\(995\) 61.2548i 1.94191i
\(996\) 6.68629i 0.211863i
\(997\) −39.2548 −1.24321 −0.621607 0.783330i \(-0.713520\pi\)
−0.621607 + 0.783330i \(0.713520\pi\)
\(998\) −10.8284 −0.342768
\(999\) 7.65685i 0.242252i
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 507.2.b.e.337.3 4
3.2 odd 2 1521.2.b.j.1351.2 4
13.2 odd 12 507.2.e.h.22.1 4
13.3 even 3 507.2.j.f.316.2 8
13.4 even 6 507.2.j.f.361.2 8
13.5 odd 4 39.2.a.b.1.2 2
13.6 odd 12 507.2.e.h.484.1 4
13.7 odd 12 507.2.e.d.484.2 4
13.8 odd 4 507.2.a.h.1.1 2
13.9 even 3 507.2.j.f.361.3 8
13.10 even 6 507.2.j.f.316.3 8
13.11 odd 12 507.2.e.d.22.2 4
13.12 even 2 inner 507.2.b.e.337.2 4
39.5 even 4 117.2.a.c.1.1 2
39.8 even 4 1521.2.a.f.1.2 2
39.38 odd 2 1521.2.b.j.1351.3 4
52.31 even 4 624.2.a.k.1.1 2
52.47 even 4 8112.2.a.bm.1.2 2
65.18 even 4 975.2.c.h.274.2 4
65.44 odd 4 975.2.a.l.1.1 2
65.57 even 4 975.2.c.h.274.3 4
91.83 even 4 1911.2.a.h.1.2 2
104.5 odd 4 2496.2.a.bf.1.2 2
104.83 even 4 2496.2.a.bi.1.2 2
117.5 even 12 1053.2.e.e.703.2 4
117.31 odd 12 1053.2.e.m.703.1 4
117.70 odd 12 1053.2.e.m.352.1 4
117.83 even 12 1053.2.e.e.352.2 4
143.109 even 4 4719.2.a.p.1.1 2
156.83 odd 4 1872.2.a.w.1.2 2
195.44 even 4 2925.2.a.v.1.2 2
195.83 odd 4 2925.2.c.u.2224.3 4
195.122 odd 4 2925.2.c.u.2224.2 4
273.83 odd 4 5733.2.a.u.1.1 2
312.5 even 4 7488.2.a.cl.1.1 2
312.83 odd 4 7488.2.a.co.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 13.5 odd 4
117.2.a.c.1.1 2 39.5 even 4
507.2.a.h.1.1 2 13.8 odd 4
507.2.b.e.337.2 4 13.12 even 2 inner
507.2.b.e.337.3 4 1.1 even 1 trivial
507.2.e.d.22.2 4 13.11 odd 12
507.2.e.d.484.2 4 13.7 odd 12
507.2.e.h.22.1 4 13.2 odd 12
507.2.e.h.484.1 4 13.6 odd 12
507.2.j.f.316.2 8 13.3 even 3
507.2.j.f.316.3 8 13.10 even 6
507.2.j.f.361.2 8 13.4 even 6
507.2.j.f.361.3 8 13.9 even 3
624.2.a.k.1.1 2 52.31 even 4
975.2.a.l.1.1 2 65.44 odd 4
975.2.c.h.274.2 4 65.18 even 4
975.2.c.h.274.3 4 65.57 even 4
1053.2.e.e.352.2 4 117.83 even 12
1053.2.e.e.703.2 4 117.5 even 12
1053.2.e.m.352.1 4 117.70 odd 12
1053.2.e.m.703.1 4 117.31 odd 12
1521.2.a.f.1.2 2 39.8 even 4
1521.2.b.j.1351.2 4 3.2 odd 2
1521.2.b.j.1351.3 4 39.38 odd 2
1872.2.a.w.1.2 2 156.83 odd 4
1911.2.a.h.1.2 2 91.83 even 4
2496.2.a.bf.1.2 2 104.5 odd 4
2496.2.a.bi.1.2 2 104.83 even 4
2925.2.a.v.1.2 2 195.44 even 4
2925.2.c.u.2224.2 4 195.122 odd 4
2925.2.c.u.2224.3 4 195.83 odd 4
4719.2.a.p.1.1 2 143.109 even 4
5733.2.a.u.1.1 2 273.83 odd 4
7488.2.a.cl.1.1 2 312.5 even 4
7488.2.a.co.1.1 2 312.83 odd 4
8112.2.a.bm.1.2 2 52.47 even 4