Properties

Label 975.2.c.h.274.2
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.h.274.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} +0.414214 q^{6} -2.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.414214i q^{2} +1.00000i q^{3} +1.82843 q^{4} +0.414214 q^{6} -2.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.82843i q^{12} -1.00000i q^{13} -1.17157 q^{14} +3.00000 q^{16} -7.65685i q^{17} +0.414214i q^{18} +2.82843 q^{19} +2.82843 q^{21} +0.828427i q^{22} -4.00000i q^{23} +1.58579 q^{24} -0.414214 q^{26} -1.00000i q^{27} -5.17157i q^{28} -2.00000 q^{29} -1.17157 q^{31} -4.41421i q^{32} -2.00000i q^{33} -3.17157 q^{34} -1.82843 q^{36} +7.65685i q^{37} -1.17157i q^{38} +1.00000 q^{39} +5.17157 q^{41} -1.17157i q^{42} -1.65685i q^{43} -3.65685 q^{44} -1.65685 q^{46} +11.6569i q^{47} +3.00000i q^{48} -1.00000 q^{49} +7.65685 q^{51} -1.82843i q^{52} -2.00000i q^{53} -0.414214 q^{54} -4.48528 q^{56} +2.82843i q^{57} +0.828427i q^{58} -7.65685 q^{59} +13.3137 q^{61} +0.485281i q^{62} +2.82843i q^{63} +4.17157 q^{64} -0.828427 q^{66} -6.82843i q^{67} -14.0000i q^{68} +4.00000 q^{69} +2.00000 q^{71} +1.58579i q^{72} +0.343146i q^{73} +3.17157 q^{74} +5.17157 q^{76} +5.65685i q^{77} -0.414214i q^{78} +11.3137 q^{79} +1.00000 q^{81} -2.14214i q^{82} +3.65685i q^{83} +5.17157 q^{84} -0.686292 q^{86} -2.00000i q^{87} +3.17157i q^{88} -14.8284 q^{89} -2.82843 q^{91} -7.31371i q^{92} -1.17157i q^{93} +4.82843 q^{94} +4.41421 q^{96} -3.65685i q^{97} +0.414214i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{6} - 4q^{9} - 8q^{11} - 16q^{14} + 12q^{16} + 12q^{24} + 4q^{26} - 8q^{29} - 16q^{31} - 24q^{34} + 4q^{36} + 4q^{39} + 32q^{41} + 8q^{44} + 16q^{46} - 4q^{49} + 8q^{51} + 4q^{54} + 16q^{56} - 8q^{59} + 8q^{61} + 28q^{64} + 8q^{66} + 16q^{69} + 8q^{71} + 24q^{74} + 32q^{76} + 4q^{81} + 32q^{84} - 48q^{86} - 48q^{89} + 8q^{94} + 12q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) 0.414214 0.169102
\(7\) − 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
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.82843i 0.527821i
\(13\) − 1.00000i − 0.277350i
\(14\) −1.17157 −0.313116
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 7.65685i − 1.85706i −0.371257 0.928530i \(-0.621073\pi\)
0.371257 0.928530i \(-0.378927\pi\)
\(18\) 0.414214i 0.0976311i
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0.828427i 0.176621i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 1.58579 0.323697
\(25\) 0 0
\(26\) −0.414214 −0.0812340
\(27\) − 1.00000i − 0.192450i
\(28\) − 5.17157i − 0.977335i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) − 2.00000i − 0.348155i
\(34\) −3.17157 −0.543920
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) 7.65685i 1.25878i 0.777090 + 0.629390i \(0.216695\pi\)
−0.777090 + 0.629390i \(0.783305\pi\)
\(38\) − 1.17157i − 0.190054i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 5.17157 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(42\) − 1.17157i − 0.180778i
\(43\) − 1.65685i − 0.252668i −0.991988 0.126334i \(-0.959679\pi\)
0.991988 0.126334i \(-0.0403211\pi\)
\(44\) −3.65685 −0.551292
\(45\) 0 0
\(46\) −1.65685 −0.244290
\(47\) 11.6569i 1.70033i 0.526519 + 0.850163i \(0.323497\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(48\) 3.00000i 0.433013i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 7.65685 1.07217
\(52\) − 1.82843i − 0.253557i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0 0
\(56\) −4.48528 −0.599371
\(57\) 2.82843i 0.374634i
\(58\) 0.828427i 0.108778i
\(59\) −7.65685 −0.996838 −0.498419 0.866936i \(-0.666086\pi\)
−0.498419 + 0.866936i \(0.666086\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0.485281i 0.0616308i
\(63\) 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 \(-0.863042\pi\)
0.908855 0.417113i \(-0.136958\pi\)
\(68\) − 14.0000i − 1.69775i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.58579i 0.186887i
\(73\) 0.343146i 0.0401622i 0.999798 + 0.0200811i \(0.00639244\pi\)
−0.999798 + 0.0200811i \(0.993608\pi\)
\(74\) 3.17157 0.368688
\(75\) 0 0
\(76\) 5.17157 0.593220
\(77\) 5.65685i 0.644658i
\(78\) − 0.414214i − 0.0469005i
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.14214i − 0.236559i
\(83\) 3.65685i 0.401392i 0.979654 + 0.200696i \(0.0643203\pi\)
−0.979654 + 0.200696i \(0.935680\pi\)
\(84\) 5.17157 0.564265
\(85\) 0 0
\(86\) −0.686292 −0.0740047
\(87\) − 2.00000i − 0.214423i
\(88\) 3.17157i 0.338091i
\(89\) −14.8284 −1.57181 −0.785905 0.618347i \(-0.787803\pi\)
−0.785905 + 0.618347i \(0.787803\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) − 7.31371i − 0.762507i
\(93\) − 1.17157i − 0.121486i
\(94\) 4.82843 0.498014
\(95\) 0 0
\(96\) 4.41421 0.450524
\(97\) − 3.65685i − 0.371297i −0.982616 0.185649i \(-0.940561\pi\)
0.982616 0.185649i \(-0.0594386\pi\)
\(98\) 0.414214i 0.0418419i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) − 3.17157i − 0.314033i
\(103\) 2.34315i 0.230877i 0.993315 + 0.115439i \(0.0368273\pi\)
−0.993315 + 0.115439i \(0.963173\pi\)
\(104\) −1.58579 −0.155499
\(105\) 0 0
\(106\) −0.828427 −0.0804640
\(107\) 11.3137i 1.09374i 0.837218 + 0.546869i \(0.184180\pi\)
−0.837218 + 0.546869i \(0.815820\pi\)
\(108\) − 1.82843i − 0.175940i
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) −7.65685 −0.726756
\(112\) − 8.48528i − 0.801784i
\(113\) − 5.31371i − 0.499872i −0.968262 0.249936i \(-0.919590\pi\)
0.968262 0.249936i \(-0.0804095\pi\)
\(114\) 1.17157 0.109728
\(115\) 0 0
\(116\) −3.65685 −0.339530
\(117\) 1.00000i 0.0924500i
\(118\) 3.17157i 0.291967i
\(119\) −21.6569 −1.98528
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 5.51472i − 0.499279i
\(123\) 5.17157i 0.466305i
\(124\) −2.14214 −0.192369
\(125\) 0 0
\(126\) 1.17157 0.104372
\(127\) − 5.65685i − 0.501965i −0.967992 0.250982i \(-0.919246\pi\)
0.967992 0.250982i \(-0.0807536\pi\)
\(128\) − 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
\(133\) − 8.00000i − 0.693688i
\(134\) −2.82843 −0.244339
\(135\) 0 0
\(136\) −12.1421 −1.04118
\(137\) 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 \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(140\) 0 0
\(141\) −11.6569 −0.981684
\(142\) − 0.828427i − 0.0695201i
\(143\) 2.00000i 0.167248i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 0.142136 0.0117632
\(147\) − 1.00000i − 0.0824786i
\(148\) 14.0000i 1.15079i
\(149\) 9.17157 0.751365 0.375682 0.926749i \(-0.377408\pi\)
0.375682 + 0.926749i \(0.377408\pi\)
\(150\) 0 0
\(151\) −3.51472 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(152\) − 4.48528i − 0.363804i
\(153\) 7.65685i 0.619020i
\(154\) 2.34315 0.188816
\(155\) 0 0
\(156\) 1.82843 0.146391
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 4.68629i − 0.372821i
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −11.3137 −0.891645
\(162\) − 0.414214i − 0.0325437i
\(163\) 18.8284i 1.47476i 0.675480 + 0.737378i \(0.263936\pi\)
−0.675480 + 0.737378i \(0.736064\pi\)
\(164\) 9.45584 0.738377
\(165\) 0 0
\(166\) 1.51472 0.117565
\(167\) 3.65685i 0.282976i 0.989940 + 0.141488i \(0.0451886\pi\)
−0.989940 + 0.141488i \(0.954811\pi\)
\(168\) − 4.48528i − 0.346047i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) − 3.02944i − 0.230992i
\(173\) − 11.6569i − 0.886254i −0.896459 0.443127i \(-0.853869\pi\)
0.896459 0.443127i \(-0.146131\pi\)
\(174\) −0.828427 −0.0628029
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) − 7.65685i − 0.575524i
\(178\) 6.14214i 0.460373i
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.17157i 0.0868428i
\(183\) 13.3137i 0.984178i
\(184\) −6.34315 −0.467623
\(185\) 0 0
\(186\) −0.485281 −0.0355826
\(187\) 15.3137i 1.11985i
\(188\) 21.3137i 1.55446i
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) 4.17157i 0.301057i
\(193\) 5.31371i 0.382489i 0.981542 + 0.191245i \(0.0612524\pi\)
−0.981542 + 0.191245i \(0.938748\pi\)
\(194\) −1.51472 −0.108750
\(195\) 0 0
\(196\) −1.82843 −0.130602
\(197\) − 0.485281i − 0.0345749i −0.999851 0.0172874i \(-0.994497\pi\)
0.999851 0.0172874i \(-0.00550304\pi\)
\(198\) − 0.828427i − 0.0588738i
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 6.82843 0.481640
\(202\) − 3.17157i − 0.223151i
\(203\) 5.65685i 0.397033i
\(204\) 14.0000 0.980196
\(205\) 0 0
\(206\) 0.970563 0.0676223
\(207\) 4.00000i 0.278019i
\(208\) − 3.00000i − 0.208013i
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 3.65685i − 0.251154i
\(213\) 2.00000i 0.137038i
\(214\) 4.68629 0.320348
\(215\) 0 0
\(216\) −1.58579 −0.107899
\(217\) 3.31371i 0.224949i
\(218\) 2.20101i 0.149071i
\(219\) −0.343146 −0.0231876
\(220\) 0 0
\(221\) −7.65685 −0.515056
\(222\) 3.17157i 0.212862i
\(223\) − 12.4853i − 0.836076i −0.908429 0.418038i \(-0.862718\pi\)
0.908429 0.418038i \(-0.137282\pi\)
\(224\) −12.4853 −0.834208
\(225\) 0 0
\(226\) −2.20101 −0.146409
\(227\) 17.3137i 1.14915i 0.818452 + 0.574576i \(0.194833\pi\)
−0.818452 + 0.574576i \(0.805167\pi\)
\(228\) 5.17157i 0.342496i
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 0 0
\(231\) −5.65685 −0.372194
\(232\) 3.17157i 0.208224i
\(233\) 6.97056i 0.456657i 0.973584 + 0.228328i \(0.0733260\pi\)
−0.973584 + 0.228328i \(0.926674\pi\)
\(234\) 0.414214 0.0270780
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) 11.3137i 0.734904i
\(238\) 8.97056i 0.581475i
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 0 0
\(241\) 0.343146 0.0221040 0.0110520 0.999939i \(-0.496482\pi\)
0.0110520 + 0.999939i \(0.496482\pi\)
\(242\) 2.89949i 0.186387i
\(243\) 1.00000i 0.0641500i
\(244\) 24.3431 1.55841
\(245\) 0 0
\(246\) 2.14214 0.136578
\(247\) − 2.82843i − 0.179969i
\(248\) 1.85786i 0.117975i
\(249\) −3.65685 −0.231744
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 5.17157i 0.325778i
\(253\) 8.00000i 0.502956i
\(254\) −2.34315 −0.147022
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 4.34315i 0.270918i 0.990783 + 0.135459i \(0.0432509\pi\)
−0.990783 + 0.135459i \(0.956749\pi\)
\(258\) − 0.686292i − 0.0427266i
\(259\) 21.6569 1.34569
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 3.31371i 0.204722i
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) −3.17157 −0.195197
\(265\) 0 0
\(266\) −3.31371 −0.203177
\(267\) − 14.8284i − 0.907485i
\(268\) − 12.4853i − 0.762660i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) − 22.9706i − 1.39279i
\(273\) − 2.82843i − 0.171184i
\(274\) 4.48528 0.270966
\(275\) 0 0
\(276\) 7.31371 0.440234
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 3.02944i − 0.181694i
\(279\) 1.17157 0.0701402
\(280\) 0 0
\(281\) 21.1716 1.26299 0.631495 0.775380i \(-0.282442\pi\)
0.631495 + 0.775380i \(0.282442\pi\)
\(282\) 4.82843i 0.287529i
\(283\) 28.9706i 1.72212i 0.508502 + 0.861061i \(0.330199\pi\)
−0.508502 + 0.861061i \(0.669801\pi\)
\(284\) 3.65685 0.216994
\(285\) 0 0
\(286\) 0.828427 0.0489859
\(287\) − 14.6274i − 0.863429i
\(288\) 4.41421i 0.260110i
\(289\) −41.6274 −2.44867
\(290\) 0 0
\(291\) 3.65685 0.214369
\(292\) 0.627417i 0.0367168i
\(293\) 2.14214i 0.125145i 0.998040 + 0.0625724i \(0.0199304\pi\)
−0.998040 + 0.0625724i \(0.980070\pi\)
\(294\) −0.414214 −0.0241574
\(295\) 0 0
\(296\) 12.1421 0.705747
\(297\) 2.00000i 0.116052i
\(298\) − 3.79899i − 0.220070i
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −4.68629 −0.270113
\(302\) 1.45584i 0.0837744i
\(303\) 7.65685i 0.439875i
\(304\) 8.48528 0.486664
\(305\) 0 0
\(306\) 3.17157 0.181307
\(307\) 22.8284i 1.30289i 0.758697 + 0.651444i \(0.225836\pi\)
−0.758697 + 0.651444i \(0.774164\pi\)
\(308\) 10.3431i 0.589355i
\(309\) −2.34315 −0.133297
\(310\) 0 0
\(311\) −10.6274 −0.602626 −0.301313 0.953525i \(-0.597425\pi\)
−0.301313 + 0.953525i \(0.597425\pi\)
\(312\) − 1.58579i − 0.0897775i
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 4.14214 0.233754
\(315\) 0 0
\(316\) 20.6863 1.16369
\(317\) − 8.48528i − 0.476581i −0.971194 0.238290i \(-0.923413\pi\)
0.971194 0.238290i \(-0.0765870\pi\)
\(318\) − 0.828427i − 0.0464559i
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −11.3137 −0.631470
\(322\) 4.68629i 0.261157i
\(323\) − 21.6569i − 1.20502i
\(324\) 1.82843 0.101579
\(325\) 0 0
\(326\) 7.79899 0.431946
\(327\) − 5.31371i − 0.293849i
\(328\) − 8.20101i − 0.452825i
\(329\) 32.9706 1.81773
\(330\) 0 0
\(331\) 26.1421 1.43690 0.718451 0.695578i \(-0.244852\pi\)
0.718451 + 0.695578i \(0.244852\pi\)
\(332\) 6.68629i 0.366958i
\(333\) − 7.65685i − 0.419593i
\(334\) 1.51472 0.0828817
\(335\) 0 0
\(336\) 8.48528 0.462910
\(337\) − 9.31371i − 0.507350i −0.967290 0.253675i \(-0.918361\pi\)
0.967290 0.253675i \(-0.0816394\pi\)
\(338\) 0.414214i 0.0225302i
\(339\) 5.31371 0.288601
\(340\) 0 0
\(341\) 2.34315 0.126888
\(342\) 1.17157i 0.0633514i
\(343\) − 16.9706i − 0.916324i
\(344\) −2.62742 −0.141661
\(345\) 0 0
\(346\) −4.82843 −0.259578
\(347\) 8.68629i 0.466305i 0.972440 + 0.233152i \(0.0749041\pi\)
−0.972440 + 0.233152i \(0.925096\pi\)
\(348\) − 3.65685i − 0.196028i
\(349\) −3.65685 −0.195747 −0.0978735 0.995199i \(-0.531204\pi\)
−0.0978735 + 0.995199i \(0.531204\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 8.82843i 0.470557i
\(353\) − 33.4558i − 1.78067i −0.455301 0.890337i \(-0.650468\pi\)
0.455301 0.890337i \(-0.349532\pi\)
\(354\) −3.17157 −0.168567
\(355\) 0 0
\(356\) −27.1127 −1.43697
\(357\) − 21.6569i − 1.14620i
\(358\) − 9.65685i − 0.510381i
\(359\) −34.9706 −1.84568 −0.922838 0.385189i \(-0.874136\pi\)
−0.922838 + 0.385189i \(0.874136\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) − 5.79899i − 0.304788i
\(363\) − 7.00000i − 0.367405i
\(364\) −5.17157 −0.271064
\(365\) 0 0
\(366\) 5.51472 0.288259
\(367\) 24.0000i 1.25279i 0.779506 + 0.626395i \(0.215470\pi\)
−0.779506 + 0.626395i \(0.784530\pi\)
\(368\) − 12.0000i − 0.625543i
\(369\) −5.17157 −0.269221
\(370\) 0 0
\(371\) −5.65685 −0.293689
\(372\) − 2.14214i − 0.111065i
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 6.34315 0.327996
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) 2.00000i 0.103005i
\(378\) 1.17157i 0.0602592i
\(379\) −0.485281 −0.0249272 −0.0124636 0.999922i \(-0.503967\pi\)
−0.0124636 + 0.999922i \(0.503967\pi\)
\(380\) 0 0
\(381\) 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 \(0.709425\pi\)
\(384\) 10.5563 0.538701
\(385\) 0 0
\(386\) 2.20101 0.112028
\(387\) 1.65685i 0.0842226i
\(388\) − 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
\(393\) − 8.00000i − 0.403547i
\(394\) −0.201010 −0.0101267
\(395\) 0 0
\(396\) 3.65685 0.183764
\(397\) − 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
\(400\) 0 0
\(401\) 26.1421 1.30548 0.652738 0.757584i \(-0.273620\pi\)
0.652738 + 0.757584i \(0.273620\pi\)
\(402\) − 2.82843i − 0.141069i
\(403\) 1.17157i 0.0583602i
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 2.34315 0.116288
\(407\) − 15.3137i − 0.759072i
\(408\) − 12.1421i − 0.601125i
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) 0 0
\(411\) −10.8284 −0.534127
\(412\) 4.28427i 0.211071i
\(413\) 21.6569i 1.06566i
\(414\) 1.65685 0.0814299
\(415\) 0 0
\(416\) −4.41421 −0.216425
\(417\) 7.31371i 0.358154i
\(418\) 2.34315i 0.114607i
\(419\) −14.6274 −0.714596 −0.357298 0.933990i \(-0.616302\pi\)
−0.357298 + 0.933990i \(0.616302\pi\)
\(420\) 0 0
\(421\) 37.3137 1.81856 0.909279 0.416186i \(-0.136634\pi\)
0.909279 + 0.416186i \(0.136634\pi\)
\(422\) 4.97056i 0.241963i
\(423\) − 11.6569i − 0.566776i
\(424\) −3.17157 −0.154025
\(425\) 0 0
\(426\) 0.828427 0.0401374
\(427\) − 37.6569i − 1.82234i
\(428\) 20.6863i 0.999910i
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 8.34315 0.401875 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(432\) − 3.00000i − 0.144338i
\(433\) − 21.3137i − 1.02427i −0.858905 0.512136i \(-0.828854\pi\)
0.858905 0.512136i \(-0.171146\pi\)
\(434\) 1.37258 0.0658861
\(435\) 0 0
\(436\) −9.71573 −0.465299
\(437\) − 11.3137i − 0.541208i
\(438\) 0.142136i 0.00679150i
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 3.17157i 0.150856i
\(443\) − 25.9411i − 1.23250i −0.787551 0.616250i \(-0.788651\pi\)
0.787551 0.616250i \(-0.211349\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) −5.17157 −0.244881
\(447\) 9.17157i 0.433801i
\(448\) − 11.7990i − 0.557450i
\(449\) 31.7990 1.50069 0.750344 0.661048i \(-0.229888\pi\)
0.750344 + 0.661048i \(0.229888\pi\)
\(450\) 0 0
\(451\) −10.3431 −0.487040
\(452\) − 9.71573i − 0.456989i
\(453\) − 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 \(0.0573141\pi\)
−0.983833 + 0.179086i \(0.942686\pi\)
\(458\) − 0.544156i − 0.0254267i
\(459\) −7.65685 −0.357391
\(460\) 0 0
\(461\) 5.17157 0.240864 0.120432 0.992722i \(-0.461572\pi\)
0.120432 + 0.992722i \(0.461572\pi\)
\(462\) 2.34315i 0.109013i
\(463\) − 24.4853i − 1.13793i −0.822363 0.568964i \(-0.807344\pi\)
0.822363 0.568964i \(-0.192656\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 2.88730 0.133752
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 1.82843i 0.0845191i
\(469\) −19.3137 −0.891824
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 12.1421i 0.558887i
\(473\) 3.31371i 0.152364i
\(474\) 4.68629 0.215248
\(475\) 0 0
\(476\) −39.5980 −1.81497
\(477\) 2.00000i 0.0915737i
\(478\) 0.828427i 0.0378914i
\(479\) −25.3137 −1.15661 −0.578306 0.815820i \(-0.696286\pi\)
−0.578306 + 0.815820i \(0.696286\pi\)
\(480\) 0 0
\(481\) 7.65685 0.349123
\(482\) − 0.142136i − 0.00647410i
\(483\) − 11.3137i − 0.514792i
\(484\) −12.7990 −0.581772
\(485\) 0 0
\(486\) 0.414214 0.0187891
\(487\) 7.79899i 0.353406i 0.984264 + 0.176703i \(0.0565432\pi\)
−0.984264 + 0.176703i \(0.943457\pi\)
\(488\) − 21.1127i − 0.955727i
\(489\) −18.8284 −0.851451
\(490\) 0 0
\(491\) 30.6274 1.38220 0.691098 0.722761i \(-0.257127\pi\)
0.691098 + 0.722761i \(0.257127\pi\)
\(492\) 9.45584i 0.426302i
\(493\) 15.3137i 0.689695i
\(494\) −1.17157 −0.0527116
\(495\) 0 0
\(496\) −3.51472 −0.157816
\(497\) − 5.65685i − 0.253745i
\(498\) 1.51472i 0.0678762i
\(499\) −26.1421 −1.17028 −0.585141 0.810931i \(-0.698961\pi\)
−0.585141 + 0.810931i \(0.698961\pi\)
\(500\) 0 0
\(501\) −3.65685 −0.163376
\(502\) 0 0
\(503\) − 7.31371i − 0.326102i −0.986618 0.163051i \(-0.947866\pi\)
0.986618 0.163051i \(-0.0521335\pi\)
\(504\) 4.48528 0.199790
\(505\) 0 0
\(506\) 3.31371 0.147312
\(507\) − 1.00000i − 0.0444116i
\(508\) − 10.3431i − 0.458903i
\(509\) −11.7990 −0.522981 −0.261491 0.965206i \(-0.584214\pi\)
−0.261491 + 0.965206i \(0.584214\pi\)
\(510\) 0 0
\(511\) 0.970563 0.0429352
\(512\) − 22.7574i − 1.00574i
\(513\) − 2.82843i − 0.124878i
\(514\) 1.79899 0.0793500
\(515\) 0 0
\(516\) 3.02944 0.133364
\(517\) − 23.3137i − 1.02534i
\(518\) − 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
\(523\) − 15.3137i − 0.669622i −0.942285 0.334811i \(-0.891328\pi\)
0.942285 0.334811i \(-0.108672\pi\)
\(524\) −14.6274 −0.639002
\(525\) 0 0
\(526\) 4.97056 0.216727
\(527\) 8.97056i 0.390764i
\(528\) − 6.00000i − 0.261116i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 7.65685 0.332279
\(532\) − 14.6274i − 0.634179i
\(533\) − 5.17157i − 0.224006i
\(534\) −6.14214 −0.265796
\(535\) 0 0
\(536\) −10.8284 −0.467717
\(537\) 23.3137i 1.00606i
\(538\) 7.45584i 0.321444i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 11.5147i 0.494600i
\(543\) 14.0000i 0.600798i
\(544\) −33.7990 −1.44912
\(545\) 0 0
\(546\) −1.17157 −0.0501387
\(547\) − 23.3137i − 0.996822i −0.866941 0.498411i \(-0.833917\pi\)
0.866941 0.498411i \(-0.166083\pi\)
\(548\) 19.7990i 0.845771i
\(549\) −13.3137 −0.568215
\(550\) 0 0
\(551\) −5.65685 −0.240990
\(552\) − 6.34315i − 0.269982i
\(553\) − 32.0000i − 1.36078i
\(554\) 0.828427 0.0351965
\(555\) 0 0
\(556\) 13.3726 0.567124
\(557\) 7.79899i 0.330454i 0.986256 + 0.165227i \(0.0528356\pi\)
−0.986256 + 0.165227i \(0.947164\pi\)
\(558\) − 0.485281i − 0.0205436i
\(559\) −1.65685 −0.0700775
\(560\) 0 0
\(561\) −15.3137 −0.646545
\(562\) − 8.76955i − 0.369921i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) −21.3137 −0.897469
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) − 2.82843i − 0.118783i
\(568\) − 3.17157i − 0.133076i
\(569\) 42.9706 1.80142 0.900710 0.434421i \(-0.143047\pi\)
0.900710 + 0.434421i \(0.143047\pi\)
\(570\) 0 0
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 3.65685i 0.152901i
\(573\) 3.31371i 0.138432i
\(574\) −6.05887 −0.252893
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) 31.9411i 1.32973i 0.746965 + 0.664863i \(0.231510\pi\)
−0.746965 + 0.664863i \(0.768490\pi\)
\(578\) 17.2426i 0.717199i
\(579\) −5.31371 −0.220830
\(580\) 0 0
\(581\) 10.3431 0.429106
\(582\) − 1.51472i − 0.0627871i
\(583\) 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 \(0.0726962\pi\)
−0.974034 + 0.226402i \(0.927304\pi\)
\(588\) − 1.82843i − 0.0754031i
\(589\) −3.31371 −0.136539
\(590\) 0 0
\(591\) 0.485281 0.0199618
\(592\) 22.9706i 0.944084i
\(593\) − 20.4853i − 0.841230i −0.907239 0.420615i \(-0.861814\pi\)
0.907239 0.420615i \(-0.138186\pi\)
\(594\) 0.828427 0.0339908
\(595\) 0 0
\(596\) 16.7696 0.686908
\(597\) − 21.6569i − 0.886356i
\(598\) 1.65685i 0.0677538i
\(599\) 23.3137 0.952572 0.476286 0.879290i \(-0.341983\pi\)
0.476286 + 0.879290i \(0.341983\pi\)
\(600\) 0 0
\(601\) −0.627417 −0.0255929 −0.0127964 0.999918i \(-0.504073\pi\)
−0.0127964 + 0.999918i \(0.504073\pi\)
\(602\) 1.94113i 0.0791144i
\(603\) 6.82843i 0.278075i
\(604\) −6.42641 −0.261487
\(605\) 0 0
\(606\) 3.17157 0.128836
\(607\) − 41.9411i − 1.70234i −0.524892 0.851169i \(-0.675894\pi\)
0.524892 0.851169i \(-0.324106\pi\)
\(608\) − 12.4853i − 0.506345i
\(609\) −5.65685 −0.229227
\(610\) 0 0
\(611\) 11.6569 0.471586
\(612\) 14.0000i 0.565916i
\(613\) − 47.6569i − 1.92484i −0.271561 0.962421i \(-0.587540\pi\)
0.271561 0.962421i \(-0.412460\pi\)
\(614\) 9.45584 0.381607
\(615\) 0 0
\(616\) 8.97056 0.361434
\(617\) 34.8284i 1.40214i 0.713093 + 0.701070i \(0.247294\pi\)
−0.713093 + 0.701070i \(0.752706\pi\)
\(618\) 0.970563i 0.0390418i
\(619\) −23.7990 −0.956562 −0.478281 0.878207i \(-0.658740\pi\)
−0.478281 + 0.878207i \(0.658740\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 4.40202i 0.176505i
\(623\) 41.9411i 1.68034i
\(624\) 3.00000 0.120096
\(625\) 0 0
\(626\) 2.48528 0.0993318
\(627\) − 5.65685i − 0.225913i
\(628\) 18.2843i 0.729622i
\(629\) 58.6274 2.33763
\(630\) 0 0
\(631\) 43.1127 1.71629 0.858145 0.513408i \(-0.171617\pi\)
0.858145 + 0.513408i \(0.171617\pi\)
\(632\) − 17.9411i − 0.713660i
\(633\) − 12.0000i − 0.476957i
\(634\) −3.51472 −0.139587
\(635\) 0 0
\(636\) 3.65685 0.145004
\(637\) 1.00000i 0.0396214i
\(638\) − 1.65685i − 0.0655955i
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 30.2843 1.19616 0.598078 0.801438i \(-0.295931\pi\)
0.598078 + 0.801438i \(0.295931\pi\)
\(642\) 4.68629i 0.184953i
\(643\) 22.8284i 0.900265i 0.892962 + 0.450133i \(0.148623\pi\)
−0.892962 + 0.450133i \(0.851377\pi\)
\(644\) −20.6863 −0.815154
\(645\) 0 0
\(646\) −8.97056 −0.352942
\(647\) − 11.3137i − 0.444788i −0.974957 0.222394i \(-0.928613\pi\)
0.974957 0.222394i \(-0.0713871\pi\)
\(648\) − 1.58579i − 0.0622956i
\(649\) 15.3137 0.601116
\(650\) 0 0
\(651\) −3.31371 −0.129874
\(652\) 34.4264i 1.34824i
\(653\) − 25.3137i − 0.990602i −0.868721 0.495301i \(-0.835058\pi\)
0.868721 0.495301i \(-0.164942\pi\)
\(654\) −2.20101 −0.0860663
\(655\) 0 0
\(656\) 15.5147 0.605748
\(657\) − 0.343146i − 0.0133874i
\(658\) − 13.6569i − 0.532400i
\(659\) 47.3137 1.84308 0.921540 0.388283i \(-0.126932\pi\)
0.921540 + 0.388283i \(0.126932\pi\)
\(660\) 0 0
\(661\) −34.9706 −1.36020 −0.680099 0.733121i \(-0.738063\pi\)
−0.680099 + 0.733121i \(0.738063\pi\)
\(662\) − 10.8284i − 0.420859i
\(663\) − 7.65685i − 0.297368i
\(664\) 5.79899 0.225044
\(665\) 0 0
\(666\) −3.17157 −0.122896
\(667\) 8.00000i 0.309761i
\(668\) 6.68629i 0.258700i
\(669\) 12.4853 0.482709
\(670\) 0 0
\(671\) −26.6274 −1.02794
\(672\) − 12.4853i − 0.481630i
\(673\) 16.6274i 0.640940i 0.947259 + 0.320470i \(0.103841\pi\)
−0.947259 + 0.320470i \(0.896159\pi\)
\(674\) −3.85786 −0.148599
\(675\) 0 0
\(676\) −1.82843 −0.0703241
\(677\) − 26.6863i − 1.02564i −0.858497 0.512819i \(-0.828601\pi\)
0.858497 0.512819i \(-0.171399\pi\)
\(678\) − 2.20101i − 0.0845293i
\(679\) −10.3431 −0.396934
\(680\) 0 0
\(681\) −17.3137 −0.663463
\(682\) − 0.970563i − 0.0371648i
\(683\) 47.9411i 1.83442i 0.398408 + 0.917208i \(0.369563\pi\)
−0.398408 + 0.917208i \(0.630437\pi\)
\(684\) −5.17157 −0.197740
\(685\) 0 0
\(686\) −7.02944 −0.268385
\(687\) 1.31371i 0.0501211i
\(688\) − 4.97056i − 0.189501i
\(689\) −2.00000 −0.0761939
\(690\) 0 0
\(691\) −5.85786 −0.222844 −0.111422 0.993773i \(-0.535540\pi\)
−0.111422 + 0.993773i \(0.535540\pi\)
\(692\) − 21.3137i − 0.810226i
\(693\) − 5.65685i − 0.214886i
\(694\) 3.59798 0.136577
\(695\) 0 0
\(696\) −3.17157 −0.120218
\(697\) − 39.5980i − 1.49988i
\(698\) 1.51472i 0.0573329i
\(699\) −6.97056 −0.263651
\(700\) 0 0
\(701\) 5.02944 0.189959 0.0949796 0.995479i \(-0.469721\pi\)
0.0949796 + 0.995479i \(0.469721\pi\)
\(702\) 0.414214i 0.0156335i
\(703\) 21.6569i 0.816804i
\(704\) −8.34315 −0.314444
\(705\) 0 0
\(706\) −13.8579 −0.521548
\(707\) − 21.6569i − 0.814490i
\(708\) − 14.0000i − 0.526152i
\(709\) 4.62742 0.173786 0.0868931 0.996218i \(-0.472306\pi\)
0.0868931 + 0.996218i \(0.472306\pi\)
\(710\) 0 0
\(711\) −11.3137 −0.424297
\(712\) 23.5147i 0.881251i
\(713\) 4.68629i 0.175503i
\(714\) −8.97056 −0.335715
\(715\) 0 0
\(716\) 42.6274 1.59306
\(717\) − 2.00000i − 0.0746914i
\(718\) 14.4853i 0.540586i
\(719\) 29.9411 1.11662 0.558308 0.829634i \(-0.311451\pi\)
0.558308 + 0.829634i \(0.311451\pi\)
\(720\) 0 0
\(721\) 6.62742 0.246818
\(722\) 4.55635i 0.169570i
\(723\) 0.343146i 0.0127617i
\(724\) 25.5980 0.951341
\(725\) 0 0
\(726\) −2.89949 −0.107610
\(727\) 10.3431i 0.383606i 0.981433 + 0.191803i \(0.0614335\pi\)
−0.981433 + 0.191803i \(0.938567\pi\)
\(728\) 4.48528i 0.166236i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −12.6863 −0.469219
\(732\) 24.3431i 0.899749i
\(733\) − 36.6274i − 1.35286i −0.736505 0.676432i \(-0.763525\pi\)
0.736505 0.676432i \(-0.236475\pi\)
\(734\) 9.94113 0.366934
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) 13.6569i 0.503057i
\(738\) 2.14214i 0.0788531i
\(739\) 18.1421 0.667369 0.333685 0.942685i \(-0.391708\pi\)
0.333685 + 0.942685i \(0.391708\pi\)
\(740\) 0 0
\(741\) 2.82843 0.103905
\(742\) 2.34315i 0.0860196i
\(743\) 2.00000i 0.0733729i 0.999327 + 0.0366864i \(0.0116803\pi\)
−0.999327 + 0.0366864i \(0.988320\pi\)
\(744\) −1.85786 −0.0681126
\(745\) 0 0
\(746\) 4.14214 0.151654
\(747\) − 3.65685i − 0.133797i
\(748\) 28.0000i 1.02378i
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −0.970563 −0.0354163 −0.0177082 0.999843i \(-0.505637\pi\)
−0.0177082 + 0.999843i \(0.505637\pi\)
\(752\) 34.9706i 1.27525i
\(753\) 0 0
\(754\) 0.828427 0.0301695
\(755\) 0 0
\(756\) −5.17157 −0.188088
\(757\) − 51.9411i − 1.88783i −0.330185 0.943916i \(-0.607111\pi\)
0.330185 0.943916i \(-0.392889\pi\)
\(758\) 0.201010i 0.00730102i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 32.4853 1.17759 0.588795 0.808282i \(-0.299602\pi\)
0.588795 + 0.808282i \(0.299602\pi\)
\(762\) − 2.34315i − 0.0848832i
\(763\) 15.0294i 0.544102i
\(764\) 6.05887 0.219202
\(765\) 0 0
\(766\) 12.8284 0.463510
\(767\) 7.65685i 0.276473i
\(768\) 3.97056i 0.143275i
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) −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 \(0.789558\pi\)
\(774\) 0.686292 0.0246682
\(775\) 0 0
\(776\) −5.79899 −0.208172
\(777\) 21.6569i 0.776935i
\(778\) − 11.1716i − 0.400520i
\(779\) 14.6274 0.524082
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 12.6863i 0.453661i
\(783\) 2.00000i 0.0714742i
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −3.31371 −0.118196
\(787\) 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
\(790\) 0 0
\(791\) −15.0294 −0.534385
\(792\) − 3.17157i − 0.112697i
\(793\) − 13.3137i − 0.472784i
\(794\) −12.8284 −0.455264
\(795\) 0 0
\(796\) −39.5980 −1.40351
\(797\) − 24.3431i − 0.862278i −0.902285 0.431139i \(-0.858112\pi\)
0.902285 0.431139i \(-0.141888\pi\)
\(798\) − 3.31371i − 0.117304i
\(799\) 89.2548 3.15761
\(800\) 0 0
\(801\) 14.8284 0.523937
\(802\) − 10.8284i − 0.382365i
\(803\) − 0.686292i − 0.0242187i
\(804\) 12.4853 0.440322
\(805\) 0 0
\(806\) 0.485281 0.0170933
\(807\) − 18.0000i − 0.633630i
\(808\) − 12.1421i − 0.427159i
\(809\) −18.6863 −0.656975 −0.328488 0.944508i \(-0.606539\pi\)
−0.328488 + 0.944508i \(0.606539\pi\)
\(810\) 0 0
\(811\) −30.1421 −1.05843 −0.529217 0.848487i \(-0.677514\pi\)
−0.529217 + 0.848487i \(0.677514\pi\)
\(812\) 10.3431i 0.362973i
\(813\) − 27.7990i − 0.974953i
\(814\) −6.34315 −0.222327
\(815\) 0 0
\(816\) 22.9706 0.804131
\(817\) − 4.68629i − 0.163953i
\(818\) − 14.4853i − 0.506466i
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) −23.7990 −0.830590 −0.415295 0.909687i \(-0.636322\pi\)
−0.415295 + 0.909687i \(0.636322\pi\)
\(822\) 4.48528i 0.156442i
\(823\) 15.0294i 0.523893i 0.965082 + 0.261947i \(0.0843644\pi\)
−0.965082 + 0.261947i \(0.915636\pi\)
\(824\) 3.71573 0.129444
\(825\) 0 0
\(826\) 8.97056 0.312126
\(827\) 26.0000i 0.904109i 0.891990 + 0.452054i \(0.149309\pi\)
−0.891990 + 0.452054i \(0.850691\pi\)
\(828\) 7.31371i 0.254169i
\(829\) 17.3137 0.601330 0.300665 0.953730i \(-0.402791\pi\)
0.300665 + 0.953730i \(0.402791\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) − 4.17157i − 0.144623i
\(833\) 7.65685i 0.265294i
\(834\) 3.02944 0.104901
\(835\) 0 0
\(836\) −10.3431 −0.357725
\(837\) 1.17157i 0.0404955i
\(838\) 6.05887i 0.209300i
\(839\) −43.2548 −1.49332 −0.746661 0.665204i \(-0.768344\pi\)
−0.746661 + 0.665204i \(0.768344\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 15.4558i − 0.532644i
\(843\) 21.1716i 0.729188i
\(844\) −21.9411 −0.755245
\(845\) 0 0
\(846\) −4.82843 −0.166005
\(847\) 19.7990i 0.680301i
\(848\) − 6.00000i − 0.206041i
\(849\) −28.9706 −0.994267
\(850\) 0 0
\(851\) 30.6274 1.04989
\(852\) 3.65685i 0.125282i
\(853\) 3.65685i 0.125208i 0.998038 + 0.0626042i \(0.0199406\pi\)
−0.998038 + 0.0626042i \(0.980059\pi\)
\(854\) −15.5980 −0.533752
\(855\) 0 0
\(856\) 17.9411 0.613215
\(857\) 49.5980i 1.69423i 0.531406 + 0.847117i \(0.321664\pi\)
−0.531406 + 0.847117i \(0.678336\pi\)
\(858\) 0.828427i 0.0282820i
\(859\) 0.686292 0.0234160 0.0117080 0.999931i \(-0.496273\pi\)
0.0117080 + 0.999931i \(0.496273\pi\)
\(860\) 0 0
\(861\) 14.6274 0.498501
\(862\) − 3.45584i − 0.117707i
\(863\) − 28.3431i − 0.964812i −0.875948 0.482406i \(-0.839763\pi\)
0.875948 0.482406i \(-0.160237\pi\)
\(864\) −4.41421 −0.150175
\(865\) 0 0
\(866\) −8.82843 −0.300002
\(867\) − 41.6274i − 1.41374i
\(868\) 6.05887i 0.205652i
\(869\) −22.6274 −0.767583
\(870\) 0 0
\(871\) −6.82843 −0.231372
\(872\) 8.42641i 0.285354i
\(873\) 3.65685i 0.123766i
\(874\) −4.68629 −0.158516
\(875\) 0 0
\(876\) −0.627417 −0.0211985
\(877\) − 42.2843i − 1.42784i −0.700228 0.713919i \(-0.746918\pi\)
0.700228 0.713919i \(-0.253082\pi\)
\(878\) 7.02944i 0.237232i
\(879\) −2.14214 −0.0722524
\(880\) 0 0
\(881\) −25.5980 −0.862418 −0.431209 0.902252i \(-0.641913\pi\)
−0.431209 + 0.902252i \(0.641913\pi\)
\(882\) − 0.414214i − 0.0139473i
\(883\) 27.5980i 0.928746i 0.885640 + 0.464373i \(0.153720\pi\)
−0.885640 + 0.464373i \(0.846280\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) −10.7452 −0.360991
\(887\) 8.00000i 0.268614i 0.990940 + 0.134307i \(0.0428808\pi\)
−0.990940 + 0.134307i \(0.957119\pi\)
\(888\) 12.1421i 0.407463i
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) − 22.8284i − 0.764352i
\(893\) 32.9706i 1.10332i
\(894\) 3.79899 0.127057
\(895\) 0 0
\(896\) −29.8579 −0.997481
\(897\) − 4.00000i − 0.133556i
\(898\) − 13.1716i − 0.439541i
\(899\) 2.34315 0.0781483
\(900\) 0 0
\(901\) −15.3137 −0.510174
\(902\) 4.28427i 0.142651i
\(903\) − 4.68629i − 0.155950i
\(904\) −8.42641 −0.280258
\(905\) 0 0
\(906\) −1.45584 −0.0483672
\(907\) 12.9706i 0.430680i 0.976539 + 0.215340i \(0.0690860\pi\)
−0.976539 + 0.215340i \(0.930914\pi\)
\(908\) 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
\(913\) − 7.31371i − 0.242048i
\(914\) 3.17157 0.104906
\(915\) 0 0
\(916\) 2.40202 0.0793650
\(917\) 22.6274i 0.747223i
\(918\) 3.17157i 0.104678i
\(919\) −3.31371 −0.109309 −0.0546546 0.998505i \(-0.517406\pi\)
−0.0546546 + 0.998505i \(0.517406\pi\)
\(920\) 0 0
\(921\) −22.8284 −0.752222
\(922\) − 2.14214i − 0.0705475i
\(923\) − 2.00000i − 0.0658308i
\(924\) −10.3431 −0.340265
\(925\) 0 0
\(926\) −10.1421 −0.333291
\(927\) − 2.34315i − 0.0769590i
\(928\) 8.82843i 0.289807i
\(929\) −11.7990 −0.387112 −0.193556 0.981089i \(-0.562002\pi\)
−0.193556 + 0.981089i \(0.562002\pi\)
\(930\) 0 0
\(931\) −2.82843 −0.0926980
\(932\) 12.7452i 0.417482i
\(933\) − 10.6274i − 0.347926i
\(934\) 3.31371 0.108428
\(935\) 0 0
\(936\) 1.58579 0.0518331
\(937\) 21.3137i 0.696289i 0.937441 + 0.348144i \(0.113188\pi\)
−0.937441 + 0.348144i \(0.886812\pi\)
\(938\) 8.00000i 0.261209i
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 34.1421 1.11300 0.556501 0.830847i \(-0.312144\pi\)
0.556501 + 0.830847i \(0.312144\pi\)
\(942\) 4.14214i 0.134958i
\(943\) − 20.6863i − 0.673638i
\(944\) −22.9706 −0.747628
\(945\) 0 0
\(946\) 1.37258 0.0446265
\(947\) 21.0294i 0.683365i 0.939815 + 0.341682i \(0.110997\pi\)
−0.939815 + 0.341682i \(0.889003\pi\)
\(948\) 20.6863i 0.671860i
\(949\) 0.343146 0.0111390
\(950\) 0 0
\(951\) 8.48528 0.275154
\(952\) 34.3431i 1.11307i
\(953\) 40.3431i 1.30684i 0.756994 + 0.653421i \(0.226667\pi\)
−0.756994 + 0.653421i \(0.773333\pi\)
\(954\) 0.828427 0.0268213
\(955\) 0 0
\(956\) −3.65685 −0.118271
\(957\) 4.00000i 0.129302i
\(958\) 10.4853i 0.338764i
\(959\) 30.6274 0.989011
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) − 3.17157i − 0.102256i
\(963\) − 11.3137i − 0.364579i
\(964\) 0.627417 0.0202077
\(965\) 0 0
\(966\) −4.68629 −0.150779
\(967\) − 18.1421i − 0.583412i −0.956508 0.291706i \(-0.905777\pi\)
0.956508 0.291706i \(-0.0942228\pi\)
\(968\) 11.1005i 0.356784i
\(969\) 21.6569 0.695718
\(970\) 0 0
\(971\) −15.3137 −0.491440 −0.245720 0.969341i \(-0.579024\pi\)
−0.245720 + 0.969341i \(0.579024\pi\)
\(972\) 1.82843i 0.0586468i
\(973\) − 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 \(-0.764522\pi\)
0.738619 0.674123i \(-0.235478\pi\)
\(978\) 7.79899i 0.249384i
\(979\) 29.6569 0.947837
\(980\) 0 0
\(981\) 5.31371 0.169654
\(982\) − 12.6863i − 0.404836i
\(983\) 25.3137i 0.807382i 0.914895 + 0.403691i \(0.132273\pi\)
−0.914895 + 0.403691i \(0.867727\pi\)
\(984\) 8.20101 0.261439
\(985\) 0 0
\(986\) 6.34315 0.202007
\(987\) 32.9706i 1.04946i
\(988\) − 5.17157i − 0.164530i
\(989\) −6.62742 −0.210740
\(990\) 0 0
\(991\) 4.68629 0.148865 0.0744325 0.997226i \(-0.476285\pi\)
0.0744325 + 0.997226i \(0.476285\pi\)
\(992\) 5.17157i 0.164198i
\(993\) 26.1421i 0.829596i
\(994\) −2.34315 −0.0743201
\(995\) 0 0
\(996\) −6.68629 −0.211863
\(997\) 39.2548i 1.24321i 0.783330 + 0.621607i \(0.213520\pi\)
−0.783330 + 0.621607i \(0.786480\pi\)
\(998\) 10.8284i 0.342768i
\(999\) 7.65685 0.242252
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 975.2.c.h.274.2 4
3.2 odd 2 2925.2.c.u.2224.3 4
5.2 odd 4 39.2.a.b.1.2 2
5.3 odd 4 975.2.a.l.1.1 2
5.4 even 2 inner 975.2.c.h.274.3 4
15.2 even 4 117.2.a.c.1.1 2
15.8 even 4 2925.2.a.v.1.2 2
15.14 odd 2 2925.2.c.u.2224.2 4
20.7 even 4 624.2.a.k.1.1 2
35.27 even 4 1911.2.a.h.1.2 2
40.27 even 4 2496.2.a.bi.1.2 2
40.37 odd 4 2496.2.a.bf.1.2 2
45.2 even 12 1053.2.e.e.352.2 4
45.7 odd 12 1053.2.e.m.352.1 4
45.22 odd 12 1053.2.e.m.703.1 4
45.32 even 12 1053.2.e.e.703.2 4
55.32 even 4 4719.2.a.p.1.1 2
60.47 odd 4 1872.2.a.w.1.2 2
65.2 even 12 507.2.j.f.316.3 8
65.7 even 12 507.2.j.f.361.3 8
65.12 odd 4 507.2.a.h.1.1 2
65.17 odd 12 507.2.e.d.484.2 4
65.22 odd 12 507.2.e.h.484.1 4
65.32 even 12 507.2.j.f.361.2 8
65.37 even 12 507.2.j.f.316.2 8
65.42 odd 12 507.2.e.h.22.1 4
65.47 even 4 507.2.b.e.337.3 4
65.57 even 4 507.2.b.e.337.2 4
65.62 odd 12 507.2.e.d.22.2 4
105.62 odd 4 5733.2.a.u.1.1 2
120.77 even 4 7488.2.a.cl.1.1 2
120.107 odd 4 7488.2.a.co.1.1 2
195.47 odd 4 1521.2.b.j.1351.2 4
195.77 even 4 1521.2.a.f.1.2 2
195.122 odd 4 1521.2.b.j.1351.3 4
260.207 even 4 8112.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 5.2 odd 4
117.2.a.c.1.1 2 15.2 even 4
507.2.a.h.1.1 2 65.12 odd 4
507.2.b.e.337.2 4 65.57 even 4
507.2.b.e.337.3 4 65.47 even 4
507.2.e.d.22.2 4 65.62 odd 12
507.2.e.d.484.2 4 65.17 odd 12
507.2.e.h.22.1 4 65.42 odd 12
507.2.e.h.484.1 4 65.22 odd 12
507.2.j.f.316.2 8 65.37 even 12
507.2.j.f.316.3 8 65.2 even 12
507.2.j.f.361.2 8 65.32 even 12
507.2.j.f.361.3 8 65.7 even 12
624.2.a.k.1.1 2 20.7 even 4
975.2.a.l.1.1 2 5.3 odd 4
975.2.c.h.274.2 4 1.1 even 1 trivial
975.2.c.h.274.3 4 5.4 even 2 inner
1053.2.e.e.352.2 4 45.2 even 12
1053.2.e.e.703.2 4 45.32 even 12
1053.2.e.m.352.1 4 45.7 odd 12
1053.2.e.m.703.1 4 45.22 odd 12
1521.2.a.f.1.2 2 195.77 even 4
1521.2.b.j.1351.2 4 195.47 odd 4
1521.2.b.j.1351.3 4 195.122 odd 4
1872.2.a.w.1.2 2 60.47 odd 4
1911.2.a.h.1.2 2 35.27 even 4
2496.2.a.bf.1.2 2 40.37 odd 4
2496.2.a.bi.1.2 2 40.27 even 4
2925.2.a.v.1.2 2 15.8 even 4
2925.2.c.u.2224.2 4 15.14 odd 2
2925.2.c.u.2224.3 4 3.2 odd 2
4719.2.a.p.1.1 2 55.32 even 4
5733.2.a.u.1.1 2 105.62 odd 4
7488.2.a.cl.1.1 2 120.77 even 4
7488.2.a.co.1.1 2 120.107 odd 4
8112.2.a.bm.1.2 2 260.207 even 4