Properties

Label 5070.2.b.f.1351.1
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.f.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} +2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000i q^{11} +1.00000 q^{12} +2.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000i q^{18} -2.00000i q^{19} -1.00000i q^{20} -2.00000i q^{21} -4.00000 q^{22} -2.00000 q^{23} -1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -2.00000i q^{28} +8.00000 q^{29} -1.00000 q^{30} +4.00000i q^{31} -1.00000i q^{32} +4.00000i q^{33} +4.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -6.00000i q^{37} -2.00000 q^{38} -1.00000 q^{40} +10.0000i q^{41} -2.00000 q^{42} -4.00000 q^{43} +4.00000i q^{44} +1.00000i q^{45} +2.00000i q^{46} -1.00000 q^{48} +3.00000 q^{49} +1.00000i q^{50} +4.00000 q^{51} +6.00000 q^{53} +1.00000i q^{54} +4.00000 q^{55} -2.00000 q^{56} +2.00000i q^{57} -8.00000i q^{58} +12.0000i q^{59} +1.00000i q^{60} -2.00000 q^{61} +4.00000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -8.00000i q^{67} +4.00000 q^{68} +2.00000 q^{69} +2.00000i q^{70} +1.00000i q^{72} -6.00000 q^{74} +1.00000 q^{75} +2.00000i q^{76} +8.00000 q^{77} -8.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000i q^{83} +2.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -8.00000 q^{87} +4.00000 q^{88} +10.0000i q^{89} +1.00000 q^{90} +2.00000 q^{92} -4.00000i q^{93} +2.00000 q^{95} +1.00000i q^{96} -8.00000i q^{97} -3.00000i q^{98} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} + 2q^{9} + 2q^{10} + 2q^{12} + 4q^{14} + 2q^{16} - 8q^{17} - 8q^{22} - 4q^{23} - 2q^{25} - 2q^{27} + 16q^{29} - 2q^{30} - 4q^{35} - 2q^{36} - 4q^{38} - 2q^{40} - 4q^{42} - 8q^{43} - 2q^{48} + 6q^{49} + 8q^{51} + 12q^{53} + 8q^{55} - 4q^{56} - 4q^{61} + 8q^{62} - 2q^{64} + 8q^{66} + 8q^{68} + 4q^{69} - 12q^{74} + 2q^{75} + 16q^{77} - 16q^{79} + 2q^{81} + 20q^{82} - 16q^{87} + 8q^{88} + 2q^{90} + 4q^{92} + 4q^{95} + O(q^{100}) \)

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
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 2.00000i − 0.436436i
\(22\) −4.00000 −0.852803
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 2.00000i − 0.377964i
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) −1.00000 −0.182574
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) 4.00000i 0.685994i
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000i 0.603023i
\(45\) 1.00000i 0.149071i
\(46\) 2.00000i 0.294884i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) 1.00000i 0.141421i
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 4.00000 0.539360
\(56\) −2.00000 −0.267261
\(57\) 2.00000i 0.264906i
\(58\) − 8.00000i − 1.05045i
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 4.00000 0.485071
\(69\) 2.00000 0.240772
\(70\) 2.00000i 0.239046i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 2.00000i 0.229416i
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 2.00000i 0.218218i
\(85\) − 4.00000i − 0.433861i
\(86\) 4.00000i 0.431331i
\(87\) −8.00000 −0.857690
\(88\) 4.00000 0.426401
\(89\) 10.0000i 1.06000i 0.847998 + 0.529999i \(0.177808\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 1.00000i 0.102062i
\(97\) − 8.00000i − 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) − 4.00000i − 0.402015i
\(100\) 1.00000 0.100000
\(101\) −20.0000 −1.99007 −0.995037 0.0995037i \(-0.968274\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) − 6.00000i − 0.582772i
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) − 4.00000i − 0.381385i
\(111\) 6.00000i 0.569495i
\(112\) 2.00000i 0.188982i
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 2.00000 0.187317
\(115\) − 2.00000i − 0.186501i
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) − 8.00000i − 0.733359i
\(120\) 1.00000 0.0912871
\(121\) −5.00000 −0.454545
\(122\) 2.00000i 0.181071i
\(123\) − 10.0000i − 0.901670i
\(124\) − 4.00000i − 0.359211i
\(125\) − 1.00000i − 0.0894427i
\(126\) 2.00000 0.178174
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 4.00000 0.346844
\(134\) −8.00000 −0.691095
\(135\) − 1.00000i − 0.0860663i
\(136\) − 4.00000i − 0.342997i
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) − 2.00000i − 0.170251i
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.00000i 0.664364i
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 6.00000i 0.493197i
\(149\) − 22.0000i − 1.80231i −0.433497 0.901155i \(-0.642720\pi\)
0.433497 0.901155i \(-0.357280\pi\)
\(150\) − 1.00000i − 0.0816497i
\(151\) 4.00000i 0.325515i 0.986666 + 0.162758i \(0.0520389\pi\)
−0.986666 + 0.162758i \(0.947961\pi\)
\(152\) 2.00000 0.162221
\(153\) −4.00000 −0.323381
\(154\) − 8.00000i − 0.644658i
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) − 4.00000i − 0.315244i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) − 10.0000i − 0.780869i
\(165\) −4.00000 −0.311400
\(166\) −12.0000 −0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) − 2.00000i − 0.152944i
\(172\) 4.00000 0.304997
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 8.00000i 0.606478i
\(175\) − 2.00000i − 0.151186i
\(176\) − 4.00000i − 0.301511i
\(177\) − 12.0000i − 0.901975i
\(178\) 10.0000 0.749532
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) − 2.00000i − 0.147442i
\(185\) 6.00000 0.441129
\(186\) −4.00000 −0.293294
\(187\) 16.0000i 1.17004i
\(188\) 0 0
\(189\) − 2.00000i − 0.145479i
\(190\) − 2.00000i − 0.145095i
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) − 12.0000i − 0.863779i −0.901927 0.431889i \(-0.857847\pi\)
0.901927 0.431889i \(-0.142153\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 10.0000i 0.712470i 0.934396 + 0.356235i \(0.115940\pi\)
−0.934396 + 0.356235i \(0.884060\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) 8.00000i 0.564276i
\(202\) 20.0000i 1.40720i
\(203\) 16.0000i 1.12298i
\(204\) −4.00000 −0.280056
\(205\) −10.0000 −0.698430
\(206\) 4.00000i 0.278693i
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) − 2.00000i − 0.138013i
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) − 4.00000i − 0.273434i
\(215\) − 4.00000i − 0.272798i
\(216\) − 1.00000i − 0.0680414i
\(217\) −8.00000 −0.543075
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) − 22.0000i − 1.47323i −0.676313 0.736614i \(-0.736423\pi\)
0.676313 0.736614i \(-0.263577\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 16.0000i 1.06430i
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) − 28.0000i − 1.85029i −0.379611 0.925146i \(-0.623942\pi\)
0.379611 0.925146i \(-0.376058\pi\)
\(230\) −2.00000 −0.131876
\(231\) −8.00000 −0.526361
\(232\) 8.00000i 0.525226i
\(233\) −28.0000 −1.83434 −0.917170 0.398495i \(-0.869533\pi\)
−0.917170 + 0.398495i \(0.869533\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 12.0000i − 0.781133i
\(237\) 8.00000 0.519656
\(238\) −8.00000 −0.518563
\(239\) − 8.00000i − 0.517477i −0.965947 0.258738i \(-0.916693\pi\)
0.965947 0.258738i \(-0.0833068\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 22.0000i 1.41714i 0.705638 + 0.708572i \(0.250660\pi\)
−0.705638 + 0.708572i \(0.749340\pi\)
\(242\) 5.00000i 0.321412i
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000i 0.191663i
\(246\) −10.0000 −0.637577
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) 12.0000i 0.760469i
\(250\) −1.00000 −0.0632456
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 8.00000i 0.502956i
\(254\) 20.0000i 1.25491i
\(255\) 4.00000i 0.250490i
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) − 14.0000i − 0.864923i
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) −4.00000 −0.246183
\(265\) 6.00000i 0.368577i
\(266\) − 4.00000i − 0.245256i
\(267\) − 10.0000i − 0.611990i
\(268\) 8.00000i 0.488678i
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 28.0000i 1.70088i 0.526073 + 0.850439i \(0.323664\pi\)
−0.526073 + 0.850439i \(0.676336\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 4.00000i 0.241209i
\(276\) −2.00000 −0.120386
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 16.0000i 0.959616i
\(279\) 4.00000i 0.239474i
\(280\) − 2.00000i − 0.119523i
\(281\) 10.0000i 0.596550i 0.954480 + 0.298275i \(0.0964112\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) − 1.00000i − 0.0589256i
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) 8.00000i 0.468968i
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 3.00000i 0.174964i
\(295\) −12.0000 −0.698667
\(296\) 6.00000 0.348743
\(297\) 4.00000i 0.232104i
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) − 8.00000i − 0.461112i
\(302\) 4.00000 0.230174
\(303\) 20.0000 1.14897
\(304\) − 2.00000i − 0.114708i
\(305\) − 2.00000i − 0.114520i
\(306\) 4.00000i 0.228665i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −8.00000 −0.455842
\(309\) 4.00000 0.227552
\(310\) 4.00000i 0.227185i
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 10.0000i 0.564333i
\(315\) −2.00000 −0.112687
\(316\) 8.00000 0.450035
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 6.00000i 0.336463i
\(319\) − 32.0000i − 1.79166i
\(320\) − 1.00000i − 0.0559017i
\(321\) −4.00000 −0.223258
\(322\) −4.00000 −0.222911
\(323\) 8.00000i 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 4.00000i − 0.221201i
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) 4.00000i 0.220193i
\(331\) − 10.0000i − 0.549650i −0.961494 0.274825i \(-0.911380\pi\)
0.961494 0.274825i \(-0.0886199\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 6.00000i − 0.328798i
\(334\) 12.0000 0.656611
\(335\) 8.00000 0.437087
\(336\) − 2.00000i − 0.109109i
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 4.00000i 0.216930i
\(341\) 16.0000 0.866449
\(342\) −2.00000 −0.108148
\(343\) 20.0000i 1.07990i
\(344\) − 4.00000i − 0.215666i
\(345\) 2.00000i 0.107676i
\(346\) 14.0000i 0.752645i
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 8.00000 0.428845
\(349\) 28.0000i 1.49881i 0.662114 + 0.749403i \(0.269659\pi\)
−0.662114 + 0.749403i \(0.730341\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) − 10.0000i − 0.529999i
\(357\) 8.00000i 0.423405i
\(358\) 2.00000i 0.105703i
\(359\) − 24.0000i − 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 15.0000 0.789474
\(362\) − 22.0000i − 1.15629i
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) − 2.00000i − 0.104542i
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) −2.00000 −0.104257
\(369\) 10.0000i 0.520579i
\(370\) − 6.00000i − 0.311925i
\(371\) 12.0000i 0.623009i
\(372\) 4.00000i 0.207390i
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 16.0000 0.827340
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) − 14.0000i − 0.719132i −0.933120 0.359566i \(-0.882925\pi\)
0.933120 0.359566i \(-0.117075\pi\)
\(380\) −2.00000 −0.102598
\(381\) 20.0000 1.02463
\(382\) 8.00000i 0.409316i
\(383\) − 20.0000i − 1.02195i −0.859595 0.510976i \(-0.829284\pi\)
0.859595 0.510976i \(-0.170716\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 8.00000i 0.407718i
\(386\) −12.0000 −0.610784
\(387\) −4.00000 −0.203331
\(388\) 8.00000i 0.406138i
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 3.00000i 0.151523i
\(393\) −14.0000 −0.706207
\(394\) 10.0000 0.503793
\(395\) − 8.00000i − 0.402524i
\(396\) 4.00000i 0.201008i
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 24.0000i 1.20301i
\(399\) −4.00000 −0.200250
\(400\) −1.00000 −0.0500000
\(401\) 30.0000i 1.49813i 0.662497 + 0.749064i \(0.269497\pi\)
−0.662497 + 0.749064i \(0.730503\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) 20.0000 0.995037
\(405\) 1.00000i 0.0496904i
\(406\) 16.0000 0.794067
\(407\) −24.0000 −1.18964
\(408\) 4.00000i 0.198030i
\(409\) − 18.0000i − 0.890043i −0.895520 0.445021i \(-0.853196\pi\)
0.895520 0.445021i \(-0.146804\pi\)
\(410\) 10.0000i 0.493865i
\(411\) − 2.00000i − 0.0986527i
\(412\) 4.00000 0.197066
\(413\) −24.0000 −1.18096
\(414\) 2.00000i 0.0982946i
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 8.00000i 0.391293i
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) −2.00000 −0.0975900
\(421\) − 16.0000i − 0.779792i −0.920859 0.389896i \(-0.872511\pi\)
0.920859 0.389896i \(-0.127489\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) − 4.00000i − 0.193574i
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 16.0000i 0.770693i 0.922772 + 0.385346i \(0.125918\pi\)
−0.922772 + 0.385346i \(0.874082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 8.00000i 0.384012i
\(435\) − 8.00000i − 0.383571i
\(436\) − 4.00000i − 0.191565i
\(437\) 4.00000i 0.191346i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 4.00000i 0.190693i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) − 6.00000i − 0.284747i
\(445\) −10.0000 −0.474045
\(446\) −22.0000 −1.04173
\(447\) 22.0000i 1.04056i
\(448\) − 2.00000i − 0.0944911i
\(449\) 18.0000i 0.849473i 0.905317 + 0.424736i \(0.139633\pi\)
−0.905317 + 0.424736i \(0.860367\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 40.0000 1.88353
\(452\) 16.0000 0.752577
\(453\) − 4.00000i − 0.187936i
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) − 16.0000i − 0.748448i −0.927338 0.374224i \(-0.877909\pi\)
0.927338 0.374224i \(-0.122091\pi\)
\(458\) −28.0000 −1.30835
\(459\) 4.00000 0.186704
\(460\) 2.00000i 0.0932505i
\(461\) 30.0000i 1.39724i 0.715493 + 0.698620i \(0.246202\pi\)
−0.715493 + 0.698620i \(0.753798\pi\)
\(462\) 8.00000i 0.372194i
\(463\) − 2.00000i − 0.0929479i −0.998920 0.0464739i \(-0.985202\pi\)
0.998920 0.0464739i \(-0.0147984\pi\)
\(464\) 8.00000 0.371391
\(465\) 4.00000 0.185496
\(466\) 28.0000i 1.29707i
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) 16.0000i 0.735681i
\(474\) − 8.00000i − 0.367452i
\(475\) 2.00000i 0.0917663i
\(476\) 8.00000i 0.366679i
\(477\) 6.00000 0.274721
\(478\) −8.00000 −0.365911
\(479\) − 24.0000i − 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 22.0000 1.00207
\(483\) 4.00000i 0.182006i
\(484\) 5.00000 0.227273
\(485\) 8.00000 0.363261
\(486\) 1.00000i 0.0453609i
\(487\) 38.0000i 1.72194i 0.508652 + 0.860972i \(0.330144\pi\)
−0.508652 + 0.860972i \(0.669856\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 12.0000i 0.542659i
\(490\) 3.00000 0.135526
\(491\) −38.0000 −1.71492 −0.857458 0.514554i \(-0.827958\pi\)
−0.857458 + 0.514554i \(0.827958\pi\)
\(492\) 10.0000i 0.450835i
\(493\) −32.0000 −1.44121
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 4.00000i 0.179605i
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) − 14.0000i − 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 12.0000i − 0.536120i
\(502\) 10.0000i 0.446322i
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) −2.00000 −0.0890871
\(505\) − 20.0000i − 0.889988i
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 38.0000i 1.68432i 0.539227 + 0.842160i \(0.318716\pi\)
−0.539227 + 0.842160i \(0.681284\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 2.00000i 0.0883022i
\(514\) − 8.00000i − 0.352865i
\(515\) − 4.00000i − 0.176261i
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 12.0000i − 0.527250i
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) − 8.00000i − 0.350150i
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −14.0000 −0.611593
\(525\) 2.00000i 0.0872872i
\(526\) 18.0000i 0.784837i
\(527\) − 16.0000i − 0.696971i
\(528\) 4.00000i 0.174078i
\(529\) −19.0000 −0.826087
\(530\) 6.00000 0.260623
\(531\) 12.0000i 0.520756i
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) 4.00000i 0.172935i
\(536\) 8.00000 0.345547
\(537\) 2.00000 0.0863064
\(538\) 4.00000i 0.172452i
\(539\) − 12.0000i − 0.516877i
\(540\) 1.00000i 0.0430331i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 28.0000 1.20270
\(543\) −22.0000 −0.944110
\(544\) 4.00000i 0.171499i
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) − 2.00000i − 0.0854358i
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) − 16.0000i − 0.681623i
\(552\) 2.00000i 0.0851257i
\(553\) − 16.0000i − 0.680389i
\(554\) 22.0000i 0.934690i
\(555\) −6.00000 −0.254686
\(556\) 16.0000 0.678551
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) 4.00000 0.169334
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) − 16.0000i − 0.675521i
\(562\) 10.0000 0.421825
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) − 16.0000i − 0.673125i
\(566\) 28.0000i 1.17693i
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 2.00000i 0.0837708i
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 20.0000i 0.834784i
\(575\) 2.00000 0.0834058
\(576\) −1.00000 −0.0416667
\(577\) − 4.00000i − 0.166522i −0.996528 0.0832611i \(-0.973466\pi\)
0.996528 0.0832611i \(-0.0265335\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 12.0000i 0.498703i
\(580\) − 8.00000i − 0.332182i
\(581\) 24.0000 0.995688
\(582\) 8.00000 0.331611
\(583\) − 24.0000i − 0.993978i
\(584\) 0 0
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 3.00000 0.123718
\(589\) 8.00000 0.329634
\(590\) 12.0000i 0.494032i
\(591\) − 10.0000i − 0.411345i
\(592\) − 6.00000i − 0.246598i
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) 4.00000 0.164122
\(595\) 8.00000 0.327968
\(596\) 22.0000i 0.901155i
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) −8.00000 −0.326056
\(603\) − 8.00000i − 0.325785i
\(604\) − 4.00000i − 0.162758i
\(605\) − 5.00000i − 0.203279i
\(606\) − 20.0000i − 0.812444i
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −2.00000 −0.0811107
\(609\) − 16.0000i − 0.648353i
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 8.00000i 0.322329i
\(617\) 38.0000i 1.52982i 0.644136 + 0.764911i \(0.277217\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) − 26.0000i − 1.04503i −0.852631 0.522514i \(-0.824994\pi\)
0.852631 0.522514i \(-0.175006\pi\)
\(620\) 4.00000 0.160644
\(621\) 2.00000 0.0802572
\(622\) − 16.0000i − 0.641542i
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 10.0000i 0.399680i
\(627\) 8.00000 0.319489
\(628\) 10.0000 0.399043
\(629\) 24.0000i 0.956943i
\(630\) 2.00000i 0.0796819i
\(631\) − 40.0000i − 1.59237i −0.605050 0.796187i \(-0.706847\pi\)
0.605050 0.796187i \(-0.293153\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 4.00000 0.158986
\(634\) −2.00000 −0.0794301
\(635\) − 20.0000i − 0.793676i
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −32.0000 −1.26689
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 4.00000i 0.157867i
\(643\) − 40.0000i − 1.57745i −0.614749 0.788723i \(-0.710743\pi\)
0.614749 0.788723i \(-0.289257\pi\)
\(644\) 4.00000i 0.157622i
\(645\) 4.00000i 0.157500i
\(646\) 8.00000 0.314756
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 12.0000i 0.469956i
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) −4.00000 −0.156412
\(655\) 14.0000i 0.547025i
\(656\) 10.0000i 0.390434i
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 4.00000 0.155700
\(661\) 16.0000i 0.622328i 0.950356 + 0.311164i \(0.100719\pi\)
−0.950356 + 0.311164i \(0.899281\pi\)
\(662\) −10.0000 −0.388661
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 4.00000i 0.155113i
\(666\) −6.00000 −0.232495
\(667\) −16.0000 −0.619522
\(668\) − 12.0000i − 0.464294i
\(669\) 22.0000i 0.850569i
\(670\) − 8.00000i − 0.309067i
\(671\) 8.00000i 0.308837i
\(672\) −2.00000 −0.0771517
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) − 26.0000i − 1.00148i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) − 16.0000i − 0.614476i
\(679\) 16.0000 0.614024
\(680\) 4.00000 0.153393
\(681\) − 12.0000i − 0.459841i
\(682\) − 16.0000i − 0.612672i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 2.00000i 0.0764719i
\(685\) −2.00000 −0.0764161
\(686\) 20.0000 0.763604
\(687\) 28.0000i 1.06827i
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 2.00000 0.0761387
\(691\) − 18.0000i − 0.684752i −0.939563 0.342376i \(-0.888768\pi\)
0.939563 0.342376i \(-0.111232\pi\)
\(692\) 14.0000 0.532200
\(693\) 8.00000 0.303895
\(694\) − 16.0000i − 0.607352i
\(695\) − 16.0000i − 0.606915i
\(696\) − 8.00000i − 0.303239i
\(697\) − 40.0000i − 1.51511i
\(698\) 28.0000 1.05982
\(699\) 28.0000 1.05906
\(700\) 2.00000i 0.0755929i
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 4.00000i 0.150756i
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) − 40.0000i − 1.50435i
\(708\) 12.0000i 0.450988i
\(709\) − 4.00000i − 0.150223i −0.997175 0.0751116i \(-0.976069\pi\)
0.997175 0.0751116i \(-0.0239313\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −10.0000 −0.374766
\(713\) − 8.00000i − 0.299602i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) 8.00000i 0.298765i
\(718\) −24.0000 −0.895672
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) − 8.00000i − 0.297936i
\(722\) − 15.0000i − 0.558242i
\(723\) − 22.0000i − 0.818189i
\(724\) −22.0000 −0.817624
\(725\) −8.00000 −0.297113
\(726\) − 5.00000i − 0.185567i
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) −2.00000 −0.0739221
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) 4.00000i 0.147643i
\(735\) − 3.00000i − 0.110657i
\(736\) 2.00000i 0.0737210i
\(737\) −32.0000 −1.17874
\(738\) 10.0000 0.368105
\(739\) − 50.0000i − 1.83928i −0.392763 0.919640i \(-0.628481\pi\)
0.392763 0.919640i \(-0.371519\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) 4.00000 0.146647
\(745\) 22.0000 0.806018
\(746\) − 14.0000i − 0.512576i
\(747\) − 12.0000i − 0.439057i
\(748\) − 16.0000i − 0.585018i
\(749\) 8.00000i 0.292314i
\(750\) 1.00000 0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 10.0000 0.364420
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) 2.00000i 0.0727393i
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −14.0000 −0.508503
\(759\) − 8.00000i − 0.290382i
\(760\) 2.00000i 0.0725476i
\(761\) 46.0000i 1.66750i 0.552143 + 0.833749i \(0.313810\pi\)
−0.552143 + 0.833749i \(0.686190\pi\)
\(762\) − 20.0000i − 0.724524i
\(763\) −8.00000 −0.289619
\(764\) 8.00000 0.289430
\(765\) − 4.00000i − 0.144620i
\(766\) −20.0000 −0.722629
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 10.0000i 0.360609i 0.983611 + 0.180305i \(0.0577084\pi\)
−0.983611 + 0.180305i \(0.942292\pi\)
\(770\) 8.00000 0.288300
\(771\) −8.00000 −0.288113
\(772\) 12.0000i 0.431889i
\(773\) 26.0000i 0.935155i 0.883952 + 0.467578i \(0.154873\pi\)
−0.883952 + 0.467578i \(0.845127\pi\)
\(774\) 4.00000i 0.143777i
\(775\) − 4.00000i − 0.143684i
\(776\) 8.00000 0.287183
\(777\) −12.0000 −0.430498
\(778\) 20.0000i 0.717035i
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) − 8.00000i − 0.286079i
\(783\) −8.00000 −0.285897
\(784\) 3.00000 0.107143
\(785\) − 10.0000i − 0.356915i
\(786\) 14.0000i 0.499363i
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) − 10.0000i − 0.356235i
\(789\) 18.0000 0.640817
\(790\) −8.00000 −0.284627
\(791\) − 32.0000i − 1.13779i
\(792\) 4.00000 0.142134
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) − 6.00000i − 0.212798i
\(796\) 24.0000 0.850657
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 0 0
\(800\) 1.00000i 0.0353553i
\(801\) 10.0000i 0.353333i
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) − 8.00000i − 0.282138i
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) − 20.0000i − 0.703598i
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 1.00000 0.0351364
\(811\) − 34.0000i − 1.19390i −0.802278 0.596951i \(-0.796379\pi\)
0.802278 0.596951i \(-0.203621\pi\)
\(812\) − 16.0000i − 0.561490i
\(813\) − 28.0000i − 0.982003i
\(814\) 24.0000i 0.841200i
\(815\) 12.0000 0.420342
\(816\) 4.00000 0.140028
\(817\) 8.00000i 0.279885i
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 42.0000i 1.46581i 0.680331 + 0.732905i \(0.261836\pi\)
−0.680331 + 0.732905i \(0.738164\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) − 4.00000i − 0.139347i
\(825\) − 4.00000i − 0.139262i
\(826\) 24.0000i 0.835067i
\(827\) 44.0000i 1.53003i 0.644013 + 0.765015i \(0.277268\pi\)
−0.644013 + 0.765015i \(0.722732\pi\)
\(828\) 2.00000 0.0695048
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) − 12.0000i − 0.416526i
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) − 16.0000i − 0.554035i
\(835\) −12.0000 −0.415277
\(836\) 8.00000 0.276686
\(837\) − 4.00000i − 0.138260i
\(838\) − 14.0000i − 0.483622i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 2.00000i 0.0690066i
\(841\) 35.0000 1.20690
\(842\) −16.0000 −0.551396
\(843\) − 10.0000i − 0.344418i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.0000i − 0.343604i
\(848\) 6.00000 0.206041
\(849\) 28.0000 0.960958
\(850\) − 4.00000i − 0.137199i
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) −4.00000 −0.136877
\(855\) 2.00000 0.0683986
\(856\) 4.00000i 0.136717i
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 4.00000i 0.136399i
\(861\) 20.0000 0.681598
\(862\) 16.0000 0.544962
\(863\) − 36.0000i − 1.22545i −0.790295 0.612727i \(-0.790072\pi\)
0.790295 0.612727i \(-0.209928\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) − 14.0000i − 0.476014i
\(866\) − 26.0000i − 0.883516i
\(867\) 1.00000 0.0339618
\(868\) 8.00000 0.271538
\(869\) 32.0000i 1.08553i
\(870\) −8.00000 −0.271225
\(871\) 0 0
\(872\) −4.00000 −0.135457
\(873\) − 8.00000i − 0.270759i
\(874\) 4.00000 0.135302
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 8.00000i 0.269987i
\(879\) − 6.00000i − 0.202375i
\(880\) 4.00000 0.134840
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 16.0000i 0.537531i
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) −6.00000 −0.201347
\(889\) − 40.0000i − 1.34156i
\(890\) 10.0000i 0.335201i
\(891\) − 4.00000i − 0.134005i
\(892\) 22.0000i 0.736614i
\(893\) 0 0
\(894\) 22.0000 0.735790
\(895\) − 2.00000i − 0.0668526i
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 32.0000i 1.06726i
\(900\) 1.00000 0.0333333
\(901\) −24.0000 −0.799556
\(902\) − 40.0000i − 1.33185i
\(903\) 8.00000i 0.266223i
\(904\) − 16.0000i − 0.532152i
\(905\) 22.0000i 0.731305i
\(906\) −4.00000 −0.132891
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) −20.0000 −0.663358
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) −48.0000 −1.58857
\(914\) −16.0000 −0.529233
\(915\) 2.00000i 0.0661180i
\(916\) 28.0000i 0.925146i
\(917\) 28.0000i 0.924641i
\(918\) − 4.00000i − 0.132020i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) 6.00000i 0.197279i
\(926\) −2.00000 −0.0657241
\(927\) −4.00000 −0.131377
\(928\) − 8.00000i − 0.262613i
\(929\) − 34.0000i − 1.11550i −0.830008 0.557752i \(-0.811664\pi\)
0.830008 0.557752i \(-0.188336\pi\)
\(930\) − 4.00000i − 0.131165i
\(931\) − 6.00000i − 0.196642i
\(932\) 28.0000 0.917170
\(933\) −16.0000 −0.523816
\(934\) 12.0000i 0.392652i
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) − 38.0000i − 1.23876i −0.785090 0.619382i \(-0.787383\pi\)
0.785090 0.619382i \(-0.212617\pi\)
\(942\) − 10.0000i − 0.325818i
\(943\) − 20.0000i − 0.651290i
\(944\) 12.0000i 0.390567i
\(945\) 2.00000 0.0650600
\(946\) 16.0000 0.520205
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) 2.00000i 0.0648544i
\(952\) 8.00000 0.259281
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) − 8.00000i − 0.258874i
\(956\) 8.00000i 0.258738i
\(957\) 32.0000i 1.03441i
\(958\) −24.0000 −0.775405
\(959\) −4.00000 −0.129167
\(960\) 1.00000i 0.0322749i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) − 22.0000i − 0.708572i
\(965\) 12.0000 0.386294
\(966\) 4.00000 0.128698
\(967\) 14.0000i 0.450210i 0.974335 + 0.225105i \(0.0722725\pi\)
−0.974335 + 0.225105i \(0.927728\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) − 8.00000i − 0.256997i
\(970\) − 8.00000i − 0.256865i
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 32.0000i − 1.02587i
\(974\) 38.0000 1.21760
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 12.0000 0.383718
\(979\) 40.0000 1.27841
\(980\) − 3.00000i − 0.0958315i
\(981\) 4.00000i 0.127710i
\(982\) 38.0000i 1.21263i
\(983\) 12.0000i 0.382741i 0.981518 + 0.191370i \(0.0612931\pi\)
−0.981518 + 0.191370i \(0.938707\pi\)
\(984\) 10.0000 0.318788
\(985\) −10.0000 −0.318626
\(986\) 32.0000i 1.01909i
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) − 4.00000i − 0.127128i
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 4.00000 0.127000
\(993\) 10.0000i 0.317340i
\(994\) 0 0
\(995\) − 24.0000i − 0.760851i
\(996\) − 12.0000i − 0.380235i
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −14.0000 −0.443162
\(999\) 6.00000i 0.189832i
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.f.1351.1 2
13.5 odd 4 390.2.a.b.1.1 1
13.8 odd 4 5070.2.a.n.1.1 1
13.12 even 2 inner 5070.2.b.f.1351.2 2
39.5 even 4 1170.2.a.j.1.1 1
52.31 even 4 3120.2.a.y.1.1 1
65.18 even 4 1950.2.e.m.1249.2 2
65.44 odd 4 1950.2.a.ba.1.1 1
65.57 even 4 1950.2.e.m.1249.1 2
156.83 odd 4 9360.2.a.v.1.1 1
195.44 even 4 5850.2.a.s.1.1 1
195.83 odd 4 5850.2.e.h.5149.1 2
195.122 odd 4 5850.2.e.h.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.b.1.1 1 13.5 odd 4
1170.2.a.j.1.1 1 39.5 even 4
1950.2.a.ba.1.1 1 65.44 odd 4
1950.2.e.m.1249.1 2 65.57 even 4
1950.2.e.m.1249.2 2 65.18 even 4
3120.2.a.y.1.1 1 52.31 even 4
5070.2.a.n.1.1 1 13.8 odd 4
5070.2.b.f.1351.1 2 1.1 even 1 trivial
5070.2.b.f.1351.2 2 13.12 even 2 inner
5850.2.a.s.1.1 1 195.44 even 4
5850.2.e.h.5149.1 2 195.83 odd 4
5850.2.e.h.5149.2 2 195.122 odd 4
9360.2.a.v.1.1 1 156.83 odd 4