Properties

Label 1950.2.e.p.1249.4
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Defining polynomial: \(x^{4} + 21 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(3.70156i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.p.1249.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.70156i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.70156i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.70156 q^{11} +1.00000i q^{12} +1.00000i q^{13} -4.70156 q^{14} +1.00000 q^{16} -0.701562i q^{17} -1.00000i q^{18} +1.70156 q^{19} +4.70156 q^{21} +4.70156i q^{22} -1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} -4.70156i q^{28} +6.40312 q^{29} -10.1047 q^{31} +1.00000i q^{32} -4.70156i q^{33} +0.701562 q^{34} +1.00000 q^{36} -1.70156i q^{37} +1.70156i q^{38} +1.00000 q^{39} -3.70156 q^{41} +4.70156i q^{42} +11.4031i q^{43} -4.70156 q^{44} +7.00000i q^{47} -1.00000i q^{48} -15.1047 q^{49} -0.701562 q^{51} -1.00000i q^{52} -2.40312i q^{53} -1.00000 q^{54} +4.70156 q^{56} -1.70156i q^{57} +6.40312i q^{58} -2.70156 q^{59} +14.1047 q^{61} -10.1047i q^{62} -4.70156i q^{63} -1.00000 q^{64} +4.70156 q^{66} -6.40312i q^{67} +0.701562i q^{68} -1.70156 q^{71} +1.00000i q^{72} +12.0000i q^{73} +1.70156 q^{74} -1.70156 q^{76} +22.1047i q^{77} +1.00000i q^{78} +5.70156 q^{79} +1.00000 q^{81} -3.70156i q^{82} +10.7016i q^{83} -4.70156 q^{84} -11.4031 q^{86} -6.40312i q^{87} -4.70156i q^{88} -11.4031 q^{89} -4.70156 q^{91} +10.1047i q^{93} -7.00000 q^{94} +1.00000 q^{96} -2.59688i q^{97} -15.1047i q^{98} -4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 6 q^{11} - 6 q^{14} + 4 q^{16} - 6 q^{19} + 6 q^{21} - 4 q^{24} - 4 q^{26} - 2 q^{31} - 10 q^{34} + 4 q^{36} + 4 q^{39} - 2 q^{41} - 6 q^{44} - 22 q^{49} + 10 q^{51} - 4 q^{54} + 6 q^{56} + 2 q^{59} + 18 q^{61} - 4 q^{64} + 6 q^{66} + 6 q^{71} - 6 q^{74} + 6 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} - 20 q^{86} - 20 q^{89} - 6 q^{91} - 28 q^{94} + 4 q^{96} - 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.70156i 1.77702i 0.458854 + 0.888512i \(0.348260\pi\)
−0.458854 + 0.888512i \(0.651740\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.70156 1.41757 0.708787 0.705422i \(-0.249243\pi\)
0.708787 + 0.705422i \(0.249243\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 1.00000i 0.277350i
\(14\) −4.70156 −1.25655
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 0.701562i − 0.170154i −0.996374 0.0850769i \(-0.972886\pi\)
0.996374 0.0850769i \(-0.0271136\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 1.70156 0.390365 0.195183 0.980767i \(-0.437470\pi\)
0.195183 + 0.980767i \(0.437470\pi\)
\(20\) 0 0
\(21\) 4.70156 1.02596
\(22\) 4.70156i 1.00238i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000i 0.192450i
\(28\) − 4.70156i − 0.888512i
\(29\) 6.40312 1.18903 0.594515 0.804084i \(-0.297344\pi\)
0.594515 + 0.804084i \(0.297344\pi\)
\(30\) 0 0
\(31\) −10.1047 −1.81486 −0.907428 0.420208i \(-0.861957\pi\)
−0.907428 + 0.420208i \(0.861957\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.70156i − 0.818437i
\(34\) 0.701562 0.120317
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 1.70156i − 0.279735i −0.990170 0.139868i \(-0.955332\pi\)
0.990170 0.139868i \(-0.0446677\pi\)
\(38\) 1.70156i 0.276030i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −3.70156 −0.578087 −0.289043 0.957316i \(-0.593337\pi\)
−0.289043 + 0.957316i \(0.593337\pi\)
\(42\) 4.70156i 0.725467i
\(43\) 11.4031i 1.73896i 0.493968 + 0.869480i \(0.335546\pi\)
−0.493968 + 0.869480i \(0.664454\pi\)
\(44\) −4.70156 −0.708787
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −15.1047 −2.15781
\(50\) 0 0
\(51\) −0.701562 −0.0982383
\(52\) − 1.00000i − 0.138675i
\(53\) − 2.40312i − 0.330095i −0.986286 0.165047i \(-0.947222\pi\)
0.986286 0.165047i \(-0.0527777\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.70156 0.628273
\(57\) − 1.70156i − 0.225377i
\(58\) 6.40312i 0.840771i
\(59\) −2.70156 −0.351713 −0.175857 0.984416i \(-0.556270\pi\)
−0.175857 + 0.984416i \(0.556270\pi\)
\(60\) 0 0
\(61\) 14.1047 1.80592 0.902960 0.429725i \(-0.141389\pi\)
0.902960 + 0.429725i \(0.141389\pi\)
\(62\) − 10.1047i − 1.28330i
\(63\) − 4.70156i − 0.592341i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.70156 0.578722
\(67\) − 6.40312i − 0.782266i −0.920334 0.391133i \(-0.872083\pi\)
0.920334 0.391133i \(-0.127917\pi\)
\(68\) 0.701562i 0.0850769i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.70156 −0.201938 −0.100969 0.994890i \(-0.532194\pi\)
−0.100969 + 0.994890i \(0.532194\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 12.0000i 1.40449i 0.711934 + 0.702247i \(0.247820\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(74\) 1.70156 0.197803
\(75\) 0 0
\(76\) −1.70156 −0.195183
\(77\) 22.1047i 2.51906i
\(78\) 1.00000i 0.113228i
\(79\) 5.70156 0.641476 0.320738 0.947168i \(-0.396069\pi\)
0.320738 + 0.947168i \(0.396069\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 3.70156i − 0.408769i
\(83\) 10.7016i 1.17465i 0.809352 + 0.587325i \(0.199819\pi\)
−0.809352 + 0.587325i \(0.800181\pi\)
\(84\) −4.70156 −0.512982
\(85\) 0 0
\(86\) −11.4031 −1.22963
\(87\) − 6.40312i − 0.686487i
\(88\) − 4.70156i − 0.501188i
\(89\) −11.4031 −1.20873 −0.604364 0.796708i \(-0.706573\pi\)
−0.604364 + 0.796708i \(0.706573\pi\)
\(90\) 0 0
\(91\) −4.70156 −0.492858
\(92\) 0 0
\(93\) 10.1047i 1.04781i
\(94\) −7.00000 −0.721995
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 2.59688i − 0.263673i −0.991271 0.131836i \(-0.957913\pi\)
0.991271 0.131836i \(-0.0420874\pi\)
\(98\) − 15.1047i − 1.52580i
\(99\) −4.70156 −0.472525
\(100\) 0 0
\(101\) 6.70156 0.666830 0.333415 0.942780i \(-0.391799\pi\)
0.333415 + 0.942780i \(0.391799\pi\)
\(102\) − 0.701562i − 0.0694650i
\(103\) − 1.40312i − 0.138254i −0.997608 0.0691270i \(-0.977979\pi\)
0.997608 0.0691270i \(-0.0220214\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 2.40312 0.233412
\(107\) 19.1047i 1.84692i 0.383695 + 0.923460i \(0.374651\pi\)
−0.383695 + 0.923460i \(0.625349\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −4.29844 −0.411716 −0.205858 0.978582i \(-0.565998\pi\)
−0.205858 + 0.978582i \(0.565998\pi\)
\(110\) 0 0
\(111\) −1.70156 −0.161505
\(112\) 4.70156i 0.444256i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) 1.70156 0.159366
\(115\) 0 0
\(116\) −6.40312 −0.594515
\(117\) − 1.00000i − 0.0924500i
\(118\) − 2.70156i − 0.248699i
\(119\) 3.29844 0.302367
\(120\) 0 0
\(121\) 11.1047 1.00952
\(122\) 14.1047i 1.27698i
\(123\) 3.70156i 0.333759i
\(124\) 10.1047 0.907428
\(125\) 0 0
\(126\) 4.70156 0.418848
\(127\) − 6.29844i − 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901515\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 11.4031 1.00399
\(130\) 0 0
\(131\) 22.5078 1.96652 0.983258 0.182217i \(-0.0583275\pi\)
0.983258 + 0.182217i \(0.0583275\pi\)
\(132\) 4.70156i 0.409218i
\(133\) 8.00000i 0.693688i
\(134\) 6.40312 0.553146
\(135\) 0 0
\(136\) −0.701562 −0.0601585
\(137\) 13.7016i 1.17060i 0.810816 + 0.585302i \(0.199024\pi\)
−0.810816 + 0.585302i \(0.800976\pi\)
\(138\) 0 0
\(139\) −9.40312 −0.797563 −0.398781 0.917046i \(-0.630567\pi\)
−0.398781 + 0.917046i \(0.630567\pi\)
\(140\) 0 0
\(141\) 7.00000 0.589506
\(142\) − 1.70156i − 0.142792i
\(143\) 4.70156i 0.393164i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 15.1047i 1.24581i
\(148\) 1.70156i 0.139868i
\(149\) −6.59688 −0.540437 −0.270219 0.962799i \(-0.587096\pi\)
−0.270219 + 0.962799i \(0.587096\pi\)
\(150\) 0 0
\(151\) −14.1047 −1.14782 −0.573912 0.818917i \(-0.694575\pi\)
−0.573912 + 0.818917i \(0.694575\pi\)
\(152\) − 1.70156i − 0.138015i
\(153\) 0.701562i 0.0567179i
\(154\) −22.1047 −1.78125
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 22.7016i − 1.81178i −0.423511 0.905891i \(-0.639203\pi\)
0.423511 0.905891i \(-0.360797\pi\)
\(158\) 5.70156i 0.453592i
\(159\) −2.40312 −0.190580
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 18.8062i 1.47302i 0.676427 + 0.736510i \(0.263527\pi\)
−0.676427 + 0.736510i \(0.736473\pi\)
\(164\) 3.70156 0.289043
\(165\) 0 0
\(166\) −10.7016 −0.830602
\(167\) − 1.10469i − 0.0854832i −0.999086 0.0427416i \(-0.986391\pi\)
0.999086 0.0427416i \(-0.0136092\pi\)
\(168\) − 4.70156i − 0.362733i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −1.70156 −0.130122
\(172\) − 11.4031i − 0.869480i
\(173\) 0.193752i 0.0147307i 0.999973 + 0.00736533i \(0.00234448\pi\)
−0.999973 + 0.00736533i \(0.997656\pi\)
\(174\) 6.40312 0.485420
\(175\) 0 0
\(176\) 4.70156 0.354394
\(177\) 2.70156i 0.203062i
\(178\) − 11.4031i − 0.854700i
\(179\) 24.2094 1.80949 0.904747 0.425950i \(-0.140060\pi\)
0.904747 + 0.425950i \(0.140060\pi\)
\(180\) 0 0
\(181\) 11.2984 0.839806 0.419903 0.907569i \(-0.362064\pi\)
0.419903 + 0.907569i \(0.362064\pi\)
\(182\) − 4.70156i − 0.348503i
\(183\) − 14.1047i − 1.04265i
\(184\) 0 0
\(185\) 0 0
\(186\) −10.1047 −0.740912
\(187\) − 3.29844i − 0.241206i
\(188\) − 7.00000i − 0.510527i
\(189\) −4.70156 −0.341988
\(190\) 0 0
\(191\) −12.8062 −0.926628 −0.463314 0.886194i \(-0.653340\pi\)
−0.463314 + 0.886194i \(0.653340\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 16.2094i − 1.16678i −0.812194 0.583388i \(-0.801727\pi\)
0.812194 0.583388i \(-0.198273\pi\)
\(194\) 2.59688 0.186445
\(195\) 0 0
\(196\) 15.1047 1.07891
\(197\) − 5.40312i − 0.384957i −0.981301 0.192478i \(-0.938347\pi\)
0.981301 0.192478i \(-0.0616525\pi\)
\(198\) − 4.70156i − 0.334125i
\(199\) −8.29844 −0.588261 −0.294130 0.955765i \(-0.595030\pi\)
−0.294130 + 0.955765i \(0.595030\pi\)
\(200\) 0 0
\(201\) −6.40312 −0.451642
\(202\) 6.70156i 0.471520i
\(203\) 30.1047i 2.11293i
\(204\) 0.701562 0.0491192
\(205\) 0 0
\(206\) 1.40312 0.0977603
\(207\) 0 0
\(208\) 1.00000i 0.0693375i
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 6.80625 0.468561 0.234281 0.972169i \(-0.424727\pi\)
0.234281 + 0.972169i \(0.424727\pi\)
\(212\) 2.40312i 0.165047i
\(213\) 1.70156i 0.116589i
\(214\) −19.1047 −1.30597
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 47.5078i − 3.22504i
\(218\) − 4.29844i − 0.291127i
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 0.701562 0.0471922
\(222\) − 1.70156i − 0.114201i
\(223\) − 11.4031i − 0.763610i −0.924243 0.381805i \(-0.875303\pi\)
0.924243 0.381805i \(-0.124697\pi\)
\(224\) −4.70156 −0.314136
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 16.1047i − 1.06891i −0.845198 0.534453i \(-0.820518\pi\)
0.845198 0.534453i \(-0.179482\pi\)
\(228\) 1.70156i 0.112689i
\(229\) −15.7016 −1.03759 −0.518794 0.854899i \(-0.673619\pi\)
−0.518794 + 0.854899i \(0.673619\pi\)
\(230\) 0 0
\(231\) 22.1047 1.45438
\(232\) − 6.40312i − 0.420386i
\(233\) 20.2094i 1.32396i 0.749521 + 0.661980i \(0.230284\pi\)
−0.749521 + 0.661980i \(0.769716\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 2.70156 0.175857
\(237\) − 5.70156i − 0.370356i
\(238\) 3.29844i 0.213806i
\(239\) −6.10469 −0.394879 −0.197440 0.980315i \(-0.563263\pi\)
−0.197440 + 0.980315i \(0.563263\pi\)
\(240\) 0 0
\(241\) −2.59688 −0.167279 −0.0836397 0.996496i \(-0.526654\pi\)
−0.0836397 + 0.996496i \(0.526654\pi\)
\(242\) 11.1047i 0.713836i
\(243\) − 1.00000i − 0.0641500i
\(244\) −14.1047 −0.902960
\(245\) 0 0
\(246\) −3.70156 −0.236003
\(247\) 1.70156i 0.108268i
\(248\) 10.1047i 0.641648i
\(249\) 10.7016 0.678184
\(250\) 0 0
\(251\) 0.298438 0.0188372 0.00941862 0.999956i \(-0.497002\pi\)
0.00941862 + 0.999956i \(0.497002\pi\)
\(252\) 4.70156i 0.296171i
\(253\) 0 0
\(254\) 6.29844 0.395199
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 15.2984i − 0.954290i −0.878824 0.477145i \(-0.841672\pi\)
0.878824 0.477145i \(-0.158328\pi\)
\(258\) 11.4031i 0.709928i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −6.40312 −0.396343
\(262\) 22.5078i 1.39054i
\(263\) − 2.00000i − 0.123325i −0.998097 0.0616626i \(-0.980360\pi\)
0.998097 0.0616626i \(-0.0196403\pi\)
\(264\) −4.70156 −0.289361
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 11.4031i 0.697860i
\(268\) 6.40312i 0.391133i
\(269\) −31.2094 −1.90287 −0.951435 0.307851i \(-0.900390\pi\)
−0.951435 + 0.307851i \(0.900390\pi\)
\(270\) 0 0
\(271\) 19.5078 1.18502 0.592508 0.805565i \(-0.298138\pi\)
0.592508 + 0.805565i \(0.298138\pi\)
\(272\) − 0.701562i − 0.0425385i
\(273\) 4.70156i 0.284551i
\(274\) −13.7016 −0.827742
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) − 9.40312i − 0.563962i
\(279\) 10.1047 0.604952
\(280\) 0 0
\(281\) −21.9109 −1.30710 −0.653548 0.756885i \(-0.726720\pi\)
−0.653548 + 0.756885i \(0.726720\pi\)
\(282\) 7.00000i 0.416844i
\(283\) − 21.4031i − 1.27228i −0.771572 0.636142i \(-0.780529\pi\)
0.771572 0.636142i \(-0.219471\pi\)
\(284\) 1.70156 0.100969
\(285\) 0 0
\(286\) −4.70156 −0.278009
\(287\) − 17.4031i − 1.02727i
\(288\) − 1.00000i − 0.0589256i
\(289\) 16.5078 0.971048
\(290\) 0 0
\(291\) −2.59688 −0.152232
\(292\) − 12.0000i − 0.702247i
\(293\) − 21.4031i − 1.25038i −0.780471 0.625192i \(-0.785021\pi\)
0.780471 0.625192i \(-0.214979\pi\)
\(294\) −15.1047 −0.880923
\(295\) 0 0
\(296\) −1.70156 −0.0989013
\(297\) 4.70156i 0.272812i
\(298\) − 6.59688i − 0.382147i
\(299\) 0 0
\(300\) 0 0
\(301\) −53.6125 −3.09017
\(302\) − 14.1047i − 0.811633i
\(303\) − 6.70156i − 0.384995i
\(304\) 1.70156 0.0975913
\(305\) 0 0
\(306\) −0.701562 −0.0401056
\(307\) − 5.70156i − 0.325405i −0.986675 0.162703i \(-0.947979\pi\)
0.986675 0.162703i \(-0.0520211\pi\)
\(308\) − 22.1047i − 1.25953i
\(309\) −1.40312 −0.0798209
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) 21.2094i 1.19882i 0.800440 + 0.599412i \(0.204599\pi\)
−0.800440 + 0.599412i \(0.795401\pi\)
\(314\) 22.7016 1.28112
\(315\) 0 0
\(316\) −5.70156 −0.320738
\(317\) 1.19375i 0.0670478i 0.999438 + 0.0335239i \(0.0106730\pi\)
−0.999438 + 0.0335239i \(0.989327\pi\)
\(318\) − 2.40312i − 0.134761i
\(319\) 30.1047 1.68554
\(320\) 0 0
\(321\) 19.1047 1.06632
\(322\) 0 0
\(323\) − 1.19375i − 0.0664221i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −18.8062 −1.04158
\(327\) 4.29844i 0.237704i
\(328\) 3.70156i 0.204385i
\(329\) −32.9109 −1.81444
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) − 10.7016i − 0.587325i
\(333\) 1.70156i 0.0932450i
\(334\) 1.10469 0.0604457
\(335\) 0 0
\(336\) 4.70156 0.256491
\(337\) 8.10469i 0.441490i 0.975332 + 0.220745i \(0.0708489\pi\)
−0.975332 + 0.220745i \(0.929151\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) −47.5078 −2.57269
\(342\) − 1.70156i − 0.0920099i
\(343\) − 38.1047i − 2.05746i
\(344\) 11.4031 0.614815
\(345\) 0 0
\(346\) −0.193752 −0.0104161
\(347\) 10.5078i 0.564089i 0.959401 + 0.282044i \(0.0910126\pi\)
−0.959401 + 0.282044i \(0.908987\pi\)
\(348\) 6.40312i 0.343243i
\(349\) 11.4031 0.610395 0.305198 0.952289i \(-0.401277\pi\)
0.305198 + 0.952289i \(0.401277\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 4.70156i 0.250594i
\(353\) − 14.5078i − 0.772173i −0.922463 0.386086i \(-0.873827\pi\)
0.922463 0.386086i \(-0.126173\pi\)
\(354\) −2.70156 −0.143586
\(355\) 0 0
\(356\) 11.4031 0.604364
\(357\) − 3.29844i − 0.174572i
\(358\) 24.2094i 1.27951i
\(359\) 34.6125 1.82678 0.913389 0.407088i \(-0.133456\pi\)
0.913389 + 0.407088i \(0.133456\pi\)
\(360\) 0 0
\(361\) −16.1047 −0.847615
\(362\) 11.2984i 0.593833i
\(363\) − 11.1047i − 0.582845i
\(364\) 4.70156 0.246429
\(365\) 0 0
\(366\) 14.1047 0.737264
\(367\) − 2.29844i − 0.119977i −0.998199 0.0599887i \(-0.980894\pi\)
0.998199 0.0599887i \(-0.0191065\pi\)
\(368\) 0 0
\(369\) 3.70156 0.192696
\(370\) 0 0
\(371\) 11.2984 0.586586
\(372\) − 10.1047i − 0.523904i
\(373\) − 5.29844i − 0.274343i −0.990547 0.137171i \(-0.956199\pi\)
0.990547 0.137171i \(-0.0438011\pi\)
\(374\) 3.29844 0.170558
\(375\) 0 0
\(376\) 7.00000 0.360997
\(377\) 6.40312i 0.329778i
\(378\) − 4.70156i − 0.241822i
\(379\) 13.8953 0.713754 0.356877 0.934151i \(-0.383841\pi\)
0.356877 + 0.934151i \(0.383841\pi\)
\(380\) 0 0
\(381\) −6.29844 −0.322679
\(382\) − 12.8062i − 0.655225i
\(383\) − 32.5078i − 1.66107i −0.556965 0.830536i \(-0.688034\pi\)
0.556965 0.830536i \(-0.311966\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 16.2094 0.825035
\(387\) − 11.4031i − 0.579653i
\(388\) 2.59688i 0.131836i
\(389\) 27.1047 1.37426 0.687131 0.726533i \(-0.258870\pi\)
0.687131 + 0.726533i \(0.258870\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 15.1047i 0.762902i
\(393\) − 22.5078i − 1.13537i
\(394\) 5.40312 0.272205
\(395\) 0 0
\(396\) 4.70156 0.236262
\(397\) − 21.9109i − 1.09968i −0.835271 0.549839i \(-0.814689\pi\)
0.835271 0.549839i \(-0.185311\pi\)
\(398\) − 8.29844i − 0.415963i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 18.2094 0.909333 0.454666 0.890662i \(-0.349759\pi\)
0.454666 + 0.890662i \(0.349759\pi\)
\(402\) − 6.40312i − 0.319359i
\(403\) − 10.1047i − 0.503350i
\(404\) −6.70156 −0.333415
\(405\) 0 0
\(406\) −30.1047 −1.49407
\(407\) − 8.00000i − 0.396545i
\(408\) 0.701562i 0.0347325i
\(409\) −29.4031 −1.45389 −0.726945 0.686695i \(-0.759061\pi\)
−0.726945 + 0.686695i \(0.759061\pi\)
\(410\) 0 0
\(411\) 13.7016 0.675848
\(412\) 1.40312i 0.0691270i
\(413\) − 12.7016i − 0.625003i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 9.40312i 0.460473i
\(418\) 8.00000i 0.391293i
\(419\) −9.91093 −0.484181 −0.242090 0.970254i \(-0.577833\pi\)
−0.242090 + 0.970254i \(0.577833\pi\)
\(420\) 0 0
\(421\) −0.596876 −0.0290899 −0.0145450 0.999894i \(-0.504630\pi\)
−0.0145450 + 0.999894i \(0.504630\pi\)
\(422\) 6.80625i 0.331323i
\(423\) − 7.00000i − 0.340352i
\(424\) −2.40312 −0.116706
\(425\) 0 0
\(426\) −1.70156 −0.0824410
\(427\) 66.3141i 3.20916i
\(428\) − 19.1047i − 0.923460i
\(429\) 4.70156 0.226994
\(430\) 0 0
\(431\) 35.3141 1.70102 0.850509 0.525960i \(-0.176294\pi\)
0.850509 + 0.525960i \(0.176294\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 11.1047i 0.533657i 0.963744 + 0.266829i \(0.0859758\pi\)
−0.963744 + 0.266829i \(0.914024\pi\)
\(434\) 47.5078 2.28045
\(435\) 0 0
\(436\) 4.29844 0.205858
\(437\) 0 0
\(438\) 12.0000i 0.573382i
\(439\) 10.2984 0.491518 0.245759 0.969331i \(-0.420963\pi\)
0.245759 + 0.969331i \(0.420963\pi\)
\(440\) 0 0
\(441\) 15.1047 0.719271
\(442\) 0.701562i 0.0333699i
\(443\) − 32.5078i − 1.54449i −0.635323 0.772246i \(-0.719133\pi\)
0.635323 0.772246i \(-0.280867\pi\)
\(444\) 1.70156 0.0807526
\(445\) 0 0
\(446\) 11.4031 0.539954
\(447\) 6.59688i 0.312022i
\(448\) − 4.70156i − 0.222128i
\(449\) −2.29844 −0.108470 −0.0542350 0.998528i \(-0.517272\pi\)
−0.0542350 + 0.998528i \(0.517272\pi\)
\(450\) 0 0
\(451\) −17.4031 −0.819481
\(452\) − 14.0000i − 0.658505i
\(453\) 14.1047i 0.662696i
\(454\) 16.1047 0.755830
\(455\) 0 0
\(456\) −1.70156 −0.0796829
\(457\) − 2.59688i − 0.121477i −0.998154 0.0607384i \(-0.980654\pi\)
0.998154 0.0607384i \(-0.0193455\pi\)
\(458\) − 15.7016i − 0.733686i
\(459\) 0.701562 0.0327461
\(460\) 0 0
\(461\) 36.2094 1.68644 0.843219 0.537570i \(-0.180658\pi\)
0.843219 + 0.537570i \(0.180658\pi\)
\(462\) 22.1047i 1.02840i
\(463\) 21.2984i 0.989822i 0.868944 + 0.494911i \(0.164799\pi\)
−0.868944 + 0.494911i \(0.835201\pi\)
\(464\) 6.40312 0.297258
\(465\) 0 0
\(466\) −20.2094 −0.936181
\(467\) 30.2984i 1.40204i 0.713139 + 0.701022i \(0.247273\pi\)
−0.713139 + 0.701022i \(0.752727\pi\)
\(468\) 1.00000i 0.0462250i
\(469\) 30.1047 1.39011
\(470\) 0 0
\(471\) −22.7016 −1.04603
\(472\) 2.70156i 0.124349i
\(473\) 53.6125i 2.46511i
\(474\) 5.70156 0.261881
\(475\) 0 0
\(476\) −3.29844 −0.151184
\(477\) 2.40312i 0.110032i
\(478\) − 6.10469i − 0.279222i
\(479\) 40.6125 1.85563 0.927816 0.373038i \(-0.121684\pi\)
0.927816 + 0.373038i \(0.121684\pi\)
\(480\) 0 0
\(481\) 1.70156 0.0775846
\(482\) − 2.59688i − 0.118284i
\(483\) 0 0
\(484\) −11.1047 −0.504758
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 7.89531i − 0.357771i −0.983870 0.178885i \(-0.942751\pi\)
0.983870 0.178885i \(-0.0572491\pi\)
\(488\) − 14.1047i − 0.638489i
\(489\) 18.8062 0.850448
\(490\) 0 0
\(491\) 33.4031 1.50746 0.753731 0.657183i \(-0.228252\pi\)
0.753731 + 0.657183i \(0.228252\pi\)
\(492\) − 3.70156i − 0.166879i
\(493\) − 4.49219i − 0.202318i
\(494\) −1.70156 −0.0765569
\(495\) 0 0
\(496\) −10.1047 −0.453714
\(497\) − 8.00000i − 0.358849i
\(498\) 10.7016i 0.479548i
\(499\) −17.2094 −0.770397 −0.385199 0.922834i \(-0.625867\pi\)
−0.385199 + 0.922834i \(0.625867\pi\)
\(500\) 0 0
\(501\) −1.10469 −0.0493537
\(502\) 0.298438i 0.0133199i
\(503\) 12.2094i 0.544389i 0.962242 + 0.272195i \(0.0877494\pi\)
−0.962242 + 0.272195i \(0.912251\pi\)
\(504\) −4.70156 −0.209424
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 6.29844i 0.279448i
\(509\) 5.40312 0.239489 0.119745 0.992805i \(-0.461792\pi\)
0.119745 + 0.992805i \(0.461792\pi\)
\(510\) 0 0
\(511\) −56.4187 −2.49582
\(512\) 1.00000i 0.0441942i
\(513\) 1.70156i 0.0751258i
\(514\) 15.2984 0.674785
\(515\) 0 0
\(516\) −11.4031 −0.501995
\(517\) 32.9109i 1.44742i
\(518\) 8.00000i 0.351500i
\(519\) 0.193752 0.00850475
\(520\) 0 0
\(521\) −36.2094 −1.58636 −0.793181 0.608986i \(-0.791576\pi\)
−0.793181 + 0.608986i \(0.791576\pi\)
\(522\) − 6.40312i − 0.280257i
\(523\) 24.8062i 1.08470i 0.840152 + 0.542351i \(0.182466\pi\)
−0.840152 + 0.542351i \(0.817534\pi\)
\(524\) −22.5078 −0.983258
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 7.08907i 0.308805i
\(528\) − 4.70156i − 0.204609i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 2.70156 0.117238
\(532\) − 8.00000i − 0.346844i
\(533\) − 3.70156i − 0.160332i
\(534\) −11.4031 −0.493461
\(535\) 0 0
\(536\) −6.40312 −0.276573
\(537\) − 24.2094i − 1.04471i
\(538\) − 31.2094i − 1.34553i
\(539\) −71.0156 −3.05886
\(540\) 0 0
\(541\) −1.79063 −0.0769851 −0.0384925 0.999259i \(-0.512256\pi\)
−0.0384925 + 0.999259i \(0.512256\pi\)
\(542\) 19.5078i 0.837932i
\(543\) − 11.2984i − 0.484862i
\(544\) 0.701562 0.0300792
\(545\) 0 0
\(546\) −4.70156 −0.201208
\(547\) − 31.6125i − 1.35165i −0.737061 0.675826i \(-0.763787\pi\)
0.737061 0.675826i \(-0.236213\pi\)
\(548\) − 13.7016i − 0.585302i
\(549\) −14.1047 −0.601973
\(550\) 0 0
\(551\) 10.8953 0.464156
\(552\) 0 0
\(553\) 26.8062i 1.13992i
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 9.40312 0.398781
\(557\) 16.8062i 0.712104i 0.934466 + 0.356052i \(0.115877\pi\)
−0.934466 + 0.356052i \(0.884123\pi\)
\(558\) 10.1047i 0.427765i
\(559\) −11.4031 −0.482301
\(560\) 0 0
\(561\) −3.29844 −0.139260
\(562\) − 21.9109i − 0.924257i
\(563\) 32.5078i 1.37004i 0.728524 + 0.685020i \(0.240207\pi\)
−0.728524 + 0.685020i \(0.759793\pi\)
\(564\) −7.00000 −0.294753
\(565\) 0 0
\(566\) 21.4031 0.899640
\(567\) 4.70156i 0.197447i
\(568\) 1.70156i 0.0713960i
\(569\) 11.2984 0.473655 0.236828 0.971552i \(-0.423892\pi\)
0.236828 + 0.971552i \(0.423892\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) − 4.70156i − 0.196582i
\(573\) 12.8062i 0.534989i
\(574\) 17.4031 0.726392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 24.2094i 1.00785i 0.863748 + 0.503925i \(0.168111\pi\)
−0.863748 + 0.503925i \(0.831889\pi\)
\(578\) 16.5078i 0.686634i
\(579\) −16.2094 −0.673639
\(580\) 0 0
\(581\) −50.3141 −2.08738
\(582\) − 2.59688i − 0.107644i
\(583\) − 11.2984i − 0.467933i
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) 21.4031 0.884155
\(587\) − 7.50781i − 0.309881i −0.987924 0.154940i \(-0.950481\pi\)
0.987924 0.154940i \(-0.0495185\pi\)
\(588\) − 15.1047i − 0.622907i
\(589\) −17.1938 −0.708456
\(590\) 0 0
\(591\) −5.40312 −0.222255
\(592\) − 1.70156i − 0.0699338i
\(593\) − 17.9109i − 0.735514i −0.929922 0.367757i \(-0.880126\pi\)
0.929922 0.367757i \(-0.119874\pi\)
\(594\) −4.70156 −0.192907
\(595\) 0 0
\(596\) 6.59688 0.270219
\(597\) 8.29844i 0.339632i
\(598\) 0 0
\(599\) −8.20937 −0.335426 −0.167713 0.985836i \(-0.553638\pi\)
−0.167713 + 0.985836i \(0.553638\pi\)
\(600\) 0 0
\(601\) 10.6125 0.432893 0.216446 0.976295i \(-0.430553\pi\)
0.216446 + 0.976295i \(0.430553\pi\)
\(602\) − 53.6125i − 2.18508i
\(603\) 6.40312i 0.260755i
\(604\) 14.1047 0.573912
\(605\) 0 0
\(606\) 6.70156 0.272232
\(607\) − 35.1047i − 1.42486i −0.701746 0.712428i \(-0.747596\pi\)
0.701746 0.712428i \(-0.252404\pi\)
\(608\) 1.70156i 0.0690075i
\(609\) 30.1047 1.21990
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) − 0.701562i − 0.0283590i
\(613\) 12.8062i 0.517240i 0.965979 + 0.258620i \(0.0832677\pi\)
−0.965979 + 0.258620i \(0.916732\pi\)
\(614\) 5.70156 0.230096
\(615\) 0 0
\(616\) 22.1047 0.890623
\(617\) − 27.1047i − 1.09119i −0.838048 0.545597i \(-0.816303\pi\)
0.838048 0.545597i \(-0.183697\pi\)
\(618\) − 1.40312i − 0.0564419i
\(619\) 13.1938 0.530302 0.265151 0.964207i \(-0.414578\pi\)
0.265151 + 0.964207i \(0.414578\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 30.0000i 1.20289i
\(623\) − 53.6125i − 2.14794i
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −21.2094 −0.847697
\(627\) − 8.00000i − 0.319489i
\(628\) 22.7016i 0.905891i
\(629\) −1.19375 −0.0475980
\(630\) 0 0
\(631\) 33.6125 1.33809 0.669046 0.743221i \(-0.266703\pi\)
0.669046 + 0.743221i \(0.266703\pi\)
\(632\) − 5.70156i − 0.226796i
\(633\) − 6.80625i − 0.270524i
\(634\) −1.19375 −0.0474099
\(635\) 0 0
\(636\) 2.40312 0.0952901
\(637\) − 15.1047i − 0.598469i
\(638\) 30.1047i 1.19186i
\(639\) 1.70156 0.0673128
\(640\) 0 0
\(641\) 5.50781 0.217545 0.108773 0.994067i \(-0.465308\pi\)
0.108773 + 0.994067i \(0.465308\pi\)
\(642\) 19.1047i 0.754002i
\(643\) 10.2984i 0.406131i 0.979165 + 0.203065i \(0.0650904\pi\)
−0.979165 + 0.203065i \(0.934910\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.19375 0.0469675
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −12.7016 −0.498580
\(650\) 0 0
\(651\) −47.5078 −1.86198
\(652\) − 18.8062i − 0.736510i
\(653\) 21.2984i 0.833472i 0.909027 + 0.416736i \(0.136826\pi\)
−0.909027 + 0.416736i \(0.863174\pi\)
\(654\) −4.29844 −0.168082
\(655\) 0 0
\(656\) −3.70156 −0.144522
\(657\) − 12.0000i − 0.468165i
\(658\) − 32.9109i − 1.28300i
\(659\) 19.3141 0.752369 0.376184 0.926545i \(-0.377236\pi\)
0.376184 + 0.926545i \(0.377236\pi\)
\(660\) 0 0
\(661\) −40.1203 −1.56050 −0.780250 0.625468i \(-0.784908\pi\)
−0.780250 + 0.625468i \(0.784908\pi\)
\(662\) − 12.0000i − 0.466393i
\(663\) − 0.701562i − 0.0272464i
\(664\) 10.7016 0.415301
\(665\) 0 0
\(666\) −1.70156 −0.0659342
\(667\) 0 0
\(668\) 1.10469i 0.0427416i
\(669\) −11.4031 −0.440870
\(670\) 0 0
\(671\) 66.3141 2.56003
\(672\) 4.70156i 0.181367i
\(673\) − 36.0156i − 1.38830i −0.719830 0.694150i \(-0.755780\pi\)
0.719830 0.694150i \(-0.244220\pi\)
\(674\) −8.10469 −0.312181
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 31.6125i 1.21497i 0.794332 + 0.607483i \(0.207821\pi\)
−0.794332 + 0.607483i \(0.792179\pi\)
\(678\) 14.0000i 0.537667i
\(679\) 12.2094 0.468553
\(680\) 0 0
\(681\) −16.1047 −0.617133
\(682\) − 47.5078i − 1.81917i
\(683\) − 44.7016i − 1.71046i −0.518251 0.855229i \(-0.673417\pi\)
0.518251 0.855229i \(-0.326583\pi\)
\(684\) 1.70156 0.0650609
\(685\) 0 0
\(686\) 38.1047 1.45484
\(687\) 15.7016i 0.599052i
\(688\) 11.4031i 0.434740i
\(689\) 2.40312 0.0915517
\(690\) 0 0
\(691\) −22.1938 −0.844290 −0.422145 0.906528i \(-0.638723\pi\)
−0.422145 + 0.906528i \(0.638723\pi\)
\(692\) − 0.193752i − 0.00736533i
\(693\) − 22.1047i − 0.839688i
\(694\) −10.5078 −0.398871
\(695\) 0 0
\(696\) −6.40312 −0.242710
\(697\) 2.59688i 0.0983637i
\(698\) 11.4031i 0.431615i
\(699\) 20.2094 0.764389
\(700\) 0 0
\(701\) 30.9109 1.16749 0.583745 0.811937i \(-0.301587\pi\)
0.583745 + 0.811937i \(0.301587\pi\)
\(702\) − 1.00000i − 0.0377426i
\(703\) − 2.89531i − 0.109199i
\(704\) −4.70156 −0.177197
\(705\) 0 0
\(706\) 14.5078 0.546009
\(707\) 31.5078i 1.18497i
\(708\) − 2.70156i − 0.101531i
\(709\) −15.6125 −0.586340 −0.293170 0.956060i \(-0.594710\pi\)
−0.293170 + 0.956060i \(0.594710\pi\)
\(710\) 0 0
\(711\) −5.70156 −0.213825
\(712\) 11.4031i 0.427350i
\(713\) 0 0
\(714\) 3.29844 0.123441
\(715\) 0 0
\(716\) −24.2094 −0.904747
\(717\) 6.10469i 0.227984i
\(718\) 34.6125i 1.29173i
\(719\) 11.0156 0.410813 0.205407 0.978677i \(-0.434148\pi\)
0.205407 + 0.978677i \(0.434148\pi\)
\(720\) 0 0
\(721\) 6.59688 0.245680
\(722\) − 16.1047i − 0.599354i
\(723\) 2.59688i 0.0965788i
\(724\) −11.2984 −0.419903
\(725\) 0 0
\(726\) 11.1047 0.412134
\(727\) 7.79063i 0.288938i 0.989509 + 0.144469i \(0.0461475\pi\)
−0.989509 + 0.144469i \(0.953853\pi\)
\(728\) 4.70156i 0.174251i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 14.1047i 0.521324i
\(733\) − 47.9109i − 1.76963i −0.465942 0.884815i \(-0.654284\pi\)
0.465942 0.884815i \(-0.345716\pi\)
\(734\) 2.29844 0.0848369
\(735\) 0 0
\(736\) 0 0
\(737\) − 30.1047i − 1.10892i
\(738\) 3.70156i 0.136256i
\(739\) −4.61250 −0.169673 −0.0848367 0.996395i \(-0.527037\pi\)
−0.0848367 + 0.996395i \(0.527037\pi\)
\(740\) 0 0
\(741\) 1.70156 0.0625084
\(742\) 11.2984i 0.414779i
\(743\) 30.6125i 1.12306i 0.827455 + 0.561532i \(0.189788\pi\)
−0.827455 + 0.561532i \(0.810212\pi\)
\(744\) 10.1047 0.370456
\(745\) 0 0
\(746\) 5.29844 0.193990
\(747\) − 10.7016i − 0.391550i
\(748\) 3.29844i 0.120603i
\(749\) −89.8219 −3.28202
\(750\) 0 0
\(751\) 8.50781 0.310454 0.155227 0.987879i \(-0.450389\pi\)
0.155227 + 0.987879i \(0.450389\pi\)
\(752\) 7.00000i 0.255264i
\(753\) − 0.298438i − 0.0108757i
\(754\) −6.40312 −0.233188
\(755\) 0 0
\(756\) 4.70156 0.170994
\(757\) − 17.8953i − 0.650416i −0.945642 0.325208i \(-0.894566\pi\)
0.945642 0.325208i \(-0.105434\pi\)
\(758\) 13.8953i 0.504701i
\(759\) 0 0
\(760\) 0 0
\(761\) −26.7172 −0.968497 −0.484249 0.874930i \(-0.660907\pi\)
−0.484249 + 0.874930i \(0.660907\pi\)
\(762\) − 6.29844i − 0.228168i
\(763\) − 20.2094i − 0.731628i
\(764\) 12.8062 0.463314
\(765\) 0 0
\(766\) 32.5078 1.17455
\(767\) − 2.70156i − 0.0975478i
\(768\) − 1.00000i − 0.0360844i
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 0 0
\(771\) −15.2984 −0.550960
\(772\) 16.2094i 0.583388i
\(773\) − 26.5969i − 0.956623i −0.878190 0.478312i \(-0.841249\pi\)
0.878190 0.478312i \(-0.158751\pi\)
\(774\) 11.4031 0.409877
\(775\) 0 0
\(776\) −2.59688 −0.0932224
\(777\) − 8.00000i − 0.286998i
\(778\) 27.1047i 0.971750i
\(779\) −6.29844 −0.225665
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 6.40312i 0.228829i
\(784\) −15.1047 −0.539453
\(785\) 0 0
\(786\) 22.5078 0.802827
\(787\) 33.8953i 1.20824i 0.796894 + 0.604119i \(0.206475\pi\)
−0.796894 + 0.604119i \(0.793525\pi\)
\(788\) 5.40312i 0.192478i
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) −65.8219 −2.34036
\(792\) 4.70156i 0.167063i
\(793\) 14.1047i 0.500872i
\(794\) 21.9109 0.777590
\(795\) 0 0
\(796\) 8.29844 0.294130
\(797\) 2.91093i 0.103111i 0.998670 + 0.0515553i \(0.0164178\pi\)
−0.998670 + 0.0515553i \(0.983582\pi\)
\(798\) 8.00000i 0.283197i
\(799\) 4.91093 0.173736
\(800\) 0 0
\(801\) 11.4031 0.402910
\(802\) 18.2094i 0.642995i
\(803\) 56.4187i 1.99097i
\(804\) 6.40312 0.225821
\(805\) 0 0
\(806\) 10.1047 0.355922
\(807\) 31.2094i 1.09862i
\(808\) − 6.70156i − 0.235760i
\(809\) −7.19375 −0.252919 −0.126459 0.991972i \(-0.540361\pi\)
−0.126459 + 0.991972i \(0.540361\pi\)
\(810\) 0 0
\(811\) 48.7016 1.71014 0.855072 0.518510i \(-0.173513\pi\)
0.855072 + 0.518510i \(0.173513\pi\)
\(812\) − 30.1047i − 1.05647i
\(813\) − 19.5078i − 0.684169i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −0.701562 −0.0245596
\(817\) 19.4031i 0.678829i
\(818\) − 29.4031i − 1.02806i
\(819\) 4.70156 0.164286
\(820\) 0 0
\(821\) −20.5969 −0.718836 −0.359418 0.933177i \(-0.617025\pi\)
−0.359418 + 0.933177i \(0.617025\pi\)
\(822\) 13.7016i 0.477897i
\(823\) − 17.1047i − 0.596232i −0.954530 0.298116i \(-0.903642\pi\)
0.954530 0.298116i \(-0.0963582\pi\)
\(824\) −1.40312 −0.0488801
\(825\) 0 0
\(826\) 12.7016 0.441944
\(827\) 20.7016i 0.719864i 0.932979 + 0.359932i \(0.117200\pi\)
−0.932979 + 0.359932i \(0.882800\pi\)
\(828\) 0 0
\(829\) 5.50781 0.191294 0.0956471 0.995415i \(-0.469508\pi\)
0.0956471 + 0.995415i \(0.469508\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) − 1.00000i − 0.0346688i
\(833\) 10.5969i 0.367160i
\(834\) −9.40312 −0.325604
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) − 10.1047i − 0.349269i
\(838\) − 9.91093i − 0.342368i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 12.0000 0.413793
\(842\) − 0.596876i − 0.0205697i
\(843\) 21.9109i 0.754653i
\(844\) −6.80625 −0.234281
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) 52.2094i 1.79394i
\(848\) − 2.40312i − 0.0825236i
\(849\) −21.4031 −0.734553
\(850\) 0 0
\(851\) 0 0
\(852\) − 1.70156i − 0.0582946i
\(853\) − 0.507811i − 0.0173871i −0.999962 0.00869355i \(-0.997233\pi\)
0.999962 0.00869355i \(-0.00276728\pi\)
\(854\) −66.3141 −2.26922
\(855\) 0 0
\(856\) 19.1047 0.652985
\(857\) − 7.61250i − 0.260038i −0.991512 0.130019i \(-0.958496\pi\)
0.991512 0.130019i \(-0.0415038\pi\)
\(858\) 4.70156i 0.160509i
\(859\) 6.20937 0.211861 0.105931 0.994374i \(-0.466218\pi\)
0.105931 + 0.994374i \(0.466218\pi\)
\(860\) 0 0
\(861\) −17.4031 −0.593097
\(862\) 35.3141i 1.20280i
\(863\) − 54.2250i − 1.84584i −0.384991 0.922920i \(-0.625796\pi\)
0.384991 0.922920i \(-0.374204\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −11.1047 −0.377353
\(867\) − 16.5078i − 0.560635i
\(868\) 47.5078i 1.61252i
\(869\) 26.8062 0.909340
\(870\) 0 0
\(871\) 6.40312 0.216962
\(872\) 4.29844i 0.145563i
\(873\) 2.59688i 0.0878909i
\(874\) 0 0
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) − 42.7172i − 1.44246i −0.692697 0.721228i \(-0.743578\pi\)
0.692697 0.721228i \(-0.256422\pi\)
\(878\) 10.2984i 0.347555i
\(879\) −21.4031 −0.721909
\(880\) 0 0
\(881\) −39.7172 −1.33811 −0.669053 0.743215i \(-0.733300\pi\)
−0.669053 + 0.743215i \(0.733300\pi\)
\(882\) 15.1047i 0.508601i
\(883\) − 45.6125i − 1.53498i −0.641059 0.767491i \(-0.721505\pi\)
0.641059 0.767491i \(-0.278495\pi\)
\(884\) −0.701562 −0.0235961
\(885\) 0 0
\(886\) 32.5078 1.09212
\(887\) − 20.0000i − 0.671534i −0.941945 0.335767i \(-0.891004\pi\)
0.941945 0.335767i \(-0.108996\pi\)
\(888\) 1.70156i 0.0571007i
\(889\) 29.6125 0.993171
\(890\) 0 0
\(891\) 4.70156 0.157508
\(892\) 11.4031i 0.381805i
\(893\) 11.9109i 0.398584i
\(894\) −6.59688 −0.220633
\(895\) 0 0
\(896\) 4.70156 0.157068
\(897\) 0 0
\(898\) − 2.29844i − 0.0766999i
\(899\) −64.7016 −2.15792
\(900\) 0 0
\(901\) −1.68594 −0.0561668
\(902\) − 17.4031i − 0.579461i
\(903\) 53.6125i 1.78411i
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) −14.1047 −0.468597
\(907\) − 25.0156i − 0.830630i −0.909678 0.415315i \(-0.863671\pi\)
0.909678 0.415315i \(-0.136329\pi\)
\(908\) 16.1047i 0.534453i
\(909\) −6.70156 −0.222277
\(910\) 0 0
\(911\) 42.2094 1.39846 0.699229 0.714897i \(-0.253527\pi\)
0.699229 + 0.714897i \(0.253527\pi\)
\(912\) − 1.70156i − 0.0563444i
\(913\) 50.3141i 1.66515i
\(914\) 2.59688 0.0858970
\(915\) 0 0
\(916\) 15.7016 0.518794
\(917\) 105.822i 3.49455i
\(918\) 0.701562i 0.0231550i
\(919\) 4.89531 0.161481 0.0807407 0.996735i \(-0.474271\pi\)
0.0807407 + 0.996735i \(0.474271\pi\)
\(920\) 0 0
\(921\) −5.70156 −0.187873
\(922\) 36.2094i 1.19249i
\(923\) − 1.70156i − 0.0560076i
\(924\) −22.1047 −0.727191
\(925\) 0 0
\(926\) −21.2984 −0.699910
\(927\) 1.40312i 0.0460846i
\(928\) 6.40312i 0.210193i
\(929\) −52.2984 −1.71586 −0.857928 0.513770i \(-0.828249\pi\)
−0.857928 + 0.513770i \(0.828249\pi\)
\(930\) 0 0
\(931\) −25.7016 −0.842335
\(932\) − 20.2094i − 0.661980i
\(933\) − 30.0000i − 0.982156i
\(934\) −30.2984 −0.991395
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 14.9109i 0.487119i 0.969886 + 0.243560i \(0.0783151\pi\)
−0.969886 + 0.243560i \(0.921685\pi\)
\(938\) 30.1047i 0.982953i
\(939\) 21.2094 0.692142
\(940\) 0 0
\(941\) 28.4187 0.926425 0.463212 0.886247i \(-0.346697\pi\)
0.463212 + 0.886247i \(0.346697\pi\)
\(942\) − 22.7016i − 0.739657i
\(943\) 0 0
\(944\) −2.70156 −0.0879284
\(945\) 0 0
\(946\) −53.6125 −1.74309
\(947\) 47.5078i 1.54380i 0.635746 + 0.771898i \(0.280693\pi\)
−0.635746 + 0.771898i \(0.719307\pi\)
\(948\) 5.70156i 0.185178i
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 1.19375 0.0387100
\(952\) − 3.29844i − 0.106903i
\(953\) 31.2984i 1.01386i 0.861988 + 0.506928i \(0.169219\pi\)
−0.861988 + 0.506928i \(0.830781\pi\)
\(954\) −2.40312 −0.0778040
\(955\) 0 0
\(956\) 6.10469 0.197440
\(957\) − 30.1047i − 0.973146i
\(958\) 40.6125i 1.31213i
\(959\) −64.4187 −2.08019
\(960\) 0 0
\(961\) 71.1047 2.29370
\(962\) 1.70156i 0.0548606i
\(963\) − 19.1047i − 0.615640i
\(964\) 2.59688 0.0836397
\(965\) 0 0
\(966\) 0 0
\(967\) 29.8953i 0.961368i 0.876894 + 0.480684i \(0.159612\pi\)
−0.876894 + 0.480684i \(0.840388\pi\)
\(968\) − 11.1047i − 0.356918i
\(969\) −1.19375 −0.0383488
\(970\) 0 0
\(971\) −36.8953 −1.18403 −0.592013 0.805928i \(-0.701667\pi\)
−0.592013 + 0.805928i \(0.701667\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 44.2094i − 1.41729i
\(974\) 7.89531 0.252982
\(975\) 0 0
\(976\) 14.1047 0.451480
\(977\) − 23.4031i − 0.748732i −0.927281 0.374366i \(-0.877860\pi\)
0.927281 0.374366i \(-0.122140\pi\)
\(978\) 18.8062i 0.601358i
\(979\) −53.6125 −1.71346
\(980\) 0 0
\(981\) 4.29844 0.137239
\(982\) 33.4031i 1.06594i
\(983\) 7.50781i 0.239462i 0.992806 + 0.119731i \(0.0382032\pi\)
−0.992806 + 0.119731i \(0.961797\pi\)
\(984\) 3.70156 0.118001
\(985\) 0 0
\(986\) 4.49219 0.143060
\(987\) 32.9109i 1.04757i
\(988\) − 1.70156i − 0.0541339i
\(989\) 0 0
\(990\) 0 0
\(991\) −9.10469 −0.289220 −0.144610 0.989489i \(-0.546193\pi\)
−0.144610 + 0.989489i \(0.546193\pi\)
\(992\) − 10.1047i − 0.320824i
\(993\) 12.0000i 0.380808i
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) −10.7016 −0.339092
\(997\) − 10.4922i − 0.332291i −0.986101 0.166145i \(-0.946868\pi\)
0.986101 0.166145i \(-0.0531321\pi\)
\(998\) − 17.2094i − 0.544753i
\(999\) 1.70156 0.0538350
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 1950.2.e.p.1249.4 4
3.2 odd 2 5850.2.e.bi.5149.2 4
5.2 odd 4 1950.2.a.bc.1.1 2
5.3 odd 4 1950.2.a.bg.1.2 yes 2
5.4 even 2 inner 1950.2.e.p.1249.1 4
15.2 even 4 5850.2.a.cj.1.1 2
15.8 even 4 5850.2.a.cg.1.2 2
15.14 odd 2 5850.2.e.bi.5149.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.bc.1.1 2 5.2 odd 4
1950.2.a.bg.1.2 yes 2 5.3 odd 4
1950.2.e.p.1249.1 4 5.4 even 2 inner
1950.2.e.p.1249.4 4 1.1 even 1 trivial
5850.2.a.cg.1.2 2 15.8 even 4
5850.2.a.cj.1.1 2 15.2 even 4
5850.2.e.bi.5149.2 4 3.2 odd 2
5850.2.e.bi.5149.3 4 15.14 odd 2