Properties

Label 1950.2.e.n.1249.2
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.n.1249.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} -7.00000 q^{19} -4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -5.00000 q^{29} +4.00000 q^{31} +1.00000i q^{32} -4.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} +9.00000i q^{37} -7.00000i q^{38} -1.00000 q^{39} -5.00000 q^{41} -4.00000i q^{42} +10.0000i q^{43} -4.00000 q^{44} +4.00000 q^{46} +3.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -4.00000 q^{51} +1.00000i q^{52} -9.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} +7.00000i q^{57} -5.00000i q^{58} +6.00000 q^{59} +4.00000 q^{61} +4.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -7.00000i q^{67} +4.00000i q^{68} -4.00000 q^{69} -15.0000 q^{71} +1.00000i q^{72} -12.0000i q^{73} -9.00000 q^{74} +7.00000 q^{76} -16.0000i q^{77} -1.00000i q^{78} -7.00000 q^{79} +1.00000 q^{81} -5.00000i q^{82} -6.00000i q^{83} +4.00000 q^{84} -10.0000 q^{86} +5.00000i q^{87} -4.00000i q^{88} -14.0000 q^{89} -4.00000 q^{91} +4.00000i q^{92} -4.00000i q^{93} -3.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} -9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{11} + 8q^{14} + 2q^{16} - 14q^{19} - 8q^{21} - 2q^{24} + 2q^{26} - 10q^{29} + 8q^{31} + 8q^{34} + 2q^{36} - 2q^{39} - 10q^{41} - 8q^{44} + 8q^{46} - 18q^{49} - 8q^{51} - 2q^{54} - 8q^{56} + 12q^{59} + 8q^{61} - 2q^{64} + 8q^{66} - 8q^{69} - 30q^{71} - 18q^{74} + 14q^{76} - 14q^{79} + 2q^{81} + 8q^{84} - 20q^{86} - 28q^{89} - 8q^{91} - 6q^{94} + 2q^{96} - 8q^{99} + O(q^{100}) \)

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
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 1.00000i − 0.277350i
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 4.00000i 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 9.00000i 1.47959i 0.672832 + 0.739795i \(0.265078\pi\)
−0.672832 + 0.739795i \(0.734922\pi\)
\(38\) − 7.00000i − 1.13555i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 1.00000i 0.138675i
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 7.00000i 0.927173i
\(58\) − 5.00000i − 0.656532i
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 7.00000i − 0.855186i −0.903971 0.427593i \(-0.859362\pi\)
0.903971 0.427593i \(-0.140638\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 12.0000i − 1.40449i −0.711934 0.702247i \(-0.752180\pi\)
0.711934 0.702247i \(-0.247820\pi\)
\(74\) −9.00000 −1.04623
\(75\) 0 0
\(76\) 7.00000 0.802955
\(77\) − 16.0000i − 1.82337i
\(78\) − 1.00000i − 0.113228i
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 5.00000i − 0.552158i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 5.00000i 0.536056i
\(88\) − 4.00000i − 0.426401i
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 4.00000i 0.417029i
\(93\) − 4.00000i − 0.414781i
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 5.00000i 0.483368i 0.970355 + 0.241684i \(0.0776998\pi\)
−0.970355 + 0.241684i \(0.922300\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) − 4.00000i − 0.377964i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −7.00000 −0.655610
\(115\) 0 0
\(116\) 5.00000 0.464238
\(117\) 1.00000i 0.0924500i
\(118\) 6.00000i 0.552345i
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 4.00000i 0.362143i
\(123\) 5.00000i 0.450835i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) − 5.00000i − 0.443678i −0.975083 0.221839i \(-0.928794\pi\)
0.975083 0.221839i \(-0.0712060\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 17.0000 1.48530 0.742648 0.669681i \(-0.233569\pi\)
0.742648 + 0.669681i \(0.233569\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 28.0000i 2.42791i
\(134\) 7.00000 0.604708
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) − 19.0000i − 1.62328i −0.584158 0.811640i \(-0.698575\pi\)
0.584158 0.811640i \(-0.301425\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) − 15.0000i − 1.25877i
\(143\) − 4.00000i − 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) 9.00000i 0.742307i
\(148\) − 9.00000i − 0.739795i
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 7.00000i 0.567775i
\(153\) 4.00000i 0.323381i
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) − 7.00000i − 0.556890i
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 1.00000i 0.0785674i
\(163\) 24.0000i 1.87983i 0.341415 + 0.939913i \(0.389094\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 5.00000 0.390434
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 9.00000i 0.696441i 0.937413 + 0.348220i \(0.113214\pi\)
−0.937413 + 0.348220i \(0.886786\pi\)
\(168\) 4.00000i 0.308607i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) − 10.0000i − 0.762493i
\(173\) − 13.0000i − 0.988372i −0.869356 0.494186i \(-0.835466\pi\)
0.869356 0.494186i \(-0.164534\pi\)
\(174\) −5.00000 −0.379049
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 6.00000i − 0.450988i
\(178\) − 14.0000i − 1.04934i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −4.00000 −0.297318 −0.148659 0.988889i \(-0.547496\pi\)
−0.148659 + 0.988889i \(0.547496\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) − 4.00000i − 0.295689i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) − 16.0000i − 1.17004i
\(188\) − 3.00000i − 0.218797i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 12.0000i 0.863779i 0.901927 + 0.431889i \(0.142153\pi\)
−0.901927 + 0.431889i \(0.857847\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 16.0000i − 1.13995i −0.821661 0.569976i \(-0.806952\pi\)
0.821661 0.569976i \(-0.193048\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 3.00000 0.212664 0.106332 0.994331i \(-0.466089\pi\)
0.106332 + 0.994331i \(0.466089\pi\)
\(200\) 0 0
\(201\) −7.00000 −0.493742
\(202\) − 10.0000i − 0.703598i
\(203\) 20.0000i 1.40372i
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000i 0.278019i
\(208\) − 1.00000i − 0.0693375i
\(209\) −28.0000 −1.93680
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 9.00000i 0.618123i
\(213\) 15.0000i 1.02778i
\(214\) −5.00000 −0.341793
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) − 16.0000i − 1.08615i
\(218\) 11.0000i 0.745014i
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 9.00000i 0.604040i
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 6.00000i 0.398234i 0.979976 + 0.199117i \(0.0638074\pi\)
−0.979976 + 0.199117i \(0.936193\pi\)
\(228\) − 7.00000i − 0.463586i
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 5.00000i 0.328266i
\(233\) − 8.00000i − 0.524097i −0.965055 0.262049i \(-0.915602\pi\)
0.965055 0.262049i \(-0.0843981\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 7.00000i 0.454699i
\(238\) − 16.0000i − 1.03713i
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 5.00000i 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) 7.00000i 0.445399i
\(248\) − 4.00000i − 0.254000i
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) − 16.0000i − 1.00591i
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 8.00000i 0.499026i 0.968371 + 0.249513i \(0.0802706\pi\)
−0.968371 + 0.249513i \(0.919729\pi\)
\(258\) 10.0000i 0.622573i
\(259\) 36.0000 2.23693
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 17.0000i 1.05026i
\(263\) − 30.0000i − 1.84988i −0.380114 0.924940i \(-0.624115\pi\)
0.380114 0.924940i \(-0.375885\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −28.0000 −1.71679
\(267\) 14.0000i 0.856786i
\(268\) 7.00000i 0.427593i
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) − 4.00000i − 0.242536i
\(273\) 4.00000i 0.242091i
\(274\) 19.0000 1.14783
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 4.00000i 0.239904i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 17.0000 1.01413 0.507067 0.861906i \(-0.330729\pi\)
0.507067 + 0.861906i \(0.330729\pi\)
\(282\) 3.00000i 0.178647i
\(283\) − 8.00000i − 0.475551i −0.971320 0.237775i \(-0.923582\pi\)
0.971320 0.237775i \(-0.0764182\pi\)
\(284\) 15.0000 0.890086
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 20.0000i 1.18056i
\(288\) − 1.00000i − 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 12.0000i 0.702247i
\(293\) 12.0000i 0.701047i 0.936554 + 0.350524i \(0.113996\pi\)
−0.936554 + 0.350524i \(0.886004\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 9.00000 0.523114
\(297\) 4.00000i 0.232104i
\(298\) 4.00000i 0.231714i
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 40.0000 2.30556
\(302\) − 4.00000i − 0.230174i
\(303\) 10.0000i 0.574485i
\(304\) −7.00000 −0.401478
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 21.0000i 1.19853i 0.800549 + 0.599267i \(0.204541\pi\)
−0.800549 + 0.599267i \(0.795459\pi\)
\(308\) 16.0000i 0.911685i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) 23.0000i 1.30004i 0.759918 + 0.650018i \(0.225239\pi\)
−0.759918 + 0.650018i \(0.774761\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 7.00000 0.393781
\(317\) − 28.0000i − 1.57264i −0.617822 0.786318i \(-0.711985\pi\)
0.617822 0.786318i \(-0.288015\pi\)
\(318\) − 9.00000i − 0.504695i
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) − 16.0000i − 0.891645i
\(323\) 28.0000i 1.55796i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −24.0000 −1.32924
\(327\) − 11.0000i − 0.608301i
\(328\) 5.00000i 0.276079i
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 6.00000i 0.329293i
\(333\) − 9.00000i − 0.493197i
\(334\) −9.00000 −0.492458
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) − 34.0000i − 1.85210i −0.377403 0.926049i \(-0.623183\pi\)
0.377403 0.926049i \(-0.376817\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 7.00000i 0.378517i
\(343\) 8.00000i 0.431959i
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 13.0000 0.698884
\(347\) 11.0000i 0.590511i 0.955418 + 0.295255i \(0.0954048\pi\)
−0.955418 + 0.295255i \(0.904595\pi\)
\(348\) − 5.00000i − 0.268028i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 4.00000i 0.213201i
\(353\) 21.0000i 1.11772i 0.829263 + 0.558859i \(0.188761\pi\)
−0.829263 + 0.558859i \(0.811239\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 16.0000i 0.846810i
\(358\) 4.00000i 0.211407i
\(359\) 15.0000 0.791670 0.395835 0.918322i \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) − 4.00000i − 0.210235i
\(363\) − 5.00000i − 0.262432i
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 7.00000i 0.365397i 0.983169 + 0.182699i \(0.0584832\pi\)
−0.983169 + 0.182699i \(0.941517\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 5.00000 0.260290
\(370\) 0 0
\(371\) −36.0000 −1.86903
\(372\) 4.00000i 0.207390i
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) 3.00000 0.154713
\(377\) 5.00000i 0.257513i
\(378\) 4.00000i 0.205738i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 10.0000i 0.511645i
\(383\) − 5.00000i − 0.255488i −0.991807 0.127744i \(-0.959226\pi\)
0.991807 0.127744i \(-0.0407736\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −12.0000 −0.610784
\(387\) − 10.0000i − 0.508329i
\(388\) 16.0000i 0.812277i
\(389\) −5.00000 −0.253510 −0.126755 0.991934i \(-0.540456\pi\)
−0.126755 + 0.991934i \(0.540456\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 9.00000i 0.454569i
\(393\) − 17.0000i − 0.857537i
\(394\) 16.0000 0.806068
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) − 15.0000i − 0.752828i −0.926451 0.376414i \(-0.877157\pi\)
0.926451 0.376414i \(-0.122843\pi\)
\(398\) 3.00000i 0.150376i
\(399\) 28.0000 1.40175
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) − 7.00000i − 0.349128i
\(403\) − 4.00000i − 0.199254i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) 36.0000i 1.78445i
\(408\) 4.00000i 0.198030i
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) −19.0000 −0.937201
\(412\) 8.00000i 0.394132i
\(413\) − 24.0000i − 1.18096i
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) − 4.00000i − 0.195881i
\(418\) − 28.0000i − 1.36952i
\(419\) −23.0000 −1.12362 −0.561812 0.827265i \(-0.689895\pi\)
−0.561812 + 0.827265i \(0.689895\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) − 16.0000i − 0.778868i
\(423\) − 3.00000i − 0.145865i
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −15.0000 −0.726752
\(427\) − 16.0000i − 0.774294i
\(428\) − 5.00000i − 0.241684i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 19.0000 0.915198 0.457599 0.889159i \(-0.348710\pi\)
0.457599 + 0.889159i \(0.348710\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 19.0000i 0.913082i 0.889702 + 0.456541i \(0.150912\pi\)
−0.889702 + 0.456541i \(0.849088\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −11.0000 −0.526804
\(437\) 28.0000i 1.33942i
\(438\) − 12.0000i − 0.573382i
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 4.00000i − 0.190261i
\(443\) − 25.0000i − 1.18779i −0.804544 0.593893i \(-0.797590\pi\)
0.804544 0.593893i \(-0.202410\pi\)
\(444\) −9.00000 −0.427121
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) − 4.00000i − 0.189194i
\(448\) 4.00000i 0.188982i
\(449\) 27.0000 1.27421 0.637104 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) − 14.0000i − 0.658505i
\(453\) 4.00000i 0.187936i
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 7.00000 0.327805
\(457\) 4.00000i 0.187112i 0.995614 + 0.0935561i \(0.0298234\pi\)
−0.995614 + 0.0935561i \(0.970177\pi\)
\(458\) 17.0000i 0.794358i
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) − 16.0000i − 0.744387i
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 3.00000i 0.138823i 0.997588 + 0.0694117i \(0.0221122\pi\)
−0.997588 + 0.0694117i \(0.977888\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) − 6.00000i − 0.276172i
\(473\) 40.0000i 1.83920i
\(474\) −7.00000 −0.321521
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 9.00000i 0.412082i
\(478\) − 16.0000i − 0.731823i
\(479\) −15.0000 −0.685367 −0.342684 0.939451i \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 0 0
\(481\) 9.00000 0.410365
\(482\) 8.00000i 0.364390i
\(483\) 16.0000i 0.728025i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 22.0000i 0.996915i 0.866914 + 0.498458i \(0.166100\pi\)
−0.866914 + 0.498458i \(0.833900\pi\)
\(488\) − 4.00000i − 0.181071i
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) − 5.00000i − 0.225417i
\(493\) 20.0000i 0.900755i
\(494\) −7.00000 −0.314945
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 60.0000i 2.69137i
\(498\) − 6.00000i − 0.268866i
\(499\) 23.0000 1.02962 0.514811 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 0 0
\(501\) 9.00000 0.402090
\(502\) − 5.00000i − 0.223161i
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 1.00000i 0.0444116i
\(508\) 5.00000i 0.221839i
\(509\) 4.00000 0.177297 0.0886484 0.996063i \(-0.471745\pi\)
0.0886484 + 0.996063i \(0.471745\pi\)
\(510\) 0 0
\(511\) −48.0000 −2.12339
\(512\) 1.00000i 0.0441942i
\(513\) − 7.00000i − 0.309058i
\(514\) −8.00000 −0.352865
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 12.0000i 0.527759i
\(518\) 36.0000i 1.58175i
\(519\) −13.0000 −0.570637
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 5.00000i 0.218844i
\(523\) 14.0000i 0.612177i 0.952003 + 0.306089i \(0.0990204\pi\)
−0.952003 + 0.306089i \(0.900980\pi\)
\(524\) −17.0000 −0.742648
\(525\) 0 0
\(526\) 30.0000 1.30806
\(527\) − 16.0000i − 0.696971i
\(528\) − 4.00000i − 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) − 28.0000i − 1.21395i
\(533\) 5.00000i 0.216574i
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) −7.00000 −0.302354
\(537\) − 4.00000i − 0.172613i
\(538\) − 5.00000i − 0.215565i
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 32.0000i 1.37452i
\(543\) 4.00000i 0.171656i
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 6.00000i 0.256541i 0.991739 + 0.128271i \(0.0409426\pi\)
−0.991739 + 0.128271i \(0.959057\pi\)
\(548\) 19.0000i 0.811640i
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 35.0000 1.49105
\(552\) 4.00000i 0.170251i
\(553\) 28.0000i 1.19068i
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) − 38.0000i − 1.61011i −0.593199 0.805056i \(-0.702135\pi\)
0.593199 0.805056i \(-0.297865\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 17.0000i 0.717102i
\(563\) − 39.0000i − 1.64365i −0.569737 0.821827i \(-0.692955\pi\)
0.569737 0.821827i \(-0.307045\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) − 4.00000i − 0.167984i
\(568\) 15.0000i 0.629386i
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 4.00000i 0.167248i
\(573\) − 10.0000i − 0.417756i
\(574\) −20.0000 −0.834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 8.00000i − 0.333044i −0.986038 0.166522i \(-0.946746\pi\)
0.986038 0.166522i \(-0.0532537\pi\)
\(578\) 1.00000i 0.0415945i
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) −24.0000 −0.995688
\(582\) − 16.0000i − 0.663221i
\(583\) − 36.0000i − 1.49097i
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) −28.0000 −1.15372
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 9.00000i 0.369898i
\(593\) − 29.0000i − 1.19089i −0.803397 0.595444i \(-0.796976\pi\)
0.803397 0.595444i \(-0.203024\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) − 3.00000i − 0.122782i
\(598\) − 4.00000i − 0.163572i
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −3.00000 −0.122373 −0.0611863 0.998126i \(-0.519488\pi\)
−0.0611863 + 0.998126i \(0.519488\pi\)
\(602\) 40.0000i 1.63028i
\(603\) 7.00000i 0.285062i
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) − 3.00000i − 0.121766i −0.998145 0.0608831i \(-0.980608\pi\)
0.998145 0.0608831i \(-0.0193917\pi\)
\(608\) − 7.00000i − 0.283887i
\(609\) 20.0000 0.810441
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) − 4.00000i − 0.161690i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) −21.0000 −0.847491
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) 7.00000i 0.281809i 0.990023 + 0.140905i \(0.0450011\pi\)
−0.990023 + 0.140905i \(0.954999\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 2.00000i 0.0801927i
\(623\) 56.0000i 2.24359i
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −23.0000 −0.919265
\(627\) 28.0000i 1.11821i
\(628\) 14.0000i 0.558661i
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 7.00000i 0.278445i
\(633\) 16.0000i 0.635943i
\(634\) 28.0000 1.11202
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 9.00000i 0.356593i
\(638\) − 20.0000i − 0.791808i
\(639\) 15.0000 0.593391
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 5.00000i 0.197334i
\(643\) 5.00000i 0.197181i 0.995128 + 0.0985904i \(0.0314334\pi\)
−0.995128 + 0.0985904i \(0.968567\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −28.0000 −1.10165
\(647\) − 36.0000i − 1.41531i −0.706560 0.707653i \(-0.749754\pi\)
0.706560 0.707653i \(-0.250246\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) − 24.0000i − 0.939913i
\(653\) 10.0000i 0.391330i 0.980671 + 0.195665i \(0.0626866\pi\)
−0.980671 + 0.195665i \(0.937313\pi\)
\(654\) 11.0000 0.430134
\(655\) 0 0
\(656\) −5.00000 −0.195217
\(657\) 12.0000i 0.468165i
\(658\) 12.0000i 0.467809i
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) −7.00000 −0.272268 −0.136134 0.990690i \(-0.543468\pi\)
−0.136134 + 0.990690i \(0.543468\pi\)
\(662\) − 28.0000i − 1.08825i
\(663\) 4.00000i 0.155347i
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 9.00000 0.348743
\(667\) 20.0000i 0.774403i
\(668\) − 9.00000i − 0.348220i
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 16.0000 0.617673
\(672\) − 4.00000i − 0.154303i
\(673\) − 19.0000i − 0.732396i −0.930537 0.366198i \(-0.880659\pi\)
0.930537 0.366198i \(-0.119341\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) − 14.0000i − 0.538064i −0.963131 0.269032i \(-0.913296\pi\)
0.963131 0.269032i \(-0.0867037\pi\)
\(678\) 14.0000i 0.537667i
\(679\) −64.0000 −2.45609
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 16.0000i 0.612672i
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −7.00000 −0.267652
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) − 17.0000i − 0.648590i
\(688\) 10.0000i 0.381246i
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) 3.00000 0.114125 0.0570627 0.998371i \(-0.481827\pi\)
0.0570627 + 0.998371i \(0.481827\pi\)
\(692\) 13.0000i 0.494186i
\(693\) 16.0000i 0.607790i
\(694\) −11.0000 −0.417554
\(695\) 0 0
\(696\) 5.00000 0.189525
\(697\) 20.0000i 0.757554i
\(698\) 34.0000i 1.28692i
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) − 63.0000i − 2.37609i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −21.0000 −0.790345
\(707\) 40.0000i 1.50435i
\(708\) 6.00000i 0.225494i
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 7.00000 0.262521
\(712\) 14.0000i 0.524672i
\(713\) − 16.0000i − 0.599205i
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 16.0000i 0.597531i
\(718\) 15.0000i 0.559795i
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 30.0000i 1.11648i
\(723\) − 8.00000i − 0.297523i
\(724\) 4.00000 0.148659
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 4.00000i 0.148250i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 40.0000 1.47945
\(732\) 4.00000i 0.147844i
\(733\) 3.00000i 0.110808i 0.998464 + 0.0554038i \(0.0176446\pi\)
−0.998464 + 0.0554038i \(0.982355\pi\)
\(734\) −7.00000 −0.258375
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) − 28.0000i − 1.03139i
\(738\) 5.00000i 0.184053i
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 7.00000 0.257151
\(742\) − 36.0000i − 1.32160i
\(743\) 21.0000i 0.770415i 0.922830 + 0.385208i \(0.125870\pi\)
−0.922830 + 0.385208i \(0.874130\pi\)
\(744\) −4.00000 −0.146647
\(745\) 0 0
\(746\) 22.0000 0.805477
\(747\) 6.00000i 0.219529i
\(748\) 16.0000i 0.585018i
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 3.00000i 0.109399i
\(753\) 5.00000i 0.182210i
\(754\) −5.00000 −0.182089
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 8.00000i 0.290765i 0.989376 + 0.145382i \(0.0464413\pi\)
−0.989376 + 0.145382i \(0.953559\pi\)
\(758\) 20.0000i 0.726433i
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 35.0000 1.26875 0.634375 0.773026i \(-0.281258\pi\)
0.634375 + 0.773026i \(0.281258\pi\)
\(762\) − 5.00000i − 0.181131i
\(763\) − 44.0000i − 1.59291i
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) 5.00000 0.180657
\(767\) − 6.00000i − 0.216647i
\(768\) − 1.00000i − 0.0360844i
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) − 12.0000i − 0.431889i
\(773\) 8.00000i 0.287740i 0.989597 + 0.143870i \(0.0459547\pi\)
−0.989597 + 0.143870i \(0.954045\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) − 36.0000i − 1.29149i
\(778\) − 5.00000i − 0.179259i
\(779\) 35.0000 1.25401
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) − 16.0000i − 0.572159i
\(783\) − 5.00000i − 0.178685i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 17.0000 0.606370
\(787\) − 32.0000i − 1.14068i −0.821410 0.570338i \(-0.806812\pi\)
0.821410 0.570338i \(-0.193188\pi\)
\(788\) 16.0000i 0.569976i
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 56.0000 1.99113
\(792\) 4.00000i 0.142134i
\(793\) − 4.00000i − 0.142044i
\(794\) 15.0000 0.532330
\(795\) 0 0
\(796\) −3.00000 −0.106332
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 28.0000i 0.991189i
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) − 30.0000i − 1.05934i
\(803\) − 48.0000i − 1.69388i
\(804\) 7.00000 0.246871
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 5.00000i 0.176008i
\(808\) 10.0000i 0.351799i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) − 20.0000i − 0.701862i
\(813\) − 32.0000i − 1.12229i
\(814\) −36.0000 −1.26180
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) − 70.0000i − 2.44899i
\(818\) 24.0000i 0.839140i
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) − 19.0000i − 0.662701i
\(823\) 19.0000i 0.662298i 0.943578 + 0.331149i \(0.107436\pi\)
−0.943578 + 0.331149i \(0.892564\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) − 4.00000i − 0.139094i −0.997579 0.0695468i \(-0.977845\pi\)
0.997579 0.0695468i \(-0.0221553\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 1.00000i 0.0346688i
\(833\) 36.0000i 1.24733i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 28.0000 0.968400
\(837\) 4.00000i 0.138260i
\(838\) − 23.0000i − 0.794522i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 14.0000i 0.482472i
\(843\) − 17.0000i − 0.585511i
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) 3.00000 0.103142
\(847\) − 20.0000i − 0.687208i
\(848\) − 9.00000i − 0.309061i
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 36.0000 1.23406
\(852\) − 15.0000i − 0.513892i
\(853\) − 19.0000i − 0.650548i −0.945620 0.325274i \(-0.894544\pi\)
0.945620 0.325274i \(-0.105456\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 5.00000 0.170896
\(857\) − 50.0000i − 1.70797i −0.520300 0.853984i \(-0.674180\pi\)
0.520300 0.853984i \(-0.325820\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) 2.00000 0.0682391 0.0341196 0.999418i \(-0.489137\pi\)
0.0341196 + 0.999418i \(0.489137\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) 19.0000i 0.647143i
\(863\) − 39.0000i − 1.32758i −0.747921 0.663788i \(-0.768948\pi\)
0.747921 0.663788i \(-0.231052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −19.0000 −0.645646
\(867\) − 1.00000i − 0.0339618i
\(868\) 16.0000i 0.543075i
\(869\) −28.0000 −0.949835
\(870\) 0 0
\(871\) −7.00000 −0.237186
\(872\) − 11.0000i − 0.372507i
\(873\) 16.0000i 0.541518i
\(874\) −28.0000 −0.947114
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 43.0000i 1.45201i 0.687691 + 0.726003i \(0.258624\pi\)
−0.687691 + 0.726003i \(0.741376\pi\)
\(878\) 7.00000i 0.236239i
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 9.00000i 0.303046i
\(883\) − 40.0000i − 1.34611i −0.739594 0.673054i \(-0.764982\pi\)
0.739594 0.673054i \(-0.235018\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 25.0000 0.839891
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) − 9.00000i − 0.302020i
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) − 2.00000i − 0.0669650i
\(893\) − 21.0000i − 0.702738i
\(894\) 4.00000 0.133780
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 4.00000i 0.133556i
\(898\) 27.0000i 0.901002i
\(899\) −20.0000 −0.667037
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) − 20.0000i − 0.665927i
\(903\) − 40.0000i − 1.33112i
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) −4.00000 −0.132891
\(907\) 50.0000i 1.66022i 0.557598 + 0.830111i \(0.311723\pi\)
−0.557598 + 0.830111i \(0.688277\pi\)
\(908\) − 6.00000i − 0.199117i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 7.00000i 0.231793i
\(913\) − 24.0000i − 0.794284i
\(914\) −4.00000 −0.132308
\(915\) 0 0
\(916\) −17.0000 −0.561696
\(917\) − 68.0000i − 2.24556i
\(918\) 4.00000i 0.132020i
\(919\) −29.0000 −0.956622 −0.478311 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) 21.0000 0.691974
\(922\) − 24.0000i − 0.790398i
\(923\) 15.0000i 0.493731i
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 8.00000i 0.262754i
\(928\) − 5.00000i − 0.164133i
\(929\) −11.0000 −0.360898 −0.180449 0.983584i \(-0.557755\pi\)
−0.180449 + 0.983584i \(0.557755\pi\)
\(930\) 0 0
\(931\) 63.0000 2.06474
\(932\) 8.00000i 0.262049i
\(933\) − 2.00000i − 0.0654771i
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) − 22.0000i − 0.718709i −0.933201 0.359354i \(-0.882997\pi\)
0.933201 0.359354i \(-0.117003\pi\)
\(938\) − 28.0000i − 0.914232i
\(939\) 23.0000 0.750577
\(940\) 0 0
\(941\) 52.0000 1.69515 0.847576 0.530674i \(-0.178061\pi\)
0.847576 + 0.530674i \(0.178061\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) 20.0000i 0.651290i
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −40.0000 −1.30051
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) − 7.00000i − 0.227349i
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −28.0000 −0.907962
\(952\) 16.0000i 0.518563i
\(953\) − 36.0000i − 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 20.0000i 0.646508i
\(958\) − 15.0000i − 0.484628i
\(959\) −76.0000 −2.45417
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 9.00000i 0.290172i
\(963\) − 5.00000i − 0.161123i
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) − 44.0000i − 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) 28.0000 0.899490
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 16.0000i − 0.512936i
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 24.0000i 0.767435i
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) − 40.0000i − 1.27645i
\(983\) 48.0000i 1.53096i 0.643458 + 0.765481i \(0.277499\pi\)
−0.643458 + 0.765481i \(0.722501\pi\)
\(984\) 5.00000 0.159394
\(985\) 0 0
\(986\) −20.0000 −0.636930
\(987\) − 12.0000i − 0.381964i
\(988\) − 7.00000i − 0.222700i
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 28.0000i 0.888553i
\(994\) −60.0000 −1.90308
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 23.0000i 0.728052i
\(999\) −9.00000 −0.284747
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.n.1249.2 2
3.2 odd 2 5850.2.e.c.5149.1 2
5.2 odd 4 1950.2.a.e.1.1 1
5.3 odd 4 1950.2.a.x.1.1 yes 1
5.4 even 2 inner 1950.2.e.n.1249.1 2
15.2 even 4 5850.2.a.by.1.1 1
15.8 even 4 5850.2.a.a.1.1 1
15.14 odd 2 5850.2.e.c.5149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.e.1.1 1 5.2 odd 4
1950.2.a.x.1.1 yes 1 5.3 odd 4
1950.2.e.n.1249.1 2 5.4 even 2 inner
1950.2.e.n.1249.2 2 1.1 even 1 trivial
5850.2.a.a.1.1 1 15.8 even 4
5850.2.a.by.1.1 1 15.2 even 4
5850.2.e.c.5149.1 2 3.2 odd 2
5850.2.e.c.5149.2 2 15.14 odd 2