Properties

Label 1950.2.e.f.1249.1
Level $1950$
Weight $2$
Character 1950.1249
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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 1249.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1249
Dual form 1950.2.e.f.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} -1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} -2.00000 q^{21} -4.00000i q^{22} +6.00000i q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} +4.00000 q^{29} -1.00000i q^{32} -4.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -6.00000i q^{38} -1.00000 q^{39} -2.00000 q^{41} +2.00000i q^{42} -4.00000i q^{43} -4.00000 q^{44} +6.00000 q^{46} -1.00000i q^{48} +3.00000 q^{49} -8.00000 q^{51} +1.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} +2.00000 q^{56} -6.00000i q^{57} -4.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} +2.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -12.0000i q^{67} +8.00000i q^{68} +6.00000 q^{69} -8.00000 q^{71} -1.00000i q^{72} -8.00000i q^{73} +2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} +1.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} +2.00000 q^{84} -4.00000 q^{86} -4.00000i q^{87} +4.00000i q^{88} +14.0000 q^{89} -2.00000 q^{91} -6.00000i q^{92} -1.00000 q^{96} +16.0000i q^{97} -3.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 4 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{21} + 2 q^{24} - 2 q^{26} + 8 q^{29} - 16 q^{34} + 2 q^{36} - 2 q^{39} - 4 q^{41} - 8 q^{44} + 12 q^{46} + 6 q^{49} - 16 q^{51} + 2 q^{54} + 4 q^{56} - 8 q^{59} - 20 q^{61} - 2 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 4 q^{74} - 12 q^{76} - 16 q^{79} + 2 q^{81} + 4 q^{84} - 8 q^{86} + 28 q^{89} - 4 q^{91} - 2 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 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
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 8.00000i − 1.94029i −0.242536 0.970143i \(-0.577979\pi\)
0.242536 0.970143i \(-0.422021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) − 4.00000i − 0.852803i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000i 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.00000i − 0.696311i
\(34\) −8.00000 −1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 1.00000i 0.138675i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) − 6.00000i − 0.794719i
\(58\) − 4.00000i − 0.525226i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 8.00000i 0.970143i
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 8.00000i − 0.936329i −0.883641 0.468165i \(-0.844915\pi\)
0.883641 0.468165i \(-0.155085\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) − 8.00000i − 0.911685i
\(78\) 1.00000i 0.113228i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) − 4.00000i − 0.428845i
\(88\) 4.00000i 0.426401i
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) − 6.00000i − 0.625543i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −16.0000 −1.59206 −0.796030 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) 8.00000i 0.792118i
\(103\) − 12.0000i − 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 2.00000i − 0.188982i
\(113\) 20.0000i 1.88144i 0.339182 + 0.940721i \(0.389850\pi\)
−0.339182 + 0.940721i \(0.610150\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 1.00000i 0.0924500i
\(118\) 4.00000i 0.368230i
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 4.00000i 0.348155i
\(133\) − 12.0000i − 1.04053i
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 8.00000 0.685994
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) − 4.00000i − 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −8.00000 −0.662085
\(147\) − 3.00000i − 0.247436i
\(148\) − 2.00000i − 0.164399i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 6.00000i 0.486664i
\(153\) 8.00000i 0.646762i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) − 22.0000i − 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) − 1.00000i − 0.0785674i
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 4.00000i 0.304997i
\(173\) 22.0000i 1.67263i 0.548250 + 0.836315i \(0.315294\pi\)
−0.548250 + 0.836315i \(0.684706\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 4.00000i 0.300658i
\(178\) − 14.0000i − 1.04934i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 10.0000i 0.739221i
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) − 32.0000i − 2.34007i
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 4.00000i − 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 16.0000i 1.12576i
\(203\) − 8.00000i − 0.561490i
\(204\) 8.00000 0.560112
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) − 6.00000i − 0.417029i
\(208\) − 1.00000i − 0.0693375i
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 10.0000i 0.686803i
\(213\) 8.00000i 0.548151i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 12.0000i 0.812743i
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) − 2.00000i − 0.134231i
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 6.00000i 0.397360i
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 4.00000i 0.262613i
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) 8.00000i 0.519656i
\(238\) 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
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) − 6.00000i − 0.381771i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 24.0000i 1.50887i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 12.0000i − 0.748539i −0.927320 0.374270i \(-0.877893\pi\)
0.927320 0.374270i \(-0.122107\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) − 10.0000i − 0.617802i
\(263\) 2.00000i 0.123325i 0.998097 + 0.0616626i \(0.0196403\pi\)
−0.998097 + 0.0616626i \(0.980360\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) −12.0000 −0.735767
\(267\) − 14.0000i − 0.856786i
\(268\) 12.0000i 0.733017i
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) − 8.00000i − 0.485071i
\(273\) 2.00000i 0.121046i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 4.00000i 0.236113i
\(288\) 1.00000i 0.0589256i
\(289\) −47.0000 −2.76471
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 8.00000i 0.468165i
\(293\) − 26.0000i − 1.51894i −0.650545 0.759468i \(-0.725459\pi\)
0.650545 0.759468i \(-0.274541\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 4.00000i 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 16.0000i 0.919176i
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 8.00000 0.457330
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 8.00000i 0.455842i
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 10.0000i 0.560772i
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) − 12.0000i − 0.668734i
\(323\) − 48.0000i − 2.67079i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 12.0000i 0.663602i
\(328\) − 2.00000i − 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) − 12.0000i − 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 4.00000 0.218870
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 1.00000i 0.0543928i
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000i 0.324443i
\(343\) − 20.0000i − 1.07990i
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) 24.0000i 1.28839i 0.764862 + 0.644194i \(0.222807\pi\)
−0.764862 + 0.644194i \(0.777193\pi\)
\(348\) 4.00000i 0.214423i
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) − 4.00000i − 0.213201i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 16.0000i 0.846810i
\(358\) − 10.0000i − 0.528516i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) − 10.0000i − 0.525588i
\(363\) − 5.00000i − 0.262432i
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 36.0000i 1.87918i 0.342296 + 0.939592i \(0.388796\pi\)
−0.342296 + 0.939592i \(0.611204\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) − 2.00000i − 0.103556i −0.998659 0.0517780i \(-0.983511\pi\)
0.998659 0.0517780i \(-0.0164888\pi\)
\(374\) −32.0000 −1.65468
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.00000i − 0.206010i
\(378\) − 2.00000i − 0.102869i
\(379\) 18.0000 0.924598 0.462299 0.886724i \(-0.347025\pi\)
0.462299 + 0.886724i \(0.347025\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) − 12.0000i − 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 4.00000i 0.203331i
\(388\) − 16.0000i − 0.812277i
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 3.00000i 0.151523i
\(393\) − 10.0000i − 0.504433i
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 12.0000i 0.598506i
\(403\) 0 0
\(404\) 16.0000 0.796030
\(405\) 0 0
\(406\) −8.00000 −0.397033
\(407\) 8.00000i 0.396545i
\(408\) − 8.00000i − 0.396059i
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 12.0000i 0.591198i
\(413\) 8.00000i 0.393654i
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) − 8.00000i − 0.391762i
\(418\) − 24.0000i − 1.17388i
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 20.0000i 0.973585i
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 20.0000i 0.967868i
\(428\) 12.0000i 0.580042i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 36.0000i 1.72211i
\(438\) 8.00000i 0.382255i
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000i 0.380521i
\(443\) 16.0000i 0.760183i 0.924949 + 0.380091i \(0.124107\pi\)
−0.924949 + 0.380091i \(0.875893\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −2.00000 −0.0947027
\(447\) − 10.0000i − 0.472984i
\(448\) 2.00000i 0.0944911i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) − 20.0000i − 0.940721i
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) − 8.00000i − 0.374224i −0.982339 0.187112i \(-0.940087\pi\)
0.982339 0.187112i \(-0.0599128\pi\)
\(458\) 4.00000i 0.186908i
\(459\) 8.00000 0.373408
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 8.00000i 0.372194i
\(463\) − 26.0000i − 1.20832i −0.796862 0.604161i \(-0.793508\pi\)
0.796862 0.604161i \(-0.206492\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) − 4.00000i − 0.184115i
\(473\) − 16.0000i − 0.735681i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 10.0000i 0.457869i
\(478\) 16.0000i 0.731823i
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) − 2.00000i − 0.0910975i
\(483\) − 12.0000i − 0.546019i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 26.0000i − 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) − 32.0000i − 1.44121i
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000i 0.717698i
\(498\) − 12.0000i − 0.537733i
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 0 0
\(501\) 4.00000 0.178707
\(502\) − 6.00000i − 0.267793i
\(503\) 10.0000i 0.445878i 0.974832 + 0.222939i \(0.0715651\pi\)
−0.974832 + 0.222939i \(0.928435\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 1.00000i 0.0444116i
\(508\) 4.00000i 0.177471i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) − 1.00000i − 0.0441942i
\(513\) 6.00000i 0.264906i
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) − 4.00000i − 0.175750i
\(519\) 22.0000 0.965693
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 36.0000i 1.57417i 0.616844 + 0.787085i \(0.288411\pi\)
−0.616844 + 0.787085i \(0.711589\pi\)
\(524\) −10.0000 −0.436852
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) 0 0
\(528\) − 4.00000i − 0.174078i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 12.0000i 0.520266i
\(533\) 2.00000i 0.0866296i
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) − 10.0000i − 0.431532i
\(538\) − 24.0000i − 1.03471i
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) − 10.0000i − 0.429141i
\(544\) −8.00000 −0.342997
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 6.00000i 0.255377i
\(553\) 16.0000i 0.680389i
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) −32.0000 −1.35104
\(562\) 6.00000i 0.253095i
\(563\) 24.0000i 1.01148i 0.862686 + 0.505740i \(0.168780\pi\)
−0.862686 + 0.505740i \(0.831220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) − 2.00000i − 0.0839921i
\(568\) − 8.00000i − 0.335673i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 0 0
\(574\) 4.00000 0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 12.0000i − 0.499567i −0.968302 0.249783i \(-0.919641\pi\)
0.968302 0.249783i \(-0.0803594\pi\)
\(578\) 47.0000i 1.95494i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) − 16.0000i − 0.663221i
\(583\) − 40.0000i − 1.65663i
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) − 36.0000i − 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 3.00000i 0.123718i
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 2.00000i 0.0821995i
\(593\) 22.0000i 0.903432i 0.892162 + 0.451716i \(0.149188\pi\)
−0.892162 + 0.451716i \(0.850812\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) − 6.00000i − 0.245358i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 8.00000i 0.326056i
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 0 0
\(606\) 16.0000 0.649956
\(607\) 28.0000i 1.13648i 0.822861 + 0.568242i \(0.192376\pi\)
−0.822861 + 0.568242i \(0.807624\pi\)
\(608\) − 6.00000i − 0.243332i
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) − 8.00000i − 0.323381i
\(613\) − 10.0000i − 0.403896i −0.979396 0.201948i \(-0.935273\pi\)
0.979396 0.201948i \(-0.0647272\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 12.0000i 0.482711i
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) − 24.0000i − 0.962312i
\(623\) − 28.0000i − 1.12180i
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) − 24.0000i − 0.958468i
\(628\) 22.0000i 0.877896i
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 20.0000i 0.794929i
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) − 3.00000i − 0.118864i
\(638\) − 16.0000i − 0.633446i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) −12.0000 −0.472866
\(645\) 0 0
\(646\) −48.0000 −1.88853
\(647\) 30.0000i 1.17942i 0.807614 + 0.589711i \(0.200758\pi\)
−0.807614 + 0.589711i \(0.799242\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000i 0.626608i
\(653\) 42.0000i 1.64359i 0.569785 + 0.821794i \(0.307026\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) −2.00000 −0.0779089 −0.0389545 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 0 0
\(661\) 48.0000 1.86698 0.933492 0.358599i \(-0.116745\pi\)
0.933492 + 0.358599i \(0.116745\pi\)
\(662\) − 10.0000i − 0.388661i
\(663\) 8.00000i 0.310694i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 24.0000i 0.929284i
\(668\) − 4.00000i − 0.154765i
\(669\) −2.00000 −0.0773245
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 2.00000i 0.0771517i
\(673\) 26.0000i 1.00223i 0.865382 + 0.501113i \(0.167076\pi\)
−0.865382 + 0.501113i \(0.832924\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 30.0000i 1.15299i 0.817099 + 0.576497i \(0.195581\pi\)
−0.817099 + 0.576497i \(0.804419\pi\)
\(678\) − 20.0000i − 0.768095i
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) − 44.0000i − 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 4.00000i 0.152610i
\(688\) − 4.00000i − 0.152499i
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) − 22.0000i − 0.836315i
\(693\) 8.00000i 0.303895i
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 16.0000i 0.606043i
\(698\) − 28.0000i − 1.05982i
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 32.0000 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(702\) − 1.00000i − 0.0377426i
\(703\) 12.0000i 0.452589i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 32.0000i 1.20348i
\(708\) − 4.00000i − 0.150329i
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 14.0000i 0.524672i
\(713\) 0 0
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 16.0000i 0.597531i
\(718\) − 24.0000i − 0.895672i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) − 17.0000i − 0.632674i
\(723\) − 2.00000i − 0.0743808i
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −32.0000 −1.18356
\(732\) − 10.0000i − 0.369611i
\(733\) 38.0000i 1.40356i 0.712393 + 0.701781i \(0.247612\pi\)
−0.712393 + 0.701781i \(0.752388\pi\)
\(734\) 36.0000 1.32878
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) − 48.0000i − 1.76810i
\(738\) − 2.00000i − 0.0736210i
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) −6.00000 −0.220416
\(742\) 20.0000i 0.734223i
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) − 12.0000i − 0.439057i
\(748\) 32.0000i 1.17004i
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) − 6.00000i − 0.218652i
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 6.00000i 0.218074i 0.994038 + 0.109037i \(0.0347767\pi\)
−0.994038 + 0.109037i \(0.965223\pi\)
\(758\) − 18.0000i − 0.653789i
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 4.00000i 0.144905i
\(763\) 24.0000i 0.868858i
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 4.00000i 0.144432i
\(768\) − 1.00000i − 0.0360844i
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 4.00000i 0.143963i
\(773\) 38.0000i 1.36677i 0.730061 + 0.683383i \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −16.0000 −0.574367
\(777\) − 4.00000i − 0.143499i
\(778\) 8.00000i 0.286814i
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) − 48.0000i − 1.71648i
\(783\) 4.00000i 0.142948i
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) 2.00000 0.0712019
\(790\) 0 0
\(791\) 40.0000 1.42224
\(792\) − 4.00000i − 0.142134i
\(793\) 10.0000i 0.355110i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 0 0
\(797\) 10.0000i 0.354218i 0.984191 + 0.177109i \(0.0566745\pi\)
−0.984191 + 0.177109i \(0.943325\pi\)
\(798\) 12.0000i 0.424795i
\(799\) 0 0
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) − 30.0000i − 1.05934i
\(803\) − 32.0000i − 1.12926i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) − 24.0000i − 0.844840i
\(808\) − 16.0000i − 0.562878i
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 8.00000i 0.280745i
\(813\) − 16.0000i − 0.561144i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) − 24.0000i − 0.839654i
\(818\) − 26.0000i − 0.909069i
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 52.0000i − 1.81261i −0.422628 0.906303i \(-0.638892\pi\)
0.422628 0.906303i \(-0.361108\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 6.00000i 0.208514i
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 1.00000i 0.0346688i
\(833\) − 24.0000i − 0.831551i
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 0 0
\(838\) 10.0000i 0.345444i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 8.00000i 0.275698i
\(843\) 6.00000i 0.206651i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.0000i − 0.343604i
\(848\) − 10.0000i − 0.343401i
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) − 8.00000i − 0.274075i
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 20.0000 0.684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 4.00000i − 0.136637i −0.997664 0.0683187i \(-0.978237\pi\)
0.997664 0.0683187i \(-0.0217635\pi\)
\(858\) 4.00000i 0.136558i
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 47.0000i 1.59620i
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) − 12.0000i − 0.406371i
\(873\) − 16.0000i − 0.541518i
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) − 32.0000i − 1.07995i
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 3.00000i 0.101015i
\(883\) − 28.0000i − 0.942275i −0.882060 0.471138i \(-0.843844\pi\)
0.882060 0.471138i \(-0.156156\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 2.00000i 0.0671534i 0.999436 + 0.0335767i \(0.0106898\pi\)
−0.999436 + 0.0335767i \(0.989310\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 2.00000i 0.0669650i
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) − 6.00000i − 0.200334i
\(898\) − 6.00000i − 0.200223i
\(899\) 0 0
\(900\) 0 0
\(901\) −80.0000 −2.66519
\(902\) 8.00000i 0.266371i
\(903\) 8.00000i 0.266223i
\(904\) −20.0000 −0.665190
\(905\) 0 0
\(906\) 0 0
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) − 4.00000i − 0.132745i
\(909\) 16.0000 0.530687
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) − 6.00000i − 0.198680i
\(913\) 48.0000i 1.58857i
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) − 20.0000i − 0.660458i
\(918\) − 8.00000i − 0.264039i
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 6.00000i 0.197599i
\(923\) 8.00000i 0.263323i
\(924\) 8.00000 0.263181
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 12.0000i 0.394132i
\(928\) − 4.00000i − 0.131306i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) − 24.0000i − 0.786146i
\(933\) − 24.0000i − 0.785725i
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) − 18.0000i − 0.588034i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949923\pi\)
\(938\) 24.0000i 0.783628i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 22.0000i 0.716799i
\(943\) − 12.0000i − 0.390774i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) − 16.0000i − 0.518563i
\(953\) 24.0000i 0.777436i 0.921357 + 0.388718i \(0.127082\pi\)
−0.921357 + 0.388718i \(0.872918\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) − 16.0000i − 0.517207i
\(958\) 32.0000i 1.03387i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 2.00000i − 0.0644826i
\(963\) 12.0000i 0.386695i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 14.0000i 0.450210i 0.974335 + 0.225105i \(0.0722725\pi\)
−0.974335 + 0.225105i \(0.927728\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 16.0000i − 0.512936i
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 16.0000i 0.511624i
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) − 42.0000i − 1.34027i
\(983\) − 52.0000i − 1.65854i −0.558846 0.829271i \(-0.688756\pi\)
0.558846 0.829271i \(-0.311244\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −32.0000 −1.01909
\(987\) 0 0
\(988\) 6.00000i 0.190885i
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) − 10.0000i − 0.317340i
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) 38.0000i 1.20287i
\(999\) −2.00000 −0.0632772
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.f.1249.1 2
3.2 odd 2 5850.2.e.i.5149.2 2
5.2 odd 4 390.2.a.e.1.1 1
5.3 odd 4 1950.2.a.h.1.1 1
5.4 even 2 inner 1950.2.e.f.1249.2 2
15.2 even 4 1170.2.a.e.1.1 1
15.8 even 4 5850.2.a.bi.1.1 1
15.14 odd 2 5850.2.e.i.5149.1 2
20.7 even 4 3120.2.a.o.1.1 1
60.47 odd 4 9360.2.a.bh.1.1 1
65.12 odd 4 5070.2.a.e.1.1 1
65.47 even 4 5070.2.b.e.1351.2 2
65.57 even 4 5070.2.b.e.1351.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.e.1.1 1 5.2 odd 4
1170.2.a.e.1.1 1 15.2 even 4
1950.2.a.h.1.1 1 5.3 odd 4
1950.2.e.f.1249.1 2 1.1 even 1 trivial
1950.2.e.f.1249.2 2 5.4 even 2 inner
3120.2.a.o.1.1 1 20.7 even 4
5070.2.a.e.1.1 1 65.12 odd 4
5070.2.b.e.1351.1 2 65.57 even 4
5070.2.b.e.1351.2 2 65.47 even 4
5850.2.a.bi.1.1 1 15.8 even 4
5850.2.e.i.5149.1 2 15.14 odd 2
5850.2.e.i.5149.2 2 3.2 odd 2
9360.2.a.bh.1.1 1 60.47 odd 4