Properties

Label 1850.2.b.e.149.1
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.e.149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} +2.00000 q^{9} +3.00000 q^{11} -1.00000i q^{12} -6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -2.00000i q^{18} +3.00000 q^{19} -4.00000 q^{21} -3.00000i q^{22} -2.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +5.00000i q^{27} -4.00000i q^{28} -1.00000i q^{32} +3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} -1.00000i q^{37} -3.00000i q^{38} +6.00000 q^{39} -3.00000 q^{41} +4.00000i q^{42} +4.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} +4.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} -3.00000 q^{51} +6.00000i q^{52} +2.00000i q^{53} +5.00000 q^{54} -4.00000 q^{56} +3.00000i q^{57} +12.0000 q^{59} +12.0000 q^{61} +8.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} +9.00000i q^{67} -3.00000i q^{68} +2.00000 q^{69} -2.00000 q^{71} +2.00000i q^{72} +9.00000i q^{73} -1.00000 q^{74} -3.00000 q^{76} +12.0000i q^{77} -6.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} +3.00000i q^{82} +7.00000i q^{83} +4.00000 q^{84} +4.00000 q^{86} +3.00000i q^{88} +3.00000 q^{89} +24.0000 q^{91} +2.00000i q^{92} +4.00000 q^{94} +1.00000 q^{96} +2.00000i q^{97} +9.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 6 q^{19} - 8 q^{21} - 2 q^{24} - 12 q^{26} + 6 q^{34} - 4 q^{36} + 12 q^{39} - 6 q^{41} - 6 q^{44} - 4 q^{46} - 18 q^{49} - 6 q^{51} + 10 q^{54} - 8 q^{56} + 24 q^{59} + 24 q^{61} - 2 q^{64} + 6 q^{66} + 4 q^{69} - 4 q^{71} - 2 q^{74} - 6 q^{76} + 4 q^{79} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 6 q^{89} + 48 q^{91} + 8 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) − 3.00000i − 0.639602i
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 5.00000i 0.962250i
\(28\) − 4.00000i − 0.755929i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 3.00000i 0.522233i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) − 1.00000i − 0.164399i
\(38\) − 3.00000i − 0.486664i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 6.00000i 0.832050i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 3.00000i 0.397360i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 8.00000i 1.00791i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 9.00000i 1.09952i 0.835321 + 0.549762i \(0.185282\pi\)
−0.835321 + 0.549762i \(0.814718\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 2.00000i 0.235702i
\(73\) 9.00000i 1.05337i 0.850060 + 0.526685i \(0.176565\pi\)
−0.850060 + 0.526685i \(0.823435\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) 12.0000i 1.36753i
\(78\) − 6.00000i − 0.679366i
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000i 0.331295i
\(83\) 7.00000i 0.768350i 0.923260 + 0.384175i \(0.125514\pi\)
−0.923260 + 0.384175i \(0.874486\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 3.00000i 0.319801i
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) 2.00000i 0.208514i
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 3.00000i 0.297044i
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 1.00000i − 0.0966736i −0.998831 0.0483368i \(-0.984608\pi\)
0.998831 0.0483368i \(-0.0153921\pi\)
\(108\) − 5.00000i − 0.481125i
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) 4.00000i 0.377964i
\(113\) − 5.00000i − 0.470360i −0.971952 0.235180i \(-0.924432\pi\)
0.971952 0.235180i \(-0.0755680\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) 0 0
\(117\) − 12.0000i − 1.10940i
\(118\) − 12.0000i − 1.10469i
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) − 12.0000i − 1.08643i
\(123\) − 3.00000i − 0.270501i
\(124\) 0 0
\(125\) 0 0
\(126\) 8.00000 0.712697
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 3.00000i − 0.261116i
\(133\) 12.0000i 1.04053i
\(134\) 9.00000 0.777482
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) − 7.00000i − 0.598050i −0.954245 0.299025i \(-0.903339\pi\)
0.954245 0.299025i \(-0.0966615\pi\)
\(138\) − 2.00000i − 0.170251i
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 2.00000i 0.167836i
\(143\) − 18.0000i − 1.50524i
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) 9.00000 0.744845
\(147\) − 9.00000i − 0.742307i
\(148\) 1.00000i 0.0821995i
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) 3.00000i 0.243332i
\(153\) 6.00000i 0.485071i
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 24.0000i 1.91541i 0.287754 + 0.957704i \(0.407091\pi\)
−0.287754 + 0.957704i \(0.592909\pi\)
\(158\) − 2.00000i − 0.159111i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) 17.0000i 1.33154i 0.746156 + 0.665771i \(0.231897\pi\)
−0.746156 + 0.665771i \(0.768103\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 7.00000 0.543305
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) − 4.00000i − 0.304997i
\(173\) − 24.0000i − 1.82469i −0.409426 0.912343i \(-0.634271\pi\)
0.409426 0.912343i \(-0.365729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 12.0000i 0.901975i
\(178\) − 3.00000i − 0.224860i
\(179\) −21.0000 −1.56961 −0.784807 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) − 24.0000i − 1.77900i
\(183\) 12.0000i 0.887066i
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000i 0.658145i
\(188\) − 4.00000i − 0.291730i
\(189\) −20.0000 −1.45479
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 11.0000i − 0.791797i −0.918294 0.395899i \(-0.870433\pi\)
0.918294 0.395899i \(-0.129567\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) − 6.00000i − 0.426401i
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −9.00000 −0.634811
\(202\) 10.0000i 0.703598i
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) − 4.00000i − 0.278019i
\(208\) − 6.00000i − 0.416025i
\(209\) 9.00000 0.622543
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) − 2.00000i − 0.137038i
\(214\) −1.00000 −0.0683586
\(215\) 0 0
\(216\) −5.00000 −0.340207
\(217\) 0 0
\(218\) 18.0000i 1.21911i
\(219\) −9.00000 −0.608164
\(220\) 0 0
\(221\) 18.0000 1.21081
\(222\) − 1.00000i − 0.0671156i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) −5.00000 −0.332595
\(227\) 28.0000i 1.85843i 0.369546 + 0.929213i \(0.379513\pi\)
−0.369546 + 0.929213i \(0.620487\pi\)
\(228\) − 3.00000i − 0.198680i
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) −12.0000 −0.784465
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 2.00000i 0.129914i
\(238\) 12.0000i 0.777844i
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −21.0000 −1.35273 −0.676364 0.736567i \(-0.736446\pi\)
−0.676364 + 0.736567i \(0.736446\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 16.0000i 1.02640i
\(244\) −12.0000 −0.768221
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) − 18.0000i − 1.14531i
\(248\) 0 0
\(249\) −7.00000 −0.443607
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) − 8.00000i − 0.503953i
\(253\) − 6.00000i − 0.377217i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) − 12.0000i − 0.741362i
\(263\) − 26.0000i − 1.60323i −0.597841 0.801614i \(-0.703975\pi\)
0.597841 0.801614i \(-0.296025\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 3.00000i 0.183597i
\(268\) − 9.00000i − 0.549762i
\(269\) 28.0000 1.70719 0.853595 0.520937i \(-0.174417\pi\)
0.853595 + 0.520937i \(0.174417\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 24.0000i 1.45255i
\(274\) −7.00000 −0.422885
\(275\) 0 0
\(276\) −2.00000 −0.120386
\(277\) 28.0000i 1.68236i 0.540758 + 0.841178i \(0.318138\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 3.00000i 0.179928i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 7.00000i − 0.416107i −0.978117 0.208053i \(-0.933287\pi\)
0.978117 0.208053i \(-0.0667128\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) −18.0000 −1.06436
\(287\) − 12.0000i − 0.708338i
\(288\) − 2.00000i − 0.117851i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) − 9.00000i − 0.526685i
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 15.0000i 0.870388i
\(298\) − 18.0000i − 1.04271i
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) − 22.0000i − 1.26596i
\(303\) − 10.0000i − 0.574485i
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 29.0000i − 1.65512i −0.561379 0.827559i \(-0.689729\pi\)
0.561379 0.827559i \(-0.310271\pi\)
\(308\) − 12.0000i − 0.683763i
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 6.00000i 0.339683i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 24.0000 1.35440
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 0 0
\(320\) 0 0
\(321\) 1.00000 0.0558146
\(322\) − 8.00000i − 0.445823i
\(323\) 9.00000i 0.500773i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 17.0000 0.941543
\(327\) − 18.0000i − 0.995402i
\(328\) − 3.00000i − 0.165647i
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) − 7.00000i − 0.384175i
\(333\) − 2.00000i − 0.109599i
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) − 9.00000i − 0.490261i −0.969490 0.245131i \(-0.921169\pi\)
0.969490 0.245131i \(-0.0788309\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 5.00000 0.271563
\(340\) 0 0
\(341\) 0 0
\(342\) − 6.00000i − 0.324443i
\(343\) − 8.00000i − 0.431959i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) − 31.0000i − 1.66417i −0.554650 0.832084i \(-0.687148\pi\)
0.554650 0.832084i \(-0.312852\pi\)
\(348\) 0 0
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) 30.0000 1.60128
\(352\) − 3.00000i − 0.159901i
\(353\) 14.0000i 0.745145i 0.928003 + 0.372572i \(0.121524\pi\)
−0.928003 + 0.372572i \(0.878476\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) − 12.0000i − 0.635107i
\(358\) 21.0000i 1.10988i
\(359\) −34.0000 −1.79445 −0.897226 0.441572i \(-0.854421\pi\)
−0.897226 + 0.441572i \(0.854421\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 10.0000i 0.525588i
\(363\) − 2.00000i − 0.104973i
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) 12.0000 0.627250
\(367\) − 2.00000i − 0.104399i −0.998637 0.0521996i \(-0.983377\pi\)
0.998637 0.0521996i \(-0.0166232\pi\)
\(368\) − 2.00000i − 0.104257i
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) − 28.0000i − 1.44979i −0.688862 0.724893i \(-0.741889\pi\)
0.688862 0.724893i \(-0.258111\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 20.0000i 1.02869i
\(379\) −1.00000 −0.0513665 −0.0256833 0.999670i \(-0.508176\pi\)
−0.0256833 + 0.999670i \(0.508176\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) − 6.00000i − 0.306987i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −11.0000 −0.559885
\(387\) 8.00000i 0.406663i
\(388\) − 2.00000i − 0.101535i
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) − 9.00000i − 0.454569i
\(393\) 12.0000i 0.605320i
\(394\) 12.0000 0.604551
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) − 30.0000i − 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) 8.00000i 0.401004i
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 9.00000i 0.448879i
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.00000i − 0.148704i
\(408\) − 3.00000i − 0.148522i
\(409\) −15.0000 −0.741702 −0.370851 0.928692i \(-0.620934\pi\)
−0.370851 + 0.928692i \(0.620934\pi\)
\(410\) 0 0
\(411\) 7.00000 0.345285
\(412\) 14.0000i 0.689730i
\(413\) 48.0000i 2.36193i
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) − 3.00000i − 0.146911i
\(418\) − 9.00000i − 0.440204i
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) − 9.00000i − 0.438113i
\(423\) 8.00000i 0.388973i
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 48.0000i 2.32288i
\(428\) 1.00000i 0.0483368i
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) 5.00000i 0.240563i
\(433\) − 5.00000i − 0.240285i −0.992757 0.120142i \(-0.961665\pi\)
0.992757 0.120142i \(-0.0383351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 18.0000 0.862044
\(437\) − 6.00000i − 0.287019i
\(438\) 9.00000i 0.430037i
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) − 18.0000i − 0.856173i
\(443\) − 5.00000i − 0.237557i −0.992921 0.118779i \(-0.962102\pi\)
0.992921 0.118779i \(-0.0378979\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 18.0000i 0.851371i
\(448\) − 4.00000i − 0.188982i
\(449\) 37.0000 1.74614 0.873069 0.487597i \(-0.162126\pi\)
0.873069 + 0.487597i \(0.162126\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 5.00000i 0.235180i
\(453\) 22.0000i 1.03365i
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −3.00000 −0.140488
\(457\) − 33.0000i − 1.54367i −0.635820 0.771837i \(-0.719338\pi\)
0.635820 0.771837i \(-0.280662\pi\)
\(458\) 0 0
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 20.0000 0.931493 0.465746 0.884918i \(-0.345786\pi\)
0.465746 + 0.884918i \(0.345786\pi\)
\(462\) 12.0000i 0.558291i
\(463\) − 6.00000i − 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) − 28.0000i − 1.29569i −0.761774 0.647843i \(-0.775671\pi\)
0.761774 0.647843i \(-0.224329\pi\)
\(468\) 12.0000i 0.554700i
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 12.0000i 0.552345i
\(473\) 12.0000i 0.551761i
\(474\) 2.00000 0.0918630
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 4.00000i 0.183147i
\(478\) − 12.0000i − 0.548867i
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 21.0000i 0.956524i
\(483\) 8.00000i 0.364013i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 16.0000 0.725775
\(487\) 28.0000i 1.26880i 0.773004 + 0.634401i \(0.218753\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(488\) 12.0000i 0.543214i
\(489\) −17.0000 −0.768767
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 3.00000i 0.135250i
\(493\) 0 0
\(494\) −18.0000 −0.809858
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) 7.00000i 0.313678i
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 9.00000i 0.401690i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) −8.00000 −0.356348
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) − 23.0000i − 1.02147i
\(508\) − 8.00000i − 0.354943i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −36.0000 −1.59255
\(512\) − 1.00000i − 0.0441942i
\(513\) 15.0000i 0.662266i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 12.0000i 0.527759i
\(518\) − 4.00000i − 0.175750i
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) −11.0000 −0.481919 −0.240959 0.970535i \(-0.577462\pi\)
−0.240959 + 0.970535i \(0.577462\pi\)
\(522\) 0 0
\(523\) − 37.0000i − 1.61790i −0.587879 0.808949i \(-0.700037\pi\)
0.587879 0.808949i \(-0.299963\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −26.0000 −1.13365
\(527\) 0 0
\(528\) 3.00000i 0.130558i
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) − 12.0000i − 0.520266i
\(533\) 18.0000i 0.779667i
\(534\) 3.00000 0.129823
\(535\) 0 0
\(536\) −9.00000 −0.388741
\(537\) − 21.0000i − 0.906217i
\(538\) − 28.0000i − 1.20717i
\(539\) −27.0000 −1.16297
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 28.0000i 1.20270i
\(543\) − 10.0000i − 0.429141i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) 5.00000i 0.213785i 0.994271 + 0.106892i \(0.0340900\pi\)
−0.994271 + 0.106892i \(0.965910\pi\)
\(548\) 7.00000i 0.299025i
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) 0 0
\(552\) 2.00000i 0.0851257i
\(553\) 8.00000i 0.340195i
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 3.00000 0.127228
\(557\) 32.0000i 1.35588i 0.735116 + 0.677942i \(0.237128\pi\)
−0.735116 + 0.677942i \(0.762872\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 6.00000i 0.253095i
\(563\) − 4.00000i − 0.168580i −0.996441 0.0842900i \(-0.973138\pi\)
0.996441 0.0842900i \(-0.0268622\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −7.00000 −0.294232
\(567\) 4.00000i 0.167984i
\(568\) − 2.00000i − 0.0839181i
\(569\) −19.0000 −0.796521 −0.398261 0.917272i \(-0.630386\pi\)
−0.398261 + 0.917272i \(0.630386\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 18.0000i 0.752618i
\(573\) 6.00000i 0.250654i
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) −2.00000 −0.0833333
\(577\) − 15.0000i − 0.624458i −0.950007 0.312229i \(-0.898924\pi\)
0.950007 0.312229i \(-0.101076\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) −28.0000 −1.16164
\(582\) 2.00000i 0.0829027i
\(583\) 6.00000i 0.248495i
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) − 13.0000i − 0.536567i −0.963340 0.268284i \(-0.913544\pi\)
0.963340 0.268284i \(-0.0864565\pi\)
\(588\) 9.00000i 0.371154i
\(589\) 0 0
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) − 1.00000i − 0.0410997i
\(593\) 9.00000i 0.369586i 0.982777 + 0.184793i \(0.0591614\pi\)
−0.982777 + 0.184793i \(0.940839\pi\)
\(594\) 15.0000 0.615457
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) − 8.00000i − 0.327418i
\(598\) 12.0000i 0.490716i
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −33.0000 −1.34610 −0.673049 0.739598i \(-0.735016\pi\)
−0.673049 + 0.739598i \(0.735016\pi\)
\(602\) 16.0000i 0.652111i
\(603\) 18.0000i 0.733017i
\(604\) −22.0000 −0.895167
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) − 26.0000i − 1.05531i −0.849460 0.527654i \(-0.823072\pi\)
0.849460 0.527654i \(-0.176928\pi\)
\(608\) − 3.00000i − 0.121666i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) − 6.00000i − 0.242536i
\(613\) 46.0000i 1.85792i 0.370177 + 0.928961i \(0.379297\pi\)
−0.370177 + 0.928961i \(0.620703\pi\)
\(614\) −29.0000 −1.17034
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) − 42.0000i − 1.69086i −0.534089 0.845428i \(-0.679345\pi\)
0.534089 0.845428i \(-0.320655\pi\)
\(618\) − 14.0000i − 0.563163i
\(619\) 16.0000 0.643094 0.321547 0.946894i \(-0.395797\pi\)
0.321547 + 0.946894i \(0.395797\pi\)
\(620\) 0 0
\(621\) 10.0000 0.401286
\(622\) − 28.0000i − 1.12270i
\(623\) 12.0000i 0.480770i
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 9.00000i 0.359425i
\(628\) − 24.0000i − 0.957704i
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 22.0000 0.875806 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) 2.00000i 0.0795557i
\(633\) 9.00000i 0.357718i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) 54.0000i 2.13956i
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) − 1.00000i − 0.0394669i
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) 42.0000i 1.65119i 0.564263 + 0.825595i \(0.309160\pi\)
−0.564263 + 0.825595i \(0.690840\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 0 0
\(652\) − 17.0000i − 0.665771i
\(653\) − 36.0000i − 1.40879i −0.709809 0.704394i \(-0.751219\pi\)
0.709809 0.704394i \(-0.248781\pi\)
\(654\) −18.0000 −0.703856
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 18.0000i 0.702247i
\(658\) 16.0000i 0.623745i
\(659\) −9.00000 −0.350590 −0.175295 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 3.00000i 0.116598i
\(663\) 18.0000i 0.699062i
\(664\) −7.00000 −0.271653
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 18.0000i 0.696441i
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 4.00000i 0.154303i
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) −9.00000 −0.346667
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 8.00000i − 0.307465i −0.988113 0.153732i \(-0.950871\pi\)
0.988113 0.153732i \(-0.0491294\pi\)
\(678\) − 5.00000i − 0.192024i
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) − 17.0000i − 0.650487i −0.945630 0.325243i \(-0.894554\pi\)
0.945630 0.325243i \(-0.105446\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 0 0
\(688\) 4.00000i 0.152499i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −1.00000 −0.0380418 −0.0190209 0.999819i \(-0.506055\pi\)
−0.0190209 + 0.999819i \(0.506055\pi\)
\(692\) 24.0000i 0.912343i
\(693\) 24.0000i 0.911685i
\(694\) −31.0000 −1.17674
\(695\) 0 0
\(696\) 0 0
\(697\) − 9.00000i − 0.340899i
\(698\) − 16.0000i − 0.605609i
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) − 30.0000i − 1.13228i
\(703\) − 3.00000i − 0.113147i
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) − 40.0000i − 1.50435i
\(708\) − 12.0000i − 0.450988i
\(709\) −32.0000 −1.20179 −0.600893 0.799330i \(-0.705188\pi\)
−0.600893 + 0.799330i \(0.705188\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 3.00000i 0.112430i
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) 21.0000 0.784807
\(717\) 12.0000i 0.448148i
\(718\) 34.0000i 1.26887i
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 56.0000 2.08555
\(722\) 10.0000i 0.372161i
\(723\) − 21.0000i − 0.780998i
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) − 44.0000i − 1.63187i −0.578144 0.815935i \(-0.696223\pi\)
0.578144 0.815935i \(-0.303777\pi\)
\(728\) 24.0000i 0.889499i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) − 12.0000i − 0.443533i
\(733\) − 40.0000i − 1.47743i −0.674016 0.738717i \(-0.735432\pi\)
0.674016 0.738717i \(-0.264568\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 27.0000i 0.994558i
\(738\) 6.00000i 0.220863i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 18.0000 0.661247
\(742\) 8.00000i 0.293689i
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −28.0000 −1.02515
\(747\) 14.0000i 0.512233i
\(748\) − 9.00000i − 0.329073i
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 4.00000i 0.145865i
\(753\) − 9.00000i − 0.327978i
\(754\) 0 0
\(755\) 0 0
\(756\) 20.0000 0.727393
\(757\) 16.0000i 0.581530i 0.956795 + 0.290765i \(0.0939098\pi\)
−0.956795 + 0.290765i \(0.906090\pi\)
\(758\) 1.00000i 0.0363216i
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) 45.0000 1.63125 0.815624 0.578582i \(-0.196394\pi\)
0.815624 + 0.578582i \(0.196394\pi\)
\(762\) 8.00000i 0.289809i
\(763\) − 72.0000i − 2.60658i
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) − 72.0000i − 2.59977i
\(768\) 1.00000i 0.0360844i
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) 11.0000i 0.395899i
\(773\) − 46.0000i − 1.65451i −0.561830 0.827253i \(-0.689903\pi\)
0.561830 0.827253i \(-0.310097\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 4.00000i 0.143499i
\(778\) − 12.0000i − 0.430221i
\(779\) −9.00000 −0.322458
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) − 6.00000i − 0.214560i
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) − 12.0000i − 0.427482i
\(789\) 26.0000 0.925625
\(790\) 0 0
\(791\) 20.0000 0.711118
\(792\) 6.00000i 0.213201i
\(793\) − 72.0000i − 2.55679i
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 16.0000i 0.566749i 0.959009 + 0.283375i \(0.0914540\pi\)
−0.959009 + 0.283375i \(0.908546\pi\)
\(798\) 12.0000i 0.424795i
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 5.00000i 0.176556i
\(803\) 27.0000i 0.952809i
\(804\) 9.00000 0.317406
\(805\) 0 0
\(806\) 0 0
\(807\) 28.0000i 0.985647i
\(808\) − 10.0000i − 0.351799i
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) − 28.0000i − 0.982003i
\(814\) −3.00000 −0.105150
\(815\) 0 0
\(816\) −3.00000 −0.105021
\(817\) 12.0000i 0.419827i
\(818\) 15.0000i 0.524463i
\(819\) 48.0000 1.67726
\(820\) 0 0
\(821\) −4.00000 −0.139601 −0.0698005 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) − 7.00000i − 0.244153i
\(823\) − 42.0000i − 1.46403i −0.681290 0.732014i \(-0.738581\pi\)
0.681290 0.732014i \(-0.261419\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) 7.00000i 0.243414i 0.992566 + 0.121707i \(0.0388368\pi\)
−0.992566 + 0.121707i \(0.961163\pi\)
\(828\) 4.00000i 0.139010i
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) −28.0000 −0.971309
\(832\) 6.00000i 0.208013i
\(833\) − 27.0000i − 0.935495i
\(834\) −3.00000 −0.103882
\(835\) 0 0
\(836\) −9.00000 −0.311272
\(837\) 0 0
\(838\) 27.0000i 0.932700i
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 32.0000i 1.10279i
\(843\) − 6.00000i − 0.206651i
\(844\) −9.00000 −0.309793
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) − 8.00000i − 0.274883i
\(848\) 2.00000i 0.0686803i
\(849\) 7.00000 0.240239
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 2.00000i 0.0685189i
\(853\) − 40.0000i − 1.36957i −0.728743 0.684787i \(-0.759895\pi\)
0.728743 0.684787i \(-0.240105\pi\)
\(854\) 48.0000 1.64253
\(855\) 0 0
\(856\) 1.00000 0.0341793
\(857\) 3.00000i 0.102478i 0.998686 + 0.0512390i \(0.0163170\pi\)
−0.998686 + 0.0512390i \(0.983683\pi\)
\(858\) − 18.0000i − 0.614510i
\(859\) −9.00000 −0.307076 −0.153538 0.988143i \(-0.549067\pi\)
−0.153538 + 0.988143i \(0.549067\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) − 2.00000i − 0.0681203i
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −5.00000 −0.169907
\(867\) 8.00000i 0.271694i
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 54.0000 1.82972
\(872\) − 18.0000i − 0.609557i
\(873\) 4.00000i 0.135379i
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) 9.00000 0.304082
\(877\) 26.0000i 0.877958i 0.898497 + 0.438979i \(0.144660\pi\)
−0.898497 + 0.438979i \(0.855340\pi\)
\(878\) 16.0000i 0.539974i
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 18.0000i 0.606092i
\(883\) 7.00000i 0.235569i 0.993039 + 0.117784i \(0.0375792\pi\)
−0.993039 + 0.117784i \(0.962421\pi\)
\(884\) −18.0000 −0.605406
\(885\) 0 0
\(886\) −5.00000 −0.167978
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) 1.00000i 0.0335578i
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 8.00000i 0.267860i
\(893\) 12.0000i 0.401565i
\(894\) 18.0000 0.602010
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) − 12.0000i − 0.400668i
\(898\) − 37.0000i − 1.23471i
\(899\) 0 0
\(900\) 0 0
\(901\) −6.00000 −0.199889
\(902\) 9.00000i 0.299667i
\(903\) − 16.0000i − 0.532447i
\(904\) 5.00000 0.166298
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 36.0000i 1.19536i 0.801735 + 0.597680i \(0.203911\pi\)
−0.801735 + 0.597680i \(0.796089\pi\)
\(908\) − 28.0000i − 0.929213i
\(909\) −20.0000 −0.663358
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 3.00000i 0.0993399i
\(913\) 21.0000i 0.694999i
\(914\) −33.0000 −1.09154
\(915\) 0 0
\(916\) 0 0
\(917\) 48.0000i 1.58510i
\(918\) 15.0000i 0.495074i
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) 29.0000 0.955582
\(922\) − 20.0000i − 0.658665i
\(923\) 12.0000i 0.394985i
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) −6.00000 −0.197172
\(927\) − 28.0000i − 0.919641i
\(928\) 0 0
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) −27.0000 −0.884889
\(932\) − 26.0000i − 0.851658i
\(933\) 28.0000i 0.916679i
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) 36.0000i 1.17544i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 24.0000i 0.781962i
\(943\) 6.00000i 0.195387i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) − 2.00000i − 0.0649570i
\(949\) 54.0000 1.75291
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) − 12.0000i − 0.388922i
\(953\) − 33.0000i − 1.06897i −0.845176 0.534487i \(-0.820505\pi\)
0.845176 0.534487i \(-0.179495\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) 16.0000i 0.516937i
\(959\) 28.0000 0.904167
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 6.00000i 0.193448i
\(963\) − 2.00000i − 0.0644491i
\(964\) 21.0000 0.676364
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) − 48.0000i − 1.54358i −0.635880 0.771788i \(-0.719363\pi\)
0.635880 0.771788i \(-0.280637\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) −9.00000 −0.289122
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) − 16.0000i − 0.513200i
\(973\) − 12.0000i − 0.384702i
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 12.0000 0.384111
\(977\) 23.0000i 0.735835i 0.929858 + 0.367918i \(0.119929\pi\)
−0.929858 + 0.367918i \(0.880071\pi\)
\(978\) 17.0000i 0.543600i
\(979\) 9.00000 0.287641
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) 20.0000i 0.638226i
\(983\) − 18.0000i − 0.574111i −0.957914 0.287055i \(-0.907324\pi\)
0.957914 0.287055i \(-0.0926764\pi\)
\(984\) 3.00000 0.0956365
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 18.0000i 0.572656i
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −12.0000 −0.381193 −0.190596 0.981669i \(-0.561042\pi\)
−0.190596 + 0.981669i \(0.561042\pi\)
\(992\) 0 0
\(993\) − 3.00000i − 0.0952021i
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 7.00000 0.221803
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) − 16.0000i − 0.506471i
\(999\) 5.00000 0.158193
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 1850.2.b.e.149.1 2
5.2 odd 4 1850.2.a.m.1.1 yes 1
5.3 odd 4 1850.2.a.c.1.1 1
5.4 even 2 inner 1850.2.b.e.149.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.c.1.1 1 5.3 odd 4
1850.2.a.m.1.1 yes 1 5.2 odd 4
1850.2.b.e.149.1 2 1.1 even 1 trivial
1850.2.b.e.149.2 2 5.4 even 2 inner