Properties

Label 1850.2.b.l.149.2
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.l.149.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} -1.26795i q^{7} +1.00000i q^{8} -4.46410 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +2.73205i q^{3} -1.00000 q^{4} +2.73205 q^{6} -1.26795i q^{7} +1.00000i q^{8} -4.46410 q^{9} +1.46410 q^{11} -2.73205i q^{12} -1.46410i q^{13} -1.26795 q^{14} +1.00000 q^{16} -1.46410i q^{17} +4.46410i q^{18} +4.19615 q^{19} +3.46410 q^{21} -1.46410i q^{22} +8.00000i q^{23} -2.73205 q^{24} -1.46410 q^{26} -4.00000i q^{27} +1.26795i q^{28} +8.92820 q^{29} -2.73205 q^{31} -1.00000i q^{32} +4.00000i q^{33} -1.46410 q^{34} +4.46410 q^{36} +1.00000i q^{37} -4.19615i q^{38} +4.00000 q^{39} -2.00000 q^{41} -3.46410i q^{42} +6.92820i q^{43} -1.46410 q^{44} +8.00000 q^{46} -1.26795i q^{47} +2.73205i q^{48} +5.39230 q^{49} +4.00000 q^{51} +1.46410i q^{52} +6.00000i q^{53} -4.00000 q^{54} +1.26795 q^{56} +11.4641i q^{57} -8.92820i q^{58} -0.196152 q^{59} +8.92820 q^{61} +2.73205i q^{62} +5.66025i q^{63} -1.00000 q^{64} +4.00000 q^{66} +13.6603i q^{67} +1.46410i q^{68} -21.8564 q^{69} -10.9282 q^{71} -4.46410i q^{72} -12.9282i q^{73} +1.00000 q^{74} -4.19615 q^{76} -1.85641i q^{77} -4.00000i q^{78} -5.26795 q^{79} -2.46410 q^{81} +2.00000i q^{82} +5.26795i q^{83} -3.46410 q^{84} +6.92820 q^{86} +24.3923i q^{87} +1.46410i q^{88} +2.00000 q^{89} -1.85641 q^{91} -8.00000i q^{92} -7.46410i q^{93} -1.26795 q^{94} +2.73205 q^{96} -2.00000i q^{97} -5.39230i q^{98} -6.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 8 q^{11} - 12 q^{14} + 4 q^{16} - 4 q^{19} - 4 q^{24} + 8 q^{26} + 8 q^{29} - 4 q^{31} + 8 q^{34} + 4 q^{36} + 16 q^{39} - 8 q^{41} + 8 q^{44} + 32 q^{46} - 20 q^{49} + 16 q^{51} - 16 q^{54} + 12 q^{56} + 20 q^{59} + 8 q^{61} - 4 q^{64} + 16 q^{66} - 32 q^{69} - 16 q^{71} + 4 q^{74} + 4 q^{76} - 28 q^{79} + 4 q^{81} + 8 q^{89} + 48 q^{91} - 12 q^{94} + 4 q^{96} - 40 q^{99}+O(q^{100}) \) 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
\(3\) 2.73205i 1.57735i 0.614810 + 0.788675i \(0.289233\pi\)
−0.614810 + 0.788675i \(0.710767\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.73205 1.11536
\(7\) − 1.26795i − 0.479240i −0.970867 0.239620i \(-0.922977\pi\)
0.970867 0.239620i \(-0.0770228\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) 1.46410 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) − 2.73205i − 0.788675i
\(13\) − 1.46410i − 0.406069i −0.979172 0.203034i \(-0.934920\pi\)
0.979172 0.203034i \(-0.0650803\pi\)
\(14\) −1.26795 −0.338874
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.46410i − 0.355097i −0.984112 0.177548i \(-0.943183\pi\)
0.984112 0.177548i \(-0.0568166\pi\)
\(18\) 4.46410i 1.05220i
\(19\) 4.19615 0.962663 0.481332 0.876539i \(-0.340153\pi\)
0.481332 + 0.876539i \(0.340153\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) − 1.46410i − 0.312148i
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) −2.73205 −0.557678
\(25\) 0 0
\(26\) −1.46410 −0.287134
\(27\) − 4.00000i − 0.769800i
\(28\) 1.26795i 0.239620i
\(29\) 8.92820 1.65793 0.828963 0.559304i \(-0.188931\pi\)
0.828963 + 0.559304i \(0.188931\pi\)
\(30\) 0 0
\(31\) −2.73205 −0.490691 −0.245345 0.969436i \(-0.578901\pi\)
−0.245345 + 0.969436i \(0.578901\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) −1.46410 −0.251091
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) 1.00000i 0.164399i
\(38\) − 4.19615i − 0.680706i
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) − 3.46410i − 0.534522i
\(43\) 6.92820i 1.05654i 0.849076 + 0.528271i \(0.177159\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(44\) −1.46410 −0.220722
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) − 1.26795i − 0.184949i −0.995715 0.0924747i \(-0.970522\pi\)
0.995715 0.0924747i \(-0.0294777\pi\)
\(48\) 2.73205i 0.394338i
\(49\) 5.39230 0.770329
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 1.46410i 0.203034i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.26795 0.169437
\(57\) 11.4641i 1.51846i
\(58\) − 8.92820i − 1.17233i
\(59\) −0.196152 −0.0255369 −0.0127684 0.999918i \(-0.504064\pi\)
−0.0127684 + 0.999918i \(0.504064\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 2.73205i 0.346971i
\(63\) 5.66025i 0.713125i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 13.6603i 1.66887i 0.551110 + 0.834433i \(0.314205\pi\)
−0.551110 + 0.834433i \(0.685795\pi\)
\(68\) 1.46410i 0.177548i
\(69\) −21.8564 −2.63120
\(70\) 0 0
\(71\) −10.9282 −1.29694 −0.648470 0.761241i \(-0.724591\pi\)
−0.648470 + 0.761241i \(0.724591\pi\)
\(72\) − 4.46410i − 0.526099i
\(73\) − 12.9282i − 1.51313i −0.653917 0.756566i \(-0.726876\pi\)
0.653917 0.756566i \(-0.273124\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −4.19615 −0.481332
\(77\) − 1.85641i − 0.211557i
\(78\) − 4.00000i − 0.452911i
\(79\) −5.26795 −0.592691 −0.296345 0.955081i \(-0.595768\pi\)
−0.296345 + 0.955081i \(0.595768\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 2.00000i 0.220863i
\(83\) 5.26795i 0.578233i 0.957294 + 0.289116i \(0.0933614\pi\)
−0.957294 + 0.289116i \(0.906639\pi\)
\(84\) −3.46410 −0.377964
\(85\) 0 0
\(86\) 6.92820 0.747087
\(87\) 24.3923i 2.61513i
\(88\) 1.46410i 0.156074i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −1.85641 −0.194604
\(92\) − 8.00000i − 0.834058i
\(93\) − 7.46410i − 0.773991i
\(94\) −1.26795 −0.130779
\(95\) 0 0
\(96\) 2.73205 0.278839
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) − 5.39230i − 0.544705i
\(99\) −6.53590 −0.656883
\(100\) 0 0
\(101\) −2.53590 −0.252331 −0.126166 0.992009i \(-0.540267\pi\)
−0.126166 + 0.992009i \(0.540267\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) 13.4641i 1.32666i 0.748328 + 0.663329i \(0.230857\pi\)
−0.748328 + 0.663329i \(0.769143\pi\)
\(104\) 1.46410 0.143567
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 6.73205i 0.650812i 0.945574 + 0.325406i \(0.105501\pi\)
−0.945574 + 0.325406i \(0.894499\pi\)
\(108\) 4.00000i 0.384900i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −2.73205 −0.259315
\(112\) − 1.26795i − 0.119810i
\(113\) 17.4641i 1.64288i 0.570292 + 0.821442i \(0.306830\pi\)
−0.570292 + 0.821442i \(0.693170\pi\)
\(114\) 11.4641 1.07371
\(115\) 0 0
\(116\) −8.92820 −0.828963
\(117\) 6.53590i 0.604244i
\(118\) 0.196152i 0.0180573i
\(119\) −1.85641 −0.170177
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) − 8.92820i − 0.808322i
\(123\) − 5.46410i − 0.492681i
\(124\) 2.73205 0.245345
\(125\) 0 0
\(126\) 5.66025 0.504256
\(127\) − 13.6603i − 1.21215i −0.795407 0.606076i \(-0.792743\pi\)
0.795407 0.606076i \(-0.207257\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −18.9282 −1.66654
\(130\) 0 0
\(131\) 12.5885 1.09986 0.549929 0.835211i \(-0.314655\pi\)
0.549929 + 0.835211i \(0.314655\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) − 5.32051i − 0.461347i
\(134\) 13.6603 1.18007
\(135\) 0 0
\(136\) 1.46410 0.125546
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 21.8564i 1.86054i
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) 10.9282i 0.917074i
\(143\) − 2.14359i − 0.179256i
\(144\) −4.46410 −0.372008
\(145\) 0 0
\(146\) −12.9282 −1.06995
\(147\) 14.7321i 1.21508i
\(148\) − 1.00000i − 0.0821995i
\(149\) 16.3923 1.34291 0.671455 0.741045i \(-0.265670\pi\)
0.671455 + 0.741045i \(0.265670\pi\)
\(150\) 0 0
\(151\) 8.39230 0.682956 0.341478 0.939890i \(-0.389073\pi\)
0.341478 + 0.939890i \(0.389073\pi\)
\(152\) 4.19615i 0.340353i
\(153\) 6.53590i 0.528396i
\(154\) −1.85641 −0.149593
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 16.9282i 1.35102i 0.737352 + 0.675509i \(0.236076\pi\)
−0.737352 + 0.675509i \(0.763924\pi\)
\(158\) 5.26795i 0.419096i
\(159\) −16.3923 −1.29999
\(160\) 0 0
\(161\) 10.1436 0.799427
\(162\) 2.46410i 0.193598i
\(163\) − 23.3205i − 1.82660i −0.407284 0.913302i \(-0.633524\pi\)
0.407284 0.913302i \(-0.366476\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 5.26795 0.408872
\(167\) 5.46410i 0.422825i 0.977397 + 0.211412i \(0.0678063\pi\)
−0.977397 + 0.211412i \(0.932194\pi\)
\(168\) 3.46410i 0.267261i
\(169\) 10.8564 0.835108
\(170\) 0 0
\(171\) −18.7321 −1.43248
\(172\) − 6.92820i − 0.528271i
\(173\) − 10.0000i − 0.760286i −0.924928 0.380143i \(-0.875875\pi\)
0.924928 0.380143i \(-0.124125\pi\)
\(174\) 24.3923 1.84918
\(175\) 0 0
\(176\) 1.46410 0.110361
\(177\) − 0.535898i − 0.0402806i
\(178\) − 2.00000i − 0.149906i
\(179\) 17.6603 1.31999 0.659995 0.751270i \(-0.270559\pi\)
0.659995 + 0.751270i \(0.270559\pi\)
\(180\) 0 0
\(181\) −1.46410 −0.108826 −0.0544129 0.998519i \(-0.517329\pi\)
−0.0544129 + 0.998519i \(0.517329\pi\)
\(182\) 1.85641i 0.137606i
\(183\) 24.3923i 1.80313i
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) −7.46410 −0.547294
\(187\) − 2.14359i − 0.156755i
\(188\) 1.26795i 0.0924747i
\(189\) −5.07180 −0.368919
\(190\) 0 0
\(191\) −5.26795 −0.381175 −0.190588 0.981670i \(-0.561039\pi\)
−0.190588 + 0.981670i \(0.561039\pi\)
\(192\) − 2.73205i − 0.197169i
\(193\) 11.8564i 0.853443i 0.904383 + 0.426721i \(0.140332\pi\)
−0.904383 + 0.426721i \(0.859668\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −5.39230 −0.385165
\(197\) 18.7846i 1.33835i 0.743106 + 0.669174i \(0.233352\pi\)
−0.743106 + 0.669174i \(0.766648\pi\)
\(198\) 6.53590i 0.464486i
\(199\) −26.0526 −1.84682 −0.923408 0.383819i \(-0.874609\pi\)
−0.923408 + 0.383819i \(0.874609\pi\)
\(200\) 0 0
\(201\) −37.3205 −2.63239
\(202\) 2.53590i 0.178425i
\(203\) − 11.3205i − 0.794544i
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 13.4641 0.938088
\(207\) − 35.7128i − 2.48221i
\(208\) − 1.46410i − 0.101517i
\(209\) 6.14359 0.424961
\(210\) 0 0
\(211\) 9.85641 0.678543 0.339272 0.940688i \(-0.389819\pi\)
0.339272 + 0.940688i \(0.389819\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) − 29.8564i − 2.04573i
\(214\) 6.73205 0.460194
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 3.46410i 0.235159i
\(218\) 2.00000i 0.135457i
\(219\) 35.3205 2.38674
\(220\) 0 0
\(221\) −2.14359 −0.144194
\(222\) 2.73205i 0.183363i
\(223\) 22.0526i 1.47675i 0.674391 + 0.738374i \(0.264406\pi\)
−0.674391 + 0.738374i \(0.735594\pi\)
\(224\) −1.26795 −0.0847184
\(225\) 0 0
\(226\) 17.4641 1.16169
\(227\) 3.60770i 0.239451i 0.992807 + 0.119726i \(0.0382015\pi\)
−0.992807 + 0.119726i \(0.961799\pi\)
\(228\) − 11.4641i − 0.759229i
\(229\) −15.8564 −1.04782 −0.523910 0.851773i \(-0.675527\pi\)
−0.523910 + 0.851773i \(0.675527\pi\)
\(230\) 0 0
\(231\) 5.07180 0.333700
\(232\) 8.92820i 0.586165i
\(233\) − 15.0718i − 0.987386i −0.869636 0.493693i \(-0.835647\pi\)
0.869636 0.493693i \(-0.164353\pi\)
\(234\) 6.53590 0.427265
\(235\) 0 0
\(236\) 0.196152 0.0127684
\(237\) − 14.3923i − 0.934881i
\(238\) 1.85641i 0.120333i
\(239\) 17.2679 1.11697 0.558485 0.829514i \(-0.311383\pi\)
0.558485 + 0.829514i \(0.311383\pi\)
\(240\) 0 0
\(241\) −8.92820 −0.575116 −0.287558 0.957763i \(-0.592843\pi\)
−0.287558 + 0.957763i \(0.592843\pi\)
\(242\) 8.85641i 0.569311i
\(243\) − 18.7321i − 1.20166i
\(244\) −8.92820 −0.571570
\(245\) 0 0
\(246\) −5.46410 −0.348378
\(247\) − 6.14359i − 0.390907i
\(248\) − 2.73205i − 0.173485i
\(249\) −14.3923 −0.912075
\(250\) 0 0
\(251\) 22.7321 1.43483 0.717417 0.696644i \(-0.245324\pi\)
0.717417 + 0.696644i \(0.245324\pi\)
\(252\) − 5.66025i − 0.356562i
\(253\) 11.7128i 0.736378i
\(254\) −13.6603 −0.857121
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 25.4641i − 1.58841i −0.607652 0.794204i \(-0.707888\pi\)
0.607652 0.794204i \(-0.292112\pi\)
\(258\) 18.9282i 1.17842i
\(259\) 1.26795 0.0787865
\(260\) 0 0
\(261\) −39.8564 −2.46705
\(262\) − 12.5885i − 0.777717i
\(263\) − 30.0526i − 1.85312i −0.376147 0.926560i \(-0.622751\pi\)
0.376147 0.926560i \(-0.377249\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −5.32051 −0.326221
\(267\) 5.46410i 0.334398i
\(268\) − 13.6603i − 0.834433i
\(269\) 0.392305 0.0239192 0.0119596 0.999928i \(-0.496193\pi\)
0.0119596 + 0.999928i \(0.496193\pi\)
\(270\) 0 0
\(271\) 16.7846 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(272\) − 1.46410i − 0.0887742i
\(273\) − 5.07180i − 0.306959i
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 21.8564 1.31560
\(277\) − 26.2487i − 1.57713i −0.614950 0.788566i \(-0.710824\pi\)
0.614950 0.788566i \(-0.289176\pi\)
\(278\) − 6.92820i − 0.415526i
\(279\) 12.1962 0.730165
\(280\) 0 0
\(281\) 4.92820 0.293992 0.146996 0.989137i \(-0.453040\pi\)
0.146996 + 0.989137i \(0.453040\pi\)
\(282\) − 3.46410i − 0.206284i
\(283\) 4.39230i 0.261095i 0.991442 + 0.130548i \(0.0416736\pi\)
−0.991442 + 0.130548i \(0.958326\pi\)
\(284\) 10.9282 0.648470
\(285\) 0 0
\(286\) −2.14359 −0.126753
\(287\) 2.53590i 0.149689i
\(288\) 4.46410i 0.263050i
\(289\) 14.8564 0.873906
\(290\) 0 0
\(291\) 5.46410 0.320311
\(292\) 12.9282i 0.756566i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 14.7321 0.859191
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) − 5.85641i − 0.339823i
\(298\) − 16.3923i − 0.949581i
\(299\) 11.7128 0.677369
\(300\) 0 0
\(301\) 8.78461 0.506336
\(302\) − 8.39230i − 0.482923i
\(303\) − 6.92820i − 0.398015i
\(304\) 4.19615 0.240666
\(305\) 0 0
\(306\) 6.53590 0.373632
\(307\) − 12.5885i − 0.718461i −0.933249 0.359231i \(-0.883039\pi\)
0.933249 0.359231i \(-0.116961\pi\)
\(308\) 1.85641i 0.105779i
\(309\) −36.7846 −2.09260
\(310\) 0 0
\(311\) 27.1244 1.53808 0.769041 0.639200i \(-0.220734\pi\)
0.769041 + 0.639200i \(0.220734\pi\)
\(312\) 4.00000i 0.226455i
\(313\) − 3.85641i − 0.217977i −0.994043 0.108988i \(-0.965239\pi\)
0.994043 0.108988i \(-0.0347612\pi\)
\(314\) 16.9282 0.955314
\(315\) 0 0
\(316\) 5.26795 0.296345
\(317\) − 31.8564i − 1.78923i −0.446834 0.894617i \(-0.647448\pi\)
0.446834 0.894617i \(-0.352552\pi\)
\(318\) 16.3923i 0.919235i
\(319\) 13.0718 0.731880
\(320\) 0 0
\(321\) −18.3923 −1.02656
\(322\) − 10.1436i − 0.565280i
\(323\) − 6.14359i − 0.341839i
\(324\) 2.46410 0.136895
\(325\) 0 0
\(326\) −23.3205 −1.29160
\(327\) − 5.46410i − 0.302166i
\(328\) − 2.00000i − 0.110432i
\(329\) −1.60770 −0.0886351
\(330\) 0 0
\(331\) −8.87564 −0.487850 −0.243925 0.969794i \(-0.578435\pi\)
−0.243925 + 0.969794i \(0.578435\pi\)
\(332\) − 5.26795i − 0.289116i
\(333\) − 4.46410i − 0.244631i
\(334\) 5.46410 0.298982
\(335\) 0 0
\(336\) 3.46410 0.188982
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) − 10.8564i − 0.590511i
\(339\) −47.7128 −2.59140
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 18.7321i 1.01291i
\(343\) − 15.7128i − 0.848412i
\(344\) −6.92820 −0.373544
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 17.0718i 0.916462i 0.888833 + 0.458231i \(0.151517\pi\)
−0.888833 + 0.458231i \(0.848483\pi\)
\(348\) − 24.3923i − 1.30756i
\(349\) −19.3205 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(350\) 0 0
\(351\) −5.85641 −0.312592
\(352\) − 1.46410i − 0.0780369i
\(353\) − 15.8564i − 0.843951i −0.906607 0.421976i \(-0.861337\pi\)
0.906607 0.421976i \(-0.138663\pi\)
\(354\) −0.535898 −0.0284827
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) − 5.07180i − 0.268428i
\(358\) − 17.6603i − 0.933373i
\(359\) −8.39230 −0.442929 −0.221464 0.975168i \(-0.571084\pi\)
−0.221464 + 0.975168i \(0.571084\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) 1.46410i 0.0769515i
\(363\) − 24.1962i − 1.26997i
\(364\) 1.85641 0.0973021
\(365\) 0 0
\(366\) 24.3923 1.27501
\(367\) − 21.6603i − 1.13066i −0.824866 0.565328i \(-0.808750\pi\)
0.824866 0.565328i \(-0.191250\pi\)
\(368\) 8.00000i 0.417029i
\(369\) 8.92820 0.464784
\(370\) 0 0
\(371\) 7.60770 0.394972
\(372\) 7.46410i 0.386996i
\(373\) − 30.7846i − 1.59397i −0.604001 0.796983i \(-0.706428\pi\)
0.604001 0.796983i \(-0.293572\pi\)
\(374\) −2.14359 −0.110843
\(375\) 0 0
\(376\) 1.26795 0.0653895
\(377\) − 13.0718i − 0.673232i
\(378\) 5.07180i 0.260865i
\(379\) −11.6077 −0.596247 −0.298124 0.954527i \(-0.596361\pi\)
−0.298124 + 0.954527i \(0.596361\pi\)
\(380\) 0 0
\(381\) 37.3205 1.91199
\(382\) 5.26795i 0.269532i
\(383\) 8.00000i 0.408781i 0.978889 + 0.204390i \(0.0655212\pi\)
−0.978889 + 0.204390i \(0.934479\pi\)
\(384\) −2.73205 −0.139419
\(385\) 0 0
\(386\) 11.8564 0.603475
\(387\) − 30.9282i − 1.57217i
\(388\) 2.00000i 0.101535i
\(389\) −15.8564 −0.803952 −0.401976 0.915650i \(-0.631676\pi\)
−0.401976 + 0.915650i \(0.631676\pi\)
\(390\) 0 0
\(391\) 11.7128 0.592342
\(392\) 5.39230i 0.272353i
\(393\) 34.3923i 1.73486i
\(394\) 18.7846 0.946355
\(395\) 0 0
\(396\) 6.53590 0.328441
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 26.0526i 1.30590i
\(399\) 14.5359 0.727705
\(400\) 0 0
\(401\) −19.0718 −0.952400 −0.476200 0.879337i \(-0.657986\pi\)
−0.476200 + 0.879337i \(0.657986\pi\)
\(402\) 37.3205i 1.86138i
\(403\) 4.00000i 0.199254i
\(404\) 2.53590 0.126166
\(405\) 0 0
\(406\) −11.3205 −0.561827
\(407\) 1.46410i 0.0725728i
\(408\) 4.00000i 0.198030i
\(409\) −16.9282 −0.837046 −0.418523 0.908206i \(-0.637452\pi\)
−0.418523 + 0.908206i \(0.637452\pi\)
\(410\) 0 0
\(411\) −5.46410 −0.269524
\(412\) − 13.4641i − 0.663329i
\(413\) 0.248711i 0.0122383i
\(414\) −35.7128 −1.75519
\(415\) 0 0
\(416\) −1.46410 −0.0717835
\(417\) 18.9282i 0.926918i
\(418\) − 6.14359i − 0.300493i
\(419\) 34.2487 1.67316 0.836580 0.547846i \(-0.184552\pi\)
0.836580 + 0.547846i \(0.184552\pi\)
\(420\) 0 0
\(421\) −27.8564 −1.35764 −0.678819 0.734306i \(-0.737508\pi\)
−0.678819 + 0.734306i \(0.737508\pi\)
\(422\) − 9.85641i − 0.479802i
\(423\) 5.66025i 0.275211i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −29.8564 −1.44655
\(427\) − 11.3205i − 0.547838i
\(428\) − 6.73205i − 0.325406i
\(429\) 5.85641 0.282750
\(430\) 0 0
\(431\) 8.19615 0.394795 0.197397 0.980324i \(-0.436751\pi\)
0.197397 + 0.980324i \(0.436751\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 34.0000i 1.63394i 0.576683 + 0.816968i \(0.304347\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 3.46410 0.166282
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 33.5692i 1.60583i
\(438\) − 35.3205i − 1.68768i
\(439\) 31.5167 1.50421 0.752104 0.659044i \(-0.229039\pi\)
0.752104 + 0.659044i \(0.229039\pi\)
\(440\) 0 0
\(441\) −24.0718 −1.14628
\(442\) 2.14359i 0.101960i
\(443\) 39.1244i 1.85885i 0.369006 + 0.929427i \(0.379698\pi\)
−0.369006 + 0.929427i \(0.620302\pi\)
\(444\) 2.73205 0.129657
\(445\) 0 0
\(446\) 22.0526 1.04422
\(447\) 44.7846i 2.11824i
\(448\) 1.26795i 0.0599050i
\(449\) −33.7128 −1.59101 −0.795503 0.605950i \(-0.792793\pi\)
−0.795503 + 0.605950i \(0.792793\pi\)
\(450\) 0 0
\(451\) −2.92820 −0.137884
\(452\) − 17.4641i − 0.821442i
\(453\) 22.9282i 1.07726i
\(454\) 3.60770 0.169318
\(455\) 0 0
\(456\) −11.4641 −0.536856
\(457\) − 4.14359i − 0.193829i −0.995293 0.0969146i \(-0.969103\pi\)
0.995293 0.0969146i \(-0.0308974\pi\)
\(458\) 15.8564i 0.740921i
\(459\) −5.85641 −0.273354
\(460\) 0 0
\(461\) −26.7846 −1.24748 −0.623742 0.781630i \(-0.714388\pi\)
−0.623742 + 0.781630i \(0.714388\pi\)
\(462\) − 5.07180i − 0.235961i
\(463\) − 5.07180i − 0.235706i −0.993031 0.117853i \(-0.962399\pi\)
0.993031 0.117853i \(-0.0376012\pi\)
\(464\) 8.92820 0.414481
\(465\) 0 0
\(466\) −15.0718 −0.698188
\(467\) 37.1769i 1.72034i 0.510005 + 0.860171i \(0.329643\pi\)
−0.510005 + 0.860171i \(0.670357\pi\)
\(468\) − 6.53590i − 0.302122i
\(469\) 17.3205 0.799787
\(470\) 0 0
\(471\) −46.2487 −2.13103
\(472\) − 0.196152i − 0.00902865i
\(473\) 10.1436i 0.466403i
\(474\) −14.3923 −0.661060
\(475\) 0 0
\(476\) 1.85641 0.0850883
\(477\) − 26.7846i − 1.22638i
\(478\) − 17.2679i − 0.789818i
\(479\) 34.0526 1.55590 0.777951 0.628325i \(-0.216259\pi\)
0.777951 + 0.628325i \(0.216259\pi\)
\(480\) 0 0
\(481\) 1.46410 0.0667573
\(482\) 8.92820i 0.406669i
\(483\) 27.7128i 1.26098i
\(484\) 8.85641 0.402564
\(485\) 0 0
\(486\) −18.7321 −0.849703
\(487\) − 16.3923i − 0.742806i −0.928472 0.371403i \(-0.878877\pi\)
0.928472 0.371403i \(-0.121123\pi\)
\(488\) 8.92820i 0.404161i
\(489\) 63.7128 2.88119
\(490\) 0 0
\(491\) −1.07180 −0.0483695 −0.0241848 0.999708i \(-0.507699\pi\)
−0.0241848 + 0.999708i \(0.507699\pi\)
\(492\) 5.46410i 0.246341i
\(493\) − 13.0718i − 0.588724i
\(494\) −6.14359 −0.276413
\(495\) 0 0
\(496\) −2.73205 −0.122673
\(497\) 13.8564i 0.621545i
\(498\) 14.3923i 0.644935i
\(499\) −40.5885 −1.81699 −0.908494 0.417897i \(-0.862767\pi\)
−0.908494 + 0.417897i \(0.862767\pi\)
\(500\) 0 0
\(501\) −14.9282 −0.666943
\(502\) − 22.7321i − 1.01458i
\(503\) 5.07180i 0.226140i 0.993587 + 0.113070i \(0.0360685\pi\)
−0.993587 + 0.113070i \(0.963932\pi\)
\(504\) −5.66025 −0.252128
\(505\) 0 0
\(506\) 11.7128 0.520698
\(507\) 29.6603i 1.31726i
\(508\) 13.6603i 0.606076i
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −16.3923 −0.725153
\(512\) − 1.00000i − 0.0441942i
\(513\) − 16.7846i − 0.741059i
\(514\) −25.4641 −1.12317
\(515\) 0 0
\(516\) 18.9282 0.833268
\(517\) − 1.85641i − 0.0816447i
\(518\) − 1.26795i − 0.0557105i
\(519\) 27.3205 1.19924
\(520\) 0 0
\(521\) 33.4641 1.46609 0.733044 0.680181i \(-0.238099\pi\)
0.733044 + 0.680181i \(0.238099\pi\)
\(522\) 39.8564i 1.74447i
\(523\) 4.78461i 0.209216i 0.994514 + 0.104608i \(0.0333589\pi\)
−0.994514 + 0.104608i \(0.966641\pi\)
\(524\) −12.5885 −0.549929
\(525\) 0 0
\(526\) −30.0526 −1.31035
\(527\) 4.00000i 0.174243i
\(528\) 4.00000i 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) 0.875644 0.0379997
\(532\) 5.32051i 0.230673i
\(533\) 2.92820i 0.126835i
\(534\) 5.46410 0.236455
\(535\) 0 0
\(536\) −13.6603 −0.590033
\(537\) 48.2487i 2.08209i
\(538\) − 0.392305i − 0.0169135i
\(539\) 7.89488 0.340057
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) − 16.7846i − 0.720961i
\(543\) − 4.00000i − 0.171656i
\(544\) −1.46410 −0.0627728
\(545\) 0 0
\(546\) −5.07180 −0.217053
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) − 2.00000i − 0.0854358i
\(549\) −39.8564 −1.70103
\(550\) 0 0
\(551\) 37.4641 1.59602
\(552\) − 21.8564i − 0.930270i
\(553\) 6.67949i 0.284041i
\(554\) −26.2487 −1.11520
\(555\) 0 0
\(556\) −6.92820 −0.293821
\(557\) − 32.1051i − 1.36034i −0.733056 0.680169i \(-0.761906\pi\)
0.733056 0.680169i \(-0.238094\pi\)
\(558\) − 12.1962i − 0.516304i
\(559\) 10.1436 0.429028
\(560\) 0 0
\(561\) 5.85641 0.247258
\(562\) − 4.92820i − 0.207884i
\(563\) − 17.0718i − 0.719490i −0.933051 0.359745i \(-0.882864\pi\)
0.933051 0.359745i \(-0.117136\pi\)
\(564\) −3.46410 −0.145865
\(565\) 0 0
\(566\) 4.39230 0.184622
\(567\) 3.12436i 0.131211i
\(568\) − 10.9282i − 0.458537i
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 2.14359i 0.0896281i
\(573\) − 14.3923i − 0.601247i
\(574\) 2.53590 0.105846
\(575\) 0 0
\(576\) 4.46410 0.186004
\(577\) − 3.60770i − 0.150190i −0.997176 0.0750952i \(-0.976074\pi\)
0.997176 0.0750952i \(-0.0239261\pi\)
\(578\) − 14.8564i − 0.617945i
\(579\) −32.3923 −1.34618
\(580\) 0 0
\(581\) 6.67949 0.277112
\(582\) − 5.46410i − 0.226494i
\(583\) 8.78461i 0.363821i
\(584\) 12.9282 0.534973
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 3.21539i 0.132713i 0.997796 + 0.0663567i \(0.0211375\pi\)
−0.997796 + 0.0663567i \(0.978862\pi\)
\(588\) − 14.7321i − 0.607540i
\(589\) −11.4641 −0.472370
\(590\) 0 0
\(591\) −51.3205 −2.11104
\(592\) 1.00000i 0.0410997i
\(593\) − 6.78461i − 0.278611i −0.990249 0.139305i \(-0.955513\pi\)
0.990249 0.139305i \(-0.0444869\pi\)
\(594\) −5.85641 −0.240291
\(595\) 0 0
\(596\) −16.3923 −0.671455
\(597\) − 71.1769i − 2.91308i
\(598\) − 11.7128i − 0.478973i
\(599\) −2.53590 −0.103614 −0.0518070 0.998657i \(-0.516498\pi\)
−0.0518070 + 0.998657i \(0.516498\pi\)
\(600\) 0 0
\(601\) 48.3923 1.97396 0.986982 0.160833i \(-0.0514180\pi\)
0.986982 + 0.160833i \(0.0514180\pi\)
\(602\) − 8.78461i − 0.358034i
\(603\) − 60.9808i − 2.48333i
\(604\) −8.39230 −0.341478
\(605\) 0 0
\(606\) −6.92820 −0.281439
\(607\) − 40.7846i − 1.65540i −0.561174 0.827698i \(-0.689650\pi\)
0.561174 0.827698i \(-0.310350\pi\)
\(608\) − 4.19615i − 0.170176i
\(609\) 30.9282 1.25327
\(610\) 0 0
\(611\) −1.85641 −0.0751022
\(612\) − 6.53590i − 0.264198i
\(613\) − 3.07180i − 0.124069i −0.998074 0.0620344i \(-0.980241\pi\)
0.998074 0.0620344i \(-0.0197588\pi\)
\(614\) −12.5885 −0.508029
\(615\) 0 0
\(616\) 1.85641 0.0747967
\(617\) 12.9282i 0.520470i 0.965545 + 0.260235i \(0.0838000\pi\)
−0.965545 + 0.260235i \(0.916200\pi\)
\(618\) 36.7846i 1.47969i
\(619\) −9.85641 −0.396162 −0.198081 0.980186i \(-0.563471\pi\)
−0.198081 + 0.980186i \(0.563471\pi\)
\(620\) 0 0
\(621\) 32.0000 1.28412
\(622\) − 27.1244i − 1.08759i
\(623\) − 2.53590i − 0.101599i
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −3.85641 −0.154133
\(627\) 16.7846i 0.670313i
\(628\) − 16.9282i − 0.675509i
\(629\) 1.46410 0.0583776
\(630\) 0 0
\(631\) −20.9808 −0.835231 −0.417615 0.908624i \(-0.637134\pi\)
−0.417615 + 0.908624i \(0.637134\pi\)
\(632\) − 5.26795i − 0.209548i
\(633\) 26.9282i 1.07030i
\(634\) −31.8564 −1.26518
\(635\) 0 0
\(636\) 16.3923 0.649997
\(637\) − 7.89488i − 0.312807i
\(638\) − 13.0718i − 0.517517i
\(639\) 48.7846 1.92989
\(640\) 0 0
\(641\) −13.4641 −0.531800 −0.265900 0.964001i \(-0.585669\pi\)
−0.265900 + 0.964001i \(0.585669\pi\)
\(642\) 18.3923i 0.725886i
\(643\) − 41.4641i − 1.63518i −0.575798 0.817592i \(-0.695308\pi\)
0.575798 0.817592i \(-0.304692\pi\)
\(644\) −10.1436 −0.399714
\(645\) 0 0
\(646\) −6.14359 −0.241716
\(647\) 19.7128i 0.774991i 0.921872 + 0.387495i \(0.126660\pi\)
−0.921872 + 0.387495i \(0.873340\pi\)
\(648\) − 2.46410i − 0.0967991i
\(649\) −0.287187 −0.0112731
\(650\) 0 0
\(651\) −9.46410 −0.370927
\(652\) 23.3205i 0.913302i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) −5.46410 −0.213663
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 57.7128i 2.25159i
\(658\) 1.60770i 0.0626745i
\(659\) −6.92820 −0.269884 −0.134942 0.990853i \(-0.543085\pi\)
−0.134942 + 0.990853i \(0.543085\pi\)
\(660\) 0 0
\(661\) −44.9282 −1.74750 −0.873752 0.486371i \(-0.838320\pi\)
−0.873752 + 0.486371i \(0.838320\pi\)
\(662\) 8.87564i 0.344962i
\(663\) − 5.85641i − 0.227444i
\(664\) −5.26795 −0.204436
\(665\) 0 0
\(666\) −4.46410 −0.172980
\(667\) 71.4256i 2.76561i
\(668\) − 5.46410i − 0.211412i
\(669\) −60.2487 −2.32935
\(670\) 0 0
\(671\) 13.0718 0.504631
\(672\) − 3.46410i − 0.133631i
\(673\) 19.0718i 0.735164i 0.929991 + 0.367582i \(0.119814\pi\)
−0.929991 + 0.367582i \(0.880186\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) − 31.8564i − 1.22434i −0.790726 0.612171i \(-0.790297\pi\)
0.790726 0.612171i \(-0.209703\pi\)
\(678\) 47.7128i 1.83240i
\(679\) −2.53590 −0.0973188
\(680\) 0 0
\(681\) −9.85641 −0.377698
\(682\) 4.00000i 0.153168i
\(683\) − 36.7846i − 1.40752i −0.710436 0.703762i \(-0.751502\pi\)
0.710436 0.703762i \(-0.248498\pi\)
\(684\) 18.7321 0.716238
\(685\) 0 0
\(686\) −15.7128 −0.599918
\(687\) − 43.3205i − 1.65278i
\(688\) 6.92820i 0.264135i
\(689\) 8.78461 0.334667
\(690\) 0 0
\(691\) 20.3923 0.775760 0.387880 0.921710i \(-0.373208\pi\)
0.387880 + 0.921710i \(0.373208\pi\)
\(692\) 10.0000i 0.380143i
\(693\) 8.28719i 0.314804i
\(694\) 17.0718 0.648037
\(695\) 0 0
\(696\) −24.3923 −0.924588
\(697\) 2.92820i 0.110914i
\(698\) 19.3205i 0.731292i
\(699\) 41.1769 1.55745
\(700\) 0 0
\(701\) 15.8564 0.598888 0.299444 0.954114i \(-0.403199\pi\)
0.299444 + 0.954114i \(0.403199\pi\)
\(702\) 5.85641i 0.221036i
\(703\) 4.19615i 0.158261i
\(704\) −1.46410 −0.0551804
\(705\) 0 0
\(706\) −15.8564 −0.596764
\(707\) 3.21539i 0.120927i
\(708\) 0.535898i 0.0201403i
\(709\) −16.1436 −0.606285 −0.303143 0.952945i \(-0.598036\pi\)
−0.303143 + 0.952945i \(0.598036\pi\)
\(710\) 0 0
\(711\) 23.5167 0.881944
\(712\) 2.00000i 0.0749532i
\(713\) − 21.8564i − 0.818529i
\(714\) −5.07180 −0.189807
\(715\) 0 0
\(716\) −17.6603 −0.659995
\(717\) 47.1769i 1.76185i
\(718\) 8.39230i 0.313198i
\(719\) −8.39230 −0.312980 −0.156490 0.987680i \(-0.550018\pi\)
−0.156490 + 0.987680i \(0.550018\pi\)
\(720\) 0 0
\(721\) 17.0718 0.635787
\(722\) 1.39230i 0.0518162i
\(723\) − 24.3923i − 0.907160i
\(724\) 1.46410 0.0544129
\(725\) 0 0
\(726\) −24.1962 −0.898003
\(727\) − 32.7846i − 1.21591i −0.793970 0.607957i \(-0.791989\pi\)
0.793970 0.607957i \(-0.208011\pi\)
\(728\) − 1.85641i − 0.0688030i
\(729\) 43.7846 1.62165
\(730\) 0 0
\(731\) 10.1436 0.375174
\(732\) − 24.3923i − 0.901566i
\(733\) − 0.143594i − 0.00530375i −0.999996 0.00265187i \(-0.999156\pi\)
0.999996 0.00265187i \(-0.000844119\pi\)
\(734\) −21.6603 −0.799495
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 20.0000i 0.736709i
\(738\) − 8.92820i − 0.328652i
\(739\) 6.92820 0.254858 0.127429 0.991848i \(-0.459327\pi\)
0.127429 + 0.991848i \(0.459327\pi\)
\(740\) 0 0
\(741\) 16.7846 0.616598
\(742\) − 7.60770i − 0.279287i
\(743\) 7.12436i 0.261367i 0.991424 + 0.130684i \(0.0417172\pi\)
−0.991424 + 0.130684i \(0.958283\pi\)
\(744\) 7.46410 0.273647
\(745\) 0 0
\(746\) −30.7846 −1.12710
\(747\) − 23.5167i − 0.860430i
\(748\) 2.14359i 0.0783775i
\(749\) 8.53590 0.311895
\(750\) 0 0
\(751\) 0.392305 0.0143154 0.00715770 0.999974i \(-0.497722\pi\)
0.00715770 + 0.999974i \(0.497722\pi\)
\(752\) − 1.26795i − 0.0462373i
\(753\) 62.1051i 2.26324i
\(754\) −13.0718 −0.476047
\(755\) 0 0
\(756\) 5.07180 0.184459
\(757\) − 53.7128i − 1.95223i −0.217265 0.976113i \(-0.569714\pi\)
0.217265 0.976113i \(-0.430286\pi\)
\(758\) 11.6077i 0.421610i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) − 37.3205i − 1.35198i
\(763\) 2.53590i 0.0918057i
\(764\) 5.26795 0.190588
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 0.287187i 0.0103697i
\(768\) 2.73205i 0.0985844i
\(769\) −20.9282 −0.754690 −0.377345 0.926073i \(-0.623163\pi\)
−0.377345 + 0.926073i \(0.623163\pi\)
\(770\) 0 0
\(771\) 69.5692 2.50547
\(772\) − 11.8564i − 0.426721i
\(773\) 38.7846i 1.39499i 0.716592 + 0.697493i \(0.245701\pi\)
−0.716592 + 0.697493i \(0.754299\pi\)
\(774\) −30.9282 −1.11169
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 3.46410i 0.124274i
\(778\) 15.8564i 0.568480i
\(779\) −8.39230 −0.300686
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) − 11.7128i − 0.418849i
\(783\) − 35.7128i − 1.27627i
\(784\) 5.39230 0.192582
\(785\) 0 0
\(786\) 34.3923 1.22673
\(787\) 11.8038i 0.420762i 0.977620 + 0.210381i \(0.0674704\pi\)
−0.977620 + 0.210381i \(0.932530\pi\)
\(788\) − 18.7846i − 0.669174i
\(789\) 82.1051 2.92302
\(790\) 0 0
\(791\) 22.1436 0.787336
\(792\) − 6.53590i − 0.232243i
\(793\) − 13.0718i − 0.464193i
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 26.0526 0.923408
\(797\) 17.4641i 0.618610i 0.950963 + 0.309305i \(0.100096\pi\)
−0.950963 + 0.309305i \(0.899904\pi\)
\(798\) − 14.5359i − 0.514565i
\(799\) −1.85641 −0.0656749
\(800\) 0 0
\(801\) −8.92820 −0.315463
\(802\) 19.0718i 0.673449i
\(803\) − 18.9282i − 0.667962i
\(804\) 37.3205 1.31619
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 1.07180i 0.0377290i
\(808\) − 2.53590i − 0.0892126i
\(809\) 39.5692 1.39118 0.695590 0.718439i \(-0.255143\pi\)
0.695590 + 0.718439i \(0.255143\pi\)
\(810\) 0 0
\(811\) −14.1436 −0.496649 −0.248324 0.968677i \(-0.579880\pi\)
−0.248324 + 0.968677i \(0.579880\pi\)
\(812\) 11.3205i 0.397272i
\(813\) 45.8564i 1.60825i
\(814\) 1.46410 0.0513167
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 29.0718i 1.01709i
\(818\) 16.9282i 0.591881i
\(819\) 8.28719 0.289578
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 5.46410i 0.190582i
\(823\) 10.4449i 0.364085i 0.983291 + 0.182043i \(0.0582709\pi\)
−0.983291 + 0.182043i \(0.941729\pi\)
\(824\) −13.4641 −0.469044
\(825\) 0 0
\(826\) 0.248711 0.00865377
\(827\) 4.39230i 0.152735i 0.997080 + 0.0763677i \(0.0243323\pi\)
−0.997080 + 0.0763677i \(0.975668\pi\)
\(828\) 35.7128i 1.24111i
\(829\) −31.5692 −1.09644 −0.548222 0.836333i \(-0.684695\pi\)
−0.548222 + 0.836333i \(0.684695\pi\)
\(830\) 0 0
\(831\) 71.7128 2.48769
\(832\) 1.46410i 0.0507586i
\(833\) − 7.89488i − 0.273541i
\(834\) 18.9282 0.655430
\(835\) 0 0
\(836\) −6.14359 −0.212481
\(837\) 10.9282i 0.377734i
\(838\) − 34.2487i − 1.18310i
\(839\) −32.7846 −1.13185 −0.565925 0.824457i \(-0.691481\pi\)
−0.565925 + 0.824457i \(0.691481\pi\)
\(840\) 0 0
\(841\) 50.7128 1.74872
\(842\) 27.8564i 0.959995i
\(843\) 13.4641i 0.463728i
\(844\) −9.85641 −0.339272
\(845\) 0 0
\(846\) 5.66025 0.194604
\(847\) 11.2295i 0.385849i
\(848\) 6.00000i 0.206041i
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 29.8564i 1.02286i
\(853\) 30.0000i 1.02718i 0.858036 + 0.513590i \(0.171685\pi\)
−0.858036 + 0.513590i \(0.828315\pi\)
\(854\) −11.3205 −0.387380
\(855\) 0 0
\(856\) −6.73205 −0.230097
\(857\) − 4.14359i − 0.141542i −0.997493 0.0707712i \(-0.977454\pi\)
0.997493 0.0707712i \(-0.0225460\pi\)
\(858\) − 5.85641i − 0.199934i
\(859\) 14.4449 0.492852 0.246426 0.969162i \(-0.420744\pi\)
0.246426 + 0.969162i \(0.420744\pi\)
\(860\) 0 0
\(861\) −6.92820 −0.236113
\(862\) − 8.19615i − 0.279162i
\(863\) − 38.4449i − 1.30868i −0.756201 0.654339i \(-0.772947\pi\)
0.756201 0.654339i \(-0.227053\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 40.5885i 1.37846i
\(868\) − 3.46410i − 0.117579i
\(869\) −7.71281 −0.261639
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) − 2.00000i − 0.0677285i
\(873\) 8.92820i 0.302174i
\(874\) 33.5692 1.13550
\(875\) 0 0
\(876\) −35.3205 −1.19337
\(877\) − 50.7846i − 1.71487i −0.514589 0.857437i \(-0.672055\pi\)
0.514589 0.857437i \(-0.327945\pi\)
\(878\) − 31.5167i − 1.06364i
\(879\) 16.3923 0.552899
\(880\) 0 0
\(881\) 47.3205 1.59427 0.797134 0.603802i \(-0.206348\pi\)
0.797134 + 0.603802i \(0.206348\pi\)
\(882\) 24.0718i 0.810540i
\(883\) 34.2487i 1.15256i 0.817252 + 0.576280i \(0.195496\pi\)
−0.817252 + 0.576280i \(0.804504\pi\)
\(884\) 2.14359 0.0720969
\(885\) 0 0
\(886\) 39.1244 1.31441
\(887\) 20.1962i 0.678120i 0.940765 + 0.339060i \(0.110109\pi\)
−0.940765 + 0.339060i \(0.889891\pi\)
\(888\) − 2.73205i − 0.0916816i
\(889\) −17.3205 −0.580911
\(890\) 0 0
\(891\) −3.60770 −0.120862
\(892\) − 22.0526i − 0.738374i
\(893\) − 5.32051i − 0.178044i
\(894\) 44.7846 1.49782
\(895\) 0 0
\(896\) 1.26795 0.0423592
\(897\) 32.0000i 1.06845i
\(898\) 33.7128i 1.12501i
\(899\) −24.3923 −0.813529
\(900\) 0 0
\(901\) 8.78461 0.292658
\(902\) 2.92820i 0.0974985i
\(903\) 24.0000i 0.798670i
\(904\) −17.4641 −0.580847
\(905\) 0 0
\(906\) 22.9282 0.761739
\(907\) 5.75129i 0.190968i 0.995431 + 0.0954842i \(0.0304399\pi\)
−0.995431 + 0.0954842i \(0.969560\pi\)
\(908\) − 3.60770i − 0.119726i
\(909\) 11.3205 0.375478
\(910\) 0 0
\(911\) −27.9090 −0.924665 −0.462333 0.886707i \(-0.652987\pi\)
−0.462333 + 0.886707i \(0.652987\pi\)
\(912\) 11.4641i 0.379614i
\(913\) 7.71281i 0.255257i
\(914\) −4.14359 −0.137058
\(915\) 0 0
\(916\) 15.8564 0.523910
\(917\) − 15.9615i − 0.527096i
\(918\) 5.85641i 0.193290i
\(919\) 13.2679 0.437669 0.218835 0.975762i \(-0.429774\pi\)
0.218835 + 0.975762i \(0.429774\pi\)
\(920\) 0 0
\(921\) 34.3923 1.13326
\(922\) 26.7846i 0.882104i
\(923\) 16.0000i 0.526646i
\(924\) −5.07180 −0.166850
\(925\) 0 0
\(926\) −5.07180 −0.166670
\(927\) − 60.1051i − 1.97411i
\(928\) − 8.92820i − 0.293083i
\(929\) 15.8564 0.520232 0.260116 0.965577i \(-0.416239\pi\)
0.260116 + 0.965577i \(0.416239\pi\)
\(930\) 0 0
\(931\) 22.6269 0.741568
\(932\) 15.0718i 0.493693i
\(933\) 74.1051i 2.42609i
\(934\) 37.1769 1.21647
\(935\) 0 0
\(936\) −6.53590 −0.213633
\(937\) 3.85641i 0.125983i 0.998014 + 0.0629917i \(0.0200642\pi\)
−0.998014 + 0.0629917i \(0.979936\pi\)
\(938\) − 17.3205i − 0.565535i
\(939\) 10.5359 0.343826
\(940\) 0 0
\(941\) 28.3923 0.925563 0.462781 0.886472i \(-0.346852\pi\)
0.462781 + 0.886472i \(0.346852\pi\)
\(942\) 46.2487i 1.50686i
\(943\) − 16.0000i − 0.521032i
\(944\) −0.196152 −0.00638422
\(945\) 0 0
\(946\) 10.1436 0.329797
\(947\) − 45.1769i − 1.46805i −0.679121 0.734026i \(-0.737639\pi\)
0.679121 0.734026i \(-0.262361\pi\)
\(948\) 14.3923i 0.467440i
\(949\) −18.9282 −0.614435
\(950\) 0 0
\(951\) 87.0333 2.82225
\(952\) − 1.85641i − 0.0601665i
\(953\) 11.8564i 0.384067i 0.981388 + 0.192033i \(0.0615082\pi\)
−0.981388 + 0.192033i \(0.938492\pi\)
\(954\) −26.7846 −0.867184
\(955\) 0 0
\(956\) −17.2679 −0.558485
\(957\) 35.7128i 1.15443i
\(958\) − 34.0526i − 1.10019i
\(959\) 2.53590 0.0818884
\(960\) 0 0
\(961\) −23.5359 −0.759223
\(962\) − 1.46410i − 0.0472045i
\(963\) − 30.0526i − 0.968430i
\(964\) 8.92820 0.287558
\(965\) 0 0
\(966\) 27.7128 0.891645
\(967\) − 23.6077i − 0.759172i −0.925156 0.379586i \(-0.876066\pi\)
0.925156 0.379586i \(-0.123934\pi\)
\(968\) − 8.85641i − 0.284656i
\(969\) 16.7846 0.539199
\(970\) 0 0
\(971\) 14.9282 0.479069 0.239534 0.970888i \(-0.423005\pi\)
0.239534 + 0.970888i \(0.423005\pi\)
\(972\) 18.7321i 0.600831i
\(973\) − 8.78461i − 0.281622i
\(974\) −16.3923 −0.525243
\(975\) 0 0
\(976\) 8.92820 0.285785
\(977\) − 34.0000i − 1.08776i −0.839164 0.543878i \(-0.816955\pi\)
0.839164 0.543878i \(-0.183045\pi\)
\(978\) − 63.7128i − 2.03731i
\(979\) 2.92820 0.0935858
\(980\) 0 0
\(981\) 8.92820 0.285056
\(982\) 1.07180i 0.0342024i
\(983\) 38.4449i 1.22620i 0.790005 + 0.613100i \(0.210078\pi\)
−0.790005 + 0.613100i \(0.789922\pi\)
\(984\) 5.46410 0.174189
\(985\) 0 0
\(986\) −13.0718 −0.416291
\(987\) − 4.39230i − 0.139809i
\(988\) 6.14359i 0.195454i
\(989\) −55.4256 −1.76243
\(990\) 0 0
\(991\) 5.94744 0.188927 0.0944633 0.995528i \(-0.469886\pi\)
0.0944633 + 0.995528i \(0.469886\pi\)
\(992\) 2.73205i 0.0867427i
\(993\) − 24.2487i − 0.769510i
\(994\) 13.8564 0.439499
\(995\) 0 0
\(996\) 14.3923 0.456038
\(997\) − 4.14359i − 0.131229i −0.997845 0.0656145i \(-0.979099\pi\)
0.997845 0.0656145i \(-0.0209007\pi\)
\(998\) 40.5885i 1.28481i
\(999\) 4.00000 0.126554
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.l.149.2 4
5.2 odd 4 1850.2.a.x.1.2 2
5.3 odd 4 370.2.a.e.1.1 2
5.4 even 2 inner 1850.2.b.l.149.3 4
15.8 even 4 3330.2.a.bd.1.2 2
20.3 even 4 2960.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.1 2 5.3 odd 4
1850.2.a.x.1.2 2 5.2 odd 4
1850.2.b.l.149.2 4 1.1 even 1 trivial
1850.2.b.l.149.3 4 5.4 even 2 inner
2960.2.a.q.1.2 2 20.3 even 4
3330.2.a.bd.1.2 2 15.8 even 4