Properties

Label 2166.2.a.u.1.3
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.2955970778\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.22668 q^{10} -1.53209 q^{11} +1.00000 q^{12} +0.467911 q^{13} -0.652704 q^{14} +2.22668 q^{15} +1.00000 q^{16} -2.10607 q^{17} +1.00000 q^{18} +2.22668 q^{20} -0.652704 q^{21} -1.53209 q^{22} +5.90167 q^{23} +1.00000 q^{24} -0.0418891 q^{25} +0.467911 q^{26} +1.00000 q^{27} -0.652704 q^{28} +8.33275 q^{29} +2.22668 q^{30} +8.63816 q^{31} +1.00000 q^{32} -1.53209 q^{33} -2.10607 q^{34} -1.45336 q^{35} +1.00000 q^{36} +4.67499 q^{37} +0.467911 q^{39} +2.22668 q^{40} -3.47565 q^{41} -0.652704 q^{42} +10.2909 q^{43} -1.53209 q^{44} +2.22668 q^{45} +5.90167 q^{46} -4.63816 q^{47} +1.00000 q^{48} -6.57398 q^{49} -0.0418891 q^{50} -2.10607 q^{51} +0.467911 q^{52} +11.9513 q^{53} +1.00000 q^{54} -3.41147 q^{55} -0.652704 q^{56} +8.33275 q^{58} -14.2344 q^{59} +2.22668 q^{60} -9.12061 q^{61} +8.63816 q^{62} -0.652704 q^{63} +1.00000 q^{64} +1.04189 q^{65} -1.53209 q^{66} -0.248970 q^{67} -2.10607 q^{68} +5.90167 q^{69} -1.45336 q^{70} +4.44831 q^{71} +1.00000 q^{72} -9.09152 q^{73} +4.67499 q^{74} -0.0418891 q^{75} +1.00000 q^{77} +0.467911 q^{78} -12.0574 q^{79} +2.22668 q^{80} +1.00000 q^{81} -3.47565 q^{82} +3.70914 q^{83} -0.652704 q^{84} -4.68954 q^{85} +10.2909 q^{86} +8.33275 q^{87} -1.53209 q^{88} -16.7665 q^{89} +2.22668 q^{90} -0.305407 q^{91} +5.90167 q^{92} +8.63816 q^{93} -4.63816 q^{94} +1.00000 q^{96} -2.46791 q^{97} -6.57398 q^{98} -1.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{12} + 6q^{13} - 3q^{14} + 3q^{16} + 6q^{17} + 3q^{18} - 3q^{21} + 6q^{23} + 3q^{24} + 3q^{25} + 6q^{26} + 3q^{27} - 3q^{28} + 6q^{29} + 9q^{31} + 3q^{32} + 6q^{34} + 9q^{35} + 3q^{36} + 9q^{37} + 6q^{39} + 9q^{41} - 3q^{42} + 15q^{43} + 6q^{46} + 3q^{47} + 3q^{48} - 12q^{49} + 3q^{50} + 6q^{51} + 6q^{52} - 3q^{53} + 3q^{54} - 3q^{56} + 6q^{58} - 12q^{59} - 33q^{61} + 9q^{62} - 3q^{63} + 3q^{64} + 12q^{67} + 6q^{68} + 6q^{69} + 9q^{70} + 15q^{71} + 3q^{72} + 3q^{73} + 9q^{74} + 3q^{75} + 3q^{77} + 6q^{78} + 15q^{79} + 3q^{81} + 9q^{82} + 27q^{83} - 3q^{84} - 27q^{85} + 15q^{86} + 6q^{87} - 15q^{89} - 3q^{91} + 6q^{92} + 9q^{93} + 3q^{94} + 3q^{96} - 12q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.22668 0.995802 0.497901 0.867234i \(-0.334104\pi\)
0.497901 + 0.867234i \(0.334104\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.652704 −0.246699 −0.123349 0.992363i \(-0.539364\pi\)
−0.123349 + 0.992363i \(0.539364\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.22668 0.704139
\(11\) −1.53209 −0.461942 −0.230971 0.972961i \(-0.574190\pi\)
−0.230971 + 0.972961i \(0.574190\pi\)
\(12\) 1.00000 0.288675
\(13\) 0.467911 0.129775 0.0648876 0.997893i \(-0.479331\pi\)
0.0648876 + 0.997893i \(0.479331\pi\)
\(14\) −0.652704 −0.174442
\(15\) 2.22668 0.574927
\(16\) 1.00000 0.250000
\(17\) −2.10607 −0.510796 −0.255398 0.966836i \(-0.582207\pi\)
−0.255398 + 0.966836i \(0.582207\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) 2.22668 0.497901
\(21\) −0.652704 −0.142432
\(22\) −1.53209 −0.326642
\(23\) 5.90167 1.23058 0.615292 0.788299i \(-0.289038\pi\)
0.615292 + 0.788299i \(0.289038\pi\)
\(24\) 1.00000 0.204124
\(25\) −0.0418891 −0.00837781
\(26\) 0.467911 0.0917649
\(27\) 1.00000 0.192450
\(28\) −0.652704 −0.123349
\(29\) 8.33275 1.54735 0.773676 0.633581i \(-0.218416\pi\)
0.773676 + 0.633581i \(0.218416\pi\)
\(30\) 2.22668 0.406535
\(31\) 8.63816 1.55146 0.775729 0.631066i \(-0.217382\pi\)
0.775729 + 0.631066i \(0.217382\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.53209 −0.266702
\(34\) −2.10607 −0.361187
\(35\) −1.45336 −0.245663
\(36\) 1.00000 0.166667
\(37\) 4.67499 0.768564 0.384282 0.923216i \(-0.374449\pi\)
0.384282 + 0.923216i \(0.374449\pi\)
\(38\) 0 0
\(39\) 0.467911 0.0749257
\(40\) 2.22668 0.352069
\(41\) −3.47565 −0.542806 −0.271403 0.962466i \(-0.587488\pi\)
−0.271403 + 0.962466i \(0.587488\pi\)
\(42\) −0.652704 −0.100714
\(43\) 10.2909 1.56934 0.784671 0.619913i \(-0.212832\pi\)
0.784671 + 0.619913i \(0.212832\pi\)
\(44\) −1.53209 −0.230971
\(45\) 2.22668 0.331934
\(46\) 5.90167 0.870154
\(47\) −4.63816 −0.676545 −0.338272 0.941048i \(-0.609842\pi\)
−0.338272 + 0.941048i \(0.609842\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.57398 −0.939140
\(50\) −0.0418891 −0.00592401
\(51\) −2.10607 −0.294908
\(52\) 0.467911 0.0648876
\(53\) 11.9513 1.64164 0.820819 0.571189i \(-0.193517\pi\)
0.820819 + 0.571189i \(0.193517\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.41147 −0.460003
\(56\) −0.652704 −0.0872212
\(57\) 0 0
\(58\) 8.33275 1.09414
\(59\) −14.2344 −1.85316 −0.926582 0.376093i \(-0.877267\pi\)
−0.926582 + 0.376093i \(0.877267\pi\)
\(60\) 2.22668 0.287463
\(61\) −9.12061 −1.16778 −0.583888 0.811835i \(-0.698469\pi\)
−0.583888 + 0.811835i \(0.698469\pi\)
\(62\) 8.63816 1.09705
\(63\) −0.652704 −0.0822329
\(64\) 1.00000 0.125000
\(65\) 1.04189 0.129230
\(66\) −1.53209 −0.188587
\(67\) −0.248970 −0.0304166 −0.0152083 0.999884i \(-0.504841\pi\)
−0.0152083 + 0.999884i \(0.504841\pi\)
\(68\) −2.10607 −0.255398
\(69\) 5.90167 0.710478
\(70\) −1.45336 −0.173710
\(71\) 4.44831 0.527917 0.263959 0.964534i \(-0.414972\pi\)
0.263959 + 0.964534i \(0.414972\pi\)
\(72\) 1.00000 0.117851
\(73\) −9.09152 −1.06408 −0.532041 0.846719i \(-0.678575\pi\)
−0.532041 + 0.846719i \(0.678575\pi\)
\(74\) 4.67499 0.543457
\(75\) −0.0418891 −0.00483693
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0.467911 0.0529805
\(79\) −12.0574 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(80\) 2.22668 0.248951
\(81\) 1.00000 0.111111
\(82\) −3.47565 −0.383822
\(83\) 3.70914 0.407131 0.203566 0.979061i \(-0.434747\pi\)
0.203566 + 0.979061i \(0.434747\pi\)
\(84\) −0.652704 −0.0712158
\(85\) −4.68954 −0.508652
\(86\) 10.2909 1.10969
\(87\) 8.33275 0.893364
\(88\) −1.53209 −0.163321
\(89\) −16.7665 −1.77725 −0.888623 0.458638i \(-0.848338\pi\)
−0.888623 + 0.458638i \(0.848338\pi\)
\(90\) 2.22668 0.234713
\(91\) −0.305407 −0.0320154
\(92\) 5.90167 0.615292
\(93\) 8.63816 0.895735
\(94\) −4.63816 −0.478389
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.46791 −0.250578 −0.125289 0.992120i \(-0.539986\pi\)
−0.125289 + 0.992120i \(0.539986\pi\)
\(98\) −6.57398 −0.664072
\(99\) −1.53209 −0.153981
\(100\) −0.0418891 −0.00418891
\(101\) −5.73917 −0.571069 −0.285534 0.958368i \(-0.592171\pi\)
−0.285534 + 0.958368i \(0.592171\pi\)
\(102\) −2.10607 −0.208532
\(103\) 13.8084 1.36058 0.680291 0.732942i \(-0.261853\pi\)
0.680291 + 0.732942i \(0.261853\pi\)
\(104\) 0.467911 0.0458825
\(105\) −1.45336 −0.141834
\(106\) 11.9513 1.16081
\(107\) −1.91353 −0.184988 −0.0924941 0.995713i \(-0.529484\pi\)
−0.0924941 + 0.995713i \(0.529484\pi\)
\(108\) 1.00000 0.0962250
\(109\) 11.8152 1.13169 0.565846 0.824511i \(-0.308550\pi\)
0.565846 + 0.824511i \(0.308550\pi\)
\(110\) −3.41147 −0.325271
\(111\) 4.67499 0.443731
\(112\) −0.652704 −0.0616747
\(113\) −5.39693 −0.507700 −0.253850 0.967244i \(-0.581697\pi\)
−0.253850 + 0.967244i \(0.581697\pi\)
\(114\) 0 0
\(115\) 13.1411 1.22542
\(116\) 8.33275 0.773676
\(117\) 0.467911 0.0432584
\(118\) −14.2344 −1.31038
\(119\) 1.37464 0.126013
\(120\) 2.22668 0.203267
\(121\) −8.65270 −0.786609
\(122\) −9.12061 −0.825742
\(123\) −3.47565 −0.313389
\(124\) 8.63816 0.775729
\(125\) −11.2267 −1.00414
\(126\) −0.652704 −0.0581475
\(127\) −9.67499 −0.858517 −0.429258 0.903182i \(-0.641225\pi\)
−0.429258 + 0.903182i \(0.641225\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.2909 0.906060
\(130\) 1.04189 0.0913797
\(131\) −1.90167 −0.166150 −0.0830750 0.996543i \(-0.526474\pi\)
−0.0830750 + 0.996543i \(0.526474\pi\)
\(132\) −1.53209 −0.133351
\(133\) 0 0
\(134\) −0.248970 −0.0215078
\(135\) 2.22668 0.191642
\(136\) −2.10607 −0.180594
\(137\) −1.77837 −0.151936 −0.0759682 0.997110i \(-0.524205\pi\)
−0.0759682 + 0.997110i \(0.524205\pi\)
\(138\) 5.90167 0.502384
\(139\) −0.751030 −0.0637015 −0.0318508 0.999493i \(-0.510140\pi\)
−0.0318508 + 0.999493i \(0.510140\pi\)
\(140\) −1.45336 −0.122832
\(141\) −4.63816 −0.390603
\(142\) 4.44831 0.373294
\(143\) −0.716881 −0.0599486
\(144\) 1.00000 0.0833333
\(145\) 18.5544 1.54086
\(146\) −9.09152 −0.752419
\(147\) −6.57398 −0.542213
\(148\) 4.67499 0.384282
\(149\) −5.97771 −0.489713 −0.244857 0.969559i \(-0.578741\pi\)
−0.244857 + 0.969559i \(0.578741\pi\)
\(150\) −0.0418891 −0.00342023
\(151\) 16.5749 1.34885 0.674424 0.738345i \(-0.264392\pi\)
0.674424 + 0.738345i \(0.264392\pi\)
\(152\) 0 0
\(153\) −2.10607 −0.170265
\(154\) 1.00000 0.0805823
\(155\) 19.2344 1.54495
\(156\) 0.467911 0.0374629
\(157\) −16.7023 −1.33299 −0.666496 0.745509i \(-0.732207\pi\)
−0.666496 + 0.745509i \(0.732207\pi\)
\(158\) −12.0574 −0.959233
\(159\) 11.9513 0.947800
\(160\) 2.22668 0.176035
\(161\) −3.85204 −0.303584
\(162\) 1.00000 0.0785674
\(163\) −15.8571 −1.24202 −0.621012 0.783801i \(-0.713278\pi\)
−0.621012 + 0.783801i \(0.713278\pi\)
\(164\) −3.47565 −0.271403
\(165\) −3.41147 −0.265583
\(166\) 3.70914 0.287885
\(167\) 2.26352 0.175156 0.0875781 0.996158i \(-0.472087\pi\)
0.0875781 + 0.996158i \(0.472087\pi\)
\(168\) −0.652704 −0.0503572
\(169\) −12.7811 −0.983158
\(170\) −4.68954 −0.359671
\(171\) 0 0
\(172\) 10.2909 0.784671
\(173\) −9.35504 −0.711250 −0.355625 0.934629i \(-0.615732\pi\)
−0.355625 + 0.934629i \(0.615732\pi\)
\(174\) 8.33275 0.631704
\(175\) 0.0273411 0.00206680
\(176\) −1.53209 −0.115486
\(177\) −14.2344 −1.06992
\(178\) −16.7665 −1.25670
\(179\) 15.7784 1.17933 0.589665 0.807648i \(-0.299260\pi\)
0.589665 + 0.807648i \(0.299260\pi\)
\(180\) 2.22668 0.165967
\(181\) −7.00269 −0.520506 −0.260253 0.965540i \(-0.583806\pi\)
−0.260253 + 0.965540i \(0.583806\pi\)
\(182\) −0.305407 −0.0226383
\(183\) −9.12061 −0.674215
\(184\) 5.90167 0.435077
\(185\) 10.4097 0.765338
\(186\) 8.63816 0.633380
\(187\) 3.22668 0.235958
\(188\) −4.63816 −0.338272
\(189\) −0.652704 −0.0474772
\(190\) 0 0
\(191\) 4.39187 0.317785 0.158892 0.987296i \(-0.449208\pi\)
0.158892 + 0.987296i \(0.449208\pi\)
\(192\) 1.00000 0.0721688
\(193\) −4.85710 −0.349621 −0.174811 0.984602i \(-0.555931\pi\)
−0.174811 + 0.984602i \(0.555931\pi\)
\(194\) −2.46791 −0.177186
\(195\) 1.04189 0.0746112
\(196\) −6.57398 −0.469570
\(197\) 20.2841 1.44518 0.722589 0.691277i \(-0.242952\pi\)
0.722589 + 0.691277i \(0.242952\pi\)
\(198\) −1.53209 −0.108881
\(199\) −22.4611 −1.59223 −0.796113 0.605148i \(-0.793114\pi\)
−0.796113 + 0.605148i \(0.793114\pi\)
\(200\) −0.0418891 −0.00296200
\(201\) −0.248970 −0.0175610
\(202\) −5.73917 −0.403807
\(203\) −5.43882 −0.381730
\(204\) −2.10607 −0.147454
\(205\) −7.73917 −0.540527
\(206\) 13.8084 0.962077
\(207\) 5.90167 0.410195
\(208\) 0.467911 0.0324438
\(209\) 0 0
\(210\) −1.45336 −0.100292
\(211\) 9.95811 0.685545 0.342772 0.939419i \(-0.388634\pi\)
0.342772 + 0.939419i \(0.388634\pi\)
\(212\) 11.9513 0.820819
\(213\) 4.44831 0.304793
\(214\) −1.91353 −0.130806
\(215\) 22.9145 1.56275
\(216\) 1.00000 0.0680414
\(217\) −5.63816 −0.382743
\(218\) 11.8152 0.800227
\(219\) −9.09152 −0.614348
\(220\) −3.41147 −0.230002
\(221\) −0.985452 −0.0662887
\(222\) 4.67499 0.313765
\(223\) −1.92127 −0.128658 −0.0643290 0.997929i \(-0.520491\pi\)
−0.0643290 + 0.997929i \(0.520491\pi\)
\(224\) −0.652704 −0.0436106
\(225\) −0.0418891 −0.00279260
\(226\) −5.39693 −0.358998
\(227\) −0.901674 −0.0598462 −0.0299231 0.999552i \(-0.509526\pi\)
−0.0299231 + 0.999552i \(0.509526\pi\)
\(228\) 0 0
\(229\) −0.354103 −0.0233998 −0.0116999 0.999932i \(-0.503724\pi\)
−0.0116999 + 0.999932i \(0.503724\pi\)
\(230\) 13.1411 0.866502
\(231\) 1.00000 0.0657952
\(232\) 8.33275 0.547072
\(233\) −16.7023 −1.09421 −0.547103 0.837065i \(-0.684269\pi\)
−0.547103 + 0.837065i \(0.684269\pi\)
\(234\) 0.467911 0.0305883
\(235\) −10.3277 −0.673705
\(236\) −14.2344 −0.926582
\(237\) −12.0574 −0.783210
\(238\) 1.37464 0.0891045
\(239\) 10.0547 0.650383 0.325192 0.945648i \(-0.394571\pi\)
0.325192 + 0.945648i \(0.394571\pi\)
\(240\) 2.22668 0.143732
\(241\) 23.0993 1.48795 0.743977 0.668205i \(-0.232937\pi\)
0.743977 + 0.668205i \(0.232937\pi\)
\(242\) −8.65270 −0.556217
\(243\) 1.00000 0.0641500
\(244\) −9.12061 −0.583888
\(245\) −14.6382 −0.935197
\(246\) −3.47565 −0.221599
\(247\) 0 0
\(248\) 8.63816 0.548523
\(249\) 3.70914 0.235057
\(250\) −11.2267 −0.710038
\(251\) 12.1625 0.767690 0.383845 0.923397i \(-0.374600\pi\)
0.383845 + 0.923397i \(0.374600\pi\)
\(252\) −0.652704 −0.0411165
\(253\) −9.04189 −0.568459
\(254\) −9.67499 −0.607063
\(255\) −4.68954 −0.293670
\(256\) 1.00000 0.0625000
\(257\) 3.94625 0.246160 0.123080 0.992397i \(-0.460723\pi\)
0.123080 + 0.992397i \(0.460723\pi\)
\(258\) 10.2909 0.640681
\(259\) −3.05138 −0.189604
\(260\) 1.04189 0.0646152
\(261\) 8.33275 0.515784
\(262\) −1.90167 −0.117486
\(263\) 9.06923 0.559233 0.279616 0.960112i \(-0.409793\pi\)
0.279616 + 0.960112i \(0.409793\pi\)
\(264\) −1.53209 −0.0942936
\(265\) 26.6117 1.63475
\(266\) 0 0
\(267\) −16.7665 −1.02609
\(268\) −0.248970 −0.0152083
\(269\) 28.0128 1.70797 0.853985 0.520298i \(-0.174179\pi\)
0.853985 + 0.520298i \(0.174179\pi\)
\(270\) 2.22668 0.135512
\(271\) −16.8357 −1.02270 −0.511349 0.859373i \(-0.670854\pi\)
−0.511349 + 0.859373i \(0.670854\pi\)
\(272\) −2.10607 −0.127699
\(273\) −0.305407 −0.0184841
\(274\) −1.77837 −0.107435
\(275\) 0.0641778 0.00387007
\(276\) 5.90167 0.355239
\(277\) −19.1506 −1.15065 −0.575325 0.817925i \(-0.695125\pi\)
−0.575325 + 0.817925i \(0.695125\pi\)
\(278\) −0.751030 −0.0450438
\(279\) 8.63816 0.517153
\(280\) −1.45336 −0.0868551
\(281\) 3.95811 0.236121 0.118061 0.993006i \(-0.462332\pi\)
0.118061 + 0.993006i \(0.462332\pi\)
\(282\) −4.63816 −0.276198
\(283\) −6.57667 −0.390942 −0.195471 0.980709i \(-0.562624\pi\)
−0.195471 + 0.980709i \(0.562624\pi\)
\(284\) 4.44831 0.263959
\(285\) 0 0
\(286\) −0.716881 −0.0423901
\(287\) 2.26857 0.133909
\(288\) 1.00000 0.0589256
\(289\) −12.5645 −0.739087
\(290\) 18.5544 1.08955
\(291\) −2.46791 −0.144672
\(292\) −9.09152 −0.532041
\(293\) −21.6955 −1.26747 −0.633733 0.773552i \(-0.718478\pi\)
−0.633733 + 0.773552i \(0.718478\pi\)
\(294\) −6.57398 −0.383402
\(295\) −31.6955 −1.84538
\(296\) 4.67499 0.271728
\(297\) −1.53209 −0.0889008
\(298\) −5.97771 −0.346280
\(299\) 2.76146 0.159699
\(300\) −0.0418891 −0.00241847
\(301\) −6.71688 −0.387155
\(302\) 16.5749 0.953779
\(303\) −5.73917 −0.329707
\(304\) 0 0
\(305\) −20.3087 −1.16287
\(306\) −2.10607 −0.120396
\(307\) −17.8007 −1.01594 −0.507969 0.861376i \(-0.669603\pi\)
−0.507969 + 0.861376i \(0.669603\pi\)
\(308\) 1.00000 0.0569803
\(309\) 13.8084 0.785532
\(310\) 19.2344 1.09244
\(311\) −1.62536 −0.0921659 −0.0460829 0.998938i \(-0.514674\pi\)
−0.0460829 + 0.998938i \(0.514674\pi\)
\(312\) 0.467911 0.0264903
\(313\) 30.0729 1.69982 0.849909 0.526929i \(-0.176657\pi\)
0.849909 + 0.526929i \(0.176657\pi\)
\(314\) −16.7023 −0.942567
\(315\) −1.45336 −0.0818877
\(316\) −12.0574 −0.678280
\(317\) 11.3327 0.636511 0.318255 0.948005i \(-0.396903\pi\)
0.318255 + 0.948005i \(0.396903\pi\)
\(318\) 11.9513 0.670196
\(319\) −12.7665 −0.714787
\(320\) 2.22668 0.124475
\(321\) −1.91353 −0.106803
\(322\) −3.85204 −0.214666
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −0.0196004 −0.00108723
\(326\) −15.8571 −0.878243
\(327\) 11.8152 0.653382
\(328\) −3.47565 −0.191911
\(329\) 3.02734 0.166903
\(330\) −3.41147 −0.187795
\(331\) 32.1242 1.76571 0.882854 0.469648i \(-0.155619\pi\)
0.882854 + 0.469648i \(0.155619\pi\)
\(332\) 3.70914 0.203566
\(333\) 4.67499 0.256188
\(334\) 2.26352 0.123854
\(335\) −0.554378 −0.0302889
\(336\) −0.652704 −0.0356079
\(337\) −30.3337 −1.65238 −0.826190 0.563391i \(-0.809497\pi\)
−0.826190 + 0.563391i \(0.809497\pi\)
\(338\) −12.7811 −0.695198
\(339\) −5.39693 −0.293121
\(340\) −4.68954 −0.254326
\(341\) −13.2344 −0.716684
\(342\) 0 0
\(343\) 8.85978 0.478383
\(344\) 10.2909 0.554846
\(345\) 13.1411 0.707496
\(346\) −9.35504 −0.502930
\(347\) 4.30541 0.231126 0.115563 0.993300i \(-0.463133\pi\)
0.115563 + 0.993300i \(0.463133\pi\)
\(348\) 8.33275 0.446682
\(349\) −2.66044 −0.142410 −0.0712052 0.997462i \(-0.522685\pi\)
−0.0712052 + 0.997462i \(0.522685\pi\)
\(350\) 0.0273411 0.00146145
\(351\) 0.467911 0.0249752
\(352\) −1.53209 −0.0816606
\(353\) 36.1925 1.92633 0.963167 0.268904i \(-0.0866613\pi\)
0.963167 + 0.268904i \(0.0866613\pi\)
\(354\) −14.2344 −0.756551
\(355\) 9.90497 0.525701
\(356\) −16.7665 −0.888623
\(357\) 1.37464 0.0727535
\(358\) 15.7784 0.833913
\(359\) 1.87164 0.0987816 0.0493908 0.998780i \(-0.484272\pi\)
0.0493908 + 0.998780i \(0.484272\pi\)
\(360\) 2.22668 0.117356
\(361\) 0 0
\(362\) −7.00269 −0.368053
\(363\) −8.65270 −0.454149
\(364\) −0.305407 −0.0160077
\(365\) −20.2439 −1.05962
\(366\) −9.12061 −0.476742
\(367\) 3.53983 0.184778 0.0923888 0.995723i \(-0.470550\pi\)
0.0923888 + 0.995723i \(0.470550\pi\)
\(368\) 5.90167 0.307646
\(369\) −3.47565 −0.180935
\(370\) 10.4097 0.541176
\(371\) −7.80066 −0.404990
\(372\) 8.63816 0.447868
\(373\) −16.5185 −0.855294 −0.427647 0.903946i \(-0.640657\pi\)
−0.427647 + 0.903946i \(0.640657\pi\)
\(374\) 3.22668 0.166848
\(375\) −11.2267 −0.579743
\(376\) −4.63816 −0.239195
\(377\) 3.89899 0.200808
\(378\) −0.652704 −0.0335715
\(379\) 3.56893 0.183323 0.0916617 0.995790i \(-0.470782\pi\)
0.0916617 + 0.995790i \(0.470782\pi\)
\(380\) 0 0
\(381\) −9.67499 −0.495665
\(382\) 4.39187 0.224708
\(383\) 12.0642 0.616451 0.308225 0.951313i \(-0.400265\pi\)
0.308225 + 0.951313i \(0.400265\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.22668 0.113482
\(386\) −4.85710 −0.247220
\(387\) 10.2909 0.523114
\(388\) −2.46791 −0.125289
\(389\) 18.8949 0.958008 0.479004 0.877813i \(-0.340998\pi\)
0.479004 + 0.877813i \(0.340998\pi\)
\(390\) 1.04189 0.0527581
\(391\) −12.4293 −0.628578
\(392\) −6.57398 −0.332036
\(393\) −1.90167 −0.0959268
\(394\) 20.2841 1.02190
\(395\) −26.8479 −1.35087
\(396\) −1.53209 −0.0769904
\(397\) 18.2422 0.915548 0.457774 0.889069i \(-0.348647\pi\)
0.457774 + 0.889069i \(0.348647\pi\)
\(398\) −22.4611 −1.12587
\(399\) 0 0
\(400\) −0.0418891 −0.00209445
\(401\) −33.8854 −1.69215 −0.846077 0.533060i \(-0.821042\pi\)
−0.846077 + 0.533060i \(0.821042\pi\)
\(402\) −0.248970 −0.0124175
\(403\) 4.04189 0.201341
\(404\) −5.73917 −0.285534
\(405\) 2.22668 0.110645
\(406\) −5.43882 −0.269924
\(407\) −7.16250 −0.355032
\(408\) −2.10607 −0.104266
\(409\) 0.172933 0.00855098 0.00427549 0.999991i \(-0.498639\pi\)
0.00427549 + 0.999991i \(0.498639\pi\)
\(410\) −7.73917 −0.382210
\(411\) −1.77837 −0.0877206
\(412\) 13.8084 0.680291
\(413\) 9.29086 0.457173
\(414\) 5.90167 0.290051
\(415\) 8.25908 0.405422
\(416\) 0.467911 0.0229412
\(417\) −0.751030 −0.0367781
\(418\) 0 0
\(419\) −36.5800 −1.78705 −0.893524 0.449015i \(-0.851775\pi\)
−0.893524 + 0.449015i \(0.851775\pi\)
\(420\) −1.45336 −0.0709169
\(421\) −36.8631 −1.79660 −0.898298 0.439386i \(-0.855196\pi\)
−0.898298 + 0.439386i \(0.855196\pi\)
\(422\) 9.95811 0.484753
\(423\) −4.63816 −0.225515
\(424\) 11.9513 0.580407
\(425\) 0.0882212 0.00427936
\(426\) 4.44831 0.215521
\(427\) 5.95306 0.288089
\(428\) −1.91353 −0.0924941
\(429\) −0.716881 −0.0346114
\(430\) 22.9145 1.10503
\(431\) 18.9172 0.911207 0.455604 0.890183i \(-0.349423\pi\)
0.455604 + 0.890183i \(0.349423\pi\)
\(432\) 1.00000 0.0481125
\(433\) −22.2668 −1.07007 −0.535037 0.844828i \(-0.679703\pi\)
−0.535037 + 0.844828i \(0.679703\pi\)
\(434\) −5.63816 −0.270640
\(435\) 18.5544 0.889614
\(436\) 11.8152 0.565846
\(437\) 0 0
\(438\) −9.09152 −0.434410
\(439\) −33.4388 −1.59595 −0.797974 0.602692i \(-0.794095\pi\)
−0.797974 + 0.602692i \(0.794095\pi\)
\(440\) −3.41147 −0.162636
\(441\) −6.57398 −0.313047
\(442\) −0.985452 −0.0468732
\(443\) 34.8949 1.65791 0.828953 0.559319i \(-0.188937\pi\)
0.828953 + 0.559319i \(0.188937\pi\)
\(444\) 4.67499 0.221865
\(445\) −37.3337 −1.76979
\(446\) −1.92127 −0.0909750
\(447\) −5.97771 −0.282736
\(448\) −0.652704 −0.0308373
\(449\) 8.18304 0.386181 0.193091 0.981181i \(-0.438149\pi\)
0.193091 + 0.981181i \(0.438149\pi\)
\(450\) −0.0418891 −0.00197467
\(451\) 5.32501 0.250745
\(452\) −5.39693 −0.253850
\(453\) 16.5749 0.778757
\(454\) −0.901674 −0.0423177
\(455\) −0.680045 −0.0318810
\(456\) 0 0
\(457\) −41.0259 −1.91911 −0.959556 0.281519i \(-0.909162\pi\)
−0.959556 + 0.281519i \(0.909162\pi\)
\(458\) −0.354103 −0.0165462
\(459\) −2.10607 −0.0983028
\(460\) 13.1411 0.612709
\(461\) −4.62267 −0.215299 −0.107650 0.994189i \(-0.534332\pi\)
−0.107650 + 0.994189i \(0.534332\pi\)
\(462\) 1.00000 0.0465242
\(463\) 5.51249 0.256187 0.128094 0.991762i \(-0.459114\pi\)
0.128094 + 0.991762i \(0.459114\pi\)
\(464\) 8.33275 0.386838
\(465\) 19.2344 0.891975
\(466\) −16.7023 −0.773721
\(467\) 11.1693 0.516854 0.258427 0.966031i \(-0.416796\pi\)
0.258427 + 0.966031i \(0.416796\pi\)
\(468\) 0.467911 0.0216292
\(469\) 0.162504 0.00750373
\(470\) −10.3277 −0.476381
\(471\) −16.7023 −0.769603
\(472\) −14.2344 −0.655192
\(473\) −15.7665 −0.724945
\(474\) −12.0574 −0.553813
\(475\) 0 0
\(476\) 1.37464 0.0630064
\(477\) 11.9513 0.547213
\(478\) 10.0547 0.459890
\(479\) 8.66456 0.395894 0.197947 0.980213i \(-0.436573\pi\)
0.197947 + 0.980213i \(0.436573\pi\)
\(480\) 2.22668 0.101634
\(481\) 2.18748 0.0997405
\(482\) 23.0993 1.05214
\(483\) −3.85204 −0.175274
\(484\) −8.65270 −0.393305
\(485\) −5.49525 −0.249527
\(486\) 1.00000 0.0453609
\(487\) 0.0942073 0.00426894 0.00213447 0.999998i \(-0.499321\pi\)
0.00213447 + 0.999998i \(0.499321\pi\)
\(488\) −9.12061 −0.412871
\(489\) −15.8571 −0.717083
\(490\) −14.6382 −0.661284
\(491\) −27.9959 −1.26344 −0.631718 0.775198i \(-0.717650\pi\)
−0.631718 + 0.775198i \(0.717650\pi\)
\(492\) −3.47565 −0.156694
\(493\) −17.5493 −0.790382
\(494\) 0 0
\(495\) −3.41147 −0.153334
\(496\) 8.63816 0.387865
\(497\) −2.90343 −0.130237
\(498\) 3.70914 0.166211
\(499\) −36.9796 −1.65543 −0.827717 0.561146i \(-0.810361\pi\)
−0.827717 + 0.561146i \(0.810361\pi\)
\(500\) −11.2267 −0.502072
\(501\) 2.26352 0.101127
\(502\) 12.1625 0.542839
\(503\) −22.5945 −1.00744 −0.503720 0.863867i \(-0.668036\pi\)
−0.503720 + 0.863867i \(0.668036\pi\)
\(504\) −0.652704 −0.0290737
\(505\) −12.7793 −0.568672
\(506\) −9.04189 −0.401961
\(507\) −12.7811 −0.567627
\(508\) −9.67499 −0.429258
\(509\) 44.4502 1.97022 0.985110 0.171927i \(-0.0549992\pi\)
0.985110 + 0.171927i \(0.0549992\pi\)
\(510\) −4.68954 −0.207656
\(511\) 5.93407 0.262508
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 3.94625 0.174062
\(515\) 30.7469 1.35487
\(516\) 10.2909 0.453030
\(517\) 7.10607 0.312524
\(518\) −3.05138 −0.134070
\(519\) −9.35504 −0.410641
\(520\) 1.04189 0.0456899
\(521\) 24.2645 1.06304 0.531522 0.847044i \(-0.321620\pi\)
0.531522 + 0.847044i \(0.321620\pi\)
\(522\) 8.33275 0.364714
\(523\) 24.2104 1.05865 0.529323 0.848420i \(-0.322446\pi\)
0.529323 + 0.848420i \(0.322446\pi\)
\(524\) −1.90167 −0.0830750
\(525\) 0.0273411 0.00119327
\(526\) 9.06923 0.395437
\(527\) −18.1925 −0.792479
\(528\) −1.53209 −0.0666756
\(529\) 11.8298 0.514337
\(530\) 26.6117 1.15594
\(531\) −14.2344 −0.617721
\(532\) 0 0
\(533\) −1.62630 −0.0704427
\(534\) −16.7665 −0.725558
\(535\) −4.26083 −0.184212
\(536\) −0.248970 −0.0107539
\(537\) 15.7784 0.680887
\(538\) 28.0128 1.20772
\(539\) 10.0719 0.433828
\(540\) 2.22668 0.0958211
\(541\) 34.2303 1.47168 0.735838 0.677158i \(-0.236789\pi\)
0.735838 + 0.677158i \(0.236789\pi\)
\(542\) −16.8357 −0.723157
\(543\) −7.00269 −0.300514
\(544\) −2.10607 −0.0902969
\(545\) 26.3087 1.12694
\(546\) −0.305407 −0.0130702
\(547\) −25.2935 −1.08147 −0.540737 0.841192i \(-0.681855\pi\)
−0.540737 + 0.841192i \(0.681855\pi\)
\(548\) −1.77837 −0.0759682
\(549\) −9.12061 −0.389258
\(550\) 0.0641778 0.00273655
\(551\) 0 0
\(552\) 5.90167 0.251192
\(553\) 7.86989 0.334662
\(554\) −19.1506 −0.813633
\(555\) 10.4097 0.441868
\(556\) −0.751030 −0.0318508
\(557\) −10.8939 −0.461591 −0.230795 0.973002i \(-0.574133\pi\)
−0.230795 + 0.973002i \(0.574133\pi\)
\(558\) 8.63816 0.365682
\(559\) 4.81521 0.203662
\(560\) −1.45336 −0.0614158
\(561\) 3.22668 0.136231
\(562\) 3.95811 0.166963
\(563\) −17.2867 −0.728549 −0.364275 0.931292i \(-0.618683\pi\)
−0.364275 + 0.931292i \(0.618683\pi\)
\(564\) −4.63816 −0.195302
\(565\) −12.0172 −0.505569
\(566\) −6.57667 −0.276438
\(567\) −0.652704 −0.0274110
\(568\) 4.44831 0.186647
\(569\) −6.47834 −0.271586 −0.135793 0.990737i \(-0.543358\pi\)
−0.135793 + 0.990737i \(0.543358\pi\)
\(570\) 0 0
\(571\) 38.5476 1.61317 0.806583 0.591121i \(-0.201314\pi\)
0.806583 + 0.591121i \(0.201314\pi\)
\(572\) −0.716881 −0.0299743
\(573\) 4.39187 0.183473
\(574\) 2.26857 0.0946883
\(575\) −0.247216 −0.0103096
\(576\) 1.00000 0.0416667
\(577\) −11.4551 −0.476883 −0.238441 0.971157i \(-0.576636\pi\)
−0.238441 + 0.971157i \(0.576636\pi\)
\(578\) −12.5645 −0.522614
\(579\) −4.85710 −0.201854
\(580\) 18.5544 0.770429
\(581\) −2.42097 −0.100439
\(582\) −2.46791 −0.102298
\(583\) −18.3105 −0.758342
\(584\) −9.09152 −0.376210
\(585\) 1.04189 0.0430768
\(586\) −21.6955 −0.896234
\(587\) 42.7478 1.76439 0.882196 0.470882i \(-0.156064\pi\)
0.882196 + 0.470882i \(0.156064\pi\)
\(588\) −6.57398 −0.271106
\(589\) 0 0
\(590\) −31.6955 −1.30488
\(591\) 20.2841 0.834374
\(592\) 4.67499 0.192141
\(593\) 32.7151 1.34345 0.671725 0.740801i \(-0.265554\pi\)
0.671725 + 0.740801i \(0.265554\pi\)
\(594\) −1.53209 −0.0628624
\(595\) 3.06088 0.125484
\(596\) −5.97771 −0.244857
\(597\) −22.4611 −0.919272
\(598\) 2.76146 0.112924
\(599\) −9.21482 −0.376507 −0.188254 0.982120i \(-0.560283\pi\)
−0.188254 + 0.982120i \(0.560283\pi\)
\(600\) −0.0418891 −0.00171011
\(601\) −9.67768 −0.394761 −0.197380 0.980327i \(-0.563243\pi\)
−0.197380 + 0.980327i \(0.563243\pi\)
\(602\) −6.71688 −0.273760
\(603\) −0.248970 −0.0101389
\(604\) 16.5749 0.674424
\(605\) −19.2668 −0.783307
\(606\) −5.73917 −0.233138
\(607\) −36.4766 −1.48054 −0.740269 0.672310i \(-0.765302\pi\)
−0.740269 + 0.672310i \(0.765302\pi\)
\(608\) 0 0
\(609\) −5.43882 −0.220392
\(610\) −20.3087 −0.822275
\(611\) −2.17024 −0.0877987
\(612\) −2.10607 −0.0851327
\(613\) −11.6895 −0.472136 −0.236068 0.971737i \(-0.575859\pi\)
−0.236068 + 0.971737i \(0.575859\pi\)
\(614\) −17.8007 −0.718376
\(615\) −7.73917 −0.312073
\(616\) 1.00000 0.0402911
\(617\) 17.8135 0.717143 0.358571 0.933502i \(-0.383264\pi\)
0.358571 + 0.933502i \(0.383264\pi\)
\(618\) 13.8084 0.555455
\(619\) 27.9864 1.12487 0.562434 0.826842i \(-0.309865\pi\)
0.562434 + 0.826842i \(0.309865\pi\)
\(620\) 19.2344 0.772473
\(621\) 5.90167 0.236826
\(622\) −1.62536 −0.0651711
\(623\) 10.9436 0.438445
\(624\) 0.467911 0.0187314
\(625\) −24.7888 −0.991552
\(626\) 30.0729 1.20195
\(627\) 0 0
\(628\) −16.7023 −0.666496
\(629\) −9.84585 −0.392580
\(630\) −1.45336 −0.0579034
\(631\) −3.76146 −0.149741 −0.0748707 0.997193i \(-0.523854\pi\)
−0.0748707 + 0.997193i \(0.523854\pi\)
\(632\) −12.0574 −0.479616
\(633\) 9.95811 0.395799
\(634\) 11.3327 0.450081
\(635\) −21.5431 −0.854913
\(636\) 11.9513 0.473900
\(637\) −3.07604 −0.121877
\(638\) −12.7665 −0.505431
\(639\) 4.44831 0.175972
\(640\) 2.22668 0.0880173
\(641\) −16.7769 −0.662649 −0.331325 0.943517i \(-0.607496\pi\)
−0.331325 + 0.943517i \(0.607496\pi\)
\(642\) −1.91353 −0.0755211
\(643\) −22.1771 −0.874578 −0.437289 0.899321i \(-0.644061\pi\)
−0.437289 + 0.899321i \(0.644061\pi\)
\(644\) −3.85204 −0.151792
\(645\) 22.9145 0.902256
\(646\) 0 0
\(647\) 16.6759 0.655598 0.327799 0.944747i \(-0.393693\pi\)
0.327799 + 0.944747i \(0.393693\pi\)
\(648\) 1.00000 0.0392837
\(649\) 21.8084 0.856055
\(650\) −0.0196004 −0.000768789 0
\(651\) −5.63816 −0.220977
\(652\) −15.8571 −0.621012
\(653\) −26.8066 −1.04903 −0.524513 0.851403i \(-0.675752\pi\)
−0.524513 + 0.851403i \(0.675752\pi\)
\(654\) 11.8152 0.462011
\(655\) −4.23442 −0.165453
\(656\) −3.47565 −0.135701
\(657\) −9.09152 −0.354694
\(658\) 3.02734 0.118018
\(659\) 17.6777 0.688625 0.344312 0.938855i \(-0.388112\pi\)
0.344312 + 0.938855i \(0.388112\pi\)
\(660\) −3.41147 −0.132791
\(661\) −8.67230 −0.337314 −0.168657 0.985675i \(-0.553943\pi\)
−0.168657 + 0.985675i \(0.553943\pi\)
\(662\) 32.1242 1.24854
\(663\) −0.985452 −0.0382718
\(664\) 3.70914 0.143943
\(665\) 0 0
\(666\) 4.67499 0.181152
\(667\) 49.1772 1.90415
\(668\) 2.26352 0.0875781
\(669\) −1.92127 −0.0742808
\(670\) −0.554378 −0.0214175
\(671\) 13.9736 0.539445
\(672\) −0.652704 −0.0251786
\(673\) −2.16487 −0.0834495 −0.0417248 0.999129i \(-0.513285\pi\)
−0.0417248 + 0.999129i \(0.513285\pi\)
\(674\) −30.3337 −1.16841
\(675\) −0.0418891 −0.00161231
\(676\) −12.7811 −0.491579
\(677\) −12.7537 −0.490165 −0.245083 0.969502i \(-0.578815\pi\)
−0.245083 + 0.969502i \(0.578815\pi\)
\(678\) −5.39693 −0.207268
\(679\) 1.61081 0.0618174
\(680\) −4.68954 −0.179836
\(681\) −0.901674 −0.0345522
\(682\) −13.2344 −0.506772
\(683\) 25.2608 0.966579 0.483289 0.875461i \(-0.339442\pi\)
0.483289 + 0.875461i \(0.339442\pi\)
\(684\) 0 0
\(685\) −3.95987 −0.151299
\(686\) 8.85978 0.338268
\(687\) −0.354103 −0.0135099
\(688\) 10.2909 0.392335
\(689\) 5.59215 0.213044
\(690\) 13.1411 0.500275
\(691\) 15.2959 0.581884 0.290942 0.956741i \(-0.406031\pi\)
0.290942 + 0.956741i \(0.406031\pi\)
\(692\) −9.35504 −0.355625
\(693\) 1.00000 0.0379869
\(694\) 4.30541 0.163431
\(695\) −1.67230 −0.0634341
\(696\) 8.33275 0.315852
\(697\) 7.31996 0.277263
\(698\) −2.66044 −0.100699
\(699\) −16.7023 −0.631740
\(700\) 0.0273411 0.00103340
\(701\) 39.8631 1.50561 0.752804 0.658245i \(-0.228701\pi\)
0.752804 + 0.658245i \(0.228701\pi\)
\(702\) 0.467911 0.0176602
\(703\) 0 0
\(704\) −1.53209 −0.0577428
\(705\) −10.3277 −0.388964
\(706\) 36.1925 1.36212
\(707\) 3.74598 0.140882
\(708\) −14.2344 −0.534962
\(709\) 23.8749 0.896642 0.448321 0.893873i \(-0.352022\pi\)
0.448321 + 0.893873i \(0.352022\pi\)
\(710\) 9.90497 0.371727
\(711\) −12.0574 −0.452187
\(712\) −16.7665 −0.628352
\(713\) 50.9796 1.90920
\(714\) 1.37464 0.0514445
\(715\) −1.59627 −0.0596970
\(716\) 15.7784 0.589665
\(717\) 10.0547 0.375499
\(718\) 1.87164 0.0698492
\(719\) −36.5066 −1.36147 −0.680734 0.732531i \(-0.738339\pi\)
−0.680734 + 0.732531i \(0.738339\pi\)
\(720\) 2.22668 0.0829835
\(721\) −9.01279 −0.335654
\(722\) 0 0
\(723\) 23.0993 0.859071
\(724\) −7.00269 −0.260253
\(725\) −0.349051 −0.0129634
\(726\) −8.65270 −0.321132
\(727\) 32.2249 1.19516 0.597578 0.801811i \(-0.296130\pi\)
0.597578 + 0.801811i \(0.296130\pi\)
\(728\) −0.305407 −0.0113191
\(729\) 1.00000 0.0370370
\(730\) −20.2439 −0.749261
\(731\) −21.6732 −0.801614
\(732\) −9.12061 −0.337108
\(733\) −6.97535 −0.257640 −0.128820 0.991668i \(-0.541119\pi\)
−0.128820 + 0.991668i \(0.541119\pi\)
\(734\) 3.53983 0.130657
\(735\) −14.6382 −0.539937
\(736\) 5.90167 0.217539
\(737\) 0.381445 0.0140507
\(738\) −3.47565 −0.127941
\(739\) 24.3696 0.896450 0.448225 0.893921i \(-0.352056\pi\)
0.448225 + 0.893921i \(0.352056\pi\)
\(740\) 10.4097 0.382669
\(741\) 0 0
\(742\) −7.80066 −0.286371
\(743\) 38.0574 1.39619 0.698095 0.716005i \(-0.254031\pi\)
0.698095 + 0.716005i \(0.254031\pi\)
\(744\) 8.63816 0.316690
\(745\) −13.3105 −0.487658
\(746\) −16.5185 −0.604784
\(747\) 3.70914 0.135710
\(748\) 3.22668 0.117979
\(749\) 1.24897 0.0456364
\(750\) −11.2267 −0.409940
\(751\) 2.93582 0.107130 0.0535648 0.998564i \(-0.482942\pi\)
0.0535648 + 0.998564i \(0.482942\pi\)
\(752\) −4.63816 −0.169136
\(753\) 12.1625 0.443226
\(754\) 3.89899 0.141993
\(755\) 36.9071 1.34319
\(756\) −0.652704 −0.0237386
\(757\) 11.3655 0.413085 0.206542 0.978438i \(-0.433779\pi\)
0.206542 + 0.978438i \(0.433779\pi\)
\(758\) 3.56893 0.129629
\(759\) −9.04189 −0.328200
\(760\) 0 0
\(761\) 10.4976 0.380538 0.190269 0.981732i \(-0.439064\pi\)
0.190269 + 0.981732i \(0.439064\pi\)
\(762\) −9.67499 −0.350488
\(763\) −7.71183 −0.279187
\(764\) 4.39187 0.158892
\(765\) −4.68954 −0.169551
\(766\) 12.0642 0.435896
\(767\) −6.66044 −0.240495
\(768\) 1.00000 0.0360844
\(769\) 7.74153 0.279167 0.139583 0.990210i \(-0.455424\pi\)
0.139583 + 0.990210i \(0.455424\pi\)
\(770\) 2.22668 0.0802440
\(771\) 3.94625 0.142121
\(772\) −4.85710 −0.174811
\(773\) 6.15333 0.221320 0.110660 0.993858i \(-0.464704\pi\)
0.110660 + 0.993858i \(0.464704\pi\)
\(774\) 10.2909 0.369897
\(775\) −0.361844 −0.0129978
\(776\) −2.46791 −0.0885928
\(777\) −3.05138 −0.109468
\(778\) 18.8949 0.677414
\(779\) 0 0
\(780\) 1.04189 0.0373056
\(781\) −6.81521 −0.243867
\(782\) −12.4293 −0.444472
\(783\) 8.33275 0.297788
\(784\) −6.57398 −0.234785
\(785\) −37.1908 −1.32740
\(786\) −1.90167 −0.0678305
\(787\) 8.05199 0.287023 0.143511 0.989649i \(-0.454161\pi\)
0.143511 + 0.989649i \(0.454161\pi\)
\(788\) 20.2841 0.722589
\(789\) 9.06923 0.322873
\(790\) −26.8479 −0.955206
\(791\) 3.52259 0.125249
\(792\) −1.53209 −0.0544404
\(793\) −4.26764 −0.151548
\(794\) 18.2422 0.647390
\(795\) 26.6117 0.943821
\(796\) −22.4611 −0.796113
\(797\) −10.2139 −0.361794 −0.180897 0.983502i \(-0.557900\pi\)
−0.180897 + 0.983502i \(0.557900\pi\)
\(798\) 0 0
\(799\) 9.76827 0.345576
\(800\) −0.0418891 −0.00148100
\(801\) −16.7665 −0.592416
\(802\) −33.8854 −1.19653
\(803\) 13.9290 0.491544
\(804\) −0.248970 −0.00878051
\(805\) −8.57728 −0.302309
\(806\) 4.04189 0.142369
\(807\) 28.0128 0.986097
\(808\) −5.73917 −0.201903
\(809\) 47.2995 1.66296 0.831482 0.555552i \(-0.187493\pi\)
0.831482 + 0.555552i \(0.187493\pi\)
\(810\) 2.22668 0.0782376
\(811\) −52.6792 −1.84982 −0.924909 0.380189i \(-0.875859\pi\)
−0.924909 + 0.380189i \(0.875859\pi\)
\(812\) −5.43882 −0.190865
\(813\) −16.8357 −0.590455
\(814\) −7.16250 −0.251046
\(815\) −35.3087 −1.23681
\(816\) −2.10607 −0.0737271
\(817\) 0 0
\(818\) 0.172933 0.00604646
\(819\) −0.305407 −0.0106718
\(820\) −7.73917 −0.270264
\(821\) −55.1516 −1.92480 −0.962402 0.271630i \(-0.912437\pi\)
−0.962402 + 0.271630i \(0.912437\pi\)
\(822\) −1.77837 −0.0620278
\(823\) −24.6435 −0.859020 −0.429510 0.903062i \(-0.641314\pi\)
−0.429510 + 0.903062i \(0.641314\pi\)
\(824\) 13.8084 0.481038
\(825\) 0.0641778 0.00223438
\(826\) 9.29086 0.323270
\(827\) −37.5562 −1.30596 −0.652979 0.757376i \(-0.726481\pi\)
−0.652979 + 0.757376i \(0.726481\pi\)
\(828\) 5.90167 0.205097
\(829\) 13.0000 0.451509 0.225754 0.974184i \(-0.427515\pi\)
0.225754 + 0.974184i \(0.427515\pi\)
\(830\) 8.25908 0.286677
\(831\) −19.1506 −0.664328
\(832\) 0.467911 0.0162219
\(833\) 13.8452 0.479709
\(834\) −0.751030 −0.0260060
\(835\) 5.04013 0.174421
\(836\) 0 0
\(837\) 8.63816 0.298578
\(838\) −36.5800 −1.26363
\(839\) 18.8425 0.650517 0.325259 0.945625i \(-0.394549\pi\)
0.325259 + 0.945625i \(0.394549\pi\)
\(840\) −1.45336 −0.0501458
\(841\) 40.4347 1.39430
\(842\) −36.8631 −1.27039
\(843\) 3.95811 0.136325
\(844\) 9.95811 0.342772
\(845\) −28.4593 −0.979031
\(846\) −4.63816 −0.159463
\(847\) 5.64765 0.194056
\(848\) 11.9513 0.410409
\(849\) −6.57667 −0.225711
\(850\) 0.0882212 0.00302596
\(851\) 27.5903 0.945783
\(852\) 4.44831 0.152397
\(853\) −11.8271 −0.404951 −0.202476 0.979287i \(-0.564899\pi\)
−0.202476 + 0.979287i \(0.564899\pi\)
\(854\) 5.95306 0.203709
\(855\) 0 0
\(856\) −1.91353 −0.0654032
\(857\) −19.7864 −0.675892 −0.337946 0.941166i \(-0.609732\pi\)
−0.337946 + 0.941166i \(0.609732\pi\)
\(858\) −0.716881 −0.0244739
\(859\) 32.1976 1.09857 0.549284 0.835636i \(-0.314901\pi\)
0.549284 + 0.835636i \(0.314901\pi\)
\(860\) 22.9145 0.781377
\(861\) 2.26857 0.0773127
\(862\) 18.9172 0.644321
\(863\) 38.2300 1.30136 0.650682 0.759350i \(-0.274483\pi\)
0.650682 + 0.759350i \(0.274483\pi\)
\(864\) 1.00000 0.0340207
\(865\) −20.8307 −0.708265
\(866\) −22.2668 −0.756657
\(867\) −12.5645 −0.426712
\(868\) −5.63816 −0.191371
\(869\) 18.4730 0.626652
\(870\) 18.5544 0.629052
\(871\) −0.116496 −0.00394732
\(872\) 11.8152 0.400113
\(873\) −2.46791 −0.0835261
\(874\) 0 0
\(875\) 7.32770 0.247721
\(876\) −9.09152 −0.307174
\(877\) 29.9249 1.01049 0.505246 0.862975i \(-0.331402\pi\)
0.505246 + 0.862975i \(0.331402\pi\)
\(878\) −33.4388 −1.12851
\(879\) −21.6955 −0.731772
\(880\) −3.41147 −0.115001
\(881\) −40.8316 −1.37565 −0.687826 0.725875i \(-0.741435\pi\)
−0.687826 + 0.725875i \(0.741435\pi\)
\(882\) −6.57398 −0.221357
\(883\) −24.5776 −0.827102 −0.413551 0.910481i \(-0.635712\pi\)
−0.413551 + 0.910481i \(0.635712\pi\)
\(884\) −0.985452 −0.0331443
\(885\) −31.6955 −1.06543
\(886\) 34.8949 1.17232
\(887\) −24.0624 −0.807937 −0.403969 0.914773i \(-0.632369\pi\)
−0.403969 + 0.914773i \(0.632369\pi\)
\(888\) 4.67499 0.156882
\(889\) 6.31490 0.211795
\(890\) −37.3337 −1.25143
\(891\) −1.53209 −0.0513269
\(892\) −1.92127 −0.0643290
\(893\) 0 0
\(894\) −5.97771 −0.199925
\(895\) 35.1334 1.17438
\(896\) −0.652704 −0.0218053
\(897\) 2.76146 0.0922024
\(898\) 8.18304 0.273072
\(899\) 71.9796 2.40065
\(900\) −0.0418891 −0.00139630
\(901\) −25.1702 −0.838542
\(902\) 5.32501 0.177303
\(903\) −6.71688 −0.223524
\(904\) −5.39693 −0.179499
\(905\) −15.5928 −0.518321
\(906\) 16.5749 0.550665
\(907\) −0.502059 −0.0166706 −0.00833530 0.999965i \(-0.502653\pi\)
−0.00833530 + 0.999965i \(0.502653\pi\)
\(908\) −0.901674 −0.0299231
\(909\) −5.73917 −0.190356
\(910\) −0.680045 −0.0225433
\(911\) 11.2243 0.371878 0.185939 0.982561i \(-0.440467\pi\)
0.185939 + 0.982561i \(0.440467\pi\)
\(912\) 0 0
\(913\) −5.68273 −0.188071
\(914\) −41.0259 −1.35702
\(915\) −20.3087 −0.671385
\(916\) −0.354103 −0.0116999
\(917\) 1.24123 0.0409890
\(918\) −2.10607 −0.0695106
\(919\) 23.5398 0.776507 0.388254 0.921553i \(-0.373078\pi\)
0.388254 + 0.921553i \(0.373078\pi\)
\(920\) 13.1411 0.433251
\(921\) −17.8007 −0.586552
\(922\) −4.62267 −0.152240
\(923\) 2.08141 0.0685106
\(924\) 1.00000 0.0328976
\(925\) −0.195831 −0.00643889
\(926\) 5.51249 0.181152
\(927\) 13.8084 0.453527
\(928\) 8.33275 0.273536
\(929\) −35.0978 −1.15152 −0.575761 0.817618i \(-0.695294\pi\)
−0.575761 + 0.817618i \(0.695294\pi\)
\(930\) 19.2344 0.630722
\(931\) 0 0
\(932\) −16.7023 −0.547103
\(933\) −1.62536 −0.0532120
\(934\) 11.1693 0.365471
\(935\) 7.18479 0.234968
\(936\) 0.467911 0.0152942
\(937\) −44.8120 −1.46394 −0.731972 0.681334i \(-0.761400\pi\)
−0.731972 + 0.681334i \(0.761400\pi\)
\(938\) 0.162504 0.00530594
\(939\) 30.0729 0.981390
\(940\) −10.3277 −0.336852
\(941\) 12.4902 0.407169 0.203584 0.979057i \(-0.434741\pi\)
0.203584 + 0.979057i \(0.434741\pi\)
\(942\) −16.7023 −0.544191
\(943\) −20.5122 −0.667968
\(944\) −14.2344 −0.463291
\(945\) −1.45336 −0.0472779
\(946\) −15.7665 −0.512613
\(947\) 51.7110 1.68038 0.840191 0.542291i \(-0.182443\pi\)
0.840191 + 0.542291i \(0.182443\pi\)
\(948\) −12.0574 −0.391605
\(949\) −4.25402 −0.138091
\(950\) 0 0
\(951\) 11.3327 0.367490
\(952\) 1.37464 0.0445523
\(953\) −39.6769 −1.28526 −0.642630 0.766177i \(-0.722157\pi\)
−0.642630 + 0.766177i \(0.722157\pi\)
\(954\) 11.9513 0.386938
\(955\) 9.77930 0.316451
\(956\) 10.0547 0.325192
\(957\) −12.7665 −0.412683
\(958\) 8.66456 0.279939
\(959\) 1.16075 0.0374825
\(960\) 2.22668 0.0718658
\(961\) 43.6177 1.40702
\(962\) 2.18748 0.0705272
\(963\) −1.91353 −0.0616628
\(964\) 23.0993 0.743977
\(965\) −10.8152 −0.348154
\(966\) −3.85204 −0.123937
\(967\) 27.4587 0.883014 0.441507 0.897258i \(-0.354444\pi\)
0.441507 + 0.897258i \(0.354444\pi\)
\(968\) −8.65270 −0.278108
\(969\) 0 0
\(970\) −5.49525 −0.176442
\(971\) 23.3019 0.747793 0.373897 0.927470i \(-0.378021\pi\)
0.373897 + 0.927470i \(0.378021\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0.490200 0.0157151
\(974\) 0.0942073 0.00301860
\(975\) −0.0196004 −0.000627714 0
\(976\) −9.12061 −0.291944
\(977\) −13.4602 −0.430629 −0.215315 0.976545i \(-0.569078\pi\)
−0.215315 + 0.976545i \(0.569078\pi\)
\(978\) −15.8571 −0.507054
\(979\) 25.6878 0.820985
\(980\) −14.6382 −0.467599
\(981\) 11.8152 0.377231
\(982\) −27.9959 −0.893384
\(983\) 45.7665 1.45973 0.729863 0.683594i \(-0.239584\pi\)
0.729863 + 0.683594i \(0.239584\pi\)
\(984\) −3.47565 −0.110800
\(985\) 45.1661 1.43911
\(986\) −17.5493 −0.558884
\(987\) 3.02734 0.0963613
\(988\) 0 0
\(989\) 60.7333 1.93121
\(990\) −3.41147 −0.108424
\(991\) −25.0256 −0.794964 −0.397482 0.917610i \(-0.630116\pi\)
−0.397482 + 0.917610i \(0.630116\pi\)
\(992\) 8.63816 0.274262
\(993\) 32.1242 1.01943
\(994\) −2.90343 −0.0920912
\(995\) −50.0137 −1.58554
\(996\) 3.70914 0.117529
\(997\) −26.7187 −0.846191 −0.423096 0.906085i \(-0.639057\pi\)
−0.423096 + 0.906085i \(0.639057\pi\)
\(998\) −36.9796 −1.17057
\(999\) 4.67499 0.147910
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 2166.2.a.u.1.3 3
3.2 odd 2 6498.2.a.bn.1.1 3
19.14 odd 18 114.2.i.d.25.1 6
19.15 odd 18 114.2.i.d.73.1 yes 6
19.18 odd 2 2166.2.a.o.1.3 3
57.14 even 18 342.2.u.a.253.1 6
57.53 even 18 342.2.u.a.73.1 6
57.56 even 2 6498.2.a.bs.1.1 3
76.15 even 18 912.2.bo.f.529.1 6
76.71 even 18 912.2.bo.f.481.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.d.25.1 6 19.14 odd 18
114.2.i.d.73.1 yes 6 19.15 odd 18
342.2.u.a.73.1 6 57.53 even 18
342.2.u.a.253.1 6 57.14 even 18
912.2.bo.f.481.1 6 76.71 even 18
912.2.bo.f.529.1 6 76.15 even 18
2166.2.a.o.1.3 3 19.18 odd 2
2166.2.a.u.1.3 3 1.1 even 1 trivial
6498.2.a.bn.1.1 3 3.2 odd 2
6498.2.a.bs.1.1 3 57.56 even 2