Properties

Label 966.2.a.p.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} -4.47214 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +4.47214 q^{17} +1.00000 q^{18} +2.47214 q^{19} +2.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -4.47214 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -2.47214 q^{31} +1.00000 q^{32} +4.47214 q^{34} +2.00000 q^{35} +1.00000 q^{36} +6.94427 q^{37} +2.47214 q^{38} -4.47214 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.94427 q^{43} +2.00000 q^{45} -1.00000 q^{46} -2.47214 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.47214 q^{51} -4.47214 q^{52} -10.9443 q^{53} +1.00000 q^{54} +1.00000 q^{56} +2.47214 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} -2.47214 q^{62} +1.00000 q^{63} +1.00000 q^{64} -8.94427 q^{65} +4.94427 q^{67} +4.47214 q^{68} -1.00000 q^{69} +2.00000 q^{70} +4.94427 q^{71} +1.00000 q^{72} +14.9443 q^{73} +6.94427 q^{74} -1.00000 q^{75} +2.47214 q^{76} -4.47214 q^{78} -4.94427 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +2.47214 q^{83} +1.00000 q^{84} +8.94427 q^{85} -4.94427 q^{86} -2.00000 q^{87} +9.41641 q^{89} +2.00000 q^{90} -4.47214 q^{91} -1.00000 q^{92} -2.47214 q^{93} -2.47214 q^{94} +4.94427 q^{95} +1.00000 q^{96} -0.472136 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 2 q^{14} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 4 q^{19} + 4 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{27} + 2 q^{28} - 4 q^{29} + 4 q^{30} + 4 q^{31} + 2 q^{32} + 4 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 4 q^{40} - 12 q^{41} + 2 q^{42} + 8 q^{43} + 4 q^{45} - 2 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 4 q^{53} + 2 q^{54} + 2 q^{56} - 4 q^{57} - 4 q^{58} + 8 q^{59} + 4 q^{60} - 12 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 8 q^{67} - 2 q^{69} + 4 q^{70} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 4 q^{74} - 2 q^{75} - 4 q^{76} + 8 q^{79} + 4 q^{80} + 2 q^{81} - 12 q^{82} - 4 q^{83} + 2 q^{84} + 8 q^{86} - 4 q^{87} - 8 q^{89} + 4 q^{90} - 2 q^{92} + 4 q^{93} + 4 q^{94} - 8 q^{95} + 2 q^{96} + 8 q^{97} + 2 q^{98} + O(q^{100}) \)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −4.47214 −0.877058
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.47214 0.766965
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 2.47214 0.401033
\(39\) −4.47214 −0.716115
\(40\) 2.00000 0.316228
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.94427 −0.753994 −0.376997 0.926214i \(-0.623043\pi\)
−0.376997 + 0.926214i \(0.623043\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) −1.00000 −0.147442
\(47\) −2.47214 −0.360598 −0.180299 0.983612i \(-0.557707\pi\)
−0.180299 + 0.983612i \(0.557707\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.47214 0.626224
\(52\) −4.47214 −0.620174
\(53\) −10.9443 −1.50331 −0.751656 0.659556i \(-0.770744\pi\)
−0.751656 + 0.659556i \(0.770744\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.47214 0.327442
\(58\) −2.00000 −0.262613
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 2.00000 0.258199
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −2.47214 −0.313962
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −8.94427 −1.10940
\(66\) 0 0
\(67\) 4.94427 0.604039 0.302019 0.953302i \(-0.402339\pi\)
0.302019 + 0.953302i \(0.402339\pi\)
\(68\) 4.47214 0.542326
\(69\) −1.00000 −0.120386
\(70\) 2.00000 0.239046
\(71\) 4.94427 0.586777 0.293389 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.9443 1.74909 0.874547 0.484940i \(-0.161159\pi\)
0.874547 + 0.484940i \(0.161159\pi\)
\(74\) 6.94427 0.807255
\(75\) −1.00000 −0.115470
\(76\) 2.47214 0.283573
\(77\) 0 0
\(78\) −4.47214 −0.506370
\(79\) −4.94427 −0.556274 −0.278137 0.960541i \(-0.589717\pi\)
−0.278137 + 0.960541i \(0.589717\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 2.47214 0.271352 0.135676 0.990753i \(-0.456679\pi\)
0.135676 + 0.990753i \(0.456679\pi\)
\(84\) 1.00000 0.109109
\(85\) 8.94427 0.970143
\(86\) −4.94427 −0.533155
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 9.41641 0.998137 0.499069 0.866562i \(-0.333676\pi\)
0.499069 + 0.866562i \(0.333676\pi\)
\(90\) 2.00000 0.210819
\(91\) −4.47214 −0.468807
\(92\) −1.00000 −0.104257
\(93\) −2.47214 −0.256349
\(94\) −2.47214 −0.254981
\(95\) 4.94427 0.507272
\(96\) 1.00000 0.102062
\(97\) −0.472136 −0.0479381 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −9.41641 −0.936968 −0.468484 0.883472i \(-0.655200\pi\)
−0.468484 + 0.883472i \(0.655200\pi\)
\(102\) 4.47214 0.442807
\(103\) −4.94427 −0.487174 −0.243587 0.969879i \(-0.578324\pi\)
−0.243587 + 0.969879i \(0.578324\pi\)
\(104\) −4.47214 −0.438529
\(105\) 2.00000 0.195180
\(106\) −10.9443 −1.06300
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.94427 −0.282010 −0.141005 0.990009i \(-0.545033\pi\)
−0.141005 + 0.990009i \(0.545033\pi\)
\(110\) 0 0
\(111\) 6.94427 0.659121
\(112\) 1.00000 0.0944911
\(113\) −18.9443 −1.78213 −0.891064 0.453878i \(-0.850040\pi\)
−0.891064 + 0.453878i \(0.850040\pi\)
\(114\) 2.47214 0.231537
\(115\) −2.00000 −0.186501
\(116\) −2.00000 −0.185695
\(117\) −4.47214 −0.413449
\(118\) 4.00000 0.368230
\(119\) 4.47214 0.409960
\(120\) 2.00000 0.182574
\(121\) −11.0000 −1.00000
\(122\) −6.00000 −0.543214
\(123\) −6.00000 −0.541002
\(124\) −2.47214 −0.222004
\(125\) −12.0000 −1.07331
\(126\) 1.00000 0.0890871
\(127\) 20.9443 1.85850 0.929252 0.369447i \(-0.120453\pi\)
0.929252 + 0.369447i \(0.120453\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.94427 −0.435319
\(130\) −8.94427 −0.784465
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 2.47214 0.214361
\(134\) 4.94427 0.427120
\(135\) 2.00000 0.172133
\(136\) 4.47214 0.383482
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 8.94427 0.758643 0.379322 0.925265i \(-0.376157\pi\)
0.379322 + 0.925265i \(0.376157\pi\)
\(140\) 2.00000 0.169031
\(141\) −2.47214 −0.208191
\(142\) 4.94427 0.414914
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) 14.9443 1.23680
\(147\) 1.00000 0.0824786
\(148\) 6.94427 0.570816
\(149\) −1.05573 −0.0864886 −0.0432443 0.999065i \(-0.513769\pi\)
−0.0432443 + 0.999065i \(0.513769\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 4.94427 0.402359 0.201180 0.979554i \(-0.435523\pi\)
0.201180 + 0.979554i \(0.435523\pi\)
\(152\) 2.47214 0.200517
\(153\) 4.47214 0.361551
\(154\) 0 0
\(155\) −4.94427 −0.397133
\(156\) −4.47214 −0.358057
\(157\) −2.94427 −0.234978 −0.117489 0.993074i \(-0.537485\pi\)
−0.117489 + 0.993074i \(0.537485\pi\)
\(158\) −4.94427 −0.393345
\(159\) −10.9443 −0.867937
\(160\) 2.00000 0.158114
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) −8.94427 −0.700569 −0.350285 0.936643i \(-0.613915\pi\)
−0.350285 + 0.936643i \(0.613915\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 2.47214 0.191875
\(167\) −15.4164 −1.19296 −0.596479 0.802629i \(-0.703434\pi\)
−0.596479 + 0.802629i \(0.703434\pi\)
\(168\) 1.00000 0.0771517
\(169\) 7.00000 0.538462
\(170\) 8.94427 0.685994
\(171\) 2.47214 0.189049
\(172\) −4.94427 −0.376997
\(173\) 0.472136 0.0358958 0.0179479 0.999839i \(-0.494287\pi\)
0.0179479 + 0.999839i \(0.494287\pi\)
\(174\) −2.00000 −0.151620
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 9.41641 0.705790
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −4.47214 −0.331497
\(183\) −6.00000 −0.443533
\(184\) −1.00000 −0.0737210
\(185\) 13.8885 1.02111
\(186\) −2.47214 −0.181266
\(187\) 0 0
\(188\) −2.47214 −0.180299
\(189\) 1.00000 0.0727393
\(190\) 4.94427 0.358695
\(191\) −3.05573 −0.221105 −0.110552 0.993870i \(-0.535262\pi\)
−0.110552 + 0.993870i \(0.535262\pi\)
\(192\) 1.00000 0.0721688
\(193\) 11.8885 0.855756 0.427878 0.903836i \(-0.359261\pi\)
0.427878 + 0.903836i \(0.359261\pi\)
\(194\) −0.472136 −0.0338974
\(195\) −8.94427 −0.640513
\(196\) 1.00000 0.0714286
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 0 0
\(199\) 3.05573 0.216615 0.108307 0.994117i \(-0.465457\pi\)
0.108307 + 0.994117i \(0.465457\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.94427 0.348742
\(202\) −9.41641 −0.662536
\(203\) −2.00000 −0.140372
\(204\) 4.47214 0.313112
\(205\) −12.0000 −0.838116
\(206\) −4.94427 −0.344484
\(207\) −1.00000 −0.0695048
\(208\) −4.47214 −0.310087
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) −0.944272 −0.0650064 −0.0325032 0.999472i \(-0.510348\pi\)
−0.0325032 + 0.999472i \(0.510348\pi\)
\(212\) −10.9443 −0.751656
\(213\) 4.94427 0.338776
\(214\) −8.00000 −0.546869
\(215\) −9.88854 −0.674393
\(216\) 1.00000 0.0680414
\(217\) −2.47214 −0.167820
\(218\) −2.94427 −0.199411
\(219\) 14.9443 1.00984
\(220\) 0 0
\(221\) −20.0000 −1.34535
\(222\) 6.94427 0.466069
\(223\) 18.4721 1.23699 0.618493 0.785790i \(-0.287744\pi\)
0.618493 + 0.785790i \(0.287744\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −18.9443 −1.26015
\(227\) 5.52786 0.366897 0.183449 0.983029i \(-0.441274\pi\)
0.183449 + 0.983029i \(0.441274\pi\)
\(228\) 2.47214 0.163721
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.47214 −0.292353
\(235\) −4.94427 −0.322529
\(236\) 4.00000 0.260378
\(237\) −4.94427 −0.321165
\(238\) 4.47214 0.289886
\(239\) 20.9443 1.35477 0.677386 0.735628i \(-0.263113\pi\)
0.677386 + 0.735628i \(0.263113\pi\)
\(240\) 2.00000 0.129099
\(241\) 20.4721 1.31873 0.659363 0.751825i \(-0.270826\pi\)
0.659363 + 0.751825i \(0.270826\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −6.00000 −0.384111
\(245\) 2.00000 0.127775
\(246\) −6.00000 −0.382546
\(247\) −11.0557 −0.703459
\(248\) −2.47214 −0.156981
\(249\) 2.47214 0.156665
\(250\) −12.0000 −0.758947
\(251\) −5.52786 −0.348916 −0.174458 0.984665i \(-0.555817\pi\)
−0.174458 + 0.984665i \(0.555817\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 20.9443 1.31416
\(255\) 8.94427 0.560112
\(256\) 1.00000 0.0625000
\(257\) −2.94427 −0.183659 −0.0918293 0.995775i \(-0.529271\pi\)
−0.0918293 + 0.995775i \(0.529271\pi\)
\(258\) −4.94427 −0.307817
\(259\) 6.94427 0.431496
\(260\) −8.94427 −0.554700
\(261\) −2.00000 −0.123797
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −21.8885 −1.34460
\(266\) 2.47214 0.151576
\(267\) 9.41641 0.576275
\(268\) 4.94427 0.302019
\(269\) 16.4721 1.00432 0.502162 0.864774i \(-0.332538\pi\)
0.502162 + 0.864774i \(0.332538\pi\)
\(270\) 2.00000 0.121716
\(271\) 15.4164 0.936480 0.468240 0.883601i \(-0.344888\pi\)
0.468240 + 0.883601i \(0.344888\pi\)
\(272\) 4.47214 0.271163
\(273\) −4.47214 −0.270666
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −11.8885 −0.714313 −0.357157 0.934044i \(-0.616254\pi\)
−0.357157 + 0.934044i \(0.616254\pi\)
\(278\) 8.94427 0.536442
\(279\) −2.47214 −0.148003
\(280\) 2.00000 0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −2.47214 −0.147214
\(283\) 18.4721 1.09805 0.549027 0.835804i \(-0.314998\pi\)
0.549027 + 0.835804i \(0.314998\pi\)
\(284\) 4.94427 0.293389
\(285\) 4.94427 0.292873
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 3.00000 0.176471
\(290\) −4.00000 −0.234888
\(291\) −0.472136 −0.0276771
\(292\) 14.9443 0.874547
\(293\) 27.8885 1.62927 0.814633 0.579977i \(-0.196938\pi\)
0.814633 + 0.579977i \(0.196938\pi\)
\(294\) 1.00000 0.0583212
\(295\) 8.00000 0.465778
\(296\) 6.94427 0.403628
\(297\) 0 0
\(298\) −1.05573 −0.0611567
\(299\) 4.47214 0.258630
\(300\) −1.00000 −0.0577350
\(301\) −4.94427 −0.284983
\(302\) 4.94427 0.284511
\(303\) −9.41641 −0.540958
\(304\) 2.47214 0.141787
\(305\) −12.0000 −0.687118
\(306\) 4.47214 0.255655
\(307\) 24.9443 1.42364 0.711822 0.702360i \(-0.247870\pi\)
0.711822 + 0.702360i \(0.247870\pi\)
\(308\) 0 0
\(309\) −4.94427 −0.281270
\(310\) −4.94427 −0.280816
\(311\) −2.47214 −0.140182 −0.0700910 0.997541i \(-0.522329\pi\)
−0.0700910 + 0.997541i \(0.522329\pi\)
\(312\) −4.47214 −0.253185
\(313\) −13.4164 −0.758340 −0.379170 0.925327i \(-0.623790\pi\)
−0.379170 + 0.925327i \(0.623790\pi\)
\(314\) −2.94427 −0.166155
\(315\) 2.00000 0.112687
\(316\) −4.94427 −0.278137
\(317\) −13.0557 −0.733283 −0.366641 0.930362i \(-0.619492\pi\)
−0.366641 + 0.930362i \(0.619492\pi\)
\(318\) −10.9443 −0.613724
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) −8.00000 −0.446516
\(322\) −1.00000 −0.0557278
\(323\) 11.0557 0.615157
\(324\) 1.00000 0.0555556
\(325\) 4.47214 0.248069
\(326\) −8.94427 −0.495377
\(327\) −2.94427 −0.162819
\(328\) −6.00000 −0.331295
\(329\) −2.47214 −0.136293
\(330\) 0 0
\(331\) −13.8885 −0.763383 −0.381692 0.924290i \(-0.624658\pi\)
−0.381692 + 0.924290i \(0.624658\pi\)
\(332\) 2.47214 0.135676
\(333\) 6.94427 0.380544
\(334\) −15.4164 −0.843548
\(335\) 9.88854 0.540269
\(336\) 1.00000 0.0545545
\(337\) −2.94427 −0.160385 −0.0801924 0.996779i \(-0.525553\pi\)
−0.0801924 + 0.996779i \(0.525553\pi\)
\(338\) 7.00000 0.380750
\(339\) −18.9443 −1.02891
\(340\) 8.94427 0.485071
\(341\) 0 0
\(342\) 2.47214 0.133678
\(343\) 1.00000 0.0539949
\(344\) −4.94427 −0.266577
\(345\) −2.00000 −0.107676
\(346\) 0.472136 0.0253822
\(347\) −0.944272 −0.0506912 −0.0253456 0.999679i \(-0.508069\pi\)
−0.0253456 + 0.999679i \(0.508069\pi\)
\(348\) −2.00000 −0.107211
\(349\) 24.4721 1.30996 0.654982 0.755645i \(-0.272676\pi\)
0.654982 + 0.755645i \(0.272676\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 4.00000 0.212598
\(355\) 9.88854 0.524829
\(356\) 9.41641 0.499069
\(357\) 4.47214 0.236691
\(358\) −20.0000 −1.05703
\(359\) −30.8328 −1.62729 −0.813647 0.581359i \(-0.802521\pi\)
−0.813647 + 0.581359i \(0.802521\pi\)
\(360\) 2.00000 0.105409
\(361\) −12.8885 −0.678344
\(362\) 2.00000 0.105118
\(363\) −11.0000 −0.577350
\(364\) −4.47214 −0.234404
\(365\) 29.8885 1.56444
\(366\) −6.00000 −0.313625
\(367\) 30.8328 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.00000 −0.312348
\(370\) 13.8885 0.722031
\(371\) −10.9443 −0.568198
\(372\) −2.47214 −0.128174
\(373\) 6.94427 0.359561 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) −2.47214 −0.127491
\(377\) 8.94427 0.460653
\(378\) 1.00000 0.0514344
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 4.94427 0.253636
\(381\) 20.9443 1.07301
\(382\) −3.05573 −0.156345
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 11.8885 0.605111
\(387\) −4.94427 −0.251331
\(388\) −0.472136 −0.0239691
\(389\) −18.9443 −0.960513 −0.480256 0.877128i \(-0.659456\pi\)
−0.480256 + 0.877128i \(0.659456\pi\)
\(390\) −8.94427 −0.452911
\(391\) −4.47214 −0.226166
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) −14.9443 −0.752882
\(395\) −9.88854 −0.497547
\(396\) 0 0
\(397\) −36.4721 −1.83048 −0.915242 0.402905i \(-0.868001\pi\)
−0.915242 + 0.402905i \(0.868001\pi\)
\(398\) 3.05573 0.153170
\(399\) 2.47214 0.123762
\(400\) −1.00000 −0.0500000
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 4.94427 0.246598
\(403\) 11.0557 0.550725
\(404\) −9.41641 −0.468484
\(405\) 2.00000 0.0993808
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 4.47214 0.221404
\(409\) −15.8885 −0.785638 −0.392819 0.919616i \(-0.628500\pi\)
−0.392819 + 0.919616i \(0.628500\pi\)
\(410\) −12.0000 −0.592638
\(411\) −14.0000 −0.690569
\(412\) −4.94427 −0.243587
\(413\) 4.00000 0.196827
\(414\) −1.00000 −0.0491473
\(415\) 4.94427 0.242705
\(416\) −4.47214 −0.219265
\(417\) 8.94427 0.438003
\(418\) 0 0
\(419\) −7.41641 −0.362315 −0.181158 0.983454i \(-0.557984\pi\)
−0.181158 + 0.983454i \(0.557984\pi\)
\(420\) 2.00000 0.0975900
\(421\) 32.8328 1.60017 0.800087 0.599884i \(-0.204787\pi\)
0.800087 + 0.599884i \(0.204787\pi\)
\(422\) −0.944272 −0.0459664
\(423\) −2.47214 −0.120199
\(424\) −10.9443 −0.531501
\(425\) −4.47214 −0.216930
\(426\) 4.94427 0.239551
\(427\) −6.00000 −0.290360
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −9.88854 −0.476868
\(431\) 17.8885 0.861661 0.430830 0.902433i \(-0.358221\pi\)
0.430830 + 0.902433i \(0.358221\pi\)
\(432\) 1.00000 0.0481125
\(433\) 25.4164 1.22143 0.610717 0.791849i \(-0.290881\pi\)
0.610717 + 0.791849i \(0.290881\pi\)
\(434\) −2.47214 −0.118666
\(435\) −4.00000 −0.191785
\(436\) −2.94427 −0.141005
\(437\) −2.47214 −0.118258
\(438\) 14.9443 0.714065
\(439\) 28.3607 1.35358 0.676791 0.736175i \(-0.263370\pi\)
0.676791 + 0.736175i \(0.263370\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −20.0000 −0.951303
\(443\) 34.8328 1.65496 0.827479 0.561497i \(-0.189775\pi\)
0.827479 + 0.561497i \(0.189775\pi\)
\(444\) 6.94427 0.329561
\(445\) 18.8328 0.892761
\(446\) 18.4721 0.874681
\(447\) −1.05573 −0.0499342
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0 0
\(452\) −18.9443 −0.891064
\(453\) 4.94427 0.232302
\(454\) 5.52786 0.259436
\(455\) −8.94427 −0.419314
\(456\) 2.47214 0.115768
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −14.0000 −0.654177
\(459\) 4.47214 0.208741
\(460\) −2.00000 −0.0932505
\(461\) 6.58359 0.306628 0.153314 0.988177i \(-0.451005\pi\)
0.153314 + 0.988177i \(0.451005\pi\)
\(462\) 0 0
\(463\) 6.11146 0.284023 0.142012 0.989865i \(-0.454643\pi\)
0.142012 + 0.989865i \(0.454643\pi\)
\(464\) −2.00000 −0.0928477
\(465\) −4.94427 −0.229285
\(466\) −6.00000 −0.277945
\(467\) 33.3050 1.54117 0.770585 0.637338i \(-0.219964\pi\)
0.770585 + 0.637338i \(0.219964\pi\)
\(468\) −4.47214 −0.206725
\(469\) 4.94427 0.228305
\(470\) −4.94427 −0.228062
\(471\) −2.94427 −0.135665
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) −4.94427 −0.227098
\(475\) −2.47214 −0.113429
\(476\) 4.47214 0.204980
\(477\) −10.9443 −0.501104
\(478\) 20.9443 0.957969
\(479\) −4.94427 −0.225910 −0.112955 0.993600i \(-0.536032\pi\)
−0.112955 + 0.993600i \(0.536032\pi\)
\(480\) 2.00000 0.0912871
\(481\) −31.0557 −1.41602
\(482\) 20.4721 0.932480
\(483\) −1.00000 −0.0455016
\(484\) −11.0000 −0.500000
\(485\) −0.944272 −0.0428772
\(486\) 1.00000 0.0453609
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −6.00000 −0.271607
\(489\) −8.94427 −0.404474
\(490\) 2.00000 0.0903508
\(491\) 40.9443 1.84779 0.923895 0.382647i \(-0.124987\pi\)
0.923895 + 0.382647i \(0.124987\pi\)
\(492\) −6.00000 −0.270501
\(493\) −8.94427 −0.402830
\(494\) −11.0557 −0.497421
\(495\) 0 0
\(496\) −2.47214 −0.111002
\(497\) 4.94427 0.221781
\(498\) 2.47214 0.110779
\(499\) 32.9443 1.47479 0.737394 0.675463i \(-0.236056\pi\)
0.737394 + 0.675463i \(0.236056\pi\)
\(500\) −12.0000 −0.536656
\(501\) −15.4164 −0.688754
\(502\) −5.52786 −0.246721
\(503\) −9.88854 −0.440908 −0.220454 0.975397i \(-0.570754\pi\)
−0.220454 + 0.975397i \(0.570754\pi\)
\(504\) 1.00000 0.0445435
\(505\) −18.8328 −0.838049
\(506\) 0 0
\(507\) 7.00000 0.310881
\(508\) 20.9443 0.929252
\(509\) 6.58359 0.291813 0.145906 0.989298i \(-0.453390\pi\)
0.145906 + 0.989298i \(0.453390\pi\)
\(510\) 8.94427 0.396059
\(511\) 14.9443 0.661096
\(512\) 1.00000 0.0441942
\(513\) 2.47214 0.109147
\(514\) −2.94427 −0.129866
\(515\) −9.88854 −0.435741
\(516\) −4.94427 −0.217659
\(517\) 0 0
\(518\) 6.94427 0.305114
\(519\) 0.472136 0.0207245
\(520\) −8.94427 −0.392232
\(521\) 33.4164 1.46400 0.732000 0.681305i \(-0.238587\pi\)
0.732000 + 0.681305i \(0.238587\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −25.3050 −1.10651 −0.553254 0.833013i \(-0.686614\pi\)
−0.553254 + 0.833013i \(0.686614\pi\)
\(524\) 12.0000 0.524222
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) −11.0557 −0.481595
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −21.8885 −0.950778
\(531\) 4.00000 0.173585
\(532\) 2.47214 0.107181
\(533\) 26.8328 1.16226
\(534\) 9.41641 0.407488
\(535\) −16.0000 −0.691740
\(536\) 4.94427 0.213560
\(537\) −20.0000 −0.863064
\(538\) 16.4721 0.710164
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 15.4164 0.662191
\(543\) 2.00000 0.0858282
\(544\) 4.47214 0.191741
\(545\) −5.88854 −0.252238
\(546\) −4.47214 −0.191390
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −14.0000 −0.598050
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −4.94427 −0.210633
\(552\) −1.00000 −0.0425628
\(553\) −4.94427 −0.210252
\(554\) −11.8885 −0.505096
\(555\) 13.8885 0.589536
\(556\) 8.94427 0.379322
\(557\) −17.0557 −0.722674 −0.361337 0.932435i \(-0.617680\pi\)
−0.361337 + 0.932435i \(0.617680\pi\)
\(558\) −2.47214 −0.104654
\(559\) 22.1115 0.935215
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) 26.4721 1.11567 0.557834 0.829953i \(-0.311633\pi\)
0.557834 + 0.829953i \(0.311633\pi\)
\(564\) −2.47214 −0.104096
\(565\) −37.8885 −1.59398
\(566\) 18.4721 0.776442
\(567\) 1.00000 0.0419961
\(568\) 4.94427 0.207457
\(569\) −7.88854 −0.330705 −0.165352 0.986235i \(-0.552876\pi\)
−0.165352 + 0.986235i \(0.552876\pi\)
\(570\) 4.94427 0.207093
\(571\) −12.9443 −0.541701 −0.270850 0.962621i \(-0.587305\pi\)
−0.270850 + 0.962621i \(0.587305\pi\)
\(572\) 0 0
\(573\) −3.05573 −0.127655
\(574\) −6.00000 −0.250435
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −2.94427 −0.122572 −0.0612858 0.998120i \(-0.519520\pi\)
−0.0612858 + 0.998120i \(0.519520\pi\)
\(578\) 3.00000 0.124784
\(579\) 11.8885 0.494071
\(580\) −4.00000 −0.166091
\(581\) 2.47214 0.102561
\(582\) −0.472136 −0.0195707
\(583\) 0 0
\(584\) 14.9443 0.618398
\(585\) −8.94427 −0.369800
\(586\) 27.8885 1.15207
\(587\) −0.944272 −0.0389743 −0.0194871 0.999810i \(-0.506203\pi\)
−0.0194871 + 0.999810i \(0.506203\pi\)
\(588\) 1.00000 0.0412393
\(589\) −6.11146 −0.251818
\(590\) 8.00000 0.329355
\(591\) −14.9443 −0.614725
\(592\) 6.94427 0.285408
\(593\) −28.8328 −1.18402 −0.592011 0.805930i \(-0.701666\pi\)
−0.592011 + 0.805930i \(0.701666\pi\)
\(594\) 0 0
\(595\) 8.94427 0.366679
\(596\) −1.05573 −0.0432443
\(597\) 3.05573 0.125063
\(598\) 4.47214 0.182879
\(599\) 33.8885 1.38465 0.692324 0.721587i \(-0.256587\pi\)
0.692324 + 0.721587i \(0.256587\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 40.8328 1.66561 0.832803 0.553570i \(-0.186735\pi\)
0.832803 + 0.553570i \(0.186735\pi\)
\(602\) −4.94427 −0.201513
\(603\) 4.94427 0.201346
\(604\) 4.94427 0.201180
\(605\) −22.0000 −0.894427
\(606\) −9.41641 −0.382515
\(607\) −28.3607 −1.15112 −0.575562 0.817758i \(-0.695217\pi\)
−0.575562 + 0.817758i \(0.695217\pi\)
\(608\) 2.47214 0.100258
\(609\) −2.00000 −0.0810441
\(610\) −12.0000 −0.485866
\(611\) 11.0557 0.447267
\(612\) 4.47214 0.180775
\(613\) 40.8328 1.64922 0.824611 0.565700i \(-0.191394\pi\)
0.824611 + 0.565700i \(0.191394\pi\)
\(614\) 24.9443 1.00667
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −4.94427 −0.198888
\(619\) −34.4721 −1.38555 −0.692776 0.721153i \(-0.743613\pi\)
−0.692776 + 0.721153i \(0.743613\pi\)
\(620\) −4.94427 −0.198567
\(621\) −1.00000 −0.0401286
\(622\) −2.47214 −0.0991236
\(623\) 9.41641 0.377260
\(624\) −4.47214 −0.179029
\(625\) −19.0000 −0.760000
\(626\) −13.4164 −0.536228
\(627\) 0 0
\(628\) −2.94427 −0.117489
\(629\) 31.0557 1.23827
\(630\) 2.00000 0.0796819
\(631\) −11.0557 −0.440122 −0.220061 0.975486i \(-0.570626\pi\)
−0.220061 + 0.975486i \(0.570626\pi\)
\(632\) −4.94427 −0.196673
\(633\) −0.944272 −0.0375314
\(634\) −13.0557 −0.518509
\(635\) 41.8885 1.66230
\(636\) −10.9443 −0.433969
\(637\) −4.47214 −0.177192
\(638\) 0 0
\(639\) 4.94427 0.195592
\(640\) 2.00000 0.0790569
\(641\) 10.0000 0.394976 0.197488 0.980305i \(-0.436722\pi\)
0.197488 + 0.980305i \(0.436722\pi\)
\(642\) −8.00000 −0.315735
\(643\) −10.4721 −0.412981 −0.206490 0.978449i \(-0.566204\pi\)
−0.206490 + 0.978449i \(0.566204\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −9.88854 −0.389361
\(646\) 11.0557 0.434982
\(647\) 12.3607 0.485948 0.242974 0.970033i \(-0.421877\pi\)
0.242974 + 0.970033i \(0.421877\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 4.47214 0.175412
\(651\) −2.47214 −0.0968906
\(652\) −8.94427 −0.350285
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −2.94427 −0.115130
\(655\) 24.0000 0.937758
\(656\) −6.00000 −0.234261
\(657\) 14.9443 0.583032
\(658\) −2.47214 −0.0963739
\(659\) 46.8328 1.82435 0.912174 0.409804i \(-0.134403\pi\)
0.912174 + 0.409804i \(0.134403\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) −13.8885 −0.539794
\(663\) −20.0000 −0.776736
\(664\) 2.47214 0.0959375
\(665\) 4.94427 0.191731
\(666\) 6.94427 0.269085
\(667\) 2.00000 0.0774403
\(668\) −15.4164 −0.596479
\(669\) 18.4721 0.714174
\(670\) 9.88854 0.382028
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −2.94427 −0.113409
\(675\) −1.00000 −0.0384900
\(676\) 7.00000 0.269231
\(677\) −4.11146 −0.158016 −0.0790080 0.996874i \(-0.525175\pi\)
−0.0790080 + 0.996874i \(0.525175\pi\)
\(678\) −18.9443 −0.727550
\(679\) −0.472136 −0.0181189
\(680\) 8.94427 0.342997
\(681\) 5.52786 0.211828
\(682\) 0 0
\(683\) 16.9443 0.648355 0.324177 0.945996i \(-0.394913\pi\)
0.324177 + 0.945996i \(0.394913\pi\)
\(684\) 2.47214 0.0945245
\(685\) −28.0000 −1.06983
\(686\) 1.00000 0.0381802
\(687\) −14.0000 −0.534133
\(688\) −4.94427 −0.188499
\(689\) 48.9443 1.86463
\(690\) −2.00000 −0.0761387
\(691\) −15.0557 −0.572747 −0.286373 0.958118i \(-0.592450\pi\)
−0.286373 + 0.958118i \(0.592450\pi\)
\(692\) 0.472136 0.0179479
\(693\) 0 0
\(694\) −0.944272 −0.0358441
\(695\) 17.8885 0.678551
\(696\) −2.00000 −0.0758098
\(697\) −26.8328 −1.01637
\(698\) 24.4721 0.926284
\(699\) −6.00000 −0.226941
\(700\) −1.00000 −0.0377964
\(701\) −20.8328 −0.786845 −0.393422 0.919358i \(-0.628709\pi\)
−0.393422 + 0.919358i \(0.628709\pi\)
\(702\) −4.47214 −0.168790
\(703\) 17.1672 0.647473
\(704\) 0 0
\(705\) −4.94427 −0.186212
\(706\) 2.00000 0.0752710
\(707\) −9.41641 −0.354140
\(708\) 4.00000 0.150329
\(709\) 6.94427 0.260798 0.130399 0.991462i \(-0.458374\pi\)
0.130399 + 0.991462i \(0.458374\pi\)
\(710\) 9.88854 0.371110
\(711\) −4.94427 −0.185425
\(712\) 9.41641 0.352895
\(713\) 2.47214 0.0925822
\(714\) 4.47214 0.167365
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 20.9443 0.782178
\(718\) −30.8328 −1.15067
\(719\) −5.52786 −0.206155 −0.103077 0.994673i \(-0.532869\pi\)
−0.103077 + 0.994673i \(0.532869\pi\)
\(720\) 2.00000 0.0745356
\(721\) −4.94427 −0.184134
\(722\) −12.8885 −0.479662
\(723\) 20.4721 0.761367
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) −11.0000 −0.408248
\(727\) 3.05573 0.113331 0.0566653 0.998393i \(-0.481953\pi\)
0.0566653 + 0.998393i \(0.481953\pi\)
\(728\) −4.47214 −0.165748
\(729\) 1.00000 0.0370370
\(730\) 29.8885 1.10622
\(731\) −22.1115 −0.817822
\(732\) −6.00000 −0.221766
\(733\) −50.9443 −1.88167 −0.940835 0.338866i \(-0.889957\pi\)
−0.940835 + 0.338866i \(0.889957\pi\)
\(734\) 30.8328 1.13806
\(735\) 2.00000 0.0737711
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) −6.00000 −0.220863
\(739\) −8.94427 −0.329020 −0.164510 0.986375i \(-0.552604\pi\)
−0.164510 + 0.986375i \(0.552604\pi\)
\(740\) 13.8885 0.510553
\(741\) −11.0557 −0.406142
\(742\) −10.9443 −0.401777
\(743\) −22.8328 −0.837655 −0.418827 0.908066i \(-0.637559\pi\)
−0.418827 + 0.908066i \(0.637559\pi\)
\(744\) −2.47214 −0.0906329
\(745\) −2.11146 −0.0773578
\(746\) 6.94427 0.254248
\(747\) 2.47214 0.0904507
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) −12.0000 −0.438178
\(751\) 17.8885 0.652762 0.326381 0.945238i \(-0.394171\pi\)
0.326381 + 0.945238i \(0.394171\pi\)
\(752\) −2.47214 −0.0901495
\(753\) −5.52786 −0.201447
\(754\) 8.94427 0.325731
\(755\) 9.88854 0.359881
\(756\) 1.00000 0.0363696
\(757\) −42.9443 −1.56084 −0.780418 0.625258i \(-0.784994\pi\)
−0.780418 + 0.625258i \(0.784994\pi\)
\(758\) 24.0000 0.871719
\(759\) 0 0
\(760\) 4.94427 0.179348
\(761\) 21.0557 0.763270 0.381635 0.924313i \(-0.375361\pi\)
0.381635 + 0.924313i \(0.375361\pi\)
\(762\) 20.9443 0.758731
\(763\) −2.94427 −0.106590
\(764\) −3.05573 −0.110552
\(765\) 8.94427 0.323381
\(766\) 16.0000 0.578103
\(767\) −17.8885 −0.645918
\(768\) 1.00000 0.0360844
\(769\) 30.3607 1.09483 0.547417 0.836860i \(-0.315611\pi\)
0.547417 + 0.836860i \(0.315611\pi\)
\(770\) 0 0
\(771\) −2.94427 −0.106035
\(772\) 11.8885 0.427878
\(773\) 5.05573 0.181842 0.0909210 0.995858i \(-0.471019\pi\)
0.0909210 + 0.995858i \(0.471019\pi\)
\(774\) −4.94427 −0.177718
\(775\) 2.47214 0.0888017
\(776\) −0.472136 −0.0169487
\(777\) 6.94427 0.249124
\(778\) −18.9443 −0.679185
\(779\) −14.8328 −0.531441
\(780\) −8.94427 −0.320256
\(781\) 0 0
\(782\) −4.47214 −0.159923
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −5.88854 −0.210171
\(786\) 12.0000 0.428026
\(787\) −0.583592 −0.0208028 −0.0104014 0.999946i \(-0.503311\pi\)
−0.0104014 + 0.999946i \(0.503311\pi\)
\(788\) −14.9443 −0.532368
\(789\) −24.0000 −0.854423
\(790\) −9.88854 −0.351819
\(791\) −18.9443 −0.673581
\(792\) 0 0
\(793\) 26.8328 0.952861
\(794\) −36.4721 −1.29435
\(795\) −21.8885 −0.776307
\(796\) 3.05573 0.108307
\(797\) −28.8328 −1.02131 −0.510655 0.859785i \(-0.670597\pi\)
−0.510655 + 0.859785i \(0.670597\pi\)
\(798\) 2.47214 0.0875127
\(799\) −11.0557 −0.391124
\(800\) −1.00000 −0.0353553
\(801\) 9.41641 0.332712
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) 4.94427 0.174371
\(805\) −2.00000 −0.0704907
\(806\) 11.0557 0.389421
\(807\) 16.4721 0.579847
\(808\) −9.41641 −0.331268
\(809\) 35.8885 1.26177 0.630887 0.775875i \(-0.282691\pi\)
0.630887 + 0.775875i \(0.282691\pi\)
\(810\) 2.00000 0.0702728
\(811\) −37.8885 −1.33045 −0.665223 0.746644i \(-0.731664\pi\)
−0.665223 + 0.746644i \(0.731664\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 15.4164 0.540677
\(814\) 0 0
\(815\) −17.8885 −0.626608
\(816\) 4.47214 0.156556
\(817\) −12.2229 −0.427626
\(818\) −15.8885 −0.555530
\(819\) −4.47214 −0.156269
\(820\) −12.0000 −0.419058
\(821\) −16.8328 −0.587469 −0.293735 0.955887i \(-0.594898\pi\)
−0.293735 + 0.955887i \(0.594898\pi\)
\(822\) −14.0000 −0.488306
\(823\) 20.9443 0.730071 0.365036 0.930994i \(-0.381057\pi\)
0.365036 + 0.930994i \(0.381057\pi\)
\(824\) −4.94427 −0.172242
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) 33.8885 1.17842 0.589210 0.807980i \(-0.299439\pi\)
0.589210 + 0.807980i \(0.299439\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 13.4164 0.465971 0.232986 0.972480i \(-0.425151\pi\)
0.232986 + 0.972480i \(0.425151\pi\)
\(830\) 4.94427 0.171618
\(831\) −11.8885 −0.412409
\(832\) −4.47214 −0.155043
\(833\) 4.47214 0.154950
\(834\) 8.94427 0.309715
\(835\) −30.8328 −1.06701
\(836\) 0 0
\(837\) −2.47214 −0.0854495
\(838\) −7.41641 −0.256196
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 2.00000 0.0690066
\(841\) −25.0000 −0.862069
\(842\) 32.8328 1.13149
\(843\) 10.0000 0.344418
\(844\) −0.944272 −0.0325032
\(845\) 14.0000 0.481615
\(846\) −2.47214 −0.0849938
\(847\) −11.0000 −0.377964
\(848\) −10.9443 −0.375828
\(849\) 18.4721 0.633962
\(850\) −4.47214 −0.153393
\(851\) −6.94427 −0.238047
\(852\) 4.94427 0.169388
\(853\) −6.36068 −0.217786 −0.108893 0.994054i \(-0.534731\pi\)
−0.108893 + 0.994054i \(0.534731\pi\)
\(854\) −6.00000 −0.205316
\(855\) 4.94427 0.169091
\(856\) −8.00000 −0.273434
\(857\) −47.8885 −1.63584 −0.817921 0.575331i \(-0.804873\pi\)
−0.817921 + 0.575331i \(0.804873\pi\)
\(858\) 0 0
\(859\) 32.9443 1.12404 0.562022 0.827122i \(-0.310024\pi\)
0.562022 + 0.827122i \(0.310024\pi\)
\(860\) −9.88854 −0.337197
\(861\) −6.00000 −0.204479
\(862\) 17.8885 0.609286
\(863\) 20.9443 0.712951 0.356476 0.934305i \(-0.383978\pi\)
0.356476 + 0.934305i \(0.383978\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0.944272 0.0321062
\(866\) 25.4164 0.863685
\(867\) 3.00000 0.101885
\(868\) −2.47214 −0.0839098
\(869\) 0 0
\(870\) −4.00000 −0.135613
\(871\) −22.1115 −0.749218
\(872\) −2.94427 −0.0997056
\(873\) −0.472136 −0.0159794
\(874\) −2.47214 −0.0836212
\(875\) −12.0000 −0.405674
\(876\) 14.9443 0.504920
\(877\) −34.7214 −1.17246 −0.586229 0.810146i \(-0.699388\pi\)
−0.586229 + 0.810146i \(0.699388\pi\)
\(878\) 28.3607 0.957127
\(879\) 27.8885 0.940657
\(880\) 0 0
\(881\) 22.3607 0.753350 0.376675 0.926345i \(-0.377067\pi\)
0.376675 + 0.926345i \(0.377067\pi\)
\(882\) 1.00000 0.0336718
\(883\) 2.83282 0.0953318 0.0476659 0.998863i \(-0.484822\pi\)
0.0476659 + 0.998863i \(0.484822\pi\)
\(884\) −20.0000 −0.672673
\(885\) 8.00000 0.268917
\(886\) 34.8328 1.17023
\(887\) −5.52786 −0.185608 −0.0928038 0.995684i \(-0.529583\pi\)
−0.0928038 + 0.995684i \(0.529583\pi\)
\(888\) 6.94427 0.233035
\(889\) 20.9443 0.702448
\(890\) 18.8328 0.631277
\(891\) 0 0
\(892\) 18.4721 0.618493
\(893\) −6.11146 −0.204512
\(894\) −1.05573 −0.0353088
\(895\) −40.0000 −1.33705
\(896\) 1.00000 0.0334077
\(897\) 4.47214 0.149320
\(898\) 18.0000 0.600668
\(899\) 4.94427 0.164901
\(900\) −1.00000 −0.0333333
\(901\) −48.9443 −1.63057
\(902\) 0 0
\(903\) −4.94427 −0.164535
\(904\) −18.9443 −0.630077
\(905\) 4.00000 0.132964
\(906\) 4.94427 0.164262
\(907\) −41.8885 −1.39089 −0.695443 0.718581i \(-0.744792\pi\)
−0.695443 + 0.718581i \(0.744792\pi\)
\(908\) 5.52786 0.183449
\(909\) −9.41641 −0.312323
\(910\) −8.94427 −0.296500
\(911\) −48.7214 −1.61421 −0.807105 0.590407i \(-0.798967\pi\)
−0.807105 + 0.590407i \(0.798967\pi\)
\(912\) 2.47214 0.0818606
\(913\) 0 0
\(914\) 2.00000 0.0661541
\(915\) −12.0000 −0.396708
\(916\) −14.0000 −0.462573
\(917\) 12.0000 0.396275
\(918\) 4.47214 0.147602
\(919\) 35.0557 1.15638 0.578191 0.815902i \(-0.303759\pi\)
0.578191 + 0.815902i \(0.303759\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 24.9443 0.821942
\(922\) 6.58359 0.216819
\(923\) −22.1115 −0.727807
\(924\) 0 0
\(925\) −6.94427 −0.228326
\(926\) 6.11146 0.200835
\(927\) −4.94427 −0.162391
\(928\) −2.00000 −0.0656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −4.94427 −0.162129
\(931\) 2.47214 0.0810210
\(932\) −6.00000 −0.196537
\(933\) −2.47214 −0.0809341
\(934\) 33.3050 1.08977
\(935\) 0 0
\(936\) −4.47214 −0.146176
\(937\) 40.2492 1.31488 0.657442 0.753505i \(-0.271638\pi\)
0.657442 + 0.753505i \(0.271638\pi\)
\(938\) 4.94427 0.161436
\(939\) −13.4164 −0.437828
\(940\) −4.94427 −0.161264
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) −2.94427 −0.0959296
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 0.944272 0.0306847 0.0153424 0.999882i \(-0.495116\pi\)
0.0153424 + 0.999882i \(0.495116\pi\)
\(948\) −4.94427 −0.160582
\(949\) −66.8328 −2.16949
\(950\) −2.47214 −0.0802067
\(951\) −13.0557 −0.423361
\(952\) 4.47214 0.144943
\(953\) −9.05573 −0.293344 −0.146672 0.989185i \(-0.546856\pi\)
−0.146672 + 0.989185i \(0.546856\pi\)
\(954\) −10.9443 −0.354334
\(955\) −6.11146 −0.197762
\(956\) 20.9443 0.677386
\(957\) 0 0
\(958\) −4.94427 −0.159742
\(959\) −14.0000 −0.452084
\(960\) 2.00000 0.0645497
\(961\) −24.8885 −0.802856
\(962\) −31.0557 −1.00128
\(963\) −8.00000 −0.257796
\(964\) 20.4721 0.659363
\(965\) 23.7771 0.765412
\(966\) −1.00000 −0.0321745
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) −11.0000 −0.353553
\(969\) 11.0557 0.355161
\(970\) −0.944272 −0.0303187
\(971\) −23.4164 −0.751468 −0.375734 0.926727i \(-0.622609\pi\)
−0.375734 + 0.926727i \(0.622609\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.94427 0.286740
\(974\) 24.0000 0.769010
\(975\) 4.47214 0.143223
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −8.94427 −0.286006
\(979\) 0 0
\(980\) 2.00000 0.0638877
\(981\) −2.94427 −0.0940034
\(982\) 40.9443 1.30658
\(983\) −22.1115 −0.705246 −0.352623 0.935765i \(-0.614710\pi\)
−0.352623 + 0.935765i \(0.614710\pi\)
\(984\) −6.00000 −0.191273
\(985\) −29.8885 −0.952328
\(986\) −8.94427 −0.284844
\(987\) −2.47214 −0.0786890
\(988\) −11.0557 −0.351730
\(989\) 4.94427 0.157219
\(990\) 0 0
\(991\) 60.9443 1.93596 0.967979 0.251030i \(-0.0807693\pi\)
0.967979 + 0.251030i \(0.0807693\pi\)
\(992\) −2.47214 −0.0784904
\(993\) −13.8885 −0.440740
\(994\) 4.94427 0.156823
\(995\) 6.11146 0.193746
\(996\) 2.47214 0.0783326
\(997\) −46.3607 −1.46826 −0.734129 0.679010i \(-0.762409\pi\)
−0.734129 + 0.679010i \(0.762409\pi\)
\(998\) 32.9443 1.04283
\(999\) 6.94427 0.219707
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 966.2.a.p.1.1 2
3.2 odd 2 2898.2.a.v.1.1 2
4.3 odd 2 7728.2.a.bd.1.1 2
7.6 odd 2 6762.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.p.1.1 2 1.1 even 1 trivial
2898.2.a.v.1.1 2 3.2 odd 2
6762.2.a.bz.1.2 2 7.6 odd 2
7728.2.a.bd.1.1 2 4.3 odd 2