Properties

Label 2541.2.a.ba.1.2
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -4.23607 q^{5} +1.61803 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -4.23607 q^{5} +1.61803 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -6.85410 q^{10} +0.618034 q^{12} +6.23607 q^{13} -1.61803 q^{14} -4.23607 q^{15} -4.85410 q^{16} -4.47214 q^{17} +1.61803 q^{18} +3.00000 q^{19} -2.61803 q^{20} -1.00000 q^{21} +8.47214 q^{23} -2.23607 q^{24} +12.9443 q^{25} +10.0902 q^{26} +1.00000 q^{27} -0.618034 q^{28} -3.00000 q^{29} -6.85410 q^{30} -3.38197 q^{32} -7.23607 q^{34} +4.23607 q^{35} +0.618034 q^{36} +3.47214 q^{37} +4.85410 q^{38} +6.23607 q^{39} +9.47214 q^{40} +1.52786 q^{41} -1.61803 q^{42} +10.9443 q^{43} -4.23607 q^{45} +13.7082 q^{46} -3.00000 q^{47} -4.85410 q^{48} +1.00000 q^{49} +20.9443 q^{50} -4.47214 q^{51} +3.85410 q^{52} +8.94427 q^{53} +1.61803 q^{54} +2.23607 q^{56} +3.00000 q^{57} -4.85410 q^{58} -1.47214 q^{59} -2.61803 q^{60} +3.52786 q^{61} -1.00000 q^{63} +4.23607 q^{64} -26.4164 q^{65} -8.70820 q^{67} -2.76393 q^{68} +8.47214 q^{69} +6.85410 q^{70} +1.52786 q^{71} -2.23607 q^{72} +12.2361 q^{73} +5.61803 q^{74} +12.9443 q^{75} +1.85410 q^{76} +10.0902 q^{78} -13.4164 q^{79} +20.5623 q^{80} +1.00000 q^{81} +2.47214 q^{82} +6.00000 q^{83} -0.618034 q^{84} +18.9443 q^{85} +17.7082 q^{86} -3.00000 q^{87} -4.47214 q^{89} -6.85410 q^{90} -6.23607 q^{91} +5.23607 q^{92} -4.85410 q^{94} -12.7082 q^{95} -3.38197 q^{96} +2.00000 q^{97} +1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{5} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{5} + q^{6} - 2 q^{7} + 2 q^{9} - 7 q^{10} - q^{12} + 8 q^{13} - q^{14} - 4 q^{15} - 3 q^{16} + q^{18} + 6 q^{19} - 3 q^{20} - 2 q^{21} + 8 q^{23} + 8 q^{25} + 9 q^{26} + 2 q^{27} + q^{28} - 6 q^{29} - 7 q^{30} - 9 q^{32} - 10 q^{34} + 4 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} + 8 q^{39} + 10 q^{40} + 12 q^{41} - q^{42} + 4 q^{43} - 4 q^{45} + 14 q^{46} - 6 q^{47} - 3 q^{48} + 2 q^{49} + 24 q^{50} + q^{52} + q^{54} + 6 q^{57} - 3 q^{58} + 6 q^{59} - 3 q^{60} + 16 q^{61} - 2 q^{63} + 4 q^{64} - 26 q^{65} - 4 q^{67} - 10 q^{68} + 8 q^{69} + 7 q^{70} + 12 q^{71} + 20 q^{73} + 9 q^{74} + 8 q^{75} - 3 q^{76} + 9 q^{78} + 21 q^{80} + 2 q^{81} - 4 q^{82} + 12 q^{83} + q^{84} + 20 q^{85} + 22 q^{86} - 6 q^{87} - 7 q^{90} - 8 q^{91} + 6 q^{92} - 3 q^{94} - 12 q^{95} - 9 q^{96} + 4 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.618034 0.309017
\(5\) −4.23607 −1.89443 −0.947214 0.320603i \(-0.896114\pi\)
−0.947214 + 0.320603i \(0.896114\pi\)
\(6\) 1.61803 0.660560
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) −6.85410 −2.16746
\(11\) 0 0
\(12\) 0.618034 0.178411
\(13\) 6.23607 1.72957 0.864787 0.502139i \(-0.167453\pi\)
0.864787 + 0.502139i \(0.167453\pi\)
\(14\) −1.61803 −0.432438
\(15\) −4.23607 −1.09375
\(16\) −4.85410 −1.21353
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 1.61803 0.381374
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −2.61803 −0.585410
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 8.47214 1.76656 0.883281 0.468844i \(-0.155329\pi\)
0.883281 + 0.468844i \(0.155329\pi\)
\(24\) −2.23607 −0.456435
\(25\) 12.9443 2.58885
\(26\) 10.0902 1.97885
\(27\) 1.00000 0.192450
\(28\) −0.618034 −0.116797
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −6.85410 −1.25138
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −7.23607 −1.24098
\(35\) 4.23607 0.716026
\(36\) 0.618034 0.103006
\(37\) 3.47214 0.570816 0.285408 0.958406i \(-0.407871\pi\)
0.285408 + 0.958406i \(0.407871\pi\)
\(38\) 4.85410 0.787439
\(39\) 6.23607 0.998570
\(40\) 9.47214 1.49768
\(41\) 1.52786 0.238612 0.119306 0.992858i \(-0.461933\pi\)
0.119306 + 0.992858i \(0.461933\pi\)
\(42\) −1.61803 −0.249668
\(43\) 10.9443 1.66899 0.834493 0.551019i \(-0.185761\pi\)
0.834493 + 0.551019i \(0.185761\pi\)
\(44\) 0 0
\(45\) −4.23607 −0.631476
\(46\) 13.7082 2.02116
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −4.85410 −0.700629
\(49\) 1.00000 0.142857
\(50\) 20.9443 2.96197
\(51\) −4.47214 −0.626224
\(52\) 3.85410 0.534468
\(53\) 8.94427 1.22859 0.614295 0.789076i \(-0.289440\pi\)
0.614295 + 0.789076i \(0.289440\pi\)
\(54\) 1.61803 0.220187
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 3.00000 0.397360
\(58\) −4.85410 −0.637375
\(59\) −1.47214 −0.191656 −0.0958279 0.995398i \(-0.530550\pi\)
−0.0958279 + 0.995398i \(0.530550\pi\)
\(60\) −2.61803 −0.337987
\(61\) 3.52786 0.451697 0.225848 0.974162i \(-0.427485\pi\)
0.225848 + 0.974162i \(0.427485\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 4.23607 0.529508
\(65\) −26.4164 −3.27655
\(66\) 0 0
\(67\) −8.70820 −1.06388 −0.531938 0.846783i \(-0.678536\pi\)
−0.531938 + 0.846783i \(0.678536\pi\)
\(68\) −2.76393 −0.335176
\(69\) 8.47214 1.01993
\(70\) 6.85410 0.819222
\(71\) 1.52786 0.181324 0.0906621 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) −2.23607 −0.263523
\(73\) 12.2361 1.43212 0.716062 0.698037i \(-0.245943\pi\)
0.716062 + 0.698037i \(0.245943\pi\)
\(74\) 5.61803 0.653083
\(75\) 12.9443 1.49468
\(76\) 1.85410 0.212680
\(77\) 0 0
\(78\) 10.0902 1.14249
\(79\) −13.4164 −1.50946 −0.754732 0.656033i \(-0.772233\pi\)
−0.754732 + 0.656033i \(0.772233\pi\)
\(80\) 20.5623 2.29894
\(81\) 1.00000 0.111111
\(82\) 2.47214 0.273002
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −0.618034 −0.0674330
\(85\) 18.9443 2.05479
\(86\) 17.7082 1.90952
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) −6.85410 −0.722486
\(91\) −6.23607 −0.653718
\(92\) 5.23607 0.545898
\(93\) 0 0
\(94\) −4.85410 −0.500662
\(95\) −12.7082 −1.30383
\(96\) −3.38197 −0.345170
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.61803 0.163446
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) −16.9443 −1.68602 −0.843009 0.537899i \(-0.819218\pi\)
−0.843009 + 0.537899i \(0.819218\pi\)
\(102\) −7.23607 −0.716477
\(103\) 6.47214 0.637719 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(104\) −13.9443 −1.36735
\(105\) 4.23607 0.413398
\(106\) 14.4721 1.40566
\(107\) 9.76393 0.943915 0.471957 0.881621i \(-0.343548\pi\)
0.471957 + 0.881621i \(0.343548\pi\)
\(108\) 0.618034 0.0594703
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 3.47214 0.329561
\(112\) 4.85410 0.458670
\(113\) −6.47214 −0.608847 −0.304424 0.952537i \(-0.598464\pi\)
−0.304424 + 0.952537i \(0.598464\pi\)
\(114\) 4.85410 0.454628
\(115\) −35.8885 −3.34662
\(116\) −1.85410 −0.172149
\(117\) 6.23607 0.576525
\(118\) −2.38197 −0.219278
\(119\) 4.47214 0.409960
\(120\) 9.47214 0.864684
\(121\) 0 0
\(122\) 5.70820 0.516797
\(123\) 1.52786 0.137763
\(124\) 0 0
\(125\) −33.6525 −3.00997
\(126\) −1.61803 −0.144146
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 13.6180 1.20368
\(129\) 10.9443 0.963589
\(130\) −42.7426 −3.74878
\(131\) 2.47214 0.215992 0.107996 0.994151i \(-0.465557\pi\)
0.107996 + 0.994151i \(0.465557\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) −14.0902 −1.21721
\(135\) −4.23607 −0.364583
\(136\) 10.0000 0.857493
\(137\) 2.94427 0.251546 0.125773 0.992059i \(-0.459859\pi\)
0.125773 + 0.992059i \(0.459859\pi\)
\(138\) 13.7082 1.16692
\(139\) −0.944272 −0.0800921 −0.0400460 0.999198i \(-0.512750\pi\)
−0.0400460 + 0.999198i \(0.512750\pi\)
\(140\) 2.61803 0.221264
\(141\) −3.00000 −0.252646
\(142\) 2.47214 0.207457
\(143\) 0 0
\(144\) −4.85410 −0.404508
\(145\) 12.7082 1.05536
\(146\) 19.7984 1.63853
\(147\) 1.00000 0.0824786
\(148\) 2.14590 0.176392
\(149\) 17.9443 1.47005 0.735026 0.678039i \(-0.237170\pi\)
0.735026 + 0.678039i \(0.237170\pi\)
\(150\) 20.9443 1.71009
\(151\) 17.4164 1.41733 0.708664 0.705547i \(-0.249298\pi\)
0.708664 + 0.705547i \(0.249298\pi\)
\(152\) −6.70820 −0.544107
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) 0 0
\(156\) 3.85410 0.308575
\(157\) −13.4164 −1.07075 −0.535373 0.844616i \(-0.679829\pi\)
−0.535373 + 0.844616i \(0.679829\pi\)
\(158\) −21.7082 −1.72701
\(159\) 8.94427 0.709327
\(160\) 14.3262 1.13259
\(161\) −8.47214 −0.667698
\(162\) 1.61803 0.127125
\(163\) 7.29180 0.571138 0.285569 0.958358i \(-0.407818\pi\)
0.285569 + 0.958358i \(0.407818\pi\)
\(164\) 0.944272 0.0737352
\(165\) 0 0
\(166\) 9.70820 0.753503
\(167\) 18.9443 1.46595 0.732976 0.680255i \(-0.238131\pi\)
0.732976 + 0.680255i \(0.238131\pi\)
\(168\) 2.23607 0.172516
\(169\) 25.8885 1.99143
\(170\) 30.6525 2.35094
\(171\) 3.00000 0.229416
\(172\) 6.76393 0.515745
\(173\) 7.52786 0.572333 0.286166 0.958180i \(-0.407619\pi\)
0.286166 + 0.958180i \(0.407619\pi\)
\(174\) −4.85410 −0.367989
\(175\) −12.9443 −0.978495
\(176\) 0 0
\(177\) −1.47214 −0.110653
\(178\) −7.23607 −0.542366
\(179\) −16.4721 −1.23119 −0.615593 0.788065i \(-0.711083\pi\)
−0.615593 + 0.788065i \(0.711083\pi\)
\(180\) −2.61803 −0.195137
\(181\) −3.41641 −0.253940 −0.126970 0.991907i \(-0.540525\pi\)
−0.126970 + 0.991907i \(0.540525\pi\)
\(182\) −10.0902 −0.747933
\(183\) 3.52786 0.260787
\(184\) −18.9443 −1.39659
\(185\) −14.7082 −1.08137
\(186\) 0 0
\(187\) 0 0
\(188\) −1.85410 −0.135224
\(189\) −1.00000 −0.0727393
\(190\) −20.5623 −1.49175
\(191\) 2.94427 0.213040 0.106520 0.994311i \(-0.466029\pi\)
0.106520 + 0.994311i \(0.466029\pi\)
\(192\) 4.23607 0.305712
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 3.23607 0.232336
\(195\) −26.4164 −1.89172
\(196\) 0.618034 0.0441453
\(197\) −1.05573 −0.0752175 −0.0376088 0.999293i \(-0.511974\pi\)
−0.0376088 + 0.999293i \(0.511974\pi\)
\(198\) 0 0
\(199\) −12.4721 −0.884126 −0.442063 0.896984i \(-0.645753\pi\)
−0.442063 + 0.896984i \(0.645753\pi\)
\(200\) −28.9443 −2.04667
\(201\) −8.70820 −0.614229
\(202\) −27.4164 −1.92901
\(203\) 3.00000 0.210559
\(204\) −2.76393 −0.193514
\(205\) −6.47214 −0.452034
\(206\) 10.4721 0.729628
\(207\) 8.47214 0.588854
\(208\) −30.2705 −2.09888
\(209\) 0 0
\(210\) 6.85410 0.472978
\(211\) −1.41641 −0.0975095 −0.0487548 0.998811i \(-0.515525\pi\)
−0.0487548 + 0.998811i \(0.515525\pi\)
\(212\) 5.52786 0.379655
\(213\) 1.52786 0.104688
\(214\) 15.7984 1.07995
\(215\) −46.3607 −3.16177
\(216\) −2.23607 −0.152145
\(217\) 0 0
\(218\) 0 0
\(219\) 12.2361 0.826837
\(220\) 0 0
\(221\) −27.8885 −1.87599
\(222\) 5.61803 0.377058
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 3.38197 0.225967
\(225\) 12.9443 0.862951
\(226\) −10.4721 −0.696596
\(227\) −15.5279 −1.03062 −0.515310 0.857004i \(-0.672323\pi\)
−0.515310 + 0.857004i \(0.672323\pi\)
\(228\) 1.85410 0.122791
\(229\) −24.4721 −1.61716 −0.808582 0.588383i \(-0.799765\pi\)
−0.808582 + 0.588383i \(0.799765\pi\)
\(230\) −58.0689 −3.82895
\(231\) 0 0
\(232\) 6.70820 0.440415
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 10.0902 0.659615
\(235\) 12.7082 0.828992
\(236\) −0.909830 −0.0592249
\(237\) −13.4164 −0.871489
\(238\) 7.23607 0.469045
\(239\) 12.7082 0.822025 0.411013 0.911630i \(-0.365175\pi\)
0.411013 + 0.911630i \(0.365175\pi\)
\(240\) 20.5623 1.32729
\(241\) 12.7082 0.818607 0.409304 0.912398i \(-0.365772\pi\)
0.409304 + 0.912398i \(0.365772\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 2.18034 0.139582
\(245\) −4.23607 −0.270632
\(246\) 2.47214 0.157618
\(247\) 18.7082 1.19037
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) −54.4508 −3.44377
\(251\) −16.4164 −1.03619 −0.518097 0.855322i \(-0.673359\pi\)
−0.518097 + 0.855322i \(0.673359\pi\)
\(252\) −0.618034 −0.0389325
\(253\) 0 0
\(254\) 3.23607 0.203049
\(255\) 18.9443 1.18634
\(256\) 13.5623 0.847644
\(257\) 0.347524 0.0216780 0.0108390 0.999941i \(-0.496550\pi\)
0.0108390 + 0.999941i \(0.496550\pi\)
\(258\) 17.7082 1.10246
\(259\) −3.47214 −0.215748
\(260\) −16.3262 −1.01251
\(261\) −3.00000 −0.185695
\(262\) 4.00000 0.247121
\(263\) 0.236068 0.0145566 0.00727829 0.999974i \(-0.497683\pi\)
0.00727829 + 0.999974i \(0.497683\pi\)
\(264\) 0 0
\(265\) −37.8885 −2.32747
\(266\) −4.85410 −0.297624
\(267\) −4.47214 −0.273690
\(268\) −5.38197 −0.328756
\(269\) −19.5279 −1.19063 −0.595317 0.803491i \(-0.702974\pi\)
−0.595317 + 0.803491i \(0.702974\pi\)
\(270\) −6.85410 −0.417127
\(271\) −3.47214 −0.210917 −0.105459 0.994424i \(-0.533631\pi\)
−0.105459 + 0.994424i \(0.533631\pi\)
\(272\) 21.7082 1.31625
\(273\) −6.23607 −0.377424
\(274\) 4.76393 0.287800
\(275\) 0 0
\(276\) 5.23607 0.315174
\(277\) −12.4721 −0.749378 −0.374689 0.927151i \(-0.622251\pi\)
−0.374689 + 0.927151i \(0.622251\pi\)
\(278\) −1.52786 −0.0916352
\(279\) 0 0
\(280\) −9.47214 −0.566068
\(281\) 16.4164 0.979321 0.489660 0.871913i \(-0.337121\pi\)
0.489660 + 0.871913i \(0.337121\pi\)
\(282\) −4.85410 −0.289058
\(283\) −18.8885 −1.12281 −0.561404 0.827542i \(-0.689738\pi\)
−0.561404 + 0.827542i \(0.689738\pi\)
\(284\) 0.944272 0.0560322
\(285\) −12.7082 −0.752769
\(286\) 0 0
\(287\) −1.52786 −0.0901870
\(288\) −3.38197 −0.199284
\(289\) 3.00000 0.176471
\(290\) 20.5623 1.20746
\(291\) 2.00000 0.117242
\(292\) 7.56231 0.442550
\(293\) 9.88854 0.577695 0.288847 0.957375i \(-0.406728\pi\)
0.288847 + 0.957375i \(0.406728\pi\)
\(294\) 1.61803 0.0943657
\(295\) 6.23607 0.363078
\(296\) −7.76393 −0.451269
\(297\) 0 0
\(298\) 29.0344 1.68192
\(299\) 52.8328 3.05540
\(300\) 8.00000 0.461880
\(301\) −10.9443 −0.630817
\(302\) 28.1803 1.62160
\(303\) −16.9443 −0.973423
\(304\) −14.5623 −0.835206
\(305\) −14.9443 −0.855707
\(306\) −7.23607 −0.413658
\(307\) 16.9443 0.967061 0.483530 0.875328i \(-0.339354\pi\)
0.483530 + 0.875328i \(0.339354\pi\)
\(308\) 0 0
\(309\) 6.47214 0.368187
\(310\) 0 0
\(311\) −17.8885 −1.01437 −0.507183 0.861838i \(-0.669313\pi\)
−0.507183 + 0.861838i \(0.669313\pi\)
\(312\) −13.9443 −0.789439
\(313\) −20.4721 −1.15715 −0.578577 0.815628i \(-0.696392\pi\)
−0.578577 + 0.815628i \(0.696392\pi\)
\(314\) −21.7082 −1.22506
\(315\) 4.23607 0.238675
\(316\) −8.29180 −0.466450
\(317\) 30.4721 1.71149 0.855743 0.517401i \(-0.173101\pi\)
0.855743 + 0.517401i \(0.173101\pi\)
\(318\) 14.4721 0.811557
\(319\) 0 0
\(320\) −17.9443 −1.00312
\(321\) 9.76393 0.544970
\(322\) −13.7082 −0.763928
\(323\) −13.4164 −0.746509
\(324\) 0.618034 0.0343352
\(325\) 80.7214 4.47762
\(326\) 11.7984 0.653451
\(327\) 0 0
\(328\) −3.41641 −0.188640
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 3.70820 0.203514
\(333\) 3.47214 0.190272
\(334\) 30.6525 1.67723
\(335\) 36.8885 2.01544
\(336\) 4.85410 0.264813
\(337\) 24.8328 1.35273 0.676365 0.736567i \(-0.263554\pi\)
0.676365 + 0.736567i \(0.263554\pi\)
\(338\) 41.8885 2.27844
\(339\) −6.47214 −0.351518
\(340\) 11.7082 0.634967
\(341\) 0 0
\(342\) 4.85410 0.262480
\(343\) −1.00000 −0.0539949
\(344\) −24.4721 −1.31945
\(345\) −35.8885 −1.93217
\(346\) 12.1803 0.654819
\(347\) −3.05573 −0.164040 −0.0820200 0.996631i \(-0.526137\pi\)
−0.0820200 + 0.996631i \(0.526137\pi\)
\(348\) −1.85410 −0.0993903
\(349\) −22.7082 −1.21554 −0.607771 0.794112i \(-0.707936\pi\)
−0.607771 + 0.794112i \(0.707936\pi\)
\(350\) −20.9443 −1.11952
\(351\) 6.23607 0.332857
\(352\) 0 0
\(353\) 32.5967 1.73495 0.867475 0.497481i \(-0.165742\pi\)
0.867475 + 0.497481i \(0.165742\pi\)
\(354\) −2.38197 −0.126600
\(355\) −6.47214 −0.343505
\(356\) −2.76393 −0.146488
\(357\) 4.47214 0.236691
\(358\) −26.6525 −1.40863
\(359\) −19.0557 −1.00572 −0.502861 0.864367i \(-0.667719\pi\)
−0.502861 + 0.864367i \(0.667719\pi\)
\(360\) 9.47214 0.499225
\(361\) −10.0000 −0.526316
\(362\) −5.52786 −0.290538
\(363\) 0 0
\(364\) −3.85410 −0.202010
\(365\) −51.8328 −2.71305
\(366\) 5.70820 0.298373
\(367\) −22.8328 −1.19186 −0.595932 0.803035i \(-0.703217\pi\)
−0.595932 + 0.803035i \(0.703217\pi\)
\(368\) −41.1246 −2.14377
\(369\) 1.52786 0.0795374
\(370\) −23.7984 −1.23722
\(371\) −8.94427 −0.464363
\(372\) 0 0
\(373\) 2.58359 0.133773 0.0668867 0.997761i \(-0.478693\pi\)
0.0668867 + 0.997761i \(0.478693\pi\)
\(374\) 0 0
\(375\) −33.6525 −1.73781
\(376\) 6.70820 0.345949
\(377\) −18.7082 −0.963522
\(378\) −1.61803 −0.0832227
\(379\) −4.23607 −0.217592 −0.108796 0.994064i \(-0.534700\pi\)
−0.108796 + 0.994064i \(0.534700\pi\)
\(380\) −7.85410 −0.402907
\(381\) 2.00000 0.102463
\(382\) 4.76393 0.243744
\(383\) 13.8885 0.709671 0.354836 0.934929i \(-0.384537\pi\)
0.354836 + 0.934929i \(0.384537\pi\)
\(384\) 13.6180 0.694942
\(385\) 0 0
\(386\) 19.4164 0.988269
\(387\) 10.9443 0.556329
\(388\) 1.23607 0.0627518
\(389\) 28.3607 1.43794 0.718972 0.695039i \(-0.244613\pi\)
0.718972 + 0.695039i \(0.244613\pi\)
\(390\) −42.7426 −2.16436
\(391\) −37.8885 −1.91611
\(392\) −2.23607 −0.112938
\(393\) 2.47214 0.124703
\(394\) −1.70820 −0.0860581
\(395\) 56.8328 2.85957
\(396\) 0 0
\(397\) 12.4721 0.625959 0.312979 0.949760i \(-0.398673\pi\)
0.312979 + 0.949760i \(0.398673\pi\)
\(398\) −20.1803 −1.01155
\(399\) −3.00000 −0.150188
\(400\) −62.8328 −3.14164
\(401\) −10.3607 −0.517388 −0.258694 0.965959i \(-0.583292\pi\)
−0.258694 + 0.965959i \(0.583292\pi\)
\(402\) −14.0902 −0.702754
\(403\) 0 0
\(404\) −10.4721 −0.521008
\(405\) −4.23607 −0.210492
\(406\) 4.85410 0.240905
\(407\) 0 0
\(408\) 10.0000 0.495074
\(409\) 2.58359 0.127750 0.0638752 0.997958i \(-0.479654\pi\)
0.0638752 + 0.997958i \(0.479654\pi\)
\(410\) −10.4721 −0.517182
\(411\) 2.94427 0.145230
\(412\) 4.00000 0.197066
\(413\) 1.47214 0.0724391
\(414\) 13.7082 0.673721
\(415\) −25.4164 −1.24764
\(416\) −21.0902 −1.03403
\(417\) −0.944272 −0.0462412
\(418\) 0 0
\(419\) 24.5279 1.19826 0.599132 0.800650i \(-0.295512\pi\)
0.599132 + 0.800650i \(0.295512\pi\)
\(420\) 2.61803 0.127747
\(421\) −16.4164 −0.800087 −0.400043 0.916496i \(-0.631005\pi\)
−0.400043 + 0.916496i \(0.631005\pi\)
\(422\) −2.29180 −0.111563
\(423\) −3.00000 −0.145865
\(424\) −20.0000 −0.971286
\(425\) −57.8885 −2.80801
\(426\) 2.47214 0.119775
\(427\) −3.52786 −0.170725
\(428\) 6.03444 0.291686
\(429\) 0 0
\(430\) −75.0132 −3.61746
\(431\) −18.5967 −0.895774 −0.447887 0.894090i \(-0.647823\pi\)
−0.447887 + 0.894090i \(0.647823\pi\)
\(432\) −4.85410 −0.233543
\(433\) 9.41641 0.452524 0.226262 0.974067i \(-0.427349\pi\)
0.226262 + 0.974067i \(0.427349\pi\)
\(434\) 0 0
\(435\) 12.7082 0.609312
\(436\) 0 0
\(437\) 25.4164 1.21583
\(438\) 19.7984 0.946003
\(439\) −9.36068 −0.446761 −0.223380 0.974731i \(-0.571709\pi\)
−0.223380 + 0.974731i \(0.571709\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −45.1246 −2.14636
\(443\) −7.88854 −0.374796 −0.187398 0.982284i \(-0.560005\pi\)
−0.187398 + 0.982284i \(0.560005\pi\)
\(444\) 2.14590 0.101840
\(445\) 18.9443 0.898045
\(446\) 9.70820 0.459697
\(447\) 17.9443 0.848735
\(448\) −4.23607 −0.200135
\(449\) 28.4721 1.34368 0.671842 0.740695i \(-0.265504\pi\)
0.671842 + 0.740695i \(0.265504\pi\)
\(450\) 20.9443 0.987322
\(451\) 0 0
\(452\) −4.00000 −0.188144
\(453\) 17.4164 0.818294
\(454\) −25.1246 −1.17916
\(455\) 26.4164 1.23842
\(456\) −6.70820 −0.314140
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −39.5967 −1.85023
\(459\) −4.47214 −0.208741
\(460\) −22.1803 −1.03416
\(461\) 26.9443 1.25492 0.627460 0.778649i \(-0.284095\pi\)
0.627460 + 0.778649i \(0.284095\pi\)
\(462\) 0 0
\(463\) −32.1246 −1.49296 −0.746479 0.665409i \(-0.768257\pi\)
−0.746479 + 0.665409i \(0.768257\pi\)
\(464\) 14.5623 0.676038
\(465\) 0 0
\(466\) 32.1803 1.49073
\(467\) 15.4721 0.715965 0.357983 0.933728i \(-0.383465\pi\)
0.357983 + 0.933728i \(0.383465\pi\)
\(468\) 3.85410 0.178156
\(469\) 8.70820 0.402107
\(470\) 20.5623 0.948468
\(471\) −13.4164 −0.618195
\(472\) 3.29180 0.151517
\(473\) 0 0
\(474\) −21.7082 −0.997091
\(475\) 38.8328 1.78177
\(476\) 2.76393 0.126685
\(477\) 8.94427 0.409530
\(478\) 20.5623 0.940498
\(479\) −28.4721 −1.30093 −0.650463 0.759538i \(-0.725425\pi\)
−0.650463 + 0.759538i \(0.725425\pi\)
\(480\) 14.3262 0.653900
\(481\) 21.6525 0.987268
\(482\) 20.5623 0.936587
\(483\) −8.47214 −0.385496
\(484\) 0 0
\(485\) −8.47214 −0.384700
\(486\) 1.61803 0.0733955
\(487\) 13.8885 0.629350 0.314675 0.949199i \(-0.398104\pi\)
0.314675 + 0.949199i \(0.398104\pi\)
\(488\) −7.88854 −0.357098
\(489\) 7.29180 0.329746
\(490\) −6.85410 −0.309637
\(491\) 20.5967 0.929518 0.464759 0.885437i \(-0.346141\pi\)
0.464759 + 0.885437i \(0.346141\pi\)
\(492\) 0.944272 0.0425711
\(493\) 13.4164 0.604245
\(494\) 30.2705 1.36193
\(495\) 0 0
\(496\) 0 0
\(497\) −1.52786 −0.0685341
\(498\) 9.70820 0.435035
\(499\) −23.2918 −1.04268 −0.521342 0.853348i \(-0.674568\pi\)
−0.521342 + 0.853348i \(0.674568\pi\)
\(500\) −20.7984 −0.930132
\(501\) 18.9443 0.846368
\(502\) −26.5623 −1.18553
\(503\) −19.4164 −0.865735 −0.432867 0.901458i \(-0.642498\pi\)
−0.432867 + 0.901458i \(0.642498\pi\)
\(504\) 2.23607 0.0996024
\(505\) 71.7771 3.19404
\(506\) 0 0
\(507\) 25.8885 1.14975
\(508\) 1.23607 0.0548416
\(509\) 26.3607 1.16842 0.584208 0.811604i \(-0.301405\pi\)
0.584208 + 0.811604i \(0.301405\pi\)
\(510\) 30.6525 1.35731
\(511\) −12.2361 −0.541292
\(512\) −5.29180 −0.233867
\(513\) 3.00000 0.132453
\(514\) 0.562306 0.0248022
\(515\) −27.4164 −1.20811
\(516\) 6.76393 0.297766
\(517\) 0 0
\(518\) −5.61803 −0.246842
\(519\) 7.52786 0.330437
\(520\) 59.0689 2.59034
\(521\) 14.2361 0.623693 0.311847 0.950132i \(-0.399052\pi\)
0.311847 + 0.950132i \(0.399052\pi\)
\(522\) −4.85410 −0.212458
\(523\) −3.00000 −0.131181 −0.0655904 0.997847i \(-0.520893\pi\)
−0.0655904 + 0.997847i \(0.520893\pi\)
\(524\) 1.52786 0.0667451
\(525\) −12.9443 −0.564934
\(526\) 0.381966 0.0166545
\(527\) 0 0
\(528\) 0 0
\(529\) 48.7771 2.12074
\(530\) −61.3050 −2.66292
\(531\) −1.47214 −0.0638853
\(532\) −1.85410 −0.0803855
\(533\) 9.52786 0.412698
\(534\) −7.23607 −0.313135
\(535\) −41.3607 −1.78818
\(536\) 19.4721 0.841068
\(537\) −16.4721 −0.710825
\(538\) −31.5967 −1.36223
\(539\) 0 0
\(540\) −2.61803 −0.112662
\(541\) 15.4164 0.662803 0.331402 0.943490i \(-0.392479\pi\)
0.331402 + 0.943490i \(0.392479\pi\)
\(542\) −5.61803 −0.241315
\(543\) −3.41641 −0.146612
\(544\) 15.1246 0.648462
\(545\) 0 0
\(546\) −10.0902 −0.431819
\(547\) 36.8328 1.57486 0.787429 0.616406i \(-0.211412\pi\)
0.787429 + 0.616406i \(0.211412\pi\)
\(548\) 1.81966 0.0777320
\(549\) 3.52786 0.150566
\(550\) 0 0
\(551\) −9.00000 −0.383413
\(552\) −18.9443 −0.806322
\(553\) 13.4164 0.570524
\(554\) −20.1803 −0.857380
\(555\) −14.7082 −0.624328
\(556\) −0.583592 −0.0247498
\(557\) −39.0000 −1.65248 −0.826242 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) 68.2492 2.88663
\(560\) −20.5623 −0.868916
\(561\) 0 0
\(562\) 26.5623 1.12046
\(563\) −2.47214 −0.104188 −0.0520941 0.998642i \(-0.516590\pi\)
−0.0520941 + 0.998642i \(0.516590\pi\)
\(564\) −1.85410 −0.0780718
\(565\) 27.4164 1.15342
\(566\) −30.5623 −1.28463
\(567\) −1.00000 −0.0419961
\(568\) −3.41641 −0.143349
\(569\) −8.11146 −0.340050 −0.170025 0.985440i \(-0.554385\pi\)
−0.170025 + 0.985440i \(0.554385\pi\)
\(570\) −20.5623 −0.861260
\(571\) 22.9443 0.960188 0.480094 0.877217i \(-0.340603\pi\)
0.480094 + 0.877217i \(0.340603\pi\)
\(572\) 0 0
\(573\) 2.94427 0.122999
\(574\) −2.47214 −0.103185
\(575\) 109.666 4.57337
\(576\) 4.23607 0.176503
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) 4.85410 0.201904
\(579\) 12.0000 0.498703
\(580\) 7.85410 0.326124
\(581\) −6.00000 −0.248922
\(582\) 3.23607 0.134139
\(583\) 0 0
\(584\) −27.3607 −1.13219
\(585\) −26.4164 −1.09218
\(586\) 16.0000 0.660954
\(587\) −0.639320 −0.0263876 −0.0131938 0.999913i \(-0.504200\pi\)
−0.0131938 + 0.999913i \(0.504200\pi\)
\(588\) 0.618034 0.0254873
\(589\) 0 0
\(590\) 10.0902 0.415406
\(591\) −1.05573 −0.0434269
\(592\) −16.8541 −0.692699
\(593\) −29.7771 −1.22280 −0.611399 0.791322i \(-0.709393\pi\)
−0.611399 + 0.791322i \(0.709393\pi\)
\(594\) 0 0
\(595\) −18.9443 −0.776639
\(596\) 11.0902 0.454271
\(597\) −12.4721 −0.510451
\(598\) 85.4853 3.49575
\(599\) 1.41641 0.0578729 0.0289364 0.999581i \(-0.490788\pi\)
0.0289364 + 0.999581i \(0.490788\pi\)
\(600\) −28.9443 −1.18164
\(601\) 9.18034 0.374474 0.187237 0.982315i \(-0.440047\pi\)
0.187237 + 0.982315i \(0.440047\pi\)
\(602\) −17.7082 −0.721733
\(603\) −8.70820 −0.354625
\(604\) 10.7639 0.437978
\(605\) 0 0
\(606\) −27.4164 −1.11372
\(607\) −43.2492 −1.75543 −0.877716 0.479181i \(-0.840934\pi\)
−0.877716 + 0.479181i \(0.840934\pi\)
\(608\) −10.1459 −0.411470
\(609\) 3.00000 0.121566
\(610\) −24.1803 −0.979033
\(611\) −18.7082 −0.756853
\(612\) −2.76393 −0.111725
\(613\) −17.5279 −0.707944 −0.353972 0.935256i \(-0.615169\pi\)
−0.353972 + 0.935256i \(0.615169\pi\)
\(614\) 27.4164 1.10644
\(615\) −6.47214 −0.260982
\(616\) 0 0
\(617\) −29.8885 −1.20327 −0.601634 0.798772i \(-0.705483\pi\)
−0.601634 + 0.798772i \(0.705483\pi\)
\(618\) 10.4721 0.421251
\(619\) 46.2492 1.85891 0.929457 0.368931i \(-0.120276\pi\)
0.929457 + 0.368931i \(0.120276\pi\)
\(620\) 0 0
\(621\) 8.47214 0.339975
\(622\) −28.9443 −1.16056
\(623\) 4.47214 0.179172
\(624\) −30.2705 −1.21179
\(625\) 77.8328 3.11331
\(626\) −33.1246 −1.32393
\(627\) 0 0
\(628\) −8.29180 −0.330879
\(629\) −15.5279 −0.619136
\(630\) 6.85410 0.273074
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 30.0000 1.19334
\(633\) −1.41641 −0.0562972
\(634\) 49.3050 1.95815
\(635\) −8.47214 −0.336206
\(636\) 5.52786 0.219194
\(637\) 6.23607 0.247082
\(638\) 0 0
\(639\) 1.52786 0.0604414
\(640\) −57.6869 −2.28028
\(641\) −38.8328 −1.53380 −0.766902 0.641764i \(-0.778203\pi\)
−0.766902 + 0.641764i \(0.778203\pi\)
\(642\) 15.7984 0.623512
\(643\) −1.41641 −0.0558577 −0.0279288 0.999610i \(-0.508891\pi\)
−0.0279288 + 0.999610i \(0.508891\pi\)
\(644\) −5.23607 −0.206330
\(645\) −46.3607 −1.82545
\(646\) −21.7082 −0.854098
\(647\) 4.88854 0.192188 0.0960942 0.995372i \(-0.469365\pi\)
0.0960942 + 0.995372i \(0.469365\pi\)
\(648\) −2.23607 −0.0878410
\(649\) 0 0
\(650\) 130.610 5.12294
\(651\) 0 0
\(652\) 4.50658 0.176491
\(653\) −44.9443 −1.75881 −0.879403 0.476079i \(-0.842058\pi\)
−0.879403 + 0.476079i \(0.842058\pi\)
\(654\) 0 0
\(655\) −10.4721 −0.409180
\(656\) −7.41641 −0.289562
\(657\) 12.2361 0.477374
\(658\) 4.85410 0.189233
\(659\) 25.7639 1.00362 0.501810 0.864978i \(-0.332668\pi\)
0.501810 + 0.864978i \(0.332668\pi\)
\(660\) 0 0
\(661\) 30.8328 1.19926 0.599629 0.800278i \(-0.295315\pi\)
0.599629 + 0.800278i \(0.295315\pi\)
\(662\) 38.8328 1.50928
\(663\) −27.8885 −1.08310
\(664\) −13.4164 −0.520658
\(665\) 12.7082 0.492803
\(666\) 5.61803 0.217694
\(667\) −25.4164 −0.984127
\(668\) 11.7082 0.453004
\(669\) 6.00000 0.231973
\(670\) 59.6869 2.30591
\(671\) 0 0
\(672\) 3.38197 0.130462
\(673\) 33.3050 1.28381 0.641906 0.766784i \(-0.278144\pi\)
0.641906 + 0.766784i \(0.278144\pi\)
\(674\) 40.1803 1.54769
\(675\) 12.9443 0.498225
\(676\) 16.0000 0.615385
\(677\) 28.3607 1.08999 0.544995 0.838439i \(-0.316532\pi\)
0.544995 + 0.838439i \(0.316532\pi\)
\(678\) −10.4721 −0.402180
\(679\) −2.00000 −0.0767530
\(680\) −42.3607 −1.62446
\(681\) −15.5279 −0.595029
\(682\) 0 0
\(683\) −9.52786 −0.364574 −0.182287 0.983245i \(-0.558350\pi\)
−0.182287 + 0.983245i \(0.558350\pi\)
\(684\) 1.85410 0.0708934
\(685\) −12.4721 −0.476536
\(686\) −1.61803 −0.0617768
\(687\) −24.4721 −0.933670
\(688\) −53.1246 −2.02536
\(689\) 55.7771 2.12494
\(690\) −58.0689 −2.21064
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 4.65248 0.176861
\(693\) 0 0
\(694\) −4.94427 −0.187682
\(695\) 4.00000 0.151729
\(696\) 6.70820 0.254274
\(697\) −6.83282 −0.258811
\(698\) −36.7426 −1.39073
\(699\) 19.8885 0.752254
\(700\) −8.00000 −0.302372
\(701\) 6.94427 0.262282 0.131141 0.991364i \(-0.458136\pi\)
0.131141 + 0.991364i \(0.458136\pi\)
\(702\) 10.0902 0.380829
\(703\) 10.4164 0.392862
\(704\) 0 0
\(705\) 12.7082 0.478619
\(706\) 52.7426 1.98500
\(707\) 16.9443 0.637255
\(708\) −0.909830 −0.0341935
\(709\) 11.5836 0.435031 0.217515 0.976057i \(-0.430205\pi\)
0.217515 + 0.976057i \(0.430205\pi\)
\(710\) −10.4721 −0.393012
\(711\) −13.4164 −0.503155
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 7.23607 0.270803
\(715\) 0 0
\(716\) −10.1803 −0.380457
\(717\) 12.7082 0.474597
\(718\) −30.8328 −1.15067
\(719\) −24.7771 −0.924029 −0.462015 0.886872i \(-0.652873\pi\)
−0.462015 + 0.886872i \(0.652873\pi\)
\(720\) 20.5623 0.766312
\(721\) −6.47214 −0.241035
\(722\) −16.1803 −0.602170
\(723\) 12.7082 0.472623
\(724\) −2.11146 −0.0784717
\(725\) −38.8328 −1.44221
\(726\) 0 0
\(727\) 44.8328 1.66276 0.831379 0.555706i \(-0.187552\pi\)
0.831379 + 0.555706i \(0.187552\pi\)
\(728\) 13.9443 0.516809
\(729\) 1.00000 0.0370370
\(730\) −83.8673 −3.10407
\(731\) −48.9443 −1.81027
\(732\) 2.18034 0.0805877
\(733\) 24.4721 0.903899 0.451949 0.892044i \(-0.350729\pi\)
0.451949 + 0.892044i \(0.350729\pi\)
\(734\) −36.9443 −1.36364
\(735\) −4.23607 −0.156250
\(736\) −28.6525 −1.05614
\(737\) 0 0
\(738\) 2.47214 0.0910006
\(739\) 51.3050 1.88728 0.943642 0.330969i \(-0.107376\pi\)
0.943642 + 0.330969i \(0.107376\pi\)
\(740\) −9.09017 −0.334161
\(741\) 18.7082 0.687263
\(742\) −14.4721 −0.531289
\(743\) −15.7639 −0.578323 −0.289161 0.957280i \(-0.593376\pi\)
−0.289161 + 0.957280i \(0.593376\pi\)
\(744\) 0 0
\(745\) −76.0132 −2.78491
\(746\) 4.18034 0.153053
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −9.76393 −0.356766
\(750\) −54.4508 −1.98826
\(751\) −18.2361 −0.665444 −0.332722 0.943025i \(-0.607967\pi\)
−0.332722 + 0.943025i \(0.607967\pi\)
\(752\) 14.5623 0.531033
\(753\) −16.4164 −0.598247
\(754\) −30.2705 −1.10239
\(755\) −73.7771 −2.68502
\(756\) −0.618034 −0.0224777
\(757\) 24.4164 0.887429 0.443715 0.896168i \(-0.353660\pi\)
0.443715 + 0.896168i \(0.353660\pi\)
\(758\) −6.85410 −0.248952
\(759\) 0 0
\(760\) 28.4164 1.03077
\(761\) 6.36068 0.230574 0.115287 0.993332i \(-0.463221\pi\)
0.115287 + 0.993332i \(0.463221\pi\)
\(762\) 3.23607 0.117230
\(763\) 0 0
\(764\) 1.81966 0.0658330
\(765\) 18.9443 0.684932
\(766\) 22.4721 0.811951
\(767\) −9.18034 −0.331483
\(768\) 13.5623 0.489388
\(769\) −14.1246 −0.509347 −0.254673 0.967027i \(-0.581968\pi\)
−0.254673 + 0.967027i \(0.581968\pi\)
\(770\) 0 0
\(771\) 0.347524 0.0125158
\(772\) 7.41641 0.266922
\(773\) −11.2918 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(774\) 17.7082 0.636508
\(775\) 0 0
\(776\) −4.47214 −0.160540
\(777\) −3.47214 −0.124562
\(778\) 45.8885 1.64518
\(779\) 4.58359 0.164224
\(780\) −16.3262 −0.584573
\(781\) 0 0
\(782\) −61.3050 −2.19226
\(783\) −3.00000 −0.107211
\(784\) −4.85410 −0.173361
\(785\) 56.8328 2.02845
\(786\) 4.00000 0.142675
\(787\) −27.0000 −0.962446 −0.481223 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) −0.652476 −0.0232435
\(789\) 0.236068 0.00840424
\(790\) 91.9574 3.27170
\(791\) 6.47214 0.230123
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) 20.1803 0.716173
\(795\) −37.8885 −1.34377
\(796\) −7.70820 −0.273210
\(797\) −14.8197 −0.524939 −0.262470 0.964940i \(-0.584537\pi\)
−0.262470 + 0.964940i \(0.584537\pi\)
\(798\) −4.85410 −0.171833
\(799\) 13.4164 0.474638
\(800\) −43.7771 −1.54775
\(801\) −4.47214 −0.158015
\(802\) −16.7639 −0.591955
\(803\) 0 0
\(804\) −5.38197 −0.189807
\(805\) 35.8885 1.26490
\(806\) 0 0
\(807\) −19.5279 −0.687413
\(808\) 37.8885 1.33291
\(809\) −31.3607 −1.10258 −0.551291 0.834313i \(-0.685865\pi\)
−0.551291 + 0.834313i \(0.685865\pi\)
\(810\) −6.85410 −0.240829
\(811\) 55.8328 1.96056 0.980278 0.197625i \(-0.0633229\pi\)
0.980278 + 0.197625i \(0.0633229\pi\)
\(812\) 1.85410 0.0650662
\(813\) −3.47214 −0.121773
\(814\) 0 0
\(815\) −30.8885 −1.08198
\(816\) 21.7082 0.759939
\(817\) 32.8328 1.14867
\(818\) 4.18034 0.146162
\(819\) −6.23607 −0.217906
\(820\) −4.00000 −0.139686
\(821\) −17.9443 −0.626259 −0.313130 0.949710i \(-0.601377\pi\)
−0.313130 + 0.949710i \(0.601377\pi\)
\(822\) 4.76393 0.166161
\(823\) −15.1803 −0.529153 −0.264577 0.964365i \(-0.585232\pi\)
−0.264577 + 0.964365i \(0.585232\pi\)
\(824\) −14.4721 −0.504161
\(825\) 0 0
\(826\) 2.38197 0.0828792
\(827\) −20.1246 −0.699801 −0.349901 0.936787i \(-0.613785\pi\)
−0.349901 + 0.936787i \(0.613785\pi\)
\(828\) 5.23607 0.181966
\(829\) −38.8328 −1.34872 −0.674360 0.738403i \(-0.735580\pi\)
−0.674360 + 0.738403i \(0.735580\pi\)
\(830\) −41.1246 −1.42746
\(831\) −12.4721 −0.432654
\(832\) 26.4164 0.915824
\(833\) −4.47214 −0.154950
\(834\) −1.52786 −0.0529056
\(835\) −80.2492 −2.77714
\(836\) 0 0
\(837\) 0 0
\(838\) 39.6869 1.37096
\(839\) −6.05573 −0.209067 −0.104533 0.994521i \(-0.533335\pi\)
−0.104533 + 0.994521i \(0.533335\pi\)
\(840\) −9.47214 −0.326820
\(841\) −20.0000 −0.689655
\(842\) −26.5623 −0.915398
\(843\) 16.4164 0.565411
\(844\) −0.875388 −0.0301321
\(845\) −109.666 −3.77261
\(846\) −4.85410 −0.166887
\(847\) 0 0
\(848\) −43.4164 −1.49093
\(849\) −18.8885 −0.648253
\(850\) −93.6656 −3.21270
\(851\) 29.4164 1.00838
\(852\) 0.944272 0.0323502
\(853\) 22.5836 0.773247 0.386624 0.922238i \(-0.373641\pi\)
0.386624 + 0.922238i \(0.373641\pi\)
\(854\) −5.70820 −0.195331
\(855\) −12.7082 −0.434611
\(856\) −21.8328 −0.746230
\(857\) −28.4721 −0.972590 −0.486295 0.873795i \(-0.661652\pi\)
−0.486295 + 0.873795i \(0.661652\pi\)
\(858\) 0 0
\(859\) 37.7771 1.28894 0.644469 0.764631i \(-0.277079\pi\)
0.644469 + 0.764631i \(0.277079\pi\)
\(860\) −28.6525 −0.977041
\(861\) −1.52786 −0.0520695
\(862\) −30.0902 −1.02488
\(863\) 7.63932 0.260045 0.130023 0.991511i \(-0.458495\pi\)
0.130023 + 0.991511i \(0.458495\pi\)
\(864\) −3.38197 −0.115057
\(865\) −31.8885 −1.08424
\(866\) 15.2361 0.517743
\(867\) 3.00000 0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 20.5623 0.697127
\(871\) −54.3050 −1.84005
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 41.1246 1.39106
\(875\) 33.6525 1.13766
\(876\) 7.56231 0.255507
\(877\) −2.83282 −0.0956574 −0.0478287 0.998856i \(-0.515230\pi\)
−0.0478287 + 0.998856i \(0.515230\pi\)
\(878\) −15.1459 −0.511149
\(879\) 9.88854 0.333532
\(880\) 0 0
\(881\) −26.4853 −0.892312 −0.446156 0.894955i \(-0.647207\pi\)
−0.446156 + 0.894955i \(0.647207\pi\)
\(882\) 1.61803 0.0544820
\(883\) 40.7082 1.36994 0.684970 0.728571i \(-0.259815\pi\)
0.684970 + 0.728571i \(0.259815\pi\)
\(884\) −17.2361 −0.579712
\(885\) 6.23607 0.209623
\(886\) −12.7639 −0.428813
\(887\) 7.41641 0.249019 0.124509 0.992218i \(-0.460264\pi\)
0.124509 + 0.992218i \(0.460264\pi\)
\(888\) −7.76393 −0.260540
\(889\) −2.00000 −0.0670778
\(890\) 30.6525 1.02747
\(891\) 0 0
\(892\) 3.70820 0.124160
\(893\) −9.00000 −0.301174
\(894\) 29.0344 0.971057
\(895\) 69.7771 2.33239
\(896\) −13.6180 −0.454947
\(897\) 52.8328 1.76404
\(898\) 46.0689 1.53734
\(899\) 0 0
\(900\) 8.00000 0.266667
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) −10.9443 −0.364203
\(904\) 14.4721 0.481336
\(905\) 14.4721 0.481070
\(906\) 28.1803 0.936229
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −9.59675 −0.318479
\(909\) −16.9443 −0.562006
\(910\) 42.7426 1.41690
\(911\) −21.3050 −0.705865 −0.352932 0.935649i \(-0.614815\pi\)
−0.352932 + 0.935649i \(0.614815\pi\)
\(912\) −14.5623 −0.482206
\(913\) 0 0
\(914\) 42.0689 1.39151
\(915\) −14.9443 −0.494042
\(916\) −15.1246 −0.499731
\(917\) −2.47214 −0.0816371
\(918\) −7.23607 −0.238826
\(919\) −23.0557 −0.760538 −0.380269 0.924876i \(-0.624169\pi\)
−0.380269 + 0.924876i \(0.624169\pi\)
\(920\) 80.2492 2.64574
\(921\) 16.9443 0.558333
\(922\) 43.5967 1.43578
\(923\) 9.52786 0.313613
\(924\) 0 0
\(925\) 44.9443 1.47776
\(926\) −51.9787 −1.70813
\(927\) 6.47214 0.212573
\(928\) 10.1459 0.333055
\(929\) −29.0689 −0.953719 −0.476860 0.878979i \(-0.658225\pi\)
−0.476860 + 0.878979i \(0.658225\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 12.2918 0.402631
\(933\) −17.8885 −0.585645
\(934\) 25.0344 0.819152
\(935\) 0 0
\(936\) −13.9443 −0.455783
\(937\) −29.4164 −0.960992 −0.480496 0.876997i \(-0.659543\pi\)
−0.480496 + 0.876997i \(0.659543\pi\)
\(938\) 14.0902 0.460060
\(939\) −20.4721 −0.668083
\(940\) 7.85410 0.256173
\(941\) −22.3607 −0.728937 −0.364469 0.931216i \(-0.618749\pi\)
−0.364469 + 0.931216i \(0.618749\pi\)
\(942\) −21.7082 −0.707292
\(943\) 12.9443 0.421523
\(944\) 7.14590 0.232579
\(945\) 4.23607 0.137799
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −8.29180 −0.269305
\(949\) 76.3050 2.47696
\(950\) 62.8328 2.03857
\(951\) 30.4721 0.988127
\(952\) −10.0000 −0.324102
\(953\) −11.3607 −0.368009 −0.184004 0.982925i \(-0.558906\pi\)
−0.184004 + 0.982925i \(0.558906\pi\)
\(954\) 14.4721 0.468553
\(955\) −12.4721 −0.403589
\(956\) 7.85410 0.254020
\(957\) 0 0
\(958\) −46.0689 −1.48842
\(959\) −2.94427 −0.0950755
\(960\) −17.9443 −0.579149
\(961\) −31.0000 −1.00000
\(962\) 35.0344 1.12956
\(963\) 9.76393 0.314638
\(964\) 7.85410 0.252964
\(965\) −50.8328 −1.63637
\(966\) −13.7082 −0.441054
\(967\) −29.5279 −0.949552 −0.474776 0.880107i \(-0.657471\pi\)
−0.474776 + 0.880107i \(0.657471\pi\)
\(968\) 0 0
\(969\) −13.4164 −0.430997
\(970\) −13.7082 −0.440144
\(971\) −23.4721 −0.753257 −0.376628 0.926364i \(-0.622917\pi\)
−0.376628 + 0.926364i \(0.622917\pi\)
\(972\) 0.618034 0.0198234
\(973\) 0.944272 0.0302720
\(974\) 22.4721 0.720054
\(975\) 80.7214 2.58515
\(976\) −17.1246 −0.548145
\(977\) 57.7771 1.84845 0.924226 0.381845i \(-0.124711\pi\)
0.924226 + 0.381845i \(0.124711\pi\)
\(978\) 11.7984 0.377270
\(979\) 0 0
\(980\) −2.61803 −0.0836300
\(981\) 0 0
\(982\) 33.3262 1.06348
\(983\) 41.8885 1.33604 0.668019 0.744145i \(-0.267143\pi\)
0.668019 + 0.744145i \(0.267143\pi\)
\(984\) −3.41641 −0.108911
\(985\) 4.47214 0.142494
\(986\) 21.7082 0.691330
\(987\) 3.00000 0.0954911
\(988\) 11.5623 0.367846
\(989\) 92.7214 2.94837
\(990\) 0 0
\(991\) −54.2361 −1.72287 −0.861433 0.507872i \(-0.830432\pi\)
−0.861433 + 0.507872i \(0.830432\pi\)
\(992\) 0 0
\(993\) 24.0000 0.761617
\(994\) −2.47214 −0.0784114
\(995\) 52.8328 1.67491
\(996\) 3.70820 0.117499
\(997\) −10.5836 −0.335186 −0.167593 0.985856i \(-0.553599\pi\)
−0.167593 + 0.985856i \(0.553599\pi\)
\(998\) −37.6869 −1.19296
\(999\) 3.47214 0.109854
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 2541.2.a.ba.1.2 yes 2
3.2 odd 2 7623.2.a.bb.1.1 2
11.10 odd 2 2541.2.a.r.1.1 2
33.32 even 2 7623.2.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.r.1.1 2 11.10 odd 2
2541.2.a.ba.1.2 yes 2 1.1 even 1 trivial
7623.2.a.bb.1.1 2 3.2 odd 2
7623.2.a.bq.1.2 2 33.32 even 2