Properties

Label 2541.2.a.ba.1.1
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.236068 q^{5} -0.618034 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} +0.236068 q^{5} -0.618034 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -0.145898 q^{10} -1.61803 q^{12} +1.76393 q^{13} +0.618034 q^{14} +0.236068 q^{15} +1.85410 q^{16} +4.47214 q^{17} -0.618034 q^{18} +3.00000 q^{19} -0.381966 q^{20} -1.00000 q^{21} -0.472136 q^{23} +2.23607 q^{24} -4.94427 q^{25} -1.09017 q^{26} +1.00000 q^{27} +1.61803 q^{28} -3.00000 q^{29} -0.145898 q^{30} -5.61803 q^{32} -2.76393 q^{34} -0.236068 q^{35} -1.61803 q^{36} -5.47214 q^{37} -1.85410 q^{38} +1.76393 q^{39} +0.527864 q^{40} +10.4721 q^{41} +0.618034 q^{42} -6.94427 q^{43} +0.236068 q^{45} +0.291796 q^{46} -3.00000 q^{47} +1.85410 q^{48} +1.00000 q^{49} +3.05573 q^{50} +4.47214 q^{51} -2.85410 q^{52} -8.94427 q^{53} -0.618034 q^{54} -2.23607 q^{56} +3.00000 q^{57} +1.85410 q^{58} +7.47214 q^{59} -0.381966 q^{60} +12.4721 q^{61} -1.00000 q^{63} -0.236068 q^{64} +0.416408 q^{65} +4.70820 q^{67} -7.23607 q^{68} -0.472136 q^{69} +0.145898 q^{70} +10.4721 q^{71} +2.23607 q^{72} +7.76393 q^{73} +3.38197 q^{74} -4.94427 q^{75} -4.85410 q^{76} -1.09017 q^{78} +13.4164 q^{79} +0.437694 q^{80} +1.00000 q^{81} -6.47214 q^{82} +6.00000 q^{83} +1.61803 q^{84} +1.05573 q^{85} +4.29180 q^{86} -3.00000 q^{87} +4.47214 q^{89} -0.145898 q^{90} -1.76393 q^{91} +0.763932 q^{92} +1.85410 q^{94} +0.708204 q^{95} -5.61803 q^{96} +2.00000 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} - 4 q^{5} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) 0.236068 0.105573 0.0527864 0.998606i \(-0.483190\pi\)
0.0527864 + 0.998606i \(0.483190\pi\)
\(6\) −0.618034 −0.252311
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) −0.145898 −0.0461370
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) 1.76393 0.489227 0.244613 0.969621i \(-0.421339\pi\)
0.244613 + 0.969621i \(0.421339\pi\)
\(14\) 0.618034 0.165177
\(15\) 0.236068 0.0609525
\(16\) 1.85410 0.463525
\(17\) 4.47214 1.08465 0.542326 0.840168i \(-0.317544\pi\)
0.542326 + 0.840168i \(0.317544\pi\)
\(18\) −0.618034 −0.145672
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −0.381966 −0.0854102
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −0.472136 −0.0984472 −0.0492236 0.998788i \(-0.515675\pi\)
−0.0492236 + 0.998788i \(0.515675\pi\)
\(24\) 2.23607 0.456435
\(25\) −4.94427 −0.988854
\(26\) −1.09017 −0.213800
\(27\) 1.00000 0.192450
\(28\) 1.61803 0.305780
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −0.145898 −0.0266372
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −2.76393 −0.474010
\(35\) −0.236068 −0.0399028
\(36\) −1.61803 −0.269672
\(37\) −5.47214 −0.899614 −0.449807 0.893126i \(-0.648507\pi\)
−0.449807 + 0.893126i \(0.648507\pi\)
\(38\) −1.85410 −0.300775
\(39\) 1.76393 0.282455
\(40\) 0.527864 0.0834626
\(41\) 10.4721 1.63547 0.817736 0.575593i \(-0.195229\pi\)
0.817736 + 0.575593i \(0.195229\pi\)
\(42\) 0.618034 0.0953647
\(43\) −6.94427 −1.05899 −0.529496 0.848313i \(-0.677619\pi\)
−0.529496 + 0.848313i \(0.677619\pi\)
\(44\) 0 0
\(45\) 0.236068 0.0351909
\(46\) 0.291796 0.0430230
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.85410 0.267617
\(49\) 1.00000 0.142857
\(50\) 3.05573 0.432145
\(51\) 4.47214 0.626224
\(52\) −2.85410 −0.395793
\(53\) −8.94427 −1.22859 −0.614295 0.789076i \(-0.710560\pi\)
−0.614295 + 0.789076i \(0.710560\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 3.00000 0.397360
\(58\) 1.85410 0.243456
\(59\) 7.47214 0.972789 0.486395 0.873739i \(-0.338312\pi\)
0.486395 + 0.873739i \(0.338312\pi\)
\(60\) −0.381966 −0.0493116
\(61\) 12.4721 1.59689 0.798447 0.602066i \(-0.205655\pi\)
0.798447 + 0.602066i \(0.205655\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −0.236068 −0.0295085
\(65\) 0.416408 0.0516490
\(66\) 0 0
\(67\) 4.70820 0.575199 0.287599 0.957751i \(-0.407143\pi\)
0.287599 + 0.957751i \(0.407143\pi\)
\(68\) −7.23607 −0.877502
\(69\) −0.472136 −0.0568385
\(70\) 0.145898 0.0174382
\(71\) 10.4721 1.24281 0.621407 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) 2.23607 0.263523
\(73\) 7.76393 0.908700 0.454350 0.890823i \(-0.349872\pi\)
0.454350 + 0.890823i \(0.349872\pi\)
\(74\) 3.38197 0.393146
\(75\) −4.94427 −0.570915
\(76\) −4.85410 −0.556804
\(77\) 0 0
\(78\) −1.09017 −0.123437
\(79\) 13.4164 1.50946 0.754732 0.656033i \(-0.227767\pi\)
0.754732 + 0.656033i \(0.227767\pi\)
\(80\) 0.437694 0.0489357
\(81\) 1.00000 0.111111
\(82\) −6.47214 −0.714728
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 1.61803 0.176542
\(85\) 1.05573 0.114510
\(86\) 4.29180 0.462796
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 4.47214 0.474045 0.237023 0.971504i \(-0.423828\pi\)
0.237023 + 0.971504i \(0.423828\pi\)
\(90\) −0.145898 −0.0153790
\(91\) −1.76393 −0.184910
\(92\) 0.763932 0.0796454
\(93\) 0 0
\(94\) 1.85410 0.191236
\(95\) 0.708204 0.0726602
\(96\) −5.61803 −0.573388
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −0.618034 −0.0624309
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) 0.944272 0.0939586 0.0469793 0.998896i \(-0.485041\pi\)
0.0469793 + 0.998896i \(0.485041\pi\)
\(102\) −2.76393 −0.273670
\(103\) −2.47214 −0.243587 −0.121793 0.992555i \(-0.538865\pi\)
−0.121793 + 0.992555i \(0.538865\pi\)
\(104\) 3.94427 0.386768
\(105\) −0.236068 −0.0230379
\(106\) 5.52786 0.536914
\(107\) 14.2361 1.37625 0.688126 0.725591i \(-0.258433\pi\)
0.688126 + 0.725591i \(0.258433\pi\)
\(108\) −1.61803 −0.155695
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −5.47214 −0.519392
\(112\) −1.85410 −0.175196
\(113\) 2.47214 0.232559 0.116279 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(114\) −1.85410 −0.173653
\(115\) −0.111456 −0.0103933
\(116\) 4.85410 0.450692
\(117\) 1.76393 0.163076
\(118\) −4.61803 −0.425124
\(119\) −4.47214 −0.409960
\(120\) 0.527864 0.0481872
\(121\) 0 0
\(122\) −7.70820 −0.697868
\(123\) 10.4721 0.944241
\(124\) 0 0
\(125\) −2.34752 −0.209969
\(126\) 0.618034 0.0550588
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 11.3820 1.00603
\(129\) −6.94427 −0.611409
\(130\) −0.257354 −0.0225715
\(131\) −6.47214 −0.565473 −0.282737 0.959198i \(-0.591242\pi\)
−0.282737 + 0.959198i \(0.591242\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) −2.90983 −0.251371
\(135\) 0.236068 0.0203175
\(136\) 10.0000 0.857493
\(137\) −14.9443 −1.27678 −0.638388 0.769715i \(-0.720398\pi\)
−0.638388 + 0.769715i \(0.720398\pi\)
\(138\) 0.291796 0.0248393
\(139\) 16.9443 1.43719 0.718597 0.695427i \(-0.244785\pi\)
0.718597 + 0.695427i \(0.244785\pi\)
\(140\) 0.381966 0.0322820
\(141\) −3.00000 −0.252646
\(142\) −6.47214 −0.543130
\(143\) 0 0
\(144\) 1.85410 0.154508
\(145\) −0.708204 −0.0588131
\(146\) −4.79837 −0.397116
\(147\) 1.00000 0.0824786
\(148\) 8.85410 0.727803
\(149\) 0.0557281 0.00456542 0.00228271 0.999997i \(-0.499273\pi\)
0.00228271 + 0.999997i \(0.499273\pi\)
\(150\) 3.05573 0.249499
\(151\) −9.41641 −0.766296 −0.383148 0.923687i \(-0.625160\pi\)
−0.383148 + 0.923687i \(0.625160\pi\)
\(152\) 6.70820 0.544107
\(153\) 4.47214 0.361551
\(154\) 0 0
\(155\) 0 0
\(156\) −2.85410 −0.228511
\(157\) 13.4164 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) −8.29180 −0.659660
\(159\) −8.94427 −0.709327
\(160\) −1.32624 −0.104848
\(161\) 0.472136 0.0372095
\(162\) −0.618034 −0.0485573
\(163\) 20.7082 1.62199 0.810996 0.585052i \(-0.198926\pi\)
0.810996 + 0.585052i \(0.198926\pi\)
\(164\) −16.9443 −1.32313
\(165\) 0 0
\(166\) −3.70820 −0.287812
\(167\) 1.05573 0.0816947 0.0408473 0.999165i \(-0.486994\pi\)
0.0408473 + 0.999165i \(0.486994\pi\)
\(168\) −2.23607 −0.172516
\(169\) −9.88854 −0.760657
\(170\) −0.652476 −0.0500426
\(171\) 3.00000 0.229416
\(172\) 11.2361 0.856742
\(173\) 16.4721 1.25235 0.626177 0.779681i \(-0.284619\pi\)
0.626177 + 0.779681i \(0.284619\pi\)
\(174\) 1.85410 0.140559
\(175\) 4.94427 0.373752
\(176\) 0 0
\(177\) 7.47214 0.561640
\(178\) −2.76393 −0.207165
\(179\) −7.52786 −0.562659 −0.281329 0.959611i \(-0.590775\pi\)
−0.281329 + 0.959611i \(0.590775\pi\)
\(180\) −0.381966 −0.0284701
\(181\) 23.4164 1.74053 0.870264 0.492586i \(-0.163948\pi\)
0.870264 + 0.492586i \(0.163948\pi\)
\(182\) 1.09017 0.0808088
\(183\) 12.4721 0.921967
\(184\) −1.05573 −0.0778293
\(185\) −1.29180 −0.0949747
\(186\) 0 0
\(187\) 0 0
\(188\) 4.85410 0.354022
\(189\) −1.00000 −0.0727393
\(190\) −0.437694 −0.0317537
\(191\) −14.9443 −1.08133 −0.540665 0.841238i \(-0.681827\pi\)
−0.540665 + 0.841238i \(0.681827\pi\)
\(192\) −0.236068 −0.0170367
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) −1.23607 −0.0887445
\(195\) 0.416408 0.0298196
\(196\) −1.61803 −0.115574
\(197\) −18.9443 −1.34972 −0.674862 0.737944i \(-0.735797\pi\)
−0.674862 + 0.737944i \(0.735797\pi\)
\(198\) 0 0
\(199\) −3.52786 −0.250084 −0.125042 0.992151i \(-0.539907\pi\)
−0.125042 + 0.992151i \(0.539907\pi\)
\(200\) −11.0557 −0.781758
\(201\) 4.70820 0.332091
\(202\) −0.583592 −0.0410614
\(203\) 3.00000 0.210559
\(204\) −7.23607 −0.506626
\(205\) 2.47214 0.172661
\(206\) 1.52786 0.106451
\(207\) −0.472136 −0.0328157
\(208\) 3.27051 0.226769
\(209\) 0 0
\(210\) 0.145898 0.0100679
\(211\) 25.4164 1.74974 0.874869 0.484360i \(-0.160947\pi\)
0.874869 + 0.484360i \(0.160947\pi\)
\(212\) 14.4721 0.993950
\(213\) 10.4721 0.717539
\(214\) −8.79837 −0.601444
\(215\) −1.63932 −0.111801
\(216\) 2.23607 0.152145
\(217\) 0 0
\(218\) 0 0
\(219\) 7.76393 0.524638
\(220\) 0 0
\(221\) 7.88854 0.530641
\(222\) 3.38197 0.226983
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 5.61803 0.375371
\(225\) −4.94427 −0.329618
\(226\) −1.52786 −0.101632
\(227\) −24.4721 −1.62427 −0.812136 0.583468i \(-0.801695\pi\)
−0.812136 + 0.583468i \(0.801695\pi\)
\(228\) −4.85410 −0.321471
\(229\) −15.5279 −1.02611 −0.513055 0.858356i \(-0.671486\pi\)
−0.513055 + 0.858356i \(0.671486\pi\)
\(230\) 0.0688837 0.00454206
\(231\) 0 0
\(232\) −6.70820 −0.440415
\(233\) −15.8885 −1.04089 −0.520447 0.853894i \(-0.674235\pi\)
−0.520447 + 0.853894i \(0.674235\pi\)
\(234\) −1.09017 −0.0712666
\(235\) −0.708204 −0.0461981
\(236\) −12.0902 −0.787003
\(237\) 13.4164 0.871489
\(238\) 2.76393 0.179159
\(239\) −0.708204 −0.0458099 −0.0229050 0.999738i \(-0.507292\pi\)
−0.0229050 + 0.999738i \(0.507292\pi\)
\(240\) 0.437694 0.0282530
\(241\) −0.708204 −0.0456194 −0.0228097 0.999740i \(-0.507261\pi\)
−0.0228097 + 0.999740i \(0.507261\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −20.1803 −1.29191
\(245\) 0.236068 0.0150818
\(246\) −6.47214 −0.412648
\(247\) 5.29180 0.336709
\(248\) 0 0
\(249\) 6.00000 0.380235
\(250\) 1.45085 0.0917598
\(251\) 10.4164 0.657478 0.328739 0.944421i \(-0.393376\pi\)
0.328739 + 0.944421i \(0.393376\pi\)
\(252\) 1.61803 0.101927
\(253\) 0 0
\(254\) −1.23607 −0.0775578
\(255\) 1.05573 0.0661123
\(256\) −6.56231 −0.410144
\(257\) 31.6525 1.97443 0.987214 0.159403i \(-0.0509569\pi\)
0.987214 + 0.159403i \(0.0509569\pi\)
\(258\) 4.29180 0.267196
\(259\) 5.47214 0.340022
\(260\) −0.673762 −0.0417850
\(261\) −3.00000 −0.185695
\(262\) 4.00000 0.247121
\(263\) −4.23607 −0.261207 −0.130604 0.991435i \(-0.541692\pi\)
−0.130604 + 0.991435i \(0.541692\pi\)
\(264\) 0 0
\(265\) −2.11146 −0.129706
\(266\) 1.85410 0.113682
\(267\) 4.47214 0.273690
\(268\) −7.61803 −0.465345
\(269\) −28.4721 −1.73598 −0.867988 0.496584i \(-0.834587\pi\)
−0.867988 + 0.496584i \(0.834587\pi\)
\(270\) −0.145898 −0.00887907
\(271\) 5.47214 0.332409 0.166204 0.986091i \(-0.446849\pi\)
0.166204 + 0.986091i \(0.446849\pi\)
\(272\) 8.29180 0.502764
\(273\) −1.76393 −0.106758
\(274\) 9.23607 0.557971
\(275\) 0 0
\(276\) 0.763932 0.0459833
\(277\) −3.52786 −0.211969 −0.105984 0.994368i \(-0.533799\pi\)
−0.105984 + 0.994368i \(0.533799\pi\)
\(278\) −10.4721 −0.628077
\(279\) 0 0
\(280\) −0.527864 −0.0315459
\(281\) −10.4164 −0.621391 −0.310695 0.950510i \(-0.600562\pi\)
−0.310695 + 0.950510i \(0.600562\pi\)
\(282\) 1.85410 0.110410
\(283\) 16.8885 1.00392 0.501960 0.864891i \(-0.332613\pi\)
0.501960 + 0.864891i \(0.332613\pi\)
\(284\) −16.9443 −1.00546
\(285\) 0.708204 0.0419504
\(286\) 0 0
\(287\) −10.4721 −0.618151
\(288\) −5.61803 −0.331046
\(289\) 3.00000 0.176471
\(290\) 0.437694 0.0257023
\(291\) 2.00000 0.117242
\(292\) −12.5623 −0.735153
\(293\) −25.8885 −1.51242 −0.756212 0.654326i \(-0.772952\pi\)
−0.756212 + 0.654326i \(0.772952\pi\)
\(294\) −0.618034 −0.0360445
\(295\) 1.76393 0.102700
\(296\) −12.2361 −0.711207
\(297\) 0 0
\(298\) −0.0344419 −0.00199516
\(299\) −0.832816 −0.0481630
\(300\) 8.00000 0.461880
\(301\) 6.94427 0.400261
\(302\) 5.81966 0.334884
\(303\) 0.944272 0.0542470
\(304\) 5.56231 0.319020
\(305\) 2.94427 0.168589
\(306\) −2.76393 −0.158003
\(307\) −0.944272 −0.0538924 −0.0269462 0.999637i \(-0.508578\pi\)
−0.0269462 + 0.999637i \(0.508578\pi\)
\(308\) 0 0
\(309\) −2.47214 −0.140635
\(310\) 0 0
\(311\) 17.8885 1.01437 0.507183 0.861838i \(-0.330687\pi\)
0.507183 + 0.861838i \(0.330687\pi\)
\(312\) 3.94427 0.223300
\(313\) −11.5279 −0.651593 −0.325797 0.945440i \(-0.605632\pi\)
−0.325797 + 0.945440i \(0.605632\pi\)
\(314\) −8.29180 −0.467933
\(315\) −0.236068 −0.0133009
\(316\) −21.7082 −1.22118
\(317\) 21.5279 1.20913 0.604563 0.796558i \(-0.293348\pi\)
0.604563 + 0.796558i \(0.293348\pi\)
\(318\) 5.52786 0.309987
\(319\) 0 0
\(320\) −0.0557281 −0.00311529
\(321\) 14.2361 0.794580
\(322\) −0.291796 −0.0162612
\(323\) 13.4164 0.746509
\(324\) −1.61803 −0.0898908
\(325\) −8.72136 −0.483774
\(326\) −12.7984 −0.708836
\(327\) 0 0
\(328\) 23.4164 1.29295
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) −9.70820 −0.532807
\(333\) −5.47214 −0.299871
\(334\) −0.652476 −0.0357019
\(335\) 1.11146 0.0607253
\(336\) −1.85410 −0.101150
\(337\) −28.8328 −1.57062 −0.785312 0.619100i \(-0.787497\pi\)
−0.785312 + 0.619100i \(0.787497\pi\)
\(338\) 6.11146 0.332419
\(339\) 2.47214 0.134268
\(340\) −1.70820 −0.0926404
\(341\) 0 0
\(342\) −1.85410 −0.100258
\(343\) −1.00000 −0.0539949
\(344\) −15.5279 −0.837206
\(345\) −0.111456 −0.00600060
\(346\) −10.1803 −0.547298
\(347\) −20.9443 −1.12435 −0.562174 0.827019i \(-0.690035\pi\)
−0.562174 + 0.827019i \(0.690035\pi\)
\(348\) 4.85410 0.260207
\(349\) −9.29180 −0.497378 −0.248689 0.968583i \(-0.580000\pi\)
−0.248689 + 0.968583i \(0.580000\pi\)
\(350\) −3.05573 −0.163336
\(351\) 1.76393 0.0941517
\(352\) 0 0
\(353\) −16.5967 −0.883356 −0.441678 0.897174i \(-0.645617\pi\)
−0.441678 + 0.897174i \(0.645617\pi\)
\(354\) −4.61803 −0.245446
\(355\) 2.47214 0.131207
\(356\) −7.23607 −0.383511
\(357\) −4.47214 −0.236691
\(358\) 4.65248 0.245891
\(359\) −36.9443 −1.94984 −0.974922 0.222547i \(-0.928563\pi\)
−0.974922 + 0.222547i \(0.928563\pi\)
\(360\) 0.527864 0.0278209
\(361\) −10.0000 −0.526316
\(362\) −14.4721 −0.760639
\(363\) 0 0
\(364\) 2.85410 0.149596
\(365\) 1.83282 0.0959340
\(366\) −7.70820 −0.402914
\(367\) 30.8328 1.60946 0.804730 0.593641i \(-0.202310\pi\)
0.804730 + 0.593641i \(0.202310\pi\)
\(368\) −0.875388 −0.0456328
\(369\) 10.4721 0.545158
\(370\) 0.798374 0.0415055
\(371\) 8.94427 0.464363
\(372\) 0 0
\(373\) 29.4164 1.52312 0.761562 0.648092i \(-0.224433\pi\)
0.761562 + 0.648092i \(0.224433\pi\)
\(374\) 0 0
\(375\) −2.34752 −0.121226
\(376\) −6.70820 −0.345949
\(377\) −5.29180 −0.272541
\(378\) 0.618034 0.0317882
\(379\) 0.236068 0.0121260 0.00606300 0.999982i \(-0.498070\pi\)
0.00606300 + 0.999982i \(0.498070\pi\)
\(380\) −1.14590 −0.0587833
\(381\) 2.00000 0.102463
\(382\) 9.23607 0.472558
\(383\) −21.8885 −1.11845 −0.559226 0.829015i \(-0.688902\pi\)
−0.559226 + 0.829015i \(0.688902\pi\)
\(384\) 11.3820 0.580834
\(385\) 0 0
\(386\) −7.41641 −0.377485
\(387\) −6.94427 −0.352997
\(388\) −3.23607 −0.164286
\(389\) −16.3607 −0.829519 −0.414760 0.909931i \(-0.636134\pi\)
−0.414760 + 0.909931i \(0.636134\pi\)
\(390\) −0.257354 −0.0130316
\(391\) −2.11146 −0.106781
\(392\) 2.23607 0.112938
\(393\) −6.47214 −0.326476
\(394\) 11.7082 0.589851
\(395\) 3.16718 0.159358
\(396\) 0 0
\(397\) 3.52786 0.177058 0.0885292 0.996074i \(-0.471783\pi\)
0.0885292 + 0.996074i \(0.471783\pi\)
\(398\) 2.18034 0.109291
\(399\) −3.00000 −0.150188
\(400\) −9.16718 −0.458359
\(401\) 34.3607 1.71589 0.857945 0.513741i \(-0.171741\pi\)
0.857945 + 0.513741i \(0.171741\pi\)
\(402\) −2.90983 −0.145129
\(403\) 0 0
\(404\) −1.52786 −0.0760141
\(405\) 0.236068 0.0117303
\(406\) −1.85410 −0.0920175
\(407\) 0 0
\(408\) 10.0000 0.495074
\(409\) 29.4164 1.45455 0.727274 0.686347i \(-0.240787\pi\)
0.727274 + 0.686347i \(0.240787\pi\)
\(410\) −1.52786 −0.0754558
\(411\) −14.9443 −0.737147
\(412\) 4.00000 0.197066
\(413\) −7.47214 −0.367680
\(414\) 0.291796 0.0143410
\(415\) 1.41641 0.0695287
\(416\) −9.90983 −0.485869
\(417\) 16.9443 0.829765
\(418\) 0 0
\(419\) 33.4721 1.63522 0.817610 0.575772i \(-0.195298\pi\)
0.817610 + 0.575772i \(0.195298\pi\)
\(420\) 0.381966 0.0186380
\(421\) 10.4164 0.507665 0.253832 0.967248i \(-0.418309\pi\)
0.253832 + 0.967248i \(0.418309\pi\)
\(422\) −15.7082 −0.764663
\(423\) −3.00000 −0.145865
\(424\) −20.0000 −0.971286
\(425\) −22.1115 −1.07256
\(426\) −6.47214 −0.313576
\(427\) −12.4721 −0.603569
\(428\) −23.0344 −1.11341
\(429\) 0 0
\(430\) 1.01316 0.0488587
\(431\) 30.5967 1.47379 0.736897 0.676005i \(-0.236290\pi\)
0.736897 + 0.676005i \(0.236290\pi\)
\(432\) 1.85410 0.0892055
\(433\) −17.4164 −0.836979 −0.418490 0.908222i \(-0.637440\pi\)
−0.418490 + 0.908222i \(0.637440\pi\)
\(434\) 0 0
\(435\) −0.708204 −0.0339558
\(436\) 0 0
\(437\) −1.41641 −0.0677560
\(438\) −4.79837 −0.229275
\(439\) 35.3607 1.68767 0.843837 0.536600i \(-0.180292\pi\)
0.843837 + 0.536600i \(0.180292\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −4.87539 −0.231899
\(443\) 27.8885 1.32502 0.662512 0.749051i \(-0.269490\pi\)
0.662512 + 0.749051i \(0.269490\pi\)
\(444\) 8.85410 0.420197
\(445\) 1.05573 0.0500463
\(446\) −3.70820 −0.175589
\(447\) 0.0557281 0.00263585
\(448\) 0.236068 0.0111532
\(449\) 19.5279 0.921577 0.460788 0.887510i \(-0.347567\pi\)
0.460788 + 0.887510i \(0.347567\pi\)
\(450\) 3.05573 0.144048
\(451\) 0 0
\(452\) −4.00000 −0.188144
\(453\) −9.41641 −0.442421
\(454\) 15.1246 0.709833
\(455\) −0.416408 −0.0195215
\(456\) 6.70820 0.314140
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 9.59675 0.448427
\(459\) 4.47214 0.208741
\(460\) 0.180340 0.00840839
\(461\) 9.05573 0.421767 0.210884 0.977511i \(-0.432366\pi\)
0.210884 + 0.977511i \(0.432366\pi\)
\(462\) 0 0
\(463\) 8.12461 0.377583 0.188791 0.982017i \(-0.439543\pi\)
0.188791 + 0.982017i \(0.439543\pi\)
\(464\) −5.56231 −0.258224
\(465\) 0 0
\(466\) 9.81966 0.454887
\(467\) 6.52786 0.302074 0.151037 0.988528i \(-0.451739\pi\)
0.151037 + 0.988528i \(0.451739\pi\)
\(468\) −2.85410 −0.131931
\(469\) −4.70820 −0.217405
\(470\) 0.437694 0.0201893
\(471\) 13.4164 0.618195
\(472\) 16.7082 0.769057
\(473\) 0 0
\(474\) −8.29180 −0.380855
\(475\) −14.8328 −0.680576
\(476\) 7.23607 0.331665
\(477\) −8.94427 −0.409530
\(478\) 0.437694 0.0200197
\(479\) −19.5279 −0.892251 −0.446125 0.894970i \(-0.647196\pi\)
−0.446125 + 0.894970i \(0.647196\pi\)
\(480\) −1.32624 −0.0605342
\(481\) −9.65248 −0.440115
\(482\) 0.437694 0.0199364
\(483\) 0.472136 0.0214829
\(484\) 0 0
\(485\) 0.472136 0.0214386
\(486\) −0.618034 −0.0280346
\(487\) −21.8885 −0.991865 −0.495932 0.868361i \(-0.665174\pi\)
−0.495932 + 0.868361i \(0.665174\pi\)
\(488\) 27.8885 1.26246
\(489\) 20.7082 0.936457
\(490\) −0.145898 −0.00659100
\(491\) −28.5967 −1.29055 −0.645277 0.763949i \(-0.723258\pi\)
−0.645277 + 0.763949i \(0.723258\pi\)
\(492\) −16.9443 −0.763907
\(493\) −13.4164 −0.604245
\(494\) −3.27051 −0.147147
\(495\) 0 0
\(496\) 0 0
\(497\) −10.4721 −0.469739
\(498\) −3.70820 −0.166169
\(499\) −36.7082 −1.64328 −0.821642 0.570003i \(-0.806942\pi\)
−0.821642 + 0.570003i \(0.806942\pi\)
\(500\) 3.79837 0.169868
\(501\) 1.05573 0.0471665
\(502\) −6.43769 −0.287328
\(503\) 7.41641 0.330681 0.165341 0.986237i \(-0.447128\pi\)
0.165341 + 0.986237i \(0.447128\pi\)
\(504\) −2.23607 −0.0996024
\(505\) 0.222912 0.00991947
\(506\) 0 0
\(507\) −9.88854 −0.439166
\(508\) −3.23607 −0.143577
\(509\) −18.3607 −0.813823 −0.406911 0.913468i \(-0.633394\pi\)
−0.406911 + 0.913468i \(0.633394\pi\)
\(510\) −0.652476 −0.0288921
\(511\) −7.76393 −0.343456
\(512\) −18.7082 −0.826794
\(513\) 3.00000 0.132453
\(514\) −19.5623 −0.862856
\(515\) −0.583592 −0.0257161
\(516\) 11.2361 0.494640
\(517\) 0 0
\(518\) −3.38197 −0.148595
\(519\) 16.4721 0.723047
\(520\) 0.931116 0.0408322
\(521\) 9.76393 0.427766 0.213883 0.976859i \(-0.431389\pi\)
0.213883 + 0.976859i \(0.431389\pi\)
\(522\) 1.85410 0.0811518
\(523\) −3.00000 −0.131181 −0.0655904 0.997847i \(-0.520893\pi\)
−0.0655904 + 0.997847i \(0.520893\pi\)
\(524\) 10.4721 0.457477
\(525\) 4.94427 0.215786
\(526\) 2.61803 0.114152
\(527\) 0 0
\(528\) 0 0
\(529\) −22.7771 −0.990308
\(530\) 1.30495 0.0566835
\(531\) 7.47214 0.324263
\(532\) 4.85410 0.210452
\(533\) 18.4721 0.800117
\(534\) −2.76393 −0.119607
\(535\) 3.36068 0.145295
\(536\) 10.5279 0.454734
\(537\) −7.52786 −0.324851
\(538\) 17.5967 0.758650
\(539\) 0 0
\(540\) −0.381966 −0.0164372
\(541\) −11.4164 −0.490830 −0.245415 0.969418i \(-0.578924\pi\)
−0.245415 + 0.969418i \(0.578924\pi\)
\(542\) −3.38197 −0.145268
\(543\) 23.4164 1.00489
\(544\) −25.1246 −1.07721
\(545\) 0 0
\(546\) 1.09017 0.0466550
\(547\) −16.8328 −0.719719 −0.359860 0.933006i \(-0.617175\pi\)
−0.359860 + 0.933006i \(0.617175\pi\)
\(548\) 24.1803 1.03293
\(549\) 12.4721 0.532298
\(550\) 0 0
\(551\) −9.00000 −0.383413
\(552\) −1.05573 −0.0449348
\(553\) −13.4164 −0.570524
\(554\) 2.18034 0.0926338
\(555\) −1.29180 −0.0548337
\(556\) −27.4164 −1.16271
\(557\) −39.0000 −1.65248 −0.826242 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) −12.2492 −0.518087
\(560\) −0.437694 −0.0184960
\(561\) 0 0
\(562\) 6.43769 0.271558
\(563\) 6.47214 0.272768 0.136384 0.990656i \(-0.456452\pi\)
0.136384 + 0.990656i \(0.456452\pi\)
\(564\) 4.85410 0.204395
\(565\) 0.583592 0.0245519
\(566\) −10.4377 −0.438729
\(567\) −1.00000 −0.0419961
\(568\) 23.4164 0.982531
\(569\) −43.8885 −1.83990 −0.919952 0.392032i \(-0.871772\pi\)
−0.919952 + 0.392032i \(0.871772\pi\)
\(570\) −0.437694 −0.0183330
\(571\) 5.05573 0.211576 0.105788 0.994389i \(-0.466264\pi\)
0.105788 + 0.994389i \(0.466264\pi\)
\(572\) 0 0
\(573\) −14.9443 −0.624306
\(574\) 6.47214 0.270142
\(575\) 2.33437 0.0973499
\(576\) −0.236068 −0.00983617
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) −1.85410 −0.0771205
\(579\) 12.0000 0.498703
\(580\) 1.14590 0.0475808
\(581\) −6.00000 −0.248922
\(582\) −1.23607 −0.0512367
\(583\) 0 0
\(584\) 17.3607 0.718390
\(585\) 0.416408 0.0172163
\(586\) 16.0000 0.660954
\(587\) −45.3607 −1.87224 −0.936118 0.351687i \(-0.885608\pi\)
−0.936118 + 0.351687i \(0.885608\pi\)
\(588\) −1.61803 −0.0667266
\(589\) 0 0
\(590\) −1.09017 −0.0448816
\(591\) −18.9443 −0.779263
\(592\) −10.1459 −0.416994
\(593\) 41.7771 1.71558 0.857790 0.514001i \(-0.171837\pi\)
0.857790 + 0.514001i \(0.171837\pi\)
\(594\) 0 0
\(595\) −1.05573 −0.0432806
\(596\) −0.0901699 −0.00369350
\(597\) −3.52786 −0.144386
\(598\) 0.514708 0.0210480
\(599\) −25.4164 −1.03849 −0.519243 0.854627i \(-0.673786\pi\)
−0.519243 + 0.854627i \(0.673786\pi\)
\(600\) −11.0557 −0.451348
\(601\) −13.1803 −0.537637 −0.268819 0.963191i \(-0.586633\pi\)
−0.268819 + 0.963191i \(0.586633\pi\)
\(602\) −4.29180 −0.174921
\(603\) 4.70820 0.191733
\(604\) 15.2361 0.619947
\(605\) 0 0
\(606\) −0.583592 −0.0237068
\(607\) 37.2492 1.51190 0.755950 0.654630i \(-0.227175\pi\)
0.755950 + 0.654630i \(0.227175\pi\)
\(608\) −16.8541 −0.683524
\(609\) 3.00000 0.121566
\(610\) −1.81966 −0.0736759
\(611\) −5.29180 −0.214083
\(612\) −7.23607 −0.292501
\(613\) −26.4721 −1.06920 −0.534600 0.845105i \(-0.679538\pi\)
−0.534600 + 0.845105i \(0.679538\pi\)
\(614\) 0.583592 0.0235519
\(615\) 2.47214 0.0996861
\(616\) 0 0
\(617\) 5.88854 0.237064 0.118532 0.992950i \(-0.462181\pi\)
0.118532 + 0.992950i \(0.462181\pi\)
\(618\) 1.52786 0.0614597
\(619\) −34.2492 −1.37659 −0.688296 0.725430i \(-0.741641\pi\)
−0.688296 + 0.725430i \(0.741641\pi\)
\(620\) 0 0
\(621\) −0.472136 −0.0189462
\(622\) −11.0557 −0.443294
\(623\) −4.47214 −0.179172
\(624\) 3.27051 0.130925
\(625\) 24.1672 0.966687
\(626\) 7.12461 0.284757
\(627\) 0 0
\(628\) −21.7082 −0.866252
\(629\) −24.4721 −0.975768
\(630\) 0.145898 0.00581272
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 30.0000 1.19334
\(633\) 25.4164 1.01021
\(634\) −13.3050 −0.528407
\(635\) 0.472136 0.0187361
\(636\) 14.4721 0.573858
\(637\) 1.76393 0.0698895
\(638\) 0 0
\(639\) 10.4721 0.414271
\(640\) 2.68692 0.106210
\(641\) 14.8328 0.585861 0.292930 0.956134i \(-0.405370\pi\)
0.292930 + 0.956134i \(0.405370\pi\)
\(642\) −8.79837 −0.347244
\(643\) 25.4164 1.00233 0.501163 0.865353i \(-0.332906\pi\)
0.501163 + 0.865353i \(0.332906\pi\)
\(644\) −0.763932 −0.0301031
\(645\) −1.63932 −0.0645482
\(646\) −8.29180 −0.326236
\(647\) −30.8885 −1.21435 −0.607177 0.794567i \(-0.707698\pi\)
−0.607177 + 0.794567i \(0.707698\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) 5.39010 0.211417
\(651\) 0 0
\(652\) −33.5066 −1.31222
\(653\) −27.0557 −1.05877 −0.529386 0.848381i \(-0.677578\pi\)
−0.529386 + 0.848381i \(0.677578\pi\)
\(654\) 0 0
\(655\) −1.52786 −0.0596986
\(656\) 19.4164 0.758083
\(657\) 7.76393 0.302900
\(658\) −1.85410 −0.0722804
\(659\) 30.2361 1.17783 0.588915 0.808195i \(-0.299555\pi\)
0.588915 + 0.808195i \(0.299555\pi\)
\(660\) 0 0
\(661\) −22.8328 −0.888094 −0.444047 0.896004i \(-0.646458\pi\)
−0.444047 + 0.896004i \(0.646458\pi\)
\(662\) −14.8328 −0.576494
\(663\) 7.88854 0.306366
\(664\) 13.4164 0.520658
\(665\) −0.708204 −0.0274630
\(666\) 3.38197 0.131049
\(667\) 1.41641 0.0548435
\(668\) −1.70820 −0.0660924
\(669\) 6.00000 0.231973
\(670\) −0.686918 −0.0265379
\(671\) 0 0
\(672\) 5.61803 0.216720
\(673\) −29.3050 −1.12962 −0.564811 0.825220i \(-0.691051\pi\)
−0.564811 + 0.825220i \(0.691051\pi\)
\(674\) 17.8197 0.686388
\(675\) −4.94427 −0.190305
\(676\) 16.0000 0.615385
\(677\) −16.3607 −0.628792 −0.314396 0.949292i \(-0.601802\pi\)
−0.314396 + 0.949292i \(0.601802\pi\)
\(678\) −1.52786 −0.0586773
\(679\) −2.00000 −0.0767530
\(680\) 2.36068 0.0905279
\(681\) −24.4721 −0.937774
\(682\) 0 0
\(683\) −18.4721 −0.706817 −0.353408 0.935469i \(-0.614977\pi\)
−0.353408 + 0.935469i \(0.614977\pi\)
\(684\) −4.85410 −0.185601
\(685\) −3.52786 −0.134793
\(686\) 0.618034 0.0235966
\(687\) −15.5279 −0.592425
\(688\) −12.8754 −0.490870
\(689\) −15.7771 −0.601059
\(690\) 0.0688837 0.00262236
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) −26.6525 −1.01318
\(693\) 0 0
\(694\) 12.9443 0.491358
\(695\) 4.00000 0.151729
\(696\) −6.70820 −0.254274
\(697\) 46.8328 1.77392
\(698\) 5.74265 0.217362
\(699\) −15.8885 −0.600960
\(700\) −8.00000 −0.302372
\(701\) −10.9443 −0.413359 −0.206680 0.978409i \(-0.566266\pi\)
−0.206680 + 0.978409i \(0.566266\pi\)
\(702\) −1.09017 −0.0411458
\(703\) −16.4164 −0.619157
\(704\) 0 0
\(705\) −0.708204 −0.0266725
\(706\) 10.2574 0.386041
\(707\) −0.944272 −0.0355130
\(708\) −12.0902 −0.454376
\(709\) 38.4164 1.44276 0.721379 0.692540i \(-0.243508\pi\)
0.721379 + 0.692540i \(0.243508\pi\)
\(710\) −1.52786 −0.0573397
\(711\) 13.4164 0.503155
\(712\) 10.0000 0.374766
\(713\) 0 0
\(714\) 2.76393 0.103438
\(715\) 0 0
\(716\) 12.1803 0.455201
\(717\) −0.708204 −0.0264484
\(718\) 22.8328 0.852113
\(719\) 46.7771 1.74449 0.872246 0.489068i \(-0.162663\pi\)
0.872246 + 0.489068i \(0.162663\pi\)
\(720\) 0.437694 0.0163119
\(721\) 2.47214 0.0920672
\(722\) 6.18034 0.230008
\(723\) −0.708204 −0.0263384
\(724\) −37.8885 −1.40812
\(725\) 14.8328 0.550877
\(726\) 0 0
\(727\) −8.83282 −0.327591 −0.163796 0.986494i \(-0.552374\pi\)
−0.163796 + 0.986494i \(0.552374\pi\)
\(728\) −3.94427 −0.146184
\(729\) 1.00000 0.0370370
\(730\) −1.13274 −0.0419247
\(731\) −31.0557 −1.14864
\(732\) −20.1803 −0.745887
\(733\) 15.5279 0.573535 0.286767 0.958000i \(-0.407419\pi\)
0.286767 + 0.958000i \(0.407419\pi\)
\(734\) −19.0557 −0.703360
\(735\) 0.236068 0.00870750
\(736\) 2.65248 0.0977716
\(737\) 0 0
\(738\) −6.47214 −0.238243
\(739\) −11.3050 −0.415859 −0.207930 0.978144i \(-0.566673\pi\)
−0.207930 + 0.978144i \(0.566673\pi\)
\(740\) 2.09017 0.0768362
\(741\) 5.29180 0.194399
\(742\) −5.52786 −0.202934
\(743\) −20.2361 −0.742389 −0.371195 0.928555i \(-0.621052\pi\)
−0.371195 + 0.928555i \(0.621052\pi\)
\(744\) 0 0
\(745\) 0.0131556 0.000481985 0
\(746\) −18.1803 −0.665630
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −14.2361 −0.520175
\(750\) 1.45085 0.0529775
\(751\) −13.7639 −0.502253 −0.251127 0.967954i \(-0.580801\pi\)
−0.251127 + 0.967954i \(0.580801\pi\)
\(752\) −5.56231 −0.202836
\(753\) 10.4164 0.379595
\(754\) 3.27051 0.119105
\(755\) −2.22291 −0.0809001
\(756\) 1.61803 0.0588473
\(757\) −2.41641 −0.0878258 −0.0439129 0.999035i \(-0.513982\pi\)
−0.0439129 + 0.999035i \(0.513982\pi\)
\(758\) −0.145898 −0.00529926
\(759\) 0 0
\(760\) 1.58359 0.0574429
\(761\) −38.3607 −1.39057 −0.695287 0.718732i \(-0.744723\pi\)
−0.695287 + 0.718732i \(0.744723\pi\)
\(762\) −1.23607 −0.0447780
\(763\) 0 0
\(764\) 24.1803 0.874814
\(765\) 1.05573 0.0381699
\(766\) 13.5279 0.488782
\(767\) 13.1803 0.475914
\(768\) −6.56231 −0.236797
\(769\) 26.1246 0.942078 0.471039 0.882112i \(-0.343879\pi\)
0.471039 + 0.882112i \(0.343879\pi\)
\(770\) 0 0
\(771\) 31.6525 1.13994
\(772\) −19.4164 −0.698812
\(773\) −24.7082 −0.888692 −0.444346 0.895855i \(-0.646564\pi\)
−0.444346 + 0.895855i \(0.646564\pi\)
\(774\) 4.29180 0.154265
\(775\) 0 0
\(776\) 4.47214 0.160540
\(777\) 5.47214 0.196312
\(778\) 10.1115 0.362513
\(779\) 31.4164 1.12561
\(780\) −0.673762 −0.0241246
\(781\) 0 0
\(782\) 1.30495 0.0466650
\(783\) −3.00000 −0.107211
\(784\) 1.85410 0.0662179
\(785\) 3.16718 0.113042
\(786\) 4.00000 0.142675
\(787\) −27.0000 −0.962446 −0.481223 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) 30.6525 1.09195
\(789\) −4.23607 −0.150808
\(790\) −1.95743 −0.0696421
\(791\) −2.47214 −0.0878990
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) −2.18034 −0.0773774
\(795\) −2.11146 −0.0748856
\(796\) 5.70820 0.202322
\(797\) −37.1803 −1.31700 −0.658498 0.752583i \(-0.728808\pi\)
−0.658498 + 0.752583i \(0.728808\pi\)
\(798\) 1.85410 0.0656345
\(799\) −13.4164 −0.474638
\(800\) 27.7771 0.982068
\(801\) 4.47214 0.158015
\(802\) −21.2361 −0.749872
\(803\) 0 0
\(804\) −7.61803 −0.268667
\(805\) 0.111456 0.00392831
\(806\) 0 0
\(807\) −28.4721 −1.00227
\(808\) 2.11146 0.0742808
\(809\) 13.3607 0.469736 0.234868 0.972027i \(-0.424534\pi\)
0.234868 + 0.972027i \(0.424534\pi\)
\(810\) −0.145898 −0.00512633
\(811\) 2.16718 0.0761001 0.0380501 0.999276i \(-0.487885\pi\)
0.0380501 + 0.999276i \(0.487885\pi\)
\(812\) −4.85410 −0.170346
\(813\) 5.47214 0.191916
\(814\) 0 0
\(815\) 4.88854 0.171238
\(816\) 8.29180 0.290271
\(817\) −20.8328 −0.728848
\(818\) −18.1803 −0.635661
\(819\) −1.76393 −0.0616368
\(820\) −4.00000 −0.139686
\(821\) −0.0557281 −0.00194492 −0.000972462 1.00000i \(-0.500310\pi\)
−0.000972462 1.00000i \(0.500310\pi\)
\(822\) 9.23607 0.322145
\(823\) 7.18034 0.250291 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(824\) −5.52786 −0.192572
\(825\) 0 0
\(826\) 4.61803 0.160682
\(827\) 20.1246 0.699801 0.349901 0.936787i \(-0.386215\pi\)
0.349901 + 0.936787i \(0.386215\pi\)
\(828\) 0.763932 0.0265485
\(829\) 14.8328 0.515165 0.257582 0.966256i \(-0.417074\pi\)
0.257582 + 0.966256i \(0.417074\pi\)
\(830\) −0.875388 −0.0303852
\(831\) −3.52786 −0.122380
\(832\) −0.416408 −0.0144363
\(833\) 4.47214 0.154950
\(834\) −10.4721 −0.362620
\(835\) 0.249224 0.00862474
\(836\) 0 0
\(837\) 0 0
\(838\) −20.6869 −0.714618
\(839\) −23.9443 −0.826648 −0.413324 0.910584i \(-0.635632\pi\)
−0.413324 + 0.910584i \(0.635632\pi\)
\(840\) −0.527864 −0.0182130
\(841\) −20.0000 −0.689655
\(842\) −6.43769 −0.221858
\(843\) −10.4164 −0.358760
\(844\) −41.1246 −1.41557
\(845\) −2.33437 −0.0803047
\(846\) 1.85410 0.0637453
\(847\) 0 0
\(848\) −16.5836 −0.569483
\(849\) 16.8885 0.579613
\(850\) 13.6656 0.468727
\(851\) 2.58359 0.0885644
\(852\) −16.9443 −0.580501
\(853\) 49.4164 1.69199 0.845993 0.533194i \(-0.179009\pi\)
0.845993 + 0.533194i \(0.179009\pi\)
\(854\) 7.70820 0.263769
\(855\) 0.708204 0.0242201
\(856\) 31.8328 1.08802
\(857\) −19.5279 −0.667059 −0.333530 0.942740i \(-0.608240\pi\)
−0.333530 + 0.942740i \(0.608240\pi\)
\(858\) 0 0
\(859\) −33.7771 −1.15246 −0.576230 0.817288i \(-0.695477\pi\)
−0.576230 + 0.817288i \(0.695477\pi\)
\(860\) 2.65248 0.0904487
\(861\) −10.4721 −0.356889
\(862\) −18.9098 −0.644071
\(863\) 52.3607 1.78238 0.891189 0.453632i \(-0.149872\pi\)
0.891189 + 0.453632i \(0.149872\pi\)
\(864\) −5.61803 −0.191129
\(865\) 3.88854 0.132214
\(866\) 10.7639 0.365773
\(867\) 3.00000 0.101885
\(868\) 0 0
\(869\) 0 0
\(870\) 0.437694 0.0148392
\(871\) 8.30495 0.281403
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0.875388 0.0296104
\(875\) 2.34752 0.0793608
\(876\) −12.5623 −0.424441
\(877\) 50.8328 1.71650 0.858251 0.513230i \(-0.171551\pi\)
0.858251 + 0.513230i \(0.171551\pi\)
\(878\) −21.8541 −0.737540
\(879\) −25.8885 −0.873199
\(880\) 0 0
\(881\) 58.4853 1.97042 0.985210 0.171353i \(-0.0548137\pi\)
0.985210 + 0.171353i \(0.0548137\pi\)
\(882\) −0.618034 −0.0208103
\(883\) 27.2918 0.918442 0.459221 0.888322i \(-0.348129\pi\)
0.459221 + 0.888322i \(0.348129\pi\)
\(884\) −12.7639 −0.429297
\(885\) 1.76393 0.0592939
\(886\) −17.2361 −0.579057
\(887\) −19.4164 −0.651939 −0.325970 0.945380i \(-0.605691\pi\)
−0.325970 + 0.945380i \(0.605691\pi\)
\(888\) −12.2361 −0.410616
\(889\) −2.00000 −0.0670778
\(890\) −0.652476 −0.0218710
\(891\) 0 0
\(892\) −9.70820 −0.325055
\(893\) −9.00000 −0.301174
\(894\) −0.0344419 −0.00115191
\(895\) −1.77709 −0.0594015
\(896\) −11.3820 −0.380245
\(897\) −0.832816 −0.0278069
\(898\) −12.0689 −0.402744
\(899\) 0 0
\(900\) 8.00000 0.266667
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) 6.94427 0.231091
\(904\) 5.52786 0.183854
\(905\) 5.52786 0.183752
\(906\) 5.81966 0.193345
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 39.5967 1.31406
\(909\) 0.944272 0.0313195
\(910\) 0.257354 0.00853121
\(911\) 41.3050 1.36849 0.684247 0.729250i \(-0.260131\pi\)
0.684247 + 0.729250i \(0.260131\pi\)
\(912\) 5.56231 0.184186
\(913\) 0 0
\(914\) −16.0689 −0.531511
\(915\) 2.94427 0.0973346
\(916\) 25.1246 0.830141
\(917\) 6.47214 0.213729
\(918\) −2.76393 −0.0912234
\(919\) −40.9443 −1.35063 −0.675313 0.737531i \(-0.735992\pi\)
−0.675313 + 0.737531i \(0.735992\pi\)
\(920\) −0.249224 −0.00821666
\(921\) −0.944272 −0.0311148
\(922\) −5.59675 −0.184319
\(923\) 18.4721 0.608018
\(924\) 0 0
\(925\) 27.0557 0.889587
\(926\) −5.02129 −0.165010
\(927\) −2.47214 −0.0811956
\(928\) 16.8541 0.553263
\(929\) 29.0689 0.953719 0.476860 0.878979i \(-0.341775\pi\)
0.476860 + 0.878979i \(0.341775\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 25.7082 0.842100
\(933\) 17.8885 0.585645
\(934\) −4.03444 −0.132011
\(935\) 0 0
\(936\) 3.94427 0.128923
\(937\) −2.58359 −0.0844023 −0.0422011 0.999109i \(-0.513437\pi\)
−0.0422011 + 0.999109i \(0.513437\pi\)
\(938\) 2.90983 0.0950093
\(939\) −11.5279 −0.376198
\(940\) 1.14590 0.0373751
\(941\) 22.3607 0.728937 0.364469 0.931216i \(-0.381251\pi\)
0.364469 + 0.931216i \(0.381251\pi\)
\(942\) −8.29180 −0.270161
\(943\) −4.94427 −0.161008
\(944\) 13.8541 0.450913
\(945\) −0.236068 −0.00767929
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −21.7082 −0.705050
\(949\) 13.6950 0.444560
\(950\) 9.16718 0.297423
\(951\) 21.5279 0.698089
\(952\) −10.0000 −0.324102
\(953\) 33.3607 1.08066 0.540329 0.841454i \(-0.318300\pi\)
0.540329 + 0.841454i \(0.318300\pi\)
\(954\) 5.52786 0.178971
\(955\) −3.52786 −0.114159
\(956\) 1.14590 0.0370610
\(957\) 0 0
\(958\) 12.0689 0.389928
\(959\) 14.9443 0.482576
\(960\) −0.0557281 −0.00179862
\(961\) −31.0000 −1.00000
\(962\) 5.96556 0.192337
\(963\) 14.2361 0.458751
\(964\) 1.14590 0.0369069
\(965\) 2.83282 0.0911916
\(966\) −0.291796 −0.00938838
\(967\) −38.4721 −1.23718 −0.618590 0.785714i \(-0.712296\pi\)
−0.618590 + 0.785714i \(0.712296\pi\)
\(968\) 0 0
\(969\) 13.4164 0.430997
\(970\) −0.291796 −0.00936901
\(971\) −14.5279 −0.466221 −0.233111 0.972450i \(-0.574890\pi\)
−0.233111 + 0.972450i \(0.574890\pi\)
\(972\) −1.61803 −0.0518985
\(973\) −16.9443 −0.543208
\(974\) 13.5279 0.433461
\(975\) −8.72136 −0.279307
\(976\) 23.1246 0.740201
\(977\) −13.7771 −0.440768 −0.220384 0.975413i \(-0.570731\pi\)
−0.220384 + 0.975413i \(0.570731\pi\)
\(978\) −12.7984 −0.409247
\(979\) 0 0
\(980\) −0.381966 −0.0122015
\(981\) 0 0
\(982\) 17.6738 0.563992
\(983\) 6.11146 0.194925 0.0974626 0.995239i \(-0.468927\pi\)
0.0974626 + 0.995239i \(0.468927\pi\)
\(984\) 23.4164 0.746488
\(985\) −4.47214 −0.142494
\(986\) 8.29180 0.264065
\(987\) 3.00000 0.0954911
\(988\) −8.56231 −0.272403
\(989\) 3.27864 0.104255
\(990\) 0 0
\(991\) −49.7639 −1.58080 −0.790402 0.612589i \(-0.790128\pi\)
−0.790402 + 0.612589i \(0.790128\pi\)
\(992\) 0 0
\(993\) 24.0000 0.761617
\(994\) 6.47214 0.205284
\(995\) −0.832816 −0.0264020
\(996\) −9.70820 −0.307616
\(997\) −37.4164 −1.18499 −0.592495 0.805574i \(-0.701857\pi\)
−0.592495 + 0.805574i \(0.701857\pi\)
\(998\) 22.6869 0.718142
\(999\) −5.47214 −0.173131
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.1.1 yes 2
3.2 odd 2 7623.2.a.bb.1.2 2
11.10 odd 2 2541.2.a.r.1.2 2
33.32 even 2 7623.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.r.1.2 2 11.10 odd 2
2541.2.a.ba.1.1 yes 2 1.1 even 1 trivial
7623.2.a.bb.1.2 2 3.2 odd 2
7623.2.a.bq.1.1 2 33.32 even 2