Properties

Label 5445.2.a.w.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{5} -2.00000 q^{7} +2.23607 q^{8} -0.618034 q^{10} -2.47214 q^{13} +1.23607 q^{14} +1.85410 q^{16} +3.47214 q^{17} -1.61803 q^{20} -3.76393 q^{23} +1.00000 q^{25} +1.52786 q^{26} +3.23607 q^{28} +8.47214 q^{29} -6.70820 q^{31} -5.61803 q^{32} -2.14590 q^{34} -2.00000 q^{35} -0.472136 q^{37} +2.23607 q^{40} +6.00000 q^{41} -6.00000 q^{43} +2.32624 q^{46} -8.23607 q^{47} -3.00000 q^{49} -0.618034 q^{50} +4.00000 q^{52} +5.94427 q^{53} -4.47214 q^{56} -5.23607 q^{58} +6.94427 q^{59} +3.47214 q^{61} +4.14590 q^{62} -0.236068 q^{64} -2.47214 q^{65} +11.4164 q^{67} -5.61803 q^{68} +1.23607 q^{70} +4.47214 q^{71} +4.94427 q^{73} +0.291796 q^{74} -2.70820 q^{79} +1.85410 q^{80} -3.70820 q^{82} +3.47214 q^{85} +3.70820 q^{86} -8.47214 q^{89} +4.94427 q^{91} +6.09017 q^{92} +5.09017 q^{94} +2.47214 q^{97} +1.85410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + 2q^{5} - 4q^{7} + q^{10} + 4q^{13} - 2q^{14} - 3q^{16} - 2q^{17} - q^{20} - 12q^{23} + 2q^{25} + 12q^{26} + 2q^{28} + 8q^{29} - 9q^{32} - 11q^{34} - 4q^{35} + 8q^{37} + 12q^{41} - 12q^{43} - 11q^{46} - 12q^{47} - 6q^{49} + q^{50} + 8q^{52} - 6q^{53} - 6q^{58} - 4q^{59} - 2q^{61} + 15q^{62} + 4q^{64} + 4q^{65} - 4q^{67} - 9q^{68} - 2q^{70} - 8q^{73} + 14q^{74} + 8q^{79} - 3q^{80} + 6q^{82} - 2q^{85} - 6q^{86} - 8q^{89} - 8q^{91} + q^{92} - q^{94} - 4q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) −0.618034 −0.195440
\(11\) 0 0
\(12\) 0 0
\(13\) −2.47214 −0.685647 −0.342824 0.939400i \(-0.611383\pi\)
−0.342824 + 0.939400i \(0.611383\pi\)
\(14\) 1.23607 0.330353
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 3.47214 0.842117 0.421058 0.907034i \(-0.361659\pi\)
0.421058 + 0.907034i \(0.361659\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.61803 −0.361803
\(21\) 0 0
\(22\) 0 0
\(23\) −3.76393 −0.784834 −0.392417 0.919787i \(-0.628361\pi\)
−0.392417 + 0.919787i \(0.628361\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.52786 0.299639
\(27\) 0 0
\(28\) 3.23607 0.611559
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −2.14590 −0.368018
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −0.472136 −0.0776187 −0.0388093 0.999247i \(-0.512356\pi\)
−0.0388093 + 0.999247i \(0.512356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 2.23607 0.353553
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.32624 0.342985
\(47\) −8.23607 −1.20135 −0.600677 0.799492i \(-0.705102\pi\)
−0.600677 + 0.799492i \(0.705102\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −0.618034 −0.0874032
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 5.94427 0.816509 0.408254 0.912868i \(-0.366138\pi\)
0.408254 + 0.912868i \(0.366138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.47214 −0.597614
\(57\) 0 0
\(58\) −5.23607 −0.687529
\(59\) 6.94427 0.904067 0.452034 0.892001i \(-0.350699\pi\)
0.452034 + 0.892001i \(0.350699\pi\)
\(60\) 0 0
\(61\) 3.47214 0.444561 0.222281 0.974983i \(-0.428650\pi\)
0.222281 + 0.974983i \(0.428650\pi\)
\(62\) 4.14590 0.526530
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) −2.47214 −0.306631
\(66\) 0 0
\(67\) 11.4164 1.39474 0.697368 0.716713i \(-0.254354\pi\)
0.697368 + 0.716713i \(0.254354\pi\)
\(68\) −5.61803 −0.681287
\(69\) 0 0
\(70\) 1.23607 0.147738
\(71\) 4.47214 0.530745 0.265372 0.964146i \(-0.414505\pi\)
0.265372 + 0.964146i \(0.414505\pi\)
\(72\) 0 0
\(73\) 4.94427 0.578683 0.289342 0.957226i \(-0.406564\pi\)
0.289342 + 0.957226i \(0.406564\pi\)
\(74\) 0.291796 0.0339206
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.70820 −0.304697 −0.152348 0.988327i \(-0.548684\pi\)
−0.152348 + 0.988327i \(0.548684\pi\)
\(80\) 1.85410 0.207295
\(81\) 0 0
\(82\) −3.70820 −0.409503
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 3.47214 0.376606
\(86\) 3.70820 0.399866
\(87\) 0 0
\(88\) 0 0
\(89\) −8.47214 −0.898045 −0.449022 0.893521i \(-0.648228\pi\)
−0.449022 + 0.893521i \(0.648228\pi\)
\(90\) 0 0
\(91\) 4.94427 0.518301
\(92\) 6.09017 0.634944
\(93\) 0 0
\(94\) 5.09017 0.525011
\(95\) 0 0
\(96\) 0 0
\(97\) 2.47214 0.251007 0.125504 0.992093i \(-0.459945\pi\)
0.125504 + 0.992093i \(0.459945\pi\)
\(98\) 1.85410 0.187293
\(99\) 0 0
\(100\) −1.61803 −0.161803
\(101\) −18.9443 −1.88503 −0.942513 0.334170i \(-0.891544\pi\)
−0.942513 + 0.334170i \(0.891544\pi\)
\(102\) 0 0
\(103\) −12.4721 −1.22892 −0.614458 0.788950i \(-0.710625\pi\)
−0.614458 + 0.788950i \(0.710625\pi\)
\(104\) −5.52786 −0.542052
\(105\) 0 0
\(106\) −3.67376 −0.356827
\(107\) 6.23607 0.602863 0.301432 0.953488i \(-0.402535\pi\)
0.301432 + 0.953488i \(0.402535\pi\)
\(108\) 0 0
\(109\) 18.9443 1.81453 0.907266 0.420557i \(-0.138165\pi\)
0.907266 + 0.420557i \(0.138165\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.70820 −0.350392
\(113\) −15.4721 −1.45550 −0.727748 0.685845i \(-0.759433\pi\)
−0.727748 + 0.685845i \(0.759433\pi\)
\(114\) 0 0
\(115\) −3.76393 −0.350988
\(116\) −13.7082 −1.27277
\(117\) 0 0
\(118\) −4.29180 −0.395092
\(119\) −6.94427 −0.636580
\(120\) 0 0
\(121\) 0 0
\(122\) −2.14590 −0.194280
\(123\) 0 0
\(124\) 10.8541 0.974727
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.47214 −0.574309 −0.287155 0.957884i \(-0.592709\pi\)
−0.287155 + 0.957884i \(0.592709\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) 1.52786 0.134003
\(131\) −4.47214 −0.390732 −0.195366 0.980730i \(-0.562590\pi\)
−0.195366 + 0.980730i \(0.562590\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −7.05573 −0.609522
\(135\) 0 0
\(136\) 7.76393 0.665752
\(137\) −16.4164 −1.40255 −0.701274 0.712892i \(-0.747385\pi\)
−0.701274 + 0.712892i \(0.747385\pi\)
\(138\) 0 0
\(139\) 8.70820 0.738620 0.369310 0.929306i \(-0.379594\pi\)
0.369310 + 0.929306i \(0.379594\pi\)
\(140\) 3.23607 0.273498
\(141\) 0 0
\(142\) −2.76393 −0.231944
\(143\) 0 0
\(144\) 0 0
\(145\) 8.47214 0.703573
\(146\) −3.05573 −0.252894
\(147\) 0 0
\(148\) 0.763932 0.0627948
\(149\) −1.52786 −0.125167 −0.0625837 0.998040i \(-0.519934\pi\)
−0.0625837 + 0.998040i \(0.519934\pi\)
\(150\) 0 0
\(151\) 6.70820 0.545906 0.272953 0.962027i \(-0.412000\pi\)
0.272953 + 0.962027i \(0.412000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.70820 −0.538816
\(156\) 0 0
\(157\) 11.4164 0.911129 0.455564 0.890203i \(-0.349438\pi\)
0.455564 + 0.890203i \(0.349438\pi\)
\(158\) 1.67376 0.133157
\(159\) 0 0
\(160\) −5.61803 −0.444145
\(161\) 7.52786 0.593279
\(162\) 0 0
\(163\) −11.4164 −0.894202 −0.447101 0.894483i \(-0.647544\pi\)
−0.447101 + 0.894483i \(0.647544\pi\)
\(164\) −9.70820 −0.758083
\(165\) 0 0
\(166\) 0 0
\(167\) −11.2918 −0.873785 −0.436893 0.899514i \(-0.643921\pi\)
−0.436893 + 0.899514i \(0.643921\pi\)
\(168\) 0 0
\(169\) −6.88854 −0.529888
\(170\) −2.14590 −0.164583
\(171\) 0 0
\(172\) 9.70820 0.740244
\(173\) 5.05573 0.384380 0.192190 0.981358i \(-0.438441\pi\)
0.192190 + 0.981358i \(0.438441\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 5.23607 0.392460
\(179\) −26.4721 −1.97862 −0.989310 0.145827i \(-0.953416\pi\)
−0.989310 + 0.145827i \(0.953416\pi\)
\(180\) 0 0
\(181\) 18.9443 1.40812 0.704058 0.710142i \(-0.251369\pi\)
0.704058 + 0.710142i \(0.251369\pi\)
\(182\) −3.05573 −0.226506
\(183\) 0 0
\(184\) −8.41641 −0.620466
\(185\) −0.472136 −0.0347121
\(186\) 0 0
\(187\) 0 0
\(188\) 13.3262 0.971916
\(189\) 0 0
\(190\) 0 0
\(191\) −18.9443 −1.37076 −0.685380 0.728186i \(-0.740364\pi\)
−0.685380 + 0.728186i \(0.740364\pi\)
\(192\) 0 0
\(193\) −2.47214 −0.177948 −0.0889741 0.996034i \(-0.528359\pi\)
−0.0889741 + 0.996034i \(0.528359\pi\)
\(194\) −1.52786 −0.109694
\(195\) 0 0
\(196\) 4.85410 0.346722
\(197\) 22.9443 1.63471 0.817356 0.576133i \(-0.195439\pi\)
0.817356 + 0.576133i \(0.195439\pi\)
\(198\) 0 0
\(199\) −19.1803 −1.35966 −0.679829 0.733371i \(-0.737946\pi\)
−0.679829 + 0.733371i \(0.737946\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) 11.7082 0.823786
\(203\) −16.9443 −1.18925
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 7.70820 0.537056
\(207\) 0 0
\(208\) −4.58359 −0.317815
\(209\) 0 0
\(210\) 0 0
\(211\) −16.7082 −1.15024 −0.575120 0.818069i \(-0.695045\pi\)
−0.575120 + 0.818069i \(0.695045\pi\)
\(212\) −9.61803 −0.660569
\(213\) 0 0
\(214\) −3.85410 −0.263461
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 13.4164 0.910765
\(218\) −11.7082 −0.792980
\(219\) 0 0
\(220\) 0 0
\(221\) −8.58359 −0.577395
\(222\) 0 0
\(223\) −15.5279 −1.03982 −0.519911 0.854220i \(-0.674035\pi\)
−0.519911 + 0.854220i \(0.674035\pi\)
\(224\) 11.2361 0.750741
\(225\) 0 0
\(226\) 9.56231 0.636075
\(227\) 15.1803 1.00755 0.503777 0.863834i \(-0.331943\pi\)
0.503777 + 0.863834i \(0.331943\pi\)
\(228\) 0 0
\(229\) 0.527864 0.0348822 0.0174411 0.999848i \(-0.494448\pi\)
0.0174411 + 0.999848i \(0.494448\pi\)
\(230\) 2.32624 0.153388
\(231\) 0 0
\(232\) 18.9443 1.24375
\(233\) −27.3607 −1.79246 −0.896229 0.443592i \(-0.853704\pi\)
−0.896229 + 0.443592i \(0.853704\pi\)
\(234\) 0 0
\(235\) −8.23607 −0.537262
\(236\) −11.2361 −0.731406
\(237\) 0 0
\(238\) 4.29180 0.278196
\(239\) 16.3607 1.05828 0.529142 0.848533i \(-0.322514\pi\)
0.529142 + 0.848533i \(0.322514\pi\)
\(240\) 0 0
\(241\) 15.0000 0.966235 0.483117 0.875556i \(-0.339504\pi\)
0.483117 + 0.875556i \(0.339504\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −5.61803 −0.359658
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) −15.0000 −0.952501
\(249\) 0 0
\(250\) −0.618034 −0.0390879
\(251\) −19.4164 −1.22555 −0.612776 0.790256i \(-0.709947\pi\)
−0.612776 + 0.790256i \(0.709947\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) 8.52786 0.531954 0.265977 0.963979i \(-0.414306\pi\)
0.265977 + 0.963979i \(0.414306\pi\)
\(258\) 0 0
\(259\) 0.944272 0.0586742
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) 2.76393 0.170756
\(263\) 0.708204 0.0436697 0.0218349 0.999762i \(-0.493049\pi\)
0.0218349 + 0.999762i \(0.493049\pi\)
\(264\) 0 0
\(265\) 5.94427 0.365154
\(266\) 0 0
\(267\) 0 0
\(268\) −18.4721 −1.12837
\(269\) 17.8885 1.09068 0.545342 0.838214i \(-0.316400\pi\)
0.545342 + 0.838214i \(0.316400\pi\)
\(270\) 0 0
\(271\) −1.76393 −0.107151 −0.0535756 0.998564i \(-0.517062\pi\)
−0.0535756 + 0.998564i \(0.517062\pi\)
\(272\) 6.43769 0.390343
\(273\) 0 0
\(274\) 10.1459 0.612936
\(275\) 0 0
\(276\) 0 0
\(277\) 6.94427 0.417241 0.208620 0.977997i \(-0.433103\pi\)
0.208620 + 0.977997i \(0.433103\pi\)
\(278\) −5.38197 −0.322789
\(279\) 0 0
\(280\) −4.47214 −0.267261
\(281\) −16.3607 −0.975996 −0.487998 0.872845i \(-0.662273\pi\)
−0.487998 + 0.872845i \(0.662273\pi\)
\(282\) 0 0
\(283\) −30.3607 −1.80476 −0.902378 0.430946i \(-0.858180\pi\)
−0.902378 + 0.430946i \(0.858180\pi\)
\(284\) −7.23607 −0.429382
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −4.94427 −0.290840
\(290\) −5.23607 −0.307472
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) −5.94427 −0.347268 −0.173634 0.984810i \(-0.555551\pi\)
−0.173634 + 0.984810i \(0.555551\pi\)
\(294\) 0 0
\(295\) 6.94427 0.404311
\(296\) −1.05573 −0.0613629
\(297\) 0 0
\(298\) 0.944272 0.0547002
\(299\) 9.30495 0.538119
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −4.14590 −0.238570
\(303\) 0 0
\(304\) 0 0
\(305\) 3.47214 0.198814
\(306\) 0 0
\(307\) −14.3607 −0.819607 −0.409804 0.912174i \(-0.634403\pi\)
−0.409804 + 0.912174i \(0.634403\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.14590 0.235471
\(311\) −26.4721 −1.50110 −0.750549 0.660815i \(-0.770211\pi\)
−0.750549 + 0.660815i \(0.770211\pi\)
\(312\) 0 0
\(313\) −3.52786 −0.199407 −0.0997033 0.995017i \(-0.531789\pi\)
−0.0997033 + 0.995017i \(0.531789\pi\)
\(314\) −7.05573 −0.398178
\(315\) 0 0
\(316\) 4.38197 0.246505
\(317\) −10.0557 −0.564786 −0.282393 0.959299i \(-0.591128\pi\)
−0.282393 + 0.959299i \(0.591128\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.236068 −0.0131966
\(321\) 0 0
\(322\) −4.65248 −0.259272
\(323\) 0 0
\(324\) 0 0
\(325\) −2.47214 −0.137129
\(326\) 7.05573 0.390781
\(327\) 0 0
\(328\) 13.4164 0.740797
\(329\) 16.4721 0.908138
\(330\) 0 0
\(331\) 19.7639 1.08632 0.543162 0.839628i \(-0.317227\pi\)
0.543162 + 0.839628i \(0.317227\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 6.97871 0.381858
\(335\) 11.4164 0.623745
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 4.25735 0.231570
\(339\) 0 0
\(340\) −5.61803 −0.304681
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −13.4164 −0.723364
\(345\) 0 0
\(346\) −3.12461 −0.167980
\(347\) −8.12461 −0.436152 −0.218076 0.975932i \(-0.569978\pi\)
−0.218076 + 0.975932i \(0.569978\pi\)
\(348\) 0 0
\(349\) −3.58359 −0.191825 −0.0959126 0.995390i \(-0.530577\pi\)
−0.0959126 + 0.995390i \(0.530577\pi\)
\(350\) 1.23607 0.0660706
\(351\) 0 0
\(352\) 0 0
\(353\) −20.5279 −1.09259 −0.546294 0.837594i \(-0.683962\pi\)
−0.546294 + 0.837594i \(0.683962\pi\)
\(354\) 0 0
\(355\) 4.47214 0.237356
\(356\) 13.7082 0.726533
\(357\) 0 0
\(358\) 16.3607 0.864689
\(359\) 19.5279 1.03064 0.515321 0.856997i \(-0.327673\pi\)
0.515321 + 0.856997i \(0.327673\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) −11.7082 −0.615370
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 4.94427 0.258795
\(366\) 0 0
\(367\) 31.4164 1.63992 0.819962 0.572419i \(-0.193995\pi\)
0.819962 + 0.572419i \(0.193995\pi\)
\(368\) −6.97871 −0.363791
\(369\) 0 0
\(370\) 0.291796 0.0151698
\(371\) −11.8885 −0.617222
\(372\) 0 0
\(373\) −37.8885 −1.96179 −0.980897 0.194527i \(-0.937683\pi\)
−0.980897 + 0.194527i \(0.937683\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −18.4164 −0.949754
\(377\) −20.9443 −1.07868
\(378\) 0 0
\(379\) 22.5967 1.16072 0.580358 0.814361i \(-0.302912\pi\)
0.580358 + 0.814361i \(0.302912\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 11.7082 0.599044
\(383\) −7.05573 −0.360531 −0.180265 0.983618i \(-0.557696\pi\)
−0.180265 + 0.983618i \(0.557696\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.52786 0.0777662
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) −8.47214 −0.429554 −0.214777 0.976663i \(-0.568903\pi\)
−0.214777 + 0.976663i \(0.568903\pi\)
\(390\) 0 0
\(391\) −13.0689 −0.660922
\(392\) −6.70820 −0.338815
\(393\) 0 0
\(394\) −14.1803 −0.714395
\(395\) −2.70820 −0.136265
\(396\) 0 0
\(397\) −0.944272 −0.0473916 −0.0236958 0.999719i \(-0.507543\pi\)
−0.0236958 + 0.999719i \(0.507543\pi\)
\(398\) 11.8541 0.594192
\(399\) 0 0
\(400\) 1.85410 0.0927051
\(401\) −17.8885 −0.893311 −0.446656 0.894706i \(-0.647385\pi\)
−0.446656 + 0.894706i \(0.647385\pi\)
\(402\) 0 0
\(403\) 16.5836 0.826088
\(404\) 30.6525 1.52502
\(405\) 0 0
\(406\) 10.4721 0.519723
\(407\) 0 0
\(408\) 0 0
\(409\) −3.00000 −0.148340 −0.0741702 0.997246i \(-0.523631\pi\)
−0.0741702 + 0.997246i \(0.523631\pi\)
\(410\) −3.70820 −0.183135
\(411\) 0 0
\(412\) 20.1803 0.994214
\(413\) −13.8885 −0.683411
\(414\) 0 0
\(415\) 0 0
\(416\) 13.8885 0.680942
\(417\) 0 0
\(418\) 0 0
\(419\) −39.7771 −1.94324 −0.971619 0.236552i \(-0.923983\pi\)
−0.971619 + 0.236552i \(0.923983\pi\)
\(420\) 0 0
\(421\) 7.58359 0.369602 0.184801 0.982776i \(-0.440836\pi\)
0.184801 + 0.982776i \(0.440836\pi\)
\(422\) 10.3262 0.502673
\(423\) 0 0
\(424\) 13.2918 0.645507
\(425\) 3.47214 0.168423
\(426\) 0 0
\(427\) −6.94427 −0.336057
\(428\) −10.0902 −0.487727
\(429\) 0 0
\(430\) 3.70820 0.178825
\(431\) −11.8885 −0.572651 −0.286326 0.958132i \(-0.592434\pi\)
−0.286326 + 0.958132i \(0.592434\pi\)
\(432\) 0 0
\(433\) −36.4721 −1.75274 −0.876369 0.481639i \(-0.840042\pi\)
−0.876369 + 0.481639i \(0.840042\pi\)
\(434\) −8.29180 −0.398019
\(435\) 0 0
\(436\) −30.6525 −1.46799
\(437\) 0 0
\(438\) 0 0
\(439\) 2.70820 0.129256 0.0646278 0.997909i \(-0.479414\pi\)
0.0646278 + 0.997909i \(0.479414\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.30495 0.252331
\(443\) −3.05573 −0.145182 −0.0725910 0.997362i \(-0.523127\pi\)
−0.0725910 + 0.997362i \(0.523127\pi\)
\(444\) 0 0
\(445\) −8.47214 −0.401618
\(446\) 9.59675 0.454419
\(447\) 0 0
\(448\) 0.472136 0.0223063
\(449\) −17.8885 −0.844213 −0.422106 0.906546i \(-0.638709\pi\)
−0.422106 + 0.906546i \(0.638709\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 25.0344 1.17752
\(453\) 0 0
\(454\) −9.38197 −0.440317
\(455\) 4.94427 0.231791
\(456\) 0 0
\(457\) −16.8328 −0.787406 −0.393703 0.919238i \(-0.628806\pi\)
−0.393703 + 0.919238i \(0.628806\pi\)
\(458\) −0.326238 −0.0152441
\(459\) 0 0
\(460\) 6.09017 0.283956
\(461\) 5.52786 0.257458 0.128729 0.991680i \(-0.458910\pi\)
0.128729 + 0.991680i \(0.458910\pi\)
\(462\) 0 0
\(463\) −35.3050 −1.64076 −0.820380 0.571819i \(-0.806238\pi\)
−0.820380 + 0.571819i \(0.806238\pi\)
\(464\) 15.7082 0.729235
\(465\) 0 0
\(466\) 16.9098 0.783333
\(467\) 27.6525 1.27960 0.639802 0.768540i \(-0.279016\pi\)
0.639802 + 0.768540i \(0.279016\pi\)
\(468\) 0 0
\(469\) −22.8328 −1.05432
\(470\) 5.09017 0.234792
\(471\) 0 0
\(472\) 15.5279 0.714728
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 11.2361 0.515004
\(477\) 0 0
\(478\) −10.1115 −0.462487
\(479\) 19.4164 0.887158 0.443579 0.896235i \(-0.353709\pi\)
0.443579 + 0.896235i \(0.353709\pi\)
\(480\) 0 0
\(481\) 1.16718 0.0532190
\(482\) −9.27051 −0.422260
\(483\) 0 0
\(484\) 0 0
\(485\) 2.47214 0.112254
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 7.76393 0.351457
\(489\) 0 0
\(490\) 1.85410 0.0837598
\(491\) −7.52786 −0.339728 −0.169864 0.985468i \(-0.554333\pi\)
−0.169864 + 0.985468i \(0.554333\pi\)
\(492\) 0 0
\(493\) 29.4164 1.32485
\(494\) 0 0
\(495\) 0 0
\(496\) −12.4377 −0.558469
\(497\) −8.94427 −0.401205
\(498\) 0 0
\(499\) −4.94427 −0.221336 −0.110668 0.993857i \(-0.535299\pi\)
−0.110668 + 0.993857i \(0.535299\pi\)
\(500\) −1.61803 −0.0723607
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −27.7639 −1.23793 −0.618966 0.785418i \(-0.712448\pi\)
−0.618966 + 0.785418i \(0.712448\pi\)
\(504\) 0 0
\(505\) −18.9443 −0.843009
\(506\) 0 0
\(507\) 0 0
\(508\) 10.4721 0.464626
\(509\) 30.3607 1.34571 0.672857 0.739773i \(-0.265067\pi\)
0.672857 + 0.739773i \(0.265067\pi\)
\(510\) 0 0
\(511\) −9.88854 −0.437443
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) −5.27051 −0.232472
\(515\) −12.4721 −0.549588
\(516\) 0 0
\(517\) 0 0
\(518\) −0.583592 −0.0256416
\(519\) 0 0
\(520\) −5.52786 −0.242413
\(521\) 26.4721 1.15977 0.579883 0.814700i \(-0.303098\pi\)
0.579883 + 0.814700i \(0.303098\pi\)
\(522\) 0 0
\(523\) −6.47214 −0.283007 −0.141503 0.989938i \(-0.545194\pi\)
−0.141503 + 0.989938i \(0.545194\pi\)
\(524\) 7.23607 0.316109
\(525\) 0 0
\(526\) −0.437694 −0.0190844
\(527\) −23.2918 −1.01461
\(528\) 0 0
\(529\) −8.83282 −0.384035
\(530\) −3.67376 −0.159578
\(531\) 0 0
\(532\) 0 0
\(533\) −14.8328 −0.642481
\(534\) 0 0
\(535\) 6.23607 0.269609
\(536\) 25.5279 1.10264
\(537\) 0 0
\(538\) −11.0557 −0.476646
\(539\) 0 0
\(540\) 0 0
\(541\) −32.8328 −1.41159 −0.705797 0.708415i \(-0.749411\pi\)
−0.705797 + 0.708415i \(0.749411\pi\)
\(542\) 1.09017 0.0468268
\(543\) 0 0
\(544\) −19.5066 −0.836338
\(545\) 18.9443 0.811483
\(546\) 0 0
\(547\) 39.4164 1.68532 0.842662 0.538443i \(-0.180987\pi\)
0.842662 + 0.538443i \(0.180987\pi\)
\(548\) 26.5623 1.13469
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.41641 0.230329
\(554\) −4.29180 −0.182341
\(555\) 0 0
\(556\) −14.0902 −0.597556
\(557\) 0.0557281 0.00236127 0.00118064 0.999999i \(-0.499624\pi\)
0.00118064 + 0.999999i \(0.499624\pi\)
\(558\) 0 0
\(559\) 14.8328 0.627361
\(560\) −3.70820 −0.156700
\(561\) 0 0
\(562\) 10.1115 0.426526
\(563\) −35.7771 −1.50782 −0.753912 0.656975i \(-0.771836\pi\)
−0.753912 + 0.656975i \(0.771836\pi\)
\(564\) 0 0
\(565\) −15.4721 −0.650918
\(566\) 18.7639 0.788707
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) 2.36068 0.0989648 0.0494824 0.998775i \(-0.484243\pi\)
0.0494824 + 0.998775i \(0.484243\pi\)
\(570\) 0 0
\(571\) 36.7082 1.53619 0.768095 0.640336i \(-0.221205\pi\)
0.768095 + 0.640336i \(0.221205\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 7.41641 0.309555
\(575\) −3.76393 −0.156967
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 3.05573 0.127102
\(579\) 0 0
\(580\) −13.7082 −0.569202
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 11.0557 0.457489
\(585\) 0 0
\(586\) 3.67376 0.151762
\(587\) −39.6525 −1.63663 −0.818316 0.574768i \(-0.805092\pi\)
−0.818316 + 0.574768i \(0.805092\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −4.29180 −0.176690
\(591\) 0 0
\(592\) −0.875388 −0.0359782
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) −6.94427 −0.284687
\(596\) 2.47214 0.101263
\(597\) 0 0
\(598\) −5.75078 −0.235167
\(599\) −0.111456 −0.00455398 −0.00227699 0.999997i \(-0.500725\pi\)
−0.00227699 + 0.999997i \(0.500725\pi\)
\(600\) 0 0
\(601\) −36.8328 −1.50244 −0.751221 0.660051i \(-0.770535\pi\)
−0.751221 + 0.660051i \(0.770535\pi\)
\(602\) −7.41641 −0.302270
\(603\) 0 0
\(604\) −10.8541 −0.441647
\(605\) 0 0
\(606\) 0 0
\(607\) −6.47214 −0.262696 −0.131348 0.991336i \(-0.541931\pi\)
−0.131348 + 0.991336i \(0.541931\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −2.14590 −0.0868849
\(611\) 20.3607 0.823705
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 8.87539 0.358182
\(615\) 0 0
\(616\) 0 0
\(617\) −26.9443 −1.08474 −0.542368 0.840141i \(-0.682472\pi\)
−0.542368 + 0.840141i \(0.682472\pi\)
\(618\) 0 0
\(619\) −4.94427 −0.198727 −0.0993635 0.995051i \(-0.531681\pi\)
−0.0993635 + 0.995051i \(0.531681\pi\)
\(620\) 10.8541 0.435911
\(621\) 0 0
\(622\) 16.3607 0.656003
\(623\) 16.9443 0.678858
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.18034 0.0871439
\(627\) 0 0
\(628\) −18.4721 −0.737118
\(629\) −1.63932 −0.0653640
\(630\) 0 0
\(631\) 42.7082 1.70019 0.850093 0.526632i \(-0.176545\pi\)
0.850093 + 0.526632i \(0.176545\pi\)
\(632\) −6.05573 −0.240884
\(633\) 0 0
\(634\) 6.21478 0.246821
\(635\) −6.47214 −0.256839
\(636\) 0 0
\(637\) 7.41641 0.293849
\(638\) 0 0
\(639\) 0 0
\(640\) 11.3820 0.449912
\(641\) 4.58359 0.181041 0.0905205 0.995895i \(-0.471147\pi\)
0.0905205 + 0.995895i \(0.471147\pi\)
\(642\) 0 0
\(643\) −37.4164 −1.47556 −0.737780 0.675042i \(-0.764126\pi\)
−0.737780 + 0.675042i \(0.764126\pi\)
\(644\) −12.1803 −0.479973
\(645\) 0 0
\(646\) 0 0
\(647\) 25.6525 1.00850 0.504251 0.863557i \(-0.331768\pi\)
0.504251 + 0.863557i \(0.331768\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.52786 0.0599278
\(651\) 0 0
\(652\) 18.4721 0.723425
\(653\) −23.8885 −0.934831 −0.467415 0.884038i \(-0.654815\pi\)
−0.467415 + 0.884038i \(0.654815\pi\)
\(654\) 0 0
\(655\) −4.47214 −0.174741
\(656\) 11.1246 0.434343
\(657\) 0 0
\(658\) −10.1803 −0.396871
\(659\) −32.8328 −1.27898 −0.639492 0.768797i \(-0.720855\pi\)
−0.639492 + 0.768797i \(0.720855\pi\)
\(660\) 0 0
\(661\) 1.05573 0.0410631 0.0205315 0.999789i \(-0.493464\pi\)
0.0205315 + 0.999789i \(0.493464\pi\)
\(662\) −12.2148 −0.474741
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −31.8885 −1.23473
\(668\) 18.2705 0.706907
\(669\) 0 0
\(670\) −7.05573 −0.272587
\(671\) 0 0
\(672\) 0 0
\(673\) 45.4164 1.75067 0.875337 0.483513i \(-0.160640\pi\)
0.875337 + 0.483513i \(0.160640\pi\)
\(674\) 14.8328 0.571339
\(675\) 0 0
\(676\) 11.1459 0.428688
\(677\) 15.8885 0.610646 0.305323 0.952249i \(-0.401236\pi\)
0.305323 + 0.952249i \(0.401236\pi\)
\(678\) 0 0
\(679\) −4.94427 −0.189744
\(680\) 7.76393 0.297733
\(681\) 0 0
\(682\) 0 0
\(683\) 35.7771 1.36897 0.684486 0.729026i \(-0.260027\pi\)
0.684486 + 0.729026i \(0.260027\pi\)
\(684\) 0 0
\(685\) −16.4164 −0.627239
\(686\) −12.3607 −0.471933
\(687\) 0 0
\(688\) −11.1246 −0.424122
\(689\) −14.6950 −0.559837
\(690\) 0 0
\(691\) 14.5967 0.555286 0.277643 0.960684i \(-0.410447\pi\)
0.277643 + 0.960684i \(0.410447\pi\)
\(692\) −8.18034 −0.310970
\(693\) 0 0
\(694\) 5.02129 0.190605
\(695\) 8.70820 0.330321
\(696\) 0 0
\(697\) 20.8328 0.789099
\(698\) 2.21478 0.0838307
\(699\) 0 0
\(700\) 3.23607 0.122312
\(701\) −44.3607 −1.67548 −0.837740 0.546070i \(-0.816123\pi\)
−0.837740 + 0.546070i \(0.816123\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 12.6869 0.477478
\(707\) 37.8885 1.42495
\(708\) 0 0
\(709\) 33.4721 1.25707 0.628536 0.777780i \(-0.283654\pi\)
0.628536 + 0.777780i \(0.283654\pi\)
\(710\) −2.76393 −0.103729
\(711\) 0 0
\(712\) −18.9443 −0.709967
\(713\) 25.2492 0.945591
\(714\) 0 0
\(715\) 0 0
\(716\) 42.8328 1.60074
\(717\) 0 0
\(718\) −12.0689 −0.450407
\(719\) 19.3050 0.719953 0.359977 0.932961i \(-0.382785\pi\)
0.359977 + 0.932961i \(0.382785\pi\)
\(720\) 0 0
\(721\) 24.9443 0.928973
\(722\) 11.7426 0.437016
\(723\) 0 0
\(724\) −30.6525 −1.13919
\(725\) 8.47214 0.314647
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 11.0557 0.409753
\(729\) 0 0
\(730\) −3.05573 −0.113098
\(731\) −20.8328 −0.770530
\(732\) 0 0
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) −19.4164 −0.716673
\(735\) 0 0
\(736\) 21.1459 0.779448
\(737\) 0 0
\(738\) 0 0
\(739\) −15.2918 −0.562518 −0.281259 0.959632i \(-0.590752\pi\)
−0.281259 + 0.959632i \(0.590752\pi\)
\(740\) 0.763932 0.0280827
\(741\) 0 0
\(742\) 7.34752 0.269736
\(743\) 46.5967 1.70947 0.854734 0.519066i \(-0.173720\pi\)
0.854734 + 0.519066i \(0.173720\pi\)
\(744\) 0 0
\(745\) −1.52786 −0.0559766
\(746\) 23.4164 0.857336
\(747\) 0 0
\(748\) 0 0
\(749\) −12.4721 −0.455722
\(750\) 0 0
\(751\) 41.5410 1.51585 0.757927 0.652340i \(-0.226212\pi\)
0.757927 + 0.652340i \(0.226212\pi\)
\(752\) −15.2705 −0.556858
\(753\) 0 0
\(754\) 12.9443 0.471403
\(755\) 6.70820 0.244137
\(756\) 0 0
\(757\) 10.9443 0.397776 0.198888 0.980022i \(-0.436267\pi\)
0.198888 + 0.980022i \(0.436267\pi\)
\(758\) −13.9656 −0.507252
\(759\) 0 0
\(760\) 0 0
\(761\) −4.58359 −0.166155 −0.0830775 0.996543i \(-0.526475\pi\)
−0.0830775 + 0.996543i \(0.526475\pi\)
\(762\) 0 0
\(763\) −37.8885 −1.37166
\(764\) 30.6525 1.10897
\(765\) 0 0
\(766\) 4.36068 0.157558
\(767\) −17.1672 −0.619871
\(768\) 0 0
\(769\) −11.0000 −0.396670 −0.198335 0.980134i \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 20.0557 0.721354 0.360677 0.932691i \(-0.382546\pi\)
0.360677 + 0.932691i \(0.382546\pi\)
\(774\) 0 0
\(775\) −6.70820 −0.240966
\(776\) 5.52786 0.198439
\(777\) 0 0
\(778\) 5.23607 0.187722
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 8.07701 0.288833
\(783\) 0 0
\(784\) −5.56231 −0.198654
\(785\) 11.4164 0.407469
\(786\) 0 0
\(787\) 6.58359 0.234680 0.117340 0.993092i \(-0.462563\pi\)
0.117340 + 0.993092i \(0.462563\pi\)
\(788\) −37.1246 −1.32251
\(789\) 0 0
\(790\) 1.67376 0.0595498
\(791\) 30.9443 1.10025
\(792\) 0 0
\(793\) −8.58359 −0.304812
\(794\) 0.583592 0.0207109
\(795\) 0 0
\(796\) 31.0344 1.09999
\(797\) −1.05573 −0.0373958 −0.0186979 0.999825i \(-0.505952\pi\)
−0.0186979 + 0.999825i \(0.505952\pi\)
\(798\) 0 0
\(799\) −28.5967 −1.01168
\(800\) −5.61803 −0.198627
\(801\) 0 0
\(802\) 11.0557 0.390391
\(803\) 0 0
\(804\) 0 0
\(805\) 7.52786 0.265322
\(806\) −10.2492 −0.361014
\(807\) 0 0
\(808\) −42.3607 −1.49024
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 46.1246 1.61965 0.809827 0.586669i \(-0.199561\pi\)
0.809827 + 0.586669i \(0.199561\pi\)
\(812\) 27.4164 0.962127
\(813\) 0 0
\(814\) 0 0
\(815\) −11.4164 −0.399899
\(816\) 0 0
\(817\) 0 0
\(818\) 1.85410 0.0648272
\(819\) 0 0
\(820\) −9.70820 −0.339025
\(821\) −23.8885 −0.833716 −0.416858 0.908972i \(-0.636869\pi\)
−0.416858 + 0.908972i \(0.636869\pi\)
\(822\) 0 0
\(823\) 28.0000 0.976019 0.488009 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) −27.8885 −0.971543
\(825\) 0 0
\(826\) 8.58359 0.298661
\(827\) 3.05573 0.106258 0.0531290 0.998588i \(-0.483081\pi\)
0.0531290 + 0.998588i \(0.483081\pi\)
\(828\) 0 0
\(829\) 28.4164 0.986943 0.493471 0.869762i \(-0.335728\pi\)
0.493471 + 0.869762i \(0.335728\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.583592 0.0202324
\(833\) −10.4164 −0.360907
\(834\) 0 0
\(835\) −11.2918 −0.390769
\(836\) 0 0
\(837\) 0 0
\(838\) 24.5836 0.849226
\(839\) −6.36068 −0.219595 −0.109798 0.993954i \(-0.535020\pi\)
−0.109798 + 0.993954i \(0.535020\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) −4.68692 −0.161522
\(843\) 0 0
\(844\) 27.0344 0.930564
\(845\) −6.88854 −0.236973
\(846\) 0 0
\(847\) 0 0
\(848\) 11.0213 0.378473
\(849\) 0 0
\(850\) −2.14590 −0.0736037
\(851\) 1.77709 0.0609178
\(852\) 0 0
\(853\) −32.8328 −1.12417 −0.562087 0.827078i \(-0.690001\pi\)
−0.562087 + 0.827078i \(0.690001\pi\)
\(854\) 4.29180 0.146862
\(855\) 0 0
\(856\) 13.9443 0.476605
\(857\) 31.4721 1.07507 0.537534 0.843242i \(-0.319356\pi\)
0.537534 + 0.843242i \(0.319356\pi\)
\(858\) 0 0
\(859\) 31.7771 1.08422 0.542110 0.840307i \(-0.317626\pi\)
0.542110 + 0.840307i \(0.317626\pi\)
\(860\) 9.70820 0.331047
\(861\) 0 0
\(862\) 7.34752 0.250258
\(863\) 27.0557 0.920988 0.460494 0.887663i \(-0.347672\pi\)
0.460494 + 0.887663i \(0.347672\pi\)
\(864\) 0 0
\(865\) 5.05573 0.171900
\(866\) 22.5410 0.765975
\(867\) 0 0
\(868\) −21.7082 −0.736824
\(869\) 0 0
\(870\) 0 0
\(871\) −28.2229 −0.956297
\(872\) 42.3607 1.43451
\(873\) 0 0
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −26.8328 −0.906080 −0.453040 0.891490i \(-0.649660\pi\)
−0.453040 + 0.891490i \(0.649660\pi\)
\(878\) −1.67376 −0.0564867
\(879\) 0 0
\(880\) 0 0
\(881\) 2.94427 0.0991950 0.0495975 0.998769i \(-0.484206\pi\)
0.0495975 + 0.998769i \(0.484206\pi\)
\(882\) 0 0
\(883\) −36.3607 −1.22363 −0.611817 0.790999i \(-0.709561\pi\)
−0.611817 + 0.790999i \(0.709561\pi\)
\(884\) 13.8885 0.467122
\(885\) 0 0
\(886\) 1.88854 0.0634469
\(887\) −39.7771 −1.33558 −0.667792 0.744348i \(-0.732760\pi\)
−0.667792 + 0.744348i \(0.732760\pi\)
\(888\) 0 0
\(889\) 12.9443 0.434137
\(890\) 5.23607 0.175513
\(891\) 0 0
\(892\) 25.1246 0.841234
\(893\) 0 0
\(894\) 0 0
\(895\) −26.4721 −0.884866
\(896\) −22.7639 −0.760490
\(897\) 0 0
\(898\) 11.0557 0.368934
\(899\) −56.8328 −1.89548
\(900\) 0 0
\(901\) 20.6393 0.687595
\(902\) 0 0
\(903\) 0 0
\(904\) −34.5967 −1.15067
\(905\) 18.9443 0.629729
\(906\) 0 0
\(907\) 14.8328 0.492516 0.246258 0.969204i \(-0.420799\pi\)
0.246258 + 0.969204i \(0.420799\pi\)
\(908\) −24.5623 −0.815129
\(909\) 0 0
\(910\) −3.05573 −0.101296
\(911\) 20.8328 0.690222 0.345111 0.938562i \(-0.387841\pi\)
0.345111 + 0.938562i \(0.387841\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 10.4033 0.344109
\(915\) 0 0
\(916\) −0.854102 −0.0282203
\(917\) 8.94427 0.295366
\(918\) 0 0
\(919\) −34.8328 −1.14903 −0.574514 0.818495i \(-0.694809\pi\)
−0.574514 + 0.818495i \(0.694809\pi\)
\(920\) −8.41641 −0.277481
\(921\) 0 0
\(922\) −3.41641 −0.112513
\(923\) −11.0557 −0.363904
\(924\) 0 0
\(925\) −0.472136 −0.0155237
\(926\) 21.8197 0.717039
\(927\) 0 0
\(928\) −47.5967 −1.56244
\(929\) 50.7214 1.66411 0.832057 0.554690i \(-0.187163\pi\)
0.832057 + 0.554690i \(0.187163\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 44.2705 1.45013
\(933\) 0 0
\(934\) −17.0902 −0.559207
\(935\) 0 0
\(936\) 0 0
\(937\) −58.2492 −1.90292 −0.951460 0.307774i \(-0.900416\pi\)
−0.951460 + 0.307774i \(0.900416\pi\)
\(938\) 14.1115 0.460755
\(939\) 0 0
\(940\) 13.3262 0.434654
\(941\) 58.4721 1.90614 0.953069 0.302754i \(-0.0979062\pi\)
0.953069 + 0.302754i \(0.0979062\pi\)
\(942\) 0 0
\(943\) −22.5836 −0.735423
\(944\) 12.8754 0.419058
\(945\) 0 0
\(946\) 0 0
\(947\) −16.3475 −0.531223 −0.265612 0.964080i \(-0.585574\pi\)
−0.265612 + 0.964080i \(0.585574\pi\)
\(948\) 0 0
\(949\) −12.2229 −0.396773
\(950\) 0 0
\(951\) 0 0
\(952\) −15.5279 −0.503261
\(953\) −19.8885 −0.644253 −0.322127 0.946697i \(-0.604398\pi\)
−0.322127 + 0.946697i \(0.604398\pi\)
\(954\) 0 0
\(955\) −18.9443 −0.613022
\(956\) −26.4721 −0.856170
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) 32.8328 1.06023
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) −0.721360 −0.0232576
\(963\) 0 0
\(964\) −24.2705 −0.781700
\(965\) −2.47214 −0.0795809
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1.52786 −0.0490568
\(971\) 9.30495 0.298610 0.149305 0.988791i \(-0.452296\pi\)
0.149305 + 0.988791i \(0.452296\pi\)
\(972\) 0 0
\(973\) −17.4164 −0.558344
\(974\) −1.23607 −0.0396062
\(975\) 0 0
\(976\) 6.43769 0.206066
\(977\) −12.5279 −0.400802 −0.200401 0.979714i \(-0.564224\pi\)
−0.200401 + 0.979714i \(0.564224\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 4.85410 0.155059
\(981\) 0 0
\(982\) 4.65248 0.148466
\(983\) −57.4296 −1.83172 −0.915859 0.401499i \(-0.868489\pi\)
−0.915859 + 0.401499i \(0.868489\pi\)
\(984\) 0 0
\(985\) 22.9443 0.731065
\(986\) −18.1803 −0.578980
\(987\) 0 0
\(988\) 0 0
\(989\) 22.5836 0.718116
\(990\) 0 0
\(991\) 5.29180 0.168099 0.0840497 0.996462i \(-0.473215\pi\)
0.0840497 + 0.996462i \(0.473215\pi\)
\(992\) 37.6869 1.19656
\(993\) 0 0
\(994\) 5.52786 0.175333
\(995\) −19.1803 −0.608058
\(996\) 0 0
\(997\) −14.5836 −0.461867 −0.230933 0.972970i \(-0.574178\pi\)
−0.230933 + 0.972970i \(0.574178\pi\)
\(998\) 3.05573 0.0967274
\(999\) 0 0
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 5445.2.a.w.1.1 yes 2
3.2 odd 2 5445.2.a.n.1.2 2
11.10 odd 2 5445.2.a.p.1.2 yes 2
33.32 even 2 5445.2.a.v.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5445.2.a.n.1.2 2 3.2 odd 2
5445.2.a.p.1.2 yes 2 11.10 odd 2
5445.2.a.v.1.1 yes 2 33.32 even 2
5445.2.a.w.1.1 yes 2 1.1 even 1 trivial