Properties

Label 3626.2.a.a.1.1
Level $3626$
Weight $2$
Character 3626.1
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9537557729\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 3626.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} +2.30278 q^{5} +3.30278 q^{6} -1.00000 q^{8} +7.90833 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.30278 q^{3} +1.00000 q^{4} +2.30278 q^{5} +3.30278 q^{6} -1.00000 q^{8} +7.90833 q^{9} -2.30278 q^{10} -2.30278 q^{11} -3.30278 q^{12} -1.30278 q^{13} -7.60555 q^{15} +1.00000 q^{16} +6.00000 q^{17} -7.90833 q^{18} -2.00000 q^{19} +2.30278 q^{20} +2.30278 q^{22} +3.90833 q^{23} +3.30278 q^{24} +0.302776 q^{25} +1.30278 q^{26} -16.2111 q^{27} -3.90833 q^{29} +7.60555 q^{30} +0.302776 q^{31} -1.00000 q^{32} +7.60555 q^{33} -6.00000 q^{34} +7.90833 q^{36} +1.00000 q^{37} +2.00000 q^{38} +4.30278 q^{39} -2.30278 q^{40} -9.90833 q^{41} +0.605551 q^{43} -2.30278 q^{44} +18.2111 q^{45} -3.90833 q^{46} -4.60555 q^{47} -3.30278 q^{48} -0.302776 q^{50} -19.8167 q^{51} -1.30278 q^{52} -6.00000 q^{53} +16.2111 q^{54} -5.30278 q^{55} +6.60555 q^{57} +3.90833 q^{58} -10.6056 q^{59} -7.60555 q^{60} -7.51388 q^{61} -0.302776 q^{62} +1.00000 q^{64} -3.00000 q^{65} -7.60555 q^{66} -3.51388 q^{67} +6.00000 q^{68} -12.9083 q^{69} +6.00000 q^{71} -7.90833 q^{72} +12.3028 q^{73} -1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -4.30278 q^{78} +9.11943 q^{79} +2.30278 q^{80} +29.8167 q^{81} +9.90833 q^{82} -2.78890 q^{83} +13.8167 q^{85} -0.605551 q^{86} +12.9083 q^{87} +2.30278 q^{88} +9.21110 q^{89} -18.2111 q^{90} +3.90833 q^{92} -1.00000 q^{93} +4.60555 q^{94} -4.60555 q^{95} +3.30278 q^{96} +16.4222 q^{97} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9} - q^{10} - q^{11} - 3 q^{12} + q^{13} - 8 q^{15} + 2 q^{16} + 12 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + q^{22} - 3 q^{23} + 3 q^{24} - 3 q^{25} - q^{26} - 18 q^{27} + 3 q^{29} + 8 q^{30} - 3 q^{31} - 2 q^{32} + 8 q^{33} - 12 q^{34} + 5 q^{36} + 2 q^{37} + 4 q^{38} + 5 q^{39} - q^{40} - 9 q^{41} - 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 3 q^{50} - 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 6 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} + 3 q^{61} + 3 q^{62} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} + 12 q^{71} - 5 q^{72} + 21 q^{73} - 2 q^{74} - 2 q^{75} - 4 q^{76} - 5 q^{78} - 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} - 20 q^{83} + 6 q^{85} + 6 q^{86} + 15 q^{87} + q^{88} + 4 q^{89} - 22 q^{90} - 3 q^{92} - 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −3.30278 −1.90686 −0.953429 0.301617i \(-0.902474\pi\)
−0.953429 + 0.301617i \(0.902474\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.30278 1.02983 0.514916 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(6\) 3.30278 1.34835
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 7.90833 2.63611
\(10\) −2.30278 −0.728202
\(11\) −2.30278 −0.694313 −0.347156 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) −3.30278 −0.953429
\(13\) −1.30278 −0.361325 −0.180662 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 0 0
\(15\) −7.60555 −1.96374
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −7.90833 −1.86401
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.30278 0.514916
\(21\) 0 0
\(22\) 2.30278 0.490953
\(23\) 3.90833 0.814942 0.407471 0.913218i \(-0.366411\pi\)
0.407471 + 0.913218i \(0.366411\pi\)
\(24\) 3.30278 0.674176
\(25\) 0.302776 0.0605551
\(26\) 1.30278 0.255495
\(27\) −16.2111 −3.11983
\(28\) 0 0
\(29\) −3.90833 −0.725758 −0.362879 0.931836i \(-0.618206\pi\)
−0.362879 + 0.931836i \(0.618206\pi\)
\(30\) 7.60555 1.38858
\(31\) 0.302776 0.0543801 0.0271901 0.999630i \(-0.491344\pi\)
0.0271901 + 0.999630i \(0.491344\pi\)
\(32\) −1.00000 −0.176777
\(33\) 7.60555 1.32396
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 4.30278 0.688996
\(40\) −2.30278 −0.364101
\(41\) −9.90833 −1.54742 −0.773710 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(42\) 0 0
\(43\) 0.605551 0.0923457 0.0461729 0.998933i \(-0.485297\pi\)
0.0461729 + 0.998933i \(0.485297\pi\)
\(44\) −2.30278 −0.347156
\(45\) 18.2111 2.71475
\(46\) −3.90833 −0.576251
\(47\) −4.60555 −0.671789 −0.335894 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) −3.30278 −0.476715
\(49\) 0 0
\(50\) −0.302776 −0.0428189
\(51\) −19.8167 −2.77489
\(52\) −1.30278 −0.180662
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 16.2111 2.20605
\(55\) −5.30278 −0.715026
\(56\) 0 0
\(57\) 6.60555 0.874927
\(58\) 3.90833 0.513188
\(59\) −10.6056 −1.38073 −0.690363 0.723464i \(-0.742549\pi\)
−0.690363 + 0.723464i \(0.742549\pi\)
\(60\) −7.60555 −0.981872
\(61\) −7.51388 −0.962054 −0.481027 0.876706i \(-0.659736\pi\)
−0.481027 + 0.876706i \(0.659736\pi\)
\(62\) −0.302776 −0.0384525
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −7.60555 −0.936179
\(67\) −3.51388 −0.429289 −0.214644 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) 6.00000 0.727607
\(69\) −12.9083 −1.55398
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −7.90833 −0.932005
\(73\) 12.3028 1.43993 0.719965 0.694010i \(-0.244158\pi\)
0.719965 + 0.694010i \(0.244158\pi\)
\(74\) −1.00000 −0.116248
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −4.30278 −0.487193
\(79\) 9.11943 1.02602 0.513008 0.858384i \(-0.328531\pi\)
0.513008 + 0.858384i \(0.328531\pi\)
\(80\) 2.30278 0.257458
\(81\) 29.8167 3.31296
\(82\) 9.90833 1.09419
\(83\) −2.78890 −0.306121 −0.153061 0.988217i \(-0.548913\pi\)
−0.153061 + 0.988217i \(0.548913\pi\)
\(84\) 0 0
\(85\) 13.8167 1.49863
\(86\) −0.605551 −0.0652983
\(87\) 12.9083 1.38392
\(88\) 2.30278 0.245477
\(89\) 9.21110 0.976375 0.488187 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(90\) −18.2111 −1.91962
\(91\) 0 0
\(92\) 3.90833 0.407471
\(93\) −1.00000 −0.103695
\(94\) 4.60555 0.475026
\(95\) −4.60555 −0.472520
\(96\) 3.30278 0.337088
\(97\) 16.4222 1.66742 0.833711 0.552201i \(-0.186212\pi\)
0.833711 + 0.552201i \(0.186212\pi\)
\(98\) 0 0
\(99\) −18.2111 −1.83028
\(100\) 0.302776 0.0302776
\(101\) 12.4222 1.23606 0.618028 0.786156i \(-0.287932\pi\)
0.618028 + 0.786156i \(0.287932\pi\)
\(102\) 19.8167 1.96214
\(103\) 0.302776 0.0298334 0.0149167 0.999889i \(-0.495252\pi\)
0.0149167 + 0.999889i \(0.495252\pi\)
\(104\) 1.30278 0.127748
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 0.697224 0.0674032 0.0337016 0.999432i \(-0.489270\pi\)
0.0337016 + 0.999432i \(0.489270\pi\)
\(108\) −16.2111 −1.55991
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 5.30278 0.505600
\(111\) −3.30278 −0.313486
\(112\) 0 0
\(113\) −3.21110 −0.302075 −0.151038 0.988528i \(-0.548261\pi\)
−0.151038 + 0.988528i \(0.548261\pi\)
\(114\) −6.60555 −0.618667
\(115\) 9.00000 0.839254
\(116\) −3.90833 −0.362879
\(117\) −10.3028 −0.952492
\(118\) 10.6056 0.976320
\(119\) 0 0
\(120\) 7.60555 0.694289
\(121\) −5.69722 −0.517929
\(122\) 7.51388 0.680275
\(123\) 32.7250 2.95071
\(124\) 0.302776 0.0271901
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) −19.2111 −1.70471 −0.852355 0.522964i \(-0.824826\pi\)
−0.852355 + 0.522964i \(0.824826\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 3.00000 0.263117
\(131\) −10.6056 −0.926611 −0.463306 0.886199i \(-0.653337\pi\)
−0.463306 + 0.886199i \(0.653337\pi\)
\(132\) 7.60555 0.661978
\(133\) 0 0
\(134\) 3.51388 0.303553
\(135\) −37.3305 −3.21290
\(136\) −6.00000 −0.514496
\(137\) 0.908327 0.0776036 0.0388018 0.999247i \(-0.487646\pi\)
0.0388018 + 0.999247i \(0.487646\pi\)
\(138\) 12.9083 1.09883
\(139\) 1.90833 0.161862 0.0809311 0.996720i \(-0.474211\pi\)
0.0809311 + 0.996720i \(0.474211\pi\)
\(140\) 0 0
\(141\) 15.2111 1.28101
\(142\) −6.00000 −0.503509
\(143\) 3.00000 0.250873
\(144\) 7.90833 0.659027
\(145\) −9.00000 −0.747409
\(146\) −12.3028 −1.01818
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 19.8167 1.62344 0.811722 0.584044i \(-0.198531\pi\)
0.811722 + 0.584044i \(0.198531\pi\)
\(150\) 1.00000 0.0816497
\(151\) −20.6056 −1.67686 −0.838428 0.545012i \(-0.816525\pi\)
−0.838428 + 0.545012i \(0.816525\pi\)
\(152\) 2.00000 0.162221
\(153\) 47.4500 3.83610
\(154\) 0 0
\(155\) 0.697224 0.0560024
\(156\) 4.30278 0.344498
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) −9.11943 −0.725503
\(159\) 19.8167 1.57156
\(160\) −2.30278 −0.182050
\(161\) 0 0
\(162\) −29.8167 −2.34262
\(163\) 8.42221 0.659678 0.329839 0.944037i \(-0.393006\pi\)
0.329839 + 0.944037i \(0.393006\pi\)
\(164\) −9.90833 −0.773710
\(165\) 17.5139 1.36345
\(166\) 2.78890 0.216460
\(167\) 5.51388 0.426677 0.213338 0.976978i \(-0.431566\pi\)
0.213338 + 0.976978i \(0.431566\pi\)
\(168\) 0 0
\(169\) −11.3028 −0.869444
\(170\) −13.8167 −1.05969
\(171\) −15.8167 −1.20953
\(172\) 0.605551 0.0461729
\(173\) 8.78890 0.668207 0.334104 0.942536i \(-0.391566\pi\)
0.334104 + 0.942536i \(0.391566\pi\)
\(174\) −12.9083 −0.978578
\(175\) 0 0
\(176\) −2.30278 −0.173578
\(177\) 35.0278 2.63285
\(178\) −9.21110 −0.690401
\(179\) −13.8167 −1.03271 −0.516353 0.856376i \(-0.672711\pi\)
−0.516353 + 0.856376i \(0.672711\pi\)
\(180\) 18.2111 1.35738
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 24.8167 1.83450
\(184\) −3.90833 −0.288126
\(185\) 2.30278 0.169303
\(186\) 1.00000 0.0733236
\(187\) −13.8167 −1.01037
\(188\) −4.60555 −0.335894
\(189\) 0 0
\(190\) 4.60555 0.334122
\(191\) −5.51388 −0.398970 −0.199485 0.979901i \(-0.563927\pi\)
−0.199485 + 0.979901i \(0.563927\pi\)
\(192\) −3.30278 −0.238357
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −16.4222 −1.17905
\(195\) 9.90833 0.709550
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 18.2111 1.29421
\(199\) −26.4222 −1.87302 −0.936510 0.350640i \(-0.885964\pi\)
−0.936510 + 0.350640i \(0.885964\pi\)
\(200\) −0.302776 −0.0214095
\(201\) 11.6056 0.818592
\(202\) −12.4222 −0.874023
\(203\) 0 0
\(204\) −19.8167 −1.38744
\(205\) −22.8167 −1.59358
\(206\) −0.302776 −0.0210954
\(207\) 30.9083 2.14828
\(208\) −1.30278 −0.0903312
\(209\) 4.60555 0.318573
\(210\) 0 0
\(211\) 10.3028 0.709272 0.354636 0.935004i \(-0.384605\pi\)
0.354636 + 0.935004i \(0.384605\pi\)
\(212\) −6.00000 −0.412082
\(213\) −19.8167 −1.35781
\(214\) −0.697224 −0.0476613
\(215\) 1.39445 0.0951006
\(216\) 16.2111 1.10303
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) −40.6333 −2.74574
\(220\) −5.30278 −0.357513
\(221\) −7.81665 −0.525805
\(222\) 3.30278 0.221668
\(223\) 5.81665 0.389512 0.194756 0.980852i \(-0.437609\pi\)
0.194756 + 0.980852i \(0.437609\pi\)
\(224\) 0 0
\(225\) 2.39445 0.159630
\(226\) 3.21110 0.213599
\(227\) −13.8167 −0.917044 −0.458522 0.888683i \(-0.651621\pi\)
−0.458522 + 0.888683i \(0.651621\pi\)
\(228\) 6.60555 0.437463
\(229\) −24.6056 −1.62598 −0.812990 0.582277i \(-0.802162\pi\)
−0.812990 + 0.582277i \(0.802162\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) 3.90833 0.256594
\(233\) 8.51388 0.557763 0.278881 0.960326i \(-0.410036\pi\)
0.278881 + 0.960326i \(0.410036\pi\)
\(234\) 10.3028 0.673514
\(235\) −10.6056 −0.691830
\(236\) −10.6056 −0.690363
\(237\) −30.1194 −1.95647
\(238\) 0 0
\(239\) −17.5139 −1.13288 −0.566439 0.824103i \(-0.691679\pi\)
−0.566439 + 0.824103i \(0.691679\pi\)
\(240\) −7.60555 −0.490936
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 5.69722 0.366231
\(243\) −49.8444 −3.19752
\(244\) −7.51388 −0.481027
\(245\) 0 0
\(246\) −32.7250 −2.08647
\(247\) 2.60555 0.165787
\(248\) −0.302776 −0.0192263
\(249\) 9.21110 0.583730
\(250\) 10.8167 0.684105
\(251\) 21.2111 1.33883 0.669416 0.742887i \(-0.266544\pi\)
0.669416 + 0.742887i \(0.266544\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 19.2111 1.20541
\(255\) −45.6333 −2.85767
\(256\) 1.00000 0.0625000
\(257\) −3.21110 −0.200303 −0.100152 0.994972i \(-0.531933\pi\)
−0.100152 + 0.994972i \(0.531933\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) −30.9083 −1.91318
\(262\) 10.6056 0.655213
\(263\) 13.8167 0.851971 0.425986 0.904730i \(-0.359927\pi\)
0.425986 + 0.904730i \(0.359927\pi\)
\(264\) −7.60555 −0.468089
\(265\) −13.8167 −0.848750
\(266\) 0 0
\(267\) −30.4222 −1.86181
\(268\) −3.51388 −0.214644
\(269\) 21.2111 1.29326 0.646632 0.762802i \(-0.276177\pi\)
0.646632 + 0.762802i \(0.276177\pi\)
\(270\) 37.3305 2.27186
\(271\) 22.4222 1.36205 0.681026 0.732259i \(-0.261534\pi\)
0.681026 + 0.732259i \(0.261534\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −0.908327 −0.0548740
\(275\) −0.697224 −0.0420442
\(276\) −12.9083 −0.776990
\(277\) 0.119429 0.00717582 0.00358791 0.999994i \(-0.498858\pi\)
0.00358791 + 0.999994i \(0.498858\pi\)
\(278\) −1.90833 −0.114454
\(279\) 2.39445 0.143352
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) −15.2111 −0.905808
\(283\) −24.6056 −1.46265 −0.731324 0.682030i \(-0.761097\pi\)
−0.731324 + 0.682030i \(0.761097\pi\)
\(284\) 6.00000 0.356034
\(285\) 15.2111 0.901028
\(286\) −3.00000 −0.177394
\(287\) 0 0
\(288\) −7.90833 −0.466003
\(289\) 19.0000 1.11765
\(290\) 9.00000 0.528498
\(291\) −54.2389 −3.17954
\(292\) 12.3028 0.719965
\(293\) −11.0278 −0.644248 −0.322124 0.946697i \(-0.604397\pi\)
−0.322124 + 0.946697i \(0.604397\pi\)
\(294\) 0 0
\(295\) −24.4222 −1.42192
\(296\) −1.00000 −0.0581238
\(297\) 37.3305 2.16614
\(298\) −19.8167 −1.14795
\(299\) −5.09167 −0.294459
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 20.6056 1.18572
\(303\) −41.0278 −2.35698
\(304\) −2.00000 −0.114708
\(305\) −17.3028 −0.990754
\(306\) −47.4500 −2.71253
\(307\) −17.9083 −1.02208 −0.511041 0.859556i \(-0.670740\pi\)
−0.511041 + 0.859556i \(0.670740\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) −0.697224 −0.0395997
\(311\) −15.9083 −0.902078 −0.451039 0.892504i \(-0.648947\pi\)
−0.451039 + 0.892504i \(0.648947\pi\)
\(312\) −4.30278 −0.243597
\(313\) 9.02776 0.510279 0.255139 0.966904i \(-0.417879\pi\)
0.255139 + 0.966904i \(0.417879\pi\)
\(314\) −7.21110 −0.406946
\(315\) 0 0
\(316\) 9.11943 0.513008
\(317\) 9.21110 0.517347 0.258674 0.965965i \(-0.416715\pi\)
0.258674 + 0.965965i \(0.416715\pi\)
\(318\) −19.8167 −1.11126
\(319\) 9.00000 0.503903
\(320\) 2.30278 0.128729
\(321\) −2.30278 −0.128528
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 29.8167 1.65648
\(325\) −0.394449 −0.0218801
\(326\) −8.42221 −0.466463
\(327\) −6.60555 −0.365288
\(328\) 9.90833 0.547096
\(329\) 0 0
\(330\) −17.5139 −0.964107
\(331\) −13.2111 −0.726148 −0.363074 0.931760i \(-0.618273\pi\)
−0.363074 + 0.931760i \(0.618273\pi\)
\(332\) −2.78890 −0.153061
\(333\) 7.90833 0.433374
\(334\) −5.51388 −0.301706
\(335\) −8.09167 −0.442095
\(336\) 0 0
\(337\) 6.11943 0.333347 0.166673 0.986012i \(-0.446697\pi\)
0.166673 + 0.986012i \(0.446697\pi\)
\(338\) 11.3028 0.614790
\(339\) 10.6056 0.576014
\(340\) 13.8167 0.749313
\(341\) −0.697224 −0.0377568
\(342\) 15.8167 0.855267
\(343\) 0 0
\(344\) −0.605551 −0.0326491
\(345\) −29.7250 −1.60034
\(346\) −8.78890 −0.472494
\(347\) 10.1833 0.546671 0.273335 0.961919i \(-0.411873\pi\)
0.273335 + 0.961919i \(0.411873\pi\)
\(348\) 12.9083 0.691959
\(349\) −28.2389 −1.51159 −0.755796 0.654807i \(-0.772750\pi\)
−0.755796 + 0.654807i \(0.772750\pi\)
\(350\) 0 0
\(351\) 21.1194 1.12727
\(352\) 2.30278 0.122738
\(353\) −10.1833 −0.542005 −0.271002 0.962579i \(-0.587355\pi\)
−0.271002 + 0.962579i \(0.587355\pi\)
\(354\) −35.0278 −1.86170
\(355\) 13.8167 0.733312
\(356\) 9.21110 0.488187
\(357\) 0 0
\(358\) 13.8167 0.730233
\(359\) 3.21110 0.169476 0.0847378 0.996403i \(-0.472995\pi\)
0.0847378 + 0.996403i \(0.472995\pi\)
\(360\) −18.2111 −0.959809
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) 18.8167 0.987618
\(364\) 0 0
\(365\) 28.3305 1.48289
\(366\) −24.8167 −1.29719
\(367\) −3.81665 −0.199228 −0.0996139 0.995026i \(-0.531761\pi\)
−0.0996139 + 0.995026i \(0.531761\pi\)
\(368\) 3.90833 0.203736
\(369\) −78.3583 −4.07917
\(370\) −2.30278 −0.119716
\(371\) 0 0
\(372\) −1.00000 −0.0518476
\(373\) −17.8167 −0.922511 −0.461256 0.887267i \(-0.652601\pi\)
−0.461256 + 0.887267i \(0.652601\pi\)
\(374\) 13.8167 0.714442
\(375\) 35.7250 1.84483
\(376\) 4.60555 0.237513
\(377\) 5.09167 0.262235
\(378\) 0 0
\(379\) 24.3305 1.24978 0.624888 0.780715i \(-0.285145\pi\)
0.624888 + 0.780715i \(0.285145\pi\)
\(380\) −4.60555 −0.236260
\(381\) 63.4500 3.25064
\(382\) 5.51388 0.282115
\(383\) 36.8444 1.88266 0.941331 0.337486i \(-0.109576\pi\)
0.941331 + 0.337486i \(0.109576\pi\)
\(384\) 3.30278 0.168544
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 4.78890 0.243433
\(388\) 16.4222 0.833711
\(389\) −37.1194 −1.88203 −0.941015 0.338365i \(-0.890126\pi\)
−0.941015 + 0.338365i \(0.890126\pi\)
\(390\) −9.90833 −0.501728
\(391\) 23.4500 1.18592
\(392\) 0 0
\(393\) 35.0278 1.76692
\(394\) 6.00000 0.302276
\(395\) 21.0000 1.05662
\(396\) −18.2111 −0.915142
\(397\) −6.18335 −0.310333 −0.155167 0.987888i \(-0.549591\pi\)
−0.155167 + 0.987888i \(0.549591\pi\)
\(398\) 26.4222 1.32443
\(399\) 0 0
\(400\) 0.302776 0.0151388
\(401\) −7.81665 −0.390345 −0.195173 0.980769i \(-0.562527\pi\)
−0.195173 + 0.980769i \(0.562527\pi\)
\(402\) −11.6056 −0.578832
\(403\) −0.394449 −0.0196489
\(404\) 12.4222 0.618028
\(405\) 68.6611 3.41180
\(406\) 0 0
\(407\) −2.30278 −0.114144
\(408\) 19.8167 0.981071
\(409\) −31.0278 −1.53422 −0.767112 0.641513i \(-0.778307\pi\)
−0.767112 + 0.641513i \(0.778307\pi\)
\(410\) 22.8167 1.12683
\(411\) −3.00000 −0.147979
\(412\) 0.302776 0.0149167
\(413\) 0 0
\(414\) −30.9083 −1.51906
\(415\) −6.42221 −0.315254
\(416\) 1.30278 0.0638738
\(417\) −6.30278 −0.308648
\(418\) −4.60555 −0.225265
\(419\) −36.1472 −1.76591 −0.882953 0.469462i \(-0.844448\pi\)
−0.882953 + 0.469462i \(0.844448\pi\)
\(420\) 0 0
\(421\) −3.72498 −0.181544 −0.0907722 0.995872i \(-0.528934\pi\)
−0.0907722 + 0.995872i \(0.528934\pi\)
\(422\) −10.3028 −0.501531
\(423\) −36.4222 −1.77091
\(424\) 6.00000 0.291386
\(425\) 1.81665 0.0881207
\(426\) 19.8167 0.960120
\(427\) 0 0
\(428\) 0.697224 0.0337016
\(429\) −9.90833 −0.478379
\(430\) −1.39445 −0.0672463
\(431\) 9.21110 0.443683 0.221842 0.975083i \(-0.428793\pi\)
0.221842 + 0.975083i \(0.428793\pi\)
\(432\) −16.2111 −0.779957
\(433\) −34.9361 −1.67892 −0.839461 0.543421i \(-0.817129\pi\)
−0.839461 + 0.543421i \(0.817129\pi\)
\(434\) 0 0
\(435\) 29.7250 1.42520
\(436\) 2.00000 0.0957826
\(437\) −7.81665 −0.373921
\(438\) 40.6333 1.94153
\(439\) −30.3305 −1.44760 −0.723799 0.690011i \(-0.757606\pi\)
−0.723799 + 0.690011i \(0.757606\pi\)
\(440\) 5.30278 0.252800
\(441\) 0 0
\(442\) 7.81665 0.371800
\(443\) 32.7250 1.55481 0.777405 0.629000i \(-0.216535\pi\)
0.777405 + 0.629000i \(0.216535\pi\)
\(444\) −3.30278 −0.156743
\(445\) 21.2111 1.00550
\(446\) −5.81665 −0.275427
\(447\) −65.4500 −3.09568
\(448\) 0 0
\(449\) −15.2111 −0.717856 −0.358928 0.933365i \(-0.616858\pi\)
−0.358928 + 0.933365i \(0.616858\pi\)
\(450\) −2.39445 −0.112875
\(451\) 22.8167 1.07439
\(452\) −3.21110 −0.151038
\(453\) 68.0555 3.19753
\(454\) 13.8167 0.648448
\(455\) 0 0
\(456\) −6.60555 −0.309333
\(457\) −2.60555 −0.121883 −0.0609413 0.998141i \(-0.519410\pi\)
−0.0609413 + 0.998141i \(0.519410\pi\)
\(458\) 24.6056 1.14974
\(459\) −97.2666 −4.54002
\(460\) 9.00000 0.419627
\(461\) −12.4222 −0.578560 −0.289280 0.957245i \(-0.593416\pi\)
−0.289280 + 0.957245i \(0.593416\pi\)
\(462\) 0 0
\(463\) 26.6972 1.24073 0.620363 0.784315i \(-0.286985\pi\)
0.620363 + 0.784315i \(0.286985\pi\)
\(464\) −3.90833 −0.181440
\(465\) −2.30278 −0.106789
\(466\) −8.51388 −0.394398
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −10.3028 −0.476246
\(469\) 0 0
\(470\) 10.6056 0.489198
\(471\) −23.8167 −1.09741
\(472\) 10.6056 0.488160
\(473\) −1.39445 −0.0641168
\(474\) 30.1194 1.38343
\(475\) −0.605551 −0.0277846
\(476\) 0 0
\(477\) −47.4500 −2.17258
\(478\) 17.5139 0.801066
\(479\) 13.1194 0.599442 0.299721 0.954027i \(-0.403106\pi\)
0.299721 + 0.954027i \(0.403106\pi\)
\(480\) 7.60555 0.347144
\(481\) −1.30278 −0.0594015
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −5.69722 −0.258965
\(485\) 37.8167 1.71717
\(486\) 49.8444 2.26099
\(487\) −37.2111 −1.68620 −0.843098 0.537760i \(-0.819271\pi\)
−0.843098 + 0.537760i \(0.819271\pi\)
\(488\) 7.51388 0.340137
\(489\) −27.8167 −1.25791
\(490\) 0 0
\(491\) 17.7250 0.799917 0.399959 0.916533i \(-0.369025\pi\)
0.399959 + 0.916533i \(0.369025\pi\)
\(492\) 32.7250 1.47536
\(493\) −23.4500 −1.05613
\(494\) −2.60555 −0.117229
\(495\) −41.9361 −1.88489
\(496\) 0.302776 0.0135950
\(497\) 0 0
\(498\) −9.21110 −0.412759
\(499\) −42.2389 −1.89087 −0.945436 0.325809i \(-0.894363\pi\)
−0.945436 + 0.325809i \(0.894363\pi\)
\(500\) −10.8167 −0.483735
\(501\) −18.2111 −0.813612
\(502\) −21.2111 −0.946698
\(503\) 6.48612 0.289202 0.144601 0.989490i \(-0.453810\pi\)
0.144601 + 0.989490i \(0.453810\pi\)
\(504\) 0 0
\(505\) 28.6056 1.27293
\(506\) 9.00000 0.400099
\(507\) 37.3305 1.65791
\(508\) −19.2111 −0.852355
\(509\) 4.18335 0.185424 0.0927118 0.995693i \(-0.470446\pi\)
0.0927118 + 0.995693i \(0.470446\pi\)
\(510\) 45.6333 2.02068
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 32.4222 1.43148
\(514\) 3.21110 0.141636
\(515\) 0.697224 0.0307234
\(516\) −2.00000 −0.0880451
\(517\) 10.6056 0.466432
\(518\) 0 0
\(519\) −29.0278 −1.27418
\(520\) 3.00000 0.131559
\(521\) −33.6333 −1.47350 −0.736751 0.676164i \(-0.763641\pi\)
−0.736751 + 0.676164i \(0.763641\pi\)
\(522\) 30.9083 1.35282
\(523\) 18.2389 0.797530 0.398765 0.917053i \(-0.369439\pi\)
0.398765 + 0.917053i \(0.369439\pi\)
\(524\) −10.6056 −0.463306
\(525\) 0 0
\(526\) −13.8167 −0.602435
\(527\) 1.81665 0.0791347
\(528\) 7.60555 0.330989
\(529\) −7.72498 −0.335869
\(530\) 13.8167 0.600157
\(531\) −83.8722 −3.63974
\(532\) 0 0
\(533\) 12.9083 0.559122
\(534\) 30.4222 1.31650
\(535\) 1.60555 0.0694140
\(536\) 3.51388 0.151776
\(537\) 45.6333 1.96922
\(538\) −21.2111 −0.914476
\(539\) 0 0
\(540\) −37.3305 −1.60645
\(541\) 25.9361 1.11508 0.557540 0.830150i \(-0.311745\pi\)
0.557540 + 0.830150i \(0.311745\pi\)
\(542\) −22.4222 −0.963116
\(543\) 66.0555 2.83471
\(544\) −6.00000 −0.257248
\(545\) 4.60555 0.197280
\(546\) 0 0
\(547\) −20.6056 −0.881030 −0.440515 0.897745i \(-0.645204\pi\)
−0.440515 + 0.897745i \(0.645204\pi\)
\(548\) 0.908327 0.0388018
\(549\) −59.4222 −2.53608
\(550\) 0.697224 0.0297297
\(551\) 7.81665 0.333001
\(552\) 12.9083 0.549415
\(553\) 0 0
\(554\) −0.119429 −0.00507407
\(555\) −7.60555 −0.322838
\(556\) 1.90833 0.0809311
\(557\) 11.5139 0.487859 0.243929 0.969793i \(-0.421564\pi\)
0.243929 + 0.969793i \(0.421564\pi\)
\(558\) −2.39445 −0.101365
\(559\) −0.788897 −0.0333668
\(560\) 0 0
\(561\) 45.6333 1.92664
\(562\) 12.0000 0.506189
\(563\) 28.0555 1.18240 0.591199 0.806525i \(-0.298655\pi\)
0.591199 + 0.806525i \(0.298655\pi\)
\(564\) 15.2111 0.640503
\(565\) −7.39445 −0.311087
\(566\) 24.6056 1.03425
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 18.4222 0.772299 0.386150 0.922436i \(-0.373805\pi\)
0.386150 + 0.922436i \(0.373805\pi\)
\(570\) −15.2111 −0.637123
\(571\) −16.6972 −0.698757 −0.349379 0.936982i \(-0.613607\pi\)
−0.349379 + 0.936982i \(0.613607\pi\)
\(572\) 3.00000 0.125436
\(573\) 18.2111 0.760780
\(574\) 0 0
\(575\) 1.18335 0.0493489
\(576\) 7.90833 0.329514
\(577\) −22.2389 −0.925816 −0.462908 0.886406i \(-0.653194\pi\)
−0.462908 + 0.886406i \(0.653194\pi\)
\(578\) −19.0000 −0.790296
\(579\) 13.2111 0.549035
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) 54.2389 2.24827
\(583\) 13.8167 0.572227
\(584\) −12.3028 −0.509092
\(585\) −23.7250 −0.980907
\(586\) 11.0278 0.455552
\(587\) 45.6333 1.88349 0.941744 0.336330i \(-0.109186\pi\)
0.941744 + 0.336330i \(0.109186\pi\)
\(588\) 0 0
\(589\) −0.605551 −0.0249513
\(590\) 24.4222 1.00545
\(591\) 19.8167 0.815148
\(592\) 1.00000 0.0410997
\(593\) −18.4861 −0.759134 −0.379567 0.925164i \(-0.623927\pi\)
−0.379567 + 0.925164i \(0.623927\pi\)
\(594\) −37.3305 −1.53169
\(595\) 0 0
\(596\) 19.8167 0.811722
\(597\) 87.2666 3.57158
\(598\) 5.09167 0.208214
\(599\) −20.7889 −0.849411 −0.424706 0.905331i \(-0.639622\pi\)
−0.424706 + 0.905331i \(0.639622\pi\)
\(600\) 1.00000 0.0408248
\(601\) 24.3028 0.991331 0.495665 0.868514i \(-0.334924\pi\)
0.495665 + 0.868514i \(0.334924\pi\)
\(602\) 0 0
\(603\) −27.7889 −1.13165
\(604\) −20.6056 −0.838428
\(605\) −13.1194 −0.533381
\(606\) 41.0278 1.66664
\(607\) 13.4861 0.547385 0.273692 0.961817i \(-0.411755\pi\)
0.273692 + 0.961817i \(0.411755\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 17.3028 0.700569
\(611\) 6.00000 0.242734
\(612\) 47.4500 1.91805
\(613\) −29.8167 −1.20428 −0.602142 0.798389i \(-0.705686\pi\)
−0.602142 + 0.798389i \(0.705686\pi\)
\(614\) 17.9083 0.722721
\(615\) 75.3583 3.03874
\(616\) 0 0
\(617\) −42.5694 −1.71378 −0.856890 0.515500i \(-0.827606\pi\)
−0.856890 + 0.515500i \(0.827606\pi\)
\(618\) 1.00000 0.0402259
\(619\) 6.30278 0.253330 0.126665 0.991946i \(-0.459573\pi\)
0.126665 + 0.991946i \(0.459573\pi\)
\(620\) 0.697224 0.0280012
\(621\) −63.3583 −2.54248
\(622\) 15.9083 0.637866
\(623\) 0 0
\(624\) 4.30278 0.172249
\(625\) −26.4222 −1.05689
\(626\) −9.02776 −0.360822
\(627\) −15.2111 −0.607473
\(628\) 7.21110 0.287754
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 14.6972 0.585087 0.292544 0.956252i \(-0.405498\pi\)
0.292544 + 0.956252i \(0.405498\pi\)
\(632\) −9.11943 −0.362751
\(633\) −34.0278 −1.35248
\(634\) −9.21110 −0.365820
\(635\) −44.2389 −1.75557
\(636\) 19.8167 0.785781
\(637\) 0 0
\(638\) −9.00000 −0.356313
\(639\) 47.4500 1.87709
\(640\) −2.30278 −0.0910252
\(641\) −20.5139 −0.810249 −0.405125 0.914261i \(-0.632772\pi\)
−0.405125 + 0.914261i \(0.632772\pi\)
\(642\) 2.30278 0.0908833
\(643\) 8.18335 0.322720 0.161360 0.986896i \(-0.448412\pi\)
0.161360 + 0.986896i \(0.448412\pi\)
\(644\) 0 0
\(645\) −4.60555 −0.181343
\(646\) 12.0000 0.472134
\(647\) −20.9361 −0.823082 −0.411541 0.911391i \(-0.635009\pi\)
−0.411541 + 0.911391i \(0.635009\pi\)
\(648\) −29.8167 −1.17131
\(649\) 24.4222 0.958655
\(650\) 0.394449 0.0154716
\(651\) 0 0
\(652\) 8.42221 0.329839
\(653\) −3.90833 −0.152945 −0.0764723 0.997072i \(-0.524366\pi\)
−0.0764723 + 0.997072i \(0.524366\pi\)
\(654\) 6.60555 0.258297
\(655\) −24.4222 −0.954255
\(656\) −9.90833 −0.386855
\(657\) 97.2944 3.79581
\(658\) 0 0
\(659\) −16.8806 −0.657574 −0.328787 0.944404i \(-0.606640\pi\)
−0.328787 + 0.944404i \(0.606640\pi\)
\(660\) 17.5139 0.681727
\(661\) 30.5139 1.18685 0.593426 0.804888i \(-0.297775\pi\)
0.593426 + 0.804888i \(0.297775\pi\)
\(662\) 13.2111 0.513464
\(663\) 25.8167 1.00264
\(664\) 2.78890 0.108230
\(665\) 0 0
\(666\) −7.90833 −0.306441
\(667\) −15.2750 −0.591451
\(668\) 5.51388 0.213338
\(669\) −19.2111 −0.742744
\(670\) 8.09167 0.312609
\(671\) 17.3028 0.667966
\(672\) 0 0
\(673\) 20.6972 0.797819 0.398910 0.916990i \(-0.369389\pi\)
0.398910 + 0.916990i \(0.369389\pi\)
\(674\) −6.11943 −0.235712
\(675\) −4.90833 −0.188922
\(676\) −11.3028 −0.434722
\(677\) −14.2389 −0.547244 −0.273622 0.961837i \(-0.588222\pi\)
−0.273622 + 0.961837i \(0.588222\pi\)
\(678\) −10.6056 −0.407304
\(679\) 0 0
\(680\) −13.8167 −0.529844
\(681\) 45.6333 1.74867
\(682\) 0.697224 0.0266981
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −15.8167 −0.604765
\(685\) 2.09167 0.0799187
\(686\) 0 0
\(687\) 81.2666 3.10051
\(688\) 0.605551 0.0230864
\(689\) 7.81665 0.297791
\(690\) 29.7250 1.13161
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 8.78890 0.334104
\(693\) 0 0
\(694\) −10.1833 −0.386555
\(695\) 4.39445 0.166691
\(696\) −12.9083 −0.489289
\(697\) −59.4500 −2.25183
\(698\) 28.2389 1.06886
\(699\) −28.1194 −1.06357
\(700\) 0 0
\(701\) 40.1194 1.51529 0.757645 0.652667i \(-0.226350\pi\)
0.757645 + 0.652667i \(0.226350\pi\)
\(702\) −21.1194 −0.797101
\(703\) −2.00000 −0.0754314
\(704\) −2.30278 −0.0867891
\(705\) 35.0278 1.31922
\(706\) 10.1833 0.383255
\(707\) 0 0
\(708\) 35.0278 1.31642
\(709\) −41.3305 −1.55220 −0.776100 0.630609i \(-0.782805\pi\)
−0.776100 + 0.630609i \(0.782805\pi\)
\(710\) −13.8167 −0.518530
\(711\) 72.1194 2.70469
\(712\) −9.21110 −0.345201
\(713\) 1.18335 0.0443167
\(714\) 0 0
\(715\) 6.90833 0.258357
\(716\) −13.8167 −0.516353
\(717\) 57.8444 2.16024
\(718\) −3.21110 −0.119837
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 18.2111 0.678688
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) 26.4222 0.982652
\(724\) −20.0000 −0.743294
\(725\) −1.18335 −0.0439484
\(726\) −18.8167 −0.698352
\(727\) −19.0917 −0.708071 −0.354035 0.935232i \(-0.615191\pi\)
−0.354035 + 0.935232i \(0.615191\pi\)
\(728\) 0 0
\(729\) 75.1749 2.78426
\(730\) −28.3305 −1.04856
\(731\) 3.63331 0.134383
\(732\) 24.8167 0.917250
\(733\) 13.6333 0.503558 0.251779 0.967785i \(-0.418984\pi\)
0.251779 + 0.967785i \(0.418984\pi\)
\(734\) 3.81665 0.140875
\(735\) 0 0
\(736\) −3.90833 −0.144063
\(737\) 8.09167 0.298061
\(738\) 78.3583 2.88441
\(739\) −2.66947 −0.0981980 −0.0490990 0.998794i \(-0.515635\pi\)
−0.0490990 + 0.998794i \(0.515635\pi\)
\(740\) 2.30278 0.0846517
\(741\) −8.60555 −0.316133
\(742\) 0 0
\(743\) −29.4500 −1.08041 −0.540207 0.841532i \(-0.681654\pi\)
−0.540207 + 0.841532i \(0.681654\pi\)
\(744\) 1.00000 0.0366618
\(745\) 45.6333 1.67188
\(746\) 17.8167 0.652314
\(747\) −22.0555 −0.806969
\(748\) −13.8167 −0.505187
\(749\) 0 0
\(750\) −35.7250 −1.30449
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) −4.60555 −0.167947
\(753\) −70.0555 −2.55296
\(754\) −5.09167 −0.185428
\(755\) −47.4500 −1.72688
\(756\) 0 0
\(757\) 5.69722 0.207069 0.103535 0.994626i \(-0.466985\pi\)
0.103535 + 0.994626i \(0.466985\pi\)
\(758\) −24.3305 −0.883725
\(759\) 29.7250 1.07895
\(760\) 4.60555 0.167061
\(761\) −16.8806 −0.611920 −0.305960 0.952044i \(-0.598977\pi\)
−0.305960 + 0.952044i \(0.598977\pi\)
\(762\) −63.4500 −2.29855
\(763\) 0 0
\(764\) −5.51388 −0.199485
\(765\) 109.267 3.95054
\(766\) −36.8444 −1.33124
\(767\) 13.8167 0.498890
\(768\) −3.30278 −0.119179
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 10.6056 0.381950
\(772\) −4.00000 −0.143963
\(773\) −22.0555 −0.793282 −0.396641 0.917974i \(-0.629824\pi\)
−0.396641 + 0.917974i \(0.629824\pi\)
\(774\) −4.78890 −0.172133
\(775\) 0.0916731 0.00329299
\(776\) −16.4222 −0.589523
\(777\) 0 0
\(778\) 37.1194 1.33080
\(779\) 19.8167 0.710005
\(780\) 9.90833 0.354775
\(781\) −13.8167 −0.494399
\(782\) −23.4500 −0.838569
\(783\) 63.3583 2.26424
\(784\) 0 0
\(785\) 16.6056 0.592678
\(786\) −35.0278 −1.24940
\(787\) −10.7889 −0.384583 −0.192291 0.981338i \(-0.561592\pi\)
−0.192291 + 0.981338i \(0.561592\pi\)
\(788\) −6.00000 −0.213741
\(789\) −45.6333 −1.62459
\(790\) −21.0000 −0.747146
\(791\) 0 0
\(792\) 18.2111 0.647103
\(793\) 9.78890 0.347614
\(794\) 6.18335 0.219439
\(795\) 45.6333 1.61845
\(796\) −26.4222 −0.936510
\(797\) −22.3305 −0.790988 −0.395494 0.918469i \(-0.629427\pi\)
−0.395494 + 0.918469i \(0.629427\pi\)
\(798\) 0 0
\(799\) −27.6333 −0.977596
\(800\) −0.302776 −0.0107047
\(801\) 72.8444 2.57383
\(802\) 7.81665 0.276016
\(803\) −28.3305 −0.999763
\(804\) 11.6056 0.409296
\(805\) 0 0
\(806\) 0.394449 0.0138939
\(807\) −70.0555 −2.46607
\(808\) −12.4222 −0.437012
\(809\) −35.4500 −1.24635 −0.623177 0.782081i \(-0.714158\pi\)
−0.623177 + 0.782081i \(0.714158\pi\)
\(810\) −68.6611 −2.41250
\(811\) 7.14719 0.250972 0.125486 0.992095i \(-0.459951\pi\)
0.125486 + 0.992095i \(0.459951\pi\)
\(812\) 0 0
\(813\) −74.0555 −2.59724
\(814\) 2.30278 0.0807122
\(815\) 19.3944 0.679358
\(816\) −19.8167 −0.693722
\(817\) −1.21110 −0.0423711
\(818\) 31.0278 1.08486
\(819\) 0 0
\(820\) −22.8167 −0.796792
\(821\) −3.21110 −0.112068 −0.0560341 0.998429i \(-0.517846\pi\)
−0.0560341 + 0.998429i \(0.517846\pi\)
\(822\) 3.00000 0.104637
\(823\) 44.8444 1.56318 0.781589 0.623794i \(-0.214410\pi\)
0.781589 + 0.623794i \(0.214410\pi\)
\(824\) −0.302776 −0.0105477
\(825\) 2.30278 0.0801724
\(826\) 0 0
\(827\) 34.6056 1.20335 0.601676 0.798740i \(-0.294500\pi\)
0.601676 + 0.798740i \(0.294500\pi\)
\(828\) 30.9083 1.07414
\(829\) 27.7250 0.962928 0.481464 0.876466i \(-0.340105\pi\)
0.481464 + 0.876466i \(0.340105\pi\)
\(830\) 6.42221 0.222918
\(831\) −0.394449 −0.0136833
\(832\) −1.30278 −0.0451656
\(833\) 0 0
\(834\) 6.30278 0.218247
\(835\) 12.6972 0.439406
\(836\) 4.60555 0.159286
\(837\) −4.90833 −0.169657
\(838\) 36.1472 1.24868
\(839\) 12.9722 0.447852 0.223926 0.974606i \(-0.428113\pi\)
0.223926 + 0.974606i \(0.428113\pi\)
\(840\) 0 0
\(841\) −13.7250 −0.473275
\(842\) 3.72498 0.128371
\(843\) 39.6333 1.36504
\(844\) 10.3028 0.354636
\(845\) −26.0278 −0.895382
\(846\) 36.4222 1.25222
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 81.2666 2.78906
\(850\) −1.81665 −0.0623107
\(851\) 3.90833 0.133976
\(852\) −19.8167 −0.678907
\(853\) −42.5416 −1.45660 −0.728299 0.685260i \(-0.759689\pi\)
−0.728299 + 0.685260i \(0.759689\pi\)
\(854\) 0 0
\(855\) −36.4222 −1.24561
\(856\) −0.697224 −0.0238306
\(857\) −42.8444 −1.46354 −0.731769 0.681553i \(-0.761305\pi\)
−0.731769 + 0.681553i \(0.761305\pi\)
\(858\) 9.90833 0.338265
\(859\) −48.0555 −1.63963 −0.819816 0.572626i \(-0.805925\pi\)
−0.819816 + 0.572626i \(0.805925\pi\)
\(860\) 1.39445 0.0475503
\(861\) 0 0
\(862\) −9.21110 −0.313731
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 16.2111 0.551513
\(865\) 20.2389 0.688142
\(866\) 34.9361 1.18718
\(867\) −62.7527 −2.13119
\(868\) 0 0
\(869\) −21.0000 −0.712376
\(870\) −29.7250 −1.00777
\(871\) 4.57779 0.155113
\(872\) −2.00000 −0.0677285
\(873\) 129.872 4.39551
\(874\) 7.81665 0.264402
\(875\) 0 0
\(876\) −40.6333 −1.37287
\(877\) −7.21110 −0.243502 −0.121751 0.992561i \(-0.538851\pi\)
−0.121751 + 0.992561i \(0.538851\pi\)
\(878\) 30.3305 1.02361
\(879\) 36.4222 1.22849
\(880\) −5.30278 −0.178757
\(881\) 28.5416 0.961592 0.480796 0.876832i \(-0.340348\pi\)
0.480796 + 0.876832i \(0.340348\pi\)
\(882\) 0 0
\(883\) 26.4222 0.889178 0.444589 0.895735i \(-0.353350\pi\)
0.444589 + 0.895735i \(0.353350\pi\)
\(884\) −7.81665 −0.262903
\(885\) 80.6611 2.71139
\(886\) −32.7250 −1.09942
\(887\) 0.422205 0.0141763 0.00708813 0.999975i \(-0.497744\pi\)
0.00708813 + 0.999975i \(0.497744\pi\)
\(888\) 3.30278 0.110834
\(889\) 0 0
\(890\) −21.2111 −0.710998
\(891\) −68.6611 −2.30023
\(892\) 5.81665 0.194756
\(893\) 9.21110 0.308238
\(894\) 65.4500 2.18897
\(895\) −31.8167 −1.06351
\(896\) 0 0
\(897\) 16.8167 0.561492
\(898\) 15.2111 0.507601
\(899\) −1.18335 −0.0394668
\(900\) 2.39445 0.0798150
\(901\) −36.0000 −1.19933
\(902\) −22.8167 −0.759711
\(903\) 0 0
\(904\) 3.21110 0.106800
\(905\) −46.0555 −1.53094
\(906\) −68.0555 −2.26099
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) −13.8167 −0.458522
\(909\) 98.2389 3.25838
\(910\) 0 0
\(911\) −17.5778 −0.582378 −0.291189 0.956665i \(-0.594051\pi\)
−0.291189 + 0.956665i \(0.594051\pi\)
\(912\) 6.60555 0.218732
\(913\) 6.42221 0.212544
\(914\) 2.60555 0.0861840
\(915\) 57.1472 1.88923
\(916\) −24.6056 −0.812990
\(917\) 0 0
\(918\) 97.2666 3.21028
\(919\) −9.57779 −0.315942 −0.157971 0.987444i \(-0.550495\pi\)
−0.157971 + 0.987444i \(0.550495\pi\)
\(920\) −9.00000 −0.296721
\(921\) 59.1472 1.94897
\(922\) 12.4222 0.409104
\(923\) −7.81665 −0.257288
\(924\) 0 0
\(925\) 0.302776 0.00995520
\(926\) −26.6972 −0.877325
\(927\) 2.39445 0.0786440
\(928\) 3.90833 0.128297
\(929\) 18.4861 0.606510 0.303255 0.952909i \(-0.401927\pi\)
0.303255 + 0.952909i \(0.401927\pi\)
\(930\) 2.30278 0.0755110
\(931\) 0 0
\(932\) 8.51388 0.278881
\(933\) 52.5416 1.72014
\(934\) 0 0
\(935\) −31.8167 −1.04052
\(936\) 10.3028 0.336757
\(937\) 18.0917 0.591029 0.295515 0.955338i \(-0.404509\pi\)
0.295515 + 0.955338i \(0.404509\pi\)
\(938\) 0 0
\(939\) −29.8167 −0.973030
\(940\) −10.6056 −0.345915
\(941\) −13.8167 −0.450410 −0.225205 0.974311i \(-0.572305\pi\)
−0.225205 + 0.974311i \(0.572305\pi\)
\(942\) 23.8167 0.775989
\(943\) −38.7250 −1.26106
\(944\) −10.6056 −0.345181
\(945\) 0 0
\(946\) 1.39445 0.0453374
\(947\) −3.63331 −0.118067 −0.0590333 0.998256i \(-0.518802\pi\)
−0.0590333 + 0.998256i \(0.518802\pi\)
\(948\) −30.1194 −0.978234
\(949\) −16.0278 −0.520283
\(950\) 0.605551 0.0196467
\(951\) −30.4222 −0.986508
\(952\) 0 0
\(953\) 49.7527 1.61165 0.805825 0.592154i \(-0.201722\pi\)
0.805825 + 0.592154i \(0.201722\pi\)
\(954\) 47.4500 1.53625
\(955\) −12.6972 −0.410873
\(956\) −17.5139 −0.566439
\(957\) −29.7250 −0.960872
\(958\) −13.1194 −0.423870
\(959\) 0 0
\(960\) −7.60555 −0.245468
\(961\) −30.9083 −0.997043
\(962\) 1.30278 0.0420032
\(963\) 5.51388 0.177682
\(964\) −8.00000 −0.257663
\(965\) −9.21110 −0.296516
\(966\) 0 0
\(967\) −6.72498 −0.216261 −0.108130 0.994137i \(-0.534486\pi\)
−0.108130 + 0.994137i \(0.534486\pi\)
\(968\) 5.69722 0.183116
\(969\) 39.6333 1.27321
\(970\) −37.8167 −1.21422
\(971\) 22.5416 0.723395 0.361698 0.932295i \(-0.382197\pi\)
0.361698 + 0.932295i \(0.382197\pi\)
\(972\) −49.8444 −1.59876
\(973\) 0 0
\(974\) 37.2111 1.19232
\(975\) 1.30278 0.0417222
\(976\) −7.51388 −0.240513
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 27.8167 0.889479
\(979\) −21.2111 −0.677910
\(980\) 0 0
\(981\) 15.8167 0.504987
\(982\) −17.7250 −0.565627
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) −32.7250 −1.04323
\(985\) −13.8167 −0.440235
\(986\) 23.4500 0.746799
\(987\) 0 0
\(988\) 2.60555 0.0828936
\(989\) 2.36669 0.0752564
\(990\) 41.9361 1.33282
\(991\) 50.6972 1.61045 0.805225 0.592969i \(-0.202044\pi\)
0.805225 + 0.592969i \(0.202044\pi\)
\(992\) −0.302776 −0.00961314
\(993\) 43.6333 1.38466
\(994\) 0 0
\(995\) −60.8444 −1.92890
\(996\) 9.21110 0.291865
\(997\) 52.4222 1.66023 0.830114 0.557594i \(-0.188275\pi\)
0.830114 + 0.557594i \(0.188275\pi\)
\(998\) 42.2389 1.33705
\(999\) −16.2111 −0.512897
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 3626.2.a.a.1.1 2
7.6 odd 2 74.2.a.a.1.2 2
21.20 even 2 666.2.a.j.1.2 2
28.27 even 2 592.2.a.f.1.1 2
35.13 even 4 1850.2.b.i.149.4 4
35.27 even 4 1850.2.b.i.149.1 4
35.34 odd 2 1850.2.a.u.1.1 2
56.13 odd 2 2368.2.a.s.1.1 2
56.27 even 2 2368.2.a.ba.1.2 2
77.76 even 2 8954.2.a.p.1.2 2
84.83 odd 2 5328.2.a.bf.1.2 2
259.258 odd 2 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 2 7.6 odd 2
592.2.a.f.1.1 2 28.27 even 2
666.2.a.j.1.2 2 21.20 even 2
1850.2.a.u.1.1 2 35.34 odd 2
1850.2.b.i.149.1 4 35.27 even 4
1850.2.b.i.149.4 4 35.13 even 4
2368.2.a.s.1.1 2 56.13 odd 2
2368.2.a.ba.1.2 2 56.27 even 2
2738.2.a.l.1.2 2 259.258 odd 2
3626.2.a.a.1.1 2 1.1 even 1 trivial
5328.2.a.bf.1.2 2 84.83 odd 2
8954.2.a.p.1.2 2 77.76 even 2