Properties

Label 2450.2.a.bs.1.2
Level $2450$
Weight $2$
Character 2450.1
Self dual yes
Analytic conductor $19.563$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.41421 q^{3} +1.00000 q^{4} +3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.41421 q^{3} +1.00000 q^{4} +3.41421 q^{6} +1.00000 q^{8} +8.65685 q^{9} -0.828427 q^{11} +3.41421 q^{12} -4.82843 q^{13} +1.00000 q^{16} +2.58579 q^{17} +8.65685 q^{18} -0.585786 q^{19} -0.828427 q^{22} +1.17157 q^{23} +3.41421 q^{24} -4.82843 q^{26} +19.3137 q^{27} -4.82843 q^{29} +2.82843 q^{31} +1.00000 q^{32} -2.82843 q^{33} +2.58579 q^{34} +8.65685 q^{36} +7.65685 q^{37} -0.585786 q^{38} -16.4853 q^{39} +3.07107 q^{41} +8.82843 q^{43} -0.828427 q^{44} +1.17157 q^{46} +5.17157 q^{47} +3.41421 q^{48} +8.82843 q^{51} -4.82843 q^{52} -6.48528 q^{53} +19.3137 q^{54} -2.00000 q^{57} -4.82843 q^{58} -8.58579 q^{59} -9.31371 q^{61} +2.82843 q^{62} +1.00000 q^{64} -2.82843 q^{66} -1.65685 q^{67} +2.58579 q^{68} +4.00000 q^{69} -4.48528 q^{71} +8.65685 q^{72} -9.41421 q^{73} +7.65685 q^{74} -0.585786 q^{76} -16.4853 q^{78} -6.82843 q^{79} +39.9706 q^{81} +3.07107 q^{82} -2.24264 q^{83} +8.82843 q^{86} -16.4853 q^{87} -0.828427 q^{88} -12.7279 q^{89} +1.17157 q^{92} +9.65685 q^{93} +5.17157 q^{94} +3.41421 q^{96} -7.75736 q^{97} -7.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 4q^{3} + 2q^{4} + 4q^{6} + 2q^{8} + 6q^{9} + 4q^{11} + 4q^{12} - 4q^{13} + 2q^{16} + 8q^{17} + 6q^{18} - 4q^{19} + 4q^{22} + 8q^{23} + 4q^{24} - 4q^{26} + 16q^{27} - 4q^{29} + 2q^{32} + 8q^{34} + 6q^{36} + 4q^{37} - 4q^{38} - 16q^{39} - 8q^{41} + 12q^{43} + 4q^{44} + 8q^{46} + 16q^{47} + 4q^{48} + 12q^{51} - 4q^{52} + 4q^{53} + 16q^{54} - 4q^{57} - 4q^{58} - 20q^{59} + 4q^{61} + 2q^{64} + 8q^{67} + 8q^{68} + 8q^{69} + 8q^{71} + 6q^{72} - 16q^{73} + 4q^{74} - 4q^{76} - 16q^{78} - 8q^{79} + 46q^{81} - 8q^{82} + 4q^{83} + 12q^{86} - 16q^{87} + 4q^{88} + 8q^{92} + 8q^{93} + 16q^{94} + 4q^{96} - 24q^{97} - 20q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 3.41421 1.97120 0.985599 0.169102i \(-0.0540867\pi\)
0.985599 + 0.169102i \(0.0540867\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 3.41421 1.39385
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 8.65685 2.88562
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 3.41421 0.985599
\(13\) −4.82843 −1.33916 −0.669582 0.742738i \(-0.733527\pi\)
−0.669582 + 0.742738i \(0.733527\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.58579 0.627145 0.313573 0.949564i \(-0.398474\pi\)
0.313573 + 0.949564i \(0.398474\pi\)
\(18\) 8.65685 2.04044
\(19\) −0.585786 −0.134389 −0.0671943 0.997740i \(-0.521405\pi\)
−0.0671943 + 0.997740i \(0.521405\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.828427 −0.176621
\(23\) 1.17157 0.244290 0.122145 0.992512i \(-0.461023\pi\)
0.122145 + 0.992512i \(0.461023\pi\)
\(24\) 3.41421 0.696923
\(25\) 0 0
\(26\) −4.82843 −0.946932
\(27\) 19.3137 3.71692
\(28\) 0 0
\(29\) −4.82843 −0.896616 −0.448308 0.893879i \(-0.647973\pi\)
−0.448308 + 0.893879i \(0.647973\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.82843 −0.492366
\(34\) 2.58579 0.443459
\(35\) 0 0
\(36\) 8.65685 1.44281
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) −0.585786 −0.0950271
\(39\) −16.4853 −2.63976
\(40\) 0 0
\(41\) 3.07107 0.479620 0.239810 0.970820i \(-0.422915\pi\)
0.239810 + 0.970820i \(0.422915\pi\)
\(42\) 0 0
\(43\) 8.82843 1.34632 0.673161 0.739496i \(-0.264936\pi\)
0.673161 + 0.739496i \(0.264936\pi\)
\(44\) −0.828427 −0.124890
\(45\) 0 0
\(46\) 1.17157 0.172739
\(47\) 5.17157 0.754351 0.377176 0.926142i \(-0.376895\pi\)
0.377176 + 0.926142i \(0.376895\pi\)
\(48\) 3.41421 0.492799
\(49\) 0 0
\(50\) 0 0
\(51\) 8.82843 1.23623
\(52\) −4.82843 −0.669582
\(53\) −6.48528 −0.890822 −0.445411 0.895326i \(-0.646942\pi\)
−0.445411 + 0.895326i \(0.646942\pi\)
\(54\) 19.3137 2.62826
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) −4.82843 −0.634004
\(59\) −8.58579 −1.11777 −0.558887 0.829244i \(-0.688771\pi\)
−0.558887 + 0.829244i \(0.688771\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 2.82843 0.359211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.82843 −0.348155
\(67\) −1.65685 −0.202417 −0.101208 0.994865i \(-0.532271\pi\)
−0.101208 + 0.994865i \(0.532271\pi\)
\(68\) 2.58579 0.313573
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −4.48528 −0.532305 −0.266152 0.963931i \(-0.585752\pi\)
−0.266152 + 0.963931i \(0.585752\pi\)
\(72\) 8.65685 1.02022
\(73\) −9.41421 −1.10185 −0.550925 0.834555i \(-0.685725\pi\)
−0.550925 + 0.834555i \(0.685725\pi\)
\(74\) 7.65685 0.890091
\(75\) 0 0
\(76\) −0.585786 −0.0671943
\(77\) 0 0
\(78\) −16.4853 −1.86659
\(79\) −6.82843 −0.768258 −0.384129 0.923279i \(-0.625498\pi\)
−0.384129 + 0.923279i \(0.625498\pi\)
\(80\) 0 0
\(81\) 39.9706 4.44117
\(82\) 3.07107 0.339143
\(83\) −2.24264 −0.246162 −0.123081 0.992397i \(-0.539277\pi\)
−0.123081 + 0.992397i \(0.539277\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.82843 0.951994
\(87\) −16.4853 −1.76741
\(88\) −0.828427 −0.0883106
\(89\) −12.7279 −1.34916 −0.674579 0.738203i \(-0.735675\pi\)
−0.674579 + 0.738203i \(0.735675\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.17157 0.122145
\(93\) 9.65685 1.00137
\(94\) 5.17157 0.533407
\(95\) 0 0
\(96\) 3.41421 0.348462
\(97\) −7.75736 −0.787641 −0.393820 0.919187i \(-0.628847\pi\)
−0.393820 + 0.919187i \(0.628847\pi\)
\(98\) 0 0
\(99\) −7.17157 −0.720770
\(100\) 0 0
\(101\) −13.3137 −1.32476 −0.662382 0.749166i \(-0.730454\pi\)
−0.662382 + 0.749166i \(0.730454\pi\)
\(102\) 8.82843 0.874145
\(103\) −14.8284 −1.46109 −0.730544 0.682865i \(-0.760734\pi\)
−0.730544 + 0.682865i \(0.760734\pi\)
\(104\) −4.82843 −0.473466
\(105\) 0 0
\(106\) −6.48528 −0.629906
\(107\) −9.65685 −0.933563 −0.466782 0.884373i \(-0.654587\pi\)
−0.466782 + 0.884373i \(0.654587\pi\)
\(108\) 19.3137 1.85846
\(109\) 2.48528 0.238047 0.119023 0.992891i \(-0.462024\pi\)
0.119023 + 0.992891i \(0.462024\pi\)
\(110\) 0 0
\(111\) 26.1421 2.48130
\(112\) 0 0
\(113\) 15.3137 1.44059 0.720296 0.693667i \(-0.244006\pi\)
0.720296 + 0.693667i \(0.244006\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −4.82843 −0.448308
\(117\) −41.7990 −3.86432
\(118\) −8.58579 −0.790386
\(119\) 0 0
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) −9.31371 −0.843224
\(123\) 10.4853 0.945426
\(124\) 2.82843 0.254000
\(125\) 0 0
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) 1.00000 0.0883883
\(129\) 30.1421 2.65387
\(130\) 0 0
\(131\) 6.24264 0.545422 0.272711 0.962096i \(-0.412080\pi\)
0.272711 + 0.962096i \(0.412080\pi\)
\(132\) −2.82843 −0.246183
\(133\) 0 0
\(134\) −1.65685 −0.143130
\(135\) 0 0
\(136\) 2.58579 0.221729
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 4.00000 0.340503
\(139\) −19.8995 −1.68785 −0.843927 0.536459i \(-0.819762\pi\)
−0.843927 + 0.536459i \(0.819762\pi\)
\(140\) 0 0
\(141\) 17.6569 1.48698
\(142\) −4.48528 −0.376396
\(143\) 4.00000 0.334497
\(144\) 8.65685 0.721405
\(145\) 0 0
\(146\) −9.41421 −0.779126
\(147\) 0 0
\(148\) 7.65685 0.629390
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −11.3137 −0.920697 −0.460348 0.887738i \(-0.652275\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(152\) −0.585786 −0.0475136
\(153\) 22.3848 1.80970
\(154\) 0 0
\(155\) 0 0
\(156\) −16.4853 −1.31988
\(157\) 6.48528 0.517582 0.258791 0.965933i \(-0.416676\pi\)
0.258791 + 0.965933i \(0.416676\pi\)
\(158\) −6.82843 −0.543240
\(159\) −22.1421 −1.75599
\(160\) 0 0
\(161\) 0 0
\(162\) 39.9706 3.14038
\(163\) 20.1421 1.57765 0.788827 0.614615i \(-0.210689\pi\)
0.788827 + 0.614615i \(0.210689\pi\)
\(164\) 3.07107 0.239810
\(165\) 0 0
\(166\) −2.24264 −0.174063
\(167\) 15.7990 1.22256 0.611281 0.791413i \(-0.290654\pi\)
0.611281 + 0.791413i \(0.290654\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) −5.07107 −0.387794
\(172\) 8.82843 0.673161
\(173\) −8.82843 −0.671213 −0.335606 0.942002i \(-0.608941\pi\)
−0.335606 + 0.942002i \(0.608941\pi\)
\(174\) −16.4853 −1.24975
\(175\) 0 0
\(176\) −0.828427 −0.0624450
\(177\) −29.3137 −2.20335
\(178\) −12.7279 −0.953998
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 2.48528 0.184730 0.0923648 0.995725i \(-0.470557\pi\)
0.0923648 + 0.995725i \(0.470557\pi\)
\(182\) 0 0
\(183\) −31.7990 −2.35065
\(184\) 1.17157 0.0863695
\(185\) 0 0
\(186\) 9.65685 0.708075
\(187\) −2.14214 −0.156648
\(188\) 5.17157 0.377176
\(189\) 0 0
\(190\) 0 0
\(191\) 10.1421 0.733859 0.366930 0.930249i \(-0.380409\pi\)
0.366930 + 0.930249i \(0.380409\pi\)
\(192\) 3.41421 0.246400
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) −7.75736 −0.556946
\(195\) 0 0
\(196\) 0 0
\(197\) 25.7990 1.83810 0.919051 0.394139i \(-0.128957\pi\)
0.919051 + 0.394139i \(0.128957\pi\)
\(198\) −7.17157 −0.509661
\(199\) −16.4853 −1.16861 −0.584305 0.811534i \(-0.698633\pi\)
−0.584305 + 0.811534i \(0.698633\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) −13.3137 −0.936749
\(203\) 0 0
\(204\) 8.82843 0.618114
\(205\) 0 0
\(206\) −14.8284 −1.03315
\(207\) 10.1421 0.704927
\(208\) −4.82843 −0.334791
\(209\) 0.485281 0.0335676
\(210\) 0 0
\(211\) 18.6274 1.28236 0.641182 0.767389i \(-0.278444\pi\)
0.641182 + 0.767389i \(0.278444\pi\)
\(212\) −6.48528 −0.445411
\(213\) −15.3137 −1.04928
\(214\) −9.65685 −0.660129
\(215\) 0 0
\(216\) 19.3137 1.31413
\(217\) 0 0
\(218\) 2.48528 0.168324
\(219\) −32.1421 −2.17196
\(220\) 0 0
\(221\) −12.4853 −0.839851
\(222\) 26.1421 1.75455
\(223\) −7.31371 −0.489762 −0.244881 0.969553i \(-0.578749\pi\)
−0.244881 + 0.969553i \(0.578749\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 15.3137 1.01865
\(227\) 18.2426 1.21081 0.605403 0.795919i \(-0.293012\pi\)
0.605403 + 0.795919i \(0.293012\pi\)
\(228\) −2.00000 −0.132453
\(229\) 16.1421 1.06670 0.533351 0.845894i \(-0.320932\pi\)
0.533351 + 0.845894i \(0.320932\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.82843 −0.317002
\(233\) −23.3137 −1.52733 −0.763666 0.645612i \(-0.776603\pi\)
−0.763666 + 0.645612i \(0.776603\pi\)
\(234\) −41.7990 −2.73249
\(235\) 0 0
\(236\) −8.58579 −0.558887
\(237\) −23.3137 −1.51439
\(238\) 0 0
\(239\) 1.65685 0.107173 0.0535865 0.998563i \(-0.482935\pi\)
0.0535865 + 0.998563i \(0.482935\pi\)
\(240\) 0 0
\(241\) 13.4142 0.864085 0.432043 0.901853i \(-0.357793\pi\)
0.432043 + 0.901853i \(0.357793\pi\)
\(242\) −10.3137 −0.662990
\(243\) 78.5269 5.03750
\(244\) −9.31371 −0.596249
\(245\) 0 0
\(246\) 10.4853 0.668517
\(247\) 2.82843 0.179969
\(248\) 2.82843 0.179605
\(249\) −7.65685 −0.485233
\(250\) 0 0
\(251\) −0.585786 −0.0369745 −0.0184873 0.999829i \(-0.505885\pi\)
−0.0184873 + 0.999829i \(0.505885\pi\)
\(252\) 0 0
\(253\) −0.970563 −0.0610188
\(254\) −2.82843 −0.177471
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.89949 −0.617514 −0.308757 0.951141i \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) 30.1421 1.87657
\(259\) 0 0
\(260\) 0 0
\(261\) −41.7990 −2.58729
\(262\) 6.24264 0.385672
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) −2.82843 −0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) −43.4558 −2.65945
\(268\) −1.65685 −0.101208
\(269\) −18.4853 −1.12707 −0.563534 0.826093i \(-0.690559\pi\)
−0.563534 + 0.826093i \(0.690559\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 2.58579 0.156786
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 8.14214 0.489214 0.244607 0.969622i \(-0.421341\pi\)
0.244607 + 0.969622i \(0.421341\pi\)
\(278\) −19.8995 −1.19349
\(279\) 24.4853 1.46590
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 17.6569 1.05145
\(283\) −2.24264 −0.133311 −0.0666556 0.997776i \(-0.521233\pi\)
−0.0666556 + 0.997776i \(0.521233\pi\)
\(284\) −4.48528 −0.266152
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 8.65685 0.510110
\(289\) −10.3137 −0.606689
\(290\) 0 0
\(291\) −26.4853 −1.55259
\(292\) −9.41421 −0.550925
\(293\) −8.34315 −0.487412 −0.243706 0.969849i \(-0.578363\pi\)
−0.243706 + 0.969849i \(0.578363\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.65685 0.445046
\(297\) −16.0000 −0.928414
\(298\) −6.00000 −0.347571
\(299\) −5.65685 −0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) −11.3137 −0.651031
\(303\) −45.4558 −2.61137
\(304\) −0.585786 −0.0335972
\(305\) 0 0
\(306\) 22.3848 1.27965
\(307\) 14.9289 0.852039 0.426020 0.904714i \(-0.359915\pi\)
0.426020 + 0.904714i \(0.359915\pi\)
\(308\) 0 0
\(309\) −50.6274 −2.88009
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) −16.4853 −0.933295
\(313\) 14.3848 0.813076 0.406538 0.913634i \(-0.366736\pi\)
0.406538 + 0.913634i \(0.366736\pi\)
\(314\) 6.48528 0.365986
\(315\) 0 0
\(316\) −6.82843 −0.384129
\(317\) −10.4853 −0.588912 −0.294456 0.955665i \(-0.595138\pi\)
−0.294456 + 0.955665i \(0.595138\pi\)
\(318\) −22.1421 −1.24167
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −32.9706 −1.84024
\(322\) 0 0
\(323\) −1.51472 −0.0842812
\(324\) 39.9706 2.22059
\(325\) 0 0
\(326\) 20.1421 1.11557
\(327\) 8.48528 0.469237
\(328\) 3.07107 0.169571
\(329\) 0 0
\(330\) 0 0
\(331\) 33.7990 1.85776 0.928880 0.370380i \(-0.120773\pi\)
0.928880 + 0.370380i \(0.120773\pi\)
\(332\) −2.24264 −0.123081
\(333\) 66.2843 3.63236
\(334\) 15.7990 0.864482
\(335\) 0 0
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 10.3137 0.560992
\(339\) 52.2843 2.83969
\(340\) 0 0
\(341\) −2.34315 −0.126888
\(342\) −5.07107 −0.274212
\(343\) 0 0
\(344\) 8.82843 0.475997
\(345\) 0 0
\(346\) −8.82843 −0.474619
\(347\) 3.17157 0.170259 0.0851295 0.996370i \(-0.472870\pi\)
0.0851295 + 0.996370i \(0.472870\pi\)
\(348\) −16.4853 −0.883704
\(349\) −2.48528 −0.133034 −0.0665170 0.997785i \(-0.521189\pi\)
−0.0665170 + 0.997785i \(0.521189\pi\)
\(350\) 0 0
\(351\) −93.2548 −4.97757
\(352\) −0.828427 −0.0441553
\(353\) −2.38478 −0.126929 −0.0634644 0.997984i \(-0.520215\pi\)
−0.0634644 + 0.997984i \(0.520215\pi\)
\(354\) −29.3137 −1.55801
\(355\) 0 0
\(356\) −12.7279 −0.674579
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −18.6569 −0.981940
\(362\) 2.48528 0.130623
\(363\) −35.2132 −1.84821
\(364\) 0 0
\(365\) 0 0
\(366\) −31.7990 −1.66216
\(367\) 24.9706 1.30345 0.651726 0.758454i \(-0.274045\pi\)
0.651726 + 0.758454i \(0.274045\pi\)
\(368\) 1.17157 0.0610725
\(369\) 26.5858 1.38400
\(370\) 0 0
\(371\) 0 0
\(372\) 9.65685 0.500685
\(373\) 30.4853 1.57847 0.789234 0.614093i \(-0.210478\pi\)
0.789234 + 0.614093i \(0.210478\pi\)
\(374\) −2.14214 −0.110767
\(375\) 0 0
\(376\) 5.17157 0.266704
\(377\) 23.3137 1.20072
\(378\) 0 0
\(379\) 34.4853 1.77139 0.885695 0.464268i \(-0.153682\pi\)
0.885695 + 0.464268i \(0.153682\pi\)
\(380\) 0 0
\(381\) −9.65685 −0.494736
\(382\) 10.1421 0.518917
\(383\) −32.4853 −1.65992 −0.829960 0.557823i \(-0.811637\pi\)
−0.829960 + 0.557823i \(0.811637\pi\)
\(384\) 3.41421 0.174231
\(385\) 0 0
\(386\) −5.65685 −0.287926
\(387\) 76.4264 3.88497
\(388\) −7.75736 −0.393820
\(389\) 28.1421 1.42686 0.713431 0.700725i \(-0.247140\pi\)
0.713431 + 0.700725i \(0.247140\pi\)
\(390\) 0 0
\(391\) 3.02944 0.153205
\(392\) 0 0
\(393\) 21.3137 1.07513
\(394\) 25.7990 1.29973
\(395\) 0 0
\(396\) −7.17157 −0.360385
\(397\) 33.7990 1.69632 0.848161 0.529738i \(-0.177710\pi\)
0.848161 + 0.529738i \(0.177710\pi\)
\(398\) −16.4853 −0.826332
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −5.65685 −0.282138
\(403\) −13.6569 −0.680296
\(404\) −13.3137 −0.662382
\(405\) 0 0
\(406\) 0 0
\(407\) −6.34315 −0.314418
\(408\) 8.82843 0.437072
\(409\) 10.5858 0.523433 0.261717 0.965145i \(-0.415711\pi\)
0.261717 + 0.965145i \(0.415711\pi\)
\(410\) 0 0
\(411\) 54.6274 2.69457
\(412\) −14.8284 −0.730544
\(413\) 0 0
\(414\) 10.1421 0.498459
\(415\) 0 0
\(416\) −4.82843 −0.236733
\(417\) −67.9411 −3.32709
\(418\) 0.485281 0.0237359
\(419\) 20.8701 1.01957 0.509785 0.860302i \(-0.329725\pi\)
0.509785 + 0.860302i \(0.329725\pi\)
\(420\) 0 0
\(421\) 17.3137 0.843819 0.421909 0.906638i \(-0.361360\pi\)
0.421909 + 0.906638i \(0.361360\pi\)
\(422\) 18.6274 0.906768
\(423\) 44.7696 2.17677
\(424\) −6.48528 −0.314953
\(425\) 0 0
\(426\) −15.3137 −0.741952
\(427\) 0 0
\(428\) −9.65685 −0.466782
\(429\) 13.6569 0.659359
\(430\) 0 0
\(431\) −22.3431 −1.07623 −0.538116 0.842871i \(-0.680864\pi\)
−0.538116 + 0.842871i \(0.680864\pi\)
\(432\) 19.3137 0.929231
\(433\) 10.5858 0.508720 0.254360 0.967110i \(-0.418135\pi\)
0.254360 + 0.967110i \(0.418135\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.48528 0.119023
\(437\) −0.686292 −0.0328298
\(438\) −32.1421 −1.53581
\(439\) 24.9706 1.19178 0.595890 0.803066i \(-0.296799\pi\)
0.595890 + 0.803066i \(0.296799\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.4853 −0.593864
\(443\) −3.02944 −0.143933 −0.0719665 0.997407i \(-0.522927\pi\)
−0.0719665 + 0.997407i \(0.522927\pi\)
\(444\) 26.1421 1.24065
\(445\) 0 0
\(446\) −7.31371 −0.346314
\(447\) −20.4853 −0.968921
\(448\) 0 0
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) −2.54416 −0.119800
\(452\) 15.3137 0.720296
\(453\) −38.6274 −1.81487
\(454\) 18.2426 0.856170
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −21.6569 −1.01306 −0.506532 0.862221i \(-0.669073\pi\)
−0.506532 + 0.862221i \(0.669073\pi\)
\(458\) 16.1421 0.754272
\(459\) 49.9411 2.33105
\(460\) 0 0
\(461\) 12.8284 0.597479 0.298740 0.954335i \(-0.403434\pi\)
0.298740 + 0.954335i \(0.403434\pi\)
\(462\) 0 0
\(463\) 16.9706 0.788689 0.394344 0.918963i \(-0.370972\pi\)
0.394344 + 0.918963i \(0.370972\pi\)
\(464\) −4.82843 −0.224154
\(465\) 0 0
\(466\) −23.3137 −1.07999
\(467\) −15.8995 −0.735741 −0.367870 0.929877i \(-0.619913\pi\)
−0.367870 + 0.929877i \(0.619913\pi\)
\(468\) −41.7990 −1.93216
\(469\) 0 0
\(470\) 0 0
\(471\) 22.1421 1.02026
\(472\) −8.58579 −0.395193
\(473\) −7.31371 −0.336285
\(474\) −23.3137 −1.07083
\(475\) 0 0
\(476\) 0 0
\(477\) −56.1421 −2.57057
\(478\) 1.65685 0.0757827
\(479\) 17.1716 0.784589 0.392295 0.919840i \(-0.371681\pi\)
0.392295 + 0.919840i \(0.371681\pi\)
\(480\) 0 0
\(481\) −36.9706 −1.68571
\(482\) 13.4142 0.611001
\(483\) 0 0
\(484\) −10.3137 −0.468805
\(485\) 0 0
\(486\) 78.5269 3.56205
\(487\) −31.7990 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(488\) −9.31371 −0.421612
\(489\) 68.7696 3.10987
\(490\) 0 0
\(491\) 32.2843 1.45697 0.728484 0.685062i \(-0.240225\pi\)
0.728484 + 0.685062i \(0.240225\pi\)
\(492\) 10.4853 0.472713
\(493\) −12.4853 −0.562309
\(494\) 2.82843 0.127257
\(495\) 0 0
\(496\) 2.82843 0.127000
\(497\) 0 0
\(498\) −7.65685 −0.343112
\(499\) 30.3431 1.35835 0.679173 0.733978i \(-0.262339\pi\)
0.679173 + 0.733978i \(0.262339\pi\)
\(500\) 0 0
\(501\) 53.9411 2.40991
\(502\) −0.585786 −0.0261449
\(503\) 17.6569 0.787280 0.393640 0.919265i \(-0.371216\pi\)
0.393640 + 0.919265i \(0.371216\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.970563 −0.0431468
\(507\) 35.2132 1.56387
\(508\) −2.82843 −0.125491
\(509\) −5.79899 −0.257036 −0.128518 0.991707i \(-0.541022\pi\)
−0.128518 + 0.991707i \(0.541022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −11.3137 −0.499512
\(514\) −9.89949 −0.436648
\(515\) 0 0
\(516\) 30.1421 1.32693
\(517\) −4.28427 −0.188422
\(518\) 0 0
\(519\) −30.1421 −1.32309
\(520\) 0 0
\(521\) −19.0711 −0.835519 −0.417759 0.908558i \(-0.637184\pi\)
−0.417759 + 0.908558i \(0.637184\pi\)
\(522\) −41.7990 −1.82949
\(523\) 23.8995 1.04505 0.522526 0.852623i \(-0.324990\pi\)
0.522526 + 0.852623i \(0.324990\pi\)
\(524\) 6.24264 0.272711
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) 7.31371 0.318590
\(528\) −2.82843 −0.123091
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) −74.3259 −3.22547
\(532\) 0 0
\(533\) −14.8284 −0.642290
\(534\) −43.4558 −1.88052
\(535\) 0 0
\(536\) −1.65685 −0.0715652
\(537\) 13.6569 0.589337
\(538\) −18.4853 −0.796957
\(539\) 0 0
\(540\) 0 0
\(541\) −14.9706 −0.643635 −0.321817 0.946802i \(-0.604294\pi\)
−0.321817 + 0.946802i \(0.604294\pi\)
\(542\) −12.0000 −0.515444
\(543\) 8.48528 0.364138
\(544\) 2.58579 0.110865
\(545\) 0 0
\(546\) 0 0
\(547\) 10.4853 0.448318 0.224159 0.974553i \(-0.428036\pi\)
0.224159 + 0.974553i \(0.428036\pi\)
\(548\) 16.0000 0.683486
\(549\) −80.6274 −3.44109
\(550\) 0 0
\(551\) 2.82843 0.120495
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) 8.14214 0.345926
\(555\) 0 0
\(556\) −19.8995 −0.843927
\(557\) −15.1716 −0.642840 −0.321420 0.946937i \(-0.604160\pi\)
−0.321420 + 0.946937i \(0.604160\pi\)
\(558\) 24.4853 1.03654
\(559\) −42.6274 −1.80295
\(560\) 0 0
\(561\) −7.31371 −0.308785
\(562\) 8.00000 0.337460
\(563\) −36.5858 −1.54191 −0.770954 0.636891i \(-0.780220\pi\)
−0.770954 + 0.636891i \(0.780220\pi\)
\(564\) 17.6569 0.743488
\(565\) 0 0
\(566\) −2.24264 −0.0942652
\(567\) 0 0
\(568\) −4.48528 −0.188198
\(569\) −29.3137 −1.22889 −0.614447 0.788958i \(-0.710621\pi\)
−0.614447 + 0.788958i \(0.710621\pi\)
\(570\) 0 0
\(571\) −2.20101 −0.0921094 −0.0460547 0.998939i \(-0.514665\pi\)
−0.0460547 + 0.998939i \(0.514665\pi\)
\(572\) 4.00000 0.167248
\(573\) 34.6274 1.44658
\(574\) 0 0
\(575\) 0 0
\(576\) 8.65685 0.360702
\(577\) 6.10051 0.253967 0.126984 0.991905i \(-0.459470\pi\)
0.126984 + 0.991905i \(0.459470\pi\)
\(578\) −10.3137 −0.428994
\(579\) −19.3137 −0.802650
\(580\) 0 0
\(581\) 0 0
\(582\) −26.4853 −1.09785
\(583\) 5.37258 0.222510
\(584\) −9.41421 −0.389563
\(585\) 0 0
\(586\) −8.34315 −0.344652
\(587\) 17.0711 0.704598 0.352299 0.935887i \(-0.385400\pi\)
0.352299 + 0.935887i \(0.385400\pi\)
\(588\) 0 0
\(589\) −1.65685 −0.0682695
\(590\) 0 0
\(591\) 88.0833 3.62326
\(592\) 7.65685 0.314695
\(593\) 3.27208 0.134368 0.0671841 0.997741i \(-0.478599\pi\)
0.0671841 + 0.997741i \(0.478599\pi\)
\(594\) −16.0000 −0.656488
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −56.2843 −2.30356
\(598\) −5.65685 −0.231326
\(599\) −10.8284 −0.442438 −0.221219 0.975224i \(-0.571003\pi\)
−0.221219 + 0.975224i \(0.571003\pi\)
\(600\) 0 0
\(601\) −6.58579 −0.268640 −0.134320 0.990938i \(-0.542885\pi\)
−0.134320 + 0.990938i \(0.542885\pi\)
\(602\) 0 0
\(603\) −14.3431 −0.584098
\(604\) −11.3137 −0.460348
\(605\) 0 0
\(606\) −45.4558 −1.84652
\(607\) 16.2843 0.660958 0.330479 0.943813i \(-0.392790\pi\)
0.330479 + 0.943813i \(0.392790\pi\)
\(608\) −0.585786 −0.0237568
\(609\) 0 0
\(610\) 0 0
\(611\) −24.9706 −1.01020
\(612\) 22.3848 0.904851
\(613\) 12.3431 0.498535 0.249267 0.968435i \(-0.419810\pi\)
0.249267 + 0.968435i \(0.419810\pi\)
\(614\) 14.9289 0.602483
\(615\) 0 0
\(616\) 0 0
\(617\) 33.3137 1.34116 0.670580 0.741837i \(-0.266045\pi\)
0.670580 + 0.741837i \(0.266045\pi\)
\(618\) −50.6274 −2.03653
\(619\) −29.0711 −1.16846 −0.584232 0.811586i \(-0.698604\pi\)
−0.584232 + 0.811586i \(0.698604\pi\)
\(620\) 0 0
\(621\) 22.6274 0.908007
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) −16.4853 −0.659939
\(625\) 0 0
\(626\) 14.3848 0.574931
\(627\) 1.65685 0.0661684
\(628\) 6.48528 0.258791
\(629\) 19.7990 0.789437
\(630\) 0 0
\(631\) −12.4853 −0.497031 −0.248516 0.968628i \(-0.579943\pi\)
−0.248516 + 0.968628i \(0.579943\pi\)
\(632\) −6.82843 −0.271620
\(633\) 63.5980 2.52779
\(634\) −10.4853 −0.416424
\(635\) 0 0
\(636\) −22.1421 −0.877993
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) −38.8284 −1.53603
\(640\) 0 0
\(641\) −24.6274 −0.972724 −0.486362 0.873757i \(-0.661676\pi\)
−0.486362 + 0.873757i \(0.661676\pi\)
\(642\) −32.9706 −1.30124
\(643\) 4.78680 0.188773 0.0943864 0.995536i \(-0.469911\pi\)
0.0943864 + 0.995536i \(0.469911\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.51472 −0.0595958
\(647\) −23.1127 −0.908654 −0.454327 0.890835i \(-0.650120\pi\)
−0.454327 + 0.890835i \(0.650120\pi\)
\(648\) 39.9706 1.57019
\(649\) 7.11270 0.279198
\(650\) 0 0
\(651\) 0 0
\(652\) 20.1421 0.788827
\(653\) −4.34315 −0.169960 −0.0849802 0.996383i \(-0.527083\pi\)
−0.0849802 + 0.996383i \(0.527083\pi\)
\(654\) 8.48528 0.331801
\(655\) 0 0
\(656\) 3.07107 0.119905
\(657\) −81.4975 −3.17952
\(658\) 0 0
\(659\) −27.1716 −1.05845 −0.529227 0.848480i \(-0.677518\pi\)
−0.529227 + 0.848480i \(0.677518\pi\)
\(660\) 0 0
\(661\) 38.2843 1.48909 0.744543 0.667575i \(-0.232668\pi\)
0.744543 + 0.667575i \(0.232668\pi\)
\(662\) 33.7990 1.31364
\(663\) −42.6274 −1.65551
\(664\) −2.24264 −0.0870313
\(665\) 0 0
\(666\) 66.2843 2.56846
\(667\) −5.65685 −0.219034
\(668\) 15.7990 0.611281
\(669\) −24.9706 −0.965418
\(670\) 0 0
\(671\) 7.71573 0.297862
\(672\) 0 0
\(673\) 48.0000 1.85026 0.925132 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) 10.3137 0.396681
\(677\) 39.4558 1.51641 0.758206 0.652015i \(-0.226076\pi\)
0.758206 + 0.652015i \(0.226076\pi\)
\(678\) 52.2843 2.00797
\(679\) 0 0
\(680\) 0 0
\(681\) 62.2843 2.38674
\(682\) −2.34315 −0.0897237
\(683\) −33.6569 −1.28784 −0.643922 0.765091i \(-0.722694\pi\)
−0.643922 + 0.765091i \(0.722694\pi\)
\(684\) −5.07107 −0.193897
\(685\) 0 0
\(686\) 0 0
\(687\) 55.1127 2.10268
\(688\) 8.82843 0.336581
\(689\) 31.3137 1.19296
\(690\) 0 0
\(691\) −1.75736 −0.0668531 −0.0334265 0.999441i \(-0.510642\pi\)
−0.0334265 + 0.999441i \(0.510642\pi\)
\(692\) −8.82843 −0.335606
\(693\) 0 0
\(694\) 3.17157 0.120391
\(695\) 0 0
\(696\) −16.4853 −0.624873
\(697\) 7.94113 0.300792
\(698\) −2.48528 −0.0940693
\(699\) −79.5980 −3.01067
\(700\) 0 0
\(701\) 2.48528 0.0938678 0.0469339 0.998898i \(-0.485055\pi\)
0.0469339 + 0.998898i \(0.485055\pi\)
\(702\) −93.2548 −3.51968
\(703\) −4.48528 −0.169166
\(704\) −0.828427 −0.0312225
\(705\) 0 0
\(706\) −2.38478 −0.0897522
\(707\) 0 0
\(708\) −29.3137 −1.10168
\(709\) 45.1127 1.69424 0.847121 0.531399i \(-0.178334\pi\)
0.847121 + 0.531399i \(0.178334\pi\)
\(710\) 0 0
\(711\) −59.1127 −2.21690
\(712\) −12.7279 −0.476999
\(713\) 3.31371 0.124099
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 5.65685 0.211259
\(718\) 28.2843 1.05556
\(719\) −41.4558 −1.54604 −0.773021 0.634380i \(-0.781255\pi\)
−0.773021 + 0.634380i \(0.781255\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.6569 −0.694336
\(723\) 45.7990 1.70328
\(724\) 2.48528 0.0923648
\(725\) 0 0
\(726\) −35.2132 −1.30688
\(727\) −3.51472 −0.130354 −0.0651768 0.997874i \(-0.520761\pi\)
−0.0651768 + 0.997874i \(0.520761\pi\)
\(728\) 0 0
\(729\) 148.196 5.48874
\(730\) 0 0
\(731\) 22.8284 0.844340
\(732\) −31.7990 −1.17532
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 24.9706 0.921680
\(735\) 0 0
\(736\) 1.17157 0.0431847
\(737\) 1.37258 0.0505597
\(738\) 26.5858 0.978636
\(739\) 3.17157 0.116668 0.0583341 0.998297i \(-0.481421\pi\)
0.0583341 + 0.998297i \(0.481421\pi\)
\(740\) 0 0
\(741\) 9.65685 0.354753
\(742\) 0 0
\(743\) −51.7990 −1.90032 −0.950160 0.311762i \(-0.899081\pi\)
−0.950160 + 0.311762i \(0.899081\pi\)
\(744\) 9.65685 0.354037
\(745\) 0 0
\(746\) 30.4853 1.11615
\(747\) −19.4142 −0.710329
\(748\) −2.14214 −0.0783242
\(749\) 0 0
\(750\) 0 0
\(751\) −39.3137 −1.43458 −0.717289 0.696776i \(-0.754617\pi\)
−0.717289 + 0.696776i \(0.754617\pi\)
\(752\) 5.17157 0.188588
\(753\) −2.00000 −0.0728841
\(754\) 23.3137 0.849035
\(755\) 0 0
\(756\) 0 0
\(757\) −3.65685 −0.132911 −0.0664553 0.997789i \(-0.521169\pi\)
−0.0664553 + 0.997789i \(0.521169\pi\)
\(758\) 34.4853 1.25256
\(759\) −3.31371 −0.120280
\(760\) 0 0
\(761\) −22.3848 −0.811448 −0.405724 0.913996i \(-0.632980\pi\)
−0.405724 + 0.913996i \(0.632980\pi\)
\(762\) −9.65685 −0.349831
\(763\) 0 0
\(764\) 10.1421 0.366930
\(765\) 0 0
\(766\) −32.4853 −1.17374
\(767\) 41.4558 1.49688
\(768\) 3.41421 0.123200
\(769\) −19.5563 −0.705220 −0.352610 0.935770i \(-0.614706\pi\)
−0.352610 + 0.935770i \(0.614706\pi\)
\(770\) 0 0
\(771\) −33.7990 −1.21724
\(772\) −5.65685 −0.203595
\(773\) −2.00000 −0.0719350 −0.0359675 0.999353i \(-0.511451\pi\)
−0.0359675 + 0.999353i \(0.511451\pi\)
\(774\) 76.4264 2.74709
\(775\) 0 0
\(776\) −7.75736 −0.278473
\(777\) 0 0
\(778\) 28.1421 1.00894
\(779\) −1.79899 −0.0644555
\(780\) 0 0
\(781\) 3.71573 0.132959
\(782\) 3.02944 0.108332
\(783\) −93.2548 −3.33266
\(784\) 0 0
\(785\) 0 0
\(786\) 21.3137 0.760235
\(787\) −1.27208 −0.0453447 −0.0226723 0.999743i \(-0.507217\pi\)
−0.0226723 + 0.999743i \(0.507217\pi\)
\(788\) 25.7990 0.919051
\(789\) −95.5980 −3.40338
\(790\) 0 0
\(791\) 0 0
\(792\) −7.17157 −0.254831
\(793\) 44.9706 1.59695
\(794\) 33.7990 1.19948
\(795\) 0 0
\(796\) −16.4853 −0.584305
\(797\) −41.7990 −1.48060 −0.740298 0.672279i \(-0.765316\pi\)
−0.740298 + 0.672279i \(0.765316\pi\)
\(798\) 0 0
\(799\) 13.3726 0.473088
\(800\) 0 0
\(801\) −110.184 −3.89315
\(802\) −6.00000 −0.211867
\(803\) 7.79899 0.275220
\(804\) −5.65685 −0.199502
\(805\) 0 0
\(806\) −13.6569 −0.481042
\(807\) −63.1127 −2.22167
\(808\) −13.3137 −0.468375
\(809\) 3.02944 0.106509 0.0532547 0.998581i \(-0.483040\pi\)
0.0532547 + 0.998581i \(0.483040\pi\)
\(810\) 0 0
\(811\) −32.5858 −1.14424 −0.572121 0.820169i \(-0.693879\pi\)
−0.572121 + 0.820169i \(0.693879\pi\)
\(812\) 0 0
\(813\) −40.9706 −1.43690
\(814\) −6.34315 −0.222327
\(815\) 0 0
\(816\) 8.82843 0.309057
\(817\) −5.17157 −0.180930
\(818\) 10.5858 0.370123
\(819\) 0 0
\(820\) 0 0
\(821\) 17.3137 0.604253 0.302126 0.953268i \(-0.402304\pi\)
0.302126 + 0.953268i \(0.402304\pi\)
\(822\) 54.6274 1.90535
\(823\) −20.2843 −0.707065 −0.353533 0.935422i \(-0.615020\pi\)
−0.353533 + 0.935422i \(0.615020\pi\)
\(824\) −14.8284 −0.516573
\(825\) 0 0
\(826\) 0 0
\(827\) 5.37258 0.186823 0.0934115 0.995628i \(-0.470223\pi\)
0.0934115 + 0.995628i \(0.470223\pi\)
\(828\) 10.1421 0.352464
\(829\) 5.02944 0.174680 0.0873398 0.996179i \(-0.472163\pi\)
0.0873398 + 0.996179i \(0.472163\pi\)
\(830\) 0 0
\(831\) 27.7990 0.964336
\(832\) −4.82843 −0.167396
\(833\) 0 0
\(834\) −67.9411 −2.35261
\(835\) 0 0
\(836\) 0.485281 0.0167838
\(837\) 54.6274 1.88820
\(838\) 20.8701 0.720944
\(839\) 42.1421 1.45491 0.727454 0.686156i \(-0.240703\pi\)
0.727454 + 0.686156i \(0.240703\pi\)
\(840\) 0 0
\(841\) −5.68629 −0.196079
\(842\) 17.3137 0.596670
\(843\) 27.3137 0.940734
\(844\) 18.6274 0.641182
\(845\) 0 0
\(846\) 44.7696 1.53921
\(847\) 0 0
\(848\) −6.48528 −0.222705
\(849\) −7.65685 −0.262783
\(850\) 0 0
\(851\) 8.97056 0.307507
\(852\) −15.3137 −0.524639
\(853\) −43.1716 −1.47817 −0.739083 0.673614i \(-0.764741\pi\)
−0.739083 + 0.673614i \(0.764741\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.65685 −0.330064
\(857\) −4.92893 −0.168369 −0.0841846 0.996450i \(-0.526829\pi\)
−0.0841846 + 0.996450i \(0.526829\pi\)
\(858\) 13.6569 0.466237
\(859\) 7.21320 0.246111 0.123056 0.992400i \(-0.460731\pi\)
0.123056 + 0.992400i \(0.460731\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −22.3431 −0.761011
\(863\) 4.97056 0.169200 0.0846000 0.996415i \(-0.473039\pi\)
0.0846000 + 0.996415i \(0.473039\pi\)
\(864\) 19.3137 0.657066
\(865\) 0 0
\(866\) 10.5858 0.359720
\(867\) −35.2132 −1.19590
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) 2.48528 0.0841622
\(873\) −67.1543 −2.27283
\(874\) −0.686292 −0.0232142
\(875\) 0 0
\(876\) −32.1421 −1.08598
\(877\) 30.2843 1.02263 0.511314 0.859394i \(-0.329159\pi\)
0.511314 + 0.859394i \(0.329159\pi\)
\(878\) 24.9706 0.842716
\(879\) −28.4853 −0.960785
\(880\) 0 0
\(881\) 2.38478 0.0803452 0.0401726 0.999193i \(-0.487209\pi\)
0.0401726 + 0.999193i \(0.487209\pi\)
\(882\) 0 0
\(883\) 41.6569 1.40186 0.700932 0.713228i \(-0.252767\pi\)
0.700932 + 0.713228i \(0.252767\pi\)
\(884\) −12.4853 −0.419925
\(885\) 0 0
\(886\) −3.02944 −0.101776
\(887\) −55.1127 −1.85050 −0.925252 0.379354i \(-0.876146\pi\)
−0.925252 + 0.379354i \(0.876146\pi\)
\(888\) 26.1421 0.877273
\(889\) 0 0
\(890\) 0 0
\(891\) −33.1127 −1.10932
\(892\) −7.31371 −0.244881
\(893\) −3.02944 −0.101376
\(894\) −20.4853 −0.685130
\(895\) 0 0
\(896\) 0 0
\(897\) −19.3137 −0.644866
\(898\) −16.6274 −0.554864
\(899\) −13.6569 −0.455482
\(900\) 0 0
\(901\) −16.7696 −0.558675
\(902\) −2.54416 −0.0847111
\(903\) 0 0
\(904\) 15.3137 0.509326
\(905\) 0 0
\(906\) −38.6274 −1.28331
\(907\) −0.284271 −0.00943907 −0.00471954 0.999989i \(-0.501502\pi\)
−0.00471954 + 0.999989i \(0.501502\pi\)
\(908\) 18.2426 0.605403
\(909\) −115.255 −3.82276
\(910\) 0 0
\(911\) 36.2843 1.20215 0.601076 0.799192i \(-0.294739\pi\)
0.601076 + 0.799192i \(0.294739\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 1.85786 0.0614863
\(914\) −21.6569 −0.716345
\(915\) 0 0
\(916\) 16.1421 0.533351
\(917\) 0 0
\(918\) 49.9411 1.64830
\(919\) 15.5147 0.511783 0.255892 0.966705i \(-0.417631\pi\)
0.255892 + 0.966705i \(0.417631\pi\)
\(920\) 0 0
\(921\) 50.9706 1.67954
\(922\) 12.8284 0.422482
\(923\) 21.6569 0.712844
\(924\) 0 0
\(925\) 0 0
\(926\) 16.9706 0.557687
\(927\) −128.368 −4.21614
\(928\) −4.82843 −0.158501
\(929\) 17.2132 0.564747 0.282373 0.959305i \(-0.408878\pi\)
0.282373 + 0.959305i \(0.408878\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −23.3137 −0.763666
\(933\) −13.6569 −0.447105
\(934\) −15.8995 −0.520247
\(935\) 0 0
\(936\) −41.7990 −1.36624
\(937\) 20.2426 0.661298 0.330649 0.943754i \(-0.392732\pi\)
0.330649 + 0.943754i \(0.392732\pi\)
\(938\) 0 0
\(939\) 49.1127 1.60273
\(940\) 0 0
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 22.1421 0.721430
\(943\) 3.59798 0.117166
\(944\) −8.58579 −0.279444
\(945\) 0 0
\(946\) −7.31371 −0.237789
\(947\) 4.82843 0.156903 0.0784514 0.996918i \(-0.475002\pi\)
0.0784514 + 0.996918i \(0.475002\pi\)
\(948\) −23.3137 −0.757194
\(949\) 45.4558 1.47556
\(950\) 0 0
\(951\) −35.7990 −1.16086
\(952\) 0 0
\(953\) −0.343146 −0.0111156 −0.00555779 0.999985i \(-0.501769\pi\)
−0.00555779 + 0.999985i \(0.501769\pi\)
\(954\) −56.1421 −1.81767
\(955\) 0 0
\(956\) 1.65685 0.0535865
\(957\) 13.6569 0.441463
\(958\) 17.1716 0.554788
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) −36.9706 −1.19198
\(963\) −83.5980 −2.69391
\(964\) 13.4142 0.432043
\(965\) 0 0
\(966\) 0 0
\(967\) 37.4558 1.20450 0.602249 0.798308i \(-0.294271\pi\)
0.602249 + 0.798308i \(0.294271\pi\)
\(968\) −10.3137 −0.331495
\(969\) −5.17157 −0.166135
\(970\) 0 0
\(971\) 33.3553 1.07042 0.535212 0.844718i \(-0.320232\pi\)
0.535212 + 0.844718i \(0.320232\pi\)
\(972\) 78.5269 2.51875
\(973\) 0 0
\(974\) −31.7990 −1.01891
\(975\) 0 0
\(976\) −9.31371 −0.298125
\(977\) 12.6863 0.405870 0.202935 0.979192i \(-0.434952\pi\)
0.202935 + 0.979192i \(0.434952\pi\)
\(978\) 68.7696 2.19901
\(979\) 10.5442 0.336993
\(980\) 0 0
\(981\) 21.5147 0.686912
\(982\) 32.2843 1.03023
\(983\) −12.2010 −0.389152 −0.194576 0.980887i \(-0.562333\pi\)
−0.194576 + 0.980887i \(0.562333\pi\)
\(984\) 10.4853 0.334259
\(985\) 0 0
\(986\) −12.4853 −0.397612
\(987\) 0 0
\(988\) 2.82843 0.0899843
\(989\) 10.3431 0.328893
\(990\) 0 0
\(991\) 44.7696 1.42215 0.711076 0.703115i \(-0.248208\pi\)
0.711076 + 0.703115i \(0.248208\pi\)
\(992\) 2.82843 0.0898027
\(993\) 115.397 3.66201
\(994\) 0 0
\(995\) 0 0
\(996\) −7.65685 −0.242617
\(997\) −18.2843 −0.579069 −0.289534 0.957168i \(-0.593500\pi\)
−0.289534 + 0.957168i \(0.593500\pi\)
\(998\) 30.3431 0.960495
\(999\) 147.882 4.67879
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 2450.2.a.bs.1.2 2
5.2 odd 4 2450.2.c.w.99.3 4
5.3 odd 4 2450.2.c.w.99.2 4
5.4 even 2 490.2.a.l.1.1 2
7.6 odd 2 2450.2.a.bn.1.1 2
15.14 odd 2 4410.2.a.by.1.2 2
20.19 odd 2 3920.2.a.ca.1.2 2
35.4 even 6 490.2.e.j.471.2 4
35.9 even 6 490.2.e.j.361.2 4
35.13 even 4 2450.2.c.t.99.1 4
35.19 odd 6 490.2.e.i.361.1 4
35.24 odd 6 490.2.e.i.471.1 4
35.27 even 4 2450.2.c.t.99.4 4
35.34 odd 2 490.2.a.m.1.2 yes 2
105.104 even 2 4410.2.a.bt.1.2 2
140.139 even 2 3920.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.1 2 5.4 even 2
490.2.a.m.1.2 yes 2 35.34 odd 2
490.2.e.i.361.1 4 35.19 odd 6
490.2.e.i.471.1 4 35.24 odd 6
490.2.e.j.361.2 4 35.9 even 6
490.2.e.j.471.2 4 35.4 even 6
2450.2.a.bn.1.1 2 7.6 odd 2
2450.2.a.bs.1.2 2 1.1 even 1 trivial
2450.2.c.t.99.1 4 35.13 even 4
2450.2.c.t.99.4 4 35.27 even 4
2450.2.c.w.99.2 4 5.3 odd 4
2450.2.c.w.99.3 4 5.2 odd 4
3920.2.a.bm.1.1 2 140.139 even 2
3920.2.a.ca.1.2 2 20.19 odd 2
4410.2.a.bt.1.2 2 105.104 even 2
4410.2.a.by.1.2 2 15.14 odd 2