Properties

Label 3330.2.a.bb.1.2
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.37228 q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.37228 q^{11} -4.74456 q^{13} -1.37228 q^{14} +1.00000 q^{16} +5.37228 q^{17} -2.00000 q^{19} -1.00000 q^{20} -3.37228 q^{22} -6.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.37228 q^{28} -8.11684 q^{29} -2.62772 q^{31} -1.00000 q^{32} -5.37228 q^{34} -1.37228 q^{35} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -5.37228 q^{41} +7.37228 q^{43} +3.37228 q^{44} +6.74456 q^{46} +8.74456 q^{47} -5.11684 q^{49} -1.00000 q^{50} -4.74456 q^{52} -1.37228 q^{53} -3.37228 q^{55} -1.37228 q^{56} +8.11684 q^{58} -12.7446 q^{59} -5.37228 q^{61} +2.62772 q^{62} +1.00000 q^{64} +4.74456 q^{65} +4.74456 q^{67} +5.37228 q^{68} +1.37228 q^{70} +6.74456 q^{71} +8.74456 q^{73} -1.00000 q^{74} -2.00000 q^{76} +4.62772 q^{77} -4.74456 q^{79} -1.00000 q^{80} +5.37228 q^{82} +0.744563 q^{83} -5.37228 q^{85} -7.37228 q^{86} -3.37228 q^{88} -10.0000 q^{89} -6.51087 q^{91} -6.74456 q^{92} -8.74456 q^{94} +2.00000 q^{95} -0.116844 q^{97} +5.11684 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 3 q^{7} - 2 q^{8} + 2 q^{10} + q^{11} + 2 q^{13} + 3 q^{14} + 2 q^{16} + 5 q^{17} - 4 q^{19} - 2 q^{20} - q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{26} - 3 q^{28} + q^{29} - 11 q^{31} - 2 q^{32} - 5 q^{34} + 3 q^{35} + 2 q^{37} + 4 q^{38} + 2 q^{40} - 5 q^{41} + 9 q^{43} + q^{44} + 2 q^{46} + 6 q^{47} + 7 q^{49} - 2 q^{50} + 2 q^{52} + 3 q^{53} - q^{55} + 3 q^{56} - q^{58} - 14 q^{59} - 5 q^{61} + 11 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{67} + 5 q^{68} - 3 q^{70} + 2 q^{71} + 6 q^{73} - 2 q^{74} - 4 q^{76} + 15 q^{77} + 2 q^{79} - 2 q^{80} + 5 q^{82} - 10 q^{83} - 5 q^{85} - 9 q^{86} - q^{88} - 20 q^{89} - 36 q^{91} - 2 q^{92} - 6 q^{94} + 4 q^{95} + 17 q^{97} - 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.37228 0.518674 0.259337 0.965787i \(-0.416496\pi\)
0.259337 + 0.965787i \(0.416496\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.37228 1.01678 0.508391 0.861127i \(-0.330241\pi\)
0.508391 + 0.861127i \(0.330241\pi\)
\(12\) 0 0
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) −1.37228 −0.366758
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.37228 1.30297 0.651485 0.758662i \(-0.274146\pi\)
0.651485 + 0.758662i \(0.274146\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.37228 −0.718973
\(23\) −6.74456 −1.40634 −0.703169 0.711022i \(-0.748232\pi\)
−0.703169 + 0.711022i \(0.748232\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.74456 0.930485
\(27\) 0 0
\(28\) 1.37228 0.259337
\(29\) −8.11684 −1.50726 −0.753630 0.657299i \(-0.771699\pi\)
−0.753630 + 0.657299i \(0.771699\pi\)
\(30\) 0 0
\(31\) −2.62772 −0.471952 −0.235976 0.971759i \(-0.575829\pi\)
−0.235976 + 0.971759i \(0.575829\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −5.37228 −0.921339
\(35\) −1.37228 −0.231958
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −5.37228 −0.839009 −0.419505 0.907753i \(-0.637796\pi\)
−0.419505 + 0.907753i \(0.637796\pi\)
\(42\) 0 0
\(43\) 7.37228 1.12426 0.562131 0.827048i \(-0.309982\pi\)
0.562131 + 0.827048i \(0.309982\pi\)
\(44\) 3.37228 0.508391
\(45\) 0 0
\(46\) 6.74456 0.994432
\(47\) 8.74456 1.27553 0.637763 0.770233i \(-0.279860\pi\)
0.637763 + 0.770233i \(0.279860\pi\)
\(48\) 0 0
\(49\) −5.11684 −0.730978
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −4.74456 −0.657952
\(53\) −1.37228 −0.188497 −0.0942487 0.995549i \(-0.530045\pi\)
−0.0942487 + 0.995549i \(0.530045\pi\)
\(54\) 0 0
\(55\) −3.37228 −0.454718
\(56\) −1.37228 −0.183379
\(57\) 0 0
\(58\) 8.11684 1.06579
\(59\) −12.7446 −1.65920 −0.829600 0.558358i \(-0.811432\pi\)
−0.829600 + 0.558358i \(0.811432\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) 2.62772 0.333721
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.74456 0.588491
\(66\) 0 0
\(67\) 4.74456 0.579641 0.289820 0.957081i \(-0.406404\pi\)
0.289820 + 0.957081i \(0.406404\pi\)
\(68\) 5.37228 0.651485
\(69\) 0 0
\(70\) 1.37228 0.164019
\(71\) 6.74456 0.800432 0.400216 0.916421i \(-0.368935\pi\)
0.400216 + 0.916421i \(0.368935\pi\)
\(72\) 0 0
\(73\) 8.74456 1.02347 0.511737 0.859142i \(-0.329002\pi\)
0.511737 + 0.859142i \(0.329002\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 4.62772 0.527377
\(78\) 0 0
\(79\) −4.74456 −0.533805 −0.266903 0.963724i \(-0.586000\pi\)
−0.266903 + 0.963724i \(0.586000\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 5.37228 0.593269
\(83\) 0.744563 0.0817264 0.0408632 0.999165i \(-0.486989\pi\)
0.0408632 + 0.999165i \(0.486989\pi\)
\(84\) 0 0
\(85\) −5.37228 −0.582706
\(86\) −7.37228 −0.794974
\(87\) 0 0
\(88\) −3.37228 −0.359486
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −6.51087 −0.682525
\(92\) −6.74456 −0.703169
\(93\) 0 0
\(94\) −8.74456 −0.901933
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −0.116844 −0.0118637 −0.00593185 0.999982i \(-0.501888\pi\)
−0.00593185 + 0.999982i \(0.501888\pi\)
\(98\) 5.11684 0.516879
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 11.4891 1.14321 0.571605 0.820529i \(-0.306321\pi\)
0.571605 + 0.820529i \(0.306321\pi\)
\(102\) 0 0
\(103\) −9.48913 −0.934991 −0.467496 0.883995i \(-0.654844\pi\)
−0.467496 + 0.883995i \(0.654844\pi\)
\(104\) 4.74456 0.465243
\(105\) 0 0
\(106\) 1.37228 0.133288
\(107\) 3.48913 0.337306 0.168653 0.985675i \(-0.446058\pi\)
0.168653 + 0.985675i \(0.446058\pi\)
\(108\) 0 0
\(109\) 0.116844 0.0111916 0.00559581 0.999984i \(-0.498219\pi\)
0.00559581 + 0.999984i \(0.498219\pi\)
\(110\) 3.37228 0.321534
\(111\) 0 0
\(112\) 1.37228 0.129668
\(113\) −17.3723 −1.63425 −0.817123 0.576463i \(-0.804433\pi\)
−0.817123 + 0.576463i \(0.804433\pi\)
\(114\) 0 0
\(115\) 6.74456 0.628934
\(116\) −8.11684 −0.753630
\(117\) 0 0
\(118\) 12.7446 1.17323
\(119\) 7.37228 0.675816
\(120\) 0 0
\(121\) 0.372281 0.0338438
\(122\) 5.37228 0.486383
\(123\) 0 0
\(124\) −2.62772 −0.235976
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.7446 −1.48584 −0.742920 0.669380i \(-0.766560\pi\)
−0.742920 + 0.669380i \(0.766560\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −4.74456 −0.416126
\(131\) 3.25544 0.284429 0.142214 0.989836i \(-0.454578\pi\)
0.142214 + 0.989836i \(0.454578\pi\)
\(132\) 0 0
\(133\) −2.74456 −0.237984
\(134\) −4.74456 −0.409868
\(135\) 0 0
\(136\) −5.37228 −0.460669
\(137\) −16.7446 −1.43058 −0.715292 0.698825i \(-0.753706\pi\)
−0.715292 + 0.698825i \(0.753706\pi\)
\(138\) 0 0
\(139\) −1.88316 −0.159727 −0.0798636 0.996806i \(-0.525448\pi\)
−0.0798636 + 0.996806i \(0.525448\pi\)
\(140\) −1.37228 −0.115979
\(141\) 0 0
\(142\) −6.74456 −0.565991
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) 8.11684 0.674067
\(146\) −8.74456 −0.723705
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 11.4891 0.941226 0.470613 0.882340i \(-0.344033\pi\)
0.470613 + 0.882340i \(0.344033\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) −4.62772 −0.372912
\(155\) 2.62772 0.211063
\(156\) 0 0
\(157\) 17.3723 1.38646 0.693229 0.720717i \(-0.256187\pi\)
0.693229 + 0.720717i \(0.256187\pi\)
\(158\) 4.74456 0.377457
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −9.25544 −0.729431
\(162\) 0 0
\(163\) 19.3723 1.51735 0.758677 0.651467i \(-0.225846\pi\)
0.758677 + 0.651467i \(0.225846\pi\)
\(164\) −5.37228 −0.419505
\(165\) 0 0
\(166\) −0.744563 −0.0577893
\(167\) −1.48913 −0.115232 −0.0576160 0.998339i \(-0.518350\pi\)
−0.0576160 + 0.998339i \(0.518350\pi\)
\(168\) 0 0
\(169\) 9.51087 0.731606
\(170\) 5.37228 0.412035
\(171\) 0 0
\(172\) 7.37228 0.562131
\(173\) −22.8614 −1.73812 −0.869060 0.494706i \(-0.835276\pi\)
−0.869060 + 0.494706i \(0.835276\pi\)
\(174\) 0 0
\(175\) 1.37228 0.103735
\(176\) 3.37228 0.254195
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 26.2337 1.96080 0.980399 0.197023i \(-0.0631272\pi\)
0.980399 + 0.197023i \(0.0631272\pi\)
\(180\) 0 0
\(181\) 7.48913 0.556662 0.278331 0.960485i \(-0.410219\pi\)
0.278331 + 0.960485i \(0.410219\pi\)
\(182\) 6.51087 0.482618
\(183\) 0 0
\(184\) 6.74456 0.497216
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 18.1168 1.32483
\(188\) 8.74456 0.637763
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) −18.6277 −1.34785 −0.673927 0.738798i \(-0.735394\pi\)
−0.673927 + 0.738798i \(0.735394\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0.116844 0.00838891
\(195\) 0 0
\(196\) −5.11684 −0.365489
\(197\) −27.4891 −1.95852 −0.979260 0.202610i \(-0.935058\pi\)
−0.979260 + 0.202610i \(0.935058\pi\)
\(198\) 0 0
\(199\) −11.2554 −0.797877 −0.398938 0.916978i \(-0.630621\pi\)
−0.398938 + 0.916978i \(0.630621\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −11.4891 −0.808372
\(203\) −11.1386 −0.781776
\(204\) 0 0
\(205\) 5.37228 0.375216
\(206\) 9.48913 0.661139
\(207\) 0 0
\(208\) −4.74456 −0.328976
\(209\) −6.74456 −0.466531
\(210\) 0 0
\(211\) −14.1168 −0.971844 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(212\) −1.37228 −0.0942487
\(213\) 0 0
\(214\) −3.48913 −0.238512
\(215\) −7.37228 −0.502785
\(216\) 0 0
\(217\) −3.60597 −0.244789
\(218\) −0.116844 −0.00791367
\(219\) 0 0
\(220\) −3.37228 −0.227359
\(221\) −25.4891 −1.71458
\(222\) 0 0
\(223\) −1.37228 −0.0918948 −0.0459474 0.998944i \(-0.514631\pi\)
−0.0459474 + 0.998944i \(0.514631\pi\)
\(224\) −1.37228 −0.0916894
\(225\) 0 0
\(226\) 17.3723 1.15559
\(227\) 11.3723 0.754805 0.377402 0.926049i \(-0.376817\pi\)
0.377402 + 0.926049i \(0.376817\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −6.74456 −0.444723
\(231\) 0 0
\(232\) 8.11684 0.532897
\(233\) −6.23369 −0.408382 −0.204191 0.978931i \(-0.565456\pi\)
−0.204191 + 0.978931i \(0.565456\pi\)
\(234\) 0 0
\(235\) −8.74456 −0.570432
\(236\) −12.7446 −0.829600
\(237\) 0 0
\(238\) −7.37228 −0.477874
\(239\) −17.6060 −1.13884 −0.569418 0.822048i \(-0.692831\pi\)
−0.569418 + 0.822048i \(0.692831\pi\)
\(240\) 0 0
\(241\) −10.2337 −0.659210 −0.329605 0.944119i \(-0.606916\pi\)
−0.329605 + 0.944119i \(0.606916\pi\)
\(242\) −0.372281 −0.0239311
\(243\) 0 0
\(244\) −5.37228 −0.343925
\(245\) 5.11684 0.326903
\(246\) 0 0
\(247\) 9.48913 0.603779
\(248\) 2.62772 0.166860
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −11.4891 −0.725187 −0.362594 0.931947i \(-0.618109\pi\)
−0.362594 + 0.931947i \(0.618109\pi\)
\(252\) 0 0
\(253\) −22.7446 −1.42994
\(254\) 16.7446 1.05065
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 20.9783 1.30859 0.654294 0.756241i \(-0.272966\pi\)
0.654294 + 0.756241i \(0.272966\pi\)
\(258\) 0 0
\(259\) 1.37228 0.0852694
\(260\) 4.74456 0.294245
\(261\) 0 0
\(262\) −3.25544 −0.201122
\(263\) −0.116844 −0.00720491 −0.00360245 0.999994i \(-0.501147\pi\)
−0.00360245 + 0.999994i \(0.501147\pi\)
\(264\) 0 0
\(265\) 1.37228 0.0842986
\(266\) 2.74456 0.168280
\(267\) 0 0
\(268\) 4.74456 0.289820
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −6.74456 −0.409703 −0.204852 0.978793i \(-0.565671\pi\)
−0.204852 + 0.978793i \(0.565671\pi\)
\(272\) 5.37228 0.325742
\(273\) 0 0
\(274\) 16.7446 1.01158
\(275\) 3.37228 0.203356
\(276\) 0 0
\(277\) 0.744563 0.0447364 0.0223682 0.999750i \(-0.492879\pi\)
0.0223682 + 0.999750i \(0.492879\pi\)
\(278\) 1.88316 0.112944
\(279\) 0 0
\(280\) 1.37228 0.0820095
\(281\) −24.7446 −1.47614 −0.738068 0.674726i \(-0.764262\pi\)
−0.738068 + 0.674726i \(0.764262\pi\)
\(282\) 0 0
\(283\) 5.48913 0.326295 0.163147 0.986602i \(-0.447835\pi\)
0.163147 + 0.986602i \(0.447835\pi\)
\(284\) 6.74456 0.400216
\(285\) 0 0
\(286\) 16.0000 0.946100
\(287\) −7.37228 −0.435172
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) −8.11684 −0.476637
\(291\) 0 0
\(292\) 8.74456 0.511737
\(293\) 18.8614 1.10190 0.550948 0.834540i \(-0.314266\pi\)
0.550948 + 0.834540i \(0.314266\pi\)
\(294\) 0 0
\(295\) 12.7446 0.742017
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −11.4891 −0.665547
\(299\) 32.0000 1.85061
\(300\) 0 0
\(301\) 10.1168 0.583125
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 5.37228 0.307616
\(306\) 0 0
\(307\) −23.4891 −1.34060 −0.670298 0.742092i \(-0.733834\pi\)
−0.670298 + 0.742092i \(0.733834\pi\)
\(308\) 4.62772 0.263689
\(309\) 0 0
\(310\) −2.62772 −0.149244
\(311\) 1.37228 0.0778149 0.0389075 0.999243i \(-0.487612\pi\)
0.0389075 + 0.999243i \(0.487612\pi\)
\(312\) 0 0
\(313\) −3.48913 −0.197217 −0.0986085 0.995126i \(-0.531439\pi\)
−0.0986085 + 0.995126i \(0.531439\pi\)
\(314\) −17.3723 −0.980375
\(315\) 0 0
\(316\) −4.74456 −0.266903
\(317\) −25.3723 −1.42505 −0.712525 0.701647i \(-0.752448\pi\)
−0.712525 + 0.701647i \(0.752448\pi\)
\(318\) 0 0
\(319\) −27.3723 −1.53255
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 9.25544 0.515785
\(323\) −10.7446 −0.597843
\(324\) 0 0
\(325\) −4.74456 −0.263181
\(326\) −19.3723 −1.07293
\(327\) 0 0
\(328\) 5.37228 0.296635
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 23.4891 1.29108 0.645540 0.763727i \(-0.276633\pi\)
0.645540 + 0.763727i \(0.276633\pi\)
\(332\) 0.744563 0.0408632
\(333\) 0 0
\(334\) 1.48913 0.0814813
\(335\) −4.74456 −0.259223
\(336\) 0 0
\(337\) −7.25544 −0.395229 −0.197614 0.980280i \(-0.563319\pi\)
−0.197614 + 0.980280i \(0.563319\pi\)
\(338\) −9.51087 −0.517323
\(339\) 0 0
\(340\) −5.37228 −0.291353
\(341\) −8.86141 −0.479872
\(342\) 0 0
\(343\) −16.6277 −0.897812
\(344\) −7.37228 −0.397487
\(345\) 0 0
\(346\) 22.8614 1.22904
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −1.37228 −0.0733515
\(351\) 0 0
\(352\) −3.37228 −0.179743
\(353\) 22.8614 1.21679 0.608395 0.793634i \(-0.291814\pi\)
0.608395 + 0.793634i \(0.291814\pi\)
\(354\) 0 0
\(355\) −6.74456 −0.357964
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −26.2337 −1.38649
\(359\) 30.9783 1.63497 0.817485 0.575950i \(-0.195368\pi\)
0.817485 + 0.575950i \(0.195368\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −7.48913 −0.393620
\(363\) 0 0
\(364\) −6.51087 −0.341263
\(365\) −8.74456 −0.457711
\(366\) 0 0
\(367\) 3.88316 0.202699 0.101350 0.994851i \(-0.467684\pi\)
0.101350 + 0.994851i \(0.467684\pi\)
\(368\) −6.74456 −0.351585
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −1.88316 −0.0977686
\(372\) 0 0
\(373\) −31.4891 −1.63045 −0.815223 0.579148i \(-0.803385\pi\)
−0.815223 + 0.579148i \(0.803385\pi\)
\(374\) −18.1168 −0.936800
\(375\) 0 0
\(376\) −8.74456 −0.450966
\(377\) 38.5109 1.98341
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 18.6277 0.953077
\(383\) 13.4891 0.689262 0.344631 0.938738i \(-0.388004\pi\)
0.344631 + 0.938738i \(0.388004\pi\)
\(384\) 0 0
\(385\) −4.62772 −0.235850
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −0.116844 −0.00593185
\(389\) −14.8614 −0.753503 −0.376752 0.926314i \(-0.622959\pi\)
−0.376752 + 0.926314i \(0.622959\pi\)
\(390\) 0 0
\(391\) −36.2337 −1.83242
\(392\) 5.11684 0.258440
\(393\) 0 0
\(394\) 27.4891 1.38488
\(395\) 4.74456 0.238725
\(396\) 0 0
\(397\) 24.9783 1.25362 0.626811 0.779171i \(-0.284360\pi\)
0.626811 + 0.779171i \(0.284360\pi\)
\(398\) 11.2554 0.564184
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 12.4674 0.621044
\(404\) 11.4891 0.571605
\(405\) 0 0
\(406\) 11.1386 0.552799
\(407\) 3.37228 0.167158
\(408\) 0 0
\(409\) 6.23369 0.308236 0.154118 0.988052i \(-0.450746\pi\)
0.154118 + 0.988052i \(0.450746\pi\)
\(410\) −5.37228 −0.265318
\(411\) 0 0
\(412\) −9.48913 −0.467496
\(413\) −17.4891 −0.860584
\(414\) 0 0
\(415\) −0.744563 −0.0365491
\(416\) 4.74456 0.232621
\(417\) 0 0
\(418\) 6.74456 0.329887
\(419\) −13.4891 −0.658987 −0.329493 0.944158i \(-0.606878\pi\)
−0.329493 + 0.944158i \(0.606878\pi\)
\(420\) 0 0
\(421\) −7.48913 −0.364998 −0.182499 0.983206i \(-0.558419\pi\)
−0.182499 + 0.983206i \(0.558419\pi\)
\(422\) 14.1168 0.687197
\(423\) 0 0
\(424\) 1.37228 0.0666439
\(425\) 5.37228 0.260594
\(426\) 0 0
\(427\) −7.37228 −0.356770
\(428\) 3.48913 0.168653
\(429\) 0 0
\(430\) 7.37228 0.355523
\(431\) 1.37228 0.0661005 0.0330502 0.999454i \(-0.489478\pi\)
0.0330502 + 0.999454i \(0.489478\pi\)
\(432\) 0 0
\(433\) 12.9783 0.623695 0.311847 0.950132i \(-0.399052\pi\)
0.311847 + 0.950132i \(0.399052\pi\)
\(434\) 3.60597 0.173092
\(435\) 0 0
\(436\) 0.116844 0.00559581
\(437\) 13.4891 0.645272
\(438\) 0 0
\(439\) −13.3723 −0.638224 −0.319112 0.947717i \(-0.603385\pi\)
−0.319112 + 0.947717i \(0.603385\pi\)
\(440\) 3.37228 0.160767
\(441\) 0 0
\(442\) 25.4891 1.21239
\(443\) 16.9783 0.806661 0.403331 0.915054i \(-0.367852\pi\)
0.403331 + 0.915054i \(0.367852\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 1.37228 0.0649794
\(447\) 0 0
\(448\) 1.37228 0.0648342
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −18.1168 −0.853089
\(452\) −17.3723 −0.817123
\(453\) 0 0
\(454\) −11.3723 −0.533728
\(455\) 6.51087 0.305235
\(456\) 0 0
\(457\) 18.8614 0.882299 0.441150 0.897434i \(-0.354571\pi\)
0.441150 + 0.897434i \(0.354571\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 6.74456 0.314467
\(461\) 39.0951 1.82084 0.910420 0.413685i \(-0.135759\pi\)
0.910420 + 0.413685i \(0.135759\pi\)
\(462\) 0 0
\(463\) −5.48913 −0.255101 −0.127551 0.991832i \(-0.540712\pi\)
−0.127551 + 0.991832i \(0.540712\pi\)
\(464\) −8.11684 −0.376815
\(465\) 0 0
\(466\) 6.23369 0.288770
\(467\) −30.3505 −1.40446 −0.702228 0.711953i \(-0.747811\pi\)
−0.702228 + 0.711953i \(0.747811\pi\)
\(468\) 0 0
\(469\) 6.51087 0.300644
\(470\) 8.74456 0.403357
\(471\) 0 0
\(472\) 12.7446 0.586616
\(473\) 24.8614 1.14313
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 7.37228 0.337908
\(477\) 0 0
\(478\) 17.6060 0.805278
\(479\) −3.25544 −0.148745 −0.0743724 0.997231i \(-0.523695\pi\)
−0.0743724 + 0.997231i \(0.523695\pi\)
\(480\) 0 0
\(481\) −4.74456 −0.216333
\(482\) 10.2337 0.466132
\(483\) 0 0
\(484\) 0.372281 0.0169219
\(485\) 0.116844 0.00530561
\(486\) 0 0
\(487\) −37.7228 −1.70938 −0.854692 0.519136i \(-0.826254\pi\)
−0.854692 + 0.519136i \(0.826254\pi\)
\(488\) 5.37228 0.243192
\(489\) 0 0
\(490\) −5.11684 −0.231155
\(491\) −14.9783 −0.675959 −0.337979 0.941153i \(-0.609743\pi\)
−0.337979 + 0.941153i \(0.609743\pi\)
\(492\) 0 0
\(493\) −43.6060 −1.96391
\(494\) −9.48913 −0.426936
\(495\) 0 0
\(496\) −2.62772 −0.117988
\(497\) 9.25544 0.415163
\(498\) 0 0
\(499\) −24.9783 −1.11818 −0.559090 0.829107i \(-0.688849\pi\)
−0.559090 + 0.829107i \(0.688849\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 11.4891 0.512785
\(503\) −29.4891 −1.31486 −0.657428 0.753518i \(-0.728355\pi\)
−0.657428 + 0.753518i \(0.728355\pi\)
\(504\) 0 0
\(505\) −11.4891 −0.511259
\(506\) 22.7446 1.01112
\(507\) 0 0
\(508\) −16.7446 −0.742920
\(509\) 11.2554 0.498888 0.249444 0.968389i \(-0.419752\pi\)
0.249444 + 0.968389i \(0.419752\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −20.9783 −0.925311
\(515\) 9.48913 0.418141
\(516\) 0 0
\(517\) 29.4891 1.29693
\(518\) −1.37228 −0.0602946
\(519\) 0 0
\(520\) −4.74456 −0.208063
\(521\) −27.0951 −1.18706 −0.593529 0.804813i \(-0.702266\pi\)
−0.593529 + 0.804813i \(0.702266\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 3.25544 0.142214
\(525\) 0 0
\(526\) 0.116844 0.00509464
\(527\) −14.1168 −0.614939
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) −1.37228 −0.0596081
\(531\) 0 0
\(532\) −2.74456 −0.118992
\(533\) 25.4891 1.10406
\(534\) 0 0
\(535\) −3.48913 −0.150848
\(536\) −4.74456 −0.204934
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) −17.2554 −0.743244
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 6.74456 0.289704
\(543\) 0 0
\(544\) −5.37228 −0.230335
\(545\) −0.116844 −0.00500505
\(546\) 0 0
\(547\) 39.3723 1.68344 0.841719 0.539916i \(-0.181544\pi\)
0.841719 + 0.539916i \(0.181544\pi\)
\(548\) −16.7446 −0.715292
\(549\) 0 0
\(550\) −3.37228 −0.143795
\(551\) 16.2337 0.691578
\(552\) 0 0
\(553\) −6.51087 −0.276871
\(554\) −0.744563 −0.0316334
\(555\) 0 0
\(556\) −1.88316 −0.0798636
\(557\) −8.74456 −0.370519 −0.185260 0.982690i \(-0.559313\pi\)
−0.185260 + 0.982690i \(0.559313\pi\)
\(558\) 0 0
\(559\) −34.9783 −1.47942
\(560\) −1.37228 −0.0579895
\(561\) 0 0
\(562\) 24.7446 1.04379
\(563\) −33.0951 −1.39479 −0.697396 0.716686i \(-0.745658\pi\)
−0.697396 + 0.716686i \(0.745658\pi\)
\(564\) 0 0
\(565\) 17.3723 0.730857
\(566\) −5.48913 −0.230725
\(567\) 0 0
\(568\) −6.74456 −0.282996
\(569\) 32.9783 1.38252 0.691260 0.722606i \(-0.257056\pi\)
0.691260 + 0.722606i \(0.257056\pi\)
\(570\) 0 0
\(571\) −27.6060 −1.15527 −0.577637 0.816294i \(-0.696025\pi\)
−0.577637 + 0.816294i \(0.696025\pi\)
\(572\) −16.0000 −0.668994
\(573\) 0 0
\(574\) 7.37228 0.307713
\(575\) −6.74456 −0.281268
\(576\) 0 0
\(577\) −7.48913 −0.311776 −0.155888 0.987775i \(-0.549824\pi\)
−0.155888 + 0.987775i \(0.549824\pi\)
\(578\) −11.8614 −0.493369
\(579\) 0 0
\(580\) 8.11684 0.337034
\(581\) 1.02175 0.0423893
\(582\) 0 0
\(583\) −4.62772 −0.191661
\(584\) −8.74456 −0.361853
\(585\) 0 0
\(586\) −18.8614 −0.779158
\(587\) −47.8397 −1.97455 −0.987277 0.159010i \(-0.949170\pi\)
−0.987277 + 0.159010i \(0.949170\pi\)
\(588\) 0 0
\(589\) 5.25544 0.216547
\(590\) −12.7446 −0.524685
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 10.2337 0.420247 0.210124 0.977675i \(-0.432613\pi\)
0.210124 + 0.977675i \(0.432613\pi\)
\(594\) 0 0
\(595\) −7.37228 −0.302234
\(596\) 11.4891 0.470613
\(597\) 0 0
\(598\) −32.0000 −1.30858
\(599\) −17.4891 −0.714586 −0.357293 0.933992i \(-0.616300\pi\)
−0.357293 + 0.933992i \(0.616300\pi\)
\(600\) 0 0
\(601\) −5.37228 −0.219140 −0.109570 0.993979i \(-0.534947\pi\)
−0.109570 + 0.993979i \(0.534947\pi\)
\(602\) −10.1168 −0.412332
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) −0.372281 −0.0151354
\(606\) 0 0
\(607\) 17.7228 0.719347 0.359673 0.933078i \(-0.382888\pi\)
0.359673 + 0.933078i \(0.382888\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) −5.37228 −0.217517
\(611\) −41.4891 −1.67847
\(612\) 0 0
\(613\) 43.0951 1.74059 0.870297 0.492527i \(-0.163927\pi\)
0.870297 + 0.492527i \(0.163927\pi\)
\(614\) 23.4891 0.947944
\(615\) 0 0
\(616\) −4.62772 −0.186456
\(617\) 31.7228 1.27711 0.638556 0.769575i \(-0.279532\pi\)
0.638556 + 0.769575i \(0.279532\pi\)
\(618\) 0 0
\(619\) 30.1168 1.21050 0.605249 0.796036i \(-0.293074\pi\)
0.605249 + 0.796036i \(0.293074\pi\)
\(620\) 2.62772 0.105532
\(621\) 0 0
\(622\) −1.37228 −0.0550235
\(623\) −13.7228 −0.549793
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.48913 0.139453
\(627\) 0 0
\(628\) 17.3723 0.693229
\(629\) 5.37228 0.214207
\(630\) 0 0
\(631\) −19.0951 −0.760164 −0.380082 0.924953i \(-0.624104\pi\)
−0.380082 + 0.924953i \(0.624104\pi\)
\(632\) 4.74456 0.188729
\(633\) 0 0
\(634\) 25.3723 1.00766
\(635\) 16.7446 0.664488
\(636\) 0 0
\(637\) 24.2772 0.961897
\(638\) 27.3723 1.08368
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 49.6060 1.95932 0.979659 0.200670i \(-0.0643117\pi\)
0.979659 + 0.200670i \(0.0643117\pi\)
\(642\) 0 0
\(643\) 3.37228 0.132990 0.0664949 0.997787i \(-0.478818\pi\)
0.0664949 + 0.997787i \(0.478818\pi\)
\(644\) −9.25544 −0.364715
\(645\) 0 0
\(646\) 10.7446 0.422739
\(647\) 13.4891 0.530312 0.265156 0.964205i \(-0.414577\pi\)
0.265156 + 0.964205i \(0.414577\pi\)
\(648\) 0 0
\(649\) −42.9783 −1.68704
\(650\) 4.74456 0.186097
\(651\) 0 0
\(652\) 19.3723 0.758677
\(653\) −0.510875 −0.0199921 −0.00999604 0.999950i \(-0.503182\pi\)
−0.00999604 + 0.999950i \(0.503182\pi\)
\(654\) 0 0
\(655\) −3.25544 −0.127200
\(656\) −5.37228 −0.209752
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 4.35053 0.169216 0.0846080 0.996414i \(-0.473036\pi\)
0.0846080 + 0.996414i \(0.473036\pi\)
\(662\) −23.4891 −0.912931
\(663\) 0 0
\(664\) −0.744563 −0.0288946
\(665\) 2.74456 0.106430
\(666\) 0 0
\(667\) 54.7446 2.11972
\(668\) −1.48913 −0.0576160
\(669\) 0 0
\(670\) 4.74456 0.183298
\(671\) −18.1168 −0.699393
\(672\) 0 0
\(673\) −8.51087 −0.328070 −0.164035 0.986455i \(-0.552451\pi\)
−0.164035 + 0.986455i \(0.552451\pi\)
\(674\) 7.25544 0.279469
\(675\) 0 0
\(676\) 9.51087 0.365803
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −0.160343 −0.00615339
\(680\) 5.37228 0.206018
\(681\) 0 0
\(682\) 8.86141 0.339321
\(683\) −8.62772 −0.330130 −0.165065 0.986283i \(-0.552783\pi\)
−0.165065 + 0.986283i \(0.552783\pi\)
\(684\) 0 0
\(685\) 16.7446 0.639777
\(686\) 16.6277 0.634849
\(687\) 0 0
\(688\) 7.37228 0.281066
\(689\) 6.51087 0.248045
\(690\) 0 0
\(691\) 22.3505 0.850254 0.425127 0.905134i \(-0.360229\pi\)
0.425127 + 0.905134i \(0.360229\pi\)
\(692\) −22.8614 −0.869060
\(693\) 0 0
\(694\) −22.9783 −0.872242
\(695\) 1.88316 0.0714322
\(696\) 0 0
\(697\) −28.8614 −1.09320
\(698\) 22.0000 0.832712
\(699\) 0 0
\(700\) 1.37228 0.0518674
\(701\) 26.4674 0.999659 0.499829 0.866124i \(-0.333396\pi\)
0.499829 + 0.866124i \(0.333396\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 3.37228 0.127098
\(705\) 0 0
\(706\) −22.8614 −0.860400
\(707\) 15.7663 0.592953
\(708\) 0 0
\(709\) 40.1168 1.50662 0.753310 0.657666i \(-0.228456\pi\)
0.753310 + 0.657666i \(0.228456\pi\)
\(710\) 6.74456 0.253119
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 17.7228 0.663725
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) 26.2337 0.980399
\(717\) 0 0
\(718\) −30.9783 −1.15610
\(719\) 34.7446 1.29575 0.647877 0.761745i \(-0.275657\pi\)
0.647877 + 0.761745i \(0.275657\pi\)
\(720\) 0 0
\(721\) −13.0217 −0.484955
\(722\) 15.0000 0.558242
\(723\) 0 0
\(724\) 7.48913 0.278331
\(725\) −8.11684 −0.301452
\(726\) 0 0
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) 6.51087 0.241309
\(729\) 0 0
\(730\) 8.74456 0.323651
\(731\) 39.6060 1.46488
\(732\) 0 0
\(733\) −29.3723 −1.08489 −0.542445 0.840091i \(-0.682501\pi\)
−0.542445 + 0.840091i \(0.682501\pi\)
\(734\) −3.88316 −0.143330
\(735\) 0 0
\(736\) 6.74456 0.248608
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) −36.8614 −1.35597 −0.677984 0.735076i \(-0.737146\pi\)
−0.677984 + 0.735076i \(0.737146\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 1.88316 0.0691328
\(743\) 5.37228 0.197090 0.0985449 0.995133i \(-0.468581\pi\)
0.0985449 + 0.995133i \(0.468581\pi\)
\(744\) 0 0
\(745\) −11.4891 −0.420929
\(746\) 31.4891 1.15290
\(747\) 0 0
\(748\) 18.1168 0.662417
\(749\) 4.78806 0.174952
\(750\) 0 0
\(751\) 10.7446 0.392075 0.196037 0.980596i \(-0.437193\pi\)
0.196037 + 0.980596i \(0.437193\pi\)
\(752\) 8.74456 0.318881
\(753\) 0 0
\(754\) −38.5109 −1.40248
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 43.9565 1.59763 0.798813 0.601579i \(-0.205462\pi\)
0.798813 + 0.601579i \(0.205462\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) −2.00000 −0.0725476
\(761\) −5.37228 −0.194745 −0.0973725 0.995248i \(-0.531044\pi\)
−0.0973725 + 0.995248i \(0.531044\pi\)
\(762\) 0 0
\(763\) 0.160343 0.00580480
\(764\) −18.6277 −0.673927
\(765\) 0 0
\(766\) −13.4891 −0.487382
\(767\) 60.4674 2.18335
\(768\) 0 0
\(769\) 46.2337 1.66723 0.833615 0.552346i \(-0.186267\pi\)
0.833615 + 0.552346i \(0.186267\pi\)
\(770\) 4.62772 0.166771
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) 24.3505 0.875828 0.437914 0.899017i \(-0.355717\pi\)
0.437914 + 0.899017i \(0.355717\pi\)
\(774\) 0 0
\(775\) −2.62772 −0.0943904
\(776\) 0.116844 0.00419445
\(777\) 0 0
\(778\) 14.8614 0.532807
\(779\) 10.7446 0.384964
\(780\) 0 0
\(781\) 22.7446 0.813864
\(782\) 36.2337 1.29571
\(783\) 0 0
\(784\) −5.11684 −0.182744
\(785\) −17.3723 −0.620043
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) −27.4891 −0.979260
\(789\) 0 0
\(790\) −4.74456 −0.168804
\(791\) −23.8397 −0.847641
\(792\) 0 0
\(793\) 25.4891 0.905145
\(794\) −24.9783 −0.886445
\(795\) 0 0
\(796\) −11.2554 −0.398938
\(797\) 27.4891 0.973715 0.486857 0.873481i \(-0.338143\pi\)
0.486857 + 0.873481i \(0.338143\pi\)
\(798\) 0 0
\(799\) 46.9783 1.66197
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 10.0000 0.353112
\(803\) 29.4891 1.04065
\(804\) 0 0
\(805\) 9.25544 0.326211
\(806\) −12.4674 −0.439145
\(807\) 0 0
\(808\) −11.4891 −0.404186
\(809\) 51.4891 1.81026 0.905131 0.425134i \(-0.139773\pi\)
0.905131 + 0.425134i \(0.139773\pi\)
\(810\) 0 0
\(811\) 49.4891 1.73780 0.868899 0.494989i \(-0.164828\pi\)
0.868899 + 0.494989i \(0.164828\pi\)
\(812\) −11.1386 −0.390888
\(813\) 0 0
\(814\) −3.37228 −0.118198
\(815\) −19.3723 −0.678581
\(816\) 0 0
\(817\) −14.7446 −0.515847
\(818\) −6.23369 −0.217956
\(819\) 0 0
\(820\) 5.37228 0.187608
\(821\) 32.7446 1.14279 0.571397 0.820674i \(-0.306402\pi\)
0.571397 + 0.820674i \(0.306402\pi\)
\(822\) 0 0
\(823\) −39.7228 −1.38465 −0.692325 0.721586i \(-0.743414\pi\)
−0.692325 + 0.721586i \(0.743414\pi\)
\(824\) 9.48913 0.330569
\(825\) 0 0
\(826\) 17.4891 0.608524
\(827\) 54.3505 1.88995 0.944977 0.327138i \(-0.106084\pi\)
0.944977 + 0.327138i \(0.106084\pi\)
\(828\) 0 0
\(829\) −40.3505 −1.40143 −0.700716 0.713440i \(-0.747136\pi\)
−0.700716 + 0.713440i \(0.747136\pi\)
\(830\) 0.744563 0.0258441
\(831\) 0 0
\(832\) −4.74456 −0.164488
\(833\) −27.4891 −0.952442
\(834\) 0 0
\(835\) 1.48913 0.0515333
\(836\) −6.74456 −0.233266
\(837\) 0 0
\(838\) 13.4891 0.465974
\(839\) 29.4891 1.01808 0.509039 0.860744i \(-0.330001\pi\)
0.509039 + 0.860744i \(0.330001\pi\)
\(840\) 0 0
\(841\) 36.8832 1.27183
\(842\) 7.48913 0.258092
\(843\) 0 0
\(844\) −14.1168 −0.485922
\(845\) −9.51087 −0.327184
\(846\) 0 0
\(847\) 0.510875 0.0175539
\(848\) −1.37228 −0.0471243
\(849\) 0 0
\(850\) −5.37228 −0.184268
\(851\) −6.74456 −0.231201
\(852\) 0 0
\(853\) 35.4891 1.21512 0.607562 0.794272i \(-0.292148\pi\)
0.607562 + 0.794272i \(0.292148\pi\)
\(854\) 7.37228 0.252274
\(855\) 0 0
\(856\) −3.48913 −0.119256
\(857\) 29.1386 0.995355 0.497678 0.867362i \(-0.334186\pi\)
0.497678 + 0.867362i \(0.334186\pi\)
\(858\) 0 0
\(859\) −19.7228 −0.672934 −0.336467 0.941695i \(-0.609232\pi\)
−0.336467 + 0.941695i \(0.609232\pi\)
\(860\) −7.37228 −0.251393
\(861\) 0 0
\(862\) −1.37228 −0.0467401
\(863\) −53.8397 −1.83272 −0.916362 0.400352i \(-0.868888\pi\)
−0.916362 + 0.400352i \(0.868888\pi\)
\(864\) 0 0
\(865\) 22.8614 0.777311
\(866\) −12.9783 −0.441019
\(867\) 0 0
\(868\) −3.60597 −0.122395
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −22.5109 −0.762752
\(872\) −0.116844 −0.00395684
\(873\) 0 0
\(874\) −13.4891 −0.456276
\(875\) −1.37228 −0.0463916
\(876\) 0 0
\(877\) 49.3723 1.66718 0.833592 0.552381i \(-0.186281\pi\)
0.833592 + 0.552381i \(0.186281\pi\)
\(878\) 13.3723 0.451293
\(879\) 0 0
\(880\) −3.37228 −0.113680
\(881\) −1.37228 −0.0462333 −0.0231167 0.999733i \(-0.507359\pi\)
−0.0231167 + 0.999733i \(0.507359\pi\)
\(882\) 0 0
\(883\) −11.3723 −0.382708 −0.191354 0.981521i \(-0.561288\pi\)
−0.191354 + 0.981521i \(0.561288\pi\)
\(884\) −25.4891 −0.857292
\(885\) 0 0
\(886\) −16.9783 −0.570395
\(887\) 25.3723 0.851918 0.425959 0.904743i \(-0.359937\pi\)
0.425959 + 0.904743i \(0.359937\pi\)
\(888\) 0 0
\(889\) −22.9783 −0.770666
\(890\) −10.0000 −0.335201
\(891\) 0 0
\(892\) −1.37228 −0.0459474
\(893\) −17.4891 −0.585251
\(894\) 0 0
\(895\) −26.2337 −0.876895
\(896\) −1.37228 −0.0458447
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 21.3288 0.711355
\(900\) 0 0
\(901\) −7.37228 −0.245606
\(902\) 18.1168 0.603225
\(903\) 0 0
\(904\) 17.3723 0.577793
\(905\) −7.48913 −0.248947
\(906\) 0 0
\(907\) 40.4674 1.34370 0.671849 0.740689i \(-0.265501\pi\)
0.671849 + 0.740689i \(0.265501\pi\)
\(908\) 11.3723 0.377402
\(909\) 0 0
\(910\) −6.51087 −0.215833
\(911\) −17.7663 −0.588624 −0.294312 0.955709i \(-0.595091\pi\)
−0.294312 + 0.955709i \(0.595091\pi\)
\(912\) 0 0
\(913\) 2.51087 0.0830978
\(914\) −18.8614 −0.623880
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 4.46738 0.147526
\(918\) 0 0
\(919\) 7.72281 0.254752 0.127376 0.991854i \(-0.459344\pi\)
0.127376 + 0.991854i \(0.459344\pi\)
\(920\) −6.74456 −0.222362
\(921\) 0 0
\(922\) −39.0951 −1.28753
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 5.48913 0.180384
\(927\) 0 0
\(928\) 8.11684 0.266448
\(929\) −31.0951 −1.02020 −0.510098 0.860116i \(-0.670391\pi\)
−0.510098 + 0.860116i \(0.670391\pi\)
\(930\) 0 0
\(931\) 10.2337 0.335396
\(932\) −6.23369 −0.204191
\(933\) 0 0
\(934\) 30.3505 0.993100
\(935\) −18.1168 −0.592484
\(936\) 0 0
\(937\) 36.9783 1.20803 0.604013 0.796974i \(-0.293567\pi\)
0.604013 + 0.796974i \(0.293567\pi\)
\(938\) −6.51087 −0.212588
\(939\) 0 0
\(940\) −8.74456 −0.285216
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 36.2337 1.17993
\(944\) −12.7446 −0.414800
\(945\) 0 0
\(946\) −24.8614 −0.808314
\(947\) −37.0951 −1.20543 −0.602714 0.797957i \(-0.705914\pi\)
−0.602714 + 0.797957i \(0.705914\pi\)
\(948\) 0 0
\(949\) −41.4891 −1.34679
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) −7.37228 −0.238937
\(953\) −46.2337 −1.49766 −0.748828 0.662764i \(-0.769383\pi\)
−0.748828 + 0.662764i \(0.769383\pi\)
\(954\) 0 0
\(955\) 18.6277 0.602779
\(956\) −17.6060 −0.569418
\(957\) 0 0
\(958\) 3.25544 0.105178
\(959\) −22.9783 −0.742006
\(960\) 0 0
\(961\) −24.0951 −0.777261
\(962\) 4.74456 0.152971
\(963\) 0 0
\(964\) −10.2337 −0.329605
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) −0.372281 −0.0119656
\(969\) 0 0
\(970\) −0.116844 −0.00375163
\(971\) 19.6060 0.629185 0.314593 0.949227i \(-0.398132\pi\)
0.314593 + 0.949227i \(0.398132\pi\)
\(972\) 0 0
\(973\) −2.58422 −0.0828463
\(974\) 37.7228 1.20872
\(975\) 0 0
\(976\) −5.37228 −0.171963
\(977\) 11.8832 0.380176 0.190088 0.981767i \(-0.439123\pi\)
0.190088 + 0.981767i \(0.439123\pi\)
\(978\) 0 0
\(979\) −33.7228 −1.07779
\(980\) 5.11684 0.163452
\(981\) 0 0
\(982\) 14.9783 0.477975
\(983\) −30.8614 −0.984326 −0.492163 0.870503i \(-0.663794\pi\)
−0.492163 + 0.870503i \(0.663794\pi\)
\(984\) 0 0
\(985\) 27.4891 0.875876
\(986\) 43.6060 1.38870
\(987\) 0 0
\(988\) 9.48913 0.301889
\(989\) −49.7228 −1.58109
\(990\) 0 0
\(991\) 25.6060 0.813400 0.406700 0.913562i \(-0.366679\pi\)
0.406700 + 0.913562i \(0.366679\pi\)
\(992\) 2.62772 0.0834302
\(993\) 0 0
\(994\) −9.25544 −0.293565
\(995\) 11.2554 0.356821
\(996\) 0 0
\(997\) 26.2337 0.830829 0.415415 0.909632i \(-0.363637\pi\)
0.415415 + 0.909632i \(0.363637\pi\)
\(998\) 24.9783 0.790673
\(999\) 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 3330.2.a.bb.1.2 2
3.2 odd 2 370.2.a.f.1.2 2
12.11 even 2 2960.2.a.o.1.1 2
15.2 even 4 1850.2.b.m.149.4 4
15.8 even 4 1850.2.b.m.149.1 4
15.14 odd 2 1850.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.f.1.2 2 3.2 odd 2
1850.2.a.q.1.1 2 15.14 odd 2
1850.2.b.m.149.1 4 15.8 even 4
1850.2.b.m.149.4 4 15.2 even 4
2960.2.a.o.1.1 2 12.11 even 2
3330.2.a.bb.1.2 2 1.1 even 1 trivial