Properties

Label 1521.2.a.m.1.1
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{2} +0.438447 q^{4} -3.56155 q^{5} -0.561553 q^{7} +2.43845 q^{8} +O(q^{10})\) \(q-1.56155 q^{2} +0.438447 q^{4} -3.56155 q^{5} -0.561553 q^{7} +2.43845 q^{8} +5.56155 q^{10} -2.00000 q^{11} +0.876894 q^{14} -4.68466 q^{16} +1.56155 q^{17} -7.12311 q^{19} -1.56155 q^{20} +3.12311 q^{22} -2.00000 q^{23} +7.68466 q^{25} -0.246211 q^{28} -6.68466 q^{29} -2.56155 q^{31} +2.43845 q^{32} -2.43845 q^{34} +2.00000 q^{35} -7.56155 q^{37} +11.1231 q^{38} -8.68466 q^{40} -1.56155 q^{41} +4.56155 q^{43} -0.876894 q^{44} +3.12311 q^{46} +8.24621 q^{47} -6.68466 q^{49} -12.0000 q^{50} +0.684658 q^{53} +7.12311 q^{55} -1.36932 q^{56} +10.4384 q^{58} -2.87689 q^{59} +3.87689 q^{61} +4.00000 q^{62} +5.56155 q^{64} -4.56155 q^{67} +0.684658 q^{68} -3.12311 q^{70} +14.0000 q^{71} +10.1231 q^{73} +11.8078 q^{74} -3.12311 q^{76} +1.12311 q^{77} +5.43845 q^{79} +16.6847 q^{80} +2.43845 q^{82} -0.876894 q^{83} -5.56155 q^{85} -7.12311 q^{86} -4.87689 q^{88} +4.87689 q^{89} -0.876894 q^{92} -12.8769 q^{94} +25.3693 q^{95} +8.56155 q^{97} +10.4384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} - 3q^{5} + 3q^{7} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} - 3q^{5} + 3q^{7} + 9q^{8} + 7q^{10} - 4q^{11} + 10q^{14} + 3q^{16} - q^{17} - 6q^{19} + q^{20} - 2q^{22} - 4q^{23} + 3q^{25} + 16q^{28} - q^{29} - q^{31} + 9q^{32} - 9q^{34} + 4q^{35} - 11q^{37} + 14q^{38} - 5q^{40} + q^{41} + 5q^{43} - 10q^{44} - 2q^{46} - q^{49} - 24q^{50} - 11q^{53} + 6q^{55} + 22q^{56} + 25q^{58} - 14q^{59} + 16q^{61} + 8q^{62} + 7q^{64} - 5q^{67} - 11q^{68} + 2q^{70} + 28q^{71} + 12q^{73} + 3q^{74} + 2q^{76} - 6q^{77} + 15q^{79} + 21q^{80} + 9q^{82} - 10q^{83} - 7q^{85} - 6q^{86} - 18q^{88} + 18q^{89} - 10q^{92} - 34q^{94} + 26q^{95} + 13q^{97} + 25q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.56155 −1.10418 −0.552092 0.833783i \(-0.686170\pi\)
−0.552092 + 0.833783i \(0.686170\pi\)
\(3\) 0 0
\(4\) 0.438447 0.219224
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) −0.561553 −0.212247 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) 2.43845 0.862121
\(9\) 0 0
\(10\) 5.56155 1.75872
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0.876894 0.234360
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 1.56155 0.378732 0.189366 0.981907i \(-0.439357\pi\)
0.189366 + 0.981907i \(0.439357\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) −1.56155 −0.349174
\(21\) 0 0
\(22\) 3.12311 0.665848
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) 0 0
\(28\) −0.246211 −0.0465296
\(29\) −6.68466 −1.24131 −0.620655 0.784084i \(-0.713133\pi\)
−0.620655 + 0.784084i \(0.713133\pi\)
\(30\) 0 0
\(31\) −2.56155 −0.460068 −0.230034 0.973183i \(-0.573884\pi\)
−0.230034 + 0.973183i \(0.573884\pi\)
\(32\) 2.43845 0.431061
\(33\) 0 0
\(34\) −2.43845 −0.418190
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −7.56155 −1.24311 −0.621556 0.783370i \(-0.713499\pi\)
−0.621556 + 0.783370i \(0.713499\pi\)
\(38\) 11.1231 1.80441
\(39\) 0 0
\(40\) −8.68466 −1.37317
\(41\) −1.56155 −0.243874 −0.121937 0.992538i \(-0.538911\pi\)
−0.121937 + 0.992538i \(0.538911\pi\)
\(42\) 0 0
\(43\) 4.56155 0.695630 0.347815 0.937563i \(-0.386924\pi\)
0.347815 + 0.937563i \(0.386924\pi\)
\(44\) −0.876894 −0.132197
\(45\) 0 0
\(46\) 3.12311 0.460477
\(47\) 8.24621 1.20283 0.601417 0.798935i \(-0.294603\pi\)
0.601417 + 0.798935i \(0.294603\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) −12.0000 −1.69706
\(51\) 0 0
\(52\) 0 0
\(53\) 0.684658 0.0940451 0.0470225 0.998894i \(-0.485027\pi\)
0.0470225 + 0.998894i \(0.485027\pi\)
\(54\) 0 0
\(55\) 7.12311 0.960479
\(56\) −1.36932 −0.182983
\(57\) 0 0
\(58\) 10.4384 1.37064
\(59\) −2.87689 −0.374540 −0.187270 0.982309i \(-0.559964\pi\)
−0.187270 + 0.982309i \(0.559964\pi\)
\(60\) 0 0
\(61\) 3.87689 0.496385 0.248193 0.968711i \(-0.420163\pi\)
0.248193 + 0.968711i \(0.420163\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 5.56155 0.695194
\(65\) 0 0
\(66\) 0 0
\(67\) −4.56155 −0.557282 −0.278641 0.960395i \(-0.589884\pi\)
−0.278641 + 0.960395i \(0.589884\pi\)
\(68\) 0.684658 0.0830270
\(69\) 0 0
\(70\) −3.12311 −0.373283
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 10.1231 1.18482 0.592410 0.805637i \(-0.298177\pi\)
0.592410 + 0.805637i \(0.298177\pi\)
\(74\) 11.8078 1.37262
\(75\) 0 0
\(76\) −3.12311 −0.358245
\(77\) 1.12311 0.127990
\(78\) 0 0
\(79\) 5.43845 0.611873 0.305937 0.952052i \(-0.401030\pi\)
0.305937 + 0.952052i \(0.401030\pi\)
\(80\) 16.6847 1.86540
\(81\) 0 0
\(82\) 2.43845 0.269281
\(83\) −0.876894 −0.0962517 −0.0481258 0.998841i \(-0.515325\pi\)
−0.0481258 + 0.998841i \(0.515325\pi\)
\(84\) 0 0
\(85\) −5.56155 −0.603235
\(86\) −7.12311 −0.768104
\(87\) 0 0
\(88\) −4.87689 −0.519879
\(89\) 4.87689 0.516950 0.258475 0.966018i \(-0.416780\pi\)
0.258475 + 0.966018i \(0.416780\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.876894 −0.0914226
\(93\) 0 0
\(94\) −12.8769 −1.32815
\(95\) 25.3693 2.60284
\(96\) 0 0
\(97\) 8.56155 0.869294 0.434647 0.900601i \(-0.356873\pi\)
0.434647 + 0.900601i \(0.356873\pi\)
\(98\) 10.4384 1.05444
\(99\) 0 0
\(100\) 3.36932 0.336932
\(101\) 7.56155 0.752403 0.376201 0.926538i \(-0.377230\pi\)
0.376201 + 0.926538i \(0.377230\pi\)
\(102\) 0 0
\(103\) −3.43845 −0.338800 −0.169400 0.985547i \(-0.554183\pi\)
−0.169400 + 0.985547i \(0.554183\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1.06913 −0.103843
\(107\) 8.24621 0.797191 0.398596 0.917127i \(-0.369498\pi\)
0.398596 + 0.917127i \(0.369498\pi\)
\(108\) 0 0
\(109\) 2.80776 0.268935 0.134468 0.990918i \(-0.457068\pi\)
0.134468 + 0.990918i \(0.457068\pi\)
\(110\) −11.1231 −1.06055
\(111\) 0 0
\(112\) 2.63068 0.248576
\(113\) −5.80776 −0.546348 −0.273174 0.961965i \(-0.588074\pi\)
−0.273174 + 0.961965i \(0.588074\pi\)
\(114\) 0 0
\(115\) 7.12311 0.664233
\(116\) −2.93087 −0.272124
\(117\) 0 0
\(118\) 4.49242 0.413561
\(119\) −0.876894 −0.0803848
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −6.05398 −0.548101
\(123\) 0 0
\(124\) −1.12311 −0.100858
\(125\) −9.56155 −0.855211
\(126\) 0 0
\(127\) 5.43845 0.482584 0.241292 0.970453i \(-0.422429\pi\)
0.241292 + 0.970453i \(0.422429\pi\)
\(128\) −13.5616 −1.19868
\(129\) 0 0
\(130\) 0 0
\(131\) −7.36932 −0.643860 −0.321930 0.946763i \(-0.604332\pi\)
−0.321930 + 0.946763i \(0.604332\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 7.12311 0.615343
\(135\) 0 0
\(136\) 3.80776 0.326513
\(137\) −5.56155 −0.475156 −0.237578 0.971369i \(-0.576353\pi\)
−0.237578 + 0.971369i \(0.576353\pi\)
\(138\) 0 0
\(139\) −17.9309 −1.52088 −0.760438 0.649410i \(-0.775016\pi\)
−0.760438 + 0.649410i \(0.775016\pi\)
\(140\) 0.876894 0.0741111
\(141\) 0 0
\(142\) −21.8617 −1.83460
\(143\) 0 0
\(144\) 0 0
\(145\) 23.8078 1.97713
\(146\) −15.8078 −1.30826
\(147\) 0 0
\(148\) −3.31534 −0.272519
\(149\) −2.43845 −0.199765 −0.0998827 0.994999i \(-0.531847\pi\)
−0.0998827 + 0.994999i \(0.531847\pi\)
\(150\) 0 0
\(151\) 9.36932 0.762464 0.381232 0.924479i \(-0.375500\pi\)
0.381232 + 0.924479i \(0.375500\pi\)
\(152\) −17.3693 −1.40884
\(153\) 0 0
\(154\) −1.75379 −0.141324
\(155\) 9.12311 0.732785
\(156\) 0 0
\(157\) 20.3693 1.62565 0.812824 0.582509i \(-0.197929\pi\)
0.812824 + 0.582509i \(0.197929\pi\)
\(158\) −8.49242 −0.675621
\(159\) 0 0
\(160\) −8.68466 −0.686583
\(161\) 1.12311 0.0885131
\(162\) 0 0
\(163\) 4.80776 0.376573 0.188287 0.982114i \(-0.439707\pi\)
0.188287 + 0.982114i \(0.439707\pi\)
\(164\) −0.684658 −0.0534628
\(165\) 0 0
\(166\) 1.36932 0.106280
\(167\) 10.2462 0.792876 0.396438 0.918062i \(-0.370246\pi\)
0.396438 + 0.918062i \(0.370246\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 8.68466 0.666083
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −20.2462 −1.53929 −0.769645 0.638471i \(-0.779567\pi\)
−0.769645 + 0.638471i \(0.779567\pi\)
\(174\) 0 0
\(175\) −4.31534 −0.326209
\(176\) 9.36932 0.706239
\(177\) 0 0
\(178\) −7.61553 −0.570808
\(179\) 4.87689 0.364516 0.182258 0.983251i \(-0.441659\pi\)
0.182258 + 0.983251i \(0.441659\pi\)
\(180\) 0 0
\(181\) −2.68466 −0.199549 −0.0997745 0.995010i \(-0.531812\pi\)
−0.0997745 + 0.995010i \(0.531812\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.87689 −0.359529
\(185\) 26.9309 1.98000
\(186\) 0 0
\(187\) −3.12311 −0.228384
\(188\) 3.61553 0.263689
\(189\) 0 0
\(190\) −39.6155 −2.87401
\(191\) 9.12311 0.660125 0.330062 0.943959i \(-0.392930\pi\)
0.330062 + 0.943959i \(0.392930\pi\)
\(192\) 0 0
\(193\) 13.4924 0.971206 0.485603 0.874180i \(-0.338600\pi\)
0.485603 + 0.874180i \(0.338600\pi\)
\(194\) −13.3693 −0.959861
\(195\) 0 0
\(196\) −2.93087 −0.209348
\(197\) 13.3693 0.952524 0.476262 0.879303i \(-0.341991\pi\)
0.476262 + 0.879303i \(0.341991\pi\)
\(198\) 0 0
\(199\) 22.1771 1.57209 0.786046 0.618168i \(-0.212125\pi\)
0.786046 + 0.618168i \(0.212125\pi\)
\(200\) 18.7386 1.32502
\(201\) 0 0
\(202\) −11.8078 −0.830791
\(203\) 3.75379 0.263464
\(204\) 0 0
\(205\) 5.56155 0.388436
\(206\) 5.36932 0.374098
\(207\) 0 0
\(208\) 0 0
\(209\) 14.2462 0.985431
\(210\) 0 0
\(211\) 19.6847 1.35515 0.677574 0.735455i \(-0.263031\pi\)
0.677574 + 0.735455i \(0.263031\pi\)
\(212\) 0.300187 0.0206169
\(213\) 0 0
\(214\) −12.8769 −0.880246
\(215\) −16.2462 −1.10798
\(216\) 0 0
\(217\) 1.43845 0.0976482
\(218\) −4.38447 −0.296954
\(219\) 0 0
\(220\) 3.12311 0.210560
\(221\) 0 0
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.36932 −0.0914913
\(225\) 0 0
\(226\) 9.06913 0.603270
\(227\) 7.12311 0.472777 0.236389 0.971659i \(-0.424036\pi\)
0.236389 + 0.971659i \(0.424036\pi\)
\(228\) 0 0
\(229\) 16.2462 1.07358 0.536790 0.843716i \(-0.319637\pi\)
0.536790 + 0.843716i \(0.319637\pi\)
\(230\) −11.1231 −0.733436
\(231\) 0 0
\(232\) −16.3002 −1.07016
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −29.3693 −1.91584
\(236\) −1.26137 −0.0821079
\(237\) 0 0
\(238\) 1.36932 0.0887596
\(239\) −25.3693 −1.64100 −0.820502 0.571643i \(-0.806306\pi\)
−0.820502 + 0.571643i \(0.806306\pi\)
\(240\) 0 0
\(241\) 17.8078 1.14710 0.573549 0.819171i \(-0.305566\pi\)
0.573549 + 0.819171i \(0.305566\pi\)
\(242\) 10.9309 0.702663
\(243\) 0 0
\(244\) 1.69981 0.108819
\(245\) 23.8078 1.52102
\(246\) 0 0
\(247\) 0 0
\(248\) −6.24621 −0.396635
\(249\) 0 0
\(250\) 14.9309 0.944311
\(251\) −18.7386 −1.18277 −0.591386 0.806389i \(-0.701419\pi\)
−0.591386 + 0.806389i \(0.701419\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −8.49242 −0.532862
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) −29.1771 −1.82002 −0.910008 0.414590i \(-0.863925\pi\)
−0.910008 + 0.414590i \(0.863925\pi\)
\(258\) 0 0
\(259\) 4.24621 0.263847
\(260\) 0 0
\(261\) 0 0
\(262\) 11.5076 0.710941
\(263\) −9.36932 −0.577737 −0.288868 0.957369i \(-0.593279\pi\)
−0.288868 + 0.957369i \(0.593279\pi\)
\(264\) 0 0
\(265\) −2.43845 −0.149793
\(266\) −6.24621 −0.382980
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 21.3693 1.30291 0.651455 0.758687i \(-0.274159\pi\)
0.651455 + 0.758687i \(0.274159\pi\)
\(270\) 0 0
\(271\) 29.9309 1.81817 0.909085 0.416610i \(-0.136782\pi\)
0.909085 + 0.416610i \(0.136782\pi\)
\(272\) −7.31534 −0.443558
\(273\) 0 0
\(274\) 8.68466 0.524659
\(275\) −15.3693 −0.926805
\(276\) 0 0
\(277\) 5.31534 0.319368 0.159684 0.987168i \(-0.448952\pi\)
0.159684 + 0.987168i \(0.448952\pi\)
\(278\) 28.0000 1.67933
\(279\) 0 0
\(280\) 4.87689 0.291450
\(281\) 17.8078 1.06232 0.531161 0.847271i \(-0.321756\pi\)
0.531161 + 0.847271i \(0.321756\pi\)
\(282\) 0 0
\(283\) −13.6847 −0.813469 −0.406734 0.913547i \(-0.633333\pi\)
−0.406734 + 0.913547i \(0.633333\pi\)
\(284\) 6.13826 0.364239
\(285\) 0 0
\(286\) 0 0
\(287\) 0.876894 0.0517614
\(288\) 0 0
\(289\) −14.5616 −0.856562
\(290\) −37.1771 −2.18311
\(291\) 0 0
\(292\) 4.43845 0.259740
\(293\) 20.4384 1.19403 0.597013 0.802231i \(-0.296354\pi\)
0.597013 + 0.802231i \(0.296354\pi\)
\(294\) 0 0
\(295\) 10.2462 0.596557
\(296\) −18.4384 −1.07171
\(297\) 0 0
\(298\) 3.80776 0.220578
\(299\) 0 0
\(300\) 0 0
\(301\) −2.56155 −0.147645
\(302\) −14.6307 −0.841901
\(303\) 0 0
\(304\) 33.3693 1.91386
\(305\) −13.8078 −0.790630
\(306\) 0 0
\(307\) −30.8078 −1.75829 −0.879146 0.476553i \(-0.841886\pi\)
−0.879146 + 0.476553i \(0.841886\pi\)
\(308\) 0.492423 0.0280584
\(309\) 0 0
\(310\) −14.2462 −0.809130
\(311\) 19.1231 1.08437 0.542186 0.840259i \(-0.317597\pi\)
0.542186 + 0.840259i \(0.317597\pi\)
\(312\) 0 0
\(313\) −13.6847 −0.773503 −0.386751 0.922184i \(-0.626403\pi\)
−0.386751 + 0.922184i \(0.626403\pi\)
\(314\) −31.8078 −1.79502
\(315\) 0 0
\(316\) 2.38447 0.134137
\(317\) 14.0540 0.789350 0.394675 0.918821i \(-0.370857\pi\)
0.394675 + 0.918821i \(0.370857\pi\)
\(318\) 0 0
\(319\) 13.3693 0.748538
\(320\) −19.8078 −1.10729
\(321\) 0 0
\(322\) −1.75379 −0.0977348
\(323\) −11.1231 −0.618906
\(324\) 0 0
\(325\) 0 0
\(326\) −7.50758 −0.415806
\(327\) 0 0
\(328\) −3.80776 −0.210249
\(329\) −4.63068 −0.255298
\(330\) 0 0
\(331\) 3.19224 0.175461 0.0877306 0.996144i \(-0.472039\pi\)
0.0877306 + 0.996144i \(0.472039\pi\)
\(332\) −0.384472 −0.0211006
\(333\) 0 0
\(334\) −16.0000 −0.875481
\(335\) 16.2462 0.887625
\(336\) 0 0
\(337\) −6.12311 −0.333547 −0.166773 0.985995i \(-0.553335\pi\)
−0.166773 + 0.985995i \(0.553335\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −2.43845 −0.132243
\(341\) 5.12311 0.277432
\(342\) 0 0
\(343\) 7.68466 0.414933
\(344\) 11.1231 0.599718
\(345\) 0 0
\(346\) 31.6155 1.69966
\(347\) −27.6155 −1.48248 −0.741240 0.671241i \(-0.765762\pi\)
−0.741240 + 0.671241i \(0.765762\pi\)
\(348\) 0 0
\(349\) −6.80776 −0.364411 −0.182206 0.983260i \(-0.558324\pi\)
−0.182206 + 0.983260i \(0.558324\pi\)
\(350\) 6.73863 0.360195
\(351\) 0 0
\(352\) −4.87689 −0.259939
\(353\) 5.31534 0.282907 0.141454 0.989945i \(-0.454822\pi\)
0.141454 + 0.989945i \(0.454822\pi\)
\(354\) 0 0
\(355\) −49.8617 −2.64639
\(356\) 2.13826 0.113328
\(357\) 0 0
\(358\) −7.61553 −0.402493
\(359\) −9.36932 −0.494494 −0.247247 0.968953i \(-0.579526\pi\)
−0.247247 + 0.968953i \(0.579526\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 4.19224 0.220339
\(363\) 0 0
\(364\) 0 0
\(365\) −36.0540 −1.88715
\(366\) 0 0
\(367\) −17.0540 −0.890210 −0.445105 0.895478i \(-0.646834\pi\)
−0.445105 + 0.895478i \(0.646834\pi\)
\(368\) 9.36932 0.488409
\(369\) 0 0
\(370\) −42.0540 −2.18628
\(371\) −0.384472 −0.0199608
\(372\) 0 0
\(373\) 28.3693 1.46891 0.734454 0.678659i \(-0.237438\pi\)
0.734454 + 0.678659i \(0.237438\pi\)
\(374\) 4.87689 0.252178
\(375\) 0 0
\(376\) 20.1080 1.03699
\(377\) 0 0
\(378\) 0 0
\(379\) −23.6847 −1.21660 −0.608300 0.793708i \(-0.708148\pi\)
−0.608300 + 0.793708i \(0.708148\pi\)
\(380\) 11.1231 0.570603
\(381\) 0 0
\(382\) −14.2462 −0.728900
\(383\) −22.7386 −1.16189 −0.580945 0.813943i \(-0.697317\pi\)
−0.580945 + 0.813943i \(0.697317\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) −21.0691 −1.07239
\(387\) 0 0
\(388\) 3.75379 0.190570
\(389\) −34.0540 −1.72661 −0.863303 0.504687i \(-0.831608\pi\)
−0.863303 + 0.504687i \(0.831608\pi\)
\(390\) 0 0
\(391\) −3.12311 −0.157942
\(392\) −16.3002 −0.823284
\(393\) 0 0
\(394\) −20.8769 −1.05176
\(395\) −19.3693 −0.974576
\(396\) 0 0
\(397\) −25.0540 −1.25742 −0.628711 0.777639i \(-0.716417\pi\)
−0.628711 + 0.777639i \(0.716417\pi\)
\(398\) −34.6307 −1.73588
\(399\) 0 0
\(400\) −36.0000 −1.80000
\(401\) 14.4384 0.721022 0.360511 0.932755i \(-0.382602\pi\)
0.360511 + 0.932755i \(0.382602\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 3.31534 0.164944
\(405\) 0 0
\(406\) −5.86174 −0.290913
\(407\) 15.1231 0.749625
\(408\) 0 0
\(409\) −6.36932 −0.314942 −0.157471 0.987524i \(-0.550334\pi\)
−0.157471 + 0.987524i \(0.550334\pi\)
\(410\) −8.68466 −0.428905
\(411\) 0 0
\(412\) −1.50758 −0.0742730
\(413\) 1.61553 0.0794949
\(414\) 0 0
\(415\) 3.12311 0.153307
\(416\) 0 0
\(417\) 0 0
\(418\) −22.2462 −1.08810
\(419\) 34.2462 1.67304 0.836518 0.547939i \(-0.184587\pi\)
0.836518 + 0.547939i \(0.184587\pi\)
\(420\) 0 0
\(421\) −31.2462 −1.52285 −0.761424 0.648255i \(-0.775499\pi\)
−0.761424 + 0.648255i \(0.775499\pi\)
\(422\) −30.7386 −1.49633
\(423\) 0 0
\(424\) 1.66950 0.0810783
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −2.17708 −0.105356
\(428\) 3.61553 0.174763
\(429\) 0 0
\(430\) 25.3693 1.22342
\(431\) −11.1231 −0.535781 −0.267891 0.963449i \(-0.586327\pi\)
−0.267891 + 0.963449i \(0.586327\pi\)
\(432\) 0 0
\(433\) 8.75379 0.420680 0.210340 0.977628i \(-0.432543\pi\)
0.210340 + 0.977628i \(0.432543\pi\)
\(434\) −2.24621 −0.107822
\(435\) 0 0
\(436\) 1.23106 0.0589569
\(437\) 14.2462 0.681489
\(438\) 0 0
\(439\) −13.6847 −0.653133 −0.326567 0.945174i \(-0.605892\pi\)
−0.326567 + 0.945174i \(0.605892\pi\)
\(440\) 17.3693 0.828050
\(441\) 0 0
\(442\) 0 0
\(443\) 34.7386 1.65048 0.825241 0.564781i \(-0.191039\pi\)
0.825241 + 0.564781i \(0.191039\pi\)
\(444\) 0 0
\(445\) −17.3693 −0.823385
\(446\) 12.4924 0.591533
\(447\) 0 0
\(448\) −3.12311 −0.147553
\(449\) −8.24621 −0.389163 −0.194581 0.980886i \(-0.562335\pi\)
−0.194581 + 0.980886i \(0.562335\pi\)
\(450\) 0 0
\(451\) 3.12311 0.147061
\(452\) −2.54640 −0.119772
\(453\) 0 0
\(454\) −11.1231 −0.522033
\(455\) 0 0
\(456\) 0 0
\(457\) −12.6155 −0.590130 −0.295065 0.955477i \(-0.595341\pi\)
−0.295065 + 0.955477i \(0.595341\pi\)
\(458\) −25.3693 −1.18543
\(459\) 0 0
\(460\) 3.12311 0.145616
\(461\) −16.1922 −0.754148 −0.377074 0.926183i \(-0.623070\pi\)
−0.377074 + 0.926183i \(0.623070\pi\)
\(462\) 0 0
\(463\) 14.3153 0.665290 0.332645 0.943052i \(-0.392059\pi\)
0.332645 + 0.943052i \(0.392059\pi\)
\(464\) 31.3153 1.45378
\(465\) 0 0
\(466\) 40.6004 1.88078
\(467\) 26.0000 1.20314 0.601568 0.798821i \(-0.294543\pi\)
0.601568 + 0.798821i \(0.294543\pi\)
\(468\) 0 0
\(469\) 2.56155 0.118282
\(470\) 45.8617 2.11544
\(471\) 0 0
\(472\) −7.01515 −0.322899
\(473\) −9.12311 −0.419481
\(474\) 0 0
\(475\) −54.7386 −2.51158
\(476\) −0.384472 −0.0176222
\(477\) 0 0
\(478\) 39.6155 1.81197
\(479\) 10.2462 0.468161 0.234081 0.972217i \(-0.424792\pi\)
0.234081 + 0.972217i \(0.424792\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −27.8078 −1.26661
\(483\) 0 0
\(484\) −3.06913 −0.139506
\(485\) −30.4924 −1.38459
\(486\) 0 0
\(487\) −7.12311 −0.322779 −0.161389 0.986891i \(-0.551597\pi\)
−0.161389 + 0.986891i \(0.551597\pi\)
\(488\) 9.45360 0.427944
\(489\) 0 0
\(490\) −37.1771 −1.67949
\(491\) 36.2462 1.63577 0.817884 0.575383i \(-0.195147\pi\)
0.817884 + 0.575383i \(0.195147\pi\)
\(492\) 0 0
\(493\) −10.4384 −0.470124
\(494\) 0 0
\(495\) 0 0
\(496\) 12.0000 0.538816
\(497\) −7.86174 −0.352647
\(498\) 0 0
\(499\) −4.49242 −0.201108 −0.100554 0.994932i \(-0.532062\pi\)
−0.100554 + 0.994932i \(0.532062\pi\)
\(500\) −4.19224 −0.187482
\(501\) 0 0
\(502\) 29.2614 1.30600
\(503\) 28.2462 1.25944 0.629718 0.776824i \(-0.283170\pi\)
0.629718 + 0.776824i \(0.283170\pi\)
\(504\) 0 0
\(505\) −26.9309 −1.19841
\(506\) −6.24621 −0.277678
\(507\) 0 0
\(508\) 2.38447 0.105794
\(509\) −13.8078 −0.612018 −0.306009 0.952029i \(-0.598994\pi\)
−0.306009 + 0.952029i \(0.598994\pi\)
\(510\) 0 0
\(511\) −5.68466 −0.251474
\(512\) 11.4233 0.504843
\(513\) 0 0
\(514\) 45.5616 2.00963
\(515\) 12.2462 0.539633
\(516\) 0 0
\(517\) −16.4924 −0.725336
\(518\) −6.63068 −0.291335
\(519\) 0 0
\(520\) 0 0
\(521\) 9.06913 0.397326 0.198663 0.980068i \(-0.436340\pi\)
0.198663 + 0.980068i \(0.436340\pi\)
\(522\) 0 0
\(523\) 33.8617 1.48067 0.740335 0.672238i \(-0.234667\pi\)
0.740335 + 0.672238i \(0.234667\pi\)
\(524\) −3.23106 −0.141149
\(525\) 0 0
\(526\) 14.6307 0.637928
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 3.80776 0.165399
\(531\) 0 0
\(532\) 1.75379 0.0760364
\(533\) 0 0
\(534\) 0 0
\(535\) −29.3693 −1.26975
\(536\) −11.1231 −0.480445
\(537\) 0 0
\(538\) −33.3693 −1.43865
\(539\) 13.3693 0.575857
\(540\) 0 0
\(541\) −19.7386 −0.848630 −0.424315 0.905515i \(-0.639485\pi\)
−0.424315 + 0.905515i \(0.639485\pi\)
\(542\) −46.7386 −2.00760
\(543\) 0 0
\(544\) 3.80776 0.163257
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 3.93087 0.168072 0.0840359 0.996463i \(-0.473219\pi\)
0.0840359 + 0.996463i \(0.473219\pi\)
\(548\) −2.43845 −0.104165
\(549\) 0 0
\(550\) 24.0000 1.02336
\(551\) 47.6155 2.02849
\(552\) 0 0
\(553\) −3.05398 −0.129868
\(554\) −8.30019 −0.352641
\(555\) 0 0
\(556\) −7.86174 −0.333412
\(557\) −42.9309 −1.81904 −0.909520 0.415661i \(-0.863550\pi\)
−0.909520 + 0.415661i \(0.863550\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −9.36932 −0.395926
\(561\) 0 0
\(562\) −27.8078 −1.17300
\(563\) 23.3693 0.984899 0.492450 0.870341i \(-0.336102\pi\)
0.492450 + 0.870341i \(0.336102\pi\)
\(564\) 0 0
\(565\) 20.6847 0.870210
\(566\) 21.3693 0.898219
\(567\) 0 0
\(568\) 34.1383 1.43241
\(569\) 8.73863 0.366343 0.183171 0.983081i \(-0.441364\pi\)
0.183171 + 0.983081i \(0.441364\pi\)
\(570\) 0 0
\(571\) −5.36932 −0.224699 −0.112349 0.993669i \(-0.535838\pi\)
−0.112349 + 0.993669i \(0.535838\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.36932 −0.0571542
\(575\) −15.3693 −0.640945
\(576\) 0 0
\(577\) 17.3153 0.720847 0.360424 0.932789i \(-0.382632\pi\)
0.360424 + 0.932789i \(0.382632\pi\)
\(578\) 22.7386 0.945802
\(579\) 0 0
\(580\) 10.4384 0.433433
\(581\) 0.492423 0.0204291
\(582\) 0 0
\(583\) −1.36932 −0.0567113
\(584\) 24.6847 1.02146
\(585\) 0 0
\(586\) −31.9157 −1.31843
\(587\) 39.3693 1.62495 0.812473 0.582999i \(-0.198121\pi\)
0.812473 + 0.582999i \(0.198121\pi\)
\(588\) 0 0
\(589\) 18.2462 0.751822
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) 35.4233 1.45589
\(593\) −17.4233 −0.715489 −0.357744 0.933820i \(-0.616454\pi\)
−0.357744 + 0.933820i \(0.616454\pi\)
\(594\) 0 0
\(595\) 3.12311 0.128035
\(596\) −1.06913 −0.0437933
\(597\) 0 0
\(598\) 0 0
\(599\) 41.6155 1.70036 0.850182 0.526489i \(-0.176492\pi\)
0.850182 + 0.526489i \(0.176492\pi\)
\(600\) 0 0
\(601\) −7.06913 −0.288356 −0.144178 0.989552i \(-0.546054\pi\)
−0.144178 + 0.989552i \(0.546054\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 4.10795 0.167150
\(605\) 24.9309 1.01358
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −17.3693 −0.704419
\(609\) 0 0
\(610\) 21.5616 0.873002
\(611\) 0 0
\(612\) 0 0
\(613\) 34.8617 1.40805 0.704026 0.710174i \(-0.251384\pi\)
0.704026 + 0.710174i \(0.251384\pi\)
\(614\) 48.1080 1.94148
\(615\) 0 0
\(616\) 2.73863 0.110343
\(617\) 9.80776 0.394846 0.197423 0.980318i \(-0.436743\pi\)
0.197423 + 0.980318i \(0.436743\pi\)
\(618\) 0 0
\(619\) −29.3002 −1.17767 −0.588837 0.808252i \(-0.700414\pi\)
−0.588837 + 0.808252i \(0.700414\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −29.8617 −1.19735
\(623\) −2.73863 −0.109721
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 21.3693 0.854090
\(627\) 0 0
\(628\) 8.93087 0.356380
\(629\) −11.8078 −0.470806
\(630\) 0 0
\(631\) 18.5616 0.738924 0.369462 0.929246i \(-0.379542\pi\)
0.369462 + 0.929246i \(0.379542\pi\)
\(632\) 13.2614 0.527509
\(633\) 0 0
\(634\) −21.9460 −0.871588
\(635\) −19.3693 −0.768648
\(636\) 0 0
\(637\) 0 0
\(638\) −20.8769 −0.826524
\(639\) 0 0
\(640\) 48.3002 1.90923
\(641\) 19.1771 0.757449 0.378725 0.925509i \(-0.376363\pi\)
0.378725 + 0.925509i \(0.376363\pi\)
\(642\) 0 0
\(643\) 31.5464 1.24407 0.622034 0.782990i \(-0.286306\pi\)
0.622034 + 0.782990i \(0.286306\pi\)
\(644\) 0.492423 0.0194042
\(645\) 0 0
\(646\) 17.3693 0.683387
\(647\) 6.38447 0.250999 0.125500 0.992094i \(-0.459947\pi\)
0.125500 + 0.992094i \(0.459947\pi\)
\(648\) 0 0
\(649\) 5.75379 0.225856
\(650\) 0 0
\(651\) 0 0
\(652\) 2.10795 0.0825537
\(653\) −23.1231 −0.904877 −0.452439 0.891796i \(-0.649446\pi\)
−0.452439 + 0.891796i \(0.649446\pi\)
\(654\) 0 0
\(655\) 26.2462 1.02552
\(656\) 7.31534 0.285616
\(657\) 0 0
\(658\) 7.23106 0.281896
\(659\) 2.24621 0.0875000 0.0437500 0.999043i \(-0.486070\pi\)
0.0437500 + 0.999043i \(0.486070\pi\)
\(660\) 0 0
\(661\) −5.63068 −0.219008 −0.109504 0.993986i \(-0.534926\pi\)
−0.109504 + 0.993986i \(0.534926\pi\)
\(662\) −4.98485 −0.193742
\(663\) 0 0
\(664\) −2.13826 −0.0829806
\(665\) −14.2462 −0.552444
\(666\) 0 0
\(667\) 13.3693 0.517662
\(668\) 4.49242 0.173817
\(669\) 0 0
\(670\) −25.3693 −0.980102
\(671\) −7.75379 −0.299332
\(672\) 0 0
\(673\) −23.2462 −0.896076 −0.448038 0.894015i \(-0.647877\pi\)
−0.448038 + 0.894015i \(0.647877\pi\)
\(674\) 9.56155 0.368297
\(675\) 0 0
\(676\) 0 0
\(677\) 15.6155 0.600153 0.300077 0.953915i \(-0.402988\pi\)
0.300077 + 0.953915i \(0.402988\pi\)
\(678\) 0 0
\(679\) −4.80776 −0.184505
\(680\) −13.5616 −0.520062
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −38.1080 −1.45816 −0.729080 0.684428i \(-0.760052\pi\)
−0.729080 + 0.684428i \(0.760052\pi\)
\(684\) 0 0
\(685\) 19.8078 0.756816
\(686\) −12.0000 −0.458162
\(687\) 0 0
\(688\) −21.3693 −0.814698
\(689\) 0 0
\(690\) 0 0
\(691\) 51.3002 1.95155 0.975776 0.218774i \(-0.0702058\pi\)
0.975776 + 0.218774i \(0.0702058\pi\)
\(692\) −8.87689 −0.337449
\(693\) 0 0
\(694\) 43.1231 1.63693
\(695\) 63.8617 2.42241
\(696\) 0 0
\(697\) −2.43845 −0.0923628
\(698\) 10.6307 0.402377
\(699\) 0 0
\(700\) −1.89205 −0.0715127
\(701\) 5.36932 0.202796 0.101398 0.994846i \(-0.467668\pi\)
0.101398 + 0.994846i \(0.467668\pi\)
\(702\) 0 0
\(703\) 53.8617 2.03143
\(704\) −11.1231 −0.419218
\(705\) 0 0
\(706\) −8.30019 −0.312382
\(707\) −4.24621 −0.159695
\(708\) 0 0
\(709\) 7.49242 0.281384 0.140692 0.990053i \(-0.455067\pi\)
0.140692 + 0.990053i \(0.455067\pi\)
\(710\) 77.8617 2.92210
\(711\) 0 0
\(712\) 11.8920 0.445673
\(713\) 5.12311 0.191862
\(714\) 0 0
\(715\) 0 0
\(716\) 2.13826 0.0799106
\(717\) 0 0
\(718\) 14.6307 0.546012
\(719\) 23.3693 0.871528 0.435764 0.900061i \(-0.356478\pi\)
0.435764 + 0.900061i \(0.356478\pi\)
\(720\) 0 0
\(721\) 1.93087 0.0719093
\(722\) −49.5616 −1.84449
\(723\) 0 0
\(724\) −1.17708 −0.0437459
\(725\) −51.3693 −1.90781
\(726\) 0 0
\(727\) −38.6695 −1.43417 −0.717086 0.696984i \(-0.754525\pi\)
−0.717086 + 0.696984i \(0.754525\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 56.3002 2.08376
\(731\) 7.12311 0.263458
\(732\) 0 0
\(733\) −20.5076 −0.757465 −0.378732 0.925506i \(-0.623640\pi\)
−0.378732 + 0.925506i \(0.623640\pi\)
\(734\) 26.6307 0.982956
\(735\) 0 0
\(736\) −4.87689 −0.179765
\(737\) 9.12311 0.336054
\(738\) 0 0
\(739\) −10.2462 −0.376913 −0.188456 0.982082i \(-0.560348\pi\)
−0.188456 + 0.982082i \(0.560348\pi\)
\(740\) 11.8078 0.434062
\(741\) 0 0
\(742\) 0.600373 0.0220404
\(743\) −12.6307 −0.463375 −0.231687 0.972790i \(-0.574425\pi\)
−0.231687 + 0.972790i \(0.574425\pi\)
\(744\) 0 0
\(745\) 8.68466 0.318181
\(746\) −44.3002 −1.62195
\(747\) 0 0
\(748\) −1.36932 −0.0500672
\(749\) −4.63068 −0.169201
\(750\) 0 0
\(751\) 44.1080 1.60952 0.804761 0.593599i \(-0.202293\pi\)
0.804761 + 0.593599i \(0.202293\pi\)
\(752\) −38.6307 −1.40872
\(753\) 0 0
\(754\) 0 0
\(755\) −33.3693 −1.21443
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 36.9848 1.34335
\(759\) 0 0
\(760\) 61.8617 2.24396
\(761\) 9.36932 0.339637 0.169819 0.985475i \(-0.445682\pi\)
0.169819 + 0.985475i \(0.445682\pi\)
\(762\) 0 0
\(763\) −1.57671 −0.0570807
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) 35.5076 1.28294
\(767\) 0 0
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 6.24621 0.225098
\(771\) 0 0
\(772\) 5.91571 0.212911
\(773\) −24.2462 −0.872076 −0.436038 0.899928i \(-0.643619\pi\)
−0.436038 + 0.899928i \(0.643619\pi\)
\(774\) 0 0
\(775\) −19.6847 −0.707094
\(776\) 20.8769 0.749437
\(777\) 0 0
\(778\) 53.1771 1.90649
\(779\) 11.1231 0.398527
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 4.87689 0.174397
\(783\) 0 0
\(784\) 31.3153 1.11841
\(785\) −72.5464 −2.58929
\(786\) 0 0
\(787\) −44.1771 −1.57474 −0.787371 0.616479i \(-0.788559\pi\)
−0.787371 + 0.616479i \(0.788559\pi\)
\(788\) 5.86174 0.208816
\(789\) 0 0
\(790\) 30.2462 1.07611
\(791\) 3.26137 0.115961
\(792\) 0 0
\(793\) 0 0
\(794\) 39.1231 1.38843
\(795\) 0 0
\(796\) 9.72348 0.344640
\(797\) 0.384472 0.0136187 0.00680935 0.999977i \(-0.497833\pi\)
0.00680935 + 0.999977i \(0.497833\pi\)
\(798\) 0 0
\(799\) 12.8769 0.455552
\(800\) 18.7386 0.662511
\(801\) 0 0
\(802\) −22.5464 −0.796141
\(803\) −20.2462 −0.714473
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 0 0
\(808\) 18.4384 0.648662
\(809\) 16.3002 0.573084 0.286542 0.958068i \(-0.407494\pi\)
0.286542 + 0.958068i \(0.407494\pi\)
\(810\) 0 0
\(811\) −2.56155 −0.0899483 −0.0449741 0.998988i \(-0.514321\pi\)
−0.0449741 + 0.998988i \(0.514321\pi\)
\(812\) 1.64584 0.0577576
\(813\) 0 0
\(814\) −23.6155 −0.827724
\(815\) −17.1231 −0.599796
\(816\) 0 0
\(817\) −32.4924 −1.13677
\(818\) 9.94602 0.347755
\(819\) 0 0
\(820\) 2.43845 0.0851543
\(821\) 6.49242 0.226587 0.113294 0.993562i \(-0.463860\pi\)
0.113294 + 0.993562i \(0.463860\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −8.38447 −0.292087
\(825\) 0 0
\(826\) −2.52273 −0.0877771
\(827\) −14.7386 −0.512513 −0.256256 0.966609i \(-0.582489\pi\)
−0.256256 + 0.966609i \(0.582489\pi\)
\(828\) 0 0
\(829\) 13.4924 0.468611 0.234306 0.972163i \(-0.424718\pi\)
0.234306 + 0.972163i \(0.424718\pi\)
\(830\) −4.87689 −0.169279
\(831\) 0 0
\(832\) 0 0
\(833\) −10.4384 −0.361671
\(834\) 0 0
\(835\) −36.4924 −1.26287
\(836\) 6.24621 0.216030
\(837\) 0 0
\(838\) −53.4773 −1.84734
\(839\) 21.6155 0.746251 0.373125 0.927781i \(-0.378286\pi\)
0.373125 + 0.927781i \(0.378286\pi\)
\(840\) 0 0
\(841\) 15.6847 0.540850
\(842\) 48.7926 1.68150
\(843\) 0 0
\(844\) 8.63068 0.297080
\(845\) 0 0
\(846\) 0 0
\(847\) 3.93087 0.135066
\(848\) −3.20739 −0.110142
\(849\) 0 0
\(850\) −18.7386 −0.642730
\(851\) 15.1231 0.518413
\(852\) 0 0
\(853\) −2.12311 −0.0726938 −0.0363469 0.999339i \(-0.511572\pi\)
−0.0363469 + 0.999339i \(0.511572\pi\)
\(854\) 3.39963 0.116333
\(855\) 0 0
\(856\) 20.1080 0.687276
\(857\) −35.5616 −1.21476 −0.607380 0.794412i \(-0.707779\pi\)
−0.607380 + 0.794412i \(0.707779\pi\)
\(858\) 0 0
\(859\) 24.5616 0.838029 0.419015 0.907979i \(-0.362376\pi\)
0.419015 + 0.907979i \(0.362376\pi\)
\(860\) −7.12311 −0.242896
\(861\) 0 0
\(862\) 17.3693 0.591601
\(863\) 30.4924 1.03797 0.518987 0.854782i \(-0.326309\pi\)
0.518987 + 0.854782i \(0.326309\pi\)
\(864\) 0 0
\(865\) 72.1080 2.45174
\(866\) −13.6695 −0.464509
\(867\) 0 0
\(868\) 0.630683 0.0214068
\(869\) −10.8769 −0.368973
\(870\) 0 0
\(871\) 0 0
\(872\) 6.84658 0.231855
\(873\) 0 0
\(874\) −22.2462 −0.752489
\(875\) 5.36932 0.181516
\(876\) 0 0
\(877\) −23.5616 −0.795617 −0.397809 0.917468i \(-0.630229\pi\)
−0.397809 + 0.917468i \(0.630229\pi\)
\(878\) 21.3693 0.721180
\(879\) 0 0
\(880\) −33.3693 −1.12488
\(881\) 9.06913 0.305547 0.152773 0.988261i \(-0.451180\pi\)
0.152773 + 0.988261i \(0.451180\pi\)
\(882\) 0 0
\(883\) 8.80776 0.296405 0.148202 0.988957i \(-0.452651\pi\)
0.148202 + 0.988957i \(0.452651\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −54.2462 −1.82244
\(887\) −24.6307 −0.827017 −0.413509 0.910500i \(-0.635697\pi\)
−0.413509 + 0.910500i \(0.635697\pi\)
\(888\) 0 0
\(889\) −3.05398 −0.102427
\(890\) 27.1231 0.909169
\(891\) 0 0
\(892\) −3.50758 −0.117442
\(893\) −58.7386 −1.96561
\(894\) 0 0
\(895\) −17.3693 −0.580592
\(896\) 7.61553 0.254417
\(897\) 0 0
\(898\) 12.8769 0.429708
\(899\) 17.1231 0.571088
\(900\) 0 0
\(901\) 1.06913 0.0356179
\(902\) −4.87689 −0.162383
\(903\) 0 0
\(904\) −14.1619 −0.471019
\(905\) 9.56155 0.317837
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 3.12311 0.103644
\(909\) 0 0
\(910\) 0 0
\(911\) −38.7386 −1.28347 −0.641734 0.766927i \(-0.721785\pi\)
−0.641734 + 0.766927i \(0.721785\pi\)
\(912\) 0 0
\(913\) 1.75379 0.0580419
\(914\) 19.6998 0.651612
\(915\) 0 0
\(916\) 7.12311 0.235354
\(917\) 4.13826 0.136657
\(918\) 0 0
\(919\) 11.5076 0.379600 0.189800 0.981823i \(-0.439216\pi\)
0.189800 + 0.981823i \(0.439216\pi\)
\(920\) 17.3693 0.572649
\(921\) 0 0
\(922\) 25.2850 0.832718
\(923\) 0 0
\(924\) 0 0
\(925\) −58.1080 −1.91058
\(926\) −22.3542 −0.734603
\(927\) 0 0
\(928\) −16.3002 −0.535080
\(929\) −7.80776 −0.256164 −0.128082 0.991764i \(-0.540882\pi\)
−0.128082 + 0.991764i \(0.540882\pi\)
\(930\) 0 0
\(931\) 47.6155 1.56054
\(932\) −11.3996 −0.373407
\(933\) 0 0
\(934\) −40.6004 −1.32848
\(935\) 11.1231 0.363764
\(936\) 0 0
\(937\) −7.56155 −0.247025 −0.123513 0.992343i \(-0.539416\pi\)
−0.123513 + 0.992343i \(0.539416\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −12.8769 −0.419998
\(941\) 30.4924 0.994025 0.497012 0.867744i \(-0.334430\pi\)
0.497012 + 0.867744i \(0.334430\pi\)
\(942\) 0 0
\(943\) 3.12311 0.101702
\(944\) 13.4773 0.438648
\(945\) 0 0
\(946\) 14.2462 0.463184
\(947\) −38.7386 −1.25884 −0.629418 0.777067i \(-0.716707\pi\)
−0.629418 + 0.777067i \(0.716707\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 85.4773 2.77325
\(951\) 0 0
\(952\) −2.13826 −0.0693014
\(953\) −30.9848 −1.00370 −0.501849 0.864955i \(-0.667347\pi\)
−0.501849 + 0.864955i \(0.667347\pi\)
\(954\) 0 0
\(955\) −32.4924 −1.05143
\(956\) −11.1231 −0.359747
\(957\) 0 0
\(958\) −16.0000 −0.516937
\(959\) 3.12311 0.100850
\(960\) 0 0
\(961\) −24.4384 −0.788337
\(962\) 0 0
\(963\) 0 0
\(964\) 7.80776 0.251471
\(965\) −48.0540 −1.54691
\(966\) 0 0
\(967\) 0.876894 0.0281990 0.0140995 0.999901i \(-0.495512\pi\)
0.0140995 + 0.999901i \(0.495512\pi\)
\(968\) −17.0691 −0.548623
\(969\) 0 0
\(970\) 47.6155 1.52884
\(971\) 12.9848 0.416704 0.208352 0.978054i \(-0.433190\pi\)
0.208352 + 0.978054i \(0.433190\pi\)
\(972\) 0 0
\(973\) 10.0691 0.322801
\(974\) 11.1231 0.356407
\(975\) 0 0
\(976\) −18.1619 −0.581349
\(977\) 61.1771 1.95723 0.978614 0.205705i \(-0.0659486\pi\)
0.978614 + 0.205705i \(0.0659486\pi\)
\(978\) 0 0
\(979\) −9.75379 −0.311732
\(980\) 10.4384 0.333444
\(981\) 0 0
\(982\) −56.6004 −1.80619
\(983\) 13.6155 0.434268 0.217134 0.976142i \(-0.430329\pi\)
0.217134 + 0.976142i \(0.430329\pi\)
\(984\) 0 0
\(985\) −47.6155 −1.51716
\(986\) 16.3002 0.519104
\(987\) 0 0
\(988\) 0 0
\(989\) −9.12311 −0.290098
\(990\) 0 0
\(991\) −50.3542 −1.59955 −0.799776 0.600298i \(-0.795049\pi\)
−0.799776 + 0.600298i \(0.795049\pi\)
\(992\) −6.24621 −0.198317
\(993\) 0 0
\(994\) 12.2765 0.389388
\(995\) −78.9848 −2.50399
\(996\) 0 0
\(997\) −20.6155 −0.652900 −0.326450 0.945214i \(-0.605853\pi\)
−0.326450 + 0.945214i \(0.605853\pi\)
\(998\) 7.01515 0.222061
\(999\) 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 1521.2.a.m.1.1 2
3.2 odd 2 507.2.a.d.1.2 2
12.11 even 2 8112.2.a.bo.1.2 2
13.4 even 6 117.2.g.c.55.1 4
13.5 odd 4 1521.2.b.h.1351.3 4
13.8 odd 4 1521.2.b.h.1351.2 4
13.10 even 6 117.2.g.c.100.1 4
13.12 even 2 1521.2.a.g.1.2 2
39.2 even 12 507.2.j.g.316.3 8
39.5 even 4 507.2.b.d.337.2 4
39.8 even 4 507.2.b.d.337.3 4
39.11 even 12 507.2.j.g.316.2 8
39.17 odd 6 39.2.e.b.16.2 4
39.20 even 12 507.2.j.g.361.3 8
39.23 odd 6 39.2.e.b.22.2 yes 4
39.29 odd 6 507.2.e.g.22.1 4
39.32 even 12 507.2.j.g.361.2 8
39.35 odd 6 507.2.e.g.484.1 4
39.38 odd 2 507.2.a.g.1.1 2
52.23 odd 6 1872.2.t.r.1153.2 4
52.43 odd 6 1872.2.t.r.289.2 4
156.23 even 6 624.2.q.h.529.1 4
156.95 even 6 624.2.q.h.289.1 4
156.155 even 2 8112.2.a.bk.1.1 2
195.17 even 12 975.2.bb.i.874.2 8
195.23 even 12 975.2.bb.i.724.2 8
195.62 even 12 975.2.bb.i.724.3 8
195.134 odd 6 975.2.i.k.601.1 4
195.173 even 12 975.2.bb.i.874.3 8
195.179 odd 6 975.2.i.k.451.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.2 4 39.17 odd 6
39.2.e.b.22.2 yes 4 39.23 odd 6
117.2.g.c.55.1 4 13.4 even 6
117.2.g.c.100.1 4 13.10 even 6
507.2.a.d.1.2 2 3.2 odd 2
507.2.a.g.1.1 2 39.38 odd 2
507.2.b.d.337.2 4 39.5 even 4
507.2.b.d.337.3 4 39.8 even 4
507.2.e.g.22.1 4 39.29 odd 6
507.2.e.g.484.1 4 39.35 odd 6
507.2.j.g.316.2 8 39.11 even 12
507.2.j.g.316.3 8 39.2 even 12
507.2.j.g.361.2 8 39.32 even 12
507.2.j.g.361.3 8 39.20 even 12
624.2.q.h.289.1 4 156.95 even 6
624.2.q.h.529.1 4 156.23 even 6
975.2.i.k.451.1 4 195.179 odd 6
975.2.i.k.601.1 4 195.134 odd 6
975.2.bb.i.724.2 8 195.23 even 12
975.2.bb.i.724.3 8 195.62 even 12
975.2.bb.i.874.2 8 195.17 even 12
975.2.bb.i.874.3 8 195.173 even 12
1521.2.a.g.1.2 2 13.12 even 2
1521.2.a.m.1.1 2 1.1 even 1 trivial
1521.2.b.h.1351.2 4 13.8 odd 4
1521.2.b.h.1351.3 4 13.5 odd 4
1872.2.t.r.289.2 4 52.43 odd 6
1872.2.t.r.1153.2 4 52.23 odd 6
8112.2.a.bk.1.1 2 156.155 even 2
8112.2.a.bo.1.2 2 12.11 even 2