Properties

Label 1521.2.b.m.1351.2
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.m.1351.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.24698i q^{2} +0.445042 q^{4} +2.80194i q^{5} +4.80194i q^{7} -3.04892i q^{8} +O(q^{10})\) \(q-1.24698i q^{2} +0.445042 q^{4} +2.80194i q^{5} +4.80194i q^{7} -3.04892i q^{8} +3.49396 q^{10} +1.46681i q^{11} +5.98792 q^{14} -2.91185 q^{16} -2.44504 q^{17} +2.54288i q^{19} +1.24698i q^{20} +1.82908 q^{22} -3.51573 q^{23} -2.85086 q^{25} +2.13706i q^{28} -1.85086 q^{29} -7.63102i q^{31} -2.46681i q^{32} +3.04892i q^{34} -13.4547 q^{35} +4.55496i q^{37} +3.17092 q^{38} +8.54288 q^{40} -1.24698i q^{41} -2.38404 q^{43} +0.652793i q^{44} +4.38404i q^{46} +12.8170i q^{47} -16.0586 q^{49} +3.55496i q^{50} +8.85086 q^{53} -4.10992 q^{55} +14.6407 q^{56} +2.30798i q^{58} +2.17629i q^{59} -7.82908 q^{61} -9.51573 q^{62} -8.89977 q^{64} -3.58211i q^{67} -1.08815 q^{68} +16.7778i q^{70} +8.83877i q^{71} +7.69202i q^{73} +5.67994 q^{74} +1.13169i q^{76} -7.04354 q^{77} -4.02177 q^{79} -8.15883i q^{80} -1.55496 q^{82} +0.652793i q^{83} -6.85086i q^{85} +2.97285i q^{86} +4.47219 q^{88} +6.29590i q^{89} -1.56465 q^{92} +15.9825 q^{94} -7.12498 q^{95} -10.0315i q^{97} +20.0248i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{4} + O(q^{10}) \) \( 6q + 2q^{4} + 2q^{10} - 2q^{14} - 10q^{16} - 14q^{17} - 10q^{22} + 4q^{23} + 10q^{25} + 16q^{29} - 36q^{35} + 40q^{38} + 14q^{40} + 6q^{43} - 34q^{49} + 26q^{53} - 26q^{55} + 14q^{56} - 26q^{61} - 32q^{62} - 8q^{64} - 14q^{68} - 14q^{74} - 30q^{77} - 18q^{79} - 10q^{82} + 14q^{88} + 34q^{92} + 64q^{94} + 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.24698i − 0.881748i −0.897569 0.440874i \(-0.854669\pi\)
0.897569 0.440874i \(-0.145331\pi\)
\(3\) 0 0
\(4\) 0.445042 0.222521
\(5\) 2.80194i 1.25306i 0.779395 + 0.626532i \(0.215526\pi\)
−0.779395 + 0.626532i \(0.784474\pi\)
\(6\) 0 0
\(7\) 4.80194i 1.81496i 0.420093 + 0.907481i \(0.361997\pi\)
−0.420093 + 0.907481i \(0.638003\pi\)
\(8\) − 3.04892i − 1.07796i
\(9\) 0 0
\(10\) 3.49396 1.10489
\(11\) 1.46681i 0.442260i 0.975244 + 0.221130i \(0.0709746\pi\)
−0.975244 + 0.221130i \(0.929025\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 5.98792 1.60034
\(15\) 0 0
\(16\) −2.91185 −0.727963
\(17\) −2.44504 −0.593010 −0.296505 0.955031i \(-0.595821\pi\)
−0.296505 + 0.955031i \(0.595821\pi\)
\(18\) 0 0
\(19\) 2.54288i 0.583376i 0.956514 + 0.291688i \(0.0942169\pi\)
−0.956514 + 0.291688i \(0.905783\pi\)
\(20\) 1.24698i 0.278833i
\(21\) 0 0
\(22\) 1.82908 0.389962
\(23\) −3.51573 −0.733080 −0.366540 0.930402i \(-0.619458\pi\)
−0.366540 + 0.930402i \(0.619458\pi\)
\(24\) 0 0
\(25\) −2.85086 −0.570171
\(26\) 0 0
\(27\) 0 0
\(28\) 2.13706i 0.403867i
\(29\) −1.85086 −0.343695 −0.171848 0.985124i \(-0.554974\pi\)
−0.171848 + 0.985124i \(0.554974\pi\)
\(30\) 0 0
\(31\) − 7.63102i − 1.37057i −0.728274 0.685286i \(-0.759677\pi\)
0.728274 0.685286i \(-0.240323\pi\)
\(32\) − 2.46681i − 0.436075i
\(33\) 0 0
\(34\) 3.04892i 0.522885i
\(35\) −13.4547 −2.27426
\(36\) 0 0
\(37\) 4.55496i 0.748831i 0.927261 + 0.374415i \(0.122157\pi\)
−0.927261 + 0.374415i \(0.877843\pi\)
\(38\) 3.17092 0.514390
\(39\) 0 0
\(40\) 8.54288 1.35075
\(41\) − 1.24698i − 0.194745i −0.995248 0.0973727i \(-0.968956\pi\)
0.995248 0.0973727i \(-0.0310439\pi\)
\(42\) 0 0
\(43\) −2.38404 −0.363563 −0.181782 0.983339i \(-0.558186\pi\)
−0.181782 + 0.983339i \(0.558186\pi\)
\(44\) 0.652793i 0.0984122i
\(45\) 0 0
\(46\) 4.38404i 0.646392i
\(47\) 12.8170i 1.86955i 0.355238 + 0.934776i \(0.384400\pi\)
−0.355238 + 0.934776i \(0.615600\pi\)
\(48\) 0 0
\(49\) −16.0586 −2.29409
\(50\) 3.55496i 0.502747i
\(51\) 0 0
\(52\) 0 0
\(53\) 8.85086 1.21576 0.607879 0.794030i \(-0.292020\pi\)
0.607879 + 0.794030i \(0.292020\pi\)
\(54\) 0 0
\(55\) −4.10992 −0.554181
\(56\) 14.6407 1.95645
\(57\) 0 0
\(58\) 2.30798i 0.303052i
\(59\) 2.17629i 0.283329i 0.989915 + 0.141665i \(0.0452454\pi\)
−0.989915 + 0.141665i \(0.954755\pi\)
\(60\) 0 0
\(61\) −7.82908 −1.00241 −0.501206 0.865328i \(-0.667110\pi\)
−0.501206 + 0.865328i \(0.667110\pi\)
\(62\) −9.51573 −1.20850
\(63\) 0 0
\(64\) −8.89977 −1.11247
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.58211i − 0.437624i −0.975767 0.218812i \(-0.929782\pi\)
0.975767 0.218812i \(-0.0702181\pi\)
\(68\) −1.08815 −0.131957
\(69\) 0 0
\(70\) 16.7778i 2.00533i
\(71\) 8.83877i 1.04897i 0.851420 + 0.524485i \(0.175742\pi\)
−0.851420 + 0.524485i \(0.824258\pi\)
\(72\) 0 0
\(73\) 7.69202i 0.900283i 0.892957 + 0.450142i \(0.148626\pi\)
−0.892957 + 0.450142i \(0.851374\pi\)
\(74\) 5.67994 0.660280
\(75\) 0 0
\(76\) 1.13169i 0.129813i
\(77\) −7.04354 −0.802686
\(78\) 0 0
\(79\) −4.02177 −0.452485 −0.226242 0.974071i \(-0.572644\pi\)
−0.226242 + 0.974071i \(0.572644\pi\)
\(80\) − 8.15883i − 0.912185i
\(81\) 0 0
\(82\) −1.55496 −0.171716
\(83\) 0.652793i 0.0716533i 0.999358 + 0.0358267i \(0.0114064\pi\)
−0.999358 + 0.0358267i \(0.988594\pi\)
\(84\) 0 0
\(85\) − 6.85086i − 0.743080i
\(86\) 2.97285i 0.320571i
\(87\) 0 0
\(88\) 4.47219 0.476737
\(89\) 6.29590i 0.667364i 0.942686 + 0.333682i \(0.108291\pi\)
−0.942686 + 0.333682i \(0.891709\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.56465 −0.163126
\(93\) 0 0
\(94\) 15.9825 1.64847
\(95\) −7.12498 −0.731008
\(96\) 0 0
\(97\) − 10.0315i − 1.01854i −0.860607 0.509270i \(-0.829915\pi\)
0.860607 0.509270i \(-0.170085\pi\)
\(98\) 20.0248i 2.02281i
\(99\) 0 0
\(100\) −1.26875 −0.126875
\(101\) 13.8877 1.38188 0.690938 0.722914i \(-0.257198\pi\)
0.690938 + 0.722914i \(0.257198\pi\)
\(102\) 0 0
\(103\) 17.4034 1.71481 0.857405 0.514642i \(-0.172075\pi\)
0.857405 + 0.514642i \(0.172075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 11.0368i − 1.07199i
\(107\) −10.5526 −1.02015 −0.510077 0.860128i \(-0.670383\pi\)
−0.510077 + 0.860128i \(0.670383\pi\)
\(108\) 0 0
\(109\) 1.07069i 0.102553i 0.998684 + 0.0512766i \(0.0163290\pi\)
−0.998684 + 0.0512766i \(0.983671\pi\)
\(110\) 5.12498i 0.488648i
\(111\) 0 0
\(112\) − 13.9825i − 1.32123i
\(113\) 16.5308 1.55509 0.777543 0.628830i \(-0.216466\pi\)
0.777543 + 0.628830i \(0.216466\pi\)
\(114\) 0 0
\(115\) − 9.85086i − 0.918597i
\(116\) −0.823708 −0.0764794
\(117\) 0 0
\(118\) 2.71379 0.249825
\(119\) − 11.7409i − 1.07629i
\(120\) 0 0
\(121\) 8.84846 0.804406
\(122\) 9.76271i 0.883874i
\(123\) 0 0
\(124\) − 3.39612i − 0.304981i
\(125\) 6.02177i 0.538604i
\(126\) 0 0
\(127\) 9.53750 0.846316 0.423158 0.906056i \(-0.360921\pi\)
0.423158 + 0.906056i \(0.360921\pi\)
\(128\) 6.16421i 0.544844i
\(129\) 0 0
\(130\) 0 0
\(131\) 5.50902 0.481326 0.240663 0.970609i \(-0.422635\pi\)
0.240663 + 0.970609i \(0.422635\pi\)
\(132\) 0 0
\(133\) −12.2107 −1.05880
\(134\) −4.46681 −0.385874
\(135\) 0 0
\(136\) 7.45473i 0.639238i
\(137\) − 16.1836i − 1.38266i −0.722540 0.691329i \(-0.757026\pi\)
0.722540 0.691329i \(-0.242974\pi\)
\(138\) 0 0
\(139\) −10.5090 −0.891364 −0.445682 0.895191i \(-0.647039\pi\)
−0.445682 + 0.895191i \(0.647039\pi\)
\(140\) −5.98792 −0.506071
\(141\) 0 0
\(142\) 11.0218 0.924926
\(143\) 0 0
\(144\) 0 0
\(145\) − 5.18598i − 0.430672i
\(146\) 9.59179 0.793823
\(147\) 0 0
\(148\) 2.02715i 0.166630i
\(149\) − 14.3502i − 1.17561i −0.809001 0.587807i \(-0.799992\pi\)
0.809001 0.587807i \(-0.200008\pi\)
\(150\) 0 0
\(151\) − 1.96615i − 0.160003i −0.996795 0.0800014i \(-0.974508\pi\)
0.996795 0.0800014i \(-0.0254925\pi\)
\(152\) 7.75302 0.628853
\(153\) 0 0
\(154\) 8.78315i 0.707767i
\(155\) 21.3817 1.71742
\(156\) 0 0
\(157\) 10.7017 0.854089 0.427045 0.904231i \(-0.359555\pi\)
0.427045 + 0.904231i \(0.359555\pi\)
\(158\) 5.01507i 0.398977i
\(159\) 0 0
\(160\) 6.91185 0.546430
\(161\) − 16.8823i − 1.33051i
\(162\) 0 0
\(163\) − 3.89977i − 0.305454i −0.988268 0.152727i \(-0.951195\pi\)
0.988268 0.152727i \(-0.0488055\pi\)
\(164\) − 0.554958i − 0.0433349i
\(165\) 0 0
\(166\) 0.814019 0.0631802
\(167\) 21.0194i 1.62653i 0.581895 + 0.813264i \(0.302312\pi\)
−0.581895 + 0.813264i \(0.697688\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −8.54288 −0.655209
\(171\) 0 0
\(172\) −1.06100 −0.0809004
\(173\) 13.2349 1.00623 0.503115 0.864219i \(-0.332187\pi\)
0.503115 + 0.864219i \(0.332187\pi\)
\(174\) 0 0
\(175\) − 13.6896i − 1.03484i
\(176\) − 4.27114i − 0.321950i
\(177\) 0 0
\(178\) 7.85086 0.588446
\(179\) 8.52781 0.637399 0.318699 0.947856i \(-0.396754\pi\)
0.318699 + 0.947856i \(0.396754\pi\)
\(180\) 0 0
\(181\) 3.63640 0.270291 0.135146 0.990826i \(-0.456850\pi\)
0.135146 + 0.990826i \(0.456850\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 10.7192i 0.790228i
\(185\) −12.7627 −0.938333
\(186\) 0 0
\(187\) − 3.58642i − 0.262265i
\(188\) 5.70410i 0.416014i
\(189\) 0 0
\(190\) 8.88471i 0.644564i
\(191\) −21.3817 −1.54712 −0.773561 0.633722i \(-0.781526\pi\)
−0.773561 + 0.633722i \(0.781526\pi\)
\(192\) 0 0
\(193\) 8.42758i 0.606631i 0.952890 + 0.303315i \(0.0980936\pi\)
−0.952890 + 0.303315i \(0.901906\pi\)
\(194\) −12.5090 −0.898096
\(195\) 0 0
\(196\) −7.14675 −0.510482
\(197\) 26.4765i 1.88637i 0.332264 + 0.943186i \(0.392187\pi\)
−0.332264 + 0.943186i \(0.607813\pi\)
\(198\) 0 0
\(199\) 14.2524 1.01032 0.505161 0.863025i \(-0.331433\pi\)
0.505161 + 0.863025i \(0.331433\pi\)
\(200\) 8.69202i 0.614619i
\(201\) 0 0
\(202\) − 17.3177i − 1.21847i
\(203\) − 8.88769i − 0.623794i
\(204\) 0 0
\(205\) 3.49396 0.244029
\(206\) − 21.7017i − 1.51203i
\(207\) 0 0
\(208\) 0 0
\(209\) −3.72992 −0.258004
\(210\) 0 0
\(211\) 1.85086 0.127418 0.0637091 0.997969i \(-0.479707\pi\)
0.0637091 + 0.997969i \(0.479707\pi\)
\(212\) 3.93900 0.270532
\(213\) 0 0
\(214\) 13.1588i 0.899519i
\(215\) − 6.67994i − 0.455568i
\(216\) 0 0
\(217\) 36.6437 2.48754
\(218\) 1.33513 0.0904261
\(219\) 0 0
\(220\) −1.82908 −0.123317
\(221\) 0 0
\(222\) 0 0
\(223\) 18.6504i 1.24892i 0.781056 + 0.624462i \(0.214682\pi\)
−0.781056 + 0.624462i \(0.785318\pi\)
\(224\) 11.8455 0.791459
\(225\) 0 0
\(226\) − 20.6136i − 1.37119i
\(227\) − 9.75733i − 0.647617i −0.946123 0.323808i \(-0.895037\pi\)
0.946123 0.323808i \(-0.104963\pi\)
\(228\) 0 0
\(229\) 2.86294i 0.189188i 0.995516 + 0.0945941i \(0.0301553\pi\)
−0.995516 + 0.0945941i \(0.969845\pi\)
\(230\) −12.2838 −0.809971
\(231\) 0 0
\(232\) 5.64310i 0.370488i
\(233\) −5.78554 −0.379024 −0.189512 0.981878i \(-0.560691\pi\)
−0.189512 + 0.981878i \(0.560691\pi\)
\(234\) 0 0
\(235\) −35.9124 −2.34267
\(236\) 0.968541i 0.0630467i
\(237\) 0 0
\(238\) −14.6407 −0.949016
\(239\) − 7.09246i − 0.458773i −0.973335 0.229386i \(-0.926328\pi\)
0.973335 0.229386i \(-0.0736720\pi\)
\(240\) 0 0
\(241\) − 3.89977i − 0.251206i −0.992081 0.125603i \(-0.959913\pi\)
0.992081 0.125603i \(-0.0400866\pi\)
\(242\) − 11.0339i − 0.709283i
\(243\) 0 0
\(244\) −3.48427 −0.223058
\(245\) − 44.9952i − 2.87464i
\(246\) 0 0
\(247\) 0 0
\(248\) −23.2664 −1.47742
\(249\) 0 0
\(250\) 7.50902 0.474912
\(251\) −2.44504 −0.154330 −0.0771648 0.997018i \(-0.524587\pi\)
−0.0771648 + 0.997018i \(0.524587\pi\)
\(252\) 0 0
\(253\) − 5.15691i − 0.324212i
\(254\) − 11.8931i − 0.746237i
\(255\) 0 0
\(256\) −10.1129 −0.632056
\(257\) −14.1304 −0.881428 −0.440714 0.897648i \(-0.645275\pi\)
−0.440714 + 0.897648i \(0.645275\pi\)
\(258\) 0 0
\(259\) −21.8726 −1.35910
\(260\) 0 0
\(261\) 0 0
\(262\) − 6.86964i − 0.424408i
\(263\) 23.7235 1.46285 0.731426 0.681921i \(-0.238855\pi\)
0.731426 + 0.681921i \(0.238855\pi\)
\(264\) 0 0
\(265\) 24.7995i 1.52342i
\(266\) 15.2265i 0.933599i
\(267\) 0 0
\(268\) − 1.59419i − 0.0973805i
\(269\) 5.91617 0.360715 0.180357 0.983601i \(-0.442275\pi\)
0.180357 + 0.983601i \(0.442275\pi\)
\(270\) 0 0
\(271\) − 3.19029i − 0.193796i −0.995294 0.0968982i \(-0.969108\pi\)
0.995294 0.0968982i \(-0.0308921\pi\)
\(272\) 7.11960 0.431689
\(273\) 0 0
\(274\) −20.1806 −1.21915
\(275\) − 4.18167i − 0.252164i
\(276\) 0 0
\(277\) −21.9366 −1.31804 −0.659022 0.752124i \(-0.729029\pi\)
−0.659022 + 0.752124i \(0.729029\pi\)
\(278\) 13.1045i 0.785958i
\(279\) 0 0
\(280\) 41.0224i 2.45155i
\(281\) − 11.9903i − 0.715282i −0.933859 0.357641i \(-0.883581\pi\)
0.933859 0.357641i \(-0.116419\pi\)
\(282\) 0 0
\(283\) 14.3666 0.854005 0.427002 0.904250i \(-0.359570\pi\)
0.427002 + 0.904250i \(0.359570\pi\)
\(284\) 3.93362i 0.233418i
\(285\) 0 0
\(286\) 0 0
\(287\) 5.98792 0.353456
\(288\) 0 0
\(289\) −11.0218 −0.648339
\(290\) −6.46681 −0.379744
\(291\) 0 0
\(292\) 3.42327i 0.200332i
\(293\) − 18.7584i − 1.09588i −0.836519 0.547939i \(-0.815413\pi\)
0.836519 0.547939i \(-0.184587\pi\)
\(294\) 0 0
\(295\) −6.09783 −0.355030
\(296\) 13.8877 0.807206
\(297\) 0 0
\(298\) −17.8944 −1.03659
\(299\) 0 0
\(300\) 0 0
\(301\) − 11.4480i − 0.659853i
\(302\) −2.45175 −0.141082
\(303\) 0 0
\(304\) − 7.40449i − 0.424676i
\(305\) − 21.9366i − 1.25609i
\(306\) 0 0
\(307\) 25.6262i 1.46257i 0.682074 + 0.731283i \(0.261078\pi\)
−0.682074 + 0.731283i \(0.738922\pi\)
\(308\) −3.13467 −0.178614
\(309\) 0 0
\(310\) − 26.6625i − 1.51433i
\(311\) 11.3817 0.645394 0.322697 0.946502i \(-0.395410\pi\)
0.322697 + 0.946502i \(0.395410\pi\)
\(312\) 0 0
\(313\) 27.5743 1.55859 0.779297 0.626655i \(-0.215576\pi\)
0.779297 + 0.626655i \(0.215576\pi\)
\(314\) − 13.3448i − 0.753091i
\(315\) 0 0
\(316\) −1.78986 −0.100687
\(317\) − 11.2597i − 0.632405i −0.948692 0.316203i \(-0.897592\pi\)
0.948692 0.316203i \(-0.102408\pi\)
\(318\) 0 0
\(319\) − 2.71486i − 0.152003i
\(320\) − 24.9366i − 1.39400i
\(321\) 0 0
\(322\) −21.0519 −1.17318
\(323\) − 6.21744i − 0.345948i
\(324\) 0 0
\(325\) 0 0
\(326\) −4.86294 −0.269333
\(327\) 0 0
\(328\) −3.80194 −0.209927
\(329\) −61.5465 −3.39317
\(330\) 0 0
\(331\) 11.9065i 0.654439i 0.944948 + 0.327220i \(0.106112\pi\)
−0.944948 + 0.327220i \(0.893888\pi\)
\(332\) 0.290520i 0.0159444i
\(333\) 0 0
\(334\) 26.2107 1.43419
\(335\) 10.0368 0.548371
\(336\) 0 0
\(337\) −17.1672 −0.935157 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 3.04892i − 0.165351i
\(341\) 11.1933 0.606150
\(342\) 0 0
\(343\) − 43.4989i − 2.34872i
\(344\) 7.26875i 0.391905i
\(345\) 0 0
\(346\) − 16.5036i − 0.887242i
\(347\) 24.2760 1.30321 0.651603 0.758560i \(-0.274097\pi\)
0.651603 + 0.758560i \(0.274097\pi\)
\(348\) 0 0
\(349\) − 4.57242i − 0.244756i −0.992484 0.122378i \(-0.960948\pi\)
0.992484 0.122378i \(-0.0390520\pi\)
\(350\) −17.0707 −0.912467
\(351\) 0 0
\(352\) 3.61835 0.192859
\(353\) − 6.07606i − 0.323396i −0.986840 0.161698i \(-0.948303\pi\)
0.986840 0.161698i \(-0.0516971\pi\)
\(354\) 0 0
\(355\) −24.7657 −1.31443
\(356\) 2.80194i 0.148502i
\(357\) 0 0
\(358\) − 10.6340i − 0.562025i
\(359\) 14.9661i 0.789883i 0.918706 + 0.394942i \(0.129235\pi\)
−0.918706 + 0.394942i \(0.870765\pi\)
\(360\) 0 0
\(361\) 12.5338 0.659673
\(362\) − 4.53452i − 0.238329i
\(363\) 0 0
\(364\) 0 0
\(365\) −21.5526 −1.12811
\(366\) 0 0
\(367\) 37.0834 1.93574 0.967868 0.251459i \(-0.0809105\pi\)
0.967868 + 0.251459i \(0.0809105\pi\)
\(368\) 10.2373 0.533656
\(369\) 0 0
\(370\) 15.9148i 0.827373i
\(371\) 42.5013i 2.20656i
\(372\) 0 0
\(373\) −36.5090 −1.89037 −0.945183 0.326542i \(-0.894117\pi\)
−0.945183 + 0.326542i \(0.894117\pi\)
\(374\) −4.47219 −0.231251
\(375\) 0 0
\(376\) 39.0780 2.01529
\(377\) 0 0
\(378\) 0 0
\(379\) 26.5851i 1.36558i 0.730613 + 0.682792i \(0.239235\pi\)
−0.730613 + 0.682792i \(0.760765\pi\)
\(380\) −3.17092 −0.162665
\(381\) 0 0
\(382\) 26.6625i 1.36417i
\(383\) − 14.3502i − 0.733261i −0.930367 0.366630i \(-0.880511\pi\)
0.930367 0.366630i \(-0.119489\pi\)
\(384\) 0 0
\(385\) − 19.7356i − 1.00582i
\(386\) 10.5090 0.534895
\(387\) 0 0
\(388\) − 4.46442i − 0.226647i
\(389\) −22.6582 −1.14881 −0.574407 0.818570i \(-0.694767\pi\)
−0.574407 + 0.818570i \(0.694767\pi\)
\(390\) 0 0
\(391\) 8.59611 0.434724
\(392\) 48.9614i 2.47292i
\(393\) 0 0
\(394\) 33.0157 1.66330
\(395\) − 11.2687i − 0.566992i
\(396\) 0 0
\(397\) 7.90754i 0.396868i 0.980114 + 0.198434i \(0.0635856\pi\)
−0.980114 + 0.198434i \(0.936414\pi\)
\(398\) − 17.7724i − 0.890850i
\(399\) 0 0
\(400\) 8.30127 0.415064
\(401\) − 2.93661i − 0.146647i −0.997308 0.0733236i \(-0.976639\pi\)
0.997308 0.0733236i \(-0.0233606\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.18060 0.307497
\(405\) 0 0
\(406\) −11.0828 −0.550029
\(407\) −6.68127 −0.331178
\(408\) 0 0
\(409\) − 11.7549i − 0.581244i −0.956838 0.290622i \(-0.906138\pi\)
0.956838 0.290622i \(-0.0938623\pi\)
\(410\) − 4.35690i − 0.215172i
\(411\) 0 0
\(412\) 7.74525 0.381581
\(413\) −10.4504 −0.514231
\(414\) 0 0
\(415\) −1.82908 −0.0897862
\(416\) 0 0
\(417\) 0 0
\(418\) 4.65114i 0.227495i
\(419\) −7.34183 −0.358672 −0.179336 0.983788i \(-0.557395\pi\)
−0.179336 + 0.983788i \(0.557395\pi\)
\(420\) 0 0
\(421\) − 25.6963i − 1.25236i −0.779677 0.626181i \(-0.784617\pi\)
0.779677 0.626181i \(-0.215383\pi\)
\(422\) − 2.30798i − 0.112351i
\(423\) 0 0
\(424\) − 26.9855i − 1.31053i
\(425\) 6.97046 0.338117
\(426\) 0 0
\(427\) − 37.5948i − 1.81934i
\(428\) −4.69633 −0.227006
\(429\) 0 0
\(430\) −8.32975 −0.401696
\(431\) 8.94198i 0.430720i 0.976535 + 0.215360i \(0.0690925\pi\)
−0.976535 + 0.215360i \(0.930907\pi\)
\(432\) 0 0
\(433\) −2.91484 −0.140078 −0.0700391 0.997544i \(-0.522312\pi\)
−0.0700391 + 0.997544i \(0.522312\pi\)
\(434\) − 45.6939i − 2.19338i
\(435\) 0 0
\(436\) 0.476501i 0.0228203i
\(437\) − 8.94007i − 0.427661i
\(438\) 0 0
\(439\) −9.05861 −0.432344 −0.216172 0.976355i \(-0.569357\pi\)
−0.216172 + 0.976355i \(0.569357\pi\)
\(440\) 12.5308i 0.597382i
\(441\) 0 0
\(442\) 0 0
\(443\) −11.2325 −0.533672 −0.266836 0.963742i \(-0.585978\pi\)
−0.266836 + 0.963742i \(0.585978\pi\)
\(444\) 0 0
\(445\) −17.6407 −0.836250
\(446\) 23.2567 1.10124
\(447\) 0 0
\(448\) − 42.7362i − 2.01909i
\(449\) − 28.7579i − 1.35717i −0.734522 0.678585i \(-0.762593\pi\)
0.734522 0.678585i \(-0.237407\pi\)
\(450\) 0 0
\(451\) 1.82908 0.0861282
\(452\) 7.35690 0.346039
\(453\) 0 0
\(454\) −12.1672 −0.571035
\(455\) 0 0
\(456\) 0 0
\(457\) 19.0761i 0.892341i 0.894948 + 0.446170i \(0.147212\pi\)
−0.894948 + 0.446170i \(0.852788\pi\)
\(458\) 3.57002 0.166816
\(459\) 0 0
\(460\) − 4.38404i − 0.204407i
\(461\) 31.7332i 1.47796i 0.673727 + 0.738981i \(0.264692\pi\)
−0.673727 + 0.738981i \(0.735308\pi\)
\(462\) 0 0
\(463\) − 36.4784i − 1.69530i −0.530559 0.847648i \(-0.678018\pi\)
0.530559 0.847648i \(-0.321982\pi\)
\(464\) 5.38942 0.250198
\(465\) 0 0
\(466\) 7.21446i 0.334203i
\(467\) 13.0000 0.601568 0.300784 0.953692i \(-0.402752\pi\)
0.300784 + 0.953692i \(0.402752\pi\)
\(468\) 0 0
\(469\) 17.2010 0.794271
\(470\) 44.7821i 2.06564i
\(471\) 0 0
\(472\) 6.63533 0.305416
\(473\) − 3.49694i − 0.160790i
\(474\) 0 0
\(475\) − 7.24937i − 0.332624i
\(476\) − 5.22521i − 0.239497i
\(477\) 0 0
\(478\) −8.84415 −0.404522
\(479\) − 5.61655i − 0.256627i −0.991734 0.128313i \(-0.959044\pi\)
0.991734 0.128313i \(-0.0409563\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −4.86294 −0.221501
\(483\) 0 0
\(484\) 3.93794 0.178997
\(485\) 28.1075 1.27630
\(486\) 0 0
\(487\) 9.75733i 0.442147i 0.975257 + 0.221073i \(0.0709561\pi\)
−0.975257 + 0.221073i \(0.929044\pi\)
\(488\) 23.8702i 1.08055i
\(489\) 0 0
\(490\) −56.1081 −2.53471
\(491\) −7.38835 −0.333432 −0.166716 0.986005i \(-0.553316\pi\)
−0.166716 + 0.986005i \(0.553316\pi\)
\(492\) 0 0
\(493\) 4.52542 0.203815
\(494\) 0 0
\(495\) 0 0
\(496\) 22.2204i 0.997726i
\(497\) −42.4432 −1.90384
\(498\) 0 0
\(499\) 43.2814i 1.93754i 0.247956 + 0.968771i \(0.420241\pi\)
−0.247956 + 0.968771i \(0.579759\pi\)
\(500\) 2.67994i 0.119851i
\(501\) 0 0
\(502\) 3.04892i 0.136080i
\(503\) −10.7670 −0.480078 −0.240039 0.970763i \(-0.577160\pi\)
−0.240039 + 0.970763i \(0.577160\pi\)
\(504\) 0 0
\(505\) 38.9124i 1.73158i
\(506\) −6.43057 −0.285874
\(507\) 0 0
\(508\) 4.24459 0.188323
\(509\) − 41.5448i − 1.84144i −0.390223 0.920720i \(-0.627602\pi\)
0.390223 0.920720i \(-0.372398\pi\)
\(510\) 0 0
\(511\) −36.9366 −1.63398
\(512\) 24.9390i 1.10216i
\(513\) 0 0
\(514\) 17.6203i 0.777197i
\(515\) 48.7633i 2.14877i
\(516\) 0 0
\(517\) −18.8001 −0.826829
\(518\) 27.2747i 1.19838i
\(519\) 0 0
\(520\) 0 0
\(521\) −25.7198 −1.12680 −0.563402 0.826183i \(-0.690508\pi\)
−0.563402 + 0.826183i \(0.690508\pi\)
\(522\) 0 0
\(523\) 8.59286 0.375739 0.187870 0.982194i \(-0.439842\pi\)
0.187870 + 0.982194i \(0.439842\pi\)
\(524\) 2.45175 0.107105
\(525\) 0 0
\(526\) − 29.5827i − 1.28987i
\(527\) 18.6582i 0.812763i
\(528\) 0 0
\(529\) −10.6396 −0.462593
\(530\) 30.9245 1.34328
\(531\) 0 0
\(532\) −5.43429 −0.235606
\(533\) 0 0
\(534\) 0 0
\(535\) − 29.5676i − 1.27832i
\(536\) −10.9215 −0.471739
\(537\) 0 0
\(538\) − 7.37734i − 0.318060i
\(539\) − 23.5550i − 1.01458i
\(540\) 0 0
\(541\) 31.3534i 1.34799i 0.738736 + 0.673995i \(0.235423\pi\)
−0.738736 + 0.673995i \(0.764577\pi\)
\(542\) −3.97823 −0.170880
\(543\) 0 0
\(544\) 6.03146i 0.258597i
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 19.9342 0.852325 0.426163 0.904647i \(-0.359865\pi\)
0.426163 + 0.904647i \(0.359865\pi\)
\(548\) − 7.20237i − 0.307670i
\(549\) 0 0
\(550\) −5.21446 −0.222345
\(551\) − 4.70650i − 0.200503i
\(552\) 0 0
\(553\) − 19.3123i − 0.821242i
\(554\) 27.3545i 1.16218i
\(555\) 0 0
\(556\) −4.67696 −0.198347
\(557\) − 33.2349i − 1.40821i −0.710097 0.704104i \(-0.751349\pi\)
0.710097 0.704104i \(-0.248651\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 39.1782 1.65558
\(561\) 0 0
\(562\) −14.9517 −0.630698
\(563\) 3.87130 0.163156 0.0815779 0.996667i \(-0.474004\pi\)
0.0815779 + 0.996667i \(0.474004\pi\)
\(564\) 0 0
\(565\) 46.3183i 1.94862i
\(566\) − 17.9148i − 0.753017i
\(567\) 0 0
\(568\) 26.9487 1.13074
\(569\) 20.1457 0.844551 0.422276 0.906468i \(-0.361231\pi\)
0.422276 + 0.906468i \(0.361231\pi\)
\(570\) 0 0
\(571\) 32.1269 1.34447 0.672234 0.740338i \(-0.265335\pi\)
0.672234 + 0.740338i \(0.265335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 7.46681i − 0.311659i
\(575\) 10.0228 0.417981
\(576\) 0 0
\(577\) 16.7506i 0.697338i 0.937246 + 0.348669i \(0.113366\pi\)
−0.937246 + 0.348669i \(0.886634\pi\)
\(578\) 13.7439i 0.571672i
\(579\) 0 0
\(580\) − 2.30798i − 0.0958336i
\(581\) −3.13467 −0.130048
\(582\) 0 0
\(583\) 12.9825i 0.537682i
\(584\) 23.4523 0.970465
\(585\) 0 0
\(586\) −23.3913 −0.966287
\(587\) − 6.73795i − 0.278105i −0.990285 0.139053i \(-0.955594\pi\)
0.990285 0.139053i \(-0.0444057\pi\)
\(588\) 0 0
\(589\) 19.4047 0.799559
\(590\) 7.60388i 0.313047i
\(591\) 0 0
\(592\) − 13.2634i − 0.545121i
\(593\) − 18.1172i − 0.743985i −0.928236 0.371992i \(-0.878675\pi\)
0.928236 0.371992i \(-0.121325\pi\)
\(594\) 0 0
\(595\) 32.8974 1.34866
\(596\) − 6.38644i − 0.261599i
\(597\) 0 0
\(598\) 0 0
\(599\) 26.7851 1.09441 0.547204 0.836999i \(-0.315692\pi\)
0.547204 + 0.836999i \(0.315692\pi\)
\(600\) 0 0
\(601\) −4.70171 −0.191787 −0.0958934 0.995392i \(-0.530571\pi\)
−0.0958934 + 0.995392i \(0.530571\pi\)
\(602\) −14.2755 −0.581824
\(603\) 0 0
\(604\) − 0.875018i − 0.0356040i
\(605\) 24.7928i 1.00797i
\(606\) 0 0
\(607\) 27.6396 1.12186 0.560929 0.827864i \(-0.310444\pi\)
0.560929 + 0.827864i \(0.310444\pi\)
\(608\) 6.27280 0.254396
\(609\) 0 0
\(610\) −27.3545 −1.10755
\(611\) 0 0
\(612\) 0 0
\(613\) − 48.1782i − 1.94590i −0.231017 0.972950i \(-0.574205\pi\)
0.231017 0.972950i \(-0.425795\pi\)
\(614\) 31.9554 1.28961
\(615\) 0 0
\(616\) 21.4752i 0.865259i
\(617\) 30.3043i 1.22000i 0.792400 + 0.610002i \(0.208831\pi\)
−0.792400 + 0.610002i \(0.791169\pi\)
\(618\) 0 0
\(619\) − 10.9041i − 0.438272i −0.975694 0.219136i \(-0.929676\pi\)
0.975694 0.219136i \(-0.0703239\pi\)
\(620\) 9.51573 0.382161
\(621\) 0 0
\(622\) − 14.1927i − 0.569075i
\(623\) −30.2325 −1.21124
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) − 34.3846i − 1.37429i
\(627\) 0 0
\(628\) 4.76271 0.190053
\(629\) − 11.1371i − 0.444064i
\(630\) 0 0
\(631\) − 10.4523i − 0.416101i −0.978118 0.208050i \(-0.933288\pi\)
0.978118 0.208050i \(-0.0667118\pi\)
\(632\) 12.2620i 0.487758i
\(633\) 0 0
\(634\) −14.0406 −0.557622
\(635\) 26.7235i 1.06049i
\(636\) 0 0
\(637\) 0 0
\(638\) −3.38537 −0.134028
\(639\) 0 0
\(640\) −17.2717 −0.682725
\(641\) −17.5942 −0.694929 −0.347464 0.937693i \(-0.612957\pi\)
−0.347464 + 0.937693i \(0.612957\pi\)
\(642\) 0 0
\(643\) − 22.6058i − 0.891486i −0.895161 0.445743i \(-0.852940\pi\)
0.895161 0.445743i \(-0.147060\pi\)
\(644\) − 7.51334i − 0.296067i
\(645\) 0 0
\(646\) −7.75302 −0.305039
\(647\) −24.7918 −0.974665 −0.487333 0.873216i \(-0.662030\pi\)
−0.487333 + 0.873216i \(0.662030\pi\)
\(648\) 0 0
\(649\) −3.19221 −0.125305
\(650\) 0 0
\(651\) 0 0
\(652\) − 1.73556i − 0.0679699i
\(653\) 21.8106 0.853513 0.426757 0.904367i \(-0.359656\pi\)
0.426757 + 0.904367i \(0.359656\pi\)
\(654\) 0 0
\(655\) 15.4359i 0.603132i
\(656\) 3.63102i 0.141768i
\(657\) 0 0
\(658\) 76.7472i 2.99192i
\(659\) −16.5526 −0.644796 −0.322398 0.946604i \(-0.604489\pi\)
−0.322398 + 0.946604i \(0.604489\pi\)
\(660\) 0 0
\(661\) − 15.9541i − 0.620541i −0.950648 0.310271i \(-0.899580\pi\)
0.950648 0.310271i \(-0.100420\pi\)
\(662\) 14.8471 0.577050
\(663\) 0 0
\(664\) 1.99031 0.0772391
\(665\) − 34.2137i − 1.32675i
\(666\) 0 0
\(667\) 6.50711 0.251956
\(668\) 9.35450i 0.361937i
\(669\) 0 0
\(670\) − 12.5157i − 0.483525i
\(671\) − 11.4838i − 0.443327i
\(672\) 0 0
\(673\) 7.38835 0.284800 0.142400 0.989809i \(-0.454518\pi\)
0.142400 + 0.989809i \(0.454518\pi\)
\(674\) 21.4071i 0.824572i
\(675\) 0 0
\(676\) 0 0
\(677\) 22.1454 0.851118 0.425559 0.904931i \(-0.360078\pi\)
0.425559 + 0.904931i \(0.360078\pi\)
\(678\) 0 0
\(679\) 48.1704 1.84861
\(680\) −20.8877 −0.801006
\(681\) 0 0
\(682\) − 13.9578i − 0.534471i
\(683\) − 9.10023i − 0.348211i −0.984727 0.174105i \(-0.944297\pi\)
0.984727 0.174105i \(-0.0557033\pi\)
\(684\) 0 0
\(685\) 45.3454 1.73256
\(686\) −54.2422 −2.07098
\(687\) 0 0
\(688\) 6.94198 0.264661
\(689\) 0 0
\(690\) 0 0
\(691\) − 13.7711i − 0.523876i −0.965085 0.261938i \(-0.915638\pi\)
0.965085 0.261938i \(-0.0843616\pi\)
\(692\) 5.89008 0.223907
\(693\) 0 0
\(694\) − 30.2717i − 1.14910i
\(695\) − 29.4456i − 1.11694i
\(696\) 0 0
\(697\) 3.04892i 0.115486i
\(698\) −5.70171 −0.215813
\(699\) 0 0
\(700\) − 6.09246i − 0.230273i
\(701\) 46.5090 1.75662 0.878311 0.478090i \(-0.158671\pi\)
0.878311 + 0.478090i \(0.158671\pi\)
\(702\) 0 0
\(703\) −11.5827 −0.436850
\(704\) − 13.0543i − 0.492002i
\(705\) 0 0
\(706\) −7.57673 −0.285154
\(707\) 66.6878i 2.50805i
\(708\) 0 0
\(709\) 7.24565i 0.272116i 0.990701 + 0.136058i \(0.0434434\pi\)
−0.990701 + 0.136058i \(0.956557\pi\)
\(710\) 30.8823i 1.15899i
\(711\) 0 0
\(712\) 19.1957 0.719388
\(713\) 26.8286i 1.00474i
\(714\) 0 0
\(715\) 0 0
\(716\) 3.79523 0.141835
\(717\) 0 0
\(718\) 18.6625 0.696478
\(719\) 25.5147 0.951536 0.475768 0.879571i \(-0.342170\pi\)
0.475768 + 0.879571i \(0.342170\pi\)
\(720\) 0 0
\(721\) 83.5701i 3.11231i
\(722\) − 15.6294i − 0.581665i
\(723\) 0 0
\(724\) 1.61835 0.0601455
\(725\) 5.27652 0.195965
\(726\) 0 0
\(727\) 14.4873 0.537303 0.268651 0.963238i \(-0.413422\pi\)
0.268651 + 0.963238i \(0.413422\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 26.8756i 0.994711i
\(731\) 5.82908 0.215596
\(732\) 0 0
\(733\) − 37.5036i − 1.38523i −0.721308 0.692614i \(-0.756459\pi\)
0.721308 0.692614i \(-0.243541\pi\)
\(734\) − 46.2422i − 1.70683i
\(735\) 0 0
\(736\) 8.67264i 0.319678i
\(737\) 5.25428 0.193544
\(738\) 0 0
\(739\) 43.1876i 1.58868i 0.607472 + 0.794341i \(0.292184\pi\)
−0.607472 + 0.794341i \(0.707816\pi\)
\(740\) −5.67994 −0.208799
\(741\) 0 0
\(742\) 52.9982 1.94563
\(743\) 13.4765i 0.494405i 0.968964 + 0.247202i \(0.0795113\pi\)
−0.968964 + 0.247202i \(0.920489\pi\)
\(744\) 0 0
\(745\) 40.2083 1.47312
\(746\) 45.5260i 1.66683i
\(747\) 0 0
\(748\) − 1.59611i − 0.0583594i
\(749\) − 50.6728i − 1.85154i
\(750\) 0 0
\(751\) −35.5894 −1.29868 −0.649338 0.760500i \(-0.724954\pi\)
−0.649338 + 0.760500i \(0.724954\pi\)
\(752\) − 37.3212i − 1.36097i
\(753\) 0 0
\(754\) 0 0
\(755\) 5.50902 0.200494
\(756\) 0 0
\(757\) −12.2107 −0.443807 −0.221903 0.975069i \(-0.571227\pi\)
−0.221903 + 0.975069i \(0.571227\pi\)
\(758\) 33.1511 1.20410
\(759\) 0 0
\(760\) 21.7235i 0.787993i
\(761\) − 30.9071i − 1.12038i −0.828364 0.560190i \(-0.810728\pi\)
0.828364 0.560190i \(-0.189272\pi\)
\(762\) 0 0
\(763\) −5.14138 −0.186130
\(764\) −9.51573 −0.344267
\(765\) 0 0
\(766\) −17.8944 −0.646551
\(767\) 0 0
\(768\) 0 0
\(769\) − 11.9892i − 0.432343i −0.976355 0.216172i \(-0.930643\pi\)
0.976355 0.216172i \(-0.0693571\pi\)
\(770\) −24.6098 −0.886877
\(771\) 0 0
\(772\) 3.75063i 0.134988i
\(773\) 43.9661i 1.58135i 0.612235 + 0.790676i \(0.290271\pi\)
−0.612235 + 0.790676i \(0.709729\pi\)
\(774\) 0 0
\(775\) 21.7549i 0.781460i
\(776\) −30.5851 −1.09794
\(777\) 0 0
\(778\) 28.2543i 1.01296i
\(779\) 3.17092 0.113610
\(780\) 0 0
\(781\) −12.9648 −0.463918
\(782\) − 10.7192i − 0.383317i
\(783\) 0 0
\(784\) 46.7603 1.67001
\(785\) 29.9855i 1.07023i
\(786\) 0 0
\(787\) 30.3763i 1.08280i 0.840766 + 0.541399i \(0.182105\pi\)
−0.840766 + 0.541399i \(0.817895\pi\)
\(788\) 11.7832i 0.419757i
\(789\) 0 0
\(790\) −14.0519 −0.499944
\(791\) 79.3798i 2.82242i
\(792\) 0 0
\(793\) 0 0
\(794\) 9.86054 0.349938
\(795\) 0 0
\(796\) 6.34290 0.224818
\(797\) 52.5763 1.86235 0.931173 0.364577i \(-0.118786\pi\)
0.931173 + 0.364577i \(0.118786\pi\)
\(798\) 0 0
\(799\) − 31.3381i − 1.10866i
\(800\) 7.03252i 0.248637i
\(801\) 0 0
\(802\) −3.66189 −0.129306
\(803\) −11.2828 −0.398160
\(804\) 0 0
\(805\) 47.3032 1.66722
\(806\) 0 0
\(807\) 0 0
\(808\) − 42.3424i − 1.48960i
\(809\) −49.4215 −1.73757 −0.868783 0.495193i \(-0.835097\pi\)
−0.868783 + 0.495193i \(0.835097\pi\)
\(810\) 0 0
\(811\) 1.36526i 0.0479406i 0.999713 + 0.0239703i \(0.00763072\pi\)
−0.999713 + 0.0239703i \(0.992369\pi\)
\(812\) − 3.95539i − 0.138807i
\(813\) 0 0
\(814\) 8.33140i 0.292016i
\(815\) 10.9269 0.382753
\(816\) 0 0
\(817\) − 6.06233i − 0.212094i
\(818\) −14.6582 −0.512511
\(819\) 0 0
\(820\) 1.55496 0.0543015
\(821\) 0.665939i 0.0232414i 0.999932 + 0.0116207i \(0.00369907\pi\)
−0.999932 + 0.0116207i \(0.996301\pi\)
\(822\) 0 0
\(823\) 10.0592 0.350642 0.175321 0.984511i \(-0.443904\pi\)
0.175321 + 0.984511i \(0.443904\pi\)
\(824\) − 53.0616i − 1.84849i
\(825\) 0 0
\(826\) 13.0315i 0.453422i
\(827\) 37.3038i 1.29718i 0.761138 + 0.648590i \(0.224641\pi\)
−0.761138 + 0.648590i \(0.775359\pi\)
\(828\) 0 0
\(829\) 42.6209 1.48028 0.740142 0.672451i \(-0.234758\pi\)
0.740142 + 0.672451i \(0.234758\pi\)
\(830\) 2.28083i 0.0791688i
\(831\) 0 0
\(832\) 0 0
\(833\) 39.2640 1.36042
\(834\) 0 0
\(835\) −58.8950 −2.03815
\(836\) −1.65997 −0.0574113
\(837\) 0 0
\(838\) 9.15511i 0.316258i
\(839\) − 4.14005i − 0.142930i −0.997443 0.0714652i \(-0.977233\pi\)
0.997443 0.0714652i \(-0.0227675\pi\)
\(840\) 0 0
\(841\) −25.5743 −0.881874
\(842\) −32.0428 −1.10427
\(843\) 0 0
\(844\) 0.823708 0.0283532
\(845\) 0 0
\(846\) 0 0
\(847\) 42.4898i 1.45997i
\(848\) −25.7724 −0.885028
\(849\) 0 0
\(850\) − 8.69202i − 0.298134i
\(851\) − 16.0140i − 0.548953i
\(852\) 0 0
\(853\) 17.3502i 0.594059i 0.954868 + 0.297030i \(0.0959960\pi\)
−0.954868 + 0.297030i \(0.904004\pi\)
\(854\) −46.8799 −1.60420
\(855\) 0 0
\(856\) 32.1739i 1.09968i
\(857\) 41.0180 1.40115 0.700575 0.713579i \(-0.252927\pi\)
0.700575 + 0.713579i \(0.252927\pi\)
\(858\) 0 0
\(859\) 6.59286 0.224945 0.112473 0.993655i \(-0.464123\pi\)
0.112473 + 0.993655i \(0.464123\pi\)
\(860\) − 2.97285i − 0.101373i
\(861\) 0 0
\(862\) 11.1505 0.379787
\(863\) − 16.6455i − 0.566619i −0.959028 0.283310i \(-0.908568\pi\)
0.959028 0.283310i \(-0.0914324\pi\)
\(864\) 0 0
\(865\) 37.0834i 1.26087i
\(866\) 3.63474i 0.123514i
\(867\) 0 0
\(868\) 16.3080 0.553529
\(869\) − 5.89918i − 0.200116i
\(870\) 0 0
\(871\) 0 0
\(872\) 3.26444 0.110548
\(873\) 0 0
\(874\) −11.1481 −0.377089
\(875\) −28.9162 −0.977545
\(876\) 0 0
\(877\) − 54.4965i − 1.84022i −0.391666 0.920108i \(-0.628101\pi\)
0.391666 0.920108i \(-0.371899\pi\)
\(878\) 11.2959i 0.381218i
\(879\) 0 0
\(880\) 11.9675 0.403424
\(881\) −9.00670 −0.303444 −0.151722 0.988423i \(-0.548482\pi\)
−0.151722 + 0.988423i \(0.548482\pi\)
\(882\) 0 0
\(883\) −18.8907 −0.635722 −0.317861 0.948137i \(-0.602965\pi\)
−0.317861 + 0.948137i \(0.602965\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 14.0067i 0.470564i
\(887\) 46.9124 1.57517 0.787583 0.616209i \(-0.211332\pi\)
0.787583 + 0.616209i \(0.211332\pi\)
\(888\) 0 0
\(889\) 45.7985i 1.53603i
\(890\) 21.9976i 0.737361i
\(891\) 0 0
\(892\) 8.30021i 0.277912i
\(893\) −32.5921 −1.09065
\(894\) 0 0
\(895\) 23.8944i 0.798702i
\(896\) −29.6002 −0.988872
\(897\) 0 0
\(898\) −35.8605 −1.19668
\(899\) 14.1239i 0.471059i
\(900\) 0 0
\(901\) −21.6407 −0.720957
\(902\) − 2.28083i − 0.0759434i
\(903\) 0 0
\(904\) − 50.4010i − 1.67631i
\(905\) 10.1890i 0.338693i
\(906\) 0 0
\(907\) 35.3013 1.17216 0.586080 0.810253i \(-0.300671\pi\)
0.586080 + 0.810253i \(0.300671\pi\)
\(908\) − 4.34242i − 0.144108i
\(909\) 0 0
\(910\) 0 0
\(911\) 9.80731 0.324931 0.162465 0.986714i \(-0.448055\pi\)
0.162465 + 0.986714i \(0.448055\pi\)
\(912\) 0 0
\(913\) −0.957524 −0.0316894
\(914\) 23.7875 0.786819
\(915\) 0 0
\(916\) 1.27413i 0.0420983i
\(917\) 26.4540i 0.873588i
\(918\) 0 0
\(919\) 18.4655 0.609120 0.304560 0.952493i \(-0.401491\pi\)
0.304560 + 0.952493i \(0.401491\pi\)
\(920\) −30.0344 −0.990206
\(921\) 0 0
\(922\) 39.5706 1.30319
\(923\) 0 0
\(924\) 0 0
\(925\) − 12.9855i − 0.426961i
\(926\) −45.4878 −1.49482
\(927\) 0 0
\(928\) 4.56571i 0.149877i
\(929\) − 25.6267i − 0.840785i −0.907342 0.420393i \(-0.861892\pi\)
0.907342 0.420393i \(-0.138108\pi\)
\(930\) 0 0
\(931\) − 40.8351i − 1.33831i
\(932\) −2.57481 −0.0843407
\(933\) 0 0
\(934\) − 16.2107i − 0.530431i
\(935\) 10.0489 0.328635
\(936\) 0 0
\(937\) 7.54932 0.246625 0.123313 0.992368i \(-0.460648\pi\)
0.123313 + 0.992368i \(0.460648\pi\)
\(938\) − 21.4494i − 0.700346i
\(939\) 0 0
\(940\) −15.9825 −0.521293
\(941\) − 12.6418i − 0.412110i −0.978540 0.206055i \(-0.933937\pi\)
0.978540 0.206055i \(-0.0660626\pi\)
\(942\) 0 0
\(943\) 4.38404i 0.142764i
\(944\) − 6.33704i − 0.206253i
\(945\) 0 0
\(946\) −4.36062 −0.141776
\(947\) − 27.9801i − 0.909233i −0.890687 0.454616i \(-0.849776\pi\)
0.890687 0.454616i \(-0.150224\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −9.03982 −0.293290
\(951\) 0 0
\(952\) −35.7972 −1.16019
\(953\) 4.00239 0.129650 0.0648251 0.997897i \(-0.479351\pi\)
0.0648251 + 0.997897i \(0.479351\pi\)
\(954\) 0 0
\(955\) − 59.9101i − 1.93864i
\(956\) − 3.15644i − 0.102087i
\(957\) 0 0
\(958\) −7.00372 −0.226280
\(959\) 77.7126 2.50947
\(960\) 0 0
\(961\) −27.2325 −0.878468
\(962\) 0 0
\(963\) 0 0
\(964\) − 1.73556i − 0.0558987i
\(965\) −23.6136 −0.760148
\(966\) 0 0
\(967\) − 12.2239i − 0.393094i −0.980494 0.196547i \(-0.937027\pi\)
0.980494 0.196547i \(-0.0629728\pi\)
\(968\) − 26.9782i − 0.867113i
\(969\) 0 0
\(970\) − 35.0495i − 1.12537i
\(971\) 23.6401 0.758648 0.379324 0.925264i \(-0.376157\pi\)
0.379324 + 0.925264i \(0.376157\pi\)
\(972\) 0 0
\(973\) − 50.4637i − 1.61779i
\(974\) 12.1672 0.389862
\(975\) 0 0
\(976\) 22.7972 0.729719
\(977\) 18.7313i 0.599266i 0.954055 + 0.299633i \(0.0968642\pi\)
−0.954055 + 0.299633i \(0.903136\pi\)
\(978\) 0 0
\(979\) −9.23490 −0.295149
\(980\) − 20.0248i − 0.639667i
\(981\) 0 0
\(982\) 9.21313i 0.294003i
\(983\) − 12.0954i − 0.385785i −0.981220 0.192892i \(-0.938213\pi\)
0.981220 0.192892i \(-0.0617868\pi\)
\(984\) 0 0
\(985\) −74.1855 −2.36375
\(986\) − 5.64310i − 0.179713i
\(987\) 0 0
\(988\) 0 0
\(989\) 8.38165 0.266521
\(990\) 0 0
\(991\) −28.5526 −0.907002 −0.453501 0.891256i \(-0.649825\pi\)
−0.453501 + 0.891256i \(0.649825\pi\)
\(992\) −18.8243 −0.597672
\(993\) 0 0
\(994\) 52.9259i 1.67871i
\(995\) 39.9342i 1.26600i
\(996\) 0 0
\(997\) −23.9347 −0.758019 −0.379010 0.925393i \(-0.623735\pi\)
−0.379010 + 0.925393i \(0.623735\pi\)
\(998\) 53.9711 1.70842
\(999\) 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.b.m.1351.2 6
3.2 odd 2 507.2.b.g.337.5 6
13.5 odd 4 1521.2.a.q.1.1 3
13.8 odd 4 1521.2.a.p.1.3 3
13.12 even 2 inner 1521.2.b.m.1351.5 6
39.2 even 12 507.2.e.k.22.1 6
39.5 even 4 507.2.a.j.1.3 3
39.8 even 4 507.2.a.k.1.1 yes 3
39.11 even 12 507.2.e.j.22.3 6
39.17 odd 6 507.2.j.h.361.2 12
39.20 even 12 507.2.e.j.484.3 6
39.23 odd 6 507.2.j.h.316.5 12
39.29 odd 6 507.2.j.h.316.2 12
39.32 even 12 507.2.e.k.484.1 6
39.35 odd 6 507.2.j.h.361.5 12
39.38 odd 2 507.2.b.g.337.2 6
156.47 odd 4 8112.2.a.cf.1.3 3
156.83 odd 4 8112.2.a.by.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.3 3 39.5 even 4
507.2.a.k.1.1 yes 3 39.8 even 4
507.2.b.g.337.2 6 39.38 odd 2
507.2.b.g.337.5 6 3.2 odd 2
507.2.e.j.22.3 6 39.11 even 12
507.2.e.j.484.3 6 39.20 even 12
507.2.e.k.22.1 6 39.2 even 12
507.2.e.k.484.1 6 39.32 even 12
507.2.j.h.316.2 12 39.29 odd 6
507.2.j.h.316.5 12 39.23 odd 6
507.2.j.h.361.2 12 39.17 odd 6
507.2.j.h.361.5 12 39.35 odd 6
1521.2.a.p.1.3 3 13.8 odd 4
1521.2.a.q.1.1 3 13.5 odd 4
1521.2.b.m.1351.2 6 1.1 even 1 trivial
1521.2.b.m.1351.5 6 13.12 even 2 inner
8112.2.a.by.1.1 3 156.83 odd 4
8112.2.a.cf.1.3 3 156.47 odd 4