Properties

Label 507.2.b.g.337.5
Level $507$
Weight $2$
Character 507.337
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.5
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 507.337
Dual form 507.2.b.g.337.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.24698i q^{2} +1.00000 q^{3} +0.445042 q^{4} -2.80194i q^{5} +1.24698i q^{6} +4.80194i q^{7} +3.04892i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.24698i q^{2} +1.00000 q^{3} +0.445042 q^{4} -2.80194i q^{5} +1.24698i q^{6} +4.80194i q^{7} +3.04892i q^{8} +1.00000 q^{9} +3.49396 q^{10} -1.46681i q^{11} +0.445042 q^{12} -5.98792 q^{14} -2.80194i q^{15} -2.91185 q^{16} +2.44504 q^{17} +1.24698i q^{18} +2.54288i q^{19} -1.24698i q^{20} +4.80194i q^{21} +1.82908 q^{22} +3.51573 q^{23} +3.04892i q^{24} -2.85086 q^{25} +1.00000 q^{27} +2.13706i q^{28} +1.85086 q^{29} +3.49396 q^{30} -7.63102i q^{31} +2.46681i q^{32} -1.46681i q^{33} +3.04892i q^{34} +13.4547 q^{35} +0.445042 q^{36} +4.55496i q^{37} -3.17092 q^{38} +8.54288 q^{40} +1.24698i q^{41} -5.98792 q^{42} -2.38404 q^{43} -0.652793i q^{44} -2.80194i q^{45} +4.38404i q^{46} -12.8170i q^{47} -2.91185 q^{48} -16.0586 q^{49} -3.55496i q^{50} +2.44504 q^{51} -8.85086 q^{53} +1.24698i q^{54} -4.10992 q^{55} -14.6407 q^{56} +2.54288i q^{57} +2.30798i q^{58} -2.17629i q^{59} -1.24698i q^{60} -7.82908 q^{61} +9.51573 q^{62} +4.80194i q^{63} -8.89977 q^{64} +1.82908 q^{66} -3.58211i q^{67} +1.08815 q^{68} +3.51573 q^{69} +16.7778i q^{70} -8.83877i q^{71} +3.04892i q^{72} +7.69202i q^{73} -5.67994 q^{74} -2.85086 q^{75} +1.13169i q^{76} +7.04354 q^{77} -4.02177 q^{79} +8.15883i q^{80} +1.00000 q^{81} -1.55496 q^{82} -0.652793i q^{83} +2.13706i q^{84} -6.85086i q^{85} -2.97285i q^{86} +1.85086 q^{87} +4.47219 q^{88} -6.29590i q^{89} +3.49396 q^{90} +1.56465 q^{92} -7.63102i q^{93} +15.9825 q^{94} +7.12498 q^{95} +2.46681i q^{96} -10.0315i q^{97} -20.0248i q^{98} -1.46681i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{4} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{4} + 6q^{9} + 2q^{10} + 2q^{12} + 2q^{14} - 10q^{16} + 14q^{17} - 10q^{22} - 4q^{23} + 10q^{25} + 6q^{27} - 16q^{29} + 2q^{30} + 36q^{35} + 2q^{36} - 40q^{38} + 14q^{40} + 2q^{42} + 6q^{43} - 10q^{48} - 34q^{49} + 14q^{51} - 26q^{53} - 26q^{55} - 14q^{56} - 26q^{61} + 32q^{62} - 8q^{64} - 10q^{66} + 14q^{68} - 4q^{69} + 14q^{74} + 10q^{75} + 30q^{77} - 18q^{79} + 6q^{81} - 10q^{82} - 16q^{87} + 14q^{88} + 2q^{90} - 34q^{92} + 64q^{94} - 6q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 \(0.145331\pi\)
−0.897569 + 0.440874i \(0.854669\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.445042 0.222521
\(5\) − 2.80194i − 1.25306i −0.779395 0.626532i \(-0.784474\pi\)
0.779395 0.626532i \(-0.215526\pi\)
\(6\) 1.24698i 0.509077i
\(7\) 4.80194i 1.81496i 0.420093 + 0.907481i \(0.361997\pi\)
−0.420093 + 0.907481i \(0.638003\pi\)
\(8\) 3.04892i 1.07796i
\(9\) 1.00000 0.333333
\(10\) 3.49396 1.10489
\(11\) − 1.46681i − 0.442260i −0.975244 0.221130i \(-0.929025\pi\)
0.975244 0.221130i \(-0.0709746\pi\)
\(12\) 0.445042 0.128473
\(13\) 0 0
\(14\) −5.98792 −1.60034
\(15\) − 2.80194i − 0.723457i
\(16\) −2.91185 −0.727963
\(17\) 2.44504 0.593010 0.296505 0.955031i \(-0.404179\pi\)
0.296505 + 0.955031i \(0.404179\pi\)
\(18\) 1.24698i 0.293916i
\(19\) 2.54288i 0.583376i 0.956514 + 0.291688i \(0.0942169\pi\)
−0.956514 + 0.291688i \(0.905783\pi\)
\(20\) − 1.24698i − 0.278833i
\(21\) 4.80194i 1.04787i
\(22\) 1.82908 0.389962
\(23\) 3.51573 0.733080 0.366540 0.930402i \(-0.380542\pi\)
0.366540 + 0.930402i \(0.380542\pi\)
\(24\) 3.04892i 0.622358i
\(25\) −2.85086 −0.570171
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.13706i 0.403867i
\(29\) 1.85086 0.343695 0.171848 0.985124i \(-0.445026\pi\)
0.171848 + 0.985124i \(0.445026\pi\)
\(30\) 3.49396 0.637907
\(31\) − 7.63102i − 1.37057i −0.728274 0.685286i \(-0.759677\pi\)
0.728274 0.685286i \(-0.240323\pi\)
\(32\) 2.46681i 0.436075i
\(33\) − 1.46681i − 0.255339i
\(34\) 3.04892i 0.522885i
\(35\) 13.4547 2.27426
\(36\) 0.445042 0.0741736
\(37\) 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 \(0.0310439\pi\)
−0.995248 + 0.0973727i \(0.968956\pi\)
\(42\) −5.98792 −0.923956
\(43\) −2.38404 −0.363563 −0.181782 0.983339i \(-0.558186\pi\)
−0.181782 + 0.983339i \(0.558186\pi\)
\(44\) − 0.652793i − 0.0984122i
\(45\) − 2.80194i − 0.417688i
\(46\) 4.38404i 0.646392i
\(47\) − 12.8170i − 1.86955i −0.355238 0.934776i \(-0.615600\pi\)
0.355238 0.934776i \(-0.384400\pi\)
\(48\) −2.91185 −0.420290
\(49\) −16.0586 −2.29409
\(50\) − 3.55496i − 0.502747i
\(51\) 2.44504 0.342374
\(52\) 0 0
\(53\) −8.85086 −1.21576 −0.607879 0.794030i \(-0.707980\pi\)
−0.607879 + 0.794030i \(0.707980\pi\)
\(54\) 1.24698i 0.169692i
\(55\) −4.10992 −0.554181
\(56\) −14.6407 −1.95645
\(57\) 2.54288i 0.336812i
\(58\) 2.30798i 0.303052i
\(59\) − 2.17629i − 0.283329i −0.989915 0.141665i \(-0.954755\pi\)
0.989915 0.141665i \(-0.0452454\pi\)
\(60\) − 1.24698i − 0.160984i
\(61\) −7.82908 −1.00241 −0.501206 0.865328i \(-0.667110\pi\)
−0.501206 + 0.865328i \(0.667110\pi\)
\(62\) 9.51573 1.20850
\(63\) 4.80194i 0.604987i
\(64\) −8.89977 −1.11247
\(65\) 0 0
\(66\) 1.82908 0.225145
\(67\) − 3.58211i − 0.437624i −0.975767 0.218812i \(-0.929782\pi\)
0.975767 0.218812i \(-0.0702181\pi\)
\(68\) 1.08815 0.131957
\(69\) 3.51573 0.423244
\(70\) 16.7778i 2.00533i
\(71\) − 8.83877i − 1.04897i −0.851420 0.524485i \(-0.824258\pi\)
0.851420 0.524485i \(-0.175742\pi\)
\(72\) 3.04892i 0.359318i
\(73\) 7.69202i 0.900283i 0.892957 + 0.450142i \(0.148626\pi\)
−0.892957 + 0.450142i \(0.851374\pi\)
\(74\) −5.67994 −0.660280
\(75\) −2.85086 −0.329188
\(76\) 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
\(81\) 1.00000 0.111111
\(82\) −1.55496 −0.171716
\(83\) − 0.652793i − 0.0716533i −0.999358 0.0358267i \(-0.988594\pi\)
0.999358 0.0358267i \(-0.0114064\pi\)
\(84\) 2.13706i 0.233173i
\(85\) − 6.85086i − 0.743080i
\(86\) − 2.97285i − 0.320571i
\(87\) 1.85086 0.198432
\(88\) 4.47219 0.476737
\(89\) − 6.29590i − 0.667364i −0.942686 0.333682i \(-0.891709\pi\)
0.942686 0.333682i \(-0.108291\pi\)
\(90\) 3.49396 0.368296
\(91\) 0 0
\(92\) 1.56465 0.163126
\(93\) − 7.63102i − 0.791300i
\(94\) 15.9825 1.64847
\(95\) 7.12498 0.731008
\(96\) 2.46681i 0.251768i
\(97\) − 10.0315i − 1.01854i −0.860607 0.509270i \(-0.829915\pi\)
0.860607 0.509270i \(-0.170085\pi\)
\(98\) − 20.0248i − 2.02281i
\(99\) − 1.46681i − 0.147420i
\(100\) −1.26875 −0.126875
\(101\) −13.8877 −1.38188 −0.690938 0.722914i \(-0.742802\pi\)
−0.690938 + 0.722914i \(0.742802\pi\)
\(102\) 3.04892i 0.301888i
\(103\) 17.4034 1.71481 0.857405 0.514642i \(-0.172075\pi\)
0.857405 + 0.514642i \(0.172075\pi\)
\(104\) 0 0
\(105\) 13.4547 1.31305
\(106\) − 11.0368i − 1.07199i
\(107\) 10.5526 1.02015 0.510077 0.860128i \(-0.329617\pi\)
0.510077 + 0.860128i \(0.329617\pi\)
\(108\) 0.445042 0.0428242
\(109\) 1.07069i 0.102553i 0.998684 + 0.0512766i \(0.0163290\pi\)
−0.998684 + 0.0512766i \(0.983671\pi\)
\(110\) − 5.12498i − 0.488648i
\(111\) 4.55496i 0.432337i
\(112\) − 13.9825i − 1.32123i
\(113\) −16.5308 −1.55509 −0.777543 0.628830i \(-0.783534\pi\)
−0.777543 + 0.628830i \(0.783534\pi\)
\(114\) −3.17092 −0.296983
\(115\) − 9.85086i − 0.918597i
\(116\) 0.823708 0.0764794
\(117\) 0 0
\(118\) 2.71379 0.249825
\(119\) 11.7409i 1.07629i
\(120\) 8.54288 0.779854
\(121\) 8.84846 0.804406
\(122\) − 9.76271i − 0.883874i
\(123\) 1.24698i 0.112436i
\(124\) − 3.39612i − 0.304981i
\(125\) − 6.02177i − 0.538604i
\(126\) −5.98792 −0.533446
\(127\) 9.53750 0.846316 0.423158 0.906056i \(-0.360921\pi\)
0.423158 + 0.906056i \(0.360921\pi\)
\(128\) − 6.16421i − 0.544844i
\(129\) −2.38404 −0.209903
\(130\) 0 0
\(131\) −5.50902 −0.481326 −0.240663 0.970609i \(-0.577365\pi\)
−0.240663 + 0.970609i \(0.577365\pi\)
\(132\) − 0.652793i − 0.0568183i
\(133\) −12.2107 −1.05880
\(134\) 4.46681 0.385874
\(135\) − 2.80194i − 0.241152i
\(136\) 7.45473i 0.639238i
\(137\) 16.1836i 1.38266i 0.722540 + 0.691329i \(0.242974\pi\)
−0.722540 + 0.691329i \(0.757026\pi\)
\(138\) 4.38404i 0.373195i
\(139\) −10.5090 −0.891364 −0.445682 0.895191i \(-0.647039\pi\)
−0.445682 + 0.895191i \(0.647039\pi\)
\(140\) 5.98792 0.506071
\(141\) − 12.8170i − 1.07939i
\(142\) 11.0218 0.924926
\(143\) 0 0
\(144\) −2.91185 −0.242654
\(145\) − 5.18598i − 0.430672i
\(146\) −9.59179 −0.793823
\(147\) −16.0586 −1.32449
\(148\) 2.02715i 0.166630i
\(149\) 14.3502i 1.17561i 0.809001 + 0.587807i \(0.200008\pi\)
−0.809001 + 0.587807i \(0.799992\pi\)
\(150\) − 3.55496i − 0.290261i
\(151\) − 1.96615i − 0.160003i −0.996795 0.0800014i \(-0.974508\pi\)
0.996795 0.0800014i \(-0.0254925\pi\)
\(152\) −7.75302 −0.628853
\(153\) 2.44504 0.197670
\(154\) 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
\(159\) −8.85086 −0.701918
\(160\) 6.91185 0.546430
\(161\) 16.8823i 1.33051i
\(162\) 1.24698i 0.0979720i
\(163\) − 3.89977i − 0.305454i −0.988268 0.152727i \(-0.951195\pi\)
0.988268 0.152727i \(-0.0488055\pi\)
\(164\) 0.554958i 0.0433349i
\(165\) −4.10992 −0.319957
\(166\) 0.814019 0.0631802
\(167\) − 21.0194i − 1.62653i −0.581895 0.813264i \(-0.697688\pi\)
0.581895 0.813264i \(-0.302312\pi\)
\(168\) −14.6407 −1.12956
\(169\) 0 0
\(170\) 8.54288 0.655209
\(171\) 2.54288i 0.194459i
\(172\) −1.06100 −0.0809004
\(173\) −13.2349 −1.00623 −0.503115 0.864219i \(-0.667813\pi\)
−0.503115 + 0.864219i \(0.667813\pi\)
\(174\) 2.30798i 0.174967i
\(175\) − 13.6896i − 1.03484i
\(176\) 4.27114i 0.321950i
\(177\) − 2.17629i − 0.163580i
\(178\) 7.85086 0.588446
\(179\) −8.52781 −0.637399 −0.318699 0.947856i \(-0.603246\pi\)
−0.318699 + 0.947856i \(0.603246\pi\)
\(180\) − 1.24698i − 0.0929444i
\(181\) 3.63640 0.270291 0.135146 0.990826i \(-0.456850\pi\)
0.135146 + 0.990826i \(0.456850\pi\)
\(182\) 0 0
\(183\) −7.82908 −0.578743
\(184\) 10.7192i 0.790228i
\(185\) 12.7627 0.938333
\(186\) 9.51573 0.697727
\(187\) − 3.58642i − 0.262265i
\(188\) − 5.70410i − 0.416014i
\(189\) 4.80194i 0.349290i
\(190\) 8.88471i 0.644564i
\(191\) 21.3817 1.54712 0.773561 0.633722i \(-0.218474\pi\)
0.773561 + 0.633722i \(0.218474\pi\)
\(192\) −8.89977 −0.642286
\(193\) 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 \(-0.607813\pi\)
0.332264 0.943186i \(-0.392187\pi\)
\(198\) 1.82908 0.129987
\(199\) 14.2524 1.01032 0.505161 0.863025i \(-0.331433\pi\)
0.505161 + 0.863025i \(0.331433\pi\)
\(200\) − 8.69202i − 0.614619i
\(201\) − 3.58211i − 0.252662i
\(202\) − 17.3177i − 1.21847i
\(203\) 8.88769i 0.623794i
\(204\) 1.08815 0.0761855
\(205\) 3.49396 0.244029
\(206\) 21.7017i 1.51203i
\(207\) 3.51573 0.244360
\(208\) 0 0
\(209\) 3.72992 0.258004
\(210\) 16.7778i 1.15778i
\(211\) 1.85086 0.127418 0.0637091 0.997969i \(-0.479707\pi\)
0.0637091 + 0.997969i \(0.479707\pi\)
\(212\) −3.93900 −0.270532
\(213\) − 8.83877i − 0.605623i
\(214\) 13.1588i 0.899519i
\(215\) 6.67994i 0.455568i
\(216\) 3.04892i 0.207453i
\(217\) 36.6437 2.48754
\(218\) −1.33513 −0.0904261
\(219\) 7.69202i 0.519779i
\(220\) −1.82908 −0.123317
\(221\) 0 0
\(222\) −5.67994 −0.381213
\(223\) 18.6504i 1.24892i 0.781056 + 0.624462i \(0.214682\pi\)
−0.781056 + 0.624462i \(0.785318\pi\)
\(224\) −11.8455 −0.791459
\(225\) −2.85086 −0.190057
\(226\) − 20.6136i − 1.37119i
\(227\) 9.75733i 0.647617i 0.946123 + 0.323808i \(0.104963\pi\)
−0.946123 + 0.323808i \(0.895037\pi\)
\(228\) 1.13169i 0.0749478i
\(229\) 2.86294i 0.189188i 0.995516 + 0.0945941i \(0.0301553\pi\)
−0.995516 + 0.0945941i \(0.969845\pi\)
\(230\) 12.2838 0.809971
\(231\) 7.04354 0.463431
\(232\) 5.64310i 0.370488i
\(233\) 5.78554 0.379024 0.189512 0.981878i \(-0.439309\pi\)
0.189512 + 0.981878i \(0.439309\pi\)
\(234\) 0 0
\(235\) −35.9124 −2.34267
\(236\) − 0.968541i − 0.0630467i
\(237\) −4.02177 −0.261242
\(238\) −14.6407 −0.949016
\(239\) 7.09246i 0.458773i 0.973335 + 0.229386i \(0.0736720\pi\)
−0.973335 + 0.229386i \(0.926328\pi\)
\(240\) 8.15883i 0.526650i
\(241\) − 3.89977i − 0.251206i −0.992081 0.125603i \(-0.959913\pi\)
0.992081 0.125603i \(-0.0400866\pi\)
\(242\) 11.0339i 0.709283i
\(243\) 1.00000 0.0641500
\(244\) −3.48427 −0.223058
\(245\) 44.9952i 2.87464i
\(246\) −1.55496 −0.0991405
\(247\) 0 0
\(248\) 23.2664 1.47742
\(249\) − 0.652793i − 0.0413691i
\(250\) 7.50902 0.474912
\(251\) 2.44504 0.154330 0.0771648 0.997018i \(-0.475413\pi\)
0.0771648 + 0.997018i \(0.475413\pi\)
\(252\) 2.13706i 0.134622i
\(253\) − 5.15691i − 0.324212i
\(254\) 11.8931i 0.746237i
\(255\) − 6.85086i − 0.429017i
\(256\) −10.1129 −0.632056
\(257\) 14.1304 0.881428 0.440714 0.897648i \(-0.354725\pi\)
0.440714 + 0.897648i \(0.354725\pi\)
\(258\) − 2.97285i − 0.185082i
\(259\) −21.8726 −1.35910
\(260\) 0 0
\(261\) 1.85086 0.114565
\(262\) − 6.86964i − 0.424408i
\(263\) −23.7235 −1.46285 −0.731426 0.681921i \(-0.761145\pi\)
−0.731426 + 0.681921i \(0.761145\pi\)
\(264\) 4.47219 0.275244
\(265\) 24.7995i 1.52342i
\(266\) − 15.2265i − 0.933599i
\(267\) − 6.29590i − 0.385303i
\(268\) − 1.59419i − 0.0973805i
\(269\) −5.91617 −0.360715 −0.180357 0.983601i \(-0.557725\pi\)
−0.180357 + 0.983601i \(0.557725\pi\)
\(270\) 3.49396 0.212636
\(271\) − 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
\(276\) 1.56465 0.0941807
\(277\) −21.9366 −1.31804 −0.659022 0.752124i \(-0.729029\pi\)
−0.659022 + 0.752124i \(0.729029\pi\)
\(278\) − 13.1045i − 0.785958i
\(279\) − 7.63102i − 0.456857i
\(280\) 41.0224i 2.45155i
\(281\) 11.9903i 0.715282i 0.933859 + 0.357641i \(0.116419\pi\)
−0.933859 + 0.357641i \(0.883581\pi\)
\(282\) 15.9825 0.951747
\(283\) 14.3666 0.854005 0.427002 0.904250i \(-0.359570\pi\)
0.427002 + 0.904250i \(0.359570\pi\)
\(284\) − 3.93362i − 0.233418i
\(285\) 7.12498 0.422047
\(286\) 0 0
\(287\) −5.98792 −0.353456
\(288\) 2.46681i 0.145358i
\(289\) −11.0218 −0.648339
\(290\) 6.46681 0.379744
\(291\) − 10.0315i − 0.588055i
\(292\) 3.42327i 0.200332i
\(293\) 18.7584i 1.09588i 0.836519 + 0.547939i \(0.184587\pi\)
−0.836519 + 0.547939i \(0.815413\pi\)
\(294\) − 20.0248i − 1.16787i
\(295\) −6.09783 −0.355030
\(296\) −13.8877 −0.807206
\(297\) − 1.46681i − 0.0851131i
\(298\) −17.8944 −1.03659
\(299\) 0 0
\(300\) −1.26875 −0.0732513
\(301\) − 11.4480i − 0.659853i
\(302\) 2.45175 0.141082
\(303\) −13.8877 −0.797827
\(304\) − 7.40449i − 0.424676i
\(305\) 21.9366i 1.25609i
\(306\) 3.04892i 0.174295i
\(307\) 25.6262i 1.46257i 0.682074 + 0.731283i \(0.261078\pi\)
−0.682074 + 0.731283i \(0.738922\pi\)
\(308\) 3.13467 0.178614
\(309\) 17.4034 0.990046
\(310\) − 26.6625i − 1.51433i
\(311\) −11.3817 −0.645394 −0.322697 0.946502i \(-0.604590\pi\)
−0.322697 + 0.946502i \(0.604590\pi\)
\(312\) 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
\(315\) 13.4547 0.758088
\(316\) −1.78986 −0.100687
\(317\) 11.2597i 0.632405i 0.948692 + 0.316203i \(0.102408\pi\)
−0.948692 + 0.316203i \(0.897592\pi\)
\(318\) − 11.0368i − 0.618915i
\(319\) − 2.71486i − 0.152003i
\(320\) 24.9366i 1.39400i
\(321\) 10.5526 0.588987
\(322\) −21.0519 −1.17318
\(323\) 6.21744i 0.345948i
\(324\) 0.445042 0.0247245
\(325\) 0 0
\(326\) 4.86294 0.269333
\(327\) 1.07069i 0.0592092i
\(328\) −3.80194 −0.209927
\(329\) 61.5465 3.39317
\(330\) − 5.12498i − 0.282121i
\(331\) 11.9065i 0.654439i 0.944948 + 0.327220i \(0.106112\pi\)
−0.944948 + 0.327220i \(0.893888\pi\)
\(332\) − 0.290520i − 0.0159444i
\(333\) 4.55496i 0.249610i
\(334\) 26.2107 1.43419
\(335\) −10.0368 −0.548371
\(336\) − 13.9825i − 0.762810i
\(337\) −17.1672 −0.935157 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(338\) 0 0
\(339\) −16.5308 −0.897830
\(340\) − 3.04892i − 0.165351i
\(341\) −11.1933 −0.606150
\(342\) −3.17092 −0.171463
\(343\) − 43.4989i − 2.34872i
\(344\) − 7.26875i − 0.391905i
\(345\) − 9.85086i − 0.530352i
\(346\) − 16.5036i − 0.887242i
\(347\) −24.2760 −1.30321 −0.651603 0.758560i \(-0.725903\pi\)
−0.651603 + 0.758560i \(0.725903\pi\)
\(348\) 0.823708 0.0441554
\(349\) − 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 \(0.0516971\pi\)
−0.986840 + 0.161698i \(0.948303\pi\)
\(354\) 2.71379 0.144236
\(355\) −24.7657 −1.31443
\(356\) − 2.80194i − 0.148502i
\(357\) 11.7409i 0.621396i
\(358\) − 10.6340i − 0.562025i
\(359\) − 14.9661i − 0.789883i −0.918706 0.394942i \(-0.870765\pi\)
0.918706 0.394942i \(-0.129235\pi\)
\(360\) 8.54288 0.450249
\(361\) 12.5338 0.659673
\(362\) 4.53452i 0.238329i
\(363\) 8.84846 0.464424
\(364\) 0 0
\(365\) 21.5526 1.12811
\(366\) − 9.76271i − 0.510305i
\(367\) 37.0834 1.93574 0.967868 0.251459i \(-0.0809105\pi\)
0.967868 + 0.251459i \(0.0809105\pi\)
\(368\) −10.2373 −0.533656
\(369\) 1.24698i 0.0649152i
\(370\) 15.9148i 0.827373i
\(371\) − 42.5013i − 2.20656i
\(372\) − 3.39612i − 0.176081i
\(373\) −36.5090 −1.89037 −0.945183 0.326542i \(-0.894117\pi\)
−0.945183 + 0.326542i \(0.894117\pi\)
\(374\) 4.47219 0.231251
\(375\) − 6.02177i − 0.310963i
\(376\) 39.0780 2.01529
\(377\) 0 0
\(378\) −5.98792 −0.307985
\(379\) 26.5851i 1.36558i 0.730613 + 0.682792i \(0.239235\pi\)
−0.730613 + 0.682792i \(0.760765\pi\)
\(380\) 3.17092 0.162665
\(381\) 9.53750 0.488621
\(382\) 26.6625i 1.36417i
\(383\) 14.3502i 0.733261i 0.930367 + 0.366630i \(0.119489\pi\)
−0.930367 + 0.366630i \(0.880511\pi\)
\(384\) − 6.16421i − 0.314566i
\(385\) − 19.7356i − 1.00582i
\(386\) −10.5090 −0.534895
\(387\) −2.38404 −0.121188
\(388\) − 4.46442i − 0.226647i
\(389\) 22.6582 1.14881 0.574407 0.818570i \(-0.305233\pi\)
0.574407 + 0.818570i \(0.305233\pi\)
\(390\) 0 0
\(391\) 8.59611 0.434724
\(392\) − 48.9614i − 2.47292i
\(393\) −5.50902 −0.277894
\(394\) 33.0157 1.66330
\(395\) 11.2687i 0.566992i
\(396\) − 0.652793i − 0.0328041i
\(397\) 7.90754i 0.396868i 0.980114 + 0.198434i \(0.0635856\pi\)
−0.980114 + 0.198434i \(0.936414\pi\)
\(398\) 17.7724i 0.890850i
\(399\) −12.2107 −0.611301
\(400\) 8.30127 0.415064
\(401\) 2.93661i 0.146647i 0.997308 + 0.0733236i \(0.0233606\pi\)
−0.997308 + 0.0733236i \(0.976639\pi\)
\(402\) 4.46681 0.222784
\(403\) 0 0
\(404\) −6.18060 −0.307497
\(405\) − 2.80194i − 0.139229i
\(406\) −11.0828 −0.550029
\(407\) 6.68127 0.331178
\(408\) 7.45473i 0.369064i
\(409\) − 11.7549i − 0.581244i −0.956838 0.290622i \(-0.906138\pi\)
0.956838 0.290622i \(-0.0938623\pi\)
\(410\) 4.35690i 0.215172i
\(411\) 16.1836i 0.798278i
\(412\) 7.74525 0.381581
\(413\) 10.4504 0.514231
\(414\) 4.38404i 0.215464i
\(415\) −1.82908 −0.0897862
\(416\) 0 0
\(417\) −10.5090 −0.514629
\(418\) 4.65114i 0.227495i
\(419\) 7.34183 0.358672 0.179336 0.983788i \(-0.442605\pi\)
0.179336 + 0.983788i \(0.442605\pi\)
\(420\) 5.98792 0.292181
\(421\) − 25.6963i − 1.25236i −0.779677 0.626181i \(-0.784617\pi\)
0.779677 0.626181i \(-0.215383\pi\)
\(422\) 2.30798i 0.112351i
\(423\) − 12.8170i − 0.623184i
\(424\) − 26.9855i − 1.31053i
\(425\) −6.97046 −0.338117
\(426\) 11.0218 0.534007
\(427\) − 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 \(-0.930907\pi\)
0.976535 0.215360i \(-0.0690925\pi\)
\(432\) −2.91185 −0.140097
\(433\) −2.91484 −0.140078 −0.0700391 0.997544i \(-0.522312\pi\)
−0.0700391 + 0.997544i \(0.522312\pi\)
\(434\) 45.6939i 2.19338i
\(435\) − 5.18598i − 0.248649i
\(436\) 0.476501i 0.0228203i
\(437\) 8.94007i 0.427661i
\(438\) −9.59179 −0.458314
\(439\) −9.05861 −0.432344 −0.216172 0.976355i \(-0.569357\pi\)
−0.216172 + 0.976355i \(0.569357\pi\)
\(440\) − 12.5308i − 0.597382i
\(441\) −16.0586 −0.764696
\(442\) 0 0
\(443\) 11.2325 0.533672 0.266836 0.963742i \(-0.414022\pi\)
0.266836 + 0.963742i \(0.414022\pi\)
\(444\) 2.02715i 0.0962041i
\(445\) −17.6407 −0.836250
\(446\) −23.2567 −1.10124
\(447\) 14.3502i 0.678741i
\(448\) − 42.7362i − 2.01909i
\(449\) 28.7579i 1.35717i 0.734522 + 0.678585i \(0.237407\pi\)
−0.734522 + 0.678585i \(0.762593\pi\)
\(450\) − 3.55496i − 0.167582i
\(451\) 1.82908 0.0861282
\(452\) −7.35690 −0.346039
\(453\) − 1.96615i − 0.0923777i
\(454\) −12.1672 −0.571035
\(455\) 0 0
\(456\) −7.75302 −0.363068
\(457\) 19.0761i 0.892341i 0.894948 + 0.446170i \(0.147212\pi\)
−0.894948 + 0.446170i \(0.852788\pi\)
\(458\) −3.57002 −0.166816
\(459\) 2.44504 0.114125
\(460\) − 4.38404i − 0.204407i
\(461\) − 31.7332i − 1.47796i −0.673727 0.738981i \(-0.735308\pi\)
0.673727 0.738981i \(-0.264692\pi\)
\(462\) 8.78315i 0.408629i
\(463\) − 36.4784i − 1.69530i −0.530559 0.847648i \(-0.678018\pi\)
0.530559 0.847648i \(-0.321982\pi\)
\(464\) −5.38942 −0.250198
\(465\) −21.3817 −0.991550
\(466\) 7.21446i 0.334203i
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 17.2010 0.794271
\(470\) − 44.7821i − 2.06564i
\(471\) 10.7017 0.493109
\(472\) 6.63533 0.305416
\(473\) 3.49694i 0.160790i
\(474\) − 5.01507i − 0.230350i
\(475\) − 7.24937i − 0.332624i
\(476\) 5.22521i 0.239497i
\(477\) −8.85086 −0.405253
\(478\) −8.84415 −0.404522
\(479\) 5.61655i 0.256627i 0.991734 + 0.128313i \(0.0409563\pi\)
−0.991734 + 0.128313i \(0.959044\pi\)
\(480\) 6.91185 0.315482
\(481\) 0 0
\(482\) 4.86294 0.221501
\(483\) 16.8823i 0.768172i
\(484\) 3.93794 0.178997
\(485\) −28.1075 −1.27630
\(486\) 1.24698i 0.0565641i
\(487\) 9.75733i 0.442147i 0.975257 + 0.221073i \(0.0709561\pi\)
−0.975257 + 0.221073i \(0.929044\pi\)
\(488\) − 23.8702i − 1.08055i
\(489\) − 3.89977i − 0.176354i
\(490\) −56.1081 −2.53471
\(491\) 7.38835 0.333432 0.166716 0.986005i \(-0.446684\pi\)
0.166716 + 0.986005i \(0.446684\pi\)
\(492\) 0.554958i 0.0250194i
\(493\) 4.52542 0.203815
\(494\) 0 0
\(495\) −4.10992 −0.184727
\(496\) 22.2204i 0.997726i
\(497\) 42.4432 1.90384
\(498\) 0.814019 0.0364771
\(499\) 43.2814i 1.93754i 0.247956 + 0.968771i \(0.420241\pi\)
−0.247956 + 0.968771i \(0.579759\pi\)
\(500\) − 2.67994i − 0.119851i
\(501\) − 21.0194i − 0.939077i
\(502\) 3.04892i 0.136080i
\(503\) 10.7670 0.480078 0.240039 0.970763i \(-0.422840\pi\)
0.240039 + 0.970763i \(0.422840\pi\)
\(504\) −14.6407 −0.652149
\(505\) 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 \(0.372398\pi\)
−0.390223 + 0.920720i \(0.627602\pi\)
\(510\) 8.54288 0.378285
\(511\) −36.9366 −1.63398
\(512\) − 24.9390i − 1.10216i
\(513\) 2.54288i 0.112271i
\(514\) 17.6203i 0.777197i
\(515\) − 48.7633i − 2.14877i
\(516\) −1.06100 −0.0467079
\(517\) −18.8001 −0.826829
\(518\) − 27.2747i − 1.19838i
\(519\) −13.2349 −0.580948
\(520\) 0 0
\(521\) 25.7198 1.12680 0.563402 0.826183i \(-0.309492\pi\)
0.563402 + 0.826183i \(0.309492\pi\)
\(522\) 2.30798i 0.101017i
\(523\) 8.59286 0.375739 0.187870 0.982194i \(-0.439842\pi\)
0.187870 + 0.982194i \(0.439842\pi\)
\(524\) −2.45175 −0.107105
\(525\) − 13.6896i − 0.597464i
\(526\) − 29.5827i − 1.28987i
\(527\) − 18.6582i − 0.812763i
\(528\) 4.27114i 0.185878i
\(529\) −10.6396 −0.462593
\(530\) −30.9245 −1.34328
\(531\) − 2.17629i − 0.0944430i
\(532\) −5.43429 −0.235606
\(533\) 0 0
\(534\) 7.85086 0.339740
\(535\) − 29.5676i − 1.27832i
\(536\) 10.9215 0.471739
\(537\) −8.52781 −0.368002
\(538\) − 7.37734i − 0.318060i
\(539\) 23.5550i 1.01458i
\(540\) − 1.24698i − 0.0536615i
\(541\) 31.3534i 1.34799i 0.738736 + 0.673995i \(0.235423\pi\)
−0.738736 + 0.673995i \(0.764577\pi\)
\(542\) 3.97823 0.170880
\(543\) 3.63640 0.156053
\(544\) 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
\(549\) −7.82908 −0.334137
\(550\) −5.21446 −0.222345
\(551\) 4.70650i 0.200503i
\(552\) 10.7192i 0.456238i
\(553\) − 19.3123i − 0.821242i
\(554\) − 27.3545i − 1.16218i
\(555\) 12.7627 0.541747
\(556\) −4.67696 −0.198347
\(557\) 33.2349i 1.40821i 0.710097 + 0.704104i \(0.248651\pi\)
−0.710097 + 0.704104i \(0.751349\pi\)
\(558\) 9.51573 0.402833
\(559\) 0 0
\(560\) −39.1782 −1.65558
\(561\) − 3.58642i − 0.151419i
\(562\) −14.9517 −0.630698
\(563\) −3.87130 −0.163156 −0.0815779 0.996667i \(-0.525996\pi\)
−0.0815779 + 0.996667i \(0.525996\pi\)
\(564\) − 5.70410i − 0.240186i
\(565\) 46.3183i 1.94862i
\(566\) 17.9148i 0.753017i
\(567\) 4.80194i 0.201662i
\(568\) 26.9487 1.13074
\(569\) −20.1457 −0.844551 −0.422276 0.906468i \(-0.638769\pi\)
−0.422276 + 0.906468i \(0.638769\pi\)
\(570\) 8.88471i 0.372139i
\(571\) 32.1269 1.34447 0.672234 0.740338i \(-0.265335\pi\)
0.672234 + 0.740338i \(0.265335\pi\)
\(572\) 0 0
\(573\) 21.3817 0.893231
\(574\) − 7.46681i − 0.311659i
\(575\) −10.0228 −0.417981
\(576\) −8.89977 −0.370824
\(577\) 16.7506i 0.697338i 0.937246 + 0.348669i \(0.113366\pi\)
−0.937246 + 0.348669i \(0.886634\pi\)
\(578\) − 13.7439i − 0.571672i
\(579\) 8.42758i 0.350238i
\(580\) − 2.30798i − 0.0958336i
\(581\) 3.13467 0.130048
\(582\) 12.5090 0.518516
\(583\) 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 \(0.0444057\pi\)
−0.990285 + 0.139053i \(0.955594\pi\)
\(588\) −7.14675 −0.294727
\(589\) 19.4047 0.799559
\(590\) − 7.60388i − 0.313047i
\(591\) − 26.4765i − 1.08910i
\(592\) − 13.2634i − 0.545121i
\(593\) 18.1172i 0.743985i 0.928236 + 0.371992i \(0.121325\pi\)
−0.928236 + 0.371992i \(0.878675\pi\)
\(594\) 1.82908 0.0750483
\(595\) 32.8974 1.34866
\(596\) 6.38644i 0.261599i
\(597\) 14.2524 0.583310
\(598\) 0 0
\(599\) −26.7851 −1.09441 −0.547204 0.836999i \(-0.684308\pi\)
−0.547204 + 0.836999i \(0.684308\pi\)
\(600\) − 8.69202i − 0.354850i
\(601\) −4.70171 −0.191787 −0.0958934 0.995392i \(-0.530571\pi\)
−0.0958934 + 0.995392i \(0.530571\pi\)
\(602\) 14.2755 0.581824
\(603\) − 3.58211i − 0.145875i
\(604\) − 0.875018i − 0.0356040i
\(605\) − 24.7928i − 1.00797i
\(606\) − 17.3177i − 0.703482i
\(607\) 27.6396 1.12186 0.560929 0.827864i \(-0.310444\pi\)
0.560929 + 0.827864i \(0.310444\pi\)
\(608\) −6.27280 −0.254396
\(609\) 8.88769i 0.360147i
\(610\) −27.3545 −1.10755
\(611\) 0 0
\(612\) 1.08815 0.0439857
\(613\) − 48.1782i − 1.94590i −0.231017 0.972950i \(-0.574205\pi\)
0.231017 0.972950i \(-0.425795\pi\)
\(614\) −31.9554 −1.28961
\(615\) 3.49396 0.140890
\(616\) 21.4752i 0.865259i
\(617\) − 30.3043i − 1.22000i −0.792400 0.610002i \(-0.791169\pi\)
0.792400 0.610002i \(-0.208831\pi\)
\(618\) 21.7017i 0.872971i
\(619\) − 10.9041i − 0.438272i −0.975694 0.219136i \(-0.929676\pi\)
0.975694 0.219136i \(-0.0703239\pi\)
\(620\) −9.51573 −0.382161
\(621\) 3.51573 0.141081
\(622\) − 14.1927i − 0.569075i
\(623\) 30.2325 1.21124
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 34.3846i 1.37429i
\(627\) 3.72992 0.148959
\(628\) 4.76271 0.190053
\(629\) 11.1371i 0.444064i
\(630\) 16.7778i 0.668443i
\(631\) − 10.4523i − 0.416101i −0.978118 0.208050i \(-0.933288\pi\)
0.978118 0.208050i \(-0.0667118\pi\)
\(632\) − 12.2620i − 0.487758i
\(633\) 1.85086 0.0735649
\(634\) −14.0406 −0.557622
\(635\) − 26.7235i − 1.06049i
\(636\) −3.93900 −0.156192
\(637\) 0 0
\(638\) 3.38537 0.134028
\(639\) − 8.83877i − 0.349656i
\(640\) −17.2717 −0.682725
\(641\) 17.5942 0.694929 0.347464 0.937693i \(-0.387043\pi\)
0.347464 + 0.937693i \(0.387043\pi\)
\(642\) 13.1588i 0.519338i
\(643\) − 22.6058i − 0.891486i −0.895161 0.445743i \(-0.852940\pi\)
0.895161 0.445743i \(-0.147060\pi\)
\(644\) 7.51334i 0.296067i
\(645\) 6.67994i 0.263022i
\(646\) −7.75302 −0.305039
\(647\) 24.7918 0.974665 0.487333 0.873216i \(-0.337970\pi\)
0.487333 + 0.873216i \(0.337970\pi\)
\(648\) 3.04892i 0.119773i
\(649\) −3.19221 −0.125305
\(650\) 0 0
\(651\) 36.6437 1.43618
\(652\) − 1.73556i − 0.0679699i
\(653\) −21.8106 −0.853513 −0.426757 0.904367i \(-0.640344\pi\)
−0.426757 + 0.904367i \(0.640344\pi\)
\(654\) −1.33513 −0.0522075
\(655\) 15.4359i 0.603132i
\(656\) − 3.63102i − 0.141768i
\(657\) 7.69202i 0.300094i
\(658\) 76.7472i 2.99192i
\(659\) 16.5526 0.644796 0.322398 0.946604i \(-0.395511\pi\)
0.322398 + 0.946604i \(0.395511\pi\)
\(660\) −1.82908 −0.0711970
\(661\) − 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
\(666\) −5.67994 −0.220093
\(667\) 6.50711 0.251956
\(668\) − 9.35450i − 0.361937i
\(669\) 18.6504i 0.721066i
\(670\) − 12.5157i − 0.483525i
\(671\) 11.4838i 0.443327i
\(672\) −11.8455 −0.456949
\(673\) 7.38835 0.284800 0.142400 0.989809i \(-0.454518\pi\)
0.142400 + 0.989809i \(0.454518\pi\)
\(674\) − 21.4071i − 0.824572i
\(675\) −2.85086 −0.109729
\(676\) 0 0
\(677\) −22.1454 −0.851118 −0.425559 0.904931i \(-0.639922\pi\)
−0.425559 + 0.904931i \(0.639922\pi\)
\(678\) − 20.6136i − 0.791659i
\(679\) 48.1704 1.84861
\(680\) 20.8877 0.801006
\(681\) 9.75733i 0.373902i
\(682\) − 13.9578i − 0.534471i
\(683\) 9.10023i 0.348211i 0.984727 + 0.174105i \(0.0557033\pi\)
−0.984727 + 0.174105i \(0.944297\pi\)
\(684\) 1.13169i 0.0432711i
\(685\) 45.3454 1.73256
\(686\) 54.2422 2.07098
\(687\) 2.86294i 0.109228i
\(688\) 6.94198 0.264661
\(689\) 0 0
\(690\) 12.2838 0.467637
\(691\) − 13.7711i − 0.523876i −0.965085 0.261938i \(-0.915638\pi\)
0.965085 0.261938i \(-0.0843616\pi\)
\(692\) −5.89008 −0.223907
\(693\) 7.04354 0.267562
\(694\) − 30.2717i − 1.14910i
\(695\) 29.4456i 1.11694i
\(696\) 5.64310i 0.213901i
\(697\) 3.04892i 0.115486i
\(698\) 5.70171 0.215813
\(699\) 5.78554 0.218829
\(700\) − 6.09246i − 0.230273i
\(701\) −46.5090 −1.75662 −0.878311 0.478090i \(-0.841329\pi\)
−0.878311 + 0.478090i \(0.841329\pi\)
\(702\) 0 0
\(703\) −11.5827 −0.436850
\(704\) 13.0543i 0.492002i
\(705\) −35.9124 −1.35254
\(706\) −7.57673 −0.285154
\(707\) − 66.6878i − 2.50805i
\(708\) − 0.968541i − 0.0364000i
\(709\) 7.24565i 0.272116i 0.990701 + 0.136058i \(0.0434434\pi\)
−0.990701 + 0.136058i \(0.956557\pi\)
\(710\) − 30.8823i − 1.15899i
\(711\) −4.02177 −0.150828
\(712\) 19.1957 0.719388
\(713\) − 26.8286i − 1.00474i
\(714\) −14.6407 −0.547915
\(715\) 0 0
\(716\) −3.79523 −0.141835
\(717\) 7.09246i 0.264873i
\(718\) 18.6625 0.696478
\(719\) −25.5147 −0.951536 −0.475768 0.879571i \(-0.657830\pi\)
−0.475768 + 0.879571i \(0.657830\pi\)
\(720\) 8.15883i 0.304062i
\(721\) 83.5701i 3.11231i
\(722\) 15.6294i 0.581665i
\(723\) − 3.89977i − 0.145034i
\(724\) 1.61835 0.0601455
\(725\) −5.27652 −0.195965
\(726\) 11.0339i 0.409505i
\(727\) 14.4873 0.537303 0.268651 0.963238i \(-0.413422\pi\)
0.268651 + 0.963238i \(0.413422\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 26.8756i 0.994711i
\(731\) −5.82908 −0.215596
\(732\) −3.48427 −0.128782
\(733\) − 37.5036i − 1.38523i −0.721308 0.692614i \(-0.756459\pi\)
0.721308 0.692614i \(-0.243541\pi\)
\(734\) 46.2422i 1.70683i
\(735\) 44.9952i 1.65967i
\(736\) 8.67264i 0.319678i
\(737\) −5.25428 −0.193544
\(738\) −1.55496 −0.0572388
\(739\) 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 \(-0.920489\pi\)
0.968964 0.247202i \(-0.0795113\pi\)
\(744\) 23.2664 0.852986
\(745\) 40.2083 1.47312
\(746\) − 45.5260i − 1.66683i
\(747\) − 0.652793i − 0.0238844i
\(748\) − 1.59611i − 0.0583594i
\(749\) 50.6728i 1.85154i
\(750\) 7.50902 0.274191
\(751\) −35.5894 −1.29868 −0.649338 0.760500i \(-0.724954\pi\)
−0.649338 + 0.760500i \(0.724954\pi\)
\(752\) 37.3212i 1.36097i
\(753\) 2.44504 0.0891023
\(754\) 0 0
\(755\) −5.50902 −0.200494
\(756\) 2.13706i 0.0777242i
\(757\) −12.2107 −0.443807 −0.221903 0.975069i \(-0.571227\pi\)
−0.221903 + 0.975069i \(0.571227\pi\)
\(758\) −33.1511 −1.20410
\(759\) − 5.15691i − 0.187184i
\(760\) 21.7235i 0.787993i
\(761\) 30.9071i 1.12038i 0.828364 + 0.560190i \(0.189272\pi\)
−0.828364 + 0.560190i \(0.810728\pi\)
\(762\) 11.8931i 0.430840i
\(763\) −5.14138 −0.186130
\(764\) 9.51573 0.344267
\(765\) − 6.85086i − 0.247693i
\(766\) −17.8944 −0.646551
\(767\) 0 0
\(768\) −10.1129 −0.364918
\(769\) − 11.9892i − 0.432343i −0.976355 0.216172i \(-0.930643\pi\)
0.976355 0.216172i \(-0.0693571\pi\)
\(770\) 24.6098 0.886877
\(771\) 14.1304 0.508892
\(772\) 3.75063i 0.134988i
\(773\) − 43.9661i − 1.58135i −0.612235 0.790676i \(-0.709729\pi\)
0.612235 0.790676i \(-0.290271\pi\)
\(774\) − 2.97285i − 0.106857i
\(775\) 21.7549i 0.781460i
\(776\) 30.5851 1.09794
\(777\) −21.8726 −0.784676
\(778\) 28.2543i 1.01296i
\(779\) −3.17092 −0.113610
\(780\) 0 0
\(781\) −12.9648 −0.463918
\(782\) 10.7192i 0.383317i
\(783\) 1.85086 0.0661442
\(784\) 46.7603 1.67001
\(785\) − 29.9855i − 1.07023i
\(786\) − 6.86964i − 0.245032i
\(787\) 30.3763i 1.08280i 0.840766 + 0.541399i \(0.182105\pi\)
−0.840766 + 0.541399i \(0.817895\pi\)
\(788\) − 11.7832i − 0.419757i
\(789\) −23.7235 −0.844578
\(790\) −14.0519 −0.499944
\(791\) − 79.3798i − 2.82242i
\(792\) 4.47219 0.158912
\(793\) 0 0
\(794\) −9.86054 −0.349938
\(795\) 24.7995i 0.879549i
\(796\) 6.34290 0.224818
\(797\) −52.5763 −1.86235 −0.931173 0.364577i \(-0.881214\pi\)
−0.931173 + 0.364577i \(0.881214\pi\)
\(798\) − 15.2265i − 0.539014i
\(799\) − 31.3381i − 1.10866i
\(800\) − 7.03252i − 0.248637i
\(801\) − 6.29590i − 0.222455i
\(802\) −3.66189 −0.129306
\(803\) 11.2828 0.398160
\(804\) − 1.59419i − 0.0562226i
\(805\) 47.3032 1.66722
\(806\) 0 0
\(807\) −5.91617 −0.208259
\(808\) − 42.3424i − 1.48960i
\(809\) 49.4215 1.73757 0.868783 0.495193i \(-0.164903\pi\)
0.868783 + 0.495193i \(0.164903\pi\)
\(810\) 3.49396 0.122765
\(811\) 1.36526i 0.0479406i 0.999713 + 0.0239703i \(0.00763072\pi\)
−0.999713 + 0.0239703i \(0.992369\pi\)
\(812\) 3.95539i 0.138807i
\(813\) − 3.19029i − 0.111888i
\(814\) 8.33140i 0.292016i
\(815\) −10.9269 −0.382753
\(816\) −7.11960 −0.249236
\(817\) − 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 \(-0.996301\pi\)
0.999932 0.0116207i \(-0.00369907\pi\)
\(822\) −20.1806 −0.703879
\(823\) 10.0592 0.350642 0.175321 0.984511i \(-0.443904\pi\)
0.175321 + 0.984511i \(0.443904\pi\)
\(824\) 53.0616i 1.84849i
\(825\) 4.18167i 0.145587i
\(826\) 13.0315i 0.453422i
\(827\) − 37.3038i − 1.29718i −0.761138 0.648590i \(-0.775359\pi\)
0.761138 0.648590i \(-0.224641\pi\)
\(828\) 1.56465 0.0543752
\(829\) 42.6209 1.48028 0.740142 0.672451i \(-0.234758\pi\)
0.740142 + 0.672451i \(0.234758\pi\)
\(830\) − 2.28083i − 0.0791688i
\(831\) −21.9366 −0.760973
\(832\) 0 0
\(833\) −39.2640 −1.36042
\(834\) − 13.1045i − 0.453773i
\(835\) −58.8950 −2.03815
\(836\) 1.65997 0.0574113
\(837\) − 7.63102i − 0.263767i
\(838\) 9.15511i 0.316258i
\(839\) 4.14005i 0.142930i 0.997443 + 0.0714652i \(0.0227675\pi\)
−0.997443 + 0.0714652i \(0.977233\pi\)
\(840\) 41.0224i 1.41541i
\(841\) −25.5743 −0.881874
\(842\) 32.0428 1.10427
\(843\) 11.9903i 0.412968i
\(844\) 0.823708 0.0283532
\(845\) 0 0
\(846\) 15.9825 0.549491
\(847\) 42.4898i 1.45997i
\(848\) 25.7724 0.885028
\(849\) 14.3666 0.493060
\(850\) − 8.69202i − 0.298134i
\(851\) 16.0140i 0.548953i
\(852\) − 3.93362i − 0.134764i
\(853\) 17.3502i 0.594059i 0.954868 + 0.297030i \(0.0959960\pi\)
−0.954868 + 0.297030i \(0.904004\pi\)
\(854\) 46.8799 1.60420
\(855\) 7.12498 0.243669
\(856\) 32.1739i 1.09968i
\(857\) −41.0180 −1.40115 −0.700575 0.713579i \(-0.747073\pi\)
−0.700575 + 0.713579i \(0.747073\pi\)
\(858\) 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
\(861\) −5.98792 −0.204068
\(862\) 11.1505 0.379787
\(863\) 16.6455i 0.566619i 0.959028 + 0.283310i \(0.0914324\pi\)
−0.959028 + 0.283310i \(0.908568\pi\)
\(864\) 2.46681i 0.0839227i
\(865\) 37.0834i 1.26087i
\(866\) − 3.63474i − 0.123514i
\(867\) −11.0218 −0.374319
\(868\) 16.3080 0.553529
\(869\) 5.89918i 0.200116i
\(870\) 6.46681 0.219245
\(871\) 0 0
\(872\) −3.26444 −0.110548
\(873\) − 10.0315i − 0.339513i
\(874\) −11.1481 −0.377089
\(875\) 28.9162 0.977545
\(876\) 3.42327i 0.115662i
\(877\) − 54.4965i − 1.84022i −0.391666 0.920108i \(-0.628101\pi\)
0.391666 0.920108i \(-0.371899\pi\)
\(878\) − 11.2959i − 0.381218i
\(879\) 18.7584i 0.632705i
\(880\) 11.9675 0.403424
\(881\) 9.00670 0.303444 0.151722 0.988423i \(-0.451518\pi\)
0.151722 + 0.988423i \(0.451518\pi\)
\(882\) − 20.0248i − 0.674269i
\(883\) −18.8907 −0.635722 −0.317861 0.948137i \(-0.602965\pi\)
−0.317861 + 0.948137i \(0.602965\pi\)
\(884\) 0 0
\(885\) −6.09783 −0.204976
\(886\) 14.0067i 0.470564i
\(887\) −46.9124 −1.57517 −0.787583 0.616209i \(-0.788668\pi\)
−0.787583 + 0.616209i \(0.788668\pi\)
\(888\) −13.8877 −0.466040
\(889\) 45.7985i 1.53603i
\(890\) − 21.9976i − 0.737361i
\(891\) − 1.46681i − 0.0491401i
\(892\) 8.30021i 0.277912i
\(893\) 32.5921 1.09065
\(894\) −17.8944 −0.598478
\(895\) 23.8944i 0.798702i
\(896\) 29.6002 0.988872
\(897\) 0 0
\(898\) −35.8605 −1.19668
\(899\) − 14.1239i − 0.471059i
\(900\) −1.26875 −0.0422917
\(901\) −21.6407 −0.720957
\(902\) 2.28083i 0.0759434i
\(903\) − 11.4480i − 0.380966i
\(904\) − 50.4010i − 1.67631i
\(905\) − 10.1890i − 0.338693i
\(906\) 2.45175 0.0814538
\(907\) 35.3013 1.17216 0.586080 0.810253i \(-0.300671\pi\)
0.586080 + 0.810253i \(0.300671\pi\)
\(908\) 4.34242i 0.144108i
\(909\) −13.8877 −0.460626
\(910\) 0 0
\(911\) −9.80731 −0.324931 −0.162465 0.986714i \(-0.551945\pi\)
−0.162465 + 0.986714i \(0.551945\pi\)
\(912\) − 7.40449i − 0.245187i
\(913\) −0.957524 −0.0316894
\(914\) −23.7875 −0.786819
\(915\) 21.9366i 0.725202i
\(916\) 1.27413i 0.0420983i
\(917\) − 26.4540i − 0.873588i
\(918\) 3.04892i 0.100629i
\(919\) 18.4655 0.609120 0.304560 0.952493i \(-0.401491\pi\)
0.304560 + 0.952493i \(0.401491\pi\)
\(920\) 30.0344 0.990206
\(921\) 25.6262i 0.844413i
\(922\) 39.5706 1.30319
\(923\) 0 0
\(924\) 3.13467 0.103123
\(925\) − 12.9855i − 0.426961i
\(926\) 45.4878 1.49482
\(927\) 17.4034 0.571603
\(928\) 4.56571i 0.149877i
\(929\) 25.6267i 0.840785i 0.907342 + 0.420393i \(0.138108\pi\)
−0.907342 + 0.420393i \(0.861892\pi\)
\(930\) − 26.6625i − 0.874297i
\(931\) − 40.8351i − 1.33831i
\(932\) 2.57481 0.0843407
\(933\) −11.3817 −0.372618
\(934\) − 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
\(939\) 27.5743 0.899854
\(940\) −15.9825 −0.521293
\(941\) 12.6418i 0.412110i 0.978540 + 0.206055i \(0.0660626\pi\)
−0.978540 + 0.206055i \(0.933937\pi\)
\(942\) 13.3448i 0.434798i
\(943\) 4.38404i 0.142764i
\(944\) 6.33704i 0.206253i
\(945\) 13.4547 0.437682
\(946\) −4.36062 −0.141776
\(947\) 27.9801i 0.909233i 0.890687 + 0.454616i \(0.150224\pi\)
−0.890687 + 0.454616i \(0.849776\pi\)
\(948\) −1.78986 −0.0581318
\(949\) 0 0
\(950\) 9.03982 0.293290
\(951\) 11.2597i 0.365119i
\(952\) −35.7972 −1.16019
\(953\) −4.00239 −0.129650 −0.0648251 0.997897i \(-0.520649\pi\)
−0.0648251 + 0.997897i \(0.520649\pi\)
\(954\) − 11.0368i − 0.357331i
\(955\) − 59.9101i − 1.93864i
\(956\) 3.15644i 0.102087i
\(957\) − 2.71486i − 0.0877589i
\(958\) −7.00372 −0.226280
\(959\) −77.7126 −2.50947
\(960\) 24.9366i 0.804826i
\(961\) −27.2325 −0.878468
\(962\) 0 0
\(963\) 10.5526 0.340052
\(964\) − 1.73556i − 0.0558987i
\(965\) 23.6136 0.760148
\(966\) −21.0519 −0.677334
\(967\) − 12.2239i − 0.393094i −0.980494 0.196547i \(-0.937027\pi\)
0.980494 0.196547i \(-0.0629728\pi\)
\(968\) 26.9782i 0.867113i
\(969\) 6.21744i 0.199733i
\(970\) − 35.0495i − 1.12537i
\(971\) −23.6401 −0.758648 −0.379324 0.925264i \(-0.623843\pi\)
−0.379324 + 0.925264i \(0.623843\pi\)
\(972\) 0.445042 0.0142747
\(973\) − 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 \(-0.903136\pi\)
0.954055 0.299633i \(-0.0968642\pi\)
\(978\) 4.86294 0.155500
\(979\) −9.23490 −0.295149
\(980\) 20.0248i 0.639667i
\(981\) 1.07069i 0.0341844i
\(982\) 9.21313i 0.294003i
\(983\) 12.0954i 0.385785i 0.981220 + 0.192892i \(0.0617868\pi\)
−0.981220 + 0.192892i \(0.938213\pi\)
\(984\) −3.80194 −0.121201
\(985\) −74.1855 −2.36375
\(986\) 5.64310i 0.179713i
\(987\) 61.5465 1.95905
\(988\) 0 0
\(989\) −8.38165 −0.266521
\(990\) − 5.12498i − 0.162883i
\(991\) −28.5526 −0.907002 −0.453501 0.891256i \(-0.649825\pi\)
−0.453501 + 0.891256i \(0.649825\pi\)
\(992\) 18.8243 0.597672
\(993\) 11.9065i 0.377841i
\(994\) 52.9259i 1.67871i
\(995\) − 39.9342i − 1.26600i
\(996\) − 0.290520i − 0.00920548i
\(997\) −23.9347 −0.758019 −0.379010 0.925393i \(-0.623735\pi\)
−0.379010 + 0.925393i \(0.623735\pi\)
\(998\) −53.9711 −1.70842
\(999\) 4.55496i 0.144112i
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 507.2.b.g.337.5 6
3.2 odd 2 1521.2.b.m.1351.2 6
13.2 odd 12 507.2.e.k.22.1 6
13.3 even 3 507.2.j.h.316.2 12
13.4 even 6 507.2.j.h.361.2 12
13.5 odd 4 507.2.a.j.1.3 3
13.6 odd 12 507.2.e.k.484.1 6
13.7 odd 12 507.2.e.j.484.3 6
13.8 odd 4 507.2.a.k.1.1 yes 3
13.9 even 3 507.2.j.h.361.5 12
13.10 even 6 507.2.j.h.316.5 12
13.11 odd 12 507.2.e.j.22.3 6
13.12 even 2 inner 507.2.b.g.337.2 6
39.5 even 4 1521.2.a.q.1.1 3
39.8 even 4 1521.2.a.p.1.3 3
39.38 odd 2 1521.2.b.m.1351.5 6
52.31 even 4 8112.2.a.by.1.1 3
52.47 even 4 8112.2.a.cf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.3 3 13.5 odd 4
507.2.a.k.1.1 yes 3 13.8 odd 4
507.2.b.g.337.2 6 13.12 even 2 inner
507.2.b.g.337.5 6 1.1 even 1 trivial
507.2.e.j.22.3 6 13.11 odd 12
507.2.e.j.484.3 6 13.7 odd 12
507.2.e.k.22.1 6 13.2 odd 12
507.2.e.k.484.1 6 13.6 odd 12
507.2.j.h.316.2 12 13.3 even 3
507.2.j.h.316.5 12 13.10 even 6
507.2.j.h.361.2 12 13.4 even 6
507.2.j.h.361.5 12 13.9 even 3
1521.2.a.p.1.3 3 39.8 even 4
1521.2.a.q.1.1 3 39.5 even 4
1521.2.b.m.1351.2 6 3.2 odd 2
1521.2.b.m.1351.5 6 39.38 odd 2
8112.2.a.by.1.1 3 52.31 even 4
8112.2.a.cf.1.3 3 52.47 even 4