Properties

Label 7098.2.a.bk.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
Defining polynomial: \(x^{2} - x - 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.17891\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -6.17891 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} +8.17891 q^{19} +2.00000 q^{20} -1.00000 q^{21} +6.17891 q^{22} +3.17891 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} +8.17891 q^{29} +2.00000 q^{30} +7.17891 q^{31} -1.00000 q^{32} +6.17891 q^{33} -5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -8.17891 q^{38} -2.00000 q^{40} -4.17891 q^{41} +1.00000 q^{42} +0.821092 q^{43} -6.17891 q^{44} +2.00000 q^{45} -3.17891 q^{46} -4.17891 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -5.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} -12.3578 q^{55} -1.00000 q^{56} -8.17891 q^{57} -8.17891 q^{58} +0.821092 q^{59} -2.00000 q^{60} -3.00000 q^{61} -7.17891 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.17891 q^{66} +15.1789 q^{67} +5.00000 q^{68} -3.17891 q^{69} -2.00000 q^{70} -9.17891 q^{71} -1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +8.17891 q^{76} -6.17891 q^{77} +1.82109 q^{79} +2.00000 q^{80} +1.00000 q^{81} +4.17891 q^{82} +5.17891 q^{83} -1.00000 q^{84} +10.0000 q^{85} -0.821092 q^{86} -8.17891 q^{87} +6.17891 q^{88} -9.00000 q^{89} -2.00000 q^{90} +3.17891 q^{92} -7.17891 q^{93} +4.17891 q^{94} +16.3578 q^{95} +1.00000 q^{96} -12.3578 q^{97} -1.00000 q^{98} -6.17891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 4q^{10} - q^{11} - 2q^{12} - 2q^{14} - 4q^{15} + 2q^{16} + 10q^{17} - 2q^{18} + 5q^{19} + 4q^{20} - 2q^{21} + q^{22} - 5q^{23} + 2q^{24} - 2q^{25} - 2q^{27} + 2q^{28} + 5q^{29} + 4q^{30} + 3q^{31} - 2q^{32} + q^{33} - 10q^{34} + 4q^{35} + 2q^{36} - 4q^{37} - 5q^{38} - 4q^{40} + 3q^{41} + 2q^{42} + 13q^{43} - q^{44} + 4q^{45} + 5q^{46} + 3q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 10q^{51} - 6q^{53} + 2q^{54} - 2q^{55} - 2q^{56} - 5q^{57} - 5q^{58} + 13q^{59} - 4q^{60} - 6q^{61} - 3q^{62} + 2q^{63} + 2q^{64} - q^{66} + 19q^{67} + 10q^{68} + 5q^{69} - 4q^{70} - 7q^{71} - 2q^{72} - 8q^{73} + 4q^{74} + 2q^{75} + 5q^{76} - q^{77} + 15q^{79} + 4q^{80} + 2q^{81} - 3q^{82} - q^{83} - 2q^{84} + 20q^{85} - 13q^{86} - 5q^{87} + q^{88} - 18q^{89} - 4q^{90} - 5q^{92} - 3q^{93} - 3q^{94} + 10q^{95} + 2q^{96} - 2q^{97} - 2q^{98} - q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −6.17891 −1.86301 −0.931505 0.363727i \(-0.881504\pi\)
−0.931505 + 0.363727i \(0.881504\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −1.00000 −0.235702
\(19\) 8.17891 1.87637 0.938185 0.346134i \(-0.112506\pi\)
0.938185 + 0.346134i \(0.112506\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 6.17891 1.31735
\(23\) 3.17891 0.662848 0.331424 0.943482i \(-0.392471\pi\)
0.331424 + 0.943482i \(0.392471\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 8.17891 1.51879 0.759393 0.650633i \(-0.225496\pi\)
0.759393 + 0.650633i \(0.225496\pi\)
\(30\) 2.00000 0.365148
\(31\) 7.17891 1.28937 0.644685 0.764448i \(-0.276989\pi\)
0.644685 + 0.764448i \(0.276989\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.17891 1.07561
\(34\) −5.00000 −0.857493
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −8.17891 −1.32679
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −4.17891 −0.652636 −0.326318 0.945260i \(-0.605808\pi\)
−0.326318 + 0.945260i \(0.605808\pi\)
\(42\) 1.00000 0.154303
\(43\) 0.821092 0.125215 0.0626077 0.998038i \(-0.480058\pi\)
0.0626077 + 0.998038i \(0.480058\pi\)
\(44\) −6.17891 −0.931505
\(45\) 2.00000 0.298142
\(46\) −3.17891 −0.468704
\(47\) −4.17891 −0.609556 −0.304778 0.952423i \(-0.598582\pi\)
−0.304778 + 0.952423i \(0.598582\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 1.00000 0.136083
\(55\) −12.3578 −1.66633
\(56\) −1.00000 −0.133631
\(57\) −8.17891 −1.08332
\(58\) −8.17891 −1.07394
\(59\) 0.821092 0.106897 0.0534485 0.998571i \(-0.482979\pi\)
0.0534485 + 0.998571i \(0.482979\pi\)
\(60\) −2.00000 −0.258199
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) −7.17891 −0.911722
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.17891 −0.760571
\(67\) 15.1789 1.85440 0.927199 0.374568i \(-0.122209\pi\)
0.927199 + 0.374568i \(0.122209\pi\)
\(68\) 5.00000 0.606339
\(69\) −3.17891 −0.382696
\(70\) −2.00000 −0.239046
\(71\) −9.17891 −1.08934 −0.544668 0.838652i \(-0.683344\pi\)
−0.544668 + 0.838652i \(0.683344\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 2.00000 0.232495
\(75\) 1.00000 0.115470
\(76\) 8.17891 0.938185
\(77\) −6.17891 −0.704152
\(78\) 0 0
\(79\) 1.82109 0.204889 0.102444 0.994739i \(-0.467334\pi\)
0.102444 + 0.994739i \(0.467334\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 4.17891 0.461483
\(83\) 5.17891 0.568459 0.284230 0.958756i \(-0.408262\pi\)
0.284230 + 0.958756i \(0.408262\pi\)
\(84\) −1.00000 −0.109109
\(85\) 10.0000 1.08465
\(86\) −0.821092 −0.0885406
\(87\) −8.17891 −0.876871
\(88\) 6.17891 0.658674
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 3.17891 0.331424
\(93\) −7.17891 −0.744418
\(94\) 4.17891 0.431021
\(95\) 16.3578 1.67828
\(96\) 1.00000 0.102062
\(97\) −12.3578 −1.25475 −0.627373 0.778719i \(-0.715870\pi\)
−0.627373 + 0.778719i \(0.715870\pi\)
\(98\) −1.00000 −0.101015
\(99\) −6.17891 −0.621004
\(100\) −1.00000 −0.100000
\(101\) −16.3578 −1.62766 −0.813832 0.581101i \(-0.802622\pi\)
−0.813832 + 0.581101i \(0.802622\pi\)
\(102\) 5.00000 0.495074
\(103\) 3.17891 0.313227 0.156614 0.987660i \(-0.449942\pi\)
0.156614 + 0.987660i \(0.449942\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 3.00000 0.291386
\(107\) 7.82109 0.756093 0.378047 0.925787i \(-0.376596\pi\)
0.378047 + 0.925787i \(0.376596\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 12.3578 1.17827
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −4.35782 −0.409949 −0.204974 0.978767i \(-0.565711\pi\)
−0.204974 + 0.978767i \(0.565711\pi\)
\(114\) 8.17891 0.766025
\(115\) 6.35782 0.592869
\(116\) 8.17891 0.759393
\(117\) 0 0
\(118\) −0.821092 −0.0755876
\(119\) 5.00000 0.458349
\(120\) 2.00000 0.182574
\(121\) 27.1789 2.47081
\(122\) 3.00000 0.271607
\(123\) 4.17891 0.376799
\(124\) 7.17891 0.644685
\(125\) −12.0000 −1.07331
\(126\) −1.00000 −0.0890871
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.821092 −0.0722931
\(130\) 0 0
\(131\) 19.5367 1.70693 0.853466 0.521149i \(-0.174496\pi\)
0.853466 + 0.521149i \(0.174496\pi\)
\(132\) 6.17891 0.537805
\(133\) 8.17891 0.709201
\(134\) −15.1789 −1.31126
\(135\) −2.00000 −0.172133
\(136\) −5.00000 −0.428746
\(137\) 16.3578 1.39754 0.698771 0.715345i \(-0.253731\pi\)
0.698771 + 0.715345i \(0.253731\pi\)
\(138\) 3.17891 0.270607
\(139\) −18.1789 −1.54191 −0.770957 0.636887i \(-0.780222\pi\)
−0.770957 + 0.636887i \(0.780222\pi\)
\(140\) 2.00000 0.169031
\(141\) 4.17891 0.351928
\(142\) 9.17891 0.770277
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 16.3578 1.35844
\(146\) 4.00000 0.331042
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) −4.82109 −0.394959 −0.197480 0.980307i \(-0.563276\pi\)
−0.197480 + 0.980307i \(0.563276\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 14.5367 1.18298 0.591491 0.806312i \(-0.298540\pi\)
0.591491 + 0.806312i \(0.298540\pi\)
\(152\) −8.17891 −0.663397
\(153\) 5.00000 0.404226
\(154\) 6.17891 0.497911
\(155\) 14.3578 1.15325
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −1.82109 −0.144878
\(159\) 3.00000 0.237915
\(160\) −2.00000 −0.158114
\(161\) 3.17891 0.250533
\(162\) −1.00000 −0.0785674
\(163\) 8.82109 0.690921 0.345461 0.938433i \(-0.387723\pi\)
0.345461 + 0.938433i \(0.387723\pi\)
\(164\) −4.17891 −0.326318
\(165\) 12.3578 0.962055
\(166\) −5.17891 −0.401961
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −10.0000 −0.766965
\(171\) 8.17891 0.625457
\(172\) 0.821092 0.0626077
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 8.17891 0.620041
\(175\) −1.00000 −0.0755929
\(176\) −6.17891 −0.465753
\(177\) −0.821092 −0.0617170
\(178\) 9.00000 0.674579
\(179\) −14.3578 −1.07315 −0.536577 0.843851i \(-0.680283\pi\)
−0.536577 + 0.843851i \(0.680283\pi\)
\(180\) 2.00000 0.149071
\(181\) 7.82109 0.581337 0.290669 0.956824i \(-0.406122\pi\)
0.290669 + 0.956824i \(0.406122\pi\)
\(182\) 0 0
\(183\) 3.00000 0.221766
\(184\) −3.17891 −0.234352
\(185\) −4.00000 −0.294086
\(186\) 7.17891 0.526383
\(187\) −30.8945 −2.25923
\(188\) −4.17891 −0.304778
\(189\) −1.00000 −0.0727393
\(190\) −16.3578 −1.18672
\(191\) 27.5367 1.99249 0.996244 0.0865933i \(-0.0275981\pi\)
0.996244 + 0.0865933i \(0.0275981\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.5367 −1.47827 −0.739133 0.673560i \(-0.764764\pi\)
−0.739133 + 0.673560i \(0.764764\pi\)
\(194\) 12.3578 0.887240
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −17.0000 −1.21120 −0.605600 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) 6.17891 0.439116
\(199\) 12.8211 0.908863 0.454432 0.890782i \(-0.349842\pi\)
0.454432 + 0.890782i \(0.349842\pi\)
\(200\) 1.00000 0.0707107
\(201\) −15.1789 −1.07064
\(202\) 16.3578 1.15093
\(203\) 8.17891 0.574047
\(204\) −5.00000 −0.350070
\(205\) −8.35782 −0.583735
\(206\) −3.17891 −0.221485
\(207\) 3.17891 0.220949
\(208\) 0 0
\(209\) −50.5367 −3.49570
\(210\) 2.00000 0.138013
\(211\) 16.3578 1.12612 0.563059 0.826417i \(-0.309624\pi\)
0.563059 + 0.826417i \(0.309624\pi\)
\(212\) −3.00000 −0.206041
\(213\) 9.17891 0.628928
\(214\) −7.82109 −0.534639
\(215\) 1.64218 0.111996
\(216\) 1.00000 0.0680414
\(217\) 7.17891 0.487336
\(218\) 4.00000 0.270914
\(219\) 4.00000 0.270295
\(220\) −12.3578 −0.833164
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −9.17891 −0.614665 −0.307333 0.951602i \(-0.599436\pi\)
−0.307333 + 0.951602i \(0.599436\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 4.35782 0.289878
\(227\) −8.35782 −0.554728 −0.277364 0.960765i \(-0.589461\pi\)
−0.277364 + 0.960765i \(0.589461\pi\)
\(228\) −8.17891 −0.541661
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) −6.35782 −0.419222
\(231\) 6.17891 0.406542
\(232\) −8.17891 −0.536972
\(233\) −0.357817 −0.0234414 −0.0117207 0.999931i \(-0.503731\pi\)
−0.0117207 + 0.999931i \(0.503731\pi\)
\(234\) 0 0
\(235\) −8.35782 −0.545204
\(236\) 0.821092 0.0534485
\(237\) −1.82109 −0.118293
\(238\) −5.00000 −0.324102
\(239\) 1.17891 0.0762572 0.0381286 0.999273i \(-0.487860\pi\)
0.0381286 + 0.999273i \(0.487860\pi\)
\(240\) −2.00000 −0.129099
\(241\) 18.3578 1.18253 0.591265 0.806477i \(-0.298629\pi\)
0.591265 + 0.806477i \(0.298629\pi\)
\(242\) −27.1789 −1.74713
\(243\) −1.00000 −0.0641500
\(244\) −3.00000 −0.192055
\(245\) 2.00000 0.127775
\(246\) −4.17891 −0.266437
\(247\) 0 0
\(248\) −7.17891 −0.455861
\(249\) −5.17891 −0.328200
\(250\) 12.0000 0.758947
\(251\) 10.8211 0.683021 0.341511 0.939878i \(-0.389061\pi\)
0.341511 + 0.939878i \(0.389061\pi\)
\(252\) 1.00000 0.0629941
\(253\) −19.6422 −1.23489
\(254\) −12.0000 −0.752947
\(255\) −10.0000 −0.626224
\(256\) 1.00000 0.0625000
\(257\) 25.7156 1.60410 0.802049 0.597259i \(-0.203743\pi\)
0.802049 + 0.597259i \(0.203743\pi\)
\(258\) 0.821092 0.0511189
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 8.17891 0.506262
\(262\) −19.5367 −1.20698
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −6.17891 −0.380286
\(265\) −6.00000 −0.368577
\(266\) −8.17891 −0.501481
\(267\) 9.00000 0.550791
\(268\) 15.1789 0.927199
\(269\) 18.3578 1.11930 0.559648 0.828730i \(-0.310936\pi\)
0.559648 + 0.828730i \(0.310936\pi\)
\(270\) 2.00000 0.121716
\(271\) −31.1789 −1.89398 −0.946992 0.321257i \(-0.895895\pi\)
−0.946992 + 0.321257i \(0.895895\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) −16.3578 −0.988212
\(275\) 6.17891 0.372602
\(276\) −3.17891 −0.191348
\(277\) 28.3578 1.70386 0.851928 0.523659i \(-0.175433\pi\)
0.851928 + 0.523659i \(0.175433\pi\)
\(278\) 18.1789 1.09030
\(279\) 7.17891 0.429790
\(280\) −2.00000 −0.119523
\(281\) −5.64218 −0.336584 −0.168292 0.985737i \(-0.553825\pi\)
−0.168292 + 0.985737i \(0.553825\pi\)
\(282\) −4.17891 −0.248850
\(283\) 30.3578 1.80458 0.902292 0.431125i \(-0.141883\pi\)
0.902292 + 0.431125i \(0.141883\pi\)
\(284\) −9.17891 −0.544668
\(285\) −16.3578 −0.968953
\(286\) 0 0
\(287\) −4.17891 −0.246673
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) −16.3578 −0.960564
\(291\) 12.3578 0.724428
\(292\) −4.00000 −0.234082
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 1.00000 0.0583212
\(295\) 1.64218 0.0956116
\(296\) 2.00000 0.116248
\(297\) 6.17891 0.358537
\(298\) 4.82109 0.279278
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0.821092 0.0473269
\(302\) −14.5367 −0.836495
\(303\) 16.3578 0.939732
\(304\) 8.17891 0.469093
\(305\) −6.00000 −0.343559
\(306\) −5.00000 −0.285831
\(307\) −24.8945 −1.42081 −0.710403 0.703795i \(-0.751487\pi\)
−0.710403 + 0.703795i \(0.751487\pi\)
\(308\) −6.17891 −0.352076
\(309\) −3.17891 −0.180842
\(310\) −14.3578 −0.815469
\(311\) −8.53673 −0.484073 −0.242037 0.970267i \(-0.577815\pi\)
−0.242037 + 0.970267i \(0.577815\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) −2.00000 −0.112867
\(315\) 2.00000 0.112687
\(316\) 1.82109 0.102444
\(317\) −7.17891 −0.403208 −0.201604 0.979467i \(-0.564615\pi\)
−0.201604 + 0.979467i \(0.564615\pi\)
\(318\) −3.00000 −0.168232
\(319\) −50.5367 −2.82951
\(320\) 2.00000 0.111803
\(321\) −7.82109 −0.436531
\(322\) −3.17891 −0.177154
\(323\) 40.8945 2.27543
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −8.82109 −0.488555
\(327\) 4.00000 0.221201
\(328\) 4.17891 0.230742
\(329\) −4.17891 −0.230391
\(330\) −12.3578 −0.680275
\(331\) 24.3578 1.33883 0.669413 0.742890i \(-0.266546\pi\)
0.669413 + 0.742890i \(0.266546\pi\)
\(332\) 5.17891 0.284230
\(333\) −2.00000 −0.109599
\(334\) −8.00000 −0.437741
\(335\) 30.3578 1.65862
\(336\) −1.00000 −0.0545545
\(337\) −0.536725 −0.0292373 −0.0146186 0.999893i \(-0.504653\pi\)
−0.0146186 + 0.999893i \(0.504653\pi\)
\(338\) 0 0
\(339\) 4.35782 0.236684
\(340\) 10.0000 0.542326
\(341\) −44.3578 −2.40211
\(342\) −8.17891 −0.442265
\(343\) 1.00000 0.0539949
\(344\) −0.821092 −0.0442703
\(345\) −6.35782 −0.342293
\(346\) −4.00000 −0.215041
\(347\) −7.82109 −0.419858 −0.209929 0.977717i \(-0.567323\pi\)
−0.209929 + 0.977717i \(0.567323\pi\)
\(348\) −8.17891 −0.438436
\(349\) 3.17891 0.170163 0.0850815 0.996374i \(-0.472885\pi\)
0.0850815 + 0.996374i \(0.472885\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 6.17891 0.329337
\(353\) 7.17891 0.382095 0.191047 0.981581i \(-0.438812\pi\)
0.191047 + 0.981581i \(0.438812\pi\)
\(354\) 0.821092 0.0436405
\(355\) −18.3578 −0.974332
\(356\) −9.00000 −0.476999
\(357\) −5.00000 −0.264628
\(358\) 14.3578 0.758834
\(359\) −12.3578 −0.652221 −0.326110 0.945332i \(-0.605738\pi\)
−0.326110 + 0.945332i \(0.605738\pi\)
\(360\) −2.00000 −0.105409
\(361\) 47.8945 2.52077
\(362\) −7.82109 −0.411067
\(363\) −27.1789 −1.42652
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) −3.00000 −0.156813
\(367\) 3.17891 0.165938 0.0829688 0.996552i \(-0.473560\pi\)
0.0829688 + 0.996552i \(0.473560\pi\)
\(368\) 3.17891 0.165712
\(369\) −4.17891 −0.217545
\(370\) 4.00000 0.207950
\(371\) −3.00000 −0.155752
\(372\) −7.17891 −0.372209
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 30.8945 1.59752
\(375\) 12.0000 0.619677
\(376\) 4.17891 0.215511
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −20.7156 −1.06409 −0.532045 0.846716i \(-0.678576\pi\)
−0.532045 + 0.846716i \(0.678576\pi\)
\(380\) 16.3578 0.839138
\(381\) −12.0000 −0.614779
\(382\) −27.5367 −1.40890
\(383\) −9.82109 −0.501834 −0.250917 0.968009i \(-0.580732\pi\)
−0.250917 + 0.968009i \(0.580732\pi\)
\(384\) 1.00000 0.0510310
\(385\) −12.3578 −0.629813
\(386\) 20.5367 1.04529
\(387\) 0.821092 0.0417384
\(388\) −12.3578 −0.627373
\(389\) −0.821092 −0.0416310 −0.0208155 0.999783i \(-0.506626\pi\)
−0.0208155 + 0.999783i \(0.506626\pi\)
\(390\) 0 0
\(391\) 15.8945 0.803822
\(392\) −1.00000 −0.0505076
\(393\) −19.5367 −0.985497
\(394\) 17.0000 0.856448
\(395\) 3.64218 0.183258
\(396\) −6.17891 −0.310502
\(397\) −3.00000 −0.150566 −0.0752828 0.997162i \(-0.523986\pi\)
−0.0752828 + 0.997162i \(0.523986\pi\)
\(398\) −12.8211 −0.642663
\(399\) −8.17891 −0.409458
\(400\) −1.00000 −0.0500000
\(401\) −32.3578 −1.61587 −0.807936 0.589270i \(-0.799415\pi\)
−0.807936 + 0.589270i \(0.799415\pi\)
\(402\) 15.1789 0.757055
\(403\) 0 0
\(404\) −16.3578 −0.813832
\(405\) 2.00000 0.0993808
\(406\) −8.17891 −0.405912
\(407\) 12.3578 0.612554
\(408\) 5.00000 0.247537
\(409\) 13.6422 0.674563 0.337281 0.941404i \(-0.390493\pi\)
0.337281 + 0.941404i \(0.390493\pi\)
\(410\) 8.35782 0.412763
\(411\) −16.3578 −0.806872
\(412\) 3.17891 0.156614
\(413\) 0.821092 0.0404033
\(414\) −3.17891 −0.156235
\(415\) 10.3578 0.508445
\(416\) 0 0
\(417\) 18.1789 0.890225
\(418\) 50.5367 2.47183
\(419\) 29.8945 1.46044 0.730222 0.683210i \(-0.239417\pi\)
0.730222 + 0.683210i \(0.239417\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 34.7156 1.69194 0.845968 0.533233i \(-0.179023\pi\)
0.845968 + 0.533233i \(0.179023\pi\)
\(422\) −16.3578 −0.796286
\(423\) −4.17891 −0.203185
\(424\) 3.00000 0.145693
\(425\) −5.00000 −0.242536
\(426\) −9.17891 −0.444720
\(427\) −3.00000 −0.145180
\(428\) 7.82109 0.378047
\(429\) 0 0
\(430\) −1.64218 −0.0791931
\(431\) 1.17891 0.0567860 0.0283930 0.999597i \(-0.490961\pi\)
0.0283930 + 0.999597i \(0.490961\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 11.6422 0.559488 0.279744 0.960075i \(-0.409750\pi\)
0.279744 + 0.960075i \(0.409750\pi\)
\(434\) −7.17891 −0.344599
\(435\) −16.3578 −0.784297
\(436\) −4.00000 −0.191565
\(437\) 26.0000 1.24375
\(438\) −4.00000 −0.191127
\(439\) −4.71563 −0.225065 −0.112532 0.993648i \(-0.535896\pi\)
−0.112532 + 0.993648i \(0.535896\pi\)
\(440\) 12.3578 0.589136
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −8.17891 −0.388592 −0.194296 0.980943i \(-0.562242\pi\)
−0.194296 + 0.980943i \(0.562242\pi\)
\(444\) 2.00000 0.0949158
\(445\) −18.0000 −0.853282
\(446\) 9.17891 0.434634
\(447\) 4.82109 0.228030
\(448\) 1.00000 0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) 25.8211 1.21587
\(452\) −4.35782 −0.204974
\(453\) −14.5367 −0.682995
\(454\) 8.35782 0.392252
\(455\) 0 0
\(456\) 8.17891 0.383012
\(457\) 31.8945 1.49196 0.745982 0.665966i \(-0.231981\pi\)
0.745982 + 0.665966i \(0.231981\pi\)
\(458\) 1.00000 0.0467269
\(459\) −5.00000 −0.233380
\(460\) 6.35782 0.296435
\(461\) 24.7156 1.15112 0.575561 0.817759i \(-0.304784\pi\)
0.575561 + 0.817759i \(0.304784\pi\)
\(462\) −6.17891 −0.287469
\(463\) 28.8945 1.34284 0.671422 0.741076i \(-0.265684\pi\)
0.671422 + 0.741076i \(0.265684\pi\)
\(464\) 8.17891 0.379696
\(465\) −14.3578 −0.665828
\(466\) 0.357817 0.0165755
\(467\) 33.8945 1.56845 0.784226 0.620475i \(-0.213060\pi\)
0.784226 + 0.620475i \(0.213060\pi\)
\(468\) 0 0
\(469\) 15.1789 0.700897
\(470\) 8.35782 0.385517
\(471\) −2.00000 −0.0921551
\(472\) −0.821092 −0.0377938
\(473\) −5.07345 −0.233277
\(474\) 1.82109 0.0836455
\(475\) −8.17891 −0.375274
\(476\) 5.00000 0.229175
\(477\) −3.00000 −0.137361
\(478\) −1.17891 −0.0539220
\(479\) 29.8211 1.36256 0.681280 0.732023i \(-0.261424\pi\)
0.681280 + 0.732023i \(0.261424\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) −18.3578 −0.836176
\(483\) −3.17891 −0.144645
\(484\) 27.1789 1.23540
\(485\) −24.7156 −1.12228
\(486\) 1.00000 0.0453609
\(487\) 38.1789 1.73005 0.865026 0.501727i \(-0.167302\pi\)
0.865026 + 0.501727i \(0.167302\pi\)
\(488\) 3.00000 0.135804
\(489\) −8.82109 −0.398904
\(490\) −2.00000 −0.0903508
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 4.17891 0.188400
\(493\) 40.8945 1.84180
\(494\) 0 0
\(495\) −12.3578 −0.555443
\(496\) 7.17891 0.322343
\(497\) −9.17891 −0.411730
\(498\) 5.17891 0.232072
\(499\) 3.17891 0.142307 0.0711537 0.997465i \(-0.477332\pi\)
0.0711537 + 0.997465i \(0.477332\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) −10.8211 −0.482969
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −32.7156 −1.45583
\(506\) 19.6422 0.873202
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) −28.7156 −1.27280 −0.636399 0.771360i \(-0.719577\pi\)
−0.636399 + 0.771360i \(0.719577\pi\)
\(510\) 10.0000 0.442807
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) −8.17891 −0.361108
\(514\) −25.7156 −1.13427
\(515\) 6.35782 0.280159
\(516\) −0.821092 −0.0361465
\(517\) 25.8211 1.13561
\(518\) 2.00000 0.0878750
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −4.17891 −0.183081 −0.0915406 0.995801i \(-0.529179\pi\)
−0.0915406 + 0.995801i \(0.529179\pi\)
\(522\) −8.17891 −0.357981
\(523\) −22.5367 −0.985462 −0.492731 0.870182i \(-0.664001\pi\)
−0.492731 + 0.870182i \(0.664001\pi\)
\(524\) 19.5367 0.853466
\(525\) 1.00000 0.0436436
\(526\) 12.0000 0.523225
\(527\) 35.8945 1.56359
\(528\) 6.17891 0.268902
\(529\) −12.8945 −0.560632
\(530\) 6.00000 0.260623
\(531\) 0.821092 0.0356323
\(532\) 8.17891 0.354601
\(533\) 0 0
\(534\) −9.00000 −0.389468
\(535\) 15.6422 0.676271
\(536\) −15.1789 −0.655629
\(537\) 14.3578 0.619586
\(538\) −18.3578 −0.791462
\(539\) −6.17891 −0.266144
\(540\) −2.00000 −0.0860663
\(541\) −2.35782 −0.101370 −0.0506852 0.998715i \(-0.516141\pi\)
−0.0506852 + 0.998715i \(0.516141\pi\)
\(542\) 31.1789 1.33925
\(543\) −7.82109 −0.335635
\(544\) −5.00000 −0.214373
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 16.3578 0.698771
\(549\) −3.00000 −0.128037
\(550\) −6.17891 −0.263470
\(551\) 66.8945 2.84980
\(552\) 3.17891 0.135303
\(553\) 1.82109 0.0774407
\(554\) −28.3578 −1.20481
\(555\) 4.00000 0.169791
\(556\) −18.1789 −0.770957
\(557\) 21.7156 0.920121 0.460060 0.887888i \(-0.347828\pi\)
0.460060 + 0.887888i \(0.347828\pi\)
\(558\) −7.17891 −0.303907
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) 30.8945 1.30437
\(562\) 5.64218 0.238001
\(563\) 4.35782 0.183660 0.0918300 0.995775i \(-0.470728\pi\)
0.0918300 + 0.995775i \(0.470728\pi\)
\(564\) 4.17891 0.175964
\(565\) −8.71563 −0.366669
\(566\) −30.3578 −1.27603
\(567\) 1.00000 0.0419961
\(568\) 9.17891 0.385138
\(569\) −26.3578 −1.10498 −0.552489 0.833520i \(-0.686322\pi\)
−0.552489 + 0.833520i \(0.686322\pi\)
\(570\) 16.3578 0.685154
\(571\) −22.8211 −0.955033 −0.477516 0.878623i \(-0.658463\pi\)
−0.477516 + 0.878623i \(0.658463\pi\)
\(572\) 0 0
\(573\) −27.5367 −1.15036
\(574\) 4.17891 0.174424
\(575\) −3.17891 −0.132570
\(576\) 1.00000 0.0416667
\(577\) 5.64218 0.234887 0.117444 0.993080i \(-0.462530\pi\)
0.117444 + 0.993080i \(0.462530\pi\)
\(578\) −8.00000 −0.332756
\(579\) 20.5367 0.853477
\(580\) 16.3578 0.679221
\(581\) 5.17891 0.214857
\(582\) −12.3578 −0.512248
\(583\) 18.5367 0.767713
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 11.1789 0.461403 0.230701 0.973025i \(-0.425898\pi\)
0.230701 + 0.973025i \(0.425898\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 58.7156 2.41934
\(590\) −1.64218 −0.0676076
\(591\) 17.0000 0.699287
\(592\) −2.00000 −0.0821995
\(593\) 33.0000 1.35515 0.677574 0.735455i \(-0.263031\pi\)
0.677574 + 0.735455i \(0.263031\pi\)
\(594\) −6.17891 −0.253524
\(595\) 10.0000 0.409960
\(596\) −4.82109 −0.197480
\(597\) −12.8211 −0.524732
\(598\) 0 0
\(599\) 28.8211 1.17760 0.588799 0.808280i \(-0.299601\pi\)
0.588799 + 0.808280i \(0.299601\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −0.821092 −0.0334652
\(603\) 15.1789 0.618133
\(604\) 14.5367 0.591491
\(605\) 54.3578 2.20996
\(606\) −16.3578 −0.664491
\(607\) 33.5367 1.36121 0.680607 0.732649i \(-0.261716\pi\)
0.680607 + 0.732649i \(0.261716\pi\)
\(608\) −8.17891 −0.331699
\(609\) −8.17891 −0.331426
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) 5.00000 0.202113
\(613\) −38.3578 −1.54926 −0.774629 0.632416i \(-0.782063\pi\)
−0.774629 + 0.632416i \(0.782063\pi\)
\(614\) 24.8945 1.00466
\(615\) 8.35782 0.337020
\(616\) 6.17891 0.248955
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 3.17891 0.127874
\(619\) 8.53673 0.343120 0.171560 0.985174i \(-0.445119\pi\)
0.171560 + 0.985174i \(0.445119\pi\)
\(620\) 14.3578 0.576624
\(621\) −3.17891 −0.127565
\(622\) 8.53673 0.342291
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −4.00000 −0.159872
\(627\) 50.5367 2.01824
\(628\) 2.00000 0.0798087
\(629\) −10.0000 −0.398726
\(630\) −2.00000 −0.0796819
\(631\) 30.8945 1.22989 0.614946 0.788569i \(-0.289178\pi\)
0.614946 + 0.788569i \(0.289178\pi\)
\(632\) −1.82109 −0.0724391
\(633\) −16.3578 −0.650165
\(634\) 7.17891 0.285111
\(635\) 24.0000 0.952411
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) 50.5367 2.00077
\(639\) −9.17891 −0.363112
\(640\) −2.00000 −0.0790569
\(641\) −41.0735 −1.62230 −0.811152 0.584836i \(-0.801159\pi\)
−0.811152 + 0.584836i \(0.801159\pi\)
\(642\) 7.82109 0.308674
\(643\) 20.1789 0.795778 0.397889 0.917433i \(-0.369743\pi\)
0.397889 + 0.917433i \(0.369743\pi\)
\(644\) 3.17891 0.125267
\(645\) −1.64218 −0.0646609
\(646\) −40.8945 −1.60897
\(647\) −8.89454 −0.349681 −0.174840 0.984597i \(-0.555941\pi\)
−0.174840 + 0.984597i \(0.555941\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.07345 −0.199150
\(650\) 0 0
\(651\) −7.17891 −0.281364
\(652\) 8.82109 0.345461
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) −4.00000 −0.156412
\(655\) 39.0735 1.52673
\(656\) −4.17891 −0.163159
\(657\) −4.00000 −0.156055
\(658\) 4.17891 0.162911
\(659\) −16.5367 −0.644179 −0.322090 0.946709i \(-0.604385\pi\)
−0.322090 + 0.946709i \(0.604385\pi\)
\(660\) 12.3578 0.481027
\(661\) 17.5367 0.682100 0.341050 0.940045i \(-0.389218\pi\)
0.341050 + 0.940045i \(0.389218\pi\)
\(662\) −24.3578 −0.946693
\(663\) 0 0
\(664\) −5.17891 −0.200981
\(665\) 16.3578 0.634329
\(666\) 2.00000 0.0774984
\(667\) 26.0000 1.00672
\(668\) 8.00000 0.309529
\(669\) 9.17891 0.354877
\(670\) −30.3578 −1.17282
\(671\) 18.5367 0.715602
\(672\) 1.00000 0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0.536725 0.0206739
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) −4.35782 −0.167361
\(679\) −12.3578 −0.474249
\(680\) −10.0000 −0.383482
\(681\) 8.35782 0.320272
\(682\) 44.3578 1.69855
\(683\) 14.3578 0.549387 0.274693 0.961532i \(-0.411424\pi\)
0.274693 + 0.961532i \(0.411424\pi\)
\(684\) 8.17891 0.312728
\(685\) 32.7156 1.25000
\(686\) −1.00000 −0.0381802
\(687\) 1.00000 0.0381524
\(688\) 0.821092 0.0313038
\(689\) 0 0
\(690\) 6.35782 0.242038
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 4.00000 0.152057
\(693\) −6.17891 −0.234717
\(694\) 7.82109 0.296885
\(695\) −36.3578 −1.37913
\(696\) 8.17891 0.310021
\(697\) −20.8945 −0.791437
\(698\) −3.17891 −0.120323
\(699\) 0.357817 0.0135339
\(700\) −1.00000 −0.0377964
\(701\) 23.7156 0.895727 0.447864 0.894102i \(-0.352185\pi\)
0.447864 + 0.894102i \(0.352185\pi\)
\(702\) 0 0
\(703\) −16.3578 −0.616947
\(704\) −6.17891 −0.232876
\(705\) 8.35782 0.314774
\(706\) −7.17891 −0.270182
\(707\) −16.3578 −0.615199
\(708\) −0.821092 −0.0308585
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 18.3578 0.688957
\(711\) 1.82109 0.0682963
\(712\) 9.00000 0.337289
\(713\) 22.8211 0.854657
\(714\) 5.00000 0.187120
\(715\) 0 0
\(716\) −14.3578 −0.536577
\(717\) −1.17891 −0.0440271
\(718\) 12.3578 0.461190
\(719\) −25.8211 −0.962964 −0.481482 0.876456i \(-0.659901\pi\)
−0.481482 + 0.876456i \(0.659901\pi\)
\(720\) 2.00000 0.0745356
\(721\) 3.17891 0.118389
\(722\) −47.8945 −1.78245
\(723\) −18.3578 −0.682735
\(724\) 7.82109 0.290669
\(725\) −8.17891 −0.303757
\(726\) 27.1789 1.00870
\(727\) 0.463275 0.0171819 0.00859096 0.999963i \(-0.497265\pi\)
0.00859096 + 0.999963i \(0.497265\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.00000 0.296093
\(731\) 4.10546 0.151846
\(732\) 3.00000 0.110883
\(733\) 25.7156 0.949829 0.474914 0.880032i \(-0.342479\pi\)
0.474914 + 0.880032i \(0.342479\pi\)
\(734\) −3.17891 −0.117336
\(735\) −2.00000 −0.0737711
\(736\) −3.17891 −0.117176
\(737\) −93.7891 −3.45477
\(738\) 4.17891 0.153828
\(739\) 4.82109 0.177347 0.0886734 0.996061i \(-0.471737\pi\)
0.0886734 + 0.996061i \(0.471737\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) −43.1789 −1.58408 −0.792040 0.610469i \(-0.790981\pi\)
−0.792040 + 0.610469i \(0.790981\pi\)
\(744\) 7.17891 0.263192
\(745\) −9.64218 −0.353262
\(746\) 20.0000 0.732252
\(747\) 5.17891 0.189486
\(748\) −30.8945 −1.12962
\(749\) 7.82109 0.285776
\(750\) −12.0000 −0.438178
\(751\) 20.8945 0.762453 0.381226 0.924482i \(-0.375502\pi\)
0.381226 + 0.924482i \(0.375502\pi\)
\(752\) −4.17891 −0.152389
\(753\) −10.8211 −0.394343
\(754\) 0 0
\(755\) 29.0735 1.05809
\(756\) −1.00000 −0.0363696
\(757\) 2.35782 0.0856963 0.0428482 0.999082i \(-0.486357\pi\)
0.0428482 + 0.999082i \(0.486357\pi\)
\(758\) 20.7156 0.752426
\(759\) 19.6422 0.712966
\(760\) −16.3578 −0.593360
\(761\) −47.4313 −1.71938 −0.859691 0.510814i \(-0.829344\pi\)
−0.859691 + 0.510814i \(0.829344\pi\)
\(762\) 12.0000 0.434714
\(763\) −4.00000 −0.144810
\(764\) 27.5367 0.996244
\(765\) 10.0000 0.361551
\(766\) 9.82109 0.354850
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 4.35782 0.157147 0.0785734 0.996908i \(-0.474963\pi\)
0.0785734 + 0.996908i \(0.474963\pi\)
\(770\) 12.3578 0.445345
\(771\) −25.7156 −0.926126
\(772\) −20.5367 −0.739133
\(773\) 30.7156 1.10476 0.552382 0.833591i \(-0.313719\pi\)
0.552382 + 0.833591i \(0.313719\pi\)
\(774\) −0.821092 −0.0295135
\(775\) −7.17891 −0.257874
\(776\) 12.3578 0.443620
\(777\) 2.00000 0.0717496
\(778\) 0.821092 0.0294376
\(779\) −34.1789 −1.22459
\(780\) 0 0
\(781\) 56.7156 2.02944
\(782\) −15.8945 −0.568388
\(783\) −8.17891 −0.292290
\(784\) 1.00000 0.0357143
\(785\) 4.00000 0.142766
\(786\) 19.5367 0.696852
\(787\) 22.1789 0.790593 0.395296 0.918554i \(-0.370642\pi\)
0.395296 + 0.918554i \(0.370642\pi\)
\(788\) −17.0000 −0.605600
\(789\) 12.0000 0.427211
\(790\) −3.64218 −0.129583
\(791\) −4.35782 −0.154946
\(792\) 6.17891 0.219558
\(793\) 0 0
\(794\) 3.00000 0.106466
\(795\) 6.00000 0.212798
\(796\) 12.8211 0.454432
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 8.17891 0.289530
\(799\) −20.8945 −0.739196
\(800\) 1.00000 0.0353553
\(801\) −9.00000 −0.317999
\(802\) 32.3578 1.14259
\(803\) 24.7156 0.872196
\(804\) −15.1789 −0.535319
\(805\) 6.35782 0.224084
\(806\) 0 0
\(807\) −18.3578 −0.646226
\(808\) 16.3578 0.575466
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −32.7156 −1.14880 −0.574401 0.818574i \(-0.694765\pi\)
−0.574401 + 0.818574i \(0.694765\pi\)
\(812\) 8.17891 0.287023
\(813\) 31.1789 1.09349
\(814\) −12.3578 −0.433141
\(815\) 17.6422 0.617979
\(816\) −5.00000 −0.175035
\(817\) 6.71563 0.234950
\(818\) −13.6422 −0.476988
\(819\) 0 0
\(820\) −8.35782 −0.291868
\(821\) −45.7156 −1.59549 −0.797743 0.602997i \(-0.793973\pi\)
−0.797743 + 0.602997i \(0.793973\pi\)
\(822\) 16.3578 0.570544
\(823\) −12.7156 −0.443239 −0.221620 0.975133i \(-0.571134\pi\)
−0.221620 + 0.975133i \(0.571134\pi\)
\(824\) −3.17891 −0.110743
\(825\) −6.17891 −0.215122
\(826\) −0.821092 −0.0285694
\(827\) −4.71563 −0.163979 −0.0819893 0.996633i \(-0.526127\pi\)
−0.0819893 + 0.996633i \(0.526127\pi\)
\(828\) 3.17891 0.110475
\(829\) 8.17891 0.284065 0.142033 0.989862i \(-0.454636\pi\)
0.142033 + 0.989862i \(0.454636\pi\)
\(830\) −10.3578 −0.359525
\(831\) −28.3578 −0.983722
\(832\) 0 0
\(833\) 5.00000 0.173240
\(834\) −18.1789 −0.629484
\(835\) 16.0000 0.553703
\(836\) −50.5367 −1.74785
\(837\) −7.17891 −0.248139
\(838\) −29.8945 −1.03269
\(839\) −34.3578 −1.18616 −0.593082 0.805142i \(-0.702089\pi\)
−0.593082 + 0.805142i \(0.702089\pi\)
\(840\) 2.00000 0.0690066
\(841\) 37.8945 1.30671
\(842\) −34.7156 −1.19638
\(843\) 5.64218 0.194327
\(844\) 16.3578 0.563059
\(845\) 0 0
\(846\) 4.17891 0.143674
\(847\) 27.1789 0.933878
\(848\) −3.00000 −0.103020
\(849\) −30.3578 −1.04188
\(850\) 5.00000 0.171499
\(851\) −6.35782 −0.217943
\(852\) 9.17891 0.314464
\(853\) 17.7156 0.606572 0.303286 0.952900i \(-0.401916\pi\)
0.303286 + 0.952900i \(0.401916\pi\)
\(854\) 3.00000 0.102658
\(855\) 16.3578 0.559426
\(856\) −7.82109 −0.267319
\(857\) 34.7156 1.18586 0.592932 0.805253i \(-0.297970\pi\)
0.592932 + 0.805253i \(0.297970\pi\)
\(858\) 0 0
\(859\) −36.8945 −1.25883 −0.629413 0.777071i \(-0.716705\pi\)
−0.629413 + 0.777071i \(0.716705\pi\)
\(860\) 1.64218 0.0559980
\(861\) 4.17891 0.142417
\(862\) −1.17891 −0.0401538
\(863\) −36.3578 −1.23763 −0.618817 0.785535i \(-0.712388\pi\)
−0.618817 + 0.785535i \(0.712388\pi\)
\(864\) 1.00000 0.0340207
\(865\) 8.00000 0.272008
\(866\) −11.6422 −0.395617
\(867\) −8.00000 −0.271694
\(868\) 7.17891 0.243668
\(869\) −11.2524 −0.381710
\(870\) 16.3578 0.554582
\(871\) 0 0
\(872\) 4.00000 0.135457
\(873\) −12.3578 −0.418249
\(874\) −26.0000 −0.879463
\(875\) −12.0000 −0.405674
\(876\) 4.00000 0.135147
\(877\) 17.6422 0.595734 0.297867 0.954607i \(-0.403725\pi\)
0.297867 + 0.954607i \(0.403725\pi\)
\(878\) 4.71563 0.159145
\(879\) 12.0000 0.404750
\(880\) −12.3578 −0.416582
\(881\) 45.5367 1.53417 0.767086 0.641545i \(-0.221706\pi\)
0.767086 + 0.641545i \(0.221706\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −37.5367 −1.26321 −0.631606 0.775290i \(-0.717604\pi\)
−0.631606 + 0.775290i \(0.717604\pi\)
\(884\) 0 0
\(885\) −1.64218 −0.0552014
\(886\) 8.17891 0.274776
\(887\) −11.4633 −0.384899 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 12.0000 0.402467
\(890\) 18.0000 0.603361
\(891\) −6.17891 −0.207001
\(892\) −9.17891 −0.307333
\(893\) −34.1789 −1.14375
\(894\) −4.82109 −0.161241
\(895\) −28.7156 −0.959858
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) 58.7156 1.95828
\(900\) −1.00000 −0.0333333
\(901\) −15.0000 −0.499722
\(902\) −25.8211 −0.859748
\(903\) −0.821092 −0.0273242
\(904\) 4.35782 0.144939
\(905\) 15.6422 0.519964
\(906\) 14.5367 0.482950
\(907\) −52.6102 −1.74689 −0.873446 0.486921i \(-0.838120\pi\)
−0.873446 + 0.486921i \(0.838120\pi\)
\(908\) −8.35782 −0.277364
\(909\) −16.3578 −0.542555
\(910\) 0 0
\(911\) −16.7156 −0.553814 −0.276907 0.960897i \(-0.589309\pi\)
−0.276907 + 0.960897i \(0.589309\pi\)
\(912\) −8.17891 −0.270831
\(913\) −32.0000 −1.05905
\(914\) −31.8945 −1.05498
\(915\) 6.00000 0.198354
\(916\) −1.00000 −0.0330409
\(917\) 19.5367 0.645159
\(918\) 5.00000 0.165025
\(919\) 21.2524 0.701051 0.350525 0.936553i \(-0.386003\pi\)
0.350525 + 0.936553i \(0.386003\pi\)
\(920\) −6.35782 −0.209611
\(921\) 24.8945 0.820303
\(922\) −24.7156 −0.813966
\(923\) 0 0
\(924\) 6.17891 0.203271
\(925\) 2.00000 0.0657596
\(926\) −28.8945 −0.949534
\(927\) 3.17891 0.104409
\(928\) −8.17891 −0.268486
\(929\) −39.0000 −1.27955 −0.639774 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) 14.3578 0.470811
\(931\) 8.17891 0.268053
\(932\) −0.357817 −0.0117207
\(933\) 8.53673 0.279480
\(934\) −33.8945 −1.10906
\(935\) −61.7891 −2.02072
\(936\) 0 0
\(937\) −47.0735 −1.53782 −0.768911 0.639355i \(-0.779201\pi\)
−0.768911 + 0.639355i \(0.779201\pi\)
\(938\) −15.1789 −0.495609
\(939\) −4.00000 −0.130535
\(940\) −8.35782 −0.272602
\(941\) 20.0000 0.651981 0.325991 0.945373i \(-0.394302\pi\)
0.325991 + 0.945373i \(0.394302\pi\)
\(942\) 2.00000 0.0651635
\(943\) −13.2844 −0.432598
\(944\) 0.821092 0.0267243
\(945\) −2.00000 −0.0650600
\(946\) 5.07345 0.164952
\(947\) −39.2524 −1.27553 −0.637765 0.770231i \(-0.720141\pi\)
−0.637765 + 0.770231i \(0.720141\pi\)
\(948\) −1.82109 −0.0591463
\(949\) 0 0
\(950\) 8.17891 0.265359
\(951\) 7.17891 0.232792
\(952\) −5.00000 −0.162051
\(953\) 42.7156 1.38370 0.691848 0.722044i \(-0.256797\pi\)
0.691848 + 0.722044i \(0.256797\pi\)
\(954\) 3.00000 0.0971286
\(955\) 55.0735 1.78213
\(956\) 1.17891 0.0381286
\(957\) 50.5367 1.63362
\(958\) −29.8211 −0.963476
\(959\) 16.3578 0.528221
\(960\) −2.00000 −0.0645497
\(961\) 20.5367 0.662475
\(962\) 0 0
\(963\) 7.82109 0.252031
\(964\) 18.3578 0.591265
\(965\) −41.0735 −1.32220
\(966\) 3.17891 0.102280
\(967\) −34.3578 −1.10487 −0.552436 0.833555i \(-0.686302\pi\)
−0.552436 + 0.833555i \(0.686302\pi\)
\(968\) −27.1789 −0.873563
\(969\) −40.8945 −1.31372
\(970\) 24.7156 0.793571
\(971\) −16.8211 −0.539815 −0.269907 0.962886i \(-0.586993\pi\)
−0.269907 + 0.962886i \(0.586993\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −18.1789 −0.582789
\(974\) −38.1789 −1.22333
\(975\) 0 0
\(976\) −3.00000 −0.0960277
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 8.82109 0.282067
\(979\) 55.6102 1.77731
\(980\) 2.00000 0.0638877
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −55.0735 −1.75657 −0.878285 0.478137i \(-0.841312\pi\)
−0.878285 + 0.478137i \(0.841312\pi\)
\(984\) −4.17891 −0.133219
\(985\) −34.0000 −1.08333
\(986\) −40.8945 −1.30235
\(987\) 4.17891 0.133016
\(988\) 0 0
\(989\) 2.61018 0.0829987
\(990\) 12.3578 0.392757
\(991\) −15.4633 −0.491207 −0.245604 0.969370i \(-0.578986\pi\)
−0.245604 + 0.969370i \(0.578986\pi\)
\(992\) −7.17891 −0.227931
\(993\) −24.3578 −0.772972
\(994\) 9.17891 0.291137
\(995\) 25.6422 0.812912
\(996\) −5.17891 −0.164100
\(997\) 57.7156 1.82787 0.913936 0.405858i \(-0.133027\pi\)
0.913936 + 0.405858i \(0.133027\pi\)
\(998\) −3.17891 −0.100627
\(999\) 2.00000 0.0632772
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 7098.2.a.bk.1.1 2
13.4 even 6 546.2.l.k.211.1 4
13.10 even 6 546.2.l.k.295.1 yes 4
13.12 even 2 7098.2.a.br.1.2 2
39.17 odd 6 1638.2.r.ba.757.2 4
39.23 odd 6 1638.2.r.ba.1387.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.k.211.1 4 13.4 even 6
546.2.l.k.295.1 yes 4 13.10 even 6
1638.2.r.ba.757.2 4 39.17 odd 6
1638.2.r.ba.1387.2 4 39.23 odd 6
7098.2.a.bk.1.1 2 1.1 even 1 trivial
7098.2.a.br.1.2 2 13.12 even 2