Properties

Label 7098.2.a.bk.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(6.17891\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +5.17891 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -1.00000 q^{18} -3.17891 q^{19} +2.00000 q^{20} -1.00000 q^{21} -5.17891 q^{22} -8.17891 q^{23} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.17891 q^{29} +2.00000 q^{30} -4.17891 q^{31} -1.00000 q^{32} -5.17891 q^{33} -5.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +3.17891 q^{38} -2.00000 q^{40} +7.17891 q^{41} +1.00000 q^{42} +12.1789 q^{43} +5.17891 q^{44} +2.00000 q^{45} +8.17891 q^{46} +7.17891 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -5.00000 q^{51} -3.00000 q^{53} +1.00000 q^{54} +10.3578 q^{55} -1.00000 q^{56} +3.17891 q^{57} +3.17891 q^{58} +12.1789 q^{59} -2.00000 q^{60} -3.00000 q^{61} +4.17891 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.17891 q^{66} +3.82109 q^{67} +5.00000 q^{68} +8.17891 q^{69} -2.00000 q^{70} +2.17891 q^{71} -1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -3.17891 q^{76} +5.17891 q^{77} +13.1789 q^{79} +2.00000 q^{80} +1.00000 q^{81} -7.17891 q^{82} -6.17891 q^{83} -1.00000 q^{84} +10.0000 q^{85} -12.1789 q^{86} +3.17891 q^{87} -5.17891 q^{88} -9.00000 q^{89} -2.00000 q^{90} -8.17891 q^{92} +4.17891 q^{93} -7.17891 q^{94} -6.35782 q^{95} +1.00000 q^{96} +10.3578 q^{97} -1.00000 q^{98} +5.17891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 4q^{10} - q^{11} - 2q^{12} - 2q^{14} - 4q^{15} + 2q^{16} + 10q^{17} - 2q^{18} + 5q^{19} + 4q^{20} - 2q^{21} + q^{22} - 5q^{23} + 2q^{24} - 2q^{25} - 2q^{27} + 2q^{28} + 5q^{29} + 4q^{30} + 3q^{31} - 2q^{32} + q^{33} - 10q^{34} + 4q^{35} + 2q^{36} - 4q^{37} - 5q^{38} - 4q^{40} + 3q^{41} + 2q^{42} + 13q^{43} - q^{44} + 4q^{45} + 5q^{46} + 3q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 10q^{51} - 6q^{53} + 2q^{54} - 2q^{55} - 2q^{56} - 5q^{57} - 5q^{58} + 13q^{59} - 4q^{60} - 6q^{61} - 3q^{62} + 2q^{63} + 2q^{64} - q^{66} + 19q^{67} + 10q^{68} + 5q^{69} - 4q^{70} - 7q^{71} - 2q^{72} - 8q^{73} + 4q^{74} + 2q^{75} + 5q^{76} - q^{77} + 15q^{79} + 4q^{80} + 2q^{81} - 3q^{82} - q^{83} - 2q^{84} + 20q^{85} - 13q^{86} - 5q^{87} + q^{88} - 18q^{89} - 4q^{90} - 5q^{92} - 3q^{93} - 3q^{94} + 10q^{95} + 2q^{96} - 2q^{97} - 2q^{98} - q^{99} + O(q^{100}) \)

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
\(11\) 5.17891 1.56150 0.780750 0.624844i \(-0.214837\pi\)
0.780750 + 0.624844i \(0.214837\pi\)
\(12\) −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
\(19\) −3.17891 −0.729292 −0.364646 0.931146i \(-0.618810\pi\)
−0.364646 + 0.931146i \(0.618810\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) −5.17891 −1.10415
\(23\) −8.17891 −1.70542 −0.852710 0.522384i \(-0.825043\pi\)
−0.852710 + 0.522384i \(0.825043\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −3.17891 −0.590308 −0.295154 0.955450i \(-0.595371\pi\)
−0.295154 + 0.955450i \(0.595371\pi\)
\(30\) 2.00000 0.365148
\(31\) −4.17891 −0.750554 −0.375277 0.926913i \(-0.622452\pi\)
−0.375277 + 0.926913i \(0.622452\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.17891 −0.901532
\(34\) −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 \(0.552568\pi\)
\(38\) 3.17891 0.515687
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 7.17891 1.12116 0.560579 0.828101i \(-0.310579\pi\)
0.560579 + 0.828101i \(0.310579\pi\)
\(42\) 1.00000 0.154303
\(43\) 12.1789 1.85727 0.928633 0.371000i \(-0.120985\pi\)
0.928633 + 0.371000i \(0.120985\pi\)
\(44\) 5.17891 0.780750
\(45\) 2.00000 0.298142
\(46\) 8.17891 1.20591
\(47\) 7.17891 1.04715 0.523576 0.851979i \(-0.324598\pi\)
0.523576 + 0.851979i \(0.324598\pi\)
\(48\) −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
\(55\) 10.3578 1.39665
\(56\) −1.00000 −0.133631
\(57\) 3.17891 0.421057
\(58\) 3.17891 0.417411
\(59\) 12.1789 1.58556 0.792779 0.609509i \(-0.208633\pi\)
0.792779 + 0.609509i \(0.208633\pi\)
\(60\) −2.00000 −0.258199
\(61\) −3.00000 −0.384111 −0.192055 0.981384i \(-0.561515\pi\)
−0.192055 + 0.981384i \(0.561515\pi\)
\(62\) 4.17891 0.530722
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.17891 0.637480
\(67\) 3.82109 0.466821 0.233410 0.972378i \(-0.425011\pi\)
0.233410 + 0.972378i \(0.425011\pi\)
\(68\) 5.00000 0.606339
\(69\) 8.17891 0.984625
\(70\) −2.00000 −0.239046
\(71\) 2.17891 0.258589 0.129294 0.991606i \(-0.458729\pi\)
0.129294 + 0.991606i \(0.458729\pi\)
\(72\) −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
\(76\) −3.17891 −0.364646
\(77\) 5.17891 0.590191
\(78\) 0 0
\(79\) 13.1789 1.48274 0.741372 0.671095i \(-0.234176\pi\)
0.741372 + 0.671095i \(0.234176\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −7.17891 −0.792778
\(83\) −6.17891 −0.678223 −0.339112 0.940746i \(-0.610126\pi\)
−0.339112 + 0.940746i \(0.610126\pi\)
\(84\) −1.00000 −0.109109
\(85\) 10.0000 1.08465
\(86\) −12.1789 −1.31329
\(87\) 3.17891 0.340815
\(88\) −5.17891 −0.552073
\(89\) −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
\(92\) −8.17891 −0.852710
\(93\) 4.17891 0.433333
\(94\) −7.17891 −0.740448
\(95\) −6.35782 −0.652298
\(96\) 1.00000 0.102062
\(97\) 10.3578 1.05168 0.525838 0.850584i \(-0.323752\pi\)
0.525838 + 0.850584i \(0.323752\pi\)
\(98\) −1.00000 −0.101015
\(99\) 5.17891 0.520500
\(100\) −1.00000 −0.100000
\(101\) 6.35782 0.632626 0.316313 0.948655i \(-0.397555\pi\)
0.316313 + 0.948655i \(0.397555\pi\)
\(102\) 5.00000 0.495074
\(103\) −8.17891 −0.805892 −0.402946 0.915224i \(-0.632014\pi\)
−0.402946 + 0.915224i \(0.632014\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 3.00000 0.291386
\(107\) 19.1789 1.85410 0.927048 0.374944i \(-0.122338\pi\)
0.927048 + 0.374944i \(0.122338\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −10.3578 −0.987579
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) 18.3578 1.72696 0.863479 0.504385i \(-0.168281\pi\)
0.863479 + 0.504385i \(0.168281\pi\)
\(114\) −3.17891 −0.297732
\(115\) −16.3578 −1.52537
\(116\) −3.17891 −0.295154
\(117\) 0 0
\(118\) −12.1789 −1.12116
\(119\) 5.00000 0.458349
\(120\) 2.00000 0.182574
\(121\) 15.8211 1.43828
\(122\) 3.00000 0.271607
\(123\) −7.17891 −0.647300
\(124\) −4.17891 −0.375277
\(125\) −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
\(129\) −12.1789 −1.07229
\(130\) 0 0
\(131\) −14.5367 −1.27008 −0.635040 0.772479i \(-0.719016\pi\)
−0.635040 + 0.772479i \(0.719016\pi\)
\(132\) −5.17891 −0.450766
\(133\) −3.17891 −0.275646
\(134\) −3.82109 −0.330092
\(135\) −2.00000 −0.172133
\(136\) −5.00000 −0.428746
\(137\) −6.35782 −0.543185 −0.271592 0.962412i \(-0.587550\pi\)
−0.271592 + 0.962412i \(0.587550\pi\)
\(138\) −8.17891 −0.696235
\(139\) −6.82109 −0.578557 −0.289279 0.957245i \(-0.593415\pi\)
−0.289279 + 0.957245i \(0.593415\pi\)
\(140\) 2.00000 0.169031
\(141\) −7.17891 −0.604573
\(142\) −2.17891 −0.182850
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.35782 −0.527988
\(146\) 4.00000 0.331042
\(147\) −1.00000 −0.0824786
\(148\) −2.00000 −0.164399
\(149\) −16.1789 −1.32543 −0.662714 0.748873i \(-0.730595\pi\)
−0.662714 + 0.748873i \(0.730595\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −19.5367 −1.58988 −0.794938 0.606691i \(-0.792497\pi\)
−0.794938 + 0.606691i \(0.792497\pi\)
\(152\) 3.17891 0.257844
\(153\) 5.00000 0.404226
\(154\) −5.17891 −0.417328
\(155\) −8.35782 −0.671316
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −13.1789 −1.04846
\(159\) 3.00000 0.237915
\(160\) −2.00000 −0.158114
\(161\) −8.17891 −0.644588
\(162\) −1.00000 −0.0785674
\(163\) 20.1789 1.58053 0.790267 0.612763i \(-0.209942\pi\)
0.790267 + 0.612763i \(0.209942\pi\)
\(164\) 7.17891 0.560579
\(165\) −10.3578 −0.806355
\(166\) 6.17891 0.479576
\(167\) 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
\(171\) −3.17891 −0.243097
\(172\) 12.1789 0.928633
\(173\) 4.00000 0.304114 0.152057 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) −3.17891 −0.240992
\(175\) −1.00000 −0.0755929
\(176\) 5.17891 0.390375
\(177\) −12.1789 −0.915423
\(178\) 9.00000 0.674579
\(179\) 8.35782 0.624693 0.312346 0.949968i \(-0.398885\pi\)
0.312346 + 0.949968i \(0.398885\pi\)
\(180\) 2.00000 0.149071
\(181\) 19.1789 1.42556 0.712779 0.701389i \(-0.247436\pi\)
0.712779 + 0.701389i \(0.247436\pi\)
\(182\) 0 0
\(183\) 3.00000 0.221766
\(184\) 8.17891 0.602957
\(185\) −4.00000 −0.294086
\(186\) −4.17891 −0.306412
\(187\) 25.8945 1.89360
\(188\) 7.17891 0.523576
\(189\) −1.00000 −0.0727393
\(190\) 6.35782 0.461245
\(191\) −6.53673 −0.472981 −0.236490 0.971634i \(-0.575997\pi\)
−0.236490 + 0.971634i \(0.575997\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 13.5367 0.974395 0.487197 0.873292i \(-0.338019\pi\)
0.487197 + 0.873292i \(0.338019\pi\)
\(194\) −10.3578 −0.743648
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −17.0000 −1.21120 −0.605600 0.795769i \(-0.707067\pi\)
−0.605600 + 0.795769i \(0.707067\pi\)
\(198\) −5.17891 −0.368049
\(199\) 24.1789 1.71400 0.856999 0.515319i \(-0.172326\pi\)
0.856999 + 0.515319i \(0.172326\pi\)
\(200\) 1.00000 0.0707107
\(201\) −3.82109 −0.269519
\(202\) −6.35782 −0.447334
\(203\) −3.17891 −0.223116
\(204\) −5.00000 −0.350070
\(205\) 14.3578 1.00279
\(206\) 8.17891 0.569852
\(207\) −8.17891 −0.568473
\(208\) 0 0
\(209\) −16.4633 −1.13879
\(210\) 2.00000 0.138013
\(211\) −6.35782 −0.437690 −0.218845 0.975760i \(-0.570229\pi\)
−0.218845 + 0.975760i \(0.570229\pi\)
\(212\) −3.00000 −0.206041
\(213\) −2.17891 −0.149296
\(214\) −19.1789 −1.31104
\(215\) 24.3578 1.66119
\(216\) 1.00000 0.0680414
\(217\) −4.17891 −0.283683
\(218\) 4.00000 0.270914
\(219\) 4.00000 0.270295
\(220\) 10.3578 0.698324
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) 2.17891 0.145910 0.0729552 0.997335i \(-0.476757\pi\)
0.0729552 + 0.997335i \(0.476757\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −18.3578 −1.22114
\(227\) 14.3578 0.952962 0.476481 0.879185i \(-0.341912\pi\)
0.476481 + 0.879185i \(0.341912\pi\)
\(228\) 3.17891 0.210528
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 16.3578 1.07860
\(231\) −5.17891 −0.340747
\(232\) 3.17891 0.208706
\(233\) 22.3578 1.46471 0.732355 0.680923i \(-0.238421\pi\)
0.732355 + 0.680923i \(0.238421\pi\)
\(234\) 0 0
\(235\) 14.3578 0.936601
\(236\) 12.1789 0.792779
\(237\) −13.1789 −0.856062
\(238\) −5.00000 −0.324102
\(239\) −10.1789 −0.658419 −0.329209 0.944257i \(-0.606782\pi\)
−0.329209 + 0.944257i \(0.606782\pi\)
\(240\) −2.00000 −0.129099
\(241\) −4.35782 −0.280712 −0.140356 0.990101i \(-0.544825\pi\)
−0.140356 + 0.990101i \(0.544825\pi\)
\(242\) −15.8211 −1.01702
\(243\) −1.00000 −0.0641500
\(244\) −3.00000 −0.192055
\(245\) 2.00000 0.127775
\(246\) 7.17891 0.457710
\(247\) 0 0
\(248\) 4.17891 0.265361
\(249\) 6.17891 0.391572
\(250\) 12.0000 0.758947
\(251\) 22.1789 1.39992 0.699960 0.714182i \(-0.253201\pi\)
0.699960 + 0.714182i \(0.253201\pi\)
\(252\) 1.00000 0.0629941
\(253\) −42.3578 −2.66301
\(254\) −12.0000 −0.752947
\(255\) −10.0000 −0.626224
\(256\) 1.00000 0.0625000
\(257\) −19.7156 −1.22983 −0.614914 0.788594i \(-0.710809\pi\)
−0.614914 + 0.788594i \(0.710809\pi\)
\(258\) 12.1789 0.758226
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −3.17891 −0.196769
\(262\) 14.5367 0.898082
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 5.17891 0.318740
\(265\) −6.00000 −0.368577
\(266\) 3.17891 0.194911
\(267\) 9.00000 0.550791
\(268\) 3.82109 0.233410
\(269\) −4.35782 −0.265701 −0.132850 0.991136i \(-0.542413\pi\)
−0.132850 + 0.991136i \(0.542413\pi\)
\(270\) 2.00000 0.121716
\(271\) −19.8211 −1.20405 −0.602023 0.798479i \(-0.705638\pi\)
−0.602023 + 0.798479i \(0.705638\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) 6.35782 0.384090
\(275\) −5.17891 −0.312300
\(276\) 8.17891 0.492312
\(277\) 5.64218 0.339006 0.169503 0.985530i \(-0.445784\pi\)
0.169503 + 0.985530i \(0.445784\pi\)
\(278\) 6.82109 0.409102
\(279\) −4.17891 −0.250185
\(280\) −2.00000 −0.119523
\(281\) −28.3578 −1.69169 −0.845843 0.533432i \(-0.820902\pi\)
−0.845843 + 0.533432i \(0.820902\pi\)
\(282\) 7.17891 0.427498
\(283\) 7.64218 0.454281 0.227140 0.973862i \(-0.427062\pi\)
0.227140 + 0.973862i \(0.427062\pi\)
\(284\) 2.17891 0.129294
\(285\) 6.35782 0.376605
\(286\) 0 0
\(287\) 7.17891 0.423758
\(288\) −1.00000 −0.0589256
\(289\) 8.00000 0.470588
\(290\) 6.35782 0.373344
\(291\) −10.3578 −0.607186
\(292\) −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
\(295\) 24.3578 1.41817
\(296\) 2.00000 0.116248
\(297\) −5.17891 −0.300511
\(298\) 16.1789 0.937219
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 12.1789 0.701981
\(302\) 19.5367 1.12421
\(303\) −6.35782 −0.365247
\(304\) −3.17891 −0.182323
\(305\) −6.00000 −0.343559
\(306\) −5.00000 −0.285831
\(307\) 31.8945 1.82032 0.910159 0.414259i \(-0.135959\pi\)
0.910159 + 0.414259i \(0.135959\pi\)
\(308\) 5.17891 0.295096
\(309\) 8.17891 0.465282
\(310\) 8.35782 0.474692
\(311\) 25.5367 1.44805 0.724027 0.689771i \(-0.242289\pi\)
0.724027 + 0.689771i \(0.242289\pi\)
\(312\) 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
\(316\) 13.1789 0.741372
\(317\) 4.17891 0.234711 0.117355 0.993090i \(-0.462558\pi\)
0.117355 + 0.993090i \(0.462558\pi\)
\(318\) −3.00000 −0.168232
\(319\) −16.4633 −0.921766
\(320\) 2.00000 0.111803
\(321\) −19.1789 −1.07046
\(322\) 8.17891 0.455793
\(323\) −15.8945 −0.884396
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.1789 −1.11761
\(327\) 4.00000 0.221201
\(328\) −7.17891 −0.396389
\(329\) 7.17891 0.395786
\(330\) 10.3578 0.570179
\(331\) 1.64218 0.0902626 0.0451313 0.998981i \(-0.485629\pi\)
0.0451313 + 0.998981i \(0.485629\pi\)
\(332\) −6.17891 −0.339112
\(333\) −2.00000 −0.109599
\(334\) −8.00000 −0.437741
\(335\) 7.64218 0.417537
\(336\) −1.00000 −0.0545545
\(337\) 33.5367 1.82686 0.913431 0.406994i \(-0.133423\pi\)
0.913431 + 0.406994i \(0.133423\pi\)
\(338\) 0 0
\(339\) −18.3578 −0.997060
\(340\) 10.0000 0.542326
\(341\) −21.6422 −1.17199
\(342\) 3.17891 0.171896
\(343\) 1.00000 0.0539949
\(344\) −12.1789 −0.656643
\(345\) 16.3578 0.880675
\(346\) −4.00000 −0.215041
\(347\) −19.1789 −1.02958 −0.514789 0.857317i \(-0.672130\pi\)
−0.514789 + 0.857317i \(0.672130\pi\)
\(348\) 3.17891 0.170407
\(349\) −8.17891 −0.437807 −0.218903 0.975747i \(-0.570248\pi\)
−0.218903 + 0.975747i \(0.570248\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −5.17891 −0.276037
\(353\) −4.17891 −0.222421 −0.111210 0.993797i \(-0.535473\pi\)
−0.111210 + 0.993797i \(0.535473\pi\)
\(354\) 12.1789 0.647302
\(355\) 4.35782 0.231289
\(356\) −9.00000 −0.476999
\(357\) −5.00000 −0.264628
\(358\) −8.35782 −0.441724
\(359\) 10.3578 0.546665 0.273332 0.961920i \(-0.411874\pi\)
0.273332 + 0.961920i \(0.411874\pi\)
\(360\) −2.00000 −0.105409
\(361\) −8.89454 −0.468134
\(362\) −19.1789 −1.00802
\(363\) −15.8211 −0.830392
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) −3.00000 −0.156813
\(367\) −8.17891 −0.426936 −0.213468 0.976950i \(-0.568476\pi\)
−0.213468 + 0.976950i \(0.568476\pi\)
\(368\) −8.17891 −0.426355
\(369\) 7.17891 0.373719
\(370\) 4.00000 0.207950
\(371\) −3.00000 −0.155752
\(372\) 4.17891 0.216666
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) −25.8945 −1.33897
\(375\) 12.0000 0.619677
\(376\) −7.17891 −0.370224
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 24.7156 1.26956 0.634778 0.772694i \(-0.281091\pi\)
0.634778 + 0.772694i \(0.281091\pi\)
\(380\) −6.35782 −0.326149
\(381\) −12.0000 −0.614779
\(382\) 6.53673 0.334448
\(383\) −21.1789 −1.08219 −0.541096 0.840961i \(-0.681990\pi\)
−0.541096 + 0.840961i \(0.681990\pi\)
\(384\) 1.00000 0.0510310
\(385\) 10.3578 0.527883
\(386\) −13.5367 −0.689001
\(387\) 12.1789 0.619089
\(388\) 10.3578 0.525838
\(389\) −12.1789 −0.617495 −0.308748 0.951144i \(-0.599910\pi\)
−0.308748 + 0.951144i \(0.599910\pi\)
\(390\) 0 0
\(391\) −40.8945 −2.06813
\(392\) −1.00000 −0.0505076
\(393\) 14.5367 0.733281
\(394\) 17.0000 0.856448
\(395\) 26.3578 1.32621
\(396\) 5.17891 0.260250
\(397\) −3.00000 −0.150566 −0.0752828 0.997162i \(-0.523986\pi\)
−0.0752828 + 0.997162i \(0.523986\pi\)
\(398\) −24.1789 −1.21198
\(399\) 3.17891 0.159144
\(400\) −1.00000 −0.0500000
\(401\) −9.64218 −0.481508 −0.240754 0.970586i \(-0.577395\pi\)
−0.240754 + 0.970586i \(0.577395\pi\)
\(402\) 3.82109 0.190579
\(403\) 0 0
\(404\) 6.35782 0.316313
\(405\) 2.00000 0.0993808
\(406\) 3.17891 0.157767
\(407\) −10.3578 −0.513418
\(408\) 5.00000 0.247537
\(409\) 36.3578 1.79778 0.898889 0.438176i \(-0.144375\pi\)
0.898889 + 0.438176i \(0.144375\pi\)
\(410\) −14.3578 −0.709082
\(411\) 6.35782 0.313608
\(412\) −8.17891 −0.402946
\(413\) 12.1789 0.599285
\(414\) 8.17891 0.401971
\(415\) −12.3578 −0.606621
\(416\) 0 0
\(417\) 6.82109 0.334030
\(418\) 16.4633 0.805245
\(419\) −26.8945 −1.31388 −0.656942 0.753941i \(-0.728150\pi\)
−0.656942 + 0.753941i \(0.728150\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −10.7156 −0.522248 −0.261124 0.965305i \(-0.584093\pi\)
−0.261124 + 0.965305i \(0.584093\pi\)
\(422\) 6.35782 0.309494
\(423\) 7.17891 0.349050
\(424\) 3.00000 0.145693
\(425\) −5.00000 −0.242536
\(426\) 2.17891 0.105568
\(427\) −3.00000 −0.145180
\(428\) 19.1789 0.927048
\(429\) 0 0
\(430\) −24.3578 −1.17464
\(431\) −10.1789 −0.490301 −0.245150 0.969485i \(-0.578837\pi\)
−0.245150 + 0.969485i \(0.578837\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.3578 1.65113 0.825566 0.564306i \(-0.190856\pi\)
0.825566 + 0.564306i \(0.190856\pi\)
\(434\) 4.17891 0.200594
\(435\) 6.35782 0.304834
\(436\) −4.00000 −0.191565
\(437\) 26.0000 1.24375
\(438\) −4.00000 −0.191127
\(439\) 40.7156 1.94325 0.971626 0.236524i \(-0.0760083\pi\)
0.971626 + 0.236524i \(0.0760083\pi\)
\(440\) −10.3578 −0.493790
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 3.17891 0.151034 0.0755172 0.997144i \(-0.475939\pi\)
0.0755172 + 0.997144i \(0.475939\pi\)
\(444\) 2.00000 0.0949158
\(445\) −18.0000 −0.853282
\(446\) −2.17891 −0.103174
\(447\) 16.1789 0.765236
\(448\) 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
\(451\) 37.1789 1.75069
\(452\) 18.3578 0.863479
\(453\) 19.5367 0.917915
\(454\) −14.3578 −0.673846
\(455\) 0 0
\(456\) −3.17891 −0.148866
\(457\) −24.8945 −1.16452 −0.582259 0.813004i \(-0.697831\pi\)
−0.582259 + 0.813004i \(0.697831\pi\)
\(458\) 1.00000 0.0467269
\(459\) −5.00000 −0.233380
\(460\) −16.3578 −0.762687
\(461\) −20.7156 −0.964823 −0.482412 0.875945i \(-0.660239\pi\)
−0.482412 + 0.875945i \(0.660239\pi\)
\(462\) 5.17891 0.240945
\(463\) −27.8945 −1.29637 −0.648185 0.761483i \(-0.724472\pi\)
−0.648185 + 0.761483i \(0.724472\pi\)
\(464\) −3.17891 −0.147577
\(465\) 8.35782 0.387584
\(466\) −22.3578 −1.03571
\(467\) −22.8945 −1.05943 −0.529717 0.848175i \(-0.677702\pi\)
−0.529717 + 0.848175i \(0.677702\pi\)
\(468\) 0 0
\(469\) 3.82109 0.176442
\(470\) −14.3578 −0.662277
\(471\) −2.00000 −0.0921551
\(472\) −12.1789 −0.560580
\(473\) 63.0735 2.90012
\(474\) 13.1789 0.605327
\(475\) 3.17891 0.145858
\(476\) 5.00000 0.229175
\(477\) −3.00000 −0.137361
\(478\) 10.1789 0.465572
\(479\) 41.1789 1.88151 0.940756 0.339084i \(-0.110117\pi\)
0.940756 + 0.339084i \(0.110117\pi\)
\(480\) 2.00000 0.0912871
\(481\) 0 0
\(482\) 4.35782 0.198493
\(483\) 8.17891 0.372153
\(484\) 15.8211 0.719141
\(485\) 20.7156 0.940648
\(486\) 1.00000 0.0453609
\(487\) 26.8211 1.21538 0.607690 0.794174i \(-0.292096\pi\)
0.607690 + 0.794174i \(0.292096\pi\)
\(488\) 3.00000 0.135804
\(489\) −20.1789 −0.912522
\(490\) −2.00000 −0.0903508
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) −7.17891 −0.323650
\(493\) −15.8945 −0.715854
\(494\) 0 0
\(495\) 10.3578 0.465549
\(496\) −4.17891 −0.187639
\(497\) 2.17891 0.0977374
\(498\) −6.17891 −0.276884
\(499\) −8.17891 −0.366138 −0.183069 0.983100i \(-0.558603\pi\)
−0.183069 + 0.983100i \(0.558603\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) −22.1789 −0.989893
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 12.7156 0.565838
\(506\) 42.3578 1.88303
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) 16.7156 0.740907 0.370454 0.928851i \(-0.379202\pi\)
0.370454 + 0.928851i \(0.379202\pi\)
\(510\) 10.0000 0.442807
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) 3.17891 0.140352
\(514\) 19.7156 0.869619
\(515\) −16.3578 −0.720812
\(516\) −12.1789 −0.536147
\(517\) 37.1789 1.63513
\(518\) 2.00000 0.0878750
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) 7.17891 0.314514 0.157257 0.987558i \(-0.449735\pi\)
0.157257 + 0.987558i \(0.449735\pi\)
\(522\) 3.17891 0.139137
\(523\) 11.5367 0.504466 0.252233 0.967667i \(-0.418835\pi\)
0.252233 + 0.967667i \(0.418835\pi\)
\(524\) −14.5367 −0.635040
\(525\) 1.00000 0.0436436
\(526\) 12.0000 0.523225
\(527\) −20.8945 −0.910181
\(528\) −5.17891 −0.225383
\(529\) 43.8945 1.90846
\(530\) 6.00000 0.260623
\(531\) 12.1789 0.528520
\(532\) −3.17891 −0.137823
\(533\) 0 0
\(534\) −9.00000 −0.389468
\(535\) 38.3578 1.65835
\(536\) −3.82109 −0.165046
\(537\) −8.35782 −0.360666
\(538\) 4.35782 0.187879
\(539\) 5.17891 0.223071
\(540\) −2.00000 −0.0860663
\(541\) 20.3578 0.875251 0.437625 0.899157i \(-0.355820\pi\)
0.437625 + 0.899157i \(0.355820\pi\)
\(542\) 19.8211 0.851389
\(543\) −19.1789 −0.823046
\(544\) −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 \(0.220443\pi\)
\(548\) −6.35782 −0.271592
\(549\) −3.00000 −0.128037
\(550\) 5.17891 0.220829
\(551\) 10.1055 0.430507
\(552\) −8.17891 −0.348117
\(553\) 13.1789 0.560424
\(554\) −5.64218 −0.239713
\(555\) 4.00000 0.169791
\(556\) −6.82109 −0.289279
\(557\) −23.7156 −1.00486 −0.502432 0.864617i \(-0.667561\pi\)
−0.502432 + 0.864617i \(0.667561\pi\)
\(558\) 4.17891 0.176907
\(559\) 0 0
\(560\) 2.00000 0.0845154
\(561\) −25.8945 −1.09327
\(562\) 28.3578 1.19620
\(563\) −18.3578 −0.773690 −0.386845 0.922145i \(-0.626435\pi\)
−0.386845 + 0.922145i \(0.626435\pi\)
\(564\) −7.17891 −0.302287
\(565\) 36.7156 1.54464
\(566\) −7.64218 −0.321225
\(567\) 1.00000 0.0419961
\(568\) −2.17891 −0.0914250
\(569\) −3.64218 −0.152688 −0.0763441 0.997082i \(-0.524325\pi\)
−0.0763441 + 0.997082i \(0.524325\pi\)
\(570\) −6.35782 −0.266300
\(571\) −34.1789 −1.43034 −0.715171 0.698949i \(-0.753651\pi\)
−0.715171 + 0.698949i \(0.753651\pi\)
\(572\) 0 0
\(573\) 6.53673 0.273076
\(574\) −7.17891 −0.299642
\(575\) 8.17891 0.341084
\(576\) 1.00000 0.0416667
\(577\) 28.3578 1.18055 0.590276 0.807202i \(-0.299019\pi\)
0.590276 + 0.807202i \(0.299019\pi\)
\(578\) −8.00000 −0.332756
\(579\) −13.5367 −0.562567
\(580\) −6.35782 −0.263994
\(581\) −6.17891 −0.256344
\(582\) 10.3578 0.429345
\(583\) −15.5367 −0.643465
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) −0.178908 −0.00738434 −0.00369217 0.999993i \(-0.501175\pi\)
−0.00369217 + 0.999993i \(0.501175\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 13.2844 0.547373
\(590\) −24.3578 −1.00280
\(591\) 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 \(0.263031\pi\)
\(594\) 5.17891 0.212493
\(595\) 10.0000 0.409960
\(596\) −16.1789 −0.662714
\(597\) −24.1789 −0.989577
\(598\) 0 0
\(599\) 40.1789 1.64167 0.820833 0.571168i \(-0.193510\pi\)
0.820833 + 0.571168i \(0.193510\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −12.1789 −0.496375
\(603\) 3.82109 0.155607
\(604\) −19.5367 −0.794938
\(605\) 31.6422 1.28644
\(606\) 6.35782 0.258269
\(607\) −0.536725 −0.0217850 −0.0108925 0.999941i \(-0.503467\pi\)
−0.0108925 + 0.999941i \(0.503467\pi\)
\(608\) 3.17891 0.128922
\(609\) 3.17891 0.128816
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) 5.00000 0.202113
\(613\) −15.6422 −0.631782 −0.315891 0.948796i \(-0.602303\pi\)
−0.315891 + 0.948796i \(0.602303\pi\)
\(614\) −31.8945 −1.28716
\(615\) −14.3578 −0.578963
\(616\) −5.17891 −0.208664
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) −8.17891 −0.329004
\(619\) −25.5367 −1.02641 −0.513204 0.858267i \(-0.671542\pi\)
−0.513204 + 0.858267i \(0.671542\pi\)
\(620\) −8.35782 −0.335658
\(621\) 8.17891 0.328208
\(622\) −25.5367 −1.02393
\(623\) −9.00000 −0.360577
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −4.00000 −0.159872
\(627\) 16.4633 0.657480
\(628\) 2.00000 0.0798087
\(629\) −10.0000 −0.398726
\(630\) −2.00000 −0.0796819
\(631\) −25.8945 −1.03085 −0.515423 0.856936i \(-0.672365\pi\)
−0.515423 + 0.856936i \(0.672365\pi\)
\(632\) −13.1789 −0.524229
\(633\) 6.35782 0.252701
\(634\) −4.17891 −0.165966
\(635\) 24.0000 0.952411
\(636\) 3.00000 0.118958
\(637\) 0 0
\(638\) 16.4633 0.651787
\(639\) 2.17891 0.0861963
\(640\) −2.00000 −0.0790569
\(641\) 27.0735 1.06934 0.534668 0.845062i \(-0.320436\pi\)
0.534668 + 0.845062i \(0.320436\pi\)
\(642\) 19.1789 0.756931
\(643\) 8.82109 0.347870 0.173935 0.984757i \(-0.444352\pi\)
0.173935 + 0.984757i \(0.444352\pi\)
\(644\) −8.17891 −0.322294
\(645\) −24.3578 −0.959088
\(646\) 15.8945 0.625362
\(647\) 47.8945 1.88293 0.941464 0.337113i \(-0.109450\pi\)
0.941464 + 0.337113i \(0.109450\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 63.0735 2.47585
\(650\) 0 0
\(651\) 4.17891 0.163784
\(652\) 20.1789 0.790267
\(653\) 21.0000 0.821794 0.410897 0.911682i \(-0.365216\pi\)
0.410897 + 0.911682i \(0.365216\pi\)
\(654\) −4.00000 −0.156412
\(655\) −29.0735 −1.13599
\(656\) 7.17891 0.280289
\(657\) −4.00000 −0.156055
\(658\) −7.17891 −0.279863
\(659\) 17.5367 0.683134 0.341567 0.939857i \(-0.389042\pi\)
0.341567 + 0.939857i \(0.389042\pi\)
\(660\) −10.3578 −0.403177
\(661\) −16.5367 −0.643204 −0.321602 0.946875i \(-0.604221\pi\)
−0.321602 + 0.946875i \(0.604221\pi\)
\(662\) −1.64218 −0.0638253
\(663\) 0 0
\(664\) 6.17891 0.239788
\(665\) −6.35782 −0.246546
\(666\) 2.00000 0.0774984
\(667\) 26.0000 1.00672
\(668\) 8.00000 0.309529
\(669\) −2.17891 −0.0842415
\(670\) −7.64218 −0.295243
\(671\) −15.5367 −0.599789
\(672\) 1.00000 0.0385758
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −33.5367 −1.29179
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 18.3578 0.705028
\(679\) 10.3578 0.397497
\(680\) −10.0000 −0.383482
\(681\) −14.3578 −0.550193
\(682\) 21.6422 0.828722
\(683\) −8.35782 −0.319803 −0.159901 0.987133i \(-0.551118\pi\)
−0.159901 + 0.987133i \(0.551118\pi\)
\(684\) −3.17891 −0.121549
\(685\) −12.7156 −0.485839
\(686\) −1.00000 −0.0381802
\(687\) 1.00000 0.0381524
\(688\) 12.1789 0.464317
\(689\) 0 0
\(690\) −16.3578 −0.622731
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 4.00000 0.152057
\(693\) 5.17891 0.196730
\(694\) 19.1789 0.728021
\(695\) −13.6422 −0.517478
\(696\) −3.17891 −0.120496
\(697\) 35.8945 1.35960
\(698\) 8.17891 0.309576
\(699\) −22.3578 −0.845650
\(700\) −1.00000 −0.0377964
\(701\) −21.7156 −0.820188 −0.410094 0.912043i \(-0.634504\pi\)
−0.410094 + 0.912043i \(0.634504\pi\)
\(702\) 0 0
\(703\) 6.35782 0.239790
\(704\) 5.17891 0.195187
\(705\) −14.3578 −0.540747
\(706\) 4.17891 0.157275
\(707\) 6.35782 0.239110
\(708\) −12.1789 −0.457711
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) −4.35782 −0.163546
\(711\) 13.1789 0.494248
\(712\) 9.00000 0.337289
\(713\) 34.1789 1.28001
\(714\) 5.00000 0.187120
\(715\) 0 0
\(716\) 8.35782 0.312346
\(717\) 10.1789 0.380138
\(718\) −10.3578 −0.386550
\(719\) −37.1789 −1.38654 −0.693270 0.720678i \(-0.743831\pi\)
−0.693270 + 0.720678i \(0.743831\pi\)
\(720\) 2.00000 0.0745356
\(721\) −8.17891 −0.304598
\(722\) 8.89454 0.331021
\(723\) 4.35782 0.162069
\(724\) 19.1789 0.712779
\(725\) 3.17891 0.118062
\(726\) 15.8211 0.587176
\(727\) 34.5367 1.28090 0.640448 0.768001i \(-0.278749\pi\)
0.640448 + 0.768001i \(0.278749\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 8.00000 0.296093
\(731\) 60.8945 2.25227
\(732\) 3.00000 0.110883
\(733\) −19.7156 −0.728214 −0.364107 0.931357i \(-0.618626\pi\)
−0.364107 + 0.931357i \(0.618626\pi\)
\(734\) 8.17891 0.301889
\(735\) −2.00000 −0.0737711
\(736\) 8.17891 0.301479
\(737\) 19.7891 0.728940
\(738\) −7.17891 −0.264259
\(739\) 16.1789 0.595151 0.297575 0.954698i \(-0.403822\pi\)
0.297575 + 0.954698i \(0.403822\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 3.00000 0.110133
\(743\) −31.8211 −1.16740 −0.583701 0.811968i \(-0.698396\pi\)
−0.583701 + 0.811968i \(0.698396\pi\)
\(744\) −4.17891 −0.153206
\(745\) −32.3578 −1.18550
\(746\) 20.0000 0.732252
\(747\) −6.17891 −0.226074
\(748\) 25.8945 0.946798
\(749\) 19.1789 0.700782
\(750\) −12.0000 −0.438178
\(751\) −35.8945 −1.30981 −0.654905 0.755711i \(-0.727291\pi\)
−0.654905 + 0.755711i \(0.727291\pi\)
\(752\) 7.17891 0.261788
\(753\) −22.1789 −0.808244
\(754\) 0 0
\(755\) −39.0735 −1.42203
\(756\) −1.00000 −0.0363696
\(757\) −20.3578 −0.739917 −0.369959 0.929048i \(-0.620628\pi\)
−0.369959 + 0.929048i \(0.620628\pi\)
\(758\) −24.7156 −0.897712
\(759\) 42.3578 1.53749
\(760\) 6.35782 0.230622
\(761\) 43.4313 1.57438 0.787191 0.616709i \(-0.211535\pi\)
0.787191 + 0.616709i \(0.211535\pi\)
\(762\) 12.0000 0.434714
\(763\) −4.00000 −0.144810
\(764\) −6.53673 −0.236490
\(765\) 10.0000 0.361551
\(766\) 21.1789 0.765225
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −18.3578 −0.662000 −0.331000 0.943631i \(-0.607386\pi\)
−0.331000 + 0.943631i \(0.607386\pi\)
\(770\) −10.3578 −0.373270
\(771\) 19.7156 0.710041
\(772\) 13.5367 0.487197
\(773\) −14.7156 −0.529285 −0.264642 0.964347i \(-0.585254\pi\)
−0.264642 + 0.964347i \(0.585254\pi\)
\(774\) −12.1789 −0.437762
\(775\) 4.17891 0.150111
\(776\) −10.3578 −0.371824
\(777\) 2.00000 0.0717496
\(778\) 12.1789 0.436635
\(779\) −22.8211 −0.817650
\(780\) 0 0
\(781\) 11.2844 0.403786
\(782\) 40.8945 1.46239
\(783\) 3.17891 0.113605
\(784\) 1.00000 0.0357143
\(785\) 4.00000 0.142766
\(786\) −14.5367 −0.518508
\(787\) 10.8211 0.385730 0.192865 0.981225i \(-0.438222\pi\)
0.192865 + 0.981225i \(0.438222\pi\)
\(788\) −17.0000 −0.605600
\(789\) 12.0000 0.427211
\(790\) −26.3578 −0.937769
\(791\) 18.3578 0.652729
\(792\) −5.17891 −0.184024
\(793\) 0 0
\(794\) 3.00000 0.106466
\(795\) 6.00000 0.212798
\(796\) 24.1789 0.856999
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) −3.17891 −0.112532
\(799\) 35.8945 1.26986
\(800\) 1.00000 0.0353553
\(801\) −9.00000 −0.317999
\(802\) 9.64218 0.340477
\(803\) −20.7156 −0.731039
\(804\) −3.82109 −0.134760
\(805\) −16.3578 −0.576537
\(806\) 0 0
\(807\) 4.35782 0.153402
\(808\) −6.35782 −0.223667
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 12.7156 0.446506 0.223253 0.974761i \(-0.428332\pi\)
0.223253 + 0.974761i \(0.428332\pi\)
\(812\) −3.17891 −0.111558
\(813\) 19.8211 0.695156
\(814\) 10.3578 0.363041
\(815\) 40.3578 1.41367
\(816\) −5.00000 −0.175035
\(817\) −38.7156 −1.35449
\(818\) −36.3578 −1.27122
\(819\) 0 0
\(820\) 14.3578 0.501397
\(821\) −0.284367 −0.00992446 −0.00496223 0.999988i \(-0.501580\pi\)
−0.00496223 + 0.999988i \(0.501580\pi\)
\(822\) −6.35782 −0.221754
\(823\) 32.7156 1.14040 0.570198 0.821508i \(-0.306867\pi\)
0.570198 + 0.821508i \(0.306867\pi\)
\(824\) 8.17891 0.284926
\(825\) 5.17891 0.180306
\(826\) −12.1789 −0.423758
\(827\) 40.7156 1.41582 0.707911 0.706302i \(-0.249638\pi\)
0.707911 + 0.706302i \(0.249638\pi\)
\(828\) −8.17891 −0.284237
\(829\) −3.17891 −0.110408 −0.0552040 0.998475i \(-0.517581\pi\)
−0.0552040 + 0.998475i \(0.517581\pi\)
\(830\) 12.3578 0.428946
\(831\) −5.64218 −0.195725
\(832\) 0 0
\(833\) 5.00000 0.173240
\(834\) −6.82109 −0.236195
\(835\) 16.0000 0.553703
\(836\) −16.4633 −0.569394
\(837\) 4.17891 0.144444
\(838\) 26.8945 0.929057
\(839\) −11.6422 −0.401933 −0.200966 0.979598i \(-0.564408\pi\)
−0.200966 + 0.979598i \(0.564408\pi\)
\(840\) 2.00000 0.0690066
\(841\) −18.8945 −0.651536
\(842\) 10.7156 0.369285
\(843\) 28.3578 0.976695
\(844\) −6.35782 −0.218845
\(845\) 0 0
\(846\) −7.17891 −0.246816
\(847\) 15.8211 0.543619
\(848\) −3.00000 −0.103020
\(849\) −7.64218 −0.262279
\(850\) 5.00000 0.171499
\(851\) 16.3578 0.560739
\(852\) −2.17891 −0.0746482
\(853\) −27.7156 −0.948965 −0.474483 0.880265i \(-0.657365\pi\)
−0.474483 + 0.880265i \(0.657365\pi\)
\(854\) 3.00000 0.102658
\(855\) −6.35782 −0.217433
\(856\) −19.1789 −0.655522
\(857\) −10.7156 −0.366039 −0.183020 0.983109i \(-0.558587\pi\)
−0.183020 + 0.983109i \(0.558587\pi\)
\(858\) 0 0
\(859\) 19.8945 0.678793 0.339397 0.940643i \(-0.389777\pi\)
0.339397 + 0.940643i \(0.389777\pi\)
\(860\) 24.3578 0.830595
\(861\) −7.17891 −0.244657
\(862\) 10.1789 0.346695
\(863\) −13.6422 −0.464385 −0.232193 0.972670i \(-0.574590\pi\)
−0.232193 + 0.972670i \(0.574590\pi\)
\(864\) 1.00000 0.0340207
\(865\) 8.00000 0.272008
\(866\) −34.3578 −1.16753
\(867\) −8.00000 −0.271694
\(868\) −4.17891 −0.141841
\(869\) 68.2524 2.31530
\(870\) −6.35782 −0.215550
\(871\) 0 0
\(872\) 4.00000 0.135457
\(873\) 10.3578 0.350559
\(874\) −26.0000 −0.879463
\(875\) −12.0000 −0.405674
\(876\) 4.00000 0.135147
\(877\) 40.3578 1.36279 0.681393 0.731917i \(-0.261374\pi\)
0.681393 + 0.731917i \(0.261374\pi\)
\(878\) −40.7156 −1.37409
\(879\) 12.0000 0.404750
\(880\) 10.3578 0.349162
\(881\) 11.4633 0.386208 0.193104 0.981178i \(-0.438145\pi\)
0.193104 + 0.981178i \(0.438145\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −3.46327 −0.116548 −0.0582742 0.998301i \(-0.518560\pi\)
−0.0582742 + 0.998301i \(0.518560\pi\)
\(884\) 0 0
\(885\) −24.3578 −0.818779
\(886\) −3.17891 −0.106798
\(887\) −45.5367 −1.52897 −0.764487 0.644639i \(-0.777008\pi\)
−0.764487 + 0.644639i \(0.777008\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 12.0000 0.402467
\(890\) 18.0000 0.603361
\(891\) 5.17891 0.173500
\(892\) 2.17891 0.0729552
\(893\) −22.8211 −0.763679
\(894\) −16.1789 −0.541104
\(895\) 16.7156 0.558742
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) 13.2844 0.443058
\(900\) −1.00000 −0.0333333
\(901\) −15.0000 −0.499722
\(902\) −37.1789 −1.23792
\(903\) −12.1789 −0.405289
\(904\) −18.3578 −0.610572
\(905\) 38.3578 1.27506
\(906\) −19.5367 −0.649064
\(907\) 49.6102 1.64728 0.823639 0.567114i \(-0.191940\pi\)
0.823639 + 0.567114i \(0.191940\pi\)
\(908\) 14.3578 0.476481
\(909\) 6.35782 0.210875
\(910\) 0 0
\(911\) 28.7156 0.951391 0.475696 0.879610i \(-0.342196\pi\)
0.475696 + 0.879610i \(0.342196\pi\)
\(912\) 3.17891 0.105264
\(913\) −32.0000 −1.05905
\(914\) 24.8945 0.823438
\(915\) 6.00000 0.198354
\(916\) −1.00000 −0.0330409
\(917\) −14.5367 −0.480045
\(918\) 5.00000 0.165025
\(919\) −58.2524 −1.92157 −0.960784 0.277298i \(-0.910561\pi\)
−0.960784 + 0.277298i \(0.910561\pi\)
\(920\) 16.3578 0.539301
\(921\) −31.8945 −1.05096
\(922\) 20.7156 0.682233
\(923\) 0 0
\(924\) −5.17891 −0.170374
\(925\) 2.00000 0.0657596
\(926\) 27.8945 0.916672
\(927\) −8.17891 −0.268631
\(928\) 3.17891 0.104353
\(929\) −39.0000 −1.27955 −0.639774 0.768563i \(-0.720972\pi\)
−0.639774 + 0.768563i \(0.720972\pi\)
\(930\) −8.35782 −0.274064
\(931\) −3.17891 −0.104185
\(932\) 22.3578 0.732355
\(933\) −25.5367 −0.836035
\(934\) 22.8945 0.749132
\(935\) 51.7891 1.69368
\(936\) 0 0
\(937\) 21.0735 0.688440 0.344220 0.938889i \(-0.388143\pi\)
0.344220 + 0.938889i \(0.388143\pi\)
\(938\) −3.82109 −0.124763
\(939\) −4.00000 −0.130535
\(940\) 14.3578 0.468300
\(941\) 20.0000 0.651981 0.325991 0.945373i \(-0.394302\pi\)
0.325991 + 0.945373i \(0.394302\pi\)
\(942\) 2.00000 0.0651635
\(943\) −58.7156 −1.91204
\(944\) 12.1789 0.396390
\(945\) −2.00000 −0.0650600
\(946\) −63.0735 −2.05069
\(947\) 40.2524 1.30803 0.654013 0.756483i \(-0.273084\pi\)
0.654013 + 0.756483i \(0.273084\pi\)
\(948\) −13.1789 −0.428031
\(949\) 0 0
\(950\) −3.17891 −0.103137
\(951\) −4.17891 −0.135510
\(952\) −5.00000 −0.162051
\(953\) −2.71563 −0.0879680 −0.0439840 0.999032i \(-0.514005\pi\)
−0.0439840 + 0.999032i \(0.514005\pi\)
\(954\) 3.00000 0.0971286
\(955\) −13.0735 −0.423047
\(956\) −10.1789 −0.329209
\(957\) 16.4633 0.532182
\(958\) −41.1789 −1.33043
\(959\) −6.35782 −0.205305
\(960\) −2.00000 −0.0645497
\(961\) −13.5367 −0.436669
\(962\) 0 0
\(963\) 19.1789 0.618032
\(964\) −4.35782 −0.140356
\(965\) 27.0735 0.871525
\(966\) −8.17891 −0.263152
\(967\) −11.6422 −0.374387 −0.187194 0.982323i \(-0.559939\pi\)
−0.187194 + 0.982323i \(0.559939\pi\)
\(968\) −15.8211 −0.508509
\(969\) 15.8945 0.510606
\(970\) −20.7156 −0.665139
\(971\) −28.1789 −0.904304 −0.452152 0.891941i \(-0.649344\pi\)
−0.452152 + 0.891941i \(0.649344\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −6.82109 −0.218674
\(974\) −26.8211 −0.859403
\(975\) 0 0
\(976\) −3.00000 −0.0960277
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 20.1789 0.645250
\(979\) −46.6102 −1.48967
\(980\) 2.00000 0.0638877
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 13.0735 0.416978 0.208489 0.978025i \(-0.433145\pi\)
0.208489 + 0.978025i \(0.433145\pi\)
\(984\) 7.17891 0.228855
\(985\) −34.0000 −1.08333
\(986\) 15.8945 0.506185
\(987\) −7.17891 −0.228507
\(988\) 0 0
\(989\) −99.6102 −3.16742
\(990\) −10.3578 −0.329193
\(991\) −49.5367 −1.57359 −0.786793 0.617217i \(-0.788260\pi\)
−0.786793 + 0.617217i \(0.788260\pi\)
\(992\) 4.17891 0.132680
\(993\) −1.64218 −0.0521131
\(994\) −2.17891 −0.0691108
\(995\) 48.3578 1.53305
\(996\) 6.17891 0.195786
\(997\) 12.2844 0.389050 0.194525 0.980898i \(-0.437683\pi\)
0.194525 + 0.980898i \(0.437683\pi\)
\(998\) 8.17891 0.258899
\(999\) 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.1.2 2
13.4 even 6 546.2.l.k.211.2 4
13.10 even 6 546.2.l.k.295.2 yes 4
13.12 even 2 7098.2.a.br.1.1 2
39.17 odd 6 1638.2.r.ba.757.1 4
39.23 odd 6 1638.2.r.ba.1387.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.k.211.2 4 13.4 even 6
546.2.l.k.295.2 yes 4 13.10 even 6
1638.2.r.ba.757.1 4 39.17 odd 6
1638.2.r.ba.1387.1 4 39.23 odd 6
7098.2.a.bk.1.2 2 1.1 even 1 trivial
7098.2.a.br.1.1 2 13.12 even 2