Properties

Label 2394.2.a.t.1.2
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2394.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} +4.82843 q^{11} +5.65685 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.82843 q^{17} -1.00000 q^{19} +2.00000 q^{20} -4.82843 q^{22} +4.00000 q^{23} -1.00000 q^{25} -5.65685 q^{26} +1.00000 q^{28} +2.00000 q^{29} -4.82843 q^{31} -1.00000 q^{32} -6.82843 q^{34} +2.00000 q^{35} -8.48528 q^{37} +1.00000 q^{38} -2.00000 q^{40} -3.65685 q^{41} +9.65685 q^{43} +4.82843 q^{44} -4.00000 q^{46} -4.48528 q^{47} +1.00000 q^{49} +1.00000 q^{50} +5.65685 q^{52} -7.65685 q^{53} +9.65685 q^{55} -1.00000 q^{56} -2.00000 q^{58} +8.00000 q^{59} -13.3137 q^{61} +4.82843 q^{62} +1.00000 q^{64} +11.3137 q^{65} -6.82843 q^{67} +6.82843 q^{68} -2.00000 q^{70} -5.65685 q^{71} -7.65685 q^{73} +8.48528 q^{74} -1.00000 q^{76} +4.82843 q^{77} +7.65685 q^{79} +2.00000 q^{80} +3.65685 q^{82} +11.6569 q^{83} +13.6569 q^{85} -9.65685 q^{86} -4.82843 q^{88} +13.3137 q^{89} +5.65685 q^{91} +4.00000 q^{92} +4.48528 q^{94} -2.00000 q^{95} -18.4853 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8} - 4 q^{10} + 4 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 2 q^{25} + 2 q^{28} + 4 q^{29} - 4 q^{31} - 2 q^{32} - 8 q^{34} + 4 q^{35} + 2 q^{38} - 4 q^{40} + 4 q^{41} + 8 q^{43} + 4 q^{44} - 8 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 16 q^{59} - 4 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{67} + 8 q^{68} - 4 q^{70} - 4 q^{73} - 2 q^{76} + 4 q^{77} + 4 q^{79} + 4 q^{80} - 4 q^{82} + 12 q^{83} + 16 q^{85} - 8 q^{86} - 4 q^{88} + 4 q^{89} + 8 q^{92} - 8 q^{94} - 4 q^{95} - 20 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −4.82843 −1.02942
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −5.65685 −1.10940
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −4.82843 −0.867211 −0.433606 0.901103i \(-0.642759\pi\)
−0.433606 + 0.901103i \(0.642759\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.82843 −1.17107
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −3.65685 −0.571105 −0.285552 0.958363i \(-0.592177\pi\)
−0.285552 + 0.958363i \(0.592177\pi\)
\(42\) 0 0
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 4.82843 0.727913
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −4.48528 −0.654246 −0.327123 0.944982i \(-0.606079\pi\)
−0.327123 + 0.944982i \(0.606079\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 5.65685 0.784465
\(53\) −7.65685 −1.05175 −0.525875 0.850562i \(-0.676262\pi\)
−0.525875 + 0.850562i \(0.676262\pi\)
\(54\) 0 0
\(55\) 9.65685 1.30213
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −13.3137 −1.70465 −0.852323 0.523016i \(-0.824807\pi\)
−0.852323 + 0.523016i \(0.824807\pi\)
\(62\) 4.82843 0.613211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 11.3137 1.40329
\(66\) 0 0
\(67\) −6.82843 −0.834225 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(68\) 6.82843 0.828068
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −7.65685 −0.896167 −0.448084 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) 8.48528 0.986394
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 4.82843 0.550250
\(78\) 0 0
\(79\) 7.65685 0.861463 0.430732 0.902480i \(-0.358256\pi\)
0.430732 + 0.902480i \(0.358256\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 3.65685 0.403832
\(83\) 11.6569 1.27951 0.639753 0.768581i \(-0.279037\pi\)
0.639753 + 0.768581i \(0.279037\pi\)
\(84\) 0 0
\(85\) 13.6569 1.48129
\(86\) −9.65685 −1.04133
\(87\) 0 0
\(88\) −4.82843 −0.514712
\(89\) 13.3137 1.41125 0.705625 0.708585i \(-0.250666\pi\)
0.705625 + 0.708585i \(0.250666\pi\)
\(90\) 0 0
\(91\) 5.65685 0.592999
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 4.48528 0.462621
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −18.4853 −1.87690 −0.938448 0.345421i \(-0.887736\pi\)
−0.938448 + 0.345421i \(0.887736\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −5.31371 −0.528734 −0.264367 0.964422i \(-0.585163\pi\)
−0.264367 + 0.964422i \(0.585163\pi\)
\(102\) 0 0
\(103\) −7.17157 −0.706636 −0.353318 0.935503i \(-0.614947\pi\)
−0.353318 + 0.935503i \(0.614947\pi\)
\(104\) −5.65685 −0.554700
\(105\) 0 0
\(106\) 7.65685 0.743699
\(107\) 5.65685 0.546869 0.273434 0.961891i \(-0.411840\pi\)
0.273434 + 0.961891i \(0.411840\pi\)
\(108\) 0 0
\(109\) −10.8284 −1.03718 −0.518588 0.855024i \(-0.673542\pi\)
−0.518588 + 0.855024i \(0.673542\pi\)
\(110\) −9.65685 −0.920745
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −11.6569 −1.09658 −0.548292 0.836287i \(-0.684722\pi\)
−0.548292 + 0.836287i \(0.684722\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 6.82843 0.625961
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 13.3137 1.20537
\(123\) 0 0
\(124\) −4.82843 −0.433606
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 4.34315 0.385392 0.192696 0.981259i \(-0.438277\pi\)
0.192696 + 0.981259i \(0.438277\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −11.3137 −0.992278
\(131\) −9.31371 −0.813742 −0.406871 0.913486i \(-0.633380\pi\)
−0.406871 + 0.913486i \(0.633380\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 6.82843 0.589886
\(135\) 0 0
\(136\) −6.82843 −0.585533
\(137\) 19.3137 1.65008 0.825041 0.565073i \(-0.191152\pi\)
0.825041 + 0.565073i \(0.191152\pi\)
\(138\) 0 0
\(139\) −17.6569 −1.49763 −0.748817 0.662776i \(-0.769378\pi\)
−0.748817 + 0.662776i \(0.769378\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 5.65685 0.474713
\(143\) 27.3137 2.28409
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 7.65685 0.633686
\(147\) 0 0
\(148\) −8.48528 −0.697486
\(149\) −10.4853 −0.858988 −0.429494 0.903070i \(-0.641308\pi\)
−0.429494 + 0.903070i \(0.641308\pi\)
\(150\) 0 0
\(151\) 7.65685 0.623106 0.311553 0.950229i \(-0.399151\pi\)
0.311553 + 0.950229i \(0.399151\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −4.82843 −0.389086
\(155\) −9.65685 −0.775657
\(156\) 0 0
\(157\) 7.65685 0.611083 0.305542 0.952179i \(-0.401162\pi\)
0.305542 + 0.952179i \(0.401162\pi\)
\(158\) −7.65685 −0.609147
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 0.686292 0.0537545 0.0268772 0.999639i \(-0.491444\pi\)
0.0268772 + 0.999639i \(0.491444\pi\)
\(164\) −3.65685 −0.285552
\(165\) 0 0
\(166\) −11.6569 −0.904747
\(167\) 5.65685 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) −13.6569 −1.04743
\(171\) 0 0
\(172\) 9.65685 0.736328
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 4.82843 0.363956
\(177\) 0 0
\(178\) −13.3137 −0.997905
\(179\) −15.3137 −1.14460 −0.572300 0.820044i \(-0.693949\pi\)
−0.572300 + 0.820044i \(0.693949\pi\)
\(180\) 0 0
\(181\) 9.65685 0.717788 0.358894 0.933378i \(-0.383154\pi\)
0.358894 + 0.933378i \(0.383154\pi\)
\(182\) −5.65685 −0.419314
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −16.9706 −1.24770
\(186\) 0 0
\(187\) 32.9706 2.41105
\(188\) −4.48528 −0.327123
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) 0 0
\(193\) 6.97056 0.501752 0.250876 0.968019i \(-0.419281\pi\)
0.250876 + 0.968019i \(0.419281\pi\)
\(194\) 18.4853 1.32717
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 12.8284 0.913988 0.456994 0.889470i \(-0.348926\pi\)
0.456994 + 0.889470i \(0.348926\pi\)
\(198\) 0 0
\(199\) 6.34315 0.449654 0.224827 0.974399i \(-0.427818\pi\)
0.224827 + 0.974399i \(0.427818\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 5.31371 0.373871
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −7.31371 −0.510812
\(206\) 7.17157 0.499667
\(207\) 0 0
\(208\) 5.65685 0.392232
\(209\) −4.82843 −0.333989
\(210\) 0 0
\(211\) −0.485281 −0.0334081 −0.0167041 0.999860i \(-0.505317\pi\)
−0.0167041 + 0.999860i \(0.505317\pi\)
\(212\) −7.65685 −0.525875
\(213\) 0 0
\(214\) −5.65685 −0.386695
\(215\) 19.3137 1.31718
\(216\) 0 0
\(217\) −4.82843 −0.327775
\(218\) 10.8284 0.733394
\(219\) 0 0
\(220\) 9.65685 0.651065
\(221\) 38.6274 2.59836
\(222\) 0 0
\(223\) −3.17157 −0.212384 −0.106192 0.994346i \(-0.533866\pi\)
−0.106192 + 0.994346i \(0.533866\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 11.6569 0.775402
\(227\) 11.3137 0.750917 0.375459 0.926839i \(-0.377485\pi\)
0.375459 + 0.926839i \(0.377485\pi\)
\(228\) 0 0
\(229\) −7.65685 −0.505979 −0.252990 0.967469i \(-0.581414\pi\)
−0.252990 + 0.967469i \(0.581414\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −21.6569 −1.41879 −0.709394 0.704812i \(-0.751031\pi\)
−0.709394 + 0.704812i \(0.751031\pi\)
\(234\) 0 0
\(235\) −8.97056 −0.585175
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) −6.82843 −0.442621
\(239\) 12.9706 0.838996 0.419498 0.907756i \(-0.362206\pi\)
0.419498 + 0.907756i \(0.362206\pi\)
\(240\) 0 0
\(241\) 22.4853 1.44840 0.724202 0.689588i \(-0.242208\pi\)
0.724202 + 0.689588i \(0.242208\pi\)
\(242\) −12.3137 −0.791555
\(243\) 0 0
\(244\) −13.3137 −0.852323
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −5.65685 −0.359937
\(248\) 4.82843 0.306605
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 2.68629 0.169557 0.0847786 0.996400i \(-0.472982\pi\)
0.0847786 + 0.996400i \(0.472982\pi\)
\(252\) 0 0
\(253\) 19.3137 1.21424
\(254\) −4.34315 −0.272513
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.9706 0.933838 0.466919 0.884300i \(-0.345364\pi\)
0.466919 + 0.884300i \(0.345364\pi\)
\(258\) 0 0
\(259\) −8.48528 −0.527250
\(260\) 11.3137 0.701646
\(261\) 0 0
\(262\) 9.31371 0.575403
\(263\) −26.6274 −1.64192 −0.820958 0.570988i \(-0.806560\pi\)
−0.820958 + 0.570988i \(0.806560\pi\)
\(264\) 0 0
\(265\) −15.3137 −0.940714
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) −6.82843 −0.417113
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) 0 0
\(271\) 20.9706 1.27387 0.636935 0.770917i \(-0.280202\pi\)
0.636935 + 0.770917i \(0.280202\pi\)
\(272\) 6.82843 0.414034
\(273\) 0 0
\(274\) −19.3137 −1.16678
\(275\) −4.82843 −0.291165
\(276\) 0 0
\(277\) 27.6569 1.66174 0.830870 0.556467i \(-0.187843\pi\)
0.830870 + 0.556467i \(0.187843\pi\)
\(278\) 17.6569 1.05899
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) 14.9706 0.893069 0.446534 0.894766i \(-0.352658\pi\)
0.446534 + 0.894766i \(0.352658\pi\)
\(282\) 0 0
\(283\) 14.3431 0.852612 0.426306 0.904579i \(-0.359815\pi\)
0.426306 + 0.904579i \(0.359815\pi\)
\(284\) −5.65685 −0.335673
\(285\) 0 0
\(286\) −27.3137 −1.61509
\(287\) −3.65685 −0.215857
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) −7.65685 −0.448084
\(293\) −6.68629 −0.390617 −0.195309 0.980742i \(-0.562571\pi\)
−0.195309 + 0.980742i \(0.562571\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 8.48528 0.493197
\(297\) 0 0
\(298\) 10.4853 0.607396
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) 9.65685 0.556612
\(302\) −7.65685 −0.440602
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −26.6274 −1.52468
\(306\) 0 0
\(307\) 0.686292 0.0391687 0.0195844 0.999808i \(-0.493766\pi\)
0.0195844 + 0.999808i \(0.493766\pi\)
\(308\) 4.82843 0.275125
\(309\) 0 0
\(310\) 9.65685 0.548472
\(311\) −2.14214 −0.121469 −0.0607347 0.998154i \(-0.519344\pi\)
−0.0607347 + 0.998154i \(0.519344\pi\)
\(312\) 0 0
\(313\) −17.3137 −0.978629 −0.489314 0.872107i \(-0.662753\pi\)
−0.489314 + 0.872107i \(0.662753\pi\)
\(314\) −7.65685 −0.432101
\(315\) 0 0
\(316\) 7.65685 0.430732
\(317\) −29.3137 −1.64642 −0.823211 0.567736i \(-0.807820\pi\)
−0.823211 + 0.567736i \(0.807820\pi\)
\(318\) 0 0
\(319\) 9.65685 0.540680
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) −6.82843 −0.379944
\(324\) 0 0
\(325\) −5.65685 −0.313786
\(326\) −0.686292 −0.0380102
\(327\) 0 0
\(328\) 3.65685 0.201916
\(329\) −4.48528 −0.247282
\(330\) 0 0
\(331\) −0.485281 −0.0266735 −0.0133367 0.999911i \(-0.504245\pi\)
−0.0133367 + 0.999911i \(0.504245\pi\)
\(332\) 11.6569 0.639753
\(333\) 0 0
\(334\) −5.65685 −0.309529
\(335\) −13.6569 −0.746154
\(336\) 0 0
\(337\) −23.6569 −1.28867 −0.644335 0.764743i \(-0.722866\pi\)
−0.644335 + 0.764743i \(0.722866\pi\)
\(338\) −19.0000 −1.03346
\(339\) 0 0
\(340\) 13.6569 0.740647
\(341\) −23.3137 −1.26251
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −9.65685 −0.520663
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 2.48528 0.133417 0.0667084 0.997773i \(-0.478750\pi\)
0.0667084 + 0.997773i \(0.478750\pi\)
\(348\) 0 0
\(349\) 10.9706 0.587241 0.293620 0.955922i \(-0.405140\pi\)
0.293620 + 0.955922i \(0.405140\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −4.82843 −0.257356
\(353\) −30.1421 −1.60430 −0.802152 0.597120i \(-0.796312\pi\)
−0.802152 + 0.597120i \(0.796312\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 13.3137 0.705625
\(357\) 0 0
\(358\) 15.3137 0.809355
\(359\) 16.9706 0.895672 0.447836 0.894116i \(-0.352195\pi\)
0.447836 + 0.894116i \(0.352195\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −9.65685 −0.507553
\(363\) 0 0
\(364\) 5.65685 0.296500
\(365\) −15.3137 −0.801556
\(366\) 0 0
\(367\) 9.65685 0.504084 0.252042 0.967716i \(-0.418898\pi\)
0.252042 + 0.967716i \(0.418898\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 16.9706 0.882258
\(371\) −7.65685 −0.397524
\(372\) 0 0
\(373\) −34.8284 −1.80335 −0.901674 0.432417i \(-0.857661\pi\)
−0.901674 + 0.432417i \(0.857661\pi\)
\(374\) −32.9706 −1.70487
\(375\) 0 0
\(376\) 4.48528 0.231311
\(377\) 11.3137 0.582686
\(378\) 0 0
\(379\) 34.8284 1.78902 0.894508 0.447052i \(-0.147526\pi\)
0.894508 + 0.447052i \(0.147526\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) 19.3137 0.988175
\(383\) −2.34315 −0.119729 −0.0598646 0.998207i \(-0.519067\pi\)
−0.0598646 + 0.998207i \(0.519067\pi\)
\(384\) 0 0
\(385\) 9.65685 0.492159
\(386\) −6.97056 −0.354792
\(387\) 0 0
\(388\) −18.4853 −0.938448
\(389\) −13.7990 −0.699637 −0.349818 0.936818i \(-0.613757\pi\)
−0.349818 + 0.936818i \(0.613757\pi\)
\(390\) 0 0
\(391\) 27.3137 1.38131
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −12.8284 −0.646287
\(395\) 15.3137 0.770516
\(396\) 0 0
\(397\) −9.31371 −0.467442 −0.233721 0.972304i \(-0.575090\pi\)
−0.233721 + 0.972304i \(0.575090\pi\)
\(398\) −6.34315 −0.317953
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 13.3137 0.664855 0.332427 0.943129i \(-0.392132\pi\)
0.332427 + 0.943129i \(0.392132\pi\)
\(402\) 0 0
\(403\) −27.3137 −1.36059
\(404\) −5.31371 −0.264367
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −40.9706 −2.03084
\(408\) 0 0
\(409\) −18.4853 −0.914038 −0.457019 0.889457i \(-0.651083\pi\)
−0.457019 + 0.889457i \(0.651083\pi\)
\(410\) 7.31371 0.361198
\(411\) 0 0
\(412\) −7.17157 −0.353318
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 23.3137 1.14442
\(416\) −5.65685 −0.277350
\(417\) 0 0
\(418\) 4.82843 0.236166
\(419\) 14.9706 0.731360 0.365680 0.930741i \(-0.380836\pi\)
0.365680 + 0.930741i \(0.380836\pi\)
\(420\) 0 0
\(421\) −18.1421 −0.884194 −0.442097 0.896967i \(-0.645765\pi\)
−0.442097 + 0.896967i \(0.645765\pi\)
\(422\) 0.485281 0.0236231
\(423\) 0 0
\(424\) 7.65685 0.371850
\(425\) −6.82843 −0.331227
\(426\) 0 0
\(427\) −13.3137 −0.644296
\(428\) 5.65685 0.273434
\(429\) 0 0
\(430\) −19.3137 −0.931390
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) −25.7990 −1.23982 −0.619910 0.784673i \(-0.712831\pi\)
−0.619910 + 0.784673i \(0.712831\pi\)
\(434\) 4.82843 0.231772
\(435\) 0 0
\(436\) −10.8284 −0.518588
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 13.7990 0.658590 0.329295 0.944227i \(-0.393189\pi\)
0.329295 + 0.944227i \(0.393189\pi\)
\(440\) −9.65685 −0.460372
\(441\) 0 0
\(442\) −38.6274 −1.83732
\(443\) −27.1716 −1.29096 −0.645480 0.763777i \(-0.723343\pi\)
−0.645480 + 0.763777i \(0.723343\pi\)
\(444\) 0 0
\(445\) 26.6274 1.26226
\(446\) 3.17157 0.150178
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 2.68629 0.126774 0.0633870 0.997989i \(-0.479810\pi\)
0.0633870 + 0.997989i \(0.479810\pi\)
\(450\) 0 0
\(451\) −17.6569 −0.831429
\(452\) −11.6569 −0.548292
\(453\) 0 0
\(454\) −11.3137 −0.530979
\(455\) 11.3137 0.530395
\(456\) 0 0
\(457\) −2.68629 −0.125659 −0.0628297 0.998024i \(-0.520012\pi\)
−0.0628297 + 0.998024i \(0.520012\pi\)
\(458\) 7.65685 0.357781
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −19.6569 −0.915511 −0.457755 0.889078i \(-0.651346\pi\)
−0.457755 + 0.889078i \(0.651346\pi\)
\(462\) 0 0
\(463\) −31.3137 −1.45527 −0.727636 0.685964i \(-0.759381\pi\)
−0.727636 + 0.685964i \(0.759381\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 21.6569 1.00323
\(467\) −34.9706 −1.61824 −0.809122 0.587640i \(-0.800057\pi\)
−0.809122 + 0.587640i \(0.800057\pi\)
\(468\) 0 0
\(469\) −6.82843 −0.315307
\(470\) 8.97056 0.413781
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) 46.6274 2.14393
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 6.82843 0.312980
\(477\) 0 0
\(478\) −12.9706 −0.593260
\(479\) 3.51472 0.160592 0.0802958 0.996771i \(-0.474414\pi\)
0.0802958 + 0.996771i \(0.474414\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) −22.4853 −1.02418
\(483\) 0 0
\(484\) 12.3137 0.559714
\(485\) −36.9706 −1.67875
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 13.3137 0.602683
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) 4.14214 0.186932 0.0934660 0.995622i \(-0.470205\pi\)
0.0934660 + 0.995622i \(0.470205\pi\)
\(492\) 0 0
\(493\) 13.6569 0.615074
\(494\) 5.65685 0.254514
\(495\) 0 0
\(496\) −4.82843 −0.216803
\(497\) −5.65685 −0.253745
\(498\) 0 0
\(499\) −39.3137 −1.75992 −0.879962 0.475045i \(-0.842432\pi\)
−0.879962 + 0.475045i \(0.842432\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −2.68629 −0.119895
\(503\) 35.7990 1.59620 0.798099 0.602526i \(-0.205839\pi\)
0.798099 + 0.602526i \(0.205839\pi\)
\(504\) 0 0
\(505\) −10.6274 −0.472914
\(506\) −19.3137 −0.858599
\(507\) 0 0
\(508\) 4.34315 0.192696
\(509\) −31.9411 −1.41577 −0.707883 0.706330i \(-0.750349\pi\)
−0.707883 + 0.706330i \(0.750349\pi\)
\(510\) 0 0
\(511\) −7.65685 −0.338719
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.9706 −0.660323
\(515\) −14.3431 −0.632035
\(516\) 0 0
\(517\) −21.6569 −0.952467
\(518\) 8.48528 0.372822
\(519\) 0 0
\(520\) −11.3137 −0.496139
\(521\) 15.6569 0.685939 0.342970 0.939346i \(-0.388567\pi\)
0.342970 + 0.939346i \(0.388567\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) −9.31371 −0.406871
\(525\) 0 0
\(526\) 26.6274 1.16101
\(527\) −32.9706 −1.43622
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 15.3137 0.665185
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −20.6863 −0.896023
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 6.82843 0.294943
\(537\) 0 0
\(538\) −17.3137 −0.746447
\(539\) 4.82843 0.207975
\(540\) 0 0
\(541\) 27.9411 1.20128 0.600641 0.799519i \(-0.294912\pi\)
0.600641 + 0.799519i \(0.294912\pi\)
\(542\) −20.9706 −0.900763
\(543\) 0 0
\(544\) −6.82843 −0.292766
\(545\) −21.6569 −0.927678
\(546\) 0 0
\(547\) 39.1127 1.67234 0.836169 0.548472i \(-0.184790\pi\)
0.836169 + 0.548472i \(0.184790\pi\)
\(548\) 19.3137 0.825041
\(549\) 0 0
\(550\) 4.82843 0.205885
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) 7.65685 0.325603
\(554\) −27.6569 −1.17503
\(555\) 0 0
\(556\) −17.6569 −0.748817
\(557\) 34.7696 1.47323 0.736617 0.676311i \(-0.236422\pi\)
0.736617 + 0.676311i \(0.236422\pi\)
\(558\) 0 0
\(559\) 54.6274 2.31049
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −14.9706 −0.631495
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −23.3137 −0.980815
\(566\) −14.3431 −0.602887
\(567\) 0 0
\(568\) 5.65685 0.237356
\(569\) −9.02944 −0.378534 −0.189267 0.981926i \(-0.560611\pi\)
−0.189267 + 0.981926i \(0.560611\pi\)
\(570\) 0 0
\(571\) 0.686292 0.0287204 0.0143602 0.999897i \(-0.495429\pi\)
0.0143602 + 0.999897i \(0.495429\pi\)
\(572\) 27.3137 1.14204
\(573\) 0 0
\(574\) 3.65685 0.152634
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 26.2843 1.09423 0.547114 0.837058i \(-0.315726\pi\)
0.547114 + 0.837058i \(0.315726\pi\)
\(578\) −29.6274 −1.23234
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 11.6569 0.483608
\(582\) 0 0
\(583\) −36.9706 −1.53116
\(584\) 7.65685 0.316843
\(585\) 0 0
\(586\) 6.68629 0.276208
\(587\) −17.3137 −0.714613 −0.357307 0.933987i \(-0.616305\pi\)
−0.357307 + 0.933987i \(0.616305\pi\)
\(588\) 0 0
\(589\) 4.82843 0.198952
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) −8.48528 −0.348743
\(593\) 19.5147 0.801373 0.400687 0.916215i \(-0.368772\pi\)
0.400687 + 0.916215i \(0.368772\pi\)
\(594\) 0 0
\(595\) 13.6569 0.559876
\(596\) −10.4853 −0.429494
\(597\) 0 0
\(598\) −22.6274 −0.925304
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −43.4558 −1.77260 −0.886300 0.463111i \(-0.846733\pi\)
−0.886300 + 0.463111i \(0.846733\pi\)
\(602\) −9.65685 −0.393584
\(603\) 0 0
\(604\) 7.65685 0.311553
\(605\) 24.6274 1.00125
\(606\) 0 0
\(607\) −44.1421 −1.79167 −0.895837 0.444383i \(-0.853423\pi\)
−0.895837 + 0.444383i \(0.853423\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 26.6274 1.07811
\(611\) −25.3726 −1.02646
\(612\) 0 0
\(613\) −49.3137 −1.99176 −0.995881 0.0906703i \(-0.971099\pi\)
−0.995881 + 0.0906703i \(0.971099\pi\)
\(614\) −0.686292 −0.0276965
\(615\) 0 0
\(616\) −4.82843 −0.194543
\(617\) 36.9706 1.48838 0.744189 0.667969i \(-0.232836\pi\)
0.744189 + 0.667969i \(0.232836\pi\)
\(618\) 0 0
\(619\) 41.6569 1.67433 0.837165 0.546950i \(-0.184211\pi\)
0.837165 + 0.546950i \(0.184211\pi\)
\(620\) −9.65685 −0.387829
\(621\) 0 0
\(622\) 2.14214 0.0858918
\(623\) 13.3137 0.533402
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 17.3137 0.691995
\(627\) 0 0
\(628\) 7.65685 0.305542
\(629\) −57.9411 −2.31026
\(630\) 0 0
\(631\) 26.6274 1.06002 0.530010 0.847991i \(-0.322188\pi\)
0.530010 + 0.847991i \(0.322188\pi\)
\(632\) −7.65685 −0.304573
\(633\) 0 0
\(634\) 29.3137 1.16420
\(635\) 8.68629 0.344705
\(636\) 0 0
\(637\) 5.65685 0.224133
\(638\) −9.65685 −0.382319
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 5.31371 0.209879 0.104939 0.994479i \(-0.466535\pi\)
0.104939 + 0.994479i \(0.466535\pi\)
\(642\) 0 0
\(643\) 47.3137 1.86587 0.932935 0.360044i \(-0.117238\pi\)
0.932935 + 0.360044i \(0.117238\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 6.82843 0.268661
\(647\) 38.1421 1.49952 0.749761 0.661709i \(-0.230168\pi\)
0.749761 + 0.661709i \(0.230168\pi\)
\(648\) 0 0
\(649\) 38.6274 1.51626
\(650\) 5.65685 0.221880
\(651\) 0 0
\(652\) 0.686292 0.0268772
\(653\) 6.48528 0.253789 0.126894 0.991916i \(-0.459499\pi\)
0.126894 + 0.991916i \(0.459499\pi\)
\(654\) 0 0
\(655\) −18.6274 −0.727833
\(656\) −3.65685 −0.142776
\(657\) 0 0
\(658\) 4.48528 0.174854
\(659\) −29.6569 −1.15527 −0.577634 0.816296i \(-0.696024\pi\)
−0.577634 + 0.816296i \(0.696024\pi\)
\(660\) 0 0
\(661\) 8.68629 0.337858 0.168929 0.985628i \(-0.445969\pi\)
0.168929 + 0.985628i \(0.445969\pi\)
\(662\) 0.485281 0.0188610
\(663\) 0 0
\(664\) −11.6569 −0.452374
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 5.65685 0.218870
\(669\) 0 0
\(670\) 13.6569 0.527610
\(671\) −64.2843 −2.48167
\(672\) 0 0
\(673\) −29.3137 −1.12996 −0.564980 0.825104i \(-0.691116\pi\)
−0.564980 + 0.825104i \(0.691116\pi\)
\(674\) 23.6569 0.911228
\(675\) 0 0
\(676\) 19.0000 0.730769
\(677\) −13.3137 −0.511687 −0.255844 0.966718i \(-0.582353\pi\)
−0.255844 + 0.966718i \(0.582353\pi\)
\(678\) 0 0
\(679\) −18.4853 −0.709400
\(680\) −13.6569 −0.523716
\(681\) 0 0
\(682\) 23.3137 0.892728
\(683\) −37.9411 −1.45178 −0.725888 0.687812i \(-0.758571\pi\)
−0.725888 + 0.687812i \(0.758571\pi\)
\(684\) 0 0
\(685\) 38.6274 1.47588
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 9.65685 0.368164
\(689\) −43.3137 −1.65012
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −2.48528 −0.0943400
\(695\) −35.3137 −1.33953
\(696\) 0 0
\(697\) −24.9706 −0.945828
\(698\) −10.9706 −0.415242
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −21.1127 −0.797416 −0.398708 0.917078i \(-0.630541\pi\)
−0.398708 + 0.917078i \(0.630541\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 4.82843 0.181978
\(705\) 0 0
\(706\) 30.1421 1.13441
\(707\) −5.31371 −0.199843
\(708\) 0 0
\(709\) −26.9706 −1.01290 −0.506450 0.862269i \(-0.669043\pi\)
−0.506450 + 0.862269i \(0.669043\pi\)
\(710\) 11.3137 0.424596
\(711\) 0 0
\(712\) −13.3137 −0.498952
\(713\) −19.3137 −0.723304
\(714\) 0 0
\(715\) 54.6274 2.04295
\(716\) −15.3137 −0.572300
\(717\) 0 0
\(718\) −16.9706 −0.633336
\(719\) 31.7990 1.18590 0.592951 0.805238i \(-0.297963\pi\)
0.592951 + 0.805238i \(0.297963\pi\)
\(720\) 0 0
\(721\) −7.17157 −0.267083
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 9.65685 0.358894
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −19.3137 −0.716306 −0.358153 0.933663i \(-0.616593\pi\)
−0.358153 + 0.933663i \(0.616593\pi\)
\(728\) −5.65685 −0.209657
\(729\) 0 0
\(730\) 15.3137 0.566786
\(731\) 65.9411 2.43892
\(732\) 0 0
\(733\) 10.9706 0.405207 0.202603 0.979261i \(-0.435060\pi\)
0.202603 + 0.979261i \(0.435060\pi\)
\(734\) −9.65685 −0.356441
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −32.9706 −1.21449
\(738\) 0 0
\(739\) 9.65685 0.355233 0.177617 0.984100i \(-0.443161\pi\)
0.177617 + 0.984100i \(0.443161\pi\)
\(740\) −16.9706 −0.623850
\(741\) 0 0
\(742\) 7.65685 0.281092
\(743\) −53.6569 −1.96848 −0.984240 0.176840i \(-0.943412\pi\)
−0.984240 + 0.176840i \(0.943412\pi\)
\(744\) 0 0
\(745\) −20.9706 −0.768302
\(746\) 34.8284 1.27516
\(747\) 0 0
\(748\) 32.9706 1.20552
\(749\) 5.65685 0.206697
\(750\) 0 0
\(751\) 40.6274 1.48252 0.741258 0.671220i \(-0.234230\pi\)
0.741258 + 0.671220i \(0.234230\pi\)
\(752\) −4.48528 −0.163561
\(753\) 0 0
\(754\) −11.3137 −0.412021
\(755\) 15.3137 0.557323
\(756\) 0 0
\(757\) −31.9411 −1.16092 −0.580460 0.814289i \(-0.697127\pi\)
−0.580460 + 0.814289i \(0.697127\pi\)
\(758\) −34.8284 −1.26503
\(759\) 0 0
\(760\) 2.00000 0.0725476
\(761\) 33.1716 1.20247 0.601234 0.799073i \(-0.294676\pi\)
0.601234 + 0.799073i \(0.294676\pi\)
\(762\) 0 0
\(763\) −10.8284 −0.392015
\(764\) −19.3137 −0.698745
\(765\) 0 0
\(766\) 2.34315 0.0846613
\(767\) 45.2548 1.63406
\(768\) 0 0
\(769\) 6.68629 0.241114 0.120557 0.992706i \(-0.461532\pi\)
0.120557 + 0.992706i \(0.461532\pi\)
\(770\) −9.65685 −0.348009
\(771\) 0 0
\(772\) 6.97056 0.250876
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 0 0
\(775\) 4.82843 0.173442
\(776\) 18.4853 0.663583
\(777\) 0 0
\(778\) 13.7990 0.494718
\(779\) 3.65685 0.131020
\(780\) 0 0
\(781\) −27.3137 −0.977361
\(782\) −27.3137 −0.976736
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 15.3137 0.546570
\(786\) 0 0
\(787\) 9.65685 0.344230 0.172115 0.985077i \(-0.444940\pi\)
0.172115 + 0.985077i \(0.444940\pi\)
\(788\) 12.8284 0.456994
\(789\) 0 0
\(790\) −15.3137 −0.544837
\(791\) −11.6569 −0.414470
\(792\) 0 0
\(793\) −75.3137 −2.67447
\(794\) 9.31371 0.330531
\(795\) 0 0
\(796\) 6.34315 0.224827
\(797\) 19.9411 0.706351 0.353175 0.935557i \(-0.385102\pi\)
0.353175 + 0.935557i \(0.385102\pi\)
\(798\) 0 0
\(799\) −30.6274 −1.08352
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −13.3137 −0.470123
\(803\) −36.9706 −1.30466
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 27.3137 0.962084
\(807\) 0 0
\(808\) 5.31371 0.186936
\(809\) 1.65685 0.0582519 0.0291259 0.999576i \(-0.490728\pi\)
0.0291259 + 0.999576i \(0.490728\pi\)
\(810\) 0 0
\(811\) 8.68629 0.305017 0.152508 0.988302i \(-0.451265\pi\)
0.152508 + 0.988302i \(0.451265\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 40.9706 1.43602
\(815\) 1.37258 0.0480795
\(816\) 0 0
\(817\) −9.65685 −0.337851
\(818\) 18.4853 0.646323
\(819\) 0 0
\(820\) −7.31371 −0.255406
\(821\) 18.4853 0.645141 0.322570 0.946545i \(-0.395453\pi\)
0.322570 + 0.946545i \(0.395453\pi\)
\(822\) 0 0
\(823\) −39.3137 −1.37039 −0.685195 0.728360i \(-0.740283\pi\)
−0.685195 + 0.728360i \(0.740283\pi\)
\(824\) 7.17157 0.249834
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 23.3137 0.810697 0.405349 0.914162i \(-0.367150\pi\)
0.405349 + 0.914162i \(0.367150\pi\)
\(828\) 0 0
\(829\) 29.9411 1.03990 0.519949 0.854197i \(-0.325951\pi\)
0.519949 + 0.854197i \(0.325951\pi\)
\(830\) −23.3137 −0.809231
\(831\) 0 0
\(832\) 5.65685 0.196116
\(833\) 6.82843 0.236591
\(834\) 0 0
\(835\) 11.3137 0.391527
\(836\) −4.82843 −0.166995
\(837\) 0 0
\(838\) −14.9706 −0.517150
\(839\) −19.3137 −0.666783 −0.333392 0.942788i \(-0.608193\pi\)
−0.333392 + 0.942788i \(0.608193\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.1421 0.625219
\(843\) 0 0
\(844\) −0.485281 −0.0167041
\(845\) 38.0000 1.30724
\(846\) 0 0
\(847\) 12.3137 0.423104
\(848\) −7.65685 −0.262937
\(849\) 0 0
\(850\) 6.82843 0.234213
\(851\) −33.9411 −1.16349
\(852\) 0 0
\(853\) 1.31371 0.0449805 0.0224903 0.999747i \(-0.492841\pi\)
0.0224903 + 0.999747i \(0.492841\pi\)
\(854\) 13.3137 0.455586
\(855\) 0 0
\(856\) −5.65685 −0.193347
\(857\) 29.3137 1.00134 0.500669 0.865639i \(-0.333088\pi\)
0.500669 + 0.865639i \(0.333088\pi\)
\(858\) 0 0
\(859\) −3.02944 −0.103363 −0.0516815 0.998664i \(-0.516458\pi\)
−0.0516815 + 0.998664i \(0.516458\pi\)
\(860\) 19.3137 0.658592
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) −28.2843 −0.962808 −0.481404 0.876499i \(-0.659873\pi\)
−0.481404 + 0.876499i \(0.659873\pi\)
\(864\) 0 0
\(865\) −4.00000 −0.136004
\(866\) 25.7990 0.876685
\(867\) 0 0
\(868\) −4.82843 −0.163887
\(869\) 36.9706 1.25414
\(870\) 0 0
\(871\) −38.6274 −1.30884
\(872\) 10.8284 0.366697
\(873\) 0 0
\(874\) 4.00000 0.135302
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −58.4264 −1.97292 −0.986460 0.164003i \(-0.947559\pi\)
−0.986460 + 0.164003i \(0.947559\pi\)
\(878\) −13.7990 −0.465693
\(879\) 0 0
\(880\) 9.65685 0.325532
\(881\) −54.1421 −1.82409 −0.912047 0.410085i \(-0.865499\pi\)
−0.912047 + 0.410085i \(0.865499\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 38.6274 1.29918
\(885\) 0 0
\(886\) 27.1716 0.912847
\(887\) −17.9411 −0.602404 −0.301202 0.953560i \(-0.597388\pi\)
−0.301202 + 0.953560i \(0.597388\pi\)
\(888\) 0 0
\(889\) 4.34315 0.145664
\(890\) −26.6274 −0.892553
\(891\) 0 0
\(892\) −3.17157 −0.106192
\(893\) 4.48528 0.150094
\(894\) 0 0
\(895\) −30.6274 −1.02376
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −2.68629 −0.0896427
\(899\) −9.65685 −0.322074
\(900\) 0 0
\(901\) −52.2843 −1.74184
\(902\) 17.6569 0.587909
\(903\) 0 0
\(904\) 11.6569 0.387701
\(905\) 19.3137 0.642009
\(906\) 0 0
\(907\) 60.0833 1.99503 0.997516 0.0704407i \(-0.0224405\pi\)
0.997516 + 0.0704407i \(0.0224405\pi\)
\(908\) 11.3137 0.375459
\(909\) 0 0
\(910\) −11.3137 −0.375046
\(911\) 30.6274 1.01473 0.507366 0.861731i \(-0.330619\pi\)
0.507366 + 0.861731i \(0.330619\pi\)
\(912\) 0 0
\(913\) 56.2843 1.86274
\(914\) 2.68629 0.0888546
\(915\) 0 0
\(916\) −7.65685 −0.252990
\(917\) −9.31371 −0.307566
\(918\) 0 0
\(919\) −10.6274 −0.350566 −0.175283 0.984518i \(-0.556084\pi\)
−0.175283 + 0.984518i \(0.556084\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) 19.6569 0.647364
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 8.48528 0.278994
\(926\) 31.3137 1.02903
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 41.4558 1.36012 0.680061 0.733155i \(-0.261953\pi\)
0.680061 + 0.733155i \(0.261953\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −21.6569 −0.709394
\(933\) 0 0
\(934\) 34.9706 1.14427
\(935\) 65.9411 2.15651
\(936\) 0 0
\(937\) −43.9411 −1.43549 −0.717747 0.696304i \(-0.754827\pi\)
−0.717747 + 0.696304i \(0.754827\pi\)
\(938\) 6.82843 0.222956
\(939\) 0 0
\(940\) −8.97056 −0.292587
\(941\) 30.6863 1.00034 0.500172 0.865926i \(-0.333270\pi\)
0.500172 + 0.865926i \(0.333270\pi\)
\(942\) 0 0
\(943\) −14.6274 −0.476334
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −46.6274 −1.51599
\(947\) −45.7990 −1.48827 −0.744134 0.668031i \(-0.767137\pi\)
−0.744134 + 0.668031i \(0.767137\pi\)
\(948\) 0 0
\(949\) −43.3137 −1.40602
\(950\) −1.00000 −0.0324443
\(951\) 0 0
\(952\) −6.82843 −0.221311
\(953\) −7.65685 −0.248030 −0.124015 0.992280i \(-0.539577\pi\)
−0.124015 + 0.992280i \(0.539577\pi\)
\(954\) 0 0
\(955\) −38.6274 −1.24995
\(956\) 12.9706 0.419498
\(957\) 0 0
\(958\) −3.51472 −0.113555
\(959\) 19.3137 0.623672
\(960\) 0 0
\(961\) −7.68629 −0.247945
\(962\) 48.0000 1.54758
\(963\) 0 0
\(964\) 22.4853 0.724202
\(965\) 13.9411 0.448781
\(966\) 0 0
\(967\) −43.3137 −1.39287 −0.696437 0.717617i \(-0.745233\pi\)
−0.696437 + 0.717617i \(0.745233\pi\)
\(968\) −12.3137 −0.395778
\(969\) 0 0
\(970\) 36.9706 1.18705
\(971\) 22.6274 0.726148 0.363074 0.931760i \(-0.381727\pi\)
0.363074 + 0.931760i \(0.381727\pi\)
\(972\) 0 0
\(973\) −17.6569 −0.566053
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −13.3137 −0.426161
\(977\) 11.3726 0.363841 0.181921 0.983313i \(-0.441769\pi\)
0.181921 + 0.983313i \(0.441769\pi\)
\(978\) 0 0
\(979\) 64.2843 2.05453
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) −4.14214 −0.132181
\(983\) 21.6569 0.690746 0.345373 0.938465i \(-0.387752\pi\)
0.345373 + 0.938465i \(0.387752\pi\)
\(984\) 0 0
\(985\) 25.6569 0.817495
\(986\) −13.6569 −0.434923
\(987\) 0 0
\(988\) −5.65685 −0.179969
\(989\) 38.6274 1.22828
\(990\) 0 0
\(991\) −54.0000 −1.71537 −0.857683 0.514178i \(-0.828097\pi\)
−0.857683 + 0.514178i \(0.828097\pi\)
\(992\) 4.82843 0.153303
\(993\) 0 0
\(994\) 5.65685 0.179425
\(995\) 12.6863 0.402182
\(996\) 0 0
\(997\) 38.2843 1.21248 0.606238 0.795284i \(-0.292678\pi\)
0.606238 + 0.795284i \(0.292678\pi\)
\(998\) 39.3137 1.24445
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2394.2.a.t.1.2 2
3.2 odd 2 2394.2.a.u.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2394.2.a.t.1.2 2 1.1 even 1 trivial
2394.2.a.u.1.1 yes 2 3.2 odd 2