Properties

Label 49.6.c.a.18.1
Level $49$
Weight $6$
Character 49.18
Analytic conductor $7.859$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 18.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 49.18
Dual form 49.6.c.a.30.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +O(q^{10})\) \(q+(-5.50000 - 9.52628i) q^{2} +(-44.5000 + 77.0763i) q^{4} +627.000 q^{8} +(121.500 + 210.444i) q^{9} +(38.0000 - 65.8179i) q^{11} +(-2024.50 - 3506.54i) q^{16} +(1336.50 - 2314.89i) q^{18} -836.000 q^{22} +(2476.00 + 4288.56i) q^{23} +(1562.50 - 2706.33i) q^{25} +7282.00 q^{29} +(-12237.5 + 21196.0i) q^{32} -21627.0 q^{36} +(4443.00 + 7695.50i) q^{37} +11748.0 q^{43} +(3382.00 + 5857.80i) q^{44} +(27236.0 - 47174.1i) q^{46} -34375.0 q^{50} +(-12275.0 + 21260.9i) q^{53} +(-40051.0 - 69370.4i) q^{58} +139657. q^{64} +(-34682.0 + 60071.0i) q^{67} -2224.00 q^{71} +(76180.5 + 131948. i) q^{72} +(48873.0 - 84650.5i) q^{74} +(-40084.0 - 69427.5i) q^{79} +(-29524.5 + 51137.9i) q^{81} +(-64614.0 - 111915. i) q^{86} +(23826.0 - 41267.8i) q^{88} -440728. q^{92} +18468.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 11 q^{2} - 89 q^{4} + 1254 q^{8} + 243 q^{9} + 76 q^{11} - 4049 q^{16} + 2673 q^{18} - 1672 q^{22} + 4952 q^{23} + 3125 q^{25} + 14564 q^{29} - 24475 q^{32} - 43254 q^{36} + 8886 q^{37} + 23496 q^{43} + 6764 q^{44} + 54472 q^{46} - 68750 q^{50} - 24550 q^{53} - 80102 q^{58} + 279314 q^{64} - 69364 q^{67} - 4448 q^{71} + 152361 q^{72} + 97746 q^{74} - 80168 q^{79} - 59049 q^{81} - 129228 q^{86} + 47652 q^{88} - 881456 q^{92} + 36936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.50000 9.52628i −0.972272 1.68402i −0.688659 0.725085i \(-0.741800\pi\)
−0.283613 0.958939i \(-0.591533\pi\)
\(3\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −44.5000 + 77.0763i −1.39062 + 2.40863i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 627.000 3.46372
\(9\) 121.500 + 210.444i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 38.0000 65.8179i 0.0946895 0.164007i −0.814789 0.579757i \(-0.803148\pi\)
0.909479 + 0.415750i \(0.136481\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2024.50 3506.54i −1.97705 3.42435i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 1336.50 2314.89i 0.972272 1.68402i
\(19\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −836.000 −0.368256
\(23\) 2476.00 + 4288.56i 0.975958 + 1.69041i 0.676737 + 0.736225i \(0.263393\pi\)
0.299220 + 0.954184i \(0.403273\pi\)
\(24\) 0 0
\(25\) 1562.50 2706.33i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7282.00 1.60789 0.803944 0.594705i \(-0.202731\pi\)
0.803944 + 0.594705i \(0.202731\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) −12237.5 + 21196.0i −2.11260 + 3.65913i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −21627.0 −2.78125
\(37\) 4443.00 + 7695.50i 0.533546 + 0.924129i 0.999232 + 0.0391791i \(0.0124743\pi\)
−0.465686 + 0.884950i \(0.654192\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 11748.0 0.968931 0.484465 0.874810i \(-0.339014\pi\)
0.484465 + 0.874810i \(0.339014\pi\)
\(44\) 3382.00 + 5857.80i 0.263355 + 0.456145i
\(45\) 0 0
\(46\) 27236.0 47174.1i 1.89779 3.28707i
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −34375.0 −1.94454
\(51\) 0 0
\(52\) 0 0
\(53\) −12275.0 + 21260.9i −0.600250 + 1.03966i 0.392533 + 0.919738i \(0.371599\pi\)
−0.992783 + 0.119925i \(0.961735\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −40051.0 69370.4i −1.56330 2.70772i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 139657. 4.26199
\(65\) 0 0
\(66\) 0 0
\(67\) −34682.0 + 60071.0i −0.943881 + 1.63485i −0.185904 + 0.982568i \(0.559521\pi\)
−0.757977 + 0.652281i \(0.773812\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2224.00 −0.0523587 −0.0261794 0.999657i \(-0.508334\pi\)
−0.0261794 + 0.999657i \(0.508334\pi\)
\(72\) 76180.5 + 131948.i 1.73186 + 2.99967i
\(73\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 48873.0 84650.5i 1.03750 1.79701i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −40084.0 69427.5i −0.722609 1.25160i −0.959951 0.280169i \(-0.909609\pi\)
0.237342 0.971426i \(-0.423724\pi\)
\(80\) 0 0
\(81\) −29524.5 + 51137.9i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −64614.0 111915.i −0.942064 1.63170i
\(87\) 0 0
\(88\) 23826.0 41267.8i 0.327978 0.568074i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −440728. −5.42877
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 18468.0 0.189379
\(100\) 139062. + 240863.i 1.39062 + 2.40863i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 270050. 2.33442
\(107\) 32450.0 + 56205.0i 0.274003 + 0.474587i 0.969883 0.243571i \(-0.0783189\pi\)
−0.695880 + 0.718158i \(0.744986\pi\)
\(108\) 0 0
\(109\) 109791. 190164.i 0.885117 1.53307i 0.0395367 0.999218i \(-0.487412\pi\)
0.845580 0.533849i \(-0.179255\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 123202. 0.907657 0.453828 0.891089i \(-0.350058\pi\)
0.453828 + 0.891089i \(0.350058\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −324049. + 561269.i −2.23597 + 3.87281i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 77637.5 + 134472.i 0.482068 + 0.834966i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −262064. −1.44178 −0.720888 0.693051i \(-0.756266\pi\)
−0.720888 + 0.693051i \(0.756266\pi\)
\(128\) −376514. 652141.i −2.03121 3.51817i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 763004. 3.67083
\(135\) 0 0
\(136\) 0 0
\(137\) 176725. 306097.i 0.804445 1.39334i −0.112219 0.993683i \(-0.535796\pi\)
0.916665 0.399657i \(-0.130871\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12232.0 + 21186.4i 0.0509069 + 0.0881733i
\(143\) 0 0
\(144\) 491954. 852088.i 1.97705 3.42435i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −790854. −2.96785
\(149\) 168509. + 291866.i 0.621810 + 1.07701i 0.989149 + 0.146919i \(0.0469356\pi\)
−0.367339 + 0.930087i \(0.619731\pi\)
\(150\) 0 0
\(151\) 130812. 226573.i 0.466880 0.808660i −0.532404 0.846490i \(-0.678711\pi\)
0.999284 + 0.0378305i \(0.0120447\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(158\) −440924. + 763703.i −1.40514 + 2.43378i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 649539. 1.94454
\(163\) −331550. 574261.i −0.977417 1.69294i −0.671716 0.740809i \(-0.734442\pi\)
−0.305702 0.952127i \(-0.598891\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −371293. −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −522786. + 905492.i −1.34742 + 2.33380i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −307724. −0.748824
\(177\) 0 0
\(178\) 0 0
\(179\) −292282. + 506247.i −0.681820 + 1.18095i 0.292605 + 0.956233i \(0.405478\pi\)
−0.974425 + 0.224713i \(0.927856\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.55245e6 + 2.68893e6i 3.38044 + 5.85510i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −501524. 868665.i −0.994737 1.72294i −0.586102 0.810237i \(-0.699338\pi\)
−0.408635 0.912698i \(-0.633995\pi\)
\(192\) 0 0
\(193\) 192951. 334201.i 0.372867 0.645824i −0.617138 0.786855i \(-0.711708\pi\)
0.990005 + 0.141030i \(0.0450415\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 385814. 0.708292 0.354146 0.935190i \(-0.384772\pi\)
0.354146 + 0.935190i \(0.384772\pi\)
\(198\) −101574. 175931.i −0.184128 0.318919i
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 979688. 1.69687e6i 1.73186 2.99967i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −601668. + 1.04212e6i −0.975958 + 1.69041i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.09705e6 −1.69637 −0.848186 0.529699i \(-0.822305\pi\)
−0.848186 + 0.529699i \(0.822305\pi\)
\(212\) −1.09248e6 1.89222e6i −1.66944 2.89156i
\(213\) 0 0
\(214\) 356950. 618256.i 0.532811 0.922855i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2.41540e6 −3.44230
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 759375. 1.00000
\(226\) −677611. 1.17366e6i −0.882489 1.52852i
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.56581e6 5.56927
\(233\) 639749. + 1.10808e6i 0.772004 + 1.33715i 0.936463 + 0.350767i \(0.114079\pi\)
−0.164458 + 0.986384i \(0.552588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 723536. 0.819342 0.409671 0.912233i \(-0.365643\pi\)
0.409671 + 0.912233i \(0.365643\pi\)
\(240\) 0 0
\(241\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(242\) 854012. 1.47919e6i 0.937402 1.62363i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 376352. 0.369652
\(254\) 1.44135e6 + 2.49649e6i 1.40180 + 2.42799i
\(255\) 0 0
\(256\) −1.90714e6 + 3.30326e6i −1.81879 + 3.15023i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 884763. + 1.53245e6i 0.803944 + 1.39247i
\(262\) 0 0
\(263\) −765776. + 1.32636e6i −0.682672 + 1.18242i 0.291490 + 0.956574i \(0.405849\pi\)
−0.974162 + 0.225849i \(0.927484\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3.08670e6 5.34632e6i −2.62517 4.54692i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −3.88795e6 −3.12856
\(275\) −118750. 205681.i −0.0946895 0.164007i
\(276\) 0 0
\(277\) 1.27572e6 2.20962e6i 0.998982 1.73029i 0.460423 0.887700i \(-0.347698\pi\)
0.538559 0.842588i \(-0.318969\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.54797e6 1.92499 0.962497 0.271294i \(-0.0874515\pi\)
0.962497 + 0.271294i \(0.0874515\pi\)
\(282\) 0 0
\(283\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(284\) 98968.0 171418.i 0.0728113 0.126113i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −5.94742e6 −4.22520
\(289\) 709928. + 1.22963e6i 0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.78576e6 + 4.82508e6i 1.84805 + 3.20092i
\(297\) 0 0
\(298\) 1.85360e6 3.21053e6i 1.20914 2.09428i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −2.87786e6 −1.81574
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 7.13495e6 4.01951
\(317\) 110857. + 192010.i 0.0619605 + 0.107319i 0.895342 0.445380i \(-0.146931\pi\)
−0.833381 + 0.552699i \(0.813598\pi\)
\(318\) 0 0
\(319\) 276716. 479286.i 0.152250 0.263705i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −2.62768e6 4.55128e6i −1.39062 2.40863i
\(325\) 0 0
\(326\) −3.64705e6 + 6.31688e6i −1.90063 + 3.29199i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.48574e6 2.57337e6i −0.745371 1.29102i −0.950021 0.312185i \(-0.898939\pi\)
0.204651 0.978835i \(-0.434394\pi\)
\(332\) 0 0
\(333\) −1.07965e6 + 1.87001e6i −0.533546 + 0.924129i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.15965e6 −1.99518 −0.997590 0.0693859i \(-0.977896\pi\)
−0.997590 + 0.0693859i \(0.977896\pi\)
\(338\) 2.04211e6 + 3.53704e6i 0.972272 + 1.68402i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 7.36600e6 3.35610
\(345\) 0 0
\(346\) 0 0
\(347\) −1.14908e6 + 1.99027e6i −0.512304 + 0.887336i 0.487595 + 0.873070i \(0.337874\pi\)
−0.999898 + 0.0142656i \(0.995459\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 930050. + 1.61089e6i 0.400083 + 0.692963i
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 6.43020e6 2.65166
\(359\) −2.13448e6 3.69703e6i −0.874091 1.51397i −0.857728 0.514104i \(-0.828124\pi\)
−0.0163636 0.999866i \(-0.505209\pi\)
\(360\) 0 0
\(361\) 1.23805e6 2.14436e6i 0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(368\) 1.00253e7 1.73644e7i 3.85904 6.68405i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −299651. 519011.i −0.111518 0.193154i 0.804865 0.593458i \(-0.202238\pi\)
−0.916382 + 0.400304i \(0.868904\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −5.59273e6 −1.99998 −0.999991 0.00429827i \(-0.998632\pi\)
−0.999991 + 0.00429827i \(0.998632\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5.51676e6 + 9.55532e6i −1.93431 + 3.35032i
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.24492e6 −1.45011
\(387\) 1.42738e6 + 2.47230e6i 0.484465 + 0.839119i
\(388\) 0 0
\(389\) 630059. 1.09129e6i 0.211109 0.365652i −0.740953 0.671557i \(-0.765626\pi\)
0.952062 + 0.305905i \(0.0989591\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −2.12198e6 3.67537e6i −0.688652 1.19278i
\(395\) 0 0
\(396\) −821826. + 1.42344e6i −0.263355 + 0.456145i
\(397\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.26531e7 −3.95410
\(401\) −3.01646e6 5.22467e6i −0.936779 1.62255i −0.771431 0.636313i \(-0.780459\pi\)
−0.165348 0.986235i \(-0.552875\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 675336. 0.202085
\(408\) 0 0
\(409\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 1.32367e7 3.79559
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 2.07477e6 0.570513 0.285257 0.958451i \(-0.407921\pi\)
0.285257 + 0.958451i \(0.407921\pi\)
\(422\) 6.03379e6 + 1.04508e7i 1.64933 + 2.85673i
\(423\) 0 0
\(424\) −7.69643e6 + 1.33306e7i −2.07910 + 3.60110i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −5.77610e6 −1.52414
\(429\) 0 0
\(430\) 0 0
\(431\) 3.66942e6 6.35563e6i 0.951491 1.64803i 0.209290 0.977854i \(-0.432885\pi\)
0.742201 0.670177i \(-0.233782\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.77140e6 + 1.69246e7i 2.46173 + 4.26384i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.55245e6 + 4.42097e6i 0.617942 + 1.07031i 0.989861 + 0.142042i \(0.0453668\pi\)
−0.371918 + 0.928265i \(0.621300\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.99808e6 0.467732 0.233866 0.972269i \(-0.424862\pi\)
0.233866 + 0.972269i \(0.424862\pi\)
\(450\) −4.17656e6 7.23402e6i −0.972272 1.68402i
\(451\) 0 0
\(452\) −5.48249e6 + 9.49595e6i −1.26221 + 2.18621i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.88984e6 + 5.00535e6i 0.647267 + 1.12110i 0.983773 + 0.179419i \(0.0574216\pi\)
−0.336505 + 0.941682i \(0.609245\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 2.88620e6 0.625711 0.312856 0.949801i \(-0.398714\pi\)
0.312856 + 0.949801i \(0.398714\pi\)
\(464\) −1.47424e7 2.55346e7i −3.17888 5.50597i
\(465\) 0 0
\(466\) 7.03724e6 1.21889e7i 1.50120 2.60015i
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 446424. 773229.i 0.0917476 0.158911i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.96565e6 −1.20050
\(478\) −3.97945e6 6.89261e6i −0.796623 1.37979i
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.38195e7 −2.68150
\(485\) 0 0
\(486\) 0 0
\(487\) 1.38073e6 2.39150e6i 0.263807 0.456928i −0.703443 0.710752i \(-0.748355\pi\)
0.967250 + 0.253824i \(0.0816883\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.82732e6 1.65244 0.826219 0.563349i \(-0.190487\pi\)
0.826219 + 0.563349i \(0.190487\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.56018e6 + 9.63052e6i 0.999626 + 1.73140i 0.523489 + 0.852032i \(0.324630\pi\)
0.476137 + 0.879371i \(0.342037\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.06994e6 3.58523e6i −0.359402 0.622503i
\(507\) 0 0
\(508\) 1.16618e7 2.01989e7i 2.00497 3.47271i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.78601e7 3.01099
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 9.73239e6 1.68570e7i 1.56330 2.70772i
\(523\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 1.68471e7 2.65497
\(527\) 0 0
\(528\) 0 0
\(529\) −9.04298e6 + 1.56629e7i −1.40499 + 2.43351i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −2.17456e7 + 3.76645e7i −3.26934 + 5.66266i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.56306e6 9.63551e6i −0.817186 1.41541i −0.907748 0.419517i \(-0.862200\pi\)
0.0905619 0.995891i \(-0.471134\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.23604e6 −0.319529 −0.159765 0.987155i \(-0.551074\pi\)
−0.159765 + 0.987155i \(0.551074\pi\)
\(548\) 1.57285e7 + 2.72426e7i 2.23736 + 3.87523i
\(549\) 0 0
\(550\) −1.30625e6 + 2.26249e6i −0.184128 + 0.318919i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −2.80660e7 −3.88513
\(555\) 0 0
\(556\) 0 0
\(557\) −3.11974e6 + 5.40355e6i −0.426070 + 0.737975i −0.996520 0.0833578i \(-0.973436\pi\)
0.570450 + 0.821332i \(0.306769\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.40139e7 2.42727e7i −1.87162 3.24174i
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −1.39445e6 −0.181356
\(569\) −5.23689e6 9.07056e6i −0.678099 1.17450i −0.975553 0.219765i \(-0.929471\pi\)
0.297454 0.954736i \(-0.403862\pi\)
\(570\) 0 0
\(571\) −3.16956e6 + 5.48984e6i −0.406826 + 0.704644i −0.994532 0.104431i \(-0.966698\pi\)
0.587706 + 0.809075i \(0.300031\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.54750e7 1.95192
\(576\) 1.69683e7 + 2.93900e7i 2.13100 + 3.69099i
\(577\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(578\) 7.80921e6 1.35260e7i 0.972272 1.68402i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 932900. + 1.61583e6i 0.113675 + 0.196890i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.79897e7 3.11591e7i 2.10970 3.65410i
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.99946e7 −3.45882
\(597\) 0 0
\(598\) 0 0
\(599\) 3.58878e6 6.21596e6i 0.408677 0.707849i −0.586065 0.810264i \(-0.699324\pi\)
0.994742 + 0.102415i \(0.0326569\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) −1.68555e7 −1.88776
\(604\) 1.16423e7 + 2.01650e7i 1.29851 + 2.24909i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8.77570e6 + 1.52000e7i −0.943258 + 1.63377i −0.184057 + 0.982916i \(0.558923\pi\)
−0.759201 + 0.650856i \(0.774410\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.14899e6 −0.756017 −0.378008 0.925802i \(-0.623391\pi\)
−0.378008 + 0.925802i \(0.623391\pi\)
\(618\) 0 0
\(619\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4.88281e6 8.45728e6i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.99786e7 1.99752 0.998760 0.0497844i \(-0.0158534\pi\)
0.998760 + 0.0497844i \(0.0158534\pi\)
\(632\) −2.51327e7 4.35311e7i −2.50291 4.33517i
\(633\) 0 0
\(634\) 1.21943e6 2.11211e6i 0.120485 0.208686i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −6.08775e6 −0.592114
\(639\) −270216. 468028.i −0.0261794 0.0453440i
\(640\) 0 0
\(641\) 5.74814e6 9.95607e6i 0.552563 0.957068i −0.445525 0.895269i \(-0.646983\pi\)
0.998089 0.0617984i \(-0.0196836\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) −1.85119e7 + 3.20635e7i −1.73186 + 2.99967i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 5.90159e7 5.43688
\(653\) −5.48862e6 9.50658e6i −0.503710 0.872451i −0.999991 0.00428929i \(-0.998635\pi\)
0.496281 0.868162i \(-0.334699\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.02232e7 −1.81400 −0.907000 0.421131i \(-0.861633\pi\)
−0.907000 + 0.421131i \(0.861633\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) −1.63431e7 + 2.83071e7i −1.44941 + 2.51044i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 2.37523e7 2.07501
\(667\) 1.80302e7 + 3.12293e7i 1.56923 + 2.71799i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.38435e6 0.117817 0.0589085 0.998263i \(-0.481238\pi\)
0.0589085 + 0.998263i \(0.481238\pi\)
\(674\) 2.28781e7 + 3.96260e7i 1.93986 + 3.35993i
\(675\) 0 0
\(676\) 1.65225e7 2.86179e7i 1.39062 2.40863i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.06574e7 1.84592e7i 0.874179 1.51412i 0.0165451 0.999863i \(-0.494733\pi\)
0.857634 0.514260i \(-0.171933\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2.37838e7 4.11948e7i −1.91563 3.31796i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 2.52798e7 1.99239
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.88600e6 0.375542 0.187771 0.982213i \(-0.439874\pi\)
0.187771 + 0.982213i \(0.439874\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 5.30697e6 9.19193e6i 0.403566 0.698997i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 7.16124e6 + 1.24036e7i 0.535023 + 0.926688i 0.999162 + 0.0409253i \(0.0130306\pi\)
−0.464139 + 0.885763i \(0.653636\pi\)
\(710\) 0 0
\(711\) 9.74041e6 1.68709e7i 0.722609 1.25160i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −2.60131e7 4.50560e7i −1.89631 3.28451i
\(717\) 0 0
\(718\) −2.34793e7 + 4.06674e7i −1.69971 + 2.94398i
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.72371e7 −1.94454
\(723\) 0 0
\(724\) 0 0
\(725\) 1.13781e7 1.97075e7i 0.803944 1.39247i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.21200e8 −8.24724
\(737\) 2.63583e6 + 4.56539e6i 0.178751 + 0.309606i
\(738\) 0 0
\(739\) 1.32447e6 2.29404e6i 0.0892133 0.154522i −0.817966 0.575267i \(-0.804898\pi\)
0.907179 + 0.420745i \(0.138231\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.49502e7 1.65807 0.829033 0.559199i \(-0.188891\pi\)
0.829033 + 0.559199i \(0.188891\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.29616e6 + 5.70912e6i −0.216851 + 0.375597i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.25544e7 2.17448e7i −0.812260 1.40688i −0.911278 0.411791i \(-0.864903\pi\)
0.0990181 0.995086i \(-0.468430\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.78870e7 −1.13448 −0.567242 0.823551i \(-0.691989\pi\)
−0.567242 + 0.823551i \(0.691989\pi\)
\(758\) 3.07600e7 + 5.32779e7i 1.94453 + 3.36802i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 8.92713e7 5.53322
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.71726e7 + 2.97439e7i 1.03704 + 1.79620i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 1.57012e7 2.71953e7i 0.942064 1.63170i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.38613e7 −0.821022
\(779\) 0 0
\(780\) 0 0
\(781\) −84512.0 + 146379.i −0.00495782 + 0.00858720i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(788\) −1.71687e7 + 2.97371e7i −0.984969 + 1.70602i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 1.15794e7 0.655956
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797