Properties

Label 320.6.d.d.161.4
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + 1475231592 x + 653157349\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.4
Root \(-1.02698 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.d.161.9

$q$-expansion

\(f(q)\) \(=\) \(q-5.98217i q^{3} +25.0000i q^{5} +86.0039 q^{7} +207.214 q^{9} +O(q^{10})\) \(q-5.98217i q^{3} +25.0000i q^{5} +86.0039 q^{7} +207.214 q^{9} +562.865i q^{11} +659.925i q^{13} +149.554 q^{15} -989.843 q^{17} -1610.12i q^{19} -514.490i q^{21} -2932.96 q^{23} -625.000 q^{25} -2693.25i q^{27} +5656.43i q^{29} +6876.28 q^{31} +3367.15 q^{33} +2150.10i q^{35} -10863.8i q^{37} +3947.78 q^{39} -18053.4 q^{41} +14898.2i q^{43} +5180.34i q^{45} -7542.31 q^{47} -9410.33 q^{49} +5921.41i q^{51} +23275.2i q^{53} -14071.6 q^{55} -9632.01 q^{57} -9083.21i q^{59} -6457.10i q^{61} +17821.2 q^{63} -16498.1 q^{65} +62015.4i q^{67} +17545.5i q^{69} +49547.0 q^{71} +2157.48 q^{73} +3738.86i q^{75} +48408.6i q^{77} +25663.1 q^{79} +34241.4 q^{81} +54850.1i q^{83} -24746.1i q^{85} +33837.7 q^{87} -115200. q^{89} +56756.1i q^{91} -41135.1i q^{93} +40253.0 q^{95} -114285. q^{97} +116633. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q + 268q^{7} - 428q^{9} - 900q^{15} + 2400q^{17} + 8108q^{23} - 7500q^{25} + 7976q^{31} - 20776q^{33} + 40984q^{39} - 56408q^{41} + 21172q^{47} - 5540q^{49} + 18400q^{55} + 39992q^{57} + 179516q^{63} + 44000q^{65} + 367704q^{71} + 58736q^{73} + 26192q^{79} + 411692q^{81} + 183200q^{87} + 87672q^{89} + 121000q^{95} - 172336q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 5.98217i − 0.383756i −0.981419 0.191878i \(-0.938542\pi\)
0.981419 0.191878i \(-0.0614578\pi\)
\(4\) 0 0
\(5\) 25.0000i 0.447214i
\(6\) 0 0
\(7\) 86.0039 0.663396 0.331698 0.943386i \(-0.392378\pi\)
0.331698 + 0.943386i \(0.392378\pi\)
\(8\) 0 0
\(9\) 207.214 0.852731
\(10\) 0 0
\(11\) 562.865i 1.40256i 0.712884 + 0.701282i \(0.247389\pi\)
−0.712884 + 0.701282i \(0.752611\pi\)
\(12\) 0 0
\(13\) 659.925i 1.08302i 0.840694 + 0.541510i \(0.182147\pi\)
−0.840694 + 0.541510i \(0.817853\pi\)
\(14\) 0 0
\(15\) 149.554 0.171621
\(16\) 0 0
\(17\) −989.843 −0.830700 −0.415350 0.909662i \(-0.636341\pi\)
−0.415350 + 0.909662i \(0.636341\pi\)
\(18\) 0 0
\(19\) − 1610.12i − 1.02323i −0.859214 0.511616i \(-0.829047\pi\)
0.859214 0.511616i \(-0.170953\pi\)
\(20\) 0 0
\(21\) − 514.490i − 0.254582i
\(22\) 0 0
\(23\) −2932.96 −1.15608 −0.578039 0.816009i \(-0.696182\pi\)
−0.578039 + 0.816009i \(0.696182\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) − 2693.25i − 0.710997i
\(28\) 0 0
\(29\) 5656.43i 1.24896i 0.781042 + 0.624478i \(0.214688\pi\)
−0.781042 + 0.624478i \(0.785312\pi\)
\(30\) 0 0
\(31\) 6876.28 1.28514 0.642568 0.766228i \(-0.277869\pi\)
0.642568 + 0.766228i \(0.277869\pi\)
\(32\) 0 0
\(33\) 3367.15 0.538243
\(34\) 0 0
\(35\) 2150.10i 0.296680i
\(36\) 0 0
\(37\) − 10863.8i − 1.30460i −0.757960 0.652302i \(-0.773804\pi\)
0.757960 0.652302i \(-0.226196\pi\)
\(38\) 0 0
\(39\) 3947.78 0.415616
\(40\) 0 0
\(41\) −18053.4 −1.67726 −0.838628 0.544704i \(-0.816642\pi\)
−0.838628 + 0.544704i \(0.816642\pi\)
\(42\) 0 0
\(43\) 14898.2i 1.22875i 0.789014 + 0.614375i \(0.210592\pi\)
−0.789014 + 0.614375i \(0.789408\pi\)
\(44\) 0 0
\(45\) 5180.34i 0.381353i
\(46\) 0 0
\(47\) −7542.31 −0.498035 −0.249017 0.968499i \(-0.580108\pi\)
−0.249017 + 0.968499i \(0.580108\pi\)
\(48\) 0 0
\(49\) −9410.33 −0.559906
\(50\) 0 0
\(51\) 5921.41i 0.318786i
\(52\) 0 0
\(53\) 23275.2i 1.13816i 0.822282 + 0.569080i \(0.192701\pi\)
−0.822282 + 0.569080i \(0.807299\pi\)
\(54\) 0 0
\(55\) −14071.6 −0.627246
\(56\) 0 0
\(57\) −9632.01 −0.392672
\(58\) 0 0
\(59\) − 9083.21i − 0.339711i −0.985469 0.169855i \(-0.945670\pi\)
0.985469 0.169855i \(-0.0543301\pi\)
\(60\) 0 0
\(61\) − 6457.10i − 0.222184i −0.993810 0.111092i \(-0.964565\pi\)
0.993810 0.111092i \(-0.0354348\pi\)
\(62\) 0 0
\(63\) 17821.2 0.565699
\(64\) 0 0
\(65\) −16498.1 −0.484341
\(66\) 0 0
\(67\) 62015.4i 1.68777i 0.536527 + 0.843883i \(0.319736\pi\)
−0.536527 + 0.843883i \(0.680264\pi\)
\(68\) 0 0
\(69\) 17545.5i 0.443652i
\(70\) 0 0
\(71\) 49547.0 1.16646 0.583232 0.812306i \(-0.301788\pi\)
0.583232 + 0.812306i \(0.301788\pi\)
\(72\) 0 0
\(73\) 2157.48 0.0473850 0.0236925 0.999719i \(-0.492458\pi\)
0.0236925 + 0.999719i \(0.492458\pi\)
\(74\) 0 0
\(75\) 3738.86i 0.0767513i
\(76\) 0 0
\(77\) 48408.6i 0.930456i
\(78\) 0 0
\(79\) 25663.1 0.462638 0.231319 0.972878i \(-0.425696\pi\)
0.231319 + 0.972878i \(0.425696\pi\)
\(80\) 0 0
\(81\) 34241.4 0.579881
\(82\) 0 0
\(83\) 54850.1i 0.873940i 0.899476 + 0.436970i \(0.143948\pi\)
−0.899476 + 0.436970i \(0.856052\pi\)
\(84\) 0 0
\(85\) − 24746.1i − 0.371500i
\(86\) 0 0
\(87\) 33837.7 0.479295
\(88\) 0 0
\(89\) −115200. −1.54162 −0.770808 0.637068i \(-0.780147\pi\)
−0.770808 + 0.637068i \(0.780147\pi\)
\(90\) 0 0
\(91\) 56756.1i 0.718471i
\(92\) 0 0
\(93\) − 41135.1i − 0.493179i
\(94\) 0 0
\(95\) 40253.0 0.457603
\(96\) 0 0
\(97\) −114285. −1.23328 −0.616639 0.787246i \(-0.711506\pi\)
−0.616639 + 0.787246i \(0.711506\pi\)
\(98\) 0 0
\(99\) 116633.i 1.19601i
\(100\) 0 0
\(101\) 56833.6i 0.554372i 0.960816 + 0.277186i \(0.0894019\pi\)
−0.960816 + 0.277186i \(0.910598\pi\)
\(102\) 0 0
\(103\) 136106. 1.26411 0.632053 0.774925i \(-0.282212\pi\)
0.632053 + 0.774925i \(0.282212\pi\)
\(104\) 0 0
\(105\) 12862.2 0.113853
\(106\) 0 0
\(107\) 51245.3i 0.432708i 0.976315 + 0.216354i \(0.0694165\pi\)
−0.976315 + 0.216354i \(0.930584\pi\)
\(108\) 0 0
\(109\) 160990.i 1.29787i 0.760843 + 0.648937i \(0.224786\pi\)
−0.760843 + 0.648937i \(0.775214\pi\)
\(110\) 0 0
\(111\) −64989.2 −0.500650
\(112\) 0 0
\(113\) −197348. −1.45390 −0.726952 0.686688i \(-0.759064\pi\)
−0.726952 + 0.686688i \(0.759064\pi\)
\(114\) 0 0
\(115\) − 73324.1i − 0.517014i
\(116\) 0 0
\(117\) 136745.i 0.923524i
\(118\) 0 0
\(119\) −85130.4 −0.551083
\(120\) 0 0
\(121\) −155766. −0.967186
\(122\) 0 0
\(123\) 107998.i 0.643658i
\(124\) 0 0
\(125\) − 15625.0i − 0.0894427i
\(126\) 0 0
\(127\) 54027.0 0.297236 0.148618 0.988895i \(-0.452518\pi\)
0.148618 + 0.988895i \(0.452518\pi\)
\(128\) 0 0
\(129\) 89123.7 0.471540
\(130\) 0 0
\(131\) 44661.0i 0.227379i 0.993516 + 0.113689i \(0.0362669\pi\)
−0.993516 + 0.113689i \(0.963733\pi\)
\(132\) 0 0
\(133\) − 138477.i − 0.678808i
\(134\) 0 0
\(135\) 67331.4 0.317968
\(136\) 0 0
\(137\) 250220. 1.13899 0.569497 0.821994i \(-0.307138\pi\)
0.569497 + 0.821994i \(0.307138\pi\)
\(138\) 0 0
\(139\) 97606.2i 0.428490i 0.976780 + 0.214245i \(0.0687290\pi\)
−0.976780 + 0.214245i \(0.931271\pi\)
\(140\) 0 0
\(141\) 45119.4i 0.191124i
\(142\) 0 0
\(143\) −371449. −1.51900
\(144\) 0 0
\(145\) −141411. −0.558550
\(146\) 0 0
\(147\) 56294.2i 0.214867i
\(148\) 0 0
\(149\) − 274545.i − 1.01309i −0.862213 0.506545i \(-0.830922\pi\)
0.862213 0.506545i \(-0.169078\pi\)
\(150\) 0 0
\(151\) 331973. 1.18484 0.592420 0.805629i \(-0.298172\pi\)
0.592420 + 0.805629i \(0.298172\pi\)
\(152\) 0 0
\(153\) −205109. −0.708364
\(154\) 0 0
\(155\) 171907.i 0.574730i
\(156\) 0 0
\(157\) 289482.i 0.937288i 0.883387 + 0.468644i \(0.155257\pi\)
−0.883387 + 0.468644i \(0.844743\pi\)
\(158\) 0 0
\(159\) 139236. 0.436776
\(160\) 0 0
\(161\) −252246. −0.766938
\(162\) 0 0
\(163\) − 232604.i − 0.685721i −0.939386 0.342861i \(-0.888604\pi\)
0.939386 0.342861i \(-0.111396\pi\)
\(164\) 0 0
\(165\) 84178.9i 0.240710i
\(166\) 0 0
\(167\) −51950.4 −0.144144 −0.0720722 0.997399i \(-0.522961\pi\)
−0.0720722 + 0.997399i \(0.522961\pi\)
\(168\) 0 0
\(169\) −64208.1 −0.172931
\(170\) 0 0
\(171\) − 333639.i − 0.872542i
\(172\) 0 0
\(173\) 364659.i 0.926342i 0.886269 + 0.463171i \(0.153288\pi\)
−0.886269 + 0.463171i \(0.846712\pi\)
\(174\) 0 0
\(175\) −53752.4 −0.132679
\(176\) 0 0
\(177\) −54337.3 −0.130366
\(178\) 0 0
\(179\) − 84272.6i − 0.196587i −0.995157 0.0982933i \(-0.968662\pi\)
0.995157 0.0982933i \(-0.0313383\pi\)
\(180\) 0 0
\(181\) − 91685.0i − 0.208018i −0.994576 0.104009i \(-0.966833\pi\)
0.994576 0.104009i \(-0.0331671\pi\)
\(182\) 0 0
\(183\) −38627.5 −0.0852645
\(184\) 0 0
\(185\) 271596. 0.583436
\(186\) 0 0
\(187\) − 557148.i − 1.16511i
\(188\) 0 0
\(189\) − 231630.i − 0.471673i
\(190\) 0 0
\(191\) 550876. 1.09262 0.546312 0.837582i \(-0.316031\pi\)
0.546312 + 0.837582i \(0.316031\pi\)
\(192\) 0 0
\(193\) 705787. 1.36389 0.681946 0.731402i \(-0.261134\pi\)
0.681946 + 0.731402i \(0.261134\pi\)
\(194\) 0 0
\(195\) 98694.6i 0.185869i
\(196\) 0 0
\(197\) − 407510.i − 0.748123i −0.927404 0.374061i \(-0.877965\pi\)
0.927404 0.374061i \(-0.122035\pi\)
\(198\) 0 0
\(199\) −123380. −0.220857 −0.110428 0.993884i \(-0.535222\pi\)
−0.110428 + 0.993884i \(0.535222\pi\)
\(200\) 0 0
\(201\) 370986. 0.647691
\(202\) 0 0
\(203\) 486475.i 0.828553i
\(204\) 0 0
\(205\) − 451335.i − 0.750092i
\(206\) 0 0
\(207\) −607750. −0.985824
\(208\) 0 0
\(209\) 906281. 1.43515
\(210\) 0 0
\(211\) − 421862.i − 0.652325i −0.945314 0.326163i \(-0.894244\pi\)
0.945314 0.326163i \(-0.105756\pi\)
\(212\) 0 0
\(213\) − 296398.i − 0.447638i
\(214\) 0 0
\(215\) −372456. −0.549514
\(216\) 0 0
\(217\) 591387. 0.852555
\(218\) 0 0
\(219\) − 12906.4i − 0.0181843i
\(220\) 0 0
\(221\) − 653222.i − 0.899664i
\(222\) 0 0
\(223\) −1.32959e6 −1.79042 −0.895208 0.445648i \(-0.852973\pi\)
−0.895208 + 0.445648i \(0.852973\pi\)
\(224\) 0 0
\(225\) −129509. −0.170546
\(226\) 0 0
\(227\) 164830.i 0.212311i 0.994350 + 0.106155i \(0.0338541\pi\)
−0.994350 + 0.106155i \(0.966146\pi\)
\(228\) 0 0
\(229\) 881880.i 1.11127i 0.831425 + 0.555637i \(0.187525\pi\)
−0.831425 + 0.555637i \(0.812475\pi\)
\(230\) 0 0
\(231\) 289588. 0.357068
\(232\) 0 0
\(233\) −525527. −0.634170 −0.317085 0.948397i \(-0.602704\pi\)
−0.317085 + 0.948397i \(0.602704\pi\)
\(234\) 0 0
\(235\) − 188558.i − 0.222728i
\(236\) 0 0
\(237\) − 153521.i − 0.177540i
\(238\) 0 0
\(239\) −322839. −0.365587 −0.182794 0.983151i \(-0.558514\pi\)
−0.182794 + 0.983151i \(0.558514\pi\)
\(240\) 0 0
\(241\) 1.17177e6 1.29956 0.649782 0.760120i \(-0.274860\pi\)
0.649782 + 0.760120i \(0.274860\pi\)
\(242\) 0 0
\(243\) − 859299.i − 0.933530i
\(244\) 0 0
\(245\) − 235258.i − 0.250397i
\(246\) 0 0
\(247\) 1.06256e6 1.10818
\(248\) 0 0
\(249\) 328122. 0.335380
\(250\) 0 0
\(251\) − 1.68512e6i − 1.68829i −0.536119 0.844143i \(-0.680110\pi\)
0.536119 0.844143i \(-0.319890\pi\)
\(252\) 0 0
\(253\) − 1.65086e6i − 1.62147i
\(254\) 0 0
\(255\) −148035. −0.142566
\(256\) 0 0
\(257\) −516986. −0.488255 −0.244127 0.969743i \(-0.578501\pi\)
−0.244127 + 0.969743i \(0.578501\pi\)
\(258\) 0 0
\(259\) − 934331.i − 0.865469i
\(260\) 0 0
\(261\) 1.17209e6i 1.06502i
\(262\) 0 0
\(263\) −101121. −0.0901467 −0.0450734 0.998984i \(-0.514352\pi\)
−0.0450734 + 0.998984i \(0.514352\pi\)
\(264\) 0 0
\(265\) −581880. −0.509001
\(266\) 0 0
\(267\) 689144.i 0.591605i
\(268\) 0 0
\(269\) 17814.9i 0.0150107i 0.999972 + 0.00750537i \(0.00238906\pi\)
−0.999972 + 0.00750537i \(0.997611\pi\)
\(270\) 0 0
\(271\) −1.82431e6 −1.50895 −0.754474 0.656330i \(-0.772108\pi\)
−0.754474 + 0.656330i \(0.772108\pi\)
\(272\) 0 0
\(273\) 339525. 0.275718
\(274\) 0 0
\(275\) − 351791.i − 0.280513i
\(276\) 0 0
\(277\) − 250172.i − 0.195903i −0.995191 0.0979513i \(-0.968771\pi\)
0.995191 0.0979513i \(-0.0312289\pi\)
\(278\) 0 0
\(279\) 1.42486e6 1.09588
\(280\) 0 0
\(281\) 1.21044e6 0.914490 0.457245 0.889341i \(-0.348836\pi\)
0.457245 + 0.889341i \(0.348836\pi\)
\(282\) 0 0
\(283\) 739043.i 0.548534i 0.961654 + 0.274267i \(0.0884352\pi\)
−0.961654 + 0.274267i \(0.911565\pi\)
\(284\) 0 0
\(285\) − 240800.i − 0.175608i
\(286\) 0 0
\(287\) −1.55266e6 −1.11269
\(288\) 0 0
\(289\) −440067. −0.309938
\(290\) 0 0
\(291\) 683674.i 0.473279i
\(292\) 0 0
\(293\) − 2.11599e6i − 1.43994i −0.694006 0.719969i \(-0.744156\pi\)
0.694006 0.719969i \(-0.255844\pi\)
\(294\) 0 0
\(295\) 227080. 0.151923
\(296\) 0 0
\(297\) 1.51594e6 0.997219
\(298\) 0 0
\(299\) − 1.93554e6i − 1.25206i
\(300\) 0 0
\(301\) 1.28131e6i 0.815148i
\(302\) 0 0
\(303\) 339988. 0.212744
\(304\) 0 0
\(305\) 161427. 0.0993637
\(306\) 0 0
\(307\) − 2.32583e6i − 1.40842i −0.709992 0.704210i \(-0.751301\pi\)
0.709992 0.704210i \(-0.248699\pi\)
\(308\) 0 0
\(309\) − 814208.i − 0.485109i
\(310\) 0 0
\(311\) 3.00391e6 1.76111 0.880554 0.473945i \(-0.157170\pi\)
0.880554 + 0.473945i \(0.157170\pi\)
\(312\) 0 0
\(313\) −1.07318e6 −0.619172 −0.309586 0.950871i \(-0.600190\pi\)
−0.309586 + 0.950871i \(0.600190\pi\)
\(314\) 0 0
\(315\) 445529.i 0.252988i
\(316\) 0 0
\(317\) 3.09103e6i 1.72765i 0.503795 + 0.863823i \(0.331937\pi\)
−0.503795 + 0.863823i \(0.668063\pi\)
\(318\) 0 0
\(319\) −3.18381e6 −1.75174
\(320\) 0 0
\(321\) 306558. 0.166054
\(322\) 0 0
\(323\) 1.59377e6i 0.849999i
\(324\) 0 0
\(325\) − 412453.i − 0.216604i
\(326\) 0 0
\(327\) 963068. 0.498067
\(328\) 0 0
\(329\) −648668. −0.330394
\(330\) 0 0
\(331\) 198204.i 0.0994356i 0.998763 + 0.0497178i \(0.0158322\pi\)
−0.998763 + 0.0497178i \(0.984168\pi\)
\(332\) 0 0
\(333\) − 2.25113e6i − 1.11248i
\(334\) 0 0
\(335\) −1.55038e6 −0.754792
\(336\) 0 0
\(337\) 2.06781e6 0.991828 0.495914 0.868372i \(-0.334833\pi\)
0.495914 + 0.868372i \(0.334833\pi\)
\(338\) 0 0
\(339\) 1.18057e6i 0.557945i
\(340\) 0 0
\(341\) 3.87042e6i 1.80249i
\(342\) 0 0
\(343\) −2.25479e6 −1.03484
\(344\) 0 0
\(345\) −438637. −0.198407
\(346\) 0 0
\(347\) − 1.97453e6i − 0.880319i −0.897920 0.440160i \(-0.854922\pi\)
0.897920 0.440160i \(-0.145078\pi\)
\(348\) 0 0
\(349\) 1.07261e6i 0.471388i 0.971827 + 0.235694i \(0.0757363\pi\)
−0.971827 + 0.235694i \(0.924264\pi\)
\(350\) 0 0
\(351\) 1.77735e6 0.770024
\(352\) 0 0
\(353\) −2.81294e6 −1.20150 −0.600750 0.799437i \(-0.705131\pi\)
−0.600750 + 0.799437i \(0.705131\pi\)
\(354\) 0 0
\(355\) 1.23867e6i 0.521658i
\(356\) 0 0
\(357\) 509264.i 0.211482i
\(358\) 0 0
\(359\) 2.16815e6 0.887877 0.443939 0.896057i \(-0.353581\pi\)
0.443939 + 0.896057i \(0.353581\pi\)
\(360\) 0 0
\(361\) −116388. −0.0470048
\(362\) 0 0
\(363\) 931820.i 0.371164i
\(364\) 0 0
\(365\) 53937.1i 0.0211912i
\(366\) 0 0
\(367\) 4.48484e6 1.73813 0.869064 0.494700i \(-0.164722\pi\)
0.869064 + 0.494700i \(0.164722\pi\)
\(368\) 0 0
\(369\) −3.74091e6 −1.43025
\(370\) 0 0
\(371\) 2.00176e6i 0.755052i
\(372\) 0 0
\(373\) 5669.92i 0.00211011i 0.999999 + 0.00105505i \(0.000335834\pi\)
−0.999999 + 0.00105505i \(0.999664\pi\)
\(374\) 0 0
\(375\) −93471.4 −0.0343242
\(376\) 0 0
\(377\) −3.73282e6 −1.35264
\(378\) 0 0
\(379\) 3.47374e6i 1.24222i 0.783723 + 0.621110i \(0.213318\pi\)
−0.783723 + 0.621110i \(0.786682\pi\)
\(380\) 0 0
\(381\) − 323199.i − 0.114066i
\(382\) 0 0
\(383\) −5.16714e6 −1.79992 −0.899960 0.435972i \(-0.856405\pi\)
−0.899960 + 0.435972i \(0.856405\pi\)
\(384\) 0 0
\(385\) −1.21021e6 −0.416112
\(386\) 0 0
\(387\) 3.08712e6i 1.04779i
\(388\) 0 0
\(389\) − 2.58648e6i − 0.866634i −0.901242 0.433317i \(-0.857343\pi\)
0.901242 0.433317i \(-0.142657\pi\)
\(390\) 0 0
\(391\) 2.90318e6 0.960354
\(392\) 0 0
\(393\) 267169. 0.0872580
\(394\) 0 0
\(395\) 641578.i 0.206898i
\(396\) 0 0
\(397\) − 3.62059e6i − 1.15293i −0.817122 0.576465i \(-0.804432\pi\)
0.817122 0.576465i \(-0.195568\pi\)
\(398\) 0 0
\(399\) −828390. −0.260497
\(400\) 0 0
\(401\) 33479.2 0.0103971 0.00519857 0.999986i \(-0.498345\pi\)
0.00519857 + 0.999986i \(0.498345\pi\)
\(402\) 0 0
\(403\) 4.53783e6i 1.39183i
\(404\) 0 0
\(405\) 856035.i 0.259331i
\(406\) 0 0
\(407\) 6.11487e6 1.82979
\(408\) 0 0
\(409\) 5.73047e6 1.69388 0.846939 0.531690i \(-0.178443\pi\)
0.846939 + 0.531690i \(0.178443\pi\)
\(410\) 0 0
\(411\) − 1.49686e6i − 0.437096i
\(412\) 0 0
\(413\) − 781191.i − 0.225363i
\(414\) 0 0
\(415\) −1.37125e6 −0.390838
\(416\) 0 0
\(417\) 583897. 0.164436
\(418\) 0 0
\(419\) − 831585.i − 0.231404i −0.993284 0.115702i \(-0.963088\pi\)
0.993284 0.115702i \(-0.0369118\pi\)
\(420\) 0 0
\(421\) 4.09206e6i 1.12522i 0.826723 + 0.562609i \(0.190202\pi\)
−0.826723 + 0.562609i \(0.809798\pi\)
\(422\) 0 0
\(423\) −1.56287e6 −0.424690
\(424\) 0 0
\(425\) 618652. 0.166140
\(426\) 0 0
\(427\) − 555336.i − 0.147396i
\(428\) 0 0
\(429\) 2.22207e6i 0.582927i
\(430\) 0 0
\(431\) 3.02709e6 0.784932 0.392466 0.919766i \(-0.371622\pi\)
0.392466 + 0.919766i \(0.371622\pi\)
\(432\) 0 0
\(433\) −4.34502e6 −1.11371 −0.556855 0.830609i \(-0.687992\pi\)
−0.556855 + 0.830609i \(0.687992\pi\)
\(434\) 0 0
\(435\) 845943.i 0.214347i
\(436\) 0 0
\(437\) 4.72243e6i 1.18294i
\(438\) 0 0
\(439\) 7.30491e6 1.80906 0.904532 0.426406i \(-0.140221\pi\)
0.904532 + 0.426406i \(0.140221\pi\)
\(440\) 0 0
\(441\) −1.94995e6 −0.477449
\(442\) 0 0
\(443\) 6.50733e6i 1.57541i 0.616053 + 0.787705i \(0.288731\pi\)
−0.616053 + 0.787705i \(0.711269\pi\)
\(444\) 0 0
\(445\) − 2.87999e6i − 0.689431i
\(446\) 0 0
\(447\) −1.64238e6 −0.388780
\(448\) 0 0
\(449\) −3.22440e6 −0.754802 −0.377401 0.926050i \(-0.623182\pi\)
−0.377401 + 0.926050i \(0.623182\pi\)
\(450\) 0 0
\(451\) − 1.01616e7i − 2.35246i
\(452\) 0 0
\(453\) − 1.98592e6i − 0.454690i
\(454\) 0 0
\(455\) −1.41890e6 −0.321310
\(456\) 0 0
\(457\) −5.81776e6 −1.30306 −0.651531 0.758622i \(-0.725873\pi\)
−0.651531 + 0.758622i \(0.725873\pi\)
\(458\) 0 0
\(459\) 2.66590e6i 0.590625i
\(460\) 0 0
\(461\) 7.60005e6i 1.66557i 0.553594 + 0.832787i \(0.313256\pi\)
−0.553594 + 0.832787i \(0.686744\pi\)
\(462\) 0 0
\(463\) −8.98478e6 −1.94785 −0.973923 0.226877i \(-0.927148\pi\)
−0.973923 + 0.226877i \(0.927148\pi\)
\(464\) 0 0
\(465\) 1.02838e6 0.220556
\(466\) 0 0
\(467\) 1.10558e6i 0.234584i 0.993097 + 0.117292i \(0.0374214\pi\)
−0.993097 + 0.117292i \(0.962579\pi\)
\(468\) 0 0
\(469\) 5.33356e6i 1.11966i
\(470\) 0 0
\(471\) 1.73173e6 0.359690
\(472\) 0 0
\(473\) −8.38569e6 −1.72340
\(474\) 0 0
\(475\) 1.00633e6i 0.204647i
\(476\) 0 0
\(477\) 4.82294e6i 0.970545i
\(478\) 0 0
\(479\) 3.48384e6 0.693777 0.346888 0.937906i \(-0.387238\pi\)
0.346888 + 0.937906i \(0.387238\pi\)
\(480\) 0 0
\(481\) 7.16931e6 1.41291
\(482\) 0 0
\(483\) 1.50898e6i 0.294317i
\(484\) 0 0
\(485\) − 2.85713e6i − 0.551539i
\(486\) 0 0
\(487\) 1.57518e6 0.300958 0.150479 0.988613i \(-0.451918\pi\)
0.150479 + 0.988613i \(0.451918\pi\)
\(488\) 0 0
\(489\) −1.39148e6 −0.263150
\(490\) 0 0
\(491\) 8.61403e6i 1.61251i 0.591568 + 0.806255i \(0.298509\pi\)
−0.591568 + 0.806255i \(0.701491\pi\)
\(492\) 0 0
\(493\) − 5.59898e6i − 1.03751i
\(494\) 0 0
\(495\) −2.91583e6 −0.534872
\(496\) 0 0
\(497\) 4.26123e6 0.773827
\(498\) 0 0
\(499\) 385806.i 0.0693613i 0.999398 + 0.0346806i \(0.0110414\pi\)
−0.999398 + 0.0346806i \(0.988959\pi\)
\(500\) 0 0
\(501\) 310776.i 0.0553163i
\(502\) 0 0
\(503\) 4.61083e6 0.812567 0.406283 0.913747i \(-0.366825\pi\)
0.406283 + 0.913747i \(0.366825\pi\)
\(504\) 0 0
\(505\) −1.42084e6 −0.247923
\(506\) 0 0
\(507\) 384104.i 0.0663634i
\(508\) 0 0
\(509\) 57696.0i 0.00987078i 0.999988 + 0.00493539i \(0.00157099\pi\)
−0.999988 + 0.00493539i \(0.998429\pi\)
\(510\) 0 0
\(511\) 185552. 0.0314350
\(512\) 0 0
\(513\) −4.33646e6 −0.727516
\(514\) 0 0
\(515\) 3.40265e6i 0.565326i
\(516\) 0 0
\(517\) − 4.24530e6i − 0.698526i
\(518\) 0 0
\(519\) 2.18145e6 0.355490
\(520\) 0 0
\(521\) 6.74455e6 1.08858 0.544288 0.838898i \(-0.316800\pi\)
0.544288 + 0.838898i \(0.316800\pi\)
\(522\) 0 0
\(523\) − 1.12870e7i − 1.80436i −0.431356 0.902182i \(-0.641965\pi\)
0.431356 0.902182i \(-0.358035\pi\)
\(524\) 0 0
\(525\) 321556.i 0.0509165i
\(526\) 0 0
\(527\) −6.80644e6 −1.06756
\(528\) 0 0
\(529\) 2.16594e6 0.336517
\(530\) 0 0
\(531\) − 1.88217e6i − 0.289682i
\(532\) 0 0
\(533\) − 1.19139e7i − 1.81650i
\(534\) 0 0
\(535\) −1.28113e6 −0.193513
\(536\) 0 0
\(537\) −504133. −0.0754413
\(538\) 0 0
\(539\) − 5.29675e6i − 0.785303i
\(540\) 0 0
\(541\) 8.45796e6i 1.24243i 0.783640 + 0.621216i \(0.213361\pi\)
−0.783640 + 0.621216i \(0.786639\pi\)
\(542\) 0 0
\(543\) −548475. −0.0798284
\(544\) 0 0
\(545\) −4.02475e6 −0.580426
\(546\) 0 0
\(547\) − 4.58748e6i − 0.655550i −0.944756 0.327775i \(-0.893701\pi\)
0.944756 0.327775i \(-0.106299\pi\)
\(548\) 0 0
\(549\) − 1.33800e6i − 0.189463i
\(550\) 0 0
\(551\) 9.10753e6 1.27797
\(552\) 0 0
\(553\) 2.20713e6 0.306913
\(554\) 0 0
\(555\) − 1.62473e6i − 0.223897i
\(556\) 0 0
\(557\) 5.35598e6i 0.731478i 0.930718 + 0.365739i \(0.119184\pi\)
−0.930718 + 0.365739i \(0.880816\pi\)
\(558\) 0 0
\(559\) −9.83171e6 −1.33076
\(560\) 0 0
\(561\) −3.33296e6 −0.447118
\(562\) 0 0
\(563\) 2.30053e6i 0.305884i 0.988235 + 0.152942i \(0.0488747\pi\)
−0.988235 + 0.152942i \(0.951125\pi\)
\(564\) 0 0
\(565\) − 4.93369e6i − 0.650206i
\(566\) 0 0
\(567\) 2.94489e6 0.384691
\(568\) 0 0
\(569\) 5.12794e6 0.663991 0.331996 0.943281i \(-0.392278\pi\)
0.331996 + 0.943281i \(0.392278\pi\)
\(570\) 0 0
\(571\) − 7.88741e6i − 1.01238i −0.862421 0.506191i \(-0.831053\pi\)
0.862421 0.506191i \(-0.168947\pi\)
\(572\) 0 0
\(573\) − 3.29543e6i − 0.419301i
\(574\) 0 0
\(575\) 1.83310e6 0.231216
\(576\) 0 0
\(577\) 6.38694e6 0.798644 0.399322 0.916811i \(-0.369246\pi\)
0.399322 + 0.916811i \(0.369246\pi\)
\(578\) 0 0
\(579\) − 4.22213e6i − 0.523402i
\(580\) 0 0
\(581\) 4.71732e6i 0.579769i
\(582\) 0 0
\(583\) −1.31008e7 −1.59634
\(584\) 0 0
\(585\) −3.41864e6 −0.413013
\(586\) 0 0
\(587\) − 2.32088e6i − 0.278009i −0.990292 0.139004i \(-0.955610\pi\)
0.990292 0.139004i \(-0.0443902\pi\)
\(588\) 0 0
\(589\) − 1.10716e7i − 1.31499i
\(590\) 0 0
\(591\) −2.43779e6 −0.287097
\(592\) 0 0
\(593\) 3.95109e6 0.461403 0.230701 0.973025i \(-0.425898\pi\)
0.230701 + 0.973025i \(0.425898\pi\)
\(594\) 0 0
\(595\) − 2.12826e6i − 0.246452i
\(596\) 0 0
\(597\) 738078.i 0.0847553i
\(598\) 0 0
\(599\) −1.10428e7 −1.25751 −0.628755 0.777603i \(-0.716435\pi\)
−0.628755 + 0.777603i \(0.716435\pi\)
\(600\) 0 0
\(601\) 2.97957e6 0.336486 0.168243 0.985746i \(-0.446191\pi\)
0.168243 + 0.985746i \(0.446191\pi\)
\(602\) 0 0
\(603\) 1.28504e7i 1.43921i
\(604\) 0 0
\(605\) − 3.89416e6i − 0.432539i
\(606\) 0 0
\(607\) 4.18732e6 0.461280 0.230640 0.973039i \(-0.425918\pi\)
0.230640 + 0.973039i \(0.425918\pi\)
\(608\) 0 0
\(609\) 2.91017e6 0.317962
\(610\) 0 0
\(611\) − 4.97736e6i − 0.539381i
\(612\) 0 0
\(613\) − 1.29891e7i − 1.39613i −0.716033 0.698066i \(-0.754044\pi\)
0.716033 0.698066i \(-0.245956\pi\)
\(614\) 0 0
\(615\) −2.69996e6 −0.287852
\(616\) 0 0
\(617\) 1.67330e7 1.76954 0.884769 0.466030i \(-0.154316\pi\)
0.884769 + 0.466030i \(0.154316\pi\)
\(618\) 0 0
\(619\) − 3.29552e6i − 0.345699i −0.984948 0.172849i \(-0.944703\pi\)
0.984948 0.172849i \(-0.0552974\pi\)
\(620\) 0 0
\(621\) 7.89922e6i 0.821968i
\(622\) 0 0
\(623\) −9.90762e6 −1.02270
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 5.42152e6i − 0.550748i
\(628\) 0 0
\(629\) 1.07535e7i 1.08373i
\(630\) 0 0
\(631\) 1.18430e7 1.18410 0.592050 0.805901i \(-0.298319\pi\)
0.592050 + 0.805901i \(0.298319\pi\)
\(632\) 0 0
\(633\) −2.52365e6 −0.250334
\(634\) 0 0
\(635\) 1.35067e6i 0.132928i
\(636\) 0 0
\(637\) − 6.21011e6i − 0.606389i
\(638\) 0 0
\(639\) 1.02668e7 0.994680
\(640\) 0 0
\(641\) 8.99649e6 0.864824 0.432412 0.901676i \(-0.357663\pi\)
0.432412 + 0.901676i \(0.357663\pi\)
\(642\) 0 0
\(643\) 9.77796e6i 0.932655i 0.884612 + 0.466327i \(0.154423\pi\)
−0.884612 + 0.466327i \(0.845577\pi\)
\(644\) 0 0
\(645\) 2.22809e6i 0.210879i
\(646\) 0 0
\(647\) 4.52186e6 0.424675 0.212337 0.977196i \(-0.431892\pi\)
0.212337 + 0.977196i \(0.431892\pi\)
\(648\) 0 0
\(649\) 5.11262e6 0.476466
\(650\) 0 0
\(651\) − 3.53777e6i − 0.327173i
\(652\) 0 0
\(653\) − 6.59880e6i − 0.605594i −0.953055 0.302797i \(-0.902080\pi\)
0.953055 0.302797i \(-0.0979204\pi\)
\(654\) 0 0
\(655\) −1.11652e6 −0.101687
\(656\) 0 0
\(657\) 447060. 0.0404066
\(658\) 0 0
\(659\) − 2.23865e6i − 0.200804i −0.994947 0.100402i \(-0.967987\pi\)
0.994947 0.100402i \(-0.0320130\pi\)
\(660\) 0 0
\(661\) − 1.30535e7i − 1.16205i −0.813887 0.581023i \(-0.802652\pi\)
0.813887 0.581023i \(-0.197348\pi\)
\(662\) 0 0
\(663\) −3.90769e6 −0.345252
\(664\) 0 0
\(665\) 3.46191e6 0.303572
\(666\) 0 0
\(667\) − 1.65901e7i − 1.44389i
\(668\) 0 0
\(669\) 7.95380e6i 0.687084i
\(670\) 0 0
\(671\) 3.63448e6 0.311627
\(672\) 0 0
\(673\) 2.18780e7 1.86196 0.930980 0.365071i \(-0.118955\pi\)
0.930980 + 0.365071i \(0.118955\pi\)
\(674\) 0 0
\(675\) 1.68328e6i 0.142199i
\(676\) 0 0
\(677\) 1.11662e7i 0.936344i 0.883637 + 0.468172i \(0.155087\pi\)
−0.883637 + 0.468172i \(0.844913\pi\)
\(678\) 0 0
\(679\) −9.82898e6 −0.818152
\(680\) 0 0
\(681\) 986043. 0.0814757
\(682\) 0 0
\(683\) − 5.63440e6i − 0.462164i −0.972934 0.231082i \(-0.925773\pi\)
0.972934 0.231082i \(-0.0742266\pi\)
\(684\) 0 0
\(685\) 6.25551e6i 0.509373i
\(686\) 0 0
\(687\) 5.27556e6 0.426458
\(688\) 0 0
\(689\) −1.53599e7 −1.23265
\(690\) 0 0
\(691\) − 8.17015e6i − 0.650931i −0.945554 0.325466i \(-0.894479\pi\)
0.945554 0.325466i \(-0.105521\pi\)
\(692\) 0 0
\(693\) 1.00309e7i 0.793428i
\(694\) 0 0
\(695\) −2.44015e6 −0.191626
\(696\) 0 0
\(697\) 1.78700e7 1.39330
\(698\) 0 0
\(699\) 3.14379e6i 0.243367i
\(700\) 0 0
\(701\) − 2.43711e7i − 1.87318i −0.350421 0.936592i \(-0.613962\pi\)
0.350421 0.936592i \(-0.386038\pi\)
\(702\) 0 0
\(703\) −1.74921e7 −1.33491
\(704\) 0 0
\(705\) −1.12798e6 −0.0854733
\(706\) 0 0
\(707\) 4.88791e6i 0.367768i
\(708\) 0 0
\(709\) − 1.61049e7i − 1.20322i −0.798791 0.601608i \(-0.794527\pi\)
0.798791 0.601608i \(-0.205473\pi\)
\(710\) 0 0
\(711\) 5.31775e6 0.394506
\(712\) 0 0
\(713\) −2.01679e7 −1.48572
\(714\) 0 0
\(715\) − 9.28622e6i − 0.679319i
\(716\) 0 0
\(717\) 1.93128e6i 0.140296i
\(718\) 0 0
\(719\) 7.41860e6 0.535180 0.267590 0.963533i \(-0.413773\pi\)
0.267590 + 0.963533i \(0.413773\pi\)
\(720\) 0 0
\(721\) 1.17056e7 0.838604
\(722\) 0 0
\(723\) − 7.00970e6i − 0.498716i
\(724\) 0 0
\(725\) − 3.53527e6i − 0.249791i
\(726\) 0 0
\(727\) −1.45557e7 −1.02140 −0.510702 0.859758i \(-0.670614\pi\)
−0.510702 + 0.859758i \(0.670614\pi\)
\(728\) 0 0
\(729\) 3.18019e6 0.221633
\(730\) 0 0
\(731\) − 1.47469e7i − 1.02072i
\(732\) 0 0
\(733\) − 1.37973e7i − 0.948492i −0.880392 0.474246i \(-0.842721\pi\)
0.880392 0.474246i \(-0.157279\pi\)
\(734\) 0 0
\(735\) −1.40736e6 −0.0960916
\(736\) 0 0
\(737\) −3.49063e7 −2.36720
\(738\) 0 0
\(739\) − 2.41030e7i − 1.62353i −0.583984 0.811765i \(-0.698507\pi\)
0.583984 0.811765i \(-0.301493\pi\)
\(740\) 0 0
\(741\) − 6.35641e6i − 0.425271i
\(742\) 0 0
\(743\) −1.09987e7 −0.730919 −0.365459 0.930827i \(-0.619088\pi\)
−0.365459 + 0.930827i \(0.619088\pi\)
\(744\) 0 0
\(745\) 6.86363e6 0.453068
\(746\) 0 0
\(747\) 1.13657e7i 0.745236i
\(748\) 0 0
\(749\) 4.40729e6i 0.287057i
\(750\) 0 0
\(751\) 1.89927e7 1.22882 0.614408 0.788988i \(-0.289395\pi\)
0.614408 + 0.788988i \(0.289395\pi\)
\(752\) 0 0
\(753\) −1.00807e7 −0.647890
\(754\) 0 0
\(755\) 8.29932e6i 0.529877i
\(756\) 0 0
\(757\) 2.30676e7i 1.46306i 0.681810 + 0.731530i \(0.261193\pi\)
−0.681810 + 0.731530i \(0.738807\pi\)
\(758\) 0 0
\(759\) −9.87575e6 −0.622251
\(760\) 0 0
\(761\) −1.19864e6 −0.0750287 −0.0375143 0.999296i \(-0.511944\pi\)
−0.0375143 + 0.999296i \(0.511944\pi\)
\(762\) 0 0
\(763\) 1.38458e7i 0.861004i
\(764\) 0 0
\(765\) − 5.12773e6i − 0.316790i
\(766\) 0 0
\(767\) 5.99424e6 0.367913
\(768\) 0 0
\(769\) 1.94617e7 1.18677 0.593384 0.804919i \(-0.297792\pi\)
0.593384 + 0.804919i \(0.297792\pi\)
\(770\) 0 0
\(771\) 3.09270e6i 0.187371i
\(772\) 0 0
\(773\) 3.46693e6i 0.208688i 0.994541 + 0.104344i \(0.0332742\pi\)
−0.994541 + 0.104344i \(0.966726\pi\)
\(774\) 0 0
\(775\) −4.29767e6 −0.257027
\(776\) 0 0
\(777\) −5.58933e6 −0.332129
\(778\) 0 0
\(779\) 2.90681e7i 1.71622i
\(780\) 0 0
\(781\) 2.78883e7i 1.63604i
\(782\) 0 0
\(783\) 1.52342e7 0.888005
\(784\) 0 0
\(785\) −7.23706e6 −0.419168
\(786\) 0 0
\(787\) 2.32482e7i 1.33799i 0.743266 + 0.668996i \(0.233276\pi\)
−0.743266 + 0.668996i \(0.766724\pi\)
\(788\) 0 0
\(789\) 604920.i 0.0345944i
\(790\) 0 0
\(791\) −1.69727e7 −0.964515
\(792\) 0 0
\(793\) 4.26120e6 0.240630
\(794\) 0 0
\(795\) 3.48090e6i 0.195332i
\(796\) 0 0
\(797\) − 9.84771e6i − 0.549148i −0.961566 0.274574i \(-0.911463\pi\)
0.961566 0.274574i \(-0.0885368\pi\)
\(798\) 0 0
\(799\) 7.46570e6 0.413717
\(800\) 0 0
\(801\) −2.38709e7 −1.31458
\(802\) 0 0
\(803\) 1.21437e6i 0.0664605i
\(804\) 0 0
\(805\) − 6.30616e6i − 0.342985i
\(806\) 0 0
\(807\) 106572. 0.00576047
\(808\) 0 0
\(809\) −2.27282e7 −1.22094 −0.610470 0.792040i \(-0.709019\pi\)
−0.610470 + 0.792040i \(0.709019\pi\)
\(810\) 0 0
\(811\) − 6.46981e6i − 0.345413i −0.984973 0.172707i \(-0.944749\pi\)
0.984973 0.172707i \(-0.0552513\pi\)
\(812\) 0 0
\(813\) 1.09133e7i 0.579068i
\(814\) 0 0
\(815\) 5.81509e6 0.306664
\(816\) 0 0
\(817\) 2.39879e7 1.25730
\(818\) 0 0
\(819\) 1.17606e7i 0.612663i
\(820\) 0 0
\(821\) 2.40123e7i 1.24330i 0.783295 + 0.621651i \(0.213538\pi\)
−0.783295 + 0.621651i \(0.786462\pi\)
\(822\) 0 0
\(823\) 1.96788e7 1.01274 0.506372 0.862315i \(-0.330986\pi\)
0.506372 + 0.862315i \(0.330986\pi\)
\(824\) 0 0
\(825\) −2.10447e6 −0.107649
\(826\) 0 0
\(827\) − 1.32667e7i − 0.674525i −0.941411 0.337263i \(-0.890499\pi\)
0.941411 0.337263i \(-0.109501\pi\)
\(828\) 0 0
\(829\) 3.47443e6i 0.175589i 0.996139 + 0.0877945i \(0.0279819\pi\)
−0.996139 + 0.0877945i \(0.972018\pi\)
\(830\) 0 0
\(831\) −1.49657e6 −0.0751788
\(832\) 0 0
\(833\) 9.31475e6 0.465113
\(834\) 0 0
\(835\) − 1.29876e6i − 0.0644633i
\(836\) 0 0
\(837\) − 1.85196e7i − 0.913728i
\(838\) 0 0
\(839\) 2.38012e7 1.16733 0.583667 0.811993i \(-0.301618\pi\)
0.583667 + 0.811993i \(0.301618\pi\)
\(840\) 0 0
\(841\) −1.14840e7 −0.559893
\(842\) 0 0
\(843\) − 7.24108e6i − 0.350941i
\(844\) 0 0
\(845\) − 1.60520e6i − 0.0773372i
\(846\) 0 0
\(847\) −1.33965e7 −0.641628
\(848\) 0 0
\(849\) 4.42108e6 0.210503
\(850\) 0 0
\(851\) 3.18632e7i 1.50822i
\(852\) 0 0
\(853\) 1.71488e7i 0.806978i 0.914984 + 0.403489i \(0.132203\pi\)
−0.914984 + 0.403489i \(0.867797\pi\)
\(854\) 0 0
\(855\) 8.34097e6 0.390213
\(856\) 0 0
\(857\) −1.72604e7 −0.802782 −0.401391 0.915907i \(-0.631473\pi\)
−0.401391 + 0.915907i \(0.631473\pi\)
\(858\) 0 0
\(859\) − 4.82808e6i − 0.223250i −0.993750 0.111625i \(-0.964394\pi\)
0.993750 0.111625i \(-0.0356056\pi\)
\(860\) 0 0
\(861\) 9.28829e6i 0.427000i
\(862\) 0 0
\(863\) 1.74637e7 0.798197 0.399098 0.916908i \(-0.369323\pi\)
0.399098 + 0.916908i \(0.369323\pi\)
\(864\) 0 0
\(865\) −9.11647e6 −0.414273
\(866\) 0 0
\(867\) 2.63256e6i 0.118941i
\(868\) 0 0
\(869\) 1.44449e7i 0.648880i
\(870\) 0 0
\(871\) −4.09255e7 −1.82788
\(872\) 0 0
\(873\) −2.36815e7 −1.05166
\(874\) 0 0
\(875\) − 1.34381e6i − 0.0593360i
\(876\) 0 0
\(877\) − 1.08269e7i − 0.475343i −0.971346 0.237671i \(-0.923616\pi\)
0.971346 0.237671i \(-0.0763842\pi\)
\(878\) 0 0
\(879\) −1.26582e7 −0.552585
\(880\) 0 0
\(881\) −3.30776e6 −0.143580 −0.0717900 0.997420i \(-0.522871\pi\)
−0.0717900 + 0.997420i \(0.522871\pi\)
\(882\) 0 0
\(883\) − 4.32516e7i − 1.86681i −0.358821 0.933407i \(-0.616821\pi\)
0.358821 0.933407i \(-0.383179\pi\)
\(884\) 0 0
\(885\) − 1.35843e6i − 0.0583015i
\(886\) 0 0
\(887\) −7.05387e6 −0.301036 −0.150518 0.988607i \(-0.548094\pi\)
−0.150518 + 0.988607i \(0.548094\pi\)
\(888\) 0 0
\(889\) 4.64653e6 0.197185
\(890\) 0 0
\(891\) 1.92733e7i 0.813321i
\(892\) 0 0
\(893\) 1.21440e7i 0.509605i
\(894\) 0 0
\(895\) 2.10681e6 0.0879162
\(896\) 0 0
\(897\) −1.15787e7 −0.480484
\(898\) 0 0
\(899\) 3.88952e7i 1.60508i
\(900\) 0 0
\(901\) − 2.30388e7i − 0.945470i
\(902\) 0 0
\(903\) 7.66498e6 0.312818
\(904\) 0 0
\(905\) 2.29212e6 0.0930287
\(906\) 0 0
\(907\) 370745.i 0.0149643i 0.999972 + 0.00748216i \(0.00238167\pi\)
−0.999972 + 0.00748216i \(0.997618\pi\)
\(908\) 0 0
\(909\) 1.17767e7i 0.472730i
\(910\) 0 0
\(911\) 4.25833e7 1.69998 0.849989 0.526800i \(-0.176608\pi\)
0.849989 + 0.526800i \(0.176608\pi\)
\(912\) 0 0
\(913\) −3.08732e7 −1.22576
\(914\) 0 0
\(915\) − 965686.i − 0.0381315i
\(916\) 0 0
\(917\) 3.84102e6i 0.150842i
\(918\) 0 0
\(919\) 1.10538e7 0.431742 0.215871 0.976422i \(-0.430741\pi\)
0.215871 + 0.976422i \(0.430741\pi\)
\(920\) 0 0
\(921\) −1.39135e7 −0.540490
\(922\) 0 0
\(923\) 3.26973e7i 1.26330i
\(924\) 0 0
\(925\) 6.78989e6i 0.260921i
\(926\) 0 0
\(927\) 2.82030e7 1.07794
\(928\) 0 0
\(929\) 1.17601e7 0.447068 0.223534 0.974696i \(-0.428241\pi\)
0.223534 + 0.974696i \(0.428241\pi\)
\(930\) 0 0
\(931\) 1.51518e7i 0.572914i
\(932\) 0 0
\(933\) − 1.79699e7i − 0.675837i
\(934\) 0 0
\(935\) 1.39287e7 0.521053
\(936\) 0 0
\(937\) 2.73257e6 0.101677 0.0508384 0.998707i \(-0.483811\pi\)
0.0508384 + 0.998707i \(0.483811\pi\)
\(938\) 0 0
\(939\) 6.41993e6i 0.237611i
\(940\) 0 0
\(941\) − 1.09153e7i − 0.401850i −0.979607 0.200925i \(-0.935605\pi\)
0.979607 0.200925i \(-0.0643947\pi\)
\(942\) 0 0
\(943\) 5.29500e7 1.93904
\(944\) 0 0
\(945\) 5.79076e6 0.210938
\(946\) 0 0
\(947\) 2.45793e7i 0.890624i 0.895375 + 0.445312i \(0.146907\pi\)
−0.895375 + 0.445312i \(0.853093\pi\)
\(948\) 0 0
\(949\) 1.42378e6i 0.0513188i
\(950\) 0 0
\(951\) 1.84911e7 0.662995
\(952\) 0 0
\(953\) 1.84534e7 0.658179 0.329090 0.944299i \(-0.393258\pi\)
0.329090 + 0.944299i \(0.393258\pi\)
\(954\) 0 0
\(955\) 1.37719e7i 0.488636i
\(956\) 0 0
\(957\) 1.90461e7i 0.672242i
\(958\) 0 0
\(959\) 2.15199e7 0.755604
\(960\) 0 0
\(961\) 1.86541e7 0.651576
\(962\) 0 0
\(963\) 1.06187e7i 0.368983i
\(964\) 0 0
\(965\) 1.76447e7i 0.609951i
\(966\) 0 0
\(967\) −4.26565e7 −1.46696 −0.733482 0.679709i \(-0.762106\pi\)
−0.733482 + 0.679709i \(0.762106\pi\)
\(968\) 0 0
\(969\) 9.53418e6 0.326193
\(970\) 0 0
\(971\) 4.69578e7i 1.59831i 0.601127 + 0.799153i \(0.294718\pi\)
−0.601127 + 0.799153i \(0.705282\pi\)
\(972\) 0 0
\(973\) 8.39451e6i 0.284258i
\(974\) 0 0
\(975\) −2.46736e6 −0.0831231
\(976\) 0 0
\(977\) −1.67491e7 −0.561376 −0.280688 0.959799i \(-0.590563\pi\)
−0.280688 + 0.959799i \(0.590563\pi\)
\(978\) 0 0
\(979\) − 6.48419e7i − 2.16221i
\(980\) 0 0
\(981\) 3.33593e7i 1.10674i
\(982\) 0 0
\(983\) −5.08679e7 −1.67904 −0.839518 0.543332i \(-0.817163\pi\)
−0.839518 + 0.543332i \(0.817163\pi\)
\(984\) 0 0
\(985\) 1.01878e7 0.334571
\(986\) 0 0
\(987\) 3.88044e6i 0.126791i
\(988\) 0 0
\(989\) − 4.36960e7i − 1.42053i
\(990\) 0 0
\(991\) −5.37887e7 −1.73983 −0.869916 0.493200i \(-0.835827\pi\)
−0.869916 + 0.493200i \(0.835827\pi\)
\(992\) 0 0
\(993\) 1.18569e6 0.0381591
\(994\) 0 0
\(995\) − 3.08449e6i − 0.0987703i
\(996\) 0 0
\(997\) − 2.20907e7i − 0.703837i −0.936031 0.351919i \(-0.885529\pi\)
0.936031 0.351919i \(-0.114471\pi\)
\(998\) 0 0
\(999\) −2.92590e7 −0.927569
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 320.6.d.d.161.4 yes 12
4.3 odd 2 320.6.d.c.161.9 yes 12
8.3 odd 2 320.6.d.c.161.4 12
8.5 even 2 inner 320.6.d.d.161.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.c.161.4 12 8.3 odd 2
320.6.d.c.161.9 yes 12 4.3 odd 2
320.6.d.d.161.4 yes 12 1.1 even 1 trivial
320.6.d.d.161.9 yes 12 8.5 even 2 inner