Properties

Label 320.6.d.a.161.8
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.8
Root \(14.2856 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.a.161.1

$q$-expansion

\(f(q)\) \(=\) \(q+27.5713i q^{3} +25.0000i q^{5} -220.359 q^{7} -517.174 q^{9} +O(q^{10})\) \(q+27.5713i q^{3} +25.0000i q^{5} -220.359 q^{7} -517.174 q^{9} -64.4115i q^{11} +703.006i q^{13} -689.281 q^{15} -207.573 q^{17} +2790.91i q^{19} -6075.56i q^{21} -3258.47 q^{23} -625.000 q^{25} -7559.32i q^{27} -7896.13i q^{29} +5051.23 q^{31} +1775.91 q^{33} -5508.96i q^{35} -4915.89i q^{37} -19382.8 q^{39} +7036.16 q^{41} +15147.0i q^{43} -12929.4i q^{45} -9305.48 q^{47} +31750.9 q^{49} -5723.05i q^{51} -4938.18i q^{53} +1610.29 q^{55} -76949.0 q^{57} +38441.5i q^{59} +27779.9i q^{61} +113964. q^{63} -17575.2 q^{65} -7780.18i q^{67} -89840.2i q^{69} +60101.8 q^{71} +36397.3 q^{73} -17232.0i q^{75} +14193.6i q^{77} +36768.7 q^{79} +82746.7 q^{81} -21132.9i q^{83} -5189.32i q^{85} +217706. q^{87} -63887.8 q^{89} -154913. i q^{91} +139269. i q^{93} -69772.9 q^{95} -14388.5 q^{97} +33312.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 320q^{7} - 1192q^{9} + O(q^{10}) \) \( 8q - 320q^{7} - 1192q^{9} - 2400q^{17} + 1760q^{23} - 5000q^{25} + 31040q^{31} + 3760q^{33} + 4480q^{39} + 21584q^{41} + 47680q^{47} + 82824q^{49} + 18400q^{55} - 106640q^{57} + 322400q^{63} - 44000q^{65} + 246720q^{71} + 46400q^{73} + 325760q^{79} - 82328q^{81} + 636320q^{87} - 78192q^{89} - 40800q^{95} + 24960q^{97} + O(q^{100}) \)

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
\(3\) 27.5713i 1.76870i 0.466828 + 0.884348i \(0.345397\pi\)
−0.466828 + 0.884348i \(0.654603\pi\)
\(4\) 0 0
\(5\) 25.0000i 0.447214i
\(6\) 0 0
\(7\) −220.359 −1.69975 −0.849875 0.526985i \(-0.823322\pi\)
−0.849875 + 0.526985i \(0.823322\pi\)
\(8\) 0 0
\(9\) −517.174 −2.12829
\(10\) 0 0
\(11\) − 64.4115i − 0.160503i −0.996775 0.0802513i \(-0.974428\pi\)
0.996775 0.0802513i \(-0.0255723\pi\)
\(12\) 0 0
\(13\) 703.006i 1.15372i 0.816843 + 0.576860i \(0.195722\pi\)
−0.816843 + 0.576860i \(0.804278\pi\)
\(14\) 0 0
\(15\) −689.281 −0.790985
\(16\) 0 0
\(17\) −207.573 −0.174200 −0.0871001 0.996200i \(-0.527760\pi\)
−0.0871001 + 0.996200i \(0.527760\pi\)
\(18\) 0 0
\(19\) 2790.91i 1.77363i 0.462127 + 0.886814i \(0.347087\pi\)
−0.462127 + 0.886814i \(0.652913\pi\)
\(20\) 0 0
\(21\) − 6075.56i − 3.00634i
\(22\) 0 0
\(23\) −3258.47 −1.28438 −0.642192 0.766544i \(-0.721975\pi\)
−0.642192 + 0.766544i \(0.721975\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) − 7559.32i − 1.99560i
\(28\) 0 0
\(29\) − 7896.13i − 1.74349i −0.489961 0.871745i \(-0.662989\pi\)
0.489961 0.871745i \(-0.337011\pi\)
\(30\) 0 0
\(31\) 5051.23 0.944046 0.472023 0.881586i \(-0.343524\pi\)
0.472023 + 0.881586i \(0.343524\pi\)
\(32\) 0 0
\(33\) 1775.91 0.283880
\(34\) 0 0
\(35\) − 5508.96i − 0.760151i
\(36\) 0 0
\(37\) − 4915.89i − 0.590334i −0.955446 0.295167i \(-0.904625\pi\)
0.955446 0.295167i \(-0.0953753\pi\)
\(38\) 0 0
\(39\) −19382.8 −2.04058
\(40\) 0 0
\(41\) 7036.16 0.653696 0.326848 0.945077i \(-0.394013\pi\)
0.326848 + 0.945077i \(0.394013\pi\)
\(42\) 0 0
\(43\) 15147.0i 1.24927i 0.780918 + 0.624633i \(0.214752\pi\)
−0.780918 + 0.624633i \(0.785248\pi\)
\(44\) 0 0
\(45\) − 12929.4i − 0.951799i
\(46\) 0 0
\(47\) −9305.48 −0.614461 −0.307230 0.951635i \(-0.599402\pi\)
−0.307230 + 0.951635i \(0.599402\pi\)
\(48\) 0 0
\(49\) 31750.9 1.88915
\(50\) 0 0
\(51\) − 5723.05i − 0.308107i
\(52\) 0 0
\(53\) − 4938.18i − 0.241478i −0.992684 0.120739i \(-0.961474\pi\)
0.992684 0.120739i \(-0.0385264\pi\)
\(54\) 0 0
\(55\) 1610.29 0.0717789
\(56\) 0 0
\(57\) −76949.0 −3.13701
\(58\) 0 0
\(59\) 38441.5i 1.43771i 0.695162 + 0.718853i \(0.255333\pi\)
−0.695162 + 0.718853i \(0.744667\pi\)
\(60\) 0 0
\(61\) 27779.9i 0.955886i 0.878391 + 0.477943i \(0.158618\pi\)
−0.878391 + 0.477943i \(0.841382\pi\)
\(62\) 0 0
\(63\) 113964. 3.61756
\(64\) 0 0
\(65\) −17575.2 −0.515960
\(66\) 0 0
\(67\) − 7780.18i − 0.211740i −0.994380 0.105870i \(-0.966237\pi\)
0.994380 0.105870i \(-0.0337627\pi\)
\(68\) 0 0
\(69\) − 89840.2i − 2.27168i
\(70\) 0 0
\(71\) 60101.8 1.41495 0.707476 0.706737i \(-0.249834\pi\)
0.707476 + 0.706737i \(0.249834\pi\)
\(72\) 0 0
\(73\) 36397.3 0.799396 0.399698 0.916647i \(-0.369115\pi\)
0.399698 + 0.916647i \(0.369115\pi\)
\(74\) 0 0
\(75\) − 17232.0i − 0.353739i
\(76\) 0 0
\(77\) 14193.6i 0.272814i
\(78\) 0 0
\(79\) 36768.7 0.662843 0.331421 0.943483i \(-0.392472\pi\)
0.331421 + 0.943483i \(0.392472\pi\)
\(80\) 0 0
\(81\) 82746.7 1.40132
\(82\) 0 0
\(83\) − 21132.9i − 0.336716i −0.985726 0.168358i \(-0.946153\pi\)
0.985726 0.168358i \(-0.0538465\pi\)
\(84\) 0 0
\(85\) − 5189.32i − 0.0779047i
\(86\) 0 0
\(87\) 217706. 3.08370
\(88\) 0 0
\(89\) −63887.8 −0.854955 −0.427477 0.904026i \(-0.640598\pi\)
−0.427477 + 0.904026i \(0.640598\pi\)
\(90\) 0 0
\(91\) − 154913.i − 1.96104i
\(92\) 0 0
\(93\) 139269.i 1.66973i
\(94\) 0 0
\(95\) −69772.9 −0.793190
\(96\) 0 0
\(97\) −14388.5 −0.155269 −0.0776346 0.996982i \(-0.524737\pi\)
−0.0776346 + 0.996982i \(0.524737\pi\)
\(98\) 0 0
\(99\) 33312.0i 0.341596i
\(100\) 0 0
\(101\) − 73761.5i − 0.719493i −0.933050 0.359747i \(-0.882863\pi\)
0.933050 0.359747i \(-0.117137\pi\)
\(102\) 0 0
\(103\) −13374.5 −0.124218 −0.0621090 0.998069i \(-0.519783\pi\)
−0.0621090 + 0.998069i \(0.519783\pi\)
\(104\) 0 0
\(105\) 151889. 1.34448
\(106\) 0 0
\(107\) 206058.i 1.73992i 0.493120 + 0.869961i \(0.335856\pi\)
−0.493120 + 0.869961i \(0.664144\pi\)
\(108\) 0 0
\(109\) − 73086.1i − 0.589208i −0.955619 0.294604i \(-0.904812\pi\)
0.955619 0.294604i \(-0.0951877\pi\)
\(110\) 0 0
\(111\) 135537. 1.04412
\(112\) 0 0
\(113\) −154909. −1.14125 −0.570626 0.821210i \(-0.693299\pi\)
−0.570626 + 0.821210i \(0.693299\pi\)
\(114\) 0 0
\(115\) − 81461.8i − 0.574394i
\(116\) 0 0
\(117\) − 363577.i − 2.45545i
\(118\) 0 0
\(119\) 45740.5 0.296097
\(120\) 0 0
\(121\) 156902. 0.974239
\(122\) 0 0
\(123\) 193996.i 1.15619i
\(124\) 0 0
\(125\) − 15625.0i − 0.0894427i
\(126\) 0 0
\(127\) −156780. −0.862545 −0.431273 0.902222i \(-0.641935\pi\)
−0.431273 + 0.902222i \(0.641935\pi\)
\(128\) 0 0
\(129\) −417621. −2.20957
\(130\) 0 0
\(131\) − 241597.i − 1.23002i −0.788518 0.615012i \(-0.789151\pi\)
0.788518 0.615012i \(-0.210849\pi\)
\(132\) 0 0
\(133\) − 615002.i − 3.01472i
\(134\) 0 0
\(135\) 188983. 0.892459
\(136\) 0 0
\(137\) −34718.4 −0.158037 −0.0790184 0.996873i \(-0.525179\pi\)
−0.0790184 + 0.996873i \(0.525179\pi\)
\(138\) 0 0
\(139\) − 148522.i − 0.652008i −0.945368 0.326004i \(-0.894298\pi\)
0.945368 0.326004i \(-0.105702\pi\)
\(140\) 0 0
\(141\) − 256564.i − 1.08680i
\(142\) 0 0
\(143\) 45281.7 0.185175
\(144\) 0 0
\(145\) 197403. 0.779712
\(146\) 0 0
\(147\) 875412.i 3.34133i
\(148\) 0 0
\(149\) − 95567.9i − 0.352652i −0.984332 0.176326i \(-0.943579\pi\)
0.984332 0.176326i \(-0.0564213\pi\)
\(150\) 0 0
\(151\) −445946. −1.59162 −0.795810 0.605546i \(-0.792955\pi\)
−0.795810 + 0.605546i \(0.792955\pi\)
\(152\) 0 0
\(153\) 107351. 0.370748
\(154\) 0 0
\(155\) 126281.i 0.422190i
\(156\) 0 0
\(157\) 534503.i 1.73062i 0.501240 + 0.865308i \(0.332877\pi\)
−0.501240 + 0.865308i \(0.667123\pi\)
\(158\) 0 0
\(159\) 136152. 0.427101
\(160\) 0 0
\(161\) 718033. 2.18313
\(162\) 0 0
\(163\) − 509438.i − 1.50184i −0.660396 0.750918i \(-0.729611\pi\)
0.660396 0.750918i \(-0.270389\pi\)
\(164\) 0 0
\(165\) 44397.7i 0.126955i
\(166\) 0 0
\(167\) −658419. −1.82689 −0.913443 0.406967i \(-0.866586\pi\)
−0.913443 + 0.406967i \(0.866586\pi\)
\(168\) 0 0
\(169\) −122925. −0.331072
\(170\) 0 0
\(171\) − 1.44339e6i − 3.77479i
\(172\) 0 0
\(173\) − 233004.i − 0.591900i −0.955203 0.295950i \(-0.904364\pi\)
0.955203 0.295950i \(-0.0956362\pi\)
\(174\) 0 0
\(175\) 137724. 0.339950
\(176\) 0 0
\(177\) −1.05988e6 −2.54287
\(178\) 0 0
\(179\) 300920.i 0.701970i 0.936381 + 0.350985i \(0.114153\pi\)
−0.936381 + 0.350985i \(0.885847\pi\)
\(180\) 0 0
\(181\) 54124.5i 0.122800i 0.998113 + 0.0613998i \(0.0195565\pi\)
−0.998113 + 0.0613998i \(0.980444\pi\)
\(182\) 0 0
\(183\) −765927. −1.69067
\(184\) 0 0
\(185\) 122897. 0.264006
\(186\) 0 0
\(187\) 13370.1i 0.0279596i
\(188\) 0 0
\(189\) 1.66576e6i 3.39202i
\(190\) 0 0
\(191\) 417470. 0.828021 0.414011 0.910272i \(-0.364128\pi\)
0.414011 + 0.910272i \(0.364128\pi\)
\(192\) 0 0
\(193\) 430941. 0.832769 0.416384 0.909189i \(-0.363297\pi\)
0.416384 + 0.909189i \(0.363297\pi\)
\(194\) 0 0
\(195\) − 484569.i − 0.912576i
\(196\) 0 0
\(197\) − 289267.i − 0.531047i −0.964104 0.265523i \(-0.914455\pi\)
0.964104 0.265523i \(-0.0855448\pi\)
\(198\) 0 0
\(199\) 812396. 1.45424 0.727119 0.686512i \(-0.240859\pi\)
0.727119 + 0.686512i \(0.240859\pi\)
\(200\) 0 0
\(201\) 214509. 0.374504
\(202\) 0 0
\(203\) 1.73998e6i 2.96349i
\(204\) 0 0
\(205\) 175904.i 0.292342i
\(206\) 0 0
\(207\) 1.68520e6 2.73354
\(208\) 0 0
\(209\) 179767. 0.284672
\(210\) 0 0
\(211\) 227948.i 0.352476i 0.984348 + 0.176238i \(0.0563928\pi\)
−0.984348 + 0.176238i \(0.943607\pi\)
\(212\) 0 0
\(213\) 1.65708e6i 2.50262i
\(214\) 0 0
\(215\) −378675. −0.558689
\(216\) 0 0
\(217\) −1.11308e6 −1.60464
\(218\) 0 0
\(219\) 1.00352e6i 1.41389i
\(220\) 0 0
\(221\) − 145925.i − 0.200978i
\(222\) 0 0
\(223\) −371367. −0.500082 −0.250041 0.968235i \(-0.580444\pi\)
−0.250041 + 0.968235i \(0.580444\pi\)
\(224\) 0 0
\(225\) 323234. 0.425658
\(226\) 0 0
\(227\) − 761652.i − 0.981052i −0.871427 0.490526i \(-0.836805\pi\)
0.871427 0.490526i \(-0.163195\pi\)
\(228\) 0 0
\(229\) − 325400.i − 0.410043i −0.978757 0.205021i \(-0.934274\pi\)
0.978757 0.205021i \(-0.0657264\pi\)
\(230\) 0 0
\(231\) −391336. −0.482525
\(232\) 0 0
\(233\) −527689. −0.636778 −0.318389 0.947960i \(-0.603142\pi\)
−0.318389 + 0.947960i \(0.603142\pi\)
\(234\) 0 0
\(235\) − 232637.i − 0.274795i
\(236\) 0 0
\(237\) 1.01376e6i 1.17237i
\(238\) 0 0
\(239\) −1.09227e6 −1.23691 −0.618454 0.785821i \(-0.712241\pi\)
−0.618454 + 0.785821i \(0.712241\pi\)
\(240\) 0 0
\(241\) −63366.4 −0.0702776 −0.0351388 0.999382i \(-0.511187\pi\)
−0.0351388 + 0.999382i \(0.511187\pi\)
\(242\) 0 0
\(243\) 444515.i 0.482915i
\(244\) 0 0
\(245\) 793772.i 0.844852i
\(246\) 0 0
\(247\) −1.96203e6 −2.04627
\(248\) 0 0
\(249\) 582661. 0.595549
\(250\) 0 0
\(251\) − 1.59819e6i − 1.60120i −0.599201 0.800599i \(-0.704515\pi\)
0.599201 0.800599i \(-0.295485\pi\)
\(252\) 0 0
\(253\) 209883.i 0.206147i
\(254\) 0 0
\(255\) 143076. 0.137790
\(256\) 0 0
\(257\) −478358. −0.451773 −0.225886 0.974154i \(-0.572528\pi\)
−0.225886 + 0.974154i \(0.572528\pi\)
\(258\) 0 0
\(259\) 1.08326e6i 1.00342i
\(260\) 0 0
\(261\) 4.08367e6i 3.71065i
\(262\) 0 0
\(263\) 720034. 0.641895 0.320947 0.947097i \(-0.395999\pi\)
0.320947 + 0.947097i \(0.395999\pi\)
\(264\) 0 0
\(265\) 123454. 0.107992
\(266\) 0 0
\(267\) − 1.76147e6i − 1.51216i
\(268\) 0 0
\(269\) − 1.74800e6i − 1.47286i −0.676514 0.736430i \(-0.736510\pi\)
0.676514 0.736430i \(-0.263490\pi\)
\(270\) 0 0
\(271\) −1.80305e6 −1.49137 −0.745683 0.666300i \(-0.767877\pi\)
−0.745683 + 0.666300i \(0.767877\pi\)
\(272\) 0 0
\(273\) 4.27116e6 3.46848
\(274\) 0 0
\(275\) 40257.2i 0.0321005i
\(276\) 0 0
\(277\) − 243112.i − 0.190374i −0.995459 0.0951869i \(-0.969655\pi\)
0.995459 0.0951869i \(-0.0303449\pi\)
\(278\) 0 0
\(279\) −2.61237e6 −2.00920
\(280\) 0 0
\(281\) −377345. −0.285084 −0.142542 0.989789i \(-0.545528\pi\)
−0.142542 + 0.989789i \(0.545528\pi\)
\(282\) 0 0
\(283\) 904709.i 0.671495i 0.941952 + 0.335748i \(0.108989\pi\)
−0.941952 + 0.335748i \(0.891011\pi\)
\(284\) 0 0
\(285\) − 1.92373e6i − 1.40291i
\(286\) 0 0
\(287\) −1.55048e6 −1.11112
\(288\) 0 0
\(289\) −1.37677e6 −0.969654
\(290\) 0 0
\(291\) − 396708.i − 0.274624i
\(292\) 0 0
\(293\) − 1.57254e6i − 1.07012i −0.844814 0.535060i \(-0.820289\pi\)
0.844814 0.535060i \(-0.179711\pi\)
\(294\) 0 0
\(295\) −961037. −0.642962
\(296\) 0 0
\(297\) −486907. −0.320299
\(298\) 0 0
\(299\) − 2.29073e6i − 1.48182i
\(300\) 0 0
\(301\) − 3.33777e6i − 2.12344i
\(302\) 0 0
\(303\) 2.03370e6 1.27256
\(304\) 0 0
\(305\) −694498. −0.427485
\(306\) 0 0
\(307\) 1.50494e6i 0.911326i 0.890152 + 0.455663i \(0.150598\pi\)
−0.890152 + 0.455663i \(0.849402\pi\)
\(308\) 0 0
\(309\) − 368752.i − 0.219704i
\(310\) 0 0
\(311\) −50915.6 −0.0298504 −0.0149252 0.999889i \(-0.504751\pi\)
−0.0149252 + 0.999889i \(0.504751\pi\)
\(312\) 0 0
\(313\) −1.05135e6 −0.606581 −0.303290 0.952898i \(-0.598085\pi\)
−0.303290 + 0.952898i \(0.598085\pi\)
\(314\) 0 0
\(315\) 2.84909e6i 1.61782i
\(316\) 0 0
\(317\) − 591005.i − 0.330326i −0.986266 0.165163i \(-0.947185\pi\)
0.986266 0.165163i \(-0.0528151\pi\)
\(318\) 0 0
\(319\) −508602. −0.279834
\(320\) 0 0
\(321\) −5.68128e6 −3.07740
\(322\) 0 0
\(323\) − 579318.i − 0.308966i
\(324\) 0 0
\(325\) − 439379.i − 0.230744i
\(326\) 0 0
\(327\) 2.01508e6 1.04213
\(328\) 0 0
\(329\) 2.05054e6 1.04443
\(330\) 0 0
\(331\) 1.47751e6i 0.741242i 0.928784 + 0.370621i \(0.120855\pi\)
−0.928784 + 0.370621i \(0.879145\pi\)
\(332\) 0 0
\(333\) 2.54237e6i 1.25640i
\(334\) 0 0
\(335\) 194505. 0.0946930
\(336\) 0 0
\(337\) 3.32321e6 1.59398 0.796990 0.603993i \(-0.206425\pi\)
0.796990 + 0.603993i \(0.206425\pi\)
\(338\) 0 0
\(339\) − 4.27104e6i − 2.01853i
\(340\) 0 0
\(341\) − 325357.i − 0.151522i
\(342\) 0 0
\(343\) −3.29301e6 −1.51133
\(344\) 0 0
\(345\) 2.24600e6 1.01593
\(346\) 0 0
\(347\) 602144.i 0.268458i 0.990950 + 0.134229i \(0.0428558\pi\)
−0.990950 + 0.134229i \(0.957144\pi\)
\(348\) 0 0
\(349\) − 1.58858e6i − 0.698147i −0.937095 0.349073i \(-0.886496\pi\)
0.937095 0.349073i \(-0.113504\pi\)
\(350\) 0 0
\(351\) 5.31425e6 2.30237
\(352\) 0 0
\(353\) 319264. 0.136368 0.0681841 0.997673i \(-0.478279\pi\)
0.0681841 + 0.997673i \(0.478279\pi\)
\(354\) 0 0
\(355\) 1.50255e6i 0.632786i
\(356\) 0 0
\(357\) 1.26112e6i 0.523705i
\(358\) 0 0
\(359\) −3.83807e6 −1.57173 −0.785863 0.618401i \(-0.787781\pi\)
−0.785863 + 0.618401i \(0.787781\pi\)
\(360\) 0 0
\(361\) −5.31310e6 −2.14576
\(362\) 0 0
\(363\) 4.32599e6i 1.72313i
\(364\) 0 0
\(365\) 909932.i 0.357501i
\(366\) 0 0
\(367\) 2.02410e6 0.784455 0.392227 0.919868i \(-0.371705\pi\)
0.392227 + 0.919868i \(0.371705\pi\)
\(368\) 0 0
\(369\) −3.63892e6 −1.39125
\(370\) 0 0
\(371\) 1.08817e6i 0.410451i
\(372\) 0 0
\(373\) 4.39562e6i 1.63587i 0.575311 + 0.817935i \(0.304881\pi\)
−0.575311 + 0.817935i \(0.695119\pi\)
\(374\) 0 0
\(375\) 430801. 0.158197
\(376\) 0 0
\(377\) 5.55103e6 2.01150
\(378\) 0 0
\(379\) − 2.04075e6i − 0.729780i −0.931051 0.364890i \(-0.881107\pi\)
0.931051 0.364890i \(-0.118893\pi\)
\(380\) 0 0
\(381\) − 4.32263e6i − 1.52558i
\(382\) 0 0
\(383\) 2.64545e6 0.921516 0.460758 0.887526i \(-0.347578\pi\)
0.460758 + 0.887526i \(0.347578\pi\)
\(384\) 0 0
\(385\) −354841. −0.122006
\(386\) 0 0
\(387\) − 7.83363e6i − 2.65880i
\(388\) 0 0
\(389\) 1.90022e6i 0.636693i 0.947974 + 0.318346i \(0.103127\pi\)
−0.947974 + 0.318346i \(0.896873\pi\)
\(390\) 0 0
\(391\) 676371. 0.223740
\(392\) 0 0
\(393\) 6.66113e6 2.17554
\(394\) 0 0
\(395\) 919217.i 0.296432i
\(396\) 0 0
\(397\) 716277.i 0.228089i 0.993476 + 0.114045i \(0.0363807\pi\)
−0.993476 + 0.114045i \(0.963619\pi\)
\(398\) 0 0
\(399\) 1.69564e7 5.33213
\(400\) 0 0
\(401\) −1.77803e6 −0.552175 −0.276088 0.961132i \(-0.589038\pi\)
−0.276088 + 0.961132i \(0.589038\pi\)
\(402\) 0 0
\(403\) 3.55105e6i 1.08917i
\(404\) 0 0
\(405\) 2.06867e6i 0.626690i
\(406\) 0 0
\(407\) −316640. −0.0947502
\(408\) 0 0
\(409\) 3.15560e6 0.932767 0.466384 0.884583i \(-0.345557\pi\)
0.466384 + 0.884583i \(0.345557\pi\)
\(410\) 0 0
\(411\) − 957230.i − 0.279519i
\(412\) 0 0
\(413\) − 8.47091e6i − 2.44374i
\(414\) 0 0
\(415\) 528323. 0.150584
\(416\) 0 0
\(417\) 4.09493e6 1.15320
\(418\) 0 0
\(419\) 1.73881e6i 0.483857i 0.970294 + 0.241929i \(0.0777800\pi\)
−0.970294 + 0.241929i \(0.922220\pi\)
\(420\) 0 0
\(421\) 4.05279e6i 1.11442i 0.830371 + 0.557210i \(0.188128\pi\)
−0.830371 + 0.557210i \(0.811872\pi\)
\(422\) 0 0
\(423\) 4.81255e6 1.30775
\(424\) 0 0
\(425\) 129733. 0.0348400
\(426\) 0 0
\(427\) − 6.12154e6i − 1.62477i
\(428\) 0 0
\(429\) 1.24847e6i 0.327519i
\(430\) 0 0
\(431\) 4.04038e6 1.04768 0.523840 0.851817i \(-0.324499\pi\)
0.523840 + 0.851817i \(0.324499\pi\)
\(432\) 0 0
\(433\) 4.95477e6 1.27000 0.634999 0.772513i \(-0.281000\pi\)
0.634999 + 0.772513i \(0.281000\pi\)
\(434\) 0 0
\(435\) 5.44265e6i 1.37907i
\(436\) 0 0
\(437\) − 9.09412e6i − 2.27802i
\(438\) 0 0
\(439\) −2.27775e6 −0.564086 −0.282043 0.959402i \(-0.591012\pi\)
−0.282043 + 0.959402i \(0.591012\pi\)
\(440\) 0 0
\(441\) −1.64207e7 −4.02065
\(442\) 0 0
\(443\) − 2.88598e6i − 0.698688i −0.936995 0.349344i \(-0.886404\pi\)
0.936995 0.349344i \(-0.113596\pi\)
\(444\) 0 0
\(445\) − 1.59720e6i − 0.382347i
\(446\) 0 0
\(447\) 2.63493e6 0.623735
\(448\) 0 0
\(449\) −4.01250e6 −0.939288 −0.469644 0.882856i \(-0.655618\pi\)
−0.469644 + 0.882856i \(0.655618\pi\)
\(450\) 0 0
\(451\) − 453210.i − 0.104920i
\(452\) 0 0
\(453\) − 1.22953e7i − 2.81509i
\(454\) 0 0
\(455\) 3.87284e6 0.877002
\(456\) 0 0
\(457\) −6.10953e6 −1.36841 −0.684206 0.729288i \(-0.739851\pi\)
−0.684206 + 0.729288i \(0.739851\pi\)
\(458\) 0 0
\(459\) 1.56911e6i 0.347634i
\(460\) 0 0
\(461\) 6.59208e6i 1.44467i 0.691541 + 0.722337i \(0.256932\pi\)
−0.691541 + 0.722337i \(0.743068\pi\)
\(462\) 0 0
\(463\) −2.37278e6 −0.514406 −0.257203 0.966357i \(-0.582801\pi\)
−0.257203 + 0.966357i \(0.582801\pi\)
\(464\) 0 0
\(465\) −3.48172e6 −0.746726
\(466\) 0 0
\(467\) − 6.30012e6i − 1.33677i −0.743816 0.668385i \(-0.766986\pi\)
0.743816 0.668385i \(-0.233014\pi\)
\(468\) 0 0
\(469\) 1.71443e6i 0.359905i
\(470\) 0 0
\(471\) −1.47369e7 −3.06094
\(472\) 0 0
\(473\) 975641. 0.200510
\(474\) 0 0
\(475\) − 1.74432e6i − 0.354726i
\(476\) 0 0
\(477\) 2.55390e6i 0.513934i
\(478\) 0 0
\(479\) −2.28649e6 −0.455334 −0.227667 0.973739i \(-0.573110\pi\)
−0.227667 + 0.973739i \(0.573110\pi\)
\(480\) 0 0
\(481\) 3.45590e6 0.681081
\(482\) 0 0
\(483\) 1.97971e7i 3.86129i
\(484\) 0 0
\(485\) − 359712.i − 0.0694385i
\(486\) 0 0
\(487\) 2.28593e6 0.436758 0.218379 0.975864i \(-0.429923\pi\)
0.218379 + 0.975864i \(0.429923\pi\)
\(488\) 0 0
\(489\) 1.40458e7 2.65629
\(490\) 0 0
\(491\) 7.08998e6i 1.32721i 0.748081 + 0.663607i \(0.230975\pi\)
−0.748081 + 0.663607i \(0.769025\pi\)
\(492\) 0 0
\(493\) 1.63902e6i 0.303716i
\(494\) 0 0
\(495\) −832799. −0.152766
\(496\) 0 0
\(497\) −1.32440e7 −2.40506
\(498\) 0 0
\(499\) − 3.60881e6i − 0.648804i −0.945919 0.324402i \(-0.894837\pi\)
0.945919 0.324402i \(-0.105163\pi\)
\(500\) 0 0
\(501\) − 1.81534e7i − 3.23121i
\(502\) 0 0
\(503\) −1.01067e7 −1.78111 −0.890554 0.454878i \(-0.849683\pi\)
−0.890554 + 0.454878i \(0.849683\pi\)
\(504\) 0 0
\(505\) 1.84404e6 0.321767
\(506\) 0 0
\(507\) − 3.38919e6i − 0.585566i
\(508\) 0 0
\(509\) 2.49518e6i 0.426882i 0.976956 + 0.213441i \(0.0684670\pi\)
−0.976956 + 0.213441i \(0.931533\pi\)
\(510\) 0 0
\(511\) −8.02045e6 −1.35877
\(512\) 0 0
\(513\) 2.10974e7 3.53945
\(514\) 0 0
\(515\) − 334363.i − 0.0555520i
\(516\) 0 0
\(517\) 599380.i 0.0986225i
\(518\) 0 0
\(519\) 6.42422e6 1.04689
\(520\) 0 0
\(521\) −9.37259e6 −1.51274 −0.756371 0.654142i \(-0.773030\pi\)
−0.756371 + 0.654142i \(0.773030\pi\)
\(522\) 0 0
\(523\) − 1.96213e6i − 0.313670i −0.987625 0.156835i \(-0.949871\pi\)
0.987625 0.156835i \(-0.0501291\pi\)
\(524\) 0 0
\(525\) 3.79723e6i 0.601268i
\(526\) 0 0
\(527\) −1.04850e6 −0.164453
\(528\) 0 0
\(529\) 4.18131e6 0.649640
\(530\) 0 0
\(531\) − 1.98809e7i − 3.05985i
\(532\) 0 0
\(533\) 4.94646e6i 0.754183i
\(534\) 0 0
\(535\) −5.15145e6 −0.778117
\(536\) 0 0
\(537\) −8.29674e6 −1.24157
\(538\) 0 0
\(539\) − 2.04512e6i − 0.303213i
\(540\) 0 0
\(541\) − 2.57325e6i − 0.377998i −0.981977 0.188999i \(-0.939476\pi\)
0.981977 0.188999i \(-0.0605243\pi\)
\(542\) 0 0
\(543\) −1.49228e6 −0.217195
\(544\) 0 0
\(545\) 1.82715e6 0.263502
\(546\) 0 0
\(547\) − 6.99737e6i − 0.999923i −0.866048 0.499962i \(-0.833347\pi\)
0.866048 0.499962i \(-0.166653\pi\)
\(548\) 0 0
\(549\) − 1.43670e7i − 2.03440i
\(550\) 0 0
\(551\) 2.20374e7 3.09230
\(552\) 0 0
\(553\) −8.10230e6 −1.12667
\(554\) 0 0
\(555\) 3.38843e6i 0.466946i
\(556\) 0 0
\(557\) − 1.11329e7i − 1.52044i −0.649665 0.760221i \(-0.725091\pi\)
0.649665 0.760221i \(-0.274909\pi\)
\(558\) 0 0
\(559\) −1.06484e7 −1.44130
\(560\) 0 0
\(561\) −368630. −0.0494520
\(562\) 0 0
\(563\) 7.19654e6i 0.956870i 0.878123 + 0.478435i \(0.158796\pi\)
−0.878123 + 0.478435i \(0.841204\pi\)
\(564\) 0 0
\(565\) − 3.87273e6i − 0.510383i
\(566\) 0 0
\(567\) −1.82339e7 −2.38190
\(568\) 0 0
\(569\) 3.80510e6 0.492703 0.246352 0.969181i \(-0.420768\pi\)
0.246352 + 0.969181i \(0.420768\pi\)
\(570\) 0 0
\(571\) − 3.56459e6i − 0.457530i −0.973482 0.228765i \(-0.926531\pi\)
0.973482 0.228765i \(-0.0734687\pi\)
\(572\) 0 0
\(573\) 1.15102e7i 1.46452i
\(574\) 0 0
\(575\) 2.03655e6 0.256877
\(576\) 0 0
\(577\) 1.33424e7 1.66837 0.834187 0.551482i \(-0.185937\pi\)
0.834187 + 0.551482i \(0.185937\pi\)
\(578\) 0 0
\(579\) 1.18816e7i 1.47292i
\(580\) 0 0
\(581\) 4.65682e6i 0.572333i
\(582\) 0 0
\(583\) −318075. −0.0387578
\(584\) 0 0
\(585\) 9.08941e6 1.09811
\(586\) 0 0
\(587\) 1.02312e7i 1.22555i 0.790257 + 0.612775i \(0.209947\pi\)
−0.790257 + 0.612775i \(0.790053\pi\)
\(588\) 0 0
\(589\) 1.40976e7i 1.67439i
\(590\) 0 0
\(591\) 7.97545e6 0.939261
\(592\) 0 0
\(593\) 1.50572e7 1.75836 0.879181 0.476487i \(-0.158090\pi\)
0.879181 + 0.476487i \(0.158090\pi\)
\(594\) 0 0
\(595\) 1.14351e6i 0.132418i
\(596\) 0 0
\(597\) 2.23988e7i 2.57211i
\(598\) 0 0
\(599\) −994780. −0.113282 −0.0566409 0.998395i \(-0.518039\pi\)
−0.0566409 + 0.998395i \(0.518039\pi\)
\(600\) 0 0
\(601\) 6.59335e6 0.744595 0.372297 0.928113i \(-0.378570\pi\)
0.372297 + 0.928113i \(0.378570\pi\)
\(602\) 0 0
\(603\) 4.02371e6i 0.450644i
\(604\) 0 0
\(605\) 3.92255e6i 0.435693i
\(606\) 0 0
\(607\) 9.09559e6 1.00198 0.500990 0.865453i \(-0.332969\pi\)
0.500990 + 0.865453i \(0.332969\pi\)
\(608\) 0 0
\(609\) −4.79734e7 −5.24152
\(610\) 0 0
\(611\) − 6.54181e6i − 0.708916i
\(612\) 0 0
\(613\) 1.08785e7i 1.16927i 0.811295 + 0.584637i \(0.198763\pi\)
−0.811295 + 0.584637i \(0.801237\pi\)
\(614\) 0 0
\(615\) −4.84989e6 −0.517064
\(616\) 0 0
\(617\) −2.35827e6 −0.249391 −0.124696 0.992195i \(-0.539795\pi\)
−0.124696 + 0.992195i \(0.539795\pi\)
\(618\) 0 0
\(619\) 2.95345e6i 0.309815i 0.987929 + 0.154907i \(0.0495079\pi\)
−0.987929 + 0.154907i \(0.950492\pi\)
\(620\) 0 0
\(621\) 2.46318e7i 2.56311i
\(622\) 0 0
\(623\) 1.40782e7 1.45321
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) 4.95640e6i 0.503498i
\(628\) 0 0
\(629\) 1.02041e6i 0.102836i
\(630\) 0 0
\(631\) −1.49811e7 −1.49785 −0.748927 0.662653i \(-0.769430\pi\)
−0.748927 + 0.662653i \(0.769430\pi\)
\(632\) 0 0
\(633\) −6.28481e6 −0.623423
\(634\) 0 0
\(635\) − 3.91950e6i − 0.385742i
\(636\) 0 0
\(637\) 2.23211e7i 2.17955i
\(638\) 0 0
\(639\) −3.10831e7 −3.01143
\(640\) 0 0
\(641\) −4.70605e6 −0.452388 −0.226194 0.974082i \(-0.572628\pi\)
−0.226194 + 0.974082i \(0.572628\pi\)
\(642\) 0 0
\(643\) 1.00759e7i 0.961074i 0.876974 + 0.480537i \(0.159558\pi\)
−0.876974 + 0.480537i \(0.840442\pi\)
\(644\) 0 0
\(645\) − 1.04405e7i − 0.988151i
\(646\) 0 0
\(647\) −1.69585e6 −0.159267 −0.0796337 0.996824i \(-0.525375\pi\)
−0.0796337 + 0.996824i \(0.525375\pi\)
\(648\) 0 0
\(649\) 2.47607e6 0.230755
\(650\) 0 0
\(651\) − 3.06891e7i − 2.83812i
\(652\) 0 0
\(653\) 9.63786e6i 0.884500i 0.896892 + 0.442250i \(0.145820\pi\)
−0.896892 + 0.442250i \(0.854180\pi\)
\(654\) 0 0
\(655\) 6.03993e6 0.550083
\(656\) 0 0
\(657\) −1.88237e7 −1.70134
\(658\) 0 0
\(659\) − 9.50539e6i − 0.852622i −0.904577 0.426311i \(-0.859813\pi\)
0.904577 0.426311i \(-0.140187\pi\)
\(660\) 0 0
\(661\) 1.34346e7i 1.19597i 0.801507 + 0.597986i \(0.204032\pi\)
−0.801507 + 0.597986i \(0.795968\pi\)
\(662\) 0 0
\(663\) 4.02334e6 0.355470
\(664\) 0 0
\(665\) 1.53750e7 1.34822
\(666\) 0 0
\(667\) 2.57293e7i 2.23931i
\(668\) 0 0
\(669\) − 1.02390e7i − 0.884493i
\(670\) 0 0
\(671\) 1.78935e6 0.153422
\(672\) 0 0
\(673\) 930663. 0.0792054 0.0396027 0.999216i \(-0.487391\pi\)
0.0396027 + 0.999216i \(0.487391\pi\)
\(674\) 0 0
\(675\) 4.72458e6i 0.399120i
\(676\) 0 0
\(677\) − 6.87737e6i − 0.576701i −0.957525 0.288350i \(-0.906893\pi\)
0.957525 0.288350i \(-0.0931068\pi\)
\(678\) 0 0
\(679\) 3.17062e6 0.263919
\(680\) 0 0
\(681\) 2.09997e7 1.73518
\(682\) 0 0
\(683\) 909803.i 0.0746269i 0.999304 + 0.0373135i \(0.0118800\pi\)
−0.999304 + 0.0373135i \(0.988120\pi\)
\(684\) 0 0
\(685\) − 867960.i − 0.0706762i
\(686\) 0 0
\(687\) 8.97169e6 0.725241
\(688\) 0 0
\(689\) 3.47157e6 0.278598
\(690\) 0 0
\(691\) − 2.04678e7i − 1.63071i −0.578961 0.815355i \(-0.696542\pi\)
0.578961 0.815355i \(-0.303458\pi\)
\(692\) 0 0
\(693\) − 7.34058e6i − 0.580627i
\(694\) 0 0
\(695\) 3.71304e6 0.291587
\(696\) 0 0
\(697\) −1.46052e6 −0.113874
\(698\) 0 0
\(699\) − 1.45490e7i − 1.12627i
\(700\) 0 0
\(701\) 1.75532e7i 1.34916i 0.738203 + 0.674578i \(0.235674\pi\)
−0.738203 + 0.674578i \(0.764326\pi\)
\(702\) 0 0
\(703\) 1.37198e7 1.04703
\(704\) 0 0
\(705\) 6.41410e6 0.486030
\(706\) 0 0
\(707\) 1.62540e7i 1.22296i
\(708\) 0 0
\(709\) 2.23525e7i 1.66997i 0.550270 + 0.834987i \(0.314525\pi\)
−0.550270 + 0.834987i \(0.685475\pi\)
\(710\) 0 0
\(711\) −1.90158e7 −1.41072
\(712\) 0 0
\(713\) −1.64593e7 −1.21252
\(714\) 0 0
\(715\) 1.13204e6i 0.0828128i
\(716\) 0 0
\(717\) − 3.01154e7i − 2.18771i
\(718\) 0 0
\(719\) 1.84982e7 1.33447 0.667234 0.744848i \(-0.267478\pi\)
0.667234 + 0.744848i \(0.267478\pi\)
\(720\) 0 0
\(721\) 2.94719e6 0.211140
\(722\) 0 0
\(723\) − 1.74709e6i − 0.124300i
\(724\) 0 0
\(725\) 4.93508e6i 0.348698i
\(726\) 0 0
\(727\) −5.77823e6 −0.405470 −0.202735 0.979234i \(-0.564983\pi\)
−0.202735 + 0.979234i \(0.564983\pi\)
\(728\) 0 0
\(729\) 7.85162e6 0.547193
\(730\) 0 0
\(731\) − 3.14410e6i − 0.217622i
\(732\) 0 0
\(733\) 1.00639e6i 0.0691839i 0.999402 + 0.0345920i \(0.0110132\pi\)
−0.999402 + 0.0345920i \(0.988987\pi\)
\(734\) 0 0
\(735\) −2.18853e7 −1.49429
\(736\) 0 0
\(737\) −501133. −0.0339848
\(738\) 0 0
\(739\) − 1.10508e7i − 0.744357i −0.928161 0.372179i \(-0.878611\pi\)
0.928161 0.372179i \(-0.121389\pi\)
\(740\) 0 0
\(741\) − 5.40956e7i − 3.61923i
\(742\) 0 0
\(743\) −6.91879e6 −0.459789 −0.229894 0.973216i \(-0.573838\pi\)
−0.229894 + 0.973216i \(0.573838\pi\)
\(744\) 0 0
\(745\) 2.38920e6 0.157711
\(746\) 0 0
\(747\) 1.09294e7i 0.716630i
\(748\) 0 0
\(749\) − 4.54066e7i − 2.95743i
\(750\) 0 0
\(751\) −1.79833e7 −1.16351 −0.581756 0.813364i \(-0.697634\pi\)
−0.581756 + 0.813364i \(0.697634\pi\)
\(752\) 0 0
\(753\) 4.40642e7 2.83203
\(754\) 0 0
\(755\) − 1.11486e7i − 0.711795i
\(756\) 0 0
\(757\) 2.59762e7i 1.64754i 0.566923 + 0.823771i \(0.308134\pi\)
−0.566923 + 0.823771i \(0.691866\pi\)
\(758\) 0 0
\(759\) −5.78674e6 −0.364611
\(760\) 0 0
\(761\) 8.18122e6 0.512102 0.256051 0.966663i \(-0.417579\pi\)
0.256051 + 0.966663i \(0.417579\pi\)
\(762\) 0 0
\(763\) 1.61052e7i 1.00151i
\(764\) 0 0
\(765\) 2.68378e6i 0.165804i
\(766\) 0 0
\(767\) −2.70246e7 −1.65871
\(768\) 0 0
\(769\) −1.85187e7 −1.12926 −0.564630 0.825344i \(-0.690981\pi\)
−0.564630 + 0.825344i \(0.690981\pi\)
\(770\) 0 0
\(771\) − 1.31889e7i − 0.799049i
\(772\) 0 0
\(773\) 1.07043e7i 0.644329i 0.946684 + 0.322164i \(0.104410\pi\)
−0.946684 + 0.322164i \(0.895590\pi\)
\(774\) 0 0
\(775\) −3.15702e6 −0.188809
\(776\) 0 0
\(777\) −2.98668e7 −1.77475
\(778\) 0 0
\(779\) 1.96373e7i 1.15941i
\(780\) 0 0
\(781\) − 3.87125e6i − 0.227104i
\(782\) 0 0
\(783\) −5.96894e7 −3.47931
\(784\) 0 0
\(785\) −1.33626e7 −0.773955
\(786\) 0 0
\(787\) 68571.9i 0.00394648i 0.999998 + 0.00197324i \(0.000628102\pi\)
−0.999998 + 0.00197324i \(0.999372\pi\)
\(788\) 0 0
\(789\) 1.98523e7i 1.13532i
\(790\) 0 0
\(791\) 3.41356e7 1.93984
\(792\) 0 0
\(793\) −1.95294e7 −1.10283
\(794\) 0 0
\(795\) 3.40379e6i 0.191005i
\(796\) 0 0
\(797\) − 2.51343e7i − 1.40159i −0.713363 0.700794i \(-0.752829\pi\)
0.713363 0.700794i \(-0.247171\pi\)
\(798\) 0 0
\(799\) 1.93157e6 0.107039
\(800\) 0 0
\(801\) 3.30411e7 1.81959
\(802\) 0 0
\(803\) − 2.34440e6i − 0.128305i
\(804\) 0 0
\(805\) 1.79508e7i 0.976325i
\(806\) 0 0
\(807\) 4.81946e7 2.60504
\(808\) 0 0
\(809\) 2.45606e7 1.31937 0.659687 0.751540i \(-0.270689\pi\)
0.659687 + 0.751540i \(0.270689\pi\)
\(810\) 0 0
\(811\) − 1.94878e6i − 0.104043i −0.998646 0.0520213i \(-0.983434\pi\)
0.998646 0.0520213i \(-0.0165664\pi\)
\(812\) 0 0
\(813\) − 4.97123e7i − 2.63778i
\(814\) 0 0
\(815\) 1.27360e7 0.671641
\(816\) 0 0
\(817\) −4.22739e7 −2.21573
\(818\) 0 0
\(819\) 8.01172e7i 4.17365i
\(820\) 0 0
\(821\) 2.02265e7i 1.04728i 0.851940 + 0.523639i \(0.175426\pi\)
−0.851940 + 0.523639i \(0.824574\pi\)
\(822\) 0 0
\(823\) −1.57978e7 −0.813010 −0.406505 0.913649i \(-0.633253\pi\)
−0.406505 + 0.913649i \(0.633253\pi\)
\(824\) 0 0
\(825\) −1.10994e6 −0.0567761
\(826\) 0 0
\(827\) − 2.99880e7i − 1.52470i −0.647167 0.762348i \(-0.724046\pi\)
0.647167 0.762348i \(-0.275954\pi\)
\(828\) 0 0
\(829\) 1.30834e7i 0.661201i 0.943771 + 0.330601i \(0.107251\pi\)
−0.943771 + 0.330601i \(0.892749\pi\)
\(830\) 0 0
\(831\) 6.70290e6 0.336713
\(832\) 0 0
\(833\) −6.59063e6 −0.329090
\(834\) 0 0
\(835\) − 1.64605e7i − 0.817008i
\(836\) 0 0
\(837\) − 3.81839e7i − 1.88394i
\(838\) 0 0
\(839\) 1.58503e7 0.777378 0.388689 0.921369i \(-0.372928\pi\)
0.388689 + 0.921369i \(0.372928\pi\)
\(840\) 0 0
\(841\) −4.18377e7 −2.03975
\(842\) 0 0
\(843\) − 1.04039e7i − 0.504227i
\(844\) 0 0
\(845\) − 3.07312e6i − 0.148060i
\(846\) 0 0
\(847\) −3.45747e7 −1.65596
\(848\) 0 0
\(849\) −2.49440e7 −1.18767
\(850\) 0 0
\(851\) 1.60183e7i 0.758215i
\(852\) 0 0
\(853\) − 8.25379e6i − 0.388402i −0.980962 0.194201i \(-0.937789\pi\)
0.980962 0.194201i \(-0.0622113\pi\)
\(854\) 0 0
\(855\) 3.60847e7 1.68814
\(856\) 0 0
\(857\) 1.20306e7 0.559546 0.279773 0.960066i \(-0.409741\pi\)
0.279773 + 0.960066i \(0.409741\pi\)
\(858\) 0 0
\(859\) − 7.28563e6i − 0.336887i −0.985711 0.168443i \(-0.946126\pi\)
0.985711 0.168443i \(-0.0538741\pi\)
\(860\) 0 0
\(861\) − 4.27486e7i − 1.96523i
\(862\) 0 0
\(863\) 2.70168e7 1.23483 0.617415 0.786638i \(-0.288180\pi\)
0.617415 + 0.786638i \(0.288180\pi\)
\(864\) 0 0
\(865\) 5.82511e6 0.264706
\(866\) 0 0
\(867\) − 3.79593e7i − 1.71502i
\(868\) 0 0
\(869\) − 2.36833e6i − 0.106388i
\(870\) 0 0
\(871\) 5.46952e6 0.244289
\(872\) 0 0
\(873\) 7.44134e6 0.330458
\(874\) 0 0
\(875\) 3.44310e6i 0.152030i
\(876\) 0 0
\(877\) 1.20510e7i 0.529082i 0.964374 + 0.264541i \(0.0852204\pi\)
−0.964374 + 0.264541i \(0.914780\pi\)
\(878\) 0 0
\(879\) 4.33569e7 1.89272
\(880\) 0 0
\(881\) 4.10240e7 1.78073 0.890366 0.455245i \(-0.150448\pi\)
0.890366 + 0.455245i \(0.150448\pi\)
\(882\) 0 0
\(883\) − 2.77485e6i − 0.119767i −0.998205 0.0598835i \(-0.980927\pi\)
0.998205 0.0598835i \(-0.0190729\pi\)
\(884\) 0 0
\(885\) − 2.64970e7i − 1.13720i
\(886\) 0 0
\(887\) 2.74930e7 1.17331 0.586655 0.809837i \(-0.300444\pi\)
0.586655 + 0.809837i \(0.300444\pi\)
\(888\) 0 0
\(889\) 3.45479e7 1.46611
\(890\) 0 0
\(891\) − 5.32984e6i − 0.224916i
\(892\) 0 0
\(893\) − 2.59708e7i − 1.08982i
\(894\) 0 0
\(895\) −7.52300e6 −0.313931
\(896\) 0 0
\(897\) 6.31582e7 2.62089
\(898\) 0 0
\(899\) − 3.98852e7i − 1.64593i
\(900\) 0 0
\(901\) 1.02503e6i 0.0420654i
\(902\) 0 0
\(903\) 9.20264e7 3.75572
\(904\) 0 0
\(905\) −1.35311e6 −0.0549177
\(906\) 0 0
\(907\) − 1.59931e7i − 0.645525i −0.946480 0.322763i \(-0.895388\pi\)
0.946480 0.322763i \(-0.104612\pi\)
\(908\) 0 0
\(909\) 3.81476e7i 1.53129i
\(910\) 0 0
\(911\) −6.65270e6 −0.265584 −0.132792 0.991144i \(-0.542394\pi\)
−0.132792 + 0.991144i \(0.542394\pi\)
\(912\) 0 0
\(913\) −1.36120e6 −0.0540438
\(914\) 0 0
\(915\) − 1.91482e7i − 0.756092i
\(916\) 0 0
\(917\) 5.32380e7i 2.09073i
\(918\) 0 0
\(919\) 2.78769e7 1.08882 0.544409 0.838820i \(-0.316754\pi\)
0.544409 + 0.838820i \(0.316754\pi\)
\(920\) 0 0
\(921\) −4.14931e7 −1.61186
\(922\) 0 0
\(923\) 4.22520e7i 1.63246i
\(924\) 0 0
\(925\) 3.07243e6i 0.118067i
\(926\) 0 0
\(927\) 6.91695e6 0.264372
\(928\) 0 0
\(929\) 2.07437e7 0.788581 0.394290 0.918986i \(-0.370990\pi\)
0.394290 + 0.918986i \(0.370990\pi\)
\(930\) 0 0
\(931\) 8.86140e7i 3.35064i
\(932\) 0 0
\(933\) − 1.40381e6i − 0.0527963i
\(934\) 0 0
\(935\) −334252. −0.0125039
\(936\) 0 0
\(937\) −1.13976e7 −0.424097 −0.212049 0.977259i \(-0.568014\pi\)
−0.212049 + 0.977259i \(0.568014\pi\)
\(938\) 0 0
\(939\) − 2.89872e7i − 1.07286i
\(940\) 0 0
\(941\) 995880.i 0.0366634i 0.999832 + 0.0183317i \(0.00583549\pi\)
−0.999832 + 0.0183317i \(0.994165\pi\)
\(942\) 0 0
\(943\) −2.29271e7 −0.839596
\(944\) 0 0
\(945\) −4.16440e7 −1.51696
\(946\) 0 0
\(947\) − 5.26122e6i − 0.190639i −0.995447 0.0953194i \(-0.969613\pi\)
0.995447 0.0953194i \(-0.0303872\pi\)
\(948\) 0 0
\(949\) 2.55875e7i 0.922280i
\(950\) 0 0
\(951\) 1.62948e7 0.584247
\(952\) 0 0
\(953\) −2.85407e7 −1.01797 −0.508983 0.860777i \(-0.669978\pi\)
−0.508983 + 0.860777i \(0.669978\pi\)
\(954\) 0 0
\(955\) 1.04367e7i 0.370302i
\(956\) 0 0
\(957\) − 1.40228e7i − 0.494942i
\(958\) 0 0
\(959\) 7.65050e6 0.268623
\(960\) 0 0
\(961\) −3.11422e6 −0.108778
\(962\) 0 0
\(963\) − 1.06568e8i − 3.70306i
\(964\) 0 0
\(965\) 1.07735e7i 0.372426i
\(966\) 0 0
\(967\) −8.62711e6 −0.296688 −0.148344 0.988936i \(-0.547394\pi\)
−0.148344 + 0.988936i \(0.547394\pi\)
\(968\) 0 0
\(969\) 1.59725e7 0.546467
\(970\) 0 0
\(971\) − 1.69141e7i − 0.575706i −0.957675 0.287853i \(-0.907058\pi\)
0.957675 0.287853i \(-0.0929415\pi\)
\(972\) 0 0
\(973\) 3.27280e7i 1.10825i
\(974\) 0 0
\(975\) 1.21142e7 0.408117
\(976\) 0 0
\(977\) −1.09583e7 −0.367288 −0.183644 0.982993i \(-0.558789\pi\)
−0.183644 + 0.982993i \(0.558789\pi\)
\(978\) 0 0
\(979\) 4.11511e6i 0.137222i
\(980\) 0 0
\(981\) 3.77982e7i 1.25400i
\(982\) 0 0
\(983\) −1.08936e7 −0.359573 −0.179787 0.983706i \(-0.557541\pi\)
−0.179787 + 0.983706i \(0.557541\pi\)
\(984\) 0 0
\(985\) 7.23167e6 0.237491
\(986\) 0 0
\(987\) 5.65360e7i 1.84728i
\(988\) 0 0
\(989\) − 4.93561e7i − 1.60454i
\(990\) 0 0
\(991\) −1.60015e6 −0.0517578 −0.0258789 0.999665i \(-0.508238\pi\)
−0.0258789 + 0.999665i \(0.508238\pi\)
\(992\) 0 0
\(993\) −4.07367e7 −1.31103
\(994\) 0 0
\(995\) 2.03099e7i 0.650355i
\(996\) 0 0
\(997\) 1.42051e7i 0.452593i 0.974059 + 0.226296i \(0.0726618\pi\)
−0.974059 + 0.226296i \(0.927338\pi\)
\(998\) 0 0
\(999\) −3.71608e7 −1.17807
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.a.161.8 yes 8
4.3 odd 2 320.6.d.b.161.1 yes 8
8.3 odd 2 320.6.d.b.161.8 yes 8
8.5 even 2 inner 320.6.d.a.161.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.a.161.1 8 8.5 even 2 inner
320.6.d.a.161.8 yes 8 1.1 even 1 trivial
320.6.d.b.161.1 yes 8 4.3 odd 2
320.6.d.b.161.8 yes 8 8.3 odd 2