Properties

Label 324.6.a.d.1.5
Level 324
Weight 6
Character 324.1
Self dual yes
Analytic conductor 51.964
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.9643576194\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 110 x^{3} + 39 x^{2} + 2214 x - 1944\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: no (minimal twist has level 36)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-6.24439\) of defining polynomial
Character \(\chi\) \(=\) 324.1

$q$-expansion

\(f(q)\) \(=\) \(q+81.4540 q^{5} -179.262 q^{7} +O(q^{10})\) \(q+81.4540 q^{5} -179.262 q^{7} +500.501 q^{11} -550.491 q^{13} -753.636 q^{17} -2570.83 q^{19} +2745.45 q^{23} +3509.75 q^{25} -3909.72 q^{29} -3104.84 q^{31} -14601.6 q^{35} -9568.10 q^{37} -2226.94 q^{41} +14286.8 q^{43} +6472.14 q^{47} +15328.0 q^{49} -13692.2 q^{53} +40767.8 q^{55} -5708.43 q^{59} -11799.2 q^{61} -44839.7 q^{65} -3543.32 q^{67} +58429.0 q^{71} -60181.3 q^{73} -89720.9 q^{77} -55623.4 q^{79} -39980.6 q^{83} -61386.6 q^{85} -103171. q^{89} +98682.3 q^{91} -209404. q^{95} -165994. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 21q^{5} - 29q^{7} + O(q^{10}) \) \( 5q - 21q^{5} - 29q^{7} + 177q^{11} + 181q^{13} - 1140q^{17} - 416q^{19} + 399q^{23} + 4778q^{25} - 6033q^{29} - 2759q^{31} - 18573q^{35} - 7586q^{37} - 18435q^{41} - 1469q^{43} - 25155q^{47} + 4056q^{49} - 58422q^{53} + 7389q^{55} - 90537q^{59} - 1403q^{61} - 148407q^{65} - 13907q^{67} - 114684q^{71} + 7600q^{73} - 211983q^{77} - 29993q^{79} - 228951q^{83} + 49662q^{85} - 299166q^{89} + 62465q^{91} - 394764q^{95} - 40541q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 81.4540 1.45709 0.728546 0.684997i \(-0.240196\pi\)
0.728546 + 0.684997i \(0.240196\pi\)
\(6\) 0 0
\(7\) −179.262 −1.38275 −0.691376 0.722496i \(-0.742995\pi\)
−0.691376 + 0.722496i \(0.742995\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 500.501 1.24716 0.623581 0.781759i \(-0.285677\pi\)
0.623581 + 0.781759i \(0.285677\pi\)
\(12\) 0 0
\(13\) −550.491 −0.903424 −0.451712 0.892164i \(-0.649187\pi\)
−0.451712 + 0.892164i \(0.649187\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −753.636 −0.632469 −0.316234 0.948681i \(-0.602419\pi\)
−0.316234 + 0.948681i \(0.602419\pi\)
\(18\) 0 0
\(19\) −2570.83 −1.63376 −0.816882 0.576805i \(-0.804299\pi\)
−0.816882 + 0.576805i \(0.804299\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2745.45 1.08217 0.541083 0.840969i \(-0.318014\pi\)
0.541083 + 0.840969i \(0.318014\pi\)
\(24\) 0 0
\(25\) 3509.75 1.12312
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3909.72 −0.863278 −0.431639 0.902046i \(-0.642065\pi\)
−0.431639 + 0.902046i \(0.642065\pi\)
\(30\) 0 0
\(31\) −3104.84 −0.580276 −0.290138 0.956985i \(-0.593701\pi\)
−0.290138 + 0.956985i \(0.593701\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −14601.6 −2.01480
\(36\) 0 0
\(37\) −9568.10 −1.14900 −0.574502 0.818503i \(-0.694804\pi\)
−0.574502 + 0.818503i \(0.694804\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2226.94 −0.206894 −0.103447 0.994635i \(-0.532987\pi\)
−0.103447 + 0.994635i \(0.532987\pi\)
\(42\) 0 0
\(43\) 14286.8 1.17832 0.589162 0.808015i \(-0.299458\pi\)
0.589162 + 0.808015i \(0.299458\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6472.14 0.427369 0.213685 0.976903i \(-0.431454\pi\)
0.213685 + 0.976903i \(0.431454\pi\)
\(48\) 0 0
\(49\) 15328.0 0.912001
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13692.2 −0.669553 −0.334777 0.942298i \(-0.608661\pi\)
−0.334777 + 0.942298i \(0.608661\pi\)
\(54\) 0 0
\(55\) 40767.8 1.81723
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5708.43 −0.213495 −0.106747 0.994286i \(-0.534044\pi\)
−0.106747 + 0.994286i \(0.534044\pi\)
\(60\) 0 0
\(61\) −11799.2 −0.406001 −0.203000 0.979179i \(-0.565069\pi\)
−0.203000 + 0.979179i \(0.565069\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −44839.7 −1.31637
\(66\) 0 0
\(67\) −3543.32 −0.0964324 −0.0482162 0.998837i \(-0.515354\pi\)
−0.0482162 + 0.998837i \(0.515354\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 58429.0 1.37557 0.687785 0.725914i \(-0.258583\pi\)
0.687785 + 0.725914i \(0.258583\pi\)
\(72\) 0 0
\(73\) −60181.3 −1.32176 −0.660882 0.750490i \(-0.729818\pi\)
−0.660882 + 0.750490i \(0.729818\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −89720.9 −1.72452
\(78\) 0 0
\(79\) −55623.4 −1.00274 −0.501372 0.865232i \(-0.667171\pi\)
−0.501372 + 0.865232i \(0.667171\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −39980.6 −0.637022 −0.318511 0.947919i \(-0.603183\pi\)
−0.318511 + 0.947919i \(0.603183\pi\)
\(84\) 0 0
\(85\) −61386.6 −0.921566
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −103171. −1.38065 −0.690327 0.723498i \(-0.742533\pi\)
−0.690327 + 0.723498i \(0.742533\pi\)
\(90\) 0 0
\(91\) 98682.3 1.24921
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −209404. −2.38054
\(96\) 0 0
\(97\) −165994. −1.79128 −0.895638 0.444783i \(-0.853281\pi\)
−0.895638 + 0.444783i \(0.853281\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −165049. −1.60994 −0.804969 0.593317i \(-0.797818\pi\)
−0.804969 + 0.593317i \(0.797818\pi\)
\(102\) 0 0
\(103\) 72860.9 0.676709 0.338354 0.941019i \(-0.390130\pi\)
0.338354 + 0.941019i \(0.390130\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −123024. −1.03880 −0.519399 0.854532i \(-0.673844\pi\)
−0.519399 + 0.854532i \(0.673844\pi\)
\(108\) 0 0
\(109\) 24274.5 0.195697 0.0978486 0.995201i \(-0.468804\pi\)
0.0978486 + 0.995201i \(0.468804\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 201372. 1.48356 0.741778 0.670646i \(-0.233983\pi\)
0.741778 + 0.670646i \(0.233983\pi\)
\(114\) 0 0
\(115\) 223628. 1.57682
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 135099. 0.874547
\(120\) 0 0
\(121\) 89449.9 0.555414
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 31339.2 0.179396
\(126\) 0 0
\(127\) −104163. −0.573067 −0.286534 0.958070i \(-0.592503\pi\)
−0.286534 + 0.958070i \(0.592503\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −178850. −0.910563 −0.455282 0.890347i \(-0.650461\pi\)
−0.455282 + 0.890347i \(0.650461\pi\)
\(132\) 0 0
\(133\) 460853. 2.25909
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −135303. −0.615892 −0.307946 0.951404i \(-0.599642\pi\)
−0.307946 + 0.951404i \(0.599642\pi\)
\(138\) 0 0
\(139\) −226166. −0.992867 −0.496434 0.868075i \(-0.665357\pi\)
−0.496434 + 0.868075i \(0.665357\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −275521. −1.12672
\(144\) 0 0
\(145\) −318462. −1.25788
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 342897. 1.26531 0.632657 0.774432i \(-0.281964\pi\)
0.632657 + 0.774432i \(0.281964\pi\)
\(150\) 0 0
\(151\) −243940. −0.870644 −0.435322 0.900275i \(-0.643366\pi\)
−0.435322 + 0.900275i \(0.643366\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −252901. −0.845516
\(156\) 0 0
\(157\) 348876. 1.12959 0.564797 0.825230i \(-0.308954\pi\)
0.564797 + 0.825230i \(0.308954\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −492156. −1.49637
\(162\) 0 0
\(163\) 303629. 0.895107 0.447553 0.894257i \(-0.352295\pi\)
0.447553 + 0.894257i \(0.352295\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 32775.5 0.0909405 0.0454703 0.998966i \(-0.485521\pi\)
0.0454703 + 0.998966i \(0.485521\pi\)
\(168\) 0 0
\(169\) −68252.8 −0.183825
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 355302. 0.902574 0.451287 0.892379i \(-0.350965\pi\)
0.451287 + 0.892379i \(0.350965\pi\)
\(174\) 0 0
\(175\) −629166. −1.55299
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 459272. 1.07137 0.535683 0.844419i \(-0.320054\pi\)
0.535683 + 0.844419i \(0.320054\pi\)
\(180\) 0 0
\(181\) −190088. −0.431279 −0.215640 0.976473i \(-0.569184\pi\)
−0.215640 + 0.976473i \(0.569184\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −779360. −1.67421
\(186\) 0 0
\(187\) −377195. −0.788791
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 688559. 1.36571 0.682854 0.730555i \(-0.260738\pi\)
0.682854 + 0.730555i \(0.260738\pi\)
\(192\) 0 0
\(193\) 344277. 0.665295 0.332648 0.943051i \(-0.392058\pi\)
0.332648 + 0.943051i \(0.392058\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 596238. 1.09460 0.547298 0.836938i \(-0.315656\pi\)
0.547298 + 0.836938i \(0.315656\pi\)
\(198\) 0 0
\(199\) −49436.6 −0.0884945 −0.0442473 0.999021i \(-0.514089\pi\)
−0.0442473 + 0.999021i \(0.514089\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 700866. 1.19370
\(204\) 0 0
\(205\) −181393. −0.301464
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.28670e6 −2.03757
\(210\) 0 0
\(211\) −151998. −0.235035 −0.117518 0.993071i \(-0.537494\pi\)
−0.117518 + 0.993071i \(0.537494\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.16372e6 1.71693
\(216\) 0 0
\(217\) 556580. 0.802377
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 414870. 0.571388
\(222\) 0 0
\(223\) −562114. −0.756942 −0.378471 0.925613i \(-0.623550\pi\)
−0.378471 + 0.925613i \(0.623550\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.26099e6 1.62422 0.812111 0.583503i \(-0.198318\pi\)
0.812111 + 0.583503i \(0.198318\pi\)
\(228\) 0 0
\(229\) 139607. 0.175922 0.0879609 0.996124i \(-0.471965\pi\)
0.0879609 + 0.996124i \(0.471965\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.18799e6 1.43358 0.716790 0.697289i \(-0.245610\pi\)
0.716790 + 0.697289i \(0.245610\pi\)
\(234\) 0 0
\(235\) 527181. 0.622716
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 329903. 0.373587 0.186794 0.982399i \(-0.440190\pi\)
0.186794 + 0.982399i \(0.440190\pi\)
\(240\) 0 0
\(241\) 168745. 0.187149 0.0935745 0.995612i \(-0.470171\pi\)
0.0935745 + 0.995612i \(0.470171\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.24853e6 1.32887
\(246\) 0 0
\(247\) 1.41522e6 1.47598
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 224796. 0.225218 0.112609 0.993639i \(-0.464079\pi\)
0.112609 + 0.993639i \(0.464079\pi\)
\(252\) 0 0
\(253\) 1.37410e6 1.34964
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −695936. −0.657259 −0.328629 0.944459i \(-0.606587\pi\)
−0.328629 + 0.944459i \(0.606587\pi\)
\(258\) 0 0
\(259\) 1.71520e6 1.58879
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 482731. 0.430344 0.215172 0.976576i \(-0.430969\pi\)
0.215172 + 0.976576i \(0.430969\pi\)
\(264\) 0 0
\(265\) −1.11529e6 −0.975601
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.23733e6 −1.88516 −0.942581 0.333978i \(-0.891609\pi\)
−0.942581 + 0.333978i \(0.891609\pi\)
\(270\) 0 0
\(271\) −1.74480e6 −1.44318 −0.721592 0.692318i \(-0.756589\pi\)
−0.721592 + 0.692318i \(0.756589\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.75663e6 1.40071
\(276\) 0 0
\(277\) −757942. −0.593521 −0.296761 0.954952i \(-0.595906\pi\)
−0.296761 + 0.954952i \(0.595906\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −430431. −0.325191 −0.162595 0.986693i \(-0.551986\pi\)
−0.162595 + 0.986693i \(0.551986\pi\)
\(282\) 0 0
\(283\) 957298. 0.710528 0.355264 0.934766i \(-0.384391\pi\)
0.355264 + 0.934766i \(0.384391\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 399206. 0.286083
\(288\) 0 0
\(289\) −851890. −0.599983
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 566294. 0.385366 0.192683 0.981261i \(-0.438281\pi\)
0.192683 + 0.981261i \(0.438281\pi\)
\(294\) 0 0
\(295\) −464974. −0.311081
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.51135e6 −0.977655
\(300\) 0 0
\(301\) −2.56109e6 −1.62933
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −961090. −0.591581
\(306\) 0 0
\(307\) −2.00565e6 −1.21453 −0.607265 0.794499i \(-0.707733\pi\)
−0.607265 + 0.794499i \(0.707733\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.11670e6 1.24096 0.620481 0.784221i \(-0.286937\pi\)
0.620481 + 0.784221i \(0.286937\pi\)
\(312\) 0 0
\(313\) 1.21334e6 0.700037 0.350018 0.936743i \(-0.386175\pi\)
0.350018 + 0.936743i \(0.386175\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.56756e6 1.43507 0.717533 0.696524i \(-0.245271\pi\)
0.717533 + 0.696524i \(0.245271\pi\)
\(318\) 0 0
\(319\) −1.95682e6 −1.07665
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.93747e6 1.03330
\(324\) 0 0
\(325\) −1.93208e6 −1.01465
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.16021e6 −0.590945
\(330\) 0 0
\(331\) 1.44226e6 0.723557 0.361778 0.932264i \(-0.382170\pi\)
0.361778 + 0.932264i \(0.382170\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −288617. −0.140511
\(336\) 0 0
\(337\) 1.58187e6 0.758744 0.379372 0.925244i \(-0.376140\pi\)
0.379372 + 0.925244i \(0.376140\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.55397e6 −0.723698
\(342\) 0 0
\(343\) 265129. 0.121681
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.64997e6 0.735618 0.367809 0.929901i \(-0.380108\pi\)
0.367809 + 0.929901i \(0.380108\pi\)
\(348\) 0 0
\(349\) 650366. 0.285821 0.142911 0.989736i \(-0.454354\pi\)
0.142911 + 0.989736i \(0.454354\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.18163e6 −0.504715 −0.252358 0.967634i \(-0.581206\pi\)
−0.252358 + 0.967634i \(0.581206\pi\)
\(354\) 0 0
\(355\) 4.75928e6 2.00433
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −985030. −0.403379 −0.201690 0.979449i \(-0.564643\pi\)
−0.201690 + 0.979449i \(0.564643\pi\)
\(360\) 0 0
\(361\) 4.13306e6 1.66918
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.90200e6 −1.92593
\(366\) 0 0
\(367\) −1.19687e6 −0.463855 −0.231928 0.972733i \(-0.574503\pi\)
−0.231928 + 0.972733i \(0.574503\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.45450e6 0.925825
\(372\) 0 0
\(373\) −2.46460e6 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.15227e6 0.779906
\(378\) 0 0
\(379\) 5.33540e6 1.90796 0.953979 0.299875i \(-0.0969449\pi\)
0.953979 + 0.299875i \(0.0969449\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.65223e6 −0.575539 −0.287770 0.957700i \(-0.592914\pi\)
−0.287770 + 0.957700i \(0.592914\pi\)
\(384\) 0 0
\(385\) −7.30813e6 −2.51278
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.66887e6 −1.89942 −0.949712 0.313123i \(-0.898625\pi\)
−0.949712 + 0.313123i \(0.898625\pi\)
\(390\) 0 0
\(391\) −2.06907e6 −0.684437
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.53074e6 −1.46109
\(396\) 0 0
\(397\) 418875. 0.133385 0.0666927 0.997774i \(-0.478755\pi\)
0.0666927 + 0.997774i \(0.478755\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.86512e6 −0.889777 −0.444889 0.895586i \(-0.646757\pi\)
−0.444889 + 0.895586i \(0.646757\pi\)
\(402\) 0 0
\(403\) 1.70918e6 0.524235
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.78884e6 −1.43299
\(408\) 0 0
\(409\) 5.30476e6 1.56804 0.784021 0.620735i \(-0.213166\pi\)
0.784021 + 0.620735i \(0.213166\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.02331e6 0.295210
\(414\) 0 0
\(415\) −3.25658e6 −0.928200
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.84273e6 −1.34758 −0.673791 0.738922i \(-0.735335\pi\)
−0.673791 + 0.738922i \(0.735335\pi\)
\(420\) 0 0
\(421\) 4.57939e6 1.25922 0.629610 0.776911i \(-0.283215\pi\)
0.629610 + 0.776911i \(0.283215\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.64507e6 −0.710338
\(426\) 0 0
\(427\) 2.11515e6 0.561398
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.09717e6 0.803105 0.401552 0.915836i \(-0.368471\pi\)
0.401552 + 0.915836i \(0.368471\pi\)
\(432\) 0 0
\(433\) −992453. −0.254384 −0.127192 0.991878i \(-0.540596\pi\)
−0.127192 + 0.991878i \(0.540596\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.05808e6 −1.76800
\(438\) 0 0
\(439\) −885373. −0.219263 −0.109631 0.993972i \(-0.534967\pi\)
−0.109631 + 0.993972i \(0.534967\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 540777. 0.130921 0.0654605 0.997855i \(-0.479148\pi\)
0.0654605 + 0.997855i \(0.479148\pi\)
\(444\) 0 0
\(445\) −8.40373e6 −2.01174
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.22193e6 −0.520132 −0.260066 0.965591i \(-0.583744\pi\)
−0.260066 + 0.965591i \(0.583744\pi\)
\(450\) 0 0
\(451\) −1.11458e6 −0.258031
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.03806e6 1.82022
\(456\) 0 0
\(457\) −577784. −0.129412 −0.0647061 0.997904i \(-0.520611\pi\)
−0.0647061 + 0.997904i \(0.520611\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.41438e6 0.748273 0.374137 0.927374i \(-0.377939\pi\)
0.374137 + 0.927374i \(0.377939\pi\)
\(462\) 0 0
\(463\) −1.85555e6 −0.402272 −0.201136 0.979563i \(-0.564463\pi\)
−0.201136 + 0.979563i \(0.564463\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.73338e6 0.792154 0.396077 0.918217i \(-0.370371\pi\)
0.396077 + 0.918217i \(0.370371\pi\)
\(468\) 0 0
\(469\) 635184. 0.133342
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.15057e6 1.46956
\(474\) 0 0
\(475\) −9.02296e6 −1.83491
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.98493e6 0.793563 0.396782 0.917913i \(-0.370127\pi\)
0.396782 + 0.917913i \(0.370127\pi\)
\(480\) 0 0
\(481\) 5.26715e6 1.03804
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.35209e7 −2.61006
\(486\) 0 0
\(487\) 4.19007e6 0.800570 0.400285 0.916391i \(-0.368911\pi\)
0.400285 + 0.916391i \(0.368911\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.19213e6 −0.784749 −0.392374 0.919806i \(-0.628346\pi\)
−0.392374 + 0.919806i \(0.628346\pi\)
\(492\) 0 0
\(493\) 2.94651e6 0.545997
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.04741e7 −1.90207
\(498\) 0 0
\(499\) −3.70338e6 −0.665805 −0.332903 0.942961i \(-0.608028\pi\)
−0.332903 + 0.942961i \(0.608028\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.03820e6 −0.711653 −0.355827 0.934552i \(-0.615801\pi\)
−0.355827 + 0.934552i \(0.615801\pi\)
\(504\) 0 0
\(505\) −1.34439e7 −2.34583
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −167961. −0.0287351 −0.0143676 0.999897i \(-0.504573\pi\)
−0.0143676 + 0.999897i \(0.504573\pi\)
\(510\) 0 0
\(511\) 1.07882e7 1.82767
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.93481e6 0.986027
\(516\) 0 0
\(517\) 3.23931e6 0.532999
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.98731e6 −0.643555 −0.321777 0.946815i \(-0.604280\pi\)
−0.321777 + 0.946815i \(0.604280\pi\)
\(522\) 0 0
\(523\) −4.41694e6 −0.706102 −0.353051 0.935604i \(-0.614856\pi\)
−0.353051 + 0.935604i \(0.614856\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.33992e6 0.367007
\(528\) 0 0
\(529\) 1.10115e6 0.171084
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.22591e6 0.186913
\(534\) 0 0
\(535\) −1.00208e7 −1.51362
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7.67167e6 1.13741
\(540\) 0 0
\(541\) 9.24640e6 1.35825 0.679125 0.734023i \(-0.262360\pi\)
0.679125 + 0.734023i \(0.262360\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.97726e6 0.285149
\(546\) 0 0
\(547\) 1.52578e6 0.218033 0.109017 0.994040i \(-0.465230\pi\)
0.109017 + 0.994040i \(0.465230\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.00512e7 1.41039
\(552\) 0 0
\(553\) 9.97118e6 1.38654
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.86988e6 −0.801662 −0.400831 0.916152i \(-0.631279\pi\)
−0.400831 + 0.916152i \(0.631279\pi\)
\(558\) 0 0
\(559\) −7.86477e6 −1.06453
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 213372. 0.0283704 0.0141852 0.999899i \(-0.495485\pi\)
0.0141852 + 0.999899i \(0.495485\pi\)
\(564\) 0 0
\(565\) 1.64026e7 2.16168
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.12903e7 −1.46192 −0.730961 0.682419i \(-0.760928\pi\)
−0.730961 + 0.682419i \(0.760928\pi\)
\(570\) 0 0
\(571\) −464683. −0.0596440 −0.0298220 0.999555i \(-0.509494\pi\)
−0.0298220 + 0.999555i \(0.509494\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.63583e6 1.21540
\(576\) 0 0
\(577\) 8.78227e6 1.09816 0.549082 0.835769i \(-0.314978\pi\)
0.549082 + 0.835769i \(0.314978\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.16702e6 0.880843
\(582\) 0 0
\(583\) −6.85298e6 −0.835041
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.00417e7 −1.20285 −0.601423 0.798931i \(-0.705399\pi\)
−0.601423 + 0.798931i \(0.705399\pi\)
\(588\) 0 0
\(589\) 7.98200e6 0.948034
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.15945e7 1.35399 0.676995 0.735988i \(-0.263282\pi\)
0.676995 + 0.735988i \(0.263282\pi\)
\(594\) 0 0
\(595\) 1.10043e7 1.27430
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.48884e6 0.169544 0.0847720 0.996400i \(-0.472984\pi\)
0.0847720 + 0.996400i \(0.472984\pi\)
\(600\) 0 0
\(601\) −9.51007e6 −1.07398 −0.536992 0.843588i \(-0.680439\pi\)
−0.536992 + 0.843588i \(0.680439\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.28605e6 0.809289
\(606\) 0 0
\(607\) 7.40405e6 0.815639 0.407819 0.913063i \(-0.366289\pi\)
0.407819 + 0.913063i \(0.366289\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.56285e6 −0.386096
\(612\) 0 0
\(613\) −7.14368e6 −0.767840 −0.383920 0.923366i \(-0.625426\pi\)
−0.383920 + 0.923366i \(0.625426\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00404e6 0.106179 0.0530895 0.998590i \(-0.483093\pi\)
0.0530895 + 0.998590i \(0.483093\pi\)
\(618\) 0 0
\(619\) −7.48656e6 −0.785336 −0.392668 0.919680i \(-0.628448\pi\)
−0.392668 + 0.919680i \(0.628448\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.84948e7 1.90910
\(624\) 0 0
\(625\) −8.41526e6 −0.861723
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.21086e6 0.726709
\(630\) 0 0
\(631\) 1.34648e7 1.34626 0.673128 0.739526i \(-0.264950\pi\)
0.673128 + 0.739526i \(0.264950\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.48452e6 −0.835012
\(636\) 0 0
\(637\) −8.43792e6 −0.823924
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.62617e7 1.56322 0.781610 0.623768i \(-0.214399\pi\)
0.781610 + 0.623768i \(0.214399\pi\)
\(642\) 0 0
\(643\) −1.13687e7 −1.08439 −0.542194 0.840253i \(-0.682406\pi\)
−0.542194 + 0.840253i \(0.682406\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.18311e7 1.11113 0.555563 0.831474i \(-0.312503\pi\)
0.555563 + 0.831474i \(0.312503\pi\)
\(648\) 0 0
\(649\) −2.85707e6 −0.266262
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.34599e6 0.490620 0.245310 0.969445i \(-0.421110\pi\)
0.245310 + 0.969445i \(0.421110\pi\)
\(654\) 0 0
\(655\) −1.45680e7 −1.32677
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.23736e7 1.10990 0.554949 0.831884i \(-0.312738\pi\)
0.554949 + 0.831884i \(0.312738\pi\)
\(660\) 0 0
\(661\) −2.09830e7 −1.86795 −0.933974 0.357342i \(-0.883683\pi\)
−0.933974 + 0.357342i \(0.883683\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.75383e7 3.29170
\(666\) 0 0
\(667\) −1.07339e7 −0.934210
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.90550e6 −0.506349
\(672\) 0 0
\(673\) −1.26175e7 −1.07383 −0.536915 0.843637i \(-0.680410\pi\)
−0.536915 + 0.843637i \(0.680410\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.99456e6 −0.670383 −0.335192 0.942150i \(-0.608801\pi\)
−0.335192 + 0.942150i \(0.608801\pi\)
\(678\) 0 0
\(679\) 2.97565e7 2.47689
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.23985e7 −1.01699 −0.508496 0.861064i \(-0.669798\pi\)
−0.508496 + 0.861064i \(0.669798\pi\)
\(684\) 0 0
\(685\) −1.10209e7 −0.897412
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.53746e6 0.604891
\(690\) 0 0
\(691\) 2.06344e7 1.64398 0.821990 0.569502i \(-0.192864\pi\)
0.821990 + 0.569502i \(0.192864\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.84222e7 −1.44670
\(696\) 0 0
\(697\) 1.67830e6 0.130854
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.28362e7 −1.75521 −0.877603 0.479388i \(-0.840859\pi\)
−0.877603 + 0.479388i \(0.840859\pi\)
\(702\) 0 0
\(703\) 2.45980e7 1.87720
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.95870e7 2.22614
\(708\) 0 0
\(709\) −1.78247e7 −1.33170 −0.665852 0.746084i \(-0.731932\pi\)
−0.665852 + 0.746084i \(0.731932\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.52418e6 −0.627955
\(714\) 0 0
\(715\) −2.24423e7 −1.64173
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.10161e7 0.794702 0.397351 0.917667i \(-0.369929\pi\)
0.397351 + 0.917667i \(0.369929\pi\)
\(720\) 0 0
\(721\) −1.30612e7 −0.935720
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.37221e7 −0.969564
\(726\) 0 0
\(727\) 2.60033e7 1.82470 0.912351 0.409410i \(-0.134265\pi\)
0.912351 + 0.409410i \(0.134265\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.07671e7 −0.745254
\(732\) 0 0
\(733\) 2.66711e7 1.83350 0.916750 0.399462i \(-0.130803\pi\)
0.916750 + 0.399462i \(0.130803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.77343e6 −0.120267
\(738\) 0 0
\(739\) 1.31065e6 0.0882829 0.0441414 0.999025i \(-0.485945\pi\)
0.0441414 + 0.999025i \(0.485945\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.05004e7 −1.36236 −0.681178 0.732118i \(-0.738532\pi\)
−0.681178 + 0.732118i \(0.738532\pi\)
\(744\) 0 0
\(745\) 2.79303e7 1.84368
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.20536e7 1.43640
\(750\) 0 0
\(751\) −1.35740e7 −0.878230 −0.439115 0.898431i \(-0.644708\pi\)
−0.439115 + 0.898431i \(0.644708\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.98699e7 −1.26861
\(756\) 0 0
\(757\) −470115. −0.0298171 −0.0149085 0.999889i \(-0.504746\pi\)
−0.0149085 + 0.999889i \(0.504746\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.05381e6 0.128558 0.0642789 0.997932i \(-0.479525\pi\)
0.0642789 + 0.997932i \(0.479525\pi\)
\(762\) 0 0
\(763\) −4.35151e6 −0.270600
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.14244e6 0.192876
\(768\) 0 0
\(769\) 2.51201e6 0.153181 0.0765905 0.997063i \(-0.475597\pi\)
0.0765905 + 0.997063i \(0.475597\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.03339e6 0.483560 0.241780 0.970331i \(-0.422269\pi\)
0.241780 + 0.970331i \(0.422269\pi\)
\(774\) 0 0
\(775\) −1.08972e7 −0.651719
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.72508e6 0.338016
\(780\) 0 0
\(781\) 2.92438e7 1.71556
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.84174e7 1.64592
\(786\) 0 0
\(787\) 3.15766e7 1.81731 0.908653 0.417553i \(-0.137112\pi\)
0.908653 + 0.417553i \(0.137112\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.60985e7 −2.05139
\(792\) 0 0
\(793\) 6.49534e6 0.366791
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.67046e7 1.48915 0.744577 0.667536i \(-0.232651\pi\)
0.744577 + 0.667536i \(0.232651\pi\)
\(798\) 0 0
\(799\) −4.87763e6 −0.270298
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.01208e7 −1.64846
\(804\) 0 0
\(805\) −4.00880e7 −2.18034
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.27526e7 −0.685057 −0.342529 0.939507i \(-0.611283\pi\)
−0.342529 + 0.939507i \(0.611283\pi\)
\(810\) 0 0
\(811\) −2.34690e7 −1.25297 −0.626487 0.779432i \(-0.715508\pi\)
−0.626487 + 0.779432i \(0.715508\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.47318e7 1.30425
\(816\) 0 0
\(817\) −3.67290e7 −1.92510
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.58076e7 0.818481 0.409240 0.912427i \(-0.365794\pi\)
0.409240 + 0.912427i \(0.365794\pi\)
\(822\) 0 0
\(823\) 1.62543e7 0.836507 0.418254 0.908330i \(-0.362642\pi\)
0.418254 + 0.908330i \(0.362642\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.39257e7 0.708031 0.354016 0.935239i \(-0.384816\pi\)
0.354016 + 0.935239i \(0.384816\pi\)
\(828\) 0 0
\(829\) 2.52491e7 1.27603 0.638013 0.770026i \(-0.279757\pi\)
0.638013 + 0.770026i \(0.279757\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.15517e7 −0.576812
\(834\) 0 0
\(835\) 2.66969e6 0.132509
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.99108e6 −0.244788 −0.122394 0.992482i \(-0.539057\pi\)
−0.122394 + 0.992482i \(0.539057\pi\)
\(840\) 0 0
\(841\) −5.22523e6 −0.254751
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.55946e6 −0.267850
\(846\) 0 0
\(847\) −1.60350e7 −0.767999
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.62687e7 −1.24341
\(852\) 0 0
\(853\) 7.52571e6 0.354140 0.177070 0.984198i \(-0.443338\pi\)
0.177070 + 0.984198i \(0.443338\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.73880e7 −1.73892 −0.869461 0.494001i \(-0.835534\pi\)
−0.869461 + 0.494001i \(0.835534\pi\)
\(858\) 0 0
\(859\) −8.43595e6 −0.390078 −0.195039 0.980796i \(-0.562483\pi\)
−0.195039 + 0.980796i \(0.562483\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.10157e7 0.503482 0.251741 0.967795i \(-0.418997\pi\)
0.251741 + 0.967795i \(0.418997\pi\)
\(864\) 0 0
\(865\) 2.89408e7 1.31513
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.78395e7 −1.25058
\(870\) 0 0
\(871\) 1.95056e6 0.0871194
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.61793e6 −0.248060
\(876\) 0 0
\(877\) −1.48438e7 −0.651698 −0.325849 0.945422i \(-0.605650\pi\)
−0.325849 + 0.945422i \(0.605650\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.90839e7 −1.26245 −0.631224 0.775600i \(-0.717447\pi\)
−0.631224 + 0.775600i \(0.717447\pi\)
\(882\) 0 0
\(883\) −3.79255e7 −1.63693 −0.818464 0.574558i \(-0.805174\pi\)
−0.818464 + 0.574558i \(0.805174\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.53153e7 −0.653607 −0.326803 0.945092i \(-0.605971\pi\)
−0.326803 + 0.945092i \(0.605971\pi\)
\(888\) 0 0
\(889\) 1.86726e7 0.792410
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.66388e7 −0.698220
\(894\) 0 0
\(895\) 3.74096e7 1.56108
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.21390e7 0.500940
\(900\) 0 0
\(901\) 1.03190e7 0.423472
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.54834e7 −0.628414
\(906\) 0 0
\(907\) −2.68468e7 −1.08361 −0.541806 0.840504i \(-0.682259\pi\)
−0.541806 + 0.840504i \(0.682259\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.76866e6 −0.190371 −0.0951854 0.995460i \(-0.530344\pi\)
−0.0951854 + 0.995460i \(0.530344\pi\)
\(912\) 0 0
\(913\) −2.00103e7 −0.794470
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.20610e7 1.25908
\(918\) 0 0
\(919\) −2.23134e7 −0.871518 −0.435759 0.900063i \(-0.643520\pi\)
−0.435759 + 0.900063i \(0.643520\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3.21646e7 −1.24272
\(924\) 0 0
\(925\) −3.35816e7 −1.29047
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.31474e7 −0.879961 −0.439980 0.898007i \(-0.645015\pi\)
−0.439980 + 0.898007i \(0.645015\pi\)
\(930\) 0 0
\(931\) −3.94057e7 −1.48999
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.07240e7 −1.14934
\(936\) 0 0
\(937\) −3.45453e6 −0.128541 −0.0642703 0.997933i \(-0.520472\pi\)
−0.0642703 + 0.997933i \(0.520472\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.54521e7 −1.30517 −0.652586 0.757715i \(-0.726316\pi\)
−0.652586 + 0.757715i \(0.726316\pi\)
\(942\) 0 0
\(943\) −6.11395e6 −0.223894
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.90255e7 −1.41408 −0.707039 0.707174i \(-0.749970\pi\)
−0.707039 + 0.707174i \(0.749970\pi\)
\(948\) 0 0
\(949\) 3.31292e7 1.19411
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.07434e6 0.0739858 0.0369929 0.999316i \(-0.488222\pi\)
0.0369929 + 0.999316i \(0.488222\pi\)
\(954\) 0 0
\(955\) 5.60859e7 1.98996
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.42547e7 0.851626
\(960\) 0 0
\(961\) −1.89891e7 −0.663280
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.80427e7 0.969397
\(966\) 0 0
\(967\) −1.14931e7 −0.395251 −0.197625 0.980278i \(-0.563323\pi\)
−0.197625 + 0.980278i \(0.563323\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.36234e7 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(972\) 0 0
\(973\) 4.05431e7 1.37289
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.03599e7 −1.01757 −0.508785 0.860894i \(-0.669905\pi\)
−0.508785 + 0.860894i \(0.669905\pi\)
\(978\) 0 0
\(979\) −5.16374e7 −1.72190
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 6.28069e6 0.207312 0.103656 0.994613i \(-0.466946\pi\)
0.103656 + 0.994613i \(0.466946\pi\)
\(984\) 0 0
\(985\) 4.85659e7 1.59493
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.92238e7 1.27514
\(990\) 0 0
\(991\) 1.30907e7 0.423428 0.211714 0.977332i \(-0.432096\pi\)
0.211714 + 0.977332i \(0.432096\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.02681e6 −0.128945
\(996\) 0 0
\(997\) −3.38507e6 −0.107852 −0.0539262 0.998545i \(-0.517174\pi\)
−0.0539262 + 0.998545i \(0.517174\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 324.6.a.d.1.5 5
3.2 odd 2 324.6.a.e.1.1 5
9.2 odd 6 36.6.e.a.13.3 10
9.4 even 3 108.6.e.a.73.1 10
9.5 odd 6 36.6.e.a.25.3 yes 10
9.7 even 3 108.6.e.a.37.1 10
36.7 odd 6 432.6.i.d.145.1 10
36.11 even 6 144.6.i.d.49.3 10
36.23 even 6 144.6.i.d.97.3 10
36.31 odd 6 432.6.i.d.289.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.6.e.a.13.3 10 9.2 odd 6
36.6.e.a.25.3 yes 10 9.5 odd 6
108.6.e.a.37.1 10 9.7 even 3
108.6.e.a.73.1 10 9.4 even 3
144.6.i.d.49.3 10 36.11 even 6
144.6.i.d.97.3 10 36.23 even 6
324.6.a.d.1.5 5 1.1 even 1 trivial
324.6.a.e.1.1 5 3.2 odd 2
432.6.i.d.145.1 10 36.7 odd 6
432.6.i.d.289.1 10 36.31 odd 6