Properties

Label 676.6.d.a.337.2
Level $676$
Weight $6$
Character 676.337
Analytic conductor $108.419$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 676.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(108.419462194\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 676.337
Dual form 676.6.d.a.337.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} +O(q^{10})\) \(q-12.0000 q^{3} +54.0000i q^{5} +88.0000i q^{7} -99.0000 q^{9} -540.000i q^{11} -648.000i q^{15} -594.000 q^{17} +836.000i q^{19} -1056.00i q^{21} +4104.00 q^{23} +209.000 q^{25} +4104.00 q^{27} -594.000 q^{29} +4256.00i q^{31} +6480.00i q^{33} -4752.00 q^{35} +298.000i q^{37} +17226.0i q^{41} +12100.0 q^{43} -5346.00i q^{45} +1296.00i q^{47} +9063.00 q^{49} +7128.00 q^{51} +19494.0 q^{53} +29160.0 q^{55} -10032.0i q^{57} +7668.00i q^{59} -34738.0 q^{61} -8712.00i q^{63} +21812.0i q^{67} -49248.0 q^{69} -46872.0i q^{71} -67562.0i q^{73} -2508.00 q^{75} +47520.0 q^{77} -76912.0 q^{79} -25191.0 q^{81} +67716.0i q^{83} -32076.0i q^{85} +7128.00 q^{87} -29754.0i q^{89} -51072.0i q^{93} -45144.0 q^{95} -122398. i q^{97} +53460.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 24 q^{3} - 198 q^{9} + O(q^{10}) \) \( 2 q - 24 q^{3} - 198 q^{9} - 1188 q^{17} + 8208 q^{23} + 418 q^{25} + 8208 q^{27} - 1188 q^{29} - 9504 q^{35} + 24200 q^{43} + 18126 q^{49} + 14256 q^{51} + 38988 q^{53} + 58320 q^{55} - 69476 q^{61} - 98496 q^{69} - 5016 q^{75} + 95040 q^{77} - 153824 q^{79} - 50382 q^{81} + 14256 q^{87} - 90288 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) −12.0000 −0.769800 −0.384900 0.922958i \(-0.625764\pi\)
−0.384900 + 0.922958i \(0.625764\pi\)
\(4\) 0 0
\(5\) 54.0000i 0.965981i 0.875625 + 0.482991i \(0.160450\pi\)
−0.875625 + 0.482991i \(0.839550\pi\)
\(6\) 0 0
\(7\) 88.0000i 0.678793i 0.940643 + 0.339397i \(0.110223\pi\)
−0.940643 + 0.339397i \(0.889777\pi\)
\(8\) 0 0
\(9\) −99.0000 −0.407407
\(10\) 0 0
\(11\) − 540.000i − 1.34559i −0.739830 0.672794i \(-0.765094\pi\)
0.739830 0.672794i \(-0.234906\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 648.000i − 0.743613i
\(16\) 0 0
\(17\) −594.000 −0.498499 −0.249249 0.968439i \(-0.580184\pi\)
−0.249249 + 0.968439i \(0.580184\pi\)
\(18\) 0 0
\(19\) 836.000i 0.531279i 0.964072 + 0.265639i \(0.0855830\pi\)
−0.964072 + 0.265639i \(0.914417\pi\)
\(20\) 0 0
\(21\) − 1056.00i − 0.522535i
\(22\) 0 0
\(23\) 4104.00 1.61766 0.808831 0.588041i \(-0.200101\pi\)
0.808831 + 0.588041i \(0.200101\pi\)
\(24\) 0 0
\(25\) 209.000 0.0668800
\(26\) 0 0
\(27\) 4104.00 1.08342
\(28\) 0 0
\(29\) −594.000 −0.131157 −0.0655785 0.997847i \(-0.520889\pi\)
−0.0655785 + 0.997847i \(0.520889\pi\)
\(30\) 0 0
\(31\) 4256.00i 0.795422i 0.917511 + 0.397711i \(0.130195\pi\)
−0.917511 + 0.397711i \(0.869805\pi\)
\(32\) 0 0
\(33\) 6480.00i 1.03583i
\(34\) 0 0
\(35\) −4752.00 −0.655702
\(36\) 0 0
\(37\) 298.000i 0.0357859i 0.999840 + 0.0178930i \(0.00569581\pi\)
−0.999840 + 0.0178930i \(0.994304\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 17226.0i 1.60039i 0.599742 + 0.800193i \(0.295270\pi\)
−0.599742 + 0.800193i \(0.704730\pi\)
\(42\) 0 0
\(43\) 12100.0 0.997963 0.498981 0.866613i \(-0.333708\pi\)
0.498981 + 0.866613i \(0.333708\pi\)
\(44\) 0 0
\(45\) − 5346.00i − 0.393548i
\(46\) 0 0
\(47\) 1296.00i 0.0855777i 0.999084 + 0.0427888i \(0.0136243\pi\)
−0.999084 + 0.0427888i \(0.986376\pi\)
\(48\) 0 0
\(49\) 9063.00 0.539240
\(50\) 0 0
\(51\) 7128.00 0.383745
\(52\) 0 0
\(53\) 19494.0 0.953260 0.476630 0.879104i \(-0.341858\pi\)
0.476630 + 0.879104i \(0.341858\pi\)
\(54\) 0 0
\(55\) 29160.0 1.29981
\(56\) 0 0
\(57\) − 10032.0i − 0.408978i
\(58\) 0 0
\(59\) 7668.00i 0.286782i 0.989666 + 0.143391i \(0.0458007\pi\)
−0.989666 + 0.143391i \(0.954199\pi\)
\(60\) 0 0
\(61\) −34738.0 −1.19531 −0.597655 0.801754i \(-0.703901\pi\)
−0.597655 + 0.801754i \(0.703901\pi\)
\(62\) 0 0
\(63\) − 8712.00i − 0.276545i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 21812.0i 0.593620i 0.954937 + 0.296810i \(0.0959228\pi\)
−0.954937 + 0.296810i \(0.904077\pi\)
\(68\) 0 0
\(69\) −49248.0 −1.24528
\(70\) 0 0
\(71\) − 46872.0i − 1.10349i −0.834014 0.551744i \(-0.813963\pi\)
0.834014 0.551744i \(-0.186037\pi\)
\(72\) 0 0
\(73\) − 67562.0i − 1.48387i −0.670473 0.741934i \(-0.733909\pi\)
0.670473 0.741934i \(-0.266091\pi\)
\(74\) 0 0
\(75\) −2508.00 −0.0514842
\(76\) 0 0
\(77\) 47520.0 0.913376
\(78\) 0 0
\(79\) −76912.0 −1.38652 −0.693260 0.720687i \(-0.743826\pi\)
−0.693260 + 0.720687i \(0.743826\pi\)
\(80\) 0 0
\(81\) −25191.0 −0.426612
\(82\) 0 0
\(83\) 67716.0i 1.07894i 0.842006 + 0.539468i \(0.181375\pi\)
−0.842006 + 0.539468i \(0.818625\pi\)
\(84\) 0 0
\(85\) − 32076.0i − 0.481541i
\(86\) 0 0
\(87\) 7128.00 0.100965
\(88\) 0 0
\(89\) − 29754.0i − 0.398172i −0.979982 0.199086i \(-0.936203\pi\)
0.979982 0.199086i \(-0.0637973\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 51072.0i − 0.612316i
\(94\) 0 0
\(95\) −45144.0 −0.513205
\(96\) 0 0
\(97\) − 122398.i − 1.32082i −0.750903 0.660412i \(-0.770382\pi\)
0.750903 0.660412i \(-0.229618\pi\)
\(98\) 0 0
\(99\) 53460.0i 0.548202i
\(100\) 0 0
\(101\) −11286.0 −0.110087 −0.0550436 0.998484i \(-0.517530\pi\)
−0.0550436 + 0.998484i \(0.517530\pi\)
\(102\) 0 0
\(103\) 27256.0 0.253145 0.126572 0.991957i \(-0.459602\pi\)
0.126572 + 0.991957i \(0.459602\pi\)
\(104\) 0 0
\(105\) 57024.0 0.504759
\(106\) 0 0
\(107\) 122364. 1.03322 0.516612 0.856220i \(-0.327193\pi\)
0.516612 + 0.856220i \(0.327193\pi\)
\(108\) 0 0
\(109\) 99902.0i 0.805393i 0.915334 + 0.402697i \(0.131927\pi\)
−0.915334 + 0.402697i \(0.868073\pi\)
\(110\) 0 0
\(111\) − 3576.00i − 0.0275480i
\(112\) 0 0
\(113\) −29646.0 −0.218409 −0.109204 0.994019i \(-0.534830\pi\)
−0.109204 + 0.994019i \(0.534830\pi\)
\(114\) 0 0
\(115\) 221616.i 1.56263i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 52272.0i − 0.338378i
\(120\) 0 0
\(121\) −130549. −0.810607
\(122\) 0 0
\(123\) − 206712.i − 1.23198i
\(124\) 0 0
\(125\) 180036.i 1.03059i
\(126\) 0 0
\(127\) −336512. −1.85136 −0.925681 0.378305i \(-0.876507\pi\)
−0.925681 + 0.378305i \(0.876507\pi\)
\(128\) 0 0
\(129\) −145200. −0.768232
\(130\) 0 0
\(131\) 100980. 0.514111 0.257056 0.966397i \(-0.417248\pi\)
0.257056 + 0.966397i \(0.417248\pi\)
\(132\) 0 0
\(133\) −73568.0 −0.360628
\(134\) 0 0
\(135\) 221616.i 1.04657i
\(136\) 0 0
\(137\) 317142.i 1.44362i 0.692092 + 0.721809i \(0.256689\pi\)
−0.692092 + 0.721809i \(0.743311\pi\)
\(138\) 0 0
\(139\) −148324. −0.651140 −0.325570 0.945518i \(-0.605556\pi\)
−0.325570 + 0.945518i \(0.605556\pi\)
\(140\) 0 0
\(141\) − 15552.0i − 0.0658777i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 32076.0i − 0.126695i
\(146\) 0 0
\(147\) −108756. −0.415107
\(148\) 0 0
\(149\) 196614.i 0.725519i 0.931883 + 0.362759i \(0.118165\pi\)
−0.931883 + 0.362759i \(0.881835\pi\)
\(150\) 0 0
\(151\) − 74360.0i − 0.265398i −0.991156 0.132699i \(-0.957636\pi\)
0.991156 0.132699i \(-0.0423643\pi\)
\(152\) 0 0
\(153\) 58806.0 0.203092
\(154\) 0 0
\(155\) −229824. −0.768362
\(156\) 0 0
\(157\) 120878. 0.391380 0.195690 0.980666i \(-0.437305\pi\)
0.195690 + 0.980666i \(0.437305\pi\)
\(158\) 0 0
\(159\) −233928. −0.733820
\(160\) 0 0
\(161\) 361152.i 1.09806i
\(162\) 0 0
\(163\) 111340.i 0.328233i 0.986441 + 0.164116i \(0.0524773\pi\)
−0.986441 + 0.164116i \(0.947523\pi\)
\(164\) 0 0
\(165\) −349920. −1.00060
\(166\) 0 0
\(167\) 491832.i 1.36466i 0.731043 + 0.682332i \(0.239034\pi\)
−0.731043 + 0.682332i \(0.760966\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 82764.0i − 0.216447i
\(172\) 0 0
\(173\) −707454. −1.79714 −0.898572 0.438826i \(-0.855395\pi\)
−0.898572 + 0.438826i \(0.855395\pi\)
\(174\) 0 0
\(175\) 18392.0i 0.0453977i
\(176\) 0 0
\(177\) − 92016.0i − 0.220765i
\(178\) 0 0
\(179\) −493668. −1.15160 −0.575801 0.817590i \(-0.695310\pi\)
−0.575801 + 0.817590i \(0.695310\pi\)
\(180\) 0 0
\(181\) 559450. 1.26930 0.634651 0.772799i \(-0.281144\pi\)
0.634651 + 0.772799i \(0.281144\pi\)
\(182\) 0 0
\(183\) 416856. 0.920149
\(184\) 0 0
\(185\) −16092.0 −0.0345685
\(186\) 0 0
\(187\) 320760.i 0.670774i
\(188\) 0 0
\(189\) 361152.i 0.735420i
\(190\) 0 0
\(191\) −724032. −1.43607 −0.718033 0.696009i \(-0.754957\pi\)
−0.718033 + 0.696009i \(0.754957\pi\)
\(192\) 0 0
\(193\) − 7106.00i − 0.0137319i −0.999976 0.00686597i \(-0.997814\pi\)
0.999976 0.00686597i \(-0.00218552\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 530442.i − 0.973806i −0.873456 0.486903i \(-0.838127\pi\)
0.873456 0.486903i \(-0.161873\pi\)
\(198\) 0 0
\(199\) −56168.0 −0.100544 −0.0502720 0.998736i \(-0.516009\pi\)
−0.0502720 + 0.998736i \(0.516009\pi\)
\(200\) 0 0
\(201\) − 261744.i − 0.456969i
\(202\) 0 0
\(203\) − 52272.0i − 0.0890285i
\(204\) 0 0
\(205\) −930204. −1.54594
\(206\) 0 0
\(207\) −406296. −0.659047
\(208\) 0 0
\(209\) 451440. 0.714882
\(210\) 0 0
\(211\) −339196. −0.524499 −0.262249 0.965000i \(-0.584464\pi\)
−0.262249 + 0.965000i \(0.584464\pi\)
\(212\) 0 0
\(213\) 562464.i 0.849465i
\(214\) 0 0
\(215\) 653400.i 0.964013i
\(216\) 0 0
\(217\) −374528. −0.539927
\(218\) 0 0
\(219\) 810744.i 1.14228i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 779360.i 1.04948i 0.851261 + 0.524742i \(0.175838\pi\)
−0.851261 + 0.524742i \(0.824162\pi\)
\(224\) 0 0
\(225\) −20691.0 −0.0272474
\(226\) 0 0
\(227\) − 744876.i − 0.959443i −0.877421 0.479722i \(-0.840738\pi\)
0.877421 0.479722i \(-0.159262\pi\)
\(228\) 0 0
\(229\) 272746.i 0.343692i 0.985124 + 0.171846i \(0.0549732\pi\)
−0.985124 + 0.171846i \(0.945027\pi\)
\(230\) 0 0
\(231\) −570240. −0.703117
\(232\) 0 0
\(233\) 153846. 0.185651 0.0928253 0.995682i \(-0.470410\pi\)
0.0928253 + 0.995682i \(0.470410\pi\)
\(234\) 0 0
\(235\) −69984.0 −0.0826664
\(236\) 0 0
\(237\) 922944. 1.06734
\(238\) 0 0
\(239\) 1.15474e6i 1.30764i 0.756650 + 0.653820i \(0.226834\pi\)
−0.756650 + 0.653820i \(0.773166\pi\)
\(240\) 0 0
\(241\) − 657074.i − 0.728738i −0.931255 0.364369i \(-0.881285\pi\)
0.931255 0.364369i \(-0.118715\pi\)
\(242\) 0 0
\(243\) −694980. −0.755017
\(244\) 0 0
\(245\) 489402.i 0.520895i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 812592.i − 0.830566i
\(250\) 0 0
\(251\) −1.34190e6 −1.34442 −0.672211 0.740359i \(-0.734655\pi\)
−0.672211 + 0.740359i \(0.734655\pi\)
\(252\) 0 0
\(253\) − 2.21616e6i − 2.17671i
\(254\) 0 0
\(255\) 384912.i 0.370690i
\(256\) 0 0
\(257\) −132354. −0.124998 −0.0624992 0.998045i \(-0.519907\pi\)
−0.0624992 + 0.998045i \(0.519907\pi\)
\(258\) 0 0
\(259\) −26224.0 −0.0242912
\(260\) 0 0
\(261\) 58806.0 0.0534343
\(262\) 0 0
\(263\) 943272. 0.840906 0.420453 0.907314i \(-0.361871\pi\)
0.420453 + 0.907314i \(0.361871\pi\)
\(264\) 0 0
\(265\) 1.05268e6i 0.920831i
\(266\) 0 0
\(267\) 357048.i 0.306513i
\(268\) 0 0
\(269\) 967518. 0.815227 0.407613 0.913155i \(-0.366361\pi\)
0.407613 + 0.913155i \(0.366361\pi\)
\(270\) 0 0
\(271\) 518320.i 0.428721i 0.976755 + 0.214360i \(0.0687667\pi\)
−0.976755 + 0.214360i \(0.931233\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 112860.i − 0.0899929i
\(276\) 0 0
\(277\) −2.22273e6 −1.74055 −0.870275 0.492566i \(-0.836059\pi\)
−0.870275 + 0.492566i \(0.836059\pi\)
\(278\) 0 0
\(279\) − 421344.i − 0.324061i
\(280\) 0 0
\(281\) 196614.i 0.148542i 0.997238 + 0.0742709i \(0.0236629\pi\)
−0.997238 + 0.0742709i \(0.976337\pi\)
\(282\) 0 0
\(283\) 1.55228e6 1.15213 0.576067 0.817403i \(-0.304587\pi\)
0.576067 + 0.817403i \(0.304587\pi\)
\(284\) 0 0
\(285\) 541728. 0.395066
\(286\) 0 0
\(287\) −1.51589e6 −1.08633
\(288\) 0 0
\(289\) −1.06702e6 −0.751499
\(290\) 0 0
\(291\) 1.46878e6i 1.01677i
\(292\) 0 0
\(293\) 1.07217e6i 0.729616i 0.931083 + 0.364808i \(0.118865\pi\)
−0.931083 + 0.364808i \(0.881135\pi\)
\(294\) 0 0
\(295\) −414072. −0.277026
\(296\) 0 0
\(297\) − 2.21616e6i − 1.45784i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.06480e6i 0.677410i
\(302\) 0 0
\(303\) 135432. 0.0847451
\(304\) 0 0
\(305\) − 1.87585e6i − 1.15465i
\(306\) 0 0
\(307\) − 1.58589e6i − 0.960346i −0.877174 0.480173i \(-0.840574\pi\)
0.877174 0.480173i \(-0.159426\pi\)
\(308\) 0 0
\(309\) −327072. −0.194871
\(310\) 0 0
\(311\) 730728. 0.428405 0.214203 0.976789i \(-0.431285\pi\)
0.214203 + 0.976789i \(0.431285\pi\)
\(312\) 0 0
\(313\) 584858. 0.337435 0.168717 0.985664i \(-0.446038\pi\)
0.168717 + 0.985664i \(0.446038\pi\)
\(314\) 0 0
\(315\) 470448. 0.267138
\(316\) 0 0
\(317\) − 2.48287e6i − 1.38773i −0.720105 0.693865i \(-0.755906\pi\)
0.720105 0.693865i \(-0.244094\pi\)
\(318\) 0 0
\(319\) 320760.i 0.176483i
\(320\) 0 0
\(321\) −1.46837e6 −0.795376
\(322\) 0 0
\(323\) − 496584.i − 0.264842i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 1.19882e6i − 0.619992i
\(328\) 0 0
\(329\) −114048. −0.0580895
\(330\) 0 0
\(331\) 377948.i 0.189610i 0.995496 + 0.0948052i \(0.0302228\pi\)
−0.995496 + 0.0948052i \(0.969777\pi\)
\(332\) 0 0
\(333\) − 29502.0i − 0.0145794i
\(334\) 0 0
\(335\) −1.17785e6 −0.573426
\(336\) 0 0
\(337\) −639122. −0.306555 −0.153278 0.988183i \(-0.548983\pi\)
−0.153278 + 0.988183i \(0.548983\pi\)
\(338\) 0 0
\(339\) 355752. 0.168131
\(340\) 0 0
\(341\) 2.29824e6 1.07031
\(342\) 0 0
\(343\) 2.27656e6i 1.04483i
\(344\) 0 0
\(345\) − 2.65939e6i − 1.20291i
\(346\) 0 0
\(347\) −2.90466e6 −1.29501 −0.647503 0.762063i \(-0.724187\pi\)
−0.647503 + 0.762063i \(0.724187\pi\)
\(348\) 0 0
\(349\) 3.99157e6i 1.75420i 0.480304 + 0.877102i \(0.340526\pi\)
−0.480304 + 0.877102i \(0.659474\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.42922e6i 0.610466i 0.952278 + 0.305233i \(0.0987344\pi\)
−0.952278 + 0.305233i \(0.901266\pi\)
\(354\) 0 0
\(355\) 2.53109e6 1.06595
\(356\) 0 0
\(357\) 627264.i 0.260483i
\(358\) 0 0
\(359\) − 1.16186e6i − 0.475794i −0.971290 0.237897i \(-0.923542\pi\)
0.971290 0.237897i \(-0.0764581\pi\)
\(360\) 0 0
\(361\) 1.77720e6 0.717743
\(362\) 0 0
\(363\) 1.56659e6 0.624005
\(364\) 0 0
\(365\) 3.64835e6 1.43339
\(366\) 0 0
\(367\) −1.08923e6 −0.422139 −0.211069 0.977471i \(-0.567695\pi\)
−0.211069 + 0.977471i \(0.567695\pi\)
\(368\) 0 0
\(369\) − 1.70537e6i − 0.652009i
\(370\) 0 0
\(371\) 1.71547e6i 0.647066i
\(372\) 0 0
\(373\) 3.50577e6 1.30470 0.652350 0.757918i \(-0.273783\pi\)
0.652350 + 0.757918i \(0.273783\pi\)
\(374\) 0 0
\(375\) − 2.16043e6i − 0.793346i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.04385e6i 1.44610i 0.690798 + 0.723048i \(0.257260\pi\)
−0.690798 + 0.723048i \(0.742740\pi\)
\(380\) 0 0
\(381\) 4.03814e6 1.42518
\(382\) 0 0
\(383\) 5.18746e6i 1.80700i 0.428591 + 0.903499i \(0.359010\pi\)
−0.428591 + 0.903499i \(0.640990\pi\)
\(384\) 0 0
\(385\) 2.56608e6i 0.882304i
\(386\) 0 0
\(387\) −1.19790e6 −0.406577
\(388\) 0 0
\(389\) 950346. 0.318425 0.159213 0.987244i \(-0.449104\pi\)
0.159213 + 0.987244i \(0.449104\pi\)
\(390\) 0 0
\(391\) −2.43778e6 −0.806403
\(392\) 0 0
\(393\) −1.21176e6 −0.395763
\(394\) 0 0
\(395\) − 4.15325e6i − 1.33935i
\(396\) 0 0
\(397\) 520738.i 0.165822i 0.996557 + 0.0829112i \(0.0264218\pi\)
−0.996557 + 0.0829112i \(0.973578\pi\)
\(398\) 0 0
\(399\) 882816. 0.277612
\(400\) 0 0
\(401\) − 764370.i − 0.237379i −0.992931 0.118690i \(-0.962131\pi\)
0.992931 0.118690i \(-0.0378693\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 1.36031e6i − 0.412099i
\(406\) 0 0
\(407\) 160920. 0.0481531
\(408\) 0 0
\(409\) 2.64051e6i 0.780511i 0.920707 + 0.390255i \(0.127613\pi\)
−0.920707 + 0.390255i \(0.872387\pi\)
\(410\) 0 0
\(411\) − 3.80570e6i − 1.11130i
\(412\) 0 0
\(413\) −674784. −0.194666
\(414\) 0 0
\(415\) −3.65666e6 −1.04223
\(416\) 0 0
\(417\) 1.77989e6 0.501248
\(418\) 0 0
\(419\) −4.98020e6 −1.38584 −0.692918 0.721016i \(-0.743675\pi\)
−0.692918 + 0.721016i \(0.743675\pi\)
\(420\) 0 0
\(421\) − 237994.i − 0.0654426i −0.999465 0.0327213i \(-0.989583\pi\)
0.999465 0.0327213i \(-0.0104174\pi\)
\(422\) 0 0
\(423\) − 128304.i − 0.0348650i
\(424\) 0 0
\(425\) −124146. −0.0333396
\(426\) 0 0
\(427\) − 3.05694e6i − 0.811368i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 3.88238e6i − 1.00671i −0.864079 0.503356i \(-0.832098\pi\)
0.864079 0.503356i \(-0.167902\pi\)
\(432\) 0 0
\(433\) 66958.0 0.0171626 0.00858129 0.999963i \(-0.497268\pi\)
0.00858129 + 0.999963i \(0.497268\pi\)
\(434\) 0 0
\(435\) 384912.i 0.0975300i
\(436\) 0 0
\(437\) 3.43094e6i 0.859429i
\(438\) 0 0
\(439\) 6.50135e6 1.61006 0.805031 0.593233i \(-0.202149\pi\)
0.805031 + 0.593233i \(0.202149\pi\)
\(440\) 0 0
\(441\) −897237. −0.219690
\(442\) 0 0
\(443\) −4.60760e6 −1.11549 −0.557745 0.830012i \(-0.688333\pi\)
−0.557745 + 0.830012i \(0.688333\pi\)
\(444\) 0 0
\(445\) 1.60672e6 0.384626
\(446\) 0 0
\(447\) − 2.35937e6i − 0.558505i
\(448\) 0 0
\(449\) − 3.77671e6i − 0.884092i −0.896992 0.442046i \(-0.854253\pi\)
0.896992 0.442046i \(-0.145747\pi\)
\(450\) 0 0
\(451\) 9.30204e6 2.15346
\(452\) 0 0
\(453\) 892320.i 0.204303i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 3.18069e6i − 0.712412i −0.934407 0.356206i \(-0.884070\pi\)
0.934407 0.356206i \(-0.115930\pi\)
\(458\) 0 0
\(459\) −2.43778e6 −0.540085
\(460\) 0 0
\(461\) 6.68547e6i 1.46514i 0.680691 + 0.732571i \(0.261680\pi\)
−0.680691 + 0.732571i \(0.738320\pi\)
\(462\) 0 0
\(463\) 4.35122e6i 0.943318i 0.881781 + 0.471659i \(0.156345\pi\)
−0.881781 + 0.471659i \(0.843655\pi\)
\(464\) 0 0
\(465\) 2.75789e6 0.591486
\(466\) 0 0
\(467\) −7.07994e6 −1.50223 −0.751117 0.660170i \(-0.770484\pi\)
−0.751117 + 0.660170i \(0.770484\pi\)
\(468\) 0 0
\(469\) −1.91946e6 −0.402945
\(470\) 0 0
\(471\) −1.45054e6 −0.301284
\(472\) 0 0
\(473\) − 6.53400e6i − 1.34285i
\(474\) 0 0
\(475\) 174724.i 0.0355319i
\(476\) 0 0
\(477\) −1.92991e6 −0.388365
\(478\) 0 0
\(479\) − 3.22186e6i − 0.641604i −0.947146 0.320802i \(-0.896048\pi\)
0.947146 0.320802i \(-0.103952\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) − 4.33382e6i − 0.845286i
\(484\) 0 0
\(485\) 6.60949e6 1.27589
\(486\) 0 0
\(487\) 2.29710e6i 0.438891i 0.975625 + 0.219446i \(0.0704248\pi\)
−0.975625 + 0.219446i \(0.929575\pi\)
\(488\) 0 0
\(489\) − 1.33608e6i − 0.252674i
\(490\) 0 0
\(491\) −2.82150e6 −0.528173 −0.264087 0.964499i \(-0.585070\pi\)
−0.264087 + 0.964499i \(0.585070\pi\)
\(492\) 0 0
\(493\) 352836. 0.0653816
\(494\) 0 0
\(495\) −2.88684e6 −0.529553
\(496\) 0 0
\(497\) 4.12474e6 0.749040
\(498\) 0 0
\(499\) − 4.13628e6i − 0.743634i −0.928306 0.371817i \(-0.878735\pi\)
0.928306 0.371817i \(-0.121265\pi\)
\(500\) 0 0
\(501\) − 5.90198e6i − 1.05052i
\(502\) 0 0
\(503\) 8.33263e6 1.46846 0.734230 0.678901i \(-0.237543\pi\)
0.734230 + 0.678901i \(0.237543\pi\)
\(504\) 0 0
\(505\) − 609444.i − 0.106342i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.34101e6i 0.742670i 0.928499 + 0.371335i \(0.121100\pi\)
−0.928499 + 0.371335i \(0.878900\pi\)
\(510\) 0 0
\(511\) 5.94546e6 1.00724
\(512\) 0 0
\(513\) 3.43094e6i 0.575599i
\(514\) 0 0
\(515\) 1.47182e6i 0.244533i
\(516\) 0 0
\(517\) 699840. 0.115152
\(518\) 0 0
\(519\) 8.48945e6 1.38344
\(520\) 0 0
\(521\) −6.74185e6 −1.08814 −0.544070 0.839040i \(-0.683117\pi\)
−0.544070 + 0.839040i \(0.683117\pi\)
\(522\) 0 0
\(523\) −7.72196e6 −1.23445 −0.617224 0.786787i \(-0.711743\pi\)
−0.617224 + 0.786787i \(0.711743\pi\)
\(524\) 0 0
\(525\) − 220704.i − 0.0349472i
\(526\) 0 0
\(527\) − 2.52806e6i − 0.396517i
\(528\) 0 0
\(529\) 1.04065e7 1.61683
\(530\) 0 0
\(531\) − 759132.i − 0.116837i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.60766e6i 0.998075i
\(536\) 0 0
\(537\) 5.92402e6 0.886504
\(538\) 0 0
\(539\) − 4.89402e6i − 0.725594i
\(540\) 0 0
\(541\) 682066.i 0.100192i 0.998744 + 0.0500960i \(0.0159527\pi\)
−0.998744 + 0.0500960i \(0.984047\pi\)
\(542\) 0 0
\(543\) −6.71340e6 −0.977109
\(544\) 0 0
\(545\) −5.39471e6 −0.777995
\(546\) 0 0
\(547\) 2.15772e6 0.308337 0.154169 0.988045i \(-0.450730\pi\)
0.154169 + 0.988045i \(0.450730\pi\)
\(548\) 0 0
\(549\) 3.43906e6 0.486978
\(550\) 0 0
\(551\) − 496584.i − 0.0696809i
\(552\) 0 0
\(553\) − 6.76826e6i − 0.941161i
\(554\) 0 0
\(555\) 193104. 0.0266109
\(556\) 0 0
\(557\) 2.67597e6i 0.365463i 0.983163 + 0.182731i \(0.0584939\pi\)
−0.983163 + 0.182731i \(0.941506\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 3.84912e6i − 0.516362i
\(562\) 0 0
\(563\) 3.55331e6 0.472457 0.236228 0.971698i \(-0.424089\pi\)
0.236228 + 0.971698i \(0.424089\pi\)
\(564\) 0 0
\(565\) − 1.60088e6i − 0.210979i
\(566\) 0 0
\(567\) − 2.21681e6i − 0.289581i
\(568\) 0 0
\(569\) 1.29225e7 1.67327 0.836633 0.547764i \(-0.184521\pi\)
0.836633 + 0.547764i \(0.184521\pi\)
\(570\) 0 0
\(571\) 6.08357e6 0.780851 0.390426 0.920634i \(-0.372328\pi\)
0.390426 + 0.920634i \(0.372328\pi\)
\(572\) 0 0
\(573\) 8.68838e6 1.10548
\(574\) 0 0
\(575\) 857736. 0.108189
\(576\) 0 0
\(577\) − 1.58241e7i − 1.97869i −0.145579 0.989347i \(-0.546505\pi\)
0.145579 0.989347i \(-0.453495\pi\)
\(578\) 0 0
\(579\) 85272.0i 0.0105709i
\(580\) 0 0
\(581\) −5.95901e6 −0.732375
\(582\) 0 0
\(583\) − 1.05268e7i − 1.28269i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.60220e6i 0.551278i 0.961261 + 0.275639i \(0.0888894\pi\)
−0.961261 + 0.275639i \(0.911111\pi\)
\(588\) 0 0
\(589\) −3.55802e6 −0.422590
\(590\) 0 0
\(591\) 6.36530e6i 0.749636i
\(592\) 0 0
\(593\) − 8.61122e6i − 1.00561i −0.864401 0.502803i \(-0.832302\pi\)
0.864401 0.502803i \(-0.167698\pi\)
\(594\) 0 0
\(595\) 2.82269e6 0.326867
\(596\) 0 0
\(597\) 674016. 0.0773988
\(598\) 0 0
\(599\) −7.98228e6 −0.908992 −0.454496 0.890749i \(-0.650181\pi\)
−0.454496 + 0.890749i \(0.650181\pi\)
\(600\) 0 0
\(601\) 1.01740e7 1.14896 0.574481 0.818518i \(-0.305204\pi\)
0.574481 + 0.818518i \(0.305204\pi\)
\(602\) 0 0
\(603\) − 2.15939e6i − 0.241845i
\(604\) 0 0
\(605\) − 7.04965e6i − 0.783031i
\(606\) 0 0
\(607\) −9.95843e6 −1.09703 −0.548516 0.836140i \(-0.684807\pi\)
−0.548516 + 0.836140i \(0.684807\pi\)
\(608\) 0 0
\(609\) 627264.i 0.0685342i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 4.19586e6i 0.450993i 0.974244 + 0.225497i \(0.0724005\pi\)
−0.974244 + 0.225497i \(0.927600\pi\)
\(614\) 0 0
\(615\) 1.11624e7 1.19007
\(616\) 0 0
\(617\) 9.12551e6i 0.965038i 0.875885 + 0.482519i \(0.160278\pi\)
−0.875885 + 0.482519i \(0.839722\pi\)
\(618\) 0 0
\(619\) − 6.45734e6i − 0.677372i −0.940900 0.338686i \(-0.890018\pi\)
0.940900 0.338686i \(-0.109982\pi\)
\(620\) 0 0
\(621\) 1.68428e7 1.75261
\(622\) 0 0
\(623\) 2.61835e6 0.270276
\(624\) 0 0
\(625\) −9.06882e6 −0.928647
\(626\) 0 0
\(627\) −5.41728e6 −0.550316
\(628\) 0 0
\(629\) − 177012.i − 0.0178392i
\(630\) 0 0
\(631\) 1.40514e7i 1.40490i 0.711733 + 0.702450i \(0.247910\pi\)
−0.711733 + 0.702450i \(0.752090\pi\)
\(632\) 0 0
\(633\) 4.07035e6 0.403759
\(634\) 0 0
\(635\) − 1.81716e7i − 1.78838i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.64033e6i 0.449569i
\(640\) 0 0
\(641\) −8.47168e6 −0.814375 −0.407188 0.913345i \(-0.633490\pi\)
−0.407188 + 0.913345i \(0.633490\pi\)
\(642\) 0 0
\(643\) 488564.i 0.0466009i 0.999729 + 0.0233004i \(0.00741743\pi\)
−0.999729 + 0.0233004i \(0.992583\pi\)
\(644\) 0 0
\(645\) − 7.84080e6i − 0.742098i
\(646\) 0 0
\(647\) −2.48119e6 −0.233023 −0.116512 0.993189i \(-0.537171\pi\)
−0.116512 + 0.993189i \(0.537171\pi\)
\(648\) 0 0
\(649\) 4.14072e6 0.385891
\(650\) 0 0
\(651\) 4.49434e6 0.415636
\(652\) 0 0
\(653\) −5.29130e6 −0.485601 −0.242800 0.970076i \(-0.578066\pi\)
−0.242800 + 0.970076i \(0.578066\pi\)
\(654\) 0 0
\(655\) 5.45292e6i 0.496622i
\(656\) 0 0
\(657\) 6.68864e6i 0.604539i
\(658\) 0 0
\(659\) 4.72468e6 0.423798 0.211899 0.977292i \(-0.432035\pi\)
0.211899 + 0.977292i \(0.432035\pi\)
\(660\) 0 0
\(661\) 6.17420e6i 0.549639i 0.961496 + 0.274819i \(0.0886180\pi\)
−0.961496 + 0.274819i \(0.911382\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.97267e6i − 0.348360i
\(666\) 0 0
\(667\) −2.43778e6 −0.212168
\(668\) 0 0
\(669\) − 9.35232e6i − 0.807893i
\(670\) 0 0
\(671\) 1.87585e7i 1.60839i
\(672\) 0 0
\(673\) 9.40925e6 0.800787 0.400394 0.916343i \(-0.368873\pi\)
0.400394 + 0.916343i \(0.368873\pi\)
\(674\) 0 0
\(675\) 857736. 0.0724593
\(676\) 0 0
\(677\) 1.50086e7 1.25854 0.629272 0.777185i \(-0.283353\pi\)
0.629272 + 0.777185i \(0.283353\pi\)
\(678\) 0 0
\(679\) 1.07710e7 0.896567
\(680\) 0 0
\(681\) 8.93851e6i 0.738580i
\(682\) 0 0
\(683\) 1.29707e7i 1.06393i 0.846768 + 0.531963i \(0.178545\pi\)
−0.846768 + 0.531963i \(0.821455\pi\)
\(684\) 0 0
\(685\) −1.71257e7 −1.39451
\(686\) 0 0
\(687\) − 3.27295e6i − 0.264574i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 2.26556e7i 1.80501i 0.430677 + 0.902506i \(0.358275\pi\)
−0.430677 + 0.902506i \(0.641725\pi\)
\(692\) 0 0
\(693\) −4.70448e6 −0.372116
\(694\) 0 0
\(695\) − 8.00950e6i − 0.628989i
\(696\) 0 0
\(697\) − 1.02322e7i − 0.797791i
\(698\) 0 0
\(699\) −1.84615e6 −0.142914
\(700\) 0 0
\(701\) −1.90169e7 −1.46166 −0.730828 0.682562i \(-0.760866\pi\)
−0.730828 + 0.682562i \(0.760866\pi\)
\(702\) 0 0
\(703\) −249128. −0.0190123
\(704\) 0 0
\(705\) 839808. 0.0636366
\(706\) 0 0
\(707\) − 993168.i − 0.0747264i
\(708\) 0 0
\(709\) − 1.51311e7i − 1.13046i −0.824933 0.565231i \(-0.808787\pi\)
0.824933 0.565231i \(-0.191213\pi\)
\(710\) 0 0
\(711\) 7.61429e6 0.564879
\(712\) 0 0
\(713\) 1.74666e7i 1.28672i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 1.38568e7i − 1.00662i
\(718\) 0 0
\(719\) 1.50323e7 1.08443 0.542217 0.840238i \(-0.317585\pi\)
0.542217 + 0.840238i \(0.317585\pi\)
\(720\) 0 0
\(721\) 2.39853e6i 0.171833i
\(722\) 0 0
\(723\) 7.88489e6i 0.560983i
\(724\) 0 0
\(725\) −124146. −0.00877178
\(726\) 0 0
\(727\) 7.41230e6 0.520136 0.260068 0.965590i \(-0.416255\pi\)
0.260068 + 0.965590i \(0.416255\pi\)
\(728\) 0 0
\(729\) 1.44612e7 1.00782
\(730\) 0 0
\(731\) −7.18740e6 −0.497483
\(732\) 0 0
\(733\) − 2.77928e6i − 0.191061i −0.995426 0.0955306i \(-0.969545\pi\)
0.995426 0.0955306i \(-0.0304548\pi\)
\(734\) 0 0
\(735\) − 5.87282e6i − 0.400985i
\(736\) 0 0
\(737\) 1.17785e7 0.798768
\(738\) 0 0
\(739\) 1.21046e7i 0.815342i 0.913129 + 0.407671i \(0.133659\pi\)
−0.913129 + 0.407671i \(0.866341\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.46926e6i 0.297005i 0.988912 + 0.148502i \(0.0474452\pi\)
−0.988912 + 0.148502i \(0.952555\pi\)
\(744\) 0 0
\(745\) −1.06172e7 −0.700838
\(746\) 0 0
\(747\) − 6.70388e6i − 0.439567i
\(748\) 0 0
\(749\) 1.07680e7i 0.701345i
\(750\) 0 0
\(751\) −2.88463e7 −1.86634 −0.933168 0.359442i \(-0.882967\pi\)
−0.933168 + 0.359442i \(0.882967\pi\)
\(752\) 0 0
\(753\) 1.61028e7 1.03494
\(754\) 0 0
\(755\) 4.01544e6 0.256369
\(756\) 0 0
\(757\) 9.60868e6 0.609430 0.304715 0.952444i \(-0.401439\pi\)
0.304715 + 0.952444i \(0.401439\pi\)
\(758\) 0 0
\(759\) 2.65939e7i 1.67563i
\(760\) 0 0
\(761\) − 4.54588e6i − 0.284549i −0.989827 0.142274i \(-0.954558\pi\)
0.989827 0.142274i \(-0.0454415\pi\)
\(762\) 0 0
\(763\) −8.79138e6 −0.546696
\(764\) 0 0
\(765\) 3.17552e6i 0.196183i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 2.15923e7i − 1.31669i −0.752716 0.658345i \(-0.771257\pi\)
0.752716 0.658345i \(-0.228743\pi\)
\(770\) 0 0
\(771\) 1.58825e6 0.0962238
\(772\) 0 0
\(773\) − 1.48400e7i − 0.893276i −0.894715 0.446638i \(-0.852621\pi\)
0.894715 0.446638i \(-0.147379\pi\)
\(774\) 0 0
\(775\) 889504.i 0.0531978i
\(776\) 0 0
\(777\) 314688. 0.0186994
\(778\) 0 0
\(779\) −1.44009e7 −0.850251
\(780\) 0 0
\(781\) −2.53109e7 −1.48484
\(782\) 0 0
\(783\) −2.43778e6 −0.142098
\(784\) 0 0
\(785\) 6.52741e6i 0.378065i
\(786\) 0 0
\(787\) 2.48785e7i 1.43182i 0.698194 + 0.715909i \(0.253987\pi\)
−0.698194 + 0.715909i \(0.746013\pi\)
\(788\) 0 0
\(789\) −1.13193e7 −0.647330
\(790\) 0 0
\(791\) − 2.60885e6i − 0.148254i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 1.26321e7i − 0.708856i
\(796\) 0 0
\(797\) −3.16080e7 −1.76259 −0.881294 0.472568i \(-0.843327\pi\)
−0.881294 + 0.472568i \(0.843327\pi\)
\(798\) 0 0
\(799\) − 769824.i − 0.0426604i
\(800\) 0 0
\(801\) 2.94565e6i 0.162218i
\(802\) 0 0
\(803\) −3.64835e7 −1.99668
\(804\) 0 0
\(805\) −1.95022e7 −1.06070
\(806\) 0 0
\(807\) −1.16102e7 −0.627562
\(808\) 0 0
\(809\) −3.10009e6 −0.166534 −0.0832669 0.996527i \(-0.526535\pi\)
−0.0832669 + 0.996527i \(0.526535\pi\)
\(810\) 0 0
\(811\) 1.87180e6i 0.0999328i 0.998751 + 0.0499664i \(0.0159114\pi\)
−0.998751 + 0.0499664i \(0.984089\pi\)
\(812\) 0 0
\(813\) − 6.21984e6i − 0.330030i
\(814\) 0 0
\(815\) −6.01236e6 −0.317067
\(816\) 0 0
\(817\) 1.01156e7i 0.530196i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 2.00184e7i − 1.03650i −0.855228 0.518252i \(-0.826583\pi\)
0.855228 0.518252i \(-0.173417\pi\)
\(822\) 0 0
\(823\) −1.53118e7 −0.787999 −0.394000 0.919111i \(-0.628909\pi\)
−0.394000 + 0.919111i \(0.628909\pi\)
\(824\) 0 0
\(825\) 1.35432e6i 0.0692766i
\(826\) 0 0
\(827\) − 9.59310e6i − 0.487748i −0.969807 0.243874i \(-0.921582\pi\)
0.969807 0.243874i \(-0.0784183\pi\)
\(828\) 0 0
\(829\) −2.52209e7 −1.27460 −0.637302 0.770615i \(-0.719949\pi\)
−0.637302 + 0.770615i \(0.719949\pi\)
\(830\) 0 0
\(831\) 2.66727e7 1.33988
\(832\) 0 0
\(833\) −5.38342e6 −0.268810
\(834\) 0 0
\(835\) −2.65589e7 −1.31824
\(836\) 0 0
\(837\) 1.74666e7i 0.861778i
\(838\) 0 0
\(839\) 1.77623e7i 0.871154i 0.900151 + 0.435577i \(0.143456\pi\)
−0.900151 + 0.435577i \(0.856544\pi\)
\(840\) 0 0
\(841\) −2.01583e7 −0.982798
\(842\) 0 0
\(843\) − 2.35937e6i − 0.114348i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.14883e7i − 0.550234i
\(848\) 0 0
\(849\) −1.86273e7 −0.886913
\(850\) 0 0
\(851\) 1.22299e6i 0.0578895i
\(852\) 0 0
\(853\) 486970.i 0.0229155i 0.999934 + 0.0114578i \(0.00364720\pi\)
−0.999934 + 0.0114578i \(0.996353\pi\)
\(854\) 0 0
\(855\) 4.46926e6 0.209084
\(856\) 0 0
\(857\) 1.92634e6 0.0895945 0.0447972 0.998996i \(-0.485736\pi\)
0.0447972 + 0.998996i \(0.485736\pi\)
\(858\) 0 0
\(859\) 2.23538e7 1.03364 0.516820 0.856094i \(-0.327116\pi\)
0.516820 + 0.856094i \(0.327116\pi\)
\(860\) 0 0
\(861\) 1.81907e7 0.836258
\(862\) 0 0
\(863\) 1.85838e7i 0.849390i 0.905337 + 0.424695i \(0.139619\pi\)
−0.905337 + 0.424695i \(0.860381\pi\)
\(864\) 0 0
\(865\) − 3.82025e7i − 1.73601i
\(866\) 0 0
\(867\) 1.28043e7 0.578504
\(868\) 0 0
\(869\) 4.15325e7i 1.86569i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.21174e7i 0.538114i
\(874\) 0 0
\(875\) −1.58432e7 −0.699555
\(876\) 0 0
\(877\) − 2.91048e7i − 1.27781i −0.769286 0.638905i \(-0.779388\pi\)
0.769286 0.638905i \(-0.220612\pi\)
\(878\) 0 0
\(879\) − 1.28660e7i − 0.561659i
\(880\) 0 0
\(881\) 3.14696e6 0.136600 0.0683001 0.997665i \(-0.478242\pi\)
0.0683001 + 0.997665i \(0.478242\pi\)
\(882\) 0 0
\(883\) −1.59995e7 −0.690566 −0.345283 0.938499i \(-0.612217\pi\)
−0.345283 + 0.938499i \(0.612217\pi\)
\(884\) 0 0
\(885\) 4.96886e6 0.213255
\(886\) 0 0
\(887\) −3.45874e7 −1.47608 −0.738039 0.674758i \(-0.764248\pi\)
−0.738039 + 0.674758i \(0.764248\pi\)
\(888\) 0 0
\(889\) − 2.96131e7i − 1.25669i
\(890\) 0 0
\(891\) 1.36031e7i 0.574044i
\(892\) 0 0
\(893\) −1.08346e6 −0.0454656
\(894\) 0 0
\(895\) − 2.66581e7i − 1.11243i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 2.52806e6i − 0.104325i
\(900\) 0 0
\(901\) −1.15794e7 −0.475199
\(902\) 0 0
\(903\) − 1.27776e7i − 0.521471i
\(904\) 0 0
\(905\) 3.02103e7i 1.22612i
\(906\) 0 0
\(907\) −1.74396e7 −0.703914 −0.351957 0.936016i \(-0.614484\pi\)
−0.351957 + 0.936016i \(0.614484\pi\)
\(908\) 0 0
\(909\) 1.11731e6 0.0448503
\(910\) 0 0
\(911\) −2.59589e6 −0.103631 −0.0518155 0.998657i \(-0.516501\pi\)
−0.0518155 + 0.998657i \(0.516501\pi\)
\(912\) 0 0
\(913\) 3.65666e7 1.45180
\(914\) 0 0
\(915\) 2.25102e7i 0.888847i
\(916\) 0 0
\(917\) 8.88624e6i 0.348975i
\(918\) 0 0
\(919\) −1.76411e7 −0.689028 −0.344514 0.938781i \(-0.611956\pi\)
−0.344514 + 0.938781i \(0.611956\pi\)
\(920\) 0 0
\(921\) 1.90307e7i 0.739275i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 62282.0i 0.00239336i
\(926\) 0 0
\(927\) −2.69834e6 −0.103133
\(928\) 0 0
\(929\) 3.96785e7i 1.50840i 0.656646 + 0.754199i \(0.271975\pi\)
−0.656646 + 0.754199i \(0.728025\pi\)
\(930\) 0 0
\(931\) 7.57667e6i 0.286486i
\(932\) 0 0
\(933\) −8.76874e6 −0.329787
\(934\) 0 0
\(935\) −1.73210e7 −0.647955
\(936\) 0 0
\(937\) 3.93413e7 1.46386 0.731930 0.681380i \(-0.238620\pi\)
0.731930 + 0.681380i \(0.238620\pi\)
\(938\) 0 0
\(939\) −7.01830e6 −0.259757
\(940\) 0 0
\(941\) 4.62506e7i 1.70272i 0.524581 + 0.851361i \(0.324222\pi\)
−0.524581 + 0.851361i \(0.675778\pi\)
\(942\) 0 0
\(943\) 7.06955e7i 2.58888i
\(944\) 0 0
\(945\) −1.95022e7 −0.710402
\(946\) 0 0
\(947\) 3.79025e7i 1.37339i 0.726947 + 0.686693i \(0.240938\pi\)
−0.726947 + 0.686693i \(0.759062\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.97944e7i 1.06828i
\(952\) 0 0
\(953\) 2.66462e7 0.950394 0.475197 0.879879i \(-0.342377\pi\)
0.475197 + 0.879879i \(0.342377\pi\)
\(954\) 0 0
\(955\) − 3.90977e7i − 1.38721i
\(956\) 0 0
\(957\) − 3.84912e6i − 0.135857i
\(958\) 0 0
\(959\) −2.79085e7 −0.979918
\(960\) 0 0
\(961\) 1.05156e7 0.367304
\(962\) 0 0
\(963\) −1.21140e7 −0.420943
\(964\) 0 0
\(965\) 383724. 0.0132648
\(966\) 0 0
\(967\) 4.09790e7i 1.40927i 0.709568 + 0.704637i \(0.248890\pi\)
−0.709568 + 0.704637i \(0.751110\pi\)
\(968\) 0 0
\(969\) 5.95901e6i 0.203875i
\(970\) 0 0
\(971\) −2.72034e7 −0.925922 −0.462961 0.886379i \(-0.653213\pi\)
−0.462961 + 0.886379i \(0.653213\pi\)
\(972\) 0 0
\(973\) − 1.30525e7i − 0.441990i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.53555e7i 0.849839i 0.905231 + 0.424919i \(0.139698\pi\)
−0.905231 + 0.424919i \(0.860302\pi\)
\(978\) 0 0
\(979\) −1.60672e7 −0.535775
\(980\) 0 0
\(981\) − 9.89030e6i − 0.328123i
\(982\) 0 0
\(983\) − 1.19139e7i − 0.393252i −0.980479 0.196626i \(-0.937002\pi\)
0.980479 0.196626i \(-0.0629984\pi\)
\(984\) 0 0
\(985\) 2.86439e7 0.940678
\(986\) 0 0
\(987\) 1.36858e6 0.0447173
\(988\) 0 0
\(989\) 4.96584e7 1.61437
\(990\) 0 0
\(991\) 2.91931e7 0.944268 0.472134 0.881527i \(-0.343484\pi\)
0.472134 + 0.881527i \(0.343484\pi\)
\(992\) 0 0
\(993\) − 4.53538e6i − 0.145962i
\(994\) 0 0
\(995\) − 3.03307e6i − 0.0971237i
\(996\) 0 0
\(997\) −1.73001e7 −0.551201 −0.275601 0.961272i \(-0.588877\pi\)
−0.275601 + 0.961272i \(0.588877\pi\)
\(998\) 0 0
\(999\) 1.22299e6i 0.0387713i
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 676.6.d.a.337.2 2
13.5 odd 4 4.6.a.a.1.1 1
13.8 odd 4 676.6.a.a.1.1 1
13.12 even 2 inner 676.6.d.a.337.1 2
39.5 even 4 36.6.a.a.1.1 1
52.31 even 4 16.6.a.b.1.1 1
65.18 even 4 100.6.c.b.49.1 2
65.44 odd 4 100.6.a.b.1.1 1
65.57 even 4 100.6.c.b.49.2 2
91.5 even 12 196.6.e.d.165.1 2
91.18 odd 12 196.6.e.g.177.1 2
91.31 even 12 196.6.e.d.177.1 2
91.44 odd 12 196.6.e.g.165.1 2
91.83 even 4 196.6.a.e.1.1 1
104.5 odd 4 64.6.a.f.1.1 1
104.83 even 4 64.6.a.b.1.1 1
117.5 even 12 324.6.e.d.217.1 2
117.31 odd 12 324.6.e.a.217.1 2
117.70 odd 12 324.6.e.a.109.1 2
117.83 even 12 324.6.e.d.109.1 2
143.109 even 4 484.6.a.a.1.1 1
156.83 odd 4 144.6.a.c.1.1 1
195.44 even 4 900.6.a.h.1.1 1
195.83 odd 4 900.6.d.a.649.2 2
195.122 odd 4 900.6.d.a.649.1 2
208.5 odd 4 256.6.b.g.129.2 2
208.83 even 4 256.6.b.c.129.2 2
208.109 odd 4 256.6.b.g.129.1 2
208.187 even 4 256.6.b.c.129.1 2
260.83 odd 4 400.6.c.f.49.2 2
260.187 odd 4 400.6.c.f.49.1 2
260.239 even 4 400.6.a.d.1.1 1
312.5 even 4 576.6.a.bc.1.1 1
312.83 odd 4 576.6.a.bd.1.1 1
364.83 odd 4 784.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.6.a.a.1.1 1 13.5 odd 4
16.6.a.b.1.1 1 52.31 even 4
36.6.a.a.1.1 1 39.5 even 4
64.6.a.b.1.1 1 104.83 even 4
64.6.a.f.1.1 1 104.5 odd 4
100.6.a.b.1.1 1 65.44 odd 4
100.6.c.b.49.1 2 65.18 even 4
100.6.c.b.49.2 2 65.57 even 4
144.6.a.c.1.1 1 156.83 odd 4
196.6.a.e.1.1 1 91.83 even 4
196.6.e.d.165.1 2 91.5 even 12
196.6.e.d.177.1 2 91.31 even 12
196.6.e.g.165.1 2 91.44 odd 12
196.6.e.g.177.1 2 91.18 odd 12
256.6.b.c.129.1 2 208.187 even 4
256.6.b.c.129.2 2 208.83 even 4
256.6.b.g.129.1 2 208.109 odd 4
256.6.b.g.129.2 2 208.5 odd 4
324.6.e.a.109.1 2 117.70 odd 12
324.6.e.a.217.1 2 117.31 odd 12
324.6.e.d.109.1 2 117.83 even 12
324.6.e.d.217.1 2 117.5 even 12
400.6.a.d.1.1 1 260.239 even 4
400.6.c.f.49.1 2 260.187 odd 4
400.6.c.f.49.2 2 260.83 odd 4
484.6.a.a.1.1 1 143.109 even 4
576.6.a.bc.1.1 1 312.5 even 4
576.6.a.bd.1.1 1 312.83 odd 4
676.6.a.a.1.1 1 13.8 odd 4
676.6.d.a.337.1 2 13.12 even 2 inner
676.6.d.a.337.2 2 1.1 even 1 trivial
784.6.a.d.1.1 1 364.83 odd 4
900.6.a.h.1.1 1 195.44 even 4
900.6.d.a.649.1 2 195.122 odd 4
900.6.d.a.649.2 2 195.83 odd 4