Properties

Label 768.7.g.e.511.2
Level $768$
Weight $7$
Character 768.511
Analytic conductor $176.682$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 768.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(176.681536220\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 511.2
Root \(-0.780776 + 1.35234i\) of defining polynomial
Character \(\chi\) \(=\) 768.511
Dual form 768.7.g.e.511.3

$q$-expansion

\(f(q)\) \(=\) \(q-15.5885i q^{3} +20.0000 q^{5} +155.727i q^{7} -243.000 q^{9} +O(q^{10})\) \(q-15.5885i q^{3} +20.0000 q^{5} +155.727i q^{7} -243.000 q^{9} -2306.46i q^{11} -3221.18 q^{13} -311.769i q^{15} -6566.73 q^{17} -3926.40i q^{19} +2427.54 q^{21} -17392.0i q^{23} -15225.0 q^{25} +3788.00i q^{27} +44713.3 q^{29} -30636.4i q^{31} -35954.2 q^{33} +3114.54i q^{35} -44158.1 q^{37} +50213.3i q^{39} -9008.18 q^{41} -25270.3i q^{43} -4860.00 q^{45} +175606. i q^{47} +93398.1 q^{49} +102365. i q^{51} +96206.3 q^{53} -46129.2i q^{55} -61206.5 q^{57} +35018.3i q^{59} +86135.7 q^{61} -37841.7i q^{63} -64423.6 q^{65} -424605. i q^{67} -271114. q^{69} -36572.4i q^{71} +375121. q^{73} +237334. i q^{75} +359179. q^{77} +520323. i q^{79} +59049.0 q^{81} +594208. i q^{83} -131335. q^{85} -697011. i q^{87} -901101. q^{89} -501625. i q^{91} -477574. q^{93} -78528.0i q^{95} -1.25744e6 q^{97} +560470. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 80 q^{5} - 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 80 q^{5} - 972 q^{9} - 5760 q^{13} + 2232 q^{17} - 11664 q^{21} - 60900 q^{25} + 50608 q^{29} - 58320 q^{33} - 212256 q^{37} + 49464 q^{41} - 19440 q^{45} - 139388 q^{49} - 256400 q^{53} + 11664 q^{57} + 80928 q^{61} - 115200 q^{65} - 443232 q^{69} - 38456 q^{73} + 1180224 q^{77} + 236196 q^{81} + 44640 q^{85} - 3319416 q^{89} - 478224 q^{93} - 2464840 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 15.5885i − 0.577350i
\(4\) 0 0
\(5\) 20.0000 0.160000 0.0800000 0.996795i \(-0.474508\pi\)
0.0800000 + 0.996795i \(0.474508\pi\)
\(6\) 0 0
\(7\) 155.727i 0.454015i 0.973893 + 0.227007i \(0.0728942\pi\)
−0.973893 + 0.227007i \(0.927106\pi\)
\(8\) 0 0
\(9\) −243.000 −0.333333
\(10\) 0 0
\(11\) − 2306.46i − 1.73288i −0.499282 0.866439i \(-0.666403\pi\)
0.499282 0.866439i \(-0.333597\pi\)
\(12\) 0 0
\(13\) −3221.18 −1.46617 −0.733086 0.680136i \(-0.761921\pi\)
−0.733086 + 0.680136i \(0.761921\pi\)
\(14\) 0 0
\(15\) − 311.769i − 0.0923760i
\(16\) 0 0
\(17\) −6566.73 −1.33660 −0.668301 0.743891i \(-0.732978\pi\)
−0.668301 + 0.743891i \(0.732978\pi\)
\(18\) 0 0
\(19\) − 3926.40i − 0.572445i −0.958163 0.286223i \(-0.907600\pi\)
0.958163 0.286223i \(-0.0923997\pi\)
\(20\) 0 0
\(21\) 2427.54 0.262126
\(22\) 0 0
\(23\) − 17392.0i − 1.42944i −0.699411 0.714720i \(-0.746554\pi\)
0.699411 0.714720i \(-0.253446\pi\)
\(24\) 0 0
\(25\) −15225.0 −0.974400
\(26\) 0 0
\(27\) 3788.00i 0.192450i
\(28\) 0 0
\(29\) 44713.3 1.83334 0.916669 0.399648i \(-0.130868\pi\)
0.916669 + 0.399648i \(0.130868\pi\)
\(30\) 0 0
\(31\) − 30636.4i − 1.02838i −0.857677 0.514188i \(-0.828093\pi\)
0.857677 0.514188i \(-0.171907\pi\)
\(32\) 0 0
\(33\) −35954.2 −1.00048
\(34\) 0 0
\(35\) 3114.54i 0.0726424i
\(36\) 0 0
\(37\) −44158.1 −0.871776 −0.435888 0.900001i \(-0.643566\pi\)
−0.435888 + 0.900001i \(0.643566\pi\)
\(38\) 0 0
\(39\) 50213.3i 0.846495i
\(40\) 0 0
\(41\) −9008.18 −0.130703 −0.0653515 0.997862i \(-0.520817\pi\)
−0.0653515 + 0.997862i \(0.520817\pi\)
\(42\) 0 0
\(43\) − 25270.3i − 0.317838i −0.987292 0.158919i \(-0.949199\pi\)
0.987292 0.158919i \(-0.0508008\pi\)
\(44\) 0 0
\(45\) −4860.00 −0.0533333
\(46\) 0 0
\(47\) 175606.i 1.69140i 0.533658 + 0.845700i \(0.320817\pi\)
−0.533658 + 0.845700i \(0.679183\pi\)
\(48\) 0 0
\(49\) 93398.1 0.793871
\(50\) 0 0
\(51\) 102365.i 0.771688i
\(52\) 0 0
\(53\) 96206.3 0.646214 0.323107 0.946363i \(-0.395273\pi\)
0.323107 + 0.946363i \(0.395273\pi\)
\(54\) 0 0
\(55\) − 46129.2i − 0.277261i
\(56\) 0 0
\(57\) −61206.5 −0.330501
\(58\) 0 0
\(59\) 35018.3i 0.170506i 0.996359 + 0.0852529i \(0.0271698\pi\)
−0.996359 + 0.0852529i \(0.972830\pi\)
\(60\) 0 0
\(61\) 86135.7 0.379484 0.189742 0.981834i \(-0.439235\pi\)
0.189742 + 0.981834i \(0.439235\pi\)
\(62\) 0 0
\(63\) − 37841.7i − 0.151338i
\(64\) 0 0
\(65\) −64423.6 −0.234588
\(66\) 0 0
\(67\) − 424605.i − 1.41176i −0.708332 0.705880i \(-0.750552\pi\)
0.708332 0.705880i \(-0.249448\pi\)
\(68\) 0 0
\(69\) −271114. −0.825287
\(70\) 0 0
\(71\) − 36572.4i − 0.102183i −0.998694 0.0510915i \(-0.983730\pi\)
0.998694 0.0510915i \(-0.0162700\pi\)
\(72\) 0 0
\(73\) 375121. 0.964280 0.482140 0.876094i \(-0.339860\pi\)
0.482140 + 0.876094i \(0.339860\pi\)
\(74\) 0 0
\(75\) 237334.i 0.562570i
\(76\) 0 0
\(77\) 359179. 0.786753
\(78\) 0 0
\(79\) 520323.i 1.05534i 0.849450 + 0.527670i \(0.176934\pi\)
−0.849450 + 0.527670i \(0.823066\pi\)
\(80\) 0 0
\(81\) 59049.0 0.111111
\(82\) 0 0
\(83\) 594208.i 1.03921i 0.854406 + 0.519606i \(0.173921\pi\)
−0.854406 + 0.519606i \(0.826079\pi\)
\(84\) 0 0
\(85\) −131335. −0.213856
\(86\) 0 0
\(87\) − 697011.i − 1.05848i
\(88\) 0 0
\(89\) −901101. −1.27821 −0.639107 0.769118i \(-0.720696\pi\)
−0.639107 + 0.769118i \(0.720696\pi\)
\(90\) 0 0
\(91\) − 501625.i − 0.665664i
\(92\) 0 0
\(93\) −477574. −0.593733
\(94\) 0 0
\(95\) − 78528.0i − 0.0915912i
\(96\) 0 0
\(97\) −1.25744e6 −1.37775 −0.688875 0.724880i \(-0.741895\pi\)
−0.688875 + 0.724880i \(0.741895\pi\)
\(98\) 0 0
\(99\) 560470.i 0.577626i
\(100\) 0 0
\(101\) 391040. 0.379539 0.189770 0.981829i \(-0.439226\pi\)
0.189770 + 0.981829i \(0.439226\pi\)
\(102\) 0 0
\(103\) − 1.36080e6i − 1.24533i −0.782490 0.622663i \(-0.786051\pi\)
0.782490 0.622663i \(-0.213949\pi\)
\(104\) 0 0
\(105\) 48550.9 0.0419401
\(106\) 0 0
\(107\) 1.28620e6i 1.04992i 0.851127 + 0.524960i \(0.175920\pi\)
−0.851127 + 0.524960i \(0.824080\pi\)
\(108\) 0 0
\(109\) −1.75763e6 −1.35721 −0.678605 0.734504i \(-0.737415\pi\)
−0.678605 + 0.734504i \(0.737415\pi\)
\(110\) 0 0
\(111\) 688357.i 0.503320i
\(112\) 0 0
\(113\) −730749. −0.506446 −0.253223 0.967408i \(-0.581491\pi\)
−0.253223 + 0.967408i \(0.581491\pi\)
\(114\) 0 0
\(115\) − 347840.i − 0.228710i
\(116\) 0 0
\(117\) 782747. 0.488724
\(118\) 0 0
\(119\) − 1.02262e6i − 0.606837i
\(120\) 0 0
\(121\) −3.54820e6 −2.00287
\(122\) 0 0
\(123\) 140424.i 0.0754614i
\(124\) 0 0
\(125\) −617000. −0.315904
\(126\) 0 0
\(127\) 2.59678e6i 1.26772i 0.773448 + 0.633860i \(0.218531\pi\)
−0.773448 + 0.633860i \(0.781469\pi\)
\(128\) 0 0
\(129\) −393925. −0.183504
\(130\) 0 0
\(131\) 605543.i 0.269359i 0.990889 + 0.134679i \(0.0430005\pi\)
−0.990889 + 0.134679i \(0.957000\pi\)
\(132\) 0 0
\(133\) 611447. 0.259899
\(134\) 0 0
\(135\) 75759.9i 0.0307920i
\(136\) 0 0
\(137\) 1.75932e6 0.684199 0.342100 0.939664i \(-0.388862\pi\)
0.342100 + 0.939664i \(0.388862\pi\)
\(138\) 0 0
\(139\) − 303680.i − 0.113076i −0.998400 0.0565382i \(-0.981994\pi\)
0.998400 0.0565382i \(-0.0180063\pi\)
\(140\) 0 0
\(141\) 2.73743e6 0.976531
\(142\) 0 0
\(143\) 7.42953e6i 2.54070i
\(144\) 0 0
\(145\) 894265. 0.293334
\(146\) 0 0
\(147\) − 1.45593e6i − 0.458341i
\(148\) 0 0
\(149\) −1.32218e6 −0.399696 −0.199848 0.979827i \(-0.564045\pi\)
−0.199848 + 0.979827i \(0.564045\pi\)
\(150\) 0 0
\(151\) − 4.67518e6i − 1.35790i −0.734184 0.678950i \(-0.762435\pi\)
0.734184 0.678950i \(-0.237565\pi\)
\(152\) 0 0
\(153\) 1.59571e6 0.445534
\(154\) 0 0
\(155\) − 612727.i − 0.164540i
\(156\) 0 0
\(157\) 2.44769e6 0.632495 0.316247 0.948677i \(-0.397577\pi\)
0.316247 + 0.948677i \(0.397577\pi\)
\(158\) 0 0
\(159\) − 1.49971e6i − 0.373092i
\(160\) 0 0
\(161\) 2.70840e6 0.648987
\(162\) 0 0
\(163\) 6.62033e6i 1.52868i 0.644813 + 0.764340i \(0.276935\pi\)
−0.644813 + 0.764340i \(0.723065\pi\)
\(164\) 0 0
\(165\) −719084. −0.160076
\(166\) 0 0
\(167\) − 7.27537e6i − 1.56209i −0.624476 0.781044i \(-0.714687\pi\)
0.624476 0.781044i \(-0.285313\pi\)
\(168\) 0 0
\(169\) 5.54920e6 1.14966
\(170\) 0 0
\(171\) 954116.i 0.190815i
\(172\) 0 0
\(173\) −3.72601e6 −0.719623 −0.359812 0.933025i \(-0.617159\pi\)
−0.359812 + 0.933025i \(0.617159\pi\)
\(174\) 0 0
\(175\) − 2.37094e6i − 0.442392i
\(176\) 0 0
\(177\) 545881. 0.0984415
\(178\) 0 0
\(179\) − 201811.i − 0.0351873i −0.999845 0.0175937i \(-0.994399\pi\)
0.999845 0.0175937i \(-0.00560053\pi\)
\(180\) 0 0
\(181\) 5.43440e6 0.916466 0.458233 0.888832i \(-0.348483\pi\)
0.458233 + 0.888832i \(0.348483\pi\)
\(182\) 0 0
\(183\) − 1.34272e6i − 0.219095i
\(184\) 0 0
\(185\) −883162. −0.139484
\(186\) 0 0
\(187\) 1.51459e7i 2.31617i
\(188\) 0 0
\(189\) −589893. −0.0873752
\(190\) 0 0
\(191\) 6.82608e6i 0.979650i 0.871821 + 0.489825i \(0.162939\pi\)
−0.871821 + 0.489825i \(0.837061\pi\)
\(192\) 0 0
\(193\) 1.30841e6 0.182001 0.0910004 0.995851i \(-0.470994\pi\)
0.0910004 + 0.995851i \(0.470994\pi\)
\(194\) 0 0
\(195\) 1.00427e6i 0.135439i
\(196\) 0 0
\(197\) −7.61375e6 −0.995863 −0.497932 0.867216i \(-0.665907\pi\)
−0.497932 + 0.867216i \(0.665907\pi\)
\(198\) 0 0
\(199\) − 6.47692e6i − 0.821882i −0.911662 0.410941i \(-0.865200\pi\)
0.911662 0.410941i \(-0.134800\pi\)
\(200\) 0 0
\(201\) −6.61894e6 −0.815080
\(202\) 0 0
\(203\) 6.96307e6i 0.832362i
\(204\) 0 0
\(205\) −180164. −0.0209125
\(206\) 0 0
\(207\) 4.22625e6i 0.476480i
\(208\) 0 0
\(209\) −9.05609e6 −0.991978
\(210\) 0 0
\(211\) 1.58058e7i 1.68256i 0.540603 + 0.841278i \(0.318196\pi\)
−0.540603 + 0.841278i \(0.681804\pi\)
\(212\) 0 0
\(213\) −570108. −0.0589954
\(214\) 0 0
\(215\) − 505406.i − 0.0508540i
\(216\) 0 0
\(217\) 4.77091e6 0.466898
\(218\) 0 0
\(219\) − 5.84756e6i − 0.556727i
\(220\) 0 0
\(221\) 2.11526e7 1.95969
\(222\) 0 0
\(223\) 2.07099e7i 1.86751i 0.357907 + 0.933757i \(0.383491\pi\)
−0.357907 + 0.933757i \(0.616509\pi\)
\(224\) 0 0
\(225\) 3.69967e6 0.324800
\(226\) 0 0
\(227\) 7.06179e6i 0.603722i 0.953352 + 0.301861i \(0.0976079\pi\)
−0.953352 + 0.301861i \(0.902392\pi\)
\(228\) 0 0
\(229\) −4.64766e6 −0.387015 −0.193507 0.981099i \(-0.561986\pi\)
−0.193507 + 0.981099i \(0.561986\pi\)
\(230\) 0 0
\(231\) − 5.59904e6i − 0.454232i
\(232\) 0 0
\(233\) −1.13777e7 −0.899469 −0.449735 0.893162i \(-0.648481\pi\)
−0.449735 + 0.893162i \(0.648481\pi\)
\(234\) 0 0
\(235\) 3.51213e6i 0.270624i
\(236\) 0 0
\(237\) 8.11104e6 0.609300
\(238\) 0 0
\(239\) − 1.93765e7i − 1.41932i −0.704542 0.709662i \(-0.748848\pi\)
0.704542 0.709662i \(-0.251152\pi\)
\(240\) 0 0
\(241\) 6.84883e6 0.489289 0.244644 0.969613i \(-0.421329\pi\)
0.244644 + 0.969613i \(0.421329\pi\)
\(242\) 0 0
\(243\) − 920483.i − 0.0641500i
\(244\) 0 0
\(245\) 1.86796e6 0.127019
\(246\) 0 0
\(247\) 1.26477e7i 0.839304i
\(248\) 0 0
\(249\) 9.26279e6 0.599989
\(250\) 0 0
\(251\) 129527.i 0.00819104i 0.999992 + 0.00409552i \(0.00130365\pi\)
−0.999992 + 0.00409552i \(0.998696\pi\)
\(252\) 0 0
\(253\) −4.01140e7 −2.47705
\(254\) 0 0
\(255\) 2.04730e6i 0.123470i
\(256\) 0 0
\(257\) 1.18895e7 0.700427 0.350213 0.936670i \(-0.386109\pi\)
0.350213 + 0.936670i \(0.386109\pi\)
\(258\) 0 0
\(259\) − 6.87661e6i − 0.395799i
\(260\) 0 0
\(261\) −1.08653e7 −0.611113
\(262\) 0 0
\(263\) 1.35674e7i 0.745811i 0.927869 + 0.372906i \(0.121638\pi\)
−0.927869 + 0.372906i \(0.878362\pi\)
\(264\) 0 0
\(265\) 1.92413e6 0.103394
\(266\) 0 0
\(267\) 1.40468e7i 0.737977i
\(268\) 0 0
\(269\) −1.71231e7 −0.879680 −0.439840 0.898076i \(-0.644965\pi\)
−0.439840 + 0.898076i \(0.644965\pi\)
\(270\) 0 0
\(271\) 320201.i 0.0160885i 0.999968 + 0.00804423i \(0.00256058\pi\)
−0.999968 + 0.00804423i \(0.997439\pi\)
\(272\) 0 0
\(273\) −7.81956e6 −0.384321
\(274\) 0 0
\(275\) 3.51159e7i 1.68852i
\(276\) 0 0
\(277\) 1.50949e7 0.710216 0.355108 0.934825i \(-0.384444\pi\)
0.355108 + 0.934825i \(0.384444\pi\)
\(278\) 0 0
\(279\) 7.44463e6i 0.342792i
\(280\) 0 0
\(281\) 4.32580e7 1.94961 0.974804 0.223063i \(-0.0716056\pi\)
0.974804 + 0.223063i \(0.0716056\pi\)
\(282\) 0 0
\(283\) 3.22002e7i 1.42069i 0.703854 + 0.710345i \(0.251461\pi\)
−0.703854 + 0.710345i \(0.748539\pi\)
\(284\) 0 0
\(285\) −1.22413e6 −0.0528802
\(286\) 0 0
\(287\) − 1.40282e6i − 0.0593411i
\(288\) 0 0
\(289\) 1.89843e7 0.786505
\(290\) 0 0
\(291\) 1.96015e7i 0.795444i
\(292\) 0 0
\(293\) 3.01307e7 1.19786 0.598931 0.800801i \(-0.295592\pi\)
0.598931 + 0.800801i \(0.295592\pi\)
\(294\) 0 0
\(295\) 700366.i 0.0272809i
\(296\) 0 0
\(297\) 8.73687e6 0.333493
\(298\) 0 0
\(299\) 5.60228e7i 2.09581i
\(300\) 0 0
\(301\) 3.93527e6 0.144303
\(302\) 0 0
\(303\) − 6.09571e6i − 0.219127i
\(304\) 0 0
\(305\) 1.72271e6 0.0607175
\(306\) 0 0
\(307\) 2.09501e7i 0.724053i 0.932168 + 0.362026i \(0.117915\pi\)
−0.932168 + 0.362026i \(0.882085\pi\)
\(308\) 0 0
\(309\) −2.12128e7 −0.718990
\(310\) 0 0
\(311\) − 4.94101e6i − 0.164261i −0.996622 0.0821305i \(-0.973828\pi\)
0.996622 0.0821305i \(-0.0261724\pi\)
\(312\) 0 0
\(313\) −2.67449e7 −0.872182 −0.436091 0.899903i \(-0.643637\pi\)
−0.436091 + 0.899903i \(0.643637\pi\)
\(314\) 0 0
\(315\) − 756834.i − 0.0242141i
\(316\) 0 0
\(317\) 4.74499e7 1.48956 0.744779 0.667311i \(-0.232555\pi\)
0.744779 + 0.667311i \(0.232555\pi\)
\(318\) 0 0
\(319\) − 1.03129e8i − 3.17695i
\(320\) 0 0
\(321\) 2.00498e7 0.606172
\(322\) 0 0
\(323\) 2.57836e7i 0.765131i
\(324\) 0 0
\(325\) 4.90425e7 1.42864
\(326\) 0 0
\(327\) 2.73987e7i 0.783585i
\(328\) 0 0
\(329\) −2.73467e7 −0.767921
\(330\) 0 0
\(331\) − 4.84318e7i − 1.33551i −0.744382 0.667754i \(-0.767256\pi\)
0.744382 0.667754i \(-0.232744\pi\)
\(332\) 0 0
\(333\) 1.07304e7 0.290592
\(334\) 0 0
\(335\) − 8.49210e6i − 0.225881i
\(336\) 0 0
\(337\) −5.96066e7 −1.55742 −0.778709 0.627386i \(-0.784125\pi\)
−0.778709 + 0.627386i \(0.784125\pi\)
\(338\) 0 0
\(339\) 1.13913e7i 0.292397i
\(340\) 0 0
\(341\) −7.06616e7 −1.78205
\(342\) 0 0
\(343\) 3.28657e7i 0.814444i
\(344\) 0 0
\(345\) −5.42229e6 −0.132046
\(346\) 0 0
\(347\) 2.95157e7i 0.706423i 0.935544 + 0.353211i \(0.114910\pi\)
−0.935544 + 0.353211i \(0.885090\pi\)
\(348\) 0 0
\(349\) −5.55247e7 −1.30620 −0.653100 0.757272i \(-0.726532\pi\)
−0.653100 + 0.757272i \(0.726532\pi\)
\(350\) 0 0
\(351\) − 1.22018e7i − 0.282165i
\(352\) 0 0
\(353\) −1.81396e7 −0.412386 −0.206193 0.978511i \(-0.566107\pi\)
−0.206193 + 0.978511i \(0.566107\pi\)
\(354\) 0 0
\(355\) − 731449.i − 0.0163493i
\(356\) 0 0
\(357\) −1.59410e7 −0.350358
\(358\) 0 0
\(359\) 8.83261e6i 0.190900i 0.995434 + 0.0954499i \(0.0304290\pi\)
−0.995434 + 0.0954499i \(0.969571\pi\)
\(360\) 0 0
\(361\) 3.16293e7 0.672307
\(362\) 0 0
\(363\) 5.53110e7i 1.15636i
\(364\) 0 0
\(365\) 7.50242e6 0.154285
\(366\) 0 0
\(367\) 4.83247e7i 0.977623i 0.872389 + 0.488811i \(0.162569\pi\)
−0.872389 + 0.488811i \(0.837431\pi\)
\(368\) 0 0
\(369\) 2.18899e6 0.0435677
\(370\) 0 0
\(371\) 1.49819e7i 0.293391i
\(372\) 0 0
\(373\) 8.98844e7 1.73204 0.866020 0.500010i \(-0.166670\pi\)
0.866020 + 0.500010i \(0.166670\pi\)
\(374\) 0 0
\(375\) 9.61808e6i 0.182387i
\(376\) 0 0
\(377\) −1.44030e8 −2.68799
\(378\) 0 0
\(379\) 8.60279e7i 1.58024i 0.612955 + 0.790118i \(0.289981\pi\)
−0.612955 + 0.790118i \(0.710019\pi\)
\(380\) 0 0
\(381\) 4.04798e7 0.731919
\(382\) 0 0
\(383\) − 6.65042e7i − 1.18373i −0.806037 0.591865i \(-0.798392\pi\)
0.806037 0.591865i \(-0.201608\pi\)
\(384\) 0 0
\(385\) 7.18357e6 0.125880
\(386\) 0 0
\(387\) 6.14069e6i 0.105946i
\(388\) 0 0
\(389\) −5.66889e7 −0.963050 −0.481525 0.876432i \(-0.659917\pi\)
−0.481525 + 0.876432i \(0.659917\pi\)
\(390\) 0 0
\(391\) 1.14208e8i 1.91059i
\(392\) 0 0
\(393\) 9.43949e6 0.155514
\(394\) 0 0
\(395\) 1.04065e7i 0.168854i
\(396\) 0 0
\(397\) −3.21267e7 −0.513446 −0.256723 0.966485i \(-0.582643\pi\)
−0.256723 + 0.966485i \(0.582643\pi\)
\(398\) 0 0
\(399\) − 9.53152e6i − 0.150053i
\(400\) 0 0
\(401\) −7.28625e7 −1.12998 −0.564991 0.825097i \(-0.691120\pi\)
−0.564991 + 0.825097i \(0.691120\pi\)
\(402\) 0 0
\(403\) 9.86853e7i 1.50778i
\(404\) 0 0
\(405\) 1.18098e6 0.0177778
\(406\) 0 0
\(407\) 1.01849e8i 1.51068i
\(408\) 0 0
\(409\) 2.07185e7 0.302823 0.151412 0.988471i \(-0.451618\pi\)
0.151412 + 0.988471i \(0.451618\pi\)
\(410\) 0 0
\(411\) − 2.74251e7i − 0.395023i
\(412\) 0 0
\(413\) −5.45330e6 −0.0774121
\(414\) 0 0
\(415\) 1.18842e7i 0.166274i
\(416\) 0 0
\(417\) −4.73390e6 −0.0652846
\(418\) 0 0
\(419\) 6.79561e7i 0.923818i 0.886927 + 0.461909i \(0.152835\pi\)
−0.886927 + 0.461909i \(0.847165\pi\)
\(420\) 0 0
\(421\) −522580. −0.00700336 −0.00350168 0.999994i \(-0.501115\pi\)
−0.00350168 + 0.999994i \(0.501115\pi\)
\(422\) 0 0
\(423\) − 4.26723e7i − 0.563800i
\(424\) 0 0
\(425\) 9.99784e7 1.30239
\(426\) 0 0
\(427\) 1.34137e7i 0.172291i
\(428\) 0 0
\(429\) 1.15815e8 1.46687
\(430\) 0 0
\(431\) − 1.10533e8i − 1.38057i −0.723538 0.690285i \(-0.757485\pi\)
0.723538 0.690285i \(-0.242515\pi\)
\(432\) 0 0
\(433\) 6.65239e7 0.819435 0.409717 0.912213i \(-0.365627\pi\)
0.409717 + 0.912213i \(0.365627\pi\)
\(434\) 0 0
\(435\) − 1.39402e7i − 0.169356i
\(436\) 0 0
\(437\) −6.82879e7 −0.818276
\(438\) 0 0
\(439\) − 1.06995e8i − 1.26465i −0.774702 0.632327i \(-0.782100\pi\)
0.774702 0.632327i \(-0.217900\pi\)
\(440\) 0 0
\(441\) −2.26957e7 −0.264624
\(442\) 0 0
\(443\) − 1.01597e7i − 0.116861i −0.998291 0.0584304i \(-0.981390\pi\)
0.998291 0.0584304i \(-0.0186096\pi\)
\(444\) 0 0
\(445\) −1.80220e7 −0.204514
\(446\) 0 0
\(447\) 2.06107e7i 0.230765i
\(448\) 0 0
\(449\) −4.39511e7 −0.485546 −0.242773 0.970083i \(-0.578057\pi\)
−0.242773 + 0.970083i \(0.578057\pi\)
\(450\) 0 0
\(451\) 2.07770e7i 0.226492i
\(452\) 0 0
\(453\) −7.28789e7 −0.783984
\(454\) 0 0
\(455\) − 1.00325e7i − 0.106506i
\(456\) 0 0
\(457\) 5.92987e6 0.0621293 0.0310646 0.999517i \(-0.490110\pi\)
0.0310646 + 0.999517i \(0.490110\pi\)
\(458\) 0 0
\(459\) − 2.48747e7i − 0.257229i
\(460\) 0 0
\(461\) 4.31145e7 0.440069 0.220035 0.975492i \(-0.429383\pi\)
0.220035 + 0.975492i \(0.429383\pi\)
\(462\) 0 0
\(463\) − 8.09555e7i − 0.815649i −0.913060 0.407825i \(-0.866288\pi\)
0.913060 0.407825i \(-0.133712\pi\)
\(464\) 0 0
\(465\) −9.55147e6 −0.0949973
\(466\) 0 0
\(467\) − 1.20309e8i − 1.18127i −0.806940 0.590633i \(-0.798878\pi\)
0.806940 0.590633i \(-0.201122\pi\)
\(468\) 0 0
\(469\) 6.61225e7 0.640960
\(470\) 0 0
\(471\) − 3.81557e7i − 0.365171i
\(472\) 0 0
\(473\) −5.82850e7 −0.550774
\(474\) 0 0
\(475\) 5.97795e7i 0.557791i
\(476\) 0 0
\(477\) −2.33781e7 −0.215405
\(478\) 0 0
\(479\) 1.54238e8i 1.40341i 0.712468 + 0.701705i \(0.247578\pi\)
−0.712468 + 0.701705i \(0.752422\pi\)
\(480\) 0 0
\(481\) 1.42241e8 1.27817
\(482\) 0 0
\(483\) − 4.22198e7i − 0.374693i
\(484\) 0 0
\(485\) −2.51487e7 −0.220440
\(486\) 0 0
\(487\) − 6.53601e7i − 0.565882i −0.959137 0.282941i \(-0.908690\pi\)
0.959137 0.282941i \(-0.0913101\pi\)
\(488\) 0 0
\(489\) 1.03201e8 0.882584
\(490\) 0 0
\(491\) 4.51212e7i 0.381185i 0.981669 + 0.190593i \(0.0610410\pi\)
−0.981669 + 0.190593i \(0.938959\pi\)
\(492\) 0 0
\(493\) −2.93620e8 −2.45044
\(494\) 0 0
\(495\) 1.12094e7i 0.0924202i
\(496\) 0 0
\(497\) 5.69532e6 0.0463926
\(498\) 0 0
\(499\) 5.03581e7i 0.405292i 0.979252 + 0.202646i \(0.0649540\pi\)
−0.979252 + 0.202646i \(0.935046\pi\)
\(500\) 0 0
\(501\) −1.13412e8 −0.901872
\(502\) 0 0
\(503\) 4.41054e6i 0.0346567i 0.999850 + 0.0173284i \(0.00551607\pi\)
−0.999850 + 0.0173284i \(0.994484\pi\)
\(504\) 0 0
\(505\) 7.82080e6 0.0607263
\(506\) 0 0
\(507\) − 8.65035e7i − 0.663758i
\(508\) 0 0
\(509\) 2.44005e7 0.185031 0.0925156 0.995711i \(-0.470509\pi\)
0.0925156 + 0.995711i \(0.470509\pi\)
\(510\) 0 0
\(511\) 5.84165e7i 0.437797i
\(512\) 0 0
\(513\) 1.48732e7 0.110167
\(514\) 0 0
\(515\) − 2.72160e7i − 0.199252i
\(516\) 0 0
\(517\) 4.05029e8 2.93099
\(518\) 0 0
\(519\) 5.80827e7i 0.415475i
\(520\) 0 0
\(521\) −6.53917e7 −0.462391 −0.231195 0.972907i \(-0.574264\pi\)
−0.231195 + 0.972907i \(0.574264\pi\)
\(522\) 0 0
\(523\) 1.19428e8i 0.834833i 0.908715 + 0.417417i \(0.137064\pi\)
−0.908715 + 0.417417i \(0.862936\pi\)
\(524\) 0 0
\(525\) −3.69594e7 −0.255415
\(526\) 0 0
\(527\) 2.01181e8i 1.37453i
\(528\) 0 0
\(529\) −1.54446e8 −1.04330
\(530\) 0 0
\(531\) − 8.50945e6i − 0.0568352i
\(532\) 0 0
\(533\) 2.90170e7 0.191633
\(534\) 0 0
\(535\) 2.57240e7i 0.167987i
\(536\) 0 0
\(537\) −3.14593e6 −0.0203154
\(538\) 0 0
\(539\) − 2.15419e8i − 1.37568i
\(540\) 0 0
\(541\) −3.06540e7 −0.193596 −0.0967979 0.995304i \(-0.530860\pi\)
−0.0967979 + 0.995304i \(0.530860\pi\)
\(542\) 0 0
\(543\) − 8.47140e7i − 0.529122i
\(544\) 0 0
\(545\) −3.51525e7 −0.217153
\(546\) 0 0
\(547\) − 1.60733e8i − 0.982071i −0.871140 0.491035i \(-0.836619\pi\)
0.871140 0.491035i \(-0.163381\pi\)
\(548\) 0 0
\(549\) −2.09310e7 −0.126495
\(550\) 0 0
\(551\) − 1.75562e8i − 1.04949i
\(552\) 0 0
\(553\) −8.10284e7 −0.479140
\(554\) 0 0
\(555\) 1.37671e7i 0.0805313i
\(556\) 0 0
\(557\) 3.11683e8 1.80363 0.901814 0.432125i \(-0.142236\pi\)
0.901814 + 0.432125i \(0.142236\pi\)
\(558\) 0 0
\(559\) 8.14003e7i 0.466005i
\(560\) 0 0
\(561\) 2.36101e8 1.33724
\(562\) 0 0
\(563\) 2.81653e8i 1.57830i 0.614201 + 0.789149i \(0.289478\pi\)
−0.614201 + 0.789149i \(0.710522\pi\)
\(564\) 0 0
\(565\) −1.46150e7 −0.0810313
\(566\) 0 0
\(567\) 9.19553e6i 0.0504461i
\(568\) 0 0
\(569\) −8.82677e7 −0.479143 −0.239571 0.970879i \(-0.577007\pi\)
−0.239571 + 0.970879i \(0.577007\pi\)
\(570\) 0 0
\(571\) − 1.66103e8i − 0.892217i −0.894979 0.446108i \(-0.852810\pi\)
0.894979 0.446108i \(-0.147190\pi\)
\(572\) 0 0
\(573\) 1.06408e8 0.565601
\(574\) 0 0
\(575\) 2.64793e8i 1.39285i
\(576\) 0 0
\(577\) −5.26455e7 −0.274053 −0.137026 0.990567i \(-0.543754\pi\)
−0.137026 + 0.990567i \(0.543754\pi\)
\(578\) 0 0
\(579\) − 2.03962e7i − 0.105078i
\(580\) 0 0
\(581\) −9.25343e7 −0.471818
\(582\) 0 0
\(583\) − 2.21896e8i − 1.11981i
\(584\) 0 0
\(585\) 1.56549e7 0.0781959
\(586\) 0 0
\(587\) 3.88069e7i 0.191865i 0.995388 + 0.0959323i \(0.0305832\pi\)
−0.995388 + 0.0959323i \(0.969417\pi\)
\(588\) 0 0
\(589\) −1.20291e8 −0.588689
\(590\) 0 0
\(591\) 1.18687e8i 0.574962i
\(592\) 0 0
\(593\) −3.26100e8 −1.56382 −0.781909 0.623392i \(-0.785754\pi\)
−0.781909 + 0.623392i \(0.785754\pi\)
\(594\) 0 0
\(595\) − 2.04523e7i − 0.0970939i
\(596\) 0 0
\(597\) −1.00965e8 −0.474514
\(598\) 0 0
\(599\) − 3.03039e8i − 1.41000i −0.709210 0.704998i \(-0.750948\pi\)
0.709210 0.704998i \(-0.249052\pi\)
\(600\) 0 0
\(601\) 3.98916e8 1.83763 0.918815 0.394689i \(-0.129148\pi\)
0.918815 + 0.394689i \(0.129148\pi\)
\(602\) 0 0
\(603\) 1.03179e8i 0.470586i
\(604\) 0 0
\(605\) −7.09641e7 −0.320459
\(606\) 0 0
\(607\) 2.46327e8i 1.10140i 0.834703 + 0.550701i \(0.185639\pi\)
−0.834703 + 0.550701i \(0.814361\pi\)
\(608\) 0 0
\(609\) 1.08543e8 0.480565
\(610\) 0 0
\(611\) − 5.65660e8i − 2.47989i
\(612\) 0 0
\(613\) −2.16670e8 −0.940628 −0.470314 0.882499i \(-0.655859\pi\)
−0.470314 + 0.882499i \(0.655859\pi\)
\(614\) 0 0
\(615\) 2.80847e6i 0.0120738i
\(616\) 0 0
\(617\) −1.27942e8 −0.544699 −0.272350 0.962198i \(-0.587801\pi\)
−0.272350 + 0.962198i \(0.587801\pi\)
\(618\) 0 0
\(619\) − 1.96738e8i − 0.829499i −0.909936 0.414749i \(-0.863869\pi\)
0.909936 0.414749i \(-0.136131\pi\)
\(620\) 0 0
\(621\) 6.58808e7 0.275096
\(622\) 0 0
\(623\) − 1.40326e8i − 0.580328i
\(624\) 0 0
\(625\) 2.25551e8 0.923855
\(626\) 0 0
\(627\) 1.41171e8i 0.572719i
\(628\) 0 0
\(629\) 2.89974e8 1.16522
\(630\) 0 0
\(631\) − 9.81163e7i − 0.390529i −0.980751 0.195264i \(-0.937443\pi\)
0.980751 0.195264i \(-0.0625565\pi\)
\(632\) 0 0
\(633\) 2.46388e8 0.971424
\(634\) 0 0
\(635\) 5.19355e7i 0.202835i
\(636\) 0 0
\(637\) −3.00852e8 −1.16395
\(638\) 0 0
\(639\) 8.88710e6i 0.0340610i
\(640\) 0 0
\(641\) 2.85929e8 1.08564 0.542819 0.839850i \(-0.317357\pi\)
0.542819 + 0.839850i \(0.317357\pi\)
\(642\) 0 0
\(643\) − 2.98146e8i − 1.12149i −0.827988 0.560745i \(-0.810515\pi\)
0.827988 0.560745i \(-0.189485\pi\)
\(644\) 0 0
\(645\) −7.87850e6 −0.0293606
\(646\) 0 0
\(647\) 3.84615e7i 0.142008i 0.997476 + 0.0710042i \(0.0226204\pi\)
−0.997476 + 0.0710042i \(0.977380\pi\)
\(648\) 0 0
\(649\) 8.07684e7 0.295466
\(650\) 0 0
\(651\) − 7.43711e7i − 0.269564i
\(652\) 0 0
\(653\) 9.67190e7 0.347354 0.173677 0.984803i \(-0.444435\pi\)
0.173677 + 0.984803i \(0.444435\pi\)
\(654\) 0 0
\(655\) 1.21109e7i 0.0430974i
\(656\) 0 0
\(657\) −9.11545e7 −0.321427
\(658\) 0 0
\(659\) − 6.49525e7i − 0.226955i −0.993541 0.113477i \(-0.963801\pi\)
0.993541 0.113477i \(-0.0361990\pi\)
\(660\) 0 0
\(661\) 3.29739e8 1.14174 0.570868 0.821042i \(-0.306607\pi\)
0.570868 + 0.821042i \(0.306607\pi\)
\(662\) 0 0
\(663\) − 3.29737e8i − 1.13143i
\(664\) 0 0
\(665\) 1.22289e7 0.0415838
\(666\) 0 0
\(667\) − 7.77653e8i − 2.62065i
\(668\) 0 0
\(669\) 3.22836e8 1.07821
\(670\) 0 0
\(671\) − 1.98669e8i − 0.657600i
\(672\) 0 0
\(673\) 5.17571e8 1.69795 0.848975 0.528433i \(-0.177220\pi\)
0.848975 + 0.528433i \(0.177220\pi\)
\(674\) 0 0
\(675\) − 5.76722e7i − 0.187523i
\(676\) 0 0
\(677\) −3.25473e8 −1.04894 −0.524469 0.851430i \(-0.675736\pi\)
−0.524469 + 0.851430i \(0.675736\pi\)
\(678\) 0 0
\(679\) − 1.95817e8i − 0.625519i
\(680\) 0 0
\(681\) 1.10082e8 0.348559
\(682\) 0 0
\(683\) − 5.37538e8i − 1.68713i −0.537031 0.843563i \(-0.680454\pi\)
0.537031 0.843563i \(-0.319546\pi\)
\(684\) 0 0
\(685\) 3.51864e7 0.109472
\(686\) 0 0
\(687\) 7.24498e7i 0.223443i
\(688\) 0 0
\(689\) −3.09898e8 −0.947461
\(690\) 0 0
\(691\) − 4.64528e8i − 1.40792i −0.710239 0.703960i \(-0.751413\pi\)
0.710239 0.703960i \(-0.248587\pi\)
\(692\) 0 0
\(693\) −8.72804e7 −0.262251
\(694\) 0 0
\(695\) − 6.07360e6i − 0.0180922i
\(696\) 0 0
\(697\) 5.91543e7 0.174698
\(698\) 0 0
\(699\) 1.77361e8i 0.519309i
\(700\) 0 0
\(701\) 3.59130e7 0.104255 0.0521275 0.998640i \(-0.483400\pi\)
0.0521275 + 0.998640i \(0.483400\pi\)
\(702\) 0 0
\(703\) 1.73382e8i 0.499044i
\(704\) 0 0
\(705\) 5.47486e7 0.156245
\(706\) 0 0
\(707\) 6.08955e7i 0.172317i
\(708\) 0 0
\(709\) −1.86788e8 −0.524094 −0.262047 0.965055i \(-0.584398\pi\)
−0.262047 + 0.965055i \(0.584398\pi\)
\(710\) 0 0
\(711\) − 1.26439e8i − 0.351780i
\(712\) 0 0
\(713\) −5.32827e8 −1.47000
\(714\) 0 0
\(715\) 1.48591e8i 0.406512i
\(716\) 0 0
\(717\) −3.02050e8 −0.819447
\(718\) 0 0
\(719\) 3.76368e7i 0.101257i 0.998718 + 0.0506286i \(0.0161225\pi\)
−0.998718 + 0.0506286i \(0.983878\pi\)
\(720\) 0 0
\(721\) 2.11914e8 0.565397
\(722\) 0 0
\(723\) − 1.06763e8i − 0.282491i
\(724\) 0 0
\(725\) −6.80760e8 −1.78640
\(726\) 0 0
\(727\) 2.58872e8i 0.673724i 0.941554 + 0.336862i \(0.109366\pi\)
−0.941554 + 0.336862i \(0.890634\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −0.0370370
\(730\) 0 0
\(731\) 1.65943e8i 0.424822i
\(732\) 0 0
\(733\) 5.31872e8 1.35050 0.675251 0.737588i \(-0.264035\pi\)
0.675251 + 0.737588i \(0.264035\pi\)
\(734\) 0 0
\(735\) − 2.91186e7i − 0.0733346i
\(736\) 0 0
\(737\) −9.79335e8 −2.44641
\(738\) 0 0
\(739\) − 1.70943e8i − 0.423564i −0.977317 0.211782i \(-0.932073\pi\)
0.977317 0.211782i \(-0.0679267\pi\)
\(740\) 0 0
\(741\) 1.97157e8 0.484572
\(742\) 0 0
\(743\) 4.23377e8i 1.03219i 0.856531 + 0.516096i \(0.172615\pi\)
−0.856531 + 0.516096i \(0.827385\pi\)
\(744\) 0 0
\(745\) −2.64435e7 −0.0639514
\(746\) 0 0
\(747\) − 1.44393e8i − 0.346404i
\(748\) 0 0
\(749\) −2.00296e8 −0.476679
\(750\) 0 0
\(751\) 3.04170e8i 0.718118i 0.933315 + 0.359059i \(0.116902\pi\)
−0.933315 + 0.359059i \(0.883098\pi\)
\(752\) 0 0
\(753\) 2.01912e6 0.00472910
\(754\) 0 0
\(755\) − 9.35037e7i − 0.217264i
\(756\) 0 0
\(757\) 3.06435e8 0.706401 0.353201 0.935548i \(-0.385093\pi\)
0.353201 + 0.935548i \(0.385093\pi\)
\(758\) 0 0
\(759\) 6.25315e8i 1.43012i
\(760\) 0 0
\(761\) −4.23923e8 −0.961907 −0.480954 0.876746i \(-0.659709\pi\)
−0.480954 + 0.876746i \(0.659709\pi\)
\(762\) 0 0
\(763\) − 2.73710e8i − 0.616193i
\(764\) 0 0
\(765\) 3.19143e7 0.0712854
\(766\) 0 0
\(767\) − 1.12800e8i − 0.249991i
\(768\) 0 0
\(769\) −5.19668e8 −1.14274 −0.571369 0.820693i \(-0.693588\pi\)
−0.571369 + 0.820693i \(0.693588\pi\)
\(770\) 0 0
\(771\) − 1.85338e8i − 0.404391i
\(772\) 0 0
\(773\) 1.89126e7 0.0409462 0.0204731 0.999790i \(-0.493483\pi\)
0.0204731 + 0.999790i \(0.493483\pi\)
\(774\) 0 0
\(775\) 4.66439e8i 1.00205i
\(776\) 0 0
\(777\) −1.07196e8 −0.228515
\(778\) 0 0
\(779\) 3.53697e7i 0.0748203i
\(780\) 0 0
\(781\) −8.43529e7 −0.177071
\(782\) 0 0
\(783\) 1.69374e8i 0.352826i
\(784\) 0 0
\(785\) 4.89537e7 0.101199
\(786\) 0 0
\(787\) 2.65036e8i 0.543726i 0.962336 + 0.271863i \(0.0876398\pi\)
−0.962336 + 0.271863i \(0.912360\pi\)
\(788\) 0 0
\(789\) 2.11495e8 0.430594
\(790\) 0 0
\(791\) − 1.13797e8i − 0.229934i
\(792\) 0 0
\(793\) −2.77459e8 −0.556390
\(794\) 0 0
\(795\) − 2.99942e7i − 0.0596947i
\(796\) 0 0
\(797\) −6.82061e8 −1.34725 −0.673626 0.739073i \(-0.735264\pi\)
−0.673626 + 0.739073i \(0.735264\pi\)
\(798\) 0 0
\(799\) − 1.15316e9i − 2.26073i
\(800\) 0 0
\(801\) 2.18968e8 0.426071
\(802\) 0 0
\(803\) − 8.65203e8i − 1.67098i
\(804\) 0 0
\(805\) 5.41681e7 0.103838
\(806\) 0 0
\(807\) 2.66922e8i 0.507884i
\(808\) 0 0
\(809\) −5.50511e8 −1.03973 −0.519865 0.854248i \(-0.674018\pi\)
−0.519865 + 0.854248i \(0.674018\pi\)
\(810\) 0 0
\(811\) 3.11549e8i 0.584068i 0.956408 + 0.292034i \(0.0943321\pi\)
−0.956408 + 0.292034i \(0.905668\pi\)
\(812\) 0 0
\(813\) 4.99143e6 0.00928867
\(814\) 0 0
\(815\) 1.32407e8i 0.244589i
\(816\) 0 0
\(817\) −9.92214e7 −0.181945
\(818\) 0 0
\(819\) 1.21895e8i 0.221888i
\(820\) 0 0
\(821\) 7.17579e8 1.29670 0.648351 0.761342i \(-0.275459\pi\)
0.648351 + 0.761342i \(0.275459\pi\)
\(822\) 0 0
\(823\) − 5.41195e8i − 0.970854i −0.874277 0.485427i \(-0.838664\pi\)
0.874277 0.485427i \(-0.161336\pi\)
\(824\) 0 0
\(825\) 5.47402e8 0.974866
\(826\) 0 0
\(827\) − 8.79442e8i − 1.55486i −0.628971 0.777429i \(-0.716524\pi\)
0.628971 0.777429i \(-0.283476\pi\)
\(828\) 0 0
\(829\) −4.29123e8 −0.753213 −0.376607 0.926373i \(-0.622909\pi\)
−0.376607 + 0.926373i \(0.622909\pi\)
\(830\) 0 0
\(831\) − 2.35306e8i − 0.410043i
\(832\) 0 0
\(833\) −6.13320e8 −1.06109
\(834\) 0 0
\(835\) − 1.45507e8i − 0.249934i
\(836\) 0 0
\(837\) 1.16050e8 0.197911
\(838\) 0 0
\(839\) − 5.86636e8i − 0.993305i −0.867949 0.496653i \(-0.834562\pi\)
0.867949 0.496653i \(-0.165438\pi\)
\(840\) 0 0
\(841\) 1.40445e9 2.36113
\(842\) 0 0
\(843\) − 6.74325e8i − 1.12561i
\(844\) 0 0
\(845\) 1.10984e8 0.183946
\(846\) 0 0
\(847\) − 5.52552e8i − 0.909332i
\(848\) 0 0
\(849\) 5.01951e8 0.820236
\(850\) 0 0
\(851\) 7.67997e8i 1.24615i
\(852\) 0 0
\(853\) −7.00029e8 −1.12790 −0.563948 0.825810i \(-0.690718\pi\)
−0.563948 + 0.825810i \(0.690718\pi\)
\(854\) 0 0
\(855\) 1.90823e7i 0.0305304i
\(856\) 0 0
\(857\) −9.52114e8 −1.51268 −0.756339 0.654180i \(-0.773014\pi\)
−0.756339 + 0.654180i \(0.773014\pi\)
\(858\) 0 0
\(859\) − 4.16968e8i − 0.657844i −0.944357 0.328922i \(-0.893315\pi\)
0.944357 0.328922i \(-0.106685\pi\)
\(860\) 0 0
\(861\) −2.18678e7 −0.0342606
\(862\) 0 0
\(863\) − 4.62686e8i − 0.719869i −0.932978 0.359935i \(-0.882799\pi\)
0.932978 0.359935i \(-0.117201\pi\)
\(864\) 0 0
\(865\) −7.45201e7 −0.115140
\(866\) 0 0
\(867\) − 2.95936e8i − 0.454089i
\(868\) 0 0
\(869\) 1.20011e9 1.82877
\(870\) 0 0
\(871\) 1.36773e9i 2.06988i
\(872\) 0 0
\(873\) 3.05557e8 0.459250
\(874\) 0 0
\(875\) − 9.60836e7i − 0.143425i
\(876\) 0 0
\(877\) 6.73329e8 0.998225 0.499113 0.866537i \(-0.333659\pi\)
0.499113 + 0.866537i \(0.333659\pi\)
\(878\) 0 0
\(879\) − 4.69692e8i − 0.691586i
\(880\) 0 0
\(881\) 1.77155e8 0.259074 0.129537 0.991575i \(-0.458651\pi\)
0.129537 + 0.991575i \(0.458651\pi\)
\(882\) 0 0
\(883\) 1.35707e8i 0.197115i 0.995131 + 0.0985574i \(0.0314228\pi\)
−0.995131 + 0.0985574i \(0.968577\pi\)
\(884\) 0 0
\(885\) 1.09176e7 0.0157506
\(886\) 0 0
\(887\) 1.09553e9i 1.56984i 0.619598 + 0.784919i \(0.287296\pi\)
−0.619598 + 0.784919i \(0.712704\pi\)
\(888\) 0 0
\(889\) −4.04389e8 −0.575564
\(890\) 0 0
\(891\) − 1.36194e8i − 0.192542i
\(892\) 0 0
\(893\) 6.89501e8 0.968234
\(894\) 0 0
\(895\) − 4.03623e6i − 0.00562997i
\(896\) 0 0
\(897\) 8.73309e8 1.21001
\(898\) 0 0
\(899\) − 1.36985e9i − 1.88536i
\(900\) 0 0
\(901\) −6.31761e8 −0.863731
\(902\) 0 0
\(903\) − 6.13448e7i − 0.0833133i
\(904\) 0 0
\(905\) 1.08688e8 0.146635
\(906\) 0 0
\(907\) 1.34159e9i 1.79803i 0.437917 + 0.899015i \(0.355716\pi\)
−0.437917 + 0.899015i \(0.644284\pi\)
\(908\) 0 0
\(909\) −9.50227e7 −0.126513
\(910\) 0 0
\(911\) − 1.16107e9i − 1.53569i −0.640638 0.767843i \(-0.721330\pi\)
0.640638 0.767843i \(-0.278670\pi\)
\(912\) 0 0
\(913\) 1.37052e9 1.80083
\(914\) 0 0
\(915\) − 2.68545e7i − 0.0350553i
\(916\) 0 0
\(917\) −9.42995e7 −0.122293
\(918\) 0 0
\(919\) 1.03538e9i 1.33399i 0.745061 + 0.666996i \(0.232420\pi\)
−0.745061 + 0.666996i \(0.767580\pi\)
\(920\) 0 0
\(921\) 3.26579e8 0.418032
\(922\) 0 0
\(923\) 1.17806e8i 0.149818i
\(924\) 0 0
\(925\) 6.72307e8 0.849459
\(926\) 0 0
\(927\) 3.30675e8i 0.415109i
\(928\) 0 0
\(929\) −1.47698e9 −1.84217 −0.921083 0.389367i \(-0.872694\pi\)
−0.921083 + 0.389367i \(0.872694\pi\)
\(930\) 0 0
\(931\) − 3.66718e8i − 0.454447i
\(932\) 0 0
\(933\) −7.70227e7 −0.0948361
\(934\) 0 0
\(935\) 3.02918e8i 0.370587i
\(936\) 0 0
\(937\) −4.90868e8 −0.596686 −0.298343 0.954459i \(-0.596434\pi\)
−0.298343 + 0.954459i \(0.596434\pi\)
\(938\) 0 0
\(939\) 4.16911e8i 0.503555i
\(940\) 0 0
\(941\) −1.54536e8 −0.185464 −0.0927321 0.995691i \(-0.529560\pi\)
−0.0927321 + 0.995691i \(0.529560\pi\)
\(942\) 0 0
\(943\) 1.56670e8i 0.186832i
\(944\) 0 0
\(945\) −1.17979e7 −0.0139800
\(946\) 0 0
\(947\) 1.24667e9i 1.46792i 0.679192 + 0.733961i \(0.262331\pi\)
−0.679192 + 0.733961i \(0.737669\pi\)
\(948\) 0 0
\(949\) −1.20833e9 −1.41380
\(950\) 0 0
\(951\) − 7.39671e8i − 0.859997i
\(952\) 0 0
\(953\) 4.21385e8 0.486856 0.243428 0.969919i \(-0.421728\pi\)
0.243428 + 0.969919i \(0.421728\pi\)
\(954\) 0 0
\(955\) 1.36522e8i 0.156744i
\(956\) 0 0
\(957\) −1.60763e9 −1.83421
\(958\) 0 0
\(959\) 2.73973e8i 0.310637i
\(960\) 0 0
\(961\) −5.10826e7 −0.0575576
\(962\) 0 0
\(963\) − 3.12546e8i − 0.349973i
\(964\) 0 0
\(965\) 2.61683e7 0.0291201
\(966\) 0 0
\(967\) − 1.35010e9i − 1.49309i −0.665334 0.746546i \(-0.731711\pi\)
0.665334 0.746546i \(-0.268289\pi\)
\(968\) 0 0
\(969\) 4.01927e8 0.441749
\(970\) 0 0
\(971\) − 2.75696e8i − 0.301143i −0.988599 0.150571i \(-0.951889\pi\)
0.988599 0.150571i \(-0.0481113\pi\)
\(972\) 0 0
\(973\) 4.72912e7 0.0513383
\(974\) 0 0
\(975\) − 7.64497e8i − 0.824825i
\(976\) 0 0
\(977\) −4.86639e8 −0.521823 −0.260911 0.965363i \(-0.584023\pi\)
−0.260911 + 0.965363i \(0.584023\pi\)
\(978\) 0 0
\(979\) 2.07836e9i 2.21499i
\(980\) 0 0
\(981\) 4.27103e8 0.452403
\(982\) 0 0
\(983\) 1.34935e9i 1.42058i 0.703909 + 0.710290i \(0.251436\pi\)
−0.703909 + 0.710290i \(0.748564\pi\)
\(984\) 0 0
\(985\) −1.52275e8 −0.159338
\(986\) 0 0
\(987\) 4.26292e8i 0.443359i
\(988\) 0 0
\(989\) −4.39501e8 −0.454330
\(990\) 0 0
\(991\) 5.76213e8i 0.592056i 0.955179 + 0.296028i \(0.0956621\pi\)
−0.955179 + 0.296028i \(0.904338\pi\)
\(992\) 0 0
\(993\) −7.54976e8 −0.771055
\(994\) 0 0
\(995\) − 1.29538e8i − 0.131501i
\(996\) 0 0
\(997\) −8.84682e7 −0.0892692 −0.0446346 0.999003i \(-0.514212\pi\)
−0.0446346 + 0.999003i \(0.514212\pi\)
\(998\) 0 0
\(999\) − 1.67271e8i − 0.167773i
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 768.7.g.e.511.2 4
4.3 odd 2 inner 768.7.g.e.511.3 4
8.3 odd 2 768.7.g.c.511.1 4
8.5 even 2 768.7.g.c.511.4 4
16.3 odd 4 384.7.b.d.319.2 8
16.5 even 4 384.7.b.d.319.3 yes 8
16.11 odd 4 384.7.b.d.319.8 yes 8
16.13 even 4 384.7.b.d.319.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.d.319.2 8 16.3 odd 4
384.7.b.d.319.3 yes 8 16.5 even 4
384.7.b.d.319.5 yes 8 16.13 even 4
384.7.b.d.319.8 yes 8 16.11 odd 4
768.7.g.c.511.1 4 8.3 odd 2
768.7.g.c.511.4 4 8.5 even 2
768.7.g.e.511.2 4 1.1 even 1 trivial
768.7.g.e.511.3 4 4.3 odd 2 inner