Properties

Label 320.6.d.b.161.3
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.3
Root \(-7.33903 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.b.161.6

$q$-expansion

\(f(q)\) \(=\) \(q-15.6781i q^{3} -25.0000i q^{5} -164.681 q^{7} -2.80186 q^{9} +O(q^{10})\) \(q-15.6781i q^{3} -25.0000i q^{5} -164.681 q^{7} -2.80186 q^{9} -744.619i q^{11} -873.876i q^{13} -391.952 q^{15} +984.464 q^{17} +1149.05i q^{19} +2581.88i q^{21} +578.171 q^{23} -625.000 q^{25} -3765.84i q^{27} -5863.72i q^{29} -9825.27 q^{31} -11674.2 q^{33} +4117.03i q^{35} -7885.71i q^{37} -13700.7 q^{39} +10485.2 q^{41} +19074.5i q^{43} +70.0464i q^{45} +2973.28 q^{47} +10312.8 q^{49} -15434.5i q^{51} +35972.1i q^{53} -18615.5 q^{55} +18014.9 q^{57} -19390.0i q^{59} +37812.2i q^{61} +461.413 q^{63} -21846.9 q^{65} -11395.7i q^{67} -9064.61i q^{69} +33932.2 q^{71} +22794.6 q^{73} +9798.79i q^{75} +122625. i q^{77} -94753.9 q^{79} -59722.0 q^{81} -34722.9i q^{83} -24611.6i q^{85} -91931.8 q^{87} -42021.6 q^{89} +143911. i q^{91} +154041. i q^{93} +28726.3 q^{95} -11466.2 q^{97} +2086.31i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 320q^{7} - 1192q^{9} + O(q^{10}) \) \( 8q + 320q^{7} - 1192q^{9} - 2400q^{17} - 1760q^{23} - 5000q^{25} - 31040q^{31} + 3760q^{33} - 4480q^{39} + 21584q^{41} - 47680q^{47} + 82824q^{49} - 18400q^{55} - 106640q^{57} - 322400q^{63} - 44000q^{65} - 246720q^{71} + 46400q^{73} - 325760q^{79} - 82328q^{81} - 636320q^{87} - 78192q^{89} + 40800q^{95} + 24960q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 15.6781i − 1.00575i −0.864360 0.502874i \(-0.832276\pi\)
0.864360 0.502874i \(-0.167724\pi\)
\(4\) 0 0
\(5\) − 25.0000i − 0.447214i
\(6\) 0 0
\(7\) −164.681 −1.27028 −0.635139 0.772398i \(-0.719057\pi\)
−0.635139 + 0.772398i \(0.719057\pi\)
\(8\) 0 0
\(9\) −2.80186 −0.0115303
\(10\) 0 0
\(11\) − 744.619i − 1.85546i −0.373249 0.927731i \(-0.621756\pi\)
0.373249 0.927731i \(-0.378244\pi\)
\(12\) 0 0
\(13\) − 873.876i − 1.43414i −0.697001 0.717070i \(-0.745483\pi\)
0.697001 0.717070i \(-0.254517\pi\)
\(14\) 0 0
\(15\) −391.952 −0.449784
\(16\) 0 0
\(17\) 984.464 0.826185 0.413093 0.910689i \(-0.364449\pi\)
0.413093 + 0.910689i \(0.364449\pi\)
\(18\) 0 0
\(19\) 1149.05i 0.730223i 0.930964 + 0.365111i \(0.118969\pi\)
−0.930964 + 0.365111i \(0.881031\pi\)
\(20\) 0 0
\(21\) 2581.88i 1.27758i
\(22\) 0 0
\(23\) 578.171 0.227896 0.113948 0.993487i \(-0.463650\pi\)
0.113948 + 0.993487i \(0.463650\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) − 3765.84i − 0.994152i
\(28\) 0 0
\(29\) − 5863.72i − 1.29473i −0.762181 0.647364i \(-0.775871\pi\)
0.762181 0.647364i \(-0.224129\pi\)
\(30\) 0 0
\(31\) −9825.27 −1.83629 −0.918143 0.396249i \(-0.870312\pi\)
−0.918143 + 0.396249i \(0.870312\pi\)
\(32\) 0 0
\(33\) −11674.2 −1.86613
\(34\) 0 0
\(35\) 4117.03i 0.568085i
\(36\) 0 0
\(37\) − 7885.71i − 0.946970i −0.880802 0.473485i \(-0.842996\pi\)
0.880802 0.473485i \(-0.157004\pi\)
\(38\) 0 0
\(39\) −13700.7 −1.44238
\(40\) 0 0
\(41\) 10485.2 0.974131 0.487066 0.873365i \(-0.338067\pi\)
0.487066 + 0.873365i \(0.338067\pi\)
\(42\) 0 0
\(43\) 19074.5i 1.57320i 0.617465 + 0.786598i \(0.288160\pi\)
−0.617465 + 0.786598i \(0.711840\pi\)
\(44\) 0 0
\(45\) 70.0464i 0.00515650i
\(46\) 0 0
\(47\) 2973.28 0.196332 0.0981659 0.995170i \(-0.468702\pi\)
0.0981659 + 0.995170i \(0.468702\pi\)
\(48\) 0 0
\(49\) 10312.8 0.613604
\(50\) 0 0
\(51\) − 15434.5i − 0.830935i
\(52\) 0 0
\(53\) 35972.1i 1.75904i 0.475860 + 0.879521i \(0.342137\pi\)
−0.475860 + 0.879521i \(0.657863\pi\)
\(54\) 0 0
\(55\) −18615.5 −0.829788
\(56\) 0 0
\(57\) 18014.9 0.734421
\(58\) 0 0
\(59\) − 19390.0i − 0.725183i −0.931948 0.362592i \(-0.881892\pi\)
0.931948 0.362592i \(-0.118108\pi\)
\(60\) 0 0
\(61\) 37812.2i 1.30109i 0.759468 + 0.650544i \(0.225459\pi\)
−0.759468 + 0.650544i \(0.774541\pi\)
\(62\) 0 0
\(63\) 461.413 0.0146466
\(64\) 0 0
\(65\) −21846.9 −0.641367
\(66\) 0 0
\(67\) − 11395.7i − 0.310137i −0.987904 0.155069i \(-0.950440\pi\)
0.987904 0.155069i \(-0.0495599\pi\)
\(68\) 0 0
\(69\) − 9064.61i − 0.229206i
\(70\) 0 0
\(71\) 33932.2 0.798851 0.399425 0.916766i \(-0.369210\pi\)
0.399425 + 0.916766i \(0.369210\pi\)
\(72\) 0 0
\(73\) 22794.6 0.500640 0.250320 0.968163i \(-0.419464\pi\)
0.250320 + 0.968163i \(0.419464\pi\)
\(74\) 0 0
\(75\) 9798.79i 0.201150i
\(76\) 0 0
\(77\) 122625.i 2.35695i
\(78\) 0 0
\(79\) −94753.9 −1.70816 −0.854082 0.520139i \(-0.825880\pi\)
−0.854082 + 0.520139i \(0.825880\pi\)
\(80\) 0 0
\(81\) −59722.0 −1.01140
\(82\) 0 0
\(83\) − 34722.9i − 0.553249i −0.960978 0.276625i \(-0.910784\pi\)
0.960978 0.276625i \(-0.0892158\pi\)
\(84\) 0 0
\(85\) − 24611.6i − 0.369481i
\(86\) 0 0
\(87\) −91931.8 −1.30217
\(88\) 0 0
\(89\) −42021.6 −0.562338 −0.281169 0.959658i \(-0.590722\pi\)
−0.281169 + 0.959658i \(0.590722\pi\)
\(90\) 0 0
\(91\) 143911.i 1.82176i
\(92\) 0 0
\(93\) 154041.i 1.84684i
\(94\) 0 0
\(95\) 28726.3 0.326566
\(96\) 0 0
\(97\) −11466.2 −0.123734 −0.0618672 0.998084i \(-0.519706\pi\)
−0.0618672 + 0.998084i \(0.519706\pi\)
\(98\) 0 0
\(99\) 2086.31i 0.0213940i
\(100\) 0 0
\(101\) 65173.4i 0.635722i 0.948137 + 0.317861i \(0.102964\pi\)
−0.948137 + 0.317861i \(0.897036\pi\)
\(102\) 0 0
\(103\) 47472.7 0.440911 0.220455 0.975397i \(-0.429246\pi\)
0.220455 + 0.975397i \(0.429246\pi\)
\(104\) 0 0
\(105\) 64547.0 0.571351
\(106\) 0 0
\(107\) − 93432.1i − 0.788927i −0.918912 0.394464i \(-0.870930\pi\)
0.918912 0.394464i \(-0.129070\pi\)
\(108\) 0 0
\(109\) 164332.i 1.32482i 0.749143 + 0.662408i \(0.230466\pi\)
−0.749143 + 0.662408i \(0.769534\pi\)
\(110\) 0 0
\(111\) −123633. −0.952414
\(112\) 0 0
\(113\) 162453. 1.19683 0.598415 0.801186i \(-0.295797\pi\)
0.598415 + 0.801186i \(0.295797\pi\)
\(114\) 0 0
\(115\) − 14454.3i − 0.101918i
\(116\) 0 0
\(117\) 2448.48i 0.0165360i
\(118\) 0 0
\(119\) −162122. −1.04948
\(120\) 0 0
\(121\) −393406. −2.44274
\(122\) 0 0
\(123\) − 164388.i − 0.979731i
\(124\) 0 0
\(125\) 15625.0i 0.0894427i
\(126\) 0 0
\(127\) −127181. −0.699700 −0.349850 0.936806i \(-0.613767\pi\)
−0.349850 + 0.936806i \(0.613767\pi\)
\(128\) 0 0
\(129\) 299052. 1.58224
\(130\) 0 0
\(131\) 192224.i 0.978655i 0.872100 + 0.489328i \(0.162758\pi\)
−0.872100 + 0.489328i \(0.837242\pi\)
\(132\) 0 0
\(133\) − 189227.i − 0.927585i
\(134\) 0 0
\(135\) −94146.1 −0.444598
\(136\) 0 0
\(137\) 354980. 1.61586 0.807928 0.589281i \(-0.200589\pi\)
0.807928 + 0.589281i \(0.200589\pi\)
\(138\) 0 0
\(139\) − 38205.1i − 0.167720i −0.996478 0.0838600i \(-0.973275\pi\)
0.996478 0.0838600i \(-0.0267248\pi\)
\(140\) 0 0
\(141\) − 46615.2i − 0.197460i
\(142\) 0 0
\(143\) −650704. −2.66099
\(144\) 0 0
\(145\) −146593. −0.579020
\(146\) 0 0
\(147\) − 161686.i − 0.617132i
\(148\) 0 0
\(149\) − 135301.i − 0.499271i −0.968340 0.249636i \(-0.919689\pi\)
0.968340 0.249636i \(-0.0803108\pi\)
\(150\) 0 0
\(151\) −219371. −0.782955 −0.391477 0.920188i \(-0.628036\pi\)
−0.391477 + 0.920188i \(0.628036\pi\)
\(152\) 0 0
\(153\) −2758.33 −0.00952614
\(154\) 0 0
\(155\) 245632.i 0.821212i
\(156\) 0 0
\(157\) − 99927.5i − 0.323546i −0.986828 0.161773i \(-0.948279\pi\)
0.986828 0.161773i \(-0.0517212\pi\)
\(158\) 0 0
\(159\) 563973. 1.76915
\(160\) 0 0
\(161\) −95213.9 −0.289491
\(162\) 0 0
\(163\) 386516.i 1.13946i 0.821833 + 0.569729i \(0.192952\pi\)
−0.821833 + 0.569729i \(0.807048\pi\)
\(164\) 0 0
\(165\) 291855.i 0.834558i
\(166\) 0 0
\(167\) 317945. 0.882187 0.441094 0.897461i \(-0.354591\pi\)
0.441094 + 0.897461i \(0.354591\pi\)
\(168\) 0 0
\(169\) −392367. −1.05676
\(170\) 0 0
\(171\) − 3219.48i − 0.00841967i
\(172\) 0 0
\(173\) − 142037.i − 0.360815i −0.983592 0.180408i \(-0.942258\pi\)
0.983592 0.180408i \(-0.0577417\pi\)
\(174\) 0 0
\(175\) 102926. 0.254055
\(176\) 0 0
\(177\) −303998. −0.729352
\(178\) 0 0
\(179\) − 461274.i − 1.07603i −0.842934 0.538017i \(-0.819173\pi\)
0.842934 0.538017i \(-0.180827\pi\)
\(180\) 0 0
\(181\) 117514.i 0.266620i 0.991074 + 0.133310i \(0.0425607\pi\)
−0.991074 + 0.133310i \(0.957439\pi\)
\(182\) 0 0
\(183\) 592822. 1.30857
\(184\) 0 0
\(185\) −197143. −0.423498
\(186\) 0 0
\(187\) − 733050.i − 1.53296i
\(188\) 0 0
\(189\) 620163.i 1.26285i
\(190\) 0 0
\(191\) −226621. −0.449486 −0.224743 0.974418i \(-0.572154\pi\)
−0.224743 + 0.974418i \(0.572154\pi\)
\(192\) 0 0
\(193\) −21404.0 −0.0413620 −0.0206810 0.999786i \(-0.506583\pi\)
−0.0206810 + 0.999786i \(0.506583\pi\)
\(194\) 0 0
\(195\) 342517.i 0.645054i
\(196\) 0 0
\(197\) − 707471.i − 1.29880i −0.760446 0.649401i \(-0.775020\pi\)
0.760446 0.649401i \(-0.224980\pi\)
\(198\) 0 0
\(199\) 387592. 0.693812 0.346906 0.937900i \(-0.387232\pi\)
0.346906 + 0.937900i \(0.387232\pi\)
\(200\) 0 0
\(201\) −178663. −0.311920
\(202\) 0 0
\(203\) 965644.i 1.64466i
\(204\) 0 0
\(205\) − 262130.i − 0.435645i
\(206\) 0 0
\(207\) −1619.95 −0.00262771
\(208\) 0 0
\(209\) 855605. 1.35490
\(210\) 0 0
\(211\) − 209622.i − 0.324139i −0.986779 0.162070i \(-0.948183\pi\)
0.986779 0.162070i \(-0.0518169\pi\)
\(212\) 0 0
\(213\) − 531991.i − 0.803443i
\(214\) 0 0
\(215\) 476863. 0.703555
\(216\) 0 0
\(217\) 1.61804e6 2.33259
\(218\) 0 0
\(219\) − 357376.i − 0.503518i
\(220\) 0 0
\(221\) − 860299.i − 1.18487i
\(222\) 0 0
\(223\) 454449. 0.611960 0.305980 0.952038i \(-0.401016\pi\)
0.305980 + 0.952038i \(0.401016\pi\)
\(224\) 0 0
\(225\) 1751.16 0.00230606
\(226\) 0 0
\(227\) − 569941.i − 0.734117i −0.930198 0.367059i \(-0.880365\pi\)
0.930198 0.367059i \(-0.119635\pi\)
\(228\) 0 0
\(229\) − 319567.i − 0.402692i −0.979520 0.201346i \(-0.935468\pi\)
0.979520 0.201346i \(-0.0645316\pi\)
\(230\) 0 0
\(231\) 1.92252e6 2.37050
\(232\) 0 0
\(233\) 313094. 0.377820 0.188910 0.981994i \(-0.439505\pi\)
0.188910 + 0.981994i \(0.439505\pi\)
\(234\) 0 0
\(235\) − 74331.9i − 0.0878023i
\(236\) 0 0
\(237\) 1.48556e6i 1.71798i
\(238\) 0 0
\(239\) 129520. 0.146671 0.0733353 0.997307i \(-0.476636\pi\)
0.0733353 + 0.997307i \(0.476636\pi\)
\(240\) 0 0
\(241\) −1.61051e6 −1.78616 −0.893081 0.449897i \(-0.851461\pi\)
−0.893081 + 0.449897i \(0.851461\pi\)
\(242\) 0 0
\(243\) 21225.8i 0.0230594i
\(244\) 0 0
\(245\) − 257821.i − 0.274412i
\(246\) 0 0
\(247\) 1.00413e6 1.04724
\(248\) 0 0
\(249\) −544388. −0.556429
\(250\) 0 0
\(251\) − 196054.i − 0.196422i −0.995166 0.0982111i \(-0.968688\pi\)
0.995166 0.0982111i \(-0.0313120\pi\)
\(252\) 0 0
\(253\) − 430517.i − 0.422853i
\(254\) 0 0
\(255\) −385862. −0.371605
\(256\) 0 0
\(257\) −1.58199e6 −1.49407 −0.747034 0.664786i \(-0.768523\pi\)
−0.747034 + 0.664786i \(0.768523\pi\)
\(258\) 0 0
\(259\) 1.29863e6i 1.20291i
\(260\) 0 0
\(261\) 16429.3i 0.0149286i
\(262\) 0 0
\(263\) 231519. 0.206394 0.103197 0.994661i \(-0.467093\pi\)
0.103197 + 0.994661i \(0.467093\pi\)
\(264\) 0 0
\(265\) 899303. 0.786668
\(266\) 0 0
\(267\) 658818.i 0.565571i
\(268\) 0 0
\(269\) − 2.34536e6i − 1.97619i −0.153841 0.988096i \(-0.549164\pi\)
0.153841 0.988096i \(-0.450836\pi\)
\(270\) 0 0
\(271\) 398969. 0.330002 0.165001 0.986293i \(-0.447237\pi\)
0.165001 + 0.986293i \(0.447237\pi\)
\(272\) 0 0
\(273\) 2.25624e6 1.83223
\(274\) 0 0
\(275\) 465387.i 0.371092i
\(276\) 0 0
\(277\) − 1.74046e6i − 1.36290i −0.731863 0.681451i \(-0.761349\pi\)
0.731863 0.681451i \(-0.238651\pi\)
\(278\) 0 0
\(279\) 27529.0 0.0211729
\(280\) 0 0
\(281\) −812193. −0.613612 −0.306806 0.951772i \(-0.599260\pi\)
−0.306806 + 0.951772i \(0.599260\pi\)
\(282\) 0 0
\(283\) − 516938.i − 0.383683i −0.981426 0.191841i \(-0.938554\pi\)
0.981426 0.191841i \(-0.0614459\pi\)
\(284\) 0 0
\(285\) − 450373.i − 0.328443i
\(286\) 0 0
\(287\) −1.72672e6 −1.23742
\(288\) 0 0
\(289\) −450688. −0.317418
\(290\) 0 0
\(291\) 179768.i 0.124446i
\(292\) 0 0
\(293\) 1.78836e6i 1.21699i 0.793558 + 0.608495i \(0.208227\pi\)
−0.793558 + 0.608495i \(0.791773\pi\)
\(294\) 0 0
\(295\) −484750. −0.324312
\(296\) 0 0
\(297\) −2.80412e6 −1.84461
\(298\) 0 0
\(299\) − 505250.i − 0.326835i
\(300\) 0 0
\(301\) − 3.14121e6i − 1.99839i
\(302\) 0 0
\(303\) 1.02179e6 0.639376
\(304\) 0 0
\(305\) 945304. 0.581865
\(306\) 0 0
\(307\) − 804925.i − 0.487427i −0.969847 0.243713i \(-0.921634\pi\)
0.969847 0.243713i \(-0.0783656\pi\)
\(308\) 0 0
\(309\) − 744280.i − 0.443445i
\(310\) 0 0
\(311\) −478186. −0.280347 −0.140173 0.990127i \(-0.544766\pi\)
−0.140173 + 0.990127i \(0.544766\pi\)
\(312\) 0 0
\(313\) 3.14478e6 1.81439 0.907194 0.420713i \(-0.138220\pi\)
0.907194 + 0.420713i \(0.138220\pi\)
\(314\) 0 0
\(315\) − 11535.3i − 0.00655018i
\(316\) 0 0
\(317\) − 1.93977e6i − 1.08418i −0.840320 0.542090i \(-0.817633\pi\)
0.840320 0.542090i \(-0.182367\pi\)
\(318\) 0 0
\(319\) −4.36624e6 −2.40232
\(320\) 0 0
\(321\) −1.46484e6 −0.793462
\(322\) 0 0
\(323\) 1.13120e6i 0.603299i
\(324\) 0 0
\(325\) 546173.i 0.286828i
\(326\) 0 0
\(327\) 2.57641e6 1.33243
\(328\) 0 0
\(329\) −489642. −0.249396
\(330\) 0 0
\(331\) − 3.59866e6i − 1.80539i −0.430281 0.902695i \(-0.641586\pi\)
0.430281 0.902695i \(-0.358414\pi\)
\(332\) 0 0
\(333\) 22094.6i 0.0109188i
\(334\) 0 0
\(335\) −284893. −0.138698
\(336\) 0 0
\(337\) 1.58986e6 0.762577 0.381289 0.924456i \(-0.375480\pi\)
0.381289 + 0.924456i \(0.375480\pi\)
\(338\) 0 0
\(339\) − 2.54695e6i − 1.20371i
\(340\) 0 0
\(341\) 7.31608e6i 3.40716i
\(342\) 0 0
\(343\) 1.06946e6 0.490830
\(344\) 0 0
\(345\) −226615. −0.102504
\(346\) 0 0
\(347\) 2.16369e6i 0.964655i 0.875991 + 0.482327i \(0.160208\pi\)
−0.875991 + 0.482327i \(0.839792\pi\)
\(348\) 0 0
\(349\) 403328.i 0.177253i 0.996065 + 0.0886267i \(0.0282478\pi\)
−0.996065 + 0.0886267i \(0.971752\pi\)
\(350\) 0 0
\(351\) −3.29088e6 −1.42575
\(352\) 0 0
\(353\) −74007.2 −0.0316109 −0.0158055 0.999875i \(-0.505031\pi\)
−0.0158055 + 0.999875i \(0.505031\pi\)
\(354\) 0 0
\(355\) − 848304.i − 0.357257i
\(356\) 0 0
\(357\) 2.54177e6i 1.05552i
\(358\) 0 0
\(359\) 4.37764e6 1.79269 0.896343 0.443361i \(-0.146214\pi\)
0.896343 + 0.443361i \(0.146214\pi\)
\(360\) 0 0
\(361\) 1.15578e6 0.466775
\(362\) 0 0
\(363\) 6.16784e6i 2.45678i
\(364\) 0 0
\(365\) − 569866.i − 0.223893i
\(366\) 0 0
\(367\) 547770. 0.212292 0.106146 0.994351i \(-0.466149\pi\)
0.106146 + 0.994351i \(0.466149\pi\)
\(368\) 0 0
\(369\) −29378.1 −0.0112320
\(370\) 0 0
\(371\) − 5.92393e6i − 2.23447i
\(372\) 0 0
\(373\) − 1.33289e6i − 0.496045i −0.968754 0.248023i \(-0.920219\pi\)
0.968754 0.248023i \(-0.0797807\pi\)
\(374\) 0 0
\(375\) 244970. 0.0899569
\(376\) 0 0
\(377\) −5.12417e6 −1.85682
\(378\) 0 0
\(379\) 3.65535e6i 1.30717i 0.756855 + 0.653583i \(0.226735\pi\)
−0.756855 + 0.653583i \(0.773265\pi\)
\(380\) 0 0
\(381\) 1.99395e6i 0.703722i
\(382\) 0 0
\(383\) −3.69148e6 −1.28589 −0.642945 0.765913i \(-0.722287\pi\)
−0.642945 + 0.765913i \(0.722287\pi\)
\(384\) 0 0
\(385\) 3.06561e6 1.05406
\(386\) 0 0
\(387\) − 53444.1i − 0.0181394i
\(388\) 0 0
\(389\) 1.20539e6i 0.403880i 0.979398 + 0.201940i \(0.0647247\pi\)
−0.979398 + 0.201940i \(0.935275\pi\)
\(390\) 0 0
\(391\) 569189. 0.188284
\(392\) 0 0
\(393\) 3.01370e6 0.984281
\(394\) 0 0
\(395\) 2.36885e6i 0.763914i
\(396\) 0 0
\(397\) 3.96565e6i 1.26281i 0.775453 + 0.631406i \(0.217522\pi\)
−0.775453 + 0.631406i \(0.782478\pi\)
\(398\) 0 0
\(399\) −2.96671e6 −0.932918
\(400\) 0 0
\(401\) 2.45029e6 0.760951 0.380475 0.924791i \(-0.375760\pi\)
0.380475 + 0.924791i \(0.375760\pi\)
\(402\) 0 0
\(403\) 8.58607e6i 2.63349i
\(404\) 0 0
\(405\) 1.49305e6i 0.452311i
\(406\) 0 0
\(407\) −5.87184e6 −1.75707
\(408\) 0 0
\(409\) −4.78101e6 −1.41323 −0.706613 0.707600i \(-0.749778\pi\)
−0.706613 + 0.707600i \(0.749778\pi\)
\(410\) 0 0
\(411\) − 5.56540e6i − 1.62514i
\(412\) 0 0
\(413\) 3.19316e6i 0.921183i
\(414\) 0 0
\(415\) −868072. −0.247420
\(416\) 0 0
\(417\) −598983. −0.168684
\(418\) 0 0
\(419\) 1.46113e6i 0.406586i 0.979118 + 0.203293i \(0.0651645\pi\)
−0.979118 + 0.203293i \(0.934836\pi\)
\(420\) 0 0
\(421\) − 5.32662e6i − 1.46469i −0.680932 0.732347i \(-0.738425\pi\)
0.680932 0.732347i \(-0.261575\pi\)
\(422\) 0 0
\(423\) −8330.70 −0.00226376
\(424\) 0 0
\(425\) −615290. −0.165237
\(426\) 0 0
\(427\) − 6.22695e6i − 1.65274i
\(428\) 0 0
\(429\) 1.02018e7i 2.67629i
\(430\) 0 0
\(431\) 363772. 0.0943270 0.0471635 0.998887i \(-0.484982\pi\)
0.0471635 + 0.998887i \(0.484982\pi\)
\(432\) 0 0
\(433\) −4.45960e6 −1.14308 −0.571540 0.820574i \(-0.693654\pi\)
−0.571540 + 0.820574i \(0.693654\pi\)
\(434\) 0 0
\(435\) 2.29830e6i 0.582348i
\(436\) 0 0
\(437\) 664349.i 0.166415i
\(438\) 0 0
\(439\) 5.13055e6 1.27058 0.635290 0.772273i \(-0.280880\pi\)
0.635290 + 0.772273i \(0.280880\pi\)
\(440\) 0 0
\(441\) −28895.1 −0.00707503
\(442\) 0 0
\(443\) − 320762.i − 0.0776557i −0.999246 0.0388279i \(-0.987638\pi\)
0.999246 0.0388279i \(-0.0123624\pi\)
\(444\) 0 0
\(445\) 1.05054e6i 0.251485i
\(446\) 0 0
\(447\) −2.12126e6 −0.502141
\(448\) 0 0
\(449\) 557835. 0.130584 0.0652920 0.997866i \(-0.479202\pi\)
0.0652920 + 0.997866i \(0.479202\pi\)
\(450\) 0 0
\(451\) − 7.80748e6i − 1.80746i
\(452\) 0 0
\(453\) 3.43931e6i 0.787456i
\(454\) 0 0
\(455\) 3.59777e6 0.814714
\(456\) 0 0
\(457\) −2.78896e6 −0.624672 −0.312336 0.949972i \(-0.601112\pi\)
−0.312336 + 0.949972i \(0.601112\pi\)
\(458\) 0 0
\(459\) − 3.70734e6i − 0.821354i
\(460\) 0 0
\(461\) − 733222.i − 0.160688i −0.996767 0.0803439i \(-0.974398\pi\)
0.996767 0.0803439i \(-0.0256019\pi\)
\(462\) 0 0
\(463\) −6.80938e6 −1.47623 −0.738117 0.674672i \(-0.764285\pi\)
−0.738117 + 0.674672i \(0.764285\pi\)
\(464\) 0 0
\(465\) 3.85103e6 0.825933
\(466\) 0 0
\(467\) 3.33064e6i 0.706701i 0.935491 + 0.353351i \(0.114958\pi\)
−0.935491 + 0.353351i \(0.885042\pi\)
\(468\) 0 0
\(469\) 1.87666e6i 0.393960i
\(470\) 0 0
\(471\) −1.56667e6 −0.325406
\(472\) 0 0
\(473\) 1.42033e7 2.91901
\(474\) 0 0
\(475\) − 718157.i − 0.146045i
\(476\) 0 0
\(477\) − 100789.i − 0.0202822i
\(478\) 0 0
\(479\) −6.58735e6 −1.31181 −0.655907 0.754842i \(-0.727713\pi\)
−0.655907 + 0.754842i \(0.727713\pi\)
\(480\) 0 0
\(481\) −6.89113e6 −1.35809
\(482\) 0 0
\(483\) 1.49277e6i 0.291156i
\(484\) 0 0
\(485\) 286655.i 0.0553357i
\(486\) 0 0
\(487\) 3.26167e6 0.623186 0.311593 0.950216i \(-0.399137\pi\)
0.311593 + 0.950216i \(0.399137\pi\)
\(488\) 0 0
\(489\) 6.05982e6 1.14601
\(490\) 0 0
\(491\) − 5.26941e6i − 0.986412i −0.869912 0.493206i \(-0.835825\pi\)
0.869912 0.493206i \(-0.164175\pi\)
\(492\) 0 0
\(493\) − 5.77262e6i − 1.06968i
\(494\) 0 0
\(495\) 52157.9 0.00956768
\(496\) 0 0
\(497\) −5.58798e6 −1.01476
\(498\) 0 0
\(499\) 4.56847e6i 0.821333i 0.911786 + 0.410666i \(0.134704\pi\)
−0.911786 + 0.410666i \(0.865296\pi\)
\(500\) 0 0
\(501\) − 4.98476e6i − 0.887259i
\(502\) 0 0
\(503\) −6.34743e6 −1.11861 −0.559304 0.828962i \(-0.688932\pi\)
−0.559304 + 0.828962i \(0.688932\pi\)
\(504\) 0 0
\(505\) 1.62933e6 0.284303
\(506\) 0 0
\(507\) 6.15155e6i 1.06283i
\(508\) 0 0
\(509\) − 5.80648e6i − 0.993388i −0.867926 0.496694i \(-0.834547\pi\)
0.867926 0.496694i \(-0.165453\pi\)
\(510\) 0 0
\(511\) −3.75384e6 −0.635951
\(512\) 0 0
\(513\) 4.32715e6 0.725953
\(514\) 0 0
\(515\) − 1.18682e6i − 0.197181i
\(516\) 0 0
\(517\) − 2.21396e6i − 0.364286i
\(518\) 0 0
\(519\) −2.22686e6 −0.362889
\(520\) 0 0
\(521\) −9.75153e6 −1.57390 −0.786952 0.617014i \(-0.788342\pi\)
−0.786952 + 0.617014i \(0.788342\pi\)
\(522\) 0 0
\(523\) 8.86264e6i 1.41680i 0.705811 + 0.708401i \(0.250583\pi\)
−0.705811 + 0.708401i \(0.749417\pi\)
\(524\) 0 0
\(525\) − 1.61368e6i − 0.255516i
\(526\) 0 0
\(527\) −9.67262e6 −1.51711
\(528\) 0 0
\(529\) −6.10206e6 −0.948063
\(530\) 0 0
\(531\) 54328.0i 0.00836156i
\(532\) 0 0
\(533\) − 9.16277e6i − 1.39704i
\(534\) 0 0
\(535\) −2.33580e6 −0.352819
\(536\) 0 0
\(537\) −7.23188e6 −1.08222
\(538\) 0 0
\(539\) − 7.67914e6i − 1.13852i
\(540\) 0 0
\(541\) 7.43686e6i 1.09244i 0.837643 + 0.546218i \(0.183933\pi\)
−0.837643 + 0.546218i \(0.816067\pi\)
\(542\) 0 0
\(543\) 1.84239e6 0.268153
\(544\) 0 0
\(545\) 4.10830e6 0.592476
\(546\) 0 0
\(547\) − 6.65237e6i − 0.950623i −0.879818 0.475311i \(-0.842335\pi\)
0.879818 0.475311i \(-0.157665\pi\)
\(548\) 0 0
\(549\) − 105944.i − 0.0150019i
\(550\) 0 0
\(551\) 6.73772e6 0.945440
\(552\) 0 0
\(553\) 1.56042e7 2.16984
\(554\) 0 0
\(555\) 3.09082e6i 0.425932i
\(556\) 0 0
\(557\) − 1.02580e6i − 0.140095i −0.997544 0.0700477i \(-0.977685\pi\)
0.997544 0.0700477i \(-0.0223152\pi\)
\(558\) 0 0
\(559\) 1.66688e7 2.25618
\(560\) 0 0
\(561\) −1.14928e7 −1.54177
\(562\) 0 0
\(563\) 3.06816e6i 0.407950i 0.978976 + 0.203975i \(0.0653861\pi\)
−0.978976 + 0.203975i \(0.934614\pi\)
\(564\) 0 0
\(565\) − 4.06133e6i − 0.535239i
\(566\) 0 0
\(567\) 9.83508e6 1.28475
\(568\) 0 0
\(569\) −1.05690e7 −1.36853 −0.684263 0.729236i \(-0.739876\pi\)
−0.684263 + 0.729236i \(0.739876\pi\)
\(570\) 0 0
\(571\) 1.22219e7i 1.56873i 0.620298 + 0.784366i \(0.287012\pi\)
−0.620298 + 0.784366i \(0.712988\pi\)
\(572\) 0 0
\(573\) 3.55298e6i 0.452070i
\(574\) 0 0
\(575\) −361357. −0.0455792
\(576\) 0 0
\(577\) 3.08742e6 0.386061 0.193031 0.981193i \(-0.438168\pi\)
0.193031 + 0.981193i \(0.438168\pi\)
\(578\) 0 0
\(579\) 335573.i 0.0415998i
\(580\) 0 0
\(581\) 5.71820e6i 0.702780i
\(582\) 0 0
\(583\) 2.67855e7 3.26384
\(584\) 0 0
\(585\) 61211.9 0.00739514
\(586\) 0 0
\(587\) − 7.45584e6i − 0.893103i −0.894758 0.446551i \(-0.852652\pi\)
0.894758 0.446551i \(-0.147348\pi\)
\(588\) 0 0
\(589\) − 1.12897e7i − 1.34090i
\(590\) 0 0
\(591\) −1.10918e7 −1.30627
\(592\) 0 0
\(593\) 1.10599e7 1.29156 0.645780 0.763523i \(-0.276532\pi\)
0.645780 + 0.763523i \(0.276532\pi\)
\(594\) 0 0
\(595\) 4.05306e6i 0.469344i
\(596\) 0 0
\(597\) − 6.07669e6i − 0.697800i
\(598\) 0 0
\(599\) −1.08444e7 −1.23492 −0.617460 0.786602i \(-0.711838\pi\)
−0.617460 + 0.786602i \(0.711838\pi\)
\(600\) 0 0
\(601\) 2.41110e6 0.272289 0.136144 0.990689i \(-0.456529\pi\)
0.136144 + 0.990689i \(0.456529\pi\)
\(602\) 0 0
\(603\) 31929.1i 0.00357597i
\(604\) 0 0
\(605\) 9.83515e6i 1.09243i
\(606\) 0 0
\(607\) 1.25003e7 1.37704 0.688521 0.725216i \(-0.258260\pi\)
0.688521 + 0.725216i \(0.258260\pi\)
\(608\) 0 0
\(609\) 1.51394e7 1.65412
\(610\) 0 0
\(611\) − 2.59828e6i − 0.281567i
\(612\) 0 0
\(613\) 4.22386e6i 0.454002i 0.973894 + 0.227001i \(0.0728921\pi\)
−0.973894 + 0.227001i \(0.927108\pi\)
\(614\) 0 0
\(615\) −4.10970e6 −0.438149
\(616\) 0 0
\(617\) 1.02123e7 1.07997 0.539984 0.841675i \(-0.318430\pi\)
0.539984 + 0.841675i \(0.318430\pi\)
\(618\) 0 0
\(619\) 458202.i 0.0480651i 0.999711 + 0.0240326i \(0.00765054\pi\)
−0.999711 + 0.0240326i \(0.992349\pi\)
\(620\) 0 0
\(621\) − 2.17730e6i − 0.226563i
\(622\) 0 0
\(623\) 6.92016e6 0.714325
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) − 1.34142e7i − 1.36269i
\(628\) 0 0
\(629\) − 7.76319e6i − 0.782373i
\(630\) 0 0
\(631\) 1.10444e7 1.10426 0.552128 0.833760i \(-0.313816\pi\)
0.552128 + 0.833760i \(0.313816\pi\)
\(632\) 0 0
\(633\) −3.28647e6 −0.326002
\(634\) 0 0
\(635\) 3.17952e6i 0.312915i
\(636\) 0 0
\(637\) − 9.01215e6i − 0.879994i
\(638\) 0 0
\(639\) −95073.1 −0.00921097
\(640\) 0 0
\(641\) −1.64212e7 −1.57856 −0.789278 0.614036i \(-0.789545\pi\)
−0.789278 + 0.614036i \(0.789545\pi\)
\(642\) 0 0
\(643\) 6.96664e6i 0.664501i 0.943191 + 0.332251i \(0.107808\pi\)
−0.943191 + 0.332251i \(0.892192\pi\)
\(644\) 0 0
\(645\) − 7.47630e6i − 0.707599i
\(646\) 0 0
\(647\) 1.37322e7 1.28967 0.644834 0.764322i \(-0.276926\pi\)
0.644834 + 0.764322i \(0.276926\pi\)
\(648\) 0 0
\(649\) −1.44381e7 −1.34555
\(650\) 0 0
\(651\) − 2.53677e7i − 2.34600i
\(652\) 0 0
\(653\) 1.32515e7i 1.21613i 0.793886 + 0.608067i \(0.208055\pi\)
−0.793886 + 0.608067i \(0.791945\pi\)
\(654\) 0 0
\(655\) 4.80560e6 0.437668
\(656\) 0 0
\(657\) −63867.3 −0.00577251
\(658\) 0 0
\(659\) − 1.01216e6i − 0.0907892i −0.998969 0.0453946i \(-0.985545\pi\)
0.998969 0.0453946i \(-0.0144545\pi\)
\(660\) 0 0
\(661\) − 2.17387e7i − 1.93522i −0.252447 0.967611i \(-0.581235\pi\)
0.252447 0.967611i \(-0.418765\pi\)
\(662\) 0 0
\(663\) −1.34878e7 −1.19168
\(664\) 0 0
\(665\) −4.73067e6 −0.414829
\(666\) 0 0
\(667\) − 3.39024e6i − 0.295063i
\(668\) 0 0
\(669\) − 7.12489e6i − 0.615478i
\(670\) 0 0
\(671\) 2.81556e7 2.41412
\(672\) 0 0
\(673\) −7.30341e6 −0.621567 −0.310783 0.950481i \(-0.600591\pi\)
−0.310783 + 0.950481i \(0.600591\pi\)
\(674\) 0 0
\(675\) 2.35365e6i 0.198830i
\(676\) 0 0
\(677\) 8.35855e6i 0.700905i 0.936580 + 0.350453i \(0.113972\pi\)
−0.936580 + 0.350453i \(0.886028\pi\)
\(678\) 0 0
\(679\) 1.88827e6 0.157177
\(680\) 0 0
\(681\) −8.93558e6 −0.738337
\(682\) 0 0
\(683\) 1.85500e7i 1.52157i 0.649004 + 0.760785i \(0.275186\pi\)
−0.649004 + 0.760785i \(0.724814\pi\)
\(684\) 0 0
\(685\) − 8.87450e6i − 0.722633i
\(686\) 0 0
\(687\) −5.01020e6 −0.405007
\(688\) 0 0
\(689\) 3.14352e7 2.52271
\(690\) 0 0
\(691\) − 858046.i − 0.0683621i −0.999416 0.0341810i \(-0.989118\pi\)
0.999416 0.0341810i \(-0.0108823\pi\)
\(692\) 0 0
\(693\) − 343576.i − 0.0271763i
\(694\) 0 0
\(695\) −955128. −0.0750066
\(696\) 0 0
\(697\) 1.03223e7 0.804813
\(698\) 0 0
\(699\) − 4.90871e6i − 0.379992i
\(700\) 0 0
\(701\) 2.28710e7i 1.75788i 0.476928 + 0.878942i \(0.341750\pi\)
−0.476928 + 0.878942i \(0.658250\pi\)
\(702\) 0 0
\(703\) 9.06108e6 0.691499
\(704\) 0 0
\(705\) −1.16538e6 −0.0883070
\(706\) 0 0
\(707\) − 1.07328e7i − 0.807543i
\(708\) 0 0
\(709\) − 1.64348e6i − 0.122786i −0.998114 0.0613930i \(-0.980446\pi\)
0.998114 0.0613930i \(-0.0195543\pi\)
\(710\) 0 0
\(711\) 265487. 0.0196956
\(712\) 0 0
\(713\) −5.68069e6 −0.418483
\(714\) 0 0
\(715\) 1.62676e7i 1.19003i
\(716\) 0 0
\(717\) − 2.03063e6i − 0.147514i
\(718\) 0 0
\(719\) 1.22486e6 0.0883614 0.0441807 0.999024i \(-0.485932\pi\)
0.0441807 + 0.999024i \(0.485932\pi\)
\(720\) 0 0
\(721\) −7.81785e6 −0.560079
\(722\) 0 0
\(723\) 2.52497e7i 1.79643i
\(724\) 0 0
\(725\) 3.66483e6i 0.258945i
\(726\) 0 0
\(727\) −1.07644e7 −0.755362 −0.377681 0.925936i \(-0.623278\pi\)
−0.377681 + 0.925936i \(0.623278\pi\)
\(728\) 0 0
\(729\) −1.41797e7 −0.988205
\(730\) 0 0
\(731\) 1.87782e7i 1.29975i
\(732\) 0 0
\(733\) − 2.45041e7i − 1.68453i −0.539065 0.842264i \(-0.681222\pi\)
0.539065 0.842264i \(-0.318778\pi\)
\(734\) 0 0
\(735\) −4.04214e6 −0.275990
\(736\) 0 0
\(737\) −8.48545e6 −0.575448
\(738\) 0 0
\(739\) − 1.80906e7i − 1.21855i −0.792960 0.609274i \(-0.791461\pi\)
0.792960 0.609274i \(-0.208539\pi\)
\(740\) 0 0
\(741\) − 1.57428e7i − 1.05326i
\(742\) 0 0
\(743\) 7.48284e6 0.497272 0.248636 0.968597i \(-0.420018\pi\)
0.248636 + 0.968597i \(0.420018\pi\)
\(744\) 0 0
\(745\) −3.38253e6 −0.223281
\(746\) 0 0
\(747\) 97288.6i 0.00637911i
\(748\) 0 0
\(749\) 1.53865e7i 1.00216i
\(750\) 0 0
\(751\) −2.46162e6 −0.159265 −0.0796326 0.996824i \(-0.525375\pi\)
−0.0796326 + 0.996824i \(0.525375\pi\)
\(752\) 0 0
\(753\) −3.07374e6 −0.197551
\(754\) 0 0
\(755\) 5.48427e6i 0.350148i
\(756\) 0 0
\(757\) 1.46510e7i 0.929239i 0.885511 + 0.464619i \(0.153809\pi\)
−0.885511 + 0.464619i \(0.846191\pi\)
\(758\) 0 0
\(759\) −6.74968e6 −0.425284
\(760\) 0 0
\(761\) 6.15721e6 0.385409 0.192705 0.981257i \(-0.438274\pi\)
0.192705 + 0.981257i \(0.438274\pi\)
\(762\) 0 0
\(763\) − 2.70624e7i − 1.68288i
\(764\) 0 0
\(765\) 68958.2i 0.00426022i
\(766\) 0 0
\(767\) −1.69445e7 −1.04001
\(768\) 0 0
\(769\) 2.10066e7 1.28097 0.640485 0.767970i \(-0.278733\pi\)
0.640485 + 0.767970i \(0.278733\pi\)
\(770\) 0 0
\(771\) 2.48025e7i 1.50266i
\(772\) 0 0
\(773\) − 3.38948e6i − 0.204025i −0.994783 0.102013i \(-0.967472\pi\)
0.994783 0.102013i \(-0.0325282\pi\)
\(774\) 0 0
\(775\) 6.14079e6 0.367257
\(776\) 0 0
\(777\) 2.03600e7 1.20983
\(778\) 0 0
\(779\) 1.20480e7i 0.711333i
\(780\) 0 0
\(781\) − 2.52665e7i − 1.48224i
\(782\) 0 0
\(783\) −2.20819e7 −1.28716
\(784\) 0 0
\(785\) −2.49819e6 −0.144694
\(786\) 0 0
\(787\) − 1.56364e7i − 0.899914i −0.893050 0.449957i \(-0.851439\pi\)
0.893050 0.449957i \(-0.148561\pi\)
\(788\) 0 0
\(789\) − 3.62977e6i − 0.207580i
\(790\) 0 0
\(791\) −2.67530e7 −1.52031
\(792\) 0 0
\(793\) 3.30432e7 1.86594
\(794\) 0 0
\(795\) − 1.40993e7i − 0.791190i
\(796\) 0 0
\(797\) − 2.17344e7i − 1.21200i −0.795466 0.605998i \(-0.792774\pi\)
0.795466 0.605998i \(-0.207226\pi\)
\(798\) 0 0
\(799\) 2.92708e6 0.162206
\(800\) 0 0
\(801\) 117739. 0.00648391
\(802\) 0 0
\(803\) − 1.69733e7i − 0.928918i
\(804\) 0 0
\(805\) 2.38035e6i 0.129464i
\(806\) 0 0
\(807\) −3.67707e7 −1.98755
\(808\) 0 0
\(809\) −3.12327e7 −1.67779 −0.838897 0.544290i \(-0.816799\pi\)
−0.838897 + 0.544290i \(0.816799\pi\)
\(810\) 0 0
\(811\) − 1.82163e7i − 0.972542i −0.873808 0.486271i \(-0.838357\pi\)
0.873808 0.486271i \(-0.161643\pi\)
\(812\) 0 0
\(813\) − 6.25507e6i − 0.331899i
\(814\) 0 0
\(815\) 9.66290e6 0.509581
\(816\) 0 0
\(817\) −2.19176e7 −1.14878
\(818\) 0 0
\(819\) − 403218.i − 0.0210053i
\(820\) 0 0
\(821\) 1.38146e7i 0.715286i 0.933858 + 0.357643i \(0.116420\pi\)
−0.933858 + 0.357643i \(0.883580\pi\)
\(822\) 0 0
\(823\) −1.00873e7 −0.519131 −0.259566 0.965725i \(-0.583579\pi\)
−0.259566 + 0.965725i \(0.583579\pi\)
\(824\) 0 0
\(825\) 7.29636e6 0.373226
\(826\) 0 0
\(827\) 1.12662e7i 0.572813i 0.958108 + 0.286406i \(0.0924607\pi\)
−0.958108 + 0.286406i \(0.907539\pi\)
\(828\) 0 0
\(829\) − 2.16824e6i − 0.109577i −0.998498 0.0547886i \(-0.982552\pi\)
0.998498 0.0547886i \(-0.0174485\pi\)
\(830\) 0 0
\(831\) −2.72871e7 −1.37074
\(832\) 0 0
\(833\) 1.01526e7 0.506951
\(834\) 0 0
\(835\) − 7.94862e6i − 0.394526i
\(836\) 0 0
\(837\) 3.70004e7i 1.82555i
\(838\) 0 0
\(839\) −1.39434e7 −0.683856 −0.341928 0.939726i \(-0.611080\pi\)
−0.341928 + 0.939726i \(0.611080\pi\)
\(840\) 0 0
\(841\) −1.38721e7 −0.676319
\(842\) 0 0
\(843\) 1.27336e7i 0.617139i
\(844\) 0 0
\(845\) 9.80917e6i 0.472596i
\(846\) 0 0
\(847\) 6.47865e7 3.10296
\(848\) 0 0
\(849\) −8.10459e6 −0.385889
\(850\) 0 0
\(851\) − 4.55929e6i − 0.215811i
\(852\) 0 0
\(853\) 2.37458e7i 1.11741i 0.829365 + 0.558707i \(0.188702\pi\)
−0.829365 + 0.558707i \(0.811298\pi\)
\(854\) 0 0
\(855\) −80486.9 −0.00376539
\(856\) 0 0
\(857\) 2.77323e7 1.28983 0.644917 0.764253i \(-0.276892\pi\)
0.644917 + 0.764253i \(0.276892\pi\)
\(858\) 0 0
\(859\) − 1.63771e7i − 0.757276i −0.925545 0.378638i \(-0.876392\pi\)
0.925545 0.378638i \(-0.123608\pi\)
\(860\) 0 0
\(861\) 2.70716e7i 1.24453i
\(862\) 0 0
\(863\) −6.12859e6 −0.280113 −0.140057 0.990143i \(-0.544728\pi\)
−0.140057 + 0.990143i \(0.544728\pi\)
\(864\) 0 0
\(865\) −3.55091e6 −0.161361
\(866\) 0 0
\(867\) 7.06593e6i 0.319243i
\(868\) 0 0
\(869\) 7.05555e7i 3.16943i
\(870\) 0 0
\(871\) −9.95843e6 −0.444780
\(872\) 0 0
\(873\) 32126.7 0.00142669
\(874\) 0 0
\(875\) − 2.57314e6i − 0.113617i
\(876\) 0 0
\(877\) − 2.27843e7i − 1.00031i −0.865935 0.500157i \(-0.833276\pi\)
0.865935 0.500157i \(-0.166724\pi\)
\(878\) 0 0
\(879\) 2.80381e7 1.22399
\(880\) 0 0
\(881\) −1.49593e7 −0.649339 −0.324669 0.945828i \(-0.605253\pi\)
−0.324669 + 0.945828i \(0.605253\pi\)
\(882\) 0 0
\(883\) − 2.85756e7i − 1.23337i −0.787211 0.616684i \(-0.788476\pi\)
0.787211 0.616684i \(-0.211524\pi\)
\(884\) 0 0
\(885\) 7.59994e6i 0.326176i
\(886\) 0 0
\(887\) −2.05017e7 −0.874944 −0.437472 0.899232i \(-0.644126\pi\)
−0.437472 + 0.899232i \(0.644126\pi\)
\(888\) 0 0
\(889\) 2.09442e7 0.888812
\(890\) 0 0
\(891\) 4.44701e7i 1.87661i
\(892\) 0 0
\(893\) 3.41645e6i 0.143366i
\(894\) 0 0
\(895\) −1.15318e7 −0.481217
\(896\) 0 0
\(897\) −7.92135e6 −0.328714
\(898\) 0 0
\(899\) 5.76127e7i 2.37749i
\(900\) 0 0
\(901\) 3.54132e7i 1.45329i
\(902\) 0 0
\(903\) −4.92482e7 −2.00988
\(904\) 0 0
\(905\) 2.93785e6 0.119236
\(906\) 0 0
\(907\) 4.27147e7i 1.72409i 0.506835 + 0.862043i \(0.330815\pi\)
−0.506835 + 0.862043i \(0.669185\pi\)
\(908\) 0 0
\(909\) − 182607.i − 0.00733004i
\(910\) 0 0
\(911\) 2.16615e7 0.864753 0.432376 0.901693i \(-0.357675\pi\)
0.432376 + 0.901693i \(0.357675\pi\)
\(912\) 0 0
\(913\) −2.58553e7 −1.02653
\(914\) 0 0
\(915\) − 1.48205e7i − 0.585210i
\(916\) 0 0
\(917\) − 3.16557e7i − 1.24316i
\(918\) 0 0
\(919\) 4.49934e7 1.75736 0.878679 0.477413i \(-0.158425\pi\)
0.878679 + 0.477413i \(0.158425\pi\)
\(920\) 0 0
\(921\) −1.26197e7 −0.490229
\(922\) 0 0
\(923\) − 2.96525e7i − 1.14566i
\(924\) 0 0
\(925\) 4.92857e6i 0.189394i
\(926\) 0 0
\(927\) −133012. −0.00508382
\(928\) 0 0
\(929\) −1.71344e7 −0.651373 −0.325686 0.945478i \(-0.605595\pi\)
−0.325686 + 0.945478i \(0.605595\pi\)
\(930\) 0 0
\(931\) 1.18500e7i 0.448068i
\(932\) 0 0
\(933\) 7.49703e6i 0.281959i
\(934\) 0 0
\(935\) −1.83262e7 −0.685558
\(936\) 0 0
\(937\) −4.36298e7 −1.62343 −0.811715 0.584053i \(-0.801466\pi\)
−0.811715 + 0.584053i \(0.801466\pi\)
\(938\) 0 0
\(939\) − 4.93041e7i − 1.82482i
\(940\) 0 0
\(941\) 930214.i 0.0342459i 0.999853 + 0.0171230i \(0.00545067\pi\)
−0.999853 + 0.0171230i \(0.994549\pi\)
\(942\) 0 0
\(943\) 6.06225e6 0.222001
\(944\) 0 0
\(945\) 1.55041e7 0.564763
\(946\) 0 0
\(947\) − 2.95330e7i − 1.07012i −0.844813 0.535061i \(-0.820289\pi\)
0.844813 0.535061i \(-0.179711\pi\)
\(948\) 0 0
\(949\) − 1.99197e7i − 0.717988i
\(950\) 0 0
\(951\) −3.04118e7 −1.09041
\(952\) 0 0
\(953\) −2.63699e7 −0.940538 −0.470269 0.882523i \(-0.655843\pi\)
−0.470269 + 0.882523i \(0.655843\pi\)
\(954\) 0 0
\(955\) 5.66552e6i 0.201016i
\(956\) 0 0
\(957\) 6.84542e7i 2.41613i
\(958\) 0 0
\(959\) −5.84585e7 −2.05258
\(960\) 0 0
\(961\) 6.79068e7 2.37195
\(962\) 0 0
\(963\) 261783.i 0.00909655i
\(964\) 0 0
\(965\) 535100.i 0.0184977i
\(966\) 0 0
\(967\) 338149. 0.0116290 0.00581449 0.999983i \(-0.498149\pi\)
0.00581449 + 0.999983i \(0.498149\pi\)
\(968\) 0 0
\(969\) 1.77350e7 0.606767
\(970\) 0 0
\(971\) − 2.00903e7i − 0.683816i −0.939734 0.341908i \(-0.888927\pi\)
0.939734 0.341908i \(-0.111073\pi\)
\(972\) 0 0
\(973\) 6.29166e6i 0.213051i
\(974\) 0 0
\(975\) 8.56293e6 0.288477
\(976\) 0 0
\(977\) 1.04075e7 0.348826 0.174413 0.984673i \(-0.444197\pi\)
0.174413 + 0.984673i \(0.444197\pi\)
\(978\) 0 0
\(979\) 3.12901e7i 1.04340i
\(980\) 0 0
\(981\) − 460435.i − 0.0152755i
\(982\) 0 0
\(983\) −2.39326e6 −0.0789963 −0.0394982 0.999220i \(-0.512576\pi\)
−0.0394982 + 0.999220i \(0.512576\pi\)
\(984\) 0 0
\(985\) −1.76868e7 −0.580842
\(986\) 0 0
\(987\) 7.67665e6i 0.250830i
\(988\) 0 0
\(989\) 1.10284e7i 0.358525i
\(990\) 0 0
\(991\) 4.23674e7 1.37040 0.685200 0.728355i \(-0.259715\pi\)
0.685200 + 0.728355i \(0.259715\pi\)
\(992\) 0 0
\(993\) −5.64201e7 −1.81577
\(994\) 0 0
\(995\) − 9.68979e6i − 0.310282i
\(996\) 0 0
\(997\) 2.35670e6i 0.0750873i 0.999295 + 0.0375436i \(0.0119533\pi\)
−0.999295 + 0.0375436i \(0.988047\pi\)
\(998\) 0 0
\(999\) −2.96963e7 −0.941432
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 320.6.d.b.161.3 yes 8
4.3 odd 2 320.6.d.a.161.6 yes 8
8.3 odd 2 320.6.d.a.161.3 8
8.5 even 2 inner 320.6.d.b.161.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.a.161.3 8 8.3 odd 2
320.6.d.a.161.6 yes 8 4.3 odd 2
320.6.d.b.161.3 yes 8 1.1 even 1 trivial
320.6.d.b.161.6 yes 8 8.5 even 2 inner