Properties

Label 387.6.a.c.1.4
Level $387$
Weight $6$
Character 387.1
Self dual yes
Analytic conductor $62.069$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.0685382676\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.58275\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.582753 q^{2} -31.6604 q^{4} -27.7074 q^{5} -103.690 q^{7} +37.0983 q^{8} +O(q^{10})\) \(q-0.582753 q^{2} -31.6604 q^{4} -27.7074 q^{5} -103.690 q^{7} +37.0983 q^{8} +16.1466 q^{10} +158.323 q^{11} -578.882 q^{13} +60.4256 q^{14} +991.514 q^{16} -253.871 q^{17} -3092.51 q^{19} +877.227 q^{20} -92.2633 q^{22} -4163.45 q^{23} -2357.30 q^{25} +337.345 q^{26} +3282.87 q^{28} -6771.63 q^{29} +6264.06 q^{31} -1764.95 q^{32} +147.944 q^{34} +2872.98 q^{35} -3294.82 q^{37} +1802.17 q^{38} -1027.90 q^{40} +6150.84 q^{41} -1849.00 q^{43} -5012.58 q^{44} +2426.26 q^{46} -8157.17 q^{47} -6055.38 q^{49} +1373.72 q^{50} +18327.6 q^{52} +30457.8 q^{53} -4386.72 q^{55} -3846.72 q^{56} +3946.19 q^{58} +45236.1 q^{59} -7251.18 q^{61} -3650.40 q^{62} -30699.9 q^{64} +16039.3 q^{65} +19685.7 q^{67} +8037.67 q^{68} -1674.24 q^{70} +48132.2 q^{71} +42502.1 q^{73} +1920.07 q^{74} +97910.0 q^{76} -16416.5 q^{77} -65151.9 q^{79} -27472.3 q^{80} -3584.42 q^{82} -70403.6 q^{83} +7034.12 q^{85} +1077.51 q^{86} +5873.52 q^{88} -29645.1 q^{89} +60024.2 q^{91} +131816. q^{92} +4753.61 q^{94} +85685.3 q^{95} -89440.4 q^{97} +3528.79 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.582753 −0.103017 −0.0515085 0.998673i \(-0.516403\pi\)
−0.0515085 + 0.998673i \(0.516403\pi\)
\(3\) 0 0
\(4\) −31.6604 −0.989387
\(5\) −27.7074 −0.495645 −0.247822 0.968805i \(-0.579715\pi\)
−0.247822 + 0.968805i \(0.579715\pi\)
\(6\) 0 0
\(7\) −103.690 −0.799819 −0.399910 0.916555i \(-0.630958\pi\)
−0.399910 + 0.916555i \(0.630958\pi\)
\(8\) 37.0983 0.204941
\(9\) 0 0
\(10\) 16.1466 0.0510599
\(11\) 158.323 0.394515 0.197257 0.980352i \(-0.436797\pi\)
0.197257 + 0.980352i \(0.436797\pi\)
\(12\) 0 0
\(13\) −578.882 −0.950017 −0.475009 0.879981i \(-0.657555\pi\)
−0.475009 + 0.879981i \(0.657555\pi\)
\(14\) 60.4256 0.0823950
\(15\) 0 0
\(16\) 991.514 0.968275
\(17\) −253.871 −0.213055 −0.106527 0.994310i \(-0.533973\pi\)
−0.106527 + 0.994310i \(0.533973\pi\)
\(18\) 0 0
\(19\) −3092.51 −1.96529 −0.982645 0.185494i \(-0.940612\pi\)
−0.982645 + 0.185494i \(0.940612\pi\)
\(20\) 877.227 0.490385
\(21\) 0 0
\(22\) −92.2633 −0.0406417
\(23\) −4163.45 −1.64109 −0.820547 0.571579i \(-0.806331\pi\)
−0.820547 + 0.571579i \(0.806331\pi\)
\(24\) 0 0
\(25\) −2357.30 −0.754336
\(26\) 337.345 0.0978680
\(27\) 0 0
\(28\) 3282.87 0.791331
\(29\) −6771.63 −1.49520 −0.747598 0.664151i \(-0.768793\pi\)
−0.747598 + 0.664151i \(0.768793\pi\)
\(30\) 0 0
\(31\) 6264.06 1.17072 0.585358 0.810775i \(-0.300954\pi\)
0.585358 + 0.810775i \(0.300954\pi\)
\(32\) −1764.95 −0.304690
\(33\) 0 0
\(34\) 147.944 0.0219483
\(35\) 2872.98 0.396426
\(36\) 0 0
\(37\) −3294.82 −0.395665 −0.197833 0.980236i \(-0.563390\pi\)
−0.197833 + 0.980236i \(0.563390\pi\)
\(38\) 1802.17 0.202459
\(39\) 0 0
\(40\) −1027.90 −0.101578
\(41\) 6150.84 0.571445 0.285723 0.958312i \(-0.407766\pi\)
0.285723 + 0.958312i \(0.407766\pi\)
\(42\) 0 0
\(43\) −1849.00 −0.152499
\(44\) −5012.58 −0.390328
\(45\) 0 0
\(46\) 2426.26 0.169061
\(47\) −8157.17 −0.538635 −0.269318 0.963051i \(-0.586798\pi\)
−0.269318 + 0.963051i \(0.586798\pi\)
\(48\) 0 0
\(49\) −6055.38 −0.360289
\(50\) 1373.72 0.0777095
\(51\) 0 0
\(52\) 18327.6 0.939935
\(53\) 30457.8 1.48939 0.744697 0.667403i \(-0.232594\pi\)
0.744697 + 0.667403i \(0.232594\pi\)
\(54\) 0 0
\(55\) −4386.72 −0.195539
\(56\) −3846.72 −0.163916
\(57\) 0 0
\(58\) 3946.19 0.154031
\(59\) 45236.1 1.69182 0.845912 0.533322i \(-0.179057\pi\)
0.845912 + 0.533322i \(0.179057\pi\)
\(60\) 0 0
\(61\) −7251.18 −0.249508 −0.124754 0.992188i \(-0.539814\pi\)
−0.124754 + 0.992188i \(0.539814\pi\)
\(62\) −3650.40 −0.120604
\(63\) 0 0
\(64\) −30699.9 −0.936887
\(65\) 16039.3 0.470871
\(66\) 0 0
\(67\) 19685.7 0.535751 0.267876 0.963454i \(-0.413678\pi\)
0.267876 + 0.963454i \(0.413678\pi\)
\(68\) 8037.67 0.210794
\(69\) 0 0
\(70\) −1674.24 −0.0408387
\(71\) 48132.2 1.13316 0.566579 0.824008i \(-0.308267\pi\)
0.566579 + 0.824008i \(0.308267\pi\)
\(72\) 0 0
\(73\) 42502.1 0.933475 0.466738 0.884396i \(-0.345429\pi\)
0.466738 + 0.884396i \(0.345429\pi\)
\(74\) 1920.07 0.0407603
\(75\) 0 0
\(76\) 97910.0 1.94443
\(77\) −16416.5 −0.315540
\(78\) 0 0
\(79\) −65151.9 −1.17452 −0.587258 0.809400i \(-0.699793\pi\)
−0.587258 + 0.809400i \(0.699793\pi\)
\(80\) −27472.3 −0.479921
\(81\) 0 0
\(82\) −3584.42 −0.0588686
\(83\) −70403.6 −1.12176 −0.560880 0.827897i \(-0.689537\pi\)
−0.560880 + 0.827897i \(0.689537\pi\)
\(84\) 0 0
\(85\) 7034.12 0.105600
\(86\) 1077.51 0.0157100
\(87\) 0 0
\(88\) 5873.52 0.0808522
\(89\) −29645.1 −0.396714 −0.198357 0.980130i \(-0.563561\pi\)
−0.198357 + 0.980130i \(0.563561\pi\)
\(90\) 0 0
\(91\) 60024.2 0.759842
\(92\) 131816. 1.62368
\(93\) 0 0
\(94\) 4753.61 0.0554887
\(95\) 85685.3 0.974086
\(96\) 0 0
\(97\) −89440.4 −0.965171 −0.482585 0.875849i \(-0.660302\pi\)
−0.482585 + 0.875849i \(0.660302\pi\)
\(98\) 3528.79 0.0371160
\(99\) 0 0
\(100\) 74633.1 0.746331
\(101\) 28633.0 0.279295 0.139647 0.990201i \(-0.455403\pi\)
0.139647 + 0.990201i \(0.455403\pi\)
\(102\) 0 0
\(103\) 30124.9 0.279790 0.139895 0.990166i \(-0.455324\pi\)
0.139895 + 0.990166i \(0.455324\pi\)
\(104\) −21475.5 −0.194697
\(105\) 0 0
\(106\) −17749.4 −0.153433
\(107\) −83073.3 −0.701459 −0.350729 0.936477i \(-0.614066\pi\)
−0.350729 + 0.936477i \(0.614066\pi\)
\(108\) 0 0
\(109\) 58783.5 0.473903 0.236951 0.971522i \(-0.423852\pi\)
0.236951 + 0.971522i \(0.423852\pi\)
\(110\) 2556.38 0.0201439
\(111\) 0 0
\(112\) −102810. −0.774445
\(113\) 232647. 1.71396 0.856981 0.515347i \(-0.172337\pi\)
0.856981 + 0.515347i \(0.172337\pi\)
\(114\) 0 0
\(115\) 115358. 0.813400
\(116\) 214392. 1.47933
\(117\) 0 0
\(118\) −26361.5 −0.174287
\(119\) 26323.9 0.170405
\(120\) 0 0
\(121\) −135985. −0.844358
\(122\) 4225.64 0.0257036
\(123\) 0 0
\(124\) −198323. −1.15829
\(125\) 151900. 0.869528
\(126\) 0 0
\(127\) 109116. 0.600313 0.300156 0.953890i \(-0.402961\pi\)
0.300156 + 0.953890i \(0.402961\pi\)
\(128\) 74368.9 0.401205
\(129\) 0 0
\(130\) −9346.94 −0.0485078
\(131\) −139424. −0.709839 −0.354920 0.934897i \(-0.615492\pi\)
−0.354920 + 0.934897i \(0.615492\pi\)
\(132\) 0 0
\(133\) 320662. 1.57188
\(134\) −11471.9 −0.0551915
\(135\) 0 0
\(136\) −9418.19 −0.0436637
\(137\) 64194.6 0.292211 0.146106 0.989269i \(-0.453326\pi\)
0.146106 + 0.989269i \(0.453326\pi\)
\(138\) 0 0
\(139\) −281632. −1.23636 −0.618181 0.786036i \(-0.712130\pi\)
−0.618181 + 0.786036i \(0.712130\pi\)
\(140\) −90959.7 −0.392219
\(141\) 0 0
\(142\) −28049.2 −0.116735
\(143\) −91650.4 −0.374796
\(144\) 0 0
\(145\) 187624. 0.741086
\(146\) −24768.2 −0.0961639
\(147\) 0 0
\(148\) 104315. 0.391466
\(149\) −225391. −0.831710 −0.415855 0.909431i \(-0.636518\pi\)
−0.415855 + 0.909431i \(0.636518\pi\)
\(150\) 0 0
\(151\) 388390. 1.38620 0.693099 0.720842i \(-0.256245\pi\)
0.693099 + 0.720842i \(0.256245\pi\)
\(152\) −114727. −0.402768
\(153\) 0 0
\(154\) 9566.78 0.0325060
\(155\) −173561. −0.580259
\(156\) 0 0
\(157\) −369612. −1.19673 −0.598366 0.801223i \(-0.704183\pi\)
−0.598366 + 0.801223i \(0.704183\pi\)
\(158\) 37967.4 0.120995
\(159\) 0 0
\(160\) 48902.2 0.151018
\(161\) 431708. 1.31258
\(162\) 0 0
\(163\) 522204. 1.53947 0.769735 0.638363i \(-0.220388\pi\)
0.769735 + 0.638363i \(0.220388\pi\)
\(164\) −194738. −0.565381
\(165\) 0 0
\(166\) 41027.9 0.115560
\(167\) 17232.5 0.0478141 0.0239071 0.999714i \(-0.492389\pi\)
0.0239071 + 0.999714i \(0.492389\pi\)
\(168\) 0 0
\(169\) −36189.0 −0.0974674
\(170\) −4099.15 −0.0108786
\(171\) 0 0
\(172\) 58540.1 0.150880
\(173\) 571643. 1.45214 0.726072 0.687618i \(-0.241344\pi\)
0.726072 + 0.687618i \(0.241344\pi\)
\(174\) 0 0
\(175\) 244429. 0.603333
\(176\) 156980. 0.381999
\(177\) 0 0
\(178\) 17275.7 0.0408683
\(179\) 191963. 0.447802 0.223901 0.974612i \(-0.428121\pi\)
0.223901 + 0.974612i \(0.428121\pi\)
\(180\) 0 0
\(181\) 110804. 0.251396 0.125698 0.992069i \(-0.459883\pi\)
0.125698 + 0.992069i \(0.459883\pi\)
\(182\) −34979.3 −0.0782767
\(183\) 0 0
\(184\) −154457. −0.336327
\(185\) 91290.9 0.196109
\(186\) 0 0
\(187\) −40193.8 −0.0840533
\(188\) 258259. 0.532919
\(189\) 0 0
\(190\) −49933.3 −0.100348
\(191\) −295668. −0.586437 −0.293218 0.956046i \(-0.594726\pi\)
−0.293218 + 0.956046i \(0.594726\pi\)
\(192\) 0 0
\(193\) −375944. −0.726491 −0.363246 0.931693i \(-0.618331\pi\)
−0.363246 + 0.931693i \(0.618331\pi\)
\(194\) 52121.6 0.0994291
\(195\) 0 0
\(196\) 191716. 0.356466
\(197\) 732825. 1.34535 0.672674 0.739939i \(-0.265146\pi\)
0.672674 + 0.739939i \(0.265146\pi\)
\(198\) 0 0
\(199\) 589939. 1.05603 0.528013 0.849236i \(-0.322937\pi\)
0.528013 + 0.849236i \(0.322937\pi\)
\(200\) −87451.8 −0.154594
\(201\) 0 0
\(202\) −16685.9 −0.0287722
\(203\) 702150. 1.19589
\(204\) 0 0
\(205\) −170424. −0.283234
\(206\) −17555.3 −0.0288231
\(207\) 0 0
\(208\) −573969. −0.919878
\(209\) −489616. −0.775336
\(210\) 0 0
\(211\) −371450. −0.574374 −0.287187 0.957875i \(-0.592720\pi\)
−0.287187 + 0.957875i \(0.592720\pi\)
\(212\) −964307. −1.47359
\(213\) 0 0
\(214\) 48411.2 0.0722622
\(215\) 51231.0 0.0755851
\(216\) 0 0
\(217\) −649520. −0.936361
\(218\) −34256.2 −0.0488201
\(219\) 0 0
\(220\) 138885. 0.193464
\(221\) 146962. 0.202406
\(222\) 0 0
\(223\) −835552. −1.12515 −0.562576 0.826746i \(-0.690190\pi\)
−0.562576 + 0.826746i \(0.690190\pi\)
\(224\) 183008. 0.243697
\(225\) 0 0
\(226\) −135576. −0.176567
\(227\) 363342. 0.468006 0.234003 0.972236i \(-0.424818\pi\)
0.234003 + 0.972236i \(0.424818\pi\)
\(228\) 0 0
\(229\) −106091. −0.133688 −0.0668438 0.997763i \(-0.521293\pi\)
−0.0668438 + 0.997763i \(0.521293\pi\)
\(230\) −67225.3 −0.0837941
\(231\) 0 0
\(232\) −251216. −0.306427
\(233\) −713760. −0.861315 −0.430658 0.902515i \(-0.641718\pi\)
−0.430658 + 0.902515i \(0.641718\pi\)
\(234\) 0 0
\(235\) 226014. 0.266972
\(236\) −1.43219e6 −1.67387
\(237\) 0 0
\(238\) −15340.3 −0.0175547
\(239\) 70400.0 0.0797220 0.0398610 0.999205i \(-0.487308\pi\)
0.0398610 + 0.999205i \(0.487308\pi\)
\(240\) 0 0
\(241\) 44600.6 0.0494650 0.0247325 0.999694i \(-0.492127\pi\)
0.0247325 + 0.999694i \(0.492127\pi\)
\(242\) 79245.5 0.0869833
\(243\) 0 0
\(244\) 229575. 0.246860
\(245\) 167779. 0.178576
\(246\) 0 0
\(247\) 1.79020e6 1.86706
\(248\) 232386. 0.239928
\(249\) 0 0
\(250\) −88520.3 −0.0895762
\(251\) −1.05989e6 −1.06188 −0.530942 0.847408i \(-0.678162\pi\)
−0.530942 + 0.847408i \(0.678162\pi\)
\(252\) 0 0
\(253\) −659170. −0.647435
\(254\) −63587.4 −0.0618425
\(255\) 0 0
\(256\) 939058. 0.895556
\(257\) 966091. 0.912400 0.456200 0.889877i \(-0.349210\pi\)
0.456200 + 0.889877i \(0.349210\pi\)
\(258\) 0 0
\(259\) 341640. 0.316460
\(260\) −507811. −0.465874
\(261\) 0 0
\(262\) 81249.8 0.0731255
\(263\) −855270. −0.762454 −0.381227 0.924481i \(-0.624498\pi\)
−0.381227 + 0.924481i \(0.624498\pi\)
\(264\) 0 0
\(265\) −843907. −0.738210
\(266\) −186867. −0.161930
\(267\) 0 0
\(268\) −623256. −0.530065
\(269\) −1.55224e6 −1.30791 −0.653955 0.756533i \(-0.726892\pi\)
−0.653955 + 0.756533i \(0.726892\pi\)
\(270\) 0 0
\(271\) −4177.31 −0.00345520 −0.00172760 0.999999i \(-0.500550\pi\)
−0.00172760 + 0.999999i \(0.500550\pi\)
\(272\) −251717. −0.206296
\(273\) 0 0
\(274\) −37409.5 −0.0301027
\(275\) −373216. −0.297597
\(276\) 0 0
\(277\) −1.59749e6 −1.25095 −0.625475 0.780244i \(-0.715095\pi\)
−0.625475 + 0.780244i \(0.715095\pi\)
\(278\) 164122. 0.127366
\(279\) 0 0
\(280\) 106583. 0.0812439
\(281\) −1.60304e6 −1.21110 −0.605549 0.795808i \(-0.707046\pi\)
−0.605549 + 0.795808i \(0.707046\pi\)
\(282\) 0 0
\(283\) 2.04664e6 1.51906 0.759530 0.650472i \(-0.225429\pi\)
0.759530 + 0.650472i \(0.225429\pi\)
\(284\) −1.52389e6 −1.12113
\(285\) 0 0
\(286\) 53409.5 0.0386103
\(287\) −637780. −0.457053
\(288\) 0 0
\(289\) −1.35541e6 −0.954608
\(290\) −109338. −0.0763446
\(291\) 0 0
\(292\) −1.34563e6 −0.923569
\(293\) −1.35028e6 −0.918869 −0.459434 0.888212i \(-0.651948\pi\)
−0.459434 + 0.888212i \(0.651948\pi\)
\(294\) 0 0
\(295\) −1.25337e6 −0.838544
\(296\) −122232. −0.0810880
\(297\) 0 0
\(298\) 131347. 0.0856803
\(299\) 2.41014e6 1.55907
\(300\) 0 0
\(301\) 191723. 0.121971
\(302\) −226335. −0.142802
\(303\) 0 0
\(304\) −3.06626e6 −1.90294
\(305\) 200911. 0.123667
\(306\) 0 0
\(307\) −361103. −0.218668 −0.109334 0.994005i \(-0.534872\pi\)
−0.109334 + 0.994005i \(0.534872\pi\)
\(308\) 519754. 0.312192
\(309\) 0 0
\(310\) 101143. 0.0597766
\(311\) −2.12425e6 −1.24539 −0.622694 0.782466i \(-0.713962\pi\)
−0.622694 + 0.782466i \(0.713962\pi\)
\(312\) 0 0
\(313\) −541717. −0.312545 −0.156272 0.987714i \(-0.549948\pi\)
−0.156272 + 0.987714i \(0.549948\pi\)
\(314\) 215392. 0.123284
\(315\) 0 0
\(316\) 2.06273e6 1.16205
\(317\) 3.06921e6 1.71545 0.857725 0.514110i \(-0.171878\pi\)
0.857725 + 0.514110i \(0.171878\pi\)
\(318\) 0 0
\(319\) −1.07211e6 −0.589877
\(320\) 850614. 0.464363
\(321\) 0 0
\(322\) −251579. −0.135218
\(323\) 785100. 0.418715
\(324\) 0 0
\(325\) 1.36460e6 0.716632
\(326\) −304316. −0.158592
\(327\) 0 0
\(328\) 228185. 0.117112
\(329\) 845817. 0.430811
\(330\) 0 0
\(331\) −282184. −0.141567 −0.0707836 0.997492i \(-0.522550\pi\)
−0.0707836 + 0.997492i \(0.522550\pi\)
\(332\) 2.22901e6 1.10986
\(333\) 0 0
\(334\) −10042.3 −0.00492567
\(335\) −545438. −0.265542
\(336\) 0 0
\(337\) 1.04584e6 0.501640 0.250820 0.968034i \(-0.419300\pi\)
0.250820 + 0.968034i \(0.419300\pi\)
\(338\) 21089.2 0.0100408
\(339\) 0 0
\(340\) −222703. −0.104479
\(341\) 991746. 0.461864
\(342\) 0 0
\(343\) 2.37060e6 1.08799
\(344\) −68594.7 −0.0312532
\(345\) 0 0
\(346\) −333127. −0.149596
\(347\) −1.90831e6 −0.850796 −0.425398 0.905006i \(-0.639866\pi\)
−0.425398 + 0.905006i \(0.639866\pi\)
\(348\) 0 0
\(349\) 1.46350e6 0.643175 0.321588 0.946880i \(-0.395784\pi\)
0.321588 + 0.946880i \(0.395784\pi\)
\(350\) −142441. −0.0621536
\(351\) 0 0
\(352\) −279433. −0.120205
\(353\) −867374. −0.370484 −0.185242 0.982693i \(-0.559307\pi\)
−0.185242 + 0.982693i \(0.559307\pi\)
\(354\) 0 0
\(355\) −1.33362e6 −0.561643
\(356\) 938575. 0.392504
\(357\) 0 0
\(358\) −111867. −0.0461313
\(359\) −3.45466e6 −1.41472 −0.707359 0.706855i \(-0.750113\pi\)
−0.707359 + 0.706855i \(0.750113\pi\)
\(360\) 0 0
\(361\) 7.08751e6 2.86237
\(362\) −64571.2 −0.0258981
\(363\) 0 0
\(364\) −1.90039e6 −0.751778
\(365\) −1.17762e6 −0.462672
\(366\) 0 0
\(367\) −2.78797e6 −1.08050 −0.540248 0.841506i \(-0.681670\pi\)
−0.540248 + 0.841506i \(0.681670\pi\)
\(368\) −4.12811e6 −1.58903
\(369\) 0 0
\(370\) −53200.0 −0.0202026
\(371\) −3.15817e6 −1.19125
\(372\) 0 0
\(373\) −1.65578e6 −0.616212 −0.308106 0.951352i \(-0.599695\pi\)
−0.308106 + 0.951352i \(0.599695\pi\)
\(374\) 23423.0 0.00865892
\(375\) 0 0
\(376\) −302617. −0.110388
\(377\) 3.91997e6 1.42046
\(378\) 0 0
\(379\) −2.35538e6 −0.842291 −0.421146 0.906993i \(-0.638372\pi\)
−0.421146 + 0.906993i \(0.638372\pi\)
\(380\) −2.71283e6 −0.963749
\(381\) 0 0
\(382\) 172301. 0.0604130
\(383\) 4.13864e6 1.44165 0.720827 0.693115i \(-0.243762\pi\)
0.720827 + 0.693115i \(0.243762\pi\)
\(384\) 0 0
\(385\) 454859. 0.156396
\(386\) 219083. 0.0748410
\(387\) 0 0
\(388\) 2.83172e6 0.954928
\(389\) 5.27781e6 1.76840 0.884198 0.467112i \(-0.154706\pi\)
0.884198 + 0.467112i \(0.154706\pi\)
\(390\) 0 0
\(391\) 1.05698e6 0.349643
\(392\) −224644. −0.0738380
\(393\) 0 0
\(394\) −427056. −0.138594
\(395\) 1.80519e6 0.582143
\(396\) 0 0
\(397\) 3.71439e6 1.18280 0.591401 0.806378i \(-0.298575\pi\)
0.591401 + 0.806378i \(0.298575\pi\)
\(398\) −343789. −0.108789
\(399\) 0 0
\(400\) −2.33730e6 −0.730405
\(401\) −3.60034e6 −1.11810 −0.559052 0.829132i \(-0.688835\pi\)
−0.559052 + 0.829132i \(0.688835\pi\)
\(402\) 0 0
\(403\) −3.62615e6 −1.11220
\(404\) −906532. −0.276331
\(405\) 0 0
\(406\) −409180. −0.123197
\(407\) −521647. −0.156096
\(408\) 0 0
\(409\) 1.65305e6 0.488627 0.244314 0.969696i \(-0.421437\pi\)
0.244314 + 0.969696i \(0.421437\pi\)
\(410\) 99314.8 0.0291779
\(411\) 0 0
\(412\) −953765. −0.276821
\(413\) −4.69053e6 −1.35315
\(414\) 0 0
\(415\) 1.95070e6 0.555994
\(416\) 1.02170e6 0.289461
\(417\) 0 0
\(418\) 285325. 0.0798728
\(419\) −1.25123e6 −0.348180 −0.174090 0.984730i \(-0.555698\pi\)
−0.174090 + 0.984730i \(0.555698\pi\)
\(420\) 0 0
\(421\) 3.12108e6 0.858221 0.429111 0.903252i \(-0.358827\pi\)
0.429111 + 0.903252i \(0.358827\pi\)
\(422\) 216464. 0.0591703
\(423\) 0 0
\(424\) 1.12993e6 0.305238
\(425\) 598451. 0.160715
\(426\) 0 0
\(427\) 751875. 0.199561
\(428\) 2.63013e6 0.694015
\(429\) 0 0
\(430\) −29855.0 −0.00778656
\(431\) −246469. −0.0639102 −0.0319551 0.999489i \(-0.510173\pi\)
−0.0319551 + 0.999489i \(0.510173\pi\)
\(432\) 0 0
\(433\) −7.56031e6 −1.93785 −0.968924 0.247357i \(-0.920438\pi\)
−0.968924 + 0.247357i \(0.920438\pi\)
\(434\) 378510. 0.0964612
\(435\) 0 0
\(436\) −1.86111e6 −0.468873
\(437\) 1.28755e7 3.22523
\(438\) 0 0
\(439\) −3.76145e6 −0.931525 −0.465763 0.884910i \(-0.654220\pi\)
−0.465763 + 0.884910i \(0.654220\pi\)
\(440\) −162740. −0.0400740
\(441\) 0 0
\(442\) −85642.2 −0.0208513
\(443\) 1.02986e6 0.249327 0.124664 0.992199i \(-0.460215\pi\)
0.124664 + 0.992199i \(0.460215\pi\)
\(444\) 0 0
\(445\) 821387. 0.196629
\(446\) 486920. 0.115910
\(447\) 0 0
\(448\) 3.18327e6 0.749340
\(449\) 4.29961e6 1.00650 0.503249 0.864141i \(-0.332138\pi\)
0.503249 + 0.864141i \(0.332138\pi\)
\(450\) 0 0
\(451\) 973820. 0.225443
\(452\) −7.36570e6 −1.69577
\(453\) 0 0
\(454\) −211739. −0.0482126
\(455\) −1.66312e6 −0.376612
\(456\) 0 0
\(457\) −1.42831e6 −0.319913 −0.159957 0.987124i \(-0.551135\pi\)
−0.159957 + 0.987124i \(0.551135\pi\)
\(458\) 61825.0 0.0137721
\(459\) 0 0
\(460\) −3.65229e6 −0.804768
\(461\) 3.06394e6 0.671472 0.335736 0.941956i \(-0.391015\pi\)
0.335736 + 0.941956i \(0.391015\pi\)
\(462\) 0 0
\(463\) −3.51890e6 −0.762877 −0.381438 0.924394i \(-0.624571\pi\)
−0.381438 + 0.924394i \(0.624571\pi\)
\(464\) −6.71416e6 −1.44776
\(465\) 0 0
\(466\) 415945. 0.0887302
\(467\) 6.90153e6 1.46438 0.732188 0.681102i \(-0.238499\pi\)
0.732188 + 0.681102i \(0.238499\pi\)
\(468\) 0 0
\(469\) −2.04121e6 −0.428504
\(470\) −131710. −0.0275027
\(471\) 0 0
\(472\) 1.67818e6 0.346724
\(473\) −292740. −0.0601629
\(474\) 0 0
\(475\) 7.28997e6 1.48249
\(476\) −833426. −0.168597
\(477\) 0 0
\(478\) −41025.8 −0.00821272
\(479\) 4.37684e6 0.871609 0.435805 0.900041i \(-0.356464\pi\)
0.435805 + 0.900041i \(0.356464\pi\)
\(480\) 0 0
\(481\) 1.90731e6 0.375889
\(482\) −25991.1 −0.00509574
\(483\) 0 0
\(484\) 4.30533e6 0.835398
\(485\) 2.47816e6 0.478382
\(486\) 0 0
\(487\) 4.08443e6 0.780385 0.390192 0.920733i \(-0.372409\pi\)
0.390192 + 0.920733i \(0.372409\pi\)
\(488\) −269006. −0.0511343
\(489\) 0 0
\(490\) −97773.6 −0.0183963
\(491\) 4.67807e6 0.875716 0.437858 0.899044i \(-0.355737\pi\)
0.437858 + 0.899044i \(0.355737\pi\)
\(492\) 0 0
\(493\) 1.71912e6 0.318559
\(494\) −1.04324e6 −0.192339
\(495\) 0 0
\(496\) 6.21090e6 1.13357
\(497\) −4.99083e6 −0.906321
\(498\) 0 0
\(499\) 9.66466e6 1.73754 0.868771 0.495214i \(-0.164910\pi\)
0.868771 + 0.495214i \(0.164910\pi\)
\(500\) −4.80922e6 −0.860300
\(501\) 0 0
\(502\) 617655. 0.109392
\(503\) 1.03319e7 1.82079 0.910396 0.413738i \(-0.135777\pi\)
0.910396 + 0.413738i \(0.135777\pi\)
\(504\) 0 0
\(505\) −793345. −0.138431
\(506\) 384133. 0.0666969
\(507\) 0 0
\(508\) −3.45464e6 −0.593942
\(509\) 4.20006e6 0.718557 0.359278 0.933230i \(-0.383023\pi\)
0.359278 + 0.933230i \(0.383023\pi\)
\(510\) 0 0
\(511\) −4.40704e6 −0.746611
\(512\) −2.92704e6 −0.493463
\(513\) 0 0
\(514\) −562992. −0.0939928
\(515\) −834681. −0.138676
\(516\) 0 0
\(517\) −1.29147e6 −0.212500
\(518\) −199092. −0.0326008
\(519\) 0 0
\(520\) 595030. 0.0965007
\(521\) −6.46853e6 −1.04403 −0.522013 0.852938i \(-0.674819\pi\)
−0.522013 + 0.852938i \(0.674819\pi\)
\(522\) 0 0
\(523\) −3.00248e6 −0.479983 −0.239991 0.970775i \(-0.577145\pi\)
−0.239991 + 0.970775i \(0.577145\pi\)
\(524\) 4.41423e6 0.702306
\(525\) 0 0
\(526\) 498411. 0.0785458
\(527\) −1.59027e6 −0.249427
\(528\) 0 0
\(529\) 1.08979e7 1.69319
\(530\) 491789. 0.0760482
\(531\) 0 0
\(532\) −1.01523e7 −1.55520
\(533\) −3.56061e6 −0.542883
\(534\) 0 0
\(535\) 2.30175e6 0.347674
\(536\) 730304. 0.109797
\(537\) 0 0
\(538\) 904572. 0.134737
\(539\) −958708. −0.142139
\(540\) 0 0
\(541\) −5.21767e6 −0.766449 −0.383224 0.923655i \(-0.625186\pi\)
−0.383224 + 0.923655i \(0.625186\pi\)
\(542\) 2434.34 0.000355945 0
\(543\) 0 0
\(544\) 448071. 0.0649157
\(545\) −1.62874e6 −0.234887
\(546\) 0 0
\(547\) −7.86187e6 −1.12346 −0.561730 0.827321i \(-0.689864\pi\)
−0.561730 + 0.827321i \(0.689864\pi\)
\(548\) −2.03243e6 −0.289110
\(549\) 0 0
\(550\) 217492. 0.0306575
\(551\) 2.09413e7 2.93850
\(552\) 0 0
\(553\) 6.75560e6 0.939401
\(554\) 930944. 0.128869
\(555\) 0 0
\(556\) 8.91659e6 1.22324
\(557\) −1.21531e7 −1.65978 −0.829889 0.557929i \(-0.811596\pi\)
−0.829889 + 0.557929i \(0.811596\pi\)
\(558\) 0 0
\(559\) 1.07035e6 0.144876
\(560\) 2.84860e6 0.383850
\(561\) 0 0
\(562\) 934177. 0.124764
\(563\) −5.15518e6 −0.685446 −0.342723 0.939437i \(-0.611349\pi\)
−0.342723 + 0.939437i \(0.611349\pi\)
\(564\) 0 0
\(565\) −6.44604e6 −0.849517
\(566\) −1.19268e6 −0.156489
\(567\) 0 0
\(568\) 1.78562e6 0.232230
\(569\) 1.42079e6 0.183972 0.0919858 0.995760i \(-0.470679\pi\)
0.0919858 + 0.995760i \(0.470679\pi\)
\(570\) 0 0
\(571\) −1.14786e7 −1.47332 −0.736661 0.676262i \(-0.763599\pi\)
−0.736661 + 0.676262i \(0.763599\pi\)
\(572\) 2.90169e6 0.370818
\(573\) 0 0
\(574\) 371668. 0.0470842
\(575\) 9.81450e6 1.23794
\(576\) 0 0
\(577\) 1.00683e7 1.25897 0.629484 0.777014i \(-0.283266\pi\)
0.629484 + 0.777014i \(0.283266\pi\)
\(578\) 789867. 0.0983409
\(579\) 0 0
\(580\) −5.94026e6 −0.733221
\(581\) 7.30015e6 0.897205
\(582\) 0 0
\(583\) 4.82218e6 0.587587
\(584\) 1.57675e6 0.191307
\(585\) 0 0
\(586\) 786877. 0.0946592
\(587\) 1.33560e7 1.59985 0.799927 0.600097i \(-0.204871\pi\)
0.799927 + 0.600097i \(0.204871\pi\)
\(588\) 0 0
\(589\) −1.93716e7 −2.30080
\(590\) 730407. 0.0863843
\(591\) 0 0
\(592\) −3.26686e6 −0.383113
\(593\) −1.20725e7 −1.40981 −0.704906 0.709301i \(-0.749011\pi\)
−0.704906 + 0.709301i \(0.749011\pi\)
\(594\) 0 0
\(595\) −729367. −0.0844606
\(596\) 7.13598e6 0.822883
\(597\) 0 0
\(598\) −1.40452e6 −0.160611
\(599\) −1.59735e7 −1.81900 −0.909502 0.415700i \(-0.863537\pi\)
−0.909502 + 0.415700i \(0.863537\pi\)
\(600\) 0 0
\(601\) 7.82358e6 0.883526 0.441763 0.897132i \(-0.354353\pi\)
0.441763 + 0.897132i \(0.354353\pi\)
\(602\) −111727. −0.0125651
\(603\) 0 0
\(604\) −1.22966e7 −1.37149
\(605\) 3.76778e6 0.418502
\(606\) 0 0
\(607\) 3.67967e6 0.405356 0.202678 0.979245i \(-0.435036\pi\)
0.202678 + 0.979245i \(0.435036\pi\)
\(608\) 5.45813e6 0.598804
\(609\) 0 0
\(610\) −117082. −0.0127398
\(611\) 4.72204e6 0.511713
\(612\) 0 0
\(613\) 4.33193e6 0.465619 0.232810 0.972522i \(-0.425208\pi\)
0.232810 + 0.972522i \(0.425208\pi\)
\(614\) 210434. 0.0225265
\(615\) 0 0
\(616\) −609025. −0.0646671
\(617\) 6.45241e6 0.682353 0.341176 0.939999i \(-0.389175\pi\)
0.341176 + 0.939999i \(0.389175\pi\)
\(618\) 0 0
\(619\) 7.05387e6 0.739947 0.369974 0.929042i \(-0.379367\pi\)
0.369974 + 0.929042i \(0.379367\pi\)
\(620\) 5.49500e6 0.574101
\(621\) 0 0
\(622\) 1.23791e6 0.128296
\(623\) 3.07390e6 0.317299
\(624\) 0 0
\(625\) 3.15781e6 0.323359
\(626\) 315687. 0.0321974
\(627\) 0 0
\(628\) 1.17021e7 1.18403
\(629\) 836461. 0.0842984
\(630\) 0 0
\(631\) 1.46818e7 1.46793 0.733966 0.679186i \(-0.237667\pi\)
0.733966 + 0.679186i \(0.237667\pi\)
\(632\) −2.41702e6 −0.240706
\(633\) 0 0
\(634\) −1.78859e6 −0.176721
\(635\) −3.02331e6 −0.297542
\(636\) 0 0
\(637\) 3.50535e6 0.342281
\(638\) 624773. 0.0607674
\(639\) 0 0
\(640\) −2.06057e6 −0.198855
\(641\) −3.24675e6 −0.312107 −0.156054 0.987749i \(-0.549877\pi\)
−0.156054 + 0.987749i \(0.549877\pi\)
\(642\) 0 0
\(643\) −7.57964e6 −0.722971 −0.361486 0.932378i \(-0.617730\pi\)
−0.361486 + 0.932378i \(0.617730\pi\)
\(644\) −1.36680e7 −1.29865
\(645\) 0 0
\(646\) −457519. −0.0431348
\(647\) −6.13290e6 −0.575977 −0.287988 0.957634i \(-0.592986\pi\)
−0.287988 + 0.957634i \(0.592986\pi\)
\(648\) 0 0
\(649\) 7.16193e6 0.667449
\(650\) −795223. −0.0738254
\(651\) 0 0
\(652\) −1.65332e7 −1.52313
\(653\) 1.13174e6 0.103864 0.0519318 0.998651i \(-0.483462\pi\)
0.0519318 + 0.998651i \(0.483462\pi\)
\(654\) 0 0
\(655\) 3.86308e6 0.351828
\(656\) 6.09864e6 0.553316
\(657\) 0 0
\(658\) −492902. −0.0443809
\(659\) −1.03411e7 −0.927586 −0.463793 0.885944i \(-0.653512\pi\)
−0.463793 + 0.885944i \(0.653512\pi\)
\(660\) 0 0
\(661\) −8.30708e6 −0.739511 −0.369756 0.929129i \(-0.620559\pi\)
−0.369756 + 0.929129i \(0.620559\pi\)
\(662\) 164444. 0.0145838
\(663\) 0 0
\(664\) −2.61185e6 −0.229894
\(665\) −8.88471e6 −0.779093
\(666\) 0 0
\(667\) 2.81933e7 2.45376
\(668\) −545587. −0.0473067
\(669\) 0 0
\(670\) 317856. 0.0273554
\(671\) −1.14803e6 −0.0984344
\(672\) 0 0
\(673\) −1.13225e7 −0.963617 −0.481809 0.876276i \(-0.660020\pi\)
−0.481809 + 0.876276i \(0.660020\pi\)
\(674\) −609468. −0.0516775
\(675\) 0 0
\(676\) 1.14576e6 0.0964331
\(677\) −2.12894e7 −1.78522 −0.892609 0.450831i \(-0.851128\pi\)
−0.892609 + 0.450831i \(0.851128\pi\)
\(678\) 0 0
\(679\) 9.27407e6 0.771962
\(680\) 260954. 0.0216417
\(681\) 0 0
\(682\) −577943. −0.0475799
\(683\) 1.18724e7 0.973841 0.486920 0.873446i \(-0.338120\pi\)
0.486920 + 0.873446i \(0.338120\pi\)
\(684\) 0 0
\(685\) −1.77866e6 −0.144833
\(686\) −1.38147e6 −0.112081
\(687\) 0 0
\(688\) −1.83331e6 −0.147661
\(689\) −1.76315e7 −1.41495
\(690\) 0 0
\(691\) −1.96927e7 −1.56896 −0.784478 0.620157i \(-0.787069\pi\)
−0.784478 + 0.620157i \(0.787069\pi\)
\(692\) −1.80985e7 −1.43673
\(693\) 0 0
\(694\) 1.11207e6 0.0876465
\(695\) 7.80329e6 0.612796
\(696\) 0 0
\(697\) −1.56152e6 −0.121749
\(698\) −852859. −0.0662581
\(699\) 0 0
\(700\) −7.73870e6 −0.596930
\(701\) 1.36097e7 1.04605 0.523027 0.852316i \(-0.324803\pi\)
0.523027 + 0.852316i \(0.324803\pi\)
\(702\) 0 0
\(703\) 1.01893e7 0.777597
\(704\) −4.86051e6 −0.369615
\(705\) 0 0
\(706\) 505464. 0.0381662
\(707\) −2.96895e6 −0.223385
\(708\) 0 0
\(709\) 1.30274e7 0.973290 0.486645 0.873600i \(-0.338220\pi\)
0.486645 + 0.873600i \(0.338220\pi\)
\(710\) 777170. 0.0578589
\(711\) 0 0
\(712\) −1.09978e6 −0.0813029
\(713\) −2.60801e7 −1.92125
\(714\) 0 0
\(715\) 2.53939e6 0.185765
\(716\) −6.07764e6 −0.443050
\(717\) 0 0
\(718\) 2.01321e6 0.145740
\(719\) 9.11723e6 0.657719 0.328860 0.944379i \(-0.393336\pi\)
0.328860 + 0.944379i \(0.393336\pi\)
\(720\) 0 0
\(721\) −3.12365e6 −0.223781
\(722\) −4.13026e6 −0.294873
\(723\) 0 0
\(724\) −3.50809e6 −0.248728
\(725\) 1.59628e7 1.12788
\(726\) 0 0
\(727\) −1.52424e7 −1.06959 −0.534797 0.844981i \(-0.679612\pi\)
−0.534797 + 0.844981i \(0.679612\pi\)
\(728\) 2.22680e6 0.155723
\(729\) 0 0
\(730\) 686262. 0.0476631
\(731\) 469408. 0.0324906
\(732\) 0 0
\(733\) −1.61394e7 −1.10950 −0.554751 0.832016i \(-0.687187\pi\)
−0.554751 + 0.832016i \(0.687187\pi\)
\(734\) 1.62470e6 0.111310
\(735\) 0 0
\(736\) 7.34828e6 0.500025
\(737\) 3.11670e6 0.211362
\(738\) 0 0
\(739\) −9.24471e6 −0.622704 −0.311352 0.950295i \(-0.600782\pi\)
−0.311352 + 0.950295i \(0.600782\pi\)
\(740\) −2.89031e6 −0.194028
\(741\) 0 0
\(742\) 1.84043e6 0.122719
\(743\) 3.89612e6 0.258917 0.129459 0.991585i \(-0.458676\pi\)
0.129459 + 0.991585i \(0.458676\pi\)
\(744\) 0 0
\(745\) 6.24501e6 0.412233
\(746\) 964909. 0.0634804
\(747\) 0 0
\(748\) 1.27255e6 0.0831613
\(749\) 8.61387e6 0.561040
\(750\) 0 0
\(751\) −2.36303e7 −1.52887 −0.764433 0.644704i \(-0.776981\pi\)
−0.764433 + 0.644704i \(0.776981\pi\)
\(752\) −8.08795e6 −0.521547
\(753\) 0 0
\(754\) −2.28437e6 −0.146332
\(755\) −1.07613e7 −0.687062
\(756\) 0 0
\(757\) 2.86009e6 0.181401 0.0907005 0.995878i \(-0.471089\pi\)
0.0907005 + 0.995878i \(0.471089\pi\)
\(758\) 1.37260e6 0.0867704
\(759\) 0 0
\(760\) 3.17878e6 0.199630
\(761\) 2.90585e7 1.81891 0.909456 0.415800i \(-0.136498\pi\)
0.909456 + 0.415800i \(0.136498\pi\)
\(762\) 0 0
\(763\) −6.09526e6 −0.379036
\(764\) 9.36097e6 0.580213
\(765\) 0 0
\(766\) −2.41180e6 −0.148515
\(767\) −2.61864e7 −1.60726
\(768\) 0 0
\(769\) −1.85731e7 −1.13258 −0.566290 0.824206i \(-0.691622\pi\)
−0.566290 + 0.824206i \(0.691622\pi\)
\(770\) −265071. −0.0161115
\(771\) 0 0
\(772\) 1.19026e7 0.718781
\(773\) −1.38698e7 −0.834872 −0.417436 0.908706i \(-0.637071\pi\)
−0.417436 + 0.908706i \(0.637071\pi\)
\(774\) 0 0
\(775\) −1.47663e7 −0.883113
\(776\) −3.31808e6 −0.197803
\(777\) 0 0
\(778\) −3.07566e6 −0.182175
\(779\) −1.90215e7 −1.12306
\(780\) 0 0
\(781\) 7.62045e6 0.447047
\(782\) −615958. −0.0360192
\(783\) 0 0
\(784\) −6.00400e6 −0.348859
\(785\) 1.02410e7 0.593154
\(786\) 0 0
\(787\) −1.95803e7 −1.12689 −0.563445 0.826154i \(-0.690524\pi\)
−0.563445 + 0.826154i \(0.690524\pi\)
\(788\) −2.32015e7 −1.33107
\(789\) 0 0
\(790\) −1.05198e6 −0.0599707
\(791\) −2.41232e7 −1.37086
\(792\) 0 0
\(793\) 4.19757e6 0.237037
\(794\) −2.16457e6 −0.121849
\(795\) 0 0
\(796\) −1.86777e7 −1.04482
\(797\) 2.66479e7 1.48600 0.742998 0.669293i \(-0.233403\pi\)
0.742998 + 0.669293i \(0.233403\pi\)
\(798\) 0 0
\(799\) 2.07087e6 0.114759
\(800\) 4.16052e6 0.229839
\(801\) 0 0
\(802\) 2.09811e6 0.115184
\(803\) 6.72907e6 0.368270
\(804\) 0 0
\(805\) −1.19615e7 −0.650573
\(806\) 2.11315e6 0.114576
\(807\) 0 0
\(808\) 1.06223e6 0.0572390
\(809\) −4.31569e6 −0.231835 −0.115917 0.993259i \(-0.536981\pi\)
−0.115917 + 0.993259i \(0.536981\pi\)
\(810\) 0 0
\(811\) 1.07890e7 0.576007 0.288003 0.957629i \(-0.407009\pi\)
0.288003 + 0.957629i \(0.407009\pi\)
\(812\) −2.22304e7 −1.18320
\(813\) 0 0
\(814\) 303991. 0.0160805
\(815\) −1.44689e7 −0.763031
\(816\) 0 0
\(817\) 5.71805e6 0.299704
\(818\) −963319. −0.0503369
\(819\) 0 0
\(820\) 5.39568e6 0.280228
\(821\) 2.12759e7 1.10162 0.550808 0.834632i \(-0.314320\pi\)
0.550808 + 0.834632i \(0.314320\pi\)
\(822\) 0 0
\(823\) 7.04803e6 0.362717 0.181359 0.983417i \(-0.441950\pi\)
0.181359 + 0.983417i \(0.441950\pi\)
\(824\) 1.11758e6 0.0573404
\(825\) 0 0
\(826\) 2.73342e6 0.139398
\(827\) −2.00838e7 −1.02113 −0.510566 0.859839i \(-0.670564\pi\)
−0.510566 + 0.859839i \(0.670564\pi\)
\(828\) 0 0
\(829\) 2.04193e7 1.03194 0.515970 0.856606i \(-0.327431\pi\)
0.515970 + 0.856606i \(0.327431\pi\)
\(830\) −1.13678e6 −0.0572769
\(831\) 0 0
\(832\) 1.77716e7 0.890059
\(833\) 1.53729e6 0.0767614
\(834\) 0 0
\(835\) −477467. −0.0236988
\(836\) 1.55014e7 0.767108
\(837\) 0 0
\(838\) 729160. 0.0358685
\(839\) 4.92445e6 0.241520 0.120760 0.992682i \(-0.461467\pi\)
0.120760 + 0.992682i \(0.461467\pi\)
\(840\) 0 0
\(841\) 2.53438e7 1.23561
\(842\) −1.81882e6 −0.0884114
\(843\) 0 0
\(844\) 1.17603e7 0.568278
\(845\) 1.00270e6 0.0483092
\(846\) 0 0
\(847\) 1.41003e7 0.675334
\(848\) 3.01994e7 1.44214
\(849\) 0 0
\(850\) −348749. −0.0165564
\(851\) 1.37178e7 0.649324
\(852\) 0 0
\(853\) −1.25088e7 −0.588632 −0.294316 0.955708i \(-0.595092\pi\)
−0.294316 + 0.955708i \(0.595092\pi\)
\(854\) −438157. −0.0205582
\(855\) 0 0
\(856\) −3.08188e6 −0.143758
\(857\) −2.21879e7 −1.03196 −0.515982 0.856600i \(-0.672573\pi\)
−0.515982 + 0.856600i \(0.672573\pi\)
\(858\) 0 0
\(859\) −1.40835e7 −0.651218 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(860\) −1.62199e6 −0.0747830
\(861\) 0 0
\(862\) 143631. 0.00658384
\(863\) 7.94491e6 0.363130 0.181565 0.983379i \(-0.441884\pi\)
0.181565 + 0.983379i \(0.441884\pi\)
\(864\) 0 0
\(865\) −1.58387e7 −0.719748
\(866\) 4.40579e6 0.199632
\(867\) 0 0
\(868\) 2.05641e7 0.926424
\(869\) −1.03151e7 −0.463364
\(870\) 0 0
\(871\) −1.13957e7 −0.508973
\(872\) 2.18077e6 0.0971221
\(873\) 0 0
\(874\) −7.50323e6 −0.332253
\(875\) −1.57505e7 −0.695465
\(876\) 0 0
\(877\) −4.03970e6 −0.177358 −0.0886788 0.996060i \(-0.528264\pi\)
−0.0886788 + 0.996060i \(0.528264\pi\)
\(878\) 2.19200e6 0.0959630
\(879\) 0 0
\(880\) −4.34950e6 −0.189336
\(881\) 1.99840e7 0.867447 0.433724 0.901046i \(-0.357199\pi\)
0.433724 + 0.901046i \(0.357199\pi\)
\(882\) 0 0
\(883\) 1.66868e7 0.720230 0.360115 0.932908i \(-0.382737\pi\)
0.360115 + 0.932908i \(0.382737\pi\)
\(884\) −4.65286e6 −0.200258
\(885\) 0 0
\(886\) −600155. −0.0256850
\(887\) 1.79964e6 0.0768027 0.0384014 0.999262i \(-0.487773\pi\)
0.0384014 + 0.999262i \(0.487773\pi\)
\(888\) 0 0
\(889\) −1.13142e7 −0.480141
\(890\) −478666. −0.0202562
\(891\) 0 0
\(892\) 2.64539e7 1.11321
\(893\) 2.52261e7 1.05858
\(894\) 0 0
\(895\) −5.31881e6 −0.221951
\(896\) −7.71131e6 −0.320892
\(897\) 0 0
\(898\) −2.50561e6 −0.103687
\(899\) −4.24179e7 −1.75045
\(900\) 0 0
\(901\) −7.73238e6 −0.317323
\(902\) −567496. −0.0232245
\(903\) 0 0
\(904\) 8.63080e6 0.351261
\(905\) −3.07008e6 −0.124603
\(906\) 0 0
\(907\) −3.64748e6 −0.147223 −0.0736113 0.997287i \(-0.523452\pi\)
−0.0736113 + 0.997287i \(0.523452\pi\)
\(908\) −1.15036e7 −0.463039
\(909\) 0 0
\(910\) 969185. 0.0387974
\(911\) −4.92223e6 −0.196502 −0.0982508 0.995162i \(-0.531325\pi\)
−0.0982508 + 0.995162i \(0.531325\pi\)
\(912\) 0 0
\(913\) −1.11465e7 −0.442551
\(914\) 832352. 0.0329565
\(915\) 0 0
\(916\) 3.35890e6 0.132269
\(917\) 1.44569e7 0.567743
\(918\) 0 0
\(919\) 1.52960e7 0.597432 0.298716 0.954342i \(-0.403442\pi\)
0.298716 + 0.954342i \(0.403442\pi\)
\(920\) 4.27959e6 0.166699
\(921\) 0 0
\(922\) −1.78552e6 −0.0691731
\(923\) −2.78629e7 −1.07652
\(924\) 0 0
\(925\) 7.76689e6 0.298464
\(926\) 2.05065e6 0.0785893
\(927\) 0 0
\(928\) 1.19516e7 0.455571
\(929\) −3.10755e7 −1.18135 −0.590674 0.806910i \(-0.701138\pi\)
−0.590674 + 0.806910i \(0.701138\pi\)
\(930\) 0 0
\(931\) 1.87263e7 0.708073
\(932\) 2.25979e7 0.852175
\(933\) 0 0
\(934\) −4.02188e6 −0.150856
\(935\) 1.11366e6 0.0416606
\(936\) 0 0
\(937\) −3.19233e7 −1.18784 −0.593922 0.804523i \(-0.702421\pi\)
−0.593922 + 0.804523i \(0.702421\pi\)
\(938\) 1.18952e6 0.0441432
\(939\) 0 0
\(940\) −7.15569e6 −0.264139
\(941\) 4.89415e7 1.80179 0.900894 0.434040i \(-0.142912\pi\)
0.900894 + 0.434040i \(0.142912\pi\)
\(942\) 0 0
\(943\) −2.56087e7 −0.937795
\(944\) 4.48522e7 1.63815
\(945\) 0 0
\(946\) 170595. 0.00619781
\(947\) 1.45580e7 0.527505 0.263753 0.964590i \(-0.415040\pi\)
0.263753 + 0.964590i \(0.415040\pi\)
\(948\) 0 0
\(949\) −2.46037e7 −0.886818
\(950\) −4.24825e6 −0.152722
\(951\) 0 0
\(952\) 976572. 0.0349230
\(953\) 2.71191e7 0.967261 0.483631 0.875272i \(-0.339318\pi\)
0.483631 + 0.875272i \(0.339318\pi\)
\(954\) 0 0
\(955\) 8.19219e6 0.290664
\(956\) −2.22889e6 −0.0788759
\(957\) 0 0
\(958\) −2.55062e6 −0.0897907
\(959\) −6.65633e6 −0.233716
\(960\) 0 0
\(961\) 1.06093e7 0.370576
\(962\) −1.11149e6 −0.0387229
\(963\) 0 0
\(964\) −1.41207e6 −0.0489401
\(965\) 1.04164e7 0.360082
\(966\) 0 0
\(967\) 1.16883e7 0.401963 0.200982 0.979595i \(-0.435587\pi\)
0.200982 + 0.979595i \(0.435587\pi\)
\(968\) −5.04480e6 −0.173044
\(969\) 0 0
\(970\) −1.44415e6 −0.0492815
\(971\) −3.07622e6 −0.104706 −0.0523528 0.998629i \(-0.516672\pi\)
−0.0523528 + 0.998629i \(0.516672\pi\)
\(972\) 0 0
\(973\) 2.92024e7 0.988865
\(974\) −2.38021e6 −0.0803930
\(975\) 0 0
\(976\) −7.18964e6 −0.241592
\(977\) −3.81186e7 −1.27762 −0.638809 0.769365i \(-0.720573\pi\)
−0.638809 + 0.769365i \(0.720573\pi\)
\(978\) 0 0
\(979\) −4.69350e6 −0.156509
\(980\) −5.31195e6 −0.176680
\(981\) 0 0
\(982\) −2.72616e6 −0.0902137
\(983\) 3.14497e7 1.03809 0.519043 0.854748i \(-0.326289\pi\)
0.519043 + 0.854748i \(0.326289\pi\)
\(984\) 0 0
\(985\) −2.03047e7 −0.666815
\(986\) −1.00182e6 −0.0328170
\(987\) 0 0
\(988\) −5.66783e7 −1.84725
\(989\) 7.69821e6 0.250264
\(990\) 0 0
\(991\) 1.08378e7 0.350555 0.175277 0.984519i \(-0.443918\pi\)
0.175277 + 0.984519i \(0.443918\pi\)
\(992\) −1.10558e7 −0.356705
\(993\) 0 0
\(994\) 2.90842e6 0.0933665
\(995\) −1.63457e7 −0.523414
\(996\) 0 0
\(997\) −1.37214e7 −0.437179 −0.218590 0.975817i \(-0.570146\pi\)
−0.218590 + 0.975817i \(0.570146\pi\)
\(998\) −5.63211e6 −0.178997
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 387.6.a.c.1.4 8
3.2 odd 2 43.6.a.a.1.5 8
12.11 even 2 688.6.a.e.1.4 8
15.14 odd 2 1075.6.a.a.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.5 8 3.2 odd 2
387.6.a.c.1.4 8 1.1 even 1 trivial
688.6.a.e.1.4 8 12.11 even 2
1075.6.a.a.1.4 8 15.14 odd 2