Properties

Label 76.6.a.b.1.1
Level $76$
Weight $6$
Character 76.1
Self dual yes
Analytic conductor $12.189$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 140 x^{2} - 84 x + 3103\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.80512\) of defining polynomial
Character \(\chi\) \(=\) 76.1

$q$-expansion

\(f(q)\) \(=\) \(q-25.7605 q^{3} -109.245 q^{5} -177.299 q^{7} +420.606 q^{9} +O(q^{10})\) \(q-25.7605 q^{3} -109.245 q^{5} -177.299 q^{7} +420.606 q^{9} +228.019 q^{11} -476.072 q^{13} +2814.22 q^{15} -1868.17 q^{17} -361.000 q^{19} +4567.32 q^{21} +1966.09 q^{23} +8809.50 q^{25} -4575.23 q^{27} -4201.04 q^{29} -3842.27 q^{31} -5873.90 q^{33} +19369.1 q^{35} -15414.3 q^{37} +12263.9 q^{39} +1927.41 q^{41} -7330.22 q^{43} -45949.2 q^{45} +5080.98 q^{47} +14628.0 q^{49} +48125.0 q^{51} -12616.1 q^{53} -24910.0 q^{55} +9299.56 q^{57} +7718.28 q^{59} -17993.0 q^{61} -74573.1 q^{63} +52008.5 q^{65} +10746.2 q^{67} -50647.5 q^{69} +50642.8 q^{71} -58036.2 q^{73} -226938. q^{75} -40427.6 q^{77} +33913.4 q^{79} +15653.1 q^{81} +30761.3 q^{83} +204088. q^{85} +108221. q^{87} +97641.2 q^{89} +84407.1 q^{91} +98978.9 q^{93} +39437.5 q^{95} +105569. q^{97} +95906.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + O(q^{10}) \) \( 4q + 10q^{3} - 110q^{5} + 30q^{7} + 878q^{9} + 706q^{11} + 788q^{13} + 2792q^{15} + 240q^{17} - 1444q^{19} + 7128q^{21} + 5884q^{23} + 11774q^{25} + 4426q^{27} + 5240q^{29} - 860q^{31} + 10748q^{33} + 20322q^{35} - 20732q^{37} + 15404q^{39} - 10204q^{41} - 12554q^{43} - 71306q^{45} - 4826q^{47} - 21376q^{49} + 34518q^{51} - 76484q^{53} - 72914q^{55} - 3610q^{57} + 23898q^{59} - 32482q^{61} - 43566q^{63} - 6076q^{65} + 5022q^{67} - 149524q^{69} + 121300q^{71} - 104700q^{73} - 95230q^{75} - 10002q^{77} + 117128q^{79} + 44924q^{81} + 92832q^{83} + 80322q^{85} + 300148q^{87} + 5988q^{89} + 165618q^{91} + 16180q^{93} + 39710q^{95} + 22972q^{97} + 441418q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −25.7605 −1.65254 −0.826270 0.563274i \(-0.809541\pi\)
−0.826270 + 0.563274i \(0.809541\pi\)
\(4\) 0 0
\(5\) −109.245 −1.95424 −0.977118 0.212696i \(-0.931776\pi\)
−0.977118 + 0.212696i \(0.931776\pi\)
\(6\) 0 0
\(7\) −177.299 −1.36761 −0.683804 0.729666i \(-0.739676\pi\)
−0.683804 + 0.729666i \(0.739676\pi\)
\(8\) 0 0
\(9\) 420.606 1.73089
\(10\) 0 0
\(11\) 228.019 0.568184 0.284092 0.958797i \(-0.408308\pi\)
0.284092 + 0.958797i \(0.408308\pi\)
\(12\) 0 0
\(13\) −476.072 −0.781293 −0.390647 0.920541i \(-0.627749\pi\)
−0.390647 + 0.920541i \(0.627749\pi\)
\(14\) 0 0
\(15\) 2814.22 3.22945
\(16\) 0 0
\(17\) −1868.17 −1.56781 −0.783905 0.620880i \(-0.786775\pi\)
−0.783905 + 0.620880i \(0.786775\pi\)
\(18\) 0 0
\(19\) −361.000 −0.229416
\(20\) 0 0
\(21\) 4567.32 2.26003
\(22\) 0 0
\(23\) 1966.09 0.774967 0.387483 0.921877i \(-0.373344\pi\)
0.387483 + 0.921877i \(0.373344\pi\)
\(24\) 0 0
\(25\) 8809.50 2.81904
\(26\) 0 0
\(27\) −4575.23 −1.20782
\(28\) 0 0
\(29\) −4201.04 −0.927602 −0.463801 0.885939i \(-0.653515\pi\)
−0.463801 + 0.885939i \(0.653515\pi\)
\(30\) 0 0
\(31\) −3842.27 −0.718097 −0.359048 0.933319i \(-0.616899\pi\)
−0.359048 + 0.933319i \(0.616899\pi\)
\(32\) 0 0
\(33\) −5873.90 −0.938948
\(34\) 0 0
\(35\) 19369.1 2.67263
\(36\) 0 0
\(37\) −15414.3 −1.85106 −0.925528 0.378678i \(-0.876379\pi\)
−0.925528 + 0.378678i \(0.876379\pi\)
\(38\) 0 0
\(39\) 12263.9 1.29112
\(40\) 0 0
\(41\) 1927.41 0.179067 0.0895334 0.995984i \(-0.471462\pi\)
0.0895334 + 0.995984i \(0.471462\pi\)
\(42\) 0 0
\(43\) −7330.22 −0.604569 −0.302284 0.953218i \(-0.597749\pi\)
−0.302284 + 0.953218i \(0.597749\pi\)
\(44\) 0 0
\(45\) −45949.2 −3.38257
\(46\) 0 0
\(47\) 5080.98 0.335508 0.167754 0.985829i \(-0.446349\pi\)
0.167754 + 0.985829i \(0.446349\pi\)
\(48\) 0 0
\(49\) 14628.0 0.870351
\(50\) 0 0
\(51\) 48125.0 2.59087
\(52\) 0 0
\(53\) −12616.1 −0.616929 −0.308465 0.951236i \(-0.599815\pi\)
−0.308465 + 0.951236i \(0.599815\pi\)
\(54\) 0 0
\(55\) −24910.0 −1.11037
\(56\) 0 0
\(57\) 9299.56 0.379119
\(58\) 0 0
\(59\) 7718.28 0.288663 0.144331 0.989529i \(-0.453897\pi\)
0.144331 + 0.989529i \(0.453897\pi\)
\(60\) 0 0
\(61\) −17993.0 −0.619127 −0.309564 0.950879i \(-0.600183\pi\)
−0.309564 + 0.950879i \(0.600183\pi\)
\(62\) 0 0
\(63\) −74573.1 −2.36718
\(64\) 0 0
\(65\) 52008.5 1.52683
\(66\) 0 0
\(67\) 10746.2 0.292460 0.146230 0.989251i \(-0.453286\pi\)
0.146230 + 0.989251i \(0.453286\pi\)
\(68\) 0 0
\(69\) −50647.5 −1.28066
\(70\) 0 0
\(71\) 50642.8 1.19226 0.596132 0.802887i \(-0.296704\pi\)
0.596132 + 0.802887i \(0.296704\pi\)
\(72\) 0 0
\(73\) −58036.2 −1.27465 −0.637326 0.770594i \(-0.719960\pi\)
−0.637326 + 0.770594i \(0.719960\pi\)
\(74\) 0 0
\(75\) −226938. −4.65858
\(76\) 0 0
\(77\) −40427.6 −0.777054
\(78\) 0 0
\(79\) 33913.4 0.611369 0.305685 0.952133i \(-0.401115\pi\)
0.305685 + 0.952133i \(0.401115\pi\)
\(80\) 0 0
\(81\) 15653.1 0.265087
\(82\) 0 0
\(83\) 30761.3 0.490128 0.245064 0.969507i \(-0.421191\pi\)
0.245064 + 0.969507i \(0.421191\pi\)
\(84\) 0 0
\(85\) 204088. 3.06387
\(86\) 0 0
\(87\) 108221. 1.53290
\(88\) 0 0
\(89\) 97641.2 1.30665 0.653323 0.757079i \(-0.273374\pi\)
0.653323 + 0.757079i \(0.273374\pi\)
\(90\) 0 0
\(91\) 84407.1 1.06850
\(92\) 0 0
\(93\) 98978.9 1.18668
\(94\) 0 0
\(95\) 39437.5 0.448333
\(96\) 0 0
\(97\) 105569. 1.13922 0.569609 0.821916i \(-0.307095\pi\)
0.569609 + 0.821916i \(0.307095\pi\)
\(98\) 0 0
\(99\) 95906.1 0.983464
\(100\) 0 0
\(101\) −70055.6 −0.683344 −0.341672 0.939819i \(-0.610993\pi\)
−0.341672 + 0.939819i \(0.610993\pi\)
\(102\) 0 0
\(103\) 29196.8 0.271171 0.135585 0.990766i \(-0.456709\pi\)
0.135585 + 0.990766i \(0.456709\pi\)
\(104\) 0 0
\(105\) −498958. −4.41663
\(106\) 0 0
\(107\) −83750.0 −0.707173 −0.353586 0.935402i \(-0.615038\pi\)
−0.353586 + 0.935402i \(0.615038\pi\)
\(108\) 0 0
\(109\) −51998.8 −0.419206 −0.209603 0.977787i \(-0.567217\pi\)
−0.209603 + 0.977787i \(0.567217\pi\)
\(110\) 0 0
\(111\) 397081. 3.05895
\(112\) 0 0
\(113\) 174906. 1.28857 0.644287 0.764784i \(-0.277154\pi\)
0.644287 + 0.764784i \(0.277154\pi\)
\(114\) 0 0
\(115\) −214785. −1.51447
\(116\) 0 0
\(117\) −200239. −1.35233
\(118\) 0 0
\(119\) 331225. 2.14415
\(120\) 0 0
\(121\) −109058. −0.677166
\(122\) 0 0
\(123\) −49651.2 −0.295915
\(124\) 0 0
\(125\) −621005. −3.55484
\(126\) 0 0
\(127\) −10574.2 −0.0581755 −0.0290878 0.999577i \(-0.509260\pi\)
−0.0290878 + 0.999577i \(0.509260\pi\)
\(128\) 0 0
\(129\) 188830. 0.999074
\(130\) 0 0
\(131\) 38671.2 0.196884 0.0984418 0.995143i \(-0.468614\pi\)
0.0984418 + 0.995143i \(0.468614\pi\)
\(132\) 0 0
\(133\) 64005.0 0.313751
\(134\) 0 0
\(135\) 499821. 2.36037
\(136\) 0 0
\(137\) 85272.3 0.388156 0.194078 0.980986i \(-0.437828\pi\)
0.194078 + 0.980986i \(0.437828\pi\)
\(138\) 0 0
\(139\) −198686. −0.872230 −0.436115 0.899891i \(-0.643646\pi\)
−0.436115 + 0.899891i \(0.643646\pi\)
\(140\) 0 0
\(141\) −130889. −0.554440
\(142\) 0 0
\(143\) −108553. −0.443919
\(144\) 0 0
\(145\) 458943. 1.81275
\(146\) 0 0
\(147\) −376825. −1.43829
\(148\) 0 0
\(149\) −81459.5 −0.300591 −0.150295 0.988641i \(-0.548023\pi\)
−0.150295 + 0.988641i \(0.548023\pi\)
\(150\) 0 0
\(151\) −517065. −1.84545 −0.922727 0.385455i \(-0.874045\pi\)
−0.922727 + 0.385455i \(0.874045\pi\)
\(152\) 0 0
\(153\) −785763. −2.71371
\(154\) 0 0
\(155\) 419749. 1.40333
\(156\) 0 0
\(157\) 169444. 0.548627 0.274314 0.961640i \(-0.411549\pi\)
0.274314 + 0.961640i \(0.411549\pi\)
\(158\) 0 0
\(159\) 324998. 1.01950
\(160\) 0 0
\(161\) −348585. −1.05985
\(162\) 0 0
\(163\) −116429. −0.343234 −0.171617 0.985164i \(-0.554899\pi\)
−0.171617 + 0.985164i \(0.554899\pi\)
\(164\) 0 0
\(165\) 641695. 1.83493
\(166\) 0 0
\(167\) −9810.91 −0.0272219 −0.0136109 0.999907i \(-0.504333\pi\)
−0.0136109 + 0.999907i \(0.504333\pi\)
\(168\) 0 0
\(169\) −144649. −0.389581
\(170\) 0 0
\(171\) −151839. −0.397093
\(172\) 0 0
\(173\) 191280. 0.485908 0.242954 0.970038i \(-0.421884\pi\)
0.242954 + 0.970038i \(0.421884\pi\)
\(174\) 0 0
\(175\) −1.56192e6 −3.85534
\(176\) 0 0
\(177\) −198827. −0.477027
\(178\) 0 0
\(179\) 631191. 1.47241 0.736204 0.676759i \(-0.236616\pi\)
0.736204 + 0.676759i \(0.236616\pi\)
\(180\) 0 0
\(181\) −119770. −0.271739 −0.135870 0.990727i \(-0.543383\pi\)
−0.135870 + 0.990727i \(0.543383\pi\)
\(182\) 0 0
\(183\) 463511. 1.02313
\(184\) 0 0
\(185\) 1.68394e6 3.61740
\(186\) 0 0
\(187\) −425978. −0.890806
\(188\) 0 0
\(189\) 811184. 1.65183
\(190\) 0 0
\(191\) −942206. −1.86880 −0.934400 0.356226i \(-0.884063\pi\)
−0.934400 + 0.356226i \(0.884063\pi\)
\(192\) 0 0
\(193\) −560559. −1.08325 −0.541624 0.840621i \(-0.682190\pi\)
−0.541624 + 0.840621i \(0.682190\pi\)
\(194\) 0 0
\(195\) −1.33977e6 −2.52315
\(196\) 0 0
\(197\) 792520. 1.45494 0.727469 0.686140i \(-0.240696\pi\)
0.727469 + 0.686140i \(0.240696\pi\)
\(198\) 0 0
\(199\) 477931. 0.855524 0.427762 0.903891i \(-0.359302\pi\)
0.427762 + 0.903891i \(0.359302\pi\)
\(200\) 0 0
\(201\) −276827. −0.483301
\(202\) 0 0
\(203\) 744840. 1.26860
\(204\) 0 0
\(205\) −210561. −0.349939
\(206\) 0 0
\(207\) 826948. 1.34138
\(208\) 0 0
\(209\) −82314.9 −0.130350
\(210\) 0 0
\(211\) −1.09610e6 −1.69490 −0.847451 0.530874i \(-0.821864\pi\)
−0.847451 + 0.530874i \(0.821864\pi\)
\(212\) 0 0
\(213\) −1.30459e6 −1.97026
\(214\) 0 0
\(215\) 800791. 1.18147
\(216\) 0 0
\(217\) 681230. 0.982075
\(218\) 0 0
\(219\) 1.49504e6 2.10641
\(220\) 0 0
\(221\) 889382. 1.22492
\(222\) 0 0
\(223\) −78676.4 −0.105945 −0.0529727 0.998596i \(-0.516870\pi\)
−0.0529727 + 0.998596i \(0.516870\pi\)
\(224\) 0 0
\(225\) 3.70533e6 4.87945
\(226\) 0 0
\(227\) −965468. −1.24358 −0.621789 0.783185i \(-0.713594\pi\)
−0.621789 + 0.783185i \(0.713594\pi\)
\(228\) 0 0
\(229\) 837329. 1.05513 0.527566 0.849514i \(-0.323105\pi\)
0.527566 + 0.849514i \(0.323105\pi\)
\(230\) 0 0
\(231\) 1.04144e6 1.28411
\(232\) 0 0
\(233\) −881429. −1.06365 −0.531824 0.846855i \(-0.678493\pi\)
−0.531824 + 0.846855i \(0.678493\pi\)
\(234\) 0 0
\(235\) −555072. −0.655662
\(236\) 0 0
\(237\) −873628. −1.01031
\(238\) 0 0
\(239\) 298682. 0.338232 0.169116 0.985596i \(-0.445909\pi\)
0.169116 + 0.985596i \(0.445909\pi\)
\(240\) 0 0
\(241\) −1.04452e6 −1.15844 −0.579219 0.815172i \(-0.696643\pi\)
−0.579219 + 0.815172i \(0.696643\pi\)
\(242\) 0 0
\(243\) 708547. 0.769756
\(244\) 0 0
\(245\) −1.59804e6 −1.70087
\(246\) 0 0
\(247\) 171862. 0.179241
\(248\) 0 0
\(249\) −792427. −0.809955
\(250\) 0 0
\(251\) 1.10873e6 1.11082 0.555408 0.831578i \(-0.312562\pi\)
0.555408 + 0.831578i \(0.312562\pi\)
\(252\) 0 0
\(253\) 448305. 0.440324
\(254\) 0 0
\(255\) −5.25743e6 −5.06317
\(256\) 0 0
\(257\) −83101.8 −0.0784834 −0.0392417 0.999230i \(-0.512494\pi\)
−0.0392417 + 0.999230i \(0.512494\pi\)
\(258\) 0 0
\(259\) 2.73294e6 2.53152
\(260\) 0 0
\(261\) −1.76698e6 −1.60558
\(262\) 0 0
\(263\) −1.75925e6 −1.56833 −0.784167 0.620550i \(-0.786909\pi\)
−0.784167 + 0.620550i \(0.786909\pi\)
\(264\) 0 0
\(265\) 1.37825e6 1.20563
\(266\) 0 0
\(267\) −2.51529e6 −2.15929
\(268\) 0 0
\(269\) 524521. 0.441959 0.220980 0.975278i \(-0.429075\pi\)
0.220980 + 0.975278i \(0.429075\pi\)
\(270\) 0 0
\(271\) 761057. 0.629498 0.314749 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(272\) 0 0
\(273\) −2.17437e6 −1.76574
\(274\) 0 0
\(275\) 2.00873e6 1.60174
\(276\) 0 0
\(277\) −349747. −0.273877 −0.136938 0.990580i \(-0.543726\pi\)
−0.136938 + 0.990580i \(0.543726\pi\)
\(278\) 0 0
\(279\) −1.61608e6 −1.24295
\(280\) 0 0
\(281\) 1.71890e6 1.29863 0.649314 0.760520i \(-0.275056\pi\)
0.649314 + 0.760520i \(0.275056\pi\)
\(282\) 0 0
\(283\) −654088. −0.485479 −0.242739 0.970092i \(-0.578046\pi\)
−0.242739 + 0.970092i \(0.578046\pi\)
\(284\) 0 0
\(285\) −1.01593e6 −0.740888
\(286\) 0 0
\(287\) −341729. −0.244893
\(288\) 0 0
\(289\) 2.07020e6 1.45803
\(290\) 0 0
\(291\) −2.71951e6 −1.88260
\(292\) 0 0
\(293\) 1.93430e6 1.31630 0.658152 0.752885i \(-0.271339\pi\)
0.658152 + 0.752885i \(0.271339\pi\)
\(294\) 0 0
\(295\) −843185. −0.564115
\(296\) 0 0
\(297\) −1.04324e6 −0.686266
\(298\) 0 0
\(299\) −935998. −0.605476
\(300\) 0 0
\(301\) 1.29964e6 0.826813
\(302\) 0 0
\(303\) 1.80467e6 1.12925
\(304\) 0 0
\(305\) 1.96565e6 1.20992
\(306\) 0 0
\(307\) 1.65795e6 1.00398 0.501992 0.864872i \(-0.332601\pi\)
0.501992 + 0.864872i \(0.332601\pi\)
\(308\) 0 0
\(309\) −752126. −0.448120
\(310\) 0 0
\(311\) −794581. −0.465840 −0.232920 0.972496i \(-0.574828\pi\)
−0.232920 + 0.972496i \(0.574828\pi\)
\(312\) 0 0
\(313\) −2.21582e6 −1.27842 −0.639211 0.769032i \(-0.720739\pi\)
−0.639211 + 0.769032i \(0.720739\pi\)
\(314\) 0 0
\(315\) 8.14675e6 4.62602
\(316\) 0 0
\(317\) −1.59381e6 −0.890815 −0.445408 0.895328i \(-0.646941\pi\)
−0.445408 + 0.895328i \(0.646941\pi\)
\(318\) 0 0
\(319\) −957916. −0.527049
\(320\) 0 0
\(321\) 2.15745e6 1.16863
\(322\) 0 0
\(323\) 674409. 0.359681
\(324\) 0 0
\(325\) −4.19396e6 −2.20250
\(326\) 0 0
\(327\) 1.33952e6 0.692754
\(328\) 0 0
\(329\) −900853. −0.458843
\(330\) 0 0
\(331\) 3.22458e6 1.61772 0.808860 0.588001i \(-0.200085\pi\)
0.808860 + 0.588001i \(0.200085\pi\)
\(332\) 0 0
\(333\) −6.48335e6 −3.20397
\(334\) 0 0
\(335\) −1.17397e6 −0.571536
\(336\) 0 0
\(337\) 1.24462e6 0.596982 0.298491 0.954412i \(-0.403517\pi\)
0.298491 + 0.954412i \(0.403517\pi\)
\(338\) 0 0
\(339\) −4.50568e6 −2.12942
\(340\) 0 0
\(341\) −876109. −0.408012
\(342\) 0 0
\(343\) 386336. 0.177309
\(344\) 0 0
\(345\) 5.53299e6 2.50272
\(346\) 0 0
\(347\) −324910. −0.144857 −0.0724284 0.997374i \(-0.523075\pi\)
−0.0724284 + 0.997374i \(0.523075\pi\)
\(348\) 0 0
\(349\) 337444. 0.148299 0.0741495 0.997247i \(-0.476376\pi\)
0.0741495 + 0.997247i \(0.476376\pi\)
\(350\) 0 0
\(351\) 2.17814e6 0.943663
\(352\) 0 0
\(353\) 2.80025e6 1.19608 0.598039 0.801467i \(-0.295947\pi\)
0.598039 + 0.801467i \(0.295947\pi\)
\(354\) 0 0
\(355\) −5.53249e6 −2.32997
\(356\) 0 0
\(357\) −8.53253e6 −3.54329
\(358\) 0 0
\(359\) −3.26867e6 −1.33855 −0.669275 0.743015i \(-0.733395\pi\)
−0.669275 + 0.743015i \(0.733395\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) 2.80940e6 1.11904
\(364\) 0 0
\(365\) 6.34017e6 2.49097
\(366\) 0 0
\(367\) −4.37614e6 −1.69600 −0.848001 0.529995i \(-0.822194\pi\)
−0.848001 + 0.529995i \(0.822194\pi\)
\(368\) 0 0
\(369\) 810681. 0.309945
\(370\) 0 0
\(371\) 2.23682e6 0.843717
\(372\) 0 0
\(373\) 385783. 0.143572 0.0717862 0.997420i \(-0.477130\pi\)
0.0717862 + 0.997420i \(0.477130\pi\)
\(374\) 0 0
\(375\) 1.59974e7 5.87451
\(376\) 0 0
\(377\) 2.00000e6 0.724729
\(378\) 0 0
\(379\) 2.75351e6 0.984665 0.492333 0.870407i \(-0.336144\pi\)
0.492333 + 0.870407i \(0.336144\pi\)
\(380\) 0 0
\(381\) 272398. 0.0961374
\(382\) 0 0
\(383\) 5.67232e6 1.97589 0.987947 0.154792i \(-0.0494709\pi\)
0.987947 + 0.154792i \(0.0494709\pi\)
\(384\) 0 0
\(385\) 4.41652e6 1.51855
\(386\) 0 0
\(387\) −3.08313e6 −1.04644
\(388\) 0 0
\(389\) 4.22486e6 1.41559 0.707797 0.706416i \(-0.249689\pi\)
0.707797 + 0.706416i \(0.249689\pi\)
\(390\) 0 0
\(391\) −3.67298e6 −1.21500
\(392\) 0 0
\(393\) −996192. −0.325358
\(394\) 0 0
\(395\) −3.70487e6 −1.19476
\(396\) 0 0
\(397\) −1.60192e6 −0.510110 −0.255055 0.966927i \(-0.582094\pi\)
−0.255055 + 0.966927i \(0.582094\pi\)
\(398\) 0 0
\(399\) −1.64880e6 −0.518486
\(400\) 0 0
\(401\) −584352. −0.181474 −0.0907369 0.995875i \(-0.528922\pi\)
−0.0907369 + 0.995875i \(0.528922\pi\)
\(402\) 0 0
\(403\) 1.82919e6 0.561044
\(404\) 0 0
\(405\) −1.71003e6 −0.518042
\(406\) 0 0
\(407\) −3.51476e6 −1.05174
\(408\) 0 0
\(409\) −1.70400e6 −0.503687 −0.251844 0.967768i \(-0.581037\pi\)
−0.251844 + 0.967768i \(0.581037\pi\)
\(410\) 0 0
\(411\) −2.19666e6 −0.641444
\(412\) 0 0
\(413\) −1.36845e6 −0.394777
\(414\) 0 0
\(415\) −3.36052e6 −0.957825
\(416\) 0 0
\(417\) 5.11827e6 1.44139
\(418\) 0 0
\(419\) −42813.6 −0.0119137 −0.00595684 0.999982i \(-0.501896\pi\)
−0.00595684 + 0.999982i \(0.501896\pi\)
\(420\) 0 0
\(421\) −4.45775e6 −1.22577 −0.612887 0.790170i \(-0.709992\pi\)
−0.612887 + 0.790170i \(0.709992\pi\)
\(422\) 0 0
\(423\) 2.13709e6 0.580727
\(424\) 0 0
\(425\) −1.64576e7 −4.41972
\(426\) 0 0
\(427\) 3.19015e6 0.846724
\(428\) 0 0
\(429\) 2.79640e6 0.733593
\(430\) 0 0
\(431\) 127317. 0.0330136 0.0165068 0.999864i \(-0.494745\pi\)
0.0165068 + 0.999864i \(0.494745\pi\)
\(432\) 0 0
\(433\) 7.00930e6 1.79661 0.898307 0.439369i \(-0.144798\pi\)
0.898307 + 0.439369i \(0.144798\pi\)
\(434\) 0 0
\(435\) −1.18226e7 −2.99565
\(436\) 0 0
\(437\) −709757. −0.177790
\(438\) 0 0
\(439\) 4.17533e6 1.03402 0.517011 0.855979i \(-0.327045\pi\)
0.517011 + 0.855979i \(0.327045\pi\)
\(440\) 0 0
\(441\) 6.15262e6 1.50648
\(442\) 0 0
\(443\) 529129. 0.128101 0.0640504 0.997947i \(-0.479598\pi\)
0.0640504 + 0.997947i \(0.479598\pi\)
\(444\) 0 0
\(445\) −1.06668e7 −2.55350
\(446\) 0 0
\(447\) 2.09844e6 0.496739
\(448\) 0 0
\(449\) −2.14414e6 −0.501924 −0.250962 0.967997i \(-0.580747\pi\)
−0.250962 + 0.967997i \(0.580747\pi\)
\(450\) 0 0
\(451\) 439487. 0.101743
\(452\) 0 0
\(453\) 1.33199e7 3.04969
\(454\) 0 0
\(455\) −9.22107e6 −2.08811
\(456\) 0 0
\(457\) −6.90778e6 −1.54721 −0.773603 0.633670i \(-0.781548\pi\)
−0.773603 + 0.633670i \(0.781548\pi\)
\(458\) 0 0
\(459\) 8.54729e6 1.89364
\(460\) 0 0
\(461\) −6.58026e6 −1.44208 −0.721042 0.692891i \(-0.756336\pi\)
−0.721042 + 0.692891i \(0.756336\pi\)
\(462\) 0 0
\(463\) 6.91768e6 1.49971 0.749856 0.661601i \(-0.230123\pi\)
0.749856 + 0.661601i \(0.230123\pi\)
\(464\) 0 0
\(465\) −1.08130e7 −2.31906
\(466\) 0 0
\(467\) 1.45033e6 0.307734 0.153867 0.988092i \(-0.450827\pi\)
0.153867 + 0.988092i \(0.450827\pi\)
\(468\) 0 0
\(469\) −1.90528e6 −0.399970
\(470\) 0 0
\(471\) −4.36497e6 −0.906629
\(472\) 0 0
\(473\) −1.67143e6 −0.343507
\(474\) 0 0
\(475\) −3.18023e6 −0.646732
\(476\) 0 0
\(477\) −5.30641e6 −1.06784
\(478\) 0 0
\(479\) 1.81138e6 0.360720 0.180360 0.983601i \(-0.442274\pi\)
0.180360 + 0.983601i \(0.442274\pi\)
\(480\) 0 0
\(481\) 7.33832e6 1.44622
\(482\) 0 0
\(483\) 8.97975e6 1.75145
\(484\) 0 0
\(485\) −1.15329e7 −2.22630
\(486\) 0 0
\(487\) 2.19793e6 0.419944 0.209972 0.977707i \(-0.432663\pi\)
0.209972 + 0.977707i \(0.432663\pi\)
\(488\) 0 0
\(489\) 2.99927e6 0.567209
\(490\) 0 0
\(491\) 7.64236e6 1.43062 0.715310 0.698808i \(-0.246286\pi\)
0.715310 + 0.698808i \(0.246286\pi\)
\(492\) 0 0
\(493\) 7.84825e6 1.45430
\(494\) 0 0
\(495\) −1.04773e7 −1.92192
\(496\) 0 0
\(497\) −8.97893e6 −1.63055
\(498\) 0 0
\(499\) −1.96363e6 −0.353028 −0.176514 0.984298i \(-0.556482\pi\)
−0.176514 + 0.984298i \(0.556482\pi\)
\(500\) 0 0
\(501\) 252734. 0.0449852
\(502\) 0 0
\(503\) −5.51813e6 −0.972460 −0.486230 0.873831i \(-0.661628\pi\)
−0.486230 + 0.873831i \(0.661628\pi\)
\(504\) 0 0
\(505\) 7.65323e6 1.33542
\(506\) 0 0
\(507\) 3.72623e6 0.643798
\(508\) 0 0
\(509\) −2.59852e6 −0.444562 −0.222281 0.974983i \(-0.571350\pi\)
−0.222281 + 0.974983i \(0.571350\pi\)
\(510\) 0 0
\(511\) 1.02898e7 1.74322
\(512\) 0 0
\(513\) 1.65166e6 0.277094
\(514\) 0 0
\(515\) −3.18961e6 −0.529932
\(516\) 0 0
\(517\) 1.15856e6 0.190630
\(518\) 0 0
\(519\) −4.92747e6 −0.802982
\(520\) 0 0
\(521\) −665007. −0.107333 −0.0536664 0.998559i \(-0.517091\pi\)
−0.0536664 + 0.998559i \(0.517091\pi\)
\(522\) 0 0
\(523\) −2.70303e6 −0.432113 −0.216057 0.976381i \(-0.569320\pi\)
−0.216057 + 0.976381i \(0.569320\pi\)
\(524\) 0 0
\(525\) 4.02359e7 6.37111
\(526\) 0 0
\(527\) 7.17800e6 1.12584
\(528\) 0 0
\(529\) −2.57085e6 −0.399427
\(530\) 0 0
\(531\) 3.24636e6 0.499643
\(532\) 0 0
\(533\) −917587. −0.139904
\(534\) 0 0
\(535\) 9.14929e6 1.38198
\(536\) 0 0
\(537\) −1.62598e7 −2.43321
\(538\) 0 0
\(539\) 3.33546e6 0.494520
\(540\) 0 0
\(541\) 9.58822e6 1.40846 0.704230 0.709972i \(-0.251292\pi\)
0.704230 + 0.709972i \(0.251292\pi\)
\(542\) 0 0
\(543\) 3.08535e6 0.449060
\(544\) 0 0
\(545\) 5.68062e6 0.819227
\(546\) 0 0
\(547\) 1.05846e7 1.51254 0.756268 0.654261i \(-0.227020\pi\)
0.756268 + 0.654261i \(0.227020\pi\)
\(548\) 0 0
\(549\) −7.56798e6 −1.07164
\(550\) 0 0
\(551\) 1.51657e6 0.212806
\(552\) 0 0
\(553\) −6.01282e6 −0.836113
\(554\) 0 0
\(555\) −4.33792e7 −5.97790
\(556\) 0 0
\(557\) −4.71496e6 −0.643932 −0.321966 0.946751i \(-0.604344\pi\)
−0.321966 + 0.946751i \(0.604344\pi\)
\(558\) 0 0
\(559\) 3.48971e6 0.472346
\(560\) 0 0
\(561\) 1.09734e7 1.47209
\(562\) 0 0
\(563\) −9.60502e6 −1.27711 −0.638554 0.769577i \(-0.720467\pi\)
−0.638554 + 0.769577i \(0.720467\pi\)
\(564\) 0 0
\(565\) −1.91077e7 −2.51818
\(566\) 0 0
\(567\) −2.77528e6 −0.362535
\(568\) 0 0
\(569\) −6.97008e6 −0.902521 −0.451260 0.892392i \(-0.649025\pi\)
−0.451260 + 0.892392i \(0.649025\pi\)
\(570\) 0 0
\(571\) −5.27949e6 −0.677644 −0.338822 0.940851i \(-0.610028\pi\)
−0.338822 + 0.940851i \(0.610028\pi\)
\(572\) 0 0
\(573\) 2.42718e7 3.08827
\(574\) 0 0
\(575\) 1.73202e7 2.18466
\(576\) 0 0
\(577\) 1.01323e7 1.26697 0.633486 0.773754i \(-0.281623\pi\)
0.633486 + 0.773754i \(0.281623\pi\)
\(578\) 0 0
\(579\) 1.44403e7 1.79011
\(580\) 0 0
\(581\) −5.45395e6 −0.670302
\(582\) 0 0
\(583\) −2.87671e6 −0.350530
\(584\) 0 0
\(585\) 2.18751e7 2.64278
\(586\) 0 0
\(587\) 7.91129e6 0.947659 0.473830 0.880617i \(-0.342871\pi\)
0.473830 + 0.880617i \(0.342871\pi\)
\(588\) 0 0
\(589\) 1.38706e6 0.164743
\(590\) 0 0
\(591\) −2.04158e7 −2.40434
\(592\) 0 0
\(593\) −8.98912e6 −1.04974 −0.524869 0.851183i \(-0.675886\pi\)
−0.524869 + 0.851183i \(0.675886\pi\)
\(594\) 0 0
\(595\) −3.61847e7 −4.19018
\(596\) 0 0
\(597\) −1.23118e7 −1.41379
\(598\) 0 0
\(599\) 9.04333e6 1.02982 0.514910 0.857244i \(-0.327825\pi\)
0.514910 + 0.857244i \(0.327825\pi\)
\(600\) 0 0
\(601\) 8.48341e6 0.958042 0.479021 0.877803i \(-0.340992\pi\)
0.479021 + 0.877803i \(0.340992\pi\)
\(602\) 0 0
\(603\) 4.51990e6 0.506215
\(604\) 0 0
\(605\) 1.19141e7 1.32334
\(606\) 0 0
\(607\) 1.37916e7 1.51930 0.759650 0.650332i \(-0.225370\pi\)
0.759650 + 0.650332i \(0.225370\pi\)
\(608\) 0 0
\(609\) −1.91875e7 −2.09640
\(610\) 0 0
\(611\) −2.41891e6 −0.262130
\(612\) 0 0
\(613\) −8.69314e6 −0.934384 −0.467192 0.884156i \(-0.654734\pi\)
−0.467192 + 0.884156i \(0.654734\pi\)
\(614\) 0 0
\(615\) 5.42415e6 0.578288
\(616\) 0 0
\(617\) 2.63565e6 0.278725 0.139362 0.990241i \(-0.455495\pi\)
0.139362 + 0.990241i \(0.455495\pi\)
\(618\) 0 0
\(619\) −1.47325e7 −1.54543 −0.772717 0.634751i \(-0.781103\pi\)
−0.772717 + 0.634751i \(0.781103\pi\)
\(620\) 0 0
\(621\) −8.99529e6 −0.936022
\(622\) 0 0
\(623\) −1.73117e7 −1.78698
\(624\) 0 0
\(625\) 4.03120e7 4.12795
\(626\) 0 0
\(627\) 2.12048e6 0.215409
\(628\) 0 0
\(629\) 2.87965e7 2.90211
\(630\) 0 0
\(631\) 2.04039e6 0.204005 0.102002 0.994784i \(-0.467475\pi\)
0.102002 + 0.994784i \(0.467475\pi\)
\(632\) 0 0
\(633\) 2.82362e7 2.80089
\(634\) 0 0
\(635\) 1.15519e6 0.113689
\(636\) 0 0
\(637\) −6.96397e6 −0.679999
\(638\) 0 0
\(639\) 2.13007e7 2.06368
\(640\) 0 0
\(641\) 5.92090e6 0.569171 0.284585 0.958651i \(-0.408144\pi\)
0.284585 + 0.958651i \(0.408144\pi\)
\(642\) 0 0
\(643\) 8.71800e6 0.831552 0.415776 0.909467i \(-0.363510\pi\)
0.415776 + 0.909467i \(0.363510\pi\)
\(644\) 0 0
\(645\) −2.06288e7 −1.95243
\(646\) 0 0
\(647\) −1.96829e7 −1.84854 −0.924269 0.381742i \(-0.875324\pi\)
−0.924269 + 0.381742i \(0.875324\pi\)
\(648\) 0 0
\(649\) 1.75992e6 0.164014
\(650\) 0 0
\(651\) −1.75489e7 −1.62292
\(652\) 0 0
\(653\) −4.87843e6 −0.447710 −0.223855 0.974622i \(-0.571864\pi\)
−0.223855 + 0.974622i \(0.571864\pi\)
\(654\) 0 0
\(655\) −4.22464e6 −0.384757
\(656\) 0 0
\(657\) −2.44104e7 −2.20628
\(658\) 0 0
\(659\) 1.08130e7 0.969908 0.484954 0.874540i \(-0.338836\pi\)
0.484954 + 0.874540i \(0.338836\pi\)
\(660\) 0 0
\(661\) 4.22290e6 0.375930 0.187965 0.982176i \(-0.439811\pi\)
0.187965 + 0.982176i \(0.439811\pi\)
\(662\) 0 0
\(663\) −2.29110e7 −2.02423
\(664\) 0 0
\(665\) −6.99224e6 −0.613143
\(666\) 0 0
\(667\) −8.25960e6 −0.718860
\(668\) 0 0
\(669\) 2.02675e6 0.175079
\(670\) 0 0
\(671\) −4.10276e6 −0.351779
\(672\) 0 0
\(673\) −4.79237e6 −0.407861 −0.203931 0.978985i \(-0.565372\pi\)
−0.203931 + 0.978985i \(0.565372\pi\)
\(674\) 0 0
\(675\) −4.03055e7 −3.40490
\(676\) 0 0
\(677\) −8.20013e6 −0.687621 −0.343810 0.939039i \(-0.611718\pi\)
−0.343810 + 0.939039i \(0.611718\pi\)
\(678\) 0 0
\(679\) −1.87173e7 −1.55800
\(680\) 0 0
\(681\) 2.48710e7 2.05506
\(682\) 0 0
\(683\) −1.11848e6 −0.0917438 −0.0458719 0.998947i \(-0.514607\pi\)
−0.0458719 + 0.998947i \(0.514607\pi\)
\(684\) 0 0
\(685\) −9.31559e6 −0.758549
\(686\) 0 0
\(687\) −2.15700e7 −1.74365
\(688\) 0 0
\(689\) 6.00617e6 0.482003
\(690\) 0 0
\(691\) −2.53615e6 −0.202060 −0.101030 0.994883i \(-0.532214\pi\)
−0.101030 + 0.994883i \(0.532214\pi\)
\(692\) 0 0
\(693\) −1.70041e7 −1.34499
\(694\) 0 0
\(695\) 2.17055e7 1.70454
\(696\) 0 0
\(697\) −3.60073e6 −0.280743
\(698\) 0 0
\(699\) 2.27061e7 1.75772
\(700\) 0 0
\(701\) −2.13234e7 −1.63893 −0.819466 0.573128i \(-0.805730\pi\)
−0.819466 + 0.573128i \(0.805730\pi\)
\(702\) 0 0
\(703\) 5.56457e6 0.424662
\(704\) 0 0
\(705\) 1.42990e7 1.08351
\(706\) 0 0
\(707\) 1.24208e7 0.934546
\(708\) 0 0
\(709\) 9.73372e6 0.727216 0.363608 0.931552i \(-0.381545\pi\)
0.363608 + 0.931552i \(0.381545\pi\)
\(710\) 0 0
\(711\) 1.42642e7 1.05821
\(712\) 0 0
\(713\) −7.55423e6 −0.556501
\(714\) 0 0
\(715\) 1.18589e7 0.867522
\(716\) 0 0
\(717\) −7.69422e6 −0.558942
\(718\) 0 0
\(719\) −1.68007e6 −0.121200 −0.0606002 0.998162i \(-0.519301\pi\)
−0.0606002 + 0.998162i \(0.519301\pi\)
\(720\) 0 0
\(721\) −5.17657e6 −0.370855
\(722\) 0 0
\(723\) 2.69073e7 1.91437
\(724\) 0 0
\(725\) −3.70091e7 −2.61495
\(726\) 0 0
\(727\) −1.18408e7 −0.830891 −0.415445 0.909618i \(-0.636374\pi\)
−0.415445 + 0.909618i \(0.636374\pi\)
\(728\) 0 0
\(729\) −2.20563e7 −1.53714
\(730\) 0 0
\(731\) 1.36941e7 0.947850
\(732\) 0 0
\(733\) −7.38510e6 −0.507688 −0.253844 0.967245i \(-0.581695\pi\)
−0.253844 + 0.967245i \(0.581695\pi\)
\(734\) 0 0
\(735\) 4.11663e7 2.81076
\(736\) 0 0
\(737\) 2.45033e6 0.166171
\(738\) 0 0
\(739\) 2.54272e7 1.71272 0.856361 0.516377i \(-0.172720\pi\)
0.856361 + 0.516377i \(0.172720\pi\)
\(740\) 0 0
\(741\) −4.42726e6 −0.296203
\(742\) 0 0
\(743\) 1.96141e7 1.30345 0.651727 0.758454i \(-0.274045\pi\)
0.651727 + 0.758454i \(0.274045\pi\)
\(744\) 0 0
\(745\) 8.89905e6 0.587426
\(746\) 0 0
\(747\) 1.29384e7 0.848356
\(748\) 0 0
\(749\) 1.48488e7 0.967135
\(750\) 0 0
\(751\) −1.53430e6 −0.0992685 −0.0496342 0.998767i \(-0.515806\pi\)
−0.0496342 + 0.998767i \(0.515806\pi\)
\(752\) 0 0
\(753\) −2.85615e7 −1.83567
\(754\) 0 0
\(755\) 5.64869e7 3.60645
\(756\) 0 0
\(757\) −1.20770e6 −0.0765984 −0.0382992 0.999266i \(-0.512194\pi\)
−0.0382992 + 0.999266i \(0.512194\pi\)
\(758\) 0 0
\(759\) −1.15486e7 −0.727653
\(760\) 0 0
\(761\) −1.52116e7 −0.952165 −0.476082 0.879401i \(-0.657944\pi\)
−0.476082 + 0.879401i \(0.657944\pi\)
\(762\) 0 0
\(763\) 9.21934e6 0.573309
\(764\) 0 0
\(765\) 8.58408e7 5.30322
\(766\) 0 0
\(767\) −3.67446e6 −0.225530
\(768\) 0 0
\(769\) −3.34827e6 −0.204176 −0.102088 0.994775i \(-0.532552\pi\)
−0.102088 + 0.994775i \(0.532552\pi\)
\(770\) 0 0
\(771\) 2.14075e6 0.129697
\(772\) 0 0
\(773\) 1.37017e7 0.824758 0.412379 0.911012i \(-0.364698\pi\)
0.412379 + 0.911012i \(0.364698\pi\)
\(774\) 0 0
\(775\) −3.38485e7 −2.02435
\(776\) 0 0
\(777\) −7.04021e7 −4.18344
\(778\) 0 0
\(779\) −695796. −0.0410807
\(780\) 0 0
\(781\) 1.15475e7 0.677426
\(782\) 0 0
\(783\) 1.92207e7 1.12038
\(784\) 0 0
\(785\) −1.85110e7 −1.07215
\(786\) 0 0
\(787\) 2.18608e7 1.25814 0.629070 0.777349i \(-0.283436\pi\)
0.629070 + 0.777349i \(0.283436\pi\)
\(788\) 0 0
\(789\) 4.53193e7 2.59173
\(790\) 0 0
\(791\) −3.10108e7 −1.76226
\(792\) 0 0
\(793\) 8.56598e6 0.483720
\(794\) 0 0
\(795\) −3.55044e7 −1.99234
\(796\) 0 0
\(797\) −8.36301e6 −0.466355 −0.233177 0.972434i \(-0.574912\pi\)
−0.233177 + 0.972434i \(0.574912\pi\)
\(798\) 0 0
\(799\) −9.49212e6 −0.526013
\(800\) 0 0
\(801\) 4.10685e7 2.26166
\(802\) 0 0
\(803\) −1.32334e7 −0.724238
\(804\) 0 0
\(805\) 3.80813e7 2.07120
\(806\) 0 0
\(807\) −1.35120e7 −0.730355
\(808\) 0 0
\(809\) −2.37307e6 −0.127479 −0.0637396 0.997967i \(-0.520303\pi\)
−0.0637396 + 0.997967i \(0.520303\pi\)
\(810\) 0 0
\(811\) −2.18207e7 −1.16498 −0.582489 0.812839i \(-0.697921\pi\)
−0.582489 + 0.812839i \(0.697921\pi\)
\(812\) 0 0
\(813\) −1.96053e7 −1.04027
\(814\) 0 0
\(815\) 1.27193e7 0.670761
\(816\) 0 0
\(817\) 2.64621e6 0.138698
\(818\) 0 0
\(819\) 3.55021e7 1.84946
\(820\) 0 0
\(821\) −5.63150e6 −0.291586 −0.145793 0.989315i \(-0.546573\pi\)
−0.145793 + 0.989315i \(0.546573\pi\)
\(822\) 0 0
\(823\) −2.74886e7 −1.41466 −0.707331 0.706882i \(-0.750101\pi\)
−0.707331 + 0.706882i \(0.750101\pi\)
\(824\) 0 0
\(825\) −5.17461e7 −2.64693
\(826\) 0 0
\(827\) 4.53327e6 0.230488 0.115244 0.993337i \(-0.463235\pi\)
0.115244 + 0.993337i \(0.463235\pi\)
\(828\) 0 0
\(829\) −2.86843e7 −1.44963 −0.724817 0.688942i \(-0.758075\pi\)
−0.724817 + 0.688942i \(0.758075\pi\)
\(830\) 0 0
\(831\) 9.00968e6 0.452592
\(832\) 0 0
\(833\) −2.73276e7 −1.36455
\(834\) 0 0
\(835\) 1.07179e6 0.0531980
\(836\) 0 0
\(837\) 1.75792e7 0.867334
\(838\) 0 0
\(839\) 2.47030e7 1.21156 0.605781 0.795632i \(-0.292861\pi\)
0.605781 + 0.795632i \(0.292861\pi\)
\(840\) 0 0
\(841\) −2.86243e6 −0.139555
\(842\) 0 0
\(843\) −4.42798e7 −2.14604
\(844\) 0 0
\(845\) 1.58022e7 0.761334
\(846\) 0 0
\(847\) 1.93360e7 0.926098
\(848\) 0 0
\(849\) 1.68497e7 0.802273
\(850\) 0 0
\(851\) −3.03059e7 −1.43451
\(852\) 0 0
\(853\) 2.47224e7 1.16337 0.581686 0.813414i \(-0.302393\pi\)
0.581686 + 0.813414i \(0.302393\pi\)
\(854\) 0 0
\(855\) 1.65876e7 0.776014
\(856\) 0 0
\(857\) −2.44852e7 −1.13881 −0.569404 0.822058i \(-0.692826\pi\)
−0.569404 + 0.822058i \(0.692826\pi\)
\(858\) 0 0
\(859\) −6.62049e6 −0.306131 −0.153066 0.988216i \(-0.548915\pi\)
−0.153066 + 0.988216i \(0.548915\pi\)
\(860\) 0 0
\(861\) 8.80312e6 0.404696
\(862\) 0 0
\(863\) 1.86778e6 0.0853687 0.0426843 0.999089i \(-0.486409\pi\)
0.0426843 + 0.999089i \(0.486409\pi\)
\(864\) 0 0
\(865\) −2.08964e7 −0.949579
\(866\) 0 0
\(867\) −5.33294e7 −2.40945
\(868\) 0 0
\(869\) 7.73290e6 0.347370
\(870\) 0 0
\(871\) −5.11594e6 −0.228497
\(872\) 0 0
\(873\) 4.44029e7 1.97186
\(874\) 0 0
\(875\) 1.10104e8 4.86162
\(876\) 0 0
\(877\) −2.99625e7 −1.31546 −0.657732 0.753252i \(-0.728484\pi\)
−0.657732 + 0.753252i \(0.728484\pi\)
\(878\) 0 0
\(879\) −4.98288e7 −2.17524
\(880\) 0 0
\(881\) 1.53722e7 0.667261 0.333631 0.942704i \(-0.391726\pi\)
0.333631 + 0.942704i \(0.391726\pi\)
\(882\) 0 0
\(883\) 3.92079e7 1.69228 0.846139 0.532962i \(-0.178921\pi\)
0.846139 + 0.532962i \(0.178921\pi\)
\(884\) 0 0
\(885\) 2.17209e7 0.932223
\(886\) 0 0
\(887\) 1.26437e7 0.539589 0.269795 0.962918i \(-0.413044\pi\)
0.269795 + 0.962918i \(0.413044\pi\)
\(888\) 0 0
\(889\) 1.87481e6 0.0795613
\(890\) 0 0
\(891\) 3.56921e6 0.150618
\(892\) 0 0
\(893\) −1.83423e6 −0.0769708
\(894\) 0 0
\(895\) −6.89546e7 −2.87743
\(896\) 0 0
\(897\) 2.41118e7 1.00057
\(898\) 0 0
\(899\) 1.61415e7 0.666108
\(900\) 0 0
\(901\) 2.35690e7 0.967228
\(902\) 0 0
\(903\) −3.34795e7 −1.36634
\(904\) 0 0
\(905\) 1.30843e7 0.531043
\(906\) 0 0
\(907\) −2.09717e7 −0.846475 −0.423238 0.906019i \(-0.639106\pi\)
−0.423238 + 0.906019i \(0.639106\pi\)
\(908\) 0 0
\(909\) −2.94658e7 −1.18279
\(910\) 0 0
\(911\) −4.60866e7 −1.83983 −0.919916 0.392114i \(-0.871744\pi\)
−0.919916 + 0.392114i \(0.871744\pi\)
\(912\) 0 0
\(913\) 7.01416e6 0.278483
\(914\) 0 0
\(915\) −5.06363e7 −1.99944
\(916\) 0 0
\(917\) −6.85637e6 −0.269260
\(918\) 0 0
\(919\) 5.84244e6 0.228195 0.114097 0.993470i \(-0.463602\pi\)
0.114097 + 0.993470i \(0.463602\pi\)
\(920\) 0 0
\(921\) −4.27098e7 −1.65912
\(922\) 0 0
\(923\) −2.41096e7 −0.931507
\(924\) 0 0
\(925\) −1.35792e8 −5.21821
\(926\) 0 0
\(927\) 1.22804e7 0.469366
\(928\) 0 0
\(929\) 2.13555e7 0.811839 0.405919 0.913909i \(-0.366951\pi\)
0.405919 + 0.913909i \(0.366951\pi\)
\(930\) 0 0
\(931\) −5.28071e6 −0.199672
\(932\) 0 0
\(933\) 2.04688e7 0.769820
\(934\) 0 0
\(935\) 4.65360e7 1.74085
\(936\) 0 0
\(937\) −1.24265e7 −0.462382 −0.231191 0.972908i \(-0.574262\pi\)
−0.231191 + 0.972908i \(0.574262\pi\)
\(938\) 0 0
\(939\) 5.70808e7 2.11264
\(940\) 0 0
\(941\) 3.44931e7 1.26987 0.634934 0.772566i \(-0.281027\pi\)
0.634934 + 0.772566i \(0.281027\pi\)
\(942\) 0 0
\(943\) 3.78946e6 0.138771
\(944\) 0 0
\(945\) −8.86179e7 −3.22806
\(946\) 0 0
\(947\) 2.32057e7 0.840853 0.420426 0.907327i \(-0.361881\pi\)
0.420426 + 0.907327i \(0.361881\pi\)
\(948\) 0 0
\(949\) 2.76294e7 0.995877
\(950\) 0 0
\(951\) 4.10573e7 1.47211
\(952\) 0 0
\(953\) −2.54048e7 −0.906116 −0.453058 0.891481i \(-0.649667\pi\)
−0.453058 + 0.891481i \(0.649667\pi\)
\(954\) 0 0
\(955\) 1.02931e8 3.65208
\(956\) 0 0
\(957\) 2.46765e7 0.870969
\(958\) 0 0
\(959\) −1.51187e7 −0.530845
\(960\) 0 0
\(961\) −1.38662e7 −0.484337
\(962\) 0 0
\(963\) −3.52258e7 −1.22404
\(964\) 0 0
\(965\) 6.12384e7 2.11692
\(966\) 0 0
\(967\) 5.24428e6 0.180352 0.0901758 0.995926i \(-0.471257\pi\)
0.0901758 + 0.995926i \(0.471257\pi\)
\(968\) 0 0
\(969\) −1.73731e7 −0.594386
\(970\) 0 0
\(971\) −4.50453e7 −1.53321 −0.766605 0.642119i \(-0.778055\pi\)
−0.766605 + 0.642119i \(0.778055\pi\)
\(972\) 0 0
\(973\) 3.52269e7 1.19287
\(974\) 0 0
\(975\) 1.08039e8 3.63972
\(976\) 0 0
\(977\) −3.97190e7 −1.33126 −0.665628 0.746284i \(-0.731836\pi\)
−0.665628 + 0.746284i \(0.731836\pi\)
\(978\) 0 0
\(979\) 2.22641e7 0.742416
\(980\) 0 0
\(981\) −2.18710e7 −0.725598
\(982\) 0 0
\(983\) −5.93498e7 −1.95900 −0.979502 0.201436i \(-0.935439\pi\)
−0.979502 + 0.201436i \(0.935439\pi\)
\(984\) 0 0
\(985\) −8.65790e7 −2.84329
\(986\) 0 0
\(987\) 2.32065e7 0.758257
\(988\) 0 0
\(989\) −1.44118e7 −0.468521
\(990\) 0 0
\(991\) 9.53405e6 0.308385 0.154193 0.988041i \(-0.450722\pi\)
0.154193 + 0.988041i \(0.450722\pi\)
\(992\) 0 0
\(993\) −8.30670e7 −2.67335
\(994\) 0 0
\(995\) −5.22116e7 −1.67190
\(996\) 0 0
\(997\) 2.39124e7 0.761878 0.380939 0.924600i \(-0.375601\pi\)
0.380939 + 0.924600i \(0.375601\pi\)
\(998\) 0 0
\(999\) 7.05240e7 2.23575
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 76.6.a.b.1.1 4
3.2 odd 2 684.6.a.e.1.4 4
4.3 odd 2 304.6.a.k.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.6.a.b.1.1 4 1.1 even 1 trivial
304.6.a.k.1.4 4 4.3 odd 2
684.6.a.e.1.4 4 3.2 odd 2