Properties

Label 441.6.a.s.1.2
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.38987\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.38987 q^{2} +56.1696 q^{4} -71.7291 q^{5} +226.949 q^{8} +O(q^{10})\) \(q+9.38987 q^{2} +56.1696 q^{4} -71.7291 q^{5} +226.949 q^{8} -673.526 q^{10} +560.610 q^{11} -533.509 q^{13} +333.597 q^{16} +1005.70 q^{17} -1368.53 q^{19} -4028.99 q^{20} +5264.05 q^{22} -3228.08 q^{23} +2020.06 q^{25} -5009.58 q^{26} +753.456 q^{29} -8206.42 q^{31} -4129.95 q^{32} +9443.39 q^{34} -2808.66 q^{37} -12850.3 q^{38} -16278.9 q^{40} +245.827 q^{41} -17504.5 q^{43} +31489.2 q^{44} -30311.2 q^{46} -16345.5 q^{47} +18968.1 q^{50} -29967.0 q^{52} +29641.7 q^{53} -40212.0 q^{55} +7074.85 q^{58} -10356.1 q^{59} -954.179 q^{61} -77057.2 q^{62} -49454.8 q^{64} +38268.1 q^{65} -19815.2 q^{67} +56489.8 q^{68} -62125.4 q^{71} -27109.6 q^{73} -26373.0 q^{74} -76869.6 q^{76} +44687.4 q^{79} -23928.6 q^{80} +2308.29 q^{82} +15606.6 q^{83} -72137.9 q^{85} -164365. q^{86} +127230. q^{88} +13635.3 q^{89} -181320. q^{92} -153482. q^{94} +98163.1 q^{95} +12919.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29} - 4294 q^{31} - 1017 q^{32} + 17940 q^{34} - 19109 q^{37} - 6828 q^{38} - 10545 q^{40} - 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} - 23160 q^{47} + 29352 q^{50} - 33424 q^{52} + 31653 q^{53} - 17889 q^{55} - 2277 q^{58} + 41097 q^{59} - 42052 q^{61} - 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} + 44748 q^{68} - 102096 q^{71} + 28577 q^{73} + 77784 q^{74} - 85192 q^{76} - 18464 q^{79} - 71511 q^{80} + 86040 q^{82} + 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} + 29322 q^{89} - 166908 q^{92} - 109938 q^{94} + 61662 q^{95} + 9791 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.38987 1.65991 0.829955 0.557831i \(-0.188366\pi\)
0.829955 + 0.557831i \(0.188366\pi\)
\(3\) 0 0
\(4\) 56.1696 1.75530
\(5\) −71.7291 −1.28313 −0.641564 0.767069i \(-0.721714\pi\)
−0.641564 + 0.767069i \(0.721714\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 226.949 1.25373
\(9\) 0 0
\(10\) −673.526 −2.12988
\(11\) 560.610 1.39694 0.698472 0.715637i \(-0.253863\pi\)
0.698472 + 0.715637i \(0.253863\pi\)
\(12\) 0 0
\(13\) −533.509 −0.875555 −0.437777 0.899083i \(-0.644234\pi\)
−0.437777 + 0.899083i \(0.644234\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 333.597 0.325778
\(17\) 1005.70 0.844007 0.422004 0.906594i \(-0.361327\pi\)
0.422004 + 0.906594i \(0.361327\pi\)
\(18\) 0 0
\(19\) −1368.53 −0.869699 −0.434850 0.900503i \(-0.643199\pi\)
−0.434850 + 0.900503i \(0.643199\pi\)
\(20\) −4028.99 −2.25228
\(21\) 0 0
\(22\) 5264.05 2.31880
\(23\) −3228.08 −1.27240 −0.636201 0.771523i \(-0.719495\pi\)
−0.636201 + 0.771523i \(0.719495\pi\)
\(24\) 0 0
\(25\) 2020.06 0.646419
\(26\) −5009.58 −1.45334
\(27\) 0 0
\(28\) 0 0
\(29\) 753.456 0.166365 0.0831827 0.996534i \(-0.473492\pi\)
0.0831827 + 0.996534i \(0.473492\pi\)
\(30\) 0 0
\(31\) −8206.42 −1.53373 −0.766866 0.641807i \(-0.778185\pi\)
−0.766866 + 0.641807i \(0.778185\pi\)
\(32\) −4129.95 −0.712968
\(33\) 0 0
\(34\) 9443.39 1.40098
\(35\) 0 0
\(36\) 0 0
\(37\) −2808.66 −0.337284 −0.168642 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(38\) −12850.3 −1.44362
\(39\) 0 0
\(40\) −16278.9 −1.60870
\(41\) 245.827 0.0228387 0.0114193 0.999935i \(-0.496365\pi\)
0.0114193 + 0.999935i \(0.496365\pi\)
\(42\) 0 0
\(43\) −17504.5 −1.44371 −0.721853 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(44\) 31489.2 2.45206
\(45\) 0 0
\(46\) −30311.2 −2.11207
\(47\) −16345.5 −1.07933 −0.539663 0.841881i \(-0.681449\pi\)
−0.539663 + 0.841881i \(0.681449\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 18968.1 1.07300
\(51\) 0 0
\(52\) −29967.0 −1.53686
\(53\) 29641.7 1.44948 0.724741 0.689021i \(-0.241959\pi\)
0.724741 + 0.689021i \(0.241959\pi\)
\(54\) 0 0
\(55\) −40212.0 −1.79246
\(56\) 0 0
\(57\) 0 0
\(58\) 7074.85 0.276151
\(59\) −10356.1 −0.387317 −0.193659 0.981069i \(-0.562035\pi\)
−0.193659 + 0.981069i \(0.562035\pi\)
\(60\) 0 0
\(61\) −954.179 −0.0328326 −0.0164163 0.999865i \(-0.505226\pi\)
−0.0164163 + 0.999865i \(0.505226\pi\)
\(62\) −77057.2 −2.54586
\(63\) 0 0
\(64\) −49454.8 −1.50924
\(65\) 38268.1 1.12345
\(66\) 0 0
\(67\) −19815.2 −0.539276 −0.269638 0.962962i \(-0.586904\pi\)
−0.269638 + 0.962962i \(0.586904\pi\)
\(68\) 56489.8 1.48149
\(69\) 0 0
\(70\) 0 0
\(71\) −62125.4 −1.46259 −0.731296 0.682060i \(-0.761084\pi\)
−0.731296 + 0.682060i \(0.761084\pi\)
\(72\) 0 0
\(73\) −27109.6 −0.595410 −0.297705 0.954658i \(-0.596221\pi\)
−0.297705 + 0.954658i \(0.596221\pi\)
\(74\) −26373.0 −0.559861
\(75\) 0 0
\(76\) −76869.6 −1.52658
\(77\) 0 0
\(78\) 0 0
\(79\) 44687.4 0.805595 0.402798 0.915289i \(-0.368038\pi\)
0.402798 + 0.915289i \(0.368038\pi\)
\(80\) −23928.6 −0.418015
\(81\) 0 0
\(82\) 2308.29 0.0379101
\(83\) 15606.6 0.248665 0.124332 0.992241i \(-0.460321\pi\)
0.124332 + 0.992241i \(0.460321\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) −164365. −2.39642
\(87\) 0 0
\(88\) 127230. 1.75139
\(89\) 13635.3 0.182469 0.0912347 0.995829i \(-0.470919\pi\)
0.0912347 + 0.995829i \(0.470919\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −181320. −2.23345
\(93\) 0 0
\(94\) −153482. −1.79159
\(95\) 98163.1 1.11594
\(96\) 0 0
\(97\) 12919.5 0.139417 0.0697086 0.997567i \(-0.477793\pi\)
0.0697086 + 0.997567i \(0.477793\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 113466. 1.13466
\(101\) 25142.9 0.245252 0.122626 0.992453i \(-0.460868\pi\)
0.122626 + 0.992453i \(0.460868\pi\)
\(102\) 0 0
\(103\) 160753. 1.49302 0.746511 0.665373i \(-0.231727\pi\)
0.746511 + 0.665373i \(0.231727\pi\)
\(104\) −121079. −1.09771
\(105\) 0 0
\(106\) 278331. 2.40601
\(107\) 94375.5 0.796893 0.398446 0.917192i \(-0.369549\pi\)
0.398446 + 0.917192i \(0.369549\pi\)
\(108\) 0 0
\(109\) −83393.5 −0.672304 −0.336152 0.941808i \(-0.609126\pi\)
−0.336152 + 0.941808i \(0.609126\pi\)
\(110\) −377586. −2.97532
\(111\) 0 0
\(112\) 0 0
\(113\) 179254. 1.32060 0.660301 0.751001i \(-0.270429\pi\)
0.660301 + 0.751001i \(0.270429\pi\)
\(114\) 0 0
\(115\) 231547. 1.63266
\(116\) 42321.3 0.292021
\(117\) 0 0
\(118\) −97242.5 −0.642911
\(119\) 0 0
\(120\) 0 0
\(121\) 153233. 0.951455
\(122\) −8959.61 −0.0544991
\(123\) 0 0
\(124\) −460951. −2.69216
\(125\) 79256.4 0.453690
\(126\) 0 0
\(127\) −143674. −0.790440 −0.395220 0.918586i \(-0.629332\pi\)
−0.395220 + 0.918586i \(0.629332\pi\)
\(128\) −332215. −1.79223
\(129\) 0 0
\(130\) 359332. 1.86482
\(131\) 52289.9 0.266219 0.133110 0.991101i \(-0.457504\pi\)
0.133110 + 0.991101i \(0.457504\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −186062. −0.895150
\(135\) 0 0
\(136\) 228243. 1.05816
\(137\) −9410.10 −0.0428344 −0.0214172 0.999771i \(-0.506818\pi\)
−0.0214172 + 0.999771i \(0.506818\pi\)
\(138\) 0 0
\(139\) −183094. −0.803781 −0.401890 0.915688i \(-0.631647\pi\)
−0.401890 + 0.915688i \(0.631647\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −583349. −2.42777
\(143\) −299090. −1.22310
\(144\) 0 0
\(145\) −54044.7 −0.213468
\(146\) −254556. −0.988328
\(147\) 0 0
\(148\) −157762. −0.592034
\(149\) 167002. 0.616247 0.308123 0.951346i \(-0.400299\pi\)
0.308123 + 0.951346i \(0.400299\pi\)
\(150\) 0 0
\(151\) 376264. 1.34292 0.671461 0.741040i \(-0.265667\pi\)
0.671461 + 0.741040i \(0.265667\pi\)
\(152\) −310586. −1.09037
\(153\) 0 0
\(154\) 0 0
\(155\) 588639. 1.96797
\(156\) 0 0
\(157\) −39075.0 −0.126517 −0.0632587 0.997997i \(-0.520149\pi\)
−0.0632587 + 0.997997i \(0.520149\pi\)
\(158\) 419608. 1.33722
\(159\) 0 0
\(160\) 296237. 0.914829
\(161\) 0 0
\(162\) 0 0
\(163\) −477919. −1.40892 −0.704458 0.709745i \(-0.748810\pi\)
−0.704458 + 0.709745i \(0.748810\pi\)
\(164\) 13808.0 0.0400887
\(165\) 0 0
\(166\) 146544. 0.412761
\(167\) 39793.4 0.110413 0.0552064 0.998475i \(-0.482418\pi\)
0.0552064 + 0.998475i \(0.482418\pi\)
\(168\) 0 0
\(169\) −86661.4 −0.233404
\(170\) −677366. −1.79763
\(171\) 0 0
\(172\) −983221. −2.53414
\(173\) 48338.2 0.122794 0.0613968 0.998113i \(-0.480445\pi\)
0.0613968 + 0.998113i \(0.480445\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 187018. 0.455094
\(177\) 0 0
\(178\) 128034. 0.302883
\(179\) 142911. 0.333374 0.166687 0.986010i \(-0.446693\pi\)
0.166687 + 0.986010i \(0.446693\pi\)
\(180\) 0 0
\(181\) 77245.3 0.175257 0.0876285 0.996153i \(-0.472071\pi\)
0.0876285 + 0.996153i \(0.472071\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −732610. −1.59525
\(185\) 201463. 0.432778
\(186\) 0 0
\(187\) 563806. 1.17903
\(188\) −918119. −1.89454
\(189\) 0 0
\(190\) 921739. 1.85235
\(191\) −272054. −0.539600 −0.269800 0.962916i \(-0.586958\pi\)
−0.269800 + 0.962916i \(0.586958\pi\)
\(192\) 0 0
\(193\) −16033.8 −0.0309844 −0.0154922 0.999880i \(-0.504932\pi\)
−0.0154922 + 0.999880i \(0.504932\pi\)
\(194\) 121312. 0.231420
\(195\) 0 0
\(196\) 0 0
\(197\) −1.03228e6 −1.89510 −0.947552 0.319603i \(-0.896451\pi\)
−0.947552 + 0.319603i \(0.896451\pi\)
\(198\) 0 0
\(199\) 881736. 1.57836 0.789180 0.614162i \(-0.210506\pi\)
0.789180 + 0.614162i \(0.210506\pi\)
\(200\) 458451. 0.810435
\(201\) 0 0
\(202\) 236089. 0.407096
\(203\) 0 0
\(204\) 0 0
\(205\) −17633.0 −0.0293049
\(206\) 1.50945e6 2.47828
\(207\) 0 0
\(208\) −177977. −0.285237
\(209\) −767210. −1.21492
\(210\) 0 0
\(211\) −372813. −0.576480 −0.288240 0.957558i \(-0.593070\pi\)
−0.288240 + 0.957558i \(0.593070\pi\)
\(212\) 1.66496e6 2.54428
\(213\) 0 0
\(214\) 886174. 1.32277
\(215\) 1.25558e6 1.85246
\(216\) 0 0
\(217\) 0 0
\(218\) −783054. −1.11596
\(219\) 0 0
\(220\) −2.25869e6 −3.14630
\(221\) −536550. −0.738975
\(222\) 0 0
\(223\) 1.08205e6 1.45708 0.728541 0.685002i \(-0.240199\pi\)
0.728541 + 0.685002i \(0.240199\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.68317e6 2.19208
\(227\) 553049. 0.712359 0.356179 0.934418i \(-0.384079\pi\)
0.356179 + 0.934418i \(0.384079\pi\)
\(228\) 0 0
\(229\) −523024. −0.659072 −0.329536 0.944143i \(-0.606892\pi\)
−0.329536 + 0.944143i \(0.606892\pi\)
\(230\) 2.17420e6 2.71006
\(231\) 0 0
\(232\) 170996. 0.208577
\(233\) 364181. 0.439468 0.219734 0.975560i \(-0.429481\pi\)
0.219734 + 0.975560i \(0.429481\pi\)
\(234\) 0 0
\(235\) 1.17245e6 1.38492
\(236\) −581698. −0.679858
\(237\) 0 0
\(238\) 0 0
\(239\) −371841. −0.421078 −0.210539 0.977585i \(-0.567522\pi\)
−0.210539 + 0.977585i \(0.567522\pi\)
\(240\) 0 0
\(241\) 1.71147e6 1.89814 0.949069 0.315067i \(-0.102027\pi\)
0.949069 + 0.315067i \(0.102027\pi\)
\(242\) 1.43883e6 1.57933
\(243\) 0 0
\(244\) −53595.8 −0.0576310
\(245\) 0 0
\(246\) 0 0
\(247\) 730121. 0.761469
\(248\) −1.86244e6 −1.92289
\(249\) 0 0
\(250\) 744207. 0.753084
\(251\) 58134.1 0.0582434 0.0291217 0.999576i \(-0.490729\pi\)
0.0291217 + 0.999576i \(0.490729\pi\)
\(252\) 0 0
\(253\) −1.80969e6 −1.77748
\(254\) −1.34908e6 −1.31206
\(255\) 0 0
\(256\) −1.53691e6 −1.46571
\(257\) −311839. −0.294509 −0.147254 0.989099i \(-0.547044\pi\)
−0.147254 + 0.989099i \(0.547044\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.14950e6 1.97199
\(261\) 0 0
\(262\) 490995. 0.441900
\(263\) −863965. −0.770206 −0.385103 0.922874i \(-0.625834\pi\)
−0.385103 + 0.922874i \(0.625834\pi\)
\(264\) 0 0
\(265\) −2.12617e6 −1.85987
\(266\) 0 0
\(267\) 0 0
\(268\) −1.11301e6 −0.946592
\(269\) −1.12069e6 −0.944290 −0.472145 0.881521i \(-0.656520\pi\)
−0.472145 + 0.881521i \(0.656520\pi\)
\(270\) 0 0
\(271\) −1.14012e6 −0.943030 −0.471515 0.881858i \(-0.656293\pi\)
−0.471515 + 0.881858i \(0.656293\pi\)
\(272\) 335498. 0.274959
\(273\) 0 0
\(274\) −88359.6 −0.0711013
\(275\) 1.13247e6 0.903012
\(276\) 0 0
\(277\) −1.98801e6 −1.55675 −0.778375 0.627799i \(-0.783956\pi\)
−0.778375 + 0.627799i \(0.783956\pi\)
\(278\) −1.71923e6 −1.33420
\(279\) 0 0
\(280\) 0 0
\(281\) −532321. −0.402168 −0.201084 0.979574i \(-0.564446\pi\)
−0.201084 + 0.979574i \(0.564446\pi\)
\(282\) 0 0
\(283\) −2.62473e6 −1.94813 −0.974067 0.226259i \(-0.927350\pi\)
−0.974067 + 0.226259i \(0.927350\pi\)
\(284\) −3.48956e6 −2.56729
\(285\) 0 0
\(286\) −2.80842e6 −2.03024
\(287\) 0 0
\(288\) 0 0
\(289\) −408424. −0.287651
\(290\) −507473. −0.354338
\(291\) 0 0
\(292\) −1.52274e6 −1.04512
\(293\) 609962. 0.415082 0.207541 0.978226i \(-0.433454\pi\)
0.207541 + 0.978226i \(0.433454\pi\)
\(294\) 0 0
\(295\) 742834. 0.496978
\(296\) −637424. −0.422863
\(297\) 0 0
\(298\) 1.56812e6 1.02291
\(299\) 1.72221e6 1.11406
\(300\) 0 0
\(301\) 0 0
\(302\) 3.53307e6 2.22913
\(303\) 0 0
\(304\) −456536. −0.283329
\(305\) 68442.3 0.0421284
\(306\) 0 0
\(307\) 1.34843e6 0.816551 0.408275 0.912859i \(-0.366130\pi\)
0.408275 + 0.912859i \(0.366130\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 5.52724e6 3.26666
\(311\) −3.33061e6 −1.95264 −0.976320 0.216330i \(-0.930591\pi\)
−0.976320 + 0.216330i \(0.930591\pi\)
\(312\) 0 0
\(313\) 2.83670e6 1.63664 0.818320 0.574763i \(-0.194906\pi\)
0.818320 + 0.574763i \(0.194906\pi\)
\(314\) −366909. −0.210007
\(315\) 0 0
\(316\) 2.51007e6 1.41406
\(317\) 1.08839e6 0.608326 0.304163 0.952620i \(-0.401623\pi\)
0.304163 + 0.952620i \(0.401623\pi\)
\(318\) 0 0
\(319\) 422395. 0.232403
\(320\) 3.54734e6 1.93655
\(321\) 0 0
\(322\) 0 0
\(323\) −1.37633e6 −0.734033
\(324\) 0 0
\(325\) −1.07772e6 −0.565975
\(326\) −4.48760e6 −2.33867
\(327\) 0 0
\(328\) 55790.4 0.0286335
\(329\) 0 0
\(330\) 0 0
\(331\) 1.30555e6 0.654971 0.327485 0.944856i \(-0.393799\pi\)
0.327485 + 0.944856i \(0.393799\pi\)
\(332\) 876619. 0.436481
\(333\) 0 0
\(334\) 373654. 0.183275
\(335\) 1.42133e6 0.691961
\(336\) 0 0
\(337\) −3.17016e6 −1.52057 −0.760285 0.649590i \(-0.774941\pi\)
−0.760285 + 0.649590i \(0.774941\pi\)
\(338\) −813739. −0.387430
\(339\) 0 0
\(340\) −4.05196e6 −1.90094
\(341\) −4.60060e6 −2.14254
\(342\) 0 0
\(343\) 0 0
\(344\) −3.97263e6 −1.81002
\(345\) 0 0
\(346\) 453889. 0.203826
\(347\) 1.71592e6 0.765019 0.382510 0.923951i \(-0.375060\pi\)
0.382510 + 0.923951i \(0.375060\pi\)
\(348\) 0 0
\(349\) 2.95822e6 1.30007 0.650034 0.759905i \(-0.274755\pi\)
0.650034 + 0.759905i \(0.274755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.31529e6 −0.995976
\(353\) 3.76980e6 1.61021 0.805103 0.593135i \(-0.202110\pi\)
0.805103 + 0.593135i \(0.202110\pi\)
\(354\) 0 0
\(355\) 4.45620e6 1.87669
\(356\) 765890. 0.320289
\(357\) 0 0
\(358\) 1.34191e6 0.553371
\(359\) −1.92987e6 −0.790300 −0.395150 0.918617i \(-0.629307\pi\)
−0.395150 + 0.918617i \(0.629307\pi\)
\(360\) 0 0
\(361\) −603234. −0.243623
\(362\) 725323. 0.290911
\(363\) 0 0
\(364\) 0 0
\(365\) 1.94455e6 0.763988
\(366\) 0 0
\(367\) −2.36742e6 −0.917509 −0.458754 0.888563i \(-0.651704\pi\)
−0.458754 + 0.888563i \(0.651704\pi\)
\(368\) −1.07688e6 −0.414521
\(369\) 0 0
\(370\) 1.89171e6 0.718373
\(371\) 0 0
\(372\) 0 0
\(373\) 3.53829e6 1.31680 0.658402 0.752666i \(-0.271233\pi\)
0.658402 + 0.752666i \(0.271233\pi\)
\(374\) 5.29406e6 1.95709
\(375\) 0 0
\(376\) −3.70960e6 −1.35318
\(377\) −401975. −0.145662
\(378\) 0 0
\(379\) 1.79847e6 0.643139 0.321569 0.946886i \(-0.395790\pi\)
0.321569 + 0.946886i \(0.395790\pi\)
\(380\) 5.51378e6 1.95880
\(381\) 0 0
\(382\) −2.55455e6 −0.895686
\(383\) 2.60815e6 0.908521 0.454261 0.890869i \(-0.349904\pi\)
0.454261 + 0.890869i \(0.349904\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −150555. −0.0514313
\(387\) 0 0
\(388\) 725683. 0.244719
\(389\) 2.82995e6 0.948211 0.474106 0.880468i \(-0.342772\pi\)
0.474106 + 0.880468i \(0.342772\pi\)
\(390\) 0 0
\(391\) −3.24648e6 −1.07392
\(392\) 0 0
\(393\) 0 0
\(394\) −9.69299e6 −3.14570
\(395\) −3.20538e6 −1.03368
\(396\) 0 0
\(397\) 2.43062e6 0.773999 0.387000 0.922080i \(-0.373511\pi\)
0.387000 + 0.922080i \(0.373511\pi\)
\(398\) 8.27939e6 2.61993
\(399\) 0 0
\(400\) 673885. 0.210589
\(401\) 2.43184e6 0.755222 0.377611 0.925964i \(-0.376746\pi\)
0.377611 + 0.925964i \(0.376746\pi\)
\(402\) 0 0
\(403\) 4.37820e6 1.34287
\(404\) 1.41227e6 0.430491
\(405\) 0 0
\(406\) 0 0
\(407\) −1.57457e6 −0.471167
\(408\) 0 0
\(409\) 4.77466e6 1.41135 0.705674 0.708537i \(-0.250644\pi\)
0.705674 + 0.708537i \(0.250644\pi\)
\(410\) −165571. −0.0486436
\(411\) 0 0
\(412\) 9.02944e6 2.62070
\(413\) 0 0
\(414\) 0 0
\(415\) −1.11945e6 −0.319069
\(416\) 2.20336e6 0.624242
\(417\) 0 0
\(418\) −7.20400e6 −2.01666
\(419\) −457181. −0.127219 −0.0636097 0.997975i \(-0.520261\pi\)
−0.0636097 + 0.997975i \(0.520261\pi\)
\(420\) 0 0
\(421\) −1.82396e6 −0.501545 −0.250773 0.968046i \(-0.580685\pi\)
−0.250773 + 0.968046i \(0.580685\pi\)
\(422\) −3.50066e6 −0.956905
\(423\) 0 0
\(424\) 6.72715e6 1.81726
\(425\) 2.03157e6 0.545582
\(426\) 0 0
\(427\) 0 0
\(428\) 5.30104e6 1.39879
\(429\) 0 0
\(430\) 1.17897e7 3.07492
\(431\) 3.38249e6 0.877087 0.438544 0.898710i \(-0.355494\pi\)
0.438544 + 0.898710i \(0.355494\pi\)
\(432\) 0 0
\(433\) 285266. 0.0731190 0.0365595 0.999331i \(-0.488360\pi\)
0.0365595 + 0.999331i \(0.488360\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.68418e6 −1.18010
\(437\) 4.41771e6 1.10661
\(438\) 0 0
\(439\) −4.35220e6 −1.07782 −0.538911 0.842363i \(-0.681164\pi\)
−0.538911 + 0.842363i \(0.681164\pi\)
\(440\) −9.12610e6 −2.24726
\(441\) 0 0
\(442\) −5.03813e6 −1.22663
\(443\) 5.10560e6 1.23605 0.618027 0.786157i \(-0.287932\pi\)
0.618027 + 0.786157i \(0.287932\pi\)
\(444\) 0 0
\(445\) −978049. −0.234132
\(446\) 1.01603e7 2.41862
\(447\) 0 0
\(448\) 0 0
\(449\) −3.04163e6 −0.712016 −0.356008 0.934483i \(-0.615862\pi\)
−0.356008 + 0.934483i \(0.615862\pi\)
\(450\) 0 0
\(451\) 137813. 0.0319043
\(452\) 1.00686e7 2.31805
\(453\) 0 0
\(454\) 5.19305e6 1.18245
\(455\) 0 0
\(456\) 0 0
\(457\) −1.74432e6 −0.390694 −0.195347 0.980734i \(-0.562583\pi\)
−0.195347 + 0.980734i \(0.562583\pi\)
\(458\) −4.91113e6 −1.09400
\(459\) 0 0
\(460\) 1.30059e7 2.86580
\(461\) −6.85701e6 −1.50273 −0.751367 0.659884i \(-0.770605\pi\)
−0.751367 + 0.659884i \(0.770605\pi\)
\(462\) 0 0
\(463\) 5.13844e6 1.11398 0.556992 0.830518i \(-0.311955\pi\)
0.556992 + 0.830518i \(0.311955\pi\)
\(464\) 251351. 0.0541982
\(465\) 0 0
\(466\) 3.41961e6 0.729477
\(467\) −4.58171e6 −0.972154 −0.486077 0.873916i \(-0.661573\pi\)
−0.486077 + 0.873916i \(0.661573\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.10091e7 2.29883
\(471\) 0 0
\(472\) −2.35031e6 −0.485591
\(473\) −9.81320e6 −2.01678
\(474\) 0 0
\(475\) −2.76450e6 −0.562190
\(476\) 0 0
\(477\) 0 0
\(478\) −3.49154e6 −0.698952
\(479\) −288523. −0.0574568 −0.0287284 0.999587i \(-0.509146\pi\)
−0.0287284 + 0.999587i \(0.509146\pi\)
\(480\) 0 0
\(481\) 1.49845e6 0.295310
\(482\) 1.60705e7 3.15074
\(483\) 0 0
\(484\) 8.60702e6 1.67009
\(485\) −926703. −0.178890
\(486\) 0 0
\(487\) −7.81685e6 −1.49351 −0.746757 0.665097i \(-0.768390\pi\)
−0.746757 + 0.665097i \(0.768390\pi\)
\(488\) −216550. −0.0411632
\(489\) 0 0
\(490\) 0 0
\(491\) −3.14467e6 −0.588669 −0.294335 0.955702i \(-0.595098\pi\)
−0.294335 + 0.955702i \(0.595098\pi\)
\(492\) 0 0
\(493\) 757751. 0.140414
\(494\) 6.85574e6 1.26397
\(495\) 0 0
\(496\) −2.73763e6 −0.499656
\(497\) 0 0
\(498\) 0 0
\(499\) 6.72563e6 1.20915 0.604577 0.796547i \(-0.293342\pi\)
0.604577 + 0.796547i \(0.293342\pi\)
\(500\) 4.45180e6 0.796362
\(501\) 0 0
\(502\) 545871. 0.0966787
\(503\) −9.45056e6 −1.66547 −0.832737 0.553669i \(-0.813227\pi\)
−0.832737 + 0.553669i \(0.813227\pi\)
\(504\) 0 0
\(505\) −1.80348e6 −0.314690
\(506\) −1.69928e7 −2.95045
\(507\) 0 0
\(508\) −8.07011e6 −1.38746
\(509\) −8.83702e6 −1.51186 −0.755930 0.654653i \(-0.772815\pi\)
−0.755930 + 0.654653i \(0.772815\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.80044e6 −0.640707
\(513\) 0 0
\(514\) −2.92813e6 −0.488858
\(515\) −1.15307e7 −1.91574
\(516\) 0 0
\(517\) −9.16344e6 −1.50776
\(518\) 0 0
\(519\) 0 0
\(520\) 8.68492e6 1.40850
\(521\) 694985. 0.112171 0.0560855 0.998426i \(-0.482138\pi\)
0.0560855 + 0.998426i \(0.482138\pi\)
\(522\) 0 0
\(523\) −3.58210e6 −0.572643 −0.286321 0.958134i \(-0.592432\pi\)
−0.286321 + 0.958134i \(0.592432\pi\)
\(524\) 2.93710e6 0.467294
\(525\) 0 0
\(526\) −8.11252e6 −1.27847
\(527\) −8.25320e6 −1.29448
\(528\) 0 0
\(529\) 3.98415e6 0.619009
\(530\) −1.99644e7 −3.08722
\(531\) 0 0
\(532\) 0 0
\(533\) −131151. −0.0199965
\(534\) 0 0
\(535\) −6.76947e6 −1.02252
\(536\) −4.49705e6 −0.676107
\(537\) 0 0
\(538\) −1.05231e7 −1.56744
\(539\) 0 0
\(540\) 0 0
\(541\) −5.16846e6 −0.759220 −0.379610 0.925147i \(-0.623942\pi\)
−0.379610 + 0.925147i \(0.623942\pi\)
\(542\) −1.07055e7 −1.56535
\(543\) 0 0
\(544\) −4.15349e6 −0.601750
\(545\) 5.98174e6 0.862653
\(546\) 0 0
\(547\) 8.47489e6 1.21106 0.605530 0.795822i \(-0.292961\pi\)
0.605530 + 0.795822i \(0.292961\pi\)
\(548\) −528562. −0.0751873
\(549\) 0 0
\(550\) 1.06337e7 1.49892
\(551\) −1.03112e6 −0.144688
\(552\) 0 0
\(553\) 0 0
\(554\) −1.86671e7 −2.58407
\(555\) 0 0
\(556\) −1.02843e7 −1.41088
\(557\) 1.04213e7 1.42325 0.711627 0.702557i \(-0.247959\pi\)
0.711627 + 0.702557i \(0.247959\pi\)
\(558\) 0 0
\(559\) 9.33880e6 1.26404
\(560\) 0 0
\(561\) 0 0
\(562\) −4.99842e6 −0.667562
\(563\) −7.24077e6 −0.962751 −0.481375 0.876515i \(-0.659863\pi\)
−0.481375 + 0.876515i \(0.659863\pi\)
\(564\) 0 0
\(565\) −1.28577e7 −1.69450
\(566\) −2.46459e7 −3.23373
\(567\) 0 0
\(568\) −1.40993e7 −1.83369
\(569\) −916536. −0.118678 −0.0593388 0.998238i \(-0.518899\pi\)
−0.0593388 + 0.998238i \(0.518899\pi\)
\(570\) 0 0
\(571\) 1.00708e7 1.29262 0.646312 0.763073i \(-0.276311\pi\)
0.646312 + 0.763073i \(0.276311\pi\)
\(572\) −1.67998e7 −2.14691
\(573\) 0 0
\(574\) 0 0
\(575\) −6.52091e6 −0.822505
\(576\) 0 0
\(577\) −1.16997e7 −1.46298 −0.731488 0.681855i \(-0.761174\pi\)
−0.731488 + 0.681855i \(0.761174\pi\)
\(578\) −3.83505e6 −0.477475
\(579\) 0 0
\(580\) −3.03567e6 −0.374701
\(581\) 0 0
\(582\) 0 0
\(583\) 1.66174e7 2.02485
\(584\) −6.15251e6 −0.746484
\(585\) 0 0
\(586\) 5.72747e6 0.688999
\(587\) −6.92367e6 −0.829357 −0.414678 0.909968i \(-0.636106\pi\)
−0.414678 + 0.909968i \(0.636106\pi\)
\(588\) 0 0
\(589\) 1.12307e7 1.33389
\(590\) 6.97511e6 0.824938
\(591\) 0 0
\(592\) −936961. −0.109880
\(593\) 1.57770e7 1.84242 0.921208 0.389069i \(-0.127203\pi\)
0.921208 + 0.389069i \(0.127203\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.38041e6 1.08170
\(597\) 0 0
\(598\) 1.61713e7 1.84924
\(599\) −1.42708e7 −1.62511 −0.812553 0.582887i \(-0.801923\pi\)
−0.812553 + 0.582887i \(0.801923\pi\)
\(600\) 0 0
\(601\) −7.63222e6 −0.861916 −0.430958 0.902372i \(-0.641824\pi\)
−0.430958 + 0.902372i \(0.641824\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.11346e7 2.35723
\(605\) −1.09912e7 −1.22084
\(606\) 0 0
\(607\) 3.56035e6 0.392212 0.196106 0.980583i \(-0.437170\pi\)
0.196106 + 0.980583i \(0.437170\pi\)
\(608\) 5.65194e6 0.620067
\(609\) 0 0
\(610\) 642664. 0.0699294
\(611\) 8.72046e6 0.945010
\(612\) 0 0
\(613\) −1.37284e7 −1.47560 −0.737800 0.675019i \(-0.764135\pi\)
−0.737800 + 0.675019i \(0.764135\pi\)
\(614\) 1.26616e7 1.35540
\(615\) 0 0
\(616\) 0 0
\(617\) 6.51173e6 0.688626 0.344313 0.938855i \(-0.388112\pi\)
0.344313 + 0.938855i \(0.388112\pi\)
\(618\) 0 0
\(619\) −8.85110e6 −0.928476 −0.464238 0.885711i \(-0.653672\pi\)
−0.464238 + 0.885711i \(0.653672\pi\)
\(620\) 3.30636e7 3.45439
\(621\) 0 0
\(622\) −3.12739e7 −3.24121
\(623\) 0 0
\(624\) 0 0
\(625\) −1.19977e7 −1.22856
\(626\) 2.66363e7 2.71667
\(627\) 0 0
\(628\) −2.19483e6 −0.222076
\(629\) −2.82467e6 −0.284670
\(630\) 0 0
\(631\) −6.89663e6 −0.689546 −0.344773 0.938686i \(-0.612044\pi\)
−0.344773 + 0.938686i \(0.612044\pi\)
\(632\) 1.01418e7 1.01000
\(633\) 0 0
\(634\) 1.02198e7 1.00977
\(635\) 1.03056e7 1.01424
\(636\) 0 0
\(637\) 0 0
\(638\) 3.96623e6 0.385768
\(639\) 0 0
\(640\) 2.38295e7 2.29967
\(641\) 1.66695e7 1.60242 0.801210 0.598383i \(-0.204190\pi\)
0.801210 + 0.598383i \(0.204190\pi\)
\(642\) 0 0
\(643\) 1.28697e7 1.22756 0.613779 0.789478i \(-0.289649\pi\)
0.613779 + 0.789478i \(0.289649\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.29235e7 −1.21843
\(647\) −1.14731e7 −1.07751 −0.538754 0.842463i \(-0.681105\pi\)
−0.538754 + 0.842463i \(0.681105\pi\)
\(648\) 0 0
\(649\) −5.80574e6 −0.541060
\(650\) −1.01196e7 −0.939467
\(651\) 0 0
\(652\) −2.68445e7 −2.47307
\(653\) 1.31801e7 1.20958 0.604791 0.796384i \(-0.293257\pi\)
0.604791 + 0.796384i \(0.293257\pi\)
\(654\) 0 0
\(655\) −3.75070e6 −0.341593
\(656\) 82007.2 0.00744034
\(657\) 0 0
\(658\) 0 0
\(659\) 1.13068e7 1.01421 0.507104 0.861885i \(-0.330716\pi\)
0.507104 + 0.861885i \(0.330716\pi\)
\(660\) 0 0
\(661\) −2.20319e6 −0.196132 −0.0980661 0.995180i \(-0.531266\pi\)
−0.0980661 + 0.995180i \(0.531266\pi\)
\(662\) 1.22589e7 1.08719
\(663\) 0 0
\(664\) 3.54192e6 0.311758
\(665\) 0 0
\(666\) 0 0
\(667\) −2.43222e6 −0.211684
\(668\) 2.23518e6 0.193808
\(669\) 0 0
\(670\) 1.33461e7 1.14859
\(671\) −534922. −0.0458653
\(672\) 0 0
\(673\) −1.89787e7 −1.61521 −0.807606 0.589723i \(-0.799237\pi\)
−0.807606 + 0.589723i \(0.799237\pi\)
\(674\) −2.97674e7 −2.52401
\(675\) 0 0
\(676\) −4.86773e6 −0.409694
\(677\) −1.96475e7 −1.64754 −0.823771 0.566923i \(-0.808134\pi\)
−0.823771 + 0.566923i \(0.808134\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.63717e7 −1.35775
\(681\) 0 0
\(682\) −4.31990e7 −3.55642
\(683\) −1.62705e7 −1.33459 −0.667295 0.744793i \(-0.732548\pi\)
−0.667295 + 0.744793i \(0.732548\pi\)
\(684\) 0 0
\(685\) 674978. 0.0549621
\(686\) 0 0
\(687\) 0 0
\(688\) −5.83944e6 −0.470328
\(689\) −1.58141e7 −1.26910
\(690\) 0 0
\(691\) −1.94467e7 −1.54935 −0.774677 0.632357i \(-0.782088\pi\)
−0.774677 + 0.632357i \(0.782088\pi\)
\(692\) 2.71514e6 0.215539
\(693\) 0 0
\(694\) 1.61122e7 1.26986
\(695\) 1.31332e7 1.03135
\(696\) 0 0
\(697\) 247229. 0.0192760
\(698\) 2.77772e7 2.15800
\(699\) 0 0
\(700\) 0 0
\(701\) −1.49625e7 −1.15003 −0.575014 0.818144i \(-0.695003\pi\)
−0.575014 + 0.818144i \(0.695003\pi\)
\(702\) 0 0
\(703\) 3.84373e6 0.293335
\(704\) −2.77248e7 −2.10832
\(705\) 0 0
\(706\) 3.53979e7 2.67280
\(707\) 0 0
\(708\) 0 0
\(709\) 1.58639e6 0.118521 0.0592603 0.998243i \(-0.481126\pi\)
0.0592603 + 0.998243i \(0.481126\pi\)
\(710\) 4.18431e7 3.11514
\(711\) 0 0
\(712\) 3.09453e6 0.228767
\(713\) 2.64910e7 1.95152
\(714\) 0 0
\(715\) 2.14535e7 1.56940
\(716\) 8.02723e6 0.585172
\(717\) 0 0
\(718\) −1.81212e7 −1.31183
\(719\) −1.81675e7 −1.31061 −0.655305 0.755364i \(-0.727460\pi\)
−0.655305 + 0.755364i \(0.727460\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5.66429e6 −0.404392
\(723\) 0 0
\(724\) 4.33884e6 0.307629
\(725\) 1.52203e6 0.107542
\(726\) 0 0
\(727\) −1.26903e7 −0.890506 −0.445253 0.895405i \(-0.646886\pi\)
−0.445253 + 0.895405i \(0.646886\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.82591e7 1.26815
\(731\) −1.76043e7 −1.21850
\(732\) 0 0
\(733\) 3.11401e6 0.214072 0.107036 0.994255i \(-0.465864\pi\)
0.107036 + 0.994255i \(0.465864\pi\)
\(734\) −2.22298e7 −1.52298
\(735\) 0 0
\(736\) 1.33318e7 0.907182
\(737\) −1.11086e7 −0.753339
\(738\) 0 0
\(739\) 9.09899e6 0.612890 0.306445 0.951888i \(-0.400861\pi\)
0.306445 + 0.951888i \(0.400861\pi\)
\(740\) 1.13161e7 0.759656
\(741\) 0 0
\(742\) 0 0
\(743\) 8.79218e6 0.584285 0.292142 0.956375i \(-0.405632\pi\)
0.292142 + 0.956375i \(0.405632\pi\)
\(744\) 0 0
\(745\) −1.19789e7 −0.790724
\(746\) 3.32241e7 2.18578
\(747\) 0 0
\(748\) 3.16687e7 2.06955
\(749\) 0 0
\(750\) 0 0
\(751\) −2.80918e7 −1.81752 −0.908759 0.417320i \(-0.862969\pi\)
−0.908759 + 0.417320i \(0.862969\pi\)
\(752\) −5.45280e6 −0.351621
\(753\) 0 0
\(754\) −3.77450e6 −0.241786
\(755\) −2.69891e7 −1.72314
\(756\) 0 0
\(757\) −4.18815e6 −0.265634 −0.132817 0.991141i \(-0.542402\pi\)
−0.132817 + 0.991141i \(0.542402\pi\)
\(758\) 1.68874e7 1.06755
\(759\) 0 0
\(760\) 2.22781e7 1.39908
\(761\) −393182. −0.0246111 −0.0123056 0.999924i \(-0.503917\pi\)
−0.0123056 + 0.999924i \(0.503917\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.52812e7 −0.947159
\(765\) 0 0
\(766\) 2.44901e7 1.50806
\(767\) 5.52508e6 0.339117
\(768\) 0 0
\(769\) −1.57517e7 −0.960530 −0.480265 0.877124i \(-0.659459\pi\)
−0.480265 + 0.877124i \(0.659459\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −900611. −0.0543869
\(773\) 2.88472e6 0.173642 0.0868210 0.996224i \(-0.472329\pi\)
0.0868210 + 0.996224i \(0.472329\pi\)
\(774\) 0 0
\(775\) −1.65775e7 −0.991433
\(776\) 2.93207e6 0.174791
\(777\) 0 0
\(778\) 2.65729e7 1.57394
\(779\) −336421. −0.0198628
\(780\) 0 0
\(781\) −3.48281e7 −2.04316
\(782\) −3.04840e7 −1.78261
\(783\) 0 0
\(784\) 0 0
\(785\) 2.80281e6 0.162338
\(786\) 0 0
\(787\) 2.97866e7 1.71429 0.857145 0.515076i \(-0.172236\pi\)
0.857145 + 0.515076i \(0.172236\pi\)
\(788\) −5.79829e7 −3.32647
\(789\) 0 0
\(790\) −3.00981e7 −1.71582
\(791\) 0 0
\(792\) 0 0
\(793\) 509063. 0.0287467
\(794\) 2.28232e7 1.28477
\(795\) 0 0
\(796\) 4.95268e7 2.77049
\(797\) 8.11321e6 0.452425 0.226213 0.974078i \(-0.427366\pi\)
0.226213 + 0.974078i \(0.427366\pi\)
\(798\) 0 0
\(799\) −1.64387e7 −0.910960
\(800\) −8.34274e6 −0.460876
\(801\) 0 0
\(802\) 2.28347e7 1.25360
\(803\) −1.51979e7 −0.831756
\(804\) 0 0
\(805\) 0 0
\(806\) 4.11107e7 2.22904
\(807\) 0 0
\(808\) 5.70617e6 0.307480
\(809\) −111597. −0.00599489 −0.00299744 0.999996i \(-0.500954\pi\)
−0.00299744 + 0.999996i \(0.500954\pi\)
\(810\) 0 0
\(811\) 209250. 0.0111716 0.00558578 0.999984i \(-0.498222\pi\)
0.00558578 + 0.999984i \(0.498222\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.47850e7 −0.782094
\(815\) 3.42807e7 1.80782
\(816\) 0 0
\(817\) 2.39554e7 1.25559
\(818\) 4.48334e7 2.34271
\(819\) 0 0
\(820\) −990437. −0.0514390
\(821\) −6.84614e6 −0.354477 −0.177238 0.984168i \(-0.556716\pi\)
−0.177238 + 0.984168i \(0.556716\pi\)
\(822\) 0 0
\(823\) −5.61210e6 −0.288819 −0.144409 0.989518i \(-0.546128\pi\)
−0.144409 + 0.989518i \(0.546128\pi\)
\(824\) 3.64828e7 1.87185
\(825\) 0 0
\(826\) 0 0
\(827\) 2.15868e7 1.09755 0.548776 0.835969i \(-0.315094\pi\)
0.548776 + 0.835969i \(0.315094\pi\)
\(828\) 0 0
\(829\) −1.92427e7 −0.972478 −0.486239 0.873826i \(-0.661632\pi\)
−0.486239 + 0.873826i \(0.661632\pi\)
\(830\) −1.05115e7 −0.529625
\(831\) 0 0
\(832\) 2.63846e7 1.32142
\(833\) 0 0
\(834\) 0 0
\(835\) −2.85434e6 −0.141674
\(836\) −4.30939e7 −2.13255
\(837\) 0 0
\(838\) −4.29287e6 −0.211173
\(839\) 2.47678e7 1.21474 0.607369 0.794420i \(-0.292225\pi\)
0.607369 + 0.794420i \(0.292225\pi\)
\(840\) 0 0
\(841\) −1.99435e7 −0.972323
\(842\) −1.71267e7 −0.832520
\(843\) 0 0
\(844\) −2.09407e7 −1.01190
\(845\) 6.21614e6 0.299488
\(846\) 0 0
\(847\) 0 0
\(848\) 9.88836e6 0.472210
\(849\) 0 0
\(850\) 1.90762e7 0.905618
\(851\) 9.06659e6 0.429161
\(852\) 0 0
\(853\) 2.91267e7 1.37062 0.685312 0.728250i \(-0.259666\pi\)
0.685312 + 0.728250i \(0.259666\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2.14185e7 0.999088
\(857\) −3.22586e6 −0.150035 −0.0750175 0.997182i \(-0.523901\pi\)
−0.0750175 + 0.997182i \(0.523901\pi\)
\(858\) 0 0
\(859\) −2.64671e7 −1.22384 −0.611918 0.790921i \(-0.709602\pi\)
−0.611918 + 0.790921i \(0.709602\pi\)
\(860\) 7.05255e7 3.25162
\(861\) 0 0
\(862\) 3.17611e7 1.45589
\(863\) −1.58510e7 −0.724487 −0.362243 0.932084i \(-0.617989\pi\)
−0.362243 + 0.932084i \(0.617989\pi\)
\(864\) 0 0
\(865\) −3.46726e6 −0.157560
\(866\) 2.67861e6 0.121371
\(867\) 0 0
\(868\) 0 0
\(869\) 2.50522e7 1.12537
\(870\) 0 0
\(871\) 1.05716e7 0.472166
\(872\) −1.89261e7 −0.842888
\(873\) 0 0
\(874\) 4.14817e7 1.83687
\(875\) 0 0
\(876\) 0 0
\(877\) −1.49132e7 −0.654743 −0.327372 0.944896i \(-0.606163\pi\)
−0.327372 + 0.944896i \(0.606163\pi\)
\(878\) −4.08665e7 −1.78909
\(879\) 0 0
\(880\) −1.34146e7 −0.583944
\(881\) −3.70679e7 −1.60901 −0.804504 0.593947i \(-0.797569\pi\)
−0.804504 + 0.593947i \(0.797569\pi\)
\(882\) 0 0
\(883\) 1.50440e7 0.649325 0.324663 0.945830i \(-0.394749\pi\)
0.324663 + 0.945830i \(0.394749\pi\)
\(884\) −3.01378e7 −1.29712
\(885\) 0 0
\(886\) 4.79409e7 2.05174
\(887\) 1.78101e7 0.760075 0.380038 0.924971i \(-0.375911\pi\)
0.380038 + 0.924971i \(0.375911\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −9.18375e6 −0.388638
\(891\) 0 0
\(892\) 6.07782e7 2.55762
\(893\) 2.23692e7 0.938690
\(894\) 0 0
\(895\) −1.02508e7 −0.427762
\(896\) 0 0
\(897\) 0 0
\(898\) −2.85605e7 −1.18188
\(899\) −6.18317e6 −0.255160
\(900\) 0 0
\(901\) 2.98106e7 1.22337
\(902\) 1.29405e6 0.0529583
\(903\) 0 0
\(904\) 4.06815e7 1.65568
\(905\) −5.54073e6 −0.224877
\(906\) 0 0
\(907\) −1.16936e6 −0.0471987 −0.0235993 0.999721i \(-0.507513\pi\)
−0.0235993 + 0.999721i \(0.507513\pi\)
\(908\) 3.10645e7 1.25040
\(909\) 0 0
\(910\) 0 0
\(911\) −2.74389e7 −1.09539 −0.547697 0.836677i \(-0.684495\pi\)
−0.547697 + 0.836677i \(0.684495\pi\)
\(912\) 0 0
\(913\) 8.74924e6 0.347371
\(914\) −1.63790e7 −0.648516
\(915\) 0 0
\(916\) −2.93781e7 −1.15687
\(917\) 0 0
\(918\) 0 0
\(919\) 1.10218e7 0.430492 0.215246 0.976560i \(-0.430945\pi\)
0.215246 + 0.976560i \(0.430945\pi\)
\(920\) 5.25495e7 2.04691
\(921\) 0 0
\(922\) −6.43864e7 −2.49440
\(923\) 3.31444e7 1.28058
\(924\) 0 0
\(925\) −5.67367e6 −0.218027
\(926\) 4.82493e7 1.84911
\(927\) 0 0
\(928\) −3.11173e6 −0.118613
\(929\) 4.25051e6 0.161585 0.0807927 0.996731i \(-0.474255\pi\)
0.0807927 + 0.996731i \(0.474255\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2.04559e7 0.771398
\(933\) 0 0
\(934\) −4.30216e7 −1.61369
\(935\) −4.04413e7 −1.51285
\(936\) 0 0
\(937\) −3.75738e7 −1.39809 −0.699046 0.715077i \(-0.746392\pi\)
−0.699046 + 0.715077i \(0.746392\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 6.58558e7 2.43094
\(941\) 6.87043e6 0.252936 0.126468 0.991971i \(-0.459636\pi\)
0.126468 + 0.991971i \(0.459636\pi\)
\(942\) 0 0
\(943\) −793550. −0.0290600
\(944\) −3.45476e6 −0.126179
\(945\) 0 0
\(946\) −9.21446e7 −3.34767
\(947\) 2.30923e7 0.836744 0.418372 0.908276i \(-0.362601\pi\)
0.418372 + 0.908276i \(0.362601\pi\)
\(948\) 0 0
\(949\) 1.44632e7 0.521314
\(950\) −2.59583e7 −0.933185
\(951\) 0 0
\(952\) 0 0
\(953\) 2.95399e7 1.05360 0.526802 0.849988i \(-0.323391\pi\)
0.526802 + 0.849988i \(0.323391\pi\)
\(954\) 0 0
\(955\) 1.95142e7 0.692376
\(956\) −2.08862e7 −0.739119
\(957\) 0 0
\(958\) −2.70919e6 −0.0953730
\(959\) 0 0
\(960\) 0 0
\(961\) 3.87161e7 1.35233
\(962\) 1.40702e7 0.490188
\(963\) 0 0
\(964\) 9.61329e7 3.33180
\(965\) 1.15009e6 0.0397569
\(966\) 0 0
\(967\) 1.49151e7 0.512931 0.256466 0.966553i \(-0.417442\pi\)
0.256466 + 0.966553i \(0.417442\pi\)
\(968\) 3.47761e7 1.19287
\(969\) 0 0
\(970\) −8.70162e6 −0.296941
\(971\) 1.54830e7 0.526995 0.263498 0.964660i \(-0.415124\pi\)
0.263498 + 0.964660i \(0.415124\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −7.33992e7 −2.47910
\(975\) 0 0
\(976\) −318311. −0.0106961
\(977\) 2.83428e7 0.949961 0.474980 0.879996i \(-0.342455\pi\)
0.474980 + 0.879996i \(0.342455\pi\)
\(978\) 0 0
\(979\) 7.64410e6 0.254900
\(980\) 0 0
\(981\) 0 0
\(982\) −2.95280e7 −0.977138
\(983\) 2.56993e7 0.848278 0.424139 0.905597i \(-0.360577\pi\)
0.424139 + 0.905597i \(0.360577\pi\)
\(984\) 0 0
\(985\) 7.40446e7 2.43166
\(986\) 7.11518e6 0.233074
\(987\) 0 0
\(988\) 4.10106e7 1.33661
\(989\) 5.65059e7 1.83697
\(990\) 0 0
\(991\) 4.46405e7 1.44392 0.721962 0.691933i \(-0.243240\pi\)
0.721962 + 0.691933i \(0.243240\pi\)
\(992\) 3.38921e7 1.09350
\(993\) 0 0
\(994\) 0 0
\(995\) −6.32461e7 −2.02524
\(996\) 0 0
\(997\) −2.30554e7 −0.734572 −0.367286 0.930108i \(-0.619713\pi\)
−0.367286 + 0.930108i \(0.619713\pi\)
\(998\) 6.31528e7 2.00709
\(999\) 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 441.6.a.s.1.2 2
3.2 odd 2 147.6.a.k.1.1 2
7.3 odd 6 63.6.e.c.37.1 4
7.5 odd 6 63.6.e.c.46.1 4
7.6 odd 2 441.6.a.t.1.2 2
21.2 odd 6 147.6.e.l.67.2 4
21.5 even 6 21.6.e.b.4.2 4
21.11 odd 6 147.6.e.l.79.2 4
21.17 even 6 21.6.e.b.16.2 yes 4
21.20 even 2 147.6.a.i.1.1 2
84.47 odd 6 336.6.q.e.193.2 4
84.59 odd 6 336.6.q.e.289.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 4 21.5 even 6
21.6.e.b.16.2 yes 4 21.17 even 6
63.6.e.c.37.1 4 7.3 odd 6
63.6.e.c.46.1 4 7.5 odd 6
147.6.a.i.1.1 2 21.20 even 2
147.6.a.k.1.1 2 3.2 odd 2
147.6.e.l.67.2 4 21.2 odd 6
147.6.e.l.79.2 4 21.11 odd 6
336.6.q.e.193.2 4 84.47 odd 6
336.6.q.e.289.2 4 84.59 odd 6
441.6.a.s.1.2 2 1.1 even 1 trivial
441.6.a.t.1.2 2 7.6 odd 2