Properties

Label 1104.6.a.i.1.2
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(177.063737074\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5333.1
Defining polynomial: \(x^{3} - x^{2} - 11 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.714018\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.00000 q^{3} -55.5168 q^{5} -2.50462 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} -55.5168 q^{5} -2.50462 q^{7} +81.0000 q^{9} -228.550 q^{11} +658.703 q^{13} +499.651 q^{15} -1443.81 q^{17} +982.167 q^{19} +22.5416 q^{21} -529.000 q^{23} -42.8875 q^{25} -729.000 q^{27} -7157.56 q^{29} +9259.98 q^{31} +2056.95 q^{33} +139.049 q^{35} +2422.50 q^{37} -5928.33 q^{39} -4075.27 q^{41} +10417.3 q^{43} -4496.86 q^{45} -9358.08 q^{47} -16800.7 q^{49} +12994.3 q^{51} -34280.3 q^{53} +12688.4 q^{55} -8839.51 q^{57} +7268.79 q^{59} +26611.7 q^{61} -202.874 q^{63} -36569.1 q^{65} -53450.8 q^{67} +4761.00 q^{69} -21673.7 q^{71} -82856.3 q^{73} +385.987 q^{75} +572.432 q^{77} +23960.0 q^{79} +6561.00 q^{81} -81187.7 q^{83} +80155.6 q^{85} +64418.0 q^{87} +100115. q^{89} -1649.80 q^{91} -83339.8 q^{93} -54526.8 q^{95} +36122.1 q^{97} -18512.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + O(q^{10}) \) \( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99} + O(q^{100}) \)

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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) −55.5168 −0.993114 −0.496557 0.868004i \(-0.665403\pi\)
−0.496557 + 0.868004i \(0.665403\pi\)
\(6\) 0 0
\(7\) −2.50462 −0.0193196 −0.00965978 0.999953i \(-0.503075\pi\)
−0.00965978 + 0.999953i \(0.503075\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −228.550 −0.569508 −0.284754 0.958601i \(-0.591912\pi\)
−0.284754 + 0.958601i \(0.591912\pi\)
\(12\) 0 0
\(13\) 658.703 1.08101 0.540507 0.841339i \(-0.318232\pi\)
0.540507 + 0.841339i \(0.318232\pi\)
\(14\) 0 0
\(15\) 499.651 0.573375
\(16\) 0 0
\(17\) −1443.81 −1.21168 −0.605839 0.795587i \(-0.707162\pi\)
−0.605839 + 0.795587i \(0.707162\pi\)
\(18\) 0 0
\(19\) 982.167 0.624168 0.312084 0.950055i \(-0.398973\pi\)
0.312084 + 0.950055i \(0.398973\pi\)
\(20\) 0 0
\(21\) 22.5416 0.0111542
\(22\) 0 0
\(23\) −529.000 −0.208514
\(24\) 0 0
\(25\) −42.8875 −0.0137240
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −7157.56 −1.58041 −0.790205 0.612842i \(-0.790026\pi\)
−0.790205 + 0.612842i \(0.790026\pi\)
\(30\) 0 0
\(31\) 9259.98 1.73064 0.865318 0.501223i \(-0.167116\pi\)
0.865318 + 0.501223i \(0.167116\pi\)
\(32\) 0 0
\(33\) 2056.95 0.328805
\(34\) 0 0
\(35\) 139.049 0.0191865
\(36\) 0 0
\(37\) 2422.50 0.290911 0.145455 0.989365i \(-0.453535\pi\)
0.145455 + 0.989365i \(0.453535\pi\)
\(38\) 0 0
\(39\) −5928.33 −0.624124
\(40\) 0 0
\(41\) −4075.27 −0.378614 −0.189307 0.981918i \(-0.560624\pi\)
−0.189307 + 0.981918i \(0.560624\pi\)
\(42\) 0 0
\(43\) 10417.3 0.859178 0.429589 0.903024i \(-0.358658\pi\)
0.429589 + 0.903024i \(0.358658\pi\)
\(44\) 0 0
\(45\) −4496.86 −0.331038
\(46\) 0 0
\(47\) −9358.08 −0.617934 −0.308967 0.951073i \(-0.599983\pi\)
−0.308967 + 0.951073i \(0.599983\pi\)
\(48\) 0 0
\(49\) −16800.7 −0.999627
\(50\) 0 0
\(51\) 12994.3 0.699563
\(52\) 0 0
\(53\) −34280.3 −1.67631 −0.838157 0.545428i \(-0.816367\pi\)
−0.838157 + 0.545428i \(0.816367\pi\)
\(54\) 0 0
\(55\) 12688.4 0.565586
\(56\) 0 0
\(57\) −8839.51 −0.360364
\(58\) 0 0
\(59\) 7268.79 0.271852 0.135926 0.990719i \(-0.456599\pi\)
0.135926 + 0.990719i \(0.456599\pi\)
\(60\) 0 0
\(61\) 26611.7 0.915687 0.457844 0.889033i \(-0.348622\pi\)
0.457844 + 0.889033i \(0.348622\pi\)
\(62\) 0 0
\(63\) −202.874 −0.00643985
\(64\) 0 0
\(65\) −36569.1 −1.07357
\(66\) 0 0
\(67\) −53450.8 −1.45468 −0.727339 0.686278i \(-0.759243\pi\)
−0.727339 + 0.686278i \(0.759243\pi\)
\(68\) 0 0
\(69\) 4761.00 0.120386
\(70\) 0 0
\(71\) −21673.7 −0.510255 −0.255128 0.966907i \(-0.582118\pi\)
−0.255128 + 0.966907i \(0.582118\pi\)
\(72\) 0 0
\(73\) −82856.3 −1.81978 −0.909889 0.414851i \(-0.863834\pi\)
−0.909889 + 0.414851i \(0.863834\pi\)
\(74\) 0 0
\(75\) 385.987 0.00792355
\(76\) 0 0
\(77\) 572.432 0.0110026
\(78\) 0 0
\(79\) 23960.0 0.431936 0.215968 0.976400i \(-0.430709\pi\)
0.215968 + 0.976400i \(0.430709\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −81187.7 −1.29359 −0.646793 0.762666i \(-0.723890\pi\)
−0.646793 + 0.762666i \(0.723890\pi\)
\(84\) 0 0
\(85\) 80155.6 1.20334
\(86\) 0 0
\(87\) 64418.0 0.912451
\(88\) 0 0
\(89\) 100115. 1.33975 0.669876 0.742473i \(-0.266347\pi\)
0.669876 + 0.742473i \(0.266347\pi\)
\(90\) 0 0
\(91\) −1649.80 −0.0208847
\(92\) 0 0
\(93\) −83339.8 −0.999183
\(94\) 0 0
\(95\) −54526.8 −0.619870
\(96\) 0 0
\(97\) 36122.1 0.389802 0.194901 0.980823i \(-0.437562\pi\)
0.194901 + 0.980823i \(0.437562\pi\)
\(98\) 0 0
\(99\) −18512.6 −0.189836
\(100\) 0 0
\(101\) −41151.8 −0.401408 −0.200704 0.979652i \(-0.564323\pi\)
−0.200704 + 0.979652i \(0.564323\pi\)
\(102\) 0 0
\(103\) 172534. 1.60244 0.801221 0.598369i \(-0.204184\pi\)
0.801221 + 0.598369i \(0.204184\pi\)
\(104\) 0 0
\(105\) −1251.44 −0.0110774
\(106\) 0 0
\(107\) −178228. −1.50493 −0.752464 0.658633i \(-0.771135\pi\)
−0.752464 + 0.658633i \(0.771135\pi\)
\(108\) 0 0
\(109\) −138352. −1.11537 −0.557686 0.830052i \(-0.688311\pi\)
−0.557686 + 0.830052i \(0.688311\pi\)
\(110\) 0 0
\(111\) −21802.5 −0.167958
\(112\) 0 0
\(113\) −13523.5 −0.0996307 −0.0498153 0.998758i \(-0.515863\pi\)
−0.0498153 + 0.998758i \(0.515863\pi\)
\(114\) 0 0
\(115\) 29368.4 0.207079
\(116\) 0 0
\(117\) 53355.0 0.360338
\(118\) 0 0
\(119\) 3616.20 0.0234091
\(120\) 0 0
\(121\) −108816. −0.675661
\(122\) 0 0
\(123\) 36677.4 0.218593
\(124\) 0 0
\(125\) 175871. 1.00674
\(126\) 0 0
\(127\) −158567. −0.872376 −0.436188 0.899856i \(-0.643672\pi\)
−0.436188 + 0.899856i \(0.643672\pi\)
\(128\) 0 0
\(129\) −93755.6 −0.496047
\(130\) 0 0
\(131\) 383514. 1.95255 0.976277 0.216526i \(-0.0694727\pi\)
0.976277 + 0.216526i \(0.0694727\pi\)
\(132\) 0 0
\(133\) −2459.96 −0.0120587
\(134\) 0 0
\(135\) 40471.7 0.191125
\(136\) 0 0
\(137\) −33171.1 −0.150994 −0.0754968 0.997146i \(-0.524054\pi\)
−0.0754968 + 0.997146i \(0.524054\pi\)
\(138\) 0 0
\(139\) 128488. 0.564060 0.282030 0.959406i \(-0.408992\pi\)
0.282030 + 0.959406i \(0.408992\pi\)
\(140\) 0 0
\(141\) 84222.7 0.356764
\(142\) 0 0
\(143\) −150547. −0.615646
\(144\) 0 0
\(145\) 397365. 1.56953
\(146\) 0 0
\(147\) 151207. 0.577135
\(148\) 0 0
\(149\) −184228. −0.679814 −0.339907 0.940459i \(-0.610396\pi\)
−0.339907 + 0.940459i \(0.610396\pi\)
\(150\) 0 0
\(151\) 19798.1 0.0706610 0.0353305 0.999376i \(-0.488752\pi\)
0.0353305 + 0.999376i \(0.488752\pi\)
\(152\) 0 0
\(153\) −116948. −0.403893
\(154\) 0 0
\(155\) −514084. −1.71872
\(156\) 0 0
\(157\) 193317. 0.625923 0.312962 0.949766i \(-0.398679\pi\)
0.312962 + 0.949766i \(0.398679\pi\)
\(158\) 0 0
\(159\) 308523. 0.967821
\(160\) 0 0
\(161\) 1324.95 0.00402841
\(162\) 0 0
\(163\) −106286. −0.313334 −0.156667 0.987651i \(-0.550075\pi\)
−0.156667 + 0.987651i \(0.550075\pi\)
\(164\) 0 0
\(165\) −114195. −0.326541
\(166\) 0 0
\(167\) 77715.5 0.215634 0.107817 0.994171i \(-0.465614\pi\)
0.107817 + 0.994171i \(0.465614\pi\)
\(168\) 0 0
\(169\) 62597.3 0.168593
\(170\) 0 0
\(171\) 79555.5 0.208056
\(172\) 0 0
\(173\) 218524. 0.555115 0.277558 0.960709i \(-0.410475\pi\)
0.277558 + 0.960709i \(0.410475\pi\)
\(174\) 0 0
\(175\) 107.417 0.000265142 0
\(176\) 0 0
\(177\) −65419.1 −0.156954
\(178\) 0 0
\(179\) 545493. 1.27250 0.636248 0.771485i \(-0.280486\pi\)
0.636248 + 0.771485i \(0.280486\pi\)
\(180\) 0 0
\(181\) 510878. 1.15910 0.579549 0.814937i \(-0.303229\pi\)
0.579549 + 0.814937i \(0.303229\pi\)
\(182\) 0 0
\(183\) −239505. −0.528672
\(184\) 0 0
\(185\) −134490. −0.288908
\(186\) 0 0
\(187\) 329982. 0.690060
\(188\) 0 0
\(189\) 1825.87 0.00371805
\(190\) 0 0
\(191\) 57685.2 0.114415 0.0572073 0.998362i \(-0.481780\pi\)
0.0572073 + 0.998362i \(0.481780\pi\)
\(192\) 0 0
\(193\) −822091. −1.58864 −0.794322 0.607497i \(-0.792174\pi\)
−0.794322 + 0.607497i \(0.792174\pi\)
\(194\) 0 0
\(195\) 329122. 0.619827
\(196\) 0 0
\(197\) 971571. 1.78365 0.891824 0.452383i \(-0.149426\pi\)
0.891824 + 0.452383i \(0.149426\pi\)
\(198\) 0 0
\(199\) −806110. −1.44298 −0.721492 0.692423i \(-0.756543\pi\)
−0.721492 + 0.692423i \(0.756543\pi\)
\(200\) 0 0
\(201\) 481057. 0.839859
\(202\) 0 0
\(203\) 17927.0 0.0305329
\(204\) 0 0
\(205\) 226246. 0.376007
\(206\) 0 0
\(207\) −42849.0 −0.0695048
\(208\) 0 0
\(209\) −224474. −0.355468
\(210\) 0 0
\(211\) 821913. 1.27092 0.635462 0.772132i \(-0.280810\pi\)
0.635462 + 0.772132i \(0.280810\pi\)
\(212\) 0 0
\(213\) 195063. 0.294596
\(214\) 0 0
\(215\) −578334. −0.853262
\(216\) 0 0
\(217\) −23192.8 −0.0334351
\(218\) 0 0
\(219\) 745707. 1.05065
\(220\) 0 0
\(221\) −951042. −1.30984
\(222\) 0 0
\(223\) −511948. −0.689388 −0.344694 0.938715i \(-0.612017\pi\)
−0.344694 + 0.938715i \(0.612017\pi\)
\(224\) 0 0
\(225\) −3473.89 −0.00457467
\(226\) 0 0
\(227\) 1.45074e6 1.86864 0.934321 0.356433i \(-0.116007\pi\)
0.934321 + 0.356433i \(0.116007\pi\)
\(228\) 0 0
\(229\) 621169. 0.782747 0.391373 0.920232i \(-0.372000\pi\)
0.391373 + 0.920232i \(0.372000\pi\)
\(230\) 0 0
\(231\) −5151.89 −0.00635238
\(232\) 0 0
\(233\) 490334. 0.591701 0.295851 0.955234i \(-0.404397\pi\)
0.295851 + 0.955234i \(0.404397\pi\)
\(234\) 0 0
\(235\) 519530. 0.613679
\(236\) 0 0
\(237\) −215640. −0.249378
\(238\) 0 0
\(239\) −109503. −0.124003 −0.0620014 0.998076i \(-0.519748\pi\)
−0.0620014 + 0.998076i \(0.519748\pi\)
\(240\) 0 0
\(241\) 1.36171e6 1.51022 0.755111 0.655597i \(-0.227583\pi\)
0.755111 + 0.655597i \(0.227583\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) 932722. 0.992744
\(246\) 0 0
\(247\) 646957. 0.674735
\(248\) 0 0
\(249\) 730690. 0.746852
\(250\) 0 0
\(251\) −8042.27 −0.00805739 −0.00402869 0.999992i \(-0.501282\pi\)
−0.00402869 + 0.999992i \(0.501282\pi\)
\(252\) 0 0
\(253\) 120903. 0.118751
\(254\) 0 0
\(255\) −721400. −0.694746
\(256\) 0 0
\(257\) 1.16730e6 1.10243 0.551214 0.834364i \(-0.314165\pi\)
0.551214 + 0.834364i \(0.314165\pi\)
\(258\) 0 0
\(259\) −6067.46 −0.00562027
\(260\) 0 0
\(261\) −579762. −0.526804
\(262\) 0 0
\(263\) 558051. 0.497490 0.248745 0.968569i \(-0.419982\pi\)
0.248745 + 0.968569i \(0.419982\pi\)
\(264\) 0 0
\(265\) 1.90313e6 1.66477
\(266\) 0 0
\(267\) −901036. −0.773506
\(268\) 0 0
\(269\) −1.21947e6 −1.02752 −0.513759 0.857935i \(-0.671747\pi\)
−0.513759 + 0.857935i \(0.671747\pi\)
\(270\) 0 0
\(271\) −1.40581e6 −1.16280 −0.581399 0.813618i \(-0.697495\pi\)
−0.581399 + 0.813618i \(0.697495\pi\)
\(272\) 0 0
\(273\) 14848.2 0.0120578
\(274\) 0 0
\(275\) 9801.94 0.00781592
\(276\) 0 0
\(277\) 1.87799e6 1.47060 0.735300 0.677742i \(-0.237041\pi\)
0.735300 + 0.677742i \(0.237041\pi\)
\(278\) 0 0
\(279\) 750058. 0.576879
\(280\) 0 0
\(281\) −2.11759e6 −1.59984 −0.799920 0.600106i \(-0.795125\pi\)
−0.799920 + 0.600106i \(0.795125\pi\)
\(282\) 0 0
\(283\) −1.16042e6 −0.861288 −0.430644 0.902522i \(-0.641714\pi\)
−0.430644 + 0.902522i \(0.641714\pi\)
\(284\) 0 0
\(285\) 490741. 0.357882
\(286\) 0 0
\(287\) 10207.0 0.00731466
\(288\) 0 0
\(289\) 664726. 0.468164
\(290\) 0 0
\(291\) −325099. −0.225052
\(292\) 0 0
\(293\) −1.27325e6 −0.866449 −0.433225 0.901286i \(-0.642624\pi\)
−0.433225 + 0.901286i \(0.642624\pi\)
\(294\) 0 0
\(295\) −403540. −0.269980
\(296\) 0 0
\(297\) 166613. 0.109602
\(298\) 0 0
\(299\) −348454. −0.225407
\(300\) 0 0
\(301\) −26091.4 −0.0165990
\(302\) 0 0
\(303\) 370367. 0.231753
\(304\) 0 0
\(305\) −1.47739e6 −0.909382
\(306\) 0 0
\(307\) 2.26175e6 1.36961 0.684807 0.728724i \(-0.259886\pi\)
0.684807 + 0.728724i \(0.259886\pi\)
\(308\) 0 0
\(309\) −1.55281e6 −0.925170
\(310\) 0 0
\(311\) −1.20263e6 −0.705066 −0.352533 0.935799i \(-0.614680\pi\)
−0.352533 + 0.935799i \(0.614680\pi\)
\(312\) 0 0
\(313\) −484311. −0.279424 −0.139712 0.990192i \(-0.544618\pi\)
−0.139712 + 0.990192i \(0.544618\pi\)
\(314\) 0 0
\(315\) 11262.9 0.00639551
\(316\) 0 0
\(317\) 796624. 0.445251 0.222626 0.974904i \(-0.428537\pi\)
0.222626 + 0.974904i \(0.428537\pi\)
\(318\) 0 0
\(319\) 1.63586e6 0.900056
\(320\) 0 0
\(321\) 1.60405e6 0.868871
\(322\) 0 0
\(323\) −1.41806e6 −0.756291
\(324\) 0 0
\(325\) −28250.1 −0.0148358
\(326\) 0 0
\(327\) 1.24517e6 0.643961
\(328\) 0 0
\(329\) 23438.5 0.0119382
\(330\) 0 0
\(331\) 262856. 0.131871 0.0659353 0.997824i \(-0.478997\pi\)
0.0659353 + 0.997824i \(0.478997\pi\)
\(332\) 0 0
\(333\) 196223. 0.0969703
\(334\) 0 0
\(335\) 2.96741e6 1.44466
\(336\) 0 0
\(337\) 1.23123e6 0.590558 0.295279 0.955411i \(-0.404587\pi\)
0.295279 + 0.955411i \(0.404587\pi\)
\(338\) 0 0
\(339\) 121712. 0.0575218
\(340\) 0 0
\(341\) −2.11637e6 −0.985611
\(342\) 0 0
\(343\) 84174.7 0.0386319
\(344\) 0 0
\(345\) −264315. −0.119557
\(346\) 0 0
\(347\) 3.70344e6 1.65113 0.825567 0.564305i \(-0.190856\pi\)
0.825567 + 0.564305i \(0.190856\pi\)
\(348\) 0 0
\(349\) 1.62884e6 0.715837 0.357918 0.933753i \(-0.383487\pi\)
0.357918 + 0.933753i \(0.383487\pi\)
\(350\) 0 0
\(351\) −480195. −0.208041
\(352\) 0 0
\(353\) 2.56589e6 1.09598 0.547989 0.836486i \(-0.315394\pi\)
0.547989 + 0.836486i \(0.315394\pi\)
\(354\) 0 0
\(355\) 1.20326e6 0.506742
\(356\) 0 0
\(357\) −32545.8 −0.0135152
\(358\) 0 0
\(359\) 3.46895e6 1.42057 0.710284 0.703916i \(-0.248567\pi\)
0.710284 + 0.703916i \(0.248567\pi\)
\(360\) 0 0
\(361\) −1.51145e6 −0.610414
\(362\) 0 0
\(363\) 979343. 0.390093
\(364\) 0 0
\(365\) 4.59992e6 1.80725
\(366\) 0 0
\(367\) −1.64135e6 −0.636115 −0.318058 0.948071i \(-0.603031\pi\)
−0.318058 + 0.948071i \(0.603031\pi\)
\(368\) 0 0
\(369\) −330097. −0.126205
\(370\) 0 0
\(371\) 85859.4 0.0323857
\(372\) 0 0
\(373\) 1.58607e6 0.590268 0.295134 0.955456i \(-0.404636\pi\)
0.295134 + 0.955456i \(0.404636\pi\)
\(374\) 0 0
\(375\) −1.58284e6 −0.581244
\(376\) 0 0
\(377\) −4.71471e6 −1.70845
\(378\) 0 0
\(379\) 966030. 0.345456 0.172728 0.984970i \(-0.444742\pi\)
0.172728 + 0.984970i \(0.444742\pi\)
\(380\) 0 0
\(381\) 1.42710e6 0.503667
\(382\) 0 0
\(383\) 3.10322e6 1.08097 0.540487 0.841352i \(-0.318240\pi\)
0.540487 + 0.841352i \(0.318240\pi\)
\(384\) 0 0
\(385\) −31779.6 −0.0109269
\(386\) 0 0
\(387\) 843800. 0.286393
\(388\) 0 0
\(389\) 624568. 0.209269 0.104635 0.994511i \(-0.466633\pi\)
0.104635 + 0.994511i \(0.466633\pi\)
\(390\) 0 0
\(391\) 763775. 0.252652
\(392\) 0 0
\(393\) −3.45163e6 −1.12731
\(394\) 0 0
\(395\) −1.33018e6 −0.428962
\(396\) 0 0
\(397\) 2.80814e6 0.894216 0.447108 0.894480i \(-0.352454\pi\)
0.447108 + 0.894480i \(0.352454\pi\)
\(398\) 0 0
\(399\) 22139.6 0.00696207
\(400\) 0 0
\(401\) 3.81287e6 1.18411 0.592053 0.805899i \(-0.298318\pi\)
0.592053 + 0.805899i \(0.298318\pi\)
\(402\) 0 0
\(403\) 6.09958e6 1.87084
\(404\) 0 0
\(405\) −364246. −0.110346
\(406\) 0 0
\(407\) −553663. −0.165676
\(408\) 0 0
\(409\) −3.86268e6 −1.14178 −0.570888 0.821028i \(-0.693401\pi\)
−0.570888 + 0.821028i \(0.693401\pi\)
\(410\) 0 0
\(411\) 298540. 0.0871762
\(412\) 0 0
\(413\) −18205.6 −0.00525206
\(414\) 0 0
\(415\) 4.50728e6 1.28468
\(416\) 0 0
\(417\) −1.15639e6 −0.325660
\(418\) 0 0
\(419\) 4.00521e6 1.11453 0.557263 0.830336i \(-0.311851\pi\)
0.557263 + 0.830336i \(0.311851\pi\)
\(420\) 0 0
\(421\) 4.89697e6 1.34655 0.673275 0.739392i \(-0.264887\pi\)
0.673275 + 0.739392i \(0.264887\pi\)
\(422\) 0 0
\(423\) −758005. −0.205978
\(424\) 0 0
\(425\) 61921.3 0.0166291
\(426\) 0 0
\(427\) −66652.2 −0.0176907
\(428\) 0 0
\(429\) 1.35492e6 0.355443
\(430\) 0 0
\(431\) −1.81874e6 −0.471605 −0.235803 0.971801i \(-0.575772\pi\)
−0.235803 + 0.971801i \(0.575772\pi\)
\(432\) 0 0
\(433\) 545672. 0.139866 0.0699330 0.997552i \(-0.477721\pi\)
0.0699330 + 0.997552i \(0.477721\pi\)
\(434\) 0 0
\(435\) −3.57628e6 −0.906168
\(436\) 0 0
\(437\) −519566. −0.130148
\(438\) 0 0
\(439\) −2.81114e6 −0.696179 −0.348090 0.937461i \(-0.613170\pi\)
−0.348090 + 0.937461i \(0.613170\pi\)
\(440\) 0 0
\(441\) −1.36086e6 −0.333209
\(442\) 0 0
\(443\) 2.81428e6 0.681332 0.340666 0.940184i \(-0.389348\pi\)
0.340666 + 0.940184i \(0.389348\pi\)
\(444\) 0 0
\(445\) −5.55807e6 −1.33053
\(446\) 0 0
\(447\) 1.65805e6 0.392491
\(448\) 0 0
\(449\) 4.00644e6 0.937871 0.468936 0.883232i \(-0.344638\pi\)
0.468936 + 0.883232i \(0.344638\pi\)
\(450\) 0 0
\(451\) 931403. 0.215624
\(452\) 0 0
\(453\) −178182. −0.0407962
\(454\) 0 0
\(455\) 91591.8 0.0207409
\(456\) 0 0
\(457\) −5.43801e6 −1.21801 −0.609003 0.793168i \(-0.708430\pi\)
−0.609003 + 0.793168i \(0.708430\pi\)
\(458\) 0 0
\(459\) 1.05254e6 0.233188
\(460\) 0 0
\(461\) 2.50966e6 0.550001 0.275000 0.961444i \(-0.411322\pi\)
0.275000 + 0.961444i \(0.411322\pi\)
\(462\) 0 0
\(463\) −108662. −0.0235573 −0.0117787 0.999931i \(-0.503749\pi\)
−0.0117787 + 0.999931i \(0.503749\pi\)
\(464\) 0 0
\(465\) 4.62676e6 0.992303
\(466\) 0 0
\(467\) −3.09125e6 −0.655906 −0.327953 0.944694i \(-0.606359\pi\)
−0.327953 + 0.944694i \(0.606359\pi\)
\(468\) 0 0
\(469\) 133874. 0.0281037
\(470\) 0 0
\(471\) −1.73985e6 −0.361377
\(472\) 0 0
\(473\) −2.38087e6 −0.489309
\(474\) 0 0
\(475\) −42122.7 −0.00856608
\(476\) 0 0
\(477\) −2.77671e6 −0.558772
\(478\) 0 0
\(479\) 7.74743e6 1.54283 0.771417 0.636330i \(-0.219548\pi\)
0.771417 + 0.636330i \(0.219548\pi\)
\(480\) 0 0
\(481\) 1.59571e6 0.314479
\(482\) 0 0
\(483\) −11924.5 −0.00232580
\(484\) 0 0
\(485\) −2.00538e6 −0.387118
\(486\) 0 0
\(487\) −6.60929e6 −1.26279 −0.631397 0.775460i \(-0.717518\pi\)
−0.631397 + 0.775460i \(0.717518\pi\)
\(488\) 0 0
\(489\) 956575. 0.180903
\(490\) 0 0
\(491\) −1.68453e6 −0.315337 −0.157669 0.987492i \(-0.550398\pi\)
−0.157669 + 0.987492i \(0.550398\pi\)
\(492\) 0 0
\(493\) 1.03341e7 1.91495
\(494\) 0 0
\(495\) 1.02776e6 0.188529
\(496\) 0 0
\(497\) 54284.5 0.00985791
\(498\) 0 0
\(499\) −6.29003e6 −1.13084 −0.565420 0.824803i \(-0.691286\pi\)
−0.565420 + 0.824803i \(0.691286\pi\)
\(500\) 0 0
\(501\) −699440. −0.124496
\(502\) 0 0
\(503\) −3.68350e6 −0.649143 −0.324572 0.945861i \(-0.605220\pi\)
−0.324572 + 0.945861i \(0.605220\pi\)
\(504\) 0 0
\(505\) 2.28462e6 0.398644
\(506\) 0 0
\(507\) −563376. −0.0973371
\(508\) 0 0
\(509\) 3.33612e6 0.570752 0.285376 0.958416i \(-0.407882\pi\)
0.285376 + 0.958416i \(0.407882\pi\)
\(510\) 0 0
\(511\) 207524. 0.0351573
\(512\) 0 0
\(513\) −716000. −0.120121
\(514\) 0 0
\(515\) −9.57854e6 −1.59141
\(516\) 0 0
\(517\) 2.13879e6 0.351918
\(518\) 0 0
\(519\) −1.96671e6 −0.320496
\(520\) 0 0
\(521\) −987399. −0.159367 −0.0796835 0.996820i \(-0.525391\pi\)
−0.0796835 + 0.996820i \(0.525391\pi\)
\(522\) 0 0
\(523\) −6.57834e6 −1.05163 −0.525814 0.850600i \(-0.676239\pi\)
−0.525814 + 0.850600i \(0.676239\pi\)
\(524\) 0 0
\(525\) −966.753 −0.000153080 0
\(526\) 0 0
\(527\) −1.33696e7 −2.09697
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) 588772. 0.0906172
\(532\) 0 0
\(533\) −2.68439e6 −0.409287
\(534\) 0 0
\(535\) 9.89463e6 1.49457
\(536\) 0 0
\(537\) −4.90943e6 −0.734676
\(538\) 0 0
\(539\) 3.83981e6 0.569295
\(540\) 0 0
\(541\) −6.47470e6 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(542\) 0 0
\(543\) −4.59790e6 −0.669206
\(544\) 0 0
\(545\) 7.68087e6 1.10769
\(546\) 0 0
\(547\) 495799. 0.0708496 0.0354248 0.999372i \(-0.488722\pi\)
0.0354248 + 0.999372i \(0.488722\pi\)
\(548\) 0 0
\(549\) 2.15554e6 0.305229
\(550\) 0 0
\(551\) −7.02992e6 −0.986442
\(552\) 0 0
\(553\) −60010.8 −0.00834481
\(554\) 0 0
\(555\) 1.21041e6 0.166801
\(556\) 0 0
\(557\) 3.94887e6 0.539306 0.269653 0.962958i \(-0.413091\pi\)
0.269653 + 0.962958i \(0.413091\pi\)
\(558\) 0 0
\(559\) 6.86190e6 0.928784
\(560\) 0 0
\(561\) −2.96984e6 −0.398406
\(562\) 0 0
\(563\) 2.70237e6 0.359314 0.179657 0.983729i \(-0.442501\pi\)
0.179657 + 0.983729i \(0.442501\pi\)
\(564\) 0 0
\(565\) 750781. 0.0989447
\(566\) 0 0
\(567\) −16432.8 −0.00214662
\(568\) 0 0
\(569\) 1.33108e7 1.72355 0.861773 0.507294i \(-0.169354\pi\)
0.861773 + 0.507294i \(0.169354\pi\)
\(570\) 0 0
\(571\) 1.10053e7 1.41257 0.706286 0.707926i \(-0.250369\pi\)
0.706286 + 0.707926i \(0.250369\pi\)
\(572\) 0 0
\(573\) −519167. −0.0660572
\(574\) 0 0
\(575\) 22687.5 0.00286165
\(576\) 0 0
\(577\) −480544. −0.0600888 −0.0300444 0.999549i \(-0.509565\pi\)
−0.0300444 + 0.999549i \(0.509565\pi\)
\(578\) 0 0
\(579\) 7.39882e6 0.917204
\(580\) 0 0
\(581\) 203345. 0.0249915
\(582\) 0 0
\(583\) 7.83477e6 0.954674
\(584\) 0 0
\(585\) −2.96210e6 −0.357857
\(586\) 0 0
\(587\) −9.12805e6 −1.09341 −0.546705 0.837325i \(-0.684118\pi\)
−0.546705 + 0.837325i \(0.684118\pi\)
\(588\) 0 0
\(589\) 9.09485e6 1.08021
\(590\) 0 0
\(591\) −8.74414e6 −1.02979
\(592\) 0 0
\(593\) −2.66114e6 −0.310765 −0.155382 0.987854i \(-0.549661\pi\)
−0.155382 + 0.987854i \(0.549661\pi\)
\(594\) 0 0
\(595\) −200760. −0.0232479
\(596\) 0 0
\(597\) 7.25499e6 0.833107
\(598\) 0 0
\(599\) −6.44915e6 −0.734405 −0.367202 0.930141i \(-0.619684\pi\)
−0.367202 + 0.930141i \(0.619684\pi\)
\(600\) 0 0
\(601\) −1.48491e6 −0.167693 −0.0838463 0.996479i \(-0.526720\pi\)
−0.0838463 + 0.996479i \(0.526720\pi\)
\(602\) 0 0
\(603\) −4.32951e6 −0.484893
\(604\) 0 0
\(605\) 6.04111e6 0.671009
\(606\) 0 0
\(607\) 6.56656e6 0.723379 0.361690 0.932299i \(-0.382200\pi\)
0.361690 + 0.932299i \(0.382200\pi\)
\(608\) 0 0
\(609\) −161343. −0.0176281
\(610\) 0 0
\(611\) −6.16420e6 −0.667996
\(612\) 0 0
\(613\) −9.33817e6 −1.00372 −0.501858 0.864950i \(-0.667350\pi\)
−0.501858 + 0.864950i \(0.667350\pi\)
\(614\) 0 0
\(615\) −2.03621e6 −0.217088
\(616\) 0 0
\(617\) 7.56128e6 0.799617 0.399809 0.916599i \(-0.369077\pi\)
0.399809 + 0.916599i \(0.369077\pi\)
\(618\) 0 0
\(619\) −1.16180e7 −1.21872 −0.609359 0.792895i \(-0.708573\pi\)
−0.609359 + 0.792895i \(0.708573\pi\)
\(620\) 0 0
\(621\) 385641. 0.0401286
\(622\) 0 0
\(623\) −250751. −0.0258834
\(624\) 0 0
\(625\) −9.62976e6 −0.986088
\(626\) 0 0
\(627\) 2.02027e6 0.205230
\(628\) 0 0
\(629\) −3.49763e6 −0.352491
\(630\) 0 0
\(631\) 8.43872e6 0.843730 0.421865 0.906659i \(-0.361376\pi\)
0.421865 + 0.906659i \(0.361376\pi\)
\(632\) 0 0
\(633\) −7.39722e6 −0.733768
\(634\) 0 0
\(635\) 8.80313e6 0.866369
\(636\) 0 0
\(637\) −1.10667e7 −1.08061
\(638\) 0 0
\(639\) −1.75557e6 −0.170085
\(640\) 0 0
\(641\) 1.32010e7 1.26900 0.634500 0.772923i \(-0.281206\pi\)
0.634500 + 0.772923i \(0.281206\pi\)
\(642\) 0 0
\(643\) 1.57321e7 1.50058 0.750290 0.661109i \(-0.229914\pi\)
0.750290 + 0.661109i \(0.229914\pi\)
\(644\) 0 0
\(645\) 5.20501e6 0.492631
\(646\) 0 0
\(647\) 6.22066e6 0.584219 0.292109 0.956385i \(-0.405643\pi\)
0.292109 + 0.956385i \(0.405643\pi\)
\(648\) 0 0
\(649\) −1.66128e6 −0.154822
\(650\) 0 0
\(651\) 208735. 0.0193038
\(652\) 0 0
\(653\) −1.57628e7 −1.44660 −0.723302 0.690532i \(-0.757376\pi\)
−0.723302 + 0.690532i \(0.757376\pi\)
\(654\) 0 0
\(655\) −2.12915e7 −1.93911
\(656\) 0 0
\(657\) −6.71136e6 −0.606593
\(658\) 0 0
\(659\) 5.77008e6 0.517569 0.258784 0.965935i \(-0.416678\pi\)
0.258784 + 0.965935i \(0.416678\pi\)
\(660\) 0 0
\(661\) −1.48192e7 −1.31924 −0.659618 0.751601i \(-0.729282\pi\)
−0.659618 + 0.751601i \(0.729282\pi\)
\(662\) 0 0
\(663\) 8.55938e6 0.756238
\(664\) 0 0
\(665\) 136569. 0.0119756
\(666\) 0 0
\(667\) 3.78635e6 0.329538
\(668\) 0 0
\(669\) 4.60753e6 0.398018
\(670\) 0 0
\(671\) −6.08209e6 −0.521491
\(672\) 0 0
\(673\) −1.74615e7 −1.48609 −0.743045 0.669242i \(-0.766619\pi\)
−0.743045 + 0.669242i \(0.766619\pi\)
\(674\) 0 0
\(675\) 31265.0 0.00264118
\(676\) 0 0
\(677\) −2.10975e7 −1.76913 −0.884564 0.466418i \(-0.845544\pi\)
−0.884564 + 0.466418i \(0.845544\pi\)
\(678\) 0 0
\(679\) −90472.2 −0.00753080
\(680\) 0 0
\(681\) −1.30567e7 −1.07886
\(682\) 0 0
\(683\) 1.43654e7 1.17833 0.589163 0.808014i \(-0.299458\pi\)
0.589163 + 0.808014i \(0.299458\pi\)
\(684\) 0 0
\(685\) 1.84155e6 0.149954
\(686\) 0 0
\(687\) −5.59052e6 −0.451919
\(688\) 0 0
\(689\) −2.25806e7 −1.81212
\(690\) 0 0
\(691\) −1.64154e7 −1.30785 −0.653923 0.756561i \(-0.726878\pi\)
−0.653923 + 0.756561i \(0.726878\pi\)
\(692\) 0 0
\(693\) 46367.0 0.00366755
\(694\) 0 0
\(695\) −7.13324e6 −0.560176
\(696\) 0 0
\(697\) 5.88391e6 0.458758
\(698\) 0 0
\(699\) −4.41301e6 −0.341619
\(700\) 0 0
\(701\) −1.07940e7 −0.829635 −0.414818 0.909905i \(-0.636155\pi\)
−0.414818 + 0.909905i \(0.636155\pi\)
\(702\) 0 0
\(703\) 2.37930e6 0.181577
\(704\) 0 0
\(705\) −4.67577e6 −0.354308
\(706\) 0 0
\(707\) 103070. 0.00775502
\(708\) 0 0
\(709\) 2.44390e7 1.82586 0.912930 0.408115i \(-0.133814\pi\)
0.912930 + 0.408115i \(0.133814\pi\)
\(710\) 0 0
\(711\) 1.94076e6 0.143979
\(712\) 0 0
\(713\) −4.89853e6 −0.360863
\(714\) 0 0
\(715\) 8.35787e6 0.611407
\(716\) 0 0
\(717\) 985527. 0.0715930
\(718\) 0 0
\(719\) 6.41280e6 0.462621 0.231310 0.972880i \(-0.425699\pi\)
0.231310 + 0.972880i \(0.425699\pi\)
\(720\) 0 0
\(721\) −432133. −0.0309585
\(722\) 0 0
\(723\) −1.22554e7 −0.871928
\(724\) 0 0
\(725\) 306970. 0.0216896
\(726\) 0 0
\(727\) 1.58788e7 1.11424 0.557122 0.830431i \(-0.311905\pi\)
0.557122 + 0.830431i \(0.311905\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.50406e7 −1.04105
\(732\) 0 0
\(733\) 2.52517e7 1.73592 0.867962 0.496631i \(-0.165430\pi\)
0.867962 + 0.496631i \(0.165430\pi\)
\(734\) 0 0
\(735\) −8.39450e6 −0.573161
\(736\) 0 0
\(737\) 1.22162e7 0.828450
\(738\) 0 0
\(739\) 1.56260e7 1.05253 0.526267 0.850319i \(-0.323591\pi\)
0.526267 + 0.850319i \(0.323591\pi\)
\(740\) 0 0
\(741\) −5.82261e6 −0.389558
\(742\) 0 0
\(743\) −1.42696e7 −0.948284 −0.474142 0.880448i \(-0.657242\pi\)
−0.474142 + 0.880448i \(0.657242\pi\)
\(744\) 0 0
\(745\) 1.02278e7 0.675133
\(746\) 0 0
\(747\) −6.57621e6 −0.431195
\(748\) 0 0
\(749\) 446393. 0.0290746
\(750\) 0 0
\(751\) −2.22359e7 −1.43865 −0.719323 0.694676i \(-0.755548\pi\)
−0.719323 + 0.694676i \(0.755548\pi\)
\(752\) 0 0
\(753\) 72380.4 0.00465193
\(754\) 0 0
\(755\) −1.09912e6 −0.0701745
\(756\) 0 0
\(757\) −1.09113e7 −0.692048 −0.346024 0.938226i \(-0.612468\pi\)
−0.346024 + 0.938226i \(0.612468\pi\)
\(758\) 0 0
\(759\) −1.08813e6 −0.0685607
\(760\) 0 0
\(761\) 2.42444e6 0.151758 0.0758788 0.997117i \(-0.475824\pi\)
0.0758788 + 0.997117i \(0.475824\pi\)
\(762\) 0 0
\(763\) 346520. 0.0215485
\(764\) 0 0
\(765\) 6.49260e6 0.401112
\(766\) 0 0
\(767\) 4.78798e6 0.293876
\(768\) 0 0
\(769\) 2.00363e6 0.122181 0.0610903 0.998132i \(-0.480542\pi\)
0.0610903 + 0.998132i \(0.480542\pi\)
\(770\) 0 0
\(771\) −1.05057e7 −0.636487
\(772\) 0 0
\(773\) 3.91181e6 0.235466 0.117733 0.993045i \(-0.462437\pi\)
0.117733 + 0.993045i \(0.462437\pi\)
\(774\) 0 0
\(775\) −397137. −0.0237512
\(776\) 0 0
\(777\) 54607.1 0.00324487
\(778\) 0 0
\(779\) −4.00260e6 −0.236319
\(780\) 0 0
\(781\) 4.95353e6 0.290594
\(782\) 0 0
\(783\) 5.21786e6 0.304150
\(784\) 0 0
\(785\) −1.07323e7 −0.621614
\(786\) 0 0
\(787\) 2.37516e7 1.36696 0.683479 0.729970i \(-0.260466\pi\)
0.683479 + 0.729970i \(0.260466\pi\)
\(788\) 0 0
\(789\) −5.02246e6 −0.287226
\(790\) 0 0
\(791\) 33871.3 0.00192482
\(792\) 0 0
\(793\) 1.75292e7 0.989872
\(794\) 0 0
\(795\) −1.71282e7 −0.961157
\(796\) 0 0
\(797\) −2.50486e7 −1.39681 −0.698405 0.715703i \(-0.746106\pi\)
−0.698405 + 0.715703i \(0.746106\pi\)
\(798\) 0 0
\(799\) 1.35113e7 0.748737
\(800\) 0 0
\(801\) 8.10932e6 0.446584
\(802\) 0 0
\(803\) 1.89368e7 1.03638
\(804\) 0 0
\(805\) −73556.7 −0.00400067
\(806\) 0 0
\(807\) 1.09752e7 0.593237
\(808\) 0 0
\(809\) −2.26709e7 −1.21786 −0.608931 0.793223i \(-0.708402\pi\)
−0.608931 + 0.793223i \(0.708402\pi\)
\(810\) 0 0
\(811\) 1.66885e7 0.890974 0.445487 0.895288i \(-0.353031\pi\)
0.445487 + 0.895288i \(0.353031\pi\)
\(812\) 0 0
\(813\) 1.26523e7 0.671342
\(814\) 0 0
\(815\) 5.90066e6 0.311176
\(816\) 0 0
\(817\) 1.02315e7 0.536272
\(818\) 0 0
\(819\) −133634. −0.00696158
\(820\) 0 0
\(821\) −2.41094e7 −1.24833 −0.624164 0.781293i \(-0.714560\pi\)
−0.624164 + 0.781293i \(0.714560\pi\)
\(822\) 0 0
\(823\) 9.03070e6 0.464753 0.232376 0.972626i \(-0.425350\pi\)
0.232376 + 0.972626i \(0.425350\pi\)
\(824\) 0 0
\(825\) −88217.4 −0.00451252
\(826\) 0 0
\(827\) −1.22307e7 −0.621852 −0.310926 0.950434i \(-0.600639\pi\)
−0.310926 + 0.950434i \(0.600639\pi\)
\(828\) 0 0
\(829\) 2.01683e7 1.01925 0.509627 0.860395i \(-0.329783\pi\)
0.509627 + 0.860395i \(0.329783\pi\)
\(830\) 0 0
\(831\) −1.69019e7 −0.849051
\(832\) 0 0
\(833\) 2.42570e7 1.21123
\(834\) 0 0
\(835\) −4.31452e6 −0.214149
\(836\) 0 0
\(837\) −6.75053e6 −0.333061
\(838\) 0 0
\(839\) −1.80995e7 −0.887690 −0.443845 0.896103i \(-0.646386\pi\)
−0.443845 + 0.896103i \(0.646386\pi\)
\(840\) 0 0
\(841\) 3.07195e7 1.49770
\(842\) 0 0
\(843\) 1.90583e7 0.923668
\(844\) 0 0
\(845\) −3.47520e6 −0.167432
\(846\) 0 0
\(847\) 272543. 0.0130535
\(848\) 0 0
\(849\) 1.04438e7 0.497265
\(850\) 0 0
\(851\) −1.28150e6 −0.0606591
\(852\) 0 0
\(853\) −1.82781e7 −0.860121 −0.430060 0.902800i \(-0.641508\pi\)
−0.430060 + 0.902800i \(0.641508\pi\)
\(854\) 0 0
\(855\) −4.41667e6 −0.206623
\(856\) 0 0
\(857\) −1.47267e7 −0.684940 −0.342470 0.939529i \(-0.611264\pi\)
−0.342470 + 0.939529i \(0.611264\pi\)
\(858\) 0 0
\(859\) 1.82990e7 0.846142 0.423071 0.906096i \(-0.360952\pi\)
0.423071 + 0.906096i \(0.360952\pi\)
\(860\) 0 0
\(861\) −91863.1 −0.00422312
\(862\) 0 0
\(863\) 3.92569e6 0.179427 0.0897137 0.995968i \(-0.471405\pi\)
0.0897137 + 0.995968i \(0.471405\pi\)
\(864\) 0 0
\(865\) −1.21317e7 −0.551293
\(866\) 0 0
\(867\) −5.98254e6 −0.270295
\(868\) 0 0
\(869\) −5.47606e6 −0.245991
\(870\) 0 0
\(871\) −3.52082e7 −1.57253
\(872\) 0 0
\(873\) 2.92589e6 0.129934
\(874\) 0 0
\(875\) −440490. −0.0194499
\(876\) 0 0
\(877\) −3.49463e7 −1.53427 −0.767135 0.641486i \(-0.778318\pi\)
−0.767135 + 0.641486i \(0.778318\pi\)
\(878\) 0 0
\(879\) 1.14592e7 0.500245
\(880\) 0 0
\(881\) −3.45541e7 −1.49989 −0.749945 0.661501i \(-0.769920\pi\)
−0.749945 + 0.661501i \(0.769920\pi\)
\(882\) 0 0
\(883\) −2.29538e6 −0.0990722 −0.0495361 0.998772i \(-0.515774\pi\)
−0.0495361 + 0.998772i \(0.515774\pi\)
\(884\) 0 0
\(885\) 3.63186e6 0.155873
\(886\) 0 0
\(887\) −2.48902e7 −1.06223 −0.531116 0.847299i \(-0.678227\pi\)
−0.531116 + 0.847299i \(0.678227\pi\)
\(888\) 0 0
\(889\) 397151. 0.0168539
\(890\) 0 0
\(891\) −1.49952e6 −0.0632786
\(892\) 0 0
\(893\) −9.19120e6 −0.385695
\(894\) 0 0
\(895\) −3.02840e7 −1.26373
\(896\) 0 0
\(897\) 3.13609e6 0.130139
\(898\) 0 0
\(899\) −6.62789e7 −2.73512
\(900\) 0 0
\(901\) 4.94943e7 2.03115
\(902\) 0 0
\(903\) 234822. 0.00958341
\(904\) 0 0
\(905\) −2.83623e7 −1.15112
\(906\) 0 0
\(907\) 1.12581e7 0.454411 0.227205 0.973847i \(-0.427041\pi\)
0.227205 + 0.973847i \(0.427041\pi\)
\(908\) 0 0
\(909\) −3.33330e6 −0.133803
\(910\) 0 0
\(911\) 2.07869e7 0.829837 0.414919 0.909859i \(-0.363810\pi\)
0.414919 + 0.909859i \(0.363810\pi\)
\(912\) 0 0
\(913\) 1.85555e7 0.736707
\(914\) 0 0
\(915\) 1.32965e7 0.525032
\(916\) 0 0
\(917\) −960558. −0.0377225
\(918\) 0 0
\(919\) 1.17836e7 0.460244 0.230122 0.973162i \(-0.426087\pi\)
0.230122 + 0.973162i \(0.426087\pi\)
\(920\) 0 0
\(921\) −2.03557e7 −0.790747
\(922\) 0 0
\(923\) −1.42766e7 −0.551594
\(924\) 0 0
\(925\) −103895. −0.00399246
\(926\) 0 0
\(927\) 1.39753e7 0.534147
\(928\) 0 0
\(929\) −2.82219e7 −1.07287 −0.536435 0.843942i \(-0.680229\pi\)
−0.536435 + 0.843942i \(0.680229\pi\)
\(930\) 0 0
\(931\) −1.65011e7 −0.623935
\(932\) 0 0
\(933\) 1.08236e7 0.407070
\(934\) 0 0
\(935\) −1.83196e7 −0.685308
\(936\) 0 0
\(937\) 1.34325e7 0.499814 0.249907 0.968270i \(-0.419600\pi\)
0.249907 + 0.968270i \(0.419600\pi\)
\(938\) 0 0
\(939\) 4.35880e6 0.161325
\(940\) 0 0
\(941\) 4.06264e7 1.49566 0.747832 0.663888i \(-0.231095\pi\)
0.747832 + 0.663888i \(0.231095\pi\)
\(942\) 0 0
\(943\) 2.15582e6 0.0789465
\(944\) 0 0
\(945\) −101366. −0.00369245
\(946\) 0 0
\(947\) 3.34341e7 1.21148 0.605738 0.795664i \(-0.292878\pi\)
0.605738 + 0.795664i \(0.292878\pi\)
\(948\) 0 0
\(949\) −5.45778e7 −1.96721
\(950\) 0 0
\(951\) −7.16962e6 −0.257066
\(952\) 0 0
\(953\) −8.95158e6 −0.319277 −0.159638 0.987176i \(-0.551033\pi\)
−0.159638 + 0.987176i \(0.551033\pi\)
\(954\) 0 0
\(955\) −3.20250e6 −0.113627
\(956\) 0 0
\(957\) −1.47227e7 −0.519648
\(958\) 0 0
\(959\) 83081.1 0.00291713
\(960\) 0 0
\(961\) 5.71181e7 1.99510
\(962\) 0 0
\(963\) −1.44365e7 −0.501643
\(964\) 0 0
\(965\) 4.56399e7 1.57771
\(966\) 0 0
\(967\) 2.94663e7 1.01335 0.506675 0.862137i \(-0.330874\pi\)
0.506675 + 0.862137i \(0.330874\pi\)
\(968\) 0 0
\(969\) 1.27626e7 0.436645
\(970\) 0 0
\(971\) 3.03617e7 1.03342 0.516712 0.856159i \(-0.327156\pi\)
0.516712 + 0.856159i \(0.327156\pi\)
\(972\) 0 0
\(973\) −321814. −0.0108974
\(974\) 0 0
\(975\) 254251. 0.00856548
\(976\) 0 0
\(977\) 3.77587e6 0.126555 0.0632777 0.997996i \(-0.479845\pi\)
0.0632777 + 0.997996i \(0.479845\pi\)
\(978\) 0 0
\(979\) −2.28813e7 −0.762999
\(980\) 0 0
\(981\) −1.12065e7 −0.371791
\(982\) 0 0
\(983\) −1.74788e7 −0.576938 −0.288469 0.957489i \(-0.593146\pi\)
−0.288469 + 0.957489i \(0.593146\pi\)
\(984\) 0 0
\(985\) −5.39385e7 −1.77137
\(986\) 0 0
\(987\) −210946. −0.00689253
\(988\) 0 0
\(989\) −5.51074e6 −0.179151
\(990\) 0 0
\(991\) −4.83380e7 −1.56352 −0.781762 0.623576i \(-0.785679\pi\)
−0.781762 + 0.623576i \(0.785679\pi\)
\(992\) 0 0
\(993\) −2.36570e6 −0.0761355
\(994\) 0 0
\(995\) 4.47526e7 1.43305
\(996\) 0 0
\(997\) −1.74335e7 −0.555452 −0.277726 0.960660i \(-0.589581\pi\)
−0.277726 + 0.960660i \(0.589581\pi\)
\(998\) 0 0
\(999\) −1.76601e6 −0.0559858
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 1104.6.a.i.1.2 3
4.3 odd 2 69.6.a.b.1.1 3
12.11 even 2 207.6.a.c.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.b.1.1 3 4.3 odd 2
207.6.a.c.1.3 3 12.11 even 2
1104.6.a.i.1.2 3 1.1 even 1 trivial