Properties

Label 207.6.a.a.1.2
Level $207$
Weight $6$
Character 207.1
Self dual yes
Analytic conductor $33.199$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.1994507013\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \( x^{2} - x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.19258\) of defining polynomial
Character \(\chi\) \(=\) 207.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.7703 q^{2} +84.0000 q^{4} -41.6148 q^{5} +0.236813 q^{7} +560.057 q^{8} +O(q^{10})\) \(q+10.7703 q^{2} +84.0000 q^{4} -41.6148 q^{5} +0.236813 q^{7} +560.057 q^{8} -448.205 q^{10} +421.598 q^{11} +254.502 q^{13} +2.55055 q^{14} +3344.00 q^{16} +975.007 q^{17} +2039.79 q^{19} -3495.65 q^{20} +4540.75 q^{22} -529.000 q^{23} -1393.21 q^{25} +2741.07 q^{26} +19.8923 q^{28} -2671.55 q^{29} +9039.14 q^{31} +18094.2 q^{32} +10501.1 q^{34} -9.85493 q^{35} -12665.3 q^{37} +21969.2 q^{38} -23306.7 q^{40} -10146.4 q^{41} +19523.2 q^{43} +35414.2 q^{44} -5697.50 q^{46} -27679.7 q^{47} -16806.9 q^{49} -15005.3 q^{50} +21378.2 q^{52} -10852.8 q^{53} -17544.7 q^{55} +132.629 q^{56} -28773.4 q^{58} -11907.7 q^{59} +39861.9 q^{61} +97354.5 q^{62} +87872.0 q^{64} -10591.1 q^{65} -28550.5 q^{67} +81900.6 q^{68} -106.141 q^{70} -52179.8 q^{71} +56918.0 q^{73} -136410. q^{74} +171342. q^{76} +99.8398 q^{77} +23178.3 q^{79} -139160. q^{80} -109281. q^{82} +18344.6 q^{83} -40574.8 q^{85} +210272. q^{86} +236119. q^{88} -47362.4 q^{89} +60.2694 q^{91} -44436.0 q^{92} -298120. q^{94} -84885.4 q^{95} -140379. q^{97} -181016. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7} + 116 q^{10} - 320 q^{11} - 288 q^{13} + 1276 q^{14} + 6688 q^{16} + 1810 q^{17} + 730 q^{19} - 7896 q^{20} + 12528 q^{22} - 1058 q^{23} - 1774 q^{25} + 8584 q^{26} - 9912 q^{28} - 8208 q^{29} + 1772 q^{31} + 1508 q^{34} + 6184 q^{35} - 23112 q^{37} + 36076 q^{38} + 6032 q^{40} - 5516 q^{41} + 10322 q^{43} - 26880 q^{44} - 42952 q^{47} - 19634 q^{49} - 10904 q^{50} - 24192 q^{52} + 25350 q^{53} + 21304 q^{55} + 66352 q^{56} + 30856 q^{58} - 18344 q^{59} + 37224 q^{61} + 175624 q^{62} + 175744 q^{64} + 17828 q^{65} - 7482 q^{67} + 152040 q^{68} - 66816 q^{70} - 126848 q^{71} + 137660 q^{73} - 23896 q^{74} + 61320 q^{76} + 87784 q^{77} + 62286 q^{79} - 314336 q^{80} - 159152 q^{82} - 83120 q^{83} - 84316 q^{85} + 309372 q^{86} + 651456 q^{88} - 69770 q^{89} + 64204 q^{91} - 88872 q^{92} - 133632 q^{94} - 16272 q^{95} - 170104 q^{97} - 150568 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.7703 1.90394 0.951972 0.306186i \(-0.0990530\pi\)
0.951972 + 0.306186i \(0.0990530\pi\)
\(3\) 0 0
\(4\) 84.0000 2.62500
\(5\) −41.6148 −0.744429 −0.372214 0.928147i \(-0.621401\pi\)
−0.372214 + 0.928147i \(0.621401\pi\)
\(6\) 0 0
\(7\) 0.236813 0.00182667 0.000913335 1.00000i \(-0.499709\pi\)
0.000913335 1.00000i \(0.499709\pi\)
\(8\) 560.057 3.09391
\(9\) 0 0
\(10\) −448.205 −1.41735
\(11\) 421.598 1.05055 0.525275 0.850933i \(-0.323963\pi\)
0.525275 + 0.850933i \(0.323963\pi\)
\(12\) 0 0
\(13\) 254.502 0.417670 0.208835 0.977951i \(-0.433033\pi\)
0.208835 + 0.977951i \(0.433033\pi\)
\(14\) 2.55055 0.00347788
\(15\) 0 0
\(16\) 3344.00 3.26562
\(17\) 975.007 0.818249 0.409125 0.912479i \(-0.365834\pi\)
0.409125 + 0.912479i \(0.365834\pi\)
\(18\) 0 0
\(19\) 2039.79 1.29629 0.648143 0.761519i \(-0.275546\pi\)
0.648143 + 0.761519i \(0.275546\pi\)
\(20\) −3495.65 −1.95413
\(21\) 0 0
\(22\) 4540.75 2.00019
\(23\) −529.000 −0.208514
\(24\) 0 0
\(25\) −1393.21 −0.445826
\(26\) 2741.07 0.795220
\(27\) 0 0
\(28\) 19.8923 0.00479501
\(29\) −2671.55 −0.589885 −0.294943 0.955515i \(-0.595301\pi\)
−0.294943 + 0.955515i \(0.595301\pi\)
\(30\) 0 0
\(31\) 9039.14 1.68936 0.844681 0.535270i \(-0.179790\pi\)
0.844681 + 0.535270i \(0.179790\pi\)
\(32\) 18094.2 3.12366
\(33\) 0 0
\(34\) 10501.1 1.55790
\(35\) −9.85493 −0.00135983
\(36\) 0 0
\(37\) −12665.3 −1.52094 −0.760471 0.649372i \(-0.775032\pi\)
−0.760471 + 0.649372i \(0.775032\pi\)
\(38\) 21969.2 2.46805
\(39\) 0 0
\(40\) −23306.7 −2.30319
\(41\) −10146.4 −0.942658 −0.471329 0.881957i \(-0.656226\pi\)
−0.471329 + 0.881957i \(0.656226\pi\)
\(42\) 0 0
\(43\) 19523.2 1.61020 0.805102 0.593137i \(-0.202111\pi\)
0.805102 + 0.593137i \(0.202111\pi\)
\(44\) 35414.2 2.75769
\(45\) 0 0
\(46\) −5697.50 −0.397000
\(47\) −27679.7 −1.82775 −0.913875 0.405995i \(-0.866925\pi\)
−0.913875 + 0.405995i \(0.866925\pi\)
\(48\) 0 0
\(49\) −16806.9 −0.999997
\(50\) −15005.3 −0.848827
\(51\) 0 0
\(52\) 21378.2 1.09638
\(53\) −10852.8 −0.530703 −0.265351 0.964152i \(-0.585488\pi\)
−0.265351 + 0.964152i \(0.585488\pi\)
\(54\) 0 0
\(55\) −17544.7 −0.782059
\(56\) 132.629 0.00565155
\(57\) 0 0
\(58\) −28773.4 −1.12311
\(59\) −11907.7 −0.445345 −0.222672 0.974893i \(-0.571478\pi\)
−0.222672 + 0.974893i \(0.571478\pi\)
\(60\) 0 0
\(61\) 39861.9 1.37162 0.685809 0.727782i \(-0.259449\pi\)
0.685809 + 0.727782i \(0.259449\pi\)
\(62\) 97354.5 3.21645
\(63\) 0 0
\(64\) 87872.0 2.68164
\(65\) −10591.1 −0.310925
\(66\) 0 0
\(67\) −28550.5 −0.777009 −0.388504 0.921447i \(-0.627008\pi\)
−0.388504 + 0.921447i \(0.627008\pi\)
\(68\) 81900.6 2.14790
\(69\) 0 0
\(70\) −106.141 −0.00258903
\(71\) −52179.8 −1.22845 −0.614223 0.789132i \(-0.710531\pi\)
−0.614223 + 0.789132i \(0.710531\pi\)
\(72\) 0 0
\(73\) 56918.0 1.25009 0.625047 0.780587i \(-0.285080\pi\)
0.625047 + 0.780587i \(0.285080\pi\)
\(74\) −136410. −2.89579
\(75\) 0 0
\(76\) 171342. 3.40275
\(77\) 99.8398 0.00191901
\(78\) 0 0
\(79\) 23178.3 0.417844 0.208922 0.977932i \(-0.433004\pi\)
0.208922 + 0.977932i \(0.433004\pi\)
\(80\) −139160. −2.43103
\(81\) 0 0
\(82\) −109281. −1.79477
\(83\) 18344.6 0.292289 0.146144 0.989263i \(-0.453314\pi\)
0.146144 + 0.989263i \(0.453314\pi\)
\(84\) 0 0
\(85\) −40574.8 −0.609128
\(86\) 210272. 3.06574
\(87\) 0 0
\(88\) 236119. 3.25030
\(89\) −47362.4 −0.633810 −0.316905 0.948457i \(-0.602644\pi\)
−0.316905 + 0.948457i \(0.602644\pi\)
\(90\) 0 0
\(91\) 60.2694 0.000762945 0
\(92\) −44436.0 −0.547350
\(93\) 0 0
\(94\) −298120. −3.47993
\(95\) −84885.4 −0.964992
\(96\) 0 0
\(97\) −140379. −1.51486 −0.757432 0.652915i \(-0.773546\pi\)
−0.757432 + 0.652915i \(0.773546\pi\)
\(98\) −181016. −1.90394
\(99\) 0 0
\(100\) −117029. −1.17029
\(101\) −87080.3 −0.849408 −0.424704 0.905332i \(-0.639622\pi\)
−0.424704 + 0.905332i \(0.639622\pi\)
\(102\) 0 0
\(103\) 17988.9 0.167075 0.0835375 0.996505i \(-0.473378\pi\)
0.0835375 + 0.996505i \(0.473378\pi\)
\(104\) 142536. 1.29223
\(105\) 0 0
\(106\) −116888. −1.01043
\(107\) 127485. 1.07646 0.538231 0.842797i \(-0.319093\pi\)
0.538231 + 0.842797i \(0.319093\pi\)
\(108\) 0 0
\(109\) −136418. −1.09978 −0.549890 0.835237i \(-0.685330\pi\)
−0.549890 + 0.835237i \(0.685330\pi\)
\(110\) −188962. −1.48900
\(111\) 0 0
\(112\) 791.902 0.00596522
\(113\) 104012. 0.766279 0.383139 0.923691i \(-0.374843\pi\)
0.383139 + 0.923691i \(0.374843\pi\)
\(114\) 0 0
\(115\) 22014.2 0.155224
\(116\) −224410. −1.54845
\(117\) 0 0
\(118\) −128249. −0.847912
\(119\) 230.894 0.00149467
\(120\) 0 0
\(121\) 16693.7 0.103655
\(122\) 429325. 2.61148
\(123\) 0 0
\(124\) 759288. 4.43458
\(125\) 188024. 1.07631
\(126\) 0 0
\(127\) 203763. 1.12103 0.560513 0.828145i \(-0.310604\pi\)
0.560513 + 0.828145i \(0.310604\pi\)
\(128\) 367397. 1.98203
\(129\) 0 0
\(130\) −114069. −0.591984
\(131\) −60674.1 −0.308905 −0.154453 0.988000i \(-0.549361\pi\)
−0.154453 + 0.988000i \(0.549361\pi\)
\(132\) 0 0
\(133\) 483.048 0.00236789
\(134\) −307498. −1.47938
\(135\) 0 0
\(136\) 546060. 2.53159
\(137\) −94446.5 −0.429917 −0.214958 0.976623i \(-0.568962\pi\)
−0.214958 + 0.976623i \(0.568962\pi\)
\(138\) 0 0
\(139\) 403508. 1.77139 0.885697 0.464263i \(-0.153681\pi\)
0.885697 + 0.464263i \(0.153681\pi\)
\(140\) −827.814 −0.00356954
\(141\) 0 0
\(142\) −561993. −2.33889
\(143\) 107298. 0.438783
\(144\) 0 0
\(145\) 111176. 0.439128
\(146\) 613026. 2.38011
\(147\) 0 0
\(148\) −1.06389e6 −3.99247
\(149\) −384528. −1.41893 −0.709467 0.704738i \(-0.751064\pi\)
−0.709467 + 0.704738i \(0.751064\pi\)
\(150\) 0 0
\(151\) −442508. −1.57935 −0.789675 0.613525i \(-0.789751\pi\)
−0.789675 + 0.613525i \(0.789751\pi\)
\(152\) 1.14240e6 4.01059
\(153\) 0 0
\(154\) 1075.31 0.00365368
\(155\) −376162. −1.25761
\(156\) 0 0
\(157\) 240724. 0.779416 0.389708 0.920938i \(-0.372576\pi\)
0.389708 + 0.920938i \(0.372576\pi\)
\(158\) 249638. 0.795552
\(159\) 0 0
\(160\) −752985. −2.32534
\(161\) −125.274 −0.000380887 0
\(162\) 0 0
\(163\) −178188. −0.525301 −0.262651 0.964891i \(-0.584597\pi\)
−0.262651 + 0.964891i \(0.584597\pi\)
\(164\) −852301. −2.47448
\(165\) 0 0
\(166\) 197577. 0.556502
\(167\) −25525.2 −0.0708236 −0.0354118 0.999373i \(-0.511274\pi\)
−0.0354118 + 0.999373i \(0.511274\pi\)
\(168\) 0 0
\(169\) −306522. −0.825552
\(170\) −437004. −1.15975
\(171\) 0 0
\(172\) 1.63995e6 4.22678
\(173\) −701865. −1.78295 −0.891474 0.453072i \(-0.850328\pi\)
−0.891474 + 0.453072i \(0.850328\pi\)
\(174\) 0 0
\(175\) −329.929 −0.000814377 0
\(176\) 1.40982e6 3.43070
\(177\) 0 0
\(178\) −510109. −1.20674
\(179\) 292917. 0.683302 0.341651 0.939827i \(-0.389014\pi\)
0.341651 + 0.939827i \(0.389014\pi\)
\(180\) 0 0
\(181\) −38932.7 −0.0883321 −0.0441660 0.999024i \(-0.514063\pi\)
−0.0441660 + 0.999024i \(0.514063\pi\)
\(182\) 649.121 0.00145260
\(183\) 0 0
\(184\) −296270. −0.645124
\(185\) 527066. 1.13223
\(186\) 0 0
\(187\) 411061. 0.859611
\(188\) −2.32510e6 −4.79784
\(189\) 0 0
\(190\) −914243. −1.83729
\(191\) −37244.8 −0.0738724 −0.0369362 0.999318i \(-0.511760\pi\)
−0.0369362 + 0.999318i \(0.511760\pi\)
\(192\) 0 0
\(193\) −42454.3 −0.0820406 −0.0410203 0.999158i \(-0.513061\pi\)
−0.0410203 + 0.999158i \(0.513061\pi\)
\(194\) −1.51193e6 −2.88421
\(195\) 0 0
\(196\) −1.41178e6 −2.62499
\(197\) −87335.1 −0.160333 −0.0801666 0.996781i \(-0.525545\pi\)
−0.0801666 + 0.996781i \(0.525545\pi\)
\(198\) 0 0
\(199\) 1.08779e6 1.94721 0.973604 0.228245i \(-0.0732989\pi\)
0.973604 + 0.228245i \(0.0732989\pi\)
\(200\) −780275. −1.37934
\(201\) 0 0
\(202\) −937883. −1.61722
\(203\) −632.657 −0.00107753
\(204\) 0 0
\(205\) 422243. 0.701742
\(206\) 193746. 0.318101
\(207\) 0 0
\(208\) 851055. 1.36395
\(209\) 859969. 1.36181
\(210\) 0 0
\(211\) −1.02777e6 −1.58923 −0.794617 0.607111i \(-0.792328\pi\)
−0.794617 + 0.607111i \(0.792328\pi\)
\(212\) −911634. −1.39310
\(213\) 0 0
\(214\) 1.37305e6 2.04952
\(215\) −812456. −1.19868
\(216\) 0 0
\(217\) 2140.58 0.00308591
\(218\) −1.46927e6 −2.09392
\(219\) 0 0
\(220\) −1.47376e6 −2.05291
\(221\) 248141. 0.341758
\(222\) 0 0
\(223\) −209627. −0.282284 −0.141142 0.989989i \(-0.545077\pi\)
−0.141142 + 0.989989i \(0.545077\pi\)
\(224\) 4284.93 0.00570589
\(225\) 0 0
\(226\) 1.12024e6 1.45895
\(227\) 347103. 0.447088 0.223544 0.974694i \(-0.428237\pi\)
0.223544 + 0.974694i \(0.428237\pi\)
\(228\) 0 0
\(229\) −474094. −0.597414 −0.298707 0.954345i \(-0.596555\pi\)
−0.298707 + 0.954345i \(0.596555\pi\)
\(230\) 237101. 0.295538
\(231\) 0 0
\(232\) −1.49622e6 −1.82505
\(233\) −67426.2 −0.0813652 −0.0406826 0.999172i \(-0.512953\pi\)
−0.0406826 + 0.999172i \(0.512953\pi\)
\(234\) 0 0
\(235\) 1.15189e6 1.36063
\(236\) −1.00024e6 −1.16903
\(237\) 0 0
\(238\) 2486.81 0.00284577
\(239\) 631686. 0.715331 0.357665 0.933850i \(-0.383573\pi\)
0.357665 + 0.933850i \(0.383573\pi\)
\(240\) 0 0
\(241\) −582036. −0.645516 −0.322758 0.946482i \(-0.604610\pi\)
−0.322758 + 0.946482i \(0.604610\pi\)
\(242\) 179797. 0.197353
\(243\) 0 0
\(244\) 3.34840e6 3.60050
\(245\) 699418. 0.744426
\(246\) 0 0
\(247\) 519130. 0.541419
\(248\) 5.06243e6 5.22673
\(249\) 0 0
\(250\) 2.02508e6 2.04924
\(251\) 1.28835e6 1.29077 0.645386 0.763856i \(-0.276696\pi\)
0.645386 + 0.763856i \(0.276696\pi\)
\(252\) 0 0
\(253\) −223025. −0.219055
\(254\) 2.19459e6 2.13437
\(255\) 0 0
\(256\) 1.14509e6 1.09204
\(257\) 204908. 0.193520 0.0967601 0.995308i \(-0.469152\pi\)
0.0967601 + 0.995308i \(0.469152\pi\)
\(258\) 0 0
\(259\) −2999.32 −0.00277826
\(260\) −889650. −0.816179
\(261\) 0 0
\(262\) −653480. −0.588138
\(263\) −418862. −0.373407 −0.186703 0.982416i \(-0.559780\pi\)
−0.186703 + 0.982416i \(0.559780\pi\)
\(264\) 0 0
\(265\) 451637. 0.395071
\(266\) 5202.58 0.00450832
\(267\) 0 0
\(268\) −2.39824e6 −2.03965
\(269\) 1.48662e6 1.25262 0.626308 0.779576i \(-0.284565\pi\)
0.626308 + 0.779576i \(0.284565\pi\)
\(270\) 0 0
\(271\) −658640. −0.544785 −0.272392 0.962186i \(-0.587815\pi\)
−0.272392 + 0.962186i \(0.587815\pi\)
\(272\) 3.26042e6 2.67209
\(273\) 0 0
\(274\) −1.01722e6 −0.818537
\(275\) −587372. −0.468362
\(276\) 0 0
\(277\) 236526. 0.185216 0.0926082 0.995703i \(-0.470480\pi\)
0.0926082 + 0.995703i \(0.470480\pi\)
\(278\) 4.34592e6 3.37263
\(279\) 0 0
\(280\) −5519.32 −0.00420718
\(281\) 1.66253e6 1.25604 0.628020 0.778197i \(-0.283866\pi\)
0.628020 + 0.778197i \(0.283866\pi\)
\(282\) 0 0
\(283\) 966812. 0.717589 0.358795 0.933417i \(-0.383188\pi\)
0.358795 + 0.933417i \(0.383188\pi\)
\(284\) −4.38310e6 −3.22467
\(285\) 0 0
\(286\) 1.15563e6 0.835418
\(287\) −2402.81 −0.00172193
\(288\) 0 0
\(289\) −469218. −0.330469
\(290\) 1.19740e6 0.836074
\(291\) 0 0
\(292\) 4.78111e6 3.28150
\(293\) 82436.0 0.0560981 0.0280490 0.999607i \(-0.491071\pi\)
0.0280490 + 0.999607i \(0.491071\pi\)
\(294\) 0 0
\(295\) 495535. 0.331528
\(296\) −7.09332e6 −4.70565
\(297\) 0 0
\(298\) −4.14149e6 −2.70157
\(299\) −134632. −0.0870902
\(300\) 0 0
\(301\) 4623.35 0.00294131
\(302\) −4.76595e6 −3.00699
\(303\) 0 0
\(304\) 6.82105e6 4.23318
\(305\) −1.65884e6 −1.02107
\(306\) 0 0
\(307\) 471104. 0.285280 0.142640 0.989775i \(-0.454441\pi\)
0.142640 + 0.989775i \(0.454441\pi\)
\(308\) 8386.54 0.00503740
\(309\) 0 0
\(310\) −4.05139e6 −2.39442
\(311\) −974141. −0.571112 −0.285556 0.958362i \(-0.592178\pi\)
−0.285556 + 0.958362i \(0.592178\pi\)
\(312\) 0 0
\(313\) 203897. 0.117639 0.0588194 0.998269i \(-0.481266\pi\)
0.0588194 + 0.998269i \(0.481266\pi\)
\(314\) 2.59267e6 1.48396
\(315\) 0 0
\(316\) 1.94698e6 1.09684
\(317\) 1.11600e6 0.623757 0.311879 0.950122i \(-0.399042\pi\)
0.311879 + 0.950122i \(0.399042\pi\)
\(318\) 0 0
\(319\) −1.12632e6 −0.619704
\(320\) −3.65678e6 −1.99629
\(321\) 0 0
\(322\) −1349.24 −0.000725187 0
\(323\) 1.98881e6 1.06068
\(324\) 0 0
\(325\) −354574. −0.186208
\(326\) −1.91914e6 −1.00014
\(327\) 0 0
\(328\) −5.68259e6 −2.91650
\(329\) −6554.91 −0.00333870
\(330\) 0 0
\(331\) 2.08998e6 1.04851 0.524254 0.851562i \(-0.324344\pi\)
0.524254 + 0.851562i \(0.324344\pi\)
\(332\) 1.54094e6 0.767258
\(333\) 0 0
\(334\) −274915. −0.134844
\(335\) 1.18812e6 0.578428
\(336\) 0 0
\(337\) 901699. 0.432501 0.216250 0.976338i \(-0.430617\pi\)
0.216250 + 0.976338i \(0.430617\pi\)
\(338\) −3.30134e6 −1.57180
\(339\) 0 0
\(340\) −3.40828e6 −1.59896
\(341\) 3.81088e6 1.77476
\(342\) 0 0
\(343\) −7960.21 −0.00365333
\(344\) 1.09341e7 4.98182
\(345\) 0 0
\(346\) −7.55932e6 −3.39463
\(347\) 1.53207e6 0.683053 0.341526 0.939872i \(-0.389056\pi\)
0.341526 + 0.939872i \(0.389056\pi\)
\(348\) 0 0
\(349\) 1.28353e6 0.564084 0.282042 0.959402i \(-0.408988\pi\)
0.282042 + 0.959402i \(0.408988\pi\)
\(350\) −3553.44 −0.00155053
\(351\) 0 0
\(352\) 7.62846e6 3.28156
\(353\) 4.45978e6 1.90492 0.952460 0.304663i \(-0.0985440\pi\)
0.952460 + 0.304663i \(0.0985440\pi\)
\(354\) 0 0
\(355\) 2.17145e6 0.914491
\(356\) −3.97844e6 −1.66375
\(357\) 0 0
\(358\) 3.15482e6 1.30097
\(359\) −4.67506e6 −1.91448 −0.957240 0.289296i \(-0.906579\pi\)
−0.957240 + 0.289296i \(0.906579\pi\)
\(360\) 0 0
\(361\) 1.68463e6 0.680356
\(362\) −419318. −0.168179
\(363\) 0 0
\(364\) 5062.63 0.00200273
\(365\) −2.36863e6 −0.930606
\(366\) 0 0
\(367\) 2.43229e6 0.942650 0.471325 0.881960i \(-0.343776\pi\)
0.471325 + 0.881960i \(0.343776\pi\)
\(368\) −1.76898e6 −0.680930
\(369\) 0 0
\(370\) 5.67668e6 2.15571
\(371\) −2570.08 −0.000969419 0
\(372\) 0 0
\(373\) 1.37458e6 0.511561 0.255780 0.966735i \(-0.417668\pi\)
0.255780 + 0.966735i \(0.417668\pi\)
\(374\) 4.42726e6 1.63665
\(375\) 0 0
\(376\) −1.55022e7 −5.65489
\(377\) −679914. −0.246377
\(378\) 0 0
\(379\) 2.01997e6 0.722350 0.361175 0.932498i \(-0.382376\pi\)
0.361175 + 0.932498i \(0.382376\pi\)
\(380\) −7.13037e6 −2.53310
\(381\) 0 0
\(382\) −401139. −0.140649
\(383\) −2.00834e6 −0.699584 −0.349792 0.936827i \(-0.613748\pi\)
−0.349792 + 0.936827i \(0.613748\pi\)
\(384\) 0 0
\(385\) −4154.82 −0.00142857
\(386\) −457247. −0.156201
\(387\) 0 0
\(388\) −1.17919e7 −3.97652
\(389\) −3.25055e6 −1.08914 −0.544569 0.838716i \(-0.683307\pi\)
−0.544569 + 0.838716i \(0.683307\pi\)
\(390\) 0 0
\(391\) −515779. −0.170617
\(392\) −9.41285e6 −3.09390
\(393\) 0 0
\(394\) −940628. −0.305265
\(395\) −964563. −0.311055
\(396\) 0 0
\(397\) 5.43707e6 1.73137 0.865683 0.500592i \(-0.166884\pi\)
0.865683 + 0.500592i \(0.166884\pi\)
\(398\) 1.17159e7 3.70737
\(399\) 0 0
\(400\) −4.65888e6 −1.45590
\(401\) −3.27208e6 −1.01616 −0.508081 0.861309i \(-0.669645\pi\)
−0.508081 + 0.861309i \(0.669645\pi\)
\(402\) 0 0
\(403\) 2.30048e6 0.705596
\(404\) −7.31474e6 −2.22970
\(405\) 0 0
\(406\) −6813.92 −0.00205155
\(407\) −5.33968e6 −1.59783
\(408\) 0 0
\(409\) −1.56299e6 −0.462007 −0.231003 0.972953i \(-0.574201\pi\)
−0.231003 + 0.972953i \(0.574201\pi\)
\(410\) 4.54769e6 1.33608
\(411\) 0 0
\(412\) 1.51107e6 0.438572
\(413\) −2819.89 −0.000813498 0
\(414\) 0 0
\(415\) −763406. −0.217588
\(416\) 4.60500e6 1.30466
\(417\) 0 0
\(418\) 9.26215e6 2.59281
\(419\) 4.76111e6 1.32487 0.662434 0.749120i \(-0.269523\pi\)
0.662434 + 0.749120i \(0.269523\pi\)
\(420\) 0 0
\(421\) −4.56989e6 −1.25661 −0.628305 0.777967i \(-0.716251\pi\)
−0.628305 + 0.777967i \(0.716251\pi\)
\(422\) −1.10694e7 −3.02581
\(423\) 0 0
\(424\) −6.07818e6 −1.64195
\(425\) −1.35839e6 −0.364796
\(426\) 0 0
\(427\) 9439.80 0.00250549
\(428\) 1.07087e7 2.82571
\(429\) 0 0
\(430\) −8.75042e6 −2.28222
\(431\) 3.28891e6 0.852824 0.426412 0.904529i \(-0.359777\pi\)
0.426412 + 0.904529i \(0.359777\pi\)
\(432\) 0 0
\(433\) 2.11805e6 0.542895 0.271447 0.962453i \(-0.412498\pi\)
0.271447 + 0.962453i \(0.412498\pi\)
\(434\) 23054.8 0.00587540
\(435\) 0 0
\(436\) −1.14591e7 −2.88692
\(437\) −1.07905e6 −0.270294
\(438\) 0 0
\(439\) 758443. 0.187829 0.0939143 0.995580i \(-0.470062\pi\)
0.0939143 + 0.995580i \(0.470062\pi\)
\(440\) −9.82605e6 −2.41962
\(441\) 0 0
\(442\) 2.67257e6 0.650688
\(443\) −650377. −0.157455 −0.0787274 0.996896i \(-0.525086\pi\)
−0.0787274 + 0.996896i \(0.525086\pi\)
\(444\) 0 0
\(445\) 1.97098e6 0.471826
\(446\) −2.25776e6 −0.537452
\(447\) 0 0
\(448\) 20809.2 0.00489847
\(449\) 7.58111e6 1.77467 0.887333 0.461129i \(-0.152555\pi\)
0.887333 + 0.461129i \(0.152555\pi\)
\(450\) 0 0
\(451\) −4.27772e6 −0.990309
\(452\) 8.73699e6 2.01148
\(453\) 0 0
\(454\) 3.73841e6 0.851231
\(455\) −2508.10 −0.000567958 0
\(456\) 0 0
\(457\) −8.04537e6 −1.80200 −0.901002 0.433815i \(-0.857167\pi\)
−0.901002 + 0.433815i \(0.857167\pi\)
\(458\) −5.10614e6 −1.13744
\(459\) 0 0
\(460\) 1.84920e6 0.407463
\(461\) −5.63578e6 −1.23510 −0.617550 0.786532i \(-0.711875\pi\)
−0.617550 + 0.786532i \(0.711875\pi\)
\(462\) 0 0
\(463\) 1.39156e6 0.301682 0.150841 0.988558i \(-0.451802\pi\)
0.150841 + 0.988558i \(0.451802\pi\)
\(464\) −8.93365e6 −1.92634
\(465\) 0 0
\(466\) −726202. −0.154915
\(467\) −4.05063e6 −0.859469 −0.429734 0.902955i \(-0.641393\pi\)
−0.429734 + 0.902955i \(0.641393\pi\)
\(468\) 0 0
\(469\) −6761.12 −0.00141934
\(470\) 1.24062e7 2.59056
\(471\) 0 0
\(472\) −6.66897e6 −1.37786
\(473\) 8.23095e6 1.69160
\(474\) 0 0
\(475\) −2.84184e6 −0.577917
\(476\) 19395.1 0.00392351
\(477\) 0 0
\(478\) 6.80347e6 1.36195
\(479\) −6.55529e6 −1.30543 −0.652714 0.757605i \(-0.726370\pi\)
−0.652714 + 0.757605i \(0.726370\pi\)
\(480\) 0 0
\(481\) −3.22336e6 −0.635252
\(482\) −6.26872e6 −1.22903
\(483\) 0 0
\(484\) 1.40227e6 0.272094
\(485\) 5.84186e6 1.12771
\(486\) 0 0
\(487\) −1.98924e6 −0.380071 −0.190035 0.981777i \(-0.560860\pi\)
−0.190035 + 0.981777i \(0.560860\pi\)
\(488\) 2.23249e7 4.24366
\(489\) 0 0
\(490\) 7.53296e6 1.41735
\(491\) −7.99205e6 −1.49608 −0.748040 0.663654i \(-0.769005\pi\)
−0.748040 + 0.663654i \(0.769005\pi\)
\(492\) 0 0
\(493\) −2.60478e6 −0.482673
\(494\) 5.59120e6 1.03083
\(495\) 0 0
\(496\) 3.02269e7 5.51682
\(497\) −12356.8 −0.00224397
\(498\) 0 0
\(499\) −1.00308e7 −1.80338 −0.901688 0.432387i \(-0.857671\pi\)
−0.901688 + 0.432387i \(0.857671\pi\)
\(500\) 1.57940e7 2.82533
\(501\) 0 0
\(502\) 1.38760e7 2.45756
\(503\) 4.82601e6 0.850488 0.425244 0.905079i \(-0.360188\pi\)
0.425244 + 0.905079i \(0.360188\pi\)
\(504\) 0 0
\(505\) 3.62383e6 0.632324
\(506\) −2.40206e6 −0.417068
\(507\) 0 0
\(508\) 1.71161e7 2.94270
\(509\) −138289. −0.0236589 −0.0118294 0.999930i \(-0.503766\pi\)
−0.0118294 + 0.999930i \(0.503766\pi\)
\(510\) 0 0
\(511\) 13478.9 0.00228351
\(512\) 576256. 0.0971494
\(513\) 0 0
\(514\) 2.20693e6 0.368451
\(515\) −748605. −0.124375
\(516\) 0 0
\(517\) −1.16697e7 −1.92014
\(518\) −32303.6 −0.00528965
\(519\) 0 0
\(520\) −5.93160e6 −0.961975
\(521\) −4.62831e6 −0.747012 −0.373506 0.927628i \(-0.621845\pi\)
−0.373506 + 0.927628i \(0.621845\pi\)
\(522\) 0 0
\(523\) −1.54914e6 −0.247649 −0.123825 0.992304i \(-0.539516\pi\)
−0.123825 + 0.992304i \(0.539516\pi\)
\(524\) −5.09662e6 −0.810876
\(525\) 0 0
\(526\) −4.51129e6 −0.710945
\(527\) 8.81323e6 1.38232
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 4.86428e6 0.752192
\(531\) 0 0
\(532\) 40576.0 0.00621570
\(533\) −2.58229e6 −0.393720
\(534\) 0 0
\(535\) −5.30526e6 −0.801349
\(536\) −1.59899e7 −2.40399
\(537\) 0 0
\(538\) 1.60113e7 2.38491
\(539\) −7.08577e6 −1.05055
\(540\) 0 0
\(541\) −6.52702e6 −0.958787 −0.479393 0.877600i \(-0.659143\pi\)
−0.479393 + 0.877600i \(0.659143\pi\)
\(542\) −7.09377e6 −1.03724
\(543\) 0 0
\(544\) 1.76419e7 2.55593
\(545\) 5.67701e6 0.818707
\(546\) 0 0
\(547\) 1.06849e7 1.52686 0.763432 0.645888i \(-0.223513\pi\)
0.763432 + 0.645888i \(0.223513\pi\)
\(548\) −7.93351e6 −1.12853
\(549\) 0 0
\(550\) −6.32619e6 −0.891735
\(551\) −5.44938e6 −0.764660
\(552\) 0 0
\(553\) 5488.93 0.000763264 0
\(554\) 2.54746e6 0.352642
\(555\) 0 0
\(556\) 3.38947e7 4.64991
\(557\) 3.07804e6 0.420374 0.210187 0.977661i \(-0.432593\pi\)
0.210187 + 0.977661i \(0.432593\pi\)
\(558\) 0 0
\(559\) 4.96871e6 0.672533
\(560\) −32954.9 −0.00444068
\(561\) 0 0
\(562\) 1.79060e7 2.39143
\(563\) −1.10366e7 −1.46745 −0.733724 0.679448i \(-0.762219\pi\)
−0.733724 + 0.679448i \(0.762219\pi\)
\(564\) 0 0
\(565\) −4.32844e6 −0.570440
\(566\) 1.04129e7 1.36625
\(567\) 0 0
\(568\) −2.92237e7 −3.80070
\(569\) 9.26308e6 1.19943 0.599715 0.800214i \(-0.295281\pi\)
0.599715 + 0.800214i \(0.295281\pi\)
\(570\) 0 0
\(571\) −1.31913e7 −1.69315 −0.846576 0.532267i \(-0.821340\pi\)
−0.846576 + 0.532267i \(0.821340\pi\)
\(572\) 9.01300e6 1.15181
\(573\) 0 0
\(574\) −25879.0 −0.00327845
\(575\) 737006. 0.0929611
\(576\) 0 0
\(577\) −1.29741e6 −0.162232 −0.0811160 0.996705i \(-0.525848\pi\)
−0.0811160 + 0.996705i \(0.525848\pi\)
\(578\) −5.05363e6 −0.629193
\(579\) 0 0
\(580\) 9.33878e6 1.15271
\(581\) 4344.23 0.000533916 0
\(582\) 0 0
\(583\) −4.57551e6 −0.557530
\(584\) 3.18773e7 3.86767
\(585\) 0 0
\(586\) 887863. 0.106808
\(587\) −4.51203e6 −0.540477 −0.270238 0.962793i \(-0.587103\pi\)
−0.270238 + 0.962793i \(0.587103\pi\)
\(588\) 0 0
\(589\) 1.84379e7 2.18990
\(590\) 5.33708e6 0.631210
\(591\) 0 0
\(592\) −4.23529e7 −4.96683
\(593\) −9.27288e6 −1.08287 −0.541437 0.840741i \(-0.682120\pi\)
−0.541437 + 0.840741i \(0.682120\pi\)
\(594\) 0 0
\(595\) −9608.63 −0.00111268
\(596\) −3.23004e7 −3.72470
\(597\) 0 0
\(598\) −1.45003e6 −0.165815
\(599\) 6.32225e6 0.719954 0.359977 0.932961i \(-0.382785\pi\)
0.359977 + 0.932961i \(0.382785\pi\)
\(600\) 0 0
\(601\) 3.18785e6 0.360008 0.180004 0.983666i \(-0.442389\pi\)
0.180004 + 0.983666i \(0.442389\pi\)
\(602\) 49795.0 0.00560009
\(603\) 0 0
\(604\) −3.71706e7 −4.14579
\(605\) −694706. −0.0771636
\(606\) 0 0
\(607\) −5.70410e6 −0.628369 −0.314185 0.949362i \(-0.601731\pi\)
−0.314185 + 0.949362i \(0.601731\pi\)
\(608\) 3.69082e7 4.04915
\(609\) 0 0
\(610\) −1.78663e7 −1.94406
\(611\) −7.04455e6 −0.763396
\(612\) 0 0
\(613\) 1.17835e7 1.26655 0.633276 0.773926i \(-0.281710\pi\)
0.633276 + 0.773926i \(0.281710\pi\)
\(614\) 5.07394e6 0.543156
\(615\) 0 0
\(616\) 55916.0 0.00593723
\(617\) −1.41628e6 −0.149774 −0.0748868 0.997192i \(-0.523860\pi\)
−0.0748868 + 0.997192i \(0.523860\pi\)
\(618\) 0 0
\(619\) −4.37752e6 −0.459200 −0.229600 0.973285i \(-0.573742\pi\)
−0.229600 + 0.973285i \(0.573742\pi\)
\(620\) −3.15976e7 −3.30123
\(621\) 0 0
\(622\) −1.04918e7 −1.08736
\(623\) −11216.0 −0.00115776
\(624\) 0 0
\(625\) −3.47084e6 −0.355414
\(626\) 2.19604e6 0.223978
\(627\) 0 0
\(628\) 2.02208e7 2.04597
\(629\) −1.23488e7 −1.24451
\(630\) 0 0
\(631\) −1.61975e7 −1.61948 −0.809739 0.586790i \(-0.800391\pi\)
−0.809739 + 0.586790i \(0.800391\pi\)
\(632\) 1.29812e7 1.29277
\(633\) 0 0
\(634\) 1.20197e7 1.18760
\(635\) −8.47956e6 −0.834525
\(636\) 0 0
\(637\) −4.27740e6 −0.417668
\(638\) −1.21308e7 −1.17988
\(639\) 0 0
\(640\) −1.52892e7 −1.47548
\(641\) 1.56439e7 1.50383 0.751915 0.659260i \(-0.229130\pi\)
0.751915 + 0.659260i \(0.229130\pi\)
\(642\) 0 0
\(643\) −1.57570e7 −1.50295 −0.751476 0.659760i \(-0.770658\pi\)
−0.751476 + 0.659760i \(0.770658\pi\)
\(644\) −10523.0 −0.000999829 0
\(645\) 0 0
\(646\) 2.14201e7 2.01948
\(647\) 660291. 0.0620119 0.0310059 0.999519i \(-0.490129\pi\)
0.0310059 + 0.999519i \(0.490129\pi\)
\(648\) 0 0
\(649\) −5.02024e6 −0.467857
\(650\) −3.81888e6 −0.354529
\(651\) 0 0
\(652\) −1.49678e7 −1.37892
\(653\) −1.24310e7 −1.14084 −0.570419 0.821354i \(-0.693219\pi\)
−0.570419 + 0.821354i \(0.693219\pi\)
\(654\) 0 0
\(655\) 2.52494e6 0.229958
\(656\) −3.39297e7 −3.07837
\(657\) 0 0
\(658\) −70598.6 −0.00635669
\(659\) 13723.9 0.00123101 0.000615507 1.00000i \(-0.499804\pi\)
0.000615507 1.00000i \(0.499804\pi\)
\(660\) 0 0
\(661\) 6.54194e6 0.582375 0.291187 0.956666i \(-0.405950\pi\)
0.291187 + 0.956666i \(0.405950\pi\)
\(662\) 2.25098e7 1.99630
\(663\) 0 0
\(664\) 1.02740e7 0.904315
\(665\) −20101.9 −0.00176272
\(666\) 0 0
\(667\) 1.41325e6 0.123000
\(668\) −2.14412e6 −0.185912
\(669\) 0 0
\(670\) 1.27965e7 1.10129
\(671\) 1.68057e7 1.44095
\(672\) 0 0
\(673\) 1.54266e7 1.31290 0.656451 0.754369i \(-0.272057\pi\)
0.656451 + 0.754369i \(0.272057\pi\)
\(674\) 9.71159e6 0.823456
\(675\) 0 0
\(676\) −2.57478e7 −2.16707
\(677\) −5.30462e6 −0.444818 −0.222409 0.974953i \(-0.571392\pi\)
−0.222409 + 0.974953i \(0.571392\pi\)
\(678\) 0 0
\(679\) −33243.6 −0.00276716
\(680\) −2.27242e7 −1.88459
\(681\) 0 0
\(682\) 4.10444e7 3.37904
\(683\) −1.52230e7 −1.24867 −0.624337 0.781155i \(-0.714631\pi\)
−0.624337 + 0.781155i \(0.714631\pi\)
\(684\) 0 0
\(685\) 3.93038e6 0.320043
\(686\) −85734.1 −0.00695574
\(687\) 0 0
\(688\) 6.52857e7 5.25832
\(689\) −2.76206e6 −0.221659
\(690\) 0 0
\(691\) 1.04711e7 0.834248 0.417124 0.908850i \(-0.363038\pi\)
0.417124 + 0.908850i \(0.363038\pi\)
\(692\) −5.89567e7 −4.68024
\(693\) 0 0
\(694\) 1.65009e7 1.30049
\(695\) −1.67919e7 −1.31868
\(696\) 0 0
\(697\) −9.89286e6 −0.771329
\(698\) 1.38241e7 1.07398
\(699\) 0 0
\(700\) −27714.0 −0.00213774
\(701\) 2.28874e7 1.75914 0.879572 0.475765i \(-0.157829\pi\)
0.879572 + 0.475765i \(0.157829\pi\)
\(702\) 0 0
\(703\) −2.58346e7 −1.97158
\(704\) 3.70466e7 2.81720
\(705\) 0 0
\(706\) 4.80333e7 3.62686
\(707\) −20621.7 −0.00155159
\(708\) 0 0
\(709\) 1.85127e7 1.38310 0.691550 0.722329i \(-0.256928\pi\)
0.691550 + 0.722329i \(0.256928\pi\)
\(710\) 2.33873e7 1.74114
\(711\) 0 0
\(712\) −2.65257e7 −1.96095
\(713\) −4.78170e6 −0.352256
\(714\) 0 0
\(715\) −4.46517e6 −0.326643
\(716\) 2.46051e7 1.79367
\(717\) 0 0
\(718\) −5.03519e7 −3.64506
\(719\) −1.25679e7 −0.906652 −0.453326 0.891345i \(-0.649763\pi\)
−0.453326 + 0.891345i \(0.649763\pi\)
\(720\) 0 0
\(721\) 4260.00 0.000305191 0
\(722\) 1.81440e7 1.29536
\(723\) 0 0
\(724\) −3.27035e6 −0.231872
\(725\) 3.72201e6 0.262986
\(726\) 0 0
\(727\) 3.64050e6 0.255462 0.127731 0.991809i \(-0.459231\pi\)
0.127731 + 0.991809i \(0.459231\pi\)
\(728\) 33754.3 0.00236048
\(729\) 0 0
\(730\) −2.55110e7 −1.77182
\(731\) 1.90353e7 1.31755
\(732\) 0 0
\(733\) −4.38735e6 −0.301608 −0.150804 0.988564i \(-0.548186\pi\)
−0.150804 + 0.988564i \(0.548186\pi\)
\(734\) 2.61966e7 1.79475
\(735\) 0 0
\(736\) −9.57181e6 −0.651327
\(737\) −1.20368e7 −0.816287
\(738\) 0 0
\(739\) 1.95857e7 1.31925 0.659625 0.751595i \(-0.270715\pi\)
0.659625 + 0.751595i \(0.270715\pi\)
\(740\) 4.42736e7 2.97211
\(741\) 0 0
\(742\) −27680.6 −0.00184572
\(743\) 2.94833e7 1.95931 0.979657 0.200678i \(-0.0643146\pi\)
0.979657 + 0.200678i \(0.0643146\pi\)
\(744\) 0 0
\(745\) 1.60021e7 1.05630
\(746\) 1.48047e7 0.973983
\(747\) 0 0
\(748\) 3.45291e7 2.25648
\(749\) 30190.0 0.00196634
\(750\) 0 0
\(751\) 1.27147e7 0.822634 0.411317 0.911492i \(-0.365069\pi\)
0.411317 + 0.911492i \(0.365069\pi\)
\(752\) −9.25609e7 −5.96875
\(753\) 0 0
\(754\) −7.32290e6 −0.469089
\(755\) 1.84149e7 1.17571
\(756\) 0 0
\(757\) 1.63980e7 1.04004 0.520022 0.854153i \(-0.325924\pi\)
0.520022 + 0.854153i \(0.325924\pi\)
\(758\) 2.17558e7 1.37531
\(759\) 0 0
\(760\) −4.75407e7 −2.98560
\(761\) −8.50863e6 −0.532596 −0.266298 0.963891i \(-0.585801\pi\)
−0.266298 + 0.963891i \(0.585801\pi\)
\(762\) 0 0
\(763\) −32305.5 −0.00200893
\(764\) −3.12856e6 −0.193915
\(765\) 0 0
\(766\) −2.16305e7 −1.33197
\(767\) −3.03053e6 −0.186007
\(768\) 0 0
\(769\) 1.70141e7 1.03751 0.518755 0.854923i \(-0.326396\pi\)
0.518755 + 0.854923i \(0.326396\pi\)
\(770\) −44748.7 −0.00271991
\(771\) 0 0
\(772\) −3.56616e6 −0.215356
\(773\) −3.63916e6 −0.219055 −0.109527 0.993984i \(-0.534934\pi\)
−0.109527 + 0.993984i \(0.534934\pi\)
\(774\) 0 0
\(775\) −1.25934e7 −0.753161
\(776\) −7.86204e7 −4.68685
\(777\) 0 0
\(778\) −3.50095e7 −2.07366
\(779\) −2.06966e7 −1.22195
\(780\) 0 0
\(781\) −2.19989e7 −1.29054
\(782\) −5.55511e6 −0.324845
\(783\) 0 0
\(784\) −5.62024e7 −3.26561
\(785\) −1.00177e7 −0.580220
\(786\) 0 0
\(787\) −5.30127e6 −0.305101 −0.152550 0.988296i \(-0.548749\pi\)
−0.152550 + 0.988296i \(0.548749\pi\)
\(788\) −7.33615e6 −0.420875
\(789\) 0 0
\(790\) −1.03887e7 −0.592232
\(791\) 24631.3 0.00139974
\(792\) 0 0
\(793\) 1.01449e7 0.572883
\(794\) 5.85591e7 3.29642
\(795\) 0 0
\(796\) 9.13743e7 5.11142
\(797\) −1.41955e7 −0.791598 −0.395799 0.918337i \(-0.629532\pi\)
−0.395799 + 0.918337i \(0.629532\pi\)
\(798\) 0 0
\(799\) −2.69879e7 −1.49555
\(800\) −2.52089e7 −1.39261
\(801\) 0 0
\(802\) −3.52414e7 −1.93472
\(803\) 2.39965e7 1.31329
\(804\) 0 0
\(805\) 5213.26 0.000283543 0
\(806\) 2.47769e7 1.34341
\(807\) 0 0
\(808\) −4.87699e7 −2.62799
\(809\) 1.58542e7 0.851675 0.425838 0.904800i \(-0.359979\pi\)
0.425838 + 0.904800i \(0.359979\pi\)
\(810\) 0 0
\(811\) −1.40129e7 −0.748127 −0.374063 0.927403i \(-0.622036\pi\)
−0.374063 + 0.927403i \(0.622036\pi\)
\(812\) −53143.1 −0.00282851
\(813\) 0 0
\(814\) −5.75101e7 −3.04217
\(815\) 7.41524e6 0.391049
\(816\) 0 0
\(817\) 3.98232e7 2.08728
\(818\) −1.68339e7 −0.879634
\(819\) 0 0
\(820\) 3.54684e7 1.84207
\(821\) −1.16279e7 −0.602065 −0.301033 0.953614i \(-0.597331\pi\)
−0.301033 + 0.953614i \(0.597331\pi\)
\(822\) 0 0
\(823\) −2.76237e7 −1.42161 −0.710807 0.703387i \(-0.751670\pi\)
−0.710807 + 0.703387i \(0.751670\pi\)
\(824\) 1.00748e7 0.516914
\(825\) 0 0
\(826\) −30371.1 −0.00154885
\(827\) 2.78146e7 1.41420 0.707098 0.707116i \(-0.250004\pi\)
0.707098 + 0.707116i \(0.250004\pi\)
\(828\) 0 0
\(829\) −1.23659e7 −0.624943 −0.312471 0.949927i \(-0.601157\pi\)
−0.312471 + 0.949927i \(0.601157\pi\)
\(830\) −8.22214e6 −0.414276
\(831\) 0 0
\(832\) 2.23636e7 1.12004
\(833\) −1.63869e7 −0.818246
\(834\) 0 0
\(835\) 1.06223e6 0.0527231
\(836\) 7.22374e7 3.57476
\(837\) 0 0
\(838\) 5.12787e7 2.52247
\(839\) 3.49502e7 1.71413 0.857067 0.515205i \(-0.172284\pi\)
0.857067 + 0.515205i \(0.172284\pi\)
\(840\) 0 0
\(841\) −1.33740e7 −0.652035
\(842\) −4.92192e7 −2.39251
\(843\) 0 0
\(844\) −8.63323e7 −4.17174
\(845\) 1.27558e7 0.614565
\(846\) 0 0
\(847\) 3953.28 0.000189343 0
\(848\) −3.62917e7 −1.73308
\(849\) 0 0
\(850\) −1.46303e7 −0.694552
\(851\) 6.69997e6 0.317138
\(852\) 0 0
\(853\) −9.99950e6 −0.470550 −0.235275 0.971929i \(-0.575599\pi\)
−0.235275 + 0.971929i \(0.575599\pi\)
\(854\) 101670. 0.00477032
\(855\) 0 0
\(856\) 7.13987e7 3.33047
\(857\) −9.71240e6 −0.451725 −0.225863 0.974159i \(-0.572520\pi\)
−0.225863 + 0.974159i \(0.572520\pi\)
\(858\) 0 0
\(859\) 3.69393e7 1.70807 0.854035 0.520216i \(-0.174149\pi\)
0.854035 + 0.520216i \(0.174149\pi\)
\(860\) −6.82463e7 −3.14654
\(861\) 0 0
\(862\) 3.54227e7 1.62373
\(863\) 3.31284e7 1.51416 0.757082 0.653319i \(-0.226624\pi\)
0.757082 + 0.653319i \(0.226624\pi\)
\(864\) 0 0
\(865\) 2.92080e7 1.32728
\(866\) 2.28121e7 1.03364
\(867\) 0 0
\(868\) 179809. 0.00810051
\(869\) 9.77194e6 0.438966
\(870\) 0 0
\(871\) −7.26615e6 −0.324533
\(872\) −7.64019e7 −3.40262
\(873\) 0 0
\(874\) −1.16217e7 −0.514625
\(875\) 44526.6 0.00196607
\(876\) 0 0
\(877\) 1.38007e7 0.605902 0.302951 0.953006i \(-0.402028\pi\)
0.302951 + 0.953006i \(0.402028\pi\)
\(878\) 8.16869e6 0.357615
\(879\) 0 0
\(880\) −5.86696e7 −2.55391
\(881\) 7.74868e6 0.336347 0.168174 0.985757i \(-0.446213\pi\)
0.168174 + 0.985757i \(0.446213\pi\)
\(882\) 0 0
\(883\) 3.79838e7 1.63944 0.819722 0.572761i \(-0.194128\pi\)
0.819722 + 0.572761i \(0.194128\pi\)
\(884\) 2.08439e7 0.897115
\(885\) 0 0
\(886\) −7.00477e6 −0.299785
\(887\) 1.51721e7 0.647495 0.323747 0.946144i \(-0.395057\pi\)
0.323747 + 0.946144i \(0.395057\pi\)
\(888\) 0 0
\(889\) 48253.7 0.00204775
\(890\) 2.12281e7 0.898330
\(891\) 0 0
\(892\) −1.76087e7 −0.740995
\(893\) −5.64607e7 −2.36929
\(894\) 0 0
\(895\) −1.21897e7 −0.508670
\(896\) 87004.5 0.00362052
\(897\) 0 0
\(898\) 8.16510e7 3.37886
\(899\) −2.41485e7 −0.996530
\(900\) 0 0
\(901\) −1.05815e7 −0.434247
\(902\) −4.60724e7 −1.88549
\(903\) 0 0
\(904\) 5.82526e7 2.37080
\(905\) 1.62018e6 0.0657569
\(906\) 0 0
\(907\) 8.48315e6 0.342404 0.171202 0.985236i \(-0.445235\pi\)
0.171202 + 0.985236i \(0.445235\pi\)
\(908\) 2.91566e7 1.17361
\(909\) 0 0
\(910\) −27013.1 −0.00108136
\(911\) −2.96302e7 −1.18288 −0.591438 0.806351i \(-0.701439\pi\)
−0.591438 + 0.806351i \(0.701439\pi\)
\(912\) 0 0
\(913\) 7.73403e6 0.307064
\(914\) −8.66513e7 −3.43091
\(915\) 0 0
\(916\) −3.98239e7 −1.56821
\(917\) −14368.4 −0.000564268 0
\(918\) 0 0
\(919\) −1.27080e7 −0.496352 −0.248176 0.968715i \(-0.579831\pi\)
−0.248176 + 0.968715i \(0.579831\pi\)
\(920\) 1.23292e7 0.480249
\(921\) 0 0
\(922\) −6.06992e7 −2.35156
\(923\) −1.32799e7 −0.513085
\(924\) 0 0
\(925\) 1.76454e7 0.678075
\(926\) 1.49875e7 0.574385
\(927\) 0 0
\(928\) −4.83394e7 −1.84260
\(929\) 4.50533e7 1.71272 0.856362 0.516376i \(-0.172719\pi\)
0.856362 + 0.516376i \(0.172719\pi\)
\(930\) 0 0
\(931\) −3.42826e7 −1.29628
\(932\) −5.66380e6 −0.213584
\(933\) 0 0
\(934\) −4.36266e7 −1.63638
\(935\) −1.71062e7 −0.639919
\(936\) 0 0
\(937\) −2.49632e7 −0.928864 −0.464432 0.885609i \(-0.653741\pi\)
−0.464432 + 0.885609i \(0.653741\pi\)
\(938\) −72819.4 −0.00270234
\(939\) 0 0
\(940\) 9.67585e7 3.57165
\(941\) 3.36141e7 1.23751 0.618754 0.785585i \(-0.287638\pi\)
0.618754 + 0.785585i \(0.287638\pi\)
\(942\) 0 0
\(943\) 5.36747e6 0.196558
\(944\) −3.98192e7 −1.45433
\(945\) 0 0
\(946\) 8.86501e7 3.22071
\(947\) 5.33926e6 0.193467 0.0967333 0.995310i \(-0.469161\pi\)
0.0967333 + 0.995310i \(0.469161\pi\)
\(948\) 0 0
\(949\) 1.44858e7 0.522127
\(950\) −3.06076e7 −1.10032
\(951\) 0 0
\(952\) 129314. 0.00462438
\(953\) 3.69391e7 1.31751 0.658755 0.752358i \(-0.271084\pi\)
0.658755 + 0.752358i \(0.271084\pi\)
\(954\) 0 0
\(955\) 1.54994e6 0.0549928
\(956\) 5.30617e7 1.87774
\(957\) 0 0
\(958\) −7.06026e7 −2.48546
\(959\) −22366.2 −0.000785317 0
\(960\) 0 0
\(961\) 5.30769e7 1.85395
\(962\) −3.47166e7 −1.20948
\(963\) 0 0
\(964\) −4.88910e7 −1.69448
\(965\) 1.76673e6 0.0610734
\(966\) 0 0
\(967\) −4.47688e7 −1.53960 −0.769802 0.638283i \(-0.779645\pi\)
−0.769802 + 0.638283i \(0.779645\pi\)
\(968\) 9.34943e6 0.320698
\(969\) 0 0
\(970\) 6.29187e7 2.14709
\(971\) 4.07416e7 1.38672 0.693361 0.720590i \(-0.256129\pi\)
0.693361 + 0.720590i \(0.256129\pi\)
\(972\) 0 0
\(973\) 95555.9 0.00323575
\(974\) −2.14247e7 −0.723633
\(975\) 0 0
\(976\) 1.33298e8 4.47919
\(977\) −1.07576e7 −0.360561 −0.180281 0.983615i \(-0.557701\pi\)
−0.180281 + 0.983615i \(0.557701\pi\)
\(978\) 0 0
\(979\) −1.99679e7 −0.665849
\(980\) 5.87511e7 1.95412
\(981\) 0 0
\(982\) −8.60770e7 −2.84845
\(983\) 2.44766e7 0.807917 0.403958 0.914777i \(-0.367634\pi\)
0.403958 + 0.914777i \(0.367634\pi\)
\(984\) 0 0
\(985\) 3.63444e6 0.119357
\(986\) −2.80543e7 −0.918982
\(987\) 0 0
\(988\) 4.36069e7 1.42123
\(989\) −1.03278e7 −0.335751
\(990\) 0 0
\(991\) −2.09086e6 −0.0676301 −0.0338151 0.999428i \(-0.510766\pi\)
−0.0338151 + 0.999428i \(0.510766\pi\)
\(992\) 1.63556e8 5.27699
\(993\) 0 0
\(994\) −133087. −0.00427239
\(995\) −4.52682e7 −1.44956
\(996\) 0 0
\(997\) 1.49624e7 0.476721 0.238360 0.971177i \(-0.423390\pi\)
0.238360 + 0.971177i \(0.423390\pi\)
\(998\) −1.08036e8 −3.43353
\(999\) 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 207.6.a.a.1.2 2
3.2 odd 2 69.6.a.a.1.1 2
12.11 even 2 1104.6.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.a.1.1 2 3.2 odd 2
207.6.a.a.1.2 2 1.1 even 1 trivial
1104.6.a.h.1.1 2 12.11 even 2