Properties

Label 69.6.a.a.1.1
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{29}) \)
Defining polynomial: \(x^{2} - x - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.19258\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} -448.205 q^{10} -421.598 q^{11} -756.000 q^{12} +254.502 q^{13} -2.55055 q^{14} -374.534 q^{15} +3344.00 q^{16} -975.007 q^{17} -872.397 q^{18} +2039.79 q^{19} +3495.65 q^{20} -2.13132 q^{21} +4540.75 q^{22} +529.000 q^{23} +5040.51 q^{24} -1393.21 q^{25} -2741.07 q^{26} -729.000 q^{27} +19.8923 q^{28} +2671.55 q^{29} +4033.85 q^{30} +9039.14 q^{31} -18094.2 q^{32} +3794.38 q^{33} +10501.1 q^{34} +9.85493 q^{35} +6804.00 q^{36} -12665.3 q^{37} -21969.2 q^{38} -2290.52 q^{39} -23306.7 q^{40} +10146.4 q^{41} +22.9550 q^{42} +19523.2 q^{43} -35414.2 q^{44} +3370.80 q^{45} -5697.50 q^{46} +27679.7 q^{47} -30096.0 q^{48} -16806.9 q^{49} +15005.3 q^{50} +8775.06 q^{51} +21378.2 q^{52} +10852.8 q^{53} +7851.57 q^{54} -17544.7 q^{55} -132.629 q^{56} -18358.1 q^{57} -28773.4 q^{58} +11907.7 q^{59} -31460.8 q^{60} +39861.9 q^{61} -97354.5 q^{62} +19.1818 q^{63} +87872.0 q^{64} +10591.1 q^{65} -40866.7 q^{66} -28550.5 q^{67} -81900.6 q^{68} -4761.00 q^{69} -106.141 q^{70} +52179.8 q^{71} -45364.6 q^{72} +56918.0 q^{73} +136410. q^{74} +12538.8 q^{75} +171342. q^{76} -99.8398 q^{77} +24669.7 q^{78} +23178.3 q^{79} +139160. q^{80} +6561.00 q^{81} -109281. q^{82} -18344.6 q^{83} -179.031 q^{84} -40574.8 q^{85} -210272. q^{86} -24043.9 q^{87} +236119. q^{88} +47362.4 q^{89} -36304.6 q^{90} +60.2694 q^{91} +44436.0 q^{92} -81352.3 q^{93} -298120. q^{94} +84885.4 q^{95} +162847. q^{96} -140379. q^{97} +181016. q^{98} -34149.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 18q^{3} + 168q^{4} + 94q^{5} - 118q^{7} + 162q^{9} + O(q^{10}) \) \( 2q - 18q^{3} + 168q^{4} + 94q^{5} - 118q^{7} + 162q^{9} + 116q^{10} + 320q^{11} - 1512q^{12} - 288q^{13} - 1276q^{14} - 846q^{15} + 6688q^{16} - 1810q^{17} + 730q^{19} + 7896q^{20} + 1062q^{21} + 12528q^{22} + 1058q^{23} - 1774q^{25} - 8584q^{26} - 1458q^{27} - 9912q^{28} + 8208q^{29} - 1044q^{30} + 1772q^{31} - 2880q^{33} + 1508q^{34} - 6184q^{35} + 13608q^{36} - 23112q^{37} - 36076q^{38} + 2592q^{39} + 6032q^{40} + 5516q^{41} + 11484q^{42} + 10322q^{43} + 26880q^{44} + 7614q^{45} + 42952q^{47} - 60192q^{48} - 19634q^{49} + 10904q^{50} + 16290q^{51} - 24192q^{52} - 25350q^{53} + 21304q^{55} - 66352q^{56} - 6570q^{57} + 30856q^{58} + 18344q^{59} - 71064q^{60} + 37224q^{61} - 175624q^{62} - 9558q^{63} + 175744q^{64} - 17828q^{65} - 112752q^{66} - 7482q^{67} - 152040q^{68} - 9522q^{69} - 66816q^{70} + 126848q^{71} + 137660q^{73} + 23896q^{74} + 15966q^{75} + 61320q^{76} - 87784q^{77} + 77256q^{78} + 62286q^{79} + 314336q^{80} + 13122q^{81} - 159152q^{82} + 83120q^{83} + 89208q^{84} - 84316q^{85} - 309372q^{86} - 73872q^{87} + 651456q^{88} + 69770q^{89} + 9396q^{90} + 64204q^{91} + 88872q^{92} - 15948q^{93} - 133632q^{94} + 16272q^{95} - 170104q^{97} + 150568q^{98} + 25920q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.7703 −1.90394 −0.951972 0.306186i \(-0.900947\pi\)
−0.951972 + 0.306186i \(0.900947\pi\)
\(3\) −9.00000 −0.577350
\(4\) 84.0000 2.62500
\(5\) 41.6148 0.744429 0.372214 0.928147i \(-0.378599\pi\)
0.372214 + 0.928147i \(0.378599\pi\)
\(6\) 96.9330 1.09924
\(7\) 0.236813 0.00182667 0.000913335 1.00000i \(-0.499709\pi\)
0.000913335 1.00000i \(0.499709\pi\)
\(8\) −560.057 −3.09391
\(9\) 81.0000 0.333333
\(10\) −448.205 −1.41735
\(11\) −421.598 −1.05055 −0.525275 0.850933i \(-0.676037\pi\)
−0.525275 + 0.850933i \(0.676037\pi\)
\(12\) −756.000 −1.51554
\(13\) 254.502 0.417670 0.208835 0.977951i \(-0.433033\pi\)
0.208835 + 0.977951i \(0.433033\pi\)
\(14\) −2.55055 −0.00347788
\(15\) −374.534 −0.429796
\(16\) 3344.00 3.26562
\(17\) −975.007 −0.818249 −0.409125 0.912479i \(-0.634166\pi\)
−0.409125 + 0.912479i \(0.634166\pi\)
\(18\) −872.397 −0.634648
\(19\) 2039.79 1.29629 0.648143 0.761519i \(-0.275546\pi\)
0.648143 + 0.761519i \(0.275546\pi\)
\(20\) 3495.65 1.95413
\(21\) −2.13132 −0.00105463
\(22\) 4540.75 2.00019
\(23\) 529.000 0.208514
\(24\) 5040.51 1.78627
\(25\) −1393.21 −0.445826
\(26\) −2741.07 −0.795220
\(27\) −729.000 −0.192450
\(28\) 19.8923 0.00479501
\(29\) 2671.55 0.589885 0.294943 0.955515i \(-0.404699\pi\)
0.294943 + 0.955515i \(0.404699\pi\)
\(30\) 4033.85 0.818308
\(31\) 9039.14 1.68936 0.844681 0.535270i \(-0.179790\pi\)
0.844681 + 0.535270i \(0.179790\pi\)
\(32\) −18094.2 −3.12366
\(33\) 3794.38 0.606535
\(34\) 10501.1 1.55790
\(35\) 9.85493 0.00135983
\(36\) 6804.00 0.875000
\(37\) −12665.3 −1.52094 −0.760471 0.649372i \(-0.775032\pi\)
−0.760471 + 0.649372i \(0.775032\pi\)
\(38\) −21969.2 −2.46805
\(39\) −2290.52 −0.241142
\(40\) −23306.7 −2.30319
\(41\) 10146.4 0.942658 0.471329 0.881957i \(-0.343774\pi\)
0.471329 + 0.881957i \(0.343774\pi\)
\(42\) 22.9550 0.00200795
\(43\) 19523.2 1.61020 0.805102 0.593137i \(-0.202111\pi\)
0.805102 + 0.593137i \(0.202111\pi\)
\(44\) −35414.2 −2.75769
\(45\) 3370.80 0.248143
\(46\) −5697.50 −0.397000
\(47\) 27679.7 1.82775 0.913875 0.405995i \(-0.133075\pi\)
0.913875 + 0.405995i \(0.133075\pi\)
\(48\) −30096.0 −1.88541
\(49\) −16806.9 −0.999997
\(50\) 15005.3 0.848827
\(51\) 8775.06 0.472416
\(52\) 21378.2 1.09638
\(53\) 10852.8 0.530703 0.265351 0.964152i \(-0.414512\pi\)
0.265351 + 0.964152i \(0.414512\pi\)
\(54\) 7851.57 0.366414
\(55\) −17544.7 −0.782059
\(56\) −132.629 −0.00565155
\(57\) −18358.1 −0.748411
\(58\) −28773.4 −1.12311
\(59\) 11907.7 0.445345 0.222672 0.974893i \(-0.428522\pi\)
0.222672 + 0.974893i \(0.428522\pi\)
\(60\) −31460.8 −1.12821
\(61\) 39861.9 1.37162 0.685809 0.727782i \(-0.259449\pi\)
0.685809 + 0.727782i \(0.259449\pi\)
\(62\) −97354.5 −3.21645
\(63\) 19.1818 0.000608890 0
\(64\) 87872.0 2.68164
\(65\) 10591.1 0.310925
\(66\) −40866.7 −1.15481
\(67\) −28550.5 −0.777009 −0.388504 0.921447i \(-0.627008\pi\)
−0.388504 + 0.921447i \(0.627008\pi\)
\(68\) −81900.6 −2.14790
\(69\) −4761.00 −0.120386
\(70\) −106.141 −0.00258903
\(71\) 52179.8 1.22845 0.614223 0.789132i \(-0.289469\pi\)
0.614223 + 0.789132i \(0.289469\pi\)
\(72\) −45364.6 −1.03130
\(73\) 56918.0 1.25009 0.625047 0.780587i \(-0.285080\pi\)
0.625047 + 0.780587i \(0.285080\pi\)
\(74\) 136410. 2.89579
\(75\) 12538.8 0.257398
\(76\) 171342. 3.40275
\(77\) −99.8398 −0.00191901
\(78\) 24669.7 0.459120
\(79\) 23178.3 0.417844 0.208922 0.977932i \(-0.433004\pi\)
0.208922 + 0.977932i \(0.433004\pi\)
\(80\) 139160. 2.43103
\(81\) 6561.00 0.111111
\(82\) −109281. −1.79477
\(83\) −18344.6 −0.292289 −0.146144 0.989263i \(-0.546686\pi\)
−0.146144 + 0.989263i \(0.546686\pi\)
\(84\) −179.031 −0.00276840
\(85\) −40574.8 −0.609128
\(86\) −210272. −3.06574
\(87\) −24043.9 −0.340571
\(88\) 236119. 3.25030
\(89\) 47362.4 0.633810 0.316905 0.948457i \(-0.397356\pi\)
0.316905 + 0.948457i \(0.397356\pi\)
\(90\) −36304.6 −0.472450
\(91\) 60.2694 0.000762945 0
\(92\) 44436.0 0.547350
\(93\) −81352.3 −0.975354
\(94\) −298120. −3.47993
\(95\) 84885.4 0.964992
\(96\) 162847. 1.80344
\(97\) −140379. −1.51486 −0.757432 0.652915i \(-0.773546\pi\)
−0.757432 + 0.652915i \(0.773546\pi\)
\(98\) 181016. 1.90394
\(99\) −34149.4 −0.350183
\(100\) −117029. −1.17029
\(101\) 87080.3 0.849408 0.424704 0.905332i \(-0.360378\pi\)
0.424704 + 0.905332i \(0.360378\pi\)
\(102\) −94510.3 −0.899454
\(103\) 17988.9 0.167075 0.0835375 0.996505i \(-0.473378\pi\)
0.0835375 + 0.996505i \(0.473378\pi\)
\(104\) −142536. −1.29223
\(105\) −88.6944 −0.000785096 0
\(106\) −116888. −1.01043
\(107\) −127485. −1.07646 −0.538231 0.842797i \(-0.680907\pi\)
−0.538231 + 0.842797i \(0.680907\pi\)
\(108\) −61236.0 −0.505181
\(109\) −136418. −1.09978 −0.549890 0.835237i \(-0.685330\pi\)
−0.549890 + 0.835237i \(0.685330\pi\)
\(110\) 188962. 1.48900
\(111\) 113988. 0.878116
\(112\) 791.902 0.00596522
\(113\) −104012. −0.766279 −0.383139 0.923691i \(-0.625157\pi\)
−0.383139 + 0.923691i \(0.625157\pi\)
\(114\) 197723. 1.42493
\(115\) 22014.2 0.155224
\(116\) 224410. 1.54845
\(117\) 20614.7 0.139223
\(118\) −128249. −0.847912
\(119\) −230.894 −0.00149467
\(120\) 209760. 1.32975
\(121\) 16693.7 0.103655
\(122\) −429325. −2.61148
\(123\) −91318.0 −0.544244
\(124\) 759288. 4.43458
\(125\) −188024. −1.07631
\(126\) −206.595 −0.00115929
\(127\) 203763. 1.12103 0.560513 0.828145i \(-0.310604\pi\)
0.560513 + 0.828145i \(0.310604\pi\)
\(128\) −367397. −1.98203
\(129\) −175709. −0.929651
\(130\) −114069. −0.591984
\(131\) 60674.1 0.308905 0.154453 0.988000i \(-0.450639\pi\)
0.154453 + 0.988000i \(0.450639\pi\)
\(132\) 318728. 1.59215
\(133\) 483.048 0.00236789
\(134\) 307498. 1.47938
\(135\) −30337.2 −0.143265
\(136\) 546060. 2.53159
\(137\) 94446.5 0.429917 0.214958 0.976623i \(-0.431038\pi\)
0.214958 + 0.976623i \(0.431038\pi\)
\(138\) 51277.5 0.229208
\(139\) 403508. 1.77139 0.885697 0.464263i \(-0.153681\pi\)
0.885697 + 0.464263i \(0.153681\pi\)
\(140\) 827.814 0.00356954
\(141\) −249117. −1.05525
\(142\) −561993. −2.33889
\(143\) −107298. −0.438783
\(144\) 270864. 1.08854
\(145\) 111176. 0.439128
\(146\) −613026. −2.38011
\(147\) 151262. 0.577348
\(148\) −1.06389e6 −3.99247
\(149\) 384528. 1.41893 0.709467 0.704738i \(-0.248936\pi\)
0.709467 + 0.704738i \(0.248936\pi\)
\(150\) −135048. −0.490070
\(151\) −442508. −1.57935 −0.789675 0.613525i \(-0.789751\pi\)
−0.789675 + 0.613525i \(0.789751\pi\)
\(152\) −1.14240e6 −4.01059
\(153\) −78975.6 −0.272750
\(154\) 1075.31 0.00365368
\(155\) 376162. 1.25761
\(156\) −192404. −0.632997
\(157\) 240724. 0.779416 0.389708 0.920938i \(-0.372576\pi\)
0.389708 + 0.920938i \(0.372576\pi\)
\(158\) −249638. −0.795552
\(159\) −97675.1 −0.306402
\(160\) −752985. −2.32534
\(161\) 125.274 0.000380887 0
\(162\) −70664.1 −0.211549
\(163\) −178188. −0.525301 −0.262651 0.964891i \(-0.584597\pi\)
−0.262651 + 0.964891i \(0.584597\pi\)
\(164\) 852301. 2.47448
\(165\) 157903. 0.451522
\(166\) 197577. 0.556502
\(167\) 25525.2 0.0708236 0.0354118 0.999373i \(-0.488726\pi\)
0.0354118 + 0.999373i \(0.488726\pi\)
\(168\) 1193.66 0.00326292
\(169\) −306522. −0.825552
\(170\) 437004. 1.15975
\(171\) 165223. 0.432095
\(172\) 1.63995e6 4.22678
\(173\) 701865. 1.78295 0.891474 0.453072i \(-0.149672\pi\)
0.891474 + 0.453072i \(0.149672\pi\)
\(174\) 258961. 0.648427
\(175\) −329.929 −0.000814377 0
\(176\) −1.40982e6 −3.43070
\(177\) −107169. −0.257120
\(178\) −510109. −1.20674
\(179\) −292917. −0.683302 −0.341651 0.939827i \(-0.610986\pi\)
−0.341651 + 0.939827i \(0.610986\pi\)
\(180\) 283147. 0.651375
\(181\) −38932.7 −0.0883321 −0.0441660 0.999024i \(-0.514063\pi\)
−0.0441660 + 0.999024i \(0.514063\pi\)
\(182\) −649.121 −0.00145260
\(183\) −358757. −0.791904
\(184\) −296270. −0.645124
\(185\) −527066. −1.13223
\(186\) 876191. 1.85702
\(187\) 411061. 0.859611
\(188\) 2.32510e6 4.79784
\(189\) −172.637 −0.000351543 0
\(190\) −914243. −1.83729
\(191\) 37244.8 0.0738724 0.0369362 0.999318i \(-0.488240\pi\)
0.0369362 + 0.999318i \(0.488240\pi\)
\(192\) −790848. −1.54825
\(193\) −42454.3 −0.0820406 −0.0410203 0.999158i \(-0.513061\pi\)
−0.0410203 + 0.999158i \(0.513061\pi\)
\(194\) 1.51193e6 2.88421
\(195\) −95319.6 −0.179513
\(196\) −1.41178e6 −2.62499
\(197\) 87335.1 0.160333 0.0801666 0.996781i \(-0.474455\pi\)
0.0801666 + 0.996781i \(0.474455\pi\)
\(198\) 367801. 0.666729
\(199\) 1.08779e6 1.94721 0.973604 0.228245i \(-0.0732989\pi\)
0.973604 + 0.228245i \(0.0732989\pi\)
\(200\) 780275. 1.37934
\(201\) 256954. 0.448606
\(202\) −937883. −1.61722
\(203\) 632.657 0.00107753
\(204\) 737105. 1.24009
\(205\) 422243. 0.701742
\(206\) −193746. −0.318101
\(207\) 42849.0 0.0695048
\(208\) 851055. 1.36395
\(209\) −859969. −1.36181
\(210\) 955.267 0.00149478
\(211\) −1.02777e6 −1.58923 −0.794617 0.607111i \(-0.792328\pi\)
−0.794617 + 0.607111i \(0.792328\pi\)
\(212\) 911634. 1.39310
\(213\) −469618. −0.709244
\(214\) 1.37305e6 2.04952
\(215\) 812456. 1.19868
\(216\) 408282. 0.595423
\(217\) 2140.58 0.00308591
\(218\) 1.46927e6 2.09392
\(219\) −512262. −0.721742
\(220\) −1.47376e6 −2.05291
\(221\) −248141. −0.341758
\(222\) −1.22769e6 −1.67188
\(223\) −209627. −0.282284 −0.141142 0.989989i \(-0.545077\pi\)
−0.141142 + 0.989989i \(0.545077\pi\)
\(224\) −4284.93 −0.00570589
\(225\) −112850. −0.148609
\(226\) 1.12024e6 1.45895
\(227\) −347103. −0.447088 −0.223544 0.974694i \(-0.571763\pi\)
−0.223544 + 0.974694i \(0.571763\pi\)
\(228\) −1.54208e6 −1.96458
\(229\) −474094. −0.597414 −0.298707 0.954345i \(-0.596555\pi\)
−0.298707 + 0.954345i \(0.596555\pi\)
\(230\) −237101. −0.295538
\(231\) 898.558 0.00110794
\(232\) −1.49622e6 −1.82505
\(233\) 67426.2 0.0813652 0.0406826 0.999172i \(-0.487047\pi\)
0.0406826 + 0.999172i \(0.487047\pi\)
\(234\) −222027. −0.265073
\(235\) 1.15189e6 1.36063
\(236\) 1.00024e6 1.16903
\(237\) −208605. −0.241243
\(238\) 2486.81 0.00284577
\(239\) −631686. −0.715331 −0.357665 0.933850i \(-0.616427\pi\)
−0.357665 + 0.933850i \(0.616427\pi\)
\(240\) −1.25244e6 −1.40355
\(241\) −582036. −0.645516 −0.322758 0.946482i \(-0.604610\pi\)
−0.322758 + 0.946482i \(0.604610\pi\)
\(242\) −179797. −0.197353
\(243\) −59049.0 −0.0641500
\(244\) 3.34840e6 3.60050
\(245\) −699418. −0.744426
\(246\) 983525. 1.03621
\(247\) 519130. 0.541419
\(248\) −5.06243e6 −5.22673
\(249\) 165101. 0.168753
\(250\) 2.02508e6 2.04924
\(251\) −1.28835e6 −1.29077 −0.645386 0.763856i \(-0.723304\pi\)
−0.645386 + 0.763856i \(0.723304\pi\)
\(252\) 1611.27 0.00159834
\(253\) −223025. −0.219055
\(254\) −2.19459e6 −2.13437
\(255\) 365173. 0.351680
\(256\) 1.14509e6 1.09204
\(257\) −204908. −0.193520 −0.0967601 0.995308i \(-0.530848\pi\)
−0.0967601 + 0.995308i \(0.530848\pi\)
\(258\) 1.89245e6 1.77000
\(259\) −2999.32 −0.00277826
\(260\) 889650. 0.816179
\(261\) 216395. 0.196628
\(262\) −653480. −0.588138
\(263\) 418862. 0.373407 0.186703 0.982416i \(-0.440220\pi\)
0.186703 + 0.982416i \(0.440220\pi\)
\(264\) −2.12507e6 −1.87656
\(265\) 451637. 0.395071
\(266\) −5202.58 −0.00450832
\(267\) −426262. −0.365930
\(268\) −2.39824e6 −2.03965
\(269\) −1.48662e6 −1.25262 −0.626308 0.779576i \(-0.715435\pi\)
−0.626308 + 0.779576i \(0.715435\pi\)
\(270\) 326742. 0.272769
\(271\) −658640. −0.544785 −0.272392 0.962186i \(-0.587815\pi\)
−0.272392 + 0.962186i \(0.587815\pi\)
\(272\) −3.26042e6 −2.67209
\(273\) −542.425 −0.000440487 0
\(274\) −1.01722e6 −0.818537
\(275\) 587372. 0.468362
\(276\) −399924. −0.316013
\(277\) 236526. 0.185216 0.0926082 0.995703i \(-0.470480\pi\)
0.0926082 + 0.995703i \(0.470480\pi\)
\(278\) −4.34592e6 −3.37263
\(279\) 732170. 0.563121
\(280\) −5519.32 −0.00420718
\(281\) −1.66253e6 −1.25604 −0.628020 0.778197i \(-0.716134\pi\)
−0.628020 + 0.778197i \(0.716134\pi\)
\(282\) 2.68308e6 2.00914
\(283\) 966812. 0.717589 0.358795 0.933417i \(-0.383188\pi\)
0.358795 + 0.933417i \(0.383188\pi\)
\(284\) 4.38310e6 3.22467
\(285\) −763968. −0.557139
\(286\) 1.15563e6 0.835418
\(287\) 2402.81 0.00172193
\(288\) −1.46563e6 −1.04122
\(289\) −469218. −0.330469
\(290\) −1.19740e6 −0.836074
\(291\) 1.26341e6 0.874607
\(292\) 4.78111e6 3.28150
\(293\) −82436.0 −0.0560981 −0.0280490 0.999607i \(-0.508929\pi\)
−0.0280490 + 0.999607i \(0.508929\pi\)
\(294\) −1.62915e6 −1.09924
\(295\) 495535. 0.331528
\(296\) 7.09332e6 4.70565
\(297\) 307345. 0.202178
\(298\) −4.14149e6 −2.70157
\(299\) 134632. 0.0870902
\(300\) 1.05326e6 0.675669
\(301\) 4623.35 0.00294131
\(302\) 4.76595e6 3.00699
\(303\) −783722. −0.490406
\(304\) 6.82105e6 4.23318
\(305\) 1.65884e6 1.02107
\(306\) 850593. 0.519300
\(307\) 471104. 0.285280 0.142640 0.989775i \(-0.454441\pi\)
0.142640 + 0.989775i \(0.454441\pi\)
\(308\) −8386.54 −0.00503740
\(309\) −161900. −0.0964608
\(310\) −4.05139e6 −2.39442
\(311\) 974141. 0.571112 0.285556 0.958362i \(-0.407822\pi\)
0.285556 + 0.958362i \(0.407822\pi\)
\(312\) 1.28282e6 0.746071
\(313\) 203897. 0.117639 0.0588194 0.998269i \(-0.481266\pi\)
0.0588194 + 0.998269i \(0.481266\pi\)
\(314\) −2.59267e6 −1.48396
\(315\) 798.249 0.000453275 0
\(316\) 1.94698e6 1.09684
\(317\) −1.11600e6 −0.623757 −0.311879 0.950122i \(-0.600958\pi\)
−0.311879 + 0.950122i \(0.600958\pi\)
\(318\) 1.05199e6 0.583371
\(319\) −1.12632e6 −0.619704
\(320\) 3.65678e6 1.99629
\(321\) 1.14736e6 0.621496
\(322\) −1349.24 −0.000725187 0
\(323\) −1.98881e6 −1.06068
\(324\) 551124. 0.291667
\(325\) −354574. −0.186208
\(326\) 1.91914e6 1.00014
\(327\) 1.22776e6 0.634958
\(328\) −5.68259e6 −2.91650
\(329\) 6554.91 0.00333870
\(330\) −1.70066e6 −0.859673
\(331\) 2.08998e6 1.04851 0.524254 0.851562i \(-0.324344\pi\)
0.524254 + 0.851562i \(0.324344\pi\)
\(332\) −1.54094e6 −0.767258
\(333\) −1.02589e6 −0.506981
\(334\) −274915. −0.134844
\(335\) −1.18812e6 −0.578428
\(336\) −7127.12 −0.00344402
\(337\) 901699. 0.432501 0.216250 0.976338i \(-0.430617\pi\)
0.216250 + 0.976338i \(0.430617\pi\)
\(338\) 3.30134e6 1.57180
\(339\) 936107. 0.442411
\(340\) −3.40828e6 −1.59896
\(341\) −3.81088e6 −1.77476
\(342\) −1.77950e6 −0.822685
\(343\) −7960.21 −0.00365333
\(344\) −1.09341e7 −4.98182
\(345\) −198128. −0.0896187
\(346\) −7.55932e6 −3.39463
\(347\) −1.53207e6 −0.683053 −0.341526 0.939872i \(-0.610944\pi\)
−0.341526 + 0.939872i \(0.610944\pi\)
\(348\) −2.01969e6 −0.893998
\(349\) 1.28353e6 0.564084 0.282042 0.959402i \(-0.408988\pi\)
0.282042 + 0.959402i \(0.408988\pi\)
\(350\) 3553.44 0.00155053
\(351\) −185532. −0.0803806
\(352\) 7.62846e6 3.28156
\(353\) −4.45978e6 −1.90492 −0.952460 0.304663i \(-0.901456\pi\)
−0.952460 + 0.304663i \(0.901456\pi\)
\(354\) 1.15425e6 0.489542
\(355\) 2.17145e6 0.914491
\(356\) 3.97844e6 1.66375
\(357\) 2078.05 0.000862949 0
\(358\) 3.15482e6 1.30097
\(359\) 4.67506e6 1.91448 0.957240 0.289296i \(-0.0934213\pi\)
0.957240 + 0.289296i \(0.0934213\pi\)
\(360\) −1.88784e6 −0.767731
\(361\) 1.68463e6 0.680356
\(362\) 419318. 0.168179
\(363\) −150243. −0.0598451
\(364\) 5062.63 0.00200273
\(365\) 2.36863e6 0.930606
\(366\) 3.86393e6 1.50774
\(367\) 2.43229e6 0.942650 0.471325 0.881960i \(-0.343776\pi\)
0.471325 + 0.881960i \(0.343776\pi\)
\(368\) 1.76898e6 0.680930
\(369\) 821862. 0.314219
\(370\) 5.67668e6 2.15571
\(371\) 2570.08 0.000969419 0
\(372\) −6.83359e6 −2.56030
\(373\) 1.37458e6 0.511561 0.255780 0.966735i \(-0.417668\pi\)
0.255780 + 0.966735i \(0.417668\pi\)
\(374\) −4.42726e6 −1.63665
\(375\) 1.69222e6 0.621410
\(376\) −1.55022e7 −5.65489
\(377\) 679914. 0.246377
\(378\) 1859.35 0.000669318 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
\(381\) −1.83387e6 −0.647225
\(382\) −401139. −0.140649
\(383\) 2.00834e6 0.699584 0.349792 0.936827i \(-0.386252\pi\)
0.349792 + 0.936827i \(0.386252\pi\)
\(384\) 3.30658e6 1.14433
\(385\) −4154.82 −0.00142857
\(386\) 457247. 0.156201
\(387\) 1.58138e6 0.536734
\(388\) −1.17919e7 −3.97652
\(389\) 3.25055e6 1.08914 0.544569 0.838716i \(-0.316693\pi\)
0.544569 + 0.838716i \(0.316693\pi\)
\(390\) 1.02662e6 0.341782
\(391\) −515779. −0.170617
\(392\) 9.41285e6 3.09390
\(393\) −546067. −0.178346
\(394\) −940628. −0.305265
\(395\) 964563. 0.311055
\(396\) −2.86855e6 −0.919231
\(397\) 5.43707e6 1.73137 0.865683 0.500592i \(-0.166884\pi\)
0.865683 + 0.500592i \(0.166884\pi\)
\(398\) −1.17159e7 −3.70737
\(399\) −4347.43 −0.00136710
\(400\) −4.65888e6 −1.45590
\(401\) 3.27208e6 1.01616 0.508081 0.861309i \(-0.330355\pi\)
0.508081 + 0.861309i \(0.330355\pi\)
\(402\) −2.76748e6 −0.854121
\(403\) 2.30048e6 0.705596
\(404\) 7.31474e6 2.22970
\(405\) 273035. 0.0827143
\(406\) −6813.92 −0.00205155
\(407\) 5.33968e6 1.59783
\(408\) −4.91454e6 −1.46161
\(409\) −1.56299e6 −0.462007 −0.231003 0.972953i \(-0.574201\pi\)
−0.231003 + 0.972953i \(0.574201\pi\)
\(410\) −4.54769e6 −1.33608
\(411\) −850019. −0.248213
\(412\) 1.51107e6 0.438572
\(413\) 2819.89 0.000813498 0
\(414\) −461498. −0.132333
\(415\) −763406. −0.217588
\(416\) −4.60500e6 −1.30466
\(417\) −3.63157e6 −1.02272
\(418\) 9.26215e6 2.59281
\(419\) −4.76111e6 −1.32487 −0.662434 0.749120i \(-0.730477\pi\)
−0.662434 + 0.749120i \(0.730477\pi\)
\(420\) −7450.33 −0.00206088
\(421\) −4.56989e6 −1.25661 −0.628305 0.777967i \(-0.716251\pi\)
−0.628305 + 0.777967i \(0.716251\pi\)
\(422\) 1.10694e7 3.02581
\(423\) 2.24206e6 0.609250
\(424\) −6.07818e6 −1.64195
\(425\) 1.35839e6 0.364796
\(426\) 5.05794e6 1.35036
\(427\) 9439.80 0.00250549
\(428\) −1.07087e7 −2.82571
\(429\) 965678. 0.253331
\(430\) −8.75042e6 −2.28222
\(431\) −3.28891e6 −0.852824 −0.426412 0.904529i \(-0.640223\pi\)
−0.426412 + 0.904529i \(0.640223\pi\)
\(432\) −2.43778e6 −0.628470
\(433\) 2.11805e6 0.542895 0.271447 0.962453i \(-0.412498\pi\)
0.271447 + 0.962453i \(0.412498\pi\)
\(434\) −23054.8 −0.00587540
\(435\) −1.00058e6 −0.253531
\(436\) −1.14591e7 −2.88692
\(437\) 1.07905e6 0.270294
\(438\) 5.51723e6 1.37416
\(439\) 758443. 0.187829 0.0939143 0.995580i \(-0.470062\pi\)
0.0939143 + 0.995580i \(0.470062\pi\)
\(440\) 9.82605e6 2.41962
\(441\) −1.36136e6 −0.333332
\(442\) 2.67257e6 0.650688
\(443\) 650377. 0.157455 0.0787274 0.996896i \(-0.474914\pi\)
0.0787274 + 0.996896i \(0.474914\pi\)
\(444\) 9.57500e6 2.30506
\(445\) 1.97098e6 0.471826
\(446\) 2.25776e6 0.537452
\(447\) −3.46075e6 −0.819222
\(448\) 20809.2 0.00489847
\(449\) −7.58111e6 −1.77467 −0.887333 0.461129i \(-0.847445\pi\)
−0.887333 + 0.461129i \(0.847445\pi\)
\(450\) 1.21543e6 0.282942
\(451\) −4.27772e6 −0.990309
\(452\) −8.73699e6 −2.01148
\(453\) 3.98257e6 0.911838
\(454\) 3.73841e6 0.851231
\(455\) 2508.10 0.000567958 0
\(456\) 1.02816e7 2.31551
\(457\) −8.04537e6 −1.80200 −0.901002 0.433815i \(-0.857167\pi\)
−0.901002 + 0.433815i \(0.857167\pi\)
\(458\) 5.10614e6 1.13744
\(459\) 710780. 0.157472
\(460\) 1.84920e6 0.407463
\(461\) 5.63578e6 1.23510 0.617550 0.786532i \(-0.288125\pi\)
0.617550 + 0.786532i \(0.288125\pi\)
\(462\) −9677.77 −0.00210945
\(463\) 1.39156e6 0.301682 0.150841 0.988558i \(-0.451802\pi\)
0.150841 + 0.988558i \(0.451802\pi\)
\(464\) 8.93365e6 1.92634
\(465\) −3.38546e6 −0.726082
\(466\) −726202. −0.154915
\(467\) 4.05063e6 0.859469 0.429734 0.902955i \(-0.358607\pi\)
0.429734 + 0.902955i \(0.358607\pi\)
\(468\) 1.73163e6 0.365461
\(469\) −6761.12 −0.00141934
\(470\) −1.24062e7 −2.59056
\(471\) −2.16651e6 −0.449996
\(472\) −6.66897e6 −1.37786
\(473\) −8.23095e6 −1.69160
\(474\) 2.24675e6 0.459312
\(475\) −2.84184e6 −0.577917
\(476\) −19395.1 −0.00392351
\(477\) 879076. 0.176901
\(478\) 6.80347e6 1.36195
\(479\) 6.55529e6 1.30543 0.652714 0.757605i \(-0.273630\pi\)
0.652714 + 0.757605i \(0.273630\pi\)
\(480\) 6.77687e6 1.34254
\(481\) −3.22336e6 −0.635252
\(482\) 6.26872e6 1.22903
\(483\) −1127.47 −0.000219905 0
\(484\) 1.40227e6 0.272094
\(485\) −5.84186e6 −1.12771
\(486\) 635977. 0.122138
\(487\) −1.98924e6 −0.380071 −0.190035 0.981777i \(-0.560860\pi\)
−0.190035 + 0.981777i \(0.560860\pi\)
\(488\) −2.23249e7 −4.24366
\(489\) 1.60369e6 0.303283
\(490\) 7.53296e6 1.41735
\(491\) 7.99205e6 1.49608 0.748040 0.663654i \(-0.230995\pi\)
0.748040 + 0.663654i \(0.230995\pi\)
\(492\) −7.67071e6 −1.42864
\(493\) −2.60478e6 −0.482673
\(494\) −5.59120e6 −1.03083
\(495\) −1.42112e6 −0.260686
\(496\) 3.02269e7 5.51682
\(497\) 12356.8 0.00224397
\(498\) −1.77819e6 −0.321296
\(499\) −1.00308e7 −1.80338 −0.901688 0.432387i \(-0.857671\pi\)
−0.901688 + 0.432387i \(0.857671\pi\)
\(500\) −1.57940e7 −2.82533
\(501\) −229727. −0.0408900
\(502\) 1.38760e7 2.45756
\(503\) −4.82601e6 −0.850488 −0.425244 0.905079i \(-0.639812\pi\)
−0.425244 + 0.905079i \(0.639812\pi\)
\(504\) −10742.9 −0.00188385
\(505\) 3.62383e6 0.632324
\(506\) 2.40206e6 0.417068
\(507\) 2.75869e6 0.476633
\(508\) 1.71161e7 2.94270
\(509\) 138289. 0.0236589 0.0118294 0.999930i \(-0.496234\pi\)
0.0118294 + 0.999930i \(0.496234\pi\)
\(510\) −3.93303e6 −0.669579
\(511\) 13478.9 0.00228351
\(512\) −576256. −0.0971494
\(513\) −1.48700e6 −0.249470
\(514\) 2.20693e6 0.368451
\(515\) 748605. 0.124375
\(516\) −1.47596e7 −2.44033
\(517\) −1.16697e7 −1.92014
\(518\) 32303.6 0.00528965
\(519\) −6.31679e6 −1.02939
\(520\) −5.93160e6 −0.961975
\(521\) 4.62831e6 0.747012 0.373506 0.927628i \(-0.378155\pi\)
0.373506 + 0.927628i \(0.378155\pi\)
\(522\) −2.33065e6 −0.374369
\(523\) −1.54914e6 −0.247649 −0.123825 0.992304i \(-0.539516\pi\)
−0.123825 + 0.992304i \(0.539516\pi\)
\(524\) 5.09662e6 0.810876
\(525\) 2969.36 0.000470181 0
\(526\) −4.51129e6 −0.710945
\(527\) −8.81323e6 −1.38232
\(528\) 1.26884e7 1.98072
\(529\) 279841. 0.0434783
\(530\) −4.86428e6 −0.752192
\(531\) 964521. 0.148448
\(532\) 40576.0 0.00621570
\(533\) 2.58229e6 0.393720
\(534\) 4.59098e6 0.696710
\(535\) −5.30526e6 −0.801349
\(536\) 1.59899e7 2.40399
\(537\) 2.63626e6 0.394505
\(538\) 1.60113e7 2.38491
\(539\) 7.08577e6 1.05055
\(540\) −2.54833e6 −0.376072
\(541\) −6.52702e6 −0.958787 −0.479393 0.877600i \(-0.659143\pi\)
−0.479393 + 0.877600i \(0.659143\pi\)
\(542\) 7.09377e6 1.03724
\(543\) 350395. 0.0509985
\(544\) 1.76419e7 2.55593
\(545\) −5.67701e6 −0.818707
\(546\) 5842.09 0.000838662 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
\(549\) 3.22881e6 0.457206
\(550\) −6.32619e6 −0.891735
\(551\) 5.44938e6 0.764660
\(552\) 2.66643e6 0.372463
\(553\) 5488.93 0.000763264 0
\(554\) −2.54746e6 −0.352642
\(555\) 4.74360e6 0.653695
\(556\) 3.38947e7 4.64991
\(557\) −3.07804e6 −0.420374 −0.210187 0.977661i \(-0.567407\pi\)
−0.210187 + 0.977661i \(0.567407\pi\)
\(558\) −7.88572e6 −1.07215
\(559\) 4.96871e6 0.672533
\(560\) 32954.9 0.00444068
\(561\) −3.69955e6 −0.496297
\(562\) 1.79060e7 2.39143
\(563\) 1.10366e7 1.46745 0.733724 0.679448i \(-0.237781\pi\)
0.733724 + 0.679448i \(0.237781\pi\)
\(564\) −2.09259e7 −2.77004
\(565\) −4.32844e6 −0.570440
\(566\) −1.04129e7 −1.36625
\(567\) 1553.73 0.000202963 0
\(568\) −2.92237e7 −3.80070
\(569\) −9.26308e6 −1.19943 −0.599715 0.800214i \(-0.704719\pi\)
−0.599715 + 0.800214i \(0.704719\pi\)
\(570\) 8.22819e6 1.06076
\(571\) −1.31913e7 −1.69315 −0.846576 0.532267i \(-0.821340\pi\)
−0.846576 + 0.532267i \(0.821340\pi\)
\(572\) −9.01300e6 −1.15181
\(573\) −335203. −0.0426503
\(574\) −25879.0 −0.00327845
\(575\) −737006. −0.0929611
\(576\) 7.11763e6 0.893880
\(577\) −1.29741e6 −0.162232 −0.0811160 0.996705i \(-0.525848\pi\)
−0.0811160 + 0.996705i \(0.525848\pi\)
\(578\) 5.05363e6 0.629193
\(579\) 382089. 0.0473661
\(580\) 9.33878e6 1.15271
\(581\) −4344.23 −0.000533916 0
\(582\) −1.36074e7 −1.66520
\(583\) −4.57551e6 −0.557530
\(584\) −3.18773e7 −3.86767
\(585\) 857876. 0.103642
\(586\) 887863. 0.106808
\(587\) 4.51203e6 0.540477 0.270238 0.962793i \(-0.412897\pi\)
0.270238 + 0.962793i \(0.412897\pi\)
\(588\) 1.27060e7 1.51554
\(589\) 1.84379e7 2.18990
\(590\) −5.33708e6 −0.631210
\(591\) −786016. −0.0925684
\(592\) −4.23529e7 −4.96683
\(593\) 9.27288e6 1.08287 0.541437 0.840741i \(-0.317880\pi\)
0.541437 + 0.840741i \(0.317880\pi\)
\(594\) −3.31020e6 −0.384936
\(595\) −9608.63 −0.00111268
\(596\) 3.23004e7 3.72470
\(597\) −9.79011e6 −1.12422
\(598\) −1.45003e6 −0.165815
\(599\) −6.32225e6 −0.719954 −0.359977 0.932961i \(-0.617215\pi\)
−0.359977 + 0.932961i \(0.617215\pi\)
\(600\) −7.02247e6 −0.796365
\(601\) 3.18785e6 0.360008 0.180004 0.983666i \(-0.442389\pi\)
0.180004 + 0.983666i \(0.442389\pi\)
\(602\) −49795.0 −0.00560009
\(603\) −2.31259e6 −0.259003
\(604\) −3.71706e7 −4.14579
\(605\) 694706. 0.0771636
\(606\) 8.44095e6 0.933705
\(607\) −5.70410e6 −0.628369 −0.314185 0.949362i \(-0.601731\pi\)
−0.314185 + 0.949362i \(0.601731\pi\)
\(608\) −3.69082e7 −4.04915
\(609\) −5693.91 −0.000622110 0
\(610\) −1.78663e7 −1.94406
\(611\) 7.04455e6 0.763396
\(612\) −6.63395e6 −0.715968
\(613\) 1.17835e7 1.26655 0.633276 0.773926i \(-0.281710\pi\)
0.633276 + 0.773926i \(0.281710\pi\)
\(614\) −5.07394e6 −0.543156
\(615\) −3.80018e6 −0.405151
\(616\) 55916.0 0.00593723
\(617\) 1.41628e6 0.149774 0.0748868 0.997192i \(-0.476140\pi\)
0.0748868 + 0.997192i \(0.476140\pi\)
\(618\) 1.74372e6 0.183656
\(619\) −4.37752e6 −0.459200 −0.229600 0.973285i \(-0.573742\pi\)
−0.229600 + 0.973285i \(0.573742\pi\)
\(620\) 3.15976e7 3.30123
\(621\) −385641. −0.0401286
\(622\) −1.04918e7 −1.08736
\(623\) 11216.0 0.00115776
\(624\) −7.65950e6 −0.787479
\(625\) −3.47084e6 −0.355414
\(626\) −2.19604e6 −0.223978
\(627\) 7.73972e6 0.786243
\(628\) 2.02208e7 2.04597
\(629\) 1.23488e7 1.24451
\(630\) −8597.41 −0.000863011 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
\(633\) 9.24989e6 0.917545
\(634\) 1.20197e7 1.18760
\(635\) 8.47956e6 0.834525
\(636\) −8.20471e6 −0.804304
\(637\) −4.27740e6 −0.417668
\(638\) 1.21308e7 1.17988
\(639\) 4.22656e6 0.409482
\(640\) −1.52892e7 −1.47548
\(641\) −1.56439e7 −1.50383 −0.751915 0.659260i \(-0.770870\pi\)
−0.751915 + 0.659260i \(0.770870\pi\)
\(642\) −1.23575e7 −1.18329
\(643\) −1.57570e7 −1.50295 −0.751476 0.659760i \(-0.770658\pi\)
−0.751476 + 0.659760i \(0.770658\pi\)
\(644\) 10523.0 0.000999829 0
\(645\) −7.31211e6 −0.692059
\(646\) 2.14201e7 2.01948
\(647\) −660291. −0.0620119 −0.0310059 0.999519i \(-0.509871\pi\)
−0.0310059 + 0.999519i \(0.509871\pi\)
\(648\) −3.67453e6 −0.343768
\(649\) −5.02024e6 −0.467857
\(650\) 3.81888e6 0.354529
\(651\) −19265.3 −0.00178165
\(652\) −1.49678e7 −1.37892
\(653\) 1.24310e7 1.14084 0.570419 0.821354i \(-0.306781\pi\)
0.570419 + 0.821354i \(0.306781\pi\)
\(654\) −1.32234e7 −1.20892
\(655\) 2.52494e6 0.229958
\(656\) 3.39297e7 3.07837
\(657\) 4.61036e6 0.416698
\(658\) −70598.6 −0.00635669
\(659\) −13723.9 −0.00123101 −0.000615507 1.00000i \(-0.500196\pi\)
−0.000615507 1.00000i \(0.500196\pi\)
\(660\) 1.32638e7 1.18525
\(661\) 6.54194e6 0.582375 0.291187 0.956666i \(-0.405950\pi\)
0.291187 + 0.956666i \(0.405950\pi\)
\(662\) −2.25098e7 −1.99630
\(663\) 2.23327e6 0.197314
\(664\) 1.02740e7 0.904315
\(665\) 20101.9 0.00176272
\(666\) 1.10492e7 0.965262
\(667\) 1.41325e6 0.123000
\(668\) 2.14412e6 0.185912
\(669\) 1.88665e6 0.162977
\(670\) 1.27965e7 1.10129
\(671\) −1.68057e7 −1.44095
\(672\) 38564.4 0.00329430
\(673\) 1.54266e7 1.31290 0.656451 0.754369i \(-0.272057\pi\)
0.656451 + 0.754369i \(0.272057\pi\)
\(674\) −9.71159e6 −0.823456
\(675\) 1.01565e6 0.0857992
\(676\) −2.57478e7 −2.16707
\(677\) 5.30462e6 0.444818 0.222409 0.974953i \(-0.428608\pi\)
0.222409 + 0.974953i \(0.428608\pi\)
\(678\) −1.00822e7 −0.842326
\(679\) −33243.6 −0.00276716
\(680\) 2.27242e7 1.88459
\(681\) 3.12392e6 0.258127
\(682\) 4.10444e7 3.37904
\(683\) 1.52230e7 1.24867 0.624337 0.781155i \(-0.285369\pi\)
0.624337 + 0.781155i \(0.285369\pi\)
\(684\) 1.38787e7 1.13425
\(685\) 3.93038e6 0.320043
\(686\) 85734.1 0.00695574
\(687\) 4.26684e6 0.344917
\(688\) 6.52857e7 5.25832
\(689\) 2.76206e6 0.221659
\(690\) 2.13391e6 0.170629
\(691\) 1.04711e7 0.834248 0.417124 0.908850i \(-0.363038\pi\)
0.417124 + 0.908850i \(0.363038\pi\)
\(692\) 5.89567e7 4.68024
\(693\) −8087.02 −0.000639669 0
\(694\) 1.65009e7 1.30049
\(695\) 1.67919e7 1.31868
\(696\) 1.34660e7 1.05369
\(697\) −9.89286e6 −0.771329
\(698\) −1.38241e7 −1.07398
\(699\) −606836. −0.0469762
\(700\) −27714.0 −0.00213774
\(701\) −2.28874e7 −1.75914 −0.879572 0.475765i \(-0.842171\pi\)
−0.879572 + 0.475765i \(0.842171\pi\)
\(702\) 1.99824e6 0.153040
\(703\) −2.58346e7 −1.97158
\(704\) −3.70466e7 −2.81720
\(705\) −1.03670e7 −0.785560
\(706\) 4.80333e7 3.62686
\(707\) 20621.7 0.00155159
\(708\) −9.00219e6 −0.674940
\(709\) 1.85127e7 1.38310 0.691550 0.722329i \(-0.256928\pi\)
0.691550 + 0.722329i \(0.256928\pi\)
\(710\) −2.33873e7 −1.74114
\(711\) 1.87745e6 0.139281
\(712\) −2.65257e7 −1.96095
\(713\) 4.78170e6 0.352256
\(714\) −22381.3 −0.00164301
\(715\) −4.46517e6 −0.326643
\(716\) −2.46051e7 −1.79367
\(717\) 5.68518e6 0.412996
\(718\) −5.03519e7 −3.64506
\(719\) 1.25679e7 0.906652 0.453326 0.891345i \(-0.350237\pi\)
0.453326 + 0.891345i \(0.350237\pi\)
\(720\) 1.12720e7 0.810342
\(721\) 4260.00 0.000305191 0
\(722\) −1.81440e7 −1.29536
\(723\) 5.23832e6 0.372689
\(724\) −3.27035e6 −0.231872
\(725\) −3.72201e6 −0.262986
\(726\) 1.61817e6 0.113942
\(727\) 3.64050e6 0.255462 0.127731 0.991809i \(-0.459231\pi\)
0.127731 + 0.991809i \(0.459231\pi\)
\(728\) −33754.3 −0.00236048
\(729\) 531441. 0.0370370
\(730\) −2.55110e7 −1.77182
\(731\) −1.90353e7 −1.31755
\(732\) −3.01356e7 −2.07875
\(733\) −4.38735e6 −0.301608 −0.150804 0.988564i \(-0.548186\pi\)
−0.150804 + 0.988564i \(0.548186\pi\)
\(734\) −2.61966e7 −1.79475
\(735\) 6.29476e6 0.429795
\(736\) −9.57181e6 −0.651327
\(737\) 1.20368e7 0.816287
\(738\) −8.85173e6 −0.598256
\(739\) 1.95857e7 1.31925 0.659625 0.751595i \(-0.270715\pi\)
0.659625 + 0.751595i \(0.270715\pi\)
\(740\) −4.42736e7 −2.97211
\(741\) −4.67217e6 −0.312589
\(742\) −27680.6 −0.00184572
\(743\) −2.94833e7 −1.95931 −0.979657 0.200678i \(-0.935685\pi\)
−0.979657 + 0.200678i \(0.935685\pi\)
\(744\) 4.55619e7 3.01766
\(745\) 1.60021e7 1.05630
\(746\) −1.48047e7 −0.973983
\(747\) −1.48591e6 −0.0974296
\(748\) 3.45291e7 2.25648
\(749\) −30190.0 −0.00196634
\(750\) −1.82258e7 −1.18313
\(751\) 1.27147e7 0.822634 0.411317 0.911492i \(-0.365069\pi\)
0.411317 + 0.911492i \(0.365069\pi\)
\(752\) 9.25609e7 5.96875
\(753\) 1.15952e7 0.745228
\(754\) −7.32290e6 −0.469089
\(755\) −1.84149e7 −1.17571
\(756\) −14501.5 −0.000922800 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
\(759\) 2.00723e6 0.126471
\(760\) −4.75407e7 −2.98560
\(761\) 8.50863e6 0.532596 0.266298 0.963891i \(-0.414199\pi\)
0.266298 + 0.963891i \(0.414199\pi\)
\(762\) 1.97513e7 1.23228
\(763\) −32305.5 −0.00200893
\(764\) 3.12856e6 0.193915
\(765\) −3.28656e6 −0.203043
\(766\) −2.16305e7 −1.33197
\(767\) 3.03053e6 0.186007
\(768\) −1.03058e7 −0.630490
\(769\) 1.70141e7 1.03751 0.518755 0.854923i \(-0.326396\pi\)
0.518755 + 0.854923i \(0.326396\pi\)
\(770\) 44748.7 0.00271991
\(771\) 1.84417e6 0.111729
\(772\) −3.56616e6 −0.215356
\(773\) 3.63916e6 0.219055 0.109527 0.993984i \(-0.465066\pi\)
0.109527 + 0.993984i \(0.465066\pi\)
\(774\) −1.70320e7 −1.02191
\(775\) −1.25934e7 −0.753161
\(776\) 7.86204e7 4.68685
\(777\) 26993.8 0.00160403
\(778\) −3.50095e7 −2.07366
\(779\) 2.06966e7 1.22195
\(780\) −8.00685e6 −0.471221
\(781\) −2.19989e7 −1.29054
\(782\) 5.55511e6 0.324845
\(783\) −1.94756e6 −0.113524
\(784\) −5.62024e7 −3.26561
\(785\) 1.00177e7 0.580220
\(786\) 5.88132e6 0.339561
\(787\) −5.30127e6 −0.305101 −0.152550 0.988296i \(-0.548749\pi\)
−0.152550 + 0.988296i \(0.548749\pi\)
\(788\) 7.33615e6 0.420875
\(789\) −3.76976e6 −0.215586
\(790\) −1.03887e7 −0.592232
\(791\) −24631.3 −0.00139974
\(792\) 1.91256e7 1.08343
\(793\) 1.01449e7 0.572883
\(794\) −5.85591e7 −3.29642
\(795\) −4.06473e6 −0.228094
\(796\) 9.13743e7 5.11142
\(797\) 1.41955e7 0.791598 0.395799 0.918337i \(-0.370468\pi\)
0.395799 + 0.918337i \(0.370468\pi\)
\(798\) 46823.2 0.00260288
\(799\) −2.69879e7 −1.49555
\(800\) 2.52089e7 1.39261
\(801\) 3.83636e6 0.211270
\(802\) −3.52414e7 −1.93472
\(803\) −2.39965e7 −1.31329
\(804\) 2.15841e7 1.17759
\(805\) 5213.26 0.000283543 0
\(806\) −2.47769e7 −1.34341
\(807\) 1.33795e7 0.723198
\(808\) −4.87699e7 −2.62799
\(809\) −1.58542e7 −0.851675 −0.425838 0.904800i \(-0.640021\pi\)
−0.425838 + 0.904800i \(0.640021\pi\)
\(810\) −2.94068e6 −0.157483
\(811\) −1.40129e7 −0.748127 −0.374063 0.927403i \(-0.622036\pi\)
−0.374063 + 0.927403i \(0.622036\pi\)
\(812\) 53143.1 0.00282851
\(813\) 5.92776e6 0.314532
\(814\) −5.75101e7 −3.04217
\(815\) −7.41524e6 −0.391049
\(816\) 2.93438e7 1.54273
\(817\) 3.98232e7 2.08728
\(818\) 1.68339e7 0.879634
\(819\) 4881.82 0.000254315 0
\(820\) 3.54684e7 1.84207
\(821\) 1.16279e7 0.602065 0.301033 0.953614i \(-0.402669\pi\)
0.301033 + 0.953614i \(0.402669\pi\)
\(822\) 9.15498e6 0.472583
\(823\) −2.76237e7 −1.42161 −0.710807 0.703387i \(-0.751670\pi\)
−0.710807 + 0.703387i \(0.751670\pi\)
\(824\) −1.00748e7 −0.516914
\(825\) −5.28635e6 −0.270409
\(826\) −30371.1 −0.00154885
\(827\) −2.78146e7 −1.41420 −0.707098 0.707116i \(-0.749996\pi\)
−0.707098 + 0.707116i \(0.749996\pi\)
\(828\) 3.59932e6 0.182450
\(829\) −1.23659e7 −0.624943 −0.312471 0.949927i \(-0.601157\pi\)
−0.312471 + 0.949927i \(0.601157\pi\)
\(830\) 8.22214e6 0.414276
\(831\) −2.12873e6 −0.106935
\(832\) 2.23636e7 1.12004
\(833\) 1.63869e7 0.818246
\(834\) 3.91132e7 1.94719
\(835\) 1.06223e6 0.0527231
\(836\) −7.22374e7 −3.57476
\(837\) −6.58953e6 −0.325118
\(838\) 5.12787e7 2.52247
\(839\) −3.49502e7 −1.71413 −0.857067 0.515205i \(-0.827716\pi\)
−0.857067 + 0.515205i \(0.827716\pi\)
\(840\) 49673.9 0.00242901
\(841\) −1.33740e7 −0.652035
\(842\) 4.92192e7 2.39251
\(843\) 1.49628e7 0.725175
\(844\) −8.63323e7 −4.17174
\(845\) −1.27558e7 −0.614565
\(846\) −2.41477e7 −1.15998
\(847\) 3953.28 0.000189343 0
\(848\) 3.62917e7 1.73308
\(849\) −8.70131e6 −0.414300
\(850\) −1.46303e7 −0.694552
\(851\) −6.69997e6 −0.317138
\(852\) −3.94479e7 −1.86177
\(853\) −9.99950e6 −0.470550 −0.235275 0.971929i \(-0.575599\pi\)
−0.235275 + 0.971929i \(0.575599\pi\)
\(854\) −101670. −0.00477032
\(855\) 6.87571e6 0.321664
\(856\) 7.13987e7 3.33047
\(857\) 9.71240e6 0.451725 0.225863 0.974159i \(-0.427480\pi\)
0.225863 + 0.974159i \(0.427480\pi\)
\(858\) −1.04007e7 −0.482329
\(859\) 3.69393e7 1.70807 0.854035 0.520216i \(-0.174149\pi\)
0.854035 + 0.520216i \(0.174149\pi\)
\(860\) 6.82463e7 3.14654
\(861\) −21625.3 −0.000994155 0
\(862\) 3.54227e7 1.62373
\(863\) −3.31284e7 −1.51416 −0.757082 0.653319i \(-0.773376\pi\)
−0.757082 + 0.653319i \(0.773376\pi\)
\(864\) 1.31906e7 0.601148
\(865\) 2.92080e7 1.32728
\(866\) −2.28121e7 −1.03364
\(867\) 4.22296e6 0.190796
\(868\) 179809. 0.00810051
\(869\) −9.77194e6 −0.438966
\(870\) 1.07766e7 0.482708
\(871\) −7.26615e6 −0.324533
\(872\) 7.64019e7 3.40262
\(873\) −1.13707e7 −0.504954
\(874\) −1.16217e7 −0.514625
\(875\) −44526.6 −0.00196607
\(876\) −4.30300e7 −1.89457
\(877\) 1.38007e7 0.605902 0.302951 0.953006i \(-0.402028\pi\)
0.302951 + 0.953006i \(0.402028\pi\)
\(878\) −8.16869e6 −0.357615
\(879\) 741924. 0.0323882
\(880\) −5.86696e7 −2.55391
\(881\) −7.74868e6 −0.336347 −0.168174 0.985757i \(-0.553787\pi\)
−0.168174 + 0.985757i \(0.553787\pi\)
\(882\) 1.46623e7 0.634646
\(883\) 3.79838e7 1.63944 0.819722 0.572761i \(-0.194128\pi\)
0.819722 + 0.572761i \(0.194128\pi\)
\(884\) −2.08439e7 −0.897115
\(885\) −4.45982e6 −0.191408
\(886\) −7.00477e6 −0.299785
\(887\) −1.51721e7 −0.647495 −0.323747 0.946144i \(-0.604943\pi\)
−0.323747 + 0.946144i \(0.604943\pi\)
\(888\) −6.38398e7 −2.71681
\(889\) 48253.7 0.00204775
\(890\) −2.12281e7 −0.898330
\(891\) −2.76610e6 −0.116728
\(892\) −1.76087e7 −0.740995
\(893\) 5.64607e7 2.36929
\(894\) 3.72734e7 1.55975
\(895\) −1.21897e7 −0.508670
\(896\) −87004.5 −0.00362052
\(897\) −1.21168e6 −0.0502815
\(898\) 8.16510e7 3.37886
\(899\) 2.41485e7 0.996530
\(900\) −9.47937e6 −0.390098
\(901\) −1.05815e7 −0.434247
\(902\) 4.60724e7 1.88549
\(903\) −41610.2 −0.00169817
\(904\) 5.82526e7 2.37080
\(905\) −1.62018e6 −0.0657569
\(906\) −4.28936e7 −1.73609
\(907\) 8.48315e6 0.342404 0.171202 0.985236i \(-0.445235\pi\)
0.171202 + 0.985236i \(0.445235\pi\)
\(908\) −2.91566e7 −1.17361
\(909\) 7.05350e6 0.283136
\(910\) −27013.1 −0.00108136
\(911\) 2.96302e7 1.18288 0.591438 0.806351i \(-0.298561\pi\)
0.591438 + 0.806351i \(0.298561\pi\)
\(912\) −6.13894e7 −2.44403
\(913\) 7.73403e6 0.307064
\(914\) 8.66513e7 3.43091
\(915\) −1.49296e7 −0.589516
\(916\) −3.98239e7 −1.56821
\(917\) 14368.4 0.000564268 0
\(918\) −7.65534e6 −0.299818
\(919\) −1.27080e7 −0.496352 −0.248176 0.968715i \(-0.579831\pi\)
−0.248176 + 0.968715i \(0.579831\pi\)
\(920\) −1.23292e7 −0.480249
\(921\) −4.23993e6 −0.164706
\(922\) −6.06992e7 −2.35156
\(923\) 1.32799e7 0.513085
\(924\) 75478.9 0.00290834
\(925\) 1.76454e7 0.678075
\(926\) −1.49875e7 −0.574385
\(927\) 1.45710e6 0.0556916
\(928\) −4.83394e7 −1.84260
\(929\) −4.50533e7 −1.71272 −0.856362 0.516376i \(-0.827281\pi\)
−0.856362 + 0.516376i \(0.827281\pi\)
\(930\) 3.64625e7 1.38242
\(931\) −3.42826e7 −1.29628
\(932\) 5.66380e6 0.213584
\(933\) −8.76727e6 −0.329731
\(934\) −4.36266e7 −1.63638
\(935\) 1.71062e7 0.639919
\(936\) −1.15454e7 −0.430744
\(937\) −2.49632e7 −0.928864 −0.464432 0.885609i \(-0.653741\pi\)
−0.464432 + 0.885609i \(0.653741\pi\)
\(938\) 72819.4 0.00270234
\(939\) −1.83508e6 −0.0679188
\(940\) 9.67585e7 3.57165
\(941\) −3.36141e7 −1.23751 −0.618754 0.785585i \(-0.712362\pi\)
−0.618754 + 0.785585i \(0.712362\pi\)
\(942\) 2.33340e7 0.856767
\(943\) 5.36747e6 0.196558
\(944\) 3.98192e7 1.45433
\(945\) −7184.24 −0.000261699 0
\(946\) 8.86501e7 3.22071
\(947\) −5.33926e6 −0.193467 −0.0967333 0.995310i \(-0.530839\pi\)
−0.0967333 + 0.995310i \(0.530839\pi\)
\(948\) −1.75228e7 −0.633262
\(949\) 1.44858e7 0.522127
\(950\) 3.06076e7 1.10032
\(951\) 1.00440e7 0.360126
\(952\) 129314. 0.00462438
\(953\) −3.69391e7 −1.31751 −0.658755 0.752358i \(-0.728916\pi\)
−0.658755 + 0.752358i \(0.728916\pi\)
\(954\) −9.46793e6 −0.336809
\(955\) 1.54994e6 0.0549928
\(956\) −5.30617e7 −1.87774
\(957\) 1.01369e7 0.357786
\(958\) −7.06026e7 −2.48546
\(959\) 22366.2 0.000785317 0
\(960\) −3.29110e7 −1.15256
\(961\) 5.30769e7 1.85395
\(962\) 3.47166e7 1.20948
\(963\) −1.03263e7 −0.358821
\(964\) −4.88910e7 −1.69448
\(965\) −1.76673e6 −0.0610734
\(966\) 12143.2 0.000418687 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
\(969\) 1.78993e7 0.612386
\(970\) 6.29187e7 2.14709
\(971\) −4.07416e7 −1.38672 −0.693361 0.720590i \(-0.743871\pi\)
−0.693361 + 0.720590i \(0.743871\pi\)
\(972\) −4.96012e6 −0.168394
\(973\) 95555.9 0.00323575
\(974\) 2.14247e7 0.723633
\(975\) 3.19116e6 0.107507
\(976\) 1.33298e8 4.47919
\(977\) 1.07576e7 0.360561 0.180281 0.983615i \(-0.442299\pi\)
0.180281 + 0.983615i \(0.442299\pi\)
\(978\) −1.72722e7 −0.577433
\(979\) −1.99679e7 −0.665849
\(980\) −5.87511e7 −1.95412
\(981\) −1.10499e7 −0.366593
\(982\) −8.60770e7 −2.84845
\(983\) −2.44766e7 −0.807917 −0.403958 0.914777i \(-0.632366\pi\)
−0.403958 + 0.914777i \(0.632366\pi\)
\(984\) 5.11433e7 1.68384
\(985\) 3.63444e6 0.119357
\(986\) 2.80543e7 0.918982
\(987\) −58994.2 −0.00192760
\(988\) 4.36069e7 1.42123
\(989\) 1.03278e7 0.335751
\(990\) 1.53060e7 0.496332
\(991\) −2.09086e6 −0.0676301 −0.0338151 0.999428i \(-0.510766\pi\)
−0.0338151 + 0.999428i \(0.510766\pi\)
\(992\) −1.63556e8 −5.27699
\(993\) −1.88098e7 −0.605357
\(994\) −133087. −0.00427239
\(995\) 4.52682e7 1.44956
\(996\) 1.38685e7 0.442977
\(997\) 1.49624e7 0.476721 0.238360 0.971177i \(-0.423390\pi\)
0.238360 + 0.971177i \(0.423390\pi\)
\(998\) 1.08036e8 3.43353
\(999\) 9.23304e6 0.292705
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 69.6.a.a.1.1 2
3.2 odd 2 207.6.a.a.1.2 2
4.3 odd 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 1.1 even 1 trivial
207.6.a.a.1.2 2 3.2 odd 2
1104.6.a.h.1.1 2 4.3 odd 2