Properties

Label 825.6.a.j.1.1
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.34253.1
Defining polynomial: \( x^{3} - x^{2} - 52x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-7.17710\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-5.17710 q^{2} -9.00000 q^{3} -5.19759 q^{4} +46.5939 q^{6} +123.437 q^{7} +192.576 q^{8} +81.0000 q^{9} -121.000 q^{11} +46.7783 q^{12} +500.053 q^{13} -639.048 q^{14} -830.662 q^{16} +422.631 q^{17} -419.345 q^{18} -932.948 q^{19} -1110.94 q^{21} +626.430 q^{22} +1225.18 q^{23} -1733.18 q^{24} -2588.83 q^{26} -729.000 q^{27} -641.576 q^{28} -2111.62 q^{29} -159.612 q^{31} -1862.00 q^{32} +1089.00 q^{33} -2188.00 q^{34} -421.005 q^{36} -5414.46 q^{37} +4829.97 q^{38} -4500.47 q^{39} -18066.7 q^{41} +5751.43 q^{42} -6815.47 q^{43} +628.908 q^{44} -6342.88 q^{46} +15098.9 q^{47} +7475.96 q^{48} -1570.23 q^{49} -3803.67 q^{51} -2599.07 q^{52} -15367.7 q^{53} +3774.11 q^{54} +23771.0 q^{56} +8396.53 q^{57} +10932.1 q^{58} +23400.5 q^{59} +10768.4 q^{61} +826.328 q^{62} +9998.42 q^{63} +36221.0 q^{64} -5637.87 q^{66} -14507.1 q^{67} -2196.66 q^{68} -11026.6 q^{69} -28114.0 q^{71} +15598.6 q^{72} +28836.7 q^{73} +28031.2 q^{74} +4849.08 q^{76} -14935.9 q^{77} +23299.4 q^{78} -8150.52 q^{79} +6561.00 q^{81} +93533.0 q^{82} +109864. q^{83} +5774.19 q^{84} +35284.4 q^{86} +19004.5 q^{87} -23301.7 q^{88} +69673.6 q^{89} +61725.2 q^{91} -6367.98 q^{92} +1436.51 q^{93} -78168.5 q^{94} +16758.0 q^{96} -91551.4 q^{97} +8129.23 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9} - 363 q^{11} - 225 q^{12} + 654 q^{13} - 728 q^{14} - 415 q^{16} + 2366 q^{17} + 567 q^{18} - 2872 q^{19} - 1548 q^{21} - 847 q^{22} - 2272 q^{23} - 2079 q^{24} + 3422 q^{26} - 2187 q^{27} - 4592 q^{28} - 7738 q^{29} + 568 q^{31} - 1001 q^{32} + 3267 q^{33} + 2506 q^{34} + 2025 q^{36} + 9126 q^{37} - 13076 q^{38} - 5886 q^{39} - 8758 q^{41} + 6552 q^{42} + 14672 q^{43} - 3025 q^{44} - 28768 q^{46} + 19392 q^{47} + 3735 q^{48} - 26629 q^{49} - 21294 q^{51} + 61506 q^{52} + 4598 q^{53} - 5103 q^{54} + 2688 q^{56} + 25848 q^{57} - 8550 q^{58} - 9348 q^{59} - 60078 q^{61} + 14096 q^{62} + 13932 q^{63} - 7087 q^{64} + 7623 q^{66} + 38468 q^{67} - 59778 q^{68} + 20448 q^{69} - 74032 q^{71} + 18711 q^{72} + 44442 q^{73} + 82542 q^{74} - 98708 q^{76} - 20812 q^{77} - 30798 q^{78} - 108116 q^{79} + 19683 q^{81} + 92230 q^{82} + 81892 q^{83} + 41328 q^{84} + 126412 q^{86} + 69642 q^{87} - 27951 q^{88} + 167342 q^{89} - 31832 q^{91} - 72960 q^{92} - 5112 q^{93} + 12728 q^{94} + 9009 q^{96} - 159702 q^{97} - 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.17710 −0.915191 −0.457596 0.889160i \(-0.651289\pi\)
−0.457596 + 0.889160i \(0.651289\pi\)
\(3\) −9.00000 −0.577350
\(4\) −5.19759 −0.162425
\(5\) 0 0
\(6\) 46.5939 0.528386
\(7\) 123.437 0.952141 0.476071 0.879407i \(-0.342061\pi\)
0.476071 + 0.879407i \(0.342061\pi\)
\(8\) 192.576 1.06384
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 46.7783 0.0937759
\(13\) 500.053 0.820649 0.410324 0.911940i \(-0.365415\pi\)
0.410324 + 0.911940i \(0.365415\pi\)
\(14\) −639.048 −0.871392
\(15\) 0 0
\(16\) −830.662 −0.811194
\(17\) 422.631 0.354682 0.177341 0.984150i \(-0.443251\pi\)
0.177341 + 0.984150i \(0.443251\pi\)
\(18\) −419.345 −0.305064
\(19\) −932.948 −0.592889 −0.296445 0.955050i \(-0.595801\pi\)
−0.296445 + 0.955050i \(0.595801\pi\)
\(20\) 0 0
\(21\) −1110.94 −0.549719
\(22\) 626.430 0.275941
\(23\) 1225.18 0.482926 0.241463 0.970410i \(-0.422373\pi\)
0.241463 + 0.970410i \(0.422373\pi\)
\(24\) −1733.18 −0.614209
\(25\) 0 0
\(26\) −2588.83 −0.751051
\(27\) −729.000 −0.192450
\(28\) −641.576 −0.154651
\(29\) −2111.62 −0.466251 −0.233126 0.972447i \(-0.574895\pi\)
−0.233126 + 0.972447i \(0.574895\pi\)
\(30\) 0 0
\(31\) −159.612 −0.0298306 −0.0149153 0.999889i \(-0.504748\pi\)
−0.0149153 + 0.999889i \(0.504748\pi\)
\(32\) −1862.00 −0.321444
\(33\) 1089.00 0.174078
\(34\) −2188.00 −0.324601
\(35\) 0 0
\(36\) −421.005 −0.0541415
\(37\) −5414.46 −0.650205 −0.325103 0.945679i \(-0.605399\pi\)
−0.325103 + 0.945679i \(0.605399\pi\)
\(38\) 4829.97 0.542607
\(39\) −4500.47 −0.473802
\(40\) 0 0
\(41\) −18066.7 −1.67849 −0.839244 0.543756i \(-0.817002\pi\)
−0.839244 + 0.543756i \(0.817002\pi\)
\(42\) 5751.43 0.503098
\(43\) −6815.47 −0.562114 −0.281057 0.959691i \(-0.590685\pi\)
−0.281057 + 0.959691i \(0.590685\pi\)
\(44\) 628.908 0.0489729
\(45\) 0 0
\(46\) −6342.88 −0.441969
\(47\) 15098.9 0.997012 0.498506 0.866886i \(-0.333882\pi\)
0.498506 + 0.866886i \(0.333882\pi\)
\(48\) 7475.96 0.468343
\(49\) −1570.23 −0.0934270
\(50\) 0 0
\(51\) −3803.67 −0.204775
\(52\) −2599.07 −0.133294
\(53\) −15367.7 −0.751484 −0.375742 0.926724i \(-0.622612\pi\)
−0.375742 + 0.926724i \(0.622612\pi\)
\(54\) 3774.11 0.176129
\(55\) 0 0
\(56\) 23771.0 1.01293
\(57\) 8396.53 0.342305
\(58\) 10932.1 0.426709
\(59\) 23400.5 0.875177 0.437588 0.899175i \(-0.355833\pi\)
0.437588 + 0.899175i \(0.355833\pi\)
\(60\) 0 0
\(61\) 10768.4 0.370532 0.185266 0.982688i \(-0.440685\pi\)
0.185266 + 0.982688i \(0.440685\pi\)
\(62\) 826.328 0.0273007
\(63\) 9998.42 0.317380
\(64\) 36221.0 1.10538
\(65\) 0 0
\(66\) −5637.87 −0.159314
\(67\) −14507.1 −0.394814 −0.197407 0.980322i \(-0.563252\pi\)
−0.197407 + 0.980322i \(0.563252\pi\)
\(68\) −2196.66 −0.0576090
\(69\) −11026.6 −0.278817
\(70\) 0 0
\(71\) −28114.0 −0.661876 −0.330938 0.943653i \(-0.607365\pi\)
−0.330938 + 0.943653i \(0.607365\pi\)
\(72\) 15598.6 0.354614
\(73\) 28836.7 0.633342 0.316671 0.948536i \(-0.397435\pi\)
0.316671 + 0.948536i \(0.397435\pi\)
\(74\) 28031.2 0.595062
\(75\) 0 0
\(76\) 4849.08 0.0962998
\(77\) −14935.9 −0.287081
\(78\) 23299.4 0.433619
\(79\) −8150.52 −0.146932 −0.0734662 0.997298i \(-0.523406\pi\)
−0.0734662 + 0.997298i \(0.523406\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 93533.0 1.53614
\(83\) 109864. 1.75049 0.875246 0.483679i \(-0.160700\pi\)
0.875246 + 0.483679i \(0.160700\pi\)
\(84\) 5774.19 0.0892879
\(85\) 0 0
\(86\) 35284.4 0.514442
\(87\) 19004.5 0.269190
\(88\) −23301.7 −0.320760
\(89\) 69673.6 0.932380 0.466190 0.884685i \(-0.345626\pi\)
0.466190 + 0.884685i \(0.345626\pi\)
\(90\) 0 0
\(91\) 61725.2 0.781374
\(92\) −6367.98 −0.0784390
\(93\) 1436.51 0.0172227
\(94\) −78168.5 −0.912457
\(95\) 0 0
\(96\) 16758.0 0.185586
\(97\) −91551.4 −0.987952 −0.493976 0.869476i \(-0.664457\pi\)
−0.493976 + 0.869476i \(0.664457\pi\)
\(98\) 8129.23 0.0855036
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 21299.3 0.207760 0.103880 0.994590i \(-0.466874\pi\)
0.103880 + 0.994590i \(0.466874\pi\)
\(102\) 19692.0 0.187409
\(103\) 6548.06 0.0608163 0.0304081 0.999538i \(-0.490319\pi\)
0.0304081 + 0.999538i \(0.490319\pi\)
\(104\) 96298.1 0.873040
\(105\) 0 0
\(106\) 79560.3 0.687752
\(107\) −127171. −1.07381 −0.536905 0.843643i \(-0.680407\pi\)
−0.536905 + 0.843643i \(0.680407\pi\)
\(108\) 3789.04 0.0312586
\(109\) 57285.2 0.461823 0.230912 0.972975i \(-0.425829\pi\)
0.230912 + 0.972975i \(0.425829\pi\)
\(110\) 0 0
\(111\) 48730.1 0.375396
\(112\) −102535. −0.772371
\(113\) 67774.2 0.499308 0.249654 0.968335i \(-0.419683\pi\)
0.249654 + 0.968335i \(0.419683\pi\)
\(114\) −43469.7 −0.313274
\(115\) 0 0
\(116\) 10975.3 0.0757307
\(117\) 40504.3 0.273550
\(118\) −121147. −0.800954
\(119\) 52168.4 0.337707
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −55749.1 −0.339108
\(123\) 162600. 0.969075
\(124\) 829.598 0.00484522
\(125\) 0 0
\(126\) −51762.9 −0.290464
\(127\) 115352. 0.634622 0.317311 0.948322i \(-0.397220\pi\)
0.317311 + 0.948322i \(0.397220\pi\)
\(128\) −127936. −0.690187
\(129\) 61339.2 0.324537
\(130\) 0 0
\(131\) 43276.4 0.220329 0.110165 0.993913i \(-0.464862\pi\)
0.110165 + 0.993913i \(0.464862\pi\)
\(132\) −5660.17 −0.0282745
\(133\) −115161. −0.564514
\(134\) 75104.6 0.361331
\(135\) 0 0
\(136\) 81388.4 0.377325
\(137\) −401439. −1.82734 −0.913668 0.406461i \(-0.866763\pi\)
−0.913668 + 0.406461i \(0.866763\pi\)
\(138\) 57085.9 0.255171
\(139\) −302018. −1.32586 −0.662928 0.748683i \(-0.730686\pi\)
−0.662928 + 0.748683i \(0.730686\pi\)
\(140\) 0 0
\(141\) −135890. −0.575625
\(142\) 145549. 0.605743
\(143\) −60506.4 −0.247435
\(144\) −67283.6 −0.270398
\(145\) 0 0
\(146\) −149290. −0.579629
\(147\) 14132.1 0.0539401
\(148\) 28142.1 0.105609
\(149\) 326942. 1.20644 0.603218 0.797576i \(-0.293885\pi\)
0.603218 + 0.797576i \(0.293885\pi\)
\(150\) 0 0
\(151\) 164681. 0.587761 0.293881 0.955842i \(-0.405053\pi\)
0.293881 + 0.955842i \(0.405053\pi\)
\(152\) −179663. −0.630740
\(153\) 34233.1 0.118227
\(154\) 77324.8 0.262734
\(155\) 0 0
\(156\) 23391.6 0.0769571
\(157\) −248620. −0.804982 −0.402491 0.915424i \(-0.631856\pi\)
−0.402491 + 0.915424i \(0.631856\pi\)
\(158\) 42196.1 0.134471
\(159\) 138309. 0.433870
\(160\) 0 0
\(161\) 151233. 0.459813
\(162\) −33967.0 −0.101688
\(163\) 417656. 1.23126 0.615629 0.788036i \(-0.288902\pi\)
0.615629 + 0.788036i \(0.288902\pi\)
\(164\) 93903.0 0.272628
\(165\) 0 0
\(166\) −568777. −1.60203
\(167\) −704955. −1.95601 −0.978004 0.208588i \(-0.933113\pi\)
−0.978004 + 0.208588i \(0.933113\pi\)
\(168\) −213939. −0.584814
\(169\) −121240. −0.326535
\(170\) 0 0
\(171\) −75568.8 −0.197630
\(172\) 35424.0 0.0913012
\(173\) 171060. 0.434545 0.217272 0.976111i \(-0.430284\pi\)
0.217272 + 0.976111i \(0.430284\pi\)
\(174\) −98388.5 −0.246361
\(175\) 0 0
\(176\) 100510. 0.244584
\(177\) −210605. −0.505284
\(178\) −360707. −0.853306
\(179\) −166327. −0.387999 −0.193999 0.981002i \(-0.562146\pi\)
−0.193999 + 0.981002i \(0.562146\pi\)
\(180\) 0 0
\(181\) −584391. −1.32589 −0.662945 0.748668i \(-0.730694\pi\)
−0.662945 + 0.748668i \(0.730694\pi\)
\(182\) −319558. −0.715107
\(183\) −96915.5 −0.213927
\(184\) 235940. 0.513756
\(185\) 0 0
\(186\) −7436.96 −0.0157621
\(187\) −51138.3 −0.106941
\(188\) −78477.8 −0.161939
\(189\) −89985.8 −0.183240
\(190\) 0 0
\(191\) −715510. −1.41916 −0.709581 0.704624i \(-0.751116\pi\)
−0.709581 + 0.704624i \(0.751116\pi\)
\(192\) −325989. −0.638189
\(193\) 922088. 1.78188 0.890941 0.454118i \(-0.150046\pi\)
0.890941 + 0.454118i \(0.150046\pi\)
\(194\) 473971. 0.904165
\(195\) 0 0
\(196\) 8161.40 0.0151749
\(197\) 613632. 1.12653 0.563265 0.826276i \(-0.309545\pi\)
0.563265 + 0.826276i \(0.309545\pi\)
\(198\) 50740.8 0.0919802
\(199\) −378985. −0.678406 −0.339203 0.940713i \(-0.610157\pi\)
−0.339203 + 0.940713i \(0.610157\pi\)
\(200\) 0 0
\(201\) 130564. 0.227946
\(202\) −110269. −0.190140
\(203\) −260652. −0.443937
\(204\) 19769.9 0.0332606
\(205\) 0 0
\(206\) −33900.0 −0.0556585
\(207\) 99239.5 0.160975
\(208\) −415375. −0.665705
\(209\) 112887. 0.178763
\(210\) 0 0
\(211\) −473721. −0.732515 −0.366257 0.930514i \(-0.619361\pi\)
−0.366257 + 0.930514i \(0.619361\pi\)
\(212\) 79875.1 0.122060
\(213\) 253026. 0.382134
\(214\) 658376. 0.982741
\(215\) 0 0
\(216\) −140388. −0.204736
\(217\) −19702.1 −0.0284029
\(218\) −296571. −0.422657
\(219\) −259530. −0.365660
\(220\) 0 0
\(221\) 211338. 0.291069
\(222\) −252281. −0.343559
\(223\) 822747. 1.10791 0.553954 0.832547i \(-0.313118\pi\)
0.553954 + 0.832547i \(0.313118\pi\)
\(224\) −229840. −0.306060
\(225\) 0 0
\(226\) −350874. −0.456962
\(227\) 556097. 0.716285 0.358143 0.933667i \(-0.383410\pi\)
0.358143 + 0.933667i \(0.383410\pi\)
\(228\) −43641.7 −0.0555987
\(229\) −634919. −0.800073 −0.400036 0.916499i \(-0.631003\pi\)
−0.400036 + 0.916499i \(0.631003\pi\)
\(230\) 0 0
\(231\) 134423. 0.165747
\(232\) −406646. −0.496017
\(233\) 906561. 1.09397 0.546987 0.837141i \(-0.315775\pi\)
0.546987 + 0.837141i \(0.315775\pi\)
\(234\) −209695. −0.250350
\(235\) 0 0
\(236\) −121626. −0.142150
\(237\) 73354.7 0.0848314
\(238\) −270081. −0.309066
\(239\) 662586. 0.750322 0.375161 0.926960i \(-0.377587\pi\)
0.375161 + 0.926960i \(0.377587\pi\)
\(240\) 0 0
\(241\) −1.31823e6 −1.46200 −0.731001 0.682376i \(-0.760947\pi\)
−0.731001 + 0.682376i \(0.760947\pi\)
\(242\) −75798.0 −0.0831992
\(243\) −59049.0 −0.0641500
\(244\) −55969.6 −0.0601836
\(245\) 0 0
\(246\) −841797. −0.886889
\(247\) −466523. −0.486554
\(248\) −30737.4 −0.0317350
\(249\) −988775. −1.01065
\(250\) 0 0
\(251\) −11073.9 −0.0110947 −0.00554737 0.999985i \(-0.501766\pi\)
−0.00554737 + 0.999985i \(0.501766\pi\)
\(252\) −51967.7 −0.0515504
\(253\) −148247. −0.145608
\(254\) −597188. −0.580800
\(255\) 0 0
\(256\) −496734. −0.473723
\(257\) 788596. 0.744769 0.372384 0.928079i \(-0.378540\pi\)
0.372384 + 0.928079i \(0.378540\pi\)
\(258\) −317560. −0.297013
\(259\) −668346. −0.619087
\(260\) 0 0
\(261\) −171041. −0.155417
\(262\) −224046. −0.201644
\(263\) −211325. −0.188391 −0.0941956 0.995554i \(-0.530028\pi\)
−0.0941956 + 0.995554i \(0.530028\pi\)
\(264\) 209715. 0.185191
\(265\) 0 0
\(266\) 596198. 0.516639
\(267\) −627062. −0.538310
\(268\) 75401.8 0.0641276
\(269\) 552342. 0.465401 0.232701 0.972548i \(-0.425244\pi\)
0.232701 + 0.972548i \(0.425244\pi\)
\(270\) 0 0
\(271\) −1.49255e6 −1.23454 −0.617271 0.786751i \(-0.711762\pi\)
−0.617271 + 0.786751i \(0.711762\pi\)
\(272\) −351063. −0.287715
\(273\) −555527. −0.451126
\(274\) 2.07829e6 1.67236
\(275\) 0 0
\(276\) 57311.8 0.0452868
\(277\) −1.01431e6 −0.794275 −0.397138 0.917759i \(-0.629996\pi\)
−0.397138 + 0.917759i \(0.629996\pi\)
\(278\) 1.56358e6 1.21341
\(279\) −12928.6 −0.00994352
\(280\) 0 0
\(281\) −658216. −0.497282 −0.248641 0.968596i \(-0.579984\pi\)
−0.248641 + 0.968596i \(0.579984\pi\)
\(282\) 703517. 0.526807
\(283\) −1.65981e6 −1.23195 −0.615974 0.787767i \(-0.711237\pi\)
−0.615974 + 0.787767i \(0.711237\pi\)
\(284\) 146125. 0.107505
\(285\) 0 0
\(286\) 313248. 0.226450
\(287\) −2.23010e6 −1.59816
\(288\) −150822. −0.107148
\(289\) −1.24124e6 −0.874201
\(290\) 0 0
\(291\) 823963. 0.570394
\(292\) −149881. −0.102870
\(293\) 410864. 0.279594 0.139797 0.990180i \(-0.455355\pi\)
0.139797 + 0.990180i \(0.455355\pi\)
\(294\) −73163.1 −0.0493655
\(295\) 0 0
\(296\) −1.04269e6 −0.691715
\(297\) 88209.0 0.0580259
\(298\) −1.69261e6 −1.10412
\(299\) 612654. 0.396312
\(300\) 0 0
\(301\) −841283. −0.535212
\(302\) −852570. −0.537914
\(303\) −191694. −0.119950
\(304\) 774965. 0.480948
\(305\) 0 0
\(306\) −177228. −0.108200
\(307\) −1.34831e6 −0.816477 −0.408238 0.912875i \(-0.633857\pi\)
−0.408238 + 0.912875i \(0.633857\pi\)
\(308\) 77630.7 0.0466291
\(309\) −58932.6 −0.0351123
\(310\) 0 0
\(311\) −2.52585e6 −1.48083 −0.740417 0.672148i \(-0.765372\pi\)
−0.740417 + 0.672148i \(0.765372\pi\)
\(312\) −866682. −0.504050
\(313\) −2.23061e6 −1.28695 −0.643476 0.765466i \(-0.722509\pi\)
−0.643476 + 0.765466i \(0.722509\pi\)
\(314\) 1.28713e6 0.736713
\(315\) 0 0
\(316\) 42363.0 0.0238654
\(317\) 120908. 0.0675780 0.0337890 0.999429i \(-0.489243\pi\)
0.0337890 + 0.999429i \(0.489243\pi\)
\(318\) −716043. −0.397074
\(319\) 255506. 0.140580
\(320\) 0 0
\(321\) 1.14454e6 0.619964
\(322\) −782948. −0.420817
\(323\) −394292. −0.210287
\(324\) −34101.4 −0.0180472
\(325\) 0 0
\(326\) −2.16225e6 −1.12684
\(327\) −515566. −0.266634
\(328\) −3.47920e6 −1.78564
\(329\) 1.86377e6 0.949296
\(330\) 0 0
\(331\) 578480. 0.290214 0.145107 0.989416i \(-0.453647\pi\)
0.145107 + 0.989416i \(0.453647\pi\)
\(332\) −571028. −0.284323
\(333\) −438571. −0.216735
\(334\) 3.64963e6 1.79012
\(335\) 0 0
\(336\) 922812. 0.445929
\(337\) 1.78710e6 0.857186 0.428593 0.903498i \(-0.359009\pi\)
0.428593 + 0.903498i \(0.359009\pi\)
\(338\) 627674. 0.298842
\(339\) −609968. −0.288276
\(340\) 0 0
\(341\) 19313.1 0.00899425
\(342\) 391228. 0.180869
\(343\) −2.26844e6 −1.04110
\(344\) −1.31249e6 −0.598000
\(345\) 0 0
\(346\) −885598. −0.397692
\(347\) 1.46282e6 0.652179 0.326089 0.945339i \(-0.394269\pi\)
0.326089 + 0.945339i \(0.394269\pi\)
\(348\) −98777.8 −0.0437231
\(349\) 1.48501e6 0.652627 0.326314 0.945262i \(-0.394193\pi\)
0.326314 + 0.945262i \(0.394193\pi\)
\(350\) 0 0
\(351\) −364538. −0.157934
\(352\) 225302. 0.0969189
\(353\) −1.34528e6 −0.574616 −0.287308 0.957838i \(-0.592760\pi\)
−0.287308 + 0.957838i \(0.592760\pi\)
\(354\) 1.09032e6 0.462431
\(355\) 0 0
\(356\) −362135. −0.151442
\(357\) −469515. −0.194975
\(358\) 861092. 0.355093
\(359\) −1.83956e6 −0.753316 −0.376658 0.926352i \(-0.622927\pi\)
−0.376658 + 0.926352i \(0.622927\pi\)
\(360\) 0 0
\(361\) −1.60571e6 −0.648483
\(362\) 3.02546e6 1.21344
\(363\) −131769. −0.0524864
\(364\) −320822. −0.126914
\(365\) 0 0
\(366\) 501742. 0.195784
\(367\) 312079. 0.120948 0.0604741 0.998170i \(-0.480739\pi\)
0.0604741 + 0.998170i \(0.480739\pi\)
\(368\) −1.01771e6 −0.391746
\(369\) −1.46340e6 −0.559496
\(370\) 0 0
\(371\) −1.89695e6 −0.715519
\(372\) −7466.38 −0.00279739
\(373\) −3.38658e6 −1.26034 −0.630172 0.776455i \(-0.717016\pi\)
−0.630172 + 0.776455i \(0.717016\pi\)
\(374\) 264748. 0.0978710
\(375\) 0 0
\(376\) 2.90768e6 1.06066
\(377\) −1.05592e6 −0.382629
\(378\) 465866. 0.167699
\(379\) −1.62370e6 −0.580641 −0.290321 0.956929i \(-0.593762\pi\)
−0.290321 + 0.956929i \(0.593762\pi\)
\(380\) 0 0
\(381\) −1.03817e6 −0.366399
\(382\) 3.70427e6 1.29881
\(383\) 3.06187e6 1.06657 0.533285 0.845935i \(-0.320957\pi\)
0.533285 + 0.845935i \(0.320957\pi\)
\(384\) 1.15142e6 0.398480
\(385\) 0 0
\(386\) −4.77375e6 −1.63076
\(387\) −552053. −0.187371
\(388\) 475847. 0.160468
\(389\) 4.21840e6 1.41343 0.706715 0.707499i \(-0.250176\pi\)
0.706715 + 0.707499i \(0.250176\pi\)
\(390\) 0 0
\(391\) 517798. 0.171285
\(392\) −302388. −0.0993915
\(393\) −389487. −0.127207
\(394\) −3.17684e6 −1.03099
\(395\) 0 0
\(396\) 50941.6 0.0163243
\(397\) −513185. −0.163417 −0.0817085 0.996656i \(-0.526038\pi\)
−0.0817085 + 0.996656i \(0.526038\pi\)
\(398\) 1.96205e6 0.620871
\(399\) 1.03645e6 0.325922
\(400\) 0 0
\(401\) −2.18458e6 −0.678432 −0.339216 0.940709i \(-0.610162\pi\)
−0.339216 + 0.940709i \(0.610162\pi\)
\(402\) −675942. −0.208614
\(403\) −79814.5 −0.0244804
\(404\) −110705. −0.0337453
\(405\) 0 0
\(406\) 1.34942e6 0.406287
\(407\) 655149. 0.196044
\(408\) −732496. −0.217849
\(409\) 1.22307e6 0.361529 0.180764 0.983526i \(-0.442143\pi\)
0.180764 + 0.983526i \(0.442143\pi\)
\(410\) 0 0
\(411\) 3.61295e6 1.05501
\(412\) −34034.1 −0.00987806
\(413\) 2.88850e6 0.833292
\(414\) −513774. −0.147323
\(415\) 0 0
\(416\) −931098. −0.263792
\(417\) 2.71816e6 0.765483
\(418\) −584426. −0.163602
\(419\) 2.42879e6 0.675858 0.337929 0.941172i \(-0.390274\pi\)
0.337929 + 0.941172i \(0.390274\pi\)
\(420\) 0 0
\(421\) −727467. −0.200036 −0.100018 0.994986i \(-0.531890\pi\)
−0.100018 + 0.994986i \(0.531890\pi\)
\(422\) 2.45250e6 0.670391
\(423\) 1.22301e6 0.332337
\(424\) −2.95945e6 −0.799460
\(425\) 0 0
\(426\) −1.30994e6 −0.349726
\(427\) 1.32922e6 0.352799
\(428\) 660981. 0.174413
\(429\) 544557. 0.142857
\(430\) 0 0
\(431\) 6.18223e6 1.60307 0.801534 0.597949i \(-0.204018\pi\)
0.801534 + 0.597949i \(0.204018\pi\)
\(432\) 605553. 0.156114
\(433\) −2.42337e6 −0.621154 −0.310577 0.950548i \(-0.600522\pi\)
−0.310577 + 0.950548i \(0.600522\pi\)
\(434\) 102000. 0.0259941
\(435\) 0 0
\(436\) −297745. −0.0750115
\(437\) −1.14303e6 −0.286321
\(438\) 1.34361e6 0.334649
\(439\) −3.79993e6 −0.941054 −0.470527 0.882385i \(-0.655936\pi\)
−0.470527 + 0.882385i \(0.655936\pi\)
\(440\) 0 0
\(441\) −127188. −0.0311423
\(442\) −1.09412e6 −0.266384
\(443\) 5.58044e6 1.35101 0.675506 0.737354i \(-0.263925\pi\)
0.675506 + 0.737354i \(0.263925\pi\)
\(444\) −253279. −0.0609736
\(445\) 0 0
\(446\) −4.25945e6 −1.01395
\(447\) −2.94247e6 −0.696537
\(448\) 4.47102e6 1.05247
\(449\) 2.69609e6 0.631130 0.315565 0.948904i \(-0.397806\pi\)
0.315565 + 0.948904i \(0.397806\pi\)
\(450\) 0 0
\(451\) 2.18607e6 0.506083
\(452\) −352263. −0.0810999
\(453\) −1.48213e6 −0.339344
\(454\) −2.87897e6 −0.655538
\(455\) 0 0
\(456\) 1.61697e6 0.364158
\(457\) −1.14257e6 −0.255912 −0.127956 0.991780i \(-0.540842\pi\)
−0.127956 + 0.991780i \(0.540842\pi\)
\(458\) 3.28704e6 0.732220
\(459\) −308098. −0.0682585
\(460\) 0 0
\(461\) 5.37091e6 1.17705 0.588526 0.808478i \(-0.299708\pi\)
0.588526 + 0.808478i \(0.299708\pi\)
\(462\) −695923. −0.151690
\(463\) 3.80227e6 0.824311 0.412155 0.911114i \(-0.364776\pi\)
0.412155 + 0.911114i \(0.364776\pi\)
\(464\) 1.75404e6 0.378220
\(465\) 0 0
\(466\) −4.69336e6 −1.00120
\(467\) −9.41694e6 −1.99810 −0.999051 0.0435609i \(-0.986130\pi\)
−0.999051 + 0.0435609i \(0.986130\pi\)
\(468\) −210525. −0.0444312
\(469\) −1.79071e6 −0.375919
\(470\) 0 0
\(471\) 2.23758e6 0.464757
\(472\) 4.50638e6 0.931049
\(473\) 824672. 0.169484
\(474\) −379765. −0.0776370
\(475\) 0 0
\(476\) −271150. −0.0548519
\(477\) −1.24479e6 −0.250495
\(478\) −3.43028e6 −0.686688
\(479\) −4.69047e6 −0.934065 −0.467033 0.884240i \(-0.654677\pi\)
−0.467033 + 0.884240i \(0.654677\pi\)
\(480\) 0 0
\(481\) −2.70751e6 −0.533590
\(482\) 6.82461e6 1.33801
\(483\) −1.36110e6 −0.265473
\(484\) −76097.9 −0.0147659
\(485\) 0 0
\(486\) 305703. 0.0587096
\(487\) −7.07177e6 −1.35116 −0.675579 0.737288i \(-0.736106\pi\)
−0.675579 + 0.737288i \(0.736106\pi\)
\(488\) 2.07373e6 0.394187
\(489\) −3.75890e6 −0.710867
\(490\) 0 0
\(491\) 906051. 0.169609 0.0848045 0.996398i \(-0.472973\pi\)
0.0848045 + 0.996398i \(0.472973\pi\)
\(492\) −845127. −0.157402
\(493\) −892434. −0.165371
\(494\) 2.41524e6 0.445290
\(495\) 0 0
\(496\) 132584. 0.0241984
\(497\) −3.47031e6 −0.630199
\(498\) 5.11899e6 0.924935
\(499\) −7.13323e6 −1.28243 −0.641217 0.767360i \(-0.721570\pi\)
−0.641217 + 0.767360i \(0.721570\pi\)
\(500\) 0 0
\(501\) 6.34460e6 1.12930
\(502\) 57330.9 0.0101538
\(503\) 2.34927e6 0.414013 0.207006 0.978340i \(-0.433628\pi\)
0.207006 + 0.978340i \(0.433628\pi\)
\(504\) 1.92545e6 0.337642
\(505\) 0 0
\(506\) 767489. 0.133259
\(507\) 1.09116e6 0.188525
\(508\) −599551. −0.103078
\(509\) 1.97148e6 0.337286 0.168643 0.985677i \(-0.446062\pi\)
0.168643 + 0.985677i \(0.446062\pi\)
\(510\) 0 0
\(511\) 3.55952e6 0.603031
\(512\) 6.66559e6 1.12373
\(513\) 680119. 0.114102
\(514\) −4.08264e6 −0.681606
\(515\) 0 0
\(516\) −318816. −0.0527128
\(517\) −1.82697e6 −0.300610
\(518\) 3.46010e6 0.566583
\(519\) −1.53954e6 −0.250884
\(520\) 0 0
\(521\) 1.20074e7 1.93801 0.969004 0.247046i \(-0.0794599\pi\)
0.969004 + 0.247046i \(0.0794599\pi\)
\(522\) 885497. 0.142236
\(523\) −4.16772e6 −0.666260 −0.333130 0.942881i \(-0.608105\pi\)
−0.333130 + 0.942881i \(0.608105\pi\)
\(524\) −224933. −0.0357869
\(525\) 0 0
\(526\) 1.09405e6 0.172414
\(527\) −67456.9 −0.0105804
\(528\) −904591. −0.141211
\(529\) −4.93528e6 −0.766783
\(530\) 0 0
\(531\) 1.89544e6 0.291726
\(532\) 598557. 0.0916910
\(533\) −9.03428e6 −1.37745
\(534\) 3.24637e6 0.492657
\(535\) 0 0
\(536\) −2.79371e6 −0.420020
\(537\) 1.49694e6 0.224011
\(538\) −2.85953e6 −0.425931
\(539\) 189998. 0.0281693
\(540\) 0 0
\(541\) 1.01106e7 1.48519 0.742594 0.669741i \(-0.233595\pi\)
0.742594 + 0.669741i \(0.233595\pi\)
\(542\) 7.72709e6 1.12984
\(543\) 5.25952e6 0.765503
\(544\) −786938. −0.114010
\(545\) 0 0
\(546\) 2.87602e6 0.412867
\(547\) 7.39087e6 1.05615 0.528077 0.849196i \(-0.322913\pi\)
0.528077 + 0.849196i \(0.322913\pi\)
\(548\) 2.08652e6 0.296805
\(549\) 872239. 0.123511
\(550\) 0 0
\(551\) 1.97003e6 0.276435
\(552\) −2.12346e6 −0.296617
\(553\) −1.00608e6 −0.139900
\(554\) 5.25119e6 0.726914
\(555\) 0 0
\(556\) 1.56977e6 0.215352
\(557\) −1.52459e6 −0.208217 −0.104108 0.994566i \(-0.533199\pi\)
−0.104108 + 0.994566i \(0.533199\pi\)
\(558\) 66932.6 0.00910023
\(559\) −3.40809e6 −0.461299
\(560\) 0 0
\(561\) 460245. 0.0617421
\(562\) 3.40765e6 0.455108
\(563\) 1.40845e7 1.87271 0.936353 0.351059i \(-0.114178\pi\)
0.936353 + 0.351059i \(0.114178\pi\)
\(564\) 706300. 0.0934957
\(565\) 0 0
\(566\) 8.59301e6 1.12747
\(567\) 809872. 0.105793
\(568\) −5.41407e6 −0.704131
\(569\) −1.30473e7 −1.68942 −0.844712 0.535221i \(-0.820228\pi\)
−0.844712 + 0.535221i \(0.820228\pi\)
\(570\) 0 0
\(571\) −1.19873e7 −1.53862 −0.769312 0.638873i \(-0.779401\pi\)
−0.769312 + 0.638873i \(0.779401\pi\)
\(572\) 314487. 0.0401895
\(573\) 6.43959e6 0.819354
\(574\) 1.15455e7 1.46262
\(575\) 0 0
\(576\) 2.93390e6 0.368459
\(577\) −568904. −0.0711376 −0.0355688 0.999367i \(-0.511324\pi\)
−0.0355688 + 0.999367i \(0.511324\pi\)
\(578\) 6.42603e6 0.800061
\(579\) −8.29879e6 −1.02877
\(580\) 0 0
\(581\) 1.35613e7 1.66671
\(582\) −4.26574e6 −0.522020
\(583\) 1.85949e6 0.226581
\(584\) 5.55325e6 0.673775
\(585\) 0 0
\(586\) −2.12708e6 −0.255882
\(587\) −2.93323e6 −0.351358 −0.175679 0.984447i \(-0.556212\pi\)
−0.175679 + 0.984447i \(0.556212\pi\)
\(588\) −73452.6 −0.00876120
\(589\) 148910. 0.0176862
\(590\) 0 0
\(591\) −5.52269e6 −0.650402
\(592\) 4.49758e6 0.527442
\(593\) −5.90557e6 −0.689644 −0.344822 0.938668i \(-0.612061\pi\)
−0.344822 + 0.938668i \(0.612061\pi\)
\(594\) −456667. −0.0531048
\(595\) 0 0
\(596\) −1.69931e6 −0.195955
\(597\) 3.41087e6 0.391678
\(598\) −3.17178e6 −0.362702
\(599\) 1.07665e7 1.22605 0.613025 0.790064i \(-0.289953\pi\)
0.613025 + 0.790064i \(0.289953\pi\)
\(600\) 0 0
\(601\) −1.19746e7 −1.35231 −0.676154 0.736760i \(-0.736355\pi\)
−0.676154 + 0.736760i \(0.736355\pi\)
\(602\) 4.35541e6 0.489822
\(603\) −1.17507e6 −0.131605
\(604\) −855944. −0.0954669
\(605\) 0 0
\(606\) 992418. 0.109777
\(607\) 3.45506e6 0.380613 0.190307 0.981725i \(-0.439052\pi\)
0.190307 + 0.981725i \(0.439052\pi\)
\(608\) 1.73715e6 0.190580
\(609\) 2.34587e6 0.256307
\(610\) 0 0
\(611\) 7.55024e6 0.818197
\(612\) −177929. −0.0192030
\(613\) 8.22419e6 0.883979 0.441990 0.897020i \(-0.354273\pi\)
0.441990 + 0.897020i \(0.354273\pi\)
\(614\) 6.98034e6 0.747233
\(615\) 0 0
\(616\) −2.87630e6 −0.305409
\(617\) −9.83567e6 −1.04014 −0.520069 0.854124i \(-0.674094\pi\)
−0.520069 + 0.854124i \(0.674094\pi\)
\(618\) 305100. 0.0321345
\(619\) 267648. 0.0280762 0.0140381 0.999901i \(-0.495531\pi\)
0.0140381 + 0.999901i \(0.495531\pi\)
\(620\) 0 0
\(621\) −893156. −0.0929391
\(622\) 1.30766e7 1.35525
\(623\) 8.60032e6 0.887758
\(624\) 3.73837e6 0.384345
\(625\) 0 0
\(626\) 1.15481e7 1.17781
\(627\) −1.01598e6 −0.103209
\(628\) 1.29222e6 0.130749
\(629\) −2.28831e6 −0.230616
\(630\) 0 0
\(631\) −4.17135e6 −0.417065 −0.208532 0.978015i \(-0.566869\pi\)
−0.208532 + 0.978015i \(0.566869\pi\)
\(632\) −1.56959e6 −0.156313
\(633\) 4.26349e6 0.422918
\(634\) −625951. −0.0618468
\(635\) 0 0
\(636\) −718876. −0.0704711
\(637\) −785197. −0.0766708
\(638\) −1.32278e6 −0.128658
\(639\) −2.27723e6 −0.220625
\(640\) 0 0
\(641\) 8.24673e6 0.792751 0.396376 0.918088i \(-0.370268\pi\)
0.396376 + 0.918088i \(0.370268\pi\)
\(642\) −5.92538e6 −0.567386
\(643\) −1.07773e7 −1.02797 −0.513986 0.857799i \(-0.671832\pi\)
−0.513986 + 0.857799i \(0.671832\pi\)
\(644\) −786046. −0.0746850
\(645\) 0 0
\(646\) 2.04129e6 0.192453
\(647\) −7.89194e6 −0.741179 −0.370590 0.928797i \(-0.620844\pi\)
−0.370590 + 0.928797i \(0.620844\pi\)
\(648\) 1.26349e6 0.118205
\(649\) −2.83147e6 −0.263876
\(650\) 0 0
\(651\) 177319. 0.0163984
\(652\) −2.17080e6 −0.199987
\(653\) −1.47858e7 −1.35695 −0.678473 0.734625i \(-0.737358\pi\)
−0.678473 + 0.734625i \(0.737358\pi\)
\(654\) 2.66914e6 0.244021
\(655\) 0 0
\(656\) 1.50073e7 1.36158
\(657\) 2.33577e6 0.211114
\(658\) −9.64891e6 −0.868788
\(659\) −475088. −0.0426148 −0.0213074 0.999773i \(-0.506783\pi\)
−0.0213074 + 0.999773i \(0.506783\pi\)
\(660\) 0 0
\(661\) 2.73604e6 0.243567 0.121784 0.992557i \(-0.461139\pi\)
0.121784 + 0.992557i \(0.461139\pi\)
\(662\) −2.99485e6 −0.265601
\(663\) −1.90204e6 −0.168049
\(664\) 2.11571e7 1.86224
\(665\) 0 0
\(666\) 2.27053e6 0.198354
\(667\) −2.58711e6 −0.225165
\(668\) 3.66407e6 0.317704
\(669\) −7.40472e6 −0.639651
\(670\) 0 0
\(671\) −1.30297e6 −0.111720
\(672\) 2.06856e6 0.176704
\(673\) −1.52861e7 −1.30094 −0.650471 0.759531i \(-0.725428\pi\)
−0.650471 + 0.759531i \(0.725428\pi\)
\(674\) −9.25202e6 −0.784489
\(675\) 0 0
\(676\) 630157. 0.0530374
\(677\) 3.93225e6 0.329739 0.164869 0.986315i \(-0.447280\pi\)
0.164869 + 0.986315i \(0.447280\pi\)
\(678\) 3.15787e6 0.263827
\(679\) −1.13009e7 −0.940670
\(680\) 0 0
\(681\) −5.00487e6 −0.413547
\(682\) −99985.7 −0.00823146
\(683\) −3.42591e6 −0.281011 −0.140506 0.990080i \(-0.544873\pi\)
−0.140506 + 0.990080i \(0.544873\pi\)
\(684\) 392776. 0.0320999
\(685\) 0 0
\(686\) 1.17439e7 0.952803
\(687\) 5.71427e6 0.461922
\(688\) 5.66135e6 0.455984
\(689\) −7.68467e6 −0.616705
\(690\) 0 0
\(691\) 1.81896e7 1.44920 0.724599 0.689171i \(-0.242025\pi\)
0.724599 + 0.689171i \(0.242025\pi\)
\(692\) −889102. −0.0705808
\(693\) −1.20981e6 −0.0956938
\(694\) −7.57316e6 −0.596868
\(695\) 0 0
\(696\) 3.65982e6 0.286376
\(697\) −7.63552e6 −0.595328
\(698\) −7.68804e6 −0.597279
\(699\) −8.15905e6 −0.631607
\(700\) 0 0
\(701\) −1.15598e6 −0.0888494 −0.0444247 0.999013i \(-0.514145\pi\)
−0.0444247 + 0.999013i \(0.514145\pi\)
\(702\) 1.88725e6 0.144540
\(703\) 5.05141e6 0.385500
\(704\) −4.38274e6 −0.333283
\(705\) 0 0
\(706\) 6.96468e6 0.525883
\(707\) 2.62913e6 0.197817
\(708\) 1.09464e6 0.0820705
\(709\) 2.42664e7 1.81296 0.906481 0.422246i \(-0.138758\pi\)
0.906481 + 0.422246i \(0.138758\pi\)
\(710\) 0 0
\(711\) −660192. −0.0489775
\(712\) 1.34174e7 0.991904
\(713\) −195553. −0.0144059
\(714\) 2.43073e6 0.178440
\(715\) 0 0
\(716\) 864499. 0.0630205
\(717\) −5.96328e6 −0.433199
\(718\) 9.52359e6 0.689429
\(719\) −2.25503e7 −1.62679 −0.813394 0.581714i \(-0.802382\pi\)
−0.813394 + 0.581714i \(0.802382\pi\)
\(720\) 0 0
\(721\) 808276. 0.0579057
\(722\) 8.31291e6 0.593486
\(723\) 1.18641e7 0.844088
\(724\) 3.03743e6 0.215357
\(725\) 0 0
\(726\) 682182. 0.0480351
\(727\) 2.17941e7 1.52933 0.764667 0.644425i \(-0.222903\pi\)
0.764667 + 0.644425i \(0.222903\pi\)
\(728\) 1.18868e7 0.831257
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −2.88043e6 −0.199372
\(732\) 503727. 0.0347470
\(733\) −1.73565e7 −1.19317 −0.596583 0.802551i \(-0.703476\pi\)
−0.596583 + 0.802551i \(0.703476\pi\)
\(734\) −1.61567e6 −0.110691
\(735\) 0 0
\(736\) −2.28129e6 −0.155233
\(737\) 1.75536e6 0.119041
\(738\) 7.57617e6 0.512046
\(739\) 1.16961e6 0.0787825 0.0393912 0.999224i \(-0.487458\pi\)
0.0393912 + 0.999224i \(0.487458\pi\)
\(740\) 0 0
\(741\) 4.19871e6 0.280912
\(742\) 9.82071e6 0.654837
\(743\) −2.98047e6 −0.198067 −0.0990336 0.995084i \(-0.531575\pi\)
−0.0990336 + 0.995084i \(0.531575\pi\)
\(744\) 276637. 0.0183222
\(745\) 0 0
\(746\) 1.75327e7 1.15346
\(747\) 8.89898e6 0.583497
\(748\) 265796. 0.0173698
\(749\) −1.56976e7 −1.02242
\(750\) 0 0
\(751\) −1.83578e7 −1.18774 −0.593868 0.804563i \(-0.702400\pi\)
−0.593868 + 0.804563i \(0.702400\pi\)
\(752\) −1.25421e7 −0.808770
\(753\) 99665.3 0.00640555
\(754\) 5.46661e6 0.350178
\(755\) 0 0
\(756\) 467709. 0.0297626
\(757\) −1.60050e7 −1.01511 −0.507557 0.861618i \(-0.669451\pi\)
−0.507557 + 0.861618i \(0.669451\pi\)
\(758\) 8.40607e6 0.531398
\(759\) 1.33422e6 0.0840665
\(760\) 0 0
\(761\) −1.67436e7 −1.04807 −0.524033 0.851698i \(-0.675573\pi\)
−0.524033 + 0.851698i \(0.675573\pi\)
\(762\) 5.37469e6 0.335325
\(763\) 7.07113e6 0.439721
\(764\) 3.71892e6 0.230507
\(765\) 0 0
\(766\) −1.58516e7 −0.976116
\(767\) 1.17015e7 0.718213
\(768\) 4.47061e6 0.273504
\(769\) −1.73863e7 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(770\) 0 0
\(771\) −7.09736e6 −0.429993
\(772\) −4.79263e6 −0.289422
\(773\) −1.79095e7 −1.07804 −0.539021 0.842292i \(-0.681206\pi\)
−0.539021 + 0.842292i \(0.681206\pi\)
\(774\) 2.85804e6 0.171481
\(775\) 0 0
\(776\) −1.76306e7 −1.05102
\(777\) 6.01511e6 0.357430
\(778\) −2.18391e7 −1.29356
\(779\) 1.68552e7 0.995157
\(780\) 0 0
\(781\) 3.40179e6 0.199563
\(782\) −2.68070e6 −0.156758
\(783\) 1.53937e6 0.0897301
\(784\) 1.30433e6 0.0757874
\(785\) 0 0
\(786\) 2.01642e6 0.116419
\(787\) 3.01291e7 1.73400 0.867000 0.498308i \(-0.166045\pi\)
0.867000 + 0.498308i \(0.166045\pi\)
\(788\) −3.18941e6 −0.182976
\(789\) 1.90192e6 0.108768
\(790\) 0 0
\(791\) 8.36587e6 0.475412
\(792\) −1.88744e6 −0.106920
\(793\) 5.38476e6 0.304077
\(794\) 2.65681e6 0.149558
\(795\) 0 0
\(796\) 1.96981e6 0.110190
\(797\) 5.51251e6 0.307400 0.153700 0.988118i \(-0.450881\pi\)
0.153700 + 0.988118i \(0.450881\pi\)
\(798\) −5.36579e6 −0.298281
\(799\) 6.38125e6 0.353622
\(800\) 0 0
\(801\) 5.64356e6 0.310793
\(802\) 1.13098e7 0.620895
\(803\) −3.48924e6 −0.190960
\(804\) −678616. −0.0370241
\(805\) 0 0
\(806\) 413208. 0.0224043
\(807\) −4.97108e6 −0.268700
\(808\) 4.10173e6 0.221024
\(809\) 3.11564e7 1.67369 0.836847 0.547436i \(-0.184396\pi\)
0.836847 + 0.547436i \(0.184396\pi\)
\(810\) 0 0
\(811\) −1.11336e7 −0.594406 −0.297203 0.954814i \(-0.596054\pi\)
−0.297203 + 0.954814i \(0.596054\pi\)
\(812\) 1.35476e6 0.0721063
\(813\) 1.34330e7 0.712763
\(814\) −3.39178e6 −0.179418
\(815\) 0 0
\(816\) 3.15957e6 0.166113
\(817\) 6.35848e6 0.333271
\(818\) −6.33195e6 −0.330868
\(819\) 4.99974e6 0.260458
\(820\) 0 0
\(821\) −3.44603e7 −1.78427 −0.892136 0.451767i \(-0.850794\pi\)
−0.892136 + 0.451767i \(0.850794\pi\)
\(822\) −1.87046e7 −0.965539
\(823\) 9.74417e6 0.501470 0.250735 0.968056i \(-0.419328\pi\)
0.250735 + 0.968056i \(0.419328\pi\)
\(824\) 1.26100e6 0.0646989
\(825\) 0 0
\(826\) −1.49541e7 −0.762622
\(827\) 9.63214e6 0.489733 0.244866 0.969557i \(-0.421256\pi\)
0.244866 + 0.969557i \(0.421256\pi\)
\(828\) −515806. −0.0261463
\(829\) −2.79310e7 −1.41156 −0.705780 0.708431i \(-0.749403\pi\)
−0.705780 + 0.708431i \(0.749403\pi\)
\(830\) 0 0
\(831\) 9.12879e6 0.458575
\(832\) 1.81124e7 0.907126
\(833\) −663626. −0.0331368
\(834\) −1.40722e7 −0.700564
\(835\) 0 0
\(836\) −586739. −0.0290355
\(837\) 116357. 0.00574090
\(838\) −1.25741e7 −0.618540
\(839\) −3.51475e7 −1.72381 −0.861906 0.507068i \(-0.830729\pi\)
−0.861906 + 0.507068i \(0.830729\pi\)
\(840\) 0 0
\(841\) −1.60522e7 −0.782610
\(842\) 3.76617e6 0.183071
\(843\) 5.92394e6 0.287106
\(844\) 2.46221e6 0.118978
\(845\) 0 0
\(846\) −6.33165e6 −0.304152
\(847\) 1.80725e6 0.0865583
\(848\) 1.27654e7 0.609599
\(849\) 1.49383e7 0.711265
\(850\) 0 0
\(851\) −6.63368e6 −0.314001
\(852\) −1.31512e6 −0.0620680
\(853\) −2.50498e7 −1.17878 −0.589389 0.807849i \(-0.700631\pi\)
−0.589389 + 0.807849i \(0.700631\pi\)
\(854\) −6.88152e6 −0.322879
\(855\) 0 0
\(856\) −2.44900e7 −1.14236
\(857\) 1.89631e7 0.881975 0.440987 0.897513i \(-0.354628\pi\)
0.440987 + 0.897513i \(0.354628\pi\)
\(858\) −2.81923e6 −0.130741
\(859\) −3.04605e7 −1.40849 −0.704245 0.709957i \(-0.748714\pi\)
−0.704245 + 0.709957i \(0.748714\pi\)
\(860\) 0 0
\(861\) 2.00709e7 0.922696
\(862\) −3.20060e7 −1.46711
\(863\) −3.46176e7 −1.58223 −0.791115 0.611667i \(-0.790499\pi\)
−0.791115 + 0.611667i \(0.790499\pi\)
\(864\) 1.35740e6 0.0618619
\(865\) 0 0
\(866\) 1.25460e7 0.568475
\(867\) 1.11712e7 0.504720
\(868\) 102403. 0.00461333
\(869\) 986213. 0.0443018
\(870\) 0 0
\(871\) −7.25430e6 −0.324004
\(872\) 1.10317e7 0.491307
\(873\) −7.41566e6 −0.329317
\(874\) 5.91758e6 0.262039
\(875\) 0 0
\(876\) 1.34893e6 0.0593922
\(877\) 3.61979e7 1.58922 0.794611 0.607119i \(-0.207675\pi\)
0.794611 + 0.607119i \(0.207675\pi\)
\(878\) 1.96727e7 0.861245
\(879\) −3.69777e6 −0.161424
\(880\) 0 0
\(881\) 2.40371e7 1.04338 0.521690 0.853135i \(-0.325302\pi\)
0.521690 + 0.853135i \(0.325302\pi\)
\(882\) 658468. 0.0285012
\(883\) −3.64938e7 −1.57513 −0.787567 0.616229i \(-0.788659\pi\)
−0.787567 + 0.616229i \(0.788659\pi\)
\(884\) −1.09845e6 −0.0472768
\(885\) 0 0
\(886\) −2.88905e7 −1.23644
\(887\) −2.29197e7 −0.978137 −0.489069 0.872245i \(-0.662663\pi\)
−0.489069 + 0.872245i \(0.662663\pi\)
\(888\) 9.38424e6 0.399362
\(889\) 1.42387e7 0.604250
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −4.27630e6 −0.179952
\(893\) −1.40865e7 −0.591117
\(894\) 1.52335e7 0.637464
\(895\) 0 0
\(896\) −1.57920e7 −0.657156
\(897\) −5.51389e6 −0.228811
\(898\) −1.39579e7 −0.577605
\(899\) 337039. 0.0139085
\(900\) 0 0
\(901\) −6.49487e6 −0.266538
\(902\) −1.13175e7 −0.463163
\(903\) 7.57155e6 0.309005
\(904\) 1.30517e7 0.531184
\(905\) 0 0
\(906\) 7.67313e6 0.310565
\(907\) 2.66664e7 1.07633 0.538167 0.842838i \(-0.319117\pi\)
0.538167 + 0.842838i \(0.319117\pi\)
\(908\) −2.89036e6 −0.116342
\(909\) 1.72524e6 0.0692533
\(910\) 0 0
\(911\) 1.73286e7 0.691778 0.345889 0.938276i \(-0.387577\pi\)
0.345889 + 0.938276i \(0.387577\pi\)
\(912\) −6.97468e6 −0.277675
\(913\) −1.32935e7 −0.527793
\(914\) 5.91518e6 0.234208
\(915\) 0 0
\(916\) 3.30005e6 0.129952
\(917\) 5.34192e6 0.209785
\(918\) 1.59505e6 0.0624696
\(919\) −1.72198e7 −0.672573 −0.336286 0.941760i \(-0.609171\pi\)
−0.336286 + 0.941760i \(0.609171\pi\)
\(920\) 0 0
\(921\) 1.21348e7 0.471393
\(922\) −2.78058e7 −1.07723
\(923\) −1.40585e7 −0.543168
\(924\) −698677. −0.0269213
\(925\) 0 0
\(926\) −1.96848e7 −0.754402
\(927\) 530393. 0.0202721
\(928\) 3.93183e6 0.149874
\(929\) 5.47137e6 0.207997 0.103998 0.994577i \(-0.466836\pi\)
0.103998 + 0.994577i \(0.466836\pi\)
\(930\) 0 0
\(931\) 1.46494e6 0.0553919
\(932\) −4.71193e6 −0.177688
\(933\) 2.27326e7 0.854960
\(934\) 4.87525e7 1.82865
\(935\) 0 0
\(936\) 7.80014e6 0.291013
\(937\) −8.04892e6 −0.299494 −0.149747 0.988724i \(-0.547846\pi\)
−0.149747 + 0.988724i \(0.547846\pi\)
\(938\) 9.27072e6 0.344038
\(939\) 2.00755e7 0.743022
\(940\) 0 0
\(941\) −1.13721e7 −0.418665 −0.209333 0.977844i \(-0.567129\pi\)
−0.209333 + 0.977844i \(0.567129\pi\)
\(942\) −1.15842e7 −0.425341
\(943\) −2.21349e7 −0.810584
\(944\) −1.94379e7 −0.709938
\(945\) 0 0
\(946\) −4.26941e6 −0.155110
\(947\) −4.97017e7 −1.80093 −0.900463 0.434932i \(-0.856772\pi\)
−0.900463 + 0.434932i \(0.856772\pi\)
\(948\) −381267. −0.0137787
\(949\) 1.44199e7 0.519751
\(950\) 0 0
\(951\) −1.08817e6 −0.0390162
\(952\) 1.00464e7 0.359266
\(953\) −2.95809e7 −1.05507 −0.527533 0.849534i \(-0.676883\pi\)
−0.527533 + 0.849534i \(0.676883\pi\)
\(954\) 6.44438e6 0.229251
\(955\) 0 0
\(956\) −3.44385e6 −0.121871
\(957\) −2.29955e6 −0.0811639
\(958\) 2.42830e7 0.854849
\(959\) −4.95526e7 −1.73988
\(960\) 0 0
\(961\) −2.86037e7 −0.999110
\(962\) 1.40171e7 0.488337
\(963\) −1.03008e7 −0.357937
\(964\) 6.85161e6 0.237465
\(965\) 0 0
\(966\) 7.04654e6 0.242959
\(967\) −3.10475e7 −1.06773 −0.533863 0.845571i \(-0.679260\pi\)
−0.533863 + 0.845571i \(0.679260\pi\)
\(968\) 2.81950e6 0.0967128
\(969\) 3.54863e6 0.121409
\(970\) 0 0
\(971\) −1.89426e7 −0.644751 −0.322376 0.946612i \(-0.604481\pi\)
−0.322376 + 0.946612i \(0.604481\pi\)
\(972\) 306912. 0.0104195
\(973\) −3.72803e7 −1.26240
\(974\) 3.66113e7 1.23657
\(975\) 0 0
\(976\) −8.94489e6 −0.300573
\(977\) −5.49466e7 −1.84164 −0.920819 0.389989i \(-0.872479\pi\)
−0.920819 + 0.389989i \(0.872479\pi\)
\(978\) 1.94602e7 0.650580
\(979\) −8.43050e6 −0.281123
\(980\) 0 0
\(981\) 4.64010e6 0.153941
\(982\) −4.69072e6 −0.155225
\(983\) 4.72203e7 1.55864 0.779318 0.626628i \(-0.215565\pi\)
0.779318 + 0.626628i \(0.215565\pi\)
\(984\) 3.13128e7 1.03094
\(985\) 0 0
\(986\) 4.62022e6 0.151346
\(987\) −1.67739e7 −0.548076
\(988\) 2.42480e6 0.0790283
\(989\) −8.35018e6 −0.271459
\(990\) 0 0
\(991\) 4.48844e7 1.45182 0.725908 0.687792i \(-0.241420\pi\)
0.725908 + 0.687792i \(0.241420\pi\)
\(992\) 297198. 0.00958885
\(993\) −5.20632e6 −0.167555
\(994\) 1.79662e7 0.576753
\(995\) 0 0
\(996\) 5.13925e6 0.164154
\(997\) −5.08797e7 −1.62109 −0.810543 0.585679i \(-0.800828\pi\)
−0.810543 + 0.585679i \(0.800828\pi\)
\(998\) 3.69295e7 1.17367
\(999\) 3.94714e6 0.125132
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 825.6.a.j.1.1 3
5.4 even 2 165.6.a.a.1.3 3
15.14 odd 2 495.6.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.3 3 5.4 even 2
495.6.a.e.1.1 3 15.14 odd 2
825.6.a.j.1.1 3 1.1 even 1 trivial