Properties

Label 165.6.a.d.1.2
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.4633302691\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.35386\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.08127 q^{2} -9.00000 q^{3} -30.8308 q^{4} +25.0000 q^{5} -9.73147 q^{6} -139.508 q^{7} -67.9374 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.08127 q^{2} -9.00000 q^{3} -30.8308 q^{4} +25.0000 q^{5} -9.73147 q^{6} -139.508 q^{7} -67.9374 q^{8} +81.0000 q^{9} +27.0319 q^{10} -121.000 q^{11} +277.478 q^{12} -646.331 q^{13} -150.846 q^{14} -225.000 q^{15} +913.128 q^{16} -1378.32 q^{17} +87.5832 q^{18} +1908.90 q^{19} -770.771 q^{20} +1255.57 q^{21} -130.834 q^{22} +343.726 q^{23} +611.436 q^{24} +625.000 q^{25} -698.861 q^{26} -729.000 q^{27} +4301.15 q^{28} +53.5092 q^{29} -243.287 q^{30} +634.133 q^{31} +3161.34 q^{32} +1089.00 q^{33} -1490.34 q^{34} -3487.70 q^{35} -2497.30 q^{36} +11674.2 q^{37} +2064.04 q^{38} +5816.98 q^{39} -1698.43 q^{40} +18866.5 q^{41} +1357.62 q^{42} +13379.3 q^{43} +3730.53 q^{44} +2025.00 q^{45} +371.662 q^{46} -2252.95 q^{47} -8218.15 q^{48} +2655.48 q^{49} +675.796 q^{50} +12404.9 q^{51} +19926.9 q^{52} -8900.71 q^{53} -788.249 q^{54} -3025.00 q^{55} +9477.80 q^{56} -17180.1 q^{57} +57.8581 q^{58} -11276.7 q^{59} +6936.94 q^{60} +11852.4 q^{61} +685.672 q^{62} -11300.1 q^{63} -25801.8 q^{64} -16158.3 q^{65} +1177.51 q^{66} -57333.6 q^{67} +42494.7 q^{68} -3093.53 q^{69} -3771.16 q^{70} +31075.8 q^{71} -5502.93 q^{72} +56010.8 q^{73} +12623.0 q^{74} -5625.00 q^{75} -58853.0 q^{76} +16880.5 q^{77} +6289.75 q^{78} -883.036 q^{79} +22828.2 q^{80} +6561.00 q^{81} +20399.9 q^{82} +93986.3 q^{83} -38710.3 q^{84} -34457.9 q^{85} +14466.7 q^{86} -481.583 q^{87} +8220.42 q^{88} +21739.9 q^{89} +2189.58 q^{90} +90168.4 q^{91} -10597.4 q^{92} -5707.20 q^{93} -2436.06 q^{94} +47722.5 q^{95} -28452.0 q^{96} -158059. q^{97} +2871.30 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} - 363 q^{11} + 180 q^{12} - 546 q^{13} - 8 q^{14} - 675 q^{15} - 1360 q^{16} - 314 q^{17} + 162 q^{18} + 1808 q^{19} - 500 q^{20} - 1368 q^{21} - 242 q^{22} + 4288 q^{23} + 216 q^{24} + 1875 q^{25} + 812 q^{26} - 2187 q^{27} + 5888 q^{28} + 5582 q^{29} - 450 q^{30} + 6328 q^{31} - 736 q^{32} + 3267 q^{33} + 11596 q^{34} + 3800 q^{35} - 1620 q^{36} + 16866 q^{37} + 9584 q^{38} + 4914 q^{39} - 600 q^{40} + 23282 q^{41} + 72 q^{42} + 20572 q^{43} + 2420 q^{44} + 6075 q^{45} + 16592 q^{46} + 3432 q^{47} + 12240 q^{48} + 11531 q^{49} + 1250 q^{50} + 2826 q^{51} + 21816 q^{52} + 16138 q^{53} - 1458 q^{54} - 9075 q^{55} + 15648 q^{56} - 16272 q^{57} + 17460 q^{58} + 21972 q^{59} + 4500 q^{60} + 8322 q^{61} + 5056 q^{62} + 12312 q^{63} + 22208 q^{64} - 13650 q^{65} + 2178 q^{66} - 84332 q^{67} + 59832 q^{68} - 38592 q^{69} - 200 q^{70} + 50528 q^{71} - 1944 q^{72} - 53838 q^{73} + 79212 q^{74} - 16875 q^{75} - 52448 q^{76} - 18392 q^{77} - 7308 q^{78} + 6364 q^{79} - 34000 q^{80} + 19683 q^{81} - 68020 q^{82} + 96272 q^{83} - 52992 q^{84} - 7850 q^{85} + 143152 q^{86} - 50238 q^{87} + 2904 q^{88} - 38938 q^{89} + 4050 q^{90} + 104968 q^{91} + 24000 q^{92} - 56952 q^{93} + 49088 q^{94} + 45200 q^{95} + 6624 q^{96} - 103242 q^{97} + 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.08127 0.191144 0.0955720 0.995423i \(-0.469532\pi\)
0.0955720 + 0.995423i \(0.469532\pi\)
\(3\) −9.00000 −0.577350
\(4\) −30.8308 −0.963464
\(5\) 25.0000 0.447214
\(6\) −9.73147 −0.110357
\(7\) −139.508 −1.07610 −0.538052 0.842912i \(-0.680839\pi\)
−0.538052 + 0.842912i \(0.680839\pi\)
\(8\) −67.9374 −0.375304
\(9\) 81.0000 0.333333
\(10\) 27.0319 0.0854822
\(11\) −121.000 −0.301511
\(12\) 277.478 0.556256
\(13\) −646.331 −1.06071 −0.530355 0.847776i \(-0.677941\pi\)
−0.530355 + 0.847776i \(0.677941\pi\)
\(14\) −150.846 −0.205691
\(15\) −225.000 −0.258199
\(16\) 913.128 0.891727
\(17\) −1378.32 −1.15672 −0.578358 0.815783i \(-0.696306\pi\)
−0.578358 + 0.815783i \(0.696306\pi\)
\(18\) 87.5832 0.0637147
\(19\) 1908.90 1.21311 0.606553 0.795043i \(-0.292552\pi\)
0.606553 + 0.795043i \(0.292552\pi\)
\(20\) −770.771 −0.430874
\(21\) 1255.57 0.621289
\(22\) −130.834 −0.0576321
\(23\) 343.726 0.135485 0.0677427 0.997703i \(-0.478420\pi\)
0.0677427 + 0.997703i \(0.478420\pi\)
\(24\) 611.436 0.216682
\(25\) 625.000 0.200000
\(26\) −698.861 −0.202748
\(27\) −729.000 −0.192450
\(28\) 4301.15 1.03679
\(29\) 53.5092 0.0118150 0.00590750 0.999983i \(-0.498120\pi\)
0.00590750 + 0.999983i \(0.498120\pi\)
\(30\) −243.287 −0.0493532
\(31\) 634.133 0.118516 0.0592579 0.998243i \(-0.481127\pi\)
0.0592579 + 0.998243i \(0.481127\pi\)
\(32\) 3161.34 0.545753
\(33\) 1089.00 0.174078
\(34\) −1490.34 −0.221099
\(35\) −3487.70 −0.481248
\(36\) −2497.30 −0.321155
\(37\) 11674.2 1.40192 0.700960 0.713201i \(-0.252755\pi\)
0.700960 + 0.713201i \(0.252755\pi\)
\(38\) 2064.04 0.231878
\(39\) 5816.98 0.612401
\(40\) −1698.43 −0.167841
\(41\) 18866.5 1.75280 0.876399 0.481585i \(-0.159939\pi\)
0.876399 + 0.481585i \(0.159939\pi\)
\(42\) 1357.62 0.118756
\(43\) 13379.3 1.10347 0.551736 0.834018i \(-0.313965\pi\)
0.551736 + 0.834018i \(0.313965\pi\)
\(44\) 3730.53 0.290495
\(45\) 2025.00 0.149071
\(46\) 371.662 0.0258972
\(47\) −2252.95 −0.148767 −0.0743835 0.997230i \(-0.523699\pi\)
−0.0743835 + 0.997230i \(0.523699\pi\)
\(48\) −8218.15 −0.514839
\(49\) 2655.48 0.157998
\(50\) 675.796 0.0382288
\(51\) 12404.9 0.667830
\(52\) 19926.9 1.02196
\(53\) −8900.71 −0.435246 −0.217623 0.976033i \(-0.569830\pi\)
−0.217623 + 0.976033i \(0.569830\pi\)
\(54\) −788.249 −0.0367857
\(55\) −3025.00 −0.134840
\(56\) 9477.80 0.403866
\(57\) −17180.1 −0.700387
\(58\) 57.8581 0.00225837
\(59\) −11276.7 −0.421746 −0.210873 0.977514i \(-0.567631\pi\)
−0.210873 + 0.977514i \(0.567631\pi\)
\(60\) 6936.94 0.248765
\(61\) 11852.4 0.407834 0.203917 0.978988i \(-0.434633\pi\)
0.203917 + 0.978988i \(0.434633\pi\)
\(62\) 685.672 0.0226536
\(63\) −11300.1 −0.358701
\(64\) −25801.8 −0.787409
\(65\) −16158.3 −0.474364
\(66\) 1177.51 0.0332739
\(67\) −57333.6 −1.56035 −0.780175 0.625561i \(-0.784870\pi\)
−0.780175 + 0.625561i \(0.784870\pi\)
\(68\) 42494.7 1.11445
\(69\) −3093.53 −0.0782226
\(70\) −3771.16 −0.0919877
\(71\) 31075.8 0.731605 0.365803 0.930692i \(-0.380794\pi\)
0.365803 + 0.930692i \(0.380794\pi\)
\(72\) −5502.93 −0.125101
\(73\) 56010.8 1.23017 0.615084 0.788462i \(-0.289122\pi\)
0.615084 + 0.788462i \(0.289122\pi\)
\(74\) 12623.0 0.267969
\(75\) −5625.00 −0.115470
\(76\) −58853.0 −1.16878
\(77\) 16880.5 0.324457
\(78\) 6289.75 0.117057
\(79\) −883.036 −0.0159188 −0.00795941 0.999968i \(-0.502534\pi\)
−0.00795941 + 0.999968i \(0.502534\pi\)
\(80\) 22828.2 0.398792
\(81\) 6561.00 0.111111
\(82\) 20399.9 0.335037
\(83\) 93986.3 1.49751 0.748754 0.662848i \(-0.230652\pi\)
0.748754 + 0.662848i \(0.230652\pi\)
\(84\) −38710.3 −0.598589
\(85\) −34457.9 −0.517299
\(86\) 14466.7 0.210922
\(87\) −481.583 −0.00682139
\(88\) 8220.42 0.113159
\(89\) 21739.9 0.290926 0.145463 0.989364i \(-0.453533\pi\)
0.145463 + 0.989364i \(0.453533\pi\)
\(90\) 2189.58 0.0284941
\(91\) 90168.4 1.14143
\(92\) −10597.4 −0.130535
\(93\) −5707.20 −0.0684251
\(94\) −2436.06 −0.0284359
\(95\) 47722.5 0.542518
\(96\) −28452.0 −0.315090
\(97\) −158059. −1.70565 −0.852825 0.522197i \(-0.825112\pi\)
−0.852825 + 0.522197i \(0.825112\pi\)
\(98\) 2871.30 0.0302004
\(99\) −9801.00 −0.100504
\(100\) −19269.3 −0.192693
\(101\) −156968. −1.53111 −0.765556 0.643369i \(-0.777536\pi\)
−0.765556 + 0.643369i \(0.777536\pi\)
\(102\) 13413.0 0.127652
\(103\) 169126. 1.57078 0.785391 0.618999i \(-0.212462\pi\)
0.785391 + 0.618999i \(0.212462\pi\)
\(104\) 43910.0 0.398089
\(105\) 31389.3 0.277849
\(106\) −9624.11 −0.0831948
\(107\) −13538.5 −0.114317 −0.0571587 0.998365i \(-0.518204\pi\)
−0.0571587 + 0.998365i \(0.518204\pi\)
\(108\) 22475.7 0.185419
\(109\) −74736.4 −0.602513 −0.301256 0.953543i \(-0.597406\pi\)
−0.301256 + 0.953543i \(0.597406\pi\)
\(110\) −3270.85 −0.0257739
\(111\) −105068. −0.809399
\(112\) −127389. −0.959590
\(113\) −205914. −1.51701 −0.758507 0.651665i \(-0.774071\pi\)
−0.758507 + 0.651665i \(0.774071\pi\)
\(114\) −18576.4 −0.133875
\(115\) 8593.15 0.0605909
\(116\) −1649.73 −0.0113833
\(117\) −52352.8 −0.353570
\(118\) −12193.2 −0.0806142
\(119\) 192286. 1.24475
\(120\) 15285.9 0.0969032
\(121\) 14641.0 0.0909091
\(122\) 12815.7 0.0779550
\(123\) −169799. −1.01198
\(124\) −19550.9 −0.114186
\(125\) 15625.0 0.0894427
\(126\) −12218.6 −0.0685636
\(127\) −214682. −1.18110 −0.590550 0.807001i \(-0.701089\pi\)
−0.590550 + 0.807001i \(0.701089\pi\)
\(128\) −129062. −0.696261
\(129\) −120414. −0.637090
\(130\) −17471.5 −0.0906719
\(131\) 253067. 1.28842 0.644209 0.764849i \(-0.277187\pi\)
0.644209 + 0.764849i \(0.277187\pi\)
\(132\) −33574.8 −0.167718
\(133\) −266307. −1.30543
\(134\) −61993.3 −0.298252
\(135\) −18225.0 −0.0860663
\(136\) 93639.2 0.434121
\(137\) 239398. 1.08973 0.544865 0.838524i \(-0.316581\pi\)
0.544865 + 0.838524i \(0.316581\pi\)
\(138\) −3344.96 −0.0149518
\(139\) −166398. −0.730484 −0.365242 0.930913i \(-0.619014\pi\)
−0.365242 + 0.930913i \(0.619014\pi\)
\(140\) 107529. 0.463665
\(141\) 20276.5 0.0858907
\(142\) 33601.5 0.139842
\(143\) 78206.1 0.319816
\(144\) 73963.4 0.297242
\(145\) 1337.73 0.00528383
\(146\) 60563.0 0.235139
\(147\) −23899.3 −0.0912203
\(148\) −359926. −1.35070
\(149\) 295512. 1.09046 0.545229 0.838287i \(-0.316443\pi\)
0.545229 + 0.838287i \(0.316443\pi\)
\(150\) −6082.17 −0.0220714
\(151\) 428373. 1.52890 0.764452 0.644681i \(-0.223010\pi\)
0.764452 + 0.644681i \(0.223010\pi\)
\(152\) −129686. −0.455284
\(153\) −111644. −0.385572
\(154\) 18252.4 0.0620181
\(155\) 15853.3 0.0530019
\(156\) −179342. −0.590027
\(157\) −289003. −0.935735 −0.467867 0.883799i \(-0.654978\pi\)
−0.467867 + 0.883799i \(0.654978\pi\)
\(158\) −954.804 −0.00304279
\(159\) 80106.4 0.251290
\(160\) 79033.4 0.244068
\(161\) −47952.5 −0.145796
\(162\) 7094.24 0.0212382
\(163\) 288223. 0.849688 0.424844 0.905267i \(-0.360329\pi\)
0.424844 + 0.905267i \(0.360329\pi\)
\(164\) −581671. −1.68876
\(165\) 27225.0 0.0778499
\(166\) 101625. 0.286240
\(167\) −180912. −0.501969 −0.250985 0.967991i \(-0.580754\pi\)
−0.250985 + 0.967991i \(0.580754\pi\)
\(168\) −85300.2 −0.233172
\(169\) 46451.1 0.125106
\(170\) −37258.5 −0.0988787
\(171\) 154621. 0.404369
\(172\) −412495. −1.06316
\(173\) 218579. 0.555255 0.277628 0.960689i \(-0.410452\pi\)
0.277628 + 0.960689i \(0.410452\pi\)
\(174\) −520.723 −0.00130387
\(175\) −87192.5 −0.215221
\(176\) −110489. −0.268866
\(177\) 101490. 0.243495
\(178\) 23506.8 0.0556089
\(179\) 310916. 0.725287 0.362643 0.931928i \(-0.381874\pi\)
0.362643 + 0.931928i \(0.381874\pi\)
\(180\) −62432.5 −0.143625
\(181\) −423637. −0.961163 −0.480582 0.876950i \(-0.659574\pi\)
−0.480582 + 0.876950i \(0.659574\pi\)
\(182\) 97496.7 0.218178
\(183\) −106672. −0.235463
\(184\) −23351.8 −0.0508483
\(185\) 291855. 0.626957
\(186\) −6171.04 −0.0130791
\(187\) 166776. 0.348763
\(188\) 69460.3 0.143332
\(189\) 101701. 0.207096
\(190\) 51601.1 0.103699
\(191\) 861089. 1.70791 0.853954 0.520348i \(-0.174198\pi\)
0.853954 + 0.520348i \(0.174198\pi\)
\(192\) 232216. 0.454611
\(193\) 430583. 0.832077 0.416038 0.909347i \(-0.363418\pi\)
0.416038 + 0.909347i \(0.363418\pi\)
\(194\) −170905. −0.326025
\(195\) 145425. 0.273874
\(196\) −81870.6 −0.152226
\(197\) −184755. −0.339181 −0.169590 0.985515i \(-0.554244\pi\)
−0.169590 + 0.985515i \(0.554244\pi\)
\(198\) −10597.6 −0.0192107
\(199\) 322993. 0.578177 0.289089 0.957302i \(-0.406648\pi\)
0.289089 + 0.957302i \(0.406648\pi\)
\(200\) −42460.9 −0.0750609
\(201\) 516002. 0.900869
\(202\) −169725. −0.292663
\(203\) −7464.96 −0.0127142
\(204\) −382452. −0.643431
\(205\) 471663. 0.783875
\(206\) 182871. 0.300246
\(207\) 27841.8 0.0451618
\(208\) −590183. −0.945864
\(209\) −230977. −0.365765
\(210\) 33940.4 0.0531091
\(211\) −394106. −0.609406 −0.304703 0.952447i \(-0.598557\pi\)
−0.304703 + 0.952447i \(0.598557\pi\)
\(212\) 274417. 0.419344
\(213\) −279682. −0.422393
\(214\) −14638.9 −0.0218511
\(215\) 334482. 0.493488
\(216\) 49526.3 0.0722274
\(217\) −88466.6 −0.127535
\(218\) −80810.6 −0.115167
\(219\) −504097. −0.710238
\(220\) 93263.3 0.129913
\(221\) 890849. 1.22694
\(222\) −113607. −0.154712
\(223\) 445038. 0.599287 0.299643 0.954051i \(-0.403132\pi\)
0.299643 + 0.954051i \(0.403132\pi\)
\(224\) −441032. −0.587286
\(225\) 50625.0 0.0666667
\(226\) −222649. −0.289968
\(227\) 1.15887e6 1.49270 0.746349 0.665555i \(-0.231805\pi\)
0.746349 + 0.665555i \(0.231805\pi\)
\(228\) 529677. 0.674798
\(229\) 885853. 1.11628 0.558140 0.829747i \(-0.311515\pi\)
0.558140 + 0.829747i \(0.311515\pi\)
\(230\) 9291.55 0.0115816
\(231\) −151924. −0.187326
\(232\) −3635.27 −0.00443422
\(233\) 750392. 0.905520 0.452760 0.891632i \(-0.350439\pi\)
0.452760 + 0.891632i \(0.350439\pi\)
\(234\) −56607.8 −0.0675828
\(235\) −56323.7 −0.0665306
\(236\) 347669. 0.406337
\(237\) 7947.32 0.00919073
\(238\) 207914. 0.237926
\(239\) 617415. 0.699169 0.349585 0.936905i \(-0.386323\pi\)
0.349585 + 0.936905i \(0.386323\pi\)
\(240\) −205454. −0.230243
\(241\) 1.20275e6 1.33393 0.666966 0.745088i \(-0.267593\pi\)
0.666966 + 0.745088i \(0.267593\pi\)
\(242\) 15830.9 0.0173767
\(243\) −59049.0 −0.0641500
\(244\) −365421. −0.392933
\(245\) 66386.9 0.0706589
\(246\) −183599. −0.193434
\(247\) −1.23378e6 −1.28675
\(248\) −43081.3 −0.0444795
\(249\) −845877. −0.864587
\(250\) 16894.9 0.0170964
\(251\) 1.55775e6 1.56068 0.780338 0.625358i \(-0.215047\pi\)
0.780338 + 0.625358i \(0.215047\pi\)
\(252\) 348393. 0.345596
\(253\) −41590.8 −0.0408504
\(254\) −232130. −0.225760
\(255\) 310121. 0.298663
\(256\) 686107. 0.654323
\(257\) 92156.0 0.0870344 0.0435172 0.999053i \(-0.486144\pi\)
0.0435172 + 0.999053i \(0.486144\pi\)
\(258\) −130200. −0.121776
\(259\) −1.62865e6 −1.50861
\(260\) 498173. 0.457033
\(261\) 4334.24 0.00393833
\(262\) 273635. 0.246274
\(263\) −862122. −0.768563 −0.384281 0.923216i \(-0.625551\pi\)
−0.384281 + 0.923216i \(0.625551\pi\)
\(264\) −73983.8 −0.0653321
\(265\) −222518. −0.194648
\(266\) −287950. −0.249525
\(267\) −195659. −0.167966
\(268\) 1.76764e6 1.50334
\(269\) −1.25052e6 −1.05368 −0.526841 0.849964i \(-0.676624\pi\)
−0.526841 + 0.849964i \(0.676624\pi\)
\(270\) −19706.2 −0.0164511
\(271\) 844654. 0.698644 0.349322 0.937003i \(-0.386412\pi\)
0.349322 + 0.937003i \(0.386412\pi\)
\(272\) −1.25858e6 −1.03147
\(273\) −811515. −0.659007
\(274\) 258855. 0.208295
\(275\) −75625.0 −0.0603023
\(276\) 95376.2 0.0753646
\(277\) −1.52990e6 −1.19802 −0.599008 0.800743i \(-0.704438\pi\)
−0.599008 + 0.800743i \(0.704438\pi\)
\(278\) −179922. −0.139628
\(279\) 51364.8 0.0395053
\(280\) 236945. 0.180615
\(281\) 1.13284e6 0.855858 0.427929 0.903812i \(-0.359243\pi\)
0.427929 + 0.903812i \(0.359243\pi\)
\(282\) 21924.5 0.0164175
\(283\) −411138. −0.305156 −0.152578 0.988291i \(-0.548758\pi\)
−0.152578 + 0.988291i \(0.548758\pi\)
\(284\) −958094. −0.704875
\(285\) −429502. −0.313223
\(286\) 84562.2 0.0611310
\(287\) −2.63203e6 −1.88619
\(288\) 256068. 0.181918
\(289\) 479901. 0.337993
\(290\) 1446.45 0.00100997
\(291\) 1.42253e6 0.984757
\(292\) −1.72686e6 −1.18522
\(293\) 314013. 0.213687 0.106843 0.994276i \(-0.465926\pi\)
0.106843 + 0.994276i \(0.465926\pi\)
\(294\) −25841.7 −0.0174362
\(295\) −281917. −0.188610
\(296\) −793115. −0.526147
\(297\) 88209.0 0.0580259
\(298\) 319529. 0.208435
\(299\) −222161. −0.143711
\(300\) 173424. 0.111251
\(301\) −1.86652e6 −1.18745
\(302\) 463189. 0.292241
\(303\) 1.41271e6 0.883988
\(304\) 1.74307e6 1.08176
\(305\) 296311. 0.182389
\(306\) −120717. −0.0736998
\(307\) 797600. 0.482991 0.241496 0.970402i \(-0.422362\pi\)
0.241496 + 0.970402i \(0.422362\pi\)
\(308\) −520439. −0.312603
\(309\) −1.52213e6 −0.906892
\(310\) 17141.8 0.0101310
\(311\) −618368. −0.362532 −0.181266 0.983434i \(-0.558020\pi\)
−0.181266 + 0.983434i \(0.558020\pi\)
\(312\) −395190. −0.229837
\(313\) −2.60175e6 −1.50108 −0.750541 0.660824i \(-0.770207\pi\)
−0.750541 + 0.660824i \(0.770207\pi\)
\(314\) −312491. −0.178860
\(315\) −282504. −0.160416
\(316\) 27224.7 0.0153372
\(317\) −729701. −0.407846 −0.203923 0.978987i \(-0.565369\pi\)
−0.203923 + 0.978987i \(0.565369\pi\)
\(318\) 86617.0 0.0480325
\(319\) −6474.61 −0.00356235
\(320\) −645046. −0.352140
\(321\) 121847. 0.0660011
\(322\) −51849.8 −0.0278681
\(323\) −2.63107e6 −1.40322
\(324\) −202281. −0.107052
\(325\) −403957. −0.212142
\(326\) 311648. 0.162413
\(327\) 672628. 0.347861
\(328\) −1.28174e6 −0.657833
\(329\) 314304. 0.160089
\(330\) 29437.7 0.0148805
\(331\) 2.87129e6 1.44048 0.720239 0.693726i \(-0.244032\pi\)
0.720239 + 0.693726i \(0.244032\pi\)
\(332\) −2.89768e6 −1.44280
\(333\) 945611. 0.467307
\(334\) −195616. −0.0959484
\(335\) −1.43334e6 −0.697810
\(336\) 1.14650e6 0.554020
\(337\) 2.16558e6 1.03872 0.519361 0.854555i \(-0.326170\pi\)
0.519361 + 0.854555i \(0.326170\pi\)
\(338\) 50226.3 0.0239133
\(339\) 1.85323e6 0.875849
\(340\) 1.06237e6 0.498399
\(341\) −76730.1 −0.0357338
\(342\) 167187. 0.0772927
\(343\) 1.97425e6 0.906081
\(344\) −908953. −0.414138
\(345\) −77338.3 −0.0349822
\(346\) 236343. 0.106134
\(347\) 392930. 0.175183 0.0875913 0.996156i \(-0.472083\pi\)
0.0875913 + 0.996156i \(0.472083\pi\)
\(348\) 14847.6 0.00657216
\(349\) 247840. 0.108920 0.0544601 0.998516i \(-0.482656\pi\)
0.0544601 + 0.998516i \(0.482656\pi\)
\(350\) −94279.0 −0.0411381
\(351\) 471175. 0.204134
\(352\) −382522. −0.164551
\(353\) −2.90847e6 −1.24230 −0.621151 0.783691i \(-0.713335\pi\)
−0.621151 + 0.783691i \(0.713335\pi\)
\(354\) 109738. 0.0465426
\(355\) 776896. 0.327184
\(356\) −670261. −0.280297
\(357\) −1.73058e6 −0.718655
\(358\) 336185. 0.138634
\(359\) 758403. 0.310573 0.155287 0.987869i \(-0.450370\pi\)
0.155287 + 0.987869i \(0.450370\pi\)
\(360\) −137573. −0.0559471
\(361\) 1.16779e6 0.471627
\(362\) −458067. −0.183721
\(363\) −131769. −0.0524864
\(364\) −2.77997e6 −1.09973
\(365\) 1.40027e6 0.550148
\(366\) −115342. −0.0450073
\(367\) −2.29924e6 −0.891085 −0.445542 0.895261i \(-0.646989\pi\)
−0.445542 + 0.895261i \(0.646989\pi\)
\(368\) 313866. 0.120816
\(369\) 1.52819e6 0.584266
\(370\) 315575. 0.119839
\(371\) 1.24172e6 0.468370
\(372\) 175958. 0.0659251
\(373\) −19163.9 −0.00713203 −0.00356601 0.999994i \(-0.501135\pi\)
−0.00356601 + 0.999994i \(0.501135\pi\)
\(374\) 180331. 0.0666640
\(375\) −140625. −0.0516398
\(376\) 153059. 0.0558329
\(377\) −34584.7 −0.0125323
\(378\) 109967. 0.0395852
\(379\) 346535. 0.123922 0.0619610 0.998079i \(-0.480265\pi\)
0.0619610 + 0.998079i \(0.480265\pi\)
\(380\) −1.47132e6 −0.522696
\(381\) 1.93214e6 0.681909
\(382\) 931073. 0.326457
\(383\) −5.25066e6 −1.82901 −0.914507 0.404571i \(-0.867421\pi\)
−0.914507 + 0.404571i \(0.867421\pi\)
\(384\) 1.16155e6 0.401987
\(385\) 422012. 0.145102
\(386\) 465578. 0.159047
\(387\) 1.08372e6 0.367824
\(388\) 4.87309e6 1.64333
\(389\) −4.34474e6 −1.45576 −0.727880 0.685704i \(-0.759494\pi\)
−0.727880 + 0.685704i \(0.759494\pi\)
\(390\) 157244. 0.0523494
\(391\) −473763. −0.156718
\(392\) −180406. −0.0592974
\(393\) −2.27760e6 −0.743869
\(394\) −199771. −0.0648324
\(395\) −22075.9 −0.00711911
\(396\) 302173. 0.0968318
\(397\) 3.90748e6 1.24429 0.622144 0.782903i \(-0.286262\pi\)
0.622144 + 0.782903i \(0.286262\pi\)
\(398\) 349244. 0.110515
\(399\) 2.39676e6 0.753689
\(400\) 570705. 0.178345
\(401\) −3.05315e6 −0.948171 −0.474085 0.880479i \(-0.657221\pi\)
−0.474085 + 0.880479i \(0.657221\pi\)
\(402\) 557940. 0.172196
\(403\) −409860. −0.125711
\(404\) 4.83945e6 1.47517
\(405\) 164025. 0.0496904
\(406\) −8071.67 −0.00243023
\(407\) −1.41258e6 −0.422695
\(408\) −842753. −0.250640
\(409\) −5.46050e6 −1.61408 −0.807038 0.590499i \(-0.798931\pi\)
−0.807038 + 0.590499i \(0.798931\pi\)
\(410\) 509997. 0.149833
\(411\) −2.15458e6 −0.629156
\(412\) −5.21428e6 −1.51339
\(413\) 1.57318e6 0.453842
\(414\) 30104.6 0.00863241
\(415\) 2.34966e6 0.669706
\(416\) −2.04327e6 −0.578886
\(417\) 1.49758e6 0.421745
\(418\) −249749. −0.0699139
\(419\) 3.02804e6 0.842609 0.421305 0.906919i \(-0.361572\pi\)
0.421305 + 0.906919i \(0.361572\pi\)
\(420\) −967759. −0.267697
\(421\) −3.31510e6 −0.911572 −0.455786 0.890089i \(-0.650642\pi\)
−0.455786 + 0.890089i \(0.650642\pi\)
\(422\) −426136. −0.116484
\(423\) −182489. −0.0495890
\(424\) 604691. 0.163350
\(425\) −861448. −0.231343
\(426\) −302413. −0.0807378
\(427\) −1.65351e6 −0.438871
\(428\) 417404. 0.110141
\(429\) −703855. −0.184646
\(430\) 361667. 0.0943273
\(431\) −3.02139e6 −0.783455 −0.391728 0.920081i \(-0.628122\pi\)
−0.391728 + 0.920081i \(0.628122\pi\)
\(432\) −665670. −0.171613
\(433\) −739663. −0.189589 −0.0947947 0.995497i \(-0.530219\pi\)
−0.0947947 + 0.995497i \(0.530219\pi\)
\(434\) −95656.7 −0.0243776
\(435\) −12039.6 −0.00305062
\(436\) 2.30419e6 0.580499
\(437\) 656138. 0.164358
\(438\) −545067. −0.135758
\(439\) 89371.4 0.0221329 0.0110664 0.999939i \(-0.496477\pi\)
0.0110664 + 0.999939i \(0.496477\pi\)
\(440\) 205511. 0.0506060
\(441\) 215094. 0.0526661
\(442\) 963252. 0.234522
\(443\) 5.06164e6 1.22541 0.612706 0.790311i \(-0.290081\pi\)
0.612706 + 0.790311i \(0.290081\pi\)
\(444\) 3.23933e6 0.779826
\(445\) 543498. 0.130106
\(446\) 481208. 0.114550
\(447\) −2.65961e6 −0.629577
\(448\) 3.59956e6 0.847334
\(449\) 6.64598e6 1.55576 0.777881 0.628411i \(-0.216294\pi\)
0.777881 + 0.628411i \(0.216294\pi\)
\(450\) 54739.5 0.0127429
\(451\) −2.28285e6 −0.528489
\(452\) 6.34850e6 1.46159
\(453\) −3.85536e6 −0.882713
\(454\) 1.25306e6 0.285320
\(455\) 2.25421e6 0.510465
\(456\) 1.16717e6 0.262858
\(457\) −176313. −0.0394905 −0.0197453 0.999805i \(-0.506286\pi\)
−0.0197453 + 0.999805i \(0.506286\pi\)
\(458\) 957850. 0.213370
\(459\) 1.00479e6 0.222610
\(460\) −264934. −0.0583772
\(461\) 8.43740e6 1.84908 0.924542 0.381081i \(-0.124448\pi\)
0.924542 + 0.381081i \(0.124448\pi\)
\(462\) −164272. −0.0358062
\(463\) 5.65739e6 1.22649 0.613244 0.789893i \(-0.289864\pi\)
0.613244 + 0.789893i \(0.289864\pi\)
\(464\) 48860.8 0.0105357
\(465\) −142680. −0.0306006
\(466\) 811379. 0.173085
\(467\) −2.54278e6 −0.539532 −0.269766 0.962926i \(-0.586946\pi\)
−0.269766 + 0.962926i \(0.586946\pi\)
\(468\) 1.61408e6 0.340652
\(469\) 7.99850e6 1.67910
\(470\) −60901.4 −0.0127169
\(471\) 2.60102e6 0.540247
\(472\) 766107. 0.158283
\(473\) −1.61889e6 −0.332710
\(474\) 8593.23 0.00175675
\(475\) 1.19306e6 0.242621
\(476\) −5.92835e6 −1.19927
\(477\) −720958. −0.145082
\(478\) 667594. 0.133642
\(479\) 6.89136e6 1.37235 0.686177 0.727434i \(-0.259287\pi\)
0.686177 + 0.727434i \(0.259287\pi\)
\(480\) −711301. −0.140913
\(481\) −7.54540e6 −1.48703
\(482\) 1.30051e6 0.254973
\(483\) 431573. 0.0841756
\(484\) −451394. −0.0875876
\(485\) −3.95147e6 −0.762790
\(486\) −63848.2 −0.0122619
\(487\) 7.89949e6 1.50930 0.754652 0.656125i \(-0.227806\pi\)
0.754652 + 0.656125i \(0.227806\pi\)
\(488\) −805223. −0.153062
\(489\) −2.59401e6 −0.490568
\(490\) 71782.4 0.0135060
\(491\) −2.29000e6 −0.428679 −0.214340 0.976759i \(-0.568760\pi\)
−0.214340 + 0.976759i \(0.568760\pi\)
\(492\) 5.23504e6 0.975005
\(493\) −73752.6 −0.0136666
\(494\) −1.33406e6 −0.245955
\(495\) −245025. −0.0449467
\(496\) 579045. 0.105684
\(497\) −4.33533e6 −0.787283
\(498\) −914625. −0.165261
\(499\) −5.96310e6 −1.07206 −0.536032 0.844198i \(-0.680077\pi\)
−0.536032 + 0.844198i \(0.680077\pi\)
\(500\) −481732. −0.0861748
\(501\) 1.62821e6 0.289812
\(502\) 1.68435e6 0.298314
\(503\) 2.17453e6 0.383218 0.191609 0.981471i \(-0.438629\pi\)
0.191609 + 0.981471i \(0.438629\pi\)
\(504\) 767702. 0.134622
\(505\) −3.92420e6 −0.684734
\(506\) −44971.1 −0.00780831
\(507\) −418060. −0.0722301
\(508\) 6.61883e6 1.13795
\(509\) −3.33531e6 −0.570613 −0.285307 0.958436i \(-0.592095\pi\)
−0.285307 + 0.958436i \(0.592095\pi\)
\(510\) 335326. 0.0570876
\(511\) −7.81395e6 −1.32379
\(512\) 4.87184e6 0.821331
\(513\) −1.39159e6 −0.233462
\(514\) 99645.9 0.0166361
\(515\) 4.22814e6 0.702475
\(516\) 3.71245e6 0.613814
\(517\) 272607. 0.0448549
\(518\) −1.76101e6 −0.288362
\(519\) −1.96721e6 −0.320577
\(520\) 1.09775e6 0.178031
\(521\) 9.81327e6 1.58387 0.791935 0.610605i \(-0.209074\pi\)
0.791935 + 0.610605i \(0.209074\pi\)
\(522\) 4686.51 0.000752789 0
\(523\) 7.96466e6 1.27325 0.636624 0.771175i \(-0.280330\pi\)
0.636624 + 0.771175i \(0.280330\pi\)
\(524\) −7.80227e6 −1.24134
\(525\) 784732. 0.124258
\(526\) −932190. −0.146906
\(527\) −874036. −0.137089
\(528\) 994397. 0.155230
\(529\) −6.31820e6 −0.981644
\(530\) −240603. −0.0372058
\(531\) −913410. −0.140582
\(532\) 8.21046e6 1.25773
\(533\) −1.21940e7 −1.85921
\(534\) −211561. −0.0321058
\(535\) −338463. −0.0511243
\(536\) 3.89509e6 0.585607
\(537\) −2.79824e6 −0.418745
\(538\) −1.35215e6 −0.201405
\(539\) −321313. −0.0476383
\(540\) 561892. 0.0829218
\(541\) −5.50616e6 −0.808827 −0.404413 0.914576i \(-0.632524\pi\)
−0.404413 + 0.914576i \(0.632524\pi\)
\(542\) 913303. 0.133542
\(543\) 3.81273e6 0.554928
\(544\) −4.35733e6 −0.631281
\(545\) −1.86841e6 −0.269452
\(546\) −877470. −0.125965
\(547\) 2.92918e6 0.418580 0.209290 0.977854i \(-0.432885\pi\)
0.209290 + 0.977854i \(0.432885\pi\)
\(548\) −7.38084e6 −1.04992
\(549\) 960047. 0.135945
\(550\) −81771.4 −0.0115264
\(551\) 102144. 0.0143328
\(552\) 210167. 0.0293573
\(553\) 123191. 0.0171303
\(554\) −1.65424e6 −0.228994
\(555\) −2.62670e6 −0.361974
\(556\) 5.13018e6 0.703795
\(557\) 4.05453e6 0.553736 0.276868 0.960908i \(-0.410704\pi\)
0.276868 + 0.960908i \(0.410704\pi\)
\(558\) 55539.4 0.00755120
\(559\) −8.64745e6 −1.17047
\(560\) −3.18472e6 −0.429142
\(561\) −1.50099e6 −0.201358
\(562\) 1.22491e6 0.163592
\(563\) −1.65845e6 −0.220512 −0.110256 0.993903i \(-0.535167\pi\)
−0.110256 + 0.993903i \(0.535167\pi\)
\(564\) −625143. −0.0827526
\(565\) −5.14785e6 −0.678430
\(566\) −444553. −0.0583288
\(567\) −915312. −0.119567
\(568\) −2.11121e6 −0.274575
\(569\) 1.05475e7 1.36575 0.682873 0.730537i \(-0.260730\pi\)
0.682873 + 0.730537i \(0.260730\pi\)
\(570\) −464410. −0.0598707
\(571\) −1.46769e7 −1.88384 −0.941918 0.335843i \(-0.890979\pi\)
−0.941918 + 0.335843i \(0.890979\pi\)
\(572\) −2.41116e6 −0.308131
\(573\) −7.74980e6 −0.986062
\(574\) −2.84595e6 −0.360534
\(575\) 214829. 0.0270971
\(576\) −2.08995e6 −0.262470
\(577\) −8.25271e6 −1.03195 −0.515973 0.856605i \(-0.672570\pi\)
−0.515973 + 0.856605i \(0.672570\pi\)
\(578\) 518905. 0.0646053
\(579\) −3.87525e6 −0.480400
\(580\) −41243.3 −0.00509078
\(581\) −1.31118e7 −1.61147
\(582\) 1.53815e6 0.188230
\(583\) 1.07699e6 0.131232
\(584\) −3.80522e6 −0.461687
\(585\) −1.30882e6 −0.158121
\(586\) 339534. 0.0408450
\(587\) 1.01063e7 1.21059 0.605296 0.796000i \(-0.293055\pi\)
0.605296 + 0.796000i \(0.293055\pi\)
\(588\) 736835. 0.0878875
\(589\) 1.21050e6 0.143772
\(590\) −304829. −0.0360517
\(591\) 1.66280e6 0.195826
\(592\) 1.06600e7 1.25013
\(593\) 1.48922e7 1.73909 0.869543 0.493857i \(-0.164413\pi\)
0.869543 + 0.493857i \(0.164413\pi\)
\(594\) 95378.1 0.0110913
\(595\) 4.80716e6 0.556667
\(596\) −9.11088e6 −1.05062
\(597\) −2.90694e6 −0.333811
\(598\) −240217. −0.0274695
\(599\) 1.59013e7 1.81078 0.905390 0.424580i \(-0.139578\pi\)
0.905390 + 0.424580i \(0.139578\pi\)
\(600\) 382148. 0.0433364
\(601\) −2.84828e6 −0.321660 −0.160830 0.986982i \(-0.551417\pi\)
−0.160830 + 0.986982i \(0.551417\pi\)
\(602\) −2.01822e6 −0.226974
\(603\) −4.64402e6 −0.520117
\(604\) −1.32071e7 −1.47304
\(605\) 366025. 0.0406558
\(606\) 1.52753e6 0.168969
\(607\) 1.32016e6 0.145430 0.0727149 0.997353i \(-0.476834\pi\)
0.0727149 + 0.997353i \(0.476834\pi\)
\(608\) 6.03467e6 0.662056
\(609\) 67184.6 0.00734052
\(610\) 320393. 0.0348625
\(611\) 1.45615e6 0.157799
\(612\) 3.44207e6 0.371485
\(613\) −978527. −0.105177 −0.0525886 0.998616i \(-0.516747\pi\)
−0.0525886 + 0.998616i \(0.516747\pi\)
\(614\) 862424. 0.0923209
\(615\) −4.24497e6 −0.452571
\(616\) −1.14681e6 −0.121770
\(617\) 1.29643e7 1.37100 0.685500 0.728073i \(-0.259584\pi\)
0.685500 + 0.728073i \(0.259584\pi\)
\(618\) −1.64584e6 −0.173347
\(619\) 1.27406e7 1.33648 0.668240 0.743946i \(-0.267048\pi\)
0.668240 + 0.743946i \(0.267048\pi\)
\(620\) −488771. −0.0510654
\(621\) −250576. −0.0260742
\(622\) −668626. −0.0692958
\(623\) −3.03289e6 −0.313067
\(624\) 5.31165e6 0.546095
\(625\) 390625. 0.0400000
\(626\) −2.81320e6 −0.286923
\(627\) 2.07879e6 0.211175
\(628\) 8.91020e6 0.901547
\(629\) −1.60908e7 −1.62162
\(630\) −305464. −0.0306626
\(631\) −1.93988e7 −1.93955 −0.969775 0.243999i \(-0.921541\pi\)
−0.969775 + 0.243999i \(0.921541\pi\)
\(632\) 59991.1 0.00597440
\(633\) 3.54695e6 0.351840
\(634\) −789006. −0.0779574
\(635\) −5.36706e6 −0.528204
\(636\) −2.46975e6 −0.242108
\(637\) −1.71632e6 −0.167590
\(638\) −7000.83 −0.000680923 0
\(639\) 2.51714e6 0.243868
\(640\) −3.22654e6 −0.311378
\(641\) 6.77844e6 0.651605 0.325803 0.945438i \(-0.394366\pi\)
0.325803 + 0.945438i \(0.394366\pi\)
\(642\) 131750. 0.0126157
\(643\) −1.74257e7 −1.66212 −0.831060 0.556182i \(-0.812266\pi\)
−0.831060 + 0.556182i \(0.812266\pi\)
\(644\) 1.47842e6 0.140470
\(645\) −3.01034e6 −0.284915
\(646\) −2.84491e6 −0.268217
\(647\) 792517. 0.0744300 0.0372150 0.999307i \(-0.488151\pi\)
0.0372150 + 0.999307i \(0.488151\pi\)
\(648\) −445737. −0.0417005
\(649\) 1.36448e6 0.127161
\(650\) −436788. −0.0405497
\(651\) 796200. 0.0736325
\(652\) −8.88616e6 −0.818644
\(653\) −9.09663e6 −0.834829 −0.417414 0.908716i \(-0.637064\pi\)
−0.417414 + 0.908716i \(0.637064\pi\)
\(654\) 727295. 0.0664915
\(655\) 6.32667e6 0.576198
\(656\) 1.72275e7 1.56302
\(657\) 4.53687e6 0.410056
\(658\) 339849. 0.0306000
\(659\) −8.15992e6 −0.731935 −0.365968 0.930628i \(-0.619262\pi\)
−0.365968 + 0.930628i \(0.619262\pi\)
\(660\) −839370. −0.0750056
\(661\) −1.85172e6 −0.164843 −0.0824216 0.996598i \(-0.526265\pi\)
−0.0824216 + 0.996598i \(0.526265\pi\)
\(662\) 3.10465e6 0.275339
\(663\) −8.01765e6 −0.708375
\(664\) −6.38518e6 −0.562022
\(665\) −6.65766e6 −0.583805
\(666\) 1.02246e6 0.0893229
\(667\) 18392.5 0.00160076
\(668\) 5.57768e6 0.483629
\(669\) −4.00534e6 −0.345998
\(670\) −1.54983e6 −0.133382
\(671\) −1.43414e6 −0.122966
\(672\) 3.96929e6 0.339070
\(673\) 1.42740e6 0.121481 0.0607405 0.998154i \(-0.480654\pi\)
0.0607405 + 0.998154i \(0.480654\pi\)
\(674\) 2.34158e6 0.198545
\(675\) −455625. −0.0384900
\(676\) −1.43213e6 −0.120535
\(677\) 941868. 0.0789802 0.0394901 0.999220i \(-0.487427\pi\)
0.0394901 + 0.999220i \(0.487427\pi\)
\(678\) 2.00385e6 0.167413
\(679\) 2.20505e7 1.83545
\(680\) 2.34098e6 0.194145
\(681\) −1.04299e7 −0.861809
\(682\) −82966.3 −0.00683031
\(683\) 9.56089e6 0.784236 0.392118 0.919915i \(-0.371743\pi\)
0.392118 + 0.919915i \(0.371743\pi\)
\(684\) −4.76709e6 −0.389595
\(685\) 5.98494e6 0.487342
\(686\) 2.13471e6 0.173192
\(687\) −7.97268e6 −0.644484
\(688\) 1.22170e7 0.983996
\(689\) 5.75281e6 0.461670
\(690\) −83623.9 −0.00668664
\(691\) −1.53765e7 −1.22507 −0.612536 0.790443i \(-0.709851\pi\)
−0.612536 + 0.790443i \(0.709851\pi\)
\(692\) −6.73897e6 −0.534968
\(693\) 1.36732e6 0.108152
\(694\) 424865. 0.0334851
\(695\) −4.15994e6 −0.326682
\(696\) 32717.5 0.00256010
\(697\) −2.60040e7 −2.02749
\(698\) 267983. 0.0208194
\(699\) −6.75352e6 −0.522802
\(700\) 2.68822e6 0.207357
\(701\) 1.72261e7 1.32401 0.662006 0.749499i \(-0.269705\pi\)
0.662006 + 0.749499i \(0.269705\pi\)
\(702\) 509470. 0.0390190
\(703\) 2.22849e7 1.70068
\(704\) 3.12202e6 0.237413
\(705\) 506914. 0.0384115
\(706\) −3.14485e6 −0.237459
\(707\) 2.18983e7 1.64764
\(708\) −3.12902e6 −0.234599
\(709\) 2.47878e7 1.85192 0.925959 0.377625i \(-0.123259\pi\)
0.925959 + 0.377625i \(0.123259\pi\)
\(710\) 840037. 0.0625393
\(711\) −71525.9 −0.00530627
\(712\) −1.47695e6 −0.109186
\(713\) 217968. 0.0160572
\(714\) −1.87123e6 −0.137367
\(715\) 1.95515e6 0.143026
\(716\) −9.58579e6 −0.698788
\(717\) −5.55673e6 −0.403665
\(718\) 820042. 0.0593642
\(719\) 7.44502e6 0.537086 0.268543 0.963268i \(-0.413458\pi\)
0.268543 + 0.963268i \(0.413458\pi\)
\(720\) 1.84908e6 0.132931
\(721\) −2.35944e7 −1.69032
\(722\) 1.26271e6 0.0901486
\(723\) −1.08248e7 −0.770146
\(724\) 1.30611e7 0.926046
\(725\) 33443.2 0.00236300
\(726\) −142478. −0.0100325
\(727\) 3.56658e6 0.250274 0.125137 0.992139i \(-0.460063\pi\)
0.125137 + 0.992139i \(0.460063\pi\)
\(728\) −6.12580e6 −0.428385
\(729\) 531441. 0.0370370
\(730\) 1.51407e6 0.105157
\(731\) −1.84409e7 −1.27641
\(732\) 3.28879e6 0.226860
\(733\) 1.38091e7 0.949305 0.474652 0.880173i \(-0.342574\pi\)
0.474652 + 0.880173i \(0.342574\pi\)
\(734\) −2.48611e6 −0.170326
\(735\) −597482. −0.0407950
\(736\) 1.08663e6 0.0739416
\(737\) 6.93737e6 0.470463
\(738\) 1.65239e6 0.111679
\(739\) −7.18607e6 −0.484039 −0.242020 0.970271i \(-0.577810\pi\)
−0.242020 + 0.970271i \(0.577810\pi\)
\(740\) −8.99814e6 −0.604051
\(741\) 1.11040e7 0.742908
\(742\) 1.34264e6 0.0895261
\(743\) 2.18843e6 0.145432 0.0727161 0.997353i \(-0.476833\pi\)
0.0727161 + 0.997353i \(0.476833\pi\)
\(744\) 387732. 0.0256803
\(745\) 7.38780e6 0.487668
\(746\) −20721.5 −0.00136324
\(747\) 7.61289e6 0.499170
\(748\) −5.14186e6 −0.336021
\(749\) 1.88873e6 0.123017
\(750\) −152054. −0.00987064
\(751\) −1.41771e6 −0.0917250 −0.0458625 0.998948i \(-0.514604\pi\)
−0.0458625 + 0.998948i \(0.514604\pi\)
\(752\) −2.05723e6 −0.132660
\(753\) −1.40197e7 −0.901056
\(754\) −37395.5 −0.00239547
\(755\) 1.07093e7 0.683746
\(756\) −3.13554e6 −0.199530
\(757\) −2.05513e7 −1.30347 −0.651734 0.758448i \(-0.725958\pi\)
−0.651734 + 0.758448i \(0.725958\pi\)
\(758\) 374699. 0.0236870
\(759\) 374318. 0.0235850
\(760\) −3.24214e6 −0.203609
\(761\) −1.30587e7 −0.817409 −0.408705 0.912667i \(-0.634019\pi\)
−0.408705 + 0.912667i \(0.634019\pi\)
\(762\) 2.08917e6 0.130343
\(763\) 1.04263e7 0.648366
\(764\) −2.65481e7 −1.64551
\(765\) −2.79109e6 −0.172433
\(766\) −5.67740e6 −0.349605
\(767\) 7.28846e6 0.447350
\(768\) −6.17497e6 −0.377774
\(769\) 1.82667e7 1.11389 0.556946 0.830549i \(-0.311973\pi\)
0.556946 + 0.830549i \(0.311973\pi\)
\(770\) 456310. 0.0277353
\(771\) −829404. −0.0502493
\(772\) −1.32752e7 −0.801676
\(773\) −1.49182e7 −0.897981 −0.448990 0.893537i \(-0.648216\pi\)
−0.448990 + 0.893537i \(0.648216\pi\)
\(774\) 1.17180e6 0.0703074
\(775\) 396333. 0.0237032
\(776\) 1.07381e7 0.640138
\(777\) 1.46578e7 0.870996
\(778\) −4.69786e6 −0.278260
\(779\) 3.60143e7 2.12633
\(780\) −4.48356e6 −0.263868
\(781\) −3.76018e6 −0.220587
\(782\) −512268. −0.0299558
\(783\) −39008.2 −0.00227380
\(784\) 2.42479e6 0.140891
\(785\) −7.22507e6 −0.418473
\(786\) −2.46271e6 −0.142186
\(787\) −1.60565e7 −0.924087 −0.462043 0.886857i \(-0.652884\pi\)
−0.462043 + 0.886857i \(0.652884\pi\)
\(788\) 5.69616e6 0.326788
\(789\) 7.75910e6 0.443730
\(790\) −23870.1 −0.00136078
\(791\) 2.87266e7 1.63246
\(792\) 665854. 0.0377195
\(793\) −7.66060e6 −0.432593
\(794\) 4.22506e6 0.237838
\(795\) 2.00266e6 0.112380
\(796\) −9.95816e6 −0.557053
\(797\) 2.10741e7 1.17518 0.587588 0.809160i \(-0.300078\pi\)
0.587588 + 0.809160i \(0.300078\pi\)
\(798\) 2.59155e6 0.144063
\(799\) 3.10528e6 0.172081
\(800\) 1.97584e6 0.109151
\(801\) 1.76093e6 0.0969755
\(802\) −3.30129e6 −0.181237
\(803\) −6.77730e6 −0.370909
\(804\) −1.59088e7 −0.867955
\(805\) −1.19881e6 −0.0652021
\(806\) −443171. −0.0240289
\(807\) 1.12547e7 0.608344
\(808\) 1.06640e7 0.574633
\(809\) 2.38181e7 1.27949 0.639744 0.768588i \(-0.279040\pi\)
0.639744 + 0.768588i \(0.279040\pi\)
\(810\) 177356. 0.00949802
\(811\) 2.93046e7 1.56453 0.782264 0.622947i \(-0.214065\pi\)
0.782264 + 0.622947i \(0.214065\pi\)
\(812\) 230151. 0.0122496
\(813\) −7.60189e6 −0.403362
\(814\) −1.52739e6 −0.0807956
\(815\) 7.20557e6 0.379992
\(816\) 1.13272e7 0.595522
\(817\) 2.55397e7 1.33863
\(818\) −5.90430e6 −0.308521
\(819\) 7.30364e6 0.380478
\(820\) −1.45418e7 −0.755236
\(821\) −3.14594e6 −0.162889 −0.0814446 0.996678i \(-0.525953\pi\)
−0.0814446 + 0.996678i \(0.525953\pi\)
\(822\) −2.32969e6 −0.120259
\(823\) 2.44990e7 1.26081 0.630404 0.776267i \(-0.282889\pi\)
0.630404 + 0.776267i \(0.282889\pi\)
\(824\) −1.14899e7 −0.589522
\(825\) 680625. 0.0348155
\(826\) 1.70104e6 0.0867492
\(827\) 1.34163e7 0.682135 0.341067 0.940039i \(-0.389212\pi\)
0.341067 + 0.940039i \(0.389212\pi\)
\(828\) −858386. −0.0435118
\(829\) 1.42349e7 0.719394 0.359697 0.933069i \(-0.382880\pi\)
0.359697 + 0.933069i \(0.382880\pi\)
\(830\) 2.54062e6 0.128010
\(831\) 1.37691e7 0.691675
\(832\) 1.66765e7 0.835213
\(833\) −3.66009e6 −0.182759
\(834\) 1.61929e6 0.0806140
\(835\) −4.52281e6 −0.224487
\(836\) 7.12121e6 0.352402
\(837\) −462283. −0.0228084
\(838\) 3.27414e6 0.161060
\(839\) 1.15794e7 0.567913 0.283957 0.958837i \(-0.408353\pi\)
0.283957 + 0.958837i \(0.408353\pi\)
\(840\) −2.13251e6 −0.104278
\(841\) −2.05083e7 −0.999860
\(842\) −3.58453e6 −0.174242
\(843\) −1.01955e7 −0.494130
\(844\) 1.21506e7 0.587140
\(845\) 1.16128e6 0.0559492
\(846\) −197320. −0.00947865
\(847\) −2.04254e6 −0.0978276
\(848\) −8.12749e6 −0.388121
\(849\) 3.70025e6 0.176182
\(850\) −931462. −0.0442199
\(851\) 4.01273e6 0.189940
\(852\) 8.62285e6 0.406960
\(853\) 6.61595e6 0.311329 0.155664 0.987810i \(-0.450248\pi\)
0.155664 + 0.987810i \(0.450248\pi\)
\(854\) −1.78790e6 −0.0838876
\(855\) 3.86552e6 0.180839
\(856\) 919772. 0.0429038
\(857\) −9.97147e6 −0.463775 −0.231887 0.972743i \(-0.574490\pi\)
−0.231887 + 0.972743i \(0.574490\pi\)
\(858\) −761060. −0.0352940
\(859\) −3.79757e7 −1.75599 −0.877996 0.478667i \(-0.841120\pi\)
−0.877996 + 0.478667i \(0.841120\pi\)
\(860\) −1.03124e7 −0.475458
\(861\) 2.36883e7 1.08899
\(862\) −3.26695e6 −0.149753
\(863\) 8.18531e6 0.374118 0.187059 0.982349i \(-0.440104\pi\)
0.187059 + 0.982349i \(0.440104\pi\)
\(864\) −2.30462e6 −0.105030
\(865\) 5.46447e6 0.248318
\(866\) −799779. −0.0362389
\(867\) −4.31911e6 −0.195140
\(868\) 2.72750e6 0.122876
\(869\) 106847. 0.00479970
\(870\) −13018.1 −0.000583108 0
\(871\) 3.70565e7 1.65508
\(872\) 5.07740e6 0.226126
\(873\) −1.28028e7 −0.568550
\(874\) 709465. 0.0314161
\(875\) −2.17981e6 −0.0962496
\(876\) 1.55417e7 0.684288
\(877\) −3.02517e7 −1.32816 −0.664081 0.747660i \(-0.731177\pi\)
−0.664081 + 0.747660i \(0.731177\pi\)
\(878\) 96635.0 0.00423056
\(879\) −2.82611e6 −0.123372
\(880\) −2.76221e6 −0.120240
\(881\) 6.11414e6 0.265397 0.132698 0.991156i \(-0.457636\pi\)
0.132698 + 0.991156i \(0.457636\pi\)
\(882\) 232575. 0.0100668
\(883\) −7.72672e6 −0.333498 −0.166749 0.985999i \(-0.553327\pi\)
−0.166749 + 0.985999i \(0.553327\pi\)
\(884\) −2.74656e7 −1.18211
\(885\) 2.53725e6 0.108894
\(886\) 5.47302e6 0.234230
\(887\) −2.48274e7 −1.05955 −0.529775 0.848138i \(-0.677724\pi\)
−0.529775 + 0.848138i \(0.677724\pi\)
\(888\) 7.13803e6 0.303771
\(889\) 2.99499e7 1.27099
\(890\) 587671. 0.0248690
\(891\) −793881. −0.0335013
\(892\) −1.37209e7 −0.577391
\(893\) −4.30065e6 −0.180470
\(894\) −2.87576e6 −0.120340
\(895\) 7.77289e6 0.324358
\(896\) 1.80051e7 0.749249
\(897\) 1.99945e6 0.0829715
\(898\) 7.18613e6 0.297375
\(899\) 33931.9 0.00140026
\(900\) −1.56081e6 −0.0642309
\(901\) 1.22680e7 0.503457
\(902\) −2.46838e6 −0.101017
\(903\) 1.67987e7 0.685575
\(904\) 1.39893e7 0.569342
\(905\) −1.05909e7 −0.429845
\(906\) −4.16870e6 −0.168725
\(907\) −3.64608e6 −0.147166 −0.0735831 0.997289i \(-0.523443\pi\)
−0.0735831 + 0.997289i \(0.523443\pi\)
\(908\) −3.57291e7 −1.43816
\(909\) −1.27144e7 −0.510371
\(910\) 2.43742e6 0.0975723
\(911\) −1.15898e7 −0.462678 −0.231339 0.972873i \(-0.574311\pi\)
−0.231339 + 0.972873i \(0.574311\pi\)
\(912\) −1.56876e7 −0.624554
\(913\) −1.13723e7 −0.451516
\(914\) −190642. −0.00754838
\(915\) −2.66680e6 −0.105302
\(916\) −2.73116e7 −1.07550
\(917\) −3.53048e7 −1.38647
\(918\) 1.08646e6 0.0425506
\(919\) −3.48544e7 −1.36135 −0.680674 0.732586i \(-0.738313\pi\)
−0.680674 + 0.732586i \(0.738313\pi\)
\(920\) −583796. −0.0227401
\(921\) −7.17840e6 −0.278855
\(922\) 9.12314e6 0.353441
\(923\) −2.00853e7 −0.776021
\(924\) 4.68395e6 0.180481
\(925\) 7.29638e6 0.280384
\(926\) 6.11719e6 0.234436
\(927\) 1.36992e7 0.523594
\(928\) 169161. 0.00644806
\(929\) −4.11213e7 −1.56325 −0.781624 0.623750i \(-0.785608\pi\)
−0.781624 + 0.623750i \(0.785608\pi\)
\(930\) −154276. −0.00584913
\(931\) 5.06903e6 0.191669
\(932\) −2.31352e7 −0.872436
\(933\) 5.56532e6 0.209308
\(934\) −2.74945e6 −0.103128
\(935\) 4.16941e6 0.155972
\(936\) 3.55671e6 0.132696
\(937\) −1.53222e6 −0.0570126 −0.0285063 0.999594i \(-0.509075\pi\)
−0.0285063 + 0.999594i \(0.509075\pi\)
\(938\) 8.64857e6 0.320950
\(939\) 2.34157e7 0.866650
\(940\) 1.73651e6 0.0640999
\(941\) −1.91551e7 −0.705196 −0.352598 0.935775i \(-0.614702\pi\)
−0.352598 + 0.935775i \(0.614702\pi\)
\(942\) 2.81242e6 0.103265
\(943\) 6.48491e6 0.237479
\(944\) −1.02970e7 −0.376082
\(945\) 2.54253e6 0.0926162
\(946\) −1.75047e6 −0.0635955
\(947\) 2.83184e7 1.02611 0.513054 0.858356i \(-0.328514\pi\)
0.513054 + 0.858356i \(0.328514\pi\)
\(948\) −245023. −0.00885494
\(949\) −3.62015e7 −1.30485
\(950\) 1.29003e6 0.0463756
\(951\) 6.56731e6 0.235470
\(952\) −1.30634e7 −0.467159
\(953\) −5.47682e7 −1.95342 −0.976711 0.214559i \(-0.931169\pi\)
−0.976711 + 0.214559i \(0.931169\pi\)
\(954\) −779553. −0.0277316
\(955\) 2.15272e7 0.763800
\(956\) −1.90354e7 −0.673624
\(957\) 58271.5 0.00205673
\(958\) 7.45145e6 0.262318
\(959\) −3.33979e7 −1.17266
\(960\) 5.80541e6 0.203308
\(961\) −2.82270e7 −0.985954
\(962\) −8.15865e6 −0.284237
\(963\) −1.09662e6 −0.0381058
\(964\) −3.70819e7 −1.28520
\(965\) 1.07646e7 0.372116
\(966\) 466648. 0.0160897
\(967\) 2.29671e7 0.789843 0.394922 0.918715i \(-0.370772\pi\)
0.394922 + 0.918715i \(0.370772\pi\)
\(968\) −994671. −0.0341186
\(969\) 2.36796e7 0.810149
\(970\) −4.27263e6 −0.145803
\(971\) 1.46281e7 0.497898 0.248949 0.968517i \(-0.419915\pi\)
0.248949 + 0.968517i \(0.419915\pi\)
\(972\) 1.82053e6 0.0618062
\(973\) 2.32138e7 0.786076
\(974\) 8.54152e6 0.288495
\(975\) 3.63561e6 0.122480
\(976\) 1.08228e7 0.363676
\(977\) −1.54714e7 −0.518552 −0.259276 0.965803i \(-0.583484\pi\)
−0.259276 + 0.965803i \(0.583484\pi\)
\(978\) −2.80483e6 −0.0937691
\(979\) −2.63053e6 −0.0877176
\(980\) −2.04676e6 −0.0680773
\(981\) −6.05365e6 −0.200838
\(982\) −2.47612e6 −0.0819395
\(983\) 2.02694e7 0.669047 0.334524 0.942387i \(-0.391425\pi\)
0.334524 + 0.942387i \(0.391425\pi\)
\(984\) 1.15357e7 0.379800
\(985\) −4.61888e6 −0.151686
\(986\) −79746.8 −0.00261229
\(987\) −2.82874e6 −0.0924272
\(988\) 3.80385e7 1.23974
\(989\) 4.59881e6 0.149505
\(990\) −264939. −0.00859129
\(991\) −4.43452e7 −1.43437 −0.717187 0.696881i \(-0.754570\pi\)
−0.717187 + 0.696881i \(0.754570\pi\)
\(992\) 2.00471e6 0.0646803
\(993\) −2.58416e7 −0.831661
\(994\) −4.68768e6 −0.150484
\(995\) 8.07483e6 0.258569
\(996\) 2.60791e7 0.832998
\(997\) 2.32293e7 0.740112 0.370056 0.929009i \(-0.379338\pi\)
0.370056 + 0.929009i \(0.379338\pi\)
\(998\) −6.44775e6 −0.204919
\(999\) −8.51050e6 −0.269800
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 165.6.a.d.1.2 3
3.2 odd 2 495.6.a.c.1.2 3
5.4 even 2 825.6.a.h.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.d.1.2 3 1.1 even 1 trivial
495.6.a.c.1.2 3 3.2 odd 2
825.6.a.h.1.2 3 5.4 even 2