Properties

Label 165.6.a.a.1.1
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $1$
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: \(1\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(7.25531\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.25531 q^{2} +9.00000 q^{3} +53.6607 q^{4} +25.0000 q^{5} -83.2977 q^{6} +36.4478 q^{7} -200.476 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.25531 q^{2} +9.00000 q^{3} +53.6607 q^{4} +25.0000 q^{5} -83.2977 q^{6} +36.4478 q^{7} -200.476 q^{8} +81.0000 q^{9} -231.383 q^{10} -121.000 q^{11} +482.946 q^{12} -878.032 q^{13} -337.336 q^{14} +225.000 q^{15} +138.327 q^{16} +155.385 q^{17} -749.680 q^{18} -1932.65 q^{19} +1341.52 q^{20} +328.030 q^{21} +1119.89 q^{22} +1927.38 q^{23} -1804.29 q^{24} +625.000 q^{25} +8126.46 q^{26} +729.000 q^{27} +1955.81 q^{28} -480.444 q^{29} -2082.44 q^{30} +1759.49 q^{31} +5134.98 q^{32} -1089.00 q^{33} -1438.14 q^{34} +911.195 q^{35} +4346.51 q^{36} -1898.87 q^{37} +17887.3 q^{38} -7902.29 q^{39} -5011.91 q^{40} -4500.03 q^{41} -3036.02 q^{42} -4475.49 q^{43} -6492.94 q^{44} +2025.00 q^{45} -17838.5 q^{46} -12371.2 q^{47} +1244.94 q^{48} -15478.6 q^{49} -5784.57 q^{50} +1398.47 q^{51} -47115.8 q^{52} +2145.12 q^{53} -6747.12 q^{54} -3025.00 q^{55} -7306.92 q^{56} -17393.8 q^{57} +4446.66 q^{58} -15857.9 q^{59} +12073.7 q^{60} -36447.7 q^{61} -16284.6 q^{62} +2952.27 q^{63} -51952.3 q^{64} -21950.8 q^{65} +10079.0 q^{66} -15668.5 q^{67} +8338.07 q^{68} +17346.4 q^{69} -8433.39 q^{70} +10689.5 q^{71} -16238.6 q^{72} +12172.6 q^{73} +17574.6 q^{74} +5625.00 q^{75} -103707. q^{76} -4410.19 q^{77} +73138.1 q^{78} -87205.6 q^{79} +3458.17 q^{80} +6561.00 q^{81} +41649.2 q^{82} +97230.6 q^{83} +17602.3 q^{84} +3884.63 q^{85} +41422.0 q^{86} -4324.00 q^{87} +24257.6 q^{88} +38639.2 q^{89} -18742.0 q^{90} -32002.4 q^{91} +103424. q^{92} +15835.4 q^{93} +114499. q^{94} -48316.2 q^{95} +46214.8 q^{96} -36754.5 q^{97} +143259. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 7 q^{2} + 27 q^{3} + 25 q^{4} + 75 q^{5} - 63 q^{6} - 172 q^{7} - 231 q^{8} + 243 q^{9} - 175 q^{10} - 363 q^{11} + 225 q^{12} - 654 q^{13} - 728 q^{14} + 675 q^{15} - 415 q^{16} - 2366 q^{17} - 567 q^{18} - 2872 q^{19} + 625 q^{20} - 1548 q^{21} + 847 q^{22} + 2272 q^{23} - 2079 q^{24} + 1875 q^{25} + 3422 q^{26} + 2187 q^{27} + 4592 q^{28} - 7738 q^{29} - 1575 q^{30} + 568 q^{31} + 1001 q^{32} - 3267 q^{33} + 2506 q^{34} - 4300 q^{35} + 2025 q^{36} - 9126 q^{37} + 13076 q^{38} - 5886 q^{39} - 5775 q^{40} - 8758 q^{41} - 6552 q^{42} - 14672 q^{43} - 3025 q^{44} + 6075 q^{45} - 28768 q^{46} - 19392 q^{47} - 3735 q^{48} - 26629 q^{49} - 4375 q^{50} - 21294 q^{51} - 61506 q^{52} - 4598 q^{53} - 5103 q^{54} - 9075 q^{55} + 2688 q^{56} - 25848 q^{57} + 8550 q^{58} - 9348 q^{59} + 5625 q^{60} - 60078 q^{61} - 14096 q^{62} - 13932 q^{63} - 7087 q^{64} - 16350 q^{65} + 7623 q^{66} - 38468 q^{67} + 59778 q^{68} + 20448 q^{69} - 18200 q^{70} - 74032 q^{71} - 18711 q^{72} - 44442 q^{73} + 82542 q^{74} + 16875 q^{75} - 98708 q^{76} + 20812 q^{77} + 30798 q^{78} - 108116 q^{79} - 10375 q^{80} + 19683 q^{81} - 92230 q^{82} - 81892 q^{83} + 41328 q^{84} - 59150 q^{85} + 126412 q^{86} - 69642 q^{87} + 27951 q^{88} + 167342 q^{89} - 14175 q^{90} - 31832 q^{91} + 72960 q^{92} + 5112 q^{93} + 12728 q^{94} - 71800 q^{95} + 9009 q^{96} + 159702 q^{97} + 163121 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.25531 −1.63612 −0.818061 0.575131i \(-0.804951\pi\)
−0.818061 + 0.575131i \(0.804951\pi\)
\(3\) 9.00000 0.577350
\(4\) 53.6607 1.67690
\(5\) 25.0000 0.447214
\(6\) −83.2977 −0.944616
\(7\) 36.4478 0.281142 0.140571 0.990071i \(-0.455106\pi\)
0.140571 + 0.990071i \(0.455106\pi\)
\(8\) −200.476 −1.10749
\(9\) 81.0000 0.333333
\(10\) −231.383 −0.731696
\(11\) −121.000 −0.301511
\(12\) 482.946 0.968156
\(13\) −878.032 −1.44096 −0.720480 0.693476i \(-0.756079\pi\)
−0.720480 + 0.693476i \(0.756079\pi\)
\(14\) −337.336 −0.459983
\(15\) 225.000 0.258199
\(16\) 138.327 0.135085
\(17\) 155.385 0.130403 0.0652014 0.997872i \(-0.479231\pi\)
0.0652014 + 0.997872i \(0.479231\pi\)
\(18\) −749.680 −0.545374
\(19\) −1932.65 −1.22820 −0.614100 0.789228i \(-0.710481\pi\)
−0.614100 + 0.789228i \(0.710481\pi\)
\(20\) 1341.52 0.749931
\(21\) 328.030 0.162318
\(22\) 1119.89 0.493309
\(23\) 1927.38 0.759709 0.379855 0.925046i \(-0.375974\pi\)
0.379855 + 0.925046i \(0.375974\pi\)
\(24\) −1804.29 −0.639407
\(25\) 625.000 0.200000
\(26\) 8126.46 2.35759
\(27\) 729.000 0.192450
\(28\) 1955.81 0.471447
\(29\) −480.444 −0.106084 −0.0530418 0.998592i \(-0.516892\pi\)
−0.0530418 + 0.998592i \(0.516892\pi\)
\(30\) −2082.44 −0.422445
\(31\) 1759.49 0.328838 0.164419 0.986391i \(-0.447425\pi\)
0.164419 + 0.986391i \(0.447425\pi\)
\(32\) 5134.98 0.886470
\(33\) −1089.00 −0.174078
\(34\) −1438.14 −0.213355
\(35\) 911.195 0.125731
\(36\) 4346.51 0.558965
\(37\) −1898.87 −0.228029 −0.114015 0.993479i \(-0.536371\pi\)
−0.114015 + 0.993479i \(0.536371\pi\)
\(38\) 17887.3 2.00949
\(39\) −7902.29 −0.831939
\(40\) −5011.91 −0.495282
\(41\) −4500.03 −0.418076 −0.209038 0.977907i \(-0.567033\pi\)
−0.209038 + 0.977907i \(0.567033\pi\)
\(42\) −3036.02 −0.265572
\(43\) −4475.49 −0.369121 −0.184561 0.982821i \(-0.559086\pi\)
−0.184561 + 0.982821i \(0.559086\pi\)
\(44\) −6492.94 −0.505603
\(45\) 2025.00 0.149071
\(46\) −17838.5 −1.24298
\(47\) −12371.2 −0.816895 −0.408448 0.912782i \(-0.633930\pi\)
−0.408448 + 0.912782i \(0.633930\pi\)
\(48\) 1244.94 0.0779912
\(49\) −15478.6 −0.920959
\(50\) −5784.57 −0.327224
\(51\) 1398.47 0.0752881
\(52\) −47115.8 −2.41634
\(53\) 2145.12 0.104897 0.0524483 0.998624i \(-0.483298\pi\)
0.0524483 + 0.998624i \(0.483298\pi\)
\(54\) −6747.12 −0.314872
\(55\) −3025.00 −0.134840
\(56\) −7306.92 −0.311361
\(57\) −17393.8 −0.709102
\(58\) 4446.66 0.173566
\(59\) −15857.9 −0.593084 −0.296542 0.955020i \(-0.595834\pi\)
−0.296542 + 0.955020i \(0.595834\pi\)
\(60\) 12073.7 0.432973
\(61\) −36447.7 −1.25414 −0.627070 0.778963i \(-0.715746\pi\)
−0.627070 + 0.778963i \(0.715746\pi\)
\(62\) −16284.6 −0.538020
\(63\) 2952.27 0.0937141
\(64\) −51952.3 −1.58546
\(65\) −21950.8 −0.644417
\(66\) 10079.0 0.284812
\(67\) −15668.5 −0.426424 −0.213212 0.977006i \(-0.568393\pi\)
−0.213212 + 0.977006i \(0.568393\pi\)
\(68\) 8338.07 0.218672
\(69\) 17346.4 0.438618
\(70\) −8433.39 −0.205711
\(71\) 10689.5 0.251659 0.125830 0.992052i \(-0.459841\pi\)
0.125830 + 0.992052i \(0.459841\pi\)
\(72\) −16238.6 −0.369162
\(73\) 12172.6 0.267347 0.133674 0.991025i \(-0.457323\pi\)
0.133674 + 0.991025i \(0.457323\pi\)
\(74\) 17574.6 0.373084
\(75\) 5625.00 0.115470
\(76\) −103707. −2.05956
\(77\) −4410.19 −0.0847676
\(78\) 73138.1 1.36115
\(79\) −87205.6 −1.57209 −0.786043 0.618171i \(-0.787874\pi\)
−0.786043 + 0.618171i \(0.787874\pi\)
\(80\) 3458.17 0.0604117
\(81\) 6561.00 0.111111
\(82\) 41649.2 0.684024
\(83\) 97230.6 1.54920 0.774600 0.632451i \(-0.217951\pi\)
0.774600 + 0.632451i \(0.217951\pi\)
\(84\) 17602.3 0.272190
\(85\) 3884.63 0.0583179
\(86\) 41422.0 0.603928
\(87\) −4324.00 −0.0612473
\(88\) 24257.6 0.333919
\(89\) 38639.2 0.517075 0.258537 0.966001i \(-0.416759\pi\)
0.258537 + 0.966001i \(0.416759\pi\)
\(90\) −18742.0 −0.243899
\(91\) −32002.4 −0.405115
\(92\) 103424. 1.27395
\(93\) 15835.4 0.189855
\(94\) 114499. 1.33654
\(95\) −48316.2 −0.549268
\(96\) 46214.8 0.511804
\(97\) −36754.5 −0.396626 −0.198313 0.980139i \(-0.563546\pi\)
−0.198313 + 0.980139i \(0.563546\pi\)
\(98\) 143259. 1.50680
\(99\) −9801.00 −0.100504
\(100\) 33537.9 0.335379
\(101\) −185487. −1.80929 −0.904647 0.426161i \(-0.859866\pi\)
−0.904647 + 0.426161i \(0.859866\pi\)
\(102\) −12943.2 −0.123181
\(103\) 36890.7 0.342629 0.171315 0.985216i \(-0.445199\pi\)
0.171315 + 0.985216i \(0.445199\pi\)
\(104\) 176025. 1.59584
\(105\) 8200.76 0.0725907
\(106\) −19853.7 −0.171624
\(107\) −124996. −1.05545 −0.527725 0.849415i \(-0.676955\pi\)
−0.527725 + 0.849415i \(0.676955\pi\)
\(108\) 39118.6 0.322719
\(109\) −150975. −1.21714 −0.608568 0.793502i \(-0.708256\pi\)
−0.608568 + 0.793502i \(0.708256\pi\)
\(110\) 27997.3 0.220615
\(111\) −17089.8 −0.131653
\(112\) 5041.71 0.0379780
\(113\) 157970. 1.16380 0.581899 0.813261i \(-0.302310\pi\)
0.581899 + 0.813261i \(0.302310\pi\)
\(114\) 160985. 1.16018
\(115\) 48184.5 0.339752
\(116\) −25781.0 −0.177891
\(117\) −71120.6 −0.480320
\(118\) 146770. 0.970359
\(119\) 5663.45 0.0366618
\(120\) −45107.1 −0.285951
\(121\) 14641.0 0.0909091
\(122\) 337335. 2.05193
\(123\) −40500.3 −0.241377
\(124\) 94415.4 0.551428
\(125\) 15625.0 0.0894427
\(126\) −27324.2 −0.153328
\(127\) 268814. 1.47891 0.739457 0.673204i \(-0.235082\pi\)
0.739457 + 0.673204i \(0.235082\pi\)
\(128\) 316515. 1.70753
\(129\) −40279.4 −0.213112
\(130\) 203161. 1.05435
\(131\) −366914. −1.86804 −0.934020 0.357220i \(-0.883725\pi\)
−0.934020 + 0.357220i \(0.883725\pi\)
\(132\) −58436.5 −0.291910
\(133\) −70440.9 −0.345299
\(134\) 145017. 0.697682
\(135\) 18225.0 0.0860663
\(136\) −31151.0 −0.144419
\(137\) −182927. −0.832678 −0.416339 0.909209i \(-0.636687\pi\)
−0.416339 + 0.909209i \(0.636687\pi\)
\(138\) −160546. −0.717633
\(139\) 8429.85 0.0370069 0.0185035 0.999829i \(-0.494110\pi\)
0.0185035 + 0.999829i \(0.494110\pi\)
\(140\) 48895.4 0.210837
\(141\) −111341. −0.471635
\(142\) −98934.8 −0.411745
\(143\) 106242. 0.434466
\(144\) 11204.5 0.0450282
\(145\) −12011.1 −0.0474420
\(146\) −112661. −0.437413
\(147\) −139307. −0.531716
\(148\) −101895. −0.382381
\(149\) 271810. 1.00300 0.501499 0.865158i \(-0.332782\pi\)
0.501499 + 0.865158i \(0.332782\pi\)
\(150\) −52061.1 −0.188923
\(151\) −236565. −0.844322 −0.422161 0.906521i \(-0.638728\pi\)
−0.422161 + 0.906521i \(0.638728\pi\)
\(152\) 387450. 1.36021
\(153\) 12586.2 0.0434676
\(154\) 40817.6 0.138690
\(155\) 43987.2 0.147061
\(156\) −424042. −1.39508
\(157\) 211824. 0.685846 0.342923 0.939364i \(-0.388583\pi\)
0.342923 + 0.939364i \(0.388583\pi\)
\(158\) 807114. 2.57213
\(159\) 19306.1 0.0605621
\(160\) 128375. 0.396441
\(161\) 70248.7 0.213586
\(162\) −60724.1 −0.181791
\(163\) −341315. −1.00620 −0.503102 0.864227i \(-0.667808\pi\)
−0.503102 + 0.864227i \(0.667808\pi\)
\(164\) −241475. −0.701071
\(165\) −27225.0 −0.0778499
\(166\) −899899. −2.53468
\(167\) −180548. −0.500958 −0.250479 0.968122i \(-0.580588\pi\)
−0.250479 + 0.968122i \(0.580588\pi\)
\(168\) −65762.3 −0.179764
\(169\) 399647. 1.07637
\(170\) −35953.4 −0.0954153
\(171\) −156545. −0.409400
\(172\) −240158. −0.618978
\(173\) −643322. −1.63423 −0.817114 0.576476i \(-0.804428\pi\)
−0.817114 + 0.576476i \(0.804428\pi\)
\(174\) 40019.9 0.100208
\(175\) 22779.9 0.0562285
\(176\) −16737.5 −0.0407296
\(177\) −142721. −0.342417
\(178\) −357618. −0.845998
\(179\) −266922. −0.622662 −0.311331 0.950302i \(-0.600775\pi\)
−0.311331 + 0.950302i \(0.600775\pi\)
\(180\) 108663. 0.249977
\(181\) 281529. 0.638745 0.319372 0.947629i \(-0.396528\pi\)
0.319372 + 0.947629i \(0.396528\pi\)
\(182\) 296192. 0.662818
\(183\) −328030. −0.724078
\(184\) −386393. −0.841366
\(185\) −47471.7 −0.101978
\(186\) −146561. −0.310626
\(187\) −18801.6 −0.0393179
\(188\) −663846. −1.36985
\(189\) 26570.5 0.0541059
\(190\) 447182. 0.898669
\(191\) 933723. 1.85197 0.925987 0.377556i \(-0.123235\pi\)
0.925987 + 0.377556i \(0.123235\pi\)
\(192\) −467571. −0.915365
\(193\) 300124. 0.579973 0.289986 0.957031i \(-0.406349\pi\)
0.289986 + 0.957031i \(0.406349\pi\)
\(194\) 340174. 0.648929
\(195\) −197557. −0.372054
\(196\) −830590. −1.54435
\(197\) −944940. −1.73476 −0.867378 0.497649i \(-0.834197\pi\)
−0.867378 + 0.497649i \(0.834197\pi\)
\(198\) 90711.2 0.164436
\(199\) −1.00821e6 −1.80475 −0.902374 0.430954i \(-0.858177\pi\)
−0.902374 + 0.430954i \(0.858177\pi\)
\(200\) −125298. −0.221497
\(201\) −141017. −0.246196
\(202\) 1.71674e6 2.96023
\(203\) −17511.1 −0.0298246
\(204\) 75042.6 0.126250
\(205\) −112501. −0.186969
\(206\) −341435. −0.560583
\(207\) 156118. 0.253236
\(208\) −121455. −0.194652
\(209\) 233851. 0.370316
\(210\) −75900.5 −0.118767
\(211\) 497479. 0.769253 0.384626 0.923072i \(-0.374330\pi\)
0.384626 + 0.923072i \(0.374330\pi\)
\(212\) 115108. 0.175901
\(213\) 96205.7 0.145295
\(214\) 1.15688e6 1.72684
\(215\) −111887. −0.165076
\(216\) −146147. −0.213136
\(217\) 64129.5 0.0924504
\(218\) 1.39732e6 1.99138
\(219\) 109553. 0.154353
\(220\) −162324. −0.226113
\(221\) −136433. −0.187905
\(222\) 158172. 0.215400
\(223\) 1.14136e6 1.53695 0.768477 0.639878i \(-0.221015\pi\)
0.768477 + 0.639878i \(0.221015\pi\)
\(224\) 187159. 0.249224
\(225\) 50625.0 0.0666667
\(226\) −1.46206e6 −1.90411
\(227\) 669451. 0.862292 0.431146 0.902282i \(-0.358109\pi\)
0.431146 + 0.902282i \(0.358109\pi\)
\(228\) −933366. −1.18909
\(229\) 588061. 0.741026 0.370513 0.928827i \(-0.379182\pi\)
0.370513 + 0.928827i \(0.379182\pi\)
\(230\) −445962. −0.555876
\(231\) −39691.7 −0.0489406
\(232\) 96317.6 0.117486
\(233\) 199417. 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(234\) 658243. 0.785862
\(235\) −309279. −0.365327
\(236\) −850947. −0.994541
\(237\) −784850. −0.907645
\(238\) −52416.9 −0.0599832
\(239\) −408055. −0.462088 −0.231044 0.972943i \(-0.574214\pi\)
−0.231044 + 0.972943i \(0.574214\pi\)
\(240\) 31123.5 0.0348787
\(241\) 1.24022e6 1.37548 0.687742 0.725956i \(-0.258602\pi\)
0.687742 + 0.725956i \(0.258602\pi\)
\(242\) −135507. −0.148738
\(243\) 59049.0 0.0641500
\(244\) −1.95581e6 −2.10306
\(245\) −386964. −0.411865
\(246\) 374842. 0.394922
\(247\) 1.69693e6 1.76979
\(248\) −352736. −0.364183
\(249\) 875075. 0.894431
\(250\) −144614. −0.146339
\(251\) 30660.6 0.0307183 0.0153591 0.999882i \(-0.495111\pi\)
0.0153591 + 0.999882i \(0.495111\pi\)
\(252\) 158421. 0.157149
\(253\) −233213. −0.229061
\(254\) −2.48796e6 −2.41968
\(255\) 34961.7 0.0336699
\(256\) −1.26697e6 −1.20828
\(257\) −687971. −0.649737 −0.324868 0.945759i \(-0.605320\pi\)
−0.324868 + 0.945759i \(0.605320\pi\)
\(258\) 372798. 0.348678
\(259\) −69209.6 −0.0641087
\(260\) −1.17790e6 −1.08062
\(261\) −38916.0 −0.0353612
\(262\) 3.39590e6 3.05634
\(263\) 1.70103e6 1.51643 0.758216 0.652004i \(-0.226071\pi\)
0.758216 + 0.652004i \(0.226071\pi\)
\(264\) 218319. 0.192788
\(265\) 53628.0 0.0469112
\(266\) 651952. 0.564952
\(267\) 347753. 0.298533
\(268\) −840785. −0.715069
\(269\) −982644. −0.827972 −0.413986 0.910283i \(-0.635864\pi\)
−0.413986 + 0.910283i \(0.635864\pi\)
\(270\) −168678. −0.140815
\(271\) −276206. −0.228459 −0.114230 0.993454i \(-0.536440\pi\)
−0.114230 + 0.993454i \(0.536440\pi\)
\(272\) 21493.9 0.0176154
\(273\) −288021. −0.233893
\(274\) 1.69305e6 1.36236
\(275\) −75625.0 −0.0603023
\(276\) 930820. 0.735517
\(277\) 148776. 0.116502 0.0582509 0.998302i \(-0.481448\pi\)
0.0582509 + 0.998302i \(0.481448\pi\)
\(278\) −78020.8 −0.0605478
\(279\) 142519. 0.109613
\(280\) −182673. −0.139245
\(281\) 357772. 0.270297 0.135148 0.990825i \(-0.456849\pi\)
0.135148 + 0.990825i \(0.456849\pi\)
\(282\) 1.03049e6 0.771652
\(283\) −492090. −0.365240 −0.182620 0.983184i \(-0.558458\pi\)
−0.182620 + 0.983184i \(0.558458\pi\)
\(284\) 573607. 0.422006
\(285\) −434846. −0.317120
\(286\) −983301. −0.710839
\(287\) −164016. −0.117539
\(288\) 415934. 0.295490
\(289\) −1.39571e6 −0.982995
\(290\) 111166. 0.0776209
\(291\) −330791. −0.228992
\(292\) 653189. 0.448314
\(293\) 498120. 0.338973 0.169487 0.985532i \(-0.445789\pi\)
0.169487 + 0.985532i \(0.445789\pi\)
\(294\) 1.28933e6 0.869952
\(295\) −396448. −0.265235
\(296\) 380678. 0.252539
\(297\) −88209.0 −0.0580259
\(298\) −2.51568e6 −1.64103
\(299\) −1.69230e6 −1.09471
\(300\) 301841. 0.193631
\(301\) −163122. −0.103776
\(302\) 2.18948e6 1.38141
\(303\) −1.66938e6 −1.04460
\(304\) −267337. −0.165911
\(305\) −911194. −0.560869
\(306\) −116489. −0.0711183
\(307\) −998760. −0.604805 −0.302402 0.953180i \(-0.597789\pi\)
−0.302402 + 0.953180i \(0.597789\pi\)
\(308\) −236654. −0.142147
\(309\) 332017. 0.197817
\(310\) −407115. −0.240610
\(311\) 1.88783e6 1.10678 0.553389 0.832923i \(-0.313334\pi\)
0.553389 + 0.832923i \(0.313334\pi\)
\(312\) 1.58422e6 0.921360
\(313\) −2.34787e6 −1.35461 −0.677303 0.735704i \(-0.736851\pi\)
−0.677303 + 0.735704i \(0.736851\pi\)
\(314\) −1.96050e6 −1.12213
\(315\) 73806.8 0.0419102
\(316\) −4.67951e6 −2.63623
\(317\) 562721. 0.314518 0.157259 0.987557i \(-0.449734\pi\)
0.157259 + 0.987557i \(0.449734\pi\)
\(318\) −178684. −0.0990870
\(319\) 58133.7 0.0319854
\(320\) −1.29881e6 −0.709038
\(321\) −1.12497e6 −0.609364
\(322\) −650173. −0.349454
\(323\) −300305. −0.160161
\(324\) 352068. 0.186322
\(325\) −548770. −0.288192
\(326\) 3.15897e6 1.64627
\(327\) −1.35878e6 −0.702713
\(328\) 902149. 0.463013
\(329\) −450902. −0.229664
\(330\) 251976. 0.127372
\(331\) −1.00593e6 −0.504657 −0.252328 0.967642i \(-0.581196\pi\)
−0.252328 + 0.967642i \(0.581196\pi\)
\(332\) 5.21746e6 2.59785
\(333\) −153808. −0.0760098
\(334\) 1.67103e6 0.819628
\(335\) −391714. −0.190703
\(336\) 45375.4 0.0219266
\(337\) −280192. −0.134394 −0.0671971 0.997740i \(-0.521406\pi\)
−0.0671971 + 0.997740i \(0.521406\pi\)
\(338\) −3.69886e6 −1.76107
\(339\) 1.42173e6 0.671919
\(340\) 208452. 0.0977931
\(341\) −212898. −0.0991485
\(342\) 1.44887e6 0.669829
\(343\) −1.17674e6 −0.540063
\(344\) 897229. 0.408796
\(345\) 433660. 0.196156
\(346\) 5.95414e6 2.67380
\(347\) 2.10913e6 0.940328 0.470164 0.882579i \(-0.344195\pi\)
0.470164 + 0.882579i \(0.344195\pi\)
\(348\) −232029. −0.102705
\(349\) 3.88469e6 1.70723 0.853617 0.520901i \(-0.174404\pi\)
0.853617 + 0.520901i \(0.174404\pi\)
\(350\) −210835. −0.0919967
\(351\) −640085. −0.277313
\(352\) −621333. −0.267281
\(353\) 1.35663e6 0.579463 0.289732 0.957108i \(-0.406434\pi\)
0.289732 + 0.957108i \(0.406434\pi\)
\(354\) 1.32093e6 0.560237
\(355\) 267238. 0.112545
\(356\) 2.07341e6 0.867081
\(357\) 50971.0 0.0211667
\(358\) 2.47045e6 1.01875
\(359\) 3.93436e6 1.61116 0.805579 0.592488i \(-0.201854\pi\)
0.805579 + 0.592488i \(0.201854\pi\)
\(360\) −405964. −0.165094
\(361\) 1.25904e6 0.508476
\(362\) −2.60564e6 −1.04506
\(363\) 131769. 0.0524864
\(364\) −1.71727e6 −0.679336
\(365\) 304315. 0.119561
\(366\) 3.03602e6 1.18468
\(367\) −2.82588e6 −1.09519 −0.547594 0.836744i \(-0.684456\pi\)
−0.547594 + 0.836744i \(0.684456\pi\)
\(368\) 266608. 0.102625
\(369\) −364502. −0.139359
\(370\) 439365. 0.166848
\(371\) 78184.9 0.0294909
\(372\) 849738. 0.318367
\(373\) 4.58790e6 1.70743 0.853713 0.520744i \(-0.174345\pi\)
0.853713 + 0.520744i \(0.174345\pi\)
\(374\) 174015. 0.0643290
\(375\) 140625. 0.0516398
\(376\) 2.48013e6 0.904699
\(377\) 421845. 0.152862
\(378\) −245918. −0.0885239
\(379\) 2.84827e6 1.01855 0.509277 0.860603i \(-0.329913\pi\)
0.509277 + 0.860603i \(0.329913\pi\)
\(380\) −2.59268e6 −0.921065
\(381\) 2.41933e6 0.853852
\(382\) −8.64190e6 −3.03006
\(383\) −2.78467e6 −0.970013 −0.485006 0.874511i \(-0.661183\pi\)
−0.485006 + 0.874511i \(0.661183\pi\)
\(384\) 2.84863e6 0.985845
\(385\) −110255. −0.0379092
\(386\) −2.77774e6 −0.948906
\(387\) −362514. −0.123040
\(388\) −1.97227e6 −0.665101
\(389\) 3.95277e6 1.32442 0.662212 0.749316i \(-0.269618\pi\)
0.662212 + 0.749316i \(0.269618\pi\)
\(390\) 1.82845e6 0.608726
\(391\) 299486. 0.0990682
\(392\) 3.10308e6 1.01995
\(393\) −3.30223e6 −1.07851
\(394\) 8.74571e6 2.83827
\(395\) −2.18014e6 −0.703059
\(396\) −525928. −0.168534
\(397\) −5.55351e6 −1.76844 −0.884221 0.467068i \(-0.845310\pi\)
−0.884221 + 0.467068i \(0.845310\pi\)
\(398\) 9.33125e6 2.95279
\(399\) −633968. −0.199359
\(400\) 86454.2 0.0270169
\(401\) −279266. −0.0867277 −0.0433639 0.999059i \(-0.513807\pi\)
−0.0433639 + 0.999059i \(0.513807\pi\)
\(402\) 1.30515e6 0.402807
\(403\) −1.54489e6 −0.473843
\(404\) −9.95334e6 −3.03400
\(405\) 164025. 0.0496904
\(406\) 162071. 0.0487967
\(407\) 229763. 0.0687534
\(408\) −280359. −0.0833805
\(409\) 5.17128e6 1.52859 0.764293 0.644869i \(-0.223088\pi\)
0.764293 + 0.644869i \(0.223088\pi\)
\(410\) 1.04123e6 0.305905
\(411\) −1.64635e6 −0.480747
\(412\) 1.97958e6 0.574554
\(413\) −577987. −0.166741
\(414\) −1.44492e6 −0.414326
\(415\) 2.43076e6 0.692824
\(416\) −4.50868e6 −1.27737
\(417\) 75868.7 0.0213660
\(418\) −2.16436e6 −0.605883
\(419\) −6.04152e6 −1.68117 −0.840585 0.541680i \(-0.817788\pi\)
−0.840585 + 0.541680i \(0.817788\pi\)
\(420\) 440058. 0.121727
\(421\) −895386. −0.246210 −0.123105 0.992394i \(-0.539285\pi\)
−0.123105 + 0.992394i \(0.539285\pi\)
\(422\) −4.60432e6 −1.25859
\(423\) −1.00207e6 −0.272298
\(424\) −430045. −0.116171
\(425\) 97115.7 0.0260806
\(426\) −890413. −0.237721
\(427\) −1.32844e6 −0.352592
\(428\) −6.70738e6 −1.76988
\(429\) 956177. 0.250839
\(430\) 1.03555e6 0.270085
\(431\) −1.60867e6 −0.417132 −0.208566 0.978008i \(-0.566880\pi\)
−0.208566 + 0.978008i \(0.566880\pi\)
\(432\) 100840. 0.0259971
\(433\) 1.86039e6 0.476853 0.238427 0.971161i \(-0.423368\pi\)
0.238427 + 0.971161i \(0.423368\pi\)
\(434\) −593538. −0.151260
\(435\) −108100. −0.0273906
\(436\) −8.10142e6 −2.04101
\(437\) −3.72495e6 −0.933075
\(438\) −1.01395e6 −0.252540
\(439\) 2.75051e6 0.681164 0.340582 0.940215i \(-0.389376\pi\)
0.340582 + 0.940215i \(0.389376\pi\)
\(440\) 606441. 0.149333
\(441\) −1.25376e6 −0.306986
\(442\) 1.26273e6 0.307436
\(443\) 3.15804e6 0.764555 0.382278 0.924048i \(-0.375140\pi\)
0.382278 + 0.924048i \(0.375140\pi\)
\(444\) −917051. −0.220768
\(445\) 965981. 0.231243
\(446\) −1.05636e7 −2.51464
\(447\) 2.44629e6 0.579081
\(448\) −1.89355e6 −0.445740
\(449\) −127508. −0.0298484 −0.0149242 0.999889i \(-0.504751\pi\)
−0.0149242 + 0.999889i \(0.504751\pi\)
\(450\) −468550. −0.109075
\(451\) 544504. 0.126055
\(452\) 8.47675e6 1.95157
\(453\) −2.12908e6 −0.487469
\(454\) −6.19598e6 −1.41082
\(455\) −800059. −0.181173
\(456\) 3.48705e6 0.785319
\(457\) 2.19209e6 0.490984 0.245492 0.969399i \(-0.421050\pi\)
0.245492 + 0.969399i \(0.421050\pi\)
\(458\) −5.44268e6 −1.21241
\(459\) 113276. 0.0250960
\(460\) 2.58561e6 0.569729
\(461\) 5.20229e6 1.14010 0.570050 0.821610i \(-0.306924\pi\)
0.570050 + 0.821610i \(0.306924\pi\)
\(462\) 367359. 0.0800728
\(463\) 2.66624e6 0.578025 0.289013 0.957325i \(-0.406673\pi\)
0.289013 + 0.957325i \(0.406673\pi\)
\(464\) −66458.3 −0.0143303
\(465\) 395885. 0.0849057
\(466\) −1.84566e6 −0.393720
\(467\) −2.35578e6 −0.499853 −0.249926 0.968265i \(-0.580406\pi\)
−0.249926 + 0.968265i \(0.580406\pi\)
\(468\) −3.81638e6 −0.805447
\(469\) −571084. −0.119886
\(470\) 2.86247e6 0.597719
\(471\) 1.90642e6 0.395973
\(472\) 3.17914e6 0.656832
\(473\) 541534. 0.111294
\(474\) 7.26403e6 1.48502
\(475\) −1.20791e6 −0.245640
\(476\) 303905. 0.0614780
\(477\) 173755. 0.0349655
\(478\) 3.77668e6 0.756032
\(479\) −4.78329e6 −0.952551 −0.476276 0.879296i \(-0.658014\pi\)
−0.476276 + 0.879296i \(0.658014\pi\)
\(480\) 1.15537e6 0.228886
\(481\) 1.66727e6 0.328581
\(482\) −1.14786e7 −2.25046
\(483\) 632239. 0.123314
\(484\) 785646. 0.152445
\(485\) −918863. −0.177377
\(486\) −546517. −0.104957
\(487\) −3.31515e6 −0.633405 −0.316702 0.948525i \(-0.602576\pi\)
−0.316702 + 0.948525i \(0.602576\pi\)
\(488\) 7.30691e6 1.38894
\(489\) −3.07183e6 −0.580932
\(490\) 3.58147e6 0.673862
\(491\) −3.02276e6 −0.565847 −0.282924 0.959142i \(-0.591304\pi\)
−0.282924 + 0.959142i \(0.591304\pi\)
\(492\) −2.17327e6 −0.404763
\(493\) −74653.9 −0.0138336
\(494\) −1.57056e7 −2.89559
\(495\) −245025. −0.0449467
\(496\) 243384. 0.0444210
\(497\) 389610. 0.0707520
\(498\) −8.09909e6 −1.46340
\(499\) 4.46530e6 0.802785 0.401392 0.915906i \(-0.368526\pi\)
0.401392 + 0.915906i \(0.368526\pi\)
\(500\) 838448. 0.149986
\(501\) −1.62493e6 −0.289228
\(502\) −283774. −0.0502589
\(503\) −1.77528e6 −0.312858 −0.156429 0.987689i \(-0.549998\pi\)
−0.156429 + 0.987689i \(0.549998\pi\)
\(504\) −591861. −0.103787
\(505\) −4.63717e6 −0.809141
\(506\) 2.15846e6 0.374772
\(507\) 3.59683e6 0.621441
\(508\) 1.44248e7 2.47999
\(509\) −485180. −0.0830058 −0.0415029 0.999138i \(-0.513215\pi\)
−0.0415029 + 0.999138i \(0.513215\pi\)
\(510\) −323581. −0.0550880
\(511\) 443664. 0.0751627
\(512\) 1.59770e6 0.269353
\(513\) −1.40890e6 −0.236367
\(514\) 6.36739e6 1.06305
\(515\) 922269. 0.153228
\(516\) −2.16142e6 −0.357367
\(517\) 1.49691e6 0.246303
\(518\) 640556. 0.104890
\(519\) −5.78989e6 −0.943522
\(520\) 4.40061e6 0.713682
\(521\) 9.92932e6 1.60260 0.801300 0.598262i \(-0.204142\pi\)
0.801300 + 0.598262i \(0.204142\pi\)
\(522\) 360179. 0.0578552
\(523\) 5.04767e6 0.806931 0.403466 0.914995i \(-0.367805\pi\)
0.403466 + 0.914995i \(0.367805\pi\)
\(524\) −1.96889e7 −3.13251
\(525\) 205019. 0.0324635
\(526\) −1.57436e7 −2.48107
\(527\) 273398. 0.0428815
\(528\) −150638. −0.0235152
\(529\) −2.72156e6 −0.422842
\(530\) −496343. −0.0767525
\(531\) −1.28449e6 −0.197695
\(532\) −3.77990e6 −0.579031
\(533\) 3.95117e6 0.602432
\(534\) −3.21856e6 −0.488437
\(535\) −3.12491e6 −0.472011
\(536\) 3.14117e6 0.472258
\(537\) −2.40230e6 −0.359494
\(538\) 9.09467e6 1.35466
\(539\) 1.87291e6 0.277680
\(540\) 977966. 0.144324
\(541\) −4.46393e6 −0.655729 −0.327864 0.944725i \(-0.606329\pi\)
−0.327864 + 0.944725i \(0.606329\pi\)
\(542\) 2.55637e6 0.373788
\(543\) 2.53376e6 0.368779
\(544\) 797900. 0.115598
\(545\) −3.77438e6 −0.544319
\(546\) 2.66572e6 0.382678
\(547\) −6.62530e6 −0.946755 −0.473377 0.880860i \(-0.656965\pi\)
−0.473377 + 0.880860i \(0.656965\pi\)
\(548\) −9.81601e6 −1.39632
\(549\) −2.95227e6 −0.418047
\(550\) 699932. 0.0986619
\(551\) 928530. 0.130292
\(552\) −3.47754e6 −0.485763
\(553\) −3.17845e6 −0.441980
\(554\) −1.37696e6 −0.190611
\(555\) −427246. −0.0588769
\(556\) 452351. 0.0620568
\(557\) −717580. −0.0980014 −0.0490007 0.998799i \(-0.515604\pi\)
−0.0490007 + 0.998799i \(0.515604\pi\)
\(558\) −1.31905e6 −0.179340
\(559\) 3.92962e6 0.531889
\(560\) 126043. 0.0169843
\(561\) −169214. −0.0227002
\(562\) −3.31129e6 −0.442238
\(563\) 1.35097e7 1.79629 0.898143 0.439703i \(-0.144916\pi\)
0.898143 + 0.439703i \(0.144916\pi\)
\(564\) −5.97461e6 −0.790882
\(565\) 3.94924e6 0.520466
\(566\) 4.55444e6 0.597577
\(567\) 239134. 0.0312380
\(568\) −2.14300e6 −0.278709
\(569\) 1.35535e7 1.75498 0.877488 0.479598i \(-0.159218\pi\)
0.877488 + 0.479598i \(0.159218\pi\)
\(570\) 4.02463e6 0.518847
\(571\) 7.82130e6 1.00390 0.501948 0.864898i \(-0.332617\pi\)
0.501948 + 0.864898i \(0.332617\pi\)
\(572\) 5.70101e6 0.728554
\(573\) 8.40351e6 1.06924
\(574\) 1.51802e6 0.192308
\(575\) 1.20461e6 0.151942
\(576\) −4.20814e6 −0.528486
\(577\) 218443. 0.0273149 0.0136574 0.999907i \(-0.495653\pi\)
0.0136574 + 0.999907i \(0.495653\pi\)
\(578\) 1.29177e7 1.60830
\(579\) 2.70112e6 0.334847
\(580\) −644524. −0.0795553
\(581\) 3.54384e6 0.435546
\(582\) 3.06157e6 0.374659
\(583\) −259559. −0.0316275
\(584\) −2.44031e6 −0.296083
\(585\) −1.77802e6 −0.214806
\(586\) −4.61026e6 −0.554602
\(587\) −5.46003e6 −0.654033 −0.327017 0.945019i \(-0.606043\pi\)
−0.327017 + 0.945019i \(0.606043\pi\)
\(588\) −7.47531e6 −0.891632
\(589\) −3.40048e6 −0.403879
\(590\) 3.66925e6 0.433958
\(591\) −8.50446e6 −1.00156
\(592\) −262664. −0.0308033
\(593\) 1.41186e7 1.64875 0.824375 0.566044i \(-0.191527\pi\)
0.824375 + 0.566044i \(0.191527\pi\)
\(594\) 816401. 0.0949374
\(595\) 141586. 0.0163956
\(596\) 1.45855e7 1.68192
\(597\) −9.07385e6 −1.04197
\(598\) 1.56628e7 1.79108
\(599\) 8.02044e6 0.913338 0.456669 0.889637i \(-0.349042\pi\)
0.456669 + 0.889637i \(0.349042\pi\)
\(600\) −1.12768e6 −0.127881
\(601\) −1.20301e7 −1.35857 −0.679286 0.733874i \(-0.737710\pi\)
−0.679286 + 0.733874i \(0.737710\pi\)
\(602\) 1.50974e6 0.169790
\(603\) −1.26915e6 −0.142141
\(604\) −1.26942e7 −1.41584
\(605\) 366025. 0.0406558
\(606\) 1.54506e7 1.70909
\(607\) −1.58863e7 −1.75005 −0.875025 0.484078i \(-0.839155\pi\)
−0.875025 + 0.484078i \(0.839155\pi\)
\(608\) −9.92412e6 −1.08876
\(609\) −157600. −0.0172192
\(610\) 8.43338e6 0.917650
\(611\) 1.08623e7 1.17711
\(612\) 675384. 0.0728907
\(613\) 1.27701e7 1.37260 0.686301 0.727318i \(-0.259233\pi\)
0.686301 + 0.727318i \(0.259233\pi\)
\(614\) 9.24383e6 0.989535
\(615\) −1.01251e6 −0.107947
\(616\) 884137. 0.0938789
\(617\) 5.32363e6 0.562982 0.281491 0.959564i \(-0.409171\pi\)
0.281491 + 0.959564i \(0.409171\pi\)
\(618\) −3.07292e6 −0.323653
\(619\) −5.79882e6 −0.608293 −0.304147 0.952625i \(-0.598371\pi\)
−0.304147 + 0.952625i \(0.598371\pi\)
\(620\) 2.36038e6 0.246606
\(621\) 1.40506e6 0.146206
\(622\) −1.74724e7 −1.81083
\(623\) 1.40832e6 0.145372
\(624\) −1.09310e6 −0.112382
\(625\) 390625. 0.0400000
\(626\) 2.17302e7 2.21630
\(627\) 2.10466e6 0.213802
\(628\) 1.13666e7 1.15009
\(629\) −295056. −0.0297357
\(630\) −683105. −0.0685703
\(631\) −8.25264e6 −0.825124 −0.412562 0.910929i \(-0.635366\pi\)
−0.412562 + 0.910929i \(0.635366\pi\)
\(632\) 1.74826e7 1.74106
\(633\) 4.47731e6 0.444128
\(634\) −5.20816e6 −0.514590
\(635\) 6.72036e6 0.661391
\(636\) 1.03598e6 0.101556
\(637\) 1.35907e7 1.32707
\(638\) −538045. −0.0523320
\(639\) 865852. 0.0838864
\(640\) 7.91287e6 0.763632
\(641\) 1.14111e7 1.09694 0.548471 0.836169i \(-0.315210\pi\)
0.548471 + 0.836169i \(0.315210\pi\)
\(642\) 1.04119e7 0.996994
\(643\) −3.26961e6 −0.311866 −0.155933 0.987768i \(-0.549838\pi\)
−0.155933 + 0.987768i \(0.549838\pi\)
\(644\) 3.76959e6 0.358162
\(645\) −1.00698e6 −0.0953067
\(646\) 2.77941e6 0.262043
\(647\) −9.95068e6 −0.934527 −0.467264 0.884118i \(-0.654760\pi\)
−0.467264 + 0.884118i \(0.654760\pi\)
\(648\) −1.31532e6 −0.123054
\(649\) 1.91881e6 0.178822
\(650\) 5.07903e6 0.471517
\(651\) 577166. 0.0533763
\(652\) −1.83152e7 −1.68730
\(653\) 1.52022e7 1.39515 0.697577 0.716509i \(-0.254261\pi\)
0.697577 + 0.716509i \(0.254261\pi\)
\(654\) 1.25759e7 1.14973
\(655\) −9.17285e6 −0.835413
\(656\) −622474. −0.0564757
\(657\) 985979. 0.0891157
\(658\) 4.17324e6 0.375758
\(659\) −9.63232e6 −0.864007 −0.432004 0.901872i \(-0.642193\pi\)
−0.432004 + 0.901872i \(0.642193\pi\)
\(660\) −1.46091e6 −0.130546
\(661\) 2.04631e7 1.82166 0.910832 0.412778i \(-0.135442\pi\)
0.910832 + 0.412778i \(0.135442\pi\)
\(662\) 9.31016e6 0.825681
\(663\) −1.22790e6 −0.108487
\(664\) −1.94924e7 −1.71572
\(665\) −1.76102e6 −0.154422
\(666\) 1.42354e6 0.124361
\(667\) −925997. −0.0805926
\(668\) −9.68832e6 −0.840054
\(669\) 1.02722e7 0.887361
\(670\) 3.62543e6 0.312013
\(671\) 4.41018e6 0.378137
\(672\) 1.68443e6 0.143890
\(673\) 1.57773e6 0.134275 0.0671376 0.997744i \(-0.478613\pi\)
0.0671376 + 0.997744i \(0.478613\pi\)
\(674\) 2.59326e6 0.219885
\(675\) 455625. 0.0384900
\(676\) 2.14454e7 1.80496
\(677\) 6.92750e6 0.580904 0.290452 0.956890i \(-0.406194\pi\)
0.290452 + 0.956890i \(0.406194\pi\)
\(678\) −1.31585e7 −1.09934
\(679\) −1.33962e6 −0.111509
\(680\) −778776. −0.0645862
\(681\) 6.02506e6 0.497845
\(682\) 1.97044e6 0.162219
\(683\) −277554. −0.0227665 −0.0113833 0.999935i \(-0.503623\pi\)
−0.0113833 + 0.999935i \(0.503623\pi\)
\(684\) −8.40029e6 −0.686521
\(685\) −4.57318e6 −0.372385
\(686\) 1.08911e7 0.883609
\(687\) 5.29255e6 0.427832
\(688\) −619080. −0.0498627
\(689\) −1.88348e6 −0.151152
\(690\) −4.01366e6 −0.320935
\(691\) −2.12446e7 −1.69259 −0.846297 0.532712i \(-0.821173\pi\)
−0.846297 + 0.532712i \(0.821173\pi\)
\(692\) −3.45211e7 −2.74043
\(693\) −357225. −0.0282559
\(694\) −1.95206e7 −1.53849
\(695\) 210746. 0.0165500
\(696\) 866858. 0.0678305
\(697\) −699238. −0.0545184
\(698\) −3.59540e7 −2.79324
\(699\) 1.79475e6 0.138935
\(700\) 1.22238e6 0.0942893
\(701\) −1.06971e7 −0.822184 −0.411092 0.911594i \(-0.634853\pi\)
−0.411092 + 0.911594i \(0.634853\pi\)
\(702\) 5.92419e6 0.453718
\(703\) 3.66985e6 0.280066
\(704\) 6.28623e6 0.478034
\(705\) −2.78351e6 −0.210921
\(706\) −1.25561e7 −0.948072
\(707\) −6.76058e6 −0.508669
\(708\) −7.65853e6 −0.574199
\(709\) 6.52521e6 0.487505 0.243752 0.969838i \(-0.421622\pi\)
0.243752 + 0.969838i \(0.421622\pi\)
\(710\) −2.47337e6 −0.184138
\(711\) −7.06365e6 −0.524029
\(712\) −7.74625e6 −0.572653
\(713\) 3.39120e6 0.249821
\(714\) −471753. −0.0346313
\(715\) 2.65605e6 0.194299
\(716\) −1.43232e7 −1.04414
\(717\) −3.67250e6 −0.266786
\(718\) −3.64137e7 −2.63605
\(719\) −2.12413e7 −1.53235 −0.766176 0.642631i \(-0.777843\pi\)
−0.766176 + 0.642631i \(0.777843\pi\)
\(720\) 280112. 0.0201372
\(721\) 1.34459e6 0.0963276
\(722\) −1.16528e7 −0.831928
\(723\) 1.11620e7 0.794135
\(724\) 1.51071e7 1.07111
\(725\) −300278. −0.0212167
\(726\) −1.21956e6 −0.0858742
\(727\) −1.06687e7 −0.748646 −0.374323 0.927298i \(-0.622125\pi\)
−0.374323 + 0.927298i \(0.622125\pi\)
\(728\) 6.41571e6 0.448659
\(729\) 531441. 0.0370370
\(730\) −2.81652e6 −0.195617
\(731\) −695424. −0.0481345
\(732\) −1.76023e7 −1.21420
\(733\) −2.30788e7 −1.58655 −0.793274 0.608864i \(-0.791625\pi\)
−0.793274 + 0.608864i \(0.791625\pi\)
\(734\) 2.61544e7 1.79186
\(735\) −3.48268e6 −0.237791
\(736\) 9.89705e6 0.673459
\(737\) 1.89589e6 0.128572
\(738\) 3.37358e6 0.228008
\(739\) −2.27313e7 −1.53113 −0.765566 0.643358i \(-0.777541\pi\)
−0.765566 + 0.643358i \(0.777541\pi\)
\(740\) −2.54736e6 −0.171006
\(741\) 1.52724e7 1.02179
\(742\) −723625. −0.0482507
\(743\) −1.49328e7 −0.992360 −0.496180 0.868220i \(-0.665264\pi\)
−0.496180 + 0.868220i \(0.665264\pi\)
\(744\) −3.17462e6 −0.210261
\(745\) 6.79525e6 0.448554
\(746\) −4.24624e7 −2.79356
\(747\) 7.87568e6 0.516400
\(748\) −1.00891e6 −0.0659321
\(749\) −4.55584e6 −0.296732
\(750\) −1.30153e6 −0.0844890
\(751\) −1.92964e7 −1.24846 −0.624232 0.781239i \(-0.714588\pi\)
−0.624232 + 0.781239i \(0.714588\pi\)
\(752\) −1.71126e6 −0.110350
\(753\) 275946. 0.0177352
\(754\) −3.90431e6 −0.250101
\(755\) −5.91412e6 −0.377592
\(756\) 1.42579e6 0.0907300
\(757\) 1.35397e7 0.858756 0.429378 0.903125i \(-0.358733\pi\)
0.429378 + 0.903125i \(0.358733\pi\)
\(758\) −2.63616e7 −1.66648
\(759\) −2.09891e6 −0.132248
\(760\) 9.68626e6 0.608306
\(761\) 2.60669e7 1.63165 0.815825 0.578299i \(-0.196283\pi\)
0.815825 + 0.578299i \(0.196283\pi\)
\(762\) −2.23916e7 −1.39701
\(763\) −5.50271e6 −0.342188
\(764\) 5.01042e7 3.10557
\(765\) 314655. 0.0194393
\(766\) 2.57730e7 1.58706
\(767\) 1.39238e7 0.854611
\(768\) −1.14027e7 −0.697598
\(769\) −1.65354e7 −1.00832 −0.504162 0.863609i \(-0.668199\pi\)
−0.504162 + 0.863609i \(0.668199\pi\)
\(770\) 1.02044e6 0.0620242
\(771\) −6.19174e6 −0.375126
\(772\) 1.61049e7 0.972554
\(773\) 8.95978e6 0.539323 0.269661 0.962955i \(-0.413088\pi\)
0.269661 + 0.962955i \(0.413088\pi\)
\(774\) 3.35518e6 0.201309
\(775\) 1.09968e6 0.0657677
\(776\) 7.36841e6 0.439258
\(777\) −622887. −0.0370132
\(778\) −3.65841e7 −2.16692
\(779\) 8.69698e6 0.513482
\(780\) −1.06011e7 −0.623897
\(781\) −1.29343e6 −0.0758781
\(782\) −2.77183e6 −0.162088
\(783\) −350244. −0.0204158
\(784\) −2.14110e6 −0.124407
\(785\) 5.29561e6 0.306720
\(786\) 3.05631e7 1.76458
\(787\) −2.31723e7 −1.33362 −0.666812 0.745226i \(-0.732341\pi\)
−0.666812 + 0.745226i \(0.732341\pi\)
\(788\) −5.07061e7 −2.90901
\(789\) 1.53093e7 0.875512
\(790\) 2.01779e7 1.15029
\(791\) 5.75765e6 0.327193
\(792\) 1.96487e6 0.111306
\(793\) 3.20023e7 1.80717
\(794\) 5.13994e7 2.89339
\(795\) 482652. 0.0270842
\(796\) −5.41010e7 −3.02638
\(797\) −3.40811e7 −1.90050 −0.950250 0.311489i \(-0.899172\pi\)
−0.950250 + 0.311489i \(0.899172\pi\)
\(798\) 5.86757e6 0.326175
\(799\) −1.92230e6 −0.106525
\(800\) 3.20936e6 0.177294
\(801\) 3.12978e6 0.172358
\(802\) 2.58470e6 0.141897
\(803\) −1.47288e6 −0.0806082
\(804\) −7.56706e6 −0.412845
\(805\) 1.75622e6 0.0955188
\(806\) 1.42984e7 0.775265
\(807\) −8.84380e6 −0.478030
\(808\) 3.71857e7 2.00377
\(809\) 1.77957e6 0.0955969 0.0477985 0.998857i \(-0.484779\pi\)
0.0477985 + 0.998857i \(0.484779\pi\)
\(810\) −1.51810e6 −0.0812996
\(811\) 1.28099e7 0.683900 0.341950 0.939718i \(-0.388913\pi\)
0.341950 + 0.939718i \(0.388913\pi\)
\(812\) −939660. −0.0500127
\(813\) −2.48585e6 −0.131901
\(814\) −2.12653e6 −0.112489
\(815\) −8.53287e6 −0.449988
\(816\) 193445. 0.0101703
\(817\) 8.64955e6 0.453355
\(818\) −4.78618e7 −2.50095
\(819\) −2.59219e6 −0.135038
\(820\) −6.03687e6 −0.313528
\(821\) 1.47980e7 0.766205 0.383102 0.923706i \(-0.374856\pi\)
0.383102 + 0.923706i \(0.374856\pi\)
\(822\) 1.52374e7 0.786561
\(823\) −1.17405e7 −0.604207 −0.302103 0.953275i \(-0.597689\pi\)
−0.302103 + 0.953275i \(0.597689\pi\)
\(824\) −7.39572e6 −0.379457
\(825\) −680625. −0.0348155
\(826\) 5.34945e6 0.272809
\(827\) 1.61763e7 0.822461 0.411231 0.911531i \(-0.365099\pi\)
0.411231 + 0.911531i \(0.365099\pi\)
\(828\) 8.37738e6 0.424651
\(829\) −1.56514e7 −0.790984 −0.395492 0.918470i \(-0.629426\pi\)
−0.395492 + 0.918470i \(0.629426\pi\)
\(830\) −2.24975e7 −1.13354
\(831\) 1.33898e6 0.0672623
\(832\) 4.56158e7 2.28458
\(833\) −2.40514e6 −0.120096
\(834\) −702188. −0.0349573
\(835\) −4.51370e6 −0.224035
\(836\) 1.25486e7 0.620982
\(837\) 1.28267e6 0.0632850
\(838\) 5.59161e7 2.75060
\(839\) 2.70692e7 1.32761 0.663806 0.747905i \(-0.268940\pi\)
0.663806 + 0.747905i \(0.268940\pi\)
\(840\) −1.64406e6 −0.0803931
\(841\) −2.02803e7 −0.988746
\(842\) 8.28707e6 0.402829
\(843\) 3.21995e6 0.156056
\(844\) 2.66951e7 1.28996
\(845\) 9.99119e6 0.481366
\(846\) 9.27442e6 0.445513
\(847\) 533632. 0.0255584
\(848\) 296727. 0.0141699
\(849\) −4.42881e6 −0.210871
\(850\) −898836. −0.0426710
\(851\) −3.65984e6 −0.173236
\(852\) 5.16246e6 0.243645
\(853\) −2.43979e7 −1.14810 −0.574050 0.818820i \(-0.694629\pi\)
−0.574050 + 0.818820i \(0.694629\pi\)
\(854\) 1.22951e7 0.576884
\(855\) −3.91362e6 −0.183089
\(856\) 2.50588e7 1.16889
\(857\) 1.33841e7 0.622496 0.311248 0.950329i \(-0.399253\pi\)
0.311248 + 0.950329i \(0.399253\pi\)
\(858\) −8.84971e6 −0.410403
\(859\) −2.60324e7 −1.20374 −0.601869 0.798595i \(-0.705577\pi\)
−0.601869 + 0.798595i \(0.705577\pi\)
\(860\) −6.00394e6 −0.276815
\(861\) −1.47615e6 −0.0678612
\(862\) 1.48887e7 0.682479
\(863\) −2.45378e7 −1.12152 −0.560762 0.827977i \(-0.689492\pi\)
−0.560762 + 0.827977i \(0.689492\pi\)
\(864\) 3.74340e6 0.170601
\(865\) −1.60830e7 −0.730849
\(866\) −1.72185e7 −0.780190
\(867\) −1.25614e7 −0.567532
\(868\) 3.44123e6 0.155030
\(869\) 1.05519e7 0.474002
\(870\) 1.00050e6 0.0448144
\(871\) 1.37575e7 0.614460
\(872\) 3.02669e7 1.34796
\(873\) −2.97712e6 −0.132209
\(874\) 3.44755e7 1.52662
\(875\) 569497. 0.0251461
\(876\) 5.87870e6 0.258834
\(877\) 1.99034e7 0.873833 0.436917 0.899502i \(-0.356070\pi\)
0.436917 + 0.899502i \(0.356070\pi\)
\(878\) −2.54568e7 −1.11447
\(879\) 4.48308e6 0.195706
\(880\) −418438. −0.0182148
\(881\) −3.90880e7 −1.69669 −0.848346 0.529442i \(-0.822401\pi\)
−0.848346 + 0.529442i \(0.822401\pi\)
\(882\) 1.16040e7 0.502267
\(883\) 2.14490e7 0.925776 0.462888 0.886417i \(-0.346813\pi\)
0.462888 + 0.886417i \(0.346813\pi\)
\(884\) −7.32109e6 −0.315098
\(885\) −3.56803e6 −0.153134
\(886\) −2.92287e7 −1.25091
\(887\) 6.08624e6 0.259741 0.129870 0.991531i \(-0.458544\pi\)
0.129870 + 0.991531i \(0.458544\pi\)
\(888\) 3.42610e6 0.145803
\(889\) 9.79769e6 0.415786
\(890\) −8.94045e6 −0.378342
\(891\) −793881. −0.0335013
\(892\) 6.12462e7 2.57731
\(893\) 2.39091e7 1.00331
\(894\) −2.26412e7 −0.947447
\(895\) −6.67306e6 −0.278463
\(896\) 1.15363e7 0.480060
\(897\) −1.52307e7 −0.632031
\(898\) 1.18012e6 0.0488356
\(899\) −845336. −0.0348843
\(900\) 2.71657e6 0.111793
\(901\) 333319. 0.0136788
\(902\) −5.03955e6 −0.206241
\(903\) −1.46810e6 −0.0599149
\(904\) −3.16691e7 −1.28889
\(905\) 7.03823e6 0.285655
\(906\) 1.97053e7 0.797560
\(907\) −2.14459e7 −0.865618 −0.432809 0.901486i \(-0.642477\pi\)
−0.432809 + 0.901486i \(0.642477\pi\)
\(908\) 3.59232e7 1.44597
\(909\) −1.50244e7 −0.603098
\(910\) 7.40479e6 0.296421
\(911\) 1.03374e7 0.412682 0.206341 0.978480i \(-0.433844\pi\)
0.206341 + 0.978480i \(0.433844\pi\)
\(912\) −2.40603e6 −0.0957888
\(913\) −1.17649e7 −0.467102
\(914\) −2.02884e7 −0.803310
\(915\) −8.20074e6 −0.323818
\(916\) 3.15557e7 1.24262
\(917\) −1.33732e7 −0.525185
\(918\) −1.04840e6 −0.0410602
\(919\) −4.75142e7 −1.85581 −0.927907 0.372811i \(-0.878394\pi\)
−0.927907 + 0.372811i \(0.878394\pi\)
\(920\) −9.65984e6 −0.376271
\(921\) −8.98884e6 −0.349184
\(922\) −4.81488e7 −1.86534
\(923\) −9.38575e6 −0.362631
\(924\) −2.12988e6 −0.0820683
\(925\) −1.18679e6 −0.0456059
\(926\) −2.46769e7 −0.945720
\(927\) 2.98815e6 0.114210
\(928\) −2.46707e6 −0.0940398
\(929\) 6.08962e6 0.231500 0.115750 0.993278i \(-0.463073\pi\)
0.115750 + 0.993278i \(0.463073\pi\)
\(930\) −3.66404e6 −0.138916
\(931\) 2.99146e7 1.13112
\(932\) 1.07008e7 0.403532
\(933\) 1.69904e7 0.638999
\(934\) 2.18034e7 0.817820
\(935\) −470040. −0.0175835
\(936\) 1.42580e7 0.531947
\(937\) −2.73781e6 −0.101872 −0.0509360 0.998702i \(-0.516220\pi\)
−0.0509360 + 0.998702i \(0.516220\pi\)
\(938\) 5.28556e6 0.196148
\(939\) −2.11308e7 −0.782082
\(940\) −1.65961e7 −0.612615
\(941\) 3.73849e7 1.37633 0.688165 0.725554i \(-0.258417\pi\)
0.688165 + 0.725554i \(0.258417\pi\)
\(942\) −1.76445e7 −0.647861
\(943\) −8.67326e6 −0.317617
\(944\) −2.19358e6 −0.0801166
\(945\) 664261. 0.0241969
\(946\) −5.01206e6 −0.182091
\(947\) −4.08261e7 −1.47932 −0.739661 0.672979i \(-0.765014\pi\)
−0.739661 + 0.672979i \(0.765014\pi\)
\(948\) −4.21156e7 −1.52203
\(949\) −1.06879e7 −0.385237
\(950\) 1.11795e7 0.401897
\(951\) 5.06449e6 0.181587
\(952\) −1.13539e6 −0.0406024
\(953\) −4.60553e7 −1.64266 −0.821330 0.570453i \(-0.806768\pi\)
−0.821330 + 0.570453i \(0.806768\pi\)
\(954\) −1.60815e6 −0.0572079
\(955\) 2.33431e7 0.828228
\(956\) −2.18965e7 −0.774873
\(957\) 523204. 0.0184668
\(958\) 4.42709e7 1.55849
\(959\) −6.66730e6 −0.234101
\(960\) −1.16893e7 −0.409363
\(961\) −2.55333e7 −0.891865
\(962\) −1.54311e7 −0.537599
\(963\) −1.01247e7 −0.351817
\(964\) 6.65509e7 2.30654
\(965\) 7.50310e6 0.259372
\(966\) −5.85156e6 −0.201757
\(967\) 1.48162e7 0.509532 0.254766 0.967003i \(-0.418002\pi\)
0.254766 + 0.967003i \(0.418002\pi\)
\(968\) −2.93517e6 −0.100680
\(969\) −2.70275e6 −0.0924689
\(970\) 8.50436e6 0.290210
\(971\) −3.27641e7 −1.11519 −0.557597 0.830112i \(-0.688277\pi\)
−0.557597 + 0.830112i \(0.688277\pi\)
\(972\) 3.16861e6 0.107573
\(973\) 307250. 0.0104042
\(974\) 3.06828e7 1.03633
\(975\) −4.93893e6 −0.166388
\(976\) −5.04170e6 −0.169415
\(977\) −4.06772e6 −0.136337 −0.0681686 0.997674i \(-0.521716\pi\)
−0.0681686 + 0.997674i \(0.521716\pi\)
\(978\) 2.84307e7 0.950476
\(979\) −4.67535e6 −0.155904
\(980\) −2.07647e7 −0.690655
\(981\) −1.22290e7 −0.405712
\(982\) 2.79765e7 0.925795
\(983\) 3.76095e7 1.24141 0.620704 0.784045i \(-0.286847\pi\)
0.620704 + 0.784045i \(0.286847\pi\)
\(984\) 8.11934e6 0.267321
\(985\) −2.36235e7 −0.775807
\(986\) 690944. 0.0226335
\(987\) −4.05812e6 −0.132596
\(988\) 9.10583e7 2.96775
\(989\) −8.62596e6 −0.280425
\(990\) 2.26778e6 0.0735382
\(991\) 1.02888e7 0.332797 0.166399 0.986059i \(-0.446786\pi\)
0.166399 + 0.986059i \(0.446786\pi\)
\(992\) 9.03495e6 0.291505
\(993\) −9.05334e6 −0.291364
\(994\) −3.60596e6 −0.115759
\(995\) −2.52052e7 −0.807108
\(996\) 4.69571e7 1.49987
\(997\) 4.37776e7 1.39481 0.697403 0.716680i \(-0.254339\pi\)
0.697403 + 0.716680i \(0.254339\pi\)
\(998\) −4.13277e7 −1.31345
\(999\) −1.38428e6 −0.0438843
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.a.1.1 3
3.2 odd 2 495.6.a.e.1.3 3
5.4 even 2 825.6.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.1 3 1.1 even 1 trivial
495.6.a.e.1.3 3 3.2 odd 2
825.6.a.j.1.3 3 5.4 even 2