Properties

Label 825.6.a.j.1.3
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 1.3
Root \(7.25531\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.25531 q^{2} -9.00000 q^{3} +53.6607 q^{4} -83.2977 q^{6} -36.4478 q^{7} +200.476 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.25531 q^{2} -9.00000 q^{3} +53.6607 q^{4} -83.2977 q^{6} -36.4478 q^{7} +200.476 q^{8} +81.0000 q^{9} -121.000 q^{11} -482.946 q^{12} +878.032 q^{13} -337.336 q^{14} +138.327 q^{16} -155.385 q^{17} +749.680 q^{18} -1932.65 q^{19} +328.030 q^{21} -1119.89 q^{22} -1927.38 q^{23} -1804.29 q^{24} +8126.46 q^{26} -729.000 q^{27} -1955.81 q^{28} -480.444 q^{29} +1759.49 q^{31} -5134.98 q^{32} +1089.00 q^{33} -1438.14 q^{34} +4346.51 q^{36} +1898.87 q^{37} -17887.3 q^{38} -7902.29 q^{39} -4500.03 q^{41} +3036.02 q^{42} +4475.49 q^{43} -6492.94 q^{44} -17838.5 q^{46} +12371.2 q^{47} -1244.94 q^{48} -15478.6 q^{49} +1398.47 q^{51} +47115.8 q^{52} -2145.12 q^{53} -6747.12 q^{54} -7306.92 q^{56} +17393.8 q^{57} -4446.66 q^{58} -15857.9 q^{59} -36447.7 q^{61} +16284.6 q^{62} -2952.27 q^{63} -51952.3 q^{64} +10079.0 q^{66} +15668.5 q^{67} -8338.07 q^{68} +17346.4 q^{69} +10689.5 q^{71} +16238.6 q^{72} -12172.6 q^{73} +17574.6 q^{74} -103707. q^{76} +4410.19 q^{77} -73138.1 q^{78} -87205.6 q^{79} +6561.00 q^{81} -41649.2 q^{82} -97230.6 q^{83} +17602.3 q^{84} +41422.0 q^{86} +4324.00 q^{87} -24257.6 q^{88} +38639.2 q^{89} -32002.4 q^{91} -103424. q^{92} -15835.4 q^{93} +114499. q^{94} +46214.8 q^{96} +36754.5 q^{97} -143259. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 27 q^{3} + 25 q^{4} - 63 q^{6} + 172 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.25531 1.63612 0.818061 0.575131i \(-0.195049\pi\)
0.818061 + 0.575131i \(0.195049\pi\)
\(3\) −9.00000 −0.577350
\(4\) 53.6607 1.67690
\(5\) 0 0
\(6\) −83.2977 −0.944616
\(7\) −36.4478 −0.281142 −0.140571 0.990071i \(-0.544894\pi\)
−0.140571 + 0.990071i \(0.544894\pi\)
\(8\) 200.476 1.10749
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −482.946 −0.968156
\(13\) 878.032 1.44096 0.720480 0.693476i \(-0.243921\pi\)
0.720480 + 0.693476i \(0.243921\pi\)
\(14\) −337.336 −0.459983
\(15\) 0 0
\(16\) 138.327 0.135085
\(17\) −155.385 −0.130403 −0.0652014 0.997872i \(-0.520769\pi\)
−0.0652014 + 0.997872i \(0.520769\pi\)
\(18\) 749.680 0.545374
\(19\) −1932.65 −1.22820 −0.614100 0.789228i \(-0.710481\pi\)
−0.614100 + 0.789228i \(0.710481\pi\)
\(20\) 0 0
\(21\) 328.030 0.162318
\(22\) −1119.89 −0.493309
\(23\) −1927.38 −0.759709 −0.379855 0.925046i \(-0.624026\pi\)
−0.379855 + 0.925046i \(0.624026\pi\)
\(24\) −1804.29 −0.639407
\(25\) 0 0
\(26\) 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 \(0.516892\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) 4346.51 0.558965
\(37\) 1898.87 0.228029 0.114015 0.993479i \(-0.463629\pi\)
0.114015 + 0.993479i \(0.463629\pi\)
\(38\) −17887.3 −2.00949
\(39\) −7902.29 −0.831939
\(40\) 0 0
\(41\) −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 \(-0.440914\pi\)
0.184561 + 0.982821i \(0.440914\pi\)
\(44\) −6492.94 −0.505603
\(45\) 0 0
\(46\) −17838.5 −1.24298
\(47\) 12371.2 0.816895 0.408448 0.912782i \(-0.366070\pi\)
0.408448 + 0.912782i \(0.366070\pi\)
\(48\) −1244.94 −0.0779912
\(49\) −15478.6 −0.920959
\(50\) 0 0
\(51\) 1398.47 0.0752881
\(52\) 47115.8 2.41634
\(53\) −2145.12 −0.104897 −0.0524483 0.998624i \(-0.516702\pi\)
−0.0524483 + 0.998624i \(0.516702\pi\)
\(54\) −6747.12 −0.314872
\(55\) 0 0
\(56\) −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 \(0.595834\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) 10079.0 0.284812
\(67\) 15668.5 0.426424 0.213212 0.977006i \(-0.431607\pi\)
0.213212 + 0.977006i \(0.431607\pi\)
\(68\) −8338.07 −0.218672
\(69\) 17346.4 0.438618
\(70\) 0 0
\(71\) 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 \(-0.542677\pi\)
−0.133674 + 0.991025i \(0.542677\pi\)
\(74\) 17574.6 0.373084
\(75\) 0 0
\(76\) −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 \(0.787874\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −41649.2 −0.684024
\(83\) −97230.6 −1.54920 −0.774600 0.632451i \(-0.782049\pi\)
−0.774600 + 0.632451i \(0.782049\pi\)
\(84\) 17602.3 0.272190
\(85\) 0 0
\(86\) 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 \(0.416759\pi\)
\(90\) 0 0
\(91\) −32002.4 −0.405115
\(92\) −103424. −1.27395
\(93\) −15835.4 −0.189855
\(94\) 114499. 1.33654
\(95\) 0 0
\(96\) 46214.8 0.511804
\(97\) 36754.5 0.396626 0.198313 0.980139i \(-0.436454\pi\)
0.198313 + 0.980139i \(0.436454\pi\)
\(98\) −143259. −1.50680
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −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 \(-0.554801\pi\)
−0.171315 + 0.985216i \(0.554801\pi\)
\(104\) 176025. 1.59584
\(105\) 0 0
\(106\) −19853.7 −0.171624
\(107\) 124996. 1.05545 0.527725 0.849415i \(-0.323045\pi\)
0.527725 + 0.849415i \(0.323045\pi\)
\(108\) −39118.6 −0.322719
\(109\) −150975. −1.21714 −0.608568 0.793502i \(-0.708256\pi\)
−0.608568 + 0.793502i \(0.708256\pi\)
\(110\) 0 0
\(111\) −17089.8 −0.131653
\(112\) −5041.71 −0.0379780
\(113\) −157970. −1.16380 −0.581899 0.813261i \(-0.697690\pi\)
−0.581899 + 0.813261i \(0.697690\pi\)
\(114\) 160985. 1.16018
\(115\) 0 0
\(116\) −25781.0 −0.177891
\(117\) 71120.6 0.480320
\(118\) −146770. −0.970359
\(119\) 5663.45 0.0366618
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −337335. −2.05193
\(123\) 40500.3 0.241377
\(124\) 94415.4 0.551428
\(125\) 0 0
\(126\) −27324.2 −0.153328
\(127\) −268814. −1.47891 −0.739457 0.673204i \(-0.764918\pi\)
−0.739457 + 0.673204i \(0.764918\pi\)
\(128\) −316515. −1.70753
\(129\) −40279.4 −0.213112
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) −31151.0 −0.144419
\(137\) 182927. 0.832678 0.416339 0.909209i \(-0.363313\pi\)
0.416339 + 0.909209i \(0.363313\pi\)
\(138\) 160546. 0.717633
\(139\) 8429.85 0.0370069 0.0185035 0.999829i \(-0.494110\pi\)
0.0185035 + 0.999829i \(0.494110\pi\)
\(140\) 0 0
\(141\) −111341. −0.471635
\(142\) 98934.8 0.411745
\(143\) −106242. −0.434466
\(144\) 11204.5 0.0450282
\(145\) 0 0
\(146\) −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 \(0.332782\pi\)
\(150\) 0 0
\(151\) −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
\(155\) 0 0
\(156\) −424042. −1.39508
\(157\) −211824. −0.685846 −0.342923 0.939364i \(-0.611417\pi\)
−0.342923 + 0.939364i \(0.611417\pi\)
\(158\) −807114. −2.57213
\(159\) 19306.1 0.0605621
\(160\) 0 0
\(161\) 70248.7 0.213586
\(162\) 60724.1 0.181791
\(163\) 341315. 1.00620 0.503102 0.864227i \(-0.332192\pi\)
0.503102 + 0.864227i \(0.332192\pi\)
\(164\) −241475. −0.701071
\(165\) 0 0
\(166\) −899899. −2.53468
\(167\) 180548. 0.500958 0.250479 0.968122i \(-0.419412\pi\)
0.250479 + 0.968122i \(0.419412\pi\)
\(168\) 65762.3 0.179764
\(169\) 399647. 1.07637
\(170\) 0 0
\(171\) −156545. −0.409400
\(172\) 240158. 0.618978
\(173\) 643322. 1.63423 0.817114 0.576476i \(-0.195572\pi\)
0.817114 + 0.576476i \(0.195572\pi\)
\(174\) 40019.9 0.100208
\(175\) 0 0
\(176\) −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 \(0.600775\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −146561. −0.310626
\(187\) 18801.6 0.0393179
\(188\) 663846. 1.36985
\(189\) 26570.5 0.0541059
\(190\) 0 0
\(191\) 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 \(-0.593651\pi\)
−0.289986 + 0.957031i \(0.593651\pi\)
\(194\) 340174. 0.648929
\(195\) 0 0
\(196\) −830590. −1.54435
\(197\) 944940. 1.73476 0.867378 0.497649i \(-0.165803\pi\)
0.867378 + 0.497649i \(0.165803\pi\)
\(198\) −90711.2 −0.164436
\(199\) −1.00821e6 −1.80475 −0.902374 0.430954i \(-0.858177\pi\)
−0.902374 + 0.430954i \(0.858177\pi\)
\(200\) 0 0
\(201\) −141017. −0.246196
\(202\) −1.71674e6 −2.96023
\(203\) 17511.1 0.0298246
\(204\) 75042.6 0.126250
\(205\) 0 0
\(206\) −341435. −0.560583
\(207\) −156118. −0.253236
\(208\) 121455. 0.194652
\(209\) 233851. 0.370316
\(210\) 0 0
\(211\) 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
\(215\) 0 0
\(216\) −146147. −0.213136
\(217\) −64129.5 −0.0924504
\(218\) −1.39732e6 −1.99138
\(219\) 109553. 0.154353
\(220\) 0 0
\(221\) −136433. −0.187905
\(222\) −158172. −0.215400
\(223\) −1.14136e6 −1.53695 −0.768477 0.639878i \(-0.778985\pi\)
−0.768477 + 0.639878i \(0.778985\pi\)
\(224\) 187159. 0.249224
\(225\) 0 0
\(226\) −1.46206e6 −1.90411
\(227\) −669451. −0.862292 −0.431146 0.902282i \(-0.641891\pi\)
−0.431146 + 0.902282i \(0.641891\pi\)
\(228\) 933366. 1.18909
\(229\) 588061. 0.741026 0.370513 0.928827i \(-0.379182\pi\)
0.370513 + 0.928827i \(0.379182\pi\)
\(230\) 0 0
\(231\) −39691.7 −0.0489406
\(232\) −96317.6 −0.117486
\(233\) −199417. −0.240642 −0.120321 0.992735i \(-0.538392\pi\)
−0.120321 + 0.992735i \(0.538392\pi\)
\(234\) 658243. 0.785862
\(235\) 0 0
\(236\) −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 \(0.574214\pi\)
\(240\) 0 0
\(241\) 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
\(245\) 0 0
\(246\) 374842. 0.394922
\(247\) −1.69693e6 −1.76979
\(248\) 352736. 0.364183
\(249\) 875075. 0.894431
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) −1.26697e6 −1.20828
\(257\) 687971. 0.649737 0.324868 0.945759i \(-0.394680\pi\)
0.324868 + 0.945759i \(0.394680\pi\)
\(258\) −372798. −0.348678
\(259\) −69209.6 −0.0641087
\(260\) 0 0
\(261\) −38916.0 −0.0353612
\(262\) −3.39590e6 −3.05634
\(263\) −1.70103e6 −1.51643 −0.758216 0.652004i \(-0.773929\pi\)
−0.758216 + 0.652004i \(0.773929\pi\)
\(264\) 218319. 0.192788
\(265\) 0 0
\(266\) 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 \(0.635864\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) 930820. 0.735517
\(277\) −148776. −0.116502 −0.0582509 0.998302i \(-0.518552\pi\)
−0.0582509 + 0.998302i \(0.518552\pi\)
\(278\) 78020.8 0.0605478
\(279\) 142519. 0.109613
\(280\) 0 0
\(281\) 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 \(-0.441542\pi\)
0.182620 + 0.983184i \(0.441542\pi\)
\(284\) 573607. 0.422006
\(285\) 0 0
\(286\) −983301. −0.710839
\(287\) 164016. 0.117539
\(288\) −415934. −0.295490
\(289\) −1.39571e6 −0.982995
\(290\) 0 0
\(291\) −330791. −0.228992
\(292\) −653189. −0.448314
\(293\) −498120. −0.338973 −0.169487 0.985532i \(-0.554211\pi\)
−0.169487 + 0.985532i \(0.554211\pi\)
\(294\) 1.28933e6 0.869952
\(295\) 0 0
\(296\) 380678. 0.252539
\(297\) 88209.0 0.0580259
\(298\) 2.51568e6 1.64103
\(299\) −1.69230e6 −1.09471
\(300\) 0 0
\(301\) −163122. −0.103776
\(302\) −2.18948e6 −1.38141
\(303\) 1.66938e6 1.04460
\(304\) −267337. −0.165911
\(305\) 0 0
\(306\) −116489. −0.0711183
\(307\) 998760. 0.604805 0.302402 0.953180i \(-0.402211\pi\)
0.302402 + 0.953180i \(0.402211\pi\)
\(308\) 236654. 0.142147
\(309\) 332017. 0.197817
\(310\) 0 0
\(311\) 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 \(-0.263149\pi\)
0.677303 + 0.735704i \(0.263149\pi\)
\(314\) −1.96050e6 −1.12213
\(315\) 0 0
\(316\) −4.67951e6 −2.63623
\(317\) −562721. −0.314518 −0.157259 0.987557i \(-0.550266\pi\)
−0.157259 + 0.987557i \(0.550266\pi\)
\(318\) 178684. 0.0990870
\(319\) 58133.7 0.0319854
\(320\) 0 0
\(321\) −1.12497e6 −0.609364
\(322\) 650173. 0.349454
\(323\) 300305. 0.160161
\(324\) 352068. 0.186322
\(325\) 0 0
\(326\) 3.15897e6 1.64627
\(327\) 1.35878e6 0.702713
\(328\) −902149. −0.463013
\(329\) −450902. −0.229664
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) 45375.4 0.0219266
\(337\) 280192. 0.134394 0.0671971 0.997740i \(-0.478594\pi\)
0.0671971 + 0.997740i \(0.478594\pi\)
\(338\) 3.69886e6 1.76107
\(339\) 1.42173e6 0.671919
\(340\) 0 0
\(341\) −212898. −0.0991485
\(342\) −1.44887e6 −0.669829
\(343\) 1.17674e6 0.540063
\(344\) 897229. 0.408796
\(345\) 0 0
\(346\) 5.95414e6 2.67380
\(347\) −2.10913e6 −0.940328 −0.470164 0.882579i \(-0.655805\pi\)
−0.470164 + 0.882579i \(0.655805\pi\)
\(348\) 232029. 0.102705
\(349\) 3.88469e6 1.70723 0.853617 0.520901i \(-0.174404\pi\)
0.853617 + 0.520901i \(0.174404\pi\)
\(350\) 0 0
\(351\) −640085. −0.277313
\(352\) 621333. 0.267281
\(353\) −1.35663e6 −0.579463 −0.289732 0.957108i \(-0.593566\pi\)
−0.289732 + 0.957108i \(0.593566\pi\)
\(354\) 1.32093e6 0.560237
\(355\) 0 0
\(356\) 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 \(0.201854\pi\)
\(360\) 0 0
\(361\) 1.25904e6 0.508476
\(362\) 2.60564e6 1.04506
\(363\) −131769. −0.0524864
\(364\) −1.71727e6 −0.679336
\(365\) 0 0
\(366\) 3.03602e6 1.18468
\(367\) 2.82588e6 1.09519 0.547594 0.836744i \(-0.315544\pi\)
0.547594 + 0.836744i \(0.315544\pi\)
\(368\) −266608. −0.102625
\(369\) −364502. −0.139359
\(370\) 0 0
\(371\) 78184.9 0.0294909
\(372\) −849738. −0.318367
\(373\) −4.58790e6 −1.70743 −0.853713 0.520744i \(-0.825655\pi\)
−0.853713 + 0.520744i \(0.825655\pi\)
\(374\) 174015. 0.0643290
\(375\) 0 0
\(376\) 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 \(0.329913\pi\)
\(380\) 0 0
\(381\) 2.41933e6 0.853852
\(382\) 8.64190e6 3.03006
\(383\) 2.78467e6 0.970013 0.485006 0.874511i \(-0.338817\pi\)
0.485006 + 0.874511i \(0.338817\pi\)
\(384\) 2.84863e6 0.985845
\(385\) 0 0
\(386\) −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 \(0.269618\pi\)
\(390\) 0 0
\(391\) 299486. 0.0990682
\(392\) −3.10308e6 −1.01995
\(393\) 3.30223e6 1.07851
\(394\) 8.74571e6 2.83827
\(395\) 0 0
\(396\) −525928. −0.168534
\(397\) 5.55351e6 1.76844 0.884221 0.467068i \(-0.154690\pi\)
0.884221 + 0.467068i \(0.154690\pi\)
\(398\) −9.33125e6 −2.95279
\(399\) −633968. −0.199359
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 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 \(0.223088\pi\)
\(410\) 0 0
\(411\) −1.64635e6 −0.480747
\(412\) −1.97958e6 −0.574554
\(413\) 577987. 0.166741
\(414\) −1.44492e6 −0.414326
\(415\) 0 0
\(416\) −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 \(0.817788\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −890413. −0.237721
\(427\) 1.32844e6 0.352592
\(428\) 6.70738e6 1.76988
\(429\) 956177. 0.250839
\(430\) 0 0
\(431\) −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 \(-0.576632\pi\)
−0.238427 + 0.971161i \(0.576632\pi\)
\(434\) −593538. −0.151260
\(435\) 0 0
\(436\) −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 \(0.389376\pi\)
\(440\) 0 0
\(441\) −1.25376e6 −0.306986
\(442\) −1.26273e6 −0.307436
\(443\) −3.15804e6 −0.764555 −0.382278 0.924048i \(-0.624860\pi\)
−0.382278 + 0.924048i \(0.624860\pi\)
\(444\) −917051. −0.220768
\(445\) 0 0
\(446\) −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 \(0.504751\pi\)
\(450\) 0 0
\(451\) 544504. 0.126055
\(452\) −8.47675e6 −1.95157
\(453\) 2.12908e6 0.487469
\(454\) −6.19598e6 −1.41082
\(455\) 0 0
\(456\) 3.48705e6 0.785319
\(457\) −2.19209e6 −0.490984 −0.245492 0.969399i \(-0.578950\pi\)
−0.245492 + 0.969399i \(0.578950\pi\)
\(458\) 5.44268e6 1.21241
\(459\) 113276. 0.0250960
\(460\) 0 0
\(461\) 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 \(-0.593327\pi\)
−0.289013 + 0.957325i \(0.593327\pi\)
\(464\) −66458.3 −0.0143303
\(465\) 0 0
\(466\) −1.84566e6 −0.393720
\(467\) 2.35578e6 0.499853 0.249926 0.968265i \(-0.419594\pi\)
0.249926 + 0.968265i \(0.419594\pi\)
\(468\) 3.81638e6 0.805447
\(469\) −571084. −0.119886
\(470\) 0 0
\(471\) 1.90642e6 0.395973
\(472\) −3.17914e6 −0.656832
\(473\) −541534. −0.111294
\(474\) 7.26403e6 1.48502
\(475\) 0 0
\(476\) 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 \(0.658014\pi\)
\(480\) 0 0
\(481\) 1.66727e6 0.328581
\(482\) 1.14786e7 2.25046
\(483\) −632239. −0.123314
\(484\) 785646. 0.152445
\(485\) 0 0
\(486\) −546517. −0.104957
\(487\) 3.31515e6 0.633405 0.316702 0.948525i \(-0.397424\pi\)
0.316702 + 0.948525i \(0.397424\pi\)
\(488\) −7.30691e6 −1.38894
\(489\) −3.07183e6 −0.580932
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.368526\pi\)
\(500\) 0 0
\(501\) −1.62493e6 −0.289228
\(502\) 283774. 0.0502589
\(503\) 1.77528e6 0.312858 0.156429 0.987689i \(-0.450002\pi\)
0.156429 + 0.987689i \(0.450002\pi\)
\(504\) −591861. −0.103787
\(505\) 0 0
\(506\) 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 \(0.513215\pi\)
\(510\) 0 0
\(511\) 443664. 0.0751627
\(512\) −1.59770e6 −0.269353
\(513\) 1.40890e6 0.236367
\(514\) 6.36739e6 1.06305
\(515\) 0 0
\(516\) −2.16142e6 −0.357367
\(517\) −1.49691e6 −0.246303
\(518\) −640556. −0.104890
\(519\) −5.78989e6 −0.943522
\(520\) 0 0
\(521\) 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 \(-0.632195\pi\)
−0.403466 + 0.914995i \(0.632195\pi\)
\(524\) −1.96889e7 −3.13251
\(525\) 0 0
\(526\) −1.57436e7 −2.48107
\(527\) −273398. −0.0428815
\(528\) 150638. 0.0235152
\(529\) −2.72156e6 −0.422842
\(530\) 0 0
\(531\) −1.28449e6 −0.197695
\(532\) 3.77990e6 0.579031
\(533\) −3.95117e6 −0.602432
\(534\) −3.21856e6 −0.488437
\(535\) 0 0
\(536\) 3.14117e6 0.472258
\(537\) 2.40230e6 0.359494
\(538\) −9.09467e6 −1.35466
\(539\) 1.87291e6 0.277680
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) 2.66572e6 0.382678
\(547\) 6.62530e6 0.946755 0.473377 0.880860i \(-0.343035\pi\)
0.473377 + 0.880860i \(0.343035\pi\)
\(548\) 9.81601e6 1.39632
\(549\) −2.95227e6 −0.418047
\(550\) 0 0
\(551\) 928530. 0.130292
\(552\) 3.47754e6 0.485763
\(553\) 3.17845e6 0.441980
\(554\) −1.37696e6 −0.190611
\(555\) 0 0
\(556\) 452351. 0.0620568
\(557\) 717580. 0.0980014 0.0490007 0.998799i \(-0.484396\pi\)
0.0490007 + 0.998799i \(0.484396\pi\)
\(558\) 1.31905e6 0.179340
\(559\) 3.92962e6 0.531889
\(560\) 0 0
\(561\) −169214. −0.0227002
\(562\) 3.31129e6 0.442238
\(563\) −1.35097e7 −1.79629 −0.898143 0.439703i \(-0.855084\pi\)
−0.898143 + 0.439703i \(0.855084\pi\)
\(564\) −5.97461e6 −0.790882
\(565\) 0 0
\(566\) 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 \(0.159218\pi\)
\(570\) 0 0
\(571\) 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
\(575\) 0 0
\(576\) −4.20814e6 −0.528486
\(577\) −218443. −0.0273149 −0.0136574 0.999907i \(-0.504347\pi\)
−0.0136574 + 0.999907i \(0.504347\pi\)
\(578\) −1.29177e7 −1.60830
\(579\) 2.70112e6 0.334847
\(580\) 0 0
\(581\) 3.54384e6 0.435546
\(582\) −3.06157e6 −0.374659
\(583\) 259559. 0.0316275
\(584\) −2.44031e6 −0.296083
\(585\) 0 0
\(586\) −4.61026e6 −0.554602
\(587\) 5.46003e6 0.654033 0.327017 0.945019i \(-0.393957\pi\)
0.327017 + 0.945019i \(0.393957\pi\)
\(588\) 7.47531e6 0.891632
\(589\) −3.40048e6 −0.403879
\(590\) 0 0
\(591\) −8.50446e6 −1.00156
\(592\) 262664. 0.0308033
\(593\) −1.41186e7 −1.64875 −0.824375 0.566044i \(-0.808473\pi\)
−0.824375 + 0.566044i \(0.808473\pi\)
\(594\) 816401. 0.0949374
\(595\) 0 0
\(596\) 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 \(0.349042\pi\)
\(600\) 0 0
\(601\) −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
\(605\) 0 0
\(606\) 1.54506e7 1.70909
\(607\) 1.58863e7 1.75005 0.875025 0.484078i \(-0.160845\pi\)
0.875025 + 0.484078i \(0.160845\pi\)
\(608\) 9.92412e6 1.08876
\(609\) −157600. −0.0172192
\(610\) 0 0
\(611\) 1.08623e7 1.17711
\(612\) −675384. −0.0728907
\(613\) −1.27701e7 −1.37260 −0.686301 0.727318i \(-0.740767\pi\)
−0.686301 + 0.727318i \(0.740767\pi\)
\(614\) 9.24383e6 0.989535
\(615\) 0 0
\(616\) 884137. 0.0938789
\(617\) −5.32363e6 −0.562982 −0.281491 0.959564i \(-0.590829\pi\)
−0.281491 + 0.959564i \(0.590829\pi\)
\(618\) 3.07292e6 0.323653
\(619\) −5.79882e6 −0.608293 −0.304147 0.952625i \(-0.598371\pi\)
−0.304147 + 0.952625i \(0.598371\pi\)
\(620\) 0 0
\(621\) 1.40506e6 0.146206
\(622\) 1.74724e7 1.81083
\(623\) −1.40832e6 −0.145372
\(624\) −1.09310e6 −0.112382
\(625\) 0 0
\(626\) 2.17302e7 2.21630
\(627\) −2.10466e6 −0.213802
\(628\) −1.13666e7 −1.15009
\(629\) −295056. −0.0297357
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) 1.03598e6 0.101556
\(637\) −1.35907e7 −1.32707
\(638\) 538045. 0.0523320
\(639\) 865852. 0.0838864
\(640\) 0 0
\(641\) 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 \(-0.450162\pi\)
0.155933 + 0.987768i \(0.450162\pi\)
\(644\) 3.76959e6 0.358162
\(645\) 0 0
\(646\) 2.77941e6 0.262043
\(647\) 9.95068e6 0.934527 0.467264 0.884118i \(-0.345240\pi\)
0.467264 + 0.884118i \(0.345240\pi\)
\(648\) 1.31532e6 0.123054
\(649\) 1.91881e6 0.178822
\(650\) 0 0
\(651\) 577166. 0.0533763
\(652\) 1.83152e7 1.68730
\(653\) −1.52022e7 −1.39515 −0.697577 0.716509i \(-0.745739\pi\)
−0.697577 + 0.716509i \(0.745739\pi\)
\(654\) 1.25759e7 1.14973
\(655\) 0 0
\(656\) −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 \(0.642193\pi\)
\(660\) 0 0
\(661\) 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
\(665\) 0 0
\(666\) 1.42354e6 0.124361
\(667\) 925997. 0.0805926
\(668\) 9.68832e6 0.840054
\(669\) 1.02722e7 0.887361
\(670\) 0 0
\(671\) 4.41018e6 0.378137
\(672\) −1.68443e6 −0.143890
\(673\) −1.57773e6 −0.134275 −0.0671376 0.997744i \(-0.521387\pi\)
−0.0671376 + 0.997744i \(0.521387\pi\)
\(674\) 2.59326e6 0.219885
\(675\) 0 0
\(676\) 2.14454e7 1.80496
\(677\) −6.92750e6 −0.580904 −0.290452 0.956890i \(-0.593806\pi\)
−0.290452 + 0.956890i \(0.593806\pi\)
\(678\) 1.31585e7 1.09934
\(679\) −1.33962e6 −0.111509
\(680\) 0 0
\(681\) 6.02506e6 0.497845
\(682\) −1.97044e6 −0.162219
\(683\) 277554. 0.0227665 0.0113833 0.999935i \(-0.496377\pi\)
0.0113833 + 0.999935i \(0.496377\pi\)
\(684\) −8.40029e6 −0.686521
\(685\) 0 0
\(686\) 1.08911e7 0.883609
\(687\) −5.29255e6 −0.427832
\(688\) 619080. 0.0498627
\(689\) −1.88348e6 −0.151152
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 866858. 0.0678305
\(697\) 699238. 0.0545184
\(698\) 3.59540e7 2.79324
\(699\) 1.79475e6 0.138935
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 \(0.421622\pi\)
\(710\) 0 0
\(711\) −7.06365e6 −0.524029
\(712\) 7.74625e6 0.572653
\(713\) −3.39120e6 −0.249821
\(714\) −471753. −0.0346313
\(715\) 0 0
\(716\) −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 \(0.777843\pi\)
\(720\) 0 0
\(721\) 1.34459e6 0.0963276
\(722\) 1.16528e7 0.831928
\(723\) −1.11620e7 −0.794135
\(724\) 1.51071e7 1.07111
\(725\) 0 0
\(726\) −1.21956e6 −0.0858742
\(727\) 1.06687e7 0.748646 0.374323 0.927298i \(-0.377875\pi\)
0.374323 + 0.927298i \(0.377875\pi\)
\(728\) −6.41571e6 −0.448659
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −695424. −0.0481345
\(732\) 1.76023e7 1.21420
\(733\) 2.30788e7 1.58655 0.793274 0.608864i \(-0.208375\pi\)
0.793274 + 0.608864i \(0.208375\pi\)
\(734\) 2.61544e7 1.79186
\(735\) 0 0
\(736\) 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 \(0.777541\pi\)
\(740\) 0 0
\(741\) 1.52724e7 1.02179
\(742\) 723625. 0.0482507
\(743\) 1.49328e7 0.992360 0.496180 0.868220i \(-0.334736\pi\)
0.496180 + 0.868220i \(0.334736\pi\)
\(744\) −3.17462e6 −0.210261
\(745\) 0 0
\(746\) −4.24624e7 −2.79356
\(747\) −7.87568e6 −0.516400
\(748\) 1.00891e6 0.0659321
\(749\) −4.55584e6 −0.296732
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 1.42579e6 0.0907300
\(757\) −1.35397e7 −0.858756 −0.429378 0.903125i \(-0.641267\pi\)
−0.429378 + 0.903125i \(0.641267\pi\)
\(758\) 2.63616e7 1.66648
\(759\) −2.09891e6 −0.132248
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) 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 \(0.668199\pi\)
\(770\) 0 0
\(771\) −6.19174e6 −0.375126
\(772\) −1.61049e7 −0.972554
\(773\) −8.95978e6 −0.539323 −0.269661 0.962955i \(-0.586912\pi\)
−0.269661 + 0.962955i \(0.586912\pi\)
\(774\) 3.35518e6 0.201309
\(775\) 0 0
\(776\) 7.36841e6 0.439258
\(777\) 622887. 0.0370132
\(778\) 3.65841e7 2.16692
\(779\) 8.69698e6 0.513482
\(780\) 0 0
\(781\) −1.29343e6 −0.0758781
\(782\) 2.77183e6 0.162088
\(783\) 350244. 0.0204158
\(784\) −2.14110e6 −0.124407
\(785\) 0 0
\(786\) 3.05631e7 1.76458
\(787\) 2.31723e7 1.33362 0.666812 0.745226i \(-0.267659\pi\)
0.666812 + 0.745226i \(0.267659\pi\)
\(788\) 5.07061e7 2.90901
\(789\) 1.53093e7 0.875512
\(790\) 0 0
\(791\) 5.75765e6 0.327193
\(792\) −1.96487e6 −0.111306
\(793\) −3.20023e7 −1.80717
\(794\) 5.13994e7 2.89339
\(795\) 0 0
\(796\) −5.41010e7 −3.02638
\(797\) 3.40811e7 1.90050 0.950250 0.311489i \(-0.100828\pi\)
0.950250 + 0.311489i \(0.100828\pi\)
\(798\) −5.86757e6 −0.326175
\(799\) −1.92230e6 −0.106525
\(800\) 0 0
\(801\) 3.12978e6 0.172358
\(802\) −2.58470e6 −0.141897
\(803\) 1.47288e6 0.0806082
\(804\) −7.56706e6 −0.412845
\(805\) 0 0
\(806\) 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 \(0.484779\pi\)
\(810\) 0 0
\(811\) 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
\(815\) 0 0
\(816\) 193445. 0.0101703
\(817\) −8.64955e6 −0.453355
\(818\) 4.78618e7 2.50095
\(819\) −2.59219e6 −0.135038
\(820\) 0 0
\(821\) 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 \(-0.402311\pi\)
0.302103 + 0.953275i \(0.402311\pi\)
\(824\) −7.39572e6 −0.379457
\(825\) 0 0
\(826\) 5.34945e6 0.272809
\(827\) −1.61763e7 −0.822461 −0.411231 0.911531i \(-0.634901\pi\)
−0.411231 + 0.911531i \(0.634901\pi\)
\(828\) −8.37738e6 −0.424651
\(829\) −1.56514e7 −0.790984 −0.395492 0.918470i \(-0.629426\pi\)
−0.395492 + 0.918470i \(0.629426\pi\)
\(830\) 0 0
\(831\) 1.33898e6 0.0672623
\(832\) −4.56158e7 −2.28458
\(833\) 2.40514e6 0.120096
\(834\) −702188. −0.0349573
\(835\) 0 0
\(836\) 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 \(0.268940\pi\)
\(840\) 0 0
\(841\) −2.02803e7 −0.988746
\(842\) −8.28707e6 −0.402829
\(843\) −3.21995e6 −0.156056
\(844\) 2.66951e7 1.28996
\(845\) 0 0
\(846\) 9.27442e6 0.445513
\(847\) −533632. −0.0255584
\(848\) −296727. −0.0141699
\(849\) −4.42881e6 −0.210871
\(850\) 0 0
\(851\) −3.65984e6 −0.173236
\(852\) −5.16246e6 −0.243645
\(853\) 2.43979e7 1.14810 0.574050 0.818820i \(-0.305371\pi\)
0.574050 + 0.818820i \(0.305371\pi\)
\(854\) 1.22951e7 0.576884
\(855\) 0 0
\(856\) 2.50588e7 1.16889
\(857\) −1.33841e7 −0.622496 −0.311248 0.950329i \(-0.600747\pi\)
−0.311248 + 0.950329i \(0.600747\pi\)
\(858\) 8.84971e6 0.410403
\(859\) −2.60324e7 −1.20374 −0.601869 0.798595i \(-0.705577\pi\)
−0.601869 + 0.798595i \(0.705577\pi\)
\(860\) 0 0
\(861\) −1.47615e6 −0.0678612
\(862\) −1.48887e7 −0.682479
\(863\) 2.45378e7 1.12152 0.560762 0.827977i \(-0.310508\pi\)
0.560762 + 0.827977i \(0.310508\pi\)
\(864\) 3.74340e6 0.170601
\(865\) 0 0
\(866\) −1.72185e7 −0.780190
\(867\) 1.25614e7 0.567532
\(868\) −3.44123e6 −0.155030
\(869\) 1.05519e7 0.474002
\(870\) 0 0
\(871\) 1.37575e7 0.614460
\(872\) −3.02669e7 −1.34796
\(873\) 2.97712e6 0.132209
\(874\) 3.44755e7 1.52662
\(875\) 0 0
\(876\) 5.87870e6 0.258834
\(877\) −1.99034e7 −0.873833 −0.436917 0.899502i \(-0.643930\pi\)
−0.436917 + 0.899502i \(0.643930\pi\)
\(878\) 2.54568e7 1.11447
\(879\) 4.48308e6 0.195706
\(880\) 0 0
\(881\) −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 \(-0.653187\pi\)
−0.462888 + 0.886417i \(0.653187\pi\)
\(884\) −7.32109e6 −0.315098
\(885\) 0 0
\(886\) −2.92287e7 −1.25091
\(887\) −6.08624e6 −0.259741 −0.129870 0.991531i \(-0.541456\pi\)
−0.129870 + 0.991531i \(0.541456\pi\)
\(888\) −3.42610e6 −0.145803
\(889\) 9.79769e6 0.415786
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −6.12462e7 −2.57731
\(893\) −2.39091e7 −1.00331
\(894\) −2.26412e7 −0.947447
\(895\) 0 0
\(896\) 1.15363e7 0.480060
\(897\) 1.52307e7 0.632031
\(898\) −1.18012e6 −0.0488356
\(899\) −845336. −0.0348843
\(900\) 0 0
\(901\) 333319. 0.0136788
\(902\) 5.03955e6 0.206241
\(903\) 1.46810e6 0.0599149
\(904\) −3.16691e7 −1.28889
\(905\) 0 0
\(906\) 1.97053e7 0.797560
\(907\) 2.14459e7 0.865618 0.432809 0.901486i \(-0.357523\pi\)
0.432809 + 0.901486i \(0.357523\pi\)
\(908\) −3.59232e7 −1.44597
\(909\) −1.50244e7 −0.603098
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) 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 \(0.878394\pi\)
\(920\) 0 0
\(921\) −8.98884e6 −0.349184
\(922\) 4.81488e7 1.86534
\(923\) 9.38575e6 0.362631
\(924\) −2.12988e6 −0.0820683
\(925\) 0 0
\(926\) −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 \(0.463073\pi\)
\(930\) 0 0
\(931\) 2.99146e7 1.13112
\(932\) −1.07008e7 −0.403532
\(933\) −1.69904e7 −0.638999
\(934\) 2.18034e7 0.817820
\(935\) 0 0
\(936\) 1.42580e7 0.531947
\(937\) 2.73781e6 0.101872 0.0509360 0.998702i \(-0.483780\pi\)
0.0509360 + 0.998702i \(0.483780\pi\)
\(938\) −5.28556e6 −0.196148
\(939\) −2.11308e7 −0.782082
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) −5.01206e6 −0.182091
\(947\) 4.08261e7 1.47932 0.739661 0.672979i \(-0.234986\pi\)
0.739661 + 0.672979i \(0.234986\pi\)
\(948\) 4.21156e7 1.52203
\(949\) −1.06879e7 −0.385237
\(950\) 0 0
\(951\) 5.06449e6 0.181587
\(952\) 1.13539e6 0.0406024
\(953\) 4.60553e7 1.64266 0.821330 0.570453i \(-0.193232\pi\)
0.821330 + 0.570453i \(0.193232\pi\)
\(954\) −1.60815e6 −0.0572079
\(955\) 0 0
\(956\) −2.18965e7 −0.774873
\(957\) −523204. −0.0184668
\(958\) −4.42709e7 −1.55849
\(959\) −6.66730e6 −0.234101
\(960\) 0 0
\(961\) −2.55333e7 −0.891865
\(962\) 1.54311e7 0.537599
\(963\) 1.01247e7 0.351817
\(964\) 6.65509e7 2.30654
\(965\) 0 0
\(966\) −5.85156e6 −0.201757
\(967\) −1.48162e7 −0.509532 −0.254766 0.967003i \(-0.581998\pi\)
−0.254766 + 0.967003i \(0.581998\pi\)
\(968\) 2.93517e6 0.100680
\(969\) −2.70275e6 −0.0924689
\(970\) 0 0
\(971\) −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
\(975\) 0 0
\(976\) −5.04170e6 −0.169415
\(977\) 4.06772e6 0.136337 0.0681686 0.997674i \(-0.478284\pi\)
0.0681686 + 0.997674i \(0.478284\pi\)
\(978\) −2.84307e7 −0.950476
\(979\) −4.67535e6 −0.155904
\(980\) 0 0
\(981\) −1.22290e7 −0.405712
\(982\) −2.79765e7 −0.925795
\(983\) −3.76095e7 −1.24141 −0.620704 0.784045i \(-0.713153\pi\)
−0.620704 + 0.784045i \(0.713153\pi\)
\(984\) 8.11934e6 0.267321
\(985\) 0 0
\(986\) 690944. 0.0226335
\(987\) 4.05812e6 0.132596
\(988\) −9.10583e7 −2.96775
\(989\) −8.62596e6 −0.280425
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) 4.69571e7 1.49987
\(997\) −4.37776e7 −1.39481 −0.697403 0.716680i \(-0.745661\pi\)
−0.697403 + 0.716680i \(0.745661\pi\)
\(998\) 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 825.6.a.j.1.3 3
5.4 even 2 165.6.a.a.1.1 3
15.14 odd 2 495.6.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.a.1.1 3 5.4 even 2
495.6.a.e.1.3 3 15.14 odd 2
825.6.a.j.1.3 3 1.1 even 1 trivial