Properties

Label 825.6.a.h.1.1
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
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: \(0\)
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: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.87740\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.55890 q^{2} +9.00000 q^{3} +11.0192 q^{4} -59.0301 q^{6} -146.487 q^{7} +137.611 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-6.55890 q^{2} +9.00000 q^{3} +11.0192 q^{4} -59.0301 q^{6} -146.487 q^{7} +137.611 q^{8} +81.0000 q^{9} -121.000 q^{11} +99.1731 q^{12} -170.238 q^{13} +960.792 q^{14} -1255.19 q^{16} -1564.81 q^{17} -531.271 q^{18} +569.786 q^{19} -1318.38 q^{21} +793.627 q^{22} -3153.25 q^{23} +1238.50 q^{24} +1116.57 q^{26} +729.000 q^{27} -1614.17 q^{28} +3982.58 q^{29} +2990.78 q^{31} +3829.14 q^{32} -1089.00 q^{33} +10263.5 q^{34} +892.558 q^{36} -7858.92 q^{37} -3737.17 q^{38} -1532.14 q^{39} -5206.60 q^{41} +8647.13 q^{42} -13874.3 q^{43} -1333.33 q^{44} +20681.9 q^{46} -6852.01 q^{47} -11296.7 q^{48} +4651.35 q^{49} -14083.3 q^{51} -1875.89 q^{52} +3834.15 q^{53} -4781.44 q^{54} -20158.2 q^{56} +5128.08 q^{57} -26121.4 q^{58} +9649.38 q^{59} -21131.8 q^{61} -19616.2 q^{62} -11865.4 q^{63} +15051.2 q^{64} +7142.65 q^{66} +43499.9 q^{67} -17243.0 q^{68} -28379.2 q^{69} -52607.2 q^{71} +11146.5 q^{72} +64367.9 q^{73} +51545.9 q^{74} +6278.61 q^{76} +17724.9 q^{77} +10049.2 q^{78} +28935.7 q^{79} +6561.00 q^{81} +34149.6 q^{82} +4648.64 q^{83} -14527.5 q^{84} +91000.1 q^{86} +35843.2 q^{87} -16650.9 q^{88} -103458. q^{89} +24937.6 q^{91} -34746.4 q^{92} +26917.0 q^{93} +44941.6 q^{94} +34462.2 q^{96} -75809.5 q^{97} -30507.8 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.55890 −1.15946 −0.579731 0.814808i \(-0.696842\pi\)
−0.579731 + 0.814808i \(0.696842\pi\)
\(3\) 9.00000 0.577350
\(4\) 11.0192 0.344351
\(5\) 0 0
\(6\) −59.0301 −0.669415
\(7\) −146.487 −1.12993 −0.564967 0.825113i \(-0.691111\pi\)
−0.564967 + 0.825113i \(0.691111\pi\)
\(8\) 137.611 0.760200
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 99.1731 0.198811
\(13\) −170.238 −0.279382 −0.139691 0.990195i \(-0.544611\pi\)
−0.139691 + 0.990195i \(0.544611\pi\)
\(14\) 960.792 1.31012
\(15\) 0 0
\(16\) −1255.19 −1.22577
\(17\) −1564.81 −1.31323 −0.656614 0.754227i \(-0.728012\pi\)
−0.656614 + 0.754227i \(0.728012\pi\)
\(18\) −531.271 −0.386487
\(19\) 569.786 0.362100 0.181050 0.983474i \(-0.442050\pi\)
0.181050 + 0.983474i \(0.442050\pi\)
\(20\) 0 0
\(21\) −1318.38 −0.652368
\(22\) 793.627 0.349591
\(23\) −3153.25 −1.24291 −0.621454 0.783451i \(-0.713458\pi\)
−0.621454 + 0.783451i \(0.713458\pi\)
\(24\) 1238.50 0.438902
\(25\) 0 0
\(26\) 1116.57 0.323932
\(27\) 729.000 0.192450
\(28\) −1614.17 −0.389094
\(29\) 3982.58 0.879366 0.439683 0.898153i \(-0.355091\pi\)
0.439683 + 0.898153i \(0.355091\pi\)
\(30\) 0 0
\(31\) 2990.78 0.558959 0.279480 0.960152i \(-0.409838\pi\)
0.279480 + 0.960152i \(0.409838\pi\)
\(32\) 3829.14 0.661037
\(33\) −1089.00 −0.174078
\(34\) 10263.5 1.52264
\(35\) 0 0
\(36\) 892.558 0.114784
\(37\) −7858.92 −0.943753 −0.471877 0.881665i \(-0.656423\pi\)
−0.471877 + 0.881665i \(0.656423\pi\)
\(38\) −3737.17 −0.419841
\(39\) −1532.14 −0.161301
\(40\) 0 0
\(41\) −5206.60 −0.483721 −0.241860 0.970311i \(-0.577758\pi\)
−0.241860 + 0.970311i \(0.577758\pi\)
\(42\) 8647.13 0.756395
\(43\) −13874.3 −1.14430 −0.572149 0.820149i \(-0.693890\pi\)
−0.572149 + 0.820149i \(0.693890\pi\)
\(44\) −1333.33 −0.103826
\(45\) 0 0
\(46\) 20681.9 1.44110
\(47\) −6852.01 −0.452453 −0.226226 0.974075i \(-0.572639\pi\)
−0.226226 + 0.974075i \(0.572639\pi\)
\(48\) −11296.7 −0.707701
\(49\) 4651.35 0.276751
\(50\) 0 0
\(51\) −14083.3 −0.758192
\(52\) −1875.89 −0.0962054
\(53\) 3834.15 0.187490 0.0937452 0.995596i \(-0.470116\pi\)
0.0937452 + 0.995596i \(0.470116\pi\)
\(54\) −4781.44 −0.223138
\(55\) 0 0
\(56\) −20158.2 −0.858976
\(57\) 5128.08 0.209058
\(58\) −26121.4 −1.01959
\(59\) 9649.38 0.360885 0.180443 0.983586i \(-0.442247\pi\)
0.180443 + 0.983586i \(0.442247\pi\)
\(60\) 0 0
\(61\) −21131.8 −0.727128 −0.363564 0.931569i \(-0.618440\pi\)
−0.363564 + 0.931569i \(0.618440\pi\)
\(62\) −19616.2 −0.648092
\(63\) −11865.4 −0.376645
\(64\) 15051.2 0.459326
\(65\) 0 0
\(66\) 7142.65 0.201836
\(67\) 43499.9 1.18386 0.591931 0.805989i \(-0.298366\pi\)
0.591931 + 0.805989i \(0.298366\pi\)
\(68\) −17243.0 −0.452211
\(69\) −28379.2 −0.717593
\(70\) 0 0
\(71\) −52607.2 −1.23851 −0.619255 0.785190i \(-0.712565\pi\)
−0.619255 + 0.785190i \(0.712565\pi\)
\(72\) 11146.5 0.253400
\(73\) 64367.9 1.41372 0.706858 0.707355i \(-0.250112\pi\)
0.706858 + 0.707355i \(0.250112\pi\)
\(74\) 51545.9 1.09425
\(75\) 0 0
\(76\) 6278.61 0.124689
\(77\) 17724.9 0.340688
\(78\) 10049.2 0.187022
\(79\) 28935.7 0.521635 0.260817 0.965388i \(-0.416008\pi\)
0.260817 + 0.965388i \(0.416008\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 34149.6 0.560855
\(83\) 4648.64 0.0740681 0.0370340 0.999314i \(-0.488209\pi\)
0.0370340 + 0.999314i \(0.488209\pi\)
\(84\) −14527.5 −0.224643
\(85\) 0 0
\(86\) 91000.1 1.32677
\(87\) 35843.2 0.507702
\(88\) −16650.9 −0.229209
\(89\) −103458. −1.38448 −0.692242 0.721666i \(-0.743377\pi\)
−0.692242 + 0.721666i \(0.743377\pi\)
\(90\) 0 0
\(91\) 24937.6 0.315683
\(92\) −34746.4 −0.427996
\(93\) 26917.0 0.322715
\(94\) 44941.6 0.524601
\(95\) 0 0
\(96\) 34462.2 0.381650
\(97\) −75809.5 −0.818077 −0.409039 0.912517i \(-0.634136\pi\)
−0.409039 + 0.912517i \(0.634136\pi\)
\(98\) −30507.8 −0.320882
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 35023.6 0.341631 0.170816 0.985303i \(-0.445360\pi\)
0.170816 + 0.985303i \(0.445360\pi\)
\(102\) 92371.1 0.879095
\(103\) −16965.7 −0.157572 −0.0787859 0.996892i \(-0.525104\pi\)
−0.0787859 + 0.996892i \(0.525104\pi\)
\(104\) −23426.6 −0.212386
\(105\) 0 0
\(106\) −25147.8 −0.217388
\(107\) −12190.4 −0.102934 −0.0514671 0.998675i \(-0.516390\pi\)
−0.0514671 + 0.998675i \(0.516390\pi\)
\(108\) 8033.02 0.0662704
\(109\) 12138.7 0.0978606 0.0489303 0.998802i \(-0.484419\pi\)
0.0489303 + 0.998802i \(0.484419\pi\)
\(110\) 0 0
\(111\) −70730.3 −0.544876
\(112\) 183869. 1.38504
\(113\) −204649. −1.50770 −0.753848 0.657048i \(-0.771805\pi\)
−0.753848 + 0.657048i \(0.771805\pi\)
\(114\) −33634.6 −0.242395
\(115\) 0 0
\(116\) 43885.0 0.302810
\(117\) −13789.3 −0.0931273
\(118\) −63289.3 −0.418433
\(119\) 229224. 1.48386
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 138601. 0.843077
\(123\) −46859.4 −0.279276
\(124\) 32956.1 0.192478
\(125\) 0 0
\(126\) 77824.2 0.436705
\(127\) 112241. 0.617506 0.308753 0.951142i \(-0.400088\pi\)
0.308753 + 0.951142i \(0.400088\pi\)
\(128\) −221252. −1.19361
\(129\) −124869. −0.660661
\(130\) 0 0
\(131\) −11679.0 −0.0594602 −0.0297301 0.999558i \(-0.509465\pi\)
−0.0297301 + 0.999558i \(0.509465\pi\)
\(132\) −11999.9 −0.0599438
\(133\) −83466.1 −0.409149
\(134\) −285311. −1.37264
\(135\) 0 0
\(136\) −215335. −0.998315
\(137\) 384935. 1.75221 0.876105 0.482121i \(-0.160133\pi\)
0.876105 + 0.482121i \(0.160133\pi\)
\(138\) 186137. 0.832021
\(139\) −310605. −1.36355 −0.681775 0.731562i \(-0.738792\pi\)
−0.681775 + 0.731562i \(0.738792\pi\)
\(140\) 0 0
\(141\) −61668.0 −0.261224
\(142\) 345046. 1.43600
\(143\) 20598.8 0.0842368
\(144\) −101671. −0.408591
\(145\) 0 0
\(146\) −422183. −1.63915
\(147\) 41862.2 0.159782
\(148\) −86599.2 −0.324982
\(149\) 285203. 1.05242 0.526209 0.850355i \(-0.323613\pi\)
0.526209 + 0.850355i \(0.323613\pi\)
\(150\) 0 0
\(151\) −522357. −1.86434 −0.932170 0.362020i \(-0.882087\pi\)
−0.932170 + 0.362020i \(0.882087\pi\)
\(152\) 78408.8 0.275268
\(153\) −126750. −0.437743
\(154\) −116256. −0.395015
\(155\) 0 0
\(156\) −16883.0 −0.0555442
\(157\) 2221.66 0.00719330 0.00359665 0.999994i \(-0.498855\pi\)
0.00359665 + 0.999994i \(0.498855\pi\)
\(158\) −189787. −0.604816
\(159\) 34507.3 0.108248
\(160\) 0 0
\(161\) 461909. 1.40440
\(162\) −43033.0 −0.128829
\(163\) −250146. −0.737436 −0.368718 0.929541i \(-0.620203\pi\)
−0.368718 + 0.929541i \(0.620203\pi\)
\(164\) −57372.7 −0.166570
\(165\) 0 0
\(166\) −30490.0 −0.0858791
\(167\) 42083.4 0.116767 0.0583834 0.998294i \(-0.481405\pi\)
0.0583834 + 0.998294i \(0.481405\pi\)
\(168\) −181423. −0.495930
\(169\) −342312. −0.921946
\(170\) 0 0
\(171\) 46152.7 0.120700
\(172\) −152884. −0.394040
\(173\) 117206. 0.297737 0.148869 0.988857i \(-0.452437\pi\)
0.148869 + 0.988857i \(0.452437\pi\)
\(174\) −235092. −0.588661
\(175\) 0 0
\(176\) 151878. 0.369585
\(177\) 86844.4 0.208357
\(178\) 678569. 1.60526
\(179\) 582985. 1.35996 0.679978 0.733233i \(-0.261989\pi\)
0.679978 + 0.733233i \(0.261989\pi\)
\(180\) 0 0
\(181\) 235291. 0.533837 0.266919 0.963719i \(-0.413994\pi\)
0.266919 + 0.963719i \(0.413994\pi\)
\(182\) −163563. −0.366022
\(183\) −190186. −0.419808
\(184\) −433921. −0.944858
\(185\) 0 0
\(186\) −176546. −0.374176
\(187\) 189342. 0.395953
\(188\) −75503.8 −0.155802
\(189\) −106789. −0.217456
\(190\) 0 0
\(191\) 591788. 1.17377 0.586885 0.809670i \(-0.300354\pi\)
0.586885 + 0.809670i \(0.300354\pi\)
\(192\) 135461. 0.265192
\(193\) 540825. 1.04511 0.522557 0.852605i \(-0.324978\pi\)
0.522557 + 0.852605i \(0.324978\pi\)
\(194\) 497227. 0.948529
\(195\) 0 0
\(196\) 51254.3 0.0952994
\(197\) −481711. −0.884344 −0.442172 0.896930i \(-0.645792\pi\)
−0.442172 + 0.896930i \(0.645792\pi\)
\(198\) 64283.8 0.116530
\(199\) −1.01427e6 −1.81560 −0.907801 0.419401i \(-0.862240\pi\)
−0.907801 + 0.419401i \(0.862240\pi\)
\(200\) 0 0
\(201\) 391499. 0.683503
\(202\) −229717. −0.396108
\(203\) −583395. −0.993625
\(204\) −155187. −0.261084
\(205\) 0 0
\(206\) 111276. 0.182699
\(207\) −255413. −0.414302
\(208\) 213681. 0.342459
\(209\) −68944.2 −0.109177
\(210\) 0 0
\(211\) 619718. 0.958270 0.479135 0.877741i \(-0.340950\pi\)
0.479135 + 0.877741i \(0.340950\pi\)
\(212\) 42249.3 0.0645625
\(213\) −473465. −0.715054
\(214\) 79955.8 0.119348
\(215\) 0 0
\(216\) 100318. 0.146301
\(217\) −438109. −0.631587
\(218\) −79616.9 −0.113466
\(219\) 579311. 0.816210
\(220\) 0 0
\(221\) 266391. 0.366892
\(222\) 463913. 0.631763
\(223\) −894252. −1.20420 −0.602099 0.798422i \(-0.705669\pi\)
−0.602099 + 0.798422i \(0.705669\pi\)
\(224\) −560918. −0.746928
\(225\) 0 0
\(226\) 1.34227e6 1.74812
\(227\) −851468. −1.09674 −0.548370 0.836236i \(-0.684751\pi\)
−0.548370 + 0.836236i \(0.684751\pi\)
\(228\) 56507.5 0.0719894
\(229\) 137840. 0.173695 0.0868475 0.996222i \(-0.472321\pi\)
0.0868475 + 0.996222i \(0.472321\pi\)
\(230\) 0 0
\(231\) 159524. 0.196696
\(232\) 548046. 0.668494
\(233\) 1.31590e6 1.58793 0.793965 0.607963i \(-0.208013\pi\)
0.793965 + 0.607963i \(0.208013\pi\)
\(234\) 90442.6 0.107977
\(235\) 0 0
\(236\) 106329. 0.124271
\(237\) 260422. 0.301166
\(238\) −1.50346e6 −1.72048
\(239\) 1.11778e6 1.26579 0.632896 0.774237i \(-0.281866\pi\)
0.632896 + 0.774237i \(0.281866\pi\)
\(240\) 0 0
\(241\) −471581. −0.523015 −0.261507 0.965201i \(-0.584220\pi\)
−0.261507 + 0.965201i \(0.584220\pi\)
\(242\) −96028.9 −0.105406
\(243\) 59049.0 0.0641500
\(244\) −232856. −0.250387
\(245\) 0 0
\(246\) 307346. 0.323810
\(247\) −96999.3 −0.101164
\(248\) 411564. 0.424921
\(249\) 41837.8 0.0427632
\(250\) 0 0
\(251\) 1.76541e6 1.76873 0.884363 0.466799i \(-0.154593\pi\)
0.884363 + 0.466799i \(0.154593\pi\)
\(252\) −130748. −0.129698
\(253\) 381543. 0.374751
\(254\) −736176. −0.715974
\(255\) 0 0
\(256\) 969531. 0.924617
\(257\) 19920.1 0.0188130 0.00940652 0.999956i \(-0.497006\pi\)
0.00940652 + 0.999956i \(0.497006\pi\)
\(258\) 819001. 0.766011
\(259\) 1.15123e6 1.06638
\(260\) 0 0
\(261\) 322589. 0.293122
\(262\) 76601.2 0.0689418
\(263\) 2.18348e6 1.94652 0.973262 0.229697i \(-0.0737734\pi\)
0.973262 + 0.229697i \(0.0737734\pi\)
\(264\) −149858. −0.132334
\(265\) 0 0
\(266\) 547446. 0.474392
\(267\) −931119. −0.799332
\(268\) 479335. 0.407664
\(269\) 839300. 0.707190 0.353595 0.935399i \(-0.384959\pi\)
0.353595 + 0.935399i \(0.384959\pi\)
\(270\) 0 0
\(271\) 1.42641e6 1.17984 0.589918 0.807463i \(-0.299160\pi\)
0.589918 + 0.807463i \(0.299160\pi\)
\(272\) 1.96414e6 1.60972
\(273\) 224438. 0.182260
\(274\) −2.52475e6 −2.03162
\(275\) 0 0
\(276\) −312717. −0.247104
\(277\) 1.30448e6 1.02150 0.510749 0.859730i \(-0.329368\pi\)
0.510749 + 0.859730i \(0.329368\pi\)
\(278\) 2.03723e6 1.58098
\(279\) 242253. 0.186320
\(280\) 0 0
\(281\) −580646. −0.438678 −0.219339 0.975649i \(-0.570390\pi\)
−0.219339 + 0.975649i \(0.570390\pi\)
\(282\) 404475. 0.302879
\(283\) 601974. 0.446799 0.223399 0.974727i \(-0.428285\pi\)
0.223399 + 0.974727i \(0.428285\pi\)
\(284\) −579691. −0.426482
\(285\) 0 0
\(286\) −135106. −0.0976693
\(287\) 762698. 0.546572
\(288\) 310160. 0.220346
\(289\) 1.02878e6 0.724567
\(290\) 0 0
\(291\) −682286. −0.472317
\(292\) 709285. 0.486815
\(293\) 2.40993e6 1.63996 0.819982 0.572389i \(-0.193983\pi\)
0.819982 + 0.572389i \(0.193983\pi\)
\(294\) −274570. −0.185261
\(295\) 0 0
\(296\) −1.08147e6 −0.717441
\(297\) −88209.0 −0.0580259
\(298\) −1.87062e6 −1.22024
\(299\) 536803. 0.347246
\(300\) 0 0
\(301\) 2.03240e6 1.29298
\(302\) 3.42609e6 2.16163
\(303\) 315213. 0.197241
\(304\) −715191. −0.443852
\(305\) 0 0
\(306\) 831340. 0.507546
\(307\) 1.83380e6 1.11047 0.555235 0.831694i \(-0.312629\pi\)
0.555235 + 0.831694i \(0.312629\pi\)
\(308\) 195315. 0.117316
\(309\) −152691. −0.0909742
\(310\) 0 0
\(311\) 2.27507e6 1.33381 0.666906 0.745142i \(-0.267618\pi\)
0.666906 + 0.745142i \(0.267618\pi\)
\(312\) −210839. −0.122621
\(313\) −159585. −0.0920728 −0.0460364 0.998940i \(-0.514659\pi\)
−0.0460364 + 0.998940i \(0.514659\pi\)
\(314\) −14571.6 −0.00834035
\(315\) 0 0
\(316\) 318849. 0.179625
\(317\) 1.24338e6 0.694955 0.347477 0.937688i \(-0.387038\pi\)
0.347477 + 0.937688i \(0.387038\pi\)
\(318\) −226330. −0.125509
\(319\) −481892. −0.265139
\(320\) 0 0
\(321\) −109714. −0.0594290
\(322\) −3.02962e6 −1.62835
\(323\) −891609. −0.475519
\(324\) 72297.2 0.0382612
\(325\) 0 0
\(326\) 1.64068e6 0.855029
\(327\) 109249. 0.0564998
\(328\) −716485. −0.367724
\(329\) 1.00373e6 0.511242
\(330\) 0 0
\(331\) −958448. −0.480838 −0.240419 0.970669i \(-0.577285\pi\)
−0.240419 + 0.970669i \(0.577285\pi\)
\(332\) 51224.5 0.0255054
\(333\) −636573. −0.314584
\(334\) −276021. −0.135387
\(335\) 0 0
\(336\) 1.65482e6 0.799655
\(337\) 1.51074e6 0.724626 0.362313 0.932056i \(-0.381987\pi\)
0.362313 + 0.932056i \(0.381987\pi\)
\(338\) 2.24519e6 1.06896
\(339\) −1.84184e6 −0.870469
\(340\) 0 0
\(341\) −361884. −0.168533
\(342\) −302711. −0.139947
\(343\) 1.78064e6 0.817224
\(344\) −1.90925e6 −0.869896
\(345\) 0 0
\(346\) −768741. −0.345215
\(347\) 1.93847e6 0.864243 0.432122 0.901815i \(-0.357765\pi\)
0.432122 + 0.901815i \(0.357765\pi\)
\(348\) 394965. 0.174828
\(349\) −1.62462e6 −0.713985 −0.356992 0.934107i \(-0.616198\pi\)
−0.356992 + 0.934107i \(0.616198\pi\)
\(350\) 0 0
\(351\) −124103. −0.0537671
\(352\) −463325. −0.199310
\(353\) 23347.5 0.00997249 0.00498624 0.999988i \(-0.498413\pi\)
0.00498624 + 0.999988i \(0.498413\pi\)
\(354\) −569604. −0.241582
\(355\) 0 0
\(356\) −1.14002e6 −0.476748
\(357\) 2.06302e6 0.856707
\(358\) −3.82374e6 −1.57682
\(359\) 3.63918e6 1.49028 0.745139 0.666909i \(-0.232383\pi\)
0.745139 + 0.666909i \(0.232383\pi\)
\(360\) 0 0
\(361\) −2.15144e6 −0.868884
\(362\) −1.54325e6 −0.618964
\(363\) 131769. 0.0524864
\(364\) 274793. 0.108706
\(365\) 0 0
\(366\) 1.24741e6 0.486751
\(367\) −2.24713e6 −0.870890 −0.435445 0.900215i \(-0.643409\pi\)
−0.435445 + 0.900215i \(0.643409\pi\)
\(368\) 3.95793e6 1.52352
\(369\) −421735. −0.161240
\(370\) 0 0
\(371\) −561651. −0.211852
\(372\) 296605. 0.111127
\(373\) −3.82745e6 −1.42442 −0.712208 0.701968i \(-0.752305\pi\)
−0.712208 + 0.701968i \(0.752305\pi\)
\(374\) −1.24188e6 −0.459092
\(375\) 0 0
\(376\) −942910. −0.343954
\(377\) −677986. −0.245679
\(378\) 700418. 0.252132
\(379\) −4.26443e6 −1.52498 −0.762489 0.647002i \(-0.776023\pi\)
−0.762489 + 0.647002i \(0.776023\pi\)
\(380\) 0 0
\(381\) 1.01017e6 0.356517
\(382\) −3.88148e6 −1.36094
\(383\) −2.40920e6 −0.839221 −0.419611 0.907704i \(-0.637833\pi\)
−0.419611 + 0.907704i \(0.637833\pi\)
\(384\) −1.99127e6 −0.689130
\(385\) 0 0
\(386\) −3.54722e6 −1.21177
\(387\) −1.12382e6 −0.381433
\(388\) −835362. −0.281706
\(389\) 3.73204e6 1.25047 0.625233 0.780438i \(-0.285004\pi\)
0.625233 + 0.780438i \(0.285004\pi\)
\(390\) 0 0
\(391\) 4.93424e6 1.63222
\(392\) 640077. 0.210386
\(393\) −105111. −0.0343293
\(394\) 3.15950e6 1.02536
\(395\) 0 0
\(396\) −107999. −0.0346086
\(397\) 4.89288e6 1.55808 0.779038 0.626977i \(-0.215708\pi\)
0.779038 + 0.626977i \(0.215708\pi\)
\(398\) 6.65250e6 2.10512
\(399\) −751195. −0.236222
\(400\) 0 0
\(401\) −3.93524e6 −1.22211 −0.611055 0.791588i \(-0.709255\pi\)
−0.611055 + 0.791588i \(0.709255\pi\)
\(402\) −2.56780e6 −0.792495
\(403\) −509144. −0.156163
\(404\) 385933. 0.117641
\(405\) 0 0
\(406\) 3.82643e6 1.15207
\(407\) 950929. 0.284552
\(408\) −1.93802e6 −0.576378
\(409\) 657037. 0.194215 0.0971073 0.995274i \(-0.469041\pi\)
0.0971073 + 0.995274i \(0.469041\pi\)
\(410\) 0 0
\(411\) 3.46442e6 1.01164
\(412\) −186949. −0.0542600
\(413\) −1.41351e6 −0.407777
\(414\) 1.67523e6 0.480368
\(415\) 0 0
\(416\) −651864. −0.184682
\(417\) −2.79544e6 −0.787246
\(418\) 452198. 0.126587
\(419\) 4.38504e6 1.22022 0.610110 0.792317i \(-0.291125\pi\)
0.610110 + 0.792317i \(0.291125\pi\)
\(420\) 0 0
\(421\) 2.34058e6 0.643604 0.321802 0.946807i \(-0.395711\pi\)
0.321802 + 0.946807i \(0.395711\pi\)
\(422\) −4.06467e6 −1.11108
\(423\) −555012. −0.150818
\(424\) 527620. 0.142530
\(425\) 0 0
\(426\) 3.10541e6 0.829078
\(427\) 3.09552e6 0.821607
\(428\) −134329. −0.0354455
\(429\) 185389. 0.0486341
\(430\) 0 0
\(431\) −2.18215e6 −0.565838 −0.282919 0.959144i \(-0.591303\pi\)
−0.282919 + 0.959144i \(0.591303\pi\)
\(432\) −915035. −0.235900
\(433\) 3.61080e6 0.925515 0.462757 0.886485i \(-0.346860\pi\)
0.462757 + 0.886485i \(0.346860\pi\)
\(434\) 2.87352e6 0.732301
\(435\) 0 0
\(436\) 133760. 0.0336984
\(437\) −1.79668e6 −0.450056
\(438\) −3.79965e6 −0.946364
\(439\) 6.48191e6 1.60525 0.802624 0.596486i \(-0.203437\pi\)
0.802624 + 0.596486i \(0.203437\pi\)
\(440\) 0 0
\(441\) 376760. 0.0922503
\(442\) −1.74723e6 −0.425397
\(443\) 6.81523e6 1.64995 0.824976 0.565167i \(-0.191188\pi\)
0.824976 + 0.565167i \(0.191188\pi\)
\(444\) −779393. −0.187629
\(445\) 0 0
\(446\) 5.86531e6 1.39622
\(447\) 2.56683e6 0.607614
\(448\) −2.20480e6 −0.519008
\(449\) −2.98356e6 −0.698423 −0.349211 0.937044i \(-0.613551\pi\)
−0.349211 + 0.937044i \(0.613551\pi\)
\(450\) 0 0
\(451\) 629999. 0.145847
\(452\) −2.25508e6 −0.519177
\(453\) −4.70122e6 −1.07638
\(454\) 5.58469e6 1.27163
\(455\) 0 0
\(456\) 705679. 0.158926
\(457\) 6.11184e6 1.36893 0.684466 0.729045i \(-0.260036\pi\)
0.684466 + 0.729045i \(0.260036\pi\)
\(458\) −904081. −0.201393
\(459\) −1.14075e6 −0.252731
\(460\) 0 0
\(461\) −3.41772e6 −0.749003 −0.374502 0.927226i \(-0.622186\pi\)
−0.374502 + 0.927226i \(0.622186\pi\)
\(462\) −1.04630e6 −0.228062
\(463\) 4.60011e6 0.997277 0.498639 0.866810i \(-0.333834\pi\)
0.498639 + 0.866810i \(0.333834\pi\)
\(464\) −4.99890e6 −1.07790
\(465\) 0 0
\(466\) −8.63083e6 −1.84114
\(467\) −5.03617e6 −1.06858 −0.534291 0.845301i \(-0.679421\pi\)
−0.534291 + 0.845301i \(0.679421\pi\)
\(468\) −151947. −0.0320685
\(469\) −6.37215e6 −1.33769
\(470\) 0 0
\(471\) 19994.9 0.00415305
\(472\) 1.32786e6 0.274345
\(473\) 1.67879e6 0.345019
\(474\) −1.70808e6 −0.349190
\(475\) 0 0
\(476\) 2.52587e6 0.510969
\(477\) 310566. 0.0624968
\(478\) −7.33142e6 −1.46764
\(479\) −2.82760e6 −0.563091 −0.281545 0.959548i \(-0.590847\pi\)
−0.281545 + 0.959548i \(0.590847\pi\)
\(480\) 0 0
\(481\) 1.33789e6 0.263668
\(482\) 3.09306e6 0.606416
\(483\) 4.15718e6 0.810833
\(484\) 161333. 0.0313046
\(485\) 0 0
\(486\) −387297. −0.0743795
\(487\) 5.68931e6 1.08702 0.543510 0.839403i \(-0.317095\pi\)
0.543510 + 0.839403i \(0.317095\pi\)
\(488\) −2.90796e6 −0.552763
\(489\) −2.25131e6 −0.425759
\(490\) 0 0
\(491\) 3.23715e6 0.605981 0.302990 0.952994i \(-0.402015\pi\)
0.302990 + 0.952994i \(0.402015\pi\)
\(492\) −516355. −0.0961690
\(493\) −6.23199e6 −1.15481
\(494\) 636209. 0.117296
\(495\) 0 0
\(496\) −3.75400e6 −0.685157
\(497\) 7.70626e6 1.39943
\(498\) −274410. −0.0495823
\(499\) −5.08564e6 −0.914312 −0.457156 0.889386i \(-0.651132\pi\)
−0.457156 + 0.889386i \(0.651132\pi\)
\(500\) 0 0
\(501\) 378751. 0.0674154
\(502\) −1.15791e7 −2.05077
\(503\) −9.02252e6 −1.59004 −0.795019 0.606584i \(-0.792539\pi\)
−0.795019 + 0.606584i \(0.792539\pi\)
\(504\) −1.63281e6 −0.286325
\(505\) 0 0
\(506\) −2.50251e6 −0.434509
\(507\) −3.08081e6 −0.532286
\(508\) 1.23681e6 0.212639
\(509\) 8.98909e6 1.53788 0.768938 0.639323i \(-0.220785\pi\)
0.768938 + 0.639323i \(0.220785\pi\)
\(510\) 0 0
\(511\) −9.42904e6 −1.59741
\(512\) 720997. 0.121551
\(513\) 415374. 0.0696861
\(514\) −130654. −0.0218130
\(515\) 0 0
\(516\) −1.37596e6 −0.227499
\(517\) 829093. 0.136420
\(518\) −7.55079e6 −1.23643
\(519\) 1.05485e6 0.171899
\(520\) 0 0
\(521\) −1.11219e7 −1.79508 −0.897539 0.440936i \(-0.854647\pi\)
−0.897539 + 0.440936i \(0.854647\pi\)
\(522\) −2.11583e6 −0.339863
\(523\) −3.67510e6 −0.587510 −0.293755 0.955881i \(-0.594905\pi\)
−0.293755 + 0.955881i \(0.594905\pi\)
\(524\) −128693. −0.0204752
\(525\) 0 0
\(526\) −1.43212e7 −2.25692
\(527\) −4.68001e6 −0.734040
\(528\) 1.36690e6 0.213380
\(529\) 3.50664e6 0.544818
\(530\) 0 0
\(531\) 781600. 0.120295
\(532\) −919733. −0.140891
\(533\) 886361. 0.135143
\(534\) 6.10712e6 0.926795
\(535\) 0 0
\(536\) 5.98605e6 0.899971
\(537\) 5.24687e6 0.785171
\(538\) −5.50489e6 −0.819960
\(539\) −562814. −0.0834436
\(540\) 0 0
\(541\) −4.31097e6 −0.633260 −0.316630 0.948549i \(-0.602551\pi\)
−0.316630 + 0.948549i \(0.602551\pi\)
\(542\) −9.35569e6 −1.36797
\(543\) 2.11762e6 0.308211
\(544\) −5.99188e6 −0.868092
\(545\) 0 0
\(546\) −1.47207e6 −0.211323
\(547\) 1.80626e6 0.258114 0.129057 0.991637i \(-0.458805\pi\)
0.129057 + 0.991637i \(0.458805\pi\)
\(548\) 4.24169e6 0.603375
\(549\) −1.71167e6 −0.242376
\(550\) 0 0
\(551\) 2.26922e6 0.318418
\(552\) −3.90529e6 −0.545514
\(553\) −4.23870e6 −0.589413
\(554\) −8.55595e6 −1.18439
\(555\) 0 0
\(556\) −3.42262e6 −0.469540
\(557\) −8.42670e6 −1.15085 −0.575426 0.817854i \(-0.695164\pi\)
−0.575426 + 0.817854i \(0.695164\pi\)
\(558\) −1.58891e6 −0.216031
\(559\) 2.36193e6 0.319696
\(560\) 0 0
\(561\) 1.70408e6 0.228604
\(562\) 3.80840e6 0.508630
\(563\) 9.19966e6 1.22321 0.611605 0.791163i \(-0.290524\pi\)
0.611605 + 0.791163i \(0.290524\pi\)
\(564\) −679534. −0.0899526
\(565\) 0 0
\(566\) −3.94829e6 −0.518046
\(567\) −961099. −0.125548
\(568\) −7.23932e6 −0.941515
\(569\) 5.18644e6 0.671566 0.335783 0.941939i \(-0.390999\pi\)
0.335783 + 0.941939i \(0.390999\pi\)
\(570\) 0 0
\(571\) −3.29390e6 −0.422786 −0.211393 0.977401i \(-0.567800\pi\)
−0.211393 + 0.977401i \(0.567800\pi\)
\(572\) 226983. 0.0290070
\(573\) 5.32609e6 0.677676
\(574\) −5.00246e6 −0.633730
\(575\) 0 0
\(576\) 1.21915e6 0.153109
\(577\) 1.02043e7 1.27598 0.637989 0.770046i \(-0.279767\pi\)
0.637989 + 0.770046i \(0.279767\pi\)
\(578\) −6.74768e6 −0.840107
\(579\) 4.86742e6 0.603396
\(580\) 0 0
\(581\) −680965. −0.0836921
\(582\) 4.47505e6 0.547633
\(583\) −463932. −0.0565305
\(584\) 8.85773e6 1.07471
\(585\) 0 0
\(586\) −1.58065e7 −1.90148
\(587\) 1.33951e7 1.60454 0.802272 0.596958i \(-0.203624\pi\)
0.802272 + 0.596958i \(0.203624\pi\)
\(588\) 461289. 0.0550212
\(589\) 1.70411e6 0.202399
\(590\) 0 0
\(591\) −4.33540e6 −0.510576
\(592\) 9.86445e6 1.15683
\(593\) −3.76550e6 −0.439730 −0.219865 0.975530i \(-0.570562\pi\)
−0.219865 + 0.975530i \(0.570562\pi\)
\(594\) 578554. 0.0672788
\(595\) 0 0
\(596\) 3.14272e6 0.362401
\(597\) −9.12843e6 −1.04824
\(598\) −3.52084e6 −0.402618
\(599\) 1.21252e6 0.138077 0.0690384 0.997614i \(-0.478007\pi\)
0.0690384 + 0.997614i \(0.478007\pi\)
\(600\) 0 0
\(601\) −7.81714e6 −0.882799 −0.441399 0.897311i \(-0.645518\pi\)
−0.441399 + 0.897311i \(0.645518\pi\)
\(602\) −1.33303e7 −1.49916
\(603\) 3.52349e6 0.394620
\(604\) −5.75597e6 −0.641987
\(605\) 0 0
\(606\) −2.06745e6 −0.228693
\(607\) −5.43853e6 −0.599114 −0.299557 0.954078i \(-0.596839\pi\)
−0.299557 + 0.954078i \(0.596839\pi\)
\(608\) 2.18179e6 0.239361
\(609\) −5.25055e6 −0.573670
\(610\) 0 0
\(611\) 1.16647e6 0.126407
\(612\) −1.39669e6 −0.150737
\(613\) −1.70637e7 −1.83409 −0.917045 0.398783i \(-0.869433\pi\)
−0.917045 + 0.398783i \(0.869433\pi\)
\(614\) −1.20277e7 −1.28755
\(615\) 0 0
\(616\) 2.43914e6 0.258991
\(617\) −7.80667e6 −0.825568 −0.412784 0.910829i \(-0.635444\pi\)
−0.412784 + 0.910829i \(0.635444\pi\)
\(618\) 1.00149e6 0.105481
\(619\) 7.31658e6 0.767506 0.383753 0.923436i \(-0.374631\pi\)
0.383753 + 0.923436i \(0.374631\pi\)
\(620\) 0 0
\(621\) −2.29872e6 −0.239198
\(622\) −1.49220e7 −1.54650
\(623\) 1.51552e7 1.56438
\(624\) 1.92313e6 0.197719
\(625\) 0 0
\(626\) 1.04670e6 0.106755
\(627\) −620497. −0.0630335
\(628\) 24481.0 0.00247702
\(629\) 1.22977e7 1.23936
\(630\) 0 0
\(631\) −1.18729e7 −1.18709 −0.593543 0.804802i \(-0.702271\pi\)
−0.593543 + 0.804802i \(0.702271\pi\)
\(632\) 3.98187e6 0.396547
\(633\) 5.57746e6 0.553257
\(634\) −8.15523e6 −0.805773
\(635\) 0 0
\(636\) 380244. 0.0372752
\(637\) −791837. −0.0773192
\(638\) 3.16068e6 0.307418
\(639\) −4.26118e6 −0.412837
\(640\) 0 0
\(641\) −1.85963e7 −1.78764 −0.893821 0.448424i \(-0.851985\pi\)
−0.893821 + 0.448424i \(0.851985\pi\)
\(642\) 719603. 0.0689057
\(643\) 1.18028e7 1.12579 0.562894 0.826529i \(-0.309688\pi\)
0.562894 + 0.826529i \(0.309688\pi\)
\(644\) 5.08988e6 0.483608
\(645\) 0 0
\(646\) 5.84798e6 0.551346
\(647\) −2.05252e7 −1.92764 −0.963821 0.266551i \(-0.914116\pi\)
−0.963821 + 0.266551i \(0.914116\pi\)
\(648\) 902865. 0.0844666
\(649\) −1.16757e6 −0.108811
\(650\) 0 0
\(651\) −3.94298e6 −0.364647
\(652\) −2.75642e6 −0.253937
\(653\) 5.78511e6 0.530919 0.265460 0.964122i \(-0.414476\pi\)
0.265460 + 0.964122i \(0.414476\pi\)
\(654\) −716552. −0.0655094
\(655\) 0 0
\(656\) 6.53528e6 0.592932
\(657\) 5.21380e6 0.471239
\(658\) −6.58335e6 −0.592765
\(659\) 1.88598e7 1.69170 0.845852 0.533418i \(-0.179093\pi\)
0.845852 + 0.533418i \(0.179093\pi\)
\(660\) 0 0
\(661\) −1.12652e7 −1.00285 −0.501426 0.865201i \(-0.667191\pi\)
−0.501426 + 0.865201i \(0.667191\pi\)
\(662\) 6.28637e6 0.557513
\(663\) 2.39751e6 0.211825
\(664\) 639704. 0.0563065
\(665\) 0 0
\(666\) 4.17522e6 0.364749
\(667\) −1.25581e7 −1.09297
\(668\) 463727. 0.0402088
\(669\) −8.04827e6 −0.695244
\(670\) 0 0
\(671\) 2.55694e6 0.219237
\(672\) −5.04826e6 −0.431239
\(673\) −1.51301e7 −1.28767 −0.643833 0.765166i \(-0.722657\pi\)
−0.643833 + 0.765166i \(0.722657\pi\)
\(674\) −9.90878e6 −0.840176
\(675\) 0 0
\(676\) −3.77201e6 −0.317473
\(677\) 1.12001e7 0.939187 0.469593 0.882883i \(-0.344401\pi\)
0.469593 + 0.882883i \(0.344401\pi\)
\(678\) 1.20805e7 1.00928
\(679\) 1.11051e7 0.924373
\(680\) 0 0
\(681\) −7.66321e6 −0.633203
\(682\) 2.37356e6 0.195407
\(683\) −2.12907e7 −1.74638 −0.873190 0.487379i \(-0.837953\pi\)
−0.873190 + 0.487379i \(0.837953\pi\)
\(684\) 508567. 0.0415631
\(685\) 0 0
\(686\) −1.16791e7 −0.947539
\(687\) 1.24056e6 0.100283
\(688\) 1.74149e7 1.40265
\(689\) −652717. −0.0523814
\(690\) 0 0
\(691\) 1.77648e6 0.141536 0.0707678 0.997493i \(-0.477455\pi\)
0.0707678 + 0.997493i \(0.477455\pi\)
\(692\) 1.29152e6 0.102526
\(693\) 1.43572e6 0.113563
\(694\) −1.27143e7 −1.00206
\(695\) 0 0
\(696\) 4.93242e6 0.385955
\(697\) 8.14735e6 0.635235
\(698\) 1.06557e7 0.827838
\(699\) 1.18431e7 0.916792
\(700\) 0 0
\(701\) 94420.1 0.00725720 0.00362860 0.999993i \(-0.498845\pi\)
0.00362860 + 0.999993i \(0.498845\pi\)
\(702\) 813983. 0.0623408
\(703\) −4.47791e6 −0.341733
\(704\) −1.82120e6 −0.138492
\(705\) 0 0
\(706\) −153134. −0.0115627
\(707\) −5.13050e6 −0.386021
\(708\) 956958. 0.0717480
\(709\) −1.46333e7 −1.09327 −0.546633 0.837372i \(-0.684091\pi\)
−0.546633 + 0.837372i \(0.684091\pi\)
\(710\) 0 0
\(711\) 2.34379e6 0.173878
\(712\) −1.42369e7 −1.05248
\(713\) −9.43067e6 −0.694734
\(714\) −1.35311e7 −0.993319
\(715\) 0 0
\(716\) 6.42405e6 0.468302
\(717\) 1.00600e7 0.730805
\(718\) −2.38690e7 −1.72792
\(719\) 1.15215e7 0.831161 0.415580 0.909556i \(-0.363578\pi\)
0.415580 + 0.909556i \(0.363578\pi\)
\(720\) 0 0
\(721\) 2.48525e6 0.178046
\(722\) 1.41111e7 1.00744
\(723\) −4.24423e6 −0.301963
\(724\) 2.59273e6 0.183827
\(725\) 0 0
\(726\) −864260. −0.0608559
\(727\) −1.73993e7 −1.22095 −0.610473 0.792037i \(-0.709021\pi\)
−0.610473 + 0.792037i \(0.709021\pi\)
\(728\) 3.43168e6 0.239982
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.17107e7 1.50272
\(732\) −2.09570e6 −0.144561
\(733\) 1.21356e7 0.834262 0.417131 0.908846i \(-0.363036\pi\)
0.417131 + 0.908846i \(0.363036\pi\)
\(734\) 1.47387e7 1.00976
\(735\) 0 0
\(736\) −1.20742e7 −0.821608
\(737\) −5.26348e6 −0.356948
\(738\) 2.76612e6 0.186952
\(739\) 1.30600e7 0.879692 0.439846 0.898073i \(-0.355033\pi\)
0.439846 + 0.898073i \(0.355033\pi\)
\(740\) 0 0
\(741\) −872994. −0.0584071
\(742\) 3.68382e6 0.245634
\(743\) 2.36143e6 0.156929 0.0784644 0.996917i \(-0.474998\pi\)
0.0784644 + 0.996917i \(0.474998\pi\)
\(744\) 3.70407e6 0.245328
\(745\) 0 0
\(746\) 2.51038e7 1.65156
\(747\) 376540. 0.0246894
\(748\) 2.08641e6 0.136347
\(749\) 1.78574e6 0.116309
\(750\) 0 0
\(751\) 4.12172e6 0.266673 0.133336 0.991071i \(-0.457431\pi\)
0.133336 + 0.991071i \(0.457431\pi\)
\(752\) 8.60058e6 0.554604
\(753\) 1.58887e7 1.02117
\(754\) 4.44685e6 0.284855
\(755\) 0 0
\(756\) −1.17673e6 −0.0748811
\(757\) −4.12364e6 −0.261542 −0.130771 0.991413i \(-0.541745\pi\)
−0.130771 + 0.991413i \(0.541745\pi\)
\(758\) 2.79700e7 1.76815
\(759\) 3.43389e6 0.216362
\(760\) 0 0
\(761\) −2.08230e7 −1.30341 −0.651706 0.758472i \(-0.725946\pi\)
−0.651706 + 0.758472i \(0.725946\pi\)
\(762\) −6.62558e6 −0.413368
\(763\) −1.77817e6 −0.110576
\(764\) 6.52105e6 0.404189
\(765\) 0 0
\(766\) 1.58017e7 0.973045
\(767\) −1.64269e6 −0.100825
\(768\) 8.72578e6 0.533828
\(769\) 1.58156e7 0.964427 0.482213 0.876054i \(-0.339833\pi\)
0.482213 + 0.876054i \(0.339833\pi\)
\(770\) 0 0
\(771\) 179281. 0.0108617
\(772\) 5.95947e6 0.359886
\(773\) 2.58350e6 0.155510 0.0777552 0.996972i \(-0.475225\pi\)
0.0777552 + 0.996972i \(0.475225\pi\)
\(774\) 7.37101e6 0.442257
\(775\) 0 0
\(776\) −1.04322e7 −0.621902
\(777\) 1.03610e7 0.615674
\(778\) −2.44781e7 −1.44987
\(779\) −2.96665e6 −0.175155
\(780\) 0 0
\(781\) 6.36547e6 0.373425
\(782\) −3.23632e7 −1.89250
\(783\) 2.90330e6 0.169234
\(784\) −5.83834e6 −0.339234
\(785\) 0 0
\(786\) 689411. 0.0398035
\(787\) 5.64885e6 0.325105 0.162552 0.986700i \(-0.448027\pi\)
0.162552 + 0.986700i \(0.448027\pi\)
\(788\) −5.30809e6 −0.304525
\(789\) 1.96513e7 1.12383
\(790\) 0 0
\(791\) 2.99784e7 1.70360
\(792\) −1.34872e6 −0.0764030
\(793\) 3.59743e6 0.203146
\(794\) −3.20920e7 −1.80653
\(795\) 0 0
\(796\) −1.11765e7 −0.625204
\(797\) 3.72876e6 0.207931 0.103965 0.994581i \(-0.466847\pi\)
0.103965 + 0.994581i \(0.466847\pi\)
\(798\) 4.92702e6 0.273890
\(799\) 1.07221e7 0.594173
\(800\) 0 0
\(801\) −8.38007e6 −0.461495
\(802\) 2.58109e7 1.41699
\(803\) −7.78852e6 −0.426252
\(804\) 4.31401e6 0.235365
\(805\) 0 0
\(806\) 3.33943e6 0.181065
\(807\) 7.55370e6 0.408296
\(808\) 4.81963e6 0.259708
\(809\) 6.35004e6 0.341119 0.170559 0.985347i \(-0.445443\pi\)
0.170559 + 0.985347i \(0.445443\pi\)
\(810\) 0 0
\(811\) 1.34156e7 0.716241 0.358120 0.933675i \(-0.383418\pi\)
0.358120 + 0.933675i \(0.383418\pi\)
\(812\) −6.42856e6 −0.342156
\(813\) 1.28377e7 0.681178
\(814\) −6.23705e6 −0.329928
\(815\) 0 0
\(816\) 1.76773e7 0.929372
\(817\) −7.90538e6 −0.414350
\(818\) −4.30945e6 −0.225184
\(819\) 2.01995e6 0.105228
\(820\) 0 0
\(821\) 2.40201e7 1.24370 0.621851 0.783135i \(-0.286381\pi\)
0.621851 + 0.783135i \(0.286381\pi\)
\(822\) −2.27228e7 −1.17296
\(823\) 1.46523e6 0.0754058 0.0377029 0.999289i \(-0.487996\pi\)
0.0377029 + 0.999289i \(0.487996\pi\)
\(824\) −2.33466e6 −0.119786
\(825\) 0 0
\(826\) 9.27105e6 0.472801
\(827\) 2.32918e7 1.18424 0.592120 0.805850i \(-0.298291\pi\)
0.592120 + 0.805850i \(0.298291\pi\)
\(828\) −2.81446e6 −0.142665
\(829\) −2.02821e6 −0.102501 −0.0512503 0.998686i \(-0.516321\pi\)
−0.0512503 + 0.998686i \(0.516321\pi\)
\(830\) 0 0
\(831\) 1.17403e7 0.589762
\(832\) −2.56229e6 −0.128327
\(833\) −7.27850e6 −0.363437
\(834\) 1.83350e7 0.912781
\(835\) 0 0
\(836\) −759712. −0.0375953
\(837\) 2.18028e6 0.107572
\(838\) −2.87610e7 −1.41480
\(839\) −3.07518e7 −1.50822 −0.754111 0.656747i \(-0.771932\pi\)
−0.754111 + 0.656747i \(0.771932\pi\)
\(840\) 0 0
\(841\) −4.65021e6 −0.226716
\(842\) −1.53517e7 −0.746234
\(843\) −5.22582e6 −0.253271
\(844\) 6.82881e6 0.329981
\(845\) 0 0
\(846\) 3.64027e6 0.174867
\(847\) −2.14471e6 −0.102721
\(848\) −4.81259e6 −0.229821
\(849\) 5.41777e6 0.257959
\(850\) 0 0
\(851\) 2.47811e7 1.17300
\(852\) −5.21722e6 −0.246229
\(853\) 2.05598e7 0.967491 0.483745 0.875209i \(-0.339276\pi\)
0.483745 + 0.875209i \(0.339276\pi\)
\(854\) −2.03032e7 −0.952621
\(855\) 0 0
\(856\) −1.67753e6 −0.0782505
\(857\) −3.20463e7 −1.49048 −0.745240 0.666796i \(-0.767665\pi\)
−0.745240 + 0.666796i \(0.767665\pi\)
\(858\) −1.21595e6 −0.0563894
\(859\) 2.34894e6 0.108615 0.0543075 0.998524i \(-0.482705\pi\)
0.0543075 + 0.998524i \(0.482705\pi\)
\(860\) 0 0
\(861\) 6.86428e6 0.315564
\(862\) 1.43125e7 0.656068
\(863\) 1.44887e7 0.662219 0.331110 0.943592i \(-0.392577\pi\)
0.331110 + 0.943592i \(0.392577\pi\)
\(864\) 2.79144e6 0.127217
\(865\) 0 0
\(866\) −2.36829e7 −1.07310
\(867\) 9.25903e6 0.418329
\(868\) −4.82763e6 −0.217488
\(869\) −3.50122e6 −0.157279
\(870\) 0 0
\(871\) −7.40533e6 −0.330749
\(872\) 1.67042e6 0.0743936
\(873\) −6.14057e6 −0.272692
\(874\) 1.17842e7 0.521823
\(875\) 0 0
\(876\) 6.38356e6 0.281063
\(877\) 3.17684e7 1.39475 0.697376 0.716705i \(-0.254351\pi\)
0.697376 + 0.716705i \(0.254351\pi\)
\(878\) −4.25143e7 −1.86122
\(879\) 2.16893e7 0.946834
\(880\) 0 0
\(881\) 1.06777e7 0.463486 0.231743 0.972777i \(-0.425557\pi\)
0.231743 + 0.972777i \(0.425557\pi\)
\(882\) −2.47113e6 −0.106961
\(883\) −1.77890e7 −0.767804 −0.383902 0.923374i \(-0.625420\pi\)
−0.383902 + 0.923374i \(0.625420\pi\)
\(884\) 2.93542e6 0.126340
\(885\) 0 0
\(886\) −4.47005e7 −1.91306
\(887\) −3.13731e7 −1.33890 −0.669450 0.742857i \(-0.733470\pi\)
−0.669450 + 0.742857i \(0.733470\pi\)
\(888\) −9.73326e6 −0.414215
\(889\) −1.64418e7 −0.697741
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −9.85397e6 −0.414667
\(893\) −3.90418e6 −0.163833
\(894\) −1.68356e7 −0.704505
\(895\) 0 0
\(896\) 3.24104e7 1.34870
\(897\) 4.83123e6 0.200482
\(898\) 1.95689e7 0.809795
\(899\) 1.19110e7 0.491529
\(900\) 0 0
\(901\) −5.99972e6 −0.246218
\(902\) −4.13210e6 −0.169104
\(903\) 1.82916e7 0.746504
\(904\) −2.81620e7 −1.14615
\(905\) 0 0
\(906\) 3.08348e7 1.24802
\(907\) 8.46355e6 0.341613 0.170807 0.985305i \(-0.445363\pi\)
0.170807 + 0.985305i \(0.445363\pi\)
\(908\) −9.38252e6 −0.377663
\(909\) 2.83691e6 0.113877
\(910\) 0 0
\(911\) −2.18411e7 −0.871923 −0.435961 0.899965i \(-0.643592\pi\)
−0.435961 + 0.899965i \(0.643592\pi\)
\(912\) −6.43672e6 −0.256258
\(913\) −562486. −0.0223324
\(914\) −4.00870e7 −1.58722
\(915\) 0 0
\(916\) 1.51889e6 0.0598120
\(917\) 1.71081e6 0.0671861
\(918\) 7.48206e6 0.293032
\(919\) 4.18752e7 1.63557 0.817784 0.575525i \(-0.195202\pi\)
0.817784 + 0.575525i \(0.195202\pi\)
\(920\) 0 0
\(921\) 1.65042e7 0.641130
\(922\) 2.24165e7 0.868440
\(923\) 8.95575e6 0.346017
\(924\) 1.75783e6 0.0677325
\(925\) 0 0
\(926\) −3.01717e7 −1.15630
\(927\) −1.37422e6 −0.0525240
\(928\) 1.52498e7 0.581293
\(929\) 8.03740e6 0.305546 0.152773 0.988261i \(-0.451180\pi\)
0.152773 + 0.988261i \(0.451180\pi\)
\(930\) 0 0
\(931\) 2.65028e6 0.100211
\(932\) 1.45002e7 0.546805
\(933\) 2.04757e7 0.770077
\(934\) 3.30317e7 1.23898
\(935\) 0 0
\(936\) −1.89755e6 −0.0707953
\(937\) 1.78192e6 0.0663038 0.0331519 0.999450i \(-0.489445\pi\)
0.0331519 + 0.999450i \(0.489445\pi\)
\(938\) 4.17943e7 1.55099
\(939\) −1.43627e6 −0.0531583
\(940\) 0 0
\(941\) 2.97245e7 1.09431 0.547154 0.837032i \(-0.315711\pi\)
0.547154 + 0.837032i \(0.315711\pi\)
\(942\) −131145. −0.00481531
\(943\) 1.64177e7 0.601220
\(944\) −1.21118e7 −0.442364
\(945\) 0 0
\(946\) −1.10110e7 −0.400036
\(947\) −931083. −0.0337375 −0.0168688 0.999858i \(-0.505370\pi\)
−0.0168688 + 0.999858i \(0.505370\pi\)
\(948\) 2.86965e6 0.103707
\(949\) −1.09579e7 −0.394967
\(950\) 0 0
\(951\) 1.11904e7 0.401232
\(952\) 3.15437e7 1.12803
\(953\) 2.74983e7 0.980784 0.490392 0.871502i \(-0.336854\pi\)
0.490392 + 0.871502i \(0.336854\pi\)
\(954\) −2.03697e6 −0.0724626
\(955\) 0 0
\(956\) 1.23171e7 0.435876
\(957\) −4.33703e6 −0.153078
\(958\) 1.85459e7 0.652882
\(959\) −5.63879e7 −1.97988
\(960\) 0 0
\(961\) −1.96844e7 −0.687565
\(962\) −8.77507e6 −0.305712
\(963\) −987424. −0.0343114
\(964\) −5.19646e6 −0.180101
\(965\) 0 0
\(966\) −2.72666e7 −0.940129
\(967\) −4.33623e6 −0.149123 −0.0745617 0.997216i \(-0.523756\pi\)
−0.0745617 + 0.997216i \(0.523756\pi\)
\(968\) 2.01476e6 0.0691091
\(969\) −8.02448e6 −0.274541
\(970\) 0 0
\(971\) −5.44123e7 −1.85204 −0.926018 0.377480i \(-0.876791\pi\)
−0.926018 + 0.377480i \(0.876791\pi\)
\(972\) 650675. 0.0220901
\(973\) 4.54995e7 1.54072
\(974\) −3.73157e7 −1.26036
\(975\) 0 0
\(976\) 2.65244e7 0.891294
\(977\) 3.16213e7 1.05985 0.529924 0.848045i \(-0.322220\pi\)
0.529924 + 0.848045i \(0.322220\pi\)
\(978\) 1.47662e7 0.493651
\(979\) 1.25184e7 0.417438
\(980\) 0 0
\(981\) 983239. 0.0326202
\(982\) −2.12321e7 −0.702611
\(983\) −1.31368e7 −0.433617 −0.216808 0.976214i \(-0.569565\pi\)
−0.216808 + 0.976214i \(0.569565\pi\)
\(984\) −6.44836e6 −0.212306
\(985\) 0 0
\(986\) 4.08750e7 1.33895
\(987\) 9.03355e6 0.295165
\(988\) −1.06886e6 −0.0348359
\(989\) 4.37491e7 1.42226
\(990\) 0 0
\(991\) −3.53335e7 −1.14288 −0.571442 0.820643i \(-0.693616\pi\)
−0.571442 + 0.820643i \(0.693616\pi\)
\(992\) 1.14521e7 0.369493
\(993\) −8.62603e6 −0.277612
\(994\) −5.05446e7 −1.62259
\(995\) 0 0
\(996\) 461020. 0.0147256
\(997\) −3.72145e7 −1.18570 −0.592850 0.805313i \(-0.701997\pi\)
−0.592850 + 0.805313i \(0.701997\pi\)
\(998\) 3.33563e7 1.06011
\(999\) −5.72915e6 −0.181625
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.h.1.1 3
5.4 even 2 165.6.a.d.1.3 3
15.14 odd 2 495.6.a.c.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.d.1.3 3 5.4 even 2
495.6.a.c.1.1 3 15.14 odd 2
825.6.a.h.1.1 3 1.1 even 1 trivial