Properties

Label 825.6.a.q.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.09383\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.09383 q^{2} -9.00000 q^{3} +18.3224 q^{4} +63.8444 q^{6} +4.62438 q^{7} +97.0268 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-7.09383 q^{2} -9.00000 q^{3} +18.3224 q^{4} +63.8444 q^{6} +4.62438 q^{7} +97.0268 q^{8} +81.0000 q^{9} -121.000 q^{11} -164.901 q^{12} -38.0995 q^{13} -32.8046 q^{14} -1274.61 q^{16} -610.281 q^{17} -574.600 q^{18} +2102.49 q^{19} -41.6194 q^{21} +858.353 q^{22} +3875.26 q^{23} -873.241 q^{24} +270.272 q^{26} -729.000 q^{27} +84.7296 q^{28} -8371.56 q^{29} -1417.68 q^{31} +5936.98 q^{32} +1089.00 q^{33} +4329.23 q^{34} +1484.11 q^{36} -3071.11 q^{37} -14914.7 q^{38} +342.896 q^{39} -591.131 q^{41} +295.241 q^{42} -19206.4 q^{43} -2217.01 q^{44} -27490.4 q^{46} +11447.9 q^{47} +11471.5 q^{48} -16785.6 q^{49} +5492.53 q^{51} -698.074 q^{52} +17463.0 q^{53} +5171.40 q^{54} +448.689 q^{56} -18922.5 q^{57} +59386.4 q^{58} -16917.0 q^{59} -2597.86 q^{61} +10056.8 q^{62} +374.575 q^{63} -1328.50 q^{64} -7725.18 q^{66} +59848.5 q^{67} -11181.8 q^{68} -34877.3 q^{69} -62258.0 q^{71} +7859.17 q^{72} +35314.1 q^{73} +21785.9 q^{74} +38522.7 q^{76} -559.550 q^{77} -2432.44 q^{78} +78372.2 q^{79} +6561.00 q^{81} +4193.38 q^{82} +39714.4 q^{83} -762.567 q^{84} +136247. q^{86} +75344.0 q^{87} -11740.2 q^{88} +54307.3 q^{89} -176.187 q^{91} +71003.9 q^{92} +12759.2 q^{93} -81209.4 q^{94} -53432.8 q^{96} +33574.1 q^{97} +119074. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.09383 −1.25402 −0.627012 0.779010i \(-0.715722\pi\)
−0.627012 + 0.779010i \(0.715722\pi\)
\(3\) −9.00000 −0.577350
\(4\) 18.3224 0.572574
\(5\) 0 0
\(6\) 63.8444 0.724011
\(7\) 4.62438 0.0356705 0.0178352 0.999841i \(-0.494323\pi\)
0.0178352 + 0.999841i \(0.494323\pi\)
\(8\) 97.0268 0.536002
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −164.901 −0.330576
\(13\) −38.0995 −0.0625261 −0.0312631 0.999511i \(-0.509953\pi\)
−0.0312631 + 0.999511i \(0.509953\pi\)
\(14\) −32.8046 −0.0447316
\(15\) 0 0
\(16\) −1274.61 −1.24473
\(17\) −610.281 −0.512163 −0.256081 0.966655i \(-0.582431\pi\)
−0.256081 + 0.966655i \(0.582431\pi\)
\(18\) −574.600 −0.418008
\(19\) 2102.49 1.33614 0.668068 0.744100i \(-0.267121\pi\)
0.668068 + 0.744100i \(0.267121\pi\)
\(20\) 0 0
\(21\) −41.6194 −0.0205944
\(22\) 858.353 0.378102
\(23\) 3875.26 1.52750 0.763750 0.645513i \(-0.223356\pi\)
0.763750 + 0.645513i \(0.223356\pi\)
\(24\) −873.241 −0.309461
\(25\) 0 0
\(26\) 270.272 0.0784092
\(27\) −729.000 −0.192450
\(28\) 84.7296 0.0204240
\(29\) −8371.56 −1.84847 −0.924233 0.381829i \(-0.875294\pi\)
−0.924233 + 0.381829i \(0.875294\pi\)
\(30\) 0 0
\(31\) −1417.68 −0.264957 −0.132479 0.991186i \(-0.542294\pi\)
−0.132479 + 0.991186i \(0.542294\pi\)
\(32\) 5936.98 1.02492
\(33\) 1089.00 0.174078
\(34\) 4329.23 0.642264
\(35\) 0 0
\(36\) 1484.11 0.190858
\(37\) −3071.11 −0.368800 −0.184400 0.982851i \(-0.559034\pi\)
−0.184400 + 0.982851i \(0.559034\pi\)
\(38\) −14914.7 −1.67555
\(39\) 342.896 0.0360995
\(40\) 0 0
\(41\) −591.131 −0.0549192 −0.0274596 0.999623i \(-0.508742\pi\)
−0.0274596 + 0.999623i \(0.508742\pi\)
\(42\) 295.241 0.0258258
\(43\) −19206.4 −1.58407 −0.792037 0.610474i \(-0.790979\pi\)
−0.792037 + 0.610474i \(0.790979\pi\)
\(44\) −2217.01 −0.172638
\(45\) 0 0
\(46\) −27490.4 −1.91552
\(47\) 11447.9 0.755929 0.377965 0.925820i \(-0.376624\pi\)
0.377965 + 0.925820i \(0.376624\pi\)
\(48\) 11471.5 0.718647
\(49\) −16785.6 −0.998728
\(50\) 0 0
\(51\) 5492.53 0.295697
\(52\) −698.074 −0.0358008
\(53\) 17463.0 0.853944 0.426972 0.904265i \(-0.359580\pi\)
0.426972 + 0.904265i \(0.359580\pi\)
\(54\) 5171.40 0.241337
\(55\) 0 0
\(56\) 448.689 0.0191194
\(57\) −18922.5 −0.771419
\(58\) 59386.4 2.31802
\(59\) −16917.0 −0.632692 −0.316346 0.948644i \(-0.602456\pi\)
−0.316346 + 0.948644i \(0.602456\pi\)
\(60\) 0 0
\(61\) −2597.86 −0.0893904 −0.0446952 0.999001i \(-0.514232\pi\)
−0.0446952 + 0.999001i \(0.514232\pi\)
\(62\) 10056.8 0.332262
\(63\) 374.575 0.0118902
\(64\) −1328.50 −0.0405426
\(65\) 0 0
\(66\) −7725.18 −0.218297
\(67\) 59848.5 1.62879 0.814397 0.580308i \(-0.197068\pi\)
0.814397 + 0.580308i \(0.197068\pi\)
\(68\) −11181.8 −0.293251
\(69\) −34877.3 −0.881902
\(70\) 0 0
\(71\) −62258.0 −1.46572 −0.732858 0.680382i \(-0.761814\pi\)
−0.732858 + 0.680382i \(0.761814\pi\)
\(72\) 7859.17 0.178667
\(73\) 35314.1 0.775605 0.387803 0.921742i \(-0.373234\pi\)
0.387803 + 0.921742i \(0.373234\pi\)
\(74\) 21785.9 0.462484
\(75\) 0 0
\(76\) 38522.7 0.765037
\(77\) −559.550 −0.0107550
\(78\) −2432.44 −0.0452696
\(79\) 78372.2 1.41284 0.706422 0.707791i \(-0.250308\pi\)
0.706422 + 0.707791i \(0.250308\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 4193.38 0.0688700
\(83\) 39714.4 0.632781 0.316390 0.948629i \(-0.397529\pi\)
0.316390 + 0.948629i \(0.397529\pi\)
\(84\) −762.567 −0.0117918
\(85\) 0 0
\(86\) 136247. 1.98646
\(87\) 75344.0 1.06721
\(88\) −11740.2 −0.161611
\(89\) 54307.3 0.726747 0.363374 0.931643i \(-0.381625\pi\)
0.363374 + 0.931643i \(0.381625\pi\)
\(90\) 0 0
\(91\) −176.187 −0.00223034
\(92\) 71003.9 0.874606
\(93\) 12759.2 0.152973
\(94\) −81209.4 −0.947953
\(95\) 0 0
\(96\) −53432.8 −0.591739
\(97\) 33574.1 0.362305 0.181153 0.983455i \(-0.442017\pi\)
0.181153 + 0.983455i \(0.442017\pi\)
\(98\) 119074. 1.25243
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −203384. −1.98387 −0.991935 0.126750i \(-0.959545\pi\)
−0.991935 + 0.126750i \(0.959545\pi\)
\(102\) −38963.1 −0.370811
\(103\) −95955.0 −0.891199 −0.445599 0.895233i \(-0.647009\pi\)
−0.445599 + 0.895233i \(0.647009\pi\)
\(104\) −3696.68 −0.0335141
\(105\) 0 0
\(106\) −123880. −1.07087
\(107\) 170934. 1.44334 0.721669 0.692239i \(-0.243375\pi\)
0.721669 + 0.692239i \(0.243375\pi\)
\(108\) −13357.0 −0.110192
\(109\) −62167.4 −0.501183 −0.250592 0.968093i \(-0.580625\pi\)
−0.250592 + 0.968093i \(0.580625\pi\)
\(110\) 0 0
\(111\) 27640.0 0.212927
\(112\) −5894.27 −0.0444002
\(113\) 75966.1 0.559660 0.279830 0.960050i \(-0.409722\pi\)
0.279830 + 0.960050i \(0.409722\pi\)
\(114\) 134233. 0.967377
\(115\) 0 0
\(116\) −153387. −1.05838
\(117\) −3086.06 −0.0208420
\(118\) 120006. 0.793410
\(119\) −2822.17 −0.0182691
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 18428.8 0.112098
\(123\) 5320.18 0.0317076
\(124\) −25975.3 −0.151707
\(125\) 0 0
\(126\) −2657.17 −0.0149105
\(127\) −100022. −0.550285 −0.275143 0.961403i \(-0.588725\pi\)
−0.275143 + 0.961403i \(0.588725\pi\)
\(128\) −180559. −0.974080
\(129\) 172858. 0.914565
\(130\) 0 0
\(131\) 322642. 1.64264 0.821319 0.570469i \(-0.193238\pi\)
0.821319 + 0.570469i \(0.193238\pi\)
\(132\) 19953.1 0.0996723
\(133\) 9722.74 0.0476606
\(134\) −424555. −2.04255
\(135\) 0 0
\(136\) −59213.6 −0.274520
\(137\) 325855. 1.48328 0.741639 0.670799i \(-0.234049\pi\)
0.741639 + 0.670799i \(0.234049\pi\)
\(138\) 247414. 1.10593
\(139\) 415444. 1.82379 0.911895 0.410422i \(-0.134619\pi\)
0.911895 + 0.410422i \(0.134619\pi\)
\(140\) 0 0
\(141\) −103031. −0.436436
\(142\) 441648. 1.83804
\(143\) 4610.04 0.0188523
\(144\) −103243. −0.414911
\(145\) 0 0
\(146\) −250512. −0.972627
\(147\) 151071. 0.576616
\(148\) −56270.0 −0.211165
\(149\) 306627. 1.13147 0.565737 0.824585i \(-0.308592\pi\)
0.565737 + 0.824585i \(0.308592\pi\)
\(150\) 0 0
\(151\) −90347.8 −0.322460 −0.161230 0.986917i \(-0.551546\pi\)
−0.161230 + 0.986917i \(0.551546\pi\)
\(152\) 203998. 0.716172
\(153\) −49432.8 −0.170721
\(154\) 3969.35 0.0134871
\(155\) 0 0
\(156\) 6282.66 0.0206696
\(157\) 81270.2 0.263137 0.131569 0.991307i \(-0.457999\pi\)
0.131569 + 0.991307i \(0.457999\pi\)
\(158\) −555959. −1.77174
\(159\) −157167. −0.493025
\(160\) 0 0
\(161\) 17920.7 0.0544866
\(162\) −46542.6 −0.139336
\(163\) −178404. −0.525940 −0.262970 0.964804i \(-0.584702\pi\)
−0.262970 + 0.964804i \(0.584702\pi\)
\(164\) −10830.9 −0.0314453
\(165\) 0 0
\(166\) −281727. −0.793522
\(167\) 626513. 1.73836 0.869178 0.494500i \(-0.164649\pi\)
0.869178 + 0.494500i \(0.164649\pi\)
\(168\) −4038.20 −0.0110386
\(169\) −369841. −0.996090
\(170\) 0 0
\(171\) 170302. 0.445379
\(172\) −351907. −0.906999
\(173\) −325238. −0.826202 −0.413101 0.910685i \(-0.635554\pi\)
−0.413101 + 0.910685i \(0.635554\pi\)
\(174\) −534478. −1.33831
\(175\) 0 0
\(176\) 154227. 0.375301
\(177\) 152253. 0.365285
\(178\) −385247. −0.911358
\(179\) 183002. 0.426897 0.213449 0.976954i \(-0.431530\pi\)
0.213449 + 0.976954i \(0.431530\pi\)
\(180\) 0 0
\(181\) −623259. −1.41407 −0.707037 0.707177i \(-0.749968\pi\)
−0.707037 + 0.707177i \(0.749968\pi\)
\(182\) 1249.84 0.00279689
\(183\) 23380.7 0.0516096
\(184\) 376004. 0.818743
\(185\) 0 0
\(186\) −90511.3 −0.191832
\(187\) 73844.0 0.154423
\(188\) 209753. 0.432826
\(189\) −3371.18 −0.00686478
\(190\) 0 0
\(191\) 179426. 0.355879 0.177939 0.984041i \(-0.443057\pi\)
0.177939 + 0.984041i \(0.443057\pi\)
\(192\) 11956.5 0.0234073
\(193\) −904409. −1.74772 −0.873859 0.486179i \(-0.838390\pi\)
−0.873859 + 0.486179i \(0.838390\pi\)
\(194\) −238169. −0.454339
\(195\) 0 0
\(196\) −307552. −0.571845
\(197\) −494917. −0.908588 −0.454294 0.890852i \(-0.650108\pi\)
−0.454294 + 0.890852i \(0.650108\pi\)
\(198\) 69526.6 0.126034
\(199\) −824334. −1.47561 −0.737804 0.675015i \(-0.764137\pi\)
−0.737804 + 0.675015i \(0.764137\pi\)
\(200\) 0 0
\(201\) −538636. −0.940385
\(202\) 1.44277e6 2.48782
\(203\) −38713.3 −0.0659356
\(204\) 100636. 0.169308
\(205\) 0 0
\(206\) 680688. 1.11758
\(207\) 313896. 0.509166
\(208\) 48561.9 0.0778283
\(209\) −254402. −0.402860
\(210\) 0 0
\(211\) 4819.05 0.00745170 0.00372585 0.999993i \(-0.498814\pi\)
0.00372585 + 0.999993i \(0.498814\pi\)
\(212\) 319964. 0.488946
\(213\) 560322. 0.846231
\(214\) −1.21257e6 −1.80998
\(215\) 0 0
\(216\) −70732.5 −0.103154
\(217\) −6555.92 −0.00945114
\(218\) 441005. 0.628495
\(219\) −317827. −0.447796
\(220\) 0 0
\(221\) 23251.4 0.0320235
\(222\) −196073. −0.267015
\(223\) −225057. −0.303062 −0.151531 0.988453i \(-0.548420\pi\)
−0.151531 + 0.988453i \(0.548420\pi\)
\(224\) 27454.9 0.0365594
\(225\) 0 0
\(226\) −538891. −0.701826
\(227\) 847198. 1.09124 0.545620 0.838033i \(-0.316294\pi\)
0.545620 + 0.838033i \(0.316294\pi\)
\(228\) −346704. −0.441694
\(229\) 901257. 1.13569 0.567845 0.823135i \(-0.307777\pi\)
0.567845 + 0.823135i \(0.307777\pi\)
\(230\) 0 0
\(231\) 5035.95 0.00620943
\(232\) −812265. −0.990782
\(233\) −439120. −0.529900 −0.264950 0.964262i \(-0.585355\pi\)
−0.264950 + 0.964262i \(0.585355\pi\)
\(234\) 21892.0 0.0261364
\(235\) 0 0
\(236\) −309959. −0.362263
\(237\) −705350. −0.815706
\(238\) 20020.0 0.0229098
\(239\) −922328. −1.04446 −0.522229 0.852805i \(-0.674899\pi\)
−0.522229 + 0.852805i \(0.674899\pi\)
\(240\) 0 0
\(241\) −891658. −0.988907 −0.494454 0.869204i \(-0.664632\pi\)
−0.494454 + 0.869204i \(0.664632\pi\)
\(242\) −103861. −0.114002
\(243\) −59049.0 −0.0641500
\(244\) −47598.9 −0.0511826
\(245\) 0 0
\(246\) −37740.4 −0.0397621
\(247\) −80104.1 −0.0835434
\(248\) −137553. −0.142018
\(249\) −357430. −0.365336
\(250\) 0 0
\(251\) −125904. −0.126141 −0.0630704 0.998009i \(-0.520089\pi\)
−0.0630704 + 0.998009i \(0.520089\pi\)
\(252\) 6863.10 0.00680799
\(253\) −468906. −0.460558
\(254\) 709541. 0.690070
\(255\) 0 0
\(256\) 1.32337e6 1.26206
\(257\) −864526. −0.816479 −0.408240 0.912875i \(-0.633857\pi\)
−0.408240 + 0.912875i \(0.633857\pi\)
\(258\) −1.22622e6 −1.14689
\(259\) −14202.0 −0.0131553
\(260\) 0 0
\(261\) −678096. −0.616155
\(262\) −2.28876e6 −2.05991
\(263\) −860845. −0.767424 −0.383712 0.923453i \(-0.625355\pi\)
−0.383712 + 0.923453i \(0.625355\pi\)
\(264\) 105662. 0.0933060
\(265\) 0 0
\(266\) −68971.4 −0.0597675
\(267\) −488766. −0.419588
\(268\) 1.09657e6 0.932605
\(269\) −169703. −0.142991 −0.0714956 0.997441i \(-0.522777\pi\)
−0.0714956 + 0.997441i \(0.522777\pi\)
\(270\) 0 0
\(271\) −368847. −0.305087 −0.152543 0.988297i \(-0.548746\pi\)
−0.152543 + 0.988297i \(0.548746\pi\)
\(272\) 777869. 0.637506
\(273\) 1585.68 0.00128768
\(274\) −2.31156e6 −1.86006
\(275\) 0 0
\(276\) −639035. −0.504954
\(277\) −423587. −0.331698 −0.165849 0.986151i \(-0.553036\pi\)
−0.165849 + 0.986151i \(0.553036\pi\)
\(278\) −2.94708e6 −2.28708
\(279\) −114832. −0.0883190
\(280\) 0 0
\(281\) 2.39124e6 1.80658 0.903290 0.429031i \(-0.141145\pi\)
0.903290 + 0.429031i \(0.141145\pi\)
\(282\) 730885. 0.547301
\(283\) 238848. 0.177279 0.0886393 0.996064i \(-0.471748\pi\)
0.0886393 + 0.996064i \(0.471748\pi\)
\(284\) −1.14071e6 −0.839230
\(285\) 0 0
\(286\) −32702.9 −0.0236413
\(287\) −2733.62 −0.00195899
\(288\) 480895. 0.341641
\(289\) −1.04741e6 −0.737690
\(290\) 0 0
\(291\) −302167. −0.209177
\(292\) 647037. 0.444091
\(293\) 289624. 0.197091 0.0985454 0.995133i \(-0.468581\pi\)
0.0985454 + 0.995133i \(0.468581\pi\)
\(294\) −1.07167e6 −0.723089
\(295\) 0 0
\(296\) −297980. −0.197678
\(297\) 88209.0 0.0580259
\(298\) −2.17516e6 −1.41890
\(299\) −147646. −0.0955086
\(300\) 0 0
\(301\) −88817.8 −0.0565046
\(302\) 640912. 0.404372
\(303\) 1.83045e6 1.14539
\(304\) −2.67985e6 −1.66313
\(305\) 0 0
\(306\) 350668. 0.214088
\(307\) −2.74983e6 −1.66517 −0.832586 0.553895i \(-0.813141\pi\)
−0.832586 + 0.553895i \(0.813141\pi\)
\(308\) −10252.3 −0.00615806
\(309\) 863595. 0.514534
\(310\) 0 0
\(311\) −669544. −0.392534 −0.196267 0.980550i \(-0.562882\pi\)
−0.196267 + 0.980550i \(0.562882\pi\)
\(312\) 33270.1 0.0193494
\(313\) −458501. −0.264533 −0.132266 0.991214i \(-0.542225\pi\)
−0.132266 + 0.991214i \(0.542225\pi\)
\(314\) −576516. −0.329980
\(315\) 0 0
\(316\) 1.43596e6 0.808958
\(317\) 1.69327e6 0.946406 0.473203 0.880953i \(-0.343098\pi\)
0.473203 + 0.880953i \(0.343098\pi\)
\(318\) 1.11492e6 0.618264
\(319\) 1.01296e6 0.557333
\(320\) 0 0
\(321\) −1.53840e6 −0.833311
\(322\) −127126. −0.0683275
\(323\) −1.28311e6 −0.684319
\(324\) 120213. 0.0636193
\(325\) 0 0
\(326\) 1.26557e6 0.659540
\(327\) 559507. 0.289358
\(328\) −57355.6 −0.0294368
\(329\) 52939.5 0.0269644
\(330\) 0 0
\(331\) −2.96369e6 −1.48684 −0.743418 0.668827i \(-0.766797\pi\)
−0.743418 + 0.668827i \(0.766797\pi\)
\(332\) 727663. 0.362314
\(333\) −248760. −0.122933
\(334\) −4.44437e6 −2.17994
\(335\) 0 0
\(336\) 53048.4 0.0256345
\(337\) −2.28507e6 −1.09604 −0.548018 0.836466i \(-0.684618\pi\)
−0.548018 + 0.836466i \(0.684618\pi\)
\(338\) 2.62359e6 1.24912
\(339\) −683695. −0.323120
\(340\) 0 0
\(341\) 171540. 0.0798875
\(342\) −1.20809e6 −0.558515
\(343\) −155345. −0.0712955
\(344\) −1.86354e6 −0.849067
\(345\) 0 0
\(346\) 2.30718e6 1.03608
\(347\) −427019. −0.190381 −0.0951905 0.995459i \(-0.530346\pi\)
−0.0951905 + 0.995459i \(0.530346\pi\)
\(348\) 1.38048e6 0.611058
\(349\) −2.34367e6 −1.02999 −0.514995 0.857193i \(-0.672206\pi\)
−0.514995 + 0.857193i \(0.672206\pi\)
\(350\) 0 0
\(351\) 27774.6 0.0120332
\(352\) −718375. −0.309026
\(353\) −1.63784e6 −0.699574 −0.349787 0.936829i \(-0.613746\pi\)
−0.349787 + 0.936829i \(0.613746\pi\)
\(354\) −1.08005e6 −0.458076
\(355\) 0 0
\(356\) 995039. 0.416117
\(357\) 25399.6 0.0105477
\(358\) −1.29819e6 −0.535339
\(359\) 2.80870e6 1.15019 0.575095 0.818087i \(-0.304965\pi\)
0.575095 + 0.818087i \(0.304965\pi\)
\(360\) 0 0
\(361\) 1.94439e6 0.785261
\(362\) 4.42129e6 1.77328
\(363\) −131769. −0.0524864
\(364\) −3228.16 −0.00127703
\(365\) 0 0
\(366\) −165859. −0.0647196
\(367\) 40056.0 0.0155239 0.00776197 0.999970i \(-0.497529\pi\)
0.00776197 + 0.999970i \(0.497529\pi\)
\(368\) −4.93943e6 −1.90133
\(369\) −47881.6 −0.0183064
\(370\) 0 0
\(371\) 80755.6 0.0304606
\(372\) 233778. 0.0875884
\(373\) 1.60363e6 0.596806 0.298403 0.954440i \(-0.403546\pi\)
0.298403 + 0.954440i \(0.403546\pi\)
\(374\) −523837. −0.193650
\(375\) 0 0
\(376\) 1.11075e6 0.405180
\(377\) 318953. 0.115577
\(378\) 23914.5 0.00860860
\(379\) 2.12659e6 0.760476 0.380238 0.924889i \(-0.375842\pi\)
0.380238 + 0.924889i \(0.375842\pi\)
\(380\) 0 0
\(381\) 900201. 0.317707
\(382\) −1.27282e6 −0.446280
\(383\) −3.69333e6 −1.28653 −0.643267 0.765642i \(-0.722421\pi\)
−0.643267 + 0.765642i \(0.722421\pi\)
\(384\) 1.62503e6 0.562386
\(385\) 0 0
\(386\) 6.41572e6 2.19168
\(387\) −1.55572e6 −0.528024
\(388\) 615157. 0.207447
\(389\) −215626. −0.0722482 −0.0361241 0.999347i \(-0.511501\pi\)
−0.0361241 + 0.999347i \(0.511501\pi\)
\(390\) 0 0
\(391\) −2.36500e6 −0.782328
\(392\) −1.62865e6 −0.535320
\(393\) −2.90377e6 −0.948378
\(394\) 3.51086e6 1.13939
\(395\) 0 0
\(396\) −179578. −0.0575458
\(397\) 4.45417e6 1.41837 0.709186 0.705021i \(-0.249062\pi\)
0.709186 + 0.705021i \(0.249062\pi\)
\(398\) 5.84769e6 1.85045
\(399\) −87504.7 −0.0275169
\(400\) 0 0
\(401\) 5.31987e6 1.65211 0.826057 0.563586i \(-0.190579\pi\)
0.826057 + 0.563586i \(0.190579\pi\)
\(402\) 3.82099e6 1.17926
\(403\) 54013.1 0.0165667
\(404\) −3.72647e6 −1.13591
\(405\) 0 0
\(406\) 274625. 0.0826848
\(407\) 371604. 0.111197
\(408\) 532923. 0.158494
\(409\) −3.81493e6 −1.12766 −0.563830 0.825891i \(-0.690673\pi\)
−0.563830 + 0.825891i \(0.690673\pi\)
\(410\) 0 0
\(411\) −2.93269e6 −0.856371
\(412\) −1.75812e6 −0.510277
\(413\) −78230.5 −0.0225684
\(414\) −2.22672e6 −0.638506
\(415\) 0 0
\(416\) −226196. −0.0640844
\(417\) −3.73899e6 −1.05297
\(418\) 1.80468e6 0.505196
\(419\) −3.59908e6 −1.00151 −0.500756 0.865588i \(-0.666945\pi\)
−0.500756 + 0.865588i \(0.666945\pi\)
\(420\) 0 0
\(421\) −2.15462e6 −0.592467 −0.296234 0.955115i \(-0.595731\pi\)
−0.296234 + 0.955115i \(0.595731\pi\)
\(422\) −34185.5 −0.00934460
\(423\) 927280. 0.251976
\(424\) 1.69438e6 0.457716
\(425\) 0 0
\(426\) −3.97483e6 −1.06119
\(427\) −12013.5 −0.00318860
\(428\) 3.13191e6 0.826417
\(429\) −41490.4 −0.0108844
\(430\) 0 0
\(431\) −2.10237e6 −0.545149 −0.272575 0.962135i \(-0.587875\pi\)
−0.272575 + 0.962135i \(0.587875\pi\)
\(432\) 929188. 0.239549
\(433\) 41848.5 0.0107265 0.00536327 0.999986i \(-0.498293\pi\)
0.00536327 + 0.999986i \(0.498293\pi\)
\(434\) 46506.5 0.0118519
\(435\) 0 0
\(436\) −1.13905e6 −0.286964
\(437\) 8.14771e6 2.04095
\(438\) 2.25461e6 0.561546
\(439\) −5.35761e6 −1.32681 −0.663407 0.748259i \(-0.730890\pi\)
−0.663407 + 0.748259i \(0.730890\pi\)
\(440\) 0 0
\(441\) −1.35963e6 −0.332909
\(442\) −164942. −0.0401582
\(443\) −4.69738e6 −1.13722 −0.568612 0.822606i \(-0.692520\pi\)
−0.568612 + 0.822606i \(0.692520\pi\)
\(444\) 506430. 0.121916
\(445\) 0 0
\(446\) 1.59652e6 0.380046
\(447\) −2.75964e6 −0.653257
\(448\) −6143.50 −0.00144617
\(449\) 7.10537e6 1.66330 0.831650 0.555300i \(-0.187397\pi\)
0.831650 + 0.555300i \(0.187397\pi\)
\(450\) 0 0
\(451\) 71526.9 0.0165588
\(452\) 1.39188e6 0.320447
\(453\) 813131. 0.186172
\(454\) −6.00987e6 −1.36844
\(455\) 0 0
\(456\) −1.83598e6 −0.413482
\(457\) −4.82566e6 −1.08085 −0.540426 0.841392i \(-0.681737\pi\)
−0.540426 + 0.841392i \(0.681737\pi\)
\(458\) −6.39336e6 −1.42418
\(459\) 444895. 0.0985657
\(460\) 0 0
\(461\) −1.03634e6 −0.227117 −0.113558 0.993531i \(-0.536225\pi\)
−0.113558 + 0.993531i \(0.536225\pi\)
\(462\) −35724.2 −0.00778677
\(463\) −457904. −0.0992709 −0.0496355 0.998767i \(-0.515806\pi\)
−0.0496355 + 0.998767i \(0.515806\pi\)
\(464\) 1.06704e7 2.30085
\(465\) 0 0
\(466\) 3.11504e6 0.664507
\(467\) −2.02783e6 −0.430268 −0.215134 0.976584i \(-0.569019\pi\)
−0.215134 + 0.976584i \(0.569019\pi\)
\(468\) −56544.0 −0.0119336
\(469\) 276762. 0.0580998
\(470\) 0 0
\(471\) −731431. −0.151922
\(472\) −1.64140e6 −0.339124
\(473\) 2.32398e6 0.477616
\(474\) 5.00363e6 1.02291
\(475\) 0 0
\(476\) −51708.9 −0.0104604
\(477\) 1.41450e6 0.284648
\(478\) 6.54284e6 1.30977
\(479\) 3.41441e6 0.679949 0.339975 0.940435i \(-0.389582\pi\)
0.339975 + 0.940435i \(0.389582\pi\)
\(480\) 0 0
\(481\) 117008. 0.0230596
\(482\) 6.32526e6 1.24011
\(483\) −161286. −0.0314579
\(484\) 268258. 0.0520522
\(485\) 0 0
\(486\) 418883. 0.0804456
\(487\) 1.21543e6 0.232224 0.116112 0.993236i \(-0.462957\pi\)
0.116112 + 0.993236i \(0.462957\pi\)
\(488\) −252062. −0.0479134
\(489\) 1.60564e6 0.303651
\(490\) 0 0
\(491\) −4.85480e6 −0.908798 −0.454399 0.890798i \(-0.650146\pi\)
−0.454399 + 0.890798i \(0.650146\pi\)
\(492\) 97478.3 0.0181550
\(493\) 5.10901e6 0.946715
\(494\) 568244. 0.104765
\(495\) 0 0
\(496\) 1.80699e6 0.329801
\(497\) −287905. −0.0522827
\(498\) 2.53555e6 0.458140
\(499\) 7.01256e6 1.26074 0.630370 0.776295i \(-0.282903\pi\)
0.630370 + 0.776295i \(0.282903\pi\)
\(500\) 0 0
\(501\) −5.63861e6 −1.00364
\(502\) 893142. 0.158184
\(503\) −7.65551e6 −1.34913 −0.674565 0.738215i \(-0.735669\pi\)
−0.674565 + 0.738215i \(0.735669\pi\)
\(504\) 36343.8 0.00637315
\(505\) 0 0
\(506\) 3.32634e6 0.577551
\(507\) 3.32857e6 0.575093
\(508\) −1.83265e6 −0.315079
\(509\) 7.70001e6 1.31734 0.658669 0.752433i \(-0.271120\pi\)
0.658669 + 0.752433i \(0.271120\pi\)
\(510\) 0 0
\(511\) 163306. 0.0276662
\(512\) −3.60985e6 −0.608574
\(513\) −1.53272e6 −0.257140
\(514\) 6.13279e6 1.02388
\(515\) 0 0
\(516\) 3.16716e6 0.523656
\(517\) −1.38520e6 −0.227921
\(518\) 100746. 0.0164970
\(519\) 2.92714e6 0.477008
\(520\) 0 0
\(521\) −3.03644e6 −0.490083 −0.245042 0.969513i \(-0.578802\pi\)
−0.245042 + 0.969513i \(0.578802\pi\)
\(522\) 4.81030e6 0.772673
\(523\) 7.75936e6 1.24043 0.620214 0.784432i \(-0.287046\pi\)
0.620214 + 0.784432i \(0.287046\pi\)
\(524\) 5.91156e6 0.940532
\(525\) 0 0
\(526\) 6.10669e6 0.962368
\(527\) 865187. 0.135701
\(528\) −1.38805e6 −0.216680
\(529\) 8.58128e6 1.33325
\(530\) 0 0
\(531\) −1.37027e6 −0.210897
\(532\) 178144. 0.0272892
\(533\) 22521.8 0.00343389
\(534\) 3.46722e6 0.526173
\(535\) 0 0
\(536\) 5.80691e6 0.873037
\(537\) −1.64702e6 −0.246469
\(538\) 1.20385e6 0.179314
\(539\) 2.03106e6 0.301128
\(540\) 0 0
\(541\) −8.80224e6 −1.29300 −0.646502 0.762912i \(-0.723769\pi\)
−0.646502 + 0.762912i \(0.723769\pi\)
\(542\) 2.61654e6 0.382586
\(543\) 5.60933e6 0.816416
\(544\) −3.62323e6 −0.524927
\(545\) 0 0
\(546\) −11248.6 −0.00161479
\(547\) −3.58640e6 −0.512496 −0.256248 0.966611i \(-0.582486\pi\)
−0.256248 + 0.966611i \(0.582486\pi\)
\(548\) 5.97043e6 0.849286
\(549\) −210426. −0.0297968
\(550\) 0 0
\(551\) −1.76012e7 −2.46980
\(552\) −3.38403e6 −0.472701
\(553\) 362423. 0.0503968
\(554\) 3.00485e6 0.415957
\(555\) 0 0
\(556\) 7.61191e6 1.04426
\(557\) 4.61032e6 0.629641 0.314821 0.949151i \(-0.398056\pi\)
0.314821 + 0.949151i \(0.398056\pi\)
\(558\) 814602. 0.110754
\(559\) 731756. 0.0990459
\(560\) 0 0
\(561\) −664596. −0.0891561
\(562\) −1.69630e7 −2.26549
\(563\) 2.16200e6 0.287464 0.143732 0.989617i \(-0.454090\pi\)
0.143732 + 0.989617i \(0.454090\pi\)
\(564\) −1.88777e6 −0.249892
\(565\) 0 0
\(566\) −1.69435e6 −0.222311
\(567\) 30340.6 0.00396338
\(568\) −6.04070e6 −0.785627
\(569\) 1.71372e6 0.221901 0.110951 0.993826i \(-0.464610\pi\)
0.110951 + 0.993826i \(0.464610\pi\)
\(570\) 0 0
\(571\) −3.80446e6 −0.488318 −0.244159 0.969735i \(-0.578512\pi\)
−0.244159 + 0.969735i \(0.578512\pi\)
\(572\) 84466.9 0.0107944
\(573\) −1.61484e6 −0.205467
\(574\) 19391.8 0.00245662
\(575\) 0 0
\(576\) −107609. −0.0135142
\(577\) −9.84742e6 −1.23135 −0.615677 0.787998i \(-0.711117\pi\)
−0.615677 + 0.787998i \(0.711117\pi\)
\(578\) 7.43017e6 0.925080
\(579\) 8.13968e6 1.00905
\(580\) 0 0
\(581\) 183655. 0.0225716
\(582\) 2.14352e6 0.262313
\(583\) −2.11302e6 −0.257474
\(584\) 3.42641e6 0.415726
\(585\) 0 0
\(586\) −2.05455e6 −0.247156
\(587\) 4.48517e6 0.537258 0.268629 0.963244i \(-0.413429\pi\)
0.268629 + 0.963244i \(0.413429\pi\)
\(588\) 2.76797e6 0.330155
\(589\) −2.98067e6 −0.354019
\(590\) 0 0
\(591\) 4.45426e6 0.524574
\(592\) 3.91446e6 0.459058
\(593\) 1.57719e7 1.84182 0.920909 0.389777i \(-0.127448\pi\)
0.920909 + 0.389777i \(0.127448\pi\)
\(594\) −625739. −0.0727658
\(595\) 0 0
\(596\) 5.61813e6 0.647853
\(597\) 7.41901e6 0.851942
\(598\) 1.04737e6 0.119770
\(599\) −8.13287e6 −0.926140 −0.463070 0.886322i \(-0.653252\pi\)
−0.463070 + 0.886322i \(0.653252\pi\)
\(600\) 0 0
\(601\) 6.56975e6 0.741930 0.370965 0.928647i \(-0.379027\pi\)
0.370965 + 0.928647i \(0.379027\pi\)
\(602\) 630058. 0.0708581
\(603\) 4.84773e6 0.542931
\(604\) −1.65539e6 −0.184632
\(605\) 0 0
\(606\) −1.29849e7 −1.43634
\(607\) −5.13693e6 −0.565890 −0.282945 0.959136i \(-0.591311\pi\)
−0.282945 + 0.959136i \(0.591311\pi\)
\(608\) 1.24825e7 1.36944
\(609\) 348420. 0.0380680
\(610\) 0 0
\(611\) −436160. −0.0472653
\(612\) −905726. −0.0977503
\(613\) −4.41922e6 −0.475001 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(614\) 1.95068e7 2.08817
\(615\) 0 0
\(616\) −54291.4 −0.00576473
\(617\) −5.70350e6 −0.603154 −0.301577 0.953442i \(-0.597513\pi\)
−0.301577 + 0.953442i \(0.597513\pi\)
\(618\) −6.12619e6 −0.645237
\(619\) 381437. 0.0400125 0.0200063 0.999800i \(-0.493631\pi\)
0.0200063 + 0.999800i \(0.493631\pi\)
\(620\) 0 0
\(621\) −2.82506e6 −0.293967
\(622\) 4.74963e6 0.492247
\(623\) 251138. 0.0259234
\(624\) −437057. −0.0449342
\(625\) 0 0
\(626\) 3.25253e6 0.331730
\(627\) 2.28962e6 0.232592
\(628\) 1.48906e6 0.150665
\(629\) 1.87424e6 0.188886
\(630\) 0 0
\(631\) −5.26994e6 −0.526905 −0.263453 0.964672i \(-0.584861\pi\)
−0.263453 + 0.964672i \(0.584861\pi\)
\(632\) 7.60420e6 0.757288
\(633\) −43371.5 −0.00430224
\(634\) −1.20117e7 −1.18682
\(635\) 0 0
\(636\) −2.87967e6 −0.282293
\(637\) 639524. 0.0624466
\(638\) −7.18575e6 −0.698909
\(639\) −5.04290e6 −0.488572
\(640\) 0 0
\(641\) −735036. −0.0706584 −0.0353292 0.999376i \(-0.511248\pi\)
−0.0353292 + 0.999376i \(0.511248\pi\)
\(642\) 1.09132e7 1.04499
\(643\) −1.45125e7 −1.38425 −0.692125 0.721778i \(-0.743325\pi\)
−0.692125 + 0.721778i \(0.743325\pi\)
\(644\) 328349. 0.0311976
\(645\) 0 0
\(646\) 9.10218e6 0.858152
\(647\) 1.87650e6 0.176233 0.0881167 0.996110i \(-0.471915\pi\)
0.0881167 + 0.996110i \(0.471915\pi\)
\(648\) 636593. 0.0595558
\(649\) 2.04695e6 0.190764
\(650\) 0 0
\(651\) 59003.3 0.00545662
\(652\) −3.26879e6 −0.301139
\(653\) −1.72496e7 −1.58306 −0.791529 0.611131i \(-0.790715\pi\)
−0.791529 + 0.611131i \(0.790715\pi\)
\(654\) −3.96904e6 −0.362862
\(655\) 0 0
\(656\) 753460. 0.0683598
\(657\) 2.86044e6 0.258535
\(658\) −375544. −0.0338139
\(659\) 7.60070e6 0.681773 0.340887 0.940104i \(-0.389273\pi\)
0.340887 + 0.940104i \(0.389273\pi\)
\(660\) 0 0
\(661\) −74179.4 −0.00660358 −0.00330179 0.999995i \(-0.501051\pi\)
−0.00330179 + 0.999995i \(0.501051\pi\)
\(662\) 2.10239e7 1.86453
\(663\) −209263. −0.0184888
\(664\) 3.85336e6 0.339172
\(665\) 0 0
\(666\) 1.76466e6 0.154161
\(667\) −3.24420e7 −2.82353
\(668\) 1.14792e7 0.995337
\(669\) 2.02552e6 0.174973
\(670\) 0 0
\(671\) 314341. 0.0269522
\(672\) −247094. −0.0211076
\(673\) −5.80762e6 −0.494265 −0.247133 0.968982i \(-0.579488\pi\)
−0.247133 + 0.968982i \(0.579488\pi\)
\(674\) 1.62099e7 1.37446
\(675\) 0 0
\(676\) −6.77637e6 −0.570335
\(677\) −1.91692e7 −1.60743 −0.803716 0.595013i \(-0.797147\pi\)
−0.803716 + 0.595013i \(0.797147\pi\)
\(678\) 4.85002e6 0.405199
\(679\) 155259. 0.0129236
\(680\) 0 0
\(681\) −7.62478e6 −0.630028
\(682\) −1.21687e6 −0.100181
\(683\) 2.11907e7 1.73817 0.869087 0.494660i \(-0.164707\pi\)
0.869087 + 0.494660i \(0.164707\pi\)
\(684\) 3.12034e6 0.255012
\(685\) 0 0
\(686\) 1.10199e6 0.0894062
\(687\) −8.11132e6 −0.655691
\(688\) 2.44806e7 1.97175
\(689\) −665332. −0.0533938
\(690\) 0 0
\(691\) 1.24582e7 0.992565 0.496282 0.868161i \(-0.334698\pi\)
0.496282 + 0.868161i \(0.334698\pi\)
\(692\) −5.95913e6 −0.473062
\(693\) −45323.6 −0.00358502
\(694\) 3.02920e6 0.238742
\(695\) 0 0
\(696\) 7.31039e6 0.572028
\(697\) 360756. 0.0281276
\(698\) 1.66256e7 1.29163
\(699\) 3.95208e6 0.305938
\(700\) 0 0
\(701\) 582671. 0.0447845 0.0223923 0.999749i \(-0.492872\pi\)
0.0223923 + 0.999749i \(0.492872\pi\)
\(702\) −197028. −0.0150899
\(703\) −6.45699e6 −0.492767
\(704\) 160749. 0.0122241
\(705\) 0 0
\(706\) 1.16185e7 0.877282
\(707\) −940525. −0.0707655
\(708\) 2.78963e6 0.209153
\(709\) −3.31741e6 −0.247847 −0.123923 0.992292i \(-0.539548\pi\)
−0.123923 + 0.992292i \(0.539548\pi\)
\(710\) 0 0
\(711\) 6.34815e6 0.470948
\(712\) 5.26926e6 0.389538
\(713\) −5.49389e6 −0.404722
\(714\) −180180. −0.0132270
\(715\) 0 0
\(716\) 3.35303e6 0.244430
\(717\) 8.30096e6 0.603018
\(718\) −1.99244e7 −1.44236
\(719\) −8.22478e6 −0.593338 −0.296669 0.954980i \(-0.595876\pi\)
−0.296669 + 0.954980i \(0.595876\pi\)
\(720\) 0 0
\(721\) −443733. −0.0317895
\(722\) −1.37931e7 −0.984736
\(723\) 8.02492e6 0.570946
\(724\) −1.14196e7 −0.809661
\(725\) 0 0
\(726\) 934746. 0.0658191
\(727\) 5.99896e6 0.420959 0.210479 0.977598i \(-0.432497\pi\)
0.210479 + 0.977598i \(0.432497\pi\)
\(728\) −17094.8 −0.00119546
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 1.17213e7 0.811303
\(732\) 428390. 0.0295503
\(733\) 1.02874e7 0.707204 0.353602 0.935396i \(-0.384957\pi\)
0.353602 + 0.935396i \(0.384957\pi\)
\(734\) −284150. −0.0194674
\(735\) 0 0
\(736\) 2.30073e7 1.56557
\(737\) −7.24167e6 −0.491100
\(738\) 339664. 0.0229567
\(739\) 4.35478e6 0.293329 0.146665 0.989186i \(-0.453146\pi\)
0.146665 + 0.989186i \(0.453146\pi\)
\(740\) 0 0
\(741\) 720937. 0.0482338
\(742\) −572866. −0.0381983
\(743\) −6.54892e6 −0.435209 −0.217604 0.976037i \(-0.569824\pi\)
−0.217604 + 0.976037i \(0.569824\pi\)
\(744\) 1.23798e6 0.0819939
\(745\) 0 0
\(746\) −1.13759e7 −0.748409
\(747\) 3.21687e6 0.210927
\(748\) 1.35300e6 0.0884185
\(749\) 790462. 0.0514845
\(750\) 0 0
\(751\) 1.42508e6 0.0922019 0.0461009 0.998937i \(-0.485320\pi\)
0.0461009 + 0.998937i \(0.485320\pi\)
\(752\) −1.45916e7 −0.940930
\(753\) 1.13314e6 0.0728275
\(754\) −2.26259e6 −0.144937
\(755\) 0 0
\(756\) −61767.9 −0.00393060
\(757\) 1.59943e7 1.01443 0.507217 0.861818i \(-0.330674\pi\)
0.507217 + 0.861818i \(0.330674\pi\)
\(758\) −1.50856e7 −0.953654
\(759\) 4.22016e6 0.265904
\(760\) 0 0
\(761\) −3.04747e7 −1.90756 −0.953778 0.300512i \(-0.902843\pi\)
−0.953778 + 0.300512i \(0.902843\pi\)
\(762\) −6.38587e6 −0.398412
\(763\) −287486. −0.0178774
\(764\) 3.28751e6 0.203767
\(765\) 0 0
\(766\) 2.61998e7 1.61334
\(767\) 644529. 0.0395598
\(768\) −1.19103e7 −0.728652
\(769\) 8.77682e6 0.535206 0.267603 0.963529i \(-0.413768\pi\)
0.267603 + 0.963529i \(0.413768\pi\)
\(770\) 0 0
\(771\) 7.78073e6 0.471394
\(772\) −1.65709e7 −1.00070
\(773\) −867299. −0.0522060 −0.0261030 0.999659i \(-0.508310\pi\)
−0.0261030 + 0.999659i \(0.508310\pi\)
\(774\) 1.10360e7 0.662155
\(775\) 0 0
\(776\) 3.25758e6 0.194196
\(777\) 127818. 0.00759520
\(778\) 1.52961e6 0.0906009
\(779\) −1.24285e6 −0.0733796
\(780\) 0 0
\(781\) 7.53322e6 0.441930
\(782\) 1.67769e7 0.981057
\(783\) 6.10287e6 0.355737
\(784\) 2.13951e7 1.24315
\(785\) 0 0
\(786\) 2.05989e7 1.18929
\(787\) −2.23177e7 −1.28444 −0.642218 0.766522i \(-0.721985\pi\)
−0.642218 + 0.766522i \(0.721985\pi\)
\(788\) −9.06806e6 −0.520234
\(789\) 7.74761e6 0.443073
\(790\) 0 0
\(791\) 351297. 0.0199633
\(792\) −950959. −0.0538702
\(793\) 98977.2 0.00558923
\(794\) −3.15971e7 −1.77867
\(795\) 0 0
\(796\) −1.51038e7 −0.844894
\(797\) −1.21353e7 −0.676716 −0.338358 0.941018i \(-0.609871\pi\)
−0.338358 + 0.941018i \(0.609871\pi\)
\(798\) 620743. 0.0345068
\(799\) −6.98644e6 −0.387159
\(800\) 0 0
\(801\) 4.39889e6 0.242249
\(802\) −3.77382e7 −2.07179
\(803\) −4.27300e6 −0.233854
\(804\) −9.86909e6 −0.538440
\(805\) 0 0
\(806\) −383160. −0.0207751
\(807\) 1.52733e6 0.0825561
\(808\) −1.97337e7 −1.06336
\(809\) −1.25683e7 −0.675160 −0.337580 0.941297i \(-0.609608\pi\)
−0.337580 + 0.941297i \(0.609608\pi\)
\(810\) 0 0
\(811\) 6.29850e6 0.336268 0.168134 0.985764i \(-0.446226\pi\)
0.168134 + 0.985764i \(0.446226\pi\)
\(812\) −709319. −0.0377530
\(813\) 3.31963e6 0.176142
\(814\) −2.63610e6 −0.139444
\(815\) 0 0
\(816\) −7.00082e6 −0.368064
\(817\) −4.03814e7 −2.11654
\(818\) 2.70624e7 1.41411
\(819\) −14271.1 −0.000743445 0
\(820\) 0 0
\(821\) 1.89136e7 0.979303 0.489651 0.871918i \(-0.337124\pi\)
0.489651 + 0.871918i \(0.337124\pi\)
\(822\) 2.08040e7 1.07391
\(823\) −2.67549e7 −1.37691 −0.688453 0.725281i \(-0.741710\pi\)
−0.688453 + 0.725281i \(0.741710\pi\)
\(824\) −9.31020e6 −0.477684
\(825\) 0 0
\(826\) 554954. 0.0283013
\(827\) 1.76676e7 0.898283 0.449142 0.893461i \(-0.351730\pi\)
0.449142 + 0.893461i \(0.351730\pi\)
\(828\) 5.75132e6 0.291535
\(829\) −2.37092e7 −1.19820 −0.599102 0.800673i \(-0.704476\pi\)
−0.599102 + 0.800673i \(0.704476\pi\)
\(830\) 0 0
\(831\) 3.81228e6 0.191506
\(832\) 50615.3 0.00253497
\(833\) 1.02439e7 0.511511
\(834\) 2.65238e7 1.32044
\(835\) 0 0
\(836\) −4.66124e6 −0.230667
\(837\) 1.03349e6 0.0509910
\(838\) 2.55312e7 1.25592
\(839\) −1.98735e7 −0.974696 −0.487348 0.873208i \(-0.662036\pi\)
−0.487348 + 0.873208i \(0.662036\pi\)
\(840\) 0 0
\(841\) 4.95719e7 2.41683
\(842\) 1.52845e7 0.742968
\(843\) −2.15211e7 −1.04303
\(844\) 88296.4 0.00426665
\(845\) 0 0
\(846\) −6.57796e6 −0.315984
\(847\) 67705.6 0.00324277
\(848\) −2.22585e7 −1.06293
\(849\) −2.14964e6 −0.102352
\(850\) 0 0
\(851\) −1.19013e7 −0.563342
\(852\) 1.02664e7 0.484530
\(853\) 5.52250e6 0.259874 0.129937 0.991522i \(-0.458522\pi\)
0.129937 + 0.991522i \(0.458522\pi\)
\(854\) 85221.6 0.00399857
\(855\) 0 0
\(856\) 1.65851e7 0.773632
\(857\) −2.02109e7 −0.940014 −0.470007 0.882663i \(-0.655749\pi\)
−0.470007 + 0.882663i \(0.655749\pi\)
\(858\) 294326. 0.0136493
\(859\) 2.05904e7 0.952097 0.476048 0.879419i \(-0.342069\pi\)
0.476048 + 0.879419i \(0.342069\pi\)
\(860\) 0 0
\(861\) 24602.6 0.00113103
\(862\) 1.49138e7 0.683630
\(863\) 1.59735e7 0.730086 0.365043 0.930991i \(-0.381054\pi\)
0.365043 + 0.930991i \(0.381054\pi\)
\(864\) −4.32806e6 −0.197246
\(865\) 0 0
\(866\) −296866. −0.0134513
\(867\) 9.42672e6 0.425905
\(868\) −120120. −0.00541148
\(869\) −9.48304e6 −0.425989
\(870\) 0 0
\(871\) −2.28020e6 −0.101842
\(872\) −6.03190e6 −0.268635
\(873\) 2.71950e6 0.120768
\(874\) −5.77984e7 −2.55940
\(875\) 0 0
\(876\) −5.82334e6 −0.256396
\(877\) −1.40128e7 −0.615215 −0.307607 0.951513i \(-0.599528\pi\)
−0.307607 + 0.951513i \(0.599528\pi\)
\(878\) 3.80060e7 1.66386
\(879\) −2.60662e6 −0.113790
\(880\) 0 0
\(881\) −2.71339e7 −1.17780 −0.588901 0.808205i \(-0.700439\pi\)
−0.588901 + 0.808205i \(0.700439\pi\)
\(882\) 9.64501e6 0.417476
\(883\) −1.60023e7 −0.690687 −0.345344 0.938476i \(-0.612238\pi\)
−0.345344 + 0.938476i \(0.612238\pi\)
\(884\) 426021. 0.0183358
\(885\) 0 0
\(886\) 3.33224e7 1.42611
\(887\) −2.23865e7 −0.955381 −0.477690 0.878528i \(-0.658526\pi\)
−0.477690 + 0.878528i \(0.658526\pi\)
\(888\) 2.68182e6 0.114129
\(889\) −462542. −0.0196289
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −4.12358e6 −0.173525
\(893\) 2.40692e7 1.01003
\(894\) 1.95764e7 0.819200
\(895\) 0 0
\(896\) −834975. −0.0347459
\(897\) 1.32881e6 0.0551419
\(898\) −5.04042e7 −2.08582
\(899\) 1.18682e7 0.489764
\(900\) 0 0
\(901\) −1.06573e7 −0.437358
\(902\) −507399. −0.0207651
\(903\) 799361. 0.0326230
\(904\) 7.37075e6 0.299979
\(905\) 0 0
\(906\) −5.76821e6 −0.233464
\(907\) 6.33552e6 0.255719 0.127860 0.991792i \(-0.459189\pi\)
0.127860 + 0.991792i \(0.459189\pi\)
\(908\) 1.55227e7 0.624816
\(909\) −1.64741e7 −0.661290
\(910\) 0 0
\(911\) 6.57264e6 0.262388 0.131194 0.991357i \(-0.458119\pi\)
0.131194 + 0.991357i \(0.458119\pi\)
\(912\) 2.41187e7 0.960211
\(913\) −4.80545e6 −0.190791
\(914\) 3.42324e7 1.35541
\(915\) 0 0
\(916\) 1.65132e7 0.650267
\(917\) 1.49202e6 0.0585937
\(918\) −3.15601e6 −0.123604
\(919\) −4.55407e7 −1.77873 −0.889366 0.457196i \(-0.848854\pi\)
−0.889366 + 0.457196i \(0.848854\pi\)
\(920\) 0 0
\(921\) 2.47484e7 0.961388
\(922\) 7.35160e6 0.284809
\(923\) 2.37200e6 0.0916455
\(924\) 92270.6 0.00355536
\(925\) 0 0
\(926\) 3.24829e6 0.124488
\(927\) −7.77235e6 −0.297066
\(928\) −4.97018e7 −1.89453
\(929\) −3.24053e7 −1.23190 −0.615952 0.787784i \(-0.711229\pi\)
−0.615952 + 0.787784i \(0.711229\pi\)
\(930\) 0 0
\(931\) −3.52917e7 −1.33444
\(932\) −8.04572e6 −0.303407
\(933\) 6.02589e6 0.226630
\(934\) 1.43851e7 0.539567
\(935\) 0 0
\(936\) −299431. −0.0111714
\(937\) −1.53794e7 −0.572257 −0.286129 0.958191i \(-0.592368\pi\)
−0.286129 + 0.958191i \(0.592368\pi\)
\(938\) −1.96330e6 −0.0728585
\(939\) 4.12651e6 0.152728
\(940\) 0 0
\(941\) −2.92624e7 −1.07730 −0.538648 0.842531i \(-0.681065\pi\)
−0.538648 + 0.842531i \(0.681065\pi\)
\(942\) 5.18865e6 0.190514
\(943\) −2.29079e6 −0.0838891
\(944\) 2.15625e7 0.787533
\(945\) 0 0
\(946\) −1.64859e7 −0.598942
\(947\) −2.32692e7 −0.843154 −0.421577 0.906793i \(-0.638523\pi\)
−0.421577 + 0.906793i \(0.638523\pi\)
\(948\) −1.29237e7 −0.467052
\(949\) −1.34545e6 −0.0484956
\(950\) 0 0
\(951\) −1.52394e7 −0.546408
\(952\) −273826. −0.00979226
\(953\) −1.42689e7 −0.508932 −0.254466 0.967082i \(-0.581900\pi\)
−0.254466 + 0.967082i \(0.581900\pi\)
\(954\) −1.00342e7 −0.356955
\(955\) 0 0
\(956\) −1.68992e7 −0.598029
\(957\) −9.11663e6 −0.321777
\(958\) −2.42212e7 −0.852672
\(959\) 1.50688e6 0.0529092
\(960\) 0 0
\(961\) −2.66193e7 −0.929798
\(962\) −830034. −0.0289173
\(963\) 1.38456e7 0.481112
\(964\) −1.63373e7 −0.566223
\(965\) 0 0
\(966\) 1.14414e6 0.0394489
\(967\) −4.25962e7 −1.46489 −0.732444 0.680827i \(-0.761620\pi\)
−0.732444 + 0.680827i \(0.761620\pi\)
\(968\) 1.42057e6 0.0487275
\(969\) 1.15480e7 0.395092
\(970\) 0 0
\(971\) −3.93095e7 −1.33798 −0.668990 0.743271i \(-0.733273\pi\)
−0.668990 + 0.743271i \(0.733273\pi\)
\(972\) −1.08192e6 −0.0367306
\(973\) 1.92117e6 0.0650555
\(974\) −8.62205e6 −0.291215
\(975\) 0 0
\(976\) 3.31125e6 0.111267
\(977\) 5.54613e7 1.85889 0.929444 0.368962i \(-0.120287\pi\)
0.929444 + 0.368962i \(0.120287\pi\)
\(978\) −1.13901e7 −0.380786
\(979\) −6.57119e6 −0.219123
\(980\) 0 0
\(981\) −5.03556e6 −0.167061
\(982\) 3.44391e7 1.13965
\(983\) 5.37070e7 1.77275 0.886375 0.462968i \(-0.153216\pi\)
0.886375 + 0.462968i \(0.153216\pi\)
\(984\) 516200. 0.0169954
\(985\) 0 0
\(986\) −3.62424e7 −1.18720
\(987\) −476455. −0.0155679
\(988\) −1.46770e6 −0.0478348
\(989\) −7.44298e7 −2.41967
\(990\) 0 0
\(991\) −803096. −0.0259767 −0.0129883 0.999916i \(-0.504134\pi\)
−0.0129883 + 0.999916i \(0.504134\pi\)
\(992\) −8.41677e6 −0.271560
\(993\) 2.66732e7 0.858425
\(994\) 2.04235e6 0.0655638
\(995\) 0 0
\(996\) −6.54896e6 −0.209182
\(997\) −6.90977e6 −0.220154 −0.110077 0.993923i \(-0.535110\pi\)
−0.110077 + 0.993923i \(0.535110\pi\)
\(998\) −4.97459e7 −1.58100
\(999\) 2.23884e6 0.0709756
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.q.1.2 yes 8
5.4 even 2 825.6.a.p.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.7 8 5.4 even 2
825.6.a.q.1.2 yes 8 1.1 even 1 trivial