Properties

Label 825.6.a.q.1.5
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 1.5
Root \(0.894159\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.89416 q^{2} -9.00000 q^{3} -28.4122 q^{4} -17.0474 q^{6} -191.987 q^{7} -114.430 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.89416 q^{2} -9.00000 q^{3} -28.4122 q^{4} -17.0474 q^{6} -191.987 q^{7} -114.430 q^{8} +81.0000 q^{9} -121.000 q^{11} +255.709 q^{12} -536.985 q^{13} -363.654 q^{14} +692.440 q^{16} +1324.89 q^{17} +153.427 q^{18} -912.564 q^{19} +1727.88 q^{21} -229.193 q^{22} +4124.56 q^{23} +1029.87 q^{24} -1017.14 q^{26} -729.000 q^{27} +5454.77 q^{28} +340.490 q^{29} +4697.47 q^{31} +4973.36 q^{32} +1089.00 q^{33} +2509.55 q^{34} -2301.38 q^{36} +8279.99 q^{37} -1728.54 q^{38} +4832.86 q^{39} -17495.0 q^{41} +3272.89 q^{42} -5947.60 q^{43} +3437.87 q^{44} +7812.57 q^{46} -628.580 q^{47} -6231.96 q^{48} +20052.0 q^{49} -11924.0 q^{51} +15256.9 q^{52} +16952.3 q^{53} -1380.84 q^{54} +21969.1 q^{56} +8213.08 q^{57} +644.942 q^{58} -28406.8 q^{59} +51510.2 q^{61} +8897.76 q^{62} -15550.9 q^{63} -12737.7 q^{64} +2062.74 q^{66} -15258.0 q^{67} -37643.0 q^{68} -37121.0 q^{69} +38398.8 q^{71} -9268.85 q^{72} -40463.3 q^{73} +15683.6 q^{74} +25927.9 q^{76} +23230.4 q^{77} +9154.22 q^{78} -27358.8 q^{79} +6561.00 q^{81} -33138.2 q^{82} +51818.0 q^{83} -49092.9 q^{84} -11265.7 q^{86} -3064.41 q^{87} +13846.1 q^{88} +20215.2 q^{89} +103094. q^{91} -117188. q^{92} -42277.2 q^{93} -1190.63 q^{94} -44760.2 q^{96} +166901. q^{97} +37981.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.89416 0.334843 0.167422 0.985885i \(-0.446456\pi\)
0.167422 + 0.985885i \(0.446456\pi\)
\(3\) −9.00000 −0.577350
\(4\) −28.4122 −0.887880
\(5\) 0 0
\(6\) −17.0474 −0.193322
\(7\) −191.987 −1.48090 −0.740452 0.672109i \(-0.765388\pi\)
−0.740452 + 0.672109i \(0.765388\pi\)
\(8\) −114.430 −0.632144
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 255.709 0.512618
\(13\) −536.985 −0.881259 −0.440630 0.897689i \(-0.645245\pi\)
−0.440630 + 0.897689i \(0.645245\pi\)
\(14\) −363.654 −0.495871
\(15\) 0 0
\(16\) 692.440 0.676211
\(17\) 1324.89 1.11188 0.555940 0.831223i \(-0.312359\pi\)
0.555940 + 0.831223i \(0.312359\pi\)
\(18\) 153.427 0.111614
\(19\) −912.564 −0.579935 −0.289968 0.957036i \(-0.593645\pi\)
−0.289968 + 0.957036i \(0.593645\pi\)
\(20\) 0 0
\(21\) 1727.88 0.855000
\(22\) −229.193 −0.100959
\(23\) 4124.56 1.62577 0.812883 0.582427i \(-0.197897\pi\)
0.812883 + 0.582427i \(0.197897\pi\)
\(24\) 1029.87 0.364968
\(25\) 0 0
\(26\) −1017.14 −0.295084
\(27\) −729.000 −0.192450
\(28\) 5454.77 1.31486
\(29\) 340.490 0.0751811 0.0375906 0.999293i \(-0.488032\pi\)
0.0375906 + 0.999293i \(0.488032\pi\)
\(30\) 0 0
\(31\) 4697.47 0.877930 0.438965 0.898504i \(-0.355345\pi\)
0.438965 + 0.898504i \(0.355345\pi\)
\(32\) 4973.36 0.858568
\(33\) 1089.00 0.174078
\(34\) 2509.55 0.372305
\(35\) 0 0
\(36\) −2301.38 −0.295960
\(37\) 8279.99 0.994318 0.497159 0.867659i \(-0.334377\pi\)
0.497159 + 0.867659i \(0.334377\pi\)
\(38\) −1728.54 −0.194187
\(39\) 4832.86 0.508795
\(40\) 0 0
\(41\) −17495.0 −1.62537 −0.812687 0.582701i \(-0.801996\pi\)
−0.812687 + 0.582701i \(0.801996\pi\)
\(42\) 3272.89 0.286291
\(43\) −5947.60 −0.490535 −0.245268 0.969455i \(-0.578876\pi\)
−0.245268 + 0.969455i \(0.578876\pi\)
\(44\) 3437.87 0.267706
\(45\) 0 0
\(46\) 7812.57 0.544377
\(47\) −628.580 −0.0415065 −0.0207532 0.999785i \(-0.506606\pi\)
−0.0207532 + 0.999785i \(0.506606\pi\)
\(48\) −6231.96 −0.390411
\(49\) 20052.0 1.19308
\(50\) 0 0
\(51\) −11924.0 −0.641944
\(52\) 15256.9 0.782453
\(53\) 16952.3 0.828969 0.414485 0.910056i \(-0.363962\pi\)
0.414485 + 0.910056i \(0.363962\pi\)
\(54\) −1380.84 −0.0644406
\(55\) 0 0
\(56\) 21969.1 0.936144
\(57\) 8213.08 0.334826
\(58\) 644.942 0.0251739
\(59\) −28406.8 −1.06241 −0.531204 0.847244i \(-0.678260\pi\)
−0.531204 + 0.847244i \(0.678260\pi\)
\(60\) 0 0
\(61\) 51510.2 1.77243 0.886215 0.463275i \(-0.153326\pi\)
0.886215 + 0.463275i \(0.153326\pi\)
\(62\) 8897.76 0.293969
\(63\) −15550.9 −0.493635
\(64\) −12737.7 −0.388725
\(65\) 0 0
\(66\) 2062.74 0.0582887
\(67\) −15258.0 −0.415251 −0.207625 0.978208i \(-0.566573\pi\)
−0.207625 + 0.978208i \(0.566573\pi\)
\(68\) −37643.0 −0.987215
\(69\) −37121.0 −0.938636
\(70\) 0 0
\(71\) 38398.8 0.904007 0.452004 0.892016i \(-0.350709\pi\)
0.452004 + 0.892016i \(0.350709\pi\)
\(72\) −9268.85 −0.210715
\(73\) −40463.3 −0.888699 −0.444349 0.895854i \(-0.646565\pi\)
−0.444349 + 0.895854i \(0.646565\pi\)
\(74\) 15683.6 0.332941
\(75\) 0 0
\(76\) 25927.9 0.514913
\(77\) 23230.4 0.446509
\(78\) 9154.22 0.170367
\(79\) −27358.8 −0.493207 −0.246604 0.969116i \(-0.579315\pi\)
−0.246604 + 0.969116i \(0.579315\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −33138.2 −0.544245
\(83\) 51818.0 0.825631 0.412815 0.910815i \(-0.364545\pi\)
0.412815 + 0.910815i \(0.364545\pi\)
\(84\) −49092.9 −0.759138
\(85\) 0 0
\(86\) −11265.7 −0.164252
\(87\) −3064.41 −0.0434058
\(88\) 13846.1 0.190599
\(89\) 20215.2 0.270523 0.135261 0.990810i \(-0.456813\pi\)
0.135261 + 0.990810i \(0.456813\pi\)
\(90\) 0 0
\(91\) 103094. 1.30506
\(92\) −117188. −1.44348
\(93\) −42277.2 −0.506873
\(94\) −1190.63 −0.0138982
\(95\) 0 0
\(96\) −44760.2 −0.495695
\(97\) 166901. 1.80106 0.900531 0.434791i \(-0.143178\pi\)
0.900531 + 0.434791i \(0.143178\pi\)
\(98\) 37981.7 0.399493
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 18494.4 0.180400 0.0901999 0.995924i \(-0.471249\pi\)
0.0901999 + 0.995924i \(0.471249\pi\)
\(102\) −22586.0 −0.214951
\(103\) −88991.8 −0.826527 −0.413263 0.910612i \(-0.635611\pi\)
−0.413263 + 0.910612i \(0.635611\pi\)
\(104\) 61447.3 0.557083
\(105\) 0 0
\(106\) 32110.3 0.277575
\(107\) −112300. −0.948248 −0.474124 0.880458i \(-0.657235\pi\)
−0.474124 + 0.880458i \(0.657235\pi\)
\(108\) 20712.5 0.170873
\(109\) −194142. −1.56514 −0.782569 0.622564i \(-0.786091\pi\)
−0.782569 + 0.622564i \(0.786091\pi\)
\(110\) 0 0
\(111\) −74519.9 −0.574070
\(112\) −132940. −1.00140
\(113\) 4270.92 0.0314648 0.0157324 0.999876i \(-0.494992\pi\)
0.0157324 + 0.999876i \(0.494992\pi\)
\(114\) 15556.9 0.112114
\(115\) 0 0
\(116\) −9674.05 −0.0667518
\(117\) −43495.8 −0.293753
\(118\) −53806.9 −0.355740
\(119\) −254362. −1.64659
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 97568.6 0.593486
\(123\) 157455. 0.938410
\(124\) −133465. −0.779496
\(125\) 0 0
\(126\) −29456.0 −0.165290
\(127\) 338246. 1.86090 0.930452 0.366414i \(-0.119415\pi\)
0.930452 + 0.366414i \(0.119415\pi\)
\(128\) −183275. −0.988730
\(129\) 53528.4 0.283211
\(130\) 0 0
\(131\) −113323. −0.576950 −0.288475 0.957487i \(-0.593148\pi\)
−0.288475 + 0.957487i \(0.593148\pi\)
\(132\) −30940.8 −0.154560
\(133\) 175201. 0.858828
\(134\) −28901.1 −0.139044
\(135\) 0 0
\(136\) −151608. −0.702868
\(137\) −28035.0 −0.127614 −0.0638072 0.997962i \(-0.520324\pi\)
−0.0638072 + 0.997962i \(0.520324\pi\)
\(138\) −70313.1 −0.314296
\(139\) −59383.8 −0.260694 −0.130347 0.991468i \(-0.541609\pi\)
−0.130347 + 0.991468i \(0.541609\pi\)
\(140\) 0 0
\(141\) 5657.22 0.0239638
\(142\) 72733.5 0.302701
\(143\) 64975.2 0.265710
\(144\) 56087.6 0.225404
\(145\) 0 0
\(146\) −76644.0 −0.297575
\(147\) −180468. −0.688823
\(148\) −235252. −0.882835
\(149\) −226515. −0.835855 −0.417928 0.908480i \(-0.637243\pi\)
−0.417928 + 0.908480i \(0.637243\pi\)
\(150\) 0 0
\(151\) 231802. 0.827323 0.413661 0.910431i \(-0.364250\pi\)
0.413661 + 0.910431i \(0.364250\pi\)
\(152\) 104425. 0.366602
\(153\) 107316. 0.370626
\(154\) 44002.1 0.149511
\(155\) 0 0
\(156\) −137312. −0.451749
\(157\) 196398. 0.635897 0.317949 0.948108i \(-0.397006\pi\)
0.317949 + 0.948108i \(0.397006\pi\)
\(158\) −51822.0 −0.165147
\(159\) −152570. −0.478606
\(160\) 0 0
\(161\) −791862. −2.40760
\(162\) 12427.6 0.0372048
\(163\) 604868. 1.78317 0.891583 0.452857i \(-0.149595\pi\)
0.891583 + 0.452857i \(0.149595\pi\)
\(164\) 497069. 1.44314
\(165\) 0 0
\(166\) 98151.6 0.276457
\(167\) −140409. −0.389587 −0.194793 0.980844i \(-0.562404\pi\)
−0.194793 + 0.980844i \(0.562404\pi\)
\(168\) −197722. −0.540483
\(169\) −82940.2 −0.223382
\(170\) 0 0
\(171\) −73917.7 −0.193312
\(172\) 168984. 0.435537
\(173\) −352914. −0.896506 −0.448253 0.893907i \(-0.647954\pi\)
−0.448253 + 0.893907i \(0.647954\pi\)
\(174\) −5804.47 −0.0145342
\(175\) 0 0
\(176\) −83785.2 −0.203885
\(177\) 255661. 0.613382
\(178\) 38290.9 0.0905827
\(179\) −784162. −1.82925 −0.914625 0.404303i \(-0.867514\pi\)
−0.914625 + 0.404303i \(0.867514\pi\)
\(180\) 0 0
\(181\) −2532.32 −0.00574542 −0.00287271 0.999996i \(-0.500914\pi\)
−0.00287271 + 0.999996i \(0.500914\pi\)
\(182\) 195277. 0.436991
\(183\) −463592. −1.02331
\(184\) −471974. −1.02772
\(185\) 0 0
\(186\) −80079.8 −0.169723
\(187\) −160312. −0.335244
\(188\) 17859.3 0.0368527
\(189\) 139959. 0.285000
\(190\) 0 0
\(191\) 185241. 0.367411 0.183706 0.982981i \(-0.441191\pi\)
0.183706 + 0.982981i \(0.441191\pi\)
\(192\) 114640. 0.224431
\(193\) −523822. −1.01226 −0.506128 0.862458i \(-0.668924\pi\)
−0.506128 + 0.862458i \(0.668924\pi\)
\(194\) 316136. 0.603074
\(195\) 0 0
\(196\) −569721. −1.05931
\(197\) 166997. 0.306580 0.153290 0.988181i \(-0.451013\pi\)
0.153290 + 0.988181i \(0.451013\pi\)
\(198\) −18564.7 −0.0336530
\(199\) −715314. −1.28045 −0.640227 0.768186i \(-0.721160\pi\)
−0.640227 + 0.768186i \(0.721160\pi\)
\(200\) 0 0
\(201\) 137322. 0.239745
\(202\) 35031.3 0.0604056
\(203\) −65369.6 −0.111336
\(204\) 338787. 0.569969
\(205\) 0 0
\(206\) −168565. −0.276757
\(207\) 334089. 0.541922
\(208\) −371830. −0.595917
\(209\) 110420. 0.174857
\(210\) 0 0
\(211\) 917870. 1.41930 0.709651 0.704553i \(-0.248852\pi\)
0.709651 + 0.704553i \(0.248852\pi\)
\(212\) −481651. −0.736025
\(213\) −345589. −0.521929
\(214\) −212715. −0.317515
\(215\) 0 0
\(216\) 83419.7 0.121656
\(217\) −901854. −1.30013
\(218\) −367735. −0.524076
\(219\) 364170. 0.513090
\(220\) 0 0
\(221\) −711446. −0.979854
\(222\) −141153. −0.192223
\(223\) 65289.2 0.0879182 0.0439591 0.999033i \(-0.486003\pi\)
0.0439591 + 0.999033i \(0.486003\pi\)
\(224\) −954821. −1.27146
\(225\) 0 0
\(226\) 8089.80 0.0105358
\(227\) −558110. −0.718879 −0.359439 0.933168i \(-0.617032\pi\)
−0.359439 + 0.933168i \(0.617032\pi\)
\(228\) −233351. −0.297285
\(229\) 980050. 1.23498 0.617490 0.786579i \(-0.288150\pi\)
0.617490 + 0.786579i \(0.288150\pi\)
\(230\) 0 0
\(231\) −209074. −0.257792
\(232\) −38962.3 −0.0475253
\(233\) −798554. −0.963640 −0.481820 0.876270i \(-0.660024\pi\)
−0.481820 + 0.876270i \(0.660024\pi\)
\(234\) −82387.9 −0.0983612
\(235\) 0 0
\(236\) 807097. 0.943292
\(237\) 246229. 0.284753
\(238\) −481802. −0.551348
\(239\) −1.31819e6 −1.49274 −0.746369 0.665532i \(-0.768205\pi\)
−0.746369 + 0.665532i \(0.768205\pi\)
\(240\) 0 0
\(241\) −1.07597e6 −1.19332 −0.596660 0.802494i \(-0.703506\pi\)
−0.596660 + 0.802494i \(0.703506\pi\)
\(242\) 27732.4 0.0304403
\(243\) −59049.0 −0.0641500
\(244\) −1.46352e6 −1.57370
\(245\) 0 0
\(246\) 298244. 0.314220
\(247\) 490033. 0.511073
\(248\) −537533. −0.554978
\(249\) −466362. −0.476678
\(250\) 0 0
\(251\) −814105. −0.815635 −0.407818 0.913063i \(-0.633710\pi\)
−0.407818 + 0.913063i \(0.633710\pi\)
\(252\) 441836. 0.438288
\(253\) −499072. −0.490187
\(254\) 640693. 0.623111
\(255\) 0 0
\(256\) 60456.1 0.0576554
\(257\) 1.89491e6 1.78960 0.894801 0.446466i \(-0.147318\pi\)
0.894801 + 0.446466i \(0.147318\pi\)
\(258\) 101391. 0.0948312
\(259\) −1.58965e6 −1.47249
\(260\) 0 0
\(261\) 27579.7 0.0250604
\(262\) −214651. −0.193188
\(263\) −1.39679e6 −1.24521 −0.622603 0.782538i \(-0.713925\pi\)
−0.622603 + 0.782538i \(0.713925\pi\)
\(264\) −124615. −0.110042
\(265\) 0 0
\(266\) 331858. 0.287573
\(267\) −181937. −0.156186
\(268\) 433513. 0.368693
\(269\) 1.06051e6 0.893585 0.446793 0.894638i \(-0.352566\pi\)
0.446793 + 0.894638i \(0.352566\pi\)
\(270\) 0 0
\(271\) −710526. −0.587702 −0.293851 0.955851i \(-0.594937\pi\)
−0.293851 + 0.955851i \(0.594937\pi\)
\(272\) 917407. 0.751865
\(273\) −927847. −0.753477
\(274\) −53102.8 −0.0427308
\(275\) 0 0
\(276\) 1.05469e6 0.833396
\(277\) −678042. −0.530955 −0.265477 0.964117i \(-0.585530\pi\)
−0.265477 + 0.964117i \(0.585530\pi\)
\(278\) −112482. −0.0872917
\(279\) 380495. 0.292643
\(280\) 0 0
\(281\) 527403. 0.398453 0.199226 0.979953i \(-0.436157\pi\)
0.199226 + 0.979953i \(0.436157\pi\)
\(282\) 10715.7 0.00802410
\(283\) 563399. 0.418167 0.209084 0.977898i \(-0.432952\pi\)
0.209084 + 0.977898i \(0.432952\pi\)
\(284\) −1.09099e6 −0.802650
\(285\) 0 0
\(286\) 123073. 0.0889711
\(287\) 3.35880e6 2.40702
\(288\) 402842. 0.286189
\(289\) 335478. 0.236276
\(290\) 0 0
\(291\) −1.50211e6 −1.03984
\(292\) 1.14965e6 0.789058
\(293\) 1.31300e6 0.893505 0.446753 0.894657i \(-0.352580\pi\)
0.446753 + 0.894657i \(0.352580\pi\)
\(294\) −341836. −0.230648
\(295\) 0 0
\(296\) −947481. −0.628552
\(297\) 88209.0 0.0580259
\(298\) −429055. −0.279880
\(299\) −2.21483e6 −1.43272
\(300\) 0 0
\(301\) 1.14186e6 0.726436
\(302\) 439070. 0.277023
\(303\) −166449. −0.104154
\(304\) −631896. −0.392159
\(305\) 0 0
\(306\) 203274. 0.124102
\(307\) 2.59454e6 1.57114 0.785570 0.618773i \(-0.212370\pi\)
0.785570 + 0.618773i \(0.212370\pi\)
\(308\) −660027. −0.396447
\(309\) 800926. 0.477196
\(310\) 0 0
\(311\) −68979.7 −0.0404408 −0.0202204 0.999796i \(-0.506437\pi\)
−0.0202204 + 0.999796i \(0.506437\pi\)
\(312\) −553026. −0.321632
\(313\) 2.10415e6 1.21399 0.606996 0.794705i \(-0.292374\pi\)
0.606996 + 0.794705i \(0.292374\pi\)
\(314\) 372008. 0.212926
\(315\) 0 0
\(316\) 777323. 0.437909
\(317\) 3.32293e6 1.85726 0.928632 0.371003i \(-0.120986\pi\)
0.928632 + 0.371003i \(0.120986\pi\)
\(318\) −288993. −0.160258
\(319\) −41199.2 −0.0226680
\(320\) 0 0
\(321\) 1.01070e6 0.547472
\(322\) −1.49991e6 −0.806169
\(323\) −1.20905e6 −0.644818
\(324\) −186412. −0.0986533
\(325\) 0 0
\(326\) 1.14572e6 0.597081
\(327\) 1.74728e6 0.903633
\(328\) 2.00195e6 1.02747
\(329\) 120679. 0.0614671
\(330\) 0 0
\(331\) −2.65460e6 −1.33177 −0.665886 0.746054i \(-0.731946\pi\)
−0.665886 + 0.746054i \(0.731946\pi\)
\(332\) −1.47226e6 −0.733061
\(333\) 670679. 0.331439
\(334\) −265957. −0.130451
\(335\) 0 0
\(336\) 1.19646e6 0.578160
\(337\) 1.56818e6 0.752179 0.376089 0.926583i \(-0.377269\pi\)
0.376089 + 0.926583i \(0.377269\pi\)
\(338\) −157102. −0.0747980
\(339\) −38438.3 −0.0181662
\(340\) 0 0
\(341\) −568394. −0.264706
\(342\) −140012. −0.0647291
\(343\) −623002. −0.285926
\(344\) 680585. 0.310089
\(345\) 0 0
\(346\) −668475. −0.300189
\(347\) −3.47302e6 −1.54840 −0.774202 0.632939i \(-0.781848\pi\)
−0.774202 + 0.632939i \(0.781848\pi\)
\(348\) 87066.4 0.0385392
\(349\) 1.54641e6 0.679611 0.339805 0.940496i \(-0.389639\pi\)
0.339805 + 0.940496i \(0.389639\pi\)
\(350\) 0 0
\(351\) 391462. 0.169598
\(352\) −601777. −0.258868
\(353\) 1.08681e6 0.464214 0.232107 0.972690i \(-0.425438\pi\)
0.232107 + 0.972690i \(0.425438\pi\)
\(354\) 484262. 0.205387
\(355\) 0 0
\(356\) −574358. −0.240192
\(357\) 2.28926e6 0.950657
\(358\) −1.48533e6 −0.612512
\(359\) −1.38446e6 −0.566948 −0.283474 0.958980i \(-0.591487\pi\)
−0.283474 + 0.958980i \(0.591487\pi\)
\(360\) 0 0
\(361\) −1.64333e6 −0.663675
\(362\) −4796.62 −0.00192382
\(363\) −131769. −0.0524864
\(364\) −2.92913e6 −1.15874
\(365\) 0 0
\(366\) −878117. −0.342649
\(367\) −3.09462e6 −1.19934 −0.599669 0.800248i \(-0.704701\pi\)
−0.599669 + 0.800248i \(0.704701\pi\)
\(368\) 2.85601e6 1.09936
\(369\) −1.41709e6 −0.541791
\(370\) 0 0
\(371\) −3.25462e6 −1.22762
\(372\) 1.20119e6 0.450043
\(373\) −3.81239e6 −1.41881 −0.709407 0.704799i \(-0.751037\pi\)
−0.709407 + 0.704799i \(0.751037\pi\)
\(374\) −303656. −0.112254
\(375\) 0 0
\(376\) 71928.5 0.0262380
\(377\) −182838. −0.0662541
\(378\) 265104. 0.0954303
\(379\) −1.79432e6 −0.641656 −0.320828 0.947138i \(-0.603961\pi\)
−0.320828 + 0.947138i \(0.603961\pi\)
\(380\) 0 0
\(381\) −3.04422e6 −1.07439
\(382\) 350875. 0.123025
\(383\) 1.79578e6 0.625541 0.312771 0.949829i \(-0.398743\pi\)
0.312771 + 0.949829i \(0.398743\pi\)
\(384\) 1.64947e6 0.570844
\(385\) 0 0
\(386\) −992202. −0.338947
\(387\) −481755. −0.163512
\(388\) −4.74201e6 −1.59913
\(389\) 3.20505e6 1.07389 0.536946 0.843617i \(-0.319578\pi\)
0.536946 + 0.843617i \(0.319578\pi\)
\(390\) 0 0
\(391\) 5.46459e6 1.80765
\(392\) −2.29456e6 −0.754195
\(393\) 1.01990e6 0.333102
\(394\) 316319. 0.102656
\(395\) 0 0
\(396\) 278468. 0.0892353
\(397\) 5.06822e6 1.61391 0.806954 0.590614i \(-0.201114\pi\)
0.806954 + 0.590614i \(0.201114\pi\)
\(398\) −1.35492e6 −0.428751
\(399\) −1.57680e6 −0.495845
\(400\) 0 0
\(401\) −5.04243e6 −1.56595 −0.782977 0.622051i \(-0.786300\pi\)
−0.782977 + 0.622051i \(0.786300\pi\)
\(402\) 260110. 0.0802770
\(403\) −2.52247e6 −0.773684
\(404\) −525465. −0.160173
\(405\) 0 0
\(406\) −123820. −0.0372801
\(407\) −1.00188e6 −0.299798
\(408\) 1.36447e6 0.405801
\(409\) −6.11825e6 −1.80850 −0.904252 0.427000i \(-0.859570\pi\)
−0.904252 + 0.427000i \(0.859570\pi\)
\(410\) 0 0
\(411\) 252315. 0.0736782
\(412\) 2.52845e6 0.733857
\(413\) 5.45373e6 1.57333
\(414\) 632818. 0.181459
\(415\) 0 0
\(416\) −2.67062e6 −0.756621
\(417\) 534454. 0.150512
\(418\) 209154. 0.0585497
\(419\) 6.32243e6 1.75934 0.879669 0.475586i \(-0.157764\pi\)
0.879669 + 0.475586i \(0.157764\pi\)
\(420\) 0 0
\(421\) 1.04093e6 0.286232 0.143116 0.989706i \(-0.454288\pi\)
0.143116 + 0.989706i \(0.454288\pi\)
\(422\) 1.73859e6 0.475244
\(423\) −50914.9 −0.0138355
\(424\) −1.93985e6 −0.524028
\(425\) 0 0
\(426\) −654601. −0.174764
\(427\) −9.88930e6 −2.62480
\(428\) 3.19070e6 0.841931
\(429\) −584777. −0.153408
\(430\) 0 0
\(431\) 2.82045e6 0.731349 0.365674 0.930743i \(-0.380838\pi\)
0.365674 + 0.930743i \(0.380838\pi\)
\(432\) −504789. −0.130137
\(433\) −2.15739e6 −0.552980 −0.276490 0.961017i \(-0.589171\pi\)
−0.276490 + 0.961017i \(0.589171\pi\)
\(434\) −1.70825e6 −0.435340
\(435\) 0 0
\(436\) 5.51599e6 1.38966
\(437\) −3.76392e6 −0.942839
\(438\) 689796. 0.171805
\(439\) 4.60164e6 1.13960 0.569798 0.821785i \(-0.307021\pi\)
0.569798 + 0.821785i \(0.307021\pi\)
\(440\) 0 0
\(441\) 1.62421e6 0.397692
\(442\) −1.34759e6 −0.328097
\(443\) 5.03529e6 1.21903 0.609516 0.792774i \(-0.291364\pi\)
0.609516 + 0.792774i \(0.291364\pi\)
\(444\) 2.11727e6 0.509705
\(445\) 0 0
\(446\) 123668. 0.0294388
\(447\) 2.03863e6 0.482581
\(448\) 2.44548e6 0.575664
\(449\) −6.46043e6 −1.51233 −0.756163 0.654383i \(-0.772928\pi\)
−0.756163 + 0.654383i \(0.772928\pi\)
\(450\) 0 0
\(451\) 2.11689e6 0.490068
\(452\) −121346. −0.0279370
\(453\) −2.08622e6 −0.477655
\(454\) −1.05715e6 −0.240712
\(455\) 0 0
\(456\) −939825. −0.211658
\(457\) −4.15028e6 −0.929580 −0.464790 0.885421i \(-0.653870\pi\)
−0.464790 + 0.885421i \(0.653870\pi\)
\(458\) 1.85637e6 0.413524
\(459\) −965845. −0.213981
\(460\) 0 0
\(461\) 2.82145e6 0.618330 0.309165 0.951008i \(-0.399950\pi\)
0.309165 + 0.951008i \(0.399950\pi\)
\(462\) −396019. −0.0863200
\(463\) −2.16321e6 −0.468971 −0.234486 0.972120i \(-0.575341\pi\)
−0.234486 + 0.972120i \(0.575341\pi\)
\(464\) 235769. 0.0508383
\(465\) 0 0
\(466\) −1.51259e6 −0.322668
\(467\) 1.87682e6 0.398226 0.199113 0.979977i \(-0.436194\pi\)
0.199113 + 0.979977i \(0.436194\pi\)
\(468\) 1.23581e6 0.260818
\(469\) 2.92934e6 0.614946
\(470\) 0 0
\(471\) −1.76758e6 −0.367135
\(472\) 3.25059e6 0.671595
\(473\) 719659. 0.147902
\(474\) 466398. 0.0953478
\(475\) 0 0
\(476\) 7.22697e6 1.46197
\(477\) 1.37313e6 0.276323
\(478\) −2.49686e6 −0.499833
\(479\) −4.84619e6 −0.965076 −0.482538 0.875875i \(-0.660285\pi\)
−0.482538 + 0.875875i \(0.660285\pi\)
\(480\) 0 0
\(481\) −4.44623e6 −0.876252
\(482\) −2.03806e6 −0.399575
\(483\) 7.12676e6 1.39003
\(484\) −415982. −0.0807164
\(485\) 0 0
\(486\) −111848. −0.0214802
\(487\) −9.13800e6 −1.74594 −0.872969 0.487776i \(-0.837808\pi\)
−0.872969 + 0.487776i \(0.837808\pi\)
\(488\) −5.89433e6 −1.12043
\(489\) −5.44382e6 −1.02951
\(490\) 0 0
\(491\) −5.14270e6 −0.962692 −0.481346 0.876531i \(-0.659852\pi\)
−0.481346 + 0.876531i \(0.659852\pi\)
\(492\) −4.47362e6 −0.833195
\(493\) 451111. 0.0835923
\(494\) 928201. 0.171129
\(495\) 0 0
\(496\) 3.25272e6 0.593666
\(497\) −7.37207e6 −1.33875
\(498\) −883365. −0.159612
\(499\) 2.18975e6 0.393679 0.196840 0.980436i \(-0.436932\pi\)
0.196840 + 0.980436i \(0.436932\pi\)
\(500\) 0 0
\(501\) 1.26368e6 0.224928
\(502\) −1.54204e6 −0.273110
\(503\) −2.77987e6 −0.489897 −0.244948 0.969536i \(-0.578771\pi\)
−0.244948 + 0.969536i \(0.578771\pi\)
\(504\) 1.77950e6 0.312048
\(505\) 0 0
\(506\) −945321. −0.164136
\(507\) 746462. 0.128970
\(508\) −9.61031e6 −1.65226
\(509\) 885490. 0.151492 0.0757459 0.997127i \(-0.475866\pi\)
0.0757459 + 0.997127i \(0.475866\pi\)
\(510\) 0 0
\(511\) 7.76843e6 1.31608
\(512\) 5.97931e6 1.00804
\(513\) 665259. 0.111609
\(514\) 3.58927e6 0.599236
\(515\) 0 0
\(516\) −1.52086e6 −0.251457
\(517\) 76058.1 0.0125147
\(518\) −3.01105e6 −0.493053
\(519\) 3.17622e6 0.517598
\(520\) 0 0
\(521\) −3.25668e6 −0.525631 −0.262816 0.964846i \(-0.584651\pi\)
−0.262816 + 0.964846i \(0.584651\pi\)
\(522\) 52240.3 0.00839130
\(523\) −2.14727e6 −0.343268 −0.171634 0.985161i \(-0.554905\pi\)
−0.171634 + 0.985161i \(0.554905\pi\)
\(524\) 3.21974e6 0.512263
\(525\) 0 0
\(526\) −2.64574e6 −0.416949
\(527\) 6.22363e6 0.976152
\(528\) 754067. 0.117713
\(529\) 1.05756e7 1.64311
\(530\) 0 0
\(531\) −2.30095e6 −0.354136
\(532\) −4.97783e6 −0.762536
\(533\) 9.39453e6 1.43238
\(534\) −344618. −0.0522979
\(535\) 0 0
\(536\) 1.74598e6 0.262498
\(537\) 7.05746e6 1.05612
\(538\) 2.00878e6 0.299211
\(539\) −2.42629e6 −0.359726
\(540\) 0 0
\(541\) 2.92326e6 0.429412 0.214706 0.976679i \(-0.431121\pi\)
0.214706 + 0.976679i \(0.431121\pi\)
\(542\) −1.34585e6 −0.196788
\(543\) 22790.9 0.00331712
\(544\) 6.58916e6 0.954625
\(545\) 0 0
\(546\) −1.75749e6 −0.252297
\(547\) −9.05734e6 −1.29429 −0.647146 0.762366i \(-0.724038\pi\)
−0.647146 + 0.762366i \(0.724038\pi\)
\(548\) 796535. 0.113306
\(549\) 4.17233e6 0.590810
\(550\) 0 0
\(551\) −310719. −0.0436002
\(552\) 4.24777e6 0.593353
\(553\) 5.25254e6 0.730393
\(554\) −1.28432e6 −0.177787
\(555\) 0 0
\(556\) 1.68722e6 0.231465
\(557\) 9.32565e6 1.27362 0.636812 0.771019i \(-0.280253\pi\)
0.636812 + 0.771019i \(0.280253\pi\)
\(558\) 720718. 0.0979896
\(559\) 3.19377e6 0.432289
\(560\) 0 0
\(561\) 1.44281e6 0.193553
\(562\) 998985. 0.133419
\(563\) −1.27073e6 −0.168960 −0.0844798 0.996425i \(-0.526923\pi\)
−0.0844798 + 0.996425i \(0.526923\pi\)
\(564\) −160734. −0.0212769
\(565\) 0 0
\(566\) 1.06717e6 0.140020
\(567\) −1.25963e6 −0.164545
\(568\) −4.39399e6 −0.571463
\(569\) 7.32601e6 0.948608 0.474304 0.880361i \(-0.342700\pi\)
0.474304 + 0.880361i \(0.342700\pi\)
\(570\) 0 0
\(571\) 1.10876e7 1.42314 0.711569 0.702617i \(-0.247985\pi\)
0.711569 + 0.702617i \(0.247985\pi\)
\(572\) −1.84609e6 −0.235918
\(573\) −1.66717e6 −0.212125
\(574\) 6.36211e6 0.805975
\(575\) 0 0
\(576\) −1.03176e6 −0.129575
\(577\) −3.23610e6 −0.404653 −0.202327 0.979318i \(-0.564850\pi\)
−0.202327 + 0.979318i \(0.564850\pi\)
\(578\) 635448. 0.0791153
\(579\) 4.71440e6 0.584426
\(580\) 0 0
\(581\) −9.94839e6 −1.22268
\(582\) −2.84523e6 −0.348185
\(583\) −2.05123e6 −0.249944
\(584\) 4.63023e6 0.561785
\(585\) 0 0
\(586\) 2.48704e6 0.299184
\(587\) 9.64747e6 1.15563 0.577814 0.816168i \(-0.303906\pi\)
0.577814 + 0.816168i \(0.303906\pi\)
\(588\) 5.12749e6 0.611592
\(589\) −4.28674e6 −0.509142
\(590\) 0 0
\(591\) −1.50298e6 −0.177004
\(592\) 5.73339e6 0.672369
\(593\) 120468. 0.0140681 0.00703406 0.999975i \(-0.497761\pi\)
0.00703406 + 0.999975i \(0.497761\pi\)
\(594\) 167082. 0.0194296
\(595\) 0 0
\(596\) 6.43578e6 0.742139
\(597\) 6.43783e6 0.739270
\(598\) −4.19523e6 −0.479737
\(599\) 1.39130e7 1.58436 0.792178 0.610290i \(-0.208947\pi\)
0.792178 + 0.610290i \(0.208947\pi\)
\(600\) 0 0
\(601\) 6.27530e6 0.708677 0.354339 0.935117i \(-0.384706\pi\)
0.354339 + 0.935117i \(0.384706\pi\)
\(602\) 2.16287e6 0.243242
\(603\) −1.23590e6 −0.138417
\(604\) −6.58600e6 −0.734563
\(605\) 0 0
\(606\) −315281. −0.0348752
\(607\) −7.85752e6 −0.865593 −0.432796 0.901492i \(-0.642473\pi\)
−0.432796 + 0.901492i \(0.642473\pi\)
\(608\) −4.53851e6 −0.497914
\(609\) 588326. 0.0642799
\(610\) 0 0
\(611\) 337538. 0.0365779
\(612\) −3.04908e6 −0.329072
\(613\) 1.19554e7 1.28503 0.642514 0.766274i \(-0.277891\pi\)
0.642514 + 0.766274i \(0.277891\pi\)
\(614\) 4.91447e6 0.526085
\(615\) 0 0
\(616\) −2.65826e6 −0.282258
\(617\) −8.19632e6 −0.866774 −0.433387 0.901208i \(-0.642682\pi\)
−0.433387 + 0.901208i \(0.642682\pi\)
\(618\) 1.51708e6 0.159786
\(619\) −3.47785e6 −0.364824 −0.182412 0.983222i \(-0.558390\pi\)
−0.182412 + 0.983222i \(0.558390\pi\)
\(620\) 0 0
\(621\) −3.00680e6 −0.312879
\(622\) −130658. −0.0135413
\(623\) −3.88106e6 −0.400618
\(624\) 3.34647e6 0.344053
\(625\) 0 0
\(626\) 3.98560e6 0.406497
\(627\) −993783. −0.100954
\(628\) −5.58008e6 −0.564601
\(629\) 1.09701e7 1.10556
\(630\) 0 0
\(631\) 1.07592e6 0.107574 0.0537868 0.998552i \(-0.482871\pi\)
0.0537868 + 0.998552i \(0.482871\pi\)
\(632\) 3.13068e6 0.311778
\(633\) −8.26083e6 −0.819435
\(634\) 6.29416e6 0.621892
\(635\) 0 0
\(636\) 4.33486e6 0.424944
\(637\) −1.07676e7 −1.05141
\(638\) −78037.9 −0.00759021
\(639\) 3.11030e6 0.301336
\(640\) 0 0
\(641\) 1.30837e7 1.25772 0.628862 0.777517i \(-0.283521\pi\)
0.628862 + 0.777517i \(0.283521\pi\)
\(642\) 1.91443e6 0.183317
\(643\) 7.84966e6 0.748727 0.374363 0.927282i \(-0.377861\pi\)
0.374363 + 0.927282i \(0.377861\pi\)
\(644\) 2.24985e7 2.13766
\(645\) 0 0
\(646\) −2.29013e6 −0.215913
\(647\) −2.02801e7 −1.90463 −0.952313 0.305123i \(-0.901302\pi\)
−0.952313 + 0.305123i \(0.901302\pi\)
\(648\) −750777. −0.0702382
\(649\) 3.43722e6 0.320328
\(650\) 0 0
\(651\) 8.11668e6 0.750630
\(652\) −1.71856e7 −1.58324
\(653\) 1.04023e7 0.954655 0.477328 0.878725i \(-0.341606\pi\)
0.477328 + 0.878725i \(0.341606\pi\)
\(654\) 3.30962e6 0.302575
\(655\) 0 0
\(656\) −1.21142e7 −1.09910
\(657\) −3.27753e6 −0.296233
\(658\) 228585. 0.0205818
\(659\) −1.05552e7 −0.946785 −0.473392 0.880852i \(-0.656971\pi\)
−0.473392 + 0.880852i \(0.656971\pi\)
\(660\) 0 0
\(661\) −4.20388e6 −0.374237 −0.187119 0.982337i \(-0.559915\pi\)
−0.187119 + 0.982337i \(0.559915\pi\)
\(662\) −5.02824e6 −0.445935
\(663\) 6.40302e6 0.565719
\(664\) −5.92955e6 −0.521917
\(665\) 0 0
\(666\) 1.27037e6 0.110980
\(667\) 1.40437e6 0.122227
\(668\) 3.98933e6 0.345906
\(669\) −587602. −0.0507596
\(670\) 0 0
\(671\) −6.23274e6 −0.534407
\(672\) 8.59339e6 0.734076
\(673\) 1.41220e7 1.20187 0.600935 0.799298i \(-0.294795\pi\)
0.600935 + 0.799298i \(0.294795\pi\)
\(674\) 2.97038e6 0.251862
\(675\) 0 0
\(676\) 2.35651e6 0.198336
\(677\) 6.70088e6 0.561901 0.280951 0.959722i \(-0.409350\pi\)
0.280951 + 0.959722i \(0.409350\pi\)
\(678\) −72808.2 −0.00608283
\(679\) −3.20428e7 −2.66720
\(680\) 0 0
\(681\) 5.02299e6 0.415045
\(682\) −1.07663e6 −0.0886350
\(683\) −7.21441e6 −0.591765 −0.295882 0.955224i \(-0.595614\pi\)
−0.295882 + 0.955224i \(0.595614\pi\)
\(684\) 2.10016e6 0.171638
\(685\) 0 0
\(686\) −1.18007e6 −0.0957405
\(687\) −8.82045e6 −0.713016
\(688\) −4.11835e6 −0.331705
\(689\) −9.10312e6 −0.730537
\(690\) 0 0
\(691\) −7.64521e6 −0.609108 −0.304554 0.952495i \(-0.598507\pi\)
−0.304554 + 0.952495i \(0.598507\pi\)
\(692\) 1.00270e7 0.795990
\(693\) 1.88166e6 0.148836
\(694\) −6.57846e6 −0.518472
\(695\) 0 0
\(696\) 350661. 0.0274387
\(697\) −2.31789e7 −1.80722
\(698\) 2.92914e6 0.227563
\(699\) 7.18699e6 0.556358
\(700\) 0 0
\(701\) 6.16637e6 0.473952 0.236976 0.971515i \(-0.423844\pi\)
0.236976 + 0.971515i \(0.423844\pi\)
\(702\) 741491. 0.0567889
\(703\) −7.55602e6 −0.576640
\(704\) 1.54127e6 0.117205
\(705\) 0 0
\(706\) 2.05860e6 0.155439
\(707\) −3.55068e6 −0.267155
\(708\) −7.26388e6 −0.544610
\(709\) −1.96108e7 −1.46514 −0.732571 0.680691i \(-0.761680\pi\)
−0.732571 + 0.680691i \(0.761680\pi\)
\(710\) 0 0
\(711\) −2.21606e6 −0.164402
\(712\) −2.31323e6 −0.171009
\(713\) 1.93750e7 1.42731
\(714\) 4.33622e6 0.318321
\(715\) 0 0
\(716\) 2.22797e7 1.62415
\(717\) 1.18637e7 0.861833
\(718\) −2.62238e6 −0.189839
\(719\) −8.44447e6 −0.609187 −0.304593 0.952483i \(-0.598521\pi\)
−0.304593 + 0.952483i \(0.598521\pi\)
\(720\) 0 0
\(721\) 1.70853e7 1.22401
\(722\) −3.11272e6 −0.222227
\(723\) 9.68372e6 0.688964
\(724\) 71948.7 0.00510125
\(725\) 0 0
\(726\) −249591. −0.0175747
\(727\) −2.39210e7 −1.67858 −0.839291 0.543682i \(-0.817030\pi\)
−0.839291 + 0.543682i \(0.817030\pi\)
\(728\) −1.17971e7 −0.824986
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −7.87991e6 −0.545416
\(732\) 1.31717e7 0.908579
\(733\) 1.40807e7 0.967973 0.483986 0.875076i \(-0.339188\pi\)
0.483986 + 0.875076i \(0.339188\pi\)
\(734\) −5.86170e6 −0.401590
\(735\) 0 0
\(736\) 2.05129e7 1.39583
\(737\) 1.84622e6 0.125203
\(738\) −2.68420e6 −0.181415
\(739\) −1.14610e7 −0.771987 −0.385993 0.922502i \(-0.626141\pi\)
−0.385993 + 0.922502i \(0.626141\pi\)
\(740\) 0 0
\(741\) −4.41030e6 −0.295068
\(742\) −6.16476e6 −0.411061
\(743\) 2.82841e7 1.87962 0.939810 0.341698i \(-0.111002\pi\)
0.939810 + 0.341698i \(0.111002\pi\)
\(744\) 4.83780e6 0.320417
\(745\) 0 0
\(746\) −7.22128e6 −0.475081
\(747\) 4.19726e6 0.275210
\(748\) 4.55480e6 0.297657
\(749\) 2.15602e7 1.40426
\(750\) 0 0
\(751\) −1.65142e7 −1.06846 −0.534228 0.845340i \(-0.679398\pi\)
−0.534228 + 0.845340i \(0.679398\pi\)
\(752\) −435254. −0.0280671
\(753\) 7.32694e6 0.470907
\(754\) −346324. −0.0221847
\(755\) 0 0
\(756\) −3.97652e6 −0.253046
\(757\) −1.68113e7 −1.06626 −0.533128 0.846034i \(-0.678984\pi\)
−0.533128 + 0.846034i \(0.678984\pi\)
\(758\) −3.39873e6 −0.214854
\(759\) 4.49164e6 0.283009
\(760\) 0 0
\(761\) 391721. 0.0245197 0.0122598 0.999925i \(-0.496097\pi\)
0.0122598 + 0.999925i \(0.496097\pi\)
\(762\) −5.76623e6 −0.359753
\(763\) 3.72727e7 2.31782
\(764\) −5.26308e6 −0.326217
\(765\) 0 0
\(766\) 3.40149e6 0.209458
\(767\) 1.52540e7 0.936258
\(768\) −544105. −0.0332874
\(769\) 2.68596e7 1.63789 0.818944 0.573874i \(-0.194560\pi\)
0.818944 + 0.573874i \(0.194560\pi\)
\(770\) 0 0
\(771\) −1.70542e7 −1.03323
\(772\) 1.48829e7 0.898762
\(773\) −1.65258e7 −0.994748 −0.497374 0.867536i \(-0.665702\pi\)
−0.497374 + 0.867536i \(0.665702\pi\)
\(774\) −912521. −0.0547508
\(775\) 0 0
\(776\) −1.90985e7 −1.13853
\(777\) 1.43069e7 0.850142
\(778\) 6.07087e6 0.359585
\(779\) 1.59653e7 0.942611
\(780\) 0 0
\(781\) −4.64626e6 −0.272568
\(782\) 1.03508e7 0.605281
\(783\) −248217. −0.0144686
\(784\) 1.38848e7 0.806771
\(785\) 0 0
\(786\) 1.93186e6 0.111537
\(787\) 368901. 0.0212311 0.0106155 0.999944i \(-0.496621\pi\)
0.0106155 + 0.999944i \(0.496621\pi\)
\(788\) −4.74475e6 −0.272206
\(789\) 1.25711e7 0.718921
\(790\) 0 0
\(791\) −819961. −0.0465963
\(792\) 1.12153e6 0.0635328
\(793\) −2.76602e7 −1.56197
\(794\) 9.60001e6 0.540406
\(795\) 0 0
\(796\) 2.03236e7 1.13689
\(797\) −1.03717e7 −0.578370 −0.289185 0.957273i \(-0.593384\pi\)
−0.289185 + 0.957273i \(0.593384\pi\)
\(798\) −2.98672e6 −0.166030
\(799\) −832799. −0.0461502
\(800\) 0 0
\(801\) 1.63743e6 0.0901742
\(802\) −9.55117e6 −0.524349
\(803\) 4.89606e6 0.267953
\(804\) −3.90161e6 −0.212865
\(805\) 0 0
\(806\) −4.77796e6 −0.259063
\(807\) −9.54463e6 −0.515912
\(808\) −2.11631e6 −0.114039
\(809\) −2.51123e7 −1.34901 −0.674505 0.738271i \(-0.735643\pi\)
−0.674505 + 0.738271i \(0.735643\pi\)
\(810\) 0 0
\(811\) −1.14510e7 −0.611350 −0.305675 0.952136i \(-0.598882\pi\)
−0.305675 + 0.952136i \(0.598882\pi\)
\(812\) 1.85729e6 0.0988530
\(813\) 6.39474e6 0.339310
\(814\) −1.89772e6 −0.100385
\(815\) 0 0
\(816\) −8.25666e6 −0.434089
\(817\) 5.42756e6 0.284479
\(818\) −1.15889e7 −0.605565
\(819\) 8.35063e6 0.435020
\(820\) 0 0
\(821\) −1.48065e7 −0.766646 −0.383323 0.923614i \(-0.625220\pi\)
−0.383323 + 0.923614i \(0.625220\pi\)
\(822\) 477925. 0.0246706
\(823\) 2.56659e6 0.132086 0.0660430 0.997817i \(-0.478963\pi\)
0.0660430 + 0.997817i \(0.478963\pi\)
\(824\) 1.01834e7 0.522484
\(825\) 0 0
\(826\) 1.03302e7 0.526817
\(827\) 3.42288e7 1.74032 0.870159 0.492772i \(-0.164016\pi\)
0.870159 + 0.492772i \(0.164016\pi\)
\(828\) −9.49220e6 −0.481162
\(829\) 7.49591e6 0.378824 0.189412 0.981898i \(-0.439342\pi\)
0.189412 + 0.981898i \(0.439342\pi\)
\(830\) 0 0
\(831\) 6.10238e6 0.306547
\(832\) 6.83998e6 0.342568
\(833\) 2.65667e7 1.32656
\(834\) 1.01234e6 0.0503979
\(835\) 0 0
\(836\) −3.13728e6 −0.155252
\(837\) −3.42446e6 −0.168958
\(838\) 1.19757e7 0.589102
\(839\) 2.91814e7 1.43120 0.715601 0.698509i \(-0.246153\pi\)
0.715601 + 0.698509i \(0.246153\pi\)
\(840\) 0 0
\(841\) −2.03952e7 −0.994348
\(842\) 1.97169e6 0.0958427
\(843\) −4.74663e6 −0.230047
\(844\) −2.60787e7 −1.26017
\(845\) 0 0
\(846\) −96441.0 −0.00463272
\(847\) −2.81088e6 −0.134628
\(848\) 1.17384e7 0.560558
\(849\) −5.07059e6 −0.241429
\(850\) 0 0
\(851\) 3.41513e7 1.61653
\(852\) 9.81894e6 0.463410
\(853\) −265247. −0.0124818 −0.00624091 0.999981i \(-0.501987\pi\)
−0.00624091 + 0.999981i \(0.501987\pi\)
\(854\) −1.87319e7 −0.878895
\(855\) 0 0
\(856\) 1.28506e7 0.599429
\(857\) 2.09223e7 0.973102 0.486551 0.873652i \(-0.338255\pi\)
0.486551 + 0.873652i \(0.338255\pi\)
\(858\) −1.10766e6 −0.0513675
\(859\) −3.16166e7 −1.46195 −0.730976 0.682404i \(-0.760935\pi\)
−0.730976 + 0.682404i \(0.760935\pi\)
\(860\) 0 0
\(861\) −3.02292e7 −1.38969
\(862\) 5.34237e6 0.244887
\(863\) −3.04614e7 −1.39227 −0.696133 0.717913i \(-0.745098\pi\)
−0.696133 + 0.717913i \(0.745098\pi\)
\(864\) −3.62558e6 −0.165232
\(865\) 0 0
\(866\) −4.08645e6 −0.185162
\(867\) −3.01930e6 −0.136414
\(868\) 2.56236e7 1.15436
\(869\) 3.31042e6 0.148708
\(870\) 0 0
\(871\) 8.19331e6 0.365944
\(872\) 2.22157e7 0.989393
\(873\) 1.35190e7 0.600354
\(874\) −7.12947e6 −0.315703
\(875\) 0 0
\(876\) −1.03469e7 −0.455563
\(877\) −3.21784e6 −0.141275 −0.0706376 0.997502i \(-0.522503\pi\)
−0.0706376 + 0.997502i \(0.522503\pi\)
\(878\) 8.71623e6 0.381586
\(879\) −1.18170e7 −0.515866
\(880\) 0 0
\(881\) −2.71105e7 −1.17679 −0.588394 0.808574i \(-0.700240\pi\)
−0.588394 + 0.808574i \(0.700240\pi\)
\(882\) 3.07652e6 0.133164
\(883\) −7.92568e6 −0.342086 −0.171043 0.985264i \(-0.554714\pi\)
−0.171043 + 0.985264i \(0.554714\pi\)
\(884\) 2.02137e7 0.869993
\(885\) 0 0
\(886\) 9.53764e6 0.408185
\(887\) −2.02221e7 −0.863011 −0.431506 0.902110i \(-0.642017\pi\)
−0.431506 + 0.902110i \(0.642017\pi\)
\(888\) 8.52733e6 0.362895
\(889\) −6.49389e7 −2.75582
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −1.85501e6 −0.0780608
\(893\) 573619. 0.0240711
\(894\) 3.86150e6 0.161589
\(895\) 0 0
\(896\) 3.51864e7 1.46421
\(897\) 1.99334e7 0.827182
\(898\) −1.22371e7 −0.506392
\(899\) 1.59944e6 0.0660038
\(900\) 0 0
\(901\) 2.24599e7 0.921714
\(902\) 4.00973e6 0.164096
\(903\) −1.02768e7 −0.419408
\(904\) −488722. −0.0198903
\(905\) 0 0
\(906\) −3.95163e6 −0.159940
\(907\) −7.88692e6 −0.318338 −0.159169 0.987251i \(-0.550882\pi\)
−0.159169 + 0.987251i \(0.550882\pi\)
\(908\) 1.58571e7 0.638278
\(909\) 1.49804e6 0.0601332
\(910\) 0 0
\(911\) −6.12818e6 −0.244644 −0.122322 0.992490i \(-0.539034\pi\)
−0.122322 + 0.992490i \(0.539034\pi\)
\(912\) 5.68706e6 0.226413
\(913\) −6.26998e6 −0.248937
\(914\) −7.86129e6 −0.311264
\(915\) 0 0
\(916\) −2.78453e7 −1.09651
\(917\) 2.17565e7 0.854408
\(918\) −1.82946e6 −0.0716502
\(919\) 2.38050e7 0.929779 0.464890 0.885369i \(-0.346094\pi\)
0.464890 + 0.885369i \(0.346094\pi\)
\(920\) 0 0
\(921\) −2.33509e7 −0.907098
\(922\) 5.34428e6 0.207044
\(923\) −2.06196e7 −0.796665
\(924\) 5.94024e6 0.228889
\(925\) 0 0
\(926\) −4.09746e6 −0.157032
\(927\) −7.20834e6 −0.275509
\(928\) 1.69338e6 0.0645481
\(929\) 3.89856e7 1.48206 0.741028 0.671474i \(-0.234339\pi\)
0.741028 + 0.671474i \(0.234339\pi\)
\(930\) 0 0
\(931\) −1.82988e7 −0.691907
\(932\) 2.26887e7 0.855596
\(933\) 620817. 0.0233485
\(934\) 3.55499e6 0.133343
\(935\) 0 0
\(936\) 4.97723e6 0.185694
\(937\) 1.66079e7 0.617967 0.308983 0.951067i \(-0.400011\pi\)
0.308983 + 0.951067i \(0.400011\pi\)
\(938\) 5.54863e6 0.205911
\(939\) −1.89374e7 −0.700899
\(940\) 0 0
\(941\) −2.15988e7 −0.795161 −0.397581 0.917567i \(-0.630150\pi\)
−0.397581 + 0.917567i \(0.630150\pi\)
\(942\) −3.34807e6 −0.122933
\(943\) −7.21589e7 −2.64248
\(944\) −1.96700e7 −0.718412
\(945\) 0 0
\(946\) 1.36315e6 0.0495240
\(947\) −4.65688e7 −1.68741 −0.843705 0.536808i \(-0.819630\pi\)
−0.843705 + 0.536808i \(0.819630\pi\)
\(948\) −6.99591e6 −0.252827
\(949\) 2.17282e7 0.783174
\(950\) 0 0
\(951\) −2.99064e7 −1.07229
\(952\) 2.91067e7 1.04088
\(953\) −2.71746e7 −0.969240 −0.484620 0.874725i \(-0.661042\pi\)
−0.484620 + 0.874725i \(0.661042\pi\)
\(954\) 2.60094e6 0.0925249
\(955\) 0 0
\(956\) 3.74527e7 1.32537
\(957\) 370793. 0.0130874
\(958\) −9.17946e6 −0.323149
\(959\) 5.38236e6 0.188985
\(960\) 0 0
\(961\) −6.56292e6 −0.229239
\(962\) −8.42186e6 −0.293407
\(963\) −9.09634e6 −0.316083
\(964\) 3.05706e7 1.05953
\(965\) 0 0
\(966\) 1.34992e7 0.465442
\(967\) 2.23871e6 0.0769897 0.0384948 0.999259i \(-0.487744\pi\)
0.0384948 + 0.999259i \(0.487744\pi\)
\(968\) −1.67537e6 −0.0574676
\(969\) 1.08814e7 0.372286
\(970\) 0 0
\(971\) −1.25051e7 −0.425638 −0.212819 0.977092i \(-0.568264\pi\)
−0.212819 + 0.977092i \(0.568264\pi\)
\(972\) 1.67771e6 0.0569575
\(973\) 1.14009e7 0.386063
\(974\) −1.73088e7 −0.584616
\(975\) 0 0
\(976\) 3.56677e7 1.19854
\(977\) −5.79995e7 −1.94396 −0.971982 0.235056i \(-0.924473\pi\)
−0.971982 + 0.235056i \(0.924473\pi\)
\(978\) −1.03115e7 −0.344725
\(979\) −2.44604e6 −0.0815656
\(980\) 0 0
\(981\) −1.57255e7 −0.521713
\(982\) −9.74109e6 −0.322351
\(983\) −3.40518e7 −1.12397 −0.561987 0.827146i \(-0.689963\pi\)
−0.561987 + 0.827146i \(0.689963\pi\)
\(984\) −1.80176e7 −0.593210
\(985\) 0 0
\(986\) 854477. 0.0279903
\(987\) −1.08611e6 −0.0354880
\(988\) −1.39229e7 −0.453772
\(989\) −2.45312e7 −0.797495
\(990\) 0 0
\(991\) 2.08858e7 0.675565 0.337782 0.941224i \(-0.390323\pi\)
0.337782 + 0.941224i \(0.390323\pi\)
\(992\) 2.33622e7 0.753763
\(993\) 2.38914e7 0.768899
\(994\) −1.39639e7 −0.448271
\(995\) 0 0
\(996\) 1.32504e7 0.423233
\(997\) −3.35610e7 −1.06929 −0.534646 0.845076i \(-0.679555\pi\)
−0.534646 + 0.845076i \(0.679555\pi\)
\(998\) 4.14773e6 0.131821
\(999\) −6.03611e6 −0.191357
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.1.5 yes 8
5.4 even 2 825.6.a.p.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.4 8 5.4 even 2
825.6.a.q.1.5 yes 8 1.1 even 1 trivial