Properties

Label 825.6.a.q.1.7
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 1.7
Root \(7.96197\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.96197 q^{2} -9.00000 q^{3} +48.3168 q^{4} -80.6577 q^{6} +191.454 q^{7} +146.231 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+8.96197 q^{2} -9.00000 q^{3} +48.3168 q^{4} -80.6577 q^{6} +191.454 q^{7} +146.231 q^{8} +81.0000 q^{9} -121.000 q^{11} -434.852 q^{12} -257.481 q^{13} +1715.80 q^{14} -235.621 q^{16} -1247.35 q^{17} +725.919 q^{18} -2954.73 q^{19} -1723.09 q^{21} -1084.40 q^{22} +374.592 q^{23} -1316.08 q^{24} -2307.53 q^{26} -729.000 q^{27} +9250.46 q^{28} +4573.39 q^{29} +2523.66 q^{31} -6791.02 q^{32} +1089.00 q^{33} -11178.7 q^{34} +3913.66 q^{36} -11378.4 q^{37} -26480.2 q^{38} +2317.32 q^{39} -7607.66 q^{41} -15442.2 q^{42} +853.530 q^{43} -5846.34 q^{44} +3357.08 q^{46} -18520.4 q^{47} +2120.59 q^{48} +19847.7 q^{49} +11226.2 q^{51} -12440.6 q^{52} +15523.8 q^{53} -6533.27 q^{54} +27996.5 q^{56} +26592.6 q^{57} +40986.5 q^{58} -13932.5 q^{59} +21313.0 q^{61} +22617.0 q^{62} +15507.8 q^{63} -53321.0 q^{64} +9759.58 q^{66} -55365.4 q^{67} -60268.1 q^{68} -3371.33 q^{69} -72039.8 q^{71} +11844.7 q^{72} +57248.2 q^{73} -101973. q^{74} -142763. q^{76} -23165.9 q^{77} +20767.8 q^{78} +75155.7 q^{79} +6561.00 q^{81} -68179.6 q^{82} +102915. q^{83} -83254.1 q^{84} +7649.30 q^{86} -41160.5 q^{87} -17694.0 q^{88} +11387.8 q^{89} -49295.7 q^{91} +18099.1 q^{92} -22712.9 q^{93} -165980. q^{94} +61119.2 q^{96} +23382.2 q^{97} +177874. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.96197 1.58427 0.792133 0.610348i \(-0.208970\pi\)
0.792133 + 0.610348i \(0.208970\pi\)
\(3\) −9.00000 −0.577350
\(4\) 48.3168 1.50990
\(5\) 0 0
\(6\) −80.6577 −0.914677
\(7\) 191.454 1.47679 0.738396 0.674367i \(-0.235583\pi\)
0.738396 + 0.674367i \(0.235583\pi\)
\(8\) 146.231 0.807820
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −434.852 −0.871742
\(13\) −257.481 −0.422558 −0.211279 0.977426i \(-0.567763\pi\)
−0.211279 + 0.977426i \(0.567763\pi\)
\(14\) 1715.80 2.33963
\(15\) 0 0
\(16\) −235.621 −0.230099
\(17\) −1247.35 −1.04681 −0.523404 0.852085i \(-0.675338\pi\)
−0.523404 + 0.852085i \(0.675338\pi\)
\(18\) 725.919 0.528089
\(19\) −2954.73 −1.87774 −0.938868 0.344278i \(-0.888124\pi\)
−0.938868 + 0.344278i \(0.888124\pi\)
\(20\) 0 0
\(21\) −1723.09 −0.852627
\(22\) −1084.40 −0.477674
\(23\) 374.592 0.147652 0.0738260 0.997271i \(-0.476479\pi\)
0.0738260 + 0.997271i \(0.476479\pi\)
\(24\) −1316.08 −0.466395
\(25\) 0 0
\(26\) −2307.53 −0.669444
\(27\) −729.000 −0.192450
\(28\) 9250.46 2.22981
\(29\) 4573.39 1.00982 0.504909 0.863173i \(-0.331526\pi\)
0.504909 + 0.863173i \(0.331526\pi\)
\(30\) 0 0
\(31\) 2523.66 0.471657 0.235829 0.971795i \(-0.424220\pi\)
0.235829 + 0.971795i \(0.424220\pi\)
\(32\) −6791.02 −1.17236
\(33\) 1089.00 0.174078
\(34\) −11178.7 −1.65842
\(35\) 0 0
\(36\) 3913.66 0.503300
\(37\) −11378.4 −1.36640 −0.683199 0.730232i \(-0.739412\pi\)
−0.683199 + 0.730232i \(0.739412\pi\)
\(38\) −26480.2 −2.97483
\(39\) 2317.32 0.243964
\(40\) 0 0
\(41\) −7607.66 −0.706791 −0.353396 0.935474i \(-0.614973\pi\)
−0.353396 + 0.935474i \(0.614973\pi\)
\(42\) −15442.2 −1.35079
\(43\) 853.530 0.0703959 0.0351980 0.999380i \(-0.488794\pi\)
0.0351980 + 0.999380i \(0.488794\pi\)
\(44\) −5846.34 −0.455252
\(45\) 0 0
\(46\) 3357.08 0.233920
\(47\) −18520.4 −1.22294 −0.611472 0.791266i \(-0.709422\pi\)
−0.611472 + 0.791266i \(0.709422\pi\)
\(48\) 2120.59 0.132848
\(49\) 19847.7 1.18092
\(50\) 0 0
\(51\) 11226.2 0.604375
\(52\) −12440.6 −0.638020
\(53\) 15523.8 0.759116 0.379558 0.925168i \(-0.376076\pi\)
0.379558 + 0.925168i \(0.376076\pi\)
\(54\) −6533.27 −0.304892
\(55\) 0 0
\(56\) 27996.5 1.19298
\(57\) 26592.6 1.08411
\(58\) 40986.5 1.59982
\(59\) −13932.5 −0.521074 −0.260537 0.965464i \(-0.583900\pi\)
−0.260537 + 0.965464i \(0.583900\pi\)
\(60\) 0 0
\(61\) 21313.0 0.733366 0.366683 0.930346i \(-0.380493\pi\)
0.366683 + 0.930346i \(0.380493\pi\)
\(62\) 22617.0 0.747231
\(63\) 15507.8 0.492264
\(64\) −53321.0 −1.62723
\(65\) 0 0
\(66\) 9759.58 0.275785
\(67\) −55365.4 −1.50679 −0.753393 0.657571i \(-0.771584\pi\)
−0.753393 + 0.657571i \(0.771584\pi\)
\(68\) −60268.1 −1.58058
\(69\) −3371.33 −0.0852469
\(70\) 0 0
\(71\) −72039.8 −1.69600 −0.848002 0.529993i \(-0.822194\pi\)
−0.848002 + 0.529993i \(0.822194\pi\)
\(72\) 11844.7 0.269273
\(73\) 57248.2 1.25735 0.628673 0.777670i \(-0.283598\pi\)
0.628673 + 0.777670i \(0.283598\pi\)
\(74\) −101973. −2.16474
\(75\) 0 0
\(76\) −142763. −2.83520
\(77\) −23165.9 −0.445270
\(78\) 20767.8 0.386504
\(79\) 75155.7 1.35486 0.677430 0.735587i \(-0.263094\pi\)
0.677430 + 0.735587i \(0.263094\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −68179.6 −1.11975
\(83\) 102915. 1.63978 0.819889 0.572523i \(-0.194035\pi\)
0.819889 + 0.572523i \(0.194035\pi\)
\(84\) −83254.1 −1.28738
\(85\) 0 0
\(86\) 7649.30 0.111526
\(87\) −41160.5 −0.583018
\(88\) −17694.0 −0.243567
\(89\) 11387.8 0.152393 0.0761965 0.997093i \(-0.475722\pi\)
0.0761965 + 0.997093i \(0.475722\pi\)
\(90\) 0 0
\(91\) −49295.7 −0.624030
\(92\) 18099.1 0.222940
\(93\) −22712.9 −0.272312
\(94\) −165980. −1.93747
\(95\) 0 0
\(96\) 61119.2 0.676861
\(97\) 23382.2 0.252323 0.126161 0.992010i \(-0.459734\pi\)
0.126161 + 0.992010i \(0.459734\pi\)
\(98\) 177874. 1.87089
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 71052.3 0.693066 0.346533 0.938038i \(-0.387359\pi\)
0.346533 + 0.938038i \(0.387359\pi\)
\(102\) 100609. 0.957491
\(103\) −133153. −1.23668 −0.618339 0.785912i \(-0.712194\pi\)
−0.618339 + 0.785912i \(0.712194\pi\)
\(104\) −37651.6 −0.341351
\(105\) 0 0
\(106\) 139124. 1.20264
\(107\) −22312.0 −0.188400 −0.0941998 0.995553i \(-0.530029\pi\)
−0.0941998 + 0.995553i \(0.530029\pi\)
\(108\) −35223.0 −0.290581
\(109\) −212605. −1.71399 −0.856993 0.515328i \(-0.827670\pi\)
−0.856993 + 0.515328i \(0.827670\pi\)
\(110\) 0 0
\(111\) 102406. 0.788891
\(112\) −45110.7 −0.339809
\(113\) −246202. −1.81383 −0.906914 0.421316i \(-0.861568\pi\)
−0.906914 + 0.421316i \(0.861568\pi\)
\(114\) 238322. 1.71752
\(115\) 0 0
\(116\) 220972. 1.52472
\(117\) −20855.9 −0.140853
\(118\) −124863. −0.825520
\(119\) −238811. −1.54592
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 191007. 1.16185
\(123\) 68468.9 0.408066
\(124\) 121935. 0.712156
\(125\) 0 0
\(126\) 138980. 0.779878
\(127\) −119855. −0.659399 −0.329699 0.944086i \(-0.606947\pi\)
−0.329699 + 0.944086i \(0.606947\pi\)
\(128\) −260549. −1.40561
\(129\) −7681.77 −0.0406431
\(130\) 0 0
\(131\) −89009.9 −0.453169 −0.226585 0.973992i \(-0.572756\pi\)
−0.226585 + 0.973992i \(0.572756\pi\)
\(132\) 52617.0 0.262840
\(133\) −565696. −2.77303
\(134\) −496183. −2.38715
\(135\) 0 0
\(136\) −182402. −0.845632
\(137\) −91980.3 −0.418691 −0.209345 0.977842i \(-0.567133\pi\)
−0.209345 + 0.977842i \(0.567133\pi\)
\(138\) −30213.7 −0.135054
\(139\) 233520. 1.02515 0.512574 0.858643i \(-0.328692\pi\)
0.512574 + 0.858643i \(0.328692\pi\)
\(140\) 0 0
\(141\) 166684. 0.706067
\(142\) −645619. −2.68692
\(143\) 31155.1 0.127406
\(144\) −19085.3 −0.0766997
\(145\) 0 0
\(146\) 513057. 1.99197
\(147\) −178629. −0.681802
\(148\) −549769. −2.06313
\(149\) −226695. −0.836520 −0.418260 0.908327i \(-0.637360\pi\)
−0.418260 + 0.908327i \(0.637360\pi\)
\(150\) 0 0
\(151\) −461783. −1.64814 −0.824072 0.566485i \(-0.808303\pi\)
−0.824072 + 0.566485i \(0.808303\pi\)
\(152\) −432074. −1.51687
\(153\) −101036. −0.348936
\(154\) −207612. −0.705426
\(155\) 0 0
\(156\) 111966. 0.368361
\(157\) 9453.41 0.0306083 0.0153042 0.999883i \(-0.495128\pi\)
0.0153042 + 0.999883i \(0.495128\pi\)
\(158\) 673543. 2.14646
\(159\) −139714. −0.438276
\(160\) 0 0
\(161\) 71717.2 0.218051
\(162\) 58799.5 0.176030
\(163\) 183701. 0.541554 0.270777 0.962642i \(-0.412719\pi\)
0.270777 + 0.962642i \(0.412719\pi\)
\(164\) −367578. −1.06719
\(165\) 0 0
\(166\) 922324. 2.59785
\(167\) 339882. 0.943054 0.471527 0.881852i \(-0.343703\pi\)
0.471527 + 0.881852i \(0.343703\pi\)
\(168\) −251969. −0.688769
\(169\) −304997. −0.821445
\(170\) 0 0
\(171\) −239333. −0.625912
\(172\) 41239.9 0.106291
\(173\) 12098.7 0.0307342 0.0153671 0.999882i \(-0.495108\pi\)
0.0153671 + 0.999882i \(0.495108\pi\)
\(174\) −368879. −0.923657
\(175\) 0 0
\(176\) 28510.2 0.0693775
\(177\) 125393. 0.300842
\(178\) 102057. 0.241431
\(179\) 366015. 0.853820 0.426910 0.904294i \(-0.359602\pi\)
0.426910 + 0.904294i \(0.359602\pi\)
\(180\) 0 0
\(181\) −187033. −0.424347 −0.212174 0.977232i \(-0.568054\pi\)
−0.212174 + 0.977232i \(0.568054\pi\)
\(182\) −441786. −0.988630
\(183\) −191817. −0.423409
\(184\) 54777.0 0.119276
\(185\) 0 0
\(186\) −203553. −0.431414
\(187\) 150930. 0.315624
\(188\) −894849. −1.84653
\(189\) −139570. −0.284209
\(190\) 0 0
\(191\) 75921.1 0.150584 0.0752920 0.997162i \(-0.476011\pi\)
0.0752920 + 0.997162i \(0.476011\pi\)
\(192\) 479889. 0.939481
\(193\) −897793. −1.73493 −0.867467 0.497495i \(-0.834253\pi\)
−0.867467 + 0.497495i \(0.834253\pi\)
\(194\) 209551. 0.399747
\(195\) 0 0
\(196\) 958976. 1.78307
\(197\) −731669. −1.34323 −0.671613 0.740902i \(-0.734398\pi\)
−0.671613 + 0.740902i \(0.734398\pi\)
\(198\) −87836.2 −0.159225
\(199\) 60771.5 0.108785 0.0543923 0.998520i \(-0.482678\pi\)
0.0543923 + 0.998520i \(0.482678\pi\)
\(200\) 0 0
\(201\) 498289. 0.869943
\(202\) 636768. 1.09800
\(203\) 875593. 1.49129
\(204\) 542413. 0.912546
\(205\) 0 0
\(206\) −1.19331e6 −1.95923
\(207\) 30342.0 0.0492173
\(208\) 60667.9 0.0972301
\(209\) 357523. 0.566159
\(210\) 0 0
\(211\) 52810.1 0.0816603 0.0408301 0.999166i \(-0.487000\pi\)
0.0408301 + 0.999166i \(0.487000\pi\)
\(212\) 750061. 1.14619
\(213\) 648358. 0.979188
\(214\) −199960. −0.298475
\(215\) 0 0
\(216\) −106602. −0.155465
\(217\) 483165. 0.696540
\(218\) −1.90536e6 −2.71541
\(219\) −515234. −0.725929
\(220\) 0 0
\(221\) 321169. 0.442337
\(222\) 917757. 1.24981
\(223\) −274535. −0.369688 −0.184844 0.982768i \(-0.559178\pi\)
−0.184844 + 0.982768i \(0.559178\pi\)
\(224\) −1.30017e6 −1.73133
\(225\) 0 0
\(226\) −2.20646e6 −2.87359
\(227\) 459993. 0.592498 0.296249 0.955111i \(-0.404264\pi\)
0.296249 + 0.955111i \(0.404264\pi\)
\(228\) 1.28487e6 1.63690
\(229\) 708694. 0.893038 0.446519 0.894774i \(-0.352664\pi\)
0.446519 + 0.894774i \(0.352664\pi\)
\(230\) 0 0
\(231\) 208493. 0.257077
\(232\) 668771. 0.815751
\(233\) 972539. 1.17359 0.586796 0.809735i \(-0.300389\pi\)
0.586796 + 0.809735i \(0.300389\pi\)
\(234\) −186910. −0.223148
\(235\) 0 0
\(236\) −673175. −0.786770
\(237\) −676402. −0.782229
\(238\) −2.14021e6 −2.44915
\(239\) 506701. 0.573795 0.286898 0.957961i \(-0.407376\pi\)
0.286898 + 0.957961i \(0.407376\pi\)
\(240\) 0 0
\(241\) 256791. 0.284798 0.142399 0.989809i \(-0.454518\pi\)
0.142399 + 0.989809i \(0.454518\pi\)
\(242\) 131212. 0.144024
\(243\) −59049.0 −0.0641500
\(244\) 1.02978e6 1.10731
\(245\) 0 0
\(246\) 613616. 0.646486
\(247\) 760786. 0.793451
\(248\) 369038. 0.381014
\(249\) −926238. −0.946726
\(250\) 0 0
\(251\) −1.58330e6 −1.58628 −0.793138 0.609042i \(-0.791554\pi\)
−0.793138 + 0.609042i \(0.791554\pi\)
\(252\) 749287. 0.743270
\(253\) −45325.6 −0.0445187
\(254\) −1.07414e6 −1.04466
\(255\) 0 0
\(256\) −628755. −0.599628
\(257\) 955020. 0.901945 0.450972 0.892538i \(-0.351077\pi\)
0.450972 + 0.892538i \(0.351077\pi\)
\(258\) −68843.7 −0.0643895
\(259\) −2.17844e6 −2.01789
\(260\) 0 0
\(261\) 370444. 0.336606
\(262\) −797704. −0.717941
\(263\) 1.31065e6 1.16841 0.584207 0.811605i \(-0.301406\pi\)
0.584207 + 0.811605i \(0.301406\pi\)
\(264\) 159246. 0.140623
\(265\) 0 0
\(266\) −5.06975e6 −4.39321
\(267\) −102490. −0.0879842
\(268\) −2.67508e6 −2.27510
\(269\) 1.14655e6 0.966076 0.483038 0.875599i \(-0.339533\pi\)
0.483038 + 0.875599i \(0.339533\pi\)
\(270\) 0 0
\(271\) −1.21682e6 −1.00648 −0.503239 0.864147i \(-0.667859\pi\)
−0.503239 + 0.864147i \(0.667859\pi\)
\(272\) 293903. 0.240869
\(273\) 443661. 0.360284
\(274\) −824325. −0.663318
\(275\) 0 0
\(276\) −162892. −0.128714
\(277\) −772459. −0.604889 −0.302445 0.953167i \(-0.597803\pi\)
−0.302445 + 0.953167i \(0.597803\pi\)
\(278\) 2.09279e6 1.62411
\(279\) 204417. 0.157219
\(280\) 0 0
\(281\) −2.06441e6 −1.55966 −0.779830 0.625991i \(-0.784695\pi\)
−0.779830 + 0.625991i \(0.784695\pi\)
\(282\) 1.49382e6 1.11860
\(283\) 2.22733e6 1.65317 0.826586 0.562810i \(-0.190280\pi\)
0.826586 + 0.562810i \(0.190280\pi\)
\(284\) −3.48074e6 −2.56080
\(285\) 0 0
\(286\) 279211. 0.201845
\(287\) −1.45652e6 −1.04378
\(288\) −550073. −0.390786
\(289\) 136031. 0.0958063
\(290\) 0 0
\(291\) −210440. −0.145679
\(292\) 2.76605e6 1.89847
\(293\) 2.69919e6 1.83681 0.918405 0.395641i \(-0.129478\pi\)
0.918405 + 0.395641i \(0.129478\pi\)
\(294\) −1.60087e6 −1.08016
\(295\) 0 0
\(296\) −1.66388e6 −1.10380
\(297\) 88209.0 0.0580259
\(298\) −2.03163e6 −1.32527
\(299\) −96450.2 −0.0623914
\(300\) 0 0
\(301\) 163412. 0.103960
\(302\) −4.13848e6 −2.61110
\(303\) −639471. −0.400142
\(304\) 696199. 0.432065
\(305\) 0 0
\(306\) −905477. −0.552808
\(307\) 1.10821e6 0.671086 0.335543 0.942025i \(-0.391080\pi\)
0.335543 + 0.942025i \(0.391080\pi\)
\(308\) −1.11931e6 −0.672313
\(309\) 1.19837e6 0.713996
\(310\) 0 0
\(311\) −2.05197e6 −1.20301 −0.601505 0.798869i \(-0.705432\pi\)
−0.601505 + 0.798869i \(0.705432\pi\)
\(312\) 338865. 0.197079
\(313\) −2.73410e6 −1.57744 −0.788721 0.614751i \(-0.789257\pi\)
−0.788721 + 0.614751i \(0.789257\pi\)
\(314\) 84721.2 0.0484917
\(315\) 0 0
\(316\) 3.63129e6 2.04570
\(317\) 2.45349e6 1.37131 0.685656 0.727926i \(-0.259515\pi\)
0.685656 + 0.727926i \(0.259515\pi\)
\(318\) −1.25211e6 −0.694346
\(319\) −553380. −0.304471
\(320\) 0 0
\(321\) 200808. 0.108773
\(322\) 642727. 0.345451
\(323\) 3.68559e6 1.96563
\(324\) 317007. 0.167767
\(325\) 0 0
\(326\) 1.64632e6 0.857966
\(327\) 1.91345e6 0.989570
\(328\) −1.11248e6 −0.570960
\(329\) −3.54581e6 −1.80604
\(330\) 0 0
\(331\) 1.15167e6 0.577773 0.288886 0.957363i \(-0.406715\pi\)
0.288886 + 0.957363i \(0.406715\pi\)
\(332\) 4.97255e6 2.47590
\(333\) −921652. −0.455466
\(334\) 3.04601e6 1.49405
\(335\) 0 0
\(336\) 405996. 0.196189
\(337\) 2.40826e6 1.15512 0.577562 0.816347i \(-0.304004\pi\)
0.577562 + 0.816347i \(0.304004\pi\)
\(338\) −2.73337e6 −1.30139
\(339\) 2.21582e6 1.04721
\(340\) 0 0
\(341\) −305363. −0.142210
\(342\) −2.14490e6 −0.991611
\(343\) 582147. 0.267176
\(344\) 124813. 0.0568672
\(345\) 0 0
\(346\) 108428. 0.0486912
\(347\) 2.02361e6 0.902202 0.451101 0.892473i \(-0.351031\pi\)
0.451101 + 0.892473i \(0.351031\pi\)
\(348\) −1.98874e6 −0.880300
\(349\) 1.64732e6 0.723960 0.361980 0.932186i \(-0.382101\pi\)
0.361980 + 0.932186i \(0.382101\pi\)
\(350\) 0 0
\(351\) 187703. 0.0813213
\(352\) 821714. 0.353479
\(353\) 2.08981e6 0.892626 0.446313 0.894877i \(-0.352737\pi\)
0.446313 + 0.894877i \(0.352737\pi\)
\(354\) 1.12376e6 0.476614
\(355\) 0 0
\(356\) 550223. 0.230099
\(357\) 2.14930e6 0.892536
\(358\) 3.28021e6 1.35268
\(359\) 887838. 0.363578 0.181789 0.983338i \(-0.441811\pi\)
0.181789 + 0.983338i \(0.441811\pi\)
\(360\) 0 0
\(361\) 6.25435e6 2.52589
\(362\) −1.67618e6 −0.672279
\(363\) −131769. −0.0524864
\(364\) −2.38181e6 −0.942224
\(365\) 0 0
\(366\) −1.71906e6 −0.670793
\(367\) 1.96673e6 0.762218 0.381109 0.924530i \(-0.375542\pi\)
0.381109 + 0.924530i \(0.375542\pi\)
\(368\) −88261.9 −0.0339746
\(369\) −616220. −0.235597
\(370\) 0 0
\(371\) 2.97209e6 1.12106
\(372\) −1.09742e6 −0.411164
\(373\) 4.81788e6 1.79302 0.896508 0.443028i \(-0.146096\pi\)
0.896508 + 0.443028i \(0.146096\pi\)
\(374\) 1.35263e6 0.500033
\(375\) 0 0
\(376\) −2.70826e6 −0.987919
\(377\) −1.17756e6 −0.426706
\(378\) −1.25082e6 −0.450263
\(379\) −2.14286e6 −0.766296 −0.383148 0.923687i \(-0.625160\pi\)
−0.383148 + 0.923687i \(0.625160\pi\)
\(380\) 0 0
\(381\) 1.07870e6 0.380704
\(382\) 680402. 0.238565
\(383\) 939889. 0.327401 0.163700 0.986510i \(-0.447657\pi\)
0.163700 + 0.986510i \(0.447657\pi\)
\(384\) 2.34494e6 0.811528
\(385\) 0 0
\(386\) −8.04599e6 −2.74860
\(387\) 69135.9 0.0234653
\(388\) 1.12976e6 0.380982
\(389\) −3.30060e6 −1.10591 −0.552953 0.833212i \(-0.686499\pi\)
−0.552953 + 0.833212i \(0.686499\pi\)
\(390\) 0 0
\(391\) −467248. −0.154563
\(392\) 2.90234e6 0.953968
\(393\) 801089. 0.261637
\(394\) −6.55719e6 −2.12803
\(395\) 0 0
\(396\) −473553. −0.151751
\(397\) −4.66064e6 −1.48412 −0.742061 0.670333i \(-0.766151\pi\)
−0.742061 + 0.670333i \(0.766151\pi\)
\(398\) 544632. 0.172344
\(399\) 5.09126e6 1.60101
\(400\) 0 0
\(401\) −1.19103e6 −0.369879 −0.184940 0.982750i \(-0.559209\pi\)
−0.184940 + 0.982750i \(0.559209\pi\)
\(402\) 4.46565e6 1.37822
\(403\) −649793. −0.199302
\(404\) 3.43302e6 1.04646
\(405\) 0 0
\(406\) 7.84704e6 2.36260
\(407\) 1.37679e6 0.411985
\(408\) 1.64161e6 0.488226
\(409\) 1.65037e6 0.487834 0.243917 0.969796i \(-0.421568\pi\)
0.243917 + 0.969796i \(0.421568\pi\)
\(410\) 0 0
\(411\) 827823. 0.241731
\(412\) −6.43351e6 −1.86726
\(413\) −2.66743e6 −0.769518
\(414\) 271924. 0.0779733
\(415\) 0 0
\(416\) 1.74856e6 0.495389
\(417\) −2.10168e6 −0.591869
\(418\) 3.20411e6 0.896946
\(419\) −1.54582e6 −0.430155 −0.215078 0.976597i \(-0.569000\pi\)
−0.215078 + 0.976597i \(0.569000\pi\)
\(420\) 0 0
\(421\) 1.55879e6 0.428629 0.214314 0.976765i \(-0.431248\pi\)
0.214314 + 0.976765i \(0.431248\pi\)
\(422\) 473282. 0.129372
\(423\) −1.50016e6 −0.407648
\(424\) 2.27006e6 0.613229
\(425\) 0 0
\(426\) 5.81057e6 1.55130
\(427\) 4.08047e6 1.08303
\(428\) −1.07805e6 −0.284465
\(429\) −280396. −0.0735578
\(430\) 0 0
\(431\) 2.66308e6 0.690543 0.345272 0.938503i \(-0.387787\pi\)
0.345272 + 0.938503i \(0.387787\pi\)
\(432\) 171768. 0.0442826
\(433\) −2.04309e6 −0.523682 −0.261841 0.965111i \(-0.584330\pi\)
−0.261841 + 0.965111i \(0.584330\pi\)
\(434\) 4.33011e6 1.10351
\(435\) 0 0
\(436\) −1.02724e7 −2.58795
\(437\) −1.10682e6 −0.277251
\(438\) −4.61751e6 −1.15007
\(439\) −4.40979e6 −1.09209 −0.546043 0.837757i \(-0.683866\pi\)
−0.546043 + 0.837757i \(0.683866\pi\)
\(440\) 0 0
\(441\) 1.60766e6 0.393639
\(442\) 2.87831e6 0.700779
\(443\) −2.98414e6 −0.722453 −0.361227 0.932478i \(-0.617642\pi\)
−0.361227 + 0.932478i \(0.617642\pi\)
\(444\) 4.94792e6 1.19115
\(445\) 0 0
\(446\) −2.46037e6 −0.585684
\(447\) 2.04025e6 0.482965
\(448\) −1.02085e7 −2.40308
\(449\) −718293. −0.168146 −0.0840729 0.996460i \(-0.526793\pi\)
−0.0840729 + 0.996460i \(0.526793\pi\)
\(450\) 0 0
\(451\) 920526. 0.213106
\(452\) −1.18957e7 −2.73870
\(453\) 4.15604e6 0.951557
\(454\) 4.12244e6 0.938674
\(455\) 0 0
\(456\) 3.88866e6 0.875767
\(457\) 2.69166e6 0.602879 0.301439 0.953485i \(-0.402533\pi\)
0.301439 + 0.953485i \(0.402533\pi\)
\(458\) 6.35129e6 1.41481
\(459\) 909320. 0.201458
\(460\) 0 0
\(461\) 1.09475e6 0.239918 0.119959 0.992779i \(-0.461724\pi\)
0.119959 + 0.992779i \(0.461724\pi\)
\(462\) 1.86851e6 0.407278
\(463\) 8.57985e6 1.86006 0.930031 0.367482i \(-0.119780\pi\)
0.930031 + 0.367482i \(0.119780\pi\)
\(464\) −1.07759e6 −0.232358
\(465\) 0 0
\(466\) 8.71586e6 1.85928
\(467\) 6.54827e6 1.38942 0.694711 0.719289i \(-0.255532\pi\)
0.694711 + 0.719289i \(0.255532\pi\)
\(468\) −1.00769e6 −0.212673
\(469\) −1.05999e7 −2.22521
\(470\) 0 0
\(471\) −85080.7 −0.0176717
\(472\) −2.03736e6 −0.420934
\(473\) −103277. −0.0212252
\(474\) −6.06189e6 −1.23926
\(475\) 0 0
\(476\) −1.15386e7 −2.33418
\(477\) 1.25743e6 0.253039
\(478\) 4.54104e6 0.909045
\(479\) −2.02247e6 −0.402756 −0.201378 0.979514i \(-0.564542\pi\)
−0.201378 + 0.979514i \(0.564542\pi\)
\(480\) 0 0
\(481\) 2.92972e6 0.577382
\(482\) 2.30135e6 0.451196
\(483\) −645455. −0.125892
\(484\) 707407. 0.137264
\(485\) 0 0
\(486\) −529195. −0.101631
\(487\) 4.51310e6 0.862289 0.431144 0.902283i \(-0.358110\pi\)
0.431144 + 0.902283i \(0.358110\pi\)
\(488\) 3.11663e6 0.592428
\(489\) −1.65331e6 −0.312666
\(490\) 0 0
\(491\) −4.42836e6 −0.828971 −0.414485 0.910056i \(-0.636038\pi\)
−0.414485 + 0.910056i \(0.636038\pi\)
\(492\) 3.30820e6 0.616140
\(493\) −5.70463e6 −1.05708
\(494\) 6.81814e6 1.25704
\(495\) 0 0
\(496\) −594629. −0.108528
\(497\) −1.37923e7 −2.50465
\(498\) −8.30092e6 −1.49987
\(499\) −1.20046e6 −0.215822 −0.107911 0.994161i \(-0.534416\pi\)
−0.107911 + 0.994161i \(0.534416\pi\)
\(500\) 0 0
\(501\) −3.05894e6 −0.544473
\(502\) −1.41895e7 −2.51308
\(503\) −3.10866e6 −0.547840 −0.273920 0.961753i \(-0.588320\pi\)
−0.273920 + 0.961753i \(0.588320\pi\)
\(504\) 2.26772e6 0.397661
\(505\) 0 0
\(506\) −406207. −0.0705295
\(507\) 2.74497e6 0.474262
\(508\) −5.79103e6 −0.995627
\(509\) 9.30479e6 1.59189 0.795943 0.605371i \(-0.206975\pi\)
0.795943 + 0.605371i \(0.206975\pi\)
\(510\) 0 0
\(511\) 1.09604e7 1.85684
\(512\) 2.70268e6 0.455637
\(513\) 2.15400e6 0.361370
\(514\) 8.55886e6 1.42892
\(515\) 0 0
\(516\) −371159. −0.0613671
\(517\) 2.24097e6 0.368732
\(518\) −1.95231e7 −3.19687
\(519\) −108888. −0.0177444
\(520\) 0 0
\(521\) −1.64699e6 −0.265826 −0.132913 0.991128i \(-0.542433\pi\)
−0.132913 + 0.991128i \(0.542433\pi\)
\(522\) 3.31991e6 0.533273
\(523\) −321424. −0.0513835 −0.0256917 0.999670i \(-0.508179\pi\)
−0.0256917 + 0.999670i \(0.508179\pi\)
\(524\) −4.30068e6 −0.684241
\(525\) 0 0
\(526\) 1.17460e7 1.85108
\(527\) −3.14789e6 −0.493735
\(528\) −256592. −0.0400551
\(529\) −6.29602e6 −0.978199
\(530\) 0 0
\(531\) −1.12853e6 −0.173691
\(532\) −2.73326e7 −4.18700
\(533\) 1.95882e6 0.298660
\(534\) −918515. −0.139390
\(535\) 0 0
\(536\) −8.09614e6 −1.21721
\(537\) −3.29413e6 −0.492953
\(538\) 1.02753e7 1.53052
\(539\) −2.40157e6 −0.356060
\(540\) 0 0
\(541\) −7.18074e6 −1.05481 −0.527407 0.849613i \(-0.676836\pi\)
−0.527407 + 0.849613i \(0.676836\pi\)
\(542\) −1.09051e7 −1.59453
\(543\) 1.68329e6 0.244997
\(544\) 8.47080e6 1.22723
\(545\) 0 0
\(546\) 3.97608e6 0.570786
\(547\) −1.18114e6 −0.168785 −0.0843927 0.996433i \(-0.526895\pi\)
−0.0843927 + 0.996433i \(0.526895\pi\)
\(548\) −4.44420e6 −0.632182
\(549\) 1.72636e6 0.244455
\(550\) 0 0
\(551\) −1.35131e7 −1.89617
\(552\) −492993. −0.0688641
\(553\) 1.43889e7 2.00085
\(554\) −6.92275e6 −0.958306
\(555\) 0 0
\(556\) 1.12829e7 1.54787
\(557\) −8.98105e6 −1.22656 −0.613280 0.789865i \(-0.710150\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(558\) 1.83197e6 0.249077
\(559\) −219767. −0.0297463
\(560\) 0 0
\(561\) −1.35837e6 −0.182226
\(562\) −1.85012e7 −2.47092
\(563\) 8.32751e6 1.10725 0.553623 0.832768i \(-0.313245\pi\)
0.553623 + 0.832768i \(0.313245\pi\)
\(564\) 8.05364e6 1.06609
\(565\) 0 0
\(566\) 1.99612e7 2.61907
\(567\) 1.25613e6 0.164088
\(568\) −1.05345e7 −1.37007
\(569\) −1.16305e7 −1.50597 −0.752985 0.658038i \(-0.771387\pi\)
−0.752985 + 0.658038i \(0.771387\pi\)
\(570\) 0 0
\(571\) −9.03556e6 −1.15975 −0.579876 0.814705i \(-0.696899\pi\)
−0.579876 + 0.814705i \(0.696899\pi\)
\(572\) 1.50532e6 0.192370
\(573\) −683290. −0.0869397
\(574\) −1.30533e7 −1.65363
\(575\) 0 0
\(576\) −4.31901e6 −0.542410
\(577\) −1.69135e6 −0.211492 −0.105746 0.994393i \(-0.533723\pi\)
−0.105746 + 0.994393i \(0.533723\pi\)
\(578\) 1.21911e6 0.151783
\(579\) 8.08013e6 1.00166
\(580\) 0 0
\(581\) 1.97036e7 2.42161
\(582\) −1.88596e6 −0.230794
\(583\) −1.87838e6 −0.228882
\(584\) 8.37147e6 1.01571
\(585\) 0 0
\(586\) 2.41900e7 2.91000
\(587\) −1.42543e7 −1.70746 −0.853731 0.520714i \(-0.825666\pi\)
−0.853731 + 0.520714i \(0.825666\pi\)
\(588\) −8.63079e6 −1.02945
\(589\) −7.45675e6 −0.885648
\(590\) 0 0
\(591\) 6.58502e6 0.775512
\(592\) 2.68100e6 0.314407
\(593\) 8.74561e6 1.02130 0.510650 0.859789i \(-0.329405\pi\)
0.510650 + 0.859789i \(0.329405\pi\)
\(594\) 790526. 0.0919285
\(595\) 0 0
\(596\) −1.09532e7 −1.26306
\(597\) −546944. −0.0628068
\(598\) −864383. −0.0988447
\(599\) −2.04663e6 −0.233062 −0.116531 0.993187i \(-0.537177\pi\)
−0.116531 + 0.993187i \(0.537177\pi\)
\(600\) 0 0
\(601\) −1.07049e7 −1.20891 −0.604456 0.796639i \(-0.706609\pi\)
−0.604456 + 0.796639i \(0.706609\pi\)
\(602\) 1.46449e6 0.164701
\(603\) −4.48460e6 −0.502262
\(604\) −2.23119e7 −2.48854
\(605\) 0 0
\(606\) −5.73091e6 −0.633931
\(607\) −5.61449e6 −0.618499 −0.309249 0.950981i \(-0.600078\pi\)
−0.309249 + 0.950981i \(0.600078\pi\)
\(608\) 2.00657e7 2.20138
\(609\) −7.88034e6 −0.860997
\(610\) 0 0
\(611\) 4.76865e6 0.516765
\(612\) −4.88172e6 −0.526859
\(613\) 1.04084e7 1.11875 0.559373 0.828916i \(-0.311042\pi\)
0.559373 + 0.828916i \(0.311042\pi\)
\(614\) 9.93178e6 1.06318
\(615\) 0 0
\(616\) −3.38758e6 −0.359698
\(617\) −9.38303e6 −0.992271 −0.496136 0.868245i \(-0.665248\pi\)
−0.496136 + 0.868245i \(0.665248\pi\)
\(618\) 1.07398e7 1.13116
\(619\) −1.43312e7 −1.50334 −0.751668 0.659542i \(-0.770750\pi\)
−0.751668 + 0.659542i \(0.770750\pi\)
\(620\) 0 0
\(621\) −273078. −0.0284156
\(622\) −1.83896e7 −1.90589
\(623\) 2.18024e6 0.225053
\(624\) −546011. −0.0561358
\(625\) 0 0
\(626\) −2.45029e7 −2.49909
\(627\) −3.21771e6 −0.326872
\(628\) 456759. 0.0462155
\(629\) 1.41929e7 1.43036
\(630\) 0 0
\(631\) −1.34563e7 −1.34541 −0.672703 0.739912i \(-0.734867\pi\)
−0.672703 + 0.739912i \(0.734867\pi\)
\(632\) 1.09901e7 1.09448
\(633\) −475291. −0.0471466
\(634\) 2.19881e7 2.17252
\(635\) 0 0
\(636\) −6.75055e6 −0.661754
\(637\) −5.11039e6 −0.499005
\(638\) −4.95937e6 −0.482364
\(639\) −5.83523e6 −0.565335
\(640\) 0 0
\(641\) 2.02992e6 0.195134 0.0975670 0.995229i \(-0.468894\pi\)
0.0975670 + 0.995229i \(0.468894\pi\)
\(642\) 1.79964e6 0.172325
\(643\) −6.50526e6 −0.620494 −0.310247 0.950656i \(-0.600412\pi\)
−0.310247 + 0.950656i \(0.600412\pi\)
\(644\) 3.46515e6 0.329236
\(645\) 0 0
\(646\) 3.30302e7 3.11408
\(647\) −1.62702e7 −1.52804 −0.764018 0.645195i \(-0.776776\pi\)
−0.764018 + 0.645195i \(0.776776\pi\)
\(648\) 959422. 0.0897578
\(649\) 1.68583e6 0.157110
\(650\) 0 0
\(651\) −4.34849e6 −0.402148
\(652\) 8.87584e6 0.817693
\(653\) 1.40371e7 1.28823 0.644116 0.764927i \(-0.277225\pi\)
0.644116 + 0.764927i \(0.277225\pi\)
\(654\) 1.71482e7 1.56774
\(655\) 0 0
\(656\) 1.79253e6 0.162632
\(657\) 4.63711e6 0.419115
\(658\) −3.17775e7 −2.86124
\(659\) 1.08544e7 0.973626 0.486813 0.873506i \(-0.338159\pi\)
0.486813 + 0.873506i \(0.338159\pi\)
\(660\) 0 0
\(661\) 7.93752e6 0.706612 0.353306 0.935508i \(-0.385057\pi\)
0.353306 + 0.935508i \(0.385057\pi\)
\(662\) 1.03212e7 0.915346
\(663\) −2.89052e6 −0.255383
\(664\) 1.50494e7 1.32465
\(665\) 0 0
\(666\) −8.25981e6 −0.721580
\(667\) 1.71315e6 0.149101
\(668\) 1.64220e7 1.42392
\(669\) 2.47081e6 0.213439
\(670\) 0 0
\(671\) −2.57888e6 −0.221118
\(672\) 1.17015e7 0.999584
\(673\) −1.11210e7 −0.946471 −0.473236 0.880936i \(-0.656914\pi\)
−0.473236 + 0.880936i \(0.656914\pi\)
\(674\) 2.15827e7 1.83002
\(675\) 0 0
\(676\) −1.47365e7 −1.24030
\(677\) 6.96129e6 0.583738 0.291869 0.956458i \(-0.405723\pi\)
0.291869 + 0.956458i \(0.405723\pi\)
\(678\) 1.98581e7 1.65907
\(679\) 4.47662e6 0.372628
\(680\) 0 0
\(681\) −4.13994e6 −0.342079
\(682\) −2.73665e6 −0.225299
\(683\) 1.53699e7 1.26072 0.630360 0.776303i \(-0.282907\pi\)
0.630360 + 0.776303i \(0.282907\pi\)
\(684\) −1.15638e7 −0.945065
\(685\) 0 0
\(686\) 5.21718e6 0.423278
\(687\) −6.37824e6 −0.515596
\(688\) −201110. −0.0161980
\(689\) −3.99708e6 −0.320770
\(690\) 0 0
\(691\) 8.12461e6 0.647302 0.323651 0.946176i \(-0.395090\pi\)
0.323651 + 0.946176i \(0.395090\pi\)
\(692\) 584569. 0.0464056
\(693\) −1.87644e6 −0.148423
\(694\) 1.81356e7 1.42933
\(695\) 0 0
\(696\) −6.01894e6 −0.470974
\(697\) 9.48943e6 0.739875
\(698\) 1.47632e7 1.14695
\(699\) −8.75285e6 −0.677574
\(700\) 0 0
\(701\) 1.69865e7 1.30560 0.652799 0.757532i \(-0.273595\pi\)
0.652799 + 0.757532i \(0.273595\pi\)
\(702\) 1.68219e6 0.128835
\(703\) 3.36202e7 2.56574
\(704\) 6.45185e6 0.490628
\(705\) 0 0
\(706\) 1.87288e7 1.41416
\(707\) 1.36032e7 1.02351
\(708\) 6.05857e6 0.454242
\(709\) 9.79250e6 0.731607 0.365804 0.930692i \(-0.380794\pi\)
0.365804 + 0.930692i \(0.380794\pi\)
\(710\) 0 0
\(711\) 6.08761e6 0.451620
\(712\) 1.66525e6 0.123106
\(713\) 945343. 0.0696411
\(714\) 1.92619e7 1.41402
\(715\) 0 0
\(716\) 1.76847e7 1.28918
\(717\) −4.56031e6 −0.331281
\(718\) 7.95678e6 0.576005
\(719\) −2.91539e6 −0.210317 −0.105159 0.994455i \(-0.533535\pi\)
−0.105159 + 0.994455i \(0.533535\pi\)
\(720\) 0 0
\(721\) −2.54926e7 −1.82632
\(722\) 5.60513e7 4.00168
\(723\) −2.31112e6 −0.164428
\(724\) −9.03683e6 −0.640722
\(725\) 0 0
\(726\) −1.18091e6 −0.0831524
\(727\) 3.44448e6 0.241706 0.120853 0.992670i \(-0.461437\pi\)
0.120853 + 0.992670i \(0.461437\pi\)
\(728\) −7.20856e6 −0.504104
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.06465e6 −0.0736910
\(732\) −9.26801e6 −0.639306
\(733\) −2.28891e7 −1.57351 −0.786755 0.617265i \(-0.788241\pi\)
−0.786755 + 0.617265i \(0.788241\pi\)
\(734\) 1.76258e7 1.20756
\(735\) 0 0
\(736\) −2.54386e6 −0.173101
\(737\) 6.69922e6 0.454313
\(738\) −5.52254e6 −0.373249
\(739\) 1.84115e7 1.24016 0.620080 0.784539i \(-0.287100\pi\)
0.620080 + 0.784539i \(0.287100\pi\)
\(740\) 0 0
\(741\) −6.84708e6 −0.458099
\(742\) 2.66358e7 1.77605
\(743\) 9.95080e6 0.661281 0.330640 0.943757i \(-0.392735\pi\)
0.330640 + 0.943757i \(0.392735\pi\)
\(744\) −3.32134e6 −0.219979
\(745\) 0 0
\(746\) 4.31777e7 2.84062
\(747\) 8.33614e6 0.546593
\(748\) 7.29244e6 0.476562
\(749\) −4.27173e6 −0.278227
\(750\) 0 0
\(751\) 484044. 0.0313173 0.0156587 0.999877i \(-0.495015\pi\)
0.0156587 + 0.999877i \(0.495015\pi\)
\(752\) 4.36381e6 0.281398
\(753\) 1.42497e7 0.915837
\(754\) −1.05532e7 −0.676016
\(755\) 0 0
\(756\) −6.74358e6 −0.429127
\(757\) −2.43101e7 −1.54187 −0.770935 0.636914i \(-0.780211\pi\)
−0.770935 + 0.636914i \(0.780211\pi\)
\(758\) −1.92043e7 −1.21402
\(759\) 407931. 0.0257029
\(760\) 0 0
\(761\) −2.78776e6 −0.174499 −0.0872495 0.996186i \(-0.527808\pi\)
−0.0872495 + 0.996186i \(0.527808\pi\)
\(762\) 9.66725e6 0.603137
\(763\) −4.07041e7 −2.53120
\(764\) 3.66827e6 0.227367
\(765\) 0 0
\(766\) 8.42325e6 0.518690
\(767\) 3.58735e6 0.220184
\(768\) 5.65880e6 0.346195
\(769\) 1.43510e7 0.875115 0.437558 0.899190i \(-0.355844\pi\)
0.437558 + 0.899190i \(0.355844\pi\)
\(770\) 0 0
\(771\) −8.59518e6 −0.520738
\(772\) −4.33785e7 −2.61958
\(773\) −7.14561e6 −0.430121 −0.215060 0.976601i \(-0.568995\pi\)
−0.215060 + 0.976601i \(0.568995\pi\)
\(774\) 619594. 0.0371753
\(775\) 0 0
\(776\) 3.41921e6 0.203831
\(777\) 1.96060e7 1.16503
\(778\) −2.95798e7 −1.75205
\(779\) 2.24786e7 1.32717
\(780\) 0 0
\(781\) 8.71682e6 0.511364
\(782\) −4.18747e6 −0.244869
\(783\) −3.33400e6 −0.194339
\(784\) −4.67653e6 −0.271728
\(785\) 0 0
\(786\) 7.17934e6 0.414503
\(787\) −7.00732e6 −0.403288 −0.201644 0.979459i \(-0.564628\pi\)
−0.201644 + 0.979459i \(0.564628\pi\)
\(788\) −3.53519e7 −2.02814
\(789\) −1.17958e7 −0.674584
\(790\) 0 0
\(791\) −4.71364e7 −2.67865
\(792\) −1.43321e6 −0.0811890
\(793\) −5.48769e6 −0.309889
\(794\) −4.17685e7 −2.35124
\(795\) 0 0
\(796\) 2.93629e6 0.164254
\(797\) 3.12519e7 1.74273 0.871367 0.490633i \(-0.163234\pi\)
0.871367 + 0.490633i \(0.163234\pi\)
\(798\) 4.56277e7 2.53642
\(799\) 2.31015e7 1.28019
\(800\) 0 0
\(801\) 922413. 0.0507977
\(802\) −1.06739e7 −0.585988
\(803\) −6.92703e6 −0.379104
\(804\) 2.40757e7 1.31353
\(805\) 0 0
\(806\) −5.82343e6 −0.315748
\(807\) −1.03189e7 −0.557764
\(808\) 1.03900e7 0.559873
\(809\) −1.13855e7 −0.611620 −0.305810 0.952093i \(-0.598927\pi\)
−0.305810 + 0.952093i \(0.598927\pi\)
\(810\) 0 0
\(811\) 857075. 0.0457580 0.0228790 0.999738i \(-0.492717\pi\)
0.0228790 + 0.999738i \(0.492717\pi\)
\(812\) 4.23059e7 2.25170
\(813\) 1.09514e7 0.581091
\(814\) 1.23387e7 0.652694
\(815\) 0 0
\(816\) −2.64513e6 −0.139066
\(817\) −2.52195e6 −0.132185
\(818\) 1.47905e7 0.772859
\(819\) −3.99295e6 −0.208010
\(820\) 0 0
\(821\) 1.28434e7 0.665001 0.332501 0.943103i \(-0.392108\pi\)
0.332501 + 0.943103i \(0.392108\pi\)
\(822\) 7.41892e6 0.382967
\(823\) 2.50854e7 1.29099 0.645493 0.763767i \(-0.276652\pi\)
0.645493 + 0.763767i \(0.276652\pi\)
\(824\) −1.94710e7 −0.999012
\(825\) 0 0
\(826\) −2.39055e7 −1.21912
\(827\) −1.78039e7 −0.905217 −0.452608 0.891709i \(-0.649506\pi\)
−0.452608 + 0.891709i \(0.649506\pi\)
\(828\) 1.46603e6 0.0743133
\(829\) −9.98789e6 −0.504763 −0.252381 0.967628i \(-0.581214\pi\)
−0.252381 + 0.967628i \(0.581214\pi\)
\(830\) 0 0
\(831\) 6.95213e6 0.349233
\(832\) 1.37291e7 0.687598
\(833\) −2.47570e7 −1.23619
\(834\) −1.88352e7 −0.937679
\(835\) 0 0
\(836\) 1.72744e7 0.854844
\(837\) −1.83975e6 −0.0907705
\(838\) −1.38536e7 −0.681480
\(839\) 1.59655e7 0.783028 0.391514 0.920172i \(-0.371951\pi\)
0.391514 + 0.920172i \(0.371951\pi\)
\(840\) 0 0
\(841\) 404715. 0.0197314
\(842\) 1.39698e7 0.679062
\(843\) 1.85797e7 0.900471
\(844\) 2.55162e6 0.123299
\(845\) 0 0
\(846\) −1.34443e7 −0.645823
\(847\) 2.80308e6 0.134254
\(848\) −3.65774e6 −0.174672
\(849\) −2.00460e7 −0.954460
\(850\) 0 0
\(851\) −4.26227e6 −0.201751
\(852\) 3.13266e7 1.47848
\(853\) 2.94930e7 1.38786 0.693932 0.720041i \(-0.255877\pi\)
0.693932 + 0.720041i \(0.255877\pi\)
\(854\) 3.65690e7 1.71581
\(855\) 0 0
\(856\) −3.26271e6 −0.152193
\(857\) 2.53036e6 0.117688 0.0588438 0.998267i \(-0.481259\pi\)
0.0588438 + 0.998267i \(0.481259\pi\)
\(858\) −2.51290e6 −0.116535
\(859\) −3.52718e7 −1.63097 −0.815484 0.578780i \(-0.803529\pi\)
−0.815484 + 0.578780i \(0.803529\pi\)
\(860\) 0 0
\(861\) 1.31086e7 0.602629
\(862\) 2.38664e7 1.09400
\(863\) 2.36369e7 1.08035 0.540174 0.841553i \(-0.318358\pi\)
0.540174 + 0.841553i \(0.318358\pi\)
\(864\) 4.95066e6 0.225620
\(865\) 0 0
\(866\) −1.83101e7 −0.829651
\(867\) −1.22428e6 −0.0553138
\(868\) 2.33450e7 1.05171
\(869\) −9.09384e6 −0.408506
\(870\) 0 0
\(871\) 1.42555e7 0.636704
\(872\) −3.10895e7 −1.38459
\(873\) 1.89396e6 0.0841076
\(874\) −9.91929e6 −0.439240
\(875\) 0 0
\(876\) −2.48945e7 −1.09608
\(877\) −3.68580e7 −1.61820 −0.809102 0.587668i \(-0.800046\pi\)
−0.809102 + 0.587668i \(0.800046\pi\)
\(878\) −3.95204e7 −1.73015
\(879\) −2.42927e7 −1.06048
\(880\) 0 0
\(881\) 8.29757e6 0.360173 0.180087 0.983651i \(-0.442362\pi\)
0.180087 + 0.983651i \(0.442362\pi\)
\(882\) 1.44078e7 0.623629
\(883\) −1.91280e7 −0.825598 −0.412799 0.910822i \(-0.635449\pi\)
−0.412799 + 0.910822i \(0.635449\pi\)
\(884\) 1.55179e7 0.667885
\(885\) 0 0
\(886\) −2.67438e7 −1.14456
\(887\) 3.13587e7 1.33829 0.669144 0.743133i \(-0.266661\pi\)
0.669144 + 0.743133i \(0.266661\pi\)
\(888\) 1.49749e7 0.637282
\(889\) −2.29468e7 −0.973795
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −1.32647e7 −0.558192
\(893\) 5.47230e7 2.29637
\(894\) 1.82847e7 0.765145
\(895\) 0 0
\(896\) −4.98831e7 −2.07579
\(897\) 868052. 0.0360217
\(898\) −6.43732e6 −0.266388
\(899\) 1.15417e7 0.476288
\(900\) 0 0
\(901\) −1.93636e7 −0.794649
\(902\) 8.24973e6 0.337616
\(903\) −1.47071e6 −0.0600214
\(904\) −3.60024e7 −1.46525
\(905\) 0 0
\(906\) 3.72463e7 1.50752
\(907\) 2.21583e7 0.894371 0.447185 0.894441i \(-0.352426\pi\)
0.447185 + 0.894441i \(0.352426\pi\)
\(908\) 2.22254e7 0.894613
\(909\) 5.75523e6 0.231022
\(910\) 0 0
\(911\) 2.70378e7 1.07938 0.539691 0.841863i \(-0.318541\pi\)
0.539691 + 0.841863i \(0.318541\pi\)
\(912\) −6.26579e6 −0.249453
\(913\) −1.24528e7 −0.494412
\(914\) 2.41226e7 0.955121
\(915\) 0 0
\(916\) 3.42419e7 1.34840
\(917\) −1.70413e7 −0.669237
\(918\) 8.14930e6 0.319164
\(919\) −4.73521e7 −1.84948 −0.924741 0.380597i \(-0.875718\pi\)
−0.924741 + 0.380597i \(0.875718\pi\)
\(920\) 0 0
\(921\) −9.97393e6 −0.387452
\(922\) 9.81111e6 0.380094
\(923\) 1.85489e7 0.716659
\(924\) 1.00737e7 0.388160
\(925\) 0 0
\(926\) 7.68923e7 2.94683
\(927\) −1.07854e7 −0.412226
\(928\) −3.10580e7 −1.18387
\(929\) −1.66983e7 −0.634793 −0.317397 0.948293i \(-0.602809\pi\)
−0.317397 + 0.948293i \(0.602809\pi\)
\(930\) 0 0
\(931\) −5.86446e7 −2.21745
\(932\) 4.69900e7 1.77201
\(933\) 1.84677e7 0.694558
\(934\) 5.86854e7 2.20122
\(935\) 0 0
\(936\) −3.04978e6 −0.113784
\(937\) −1.68775e7 −0.628000 −0.314000 0.949423i \(-0.601669\pi\)
−0.314000 + 0.949423i \(0.601669\pi\)
\(938\) −9.49963e7 −3.52533
\(939\) 2.46069e7 0.910737
\(940\) 0 0
\(941\) 1.00920e7 0.371538 0.185769 0.982593i \(-0.440522\pi\)
0.185769 + 0.982593i \(0.440522\pi\)
\(942\) −762490. −0.0279967
\(943\) −2.84977e6 −0.104359
\(944\) 3.28280e6 0.119899
\(945\) 0 0
\(946\) −925566. −0.0336263
\(947\) 4.25599e7 1.54215 0.771073 0.636747i \(-0.219720\pi\)
0.771073 + 0.636747i \(0.219720\pi\)
\(948\) −3.26816e7 −1.18109
\(949\) −1.47403e7 −0.531301
\(950\) 0 0
\(951\) −2.20814e7 −0.791727
\(952\) −3.49215e7 −1.24882
\(953\) −1.84695e7 −0.658754 −0.329377 0.944199i \(-0.606839\pi\)
−0.329377 + 0.944199i \(0.606839\pi\)
\(954\) 1.12690e7 0.400881
\(955\) 0 0
\(956\) 2.44822e7 0.866374
\(957\) 4.98042e6 0.175787
\(958\) −1.81253e7 −0.638074
\(959\) −1.76100e7 −0.618320
\(960\) 0 0
\(961\) −2.22603e7 −0.777539
\(962\) 2.62561e7 0.914728
\(963\) −1.80728e6 −0.0627998
\(964\) 1.24073e7 0.430017
\(965\) 0 0
\(966\) −5.78454e6 −0.199446
\(967\) −3.43843e7 −1.18248 −0.591241 0.806495i \(-0.701362\pi\)
−0.591241 + 0.806495i \(0.701362\pi\)
\(968\) 2.14097e6 0.0734382
\(969\) −3.31704e7 −1.13486
\(970\) 0 0
\(971\) −2.08153e7 −0.708492 −0.354246 0.935152i \(-0.615262\pi\)
−0.354246 + 0.935152i \(0.615262\pi\)
\(972\) −2.85306e6 −0.0968602
\(973\) 4.47083e7 1.51393
\(974\) 4.04463e7 1.36610
\(975\) 0 0
\(976\) −5.02181e6 −0.168747
\(977\) 1.72220e7 0.577228 0.288614 0.957446i \(-0.406806\pi\)
0.288614 + 0.957446i \(0.406806\pi\)
\(978\) −1.48169e7 −0.495347
\(979\) −1.37793e6 −0.0459482
\(980\) 0 0
\(981\) −1.72210e7 −0.571329
\(982\) −3.96868e7 −1.31331
\(983\) −3.85176e7 −1.27138 −0.635690 0.771945i \(-0.719284\pi\)
−0.635690 + 0.771945i \(0.719284\pi\)
\(984\) 1.00123e7 0.329644
\(985\) 0 0
\(986\) −5.11247e7 −1.67470
\(987\) 3.19123e7 1.04271
\(988\) 3.67588e7 1.19803
\(989\) 319726. 0.0103941
\(990\) 0 0
\(991\) −1.18485e7 −0.383249 −0.191624 0.981468i \(-0.561376\pi\)
−0.191624 + 0.981468i \(0.561376\pi\)
\(992\) −1.71382e7 −0.552952
\(993\) −1.03650e7 −0.333577
\(994\) −1.23606e8 −3.96803
\(995\) 0 0
\(996\) −4.47529e7 −1.42946
\(997\) 2.06293e7 0.657273 0.328636 0.944457i \(-0.393411\pi\)
0.328636 + 0.944457i \(0.393411\pi\)
\(998\) −1.07584e7 −0.341919
\(999\) 8.29487e6 0.262964
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.1.7 yes 8
5.4 even 2 825.6.a.p.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.2 8 5.4 even 2
825.6.a.q.1.7 yes 8 1.1 even 1 trivial