Properties

Label 825.6.a.v.1.11
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(8.39855\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.39855 q^{2} -9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} -150.852 q^{7} -68.5217 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+7.39855 q^{2} -9.00000 q^{3} +22.7385 q^{4} -66.5869 q^{6} -150.852 q^{7} -68.5217 q^{8} +81.0000 q^{9} +121.000 q^{11} -204.646 q^{12} +868.426 q^{13} -1116.08 q^{14} -1234.59 q^{16} +2317.87 q^{17} +599.282 q^{18} -2655.24 q^{19} +1357.67 q^{21} +895.224 q^{22} +2537.58 q^{23} +616.695 q^{24} +6425.09 q^{26} -729.000 q^{27} -3430.14 q^{28} -819.917 q^{29} +7303.36 q^{31} -6941.50 q^{32} -1089.00 q^{33} +17148.9 q^{34} +1841.82 q^{36} -2993.84 q^{37} -19644.9 q^{38} -7815.84 q^{39} -4567.72 q^{41} +10044.8 q^{42} +1022.10 q^{43} +2751.36 q^{44} +18774.4 q^{46} -24499.3 q^{47} +11111.3 q^{48} +5949.25 q^{49} -20860.8 q^{51} +19746.7 q^{52} -13318.9 q^{53} -5393.54 q^{54} +10336.6 q^{56} +23897.2 q^{57} -6066.20 q^{58} -29760.6 q^{59} -17257.1 q^{61} +54034.2 q^{62} -12219.0 q^{63} -11850.0 q^{64} -8057.02 q^{66} -18645.0 q^{67} +52704.9 q^{68} -22838.2 q^{69} +47722.0 q^{71} -5550.26 q^{72} -19156.3 q^{73} -22150.1 q^{74} -60376.2 q^{76} -18253.1 q^{77} -57825.8 q^{78} -3713.15 q^{79} +6561.00 q^{81} -33794.5 q^{82} -51919.0 q^{83} +30871.3 q^{84} +7562.07 q^{86} +7379.26 q^{87} -8291.13 q^{88} -52966.5 q^{89} -131004. q^{91} +57700.8 q^{92} -65730.2 q^{93} -181259. q^{94} +62473.5 q^{96} -114526. q^{97} +44015.8 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.39855 1.30789 0.653945 0.756542i \(-0.273113\pi\)
0.653945 + 0.756542i \(0.273113\pi\)
\(3\) −9.00000 −0.577350
\(4\) 22.7385 0.710578
\(5\) 0 0
\(6\) −66.5869 −0.755111
\(7\) −150.852 −1.16360 −0.581802 0.813330i \(-0.697652\pi\)
−0.581802 + 0.813330i \(0.697652\pi\)
\(8\) −68.5217 −0.378533
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −204.646 −0.410252
\(13\) 868.426 1.42520 0.712598 0.701573i \(-0.247518\pi\)
0.712598 + 0.701573i \(0.247518\pi\)
\(14\) −1116.08 −1.52187
\(15\) 0 0
\(16\) −1234.59 −1.20566
\(17\) 2317.87 1.94521 0.972606 0.232458i \(-0.0746769\pi\)
0.972606 + 0.232458i \(0.0746769\pi\)
\(18\) 599.282 0.435964
\(19\) −2655.24 −1.68741 −0.843705 0.536808i \(-0.819630\pi\)
−0.843705 + 0.536808i \(0.819630\pi\)
\(20\) 0 0
\(21\) 1357.67 0.671807
\(22\) 895.224 0.394344
\(23\) 2537.58 1.00023 0.500116 0.865959i \(-0.333291\pi\)
0.500116 + 0.865959i \(0.333291\pi\)
\(24\) 616.695 0.218546
\(25\) 0 0
\(26\) 6425.09 1.86400
\(27\) −729.000 −0.192450
\(28\) −3430.14 −0.826831
\(29\) −819.917 −0.181040 −0.0905201 0.995895i \(-0.528853\pi\)
−0.0905201 + 0.995895i \(0.528853\pi\)
\(30\) 0 0
\(31\) 7303.36 1.36495 0.682477 0.730907i \(-0.260903\pi\)
0.682477 + 0.730907i \(0.260903\pi\)
\(32\) −6941.50 −1.19833
\(33\) −1089.00 −0.174078
\(34\) 17148.9 2.54413
\(35\) 0 0
\(36\) 1841.82 0.236859
\(37\) −2993.84 −0.359521 −0.179761 0.983710i \(-0.557532\pi\)
−0.179761 + 0.983710i \(0.557532\pi\)
\(38\) −19644.9 −2.20695
\(39\) −7815.84 −0.822837
\(40\) 0 0
\(41\) −4567.72 −0.424365 −0.212182 0.977230i \(-0.568057\pi\)
−0.212182 + 0.977230i \(0.568057\pi\)
\(42\) 10044.8 0.878650
\(43\) 1022.10 0.0842992 0.0421496 0.999111i \(-0.486579\pi\)
0.0421496 + 0.999111i \(0.486579\pi\)
\(44\) 2751.36 0.214247
\(45\) 0 0
\(46\) 18774.4 1.30819
\(47\) −24499.3 −1.61774 −0.808871 0.587987i \(-0.799921\pi\)
−0.808871 + 0.587987i \(0.799921\pi\)
\(48\) 11111.3 0.696086
\(49\) 5949.25 0.353974
\(50\) 0 0
\(51\) −20860.8 −1.12307
\(52\) 19746.7 1.01271
\(53\) −13318.9 −0.651297 −0.325649 0.945491i \(-0.605583\pi\)
−0.325649 + 0.945491i \(0.605583\pi\)
\(54\) −5393.54 −0.251704
\(55\) 0 0
\(56\) 10336.6 0.440462
\(57\) 23897.2 0.974226
\(58\) −6066.20 −0.236781
\(59\) −29760.6 −1.11304 −0.556522 0.830833i \(-0.687864\pi\)
−0.556522 + 0.830833i \(0.687864\pi\)
\(60\) 0 0
\(61\) −17257.1 −0.593805 −0.296903 0.954908i \(-0.595954\pi\)
−0.296903 + 0.954908i \(0.595954\pi\)
\(62\) 54034.2 1.78521
\(63\) −12219.0 −0.387868
\(64\) −11850.0 −0.361634
\(65\) 0 0
\(66\) −8057.02 −0.227675
\(67\) −18645.0 −0.507429 −0.253715 0.967279i \(-0.581652\pi\)
−0.253715 + 0.967279i \(0.581652\pi\)
\(68\) 52704.9 1.38223
\(69\) −22838.2 −0.577484
\(70\) 0 0
\(71\) 47722.0 1.12350 0.561750 0.827307i \(-0.310128\pi\)
0.561750 + 0.827307i \(0.310128\pi\)
\(72\) −5550.26 −0.126178
\(73\) −19156.3 −0.420730 −0.210365 0.977623i \(-0.567465\pi\)
−0.210365 + 0.977623i \(0.567465\pi\)
\(74\) −22150.1 −0.470214
\(75\) 0 0
\(76\) −60376.2 −1.19904
\(77\) −18253.1 −0.350840
\(78\) −57825.8 −1.07618
\(79\) −3713.15 −0.0669383 −0.0334692 0.999440i \(-0.510656\pi\)
−0.0334692 + 0.999440i \(0.510656\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −33794.5 −0.555023
\(83\) −51919.0 −0.827239 −0.413619 0.910450i \(-0.635736\pi\)
−0.413619 + 0.910450i \(0.635736\pi\)
\(84\) 30871.3 0.477371
\(85\) 0 0
\(86\) 7562.07 0.110254
\(87\) 7379.26 0.104524
\(88\) −8291.13 −0.114132
\(89\) −52966.5 −0.708803 −0.354402 0.935093i \(-0.615315\pi\)
−0.354402 + 0.935093i \(0.615315\pi\)
\(90\) 0 0
\(91\) −131004. −1.65836
\(92\) 57700.8 0.710742
\(93\) −65730.2 −0.788057
\(94\) −181259. −2.11583
\(95\) 0 0
\(96\) 62473.5 0.691859
\(97\) −114526. −1.23588 −0.617940 0.786225i \(-0.712032\pi\)
−0.617940 + 0.786225i \(0.712032\pi\)
\(98\) 44015.8 0.462960
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 154646. 1.50846 0.754232 0.656608i \(-0.228009\pi\)
0.754232 + 0.656608i \(0.228009\pi\)
\(102\) −154340. −1.46885
\(103\) 117955. 1.09552 0.547762 0.836634i \(-0.315480\pi\)
0.547762 + 0.836634i \(0.315480\pi\)
\(104\) −59506.1 −0.539483
\(105\) 0 0
\(106\) −98540.6 −0.851826
\(107\) −233086. −1.96815 −0.984073 0.177763i \(-0.943114\pi\)
−0.984073 + 0.177763i \(0.943114\pi\)
\(108\) −16576.4 −0.136751
\(109\) −18774.3 −0.151355 −0.0756775 0.997132i \(-0.524112\pi\)
−0.0756775 + 0.997132i \(0.524112\pi\)
\(110\) 0 0
\(111\) 26944.6 0.207570
\(112\) 186240. 1.40291
\(113\) −165056. −1.21600 −0.608001 0.793936i \(-0.708029\pi\)
−0.608001 + 0.793936i \(0.708029\pi\)
\(114\) 176804. 1.27418
\(115\) 0 0
\(116\) −18643.7 −0.128643
\(117\) 70342.5 0.475065
\(118\) −220186. −1.45574
\(119\) −349655. −2.26346
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −127678. −0.776632
\(123\) 41109.4 0.245007
\(124\) 166067. 0.969907
\(125\) 0 0
\(126\) −90402.8 −0.507289
\(127\) −203866. −1.12159 −0.560797 0.827953i \(-0.689505\pi\)
−0.560797 + 0.827953i \(0.689505\pi\)
\(128\) 134455. 0.725357
\(129\) −9198.92 −0.0486701
\(130\) 0 0
\(131\) 82311.6 0.419067 0.209533 0.977802i \(-0.432806\pi\)
0.209533 + 0.977802i \(0.432806\pi\)
\(132\) −24762.2 −0.123696
\(133\) 400548. 1.96348
\(134\) −137946. −0.663662
\(135\) 0 0
\(136\) −158825. −0.736326
\(137\) −202042. −0.919687 −0.459844 0.888000i \(-0.652095\pi\)
−0.459844 + 0.888000i \(0.652095\pi\)
\(138\) −168970. −0.755286
\(139\) −200312. −0.879367 −0.439683 0.898153i \(-0.644909\pi\)
−0.439683 + 0.898153i \(0.644909\pi\)
\(140\) 0 0
\(141\) 220494. 0.934003
\(142\) 353074. 1.46942
\(143\) 105080. 0.429713
\(144\) −100002. −0.401886
\(145\) 0 0
\(146\) −141728. −0.550269
\(147\) −53543.2 −0.204367
\(148\) −68075.4 −0.255468
\(149\) 80425.4 0.296775 0.148388 0.988929i \(-0.452592\pi\)
0.148388 + 0.988929i \(0.452592\pi\)
\(150\) 0 0
\(151\) 395216. 1.41056 0.705280 0.708928i \(-0.250821\pi\)
0.705280 + 0.708928i \(0.250821\pi\)
\(152\) 181942. 0.638739
\(153\) 187748. 0.648404
\(154\) −135046. −0.458860
\(155\) 0 0
\(156\) −177720. −0.584690
\(157\) 510056. 1.65146 0.825732 0.564063i \(-0.190763\pi\)
0.825732 + 0.564063i \(0.190763\pi\)
\(158\) −27471.9 −0.0875480
\(159\) 119870. 0.376027
\(160\) 0 0
\(161\) −382799. −1.16387
\(162\) 48541.9 0.145321
\(163\) −307324. −0.905998 −0.452999 0.891511i \(-0.649646\pi\)
−0.452999 + 0.891511i \(0.649646\pi\)
\(164\) −103863. −0.301544
\(165\) 0 0
\(166\) −384125. −1.08194
\(167\) −214345. −0.594733 −0.297367 0.954763i \(-0.596108\pi\)
−0.297367 + 0.954763i \(0.596108\pi\)
\(168\) −93029.6 −0.254301
\(169\) 382871. 1.03118
\(170\) 0 0
\(171\) −215075. −0.562470
\(172\) 23241.1 0.0599011
\(173\) −366833. −0.931865 −0.465933 0.884820i \(-0.654281\pi\)
−0.465933 + 0.884820i \(0.654281\pi\)
\(174\) 54595.8 0.136705
\(175\) 0 0
\(176\) −149386. −0.363519
\(177\) 267846. 0.642616
\(178\) −391875. −0.927037
\(179\) 678162. 1.58198 0.790990 0.611829i \(-0.209566\pi\)
0.790990 + 0.611829i \(0.209566\pi\)
\(180\) 0 0
\(181\) −270856. −0.614529 −0.307265 0.951624i \(-0.599414\pi\)
−0.307265 + 0.951624i \(0.599414\pi\)
\(182\) −969236. −2.16896
\(183\) 155314. 0.342834
\(184\) −173879. −0.378620
\(185\) 0 0
\(186\) −486308. −1.03069
\(187\) 280462. 0.586504
\(188\) −557077. −1.14953
\(189\) 109971. 0.223936
\(190\) 0 0
\(191\) 81645.4 0.161938 0.0809689 0.996717i \(-0.474199\pi\)
0.0809689 + 0.996717i \(0.474199\pi\)
\(192\) 106650. 0.208789
\(193\) 170728. 0.329922 0.164961 0.986300i \(-0.447250\pi\)
0.164961 + 0.986300i \(0.447250\pi\)
\(194\) −847329. −1.61640
\(195\) 0 0
\(196\) 135277. 0.251526
\(197\) 173263. 0.318082 0.159041 0.987272i \(-0.449160\pi\)
0.159041 + 0.987272i \(0.449160\pi\)
\(198\) 72513.2 0.131448
\(199\) 42534.0 0.0761384 0.0380692 0.999275i \(-0.487879\pi\)
0.0380692 + 0.999275i \(0.487879\pi\)
\(200\) 0 0
\(201\) 167805. 0.292964
\(202\) 1.14416e6 1.97291
\(203\) 123686. 0.210659
\(204\) −474344. −0.798028
\(205\) 0 0
\(206\) 872693. 1.43283
\(207\) 205544. 0.333410
\(208\) −1.07215e6 −1.71830
\(209\) −321284. −0.508773
\(210\) 0 0
\(211\) −625694. −0.967511 −0.483755 0.875203i \(-0.660728\pi\)
−0.483755 + 0.875203i \(0.660728\pi\)
\(212\) −302852. −0.462797
\(213\) −429498. −0.648653
\(214\) −1.72450e6 −2.57412
\(215\) 0 0
\(216\) 49952.3 0.0728486
\(217\) −1.10172e6 −1.58827
\(218\) −138902. −0.197956
\(219\) 172406. 0.242909
\(220\) 0 0
\(221\) 2.01290e6 2.77231
\(222\) 199351. 0.271478
\(223\) 42469.9 0.0571899 0.0285950 0.999591i \(-0.490897\pi\)
0.0285950 + 0.999591i \(0.490897\pi\)
\(224\) 1.04714e6 1.39439
\(225\) 0 0
\(226\) −1.22117e6 −1.59040
\(227\) −21815.5 −0.0280996 −0.0140498 0.999901i \(-0.504472\pi\)
−0.0140498 + 0.999901i \(0.504472\pi\)
\(228\) 543386. 0.692263
\(229\) 58861.1 0.0741720 0.0370860 0.999312i \(-0.488192\pi\)
0.0370860 + 0.999312i \(0.488192\pi\)
\(230\) 0 0
\(231\) 164278. 0.202557
\(232\) 56182.1 0.0685296
\(233\) −418005. −0.504419 −0.252210 0.967673i \(-0.581157\pi\)
−0.252210 + 0.967673i \(0.581157\pi\)
\(234\) 520432. 0.621333
\(235\) 0 0
\(236\) −676712. −0.790904
\(237\) 33418.3 0.0386468
\(238\) −2.58694e6 −2.96035
\(239\) −1.29252e6 −1.46367 −0.731834 0.681483i \(-0.761335\pi\)
−0.731834 + 0.681483i \(0.761335\pi\)
\(240\) 0 0
\(241\) 496148. 0.550261 0.275131 0.961407i \(-0.411279\pi\)
0.275131 + 0.961407i \(0.411279\pi\)
\(242\) 108322. 0.118899
\(243\) −59049.0 −0.0641500
\(244\) −392401. −0.421945
\(245\) 0 0
\(246\) 304150. 0.320443
\(247\) −2.30588e6 −2.40489
\(248\) −500439. −0.516680
\(249\) 467271. 0.477606
\(250\) 0 0
\(251\) −1.58588e6 −1.58886 −0.794431 0.607354i \(-0.792231\pi\)
−0.794431 + 0.607354i \(0.792231\pi\)
\(252\) −277841. −0.275610
\(253\) 307047. 0.301581
\(254\) −1.50831e6 −1.46692
\(255\) 0 0
\(256\) 1.37397e6 1.31032
\(257\) −1.41881e6 −1.33995 −0.669977 0.742381i \(-0.733696\pi\)
−0.669977 + 0.742381i \(0.733696\pi\)
\(258\) −68058.6 −0.0636552
\(259\) 451626. 0.418340
\(260\) 0 0
\(261\) −66413.3 −0.0603467
\(262\) 608987. 0.548093
\(263\) −1.62046e6 −1.44461 −0.722304 0.691575i \(-0.756917\pi\)
−0.722304 + 0.691575i \(0.756917\pi\)
\(264\) 74620.1 0.0658941
\(265\) 0 0
\(266\) 2.96347e6 2.56801
\(267\) 476698. 0.409228
\(268\) −423959. −0.360568
\(269\) 1.25821e6 1.06017 0.530083 0.847946i \(-0.322161\pi\)
0.530083 + 0.847946i \(0.322161\pi\)
\(270\) 0 0
\(271\) −2.33387e6 −1.93043 −0.965214 0.261462i \(-0.915796\pi\)
−0.965214 + 0.261462i \(0.915796\pi\)
\(272\) −2.86163e6 −2.34526
\(273\) 1.17903e6 0.957457
\(274\) −1.49482e6 −1.20285
\(275\) 0 0
\(276\) −519307. −0.410347
\(277\) 952978. 0.746248 0.373124 0.927781i \(-0.378287\pi\)
0.373124 + 0.927781i \(0.378287\pi\)
\(278\) −1.48202e6 −1.15012
\(279\) 591572. 0.454985
\(280\) 0 0
\(281\) −194035. −0.146593 −0.0732966 0.997310i \(-0.523352\pi\)
−0.0732966 + 0.997310i \(0.523352\pi\)
\(282\) 1.63133e6 1.22157
\(283\) 589829. 0.437784 0.218892 0.975749i \(-0.429756\pi\)
0.218892 + 0.975749i \(0.429756\pi\)
\(284\) 1.08513e6 0.798334
\(285\) 0 0
\(286\) 777436. 0.562017
\(287\) 689048. 0.493793
\(288\) −562261. −0.399445
\(289\) 3.95267e6 2.78385
\(290\) 0 0
\(291\) 1.03074e6 0.713535
\(292\) −435584. −0.298961
\(293\) 2.18674e6 1.48809 0.744045 0.668130i \(-0.232905\pi\)
0.744045 + 0.668130i \(0.232905\pi\)
\(294\) −396142. −0.267290
\(295\) 0 0
\(296\) 205143. 0.136090
\(297\) −88209.0 −0.0580259
\(298\) 595031. 0.388150
\(299\) 2.20370e6 1.42553
\(300\) 0 0
\(301\) −154186. −0.0980908
\(302\) 2.92402e6 1.84486
\(303\) −1.39181e6 −0.870913
\(304\) 3.27814e6 2.03444
\(305\) 0 0
\(306\) 1.38906e6 0.848042
\(307\) −2.25874e6 −1.36780 −0.683898 0.729578i \(-0.739717\pi\)
−0.683898 + 0.729578i \(0.739717\pi\)
\(308\) −415047. −0.249299
\(309\) −1.06159e6 −0.632501
\(310\) 0 0
\(311\) −3.00436e6 −1.76137 −0.880687 0.473699i \(-0.842918\pi\)
−0.880687 + 0.473699i \(0.842918\pi\)
\(312\) 535554. 0.311471
\(313\) −2.76965e6 −1.59795 −0.798976 0.601363i \(-0.794624\pi\)
−0.798976 + 0.601363i \(0.794624\pi\)
\(314\) 3.77367e6 2.15993
\(315\) 0 0
\(316\) −84431.4 −0.0475649
\(317\) −2.00320e6 −1.11963 −0.559817 0.828616i \(-0.689129\pi\)
−0.559817 + 0.828616i \(0.689129\pi\)
\(318\) 886866. 0.491802
\(319\) −99210.0 −0.0545857
\(320\) 0 0
\(321\) 2.09778e6 1.13631
\(322\) −2.83215e6 −1.52222
\(323\) −6.15451e6 −3.28237
\(324\) 149187. 0.0789531
\(325\) 0 0
\(326\) −2.27375e6 −1.18495
\(327\) 168968. 0.0873849
\(328\) 312988. 0.160636
\(329\) 3.69576e6 1.88241
\(330\) 0 0
\(331\) 1.09643e6 0.550059 0.275029 0.961436i \(-0.411312\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(332\) −1.18056e6 −0.587817
\(333\) −242501. −0.119840
\(334\) −1.58584e6 −0.777846
\(335\) 0 0
\(336\) −1.67616e6 −0.809969
\(337\) −594378. −0.285094 −0.142547 0.989788i \(-0.545529\pi\)
−0.142547 + 0.989788i \(0.545529\pi\)
\(338\) 2.83269e6 1.34868
\(339\) 1.48550e6 0.702060
\(340\) 0 0
\(341\) 883706. 0.411549
\(342\) −1.59124e6 −0.735649
\(343\) 1.63791e6 0.751718
\(344\) −70036.2 −0.0319100
\(345\) 0 0
\(346\) −2.71403e6 −1.21878
\(347\) 822560. 0.366728 0.183364 0.983045i \(-0.441301\pi\)
0.183364 + 0.983045i \(0.441301\pi\)
\(348\) 167793. 0.0742722
\(349\) 956928. 0.420548 0.210274 0.977642i \(-0.432564\pi\)
0.210274 + 0.977642i \(0.432564\pi\)
\(350\) 0 0
\(351\) −633083. −0.274279
\(352\) −839921. −0.361312
\(353\) −218804. −0.0934583 −0.0467291 0.998908i \(-0.514880\pi\)
−0.0467291 + 0.998908i \(0.514880\pi\)
\(354\) 1.98167e6 0.840472
\(355\) 0 0
\(356\) −1.20438e6 −0.503660
\(357\) 3.14689e6 1.30681
\(358\) 5.01741e6 2.06906
\(359\) −2.72858e6 −1.11738 −0.558689 0.829378i \(-0.688695\pi\)
−0.558689 + 0.829378i \(0.688695\pi\)
\(360\) 0 0
\(361\) 4.57422e6 1.84735
\(362\) −2.00394e6 −0.803737
\(363\) −131769. −0.0524864
\(364\) −2.97882e6 −1.17840
\(365\) 0 0
\(366\) 1.14910e6 0.448389
\(367\) 1.94374e6 0.753307 0.376654 0.926354i \(-0.377075\pi\)
0.376654 + 0.926354i \(0.377075\pi\)
\(368\) −3.13288e6 −1.20594
\(369\) −369985. −0.141455
\(370\) 0 0
\(371\) 2.00918e6 0.757852
\(372\) −1.49461e6 −0.559976
\(373\) 4.88061e6 1.81636 0.908180 0.418580i \(-0.137472\pi\)
0.908180 + 0.418580i \(0.137472\pi\)
\(374\) 2.07501e6 0.767083
\(375\) 0 0
\(376\) 1.67873e6 0.612368
\(377\) −712038. −0.258018
\(378\) 813625. 0.292883
\(379\) −4.08830e6 −1.46199 −0.730995 0.682383i \(-0.760944\pi\)
−0.730995 + 0.682383i \(0.760944\pi\)
\(380\) 0 0
\(381\) 1.83479e6 0.647553
\(382\) 604057. 0.211797
\(383\) −3.04558e6 −1.06090 −0.530448 0.847717i \(-0.677976\pi\)
−0.530448 + 0.847717i \(0.677976\pi\)
\(384\) −1.21009e6 −0.418785
\(385\) 0 0
\(386\) 1.26314e6 0.431502
\(387\) 82790.3 0.0280997
\(388\) −2.60416e6 −0.878189
\(389\) 2.13646e6 0.715847 0.357923 0.933751i \(-0.383485\pi\)
0.357923 + 0.933751i \(0.383485\pi\)
\(390\) 0 0
\(391\) 5.88179e6 1.94566
\(392\) −407653. −0.133991
\(393\) −740805. −0.241948
\(394\) 1.28189e6 0.416017
\(395\) 0 0
\(396\) 222860. 0.0714158
\(397\) 2.27058e6 0.723038 0.361519 0.932365i \(-0.382258\pi\)
0.361519 + 0.932365i \(0.382258\pi\)
\(398\) 314690. 0.0995807
\(399\) −3.60493e6 −1.13361
\(400\) 0 0
\(401\) 981140. 0.304698 0.152349 0.988327i \(-0.451316\pi\)
0.152349 + 0.988327i \(0.451316\pi\)
\(402\) 1.24151e6 0.383165
\(403\) 6.34243e6 1.94533
\(404\) 3.51642e6 1.07188
\(405\) 0 0
\(406\) 915096. 0.275519
\(407\) −362255. −0.108400
\(408\) 1.42942e6 0.425118
\(409\) −1.98079e6 −0.585505 −0.292752 0.956188i \(-0.594571\pi\)
−0.292752 + 0.956188i \(0.594571\pi\)
\(410\) 0 0
\(411\) 1.81838e6 0.530982
\(412\) 2.68211e6 0.778456
\(413\) 4.48945e6 1.29514
\(414\) 1.52073e6 0.436064
\(415\) 0 0
\(416\) −6.02818e6 −1.70786
\(417\) 1.80281e6 0.507703
\(418\) −2.37704e6 −0.665419
\(419\) 153079. 0.0425972 0.0212986 0.999773i \(-0.493220\pi\)
0.0212986 + 0.999773i \(0.493220\pi\)
\(420\) 0 0
\(421\) 848256. 0.233250 0.116625 0.993176i \(-0.462792\pi\)
0.116625 + 0.993176i \(0.462792\pi\)
\(422\) −4.62923e6 −1.26540
\(423\) −1.98444e6 −0.539247
\(424\) 912635. 0.246537
\(425\) 0 0
\(426\) −3.17766e6 −0.848367
\(427\) 2.60327e6 0.690954
\(428\) −5.30003e6 −1.39852
\(429\) −945716. −0.248095
\(430\) 0 0
\(431\) −492966. −0.127827 −0.0639137 0.997955i \(-0.520358\pi\)
−0.0639137 + 0.997955i \(0.520358\pi\)
\(432\) 900018. 0.232029
\(433\) −5.12774e6 −1.31434 −0.657168 0.753744i \(-0.728246\pi\)
−0.657168 + 0.753744i \(0.728246\pi\)
\(434\) −8.15116e6 −2.07728
\(435\) 0 0
\(436\) −426899. −0.107550
\(437\) −6.73790e6 −1.68780
\(438\) 1.27556e6 0.317698
\(439\) −3.98100e6 −0.985895 −0.492947 0.870059i \(-0.664081\pi\)
−0.492947 + 0.870059i \(0.664081\pi\)
\(440\) 0 0
\(441\) 481889. 0.117991
\(442\) 1.48925e7 3.62588
\(443\) 3.09112e6 0.748353 0.374177 0.927357i \(-0.377925\pi\)
0.374177 + 0.927357i \(0.377925\pi\)
\(444\) 612679. 0.147494
\(445\) 0 0
\(446\) 314216. 0.0747981
\(447\) −723829. −0.171343
\(448\) 1.78760e6 0.420799
\(449\) −1.51168e6 −0.353870 −0.176935 0.984223i \(-0.556618\pi\)
−0.176935 + 0.984223i \(0.556618\pi\)
\(450\) 0 0
\(451\) −552694. −0.127951
\(452\) −3.75312e6 −0.864065
\(453\) −3.55694e6 −0.814388
\(454\) −161403. −0.0367512
\(455\) 0 0
\(456\) −1.63748e6 −0.368776
\(457\) 5.58801e6 1.25160 0.625802 0.779982i \(-0.284772\pi\)
0.625802 + 0.779982i \(0.284772\pi\)
\(458\) 435487. 0.0970088
\(459\) −1.68973e6 −0.374356
\(460\) 0 0
\(461\) −819486. −0.179593 −0.0897964 0.995960i \(-0.528622\pi\)
−0.0897964 + 0.995960i \(0.528622\pi\)
\(462\) 1.21542e6 0.264923
\(463\) 987740. 0.214136 0.107068 0.994252i \(-0.465854\pi\)
0.107068 + 0.994252i \(0.465854\pi\)
\(464\) 1.01226e6 0.218272
\(465\) 0 0
\(466\) −3.09263e6 −0.659725
\(467\) 3.26318e6 0.692387 0.346193 0.938163i \(-0.387474\pi\)
0.346193 + 0.938163i \(0.387474\pi\)
\(468\) 1.59948e6 0.337571
\(469\) 2.81263e6 0.590447
\(470\) 0 0
\(471\) −4.59051e6 −0.953473
\(472\) 2.03925e6 0.421323
\(473\) 123674. 0.0254172
\(474\) 247247. 0.0505459
\(475\) 0 0
\(476\) −7.95063e6 −1.60836
\(477\) −1.07883e6 −0.217099
\(478\) −9.56277e6 −1.91432
\(479\) 5.61817e6 1.11881 0.559405 0.828895i \(-0.311030\pi\)
0.559405 + 0.828895i \(0.311030\pi\)
\(480\) 0 0
\(481\) −2.59993e6 −0.512388
\(482\) 3.67078e6 0.719681
\(483\) 3.44519e6 0.671962
\(484\) 332914. 0.0645980
\(485\) 0 0
\(486\) −436877. −0.0839012
\(487\) 4.31761e6 0.824938 0.412469 0.910972i \(-0.364666\pi\)
0.412469 + 0.910972i \(0.364666\pi\)
\(488\) 1.18249e6 0.224775
\(489\) 2.76591e6 0.523078
\(490\) 0 0
\(491\) 8.77622e6 1.64287 0.821436 0.570301i \(-0.193173\pi\)
0.821436 + 0.570301i \(0.193173\pi\)
\(492\) 934767. 0.174097
\(493\) −1.90046e6 −0.352162
\(494\) −1.70602e7 −3.14533
\(495\) 0 0
\(496\) −9.01667e6 −1.64567
\(497\) −7.19895e6 −1.30731
\(498\) 3.45712e6 0.624657
\(499\) −9.08110e6 −1.63263 −0.816313 0.577609i \(-0.803986\pi\)
−0.816313 + 0.577609i \(0.803986\pi\)
\(500\) 0 0
\(501\) 1.92911e6 0.343370
\(502\) −1.17332e7 −2.07806
\(503\) −4.35992e6 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(504\) 837266. 0.146821
\(505\) 0 0
\(506\) 2.27170e6 0.394435
\(507\) −3.44584e6 −0.595354
\(508\) −4.63561e6 −0.796980
\(509\) 5.30215e6 0.907105 0.453552 0.891230i \(-0.350156\pi\)
0.453552 + 0.891230i \(0.350156\pi\)
\(510\) 0 0
\(511\) 2.88975e6 0.489563
\(512\) 5.86284e6 0.988400
\(513\) 1.93567e6 0.324742
\(514\) −1.04971e7 −1.75251
\(515\) 0 0
\(516\) −209170. −0.0345839
\(517\) −2.96442e6 −0.487767
\(518\) 3.34138e6 0.547143
\(519\) 3.30150e6 0.538013
\(520\) 0 0
\(521\) 1.80284e6 0.290981 0.145490 0.989360i \(-0.453524\pi\)
0.145490 + 0.989360i \(0.453524\pi\)
\(522\) −491362. −0.0789269
\(523\) −1.48512e6 −0.237414 −0.118707 0.992929i \(-0.537875\pi\)
−0.118707 + 0.992929i \(0.537875\pi\)
\(524\) 1.87164e6 0.297779
\(525\) 0 0
\(526\) −1.19891e7 −1.88939
\(527\) 1.69282e7 2.65513
\(528\) 1.34447e6 0.209878
\(529\) 2976.47 0.000462448 0
\(530\) 0 0
\(531\) −2.41061e6 −0.371015
\(532\) 9.10786e6 1.39520
\(533\) −3.96672e6 −0.604803
\(534\) 3.52687e6 0.535225
\(535\) 0 0
\(536\) 1.27759e6 0.192078
\(537\) −6.10346e6 −0.913356
\(538\) 9.30896e6 1.38658
\(539\) 719859. 0.106727
\(540\) 0 0
\(541\) −2.18504e6 −0.320972 −0.160486 0.987038i \(-0.551306\pi\)
−0.160486 + 0.987038i \(0.551306\pi\)
\(542\) −1.72673e7 −2.52479
\(543\) 2.43771e6 0.354799
\(544\) −1.60895e7 −2.33102
\(545\) 0 0
\(546\) 8.72313e6 1.25225
\(547\) −6.00853e6 −0.858618 −0.429309 0.903158i \(-0.641243\pi\)
−0.429309 + 0.903158i \(0.641243\pi\)
\(548\) −4.59413e6 −0.653509
\(549\) −1.39783e6 −0.197935
\(550\) 0 0
\(551\) 2.17708e6 0.305489
\(552\) 1.56491e6 0.218596
\(553\) 560135. 0.0778897
\(554\) 7.05065e6 0.976011
\(555\) 0 0
\(556\) −4.55479e6 −0.624858
\(557\) −8.40252e6 −1.14755 −0.573775 0.819013i \(-0.694522\pi\)
−0.573775 + 0.819013i \(0.694522\pi\)
\(558\) 4.37677e6 0.595071
\(559\) 887620. 0.120143
\(560\) 0 0
\(561\) −2.52416e6 −0.338618
\(562\) −1.43558e6 −0.191728
\(563\) 1.68040e6 0.223430 0.111715 0.993740i \(-0.464366\pi\)
0.111715 + 0.993740i \(0.464366\pi\)
\(564\) 5.01369e6 0.663682
\(565\) 0 0
\(566\) 4.36387e6 0.572573
\(567\) −989738. −0.129289
\(568\) −3.27000e6 −0.425281
\(569\) 1.43427e7 1.85716 0.928580 0.371131i \(-0.121030\pi\)
0.928580 + 0.371131i \(0.121030\pi\)
\(570\) 0 0
\(571\) 9.83718e6 1.26264 0.631321 0.775521i \(-0.282513\pi\)
0.631321 + 0.775521i \(0.282513\pi\)
\(572\) 2.38935e6 0.305344
\(573\) −734809. −0.0934949
\(574\) 5.09795e6 0.645827
\(575\) 0 0
\(576\) −959852. −0.120545
\(577\) −4.88454e6 −0.610779 −0.305390 0.952228i \(-0.598787\pi\)
−0.305390 + 0.952228i \(0.598787\pi\)
\(578\) 2.92440e7 3.64097
\(579\) −1.53655e6 −0.190481
\(580\) 0 0
\(581\) 7.83207e6 0.962578
\(582\) 7.62596e6 0.933226
\(583\) −1.61159e6 −0.196374
\(584\) 1.31262e6 0.159260
\(585\) 0 0
\(586\) 1.61787e7 1.94626
\(587\) −3.75376e6 −0.449646 −0.224823 0.974400i \(-0.572180\pi\)
−0.224823 + 0.974400i \(0.572180\pi\)
\(588\) −1.21749e6 −0.145219
\(589\) −1.93922e7 −2.30324
\(590\) 0 0
\(591\) −1.55936e6 −0.183645
\(592\) 3.69617e6 0.433459
\(593\) −7.63599e6 −0.891720 −0.445860 0.895103i \(-0.647102\pi\)
−0.445860 + 0.895103i \(0.647102\pi\)
\(594\) −652618. −0.0758915
\(595\) 0 0
\(596\) 1.82875e6 0.210882
\(597\) −382806. −0.0439585
\(598\) 1.63042e7 1.86443
\(599\) −1.06753e6 −0.121567 −0.0607834 0.998151i \(-0.519360\pi\)
−0.0607834 + 0.998151i \(0.519360\pi\)
\(600\) 0 0
\(601\) −4.83197e6 −0.545680 −0.272840 0.962059i \(-0.587963\pi\)
−0.272840 + 0.962059i \(0.587963\pi\)
\(602\) −1.14075e6 −0.128292
\(603\) −1.51025e6 −0.169143
\(604\) 8.98661e6 1.00231
\(605\) 0 0
\(606\) −1.02974e7 −1.13906
\(607\) 8.95712e6 0.986726 0.493363 0.869824i \(-0.335767\pi\)
0.493363 + 0.869824i \(0.335767\pi\)
\(608\) 1.84314e7 2.02208
\(609\) −1.11317e6 −0.121624
\(610\) 0 0
\(611\) −2.12758e7 −2.30560
\(612\) 4.26910e6 0.460742
\(613\) 8.42569e6 0.905637 0.452818 0.891603i \(-0.350419\pi\)
0.452818 + 0.891603i \(0.350419\pi\)
\(614\) −1.67114e7 −1.78893
\(615\) 0 0
\(616\) 1.25073e6 0.132804
\(617\) 4.84242e6 0.512094 0.256047 0.966664i \(-0.417580\pi\)
0.256047 + 0.966664i \(0.417580\pi\)
\(618\) −7.85424e6 −0.827243
\(619\) −4.22402e6 −0.443097 −0.221549 0.975149i \(-0.571111\pi\)
−0.221549 + 0.975149i \(0.571111\pi\)
\(620\) 0 0
\(621\) −1.84990e6 −0.192495
\(622\) −2.22279e7 −2.30368
\(623\) 7.99008e6 0.824767
\(624\) 9.64938e6 0.992059
\(625\) 0 0
\(626\) −2.04914e7 −2.08995
\(627\) 2.89156e6 0.293740
\(628\) 1.15979e7 1.17349
\(629\) −6.93934e6 −0.699345
\(630\) 0 0
\(631\) 1.18298e7 1.18278 0.591392 0.806384i \(-0.298579\pi\)
0.591392 + 0.806384i \(0.298579\pi\)
\(632\) 254431. 0.0253383
\(633\) 5.63125e6 0.558593
\(634\) −1.48208e7 −1.46436
\(635\) 0 0
\(636\) 2.72567e6 0.267196
\(637\) 5.16648e6 0.504483
\(638\) −734010. −0.0713921
\(639\) 3.86549e6 0.374500
\(640\) 0 0
\(641\) −2.19467e6 −0.210972 −0.105486 0.994421i \(-0.533640\pi\)
−0.105486 + 0.994421i \(0.533640\pi\)
\(642\) 1.55205e7 1.48617
\(643\) −467202. −0.0445633 −0.0222816 0.999752i \(-0.507093\pi\)
−0.0222816 + 0.999752i \(0.507093\pi\)
\(644\) −8.70426e6 −0.827022
\(645\) 0 0
\(646\) −4.55345e7 −4.29298
\(647\) −944645. −0.0887172 −0.0443586 0.999016i \(-0.514124\pi\)
−0.0443586 + 0.999016i \(0.514124\pi\)
\(648\) −449571. −0.0420592
\(649\) −3.60104e6 −0.335595
\(650\) 0 0
\(651\) 9.91552e6 0.916986
\(652\) −6.98808e6 −0.643782
\(653\) −917543. −0.0842061 −0.0421030 0.999113i \(-0.513406\pi\)
−0.0421030 + 0.999113i \(0.513406\pi\)
\(654\) 1.25012e6 0.114290
\(655\) 0 0
\(656\) 5.63927e6 0.511638
\(657\) −1.55166e6 −0.140243
\(658\) 2.73433e7 2.46199
\(659\) −5.17061e6 −0.463798 −0.231899 0.972740i \(-0.574494\pi\)
−0.231899 + 0.972740i \(0.574494\pi\)
\(660\) 0 0
\(661\) −1.78425e6 −0.158837 −0.0794185 0.996841i \(-0.525306\pi\)
−0.0794185 + 0.996841i \(0.525306\pi\)
\(662\) 8.11195e6 0.719417
\(663\) −1.81161e7 −1.60059
\(664\) 3.55758e6 0.313137
\(665\) 0 0
\(666\) −1.79416e6 −0.156738
\(667\) −2.08061e6 −0.181082
\(668\) −4.87388e6 −0.422604
\(669\) −382229. −0.0330186
\(670\) 0 0
\(671\) −2.08811e6 −0.179039
\(672\) −9.42423e6 −0.805050
\(673\) −9.10656e6 −0.775026 −0.387513 0.921864i \(-0.626666\pi\)
−0.387513 + 0.921864i \(0.626666\pi\)
\(674\) −4.39753e6 −0.372872
\(675\) 0 0
\(676\) 8.70591e6 0.732736
\(677\) −657297. −0.0551175 −0.0275588 0.999620i \(-0.508773\pi\)
−0.0275588 + 0.999620i \(0.508773\pi\)
\(678\) 1.09906e7 0.918217
\(679\) 1.72765e7 1.43807
\(680\) 0 0
\(681\) 196339. 0.0162233
\(682\) 6.53814e6 0.538262
\(683\) −1.92866e7 −1.58199 −0.790997 0.611821i \(-0.790437\pi\)
−0.790997 + 0.611821i \(0.790437\pi\)
\(684\) −4.89047e6 −0.399678
\(685\) 0 0
\(686\) 1.21182e7 0.983165
\(687\) −529750. −0.0428232
\(688\) −1.26188e6 −0.101636
\(689\) −1.15665e7 −0.928226
\(690\) 0 0
\(691\) 7.09653e6 0.565394 0.282697 0.959209i \(-0.408771\pi\)
0.282697 + 0.959209i \(0.408771\pi\)
\(692\) −8.34123e6 −0.662163
\(693\) −1.47850e6 −0.116947
\(694\) 6.08575e6 0.479640
\(695\) 0 0
\(696\) −505639. −0.0395656
\(697\) −1.05874e7 −0.825480
\(698\) 7.07987e6 0.550031
\(699\) 3.76204e6 0.291227
\(700\) 0 0
\(701\) 1.57310e7 1.20910 0.604550 0.796567i \(-0.293353\pi\)
0.604550 + 0.796567i \(0.293353\pi\)
\(702\) −4.68389e6 −0.358727
\(703\) 7.94938e6 0.606659
\(704\) −1.43385e6 −0.109037
\(705\) 0 0
\(706\) −1.61883e6 −0.122233
\(707\) −2.33286e7 −1.75526
\(708\) 6.09041e6 0.456629
\(709\) 2.46071e7 1.83842 0.919210 0.393767i \(-0.128828\pi\)
0.919210 + 0.393767i \(0.128828\pi\)
\(710\) 0 0
\(711\) −300765. −0.0223128
\(712\) 3.62935e6 0.268305
\(713\) 1.85329e7 1.36527
\(714\) 2.32824e7 1.70916
\(715\) 0 0
\(716\) 1.54204e7 1.12412
\(717\) 1.16327e7 0.845049
\(718\) −2.01875e7 −1.46141
\(719\) −8.76856e6 −0.632566 −0.316283 0.948665i \(-0.602435\pi\)
−0.316283 + 0.948665i \(0.602435\pi\)
\(720\) 0 0
\(721\) −1.77937e7 −1.27476
\(722\) 3.38426e7 2.41613
\(723\) −4.46533e6 −0.317693
\(724\) −6.15887e6 −0.436671
\(725\) 0 0
\(726\) −974899. −0.0686465
\(727\) 1.62173e7 1.13800 0.568999 0.822338i \(-0.307331\pi\)
0.568999 + 0.822338i \(0.307331\pi\)
\(728\) 8.97659e6 0.627745
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.36910e6 0.163980
\(732\) 3.53161e6 0.243610
\(733\) 1.12288e7 0.771919 0.385960 0.922516i \(-0.373870\pi\)
0.385960 + 0.922516i \(0.373870\pi\)
\(734\) 1.43808e7 0.985243
\(735\) 0 0
\(736\) −1.76146e7 −1.19861
\(737\) −2.25605e6 −0.152996
\(738\) −2.73735e6 −0.185008
\(739\) 8.22290e6 0.553878 0.276939 0.960888i \(-0.410680\pi\)
0.276939 + 0.960888i \(0.410680\pi\)
\(740\) 0 0
\(741\) 2.07529e7 1.38846
\(742\) 1.48650e7 0.991188
\(743\) 2.24984e6 0.149513 0.0747566 0.997202i \(-0.476182\pi\)
0.0747566 + 0.997202i \(0.476182\pi\)
\(744\) 4.50395e6 0.298305
\(745\) 0 0
\(746\) 3.61094e7 2.37560
\(747\) −4.20544e6 −0.275746
\(748\) 6.37729e6 0.416757
\(749\) 3.51615e7 2.29014
\(750\) 0 0
\(751\) −171752. −0.0111122 −0.00555612 0.999985i \(-0.501769\pi\)
−0.00555612 + 0.999985i \(0.501769\pi\)
\(752\) 3.02467e7 1.95044
\(753\) 1.42729e7 0.917330
\(754\) −5.26804e6 −0.337459
\(755\) 0 0
\(756\) 2.50057e6 0.159124
\(757\) −2.62239e7 −1.66325 −0.831624 0.555339i \(-0.812589\pi\)
−0.831624 + 0.555339i \(0.812589\pi\)
\(758\) −3.02475e7 −1.91212
\(759\) −2.76343e6 −0.174118
\(760\) 0 0
\(761\) −2.29774e7 −1.43827 −0.719134 0.694871i \(-0.755461\pi\)
−0.719134 + 0.694871i \(0.755461\pi\)
\(762\) 1.35748e7 0.846928
\(763\) 2.83213e6 0.176117
\(764\) 1.85649e6 0.115069
\(765\) 0 0
\(766\) −2.25329e7 −1.38754
\(767\) −2.58449e7 −1.58631
\(768\) −1.23657e7 −0.756515
\(769\) −2.94748e6 −0.179736 −0.0898680 0.995954i \(-0.528645\pi\)
−0.0898680 + 0.995954i \(0.528645\pi\)
\(770\) 0 0
\(771\) 1.27693e7 0.773623
\(772\) 3.88209e6 0.234435
\(773\) 1.27928e7 0.770049 0.385024 0.922906i \(-0.374193\pi\)
0.385024 + 0.922906i \(0.374193\pi\)
\(774\) 612528. 0.0367514
\(775\) 0 0
\(776\) 7.84754e6 0.467821
\(777\) −4.06463e6 −0.241529
\(778\) 1.58067e7 0.936249
\(779\) 1.21284e7 0.716077
\(780\) 0 0
\(781\) 5.77437e6 0.338748
\(782\) 4.35167e7 2.54471
\(783\) 597720. 0.0348412
\(784\) −7.34490e6 −0.426772
\(785\) 0 0
\(786\) −5.48088e6 −0.316442
\(787\) 1.94779e6 0.112100 0.0560499 0.998428i \(-0.482149\pi\)
0.0560499 + 0.998428i \(0.482149\pi\)
\(788\) 3.93973e6 0.226022
\(789\) 1.45842e7 0.834045
\(790\) 0 0
\(791\) 2.48989e7 1.41495
\(792\) −671581. −0.0380440
\(793\) −1.49865e7 −0.846289
\(794\) 1.67990e7 0.945655
\(795\) 0 0
\(796\) 967160. 0.0541023
\(797\) 2.72445e7 1.51926 0.759630 0.650355i \(-0.225380\pi\)
0.759630 + 0.650355i \(0.225380\pi\)
\(798\) −2.66713e7 −1.48264
\(799\) −5.67862e7 −3.14685
\(800\) 0 0
\(801\) −4.29028e6 −0.236268
\(802\) 7.25901e6 0.398512
\(803\) −2.31791e6 −0.126855
\(804\) 3.81563e6 0.208174
\(805\) 0 0
\(806\) 4.69247e7 2.54428
\(807\) −1.13239e7 −0.612087
\(808\) −1.05966e7 −0.571003
\(809\) 1.56700e7 0.841780 0.420890 0.907112i \(-0.361718\pi\)
0.420890 + 0.907112i \(0.361718\pi\)
\(810\) 0 0
\(811\) 1.41672e7 0.756365 0.378182 0.925731i \(-0.376549\pi\)
0.378182 + 0.925731i \(0.376549\pi\)
\(812\) 2.81243e6 0.149690
\(813\) 2.10048e7 1.11453
\(814\) −2.68016e6 −0.141775
\(815\) 0 0
\(816\) 2.57547e7 1.35404
\(817\) −2.71393e6 −0.142247
\(818\) −1.46550e7 −0.765776
\(819\) −1.06113e7 −0.552788
\(820\) 0 0
\(821\) 6.51336e6 0.337246 0.168623 0.985681i \(-0.446068\pi\)
0.168623 + 0.985681i \(0.446068\pi\)
\(822\) 1.34533e7 0.694466
\(823\) −2.92352e7 −1.50455 −0.752275 0.658849i \(-0.771044\pi\)
−0.752275 + 0.658849i \(0.771044\pi\)
\(824\) −8.08246e6 −0.414692
\(825\) 0 0
\(826\) 3.32154e7 1.69390
\(827\) 2.56735e7 1.30533 0.652666 0.757646i \(-0.273651\pi\)
0.652666 + 0.757646i \(0.273651\pi\)
\(828\) 4.67376e6 0.236914
\(829\) −9.02846e6 −0.456276 −0.228138 0.973629i \(-0.573264\pi\)
−0.228138 + 0.973629i \(0.573264\pi\)
\(830\) 0 0
\(831\) −8.57680e6 −0.430847
\(832\) −1.02909e7 −0.515399
\(833\) 1.37896e7 0.688555
\(834\) 1.33382e7 0.664019
\(835\) 0 0
\(836\) −7.30552e6 −0.361523
\(837\) −5.32415e6 −0.262686
\(838\) 1.13256e6 0.0557124
\(839\) 3.00826e7 1.47540 0.737701 0.675128i \(-0.235912\pi\)
0.737701 + 0.675128i \(0.235912\pi\)
\(840\) 0 0
\(841\) −1.98389e7 −0.967224
\(842\) 6.27586e6 0.305065
\(843\) 1.74631e6 0.0846356
\(844\) −1.42273e7 −0.687492
\(845\) 0 0
\(846\) −1.46820e7 −0.705276
\(847\) −2.20862e6 −0.105782
\(848\) 1.64434e7 0.785241
\(849\) −5.30846e6 −0.252755
\(850\) 0 0
\(851\) −7.59711e6 −0.359604
\(852\) −9.76614e6 −0.460919
\(853\) −2.48619e7 −1.16994 −0.584968 0.811057i \(-0.698893\pi\)
−0.584968 + 0.811057i \(0.698893\pi\)
\(854\) 1.92604e7 0.903692
\(855\) 0 0
\(856\) 1.59715e7 0.745008
\(857\) −8.27535e6 −0.384888 −0.192444 0.981308i \(-0.561641\pi\)
−0.192444 + 0.981308i \(0.561641\pi\)
\(858\) −6.99693e6 −0.324481
\(859\) 1.89682e7 0.877088 0.438544 0.898710i \(-0.355494\pi\)
0.438544 + 0.898710i \(0.355494\pi\)
\(860\) 0 0
\(861\) −6.20143e6 −0.285091
\(862\) −3.64723e6 −0.167184
\(863\) −1.70264e7 −0.778209 −0.389104 0.921194i \(-0.627215\pi\)
−0.389104 + 0.921194i \(0.627215\pi\)
\(864\) 5.06035e6 0.230620
\(865\) 0 0
\(866\) −3.79378e7 −1.71901
\(867\) −3.55741e7 −1.60726
\(868\) −2.50515e7 −1.12859
\(869\) −449291. −0.0201827
\(870\) 0 0
\(871\) −1.61918e7 −0.723186
\(872\) 1.28645e6 0.0572928
\(873\) −9.27664e6 −0.411960
\(874\) −4.98506e7 −2.20746
\(875\) 0 0
\(876\) 3.92026e6 0.172605
\(877\) −2.69746e7 −1.18428 −0.592142 0.805834i \(-0.701717\pi\)
−0.592142 + 0.805834i \(0.701717\pi\)
\(878\) −2.94536e7 −1.28944
\(879\) −1.96807e7 −0.859149
\(880\) 0 0
\(881\) 2.13740e7 0.927781 0.463890 0.885893i \(-0.346453\pi\)
0.463890 + 0.885893i \(0.346453\pi\)
\(882\) 3.56528e6 0.154320
\(883\) 3.26501e6 0.140923 0.0704616 0.997514i \(-0.477553\pi\)
0.0704616 + 0.997514i \(0.477553\pi\)
\(884\) 4.57703e7 1.96994
\(885\) 0 0
\(886\) 2.28698e7 0.978764
\(887\) 6.68667e6 0.285365 0.142682 0.989769i \(-0.454427\pi\)
0.142682 + 0.989769i \(0.454427\pi\)
\(888\) −1.84629e6 −0.0785718
\(889\) 3.07536e7 1.30509
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 965702. 0.0406379
\(893\) 6.50516e7 2.72979
\(894\) −5.35528e6 −0.224098
\(895\) 0 0
\(896\) −2.02828e7 −0.844029
\(897\) −1.98333e7 −0.823027
\(898\) −1.11842e7 −0.462824
\(899\) −5.98815e6 −0.247112
\(900\) 0 0
\(901\) −3.08715e7 −1.26691
\(902\) −4.08913e6 −0.167346
\(903\) 1.38767e6 0.0566328
\(904\) 1.13099e7 0.460297
\(905\) 0 0
\(906\) −2.63162e7 −1.06513
\(907\) −2.82586e7 −1.14060 −0.570299 0.821437i \(-0.693173\pi\)
−0.570299 + 0.821437i \(0.693173\pi\)
\(908\) −496050. −0.0199669
\(909\) 1.25263e7 0.502822
\(910\) 0 0
\(911\) −9.75713e6 −0.389517 −0.194758 0.980851i \(-0.562392\pi\)
−0.194758 + 0.980851i \(0.562392\pi\)
\(912\) −2.95033e7 −1.17458
\(913\) −6.28219e6 −0.249422
\(914\) 4.13432e7 1.63696
\(915\) 0 0
\(916\) 1.33841e6 0.0527050
\(917\) −1.24169e7 −0.487628
\(918\) −1.25015e7 −0.489617
\(919\) −3.69755e7 −1.44419 −0.722097 0.691792i \(-0.756821\pi\)
−0.722097 + 0.691792i \(0.756821\pi\)
\(920\) 0 0
\(921\) 2.03287e7 0.789697
\(922\) −6.06300e6 −0.234888
\(923\) 4.14431e7 1.60121
\(924\) 3.73542e6 0.143933
\(925\) 0 0
\(926\) 7.30784e6 0.280067
\(927\) 9.55433e6 0.365175
\(928\) 5.69145e6 0.216947
\(929\) −5.05419e7 −1.92137 −0.960687 0.277633i \(-0.910450\pi\)
−0.960687 + 0.277633i \(0.910450\pi\)
\(930\) 0 0
\(931\) −1.57967e7 −0.597299
\(932\) −9.50480e6 −0.358429
\(933\) 2.70393e7 1.01693
\(934\) 2.41428e7 0.905566
\(935\) 0 0
\(936\) −4.81999e6 −0.179828
\(937\) −1.08317e7 −0.403040 −0.201520 0.979484i \(-0.564588\pi\)
−0.201520 + 0.979484i \(0.564588\pi\)
\(938\) 2.08094e7 0.772240
\(939\) 2.49268e7 0.922578
\(940\) 0 0
\(941\) 1.45659e6 0.0536246 0.0268123 0.999640i \(-0.491464\pi\)
0.0268123 + 0.999640i \(0.491464\pi\)
\(942\) −3.39631e7 −1.24704
\(943\) −1.15909e7 −0.424463
\(944\) 3.67423e7 1.34195
\(945\) 0 0
\(946\) 915011. 0.0332429
\(947\) 4.32433e7 1.56691 0.783455 0.621449i \(-0.213456\pi\)
0.783455 + 0.621449i \(0.213456\pi\)
\(948\) 759883. 0.0274616
\(949\) −1.66358e7 −0.599623
\(950\) 0 0
\(951\) 1.80288e7 0.646421
\(952\) 2.39590e7 0.856792
\(953\) −1.69154e7 −0.603322 −0.301661 0.953415i \(-0.597541\pi\)
−0.301661 + 0.953415i \(0.597541\pi\)
\(954\) −7.98179e6 −0.283942
\(955\) 0 0
\(956\) −2.93900e7 −1.04005
\(957\) 892890. 0.0315151
\(958\) 4.15663e7 1.46328
\(959\) 3.04784e7 1.07015
\(960\) 0 0
\(961\) 2.47099e7 0.863102
\(962\) −1.92357e7 −0.670147
\(963\) −1.88800e7 −0.656049
\(964\) 1.12817e7 0.391003
\(965\) 0 0
\(966\) 2.54894e7 0.878853
\(967\) −2.51066e7 −0.863421 −0.431710 0.902012i \(-0.642090\pi\)
−0.431710 + 0.902012i \(0.642090\pi\)
\(968\) −1.00323e6 −0.0344121
\(969\) 5.53906e7 1.89508
\(970\) 0 0
\(971\) −3.10806e7 −1.05789 −0.528947 0.848655i \(-0.677413\pi\)
−0.528947 + 0.848655i \(0.677413\pi\)
\(972\) −1.34269e6 −0.0455836
\(973\) 3.02174e7 1.02323
\(974\) 3.19441e7 1.07893
\(975\) 0 0
\(976\) 2.13055e7 0.715925
\(977\) 1.72964e7 0.579722 0.289861 0.957069i \(-0.406391\pi\)
0.289861 + 0.957069i \(0.406391\pi\)
\(978\) 2.04637e7 0.684129
\(979\) −6.40894e6 −0.213712
\(980\) 0 0
\(981\) −1.52072e6 −0.0504517
\(982\) 6.49313e7 2.14870
\(983\) −2.02617e7 −0.668793 −0.334396 0.942433i \(-0.608532\pi\)
−0.334396 + 0.942433i \(0.608532\pi\)
\(984\) −2.81689e6 −0.0927432
\(985\) 0 0
\(986\) −1.40607e7 −0.460589
\(987\) −3.32619e7 −1.08681
\(988\) −5.24323e7 −1.70886
\(989\) 2.59367e6 0.0843186
\(990\) 0 0
\(991\) 1.64795e7 0.533041 0.266520 0.963829i \(-0.414126\pi\)
0.266520 + 0.963829i \(0.414126\pi\)
\(992\) −5.06962e7 −1.63567
\(993\) −9.86783e6 −0.317577
\(994\) −5.32618e7 −1.70982
\(995\) 0 0
\(996\) 1.06250e7 0.339377
\(997\) −364567. −0.0116155 −0.00580777 0.999983i \(-0.501849\pi\)
−0.00580777 + 0.999983i \(0.501849\pi\)
\(998\) −6.71869e7 −2.13530
\(999\) 2.18251e6 0.0691899
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.v.1.11 13
5.2 odd 4 165.6.c.b.34.21 yes 26
5.3 odd 4 165.6.c.b.34.6 26
5.4 even 2 825.6.a.y.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.6 26 5.3 odd 4
165.6.c.b.34.21 yes 26 5.2 odd 4
825.6.a.v.1.11 13 1.1 even 1 trivial
825.6.a.y.1.3 13 5.4 even 2