Properties

Label 825.6.a.t.1.9
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-9.20891\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.20891 q^{2} +9.00000 q^{3} +52.8039 q^{4} +82.8802 q^{6} -195.275 q^{7} +191.582 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+9.20891 q^{2} +9.00000 q^{3} +52.8039 q^{4} +82.8802 q^{6} -195.275 q^{7} +191.582 q^{8} +81.0000 q^{9} +121.000 q^{11} +475.236 q^{12} -457.051 q^{13} -1798.27 q^{14} +74.5305 q^{16} +1237.33 q^{17} +745.921 q^{18} +2140.85 q^{19} -1757.47 q^{21} +1114.28 q^{22} +2272.80 q^{23} +1724.23 q^{24} -4208.94 q^{26} +729.000 q^{27} -10311.3 q^{28} +6561.91 q^{29} +6349.26 q^{31} -5444.27 q^{32} +1089.00 q^{33} +11394.5 q^{34} +4277.12 q^{36} -3376.37 q^{37} +19714.9 q^{38} -4113.46 q^{39} +3970.52 q^{41} -16184.4 q^{42} +3901.82 q^{43} +6389.28 q^{44} +20930.0 q^{46} -1267.66 q^{47} +670.775 q^{48} +21325.2 q^{49} +11136.0 q^{51} -24134.1 q^{52} +10563.3 q^{53} +6713.29 q^{54} -37411.0 q^{56} +19267.6 q^{57} +60428.0 q^{58} +43579.8 q^{59} +49086.8 q^{61} +58469.8 q^{62} -15817.2 q^{63} -52520.7 q^{64} +10028.5 q^{66} +14785.3 q^{67} +65336.1 q^{68} +20455.2 q^{69} +8961.61 q^{71} +15518.1 q^{72} -83352.2 q^{73} -31092.6 q^{74} +113045. q^{76} -23628.2 q^{77} -37880.5 q^{78} -17858.5 q^{79} +6561.00 q^{81} +36564.1 q^{82} -66830.4 q^{83} -92801.5 q^{84} +35931.5 q^{86} +59057.2 q^{87} +23181.4 q^{88} -117907. q^{89} +89250.5 q^{91} +120013. q^{92} +57143.4 q^{93} -11673.8 q^{94} -48998.4 q^{96} +156797. q^{97} +196382. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.20891 1.62792 0.813960 0.580921i \(-0.197307\pi\)
0.813960 + 0.580921i \(0.197307\pi\)
\(3\) 9.00000 0.577350
\(4\) 52.8039 1.65012
\(5\) 0 0
\(6\) 82.8802 0.939880
\(7\) −195.275 −1.50626 −0.753131 0.657870i \(-0.771458\pi\)
−0.753131 + 0.657870i \(0.771458\pi\)
\(8\) 191.582 1.05835
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 475.236 0.952699
\(13\) −457.051 −0.750078 −0.375039 0.927009i \(-0.622371\pi\)
−0.375039 + 0.927009i \(0.622371\pi\)
\(14\) −1798.27 −2.45208
\(15\) 0 0
\(16\) 74.5305 0.0727837
\(17\) 1237.33 1.03840 0.519200 0.854653i \(-0.326230\pi\)
0.519200 + 0.854653i \(0.326230\pi\)
\(18\) 745.921 0.542640
\(19\) 2140.85 1.36051 0.680255 0.732976i \(-0.261869\pi\)
0.680255 + 0.732976i \(0.261869\pi\)
\(20\) 0 0
\(21\) −1757.47 −0.869641
\(22\) 1114.28 0.490836
\(23\) 2272.80 0.895861 0.447931 0.894068i \(-0.352161\pi\)
0.447931 + 0.894068i \(0.352161\pi\)
\(24\) 1724.23 0.611038
\(25\) 0 0
\(26\) −4208.94 −1.22107
\(27\) 729.000 0.192450
\(28\) −10311.3 −2.48552
\(29\) 6561.91 1.44889 0.724445 0.689333i \(-0.242096\pi\)
0.724445 + 0.689333i \(0.242096\pi\)
\(30\) 0 0
\(31\) 6349.26 1.18664 0.593320 0.804967i \(-0.297817\pi\)
0.593320 + 0.804967i \(0.297817\pi\)
\(32\) −5444.27 −0.939863
\(33\) 1089.00 0.174078
\(34\) 11394.5 1.69043
\(35\) 0 0
\(36\) 4277.12 0.550041
\(37\) −3376.37 −0.405457 −0.202729 0.979235i \(-0.564981\pi\)
−0.202729 + 0.979235i \(0.564981\pi\)
\(38\) 19714.9 2.21480
\(39\) −4113.46 −0.433058
\(40\) 0 0
\(41\) 3970.52 0.368882 0.184441 0.982844i \(-0.440953\pi\)
0.184441 + 0.982844i \(0.440953\pi\)
\(42\) −16184.4 −1.41571
\(43\) 3901.82 0.321807 0.160904 0.986970i \(-0.448559\pi\)
0.160904 + 0.986970i \(0.448559\pi\)
\(44\) 6389.28 0.497531
\(45\) 0 0
\(46\) 20930.0 1.45839
\(47\) −1267.66 −0.0837065 −0.0418532 0.999124i \(-0.513326\pi\)
−0.0418532 + 0.999124i \(0.513326\pi\)
\(48\) 670.775 0.0420217
\(49\) 21325.2 1.26883
\(50\) 0 0
\(51\) 11136.0 0.599521
\(52\) −24134.1 −1.23772
\(53\) 10563.3 0.516546 0.258273 0.966072i \(-0.416847\pi\)
0.258273 + 0.966072i \(0.416847\pi\)
\(54\) 6713.29 0.313293
\(55\) 0 0
\(56\) −37411.0 −1.59415
\(57\) 19267.6 0.785490
\(58\) 60428.0 2.35868
\(59\) 43579.8 1.62988 0.814938 0.579548i \(-0.196771\pi\)
0.814938 + 0.579548i \(0.196771\pi\)
\(60\) 0 0
\(61\) 49086.8 1.68904 0.844521 0.535522i \(-0.179885\pi\)
0.844521 + 0.535522i \(0.179885\pi\)
\(62\) 58469.8 1.93176
\(63\) −15817.2 −0.502088
\(64\) −52520.7 −1.60280
\(65\) 0 0
\(66\) 10028.5 0.283384
\(67\) 14785.3 0.402386 0.201193 0.979552i \(-0.435518\pi\)
0.201193 + 0.979552i \(0.435518\pi\)
\(68\) 65336.1 1.71349
\(69\) 20455.2 0.517226
\(70\) 0 0
\(71\) 8961.61 0.210979 0.105490 0.994420i \(-0.466359\pi\)
0.105490 + 0.994420i \(0.466359\pi\)
\(72\) 15518.1 0.352783
\(73\) −83352.2 −1.83067 −0.915335 0.402694i \(-0.868074\pi\)
−0.915335 + 0.402694i \(0.868074\pi\)
\(74\) −31092.6 −0.660052
\(75\) 0 0
\(76\) 113045. 2.24501
\(77\) −23628.2 −0.454155
\(78\) −37880.5 −0.704983
\(79\) −17858.5 −0.321942 −0.160971 0.986959i \(-0.551463\pi\)
−0.160971 + 0.986959i \(0.551463\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 36564.1 0.600511
\(83\) −66830.4 −1.06483 −0.532413 0.846485i \(-0.678715\pi\)
−0.532413 + 0.846485i \(0.678715\pi\)
\(84\) −92801.5 −1.43502
\(85\) 0 0
\(86\) 35931.5 0.523877
\(87\) 59057.2 0.836517
\(88\) 23181.4 0.319104
\(89\) −117907. −1.57784 −0.788921 0.614495i \(-0.789360\pi\)
−0.788921 + 0.614495i \(0.789360\pi\)
\(90\) 0 0
\(91\) 89250.5 1.12981
\(92\) 120013. 1.47828
\(93\) 57143.4 0.685107
\(94\) −11673.8 −0.136267
\(95\) 0 0
\(96\) −48998.4 −0.542630
\(97\) 156797. 1.69203 0.846017 0.533155i \(-0.178994\pi\)
0.846017 + 0.533155i \(0.178994\pi\)
\(98\) 196382. 2.06555
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −89018.4 −0.868313 −0.434156 0.900838i \(-0.642953\pi\)
−0.434156 + 0.900838i \(0.642953\pi\)
\(102\) 102550. 0.975972
\(103\) 100268. 0.931253 0.465626 0.884981i \(-0.345829\pi\)
0.465626 + 0.884981i \(0.345829\pi\)
\(104\) −87562.6 −0.793844
\(105\) 0 0
\(106\) 97276.1 0.840895
\(107\) −132681. −1.12033 −0.560167 0.828380i \(-0.689263\pi\)
−0.560167 + 0.828380i \(0.689263\pi\)
\(108\) 38494.1 0.317566
\(109\) 88547.8 0.713857 0.356929 0.934132i \(-0.383824\pi\)
0.356929 + 0.934132i \(0.383824\pi\)
\(110\) 0 0
\(111\) −30387.3 −0.234091
\(112\) −14553.9 −0.109631
\(113\) 92355.2 0.680401 0.340201 0.940353i \(-0.389505\pi\)
0.340201 + 0.940353i \(0.389505\pi\)
\(114\) 177434. 1.27872
\(115\) 0 0
\(116\) 346495. 2.39085
\(117\) −37021.1 −0.250026
\(118\) 401322. 2.65331
\(119\) −241620. −1.56410
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 452036. 2.74963
\(123\) 35734.7 0.212974
\(124\) 335266. 1.95810
\(125\) 0 0
\(126\) −145660. −0.817358
\(127\) −70624.6 −0.388550 −0.194275 0.980947i \(-0.562235\pi\)
−0.194275 + 0.980947i \(0.562235\pi\)
\(128\) −309442. −1.66938
\(129\) 35116.4 0.185796
\(130\) 0 0
\(131\) 155328. 0.790809 0.395404 0.918507i \(-0.370604\pi\)
0.395404 + 0.918507i \(0.370604\pi\)
\(132\) 57503.5 0.287250
\(133\) −418053. −2.04928
\(134\) 136156. 0.655053
\(135\) 0 0
\(136\) 237051. 1.09899
\(137\) 70583.9 0.321295 0.160647 0.987012i \(-0.448642\pi\)
0.160647 + 0.987012i \(0.448642\pi\)
\(138\) 188370. 0.842002
\(139\) −400133. −1.75658 −0.878290 0.478129i \(-0.841315\pi\)
−0.878290 + 0.478129i \(0.841315\pi\)
\(140\) 0 0
\(141\) −11409.0 −0.0483279
\(142\) 82526.6 0.343458
\(143\) −55303.2 −0.226157
\(144\) 6036.97 0.0242612
\(145\) 0 0
\(146\) −767583. −2.98018
\(147\) 191927. 0.732558
\(148\) −178285. −0.669055
\(149\) 27434.0 0.101233 0.0506166 0.998718i \(-0.483881\pi\)
0.0506166 + 0.998718i \(0.483881\pi\)
\(150\) 0 0
\(151\) 111764. 0.398894 0.199447 0.979909i \(-0.436085\pi\)
0.199447 + 0.979909i \(0.436085\pi\)
\(152\) 410147. 1.43989
\(153\) 100224. 0.346133
\(154\) −217590. −0.739329
\(155\) 0 0
\(156\) −217207. −0.714598
\(157\) 231260. 0.748775 0.374388 0.927272i \(-0.377853\pi\)
0.374388 + 0.927272i \(0.377853\pi\)
\(158\) −164458. −0.524096
\(159\) 95069.4 0.298228
\(160\) 0 0
\(161\) −443819. −1.34940
\(162\) 60419.6 0.180880
\(163\) −219113. −0.645949 −0.322974 0.946408i \(-0.604683\pi\)
−0.322974 + 0.946408i \(0.604683\pi\)
\(164\) 209659. 0.608701
\(165\) 0 0
\(166\) −615435. −1.73345
\(167\) 138345. 0.383859 0.191929 0.981409i \(-0.438526\pi\)
0.191929 + 0.981409i \(0.438526\pi\)
\(168\) −336699. −0.920384
\(169\) −162397. −0.437383
\(170\) 0 0
\(171\) 173409. 0.453503
\(172\) 206031. 0.531022
\(173\) −314475. −0.798860 −0.399430 0.916764i \(-0.630792\pi\)
−0.399430 + 0.916764i \(0.630792\pi\)
\(174\) 543852. 1.36178
\(175\) 0 0
\(176\) 9018.19 0.0219451
\(177\) 392218. 0.941010
\(178\) −1.08579e6 −2.56860
\(179\) −725112. −1.69150 −0.845751 0.533578i \(-0.820847\pi\)
−0.845751 + 0.533578i \(0.820847\pi\)
\(180\) 0 0
\(181\) 843502. 1.91377 0.956885 0.290466i \(-0.0938102\pi\)
0.956885 + 0.290466i \(0.0938102\pi\)
\(182\) 821899. 1.83925
\(183\) 441782. 0.975169
\(184\) 435426. 0.948134
\(185\) 0 0
\(186\) 526228. 1.11530
\(187\) 149717. 0.313089
\(188\) −66937.6 −0.138126
\(189\) −142355. −0.289880
\(190\) 0 0
\(191\) −659387. −1.30785 −0.653923 0.756561i \(-0.726878\pi\)
−0.653923 + 0.756561i \(0.726878\pi\)
\(192\) −472686. −0.925380
\(193\) −350322. −0.676978 −0.338489 0.940970i \(-0.609916\pi\)
−0.338489 + 0.940970i \(0.609916\pi\)
\(194\) 1.44393e6 2.75450
\(195\) 0 0
\(196\) 1.12605e6 2.09372
\(197\) 504269. 0.925757 0.462878 0.886422i \(-0.346817\pi\)
0.462878 + 0.886422i \(0.346817\pi\)
\(198\) 90256.5 0.163612
\(199\) −939943. −1.68255 −0.841276 0.540605i \(-0.818195\pi\)
−0.841276 + 0.540605i \(0.818195\pi\)
\(200\) 0 0
\(201\) 133068. 0.232318
\(202\) −819762. −1.41354
\(203\) −1.28137e6 −2.18241
\(204\) 588025. 0.989283
\(205\) 0 0
\(206\) 923355. 1.51601
\(207\) 184096. 0.298620
\(208\) −34064.2 −0.0545934
\(209\) 259042. 0.410209
\(210\) 0 0
\(211\) 666610. 1.03078 0.515390 0.856956i \(-0.327647\pi\)
0.515390 + 0.856956i \(0.327647\pi\)
\(212\) 557782. 0.852364
\(213\) 80654.5 0.121809
\(214\) −1.22184e6 −1.82381
\(215\) 0 0
\(216\) 139663. 0.203679
\(217\) −1.23985e6 −1.78739
\(218\) 815428. 1.16210
\(219\) −750170. −1.05694
\(220\) 0 0
\(221\) −565525. −0.778881
\(222\) −279834. −0.381081
\(223\) −459829. −0.619205 −0.309602 0.950866i \(-0.600196\pi\)
−0.309602 + 0.950866i \(0.600196\pi\)
\(224\) 1.06313e6 1.41568
\(225\) 0 0
\(226\) 850490. 1.10764
\(227\) 1.11522e6 1.43647 0.718236 0.695799i \(-0.244950\pi\)
0.718236 + 0.695799i \(0.244950\pi\)
\(228\) 1.01741e6 1.29616
\(229\) 1.19306e6 1.50340 0.751700 0.659505i \(-0.229234\pi\)
0.751700 + 0.659505i \(0.229234\pi\)
\(230\) 0 0
\(231\) −212654. −0.262207
\(232\) 1.25714e6 1.53343
\(233\) 675874. 0.815598 0.407799 0.913072i \(-0.366296\pi\)
0.407799 + 0.913072i \(0.366296\pi\)
\(234\) −340924. −0.407022
\(235\) 0 0
\(236\) 2.30118e6 2.68950
\(237\) −160727. −0.185873
\(238\) −2.22506e6 −2.54624
\(239\) 436348. 0.494127 0.247063 0.968999i \(-0.420534\pi\)
0.247063 + 0.968999i \(0.420534\pi\)
\(240\) 0 0
\(241\) 597010. 0.662123 0.331062 0.943609i \(-0.392593\pi\)
0.331062 + 0.943609i \(0.392593\pi\)
\(242\) 134828. 0.147993
\(243\) 59049.0 0.0641500
\(244\) 2.59198e6 2.78713
\(245\) 0 0
\(246\) 329077. 0.346705
\(247\) −978476. −1.02049
\(248\) 1.21640e6 1.25588
\(249\) −601474. −0.614778
\(250\) 0 0
\(251\) −75812.0 −0.0759545 −0.0379773 0.999279i \(-0.512091\pi\)
−0.0379773 + 0.999279i \(0.512091\pi\)
\(252\) −835213. −0.828507
\(253\) 275008. 0.270112
\(254\) −650376. −0.632528
\(255\) 0 0
\(256\) −1.16896e6 −1.11480
\(257\) −1.20303e6 −1.13617 −0.568087 0.822969i \(-0.692316\pi\)
−0.568087 + 0.822969i \(0.692316\pi\)
\(258\) 323383. 0.302460
\(259\) 659319. 0.610725
\(260\) 0 0
\(261\) 531515. 0.482963
\(262\) 1.43040e6 1.28737
\(263\) 332575. 0.296483 0.148242 0.988951i \(-0.452639\pi\)
0.148242 + 0.988951i \(0.452639\pi\)
\(264\) 208632. 0.184235
\(265\) 0 0
\(266\) −3.84981e6 −3.33607
\(267\) −1.06116e6 −0.910968
\(268\) 780722. 0.663987
\(269\) 1.60565e6 1.35291 0.676455 0.736484i \(-0.263515\pi\)
0.676455 + 0.736484i \(0.263515\pi\)
\(270\) 0 0
\(271\) −973760. −0.805432 −0.402716 0.915325i \(-0.631934\pi\)
−0.402716 + 0.915325i \(0.631934\pi\)
\(272\) 92219.2 0.0755786
\(273\) 803254. 0.652299
\(274\) 650000. 0.523043
\(275\) 0 0
\(276\) 1.08011e6 0.853486
\(277\) −430301. −0.336956 −0.168478 0.985705i \(-0.553885\pi\)
−0.168478 + 0.985705i \(0.553885\pi\)
\(278\) −3.68479e6 −2.85957
\(279\) 514290. 0.395547
\(280\) 0 0
\(281\) −566489. −0.427982 −0.213991 0.976836i \(-0.568646\pi\)
−0.213991 + 0.976836i \(0.568646\pi\)
\(282\) −105064. −0.0786740
\(283\) −962280. −0.714225 −0.357113 0.934061i \(-0.616239\pi\)
−0.357113 + 0.934061i \(0.616239\pi\)
\(284\) 473208. 0.348142
\(285\) 0 0
\(286\) −509282. −0.368165
\(287\) −775342. −0.555633
\(288\) −440986. −0.313288
\(289\) 111140. 0.0782753
\(290\) 0 0
\(291\) 1.41118e6 0.976897
\(292\) −4.40133e6 −3.02083
\(293\) −762793. −0.519084 −0.259542 0.965732i \(-0.583572\pi\)
−0.259542 + 0.965732i \(0.583572\pi\)
\(294\) 1.76744e6 1.19255
\(295\) 0 0
\(296\) −646850. −0.429115
\(297\) 88209.0 0.0580259
\(298\) 252637. 0.164800
\(299\) −1.03878e6 −0.671966
\(300\) 0 0
\(301\) −761926. −0.484726
\(302\) 1.02922e6 0.649368
\(303\) −801165. −0.501321
\(304\) 159558. 0.0990229
\(305\) 0 0
\(306\) 922954. 0.563478
\(307\) 3.02419e6 1.83132 0.915658 0.401958i \(-0.131670\pi\)
0.915658 + 0.401958i \(0.131670\pi\)
\(308\) −1.24766e6 −0.749412
\(309\) 902408. 0.537659
\(310\) 0 0
\(311\) 2.33903e6 1.37131 0.685654 0.727928i \(-0.259516\pi\)
0.685654 + 0.727928i \(0.259516\pi\)
\(312\) −788063. −0.458326
\(313\) 1.42601e6 0.822740 0.411370 0.911469i \(-0.365050\pi\)
0.411370 + 0.911469i \(0.365050\pi\)
\(314\) 2.12965e6 1.21895
\(315\) 0 0
\(316\) −943001. −0.531244
\(317\) −1.60632e6 −0.897809 −0.448904 0.893580i \(-0.648186\pi\)
−0.448904 + 0.893580i \(0.648186\pi\)
\(318\) 875485. 0.485491
\(319\) 793991. 0.436857
\(320\) 0 0
\(321\) −1.19412e6 −0.646825
\(322\) −4.08709e6 −2.19672
\(323\) 2.64894e6 1.41275
\(324\) 346447. 0.183347
\(325\) 0 0
\(326\) −2.01779e6 −1.05155
\(327\) 796930. 0.412146
\(328\) 760678. 0.390406
\(329\) 247542. 0.126084
\(330\) 0 0
\(331\) 3.11538e6 1.56294 0.781469 0.623945i \(-0.214471\pi\)
0.781469 + 0.623945i \(0.214471\pi\)
\(332\) −3.52891e6 −1.75710
\(333\) −273486. −0.135152
\(334\) 1.27400e6 0.624892
\(335\) 0 0
\(336\) −130985. −0.0632957
\(337\) −1.61111e6 −0.772768 −0.386384 0.922338i \(-0.626276\pi\)
−0.386384 + 0.922338i \(0.626276\pi\)
\(338\) −1.49550e6 −0.712025
\(339\) 831197. 0.392830
\(340\) 0 0
\(341\) 768261. 0.357786
\(342\) 1.59690e6 0.738267
\(343\) −882288. −0.404925
\(344\) 747517. 0.340584
\(345\) 0 0
\(346\) −2.89597e6 −1.30048
\(347\) 847409. 0.377806 0.188903 0.981996i \(-0.439507\pi\)
0.188903 + 0.981996i \(0.439507\pi\)
\(348\) 3.11845e6 1.38036
\(349\) 2.75989e6 1.21291 0.606455 0.795118i \(-0.292591\pi\)
0.606455 + 0.795118i \(0.292591\pi\)
\(350\) 0 0
\(351\) −333190. −0.144353
\(352\) −658756. −0.283379
\(353\) −4.12207e6 −1.76067 −0.880336 0.474351i \(-0.842683\pi\)
−0.880336 + 0.474351i \(0.842683\pi\)
\(354\) 3.61190e6 1.53189
\(355\) 0 0
\(356\) −6.22594e6 −2.60363
\(357\) −2.17458e6 −0.903036
\(358\) −6.67749e6 −2.75363
\(359\) 585248. 0.239664 0.119832 0.992794i \(-0.461764\pi\)
0.119832 + 0.992794i \(0.461764\pi\)
\(360\) 0 0
\(361\) 2.10712e6 0.850985
\(362\) 7.76773e6 3.11547
\(363\) 131769. 0.0524864
\(364\) 4.71278e6 1.86433
\(365\) 0 0
\(366\) 4.06832e6 1.58750
\(367\) 3.47515e6 1.34681 0.673407 0.739272i \(-0.264830\pi\)
0.673407 + 0.739272i \(0.264830\pi\)
\(368\) 169393. 0.0652041
\(369\) 321612. 0.122961
\(370\) 0 0
\(371\) −2.06274e6 −0.778053
\(372\) 3.01740e6 1.13051
\(373\) 2.08904e6 0.777453 0.388726 0.921353i \(-0.372915\pi\)
0.388726 + 0.921353i \(0.372915\pi\)
\(374\) 1.37873e6 0.509685
\(375\) 0 0
\(376\) −242861. −0.0885906
\(377\) −2.99913e6 −1.08678
\(378\) −1.31094e6 −0.471902
\(379\) −1.41198e6 −0.504931 −0.252466 0.967606i \(-0.581241\pi\)
−0.252466 + 0.967606i \(0.581241\pi\)
\(380\) 0 0
\(381\) −635622. −0.224329
\(382\) −6.07223e6 −2.12907
\(383\) −2.02171e6 −0.704241 −0.352121 0.935955i \(-0.614539\pi\)
−0.352121 + 0.935955i \(0.614539\pi\)
\(384\) −2.78498e6 −0.963814
\(385\) 0 0
\(386\) −3.22608e6 −1.10207
\(387\) 316047. 0.107269
\(388\) 8.27952e6 2.79207
\(389\) −2.12481e6 −0.711943 −0.355971 0.934497i \(-0.615850\pi\)
−0.355971 + 0.934497i \(0.615850\pi\)
\(390\) 0 0
\(391\) 2.81221e6 0.930263
\(392\) 4.08551e6 1.34286
\(393\) 1.39795e6 0.456574
\(394\) 4.64377e6 1.50706
\(395\) 0 0
\(396\) 517531. 0.165844
\(397\) 2.11404e6 0.673188 0.336594 0.941650i \(-0.390725\pi\)
0.336594 + 0.941650i \(0.390725\pi\)
\(398\) −8.65584e6 −2.73906
\(399\) −3.76248e6 −1.18315
\(400\) 0 0
\(401\) 1.00967e6 0.313558 0.156779 0.987634i \(-0.449889\pi\)
0.156779 + 0.987634i \(0.449889\pi\)
\(402\) 1.22541e6 0.378195
\(403\) −2.90194e6 −0.890073
\(404\) −4.70052e6 −1.43282
\(405\) 0 0
\(406\) −1.18001e7 −3.55279
\(407\) −408540. −0.122250
\(408\) 2.13345e6 0.634502
\(409\) −190771. −0.0563904 −0.0281952 0.999602i \(-0.508976\pi\)
−0.0281952 + 0.999602i \(0.508976\pi\)
\(410\) 0 0
\(411\) 635255. 0.185500
\(412\) 5.29453e6 1.53668
\(413\) −8.51002e6 −2.45502
\(414\) 1.69533e6 0.486130
\(415\) 0 0
\(416\) 2.48831e6 0.704970
\(417\) −3.60120e6 −1.01416
\(418\) 2.38550e6 0.667787
\(419\) −4.82015e6 −1.34130 −0.670650 0.741774i \(-0.733985\pi\)
−0.670650 + 0.741774i \(0.733985\pi\)
\(420\) 0 0
\(421\) −1.47210e6 −0.404791 −0.202395 0.979304i \(-0.564873\pi\)
−0.202395 + 0.979304i \(0.564873\pi\)
\(422\) 6.13875e6 1.67803
\(423\) −102681. −0.0279022
\(424\) 2.02373e6 0.546685
\(425\) 0 0
\(426\) 742739. 0.198295
\(427\) −9.58542e6 −2.54414
\(428\) −7.00605e6 −1.84869
\(429\) −497729. −0.130572
\(430\) 0 0
\(431\) −3.75922e6 −0.974774 −0.487387 0.873186i \(-0.662050\pi\)
−0.487387 + 0.873186i \(0.662050\pi\)
\(432\) 54332.8 0.0140072
\(433\) −3.03901e6 −0.778955 −0.389478 0.921036i \(-0.627344\pi\)
−0.389478 + 0.921036i \(0.627344\pi\)
\(434\) −1.14177e7 −2.90973
\(435\) 0 0
\(436\) 4.67567e6 1.17795
\(437\) 4.86571e6 1.21883
\(438\) −6.90824e6 −1.72061
\(439\) −395912. −0.0980477 −0.0490239 0.998798i \(-0.515611\pi\)
−0.0490239 + 0.998798i \(0.515611\pi\)
\(440\) 0 0
\(441\) 1.72734e6 0.422943
\(442\) −5.20787e6 −1.26796
\(443\) 6.25166e6 1.51351 0.756757 0.653697i \(-0.226783\pi\)
0.756757 + 0.653697i \(0.226783\pi\)
\(444\) −1.60457e6 −0.386279
\(445\) 0 0
\(446\) −4.23452e6 −1.00802
\(447\) 246906. 0.0584470
\(448\) 1.02560e7 2.41425
\(449\) −3.94443e6 −0.923355 −0.461678 0.887048i \(-0.652752\pi\)
−0.461678 + 0.887048i \(0.652752\pi\)
\(450\) 0 0
\(451\) 480433. 0.111222
\(452\) 4.87672e6 1.12275
\(453\) 1.00587e6 0.230302
\(454\) 1.02700e7 2.33846
\(455\) 0 0
\(456\) 3.69132e6 0.831323
\(457\) 6.70635e6 1.50209 0.751045 0.660251i \(-0.229550\pi\)
0.751045 + 0.660251i \(0.229550\pi\)
\(458\) 1.09868e7 2.44741
\(459\) 902017. 0.199840
\(460\) 0 0
\(461\) −4.66837e6 −1.02309 −0.511544 0.859257i \(-0.670926\pi\)
−0.511544 + 0.859257i \(0.670926\pi\)
\(462\) −1.95831e6 −0.426852
\(463\) 5.02305e6 1.08897 0.544484 0.838771i \(-0.316725\pi\)
0.544484 + 0.838771i \(0.316725\pi\)
\(464\) 489063. 0.105456
\(465\) 0 0
\(466\) 6.22406e6 1.32773
\(467\) 1.73795e6 0.368762 0.184381 0.982855i \(-0.440972\pi\)
0.184381 + 0.982855i \(0.440972\pi\)
\(468\) −1.95486e6 −0.412574
\(469\) −2.88719e6 −0.606099
\(470\) 0 0
\(471\) 2.08134e6 0.432306
\(472\) 8.34908e6 1.72498
\(473\) 472120. 0.0970285
\(474\) −1.48012e6 −0.302587
\(475\) 0 0
\(476\) −1.27585e7 −2.58096
\(477\) 855625. 0.172182
\(478\) 4.01829e6 0.804398
\(479\) −761980. −0.151742 −0.0758708 0.997118i \(-0.524174\pi\)
−0.0758708 + 0.997118i \(0.524174\pi\)
\(480\) 0 0
\(481\) 1.54317e6 0.304125
\(482\) 5.49781e6 1.07788
\(483\) −3.99438e6 −0.779078
\(484\) 773103. 0.150011
\(485\) 0 0
\(486\) 543777. 0.104431
\(487\) 840046. 0.160502 0.0802510 0.996775i \(-0.474428\pi\)
0.0802510 + 0.996775i \(0.474428\pi\)
\(488\) 9.40413e6 1.78760
\(489\) −1.97201e6 −0.372939
\(490\) 0 0
\(491\) 1.05708e7 1.97882 0.989409 0.145156i \(-0.0463684\pi\)
0.989409 + 0.145156i \(0.0463684\pi\)
\(492\) 1.88693e6 0.351434
\(493\) 8.11928e6 1.50453
\(494\) −9.01069e6 −1.66127
\(495\) 0 0
\(496\) 473214. 0.0863681
\(497\) −1.74997e6 −0.317790
\(498\) −5.53891e6 −1.00081
\(499\) 2.87438e6 0.516765 0.258382 0.966043i \(-0.416811\pi\)
0.258382 + 0.966043i \(0.416811\pi\)
\(500\) 0 0
\(501\) 1.24510e6 0.221621
\(502\) −698145. −0.123648
\(503\) 1.40949e6 0.248395 0.124197 0.992258i \(-0.460364\pi\)
0.124197 + 0.992258i \(0.460364\pi\)
\(504\) −3.03029e6 −0.531384
\(505\) 0 0
\(506\) 2.53253e6 0.439721
\(507\) −1.46158e6 −0.252523
\(508\) −3.72926e6 −0.641156
\(509\) −7.60203e6 −1.30057 −0.650287 0.759688i \(-0.725352\pi\)
−0.650287 + 0.759688i \(0.725352\pi\)
\(510\) 0 0
\(511\) 1.62766e7 2.75747
\(512\) −862682. −0.145437
\(513\) 1.56068e6 0.261830
\(514\) −1.10786e7 −1.84960
\(515\) 0 0
\(516\) 1.85428e6 0.306586
\(517\) −153387. −0.0252384
\(518\) 6.07160e6 0.994212
\(519\) −2.83027e6 −0.461222
\(520\) 0 0
\(521\) −2.35016e6 −0.379318 −0.189659 0.981850i \(-0.560738\pi\)
−0.189659 + 0.981850i \(0.560738\pi\)
\(522\) 4.89467e6 0.786225
\(523\) −8.59353e6 −1.37378 −0.686891 0.726761i \(-0.741025\pi\)
−0.686891 + 0.726761i \(0.741025\pi\)
\(524\) 8.20193e6 1.30493
\(525\) 0 0
\(526\) 3.06265e6 0.482651
\(527\) 7.85616e6 1.23221
\(528\) 81163.7 0.0126700
\(529\) −1.27074e6 −0.197432
\(530\) 0 0
\(531\) 3.52996e6 0.543292
\(532\) −2.20748e7 −3.38157
\(533\) −1.81473e6 −0.276690
\(534\) −9.77213e6 −1.48298
\(535\) 0 0
\(536\) 2.83259e6 0.425865
\(537\) −6.52601e6 −0.976589
\(538\) 1.47862e7 2.20243
\(539\) 2.58035e6 0.382566
\(540\) 0 0
\(541\) −3.76377e6 −0.552878 −0.276439 0.961031i \(-0.589154\pi\)
−0.276439 + 0.961031i \(0.589154\pi\)
\(542\) −8.96726e6 −1.31118
\(543\) 7.59152e6 1.10492
\(544\) −6.73638e6 −0.975954
\(545\) 0 0
\(546\) 7.39709e6 1.06189
\(547\) −8.53624e6 −1.21983 −0.609914 0.792468i \(-0.708796\pi\)
−0.609914 + 0.792468i \(0.708796\pi\)
\(548\) 3.72711e6 0.530176
\(549\) 3.97603e6 0.563014
\(550\) 0 0
\(551\) 1.40480e7 1.97123
\(552\) 3.91883e6 0.547405
\(553\) 3.48732e6 0.484930
\(554\) −3.96260e6 −0.548537
\(555\) 0 0
\(556\) −2.11286e7 −2.89857
\(557\) 5.80583e6 0.792915 0.396457 0.918053i \(-0.370239\pi\)
0.396457 + 0.918053i \(0.370239\pi\)
\(558\) 4.73605e6 0.643919
\(559\) −1.78333e6 −0.241380
\(560\) 0 0
\(561\) 1.34746e6 0.180762
\(562\) −5.21674e6 −0.696720
\(563\) −7.65501e6 −1.01783 −0.508914 0.860817i \(-0.669953\pi\)
−0.508914 + 0.860817i \(0.669953\pi\)
\(564\) −602438. −0.0797471
\(565\) 0 0
\(566\) −8.86154e6 −1.16270
\(567\) −1.28120e6 −0.167363
\(568\) 1.71688e6 0.223290
\(569\) 1.92933e6 0.249819 0.124909 0.992168i \(-0.460136\pi\)
0.124909 + 0.992168i \(0.460136\pi\)
\(570\) 0 0
\(571\) −1.05928e7 −1.35962 −0.679812 0.733386i \(-0.737939\pi\)
−0.679812 + 0.733386i \(0.737939\pi\)
\(572\) −2.92023e6 −0.373187
\(573\) −5.93448e6 −0.755086
\(574\) −7.14005e6 −0.904527
\(575\) 0 0
\(576\) −4.25418e6 −0.534268
\(577\) −3.01519e6 −0.377030 −0.188515 0.982070i \(-0.560367\pi\)
−0.188515 + 0.982070i \(0.560367\pi\)
\(578\) 1.02348e6 0.127426
\(579\) −3.15290e6 −0.390853
\(580\) 0 0
\(581\) 1.30503e7 1.60391
\(582\) 1.29954e7 1.59031
\(583\) 1.27816e6 0.155744
\(584\) −1.59687e7 −1.93749
\(585\) 0 0
\(586\) −7.02449e6 −0.845027
\(587\) 2.91411e6 0.349069 0.174534 0.984651i \(-0.444158\pi\)
0.174534 + 0.984651i \(0.444158\pi\)
\(588\) 1.01345e7 1.20881
\(589\) 1.35928e7 1.61444
\(590\) 0 0
\(591\) 4.53842e6 0.534486
\(592\) −251642. −0.0295107
\(593\) −1.55954e7 −1.82121 −0.910607 0.413274i \(-0.864385\pi\)
−0.910607 + 0.413274i \(0.864385\pi\)
\(594\) 812308. 0.0944615
\(595\) 0 0
\(596\) 1.44862e6 0.167047
\(597\) −8.45948e6 −0.971422
\(598\) −9.56606e6 −1.09391
\(599\) −3.70805e6 −0.422259 −0.211130 0.977458i \(-0.567714\pi\)
−0.211130 + 0.977458i \(0.567714\pi\)
\(600\) 0 0
\(601\) 7.22245e6 0.815640 0.407820 0.913062i \(-0.366289\pi\)
0.407820 + 0.913062i \(0.366289\pi\)
\(602\) −7.01651e6 −0.789096
\(603\) 1.19761e6 0.134129
\(604\) 5.90156e6 0.658225
\(605\) 0 0
\(606\) −7.37785e6 −0.816110
\(607\) −1.06431e7 −1.17246 −0.586228 0.810146i \(-0.699388\pi\)
−0.586228 + 0.810146i \(0.699388\pi\)
\(608\) −1.16553e7 −1.27869
\(609\) −1.15324e7 −1.26001
\(610\) 0 0
\(611\) 579386. 0.0627863
\(612\) 5.29223e6 0.571163
\(613\) −7.11493e6 −0.764750 −0.382375 0.924007i \(-0.624894\pi\)
−0.382375 + 0.924007i \(0.624894\pi\)
\(614\) 2.78495e7 2.98124
\(615\) 0 0
\(616\) −4.52673e6 −0.480655
\(617\) −1.83690e6 −0.194255 −0.0971277 0.995272i \(-0.530966\pi\)
−0.0971277 + 0.995272i \(0.530966\pi\)
\(618\) 8.31019e6 0.875266
\(619\) 1.10798e7 1.16226 0.581132 0.813810i \(-0.302610\pi\)
0.581132 + 0.813810i \(0.302610\pi\)
\(620\) 0 0
\(621\) 1.65687e6 0.172409
\(622\) 2.15399e7 2.23238
\(623\) 2.30242e7 2.37665
\(624\) −306578. −0.0315195
\(625\) 0 0
\(626\) 1.31320e7 1.33935
\(627\) 2.33138e6 0.236834
\(628\) 1.22114e7 1.23557
\(629\) −4.17769e6 −0.421027
\(630\) 0 0
\(631\) 1.34216e7 1.34193 0.670966 0.741488i \(-0.265880\pi\)
0.670966 + 0.741488i \(0.265880\pi\)
\(632\) −3.42137e6 −0.340727
\(633\) 5.99949e6 0.595121
\(634\) −1.47924e7 −1.46156
\(635\) 0 0
\(636\) 5.02004e6 0.492112
\(637\) −9.74670e6 −0.951720
\(638\) 7.31179e6 0.711168
\(639\) 725890. 0.0703265
\(640\) 0 0
\(641\) 8.26487e6 0.794495 0.397247 0.917712i \(-0.369965\pi\)
0.397247 + 0.917712i \(0.369965\pi\)
\(642\) −1.09966e7 −1.05298
\(643\) −9.43394e6 −0.899841 −0.449920 0.893069i \(-0.648548\pi\)
−0.449920 + 0.893069i \(0.648548\pi\)
\(644\) −2.34354e7 −2.22668
\(645\) 0 0
\(646\) 2.43939e7 2.29985
\(647\) −9.83647e6 −0.923802 −0.461901 0.886932i \(-0.652832\pi\)
−0.461901 + 0.886932i \(0.652832\pi\)
\(648\) 1.25697e6 0.117594
\(649\) 5.27315e6 0.491426
\(650\) 0 0
\(651\) −1.11587e7 −1.03195
\(652\) −1.15700e7 −1.06590
\(653\) −2.07669e7 −1.90585 −0.952924 0.303208i \(-0.901942\pi\)
−0.952924 + 0.303208i \(0.901942\pi\)
\(654\) 7.33885e6 0.670940
\(655\) 0 0
\(656\) 295925. 0.0268486
\(657\) −6.75153e6 −0.610223
\(658\) 2.27959e6 0.205255
\(659\) −1.74284e7 −1.56331 −0.781654 0.623713i \(-0.785624\pi\)
−0.781654 + 0.623713i \(0.785624\pi\)
\(660\) 0 0
\(661\) −1.91510e7 −1.70486 −0.852428 0.522844i \(-0.824871\pi\)
−0.852428 + 0.522844i \(0.824871\pi\)
\(662\) 2.86893e7 2.54434
\(663\) −5.08972e6 −0.449687
\(664\) −1.28035e7 −1.12696
\(665\) 0 0
\(666\) −2.51850e6 −0.220017
\(667\) 1.49139e7 1.29800
\(668\) 7.30515e6 0.633415
\(669\) −4.13846e6 −0.357498
\(670\) 0 0
\(671\) 5.93951e6 0.509265
\(672\) 9.56815e6 0.817343
\(673\) −342635. −0.0291605 −0.0145802 0.999894i \(-0.504641\pi\)
−0.0145802 + 0.999894i \(0.504641\pi\)
\(674\) −1.48365e7 −1.25801
\(675\) 0 0
\(676\) −8.57522e6 −0.721737
\(677\) −4.92746e6 −0.413191 −0.206596 0.978426i \(-0.566238\pi\)
−0.206596 + 0.978426i \(0.566238\pi\)
\(678\) 7.65441e6 0.639496
\(679\) −3.06185e7 −2.54865
\(680\) 0 0
\(681\) 1.00370e7 0.829348
\(682\) 7.07484e6 0.582446
\(683\) −1.73541e7 −1.42348 −0.711740 0.702443i \(-0.752093\pi\)
−0.711740 + 0.702443i \(0.752093\pi\)
\(684\) 9.15666e6 0.748336
\(685\) 0 0
\(686\) −8.12491e6 −0.659186
\(687\) 1.07376e7 0.867988
\(688\) 290805. 0.0234223
\(689\) −4.82795e6 −0.387449
\(690\) 0 0
\(691\) 1.50935e7 1.20253 0.601264 0.799051i \(-0.294664\pi\)
0.601264 + 0.799051i \(0.294664\pi\)
\(692\) −1.66055e7 −1.31822
\(693\) −1.91389e6 −0.151385
\(694\) 7.80371e6 0.615039
\(695\) 0 0
\(696\) 1.13143e7 0.885326
\(697\) 4.91286e6 0.383047
\(698\) 2.54156e7 1.97452
\(699\) 6.08287e6 0.470886
\(700\) 0 0
\(701\) 1.70526e7 1.31068 0.655338 0.755336i \(-0.272526\pi\)
0.655338 + 0.755336i \(0.272526\pi\)
\(702\) −3.06832e6 −0.234994
\(703\) −7.22828e6 −0.551628
\(704\) −6.35501e6 −0.483264
\(705\) 0 0
\(706\) −3.79597e7 −2.86623
\(707\) 1.73830e7 1.30791
\(708\) 2.07106e7 1.55278
\(709\) 4.47464e6 0.334305 0.167152 0.985931i \(-0.446543\pi\)
0.167152 + 0.985931i \(0.446543\pi\)
\(710\) 0 0
\(711\) −1.44654e6 −0.107314
\(712\) −2.25888e7 −1.66991
\(713\) 1.44306e7 1.06307
\(714\) −2.00255e7 −1.47007
\(715\) 0 0
\(716\) −3.82888e7 −2.79119
\(717\) 3.92713e6 0.285284
\(718\) 5.38949e6 0.390155
\(719\) 2.05300e7 1.48104 0.740519 0.672036i \(-0.234580\pi\)
0.740519 + 0.672036i \(0.234580\pi\)
\(720\) 0 0
\(721\) −1.95797e7 −1.40271
\(722\) 1.94043e7 1.38534
\(723\) 5.37309e6 0.382277
\(724\) 4.45403e7 3.15796
\(725\) 0 0
\(726\) 1.21345e6 0.0854436
\(727\) −1.49603e7 −1.04979 −0.524897 0.851166i \(-0.675896\pi\)
−0.524897 + 0.851166i \(0.675896\pi\)
\(728\) 1.70987e7 1.19574
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 4.82785e6 0.334165
\(732\) 2.33278e7 1.60915
\(733\) −2.26287e7 −1.55561 −0.777804 0.628507i \(-0.783666\pi\)
−0.777804 + 0.628507i \(0.783666\pi\)
\(734\) 3.20023e7 2.19251
\(735\) 0 0
\(736\) −1.23737e7 −0.841987
\(737\) 1.78902e6 0.121324
\(738\) 2.96169e6 0.200170
\(739\) −1.39790e7 −0.941597 −0.470798 0.882241i \(-0.656034\pi\)
−0.470798 + 0.882241i \(0.656034\pi\)
\(740\) 0 0
\(741\) −8.80628e6 −0.589179
\(742\) −1.89956e7 −1.26661
\(743\) 5.66041e6 0.376163 0.188081 0.982153i \(-0.439773\pi\)
0.188081 + 0.982153i \(0.439773\pi\)
\(744\) 1.09476e7 0.725082
\(745\) 0 0
\(746\) 1.92377e7 1.26563
\(747\) −5.41326e6 −0.354942
\(748\) 7.90567e6 0.516636
\(749\) 2.59091e7 1.68752
\(750\) 0 0
\(751\) −1.77912e7 −1.15108 −0.575540 0.817774i \(-0.695208\pi\)
−0.575540 + 0.817774i \(0.695208\pi\)
\(752\) −94479.5 −0.00609247
\(753\) −682308. −0.0438524
\(754\) −2.76187e7 −1.76919
\(755\) 0 0
\(756\) −7.51692e6 −0.478338
\(757\) −2.55861e7 −1.62280 −0.811398 0.584494i \(-0.801293\pi\)
−0.811398 + 0.584494i \(0.801293\pi\)
\(758\) −1.30028e7 −0.821987
\(759\) 2.47507e6 0.155949
\(760\) 0 0
\(761\) −8.89006e6 −0.556472 −0.278236 0.960513i \(-0.589750\pi\)
−0.278236 + 0.960513i \(0.589750\pi\)
\(762\) −5.85338e6 −0.365190
\(763\) −1.72911e7 −1.07526
\(764\) −3.48182e7 −2.15811
\(765\) 0 0
\(766\) −1.86177e7 −1.14645
\(767\) −1.99182e7 −1.22253
\(768\) −1.05206e7 −0.643633
\(769\) −2.70789e7 −1.65126 −0.825629 0.564213i \(-0.809180\pi\)
−0.825629 + 0.564213i \(0.809180\pi\)
\(770\) 0 0
\(771\) −1.08273e7 −0.655970
\(772\) −1.84984e7 −1.11710
\(773\) 3.17581e7 1.91164 0.955820 0.293953i \(-0.0949708\pi\)
0.955820 + 0.293953i \(0.0949708\pi\)
\(774\) 2.91045e6 0.174626
\(775\) 0 0
\(776\) 3.00395e7 1.79076
\(777\) 5.93387e6 0.352602
\(778\) −1.95671e7 −1.15899
\(779\) 8.50027e6 0.501868
\(780\) 0 0
\(781\) 1.08435e6 0.0636127
\(782\) 2.58974e7 1.51439
\(783\) 4.78363e6 0.278839
\(784\) 1.58938e6 0.0923500
\(785\) 0 0
\(786\) 1.28736e7 0.743265
\(787\) −4.19625e6 −0.241504 −0.120752 0.992683i \(-0.538531\pi\)
−0.120752 + 0.992683i \(0.538531\pi\)
\(788\) 2.66274e7 1.52761
\(789\) 2.99318e6 0.171175
\(790\) 0 0
\(791\) −1.80346e7 −1.02486
\(792\) 1.87769e6 0.106368
\(793\) −2.24352e7 −1.26691
\(794\) 1.94680e7 1.09590
\(795\) 0 0
\(796\) −4.96327e7 −2.77642
\(797\) 320849. 0.0178918 0.00894591 0.999960i \(-0.497152\pi\)
0.00894591 + 0.999960i \(0.497152\pi\)
\(798\) −3.46483e7 −1.92608
\(799\) −1.56852e6 −0.0869208
\(800\) 0 0
\(801\) −9.55044e6 −0.525947
\(802\) 9.29794e6 0.510447
\(803\) −1.00856e7 −0.551968
\(804\) 7.02650e6 0.383353
\(805\) 0 0
\(806\) −2.67237e7 −1.44897
\(807\) 1.44508e7 0.781103
\(808\) −1.70543e7 −0.918978
\(809\) 2.32915e7 1.25120 0.625600 0.780144i \(-0.284854\pi\)
0.625600 + 0.780144i \(0.284854\pi\)
\(810\) 0 0
\(811\) 1.65090e7 0.881389 0.440695 0.897657i \(-0.354732\pi\)
0.440695 + 0.897657i \(0.354732\pi\)
\(812\) −6.76616e7 −3.60124
\(813\) −8.76384e6 −0.465016
\(814\) −3.76221e6 −0.199013
\(815\) 0 0
\(816\) 829973. 0.0436353
\(817\) 8.35319e6 0.437822
\(818\) −1.75680e6 −0.0917990
\(819\) 7.22929e6 0.376605
\(820\) 0 0
\(821\) −3.37457e7 −1.74727 −0.873637 0.486578i \(-0.838245\pi\)
−0.873637 + 0.486578i \(0.838245\pi\)
\(822\) 5.85000e6 0.301979
\(823\) 1.19897e7 0.617036 0.308518 0.951219i \(-0.400167\pi\)
0.308518 + 0.951219i \(0.400167\pi\)
\(824\) 1.92094e7 0.985590
\(825\) 0 0
\(826\) −7.83680e7 −3.99658
\(827\) −2.29575e7 −1.16724 −0.583620 0.812027i \(-0.698364\pi\)
−0.583620 + 0.812027i \(0.698364\pi\)
\(828\) 9.72102e6 0.492761
\(829\) 2.33662e7 1.18087 0.590435 0.807085i \(-0.298956\pi\)
0.590435 + 0.807085i \(0.298956\pi\)
\(830\) 0 0
\(831\) −3.87271e6 −0.194542
\(832\) 2.40046e7 1.20223
\(833\) 2.63864e7 1.31755
\(834\) −3.31631e7 −1.65097
\(835\) 0 0
\(836\) 1.36785e7 0.676895
\(837\) 4.62861e6 0.228369
\(838\) −4.43883e7 −2.18353
\(839\) 1.72966e7 0.848315 0.424157 0.905588i \(-0.360570\pi\)
0.424157 + 0.905588i \(0.360570\pi\)
\(840\) 0 0
\(841\) 2.25475e7 1.09928
\(842\) −1.35564e7 −0.658967
\(843\) −5.09840e6 −0.247096
\(844\) 3.51997e7 1.70091
\(845\) 0 0
\(846\) −945576. −0.0454225
\(847\) −2.85902e6 −0.136933
\(848\) 787286. 0.0375961
\(849\) −8.66052e6 −0.412358
\(850\) 0 0
\(851\) −7.67379e6 −0.363234
\(852\) 4.25887e6 0.201000
\(853\) 9.73772e6 0.458231 0.229116 0.973399i \(-0.426417\pi\)
0.229116 + 0.973399i \(0.426417\pi\)
\(854\) −8.82712e7 −4.14166
\(855\) 0 0
\(856\) −2.54191e7 −1.18570
\(857\) −1.49824e7 −0.696832 −0.348416 0.937340i \(-0.613280\pi\)
−0.348416 + 0.937340i \(0.613280\pi\)
\(858\) −4.58353e6 −0.212560
\(859\) −2.41887e7 −1.11848 −0.559242 0.829004i \(-0.688908\pi\)
−0.559242 + 0.829004i \(0.688908\pi\)
\(860\) 0 0
\(861\) −6.97808e6 −0.320795
\(862\) −3.46183e7 −1.58685
\(863\) 1.77029e7 0.809126 0.404563 0.914510i \(-0.367424\pi\)
0.404563 + 0.914510i \(0.367424\pi\)
\(864\) −3.96887e6 −0.180877
\(865\) 0 0
\(866\) −2.79860e7 −1.26808
\(867\) 1.00026e6 0.0451923
\(868\) −6.54690e7 −2.94942
\(869\) −2.16088e6 −0.0970692
\(870\) 0 0
\(871\) −6.75764e6 −0.301821
\(872\) 1.69641e7 0.755510
\(873\) 1.27006e7 0.564012
\(874\) 4.48078e7 1.98415
\(875\) 0 0
\(876\) −3.96119e7 −1.74408
\(877\) −3.95853e7 −1.73794 −0.868970 0.494864i \(-0.835218\pi\)
−0.868970 + 0.494864i \(0.835218\pi\)
\(878\) −3.64592e6 −0.159614
\(879\) −6.86514e6 −0.299693
\(880\) 0 0
\(881\) −1.06184e7 −0.460914 −0.230457 0.973082i \(-0.574022\pi\)
−0.230457 + 0.973082i \(0.574022\pi\)
\(882\) 1.59069e7 0.688517
\(883\) 798967. 0.0344848 0.0172424 0.999851i \(-0.494511\pi\)
0.0172424 + 0.999851i \(0.494511\pi\)
\(884\) −2.98619e7 −1.28525
\(885\) 0 0
\(886\) 5.75710e7 2.46388
\(887\) −2.53565e7 −1.08213 −0.541065 0.840981i \(-0.681979\pi\)
−0.541065 + 0.840981i \(0.681979\pi\)
\(888\) −5.82165e6 −0.247750
\(889\) 1.37912e7 0.585259
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −2.42808e7 −1.02176
\(893\) −2.71387e6 −0.113883
\(894\) 2.27373e6 0.0951470
\(895\) 0 0
\(896\) 6.04261e7 2.51452
\(897\) −9.34905e6 −0.387960
\(898\) −3.63239e7 −1.50315
\(899\) 4.16633e7 1.71931
\(900\) 0 0
\(901\) 1.30703e7 0.536381
\(902\) 4.42426e6 0.181061
\(903\) −6.85734e6 −0.279857
\(904\) 1.76936e7 0.720102
\(905\) 0 0
\(906\) 9.26298e6 0.374913
\(907\) 2.39197e6 0.0965468 0.0482734 0.998834i \(-0.484628\pi\)
0.0482734 + 0.998834i \(0.484628\pi\)
\(908\) 5.88882e7 2.37036
\(909\) −7.21049e6 −0.289438
\(910\) 0 0
\(911\) 2.84437e7 1.13551 0.567754 0.823198i \(-0.307813\pi\)
0.567754 + 0.823198i \(0.307813\pi\)
\(912\) 1.43603e6 0.0571709
\(913\) −8.08648e6 −0.321057
\(914\) 6.17582e7 2.44528
\(915\) 0 0
\(916\) 6.29984e7 2.48079
\(917\) −3.03316e7 −1.19117
\(918\) 8.30659e6 0.325324
\(919\) −1.05298e7 −0.411273 −0.205636 0.978628i \(-0.565926\pi\)
−0.205636 + 0.978628i \(0.565926\pi\)
\(920\) 0 0
\(921\) 2.72177e7 1.05731
\(922\) −4.29906e7 −1.66550
\(923\) −4.09591e6 −0.158251
\(924\) −1.12290e7 −0.432673
\(925\) 0 0
\(926\) 4.62568e7 1.77275
\(927\) 8.12168e6 0.310418
\(928\) −3.57248e7 −1.36176
\(929\) 1.19595e7 0.454647 0.227323 0.973819i \(-0.427003\pi\)
0.227323 + 0.973819i \(0.427003\pi\)
\(930\) 0 0
\(931\) 4.56540e7 1.72625
\(932\) 3.56888e7 1.34584
\(933\) 2.10513e7 0.791725
\(934\) 1.60047e7 0.600315
\(935\) 0 0
\(936\) −7.09257e6 −0.264615
\(937\) −2.02229e7 −0.752480 −0.376240 0.926522i \(-0.622783\pi\)
−0.376240 + 0.926522i \(0.622783\pi\)
\(938\) −2.65879e7 −0.986681
\(939\) 1.28341e7 0.475009
\(940\) 0 0
\(941\) 2.13506e7 0.786026 0.393013 0.919533i \(-0.371433\pi\)
0.393013 + 0.919533i \(0.371433\pi\)
\(942\) 1.91669e7 0.703759
\(943\) 9.02418e6 0.330467
\(944\) 3.24802e6 0.118628
\(945\) 0 0
\(946\) 4.34771e6 0.157955
\(947\) 3.13109e6 0.113454 0.0567271 0.998390i \(-0.481934\pi\)
0.0567271 + 0.998390i \(0.481934\pi\)
\(948\) −8.48701e6 −0.306714
\(949\) 3.80962e7 1.37314
\(950\) 0 0
\(951\) −1.44569e7 −0.518350
\(952\) −4.62900e7 −1.65537
\(953\) 4.65632e7 1.66077 0.830387 0.557188i \(-0.188120\pi\)
0.830387 + 0.557188i \(0.188120\pi\)
\(954\) 7.87937e6 0.280298
\(955\) 0 0
\(956\) 2.30409e7 0.815370
\(957\) 7.14592e6 0.252219
\(958\) −7.01700e6 −0.247023
\(959\) −1.37832e7 −0.483955
\(960\) 0 0
\(961\) 1.16840e7 0.408116
\(962\) 1.42109e7 0.495090
\(963\) −1.07471e7 −0.373445
\(964\) 3.15245e7 1.09258
\(965\) 0 0
\(966\) −3.67838e7 −1.26828
\(967\) 3.64320e7 1.25290 0.626450 0.779462i \(-0.284507\pi\)
0.626450 + 0.779462i \(0.284507\pi\)
\(968\) 2.80495e6 0.0962135
\(969\) 2.38405e7 0.815653
\(970\) 0 0
\(971\) 2.45693e7 0.836267 0.418134 0.908386i \(-0.362684\pi\)
0.418134 + 0.908386i \(0.362684\pi\)
\(972\) 3.11802e6 0.105855
\(973\) 7.81359e7 2.64587
\(974\) 7.73591e6 0.261285
\(975\) 0 0
\(976\) 3.65847e6 0.122935
\(977\) −4.28557e7 −1.43639 −0.718194 0.695842i \(-0.755031\pi\)
−0.718194 + 0.695842i \(0.755031\pi\)
\(978\) −1.81601e7 −0.607115
\(979\) −1.42667e7 −0.475737
\(980\) 0 0
\(981\) 7.17237e6 0.237952
\(982\) 9.73459e7 3.22136
\(983\) −3.25942e7 −1.07586 −0.537931 0.842989i \(-0.680794\pi\)
−0.537931 + 0.842989i \(0.680794\pi\)
\(984\) 6.84611e6 0.225401
\(985\) 0 0
\(986\) 7.47697e7 2.44925
\(987\) 2.22788e6 0.0727946
\(988\) −5.16674e7 −1.68393
\(989\) 8.86804e6 0.288295
\(990\) 0 0
\(991\) −1.81427e7 −0.586836 −0.293418 0.955984i \(-0.594793\pi\)
−0.293418 + 0.955984i \(0.594793\pi\)
\(992\) −3.45671e7 −1.11528
\(993\) 2.80384e7 0.902362
\(994\) −1.61154e7 −0.517337
\(995\) 0 0
\(996\) −3.17602e7 −1.01446
\(997\) −9.28807e6 −0.295929 −0.147965 0.988993i \(-0.547272\pi\)
−0.147965 + 0.988993i \(0.547272\pi\)
\(998\) 2.64699e7 0.841252
\(999\) −2.46137e6 −0.0780303
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.t.1.9 10
5.4 even 2 825.6.a.u.1.2 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.9 10 1.1 even 1 trivial
825.6.a.u.1.2 yes 10 5.4 even 2