Properties

Label 825.6.a.s.1.4
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.92452\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.92452 q^{2} -9.00000 q^{3} -28.2962 q^{4} +17.3207 q^{6} +59.1120 q^{7} +116.041 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-1.92452 q^{2} -9.00000 q^{3} -28.2962 q^{4} +17.3207 q^{6} +59.1120 q^{7} +116.041 q^{8} +81.0000 q^{9} -121.000 q^{11} +254.666 q^{12} -599.621 q^{13} -113.762 q^{14} +682.155 q^{16} +2271.97 q^{17} -155.886 q^{18} -2998.77 q^{19} -532.008 q^{21} +232.867 q^{22} +1571.78 q^{23} -1044.37 q^{24} +1153.98 q^{26} -729.000 q^{27} -1672.65 q^{28} +9020.65 q^{29} +1838.01 q^{31} -5026.14 q^{32} +1089.00 q^{33} -4372.45 q^{34} -2291.99 q^{36} -3881.95 q^{37} +5771.20 q^{38} +5396.59 q^{39} -2147.54 q^{41} +1023.86 q^{42} -2613.13 q^{43} +3423.84 q^{44} -3024.92 q^{46} +8379.51 q^{47} -6139.40 q^{48} -13312.8 q^{49} -20447.7 q^{51} +16967.0 q^{52} -11261.5 q^{53} +1402.98 q^{54} +6859.43 q^{56} +26988.9 q^{57} -17360.4 q^{58} -2505.76 q^{59} -42835.6 q^{61} -3537.29 q^{62} +4788.07 q^{63} -12156.1 q^{64} -2095.80 q^{66} -8325.99 q^{67} -64288.2 q^{68} -14146.0 q^{69} +13440.4 q^{71} +9399.34 q^{72} -19484.1 q^{73} +7470.89 q^{74} +84853.9 q^{76} -7152.55 q^{77} -10385.8 q^{78} -20416.2 q^{79} +6561.00 q^{81} +4132.99 q^{82} -82162.1 q^{83} +15053.8 q^{84} +5029.03 q^{86} -81185.9 q^{87} -14041.0 q^{88} +38458.1 q^{89} -35444.8 q^{91} -44475.4 q^{92} -16542.1 q^{93} -16126.5 q^{94} +45235.3 q^{96} -139725. q^{97} +25620.7 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92452 −0.340210 −0.170105 0.985426i \(-0.554411\pi\)
−0.170105 + 0.985426i \(0.554411\pi\)
\(3\) −9.00000 −0.577350
\(4\) −28.2962 −0.884257
\(5\) 0 0
\(6\) 17.3207 0.196420
\(7\) 59.1120 0.455964 0.227982 0.973665i \(-0.426787\pi\)
0.227982 + 0.973665i \(0.426787\pi\)
\(8\) 116.041 0.641044
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 254.666 0.510526
\(13\) −599.621 −0.984053 −0.492027 0.870580i \(-0.663744\pi\)
−0.492027 + 0.870580i \(0.663744\pi\)
\(14\) −113.762 −0.155124
\(15\) 0 0
\(16\) 682.155 0.666167
\(17\) 2271.97 1.90669 0.953346 0.301880i \(-0.0976143\pi\)
0.953346 + 0.301880i \(0.0976143\pi\)
\(18\) −155.886 −0.113403
\(19\) −2998.77 −1.90572 −0.952861 0.303408i \(-0.901875\pi\)
−0.952861 + 0.303408i \(0.901875\pi\)
\(20\) 0 0
\(21\) −532.008 −0.263251
\(22\) 232.867 0.102577
\(23\) 1571.78 0.619544 0.309772 0.950811i \(-0.399747\pi\)
0.309772 + 0.950811i \(0.399747\pi\)
\(24\) −1044.37 −0.370107
\(25\) 0 0
\(26\) 1153.98 0.334785
\(27\) −729.000 −0.192450
\(28\) −1672.65 −0.403189
\(29\) 9020.65 1.99179 0.995894 0.0905310i \(-0.0288564\pi\)
0.995894 + 0.0905310i \(0.0288564\pi\)
\(30\) 0 0
\(31\) 1838.01 0.343514 0.171757 0.985139i \(-0.445056\pi\)
0.171757 + 0.985139i \(0.445056\pi\)
\(32\) −5026.14 −0.867681
\(33\) 1089.00 0.174078
\(34\) −4372.45 −0.648676
\(35\) 0 0
\(36\) −2291.99 −0.294752
\(37\) −3881.95 −0.466171 −0.233086 0.972456i \(-0.574882\pi\)
−0.233086 + 0.972456i \(0.574882\pi\)
\(38\) 5771.20 0.648346
\(39\) 5396.59 0.568143
\(40\) 0 0
\(41\) −2147.54 −0.199518 −0.0997590 0.995012i \(-0.531807\pi\)
−0.0997590 + 0.995012i \(0.531807\pi\)
\(42\) 1023.86 0.0895606
\(43\) −2613.13 −0.215521 −0.107761 0.994177i \(-0.534368\pi\)
−0.107761 + 0.994177i \(0.534368\pi\)
\(44\) 3423.84 0.266614
\(45\) 0 0
\(46\) −3024.92 −0.210775
\(47\) 8379.51 0.553317 0.276659 0.960968i \(-0.410773\pi\)
0.276659 + 0.960968i \(0.410773\pi\)
\(48\) −6139.40 −0.384612
\(49\) −13312.8 −0.792097
\(50\) 0 0
\(51\) −20447.7 −1.10083
\(52\) 16967.0 0.870156
\(53\) −11261.5 −0.550690 −0.275345 0.961345i \(-0.588792\pi\)
−0.275345 + 0.961345i \(0.588792\pi\)
\(54\) 1402.98 0.0654735
\(55\) 0 0
\(56\) 6859.43 0.292293
\(57\) 26988.9 1.10027
\(58\) −17360.4 −0.677626
\(59\) −2505.76 −0.0937152 −0.0468576 0.998902i \(-0.514921\pi\)
−0.0468576 + 0.998902i \(0.514921\pi\)
\(60\) 0 0
\(61\) −42835.6 −1.47394 −0.736970 0.675925i \(-0.763744\pi\)
−0.736970 + 0.675925i \(0.763744\pi\)
\(62\) −3537.29 −0.116867
\(63\) 4788.07 0.151988
\(64\) −12156.1 −0.370974
\(65\) 0 0
\(66\) −2095.80 −0.0592230
\(67\) −8325.99 −0.226594 −0.113297 0.993561i \(-0.536141\pi\)
−0.113297 + 0.993561i \(0.536141\pi\)
\(68\) −64288.2 −1.68601
\(69\) −14146.0 −0.357694
\(70\) 0 0
\(71\) 13440.4 0.316422 0.158211 0.987405i \(-0.449427\pi\)
0.158211 + 0.987405i \(0.449427\pi\)
\(72\) 9399.34 0.213681
\(73\) −19484.1 −0.427931 −0.213966 0.976841i \(-0.568638\pi\)
−0.213966 + 0.976841i \(0.568638\pi\)
\(74\) 7470.89 0.158596
\(75\) 0 0
\(76\) 84853.9 1.68515
\(77\) −7152.55 −0.137478
\(78\) −10385.8 −0.193288
\(79\) −20416.2 −0.368050 −0.184025 0.982922i \(-0.558913\pi\)
−0.184025 + 0.982922i \(0.558913\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 4132.99 0.0678781
\(83\) −82162.1 −1.30911 −0.654555 0.756014i \(-0.727144\pi\)
−0.654555 + 0.756014i \(0.727144\pi\)
\(84\) 15053.8 0.232781
\(85\) 0 0
\(86\) 5029.03 0.0733226
\(87\) −81185.9 −1.14996
\(88\) −14041.0 −0.193282
\(89\) 38458.1 0.514651 0.257325 0.966325i \(-0.417159\pi\)
0.257325 + 0.966325i \(0.417159\pi\)
\(90\) 0 0
\(91\) −35444.8 −0.448693
\(92\) −44475.4 −0.547836
\(93\) −16542.1 −0.198328
\(94\) −16126.5 −0.188244
\(95\) 0 0
\(96\) 45235.3 0.500956
\(97\) −139725. −1.50780 −0.753901 0.656988i \(-0.771830\pi\)
−0.753901 + 0.656988i \(0.771830\pi\)
\(98\) 25620.7 0.269480
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −129073. −1.25902 −0.629511 0.776991i \(-0.716745\pi\)
−0.629511 + 0.776991i \(0.716745\pi\)
\(102\) 39352.1 0.374513
\(103\) 26675.0 0.247749 0.123875 0.992298i \(-0.460468\pi\)
0.123875 + 0.992298i \(0.460468\pi\)
\(104\) −69580.8 −0.630821
\(105\) 0 0
\(106\) 21673.0 0.187350
\(107\) 116793. 0.986180 0.493090 0.869978i \(-0.335867\pi\)
0.493090 + 0.869978i \(0.335867\pi\)
\(108\) 20627.9 0.170175
\(109\) 148976. 1.20102 0.600509 0.799618i \(-0.294965\pi\)
0.600509 + 0.799618i \(0.294965\pi\)
\(110\) 0 0
\(111\) 34937.5 0.269144
\(112\) 40323.5 0.303748
\(113\) −155580. −1.14619 −0.573095 0.819489i \(-0.694258\pi\)
−0.573095 + 0.819489i \(0.694258\pi\)
\(114\) −51940.8 −0.374323
\(115\) 0 0
\(116\) −255250. −1.76125
\(117\) −48569.3 −0.328018
\(118\) 4822.39 0.0318829
\(119\) 134301. 0.869382
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 82437.9 0.501450
\(123\) 19327.9 0.115192
\(124\) −52008.8 −0.303754
\(125\) 0 0
\(126\) −9214.74 −0.0517079
\(127\) −193163. −1.06271 −0.531354 0.847150i \(-0.678316\pi\)
−0.531354 + 0.847150i \(0.678316\pi\)
\(128\) 184231. 0.993890
\(129\) 23518.2 0.124431
\(130\) 0 0
\(131\) 337105. 1.71628 0.858138 0.513419i \(-0.171621\pi\)
0.858138 + 0.513419i \(0.171621\pi\)
\(132\) −30814.6 −0.153929
\(133\) −177263. −0.868940
\(134\) 16023.5 0.0770897
\(135\) 0 0
\(136\) 263642. 1.22227
\(137\) 349973. 1.59306 0.796531 0.604598i \(-0.206666\pi\)
0.796531 + 0.604598i \(0.206666\pi\)
\(138\) 27224.3 0.121691
\(139\) 343788. 1.50923 0.754613 0.656171i \(-0.227825\pi\)
0.754613 + 0.656171i \(0.227825\pi\)
\(140\) 0 0
\(141\) −75415.6 −0.319458
\(142\) −25866.3 −0.107650
\(143\) 72554.2 0.296703
\(144\) 55254.6 0.222056
\(145\) 0 0
\(146\) 37497.6 0.145587
\(147\) 119815. 0.457317
\(148\) 109844. 0.412215
\(149\) 447504. 1.65132 0.825661 0.564167i \(-0.190803\pi\)
0.825661 + 0.564167i \(0.190803\pi\)
\(150\) 0 0
\(151\) 496536. 1.77218 0.886091 0.463510i \(-0.153410\pi\)
0.886091 + 0.463510i \(0.153410\pi\)
\(152\) −347981. −1.22165
\(153\) 184030. 0.635564
\(154\) 13765.2 0.0467715
\(155\) 0 0
\(156\) −152703. −0.502385
\(157\) 365502. 1.18342 0.591712 0.806150i \(-0.298452\pi\)
0.591712 + 0.806150i \(0.298452\pi\)
\(158\) 39291.3 0.125214
\(159\) 101354. 0.317941
\(160\) 0 0
\(161\) 92911.0 0.282490
\(162\) −12626.8 −0.0378011
\(163\) −536870. −1.58271 −0.791353 0.611359i \(-0.790623\pi\)
−0.791353 + 0.611359i \(0.790623\pi\)
\(164\) 60767.3 0.176425
\(165\) 0 0
\(166\) 158123. 0.445373
\(167\) −158899. −0.440890 −0.220445 0.975399i \(-0.570751\pi\)
−0.220445 + 0.975399i \(0.570751\pi\)
\(168\) −61734.9 −0.168755
\(169\) −11747.5 −0.0316395
\(170\) 0 0
\(171\) −242901. −0.635240
\(172\) 73941.8 0.190576
\(173\) 19966.6 0.0507211 0.0253606 0.999678i \(-0.491927\pi\)
0.0253606 + 0.999678i \(0.491927\pi\)
\(174\) 156244. 0.391228
\(175\) 0 0
\(176\) −82540.8 −0.200857
\(177\) 22551.9 0.0541065
\(178\) −74013.3 −0.175089
\(179\) 153145. 0.357247 0.178624 0.983917i \(-0.442836\pi\)
0.178624 + 0.983917i \(0.442836\pi\)
\(180\) 0 0
\(181\) −790865. −1.79435 −0.897173 0.441680i \(-0.854383\pi\)
−0.897173 + 0.441680i \(0.854383\pi\)
\(182\) 68214.2 0.152650
\(183\) 385520. 0.850980
\(184\) 182391. 0.397155
\(185\) 0 0
\(186\) 31835.6 0.0674731
\(187\) −274908. −0.574889
\(188\) −237109. −0.489274
\(189\) −43092.6 −0.0877503
\(190\) 0 0
\(191\) 612916. 1.21568 0.607838 0.794061i \(-0.292037\pi\)
0.607838 + 0.794061i \(0.292037\pi\)
\(192\) 109405. 0.214182
\(193\) −884585. −1.70941 −0.854705 0.519114i \(-0.826262\pi\)
−0.854705 + 0.519114i \(0.826262\pi\)
\(194\) 268903. 0.512970
\(195\) 0 0
\(196\) 376701. 0.700417
\(197\) 14105.8 0.0258960 0.0129480 0.999916i \(-0.495878\pi\)
0.0129480 + 0.999916i \(0.495878\pi\)
\(198\) 18862.2 0.0341924
\(199\) −104501. −0.187063 −0.0935316 0.995616i \(-0.529816\pi\)
−0.0935316 + 0.995616i \(0.529816\pi\)
\(200\) 0 0
\(201\) 74933.9 0.130824
\(202\) 248404. 0.428332
\(203\) 533229. 0.908183
\(204\) 578594. 0.973416
\(205\) 0 0
\(206\) −51336.7 −0.0842868
\(207\) 127314. 0.206515
\(208\) −409035. −0.655544
\(209\) 362851. 0.574597
\(210\) 0 0
\(211\) 1.08804e6 1.68244 0.841221 0.540692i \(-0.181837\pi\)
0.841221 + 0.540692i \(0.181837\pi\)
\(212\) 318658. 0.486952
\(213\) −120964. −0.182686
\(214\) −224770. −0.335509
\(215\) 0 0
\(216\) −84594.1 −0.123369
\(217\) 108648. 0.156630
\(218\) −286707. −0.408599
\(219\) 175357. 0.247066
\(220\) 0 0
\(221\) −1.36232e6 −1.87629
\(222\) −67238.0 −0.0915656
\(223\) 818867. 1.10268 0.551342 0.834279i \(-0.314116\pi\)
0.551342 + 0.834279i \(0.314116\pi\)
\(224\) −297105. −0.395631
\(225\) 0 0
\(226\) 299416. 0.389946
\(227\) 409868. 0.527934 0.263967 0.964532i \(-0.414969\pi\)
0.263967 + 0.964532i \(0.414969\pi\)
\(228\) −763685. −0.972920
\(229\) −29456.0 −0.0371180 −0.0185590 0.999828i \(-0.505908\pi\)
−0.0185590 + 0.999828i \(0.505908\pi\)
\(230\) 0 0
\(231\) 64372.9 0.0793731
\(232\) 1.04677e6 1.27682
\(233\) 211115. 0.254758 0.127379 0.991854i \(-0.459344\pi\)
0.127379 + 0.991854i \(0.459344\pi\)
\(234\) 93472.6 0.111595
\(235\) 0 0
\(236\) 70903.6 0.0828683
\(237\) 183745. 0.212494
\(238\) −258464. −0.295773
\(239\) 879085. 0.995488 0.497744 0.867324i \(-0.334162\pi\)
0.497744 + 0.867324i \(0.334162\pi\)
\(240\) 0 0
\(241\) −830172. −0.920715 −0.460358 0.887734i \(-0.652279\pi\)
−0.460358 + 0.887734i \(0.652279\pi\)
\(242\) −28176.9 −0.0309282
\(243\) −59049.0 −0.0641500
\(244\) 1.21208e6 1.30334
\(245\) 0 0
\(246\) −37196.9 −0.0391894
\(247\) 1.79813e6 1.87533
\(248\) 213285. 0.220207
\(249\) 739458. 0.755815
\(250\) 0 0
\(251\) −96982.6 −0.0971649 −0.0485825 0.998819i \(-0.515470\pi\)
−0.0485825 + 0.998819i \(0.515470\pi\)
\(252\) −135484. −0.134396
\(253\) −190185. −0.186800
\(254\) 371745. 0.361544
\(255\) 0 0
\(256\) 34437.4 0.0328421
\(257\) 369343. 0.348817 0.174408 0.984673i \(-0.444199\pi\)
0.174408 + 0.984673i \(0.444199\pi\)
\(258\) −45261.3 −0.0423328
\(259\) −229470. −0.212557
\(260\) 0 0
\(261\) 730673. 0.663929
\(262\) −648765. −0.583895
\(263\) −988642. −0.881353 −0.440676 0.897666i \(-0.645261\pi\)
−0.440676 + 0.897666i \(0.645261\pi\)
\(264\) 126369. 0.111591
\(265\) 0 0
\(266\) 341147. 0.295622
\(267\) −346123. −0.297134
\(268\) 235594. 0.200367
\(269\) 1.36005e6 1.14598 0.572988 0.819564i \(-0.305784\pi\)
0.572988 + 0.819564i \(0.305784\pi\)
\(270\) 0 0
\(271\) −219383. −0.181460 −0.0907298 0.995876i \(-0.528920\pi\)
−0.0907298 + 0.995876i \(0.528920\pi\)
\(272\) 1.54984e6 1.27018
\(273\) 319003. 0.259053
\(274\) −673529. −0.541976
\(275\) 0 0
\(276\) 400279. 0.316293
\(277\) −1.78416e6 −1.39713 −0.698563 0.715549i \(-0.746177\pi\)
−0.698563 + 0.715549i \(0.746177\pi\)
\(278\) −661627. −0.513454
\(279\) 148879. 0.114505
\(280\) 0 0
\(281\) −1.01028e6 −0.763264 −0.381632 0.924314i \(-0.624638\pi\)
−0.381632 + 0.924314i \(0.624638\pi\)
\(282\) 145139. 0.108683
\(283\) 270967. 0.201118 0.100559 0.994931i \(-0.467937\pi\)
0.100559 + 0.994931i \(0.467937\pi\)
\(284\) −380313. −0.279798
\(285\) 0 0
\(286\) −139632. −0.100941
\(287\) −126945. −0.0909730
\(288\) −407118. −0.289227
\(289\) 3.74199e6 2.63547
\(290\) 0 0
\(291\) 1.25752e6 0.870530
\(292\) 551327. 0.378401
\(293\) −874428. −0.595052 −0.297526 0.954714i \(-0.596162\pi\)
−0.297526 + 0.954714i \(0.596162\pi\)
\(294\) −230586. −0.155584
\(295\) 0 0
\(296\) −450466. −0.298836
\(297\) 88209.0 0.0580259
\(298\) −861231. −0.561796
\(299\) −942472. −0.609664
\(300\) 0 0
\(301\) −154467. −0.0982700
\(302\) −955594. −0.602915
\(303\) 1.16166e6 0.726897
\(304\) −2.04563e6 −1.26953
\(305\) 0 0
\(306\) −354169. −0.216225
\(307\) 2.20038e6 1.33245 0.666225 0.745750i \(-0.267909\pi\)
0.666225 + 0.745750i \(0.267909\pi\)
\(308\) 202390. 0.121566
\(309\) −240075. −0.143038
\(310\) 0 0
\(311\) 420029. 0.246251 0.123126 0.992391i \(-0.460708\pi\)
0.123126 + 0.992391i \(0.460708\pi\)
\(312\) 626227. 0.364205
\(313\) 342934. 0.197856 0.0989282 0.995095i \(-0.468459\pi\)
0.0989282 + 0.995095i \(0.468459\pi\)
\(314\) −703415. −0.402613
\(315\) 0 0
\(316\) 577700. 0.325450
\(317\) 1.83293e6 1.02446 0.512232 0.858847i \(-0.328819\pi\)
0.512232 + 0.858847i \(0.328819\pi\)
\(318\) −195057. −0.108167
\(319\) −1.09150e6 −0.600546
\(320\) 0 0
\(321\) −1.05113e6 −0.569371
\(322\) −178809. −0.0961059
\(323\) −6.81312e6 −3.63362
\(324\) −185652. −0.0982508
\(325\) 0 0
\(326\) 1.03322e6 0.538453
\(327\) −1.34078e6 −0.693408
\(328\) −249204. −0.127900
\(329\) 495329. 0.252293
\(330\) 0 0
\(331\) 1.42543e6 0.715114 0.357557 0.933891i \(-0.383610\pi\)
0.357557 + 0.933891i \(0.383610\pi\)
\(332\) 2.32488e6 1.15759
\(333\) −314438. −0.155390
\(334\) 305805. 0.149995
\(335\) 0 0
\(336\) −362912. −0.175369
\(337\) 3.93186e6 1.88592 0.942961 0.332903i \(-0.108028\pi\)
0.942961 + 0.332903i \(0.108028\pi\)
\(338\) 22608.3 0.0107641
\(339\) 1.40022e6 0.661754
\(340\) 0 0
\(341\) −222399. −0.103573
\(342\) 467467. 0.216115
\(343\) −1.78044e6 −0.817131
\(344\) −303231. −0.138159
\(345\) 0 0
\(346\) −38426.1 −0.0172558
\(347\) 338208. 0.150786 0.0753929 0.997154i \(-0.475979\pi\)
0.0753929 + 0.997154i \(0.475979\pi\)
\(348\) 2.29725e6 1.01686
\(349\) −2.82994e6 −1.24369 −0.621846 0.783139i \(-0.713617\pi\)
−0.621846 + 0.783139i \(0.713617\pi\)
\(350\) 0 0
\(351\) 437124. 0.189381
\(352\) 608163. 0.261616
\(353\) 4.21459e6 1.80019 0.900096 0.435691i \(-0.143496\pi\)
0.900096 + 0.435691i \(0.143496\pi\)
\(354\) −43401.5 −0.0184076
\(355\) 0 0
\(356\) −1.08822e6 −0.455083
\(357\) −1.20871e6 −0.501938
\(358\) −294730. −0.121539
\(359\) 1.88776e6 0.773057 0.386529 0.922277i \(-0.373674\pi\)
0.386529 + 0.922277i \(0.373674\pi\)
\(360\) 0 0
\(361\) 6.51653e6 2.63177
\(362\) 1.52204e6 0.610455
\(363\) −131769. −0.0524864
\(364\) 1.00295e6 0.396760
\(365\) 0 0
\(366\) −741941. −0.289512
\(367\) 376896. 0.146068 0.0730342 0.997329i \(-0.476732\pi\)
0.0730342 + 0.997329i \(0.476732\pi\)
\(368\) 1.07220e6 0.412720
\(369\) −173951. −0.0665060
\(370\) 0 0
\(371\) −665691. −0.251095
\(372\) 468079. 0.175373
\(373\) −2.55080e6 −0.949300 −0.474650 0.880175i \(-0.657425\pi\)
−0.474650 + 0.880175i \(0.657425\pi\)
\(374\) 529067. 0.195583
\(375\) 0 0
\(376\) 972369. 0.354700
\(377\) −5.40897e6 −1.96002
\(378\) 82932.6 0.0298535
\(379\) 3.97187e6 1.42035 0.710177 0.704023i \(-0.248615\pi\)
0.710177 + 0.704023i \(0.248615\pi\)
\(380\) 0 0
\(381\) 1.73846e6 0.613554
\(382\) −1.17957e6 −0.413585
\(383\) 1.85461e6 0.646035 0.323018 0.946393i \(-0.395303\pi\)
0.323018 + 0.946393i \(0.395303\pi\)
\(384\) −1.65808e6 −0.573822
\(385\) 0 0
\(386\) 1.70240e6 0.581559
\(387\) −211664. −0.0718405
\(388\) 3.95369e6 1.33329
\(389\) −992340. −0.332496 −0.166248 0.986084i \(-0.553165\pi\)
−0.166248 + 0.986084i \(0.553165\pi\)
\(390\) 0 0
\(391\) 3.57104e6 1.18128
\(392\) −1.54483e6 −0.507769
\(393\) −3.03395e6 −0.990892
\(394\) −27146.9 −0.00881009
\(395\) 0 0
\(396\) 277331. 0.0888712
\(397\) 148359. 0.0472430 0.0236215 0.999721i \(-0.492480\pi\)
0.0236215 + 0.999721i \(0.492480\pi\)
\(398\) 201115. 0.0636408
\(399\) 1.59537e6 0.501683
\(400\) 0 0
\(401\) 3.81078e6 1.18346 0.591729 0.806137i \(-0.298445\pi\)
0.591729 + 0.806137i \(0.298445\pi\)
\(402\) −144212. −0.0445077
\(403\) −1.10211e6 −0.338036
\(404\) 3.65229e6 1.11330
\(405\) 0 0
\(406\) −1.02621e6 −0.308973
\(407\) 469716. 0.140556
\(408\) −2.37278e6 −0.705679
\(409\) −74545.8 −0.0220351 −0.0110175 0.999939i \(-0.503507\pi\)
−0.0110175 + 0.999939i \(0.503507\pi\)
\(410\) 0 0
\(411\) −3.14975e6 −0.919754
\(412\) −754803. −0.219074
\(413\) −148121. −0.0427307
\(414\) −245019. −0.0702584
\(415\) 0 0
\(416\) 3.01378e6 0.853844
\(417\) −3.09409e6 −0.871351
\(418\) −698315. −0.195484
\(419\) 1.48992e6 0.414599 0.207299 0.978278i \(-0.433533\pi\)
0.207299 + 0.978278i \(0.433533\pi\)
\(420\) 0 0
\(421\) −2.60498e6 −0.716307 −0.358154 0.933663i \(-0.616594\pi\)
−0.358154 + 0.933663i \(0.616594\pi\)
\(422\) −2.09396e6 −0.572384
\(423\) 678740. 0.184439
\(424\) −1.30680e6 −0.353016
\(425\) 0 0
\(426\) 232797. 0.0621517
\(427\) −2.53210e6 −0.672064
\(428\) −3.30479e6 −0.872037
\(429\) −652987. −0.171302
\(430\) 0 0
\(431\) −6.30280e6 −1.63433 −0.817166 0.576402i \(-0.804456\pi\)
−0.817166 + 0.576402i \(0.804456\pi\)
\(432\) −497291. −0.128204
\(433\) 4.19051e6 1.07411 0.537053 0.843549i \(-0.319538\pi\)
0.537053 + 0.843549i \(0.319538\pi\)
\(434\) −209096. −0.0532871
\(435\) 0 0
\(436\) −4.21545e6 −1.06201
\(437\) −4.71341e6 −1.18068
\(438\) −337478. −0.0840545
\(439\) −2.10556e6 −0.521442 −0.260721 0.965414i \(-0.583960\pi\)
−0.260721 + 0.965414i \(0.583960\pi\)
\(440\) 0 0
\(441\) −1.07833e6 −0.264032
\(442\) 2.62182e6 0.638332
\(443\) −5.79037e6 −1.40184 −0.700918 0.713242i \(-0.747226\pi\)
−0.700918 + 0.713242i \(0.747226\pi\)
\(444\) −988600. −0.237993
\(445\) 0 0
\(446\) −1.57593e6 −0.375145
\(447\) −4.02754e6 −0.953391
\(448\) −718569. −0.169150
\(449\) −2.51876e6 −0.589618 −0.294809 0.955556i \(-0.595256\pi\)
−0.294809 + 0.955556i \(0.595256\pi\)
\(450\) 0 0
\(451\) 259853. 0.0601569
\(452\) 4.40232e6 1.01353
\(453\) −4.46883e6 −1.02317
\(454\) −788800. −0.179609
\(455\) 0 0
\(456\) 3.13183e6 0.705320
\(457\) 6.44679e6 1.44395 0.721977 0.691917i \(-0.243234\pi\)
0.721977 + 0.691917i \(0.243234\pi\)
\(458\) 56688.6 0.0126279
\(459\) −1.65627e6 −0.366943
\(460\) 0 0
\(461\) 2.26707e6 0.496836 0.248418 0.968653i \(-0.420089\pi\)
0.248418 + 0.968653i \(0.420089\pi\)
\(462\) −123887. −0.0270035
\(463\) 8.82820e6 1.91390 0.956951 0.290251i \(-0.0937386\pi\)
0.956951 + 0.290251i \(0.0937386\pi\)
\(464\) 6.15349e6 1.32686
\(465\) 0 0
\(466\) −406294. −0.0866714
\(467\) −3.46435e6 −0.735072 −0.367536 0.930009i \(-0.619799\pi\)
−0.367536 + 0.930009i \(0.619799\pi\)
\(468\) 1.37433e6 0.290052
\(469\) −492165. −0.103319
\(470\) 0 0
\(471\) −3.28951e6 −0.683250
\(472\) −290772. −0.0600755
\(473\) 316189. 0.0649822
\(474\) −353622. −0.0722925
\(475\) 0 0
\(476\) −3.80020e6 −0.768757
\(477\) −912183. −0.183563
\(478\) −1.69182e6 −0.338675
\(479\) −5.50564e6 −1.09640 −0.548200 0.836347i \(-0.684687\pi\)
−0.548200 + 0.836347i \(0.684687\pi\)
\(480\) 0 0
\(481\) 2.32770e6 0.458737
\(482\) 1.59768e6 0.313237
\(483\) −836199. −0.163095
\(484\) −414285. −0.0803870
\(485\) 0 0
\(486\) 113641. 0.0218245
\(487\) 1.34130e6 0.256272 0.128136 0.991757i \(-0.459101\pi\)
0.128136 + 0.991757i \(0.459101\pi\)
\(488\) −4.97069e6 −0.944860
\(489\) 4.83183e6 0.913776
\(490\) 0 0
\(491\) 6.76450e6 1.26629 0.633144 0.774034i \(-0.281764\pi\)
0.633144 + 0.774034i \(0.281764\pi\)
\(492\) −546906. −0.101859
\(493\) 2.04947e7 3.79772
\(494\) −3.46053e6 −0.638007
\(495\) 0 0
\(496\) 1.25381e6 0.228838
\(497\) 794489. 0.144277
\(498\) −1.42310e6 −0.257136
\(499\) 6.79946e6 1.22243 0.611214 0.791466i \(-0.290682\pi\)
0.611214 + 0.791466i \(0.290682\pi\)
\(500\) 0 0
\(501\) 1.43009e6 0.254548
\(502\) 186645. 0.0330565
\(503\) 5.08998e6 0.897007 0.448503 0.893781i \(-0.351957\pi\)
0.448503 + 0.893781i \(0.351957\pi\)
\(504\) 555614. 0.0974309
\(505\) 0 0
\(506\) 366016. 0.0635511
\(507\) 105728. 0.0182670
\(508\) 5.46577e6 0.939706
\(509\) −3.53565e6 −0.604889 −0.302444 0.953167i \(-0.597803\pi\)
−0.302444 + 0.953167i \(0.597803\pi\)
\(510\) 0 0
\(511\) −1.15175e6 −0.195121
\(512\) −5.96167e6 −1.00506
\(513\) 2.18610e6 0.366756
\(514\) −710808. −0.118671
\(515\) 0 0
\(516\) −665476. −0.110029
\(517\) −1.01392e6 −0.166831
\(518\) 441619. 0.0723141
\(519\) −179699. −0.0292839
\(520\) 0 0
\(521\) −3.72572e6 −0.601334 −0.300667 0.953729i \(-0.597209\pi\)
−0.300667 + 0.953729i \(0.597209\pi\)
\(522\) −1.40619e6 −0.225875
\(523\) 1.46844e6 0.234748 0.117374 0.993088i \(-0.462552\pi\)
0.117374 + 0.993088i \(0.462552\pi\)
\(524\) −9.53880e6 −1.51763
\(525\) 0 0
\(526\) 1.90266e6 0.299845
\(527\) 4.17591e6 0.654975
\(528\) 742867. 0.115965
\(529\) −3.96585e6 −0.616165
\(530\) 0 0
\(531\) −202967. −0.0312384
\(532\) 5.01588e6 0.768366
\(533\) 1.28771e6 0.196336
\(534\) 666120. 0.101088
\(535\) 0 0
\(536\) −966158. −0.145257
\(537\) −1.37830e6 −0.206257
\(538\) −2.61745e6 −0.389873
\(539\) 1.61085e6 0.238826
\(540\) 0 0
\(541\) −158155. −0.0232322 −0.0116161 0.999933i \(-0.503698\pi\)
−0.0116161 + 0.999933i \(0.503698\pi\)
\(542\) 422207. 0.0617344
\(543\) 7.11779e6 1.03597
\(544\) −1.14193e7 −1.65440
\(545\) 0 0
\(546\) −613928. −0.0881324
\(547\) −7.82586e6 −1.11831 −0.559157 0.829062i \(-0.688875\pi\)
−0.559157 + 0.829062i \(0.688875\pi\)
\(548\) −9.90290e6 −1.40868
\(549\) −3.46968e6 −0.491314
\(550\) 0 0
\(551\) −2.70509e7 −3.79579
\(552\) −1.64152e6 −0.229297
\(553\) −1.20684e6 −0.167817
\(554\) 3.43366e6 0.475316
\(555\) 0 0
\(556\) −9.72791e6 −1.33454
\(557\) −71452.4 −0.00975841 −0.00487920 0.999988i \(-0.501553\pi\)
−0.00487920 + 0.999988i \(0.501553\pi\)
\(558\) −286520. −0.0389556
\(559\) 1.56689e6 0.212085
\(560\) 0 0
\(561\) 2.47418e6 0.331912
\(562\) 1.94430e6 0.259670
\(563\) 3.87924e6 0.515793 0.257897 0.966173i \(-0.416971\pi\)
0.257897 + 0.966173i \(0.416971\pi\)
\(564\) 2.13398e6 0.282483
\(565\) 0 0
\(566\) −521481. −0.0684223
\(567\) 387834. 0.0506626
\(568\) 1.55964e6 0.202840
\(569\) 8.73077e6 1.13050 0.565252 0.824918i \(-0.308779\pi\)
0.565252 + 0.824918i \(0.308779\pi\)
\(570\) 0 0
\(571\) 441497. 0.0566680 0.0283340 0.999599i \(-0.490980\pi\)
0.0283340 + 0.999599i \(0.490980\pi\)
\(572\) −2.05301e6 −0.262362
\(573\) −5.51625e6 −0.701871
\(574\) 244309. 0.0309499
\(575\) 0 0
\(576\) −984641. −0.123658
\(577\) 1.12074e7 1.40141 0.700703 0.713453i \(-0.252870\pi\)
0.700703 + 0.713453i \(0.252870\pi\)
\(578\) −7.20154e6 −0.896615
\(579\) 7.96126e6 0.986928
\(580\) 0 0
\(581\) −4.85676e6 −0.596907
\(582\) −2.42013e6 −0.296163
\(583\) 1.36264e6 0.166039
\(584\) −2.26096e6 −0.274323
\(585\) 0 0
\(586\) 1.68285e6 0.202443
\(587\) −9.06478e6 −1.08583 −0.542915 0.839788i \(-0.682679\pi\)
−0.542915 + 0.839788i \(0.682679\pi\)
\(588\) −3.39031e6 −0.404386
\(589\) −5.51178e6 −0.654641
\(590\) 0 0
\(591\) −126952. −0.0149511
\(592\) −2.64809e6 −0.310548
\(593\) −3789.68 −0.000442554 0 −0.000221277 1.00000i \(-0.500070\pi\)
−0.000221277 1.00000i \(0.500070\pi\)
\(594\) −169760. −0.0197410
\(595\) 0 0
\(596\) −1.26627e7 −1.46019
\(597\) 940510. 0.108001
\(598\) 1.81381e6 0.207414
\(599\) −6.84765e6 −0.779785 −0.389892 0.920860i \(-0.627488\pi\)
−0.389892 + 0.920860i \(0.627488\pi\)
\(600\) 0 0
\(601\) −8.82312e6 −0.996405 −0.498203 0.867061i \(-0.666006\pi\)
−0.498203 + 0.867061i \(0.666006\pi\)
\(602\) 297276. 0.0334325
\(603\) −674405. −0.0755314
\(604\) −1.40501e7 −1.56707
\(605\) 0 0
\(606\) −2.23564e6 −0.247298
\(607\) −1.19124e7 −1.31228 −0.656142 0.754638i \(-0.727813\pi\)
−0.656142 + 0.754638i \(0.727813\pi\)
\(608\) 1.50723e7 1.65356
\(609\) −4.79906e6 −0.524340
\(610\) 0 0
\(611\) −5.02453e6 −0.544493
\(612\) −5.20734e6 −0.562002
\(613\) 5.84838e6 0.628615 0.314307 0.949321i \(-0.398228\pi\)
0.314307 + 0.949321i \(0.398228\pi\)
\(614\) −4.23467e6 −0.453313
\(615\) 0 0
\(616\) −829991. −0.0881295
\(617\) 1.65932e6 0.175476 0.0877381 0.996144i \(-0.472036\pi\)
0.0877381 + 0.996144i \(0.472036\pi\)
\(618\) 462030. 0.0486630
\(619\) −6.13215e6 −0.643259 −0.321630 0.946866i \(-0.604231\pi\)
−0.321630 + 0.946866i \(0.604231\pi\)
\(620\) 0 0
\(621\) −1.14583e6 −0.119231
\(622\) −808355. −0.0837772
\(623\) 2.27333e6 0.234662
\(624\) 3.68131e6 0.378479
\(625\) 0 0
\(626\) −659984. −0.0673128
\(627\) −3.26566e6 −0.331744
\(628\) −1.03423e7 −1.04645
\(629\) −8.81967e6 −0.888845
\(630\) 0 0
\(631\) 1.82127e7 1.82097 0.910483 0.413547i \(-0.135710\pi\)
0.910483 + 0.413547i \(0.135710\pi\)
\(632\) −2.36912e6 −0.235936
\(633\) −9.79239e6 −0.971358
\(634\) −3.52750e6 −0.348533
\(635\) 0 0
\(636\) −2.86793e6 −0.281142
\(637\) 7.98262e6 0.779466
\(638\) 2.10061e6 0.204312
\(639\) 1.08867e6 0.105474
\(640\) 0 0
\(641\) 1.32331e7 1.27209 0.636044 0.771653i \(-0.280570\pi\)
0.636044 + 0.771653i \(0.280570\pi\)
\(642\) 2.02293e6 0.193706
\(643\) −9.98593e6 −0.952491 −0.476246 0.879312i \(-0.658003\pi\)
−0.476246 + 0.879312i \(0.658003\pi\)
\(644\) −2.62903e6 −0.249793
\(645\) 0 0
\(646\) 1.31120e7 1.23620
\(647\) −1.83105e7 −1.71965 −0.859823 0.510593i \(-0.829426\pi\)
−0.859823 + 0.510593i \(0.829426\pi\)
\(648\) 761347. 0.0712271
\(649\) 303197. 0.0282562
\(650\) 0 0
\(651\) −977836. −0.0904302
\(652\) 1.51914e7 1.39952
\(653\) 1.10272e7 1.01201 0.506004 0.862531i \(-0.331122\pi\)
0.506004 + 0.862531i \(0.331122\pi\)
\(654\) 2.58036e6 0.235905
\(655\) 0 0
\(656\) −1.46496e6 −0.132912
\(657\) −1.57821e6 −0.142644
\(658\) −953271. −0.0858325
\(659\) 8.75231e6 0.785072 0.392536 0.919737i \(-0.371598\pi\)
0.392536 + 0.919737i \(0.371598\pi\)
\(660\) 0 0
\(661\) −1.04678e6 −0.0931866 −0.0465933 0.998914i \(-0.514836\pi\)
−0.0465933 + 0.998914i \(0.514836\pi\)
\(662\) −2.74326e6 −0.243289
\(663\) 1.22609e7 1.08327
\(664\) −9.53419e6 −0.839196
\(665\) 0 0
\(666\) 605142. 0.0528654
\(667\) 1.41785e7 1.23400
\(668\) 4.49625e6 0.389860
\(669\) −7.36980e6 −0.636635
\(670\) 0 0
\(671\) 5.18310e6 0.444410
\(672\) 2.67395e6 0.228418
\(673\) −5.21480e6 −0.443813 −0.221906 0.975068i \(-0.571228\pi\)
−0.221906 + 0.975068i \(0.571228\pi\)
\(674\) −7.56695e6 −0.641610
\(675\) 0 0
\(676\) 332410. 0.0279774
\(677\) −1.81772e7 −1.52425 −0.762125 0.647430i \(-0.775844\pi\)
−0.762125 + 0.647430i \(0.775844\pi\)
\(678\) −2.69475e6 −0.225135
\(679\) −8.25942e6 −0.687503
\(680\) 0 0
\(681\) −3.68881e6 −0.304803
\(682\) 428012. 0.0352367
\(683\) 7.22749e6 0.592837 0.296419 0.955058i \(-0.404208\pi\)
0.296419 + 0.955058i \(0.404208\pi\)
\(684\) 6.87317e6 0.561716
\(685\) 0 0
\(686\) 3.42649e6 0.277996
\(687\) 265104. 0.0214301
\(688\) −1.78256e6 −0.143573
\(689\) 6.75264e6 0.541908
\(690\) 0 0
\(691\) 1.05510e7 0.840619 0.420309 0.907381i \(-0.361922\pi\)
0.420309 + 0.907381i \(0.361922\pi\)
\(692\) −564979. −0.0448505
\(693\) −579356. −0.0458261
\(694\) −650888. −0.0512989
\(695\) 0 0
\(696\) −9.42091e6 −0.737174
\(697\) −4.87915e6 −0.380419
\(698\) 5.44627e6 0.423117
\(699\) −1.90003e6 −0.147085
\(700\) 0 0
\(701\) 2.26639e7 1.74197 0.870985 0.491310i \(-0.163482\pi\)
0.870985 + 0.491310i \(0.163482\pi\)
\(702\) −841253. −0.0644294
\(703\) 1.16411e7 0.888392
\(704\) 1.47088e6 0.111853
\(705\) 0 0
\(706\) −8.11107e6 −0.612444
\(707\) −7.62978e6 −0.574069
\(708\) −638132. −0.0478440
\(709\) −1.63112e7 −1.21863 −0.609313 0.792930i \(-0.708555\pi\)
−0.609313 + 0.792930i \(0.708555\pi\)
\(710\) 0 0
\(711\) −1.65371e6 −0.122683
\(712\) 4.46272e6 0.329913
\(713\) 2.88895e6 0.212822
\(714\) 2.32618e6 0.170765
\(715\) 0 0
\(716\) −4.33341e6 −0.315898
\(717\) −7.91176e6 −0.574745
\(718\) −3.63304e6 −0.263002
\(719\) 9.77281e6 0.705013 0.352507 0.935809i \(-0.385329\pi\)
0.352507 + 0.935809i \(0.385329\pi\)
\(720\) 0 0
\(721\) 1.57681e6 0.112965
\(722\) −1.25412e7 −0.895357
\(723\) 7.47155e6 0.531575
\(724\) 2.23785e7 1.58666
\(725\) 0 0
\(726\) 253592. 0.0178564
\(727\) 1.05257e7 0.738611 0.369306 0.929308i \(-0.379596\pi\)
0.369306 + 0.929308i \(0.379596\pi\)
\(728\) −4.11306e6 −0.287631
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −5.93696e6 −0.410933
\(732\) −1.09088e7 −0.752485
\(733\) 1.34590e6 0.0925238 0.0462619 0.998929i \(-0.485269\pi\)
0.0462619 + 0.998929i \(0.485269\pi\)
\(734\) −725343. −0.0496939
\(735\) 0 0
\(736\) −7.89999e6 −0.537566
\(737\) 1.00744e6 0.0683207
\(738\) 334772. 0.0226260
\(739\) −1.89035e7 −1.27330 −0.636651 0.771152i \(-0.719681\pi\)
−0.636651 + 0.771152i \(0.719681\pi\)
\(740\) 0 0
\(741\) −1.61831e7 −1.08272
\(742\) 1.28113e6 0.0854250
\(743\) −1.35078e7 −0.897659 −0.448830 0.893617i \(-0.648159\pi\)
−0.448830 + 0.893617i \(0.648159\pi\)
\(744\) −1.91957e6 −0.127137
\(745\) 0 0
\(746\) 4.90906e6 0.322962
\(747\) −6.65513e6 −0.436370
\(748\) 7.77887e6 0.508350
\(749\) 6.90385e6 0.449662
\(750\) 0 0
\(751\) 6.35210e6 0.410977 0.205488 0.978660i \(-0.434122\pi\)
0.205488 + 0.978660i \(0.434122\pi\)
\(752\) 5.71613e6 0.368602
\(753\) 872843. 0.0560982
\(754\) 1.04097e7 0.666820
\(755\) 0 0
\(756\) 1.21936e6 0.0775938
\(757\) −8.58261e6 −0.544352 −0.272176 0.962248i \(-0.587743\pi\)
−0.272176 + 0.962248i \(0.587743\pi\)
\(758\) −7.64394e6 −0.483219
\(759\) 1.71167e6 0.107849
\(760\) 0 0
\(761\) 1.15085e7 0.720372 0.360186 0.932880i \(-0.382713\pi\)
0.360186 + 0.932880i \(0.382713\pi\)
\(762\) −3.34571e6 −0.208737
\(763\) 8.80626e6 0.547621
\(764\) −1.73432e7 −1.07497
\(765\) 0 0
\(766\) −3.56924e6 −0.219788
\(767\) 1.50251e6 0.0922207
\(768\) −309937. −0.0189614
\(769\) −2.72489e7 −1.66162 −0.830812 0.556554i \(-0.812123\pi\)
−0.830812 + 0.556554i \(0.812123\pi\)
\(770\) 0 0
\(771\) −3.32409e6 −0.201389
\(772\) 2.50304e7 1.51156
\(773\) 1.35064e7 0.813003 0.406502 0.913650i \(-0.366749\pi\)
0.406502 + 0.913650i \(0.366749\pi\)
\(774\) 407351. 0.0244409
\(775\) 0 0
\(776\) −1.62139e7 −0.966567
\(777\) 2.06523e6 0.122720
\(778\) 1.90978e6 0.113119
\(779\) 6.43999e6 0.380226
\(780\) 0 0
\(781\) −1.62629e6 −0.0954047
\(782\) −6.87253e6 −0.401883
\(783\) −6.57606e6 −0.383320
\(784\) −9.08138e6 −0.527669
\(785\) 0 0
\(786\) 5.83889e6 0.337112
\(787\) 7.81643e6 0.449854 0.224927 0.974376i \(-0.427786\pi\)
0.224927 + 0.974376i \(0.427786\pi\)
\(788\) −399141. −0.0228987
\(789\) 8.89778e6 0.508849
\(790\) 0 0
\(791\) −9.19662e6 −0.522621
\(792\) −1.13732e6 −0.0644273
\(793\) 2.56851e7 1.45044
\(794\) −285520. −0.0160726
\(795\) 0 0
\(796\) 2.95699e6 0.165412
\(797\) −2.29316e7 −1.27876 −0.639379 0.768891i \(-0.720809\pi\)
−0.639379 + 0.768891i \(0.720809\pi\)
\(798\) −3.07032e6 −0.170678
\(799\) 1.90380e7 1.05501
\(800\) 0 0
\(801\) 3.11510e6 0.171550
\(802\) −7.33393e6 −0.402625
\(803\) 2.35758e6 0.129026
\(804\) −2.12035e6 −0.115682
\(805\) 0 0
\(806\) 2.12103e6 0.115003
\(807\) −1.22405e7 −0.661629
\(808\) −1.49778e7 −0.807088
\(809\) −1.38109e7 −0.741908 −0.370954 0.928651i \(-0.620969\pi\)
−0.370954 + 0.928651i \(0.620969\pi\)
\(810\) 0 0
\(811\) 1.89793e7 1.01328 0.506638 0.862159i \(-0.330888\pi\)
0.506638 + 0.862159i \(0.330888\pi\)
\(812\) −1.50884e7 −0.803067
\(813\) 1.97445e6 0.104766
\(814\) −903977. −0.0478186
\(815\) 0 0
\(816\) −1.39485e7 −0.733336
\(817\) 7.83619e6 0.410724
\(818\) 143465. 0.00749656
\(819\) −2.87103e6 −0.149564
\(820\) 0 0
\(821\) −3.65447e6 −0.189220 −0.0946098 0.995514i \(-0.530160\pi\)
−0.0946098 + 0.995514i \(0.530160\pi\)
\(822\) 6.06176e6 0.312910
\(823\) 1.37132e7 0.705731 0.352865 0.935674i \(-0.385207\pi\)
0.352865 + 0.935674i \(0.385207\pi\)
\(824\) 3.09541e6 0.158818
\(825\) 0 0
\(826\) 285061. 0.0145374
\(827\) 1.96311e7 0.998116 0.499058 0.866569i \(-0.333679\pi\)
0.499058 + 0.866569i \(0.333679\pi\)
\(828\) −3.60251e6 −0.182612
\(829\) −1.78813e7 −0.903676 −0.451838 0.892100i \(-0.649232\pi\)
−0.451838 + 0.892100i \(0.649232\pi\)
\(830\) 0 0
\(831\) 1.60575e7 0.806631
\(832\) 7.28903e6 0.365058
\(833\) −3.02462e7 −1.51028
\(834\) 5.95465e6 0.296443
\(835\) 0 0
\(836\) −1.02673e7 −0.508091
\(837\) −1.33991e6 −0.0661092
\(838\) −2.86738e6 −0.141051
\(839\) 2.96975e6 0.145651 0.0728257 0.997345i \(-0.476798\pi\)
0.0728257 + 0.997345i \(0.476798\pi\)
\(840\) 0 0
\(841\) 6.08610e7 2.96722
\(842\) 5.01334e6 0.243695
\(843\) 9.09249e6 0.440670
\(844\) −3.07875e7 −1.48771
\(845\) 0 0
\(846\) −1.30625e6 −0.0627481
\(847\) 865458. 0.0414513
\(848\) −7.68211e6 −0.366852
\(849\) −2.43870e6 −0.116115
\(850\) 0 0
\(851\) −6.10157e6 −0.288814
\(852\) 3.42281e6 0.161542
\(853\) −7.98055e6 −0.375544 −0.187772 0.982213i \(-0.560127\pi\)
−0.187772 + 0.982213i \(0.560127\pi\)
\(854\) 4.87307e6 0.228643
\(855\) 0 0
\(856\) 1.35528e7 0.632184
\(857\) 1.41979e7 0.660347 0.330173 0.943920i \(-0.392893\pi\)
0.330173 + 0.943920i \(0.392893\pi\)
\(858\) 1.25669e6 0.0582786
\(859\) −1.26770e7 −0.586182 −0.293091 0.956084i \(-0.594684\pi\)
−0.293091 + 0.956084i \(0.594684\pi\)
\(860\) 0 0
\(861\) 1.14251e6 0.0525233
\(862\) 1.21299e7 0.556017
\(863\) 3.18697e7 1.45664 0.728318 0.685239i \(-0.240302\pi\)
0.728318 + 0.685239i \(0.240302\pi\)
\(864\) 3.66406e6 0.166985
\(865\) 0 0
\(866\) −8.06472e6 −0.365422
\(867\) −3.36780e7 −1.52159
\(868\) −3.07434e6 −0.138501
\(869\) 2.47036e6 0.110971
\(870\) 0 0
\(871\) 4.99244e6 0.222981
\(872\) 1.72873e7 0.769905
\(873\) −1.13177e7 −0.502601
\(874\) 9.07105e6 0.401679
\(875\) 0 0
\(876\) −4.96195e6 −0.218470
\(877\) −452817. −0.0198804 −0.00994018 0.999951i \(-0.503164\pi\)
−0.00994018 + 0.999951i \(0.503164\pi\)
\(878\) 4.05219e6 0.177400
\(879\) 7.86985e6 0.343554
\(880\) 0 0
\(881\) 3.26083e7 1.41543 0.707715 0.706498i \(-0.249726\pi\)
0.707715 + 0.706498i \(0.249726\pi\)
\(882\) 2.07528e6 0.0898265
\(883\) −2.36123e7 −1.01915 −0.509573 0.860427i \(-0.670197\pi\)
−0.509573 + 0.860427i \(0.670197\pi\)
\(884\) 3.85486e7 1.65912
\(885\) 0 0
\(886\) 1.11437e7 0.476919
\(887\) 3.31071e7 1.41290 0.706451 0.707762i \(-0.250295\pi\)
0.706451 + 0.707762i \(0.250295\pi\)
\(888\) 4.05420e6 0.172533
\(889\) −1.14182e7 −0.484556
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) −2.31708e7 −0.975056
\(893\) −2.51282e7 −1.05447
\(894\) 7.75108e6 0.324353
\(895\) 0 0
\(896\) 1.08903e7 0.453178
\(897\) 8.48225e6 0.351990
\(898\) 4.84740e6 0.200594
\(899\) 1.65801e7 0.684206
\(900\) 0 0
\(901\) −2.55858e7 −1.05000
\(902\) −500091. −0.0204660
\(903\) 1.39021e6 0.0567362
\(904\) −1.80537e7 −0.734758
\(905\) 0 0
\(906\) 8.60034e6 0.348093
\(907\) 8.76937e6 0.353957 0.176978 0.984215i \(-0.443368\pi\)
0.176978 + 0.984215i \(0.443368\pi\)
\(908\) −1.15977e7 −0.466829
\(909\) −1.04549e7 −0.419674
\(910\) 0 0
\(911\) 1.15460e7 0.460932 0.230466 0.973080i \(-0.425975\pi\)
0.230466 + 0.973080i \(0.425975\pi\)
\(912\) 1.84107e7 0.732963
\(913\) 9.94161e6 0.394711
\(914\) −1.24070e7 −0.491248
\(915\) 0 0
\(916\) 833493. 0.0328219
\(917\) 1.99269e7 0.782560
\(918\) 3.18752e6 0.124838
\(919\) −2.95602e7 −1.15456 −0.577282 0.816545i \(-0.695887\pi\)
−0.577282 + 0.816545i \(0.695887\pi\)
\(920\) 0 0
\(921\) −1.98034e7 −0.769291
\(922\) −4.36302e6 −0.169029
\(923\) −8.05915e6 −0.311376
\(924\) −1.82151e6 −0.0701862
\(925\) 0 0
\(926\) −1.69900e7 −0.651129
\(927\) 2.16068e6 0.0825830
\(928\) −4.53391e7 −1.72824
\(929\) 4.03347e7 1.53334 0.766672 0.642039i \(-0.221911\pi\)
0.766672 + 0.642039i \(0.221911\pi\)
\(930\) 0 0
\(931\) 3.99220e7 1.50952
\(932\) −5.97374e6 −0.225272
\(933\) −3.78026e6 −0.142173
\(934\) 6.66722e6 0.250079
\(935\) 0 0
\(936\) −5.63605e6 −0.210274
\(937\) 2.52134e6 0.0938172 0.0469086 0.998899i \(-0.485063\pi\)
0.0469086 + 0.998899i \(0.485063\pi\)
\(938\) 947182. 0.0351501
\(939\) −3.08641e6 −0.114232
\(940\) 0 0
\(941\) 3.52049e7 1.29607 0.648036 0.761609i \(-0.275590\pi\)
0.648036 + 0.761609i \(0.275590\pi\)
\(942\) 6.33074e6 0.232449
\(943\) −3.37546e6 −0.123610
\(944\) −1.70932e6 −0.0624300
\(945\) 0 0
\(946\) −608512. −0.0221076
\(947\) −3.15673e7 −1.14383 −0.571917 0.820312i \(-0.693800\pi\)
−0.571917 + 0.820312i \(0.693800\pi\)
\(948\) −5.19930e6 −0.187899
\(949\) 1.16831e7 0.421107
\(950\) 0 0
\(951\) −1.64963e7 −0.591475
\(952\) 1.55844e7 0.557312
\(953\) 2.68995e7 0.959427 0.479713 0.877425i \(-0.340741\pi\)
0.479713 + 0.877425i \(0.340741\pi\)
\(954\) 1.75551e6 0.0624501
\(955\) 0 0
\(956\) −2.48748e7 −0.880267
\(957\) 9.82349e6 0.346726
\(958\) 1.05957e7 0.373007
\(959\) 2.06876e7 0.726378
\(960\) 0 0
\(961\) −2.52509e7 −0.881998
\(962\) −4.47970e6 −0.156067
\(963\) 9.46021e6 0.328727
\(964\) 2.34907e7 0.814149
\(965\) 0 0
\(966\) 1.60928e6 0.0554867
\(967\) 2.96256e7 1.01883 0.509414 0.860522i \(-0.329862\pi\)
0.509414 + 0.860522i \(0.329862\pi\)
\(968\) 1.69896e6 0.0582767
\(969\) 6.13181e7 2.09787
\(970\) 0 0
\(971\) −3.86746e7 −1.31637 −0.658184 0.752857i \(-0.728675\pi\)
−0.658184 + 0.752857i \(0.728675\pi\)
\(972\) 1.67086e6 0.0567251
\(973\) 2.03220e7 0.688152
\(974\) −2.58135e6 −0.0871865
\(975\) 0 0
\(976\) −2.92205e7 −0.981891
\(977\) 9.61017e6 0.322103 0.161051 0.986946i \(-0.448511\pi\)
0.161051 + 0.986946i \(0.448511\pi\)
\(978\) −9.29896e6 −0.310876
\(979\) −4.65343e6 −0.155173
\(980\) 0 0
\(981\) 1.20670e7 0.400339
\(982\) −1.30184e7 −0.430804
\(983\) 2.35665e7 0.777877 0.388939 0.921264i \(-0.372842\pi\)
0.388939 + 0.921264i \(0.372842\pi\)
\(984\) 2.24283e6 0.0738429
\(985\) 0 0
\(986\) −3.94424e7 −1.29202
\(987\) −4.45797e6 −0.145661
\(988\) −5.08802e7 −1.65827
\(989\) −4.10727e6 −0.133525
\(990\) 0 0
\(991\) −2.88338e7 −0.932649 −0.466325 0.884614i \(-0.654422\pi\)
−0.466325 + 0.884614i \(0.654422\pi\)
\(992\) −9.23811e6 −0.298060
\(993\) −1.28289e7 −0.412871
\(994\) −1.52901e6 −0.0490845
\(995\) 0 0
\(996\) −2.09239e7 −0.668335
\(997\) 4.93391e7 1.57200 0.786002 0.618225i \(-0.212148\pi\)
0.786002 + 0.618225i \(0.212148\pi\)
\(998\) −1.30857e7 −0.415882
\(999\) 2.82994e6 0.0897147
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.s.1.4 yes 9
5.4 even 2 825.6.a.r.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.6 9 5.4 even 2
825.6.a.s.1.4 yes 9 1.1 even 1 trivial