Properties

Label 825.6.a.s.1.8
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 1.8
Root \(8.36252\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.36252 q^{2} -9.00000 q^{3} +37.9318 q^{4} -75.2627 q^{6} -15.4419 q^{7} +49.6048 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+8.36252 q^{2} -9.00000 q^{3} +37.9318 q^{4} -75.2627 q^{6} -15.4419 q^{7} +49.6048 q^{8} +81.0000 q^{9} -121.000 q^{11} -341.386 q^{12} +624.673 q^{13} -129.133 q^{14} -798.996 q^{16} +1254.35 q^{17} +677.364 q^{18} -440.239 q^{19} +138.977 q^{21} -1011.87 q^{22} +2534.04 q^{23} -446.443 q^{24} +5223.84 q^{26} -729.000 q^{27} -585.737 q^{28} -3653.61 q^{29} -8001.02 q^{31} -8268.98 q^{32} +1089.00 q^{33} +10489.5 q^{34} +3072.48 q^{36} -12096.5 q^{37} -3681.51 q^{38} -5622.06 q^{39} +13260.6 q^{41} +1162.20 q^{42} +4651.51 q^{43} -4589.75 q^{44} +21191.0 q^{46} +22955.5 q^{47} +7190.97 q^{48} -16568.5 q^{49} -11289.2 q^{51} +23695.0 q^{52} +13857.7 q^{53} -6096.28 q^{54} -765.990 q^{56} +3962.15 q^{57} -30553.4 q^{58} +46429.1 q^{59} +55964.9 q^{61} -66908.7 q^{62} -1250.79 q^{63} -43581.6 q^{64} +9106.79 q^{66} +3063.93 q^{67} +47579.8 q^{68} -22806.4 q^{69} +62515.4 q^{71} +4017.99 q^{72} +60834.6 q^{73} -101158. q^{74} -16699.1 q^{76} +1868.46 q^{77} -47014.6 q^{78} +53833.8 q^{79} +6561.00 q^{81} +110892. q^{82} -53585.5 q^{83} +5271.64 q^{84} +38898.4 q^{86} +32882.5 q^{87} -6002.18 q^{88} +3381.86 q^{89} -9646.11 q^{91} +96120.8 q^{92} +72009.2 q^{93} +191966. q^{94} +74420.8 q^{96} +30289.8 q^{97} -138555. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.36252 1.47830 0.739150 0.673541i \(-0.235228\pi\)
0.739150 + 0.673541i \(0.235228\pi\)
\(3\) −9.00000 −0.577350
\(4\) 37.9318 1.18537
\(5\) 0 0
\(6\) −75.2627 −0.853496
\(7\) −15.4419 −0.119112 −0.0595558 0.998225i \(-0.518968\pi\)
−0.0595558 + 0.998225i \(0.518968\pi\)
\(8\) 49.6048 0.274030
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −341.386 −0.684373
\(13\) 624.673 1.02517 0.512583 0.858638i \(-0.328689\pi\)
0.512583 + 0.858638i \(0.328689\pi\)
\(14\) −129.133 −0.176083
\(15\) 0 0
\(16\) −798.996 −0.780270
\(17\) 1254.35 1.05268 0.526341 0.850274i \(-0.323564\pi\)
0.526341 + 0.850274i \(0.323564\pi\)
\(18\) 677.364 0.492766
\(19\) −440.239 −0.279772 −0.139886 0.990168i \(-0.544674\pi\)
−0.139886 + 0.990168i \(0.544674\pi\)
\(20\) 0 0
\(21\) 138.977 0.0687692
\(22\) −1011.87 −0.445724
\(23\) 2534.04 0.998836 0.499418 0.866361i \(-0.333547\pi\)
0.499418 + 0.866361i \(0.333547\pi\)
\(24\) −446.443 −0.158212
\(25\) 0 0
\(26\) 5223.84 1.51550
\(27\) −729.000 −0.192450
\(28\) −585.737 −0.141191
\(29\) −3653.61 −0.806729 −0.403364 0.915039i \(-0.632159\pi\)
−0.403364 + 0.915039i \(0.632159\pi\)
\(30\) 0 0
\(31\) −8001.02 −1.49534 −0.747672 0.664068i \(-0.768828\pi\)
−0.747672 + 0.664068i \(0.768828\pi\)
\(32\) −8268.98 −1.42750
\(33\) 1089.00 0.174078
\(34\) 10489.5 1.55618
\(35\) 0 0
\(36\) 3072.48 0.395123
\(37\) −12096.5 −1.45264 −0.726318 0.687359i \(-0.758770\pi\)
−0.726318 + 0.687359i \(0.758770\pi\)
\(38\) −3681.51 −0.413587
\(39\) −5622.06 −0.591880
\(40\) 0 0
\(41\) 13260.6 1.23198 0.615990 0.787754i \(-0.288756\pi\)
0.615990 + 0.787754i \(0.288756\pi\)
\(42\) 1162.20 0.101661
\(43\) 4651.51 0.383639 0.191820 0.981430i \(-0.438561\pi\)
0.191820 + 0.981430i \(0.438561\pi\)
\(44\) −4589.75 −0.357402
\(45\) 0 0
\(46\) 21191.0 1.47658
\(47\) 22955.5 1.51580 0.757901 0.652370i \(-0.226225\pi\)
0.757901 + 0.652370i \(0.226225\pi\)
\(48\) 7190.97 0.450489
\(49\) −16568.5 −0.985812
\(50\) 0 0
\(51\) −11289.2 −0.607766
\(52\) 23695.0 1.21520
\(53\) 13857.7 0.677645 0.338822 0.940850i \(-0.389971\pi\)
0.338822 + 0.940850i \(0.389971\pi\)
\(54\) −6096.28 −0.284499
\(55\) 0 0
\(56\) −765.990 −0.0326402
\(57\) 3962.15 0.161527
\(58\) −30553.4 −1.19259
\(59\) 46429.1 1.73644 0.868220 0.496179i \(-0.165264\pi\)
0.868220 + 0.496179i \(0.165264\pi\)
\(60\) 0 0
\(61\) 55964.9 1.92571 0.962855 0.270018i \(-0.0870297\pi\)
0.962855 + 0.270018i \(0.0870297\pi\)
\(62\) −66908.7 −2.21057
\(63\) −1250.79 −0.0397039
\(64\) −43581.6 −1.33001
\(65\) 0 0
\(66\) 9106.79 0.257339
\(67\) 3063.93 0.0833858 0.0416929 0.999130i \(-0.486725\pi\)
0.0416929 + 0.999130i \(0.486725\pi\)
\(68\) 47579.8 1.24782
\(69\) −22806.4 −0.576678
\(70\) 0 0
\(71\) 62515.4 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(72\) 4017.99 0.0913435
\(73\) 60834.6 1.33612 0.668058 0.744110i \(-0.267126\pi\)
0.668058 + 0.744110i \(0.267126\pi\)
\(74\) −101158. −2.14743
\(75\) 0 0
\(76\) −16699.1 −0.331633
\(77\) 1868.46 0.0359135
\(78\) −47014.6 −0.874976
\(79\) 53833.8 0.970481 0.485241 0.874381i \(-0.338732\pi\)
0.485241 + 0.874381i \(0.338732\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 110892. 1.82124
\(83\) −53585.5 −0.853792 −0.426896 0.904301i \(-0.640393\pi\)
−0.426896 + 0.904301i \(0.640393\pi\)
\(84\) 5271.64 0.0815168
\(85\) 0 0
\(86\) 38898.4 0.567134
\(87\) 32882.5 0.465765
\(88\) −6002.18 −0.0826233
\(89\) 3381.86 0.0452564 0.0226282 0.999744i \(-0.492797\pi\)
0.0226282 + 0.999744i \(0.492797\pi\)
\(90\) 0 0
\(91\) −9646.11 −0.122109
\(92\) 96120.8 1.18399
\(93\) 72009.2 0.863337
\(94\) 191966. 2.24081
\(95\) 0 0
\(96\) 74420.8 0.824169
\(97\) 30289.8 0.326865 0.163432 0.986555i \(-0.447743\pi\)
0.163432 + 0.986555i \(0.447743\pi\)
\(98\) −138555. −1.45733
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 97950.2 0.955437 0.477718 0.878513i \(-0.341464\pi\)
0.477718 + 0.878513i \(0.341464\pi\)
\(102\) −94405.9 −0.898460
\(103\) −87201.9 −0.809903 −0.404951 0.914338i \(-0.632712\pi\)
−0.404951 + 0.914338i \(0.632712\pi\)
\(104\) 30986.8 0.280927
\(105\) 0 0
\(106\) 115885. 1.00176
\(107\) 12956.6 0.109403 0.0547017 0.998503i \(-0.482579\pi\)
0.0547017 + 0.998503i \(0.482579\pi\)
\(108\) −27652.3 −0.228124
\(109\) −130438. −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(110\) 0 0
\(111\) 108869. 0.838679
\(112\) 12338.0 0.0929393
\(113\) 9112.49 0.0671338 0.0335669 0.999436i \(-0.489313\pi\)
0.0335669 + 0.999436i \(0.489313\pi\)
\(114\) 33133.6 0.238785
\(115\) 0 0
\(116\) −138588. −0.956271
\(117\) 50598.5 0.341722
\(118\) 388264. 2.56698
\(119\) −19369.5 −0.125387
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 468008. 2.84678
\(123\) −119346. −0.711284
\(124\) −303493. −1.77253
\(125\) 0 0
\(126\) −10459.8 −0.0586942
\(127\) 67808.2 0.373055 0.186528 0.982450i \(-0.440277\pi\)
0.186528 + 0.982450i \(0.440277\pi\)
\(128\) −99845.2 −0.538645
\(129\) −41863.6 −0.221494
\(130\) 0 0
\(131\) 270464. 1.37699 0.688497 0.725239i \(-0.258271\pi\)
0.688497 + 0.725239i \(0.258271\pi\)
\(132\) 41307.7 0.206346
\(133\) 6798.11 0.0333241
\(134\) 25622.2 0.123269
\(135\) 0 0
\(136\) 62221.9 0.288467
\(137\) 27960.6 0.127276 0.0636378 0.997973i \(-0.479730\pi\)
0.0636378 + 0.997973i \(0.479730\pi\)
\(138\) −190719. −0.852503
\(139\) 31442.3 0.138031 0.0690157 0.997616i \(-0.478014\pi\)
0.0690157 + 0.997616i \(0.478014\pi\)
\(140\) 0 0
\(141\) −206600. −0.875148
\(142\) 522787. 2.17572
\(143\) −75585.4 −0.309099
\(144\) −64718.7 −0.260090
\(145\) 0 0
\(146\) 508731. 1.97518
\(147\) 149117. 0.569159
\(148\) −458843. −1.72191
\(149\) −120047. −0.442982 −0.221491 0.975162i \(-0.571092\pi\)
−0.221491 + 0.975162i \(0.571092\pi\)
\(150\) 0 0
\(151\) −110901. −0.395816 −0.197908 0.980221i \(-0.563415\pi\)
−0.197908 + 0.980221i \(0.563415\pi\)
\(152\) −21838.0 −0.0766661
\(153\) 101603. 0.350894
\(154\) 15625.1 0.0530909
\(155\) 0 0
\(156\) −213255. −0.701596
\(157\) 531986. 1.72247 0.861234 0.508208i \(-0.169692\pi\)
0.861234 + 0.508208i \(0.169692\pi\)
\(158\) 450186. 1.43466
\(159\) −124719. −0.391238
\(160\) 0 0
\(161\) −39130.3 −0.118973
\(162\) 54866.5 0.164255
\(163\) 459073. 1.35336 0.676680 0.736278i \(-0.263418\pi\)
0.676680 + 0.736278i \(0.263418\pi\)
\(164\) 502999. 1.46035
\(165\) 0 0
\(166\) −448110. −1.26216
\(167\) 286383. 0.794613 0.397306 0.917686i \(-0.369945\pi\)
0.397306 + 0.917686i \(0.369945\pi\)
\(168\) 6893.91 0.0188448
\(169\) 18923.3 0.0509659
\(170\) 0 0
\(171\) −35659.4 −0.0932574
\(172\) 176440. 0.454754
\(173\) −656155. −1.66683 −0.833415 0.552647i \(-0.813618\pi\)
−0.833415 + 0.552647i \(0.813618\pi\)
\(174\) 274981. 0.688540
\(175\) 0 0
\(176\) 96678.5 0.235260
\(177\) −417862. −1.00253
\(178\) 28280.8 0.0669025
\(179\) −220000. −0.513203 −0.256601 0.966517i \(-0.582603\pi\)
−0.256601 + 0.966517i \(0.582603\pi\)
\(180\) 0 0
\(181\) −426138. −0.966837 −0.483419 0.875389i \(-0.660605\pi\)
−0.483419 + 0.875389i \(0.660605\pi\)
\(182\) −80665.8 −0.180514
\(183\) −503684. −1.11181
\(184\) 125701. 0.273711
\(185\) 0 0
\(186\) 602179. 1.27627
\(187\) −151777. −0.317396
\(188\) 870744. 1.79678
\(189\) 11257.1 0.0229231
\(190\) 0 0
\(191\) 163125. 0.323546 0.161773 0.986828i \(-0.448279\pi\)
0.161773 + 0.986828i \(0.448279\pi\)
\(192\) 392235. 0.767879
\(193\) 212660. 0.410953 0.205476 0.978662i \(-0.434126\pi\)
0.205476 + 0.978662i \(0.434126\pi\)
\(194\) 253300. 0.483204
\(195\) 0 0
\(196\) −628475. −1.16855
\(197\) −729671. −1.33956 −0.669779 0.742561i \(-0.733611\pi\)
−0.669779 + 0.742561i \(0.733611\pi\)
\(198\) −81961.1 −0.148575
\(199\) 642534. 1.15017 0.575087 0.818092i \(-0.304968\pi\)
0.575087 + 0.818092i \(0.304968\pi\)
\(200\) 0 0
\(201\) −27575.4 −0.0481428
\(202\) 819111. 1.41242
\(203\) 56418.6 0.0960908
\(204\) −428218. −0.720427
\(205\) 0 0
\(206\) −729228. −1.19728
\(207\) 205257. 0.332945
\(208\) −499111. −0.799906
\(209\) 53268.9 0.0843545
\(210\) 0 0
\(211\) 460769. 0.712487 0.356244 0.934393i \(-0.384057\pi\)
0.356244 + 0.934393i \(0.384057\pi\)
\(212\) 525648. 0.803259
\(213\) −562639. −0.849729
\(214\) 108350. 0.161731
\(215\) 0 0
\(216\) −36161.9 −0.0527372
\(217\) 123551. 0.178113
\(218\) −1.09079e6 −1.55454
\(219\) −547512. −0.771406
\(220\) 0 0
\(221\) 783560. 1.07917
\(222\) 910418. 1.23982
\(223\) 841893. 1.13369 0.566846 0.823824i \(-0.308164\pi\)
0.566846 + 0.823824i \(0.308164\pi\)
\(224\) 127688. 0.170032
\(225\) 0 0
\(226\) 76203.4 0.0992438
\(227\) −415368. −0.535018 −0.267509 0.963555i \(-0.586200\pi\)
−0.267509 + 0.963555i \(0.586200\pi\)
\(228\) 150291. 0.191468
\(229\) 1.33133e6 1.67763 0.838815 0.544417i \(-0.183249\pi\)
0.838815 + 0.544417i \(0.183249\pi\)
\(230\) 0 0
\(231\) −16816.2 −0.0207347
\(232\) −181237. −0.221068
\(233\) −53370.5 −0.0644039 −0.0322019 0.999481i \(-0.510252\pi\)
−0.0322019 + 0.999481i \(0.510252\pi\)
\(234\) 423131. 0.505168
\(235\) 0 0
\(236\) 1.76114e6 2.05832
\(237\) −484504. −0.560308
\(238\) −161978. −0.185359
\(239\) 160686. 0.181963 0.0909815 0.995853i \(-0.471000\pi\)
0.0909815 + 0.995853i \(0.471000\pi\)
\(240\) 0 0
\(241\) 496394. 0.550534 0.275267 0.961368i \(-0.411234\pi\)
0.275267 + 0.961368i \(0.411234\pi\)
\(242\) 122436. 0.134391
\(243\) −59049.0 −0.0641500
\(244\) 2.12285e6 2.28268
\(245\) 0 0
\(246\) −998030. −1.05149
\(247\) −275005. −0.286813
\(248\) −396889. −0.409770
\(249\) 482270. 0.492937
\(250\) 0 0
\(251\) 1.78702e6 1.79038 0.895189 0.445686i \(-0.147040\pi\)
0.895189 + 0.445686i \(0.147040\pi\)
\(252\) −47444.7 −0.0470638
\(253\) −306619. −0.301160
\(254\) 567048. 0.551487
\(255\) 0 0
\(256\) 559655. 0.533728
\(257\) −1.37898e6 −1.30234 −0.651171 0.758931i \(-0.725722\pi\)
−0.651171 + 0.758931i \(0.725722\pi\)
\(258\) −350086. −0.327435
\(259\) 186793. 0.173026
\(260\) 0 0
\(261\) −295943. −0.268910
\(262\) 2.26177e6 2.03561
\(263\) 1.92969e6 1.72027 0.860136 0.510065i \(-0.170379\pi\)
0.860136 + 0.510065i \(0.170379\pi\)
\(264\) 54019.6 0.0477026
\(265\) 0 0
\(266\) 56849.3 0.0492630
\(267\) −30436.7 −0.0261288
\(268\) 116220. 0.0988429
\(269\) 1.14161e6 0.961916 0.480958 0.876744i \(-0.340289\pi\)
0.480958 + 0.876744i \(0.340289\pi\)
\(270\) 0 0
\(271\) −584178. −0.483195 −0.241597 0.970377i \(-0.577671\pi\)
−0.241597 + 0.970377i \(0.577671\pi\)
\(272\) −1.00222e6 −0.821376
\(273\) 86815.0 0.0704998
\(274\) 233821. 0.188151
\(275\) 0 0
\(276\) −865087. −0.683577
\(277\) −1.95194e6 −1.52850 −0.764252 0.644918i \(-0.776892\pi\)
−0.764252 + 0.644918i \(0.776892\pi\)
\(278\) 262937. 0.204052
\(279\) −648083. −0.498448
\(280\) 0 0
\(281\) 130490. 0.0985848 0.0492924 0.998784i \(-0.484303\pi\)
0.0492924 + 0.998784i \(0.484303\pi\)
\(282\) −1.72769e6 −1.29373
\(283\) −1.53762e6 −1.14125 −0.570626 0.821210i \(-0.693300\pi\)
−0.570626 + 0.821210i \(0.693300\pi\)
\(284\) 2.37132e6 1.74460
\(285\) 0 0
\(286\) −632085. −0.456941
\(287\) −204768. −0.146743
\(288\) −669787. −0.475834
\(289\) 153542. 0.108139
\(290\) 0 0
\(291\) −272609. −0.188715
\(292\) 2.30757e6 1.58379
\(293\) −2.19926e6 −1.49661 −0.748303 0.663357i \(-0.769131\pi\)
−0.748303 + 0.663357i \(0.769131\pi\)
\(294\) 1.24699e6 0.841387
\(295\) 0 0
\(296\) −600046. −0.398066
\(297\) 88209.0 0.0580259
\(298\) −1.00390e6 −0.654861
\(299\) 1.58295e6 1.02397
\(300\) 0 0
\(301\) −71828.0 −0.0456959
\(302\) −927413. −0.585135
\(303\) −881552. −0.551622
\(304\) 351749. 0.218298
\(305\) 0 0
\(306\) 849653. 0.518726
\(307\) 136632. 0.0827385 0.0413693 0.999144i \(-0.486828\pi\)
0.0413693 + 0.999144i \(0.486828\pi\)
\(308\) 70874.2 0.0425708
\(309\) 784817. 0.467597
\(310\) 0 0
\(311\) −2.03310e6 −1.19195 −0.595975 0.803003i \(-0.703234\pi\)
−0.595975 + 0.803003i \(0.703234\pi\)
\(312\) −278881. −0.162193
\(313\) −814182. −0.469743 −0.234872 0.972026i \(-0.575467\pi\)
−0.234872 + 0.972026i \(0.575467\pi\)
\(314\) 4.44875e6 2.54632
\(315\) 0 0
\(316\) 2.04201e6 1.15038
\(317\) −498935. −0.278866 −0.139433 0.990231i \(-0.544528\pi\)
−0.139433 + 0.990231i \(0.544528\pi\)
\(318\) −1.04297e6 −0.578367
\(319\) 442087. 0.243238
\(320\) 0 0
\(321\) −116609. −0.0631641
\(322\) −327228. −0.175878
\(323\) −552215. −0.294511
\(324\) 248871. 0.131708
\(325\) 0 0
\(326\) 3.83901e6 2.00067
\(327\) 1.17395e6 0.607126
\(328\) 657790. 0.337600
\(329\) −354476. −0.180550
\(330\) 0 0
\(331\) 166599. 0.0835799 0.0417900 0.999126i \(-0.486694\pi\)
0.0417900 + 0.999126i \(0.486694\pi\)
\(332\) −2.03259e6 −1.01206
\(333\) −979819. −0.484212
\(334\) 2.39488e6 1.17468
\(335\) 0 0
\(336\) −111042. −0.0536585
\(337\) 200668. 0.0962504 0.0481252 0.998841i \(-0.484675\pi\)
0.0481252 + 0.998841i \(0.484675\pi\)
\(338\) 158246. 0.0753428
\(339\) −82012.4 −0.0387597
\(340\) 0 0
\(341\) 968124. 0.450863
\(342\) −298202. −0.137862
\(343\) 515380. 0.236533
\(344\) 230737. 0.105129
\(345\) 0 0
\(346\) −5.48711e6 −2.46407
\(347\) −1.38809e6 −0.618862 −0.309431 0.950922i \(-0.600139\pi\)
−0.309431 + 0.950922i \(0.600139\pi\)
\(348\) 1.24729e6 0.552103
\(349\) −4.11254e6 −1.80737 −0.903684 0.428200i \(-0.859148\pi\)
−0.903684 + 0.428200i \(0.859148\pi\)
\(350\) 0 0
\(351\) −455387. −0.197293
\(352\) 1.00055e6 0.430408
\(353\) 1.14899e6 0.490770 0.245385 0.969426i \(-0.421086\pi\)
0.245385 + 0.969426i \(0.421086\pi\)
\(354\) −3.49438e6 −1.48205
\(355\) 0 0
\(356\) 128280. 0.0536455
\(357\) 174326. 0.0723921
\(358\) −1.83975e6 −0.758668
\(359\) 1.96193e6 0.803429 0.401714 0.915765i \(-0.368415\pi\)
0.401714 + 0.915765i \(0.368415\pi\)
\(360\) 0 0
\(361\) −2.28229e6 −0.921728
\(362\) −3.56359e6 −1.42927
\(363\) −131769. −0.0524864
\(364\) −365894. −0.144745
\(365\) 0 0
\(366\) −4.21207e6 −1.64359
\(367\) −2.98170e6 −1.15558 −0.577788 0.816187i \(-0.696084\pi\)
−0.577788 + 0.816187i \(0.696084\pi\)
\(368\) −2.02469e6 −0.779362
\(369\) 1.07411e6 0.410660
\(370\) 0 0
\(371\) −213989. −0.0807154
\(372\) 2.73144e6 1.02337
\(373\) −4.00552e6 −1.49069 −0.745344 0.666680i \(-0.767715\pi\)
−0.745344 + 0.666680i \(0.767715\pi\)
\(374\) −1.26924e6 −0.469206
\(375\) 0 0
\(376\) 1.13870e6 0.415376
\(377\) −2.28231e6 −0.827031
\(378\) 94137.9 0.0338871
\(379\) −2.50361e6 −0.895299 −0.447650 0.894209i \(-0.647739\pi\)
−0.447650 + 0.894209i \(0.647739\pi\)
\(380\) 0 0
\(381\) −610274. −0.215384
\(382\) 1.36414e6 0.478299
\(383\) −2.05384e6 −0.715434 −0.357717 0.933830i \(-0.616445\pi\)
−0.357717 + 0.933830i \(0.616445\pi\)
\(384\) 898607. 0.310987
\(385\) 0 0
\(386\) 1.77837e6 0.607511
\(387\) 376773. 0.127880
\(388\) 1.14895e6 0.387455
\(389\) −3.29367e6 −1.10359 −0.551794 0.833981i \(-0.686056\pi\)
−0.551794 + 0.833981i \(0.686056\pi\)
\(390\) 0 0
\(391\) 3.17858e6 1.05146
\(392\) −821880. −0.270143
\(393\) −2.43418e6 −0.795008
\(394\) −6.10189e6 −1.98027
\(395\) 0 0
\(396\) −371770. −0.119134
\(397\) 2.90741e6 0.925827 0.462914 0.886403i \(-0.346804\pi\)
0.462914 + 0.886403i \(0.346804\pi\)
\(398\) 5.37320e6 1.70030
\(399\) −61183.0 −0.0192397
\(400\) 0 0
\(401\) −72188.2 −0.0224184 −0.0112092 0.999937i \(-0.503568\pi\)
−0.0112092 + 0.999937i \(0.503568\pi\)
\(402\) −230600. −0.0711695
\(403\) −4.99802e6 −1.53298
\(404\) 3.71543e6 1.13255
\(405\) 0 0
\(406\) 471802. 0.142051
\(407\) 1.46368e6 0.437986
\(408\) −559997. −0.166546
\(409\) −3.53120e6 −1.04379 −0.521895 0.853009i \(-0.674775\pi\)
−0.521895 + 0.853009i \(0.674775\pi\)
\(410\) 0 0
\(411\) −251645. −0.0734826
\(412\) −3.30772e6 −0.960033
\(413\) −716951. −0.206830
\(414\) 1.71647e6 0.492193
\(415\) 0 0
\(416\) −5.16541e6 −1.46343
\(417\) −282981. −0.0796925
\(418\) 445463. 0.124701
\(419\) 2.48255e6 0.690817 0.345409 0.938452i \(-0.387740\pi\)
0.345409 + 0.938452i \(0.387740\pi\)
\(420\) 0 0
\(421\) 3.38758e6 0.931504 0.465752 0.884915i \(-0.345784\pi\)
0.465752 + 0.884915i \(0.345784\pi\)
\(422\) 3.85319e6 1.05327
\(423\) 1.85940e6 0.505267
\(424\) 687409. 0.185695
\(425\) 0 0
\(426\) −4.70508e6 −1.25615
\(427\) −864202. −0.229375
\(428\) 491466. 0.129683
\(429\) 680269. 0.178459
\(430\) 0 0
\(431\) 5.13279e6 1.33095 0.665473 0.746422i \(-0.268230\pi\)
0.665473 + 0.746422i \(0.268230\pi\)
\(432\) 582468. 0.150163
\(433\) −4.52419e6 −1.15963 −0.579817 0.814747i \(-0.696876\pi\)
−0.579817 + 0.814747i \(0.696876\pi\)
\(434\) 1.03319e6 0.263304
\(435\) 0 0
\(436\) −4.94776e6 −1.24650
\(437\) −1.11558e6 −0.279447
\(438\) −4.57858e6 −1.14037
\(439\) 2.94510e6 0.729355 0.364677 0.931134i \(-0.381179\pi\)
0.364677 + 0.931134i \(0.381179\pi\)
\(440\) 0 0
\(441\) −1.34205e6 −0.328604
\(442\) 6.55254e6 1.59534
\(443\) 1.33910e6 0.324192 0.162096 0.986775i \(-0.448175\pi\)
0.162096 + 0.986775i \(0.448175\pi\)
\(444\) 4.12959e6 0.994144
\(445\) 0 0
\(446\) 7.04035e6 1.67594
\(447\) 1.08043e6 0.255756
\(448\) 672981. 0.158419
\(449\) −4.92158e6 −1.15210 −0.576048 0.817416i \(-0.695406\pi\)
−0.576048 + 0.817416i \(0.695406\pi\)
\(450\) 0 0
\(451\) −1.60453e6 −0.371456
\(452\) 345653. 0.0795783
\(453\) 998110. 0.228525
\(454\) −3.47352e6 −0.790916
\(455\) 0 0
\(456\) 196542. 0.0442632
\(457\) 4.92247e6 1.10253 0.551267 0.834329i \(-0.314144\pi\)
0.551267 + 0.834329i \(0.314144\pi\)
\(458\) 1.11333e7 2.48004
\(459\) −914423. −0.202589
\(460\) 0 0
\(461\) 8.40393e6 1.84175 0.920874 0.389861i \(-0.127477\pi\)
0.920874 + 0.389861i \(0.127477\pi\)
\(462\) −140626. −0.0306521
\(463\) 2.90332e6 0.629422 0.314711 0.949187i \(-0.398092\pi\)
0.314711 + 0.949187i \(0.398092\pi\)
\(464\) 2.91922e6 0.629466
\(465\) 0 0
\(466\) −446312. −0.0952082
\(467\) −3.02045e6 −0.640883 −0.320442 0.947268i \(-0.603831\pi\)
−0.320442 + 0.947268i \(0.603831\pi\)
\(468\) 1.91929e6 0.405067
\(469\) −47312.8 −0.00993222
\(470\) 0 0
\(471\) −4.78788e6 −0.994468
\(472\) 2.30310e6 0.475837
\(473\) −562833. −0.115672
\(474\) −4.05168e6 −0.828302
\(475\) 0 0
\(476\) −734721. −0.148629
\(477\) 1.12248e6 0.225882
\(478\) 1.34374e6 0.268996
\(479\) −731965. −0.145764 −0.0728822 0.997341i \(-0.523220\pi\)
−0.0728822 + 0.997341i \(0.523220\pi\)
\(480\) 0 0
\(481\) −7.55638e6 −1.48919
\(482\) 4.15111e6 0.813854
\(483\) 352173. 0.0686891
\(484\) 555359. 0.107761
\(485\) 0 0
\(486\) −493799. −0.0948329
\(487\) −2.93463e6 −0.560700 −0.280350 0.959898i \(-0.590451\pi\)
−0.280350 + 0.959898i \(0.590451\pi\)
\(488\) 2.77613e6 0.527703
\(489\) −4.13166e6 −0.781362
\(490\) 0 0
\(491\) −6.20616e6 −1.16177 −0.580884 0.813987i \(-0.697293\pi\)
−0.580884 + 0.813987i \(0.697293\pi\)
\(492\) −4.52699e6 −0.843134
\(493\) −4.58292e6 −0.849229
\(494\) −2.29974e6 −0.423995
\(495\) 0 0
\(496\) 6.39279e6 1.16677
\(497\) −965354. −0.175306
\(498\) 4.03299e6 0.728708
\(499\) −6.62174e6 −1.19048 −0.595239 0.803549i \(-0.702942\pi\)
−0.595239 + 0.803549i \(0.702942\pi\)
\(500\) 0 0
\(501\) −2.57744e6 −0.458770
\(502\) 1.49440e7 2.64672
\(503\) 2.50211e6 0.440946 0.220473 0.975393i \(-0.429240\pi\)
0.220473 + 0.975393i \(0.429240\pi\)
\(504\) −62045.2 −0.0108801
\(505\) 0 0
\(506\) −2.56411e6 −0.445205
\(507\) −170309. −0.0294252
\(508\) 2.57209e6 0.442208
\(509\) 4.26240e6 0.729222 0.364611 0.931160i \(-0.381202\pi\)
0.364611 + 0.931160i \(0.381202\pi\)
\(510\) 0 0
\(511\) −939400. −0.159147
\(512\) 7.87517e6 1.32765
\(513\) 320934. 0.0538422
\(514\) −1.15317e7 −1.92525
\(515\) 0 0
\(516\) −1.58796e6 −0.262552
\(517\) −2.77762e6 −0.457031
\(518\) 1.56206e6 0.255784
\(519\) 5.90540e6 0.962345
\(520\) 0 0
\(521\) 5.90104e6 0.952433 0.476216 0.879328i \(-0.342008\pi\)
0.476216 + 0.879328i \(0.342008\pi\)
\(522\) −2.47483e6 −0.397529
\(523\) 1.27490e6 0.203809 0.101904 0.994794i \(-0.467506\pi\)
0.101904 + 0.994794i \(0.467506\pi\)
\(524\) 1.02592e7 1.63225
\(525\) 0 0
\(526\) 1.61370e7 2.54308
\(527\) −1.00361e7 −1.57412
\(528\) −870107. −0.135828
\(529\) −14971.4 −0.00232607
\(530\) 0 0
\(531\) 3.76075e6 0.578813
\(532\) 257864. 0.0395014
\(533\) 8.28355e6 1.26299
\(534\) −254528. −0.0386262
\(535\) 0 0
\(536\) 151986. 0.0228502
\(537\) 1.98000e6 0.296298
\(538\) 9.54674e6 1.42200
\(539\) 2.00479e6 0.297234
\(540\) 0 0
\(541\) 1.18799e7 1.74510 0.872551 0.488523i \(-0.162464\pi\)
0.872551 + 0.488523i \(0.162464\pi\)
\(542\) −4.88521e6 −0.714307
\(543\) 3.83524e6 0.558204
\(544\) −1.03722e7 −1.50271
\(545\) 0 0
\(546\) 725992. 0.104220
\(547\) 1.02980e7 1.47158 0.735788 0.677212i \(-0.236812\pi\)
0.735788 + 0.677212i \(0.236812\pi\)
\(548\) 1.06060e6 0.150868
\(549\) 4.53316e6 0.641904
\(550\) 0 0
\(551\) 1.60846e6 0.225700
\(552\) −1.13131e6 −0.158027
\(553\) −831294. −0.115596
\(554\) −1.63231e7 −2.25959
\(555\) 0 0
\(556\) 1.19266e6 0.163618
\(557\) −9.12350e6 −1.24602 −0.623008 0.782215i \(-0.714090\pi\)
−0.623008 + 0.782215i \(0.714090\pi\)
\(558\) −5.41961e6 −0.736855
\(559\) 2.90567e6 0.393294
\(560\) 0 0
\(561\) 1.36599e6 0.183248
\(562\) 1.09122e6 0.145738
\(563\) −1.08869e7 −1.44754 −0.723771 0.690040i \(-0.757593\pi\)
−0.723771 + 0.690040i \(0.757593\pi\)
\(564\) −7.83669e6 −1.03737
\(565\) 0 0
\(566\) −1.28584e7 −1.68711
\(567\) −101314. −0.0132346
\(568\) 3.10106e6 0.403311
\(569\) −8.89191e6 −1.15137 −0.575684 0.817672i \(-0.695264\pi\)
−0.575684 + 0.817672i \(0.695264\pi\)
\(570\) 0 0
\(571\) 1.47453e7 1.89262 0.946311 0.323258i \(-0.104778\pi\)
0.946311 + 0.323258i \(0.104778\pi\)
\(572\) −2.86709e6 −0.366397
\(573\) −1.46812e6 −0.186800
\(574\) −1.71238e6 −0.216931
\(575\) 0 0
\(576\) −3.53011e6 −0.443335
\(577\) 6.49447e6 0.812090 0.406045 0.913853i \(-0.366908\pi\)
0.406045 + 0.913853i \(0.366908\pi\)
\(578\) 1.28400e6 0.159862
\(579\) −1.91394e6 −0.237264
\(580\) 0 0
\(581\) 827460. 0.101697
\(582\) −2.27970e6 −0.278978
\(583\) −1.67678e6 −0.204318
\(584\) 3.01769e6 0.366136
\(585\) 0 0
\(586\) −1.83914e7 −2.21243
\(587\) 2.10642e6 0.252319 0.126160 0.992010i \(-0.459735\pi\)
0.126160 + 0.992010i \(0.459735\pi\)
\(588\) 5.65627e6 0.674663
\(589\) 3.52236e6 0.418356
\(590\) 0 0
\(591\) 6.56704e6 0.773394
\(592\) 9.66508e6 1.13345
\(593\) 9.24728e6 1.07988 0.539942 0.841702i \(-0.318446\pi\)
0.539942 + 0.841702i \(0.318446\pi\)
\(594\) 737650. 0.0857796
\(595\) 0 0
\(596\) −4.55361e6 −0.525098
\(597\) −5.78280e6 −0.664053
\(598\) 1.32374e7 1.51374
\(599\) 1.30180e6 0.148245 0.0741223 0.997249i \(-0.476384\pi\)
0.0741223 + 0.997249i \(0.476384\pi\)
\(600\) 0 0
\(601\) 2.74661e6 0.310178 0.155089 0.987901i \(-0.450434\pi\)
0.155089 + 0.987901i \(0.450434\pi\)
\(602\) −600663. −0.0675523
\(603\) 248178. 0.0277953
\(604\) −4.20668e6 −0.469188
\(605\) 0 0
\(606\) −7.37200e6 −0.815462
\(607\) 2.68332e6 0.295598 0.147799 0.989017i \(-0.452781\pi\)
0.147799 + 0.989017i \(0.452781\pi\)
\(608\) 3.64033e6 0.399375
\(609\) −507767. −0.0554781
\(610\) 0 0
\(611\) 1.43397e7 1.55395
\(612\) 3.85397e6 0.415939
\(613\) −2.54616e6 −0.273675 −0.136837 0.990594i \(-0.543694\pi\)
−0.136837 + 0.990594i \(0.543694\pi\)
\(614\) 1.14259e6 0.122312
\(615\) 0 0
\(616\) 92684.8 0.00984140
\(617\) −7.90825e6 −0.836310 −0.418155 0.908376i \(-0.637323\pi\)
−0.418155 + 0.908376i \(0.637323\pi\)
\(618\) 6.56305e6 0.691249
\(619\) −1.80718e7 −1.89573 −0.947864 0.318675i \(-0.896762\pi\)
−0.947864 + 0.318675i \(0.896762\pi\)
\(620\) 0 0
\(621\) −1.84732e6 −0.192226
\(622\) −1.70018e7 −1.76206
\(623\) −52222.1 −0.00539057
\(624\) 4.49200e6 0.461826
\(625\) 0 0
\(626\) −6.80862e6 −0.694421
\(627\) −479420. −0.0487021
\(628\) 2.01792e7 2.04176
\(629\) −1.51733e7 −1.52916
\(630\) 0 0
\(631\) 8.68542e6 0.868395 0.434198 0.900818i \(-0.357032\pi\)
0.434198 + 0.900818i \(0.357032\pi\)
\(632\) 2.67041e6 0.265941
\(633\) −4.14692e6 −0.411355
\(634\) −4.17236e6 −0.412248
\(635\) 0 0
\(636\) −4.73083e6 −0.463762
\(637\) −1.03499e7 −1.01062
\(638\) 3.69696e6 0.359578
\(639\) 5.06375e6 0.490591
\(640\) 0 0
\(641\) 1.20048e7 1.15402 0.577008 0.816739i \(-0.304220\pi\)
0.577008 + 0.816739i \(0.304220\pi\)
\(642\) −975147. −0.0933754
\(643\) −2.64892e6 −0.252663 −0.126331 0.991988i \(-0.540320\pi\)
−0.126331 + 0.991988i \(0.540320\pi\)
\(644\) −1.48428e6 −0.141027
\(645\) 0 0
\(646\) −4.61791e6 −0.435375
\(647\) 1.54484e7 1.45085 0.725425 0.688301i \(-0.241643\pi\)
0.725425 + 0.688301i \(0.241643\pi\)
\(648\) 325457. 0.0304478
\(649\) −5.61792e6 −0.523557
\(650\) 0 0
\(651\) −1.11196e6 −0.102834
\(652\) 1.74135e7 1.60423
\(653\) 2.03688e7 1.86932 0.934658 0.355548i \(-0.115706\pi\)
0.934658 + 0.355548i \(0.115706\pi\)
\(654\) 9.81715e6 0.897514
\(655\) 0 0
\(656\) −1.05952e7 −0.961277
\(657\) 4.92761e6 0.445372
\(658\) −2.96431e6 −0.266906
\(659\) −6.76191e6 −0.606535 −0.303268 0.952905i \(-0.598078\pi\)
−0.303268 + 0.952905i \(0.598078\pi\)
\(660\) 0 0
\(661\) 1.49341e7 1.32946 0.664728 0.747085i \(-0.268547\pi\)
0.664728 + 0.747085i \(0.268547\pi\)
\(662\) 1.39319e6 0.123556
\(663\) −7.05204e6 −0.623061
\(664\) −2.65810e6 −0.233965
\(665\) 0 0
\(666\) −8.19376e6 −0.715810
\(667\) −9.25841e6 −0.805790
\(668\) 1.08630e7 0.941909
\(669\) −7.57704e6 −0.654537
\(670\) 0 0
\(671\) −6.77175e6 −0.580624
\(672\) −1.14920e6 −0.0981682
\(673\) 6.00776e6 0.511299 0.255649 0.966770i \(-0.417711\pi\)
0.255649 + 0.966770i \(0.417711\pi\)
\(674\) 1.67809e6 0.142287
\(675\) 0 0
\(676\) 717794. 0.0604133
\(677\) −2.14516e7 −1.79882 −0.899409 0.437108i \(-0.856003\pi\)
−0.899409 + 0.437108i \(0.856003\pi\)
\(678\) −685831. −0.0572984
\(679\) −467731. −0.0389334
\(680\) 0 0
\(681\) 3.73831e6 0.308893
\(682\) 8.09596e6 0.666511
\(683\) 5.40024e6 0.442956 0.221478 0.975165i \(-0.428912\pi\)
0.221478 + 0.975165i \(0.428912\pi\)
\(684\) −1.35262e6 −0.110544
\(685\) 0 0
\(686\) 4.30988e6 0.349667
\(687\) −1.19819e7 −0.968580
\(688\) −3.71654e6 −0.299342
\(689\) 8.65654e6 0.694698
\(690\) 0 0
\(691\) −1.55798e7 −1.24127 −0.620635 0.784100i \(-0.713125\pi\)
−0.620635 + 0.784100i \(0.713125\pi\)
\(692\) −2.48892e7 −1.97581
\(693\) 151346. 0.0119712
\(694\) −1.16079e7 −0.914864
\(695\) 0 0
\(696\) 1.63113e6 0.127634
\(697\) 1.66335e7 1.29688
\(698\) −3.43912e7 −2.67183
\(699\) 480335. 0.0371836
\(700\) 0 0
\(701\) −1.49222e7 −1.14693 −0.573465 0.819230i \(-0.694401\pi\)
−0.573465 + 0.819230i \(0.694401\pi\)
\(702\) −3.80818e6 −0.291659
\(703\) 5.32536e6 0.406407
\(704\) 5.27338e6 0.401012
\(705\) 0 0
\(706\) 9.60842e6 0.725504
\(707\) −1.51253e6 −0.113804
\(708\) −1.58502e7 −1.18837
\(709\) 7.29943e6 0.545347 0.272674 0.962107i \(-0.412092\pi\)
0.272674 + 0.962107i \(0.412092\pi\)
\(710\) 0 0
\(711\) 4.36054e6 0.323494
\(712\) 167756. 0.0124016
\(713\) −2.02749e7 −1.49360
\(714\) 1.45780e6 0.107017
\(715\) 0 0
\(716\) −8.34498e6 −0.608335
\(717\) −1.44617e6 −0.105056
\(718\) 1.64067e7 1.18771
\(719\) −80423.4 −0.00580176 −0.00290088 0.999996i \(-0.500923\pi\)
−0.00290088 + 0.999996i \(0.500923\pi\)
\(720\) 0 0
\(721\) 1.34656e6 0.0964689
\(722\) −1.90857e7 −1.36259
\(723\) −4.46755e6 −0.317851
\(724\) −1.61642e7 −1.14606
\(725\) 0 0
\(726\) −1.10192e6 −0.0775906
\(727\) −1.75871e7 −1.23412 −0.617061 0.786916i \(-0.711677\pi\)
−0.617061 + 0.786916i \(0.711677\pi\)
\(728\) −478493. −0.0334617
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 5.83464e6 0.403850
\(732\) −1.91056e7 −1.31790
\(733\) 1.17001e7 0.804319 0.402160 0.915570i \(-0.368260\pi\)
0.402160 + 0.915570i \(0.368260\pi\)
\(734\) −2.49345e7 −1.70829
\(735\) 0 0
\(736\) −2.09539e7 −1.42584
\(737\) −370736. −0.0251418
\(738\) 8.98227e6 0.607079
\(739\) −1.55729e7 −1.04896 −0.524480 0.851422i \(-0.675740\pi\)
−0.524480 + 0.851422i \(0.675740\pi\)
\(740\) 0 0
\(741\) 2.47505e6 0.165592
\(742\) −1.78949e6 −0.119322
\(743\) 320800. 0.0213188 0.0106594 0.999943i \(-0.496607\pi\)
0.0106594 + 0.999943i \(0.496607\pi\)
\(744\) 3.57200e6 0.236581
\(745\) 0 0
\(746\) −3.34963e7 −2.20368
\(747\) −4.34043e6 −0.284597
\(748\) −5.75716e6 −0.376231
\(749\) −200074. −0.0130312
\(750\) 0 0
\(751\) −1.56465e7 −1.01232 −0.506159 0.862440i \(-0.668935\pi\)
−0.506159 + 0.862440i \(0.668935\pi\)
\(752\) −1.83414e7 −1.18273
\(753\) −1.60832e7 −1.03368
\(754\) −1.90859e7 −1.22260
\(755\) 0 0
\(756\) 427003. 0.0271723
\(757\) 1.55874e7 0.988629 0.494314 0.869283i \(-0.335419\pi\)
0.494314 + 0.869283i \(0.335419\pi\)
\(758\) −2.09365e7 −1.32352
\(759\) 2.75957e6 0.173875
\(760\) 0 0
\(761\) 9.31514e6 0.583079 0.291540 0.956559i \(-0.405832\pi\)
0.291540 + 0.956559i \(0.405832\pi\)
\(762\) −5.10343e6 −0.318401
\(763\) 2.01421e6 0.125255
\(764\) 6.18762e6 0.383522
\(765\) 0 0
\(766\) −1.71753e7 −1.05763
\(767\) 2.90030e7 1.78014
\(768\) −5.03689e6 −0.308148
\(769\) 1.43630e7 0.875848 0.437924 0.899012i \(-0.355714\pi\)
0.437924 + 0.899012i \(0.355714\pi\)
\(770\) 0 0
\(771\) 1.24108e7 0.751907
\(772\) 8.06656e6 0.487131
\(773\) −4.14465e6 −0.249482 −0.124741 0.992189i \(-0.539810\pi\)
−0.124741 + 0.992189i \(0.539810\pi\)
\(774\) 3.15077e6 0.189045
\(775\) 0 0
\(776\) 1.50252e6 0.0895708
\(777\) −1.68114e6 −0.0998965
\(778\) −2.75434e7 −1.63143
\(779\) −5.83784e6 −0.344674
\(780\) 0 0
\(781\) −7.56437e6 −0.443757
\(782\) 2.65810e7 1.55437
\(783\) 2.66348e6 0.155255
\(784\) 1.32382e7 0.769200
\(785\) 0 0
\(786\) −2.03559e7 −1.17526
\(787\) −1.44908e7 −0.833980 −0.416990 0.908911i \(-0.636915\pi\)
−0.416990 + 0.908911i \(0.636915\pi\)
\(788\) −2.76777e7 −1.58787
\(789\) −1.73672e7 −0.993199
\(790\) 0 0
\(791\) −140714. −0.00799642
\(792\) −486177. −0.0275411
\(793\) 3.49597e7 1.97417
\(794\) 2.43133e7 1.36865
\(795\) 0 0
\(796\) 2.43725e7 1.36338
\(797\) 4.19488e6 0.233923 0.116962 0.993136i \(-0.462685\pi\)
0.116962 + 0.993136i \(0.462685\pi\)
\(798\) −511644. −0.0284420
\(799\) 2.87943e7 1.59566
\(800\) 0 0
\(801\) 273930. 0.0150855
\(802\) −603676. −0.0331412
\(803\) −7.36099e6 −0.402854
\(804\) −1.04598e6 −0.0570670
\(805\) 0 0
\(806\) −4.17961e7 −2.26620
\(807\) −1.02745e7 −0.555363
\(808\) 4.85880e6 0.261819
\(809\) 2.08215e7 1.11851 0.559257 0.828994i \(-0.311086\pi\)
0.559257 + 0.828994i \(0.311086\pi\)
\(810\) 0 0
\(811\) 3.14240e7 1.67768 0.838839 0.544379i \(-0.183235\pi\)
0.838839 + 0.544379i \(0.183235\pi\)
\(812\) 2.14006e6 0.113903
\(813\) 5.25761e6 0.278973
\(814\) 1.22401e7 0.647474
\(815\) 0 0
\(816\) 9.02000e6 0.474222
\(817\) −2.04778e6 −0.107332
\(818\) −2.95297e7 −1.54304
\(819\) −781335. −0.0407031
\(820\) 0 0
\(821\) −1.81350e7 −0.938990 −0.469495 0.882935i \(-0.655564\pi\)
−0.469495 + 0.882935i \(0.655564\pi\)
\(822\) −2.10439e6 −0.108629
\(823\) 1.05426e7 0.542561 0.271281 0.962500i \(-0.412553\pi\)
0.271281 + 0.962500i \(0.412553\pi\)
\(824\) −4.32563e6 −0.221938
\(825\) 0 0
\(826\) −5.99552e6 −0.305757
\(827\) −2.74783e7 −1.39710 −0.698548 0.715563i \(-0.746170\pi\)
−0.698548 + 0.715563i \(0.746170\pi\)
\(828\) 7.78578e6 0.394663
\(829\) 2.35793e7 1.19164 0.595819 0.803119i \(-0.296828\pi\)
0.595819 + 0.803119i \(0.296828\pi\)
\(830\) 0 0
\(831\) 1.75674e7 0.882482
\(832\) −2.72243e7 −1.36348
\(833\) −2.07828e7 −1.03775
\(834\) −2.36644e6 −0.117809
\(835\) 0 0
\(836\) 2.02059e6 0.0999912
\(837\) 5.83274e6 0.287779
\(838\) 2.07604e7 1.02123
\(839\) −3.78658e7 −1.85713 −0.928565 0.371170i \(-0.878957\pi\)
−0.928565 + 0.371170i \(0.878957\pi\)
\(840\) 0 0
\(841\) −7.16226e6 −0.349188
\(842\) 2.83288e7 1.37704
\(843\) −1.17441e6 −0.0569180
\(844\) 1.74778e7 0.844560
\(845\) 0 0
\(846\) 1.55492e7 0.746936
\(847\) −226084. −0.0108283
\(848\) −1.10723e7 −0.528746
\(849\) 1.38385e7 0.658903
\(850\) 0 0
\(851\) −3.06531e7 −1.45094
\(852\) −2.13419e7 −1.00724
\(853\) −2.10629e7 −0.991166 −0.495583 0.868561i \(-0.665046\pi\)
−0.495583 + 0.868561i \(0.665046\pi\)
\(854\) −7.22691e6 −0.339084
\(855\) 0 0
\(856\) 642708. 0.0299799
\(857\) −6.20639e6 −0.288660 −0.144330 0.989530i \(-0.546103\pi\)
−0.144330 + 0.989530i \(0.546103\pi\)
\(858\) 5.68876e6 0.263815
\(859\) −1.19855e7 −0.554206 −0.277103 0.960840i \(-0.589374\pi\)
−0.277103 + 0.960840i \(0.589374\pi\)
\(860\) 0 0
\(861\) 1.84292e6 0.0847223
\(862\) 4.29231e7 1.96754
\(863\) −3.35712e7 −1.53440 −0.767202 0.641406i \(-0.778352\pi\)
−0.767202 + 0.641406i \(0.778352\pi\)
\(864\) 6.02809e6 0.274723
\(865\) 0 0
\(866\) −3.78336e7 −1.71429
\(867\) −1.38188e6 −0.0624341
\(868\) 4.68650e6 0.211130
\(869\) −6.51389e6 −0.292611
\(870\) 0 0
\(871\) 1.91396e6 0.0854843
\(872\) −6.47037e6 −0.288163
\(873\) 2.45348e6 0.108955
\(874\) −9.32910e6 −0.413106
\(875\) 0 0
\(876\) −2.07681e7 −0.914401
\(877\) −2.91315e7 −1.27898 −0.639490 0.768799i \(-0.720854\pi\)
−0.639490 + 0.768799i \(0.720854\pi\)
\(878\) 2.46285e7 1.07820
\(879\) 1.97933e7 0.864065
\(880\) 0 0
\(881\) 1.56250e7 0.678237 0.339119 0.940744i \(-0.389871\pi\)
0.339119 + 0.940744i \(0.389871\pi\)
\(882\) −1.12229e7 −0.485775
\(883\) 3.67889e6 0.158787 0.0793935 0.996843i \(-0.474702\pi\)
0.0793935 + 0.996843i \(0.474702\pi\)
\(884\) 2.97218e7 1.27922
\(885\) 0 0
\(886\) 1.11982e7 0.479253
\(887\) −2.49631e7 −1.06534 −0.532672 0.846322i \(-0.678812\pi\)
−0.532672 + 0.846322i \(0.678812\pi\)
\(888\) 5.40041e6 0.229824
\(889\) −1.04708e6 −0.0444352
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 3.19345e7 1.34384
\(893\) −1.01059e7 −0.424079
\(894\) 9.03508e6 0.378084
\(895\) 0 0
\(896\) 1.54180e6 0.0641589
\(897\) −1.42465e7 −0.591191
\(898\) −4.11568e7 −1.70314
\(899\) 2.92326e7 1.20634
\(900\) 0 0
\(901\) 1.73825e7 0.713344
\(902\) −1.34180e7 −0.549123
\(903\) 646452. 0.0263826
\(904\) 452023. 0.0183967
\(905\) 0 0
\(906\) 8.34672e6 0.337828
\(907\) −1.41749e7 −0.572139 −0.286070 0.958209i \(-0.592349\pi\)
−0.286070 + 0.958209i \(0.592349\pi\)
\(908\) −1.57556e7 −0.634193
\(909\) 7.93397e6 0.318479
\(910\) 0 0
\(911\) 3.03213e6 0.121047 0.0605233 0.998167i \(-0.480723\pi\)
0.0605233 + 0.998167i \(0.480723\pi\)
\(912\) −3.16574e6 −0.126034
\(913\) 6.48385e6 0.257428
\(914\) 4.11642e7 1.62988
\(915\) 0 0
\(916\) 5.04996e7 1.98861
\(917\) −4.17647e6 −0.164016
\(918\) −7.64688e6 −0.299487
\(919\) 1.47606e6 0.0576521 0.0288260 0.999584i \(-0.490823\pi\)
0.0288260 + 0.999584i \(0.490823\pi\)
\(920\) 0 0
\(921\) −1.22969e6 −0.0477691
\(922\) 7.02780e7 2.72265
\(923\) 3.90517e7 1.50881
\(924\) −637868. −0.0245782
\(925\) 0 0
\(926\) 2.42791e7 0.930475
\(927\) −7.06335e6 −0.269968
\(928\) 3.02117e7 1.15161
\(929\) −2.51584e7 −0.956409 −0.478205 0.878248i \(-0.658712\pi\)
−0.478205 + 0.878248i \(0.658712\pi\)
\(930\) 0 0
\(931\) 7.29412e6 0.275803
\(932\) −2.02444e6 −0.0763423
\(933\) 1.82979e7 0.688172
\(934\) −2.52585e7 −0.947417
\(935\) 0 0
\(936\) 2.50993e6 0.0936422
\(937\) 3.23202e7 1.20261 0.601305 0.799019i \(-0.294648\pi\)
0.601305 + 0.799019i \(0.294648\pi\)
\(938\) −395654. −0.0146828
\(939\) 7.32764e6 0.271207
\(940\) 0 0
\(941\) −2.88348e7 −1.06156 −0.530778 0.847511i \(-0.678100\pi\)
−0.530778 + 0.847511i \(0.678100\pi\)
\(942\) −4.00387e7 −1.47012
\(943\) 3.36030e7 1.23055
\(944\) −3.70966e7 −1.35489
\(945\) 0 0
\(946\) −4.70671e6 −0.170997
\(947\) 1.94005e7 0.702973 0.351487 0.936193i \(-0.385676\pi\)
0.351487 + 0.936193i \(0.385676\pi\)
\(948\) −1.83781e7 −0.664171
\(949\) 3.80018e7 1.36974
\(950\) 0 0
\(951\) 4.49042e6 0.161004
\(952\) −960821. −0.0343598
\(953\) 1.51638e6 0.0540848 0.0270424 0.999634i \(-0.491391\pi\)
0.0270424 + 0.999634i \(0.491391\pi\)
\(954\) 9.38672e6 0.333921
\(955\) 0 0
\(956\) 6.09510e6 0.215693
\(957\) −3.97879e6 −0.140433
\(958\) −6.12108e6 −0.215484
\(959\) −431763. −0.0151600
\(960\) 0 0
\(961\) 3.53872e7 1.23605
\(962\) −6.31904e7 −2.20147
\(963\) 1.04948e6 0.0364678
\(964\) 1.88291e7 0.652586
\(965\) 0 0
\(966\) 2.94505e6 0.101543
\(967\) 4.41852e6 0.151954 0.0759768 0.997110i \(-0.475793\pi\)
0.0759768 + 0.997110i \(0.475793\pi\)
\(968\) 726264. 0.0249119
\(969\) 4.96993e6 0.170036
\(970\) 0 0
\(971\) −2.45791e6 −0.0836600 −0.0418300 0.999125i \(-0.513319\pi\)
−0.0418300 + 0.999125i \(0.513319\pi\)
\(972\) −2.23983e6 −0.0760414
\(973\) −485528. −0.0164412
\(974\) −2.45409e7 −0.828883
\(975\) 0 0
\(976\) −4.47157e7 −1.50257
\(977\) −1.93483e7 −0.648495 −0.324248 0.945972i \(-0.605111\pi\)
−0.324248 + 0.945972i \(0.605111\pi\)
\(978\) −3.45511e7 −1.15509
\(979\) −409205. −0.0136453
\(980\) 0 0
\(981\) −1.05655e7 −0.350524
\(982\) −5.18992e7 −1.71744
\(983\) 5.63062e6 0.185854 0.0929271 0.995673i \(-0.470378\pi\)
0.0929271 + 0.995673i \(0.470378\pi\)
\(984\) −5.92011e6 −0.194914
\(985\) 0 0
\(986\) −3.83248e7 −1.25541
\(987\) 3.19028e6 0.104240
\(988\) −1.04314e7 −0.339979
\(989\) 1.17871e7 0.383193
\(990\) 0 0
\(991\) 1.17881e7 0.381293 0.190647 0.981659i \(-0.438942\pi\)
0.190647 + 0.981659i \(0.438942\pi\)
\(992\) 6.61603e7 2.13461
\(993\) −1.49939e6 −0.0482549
\(994\) −8.07280e6 −0.259154
\(995\) 0 0
\(996\) 1.82934e7 0.584312
\(997\) −2.55288e7 −0.813379 −0.406689 0.913567i \(-0.633317\pi\)
−0.406689 + 0.913567i \(0.633317\pi\)
\(998\) −5.53745e7 −1.75988
\(999\) 8.81837e6 0.279560
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.1.8 yes 9
5.4 even 2 825.6.a.r.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.r.1.2 9 5.4 even 2
825.6.a.s.1.8 yes 9 1.1 even 1 trivial