Properties

Label 825.6.a.y.1.8
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-3.97394\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.97394 q^{2} +9.00000 q^{3} -7.25990 q^{4} +44.7655 q^{6} -155.412 q^{7} -195.276 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.97394 q^{2} +9.00000 q^{3} -7.25990 q^{4} +44.7655 q^{6} -155.412 q^{7} -195.276 q^{8} +81.0000 q^{9} +121.000 q^{11} -65.3391 q^{12} -1173.30 q^{13} -773.008 q^{14} -738.977 q^{16} -898.118 q^{17} +402.889 q^{18} -2407.95 q^{19} -1398.70 q^{21} +601.847 q^{22} +4121.12 q^{23} -1757.49 q^{24} -5835.95 q^{26} +729.000 q^{27} +1128.27 q^{28} +75.4479 q^{29} +3595.47 q^{31} +2573.22 q^{32} +1089.00 q^{33} -4467.18 q^{34} -588.052 q^{36} +16063.1 q^{37} -11977.0 q^{38} -10559.7 q^{39} +15062.8 q^{41} -6957.07 q^{42} +12867.8 q^{43} -878.448 q^{44} +20498.2 q^{46} +6698.84 q^{47} -6650.79 q^{48} +7345.75 q^{49} -8083.06 q^{51} +8518.07 q^{52} -22850.8 q^{53} +3626.00 q^{54} +30348.2 q^{56} -21671.6 q^{57} +375.274 q^{58} -13021.3 q^{59} +882.310 q^{61} +17883.6 q^{62} -12588.3 q^{63} +36446.3 q^{64} +5416.62 q^{66} +14191.7 q^{67} +6520.24 q^{68} +37090.1 q^{69} -78746.0 q^{71} -15817.4 q^{72} -34227.3 q^{73} +79896.8 q^{74} +17481.5 q^{76} -18804.8 q^{77} -52523.5 q^{78} -48694.6 q^{79} +6561.00 q^{81} +74921.3 q^{82} +95905.3 q^{83} +10154.5 q^{84} +64003.6 q^{86} +679.031 q^{87} -23628.5 q^{88} +1342.55 q^{89} +182345. q^{91} -29918.9 q^{92} +32359.2 q^{93} +33319.6 q^{94} +23159.0 q^{96} -80938.9 q^{97} +36537.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.97394 0.879277 0.439639 0.898175i \(-0.355107\pi\)
0.439639 + 0.898175i \(0.355107\pi\)
\(3\) 9.00000 0.577350
\(4\) −7.25990 −0.226872
\(5\) 0 0
\(6\) 44.7655 0.507651
\(7\) −155.412 −1.19878 −0.599388 0.800459i \(-0.704589\pi\)
−0.599388 + 0.800459i \(0.704589\pi\)
\(8\) −195.276 −1.07876
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −65.3391 −0.130985
\(13\) −1173.30 −1.92554 −0.962770 0.270323i \(-0.912869\pi\)
−0.962770 + 0.270323i \(0.912869\pi\)
\(14\) −773.008 −1.05406
\(15\) 0 0
\(16\) −738.977 −0.721657
\(17\) −898.118 −0.753721 −0.376861 0.926270i \(-0.622996\pi\)
−0.376861 + 0.926270i \(0.622996\pi\)
\(18\) 402.889 0.293092
\(19\) −2407.95 −1.53026 −0.765128 0.643879i \(-0.777324\pi\)
−0.765128 + 0.643879i \(0.777324\pi\)
\(20\) 0 0
\(21\) −1398.70 −0.692114
\(22\) 601.847 0.265112
\(23\) 4121.12 1.62441 0.812205 0.583372i \(-0.198267\pi\)
0.812205 + 0.583372i \(0.198267\pi\)
\(24\) −1757.49 −0.622823
\(25\) 0 0
\(26\) −5835.95 −1.69308
\(27\) 729.000 0.192450
\(28\) 1128.27 0.271969
\(29\) 75.4479 0.0166591 0.00832957 0.999965i \(-0.497349\pi\)
0.00832957 + 0.999965i \(0.497349\pi\)
\(30\) 0 0
\(31\) 3595.47 0.671972 0.335986 0.941867i \(-0.390931\pi\)
0.335986 + 0.941867i \(0.390931\pi\)
\(32\) 2573.22 0.444224
\(33\) 1089.00 0.174078
\(34\) −4467.18 −0.662730
\(35\) 0 0
\(36\) −588.052 −0.0756240
\(37\) 16063.1 1.92897 0.964483 0.264145i \(-0.0850898\pi\)
0.964483 + 0.264145i \(0.0850898\pi\)
\(38\) −11977.0 −1.34552
\(39\) −10559.7 −1.11171
\(40\) 0 0
\(41\) 15062.8 1.39941 0.699705 0.714432i \(-0.253315\pi\)
0.699705 + 0.714432i \(0.253315\pi\)
\(42\) −6957.07 −0.608560
\(43\) 12867.8 1.06129 0.530643 0.847595i \(-0.321950\pi\)
0.530643 + 0.847595i \(0.321950\pi\)
\(44\) −878.448 −0.0684045
\(45\) 0 0
\(46\) 20498.2 1.42831
\(47\) 6698.84 0.442339 0.221169 0.975235i \(-0.429013\pi\)
0.221169 + 0.975235i \(0.429013\pi\)
\(48\) −6650.79 −0.416649
\(49\) 7345.75 0.437065
\(50\) 0 0
\(51\) −8083.06 −0.435161
\(52\) 8518.07 0.436851
\(53\) −22850.8 −1.11741 −0.558705 0.829367i \(-0.688702\pi\)
−0.558705 + 0.829367i \(0.688702\pi\)
\(54\) 3626.00 0.169217
\(55\) 0 0
\(56\) 30348.2 1.29319
\(57\) −21671.6 −0.883493
\(58\) 375.274 0.0146480
\(59\) −13021.3 −0.486995 −0.243498 0.969901i \(-0.578295\pi\)
−0.243498 + 0.969901i \(0.578295\pi\)
\(60\) 0 0
\(61\) 882.310 0.0303596 0.0151798 0.999885i \(-0.495168\pi\)
0.0151798 + 0.999885i \(0.495168\pi\)
\(62\) 17883.6 0.590849
\(63\) −12588.3 −0.399592
\(64\) 36446.3 1.11225
\(65\) 0 0
\(66\) 5416.62 0.153062
\(67\) 14191.7 0.386231 0.193115 0.981176i \(-0.438141\pi\)
0.193115 + 0.981176i \(0.438141\pi\)
\(68\) 6520.24 0.170998
\(69\) 37090.1 0.937854
\(70\) 0 0
\(71\) −78746.0 −1.85388 −0.926942 0.375205i \(-0.877572\pi\)
−0.926942 + 0.375205i \(0.877572\pi\)
\(72\) −15817.4 −0.359587
\(73\) −34227.3 −0.751736 −0.375868 0.926673i \(-0.622655\pi\)
−0.375868 + 0.926673i \(0.622655\pi\)
\(74\) 79896.8 1.69610
\(75\) 0 0
\(76\) 17481.5 0.347172
\(77\) −18804.8 −0.361445
\(78\) −52523.5 −0.977502
\(79\) −48694.6 −0.877835 −0.438917 0.898527i \(-0.644638\pi\)
−0.438917 + 0.898527i \(0.644638\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 74921.3 1.23047
\(83\) 95905.3 1.52809 0.764043 0.645166i \(-0.223212\pi\)
0.764043 + 0.645166i \(0.223212\pi\)
\(84\) 10154.5 0.157021
\(85\) 0 0
\(86\) 64003.6 0.933165
\(87\) 679.031 0.00961815
\(88\) −23628.5 −0.325258
\(89\) 1342.55 0.0179662 0.00898308 0.999960i \(-0.497141\pi\)
0.00898308 + 0.999960i \(0.497141\pi\)
\(90\) 0 0
\(91\) 182345. 2.30829
\(92\) −29918.9 −0.368533
\(93\) 32359.2 0.387963
\(94\) 33319.6 0.388938
\(95\) 0 0
\(96\) 23159.0 0.256473
\(97\) −80938.9 −0.873430 −0.436715 0.899600i \(-0.643858\pi\)
−0.436715 + 0.899600i \(0.643858\pi\)
\(98\) 36537.3 0.384301
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 19225.2 0.187529 0.0937644 0.995594i \(-0.470110\pi\)
0.0937644 + 0.995594i \(0.470110\pi\)
\(102\) −40204.7 −0.382627
\(103\) 126651. 1.17630 0.588148 0.808753i \(-0.299857\pi\)
0.588148 + 0.808753i \(0.299857\pi\)
\(104\) 229119. 2.07720
\(105\) 0 0
\(106\) −113659. −0.982513
\(107\) −77773.3 −0.656706 −0.328353 0.944555i \(-0.606494\pi\)
−0.328353 + 0.944555i \(0.606494\pi\)
\(108\) −5292.47 −0.0436615
\(109\) 68872.6 0.555240 0.277620 0.960691i \(-0.410454\pi\)
0.277620 + 0.960691i \(0.410454\pi\)
\(110\) 0 0
\(111\) 144568. 1.11369
\(112\) 114846. 0.865106
\(113\) 43794.9 0.322647 0.161323 0.986902i \(-0.448424\pi\)
0.161323 + 0.986902i \(0.448424\pi\)
\(114\) −107793. −0.776835
\(115\) 0 0
\(116\) −547.745 −0.00377949
\(117\) −95037.7 −0.641846
\(118\) −64767.3 −0.428204
\(119\) 139578. 0.903543
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 4388.56 0.0266945
\(123\) 135565. 0.807949
\(124\) −26102.7 −0.152452
\(125\) 0 0
\(126\) −62613.6 −0.351352
\(127\) 209089. 1.15033 0.575165 0.818037i \(-0.304938\pi\)
0.575165 + 0.818037i \(0.304938\pi\)
\(128\) 98938.8 0.533755
\(129\) 115810. 0.612734
\(130\) 0 0
\(131\) −85677.3 −0.436202 −0.218101 0.975926i \(-0.569986\pi\)
−0.218101 + 0.975926i \(0.569986\pi\)
\(132\) −7906.03 −0.0394933
\(133\) 374224. 1.83443
\(134\) 70588.6 0.339604
\(135\) 0 0
\(136\) 175381. 0.813085
\(137\) −115510. −0.525795 −0.262898 0.964824i \(-0.584678\pi\)
−0.262898 + 0.964824i \(0.584678\pi\)
\(138\) 184484. 0.824633
\(139\) −120621. −0.529523 −0.264761 0.964314i \(-0.585293\pi\)
−0.264761 + 0.964314i \(0.585293\pi\)
\(140\) 0 0
\(141\) 60289.5 0.255384
\(142\) −391678. −1.63008
\(143\) −141970. −0.580572
\(144\) −59857.1 −0.240552
\(145\) 0 0
\(146\) −170245. −0.660984
\(147\) 66111.7 0.252339
\(148\) −116616. −0.437628
\(149\) 49190.8 0.181517 0.0907587 0.995873i \(-0.471071\pi\)
0.0907587 + 0.995873i \(0.471071\pi\)
\(150\) 0 0
\(151\) −490808. −1.75174 −0.875870 0.482547i \(-0.839712\pi\)
−0.875870 + 0.482547i \(0.839712\pi\)
\(152\) 470216. 1.65078
\(153\) −72747.5 −0.251240
\(154\) −93534.0 −0.317810
\(155\) 0 0
\(156\) 76662.7 0.252216
\(157\) 273703. 0.886199 0.443099 0.896473i \(-0.353879\pi\)
0.443099 + 0.896473i \(0.353879\pi\)
\(158\) −242204. −0.771860
\(159\) −205657. −0.645137
\(160\) 0 0
\(161\) −640470. −1.94730
\(162\) 32634.0 0.0976974
\(163\) −64881.0 −0.191271 −0.0956354 0.995416i \(-0.530488\pi\)
−0.0956354 + 0.995416i \(0.530488\pi\)
\(164\) −109354. −0.317487
\(165\) 0 0
\(166\) 477028. 1.34361
\(167\) −379513. −1.05302 −0.526508 0.850170i \(-0.676499\pi\)
−0.526508 + 0.850170i \(0.676499\pi\)
\(168\) 273134. 0.746625
\(169\) 1.00535e6 2.70770
\(170\) 0 0
\(171\) −195044. −0.510085
\(172\) −93418.8 −0.240776
\(173\) 509354. 1.29391 0.646955 0.762528i \(-0.276042\pi\)
0.646955 + 0.762528i \(0.276042\pi\)
\(174\) 3377.46 0.00845702
\(175\) 0 0
\(176\) −89416.2 −0.217588
\(177\) −117192. −0.281167
\(178\) 6677.76 0.0157972
\(179\) 706267. 1.64754 0.823771 0.566923i \(-0.191866\pi\)
0.823771 + 0.566923i \(0.191866\pi\)
\(180\) 0 0
\(181\) 250646. 0.568674 0.284337 0.958724i \(-0.408227\pi\)
0.284337 + 0.958724i \(0.408227\pi\)
\(182\) 906974. 2.02963
\(183\) 7940.79 0.0175281
\(184\) −804758. −1.75235
\(185\) 0 0
\(186\) 160953. 0.341127
\(187\) −108672. −0.227256
\(188\) −48632.9 −0.100354
\(189\) −113295. −0.230705
\(190\) 0 0
\(191\) −592850. −1.17588 −0.587938 0.808906i \(-0.700060\pi\)
−0.587938 + 0.808906i \(0.700060\pi\)
\(192\) 328017. 0.642160
\(193\) 965415. 1.86561 0.932804 0.360383i \(-0.117354\pi\)
0.932804 + 0.360383i \(0.117354\pi\)
\(194\) −402585. −0.767986
\(195\) 0 0
\(196\) −53329.4 −0.0991577
\(197\) 175247. 0.321725 0.160862 0.986977i \(-0.448573\pi\)
0.160862 + 0.986977i \(0.448573\pi\)
\(198\) 48749.6 0.0883707
\(199\) −48708.5 −0.0871911 −0.0435955 0.999049i \(-0.513881\pi\)
−0.0435955 + 0.999049i \(0.513881\pi\)
\(200\) 0 0
\(201\) 127725. 0.222990
\(202\) 95625.1 0.164890
\(203\) −11725.5 −0.0199706
\(204\) 58682.2 0.0987259
\(205\) 0 0
\(206\) 629956. 1.03429
\(207\) 333811. 0.541470
\(208\) 867045. 1.38958
\(209\) −291362. −0.461389
\(210\) 0 0
\(211\) 662206. 1.02397 0.511985 0.858995i \(-0.328910\pi\)
0.511985 + 0.858995i \(0.328910\pi\)
\(212\) 165895. 0.253509
\(213\) −708714. −1.07034
\(214\) −386840. −0.577427
\(215\) 0 0
\(216\) −142357. −0.207608
\(217\) −558777. −0.805544
\(218\) 342568. 0.488209
\(219\) −308046. −0.434015
\(220\) 0 0
\(221\) 1.05377e6 1.45132
\(222\) 719071. 0.979241
\(223\) 44174.7 0.0594855 0.0297428 0.999558i \(-0.490531\pi\)
0.0297428 + 0.999558i \(0.490531\pi\)
\(224\) −399908. −0.532525
\(225\) 0 0
\(226\) 217833. 0.283696
\(227\) −585184. −0.753750 −0.376875 0.926264i \(-0.623001\pi\)
−0.376875 + 0.926264i \(0.623001\pi\)
\(228\) 157333. 0.200440
\(229\) 914300. 1.15213 0.576063 0.817405i \(-0.304588\pi\)
0.576063 + 0.817405i \(0.304588\pi\)
\(230\) 0 0
\(231\) −169243. −0.208680
\(232\) −14733.2 −0.0179712
\(233\) −702641. −0.847898 −0.423949 0.905686i \(-0.639356\pi\)
−0.423949 + 0.905686i \(0.639356\pi\)
\(234\) −472712. −0.564361
\(235\) 0 0
\(236\) 94533.5 0.110486
\(237\) −438251. −0.506818
\(238\) 694252. 0.794465
\(239\) 572014. 0.647757 0.323879 0.946099i \(-0.395013\pi\)
0.323879 + 0.946099i \(0.395013\pi\)
\(240\) 0 0
\(241\) −485324. −0.538256 −0.269128 0.963104i \(-0.586736\pi\)
−0.269128 + 0.963104i \(0.586736\pi\)
\(242\) 72823.5 0.0799343
\(243\) 59049.0 0.0641500
\(244\) −6405.48 −0.00688775
\(245\) 0 0
\(246\) 674291. 0.710411
\(247\) 2.82526e6 2.94657
\(248\) −702110. −0.724897
\(249\) 863148. 0.882240
\(250\) 0 0
\(251\) 25755.4 0.0258039 0.0129019 0.999917i \(-0.495893\pi\)
0.0129019 + 0.999917i \(0.495893\pi\)
\(252\) 91390.1 0.0906562
\(253\) 498656. 0.489778
\(254\) 1.04000e6 1.01146
\(255\) 0 0
\(256\) −674166. −0.642935
\(257\) −243229. −0.229711 −0.114856 0.993382i \(-0.536641\pi\)
−0.114856 + 0.993382i \(0.536641\pi\)
\(258\) 576032. 0.538763
\(259\) −2.49639e6 −2.31240
\(260\) 0 0
\(261\) 6111.28 0.00555304
\(262\) −426154. −0.383542
\(263\) −46278.0 −0.0412558 −0.0206279 0.999787i \(-0.506567\pi\)
−0.0206279 + 0.999787i \(0.506567\pi\)
\(264\) −212656. −0.187788
\(265\) 0 0
\(266\) 1.86137e6 1.61298
\(267\) 12082.9 0.0103728
\(268\) −103030. −0.0876249
\(269\) 174634. 0.147146 0.0735729 0.997290i \(-0.476560\pi\)
0.0735729 + 0.997290i \(0.476560\pi\)
\(270\) 0 0
\(271\) −1.78096e6 −1.47309 −0.736546 0.676387i \(-0.763545\pi\)
−0.736546 + 0.676387i \(0.763545\pi\)
\(272\) 663688. 0.543929
\(273\) 1.64111e6 1.33269
\(274\) −574538. −0.462320
\(275\) 0 0
\(276\) −269270. −0.212773
\(277\) −545088. −0.426842 −0.213421 0.976960i \(-0.568461\pi\)
−0.213421 + 0.976960i \(0.568461\pi\)
\(278\) −599960. −0.465597
\(279\) 291233. 0.223991
\(280\) 0 0
\(281\) 85469.5 0.0645722 0.0322861 0.999479i \(-0.489721\pi\)
0.0322861 + 0.999479i \(0.489721\pi\)
\(282\) 299877. 0.224554
\(283\) 1.51069e6 1.12127 0.560634 0.828064i \(-0.310558\pi\)
0.560634 + 0.828064i \(0.310558\pi\)
\(284\) 571688. 0.420594
\(285\) 0 0
\(286\) −706150. −0.510484
\(287\) −2.34093e6 −1.67758
\(288\) 208431. 0.148075
\(289\) −613242. −0.431904
\(290\) 0 0
\(291\) −728450. −0.504275
\(292\) 248487. 0.170548
\(293\) 1.05179e6 0.715746 0.357873 0.933770i \(-0.383502\pi\)
0.357873 + 0.933770i \(0.383502\pi\)
\(294\) 328836. 0.221876
\(295\) 0 0
\(296\) −3.13674e6 −2.08089
\(297\) 88209.0 0.0580259
\(298\) 244672. 0.159604
\(299\) −4.83533e6 −3.12787
\(300\) 0 0
\(301\) −1.99980e6 −1.27224
\(302\) −2.44125e6 −1.54026
\(303\) 173027. 0.108270
\(304\) 1.77942e6 1.10432
\(305\) 0 0
\(306\) −361842. −0.220910
\(307\) 1.68373e6 1.01959 0.509796 0.860295i \(-0.329721\pi\)
0.509796 + 0.860295i \(0.329721\pi\)
\(308\) 136521. 0.0820016
\(309\) 1.13986e6 0.679135
\(310\) 0 0
\(311\) −2.72412e6 −1.59707 −0.798536 0.601947i \(-0.794392\pi\)
−0.798536 + 0.601947i \(0.794392\pi\)
\(312\) 2.06207e6 1.19927
\(313\) −1.69158e6 −0.975959 −0.487979 0.872855i \(-0.662266\pi\)
−0.487979 + 0.872855i \(0.662266\pi\)
\(314\) 1.36138e6 0.779214
\(315\) 0 0
\(316\) 353518. 0.199156
\(317\) 371370. 0.207567 0.103784 0.994600i \(-0.466905\pi\)
0.103784 + 0.994600i \(0.466905\pi\)
\(318\) −1.02293e6 −0.567254
\(319\) 9129.20 0.00502292
\(320\) 0 0
\(321\) −699960. −0.379149
\(322\) −3.18566e6 −1.71222
\(323\) 2.16262e6 1.15339
\(324\) −47632.2 −0.0252080
\(325\) 0 0
\(326\) −322714. −0.168180
\(327\) 619854. 0.320568
\(328\) −2.94140e6 −1.50963
\(329\) −1.04108e6 −0.530265
\(330\) 0 0
\(331\) 1.54525e6 0.775229 0.387614 0.921822i \(-0.373299\pi\)
0.387614 + 0.921822i \(0.373299\pi\)
\(332\) −696263. −0.346680
\(333\) 1.30111e6 0.642989
\(334\) −1.88767e6 −0.925893
\(335\) 0 0
\(336\) 1.03361e6 0.499469
\(337\) 1.56563e6 0.750956 0.375478 0.926831i \(-0.377479\pi\)
0.375478 + 0.926831i \(0.377479\pi\)
\(338\) 5.00055e6 2.38082
\(339\) 394154. 0.186280
\(340\) 0 0
\(341\) 435052. 0.202607
\(342\) −970138. −0.448506
\(343\) 1.47039e6 0.674833
\(344\) −2.51277e6 −1.14487
\(345\) 0 0
\(346\) 2.53350e6 1.13771
\(347\) −2.21769e6 −0.988727 −0.494364 0.869255i \(-0.664599\pi\)
−0.494364 + 0.869255i \(0.664599\pi\)
\(348\) −4929.70 −0.00218209
\(349\) −3.10101e6 −1.36283 −0.681413 0.731899i \(-0.738634\pi\)
−0.681413 + 0.731899i \(0.738634\pi\)
\(350\) 0 0
\(351\) −855339. −0.370570
\(352\) 311359. 0.133938
\(353\) 2.34837e6 1.00307 0.501534 0.865138i \(-0.332769\pi\)
0.501534 + 0.865138i \(0.332769\pi\)
\(354\) −582905. −0.247224
\(355\) 0 0
\(356\) −9746.77 −0.00407602
\(357\) 1.25620e6 0.521661
\(358\) 3.51293e6 1.44865
\(359\) 1.76347e6 0.722158 0.361079 0.932535i \(-0.382408\pi\)
0.361079 + 0.932535i \(0.382408\pi\)
\(360\) 0 0
\(361\) 3.32213e6 1.34168
\(362\) 1.24670e6 0.500022
\(363\) 131769. 0.0524864
\(364\) −1.32381e6 −0.523686
\(365\) 0 0
\(366\) 39497.0 0.0154121
\(367\) 2.99827e6 1.16200 0.580999 0.813904i \(-0.302662\pi\)
0.580999 + 0.813904i \(0.302662\pi\)
\(368\) −3.04541e6 −1.17227
\(369\) 1.22008e6 0.466470
\(370\) 0 0
\(371\) 3.55128e6 1.33952
\(372\) −234925. −0.0880179
\(373\) 2.01705e6 0.750663 0.375331 0.926891i \(-0.377529\pi\)
0.375331 + 0.926891i \(0.377529\pi\)
\(374\) −540529. −0.199821
\(375\) 0 0
\(376\) −1.30813e6 −0.477177
\(377\) −88523.4 −0.0320778
\(378\) −563523. −0.202853
\(379\) 3.78909e6 1.35499 0.677497 0.735526i \(-0.263065\pi\)
0.677497 + 0.735526i \(0.263065\pi\)
\(380\) 0 0
\(381\) 1.88180e6 0.664144
\(382\) −2.94880e6 −1.03392
\(383\) 2.72827e6 0.950366 0.475183 0.879887i \(-0.342382\pi\)
0.475183 + 0.879887i \(0.342382\pi\)
\(384\) 890449. 0.308163
\(385\) 0 0
\(386\) 4.80192e6 1.64039
\(387\) 1.04229e6 0.353762
\(388\) 587608. 0.198157
\(389\) 432266. 0.144836 0.0724181 0.997374i \(-0.476928\pi\)
0.0724181 + 0.997374i \(0.476928\pi\)
\(390\) 0 0
\(391\) −3.70125e6 −1.22435
\(392\) −1.43445e6 −0.471488
\(393\) −771095. −0.251841
\(394\) 871667. 0.282885
\(395\) 0 0
\(396\) −71154.3 −0.0228015
\(397\) 3.90462e6 1.24338 0.621688 0.783265i \(-0.286447\pi\)
0.621688 + 0.783265i \(0.286447\pi\)
\(398\) −242273. −0.0766651
\(399\) 3.36801e6 1.05911
\(400\) 0 0
\(401\) 4.60863e6 1.43124 0.715618 0.698492i \(-0.246145\pi\)
0.715618 + 0.698492i \(0.246145\pi\)
\(402\) 635297. 0.196070
\(403\) −4.21858e6 −1.29391
\(404\) −139573. −0.0425450
\(405\) 0 0
\(406\) −58321.9 −0.0175597
\(407\) 1.94363e6 0.581605
\(408\) 1.57843e6 0.469435
\(409\) −3.86604e6 −1.14277 −0.571384 0.820683i \(-0.693593\pi\)
−0.571384 + 0.820683i \(0.693593\pi\)
\(410\) 0 0
\(411\) −1.03959e6 −0.303568
\(412\) −919476. −0.266868
\(413\) 2.02366e6 0.583799
\(414\) 1.66036e6 0.476102
\(415\) 0 0
\(416\) −3.01917e6 −0.855370
\(417\) −1.08559e6 −0.305720
\(418\) −1.44922e6 −0.405689
\(419\) 3.60488e6 1.00313 0.501563 0.865121i \(-0.332759\pi\)
0.501563 + 0.865121i \(0.332759\pi\)
\(420\) 0 0
\(421\) 2.69664e6 0.741510 0.370755 0.928731i \(-0.379099\pi\)
0.370755 + 0.928731i \(0.379099\pi\)
\(422\) 3.29377e6 0.900353
\(423\) 542606. 0.147446
\(424\) 4.46223e6 1.20542
\(425\) 0 0
\(426\) −3.52510e6 −0.941126
\(427\) −137121. −0.0363944
\(428\) 564626. 0.148988
\(429\) −1.27773e6 −0.335193
\(430\) 0 0
\(431\) −337978. −0.0876385 −0.0438193 0.999039i \(-0.513953\pi\)
−0.0438193 + 0.999039i \(0.513953\pi\)
\(432\) −538714. −0.138883
\(433\) −5.43565e6 −1.39326 −0.696629 0.717431i \(-0.745318\pi\)
−0.696629 + 0.717431i \(0.745318\pi\)
\(434\) −2.77932e6 −0.708296
\(435\) 0 0
\(436\) −500008. −0.125968
\(437\) −9.92346e6 −2.48576
\(438\) −1.53220e6 −0.381620
\(439\) −170878. −0.0423180 −0.0211590 0.999776i \(-0.506736\pi\)
−0.0211590 + 0.999776i \(0.506736\pi\)
\(440\) 0 0
\(441\) 595006. 0.145688
\(442\) 5.24137e6 1.27611
\(443\) −6.94673e6 −1.68179 −0.840893 0.541201i \(-0.817970\pi\)
−0.840893 + 0.541201i \(0.817970\pi\)
\(444\) −1.04955e6 −0.252665
\(445\) 0 0
\(446\) 219722. 0.0523043
\(447\) 442717. 0.104799
\(448\) −5.66418e6 −1.33334
\(449\) −1.91825e6 −0.449045 −0.224522 0.974469i \(-0.572082\pi\)
−0.224522 + 0.974469i \(0.572082\pi\)
\(450\) 0 0
\(451\) 1.82259e6 0.421938
\(452\) −317946. −0.0731995
\(453\) −4.41728e6 −1.01137
\(454\) −2.91067e6 −0.662755
\(455\) 0 0
\(456\) 4.23195e6 0.953077
\(457\) 1.14248e6 0.255892 0.127946 0.991781i \(-0.459162\pi\)
0.127946 + 0.991781i \(0.459162\pi\)
\(458\) 4.54768e6 1.01304
\(459\) −654728. −0.145054
\(460\) 0 0
\(461\) 3.63295e6 0.796172 0.398086 0.917348i \(-0.369675\pi\)
0.398086 + 0.917348i \(0.369675\pi\)
\(462\) −841806. −0.183488
\(463\) 6.29651e6 1.36505 0.682523 0.730864i \(-0.260883\pi\)
0.682523 + 0.730864i \(0.260883\pi\)
\(464\) −55754.3 −0.0120222
\(465\) 0 0
\(466\) −3.49490e6 −0.745537
\(467\) −46216.3 −0.00980624 −0.00490312 0.999988i \(-0.501561\pi\)
−0.00490312 + 0.999988i \(0.501561\pi\)
\(468\) 689964. 0.145617
\(469\) −2.20555e6 −0.463004
\(470\) 0 0
\(471\) 2.46333e6 0.511647
\(472\) 2.54276e6 0.525351
\(473\) 1.55700e6 0.319990
\(474\) −2.17983e6 −0.445633
\(475\) 0 0
\(476\) −1.01332e6 −0.204989
\(477\) −1.85092e6 −0.372470
\(478\) 2.84517e6 0.569558
\(479\) 1.87454e6 0.373299 0.186649 0.982427i \(-0.440237\pi\)
0.186649 + 0.982427i \(0.440237\pi\)
\(480\) 0 0
\(481\) −1.88469e7 −3.71430
\(482\) −2.41397e6 −0.473276
\(483\) −5.76423e6 −1.12428
\(484\) −106292. −0.0206247
\(485\) 0 0
\(486\) 293706. 0.0564056
\(487\) 2.93849e6 0.561439 0.280719 0.959790i \(-0.409427\pi\)
0.280719 + 0.959790i \(0.409427\pi\)
\(488\) −172294. −0.0327508
\(489\) −583929. −0.110430
\(490\) 0 0
\(491\) 7.56455e6 1.41605 0.708026 0.706186i \(-0.249586\pi\)
0.708026 + 0.706186i \(0.249586\pi\)
\(492\) −984187. −0.183301
\(493\) −67761.1 −0.0125563
\(494\) 1.40527e7 2.59085
\(495\) 0 0
\(496\) −2.65697e6 −0.484933
\(497\) 1.22380e7 2.22239
\(498\) 4.29325e6 0.775734
\(499\) −3.96894e6 −0.713548 −0.356774 0.934191i \(-0.616123\pi\)
−0.356774 + 0.934191i \(0.616123\pi\)
\(500\) 0 0
\(501\) −3.41561e6 −0.607959
\(502\) 128106. 0.0226887
\(503\) −7.67415e6 −1.35242 −0.676208 0.736711i \(-0.736378\pi\)
−0.676208 + 0.736711i \(0.736378\pi\)
\(504\) 2.45821e6 0.431064
\(505\) 0 0
\(506\) 2.48028e6 0.430651
\(507\) 9.04815e6 1.56329
\(508\) −1.51797e6 −0.260978
\(509\) −2.34465e6 −0.401129 −0.200564 0.979681i \(-0.564278\pi\)
−0.200564 + 0.979681i \(0.564278\pi\)
\(510\) 0 0
\(511\) 5.31932e6 0.901164
\(512\) −6.51930e6 −1.09907
\(513\) −1.75540e6 −0.294498
\(514\) −1.20981e6 −0.201980
\(515\) 0 0
\(516\) −840769. −0.139012
\(517\) 810559. 0.133370
\(518\) −1.24169e7 −2.03324
\(519\) 4.58418e6 0.747040
\(520\) 0 0
\(521\) −4.60694e6 −0.743563 −0.371782 0.928320i \(-0.621253\pi\)
−0.371782 + 0.928320i \(0.621253\pi\)
\(522\) 30397.2 0.00488266
\(523\) 3.79276e6 0.606319 0.303159 0.952940i \(-0.401959\pi\)
0.303159 + 0.952940i \(0.401959\pi\)
\(524\) 622008. 0.0989619
\(525\) 0 0
\(526\) −230184. −0.0362753
\(527\) −3.22915e6 −0.506480
\(528\) −804746. −0.125624
\(529\) 1.05473e7 1.63871
\(530\) 0 0
\(531\) −1.05473e6 −0.162332
\(532\) −2.71683e6 −0.416181
\(533\) −1.76732e7 −2.69462
\(534\) 60099.9 0.00912053
\(535\) 0 0
\(536\) −2.77130e6 −0.416650
\(537\) 6.35640e6 0.951208
\(538\) 868619. 0.129382
\(539\) 888835. 0.131780
\(540\) 0 0
\(541\) −8.95976e6 −1.31614 −0.658072 0.752955i \(-0.728628\pi\)
−0.658072 + 0.752955i \(0.728628\pi\)
\(542\) −8.85837e6 −1.29526
\(543\) 2.25581e6 0.328324
\(544\) −2.31105e6 −0.334821
\(545\) 0 0
\(546\) 8.16276e6 1.17181
\(547\) −2.79065e6 −0.398783 −0.199391 0.979920i \(-0.563896\pi\)
−0.199391 + 0.979920i \(0.563896\pi\)
\(548\) 838588. 0.119288
\(549\) 71467.1 0.0101199
\(550\) 0 0
\(551\) −181675. −0.0254927
\(552\) −7.24282e6 −1.01172
\(553\) 7.56770e6 1.05233
\(554\) −2.71124e6 −0.375313
\(555\) 0 0
\(556\) 875694. 0.120134
\(557\) −1.22903e7 −1.67851 −0.839256 0.543736i \(-0.817009\pi\)
−0.839256 + 0.543736i \(0.817009\pi\)
\(558\) 1.44858e6 0.196950
\(559\) −1.50978e7 −2.04355
\(560\) 0 0
\(561\) −978050. −0.131206
\(562\) 425120. 0.0567768
\(563\) −1.13900e7 −1.51444 −0.757220 0.653160i \(-0.773443\pi\)
−0.757220 + 0.653160i \(0.773443\pi\)
\(564\) −437696. −0.0579395
\(565\) 0 0
\(566\) 7.51408e6 0.985905
\(567\) −1.01966e6 −0.133197
\(568\) 1.53772e7 1.99990
\(569\) 4.71367e6 0.610350 0.305175 0.952296i \(-0.401285\pi\)
0.305175 + 0.952296i \(0.401285\pi\)
\(570\) 0 0
\(571\) 7.49421e6 0.961913 0.480957 0.876744i \(-0.340289\pi\)
0.480957 + 0.876744i \(0.340289\pi\)
\(572\) 1.03069e6 0.131715
\(573\) −5.33565e6 −0.678893
\(574\) −1.16436e7 −1.47506
\(575\) 0 0
\(576\) 2.95215e6 0.370751
\(577\) 5.77654e6 0.722318 0.361159 0.932504i \(-0.382381\pi\)
0.361159 + 0.932504i \(0.382381\pi\)
\(578\) −3.05023e6 −0.379763
\(579\) 8.68873e6 1.07711
\(580\) 0 0
\(581\) −1.49048e7 −1.83183
\(582\) −3.62327e6 −0.443397
\(583\) −2.76495e6 −0.336912
\(584\) 6.68379e6 0.810943
\(585\) 0 0
\(586\) 5.23153e6 0.629339
\(587\) 1.19273e6 0.142872 0.0714358 0.997445i \(-0.477242\pi\)
0.0714358 + 0.997445i \(0.477242\pi\)
\(588\) −479965. −0.0572487
\(589\) −8.65771e6 −1.02829
\(590\) 0 0
\(591\) 1.57722e6 0.185748
\(592\) −1.18702e7 −1.39205
\(593\) 1.14801e7 1.34064 0.670318 0.742074i \(-0.266158\pi\)
0.670318 + 0.742074i \(0.266158\pi\)
\(594\) 438746. 0.0510208
\(595\) 0 0
\(596\) −357120. −0.0411812
\(597\) −438376. −0.0503398
\(598\) −2.40506e7 −2.75026
\(599\) 1.07989e7 1.22974 0.614869 0.788630i \(-0.289209\pi\)
0.614869 + 0.788630i \(0.289209\pi\)
\(600\) 0 0
\(601\) 6.43742e6 0.726986 0.363493 0.931597i \(-0.381584\pi\)
0.363493 + 0.931597i \(0.381584\pi\)
\(602\) −9.94690e6 −1.11866
\(603\) 1.14953e6 0.128744
\(604\) 3.56322e6 0.397421
\(605\) 0 0
\(606\) 860626. 0.0951991
\(607\) −2.78009e6 −0.306258 −0.153129 0.988206i \(-0.548935\pi\)
−0.153129 + 0.988206i \(0.548935\pi\)
\(608\) −6.19619e6 −0.679776
\(609\) −105529. −0.0115300
\(610\) 0 0
\(611\) −7.85978e6 −0.851740
\(612\) 528140. 0.0569994
\(613\) 1.25907e7 1.35331 0.676657 0.736299i \(-0.263428\pi\)
0.676657 + 0.736299i \(0.263428\pi\)
\(614\) 8.37477e6 0.896504
\(615\) 0 0
\(616\) 3.67213e6 0.389912
\(617\) −1.20724e6 −0.127668 −0.0638339 0.997961i \(-0.520333\pi\)
−0.0638339 + 0.997961i \(0.520333\pi\)
\(618\) 5.66960e6 0.597147
\(619\) 1.03935e7 1.09028 0.545138 0.838346i \(-0.316477\pi\)
0.545138 + 0.838346i \(0.316477\pi\)
\(620\) 0 0
\(621\) 3.00430e6 0.312618
\(622\) −1.35496e7 −1.40427
\(623\) −208648. −0.0215374
\(624\) 7.80341e6 0.802274
\(625\) 0 0
\(626\) −8.41381e6 −0.858138
\(627\) −2.62226e6 −0.266383
\(628\) −1.98706e6 −0.201054
\(629\) −1.44265e7 −1.45390
\(630\) 0 0
\(631\) 389320. 0.0389254 0.0194627 0.999811i \(-0.493804\pi\)
0.0194627 + 0.999811i \(0.493804\pi\)
\(632\) 9.50890e6 0.946973
\(633\) 5.95985e6 0.591189
\(634\) 1.84717e6 0.182509
\(635\) 0 0
\(636\) 1.49305e6 0.146363
\(637\) −8.61880e6 −0.841585
\(638\) 45408.1 0.00441654
\(639\) −6.37843e6 −0.617961
\(640\) 0 0
\(641\) 9.85176e6 0.947041 0.473520 0.880783i \(-0.342983\pi\)
0.473520 + 0.880783i \(0.342983\pi\)
\(642\) −3.48156e6 −0.333377
\(643\) −9.41361e6 −0.897901 −0.448951 0.893557i \(-0.648202\pi\)
−0.448951 + 0.893557i \(0.648202\pi\)
\(644\) 4.64975e6 0.441789
\(645\) 0 0
\(646\) 1.07568e7 1.01415
\(647\) 8.37846e6 0.786871 0.393435 0.919352i \(-0.371287\pi\)
0.393435 + 0.919352i \(0.371287\pi\)
\(648\) −1.28121e6 −0.119862
\(649\) −1.57558e6 −0.146835
\(650\) 0 0
\(651\) −5.02899e6 −0.465081
\(652\) 471030. 0.0433940
\(653\) 1.87196e7 1.71796 0.858979 0.512011i \(-0.171099\pi\)
0.858979 + 0.512011i \(0.171099\pi\)
\(654\) 3.08312e6 0.281868
\(655\) 0 0
\(656\) −1.11310e7 −1.00989
\(657\) −2.77241e6 −0.250579
\(658\) −5.17826e6 −0.466250
\(659\) −7.53086e6 −0.675509 −0.337755 0.941234i \(-0.609667\pi\)
−0.337755 + 0.941234i \(0.609667\pi\)
\(660\) 0 0
\(661\) −1.12673e7 −1.00304 −0.501518 0.865147i \(-0.667225\pi\)
−0.501518 + 0.865147i \(0.667225\pi\)
\(662\) 7.68600e6 0.681641
\(663\) 9.48389e6 0.837920
\(664\) −1.87281e7 −1.64844
\(665\) 0 0
\(666\) 6.47164e6 0.565365
\(667\) 310930. 0.0270613
\(668\) 2.75522e6 0.238900
\(669\) 397572. 0.0343440
\(670\) 0 0
\(671\) 106759. 0.00915377
\(672\) −3.59917e6 −0.307453
\(673\) −747965. −0.0636566 −0.0318283 0.999493i \(-0.510133\pi\)
−0.0318283 + 0.999493i \(0.510133\pi\)
\(674\) 7.78735e6 0.660298
\(675\) 0 0
\(676\) −7.29874e6 −0.614301
\(677\) 1.52419e7 1.27811 0.639056 0.769160i \(-0.279325\pi\)
0.639056 + 0.769160i \(0.279325\pi\)
\(678\) 1.96050e6 0.163792
\(679\) 1.25788e7 1.04705
\(680\) 0 0
\(681\) −5.26665e6 −0.435178
\(682\) 2.16392e6 0.178148
\(683\) −891262. −0.0731061 −0.0365531 0.999332i \(-0.511638\pi\)
−0.0365531 + 0.999332i \(0.511638\pi\)
\(684\) 1.41600e6 0.115724
\(685\) 0 0
\(686\) 7.31362e6 0.593366
\(687\) 8.22870e6 0.665180
\(688\) −9.50899e6 −0.765885
\(689\) 2.68110e7 2.15162
\(690\) 0 0
\(691\) −2.23692e6 −0.178220 −0.0891100 0.996022i \(-0.528402\pi\)
−0.0891100 + 0.996022i \(0.528402\pi\)
\(692\) −3.69786e6 −0.293552
\(693\) −1.52319e6 −0.120482
\(694\) −1.10306e7 −0.869365
\(695\) 0 0
\(696\) −132599. −0.0103757
\(697\) −1.35281e7 −1.05476
\(698\) −1.54243e7 −1.19830
\(699\) −6.32377e6 −0.489534
\(700\) 0 0
\(701\) −9.23251e6 −0.709618 −0.354809 0.934939i \(-0.615454\pi\)
−0.354809 + 0.934939i \(0.615454\pi\)
\(702\) −4.25441e6 −0.325834
\(703\) −3.86791e7 −2.95181
\(704\) 4.41000e6 0.335357
\(705\) 0 0
\(706\) 1.16807e7 0.881975
\(707\) −2.98782e6 −0.224805
\(708\) 850801. 0.0637889
\(709\) −1.91122e7 −1.42789 −0.713945 0.700201i \(-0.753094\pi\)
−0.713945 + 0.700201i \(0.753094\pi\)
\(710\) 0 0
\(711\) −3.94426e6 −0.292612
\(712\) −262168. −0.0193812
\(713\) 1.48174e7 1.09156
\(714\) 6.24827e6 0.458685
\(715\) 0 0
\(716\) −5.12743e6 −0.373781
\(717\) 5.14813e6 0.373983
\(718\) 8.77140e6 0.634977
\(719\) 3.98132e6 0.287213 0.143607 0.989635i \(-0.454130\pi\)
0.143607 + 0.989635i \(0.454130\pi\)
\(720\) 0 0
\(721\) −1.96831e7 −1.41012
\(722\) 1.65241e7 1.17971
\(723\) −4.36792e6 −0.310762
\(724\) −1.81966e6 −0.129016
\(725\) 0 0
\(726\) 655411. 0.0461501
\(727\) 2.31098e6 0.162166 0.0810832 0.996707i \(-0.474162\pi\)
0.0810832 + 0.996707i \(0.474162\pi\)
\(728\) −3.56077e7 −2.49009
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.15568e7 −0.799914
\(732\) −57649.3 −0.00397664
\(733\) −1.76034e7 −1.21015 −0.605073 0.796170i \(-0.706856\pi\)
−0.605073 + 0.796170i \(0.706856\pi\)
\(734\) 1.49132e7 1.02172
\(735\) 0 0
\(736\) 1.06045e7 0.721601
\(737\) 1.71719e6 0.116453
\(738\) 6.06862e6 0.410156
\(739\) −8.91956e6 −0.600803 −0.300402 0.953813i \(-0.597121\pi\)
−0.300402 + 0.953813i \(0.597121\pi\)
\(740\) 0 0
\(741\) 2.54273e7 1.70120
\(742\) 1.76639e7 1.17781
\(743\) −3.22802e6 −0.214519 −0.107259 0.994231i \(-0.534207\pi\)
−0.107259 + 0.994231i \(0.534207\pi\)
\(744\) −6.31899e6 −0.418519
\(745\) 0 0
\(746\) 1.00327e7 0.660040
\(747\) 7.76833e6 0.509362
\(748\) 788950. 0.0515579
\(749\) 1.20869e7 0.787244
\(750\) 0 0
\(751\) −2.04930e6 −0.132588 −0.0662942 0.997800i \(-0.521118\pi\)
−0.0662942 + 0.997800i \(0.521118\pi\)
\(752\) −4.95029e6 −0.319217
\(753\) 231799. 0.0148979
\(754\) −440310. −0.0282053
\(755\) 0 0
\(756\) 822511. 0.0523404
\(757\) −2.62464e6 −0.166468 −0.0832340 0.996530i \(-0.526525\pi\)
−0.0832340 + 0.996530i \(0.526525\pi\)
\(758\) 1.88467e7 1.19141
\(759\) 4.48790e6 0.282773
\(760\) 0 0
\(761\) 1.90240e7 1.19080 0.595401 0.803428i \(-0.296993\pi\)
0.595401 + 0.803428i \(0.296993\pi\)
\(762\) 9.35999e6 0.583966
\(763\) −1.07036e7 −0.665608
\(764\) 4.30404e6 0.266773
\(765\) 0 0
\(766\) 1.35703e7 0.835635
\(767\) 1.52780e7 0.937729
\(768\) −6.06749e6 −0.371198
\(769\) −1.84565e7 −1.12547 −0.562735 0.826637i \(-0.690251\pi\)
−0.562735 + 0.826637i \(0.690251\pi\)
\(770\) 0 0
\(771\) −2.18906e6 −0.132624
\(772\) −7.00881e6 −0.423254
\(773\) 6.98835e6 0.420655 0.210327 0.977631i \(-0.432547\pi\)
0.210327 + 0.977631i \(0.432547\pi\)
\(774\) 5.18429e6 0.311055
\(775\) 0 0
\(776\) 1.58055e7 0.942221
\(777\) −2.24675e7 −1.33506
\(778\) 2.15007e6 0.127351
\(779\) −3.62704e7 −2.14145
\(780\) 0 0
\(781\) −9.52826e6 −0.558967
\(782\) −1.84098e7 −1.07655
\(783\) 55001.5 0.00320605
\(784\) −5.42834e6 −0.315411
\(785\) 0 0
\(786\) −3.83538e6 −0.221438
\(787\) 1.02434e7 0.589533 0.294766 0.955569i \(-0.404758\pi\)
0.294766 + 0.955569i \(0.404758\pi\)
\(788\) −1.27227e6 −0.0729903
\(789\) −416502. −0.0238190
\(790\) 0 0
\(791\) −6.80623e6 −0.386781
\(792\) −1.91390e6 −0.108419
\(793\) −1.03522e6 −0.0584587
\(794\) 1.94213e7 1.09327
\(795\) 0 0
\(796\) 353619. 0.0197812
\(797\) −1.36372e7 −0.760466 −0.380233 0.924891i \(-0.624156\pi\)
−0.380233 + 0.924891i \(0.624156\pi\)
\(798\) 1.67523e7 0.931252
\(799\) −6.01634e6 −0.333400
\(800\) 0 0
\(801\) 108746. 0.00598872
\(802\) 2.29231e7 1.25845
\(803\) −4.14150e6 −0.226657
\(804\) −927271. −0.0505902
\(805\) 0 0
\(806\) −2.09830e7 −1.13770
\(807\) 1.57171e6 0.0849547
\(808\) −3.75423e6 −0.202299
\(809\) −2.94019e7 −1.57944 −0.789722 0.613465i \(-0.789775\pi\)
−0.789722 + 0.613465i \(0.789775\pi\)
\(810\) 0 0
\(811\) −1.42304e7 −0.759737 −0.379869 0.925040i \(-0.624031\pi\)
−0.379869 + 0.925040i \(0.624031\pi\)
\(812\) 85125.8 0.00453076
\(813\) −1.60286e7 −0.850490
\(814\) 9.66752e6 0.511392
\(815\) 0 0
\(816\) 5.97319e6 0.314037
\(817\) −3.09850e7 −1.62404
\(818\) −1.92294e7 −1.00481
\(819\) 1.47699e7 0.769430
\(820\) 0 0
\(821\) 2.28257e7 1.18186 0.590931 0.806722i \(-0.298761\pi\)
0.590931 + 0.806722i \(0.298761\pi\)
\(822\) −5.17084e6 −0.266920
\(823\) 1.12391e6 0.0578405 0.0289202 0.999582i \(-0.490793\pi\)
0.0289202 + 0.999582i \(0.490793\pi\)
\(824\) −2.47320e7 −1.26894
\(825\) 0 0
\(826\) 1.00656e7 0.513321
\(827\) 2.69107e7 1.36824 0.684119 0.729370i \(-0.260187\pi\)
0.684119 + 0.729370i \(0.260187\pi\)
\(828\) −2.42343e6 −0.122844
\(829\) −6.83042e6 −0.345192 −0.172596 0.984993i \(-0.555216\pi\)
−0.172596 + 0.984993i \(0.555216\pi\)
\(830\) 0 0
\(831\) −4.90580e6 −0.246438
\(832\) −4.27626e7 −2.14169
\(833\) −6.59734e6 −0.329425
\(834\) −5.39964e6 −0.268813
\(835\) 0 0
\(836\) 2.11526e6 0.104676
\(837\) 2.62110e6 0.129321
\(838\) 1.79305e7 0.882026
\(839\) −1.37641e7 −0.675061 −0.337531 0.941315i \(-0.609592\pi\)
−0.337531 + 0.941315i \(0.609592\pi\)
\(840\) 0 0
\(841\) −2.05055e7 −0.999722
\(842\) 1.34129e7 0.651993
\(843\) 769225. 0.0372808
\(844\) −4.80755e6 −0.232310
\(845\) 0 0
\(846\) 2.69889e6 0.129646
\(847\) −2.27538e6 −0.108980
\(848\) 1.68862e7 0.806387
\(849\) 1.35962e7 0.647364
\(850\) 0 0
\(851\) 6.61979e7 3.13343
\(852\) 5.14519e6 0.242830
\(853\) 2.38141e7 1.12063 0.560313 0.828281i \(-0.310681\pi\)
0.560313 + 0.828281i \(0.310681\pi\)
\(854\) −682032. −0.0320008
\(855\) 0 0
\(856\) 1.51873e7 0.708428
\(857\) 1.96463e7 0.913751 0.456875 0.889531i \(-0.348969\pi\)
0.456875 + 0.889531i \(0.348969\pi\)
\(858\) −6.35535e6 −0.294728
\(859\) −4.10315e7 −1.89729 −0.948646 0.316339i \(-0.897546\pi\)
−0.948646 + 0.316339i \(0.897546\pi\)
\(860\) 0 0
\(861\) −2.10683e7 −0.968550
\(862\) −1.68108e6 −0.0770585
\(863\) 1.10401e7 0.504599 0.252300 0.967649i \(-0.418813\pi\)
0.252300 + 0.967649i \(0.418813\pi\)
\(864\) 1.87588e6 0.0854909
\(865\) 0 0
\(866\) −2.70366e7 −1.22506
\(867\) −5.51918e6 −0.249360
\(868\) 4.05667e6 0.182755
\(869\) −5.89204e6 −0.264677
\(870\) 0 0
\(871\) −1.66512e7 −0.743702
\(872\) −1.34492e7 −0.598970
\(873\) −6.55605e6 −0.291143
\(874\) −4.93587e7 −2.18567
\(875\) 0 0
\(876\) 2.23638e6 0.0984658
\(877\) 2.06101e7 0.904861 0.452430 0.891800i \(-0.350557\pi\)
0.452430 + 0.891800i \(0.350557\pi\)
\(878\) −849937. −0.0372092
\(879\) 9.46608e6 0.413236
\(880\) 0 0
\(881\) −7.17874e6 −0.311608 −0.155804 0.987788i \(-0.549797\pi\)
−0.155804 + 0.987788i \(0.549797\pi\)
\(882\) 2.95952e6 0.128100
\(883\) −1.55190e6 −0.0669824 −0.0334912 0.999439i \(-0.510663\pi\)
−0.0334912 + 0.999439i \(0.510663\pi\)
\(884\) −7.65023e6 −0.329264
\(885\) 0 0
\(886\) −3.45526e7 −1.47876
\(887\) 3.39646e7 1.44950 0.724748 0.689014i \(-0.241956\pi\)
0.724748 + 0.689014i \(0.241956\pi\)
\(888\) −2.82307e7 −1.20140
\(889\) −3.24949e7 −1.37899
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −320704. −0.0134956
\(893\) −1.61305e7 −0.676891
\(894\) 2.20205e6 0.0921474
\(895\) 0 0
\(896\) −1.53762e7 −0.639853
\(897\) −4.35180e7 −1.80587
\(898\) −9.54127e6 −0.394835
\(899\) 271271. 0.0111945
\(900\) 0 0
\(901\) 2.05227e7 0.842216
\(902\) 9.06547e6 0.371000
\(903\) −1.79982e7 −0.734531
\(904\) −8.55211e6 −0.348058
\(905\) 0 0
\(906\) −2.19713e7 −0.889272
\(907\) 2.58319e7 1.04265 0.521324 0.853359i \(-0.325438\pi\)
0.521324 + 0.853359i \(0.325438\pi\)
\(908\) 4.24837e6 0.171005
\(909\) 1.55724e6 0.0625096
\(910\) 0 0
\(911\) 549431. 0.0219340 0.0109670 0.999940i \(-0.496509\pi\)
0.0109670 + 0.999940i \(0.496509\pi\)
\(912\) 1.60148e7 0.637579
\(913\) 1.16045e7 0.460735
\(914\) 5.68262e6 0.225000
\(915\) 0 0
\(916\) −6.63773e6 −0.261385
\(917\) 1.33152e7 0.522908
\(918\) −3.25658e6 −0.127542
\(919\) 3.84177e7 1.50052 0.750261 0.661142i \(-0.229928\pi\)
0.750261 + 0.661142i \(0.229928\pi\)
\(920\) 0 0
\(921\) 1.51536e7 0.588662
\(922\) 1.80701e7 0.700055
\(923\) 9.23930e7 3.56973
\(924\) 1.22869e6 0.0473437
\(925\) 0 0
\(926\) 3.13185e7 1.20025
\(927\) 1.02588e7 0.392099
\(928\) 194144. 0.00740038
\(929\) 1.50560e7 0.572360 0.286180 0.958176i \(-0.407614\pi\)
0.286180 + 0.958176i \(0.407614\pi\)
\(930\) 0 0
\(931\) −1.76882e7 −0.668821
\(932\) 5.10110e6 0.192364
\(933\) −2.45170e7 −0.922070
\(934\) −229877. −0.00862240
\(935\) 0 0
\(936\) 1.85586e7 0.692398
\(937\) −1.83320e7 −0.682119 −0.341059 0.940042i \(-0.610786\pi\)
−0.341059 + 0.940042i \(0.610786\pi\)
\(938\) −1.09703e7 −0.407109
\(939\) −1.52242e7 −0.563470
\(940\) 0 0
\(941\) −3.82211e7 −1.40711 −0.703557 0.710639i \(-0.748406\pi\)
−0.703557 + 0.710639i \(0.748406\pi\)
\(942\) 1.22525e7 0.449879
\(943\) 6.20754e7 2.27321
\(944\) 9.62245e6 0.351444
\(945\) 0 0
\(946\) 7.74444e6 0.281360
\(947\) −3.35057e7 −1.21407 −0.607035 0.794675i \(-0.707641\pi\)
−0.607035 + 0.794675i \(0.707641\pi\)
\(948\) 3.18166e6 0.114983
\(949\) 4.01590e7 1.44750
\(950\) 0 0
\(951\) 3.34233e6 0.119839
\(952\) −2.72563e7 −0.974707
\(953\) 3.29329e7 1.17462 0.587310 0.809362i \(-0.300187\pi\)
0.587310 + 0.809362i \(0.300187\pi\)
\(954\) −9.20636e6 −0.327504
\(955\) 0 0
\(956\) −4.15277e6 −0.146958
\(957\) 82162.8 0.00289998
\(958\) 9.32386e6 0.328233
\(959\) 1.79515e7 0.630311
\(960\) 0 0
\(961\) −1.57018e7 −0.548454
\(962\) −9.37433e7 −3.26590
\(963\) −6.29964e6 −0.218902
\(964\) 3.52340e6 0.122115
\(965\) 0 0
\(966\) −2.86709e7 −0.988551
\(967\) −2.24586e7 −0.772353 −0.386177 0.922425i \(-0.626204\pi\)
−0.386177 + 0.922425i \(0.626204\pi\)
\(968\) −2.85904e6 −0.0980691
\(969\) 1.94636e7 0.665908
\(970\) 0 0
\(971\) 3.76102e7 1.28014 0.640069 0.768317i \(-0.278906\pi\)
0.640069 + 0.768317i \(0.278906\pi\)
\(972\) −428690. −0.0145538
\(973\) 1.87458e7 0.634780
\(974\) 1.46159e7 0.493660
\(975\) 0 0
\(976\) −652007. −0.0219092
\(977\) 1.90408e7 0.638187 0.319094 0.947723i \(-0.396622\pi\)
0.319094 + 0.947723i \(0.396622\pi\)
\(978\) −2.90443e6 −0.0970987
\(979\) 162448. 0.00541700
\(980\) 0 0
\(981\) 5.57868e6 0.185080
\(982\) 3.76256e7 1.24510
\(983\) 3.59407e7 1.18632 0.593162 0.805083i \(-0.297880\pi\)
0.593162 + 0.805083i \(0.297880\pi\)
\(984\) −2.64726e7 −0.871584
\(985\) 0 0
\(986\) −337040. −0.0110405
\(987\) −9.36969e6 −0.306149
\(988\) −2.05111e7 −0.668493
\(989\) 5.30297e7 1.72396
\(990\) 0 0
\(991\) 2.52119e7 0.815495 0.407748 0.913095i \(-0.366314\pi\)
0.407748 + 0.913095i \(0.366314\pi\)
\(992\) 9.25192e6 0.298506
\(993\) 1.39073e7 0.447578
\(994\) 6.08713e7 1.95410
\(995\) 0 0
\(996\) −6.26637e6 −0.200156
\(997\) −4.86935e6 −0.155143 −0.0775716 0.996987i \(-0.524717\pi\)
−0.0775716 + 0.996987i \(0.524717\pi\)
\(998\) −1.97413e7 −0.627406
\(999\) 1.17100e7 0.371230
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.y.1.8 13
5.2 odd 4 165.6.c.b.34.18 yes 26
5.3 odd 4 165.6.c.b.34.9 26
5.4 even 2 825.6.a.v.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.9 26 5.3 odd 4
165.6.c.b.34.18 yes 26 5.2 odd 4
825.6.a.v.1.6 13 5.4 even 2
825.6.a.y.1.8 13 1.1 even 1 trivial