Properties

Label 825.6.a.v.1.5
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

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

Embedding invariants

Embedding label 1.5
Root \(-4.09558\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.09558 q^{2} -9.00000 q^{3} -6.03505 q^{4} +45.8602 q^{6} -170.277 q^{7} +193.811 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-5.09558 q^{2} -9.00000 q^{3} -6.03505 q^{4} +45.8602 q^{6} -170.277 q^{7} +193.811 q^{8} +81.0000 q^{9} +121.000 q^{11} +54.3155 q^{12} -364.335 q^{13} +867.659 q^{14} -794.456 q^{16} -1307.63 q^{17} -412.742 q^{18} +619.655 q^{19} +1532.49 q^{21} -616.565 q^{22} +922.189 q^{23} -1744.30 q^{24} +1856.50 q^{26} -729.000 q^{27} +1027.63 q^{28} -3433.99 q^{29} +1479.05 q^{31} -2153.73 q^{32} -1089.00 q^{33} +6663.11 q^{34} -488.839 q^{36} -5289.91 q^{37} -3157.50 q^{38} +3279.02 q^{39} -7669.49 q^{41} -7808.93 q^{42} +17366.5 q^{43} -730.242 q^{44} -4699.09 q^{46} +6946.36 q^{47} +7150.11 q^{48} +12187.2 q^{49} +11768.6 q^{51} +2198.78 q^{52} -17379.4 q^{53} +3714.68 q^{54} -33001.5 q^{56} -5576.90 q^{57} +17498.2 q^{58} +34047.9 q^{59} +48554.5 q^{61} -7536.59 q^{62} -13792.4 q^{63} +36397.1 q^{64} +5549.09 q^{66} -34368.8 q^{67} +7891.59 q^{68} -8299.70 q^{69} +48448.3 q^{71} +15698.7 q^{72} +59439.7 q^{73} +26955.1 q^{74} -3739.65 q^{76} -20603.5 q^{77} -16708.5 q^{78} +3591.22 q^{79} +6561.00 q^{81} +39080.5 q^{82} -64310.2 q^{83} -9248.67 q^{84} -88492.6 q^{86} +30905.9 q^{87} +23451.1 q^{88} +26956.3 q^{89} +62037.9 q^{91} -5565.46 q^{92} -13311.4 q^{93} -35395.8 q^{94} +19383.5 q^{96} -16545.8 q^{97} -62100.8 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.09558 −0.900780 −0.450390 0.892832i \(-0.648715\pi\)
−0.450390 + 0.892832i \(0.648715\pi\)
\(3\) −9.00000 −0.577350
\(4\) −6.03505 −0.188595
\(5\) 0 0
\(6\) 45.8602 0.520066
\(7\) −170.277 −1.31344 −0.656720 0.754134i \(-0.728057\pi\)
−0.656720 + 0.754134i \(0.728057\pi\)
\(8\) 193.811 1.07066
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 54.3155 0.108886
\(13\) −364.335 −0.597920 −0.298960 0.954266i \(-0.596640\pi\)
−0.298960 + 0.954266i \(0.596640\pi\)
\(14\) 867.659 1.18312
\(15\) 0 0
\(16\) −794.456 −0.775836
\(17\) −1307.63 −1.09739 −0.548695 0.836023i \(-0.684875\pi\)
−0.548695 + 0.836023i \(0.684875\pi\)
\(18\) −412.742 −0.300260
\(19\) 619.655 0.393791 0.196896 0.980424i \(-0.436914\pi\)
0.196896 + 0.980424i \(0.436914\pi\)
\(20\) 0 0
\(21\) 1532.49 0.758315
\(22\) −616.565 −0.271595
\(23\) 922.189 0.363497 0.181748 0.983345i \(-0.441824\pi\)
0.181748 + 0.983345i \(0.441824\pi\)
\(24\) −1744.30 −0.618148
\(25\) 0 0
\(26\) 1856.50 0.538594
\(27\) −729.000 −0.192450
\(28\) 1027.63 0.247709
\(29\) −3433.99 −0.758235 −0.379118 0.925349i \(-0.623772\pi\)
−0.379118 + 0.925349i \(0.623772\pi\)
\(30\) 0 0
\(31\) 1479.05 0.276425 0.138212 0.990403i \(-0.455864\pi\)
0.138212 + 0.990403i \(0.455864\pi\)
\(32\) −2153.73 −0.371805
\(33\) −1089.00 −0.174078
\(34\) 6663.11 0.988507
\(35\) 0 0
\(36\) −488.839 −0.0628651
\(37\) −5289.91 −0.635249 −0.317624 0.948217i \(-0.602885\pi\)
−0.317624 + 0.948217i \(0.602885\pi\)
\(38\) −3157.50 −0.354719
\(39\) 3279.02 0.345209
\(40\) 0 0
\(41\) −7669.49 −0.712536 −0.356268 0.934384i \(-0.615951\pi\)
−0.356268 + 0.934384i \(0.615951\pi\)
\(42\) −7808.93 −0.683075
\(43\) 17366.5 1.43233 0.716164 0.697933i \(-0.245896\pi\)
0.716164 + 0.697933i \(0.245896\pi\)
\(44\) −730.242 −0.0568637
\(45\) 0 0
\(46\) −4699.09 −0.327430
\(47\) 6946.36 0.458683 0.229342 0.973346i \(-0.426343\pi\)
0.229342 + 0.973346i \(0.426343\pi\)
\(48\) 7150.11 0.447929
\(49\) 12187.2 0.725126
\(50\) 0 0
\(51\) 11768.6 0.633578
\(52\) 2198.78 0.112765
\(53\) −17379.4 −0.849857 −0.424929 0.905227i \(-0.639701\pi\)
−0.424929 + 0.905227i \(0.639701\pi\)
\(54\) 3714.68 0.173355
\(55\) 0 0
\(56\) −33001.5 −1.40625
\(57\) −5576.90 −0.227356
\(58\) 17498.2 0.683003
\(59\) 34047.9 1.27339 0.636693 0.771117i \(-0.280302\pi\)
0.636693 + 0.771117i \(0.280302\pi\)
\(60\) 0 0
\(61\) 48554.5 1.67073 0.835363 0.549699i \(-0.185258\pi\)
0.835363 + 0.549699i \(0.185258\pi\)
\(62\) −7536.59 −0.248998
\(63\) −13792.4 −0.437813
\(64\) 36397.1 1.11075
\(65\) 0 0
\(66\) 5549.09 0.156806
\(67\) −34368.8 −0.935358 −0.467679 0.883898i \(-0.654910\pi\)
−0.467679 + 0.883898i \(0.654910\pi\)
\(68\) 7891.59 0.206963
\(69\) −8299.70 −0.209865
\(70\) 0 0
\(71\) 48448.3 1.14060 0.570299 0.821437i \(-0.306827\pi\)
0.570299 + 0.821437i \(0.306827\pi\)
\(72\) 15698.7 0.356888
\(73\) 59439.7 1.30548 0.652739 0.757583i \(-0.273620\pi\)
0.652739 + 0.757583i \(0.273620\pi\)
\(74\) 26955.1 0.572219
\(75\) 0 0
\(76\) −3739.65 −0.0742673
\(77\) −20603.5 −0.396017
\(78\) −16708.5 −0.310957
\(79\) 3591.22 0.0647402 0.0323701 0.999476i \(-0.489694\pi\)
0.0323701 + 0.999476i \(0.489694\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 39080.5 0.641838
\(83\) −64310.2 −1.02467 −0.512336 0.858785i \(-0.671220\pi\)
−0.512336 + 0.858785i \(0.671220\pi\)
\(84\) −9248.67 −0.143015
\(85\) 0 0
\(86\) −88492.6 −1.29021
\(87\) 30905.9 0.437767
\(88\) 23451.1 0.322817
\(89\) 26956.3 0.360732 0.180366 0.983600i \(-0.442272\pi\)
0.180366 + 0.983600i \(0.442272\pi\)
\(90\) 0 0
\(91\) 62037.9 0.785332
\(92\) −5565.46 −0.0685538
\(93\) −13311.4 −0.159594
\(94\) −35395.8 −0.413173
\(95\) 0 0
\(96\) 19383.5 0.214662
\(97\) −16545.8 −0.178549 −0.0892745 0.996007i \(-0.528455\pi\)
−0.0892745 + 0.996007i \(0.528455\pi\)
\(98\) −62100.8 −0.653179
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 137886. 1.34498 0.672492 0.740104i \(-0.265224\pi\)
0.672492 + 0.740104i \(0.265224\pi\)
\(102\) −59968.0 −0.570715
\(103\) 63842.9 0.592952 0.296476 0.955040i \(-0.404188\pi\)
0.296476 + 0.955040i \(0.404188\pi\)
\(104\) −70612.1 −0.640171
\(105\) 0 0
\(106\) 88558.3 0.765534
\(107\) 37586.9 0.317378 0.158689 0.987329i \(-0.449273\pi\)
0.158689 + 0.987329i \(0.449273\pi\)
\(108\) 4399.55 0.0362952
\(109\) −148487. −1.19708 −0.598538 0.801095i \(-0.704251\pi\)
−0.598538 + 0.801095i \(0.704251\pi\)
\(110\) 0 0
\(111\) 47609.2 0.366761
\(112\) 135277. 1.01901
\(113\) −189126. −1.39333 −0.696666 0.717395i \(-0.745334\pi\)
−0.696666 + 0.717395i \(0.745334\pi\)
\(114\) 28417.5 0.204797
\(115\) 0 0
\(116\) 20724.3 0.143000
\(117\) −29511.2 −0.199307
\(118\) −173494. −1.14704
\(119\) 222658. 1.44136
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −247413. −1.50496
\(123\) 69025.4 0.411383
\(124\) −8926.12 −0.0521325
\(125\) 0 0
\(126\) 70280.4 0.394374
\(127\) 308080. 1.69494 0.847469 0.530846i \(-0.178126\pi\)
0.847469 + 0.530846i \(0.178126\pi\)
\(128\) −116545. −0.628737
\(129\) −156299. −0.826954
\(130\) 0 0
\(131\) 64949.6 0.330673 0.165336 0.986237i \(-0.447129\pi\)
0.165336 + 0.986237i \(0.447129\pi\)
\(132\) 6572.17 0.0328303
\(133\) −105513. −0.517222
\(134\) 175129. 0.842552
\(135\) 0 0
\(136\) −253432. −1.17493
\(137\) 306828. 1.39667 0.698334 0.715772i \(-0.253925\pi\)
0.698334 + 0.715772i \(0.253925\pi\)
\(138\) 42291.8 0.189042
\(139\) 416315. 1.82762 0.913808 0.406147i \(-0.133128\pi\)
0.913808 + 0.406147i \(0.133128\pi\)
\(140\) 0 0
\(141\) −62517.3 −0.264821
\(142\) −246872. −1.02743
\(143\) −44084.6 −0.180280
\(144\) −64351.0 −0.258612
\(145\) 0 0
\(146\) −302880. −1.17595
\(147\) −109685. −0.418651
\(148\) 31924.9 0.119805
\(149\) 478891. 1.76714 0.883570 0.468300i \(-0.155133\pi\)
0.883570 + 0.468300i \(0.155133\pi\)
\(150\) 0 0
\(151\) 251861. 0.898914 0.449457 0.893302i \(-0.351618\pi\)
0.449457 + 0.893302i \(0.351618\pi\)
\(152\) 120096. 0.421618
\(153\) −105918. −0.365797
\(154\) 104987. 0.356724
\(155\) 0 0
\(156\) −19789.1 −0.0651049
\(157\) 35343.5 0.114435 0.0572177 0.998362i \(-0.481777\pi\)
0.0572177 + 0.998362i \(0.481777\pi\)
\(158\) −18299.3 −0.0583167
\(159\) 156415. 0.490665
\(160\) 0 0
\(161\) −157027. −0.477431
\(162\) −33432.1 −0.100087
\(163\) −482364. −1.42202 −0.711010 0.703182i \(-0.751762\pi\)
−0.711010 + 0.703182i \(0.751762\pi\)
\(164\) 46285.8 0.134381
\(165\) 0 0
\(166\) 327698. 0.923003
\(167\) −285420. −0.791941 −0.395970 0.918263i \(-0.629592\pi\)
−0.395970 + 0.918263i \(0.629592\pi\)
\(168\) 297013. 0.811900
\(169\) −238553. −0.642492
\(170\) 0 0
\(171\) 50192.1 0.131264
\(172\) −104808. −0.270130
\(173\) −473489. −1.20280 −0.601402 0.798947i \(-0.705391\pi\)
−0.601402 + 0.798947i \(0.705391\pi\)
\(174\) −157484. −0.394332
\(175\) 0 0
\(176\) −96129.2 −0.233923
\(177\) −306431. −0.735190
\(178\) −137358. −0.324940
\(179\) 487786. 1.13788 0.568940 0.822379i \(-0.307354\pi\)
0.568940 + 0.822379i \(0.307354\pi\)
\(180\) 0 0
\(181\) −129180. −0.293088 −0.146544 0.989204i \(-0.546815\pi\)
−0.146544 + 0.989204i \(0.546815\pi\)
\(182\) −316119. −0.707411
\(183\) −436991. −0.964594
\(184\) 178730. 0.389182
\(185\) 0 0
\(186\) 67829.3 0.143759
\(187\) −158223. −0.330876
\(188\) −41921.7 −0.0865056
\(189\) 124132. 0.252772
\(190\) 0 0
\(191\) −386800. −0.767191 −0.383595 0.923501i \(-0.625314\pi\)
−0.383595 + 0.923501i \(0.625314\pi\)
\(192\) −327574. −0.641292
\(193\) −618444. −1.19511 −0.597554 0.801828i \(-0.703861\pi\)
−0.597554 + 0.801828i \(0.703861\pi\)
\(194\) 84310.3 0.160833
\(195\) 0 0
\(196\) −73550.3 −0.136755
\(197\) 870016. 1.59721 0.798605 0.601856i \(-0.205572\pi\)
0.798605 + 0.601856i \(0.205572\pi\)
\(198\) −49941.8 −0.0905318
\(199\) −903963. −1.61815 −0.809074 0.587707i \(-0.800031\pi\)
−0.809074 + 0.587707i \(0.800031\pi\)
\(200\) 0 0
\(201\) 309320. 0.540029
\(202\) −702610. −1.21153
\(203\) 584729. 0.995897
\(204\) −71024.3 −0.119490
\(205\) 0 0
\(206\) −325317. −0.534120
\(207\) 74697.3 0.121166
\(208\) 289449. 0.463888
\(209\) 74978.3 0.118733
\(210\) 0 0
\(211\) 414396. 0.640780 0.320390 0.947286i \(-0.396186\pi\)
0.320390 + 0.947286i \(0.396186\pi\)
\(212\) 104886. 0.160279
\(213\) −436034. −0.658524
\(214\) −191527. −0.285888
\(215\) 0 0
\(216\) −141288. −0.206049
\(217\) −251847. −0.363068
\(218\) 756626. 1.07830
\(219\) −534957. −0.753718
\(220\) 0 0
\(221\) 476414. 0.656151
\(222\) −242596. −0.330371
\(223\) −526103. −0.708449 −0.354224 0.935160i \(-0.615255\pi\)
−0.354224 + 0.935160i \(0.615255\pi\)
\(224\) 366729. 0.488344
\(225\) 0 0
\(226\) 963706. 1.25509
\(227\) −582127. −0.749813 −0.374907 0.927063i \(-0.622325\pi\)
−0.374907 + 0.927063i \(0.622325\pi\)
\(228\) 33656.9 0.0428782
\(229\) −59916.3 −0.0755016 −0.0377508 0.999287i \(-0.512019\pi\)
−0.0377508 + 0.999287i \(0.512019\pi\)
\(230\) 0 0
\(231\) 185431. 0.228641
\(232\) −665544. −0.811814
\(233\) 282811. 0.341276 0.170638 0.985334i \(-0.445417\pi\)
0.170638 + 0.985334i \(0.445417\pi\)
\(234\) 150377. 0.179531
\(235\) 0 0
\(236\) −205481. −0.240155
\(237\) −32321.0 −0.0373778
\(238\) −1.13457e6 −1.29834
\(239\) −1.69607e6 −1.92065 −0.960327 0.278875i \(-0.910039\pi\)
−0.960327 + 0.278875i \(0.910039\pi\)
\(240\) 0 0
\(241\) −1.04493e6 −1.15890 −0.579449 0.815008i \(-0.696732\pi\)
−0.579449 + 0.815008i \(0.696732\pi\)
\(242\) −74604.4 −0.0818891
\(243\) −59049.0 −0.0641500
\(244\) −293029. −0.315091
\(245\) 0 0
\(246\) −351724. −0.370565
\(247\) −225762. −0.235456
\(248\) 286655. 0.295958
\(249\) 578792. 0.591594
\(250\) 0 0
\(251\) −1.92822e6 −1.93184 −0.965921 0.258837i \(-0.916661\pi\)
−0.965921 + 0.258837i \(0.916661\pi\)
\(252\) 83238.0 0.0825696
\(253\) 111585. 0.109598
\(254\) −1.56984e6 −1.52677
\(255\) 0 0
\(256\) −570842. −0.544397
\(257\) 686476. 0.648324 0.324162 0.946002i \(-0.394918\pi\)
0.324162 + 0.946002i \(0.394918\pi\)
\(258\) 796434. 0.744904
\(259\) 900748. 0.834361
\(260\) 0 0
\(261\) −278153. −0.252745
\(262\) −330956. −0.297863
\(263\) −507368. −0.452307 −0.226154 0.974092i \(-0.572615\pi\)
−0.226154 + 0.974092i \(0.572615\pi\)
\(264\) −211060. −0.186378
\(265\) 0 0
\(266\) 537650. 0.465903
\(267\) −242606. −0.208269
\(268\) 207418. 0.176404
\(269\) 717068. 0.604198 0.302099 0.953277i \(-0.402313\pi\)
0.302099 + 0.953277i \(0.402313\pi\)
\(270\) 0 0
\(271\) −315074. −0.260609 −0.130304 0.991474i \(-0.541595\pi\)
−0.130304 + 0.991474i \(0.541595\pi\)
\(272\) 1.03885e6 0.851395
\(273\) −558341. −0.453412
\(274\) −1.56347e6 −1.25809
\(275\) 0 0
\(276\) 50089.1 0.0395795
\(277\) −1.42881e6 −1.11885 −0.559427 0.828879i \(-0.688979\pi\)
−0.559427 + 0.828879i \(0.688979\pi\)
\(278\) −2.12137e6 −1.64628
\(279\) 119803. 0.0921416
\(280\) 0 0
\(281\) 432306. 0.326607 0.163303 0.986576i \(-0.447785\pi\)
0.163303 + 0.986576i \(0.447785\pi\)
\(282\) 318562. 0.238545
\(283\) 257624. 0.191214 0.0956071 0.995419i \(-0.469521\pi\)
0.0956071 + 0.995419i \(0.469521\pi\)
\(284\) −292388. −0.215112
\(285\) 0 0
\(286\) 224637. 0.162392
\(287\) 1.30594e6 0.935873
\(288\) −174452. −0.123935
\(289\) 290027. 0.204265
\(290\) 0 0
\(291\) 148912. 0.103085
\(292\) −358722. −0.246207
\(293\) −993168. −0.675855 −0.337928 0.941172i \(-0.609726\pi\)
−0.337928 + 0.941172i \(0.609726\pi\)
\(294\) 558907. 0.377113
\(295\) 0 0
\(296\) −1.02524e6 −0.680137
\(297\) −88209.0 −0.0580259
\(298\) −2.44023e6 −1.59180
\(299\) −335986. −0.217342
\(300\) 0 0
\(301\) −2.95712e6 −1.88128
\(302\) −1.28338e6 −0.809723
\(303\) −1.24098e6 −0.776527
\(304\) −492289. −0.305518
\(305\) 0 0
\(306\) 539712. 0.329502
\(307\) −1.93257e6 −1.17028 −0.585139 0.810933i \(-0.698960\pi\)
−0.585139 + 0.810933i \(0.698960\pi\)
\(308\) 124343. 0.0746870
\(309\) −574587. −0.342341
\(310\) 0 0
\(311\) 2.35905e6 1.38305 0.691523 0.722354i \(-0.256940\pi\)
0.691523 + 0.722354i \(0.256940\pi\)
\(312\) 635509. 0.369603
\(313\) 133954. 0.0772849 0.0386425 0.999253i \(-0.487697\pi\)
0.0386425 + 0.999253i \(0.487697\pi\)
\(314\) −180096. −0.103081
\(315\) 0 0
\(316\) −21673.2 −0.0122097
\(317\) 1.69094e6 0.945102 0.472551 0.881303i \(-0.343333\pi\)
0.472551 + 0.881303i \(0.343333\pi\)
\(318\) −797025. −0.441981
\(319\) −415513. −0.228616
\(320\) 0 0
\(321\) −338282. −0.183238
\(322\) 800146. 0.430060
\(323\) −810277. −0.432143
\(324\) −39596.0 −0.0209550
\(325\) 0 0
\(326\) 2.45792e6 1.28093
\(327\) 1.33638e6 0.691132
\(328\) −1.48643e6 −0.762886
\(329\) −1.18280e6 −0.602453
\(330\) 0 0
\(331\) 313289. 0.157172 0.0785860 0.996907i \(-0.474959\pi\)
0.0785860 + 0.996907i \(0.474959\pi\)
\(332\) 388115. 0.193248
\(333\) −428482. −0.211750
\(334\) 1.45438e6 0.713364
\(335\) 0 0
\(336\) −1.21750e6 −0.588328
\(337\) 1.93465e6 0.927957 0.463979 0.885846i \(-0.346421\pi\)
0.463979 + 0.885846i \(0.346421\pi\)
\(338\) 1.21556e6 0.578744
\(339\) 1.70213e6 0.804441
\(340\) 0 0
\(341\) 178964. 0.0833452
\(342\) −255758. −0.118240
\(343\) 786647. 0.361031
\(344\) 3.36582e6 1.53354
\(345\) 0 0
\(346\) 2.41270e6 1.08346
\(347\) 360768. 0.160844 0.0804218 0.996761i \(-0.474373\pi\)
0.0804218 + 0.996761i \(0.474373\pi\)
\(348\) −186519. −0.0825609
\(349\) −1.67266e6 −0.735095 −0.367547 0.930005i \(-0.619802\pi\)
−0.367547 + 0.930005i \(0.619802\pi\)
\(350\) 0 0
\(351\) 265600. 0.115070
\(352\) −260601. −0.112103
\(353\) −268287. −0.114594 −0.0572971 0.998357i \(-0.518248\pi\)
−0.0572971 + 0.998357i \(0.518248\pi\)
\(354\) 1.56144e6 0.662245
\(355\) 0 0
\(356\) −162682. −0.0680324
\(357\) −2.00392e6 −0.832167
\(358\) −2.48555e6 −1.02498
\(359\) −2.33979e6 −0.958165 −0.479083 0.877770i \(-0.659031\pi\)
−0.479083 + 0.877770i \(0.659031\pi\)
\(360\) 0 0
\(361\) −2.09213e6 −0.844928
\(362\) 658247. 0.264008
\(363\) −131769. −0.0524864
\(364\) −374402. −0.148110
\(365\) 0 0
\(366\) 2.22672e6 0.868887
\(367\) −1.87216e6 −0.725569 −0.362785 0.931873i \(-0.618174\pi\)
−0.362785 + 0.931873i \(0.618174\pi\)
\(368\) −732639. −0.282014
\(369\) −621229. −0.237512
\(370\) 0 0
\(371\) 2.95931e6 1.11624
\(372\) 80335.1 0.0300987
\(373\) 1.51377e6 0.563363 0.281681 0.959508i \(-0.409108\pi\)
0.281681 + 0.959508i \(0.409108\pi\)
\(374\) 806236. 0.298046
\(375\) 0 0
\(376\) 1.34628e6 0.491095
\(377\) 1.25112e6 0.453364
\(378\) −632524. −0.227692
\(379\) −2.85481e6 −1.02089 −0.510446 0.859910i \(-0.670520\pi\)
−0.510446 + 0.859910i \(0.670520\pi\)
\(380\) 0 0
\(381\) −2.77272e6 −0.978572
\(382\) 1.97097e6 0.691070
\(383\) −3.23968e6 −1.12851 −0.564254 0.825601i \(-0.690836\pi\)
−0.564254 + 0.825601i \(0.690836\pi\)
\(384\) 1.04891e6 0.363002
\(385\) 0 0
\(386\) 3.15133e6 1.07653
\(387\) 1.40669e6 0.477442
\(388\) 99854.6 0.0336735
\(389\) −558246. −0.187047 −0.0935237 0.995617i \(-0.529813\pi\)
−0.0935237 + 0.995617i \(0.529813\pi\)
\(390\) 0 0
\(391\) −1.20588e6 −0.398897
\(392\) 2.36201e6 0.776365
\(393\) −584547. −0.190914
\(394\) −4.43324e6 −1.43873
\(395\) 0 0
\(396\) −59149.6 −0.0189546
\(397\) −1.57535e6 −0.501651 −0.250826 0.968032i \(-0.580702\pi\)
−0.250826 + 0.968032i \(0.580702\pi\)
\(398\) 4.60622e6 1.45760
\(399\) 949616. 0.298618
\(400\) 0 0
\(401\) 2.98866e6 0.928145 0.464073 0.885797i \(-0.346388\pi\)
0.464073 + 0.885797i \(0.346388\pi\)
\(402\) −1.57616e6 −0.486448
\(403\) −538868. −0.165280
\(404\) −832150. −0.253658
\(405\) 0 0
\(406\) −2.97953e6 −0.897084
\(407\) −640079. −0.191535
\(408\) 2.28089e6 0.678349
\(409\) 6.60301e6 1.95179 0.975897 0.218232i \(-0.0700291\pi\)
0.975897 + 0.218232i \(0.0700291\pi\)
\(410\) 0 0
\(411\) −2.76145e6 −0.806367
\(412\) −385296. −0.111828
\(413\) −5.79757e6 −1.67252
\(414\) −380626. −0.109143
\(415\) 0 0
\(416\) 784678. 0.222310
\(417\) −3.74683e6 −1.05517
\(418\) −382058. −0.106952
\(419\) −2.41796e6 −0.672842 −0.336421 0.941712i \(-0.609217\pi\)
−0.336421 + 0.941712i \(0.609217\pi\)
\(420\) 0 0
\(421\) −315653. −0.0867971 −0.0433986 0.999058i \(-0.513819\pi\)
−0.0433986 + 0.999058i \(0.513819\pi\)
\(422\) −2.11159e6 −0.577202
\(423\) 562656. 0.152894
\(424\) −3.36832e6 −0.909911
\(425\) 0 0
\(426\) 2.22185e6 0.593186
\(427\) −8.26771e6 −2.19440
\(428\) −226839. −0.0598561
\(429\) 396761. 0.104084
\(430\) 0 0
\(431\) −1.62617e6 −0.421669 −0.210835 0.977522i \(-0.567618\pi\)
−0.210835 + 0.977522i \(0.567618\pi\)
\(432\) 579159. 0.149310
\(433\) 756561. 0.193921 0.0969603 0.995288i \(-0.469088\pi\)
0.0969603 + 0.995288i \(0.469088\pi\)
\(434\) 1.28331e6 0.327044
\(435\) 0 0
\(436\) 896126. 0.225763
\(437\) 571439. 0.143142
\(438\) 2.72592e6 0.678934
\(439\) −3.70064e6 −0.916463 −0.458232 0.888833i \(-0.651517\pi\)
−0.458232 + 0.888833i \(0.651517\pi\)
\(440\) 0 0
\(441\) 987162. 0.241709
\(442\) −2.42761e6 −0.591048
\(443\) 6.39510e6 1.54824 0.774120 0.633039i \(-0.218193\pi\)
0.774120 + 0.633039i \(0.218193\pi\)
\(444\) −287324. −0.0691694
\(445\) 0 0
\(446\) 2.68080e6 0.638157
\(447\) −4.31002e6 −1.02026
\(448\) −6.19758e6 −1.45891
\(449\) −7.10566e6 −1.66337 −0.831684 0.555249i \(-0.812623\pi\)
−0.831684 + 0.555249i \(0.812623\pi\)
\(450\) 0 0
\(451\) −928008. −0.214838
\(452\) 1.14138e6 0.262776
\(453\) −2.26675e6 −0.518988
\(454\) 2.96628e6 0.675417
\(455\) 0 0
\(456\) −1.08086e6 −0.243421
\(457\) 2.49947e6 0.559832 0.279916 0.960025i \(-0.409693\pi\)
0.279916 + 0.960025i \(0.409693\pi\)
\(458\) 305308. 0.0680103
\(459\) 953259. 0.211193
\(460\) 0 0
\(461\) −5.72233e6 −1.25407 −0.627033 0.778993i \(-0.715731\pi\)
−0.627033 + 0.778993i \(0.715731\pi\)
\(462\) −944881. −0.205955
\(463\) 5.64981e6 1.22485 0.612423 0.790530i \(-0.290195\pi\)
0.612423 + 0.790530i \(0.290195\pi\)
\(464\) 2.72815e6 0.588266
\(465\) 0 0
\(466\) −1.44108e6 −0.307415
\(467\) −8.10977e6 −1.72074 −0.860372 0.509666i \(-0.829769\pi\)
−0.860372 + 0.509666i \(0.829769\pi\)
\(468\) 178101. 0.0375883
\(469\) 5.85222e6 1.22854
\(470\) 0 0
\(471\) −318092. −0.0660693
\(472\) 6.59885e6 1.36337
\(473\) 2.10135e6 0.431863
\(474\) 164694. 0.0336691
\(475\) 0 0
\(476\) −1.34375e6 −0.271833
\(477\) −1.40773e6 −0.283286
\(478\) 8.64247e6 1.73009
\(479\) 5.82406e6 1.15981 0.579905 0.814684i \(-0.303090\pi\)
0.579905 + 0.814684i \(0.303090\pi\)
\(480\) 0 0
\(481\) 1.92730e6 0.379828
\(482\) 5.32454e6 1.04391
\(483\) 1.41325e6 0.275645
\(484\) −88359.2 −0.0171450
\(485\) 0 0
\(486\) 300889. 0.0577851
\(487\) 7.82723e6 1.49550 0.747749 0.663981i \(-0.231135\pi\)
0.747749 + 0.663981i \(0.231135\pi\)
\(488\) 9.41039e6 1.78878
\(489\) 4.34127e6 0.821004
\(490\) 0 0
\(491\) −5.95082e6 −1.11397 −0.556985 0.830523i \(-0.688042\pi\)
−0.556985 + 0.830523i \(0.688042\pi\)
\(492\) −416572. −0.0775849
\(493\) 4.49037e6 0.832080
\(494\) 1.15039e6 0.212094
\(495\) 0 0
\(496\) −1.17504e6 −0.214460
\(497\) −8.24962e6 −1.49811
\(498\) −2.94928e6 −0.532896
\(499\) −1.80923e6 −0.325269 −0.162634 0.986686i \(-0.551999\pi\)
−0.162634 + 0.986686i \(0.551999\pi\)
\(500\) 0 0
\(501\) 2.56878e6 0.457227
\(502\) 9.82539e6 1.74016
\(503\) 2.53156e6 0.446137 0.223068 0.974803i \(-0.428393\pi\)
0.223068 + 0.974803i \(0.428393\pi\)
\(504\) −2.67312e6 −0.468751
\(505\) 0 0
\(506\) −568590. −0.0987240
\(507\) 2.14697e6 0.370943
\(508\) −1.85928e6 −0.319657
\(509\) 8.24104e6 1.40990 0.704948 0.709259i \(-0.250970\pi\)
0.704948 + 0.709259i \(0.250970\pi\)
\(510\) 0 0
\(511\) −1.01212e7 −1.71467
\(512\) 6.63821e6 1.11912
\(513\) −451729. −0.0757852
\(514\) −3.49799e6 −0.583998
\(515\) 0 0
\(516\) 943272. 0.155960
\(517\) 840510. 0.138298
\(518\) −4.58984e6 −0.751576
\(519\) 4.26140e6 0.694439
\(520\) 0 0
\(521\) 5.59646e6 0.903273 0.451637 0.892202i \(-0.350840\pi\)
0.451637 + 0.892202i \(0.350840\pi\)
\(522\) 1.41735e6 0.227668
\(523\) 1.34040e6 0.214279 0.107140 0.994244i \(-0.465831\pi\)
0.107140 + 0.994244i \(0.465831\pi\)
\(524\) −391974. −0.0623634
\(525\) 0 0
\(526\) 2.58533e6 0.407429
\(527\) −1.93404e6 −0.303346
\(528\) 865163. 0.135056
\(529\) −5.58591e6 −0.867870
\(530\) 0 0
\(531\) 2.75788e6 0.424462
\(532\) 636776. 0.0975456
\(533\) 2.79427e6 0.426039
\(534\) 1.23622e6 0.187604
\(535\) 0 0
\(536\) −6.66105e6 −1.00145
\(537\) −4.39007e6 −0.656955
\(538\) −3.65388e6 −0.544250
\(539\) 1.47465e6 0.218634
\(540\) 0 0
\(541\) 2.97220e6 0.436602 0.218301 0.975882i \(-0.429949\pi\)
0.218301 + 0.975882i \(0.429949\pi\)
\(542\) 1.60548e6 0.234751
\(543\) 1.16262e6 0.169215
\(544\) 2.81627e6 0.408015
\(545\) 0 0
\(546\) 2.84507e6 0.408424
\(547\) 4.70253e6 0.671991 0.335995 0.941864i \(-0.390927\pi\)
0.335995 + 0.941864i \(0.390927\pi\)
\(548\) −1.85172e6 −0.263405
\(549\) 3.93292e6 0.556908
\(550\) 0 0
\(551\) −2.12789e6 −0.298586
\(552\) −1.60857e6 −0.224694
\(553\) −611501. −0.0850324
\(554\) 7.28059e6 1.00784
\(555\) 0 0
\(556\) −2.51248e6 −0.344680
\(557\) 5.61923e6 0.767431 0.383715 0.923451i \(-0.374644\pi\)
0.383715 + 0.923451i \(0.374644\pi\)
\(558\) −610464. −0.0829993
\(559\) −6.32724e6 −0.856417
\(560\) 0 0
\(561\) 1.42400e6 0.191031
\(562\) −2.20285e6 −0.294201
\(563\) 1.12518e7 1.49607 0.748036 0.663658i \(-0.230997\pi\)
0.748036 + 0.663658i \(0.230997\pi\)
\(564\) 377295. 0.0499440
\(565\) 0 0
\(566\) −1.31274e6 −0.172242
\(567\) −1.11719e6 −0.145938
\(568\) 9.38979e6 1.22120
\(569\) 3.62036e6 0.468782 0.234391 0.972142i \(-0.424690\pi\)
0.234391 + 0.972142i \(0.424690\pi\)
\(570\) 0 0
\(571\) −6.01771e6 −0.772397 −0.386199 0.922416i \(-0.626212\pi\)
−0.386199 + 0.922416i \(0.626212\pi\)
\(572\) 266053. 0.0339999
\(573\) 3.48120e6 0.442938
\(574\) −6.65450e6 −0.843016
\(575\) 0 0
\(576\) 2.94816e6 0.370250
\(577\) −3.81655e6 −0.477233 −0.238617 0.971114i \(-0.576694\pi\)
−0.238617 + 0.971114i \(0.576694\pi\)
\(578\) −1.47785e6 −0.183998
\(579\) 5.56600e6 0.689996
\(580\) 0 0
\(581\) 1.09505e7 1.34584
\(582\) −758793. −0.0928572
\(583\) −2.10291e6 −0.256242
\(584\) 1.15200e7 1.39773
\(585\) 0 0
\(586\) 5.06077e6 0.608797
\(587\) −4.77208e6 −0.571627 −0.285813 0.958285i \(-0.592264\pi\)
−0.285813 + 0.958285i \(0.592264\pi\)
\(588\) 661953. 0.0789558
\(589\) 916498. 0.108854
\(590\) 0 0
\(591\) −7.83015e6 −0.922149
\(592\) 4.20260e6 0.492849
\(593\) −1.97557e6 −0.230704 −0.115352 0.993325i \(-0.536800\pi\)
−0.115352 + 0.993325i \(0.536800\pi\)
\(594\) 449476. 0.0522686
\(595\) 0 0
\(596\) −2.89013e6 −0.333274
\(597\) 8.13567e6 0.934238
\(598\) 1.71204e6 0.195777
\(599\) −7989.43 −0.000909806 0 −0.000454903 1.00000i \(-0.500145\pi\)
−0.000454903 1.00000i \(0.500145\pi\)
\(600\) 0 0
\(601\) 1.62885e7 1.83948 0.919741 0.392526i \(-0.128399\pi\)
0.919741 + 0.392526i \(0.128399\pi\)
\(602\) 1.50682e7 1.69462
\(603\) −2.78388e6 −0.311786
\(604\) −1.51999e6 −0.169531
\(605\) 0 0
\(606\) 6.32349e6 0.699480
\(607\) −1.05514e7 −1.16235 −0.581176 0.813778i \(-0.697407\pi\)
−0.581176 + 0.813778i \(0.697407\pi\)
\(608\) −1.33457e6 −0.146414
\(609\) −5.26256e6 −0.574981
\(610\) 0 0
\(611\) −2.53081e6 −0.274256
\(612\) 639219. 0.0689876
\(613\) −1.45442e7 −1.56329 −0.781645 0.623723i \(-0.785619\pi\)
−0.781645 + 0.623723i \(0.785619\pi\)
\(614\) 9.84757e6 1.05416
\(615\) 0 0
\(616\) −3.99318e6 −0.424001
\(617\) 1.74916e7 1.84977 0.924885 0.380246i \(-0.124161\pi\)
0.924885 + 0.380246i \(0.124161\pi\)
\(618\) 2.92785e6 0.308374
\(619\) −1.56180e7 −1.63832 −0.819160 0.573565i \(-0.805560\pi\)
−0.819160 + 0.573565i \(0.805560\pi\)
\(620\) 0 0
\(621\) −672276. −0.0699549
\(622\) −1.20208e7 −1.24582
\(623\) −4.59003e6 −0.473800
\(624\) −2.60504e6 −0.267826
\(625\) 0 0
\(626\) −682573. −0.0696167
\(627\) −674805. −0.0685503
\(628\) −213300. −0.0215820
\(629\) 6.91722e6 0.697115
\(630\) 0 0
\(631\) 9.11463e6 0.911309 0.455655 0.890157i \(-0.349405\pi\)
0.455655 + 0.890157i \(0.349405\pi\)
\(632\) 696016. 0.0693149
\(633\) −3.72956e6 −0.369955
\(634\) −8.61630e6 −0.851329
\(635\) 0 0
\(636\) −943972. −0.0925372
\(637\) −4.44022e6 −0.433567
\(638\) 2.11728e6 0.205933
\(639\) 3.92431e6 0.380199
\(640\) 0 0
\(641\) 1.47144e7 1.41448 0.707242 0.706971i \(-0.249939\pi\)
0.707242 + 0.706971i \(0.249939\pi\)
\(642\) 1.72374e6 0.165057
\(643\) 1.99364e7 1.90160 0.950799 0.309807i \(-0.100265\pi\)
0.950799 + 0.309807i \(0.100265\pi\)
\(644\) 947669. 0.0900413
\(645\) 0 0
\(646\) 4.12883e6 0.389266
\(647\) 1.17295e7 1.10159 0.550794 0.834641i \(-0.314325\pi\)
0.550794 + 0.834641i \(0.314325\pi\)
\(648\) 1.27159e6 0.118963
\(649\) 4.11980e6 0.383940
\(650\) 0 0
\(651\) 2.26662e6 0.209617
\(652\) 2.91109e6 0.268187
\(653\) 1.62173e7 1.48832 0.744158 0.668004i \(-0.232851\pi\)
0.744158 + 0.668004i \(0.232851\pi\)
\(654\) −6.80964e6 −0.622558
\(655\) 0 0
\(656\) 6.09307e6 0.552811
\(657\) 4.81462e6 0.435159
\(658\) 6.02708e6 0.542678
\(659\) −806061. −0.0723027 −0.0361513 0.999346i \(-0.511510\pi\)
−0.0361513 + 0.999346i \(0.511510\pi\)
\(660\) 0 0
\(661\) −1.54634e7 −1.37658 −0.688291 0.725435i \(-0.741639\pi\)
−0.688291 + 0.725435i \(0.741639\pi\)
\(662\) −1.59639e6 −0.141577
\(663\) −4.28773e6 −0.378829
\(664\) −1.24640e7 −1.09708
\(665\) 0 0
\(666\) 2.18337e6 0.190740
\(667\) −3.16679e6 −0.275616
\(668\) 1.72252e6 0.149356
\(669\) 4.73493e6 0.409023
\(670\) 0 0
\(671\) 5.87510e6 0.503743
\(672\) −3.30057e6 −0.281945
\(673\) 3.23414e6 0.275246 0.137623 0.990485i \(-0.456054\pi\)
0.137623 + 0.990485i \(0.456054\pi\)
\(674\) −9.85817e6 −0.835885
\(675\) 0 0
\(676\) 1.43968e6 0.121171
\(677\) 8.24825e6 0.691656 0.345828 0.938298i \(-0.387598\pi\)
0.345828 + 0.938298i \(0.387598\pi\)
\(678\) −8.67336e6 −0.724624
\(679\) 2.81736e6 0.234514
\(680\) 0 0
\(681\) 5.23914e6 0.432905
\(682\) −911928. −0.0750757
\(683\) −1.23891e7 −1.01622 −0.508111 0.861291i \(-0.669656\pi\)
−0.508111 + 0.861291i \(0.669656\pi\)
\(684\) −302912. −0.0247558
\(685\) 0 0
\(686\) −4.00842e6 −0.325210
\(687\) 539247. 0.0435909
\(688\) −1.37970e7 −1.11125
\(689\) 6.33194e6 0.508146
\(690\) 0 0
\(691\) −1.75824e7 −1.40082 −0.700410 0.713741i \(-0.746999\pi\)
−0.700410 + 0.713741i \(0.746999\pi\)
\(692\) 2.85753e6 0.226843
\(693\) −1.66888e6 −0.132006
\(694\) −1.83832e6 −0.144885
\(695\) 0 0
\(696\) 5.98989e6 0.468701
\(697\) 1.00288e7 0.781930
\(698\) 8.52316e6 0.662158
\(699\) −2.54530e6 −0.197036
\(700\) 0 0
\(701\) −1.26176e7 −0.969797 −0.484898 0.874571i \(-0.661143\pi\)
−0.484898 + 0.874571i \(0.661143\pi\)
\(702\) −1.35339e6 −0.103652
\(703\) −3.27792e6 −0.250155
\(704\) 4.40405e6 0.334904
\(705\) 0 0
\(706\) 1.36708e6 0.103224
\(707\) −2.34788e7 −1.76656
\(708\) 1.84933e6 0.138653
\(709\) 5.87588e6 0.438993 0.219496 0.975613i \(-0.429559\pi\)
0.219496 + 0.975613i \(0.429559\pi\)
\(710\) 0 0
\(711\) 290889. 0.0215801
\(712\) 5.22441e6 0.386222
\(713\) 1.36396e6 0.100479
\(714\) 1.02112e7 0.749600
\(715\) 0 0
\(716\) −2.94381e6 −0.214599
\(717\) 1.52646e7 1.10889
\(718\) 1.19226e7 0.863096
\(719\) −9.06132e6 −0.653686 −0.326843 0.945079i \(-0.605985\pi\)
−0.326843 + 0.945079i \(0.605985\pi\)
\(720\) 0 0
\(721\) −1.08710e7 −0.778808
\(722\) 1.06606e7 0.761095
\(723\) 9.40439e6 0.669090
\(724\) 779608. 0.0552752
\(725\) 0 0
\(726\) 671440. 0.0472787
\(727\) −8.13363e6 −0.570753 −0.285376 0.958415i \(-0.592119\pi\)
−0.285376 + 0.958415i \(0.592119\pi\)
\(728\) 1.20236e7 0.840826
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −2.27089e7 −1.57182
\(732\) 2.63726e6 0.181918
\(733\) −1.02344e7 −0.703561 −0.351780 0.936083i \(-0.614424\pi\)
−0.351780 + 0.936083i \(0.614424\pi\)
\(734\) 9.53976e6 0.653578
\(735\) 0 0
\(736\) −1.98614e6 −0.135150
\(737\) −4.15863e6 −0.282021
\(738\) 3.16552e6 0.213946
\(739\) −1.70091e7 −1.14570 −0.572850 0.819660i \(-0.694162\pi\)
−0.572850 + 0.819660i \(0.694162\pi\)
\(740\) 0 0
\(741\) 2.03186e6 0.135940
\(742\) −1.50794e7 −1.00548
\(743\) 1.55671e7 1.03451 0.517257 0.855830i \(-0.326953\pi\)
0.517257 + 0.855830i \(0.326953\pi\)
\(744\) −2.57989e6 −0.170871
\(745\) 0 0
\(746\) −7.71354e6 −0.507466
\(747\) −5.20912e6 −0.341557
\(748\) 954882. 0.0624016
\(749\) −6.40018e6 −0.416857
\(750\) 0 0
\(751\) −2.29900e7 −1.48744 −0.743719 0.668493i \(-0.766940\pi\)
−0.743719 + 0.668493i \(0.766940\pi\)
\(752\) −5.51858e6 −0.355863
\(753\) 1.73540e7 1.11535
\(754\) −6.37520e6 −0.408381
\(755\) 0 0
\(756\) −749142. −0.0476716
\(757\) −1.66179e7 −1.05399 −0.526994 0.849869i \(-0.676681\pi\)
−0.526994 + 0.849869i \(0.676681\pi\)
\(758\) 1.45469e7 0.919599
\(759\) −1.00426e6 −0.0632766
\(760\) 0 0
\(761\) −2.43312e7 −1.52300 −0.761502 0.648162i \(-0.775538\pi\)
−0.761502 + 0.648162i \(0.775538\pi\)
\(762\) 1.41286e7 0.881478
\(763\) 2.52839e7 1.57229
\(764\) 2.33436e6 0.144689
\(765\) 0 0
\(766\) 1.65080e7 1.01654
\(767\) −1.24049e7 −0.761383
\(768\) 5.13758e6 0.314308
\(769\) −1.81242e7 −1.10520 −0.552602 0.833445i \(-0.686365\pi\)
−0.552602 + 0.833445i \(0.686365\pi\)
\(770\) 0 0
\(771\) −6.17828e6 −0.374310
\(772\) 3.73235e6 0.225392
\(773\) −2.02819e7 −1.22084 −0.610422 0.792077i \(-0.709000\pi\)
−0.610422 + 0.792077i \(0.709000\pi\)
\(774\) −7.16790e6 −0.430071
\(775\) 0 0
\(776\) −3.20675e6 −0.191166
\(777\) −8.10673e6 −0.481719
\(778\) 2.84459e6 0.168489
\(779\) −4.75244e6 −0.280590
\(780\) 0 0
\(781\) 5.86224e6 0.343903
\(782\) 6.14465e6 0.359319
\(783\) 2.50338e6 0.145922
\(784\) −9.68219e6 −0.562579
\(785\) 0 0
\(786\) 2.97860e6 0.171971
\(787\) 1.37897e7 0.793630 0.396815 0.917899i \(-0.370115\pi\)
0.396815 + 0.917899i \(0.370115\pi\)
\(788\) −5.25060e6 −0.301226
\(789\) 4.56631e6 0.261140
\(790\) 0 0
\(791\) 3.22037e7 1.83006
\(792\) 1.89954e6 0.107606
\(793\) −1.76901e7 −0.998960
\(794\) 8.02735e6 0.451878
\(795\) 0 0
\(796\) 5.45547e6 0.305175
\(797\) −3.04194e6 −0.169631 −0.0848155 0.996397i \(-0.527030\pi\)
−0.0848155 + 0.996397i \(0.527030\pi\)
\(798\) −4.83885e6 −0.268989
\(799\) −9.08324e6 −0.503354
\(800\) 0 0
\(801\) 2.18346e6 0.120244
\(802\) −1.52290e7 −0.836055
\(803\) 7.19220e6 0.393616
\(804\) −1.86676e6 −0.101847
\(805\) 0 0
\(806\) 2.74585e6 0.148881
\(807\) −6.45361e6 −0.348834
\(808\) 2.67238e7 1.44002
\(809\) −6.70668e6 −0.360277 −0.180138 0.983641i \(-0.557655\pi\)
−0.180138 + 0.983641i \(0.557655\pi\)
\(810\) 0 0
\(811\) −8.62214e6 −0.460323 −0.230162 0.973152i \(-0.573926\pi\)
−0.230162 + 0.973152i \(0.573926\pi\)
\(812\) −3.52887e6 −0.187822
\(813\) 2.83566e6 0.150463
\(814\) 3.26157e6 0.172531
\(815\) 0 0
\(816\) −9.34966e6 −0.491553
\(817\) 1.07613e7 0.564038
\(818\) −3.36462e7 −1.75814
\(819\) 5.02507e6 0.261777
\(820\) 0 0
\(821\) 2.68361e7 1.38951 0.694754 0.719247i \(-0.255513\pi\)
0.694754 + 0.719247i \(0.255513\pi\)
\(822\) 1.40712e7 0.726359
\(823\) −1.23875e6 −0.0637508 −0.0318754 0.999492i \(-0.510148\pi\)
−0.0318754 + 0.999492i \(0.510148\pi\)
\(824\) 1.23734e7 0.634852
\(825\) 0 0
\(826\) 2.95420e7 1.50657
\(827\) −2.59198e7 −1.31786 −0.658929 0.752205i \(-0.728990\pi\)
−0.658929 + 0.752205i \(0.728990\pi\)
\(828\) −450802. −0.0228513
\(829\) −6.40573e6 −0.323729 −0.161865 0.986813i \(-0.551751\pi\)
−0.161865 + 0.986813i \(0.551751\pi\)
\(830\) 0 0
\(831\) 1.28593e7 0.645971
\(832\) −1.32607e7 −0.664140
\(833\) −1.59363e7 −0.795746
\(834\) 1.90923e7 0.950480
\(835\) 0 0
\(836\) −452498. −0.0223924
\(837\) −1.07822e6 −0.0531980
\(838\) 1.23209e7 0.606083
\(839\) −2.24119e7 −1.09919 −0.549597 0.835430i \(-0.685219\pi\)
−0.549597 + 0.835430i \(0.685219\pi\)
\(840\) 0 0
\(841\) −8.71887e6 −0.425080
\(842\) 1.60844e6 0.0781851
\(843\) −3.89075e6 −0.188566
\(844\) −2.50090e6 −0.120848
\(845\) 0 0
\(846\) −2.86706e6 −0.137724
\(847\) −2.49302e6 −0.119404
\(848\) 1.38072e7 0.659350
\(849\) −2.31862e6 −0.110398
\(850\) 0 0
\(851\) −4.87829e6 −0.230911
\(852\) 2.63149e6 0.124195
\(853\) 1.15120e7 0.541725 0.270862 0.962618i \(-0.412691\pi\)
0.270862 + 0.962618i \(0.412691\pi\)
\(854\) 4.21288e7 1.97667
\(855\) 0 0
\(856\) 7.28474e6 0.339805
\(857\) −1.03784e7 −0.482701 −0.241350 0.970438i \(-0.577590\pi\)
−0.241350 + 0.970438i \(0.577590\pi\)
\(858\) −2.02173e6 −0.0937572
\(859\) 2.03041e7 0.938860 0.469430 0.882970i \(-0.344459\pi\)
0.469430 + 0.882970i \(0.344459\pi\)
\(860\) 0 0
\(861\) −1.17534e7 −0.540327
\(862\) 8.28627e6 0.379831
\(863\) −3.21077e7 −1.46752 −0.733758 0.679411i \(-0.762235\pi\)
−0.733758 + 0.679411i \(0.762235\pi\)
\(864\) 1.57007e6 0.0715539
\(865\) 0 0
\(866\) −3.85512e6 −0.174680
\(867\) −2.61024e6 −0.117932
\(868\) 1.51991e6 0.0684729
\(869\) 434537. 0.0195199
\(870\) 0 0
\(871\) 1.25218e7 0.559269
\(872\) −2.87783e7 −1.28166
\(873\) −1.34021e6 −0.0595164
\(874\) −2.91182e6 −0.128939
\(875\) 0 0
\(876\) 3.22850e6 0.142148
\(877\) 3.06108e7 1.34393 0.671964 0.740584i \(-0.265451\pi\)
0.671964 + 0.740584i \(0.265451\pi\)
\(878\) 1.88569e7 0.825532
\(879\) 8.93851e6 0.390205
\(880\) 0 0
\(881\) 4.04714e7 1.75675 0.878373 0.477976i \(-0.158629\pi\)
0.878373 + 0.477976i \(0.158629\pi\)
\(882\) −5.03016e6 −0.217726
\(883\) −3.01707e7 −1.30222 −0.651108 0.758985i \(-0.725696\pi\)
−0.651108 + 0.758985i \(0.725696\pi\)
\(884\) −2.87518e6 −0.123747
\(885\) 0 0
\(886\) −3.25868e7 −1.39462
\(887\) −1.68446e7 −0.718874 −0.359437 0.933169i \(-0.617031\pi\)
−0.359437 + 0.933169i \(0.617031\pi\)
\(888\) 9.22716e6 0.392677
\(889\) −5.24588e7 −2.22620
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 3.17506e6 0.133610
\(893\) 4.30435e6 0.180626
\(894\) 2.19620e7 0.919028
\(895\) 0 0
\(896\) 1.98449e7 0.825809
\(897\) 3.02387e6 0.125482
\(898\) 3.62075e7 1.49833
\(899\) −5.07902e6 −0.209595
\(900\) 0 0
\(901\) 2.27258e7 0.932625
\(902\) 4.72874e6 0.193521
\(903\) 2.66141e7 1.08616
\(904\) −3.66546e7 −1.49179
\(905\) 0 0
\(906\) 1.15504e7 0.467494
\(907\) 3.35783e7 1.35532 0.677659 0.735377i \(-0.262995\pi\)
0.677659 + 0.735377i \(0.262995\pi\)
\(908\) 3.51317e6 0.141411
\(909\) 1.11688e7 0.448328
\(910\) 0 0
\(911\) −1.06806e7 −0.426385 −0.213192 0.977010i \(-0.568386\pi\)
−0.213192 + 0.977010i \(0.568386\pi\)
\(912\) 4.43060e6 0.176391
\(913\) −7.78153e6 −0.308950
\(914\) −1.27363e7 −0.504285
\(915\) 0 0
\(916\) 361598. 0.0142393
\(917\) −1.10594e7 −0.434319
\(918\) −4.85741e6 −0.190238
\(919\) −2.12108e7 −0.828453 −0.414226 0.910174i \(-0.635948\pi\)
−0.414226 + 0.910174i \(0.635948\pi\)
\(920\) 0 0
\(921\) 1.73931e7 0.675661
\(922\) 2.91586e7 1.12964
\(923\) −1.76514e7 −0.681986
\(924\) −1.11909e6 −0.0431206
\(925\) 0 0
\(926\) −2.87891e7 −1.10332
\(927\) 5.17128e6 0.197651
\(928\) 7.39587e6 0.281916
\(929\) 4.20462e7 1.59841 0.799203 0.601061i \(-0.205255\pi\)
0.799203 + 0.601061i \(0.205255\pi\)
\(930\) 0 0
\(931\) 7.55186e6 0.285548
\(932\) −1.70678e6 −0.0643631
\(933\) −2.12315e7 −0.798503
\(934\) 4.13240e7 1.55001
\(935\) 0 0
\(936\) −5.71958e6 −0.213390
\(937\) 5.84174e6 0.217367 0.108683 0.994076i \(-0.465337\pi\)
0.108683 + 0.994076i \(0.465337\pi\)
\(938\) −2.98204e7 −1.10664
\(939\) −1.20559e6 −0.0446205
\(940\) 0 0
\(941\) −1.24009e7 −0.456540 −0.228270 0.973598i \(-0.573307\pi\)
−0.228270 + 0.973598i \(0.573307\pi\)
\(942\) 1.62086e6 0.0595139
\(943\) −7.07272e6 −0.259004
\(944\) −2.70496e7 −0.987940
\(945\) 0 0
\(946\) −1.07076e7 −0.389013
\(947\) −5.45771e7 −1.97759 −0.988793 0.149293i \(-0.952300\pi\)
−0.988793 + 0.149293i \(0.952300\pi\)
\(948\) 195059. 0.00704928
\(949\) −2.16560e7 −0.780571
\(950\) 0 0
\(951\) −1.52184e7 −0.545655
\(952\) 4.31535e7 1.54321
\(953\) −1.12765e7 −0.402198 −0.201099 0.979571i \(-0.564451\pi\)
−0.201099 + 0.979571i \(0.564451\pi\)
\(954\) 7.17322e6 0.255178
\(955\) 0 0
\(956\) 1.02359e7 0.362227
\(957\) 3.73961e6 0.131992
\(958\) −2.96770e7 −1.04473
\(959\) −5.22456e7 −1.83444
\(960\) 0 0
\(961\) −2.64416e7 −0.923589
\(962\) −9.82071e6 −0.342141
\(963\) 3.04454e6 0.105793
\(964\) 6.30622e6 0.218563
\(965\) 0 0
\(966\) −7.20131e6 −0.248295
\(967\) 2.71697e6 0.0934370 0.0467185 0.998908i \(-0.485124\pi\)
0.0467185 + 0.998908i \(0.485124\pi\)
\(968\) 2.83758e6 0.0973330
\(969\) 7.29249e6 0.249498
\(970\) 0 0
\(971\) −3.01426e6 −0.102597 −0.0512983 0.998683i \(-0.516336\pi\)
−0.0512983 + 0.998683i \(0.516336\pi\)
\(972\) 356364. 0.0120984
\(973\) −7.08888e7 −2.40046
\(974\) −3.98843e7 −1.34711
\(975\) 0 0
\(976\) −3.85744e7 −1.29621
\(977\) 1.75203e7 0.587227 0.293613 0.955924i \(-0.405142\pi\)
0.293613 + 0.955924i \(0.405142\pi\)
\(978\) −2.21213e7 −0.739544
\(979\) 3.26171e6 0.108765
\(980\) 0 0
\(981\) −1.20274e7 −0.399025
\(982\) 3.03229e7 1.00344
\(983\) −2.19475e7 −0.724439 −0.362219 0.932093i \(-0.617981\pi\)
−0.362219 + 0.932093i \(0.617981\pi\)
\(984\) 1.33779e7 0.440452
\(985\) 0 0
\(986\) −2.28810e7 −0.749521
\(987\) 1.06452e7 0.347827
\(988\) 1.36249e6 0.0444059
\(989\) 1.60152e7 0.520646
\(990\) 0 0
\(991\) −4.54381e7 −1.46972 −0.734862 0.678216i \(-0.762753\pi\)
−0.734862 + 0.678216i \(0.762753\pi\)
\(992\) −3.18546e6 −0.102776
\(993\) −2.81960e6 −0.0907433
\(994\) 4.20366e7 1.34946
\(995\) 0 0
\(996\) −3.49304e6 −0.111572
\(997\) 1.38771e7 0.442141 0.221070 0.975258i \(-0.429045\pi\)
0.221070 + 0.975258i \(0.429045\pi\)
\(998\) 9.21907e6 0.292995
\(999\) 3.85634e6 0.122254
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.6.a.v.1.5 13
5.2 odd 4 165.6.c.b.34.8 26
5.3 odd 4 165.6.c.b.34.19 yes 26
5.4 even 2 825.6.a.y.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.8 26 5.2 odd 4
165.6.c.b.34.19 yes 26 5.3 odd 4
825.6.a.v.1.5 13 1.1 even 1 trivial
825.6.a.y.1.9 13 5.4 even 2