Properties

Label 325.6.a.j.1.5
Level $325$
Weight $6$
Character 325.1
Self dual yes
Analytic conductor $52.125$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5 x^{10} - 257 x^{9} + 1165 x^{8} + 22234 x^{7} - 90282 x^{6} - 751180 x^{5} + 2564400 x^{4} + \cdots + 44115200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.89241\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.89241 q^{2} +3.45730 q^{3} -23.6340 q^{4} -9.99993 q^{6} +148.288 q^{7} +160.916 q^{8} -231.047 q^{9} -712.658 q^{11} -81.7096 q^{12} -169.000 q^{13} -428.910 q^{14} +290.850 q^{16} +1131.92 q^{17} +668.283 q^{18} +1400.39 q^{19} +512.676 q^{21} +2061.30 q^{22} -897.571 q^{23} +556.336 q^{24} +488.818 q^{26} -1638.92 q^{27} -3504.63 q^{28} +3238.07 q^{29} -7976.53 q^{31} -5990.58 q^{32} -2463.87 q^{33} -3273.98 q^{34} +5460.56 q^{36} -4978.35 q^{37} -4050.50 q^{38} -584.284 q^{39} -15559.9 q^{41} -1482.87 q^{42} +2308.61 q^{43} +16842.9 q^{44} +2596.15 q^{46} +7602.52 q^{47} +1005.56 q^{48} +5182.35 q^{49} +3913.39 q^{51} +3994.14 q^{52} +14138.1 q^{53} +4740.44 q^{54} +23862.0 q^{56} +4841.56 q^{57} -9365.83 q^{58} +49808.7 q^{59} -2516.69 q^{61} +23071.4 q^{62} -34261.5 q^{63} +8020.03 q^{64} +7126.54 q^{66} +38549.3 q^{67} -26751.8 q^{68} -3103.17 q^{69} +68008.5 q^{71} -37179.2 q^{72} +21305.2 q^{73} +14399.4 q^{74} -33096.7 q^{76} -105679. q^{77} +1689.99 q^{78} +3984.60 q^{79} +50478.2 q^{81} +45005.8 q^{82} +13876.6 q^{83} -12116.6 q^{84} -6677.44 q^{86} +11195.0 q^{87} -114678. q^{88} +89289.8 q^{89} -25060.7 q^{91} +21213.1 q^{92} -27577.2 q^{93} -21989.6 q^{94} -20711.2 q^{96} +147549. q^{97} -14989.5 q^{98} +164658. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 5 q^{2} - 11 q^{3} + 187 q^{4} + 351 q^{6} - 208 q^{7} - 165 q^{8} + 1372 q^{9} + 1276 q^{11} - 1533 q^{12} - 1859 q^{13} + 578 q^{14} + 5707 q^{16} - 2218 q^{17} + 6776 q^{18} + 3520 q^{19} + 1706 q^{21}+ \cdots + 426698 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.89241 −0.511311 −0.255656 0.966768i \(-0.582291\pi\)
−0.255656 + 0.966768i \(0.582291\pi\)
\(3\) 3.45730 0.221786 0.110893 0.993832i \(-0.464629\pi\)
0.110893 + 0.993832i \(0.464629\pi\)
\(4\) −23.6340 −0.738561
\(5\) 0 0
\(6\) −9.99993 −0.113402
\(7\) 148.288 1.14383 0.571914 0.820313i \(-0.306201\pi\)
0.571914 + 0.820313i \(0.306201\pi\)
\(8\) 160.916 0.888945
\(9\) −231.047 −0.950811
\(10\) 0 0
\(11\) −712.658 −1.77582 −0.887912 0.460014i \(-0.847844\pi\)
−0.887912 + 0.460014i \(0.847844\pi\)
\(12\) −81.7096 −0.163802
\(13\) −169.000 −0.277350
\(14\) −428.910 −0.584852
\(15\) 0 0
\(16\) 290.850 0.284033
\(17\) 1131.92 0.949935 0.474968 0.880003i \(-0.342460\pi\)
0.474968 + 0.880003i \(0.342460\pi\)
\(18\) 668.283 0.486160
\(19\) 1400.39 0.889947 0.444974 0.895544i \(-0.353213\pi\)
0.444974 + 0.895544i \(0.353213\pi\)
\(20\) 0 0
\(21\) 512.676 0.253685
\(22\) 2061.30 0.907998
\(23\) −897.571 −0.353793 −0.176896 0.984229i \(-0.556606\pi\)
−0.176896 + 0.984229i \(0.556606\pi\)
\(24\) 556.336 0.197156
\(25\) 0 0
\(26\) 488.818 0.141812
\(27\) −1638.92 −0.432662
\(28\) −3504.63 −0.844787
\(29\) 3238.07 0.714975 0.357488 0.933918i \(-0.383633\pi\)
0.357488 + 0.933918i \(0.383633\pi\)
\(30\) 0 0
\(31\) −7976.53 −1.49077 −0.745383 0.666636i \(-0.767733\pi\)
−0.745383 + 0.666636i \(0.767733\pi\)
\(32\) −5990.58 −1.03417
\(33\) −2463.87 −0.393852
\(34\) −3273.98 −0.485712
\(35\) 0 0
\(36\) 5460.56 0.702232
\(37\) −4978.35 −0.597835 −0.298917 0.954279i \(-0.596626\pi\)
−0.298917 + 0.954279i \(0.596626\pi\)
\(38\) −4050.50 −0.455040
\(39\) −584.284 −0.0615123
\(40\) 0 0
\(41\) −15559.9 −1.44560 −0.722800 0.691057i \(-0.757145\pi\)
−0.722800 + 0.691057i \(0.757145\pi\)
\(42\) −1482.87 −0.129712
\(43\) 2308.61 0.190405 0.0952026 0.995458i \(-0.469650\pi\)
0.0952026 + 0.995458i \(0.469650\pi\)
\(44\) 16842.9 1.31155
\(45\) 0 0
\(46\) 2596.15 0.180898
\(47\) 7602.52 0.502011 0.251005 0.967986i \(-0.419239\pi\)
0.251005 + 0.967986i \(0.419239\pi\)
\(48\) 1005.56 0.0629946
\(49\) 5182.35 0.308344
\(50\) 0 0
\(51\) 3913.39 0.210682
\(52\) 3994.14 0.204840
\(53\) 14138.1 0.691357 0.345678 0.938353i \(-0.387649\pi\)
0.345678 + 0.938353i \(0.387649\pi\)
\(54\) 4740.44 0.221225
\(55\) 0 0
\(56\) 23862.0 1.01680
\(57\) 4841.56 0.197378
\(58\) −9365.83 −0.365575
\(59\) 49808.7 1.86284 0.931419 0.363948i \(-0.118572\pi\)
0.931419 + 0.363948i \(0.118572\pi\)
\(60\) 0 0
\(61\) −2516.69 −0.0865973 −0.0432986 0.999062i \(-0.513787\pi\)
−0.0432986 + 0.999062i \(0.513787\pi\)
\(62\) 23071.4 0.762245
\(63\) −34261.5 −1.08757
\(64\) 8020.03 0.244752
\(65\) 0 0
\(66\) 7126.54 0.201381
\(67\) 38549.3 1.04913 0.524566 0.851370i \(-0.324228\pi\)
0.524566 + 0.851370i \(0.324228\pi\)
\(68\) −26751.8 −0.701585
\(69\) −3103.17 −0.0784663
\(70\) 0 0
\(71\) 68008.5 1.60110 0.800548 0.599269i \(-0.204542\pi\)
0.800548 + 0.599269i \(0.204542\pi\)
\(72\) −37179.2 −0.845219
\(73\) 21305.2 0.467927 0.233963 0.972245i \(-0.424830\pi\)
0.233963 + 0.972245i \(0.424830\pi\)
\(74\) 14399.4 0.305680
\(75\) 0 0
\(76\) −33096.7 −0.657280
\(77\) −105679. −2.03124
\(78\) 1689.99 0.0314519
\(79\) 3984.60 0.0718318 0.0359159 0.999355i \(-0.488565\pi\)
0.0359159 + 0.999355i \(0.488565\pi\)
\(80\) 0 0
\(81\) 50478.2 0.854853
\(82\) 45005.8 0.739152
\(83\) 13876.6 0.221100 0.110550 0.993871i \(-0.464739\pi\)
0.110550 + 0.993871i \(0.464739\pi\)
\(84\) −12116.6 −0.187362
\(85\) 0 0
\(86\) −6677.44 −0.0973563
\(87\) 11195.0 0.158571
\(88\) −114678. −1.57861
\(89\) 89289.8 1.19489 0.597443 0.801911i \(-0.296183\pi\)
0.597443 + 0.801911i \(0.296183\pi\)
\(90\) 0 0
\(91\) −25060.7 −0.317241
\(92\) 21213.1 0.261298
\(93\) −27577.2 −0.330631
\(94\) −21989.6 −0.256684
\(95\) 0 0
\(96\) −20711.2 −0.229365
\(97\) 147549. 1.59223 0.796116 0.605145i \(-0.206885\pi\)
0.796116 + 0.605145i \(0.206885\pi\)
\(98\) −14989.5 −0.157660
\(99\) 164658. 1.68847
\(100\) 0 0
\(101\) −10084.2 −0.0983642 −0.0491821 0.998790i \(-0.515661\pi\)
−0.0491821 + 0.998790i \(0.515661\pi\)
\(102\) −11319.1 −0.107724
\(103\) −175080. −1.62608 −0.813041 0.582206i \(-0.802190\pi\)
−0.813041 + 0.582206i \(0.802190\pi\)
\(104\) −27194.9 −0.246549
\(105\) 0 0
\(106\) −40893.3 −0.353498
\(107\) 225117. 1.90086 0.950428 0.310946i \(-0.100646\pi\)
0.950428 + 0.310946i \(0.100646\pi\)
\(108\) 38734.2 0.319547
\(109\) 25083.5 0.202219 0.101109 0.994875i \(-0.467761\pi\)
0.101109 + 0.994875i \(0.467761\pi\)
\(110\) 0 0
\(111\) −17211.6 −0.132591
\(112\) 43129.6 0.324885
\(113\) 39258.9 0.289229 0.144615 0.989488i \(-0.453806\pi\)
0.144615 + 0.989488i \(0.453806\pi\)
\(114\) −14003.8 −0.100921
\(115\) 0 0
\(116\) −76528.3 −0.528053
\(117\) 39047.0 0.263708
\(118\) −144067. −0.952490
\(119\) 167850. 1.08656
\(120\) 0 0
\(121\) 346831. 2.15355
\(122\) 7279.29 0.0442782
\(123\) −53795.4 −0.320614
\(124\) 188517. 1.10102
\(125\) 0 0
\(126\) 99098.4 0.556084
\(127\) 267258. 1.47035 0.735177 0.677875i \(-0.237099\pi\)
0.735177 + 0.677875i \(0.237099\pi\)
\(128\) 168501. 0.909031
\(129\) 7981.55 0.0422292
\(130\) 0 0
\(131\) −157114. −0.799903 −0.399952 0.916536i \(-0.630973\pi\)
−0.399952 + 0.916536i \(0.630973\pi\)
\(132\) 58231.1 0.290884
\(133\) 207661. 1.01795
\(134\) −111501. −0.536433
\(135\) 0 0
\(136\) 182145. 0.844441
\(137\) −132383. −0.602603 −0.301302 0.953529i \(-0.597421\pi\)
−0.301302 + 0.953529i \(0.597421\pi\)
\(138\) 8975.65 0.0401207
\(139\) −285040. −1.25132 −0.625661 0.780095i \(-0.715171\pi\)
−0.625661 + 0.780095i \(0.715171\pi\)
\(140\) 0 0
\(141\) 26284.2 0.111339
\(142\) −196709. −0.818658
\(143\) 120439. 0.492525
\(144\) −67200.1 −0.270062
\(145\) 0 0
\(146\) −61623.4 −0.239256
\(147\) 17916.9 0.0683864
\(148\) 117658. 0.441537
\(149\) −156686. −0.578180 −0.289090 0.957302i \(-0.593353\pi\)
−0.289090 + 0.957302i \(0.593353\pi\)
\(150\) 0 0
\(151\) 422557. 1.50814 0.754072 0.656792i \(-0.228087\pi\)
0.754072 + 0.656792i \(0.228087\pi\)
\(152\) 225345. 0.791115
\(153\) −261527. −0.903209
\(154\) 305666. 1.03859
\(155\) 0 0
\(156\) 13808.9 0.0454306
\(157\) 129540. 0.419427 0.209713 0.977763i \(-0.432747\pi\)
0.209713 + 0.977763i \(0.432747\pi\)
\(158\) −11525.1 −0.0367284
\(159\) 48879.7 0.153333
\(160\) 0 0
\(161\) −133099. −0.404679
\(162\) −146004. −0.437096
\(163\) −616763. −1.81823 −0.909116 0.416542i \(-0.863242\pi\)
−0.909116 + 0.416542i \(0.863242\pi\)
\(164\) 367743. 1.06766
\(165\) 0 0
\(166\) −40136.9 −0.113051
\(167\) −512670. −1.42248 −0.711241 0.702948i \(-0.751867\pi\)
−0.711241 + 0.702948i \(0.751867\pi\)
\(168\) 82498.0 0.225512
\(169\) 28561.0 0.0769231
\(170\) 0 0
\(171\) −323556. −0.846172
\(172\) −54561.5 −0.140626
\(173\) 462817. 1.17569 0.587846 0.808973i \(-0.299976\pi\)
0.587846 + 0.808973i \(0.299976\pi\)
\(174\) −32380.5 −0.0810793
\(175\) 0 0
\(176\) −207277. −0.504393
\(177\) 172204. 0.413151
\(178\) −258263. −0.610959
\(179\) 413926. 0.965584 0.482792 0.875735i \(-0.339623\pi\)
0.482792 + 0.875735i \(0.339623\pi\)
\(180\) 0 0
\(181\) −167089. −0.379099 −0.189549 0.981871i \(-0.560703\pi\)
−0.189549 + 0.981871i \(0.560703\pi\)
\(182\) 72485.8 0.162209
\(183\) −8700.93 −0.0192061
\(184\) −144434. −0.314503
\(185\) 0 0
\(186\) 79764.7 0.169055
\(187\) −806673. −1.68692
\(188\) −179678. −0.370766
\(189\) −243033. −0.494892
\(190\) 0 0
\(191\) 30723.8 0.0609385 0.0304692 0.999536i \(-0.490300\pi\)
0.0304692 + 0.999536i \(0.490300\pi\)
\(192\) 27727.6 0.0542825
\(193\) 1.01259e6 1.95677 0.978387 0.206784i \(-0.0662996\pi\)
0.978387 + 0.206784i \(0.0662996\pi\)
\(194\) −426772. −0.814125
\(195\) 0 0
\(196\) −122479. −0.227731
\(197\) 422275. 0.775229 0.387614 0.921822i \(-0.373299\pi\)
0.387614 + 0.921822i \(0.373299\pi\)
\(198\) −476258. −0.863335
\(199\) 553506. 0.990808 0.495404 0.868663i \(-0.335020\pi\)
0.495404 + 0.868663i \(0.335020\pi\)
\(200\) 0 0
\(201\) 133277. 0.232683
\(202\) 29167.6 0.0502947
\(203\) 480167. 0.817809
\(204\) −92488.9 −0.155602
\(205\) 0 0
\(206\) 506402. 0.831434
\(207\) 207381. 0.336390
\(208\) −49153.7 −0.0787767
\(209\) −997998. −1.58039
\(210\) 0 0
\(211\) −758171. −1.17236 −0.586180 0.810181i \(-0.699369\pi\)
−0.586180 + 0.810181i \(0.699369\pi\)
\(212\) −334140. −0.510609
\(213\) 235126. 0.355100
\(214\) −651132. −0.971928
\(215\) 0 0
\(216\) −263729. −0.384613
\(217\) −1.18282e6 −1.70518
\(218\) −72551.8 −0.103397
\(219\) 73658.4 0.103780
\(220\) 0 0
\(221\) −191295. −0.263465
\(222\) 49783.2 0.0677954
\(223\) 142765. 0.192247 0.0961236 0.995369i \(-0.469356\pi\)
0.0961236 + 0.995369i \(0.469356\pi\)
\(224\) −888332. −1.18292
\(225\) 0 0
\(226\) −113553. −0.147886
\(227\) −578969. −0.745746 −0.372873 0.927882i \(-0.621627\pi\)
−0.372873 + 0.927882i \(0.621627\pi\)
\(228\) −114425. −0.145775
\(229\) −242404. −0.305457 −0.152729 0.988268i \(-0.548806\pi\)
−0.152729 + 0.988268i \(0.548806\pi\)
\(230\) 0 0
\(231\) −365363. −0.450500
\(232\) 521058. 0.635574
\(233\) 540278. 0.651970 0.325985 0.945375i \(-0.394304\pi\)
0.325985 + 0.945375i \(0.394304\pi\)
\(234\) −112940. −0.134837
\(235\) 0 0
\(236\) −1.17718e6 −1.37582
\(237\) 13776.0 0.0159313
\(238\) −485493. −0.555572
\(239\) −223034. −0.252567 −0.126284 0.991994i \(-0.540305\pi\)
−0.126284 + 0.991994i \(0.540305\pi\)
\(240\) 0 0
\(241\) 1.31596e6 1.45948 0.729742 0.683723i \(-0.239640\pi\)
0.729742 + 0.683723i \(0.239640\pi\)
\(242\) −1.00318e6 −1.10113
\(243\) 572776. 0.622256
\(244\) 59479.2 0.0639574
\(245\) 0 0
\(246\) 155598. 0.163933
\(247\) −236666. −0.246827
\(248\) −1.28355e6 −1.32521
\(249\) 47975.6 0.0490368
\(250\) 0 0
\(251\) 475886. 0.476781 0.238390 0.971169i \(-0.423380\pi\)
0.238390 + 0.971169i \(0.423380\pi\)
\(252\) 809735. 0.803233
\(253\) 639661. 0.628274
\(254\) −773021. −0.751808
\(255\) 0 0
\(256\) −744016. −0.709549
\(257\) 184678. 0.174415 0.0872074 0.996190i \(-0.472206\pi\)
0.0872074 + 0.996190i \(0.472206\pi\)
\(258\) −23085.9 −0.0215923
\(259\) −738230. −0.683821
\(260\) 0 0
\(261\) −748146. −0.679806
\(262\) 454439. 0.408999
\(263\) −541222. −0.482487 −0.241244 0.970465i \(-0.577555\pi\)
−0.241244 + 0.970465i \(0.577555\pi\)
\(264\) −396477. −0.350113
\(265\) 0 0
\(266\) −600641. −0.520488
\(267\) 308702. 0.265009
\(268\) −911073. −0.774848
\(269\) 198419. 0.167187 0.0835934 0.996500i \(-0.473360\pi\)
0.0835934 + 0.996500i \(0.473360\pi\)
\(270\) 0 0
\(271\) −258403. −0.213734 −0.106867 0.994273i \(-0.534082\pi\)
−0.106867 + 0.994273i \(0.534082\pi\)
\(272\) 329219. 0.269813
\(273\) −86642.3 −0.0703596
\(274\) 382907. 0.308118
\(275\) 0 0
\(276\) 73340.2 0.0579521
\(277\) −2.19835e6 −1.72146 −0.860729 0.509063i \(-0.829992\pi\)
−0.860729 + 0.509063i \(0.829992\pi\)
\(278\) 824454. 0.639815
\(279\) 1.84295e6 1.41744
\(280\) 0 0
\(281\) −842.387 −0.000636423 0 −0.000318211 1.00000i \(-0.500101\pi\)
−0.000318211 1.00000i \(0.500101\pi\)
\(282\) −76024.7 −0.0569288
\(283\) 1.93205e6 1.43401 0.717006 0.697067i \(-0.245512\pi\)
0.717006 + 0.697067i \(0.245512\pi\)
\(284\) −1.60731e6 −1.18251
\(285\) 0 0
\(286\) −348360. −0.251833
\(287\) −2.30735e6 −1.65352
\(288\) 1.38411e6 0.983305
\(289\) −138611. −0.0976230
\(290\) 0 0
\(291\) 510120. 0.353134
\(292\) −503526. −0.345593
\(293\) 1.01321e6 0.689493 0.344747 0.938696i \(-0.387965\pi\)
0.344747 + 0.938696i \(0.387965\pi\)
\(294\) −51823.1 −0.0349667
\(295\) 0 0
\(296\) −801098. −0.531443
\(297\) 1.16799e6 0.768332
\(298\) 453199. 0.295630
\(299\) 151689. 0.0981245
\(300\) 0 0
\(301\) 342339. 0.217791
\(302\) −1.22221e6 −0.771131
\(303\) −34864.0 −0.0218158
\(304\) 407303. 0.252775
\(305\) 0 0
\(306\) 756444. 0.461821
\(307\) 386274. 0.233910 0.116955 0.993137i \(-0.462687\pi\)
0.116955 + 0.993137i \(0.462687\pi\)
\(308\) 2.49761e6 1.50019
\(309\) −605303. −0.360642
\(310\) 0 0
\(311\) 1.37666e6 0.807099 0.403550 0.914958i \(-0.367776\pi\)
0.403550 + 0.914958i \(0.367776\pi\)
\(312\) −94020.8 −0.0546811
\(313\) −541373. −0.312346 −0.156173 0.987730i \(-0.549916\pi\)
−0.156173 + 0.987730i \(0.549916\pi\)
\(314\) −374684. −0.214458
\(315\) 0 0
\(316\) −94171.8 −0.0530522
\(317\) 1.91853e6 1.07231 0.536154 0.844120i \(-0.319876\pi\)
0.536154 + 0.844120i \(0.319876\pi\)
\(318\) −141380. −0.0784009
\(319\) −2.30764e6 −1.26967
\(320\) 0 0
\(321\) 778297. 0.421583
\(322\) 384977. 0.206917
\(323\) 1.58513e6 0.845392
\(324\) −1.19300e6 −0.631361
\(325\) 0 0
\(326\) 1.78393e6 0.929683
\(327\) 86721.2 0.0448493
\(328\) −2.50385e6 −1.28506
\(329\) 1.12736e6 0.574215
\(330\) 0 0
\(331\) −2.40144e6 −1.20477 −0.602383 0.798207i \(-0.705782\pi\)
−0.602383 + 0.798207i \(0.705782\pi\)
\(332\) −327959. −0.163296
\(333\) 1.15023e6 0.568428
\(334\) 1.48285e6 0.727331
\(335\) 0 0
\(336\) 149112. 0.0720550
\(337\) 3.84325e6 1.84342 0.921709 0.387882i \(-0.126793\pi\)
0.921709 + 0.387882i \(0.126793\pi\)
\(338\) −82610.2 −0.0393316
\(339\) 135730. 0.0641469
\(340\) 0 0
\(341\) 5.68454e6 2.64734
\(342\) 935856. 0.432657
\(343\) −1.72380e6 −0.791136
\(344\) 371493. 0.169260
\(345\) 0 0
\(346\) −1.33866e6 −0.601144
\(347\) −1.02696e6 −0.457858 −0.228929 0.973443i \(-0.573522\pi\)
−0.228929 + 0.973443i \(0.573522\pi\)
\(348\) −264581. −0.117115
\(349\) 1.24908e6 0.548944 0.274472 0.961595i \(-0.411497\pi\)
0.274472 + 0.961595i \(0.411497\pi\)
\(350\) 0 0
\(351\) 276978. 0.119999
\(352\) 4.26924e6 1.83651
\(353\) −3.13090e6 −1.33731 −0.668654 0.743573i \(-0.733129\pi\)
−0.668654 + 0.743573i \(0.733129\pi\)
\(354\) −498084. −0.211249
\(355\) 0 0
\(356\) −2.11027e6 −0.882497
\(357\) 580309. 0.240984
\(358\) −1.19724e6 −0.493714
\(359\) −2.63535e6 −1.07920 −0.539600 0.841921i \(-0.681425\pi\)
−0.539600 + 0.841921i \(0.681425\pi\)
\(360\) 0 0
\(361\) −515013. −0.207994
\(362\) 483291. 0.193837
\(363\) 1.19910e6 0.477626
\(364\) 592283. 0.234302
\(365\) 0 0
\(366\) 25166.7 0.00982027
\(367\) −2.23239e6 −0.865175 −0.432588 0.901592i \(-0.642399\pi\)
−0.432588 + 0.901592i \(0.642399\pi\)
\(368\) −261059. −0.100489
\(369\) 3.59508e6 1.37449
\(370\) 0 0
\(371\) 2.09651e6 0.790794
\(372\) 651759. 0.244191
\(373\) −3.72332e6 −1.38566 −0.692832 0.721099i \(-0.743637\pi\)
−0.692832 + 0.721099i \(0.743637\pi\)
\(374\) 2.33323e6 0.862539
\(375\) 0 0
\(376\) 1.22337e6 0.446260
\(377\) −547233. −0.198298
\(378\) 702951. 0.253044
\(379\) −2.11153e6 −0.755089 −0.377545 0.925991i \(-0.623232\pi\)
−0.377545 + 0.925991i \(0.623232\pi\)
\(380\) 0 0
\(381\) 923992. 0.326104
\(382\) −88865.9 −0.0311585
\(383\) −414336. −0.144330 −0.0721648 0.997393i \(-0.522991\pi\)
−0.0721648 + 0.997393i \(0.522991\pi\)
\(384\) 582560. 0.201610
\(385\) 0 0
\(386\) −2.92883e6 −1.00052
\(387\) −533397. −0.181039
\(388\) −3.48716e6 −1.17596
\(389\) 4.10572e6 1.37567 0.687837 0.725865i \(-0.258560\pi\)
0.687837 + 0.725865i \(0.258560\pi\)
\(390\) 0 0
\(391\) −1.01598e6 −0.336080
\(392\) 833924. 0.274101
\(393\) −543191. −0.177407
\(394\) −1.22139e6 −0.396383
\(395\) 0 0
\(396\) −3.89151e6 −1.24704
\(397\) −5.78249e6 −1.84136 −0.920681 0.390317i \(-0.872365\pi\)
−0.920681 + 0.390317i \(0.872365\pi\)
\(398\) −1.60097e6 −0.506611
\(399\) 717945. 0.225766
\(400\) 0 0
\(401\) −3.48722e6 −1.08298 −0.541488 0.840708i \(-0.682139\pi\)
−0.541488 + 0.840708i \(0.682139\pi\)
\(402\) −385491. −0.118973
\(403\) 1.34803e6 0.413464
\(404\) 238329. 0.0726479
\(405\) 0 0
\(406\) −1.38884e6 −0.418155
\(407\) 3.54786e6 1.06165
\(408\) 629729. 0.187285
\(409\) −3.23381e6 −0.955886 −0.477943 0.878391i \(-0.658618\pi\)
−0.477943 + 0.878391i \(0.658618\pi\)
\(410\) 0 0
\(411\) −457688. −0.133649
\(412\) 4.13782e6 1.20096
\(413\) 7.38603e6 2.13077
\(414\) −599832. −0.172000
\(415\) 0 0
\(416\) 1.01241e6 0.286829
\(417\) −985470. −0.277526
\(418\) 2.88662e6 0.808070
\(419\) 3.63720e6 1.01212 0.506060 0.862498i \(-0.331101\pi\)
0.506060 + 0.862498i \(0.331101\pi\)
\(420\) 0 0
\(421\) 2.11596e6 0.581839 0.290920 0.956747i \(-0.406039\pi\)
0.290920 + 0.956747i \(0.406039\pi\)
\(422\) 2.19294e6 0.599441
\(423\) −1.75654e6 −0.477318
\(424\) 2.27505e6 0.614578
\(425\) 0 0
\(426\) −680081. −0.181567
\(427\) −373194. −0.0990525
\(428\) −5.32041e6 −1.40390
\(429\) 416395. 0.109235
\(430\) 0 0
\(431\) 6.14425e6 1.59322 0.796609 0.604495i \(-0.206625\pi\)
0.796609 + 0.604495i \(0.206625\pi\)
\(432\) −476681. −0.122890
\(433\) −6.08667e6 −1.56013 −0.780064 0.625700i \(-0.784813\pi\)
−0.780064 + 0.625700i \(0.784813\pi\)
\(434\) 3.42121e6 0.871878
\(435\) 0 0
\(436\) −592822. −0.149351
\(437\) −1.25695e6 −0.314857
\(438\) −213050. −0.0530636
\(439\) 5.46357e6 1.35305 0.676527 0.736418i \(-0.263484\pi\)
0.676527 + 0.736418i \(0.263484\pi\)
\(440\) 0 0
\(441\) −1.19737e6 −0.293177
\(442\) 553303. 0.134712
\(443\) −4.25637e6 −1.03046 −0.515229 0.857052i \(-0.672293\pi\)
−0.515229 + 0.857052i \(0.672293\pi\)
\(444\) 406779. 0.0979268
\(445\) 0 0
\(446\) −412935. −0.0982981
\(447\) −541709. −0.128232
\(448\) 1.18927e6 0.279954
\(449\) 3.07736e6 0.720382 0.360191 0.932879i \(-0.382712\pi\)
0.360191 + 0.932879i \(0.382712\pi\)
\(450\) 0 0
\(451\) 1.10889e7 2.56713
\(452\) −927843. −0.213613
\(453\) 1.46091e6 0.334485
\(454\) 1.67462e6 0.381308
\(455\) 0 0
\(456\) 779086. 0.175458
\(457\) −1.92830e6 −0.431901 −0.215951 0.976404i \(-0.569285\pi\)
−0.215951 + 0.976404i \(0.569285\pi\)
\(458\) 701132. 0.156184
\(459\) −1.85513e6 −0.411001
\(460\) 0 0
\(461\) 1.53125e6 0.335578 0.167789 0.985823i \(-0.446337\pi\)
0.167789 + 0.985823i \(0.446337\pi\)
\(462\) 1.05678e6 0.230346
\(463\) −2.13296e6 −0.462412 −0.231206 0.972905i \(-0.574267\pi\)
−0.231206 + 0.972905i \(0.574267\pi\)
\(464\) 941792. 0.203077
\(465\) 0 0
\(466\) −1.56271e6 −0.333359
\(467\) −2.47557e6 −0.525270 −0.262635 0.964895i \(-0.584592\pi\)
−0.262635 + 0.964895i \(0.584592\pi\)
\(468\) −922834. −0.194764
\(469\) 5.71641e6 1.20003
\(470\) 0 0
\(471\) 447860. 0.0930230
\(472\) 8.01503e6 1.65596
\(473\) −1.64525e6 −0.338126
\(474\) −39845.7 −0.00814584
\(475\) 0 0
\(476\) −3.96697e6 −0.802493
\(477\) −3.26657e6 −0.657349
\(478\) 645108. 0.129140
\(479\) 6.52504e6 1.29940 0.649702 0.760189i \(-0.274894\pi\)
0.649702 + 0.760189i \(0.274894\pi\)
\(480\) 0 0
\(481\) 841341. 0.165810
\(482\) −3.80629e6 −0.746250
\(483\) −460163. −0.0897520
\(484\) −8.19699e6 −1.59053
\(485\) 0 0
\(486\) −1.65671e6 −0.318167
\(487\) 5.50061e6 1.05097 0.525483 0.850804i \(-0.323885\pi\)
0.525483 + 0.850804i \(0.323885\pi\)
\(488\) −404976. −0.0769803
\(489\) −2.13233e6 −0.403258
\(490\) 0 0
\(491\) 8.54562e6 1.59970 0.799852 0.600197i \(-0.204911\pi\)
0.799852 + 0.600197i \(0.204911\pi\)
\(492\) 1.27140e6 0.236793
\(493\) 3.66524e6 0.679180
\(494\) 684534. 0.126205
\(495\) 0 0
\(496\) −2.31997e6 −0.423427
\(497\) 1.00848e7 1.83138
\(498\) −138765. −0.0250730
\(499\) 2.98189e6 0.536094 0.268047 0.963406i \(-0.413622\pi\)
0.268047 + 0.963406i \(0.413622\pi\)
\(500\) 0 0
\(501\) −1.77245e6 −0.315486
\(502\) −1.37646e6 −0.243783
\(503\) 4.24939e6 0.748871 0.374436 0.927253i \(-0.377836\pi\)
0.374436 + 0.927253i \(0.377836\pi\)
\(504\) −5.51324e6 −0.966786
\(505\) 0 0
\(506\) −1.85016e6 −0.321243
\(507\) 98743.9 0.0170604
\(508\) −6.31637e6 −1.08595
\(509\) −1.22747e6 −0.209999 −0.105000 0.994472i \(-0.533484\pi\)
−0.105000 + 0.994472i \(0.533484\pi\)
\(510\) 0 0
\(511\) 3.15930e6 0.535228
\(512\) −3.24004e6 −0.546230
\(513\) −2.29513e6 −0.385047
\(514\) −534166. −0.0891802
\(515\) 0 0
\(516\) −188635. −0.0311888
\(517\) −5.41800e6 −0.891483
\(518\) 2.13527e6 0.349645
\(519\) 1.60010e6 0.260752
\(520\) 0 0
\(521\) −7.91981e6 −1.27826 −0.639131 0.769097i \(-0.720706\pi\)
−0.639131 + 0.769097i \(0.720706\pi\)
\(522\) 2.16395e6 0.347592
\(523\) 4.76548e6 0.761821 0.380910 0.924612i \(-0.375611\pi\)
0.380910 + 0.924612i \(0.375611\pi\)
\(524\) 3.71323e6 0.590777
\(525\) 0 0
\(526\) 1.56544e6 0.246701
\(527\) −9.02880e6 −1.41613
\(528\) −716618. −0.111867
\(529\) −5.63071e6 −0.874831
\(530\) 0 0
\(531\) −1.15082e7 −1.77121
\(532\) −4.90784e6 −0.751816
\(533\) 2.62963e6 0.400938
\(534\) −892892. −0.135502
\(535\) 0 0
\(536\) 6.20322e6 0.932621
\(537\) 1.43107e6 0.214153
\(538\) −573909. −0.0854845
\(539\) −3.69324e6 −0.547565
\(540\) 0 0
\(541\) −5.85407e6 −0.859934 −0.429967 0.902845i \(-0.641475\pi\)
−0.429967 + 0.902845i \(0.641475\pi\)
\(542\) 747408. 0.109285
\(543\) −577678. −0.0840787
\(544\) −6.78087e6 −0.982399
\(545\) 0 0
\(546\) 250605. 0.0359756
\(547\) 587042. 0.0838882 0.0419441 0.999120i \(-0.486645\pi\)
0.0419441 + 0.999120i \(0.486645\pi\)
\(548\) 3.12874e6 0.445059
\(549\) 581473. 0.0823377
\(550\) 0 0
\(551\) 4.53455e6 0.636290
\(552\) −499351. −0.0697522
\(553\) 590868. 0.0821633
\(554\) 6.35852e6 0.880201
\(555\) 0 0
\(556\) 6.73663e6 0.924178
\(557\) −3.23702e6 −0.442086 −0.221043 0.975264i \(-0.570946\pi\)
−0.221043 + 0.975264i \(0.570946\pi\)
\(558\) −5.33058e6 −0.724751
\(559\) −390155. −0.0528089
\(560\) 0 0
\(561\) −2.78891e6 −0.374134
\(562\) 2436.53 0.000325410 0
\(563\) 1.09630e7 1.45767 0.728835 0.684689i \(-0.240062\pi\)
0.728835 + 0.684689i \(0.240062\pi\)
\(564\) −621200. −0.0822306
\(565\) 0 0
\(566\) −5.58829e6 −0.733226
\(567\) 7.48531e6 0.977805
\(568\) 1.09437e7 1.42329
\(569\) 4.35413e6 0.563795 0.281897 0.959445i \(-0.409036\pi\)
0.281897 + 0.959445i \(0.409036\pi\)
\(570\) 0 0
\(571\) 4.59961e6 0.590379 0.295189 0.955439i \(-0.404617\pi\)
0.295189 + 0.955439i \(0.404617\pi\)
\(572\) −2.84646e6 −0.363760
\(573\) 106221. 0.0135153
\(574\) 6.67382e6 0.845463
\(575\) 0 0
\(576\) −1.85300e6 −0.232713
\(577\) −7.57061e6 −0.946654 −0.473327 0.880887i \(-0.656947\pi\)
−0.473327 + 0.880887i \(0.656947\pi\)
\(578\) 400919. 0.0499157
\(579\) 3.50083e6 0.433985
\(580\) 0 0
\(581\) 2.05773e6 0.252900
\(582\) −1.47548e6 −0.180561
\(583\) −1.00757e7 −1.22773
\(584\) 3.42835e6 0.415962
\(585\) 0 0
\(586\) −2.93062e6 −0.352546
\(587\) 1.48215e7 1.77541 0.887703 0.460416i \(-0.152300\pi\)
0.887703 + 0.460416i \(0.152300\pi\)
\(588\) −423448. −0.0505076
\(589\) −1.11702e7 −1.32670
\(590\) 0 0
\(591\) 1.45993e6 0.171935
\(592\) −1.44795e6 −0.169805
\(593\) 1.00021e6 0.116803 0.0584017 0.998293i \(-0.481400\pi\)
0.0584017 + 0.998293i \(0.481400\pi\)
\(594\) −3.37831e6 −0.392856
\(595\) 0 0
\(596\) 3.70310e6 0.427021
\(597\) 1.91363e6 0.219747
\(598\) −438749. −0.0501721
\(599\) −8.35745e6 −0.951715 −0.475857 0.879522i \(-0.657862\pi\)
−0.475857 + 0.879522i \(0.657862\pi\)
\(600\) 0 0
\(601\) −6.04104e6 −0.682222 −0.341111 0.940023i \(-0.610803\pi\)
−0.341111 + 0.940023i \(0.610803\pi\)
\(602\) −990185. −0.111359
\(603\) −8.90671e6 −0.997526
\(604\) −9.98669e6 −1.11386
\(605\) 0 0
\(606\) 100841. 0.0111547
\(607\) −137224. −0.0151168 −0.00755838 0.999971i \(-0.502406\pi\)
−0.00755838 + 0.999971i \(0.502406\pi\)
\(608\) −8.38914e6 −0.920361
\(609\) 1.66008e6 0.181378
\(610\) 0 0
\(611\) −1.28483e6 −0.139233
\(612\) 6.18092e6 0.667075
\(613\) 1.09895e7 1.18121 0.590606 0.806960i \(-0.298889\pi\)
0.590606 + 0.806960i \(0.298889\pi\)
\(614\) −1.11726e6 −0.119601
\(615\) 0 0
\(616\) −1.70054e7 −1.80566
\(617\) 3.42268e6 0.361954 0.180977 0.983487i \(-0.442074\pi\)
0.180977 + 0.983487i \(0.442074\pi\)
\(618\) 1.75078e6 0.184400
\(619\) −1.22543e7 −1.28547 −0.642736 0.766088i \(-0.722201\pi\)
−0.642736 + 0.766088i \(0.722201\pi\)
\(620\) 0 0
\(621\) 1.47105e6 0.153073
\(622\) −3.98188e6 −0.412679
\(623\) 1.32406e7 1.36675
\(624\) −169939. −0.0174715
\(625\) 0 0
\(626\) 1.56587e6 0.159706
\(627\) −3.45038e6 −0.350508
\(628\) −3.06155e6 −0.309772
\(629\) −5.63510e6 −0.567904
\(630\) 0 0
\(631\) 7.99788e6 0.799653 0.399827 0.916591i \(-0.369070\pi\)
0.399827 + 0.916591i \(0.369070\pi\)
\(632\) 641187. 0.0638546
\(633\) −2.62122e6 −0.260013
\(634\) −5.54917e6 −0.548283
\(635\) 0 0
\(636\) −1.15522e6 −0.113246
\(637\) −875816. −0.0855194
\(638\) 6.67463e6 0.649196
\(639\) −1.57132e7 −1.52234
\(640\) 0 0
\(641\) 8.18872e6 0.787175 0.393587 0.919287i \(-0.371234\pi\)
0.393587 + 0.919287i \(0.371234\pi\)
\(642\) −2.25116e6 −0.215560
\(643\) −1.64743e7 −1.57137 −0.785686 0.618626i \(-0.787690\pi\)
−0.785686 + 0.618626i \(0.787690\pi\)
\(644\) 3.14566e6 0.298880
\(645\) 0 0
\(646\) −4.58485e6 −0.432258
\(647\) −3.65606e6 −0.343362 −0.171681 0.985153i \(-0.554920\pi\)
−0.171681 + 0.985153i \(0.554920\pi\)
\(648\) 8.12277e6 0.759917
\(649\) −3.54966e7 −3.30807
\(650\) 0 0
\(651\) −4.08938e6 −0.378185
\(652\) 1.45766e7 1.34288
\(653\) 1.22553e7 1.12471 0.562355 0.826896i \(-0.309896\pi\)
0.562355 + 0.826896i \(0.309896\pi\)
\(654\) −250833. −0.0229319
\(655\) 0 0
\(656\) −4.52561e6 −0.410599
\(657\) −4.92250e6 −0.444910
\(658\) −3.26080e6 −0.293602
\(659\) 708926. 0.0635898 0.0317949 0.999494i \(-0.489878\pi\)
0.0317949 + 0.999494i \(0.489878\pi\)
\(660\) 0 0
\(661\) 1.69284e7 1.50700 0.753500 0.657448i \(-0.228364\pi\)
0.753500 + 0.657448i \(0.228364\pi\)
\(662\) 6.94597e6 0.616010
\(663\) −661363. −0.0584327
\(664\) 2.23297e6 0.196546
\(665\) 0 0
\(666\) −3.32695e6 −0.290644
\(667\) −2.90639e6 −0.252953
\(668\) 1.21164e7 1.05059
\(669\) 493582. 0.0426377
\(670\) 0 0
\(671\) 1.79354e6 0.153781
\(672\) −3.07123e6 −0.262355
\(673\) −1.00330e7 −0.853873 −0.426937 0.904281i \(-0.640407\pi\)
−0.426937 + 0.904281i \(0.640407\pi\)
\(674\) −1.11163e7 −0.942560
\(675\) 0 0
\(676\) −675009. −0.0568124
\(677\) −1.54623e7 −1.29659 −0.648296 0.761388i \(-0.724518\pi\)
−0.648296 + 0.761388i \(0.724518\pi\)
\(678\) −392586. −0.0327990
\(679\) 2.18797e7 1.82124
\(680\) 0 0
\(681\) −2.00167e6 −0.165396
\(682\) −1.64420e7 −1.35361
\(683\) 9.19066e6 0.753868 0.376934 0.926240i \(-0.376978\pi\)
0.376934 + 0.926240i \(0.376978\pi\)
\(684\) 7.64689e6 0.624949
\(685\) 0 0
\(686\) 4.98593e6 0.404516
\(687\) −838062. −0.0677461
\(688\) 671459. 0.0540814
\(689\) −2.38934e6 −0.191748
\(690\) 0 0
\(691\) −1.38192e7 −1.10100 −0.550502 0.834834i \(-0.685564\pi\)
−0.550502 + 0.834834i \(0.685564\pi\)
\(692\) −1.09382e7 −0.868320
\(693\) 2.44168e7 1.93132
\(694\) 2.97040e6 0.234108
\(695\) 0 0
\(696\) 1.80145e6 0.140961
\(697\) −1.76126e7 −1.37323
\(698\) −3.61286e6 −0.280681
\(699\) 1.86790e6 0.144598
\(700\) 0 0
\(701\) 1.58946e7 1.22167 0.610834 0.791758i \(-0.290834\pi\)
0.610834 + 0.791758i \(0.290834\pi\)
\(702\) −801134. −0.0613568
\(703\) −6.97162e6 −0.532041
\(704\) −5.71554e6 −0.434636
\(705\) 0 0
\(706\) 9.05584e6 0.683781
\(707\) −1.49536e6 −0.112512
\(708\) −4.06985e6 −0.305137
\(709\) 1.20549e7 0.900635 0.450317 0.892869i \(-0.351311\pi\)
0.450317 + 0.892869i \(0.351311\pi\)
\(710\) 0 0
\(711\) −920630. −0.0682985
\(712\) 1.43682e7 1.06219
\(713\) 7.15950e6 0.527423
\(714\) −1.67849e6 −0.123218
\(715\) 0 0
\(716\) −9.78271e6 −0.713143
\(717\) −771097. −0.0560159
\(718\) 7.62252e6 0.551807
\(719\) 1.21480e7 0.876357 0.438178 0.898888i \(-0.355624\pi\)
0.438178 + 0.898888i \(0.355624\pi\)
\(720\) 0 0
\(721\) −2.59622e7 −1.85996
\(722\) 1.48963e6 0.106350
\(723\) 4.54966e6 0.323693
\(724\) 3.94898e6 0.279988
\(725\) 0 0
\(726\) −3.46829e6 −0.244216
\(727\) 1.37550e7 0.965215 0.482607 0.875837i \(-0.339690\pi\)
0.482607 + 0.875837i \(0.339690\pi\)
\(728\) −4.03267e6 −0.282010
\(729\) −1.02859e7 −0.716845
\(730\) 0 0
\(731\) 2.61316e6 0.180873
\(732\) 205637. 0.0141848
\(733\) −5.10914e6 −0.351227 −0.175614 0.984459i \(-0.556191\pi\)
−0.175614 + 0.984459i \(0.556191\pi\)
\(734\) 6.45698e6 0.442374
\(735\) 0 0
\(736\) 5.37697e6 0.365884
\(737\) −2.74725e7 −1.86307
\(738\) −1.03985e7 −0.702794
\(739\) 4.07246e6 0.274312 0.137156 0.990549i \(-0.456204\pi\)
0.137156 + 0.990549i \(0.456204\pi\)
\(740\) 0 0
\(741\) −818224. −0.0547427
\(742\) −6.06398e6 −0.404342
\(743\) −1.86637e7 −1.24029 −0.620147 0.784485i \(-0.712927\pi\)
−0.620147 + 0.784485i \(0.712927\pi\)
\(744\) −4.43763e6 −0.293913
\(745\) 0 0
\(746\) 1.07694e7 0.708505
\(747\) −3.20615e6 −0.210224
\(748\) 1.90649e7 1.24589
\(749\) 3.33822e7 2.17425
\(750\) 0 0
\(751\) −2.20898e7 −1.42919 −0.714597 0.699536i \(-0.753390\pi\)
−0.714597 + 0.699536i \(0.753390\pi\)
\(752\) 2.21119e6 0.142588
\(753\) 1.64528e6 0.105743
\(754\) 1.58282e6 0.101392
\(755\) 0 0
\(756\) 5.74382e6 0.365508
\(757\) 2.27912e7 1.44553 0.722765 0.691094i \(-0.242871\pi\)
0.722765 + 0.691094i \(0.242871\pi\)
\(758\) 6.10740e6 0.386085
\(759\) 2.21150e6 0.139342
\(760\) 0 0
\(761\) 2.05216e7 1.28455 0.642274 0.766475i \(-0.277991\pi\)
0.642274 + 0.766475i \(0.277991\pi\)
\(762\) −2.67257e6 −0.166740
\(763\) 3.71958e6 0.231304
\(764\) −726125. −0.0450068
\(765\) 0 0
\(766\) 1.19843e6 0.0737973
\(767\) −8.41767e6 −0.516658
\(768\) −2.57229e6 −0.157368
\(769\) −2.11516e7 −1.28982 −0.644908 0.764260i \(-0.723104\pi\)
−0.644908 + 0.764260i \(0.723104\pi\)
\(770\) 0 0
\(771\) 638488. 0.0386827
\(772\) −2.39315e7 −1.44520
\(773\) 6.75134e6 0.406389 0.203194 0.979138i \(-0.434868\pi\)
0.203194 + 0.979138i \(0.434868\pi\)
\(774\) 1.54280e6 0.0925675
\(775\) 0 0
\(776\) 2.37430e7 1.41541
\(777\) −2.55228e6 −0.151662
\(778\) −1.18754e7 −0.703398
\(779\) −2.17900e7 −1.28651
\(780\) 0 0
\(781\) −4.84668e7 −2.84326
\(782\) 2.93863e6 0.171842
\(783\) −5.30694e6 −0.309343
\(784\) 1.50729e6 0.0875801
\(785\) 0 0
\(786\) 1.57113e6 0.0907103
\(787\) 1.78318e7 1.02626 0.513132 0.858310i \(-0.328485\pi\)
0.513132 + 0.858310i \(0.328485\pi\)
\(788\) −9.98003e6 −0.572554
\(789\) −1.87117e6 −0.107009
\(790\) 0 0
\(791\) 5.82163e6 0.330829
\(792\) 2.64961e7 1.50096
\(793\) 425320. 0.0240178
\(794\) 1.67254e7 0.941508
\(795\) 0 0
\(796\) −1.30815e7 −0.731772
\(797\) −1.22612e7 −0.683732 −0.341866 0.939749i \(-0.611059\pi\)
−0.341866 + 0.939749i \(0.611059\pi\)
\(798\) −2.07659e6 −0.115437
\(799\) 8.60546e6 0.476878
\(800\) 0 0
\(801\) −2.06301e7 −1.13611
\(802\) 1.00865e7 0.553738
\(803\) −1.51833e7 −0.830956
\(804\) −3.14985e6 −0.171850
\(805\) 0 0
\(806\) −3.89907e6 −0.211409
\(807\) 685993. 0.0370797
\(808\) −1.62271e6 −0.0874404
\(809\) −1.86542e7 −1.00208 −0.501042 0.865423i \(-0.667050\pi\)
−0.501042 + 0.865423i \(0.667050\pi\)
\(810\) 0 0
\(811\) 2.65131e6 0.141550 0.0707748 0.997492i \(-0.477453\pi\)
0.0707748 + 0.997492i \(0.477453\pi\)
\(812\) −1.13482e7 −0.604002
\(813\) −893377. −0.0474033
\(814\) −1.02619e7 −0.542833
\(815\) 0 0
\(816\) 1.13821e6 0.0598408
\(817\) 3.23295e6 0.169451
\(818\) 9.35351e6 0.488755
\(819\) 5.79020e6 0.301636
\(820\) 0 0
\(821\) 2.04948e7 1.06117 0.530587 0.847630i \(-0.321972\pi\)
0.530587 + 0.847630i \(0.321972\pi\)
\(822\) 1.32382e6 0.0683361
\(823\) −1.26841e7 −0.652768 −0.326384 0.945237i \(-0.605830\pi\)
−0.326384 + 0.945237i \(0.605830\pi\)
\(824\) −2.81732e7 −1.44550
\(825\) 0 0
\(826\) −2.13635e7 −1.08949
\(827\) −9.90644e6 −0.503679 −0.251839 0.967769i \(-0.581035\pi\)
−0.251839 + 0.967769i \(0.581035\pi\)
\(828\) −4.90124e6 −0.248445
\(829\) 8.05866e6 0.407265 0.203632 0.979047i \(-0.434725\pi\)
0.203632 + 0.979047i \(0.434725\pi\)
\(830\) 0 0
\(831\) −7.60034e6 −0.381795
\(832\) −1.35538e6 −0.0678819
\(833\) 5.86601e6 0.292907
\(834\) 2.85039e6 0.141902
\(835\) 0 0
\(836\) 2.35866e7 1.16721
\(837\) 1.30729e7 0.644998
\(838\) −1.05203e7 −0.517508
\(839\) 2.15392e7 1.05639 0.528196 0.849122i \(-0.322869\pi\)
0.528196 + 0.849122i \(0.322869\pi\)
\(840\) 0 0
\(841\) −1.00261e7 −0.488811
\(842\) −6.12024e6 −0.297501
\(843\) −2912.38 −0.000141150 0
\(844\) 1.79186e7 0.865860
\(845\) 0 0
\(846\) 5.08064e6 0.244058
\(847\) 5.14309e7 2.46329
\(848\) 4.11207e6 0.196368
\(849\) 6.67968e6 0.318044
\(850\) 0 0
\(851\) 4.46842e6 0.211510
\(852\) −5.55695e6 −0.262263
\(853\) −1.55629e7 −0.732351 −0.366175 0.930546i \(-0.619333\pi\)
−0.366175 + 0.930546i \(0.619333\pi\)
\(854\) 1.07943e6 0.0506466
\(855\) 0 0
\(856\) 3.62250e7 1.68976
\(857\) 3.66500e7 1.70460 0.852299 0.523055i \(-0.175208\pi\)
0.852299 + 0.523055i \(0.175208\pi\)
\(858\) −1.20438e6 −0.0558531
\(859\) 3.16288e7 1.46251 0.731257 0.682102i \(-0.238934\pi\)
0.731257 + 0.682102i \(0.238934\pi\)
\(860\) 0 0
\(861\) −7.97721e6 −0.366727
\(862\) −1.77717e7 −0.814630
\(863\) −1.63149e7 −0.745691 −0.372845 0.927894i \(-0.621618\pi\)
−0.372845 + 0.927894i \(0.621618\pi\)
\(864\) 9.81810e6 0.447448
\(865\) 0 0
\(866\) 1.76052e7 0.797710
\(867\) −479219. −0.0216514
\(868\) 2.79548e7 1.25938
\(869\) −2.83966e6 −0.127561
\(870\) 0 0
\(871\) −6.51484e6 −0.290977
\(872\) 4.03634e6 0.179762
\(873\) −3.40907e7 −1.51391
\(874\) 3.63561e6 0.160990
\(875\) 0 0
\(876\) −1.74084e6 −0.0766475
\(877\) 2.64601e7 1.16170 0.580848 0.814012i \(-0.302721\pi\)
0.580848 + 0.814012i \(0.302721\pi\)
\(878\) −1.58029e7 −0.691831
\(879\) 3.50297e6 0.152920
\(880\) 0 0
\(881\) 1.51155e7 0.656119 0.328059 0.944657i \(-0.393605\pi\)
0.328059 + 0.944657i \(0.393605\pi\)
\(882\) 3.46328e6 0.149905
\(883\) 2.08156e7 0.898437 0.449218 0.893422i \(-0.351703\pi\)
0.449218 + 0.893422i \(0.351703\pi\)
\(884\) 4.52105e6 0.194585
\(885\) 0 0
\(886\) 1.23112e7 0.526885
\(887\) 1.01116e7 0.431529 0.215764 0.976445i \(-0.430776\pi\)
0.215764 + 0.976445i \(0.430776\pi\)
\(888\) −2.76964e6 −0.117866
\(889\) 3.96312e7 1.68183
\(890\) 0 0
\(891\) −3.59737e7 −1.51807
\(892\) −3.37410e6 −0.141986
\(893\) 1.06465e7 0.446763
\(894\) 1.56684e6 0.0655665
\(895\) 0 0
\(896\) 2.49867e7 1.03978
\(897\) 524436. 0.0217626
\(898\) −8.90100e6 −0.368339
\(899\) −2.58285e7 −1.06586
\(900\) 0 0
\(901\) 1.60032e7 0.656744
\(902\) −3.20737e7 −1.31260
\(903\) 1.18357e6 0.0483030
\(904\) 6.31740e6 0.257109
\(905\) 0 0
\(906\) −4.22554e6 −0.171026
\(907\) −3.04365e6 −0.122850 −0.0614251 0.998112i \(-0.519565\pi\)
−0.0614251 + 0.998112i \(0.519565\pi\)
\(908\) 1.36833e7 0.550779
\(909\) 2.32992e6 0.0935257
\(910\) 0 0
\(911\) −2.90597e7 −1.16010 −0.580050 0.814581i \(-0.696967\pi\)
−0.580050 + 0.814581i \(0.696967\pi\)
\(912\) 1.40817e6 0.0560618
\(913\) −9.88928e6 −0.392634
\(914\) 5.57744e6 0.220836
\(915\) 0 0
\(916\) 5.72896e6 0.225599
\(917\) −2.32982e7 −0.914952
\(918\) 5.36581e6 0.210149
\(919\) 3.38583e7 1.32244 0.661220 0.750192i \(-0.270039\pi\)
0.661220 + 0.750192i \(0.270039\pi\)
\(920\) 0 0
\(921\) 1.33546e6 0.0518780
\(922\) −4.42900e6 −0.171585
\(923\) −1.14934e7 −0.444064
\(924\) 8.63497e6 0.332722
\(925\) 0 0
\(926\) 6.16939e6 0.236437
\(927\) 4.04516e7 1.54610
\(928\) −1.93979e7 −0.739409
\(929\) 3.78136e7 1.43750 0.718751 0.695267i \(-0.244714\pi\)
0.718751 + 0.695267i \(0.244714\pi\)
\(930\) 0 0
\(931\) 7.25729e6 0.274410
\(932\) −1.27689e7 −0.481519
\(933\) 4.75954e6 0.179003
\(934\) 7.16037e6 0.268577
\(935\) 0 0
\(936\) 6.28329e6 0.234422
\(937\) 2.00563e7 0.746281 0.373140 0.927775i \(-0.378281\pi\)
0.373140 + 0.927775i \(0.378281\pi\)
\(938\) −1.65342e7 −0.613587
\(939\) −1.87169e6 −0.0692739
\(940\) 0 0
\(941\) −2.54644e7 −0.937473 −0.468737 0.883338i \(-0.655291\pi\)
−0.468737 + 0.883338i \(0.655291\pi\)
\(942\) −1.29540e6 −0.0475637
\(943\) 1.39662e7 0.511443
\(944\) 1.44869e7 0.529108
\(945\) 0 0
\(946\) 4.75874e6 0.172888
\(947\) −1.42435e7 −0.516110 −0.258055 0.966130i \(-0.583082\pi\)
−0.258055 + 0.966130i \(0.583082\pi\)
\(948\) −325580. −0.0117662
\(949\) −3.60058e6 −0.129780
\(950\) 0 0
\(951\) 6.63292e6 0.237823
\(952\) 2.70099e7 0.965896
\(953\) 3.95389e7 1.41024 0.705119 0.709089i \(-0.250894\pi\)
0.705119 + 0.709089i \(0.250894\pi\)
\(954\) 9.44827e6 0.336110
\(955\) 0 0
\(956\) 5.27119e6 0.186536
\(957\) −7.97819e6 −0.281595
\(958\) −1.88731e7 −0.664399
\(959\) −1.96308e7 −0.689275
\(960\) 0 0
\(961\) 3.49958e7 1.22238
\(962\) −2.43351e6 −0.0847803
\(963\) −5.20127e7 −1.80735
\(964\) −3.11013e7 −1.07792
\(965\) 0 0
\(966\) 1.33098e6 0.0458912
\(967\) −4.28774e7 −1.47456 −0.737279 0.675588i \(-0.763890\pi\)
−0.737279 + 0.675588i \(0.763890\pi\)
\(968\) 5.58108e7 1.91439
\(969\) 5.48027e6 0.187496
\(970\) 0 0
\(971\) 1.56895e7 0.534023 0.267011 0.963693i \(-0.413964\pi\)
0.267011 + 0.963693i \(0.413964\pi\)
\(972\) −1.35370e7 −0.459574
\(973\) −4.22681e7 −1.43130
\(974\) −1.59100e7 −0.537370
\(975\) 0 0
\(976\) −731978. −0.0245965
\(977\) 2.22874e7 0.747004 0.373502 0.927629i \(-0.378157\pi\)
0.373502 + 0.927629i \(0.378157\pi\)
\(978\) 6.16759e6 0.206190
\(979\) −6.36331e7 −2.12191
\(980\) 0 0
\(981\) −5.79547e6 −0.192272
\(982\) −2.47174e7 −0.817946
\(983\) −2.46890e6 −0.0814929 −0.0407465 0.999170i \(-0.512974\pi\)
−0.0407465 + 0.999170i \(0.512974\pi\)
\(984\) −8.65656e6 −0.285008
\(985\) 0 0
\(986\) −1.06014e7 −0.347272
\(987\) 3.89763e6 0.127353
\(988\) 5.59334e6 0.182297
\(989\) −2.07214e6 −0.0673640
\(990\) 0 0
\(991\) 1.73304e7 0.560563 0.280281 0.959918i \(-0.409572\pi\)
0.280281 + 0.959918i \(0.409572\pi\)
\(992\) 4.77840e7 1.54171
\(993\) −8.30251e6 −0.267200
\(994\) −2.91695e7 −0.936405
\(995\) 0 0
\(996\) −1.13385e6 −0.0362166
\(997\) 5.90647e7 1.88187 0.940936 0.338586i \(-0.109948\pi\)
0.940936 + 0.338586i \(0.109948\pi\)
\(998\) −8.62487e6 −0.274111
\(999\) 8.15913e6 0.258661
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 325.6.a.j.1.5 11
5.2 odd 4 325.6.b.i.274.9 22
5.3 odd 4 325.6.b.i.274.14 22
5.4 even 2 325.6.a.k.1.7 yes 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
325.6.a.j.1.5 11 1.1 even 1 trivial
325.6.a.k.1.7 yes 11 5.4 even 2
325.6.b.i.274.9 22 5.2 odd 4
325.6.b.i.274.14 22 5.3 odd 4