Properties

Label 531.6.a.c.1.9
Level $531$
Weight $6$
Character 531.1
Self dual yes
Analytic conductor $85.164$
Analytic rank $1$
Dimension $12$
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] = [531,6,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, 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("531.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 531.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: \(85.1638083207\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + 15565376 x^{4} + 6775664 x^{3} - 75006848 x^{2} + \cdots + 49172480 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(4.10256\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.10256 q^{2} -27.5792 q^{4} +81.1594 q^{5} -97.6213 q^{7} -125.269 q^{8} +O(q^{10})\) \(q+2.10256 q^{2} -27.5792 q^{4} +81.1594 q^{5} -97.6213 q^{7} -125.269 q^{8} +170.642 q^{10} +297.163 q^{11} -814.611 q^{13} -205.254 q^{14} +619.151 q^{16} +1275.83 q^{17} +801.964 q^{19} -2238.32 q^{20} +624.802 q^{22} -255.443 q^{23} +3461.85 q^{25} -1712.77 q^{26} +2692.32 q^{28} -7410.46 q^{29} -2821.92 q^{31} +5310.40 q^{32} +2682.51 q^{34} -7922.89 q^{35} +4781.71 q^{37} +1686.18 q^{38} -10166.7 q^{40} +8224.42 q^{41} -1932.60 q^{43} -8195.53 q^{44} -537.083 q^{46} -23212.7 q^{47} -7277.08 q^{49} +7278.75 q^{50} +22466.3 q^{52} -15252.2 q^{53} +24117.6 q^{55} +12228.9 q^{56} -15580.9 q^{58} +3481.00 q^{59} +9488.35 q^{61} -5933.26 q^{62} -8647.40 q^{64} -66113.3 q^{65} +28014.3 q^{67} -35186.5 q^{68} -16658.3 q^{70} +26155.1 q^{71} +7782.60 q^{73} +10053.8 q^{74} -22117.6 q^{76} -29009.4 q^{77} -51295.8 q^{79} +50249.9 q^{80} +17292.3 q^{82} -17726.9 q^{83} +103546. q^{85} -4063.40 q^{86} -37225.2 q^{88} -104423. q^{89} +79523.3 q^{91} +7044.92 q^{92} -48806.1 q^{94} +65086.9 q^{95} -144515. q^{97} -15300.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.10256 0.371683 0.185842 0.982580i \(-0.440499\pi\)
0.185842 + 0.982580i \(0.440499\pi\)
\(3\) 0 0
\(4\) −27.5792 −0.861852
\(5\) 81.1594 1.45182 0.725912 0.687788i \(-0.241418\pi\)
0.725912 + 0.687788i \(0.241418\pi\)
\(6\) 0 0
\(7\) −97.6213 −0.753008 −0.376504 0.926415i \(-0.622874\pi\)
−0.376504 + 0.926415i \(0.622874\pi\)
\(8\) −125.269 −0.692019
\(9\) 0 0
\(10\) 170.642 0.539619
\(11\) 297.163 0.740479 0.370239 0.928936i \(-0.379276\pi\)
0.370239 + 0.928936i \(0.379276\pi\)
\(12\) 0 0
\(13\) −814.611 −1.33688 −0.668439 0.743767i \(-0.733037\pi\)
−0.668439 + 0.743767i \(0.733037\pi\)
\(14\) −205.254 −0.279880
\(15\) 0 0
\(16\) 619.151 0.604640
\(17\) 1275.83 1.07071 0.535355 0.844627i \(-0.320178\pi\)
0.535355 + 0.844627i \(0.320178\pi\)
\(18\) 0 0
\(19\) 801.964 0.509649 0.254824 0.966987i \(-0.417982\pi\)
0.254824 + 0.966987i \(0.417982\pi\)
\(20\) −2238.32 −1.25126
\(21\) 0 0
\(22\) 624.802 0.275224
\(23\) −255.443 −0.100687 −0.0503436 0.998732i \(-0.516032\pi\)
−0.0503436 + 0.998732i \(0.516032\pi\)
\(24\) 0 0
\(25\) 3461.85 1.10779
\(26\) −1712.77 −0.496895
\(27\) 0 0
\(28\) 2692.32 0.648981
\(29\) −7410.46 −1.63625 −0.818125 0.575040i \(-0.804987\pi\)
−0.818125 + 0.575040i \(0.804987\pi\)
\(30\) 0 0
\(31\) −2821.92 −0.527401 −0.263700 0.964605i \(-0.584943\pi\)
−0.263700 + 0.964605i \(0.584943\pi\)
\(32\) 5310.40 0.916754
\(33\) 0 0
\(34\) 2682.51 0.397965
\(35\) −7922.89 −1.09323
\(36\) 0 0
\(37\) 4781.71 0.574221 0.287111 0.957897i \(-0.407305\pi\)
0.287111 + 0.957897i \(0.407305\pi\)
\(38\) 1686.18 0.189428
\(39\) 0 0
\(40\) −10166.7 −1.00469
\(41\) 8224.42 0.764092 0.382046 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(42\) 0 0
\(43\) −1932.60 −0.159393 −0.0796967 0.996819i \(-0.525395\pi\)
−0.0796967 + 0.996819i \(0.525395\pi\)
\(44\) −8195.53 −0.638183
\(45\) 0 0
\(46\) −537.083 −0.0374237
\(47\) −23212.7 −1.53278 −0.766392 0.642373i \(-0.777950\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(48\) 0 0
\(49\) −7277.08 −0.432979
\(50\) 7278.75 0.411748
\(51\) 0 0
\(52\) 22466.3 1.15219
\(53\) −15252.2 −0.745836 −0.372918 0.927864i \(-0.621643\pi\)
−0.372918 + 0.927864i \(0.621643\pi\)
\(54\) 0 0
\(55\) 24117.6 1.07505
\(56\) 12228.9 0.521096
\(57\) 0 0
\(58\) −15580.9 −0.608167
\(59\) 3481.00 0.130189
\(60\) 0 0
\(61\) 9488.35 0.326487 0.163244 0.986586i \(-0.447804\pi\)
0.163244 + 0.986586i \(0.447804\pi\)
\(62\) −5933.26 −0.196026
\(63\) 0 0
\(64\) −8647.40 −0.263898
\(65\) −66113.3 −1.94091
\(66\) 0 0
\(67\) 28014.3 0.762417 0.381209 0.924489i \(-0.375508\pi\)
0.381209 + 0.924489i \(0.375508\pi\)
\(68\) −35186.5 −0.922793
\(69\) 0 0
\(70\) −16658.3 −0.406337
\(71\) 26155.1 0.615758 0.307879 0.951425i \(-0.400381\pi\)
0.307879 + 0.951425i \(0.400381\pi\)
\(72\) 0 0
\(73\) 7782.60 0.170930 0.0854649 0.996341i \(-0.472762\pi\)
0.0854649 + 0.996341i \(0.472762\pi\)
\(74\) 10053.8 0.213428
\(75\) 0 0
\(76\) −22117.6 −0.439241
\(77\) −29009.4 −0.557586
\(78\) 0 0
\(79\) −51295.8 −0.924728 −0.462364 0.886690i \(-0.652999\pi\)
−0.462364 + 0.886690i \(0.652999\pi\)
\(80\) 50249.9 0.877830
\(81\) 0 0
\(82\) 17292.3 0.284000
\(83\) −17726.9 −0.282447 −0.141224 0.989978i \(-0.545104\pi\)
−0.141224 + 0.989978i \(0.545104\pi\)
\(84\) 0 0
\(85\) 103546. 1.55448
\(86\) −4063.40 −0.0592438
\(87\) 0 0
\(88\) −37225.2 −0.512426
\(89\) −104423. −1.39740 −0.698701 0.715414i \(-0.746238\pi\)
−0.698701 + 0.715414i \(0.746238\pi\)
\(90\) 0 0
\(91\) 79523.3 1.00668
\(92\) 7044.92 0.0867774
\(93\) 0 0
\(94\) −48806.1 −0.569710
\(95\) 65086.9 0.739920
\(96\) 0 0
\(97\) −144515. −1.55949 −0.779746 0.626096i \(-0.784652\pi\)
−0.779746 + 0.626096i \(0.784652\pi\)
\(98\) −15300.5 −0.160931
\(99\) 0 0
\(100\) −95475.3 −0.954753
\(101\) −5054.57 −0.0493038 −0.0246519 0.999696i \(-0.507848\pi\)
−0.0246519 + 0.999696i \(0.507848\pi\)
\(102\) 0 0
\(103\) −12446.5 −0.115599 −0.0577997 0.998328i \(-0.518408\pi\)
−0.0577997 + 0.998328i \(0.518408\pi\)
\(104\) 102045. 0.925145
\(105\) 0 0
\(106\) −32068.7 −0.277215
\(107\) −62239.2 −0.525538 −0.262769 0.964859i \(-0.584636\pi\)
−0.262769 + 0.964859i \(0.584636\pi\)
\(108\) 0 0
\(109\) −109466. −0.882496 −0.441248 0.897385i \(-0.645464\pi\)
−0.441248 + 0.897385i \(0.645464\pi\)
\(110\) 50708.6 0.399576
\(111\) 0 0
\(112\) −60442.3 −0.455298
\(113\) 132747. 0.977977 0.488988 0.872290i \(-0.337366\pi\)
0.488988 + 0.872290i \(0.337366\pi\)
\(114\) 0 0
\(115\) −20731.6 −0.146180
\(116\) 204375. 1.41021
\(117\) 0 0
\(118\) 7319.00 0.0483890
\(119\) −124549. −0.806253
\(120\) 0 0
\(121\) −72745.3 −0.451691
\(122\) 19949.8 0.121350
\(123\) 0 0
\(124\) 77826.5 0.454541
\(125\) 27338.9 0.156497
\(126\) 0 0
\(127\) −223789. −1.23120 −0.615601 0.788058i \(-0.711087\pi\)
−0.615601 + 0.788058i \(0.711087\pi\)
\(128\) −188115. −1.01484
\(129\) 0 0
\(130\) −139007. −0.721404
\(131\) −247877. −1.26200 −0.630998 0.775785i \(-0.717354\pi\)
−0.630998 + 0.775785i \(0.717354\pi\)
\(132\) 0 0
\(133\) −78288.8 −0.383769
\(134\) 58901.7 0.283378
\(135\) 0 0
\(136\) −159822. −0.740952
\(137\) −202795. −0.923114 −0.461557 0.887111i \(-0.652709\pi\)
−0.461557 + 0.887111i \(0.652709\pi\)
\(138\) 0 0
\(139\) −192882. −0.846750 −0.423375 0.905955i \(-0.639155\pi\)
−0.423375 + 0.905955i \(0.639155\pi\)
\(140\) 218507. 0.942206
\(141\) 0 0
\(142\) 54992.6 0.228867
\(143\) −242072. −0.989930
\(144\) 0 0
\(145\) −601428. −2.37555
\(146\) 16363.4 0.0635317
\(147\) 0 0
\(148\) −131876. −0.494893
\(149\) −211494. −0.780429 −0.390214 0.920724i \(-0.627599\pi\)
−0.390214 + 0.920724i \(0.627599\pi\)
\(150\) 0 0
\(151\) −122664. −0.437797 −0.218899 0.975748i \(-0.570246\pi\)
−0.218899 + 0.975748i \(0.570246\pi\)
\(152\) −100461. −0.352687
\(153\) 0 0
\(154\) −60994.0 −0.207246
\(155\) −229026. −0.765693
\(156\) 0 0
\(157\) 120555. 0.390333 0.195166 0.980770i \(-0.437475\pi\)
0.195166 + 0.980770i \(0.437475\pi\)
\(158\) −107852. −0.343706
\(159\) 0 0
\(160\) 430989. 1.33096
\(161\) 24936.7 0.0758182
\(162\) 0 0
\(163\) −558933. −1.64775 −0.823875 0.566772i \(-0.808192\pi\)
−0.823875 + 0.566772i \(0.808192\pi\)
\(164\) −226823. −0.658534
\(165\) 0 0
\(166\) −37271.8 −0.104981
\(167\) −225611. −0.625992 −0.312996 0.949754i \(-0.601333\pi\)
−0.312996 + 0.949754i \(0.601333\pi\)
\(168\) 0 0
\(169\) 292297. 0.787242
\(170\) 217711. 0.577775
\(171\) 0 0
\(172\) 53299.6 0.137373
\(173\) −151994. −0.386110 −0.193055 0.981188i \(-0.561840\pi\)
−0.193055 + 0.981188i \(0.561840\pi\)
\(174\) 0 0
\(175\) −337951. −0.834177
\(176\) 183989. 0.447723
\(177\) 0 0
\(178\) −219556. −0.519391
\(179\) 801580. 1.86988 0.934942 0.354801i \(-0.115452\pi\)
0.934942 + 0.354801i \(0.115452\pi\)
\(180\) 0 0
\(181\) 678519. 1.53945 0.769725 0.638375i \(-0.220393\pi\)
0.769725 + 0.638375i \(0.220393\pi\)
\(182\) 167202. 0.374166
\(183\) 0 0
\(184\) 31999.0 0.0696774
\(185\) 388081. 0.833668
\(186\) 0 0
\(187\) 379130. 0.792838
\(188\) 640189. 1.32103
\(189\) 0 0
\(190\) 136849. 0.275016
\(191\) 564563. 1.11977 0.559886 0.828570i \(-0.310845\pi\)
0.559886 + 0.828570i \(0.310845\pi\)
\(192\) 0 0
\(193\) 774264. 1.49622 0.748111 0.663574i \(-0.230961\pi\)
0.748111 + 0.663574i \(0.230961\pi\)
\(194\) −303851. −0.579637
\(195\) 0 0
\(196\) 200696. 0.373164
\(197\) −530628. −0.974147 −0.487073 0.873361i \(-0.661936\pi\)
−0.487073 + 0.873361i \(0.661936\pi\)
\(198\) 0 0
\(199\) 128471. 0.229970 0.114985 0.993367i \(-0.463318\pi\)
0.114985 + 0.993367i \(0.463318\pi\)
\(200\) −433662. −0.766614
\(201\) 0 0
\(202\) −10627.5 −0.0183254
\(203\) 723418. 1.23211
\(204\) 0 0
\(205\) 667489. 1.10933
\(206\) −26169.6 −0.0429664
\(207\) 0 0
\(208\) −504367. −0.808329
\(209\) 238314. 0.377384
\(210\) 0 0
\(211\) −447265. −0.691607 −0.345803 0.938307i \(-0.612394\pi\)
−0.345803 + 0.938307i \(0.612394\pi\)
\(212\) 420645. 0.642800
\(213\) 0 0
\(214\) −130861. −0.195334
\(215\) −156848. −0.231411
\(216\) 0 0
\(217\) 275480. 0.397137
\(218\) −230158. −0.328009
\(219\) 0 0
\(220\) −665144. −0.926529
\(221\) −1.03931e6 −1.43141
\(222\) 0 0
\(223\) 523227. 0.704576 0.352288 0.935892i \(-0.385404\pi\)
0.352288 + 0.935892i \(0.385404\pi\)
\(224\) −518409. −0.690323
\(225\) 0 0
\(226\) 279108. 0.363498
\(227\) −360000. −0.463700 −0.231850 0.972752i \(-0.574478\pi\)
−0.231850 + 0.972752i \(0.574478\pi\)
\(228\) 0 0
\(229\) −1.17199e6 −1.47684 −0.738421 0.674340i \(-0.764428\pi\)
−0.738421 + 0.674340i \(0.764428\pi\)
\(230\) −43589.4 −0.0543327
\(231\) 0 0
\(232\) 928299. 1.13232
\(233\) −163113. −0.196834 −0.0984170 0.995145i \(-0.531378\pi\)
−0.0984170 + 0.995145i \(0.531378\pi\)
\(234\) 0 0
\(235\) −1.88393e6 −2.22533
\(236\) −96003.4 −0.112204
\(237\) 0 0
\(238\) −261871. −0.299671
\(239\) −1.32642e6 −1.50205 −0.751026 0.660273i \(-0.770441\pi\)
−0.751026 + 0.660273i \(0.770441\pi\)
\(240\) 0 0
\(241\) 1.52608e6 1.69252 0.846259 0.532771i \(-0.178849\pi\)
0.846259 + 0.532771i \(0.178849\pi\)
\(242\) −152951. −0.167886
\(243\) 0 0
\(244\) −261682. −0.281384
\(245\) −590604. −0.628610
\(246\) 0 0
\(247\) −653288. −0.681338
\(248\) 353499. 0.364971
\(249\) 0 0
\(250\) 57481.5 0.0581672
\(251\) 739184. 0.740573 0.370287 0.928918i \(-0.379259\pi\)
0.370287 + 0.928918i \(0.379259\pi\)
\(252\) 0 0
\(253\) −75908.1 −0.0745567
\(254\) −470529. −0.457617
\(255\) 0 0
\(256\) −118805. −0.113301
\(257\) −385343. −0.363927 −0.181964 0.983305i \(-0.558245\pi\)
−0.181964 + 0.983305i \(0.558245\pi\)
\(258\) 0 0
\(259\) −466797. −0.432393
\(260\) 1.82336e6 1.67278
\(261\) 0 0
\(262\) −521175. −0.469063
\(263\) 907092. 0.808653 0.404326 0.914615i \(-0.367506\pi\)
0.404326 + 0.914615i \(0.367506\pi\)
\(264\) 0 0
\(265\) −1.23786e6 −1.08282
\(266\) −164607. −0.142641
\(267\) 0 0
\(268\) −772613. −0.657091
\(269\) −1.65731e6 −1.39644 −0.698220 0.715883i \(-0.746024\pi\)
−0.698220 + 0.715883i \(0.746024\pi\)
\(270\) 0 0
\(271\) 2.23356e6 1.84746 0.923730 0.383045i \(-0.125125\pi\)
0.923730 + 0.383045i \(0.125125\pi\)
\(272\) 789934. 0.647394
\(273\) 0 0
\(274\) −426388. −0.343106
\(275\) 1.02873e6 0.820297
\(276\) 0 0
\(277\) −785622. −0.615197 −0.307599 0.951516i \(-0.599525\pi\)
−0.307599 + 0.951516i \(0.599525\pi\)
\(278\) −405546. −0.314723
\(279\) 0 0
\(280\) 992491. 0.756539
\(281\) −1.77758e6 −1.34296 −0.671480 0.741023i \(-0.734341\pi\)
−0.671480 + 0.741023i \(0.734341\pi\)
\(282\) 0 0
\(283\) 797157. 0.591667 0.295834 0.955239i \(-0.404403\pi\)
0.295834 + 0.955239i \(0.404403\pi\)
\(284\) −721338. −0.530692
\(285\) 0 0
\(286\) −508970. −0.367940
\(287\) −802879. −0.575367
\(288\) 0 0
\(289\) 207895. 0.146420
\(290\) −1.26454e6 −0.882952
\(291\) 0 0
\(292\) −214638. −0.147316
\(293\) 142245. 0.0967986 0.0483993 0.998828i \(-0.484588\pi\)
0.0483993 + 0.998828i \(0.484588\pi\)
\(294\) 0 0
\(295\) 282516. 0.189011
\(296\) −599000. −0.397372
\(297\) 0 0
\(298\) −444679. −0.290072
\(299\) 208086. 0.134606
\(300\) 0 0
\(301\) 188663. 0.120024
\(302\) −257907. −0.162722
\(303\) 0 0
\(304\) 496537. 0.308154
\(305\) 770069. 0.474002
\(306\) 0 0
\(307\) 2.26783e6 1.37330 0.686649 0.726989i \(-0.259081\pi\)
0.686649 + 0.726989i \(0.259081\pi\)
\(308\) 800058. 0.480557
\(309\) 0 0
\(310\) −481540. −0.284595
\(311\) −2.65296e6 −1.55536 −0.777678 0.628663i \(-0.783602\pi\)
−0.777678 + 0.628663i \(0.783602\pi\)
\(312\) 0 0
\(313\) 2.58601e6 1.49200 0.746000 0.665946i \(-0.231972\pi\)
0.746000 + 0.665946i \(0.231972\pi\)
\(314\) 253473. 0.145080
\(315\) 0 0
\(316\) 1.41470e6 0.796979
\(317\) 1.84274e6 1.02995 0.514974 0.857206i \(-0.327802\pi\)
0.514974 + 0.857206i \(0.327802\pi\)
\(318\) 0 0
\(319\) −2.20211e6 −1.21161
\(320\) −701818. −0.383133
\(321\) 0 0
\(322\) 52430.8 0.0281804
\(323\) 1.02317e6 0.545686
\(324\) 0 0
\(325\) −2.82006e6 −1.48098
\(326\) −1.17519e6 −0.612441
\(327\) 0 0
\(328\) −1.03026e6 −0.528766
\(329\) 2.26605e6 1.15420
\(330\) 0 0
\(331\) −153074. −0.0767949 −0.0383974 0.999263i \(-0.512225\pi\)
−0.0383974 + 0.999263i \(0.512225\pi\)
\(332\) 488895. 0.243428
\(333\) 0 0
\(334\) −474360. −0.232671
\(335\) 2.27362e6 1.10690
\(336\) 0 0
\(337\) 630168. 0.302260 0.151130 0.988514i \(-0.451709\pi\)
0.151130 + 0.988514i \(0.451709\pi\)
\(338\) 614572. 0.292605
\(339\) 0 0
\(340\) −2.85572e6 −1.33973
\(341\) −838570. −0.390529
\(342\) 0 0
\(343\) 2.35112e6 1.07904
\(344\) 242094. 0.110303
\(345\) 0 0
\(346\) −319576. −0.143511
\(347\) 985059. 0.439176 0.219588 0.975593i \(-0.429529\pi\)
0.219588 + 0.975593i \(0.429529\pi\)
\(348\) 0 0
\(349\) −2.32668e6 −1.02252 −0.511262 0.859425i \(-0.670822\pi\)
−0.511262 + 0.859425i \(0.670822\pi\)
\(350\) −710561. −0.310050
\(351\) 0 0
\(352\) 1.57805e6 0.678837
\(353\) −898841. −0.383925 −0.191962 0.981402i \(-0.561485\pi\)
−0.191962 + 0.981402i \(0.561485\pi\)
\(354\) 0 0
\(355\) 2.12273e6 0.893973
\(356\) 2.87991e6 1.20435
\(357\) 0 0
\(358\) 1.68537e6 0.695004
\(359\) 1.45729e6 0.596774 0.298387 0.954445i \(-0.403551\pi\)
0.298387 + 0.954445i \(0.403551\pi\)
\(360\) 0 0
\(361\) −1.83295e6 −0.740258
\(362\) 1.42663e6 0.572188
\(363\) 0 0
\(364\) −2.19319e6 −0.867608
\(365\) 631632. 0.248160
\(366\) 0 0
\(367\) 4.24268e6 1.64428 0.822139 0.569287i \(-0.192781\pi\)
0.822139 + 0.569287i \(0.192781\pi\)
\(368\) −158158. −0.0608794
\(369\) 0 0
\(370\) 815963. 0.309861
\(371\) 1.48894e6 0.561620
\(372\) 0 0
\(373\) −2.66753e6 −0.992742 −0.496371 0.868110i \(-0.665334\pi\)
−0.496371 + 0.868110i \(0.665334\pi\)
\(374\) 797144. 0.294685
\(375\) 0 0
\(376\) 2.90783e6 1.06072
\(377\) 6.03664e6 2.18747
\(378\) 0 0
\(379\) −3.11327e6 −1.11332 −0.556659 0.830741i \(-0.687917\pi\)
−0.556659 + 0.830741i \(0.687917\pi\)
\(380\) −1.79505e6 −0.637701
\(381\) 0 0
\(382\) 1.18703e6 0.416200
\(383\) 1.39854e6 0.487167 0.243584 0.969880i \(-0.421677\pi\)
0.243584 + 0.969880i \(0.421677\pi\)
\(384\) 0 0
\(385\) −2.35439e6 −0.809517
\(386\) 1.62794e6 0.556120
\(387\) 0 0
\(388\) 3.98561e6 1.34405
\(389\) −5.04752e6 −1.69124 −0.845618 0.533789i \(-0.820768\pi\)
−0.845618 + 0.533789i \(0.820768\pi\)
\(390\) 0 0
\(391\) −325902. −0.107807
\(392\) 911591. 0.299630
\(393\) 0 0
\(394\) −1.11568e6 −0.362074
\(395\) −4.16314e6 −1.34254
\(396\) 0 0
\(397\) 5.24113e6 1.66897 0.834486 0.551030i \(-0.185765\pi\)
0.834486 + 0.551030i \(0.185765\pi\)
\(398\) 270117. 0.0854761
\(399\) 0 0
\(400\) 2.14341e6 0.669816
\(401\) 4.27332e6 1.32710 0.663551 0.748131i \(-0.269049\pi\)
0.663551 + 0.748131i \(0.269049\pi\)
\(402\) 0 0
\(403\) 2.29877e6 0.705070
\(404\) 139401. 0.0424926
\(405\) 0 0
\(406\) 1.52103e6 0.457955
\(407\) 1.42095e6 0.425199
\(408\) 0 0
\(409\) 6.70311e6 1.98138 0.990691 0.136133i \(-0.0434676\pi\)
0.990691 + 0.136133i \(0.0434676\pi\)
\(410\) 1.40344e6 0.412318
\(411\) 0 0
\(412\) 343266. 0.0996296
\(413\) −339820. −0.0980333
\(414\) 0 0
\(415\) −1.43871e6 −0.410064
\(416\) −4.32591e6 −1.22559
\(417\) 0 0
\(418\) 501069. 0.140267
\(419\) 1.48858e6 0.414225 0.207112 0.978317i \(-0.433593\pi\)
0.207112 + 0.978317i \(0.433593\pi\)
\(420\) 0 0
\(421\) 4.24222e6 1.16651 0.583255 0.812289i \(-0.301779\pi\)
0.583255 + 0.812289i \(0.301779\pi\)
\(422\) −940402. −0.257059
\(423\) 0 0
\(424\) 1.91063e6 0.516133
\(425\) 4.41675e6 1.18613
\(426\) 0 0
\(427\) −926266. −0.245848
\(428\) 1.71651e6 0.452936
\(429\) 0 0
\(430\) −329783. −0.0860116
\(431\) −6.78108e6 −1.75835 −0.879176 0.476497i \(-0.841906\pi\)
−0.879176 + 0.476497i \(0.841906\pi\)
\(432\) 0 0
\(433\) −2.89130e6 −0.741094 −0.370547 0.928814i \(-0.620830\pi\)
−0.370547 + 0.928814i \(0.620830\pi\)
\(434\) 579212. 0.147609
\(435\) 0 0
\(436\) 3.01899e6 0.760580
\(437\) −204856. −0.0513151
\(438\) 0 0
\(439\) −1.15482e6 −0.285991 −0.142996 0.989723i \(-0.545674\pi\)
−0.142996 + 0.989723i \(0.545674\pi\)
\(440\) −3.02118e6 −0.743952
\(441\) 0 0
\(442\) −2.18521e6 −0.532030
\(443\) 339517. 0.0821964 0.0410982 0.999155i \(-0.486914\pi\)
0.0410982 + 0.999155i \(0.486914\pi\)
\(444\) 0 0
\(445\) −8.47492e6 −2.02878
\(446\) 1.10011e6 0.261879
\(447\) 0 0
\(448\) 844170. 0.198717
\(449\) 7.65479e6 1.79191 0.895957 0.444141i \(-0.146491\pi\)
0.895957 + 0.444141i \(0.146491\pi\)
\(450\) 0 0
\(451\) 2.44399e6 0.565794
\(452\) −3.66106e6 −0.842871
\(453\) 0 0
\(454\) −756920. −0.172350
\(455\) 6.45407e6 1.46152
\(456\) 0 0
\(457\) −4.53237e6 −1.01516 −0.507581 0.861604i \(-0.669460\pi\)
−0.507581 + 0.861604i \(0.669460\pi\)
\(458\) −2.46417e6 −0.548917
\(459\) 0 0
\(460\) 571762. 0.125985
\(461\) 999991. 0.219151 0.109576 0.993978i \(-0.465051\pi\)
0.109576 + 0.993978i \(0.465051\pi\)
\(462\) 0 0
\(463\) 7.61544e6 1.65098 0.825492 0.564414i \(-0.190898\pi\)
0.825492 + 0.564414i \(0.190898\pi\)
\(464\) −4.58819e6 −0.989342
\(465\) 0 0
\(466\) −342955. −0.0731599
\(467\) 2.99420e6 0.635315 0.317658 0.948206i \(-0.397104\pi\)
0.317658 + 0.948206i \(0.397104\pi\)
\(468\) 0 0
\(469\) −2.73479e6 −0.574106
\(470\) −3.96107e6 −0.827119
\(471\) 0 0
\(472\) −436061. −0.0900932
\(473\) −574296. −0.118027
\(474\) 0 0
\(475\) 2.77628e6 0.564585
\(476\) 3.43496e6 0.694870
\(477\) 0 0
\(478\) −2.78887e6 −0.558288
\(479\) −6.42416e6 −1.27932 −0.639658 0.768660i \(-0.720924\pi\)
−0.639658 + 0.768660i \(0.720924\pi\)
\(480\) 0 0
\(481\) −3.89523e6 −0.767664
\(482\) 3.20866e6 0.629081
\(483\) 0 0
\(484\) 2.00626e6 0.389291
\(485\) −1.17287e7 −2.26411
\(486\) 0 0
\(487\) 9.44440e6 1.80448 0.902240 0.431234i \(-0.141922\pi\)
0.902240 + 0.431234i \(0.141922\pi\)
\(488\) −1.18860e6 −0.225935
\(489\) 0 0
\(490\) −1.24178e6 −0.233644
\(491\) 7.87399e6 1.47398 0.736989 0.675905i \(-0.236247\pi\)
0.736989 + 0.675905i \(0.236247\pi\)
\(492\) 0 0
\(493\) −9.45451e6 −1.75195
\(494\) −1.37358e6 −0.253242
\(495\) 0 0
\(496\) −1.74720e6 −0.318887
\(497\) −2.55329e6 −0.463671
\(498\) 0 0
\(499\) 8.14323e6 1.46401 0.732007 0.681297i \(-0.238584\pi\)
0.732007 + 0.681297i \(0.238584\pi\)
\(500\) −753985. −0.134877
\(501\) 0 0
\(502\) 1.55418e6 0.275259
\(503\) −3.34061e6 −0.588717 −0.294358 0.955695i \(-0.595106\pi\)
−0.294358 + 0.955695i \(0.595106\pi\)
\(504\) 0 0
\(505\) −410226. −0.0715805
\(506\) −159601. −0.0277115
\(507\) 0 0
\(508\) 6.17193e6 1.06111
\(509\) 1.19251e6 0.204017 0.102009 0.994784i \(-0.467473\pi\)
0.102009 + 0.994784i \(0.467473\pi\)
\(510\) 0 0
\(511\) −759748. −0.128711
\(512\) 5.76987e6 0.972728
\(513\) 0 0
\(514\) −810206. −0.135266
\(515\) −1.01015e6 −0.167830
\(516\) 0 0
\(517\) −6.89795e6 −1.13499
\(518\) −981468. −0.160713
\(519\) 0 0
\(520\) 8.28194e6 1.34315
\(521\) −9.55161e6 −1.54164 −0.770819 0.637055i \(-0.780153\pi\)
−0.770819 + 0.637055i \(0.780153\pi\)
\(522\) 0 0
\(523\) 1.46152e6 0.233642 0.116821 0.993153i \(-0.462730\pi\)
0.116821 + 0.993153i \(0.462730\pi\)
\(524\) 6.83626e6 1.08765
\(525\) 0 0
\(526\) 1.90721e6 0.300563
\(527\) −3.60030e6 −0.564693
\(528\) 0 0
\(529\) −6.37109e6 −0.989862
\(530\) −2.60268e6 −0.402467
\(531\) 0 0
\(532\) 2.15915e6 0.330752
\(533\) −6.69970e6 −1.02150
\(534\) 0 0
\(535\) −5.05129e6 −0.762989
\(536\) −3.50932e6 −0.527607
\(537\) 0 0
\(538\) −3.48459e6 −0.519034
\(539\) −2.16248e6 −0.320612
\(540\) 0 0
\(541\) −1.06338e7 −1.56205 −0.781025 0.624500i \(-0.785303\pi\)
−0.781025 + 0.624500i \(0.785303\pi\)
\(542\) 4.69619e6 0.686670
\(543\) 0 0
\(544\) 6.77519e6 0.981577
\(545\) −8.88419e6 −1.28123
\(546\) 0 0
\(547\) 1.81124e6 0.258826 0.129413 0.991591i \(-0.458691\pi\)
0.129413 + 0.991591i \(0.458691\pi\)
\(548\) 5.59293e6 0.795587
\(549\) 0 0
\(550\) 2.16297e6 0.304891
\(551\) −5.94292e6 −0.833913
\(552\) 0 0
\(553\) 5.00756e6 0.696328
\(554\) −1.65182e6 −0.228659
\(555\) 0 0
\(556\) 5.31954e6 0.729772
\(557\) 1.77561e6 0.242499 0.121250 0.992622i \(-0.461310\pi\)
0.121250 + 0.992622i \(0.461310\pi\)
\(558\) 0 0
\(559\) 1.57431e6 0.213089
\(560\) −4.90546e6 −0.661013
\(561\) 0 0
\(562\) −3.73746e6 −0.499156
\(563\) 7.13593e6 0.948810 0.474405 0.880307i \(-0.342663\pi\)
0.474405 + 0.880307i \(0.342663\pi\)
\(564\) 0 0
\(565\) 1.07737e7 1.41985
\(566\) 1.67607e6 0.219913
\(567\) 0 0
\(568\) −3.27642e6 −0.426117
\(569\) −3.82803e6 −0.495672 −0.247836 0.968802i \(-0.579719\pi\)
−0.247836 + 0.968802i \(0.579719\pi\)
\(570\) 0 0
\(571\) 1.15161e7 1.47814 0.739070 0.673628i \(-0.235265\pi\)
0.739070 + 0.673628i \(0.235265\pi\)
\(572\) 6.67616e6 0.853172
\(573\) 0 0
\(574\) −1.68810e6 −0.213854
\(575\) −884305. −0.111541
\(576\) 0 0
\(577\) −1.06710e7 −1.33434 −0.667169 0.744906i \(-0.732494\pi\)
−0.667169 + 0.744906i \(0.732494\pi\)
\(578\) 437112. 0.0544218
\(579\) 0 0
\(580\) 1.65869e7 2.04737
\(581\) 1.73052e6 0.212685
\(582\) 0 0
\(583\) −4.53239e6 −0.552276
\(584\) −974918. −0.118287
\(585\) 0 0
\(586\) 299079. 0.0359784
\(587\) −3.86670e6 −0.463176 −0.231588 0.972814i \(-0.574392\pi\)
−0.231588 + 0.972814i \(0.574392\pi\)
\(588\) 0 0
\(589\) −2.26308e6 −0.268789
\(590\) 594006. 0.0702524
\(591\) 0 0
\(592\) 2.96060e6 0.347197
\(593\) 5.82698e6 0.680467 0.340234 0.940341i \(-0.389494\pi\)
0.340234 + 0.940341i \(0.389494\pi\)
\(594\) 0 0
\(595\) −1.01083e7 −1.17054
\(596\) 5.83286e6 0.672614
\(597\) 0 0
\(598\) 437514. 0.0500309
\(599\) 8.74348e6 0.995674 0.497837 0.867271i \(-0.334128\pi\)
0.497837 + 0.867271i \(0.334128\pi\)
\(600\) 0 0
\(601\) −5.10457e6 −0.576465 −0.288233 0.957560i \(-0.593068\pi\)
−0.288233 + 0.957560i \(0.593068\pi\)
\(602\) 396674. 0.0446111
\(603\) 0 0
\(604\) 3.38297e6 0.377316
\(605\) −5.90397e6 −0.655776
\(606\) 0 0
\(607\) 1.23037e6 0.135539 0.0677697 0.997701i \(-0.478412\pi\)
0.0677697 + 0.997701i \(0.478412\pi\)
\(608\) 4.25875e6 0.467222
\(609\) 0 0
\(610\) 1.61912e6 0.176179
\(611\) 1.89093e7 2.04915
\(612\) 0 0
\(613\) −1.66177e7 −1.78616 −0.893079 0.449900i \(-0.851460\pi\)
−0.893079 + 0.449900i \(0.851460\pi\)
\(614\) 4.76824e6 0.510432
\(615\) 0 0
\(616\) 3.63398e6 0.385860
\(617\) 2.06757e6 0.218649 0.109324 0.994006i \(-0.465131\pi\)
0.109324 + 0.994006i \(0.465131\pi\)
\(618\) 0 0
\(619\) −1.03780e7 −1.08864 −0.544321 0.838877i \(-0.683213\pi\)
−0.544321 + 0.838877i \(0.683213\pi\)
\(620\) 6.31635e6 0.659914
\(621\) 0 0
\(622\) −5.57800e6 −0.578100
\(623\) 1.01939e7 1.05225
\(624\) 0 0
\(625\) −8.59949e6 −0.880587
\(626\) 5.43723e6 0.554551
\(627\) 0 0
\(628\) −3.32481e6 −0.336409
\(629\) 6.10067e6 0.614824
\(630\) 0 0
\(631\) −1.57468e6 −0.157441 −0.0787205 0.996897i \(-0.525083\pi\)
−0.0787205 + 0.996897i \(0.525083\pi\)
\(632\) 6.42577e6 0.639930
\(633\) 0 0
\(634\) 3.87446e6 0.382814
\(635\) −1.81626e7 −1.78749
\(636\) 0 0
\(637\) 5.92799e6 0.578840
\(638\) −4.63007e6 −0.450335
\(639\) 0 0
\(640\) −1.52673e7 −1.47337
\(641\) −1.37514e7 −1.32191 −0.660954 0.750426i \(-0.729848\pi\)
−0.660954 + 0.750426i \(0.729848\pi\)
\(642\) 0 0
\(643\) −1.81725e7 −1.73336 −0.866678 0.498867i \(-0.833750\pi\)
−0.866678 + 0.498867i \(0.833750\pi\)
\(644\) −687734. −0.0653440
\(645\) 0 0
\(646\) 2.15128e6 0.202822
\(647\) −1.41770e7 −1.33144 −0.665721 0.746201i \(-0.731876\pi\)
−0.665721 + 0.746201i \(0.731876\pi\)
\(648\) 0 0
\(649\) 1.03442e6 0.0964021
\(650\) −5.92935e6 −0.550457
\(651\) 0 0
\(652\) 1.54150e7 1.42012
\(653\) 7.55832e6 0.693653 0.346826 0.937929i \(-0.387259\pi\)
0.346826 + 0.937929i \(0.387259\pi\)
\(654\) 0 0
\(655\) −2.01175e7 −1.83220
\(656\) 5.09216e6 0.462000
\(657\) 0 0
\(658\) 4.76451e6 0.428996
\(659\) 1.57669e7 1.41427 0.707134 0.707080i \(-0.249988\pi\)
0.707134 + 0.707080i \(0.249988\pi\)
\(660\) 0 0
\(661\) −4.61947e6 −0.411234 −0.205617 0.978633i \(-0.565920\pi\)
−0.205617 + 0.978633i \(0.565920\pi\)
\(662\) −321848. −0.0285434
\(663\) 0 0
\(664\) 2.22063e6 0.195459
\(665\) −6.35387e6 −0.557166
\(666\) 0 0
\(667\) 1.89295e6 0.164749
\(668\) 6.22217e6 0.539512
\(669\) 0 0
\(670\) 4.78043e6 0.411415
\(671\) 2.81959e6 0.241757
\(672\) 0 0
\(673\) −107564. −0.00915436 −0.00457718 0.999990i \(-0.501457\pi\)
−0.00457718 + 0.999990i \(0.501457\pi\)
\(674\) 1.32496e6 0.112345
\(675\) 0 0
\(676\) −8.06134e6 −0.678486
\(677\) −1.09050e7 −0.914441 −0.457221 0.889353i \(-0.651155\pi\)
−0.457221 + 0.889353i \(0.651155\pi\)
\(678\) 0 0
\(679\) 1.41077e7 1.17431
\(680\) −1.29711e7 −1.07573
\(681\) 0 0
\(682\) −1.76314e6 −0.145153
\(683\) 1.14750e7 0.941237 0.470619 0.882337i \(-0.344031\pi\)
0.470619 + 0.882337i \(0.344031\pi\)
\(684\) 0 0
\(685\) −1.64587e7 −1.34020
\(686\) 4.94337e6 0.401063
\(687\) 0 0
\(688\) −1.19657e6 −0.0963755
\(689\) 1.24246e7 0.997092
\(690\) 0 0
\(691\) −1.11344e7 −0.887101 −0.443550 0.896249i \(-0.646281\pi\)
−0.443550 + 0.896249i \(0.646281\pi\)
\(692\) 4.19188e6 0.332769
\(693\) 0 0
\(694\) 2.07114e6 0.163234
\(695\) −1.56542e7 −1.22933
\(696\) 0 0
\(697\) 1.04930e7 0.818121
\(698\) −4.89198e6 −0.380055
\(699\) 0 0
\(700\) 9.32043e6 0.718937
\(701\) 1.80281e7 1.38565 0.692827 0.721104i \(-0.256365\pi\)
0.692827 + 0.721104i \(0.256365\pi\)
\(702\) 0 0
\(703\) 3.83476e6 0.292651
\(704\) −2.56969e6 −0.195411
\(705\) 0 0
\(706\) −1.88986e6 −0.142698
\(707\) 493434. 0.0371262
\(708\) 0 0
\(709\) −2.61098e7 −1.95069 −0.975343 0.220694i \(-0.929168\pi\)
−0.975343 + 0.220694i \(0.929168\pi\)
\(710\) 4.46317e6 0.332275
\(711\) 0 0
\(712\) 1.30810e7 0.967029
\(713\) 720840. 0.0531025
\(714\) 0 0
\(715\) −1.96464e7 −1.43720
\(716\) −2.21070e7 −1.61156
\(717\) 0 0
\(718\) 3.06404e6 0.221811
\(719\) 1.47690e7 1.06544 0.532720 0.846291i \(-0.321170\pi\)
0.532720 + 0.846291i \(0.321170\pi\)
\(720\) 0 0
\(721\) 1.21505e6 0.0870473
\(722\) −3.85389e6 −0.275142
\(723\) 0 0
\(724\) −1.87131e7 −1.32678
\(725\) −2.56539e7 −1.81263
\(726\) 0 0
\(727\) 685611. 0.0481107 0.0240554 0.999711i \(-0.492342\pi\)
0.0240554 + 0.999711i \(0.492342\pi\)
\(728\) −9.96180e6 −0.696641
\(729\) 0 0
\(730\) 1.32804e6 0.0922369
\(731\) −2.46567e6 −0.170664
\(732\) 0 0
\(733\) −1.49501e7 −1.02774 −0.513871 0.857867i \(-0.671789\pi\)
−0.513871 + 0.857867i \(0.671789\pi\)
\(734\) 8.92048e6 0.611150
\(735\) 0 0
\(736\) −1.35650e6 −0.0923053
\(737\) 8.32481e6 0.564554
\(738\) 0 0
\(739\) 1.74384e7 1.17461 0.587307 0.809365i \(-0.300188\pi\)
0.587307 + 0.809365i \(0.300188\pi\)
\(740\) −1.07030e7 −0.718498
\(741\) 0 0
\(742\) 3.13059e6 0.208745
\(743\) −1.74179e7 −1.15751 −0.578753 0.815503i \(-0.696460\pi\)
−0.578753 + 0.815503i \(0.696460\pi\)
\(744\) 0 0
\(745\) −1.71648e7 −1.13304
\(746\) −5.60863e6 −0.368986
\(747\) 0 0
\(748\) −1.04561e7 −0.683309
\(749\) 6.07587e6 0.395734
\(750\) 0 0
\(751\) 1.11888e7 0.723908 0.361954 0.932196i \(-0.382110\pi\)
0.361954 + 0.932196i \(0.382110\pi\)
\(752\) −1.43722e7 −0.926782
\(753\) 0 0
\(754\) 1.26924e7 0.813045
\(755\) −9.95530e6 −0.635605
\(756\) 0 0
\(757\) 1.99140e7 1.26305 0.631523 0.775357i \(-0.282430\pi\)
0.631523 + 0.775357i \(0.282430\pi\)
\(758\) −6.54583e6 −0.413801
\(759\) 0 0
\(760\) −8.15337e6 −0.512039
\(761\) −7.12189e6 −0.445793 −0.222897 0.974842i \(-0.571551\pi\)
−0.222897 + 0.974842i \(0.571551\pi\)
\(762\) 0 0
\(763\) 1.06862e7 0.664526
\(764\) −1.55702e7 −0.965076
\(765\) 0 0
\(766\) 2.94051e6 0.181072
\(767\) −2.83566e6 −0.174047
\(768\) 0 0
\(769\) −1.73482e7 −1.05788 −0.528941 0.848658i \(-0.677411\pi\)
−0.528941 + 0.848658i \(0.677411\pi\)
\(770\) −4.95024e6 −0.300884
\(771\) 0 0
\(772\) −2.13536e7 −1.28952
\(773\) −1.42390e7 −0.857096 −0.428548 0.903519i \(-0.640975\pi\)
−0.428548 + 0.903519i \(0.640975\pi\)
\(774\) 0 0
\(775\) −9.76908e6 −0.584251
\(776\) 1.81032e7 1.07920
\(777\) 0 0
\(778\) −1.06127e7 −0.628604
\(779\) 6.59569e6 0.389418
\(780\) 0 0
\(781\) 7.77232e6 0.455956
\(782\) −685229. −0.0400699
\(783\) 0 0
\(784\) −4.50561e6 −0.261796
\(785\) 9.78415e6 0.566694
\(786\) 0 0
\(787\) 1.80722e7 1.04010 0.520049 0.854137i \(-0.325914\pi\)
0.520049 + 0.854137i \(0.325914\pi\)
\(788\) 1.46343e7 0.839570
\(789\) 0 0
\(790\) −8.75324e6 −0.499001
\(791\) −1.29589e7 −0.736424
\(792\) 0 0
\(793\) −7.72931e6 −0.436474
\(794\) 1.10198e7 0.620329
\(795\) 0 0
\(796\) −3.54313e6 −0.198200
\(797\) −1.94132e7 −1.08256 −0.541280 0.840843i \(-0.682060\pi\)
−0.541280 + 0.840843i \(0.682060\pi\)
\(798\) 0 0
\(799\) −2.96156e7 −1.64117
\(800\) 1.83838e7 1.01557
\(801\) 0 0
\(802\) 8.98490e6 0.493261
\(803\) 2.31270e6 0.126570
\(804\) 0 0
\(805\) 2.02384e6 0.110075
\(806\) 4.83329e6 0.262063
\(807\) 0 0
\(808\) 633180. 0.0341192
\(809\) 2.24002e7 1.20332 0.601659 0.798753i \(-0.294507\pi\)
0.601659 + 0.798753i \(0.294507\pi\)
\(810\) 0 0
\(811\) −1.60987e7 −0.859486 −0.429743 0.902951i \(-0.641396\pi\)
−0.429743 + 0.902951i \(0.641396\pi\)
\(812\) −1.99513e7 −1.06190
\(813\) 0 0
\(814\) 2.98762e6 0.158039
\(815\) −4.53627e7 −2.39224
\(816\) 0 0
\(817\) −1.54987e6 −0.0812346
\(818\) 1.40937e7 0.736446
\(819\) 0 0
\(820\) −1.84089e7 −0.956075
\(821\) −3.03172e7 −1.56975 −0.784877 0.619651i \(-0.787274\pi\)
−0.784877 + 0.619651i \(0.787274\pi\)
\(822\) 0 0
\(823\) 7.97674e6 0.410512 0.205256 0.978708i \(-0.434197\pi\)
0.205256 + 0.978708i \(0.434197\pi\)
\(824\) 1.55916e6 0.0799970
\(825\) 0 0
\(826\) −714491. −0.0364373
\(827\) −1.52413e7 −0.774923 −0.387461 0.921886i \(-0.626648\pi\)
−0.387461 + 0.921886i \(0.626648\pi\)
\(828\) 0 0
\(829\) 1.31859e7 0.666384 0.333192 0.942859i \(-0.391874\pi\)
0.333192 + 0.942859i \(0.391874\pi\)
\(830\) −3.02496e6 −0.152414
\(831\) 0 0
\(832\) 7.04426e6 0.352799
\(833\) −9.28435e6 −0.463595
\(834\) 0 0
\(835\) −1.83104e7 −0.908830
\(836\) −6.57252e6 −0.325249
\(837\) 0 0
\(838\) 3.12982e6 0.153960
\(839\) −1.87408e7 −0.919144 −0.459572 0.888141i \(-0.651997\pi\)
−0.459572 + 0.888141i \(0.651997\pi\)
\(840\) 0 0
\(841\) 3.44037e7 1.67732
\(842\) 8.91952e6 0.433572
\(843\) 0 0
\(844\) 1.23352e7 0.596062
\(845\) 2.37227e7 1.14294
\(846\) 0 0
\(847\) 7.10149e6 0.340127
\(848\) −9.44343e6 −0.450962
\(849\) 0 0
\(850\) 9.28647e6 0.440863
\(851\) −1.22145e6 −0.0578167
\(852\) 0 0
\(853\) 3.32001e7 1.56231 0.781153 0.624339i \(-0.214632\pi\)
0.781153 + 0.624339i \(0.214632\pi\)
\(854\) −1.94753e6 −0.0913774
\(855\) 0 0
\(856\) 7.79663e6 0.363682
\(857\) −1.05094e7 −0.488795 −0.244397 0.969675i \(-0.578590\pi\)
−0.244397 + 0.969675i \(0.578590\pi\)
\(858\) 0 0
\(859\) 1.47534e7 0.682198 0.341099 0.940027i \(-0.389201\pi\)
0.341099 + 0.940027i \(0.389201\pi\)
\(860\) 4.32576e6 0.199442
\(861\) 0 0
\(862\) −1.42576e7 −0.653550
\(863\) −3.15055e7 −1.43999 −0.719994 0.693980i \(-0.755856\pi\)
−0.719994 + 0.693980i \(0.755856\pi\)
\(864\) 0 0
\(865\) −1.23357e7 −0.560564
\(866\) −6.07912e6 −0.275452
\(867\) 0 0
\(868\) −7.59752e6 −0.342273
\(869\) −1.52432e7 −0.684742
\(870\) 0 0
\(871\) −2.28207e7 −1.01926
\(872\) 1.37127e7 0.610704
\(873\) 0 0
\(874\) −430721. −0.0190729
\(875\) −2.66885e6 −0.117843
\(876\) 0 0
\(877\) 1.78432e7 0.783380 0.391690 0.920097i \(-0.371891\pi\)
0.391690 + 0.920097i \(0.371891\pi\)
\(878\) −2.42808e6 −0.106298
\(879\) 0 0
\(880\) 1.49324e7 0.650015
\(881\) 4.44430e6 0.192914 0.0964569 0.995337i \(-0.469249\pi\)
0.0964569 + 0.995337i \(0.469249\pi\)
\(882\) 0 0
\(883\) 2.25724e7 0.974264 0.487132 0.873328i \(-0.338043\pi\)
0.487132 + 0.873328i \(0.338043\pi\)
\(884\) 2.86633e7 1.23366
\(885\) 0 0
\(886\) 713855. 0.0305510
\(887\) 1.77218e7 0.756307 0.378153 0.925743i \(-0.376559\pi\)
0.378153 + 0.925743i \(0.376559\pi\)
\(888\) 0 0
\(889\) 2.18466e7 0.927104
\(890\) −1.78190e7 −0.754064
\(891\) 0 0
\(892\) −1.44302e7 −0.607240
\(893\) −1.86158e7 −0.781182
\(894\) 0 0
\(895\) 6.50558e7 2.71474
\(896\) 1.83640e7 0.764182
\(897\) 0 0
\(898\) 1.60946e7 0.666025
\(899\) 2.09117e7 0.862960
\(900\) 0 0
\(901\) −1.94593e7 −0.798574
\(902\) 5.13863e6 0.210296
\(903\) 0 0
\(904\) −1.66291e7 −0.676779
\(905\) 5.50683e7 2.23501
\(906\) 0 0
\(907\) −309663. −0.0124989 −0.00624944 0.999980i \(-0.501989\pi\)
−0.00624944 + 0.999980i \(0.501989\pi\)
\(908\) 9.92852e6 0.399641
\(909\) 0 0
\(910\) 1.35701e7 0.543223
\(911\) −3.21108e7 −1.28190 −0.640952 0.767581i \(-0.721460\pi\)
−0.640952 + 0.767581i \(0.721460\pi\)
\(912\) 0 0
\(913\) −5.26777e6 −0.209146
\(914\) −9.52958e6 −0.377318
\(915\) 0 0
\(916\) 3.23225e7 1.27282
\(917\) 2.41981e7 0.950292
\(918\) 0 0
\(919\) −1.92631e7 −0.752380 −0.376190 0.926543i \(-0.622766\pi\)
−0.376190 + 0.926543i \(0.622766\pi\)
\(920\) 2.59702e6 0.101159
\(921\) 0 0
\(922\) 2.10254e6 0.0814549
\(923\) −2.13062e7 −0.823194
\(924\) 0 0
\(925\) 1.65536e7 0.636118
\(926\) 1.60119e7 0.613643
\(927\) 0 0
\(928\) −3.93525e7 −1.50004
\(929\) 2.58406e7 0.982344 0.491172 0.871063i \(-0.336569\pi\)
0.491172 + 0.871063i \(0.336569\pi\)
\(930\) 0 0
\(931\) −5.83596e6 −0.220667
\(932\) 4.49855e6 0.169642
\(933\) 0 0
\(934\) 6.29549e6 0.236136
\(935\) 3.07700e7 1.15106
\(936\) 0 0
\(937\) −3.94413e7 −1.46758 −0.733791 0.679375i \(-0.762251\pi\)
−0.733791 + 0.679375i \(0.762251\pi\)
\(938\) −5.75006e6 −0.213386
\(939\) 0 0
\(940\) 5.19574e7 1.91791
\(941\) 3.84459e7 1.41539 0.707695 0.706518i \(-0.249735\pi\)
0.707695 + 0.706518i \(0.249735\pi\)
\(942\) 0 0
\(943\) −2.10087e6 −0.0769342
\(944\) 2.15526e6 0.0787174
\(945\) 0 0
\(946\) −1.20749e6 −0.0438688
\(947\) 1.88805e7 0.684129 0.342065 0.939676i \(-0.388874\pi\)
0.342065 + 0.939676i \(0.388874\pi\)
\(948\) 0 0
\(949\) −6.33979e6 −0.228512
\(950\) 5.83729e6 0.209847
\(951\) 0 0
\(952\) 1.56021e7 0.557942
\(953\) 1.50588e7 0.537103 0.268552 0.963265i \(-0.413455\pi\)
0.268552 + 0.963265i \(0.413455\pi\)
\(954\) 0 0
\(955\) 4.58196e7 1.62571
\(956\) 3.65816e7 1.29455
\(957\) 0 0
\(958\) −1.35072e7 −0.475500
\(959\) 1.97971e7 0.695112
\(960\) 0 0
\(961\) −2.06659e7 −0.721848
\(962\) −8.18996e6 −0.285328
\(963\) 0 0
\(964\) −4.20880e7 −1.45870
\(965\) 6.28388e7 2.17225
\(966\) 0 0
\(967\) 1.72922e7 0.594681 0.297340 0.954772i \(-0.403900\pi\)
0.297340 + 0.954772i \(0.403900\pi\)
\(968\) 9.11272e6 0.312579
\(969\) 0 0
\(970\) −2.46604e7 −0.841531
\(971\) −2.51318e7 −0.855411 −0.427705 0.903918i \(-0.640678\pi\)
−0.427705 + 0.903918i \(0.640678\pi\)
\(972\) 0 0
\(973\) 1.88294e7 0.637609
\(974\) 1.98574e7 0.670695
\(975\) 0 0
\(976\) 5.87472e6 0.197407
\(977\) −3.05606e7 −1.02430 −0.512148 0.858897i \(-0.671150\pi\)
−0.512148 + 0.858897i \(0.671150\pi\)
\(978\) 0 0
\(979\) −3.10306e7 −1.03475
\(980\) 1.62884e7 0.541768
\(981\) 0 0
\(982\) 1.65555e7 0.547853
\(983\) −2.31760e7 −0.764989 −0.382495 0.923958i \(-0.624935\pi\)
−0.382495 + 0.923958i \(0.624935\pi\)
\(984\) 0 0
\(985\) −4.30654e7 −1.41429
\(986\) −1.98787e7 −0.651171
\(987\) 0 0
\(988\) 1.80172e7 0.587212
\(989\) 493668. 0.0160489
\(990\) 0 0
\(991\) −3.50375e7 −1.13331 −0.566655 0.823955i \(-0.691763\pi\)
−0.566655 + 0.823955i \(0.691763\pi\)
\(992\) −1.49855e7 −0.483497
\(993\) 0 0
\(994\) −5.36845e6 −0.172339
\(995\) 1.04266e7 0.333876
\(996\) 0 0
\(997\) −5.65148e6 −0.180063 −0.0900314 0.995939i \(-0.528697\pi\)
−0.0900314 + 0.995939i \(0.528697\pi\)
\(998\) 1.71216e7 0.544150
\(999\) 0 0
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 531.6.a.c.1.9 12
3.2 odd 2 177.6.a.c.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.c.1.4 12 3.2 odd 2
531.6.a.c.1.9 12 1.1 even 1 trivial