Properties

Label 160.6.a.g.1.3
Level $160$
Weight $6$
Character 160.1
Self dual yes
Analytic conductor $25.661$
Analytic rank $0$
Dimension $3$
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] = [160,6,Mod(1,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("160.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.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: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.39180.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 36x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(6.80681\) of defining polynomial
Character \(\chi\) \(=\) 160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+29.2272 q^{3} -25.0000 q^{5} +44.5253 q^{7} +611.232 q^{9} +O(q^{10})\) \(q+29.2272 q^{3} -25.0000 q^{5} +44.5253 q^{7} +611.232 q^{9} -349.258 q^{11} +255.040 q^{13} -730.681 q^{15} +1505.12 q^{17} +2431.11 q^{19} +1301.35 q^{21} +1435.50 q^{23} +625.000 q^{25} +10762.4 q^{27} +2872.04 q^{29} -8940.96 q^{31} -10207.8 q^{33} -1113.13 q^{35} -14536.6 q^{37} +7454.11 q^{39} +7504.01 q^{41} +13490.7 q^{43} -15280.8 q^{45} +8449.69 q^{47} -14824.5 q^{49} +43990.5 q^{51} -28317.3 q^{53} +8731.44 q^{55} +71054.8 q^{57} -3530.13 q^{59} +45644.9 q^{61} +27215.3 q^{63} -6376.00 q^{65} -69849.6 q^{67} +41955.6 q^{69} +60090.2 q^{71} -49204.6 q^{73} +18267.0 q^{75} -15550.8 q^{77} -981.509 q^{79} +166026. q^{81} -38372.3 q^{83} -37628.0 q^{85} +83941.9 q^{87} +39418.6 q^{89} +11355.7 q^{91} -261320. q^{93} -60777.9 q^{95} +51340.9 q^{97} -213477. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 10 q^{3} - 75 q^{5} - 6 q^{7} + 467 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 10 q^{3} - 75 q^{5} - 6 q^{7} + 467 q^{9} - 396 q^{11} - 354 q^{13} - 250 q^{15} + 1158 q^{17} + 3192 q^{19} - 820 q^{21} + 6126 q^{23} + 1875 q^{25} + 13804 q^{27} + 426 q^{29} + 3276 q^{31} - 2408 q^{33} + 150 q^{35} - 11562 q^{37} + 21348 q^{39} + 12450 q^{41} + 26346 q^{43} - 11675 q^{45} + 36762 q^{47} - 3849 q^{49} + 71444 q^{51} - 21162 q^{53} + 9900 q^{55} + 69136 q^{57} + 35040 q^{59} - 24138 q^{61} + 80986 q^{63} + 8850 q^{65} - 9570 q^{67} - 27036 q^{69} + 88092 q^{71} + 66750 q^{73} + 6250 q^{75} - 136488 q^{77} - 92952 q^{79} + 151391 q^{81} + 30258 q^{83} - 28950 q^{85} - 26228 q^{87} + 172686 q^{89} - 106812 q^{91} - 318232 q^{93} - 79800 q^{95} + 170910 q^{97} - 351436 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 29.2272 1.87493 0.937464 0.348081i \(-0.113167\pi\)
0.937464 + 0.348081i \(0.113167\pi\)
\(4\) 0 0
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) 44.5253 0.343449 0.171724 0.985145i \(-0.445066\pi\)
0.171724 + 0.985145i \(0.445066\pi\)
\(8\) 0 0
\(9\) 611.232 2.51536
\(10\) 0 0
\(11\) −349.258 −0.870290 −0.435145 0.900360i \(-0.643303\pi\)
−0.435145 + 0.900360i \(0.643303\pi\)
\(12\) 0 0
\(13\) 255.040 0.418552 0.209276 0.977857i \(-0.432889\pi\)
0.209276 + 0.977857i \(0.432889\pi\)
\(14\) 0 0
\(15\) −730.681 −0.838494
\(16\) 0 0
\(17\) 1505.12 1.26313 0.631566 0.775322i \(-0.282412\pi\)
0.631566 + 0.775322i \(0.282412\pi\)
\(18\) 0 0
\(19\) 2431.11 1.54498 0.772488 0.635030i \(-0.219012\pi\)
0.772488 + 0.635030i \(0.219012\pi\)
\(20\) 0 0
\(21\) 1301.35 0.643942
\(22\) 0 0
\(23\) 1435.50 0.565825 0.282913 0.959146i \(-0.408699\pi\)
0.282913 + 0.959146i \(0.408699\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 10762.4 2.84119
\(28\) 0 0
\(29\) 2872.04 0.634156 0.317078 0.948400i \(-0.397298\pi\)
0.317078 + 0.948400i \(0.397298\pi\)
\(30\) 0 0
\(31\) −8940.96 −1.67101 −0.835507 0.549480i \(-0.814826\pi\)
−0.835507 + 0.549480i \(0.814826\pi\)
\(32\) 0 0
\(33\) −10207.8 −1.63173
\(34\) 0 0
\(35\) −1113.13 −0.153595
\(36\) 0 0
\(37\) −14536.6 −1.74566 −0.872830 0.488024i \(-0.837718\pi\)
−0.872830 + 0.488024i \(0.837718\pi\)
\(38\) 0 0
\(39\) 7454.11 0.784756
\(40\) 0 0
\(41\) 7504.01 0.697162 0.348581 0.937279i \(-0.386664\pi\)
0.348581 + 0.937279i \(0.386664\pi\)
\(42\) 0 0
\(43\) 13490.7 1.11266 0.556331 0.830961i \(-0.312209\pi\)
0.556331 + 0.830961i \(0.312209\pi\)
\(44\) 0 0
\(45\) −15280.8 −1.12490
\(46\) 0 0
\(47\) 8449.69 0.557951 0.278975 0.960298i \(-0.410005\pi\)
0.278975 + 0.960298i \(0.410005\pi\)
\(48\) 0 0
\(49\) −14824.5 −0.882043
\(50\) 0 0
\(51\) 43990.5 2.36828
\(52\) 0 0
\(53\) −28317.3 −1.38472 −0.692360 0.721553i \(-0.743429\pi\)
−0.692360 + 0.721553i \(0.743429\pi\)
\(54\) 0 0
\(55\) 8731.44 0.389206
\(56\) 0 0
\(57\) 71054.8 2.89672
\(58\) 0 0
\(59\) −3530.13 −0.132026 −0.0660131 0.997819i \(-0.521028\pi\)
−0.0660131 + 0.997819i \(0.521028\pi\)
\(60\) 0 0
\(61\) 45644.9 1.57061 0.785304 0.619110i \(-0.212506\pi\)
0.785304 + 0.619110i \(0.212506\pi\)
\(62\) 0 0
\(63\) 27215.3 0.863896
\(64\) 0 0
\(65\) −6376.00 −0.187182
\(66\) 0 0
\(67\) −69849.6 −1.90098 −0.950489 0.310759i \(-0.899417\pi\)
−0.950489 + 0.310759i \(0.899417\pi\)
\(68\) 0 0
\(69\) 41955.6 1.06088
\(70\) 0 0
\(71\) 60090.2 1.41468 0.707339 0.706874i \(-0.249895\pi\)
0.707339 + 0.706874i \(0.249895\pi\)
\(72\) 0 0
\(73\) −49204.6 −1.08068 −0.540342 0.841445i \(-0.681705\pi\)
−0.540342 + 0.841445i \(0.681705\pi\)
\(74\) 0 0
\(75\) 18267.0 0.374986
\(76\) 0 0
\(77\) −15550.8 −0.298900
\(78\) 0 0
\(79\) −981.509 −0.0176940 −0.00884702 0.999961i \(-0.502816\pi\)
−0.00884702 + 0.999961i \(0.502816\pi\)
\(80\) 0 0
\(81\) 166026. 2.81167
\(82\) 0 0
\(83\) −38372.3 −0.611396 −0.305698 0.952129i \(-0.598890\pi\)
−0.305698 + 0.952129i \(0.598890\pi\)
\(84\) 0 0
\(85\) −37628.0 −0.564890
\(86\) 0 0
\(87\) 83941.9 1.18900
\(88\) 0 0
\(89\) 39418.6 0.527504 0.263752 0.964591i \(-0.415040\pi\)
0.263752 + 0.964591i \(0.415040\pi\)
\(90\) 0 0
\(91\) 11355.7 0.143751
\(92\) 0 0
\(93\) −261320. −3.13303
\(94\) 0 0
\(95\) −60777.9 −0.690934
\(96\) 0 0
\(97\) 51340.9 0.554031 0.277016 0.960865i \(-0.410655\pi\)
0.277016 + 0.960865i \(0.410655\pi\)
\(98\) 0 0
\(99\) −213477. −2.18909
\(100\) 0 0
\(101\) −51557.0 −0.502903 −0.251451 0.967870i \(-0.580908\pi\)
−0.251451 + 0.967870i \(0.580908\pi\)
\(102\) 0 0
\(103\) 6370.92 0.0591710 0.0295855 0.999562i \(-0.490581\pi\)
0.0295855 + 0.999562i \(0.490581\pi\)
\(104\) 0 0
\(105\) −32533.8 −0.287979
\(106\) 0 0
\(107\) −5654.13 −0.0477426 −0.0238713 0.999715i \(-0.507599\pi\)
−0.0238713 + 0.999715i \(0.507599\pi\)
\(108\) 0 0
\(109\) −217953. −1.75710 −0.878550 0.477650i \(-0.841489\pi\)
−0.878550 + 0.477650i \(0.841489\pi\)
\(110\) 0 0
\(111\) −424866. −3.27299
\(112\) 0 0
\(113\) −129389. −0.953240 −0.476620 0.879109i \(-0.658138\pi\)
−0.476620 + 0.879109i \(0.658138\pi\)
\(114\) 0 0
\(115\) −35887.4 −0.253045
\(116\) 0 0
\(117\) 155889. 1.05281
\(118\) 0 0
\(119\) 67015.9 0.433821
\(120\) 0 0
\(121\) −39070.1 −0.242595
\(122\) 0 0
\(123\) 219322. 1.30713
\(124\) 0 0
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) 106258. 0.584594 0.292297 0.956328i \(-0.405580\pi\)
0.292297 + 0.956328i \(0.405580\pi\)
\(128\) 0 0
\(129\) 394296. 2.08616
\(130\) 0 0
\(131\) 132808. 0.676155 0.338077 0.941118i \(-0.390223\pi\)
0.338077 + 0.941118i \(0.390223\pi\)
\(132\) 0 0
\(133\) 108246. 0.530619
\(134\) 0 0
\(135\) −269060. −1.27062
\(136\) 0 0
\(137\) −301171. −1.37092 −0.685460 0.728110i \(-0.740399\pi\)
−0.685460 + 0.728110i \(0.740399\pi\)
\(138\) 0 0
\(139\) −139021. −0.610298 −0.305149 0.952305i \(-0.598706\pi\)
−0.305149 + 0.952305i \(0.598706\pi\)
\(140\) 0 0
\(141\) 246961. 1.04612
\(142\) 0 0
\(143\) −89074.6 −0.364262
\(144\) 0 0
\(145\) −71801.1 −0.283603
\(146\) 0 0
\(147\) −433279. −1.65377
\(148\) 0 0
\(149\) −195083. −0.719869 −0.359934 0.932978i \(-0.617201\pi\)
−0.359934 + 0.932978i \(0.617201\pi\)
\(150\) 0 0
\(151\) −161597. −0.576755 −0.288378 0.957517i \(-0.593116\pi\)
−0.288378 + 0.957517i \(0.593116\pi\)
\(152\) 0 0
\(153\) 919977. 3.17723
\(154\) 0 0
\(155\) 223524. 0.747300
\(156\) 0 0
\(157\) 108055. 0.349860 0.174930 0.984581i \(-0.444030\pi\)
0.174930 + 0.984581i \(0.444030\pi\)
\(158\) 0 0
\(159\) −827636. −2.59625
\(160\) 0 0
\(161\) 63915.9 0.194332
\(162\) 0 0
\(163\) −35513.9 −0.104696 −0.0523479 0.998629i \(-0.516670\pi\)
−0.0523479 + 0.998629i \(0.516670\pi\)
\(164\) 0 0
\(165\) 255196. 0.729733
\(166\) 0 0
\(167\) 377413. 1.04719 0.523595 0.851967i \(-0.324591\pi\)
0.523595 + 0.851967i \(0.324591\pi\)
\(168\) 0 0
\(169\) −306248. −0.824814
\(170\) 0 0
\(171\) 1.48598e6 3.88617
\(172\) 0 0
\(173\) −142405. −0.361752 −0.180876 0.983506i \(-0.557893\pi\)
−0.180876 + 0.983506i \(0.557893\pi\)
\(174\) 0 0
\(175\) 27828.3 0.0686897
\(176\) 0 0
\(177\) −103176. −0.247540
\(178\) 0 0
\(179\) −457275. −1.06671 −0.533353 0.845893i \(-0.679068\pi\)
−0.533353 + 0.845893i \(0.679068\pi\)
\(180\) 0 0
\(181\) −871860. −1.97811 −0.989055 0.147549i \(-0.952862\pi\)
−0.989055 + 0.147549i \(0.952862\pi\)
\(182\) 0 0
\(183\) 1.33408e6 2.94478
\(184\) 0 0
\(185\) 363416. 0.780683
\(186\) 0 0
\(187\) −525675. −1.09929
\(188\) 0 0
\(189\) 479199. 0.975802
\(190\) 0 0
\(191\) 330154. 0.654837 0.327418 0.944879i \(-0.393821\pi\)
0.327418 + 0.944879i \(0.393821\pi\)
\(192\) 0 0
\(193\) −152914. −0.295498 −0.147749 0.989025i \(-0.547203\pi\)
−0.147749 + 0.989025i \(0.547203\pi\)
\(194\) 0 0
\(195\) −186353. −0.350953
\(196\) 0 0
\(197\) −714867. −1.31238 −0.656190 0.754596i \(-0.727833\pi\)
−0.656190 + 0.754596i \(0.727833\pi\)
\(198\) 0 0
\(199\) 292872. 0.524258 0.262129 0.965033i \(-0.415575\pi\)
0.262129 + 0.965033i \(0.415575\pi\)
\(200\) 0 0
\(201\) −2.04151e6 −3.56420
\(202\) 0 0
\(203\) 127879. 0.217800
\(204\) 0 0
\(205\) −187600. −0.311780
\(206\) 0 0
\(207\) 877421. 1.42325
\(208\) 0 0
\(209\) −849085. −1.34458
\(210\) 0 0
\(211\) 365716. 0.565507 0.282754 0.959193i \(-0.408752\pi\)
0.282754 + 0.959193i \(0.408752\pi\)
\(212\) 0 0
\(213\) 1.75627e6 2.65242
\(214\) 0 0
\(215\) −337267. −0.497597
\(216\) 0 0
\(217\) −398099. −0.573907
\(218\) 0 0
\(219\) −1.43812e6 −2.02621
\(220\) 0 0
\(221\) 383866. 0.528687
\(222\) 0 0
\(223\) 748539. 1.00798 0.503990 0.863709i \(-0.331865\pi\)
0.503990 + 0.863709i \(0.331865\pi\)
\(224\) 0 0
\(225\) 382020. 0.503072
\(226\) 0 0
\(227\) 424142. 0.546319 0.273159 0.961969i \(-0.411931\pi\)
0.273159 + 0.961969i \(0.411931\pi\)
\(228\) 0 0
\(229\) 1.02868e6 1.29625 0.648126 0.761533i \(-0.275553\pi\)
0.648126 + 0.761533i \(0.275553\pi\)
\(230\) 0 0
\(231\) −454507. −0.560416
\(232\) 0 0
\(233\) 614128. 0.741087 0.370544 0.928815i \(-0.379171\pi\)
0.370544 + 0.928815i \(0.379171\pi\)
\(234\) 0 0
\(235\) −211242. −0.249523
\(236\) 0 0
\(237\) −28686.8 −0.0331750
\(238\) 0 0
\(239\) 1.36204e6 1.54239 0.771197 0.636597i \(-0.219658\pi\)
0.771197 + 0.636597i \(0.219658\pi\)
\(240\) 0 0
\(241\) 354377. 0.393027 0.196514 0.980501i \(-0.437038\pi\)
0.196514 + 0.980501i \(0.437038\pi\)
\(242\) 0 0
\(243\) 2.23722e6 2.43049
\(244\) 0 0
\(245\) 370612. 0.394462
\(246\) 0 0
\(247\) 620031. 0.646653
\(248\) 0 0
\(249\) −1.12152e6 −1.14632
\(250\) 0 0
\(251\) −826955. −0.828509 −0.414255 0.910161i \(-0.635958\pi\)
−0.414255 + 0.910161i \(0.635958\pi\)
\(252\) 0 0
\(253\) −501358. −0.492432
\(254\) 0 0
\(255\) −1.09976e6 −1.05913
\(256\) 0 0
\(257\) 1.47307e6 1.39121 0.695603 0.718426i \(-0.255137\pi\)
0.695603 + 0.718426i \(0.255137\pi\)
\(258\) 0 0
\(259\) −647248. −0.599545
\(260\) 0 0
\(261\) 1.75548e6 1.59513
\(262\) 0 0
\(263\) 669755. 0.597072 0.298536 0.954398i \(-0.403502\pi\)
0.298536 + 0.954398i \(0.403502\pi\)
\(264\) 0 0
\(265\) 707932. 0.619265
\(266\) 0 0
\(267\) 1.15210e6 0.989032
\(268\) 0 0
\(269\) −161730. −0.136273 −0.0681367 0.997676i \(-0.521705\pi\)
−0.0681367 + 0.997676i \(0.521705\pi\)
\(270\) 0 0
\(271\) −709813. −0.587112 −0.293556 0.955942i \(-0.594839\pi\)
−0.293556 + 0.955942i \(0.594839\pi\)
\(272\) 0 0
\(273\) 331897. 0.269523
\(274\) 0 0
\(275\) −218286. −0.174058
\(276\) 0 0
\(277\) 1.25136e6 0.979904 0.489952 0.871749i \(-0.337014\pi\)
0.489952 + 0.871749i \(0.337014\pi\)
\(278\) 0 0
\(279\) −5.46500e6 −4.20320
\(280\) 0 0
\(281\) −1.14718e6 −0.866692 −0.433346 0.901228i \(-0.642667\pi\)
−0.433346 + 0.901228i \(0.642667\pi\)
\(282\) 0 0
\(283\) −1.01151e6 −0.750766 −0.375383 0.926870i \(-0.622489\pi\)
−0.375383 + 0.926870i \(0.622489\pi\)
\(284\) 0 0
\(285\) −1.77637e6 −1.29545
\(286\) 0 0
\(287\) 334118. 0.239439
\(288\) 0 0
\(289\) 845528. 0.595502
\(290\) 0 0
\(291\) 1.50055e6 1.03877
\(292\) 0 0
\(293\) −161999. −0.110241 −0.0551204 0.998480i \(-0.517554\pi\)
−0.0551204 + 0.998480i \(0.517554\pi\)
\(294\) 0 0
\(295\) 88253.2 0.0590439
\(296\) 0 0
\(297\) −3.75885e6 −2.47266
\(298\) 0 0
\(299\) 366109. 0.236828
\(300\) 0 0
\(301\) 600677. 0.382142
\(302\) 0 0
\(303\) −1.50687e6 −0.942907
\(304\) 0 0
\(305\) −1.14112e6 −0.702398
\(306\) 0 0
\(307\) −1.20761e6 −0.731276 −0.365638 0.930757i \(-0.619149\pi\)
−0.365638 + 0.930757i \(0.619149\pi\)
\(308\) 0 0
\(309\) 186204. 0.110941
\(310\) 0 0
\(311\) 2.90780e6 1.70476 0.852382 0.522920i \(-0.175157\pi\)
0.852382 + 0.522920i \(0.175157\pi\)
\(312\) 0 0
\(313\) −1.27324e6 −0.734597 −0.367298 0.930103i \(-0.619717\pi\)
−0.367298 + 0.930103i \(0.619717\pi\)
\(314\) 0 0
\(315\) −680382. −0.386346
\(316\) 0 0
\(317\) 1.34305e6 0.750659 0.375330 0.926891i \(-0.377530\pi\)
0.375330 + 0.926891i \(0.377530\pi\)
\(318\) 0 0
\(319\) −1.00308e6 −0.551899
\(320\) 0 0
\(321\) −165255. −0.0895140
\(322\) 0 0
\(323\) 3.65912e6 1.95151
\(324\) 0 0
\(325\) 159400. 0.0837105
\(326\) 0 0
\(327\) −6.37016e6 −3.29444
\(328\) 0 0
\(329\) 376225. 0.191627
\(330\) 0 0
\(331\) −1.59250e6 −0.798929 −0.399464 0.916749i \(-0.630804\pi\)
−0.399464 + 0.916749i \(0.630804\pi\)
\(332\) 0 0
\(333\) −8.88526e6 −4.39096
\(334\) 0 0
\(335\) 1.74624e6 0.850143
\(336\) 0 0
\(337\) 1.34715e6 0.646163 0.323082 0.946371i \(-0.395281\pi\)
0.323082 + 0.946371i \(0.395281\pi\)
\(338\) 0 0
\(339\) −3.78169e6 −1.78726
\(340\) 0 0
\(341\) 3.12270e6 1.45427
\(342\) 0 0
\(343\) −1.40840e6 −0.646385
\(344\) 0 0
\(345\) −1.04889e6 −0.474441
\(346\) 0 0
\(347\) −1.61456e6 −0.719832 −0.359916 0.932985i \(-0.617195\pi\)
−0.359916 + 0.932985i \(0.617195\pi\)
\(348\) 0 0
\(349\) −1.78417e6 −0.784103 −0.392052 0.919943i \(-0.628235\pi\)
−0.392052 + 0.919943i \(0.628235\pi\)
\(350\) 0 0
\(351\) 2.74484e6 1.18919
\(352\) 0 0
\(353\) 1.02848e6 0.439299 0.219650 0.975579i \(-0.429509\pi\)
0.219650 + 0.975579i \(0.429509\pi\)
\(354\) 0 0
\(355\) −1.50226e6 −0.632664
\(356\) 0 0
\(357\) 1.95869e6 0.813383
\(358\) 0 0
\(359\) −719588. −0.294678 −0.147339 0.989086i \(-0.547071\pi\)
−0.147339 + 0.989086i \(0.547071\pi\)
\(360\) 0 0
\(361\) 3.43422e6 1.38695
\(362\) 0 0
\(363\) −1.14191e6 −0.454848
\(364\) 0 0
\(365\) 1.23012e6 0.483297
\(366\) 0 0
\(367\) −412535. −0.159881 −0.0799403 0.996800i \(-0.525473\pi\)
−0.0799403 + 0.996800i \(0.525473\pi\)
\(368\) 0 0
\(369\) 4.58669e6 1.75361
\(370\) 0 0
\(371\) −1.26083e6 −0.475580
\(372\) 0 0
\(373\) −4.01227e6 −1.49320 −0.746600 0.665273i \(-0.768315\pi\)
−0.746600 + 0.665273i \(0.768315\pi\)
\(374\) 0 0
\(375\) −456676. −0.167699
\(376\) 0 0
\(377\) 732485. 0.265427
\(378\) 0 0
\(379\) 725543. 0.259457 0.129728 0.991550i \(-0.458589\pi\)
0.129728 + 0.991550i \(0.458589\pi\)
\(380\) 0 0
\(381\) 3.10564e6 1.09607
\(382\) 0 0
\(383\) 4.16948e6 1.45240 0.726199 0.687485i \(-0.241285\pi\)
0.726199 + 0.687485i \(0.241285\pi\)
\(384\) 0 0
\(385\) 388770. 0.133672
\(386\) 0 0
\(387\) 8.24594e6 2.79874
\(388\) 0 0
\(389\) 710805. 0.238164 0.119082 0.992884i \(-0.462005\pi\)
0.119082 + 0.992884i \(0.462005\pi\)
\(390\) 0 0
\(391\) 2.16059e6 0.714712
\(392\) 0 0
\(393\) 3.88161e6 1.26774
\(394\) 0 0
\(395\) 24537.7 0.00791301
\(396\) 0 0
\(397\) 2.60119e6 0.828315 0.414157 0.910205i \(-0.364076\pi\)
0.414157 + 0.910205i \(0.364076\pi\)
\(398\) 0 0
\(399\) 3.16374e6 0.994874
\(400\) 0 0
\(401\) 1.82517e6 0.566817 0.283409 0.958999i \(-0.408535\pi\)
0.283409 + 0.958999i \(0.408535\pi\)
\(402\) 0 0
\(403\) −2.28030e6 −0.699407
\(404\) 0 0
\(405\) −4.15066e6 −1.25742
\(406\) 0 0
\(407\) 5.07703e6 1.51923
\(408\) 0 0
\(409\) −527537. −0.155935 −0.0779677 0.996956i \(-0.524843\pi\)
−0.0779677 + 0.996956i \(0.524843\pi\)
\(410\) 0 0
\(411\) −8.80241e6 −2.57038
\(412\) 0 0
\(413\) −157180. −0.0453442
\(414\) 0 0
\(415\) 959307. 0.273425
\(416\) 0 0
\(417\) −4.06319e6 −1.14427
\(418\) 0 0
\(419\) 2.11729e6 0.589175 0.294587 0.955625i \(-0.404818\pi\)
0.294587 + 0.955625i \(0.404818\pi\)
\(420\) 0 0
\(421\) 1.67556e6 0.460738 0.230369 0.973103i \(-0.426007\pi\)
0.230369 + 0.973103i \(0.426007\pi\)
\(422\) 0 0
\(423\) 5.16472e6 1.40345
\(424\) 0 0
\(425\) 940700. 0.252626
\(426\) 0 0
\(427\) 2.03235e6 0.539423
\(428\) 0 0
\(429\) −2.60341e6 −0.682965
\(430\) 0 0
\(431\) 1.44216e6 0.373957 0.186978 0.982364i \(-0.440131\pi\)
0.186978 + 0.982364i \(0.440131\pi\)
\(432\) 0 0
\(433\) 5.49118e6 1.40749 0.703746 0.710452i \(-0.251509\pi\)
0.703746 + 0.710452i \(0.251509\pi\)
\(434\) 0 0
\(435\) −2.09855e6 −0.531735
\(436\) 0 0
\(437\) 3.48986e6 0.874186
\(438\) 0 0
\(439\) −3.57088e6 −0.884329 −0.442165 0.896934i \(-0.645789\pi\)
−0.442165 + 0.896934i \(0.645789\pi\)
\(440\) 0 0
\(441\) −9.06121e6 −2.21865
\(442\) 0 0
\(443\) −2.15711e6 −0.522231 −0.261116 0.965308i \(-0.584090\pi\)
−0.261116 + 0.965308i \(0.584090\pi\)
\(444\) 0 0
\(445\) −985464. −0.235907
\(446\) 0 0
\(447\) −5.70174e6 −1.34970
\(448\) 0 0
\(449\) −4.15945e6 −0.973689 −0.486845 0.873489i \(-0.661852\pi\)
−0.486845 + 0.873489i \(0.661852\pi\)
\(450\) 0 0
\(451\) −2.62083e6 −0.606733
\(452\) 0 0
\(453\) −4.72304e6 −1.08138
\(454\) 0 0
\(455\) −283893. −0.0642875
\(456\) 0 0
\(457\) 6.32204e6 1.41601 0.708006 0.706207i \(-0.249595\pi\)
0.708006 + 0.706207i \(0.249595\pi\)
\(458\) 0 0
\(459\) 1.61987e7 3.58880
\(460\) 0 0
\(461\) −8.12007e6 −1.77954 −0.889769 0.456411i \(-0.849135\pi\)
−0.889769 + 0.456411i \(0.849135\pi\)
\(462\) 0 0
\(463\) −5.51746e6 −1.19615 −0.598076 0.801439i \(-0.704068\pi\)
−0.598076 + 0.801439i \(0.704068\pi\)
\(464\) 0 0
\(465\) 6.53299e6 1.40113
\(466\) 0 0
\(467\) 7.98287e6 1.69382 0.846909 0.531737i \(-0.178461\pi\)
0.846909 + 0.531737i \(0.178461\pi\)
\(468\) 0 0
\(469\) −3.11007e6 −0.652888
\(470\) 0 0
\(471\) 3.15814e6 0.655963
\(472\) 0 0
\(473\) −4.71172e6 −0.968338
\(474\) 0 0
\(475\) 1.51945e6 0.308995
\(476\) 0 0
\(477\) −1.73084e7 −3.48307
\(478\) 0 0
\(479\) 7.21796e6 1.43739 0.718697 0.695323i \(-0.244739\pi\)
0.718697 + 0.695323i \(0.244739\pi\)
\(480\) 0 0
\(481\) −3.70742e6 −0.730650
\(482\) 0 0
\(483\) 1.86809e6 0.364358
\(484\) 0 0
\(485\) −1.28352e6 −0.247770
\(486\) 0 0
\(487\) 1.74447e6 0.333305 0.166652 0.986016i \(-0.446704\pi\)
0.166652 + 0.986016i \(0.446704\pi\)
\(488\) 0 0
\(489\) −1.03797e6 −0.196297
\(490\) 0 0
\(491\) 9.10988e6 1.70533 0.852666 0.522457i \(-0.174985\pi\)
0.852666 + 0.522457i \(0.174985\pi\)
\(492\) 0 0
\(493\) 4.32277e6 0.801022
\(494\) 0 0
\(495\) 5.33694e6 0.978992
\(496\) 0 0
\(497\) 2.67553e6 0.485869
\(498\) 0 0
\(499\) −5.94465e6 −1.06875 −0.534374 0.845248i \(-0.679452\pi\)
−0.534374 + 0.845248i \(0.679452\pi\)
\(500\) 0 0
\(501\) 1.10307e7 1.96341
\(502\) 0 0
\(503\) −2.53195e6 −0.446207 −0.223103 0.974795i \(-0.571619\pi\)
−0.223103 + 0.974795i \(0.571619\pi\)
\(504\) 0 0
\(505\) 1.28892e6 0.224905
\(506\) 0 0
\(507\) −8.95078e6 −1.54647
\(508\) 0 0
\(509\) 4.91009e6 0.840030 0.420015 0.907517i \(-0.362025\pi\)
0.420015 + 0.907517i \(0.362025\pi\)
\(510\) 0 0
\(511\) −2.19085e6 −0.371160
\(512\) 0 0
\(513\) 2.61647e7 4.38957
\(514\) 0 0
\(515\) −159273. −0.0264621
\(516\) 0 0
\(517\) −2.95112e6 −0.485579
\(518\) 0 0
\(519\) −4.16211e6 −0.678259
\(520\) 0 0
\(521\) −5.90160e6 −0.952522 −0.476261 0.879304i \(-0.658008\pi\)
−0.476261 + 0.879304i \(0.658008\pi\)
\(522\) 0 0
\(523\) 6.73830e6 1.07720 0.538600 0.842562i \(-0.318953\pi\)
0.538600 + 0.842562i \(0.318953\pi\)
\(524\) 0 0
\(525\) 813345. 0.128788
\(526\) 0 0
\(527\) −1.34572e7 −2.11071
\(528\) 0 0
\(529\) −4.37569e6 −0.679842
\(530\) 0 0
\(531\) −2.15773e6 −0.332093
\(532\) 0 0
\(533\) 1.91382e6 0.291799
\(534\) 0 0
\(535\) 141353. 0.0213511
\(536\) 0 0
\(537\) −1.33649e7 −2.00000
\(538\) 0 0
\(539\) 5.17757e6 0.767634
\(540\) 0 0
\(541\) −6.56711e6 −0.964675 −0.482337 0.875986i \(-0.660212\pi\)
−0.482337 + 0.875986i \(0.660212\pi\)
\(542\) 0 0
\(543\) −2.54821e7 −3.70881
\(544\) 0 0
\(545\) 5.44882e6 0.785799
\(546\) 0 0
\(547\) −1.27471e7 −1.82156 −0.910778 0.412896i \(-0.864517\pi\)
−0.910778 + 0.412896i \(0.864517\pi\)
\(548\) 0 0
\(549\) 2.78997e7 3.95064
\(550\) 0 0
\(551\) 6.98226e6 0.979755
\(552\) 0 0
\(553\) −43702.0 −0.00607699
\(554\) 0 0
\(555\) 1.06217e7 1.46373
\(556\) 0 0
\(557\) 1.22479e7 1.67273 0.836364 0.548175i \(-0.184677\pi\)
0.836364 + 0.548175i \(0.184677\pi\)
\(558\) 0 0
\(559\) 3.44066e6 0.465707
\(560\) 0 0
\(561\) −1.53640e7 −2.06109
\(562\) 0 0
\(563\) −2.23945e6 −0.297763 −0.148882 0.988855i \(-0.547567\pi\)
−0.148882 + 0.988855i \(0.547567\pi\)
\(564\) 0 0
\(565\) 3.23473e6 0.426302
\(566\) 0 0
\(567\) 7.39237e6 0.965664
\(568\) 0 0
\(569\) −3.95223e6 −0.511755 −0.255877 0.966709i \(-0.582364\pi\)
−0.255877 + 0.966709i \(0.582364\pi\)
\(570\) 0 0
\(571\) −6.99947e6 −0.898411 −0.449206 0.893428i \(-0.648293\pi\)
−0.449206 + 0.893428i \(0.648293\pi\)
\(572\) 0 0
\(573\) 9.64949e6 1.22777
\(574\) 0 0
\(575\) 897185. 0.113165
\(576\) 0 0
\(577\) 3.20338e6 0.400561 0.200280 0.979739i \(-0.435815\pi\)
0.200280 + 0.979739i \(0.435815\pi\)
\(578\) 0 0
\(579\) −4.46927e6 −0.554039
\(580\) 0 0
\(581\) −1.70854e6 −0.209983
\(582\) 0 0
\(583\) 9.89002e6 1.20511
\(584\) 0 0
\(585\) −3.89721e6 −0.470831
\(586\) 0 0
\(587\) 1.35926e7 1.62820 0.814102 0.580722i \(-0.197230\pi\)
0.814102 + 0.580722i \(0.197230\pi\)
\(588\) 0 0
\(589\) −2.17365e7 −2.58168
\(590\) 0 0
\(591\) −2.08936e7 −2.46062
\(592\) 0 0
\(593\) 713635. 0.0833373 0.0416687 0.999131i \(-0.486733\pi\)
0.0416687 + 0.999131i \(0.486733\pi\)
\(594\) 0 0
\(595\) −1.67540e6 −0.194011
\(596\) 0 0
\(597\) 8.55984e6 0.982946
\(598\) 0 0
\(599\) 5.09641e6 0.580359 0.290180 0.956972i \(-0.406285\pi\)
0.290180 + 0.956972i \(0.406285\pi\)
\(600\) 0 0
\(601\) 136310. 0.0153937 0.00769683 0.999970i \(-0.497550\pi\)
0.00769683 + 0.999970i \(0.497550\pi\)
\(602\) 0 0
\(603\) −4.26943e7 −4.78164
\(604\) 0 0
\(605\) 976753. 0.108492
\(606\) 0 0
\(607\) 6.23096e6 0.686409 0.343204 0.939261i \(-0.388488\pi\)
0.343204 + 0.939261i \(0.388488\pi\)
\(608\) 0 0
\(609\) 3.73754e6 0.408359
\(610\) 0 0
\(611\) 2.15501e6 0.233532
\(612\) 0 0
\(613\) −1.37774e7 −1.48087 −0.740435 0.672128i \(-0.765380\pi\)
−0.740435 + 0.672128i \(0.765380\pi\)
\(614\) 0 0
\(615\) −5.48304e6 −0.584566
\(616\) 0 0
\(617\) −8.85147e6 −0.936058 −0.468029 0.883713i \(-0.655036\pi\)
−0.468029 + 0.883713i \(0.655036\pi\)
\(618\) 0 0
\(619\) 1.22847e7 1.28866 0.644329 0.764748i \(-0.277137\pi\)
0.644329 + 0.764748i \(0.277137\pi\)
\(620\) 0 0
\(621\) 1.54494e7 1.60762
\(622\) 0 0
\(623\) 1.75512e6 0.181170
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) −2.48164e7 −2.52099
\(628\) 0 0
\(629\) −2.18794e7 −2.20500
\(630\) 0 0
\(631\) −1.75131e7 −1.75101 −0.875506 0.483207i \(-0.839472\pi\)
−0.875506 + 0.483207i \(0.839472\pi\)
\(632\) 0 0
\(633\) 1.06889e7 1.06029
\(634\) 0 0
\(635\) −2.65646e6 −0.261438
\(636\) 0 0
\(637\) −3.78084e6 −0.369181
\(638\) 0 0
\(639\) 3.67291e7 3.55842
\(640\) 0 0
\(641\) −1.24879e7 −1.20045 −0.600223 0.799832i \(-0.704922\pi\)
−0.600223 + 0.799832i \(0.704922\pi\)
\(642\) 0 0
\(643\) 5.50652e6 0.525231 0.262615 0.964901i \(-0.415415\pi\)
0.262615 + 0.964901i \(0.415415\pi\)
\(644\) 0 0
\(645\) −9.85739e6 −0.932959
\(646\) 0 0
\(647\) 2.74623e6 0.257915 0.128957 0.991650i \(-0.458837\pi\)
0.128957 + 0.991650i \(0.458837\pi\)
\(648\) 0 0
\(649\) 1.23292e6 0.114901
\(650\) 0 0
\(651\) −1.16353e7 −1.07604
\(652\) 0 0
\(653\) −1.24705e7 −1.14446 −0.572232 0.820092i \(-0.693922\pi\)
−0.572232 + 0.820092i \(0.693922\pi\)
\(654\) 0 0
\(655\) −3.32020e6 −0.302386
\(656\) 0 0
\(657\) −3.00755e7 −2.71831
\(658\) 0 0
\(659\) −1.75360e7 −1.57296 −0.786478 0.617618i \(-0.788098\pi\)
−0.786478 + 0.617618i \(0.788098\pi\)
\(660\) 0 0
\(661\) 9.04528e6 0.805227 0.402613 0.915370i \(-0.368102\pi\)
0.402613 + 0.915370i \(0.368102\pi\)
\(662\) 0 0
\(663\) 1.12193e7 0.991250
\(664\) 0 0
\(665\) −2.70615e6 −0.237300
\(666\) 0 0
\(667\) 4.12281e6 0.358821
\(668\) 0 0
\(669\) 2.18777e7 1.88989
\(670\) 0 0
\(671\) −1.59418e7 −1.36689
\(672\) 0 0
\(673\) −9.09208e6 −0.773794 −0.386897 0.922123i \(-0.626453\pi\)
−0.386897 + 0.922123i \(0.626453\pi\)
\(674\) 0 0
\(675\) 6.72651e6 0.568238
\(676\) 0 0
\(677\) 1.55584e7 1.30465 0.652325 0.757940i \(-0.273794\pi\)
0.652325 + 0.757940i \(0.273794\pi\)
\(678\) 0 0
\(679\) 2.28597e6 0.190281
\(680\) 0 0
\(681\) 1.23965e7 1.02431
\(682\) 0 0
\(683\) 5.22623e6 0.428684 0.214342 0.976759i \(-0.431239\pi\)
0.214342 + 0.976759i \(0.431239\pi\)
\(684\) 0 0
\(685\) 7.52928e6 0.613094
\(686\) 0 0
\(687\) 3.00653e7 2.43038
\(688\) 0 0
\(689\) −7.22203e6 −0.579577
\(690\) 0 0
\(691\) 2.01073e7 1.60199 0.800994 0.598672i \(-0.204305\pi\)
0.800994 + 0.598672i \(0.204305\pi\)
\(692\) 0 0
\(693\) −9.50515e6 −0.751840
\(694\) 0 0
\(695\) 3.47551e6 0.272934
\(696\) 0 0
\(697\) 1.12944e7 0.880608
\(698\) 0 0
\(699\) 1.79493e7 1.38949
\(700\) 0 0
\(701\) 1.57869e7 1.21339 0.606695 0.794934i \(-0.292495\pi\)
0.606695 + 0.794934i \(0.292495\pi\)
\(702\) 0 0
\(703\) −3.53402e7 −2.69700
\(704\) 0 0
\(705\) −6.17403e6 −0.467838
\(706\) 0 0
\(707\) −2.29559e6 −0.172721
\(708\) 0 0
\(709\) −5.13552e6 −0.383680 −0.191840 0.981426i \(-0.561445\pi\)
−0.191840 + 0.981426i \(0.561445\pi\)
\(710\) 0 0
\(711\) −599930. −0.0445068
\(712\) 0 0
\(713\) −1.28347e7 −0.945502
\(714\) 0 0
\(715\) 2.22687e6 0.162903
\(716\) 0 0
\(717\) 3.98087e7 2.89188
\(718\) 0 0
\(719\) 2.90785e6 0.209773 0.104886 0.994484i \(-0.466552\pi\)
0.104886 + 0.994484i \(0.466552\pi\)
\(720\) 0 0
\(721\) 283667. 0.0203222
\(722\) 0 0
\(723\) 1.03575e7 0.736898
\(724\) 0 0
\(725\) 1.79503e6 0.126831
\(726\) 0 0
\(727\) −7.85123e6 −0.550937 −0.275468 0.961310i \(-0.588833\pi\)
−0.275468 + 0.961310i \(0.588833\pi\)
\(728\) 0 0
\(729\) 2.50435e7 1.74533
\(730\) 0 0
\(731\) 2.03051e7 1.40544
\(732\) 0 0
\(733\) −1.41173e7 −0.970491 −0.485245 0.874378i \(-0.661270\pi\)
−0.485245 + 0.874378i \(0.661270\pi\)
\(734\) 0 0
\(735\) 1.08320e7 0.739588
\(736\) 0 0
\(737\) 2.43955e7 1.65440
\(738\) 0 0
\(739\) −4.20049e6 −0.282937 −0.141468 0.989943i \(-0.545182\pi\)
−0.141468 + 0.989943i \(0.545182\pi\)
\(740\) 0 0
\(741\) 1.81218e7 1.21243
\(742\) 0 0
\(743\) 1.18490e6 0.0787425 0.0393712 0.999225i \(-0.487465\pi\)
0.0393712 + 0.999225i \(0.487465\pi\)
\(744\) 0 0
\(745\) 4.87707e6 0.321935
\(746\) 0 0
\(747\) −2.34544e7 −1.53788
\(748\) 0 0
\(749\) −251752. −0.0163971
\(750\) 0 0
\(751\) −3.94605e6 −0.255307 −0.127653 0.991819i \(-0.540745\pi\)
−0.127653 + 0.991819i \(0.540745\pi\)
\(752\) 0 0
\(753\) −2.41696e7 −1.55340
\(754\) 0 0
\(755\) 4.03993e6 0.257933
\(756\) 0 0
\(757\) −1.22109e6 −0.0774479 −0.0387239 0.999250i \(-0.512329\pi\)
−0.0387239 + 0.999250i \(0.512329\pi\)
\(758\) 0 0
\(759\) −1.46533e7 −0.923276
\(760\) 0 0
\(761\) 1.91652e7 1.19964 0.599822 0.800134i \(-0.295238\pi\)
0.599822 + 0.800134i \(0.295238\pi\)
\(762\) 0 0
\(763\) −9.70442e6 −0.603473
\(764\) 0 0
\(765\) −2.29994e7 −1.42090
\(766\) 0 0
\(767\) −900323. −0.0552599
\(768\) 0 0
\(769\) 9.50951e6 0.579886 0.289943 0.957044i \(-0.406364\pi\)
0.289943 + 0.957044i \(0.406364\pi\)
\(770\) 0 0
\(771\) 4.30539e7 2.60841
\(772\) 0 0
\(773\) 526263. 0.0316777 0.0158389 0.999875i \(-0.494958\pi\)
0.0158389 + 0.999875i \(0.494958\pi\)
\(774\) 0 0
\(775\) −5.58810e6 −0.334203
\(776\) 0 0
\(777\) −1.89173e7 −1.12410
\(778\) 0 0
\(779\) 1.82431e7 1.07710
\(780\) 0 0
\(781\) −2.09870e7 −1.23118
\(782\) 0 0
\(783\) 3.09101e7 1.80176
\(784\) 0 0
\(785\) −2.70137e6 −0.156462
\(786\) 0 0
\(787\) 2.05915e7 1.18509 0.592544 0.805538i \(-0.298123\pi\)
0.592544 + 0.805538i \(0.298123\pi\)
\(788\) 0 0
\(789\) 1.95751e7 1.11947
\(790\) 0 0
\(791\) −5.76109e6 −0.327389
\(792\) 0 0
\(793\) 1.16413e7 0.657382
\(794\) 0 0
\(795\) 2.06909e7 1.16108
\(796\) 0 0
\(797\) −2.08952e7 −1.16520 −0.582599 0.812760i \(-0.697964\pi\)
−0.582599 + 0.812760i \(0.697964\pi\)
\(798\) 0 0
\(799\) 1.27178e7 0.704766
\(800\) 0 0
\(801\) 2.40939e7 1.32686
\(802\) 0 0
\(803\) 1.71851e7 0.940509
\(804\) 0 0
\(805\) −1.59790e6 −0.0869079
\(806\) 0 0
\(807\) −4.72694e6 −0.255503
\(808\) 0 0
\(809\) −3.38823e7 −1.82013 −0.910063 0.414470i \(-0.863967\pi\)
−0.910063 + 0.414470i \(0.863967\pi\)
\(810\) 0 0
\(811\) 8.25428e6 0.440684 0.220342 0.975423i \(-0.429283\pi\)
0.220342 + 0.975423i \(0.429283\pi\)
\(812\) 0 0
\(813\) −2.07459e7 −1.10079
\(814\) 0 0
\(815\) 887847. 0.0468214
\(816\) 0 0
\(817\) 3.27974e7 1.71903
\(818\) 0 0
\(819\) 6.94098e6 0.361586
\(820\) 0 0
\(821\) 1.99898e6 0.103503 0.0517513 0.998660i \(-0.483520\pi\)
0.0517513 + 0.998660i \(0.483520\pi\)
\(822\) 0 0
\(823\) 1.28086e7 0.659176 0.329588 0.944125i \(-0.393090\pi\)
0.329588 + 0.944125i \(0.393090\pi\)
\(824\) 0 0
\(825\) −6.37990e6 −0.326347
\(826\) 0 0
\(827\) −2.83308e7 −1.44044 −0.720221 0.693745i \(-0.755960\pi\)
−0.720221 + 0.693745i \(0.755960\pi\)
\(828\) 0 0
\(829\) 7.91663e6 0.400087 0.200043 0.979787i \(-0.435892\pi\)
0.200043 + 0.979787i \(0.435892\pi\)
\(830\) 0 0
\(831\) 3.65739e7 1.83725
\(832\) 0 0
\(833\) −2.23126e7 −1.11414
\(834\) 0 0
\(835\) −9.43532e6 −0.468317
\(836\) 0 0
\(837\) −9.62263e7 −4.74767
\(838\) 0 0
\(839\) −2.18676e7 −1.07250 −0.536249 0.844060i \(-0.680159\pi\)
−0.536249 + 0.844060i \(0.680159\pi\)
\(840\) 0 0
\(841\) −1.22625e7 −0.597847
\(842\) 0 0
\(843\) −3.35289e7 −1.62499
\(844\) 0 0
\(845\) 7.65619e6 0.368868
\(846\) 0 0
\(847\) −1.73961e6 −0.0833188
\(848\) 0 0
\(849\) −2.95637e7 −1.40763
\(850\) 0 0
\(851\) −2.08673e7 −0.987739
\(852\) 0 0
\(853\) −6.73771e6 −0.317059 −0.158529 0.987354i \(-0.550675\pi\)
−0.158529 + 0.987354i \(0.550675\pi\)
\(854\) 0 0
\(855\) −3.71494e7 −1.73795
\(856\) 0 0
\(857\) 2.01368e7 0.936565 0.468283 0.883579i \(-0.344873\pi\)
0.468283 + 0.883579i \(0.344873\pi\)
\(858\) 0 0
\(859\) −2.31020e7 −1.06823 −0.534117 0.845410i \(-0.679356\pi\)
−0.534117 + 0.845410i \(0.679356\pi\)
\(860\) 0 0
\(861\) 9.76535e6 0.448932
\(862\) 0 0
\(863\) −2.79582e7 −1.27785 −0.638927 0.769267i \(-0.720622\pi\)
−0.638927 + 0.769267i \(0.720622\pi\)
\(864\) 0 0
\(865\) 3.56013e6 0.161780
\(866\) 0 0
\(867\) 2.47125e7 1.11652
\(868\) 0 0
\(869\) 342800. 0.0153989
\(870\) 0 0
\(871\) −1.78144e7 −0.795658
\(872\) 0 0
\(873\) 3.13812e7 1.39359
\(874\) 0 0
\(875\) −695708. −0.0307190
\(876\) 0 0
\(877\) 3.75058e7 1.64664 0.823322 0.567574i \(-0.192118\pi\)
0.823322 + 0.567574i \(0.192118\pi\)
\(878\) 0 0
\(879\) −4.73477e6 −0.206693
\(880\) 0 0
\(881\) −1.94116e6 −0.0842601 −0.0421300 0.999112i \(-0.513414\pi\)
−0.0421300 + 0.999112i \(0.513414\pi\)
\(882\) 0 0
\(883\) 3.90752e7 1.68655 0.843276 0.537481i \(-0.180624\pi\)
0.843276 + 0.537481i \(0.180624\pi\)
\(884\) 0 0
\(885\) 2.57940e6 0.110703
\(886\) 0 0
\(887\) 1.87662e7 0.800878 0.400439 0.916323i \(-0.368858\pi\)
0.400439 + 0.916323i \(0.368858\pi\)
\(888\) 0 0
\(889\) 4.73119e6 0.200778
\(890\) 0 0
\(891\) −5.79859e7 −2.44697
\(892\) 0 0
\(893\) 2.05422e7 0.862020
\(894\) 0 0
\(895\) 1.14319e7 0.477045
\(896\) 0 0
\(897\) 1.07004e7 0.444035
\(898\) 0 0
\(899\) −2.56788e7 −1.05968
\(900\) 0 0
\(901\) −4.26209e7 −1.74908
\(902\) 0 0
\(903\) 1.75561e7 0.716489
\(904\) 0 0
\(905\) 2.17965e7 0.884637
\(906\) 0 0
\(907\) 2.79416e7 1.12780 0.563901 0.825842i \(-0.309300\pi\)
0.563901 + 0.825842i \(0.309300\pi\)
\(908\) 0 0
\(909\) −3.15133e7 −1.26498
\(910\) 0 0
\(911\) −1.32374e7 −0.528452 −0.264226 0.964461i \(-0.585116\pi\)
−0.264226 + 0.964461i \(0.585116\pi\)
\(912\) 0 0
\(913\) 1.34018e7 0.532092
\(914\) 0 0
\(915\) −3.33519e7 −1.31695
\(916\) 0 0
\(917\) 5.91332e6 0.232224
\(918\) 0 0
\(919\) 2.80433e7 1.09532 0.547660 0.836701i \(-0.315519\pi\)
0.547660 + 0.836701i \(0.315519\pi\)
\(920\) 0 0
\(921\) −3.52951e7 −1.37109
\(922\) 0 0
\(923\) 1.53254e7 0.592117
\(924\) 0 0
\(925\) −9.08540e6 −0.349132
\(926\) 0 0
\(927\) 3.89411e6 0.148836
\(928\) 0 0
\(929\) 1.96341e7 0.746400 0.373200 0.927751i \(-0.378261\pi\)
0.373200 + 0.927751i \(0.378261\pi\)
\(930\) 0 0
\(931\) −3.60401e7 −1.36273
\(932\) 0 0
\(933\) 8.49871e7 3.19631
\(934\) 0 0
\(935\) 1.31419e7 0.491618
\(936\) 0 0
\(937\) 2.71944e7 1.01188 0.505942 0.862567i \(-0.331145\pi\)
0.505942 + 0.862567i \(0.331145\pi\)
\(938\) 0 0
\(939\) −3.72133e7 −1.37732
\(940\) 0 0
\(941\) 2.74326e7 1.00993 0.504966 0.863139i \(-0.331505\pi\)
0.504966 + 0.863139i \(0.331505\pi\)
\(942\) 0 0
\(943\) 1.07720e7 0.394472
\(944\) 0 0
\(945\) −1.19800e7 −0.436392
\(946\) 0 0
\(947\) −1.42413e7 −0.516029 −0.258014 0.966141i \(-0.583068\pi\)
−0.258014 + 0.966141i \(0.583068\pi\)
\(948\) 0 0
\(949\) −1.25491e7 −0.452323
\(950\) 0 0
\(951\) 3.92535e7 1.40743
\(952\) 0 0
\(953\) 3.25786e7 1.16198 0.580992 0.813910i \(-0.302665\pi\)
0.580992 + 0.813910i \(0.302665\pi\)
\(954\) 0 0
\(955\) −8.25385e6 −0.292852
\(956\) 0 0
\(957\) −2.93173e7 −1.03477
\(958\) 0 0
\(959\) −1.34097e7 −0.470841
\(960\) 0 0
\(961\) 5.13117e7 1.79229
\(962\) 0 0
\(963\) −3.45598e6 −0.120090
\(964\) 0 0
\(965\) 3.82286e6 0.132151
\(966\) 0 0
\(967\) 4.03932e6 0.138913 0.0694564 0.997585i \(-0.477874\pi\)
0.0694564 + 0.997585i \(0.477874\pi\)
\(968\) 0 0
\(969\) 1.06946e8 3.65894
\(970\) 0 0
\(971\) −2.98023e7 −1.01438 −0.507191 0.861834i \(-0.669316\pi\)
−0.507191 + 0.861834i \(0.669316\pi\)
\(972\) 0 0
\(973\) −6.18993e6 −0.209606
\(974\) 0 0
\(975\) 4.65882e6 0.156951
\(976\) 0 0
\(977\) −2.70486e7 −0.906586 −0.453293 0.891362i \(-0.649751\pi\)
−0.453293 + 0.891362i \(0.649751\pi\)
\(978\) 0 0
\(979\) −1.37672e7 −0.459082
\(980\) 0 0
\(981\) −1.33220e8 −4.41974
\(982\) 0 0
\(983\) −3.15761e7 −1.04226 −0.521128 0.853479i \(-0.674489\pi\)
−0.521128 + 0.853479i \(0.674489\pi\)
\(984\) 0 0
\(985\) 1.78717e7 0.586914
\(986\) 0 0
\(987\) 1.09960e7 0.359288
\(988\) 0 0
\(989\) 1.93658e7 0.629572
\(990\) 0 0
\(991\) 6.49963e6 0.210235 0.105117 0.994460i \(-0.466478\pi\)
0.105117 + 0.994460i \(0.466478\pi\)
\(992\) 0 0
\(993\) −4.65443e7 −1.49794
\(994\) 0 0
\(995\) −7.32180e6 −0.234455
\(996\) 0 0
\(997\) −1.34950e7 −0.429966 −0.214983 0.976618i \(-0.568970\pi\)
−0.214983 + 0.976618i \(0.568970\pi\)
\(998\) 0 0
\(999\) −1.56449e8 −4.95975
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 160.6.a.g.1.3 yes 3
4.3 odd 2 160.6.a.f.1.1 3
5.2 odd 4 800.6.c.j.449.1 6
5.3 odd 4 800.6.c.j.449.6 6
5.4 even 2 800.6.a.n.1.1 3
8.3 odd 2 320.6.a.y.1.3 3
8.5 even 2 320.6.a.x.1.1 3
20.3 even 4 800.6.c.k.449.1 6
20.7 even 4 800.6.c.k.449.6 6
20.19 odd 2 800.6.a.o.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.a.f.1.1 3 4.3 odd 2
160.6.a.g.1.3 yes 3 1.1 even 1 trivial
320.6.a.x.1.1 3 8.5 even 2
320.6.a.y.1.3 3 8.3 odd 2
800.6.a.n.1.1 3 5.4 even 2
800.6.a.o.1.3 3 20.19 odd 2
800.6.c.j.449.1 6 5.2 odd 4
800.6.c.j.449.6 6 5.3 odd 4
800.6.c.k.449.1 6 20.3 even 4
800.6.c.k.449.6 6 20.7 even 4