Properties

Label 192.6.a.r.1.1
Level $192$
Weight $6$
Character 192.1
Self dual yes
Analytic conductor $30.794$
Analytic rank $0$
Dimension $2$
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] = [192,6,Mod(1,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, 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("192.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 192.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: \(30.7936934041\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.56776\) of defining polynomial
Character \(\chi\) \(=\) 192.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} -107.084 q^{5} -149.084 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+9.00000 q^{3} -107.084 q^{5} -149.084 q^{7} +81.0000 q^{9} -434.505 q^{11} +392.505 q^{13} -963.758 q^{15} +803.495 q^{17} +854.842 q^{19} -1341.76 q^{21} +4592.53 q^{23} +8342.03 q^{25} +729.000 q^{27} -6798.60 q^{29} +4798.58 q^{31} -3910.55 q^{33} +15964.6 q^{35} +909.075 q^{37} +3532.55 q^{39} -2372.21 q^{41} -8664.95 q^{43} -8673.82 q^{45} -16878.7 q^{47} +5419.11 q^{49} +7231.45 q^{51} -10831.0 q^{53} +46528.7 q^{55} +7693.58 q^{57} +8305.81 q^{59} +35998.0 q^{61} -12075.8 q^{63} -42031.1 q^{65} +25077.1 q^{67} +41332.7 q^{69} +57850.9 q^{71} -68092.5 q^{73} +75078.3 q^{75} +64777.9 q^{77} +98723.8 q^{79} +6561.00 q^{81} +34859.3 q^{83} -86041.6 q^{85} -61187.4 q^{87} +34678.2 q^{89} -58516.4 q^{91} +43187.2 q^{93} -91540.1 q^{95} +39891.1 q^{97} -35194.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} - 36 q^{5} - 120 q^{7} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 18 q^{3} - 36 q^{5} - 120 q^{7} + 162 q^{9} + 200 q^{11} - 284 q^{13} - 324 q^{15} + 2676 q^{17} - 72 q^{19} - 1080 q^{21} + 3840 q^{23} + 10270 q^{25} + 1458 q^{27} - 10212 q^{29} + 10488 q^{31} + 1800 q^{33} + 18032 q^{35} - 13148 q^{37} - 2556 q^{39} + 4164 q^{41} + 5832 q^{43} - 2916 q^{45} + 1520 q^{47} - 10542 q^{49} + 24084 q^{51} - 9012 q^{53} + 91632 q^{55} - 648 q^{57} + 55096 q^{59} + 63444 q^{61} - 9720 q^{63} - 90120 q^{65} - 36792 q^{67} + 34560 q^{69} + 37664 q^{71} - 37836 q^{73} + 92430 q^{75} + 83232 q^{77} + 144888 q^{79} + 13122 q^{81} + 109272 q^{83} + 47064 q^{85} - 91908 q^{87} - 32556 q^{89} - 78192 q^{91} + 94392 q^{93} - 157424 q^{95} + 69092 q^{97} + 16200 q^{99}+O(q^{100}) \) 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
\(3\) 9.00000 0.577350
\(4\) 0 0
\(5\) −107.084 −1.91558 −0.957790 0.287467i \(-0.907187\pi\)
−0.957790 + 0.287467i \(0.907187\pi\)
\(6\) 0 0
\(7\) −149.084 −1.14997 −0.574985 0.818164i \(-0.694992\pi\)
−0.574985 + 0.818164i \(0.694992\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −434.505 −1.08271 −0.541357 0.840793i \(-0.682089\pi\)
−0.541357 + 0.840793i \(0.682089\pi\)
\(12\) 0 0
\(13\) 392.505 0.644150 0.322075 0.946714i \(-0.395620\pi\)
0.322075 + 0.946714i \(0.395620\pi\)
\(14\) 0 0
\(15\) −963.758 −1.10596
\(16\) 0 0
\(17\) 803.495 0.674312 0.337156 0.941449i \(-0.390535\pi\)
0.337156 + 0.941449i \(0.390535\pi\)
\(18\) 0 0
\(19\) 854.842 0.543253 0.271626 0.962403i \(-0.412438\pi\)
0.271626 + 0.962403i \(0.412438\pi\)
\(20\) 0 0
\(21\) −1341.76 −0.663936
\(22\) 0 0
\(23\) 4592.53 1.81022 0.905112 0.425174i \(-0.139787\pi\)
0.905112 + 0.425174i \(0.139787\pi\)
\(24\) 0 0
\(25\) 8342.03 2.66945
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −6798.60 −1.50115 −0.750576 0.660784i \(-0.770224\pi\)
−0.750576 + 0.660784i \(0.770224\pi\)
\(30\) 0 0
\(31\) 4798.58 0.896826 0.448413 0.893826i \(-0.351989\pi\)
0.448413 + 0.893826i \(0.351989\pi\)
\(32\) 0 0
\(33\) −3910.55 −0.625105
\(34\) 0 0
\(35\) 15964.6 2.20286
\(36\) 0 0
\(37\) 909.075 0.109168 0.0545840 0.998509i \(-0.482617\pi\)
0.0545840 + 0.998509i \(0.482617\pi\)
\(38\) 0 0
\(39\) 3532.55 0.371900
\(40\) 0 0
\(41\) −2372.21 −0.220391 −0.110195 0.993910i \(-0.535148\pi\)
−0.110195 + 0.993910i \(0.535148\pi\)
\(42\) 0 0
\(43\) −8664.95 −0.714652 −0.357326 0.933980i \(-0.616312\pi\)
−0.357326 + 0.933980i \(0.616312\pi\)
\(44\) 0 0
\(45\) −8673.82 −0.638527
\(46\) 0 0
\(47\) −16878.7 −1.11454 −0.557268 0.830333i \(-0.688150\pi\)
−0.557268 + 0.830333i \(0.688150\pi\)
\(48\) 0 0
\(49\) 5419.11 0.322432
\(50\) 0 0
\(51\) 7231.45 0.389314
\(52\) 0 0
\(53\) −10831.0 −0.529637 −0.264818 0.964298i \(-0.585312\pi\)
−0.264818 + 0.964298i \(0.585312\pi\)
\(54\) 0 0
\(55\) 46528.7 2.07402
\(56\) 0 0
\(57\) 7693.58 0.313647
\(58\) 0 0
\(59\) 8305.81 0.310636 0.155318 0.987865i \(-0.450360\pi\)
0.155318 + 0.987865i \(0.450360\pi\)
\(60\) 0 0
\(61\) 35998.0 1.23867 0.619333 0.785128i \(-0.287403\pi\)
0.619333 + 0.785128i \(0.287403\pi\)
\(62\) 0 0
\(63\) −12075.8 −0.383323
\(64\) 0 0
\(65\) −42031.1 −1.23392
\(66\) 0 0
\(67\) 25077.1 0.682481 0.341240 0.939976i \(-0.389153\pi\)
0.341240 + 0.939976i \(0.389153\pi\)
\(68\) 0 0
\(69\) 41332.7 1.04513
\(70\) 0 0
\(71\) 57850.9 1.36196 0.680980 0.732302i \(-0.261554\pi\)
0.680980 + 0.732302i \(0.261554\pi\)
\(72\) 0 0
\(73\) −68092.5 −1.49552 −0.747760 0.663969i \(-0.768871\pi\)
−0.747760 + 0.663969i \(0.768871\pi\)
\(74\) 0 0
\(75\) 75078.3 1.54121
\(76\) 0 0
\(77\) 64777.9 1.24509
\(78\) 0 0
\(79\) 98723.8 1.77973 0.889865 0.456223i \(-0.150798\pi\)
0.889865 + 0.456223i \(0.150798\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 34859.3 0.555422 0.277711 0.960665i \(-0.410424\pi\)
0.277711 + 0.960665i \(0.410424\pi\)
\(84\) 0 0
\(85\) −86041.6 −1.29170
\(86\) 0 0
\(87\) −61187.4 −0.866690
\(88\) 0 0
\(89\) 34678.2 0.464068 0.232034 0.972708i \(-0.425462\pi\)
0.232034 + 0.972708i \(0.425462\pi\)
\(90\) 0 0
\(91\) −58516.4 −0.740754
\(92\) 0 0
\(93\) 43187.2 0.517783
\(94\) 0 0
\(95\) −91540.1 −1.04064
\(96\) 0 0
\(97\) 39891.1 0.430473 0.215237 0.976562i \(-0.430948\pi\)
0.215237 + 0.976562i \(0.430948\pi\)
\(98\) 0 0
\(99\) −35194.9 −0.360904
\(100\) 0 0
\(101\) −35341.5 −0.344732 −0.172366 0.985033i \(-0.555141\pi\)
−0.172366 + 0.985033i \(0.555141\pi\)
\(102\) 0 0
\(103\) −37097.4 −0.344549 −0.172274 0.985049i \(-0.555112\pi\)
−0.172274 + 0.985049i \(0.555112\pi\)
\(104\) 0 0
\(105\) 143681. 1.27182
\(106\) 0 0
\(107\) 153566. 1.29669 0.648345 0.761347i \(-0.275462\pi\)
0.648345 + 0.761347i \(0.275462\pi\)
\(108\) 0 0
\(109\) 106749. 0.860590 0.430295 0.902688i \(-0.358410\pi\)
0.430295 + 0.902688i \(0.358410\pi\)
\(110\) 0 0
\(111\) 8181.68 0.0630282
\(112\) 0 0
\(113\) −102232. −0.753168 −0.376584 0.926383i \(-0.622901\pi\)
−0.376584 + 0.926383i \(0.622901\pi\)
\(114\) 0 0
\(115\) −491787. −3.46763
\(116\) 0 0
\(117\) 31792.9 0.214717
\(118\) 0 0
\(119\) −119788. −0.775438
\(120\) 0 0
\(121\) 27743.9 0.172268
\(122\) 0 0
\(123\) −21349.9 −0.127243
\(124\) 0 0
\(125\) −558662. −3.19797
\(126\) 0 0
\(127\) 47290.4 0.260174 0.130087 0.991503i \(-0.458474\pi\)
0.130087 + 0.991503i \(0.458474\pi\)
\(128\) 0 0
\(129\) −77984.5 −0.412605
\(130\) 0 0
\(131\) −82047.2 −0.417720 −0.208860 0.977946i \(-0.566975\pi\)
−0.208860 + 0.977946i \(0.566975\pi\)
\(132\) 0 0
\(133\) −127444. −0.624725
\(134\) 0 0
\(135\) −78064.4 −0.368654
\(136\) 0 0
\(137\) −68984.2 −0.314013 −0.157007 0.987598i \(-0.550184\pi\)
−0.157007 + 0.987598i \(0.550184\pi\)
\(138\) 0 0
\(139\) 437887. 1.92232 0.961158 0.275998i \(-0.0890084\pi\)
0.961158 + 0.275998i \(0.0890084\pi\)
\(140\) 0 0
\(141\) −151908. −0.643477
\(142\) 0 0
\(143\) −170546. −0.697430
\(144\) 0 0
\(145\) 728023. 2.87558
\(146\) 0 0
\(147\) 48772.0 0.186156
\(148\) 0 0
\(149\) 27672.3 0.102113 0.0510564 0.998696i \(-0.483741\pi\)
0.0510564 + 0.998696i \(0.483741\pi\)
\(150\) 0 0
\(151\) 84706.4 0.302325 0.151162 0.988509i \(-0.451698\pi\)
0.151162 + 0.988509i \(0.451698\pi\)
\(152\) 0 0
\(153\) 65083.1 0.224771
\(154\) 0 0
\(155\) −513852. −1.71794
\(156\) 0 0
\(157\) 565956. 1.83246 0.916229 0.400656i \(-0.131218\pi\)
0.916229 + 0.400656i \(0.131218\pi\)
\(158\) 0 0
\(159\) −97478.8 −0.305786
\(160\) 0 0
\(161\) −684673. −2.08170
\(162\) 0 0
\(163\) 26326.8 0.0776121 0.0388060 0.999247i \(-0.487645\pi\)
0.0388060 + 0.999247i \(0.487645\pi\)
\(164\) 0 0
\(165\) 418758. 1.19744
\(166\) 0 0
\(167\) 268359. 0.744603 0.372301 0.928112i \(-0.378569\pi\)
0.372301 + 0.928112i \(0.378569\pi\)
\(168\) 0 0
\(169\) −217233. −0.585070
\(170\) 0 0
\(171\) 69242.2 0.181084
\(172\) 0 0
\(173\) −103090. −0.261879 −0.130939 0.991390i \(-0.541799\pi\)
−0.130939 + 0.991390i \(0.541799\pi\)
\(174\) 0 0
\(175\) −1.24367e6 −3.06979
\(176\) 0 0
\(177\) 74752.3 0.179346
\(178\) 0 0
\(179\) −671994. −1.56759 −0.783796 0.621019i \(-0.786719\pi\)
−0.783796 + 0.621019i \(0.786719\pi\)
\(180\) 0 0
\(181\) 445972. 1.01184 0.505919 0.862581i \(-0.331154\pi\)
0.505919 + 0.862581i \(0.331154\pi\)
\(182\) 0 0
\(183\) 323982. 0.715144
\(184\) 0 0
\(185\) −97347.6 −0.209120
\(186\) 0 0
\(187\) −349123. −0.730086
\(188\) 0 0
\(189\) −108682. −0.221312
\(190\) 0 0
\(191\) −420397. −0.833827 −0.416914 0.908946i \(-0.636888\pi\)
−0.416914 + 0.908946i \(0.636888\pi\)
\(192\) 0 0
\(193\) −148098. −0.286190 −0.143095 0.989709i \(-0.545705\pi\)
−0.143095 + 0.989709i \(0.545705\pi\)
\(194\) 0 0
\(195\) −378280. −0.712405
\(196\) 0 0
\(197\) −399632. −0.733660 −0.366830 0.930288i \(-0.619557\pi\)
−0.366830 + 0.930288i \(0.619557\pi\)
\(198\) 0 0
\(199\) 639043. 1.14393 0.571963 0.820280i \(-0.306182\pi\)
0.571963 + 0.820280i \(0.306182\pi\)
\(200\) 0 0
\(201\) 225694. 0.394030
\(202\) 0 0
\(203\) 1.01356e6 1.72628
\(204\) 0 0
\(205\) 254026. 0.422177
\(206\) 0 0
\(207\) 371995. 0.603408
\(208\) 0 0
\(209\) −371434. −0.588187
\(210\) 0 0
\(211\) 840471. 1.29962 0.649810 0.760097i \(-0.274848\pi\)
0.649810 + 0.760097i \(0.274848\pi\)
\(212\) 0 0
\(213\) 520658. 0.786328
\(214\) 0 0
\(215\) 927879. 1.36897
\(216\) 0 0
\(217\) −715392. −1.03132
\(218\) 0 0
\(219\) −612832. −0.863439
\(220\) 0 0
\(221\) 315376. 0.434358
\(222\) 0 0
\(223\) −436150. −0.587319 −0.293659 0.955910i \(-0.594873\pi\)
−0.293659 + 0.955910i \(0.594873\pi\)
\(224\) 0 0
\(225\) 675705. 0.889817
\(226\) 0 0
\(227\) 656896. 0.846119 0.423060 0.906102i \(-0.360956\pi\)
0.423060 + 0.906102i \(0.360956\pi\)
\(228\) 0 0
\(229\) −769622. −0.969815 −0.484908 0.874565i \(-0.661147\pi\)
−0.484908 + 0.874565i \(0.661147\pi\)
\(230\) 0 0
\(231\) 583001. 0.718852
\(232\) 0 0
\(233\) −1.11414e6 −1.34446 −0.672232 0.740341i \(-0.734664\pi\)
−0.672232 + 0.740341i \(0.734664\pi\)
\(234\) 0 0
\(235\) 1.80744e6 2.13498
\(236\) 0 0
\(237\) 888515. 1.02753
\(238\) 0 0
\(239\) 742802. 0.841160 0.420580 0.907256i \(-0.361827\pi\)
0.420580 + 0.907256i \(0.361827\pi\)
\(240\) 0 0
\(241\) 427097. 0.473679 0.236840 0.971549i \(-0.423888\pi\)
0.236840 + 0.971549i \(0.423888\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) −580301. −0.617644
\(246\) 0 0
\(247\) 335530. 0.349936
\(248\) 0 0
\(249\) 313734. 0.320673
\(250\) 0 0
\(251\) 714389. 0.715732 0.357866 0.933773i \(-0.383505\pi\)
0.357866 + 0.933773i \(0.383505\pi\)
\(252\) 0 0
\(253\) −1.99548e6 −1.95995
\(254\) 0 0
\(255\) −774374. −0.745763
\(256\) 0 0
\(257\) 2.01289e6 1.90102 0.950511 0.310691i \(-0.100561\pi\)
0.950511 + 0.310691i \(0.100561\pi\)
\(258\) 0 0
\(259\) −135529. −0.125540
\(260\) 0 0
\(261\) −550687. −0.500384
\(262\) 0 0
\(263\) 1.13759e6 1.01413 0.507067 0.861907i \(-0.330730\pi\)
0.507067 + 0.861907i \(0.330730\pi\)
\(264\) 0 0
\(265\) 1.15983e6 1.01456
\(266\) 0 0
\(267\) 312104. 0.267930
\(268\) 0 0
\(269\) −1.15018e6 −0.969138 −0.484569 0.874753i \(-0.661023\pi\)
−0.484569 + 0.874753i \(0.661023\pi\)
\(270\) 0 0
\(271\) −271676. −0.224713 −0.112356 0.993668i \(-0.535840\pi\)
−0.112356 + 0.993668i \(0.535840\pi\)
\(272\) 0 0
\(273\) −526647. −0.427674
\(274\) 0 0
\(275\) −3.62466e6 −2.89025
\(276\) 0 0
\(277\) 1.20183e6 0.941115 0.470557 0.882369i \(-0.344053\pi\)
0.470557 + 0.882369i \(0.344053\pi\)
\(278\) 0 0
\(279\) 388685. 0.298942
\(280\) 0 0
\(281\) 1.14613e6 0.865900 0.432950 0.901418i \(-0.357473\pi\)
0.432950 + 0.901418i \(0.357473\pi\)
\(282\) 0 0
\(283\) 552547. 0.410113 0.205056 0.978750i \(-0.434262\pi\)
0.205056 + 0.978750i \(0.434262\pi\)
\(284\) 0 0
\(285\) −823861. −0.600817
\(286\) 0 0
\(287\) 353659. 0.253443
\(288\) 0 0
\(289\) −774253. −0.545304
\(290\) 0 0
\(291\) 359019. 0.248534
\(292\) 0 0
\(293\) −601434. −0.409278 −0.204639 0.978837i \(-0.565602\pi\)
−0.204639 + 0.978837i \(0.565602\pi\)
\(294\) 0 0
\(295\) −889421. −0.595048
\(296\) 0 0
\(297\) −316754. −0.208368
\(298\) 0 0
\(299\) 1.80259e6 1.16606
\(300\) 0 0
\(301\) 1.29181e6 0.821829
\(302\) 0 0
\(303\) −318074. −0.199031
\(304\) 0 0
\(305\) −3.85482e6 −2.37277
\(306\) 0 0
\(307\) −561644. −0.340107 −0.170053 0.985435i \(-0.554394\pi\)
−0.170053 + 0.985435i \(0.554394\pi\)
\(308\) 0 0
\(309\) −333877. −0.198925
\(310\) 0 0
\(311\) −1.83235e6 −1.07425 −0.537126 0.843502i \(-0.680490\pi\)
−0.537126 + 0.843502i \(0.680490\pi\)
\(312\) 0 0
\(313\) 2.63782e6 1.52190 0.760948 0.648813i \(-0.224734\pi\)
0.760948 + 0.648813i \(0.224734\pi\)
\(314\) 0 0
\(315\) 1.29313e6 0.734287
\(316\) 0 0
\(317\) 2.11403e6 1.18158 0.590791 0.806825i \(-0.298816\pi\)
0.590791 + 0.806825i \(0.298816\pi\)
\(318\) 0 0
\(319\) 2.95403e6 1.62532
\(320\) 0 0
\(321\) 1.38209e6 0.748644
\(322\) 0 0
\(323\) 686861. 0.366322
\(324\) 0 0
\(325\) 3.27429e6 1.71953
\(326\) 0 0
\(327\) 960738. 0.496862
\(328\) 0 0
\(329\) 2.51634e6 1.28168
\(330\) 0 0
\(331\) −432931. −0.217194 −0.108597 0.994086i \(-0.534636\pi\)
−0.108597 + 0.994086i \(0.534636\pi\)
\(332\) 0 0
\(333\) 73635.1 0.0363893
\(334\) 0 0
\(335\) −2.68536e6 −1.30735
\(336\) 0 0
\(337\) −188395. −0.0903640 −0.0451820 0.998979i \(-0.514387\pi\)
−0.0451820 + 0.998979i \(0.514387\pi\)
\(338\) 0 0
\(339\) −920090. −0.434842
\(340\) 0 0
\(341\) −2.08501e6 −0.971006
\(342\) 0 0
\(343\) 1.69776e6 0.779184
\(344\) 0 0
\(345\) −4.42608e6 −2.00204
\(346\) 0 0
\(347\) 2.75126e6 1.22661 0.613306 0.789845i \(-0.289839\pi\)
0.613306 + 0.789845i \(0.289839\pi\)
\(348\) 0 0
\(349\) −2.36608e6 −1.03984 −0.519919 0.854216i \(-0.674038\pi\)
−0.519919 + 0.854216i \(0.674038\pi\)
\(350\) 0 0
\(351\) 286136. 0.123967
\(352\) 0 0
\(353\) −67235.9 −0.0287187 −0.0143593 0.999897i \(-0.504571\pi\)
−0.0143593 + 0.999897i \(0.504571\pi\)
\(354\) 0 0
\(355\) −6.19492e6 −2.60894
\(356\) 0 0
\(357\) −1.07810e6 −0.447700
\(358\) 0 0
\(359\) −880277. −0.360482 −0.180241 0.983622i \(-0.557688\pi\)
−0.180241 + 0.983622i \(0.557688\pi\)
\(360\) 0 0
\(361\) −1.74534e6 −0.704876
\(362\) 0 0
\(363\) 249695. 0.0994589
\(364\) 0 0
\(365\) 7.29163e6 2.86479
\(366\) 0 0
\(367\) −2.71882e6 −1.05370 −0.526848 0.849959i \(-0.676626\pi\)
−0.526848 + 0.849959i \(0.676626\pi\)
\(368\) 0 0
\(369\) −192149. −0.0734637
\(370\) 0 0
\(371\) 1.61473e6 0.609066
\(372\) 0 0
\(373\) −111486. −0.0414904 −0.0207452 0.999785i \(-0.506604\pi\)
−0.0207452 + 0.999785i \(0.506604\pi\)
\(374\) 0 0
\(375\) −5.02796e6 −1.84635
\(376\) 0 0
\(377\) −2.66849e6 −0.966967
\(378\) 0 0
\(379\) −2.62051e6 −0.937104 −0.468552 0.883436i \(-0.655224\pi\)
−0.468552 + 0.883436i \(0.655224\pi\)
\(380\) 0 0
\(381\) 425614. 0.150212
\(382\) 0 0
\(383\) −1.82442e6 −0.635519 −0.317760 0.948171i \(-0.602930\pi\)
−0.317760 + 0.948171i \(0.602930\pi\)
\(384\) 0 0
\(385\) −6.93669e6 −2.38507
\(386\) 0 0
\(387\) −701861. −0.238217
\(388\) 0 0
\(389\) 162259. 0.0543668 0.0271834 0.999630i \(-0.491346\pi\)
0.0271834 + 0.999630i \(0.491346\pi\)
\(390\) 0 0
\(391\) 3.69007e6 1.22065
\(392\) 0 0
\(393\) −738425. −0.241171
\(394\) 0 0
\(395\) −1.05718e7 −3.40922
\(396\) 0 0
\(397\) −2.92783e6 −0.932330 −0.466165 0.884698i \(-0.654365\pi\)
−0.466165 + 0.884698i \(0.654365\pi\)
\(398\) 0 0
\(399\) −1.14699e6 −0.360685
\(400\) 0 0
\(401\) 3.92259e6 1.21818 0.609090 0.793101i \(-0.291535\pi\)
0.609090 + 0.793101i \(0.291535\pi\)
\(402\) 0 0
\(403\) 1.88347e6 0.577691
\(404\) 0 0
\(405\) −702580. −0.212842
\(406\) 0 0
\(407\) −394998. −0.118198
\(408\) 0 0
\(409\) −151551. −0.0447973 −0.0223986 0.999749i \(-0.507130\pi\)
−0.0223986 + 0.999749i \(0.507130\pi\)
\(410\) 0 0
\(411\) −620858. −0.181296
\(412\) 0 0
\(413\) −1.23826e6 −0.357222
\(414\) 0 0
\(415\) −3.73288e6 −1.06396
\(416\) 0 0
\(417\) 3.94098e6 1.10985
\(418\) 0 0
\(419\) 3.13904e6 0.873497 0.436748 0.899584i \(-0.356130\pi\)
0.436748 + 0.899584i \(0.356130\pi\)
\(420\) 0 0
\(421\) −2.32328e6 −0.638847 −0.319423 0.947612i \(-0.603489\pi\)
−0.319423 + 0.947612i \(0.603489\pi\)
\(422\) 0 0
\(423\) −1.36717e6 −0.371512
\(424\) 0 0
\(425\) 6.70278e6 1.80004
\(426\) 0 0
\(427\) −5.36674e6 −1.42443
\(428\) 0 0
\(429\) −1.53491e6 −0.402661
\(430\) 0 0
\(431\) 998677. 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(432\) 0 0
\(433\) 4.18444e6 1.07255 0.536275 0.844044i \(-0.319831\pi\)
0.536275 + 0.844044i \(0.319831\pi\)
\(434\) 0 0
\(435\) 6.55221e6 1.66022
\(436\) 0 0
\(437\) 3.92589e6 0.983409
\(438\) 0 0
\(439\) −3.30452e6 −0.818364 −0.409182 0.912453i \(-0.634186\pi\)
−0.409182 + 0.912453i \(0.634186\pi\)
\(440\) 0 0
\(441\) 438948. 0.107477
\(442\) 0 0
\(443\) 4.11097e6 0.995257 0.497629 0.867390i \(-0.334204\pi\)
0.497629 + 0.867390i \(0.334204\pi\)
\(444\) 0 0
\(445\) −3.71349e6 −0.888959
\(446\) 0 0
\(447\) 249051. 0.0589548
\(448\) 0 0
\(449\) 7.22741e6 1.69187 0.845934 0.533287i \(-0.179043\pi\)
0.845934 + 0.533287i \(0.179043\pi\)
\(450\) 0 0
\(451\) 1.03074e6 0.238620
\(452\) 0 0
\(453\) 762357. 0.174547
\(454\) 0 0
\(455\) 6.26618e6 1.41897
\(456\) 0 0
\(457\) −6.11469e6 −1.36957 −0.684784 0.728746i \(-0.740103\pi\)
−0.684784 + 0.728746i \(0.740103\pi\)
\(458\) 0 0
\(459\) 585748. 0.129771
\(460\) 0 0
\(461\) 3.91568e6 0.858134 0.429067 0.903273i \(-0.358842\pi\)
0.429067 + 0.903273i \(0.358842\pi\)
\(462\) 0 0
\(463\) 6.39830e6 1.38711 0.693557 0.720402i \(-0.256043\pi\)
0.693557 + 0.720402i \(0.256043\pi\)
\(464\) 0 0
\(465\) −4.62467e6 −0.991855
\(466\) 0 0
\(467\) 3.54919e6 0.753074 0.376537 0.926402i \(-0.377115\pi\)
0.376537 + 0.926402i \(0.377115\pi\)
\(468\) 0 0
\(469\) −3.73860e6 −0.784833
\(470\) 0 0
\(471\) 5.09361e6 1.05797
\(472\) 0 0
\(473\) 3.76497e6 0.773764
\(474\) 0 0
\(475\) 7.13112e6 1.45019
\(476\) 0 0
\(477\) −877309. −0.176546
\(478\) 0 0
\(479\) −5.52909e6 −1.10107 −0.550535 0.834812i \(-0.685576\pi\)
−0.550535 + 0.834812i \(0.685576\pi\)
\(480\) 0 0
\(481\) 356817. 0.0703206
\(482\) 0 0
\(483\) −6.16206e6 −1.20187
\(484\) 0 0
\(485\) −4.27170e6 −0.824606
\(486\) 0 0
\(487\) 1.29274e6 0.246996 0.123498 0.992345i \(-0.460589\pi\)
0.123498 + 0.992345i \(0.460589\pi\)
\(488\) 0 0
\(489\) 236941. 0.0448094
\(490\) 0 0
\(491\) −5.06222e6 −0.947626 −0.473813 0.880625i \(-0.657123\pi\)
−0.473813 + 0.880625i \(0.657123\pi\)
\(492\) 0 0
\(493\) −5.46264e6 −1.01224
\(494\) 0 0
\(495\) 3.76882e6 0.691342
\(496\) 0 0
\(497\) −8.62466e6 −1.56621
\(498\) 0 0
\(499\) 2.89211e6 0.519951 0.259976 0.965615i \(-0.416285\pi\)
0.259976 + 0.965615i \(0.416285\pi\)
\(500\) 0 0
\(501\) 2.41523e6 0.429897
\(502\) 0 0
\(503\) −4.20037e6 −0.740231 −0.370115 0.928986i \(-0.620682\pi\)
−0.370115 + 0.928986i \(0.620682\pi\)
\(504\) 0 0
\(505\) 3.78452e6 0.660363
\(506\) 0 0
\(507\) −1.95509e6 −0.337791
\(508\) 0 0
\(509\) 3.09356e6 0.529255 0.264627 0.964351i \(-0.414751\pi\)
0.264627 + 0.964351i \(0.414751\pi\)
\(510\) 0 0
\(511\) 1.01515e7 1.71980
\(512\) 0 0
\(513\) 623180. 0.104549
\(514\) 0 0
\(515\) 3.97255e6 0.660011
\(516\) 0 0
\(517\) 7.33388e6 1.20672
\(518\) 0 0
\(519\) −927807. −0.151196
\(520\) 0 0
\(521\) 6.99447e6 1.12891 0.564456 0.825463i \(-0.309086\pi\)
0.564456 + 0.825463i \(0.309086\pi\)
\(522\) 0 0
\(523\) −1.93493e6 −0.309322 −0.154661 0.987968i \(-0.549429\pi\)
−0.154661 + 0.987968i \(0.549429\pi\)
\(524\) 0 0
\(525\) −1.11930e7 −1.77234
\(526\) 0 0
\(527\) 3.85563e6 0.604741
\(528\) 0 0
\(529\) 1.46550e7 2.27691
\(530\) 0 0
\(531\) 672770. 0.103545
\(532\) 0 0
\(533\) −931106. −0.141965
\(534\) 0 0
\(535\) −1.64445e7 −2.48391
\(536\) 0 0
\(537\) −6.04795e6 −0.905049
\(538\) 0 0
\(539\) −2.35463e6 −0.349101
\(540\) 0 0
\(541\) 8.23886e6 1.21025 0.605124 0.796131i \(-0.293124\pi\)
0.605124 + 0.796131i \(0.293124\pi\)
\(542\) 0 0
\(543\) 4.01375e6 0.584185
\(544\) 0 0
\(545\) −1.14311e7 −1.64853
\(546\) 0 0
\(547\) 9.16514e6 1.30970 0.654849 0.755760i \(-0.272732\pi\)
0.654849 + 0.755760i \(0.272732\pi\)
\(548\) 0 0
\(549\) 2.91584e6 0.412889
\(550\) 0 0
\(551\) −5.81173e6 −0.815505
\(552\) 0 0
\(553\) −1.47182e7 −2.04664
\(554\) 0 0
\(555\) −876129. −0.120736
\(556\) 0 0
\(557\) 6.54035e6 0.893230 0.446615 0.894726i \(-0.352629\pi\)
0.446615 + 0.894726i \(0.352629\pi\)
\(558\) 0 0
\(559\) −3.40104e6 −0.460344
\(560\) 0 0
\(561\) −3.14210e6 −0.421515
\(562\) 0 0
\(563\) −4.50619e6 −0.599155 −0.299577 0.954072i \(-0.596846\pi\)
−0.299577 + 0.954072i \(0.596846\pi\)
\(564\) 0 0
\(565\) 1.09475e7 1.44275
\(566\) 0 0
\(567\) −978142. −0.127774
\(568\) 0 0
\(569\) 3.55292e6 0.460050 0.230025 0.973185i \(-0.426119\pi\)
0.230025 + 0.973185i \(0.426119\pi\)
\(570\) 0 0
\(571\) 1.19688e7 1.53624 0.768122 0.640304i \(-0.221192\pi\)
0.768122 + 0.640304i \(0.221192\pi\)
\(572\) 0 0
\(573\) −3.78357e6 −0.481410
\(574\) 0 0
\(575\) 3.83110e7 4.83230
\(576\) 0 0
\(577\) −1.21327e7 −1.51711 −0.758554 0.651611i \(-0.774094\pi\)
−0.758554 + 0.651611i \(0.774094\pi\)
\(578\) 0 0
\(579\) −1.33288e6 −0.165232
\(580\) 0 0
\(581\) −5.19697e6 −0.638719
\(582\) 0 0
\(583\) 4.70612e6 0.573445
\(584\) 0 0
\(585\) −3.40452e6 −0.411307
\(586\) 0 0
\(587\) 1.03620e7 1.24122 0.620612 0.784118i \(-0.286884\pi\)
0.620612 + 0.784118i \(0.286884\pi\)
\(588\) 0 0
\(589\) 4.10203e6 0.487204
\(590\) 0 0
\(591\) −3.59669e6 −0.423579
\(592\) 0 0
\(593\) −1.01261e7 −1.18251 −0.591254 0.806486i \(-0.701367\pi\)
−0.591254 + 0.806486i \(0.701367\pi\)
\(594\) 0 0
\(595\) 1.28274e7 1.48541
\(596\) 0 0
\(597\) 5.75139e6 0.660446
\(598\) 0 0
\(599\) −313236. −0.0356701 −0.0178350 0.999841i \(-0.505677\pi\)
−0.0178350 + 0.999841i \(0.505677\pi\)
\(600\) 0 0
\(601\) −4.89443e6 −0.552734 −0.276367 0.961052i \(-0.589131\pi\)
−0.276367 + 0.961052i \(0.589131\pi\)
\(602\) 0 0
\(603\) 2.03125e6 0.227494
\(604\) 0 0
\(605\) −2.97094e6 −0.329993
\(606\) 0 0
\(607\) −7.48909e6 −0.825006 −0.412503 0.910956i \(-0.635345\pi\)
−0.412503 + 0.910956i \(0.635345\pi\)
\(608\) 0 0
\(609\) 9.12208e6 0.996668
\(610\) 0 0
\(611\) −6.62497e6 −0.717928
\(612\) 0 0
\(613\) −1.20289e7 −1.29293 −0.646466 0.762942i \(-0.723754\pi\)
−0.646466 + 0.762942i \(0.723754\pi\)
\(614\) 0 0
\(615\) 2.28624e6 0.243744
\(616\) 0 0
\(617\) −4.27115e6 −0.451681 −0.225841 0.974164i \(-0.572513\pi\)
−0.225841 + 0.974164i \(0.572513\pi\)
\(618\) 0 0
\(619\) −1.05328e7 −1.10488 −0.552442 0.833552i \(-0.686304\pi\)
−0.552442 + 0.833552i \(0.686304\pi\)
\(620\) 0 0
\(621\) 3.34795e6 0.348378
\(622\) 0 0
\(623\) −5.16997e6 −0.533664
\(624\) 0 0
\(625\) 3.37550e7 3.45651
\(626\) 0 0
\(627\) −3.34290e6 −0.339590
\(628\) 0 0
\(629\) 730437. 0.0736133
\(630\) 0 0
\(631\) −1.47931e7 −1.47906 −0.739532 0.673122i \(-0.764953\pi\)
−0.739532 + 0.673122i \(0.764953\pi\)
\(632\) 0 0
\(633\) 7.56424e6 0.750336
\(634\) 0 0
\(635\) −5.06406e6 −0.498384
\(636\) 0 0
\(637\) 2.12703e6 0.207694
\(638\) 0 0
\(639\) 4.68592e6 0.453986
\(640\) 0 0
\(641\) −1.86457e6 −0.179240 −0.0896198 0.995976i \(-0.528565\pi\)
−0.0896198 + 0.995976i \(0.528565\pi\)
\(642\) 0 0
\(643\) 6.14368e6 0.586004 0.293002 0.956112i \(-0.405346\pi\)
0.293002 + 0.956112i \(0.405346\pi\)
\(644\) 0 0
\(645\) 8.35092e6 0.790378
\(646\) 0 0
\(647\) 5.58318e6 0.524350 0.262175 0.965020i \(-0.415560\pi\)
0.262175 + 0.965020i \(0.415560\pi\)
\(648\) 0 0
\(649\) −3.60892e6 −0.336330
\(650\) 0 0
\(651\) −6.43853e6 −0.595435
\(652\) 0 0
\(653\) −9.55163e6 −0.876586 −0.438293 0.898832i \(-0.644417\pi\)
−0.438293 + 0.898832i \(0.644417\pi\)
\(654\) 0 0
\(655\) 8.78596e6 0.800177
\(656\) 0 0
\(657\) −5.51549e6 −0.498507
\(658\) 0 0
\(659\) 2.48421e6 0.222831 0.111415 0.993774i \(-0.464462\pi\)
0.111415 + 0.993774i \(0.464462\pi\)
\(660\) 0 0
\(661\) −1.32567e7 −1.18013 −0.590066 0.807355i \(-0.700898\pi\)
−0.590066 + 0.807355i \(0.700898\pi\)
\(662\) 0 0
\(663\) 2.83838e6 0.250777
\(664\) 0 0
\(665\) 1.36472e7 1.19671
\(666\) 0 0
\(667\) −3.12228e7 −2.71742
\(668\) 0 0
\(669\) −3.92535e6 −0.339089
\(670\) 0 0
\(671\) −1.56413e7 −1.34112
\(672\) 0 0
\(673\) −1.13191e7 −0.963327 −0.481663 0.876356i \(-0.659967\pi\)
−0.481663 + 0.876356i \(0.659967\pi\)
\(674\) 0 0
\(675\) 6.08134e6 0.513736
\(676\) 0 0
\(677\) 5.27180e6 0.442066 0.221033 0.975266i \(-0.429057\pi\)
0.221033 + 0.975266i \(0.429057\pi\)
\(678\) 0 0
\(679\) −5.94713e6 −0.495031
\(680\) 0 0
\(681\) 5.91206e6 0.488507
\(682\) 0 0
\(683\) −9.64351e6 −0.791013 −0.395506 0.918463i \(-0.629431\pi\)
−0.395506 + 0.918463i \(0.629431\pi\)
\(684\) 0 0
\(685\) 7.38712e6 0.601518
\(686\) 0 0
\(687\) −6.92660e6 −0.559923
\(688\) 0 0
\(689\) −4.25122e6 −0.341166
\(690\) 0 0
\(691\) −1.01345e7 −0.807437 −0.403719 0.914883i \(-0.632283\pi\)
−0.403719 + 0.914883i \(0.632283\pi\)
\(692\) 0 0
\(693\) 5.24701e6 0.415029
\(694\) 0 0
\(695\) −4.68908e7 −3.68235
\(696\) 0 0
\(697\) −1.90606e6 −0.148612
\(698\) 0 0
\(699\) −1.00272e7 −0.776227
\(700\) 0 0
\(701\) 2.04002e7 1.56797 0.783987 0.620777i \(-0.213183\pi\)
0.783987 + 0.620777i \(0.213183\pi\)
\(702\) 0 0
\(703\) 777116. 0.0593059
\(704\) 0 0
\(705\) 1.62670e7 1.23263
\(706\) 0 0
\(707\) 5.26887e6 0.396432
\(708\) 0 0
\(709\) −1.99117e7 −1.48762 −0.743810 0.668391i \(-0.766983\pi\)
−0.743810 + 0.668391i \(0.766983\pi\)
\(710\) 0 0
\(711\) 7.99663e6 0.593244
\(712\) 0 0
\(713\) 2.20376e7 1.62346
\(714\) 0 0
\(715\) 1.82628e7 1.33598
\(716\) 0 0
\(717\) 6.68522e6 0.485644
\(718\) 0 0
\(719\) −1.41435e7 −1.02032 −0.510159 0.860080i \(-0.670413\pi\)
−0.510159 + 0.860080i \(0.670413\pi\)
\(720\) 0 0
\(721\) 5.53064e6 0.396221
\(722\) 0 0
\(723\) 3.84388e6 0.273479
\(724\) 0 0
\(725\) −5.67141e7 −4.00725
\(726\) 0 0
\(727\) 1.28420e7 0.901146 0.450573 0.892740i \(-0.351220\pi\)
0.450573 + 0.892740i \(0.351220\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −6.96224e6 −0.481899
\(732\) 0 0
\(733\) 1.88331e6 0.129468 0.0647339 0.997903i \(-0.479380\pi\)
0.0647339 + 0.997903i \(0.479380\pi\)
\(734\) 0 0
\(735\) −5.22271e6 −0.356597
\(736\) 0 0
\(737\) −1.08961e7 −0.738931
\(738\) 0 0
\(739\) −1.31690e6 −0.0887036 −0.0443518 0.999016i \(-0.514122\pi\)
−0.0443518 + 0.999016i \(0.514122\pi\)
\(740\) 0 0
\(741\) 3.01977e6 0.202036
\(742\) 0 0
\(743\) 2.87356e7 1.90962 0.954812 0.297210i \(-0.0960563\pi\)
0.954812 + 0.297210i \(0.0960563\pi\)
\(744\) 0 0
\(745\) −2.96327e6 −0.195605
\(746\) 0 0
\(747\) 2.82360e6 0.185141
\(748\) 0 0
\(749\) −2.28943e7 −1.49115
\(750\) 0 0
\(751\) −7.33862e6 −0.474804 −0.237402 0.971411i \(-0.576296\pi\)
−0.237402 + 0.971411i \(0.576296\pi\)
\(752\) 0 0
\(753\) 6.42950e6 0.413228
\(754\) 0 0
\(755\) −9.07072e6 −0.579128
\(756\) 0 0
\(757\) 2.05944e7 1.30620 0.653100 0.757272i \(-0.273468\pi\)
0.653100 + 0.757272i \(0.273468\pi\)
\(758\) 0 0
\(759\) −1.79593e7 −1.13158
\(760\) 0 0
\(761\) −2.08295e7 −1.30382 −0.651911 0.758296i \(-0.726032\pi\)
−0.651911 + 0.758296i \(0.726032\pi\)
\(762\) 0 0
\(763\) −1.59145e7 −0.989653
\(764\) 0 0
\(765\) −6.96937e6 −0.430566
\(766\) 0 0
\(767\) 3.26007e6 0.200096
\(768\) 0 0
\(769\) 7.24343e6 0.441701 0.220851 0.975308i \(-0.429117\pi\)
0.220851 + 0.975308i \(0.429117\pi\)
\(770\) 0 0
\(771\) 1.81160e7 1.09756
\(772\) 0 0
\(773\) 7.56360e6 0.455281 0.227641 0.973745i \(-0.426899\pi\)
0.227641 + 0.973745i \(0.426899\pi\)
\(774\) 0 0
\(775\) 4.00299e7 2.39403
\(776\) 0 0
\(777\) −1.21976e6 −0.0724806
\(778\) 0 0
\(779\) −2.02787e6 −0.119728
\(780\) 0 0
\(781\) −2.51365e7 −1.47461
\(782\) 0 0
\(783\) −4.95618e6 −0.288897
\(784\) 0 0
\(785\) −6.06050e7 −3.51022
\(786\) 0 0
\(787\) −1.74475e7 −1.00415 −0.502074 0.864825i \(-0.667429\pi\)
−0.502074 + 0.864825i \(0.667429\pi\)
\(788\) 0 0
\(789\) 1.02383e7 0.585510
\(790\) 0 0
\(791\) 1.52412e7 0.866121
\(792\) 0 0
\(793\) 1.41294e7 0.797887
\(794\) 0 0
\(795\) 1.04384e7 0.585758
\(796\) 0 0
\(797\) 2.86812e7 1.59938 0.799689 0.600414i \(-0.204998\pi\)
0.799689 + 0.600414i \(0.204998\pi\)
\(798\) 0 0
\(799\) −1.35619e7 −0.751544
\(800\) 0 0
\(801\) 2.80893e6 0.154689
\(802\) 0 0
\(803\) 2.95866e7 1.61922
\(804\) 0 0
\(805\) 7.33177e7 3.98767
\(806\) 0 0
\(807\) −1.03516e7 −0.559532
\(808\) 0 0
\(809\) 1.18731e7 0.637812 0.318906 0.947786i \(-0.396685\pi\)
0.318906 + 0.947786i \(0.396685\pi\)
\(810\) 0 0
\(811\) 7.63302e6 0.407516 0.203758 0.979021i \(-0.434684\pi\)
0.203758 + 0.979021i \(0.434684\pi\)
\(812\) 0 0
\(813\) −2.44508e6 −0.129738
\(814\) 0 0
\(815\) −2.81919e6 −0.148672
\(816\) 0 0
\(817\) −7.40717e6 −0.388237
\(818\) 0 0
\(819\) −4.73983e6 −0.246918
\(820\) 0 0
\(821\) −1.42730e7 −0.739021 −0.369511 0.929227i \(-0.620475\pi\)
−0.369511 + 0.929227i \(0.620475\pi\)
\(822\) 0 0
\(823\) −1.71695e7 −0.883604 −0.441802 0.897113i \(-0.645661\pi\)
−0.441802 + 0.897113i \(0.645661\pi\)
\(824\) 0 0
\(825\) −3.26219e7 −1.66869
\(826\) 0 0
\(827\) −2.21323e7 −1.12529 −0.562644 0.826699i \(-0.690216\pi\)
−0.562644 + 0.826699i \(0.690216\pi\)
\(828\) 0 0
\(829\) −1.55903e7 −0.787896 −0.393948 0.919133i \(-0.628891\pi\)
−0.393948 + 0.919133i \(0.628891\pi\)
\(830\) 0 0
\(831\) 1.08164e7 0.543353
\(832\) 0 0
\(833\) 4.35422e6 0.217419
\(834\) 0 0
\(835\) −2.87370e7 −1.42635
\(836\) 0 0
\(837\) 3.49816e6 0.172594
\(838\) 0 0
\(839\) −6.48897e6 −0.318252 −0.159126 0.987258i \(-0.550868\pi\)
−0.159126 + 0.987258i \(0.550868\pi\)
\(840\) 0 0
\(841\) 2.57098e7 1.25346
\(842\) 0 0
\(843\) 1.03152e7 0.499928
\(844\) 0 0
\(845\) 2.32622e7 1.12075
\(846\) 0 0
\(847\) −4.13618e6 −0.198103
\(848\) 0 0
\(849\) 4.97292e6 0.236779
\(850\) 0 0
\(851\) 4.17495e6 0.197619
\(852\) 0 0
\(853\) 2.78176e7 1.30902 0.654512 0.756052i \(-0.272874\pi\)
0.654512 + 0.756052i \(0.272874\pi\)
\(854\) 0 0
\(855\) −7.41475e6 −0.346882
\(856\) 0 0
\(857\) −1.23818e7 −0.575880 −0.287940 0.957648i \(-0.592970\pi\)
−0.287940 + 0.957648i \(0.592970\pi\)
\(858\) 0 0
\(859\) −1.64820e7 −0.762126 −0.381063 0.924549i \(-0.624442\pi\)
−0.381063 + 0.924549i \(0.624442\pi\)
\(860\) 0 0
\(861\) 3.18293e6 0.146325
\(862\) 0 0
\(863\) 6.92492e6 0.316510 0.158255 0.987398i \(-0.449413\pi\)
0.158255 + 0.987398i \(0.449413\pi\)
\(864\) 0 0
\(865\) 1.10393e7 0.501650
\(866\) 0 0
\(867\) −6.96828e6 −0.314831
\(868\) 0 0
\(869\) −4.28960e7 −1.92694
\(870\) 0 0
\(871\) 9.84290e6 0.439620
\(872\) 0 0
\(873\) 3.23118e6 0.143491
\(874\) 0 0
\(875\) 8.32877e7 3.67757
\(876\) 0 0
\(877\) 1.07951e7 0.473943 0.236972 0.971517i \(-0.423845\pi\)
0.236972 + 0.971517i \(0.423845\pi\)
\(878\) 0 0
\(879\) −5.41290e6 −0.236297
\(880\) 0 0
\(881\) 1.27616e7 0.553942 0.276971 0.960878i \(-0.410669\pi\)
0.276971 + 0.960878i \(0.410669\pi\)
\(882\) 0 0
\(883\) 6.93416e6 0.299290 0.149645 0.988740i \(-0.452187\pi\)
0.149645 + 0.988740i \(0.452187\pi\)
\(884\) 0 0
\(885\) −8.00479e6 −0.343551
\(886\) 0 0
\(887\) −2.18558e7 −0.932734 −0.466367 0.884591i \(-0.654437\pi\)
−0.466367 + 0.884591i \(0.654437\pi\)
\(888\) 0 0
\(889\) −7.05026e6 −0.299192
\(890\) 0 0
\(891\) −2.85079e6 −0.120301
\(892\) 0 0
\(893\) −1.44286e7 −0.605474
\(894\) 0 0
\(895\) 7.19600e7 3.00285
\(896\) 0 0
\(897\) 1.62233e7 0.673223
\(898\) 0 0
\(899\) −3.26236e7 −1.34627
\(900\) 0 0
\(901\) −8.70263e6 −0.357140
\(902\) 0 0
\(903\) 1.16263e7 0.474483
\(904\) 0 0
\(905\) −4.77565e7 −1.93826
\(906\) 0 0
\(907\) 4.22111e7 1.70376 0.851880 0.523738i \(-0.175463\pi\)
0.851880 + 0.523738i \(0.175463\pi\)
\(908\) 0 0
\(909\) −2.86266e6 −0.114911
\(910\) 0 0
\(911\) −1.45953e7 −0.582661 −0.291331 0.956622i \(-0.594098\pi\)
−0.291331 + 0.956622i \(0.594098\pi\)
\(912\) 0 0
\(913\) −1.51466e7 −0.601363
\(914\) 0 0
\(915\) −3.46934e7 −1.36992
\(916\) 0 0
\(917\) 1.22319e7 0.480366
\(918\) 0 0
\(919\) 4.21908e6 0.164789 0.0823946 0.996600i \(-0.473743\pi\)
0.0823946 + 0.996600i \(0.473743\pi\)
\(920\) 0 0
\(921\) −5.05480e6 −0.196361
\(922\) 0 0
\(923\) 2.27068e7 0.877307
\(924\) 0 0
\(925\) 7.58354e6 0.291419
\(926\) 0 0
\(927\) −3.00489e6 −0.114850
\(928\) 0 0
\(929\) 2.76343e7 1.05053 0.525265 0.850939i \(-0.323966\pi\)
0.525265 + 0.850939i \(0.323966\pi\)
\(930\) 0 0
\(931\) 4.63248e6 0.175162
\(932\) 0 0
\(933\) −1.64911e7 −0.620220
\(934\) 0 0
\(935\) 3.73855e7 1.39854
\(936\) 0 0
\(937\) −2.61923e7 −0.974597 −0.487298 0.873236i \(-0.662018\pi\)
−0.487298 + 0.873236i \(0.662018\pi\)
\(938\) 0 0
\(939\) 2.37404e7 0.878667
\(940\) 0 0
\(941\) 3.59191e7 1.32236 0.661182 0.750226i \(-0.270055\pi\)
0.661182 + 0.750226i \(0.270055\pi\)
\(942\) 0 0
\(943\) −1.08944e7 −0.398957
\(944\) 0 0
\(945\) 1.16382e7 0.423941
\(946\) 0 0
\(947\) 3.21442e7 1.16474 0.582369 0.812924i \(-0.302126\pi\)
0.582369 + 0.812924i \(0.302126\pi\)
\(948\) 0 0
\(949\) −2.67267e7 −0.963339
\(950\) 0 0
\(951\) 1.90263e7 0.682187
\(952\) 0 0
\(953\) 5.06599e7 1.80689 0.903446 0.428701i \(-0.141029\pi\)
0.903446 + 0.428701i \(0.141029\pi\)
\(954\) 0 0
\(955\) 4.50179e7 1.59726
\(956\) 0 0
\(957\) 2.65863e7 0.938377
\(958\) 0 0
\(959\) 1.02845e7 0.361106
\(960\) 0 0
\(961\) −5.60279e6 −0.195702
\(962\) 0 0
\(963\) 1.24389e7 0.432230
\(964\) 0 0
\(965\) 1.58589e7 0.548221
\(966\) 0 0
\(967\) 4.48263e7 1.54158 0.770790 0.637089i \(-0.219862\pi\)
0.770790 + 0.637089i \(0.219862\pi\)
\(968\) 0 0
\(969\) 6.18175e6 0.211496
\(970\) 0 0
\(971\) −2.77374e7 −0.944100 −0.472050 0.881572i \(-0.656486\pi\)
−0.472050 + 0.881572i \(0.656486\pi\)
\(972\) 0 0
\(973\) −6.52820e7 −2.21061
\(974\) 0 0
\(975\) 2.94686e7 0.992770
\(976\) 0 0
\(977\) 3.60277e7 1.20754 0.603768 0.797160i \(-0.293666\pi\)
0.603768 + 0.797160i \(0.293666\pi\)
\(978\) 0 0
\(979\) −1.50679e7 −0.502452
\(980\) 0 0
\(981\) 8.64664e6 0.286863
\(982\) 0 0
\(983\) 2.49497e7 0.823534 0.411767 0.911289i \(-0.364912\pi\)
0.411767 + 0.911289i \(0.364912\pi\)
\(984\) 0 0
\(985\) 4.27943e7 1.40539
\(986\) 0 0
\(987\) 2.26471e7 0.739980
\(988\) 0 0
\(989\) −3.97940e7 −1.29368
\(990\) 0 0
\(991\) −1.88166e7 −0.608636 −0.304318 0.952571i \(-0.598428\pi\)
−0.304318 + 0.952571i \(0.598428\pi\)
\(992\) 0 0
\(993\) −3.89638e6 −0.125397
\(994\) 0 0
\(995\) −6.84315e7 −2.19128
\(996\) 0 0
\(997\) −4.25880e7 −1.35691 −0.678453 0.734644i \(-0.737349\pi\)
−0.678453 + 0.734644i \(0.737349\pi\)
\(998\) 0 0
\(999\) 662716. 0.0210094
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 192.6.a.r.1.1 2
3.2 odd 2 576.6.a.bm.1.2 2
4.3 odd 2 192.6.a.q.1.1 2
8.3 odd 2 96.6.a.h.1.2 yes 2
8.5 even 2 96.6.a.g.1.2 2
12.11 even 2 576.6.a.bn.1.2 2
16.3 odd 4 768.6.d.s.385.2 4
16.5 even 4 768.6.d.z.385.1 4
16.11 odd 4 768.6.d.s.385.3 4
16.13 even 4 768.6.d.z.385.4 4
24.5 odd 2 288.6.a.n.1.1 2
24.11 even 2 288.6.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.6.a.g.1.2 2 8.5 even 2
96.6.a.h.1.2 yes 2 8.3 odd 2
192.6.a.q.1.1 2 4.3 odd 2
192.6.a.r.1.1 2 1.1 even 1 trivial
288.6.a.n.1.1 2 24.5 odd 2
288.6.a.o.1.1 2 24.11 even 2
576.6.a.bm.1.2 2 3.2 odd 2
576.6.a.bn.1.2 2 12.11 even 2
768.6.d.s.385.2 4 16.3 odd 4
768.6.d.s.385.3 4 16.11 odd 4
768.6.d.z.385.1 4 16.5 even 4
768.6.d.z.385.4 4 16.13 even 4