Properties

Label 495.6.a.l.1.1
Level $495$
Weight $6$
Character 495.1
Self dual yes
Analytic conductor $79.390$
Analytic rank $1$
Dimension $6$
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] = [495,6,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, 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("495.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(79.3899908074\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 149x^{4} + 297x^{3} + 4052x^{2} - 6522x - 20692 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-10.2169\) of defining polynomial
Character \(\chi\) \(=\) 495.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-11.2169 q^{2} +93.8183 q^{4} +25.0000 q^{5} -132.518 q^{7} -693.408 q^{8} +O(q^{10})\) \(q-11.2169 q^{2} +93.8183 q^{4} +25.0000 q^{5} -132.518 q^{7} -693.408 q^{8} -280.422 q^{10} -121.000 q^{11} -916.014 q^{13} +1486.43 q^{14} +4775.68 q^{16} +607.561 q^{17} +2342.75 q^{19} +2345.46 q^{20} +1357.24 q^{22} -15.5282 q^{23} +625.000 q^{25} +10274.8 q^{26} -12432.6 q^{28} -837.103 q^{29} +2763.55 q^{31} -31379.2 q^{32} -6814.93 q^{34} -3312.94 q^{35} +5914.18 q^{37} -26278.4 q^{38} -17335.2 q^{40} +9477.40 q^{41} -3470.90 q^{43} -11352.0 q^{44} +174.178 q^{46} +5638.78 q^{47} +753.904 q^{49} -7010.55 q^{50} -85938.8 q^{52} +10588.9 q^{53} -3025.00 q^{55} +91888.7 q^{56} +9389.68 q^{58} -29479.8 q^{59} -32344.0 q^{61} -30998.4 q^{62} +199154. q^{64} -22900.3 q^{65} +58356.9 q^{67} +57000.3 q^{68} +37160.8 q^{70} +1075.01 q^{71} +28035.0 q^{73} -66338.6 q^{74} +219793. q^{76} +16034.6 q^{77} +23231.7 q^{79} +119392. q^{80} -106307. q^{82} -23943.0 q^{83} +15189.0 q^{85} +38932.6 q^{86} +83902.3 q^{88} -95010.4 q^{89} +121388. q^{91} -1456.83 q^{92} -63249.5 q^{94} +58568.8 q^{95} -33677.1 q^{97} -8456.45 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 115 q^{4} + 150 q^{5} - 66 q^{7} - 249 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} + 115 q^{4} + 150 q^{5} - 66 q^{7} - 249 q^{8} - 75 q^{10} - 726 q^{11} - 972 q^{13} - 2017 q^{14} + 6675 q^{16} - 712 q^{17} + 610 q^{19} + 2875 q^{20} + 363 q^{22} - 7100 q^{23} + 3750 q^{25} + 7742 q^{26} - 33171 q^{28} - 10964 q^{29} + 8868 q^{31} - 10861 q^{32} - 42673 q^{34} - 1650 q^{35} - 15506 q^{37} - 26185 q^{38} - 6225 q^{40} - 160 q^{41} + 7032 q^{43} - 13915 q^{44} + 1650 q^{46} + 46320 q^{47} + 48512 q^{49} - 1875 q^{50} - 145838 q^{52} - 30518 q^{53} - 18150 q^{55} + 21297 q^{56} + 51477 q^{58} - 120532 q^{59} - 9228 q^{61} + 81355 q^{62} + 302839 q^{64} - 24300 q^{65} + 47432 q^{67} - 78111 q^{68} - 50425 q^{70} - 163948 q^{71} + 78528 q^{73} - 133501 q^{74} - 29355 q^{76} + 7986 q^{77} - 36712 q^{79} + 166875 q^{80} - 128902 q^{82} - 133792 q^{83} - 17800 q^{85} - 83996 q^{86} + 30129 q^{88} - 254946 q^{89} + 167844 q^{91} + 5134 q^{92} + 31414 q^{94} + 15250 q^{95} + 17404 q^{97} + 351884 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.2169 −1.98288 −0.991441 0.130556i \(-0.958324\pi\)
−0.991441 + 0.130556i \(0.958324\pi\)
\(3\) 0 0
\(4\) 93.8183 2.93182
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −132.518 −1.02218 −0.511091 0.859527i \(-0.670759\pi\)
−0.511091 + 0.859527i \(0.670759\pi\)
\(8\) −693.408 −3.83057
\(9\) 0 0
\(10\) −280.422 −0.886772
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) −916.014 −1.50329 −0.751646 0.659566i \(-0.770740\pi\)
−0.751646 + 0.659566i \(0.770740\pi\)
\(14\) 1486.43 2.02687
\(15\) 0 0
\(16\) 4775.68 4.66375
\(17\) 607.561 0.509879 0.254940 0.966957i \(-0.417944\pi\)
0.254940 + 0.966957i \(0.417944\pi\)
\(18\) 0 0
\(19\) 2342.75 1.48882 0.744411 0.667722i \(-0.232731\pi\)
0.744411 + 0.667722i \(0.232731\pi\)
\(20\) 2345.46 1.31115
\(21\) 0 0
\(22\) 1357.24 0.597861
\(23\) −15.5282 −0.00612072 −0.00306036 0.999995i \(-0.500974\pi\)
−0.00306036 + 0.999995i \(0.500974\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 10274.8 2.98085
\(27\) 0 0
\(28\) −12432.6 −2.99686
\(29\) −837.103 −0.184835 −0.0924174 0.995720i \(-0.529459\pi\)
−0.0924174 + 0.995720i \(0.529459\pi\)
\(30\) 0 0
\(31\) 2763.55 0.516492 0.258246 0.966079i \(-0.416856\pi\)
0.258246 + 0.966079i \(0.416856\pi\)
\(32\) −31379.2 −5.41710
\(33\) 0 0
\(34\) −6814.93 −1.01103
\(35\) −3312.94 −0.457134
\(36\) 0 0
\(37\) 5914.18 0.710216 0.355108 0.934825i \(-0.384444\pi\)
0.355108 + 0.934825i \(0.384444\pi\)
\(38\) −26278.4 −2.95216
\(39\) 0 0
\(40\) −17335.2 −1.71308
\(41\) 9477.40 0.880500 0.440250 0.897875i \(-0.354890\pi\)
0.440250 + 0.897875i \(0.354890\pi\)
\(42\) 0 0
\(43\) −3470.90 −0.286267 −0.143133 0.989703i \(-0.545718\pi\)
−0.143133 + 0.989703i \(0.545718\pi\)
\(44\) −11352.0 −0.883977
\(45\) 0 0
\(46\) 174.178 0.0121367
\(47\) 5638.78 0.372341 0.186170 0.982517i \(-0.440392\pi\)
0.186170 + 0.982517i \(0.440392\pi\)
\(48\) 0 0
\(49\) 753.904 0.0448566
\(50\) −7010.55 −0.396576
\(51\) 0 0
\(52\) −85938.8 −4.40739
\(53\) 10588.9 0.517797 0.258898 0.965905i \(-0.416641\pi\)
0.258898 + 0.965905i \(0.416641\pi\)
\(54\) 0 0
\(55\) −3025.00 −0.134840
\(56\) 91888.7 3.91554
\(57\) 0 0
\(58\) 9389.68 0.366506
\(59\) −29479.8 −1.10254 −0.551270 0.834327i \(-0.685856\pi\)
−0.551270 + 0.834327i \(0.685856\pi\)
\(60\) 0 0
\(61\) −32344.0 −1.11294 −0.556468 0.830869i \(-0.687844\pi\)
−0.556468 + 0.830869i \(0.687844\pi\)
\(62\) −30998.4 −1.02414
\(63\) 0 0
\(64\) 199154. 6.07771
\(65\) −22900.3 −0.672293
\(66\) 0 0
\(67\) 58356.9 1.58820 0.794100 0.607787i \(-0.207942\pi\)
0.794100 + 0.607787i \(0.207942\pi\)
\(68\) 57000.3 1.49487
\(69\) 0 0
\(70\) 37160.8 0.906442
\(71\) 1075.01 0.0253086 0.0126543 0.999920i \(-0.495972\pi\)
0.0126543 + 0.999920i \(0.495972\pi\)
\(72\) 0 0
\(73\) 28035.0 0.615734 0.307867 0.951429i \(-0.400385\pi\)
0.307867 + 0.951429i \(0.400385\pi\)
\(74\) −66338.6 −1.40827
\(75\) 0 0
\(76\) 219793. 4.36496
\(77\) 16034.6 0.308200
\(78\) 0 0
\(79\) 23231.7 0.418806 0.209403 0.977829i \(-0.432848\pi\)
0.209403 + 0.977829i \(0.432848\pi\)
\(80\) 119392. 2.08569
\(81\) 0 0
\(82\) −106307. −1.74593
\(83\) −23943.0 −0.381491 −0.190745 0.981640i \(-0.561090\pi\)
−0.190745 + 0.981640i \(0.561090\pi\)
\(84\) 0 0
\(85\) 15189.0 0.228025
\(86\) 38932.6 0.567633
\(87\) 0 0
\(88\) 83902.3 1.15496
\(89\) −95010.4 −1.27144 −0.635721 0.771919i \(-0.719297\pi\)
−0.635721 + 0.771919i \(0.719297\pi\)
\(90\) 0 0
\(91\) 121388. 1.53664
\(92\) −1456.83 −0.0179449
\(93\) 0 0
\(94\) −63249.5 −0.738307
\(95\) 58568.8 0.665821
\(96\) 0 0
\(97\) −33677.1 −0.363418 −0.181709 0.983352i \(-0.558163\pi\)
−0.181709 + 0.983352i \(0.558163\pi\)
\(98\) −8456.45 −0.0889453
\(99\) 0 0
\(100\) 58636.4 0.586364
\(101\) 78905.4 0.769668 0.384834 0.922986i \(-0.374259\pi\)
0.384834 + 0.922986i \(0.374259\pi\)
\(102\) 0 0
\(103\) −56918.2 −0.528638 −0.264319 0.964435i \(-0.585147\pi\)
−0.264319 + 0.964435i \(0.585147\pi\)
\(104\) 635171. 5.75847
\(105\) 0 0
\(106\) −118774. −1.02673
\(107\) −190035. −1.60463 −0.802313 0.596904i \(-0.796397\pi\)
−0.802313 + 0.596904i \(0.796397\pi\)
\(108\) 0 0
\(109\) 108794. 0.877078 0.438539 0.898712i \(-0.355496\pi\)
0.438539 + 0.898712i \(0.355496\pi\)
\(110\) 33931.0 0.267372
\(111\) 0 0
\(112\) −632862. −4.76720
\(113\) 208092. 1.53306 0.766532 0.642207i \(-0.221981\pi\)
0.766532 + 0.642207i \(0.221981\pi\)
\(114\) 0 0
\(115\) −388.206 −0.00273727
\(116\) −78535.5 −0.541903
\(117\) 0 0
\(118\) 330671. 2.18621
\(119\) −80512.5 −0.521189
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 362799. 2.20682
\(123\) 0 0
\(124\) 259272. 1.51426
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −308832. −1.69908 −0.849538 0.527527i \(-0.823119\pi\)
−0.849538 + 0.527527i \(0.823119\pi\)
\(128\) −1.22976e6 −6.63429
\(129\) 0 0
\(130\) 256870. 1.33308
\(131\) 63060.0 0.321052 0.160526 0.987032i \(-0.448681\pi\)
0.160526 + 0.987032i \(0.448681\pi\)
\(132\) 0 0
\(133\) −310456. −1.52185
\(134\) −654582. −3.14921
\(135\) 0 0
\(136\) −421287. −1.95313
\(137\) 41313.4 0.188057 0.0940284 0.995570i \(-0.470026\pi\)
0.0940284 + 0.995570i \(0.470026\pi\)
\(138\) 0 0
\(139\) −257986. −1.13256 −0.566278 0.824214i \(-0.691617\pi\)
−0.566278 + 0.824214i \(0.691617\pi\)
\(140\) −310814. −1.34023
\(141\) 0 0
\(142\) −12058.3 −0.0501839
\(143\) 110838. 0.453260
\(144\) 0 0
\(145\) −20927.6 −0.0826607
\(146\) −314465. −1.22093
\(147\) 0 0
\(148\) 554858. 2.08222
\(149\) −233787. −0.862689 −0.431344 0.902187i \(-0.641961\pi\)
−0.431344 + 0.902187i \(0.641961\pi\)
\(150\) 0 0
\(151\) −398038. −1.42063 −0.710317 0.703882i \(-0.751448\pi\)
−0.710317 + 0.703882i \(0.751448\pi\)
\(152\) −1.62448e6 −5.70304
\(153\) 0 0
\(154\) −179858. −0.611123
\(155\) 69088.8 0.230982
\(156\) 0 0
\(157\) 283315. 0.917318 0.458659 0.888612i \(-0.348330\pi\)
0.458659 + 0.888612i \(0.348330\pi\)
\(158\) −260587. −0.830444
\(159\) 0 0
\(160\) −784479. −2.42260
\(161\) 2057.76 0.00625649
\(162\) 0 0
\(163\) −491644. −1.44938 −0.724688 0.689077i \(-0.758016\pi\)
−0.724688 + 0.689077i \(0.758016\pi\)
\(164\) 889153. 2.58147
\(165\) 0 0
\(166\) 268566. 0.756451
\(167\) 454917. 1.26224 0.631118 0.775687i \(-0.282596\pi\)
0.631118 + 0.775687i \(0.282596\pi\)
\(168\) 0 0
\(169\) 467788. 1.25989
\(170\) −170373. −0.452146
\(171\) 0 0
\(172\) −325634. −0.839282
\(173\) 275876. 0.700807 0.350404 0.936599i \(-0.386044\pi\)
0.350404 + 0.936599i \(0.386044\pi\)
\(174\) 0 0
\(175\) −82823.5 −0.204436
\(176\) −577858. −1.40617
\(177\) 0 0
\(178\) 1.06572e6 2.52112
\(179\) 19479.7 0.0454411 0.0227206 0.999742i \(-0.492767\pi\)
0.0227206 + 0.999742i \(0.492767\pi\)
\(180\) 0 0
\(181\) −542896. −1.23174 −0.615871 0.787847i \(-0.711196\pi\)
−0.615871 + 0.787847i \(0.711196\pi\)
\(182\) −1.36159e6 −3.04697
\(183\) 0 0
\(184\) 10767.4 0.0234459
\(185\) 147855. 0.317618
\(186\) 0 0
\(187\) −73514.8 −0.153734
\(188\) 529020. 1.09164
\(189\) 0 0
\(190\) −656959. −1.32024
\(191\) −242935. −0.481845 −0.240922 0.970544i \(-0.577450\pi\)
−0.240922 + 0.970544i \(0.577450\pi\)
\(192\) 0 0
\(193\) −207159. −0.400322 −0.200161 0.979763i \(-0.564147\pi\)
−0.200161 + 0.979763i \(0.564147\pi\)
\(194\) 377752. 0.720614
\(195\) 0 0
\(196\) 70730.0 0.131511
\(197\) −253518. −0.465417 −0.232709 0.972547i \(-0.574759\pi\)
−0.232709 + 0.972547i \(0.574759\pi\)
\(198\) 0 0
\(199\) 756118. 1.35350 0.676748 0.736215i \(-0.263389\pi\)
0.676748 + 0.736215i \(0.263389\pi\)
\(200\) −433380. −0.766114
\(201\) 0 0
\(202\) −885072. −1.52616
\(203\) 110931. 0.188935
\(204\) 0 0
\(205\) 236935. 0.393772
\(206\) 638444. 1.04823
\(207\) 0 0
\(208\) −4.37459e6 −7.01098
\(209\) −283473. −0.448896
\(210\) 0 0
\(211\) 1.01855e6 1.57499 0.787494 0.616322i \(-0.211378\pi\)
0.787494 + 0.616322i \(0.211378\pi\)
\(212\) 993428. 1.51809
\(213\) 0 0
\(214\) 2.13160e6 3.18178
\(215\) −86772.4 −0.128022
\(216\) 0 0
\(217\) −366219. −0.527949
\(218\) −1.22033e6 −1.73914
\(219\) 0 0
\(220\) −283800. −0.395327
\(221\) −556534. −0.766498
\(222\) 0 0
\(223\) −612261. −0.824469 −0.412235 0.911078i \(-0.635252\pi\)
−0.412235 + 0.911078i \(0.635252\pi\)
\(224\) 4.15829e6 5.53726
\(225\) 0 0
\(226\) −2.33415e6 −3.03988
\(227\) 625281. 0.805398 0.402699 0.915332i \(-0.368072\pi\)
0.402699 + 0.915332i \(0.368072\pi\)
\(228\) 0 0
\(229\) 1.11167e6 1.40083 0.700415 0.713736i \(-0.252998\pi\)
0.700415 + 0.713736i \(0.252998\pi\)
\(230\) 4354.46 0.00542768
\(231\) 0 0
\(232\) 580454. 0.708023
\(233\) −927589. −1.11935 −0.559675 0.828712i \(-0.689074\pi\)
−0.559675 + 0.828712i \(0.689074\pi\)
\(234\) 0 0
\(235\) 140969. 0.166516
\(236\) −2.76574e6 −3.23245
\(237\) 0 0
\(238\) 903098. 1.03346
\(239\) 225927. 0.255843 0.127921 0.991784i \(-0.459169\pi\)
0.127921 + 0.991784i \(0.459169\pi\)
\(240\) 0 0
\(241\) −1.32668e6 −1.47138 −0.735688 0.677320i \(-0.763141\pi\)
−0.735688 + 0.677320i \(0.763141\pi\)
\(242\) −164226. −0.180262
\(243\) 0 0
\(244\) −3.03446e6 −3.26293
\(245\) 18847.6 0.0200605
\(246\) 0 0
\(247\) −2.14599e6 −2.23813
\(248\) −1.91627e6 −1.97846
\(249\) 0 0
\(250\) −175264. −0.177354
\(251\) −717486. −0.718834 −0.359417 0.933177i \(-0.617025\pi\)
−0.359417 + 0.933177i \(0.617025\pi\)
\(252\) 0 0
\(253\) 1878.92 0.00184547
\(254\) 3.46413e6 3.36907
\(255\) 0 0
\(256\) 7.42108e6 7.07730
\(257\) −716981. −0.677135 −0.338567 0.940942i \(-0.609942\pi\)
−0.338567 + 0.940942i \(0.609942\pi\)
\(258\) 0 0
\(259\) −783733. −0.725970
\(260\) −2.14847e6 −1.97104
\(261\) 0 0
\(262\) −707336. −0.636608
\(263\) 1.58342e6 1.41158 0.705791 0.708420i \(-0.250592\pi\)
0.705791 + 0.708420i \(0.250592\pi\)
\(264\) 0 0
\(265\) 264721. 0.231566
\(266\) 3.48235e6 3.01764
\(267\) 0 0
\(268\) 5.47494e6 4.65632
\(269\) −148436. −0.125072 −0.0625359 0.998043i \(-0.519919\pi\)
−0.0625359 + 0.998043i \(0.519919\pi\)
\(270\) 0 0
\(271\) −703775. −0.582117 −0.291059 0.956705i \(-0.594007\pi\)
−0.291059 + 0.956705i \(0.594007\pi\)
\(272\) 2.90152e6 2.37795
\(273\) 0 0
\(274\) −463407. −0.372894
\(275\) −75625.0 −0.0603023
\(276\) 0 0
\(277\) −917597. −0.718542 −0.359271 0.933233i \(-0.616975\pi\)
−0.359271 + 0.933233i \(0.616975\pi\)
\(278\) 2.89380e6 2.24572
\(279\) 0 0
\(280\) 2.29722e6 1.75108
\(281\) −2.27113e6 −1.71584 −0.857921 0.513782i \(-0.828244\pi\)
−0.857921 + 0.513782i \(0.828244\pi\)
\(282\) 0 0
\(283\) −576587. −0.427956 −0.213978 0.976839i \(-0.568642\pi\)
−0.213978 + 0.976839i \(0.568642\pi\)
\(284\) 100856. 0.0742001
\(285\) 0 0
\(286\) −1.24325e6 −0.898761
\(287\) −1.25592e6 −0.900032
\(288\) 0 0
\(289\) −1.05073e6 −0.740023
\(290\) 234742. 0.163906
\(291\) 0 0
\(292\) 2.63019e6 1.80522
\(293\) 954488. 0.649533 0.324767 0.945794i \(-0.394714\pi\)
0.324767 + 0.945794i \(0.394714\pi\)
\(294\) 0 0
\(295\) −736994. −0.493071
\(296\) −4.10094e6 −2.72053
\(297\) 0 0
\(298\) 2.62236e6 1.71061
\(299\) 14224.1 0.00920123
\(300\) 0 0
\(301\) 459955. 0.292617
\(302\) 4.46474e6 2.81695
\(303\) 0 0
\(304\) 1.11882e7 6.94349
\(305\) −808601. −0.497720
\(306\) 0 0
\(307\) 252381. 0.152831 0.0764153 0.997076i \(-0.475653\pi\)
0.0764153 + 0.997076i \(0.475653\pi\)
\(308\) 1.50434e6 0.903586
\(309\) 0 0
\(310\) −774961. −0.458010
\(311\) −784287. −0.459805 −0.229903 0.973214i \(-0.573841\pi\)
−0.229903 + 0.973214i \(0.573841\pi\)
\(312\) 0 0
\(313\) −2.16086e6 −1.24671 −0.623356 0.781938i \(-0.714231\pi\)
−0.623356 + 0.781938i \(0.714231\pi\)
\(314\) −3.17790e6 −1.81893
\(315\) 0 0
\(316\) 2.17956e6 1.22787
\(317\) −3.39464e6 −1.89734 −0.948670 0.316269i \(-0.897570\pi\)
−0.948670 + 0.316269i \(0.897570\pi\)
\(318\) 0 0
\(319\) 101289. 0.0557298
\(320\) 4.97886e6 2.71804
\(321\) 0 0
\(322\) −23081.7 −0.0124059
\(323\) 1.42336e6 0.759119
\(324\) 0 0
\(325\) −572509. −0.300659
\(326\) 5.51470e6 2.87394
\(327\) 0 0
\(328\) −6.57170e6 −3.37282
\(329\) −747237. −0.380600
\(330\) 0 0
\(331\) −1.50736e6 −0.756216 −0.378108 0.925762i \(-0.623425\pi\)
−0.378108 + 0.925762i \(0.623425\pi\)
\(332\) −2.24629e6 −1.11846
\(333\) 0 0
\(334\) −5.10274e6 −2.50287
\(335\) 1.45892e6 0.710265
\(336\) 0 0
\(337\) −301876. −0.144795 −0.0723976 0.997376i \(-0.523065\pi\)
−0.0723976 + 0.997376i \(0.523065\pi\)
\(338\) −5.24712e6 −2.49821
\(339\) 0 0
\(340\) 1.42501e6 0.668528
\(341\) −334390. −0.155728
\(342\) 0 0
\(343\) 2.12732e6 0.976331
\(344\) 2.40675e6 1.09656
\(345\) 0 0
\(346\) −3.09447e6 −1.38962
\(347\) −325201. −0.144987 −0.0724934 0.997369i \(-0.523096\pi\)
−0.0724934 + 0.997369i \(0.523096\pi\)
\(348\) 0 0
\(349\) −2.73891e6 −1.20369 −0.601845 0.798613i \(-0.705568\pi\)
−0.601845 + 0.798613i \(0.705568\pi\)
\(350\) 929020. 0.405373
\(351\) 0 0
\(352\) 3.79688e6 1.63332
\(353\) −2.86184e6 −1.22238 −0.611192 0.791482i \(-0.709310\pi\)
−0.611192 + 0.791482i \(0.709310\pi\)
\(354\) 0 0
\(355\) 26875.3 0.0113183
\(356\) −8.91371e6 −3.72764
\(357\) 0 0
\(358\) −218501. −0.0901043
\(359\) 3.49224e6 1.43011 0.715053 0.699071i \(-0.246403\pi\)
0.715053 + 0.699071i \(0.246403\pi\)
\(360\) 0 0
\(361\) 3.01239e6 1.21659
\(362\) 6.08959e6 2.44240
\(363\) 0 0
\(364\) 1.13884e7 4.50515
\(365\) 700875. 0.275365
\(366\) 0 0
\(367\) −420097. −0.162811 −0.0814056 0.996681i \(-0.525941\pi\)
−0.0814056 + 0.996681i \(0.525941\pi\)
\(368\) −74157.9 −0.0285455
\(369\) 0 0
\(370\) −1.65847e6 −0.629799
\(371\) −1.40321e6 −0.529283
\(372\) 0 0
\(373\) −281554. −0.104783 −0.0523914 0.998627i \(-0.516684\pi\)
−0.0523914 + 0.998627i \(0.516684\pi\)
\(374\) 824607. 0.304837
\(375\) 0 0
\(376\) −3.90997e6 −1.42628
\(377\) 766798. 0.277861
\(378\) 0 0
\(379\) −1.41256e6 −0.505138 −0.252569 0.967579i \(-0.581275\pi\)
−0.252569 + 0.967579i \(0.581275\pi\)
\(380\) 5.49483e6 1.95207
\(381\) 0 0
\(382\) 2.72497e6 0.955441
\(383\) −2.23548e6 −0.778708 −0.389354 0.921088i \(-0.627302\pi\)
−0.389354 + 0.921088i \(0.627302\pi\)
\(384\) 0 0
\(385\) 400866. 0.137831
\(386\) 2.32367e6 0.793792
\(387\) 0 0
\(388\) −3.15953e6 −1.06548
\(389\) −5.10434e6 −1.71027 −0.855136 0.518403i \(-0.826527\pi\)
−0.855136 + 0.518403i \(0.826527\pi\)
\(390\) 0 0
\(391\) −9434.34 −0.00312083
\(392\) −522763. −0.171826
\(393\) 0 0
\(394\) 2.84367e6 0.922867
\(395\) 580793. 0.187296
\(396\) 0 0
\(397\) −252110. −0.0802813 −0.0401406 0.999194i \(-0.512781\pi\)
−0.0401406 + 0.999194i \(0.512781\pi\)
\(398\) −8.48128e6 −2.68382
\(399\) 0 0
\(400\) 2.98480e6 0.932750
\(401\) −1.61321e6 −0.500991 −0.250496 0.968118i \(-0.580594\pi\)
−0.250496 + 0.968118i \(0.580594\pi\)
\(402\) 0 0
\(403\) −2.53145e6 −0.776439
\(404\) 7.40277e6 2.25653
\(405\) 0 0
\(406\) −1.24430e6 −0.374636
\(407\) −715616. −0.214138
\(408\) 0 0
\(409\) 1.08687e6 0.321270 0.160635 0.987014i \(-0.448646\pi\)
0.160635 + 0.987014i \(0.448646\pi\)
\(410\) −2.65767e6 −0.780803
\(411\) 0 0
\(412\) −5.33997e6 −1.54987
\(413\) 3.90659e6 1.12700
\(414\) 0 0
\(415\) −598576. −0.170608
\(416\) 2.87438e7 8.14348
\(417\) 0 0
\(418\) 3.17968e6 0.890109
\(419\) 740488. 0.206055 0.103027 0.994679i \(-0.467147\pi\)
0.103027 + 0.994679i \(0.467147\pi\)
\(420\) 0 0
\(421\) 6.54748e6 1.80040 0.900199 0.435478i \(-0.143421\pi\)
0.900199 + 0.435478i \(0.143421\pi\)
\(422\) −1.14250e7 −3.12302
\(423\) 0 0
\(424\) −7.34239e6 −1.98346
\(425\) 379725. 0.101976
\(426\) 0 0
\(427\) 4.28615e6 1.13762
\(428\) −1.78287e7 −4.70447
\(429\) 0 0
\(430\) 973315. 0.253853
\(431\) 2.75325e6 0.713925 0.356963 0.934119i \(-0.383812\pi\)
0.356963 + 0.934119i \(0.383812\pi\)
\(432\) 0 0
\(433\) −1.26541e6 −0.324349 −0.162175 0.986762i \(-0.551851\pi\)
−0.162175 + 0.986762i \(0.551851\pi\)
\(434\) 4.10784e6 1.04686
\(435\) 0 0
\(436\) 1.02069e7 2.57144
\(437\) −36378.8 −0.00911266
\(438\) 0 0
\(439\) −1.77205e6 −0.438850 −0.219425 0.975629i \(-0.570418\pi\)
−0.219425 + 0.975629i \(0.570418\pi\)
\(440\) 2.09756e6 0.516514
\(441\) 0 0
\(442\) 6.24257e6 1.51987
\(443\) −3.87503e6 −0.938135 −0.469068 0.883162i \(-0.655410\pi\)
−0.469068 + 0.883162i \(0.655410\pi\)
\(444\) 0 0
\(445\) −2.37526e6 −0.568606
\(446\) 6.86766e6 1.63483
\(447\) 0 0
\(448\) −2.63915e7 −6.21253
\(449\) −5.14887e6 −1.20530 −0.602651 0.798005i \(-0.705889\pi\)
−0.602651 + 0.798005i \(0.705889\pi\)
\(450\) 0 0
\(451\) −1.14677e6 −0.265481
\(452\) 1.95229e7 4.49467
\(453\) 0 0
\(454\) −7.01370e6 −1.59701
\(455\) 3.03470e6 0.687206
\(456\) 0 0
\(457\) −2.37155e6 −0.531179 −0.265590 0.964086i \(-0.585567\pi\)
−0.265590 + 0.964086i \(0.585567\pi\)
\(458\) −1.24694e7 −2.77768
\(459\) 0 0
\(460\) −36420.8 −0.00802518
\(461\) 2.82106e6 0.618244 0.309122 0.951022i \(-0.399965\pi\)
0.309122 + 0.951022i \(0.399965\pi\)
\(462\) 0 0
\(463\) 5.26742e6 1.14194 0.570972 0.820969i \(-0.306566\pi\)
0.570972 + 0.820969i \(0.306566\pi\)
\(464\) −3.99774e6 −0.862024
\(465\) 0 0
\(466\) 1.04047e7 2.21954
\(467\) −1.18996e6 −0.252487 −0.126243 0.991999i \(-0.540292\pi\)
−0.126243 + 0.991999i \(0.540292\pi\)
\(468\) 0 0
\(469\) −7.73332e6 −1.62343
\(470\) −1.58124e6 −0.330181
\(471\) 0 0
\(472\) 2.04415e7 4.22336
\(473\) 419979. 0.0863126
\(474\) 0 0
\(475\) 1.46422e6 0.297764
\(476\) −7.55354e6 −1.52803
\(477\) 0 0
\(478\) −2.53419e6 −0.507306
\(479\) 5.62901e6 1.12097 0.560484 0.828165i \(-0.310615\pi\)
0.560484 + 0.828165i \(0.310615\pi\)
\(480\) 0 0
\(481\) −5.41747e6 −1.06766
\(482\) 1.48812e7 2.91757
\(483\) 0 0
\(484\) 1.37359e6 0.266529
\(485\) −841929. −0.162525
\(486\) 0 0
\(487\) −1.09746e6 −0.209684 −0.104842 0.994489i \(-0.533434\pi\)
−0.104842 + 0.994489i \(0.533434\pi\)
\(488\) 2.24276e7 4.26318
\(489\) 0 0
\(490\) −211411. −0.0397775
\(491\) −1.33860e6 −0.250580 −0.125290 0.992120i \(-0.539986\pi\)
−0.125290 + 0.992120i \(0.539986\pi\)
\(492\) 0 0
\(493\) −508591. −0.0942435
\(494\) 2.40713e7 4.43796
\(495\) 0 0
\(496\) 1.31978e7 2.40879
\(497\) −142458. −0.0258700
\(498\) 0 0
\(499\) 1.29736e6 0.233243 0.116622 0.993176i \(-0.462794\pi\)
0.116622 + 0.993176i \(0.462794\pi\)
\(500\) 1.46591e6 0.262230
\(501\) 0 0
\(502\) 8.04795e6 1.42536
\(503\) −5.44945e6 −0.960357 −0.480179 0.877171i \(-0.659428\pi\)
−0.480179 + 0.877171i \(0.659428\pi\)
\(504\) 0 0
\(505\) 1.97264e6 0.344206
\(506\) −21075.6 −0.00365934
\(507\) 0 0
\(508\) −2.89741e7 −4.98139
\(509\) −9.81233e6 −1.67872 −0.839359 0.543578i \(-0.817069\pi\)
−0.839359 + 0.543578i \(0.817069\pi\)
\(510\) 0 0
\(511\) −3.71513e6 −0.629393
\(512\) −4.38892e7 −7.39916
\(513\) 0 0
\(514\) 8.04229e6 1.34268
\(515\) −1.42295e6 −0.236414
\(516\) 0 0
\(517\) −682292. −0.112265
\(518\) 8.79103e6 1.43951
\(519\) 0 0
\(520\) 1.58793e7 2.57527
\(521\) −8.84276e6 −1.42723 −0.713614 0.700539i \(-0.752943\pi\)
−0.713614 + 0.700539i \(0.752943\pi\)
\(522\) 0 0
\(523\) 5.57320e6 0.890944 0.445472 0.895296i \(-0.353036\pi\)
0.445472 + 0.895296i \(0.353036\pi\)
\(524\) 5.91618e6 0.941267
\(525\) 0 0
\(526\) −1.77610e7 −2.79900
\(527\) 1.67903e6 0.263348
\(528\) 0 0
\(529\) −6.43610e6 −0.999963
\(530\) −2.96935e6 −0.459168
\(531\) 0 0
\(532\) −2.91264e7 −4.46178
\(533\) −8.68143e6 −1.32365
\(534\) 0 0
\(535\) −4.75087e6 −0.717610
\(536\) −4.04651e7 −6.08372
\(537\) 0 0
\(538\) 1.66499e6 0.248003
\(539\) −91222.4 −0.0135248
\(540\) 0 0
\(541\) −7.05506e6 −1.03635 −0.518176 0.855274i \(-0.673389\pi\)
−0.518176 + 0.855274i \(0.673389\pi\)
\(542\) 7.89415e6 1.15427
\(543\) 0 0
\(544\) −1.90648e7 −2.76207
\(545\) 2.71985e6 0.392241
\(546\) 0 0
\(547\) −1.22859e7 −1.75565 −0.877823 0.478985i \(-0.841005\pi\)
−0.877823 + 0.478985i \(0.841005\pi\)
\(548\) 3.87595e6 0.551349
\(549\) 0 0
\(550\) 848276. 0.119572
\(551\) −1.96113e6 −0.275186
\(552\) 0 0
\(553\) −3.07861e6 −0.428097
\(554\) 1.02926e7 1.42478
\(555\) 0 0
\(556\) −2.42038e7 −3.32045
\(557\) 8.42444e6 1.15054 0.575272 0.817962i \(-0.304896\pi\)
0.575272 + 0.817962i \(0.304896\pi\)
\(558\) 0 0
\(559\) 3.17939e6 0.430342
\(560\) −1.58215e7 −2.13196
\(561\) 0 0
\(562\) 2.54750e7 3.40231
\(563\) −7.88182e6 −1.04799 −0.523993 0.851723i \(-0.675558\pi\)
−0.523993 + 0.851723i \(0.675558\pi\)
\(564\) 0 0
\(565\) 5.20231e6 0.685607
\(566\) 6.46751e6 0.848586
\(567\) 0 0
\(568\) −745421. −0.0969462
\(569\) −7.11408e6 −0.921167 −0.460583 0.887616i \(-0.652360\pi\)
−0.460583 + 0.887616i \(0.652360\pi\)
\(570\) 0 0
\(571\) −71565.4 −0.00918572 −0.00459286 0.999989i \(-0.501462\pi\)
−0.00459286 + 0.999989i \(0.501462\pi\)
\(572\) 1.03986e7 1.32888
\(573\) 0 0
\(574\) 1.40875e7 1.78466
\(575\) −9705.15 −0.00122414
\(576\) 0 0
\(577\) −2.17667e6 −0.272179 −0.136089 0.990697i \(-0.543453\pi\)
−0.136089 + 0.990697i \(0.543453\pi\)
\(578\) 1.17859e7 1.46738
\(579\) 0 0
\(580\) −1.96339e6 −0.242346
\(581\) 3.17287e6 0.389953
\(582\) 0 0
\(583\) −1.28125e6 −0.156122
\(584\) −1.94397e7 −2.35861
\(585\) 0 0
\(586\) −1.07064e7 −1.28795
\(587\) 852228. 0.102085 0.0510423 0.998696i \(-0.483746\pi\)
0.0510423 + 0.998696i \(0.483746\pi\)
\(588\) 0 0
\(589\) 6.47432e6 0.768964
\(590\) 8.26677e6 0.977701
\(591\) 0 0
\(592\) 2.82442e7 3.31227
\(593\) −9.10178e6 −1.06289 −0.531446 0.847092i \(-0.678351\pi\)
−0.531446 + 0.847092i \(0.678351\pi\)
\(594\) 0 0
\(595\) −2.01281e6 −0.233083
\(596\) −2.19335e7 −2.52925
\(597\) 0 0
\(598\) −159550. −0.0182450
\(599\) 3.68657e6 0.419813 0.209906 0.977721i \(-0.432684\pi\)
0.209906 + 0.977721i \(0.432684\pi\)
\(600\) 0 0
\(601\) 1.18735e7 1.34088 0.670442 0.741962i \(-0.266104\pi\)
0.670442 + 0.741962i \(0.266104\pi\)
\(602\) −5.15926e6 −0.580224
\(603\) 0 0
\(604\) −3.73432e7 −4.16504
\(605\) 366025. 0.0406558
\(606\) 0 0
\(607\) 8.77821e6 0.967017 0.483509 0.875340i \(-0.339362\pi\)
0.483509 + 0.875340i \(0.339362\pi\)
\(608\) −7.35137e7 −8.06509
\(609\) 0 0
\(610\) 9.06998e6 0.986919
\(611\) −5.16520e6 −0.559737
\(612\) 0 0
\(613\) 1.52076e7 1.63459 0.817294 0.576220i \(-0.195473\pi\)
0.817294 + 0.576220i \(0.195473\pi\)
\(614\) −2.83092e6 −0.303045
\(615\) 0 0
\(616\) −1.11185e7 −1.18058
\(617\) 1.13759e7 1.20302 0.601511 0.798865i \(-0.294566\pi\)
0.601511 + 0.798865i \(0.294566\pi\)
\(618\) 0 0
\(619\) 7.14541e6 0.749550 0.374775 0.927116i \(-0.377720\pi\)
0.374775 + 0.927116i \(0.377720\pi\)
\(620\) 6.48179e6 0.677198
\(621\) 0 0
\(622\) 8.79725e6 0.911739
\(623\) 1.25905e7 1.29964
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 2.42381e7 2.47208
\(627\) 0 0
\(628\) 2.65801e7 2.68941
\(629\) 3.59322e6 0.362124
\(630\) 0 0
\(631\) 4.11229e6 0.411159 0.205580 0.978640i \(-0.434092\pi\)
0.205580 + 0.978640i \(0.434092\pi\)
\(632\) −1.61090e7 −1.60427
\(633\) 0 0
\(634\) 3.80772e7 3.76220
\(635\) −7.72080e6 −0.759850
\(636\) 0 0
\(637\) −690587. −0.0674326
\(638\) −1.13615e6 −0.110506
\(639\) 0 0
\(640\) −3.07439e7 −2.96694
\(641\) −2.34166e6 −0.225102 −0.112551 0.993646i \(-0.535902\pi\)
−0.112551 + 0.993646i \(0.535902\pi\)
\(642\) 0 0
\(643\) −3.71655e6 −0.354497 −0.177248 0.984166i \(-0.556720\pi\)
−0.177248 + 0.984166i \(0.556720\pi\)
\(644\) 193056. 0.0183429
\(645\) 0 0
\(646\) −1.59657e7 −1.50524
\(647\) −8.96177e6 −0.841653 −0.420826 0.907141i \(-0.638260\pi\)
−0.420826 + 0.907141i \(0.638260\pi\)
\(648\) 0 0
\(649\) 3.56705e6 0.332428
\(650\) 6.42176e6 0.596170
\(651\) 0 0
\(652\) −4.61251e7 −4.24931
\(653\) 1.79019e7 1.64292 0.821460 0.570267i \(-0.193160\pi\)
0.821460 + 0.570267i \(0.193160\pi\)
\(654\) 0 0
\(655\) 1.57650e6 0.143579
\(656\) 4.52610e7 4.10643
\(657\) 0 0
\(658\) 8.38166e6 0.754685
\(659\) −1.90122e7 −1.70537 −0.852686 0.522423i \(-0.825028\pi\)
−0.852686 + 0.522423i \(0.825028\pi\)
\(660\) 0 0
\(661\) −4.58472e6 −0.408140 −0.204070 0.978956i \(-0.565417\pi\)
−0.204070 + 0.978956i \(0.565417\pi\)
\(662\) 1.69078e7 1.49949
\(663\) 0 0
\(664\) 1.66023e7 1.46133
\(665\) −7.76140e6 −0.680590
\(666\) 0 0
\(667\) 12998.7 0.00113132
\(668\) 4.26795e7 3.70065
\(669\) 0 0
\(670\) −1.63646e7 −1.40837
\(671\) 3.91363e6 0.335563
\(672\) 0 0
\(673\) −1.51723e7 −1.29126 −0.645630 0.763651i \(-0.723405\pi\)
−0.645630 + 0.763651i \(0.723405\pi\)
\(674\) 3.38611e6 0.287112
\(675\) 0 0
\(676\) 4.38871e7 3.69377
\(677\) 1.08298e7 0.908129 0.454064 0.890969i \(-0.349974\pi\)
0.454064 + 0.890969i \(0.349974\pi\)
\(678\) 0 0
\(679\) 4.46281e6 0.371479
\(680\) −1.05322e7 −0.873466
\(681\) 0 0
\(682\) 3.75081e6 0.308791
\(683\) 1.41968e7 1.16450 0.582248 0.813012i \(-0.302173\pi\)
0.582248 + 0.813012i \(0.302173\pi\)
\(684\) 0 0
\(685\) 1.03283e6 0.0841016
\(686\) −2.38618e7 −1.93595
\(687\) 0 0
\(688\) −1.65759e7 −1.33508
\(689\) −9.69954e6 −0.778400
\(690\) 0 0
\(691\) 9.99651e6 0.796440 0.398220 0.917290i \(-0.369628\pi\)
0.398220 + 0.917290i \(0.369628\pi\)
\(692\) 2.58822e7 2.05464
\(693\) 0 0
\(694\) 3.64774e6 0.287492
\(695\) −6.44966e6 −0.506494
\(696\) 0 0
\(697\) 5.75809e6 0.448949
\(698\) 3.07221e7 2.38678
\(699\) 0 0
\(700\) −7.77035e6 −0.599371
\(701\) 3.82320e6 0.293854 0.146927 0.989147i \(-0.453062\pi\)
0.146927 + 0.989147i \(0.453062\pi\)
\(702\) 0 0
\(703\) 1.38555e7 1.05738
\(704\) −2.40977e7 −1.83250
\(705\) 0 0
\(706\) 3.21009e7 2.42385
\(707\) −1.04564e7 −0.786741
\(708\) 0 0
\(709\) 5.38358e6 0.402213 0.201106 0.979569i \(-0.435546\pi\)
0.201106 + 0.979569i \(0.435546\pi\)
\(710\) −301457. −0.0224429
\(711\) 0 0
\(712\) 6.58810e7 4.87035
\(713\) −42913.1 −0.00316130
\(714\) 0 0
\(715\) 2.77094e6 0.202704
\(716\) 1.82755e6 0.133225
\(717\) 0 0
\(718\) −3.91720e7 −2.83573
\(719\) 1.66113e7 1.19834 0.599171 0.800621i \(-0.295497\pi\)
0.599171 + 0.800621i \(0.295497\pi\)
\(720\) 0 0
\(721\) 7.54266e6 0.540364
\(722\) −3.37896e7 −2.41235
\(723\) 0 0
\(724\) −5.09335e7 −3.61125
\(725\) −523189. −0.0369670
\(726\) 0 0
\(727\) 1.47622e7 1.03589 0.517947 0.855413i \(-0.326697\pi\)
0.517947 + 0.855413i \(0.326697\pi\)
\(728\) −8.41713e7 −5.88621
\(729\) 0 0
\(730\) −7.86163e6 −0.546016
\(731\) −2.10878e6 −0.145961
\(732\) 0 0
\(733\) −8.48819e6 −0.583519 −0.291760 0.956492i \(-0.594241\pi\)
−0.291760 + 0.956492i \(0.594241\pi\)
\(734\) 4.71217e6 0.322835
\(735\) 0 0
\(736\) 487263. 0.0331565
\(737\) −7.06119e6 −0.478861
\(738\) 0 0
\(739\) 2.60406e7 1.75404 0.877022 0.480450i \(-0.159527\pi\)
0.877022 + 0.480450i \(0.159527\pi\)
\(740\) 1.38715e7 0.931199
\(741\) 0 0
\(742\) 1.57396e7 1.04951
\(743\) 791476. 0.0525975 0.0262988 0.999654i \(-0.491628\pi\)
0.0262988 + 0.999654i \(0.491628\pi\)
\(744\) 0 0
\(745\) −5.84467e6 −0.385806
\(746\) 3.15816e6 0.207772
\(747\) 0 0
\(748\) −6.89703e6 −0.450722
\(749\) 2.51829e7 1.64022
\(750\) 0 0
\(751\) −3.80752e6 −0.246344 −0.123172 0.992385i \(-0.539307\pi\)
−0.123172 + 0.992385i \(0.539307\pi\)
\(752\) 2.69290e7 1.73650
\(753\) 0 0
\(754\) −8.60107e6 −0.550965
\(755\) −9.95095e6 −0.635327
\(756\) 0 0
\(757\) −2.46270e7 −1.56197 −0.780983 0.624552i \(-0.785281\pi\)
−0.780983 + 0.624552i \(0.785281\pi\)
\(758\) 1.58445e7 1.00163
\(759\) 0 0
\(760\) −4.06121e7 −2.55048
\(761\) 1.49393e7 0.935123 0.467561 0.883961i \(-0.345133\pi\)
0.467561 + 0.883961i \(0.345133\pi\)
\(762\) 0 0
\(763\) −1.44171e7 −0.896534
\(764\) −2.27918e7 −1.41268
\(765\) 0 0
\(766\) 2.50752e7 1.54409
\(767\) 2.70039e7 1.65744
\(768\) 0 0
\(769\) 9.19320e6 0.560597 0.280299 0.959913i \(-0.409566\pi\)
0.280299 + 0.959913i \(0.409566\pi\)
\(770\) −4.49646e6 −0.273303
\(771\) 0 0
\(772\) −1.94353e7 −1.17367
\(773\) 2.65966e7 1.60095 0.800474 0.599367i \(-0.204581\pi\)
0.800474 + 0.599367i \(0.204581\pi\)
\(774\) 0 0
\(775\) 1.72722e6 0.103298
\(776\) 2.33520e7 1.39210
\(777\) 0 0
\(778\) 5.72547e7 3.39127
\(779\) 2.22032e7 1.31091
\(780\) 0 0
\(781\) −130076. −0.00763082
\(782\) 105824. 0.00618823
\(783\) 0 0
\(784\) 3.60041e6 0.209200
\(785\) 7.08286e6 0.410237
\(786\) 0 0
\(787\) −1.01592e7 −0.584684 −0.292342 0.956314i \(-0.594435\pi\)
−0.292342 + 0.956314i \(0.594435\pi\)
\(788\) −2.37846e7 −1.36452
\(789\) 0 0
\(790\) −6.51468e6 −0.371386
\(791\) −2.75759e7 −1.56707
\(792\) 0 0
\(793\) 2.96276e7 1.67307
\(794\) 2.82789e6 0.159188
\(795\) 0 0
\(796\) 7.09377e7 3.96821
\(797\) 1.13291e7 0.631756 0.315878 0.948800i \(-0.397701\pi\)
0.315878 + 0.948800i \(0.397701\pi\)
\(798\) 0 0
\(799\) 3.42590e6 0.189849
\(800\) −1.96120e7 −1.08342
\(801\) 0 0
\(802\) 1.80952e7 0.993406
\(803\) −3.39223e6 −0.185651
\(804\) 0 0
\(805\) 51444.1 0.00279799
\(806\) 2.83950e7 1.53959
\(807\) 0 0
\(808\) −5.47136e7 −2.94827
\(809\) −2.78519e7 −1.49618 −0.748091 0.663597i \(-0.769029\pi\)
−0.748091 + 0.663597i \(0.769029\pi\)
\(810\) 0 0
\(811\) −3.02449e7 −1.61473 −0.807365 0.590052i \(-0.799107\pi\)
−0.807365 + 0.590052i \(0.799107\pi\)
\(812\) 1.04073e7 0.553923
\(813\) 0 0
\(814\) 8.02697e6 0.424611
\(815\) −1.22911e7 −0.648181
\(816\) 0 0
\(817\) −8.13146e6 −0.426200
\(818\) −1.21913e7 −0.637040
\(819\) 0 0
\(820\) 2.22288e7 1.15447
\(821\) 1.98523e7 1.02790 0.513952 0.857819i \(-0.328181\pi\)
0.513952 + 0.857819i \(0.328181\pi\)
\(822\) 0 0
\(823\) 1.25066e7 0.643636 0.321818 0.946802i \(-0.395706\pi\)
0.321818 + 0.946802i \(0.395706\pi\)
\(824\) 3.94675e7 2.02498
\(825\) 0 0
\(826\) −4.38197e7 −2.23470
\(827\) 8.51692e6 0.433031 0.216515 0.976279i \(-0.430531\pi\)
0.216515 + 0.976279i \(0.430531\pi\)
\(828\) 0 0
\(829\) 2.94676e6 0.148922 0.0744609 0.997224i \(-0.476276\pi\)
0.0744609 + 0.997224i \(0.476276\pi\)
\(830\) 6.71415e6 0.338295
\(831\) 0 0
\(832\) −1.82428e8 −9.13658
\(833\) 458043. 0.0228714
\(834\) 0 0
\(835\) 1.13729e7 0.564489
\(836\) −2.65950e7 −1.31608
\(837\) 0 0
\(838\) −8.30596e6 −0.408583
\(839\) 5.40609e6 0.265142 0.132571 0.991174i \(-0.457677\pi\)
0.132571 + 0.991174i \(0.457677\pi\)
\(840\) 0 0
\(841\) −1.98104e7 −0.965836
\(842\) −7.34422e7 −3.56998
\(843\) 0 0
\(844\) 9.55589e7 4.61758
\(845\) 1.16947e7 0.563440
\(846\) 0 0
\(847\) −1.94019e6 −0.0929257
\(848\) 5.05690e7 2.41488
\(849\) 0 0
\(850\) −4.25933e6 −0.202206
\(851\) −91836.8 −0.00434703
\(852\) 0 0
\(853\) −473034. −0.0222597 −0.0111299 0.999938i \(-0.503543\pi\)
−0.0111299 + 0.999938i \(0.503543\pi\)
\(854\) −4.80773e7 −2.25577
\(855\) 0 0
\(856\) 1.31772e8 6.14663
\(857\) −6.34036e6 −0.294891 −0.147446 0.989070i \(-0.547105\pi\)
−0.147446 + 0.989070i \(0.547105\pi\)
\(858\) 0 0
\(859\) −2.40671e7 −1.11286 −0.556430 0.830895i \(-0.687829\pi\)
−0.556430 + 0.830895i \(0.687829\pi\)
\(860\) −8.14084e6 −0.375338
\(861\) 0 0
\(862\) −3.08829e7 −1.41563
\(863\) 1.06639e6 0.0487402 0.0243701 0.999703i \(-0.492242\pi\)
0.0243701 + 0.999703i \(0.492242\pi\)
\(864\) 0 0
\(865\) 6.89690e6 0.313411
\(866\) 1.41940e7 0.643146
\(867\) 0 0
\(868\) −3.43581e7 −1.54785
\(869\) −2.81104e6 −0.126275
\(870\) 0 0
\(871\) −5.34557e7 −2.38753
\(872\) −7.54385e7 −3.35971
\(873\) 0 0
\(874\) 408057. 0.0180693
\(875\) −2.07059e6 −0.0914268
\(876\) 0 0
\(877\) −1.28432e7 −0.563864 −0.281932 0.959434i \(-0.590975\pi\)
−0.281932 + 0.959434i \(0.590975\pi\)
\(878\) 1.98769e7 0.870187
\(879\) 0 0
\(880\) −1.44464e7 −0.628860
\(881\) −2.79947e6 −0.121517 −0.0607583 0.998153i \(-0.519352\pi\)
−0.0607583 + 0.998153i \(0.519352\pi\)
\(882\) 0 0
\(883\) 2.81700e7 1.21587 0.607933 0.793988i \(-0.291999\pi\)
0.607933 + 0.793988i \(0.291999\pi\)
\(884\) −5.22130e7 −2.24723
\(885\) 0 0
\(886\) 4.34657e7 1.86021
\(887\) −4.42238e7 −1.88733 −0.943664 0.330905i \(-0.892646\pi\)
−0.943664 + 0.330905i \(0.892646\pi\)
\(888\) 0 0
\(889\) 4.09257e7 1.73677
\(890\) 2.66430e7 1.12748
\(891\) 0 0
\(892\) −5.74413e7 −2.41720
\(893\) 1.32103e7 0.554348
\(894\) 0 0
\(895\) 486992. 0.0203219
\(896\) 1.62964e8 6.78145
\(897\) 0 0
\(898\) 5.77542e7 2.38997
\(899\) −2.31338e6 −0.0954657
\(900\) 0 0
\(901\) 6.43337e6 0.264014
\(902\) 1.28631e7 0.526417
\(903\) 0 0
\(904\) −1.44293e8 −5.87251
\(905\) −1.35724e7 −0.550852
\(906\) 0 0
\(907\) 6.22787e6 0.251375 0.125687 0.992070i \(-0.459886\pi\)
0.125687 + 0.992070i \(0.459886\pi\)
\(908\) 5.86628e7 2.36128
\(909\) 0 0
\(910\) −3.40398e7 −1.36265
\(911\) −1.08864e7 −0.434599 −0.217300 0.976105i \(-0.569725\pi\)
−0.217300 + 0.976105i \(0.569725\pi\)
\(912\) 0 0
\(913\) 2.89711e6 0.115024
\(914\) 2.66013e7 1.05327
\(915\) 0 0
\(916\) 1.04295e8 4.10698
\(917\) −8.35655e6 −0.328174
\(918\) 0 0
\(919\) −1.61046e7 −0.629014 −0.314507 0.949255i \(-0.601839\pi\)
−0.314507 + 0.949255i \(0.601839\pi\)
\(920\) 269185. 0.0104853
\(921\) 0 0
\(922\) −3.16435e7 −1.22590
\(923\) −984726. −0.0380462
\(924\) 0 0
\(925\) 3.69636e6 0.142043
\(926\) −5.90839e7 −2.26434
\(927\) 0 0
\(928\) 2.62676e7 1.00127
\(929\) −1.97673e6 −0.0751465 −0.0375732 0.999294i \(-0.511963\pi\)
−0.0375732 + 0.999294i \(0.511963\pi\)
\(930\) 0 0
\(931\) 1.76621e6 0.0667834
\(932\) −8.70248e7 −3.28173
\(933\) 0 0
\(934\) 1.33476e7 0.500652
\(935\) −1.83787e6 −0.0687521
\(936\) 0 0
\(937\) 1.38945e7 0.517003 0.258502 0.966011i \(-0.416771\pi\)
0.258502 + 0.966011i \(0.416771\pi\)
\(938\) 8.67436e7 3.21907
\(939\) 0 0
\(940\) 1.32255e7 0.488194
\(941\) −4.33771e7 −1.59693 −0.798466 0.602040i \(-0.794355\pi\)
−0.798466 + 0.602040i \(0.794355\pi\)
\(942\) 0 0
\(943\) −147167. −0.00538930
\(944\) −1.40786e8 −5.14197
\(945\) 0 0
\(946\) −4.71085e6 −0.171148
\(947\) 1.79958e7 0.652075 0.326037 0.945357i \(-0.394286\pi\)
0.326037 + 0.945357i \(0.394286\pi\)
\(948\) 0 0
\(949\) −2.56804e7 −0.925629
\(950\) −1.64240e7 −0.590431
\(951\) 0 0
\(952\) 5.58280e7 1.99645
\(953\) −2.95957e7 −1.05559 −0.527796 0.849371i \(-0.676982\pi\)
−0.527796 + 0.849371i \(0.676982\pi\)
\(954\) 0 0
\(955\) −6.07338e6 −0.215487
\(956\) 2.11961e7 0.750085
\(957\) 0 0
\(958\) −6.31399e7 −2.22275
\(959\) −5.47475e6 −0.192228
\(960\) 0 0
\(961\) −2.09919e7 −0.733236
\(962\) 6.07671e7 2.11705
\(963\) 0 0
\(964\) −1.24467e8 −4.31381
\(965\) −5.17897e6 −0.179030
\(966\) 0 0
\(967\) 5.50973e7 1.89480 0.947402 0.320046i \(-0.103699\pi\)
0.947402 + 0.320046i \(0.103699\pi\)
\(968\) −1.01522e7 −0.348234
\(969\) 0 0
\(970\) 9.44381e6 0.322268
\(971\) −1.86888e7 −0.636112 −0.318056 0.948072i \(-0.603030\pi\)
−0.318056 + 0.948072i \(0.603030\pi\)
\(972\) 0 0
\(973\) 3.41877e7 1.15768
\(974\) 1.23100e7 0.415778
\(975\) 0 0
\(976\) −1.54465e8 −5.19045
\(977\) 7.09223e6 0.237710 0.118855 0.992912i \(-0.462078\pi\)
0.118855 + 0.992912i \(0.462078\pi\)
\(978\) 0 0
\(979\) 1.14963e7 0.383354
\(980\) 1.76825e6 0.0588137
\(981\) 0 0
\(982\) 1.50149e7 0.496870
\(983\) −2.97902e7 −0.983307 −0.491653 0.870791i \(-0.663607\pi\)
−0.491653 + 0.870791i \(0.663607\pi\)
\(984\) 0 0
\(985\) −6.33794e6 −0.208141
\(986\) 5.70480e6 0.186874
\(987\) 0 0
\(988\) −2.01333e8 −6.56181
\(989\) 53896.9 0.00175216
\(990\) 0 0
\(991\) −2.60948e7 −0.844054 −0.422027 0.906583i \(-0.638681\pi\)
−0.422027 + 0.906583i \(0.638681\pi\)
\(992\) −8.67180e7 −2.79789
\(993\) 0 0
\(994\) 1.59793e6 0.0512971
\(995\) 1.89030e7 0.605302
\(996\) 0 0
\(997\) −2.68998e7 −0.857061 −0.428531 0.903527i \(-0.640969\pi\)
−0.428531 + 0.903527i \(0.640969\pi\)
\(998\) −1.45523e7 −0.462494
\(999\) 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 495.6.a.l.1.1 6
3.2 odd 2 55.6.a.d.1.6 6
12.11 even 2 880.6.a.u.1.3 6
15.2 even 4 275.6.b.f.199.12 12
15.8 even 4 275.6.b.f.199.1 12
15.14 odd 2 275.6.a.f.1.1 6
33.32 even 2 605.6.a.e.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.d.1.6 6 3.2 odd 2
275.6.a.f.1.1 6 15.14 odd 2
275.6.b.f.199.1 12 15.8 even 4
275.6.b.f.199.12 12 15.2 even 4
495.6.a.l.1.1 6 1.1 even 1 trivial
605.6.a.e.1.1 6 33.32 even 2
880.6.a.u.1.3 6 12.11 even 2