Properties

Label 495.6.a.d.1.1
Level $495$
Weight $6$
Character 495.1
Self dual yes
Analytic conductor $79.390$
Analytic rank $1$
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] = [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: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.76300\) 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)
 
\(f(q)\) \(=\) \(q-7.52601 q^{2} +24.6408 q^{4} +25.0000 q^{5} -234.126 q^{7} +55.3856 q^{8} +O(q^{10})\) \(q-7.52601 q^{2} +24.6408 q^{4} +25.0000 q^{5} -234.126 q^{7} +55.3856 q^{8} -188.150 q^{10} -121.000 q^{11} +236.184 q^{13} +1762.04 q^{14} -1205.34 q^{16} +608.015 q^{17} -1799.81 q^{19} +616.019 q^{20} +910.647 q^{22} +4773.40 q^{23} +625.000 q^{25} -1777.52 q^{26} -5769.05 q^{28} -2804.99 q^{29} +10258.3 q^{31} +7299.04 q^{32} -4575.93 q^{34} -5853.16 q^{35} -7629.84 q^{37} +13545.4 q^{38} +1384.64 q^{40} -8355.48 q^{41} -11967.9 q^{43} -2981.53 q^{44} -35924.7 q^{46} -8385.48 q^{47} +38008.1 q^{49} -4703.75 q^{50} +5819.76 q^{52} -3453.00 q^{53} -3025.00 q^{55} -12967.2 q^{56} +21110.4 q^{58} +51032.4 q^{59} +9988.24 q^{61} -77204.1 q^{62} -16361.8 q^{64} +5904.60 q^{65} -36567.6 q^{67} +14982.0 q^{68} +44050.9 q^{70} +30522.3 q^{71} +83879.5 q^{73} +57422.2 q^{74} -44348.7 q^{76} +28329.3 q^{77} +103505. q^{79} -30133.4 q^{80} +62883.4 q^{82} +4341.54 q^{83} +15200.4 q^{85} +90070.3 q^{86} -6701.65 q^{88} +45464.1 q^{89} -55296.9 q^{91} +117620. q^{92} +63109.2 q^{94} -44995.2 q^{95} -183170. q^{97} -286049. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 28 q^{4} + 75 q^{5} - 232 q^{7} - 24 q^{8} + 50 q^{10} - 363 q^{11} + 450 q^{13} + 1504 q^{14} - 1360 q^{16} + 334 q^{17} - 4036 q^{19} + 700 q^{20} - 242 q^{22} + 7060 q^{23} + 1875 q^{25} - 2932 q^{26} - 8320 q^{28} - 4042 q^{29} - 608 q^{31} + 3104 q^{32} - 3644 q^{34} - 5800 q^{35} + 2250 q^{37} + 12632 q^{38} - 600 q^{40} - 10654 q^{41} - 35528 q^{43} - 3388 q^{44} - 41800 q^{46} + 2100 q^{47} + 7667 q^{49} + 1250 q^{50} - 14520 q^{52} + 12826 q^{53} - 9075 q^{55} - 17088 q^{56} - 17196 q^{58} + 81876 q^{59} - 62298 q^{61} - 109184 q^{62} - 72256 q^{64} + 11250 q^{65} - 46148 q^{67} + 35832 q^{68} + 37600 q^{70} + 64724 q^{71} + 810 q^{73} + 44796 q^{74} + 44656 q^{76} + 28072 q^{77} + 43876 q^{79} - 34000 q^{80} + 56060 q^{82} + 101024 q^{83} + 8350 q^{85} - 24128 q^{86} + 2904 q^{88} - 60022 q^{89} - 28568 q^{91} - 38256 q^{92} + 74552 q^{94} - 100900 q^{95} - 319746 q^{97} - 431134 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.52601 −1.33042 −0.665211 0.746655i \(-0.731659\pi\)
−0.665211 + 0.746655i \(0.731659\pi\)
\(3\) 0 0
\(4\) 24.6408 0.770024
\(5\) 25.0000 0.447214
\(6\) 0 0
\(7\) −234.126 −1.80595 −0.902973 0.429696i \(-0.858621\pi\)
−0.902973 + 0.429696i \(0.858621\pi\)
\(8\) 55.3856 0.305965
\(9\) 0 0
\(10\) −188.150 −0.594983
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) 236.184 0.387608 0.193804 0.981040i \(-0.437917\pi\)
0.193804 + 0.981040i \(0.437917\pi\)
\(14\) 1762.04 2.40267
\(15\) 0 0
\(16\) −1205.34 −1.17709
\(17\) 608.015 0.510261 0.255130 0.966907i \(-0.417882\pi\)
0.255130 + 0.966907i \(0.417882\pi\)
\(18\) 0 0
\(19\) −1799.81 −1.14378 −0.571890 0.820331i \(-0.693789\pi\)
−0.571890 + 0.820331i \(0.693789\pi\)
\(20\) 616.019 0.344365
\(21\) 0 0
\(22\) 910.647 0.401137
\(23\) 4773.40 1.88152 0.940759 0.339075i \(-0.110114\pi\)
0.940759 + 0.339075i \(0.110114\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) −1777.52 −0.515682
\(27\) 0 0
\(28\) −5769.05 −1.39062
\(29\) −2804.99 −0.619351 −0.309676 0.950842i \(-0.600220\pi\)
−0.309676 + 0.950842i \(0.600220\pi\)
\(30\) 0 0
\(31\) 10258.3 1.91722 0.958610 0.284724i \(-0.0919019\pi\)
0.958610 + 0.284724i \(0.0919019\pi\)
\(32\) 7299.04 1.26006
\(33\) 0 0
\(34\) −4575.93 −0.678862
\(35\) −5853.16 −0.807644
\(36\) 0 0
\(37\) −7629.84 −0.916244 −0.458122 0.888889i \(-0.651478\pi\)
−0.458122 + 0.888889i \(0.651478\pi\)
\(38\) 13545.4 1.52171
\(39\) 0 0
\(40\) 1384.64 0.136832
\(41\) −8355.48 −0.776268 −0.388134 0.921603i \(-0.626880\pi\)
−0.388134 + 0.921603i \(0.626880\pi\)
\(42\) 0 0
\(43\) −11967.9 −0.987065 −0.493533 0.869727i \(-0.664295\pi\)
−0.493533 + 0.869727i \(0.664295\pi\)
\(44\) −2981.53 −0.232171
\(45\) 0 0
\(46\) −35924.7 −2.50321
\(47\) −8385.48 −0.553711 −0.276856 0.960912i \(-0.589292\pi\)
−0.276856 + 0.960912i \(0.589292\pi\)
\(48\) 0 0
\(49\) 38008.1 2.26144
\(50\) −4703.75 −0.266085
\(51\) 0 0
\(52\) 5819.76 0.298467
\(53\) −3453.00 −0.168852 −0.0844261 0.996430i \(-0.526906\pi\)
−0.0844261 + 0.996430i \(0.526906\pi\)
\(54\) 0 0
\(55\) −3025.00 −0.134840
\(56\) −12967.2 −0.552556
\(57\) 0 0
\(58\) 21110.4 0.823999
\(59\) 51032.4 1.90860 0.954302 0.298842i \(-0.0966005\pi\)
0.954302 + 0.298842i \(0.0966005\pi\)
\(60\) 0 0
\(61\) 9988.24 0.343688 0.171844 0.985124i \(-0.445028\pi\)
0.171844 + 0.985124i \(0.445028\pi\)
\(62\) −77204.1 −2.55071
\(63\) 0 0
\(64\) −16361.8 −0.499323
\(65\) 5904.60 0.173343
\(66\) 0 0
\(67\) −36567.6 −0.995198 −0.497599 0.867407i \(-0.665785\pi\)
−0.497599 + 0.867407i \(0.665785\pi\)
\(68\) 14982.0 0.392913
\(69\) 0 0
\(70\) 44050.9 1.07451
\(71\) 30522.3 0.718573 0.359286 0.933227i \(-0.383020\pi\)
0.359286 + 0.933227i \(0.383020\pi\)
\(72\) 0 0
\(73\) 83879.5 1.84225 0.921125 0.389267i \(-0.127272\pi\)
0.921125 + 0.389267i \(0.127272\pi\)
\(74\) 57422.2 1.21899
\(75\) 0 0
\(76\) −44348.7 −0.880738
\(77\) 28329.3 0.544513
\(78\) 0 0
\(79\) 103505. 1.86592 0.932961 0.359978i \(-0.117216\pi\)
0.932961 + 0.359978i \(0.117216\pi\)
\(80\) −30133.4 −0.526409
\(81\) 0 0
\(82\) 62883.4 1.03276
\(83\) 4341.54 0.0691749 0.0345875 0.999402i \(-0.488988\pi\)
0.0345875 + 0.999402i \(0.488988\pi\)
\(84\) 0 0
\(85\) 15200.4 0.228196
\(86\) 90070.3 1.31321
\(87\) 0 0
\(88\) −6701.65 −0.0922519
\(89\) 45464.1 0.608406 0.304203 0.952607i \(-0.401610\pi\)
0.304203 + 0.952607i \(0.401610\pi\)
\(90\) 0 0
\(91\) −55296.9 −0.699999
\(92\) 117620. 1.44881
\(93\) 0 0
\(94\) 63109.2 0.736670
\(95\) −44995.2 −0.511514
\(96\) 0 0
\(97\) −183170. −1.97663 −0.988314 0.152432i \(-0.951289\pi\)
−0.988314 + 0.152432i \(0.951289\pi\)
\(98\) −286049. −3.00868
\(99\) 0 0
\(100\) 15400.5 0.154005
\(101\) 28833.3 0.281249 0.140625 0.990063i \(-0.455089\pi\)
0.140625 + 0.990063i \(0.455089\pi\)
\(102\) 0 0
\(103\) −59444.6 −0.552102 −0.276051 0.961143i \(-0.589026\pi\)
−0.276051 + 0.961143i \(0.589026\pi\)
\(104\) 13081.2 0.118594
\(105\) 0 0
\(106\) 25987.3 0.224645
\(107\) −13840.6 −0.116868 −0.0584342 0.998291i \(-0.518611\pi\)
−0.0584342 + 0.998291i \(0.518611\pi\)
\(108\) 0 0
\(109\) −13278.1 −0.107046 −0.0535231 0.998567i \(-0.517045\pi\)
−0.0535231 + 0.998567i \(0.517045\pi\)
\(110\) 22766.2 0.179394
\(111\) 0 0
\(112\) 282201. 2.12576
\(113\) −130086. −0.958370 −0.479185 0.877714i \(-0.659068\pi\)
−0.479185 + 0.877714i \(0.659068\pi\)
\(114\) 0 0
\(115\) 119335. 0.841441
\(116\) −69117.2 −0.476915
\(117\) 0 0
\(118\) −384070. −2.53925
\(119\) −142352. −0.921504
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −75171.5 −0.457250
\(123\) 0 0
\(124\) 252773. 1.47631
\(125\) 15625.0 0.0894427
\(126\) 0 0
\(127\) −274904. −1.51242 −0.756209 0.654330i \(-0.772951\pi\)
−0.756209 + 0.654330i \(0.772951\pi\)
\(128\) −110430. −0.595748
\(129\) 0 0
\(130\) −44438.1 −0.230620
\(131\) 119374. 0.607761 0.303881 0.952710i \(-0.401718\pi\)
0.303881 + 0.952710i \(0.401718\pi\)
\(132\) 0 0
\(133\) 421382. 2.06560
\(134\) 275208. 1.32403
\(135\) 0 0
\(136\) 33675.3 0.156122
\(137\) 132278. 0.602123 0.301061 0.953605i \(-0.402659\pi\)
0.301061 + 0.953605i \(0.402659\pi\)
\(138\) 0 0
\(139\) −102326. −0.449209 −0.224604 0.974450i \(-0.572109\pi\)
−0.224604 + 0.974450i \(0.572109\pi\)
\(140\) −144226. −0.621905
\(141\) 0 0
\(142\) −229711. −0.956005
\(143\) −28578.3 −0.116868
\(144\) 0 0
\(145\) −70124.8 −0.276982
\(146\) −631278. −2.45097
\(147\) 0 0
\(148\) −188005. −0.705530
\(149\) −371696. −1.37158 −0.685791 0.727798i \(-0.740544\pi\)
−0.685791 + 0.727798i \(0.740544\pi\)
\(150\) 0 0
\(151\) −111725. −0.398757 −0.199379 0.979923i \(-0.563892\pi\)
−0.199379 + 0.979923i \(0.563892\pi\)
\(152\) −99683.4 −0.349956
\(153\) 0 0
\(154\) −213206. −0.724433
\(155\) 256458. 0.857406
\(156\) 0 0
\(157\) −244726. −0.792375 −0.396187 0.918170i \(-0.629667\pi\)
−0.396187 + 0.918170i \(0.629667\pi\)
\(158\) −778979. −2.48246
\(159\) 0 0
\(160\) 182476. 0.563515
\(161\) −1.11758e6 −3.39792
\(162\) 0 0
\(163\) 435623. 1.28423 0.642114 0.766610i \(-0.278058\pi\)
0.642114 + 0.766610i \(0.278058\pi\)
\(164\) −205885. −0.597745
\(165\) 0 0
\(166\) −32674.5 −0.0920319
\(167\) −288486. −0.800450 −0.400225 0.916417i \(-0.631068\pi\)
−0.400225 + 0.916417i \(0.631068\pi\)
\(168\) 0 0
\(169\) −315510. −0.849760
\(170\) −114398. −0.303597
\(171\) 0 0
\(172\) −294898. −0.760064
\(173\) −539633. −1.37083 −0.685414 0.728154i \(-0.740379\pi\)
−0.685414 + 0.728154i \(0.740379\pi\)
\(174\) 0 0
\(175\) −146329. −0.361189
\(176\) 145846. 0.354905
\(177\) 0 0
\(178\) −342163. −0.809438
\(179\) 378874. 0.883816 0.441908 0.897060i \(-0.354302\pi\)
0.441908 + 0.897060i \(0.354302\pi\)
\(180\) 0 0
\(181\) −621633. −1.41038 −0.705192 0.709016i \(-0.749139\pi\)
−0.705192 + 0.709016i \(0.749139\pi\)
\(182\) 416165. 0.931294
\(183\) 0 0
\(184\) 264378. 0.575679
\(185\) −190746. −0.409757
\(186\) 0 0
\(187\) −73569.8 −0.153849
\(188\) −206625. −0.426371
\(189\) 0 0
\(190\) 338634. 0.680529
\(191\) −8509.61 −0.0168782 −0.00843911 0.999964i \(-0.502686\pi\)
−0.00843911 + 0.999964i \(0.502686\pi\)
\(192\) 0 0
\(193\) −564588. −1.09103 −0.545517 0.838100i \(-0.683667\pi\)
−0.545517 + 0.838100i \(0.683667\pi\)
\(194\) 1.37854e6 2.62975
\(195\) 0 0
\(196\) 936549. 1.74137
\(197\) 432994. 0.794908 0.397454 0.917622i \(-0.369894\pi\)
0.397454 + 0.917622i \(0.369894\pi\)
\(198\) 0 0
\(199\) −614871. −1.10066 −0.550328 0.834949i \(-0.685497\pi\)
−0.550328 + 0.834949i \(0.685497\pi\)
\(200\) 34616.0 0.0611930
\(201\) 0 0
\(202\) −217000. −0.374180
\(203\) 656723. 1.11852
\(204\) 0 0
\(205\) −208887. −0.347158
\(206\) 447380. 0.734528
\(207\) 0 0
\(208\) −284682. −0.456248
\(209\) 217777. 0.344862
\(210\) 0 0
\(211\) 72348.2 0.111872 0.0559360 0.998434i \(-0.482186\pi\)
0.0559360 + 0.998434i \(0.482186\pi\)
\(212\) −85084.6 −0.130020
\(213\) 0 0
\(214\) 104165. 0.155484
\(215\) −299197. −0.441429
\(216\) 0 0
\(217\) −2.40174e6 −3.46240
\(218\) 99931.4 0.142417
\(219\) 0 0
\(220\) −74538.3 −0.103830
\(221\) 143604. 0.197781
\(222\) 0 0
\(223\) 370371. 0.498740 0.249370 0.968408i \(-0.419776\pi\)
0.249370 + 0.968408i \(0.419776\pi\)
\(224\) −1.70890e6 −2.27560
\(225\) 0 0
\(226\) 979025. 1.27504
\(227\) −325481. −0.419239 −0.209619 0.977783i \(-0.567222\pi\)
−0.209619 + 0.977783i \(0.567222\pi\)
\(228\) 0 0
\(229\) −57345.0 −0.0722614 −0.0361307 0.999347i \(-0.511503\pi\)
−0.0361307 + 0.999347i \(0.511503\pi\)
\(230\) −898116. −1.11947
\(231\) 0 0
\(232\) −155356. −0.189500
\(233\) 833891. 1.00628 0.503141 0.864205i \(-0.332178\pi\)
0.503141 + 0.864205i \(0.332178\pi\)
\(234\) 0 0
\(235\) −209637. −0.247627
\(236\) 1.25748e6 1.46967
\(237\) 0 0
\(238\) 1.07134e6 1.22599
\(239\) 813986. 0.921769 0.460885 0.887460i \(-0.347532\pi\)
0.460885 + 0.887460i \(0.347532\pi\)
\(240\) 0 0
\(241\) −1.18456e6 −1.31376 −0.656878 0.753997i \(-0.728123\pi\)
−0.656878 + 0.753997i \(0.728123\pi\)
\(242\) −110188. −0.120948
\(243\) 0 0
\(244\) 246118. 0.264648
\(245\) 950202. 1.01135
\(246\) 0 0
\(247\) −425086. −0.443338
\(248\) 568163. 0.586602
\(249\) 0 0
\(250\) −117594. −0.118997
\(251\) 601749. 0.602880 0.301440 0.953485i \(-0.402533\pi\)
0.301440 + 0.953485i \(0.402533\pi\)
\(252\) 0 0
\(253\) −577582. −0.567299
\(254\) 2.06893e6 2.01216
\(255\) 0 0
\(256\) 1.35468e6 1.29192
\(257\) −904570. −0.854298 −0.427149 0.904181i \(-0.640482\pi\)
−0.427149 + 0.904181i \(0.640482\pi\)
\(258\) 0 0
\(259\) 1.78634e6 1.65469
\(260\) 145494. 0.133479
\(261\) 0 0
\(262\) −898413. −0.808580
\(263\) −1.47174e6 −1.31202 −0.656011 0.754751i \(-0.727758\pi\)
−0.656011 + 0.754751i \(0.727758\pi\)
\(264\) 0 0
\(265\) −86325.0 −0.0755130
\(266\) −3.17133e6 −2.74813
\(267\) 0 0
\(268\) −901054. −0.766327
\(269\) −897427. −0.756168 −0.378084 0.925771i \(-0.623417\pi\)
−0.378084 + 0.925771i \(0.623417\pi\)
\(270\) 0 0
\(271\) 134225. 0.111022 0.0555111 0.998458i \(-0.482321\pi\)
0.0555111 + 0.998458i \(0.482321\pi\)
\(272\) −732863. −0.600621
\(273\) 0 0
\(274\) −995523. −0.801078
\(275\) −75625.0 −0.0603023
\(276\) 0 0
\(277\) −2.01306e6 −1.57637 −0.788184 0.615439i \(-0.788979\pi\)
−0.788184 + 0.615439i \(0.788979\pi\)
\(278\) 770105. 0.597637
\(279\) 0 0
\(280\) −324180. −0.247111
\(281\) −1.45204e6 −1.09702 −0.548509 0.836145i \(-0.684804\pi\)
−0.548509 + 0.836145i \(0.684804\pi\)
\(282\) 0 0
\(283\) −2.24416e6 −1.66566 −0.832832 0.553525i \(-0.813282\pi\)
−0.832832 + 0.553525i \(0.813282\pi\)
\(284\) 752092. 0.553318
\(285\) 0 0
\(286\) 215080. 0.155484
\(287\) 1.95624e6 1.40190
\(288\) 0 0
\(289\) −1.05017e6 −0.739634
\(290\) 527760. 0.368503
\(291\) 0 0
\(292\) 2.06686e6 1.41858
\(293\) −148693. −0.101187 −0.0505933 0.998719i \(-0.516111\pi\)
−0.0505933 + 0.998719i \(0.516111\pi\)
\(294\) 0 0
\(295\) 1.27581e6 0.853554
\(296\) −422583. −0.280338
\(297\) 0 0
\(298\) 2.79739e6 1.82478
\(299\) 1.12740e6 0.729291
\(300\) 0 0
\(301\) 2.80199e6 1.78259
\(302\) 840845. 0.530516
\(303\) 0 0
\(304\) 2.16938e6 1.34633
\(305\) 249706. 0.153702
\(306\) 0 0
\(307\) 2.02304e6 1.22507 0.612533 0.790445i \(-0.290151\pi\)
0.612533 + 0.790445i \(0.290151\pi\)
\(308\) 698055. 0.419289
\(309\) 0 0
\(310\) −1.93010e6 −1.14071
\(311\) −1.87652e6 −1.10015 −0.550074 0.835116i \(-0.685401\pi\)
−0.550074 + 0.835116i \(0.685401\pi\)
\(312\) 0 0
\(313\) −89494.6 −0.0516340 −0.0258170 0.999667i \(-0.508219\pi\)
−0.0258170 + 0.999667i \(0.508219\pi\)
\(314\) 1.84181e6 1.05419
\(315\) 0 0
\(316\) 2.55044e6 1.43680
\(317\) −260115. −0.145384 −0.0726920 0.997354i \(-0.523159\pi\)
−0.0726920 + 0.997354i \(0.523159\pi\)
\(318\) 0 0
\(319\) 339404. 0.186741
\(320\) −409045. −0.223304
\(321\) 0 0
\(322\) 8.41090e6 4.52067
\(323\) −1.09431e6 −0.583626
\(324\) 0 0
\(325\) 147615. 0.0775215
\(326\) −3.27850e6 −1.70856
\(327\) 0 0
\(328\) −462773. −0.237511
\(329\) 1.96326e6 0.999973
\(330\) 0 0
\(331\) 2.19339e6 1.10039 0.550193 0.835037i \(-0.314554\pi\)
0.550193 + 0.835037i \(0.314554\pi\)
\(332\) 106979. 0.0532664
\(333\) 0 0
\(334\) 2.17115e6 1.06494
\(335\) −914190. −0.445066
\(336\) 0 0
\(337\) 933037. 0.447532 0.223766 0.974643i \(-0.428165\pi\)
0.223766 + 0.974643i \(0.428165\pi\)
\(338\) 2.37453e6 1.13054
\(339\) 0 0
\(340\) 374549. 0.175716
\(341\) −1.24126e6 −0.578063
\(342\) 0 0
\(343\) −4.96373e6 −2.27810
\(344\) −662848. −0.302007
\(345\) 0 0
\(346\) 4.06128e6 1.82378
\(347\) 1.72984e6 0.771229 0.385615 0.922660i \(-0.373989\pi\)
0.385615 + 0.922660i \(0.373989\pi\)
\(348\) 0 0
\(349\) 1.01770e6 0.447258 0.223629 0.974674i \(-0.428210\pi\)
0.223629 + 0.974674i \(0.428210\pi\)
\(350\) 1.10127e6 0.480534
\(351\) 0 0
\(352\) −883183. −0.379922
\(353\) 3.29128e6 1.40581 0.702907 0.711282i \(-0.251885\pi\)
0.702907 + 0.711282i \(0.251885\pi\)
\(354\) 0 0
\(355\) 763056. 0.321355
\(356\) 1.12027e6 0.468488
\(357\) 0 0
\(358\) −2.85140e6 −1.17585
\(359\) −2.32128e6 −0.950584 −0.475292 0.879828i \(-0.657658\pi\)
−0.475292 + 0.879828i \(0.657658\pi\)
\(360\) 0 0
\(361\) 763211. 0.308231
\(362\) 4.67841e6 1.87641
\(363\) 0 0
\(364\) −1.36256e6 −0.539016
\(365\) 2.09699e6 0.823879
\(366\) 0 0
\(367\) −1.81399e6 −0.703024 −0.351512 0.936183i \(-0.614332\pi\)
−0.351512 + 0.936183i \(0.614332\pi\)
\(368\) −5.75356e6 −2.21471
\(369\) 0 0
\(370\) 1.43556e6 0.545149
\(371\) 808438. 0.304938
\(372\) 0 0
\(373\) −160496. −0.0597298 −0.0298649 0.999554i \(-0.509508\pi\)
−0.0298649 + 0.999554i \(0.509508\pi\)
\(374\) 553687. 0.204685
\(375\) 0 0
\(376\) −464435. −0.169416
\(377\) −662495. −0.240065
\(378\) 0 0
\(379\) −622442. −0.222587 −0.111294 0.993788i \(-0.535499\pi\)
−0.111294 + 0.993788i \(0.535499\pi\)
\(380\) −1.10872e6 −0.393878
\(381\) 0 0
\(382\) 64043.4 0.0224552
\(383\) −1.49075e6 −0.519289 −0.259644 0.965704i \(-0.583605\pi\)
−0.259644 + 0.965704i \(0.583605\pi\)
\(384\) 0 0
\(385\) 708232. 0.243514
\(386\) 4.24909e6 1.45154
\(387\) 0 0
\(388\) −4.51345e6 −1.52205
\(389\) 1.66085e6 0.556490 0.278245 0.960510i \(-0.410247\pi\)
0.278245 + 0.960510i \(0.410247\pi\)
\(390\) 0 0
\(391\) 2.90230e6 0.960065
\(392\) 2.10510e6 0.691923
\(393\) 0 0
\(394\) −3.25872e6 −1.05756
\(395\) 2.58762e6 0.834465
\(396\) 0 0
\(397\) 2.46357e6 0.784493 0.392247 0.919860i \(-0.371698\pi\)
0.392247 + 0.919860i \(0.371698\pi\)
\(398\) 4.62752e6 1.46434
\(399\) 0 0
\(400\) −753336. −0.235417
\(401\) 664739. 0.206438 0.103219 0.994659i \(-0.467086\pi\)
0.103219 + 0.994659i \(0.467086\pi\)
\(402\) 0 0
\(403\) 2.42285e6 0.743129
\(404\) 710475. 0.216569
\(405\) 0 0
\(406\) −4.94250e6 −1.48810
\(407\) 923210. 0.276258
\(408\) 0 0
\(409\) −111122. −0.0328468 −0.0164234 0.999865i \(-0.505228\pi\)
−0.0164234 + 0.999865i \(0.505228\pi\)
\(410\) 1.57208e6 0.461866
\(411\) 0 0
\(412\) −1.46476e6 −0.425132
\(413\) −1.19480e7 −3.44684
\(414\) 0 0
\(415\) 108539. 0.0309360
\(416\) 1.72392e6 0.488408
\(417\) 0 0
\(418\) −1.63899e6 −0.458813
\(419\) 5.34446e6 1.48720 0.743598 0.668627i \(-0.233118\pi\)
0.743598 + 0.668627i \(0.233118\pi\)
\(420\) 0 0
\(421\) −483590. −0.132976 −0.0664878 0.997787i \(-0.521179\pi\)
−0.0664878 + 0.997787i \(0.521179\pi\)
\(422\) −544493. −0.148837
\(423\) 0 0
\(424\) −191246. −0.0516629
\(425\) 380010. 0.102052
\(426\) 0 0
\(427\) −2.33851e6 −0.620682
\(428\) −341044. −0.0899915
\(429\) 0 0
\(430\) 2.25176e6 0.587287
\(431\) 241542. 0.0626325 0.0313162 0.999510i \(-0.490030\pi\)
0.0313162 + 0.999510i \(0.490030\pi\)
\(432\) 0 0
\(433\) −1.03998e6 −0.266567 −0.133284 0.991078i \(-0.542552\pi\)
−0.133284 + 0.991078i \(0.542552\pi\)
\(434\) 1.80755e7 4.60645
\(435\) 0 0
\(436\) −327184. −0.0824281
\(437\) −8.59121e6 −2.15204
\(438\) 0 0
\(439\) −3.32986e6 −0.824642 −0.412321 0.911039i \(-0.635282\pi\)
−0.412321 + 0.911039i \(0.635282\pi\)
\(440\) −167541. −0.0412563
\(441\) 0 0
\(442\) −1.08076e6 −0.263132
\(443\) −5.12280e6 −1.24022 −0.620109 0.784515i \(-0.712912\pi\)
−0.620109 + 0.784515i \(0.712912\pi\)
\(444\) 0 0
\(445\) 1.13660e6 0.272088
\(446\) −2.78741e6 −0.663535
\(447\) 0 0
\(448\) 3.83073e6 0.901750
\(449\) −5.82328e6 −1.36318 −0.681588 0.731736i \(-0.738710\pi\)
−0.681588 + 0.731736i \(0.738710\pi\)
\(450\) 0 0
\(451\) 1.01101e6 0.234054
\(452\) −3.20541e6 −0.737968
\(453\) 0 0
\(454\) 2.44957e6 0.557765
\(455\) −1.38242e6 −0.313049
\(456\) 0 0
\(457\) −5.32427e6 −1.19253 −0.596266 0.802787i \(-0.703349\pi\)
−0.596266 + 0.802787i \(0.703349\pi\)
\(458\) 431579. 0.0961382
\(459\) 0 0
\(460\) 2.94051e6 0.647930
\(461\) −673694. −0.147642 −0.0738211 0.997272i \(-0.523519\pi\)
−0.0738211 + 0.997272i \(0.523519\pi\)
\(462\) 0 0
\(463\) −1.92230e6 −0.416742 −0.208371 0.978050i \(-0.566816\pi\)
−0.208371 + 0.978050i \(0.566816\pi\)
\(464\) 3.38096e6 0.729030
\(465\) 0 0
\(466\) −6.27587e6 −1.33878
\(467\) 7.94668e6 1.68614 0.843070 0.537804i \(-0.180746\pi\)
0.843070 + 0.537804i \(0.180746\pi\)
\(468\) 0 0
\(469\) 8.56144e6 1.79727
\(470\) 1.57773e6 0.329449
\(471\) 0 0
\(472\) 2.82646e6 0.583966
\(473\) 1.44811e6 0.297611
\(474\) 0 0
\(475\) −1.12488e6 −0.228756
\(476\) −3.50767e6 −0.709580
\(477\) 0 0
\(478\) −6.12606e6 −1.22634
\(479\) 1.80345e6 0.359141 0.179570 0.983745i \(-0.442529\pi\)
0.179570 + 0.983745i \(0.442529\pi\)
\(480\) 0 0
\(481\) −1.80205e6 −0.355143
\(482\) 8.91501e6 1.74785
\(483\) 0 0
\(484\) 360766. 0.0700022
\(485\) −4.57925e6 −0.883975
\(486\) 0 0
\(487\) 3.04965e6 0.582678 0.291339 0.956620i \(-0.405899\pi\)
0.291339 + 0.956620i \(0.405899\pi\)
\(488\) 553204. 0.105156
\(489\) 0 0
\(490\) −7.15123e6 −1.34552
\(491\) 7.53319e6 1.41018 0.705091 0.709117i \(-0.250906\pi\)
0.705091 + 0.709117i \(0.250906\pi\)
\(492\) 0 0
\(493\) −1.70548e6 −0.316031
\(494\) 3.19920e6 0.589826
\(495\) 0 0
\(496\) −1.23647e7 −2.25673
\(497\) −7.14606e6 −1.29770
\(498\) 0 0
\(499\) 6.01840e6 1.08201 0.541003 0.841021i \(-0.318045\pi\)
0.541003 + 0.841021i \(0.318045\pi\)
\(500\) 385012. 0.0688731
\(501\) 0 0
\(502\) −4.52876e6 −0.802085
\(503\) −8.91752e6 −1.57153 −0.785767 0.618522i \(-0.787732\pi\)
−0.785767 + 0.618522i \(0.787732\pi\)
\(504\) 0 0
\(505\) 720833. 0.125778
\(506\) 4.34688e6 0.754748
\(507\) 0 0
\(508\) −6.77385e6 −1.16460
\(509\) −9.71294e6 −1.66171 −0.830857 0.556486i \(-0.812149\pi\)
−0.830857 + 0.556486i \(0.812149\pi\)
\(510\) 0 0
\(511\) −1.96384e7 −3.32701
\(512\) −6.66153e6 −1.12305
\(513\) 0 0
\(514\) 6.80780e6 1.13658
\(515\) −1.48611e6 −0.246907
\(516\) 0 0
\(517\) 1.01464e6 0.166950
\(518\) −1.34440e7 −2.20143
\(519\) 0 0
\(520\) 327030. 0.0530370
\(521\) 591081. 0.0954009 0.0477004 0.998862i \(-0.484811\pi\)
0.0477004 + 0.998862i \(0.484811\pi\)
\(522\) 0 0
\(523\) 2.83152e6 0.452653 0.226327 0.974051i \(-0.427328\pi\)
0.226327 + 0.974051i \(0.427328\pi\)
\(524\) 2.94148e6 0.467991
\(525\) 0 0
\(526\) 1.10763e7 1.74554
\(527\) 6.23721e6 0.978282
\(528\) 0 0
\(529\) 1.63490e7 2.54011
\(530\) 649682. 0.100464
\(531\) 0 0
\(532\) 1.03832e7 1.59057
\(533\) −1.97343e6 −0.300887
\(534\) 0 0
\(535\) −346016. −0.0522651
\(536\) −2.02532e6 −0.304496
\(537\) 0 0
\(538\) 6.75404e6 1.00602
\(539\) −4.59898e6 −0.681851
\(540\) 0 0
\(541\) 9.93914e6 1.46001 0.730004 0.683442i \(-0.239518\pi\)
0.730004 + 0.683442i \(0.239518\pi\)
\(542\) −1.01018e6 −0.147706
\(543\) 0 0
\(544\) 4.43793e6 0.642958
\(545\) −331953. −0.0478725
\(546\) 0 0
\(547\) −4.34846e6 −0.621394 −0.310697 0.950509i \(-0.600563\pi\)
−0.310697 + 0.950509i \(0.600563\pi\)
\(548\) 3.25942e6 0.463649
\(549\) 0 0
\(550\) 569154. 0.0802275
\(551\) 5.04845e6 0.708401
\(552\) 0 0
\(553\) −2.42332e7 −3.36976
\(554\) 1.51503e7 2.09724
\(555\) 0 0
\(556\) −2.52139e6 −0.345902
\(557\) −1.07234e7 −1.46452 −0.732261 0.681024i \(-0.761535\pi\)
−0.732261 + 0.681024i \(0.761535\pi\)
\(558\) 0 0
\(559\) −2.82662e6 −0.382594
\(560\) 7.05503e6 0.950667
\(561\) 0 0
\(562\) 1.09281e7 1.45950
\(563\) −1.67966e6 −0.223332 −0.111666 0.993746i \(-0.535619\pi\)
−0.111666 + 0.993746i \(0.535619\pi\)
\(564\) 0 0
\(565\) −3.25214e6 −0.428596
\(566\) 1.68896e7 2.21604
\(567\) 0 0
\(568\) 1.69049e6 0.219858
\(569\) −1.04510e7 −1.35325 −0.676626 0.736327i \(-0.736559\pi\)
−0.676626 + 0.736327i \(0.736559\pi\)
\(570\) 0 0
\(571\) 670155. 0.0860171 0.0430085 0.999075i \(-0.486306\pi\)
0.0430085 + 0.999075i \(0.486306\pi\)
\(572\) −704191. −0.0899913
\(573\) 0 0
\(574\) −1.47226e7 −1.86512
\(575\) 2.98338e6 0.376304
\(576\) 0 0
\(577\) −7.87543e6 −0.984769 −0.492385 0.870378i \(-0.663875\pi\)
−0.492385 + 0.870378i \(0.663875\pi\)
\(578\) 7.90362e6 0.984026
\(579\) 0 0
\(580\) −1.72793e6 −0.213283
\(581\) −1.01647e6 −0.124926
\(582\) 0 0
\(583\) 417813. 0.0509109
\(584\) 4.64571e6 0.563664
\(585\) 0 0
\(586\) 1.11907e6 0.134621
\(587\) 9.13414e6 1.09414 0.547069 0.837087i \(-0.315743\pi\)
0.547069 + 0.837087i \(0.315743\pi\)
\(588\) 0 0
\(589\) −1.84630e7 −2.19288
\(590\) −9.60176e6 −1.13559
\(591\) 0 0
\(592\) 9.19652e6 1.07850
\(593\) −1.25096e7 −1.46085 −0.730425 0.682993i \(-0.760678\pi\)
−0.730425 + 0.682993i \(0.760678\pi\)
\(594\) 0 0
\(595\) −3.55881e6 −0.412109
\(596\) −9.15887e6 −1.05615
\(597\) 0 0
\(598\) −8.48484e6 −0.970265
\(599\) −1.59560e7 −1.81700 −0.908502 0.417881i \(-0.862773\pi\)
−0.908502 + 0.417881i \(0.862773\pi\)
\(600\) 0 0
\(601\) 763693. 0.0862448 0.0431224 0.999070i \(-0.486269\pi\)
0.0431224 + 0.999070i \(0.486269\pi\)
\(602\) −2.10878e7 −2.37159
\(603\) 0 0
\(604\) −2.75300e6 −0.307053
\(605\) 366025. 0.0406558
\(606\) 0 0
\(607\) −8.51753e6 −0.938301 −0.469150 0.883118i \(-0.655440\pi\)
−0.469150 + 0.883118i \(0.655440\pi\)
\(608\) −1.31369e7 −1.44123
\(609\) 0 0
\(610\) −1.87929e6 −0.204488
\(611\) −1.98052e6 −0.214623
\(612\) 0 0
\(613\) 9.58182e6 1.02990 0.514952 0.857219i \(-0.327810\pi\)
0.514952 + 0.857219i \(0.327810\pi\)
\(614\) −1.52254e7 −1.62985
\(615\) 0 0
\(616\) 1.56903e6 0.166602
\(617\) 1.12974e7 1.19472 0.597362 0.801972i \(-0.296216\pi\)
0.597362 + 0.801972i \(0.296216\pi\)
\(618\) 0 0
\(619\) −4.12945e6 −0.433177 −0.216588 0.976263i \(-0.569493\pi\)
−0.216588 + 0.976263i \(0.569493\pi\)
\(620\) 6.31932e6 0.660224
\(621\) 0 0
\(622\) 1.41227e7 1.46366
\(623\) −1.06443e7 −1.09875
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 673537. 0.0686951
\(627\) 0 0
\(628\) −6.03023e6 −0.610148
\(629\) −4.63906e6 −0.467523
\(630\) 0 0
\(631\) 1.29753e7 1.29731 0.648656 0.761082i \(-0.275332\pi\)
0.648656 + 0.761082i \(0.275332\pi\)
\(632\) 5.73268e6 0.570907
\(633\) 0 0
\(634\) 1.95762e6 0.193422
\(635\) −6.87260e6 −0.676374
\(636\) 0 0
\(637\) 8.97691e6 0.876553
\(638\) −2.55436e6 −0.248445
\(639\) 0 0
\(640\) −2.76075e6 −0.266427
\(641\) 130086. 0.0125051 0.00625255 0.999980i \(-0.498010\pi\)
0.00625255 + 0.999980i \(0.498010\pi\)
\(642\) 0 0
\(643\) −6.47507e6 −0.617614 −0.308807 0.951125i \(-0.599930\pi\)
−0.308807 + 0.951125i \(0.599930\pi\)
\(644\) −2.75380e7 −2.61648
\(645\) 0 0
\(646\) 8.23579e6 0.776469
\(647\) 5.42540e6 0.509531 0.254766 0.967003i \(-0.418002\pi\)
0.254766 + 0.967003i \(0.418002\pi\)
\(648\) 0 0
\(649\) −6.17492e6 −0.575466
\(650\) −1.11095e6 −0.103136
\(651\) 0 0
\(652\) 1.07341e7 0.988886
\(653\) −9.15697e6 −0.840367 −0.420183 0.907439i \(-0.638034\pi\)
−0.420183 + 0.907439i \(0.638034\pi\)
\(654\) 0 0
\(655\) 2.98436e6 0.271799
\(656\) 1.00712e7 0.913735
\(657\) 0 0
\(658\) −1.47755e7 −1.33039
\(659\) 4.88080e6 0.437802 0.218901 0.975747i \(-0.429753\pi\)
0.218901 + 0.975747i \(0.429753\pi\)
\(660\) 0 0
\(661\) 1.75690e7 1.56402 0.782010 0.623266i \(-0.214194\pi\)
0.782010 + 0.623266i \(0.214194\pi\)
\(662\) −1.65074e7 −1.46398
\(663\) 0 0
\(664\) 240459. 0.0211651
\(665\) 1.05346e7 0.923767
\(666\) 0 0
\(667\) −1.33894e7 −1.16532
\(668\) −7.10853e6 −0.616366
\(669\) 0 0
\(670\) 6.88020e6 0.592126
\(671\) −1.20858e6 −0.103626
\(672\) 0 0
\(673\) 1.60675e7 1.36745 0.683725 0.729740i \(-0.260359\pi\)
0.683725 + 0.729740i \(0.260359\pi\)
\(674\) −7.02205e6 −0.595407
\(675\) 0 0
\(676\) −7.77441e6 −0.654336
\(677\) 1.51931e7 1.27401 0.637007 0.770858i \(-0.280172\pi\)
0.637007 + 0.770858i \(0.280172\pi\)
\(678\) 0 0
\(679\) 4.28849e7 3.56968
\(680\) 841882. 0.0698198
\(681\) 0 0
\(682\) 9.34170e6 0.769068
\(683\) 9.75866e6 0.800457 0.400229 0.916415i \(-0.368931\pi\)
0.400229 + 0.916415i \(0.368931\pi\)
\(684\) 0 0
\(685\) 3.30694e6 0.269278
\(686\) 3.73571e7 3.03084
\(687\) 0 0
\(688\) 1.44253e7 1.16186
\(689\) −815544. −0.0654484
\(690\) 0 0
\(691\) −1.74840e7 −1.39298 −0.696490 0.717567i \(-0.745256\pi\)
−0.696490 + 0.717567i \(0.745256\pi\)
\(692\) −1.32970e7 −1.05557
\(693\) 0 0
\(694\) −1.30188e7 −1.02606
\(695\) −2.55815e6 −0.200892
\(696\) 0 0
\(697\) −5.08026e6 −0.396099
\(698\) −7.65925e6 −0.595042
\(699\) 0 0
\(700\) −3.60566e6 −0.278125
\(701\) −4.31053e6 −0.331311 −0.165656 0.986184i \(-0.552974\pi\)
−0.165656 + 0.986184i \(0.552974\pi\)
\(702\) 0 0
\(703\) 1.37322e7 1.04798
\(704\) 1.97978e6 0.150551
\(705\) 0 0
\(706\) −2.47702e7 −1.87033
\(707\) −6.75064e6 −0.507921
\(708\) 0 0
\(709\) −1.44521e6 −0.107973 −0.0539866 0.998542i \(-0.517193\pi\)
−0.0539866 + 0.998542i \(0.517193\pi\)
\(710\) −5.74277e6 −0.427539
\(711\) 0 0
\(712\) 2.51806e6 0.186151
\(713\) 4.89671e7 3.60728
\(714\) 0 0
\(715\) −714457. −0.0522650
\(716\) 9.33574e6 0.680559
\(717\) 0 0
\(718\) 1.74699e7 1.26468
\(719\) 4.22567e6 0.304841 0.152420 0.988316i \(-0.451293\pi\)
0.152420 + 0.988316i \(0.451293\pi\)
\(720\) 0 0
\(721\) 1.39175e7 0.997066
\(722\) −5.74393e6 −0.410078
\(723\) 0 0
\(724\) −1.53175e7 −1.08603
\(725\) −1.75312e6 −0.123870
\(726\) 0 0
\(727\) 1.62930e7 1.14331 0.571657 0.820493i \(-0.306301\pi\)
0.571657 + 0.820493i \(0.306301\pi\)
\(728\) −3.06265e6 −0.214175
\(729\) 0 0
\(730\) −1.57819e7 −1.09611
\(731\) −7.27665e6 −0.503661
\(732\) 0 0
\(733\) −3.00376e6 −0.206493 −0.103247 0.994656i \(-0.532923\pi\)
−0.103247 + 0.994656i \(0.532923\pi\)
\(734\) 1.36521e7 0.935319
\(735\) 0 0
\(736\) 3.48412e7 2.37082
\(737\) 4.42468e6 0.300064
\(738\) 0 0
\(739\) 5.58694e6 0.376325 0.188163 0.982138i \(-0.439747\pi\)
0.188163 + 0.982138i \(0.439747\pi\)
\(740\) −4.70013e6 −0.315522
\(741\) 0 0
\(742\) −6.08431e6 −0.405697
\(743\) 8.33428e6 0.553855 0.276927 0.960891i \(-0.410684\pi\)
0.276927 + 0.960891i \(0.410684\pi\)
\(744\) 0 0
\(745\) −9.29240e6 −0.613390
\(746\) 1.20789e6 0.0794659
\(747\) 0 0
\(748\) −1.81282e6 −0.118468
\(749\) 3.24046e6 0.211058
\(750\) 0 0
\(751\) −1.55555e7 −1.00643 −0.503215 0.864162i \(-0.667849\pi\)
−0.503215 + 0.864162i \(0.667849\pi\)
\(752\) 1.01073e7 0.651766
\(753\) 0 0
\(754\) 4.98594e6 0.319388
\(755\) −2.79313e6 −0.178330
\(756\) 0 0
\(757\) 2.42999e7 1.54122 0.770611 0.637306i \(-0.219951\pi\)
0.770611 + 0.637306i \(0.219951\pi\)
\(758\) 4.68450e6 0.296135
\(759\) 0 0
\(760\) −2.49209e6 −0.156505
\(761\) −1.84687e7 −1.15605 −0.578023 0.816021i \(-0.696175\pi\)
−0.578023 + 0.816021i \(0.696175\pi\)
\(762\) 0 0
\(763\) 3.10876e6 0.193320
\(764\) −209684. −0.0129966
\(765\) 0 0
\(766\) 1.12194e7 0.690873
\(767\) 1.20530e7 0.739790
\(768\) 0 0
\(769\) −1.51725e6 −0.0925210 −0.0462605 0.998929i \(-0.514730\pi\)
−0.0462605 + 0.998929i \(0.514730\pi\)
\(770\) −5.33016e6 −0.323976
\(771\) 0 0
\(772\) −1.39119e7 −0.840122
\(773\) −3.28907e6 −0.197981 −0.0989907 0.995088i \(-0.531561\pi\)
−0.0989907 + 0.995088i \(0.531561\pi\)
\(774\) 0 0
\(775\) 6.41145e6 0.383444
\(776\) −1.01450e7 −0.604779
\(777\) 0 0
\(778\) −1.24996e7 −0.740367
\(779\) 1.50383e7 0.887879
\(780\) 0 0
\(781\) −3.69319e6 −0.216658
\(782\) −2.18427e7 −1.27729
\(783\) 0 0
\(784\) −4.58126e7 −2.66192
\(785\) −6.11815e6 −0.354361
\(786\) 0 0
\(787\) −1.46769e7 −0.844692 −0.422346 0.906435i \(-0.638793\pi\)
−0.422346 + 0.906435i \(0.638793\pi\)
\(788\) 1.06693e7 0.612098
\(789\) 0 0
\(790\) −1.94745e7 −1.11019
\(791\) 3.04564e7 1.73076
\(792\) 0 0
\(793\) 2.35906e6 0.133216
\(794\) −1.85409e7 −1.04371
\(795\) 0 0
\(796\) −1.51509e7 −0.847531
\(797\) −4.95422e6 −0.276267 −0.138134 0.990414i \(-0.544110\pi\)
−0.138134 + 0.990414i \(0.544110\pi\)
\(798\) 0 0
\(799\) −5.09850e6 −0.282537
\(800\) 4.56190e6 0.252012
\(801\) 0 0
\(802\) −5.00283e6 −0.274650
\(803\) −1.01494e7 −0.555459
\(804\) 0 0
\(805\) −2.79395e7 −1.51960
\(806\) −1.82344e7 −0.988675
\(807\) 0 0
\(808\) 1.59695e6 0.0860524
\(809\) −1.90776e7 −1.02483 −0.512414 0.858738i \(-0.671249\pi\)
−0.512414 + 0.858738i \(0.671249\pi\)
\(810\) 0 0
\(811\) 2.54611e7 1.35933 0.679665 0.733523i \(-0.262125\pi\)
0.679665 + 0.733523i \(0.262125\pi\)
\(812\) 1.61822e7 0.861284
\(813\) 0 0
\(814\) −6.94809e6 −0.367540
\(815\) 1.08906e7 0.574324
\(816\) 0 0
\(817\) 2.15399e7 1.12899
\(818\) 836308. 0.0437002
\(819\) 0 0
\(820\) −5.14714e6 −0.267320
\(821\) 3.75556e7 1.94454 0.972269 0.233867i \(-0.0751380\pi\)
0.972269 + 0.233867i \(0.0751380\pi\)
\(822\) 0 0
\(823\) −3.42815e7 −1.76425 −0.882126 0.471014i \(-0.843888\pi\)
−0.882126 + 0.471014i \(0.843888\pi\)
\(824\) −3.29237e6 −0.168924
\(825\) 0 0
\(826\) 8.99209e7 4.58575
\(827\) −2.92800e7 −1.48870 −0.744351 0.667789i \(-0.767241\pi\)
−0.744351 + 0.667789i \(0.767241\pi\)
\(828\) 0 0
\(829\) −7.40301e6 −0.374130 −0.187065 0.982348i \(-0.559897\pi\)
−0.187065 + 0.982348i \(0.559897\pi\)
\(830\) −816862. −0.0411579
\(831\) 0 0
\(832\) −3.86440e6 −0.193541
\(833\) 2.31095e7 1.15393
\(834\) 0 0
\(835\) −7.21216e6 −0.357972
\(836\) 5.36619e6 0.265552
\(837\) 0 0
\(838\) −4.02224e7 −1.97860
\(839\) 3.44573e7 1.68996 0.844981 0.534797i \(-0.179612\pi\)
0.844981 + 0.534797i \(0.179612\pi\)
\(840\) 0 0
\(841\) −1.26432e7 −0.616404
\(842\) 3.63950e6 0.176914
\(843\) 0 0
\(844\) 1.78271e6 0.0861441
\(845\) −7.88775e6 −0.380024
\(846\) 0 0
\(847\) −3.42784e6 −0.164177
\(848\) 4.16203e6 0.198754
\(849\) 0 0
\(850\) −2.85995e6 −0.135772
\(851\) −3.64203e7 −1.72393
\(852\) 0 0
\(853\) −1.70125e7 −0.800561 −0.400281 0.916393i \(-0.631087\pi\)
−0.400281 + 0.916393i \(0.631087\pi\)
\(854\) 1.75996e7 0.825769
\(855\) 0 0
\(856\) −766572. −0.0357576
\(857\) −2.11650e7 −0.984387 −0.492194 0.870486i \(-0.663805\pi\)
−0.492194 + 0.870486i \(0.663805\pi\)
\(858\) 0 0
\(859\) −3.17802e7 −1.46952 −0.734758 0.678330i \(-0.762704\pi\)
−0.734758 + 0.678330i \(0.762704\pi\)
\(860\) −7.37244e6 −0.339911
\(861\) 0 0
\(862\) −1.81785e6 −0.0833277
\(863\) 3.53199e7 1.61433 0.807166 0.590325i \(-0.201000\pi\)
0.807166 + 0.590325i \(0.201000\pi\)
\(864\) 0 0
\(865\) −1.34908e7 −0.613053
\(866\) 7.82692e6 0.354647
\(867\) 0 0
\(868\) −5.91807e7 −2.66613
\(869\) −1.25241e7 −0.562597
\(870\) 0 0
\(871\) −8.63669e6 −0.385746
\(872\) −735417. −0.0327524
\(873\) 0 0
\(874\) 6.46575e7 2.86313
\(875\) −3.65822e6 −0.161529
\(876\) 0 0
\(877\) 256700. 0.0112701 0.00563503 0.999984i \(-0.498206\pi\)
0.00563503 + 0.999984i \(0.498206\pi\)
\(878\) 2.50606e7 1.09712
\(879\) 0 0
\(880\) 3.64614e6 0.158718
\(881\) −3.37169e7 −1.46355 −0.731776 0.681545i \(-0.761308\pi\)
−0.731776 + 0.681545i \(0.761308\pi\)
\(882\) 0 0
\(883\) −2.45722e7 −1.06058 −0.530289 0.847817i \(-0.677917\pi\)
−0.530289 + 0.847817i \(0.677917\pi\)
\(884\) 3.53850e6 0.152296
\(885\) 0 0
\(886\) 3.85542e7 1.65001
\(887\) −220190. −0.00939697 −0.00469849 0.999989i \(-0.501496\pi\)
−0.00469849 + 0.999989i \(0.501496\pi\)
\(888\) 0 0
\(889\) 6.43623e7 2.73135
\(890\) −8.55408e6 −0.361991
\(891\) 0 0
\(892\) 9.12622e6 0.384042
\(893\) 1.50923e7 0.633323
\(894\) 0 0
\(895\) 9.47184e6 0.395254
\(896\) 2.58546e7 1.07589
\(897\) 0 0
\(898\) 4.38260e7 1.81360
\(899\) −2.87745e7 −1.18743
\(900\) 0 0
\(901\) −2.09948e6 −0.0861587
\(902\) −7.60889e6 −0.311390
\(903\) 0 0
\(904\) −7.20487e6 −0.293228
\(905\) −1.55408e7 −0.630743
\(906\) 0 0
\(907\) −6.02862e6 −0.243332 −0.121666 0.992571i \(-0.538824\pi\)
−0.121666 + 0.992571i \(0.538824\pi\)
\(908\) −8.02011e6 −0.322824
\(909\) 0 0
\(910\) 1.04041e7 0.416487
\(911\) −1.68076e7 −0.670979 −0.335490 0.942044i \(-0.608902\pi\)
−0.335490 + 0.942044i \(0.608902\pi\)
\(912\) 0 0
\(913\) −525327. −0.0208570
\(914\) 4.00705e7 1.58657
\(915\) 0 0
\(916\) −1.41302e6 −0.0556430
\(917\) −2.79487e7 −1.09758
\(918\) 0 0
\(919\) 4.57189e7 1.78569 0.892846 0.450362i \(-0.148705\pi\)
0.892846 + 0.450362i \(0.148705\pi\)
\(920\) 6.60944e6 0.257451
\(921\) 0 0
\(922\) 5.07022e6 0.196426
\(923\) 7.20887e6 0.278524
\(924\) 0 0
\(925\) −4.76865e6 −0.183249
\(926\) 1.44672e7 0.554443
\(927\) 0 0
\(928\) −2.04737e7 −0.780418
\(929\) 2.84053e7 1.07984 0.539921 0.841716i \(-0.318454\pi\)
0.539921 + 0.841716i \(0.318454\pi\)
\(930\) 0 0
\(931\) −6.84073e7 −2.58659
\(932\) 2.05477e7 0.774861
\(933\) 0 0
\(934\) −5.98068e7 −2.24328
\(935\) −1.83925e6 −0.0688035
\(936\) 0 0
\(937\) −8.99373e6 −0.334650 −0.167325 0.985902i \(-0.553513\pi\)
−0.167325 + 0.985902i \(0.553513\pi\)
\(938\) −6.44334e7 −2.39113
\(939\) 0 0
\(940\) −5.16562e6 −0.190679
\(941\) 2.96562e7 1.09180 0.545898 0.837852i \(-0.316189\pi\)
0.545898 + 0.837852i \(0.316189\pi\)
\(942\) 0 0
\(943\) −3.98841e7 −1.46056
\(944\) −6.15113e7 −2.24659
\(945\) 0 0
\(946\) −1.08985e7 −0.395949
\(947\) 1.62357e7 0.588297 0.294148 0.955760i \(-0.404964\pi\)
0.294148 + 0.955760i \(0.404964\pi\)
\(948\) 0 0
\(949\) 1.98110e7 0.714070
\(950\) 8.46586e6 0.304342
\(951\) 0 0
\(952\) −7.88427e6 −0.281948
\(953\) −2.13990e7 −0.763241 −0.381621 0.924319i \(-0.624634\pi\)
−0.381621 + 0.924319i \(0.624634\pi\)
\(954\) 0 0
\(955\) −212740. −0.00754817
\(956\) 2.00572e7 0.709785
\(957\) 0 0
\(958\) −1.35728e7 −0.477809
\(959\) −3.09697e7 −1.08740
\(960\) 0 0
\(961\) 7.66039e7 2.67573
\(962\) 1.35622e7 0.472490
\(963\) 0 0
\(964\) −2.91885e7 −1.01162
\(965\) −1.41147e7 −0.487925
\(966\) 0 0
\(967\) −6.31736e6 −0.217255 −0.108627 0.994083i \(-0.534646\pi\)
−0.108627 + 0.994083i \(0.534646\pi\)
\(968\) 810900. 0.0278150
\(969\) 0 0
\(970\) 3.44635e7 1.17606
\(971\) −4.14293e7 −1.41013 −0.705065 0.709142i \(-0.749082\pi\)
−0.705065 + 0.709142i \(0.749082\pi\)
\(972\) 0 0
\(973\) 2.39572e7 0.811247
\(974\) −2.29517e7 −0.775207
\(975\) 0 0
\(976\) −1.20392e7 −0.404550
\(977\) 333761. 0.0111866 0.00559331 0.999984i \(-0.498220\pi\)
0.00559331 + 0.999984i \(0.498220\pi\)
\(978\) 0 0
\(979\) −5.50116e6 −0.183441
\(980\) 2.34137e7 0.778763
\(981\) 0 0
\(982\) −5.66948e7 −1.87614
\(983\) 3.43832e7 1.13491 0.567457 0.823403i \(-0.307927\pi\)
0.567457 + 0.823403i \(0.307927\pi\)
\(984\) 0 0
\(985\) 1.08249e7 0.355493
\(986\) 1.28354e7 0.420454
\(987\) 0 0
\(988\) −1.04745e7 −0.341381
\(989\) −5.71275e7 −1.85718
\(990\) 0 0
\(991\) 4.26710e6 0.138022 0.0690111 0.997616i \(-0.478016\pi\)
0.0690111 + 0.997616i \(0.478016\pi\)
\(992\) 7.48758e7 2.41581
\(993\) 0 0
\(994\) 5.37813e7 1.72649
\(995\) −1.53718e7 −0.492228
\(996\) 0 0
\(997\) 2.61381e7 0.832790 0.416395 0.909184i \(-0.363293\pi\)
0.416395 + 0.909184i \(0.363293\pi\)
\(998\) −4.52945e7 −1.43953
\(999\) 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.d.1.1 3
3.2 odd 2 165.6.a.b.1.3 3
15.14 odd 2 825.6.a.i.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.b.1.3 3 3.2 odd 2
495.6.a.d.1.1 3 1.1 even 1 trivial
825.6.a.i.1.1 3 15.14 odd 2