Properties

Label 165.6.a.b.1.3
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
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] = [165,6,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(26.4633302691\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.76300\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.52601 q^{2} +9.00000 q^{3} +24.6408 q^{4} -25.0000 q^{5} +67.7341 q^{6} -234.126 q^{7} -55.3856 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+7.52601 q^{2} +9.00000 q^{3} +24.6408 q^{4} -25.0000 q^{5} +67.7341 q^{6} -234.126 q^{7} -55.3856 q^{8} +81.0000 q^{9} -188.150 q^{10} +121.000 q^{11} +221.767 q^{12} +236.184 q^{13} -1762.04 q^{14} -225.000 q^{15} -1205.34 q^{16} -608.015 q^{17} +609.607 q^{18} -1799.81 q^{19} -616.019 q^{20} -2107.14 q^{21} +910.647 q^{22} -4773.40 q^{23} -498.470 q^{24} +625.000 q^{25} +1777.52 q^{26} +729.000 q^{27} -5769.05 q^{28} +2804.99 q^{29} -1693.35 q^{30} +10258.3 q^{31} -7299.04 q^{32} +1089.00 q^{33} -4575.93 q^{34} +5853.16 q^{35} +1995.90 q^{36} -7629.84 q^{37} -13545.4 q^{38} +2125.66 q^{39} +1384.64 q^{40} +8355.48 q^{41} -15858.3 q^{42} -11967.9 q^{43} +2981.53 q^{44} -2025.00 q^{45} -35924.7 q^{46} +8385.48 q^{47} -10848.0 q^{48} +38008.1 q^{49} +4703.75 q^{50} -5472.14 q^{51} +5819.76 q^{52} +3453.00 q^{53} +5486.46 q^{54} -3025.00 q^{55} +12967.2 q^{56} -16198.3 q^{57} +21110.4 q^{58} -51032.4 q^{59} -5544.17 q^{60} +9988.24 q^{61} +77204.1 q^{62} -18964.2 q^{63} -16361.8 q^{64} -5904.60 q^{65} +8195.82 q^{66} -36567.6 q^{67} -14982.0 q^{68} -42960.6 q^{69} +44050.9 q^{70} -30522.3 q^{71} -4486.23 q^{72} +83879.5 q^{73} -57422.2 q^{74} +5625.00 q^{75} -44348.7 q^{76} -28329.3 q^{77} +15997.7 q^{78} +103505. q^{79} +30133.4 q^{80} +6561.00 q^{81} +62883.4 q^{82} -4341.54 q^{83} -51921.5 q^{84} +15200.4 q^{85} -90070.3 q^{86} +25244.9 q^{87} -6701.65 q^{88} -45464.1 q^{89} -15240.2 q^{90} -55296.9 q^{91} -117620. q^{92} +92324.8 q^{93} +63109.2 q^{94} +44995.2 q^{95} -65691.3 q^{96} -183170. q^{97} +286049. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 252 q^{12} + 450 q^{13} - 1504 q^{14} - 675 q^{15} - 1360 q^{16} - 334 q^{17} - 162 q^{18} - 4036 q^{19} - 700 q^{20} - 2088 q^{21} - 242 q^{22} - 7060 q^{23} + 216 q^{24} + 1875 q^{25} + 2932 q^{26} + 2187 q^{27} - 8320 q^{28} + 4042 q^{29} + 450 q^{30} - 608 q^{31} - 3104 q^{32} + 3267 q^{33} - 3644 q^{34} + 5800 q^{35} + 2268 q^{36} + 2250 q^{37} - 12632 q^{38} + 4050 q^{39} - 600 q^{40} + 10654 q^{41} - 13536 q^{42} - 35528 q^{43} + 3388 q^{44} - 6075 q^{45} - 41800 q^{46} - 2100 q^{47} - 12240 q^{48} + 7667 q^{49} - 1250 q^{50} - 3006 q^{51} - 14520 q^{52} - 12826 q^{53} - 1458 q^{54} - 9075 q^{55} + 17088 q^{56} - 36324 q^{57} - 17196 q^{58} - 81876 q^{59} - 6300 q^{60} - 62298 q^{61} + 109184 q^{62} - 18792 q^{63} - 72256 q^{64} - 11250 q^{65} - 2178 q^{66} - 46148 q^{67} - 35832 q^{68} - 63540 q^{69} + 37600 q^{70} - 64724 q^{71} + 1944 q^{72} + 810 q^{73} - 44796 q^{74} + 16875 q^{75} + 44656 q^{76} - 28072 q^{77} + 26388 q^{78} + 43876 q^{79} + 34000 q^{80} + 19683 q^{81} + 56060 q^{82} - 101024 q^{83} - 74880 q^{84} + 8350 q^{85} + 24128 q^{86} + 36378 q^{87} + 2904 q^{88} + 60022 q^{89} + 4050 q^{90} - 28568 q^{91} + 38256 q^{92} - 5472 q^{93} + 74552 q^{94} + 100900 q^{95} - 27936 q^{96} - 319746 q^{97} + 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.52601 1.33042 0.665211 0.746655i \(-0.268341\pi\)
0.665211 + 0.746655i \(0.268341\pi\)
\(3\) 9.00000 0.577350
\(4\) 24.6408 0.770024
\(5\) −25.0000 −0.447214
\(6\) 67.7341 0.768120
\(7\) −234.126 −1.80595 −0.902973 0.429696i \(-0.858621\pi\)
−0.902973 + 0.429696i \(0.858621\pi\)
\(8\) −55.3856 −0.305965
\(9\) 81.0000 0.333333
\(10\) −188.150 −0.594983
\(11\) 121.000 0.301511
\(12\) 221.767 0.444574
\(13\) 236.184 0.387608 0.193804 0.981040i \(-0.437917\pi\)
0.193804 + 0.981040i \(0.437917\pi\)
\(14\) −1762.04 −2.40267
\(15\) −225.000 −0.258199
\(16\) −1205.34 −1.17709
\(17\) −608.015 −0.510261 −0.255130 0.966907i \(-0.582118\pi\)
−0.255130 + 0.966907i \(0.582118\pi\)
\(18\) 609.607 0.443474
\(19\) −1799.81 −1.14378 −0.571890 0.820331i \(-0.693789\pi\)
−0.571890 + 0.820331i \(0.693789\pi\)
\(20\) −616.019 −0.344365
\(21\) −2107.14 −1.04266
\(22\) 910.647 0.401137
\(23\) −4773.40 −1.88152 −0.940759 0.339075i \(-0.889886\pi\)
−0.940759 + 0.339075i \(0.889886\pi\)
\(24\) −498.470 −0.176649
\(25\) 625.000 0.200000
\(26\) 1777.52 0.515682
\(27\) 729.000 0.192450
\(28\) −5769.05 −1.39062
\(29\) 2804.99 0.619351 0.309676 0.950842i \(-0.399780\pi\)
0.309676 + 0.950842i \(0.399780\pi\)
\(30\) −1693.35 −0.343514
\(31\) 10258.3 1.91722 0.958610 0.284724i \(-0.0919019\pi\)
0.958610 + 0.284724i \(0.0919019\pi\)
\(32\) −7299.04 −1.26006
\(33\) 1089.00 0.174078
\(34\) −4575.93 −0.678862
\(35\) 5853.16 0.807644
\(36\) 1995.90 0.256675
\(37\) −7629.84 −0.916244 −0.458122 0.888889i \(-0.651478\pi\)
−0.458122 + 0.888889i \(0.651478\pi\)
\(38\) −13545.4 −1.52171
\(39\) 2125.66 0.223785
\(40\) 1384.64 0.136832
\(41\) 8355.48 0.776268 0.388134 0.921603i \(-0.373120\pi\)
0.388134 + 0.921603i \(0.373120\pi\)
\(42\) −15858.3 −1.38718
\(43\) −11967.9 −0.987065 −0.493533 0.869727i \(-0.664295\pi\)
−0.493533 + 0.869727i \(0.664295\pi\)
\(44\) 2981.53 0.232171
\(45\) −2025.00 −0.149071
\(46\) −35924.7 −2.50321
\(47\) 8385.48 0.553711 0.276856 0.960912i \(-0.410708\pi\)
0.276856 + 0.960912i \(0.410708\pi\)
\(48\) −10848.0 −0.679591
\(49\) 38008.1 2.26144
\(50\) 4703.75 0.266085
\(51\) −5472.14 −0.294599
\(52\) 5819.76 0.298467
\(53\) 3453.00 0.168852 0.0844261 0.996430i \(-0.473094\pi\)
0.0844261 + 0.996430i \(0.473094\pi\)
\(54\) 5486.46 0.256040
\(55\) −3025.00 −0.134840
\(56\) 12967.2 0.552556
\(57\) −16198.3 −0.660361
\(58\) 21110.4 0.823999
\(59\) −51032.4 −1.90860 −0.954302 0.298842i \(-0.903400\pi\)
−0.954302 + 0.298842i \(0.903400\pi\)
\(60\) −5544.17 −0.198819
\(61\) 9988.24 0.343688 0.171844 0.985124i \(-0.445028\pi\)
0.171844 + 0.985124i \(0.445028\pi\)
\(62\) 77204.1 2.55071
\(63\) −18964.2 −0.601982
\(64\) −16361.8 −0.499323
\(65\) −5904.60 −0.173343
\(66\) 8195.82 0.231597
\(67\) −36567.6 −0.995198 −0.497599 0.867407i \(-0.665785\pi\)
−0.497599 + 0.867407i \(0.665785\pi\)
\(68\) −14982.0 −0.392913
\(69\) −42960.6 −1.08630
\(70\) 44050.9 1.07451
\(71\) −30522.3 −0.718573 −0.359286 0.933227i \(-0.616980\pi\)
−0.359286 + 0.933227i \(0.616980\pi\)
\(72\) −4486.23 −0.101988
\(73\) 83879.5 1.84225 0.921125 0.389267i \(-0.127272\pi\)
0.921125 + 0.389267i \(0.127272\pi\)
\(74\) −57422.2 −1.21899
\(75\) 5625.00 0.115470
\(76\) −44348.7 −0.880738
\(77\) −28329.3 −0.544513
\(78\) 15997.7 0.297729
\(79\) 103505. 1.86592 0.932961 0.359978i \(-0.117216\pi\)
0.932961 + 0.359978i \(0.117216\pi\)
\(80\) 30133.4 0.526409
\(81\) 6561.00 0.111111
\(82\) 62883.4 1.03276
\(83\) −4341.54 −0.0691749 −0.0345875 0.999402i \(-0.511012\pi\)
−0.0345875 + 0.999402i \(0.511012\pi\)
\(84\) −51921.5 −0.802876
\(85\) 15200.4 0.228196
\(86\) −90070.3 −1.31321
\(87\) 25244.9 0.357583
\(88\) −6701.65 −0.0922519
\(89\) −45464.1 −0.608406 −0.304203 0.952607i \(-0.598390\pi\)
−0.304203 + 0.952607i \(0.598390\pi\)
\(90\) −15240.2 −0.198328
\(91\) −55296.9 −0.699999
\(92\) −117620. −1.44881
\(93\) 92324.8 1.10691
\(94\) 63109.2 0.736670
\(95\) 44995.2 0.511514
\(96\) −65691.3 −0.727495
\(97\) −183170. −1.97663 −0.988314 0.152432i \(-0.951289\pi\)
−0.988314 + 0.152432i \(0.951289\pi\)
\(98\) 286049. 3.00868
\(99\) 9801.00 0.100504
\(100\) 15400.5 0.154005
\(101\) −28833.3 −0.281249 −0.140625 0.990063i \(-0.544911\pi\)
−0.140625 + 0.990063i \(0.544911\pi\)
\(102\) −41183.3 −0.391941
\(103\) −59444.6 −0.552102 −0.276051 0.961143i \(-0.589026\pi\)
−0.276051 + 0.961143i \(0.589026\pi\)
\(104\) −13081.2 −0.118594
\(105\) 52678.4 0.466293
\(106\) 25987.3 0.224645
\(107\) 13840.6 0.116868 0.0584342 0.998291i \(-0.481389\pi\)
0.0584342 + 0.998291i \(0.481389\pi\)
\(108\) 17963.1 0.148191
\(109\) −13278.1 −0.107046 −0.0535231 0.998567i \(-0.517045\pi\)
−0.0535231 + 0.998567i \(0.517045\pi\)
\(110\) −22766.2 −0.179394
\(111\) −68668.5 −0.528993
\(112\) 282201. 2.12576
\(113\) 130086. 0.958370 0.479185 0.877714i \(-0.340932\pi\)
0.479185 + 0.877714i \(0.340932\pi\)
\(114\) −121908. −0.878560
\(115\) 119335. 0.841441
\(116\) 69117.2 0.476915
\(117\) 19130.9 0.129203
\(118\) −384070. −2.53925
\(119\) 142352. 0.921504
\(120\) 12461.8 0.0789998
\(121\) 14641.0 0.0909091
\(122\) 75171.5 0.457250
\(123\) 75199.3 0.448178
\(124\) 252773. 1.47631
\(125\) −15625.0 −0.0894427
\(126\) −142725. −0.800891
\(127\) −274904. −1.51242 −0.756209 0.654330i \(-0.772951\pi\)
−0.756209 + 0.654330i \(0.772951\pi\)
\(128\) 110430. 0.595748
\(129\) −107711. −0.569882
\(130\) −44438.1 −0.230620
\(131\) −119374. −0.607761 −0.303881 0.952710i \(-0.598282\pi\)
−0.303881 + 0.952710i \(0.598282\pi\)
\(132\) 26833.8 0.134044
\(133\) 421382. 2.06560
\(134\) −275208. −1.32403
\(135\) −18225.0 −0.0860663
\(136\) 33675.3 0.156122
\(137\) −132278. −0.602123 −0.301061 0.953605i \(-0.597341\pi\)
−0.301061 + 0.953605i \(0.597341\pi\)
\(138\) −323322. −1.44523
\(139\) −102326. −0.449209 −0.224604 0.974450i \(-0.572109\pi\)
−0.224604 + 0.974450i \(0.572109\pi\)
\(140\) 144226. 0.621905
\(141\) 75469.3 0.319685
\(142\) −229711. −0.956005
\(143\) 28578.3 0.116868
\(144\) −97632.3 −0.392362
\(145\) −70124.8 −0.276982
\(146\) 631278. 2.45097
\(147\) 342073. 1.30565
\(148\) −188005. −0.705530
\(149\) 371696. 1.37158 0.685791 0.727798i \(-0.259456\pi\)
0.685791 + 0.727798i \(0.259456\pi\)
\(150\) 42333.8 0.153624
\(151\) −111725. −0.398757 −0.199379 0.979923i \(-0.563892\pi\)
−0.199379 + 0.979923i \(0.563892\pi\)
\(152\) 99683.4 0.349956
\(153\) −49249.2 −0.170087
\(154\) −213206. −0.724433
\(155\) −256458. −0.857406
\(156\) 52377.8 0.172320
\(157\) −244726. −0.792375 −0.396187 0.918170i \(-0.629667\pi\)
−0.396187 + 0.918170i \(0.629667\pi\)
\(158\) 778979. 2.48246
\(159\) 31077.0 0.0974869
\(160\) 182476. 0.563515
\(161\) 1.11758e6 3.39792
\(162\) 49378.1 0.147825
\(163\) 435623. 1.28423 0.642114 0.766610i \(-0.278058\pi\)
0.642114 + 0.766610i \(0.278058\pi\)
\(164\) 205885. 0.597745
\(165\) −27225.0 −0.0778499
\(166\) −32674.5 −0.0920319
\(167\) 288486. 0.800450 0.400225 0.916417i \(-0.368932\pi\)
0.400225 + 0.916417i \(0.368932\pi\)
\(168\) 116705. 0.319019
\(169\) −315510. −0.849760
\(170\) 114398. 0.303597
\(171\) −145784. −0.381260
\(172\) −294898. −0.760064
\(173\) 539633. 1.37083 0.685414 0.728154i \(-0.259621\pi\)
0.685414 + 0.728154i \(0.259621\pi\)
\(174\) 189994. 0.475736
\(175\) −146329. −0.361189
\(176\) −145846. −0.354905
\(177\) −459292. −1.10193
\(178\) −342163. −0.809438
\(179\) −378874. −0.883816 −0.441908 0.897060i \(-0.645698\pi\)
−0.441908 + 0.897060i \(0.645698\pi\)
\(180\) −49897.6 −0.114788
\(181\) −621633. −1.41038 −0.705192 0.709016i \(-0.749139\pi\)
−0.705192 + 0.709016i \(0.749139\pi\)
\(182\) −416165. −0.931294
\(183\) 89894.1 0.198428
\(184\) 264378. 0.575679
\(185\) 190746. 0.409757
\(186\) 694837. 1.47265
\(187\) −73569.8 −0.153849
\(188\) 206625. 0.426371
\(189\) −170678. −0.347555
\(190\) 338634. 0.680529
\(191\) 8509.61 0.0168782 0.00843911 0.999964i \(-0.497314\pi\)
0.00843911 + 0.999964i \(0.497314\pi\)
\(192\) −147256. −0.288284
\(193\) −564588. −1.09103 −0.545517 0.838100i \(-0.683667\pi\)
−0.545517 + 0.838100i \(0.683667\pi\)
\(194\) −1.37854e6 −2.62975
\(195\) −53141.4 −0.100080
\(196\) 936549. 1.74137
\(197\) −432994. −0.794908 −0.397454 0.917622i \(-0.630106\pi\)
−0.397454 + 0.917622i \(0.630106\pi\)
\(198\) 73762.4 0.133712
\(199\) −614871. −1.10066 −0.550328 0.834949i \(-0.685497\pi\)
−0.550328 + 0.834949i \(0.685497\pi\)
\(200\) −34616.0 −0.0611930
\(201\) −329108. −0.574578
\(202\) −217000. −0.374180
\(203\) −656723. −1.11852
\(204\) −134838. −0.226849
\(205\) −208887. −0.347158
\(206\) −447380. −0.734528
\(207\) −386646. −0.627173
\(208\) −284682. −0.456248
\(209\) −217777. −0.344862
\(210\) 396458. 0.620367
\(211\) 72348.2 0.111872 0.0559360 0.998434i \(-0.482186\pi\)
0.0559360 + 0.998434i \(0.482186\pi\)
\(212\) 85084.6 0.130020
\(213\) −274700. −0.414868
\(214\) 104165. 0.155484
\(215\) 299197. 0.441429
\(216\) −40376.1 −0.0588830
\(217\) −2.40174e6 −3.46240
\(218\) −99931.4 −0.142417
\(219\) 754915. 1.06362
\(220\) −74538.3 −0.103830
\(221\) −143604. −0.197781
\(222\) −516800. −0.703785
\(223\) 370371. 0.498740 0.249370 0.968408i \(-0.419776\pi\)
0.249370 + 0.968408i \(0.419776\pi\)
\(224\) 1.70890e6 2.27560
\(225\) 50625.0 0.0666667
\(226\) 979025. 1.27504
\(227\) 325481. 0.419239 0.209619 0.977783i \(-0.432778\pi\)
0.209619 + 0.977783i \(0.432778\pi\)
\(228\) −399138. −0.508494
\(229\) −57345.0 −0.0722614 −0.0361307 0.999347i \(-0.511503\pi\)
−0.0361307 + 0.999347i \(0.511503\pi\)
\(230\) 898116. 1.11947
\(231\) −254963. −0.314375
\(232\) −155356. −0.189500
\(233\) −833891. −1.00628 −0.503141 0.864205i \(-0.667822\pi\)
−0.503141 + 0.864205i \(0.667822\pi\)
\(234\) 143979. 0.171894
\(235\) −209637. −0.247627
\(236\) −1.25748e6 −1.46967
\(237\) 931545. 1.07729
\(238\) 1.07134e6 1.22599
\(239\) −813986. −0.921769 −0.460885 0.887460i \(-0.652468\pi\)
−0.460885 + 0.887460i \(0.652468\pi\)
\(240\) 271201. 0.303923
\(241\) −1.18456e6 −1.31376 −0.656878 0.753997i \(-0.728123\pi\)
−0.656878 + 0.753997i \(0.728123\pi\)
\(242\) 110188. 0.120948
\(243\) 59049.0 0.0641500
\(244\) 246118. 0.264648
\(245\) −950202. −1.01135
\(246\) 565950. 0.596267
\(247\) −425086. −0.443338
\(248\) −568163. −0.586602
\(249\) −39073.9 −0.0399382
\(250\) −117594. −0.118997
\(251\) −601749. −0.602880 −0.301440 0.953485i \(-0.597467\pi\)
−0.301440 + 0.953485i \(0.597467\pi\)
\(252\) −467293. −0.463541
\(253\) −577582. −0.567299
\(254\) −2.06893e6 −2.01216
\(255\) 136803. 0.131749
\(256\) 1.35468e6 1.29192
\(257\) 904570. 0.854298 0.427149 0.904181i \(-0.359518\pi\)
0.427149 + 0.904181i \(0.359518\pi\)
\(258\) −810633. −0.758184
\(259\) 1.78634e6 1.65469
\(260\) −145494. −0.133479
\(261\) 227204. 0.206450
\(262\) −898413. −0.808580
\(263\) 1.47174e6 1.31202 0.656011 0.754751i \(-0.272242\pi\)
0.656011 + 0.754751i \(0.272242\pi\)
\(264\) −60314.9 −0.0532617
\(265\) −86325.0 −0.0755130
\(266\) 3.17133e6 2.74813
\(267\) −409177. −0.351264
\(268\) −901054. −0.766327
\(269\) 897427. 0.756168 0.378084 0.925771i \(-0.376583\pi\)
0.378084 + 0.925771i \(0.376583\pi\)
\(270\) −137161. −0.114505
\(271\) 134225. 0.111022 0.0555111 0.998458i \(-0.482321\pi\)
0.0555111 + 0.998458i \(0.482321\pi\)
\(272\) 732863. 0.600621
\(273\) −497672. −0.404145
\(274\) −995523. −0.801078
\(275\) 75625.0 0.0603023
\(276\) −1.05858e6 −0.836474
\(277\) −2.01306e6 −1.57637 −0.788184 0.615439i \(-0.788979\pi\)
−0.788184 + 0.615439i \(0.788979\pi\)
\(278\) −770105. −0.597637
\(279\) 830923. 0.639073
\(280\) −324180. −0.247111
\(281\) 1.45204e6 1.09702 0.548509 0.836145i \(-0.315196\pi\)
0.548509 + 0.836145i \(0.315196\pi\)
\(282\) 567983. 0.425316
\(283\) −2.24416e6 −1.66566 −0.832832 0.553525i \(-0.813282\pi\)
−0.832832 + 0.553525i \(0.813282\pi\)
\(284\) −752092. −0.553318
\(285\) 404957. 0.295323
\(286\) 215080. 0.155484
\(287\) −1.95624e6 −1.40190
\(288\) −591222. −0.420019
\(289\) −1.05017e6 −0.739634
\(290\) −527760. −0.368503
\(291\) −1.64853e6 −1.14121
\(292\) 2.06686e6 1.41858
\(293\) 148693. 0.101187 0.0505933 0.998719i \(-0.483889\pi\)
0.0505933 + 0.998719i \(0.483889\pi\)
\(294\) 2.57444e6 1.73706
\(295\) 1.27581e6 0.853554
\(296\) 422583. 0.280338
\(297\) 88209.0 0.0580259
\(298\) 2.79739e6 1.82478
\(299\) −1.12740e6 −0.729291
\(300\) 138604. 0.0889147
\(301\) 2.80199e6 1.78259
\(302\) −840845. −0.530516
\(303\) −259500. −0.162379
\(304\) 2.16938e6 1.34633
\(305\) −249706. −0.153702
\(306\) −370650. −0.226287
\(307\) 2.02304e6 1.22507 0.612533 0.790445i \(-0.290151\pi\)
0.612533 + 0.790445i \(0.290151\pi\)
\(308\) −698055. −0.419289
\(309\) −535001. −0.318756
\(310\) −1.93010e6 −1.14071
\(311\) 1.87652e6 1.10015 0.550074 0.835116i \(-0.314599\pi\)
0.550074 + 0.835116i \(0.314599\pi\)
\(312\) −117731. −0.0684705
\(313\) −89494.6 −0.0516340 −0.0258170 0.999667i \(-0.508219\pi\)
−0.0258170 + 0.999667i \(0.508219\pi\)
\(314\) −1.84181e6 −1.05419
\(315\) 474106. 0.269215
\(316\) 2.55044e6 1.43680
\(317\) 260115. 0.145384 0.0726920 0.997354i \(-0.476841\pi\)
0.0726920 + 0.997354i \(0.476841\pi\)
\(318\) 233886. 0.129699
\(319\) 339404. 0.186741
\(320\) 409045. 0.223304
\(321\) 124566. 0.0674740
\(322\) 8.41090e6 4.52067
\(323\) 1.09431e6 0.583626
\(324\) 161668. 0.0855582
\(325\) 147615. 0.0775215
\(326\) 3.27850e6 1.70856
\(327\) −119503. −0.0618031
\(328\) −462773. −0.237511
\(329\) −1.96326e6 −0.999973
\(330\) −204896. −0.103573
\(331\) 2.19339e6 1.10039 0.550193 0.835037i \(-0.314554\pi\)
0.550193 + 0.835037i \(0.314554\pi\)
\(332\) −106979. −0.0532664
\(333\) −618017. −0.305415
\(334\) 2.17115e6 1.06494
\(335\) 914190. 0.445066
\(336\) 2.53981e6 1.22731
\(337\) 933037. 0.447532 0.223766 0.974643i \(-0.428165\pi\)
0.223766 + 0.974643i \(0.428165\pi\)
\(338\) −2.37453e6 −1.13054
\(339\) 1.17077e6 0.553315
\(340\) 374549. 0.175716
\(341\) 1.24126e6 0.578063
\(342\) −1.09717e6 −0.507237
\(343\) −4.96373e6 −2.27810
\(344\) 662848. 0.302007
\(345\) 1.07402e6 0.485806
\(346\) 4.06128e6 1.82378
\(347\) −1.72984e6 −0.771229 −0.385615 0.922660i \(-0.626011\pi\)
−0.385615 + 0.922660i \(0.626011\pi\)
\(348\) 622055. 0.275347
\(349\) 1.01770e6 0.447258 0.223629 0.974674i \(-0.428210\pi\)
0.223629 + 0.974674i \(0.428210\pi\)
\(350\) −1.10127e6 −0.480534
\(351\) 172178. 0.0745951
\(352\) −883183. −0.379922
\(353\) −3.29128e6 −1.40581 −0.702907 0.711282i \(-0.748115\pi\)
−0.702907 + 0.711282i \(0.748115\pi\)
\(354\) −3.45663e6 −1.46604
\(355\) 763056. 0.321355
\(356\) −1.12027e6 −0.468488
\(357\) 1.28117e6 0.532030
\(358\) −2.85140e6 −1.17585
\(359\) 2.32128e6 0.950584 0.475292 0.879828i \(-0.342342\pi\)
0.475292 + 0.879828i \(0.342342\pi\)
\(360\) 112156. 0.0456106
\(361\) 763211. 0.308231
\(362\) −4.67841e6 −1.87641
\(363\) 131769. 0.0524864
\(364\) −1.36256e6 −0.539016
\(365\) −2.09699e6 −0.823879
\(366\) 676544. 0.263993
\(367\) −1.81399e6 −0.703024 −0.351512 0.936183i \(-0.614332\pi\)
−0.351512 + 0.936183i \(0.614332\pi\)
\(368\) 5.75356e6 2.21471
\(369\) 676794. 0.258756
\(370\) 1.43556e6 0.545149
\(371\) −808438. −0.304938
\(372\) 2.27496e6 0.852345
\(373\) −160496. −0.0597298 −0.0298649 0.999554i \(-0.509508\pi\)
−0.0298649 + 0.999554i \(0.509508\pi\)
\(374\) −553687. −0.204685
\(375\) −140625. −0.0516398
\(376\) −464435. −0.169416
\(377\) 662495. 0.240065
\(378\) −1.28452e6 −0.462395
\(379\) −622442. −0.222587 −0.111294 0.993788i \(-0.535499\pi\)
−0.111294 + 0.993788i \(0.535499\pi\)
\(380\) 1.10872e6 0.393878
\(381\) −2.47414e6 −0.873195
\(382\) 64043.4 0.0224552
\(383\) 1.49075e6 0.519289 0.259644 0.965704i \(-0.416395\pi\)
0.259644 + 0.965704i \(0.416395\pi\)
\(384\) 993871. 0.343955
\(385\) 708232. 0.243514
\(386\) −4.24909e6 −1.45154
\(387\) −969398. −0.329022
\(388\) −4.51345e6 −1.52205
\(389\) −1.66085e6 −0.556490 −0.278245 0.960510i \(-0.589753\pi\)
−0.278245 + 0.960510i \(0.589753\pi\)
\(390\) −399943. −0.133149
\(391\) 2.90230e6 0.960065
\(392\) −2.10510e6 −0.691923
\(393\) −1.07437e6 −0.350891
\(394\) −3.25872e6 −1.05756
\(395\) −2.58762e6 −0.834465
\(396\) 241504. 0.0773903
\(397\) 2.46357e6 0.784493 0.392247 0.919860i \(-0.371698\pi\)
0.392247 + 0.919860i \(0.371698\pi\)
\(398\) −4.62752e6 −1.46434
\(399\) 3.79244e6 1.19258
\(400\) −753336. −0.235417
\(401\) −664739. −0.206438 −0.103219 0.994659i \(-0.532914\pi\)
−0.103219 + 0.994659i \(0.532914\pi\)
\(402\) −2.47687e6 −0.764431
\(403\) 2.42285e6 0.743129
\(404\) −710475. −0.216569
\(405\) −164025. −0.0496904
\(406\) −4.94250e6 −1.48810
\(407\) −923210. −0.276258
\(408\) 303077. 0.0901370
\(409\) −111122. −0.0328468 −0.0164234 0.999865i \(-0.505228\pi\)
−0.0164234 + 0.999865i \(0.505228\pi\)
\(410\) −1.57208e6 −0.461866
\(411\) −1.19050e6 −0.347636
\(412\) −1.46476e6 −0.425132
\(413\) 1.19480e7 3.44684
\(414\) −2.90990e6 −0.834405
\(415\) 108539. 0.0309360
\(416\) −1.72392e6 −0.488408
\(417\) −920932. −0.259351
\(418\) −1.63899e6 −0.458813
\(419\) −5.34446e6 −1.48720 −0.743598 0.668627i \(-0.766882\pi\)
−0.743598 + 0.668627i \(0.766882\pi\)
\(420\) 1.29804e6 0.359057
\(421\) −483590. −0.132976 −0.0664878 0.997787i \(-0.521179\pi\)
−0.0664878 + 0.997787i \(0.521179\pi\)
\(422\) 544493. 0.148837
\(423\) 679224. 0.184570
\(424\) −191246. −0.0516629
\(425\) −380010. −0.102052
\(426\) −2.06740e6 −0.551950
\(427\) −2.33851e6 −0.620682
\(428\) 341044. 0.0899915
\(429\) 257205. 0.0674738
\(430\) 2.25176e6 0.587287
\(431\) −241542. −0.0626325 −0.0313162 0.999510i \(-0.509970\pi\)
−0.0313162 + 0.999510i \(0.509970\pi\)
\(432\) −878691. −0.226530
\(433\) −1.03998e6 −0.266567 −0.133284 0.991078i \(-0.542552\pi\)
−0.133284 + 0.991078i \(0.542552\pi\)
\(434\) −1.80755e7 −4.60645
\(435\) −631124. −0.159916
\(436\) −327184. −0.0824281
\(437\) 8.59121e6 2.15204
\(438\) 5.68150e6 1.41507
\(439\) −3.32986e6 −0.824642 −0.412321 0.911039i \(-0.635282\pi\)
−0.412321 + 0.911039i \(0.635282\pi\)
\(440\) 167541. 0.0412563
\(441\) 3.07866e6 0.753815
\(442\) −1.08076e6 −0.263132
\(443\) 5.12280e6 1.24022 0.620109 0.784515i \(-0.287088\pi\)
0.620109 + 0.784515i \(0.287088\pi\)
\(444\) −1.69205e6 −0.407338
\(445\) 1.13660e6 0.272088
\(446\) 2.78741e6 0.663535
\(447\) 3.34526e6 0.791884
\(448\) 3.83073e6 0.901750
\(449\) 5.82328e6 1.36318 0.681588 0.731736i \(-0.261290\pi\)
0.681588 + 0.731736i \(0.261290\pi\)
\(450\) 381004. 0.0886948
\(451\) 1.01101e6 0.234054
\(452\) 3.20541e6 0.737968
\(453\) −1.00553e6 −0.230223
\(454\) 2.44957e6 0.557765
\(455\) 1.38242e6 0.313049
\(456\) 897151. 0.202047
\(457\) −5.32427e6 −1.19253 −0.596266 0.802787i \(-0.703349\pi\)
−0.596266 + 0.802787i \(0.703349\pi\)
\(458\) −431579. −0.0961382
\(459\) −443243. −0.0981997
\(460\) 2.94051e6 0.647930
\(461\) 673694. 0.147642 0.0738211 0.997272i \(-0.476481\pi\)
0.0738211 + 0.997272i \(0.476481\pi\)
\(462\) −1.91886e6 −0.418252
\(463\) −1.92230e6 −0.416742 −0.208371 0.978050i \(-0.566816\pi\)
−0.208371 + 0.978050i \(0.566816\pi\)
\(464\) −3.38096e6 −0.729030
\(465\) −2.30812e6 −0.495024
\(466\) −6.27587e6 −1.33878
\(467\) −7.94668e6 −1.68614 −0.843070 0.537804i \(-0.819254\pi\)
−0.843070 + 0.537804i \(0.819254\pi\)
\(468\) 471401. 0.0994891
\(469\) 8.56144e6 1.79727
\(470\) −1.57773e6 −0.329449
\(471\) −2.20253e6 −0.457478
\(472\) 2.82646e6 0.583966
\(473\) −1.44811e6 −0.297611
\(474\) 7.01081e6 1.43325
\(475\) −1.12488e6 −0.228756
\(476\) 3.50767e6 0.709580
\(477\) 279693. 0.0562841
\(478\) −6.12606e6 −1.22634
\(479\) −1.80345e6 −0.359141 −0.179570 0.983745i \(-0.557471\pi\)
−0.179570 + 0.983745i \(0.557471\pi\)
\(480\) 1.64228e6 0.325346
\(481\) −1.80205e6 −0.355143
\(482\) −8.91501e6 −1.74785
\(483\) 1.00582e7 1.96179
\(484\) 360766. 0.0700022
\(485\) 4.57925e6 0.883975
\(486\) 444403. 0.0853466
\(487\) 3.04965e6 0.582678 0.291339 0.956620i \(-0.405899\pi\)
0.291339 + 0.956620i \(0.405899\pi\)
\(488\) −553204. −0.105156
\(489\) 3.92061e6 0.741449
\(490\) −7.15123e6 −1.34552
\(491\) −7.53319e6 −1.41018 −0.705091 0.709117i \(-0.749094\pi\)
−0.705091 + 0.709117i \(0.749094\pi\)
\(492\) 1.85297e6 0.345108
\(493\) −1.70548e6 −0.316031
\(494\) −3.19920e6 −0.589826
\(495\) −245025. −0.0449467
\(496\) −1.23647e7 −2.25673
\(497\) 7.14606e6 1.29770
\(498\) −294070. −0.0531346
\(499\) 6.01840e6 1.08201 0.541003 0.841021i \(-0.318045\pi\)
0.541003 + 0.841021i \(0.318045\pi\)
\(500\) −385012. −0.0688731
\(501\) 2.59638e6 0.462140
\(502\) −4.52876e6 −0.802085
\(503\) 8.91752e6 1.57153 0.785767 0.618522i \(-0.212268\pi\)
0.785767 + 0.618522i \(0.212268\pi\)
\(504\) 1.05034e6 0.184185
\(505\) 720833. 0.125778
\(506\) −4.34688e6 −0.754748
\(507\) −2.83959e6 −0.490609
\(508\) −6.77385e6 −1.16460
\(509\) 9.71294e6 1.66171 0.830857 0.556486i \(-0.187851\pi\)
0.830857 + 0.556486i \(0.187851\pi\)
\(510\) 1.02958e6 0.175282
\(511\) −1.96384e7 −3.32701
\(512\) 6.66153e6 1.12305
\(513\) −1.31206e6 −0.220120
\(514\) 6.80780e6 1.13658
\(515\) 1.48611e6 0.246907
\(516\) −2.65408e6 −0.438823
\(517\) 1.01464e6 0.166950
\(518\) 1.34440e7 2.20143
\(519\) 4.85669e6 0.791448
\(520\) 327030. 0.0530370
\(521\) −591081. −0.0954009 −0.0477004 0.998862i \(-0.515189\pi\)
−0.0477004 + 0.998862i \(0.515189\pi\)
\(522\) 1.70994e6 0.274666
\(523\) 2.83152e6 0.452653 0.226327 0.974051i \(-0.427328\pi\)
0.226327 + 0.974051i \(0.427328\pi\)
\(524\) −2.94148e6 −0.467991
\(525\) −1.31696e6 −0.208533
\(526\) 1.10763e7 1.74554
\(527\) −6.23721e6 −0.978282
\(528\) −1.31261e6 −0.204905
\(529\) 1.63490e7 2.54011
\(530\) −649682. −0.100464
\(531\) −4.13363e6 −0.636202
\(532\) 1.03832e7 1.59057
\(533\) 1.97343e6 0.300887
\(534\) −3.07947e6 −0.467329
\(535\) −346016. −0.0522651
\(536\) 2.02532e6 0.304496
\(537\) −3.40986e6 −0.510271
\(538\) 6.75404e6 1.00602
\(539\) 4.59898e6 0.681851
\(540\) −449078. −0.0662731
\(541\) 9.93914e6 1.46001 0.730004 0.683442i \(-0.239518\pi\)
0.730004 + 0.683442i \(0.239518\pi\)
\(542\) 1.01018e6 0.147706
\(543\) −5.59469e6 −0.814286
\(544\) 4.43793e6 0.642958
\(545\) 331953. 0.0478725
\(546\) −3.74548e6 −0.537683
\(547\) −4.34846e6 −0.621394 −0.310697 0.950509i \(-0.600563\pi\)
−0.310697 + 0.950509i \(0.600563\pi\)
\(548\) −3.25942e6 −0.463649
\(549\) 809047. 0.114563
\(550\) 569154. 0.0802275
\(551\) −5.04845e6 −0.708401
\(552\) 2.37940e6 0.332368
\(553\) −2.42332e7 −3.36976
\(554\) −1.51503e7 −2.09724
\(555\) 1.71671e6 0.236573
\(556\) −2.52139e6 −0.345902
\(557\) 1.07234e7 1.46452 0.732261 0.681024i \(-0.238465\pi\)
0.732261 + 0.681024i \(0.238465\pi\)
\(558\) 6.25354e6 0.850237
\(559\) −2.82662e6 −0.382594
\(560\) −7.05503e6 −0.950667
\(561\) −662129. −0.0888250
\(562\) 1.09281e7 1.45950
\(563\) 1.67966e6 0.223332 0.111666 0.993746i \(-0.464381\pi\)
0.111666 + 0.993746i \(0.464381\pi\)
\(564\) 1.85962e6 0.246165
\(565\) −3.25214e6 −0.428596
\(566\) −1.68896e7 −2.21604
\(567\) −1.53610e6 −0.200661
\(568\) 1.69049e6 0.219858
\(569\) 1.04510e7 1.35325 0.676626 0.736327i \(-0.263441\pi\)
0.676626 + 0.736327i \(0.263441\pi\)
\(570\) 3.04771e6 0.392904
\(571\) 670155. 0.0860171 0.0430085 0.999075i \(-0.486306\pi\)
0.0430085 + 0.999075i \(0.486306\pi\)
\(572\) 704191. 0.0899913
\(573\) 76586.5 0.00974464
\(574\) −1.47226e7 −1.86512
\(575\) −2.98338e6 −0.376304
\(576\) −1.32531e6 −0.166441
\(577\) −7.87543e6 −0.984769 −0.492385 0.870378i \(-0.663875\pi\)
−0.492385 + 0.870378i \(0.663875\pi\)
\(578\) −7.90362e6 −0.984026
\(579\) −5.08129e6 −0.629909
\(580\) −1.72793e6 −0.213283
\(581\) 1.01647e6 0.124926
\(582\) −1.24068e7 −1.51829
\(583\) 417813. 0.0509109
\(584\) −4.64571e6 −0.563664
\(585\) −478273. −0.0577811
\(586\) 1.11907e6 0.134621
\(587\) −9.13414e6 −1.09414 −0.547069 0.837087i \(-0.684257\pi\)
−0.547069 + 0.837087i \(0.684257\pi\)
\(588\) 8.42894e6 1.00538
\(589\) −1.84630e7 −2.19288
\(590\) 9.60176e6 1.13559
\(591\) −3.89695e6 −0.458940
\(592\) 9.19652e6 1.07850
\(593\) 1.25096e7 1.46085 0.730425 0.682993i \(-0.239322\pi\)
0.730425 + 0.682993i \(0.239322\pi\)
\(594\) 663862. 0.0771989
\(595\) −3.55881e6 −0.412109
\(596\) 9.15887e6 1.05615
\(597\) −5.53384e6 −0.635464
\(598\) −8.48484e6 −0.970265
\(599\) 1.59560e7 1.81700 0.908502 0.417881i \(-0.137227\pi\)
0.908502 + 0.417881i \(0.137227\pi\)
\(600\) −311544. −0.0353298
\(601\) 763693. 0.0862448 0.0431224 0.999070i \(-0.486269\pi\)
0.0431224 + 0.999070i \(0.486269\pi\)
\(602\) 2.10878e7 2.37159
\(603\) −2.96198e6 −0.331733
\(604\) −2.75300e6 −0.307053
\(605\) −366025. −0.0406558
\(606\) −1.95300e6 −0.216033
\(607\) −8.51753e6 −0.938301 −0.469150 0.883118i \(-0.655440\pi\)
−0.469150 + 0.883118i \(0.655440\pi\)
\(608\) 1.31369e7 1.44123
\(609\) −5.91050e6 −0.645775
\(610\) −1.87929e6 −0.204488
\(611\) 1.98052e6 0.214623
\(612\) −1.21354e6 −0.130971
\(613\) 9.58182e6 1.02990 0.514952 0.857219i \(-0.327810\pi\)
0.514952 + 0.857219i \(0.327810\pi\)
\(614\) 1.52254e7 1.62985
\(615\) −1.87998e6 −0.200431
\(616\) 1.56903e6 0.166602
\(617\) −1.12974e7 −1.19472 −0.597362 0.801972i \(-0.703784\pi\)
−0.597362 + 0.801972i \(0.703784\pi\)
\(618\) −4.02642e6 −0.424080
\(619\) −4.12945e6 −0.433177 −0.216588 0.976263i \(-0.569493\pi\)
−0.216588 + 0.976263i \(0.569493\pi\)
\(620\) −6.31932e6 −0.660224
\(621\) −3.47981e6 −0.362098
\(622\) 1.41227e7 1.46366
\(623\) 1.06443e7 1.09875
\(624\) −2.56213e6 −0.263415
\(625\) 390625. 0.0400000
\(626\) −673537. −0.0686951
\(627\) −1.95999e6 −0.199106
\(628\) −6.03023e6 −0.610148
\(629\) 4.63906e6 0.467523
\(630\) 3.56812e6 0.358169
\(631\) 1.29753e7 1.29731 0.648656 0.761082i \(-0.275332\pi\)
0.648656 + 0.761082i \(0.275332\pi\)
\(632\) −5.73268e6 −0.570907
\(633\) 651133. 0.0645893
\(634\) 1.95762e6 0.193422
\(635\) 6.87260e6 0.676374
\(636\) 765761. 0.0750672
\(637\) 8.97691e6 0.876553
\(638\) 2.55436e6 0.248445
\(639\) −2.47230e6 −0.239524
\(640\) −2.76075e6 −0.266427
\(641\) −130086. −0.0125051 −0.00625255 0.999980i \(-0.501990\pi\)
−0.00625255 + 0.999980i \(0.501990\pi\)
\(642\) 937483. 0.0897689
\(643\) −6.47507e6 −0.617614 −0.308807 0.951125i \(-0.599930\pi\)
−0.308807 + 0.951125i \(0.599930\pi\)
\(644\) 2.75380e7 2.61648
\(645\) 2.69277e6 0.254859
\(646\) 8.23579e6 0.776469
\(647\) −5.42540e6 −0.509531 −0.254766 0.967003i \(-0.581998\pi\)
−0.254766 + 0.967003i \(0.581998\pi\)
\(648\) −363385. −0.0339961
\(649\) −6.17492e6 −0.575466
\(650\) 1.11095e6 0.103136
\(651\) −2.16157e7 −1.99902
\(652\) 1.07341e7 0.988886
\(653\) 9.15697e6 0.840367 0.420183 0.907439i \(-0.361966\pi\)
0.420183 + 0.907439i \(0.361966\pi\)
\(654\) −899382. −0.0822243
\(655\) 2.98436e6 0.271799
\(656\) −1.00712e7 −0.913735
\(657\) 6.79424e6 0.614083
\(658\) −1.47755e7 −1.33039
\(659\) −4.88080e6 −0.437802 −0.218901 0.975747i \(-0.570247\pi\)
−0.218901 + 0.975747i \(0.570247\pi\)
\(660\) −670845. −0.0599463
\(661\) 1.75690e7 1.56402 0.782010 0.623266i \(-0.214194\pi\)
0.782010 + 0.623266i \(0.214194\pi\)
\(662\) 1.65074e7 1.46398
\(663\) −1.29243e6 −0.114189
\(664\) 240459. 0.0211651
\(665\) −1.05346e7 −0.923767
\(666\) −4.65120e6 −0.406330
\(667\) −1.33894e7 −1.16532
\(668\) 7.10853e6 0.616366
\(669\) 3.33333e6 0.287948
\(670\) 6.88020e6 0.592126
\(671\) 1.20858e6 0.103626
\(672\) 1.53801e7 1.31382
\(673\) 1.60675e7 1.36745 0.683725 0.729740i \(-0.260359\pi\)
0.683725 + 0.729740i \(0.260359\pi\)
\(674\) 7.02205e6 0.595407
\(675\) 455625. 0.0384900
\(676\) −7.77441e6 −0.654336
\(677\) −1.51931e7 −1.27401 −0.637007 0.770858i \(-0.719828\pi\)
−0.637007 + 0.770858i \(0.719828\pi\)
\(678\) 8.81123e6 0.736143
\(679\) 4.28849e7 3.56968
\(680\) −841882. −0.0698198
\(681\) 2.92933e6 0.242048
\(682\) 9.34170e6 0.769068
\(683\) −9.75866e6 −0.800457 −0.400229 0.916415i \(-0.631069\pi\)
−0.400229 + 0.916415i \(0.631069\pi\)
\(684\) −3.59224e6 −0.293579
\(685\) 3.30694e6 0.269278
\(686\) −3.73571e7 −3.03084
\(687\) −516105. −0.0417202
\(688\) 1.44253e7 1.16186
\(689\) 815544. 0.0654484
\(690\) 8.08305e6 0.646327
\(691\) −1.74840e7 −1.39298 −0.696490 0.717567i \(-0.745256\pi\)
−0.696490 + 0.717567i \(0.745256\pi\)
\(692\) 1.32970e7 1.05557
\(693\) −2.29467e6 −0.181504
\(694\) −1.30188e7 −1.02606
\(695\) 2.55815e6 0.200892
\(696\) −1.39821e6 −0.109408
\(697\) −5.08026e6 −0.396099
\(698\) 7.65925e6 0.595042
\(699\) −7.50502e6 −0.580977
\(700\) −3.60566e6 −0.278125
\(701\) 4.31053e6 0.331311 0.165656 0.986184i \(-0.447026\pi\)
0.165656 + 0.986184i \(0.447026\pi\)
\(702\) 1.29581e6 0.0992431
\(703\) 1.37322e7 1.04798
\(704\) −1.97978e6 −0.150551
\(705\) −1.88673e6 −0.142968
\(706\) −2.47702e7 −1.87033
\(707\) 6.75064e6 0.507921
\(708\) −1.13173e7 −0.848516
\(709\) −1.44521e6 −0.107973 −0.0539866 0.998542i \(-0.517193\pi\)
−0.0539866 + 0.998542i \(0.517193\pi\)
\(710\) 5.74277e6 0.427539
\(711\) 8.38390e6 0.621974
\(712\) 2.51806e6 0.186151
\(713\) −4.89671e7 −3.60728
\(714\) 9.64210e6 0.707825
\(715\) −714457. −0.0522650
\(716\) −9.33574e6 −0.680559
\(717\) −7.32587e6 −0.532184
\(718\) 1.74699e7 1.26468
\(719\) −4.22567e6 −0.304841 −0.152420 0.988316i \(-0.548707\pi\)
−0.152420 + 0.988316i \(0.548707\pi\)
\(720\) 2.44081e6 0.175470
\(721\) 1.39175e7 0.997066
\(722\) 5.74393e6 0.410078
\(723\) −1.06610e7 −0.758498
\(724\) −1.53175e7 −1.08603
\(725\) 1.75312e6 0.123870
\(726\) 991694. 0.0698291
\(727\) 1.62930e7 1.14331 0.571657 0.820493i \(-0.306301\pi\)
0.571657 + 0.820493i \(0.306301\pi\)
\(728\) 3.06265e6 0.214175
\(729\) 531441. 0.0370370
\(730\) −1.57819e7 −1.09611
\(731\) 7.27665e6 0.503661
\(732\) 2.21506e6 0.152795
\(733\) −3.00376e6 −0.206493 −0.103247 0.994656i \(-0.532923\pi\)
−0.103247 + 0.994656i \(0.532923\pi\)
\(734\) −1.36521e7 −0.935319
\(735\) −8.55182e6 −0.583902
\(736\) 3.48412e7 2.37082
\(737\) −4.42468e6 −0.300064
\(738\) 5.09355e6 0.344255
\(739\) 5.58694e6 0.376325 0.188163 0.982138i \(-0.439747\pi\)
0.188163 + 0.982138i \(0.439747\pi\)
\(740\) 4.70013e6 0.315522
\(741\) −3.82578e6 −0.255961
\(742\) −6.08431e6 −0.405697
\(743\) −8.33428e6 −0.553855 −0.276927 0.960891i \(-0.589316\pi\)
−0.276927 + 0.960891i \(0.589316\pi\)
\(744\) −5.11346e6 −0.338675
\(745\) −9.29240e6 −0.613390
\(746\) −1.20789e6 −0.0794659
\(747\) −351665. −0.0230583
\(748\) −1.81282e6 −0.118468
\(749\) −3.24046e6 −0.211058
\(750\) −1.05834e6 −0.0687027
\(751\) −1.55555e7 −1.00643 −0.503215 0.864162i \(-0.667849\pi\)
−0.503215 + 0.864162i \(0.667849\pi\)
\(752\) −1.01073e7 −0.651766
\(753\) −5.41574e6 −0.348073
\(754\) 4.98594e6 0.319388
\(755\) 2.79313e6 0.178330
\(756\) −4.20564e6 −0.267625
\(757\) 2.42999e7 1.54122 0.770611 0.637306i \(-0.219951\pi\)
0.770611 + 0.637306i \(0.219951\pi\)
\(758\) −4.68450e6 −0.296135
\(759\) −5.19824e6 −0.327530
\(760\) −2.49209e6 −0.156505
\(761\) 1.84687e7 1.15605 0.578023 0.816021i \(-0.303825\pi\)
0.578023 + 0.816021i \(0.303825\pi\)
\(762\) −1.86204e7 −1.16172
\(763\) 3.10876e6 0.193320
\(764\) 209684. 0.0129966
\(765\) 1.23123e6 0.0760652
\(766\) 1.12194e7 0.690873
\(767\) −1.20530e7 −0.739790
\(768\) 1.21921e7 0.745890
\(769\) −1.51725e6 −0.0925210 −0.0462605 0.998929i \(-0.514730\pi\)
−0.0462605 + 0.998929i \(0.514730\pi\)
\(770\) 5.33016e6 0.323976
\(771\) 8.14113e6 0.493229
\(772\) −1.39119e7 −0.840122
\(773\) 3.28907e6 0.197981 0.0989907 0.995088i \(-0.468439\pi\)
0.0989907 + 0.995088i \(0.468439\pi\)
\(774\) −7.29569e6 −0.437738
\(775\) 6.41145e6 0.383444
\(776\) 1.01450e7 0.604779
\(777\) 1.60771e7 0.955334
\(778\) −1.24996e7 −0.740367
\(779\) −1.50383e7 −0.887879
\(780\) −1.30945e6 −0.0770639
\(781\) −3.69319e6 −0.216658
\(782\) 2.18427e7 1.27729
\(783\) 2.04484e6 0.119194
\(784\) −4.58126e7 −2.66192
\(785\) 6.11815e6 0.354361
\(786\) −8.08572e6 −0.466834
\(787\) −1.46769e7 −0.844692 −0.422346 0.906435i \(-0.638793\pi\)
−0.422346 + 0.906435i \(0.638793\pi\)
\(788\) −1.06693e7 −0.612098
\(789\) 1.32457e7 0.757497
\(790\) −1.94745e7 −1.11019
\(791\) −3.04564e7 −1.73076
\(792\) −542834. −0.0307506
\(793\) 2.35906e6 0.133216
\(794\) 1.85409e7 1.04371
\(795\) −776925. −0.0435975
\(796\) −1.51509e7 −0.847531
\(797\) 4.95422e6 0.276267 0.138134 0.990414i \(-0.455890\pi\)
0.138134 + 0.990414i \(0.455890\pi\)
\(798\) 2.85419e7 1.58663
\(799\) −5.09850e6 −0.282537
\(800\) −4.56190e6 −0.252012
\(801\) −3.68259e6 −0.202802
\(802\) −5.00283e6 −0.274650
\(803\) 1.01494e7 0.555459
\(804\) −8.10949e6 −0.442439
\(805\) −2.79395e7 −1.51960
\(806\) 1.82344e7 0.988675
\(807\) 8.07685e6 0.436574
\(808\) 1.59695e6 0.0860524
\(809\) 1.90776e7 1.02483 0.512414 0.858738i \(-0.328751\pi\)
0.512414 + 0.858738i \(0.328751\pi\)
\(810\) −1.23445e6 −0.0661092
\(811\) 2.54611e7 1.35933 0.679665 0.733523i \(-0.262125\pi\)
0.679665 + 0.733523i \(0.262125\pi\)
\(812\) −1.61822e7 −0.861284
\(813\) 1.20802e6 0.0640987
\(814\) −6.94809e6 −0.367540
\(815\) −1.08906e7 −0.574324
\(816\) 6.59577e6 0.346769
\(817\) 2.15399e7 1.12899
\(818\) −836308. −0.0437002
\(819\) −4.47905e6 −0.233333
\(820\) −5.14714e6 −0.267320
\(821\) −3.75556e7 −1.94454 −0.972269 0.233867i \(-0.924862\pi\)
−0.972269 + 0.233867i \(0.924862\pi\)
\(822\) −8.95970e6 −0.462502
\(823\) −3.42815e7 −1.76425 −0.882126 0.471014i \(-0.843888\pi\)
−0.882126 + 0.471014i \(0.843888\pi\)
\(824\) 3.29237e6 0.168924
\(825\) 680625. 0.0348155
\(826\) 8.99209e7 4.58575
\(827\) 2.92800e7 1.48870 0.744351 0.667789i \(-0.232759\pi\)
0.744351 + 0.667789i \(0.232759\pi\)
\(828\) −9.52725e6 −0.482938
\(829\) −7.40301e6 −0.374130 −0.187065 0.982348i \(-0.559897\pi\)
−0.187065 + 0.982348i \(0.559897\pi\)
\(830\) 816862. 0.0411579
\(831\) −1.81176e7 −0.910117
\(832\) −3.86440e6 −0.193541
\(833\) −2.31095e7 −1.15393
\(834\) −6.93094e6 −0.345046
\(835\) −7.21216e6 −0.357972
\(836\) −5.36619e6 −0.265552
\(837\) 7.47831e6 0.368969
\(838\) −4.02224e7 −1.97860
\(839\) −3.44573e7 −1.68996 −0.844981 0.534797i \(-0.820388\pi\)
−0.844981 + 0.534797i \(0.820388\pi\)
\(840\) −2.91762e6 −0.142669
\(841\) −1.26432e7 −0.616404
\(842\) −3.63950e6 −0.176914
\(843\) 1.30684e7 0.633364
\(844\) 1.78271e6 0.0861441
\(845\) 7.88775e6 0.380024
\(846\) 5.11184e6 0.245557
\(847\) −3.42784e6 −0.164177
\(848\) −4.16203e6 −0.198754
\(849\) −2.01974e7 −0.961672
\(850\) −2.85995e6 −0.135772
\(851\) 3.64203e7 1.72393
\(852\) −6.76883e6 −0.319459
\(853\) −1.70125e7 −0.800561 −0.400281 0.916393i \(-0.631087\pi\)
−0.400281 + 0.916393i \(0.631087\pi\)
\(854\) −1.75996e7 −0.825769
\(855\) 3.64461e6 0.170505
\(856\) −766572. −0.0357576
\(857\) 2.11650e7 0.984387 0.492194 0.870486i \(-0.336195\pi\)
0.492194 + 0.870486i \(0.336195\pi\)
\(858\) 1.93572e6 0.0897687
\(859\) −3.17802e7 −1.46952 −0.734758 0.678330i \(-0.762704\pi\)
−0.734758 + 0.678330i \(0.762704\pi\)
\(860\) 7.37244e6 0.339911
\(861\) −1.76061e7 −0.809386
\(862\) −1.81785e6 −0.0833277
\(863\) −3.53199e7 −1.61433 −0.807166 0.590325i \(-0.799000\pi\)
−0.807166 + 0.590325i \(0.799000\pi\)
\(864\) −5.32100e6 −0.242498
\(865\) −1.34908e7 −0.613053
\(866\) −7.82692e6 −0.354647
\(867\) −9.45157e6 −0.427028
\(868\) −5.91807e7 −2.66613
\(869\) 1.25241e7 0.562597
\(870\) −4.74984e6 −0.212756
\(871\) −8.63669e6 −0.385746
\(872\) 735417. 0.0327524
\(873\) −1.48368e7 −0.658876
\(874\) 6.46575e7 2.86313
\(875\) 3.65822e6 0.161529
\(876\) 1.86017e7 0.819016
\(877\) 256700. 0.0112701 0.00563503 0.999984i \(-0.498206\pi\)
0.00563503 + 0.999984i \(0.498206\pi\)
\(878\) −2.50606e7 −1.09712
\(879\) 1.33824e6 0.0584201
\(880\) 3.64614e6 0.158718
\(881\) 3.37169e7 1.46355 0.731776 0.681545i \(-0.238692\pi\)
0.731776 + 0.681545i \(0.238692\pi\)
\(882\) 2.31700e7 1.00289
\(883\) −2.45722e7 −1.06058 −0.530289 0.847817i \(-0.677917\pi\)
−0.530289 + 0.847817i \(0.677917\pi\)
\(884\) −3.53850e6 −0.152296
\(885\) 1.14823e7 0.492800
\(886\) 3.85542e7 1.65001
\(887\) 220190. 0.00939697 0.00469849 0.999989i \(-0.498504\pi\)
0.00469849 + 0.999989i \(0.498504\pi\)
\(888\) 3.80325e6 0.161853
\(889\) 6.43623e7 2.73135
\(890\) 8.55408e6 0.361991
\(891\) 793881. 0.0335013
\(892\) 9.12622e6 0.384042
\(893\) −1.50923e7 −0.633323
\(894\) 2.51765e7 1.05354
\(895\) 9.47184e6 0.395254
\(896\) −2.58546e7 −1.07589
\(897\) −1.01466e7 −0.421056
\(898\) 4.38260e7 1.81360
\(899\) 2.87745e7 1.18743
\(900\) 1.24744e6 0.0513349
\(901\) −2.09948e6 −0.0861587
\(902\) 7.60889e6 0.311390
\(903\) 2.52179e7 1.02918
\(904\) −7.20487e6 −0.293228
\(905\) 1.55408e7 0.630743
\(906\) −7.56760e6 −0.306293
\(907\) −6.02862e6 −0.243332 −0.121666 0.992571i \(-0.538824\pi\)
−0.121666 + 0.992571i \(0.538824\pi\)
\(908\) 8.02011e6 0.322824
\(909\) −2.33550e6 −0.0937497
\(910\) 1.04041e7 0.416487
\(911\) 1.68076e7 0.670979 0.335490 0.942044i \(-0.391098\pi\)
0.335490 + 0.942044i \(0.391098\pi\)
\(912\) 1.95244e7 0.777303
\(913\) −525327. −0.0208570
\(914\) −4.00705e7 −1.58657
\(915\) −2.24735e6 −0.0887398
\(916\) −1.41302e6 −0.0556430
\(917\) 2.79487e7 1.09758
\(918\) −3.33585e6 −0.130647
\(919\) 4.57189e7 1.78569 0.892846 0.450362i \(-0.148705\pi\)
0.892846 + 0.450362i \(0.148705\pi\)
\(920\) −6.60944e6 −0.257451
\(921\) 1.82074e7 0.707292
\(922\) 5.07022e6 0.196426
\(923\) −7.20887e6 −0.278524
\(924\) −6.28250e6 −0.242076
\(925\) −4.76865e6 −0.183249
\(926\) −1.44672e7 −0.554443
\(927\) −4.81501e6 −0.184034
\(928\) −2.04737e7 −0.780418
\(929\) −2.84053e7 −1.07984 −0.539921 0.841716i \(-0.681546\pi\)
−0.539921 + 0.841716i \(0.681546\pi\)
\(930\) −1.73709e7 −0.658591
\(931\) −6.84073e7 −2.58659
\(932\) −2.05477e7 −0.774861
\(933\) 1.68886e7 0.635171
\(934\) −5.98068e7 −2.24328
\(935\) 1.83925e6 0.0688035
\(936\) −1.05958e6 −0.0395315
\(937\) −8.99373e6 −0.334650 −0.167325 0.985902i \(-0.553513\pi\)
−0.167325 + 0.985902i \(0.553513\pi\)
\(938\) 6.44334e7 2.39113
\(939\) −805452. −0.0298109
\(940\) −5.16562e6 −0.190679
\(941\) −2.96562e7 −1.09180 −0.545898 0.837852i \(-0.683811\pi\)
−0.545898 + 0.837852i \(0.683811\pi\)
\(942\) −1.65763e7 −0.608639
\(943\) −3.98841e7 −1.46056
\(944\) 6.15113e7 2.24659
\(945\) 4.26695e6 0.155431
\(946\) −1.08985e7 −0.395949
\(947\) −1.62357e7 −0.588297 −0.294148 0.955760i \(-0.595036\pi\)
−0.294148 + 0.955760i \(0.595036\pi\)
\(948\) 2.29540e7 0.829540
\(949\) 1.98110e7 0.714070
\(950\) −8.46586e6 −0.304342
\(951\) 2.34103e6 0.0839375
\(952\) −7.88427e6 −0.281948
\(953\) 2.13990e7 0.763241 0.381621 0.924319i \(-0.375366\pi\)
0.381621 + 0.924319i \(0.375366\pi\)
\(954\) 2.10497e6 0.0748816
\(955\) −212740. −0.00754817
\(956\) −2.00572e7 −0.709785
\(957\) 3.05464e6 0.107815
\(958\) −1.35728e7 −0.477809
\(959\) 3.09697e7 1.08740
\(960\) 3.68141e6 0.128925
\(961\) 7.66039e7 2.67573
\(962\) −1.35622e7 −0.472490
\(963\) 1.12109e6 0.0389561
\(964\) −2.91885e7 −1.01162
\(965\) 1.41147e7 0.487925
\(966\) 7.56981e7 2.61001
\(967\) −6.31736e6 −0.217255 −0.108627 0.994083i \(-0.534646\pi\)
−0.108627 + 0.994083i \(0.534646\pi\)
\(968\) −810900. −0.0278150
\(969\) 9.84880e6 0.336956
\(970\) 3.44635e7 1.17606
\(971\) 4.14293e7 1.41013 0.705065 0.709142i \(-0.250918\pi\)
0.705065 + 0.709142i \(0.250918\pi\)
\(972\) 1.45501e6 0.0493971
\(973\) 2.39572e7 0.811247
\(974\) 2.29517e7 0.775207
\(975\) 1.32854e6 0.0447571
\(976\) −1.20392e7 −0.404550
\(977\) −333761. −0.0111866 −0.00559331 0.999984i \(-0.501780\pi\)
−0.00559331 + 0.999984i \(0.501780\pi\)
\(978\) 2.95065e7 0.986440
\(979\) −5.50116e6 −0.183441
\(980\) −2.34137e7 −0.778763
\(981\) −1.07553e6 −0.0356820
\(982\) −5.66948e7 −1.87614
\(983\) −3.43832e7 −1.13491 −0.567457 0.823403i \(-0.692073\pi\)
−0.567457 + 0.823403i \(0.692073\pi\)
\(984\) −4.16496e6 −0.137127
\(985\) 1.08249e7 0.355493
\(986\) −1.28354e7 −0.420454
\(987\) −1.76693e7 −0.577335
\(988\) −1.04745e7 −0.341381
\(989\) 5.71275e7 1.85718
\(990\) −1.84406e6 −0.0597980
\(991\) 4.26710e6 0.138022 0.0690111 0.997616i \(-0.478016\pi\)
0.0690111 + 0.997616i \(0.478016\pi\)
\(992\) −7.48758e7 −2.41581
\(993\) 1.97405e7 0.635308
\(994\) 5.37813e7 1.72649
\(995\) 1.53718e7 0.492228
\(996\) −962811. −0.0307534
\(997\) 2.61381e7 0.832790 0.416395 0.909184i \(-0.363293\pi\)
0.416395 + 0.909184i \(0.363293\pi\)
\(998\) 4.52945e7 1.43953
\(999\) −5.56215e6 −0.176331
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 165.6.a.b.1.3 3
3.2 odd 2 495.6.a.d.1.1 3
5.4 even 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 1.1 even 1 trivial
495.6.a.d.1.1 3 3.2 odd 2
825.6.a.i.1.1 3 5.4 even 2