Properties

Label 165.6.a.c.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: 3.3.18257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 26x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.78415\) 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)
 
\(f(q)\) \(=\) \(q+5.78415 q^{2} -9.00000 q^{3} +1.45634 q^{4} +25.0000 q^{5} -52.0573 q^{6} +17.6498 q^{7} -176.669 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.78415 q^{2} -9.00000 q^{3} +1.45634 q^{4} +25.0000 q^{5} -52.0573 q^{6} +17.6498 q^{7} -176.669 q^{8} +81.0000 q^{9} +144.604 q^{10} +121.000 q^{11} -13.1070 q^{12} +674.396 q^{13} +102.089 q^{14} -225.000 q^{15} -1068.48 q^{16} -2117.62 q^{17} +468.516 q^{18} -2307.79 q^{19} +36.4084 q^{20} -158.848 q^{21} +699.882 q^{22} -3072.47 q^{23} +1590.02 q^{24} +625.000 q^{25} +3900.80 q^{26} -729.000 q^{27} +25.7040 q^{28} -1437.44 q^{29} -1301.43 q^{30} -5157.66 q^{31} -526.846 q^{32} -1089.00 q^{33} -12248.6 q^{34} +441.244 q^{35} +117.963 q^{36} +6928.88 q^{37} -13348.6 q^{38} -6069.56 q^{39} -4416.72 q^{40} +2844.78 q^{41} -918.799 q^{42} -11665.3 q^{43} +176.217 q^{44} +2025.00 q^{45} -17771.6 q^{46} +3451.40 q^{47} +9616.34 q^{48} -16495.5 q^{49} +3615.09 q^{50} +19058.6 q^{51} +982.146 q^{52} -18167.6 q^{53} -4216.64 q^{54} +3025.00 q^{55} -3118.17 q^{56} +20770.1 q^{57} -8314.36 q^{58} -8976.88 q^{59} -327.675 q^{60} -378.820 q^{61} -29832.7 q^{62} +1429.63 q^{63} +31144.1 q^{64} +16859.9 q^{65} -6298.93 q^{66} -12233.7 q^{67} -3083.97 q^{68} +27652.2 q^{69} +2552.22 q^{70} +55867.0 q^{71} -14310.2 q^{72} +19739.6 q^{73} +40077.6 q^{74} -5625.00 q^{75} -3360.92 q^{76} +2135.62 q^{77} -35107.2 q^{78} -13495.8 q^{79} -26712.0 q^{80} +6561.00 q^{81} +16454.6 q^{82} +42078.3 q^{83} -231.336 q^{84} -52940.5 q^{85} -67474.0 q^{86} +12937.0 q^{87} -21376.9 q^{88} -81223.2 q^{89} +11712.9 q^{90} +11902.9 q^{91} -4474.55 q^{92} +46419.0 q^{93} +19963.4 q^{94} -57694.8 q^{95} +4741.62 q^{96} +152101. q^{97} -95412.3 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 378 q^{12} + 290 q^{13} + 916 q^{14} - 675 q^{15} - 590 q^{16} + 434 q^{17} + 162 q^{18} - 2856 q^{19} - 1050 q^{20} + 612 q^{21} + 242 q^{22} - 640 q^{23} + 216 q^{24} + 1875 q^{25} + 2132 q^{26} - 2187 q^{27} - 580 q^{28} - 4538 q^{29} - 450 q^{30} - 14968 q^{31} - 2496 q^{32} - 3267 q^{33} - 13704 q^{34} - 1700 q^{35} - 3402 q^{36} - 6190 q^{37} - 11668 q^{38} - 2610 q^{39} - 600 q^{40} - 8926 q^{41} - 8244 q^{42} - 33592 q^{43} - 5082 q^{44} + 6075 q^{45} - 35680 q^{46} - 24640 q^{47} + 5310 q^{48} - 14693 q^{49} + 1250 q^{50} - 3906 q^{51} + 18780 q^{52} - 22934 q^{53} - 1458 q^{54} + 9075 q^{55} - 40012 q^{56} + 25704 q^{57} - 32304 q^{58} - 13756 q^{59} + 9450 q^{60} + 24602 q^{61} - 7704 q^{62} - 5508 q^{63} + 35474 q^{64} + 7250 q^{65} - 2178 q^{66} + 16868 q^{67} - 71288 q^{68} + 5760 q^{69} + 22900 q^{70} + 4856 q^{71} - 1944 q^{72} + 1910 q^{73} + 29404 q^{74} - 16875 q^{75} + 6116 q^{76} - 8228 q^{77} - 19188 q^{78} - 36844 q^{79} - 14750 q^{80} + 19683 q^{81} + 84000 q^{82} - 48796 q^{83} + 5220 q^{84} + 10850 q^{85} - 83492 q^{86} + 40842 q^{87} - 2904 q^{88} - 188978 q^{89} + 4050 q^{90} - 93208 q^{91} - 6976 q^{92} + 134712 q^{93} + 70472 q^{94} - 71400 q^{95} + 22464 q^{96} + 247526 q^{97} - 154654 q^{98} + 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.78415 1.02250 0.511251 0.859431i \(-0.329182\pi\)
0.511251 + 0.859431i \(0.329182\pi\)
\(3\) −9.00000 −0.577350
\(4\) 1.45634 0.0455105
\(5\) 25.0000 0.447214
\(6\) −52.0573 −0.590342
\(7\) 17.6498 0.136143 0.0680713 0.997680i \(-0.478315\pi\)
0.0680713 + 0.997680i \(0.478315\pi\)
\(8\) −176.669 −0.975968
\(9\) 81.0000 0.333333
\(10\) 144.604 0.457277
\(11\) 121.000 0.301511
\(12\) −13.1070 −0.0262755
\(13\) 674.396 1.10677 0.553384 0.832926i \(-0.313336\pi\)
0.553384 + 0.832926i \(0.313336\pi\)
\(14\) 102.089 0.139206
\(15\) −225.000 −0.258199
\(16\) −1068.48 −1.04344
\(17\) −2117.62 −1.77716 −0.888579 0.458724i \(-0.848307\pi\)
−0.888579 + 0.458724i \(0.848307\pi\)
\(18\) 468.516 0.340834
\(19\) −2307.79 −1.46660 −0.733302 0.679903i \(-0.762022\pi\)
−0.733302 + 0.679903i \(0.762022\pi\)
\(20\) 36.4084 0.0203529
\(21\) −158.848 −0.0786019
\(22\) 699.882 0.308296
\(23\) −3072.47 −1.21107 −0.605534 0.795820i \(-0.707040\pi\)
−0.605534 + 0.795820i \(0.707040\pi\)
\(24\) 1590.02 0.563475
\(25\) 625.000 0.200000
\(26\) 3900.80 1.13167
\(27\) −729.000 −0.192450
\(28\) 25.7040 0.00619591
\(29\) −1437.44 −0.317391 −0.158696 0.987328i \(-0.550729\pi\)
−0.158696 + 0.987328i \(0.550729\pi\)
\(30\) −1301.43 −0.264009
\(31\) −5157.66 −0.963937 −0.481969 0.876188i \(-0.660078\pi\)
−0.481969 + 0.876188i \(0.660078\pi\)
\(32\) −526.846 −0.0909513
\(33\) −1089.00 −0.174078
\(34\) −12248.6 −1.81715
\(35\) 441.244 0.0608848
\(36\) 117.963 0.0151702
\(37\) 6928.88 0.832067 0.416034 0.909349i \(-0.363420\pi\)
0.416034 + 0.909349i \(0.363420\pi\)
\(38\) −13348.6 −1.49961
\(39\) −6069.56 −0.638993
\(40\) −4416.72 −0.436466
\(41\) 2844.78 0.264295 0.132147 0.991230i \(-0.457813\pi\)
0.132147 + 0.991230i \(0.457813\pi\)
\(42\) −918.799 −0.0803706
\(43\) −11665.3 −0.962113 −0.481057 0.876689i \(-0.659747\pi\)
−0.481057 + 0.876689i \(0.659747\pi\)
\(44\) 176.217 0.0137219
\(45\) 2025.00 0.149071
\(46\) −17771.6 −1.23832
\(47\) 3451.40 0.227903 0.113952 0.993486i \(-0.463649\pi\)
0.113952 + 0.993486i \(0.463649\pi\)
\(48\) 9616.34 0.602430
\(49\) −16495.5 −0.981465
\(50\) 3615.09 0.204500
\(51\) 19058.6 1.02604
\(52\) 982.146 0.0503695
\(53\) −18167.6 −0.888401 −0.444200 0.895927i \(-0.646512\pi\)
−0.444200 + 0.895927i \(0.646512\pi\)
\(54\) −4216.64 −0.196781
\(55\) 3025.00 0.134840
\(56\) −3118.17 −0.132871
\(57\) 20770.1 0.846744
\(58\) −8314.36 −0.324533
\(59\) −8976.88 −0.335734 −0.167867 0.985810i \(-0.553688\pi\)
−0.167867 + 0.985810i \(0.553688\pi\)
\(60\) −327.675 −0.0117508
\(61\) −378.820 −0.0130349 −0.00651746 0.999979i \(-0.502075\pi\)
−0.00651746 + 0.999979i \(0.502075\pi\)
\(62\) −29832.7 −0.985628
\(63\) 1429.63 0.0453808
\(64\) 31144.1 0.950441
\(65\) 16859.9 0.494962
\(66\) −6298.93 −0.177995
\(67\) −12233.7 −0.332945 −0.166473 0.986046i \(-0.553238\pi\)
−0.166473 + 0.986046i \(0.553238\pi\)
\(68\) −3083.97 −0.0808793
\(69\) 27652.2 0.699210
\(70\) 2552.22 0.0622548
\(71\) 55867.0 1.31525 0.657627 0.753344i \(-0.271560\pi\)
0.657627 + 0.753344i \(0.271560\pi\)
\(72\) −14310.2 −0.325323
\(73\) 19739.6 0.433541 0.216771 0.976223i \(-0.430448\pi\)
0.216771 + 0.976223i \(0.430448\pi\)
\(74\) 40077.6 0.850791
\(75\) −5625.00 −0.115470
\(76\) −3360.92 −0.0667459
\(77\) 2135.62 0.0410485
\(78\) −35107.2 −0.653371
\(79\) −13495.8 −0.243293 −0.121647 0.992573i \(-0.538817\pi\)
−0.121647 + 0.992573i \(0.538817\pi\)
\(80\) −26712.0 −0.466640
\(81\) 6561.00 0.111111
\(82\) 16454.6 0.270242
\(83\) 42078.3 0.670444 0.335222 0.942139i \(-0.391189\pi\)
0.335222 + 0.942139i \(0.391189\pi\)
\(84\) −231.336 −0.00357721
\(85\) −52940.5 −0.794769
\(86\) −67474.0 −0.983763
\(87\) 12937.0 0.183246
\(88\) −21376.9 −0.294265
\(89\) −81223.2 −1.08694 −0.543469 0.839429i \(-0.682890\pi\)
−0.543469 + 0.839429i \(0.682890\pi\)
\(90\) 11712.9 0.152426
\(91\) 11902.9 0.150678
\(92\) −4474.55 −0.0551163
\(93\) 46419.0 0.556530
\(94\) 19963.4 0.233031
\(95\) −57694.8 −0.655885
\(96\) 4741.62 0.0525108
\(97\) 152101. 1.64136 0.820680 0.571388i \(-0.193595\pi\)
0.820680 + 0.571388i \(0.193595\pi\)
\(98\) −95412.3 −1.00355
\(99\) 9801.00 0.100504
\(100\) 910.210 0.00910210
\(101\) 95207.6 0.928684 0.464342 0.885656i \(-0.346291\pi\)
0.464342 + 0.885656i \(0.346291\pi\)
\(102\) 110238. 1.04913
\(103\) −150824. −1.40081 −0.700403 0.713748i \(-0.746996\pi\)
−0.700403 + 0.713748i \(0.746996\pi\)
\(104\) −119145. −1.08017
\(105\) −3971.20 −0.0351518
\(106\) −105084. −0.908392
\(107\) 192242. 1.62327 0.811633 0.584168i \(-0.198579\pi\)
0.811633 + 0.584168i \(0.198579\pi\)
\(108\) −1061.67 −0.00875850
\(109\) 96302.8 0.776377 0.388188 0.921580i \(-0.373101\pi\)
0.388188 + 0.921580i \(0.373101\pi\)
\(110\) 17497.0 0.137874
\(111\) −62359.9 −0.480394
\(112\) −18858.5 −0.142056
\(113\) −210371. −1.54985 −0.774926 0.632052i \(-0.782213\pi\)
−0.774926 + 0.632052i \(0.782213\pi\)
\(114\) 120138. 0.865798
\(115\) −76811.8 −0.541606
\(116\) −2093.39 −0.0144446
\(117\) 54626.1 0.368923
\(118\) −51923.6 −0.343289
\(119\) −37375.5 −0.241947
\(120\) 39750.5 0.251994
\(121\) 14641.0 0.0909091
\(122\) −2191.15 −0.0133282
\(123\) −25603.0 −0.152591
\(124\) −7511.29 −0.0438692
\(125\) 15625.0 0.0894427
\(126\) 8269.19 0.0464020
\(127\) −63529.8 −0.349517 −0.174758 0.984611i \(-0.555914\pi\)
−0.174758 + 0.984611i \(0.555914\pi\)
\(128\) 197001. 1.06278
\(129\) 104988. 0.555476
\(130\) 97520.1 0.506099
\(131\) −88396.0 −0.450043 −0.225022 0.974354i \(-0.572245\pi\)
−0.225022 + 0.974354i \(0.572245\pi\)
\(132\) −1585.95 −0.00792236
\(133\) −40732.0 −0.199667
\(134\) −70761.8 −0.340437
\(135\) −18225.0 −0.0860663
\(136\) 374118. 1.73445
\(137\) −42642.4 −0.194106 −0.0970532 0.995279i \(-0.530942\pi\)
−0.0970532 + 0.995279i \(0.530942\pi\)
\(138\) 159945. 0.714944
\(139\) 169503. 0.744114 0.372057 0.928210i \(-0.378653\pi\)
0.372057 + 0.928210i \(0.378653\pi\)
\(140\) 642.599 0.00277090
\(141\) −31062.6 −0.131580
\(142\) 323143. 1.34485
\(143\) 81601.9 0.333703
\(144\) −86547.0 −0.347813
\(145\) −35936.0 −0.141942
\(146\) 114176. 0.443297
\(147\) 148459. 0.566649
\(148\) 10090.8 0.0378678
\(149\) −72462.8 −0.267393 −0.133696 0.991022i \(-0.542685\pi\)
−0.133696 + 0.991022i \(0.542685\pi\)
\(150\) −32535.8 −0.118068
\(151\) −61316.2 −0.218843 −0.109422 0.993995i \(-0.534900\pi\)
−0.109422 + 0.993995i \(0.534900\pi\)
\(152\) 407716. 1.43136
\(153\) −171527. −0.592386
\(154\) 12352.7 0.0419722
\(155\) −128942. −0.431086
\(156\) −8839.32 −0.0290809
\(157\) 505789. 1.63765 0.818824 0.574045i \(-0.194627\pi\)
0.818824 + 0.574045i \(0.194627\pi\)
\(158\) −78061.5 −0.248768
\(159\) 163509. 0.512919
\(160\) −13171.2 −0.0406747
\(161\) −54228.4 −0.164878
\(162\) 37949.8 0.113611
\(163\) 387702. 1.14295 0.571477 0.820618i \(-0.306371\pi\)
0.571477 + 0.820618i \(0.306371\pi\)
\(164\) 4142.95 0.0120282
\(165\) −27225.0 −0.0778499
\(166\) 243387. 0.685530
\(167\) −635422. −1.76308 −0.881538 0.472113i \(-0.843491\pi\)
−0.881538 + 0.472113i \(0.843491\pi\)
\(168\) 28063.5 0.0767129
\(169\) 83516.8 0.224935
\(170\) −306216. −0.812653
\(171\) −186931. −0.488868
\(172\) −16988.6 −0.0437862
\(173\) −679820. −1.72695 −0.863473 0.504395i \(-0.831716\pi\)
−0.863473 + 0.504395i \(0.831716\pi\)
\(174\) 74829.3 0.187369
\(175\) 11031.1 0.0272285
\(176\) −129286. −0.314609
\(177\) 80792.0 0.193836
\(178\) −469807. −1.11140
\(179\) −63083.3 −0.147157 −0.0735787 0.997289i \(-0.523442\pi\)
−0.0735787 + 0.997289i \(0.523442\pi\)
\(180\) 2949.08 0.00678430
\(181\) 523614. 1.18800 0.593998 0.804466i \(-0.297549\pi\)
0.593998 + 0.804466i \(0.297549\pi\)
\(182\) 68848.3 0.154069
\(183\) 3409.38 0.00752571
\(184\) 542810. 1.18196
\(185\) 173222. 0.372112
\(186\) 268494. 0.569053
\(187\) −256232. −0.535833
\(188\) 5026.39 0.0103720
\(189\) −12866.7 −0.0262006
\(190\) −333715. −0.670644
\(191\) −684311. −1.35728 −0.678641 0.734470i \(-0.737431\pi\)
−0.678641 + 0.734470i \(0.737431\pi\)
\(192\) −280297. −0.548738
\(193\) 527417. 1.01920 0.509601 0.860411i \(-0.329793\pi\)
0.509601 + 0.860411i \(0.329793\pi\)
\(194\) 879777. 1.67829
\(195\) −151739. −0.285766
\(196\) −24023.0 −0.0446669
\(197\) −467139. −0.857591 −0.428796 0.903402i \(-0.641062\pi\)
−0.428796 + 0.903402i \(0.641062\pi\)
\(198\) 56690.4 0.102765
\(199\) −436613. −0.781563 −0.390782 0.920483i \(-0.627795\pi\)
−0.390782 + 0.920483i \(0.627795\pi\)
\(200\) −110418. −0.195194
\(201\) 110104. 0.192226
\(202\) 550694. 0.949581
\(203\) −25370.5 −0.0432104
\(204\) 27755.7 0.0466957
\(205\) 71119.4 0.118196
\(206\) −872389. −1.43233
\(207\) −248870. −0.403689
\(208\) −720580. −1.15484
\(209\) −279243. −0.442198
\(210\) −22970.0 −0.0359428
\(211\) −747312. −1.15557 −0.577785 0.816189i \(-0.696083\pi\)
−0.577785 + 0.816189i \(0.696083\pi\)
\(212\) −26458.2 −0.0404315
\(213\) −502803. −0.759362
\(214\) 1.11196e6 1.65979
\(215\) −291634. −0.430270
\(216\) 128792. 0.187825
\(217\) −91031.6 −0.131233
\(218\) 557029. 0.793847
\(219\) −177656. −0.250305
\(220\) 4405.41 0.00613663
\(221\) −1.42812e6 −1.96690
\(222\) −360699. −0.491204
\(223\) 870654. 1.17242 0.586211 0.810159i \(-0.300619\pi\)
0.586211 + 0.810159i \(0.300619\pi\)
\(224\) −9298.71 −0.0123823
\(225\) 50625.0 0.0666667
\(226\) −1.21682e6 −1.58473
\(227\) 82323.8 0.106038 0.0530189 0.998594i \(-0.483116\pi\)
0.0530189 + 0.998594i \(0.483116\pi\)
\(228\) 30248.3 0.0385357
\(229\) −340134. −0.428609 −0.214305 0.976767i \(-0.568748\pi\)
−0.214305 + 0.976767i \(0.568748\pi\)
\(230\) −444291. −0.553793
\(231\) −19220.6 −0.0236994
\(232\) 253951. 0.309763
\(233\) −248384. −0.299732 −0.149866 0.988706i \(-0.547884\pi\)
−0.149866 + 0.988706i \(0.547884\pi\)
\(234\) 315965. 0.377224
\(235\) 86284.9 0.101921
\(236\) −13073.4 −0.0152794
\(237\) 121462. 0.140465
\(238\) −216185. −0.247391
\(239\) −1.37978e6 −1.56248 −0.781242 0.624229i \(-0.785413\pi\)
−0.781242 + 0.624229i \(0.785413\pi\)
\(240\) 240408. 0.269415
\(241\) −1.53382e6 −1.70110 −0.850552 0.525891i \(-0.823732\pi\)
−0.850552 + 0.525891i \(0.823732\pi\)
\(242\) 84685.7 0.0929547
\(243\) −59049.0 −0.0641500
\(244\) −551.689 −0.000593225 0
\(245\) −412387. −0.438925
\(246\) −148091. −0.156024
\(247\) −1.55637e6 −1.62319
\(248\) 911199. 0.940772
\(249\) −378704. −0.387081
\(250\) 90377.3 0.0914554
\(251\) 338591. 0.339227 0.169614 0.985511i \(-0.445748\pi\)
0.169614 + 0.985511i \(0.445748\pi\)
\(252\) 2082.02 0.00206530
\(253\) −371769. −0.365151
\(254\) −367466. −0.357382
\(255\) 476465. 0.458860
\(256\) 142872. 0.136253
\(257\) 1.60985e6 1.52038 0.760189 0.649702i \(-0.225106\pi\)
0.760189 + 0.649702i \(0.225106\pi\)
\(258\) 607266. 0.567976
\(259\) 122293. 0.113280
\(260\) 24553.7 0.0225259
\(261\) −116433. −0.105797
\(262\) −511295. −0.460170
\(263\) 458118. 0.408402 0.204201 0.978929i \(-0.434540\pi\)
0.204201 + 0.978929i \(0.434540\pi\)
\(264\) 192393. 0.169894
\(265\) −454191. −0.397305
\(266\) −235600. −0.204160
\(267\) 731009. 0.627544
\(268\) −17816.4 −0.0151525
\(269\) −692380. −0.583397 −0.291698 0.956510i \(-0.594220\pi\)
−0.291698 + 0.956510i \(0.594220\pi\)
\(270\) −105416. −0.0880030
\(271\) −486690. −0.402558 −0.201279 0.979534i \(-0.564510\pi\)
−0.201279 + 0.979534i \(0.564510\pi\)
\(272\) 2.26264e6 1.85436
\(273\) −107126. −0.0869941
\(274\) −246650. −0.198474
\(275\) 75625.0 0.0603023
\(276\) 40270.9 0.0318214
\(277\) 2.38271e6 1.86583 0.932915 0.360098i \(-0.117257\pi\)
0.932915 + 0.360098i \(0.117257\pi\)
\(278\) 980428. 0.760858
\(279\) −417771. −0.321312
\(280\) −77954.1 −0.0594216
\(281\) 1.12154e6 0.847320 0.423660 0.905821i \(-0.360745\pi\)
0.423660 + 0.905821i \(0.360745\pi\)
\(282\) −179670. −0.134541
\(283\) 965956. 0.716954 0.358477 0.933539i \(-0.383296\pi\)
0.358477 + 0.933539i \(0.383296\pi\)
\(284\) 81361.1 0.0598578
\(285\) 519254. 0.378676
\(286\) 471997. 0.341212
\(287\) 50209.6 0.0359817
\(288\) −42674.5 −0.0303171
\(289\) 3.06446e6 2.15829
\(290\) −207859. −0.145136
\(291\) −1.36891e6 −0.947640
\(292\) 28747.4 0.0197307
\(293\) 2.20819e6 1.50269 0.751343 0.659912i \(-0.229406\pi\)
0.751343 + 0.659912i \(0.229406\pi\)
\(294\) 858711. 0.579400
\(295\) −224422. −0.150145
\(296\) −1.22412e6 −0.812071
\(297\) −88209.0 −0.0580259
\(298\) −419135. −0.273410
\(299\) −2.07206e6 −1.34037
\(300\) −8191.89 −0.00525510
\(301\) −205891. −0.130985
\(302\) −354662. −0.223768
\(303\) −856868. −0.536176
\(304\) 2.46584e6 1.53031
\(305\) −9470.49 −0.00582939
\(306\) −992139. −0.605716
\(307\) 805480. 0.487763 0.243881 0.969805i \(-0.421579\pi\)
0.243881 + 0.969805i \(0.421579\pi\)
\(308\) 3110.18 0.00186814
\(309\) 1.35742e6 0.808756
\(310\) −745817. −0.440786
\(311\) −870766. −0.510505 −0.255253 0.966874i \(-0.582159\pi\)
−0.255253 + 0.966874i \(0.582159\pi\)
\(312\) 1.07230e6 0.623636
\(313\) −3.16630e6 −1.82680 −0.913400 0.407063i \(-0.866553\pi\)
−0.913400 + 0.407063i \(0.866553\pi\)
\(314\) 2.92556e6 1.67450
\(315\) 35740.8 0.0202949
\(316\) −19654.4 −0.0110724
\(317\) 1.72288e6 0.962955 0.481478 0.876458i \(-0.340100\pi\)
0.481478 + 0.876458i \(0.340100\pi\)
\(318\) 945759. 0.524460
\(319\) −173930. −0.0956970
\(320\) 778602. 0.425050
\(321\) −1.73018e6 −0.937193
\(322\) −313665. −0.168588
\(323\) 4.88703e6 2.60639
\(324\) 9555.02 0.00505672
\(325\) 421497. 0.221354
\(326\) 2.24252e6 1.16867
\(327\) −866725. −0.448241
\(328\) −502584. −0.257943
\(329\) 60916.3 0.0310273
\(330\) −157473. −0.0796017
\(331\) −3.20493e6 −1.60786 −0.803932 0.594722i \(-0.797262\pi\)
−0.803932 + 0.594722i \(0.797262\pi\)
\(332\) 61280.1 0.0305122
\(333\) 561239. 0.277356
\(334\) −3.67537e6 −1.80275
\(335\) −305844. −0.148898
\(336\) 169726. 0.0820163
\(337\) 2.44638e6 1.17341 0.586703 0.809802i \(-0.300425\pi\)
0.586703 + 0.809802i \(0.300425\pi\)
\(338\) 483073. 0.229996
\(339\) 1.89334e6 0.894807
\(340\) −77099.2 −0.0361703
\(341\) −624077. −0.290638
\(342\) −1.08124e6 −0.499869
\(343\) −587781. −0.269762
\(344\) 2.06090e6 0.938992
\(345\) 691306. 0.312696
\(346\) −3.93218e6 −1.76581
\(347\) −3.05055e6 −1.36005 −0.680025 0.733189i \(-0.738031\pi\)
−0.680025 + 0.733189i \(0.738031\pi\)
\(348\) 18840.5 0.00833960
\(349\) −3.08755e6 −1.35691 −0.678454 0.734643i \(-0.737350\pi\)
−0.678454 + 0.734643i \(0.737350\pi\)
\(350\) 63805.5 0.0278412
\(351\) −491635. −0.212998
\(352\) −63748.4 −0.0274228
\(353\) 2.98932e6 1.27684 0.638418 0.769690i \(-0.279589\pi\)
0.638418 + 0.769690i \(0.279589\pi\)
\(354\) 467312. 0.198198
\(355\) 1.39668e6 0.588200
\(356\) −118288. −0.0494671
\(357\) 336380. 0.139688
\(358\) −364883. −0.150469
\(359\) −2.29991e6 −0.941836 −0.470918 0.882177i \(-0.656077\pi\)
−0.470918 + 0.882177i \(0.656077\pi\)
\(360\) −357755. −0.145489
\(361\) 2.84981e6 1.15093
\(362\) 3.02866e6 1.21473
\(363\) −131769. −0.0524864
\(364\) 17334.7 0.00685743
\(365\) 493489. 0.193885
\(366\) 19720.3 0.00769505
\(367\) 3.58207e6 1.38825 0.694127 0.719852i \(-0.255790\pi\)
0.694127 + 0.719852i \(0.255790\pi\)
\(368\) 3.28288e6 1.26368
\(369\) 230427. 0.0880982
\(370\) 1.00194e6 0.380485
\(371\) −320655. −0.120949
\(372\) 67601.6 0.0253279
\(373\) 511254. 0.190268 0.0951338 0.995464i \(-0.469672\pi\)
0.0951338 + 0.995464i \(0.469672\pi\)
\(374\) −1.48208e6 −0.547891
\(375\) −140625. −0.0516398
\(376\) −609755. −0.222426
\(377\) −969404. −0.351278
\(378\) −74422.7 −0.0267902
\(379\) 2.45209e6 0.876877 0.438438 0.898761i \(-0.355532\pi\)
0.438438 + 0.898761i \(0.355532\pi\)
\(380\) −84023.0 −0.0298497
\(381\) 571768. 0.201794
\(382\) −3.95815e6 −1.38782
\(383\) −1.46315e6 −0.509674 −0.254837 0.966984i \(-0.582022\pi\)
−0.254837 + 0.966984i \(0.582022\pi\)
\(384\) −1.77301e6 −0.613596
\(385\) 53390.5 0.0183575
\(386\) 3.05065e6 1.04214
\(387\) −944893. −0.320704
\(388\) 221511. 0.0746991
\(389\) −803978. −0.269383 −0.134691 0.990888i \(-0.543004\pi\)
−0.134691 + 0.990888i \(0.543004\pi\)
\(390\) −877681. −0.292197
\(391\) 6.50633e6 2.15226
\(392\) 2.91424e6 0.957878
\(393\) 795564. 0.259833
\(394\) −2.70200e6 −0.876889
\(395\) −337394. −0.108804
\(396\) 14273.5 0.00457397
\(397\) −2.87304e6 −0.914883 −0.457441 0.889240i \(-0.651234\pi\)
−0.457441 + 0.889240i \(0.651234\pi\)
\(398\) −2.52543e6 −0.799150
\(399\) 366588. 0.115278
\(400\) −667801. −0.208688
\(401\) −3.45072e6 −1.07164 −0.535820 0.844332i \(-0.679998\pi\)
−0.535820 + 0.844332i \(0.679998\pi\)
\(402\) 636856. 0.196551
\(403\) −3.47831e6 −1.06685
\(404\) 138654. 0.0422649
\(405\) 164025. 0.0496904
\(406\) −146747. −0.0441827
\(407\) 838394. 0.250878
\(408\) −3.36706e6 −1.00138
\(409\) −1.17776e6 −0.348136 −0.174068 0.984734i \(-0.555691\pi\)
−0.174068 + 0.984734i \(0.555691\pi\)
\(410\) 411365. 0.120856
\(411\) 383781. 0.112067
\(412\) −219651. −0.0637513
\(413\) −158440. −0.0457077
\(414\) −1.43950e6 −0.412773
\(415\) 1.05196e6 0.299832
\(416\) −355303. −0.100662
\(417\) −1.52552e6 −0.429614
\(418\) −1.61518e6 −0.452148
\(419\) −443011. −0.123276 −0.0616381 0.998099i \(-0.519632\pi\)
−0.0616381 + 0.998099i \(0.519632\pi\)
\(420\) −5783.39 −0.00159978
\(421\) −3.41894e6 −0.940127 −0.470063 0.882633i \(-0.655769\pi\)
−0.470063 + 0.882633i \(0.655769\pi\)
\(422\) −4.32256e6 −1.18157
\(423\) 279563. 0.0759677
\(424\) 3.20966e6 0.867050
\(425\) −1.32351e6 −0.355432
\(426\) −2.90829e6 −0.776450
\(427\) −6686.08 −0.00177461
\(428\) 279969. 0.0738756
\(429\) −734417. −0.192664
\(430\) −1.68685e6 −0.439952
\(431\) −4.24640e6 −1.10110 −0.550551 0.834801i \(-0.685583\pi\)
−0.550551 + 0.834801i \(0.685583\pi\)
\(432\) 778923. 0.200810
\(433\) −1.03407e6 −0.265050 −0.132525 0.991180i \(-0.542309\pi\)
−0.132525 + 0.991180i \(0.542309\pi\)
\(434\) −526540. −0.134186
\(435\) 323424. 0.0819500
\(436\) 140249. 0.0353333
\(437\) 7.09063e6 1.77616
\(438\) −1.02759e6 −0.255937
\(439\) 3.55155e6 0.879542 0.439771 0.898110i \(-0.355060\pi\)
0.439771 + 0.898110i \(0.355060\pi\)
\(440\) −534424. −0.131599
\(441\) −1.33613e6 −0.327155
\(442\) −8.26043e6 −2.01116
\(443\) 2.35135e6 0.569257 0.284629 0.958638i \(-0.408130\pi\)
0.284629 + 0.958638i \(0.408130\pi\)
\(444\) −90816.9 −0.0218630
\(445\) −2.03058e6 −0.486094
\(446\) 5.03599e6 1.19880
\(447\) 652165. 0.154379
\(448\) 549685. 0.129395
\(449\) 3.01947e6 0.706831 0.353415 0.935467i \(-0.385020\pi\)
0.353415 + 0.935467i \(0.385020\pi\)
\(450\) 292822. 0.0681668
\(451\) 344218. 0.0796878
\(452\) −306371. −0.0705345
\(453\) 551846. 0.126349
\(454\) 476173. 0.108424
\(455\) 297573. 0.0673853
\(456\) −3.66944e6 −0.826395
\(457\) 3.80671e6 0.852626 0.426313 0.904576i \(-0.359812\pi\)
0.426313 + 0.904576i \(0.359812\pi\)
\(458\) −1.96738e6 −0.438254
\(459\) 1.54375e6 0.342014
\(460\) −111864. −0.0246487
\(461\) 7.43481e6 1.62936 0.814680 0.579910i \(-0.196912\pi\)
0.814680 + 0.579910i \(0.196912\pi\)
\(462\) −111175. −0.0242327
\(463\) −108747. −0.0235757 −0.0117879 0.999931i \(-0.503752\pi\)
−0.0117879 + 0.999931i \(0.503752\pi\)
\(464\) 1.53588e6 0.331178
\(465\) 1.16047e6 0.248888
\(466\) −1.43669e6 −0.306477
\(467\) −6.85473e6 −1.45445 −0.727224 0.686400i \(-0.759190\pi\)
−0.727224 + 0.686400i \(0.759190\pi\)
\(468\) 79553.9 0.0167898
\(469\) −215923. −0.0453280
\(470\) 499085. 0.104215
\(471\) −4.55210e6 −0.945496
\(472\) 1.58594e6 0.327666
\(473\) −1.41151e6 −0.290088
\(474\) 702554. 0.143626
\(475\) −1.44237e6 −0.293321
\(476\) −54431.3 −0.0110111
\(477\) −1.47158e6 −0.296134
\(478\) −7.98085e6 −1.59764
\(479\) −9.48659e6 −1.88917 −0.944585 0.328266i \(-0.893536\pi\)
−0.944585 + 0.328266i \(0.893536\pi\)
\(480\) 118540. 0.0234835
\(481\) 4.67281e6 0.920905
\(482\) −8.87182e6 −1.73938
\(483\) 488056. 0.0951922
\(484\) 21322.2 0.00413732
\(485\) 3.80253e6 0.734038
\(486\) −341548. −0.0655935
\(487\) 9.79771e6 1.87198 0.935992 0.352021i \(-0.114506\pi\)
0.935992 + 0.352021i \(0.114506\pi\)
\(488\) 66925.7 0.0127216
\(489\) −3.48932e6 −0.659885
\(490\) −2.38531e6 −0.448801
\(491\) 9.17094e6 1.71676 0.858381 0.513012i \(-0.171470\pi\)
0.858381 + 0.513012i \(0.171470\pi\)
\(492\) −37286.5 −0.00694447
\(493\) 3.04395e6 0.564054
\(494\) −9.00225e6 −1.65972
\(495\) 245025. 0.0449467
\(496\) 5.51087e6 1.00581
\(497\) 986040. 0.179062
\(498\) −2.19048e6 −0.395791
\(499\) −1.91031e6 −0.343441 −0.171720 0.985146i \(-0.554933\pi\)
−0.171720 + 0.985146i \(0.554933\pi\)
\(500\) 22755.2 0.00407058
\(501\) 5.71880e6 1.01791
\(502\) 1.95846e6 0.346861
\(503\) 7.08413e6 1.24844 0.624219 0.781250i \(-0.285417\pi\)
0.624219 + 0.781250i \(0.285417\pi\)
\(504\) −252571. −0.0442902
\(505\) 2.38019e6 0.415320
\(506\) −2.15037e6 −0.373367
\(507\) −751651. −0.129866
\(508\) −92520.7 −0.0159067
\(509\) −7.51321e6 −1.28538 −0.642689 0.766127i \(-0.722181\pi\)
−0.642689 + 0.766127i \(0.722181\pi\)
\(510\) 2.75594e6 0.469186
\(511\) 348398. 0.0590234
\(512\) −5.47764e6 −0.923461
\(513\) 1.68238e6 0.282248
\(514\) 9.31158e6 1.55459
\(515\) −3.77060e6 −0.626459
\(516\) 152898. 0.0252800
\(517\) 417619. 0.0687154
\(518\) 707361. 0.115829
\(519\) 6.11838e6 0.997053
\(520\) −2.97862e6 −0.483066
\(521\) −1.03994e7 −1.67847 −0.839235 0.543770i \(-0.816997\pi\)
−0.839235 + 0.543770i \(0.816997\pi\)
\(522\) −673463. −0.108178
\(523\) −9.14037e6 −1.46120 −0.730600 0.682806i \(-0.760759\pi\)
−0.730600 + 0.682806i \(0.760759\pi\)
\(524\) −128734. −0.0204817
\(525\) −99279.9 −0.0157204
\(526\) 2.64982e6 0.417592
\(527\) 1.09220e7 1.71307
\(528\) 1.16358e6 0.181639
\(529\) 3.00374e6 0.466684
\(530\) −2.62711e6 −0.406245
\(531\) −727128. −0.111911
\(532\) −59319.5 −0.00908695
\(533\) 1.91850e6 0.292513
\(534\) 4.22826e6 0.641665
\(535\) 4.80606e6 0.725947
\(536\) 2.16132e6 0.324944
\(537\) 567750. 0.0849613
\(538\) −4.00483e6 −0.596524
\(539\) −1.99595e6 −0.295923
\(540\) −26541.7 −0.00391692
\(541\) 6.47380e6 0.950969 0.475484 0.879724i \(-0.342273\pi\)
0.475484 + 0.879724i \(0.342273\pi\)
\(542\) −2.81508e6 −0.411617
\(543\) −4.71253e6 −0.685890
\(544\) 1.11566e6 0.161635
\(545\) 2.40757e6 0.347206
\(546\) −619634. −0.0889516
\(547\) −226463. −0.0323616 −0.0161808 0.999869i \(-0.505151\pi\)
−0.0161808 + 0.999869i \(0.505151\pi\)
\(548\) −62101.6 −0.00883388
\(549\) −30684.4 −0.00434497
\(550\) 437426. 0.0616592
\(551\) 3.31731e6 0.465487
\(552\) −4.88529e6 −0.682406
\(553\) −238197. −0.0331226
\(554\) 1.37819e7 1.90781
\(555\) −1.55900e6 −0.214839
\(556\) 246853. 0.0338650
\(557\) −5.45555e6 −0.745076 −0.372538 0.928017i \(-0.621512\pi\)
−0.372538 + 0.928017i \(0.621512\pi\)
\(558\) −2.41645e6 −0.328543
\(559\) −7.86706e6 −1.06484
\(560\) −471461. −0.0635296
\(561\) 2.30609e6 0.309364
\(562\) 6.48712e6 0.866386
\(563\) −1.30266e7 −1.73205 −0.866024 0.500002i \(-0.833333\pi\)
−0.866024 + 0.500002i \(0.833333\pi\)
\(564\) −45237.5 −0.00598827
\(565\) −5.25928e6 −0.693115
\(566\) 5.58723e6 0.733087
\(567\) 115800. 0.0151269
\(568\) −9.86997e6 −1.28365
\(569\) 3.02736e6 0.391998 0.195999 0.980604i \(-0.437205\pi\)
0.195999 + 0.980604i \(0.437205\pi\)
\(570\) 3.00344e6 0.387197
\(571\) 9.79560e6 1.25731 0.628653 0.777686i \(-0.283607\pi\)
0.628653 + 0.777686i \(0.283607\pi\)
\(572\) 118840. 0.0151870
\(573\) 6.15880e6 0.783627
\(574\) 290420. 0.0367914
\(575\) −1.92029e6 −0.242213
\(576\) 2.52267e6 0.316814
\(577\) 56282.7 0.00703778 0.00351889 0.999994i \(-0.498880\pi\)
0.00351889 + 0.999994i \(0.498880\pi\)
\(578\) 1.77253e7 2.20686
\(579\) −4.74675e6 −0.588437
\(580\) −52334.9 −0.00645983
\(581\) 742671. 0.0912759
\(582\) −7.91799e6 −0.968963
\(583\) −2.19829e6 −0.267863
\(584\) −3.48737e6 −0.423122
\(585\) 1.36565e6 0.164987
\(586\) 1.27725e7 1.53650
\(587\) −7.92842e6 −0.949711 −0.474855 0.880064i \(-0.657500\pi\)
−0.474855 + 0.880064i \(0.657500\pi\)
\(588\) 216207. 0.0257885
\(589\) 1.19028e7 1.41371
\(590\) −1.29809e6 −0.153523
\(591\) 4.20425e6 0.495130
\(592\) −7.40338e6 −0.868212
\(593\) −5.11910e6 −0.597801 −0.298900 0.954284i \(-0.596620\pi\)
−0.298900 + 0.954284i \(0.596620\pi\)
\(594\) −510214. −0.0593316
\(595\) −934388. −0.108202
\(596\) −105530. −0.0121692
\(597\) 3.92952e6 0.451236
\(598\) −1.19851e7 −1.37053
\(599\) −1.00831e7 −1.14822 −0.574110 0.818778i \(-0.694652\pi\)
−0.574110 + 0.818778i \(0.694652\pi\)
\(600\) 993763. 0.112695
\(601\) 1.32233e7 1.49332 0.746659 0.665207i \(-0.231657\pi\)
0.746659 + 0.665207i \(0.231657\pi\)
\(602\) −1.19090e6 −0.133932
\(603\) −990934. −0.110982
\(604\) −89296.9 −0.00995965
\(605\) 366025. 0.0406558
\(606\) −4.95625e6 −0.548241
\(607\) 937160. 0.103239 0.0516193 0.998667i \(-0.483562\pi\)
0.0516193 + 0.998667i \(0.483562\pi\)
\(608\) 1.21585e6 0.133390
\(609\) 228334. 0.0249475
\(610\) −54778.7 −0.00596056
\(611\) 2.32761e6 0.252236
\(612\) −249801. −0.0269598
\(613\) 3.05919e6 0.328818 0.164409 0.986392i \(-0.447428\pi\)
0.164409 + 0.986392i \(0.447428\pi\)
\(614\) 4.65901e6 0.498738
\(615\) −640074. −0.0682406
\(616\) −377298. −0.0400620
\(617\) 8.74474e6 0.924771 0.462385 0.886679i \(-0.346994\pi\)
0.462385 + 0.886679i \(0.346994\pi\)
\(618\) 7.85150e6 0.826954
\(619\) −4.14486e6 −0.434793 −0.217397 0.976083i \(-0.569757\pi\)
−0.217397 + 0.976083i \(0.569757\pi\)
\(620\) −187782. −0.0196189
\(621\) 2.23983e6 0.233070
\(622\) −5.03664e6 −0.521993
\(623\) −1.43357e6 −0.147979
\(624\) 6.48522e6 0.666750
\(625\) 390625. 0.0400000
\(626\) −1.83143e7 −1.86791
\(627\) 2.51319e6 0.255303
\(628\) 736598. 0.0745301
\(629\) −1.46727e7 −1.47872
\(630\) 206730. 0.0207516
\(631\) −4.44508e6 −0.444433 −0.222216 0.974997i \(-0.571329\pi\)
−0.222216 + 0.974997i \(0.571329\pi\)
\(632\) 2.38429e6 0.237446
\(633\) 6.72581e6 0.667168
\(634\) 9.96537e6 0.984624
\(635\) −1.58825e6 −0.156309
\(636\) 238124. 0.0233432
\(637\) −1.11245e7 −1.08625
\(638\) −1.00604e6 −0.0978504
\(639\) 4.52523e6 0.438418
\(640\) 4.92502e6 0.475289
\(641\) 1.08278e7 1.04086 0.520432 0.853903i \(-0.325771\pi\)
0.520432 + 0.853903i \(0.325771\pi\)
\(642\) −1.00076e7 −0.958282
\(643\) −1.23064e7 −1.17383 −0.586915 0.809649i \(-0.699658\pi\)
−0.586915 + 0.809649i \(0.699658\pi\)
\(644\) −78974.7 −0.00750367
\(645\) 2.62470e6 0.248417
\(646\) 2.82673e7 2.66504
\(647\) −1.26477e7 −1.18782 −0.593912 0.804530i \(-0.702417\pi\)
−0.593912 + 0.804530i \(0.702417\pi\)
\(648\) −1.15913e6 −0.108441
\(649\) −1.08620e6 −0.101228
\(650\) 2.43800e6 0.226334
\(651\) 819284. 0.0757673
\(652\) 564624. 0.0520164
\(653\) −578855. −0.0531235 −0.0265618 0.999647i \(-0.508456\pi\)
−0.0265618 + 0.999647i \(0.508456\pi\)
\(654\) −5.01326e6 −0.458328
\(655\) −2.20990e6 −0.201265
\(656\) −3.03959e6 −0.275775
\(657\) 1.59890e6 0.144514
\(658\) 352349. 0.0317255
\(659\) 1.71481e7 1.53816 0.769081 0.639151i \(-0.220714\pi\)
0.769081 + 0.639151i \(0.220714\pi\)
\(660\) −39648.7 −0.00354299
\(661\) 6.99329e6 0.622555 0.311278 0.950319i \(-0.399243\pi\)
0.311278 + 0.950319i \(0.399243\pi\)
\(662\) −1.85378e7 −1.64404
\(663\) 1.28530e7 1.13559
\(664\) −7.43392e6 −0.654332
\(665\) −1.01830e6 −0.0892939
\(666\) 3.24629e6 0.283597
\(667\) 4.41649e6 0.384382
\(668\) −925387. −0.0802384
\(669\) −7.83589e6 −0.676898
\(670\) −1.76904e6 −0.152248
\(671\) −45837.2 −0.00393017
\(672\) 83688.4 0.00714895
\(673\) 1.98109e7 1.68604 0.843018 0.537885i \(-0.180777\pi\)
0.843018 + 0.537885i \(0.180777\pi\)
\(674\) 1.41502e7 1.19981
\(675\) −455625. −0.0384900
\(676\) 121628. 0.0102369
\(677\) −2.85012e6 −0.238996 −0.119498 0.992834i \(-0.538129\pi\)
−0.119498 + 0.992834i \(0.538129\pi\)
\(678\) 1.09514e7 0.914942
\(679\) 2.68455e6 0.223459
\(680\) 9.35295e6 0.775669
\(681\) −740914. −0.0612210
\(682\) −3.60975e6 −0.297178
\(683\) 4.58865e6 0.376386 0.188193 0.982132i \(-0.439737\pi\)
0.188193 + 0.982132i \(0.439737\pi\)
\(684\) −272235. −0.0222486
\(685\) −1.06606e6 −0.0868070
\(686\) −3.39981e6 −0.275832
\(687\) 3.06121e6 0.247458
\(688\) 1.24642e7 1.00391
\(689\) −1.22522e7 −0.983254
\(690\) 3.99862e6 0.319733
\(691\) −2.65609e6 −0.211616 −0.105808 0.994387i \(-0.533743\pi\)
−0.105808 + 0.994387i \(0.533743\pi\)
\(692\) −990046. −0.0785942
\(693\) 172985. 0.0136828
\(694\) −1.76448e7 −1.39065
\(695\) 4.23757e6 0.332778
\(696\) −2.28556e6 −0.178842
\(697\) −6.02416e6 −0.469693
\(698\) −1.78588e7 −1.38744
\(699\) 2.23546e6 0.173051
\(700\) 16065.0 0.00123918
\(701\) −7.90907e6 −0.607898 −0.303949 0.952688i \(-0.598305\pi\)
−0.303949 + 0.952688i \(0.598305\pi\)
\(702\) −2.84369e6 −0.217790
\(703\) −1.59904e7 −1.22031
\(704\) 3.76843e6 0.286569
\(705\) −776564. −0.0588443
\(706\) 1.72906e7 1.30557
\(707\) 1.68039e6 0.126433
\(708\) 117660. 0.00882158
\(709\) −2.38057e7 −1.77855 −0.889273 0.457378i \(-0.848789\pi\)
−0.889273 + 0.457378i \(0.848789\pi\)
\(710\) 8.07857e6 0.601435
\(711\) −1.09316e6 −0.0810978
\(712\) 1.43496e7 1.06082
\(713\) 1.58468e7 1.16739
\(714\) 1.94567e6 0.142831
\(715\) 2.04005e6 0.149237
\(716\) −91870.5 −0.00669720
\(717\) 1.24180e7 0.902100
\(718\) −1.33030e7 −0.963029
\(719\) −2.00866e7 −1.44905 −0.724527 0.689246i \(-0.757942\pi\)
−0.724527 + 0.689246i \(0.757942\pi\)
\(720\) −2.16368e6 −0.155547
\(721\) −2.66201e6 −0.190709
\(722\) 1.64837e7 1.17683
\(723\) 1.38043e7 0.982133
\(724\) 762558. 0.0540663
\(725\) −898400. −0.0634782
\(726\) −762171. −0.0536674
\(727\) −1.00897e7 −0.708012 −0.354006 0.935243i \(-0.615181\pi\)
−0.354006 + 0.935243i \(0.615181\pi\)
\(728\) −2.10288e6 −0.147057
\(729\) 531441. 0.0370370
\(730\) 2.85441e6 0.198248
\(731\) 2.47028e7 1.70983
\(732\) 4965.20 0.000342499 0
\(733\) −1.48230e7 −1.01901 −0.509504 0.860468i \(-0.670171\pi\)
−0.509504 + 0.860468i \(0.670171\pi\)
\(734\) 2.07192e7 1.41949
\(735\) 3.71148e6 0.253413
\(736\) 1.61872e6 0.110148
\(737\) −1.48028e6 −0.100387
\(738\) 1.33282e6 0.0900806
\(739\) −1.85393e7 −1.24877 −0.624383 0.781118i \(-0.714650\pi\)
−0.624383 + 0.781118i \(0.714650\pi\)
\(740\) 252269. 0.0169350
\(741\) 1.40073e7 0.937149
\(742\) −1.85471e6 −0.123671
\(743\) −8.68829e6 −0.577381 −0.288690 0.957423i \(-0.593220\pi\)
−0.288690 + 0.957423i \(0.593220\pi\)
\(744\) −8.20079e6 −0.543155
\(745\) −1.81157e6 −0.119582
\(746\) 2.95717e6 0.194549
\(747\) 3.40834e6 0.223481
\(748\) −373160. −0.0243860
\(749\) 3.39303e6 0.220995
\(750\) −813395. −0.0528018
\(751\) −1.37377e7 −0.888821 −0.444411 0.895823i \(-0.646587\pi\)
−0.444411 + 0.895823i \(0.646587\pi\)
\(752\) −3.68776e6 −0.237803
\(753\) −3.04732e6 −0.195853
\(754\) −5.60717e6 −0.359183
\(755\) −1.53291e6 −0.0978696
\(756\) −18738.2 −0.00119240
\(757\) 5.59097e6 0.354607 0.177304 0.984156i \(-0.443263\pi\)
0.177304 + 0.984156i \(0.443263\pi\)
\(758\) 1.41832e7 0.896608
\(759\) 3.34592e6 0.210820
\(760\) 1.01929e7 0.640123
\(761\) −2.44147e7 −1.52823 −0.764117 0.645078i \(-0.776825\pi\)
−0.764117 + 0.645078i \(0.776825\pi\)
\(762\) 3.30719e6 0.206334
\(763\) 1.69972e6 0.105698
\(764\) −996586. −0.0617706
\(765\) −4.28818e6 −0.264923
\(766\) −8.46309e6 −0.521143
\(767\) −6.05397e6 −0.371580
\(768\) −1.28584e6 −0.0786657
\(769\) 1.99742e7 1.21802 0.609008 0.793164i \(-0.291568\pi\)
0.609008 + 0.793164i \(0.291568\pi\)
\(770\) 308819. 0.0187705
\(771\) −1.44886e7 −0.877790
\(772\) 768096. 0.0463844
\(773\) −2.64430e7 −1.59170 −0.795852 0.605492i \(-0.792976\pi\)
−0.795852 + 0.605492i \(0.792976\pi\)
\(774\) −5.46540e6 −0.327921
\(775\) −3.22354e6 −0.192787
\(776\) −2.68716e7 −1.60191
\(777\) −1.10064e6 −0.0654021
\(778\) −4.65032e6 −0.275445
\(779\) −6.56515e6 −0.387616
\(780\) −220983. −0.0130054
\(781\) 6.75991e6 0.396564
\(782\) 3.76336e7 2.20069
\(783\) 1.04789e6 0.0610819
\(784\) 1.76251e7 1.02410
\(785\) 1.26447e7 0.732378
\(786\) 4.60166e6 0.265679
\(787\) −2.87892e6 −0.165689 −0.0828443 0.996562i \(-0.526400\pi\)
−0.0828443 + 0.996562i \(0.526400\pi\)
\(788\) −680310. −0.0390294
\(789\) −4.12306e6 −0.235791
\(790\) −1.95154e6 −0.111252
\(791\) −3.71300e6 −0.211001
\(792\) −1.73153e6 −0.0980884
\(793\) −255474. −0.0144266
\(794\) −1.66181e7 −0.935469
\(795\) 4.08772e6 0.229384
\(796\) −635855. −0.0355693
\(797\) −2.55634e7 −1.42552 −0.712759 0.701409i \(-0.752554\pi\)
−0.712759 + 0.701409i \(0.752554\pi\)
\(798\) 2.12040e6 0.117872
\(799\) −7.30875e6 −0.405020
\(800\) −329279. −0.0181903
\(801\) −6.57908e6 −0.362313
\(802\) −1.99595e7 −1.09575
\(803\) 2.38849e6 0.130718
\(804\) 160348. 0.00874829
\(805\) −1.35571e6 −0.0737356
\(806\) −2.01190e7 −1.09086
\(807\) 6.23142e6 0.336824
\(808\) −1.68202e7 −0.906365
\(809\) −2.32458e6 −0.124874 −0.0624371 0.998049i \(-0.519887\pi\)
−0.0624371 + 0.998049i \(0.519887\pi\)
\(810\) 948744. 0.0508085
\(811\) 1.44240e7 0.770076 0.385038 0.922901i \(-0.374188\pi\)
0.385038 + 0.922901i \(0.374188\pi\)
\(812\) −36947.9 −0.00196653
\(813\) 4.38021e6 0.232417
\(814\) 4.84939e6 0.256523
\(815\) 9.69255e6 0.511145
\(816\) −2.03638e7 −1.07061
\(817\) 2.69212e7 1.41104
\(818\) −6.81233e6 −0.355969
\(819\) 964137. 0.0502260
\(820\) 103574. 0.00537916
\(821\) 9.95289e6 0.515337 0.257668 0.966233i \(-0.417046\pi\)
0.257668 + 0.966233i \(0.417046\pi\)
\(822\) 2.21985e6 0.114589
\(823\) 2.46452e7 1.26833 0.634165 0.773198i \(-0.281344\pi\)
0.634165 + 0.773198i \(0.281344\pi\)
\(824\) 2.66460e7 1.36714
\(825\) −680625. −0.0348155
\(826\) −916439. −0.0467362
\(827\) −2.09014e7 −1.06270 −0.531351 0.847152i \(-0.678315\pi\)
−0.531351 + 0.847152i \(0.678315\pi\)
\(828\) −362438. −0.0183721
\(829\) −8.31387e6 −0.420162 −0.210081 0.977684i \(-0.567373\pi\)
−0.210081 + 0.977684i \(0.567373\pi\)
\(830\) 6.08467e6 0.306579
\(831\) −2.14444e7 −1.07724
\(832\) 2.10034e7 1.05192
\(833\) 3.49312e7 1.74422
\(834\) −8.82385e6 −0.439282
\(835\) −1.58855e7 −0.788472
\(836\) −406671. −0.0201246
\(837\) 3.75994e6 0.185510
\(838\) −2.56244e6 −0.126050
\(839\) −4.03176e7 −1.97738 −0.988690 0.149977i \(-0.952080\pi\)
−0.988690 + 0.149977i \(0.952080\pi\)
\(840\) 701587. 0.0343071
\(841\) −1.84449e7 −0.899263
\(842\) −1.97757e7 −0.961282
\(843\) −1.00938e7 −0.489200
\(844\) −1.08834e6 −0.0525905
\(845\) 2.08792e6 0.100594
\(846\) 1.61703e6 0.0776771
\(847\) 258410. 0.0123766
\(848\) 1.94118e7 0.926992
\(849\) −8.69361e6 −0.413934
\(850\) −7.65539e6 −0.363430
\(851\) −2.12888e7 −1.00769
\(852\) −732250. −0.0345589
\(853\) −2.11646e7 −0.995951 −0.497976 0.867191i \(-0.665923\pi\)
−0.497976 + 0.867191i \(0.665923\pi\)
\(854\) −38673.3 −0.00181454
\(855\) −4.67328e6 −0.218628
\(856\) −3.39633e7 −1.58425
\(857\) −1.03622e7 −0.481946 −0.240973 0.970532i \(-0.577467\pi\)
−0.240973 + 0.970532i \(0.577467\pi\)
\(858\) −4.24797e6 −0.196999
\(859\) −1.77409e7 −0.820336 −0.410168 0.912010i \(-0.634530\pi\)
−0.410168 + 0.912010i \(0.634530\pi\)
\(860\) −424716. −0.0195818
\(861\) −451886. −0.0207741
\(862\) −2.45618e7 −1.12588
\(863\) −2.62577e6 −0.120014 −0.0600068 0.998198i \(-0.519112\pi\)
−0.0600068 + 0.998198i \(0.519112\pi\)
\(864\) 384071. 0.0175036
\(865\) −1.69955e7 −0.772314
\(866\) −5.98118e6 −0.271014
\(867\) −2.75802e7 −1.24609
\(868\) −132572. −0.00597247
\(869\) −1.63299e6 −0.0733557
\(870\) 1.87073e6 0.0837941
\(871\) −8.25039e6 −0.368493
\(872\) −1.70137e7 −0.757718
\(873\) 1.23202e7 0.547120
\(874\) 4.10132e7 1.81612
\(875\) 275778. 0.0121770
\(876\) −258727. −0.0113915
\(877\) 7.30171e6 0.320572 0.160286 0.987071i \(-0.448758\pi\)
0.160286 + 0.987071i \(0.448758\pi\)
\(878\) 2.05427e7 0.899334
\(879\) −1.98737e7 −0.867576
\(880\) −3.23216e6 −0.140697
\(881\) 6.08524e6 0.264142 0.132071 0.991240i \(-0.457837\pi\)
0.132071 + 0.991240i \(0.457837\pi\)
\(882\) −7.72840e6 −0.334517
\(883\) 1.66311e7 0.717828 0.358914 0.933371i \(-0.383147\pi\)
0.358914 + 0.933371i \(0.383147\pi\)
\(884\) −2.07981e6 −0.0895146
\(885\) 2.01980e6 0.0866862
\(886\) 1.36006e7 0.582067
\(887\) −1.47741e7 −0.630511 −0.315256 0.949007i \(-0.602090\pi\)
−0.315256 + 0.949007i \(0.602090\pi\)
\(888\) 1.10171e7 0.468849
\(889\) −1.12129e6 −0.0475841
\(890\) −1.17452e7 −0.497032
\(891\) 793881. 0.0335013
\(892\) 1.26796e6 0.0533574
\(893\) −7.96511e6 −0.334244
\(894\) 3.77222e6 0.157853
\(895\) −1.57708e6 −0.0658108
\(896\) 3.47702e6 0.144689
\(897\) 1.86486e7 0.773863
\(898\) 1.74651e7 0.722736
\(899\) 7.41383e6 0.305945
\(900\) 73727.0 0.00303403
\(901\) 3.84722e7 1.57883
\(902\) 1.99101e6 0.0814810
\(903\) 1.85301e6 0.0756240
\(904\) 3.71661e7 1.51261
\(905\) 1.30904e7 0.531288
\(906\) 3.19196e6 0.129192
\(907\) −9.23123e6 −0.372599 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(908\) 119891. 0.00482583
\(909\) 7.71181e6 0.309561
\(910\) 1.72121e6 0.0689016
\(911\) −1.55931e7 −0.622495 −0.311247 0.950329i \(-0.600747\pi\)
−0.311247 + 0.950329i \(0.600747\pi\)
\(912\) −2.21925e7 −0.883526
\(913\) 5.09147e6 0.202146
\(914\) 2.20185e7 0.871812
\(915\) 85234.5 0.00336560
\(916\) −495349. −0.0195062
\(917\) −1.56017e6 −0.0612700
\(918\) 8.92925e6 0.349710
\(919\) −4.21091e6 −0.164470 −0.0822351 0.996613i \(-0.526206\pi\)
−0.0822351 + 0.996613i \(0.526206\pi\)
\(920\) 1.35703e7 0.528590
\(921\) −7.24932e6 −0.281610
\(922\) 4.30040e7 1.66602
\(923\) 3.76765e7 1.45568
\(924\) −27991.6 −0.00107857
\(925\) 4.33055e6 0.166413
\(926\) −629009. −0.0241062
\(927\) −1.22168e7 −0.466935
\(928\) 757310. 0.0288671
\(929\) 4.10363e7 1.56002 0.780008 0.625770i \(-0.215215\pi\)
0.780008 + 0.625770i \(0.215215\pi\)
\(930\) 6.71235e6 0.254488
\(931\) 3.80682e7 1.43942
\(932\) −361730. −0.0136410
\(933\) 7.83689e6 0.294740
\(934\) −3.96488e7 −1.48718
\(935\) −6.40581e6 −0.239632
\(936\) −9.65073e6 −0.360056
\(937\) 2.41565e7 0.898844 0.449422 0.893320i \(-0.351630\pi\)
0.449422 + 0.893320i \(0.351630\pi\)
\(938\) −1.24893e6 −0.0463479
\(939\) 2.84967e7 1.05470
\(940\) 125660. 0.00463849
\(941\) 3.13080e7 1.15261 0.576304 0.817236i \(-0.304495\pi\)
0.576304 + 0.817236i \(0.304495\pi\)
\(942\) −2.63300e7 −0.966772
\(943\) −8.74049e6 −0.320079
\(944\) 9.59164e6 0.350318
\(945\) −321667. −0.0117173
\(946\) −8.16436e6 −0.296616
\(947\) 5699.21 0.000206509 0 0.000103255 1.00000i \(-0.499967\pi\)
0.000103255 1.00000i \(0.499967\pi\)
\(948\) 176889. 0.00639265
\(949\) 1.33123e7 0.479829
\(950\) −8.34288e6 −0.299921
\(951\) −1.55059e7 −0.555963
\(952\) 6.60310e6 0.236132
\(953\) −4.24159e7 −1.51285 −0.756427 0.654078i \(-0.773057\pi\)
−0.756427 + 0.654078i \(0.773057\pi\)
\(954\) −8.51183e6 −0.302797
\(955\) −1.71078e7 −0.606995
\(956\) −2.00942e6 −0.0711094
\(957\) 1.56537e6 0.0552507
\(958\) −5.48718e7 −1.93168
\(959\) −752628. −0.0264261
\(960\) −7.00741e6 −0.245403
\(961\) −2.02765e6 −0.0708247
\(962\) 2.70282e7 0.941628
\(963\) 1.55716e7 0.541089
\(964\) −2.23375e6 −0.0774180
\(965\) 1.31854e7 0.455801
\(966\) 2.82298e6 0.0973342
\(967\) 1.43147e7 0.492284 0.246142 0.969234i \(-0.420837\pi\)
0.246142 + 0.969234i \(0.420837\pi\)
\(968\) −2.58661e6 −0.0887243
\(969\) −4.39833e7 −1.50480
\(970\) 2.19944e7 0.750556
\(971\) −1.06931e7 −0.363960 −0.181980 0.983302i \(-0.558251\pi\)
−0.181980 + 0.983302i \(0.558251\pi\)
\(972\) −85995.1 −0.00291950
\(973\) 2.99168e6 0.101306
\(974\) 5.66714e7 1.91411
\(975\) −3.79348e6 −0.127799
\(976\) 404762. 0.0136011
\(977\) 5.44429e7 1.82476 0.912378 0.409349i \(-0.134244\pi\)
0.912378 + 0.409349i \(0.134244\pi\)
\(978\) −2.01827e7 −0.674734
\(979\) −9.82800e6 −0.327724
\(980\) −600574. −0.0199757
\(981\) 7.80052e6 0.258792
\(982\) 5.30461e7 1.75539
\(983\) 4.16457e7 1.37463 0.687316 0.726359i \(-0.258789\pi\)
0.687316 + 0.726359i \(0.258789\pi\)
\(984\) 4.52325e6 0.148923
\(985\) −1.16785e7 −0.383526
\(986\) 1.76067e7 0.576746
\(987\) −548247. −0.0179136
\(988\) −2.26659e6 −0.0738722
\(989\) 3.58414e7 1.16518
\(990\) 1.41726e6 0.0459581
\(991\) −9.62055e6 −0.311183 −0.155591 0.987821i \(-0.549728\pi\)
−0.155591 + 0.987821i \(0.549728\pi\)
\(992\) 2.71730e6 0.0876714
\(993\) 2.88444e7 0.928300
\(994\) 5.70340e6 0.183091
\(995\) −1.09153e7 −0.349526
\(996\) −551520. −0.0176162
\(997\) 3.05865e7 0.974523 0.487262 0.873256i \(-0.337996\pi\)
0.487262 + 0.873256i \(0.337996\pi\)
\(998\) −1.10495e7 −0.351169
\(999\) −5.05115e6 −0.160131
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.c.1.3 3
3.2 odd 2 495.6.a.b.1.1 3
5.4 even 2 825.6.a.g.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.c.1.3 3 1.1 even 1 trivial
495.6.a.b.1.1 3 3.2 odd 2
825.6.a.g.1.1 3 5.4 even 2