Properties

Label 165.6.a.c.1.1
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 1.1
Root \(5.47894\) 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)
 
\(f(q)\) \(=\) \(q-4.47894 q^{2} -9.00000 q^{3} -11.9391 q^{4} +25.0000 q^{5} +40.3105 q^{6} -168.818 q^{7} +196.801 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.47894 q^{2} -9.00000 q^{3} -11.9391 q^{4} +25.0000 q^{5} +40.3105 q^{6} -168.818 q^{7} +196.801 q^{8} +81.0000 q^{9} -111.974 q^{10} +121.000 q^{11} +107.452 q^{12} +290.259 q^{13} +756.126 q^{14} -225.000 q^{15} -499.408 q^{16} +623.964 q^{17} -362.794 q^{18} -398.456 q^{19} -298.477 q^{20} +1519.36 q^{21} -541.952 q^{22} +3788.06 q^{23} -1771.21 q^{24} +625.000 q^{25} -1300.06 q^{26} -729.000 q^{27} +2015.53 q^{28} +4220.43 q^{29} +1007.76 q^{30} -5594.57 q^{31} -4060.80 q^{32} -1089.00 q^{33} -2794.70 q^{34} -4220.45 q^{35} -967.065 q^{36} +301.266 q^{37} +1784.66 q^{38} -2612.33 q^{39} +4920.01 q^{40} -14636.2 q^{41} -6805.14 q^{42} +151.418 q^{43} -1444.63 q^{44} +2025.00 q^{45} -16966.5 q^{46} -13535.0 q^{47} +4494.67 q^{48} +11692.5 q^{49} -2799.34 q^{50} -5615.68 q^{51} -3465.43 q^{52} -18116.1 q^{53} +3265.15 q^{54} +3025.00 q^{55} -33223.5 q^{56} +3586.10 q^{57} -18903.0 q^{58} -50582.4 q^{59} +2686.29 q^{60} +5984.35 q^{61} +25057.8 q^{62} -13674.3 q^{63} +34169.1 q^{64} +7256.49 q^{65} +4877.57 q^{66} +22450.7 q^{67} -7449.56 q^{68} -34092.6 q^{69} +18903.2 q^{70} +10017.6 q^{71} +15940.8 q^{72} -476.148 q^{73} -1349.35 q^{74} -5625.00 q^{75} +4757.19 q^{76} -20427.0 q^{77} +11700.5 q^{78} -85024.7 q^{79} -12485.2 q^{80} +6561.00 q^{81} +65554.5 q^{82} -25643.4 q^{83} -18139.8 q^{84} +15599.1 q^{85} -678.194 q^{86} -37983.8 q^{87} +23812.9 q^{88} +1807.91 q^{89} -9069.86 q^{90} -49001.0 q^{91} -45226.0 q^{92} +50351.2 q^{93} +60622.4 q^{94} -9961.39 q^{95} +36547.2 q^{96} +11688.5 q^{97} -52370.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} - 42 q^{4} + 75 q^{5} - 18 q^{6} - 68 q^{7} - 24 q^{8} + 243 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.47894 −0.791773 −0.395886 0.918300i \(-0.629563\pi\)
−0.395886 + 0.918300i \(0.629563\pi\)
\(3\) −9.00000 −0.577350
\(4\) −11.9391 −0.373096
\(5\) 25.0000 0.447214
\(6\) 40.3105 0.457130
\(7\) −168.818 −1.30219 −0.651094 0.758997i \(-0.725690\pi\)
−0.651094 + 0.758997i \(0.725690\pi\)
\(8\) 196.801 1.08718
\(9\) 81.0000 0.333333
\(10\) −111.974 −0.354091
\(11\) 121.000 0.301511
\(12\) 107.452 0.215407
\(13\) 290.259 0.476352 0.238176 0.971222i \(-0.423451\pi\)
0.238176 + 0.971222i \(0.423451\pi\)
\(14\) 756.126 1.03104
\(15\) −225.000 −0.258199
\(16\) −499.408 −0.487703
\(17\) 623.964 0.523645 0.261823 0.965116i \(-0.415676\pi\)
0.261823 + 0.965116i \(0.415676\pi\)
\(18\) −362.794 −0.263924
\(19\) −398.456 −0.253219 −0.126609 0.991953i \(-0.540409\pi\)
−0.126609 + 0.991953i \(0.540409\pi\)
\(20\) −298.477 −0.166854
\(21\) 1519.36 0.751819
\(22\) −541.952 −0.238728
\(23\) 3788.06 1.49313 0.746565 0.665312i \(-0.231702\pi\)
0.746565 + 0.665312i \(0.231702\pi\)
\(24\) −1771.21 −0.627684
\(25\) 625.000 0.200000
\(26\) −1300.06 −0.377162
\(27\) −729.000 −0.192450
\(28\) 2015.53 0.485841
\(29\) 4220.43 0.931883 0.465941 0.884816i \(-0.345716\pi\)
0.465941 + 0.884816i \(0.345716\pi\)
\(30\) 1007.76 0.204435
\(31\) −5594.57 −1.04559 −0.522797 0.852457i \(-0.675111\pi\)
−0.522797 + 0.852457i \(0.675111\pi\)
\(32\) −4060.80 −0.701030
\(33\) −1089.00 −0.174078
\(34\) −2794.70 −0.414608
\(35\) −4220.45 −0.582356
\(36\) −967.065 −0.124365
\(37\) 301.266 0.0361781 0.0180890 0.999836i \(-0.494242\pi\)
0.0180890 + 0.999836i \(0.494242\pi\)
\(38\) 1784.66 0.200492
\(39\) −2612.33 −0.275022
\(40\) 4920.01 0.486202
\(41\) −14636.2 −1.35978 −0.679889 0.733315i \(-0.737972\pi\)
−0.679889 + 0.733315i \(0.737972\pi\)
\(42\) −6805.14 −0.595269
\(43\) 151.418 0.0124884 0.00624421 0.999981i \(-0.498012\pi\)
0.00624421 + 0.999981i \(0.498012\pi\)
\(44\) −1444.63 −0.112493
\(45\) 2025.00 0.149071
\(46\) −16966.5 −1.18222
\(47\) −13535.0 −0.893743 −0.446872 0.894598i \(-0.647462\pi\)
−0.446872 + 0.894598i \(0.647462\pi\)
\(48\) 4494.67 0.281575
\(49\) 11692.5 0.695694
\(50\) −2799.34 −0.158355
\(51\) −5615.68 −0.302327
\(52\) −3465.43 −0.177725
\(53\) −18116.1 −0.885880 −0.442940 0.896551i \(-0.646065\pi\)
−0.442940 + 0.896551i \(0.646065\pi\)
\(54\) 3265.15 0.152377
\(55\) 3025.00 0.134840
\(56\) −33223.5 −1.41571
\(57\) 3586.10 0.146196
\(58\) −18903.0 −0.737839
\(59\) −50582.4 −1.89177 −0.945887 0.324495i \(-0.894806\pi\)
−0.945887 + 0.324495i \(0.894806\pi\)
\(60\) 2686.29 0.0963330
\(61\) 5984.35 0.205917 0.102958 0.994686i \(-0.467169\pi\)
0.102958 + 0.994686i \(0.467169\pi\)
\(62\) 25057.8 0.827872
\(63\) −13674.3 −0.434063
\(64\) 34169.1 1.04276
\(65\) 7256.49 0.213031
\(66\) 4877.57 0.137830
\(67\) 22450.7 0.611002 0.305501 0.952192i \(-0.401176\pi\)
0.305501 + 0.952192i \(0.401176\pi\)
\(68\) −7449.56 −0.195370
\(69\) −34092.6 −0.862059
\(70\) 18903.2 0.461094
\(71\) 10017.6 0.235841 0.117921 0.993023i \(-0.462377\pi\)
0.117921 + 0.993023i \(0.462377\pi\)
\(72\) 15940.8 0.362393
\(73\) −476.148 −0.0104577 −0.00522883 0.999986i \(-0.501664\pi\)
−0.00522883 + 0.999986i \(0.501664\pi\)
\(74\) −1349.35 −0.0286448
\(75\) −5625.00 −0.115470
\(76\) 4757.19 0.0944750
\(77\) −20427.0 −0.392624
\(78\) 11700.5 0.217755
\(79\) −85024.7 −1.53277 −0.766386 0.642380i \(-0.777947\pi\)
−0.766386 + 0.642380i \(0.777947\pi\)
\(80\) −12485.2 −0.218107
\(81\) 6561.00 0.111111
\(82\) 65554.5 1.07663
\(83\) −25643.4 −0.408583 −0.204292 0.978910i \(-0.565489\pi\)
−0.204292 + 0.978910i \(0.565489\pi\)
\(84\) −18139.8 −0.280501
\(85\) 15599.1 0.234181
\(86\) −678.194 −0.00988799
\(87\) −37983.8 −0.538023
\(88\) 23812.9 0.327797
\(89\) 1807.91 0.0241937 0.0120968 0.999927i \(-0.496149\pi\)
0.0120968 + 0.999927i \(0.496149\pi\)
\(90\) −9069.86 −0.118030
\(91\) −49001.0 −0.620300
\(92\) −45226.0 −0.557081
\(93\) 50351.2 0.603674
\(94\) 60622.4 0.707642
\(95\) −9961.39 −0.113243
\(96\) 36547.2 0.404740
\(97\) 11688.5 0.126133 0.0630666 0.998009i \(-0.479912\pi\)
0.0630666 + 0.998009i \(0.479912\pi\)
\(98\) −52370.1 −0.550831
\(99\) 9801.00 0.100504
\(100\) −7461.92 −0.0746192
\(101\) −135573. −1.32242 −0.661210 0.750200i \(-0.729957\pi\)
−0.661210 + 0.750200i \(0.729957\pi\)
\(102\) 25152.3 0.239374
\(103\) −88490.5 −0.821871 −0.410935 0.911664i \(-0.634798\pi\)
−0.410935 + 0.911664i \(0.634798\pi\)
\(104\) 57123.2 0.517880
\(105\) 37984.1 0.336224
\(106\) 81140.9 0.701416
\(107\) 218742. 1.84703 0.923514 0.383564i \(-0.125304\pi\)
0.923514 + 0.383564i \(0.125304\pi\)
\(108\) 8703.59 0.0718024
\(109\) 238399. 1.92193 0.960967 0.276663i \(-0.0892287\pi\)
0.960967 + 0.276663i \(0.0892287\pi\)
\(110\) −13548.8 −0.106763
\(111\) −2711.39 −0.0208874
\(112\) 84309.1 0.635081
\(113\) 187225. 1.37933 0.689664 0.724130i \(-0.257758\pi\)
0.689664 + 0.724130i \(0.257758\pi\)
\(114\) −16061.9 −0.115754
\(115\) 94701.6 0.667748
\(116\) −50388.0 −0.347682
\(117\) 23511.0 0.158784
\(118\) 226556. 1.49786
\(119\) −105336. −0.681885
\(120\) −44280.1 −0.280709
\(121\) 14641.0 0.0909091
\(122\) −26803.5 −0.163039
\(123\) 131725. 0.785068
\(124\) 66794.1 0.390107
\(125\) 15625.0 0.0894427
\(126\) 61246.2 0.343679
\(127\) 27727.6 0.152547 0.0762733 0.997087i \(-0.475698\pi\)
0.0762733 + 0.997087i \(0.475698\pi\)
\(128\) −23096.0 −0.124598
\(129\) −1362.77 −0.00721019
\(130\) −32501.4 −0.168672
\(131\) −324623. −1.65273 −0.826363 0.563138i \(-0.809594\pi\)
−0.826363 + 0.563138i \(0.809594\pi\)
\(132\) 13001.7 0.0649477
\(133\) 67266.5 0.329739
\(134\) −100555. −0.483775
\(135\) −18225.0 −0.0860663
\(136\) 122797. 0.569297
\(137\) −59748.8 −0.271974 −0.135987 0.990711i \(-0.543421\pi\)
−0.135987 + 0.990711i \(0.543421\pi\)
\(138\) 152699. 0.682555
\(139\) −117646. −0.516462 −0.258231 0.966083i \(-0.583140\pi\)
−0.258231 + 0.966083i \(0.583140\pi\)
\(140\) 50388.3 0.217275
\(141\) 121815. 0.516003
\(142\) −44868.5 −0.186733
\(143\) 35121.4 0.143626
\(144\) −40452.0 −0.162568
\(145\) 105511. 0.416751
\(146\) 2132.64 0.00828009
\(147\) −105233. −0.401659
\(148\) −3596.84 −0.0134979
\(149\) 52786.3 0.194785 0.0973925 0.995246i \(-0.468950\pi\)
0.0973925 + 0.995246i \(0.468950\pi\)
\(150\) 25194.0 0.0914260
\(151\) −430119. −1.53513 −0.767567 0.640969i \(-0.778533\pi\)
−0.767567 + 0.640969i \(0.778533\pi\)
\(152\) −78416.3 −0.275294
\(153\) 50541.1 0.174548
\(154\) 91491.3 0.310869
\(155\) −139864. −0.467604
\(156\) 31188.9 0.102610
\(157\) −150989. −0.488873 −0.244436 0.969665i \(-0.578603\pi\)
−0.244436 + 0.969665i \(0.578603\pi\)
\(158\) 380821. 1.21361
\(159\) 163045. 0.511463
\(160\) −101520. −0.313510
\(161\) −639493. −1.94434
\(162\) −29386.3 −0.0879747
\(163\) 142404. 0.419812 0.209906 0.977722i \(-0.432684\pi\)
0.209906 + 0.977722i \(0.432684\pi\)
\(164\) 174742. 0.507328
\(165\) −27225.0 −0.0778499
\(166\) 114855. 0.323505
\(167\) 55468.4 0.153906 0.0769528 0.997035i \(-0.475481\pi\)
0.0769528 + 0.997035i \(0.475481\pi\)
\(168\) 299011. 0.817362
\(169\) −287042. −0.773089
\(170\) −69867.5 −0.185418
\(171\) −32274.9 −0.0844063
\(172\) −1807.80 −0.00465938
\(173\) 133612. 0.339414 0.169707 0.985495i \(-0.445718\pi\)
0.169707 + 0.985495i \(0.445718\pi\)
\(174\) 170127. 0.425992
\(175\) −105511. −0.260438
\(176\) −60428.4 −0.147048
\(177\) 455242. 1.09222
\(178\) −8097.52 −0.0191559
\(179\) −188254. −0.439150 −0.219575 0.975596i \(-0.570467\pi\)
−0.219575 + 0.975596i \(0.570467\pi\)
\(180\) −24176.6 −0.0556179
\(181\) −652252. −1.47986 −0.739928 0.672686i \(-0.765140\pi\)
−0.739928 + 0.672686i \(0.765140\pi\)
\(182\) 219473. 0.491136
\(183\) −53859.1 −0.118886
\(184\) 745493. 1.62330
\(185\) 7531.65 0.0161793
\(186\) −225520. −0.477972
\(187\) 75499.7 0.157885
\(188\) 161595. 0.333452
\(189\) 123068. 0.250606
\(190\) 44616.5 0.0896626
\(191\) −705620. −1.39955 −0.699774 0.714365i \(-0.746716\pi\)
−0.699774 + 0.714365i \(0.746716\pi\)
\(192\) −307522. −0.602037
\(193\) −87143.5 −0.168400 −0.0841999 0.996449i \(-0.526833\pi\)
−0.0841999 + 0.996449i \(0.526833\pi\)
\(194\) −52352.1 −0.0998688
\(195\) −65308.4 −0.122994
\(196\) −139598. −0.259561
\(197\) −667596. −1.22560 −0.612800 0.790238i \(-0.709957\pi\)
−0.612800 + 0.790238i \(0.709957\pi\)
\(198\) −43898.1 −0.0795761
\(199\) −698609. −1.25055 −0.625276 0.780404i \(-0.715013\pi\)
−0.625276 + 0.780404i \(0.715013\pi\)
\(200\) 123000. 0.217436
\(201\) −202056. −0.352762
\(202\) 607224. 1.04706
\(203\) −712484. −1.21349
\(204\) 67046.0 0.112797
\(205\) −365904. −0.608111
\(206\) 396344. 0.650735
\(207\) 306833. 0.497710
\(208\) −144958. −0.232318
\(209\) −48213.1 −0.0763484
\(210\) −170128. −0.266213
\(211\) −558238. −0.863203 −0.431602 0.902064i \(-0.642051\pi\)
−0.431602 + 0.902064i \(0.642051\pi\)
\(212\) 216289. 0.330518
\(213\) −90158.8 −0.136163
\(214\) −979735. −1.46243
\(215\) 3785.46 0.00558499
\(216\) −143468. −0.209228
\(217\) 944465. 1.36156
\(218\) −1.06778e6 −1.52173
\(219\) 4285.33 0.00603773
\(220\) −36115.7 −0.0503083
\(221\) 181111. 0.249440
\(222\) 12144.2 0.0165381
\(223\) −382897. −0.515608 −0.257804 0.966197i \(-0.582999\pi\)
−0.257804 + 0.966197i \(0.582999\pi\)
\(224\) 685536. 0.912873
\(225\) 50625.0 0.0666667
\(226\) −838569. −1.09211
\(227\) −404735. −0.521322 −0.260661 0.965430i \(-0.583941\pi\)
−0.260661 + 0.965430i \(0.583941\pi\)
\(228\) −42814.7 −0.0545452
\(229\) 664894. 0.837845 0.418922 0.908022i \(-0.362408\pi\)
0.418922 + 0.908022i \(0.362408\pi\)
\(230\) −424163. −0.528705
\(231\) 183843. 0.226682
\(232\) 830582. 1.01312
\(233\) 61531.4 0.0742518 0.0371259 0.999311i \(-0.488180\pi\)
0.0371259 + 0.999311i \(0.488180\pi\)
\(234\) −105304. −0.125721
\(235\) −338374. −0.399694
\(236\) 603907. 0.705814
\(237\) 765223. 0.884946
\(238\) 471796. 0.539898
\(239\) 1.71207e6 1.93877 0.969384 0.245549i \(-0.0789682\pi\)
0.969384 + 0.245549i \(0.0789682\pi\)
\(240\) 112367. 0.125924
\(241\) 1.31915e6 1.46302 0.731512 0.681828i \(-0.238815\pi\)
0.731512 + 0.681828i \(0.238815\pi\)
\(242\) −65576.2 −0.0719793
\(243\) −59049.0 −0.0641500
\(244\) −71447.6 −0.0768268
\(245\) 292313. 0.311124
\(246\) −589991. −0.621595
\(247\) −115656. −0.120621
\(248\) −1.10102e6 −1.13675
\(249\) 230791. 0.235896
\(250\) −69983.5 −0.0708183
\(251\) −237992. −0.238440 −0.119220 0.992868i \(-0.538039\pi\)
−0.119220 + 0.992868i \(0.538039\pi\)
\(252\) 163258. 0.161947
\(253\) 458356. 0.450196
\(254\) −124190. −0.120782
\(255\) −140392. −0.135205
\(256\) −989967. −0.944106
\(257\) −1.90368e6 −1.79789 −0.898943 0.438066i \(-0.855664\pi\)
−0.898943 + 0.438066i \(0.855664\pi\)
\(258\) 6103.75 0.00570883
\(259\) −50859.1 −0.0471107
\(260\) −86635.7 −0.0794811
\(261\) 341854. 0.310628
\(262\) 1.45397e6 1.30858
\(263\) −907019. −0.808588 −0.404294 0.914629i \(-0.632483\pi\)
−0.404294 + 0.914629i \(0.632483\pi\)
\(264\) −214316. −0.189254
\(265\) −452902. −0.396178
\(266\) −301283. −0.261078
\(267\) −16271.2 −0.0139682
\(268\) −268041. −0.227963
\(269\) 698485. 0.588541 0.294270 0.955722i \(-0.404923\pi\)
0.294270 + 0.955722i \(0.404923\pi\)
\(270\) 81628.7 0.0681449
\(271\) 1.19097e6 0.985098 0.492549 0.870285i \(-0.336065\pi\)
0.492549 + 0.870285i \(0.336065\pi\)
\(272\) −311613. −0.255383
\(273\) 441009. 0.358130
\(274\) 267611. 0.215342
\(275\) 75625.0 0.0603023
\(276\) 407034. 0.321631
\(277\) −1.15110e6 −0.901392 −0.450696 0.892677i \(-0.648824\pi\)
−0.450696 + 0.892677i \(0.648824\pi\)
\(278\) 526928. 0.408921
\(279\) −453161. −0.348531
\(280\) −830587. −0.633126
\(281\) −1.54201e6 −1.16498 −0.582492 0.812836i \(-0.697922\pi\)
−0.582492 + 0.812836i \(0.697922\pi\)
\(282\) −545601. −0.408557
\(283\) 1.28590e6 0.954421 0.477211 0.878789i \(-0.341648\pi\)
0.477211 + 0.878789i \(0.341648\pi\)
\(284\) −119601. −0.0879915
\(285\) 89652.5 0.0653808
\(286\) −157307. −0.113719
\(287\) 2.47085e6 1.77069
\(288\) −328925. −0.233677
\(289\) −1.03053e6 −0.725795
\(290\) −472576. −0.329972
\(291\) −105196. −0.0728230
\(292\) 5684.77 0.00390171
\(293\) 1.08802e6 0.740404 0.370202 0.928951i \(-0.379288\pi\)
0.370202 + 0.928951i \(0.379288\pi\)
\(294\) 471331. 0.318023
\(295\) −1.26456e6 −0.846027
\(296\) 59289.3 0.0393321
\(297\) −88209.0 −0.0580259
\(298\) −236427. −0.154225
\(299\) 1.09952e6 0.711255
\(300\) 67157.3 0.0430814
\(301\) −25562.2 −0.0162623
\(302\) 1.92648e6 1.21548
\(303\) 1.22016e6 0.763500
\(304\) 198992. 0.123496
\(305\) 149609. 0.0920889
\(306\) −226371. −0.138203
\(307\) −580340. −0.351428 −0.175714 0.984441i \(-0.556223\pi\)
−0.175714 + 0.984441i \(0.556223\pi\)
\(308\) 243879. 0.146487
\(309\) 796414. 0.474507
\(310\) 626444. 0.370236
\(311\) −1.68606e6 −0.988492 −0.494246 0.869322i \(-0.664556\pi\)
−0.494246 + 0.869322i \(0.664556\pi\)
\(312\) −514109. −0.298998
\(313\) −1.50328e6 −0.867319 −0.433659 0.901077i \(-0.642778\pi\)
−0.433659 + 0.901077i \(0.642778\pi\)
\(314\) 676270. 0.387076
\(315\) −341857. −0.194119
\(316\) 1.01512e6 0.571871
\(317\) 1.31016e6 0.732281 0.366140 0.930560i \(-0.380679\pi\)
0.366140 + 0.930560i \(0.380679\pi\)
\(318\) −730269. −0.404962
\(319\) 510671. 0.280973
\(320\) 854228. 0.466336
\(321\) −1.96868e6 −1.06638
\(322\) 2.86425e6 1.53947
\(323\) −248622. −0.132597
\(324\) −78332.3 −0.0414551
\(325\) 181412. 0.0952704
\(326\) −637821. −0.332395
\(327\) −2.14559e6 −1.10963
\(328\) −2.88041e6 −1.47832
\(329\) 2.28495e6 1.16382
\(330\) 121939. 0.0616394
\(331\) 2.91035e6 1.46007 0.730037 0.683408i \(-0.239503\pi\)
0.730037 + 0.683408i \(0.239503\pi\)
\(332\) 306159. 0.152441
\(333\) 24402.5 0.0120594
\(334\) −248440. −0.121858
\(335\) 561267. 0.273248
\(336\) −758782. −0.366664
\(337\) 2.47133e6 1.18538 0.592688 0.805432i \(-0.298067\pi\)
0.592688 + 0.805432i \(0.298067\pi\)
\(338\) 1.28565e6 0.612111
\(339\) −1.68502e6 −0.796355
\(340\) −186239. −0.0873722
\(341\) −676944. −0.315258
\(342\) 144557. 0.0668306
\(343\) 863415. 0.396264
\(344\) 29799.2 0.0135772
\(345\) −852314. −0.385525
\(346\) −598440. −0.268739
\(347\) 1.75988e6 0.784619 0.392309 0.919833i \(-0.371676\pi\)
0.392309 + 0.919833i \(0.371676\pi\)
\(348\) 453492. 0.200734
\(349\) −386142. −0.169700 −0.0848502 0.996394i \(-0.527041\pi\)
−0.0848502 + 0.996394i \(0.527041\pi\)
\(350\) 472579. 0.206207
\(351\) −211599. −0.0916740
\(352\) −491357. −0.211368
\(353\) −2.44184e6 −1.04299 −0.521495 0.853254i \(-0.674625\pi\)
−0.521495 + 0.853254i \(0.674625\pi\)
\(354\) −2.03900e6 −0.864787
\(355\) 250441. 0.105471
\(356\) −21584.8 −0.00902657
\(357\) 948028. 0.393686
\(358\) 843180. 0.347707
\(359\) 3.77524e6 1.54600 0.772998 0.634408i \(-0.218756\pi\)
0.772998 + 0.634408i \(0.218756\pi\)
\(360\) 398521. 0.162067
\(361\) −2.31733e6 −0.935880
\(362\) 2.92140e6 1.17171
\(363\) −131769. −0.0524864
\(364\) 585027. 0.231431
\(365\) −11903.7 −0.00467681
\(366\) 241232. 0.0941309
\(367\) −3.98351e6 −1.54383 −0.771917 0.635723i \(-0.780702\pi\)
−0.771917 + 0.635723i \(0.780702\pi\)
\(368\) −1.89179e6 −0.728204
\(369\) −1.18553e6 −0.453259
\(370\) −33733.8 −0.0128104
\(371\) 3.05832e6 1.15358
\(372\) −601147. −0.225228
\(373\) 3.13497e6 1.16670 0.583352 0.812219i \(-0.301741\pi\)
0.583352 + 0.812219i \(0.301741\pi\)
\(374\) −338159. −0.125009
\(375\) −140625. −0.0516398
\(376\) −2.66369e6 −0.971660
\(377\) 1.22502e6 0.443904
\(378\) −551216. −0.198423
\(379\) −2.50995e6 −0.897567 −0.448784 0.893640i \(-0.648143\pi\)
−0.448784 + 0.893640i \(0.648143\pi\)
\(380\) 118930. 0.0422505
\(381\) −249548. −0.0880728
\(382\) 3.16043e6 1.10812
\(383\) −4.20287e6 −1.46403 −0.732014 0.681290i \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(384\) 207864. 0.0719368
\(385\) −510675. −0.175587
\(386\) 390311. 0.133334
\(387\) 12264.9 0.00416281
\(388\) −139550. −0.0470598
\(389\) −1.91171e6 −0.640543 −0.320271 0.947326i \(-0.603774\pi\)
−0.320271 + 0.947326i \(0.603774\pi\)
\(390\) 292512. 0.0973829
\(391\) 2.36362e6 0.781871
\(392\) 2.30110e6 0.756344
\(393\) 2.92161e6 0.954202
\(394\) 2.99013e6 0.970396
\(395\) −2.12562e6 −0.685477
\(396\) −117015. −0.0374976
\(397\) 3.62426e6 1.15410 0.577049 0.816709i \(-0.304204\pi\)
0.577049 + 0.816709i \(0.304204\pi\)
\(398\) 3.12903e6 0.990152
\(399\) −605399. −0.190375
\(400\) −312130. −0.0975406
\(401\) −2.22052e6 −0.689593 −0.344797 0.938677i \(-0.612052\pi\)
−0.344797 + 0.938677i \(0.612052\pi\)
\(402\) 904998. 0.279307
\(403\) −1.62388e6 −0.498070
\(404\) 1.61862e6 0.493390
\(405\) 164025. 0.0496904
\(406\) 3.19117e6 0.960805
\(407\) 36453.2 0.0109081
\(408\) −1.10517e6 −0.328684
\(409\) 20564.7 0.00607876 0.00303938 0.999995i \(-0.499033\pi\)
0.00303938 + 0.999995i \(0.499033\pi\)
\(410\) 1.63886e6 0.481485
\(411\) 537739. 0.157024
\(412\) 1.05649e6 0.306637
\(413\) 8.53922e6 2.46345
\(414\) −1.37429e6 −0.394073
\(415\) −641086. −0.182724
\(416\) −1.17869e6 −0.333937
\(417\) 1.05881e6 0.298180
\(418\) 215944. 0.0604505
\(419\) −2.30684e6 −0.641923 −0.320962 0.947092i \(-0.604006\pi\)
−0.320962 + 0.947092i \(0.604006\pi\)
\(420\) −453495. −0.125444
\(421\) 5.00382e6 1.37593 0.687965 0.725744i \(-0.258504\pi\)
0.687965 + 0.725744i \(0.258504\pi\)
\(422\) 2.50032e6 0.683461
\(423\) −1.09633e6 −0.297914
\(424\) −3.56526e6 −0.963111
\(425\) 389978. 0.104729
\(426\) 403816. 0.107810
\(427\) −1.01027e6 −0.268143
\(428\) −2.61158e6 −0.689119
\(429\) −316092. −0.0829222
\(430\) −16954.9 −0.00442204
\(431\) −2.70202e6 −0.700641 −0.350320 0.936630i \(-0.613927\pi\)
−0.350320 + 0.936630i \(0.613927\pi\)
\(432\) 364068. 0.0938585
\(433\) −3.92321e6 −1.00559 −0.502796 0.864405i \(-0.667695\pi\)
−0.502796 + 0.864405i \(0.667695\pi\)
\(434\) −4.23020e6 −1.07805
\(435\) −949596. −0.240611
\(436\) −2.84627e6 −0.717066
\(437\) −1.50938e6 −0.378089
\(438\) −19193.7 −0.00478051
\(439\) −53261.3 −0.0131902 −0.00659508 0.999978i \(-0.502099\pi\)
−0.00659508 + 0.999978i \(0.502099\pi\)
\(440\) 595322. 0.146595
\(441\) 947095. 0.231898
\(442\) −811188. −0.197499
\(443\) 3.30457e6 0.800029 0.400015 0.916509i \(-0.369005\pi\)
0.400015 + 0.916509i \(0.369005\pi\)
\(444\) 32371.5 0.00779302
\(445\) 45197.7 0.0108197
\(446\) 1.71497e6 0.408244
\(447\) −475077. −0.112459
\(448\) −5.76837e6 −1.35787
\(449\) 3.04596e6 0.713031 0.356516 0.934289i \(-0.383965\pi\)
0.356516 + 0.934289i \(0.383965\pi\)
\(450\) −226746. −0.0527848
\(451\) −1.77098e6 −0.409988
\(452\) −2.23529e6 −0.514622
\(453\) 3.87107e6 0.886309
\(454\) 1.81279e6 0.412769
\(455\) −1.22503e6 −0.277407
\(456\) 705747. 0.158941
\(457\) −2.67154e6 −0.598371 −0.299186 0.954195i \(-0.596715\pi\)
−0.299186 + 0.954195i \(0.596715\pi\)
\(458\) −2.97802e6 −0.663382
\(459\) −454870. −0.100776
\(460\) −1.13065e6 −0.249134
\(461\) −3.30382e6 −0.724042 −0.362021 0.932170i \(-0.617913\pi\)
−0.362021 + 0.932170i \(0.617913\pi\)
\(462\) −823421. −0.179480
\(463\) −7.66989e6 −1.66279 −0.831394 0.555684i \(-0.812457\pi\)
−0.831394 + 0.555684i \(0.812457\pi\)
\(464\) −2.10771e6 −0.454482
\(465\) 1.25878e6 0.269971
\(466\) −275596. −0.0587905
\(467\) −7.91117e6 −1.67860 −0.839302 0.543665i \(-0.817036\pi\)
−0.839302 + 0.543665i \(0.817036\pi\)
\(468\) −280700. −0.0592417
\(469\) −3.79008e6 −0.795640
\(470\) 1.51556e6 0.316467
\(471\) 1.35890e6 0.282251
\(472\) −9.95465e6 −2.05670
\(473\) 18321.6 0.00376540
\(474\) −3.42739e6 −0.700676
\(475\) −249035. −0.0506438
\(476\) 1.25762e6 0.254409
\(477\) −1.46740e6 −0.295293
\(478\) −7.66825e6 −1.53506
\(479\) 5.44099e6 1.08352 0.541762 0.840532i \(-0.317757\pi\)
0.541762 + 0.840532i \(0.317757\pi\)
\(480\) 913680. 0.181005
\(481\) 87445.3 0.0172335
\(482\) −5.90840e6 −1.15838
\(483\) 5.75544e6 1.12256
\(484\) −174800. −0.0339178
\(485\) 292212. 0.0564085
\(486\) 264477. 0.0507922
\(487\) −8.50740e6 −1.62545 −0.812727 0.582645i \(-0.802018\pi\)
−0.812727 + 0.582645i \(0.802018\pi\)
\(488\) 1.17772e6 0.223869
\(489\) −1.28164e6 −0.242378
\(490\) −1.30925e6 −0.246339
\(491\) −334168. −0.0625549 −0.0312775 0.999511i \(-0.509958\pi\)
−0.0312775 + 0.999511i \(0.509958\pi\)
\(492\) −1.57268e6 −0.292906
\(493\) 2.63339e6 0.487976
\(494\) 518014. 0.0955046
\(495\) 245025. 0.0449467
\(496\) 2.79397e6 0.509939
\(497\) −1.69116e6 −0.307110
\(498\) −1.03370e6 −0.186776
\(499\) −3.50055e6 −0.629339 −0.314669 0.949201i \(-0.601894\pi\)
−0.314669 + 0.949201i \(0.601894\pi\)
\(500\) −186548. −0.0333707
\(501\) −499216. −0.0888574
\(502\) 1.06595e6 0.188790
\(503\) 6.56301e6 1.15660 0.578300 0.815824i \(-0.303716\pi\)
0.578300 + 0.815824i \(0.303716\pi\)
\(504\) −2.69110e6 −0.471904
\(505\) −3.38932e6 −0.591405
\(506\) −2.05295e6 −0.356453
\(507\) 2.58338e6 0.446343
\(508\) −331042. −0.0569146
\(509\) 109860. 0.0187952 0.00939760 0.999956i \(-0.497009\pi\)
0.00939760 + 0.999956i \(0.497009\pi\)
\(510\) 628807. 0.107051
\(511\) 80382.3 0.0136178
\(512\) 5.17308e6 0.872115
\(513\) 290474. 0.0487320
\(514\) 8.52649e6 1.42352
\(515\) −2.21226e6 −0.367552
\(516\) 16270.2 0.00269010
\(517\) −1.63773e6 −0.269474
\(518\) 227795. 0.0373009
\(519\) −1.20251e6 −0.195961
\(520\) 1.42808e6 0.231603
\(521\) −914334. −0.147574 −0.0737872 0.997274i \(-0.523509\pi\)
−0.0737872 + 0.997274i \(0.523509\pi\)
\(522\) −1.53115e6 −0.245946
\(523\) 4.00814e6 0.640750 0.320375 0.947291i \(-0.396191\pi\)
0.320375 + 0.947291i \(0.396191\pi\)
\(524\) 3.87570e6 0.616626
\(525\) 949601. 0.150364
\(526\) 4.06249e6 0.640218
\(527\) −3.49081e6 −0.547520
\(528\) 543855. 0.0848982
\(529\) 7.91308e6 1.22944
\(530\) 2.02852e6 0.313683
\(531\) −4.09717e6 −0.630592
\(532\) −803100. −0.123024
\(533\) −4.24829e6 −0.647732
\(534\) 72877.7 0.0110597
\(535\) 5.46856e6 0.826016
\(536\) 4.41831e6 0.664269
\(537\) 1.69429e6 0.253543
\(538\) −3.12847e6 −0.465990
\(539\) 1.41480e6 0.209760
\(540\) 217590. 0.0321110
\(541\) −1.87833e6 −0.275918 −0.137959 0.990438i \(-0.544054\pi\)
−0.137959 + 0.990438i \(0.544054\pi\)
\(542\) −5.33431e6 −0.779974
\(543\) 5.87027e6 0.854395
\(544\) −2.53379e6 −0.367091
\(545\) 5.95998e6 0.859515
\(546\) −1.97525e6 −0.283558
\(547\) 87797.6 0.0125463 0.00627313 0.999980i \(-0.498003\pi\)
0.00627313 + 0.999980i \(0.498003\pi\)
\(548\) 713346. 0.101473
\(549\) 484732. 0.0686390
\(550\) −338720. −0.0477457
\(551\) −1.68165e6 −0.235970
\(552\) −6.70944e6 −0.937213
\(553\) 1.43537e7 1.99596
\(554\) 5.15571e6 0.713698
\(555\) −67784.8 −0.00934114
\(556\) 1.40458e6 0.192690
\(557\) −3.46548e6 −0.473288 −0.236644 0.971596i \(-0.576048\pi\)
−0.236644 + 0.971596i \(0.576048\pi\)
\(558\) 2.02968e6 0.275957
\(559\) 43950.6 0.00594888
\(560\) 2.10773e6 0.284017
\(561\) −679497. −0.0911550
\(562\) 6.90655e6 0.922403
\(563\) −2.85737e6 −0.379923 −0.189961 0.981792i \(-0.560836\pi\)
−0.189961 + 0.981792i \(0.560836\pi\)
\(564\) −1.45436e6 −0.192519
\(565\) 4.68062e6 0.616854
\(566\) −5.75946e6 −0.755685
\(567\) −1.10762e6 −0.144688
\(568\) 1.97148e6 0.256402
\(569\) −5.48030e6 −0.709617 −0.354808 0.934939i \(-0.615454\pi\)
−0.354808 + 0.934939i \(0.615454\pi\)
\(570\) −401549. −0.0517667
\(571\) 8.81180e6 1.13103 0.565515 0.824738i \(-0.308677\pi\)
0.565515 + 0.824738i \(0.308677\pi\)
\(572\) −419317. −0.0535861
\(573\) 6.35058e6 0.808029
\(574\) −1.10668e7 −1.40198
\(575\) 2.36754e6 0.298626
\(576\) 2.76770e6 0.347586
\(577\) 502598. 0.0628465 0.0314233 0.999506i \(-0.489996\pi\)
0.0314233 + 0.999506i \(0.489996\pi\)
\(578\) 4.61566e6 0.574665
\(579\) 784291. 0.0972257
\(580\) −1.25970e6 −0.155488
\(581\) 4.32907e6 0.532053
\(582\) 471169. 0.0576593
\(583\) −2.19205e6 −0.267103
\(584\) −93706.2 −0.0113694
\(585\) 587775. 0.0710104
\(586\) −4.87319e6 −0.586232
\(587\) −7.45878e6 −0.893455 −0.446728 0.894670i \(-0.647411\pi\)
−0.446728 + 0.894670i \(0.647411\pi\)
\(588\) 1.25638e6 0.149857
\(589\) 2.22919e6 0.264764
\(590\) 5.66389e6 0.669861
\(591\) 6.00837e6 0.707600
\(592\) −150455. −0.0176442
\(593\) 6.69352e6 0.781660 0.390830 0.920463i \(-0.372188\pi\)
0.390830 + 0.920463i \(0.372188\pi\)
\(594\) 395083. 0.0459433
\(595\) −2.63341e6 −0.304948
\(596\) −630220. −0.0726736
\(597\) 6.28748e6 0.722006
\(598\) −4.92469e6 −0.563153
\(599\) −1.64186e7 −1.86968 −0.934842 0.355065i \(-0.884459\pi\)
−0.934842 + 0.355065i \(0.884459\pi\)
\(600\) −1.10700e6 −0.125537
\(601\) 1.42657e7 1.61105 0.805523 0.592564i \(-0.201884\pi\)
0.805523 + 0.592564i \(0.201884\pi\)
\(602\) 114491. 0.0128760
\(603\) 1.81851e6 0.203667
\(604\) 5.13522e6 0.572752
\(605\) 366025. 0.0406558
\(606\) −5.46501e6 −0.604518
\(607\) 2.38734e6 0.262992 0.131496 0.991317i \(-0.458022\pi\)
0.131496 + 0.991317i \(0.458022\pi\)
\(608\) 1.61805e6 0.177514
\(609\) 6.41235e6 0.700607
\(610\) −670089. −0.0729134
\(611\) −3.92865e6 −0.425736
\(612\) −603414. −0.0651234
\(613\) −1.36125e6 −0.146314 −0.0731570 0.997320i \(-0.523307\pi\)
−0.0731570 + 0.997320i \(0.523307\pi\)
\(614\) 2.59931e6 0.278251
\(615\) 3.29314e6 0.351093
\(616\) −4.02004e6 −0.426853
\(617\) −1.33759e6 −0.141453 −0.0707264 0.997496i \(-0.522532\pi\)
−0.0707264 + 0.997496i \(0.522532\pi\)
\(618\) −3.56709e6 −0.375702
\(619\) 1.70842e7 1.79212 0.896062 0.443930i \(-0.146416\pi\)
0.896062 + 0.443930i \(0.146416\pi\)
\(620\) 1.66985e6 0.174461
\(621\) −2.76150e6 −0.287353
\(622\) 7.55179e6 0.782661
\(623\) −305208. −0.0315047
\(624\) 1.30462e6 0.134129
\(625\) 390625. 0.0400000
\(626\) 6.73310e6 0.686719
\(627\) 433918. 0.0440797
\(628\) 1.80267e6 0.182397
\(629\) 187979. 0.0189445
\(630\) 1.53116e6 0.153698
\(631\) 1.70136e7 1.70107 0.850534 0.525920i \(-0.176279\pi\)
0.850534 + 0.525920i \(0.176279\pi\)
\(632\) −1.67329e7 −1.66640
\(633\) 5.02414e6 0.498371
\(634\) −5.86815e6 −0.579800
\(635\) 693189. 0.0682209
\(636\) −1.94661e6 −0.190825
\(637\) 3.39387e6 0.331395
\(638\) −2.28727e6 −0.222467
\(639\) 811429. 0.0786138
\(640\) −577400. −0.0557220
\(641\) 1.04366e7 1.00326 0.501632 0.865081i \(-0.332733\pi\)
0.501632 + 0.865081i \(0.332733\pi\)
\(642\) 8.81761e6 0.844332
\(643\) 1.32098e7 1.25999 0.629996 0.776598i \(-0.283056\pi\)
0.629996 + 0.776598i \(0.283056\pi\)
\(644\) 7.63496e6 0.725424
\(645\) −34069.1 −0.00322450
\(646\) 1.11356e6 0.104987
\(647\) 3.14204e6 0.295087 0.147544 0.989056i \(-0.452863\pi\)
0.147544 + 0.989056i \(0.452863\pi\)
\(648\) 1.29121e6 0.120798
\(649\) −6.12047e6 −0.570391
\(650\) −812534. −0.0754325
\(651\) −8.50019e6 −0.786097
\(652\) −1.70018e6 −0.156630
\(653\) 9.82598e6 0.901764 0.450882 0.892583i \(-0.351109\pi\)
0.450882 + 0.892583i \(0.351109\pi\)
\(654\) 9.60998e6 0.878574
\(655\) −8.11557e6 −0.739122
\(656\) 7.30942e6 0.663167
\(657\) −38568.0 −0.00348589
\(658\) −1.02341e7 −0.921482
\(659\) −1.78524e7 −1.60134 −0.800670 0.599105i \(-0.795523\pi\)
−0.800670 + 0.599105i \(0.795523\pi\)
\(660\) 325041. 0.0290455
\(661\) 1.02807e7 0.915207 0.457603 0.889156i \(-0.348708\pi\)
0.457603 + 0.889156i \(0.348708\pi\)
\(662\) −1.30353e7 −1.15605
\(663\) −1.63000e6 −0.144014
\(664\) −5.04664e6 −0.444204
\(665\) 1.68166e6 0.147464
\(666\) −109298. −0.00954827
\(667\) 1.59872e7 1.39142
\(668\) −662242. −0.0574216
\(669\) 3.44607e6 0.297686
\(670\) −2.51388e6 −0.216351
\(671\) 724106. 0.0620863
\(672\) −6.16983e6 −0.527047
\(673\) −2.22873e7 −1.89679 −0.948394 0.317094i \(-0.897293\pi\)
−0.948394 + 0.317094i \(0.897293\pi\)
\(674\) −1.10690e7 −0.938549
\(675\) −455625. −0.0384900
\(676\) 3.42702e6 0.288436
\(677\) −1.56465e7 −1.31204 −0.656018 0.754745i \(-0.727761\pi\)
−0.656018 + 0.754745i \(0.727761\pi\)
\(678\) 7.54712e6 0.630532
\(679\) −1.97323e6 −0.164249
\(680\) 3.06991e6 0.254597
\(681\) 3.64262e6 0.300986
\(682\) 3.03199e6 0.249613
\(683\) −1.82208e7 −1.49457 −0.747283 0.664506i \(-0.768642\pi\)
−0.747283 + 0.664506i \(0.768642\pi\)
\(684\) 385333. 0.0314917
\(685\) −1.49372e6 −0.121631
\(686\) −3.86719e6 −0.313751
\(687\) −5.98404e6 −0.483730
\(688\) −75619.5 −0.00609064
\(689\) −5.25837e6 −0.421991
\(690\) 3.81747e6 0.305248
\(691\) 1.07957e7 0.860116 0.430058 0.902801i \(-0.358493\pi\)
0.430058 + 0.902801i \(0.358493\pi\)
\(692\) −1.59520e6 −0.126634
\(693\) −1.65459e6 −0.130875
\(694\) −7.88239e6 −0.621240
\(695\) −2.94114e6 −0.230969
\(696\) −7.47524e6 −0.584927
\(697\) −9.13244e6 −0.712041
\(698\) 1.72951e6 0.134364
\(699\) −553783. −0.0428693
\(700\) 1.25971e6 0.0971683
\(701\) −7.59781e6 −0.583973 −0.291987 0.956422i \(-0.594316\pi\)
−0.291987 + 0.956422i \(0.594316\pi\)
\(702\) 947740. 0.0725849
\(703\) −120041. −0.00916098
\(704\) 4.13447e6 0.314404
\(705\) 3.04537e6 0.230764
\(706\) 1.09369e7 0.825811
\(707\) 2.28872e7 1.72204
\(708\) −5.43516e6 −0.407502
\(709\) 1.26638e7 0.946122 0.473061 0.881030i \(-0.343149\pi\)
0.473061 + 0.881030i \(0.343149\pi\)
\(710\) −1.12171e6 −0.0835094
\(711\) −6.88700e6 −0.510924
\(712\) 355798. 0.0263029
\(713\) −2.11926e7 −1.56121
\(714\) −4.24616e6 −0.311710
\(715\) 878035. 0.0642313
\(716\) 2.24758e6 0.163845
\(717\) −1.54086e7 −1.11935
\(718\) −1.69091e7 −1.22408
\(719\) −8.16704e6 −0.589172 −0.294586 0.955625i \(-0.595182\pi\)
−0.294586 + 0.955625i \(0.595182\pi\)
\(720\) −1.01130e6 −0.0727025
\(721\) 1.49388e7 1.07023
\(722\) 1.03792e7 0.741004
\(723\) −1.18723e7 −0.844677
\(724\) 7.78729e6 0.552128
\(725\) 2.63777e6 0.186377
\(726\) 590186. 0.0415573
\(727\) 1.99934e6 0.140298 0.0701488 0.997537i \(-0.477653\pi\)
0.0701488 + 0.997537i \(0.477653\pi\)
\(728\) −9.64343e6 −0.674377
\(729\) 531441. 0.0370370
\(730\) 53316.0 0.00370297
\(731\) 94479.6 0.00653950
\(732\) 643028. 0.0443560
\(733\) 2.26768e7 1.55892 0.779458 0.626455i \(-0.215495\pi\)
0.779458 + 0.626455i \(0.215495\pi\)
\(734\) 1.78419e7 1.22237
\(735\) −2.63082e6 −0.179627
\(736\) −1.53826e7 −1.04673
\(737\) 2.71653e6 0.184224
\(738\) 5.30992e6 0.358878
\(739\) −1.49990e7 −1.01030 −0.505151 0.863031i \(-0.668563\pi\)
−0.505151 + 0.863031i \(0.668563\pi\)
\(740\) −89920.9 −0.00603645
\(741\) 1.04090e6 0.0696407
\(742\) −1.36981e7 −0.913375
\(743\) −8.72196e6 −0.579618 −0.289809 0.957084i \(-0.593592\pi\)
−0.289809 + 0.957084i \(0.593592\pi\)
\(744\) 9.90914e6 0.656302
\(745\) 1.31966e6 0.0871105
\(746\) −1.40413e7 −0.923765
\(747\) −2.07712e6 −0.136194
\(748\) −901396. −0.0589063
\(749\) −3.69277e7 −2.40518
\(750\) 629851. 0.0408870
\(751\) 3.03235e7 1.96191 0.980957 0.194227i \(-0.0622198\pi\)
0.980957 + 0.194227i \(0.0622198\pi\)
\(752\) 6.75947e6 0.435881
\(753\) 2.14193e6 0.137663
\(754\) −5.48679e6 −0.351471
\(755\) −1.07530e7 −0.686532
\(756\) −1.46932e6 −0.0935002
\(757\) 999401. 0.0633870 0.0316935 0.999498i \(-0.489910\pi\)
0.0316935 + 0.999498i \(0.489910\pi\)
\(758\) 1.12419e7 0.710669
\(759\) −4.12520e6 −0.259921
\(760\) −1.96041e6 −0.123115
\(761\) 1.23318e7 0.771904 0.385952 0.922519i \(-0.373873\pi\)
0.385952 + 0.922519i \(0.373873\pi\)
\(762\) 1.11771e6 0.0697336
\(763\) −4.02461e7 −2.50272
\(764\) 8.42445e6 0.522166
\(765\) 1.26353e6 0.0780605
\(766\) 1.88244e7 1.15918
\(767\) −1.46820e7 −0.901150
\(768\) 8.90970e6 0.545080
\(769\) −3.88179e6 −0.236710 −0.118355 0.992971i \(-0.537762\pi\)
−0.118355 + 0.992971i \(0.537762\pi\)
\(770\) 2.28728e6 0.139025
\(771\) 1.71332e7 1.03801
\(772\) 1.04041e6 0.0628293
\(773\) 2.35138e7 1.41539 0.707693 0.706520i \(-0.249736\pi\)
0.707693 + 0.706520i \(0.249736\pi\)
\(774\) −54933.7 −0.00329600
\(775\) −3.49661e6 −0.209119
\(776\) 2.30030e6 0.137129
\(777\) 457732. 0.0271994
\(778\) 8.56244e6 0.507164
\(779\) 5.83186e6 0.344321
\(780\) 779722. 0.0458884
\(781\) 1.21214e6 0.0711088
\(782\) −1.05865e7 −0.619064
\(783\) −3.07669e6 −0.179341
\(784\) −5.83934e6 −0.339292
\(785\) −3.77472e6 −0.218631
\(786\) −1.30857e7 −0.755511
\(787\) 2.16876e6 0.124817 0.0624086 0.998051i \(-0.480122\pi\)
0.0624086 + 0.998051i \(0.480122\pi\)
\(788\) 7.97049e6 0.457266
\(789\) 8.16317e6 0.466838
\(790\) 9.52052e6 0.542742
\(791\) −3.16069e7 −1.79614
\(792\) 1.92884e6 0.109266
\(793\) 1.73701e6 0.0980890
\(794\) −1.62328e7 −0.913784
\(795\) 4.07612e6 0.228733
\(796\) 8.34075e6 0.466576
\(797\) −1.85869e7 −1.03648 −0.518241 0.855235i \(-0.673413\pi\)
−0.518241 + 0.855235i \(0.673413\pi\)
\(798\) 2.71155e6 0.150733
\(799\) −8.44534e6 −0.468005
\(800\) −2.53800e6 −0.140206
\(801\) 146441. 0.00806456
\(802\) 9.94556e6 0.546001
\(803\) −57613.9 −0.00315310
\(804\) 2.41237e6 0.131614
\(805\) −1.59873e7 −0.869534
\(806\) 7.27326e6 0.394359
\(807\) −6.28637e6 −0.339794
\(808\) −2.66808e7 −1.43771
\(809\) 2.99966e7 1.61139 0.805694 0.592332i \(-0.201793\pi\)
0.805694 + 0.592332i \(0.201793\pi\)
\(810\) −734658. −0.0393435
\(811\) −1.54479e6 −0.0824740 −0.0412370 0.999149i \(-0.513130\pi\)
−0.0412370 + 0.999149i \(0.513130\pi\)
\(812\) 8.50640e6 0.452747
\(813\) −1.07188e7 −0.568747
\(814\) −163272. −0.00863674
\(815\) 3.56011e6 0.187746
\(816\) 2.80451e6 0.147446
\(817\) −60333.5 −0.00316230
\(818\) −92108.3 −0.00481299
\(819\) −3.96908e6 −0.206767
\(820\) 4.36856e6 0.226884
\(821\) −2.60445e7 −1.34852 −0.674261 0.738493i \(-0.735538\pi\)
−0.674261 + 0.738493i \(0.735538\pi\)
\(822\) −2.40850e6 −0.124328
\(823\) −9.56891e6 −0.492451 −0.246226 0.969213i \(-0.579190\pi\)
−0.246226 + 0.969213i \(0.579190\pi\)
\(824\) −1.74150e7 −0.893521
\(825\) −680625. −0.0348155
\(826\) −3.82467e7 −1.95049
\(827\) 1.07164e7 0.544860 0.272430 0.962176i \(-0.412173\pi\)
0.272430 + 0.962176i \(0.412173\pi\)
\(828\) −3.66330e6 −0.185694
\(829\) −1.64337e7 −0.830519 −0.415259 0.909703i \(-0.636309\pi\)
−0.415259 + 0.909703i \(0.636309\pi\)
\(830\) 2.87139e6 0.144676
\(831\) 1.03599e7 0.520419
\(832\) 9.91791e6 0.496720
\(833\) 7.29572e6 0.364297
\(834\) −4.74235e6 −0.236091
\(835\) 1.38671e6 0.0688287
\(836\) 575620. 0.0284853
\(837\) 4.07845e6 0.201225
\(838\) 1.03322e7 0.508257
\(839\) 9.64356e6 0.472969 0.236484 0.971635i \(-0.424005\pi\)
0.236484 + 0.971635i \(0.424005\pi\)
\(840\) 7.47528e6 0.365535
\(841\) −2.69916e6 −0.131595
\(842\) −2.24118e7 −1.08942
\(843\) 1.38781e7 0.672604
\(844\) 6.66485e6 0.322058
\(845\) −7.17606e6 −0.345736
\(846\) 4.91041e6 0.235881
\(847\) −2.47166e6 −0.118381
\(848\) 9.04732e6 0.432046
\(849\) −1.15731e7 −0.551035
\(850\) −1.74669e6 −0.0829216
\(851\) 1.14121e6 0.0540186
\(852\) 1.07641e6 0.0508019
\(853\) −2.52742e7 −1.18934 −0.594669 0.803971i \(-0.702717\pi\)
−0.594669 + 0.803971i \(0.702717\pi\)
\(854\) 4.52492e6 0.212308
\(855\) −806873. −0.0377476
\(856\) 4.30486e7 2.00805
\(857\) −262739. −0.0122200 −0.00611002 0.999981i \(-0.501945\pi\)
−0.00611002 + 0.999981i \(0.501945\pi\)
\(858\) 1.41576e6 0.0656555
\(859\) −9.43849e6 −0.436435 −0.218218 0.975900i \(-0.570024\pi\)
−0.218218 + 0.975900i \(0.570024\pi\)
\(860\) −45194.9 −0.00208374
\(861\) −2.22376e7 −1.02231
\(862\) 1.21022e7 0.554748
\(863\) 3.78002e7 1.72769 0.863847 0.503754i \(-0.168048\pi\)
0.863847 + 0.503754i \(0.168048\pi\)
\(864\) 2.96032e6 0.134913
\(865\) 3.34030e6 0.151791
\(866\) 1.75718e7 0.796200
\(867\) 9.27473e6 0.419038
\(868\) −1.12760e7 −0.507993
\(869\) −1.02880e7 −0.462148
\(870\) 4.25318e6 0.190509
\(871\) 6.51653e6 0.291052
\(872\) 4.69171e7 2.08949
\(873\) 946768. 0.0420444
\(874\) 6.76041e6 0.299360
\(875\) −2.63778e6 −0.116471
\(876\) −51162.9 −0.00225266
\(877\) 2.42348e7 1.06400 0.531999 0.846745i \(-0.321441\pi\)
0.531999 + 0.846745i \(0.321441\pi\)
\(878\) 238554. 0.0104436
\(879\) −9.79220e6 −0.427472
\(880\) −1.51071e6 −0.0657619
\(881\) 1.97060e7 0.855377 0.427689 0.903926i \(-0.359328\pi\)
0.427689 + 0.903926i \(0.359328\pi\)
\(882\) −4.24198e6 −0.183610
\(883\) −2.09867e7 −0.905822 −0.452911 0.891556i \(-0.649614\pi\)
−0.452911 + 0.891556i \(0.649614\pi\)
\(884\) −2.16230e6 −0.0930649
\(885\) 1.13810e7 0.488454
\(886\) −1.48010e7 −0.633441
\(887\) −3.47934e7 −1.48487 −0.742434 0.669919i \(-0.766329\pi\)
−0.742434 + 0.669919i \(0.766329\pi\)
\(888\) −533604. −0.0227084
\(889\) −4.68091e6 −0.198644
\(890\) −202438. −0.00856677
\(891\) 793881. 0.0335013
\(892\) 4.57143e6 0.192371
\(893\) 5.39309e6 0.226313
\(894\) 2.12784e6 0.0890421
\(895\) −4.70636e6 −0.196394
\(896\) 3.89902e6 0.162250
\(897\) −9.89569e6 −0.410644
\(898\) −1.36427e7 −0.564559
\(899\) −2.36115e7 −0.974370
\(900\) −604416. −0.0248731
\(901\) −1.13038e7 −0.463887
\(902\) 7.93210e6 0.324617
\(903\) 230059. 0.00938903
\(904\) 3.68460e7 1.49958
\(905\) −1.63063e7 −0.661811
\(906\) −1.73383e7 −0.701756
\(907\) −4.14396e7 −1.67262 −0.836310 0.548257i \(-0.815291\pi\)
−0.836310 + 0.548257i \(0.815291\pi\)
\(908\) 4.83217e6 0.194503
\(909\) −1.09814e7 −0.440807
\(910\) 5.48682e6 0.219643
\(911\) −4.55462e7 −1.81826 −0.909131 0.416511i \(-0.863253\pi\)
−0.909131 + 0.416511i \(0.863253\pi\)
\(912\) −1.79093e6 −0.0713002
\(913\) −3.10285e6 −0.123193
\(914\) 1.19657e7 0.473774
\(915\) −1.34648e6 −0.0531675
\(916\) −7.93822e6 −0.312597
\(917\) 5.48022e7 2.15216
\(918\) 2.03734e6 0.0797914
\(919\) 8.41097e6 0.328517 0.164258 0.986417i \(-0.447477\pi\)
0.164258 + 0.986417i \(0.447477\pi\)
\(920\) 1.86373e7 0.725962
\(921\) 5.22306e6 0.202897
\(922\) 1.47976e7 0.573277
\(923\) 2.90772e6 0.112343
\(924\) −2.19491e6 −0.0845741
\(925\) 188291. 0.00723562
\(926\) 3.43530e7 1.31655
\(927\) −7.16773e6 −0.273957
\(928\) −1.71383e7 −0.653278
\(929\) 1.89074e6 0.0718773 0.0359386 0.999354i \(-0.488558\pi\)
0.0359386 + 0.999354i \(0.488558\pi\)
\(930\) −5.63800e6 −0.213756
\(931\) −4.65895e6 −0.176163
\(932\) −734628. −0.0277031
\(933\) 1.51746e7 0.570706
\(934\) 3.54337e7 1.32907
\(935\) 1.88749e6 0.0706083
\(936\) 4.62698e6 0.172627
\(937\) 3.58861e7 1.33530 0.667648 0.744477i \(-0.267301\pi\)
0.667648 + 0.744477i \(0.267301\pi\)
\(938\) 1.69756e7 0.629966
\(939\) 1.35295e7 0.500747
\(940\) 4.03988e6 0.149124
\(941\) 3.25537e7 1.19847 0.599235 0.800573i \(-0.295472\pi\)
0.599235 + 0.800573i \(0.295472\pi\)
\(942\) −6.08643e6 −0.223478
\(943\) −5.54427e7 −2.03032
\(944\) 2.52612e7 0.922624
\(945\) 3.07671e6 0.112075
\(946\) −82061.5 −0.00298134
\(947\) 6.44146e6 0.233405 0.116702 0.993167i \(-0.462768\pi\)
0.116702 + 0.993167i \(0.462768\pi\)
\(948\) −9.13605e6 −0.330170
\(949\) −138206. −0.00498153
\(950\) 1.11541e6 0.0400984
\(951\) −1.17915e7 −0.422782
\(952\) −2.07303e7 −0.741332
\(953\) 2.40023e7 0.856093 0.428046 0.903757i \(-0.359202\pi\)
0.428046 + 0.903757i \(0.359202\pi\)
\(954\) 6.57242e6 0.233805
\(955\) −1.76405e7 −0.625897
\(956\) −2.04405e7 −0.723347
\(957\) −4.59604e6 −0.162220
\(958\) −2.43699e7 −0.857905
\(959\) 1.00867e7 0.354162
\(960\) −7.68806e6 −0.269239
\(961\) 2.67012e6 0.0932656
\(962\) −391662. −0.0136450
\(963\) 1.77181e7 0.615676
\(964\) −1.57494e7 −0.545849
\(965\) −2.17859e6 −0.0753107
\(966\) −2.57783e7 −0.888815
\(967\) 4.13239e7 1.42113 0.710567 0.703630i \(-0.248439\pi\)
0.710567 + 0.703630i \(0.248439\pi\)
\(968\) 2.88136e6 0.0988345
\(969\) 2.23760e6 0.0765549
\(970\) −1.30880e6 −0.0446627
\(971\) 8.46004e6 0.287955 0.143977 0.989581i \(-0.454011\pi\)
0.143977 + 0.989581i \(0.454011\pi\)
\(972\) 704991. 0.0239341
\(973\) 1.98607e7 0.672531
\(974\) 3.81042e7 1.28699
\(975\) −1.63271e6 −0.0550044
\(976\) −2.98863e6 −0.100426
\(977\) 1.33050e7 0.445942 0.222971 0.974825i \(-0.428425\pi\)
0.222971 + 0.974825i \(0.428425\pi\)
\(978\) 5.74039e6 0.191909
\(979\) 218757. 0.00729467
\(980\) −3.48995e6 −0.116079
\(981\) 1.93103e7 0.640645
\(982\) 1.49672e6 0.0495293
\(983\) 2.90239e7 0.958014 0.479007 0.877811i \(-0.340997\pi\)
0.479007 + 0.877811i \(0.340997\pi\)
\(984\) 2.59237e7 0.853510
\(985\) −1.66899e7 −0.548105
\(986\) −1.17948e7 −0.386366
\(987\) −2.05645e7 −0.671933
\(988\) 1.38082e6 0.0450033
\(989\) 573582. 0.0186468
\(990\) −1.09745e6 −0.0355875
\(991\) −4.88551e7 −1.58025 −0.790125 0.612946i \(-0.789984\pi\)
−0.790125 + 0.612946i \(0.789984\pi\)
\(992\) 2.27184e7 0.732992
\(993\) −2.61931e7 −0.842974
\(994\) 7.57460e6 0.243161
\(995\) −1.74652e7 −0.559264
\(996\) −2.75543e6 −0.0880118
\(997\) −2.19218e7 −0.698453 −0.349227 0.937038i \(-0.613556\pi\)
−0.349227 + 0.937038i \(0.613556\pi\)
\(998\) 1.56787e7 0.498293
\(999\) −219623. −0.00696248
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.1.1 3
3.2 odd 2 495.6.a.b.1.3 3
5.4 even 2 825.6.a.g.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.c.1.1 3 1.1 even 1 trivial
495.6.a.b.1.3 3 3.2 odd 2
825.6.a.g.1.3 3 5.4 even 2