Properties

Label 165.6.a.d.1.3
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) 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 \(-1.87740\) 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)
 
\(f(q)\) \(=\) \(q+6.55890 q^{2} -9.00000 q^{3} +11.0192 q^{4} +25.0000 q^{5} -59.0301 q^{6} +146.487 q^{7} -137.611 q^{8} +81.0000 q^{9} +163.973 q^{10} -121.000 q^{11} -99.1731 q^{12} +170.238 q^{13} +960.792 q^{14} -225.000 q^{15} -1255.19 q^{16} +1564.81 q^{17} +531.271 q^{18} +569.786 q^{19} +275.481 q^{20} -1318.38 q^{21} -793.627 q^{22} +3153.25 q^{23} +1238.50 q^{24} +625.000 q^{25} +1116.57 q^{26} -729.000 q^{27} +1614.17 q^{28} +3982.58 q^{29} -1475.75 q^{30} +2990.78 q^{31} -3829.14 q^{32} +1089.00 q^{33} +10263.5 q^{34} +3662.17 q^{35} +892.558 q^{36} +7858.92 q^{37} +3737.17 q^{38} -1532.14 q^{39} -3440.27 q^{40} -5206.60 q^{41} -8647.13 q^{42} +13874.3 q^{43} -1333.33 q^{44} +2025.00 q^{45} +20681.9 q^{46} +6852.01 q^{47} +11296.7 q^{48} +4651.35 q^{49} +4099.32 q^{50} -14083.3 q^{51} +1875.89 q^{52} -3834.15 q^{53} -4781.44 q^{54} -3025.00 q^{55} -20158.2 q^{56} -5128.08 q^{57} +26121.4 q^{58} +9649.38 q^{59} -2479.33 q^{60} -21131.8 q^{61} +19616.2 q^{62} +11865.4 q^{63} +15051.2 q^{64} +4255.95 q^{65} +7142.65 q^{66} -43499.9 q^{67} +17243.0 q^{68} -28379.2 q^{69} +24019.8 q^{70} -52607.2 q^{71} -11146.5 q^{72} -64367.9 q^{73} +51545.9 q^{74} -5625.00 q^{75} +6278.61 q^{76} -17724.9 q^{77} -10049.2 q^{78} +28935.7 q^{79} -31379.8 q^{80} +6561.00 q^{81} -34149.6 q^{82} -4648.64 q^{83} -14527.5 q^{84} +39120.3 q^{85} +91000.1 q^{86} -35843.2 q^{87} +16650.9 q^{88} -103458. q^{89} +13281.8 q^{90} +24937.6 q^{91} +34746.4 q^{92} -26917.0 q^{93} +44941.6 q^{94} +14244.7 q^{95} +34462.2 q^{96} +75809.5 q^{97} +30507.8 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 27 q^{3} - 20 q^{4} + 75 q^{5} - 18 q^{6} + 152 q^{7} - 24 q^{8} + 243 q^{9} + 50 q^{10} - 363 q^{11} + 180 q^{12} - 546 q^{13} - 8 q^{14} - 675 q^{15} - 1360 q^{16} - 314 q^{17} + 162 q^{18}+ \cdots - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.55890 1.15946 0.579731 0.814808i \(-0.303158\pi\)
0.579731 + 0.814808i \(0.303158\pi\)
\(3\) −9.00000 −0.577350
\(4\) 11.0192 0.344351
\(5\) 25.0000 0.447214
\(6\) −59.0301 −0.669415
\(7\) 146.487 1.12993 0.564967 0.825113i \(-0.308889\pi\)
0.564967 + 0.825113i \(0.308889\pi\)
\(8\) −137.611 −0.760200
\(9\) 81.0000 0.333333
\(10\) 163.973 0.518527
\(11\) −121.000 −0.301511
\(12\) −99.1731 −0.198811
\(13\) 170.238 0.279382 0.139691 0.990195i \(-0.455389\pi\)
0.139691 + 0.990195i \(0.455389\pi\)
\(14\) 960.792 1.31012
\(15\) −225.000 −0.258199
\(16\) −1255.19 −1.22577
\(17\) 1564.81 1.31323 0.656614 0.754227i \(-0.271988\pi\)
0.656614 + 0.754227i \(0.271988\pi\)
\(18\) 531.271 0.386487
\(19\) 569.786 0.362100 0.181050 0.983474i \(-0.442050\pi\)
0.181050 + 0.983474i \(0.442050\pi\)
\(20\) 275.481 0.153998
\(21\) −1318.38 −0.652368
\(22\) −793.627 −0.349591
\(23\) 3153.25 1.24291 0.621454 0.783451i \(-0.286542\pi\)
0.621454 + 0.783451i \(0.286542\pi\)
\(24\) 1238.50 0.438902
\(25\) 625.000 0.200000
\(26\) 1116.57 0.323932
\(27\) −729.000 −0.192450
\(28\) 1614.17 0.389094
\(29\) 3982.58 0.879366 0.439683 0.898153i \(-0.355091\pi\)
0.439683 + 0.898153i \(0.355091\pi\)
\(30\) −1475.75 −0.299372
\(31\) 2990.78 0.558959 0.279480 0.960152i \(-0.409838\pi\)
0.279480 + 0.960152i \(0.409838\pi\)
\(32\) −3829.14 −0.661037
\(33\) 1089.00 0.174078
\(34\) 10263.5 1.52264
\(35\) 3662.17 0.505322
\(36\) 892.558 0.114784
\(37\) 7858.92 0.943753 0.471877 0.881665i \(-0.343577\pi\)
0.471877 + 0.881665i \(0.343577\pi\)
\(38\) 3737.17 0.419841
\(39\) −1532.14 −0.161301
\(40\) −3440.27 −0.339972
\(41\) −5206.60 −0.483721 −0.241860 0.970311i \(-0.577758\pi\)
−0.241860 + 0.970311i \(0.577758\pi\)
\(42\) −8647.13 −0.756395
\(43\) 13874.3 1.14430 0.572149 0.820149i \(-0.306110\pi\)
0.572149 + 0.820149i \(0.306110\pi\)
\(44\) −1333.33 −0.103826
\(45\) 2025.00 0.149071
\(46\) 20681.9 1.44110
\(47\) 6852.01 0.452453 0.226226 0.974075i \(-0.427361\pi\)
0.226226 + 0.974075i \(0.427361\pi\)
\(48\) 11296.7 0.707701
\(49\) 4651.35 0.276751
\(50\) 4099.32 0.231892
\(51\) −14083.3 −0.758192
\(52\) 1875.89 0.0962054
\(53\) −3834.15 −0.187490 −0.0937452 0.995596i \(-0.529884\pi\)
−0.0937452 + 0.995596i \(0.529884\pi\)
\(54\) −4781.44 −0.223138
\(55\) −3025.00 −0.134840
\(56\) −20158.2 −0.858976
\(57\) −5128.08 −0.209058
\(58\) 26121.4 1.01959
\(59\) 9649.38 0.360885 0.180443 0.983586i \(-0.442247\pi\)
0.180443 + 0.983586i \(0.442247\pi\)
\(60\) −2479.33 −0.0889110
\(61\) −21131.8 −0.727128 −0.363564 0.931569i \(-0.618440\pi\)
−0.363564 + 0.931569i \(0.618440\pi\)
\(62\) 19616.2 0.648092
\(63\) 11865.4 0.376645
\(64\) 15051.2 0.459326
\(65\) 4255.95 0.124943
\(66\) 7142.65 0.201836
\(67\) −43499.9 −1.18386 −0.591931 0.805989i \(-0.701634\pi\)
−0.591931 + 0.805989i \(0.701634\pi\)
\(68\) 17243.0 0.452211
\(69\) −28379.2 −0.717593
\(70\) 24019.8 0.585901
\(71\) −52607.2 −1.23851 −0.619255 0.785190i \(-0.712565\pi\)
−0.619255 + 0.785190i \(0.712565\pi\)
\(72\) −11146.5 −0.253400
\(73\) −64367.9 −1.41372 −0.706858 0.707355i \(-0.749888\pi\)
−0.706858 + 0.707355i \(0.749888\pi\)
\(74\) 51545.9 1.09425
\(75\) −5625.00 −0.115470
\(76\) 6278.61 0.124689
\(77\) −17724.9 −0.340688
\(78\) −10049.2 −0.187022
\(79\) 28935.7 0.521635 0.260817 0.965388i \(-0.416008\pi\)
0.260817 + 0.965388i \(0.416008\pi\)
\(80\) −31379.8 −0.548183
\(81\) 6561.00 0.111111
\(82\) −34149.6 −0.560855
\(83\) −4648.64 −0.0740681 −0.0370340 0.999314i \(-0.511791\pi\)
−0.0370340 + 0.999314i \(0.511791\pi\)
\(84\) −14527.5 −0.224643
\(85\) 39120.3 0.587293
\(86\) 91000.1 1.32677
\(87\) −35843.2 −0.507702
\(88\) 16650.9 0.229209
\(89\) −103458. −1.38448 −0.692242 0.721666i \(-0.743377\pi\)
−0.692242 + 0.721666i \(0.743377\pi\)
\(90\) 13281.8 0.172842
\(91\) 24937.6 0.315683
\(92\) 34746.4 0.427996
\(93\) −26917.0 −0.322715
\(94\) 44941.6 0.524601
\(95\) 14244.7 0.161936
\(96\) 34462.2 0.381650
\(97\) 75809.5 0.818077 0.409039 0.912517i \(-0.365864\pi\)
0.409039 + 0.912517i \(0.365864\pi\)
\(98\) 30507.8 0.320882
\(99\) −9801.00 −0.100504
\(100\) 6887.02 0.0688702
\(101\) 35023.6 0.341631 0.170816 0.985303i \(-0.445360\pi\)
0.170816 + 0.985303i \(0.445360\pi\)
\(102\) −92371.1 −0.879095
\(103\) 16965.7 0.157572 0.0787859 0.996892i \(-0.474896\pi\)
0.0787859 + 0.996892i \(0.474896\pi\)
\(104\) −23426.6 −0.212386
\(105\) −32959.5 −0.291748
\(106\) −25147.8 −0.217388
\(107\) 12190.4 0.102934 0.0514671 0.998675i \(-0.483610\pi\)
0.0514671 + 0.998675i \(0.483610\pi\)
\(108\) −8033.02 −0.0662704
\(109\) 12138.7 0.0978606 0.0489303 0.998802i \(-0.484419\pi\)
0.0489303 + 0.998802i \(0.484419\pi\)
\(110\) −19840.7 −0.156342
\(111\) −70730.3 −0.544876
\(112\) −183869. −1.38504
\(113\) 204649. 1.50770 0.753848 0.657048i \(-0.228195\pi\)
0.753848 + 0.657048i \(0.228195\pi\)
\(114\) −33634.6 −0.242395
\(115\) 78831.2 0.555845
\(116\) 43885.0 0.302810
\(117\) 13789.3 0.0931273
\(118\) 63289.3 0.418433
\(119\) 229224. 1.48386
\(120\) 30962.4 0.196283
\(121\) 14641.0 0.0909091
\(122\) −138601. −0.843077
\(123\) 46859.4 0.279276
\(124\) 32956.1 0.192478
\(125\) 15625.0 0.0894427
\(126\) 77824.2 0.436705
\(127\) −112241. −0.617506 −0.308753 0.951142i \(-0.599912\pi\)
−0.308753 + 0.951142i \(0.599912\pi\)
\(128\) 221252. 1.19361
\(129\) −124869. −0.660661
\(130\) 27914.4 0.144867
\(131\) −11679.0 −0.0594602 −0.0297301 0.999558i \(-0.509465\pi\)
−0.0297301 + 0.999558i \(0.509465\pi\)
\(132\) 11999.9 0.0599438
\(133\) 83466.1 0.409149
\(134\) −285311. −1.37264
\(135\) −18225.0 −0.0860663
\(136\) −215335. −0.998315
\(137\) −384935. −1.75221 −0.876105 0.482121i \(-0.839867\pi\)
−0.876105 + 0.482121i \(0.839867\pi\)
\(138\) −186137. −0.832021
\(139\) −310605. −1.36355 −0.681775 0.731562i \(-0.738792\pi\)
−0.681775 + 0.731562i \(0.738792\pi\)
\(140\) 40354.3 0.174008
\(141\) −61668.0 −0.261224
\(142\) −345046. −1.43600
\(143\) −20598.8 −0.0842368
\(144\) −101671. −0.408591
\(145\) 99564.5 0.393264
\(146\) −422183. −1.63915
\(147\) −41862.2 −0.159782
\(148\) 86599.2 0.324982
\(149\) 285203. 1.05242 0.526209 0.850355i \(-0.323613\pi\)
0.526209 + 0.850355i \(0.323613\pi\)
\(150\) −36893.8 −0.133883
\(151\) −522357. −1.86434 −0.932170 0.362020i \(-0.882087\pi\)
−0.932170 + 0.362020i \(0.882087\pi\)
\(152\) −78408.8 −0.275268
\(153\) 126750. 0.437743
\(154\) −116256. −0.395015
\(155\) 74769.5 0.249974
\(156\) −16883.0 −0.0555442
\(157\) −2221.66 −0.00719330 −0.00359665 0.999994i \(-0.501145\pi\)
−0.00359665 + 0.999994i \(0.501145\pi\)
\(158\) 189787. 0.604816
\(159\) 34507.3 0.108248
\(160\) −95728.4 −0.295625
\(161\) 461909. 1.40440
\(162\) 43033.0 0.128829
\(163\) 250146. 0.737436 0.368718 0.929541i \(-0.379797\pi\)
0.368718 + 0.929541i \(0.379797\pi\)
\(164\) −57372.7 −0.166570
\(165\) 27225.0 0.0778499
\(166\) −30490.0 −0.0858791
\(167\) −42083.4 −0.116767 −0.0583834 0.998294i \(-0.518595\pi\)
−0.0583834 + 0.998294i \(0.518595\pi\)
\(168\) 181423. 0.495930
\(169\) −342312. −0.921946
\(170\) 256586. 0.680944
\(171\) 46152.7 0.120700
\(172\) 152884. 0.394040
\(173\) −117206. −0.297737 −0.148869 0.988857i \(-0.547563\pi\)
−0.148869 + 0.988857i \(0.547563\pi\)
\(174\) −235092. −0.588661
\(175\) 91554.2 0.225987
\(176\) 151878. 0.369585
\(177\) −86844.4 −0.208357
\(178\) −678569. −1.60526
\(179\) 582985. 1.35996 0.679978 0.733233i \(-0.261989\pi\)
0.679978 + 0.733233i \(0.261989\pi\)
\(180\) 22313.9 0.0513328
\(181\) 235291. 0.533837 0.266919 0.963719i \(-0.413994\pi\)
0.266919 + 0.963719i \(0.413994\pi\)
\(182\) 163563. 0.366022
\(183\) 190186. 0.419808
\(184\) −433921. −0.944858
\(185\) 196473. 0.422059
\(186\) −176546. −0.374176
\(187\) −189342. −0.395953
\(188\) 75503.8 0.155802
\(189\) −106789. −0.217456
\(190\) 93429.4 0.187758
\(191\) 591788. 1.17377 0.586885 0.809670i \(-0.300354\pi\)
0.586885 + 0.809670i \(0.300354\pi\)
\(192\) −135461. −0.265192
\(193\) −540825. −1.04511 −0.522557 0.852605i \(-0.675022\pi\)
−0.522557 + 0.852605i \(0.675022\pi\)
\(194\) 497227. 0.948529
\(195\) −38303.5 −0.0721361
\(196\) 51254.3 0.0952994
\(197\) 481711. 0.884344 0.442172 0.896930i \(-0.354208\pi\)
0.442172 + 0.896930i \(0.354208\pi\)
\(198\) −64283.8 −0.116530
\(199\) −1.01427e6 −1.81560 −0.907801 0.419401i \(-0.862240\pi\)
−0.907801 + 0.419401i \(0.862240\pi\)
\(200\) −86006.8 −0.152040
\(201\) 391499. 0.683503
\(202\) 229717. 0.396108
\(203\) 583395. 0.993625
\(204\) −155187. −0.261084
\(205\) −130165. −0.216326
\(206\) 111276. 0.182699
\(207\) 255413. 0.414302
\(208\) −213681. −0.342459
\(209\) −68944.2 −0.109177
\(210\) −216178. −0.338270
\(211\) 619718. 0.958270 0.479135 0.877741i \(-0.340950\pi\)
0.479135 + 0.877741i \(0.340950\pi\)
\(212\) −42249.3 −0.0645625
\(213\) 473465. 0.715054
\(214\) 79955.8 0.119348
\(215\) 346857. 0.511746
\(216\) 100318. 0.146301
\(217\) 438109. 0.631587
\(218\) 79616.9 0.113466
\(219\) 579311. 0.816210
\(220\) −33333.2 −0.0464323
\(221\) 266391. 0.366892
\(222\) −463913. −0.631763
\(223\) 894252. 1.20420 0.602099 0.798422i \(-0.294331\pi\)
0.602099 + 0.798422i \(0.294331\pi\)
\(224\) −560918. −0.746928
\(225\) 50625.0 0.0666667
\(226\) 1.34227e6 1.74812
\(227\) 851468. 1.09674 0.548370 0.836236i \(-0.315249\pi\)
0.548370 + 0.836236i \(0.315249\pi\)
\(228\) −56507.5 −0.0719894
\(229\) 137840. 0.173695 0.0868475 0.996222i \(-0.472321\pi\)
0.0868475 + 0.996222i \(0.472321\pi\)
\(230\) 517047. 0.644481
\(231\) 159524. 0.196696
\(232\) −548046. −0.668494
\(233\) −1.31590e6 −1.58793 −0.793965 0.607963i \(-0.791987\pi\)
−0.793965 + 0.607963i \(0.791987\pi\)
\(234\) 90442.6 0.107977
\(235\) 171300. 0.202343
\(236\) 106329. 0.124271
\(237\) −260422. −0.301166
\(238\) 1.50346e6 1.72048
\(239\) 1.11778e6 1.26579 0.632896 0.774237i \(-0.281866\pi\)
0.632896 + 0.774237i \(0.281866\pi\)
\(240\) 282418. 0.316493
\(241\) −471581. −0.523015 −0.261507 0.965201i \(-0.584220\pi\)
−0.261507 + 0.965201i \(0.584220\pi\)
\(242\) 96028.9 0.105406
\(243\) −59049.0 −0.0641500
\(244\) −232856. −0.250387
\(245\) 116284. 0.123767
\(246\) 307346. 0.323810
\(247\) 96999.3 0.101164
\(248\) −411564. −0.424921
\(249\) 41837.8 0.0427632
\(250\) 102483. 0.103705
\(251\) 1.76541e6 1.76873 0.884363 0.466799i \(-0.154593\pi\)
0.884363 + 0.466799i \(0.154593\pi\)
\(252\) 130748. 0.129698
\(253\) −381543. −0.374751
\(254\) −736176. −0.715974
\(255\) −352083. −0.339074
\(256\) 969531. 0.924617
\(257\) −19920.1 −0.0188130 −0.00940652 0.999956i \(-0.502994\pi\)
−0.00940652 + 0.999956i \(0.502994\pi\)
\(258\) −819001. −0.766011
\(259\) 1.15123e6 1.06638
\(260\) 46897.3 0.0430244
\(261\) 322589. 0.293122
\(262\) −76601.2 −0.0689418
\(263\) −2.18348e6 −1.94652 −0.973262 0.229697i \(-0.926227\pi\)
−0.973262 + 0.229697i \(0.926227\pi\)
\(264\) −149858. −0.132334
\(265\) −95853.7 −0.0838483
\(266\) 547446. 0.474392
\(267\) 931119. 0.799332
\(268\) −479335. −0.407664
\(269\) 839300. 0.707190 0.353595 0.935399i \(-0.384959\pi\)
0.353595 + 0.935399i \(0.384959\pi\)
\(270\) −119536. −0.0997906
\(271\) 1.42641e6 1.17984 0.589918 0.807463i \(-0.299160\pi\)
0.589918 + 0.807463i \(0.299160\pi\)
\(272\) −1.96414e6 −1.60972
\(273\) −224438. −0.182260
\(274\) −2.52475e6 −2.03162
\(275\) −75625.0 −0.0603023
\(276\) −312717. −0.247104
\(277\) −1.30448e6 −1.02150 −0.510749 0.859730i \(-0.670632\pi\)
−0.510749 + 0.859730i \(0.670632\pi\)
\(278\) −2.03723e6 −1.58098
\(279\) 242253. 0.186320
\(280\) −503954. −0.384146
\(281\) −580646. −0.438678 −0.219339 0.975649i \(-0.570390\pi\)
−0.219339 + 0.975649i \(0.570390\pi\)
\(282\) −404475. −0.302879
\(283\) −601974. −0.446799 −0.223399 0.974727i \(-0.571715\pi\)
−0.223399 + 0.974727i \(0.571715\pi\)
\(284\) −579691. −0.426482
\(285\) −128202. −0.0934937
\(286\) −135106. −0.0976693
\(287\) −762698. −0.546572
\(288\) −310160. −0.220346
\(289\) 1.02878e6 0.724567
\(290\) 653034. 0.455975
\(291\) −682286. −0.472317
\(292\) −709285. −0.486815
\(293\) −2.40993e6 −1.63996 −0.819982 0.572389i \(-0.806017\pi\)
−0.819982 + 0.572389i \(0.806017\pi\)
\(294\) −274570. −0.185261
\(295\) 241234. 0.161393
\(296\) −1.08147e6 −0.717441
\(297\) 88209.0 0.0580259
\(298\) 1.87062e6 1.22024
\(299\) 536803. 0.347246
\(300\) −61983.2 −0.0397622
\(301\) 2.03240e6 1.29298
\(302\) −3.42609e6 −2.16163
\(303\) −315213. −0.197241
\(304\) −715191. −0.443852
\(305\) −528294. −0.325182
\(306\) 831340. 0.507546
\(307\) −1.83380e6 −1.11047 −0.555235 0.831694i \(-0.687371\pi\)
−0.555235 + 0.831694i \(0.687371\pi\)
\(308\) −195315. −0.117316
\(309\) −152691. −0.0909742
\(310\) 490406. 0.289835
\(311\) 2.27507e6 1.33381 0.666906 0.745142i \(-0.267618\pi\)
0.666906 + 0.745142i \(0.267618\pi\)
\(312\) 210839. 0.122621
\(313\) 159585. 0.0920728 0.0460364 0.998940i \(-0.485341\pi\)
0.0460364 + 0.998940i \(0.485341\pi\)
\(314\) −14571.6 −0.00834035
\(315\) 296636. 0.168441
\(316\) 318849. 0.179625
\(317\) −1.24338e6 −0.694955 −0.347477 0.937688i \(-0.612962\pi\)
−0.347477 + 0.937688i \(0.612962\pi\)
\(318\) 226330. 0.125509
\(319\) −481892. −0.265139
\(320\) 376280. 0.205417
\(321\) −109714. −0.0594290
\(322\) 3.02962e6 1.62835
\(323\) 891609. 0.475519
\(324\) 72297.2 0.0382612
\(325\) 106399. 0.0558764
\(326\) 1.64068e6 0.855029
\(327\) −109249. −0.0564998
\(328\) 716485. 0.367724
\(329\) 1.00373e6 0.511242
\(330\) 178566. 0.0902640
\(331\) −958448. −0.480838 −0.240419 0.970669i \(-0.577285\pi\)
−0.240419 + 0.970669i \(0.577285\pi\)
\(332\) −51224.5 −0.0255054
\(333\) 636573. 0.314584
\(334\) −276021. −0.135387
\(335\) −1.08750e6 −0.529439
\(336\) 1.65482e6 0.799655
\(337\) −1.51074e6 −0.724626 −0.362313 0.932056i \(-0.618013\pi\)
−0.362313 + 0.932056i \(0.618013\pi\)
\(338\) −2.24519e6 −1.06896
\(339\) −1.84184e6 −0.870469
\(340\) 431076. 0.202235
\(341\) −361884. −0.168533
\(342\) 302711. 0.139947
\(343\) −1.78064e6 −0.817224
\(344\) −1.90925e6 −0.869896
\(345\) −709481. −0.320917
\(346\) −768741. −0.345215
\(347\) −1.93847e6 −0.864243 −0.432122 0.901815i \(-0.642235\pi\)
−0.432122 + 0.901815i \(0.642235\pi\)
\(348\) −394965. −0.174828
\(349\) −1.62462e6 −0.713985 −0.356992 0.934107i \(-0.616198\pi\)
−0.356992 + 0.934107i \(0.616198\pi\)
\(350\) 600495. 0.262023
\(351\) −124103. −0.0537671
\(352\) 463325. 0.199310
\(353\) −23347.5 −0.00997249 −0.00498624 0.999988i \(-0.501587\pi\)
−0.00498624 + 0.999988i \(0.501587\pi\)
\(354\) −569604. −0.241582
\(355\) −1.31518e6 −0.553878
\(356\) −1.14002e6 −0.476748
\(357\) −2.06302e6 −0.856707
\(358\) 3.82374e6 1.57682
\(359\) 3.63918e6 1.49028 0.745139 0.666909i \(-0.232383\pi\)
0.745139 + 0.666909i \(0.232383\pi\)
\(360\) −278662. −0.113324
\(361\) −2.15144e6 −0.868884
\(362\) 1.54325e6 0.618964
\(363\) −131769. −0.0524864
\(364\) 274793. 0.108706
\(365\) −1.60920e6 −0.632233
\(366\) 1.24741e6 0.486751
\(367\) 2.24713e6 0.870890 0.435445 0.900215i \(-0.356591\pi\)
0.435445 + 0.900215i \(0.356591\pi\)
\(368\) −3.95793e6 −1.52352
\(369\) −421735. −0.161240
\(370\) 1.28865e6 0.489362
\(371\) −561651. −0.211852
\(372\) −296605. −0.111127
\(373\) 3.82745e6 1.42442 0.712208 0.701968i \(-0.247695\pi\)
0.712208 + 0.701968i \(0.247695\pi\)
\(374\) −1.24188e6 −0.459092
\(375\) −140625. −0.0516398
\(376\) −942910. −0.343954
\(377\) 677986. 0.245679
\(378\) −700418. −0.252132
\(379\) −4.26443e6 −1.52498 −0.762489 0.647002i \(-0.776023\pi\)
−0.762489 + 0.647002i \(0.776023\pi\)
\(380\) 156965. 0.0557628
\(381\) 1.01017e6 0.356517
\(382\) 3.88148e6 1.36094
\(383\) 2.40920e6 0.839221 0.419611 0.907704i \(-0.362167\pi\)
0.419611 + 0.907704i \(0.362167\pi\)
\(384\) −1.99127e6 −0.689130
\(385\) −443122. −0.152360
\(386\) −3.54722e6 −1.21177
\(387\) 1.12382e6 0.381433
\(388\) 835362. 0.281706
\(389\) 3.73204e6 1.25047 0.625233 0.780438i \(-0.285004\pi\)
0.625233 + 0.780438i \(0.285004\pi\)
\(390\) −251229. −0.0836390
\(391\) 4.93424e6 1.63222
\(392\) −640077. −0.210386
\(393\) 105111. 0.0343293
\(394\) 3.15950e6 1.02536
\(395\) 723393. 0.233282
\(396\) −107999. −0.0346086
\(397\) −4.89288e6 −1.55808 −0.779038 0.626977i \(-0.784292\pi\)
−0.779038 + 0.626977i \(0.784292\pi\)
\(398\) −6.65250e6 −2.10512
\(399\) −751195. −0.236222
\(400\) −784495. −0.245155
\(401\) −3.93524e6 −1.22211 −0.611055 0.791588i \(-0.709255\pi\)
−0.611055 + 0.791588i \(0.709255\pi\)
\(402\) 2.56780e6 0.792495
\(403\) 509144. 0.156163
\(404\) 385933. 0.117641
\(405\) 164025. 0.0496904
\(406\) 3.82643e6 1.15207
\(407\) −950929. −0.284552
\(408\) 1.93802e6 0.576378
\(409\) 657037. 0.194215 0.0971073 0.995274i \(-0.469041\pi\)
0.0971073 + 0.995274i \(0.469041\pi\)
\(410\) −853740. −0.250822
\(411\) 3.46442e6 1.01164
\(412\) 186949. 0.0542600
\(413\) 1.41351e6 0.407777
\(414\) 1.67523e6 0.480368
\(415\) −116216. −0.0331243
\(416\) −651864. −0.184682
\(417\) 2.79544e6 0.787246
\(418\) −452198. −0.126587
\(419\) 4.38504e6 1.22022 0.610110 0.792317i \(-0.291125\pi\)
0.610110 + 0.792317i \(0.291125\pi\)
\(420\) −363188. −0.100464
\(421\) 2.34058e6 0.643604 0.321802 0.946807i \(-0.395711\pi\)
0.321802 + 0.946807i \(0.395711\pi\)
\(422\) 4.06467e6 1.11108
\(423\) 555012. 0.150818
\(424\) 527620. 0.142530
\(425\) 978008. 0.262646
\(426\) 3.10541e6 0.829078
\(427\) −3.09552e6 −0.821607
\(428\) 134329. 0.0354455
\(429\) 185389. 0.0486341
\(430\) 2.27500e6 0.593350
\(431\) −2.18215e6 −0.565838 −0.282919 0.959144i \(-0.591303\pi\)
−0.282919 + 0.959144i \(0.591303\pi\)
\(432\) 915035. 0.235900
\(433\) −3.61080e6 −0.925515 −0.462757 0.886485i \(-0.653140\pi\)
−0.462757 + 0.886485i \(0.653140\pi\)
\(434\) 2.87352e6 0.732301
\(435\) −896080. −0.227051
\(436\) 133760. 0.0336984
\(437\) 1.79668e6 0.450056
\(438\) 3.79965e6 0.946364
\(439\) 6.48191e6 1.60525 0.802624 0.596486i \(-0.203437\pi\)
0.802624 + 0.596486i \(0.203437\pi\)
\(440\) 416273. 0.102505
\(441\) 376760. 0.0922503
\(442\) 1.74723e6 0.425397
\(443\) −6.81523e6 −1.64995 −0.824976 0.565167i \(-0.808812\pi\)
−0.824976 + 0.565167i \(0.808812\pi\)
\(444\) −779393. −0.187629
\(445\) −2.58644e6 −0.619160
\(446\) 5.86531e6 1.39622
\(447\) −2.56683e6 −0.607614
\(448\) 2.20480e6 0.519008
\(449\) −2.98356e6 −0.698423 −0.349211 0.937044i \(-0.613551\pi\)
−0.349211 + 0.937044i \(0.613551\pi\)
\(450\) 332045. 0.0772974
\(451\) 629999. 0.145847
\(452\) 2.25508e6 0.519177
\(453\) 4.70122e6 1.07638
\(454\) 5.58469e6 1.27163
\(455\) 623440. 0.141178
\(456\) 705679. 0.158926
\(457\) −6.11184e6 −1.36893 −0.684466 0.729045i \(-0.739964\pi\)
−0.684466 + 0.729045i \(0.739964\pi\)
\(458\) 904081. 0.201393
\(459\) −1.14075e6 −0.252731
\(460\) 868659. 0.191406
\(461\) −3.41772e6 −0.749003 −0.374502 0.927226i \(-0.622186\pi\)
−0.374502 + 0.927226i \(0.622186\pi\)
\(462\) 1.04630e6 0.228062
\(463\) −4.60011e6 −0.997277 −0.498639 0.866810i \(-0.666166\pi\)
−0.498639 + 0.866810i \(0.666166\pi\)
\(464\) −4.99890e6 −1.07790
\(465\) −672925. −0.144323
\(466\) −8.63083e6 −1.84114
\(467\) 5.03617e6 1.06858 0.534291 0.845301i \(-0.320579\pi\)
0.534291 + 0.845301i \(0.320579\pi\)
\(468\) 151947. 0.0320685
\(469\) −6.37215e6 −1.33769
\(470\) 1.12354e6 0.234609
\(471\) 19994.9 0.00415305
\(472\) −1.32786e6 −0.274345
\(473\) −1.67879e6 −0.345019
\(474\) −1.70808e6 −0.349190
\(475\) 356117. 0.0724199
\(476\) 2.52587e6 0.510969
\(477\) −310566. −0.0624968
\(478\) 7.33142e6 1.46764
\(479\) −2.82760e6 −0.563091 −0.281545 0.959548i \(-0.590847\pi\)
−0.281545 + 0.959548i \(0.590847\pi\)
\(480\) 861556. 0.170679
\(481\) 1.33789e6 0.263668
\(482\) −3.09306e6 −0.606416
\(483\) −4.15718e6 −0.810833
\(484\) 161333. 0.0313046
\(485\) 1.89524e6 0.365855
\(486\) −387297. −0.0743795
\(487\) −5.68931e6 −1.08702 −0.543510 0.839403i \(-0.682905\pi\)
−0.543510 + 0.839403i \(0.682905\pi\)
\(488\) 2.90796e6 0.552763
\(489\) −2.25131e6 −0.425759
\(490\) 762695. 0.143503
\(491\) 3.23715e6 0.605981 0.302990 0.952994i \(-0.402015\pi\)
0.302990 + 0.952994i \(0.402015\pi\)
\(492\) 516355. 0.0961690
\(493\) 6.23199e6 1.15481
\(494\) 636209. 0.117296
\(495\) −245025. −0.0449467
\(496\) −3.75400e6 −0.685157
\(497\) −7.70626e6 −1.39943
\(498\) 274410. 0.0495823
\(499\) −5.08564e6 −0.914312 −0.457156 0.889386i \(-0.651132\pi\)
−0.457156 + 0.889386i \(0.651132\pi\)
\(500\) 172175. 0.0307997
\(501\) 378751. 0.0674154
\(502\) 1.15791e7 2.05077
\(503\) 9.02252e6 1.59004 0.795019 0.606584i \(-0.207461\pi\)
0.795019 + 0.606584i \(0.207461\pi\)
\(504\) −1.63281e6 −0.286325
\(505\) 875591. 0.152782
\(506\) −2.50251e6 −0.434509
\(507\) 3.08081e6 0.532286
\(508\) −1.23681e6 −0.212639
\(509\) 8.98909e6 1.53788 0.768938 0.639323i \(-0.220785\pi\)
0.768938 + 0.639323i \(0.220785\pi\)
\(510\) −2.30928e6 −0.393143
\(511\) −9.42904e6 −1.59741
\(512\) −720997. −0.121551
\(513\) −415374. −0.0696861
\(514\) −130654. −0.0218130
\(515\) 424143. 0.0704683
\(516\) −1.37596e6 −0.227499
\(517\) −829093. −0.136420
\(518\) 7.55079e6 1.23643
\(519\) 1.05485e6 0.171899
\(520\) −585665. −0.0949819
\(521\) −1.11219e7 −1.79508 −0.897539 0.440936i \(-0.854647\pi\)
−0.897539 + 0.440936i \(0.854647\pi\)
\(522\) 2.11583e6 0.339863
\(523\) 3.67510e6 0.587510 0.293755 0.955881i \(-0.405095\pi\)
0.293755 + 0.955881i \(0.405095\pi\)
\(524\) −128693. −0.0204752
\(525\) −823988. −0.130474
\(526\) −1.43212e7 −2.25692
\(527\) 4.68001e6 0.734040
\(528\) −1.36690e6 −0.213380
\(529\) 3.50664e6 0.544818
\(530\) −628695. −0.0972188
\(531\) 781600. 0.120295
\(532\) 919733. 0.140891
\(533\) −886361. −0.135143
\(534\) 6.10712e6 0.926795
\(535\) 304761. 0.0460335
\(536\) 5.98605e6 0.899971
\(537\) −5.24687e6 −0.785171
\(538\) 5.50489e6 0.819960
\(539\) −562814. −0.0834436
\(540\) −200825. −0.0296370
\(541\) −4.31097e6 −0.633260 −0.316630 0.948549i \(-0.602551\pi\)
−0.316630 + 0.948549i \(0.602551\pi\)
\(542\) 9.35569e6 1.36797
\(543\) −2.11762e6 −0.308211
\(544\) −5.99188e6 −0.868092
\(545\) 303469. 0.0437646
\(546\) −1.47207e6 −0.211323
\(547\) −1.80626e6 −0.258114 −0.129057 0.991637i \(-0.541195\pi\)
−0.129057 + 0.991637i \(0.541195\pi\)
\(548\) −4.24169e6 −0.603375
\(549\) −1.71167e6 −0.242376
\(550\) −496017. −0.0699182
\(551\) 2.26922e6 0.318418
\(552\) 3.90529e6 0.545514
\(553\) 4.23870e6 0.589413
\(554\) −8.55595e6 −1.18439
\(555\) −1.76826e6 −0.243676
\(556\) −3.42262e6 −0.469540
\(557\) 8.42670e6 1.15085 0.575426 0.817854i \(-0.304836\pi\)
0.575426 + 0.817854i \(0.304836\pi\)
\(558\) 1.58891e6 0.216031
\(559\) 2.36193e6 0.319696
\(560\) −4.59672e6 −0.619410
\(561\) 1.70408e6 0.228604
\(562\) −3.80840e6 −0.508630
\(563\) −9.19966e6 −1.22321 −0.611605 0.791163i \(-0.709476\pi\)
−0.611605 + 0.791163i \(0.709476\pi\)
\(564\) −679534. −0.0899526
\(565\) 5.11623e6 0.674262
\(566\) −3.94829e6 −0.518046
\(567\) 961099. 0.125548
\(568\) 7.23932e6 0.941515
\(569\) 5.18644e6 0.671566 0.335783 0.941939i \(-0.390999\pi\)
0.335783 + 0.941939i \(0.390999\pi\)
\(570\) −840864. −0.108402
\(571\) −3.29390e6 −0.422786 −0.211393 0.977401i \(-0.567800\pi\)
−0.211393 + 0.977401i \(0.567800\pi\)
\(572\) −226983. −0.0290070
\(573\) −5.32609e6 −0.677676
\(574\) −5.00246e6 −0.633730
\(575\) 1.97078e6 0.248581
\(576\) 1.21915e6 0.153109
\(577\) −1.02043e7 −1.27598 −0.637989 0.770046i \(-0.720233\pi\)
−0.637989 + 0.770046i \(0.720233\pi\)
\(578\) 6.74768e6 0.840107
\(579\) 4.86742e6 0.603396
\(580\) 1.09712e6 0.135421
\(581\) −680965. −0.0836921
\(582\) −4.47505e6 −0.547633
\(583\) 463932. 0.0565305
\(584\) 8.85773e6 1.07471
\(585\) 344732. 0.0416478
\(586\) −1.58065e7 −1.90148
\(587\) −1.33951e7 −1.60454 −0.802272 0.596958i \(-0.796376\pi\)
−0.802272 + 0.596958i \(0.796376\pi\)
\(588\) −461289. −0.0550212
\(589\) 1.70411e6 0.202399
\(590\) 1.58223e6 0.187129
\(591\) −4.33540e6 −0.510576
\(592\) −9.86445e6 −1.15683
\(593\) 3.76550e6 0.439730 0.219865 0.975530i \(-0.429438\pi\)
0.219865 + 0.975530i \(0.429438\pi\)
\(594\) 578554. 0.0672788
\(595\) 5.73061e6 0.663603
\(596\) 3.14272e6 0.362401
\(597\) 9.12843e6 1.04824
\(598\) 3.52084e6 0.402618
\(599\) 1.21252e6 0.138077 0.0690384 0.997614i \(-0.478007\pi\)
0.0690384 + 0.997614i \(0.478007\pi\)
\(600\) 774061. 0.0877803
\(601\) −7.81714e6 −0.882799 −0.441399 0.897311i \(-0.645518\pi\)
−0.441399 + 0.897311i \(0.645518\pi\)
\(602\) 1.33303e7 1.49916
\(603\) −3.52349e6 −0.394620
\(604\) −5.75597e6 −0.641987
\(605\) 366025. 0.0406558
\(606\) −2.06745e6 −0.228693
\(607\) 5.43853e6 0.599114 0.299557 0.954078i \(-0.403161\pi\)
0.299557 + 0.954078i \(0.403161\pi\)
\(608\) −2.18179e6 −0.239361
\(609\) −5.25055e6 −0.573670
\(610\) −3.46503e6 −0.377036
\(611\) 1.16647e6 0.126407
\(612\) 1.39669e6 0.150737
\(613\) 1.70637e7 1.83409 0.917045 0.398783i \(-0.130567\pi\)
0.917045 + 0.398783i \(0.130567\pi\)
\(614\) −1.20277e7 −1.28755
\(615\) 1.17149e6 0.124896
\(616\) 2.43914e6 0.258991
\(617\) 7.80667e6 0.825568 0.412784 0.910829i \(-0.364556\pi\)
0.412784 + 0.910829i \(0.364556\pi\)
\(618\) −1.00149e6 −0.105481
\(619\) 7.31658e6 0.767506 0.383753 0.923436i \(-0.374631\pi\)
0.383753 + 0.923436i \(0.374631\pi\)
\(620\) 823902. 0.0860788
\(621\) −2.29872e6 −0.239198
\(622\) 1.49220e7 1.54650
\(623\) −1.51552e7 −1.56438
\(624\) 1.92313e6 0.197719
\(625\) 390625. 0.0400000
\(626\) 1.04670e6 0.106755
\(627\) 620497. 0.0630335
\(628\) −24481.0 −0.00247702
\(629\) 1.22977e7 1.23936
\(630\) 1.94560e6 0.195300
\(631\) −1.18729e7 −1.18709 −0.593543 0.804802i \(-0.702271\pi\)
−0.593543 + 0.804802i \(0.702271\pi\)
\(632\) −3.98187e6 −0.396547
\(633\) −5.57746e6 −0.553257
\(634\) −8.15523e6 −0.805773
\(635\) −2.80602e6 −0.276157
\(636\) 380244. 0.0372752
\(637\) 791837. 0.0773192
\(638\) −3.16068e6 −0.307418
\(639\) −4.26118e6 −0.412837
\(640\) 5.53129e6 0.533798
\(641\) −1.85963e7 −1.78764 −0.893821 0.448424i \(-0.851985\pi\)
−0.893821 + 0.448424i \(0.851985\pi\)
\(642\) −719603. −0.0689057
\(643\) −1.18028e7 −1.12579 −0.562894 0.826529i \(-0.690312\pi\)
−0.562894 + 0.826529i \(0.690312\pi\)
\(644\) 5.08988e6 0.483608
\(645\) −3.12171e6 −0.295457
\(646\) 5.84798e6 0.551346
\(647\) 2.05252e7 1.92764 0.963821 0.266551i \(-0.0858842\pi\)
0.963821 + 0.266551i \(0.0858842\pi\)
\(648\) −902865. −0.0844666
\(649\) −1.16757e6 −0.108811
\(650\) 697859. 0.0647865
\(651\) −3.94298e6 −0.364647
\(652\) 2.75642e6 0.253937
\(653\) −5.78511e6 −0.530919 −0.265460 0.964122i \(-0.585524\pi\)
−0.265460 + 0.964122i \(0.585524\pi\)
\(654\) −716552. −0.0655094
\(655\) −291974. −0.0265914
\(656\) 6.53528e6 0.592932
\(657\) −5.21380e6 −0.471239
\(658\) 6.58335e6 0.592765
\(659\) 1.88598e7 1.69170 0.845852 0.533418i \(-0.179093\pi\)
0.845852 + 0.533418i \(0.179093\pi\)
\(660\) 299999. 0.0268077
\(661\) −1.12652e7 −1.00285 −0.501426 0.865201i \(-0.667191\pi\)
−0.501426 + 0.865201i \(0.667191\pi\)
\(662\) −6.28637e6 −0.557513
\(663\) −2.39751e6 −0.211825
\(664\) 639704. 0.0563065
\(665\) 2.08665e6 0.182977
\(666\) 4.17522e6 0.364749
\(667\) 1.25581e7 1.09297
\(668\) −463727. −0.0402088
\(669\) −8.04827e6 −0.695244
\(670\) −7.13279e6 −0.613864
\(671\) 2.55694e6 0.219237
\(672\) 5.04826e6 0.431239
\(673\) 1.51301e7 1.28767 0.643833 0.765166i \(-0.277343\pi\)
0.643833 + 0.765166i \(0.277343\pi\)
\(674\) −9.90878e6 −0.840176
\(675\) −455625. −0.0384900
\(676\) −3.77201e6 −0.317473
\(677\) −1.12001e7 −0.939187 −0.469593 0.882883i \(-0.655599\pi\)
−0.469593 + 0.882883i \(0.655599\pi\)
\(678\) −1.20805e7 −1.00928
\(679\) 1.11051e7 0.924373
\(680\) −5.38338e6 −0.446460
\(681\) −7.66321e6 −0.633203
\(682\) −2.37356e6 −0.195407
\(683\) 2.12907e7 1.74638 0.873190 0.487379i \(-0.162047\pi\)
0.873190 + 0.487379i \(0.162047\pi\)
\(684\) 508567. 0.0415631
\(685\) −9.62338e6 −0.783612
\(686\) −1.16791e7 −0.947539
\(687\) −1.24056e6 −0.100283
\(688\) −1.74149e7 −1.40265
\(689\) −652717. −0.0523814
\(690\) −4.65342e6 −0.372091
\(691\) 1.77648e6 0.141536 0.0707678 0.997493i \(-0.477455\pi\)
0.0707678 + 0.997493i \(0.477455\pi\)
\(692\) −1.29152e6 −0.102526
\(693\) −1.43572e6 −0.113563
\(694\) −1.27143e7 −1.00206
\(695\) −7.76512e6 −0.609798
\(696\) 4.93242e6 0.385955
\(697\) −8.14735e6 −0.635235
\(698\) −1.06557e7 −0.827838
\(699\) 1.18431e7 0.916792
\(700\) 1.00886e6 0.0778188
\(701\) 94420.1 0.00725720 0.00362860 0.999993i \(-0.498845\pi\)
0.00362860 + 0.999993i \(0.498845\pi\)
\(702\) −813983. −0.0623408
\(703\) 4.47791e6 0.341733
\(704\) −1.82120e6 −0.138492
\(705\) −1.54170e6 −0.116823
\(706\) −153134. −0.0115627
\(707\) 5.13050e6 0.386021
\(708\) −956958. −0.0717480
\(709\) −1.46333e7 −1.09327 −0.546633 0.837372i \(-0.684091\pi\)
−0.546633 + 0.837372i \(0.684091\pi\)
\(710\) −8.62614e6 −0.642201
\(711\) 2.34379e6 0.173878
\(712\) 1.42369e7 1.05248
\(713\) 9.43067e6 0.694734
\(714\) −1.35311e7 −0.993319
\(715\) −514970. −0.0376718
\(716\) 6.42405e6 0.468302
\(717\) −1.00600e7 −0.730805
\(718\) 2.38690e7 1.72792
\(719\) 1.15215e7 0.831161 0.415580 0.909556i \(-0.363578\pi\)
0.415580 + 0.909556i \(0.363578\pi\)
\(720\) −2.54176e6 −0.182728
\(721\) 2.48525e6 0.178046
\(722\) −1.41111e7 −1.00744
\(723\) 4.24423e6 0.301963
\(724\) 2.59273e6 0.183827
\(725\) 2.48911e6 0.175873
\(726\) −864260. −0.0608559
\(727\) 1.73993e7 1.22095 0.610473 0.792037i \(-0.290979\pi\)
0.610473 + 0.792037i \(0.290979\pi\)
\(728\) −3.43168e6 −0.239982
\(729\) 531441. 0.0370370
\(730\) −1.05546e7 −0.733050
\(731\) 2.17107e7 1.50272
\(732\) 2.09570e6 0.144561
\(733\) −1.21356e7 −0.834262 −0.417131 0.908846i \(-0.636964\pi\)
−0.417131 + 0.908846i \(0.636964\pi\)
\(734\) 1.47387e7 1.00976
\(735\) −1.04655e6 −0.0714568
\(736\) −1.20742e7 −0.821608
\(737\) 5.26348e6 0.356948
\(738\) −2.76612e6 −0.186952
\(739\) 1.30600e7 0.879692 0.439846 0.898073i \(-0.355033\pi\)
0.439846 + 0.898073i \(0.355033\pi\)
\(740\) 2.16498e6 0.145337
\(741\) −872994. −0.0584071
\(742\) −3.68382e6 −0.245634
\(743\) −2.36143e6 −0.156929 −0.0784644 0.996917i \(-0.525002\pi\)
−0.0784644 + 0.996917i \(0.525002\pi\)
\(744\) 3.70407e6 0.245328
\(745\) 7.13008e6 0.470656
\(746\) 2.51038e7 1.65156
\(747\) −376540. −0.0246894
\(748\) −2.08641e6 −0.136347
\(749\) 1.78574e6 0.116309
\(750\) −922346. −0.0598743
\(751\) 4.12172e6 0.266673 0.133336 0.991071i \(-0.457431\pi\)
0.133336 + 0.991071i \(0.457431\pi\)
\(752\) −8.60058e6 −0.554604
\(753\) −1.58887e7 −1.02117
\(754\) 4.44685e6 0.284855
\(755\) −1.30589e7 −0.833759
\(756\) −1.17673e6 −0.0748811
\(757\) 4.12364e6 0.261542 0.130771 0.991413i \(-0.458255\pi\)
0.130771 + 0.991413i \(0.458255\pi\)
\(758\) −2.79700e7 −1.76815
\(759\) 3.43389e6 0.216362
\(760\) −1.96022e6 −0.123104
\(761\) −2.08230e7 −1.30341 −0.651706 0.758472i \(-0.725946\pi\)
−0.651706 + 0.758472i \(0.725946\pi\)
\(762\) 6.62558e6 0.413368
\(763\) 1.77817e6 0.110576
\(764\) 6.52105e6 0.404189
\(765\) 3.16875e6 0.195764
\(766\) 1.58017e7 0.973045
\(767\) 1.64269e6 0.100825
\(768\) −8.72578e6 −0.533828
\(769\) 1.58156e7 0.964427 0.482213 0.876054i \(-0.339833\pi\)
0.482213 + 0.876054i \(0.339833\pi\)
\(770\) −2.90640e6 −0.176656
\(771\) 179281. 0.0108617
\(772\) −5.95947e6 −0.359886
\(773\) −2.58350e6 −0.155510 −0.0777552 0.996972i \(-0.524775\pi\)
−0.0777552 + 0.996972i \(0.524775\pi\)
\(774\) 7.37101e6 0.442257
\(775\) 1.86924e6 0.111792
\(776\) −1.04322e7 −0.621902
\(777\) −1.03610e7 −0.615674
\(778\) 2.44781e7 1.44987
\(779\) −2.96665e6 −0.175155
\(780\) −422076. −0.0248401
\(781\) 6.36547e6 0.373425
\(782\) 3.23632e7 1.89250
\(783\) −2.90330e6 −0.169234
\(784\) −5.83834e6 −0.339234
\(785\) −55541.5 −0.00321694
\(786\) 689411. 0.0398035
\(787\) −5.64885e6 −0.325105 −0.162552 0.986700i \(-0.551973\pi\)
−0.162552 + 0.986700i \(0.551973\pi\)
\(788\) 5.30809e6 0.304525
\(789\) 1.96513e7 1.12383
\(790\) 4.74467e6 0.270482
\(791\) 2.99784e7 1.70360
\(792\) 1.34872e6 0.0764030
\(793\) −3.59743e6 −0.203146
\(794\) −3.20920e7 −1.80653
\(795\) 862683. 0.0484098
\(796\) −1.11765e7 −0.625204
\(797\) −3.72876e6 −0.207931 −0.103965 0.994581i \(-0.533153\pi\)
−0.103965 + 0.994581i \(0.533153\pi\)
\(798\) −4.92702e6 −0.273890
\(799\) 1.07221e7 0.594173
\(800\) −2.39321e6 −0.132207
\(801\) −8.38007e6 −0.461495
\(802\) −2.58109e7 −1.41699
\(803\) 7.78852e6 0.426252
\(804\) 4.31401e6 0.235365
\(805\) 1.15477e7 0.628068
\(806\) 3.33943e6 0.181065
\(807\) −7.55370e6 −0.408296
\(808\) −4.81963e6 −0.259708
\(809\) 6.35004e6 0.341119 0.170559 0.985347i \(-0.445443\pi\)
0.170559 + 0.985347i \(0.445443\pi\)
\(810\) 1.07582e6 0.0576141
\(811\) 1.34156e7 0.716241 0.358120 0.933675i \(-0.383418\pi\)
0.358120 + 0.933675i \(0.383418\pi\)
\(812\) 6.42856e6 0.342156
\(813\) −1.28377e7 −0.681178
\(814\) −6.23705e6 −0.329928
\(815\) 6.25365e6 0.329791
\(816\) 1.76773e7 0.929372
\(817\) 7.90538e6 0.414350
\(818\) 4.30945e6 0.225184
\(819\) 2.01995e6 0.105228
\(820\) −1.43432e6 −0.0744922
\(821\) 2.40201e7 1.24370 0.621851 0.783135i \(-0.286381\pi\)
0.621851 + 0.783135i \(0.286381\pi\)
\(822\) 2.27228e7 1.17296
\(823\) −1.46523e6 −0.0754058 −0.0377029 0.999289i \(-0.512004\pi\)
−0.0377029 + 0.999289i \(0.512004\pi\)
\(824\) −2.33466e6 −0.119786
\(825\) 680625. 0.0348155
\(826\) 9.27105e6 0.472801
\(827\) −2.32918e7 −1.18424 −0.592120 0.805850i \(-0.701709\pi\)
−0.592120 + 0.805850i \(0.701709\pi\)
\(828\) 2.81446e6 0.142665
\(829\) −2.02821e6 −0.102501 −0.0512503 0.998686i \(-0.516321\pi\)
−0.0512503 + 0.998686i \(0.516321\pi\)
\(830\) −762250. −0.0384063
\(831\) 1.17403e7 0.589762
\(832\) 2.56229e6 0.128327
\(833\) 7.27850e6 0.363437
\(834\) 1.83350e7 0.912781
\(835\) −1.05208e6 −0.0522197
\(836\) −759712. −0.0375953
\(837\) −2.18028e6 −0.107572
\(838\) 2.87610e7 1.41480
\(839\) −3.07518e7 −1.50822 −0.754111 0.656747i \(-0.771932\pi\)
−0.754111 + 0.656747i \(0.771932\pi\)
\(840\) 4.53559e6 0.221787
\(841\) −4.65021e6 −0.226716
\(842\) 1.53517e7 0.746234
\(843\) 5.22582e6 0.253271
\(844\) 6.82881e6 0.329981
\(845\) −8.55780e6 −0.412307
\(846\) 3.64027e6 0.174867
\(847\) 2.14471e6 0.102721
\(848\) 4.81259e6 0.229821
\(849\) 5.41777e6 0.257959
\(850\) 6.41466e6 0.304527
\(851\) 2.47811e7 1.17300
\(852\) 5.21722e6 0.246229
\(853\) −2.05598e7 −0.967491 −0.483745 0.875209i \(-0.660724\pi\)
−0.483745 + 0.875209i \(0.660724\pi\)
\(854\) −2.03032e7 −0.952621
\(855\) 1.15382e6 0.0539786
\(856\) −1.67753e6 −0.0782505
\(857\) 3.20463e7 1.49048 0.745240 0.666796i \(-0.232335\pi\)
0.745240 + 0.666796i \(0.232335\pi\)
\(858\) 1.21595e6 0.0563894
\(859\) 2.34894e6 0.108615 0.0543075 0.998524i \(-0.482705\pi\)
0.0543075 + 0.998524i \(0.482705\pi\)
\(860\) 3.82210e6 0.176220
\(861\) 6.86428e6 0.315564
\(862\) −1.43125e7 −0.656068
\(863\) −1.44887e7 −0.662219 −0.331110 0.943592i \(-0.607423\pi\)
−0.331110 + 0.943592i \(0.607423\pi\)
\(864\) 2.79144e6 0.127217
\(865\) −2.93014e6 −0.133152
\(866\) −2.36829e7 −1.07310
\(867\) −9.25903e6 −0.418329
\(868\) 4.82763e6 0.217488
\(869\) −3.50122e6 −0.157279
\(870\) −5.87731e6 −0.263257
\(871\) −7.40533e6 −0.330749
\(872\) −1.67042e6 −0.0743936
\(873\) 6.14057e6 0.272692
\(874\) 1.17842e7 0.521823
\(875\) 2.28885e6 0.101064
\(876\) 6.38356e6 0.281063
\(877\) −3.17684e7 −1.39475 −0.697376 0.716705i \(-0.745649\pi\)
−0.697376 + 0.716705i \(0.745649\pi\)
\(878\) 4.25143e7 1.86122
\(879\) 2.16893e7 0.946834
\(880\) 3.79696e6 0.165283
\(881\) 1.06777e7 0.463486 0.231743 0.972777i \(-0.425557\pi\)
0.231743 + 0.972777i \(0.425557\pi\)
\(882\) 2.47113e6 0.106961
\(883\) 1.77890e7 0.767804 0.383902 0.923374i \(-0.374580\pi\)
0.383902 + 0.923374i \(0.374580\pi\)
\(884\) 2.93542e6 0.126340
\(885\) −2.17111e6 −0.0931802
\(886\) −4.47005e7 −1.91306
\(887\) 3.13731e7 1.33890 0.669450 0.742857i \(-0.266530\pi\)
0.669450 + 0.742857i \(0.266530\pi\)
\(888\) 9.73326e6 0.414215
\(889\) −1.64418e7 −0.697741
\(890\) −1.69642e7 −0.717892
\(891\) −793881. −0.0335013
\(892\) 9.85397e6 0.414667
\(893\) 3.90418e6 0.163833
\(894\) −1.68356e7 −0.704505
\(895\) 1.45746e7 0.608191
\(896\) 3.24104e7 1.34870
\(897\) −4.83123e6 −0.200482
\(898\) −1.95689e7 −0.809795
\(899\) 1.19110e7 0.491529
\(900\) 557849. 0.0229567
\(901\) −5.99972e6 −0.246218
\(902\) 4.13210e6 0.169104
\(903\) −1.82916e7 −0.746504
\(904\) −2.81620e7 −1.14615
\(905\) 5.88228e6 0.238739
\(906\) 3.08348e7 1.24802
\(907\) −8.46355e6 −0.341613 −0.170807 0.985305i \(-0.554637\pi\)
−0.170807 + 0.985305i \(0.554637\pi\)
\(908\) 9.38252e6 0.377663
\(909\) 2.83691e6 0.113877
\(910\) 4.08908e6 0.163690
\(911\) −2.18411e7 −0.871923 −0.435961 0.899965i \(-0.643592\pi\)
−0.435961 + 0.899965i \(0.643592\pi\)
\(912\) 6.43672e6 0.256258
\(913\) 562486. 0.0223324
\(914\) −4.00870e7 −1.58722
\(915\) 4.75464e6 0.187744
\(916\) 1.51889e6 0.0598120
\(917\) −1.71081e6 −0.0671861
\(918\) −7.48206e6 −0.293032
\(919\) 4.18752e7 1.63557 0.817784 0.575525i \(-0.195202\pi\)
0.817784 + 0.575525i \(0.195202\pi\)
\(920\) −1.08480e7 −0.422553
\(921\) 1.65042e7 0.641130
\(922\) −2.24165e7 −0.868440
\(923\) −8.95575e6 −0.346017
\(924\) 1.75783e6 0.0677325
\(925\) 4.91183e6 0.188751
\(926\) −3.01717e7 −1.15630
\(927\) 1.37422e6 0.0525240
\(928\) −1.52498e7 −0.581293
\(929\) 8.03740e6 0.305546 0.152773 0.988261i \(-0.451180\pi\)
0.152773 + 0.988261i \(0.451180\pi\)
\(930\) −4.41365e6 −0.167337
\(931\) 2.65028e6 0.100211
\(932\) −1.45002e7 −0.546805
\(933\) −2.04757e7 −0.770077
\(934\) 3.30317e7 1.23898
\(935\) −4.73356e6 −0.177076
\(936\) −1.89755e6 −0.0707953
\(937\) −1.78192e6 −0.0663038 −0.0331519 0.999450i \(-0.510555\pi\)
−0.0331519 + 0.999450i \(0.510555\pi\)
\(938\) −4.17943e7 −1.55099
\(939\) −1.43627e6 −0.0531583
\(940\) 1.88760e6 0.0696770
\(941\) 2.97245e7 1.09431 0.547154 0.837032i \(-0.315711\pi\)
0.547154 + 0.837032i \(0.315711\pi\)
\(942\) 131145. 0.00481531
\(943\) −1.64177e7 −0.601220
\(944\) −1.21118e7 −0.442364
\(945\) −2.66972e6 −0.0972492
\(946\) −1.10110e7 −0.400036
\(947\) 931083. 0.0337375 0.0168688 0.999858i \(-0.494630\pi\)
0.0168688 + 0.999858i \(0.494630\pi\)
\(948\) −2.86965e6 −0.103707
\(949\) −1.09579e7 −0.394967
\(950\) 2.33573e6 0.0839681
\(951\) 1.11904e7 0.401232
\(952\) −3.15437e7 −1.12803
\(953\) −2.74983e7 −0.980784 −0.490392 0.871502i \(-0.663146\pi\)
−0.490392 + 0.871502i \(0.663146\pi\)
\(954\) −2.03697e6 −0.0724626
\(955\) 1.47947e7 0.524926
\(956\) 1.23171e7 0.435876
\(957\) 4.33703e6 0.153078
\(958\) −1.85459e7 −0.652882
\(959\) −5.63879e7 −1.97988
\(960\) −3.38652e6 −0.118598
\(961\) −1.96844e7 −0.687565
\(962\) 8.77507e6 0.305712
\(963\) 987424. 0.0343114
\(964\) −5.19646e6 −0.180101
\(965\) −1.35206e7 −0.467389
\(966\) −2.72666e7 −0.940129
\(967\) 4.33623e6 0.149123 0.0745617 0.997216i \(-0.476244\pi\)
0.0745617 + 0.997216i \(0.476244\pi\)
\(968\) −2.01476e6 −0.0691091
\(969\) −8.02448e6 −0.274541
\(970\) 1.24307e7 0.424195
\(971\) −5.44123e7 −1.85204 −0.926018 0.377480i \(-0.876791\pi\)
−0.926018 + 0.377480i \(0.876791\pi\)
\(972\) −650675. −0.0220901
\(973\) −4.54995e7 −1.54072
\(974\) −3.73157e7 −1.26036
\(975\) −957589. −0.0322602
\(976\) 2.65244e7 0.891294
\(977\) −3.16213e7 −1.05985 −0.529924 0.848045i \(-0.677780\pi\)
−0.529924 + 0.848045i \(0.677780\pi\)
\(978\) −1.47662e7 −0.493651
\(979\) 1.25184e7 0.417438
\(980\) 1.28136e6 0.0426192
\(981\) 983239. 0.0326202
\(982\) 2.12321e7 0.702611
\(983\) 1.31368e7 0.433617 0.216808 0.976214i \(-0.430435\pi\)
0.216808 + 0.976214i \(0.430435\pi\)
\(984\) −6.44836e6 −0.212306
\(985\) 1.20428e7 0.395491
\(986\) 4.08750e7 1.33895
\(987\) −9.03355e6 −0.295165
\(988\) 1.06886e6 0.0348359
\(989\) 4.37491e7 1.42226
\(990\) −1.60710e6 −0.0521139
\(991\) −3.53335e7 −1.14288 −0.571442 0.820643i \(-0.693616\pi\)
−0.571442 + 0.820643i \(0.693616\pi\)
\(992\) −1.14521e7 −0.369493
\(993\) 8.62603e6 0.277612
\(994\) −5.05446e7 −1.62259
\(995\) −2.53567e7 −0.811962
\(996\) 461020. 0.0147256
\(997\) 3.72145e7 1.18570 0.592850 0.805313i \(-0.298003\pi\)
0.592850 + 0.805313i \(0.298003\pi\)
\(998\) −3.33563e7 −1.06011
\(999\) −5.72915e6 −0.181625
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.d.1.3 3
3.2 odd 2 495.6.a.c.1.1 3
5.4 even 2 825.6.a.h.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.d.1.3 3 1.1 even 1 trivial
495.6.a.c.1.1 3 3.2 odd 2
825.6.a.h.1.1 3 5.4 even 2