Properties

Label 242.6.a.d.1.1
Level $242$
Weight $6$
Character 242.1
Self dual yes
Analytic conductor $38.813$
Analytic rank $1$
Dimension $1$
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] = [242,6,Mod(1,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 242.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: \(38.8128843947\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 242.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +1.00000 q^{3} +16.0000 q^{4} -51.0000 q^{5} +4.00000 q^{6} +166.000 q^{7} +64.0000 q^{8} -242.000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +1.00000 q^{3} +16.0000 q^{4} -51.0000 q^{5} +4.00000 q^{6} +166.000 q^{7} +64.0000 q^{8} -242.000 q^{9} -204.000 q^{10} +16.0000 q^{12} -692.000 q^{13} +664.000 q^{14} -51.0000 q^{15} +256.000 q^{16} +738.000 q^{17} -968.000 q^{18} -1424.00 q^{19} -816.000 q^{20} +166.000 q^{21} -1779.00 q^{23} +64.0000 q^{24} -524.000 q^{25} -2768.00 q^{26} -485.000 q^{27} +2656.00 q^{28} +2064.00 q^{29} -204.000 q^{30} +6245.00 q^{31} +1024.00 q^{32} +2952.00 q^{34} -8466.00 q^{35} -3872.00 q^{36} -14785.0 q^{37} -5696.00 q^{38} -692.000 q^{39} -3264.00 q^{40} -5304.00 q^{41} +664.000 q^{42} -17798.0 q^{43} +12342.0 q^{45} -7116.00 q^{46} -17184.0 q^{47} +256.000 q^{48} +10749.0 q^{49} -2096.00 q^{50} +738.000 q^{51} -11072.0 q^{52} -30726.0 q^{53} -1940.00 q^{54} +10624.0 q^{56} -1424.00 q^{57} +8256.00 q^{58} -34989.0 q^{59} -816.000 q^{60} +45940.0 q^{61} +24980.0 q^{62} -40172.0 q^{63} +4096.00 q^{64} +35292.0 q^{65} +25343.0 q^{67} +11808.0 q^{68} -1779.00 q^{69} -33864.0 q^{70} +13311.0 q^{71} -15488.0 q^{72} +53260.0 q^{73} -59140.0 q^{74} -524.000 q^{75} -22784.0 q^{76} -2768.00 q^{78} -77234.0 q^{79} -13056.0 q^{80} +58321.0 q^{81} -21216.0 q^{82} -55014.0 q^{83} +2656.00 q^{84} -37638.0 q^{85} -71192.0 q^{86} +2064.00 q^{87} +125415. q^{89} +49368.0 q^{90} -114872. q^{91} -28464.0 q^{92} +6245.00 q^{93} -68736.0 q^{94} +72624.0 q^{95} +1024.00 q^{96} -88807.0 q^{97} +42996.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 1.00000 0.0641500 0.0320750 0.999485i \(-0.489788\pi\)
0.0320750 + 0.999485i \(0.489788\pi\)
\(4\) 16.0000 0.500000
\(5\) −51.0000 −0.912316 −0.456158 0.889899i \(-0.650775\pi\)
−0.456158 + 0.889899i \(0.650775\pi\)
\(6\) 4.00000 0.0453609
\(7\) 166.000 1.28045 0.640226 0.768187i \(-0.278841\pi\)
0.640226 + 0.768187i \(0.278841\pi\)
\(8\) 64.0000 0.353553
\(9\) −242.000 −0.995885
\(10\) −204.000 −0.645105
\(11\) 0 0
\(12\) 16.0000 0.0320750
\(13\) −692.000 −1.13566 −0.567829 0.823146i \(-0.692217\pi\)
−0.567829 + 0.823146i \(0.692217\pi\)
\(14\) 664.000 0.905416
\(15\) −51.0000 −0.0585251
\(16\) 256.000 0.250000
\(17\) 738.000 0.619347 0.309674 0.950843i \(-0.399780\pi\)
0.309674 + 0.950843i \(0.399780\pi\)
\(18\) −968.000 −0.704197
\(19\) −1424.00 −0.904953 −0.452476 0.891776i \(-0.649459\pi\)
−0.452476 + 0.891776i \(0.649459\pi\)
\(20\) −816.000 −0.456158
\(21\) 166.000 0.0821410
\(22\) 0 0
\(23\) −1779.00 −0.701223 −0.350612 0.936521i \(-0.614026\pi\)
−0.350612 + 0.936521i \(0.614026\pi\)
\(24\) 64.0000 0.0226805
\(25\) −524.000 −0.167680
\(26\) −2768.00 −0.803032
\(27\) −485.000 −0.128036
\(28\) 2656.00 0.640226
\(29\) 2064.00 0.455737 0.227869 0.973692i \(-0.426824\pi\)
0.227869 + 0.973692i \(0.426824\pi\)
\(30\) −204.000 −0.0413835
\(31\) 6245.00 1.16715 0.583577 0.812058i \(-0.301653\pi\)
0.583577 + 0.812058i \(0.301653\pi\)
\(32\) 1024.00 0.176777
\(33\) 0 0
\(34\) 2952.00 0.437944
\(35\) −8466.00 −1.16818
\(36\) −3872.00 −0.497942
\(37\) −14785.0 −1.77549 −0.887743 0.460340i \(-0.847727\pi\)
−0.887743 + 0.460340i \(0.847727\pi\)
\(38\) −5696.00 −0.639898
\(39\) −692.000 −0.0728525
\(40\) −3264.00 −0.322552
\(41\) −5304.00 −0.492770 −0.246385 0.969172i \(-0.579243\pi\)
−0.246385 + 0.969172i \(0.579243\pi\)
\(42\) 664.000 0.0580824
\(43\) −17798.0 −1.46791 −0.733956 0.679197i \(-0.762328\pi\)
−0.733956 + 0.679197i \(0.762328\pi\)
\(44\) 0 0
\(45\) 12342.0 0.908561
\(46\) −7116.00 −0.495840
\(47\) −17184.0 −1.13470 −0.567348 0.823478i \(-0.692031\pi\)
−0.567348 + 0.823478i \(0.692031\pi\)
\(48\) 256.000 0.0160375
\(49\) 10749.0 0.639555
\(50\) −2096.00 −0.118568
\(51\) 738.000 0.0397311
\(52\) −11072.0 −0.567829
\(53\) −30726.0 −1.50251 −0.751253 0.660014i \(-0.770550\pi\)
−0.751253 + 0.660014i \(0.770550\pi\)
\(54\) −1940.00 −0.0905352
\(55\) 0 0
\(56\) 10624.0 0.452708
\(57\) −1424.00 −0.0580528
\(58\) 8256.00 0.322255
\(59\) −34989.0 −1.30858 −0.654292 0.756242i \(-0.727033\pi\)
−0.654292 + 0.756242i \(0.727033\pi\)
\(60\) −816.000 −0.0292625
\(61\) 45940.0 1.58076 0.790381 0.612616i \(-0.209883\pi\)
0.790381 + 0.612616i \(0.209883\pi\)
\(62\) 24980.0 0.825303
\(63\) −40172.0 −1.27518
\(64\) 4096.00 0.125000
\(65\) 35292.0 1.03608
\(66\) 0 0
\(67\) 25343.0 0.689717 0.344859 0.938655i \(-0.387927\pi\)
0.344859 + 0.938655i \(0.387927\pi\)
\(68\) 11808.0 0.309674
\(69\) −1779.00 −0.0449835
\(70\) −33864.0 −0.826025
\(71\) 13311.0 0.313375 0.156688 0.987648i \(-0.449918\pi\)
0.156688 + 0.987648i \(0.449918\pi\)
\(72\) −15488.0 −0.352098
\(73\) 53260.0 1.16975 0.584876 0.811123i \(-0.301143\pi\)
0.584876 + 0.811123i \(0.301143\pi\)
\(74\) −59140.0 −1.25546
\(75\) −524.000 −0.0107567
\(76\) −22784.0 −0.452476
\(77\) 0 0
\(78\) −2768.00 −0.0515145
\(79\) −77234.0 −1.39233 −0.696163 0.717884i \(-0.745111\pi\)
−0.696163 + 0.717884i \(0.745111\pi\)
\(80\) −13056.0 −0.228079
\(81\) 58321.0 0.987671
\(82\) −21216.0 −0.348441
\(83\) −55014.0 −0.876553 −0.438276 0.898840i \(-0.644411\pi\)
−0.438276 + 0.898840i \(0.644411\pi\)
\(84\) 2656.00 0.0410705
\(85\) −37638.0 −0.565040
\(86\) −71192.0 −1.03797
\(87\) 2064.00 0.0292356
\(88\) 0 0
\(89\) 125415. 1.67832 0.839159 0.543886i \(-0.183047\pi\)
0.839159 + 0.543886i \(0.183047\pi\)
\(90\) 49368.0 0.642450
\(91\) −114872. −1.45416
\(92\) −28464.0 −0.350612
\(93\) 6245.00 0.0748730
\(94\) −68736.0 −0.802351
\(95\) 72624.0 0.825603
\(96\) 1024.00 0.0113402
\(97\) −88807.0 −0.958336 −0.479168 0.877723i \(-0.659062\pi\)
−0.479168 + 0.877723i \(0.659062\pi\)
\(98\) 42996.0 0.452234
\(99\) 0 0
\(100\) −8384.00 −0.0838400
\(101\) −1482.00 −0.0144559 −0.00722794 0.999974i \(-0.502301\pi\)
−0.00722794 + 0.999974i \(0.502301\pi\)
\(102\) 2952.00 0.0280942
\(103\) −117496. −1.09126 −0.545632 0.838025i \(-0.683710\pi\)
−0.545632 + 0.838025i \(0.683710\pi\)
\(104\) −44288.0 −0.401516
\(105\) −8466.00 −0.0749385
\(106\) −122904. −1.06243
\(107\) 79362.0 0.670121 0.335060 0.942197i \(-0.391243\pi\)
0.335060 + 0.942197i \(0.391243\pi\)
\(108\) −7760.00 −0.0640180
\(109\) −87842.0 −0.708167 −0.354084 0.935214i \(-0.615207\pi\)
−0.354084 + 0.935214i \(0.615207\pi\)
\(110\) 0 0
\(111\) −14785.0 −0.113897
\(112\) 42496.0 0.320113
\(113\) −47247.0 −0.348079 −0.174040 0.984739i \(-0.555682\pi\)
−0.174040 + 0.984739i \(0.555682\pi\)
\(114\) −5696.00 −0.0410495
\(115\) 90729.0 0.639737
\(116\) 33024.0 0.227869
\(117\) 167464. 1.13098
\(118\) −139956. −0.925308
\(119\) 122508. 0.793044
\(120\) −3264.00 −0.0206917
\(121\) 0 0
\(122\) 183760. 1.11777
\(123\) −5304.00 −0.0316112
\(124\) 99920.0 0.583577
\(125\) 186099. 1.06529
\(126\) −160688. −0.901690
\(127\) 239416. 1.31718 0.658588 0.752504i \(-0.271154\pi\)
0.658588 + 0.752504i \(0.271154\pi\)
\(128\) 16384.0 0.0883883
\(129\) −17798.0 −0.0941666
\(130\) 141168. 0.732618
\(131\) 98142.0 0.499662 0.249831 0.968289i \(-0.419625\pi\)
0.249831 + 0.968289i \(0.419625\pi\)
\(132\) 0 0
\(133\) −236384. −1.15875
\(134\) 101372. 0.487704
\(135\) 24735.0 0.116809
\(136\) 47232.0 0.218972
\(137\) 400137. 1.82141 0.910704 0.413059i \(-0.135540\pi\)
0.910704 + 0.413059i \(0.135540\pi\)
\(138\) −7116.00 −0.0318081
\(139\) −205766. −0.903310 −0.451655 0.892193i \(-0.649166\pi\)
−0.451655 + 0.892193i \(0.649166\pi\)
\(140\) −135456. −0.584088
\(141\) −17184.0 −0.0727908
\(142\) 53244.0 0.221590
\(143\) 0 0
\(144\) −61952.0 −0.248971
\(145\) −105264. −0.415776
\(146\) 213040. 0.827140
\(147\) 10749.0 0.0410275
\(148\) −236560. −0.887743
\(149\) −87726.0 −0.323715 −0.161857 0.986814i \(-0.551748\pi\)
−0.161857 + 0.986814i \(0.551748\pi\)
\(150\) −2096.00 −0.00760612
\(151\) 432778. 1.54462 0.772312 0.635243i \(-0.219100\pi\)
0.772312 + 0.635243i \(0.219100\pi\)
\(152\) −91136.0 −0.319949
\(153\) −178596. −0.616798
\(154\) 0 0
\(155\) −318495. −1.06481
\(156\) −11072.0 −0.0364263
\(157\) −34075.0 −0.110328 −0.0551641 0.998477i \(-0.517568\pi\)
−0.0551641 + 0.998477i \(0.517568\pi\)
\(158\) −308936. −0.984523
\(159\) −30726.0 −0.0963858
\(160\) −52224.0 −0.161276
\(161\) −295314. −0.897882
\(162\) 233284. 0.698389
\(163\) 45020.0 0.132720 0.0663600 0.997796i \(-0.478861\pi\)
0.0663600 + 0.997796i \(0.478861\pi\)
\(164\) −84864.0 −0.246385
\(165\) 0 0
\(166\) −220056. −0.619816
\(167\) −482556. −1.33893 −0.669463 0.742845i \(-0.733476\pi\)
−0.669463 + 0.742845i \(0.733476\pi\)
\(168\) 10624.0 0.0290412
\(169\) 107571. 0.289720
\(170\) −150552. −0.399544
\(171\) 344608. 0.901229
\(172\) −284768. −0.733956
\(173\) 766254. 1.94651 0.973257 0.229719i \(-0.0737808\pi\)
0.973257 + 0.229719i \(0.0737808\pi\)
\(174\) 8256.00 0.0206727
\(175\) −86984.0 −0.214706
\(176\) 0 0
\(177\) −34989.0 −0.0839457
\(178\) 501660. 1.18675
\(179\) 303399. 0.707753 0.353876 0.935292i \(-0.384863\pi\)
0.353876 + 0.935292i \(0.384863\pi\)
\(180\) 197472. 0.454281
\(181\) −285181. −0.647030 −0.323515 0.946223i \(-0.604865\pi\)
−0.323515 + 0.946223i \(0.604865\pi\)
\(182\) −459488. −1.02824
\(183\) 45940.0 0.101406
\(184\) −113856. −0.247920
\(185\) 754035. 1.61980
\(186\) 24980.0 0.0529432
\(187\) 0 0
\(188\) −274944. −0.567348
\(189\) −80510.0 −0.163944
\(190\) 290496. 0.583789
\(191\) 767067. 1.52142 0.760711 0.649090i \(-0.224850\pi\)
0.760711 + 0.649090i \(0.224850\pi\)
\(192\) 4096.00 0.00801875
\(193\) −411668. −0.795525 −0.397763 0.917488i \(-0.630213\pi\)
−0.397763 + 0.917488i \(0.630213\pi\)
\(194\) −355228. −0.677646
\(195\) 35292.0 0.0664645
\(196\) 171984. 0.319777
\(197\) 759258. 1.39387 0.696937 0.717132i \(-0.254545\pi\)
0.696937 + 0.717132i \(0.254545\pi\)
\(198\) 0 0
\(199\) −46600.0 −0.0834167 −0.0417084 0.999130i \(-0.513280\pi\)
−0.0417084 + 0.999130i \(0.513280\pi\)
\(200\) −33536.0 −0.0592838
\(201\) 25343.0 0.0442454
\(202\) −5928.00 −0.0102219
\(203\) 342624. 0.583549
\(204\) 11808.0 0.0198656
\(205\) 270504. 0.449561
\(206\) −469984. −0.771641
\(207\) 430518. 0.698338
\(208\) −177152. −0.283915
\(209\) 0 0
\(210\) −33864.0 −0.0529895
\(211\) 932428. 1.44181 0.720907 0.693032i \(-0.243726\pi\)
0.720907 + 0.693032i \(0.243726\pi\)
\(212\) −491616. −0.751253
\(213\) 13311.0 0.0201030
\(214\) 317448. 0.473847
\(215\) 907698. 1.33920
\(216\) −31040.0 −0.0452676
\(217\) 1.03667e6 1.49448
\(218\) −351368. −0.500750
\(219\) 53260.0 0.0750397
\(220\) 0 0
\(221\) −510696. −0.703367
\(222\) −59140.0 −0.0805376
\(223\) 169745. 0.228578 0.114289 0.993448i \(-0.463541\pi\)
0.114289 + 0.993448i \(0.463541\pi\)
\(224\) 169984. 0.226354
\(225\) 126808. 0.166990
\(226\) −188988. −0.246129
\(227\) −198078. −0.255136 −0.127568 0.991830i \(-0.540717\pi\)
−0.127568 + 0.991830i \(0.540717\pi\)
\(228\) −22784.0 −0.0290264
\(229\) −849997. −1.07110 −0.535548 0.844505i \(-0.679895\pi\)
−0.535548 + 0.844505i \(0.679895\pi\)
\(230\) 362916. 0.452362
\(231\) 0 0
\(232\) 132096. 0.161128
\(233\) 401832. 0.484903 0.242451 0.970164i \(-0.422048\pi\)
0.242451 + 0.970164i \(0.422048\pi\)
\(234\) 669856. 0.799727
\(235\) 876384. 1.03520
\(236\) −559824. −0.654292
\(237\) −77234.0 −0.0893177
\(238\) 490032. 0.560766
\(239\) −855174. −0.968411 −0.484206 0.874954i \(-0.660891\pi\)
−0.484206 + 0.874954i \(0.660891\pi\)
\(240\) −13056.0 −0.0146313
\(241\) −1.12546e6 −1.24821 −0.624107 0.781339i \(-0.714537\pi\)
−0.624107 + 0.781339i \(0.714537\pi\)
\(242\) 0 0
\(243\) 176176. 0.191395
\(244\) 735040. 0.790381
\(245\) −548199. −0.583476
\(246\) −21216.0 −0.0223525
\(247\) 985408. 1.02772
\(248\) 399680. 0.412651
\(249\) −55014.0 −0.0562309
\(250\) 744396. 0.753276
\(251\) −1.19751e6 −1.19976 −0.599882 0.800088i \(-0.704786\pi\)
−0.599882 + 0.800088i \(0.704786\pi\)
\(252\) −642752. −0.637591
\(253\) 0 0
\(254\) 957664. 0.931384
\(255\) −37638.0 −0.0362473
\(256\) 65536.0 0.0625000
\(257\) 37758.0 0.0356596 0.0178298 0.999841i \(-0.494324\pi\)
0.0178298 + 0.999841i \(0.494324\pi\)
\(258\) −71192.0 −0.0665858
\(259\) −2.45431e6 −2.27342
\(260\) 564672. 0.518040
\(261\) −499488. −0.453862
\(262\) 392568. 0.353315
\(263\) 631254. 0.562749 0.281375 0.959598i \(-0.409210\pi\)
0.281375 + 0.959598i \(0.409210\pi\)
\(264\) 0 0
\(265\) 1.56703e6 1.37076
\(266\) −945536. −0.819359
\(267\) 125415. 0.107664
\(268\) 405488. 0.344859
\(269\) −1.08034e6 −0.910292 −0.455146 0.890417i \(-0.650413\pi\)
−0.455146 + 0.890417i \(0.650413\pi\)
\(270\) 98940.0 0.0825967
\(271\) 816100. 0.675025 0.337513 0.941321i \(-0.390414\pi\)
0.337513 + 0.941321i \(0.390414\pi\)
\(272\) 188928. 0.154837
\(273\) −114872. −0.0932841
\(274\) 1.60055e6 1.28793
\(275\) 0 0
\(276\) −28464.0 −0.0224917
\(277\) −1.68820e6 −1.32198 −0.660989 0.750396i \(-0.729863\pi\)
−0.660989 + 0.750396i \(0.729863\pi\)
\(278\) −823064. −0.638736
\(279\) −1.51129e6 −1.16235
\(280\) −541824. −0.413012
\(281\) 879042. 0.664116 0.332058 0.943259i \(-0.392257\pi\)
0.332058 + 0.943259i \(0.392257\pi\)
\(282\) −68736.0 −0.0514709
\(283\) −1.54027e6 −1.14322 −0.571611 0.820525i \(-0.693681\pi\)
−0.571611 + 0.820525i \(0.693681\pi\)
\(284\) 212976. 0.156688
\(285\) 72624.0 0.0529624
\(286\) 0 0
\(287\) −880464. −0.630967
\(288\) −247808. −0.176049
\(289\) −875213. −0.616409
\(290\) −421056. −0.293998
\(291\) −88807.0 −0.0614773
\(292\) 852160. 0.584876
\(293\) −720840. −0.490535 −0.245267 0.969455i \(-0.578876\pi\)
−0.245267 + 0.969455i \(0.578876\pi\)
\(294\) 42996.0 0.0290108
\(295\) 1.78444e6 1.19384
\(296\) −946240. −0.627729
\(297\) 0 0
\(298\) −350904. −0.228901
\(299\) 1.23107e6 0.796350
\(300\) −8384.00 −0.00537834
\(301\) −2.95447e6 −1.87959
\(302\) 1.73111e6 1.09221
\(303\) −1482.00 −0.000927346 0
\(304\) −364544. −0.226238
\(305\) −2.34294e6 −1.44215
\(306\) −714384. −0.436142
\(307\) 1.03905e6 0.629201 0.314601 0.949224i \(-0.398129\pi\)
0.314601 + 0.949224i \(0.398129\pi\)
\(308\) 0 0
\(309\) −117496. −0.0700046
\(310\) −1.27398e6 −0.752937
\(311\) −1.25135e6 −0.733630 −0.366815 0.930294i \(-0.619552\pi\)
−0.366815 + 0.930294i \(0.619552\pi\)
\(312\) −44288.0 −0.0257573
\(313\) −1.44336e6 −0.832749 −0.416375 0.909193i \(-0.636699\pi\)
−0.416375 + 0.909193i \(0.636699\pi\)
\(314\) −136300. −0.0780139
\(315\) 2.04877e6 1.16337
\(316\) −1.23574e6 −0.696163
\(317\) −2.01208e6 −1.12460 −0.562298 0.826934i \(-0.690083\pi\)
−0.562298 + 0.826934i \(0.690083\pi\)
\(318\) −122904. −0.0681551
\(319\) 0 0
\(320\) −208896. −0.114039
\(321\) 79362.0 0.0429883
\(322\) −1.18126e6 −0.634899
\(323\) −1.05091e6 −0.560480
\(324\) 933136. 0.493836
\(325\) 362608. 0.190427
\(326\) 180080. 0.0938472
\(327\) −87842.0 −0.0454290
\(328\) −339456. −0.174220
\(329\) −2.85254e6 −1.45292
\(330\) 0 0
\(331\) 2.01734e6 1.01207 0.506033 0.862514i \(-0.331112\pi\)
0.506033 + 0.862514i \(0.331112\pi\)
\(332\) −880224. −0.438276
\(333\) 3.57797e6 1.76818
\(334\) −1.93022e6 −0.946764
\(335\) −1.29249e6 −0.629240
\(336\) 42496.0 0.0205352
\(337\) −264122. −0.126686 −0.0633432 0.997992i \(-0.520176\pi\)
−0.0633432 + 0.997992i \(0.520176\pi\)
\(338\) 430284. 0.204863
\(339\) −47247.0 −0.0223293
\(340\) −602208. −0.282520
\(341\) 0 0
\(342\) 1.37843e6 0.637265
\(343\) −1.00563e6 −0.461532
\(344\) −1.13907e6 −0.518985
\(345\) 90729.0 0.0410392
\(346\) 3.06502e6 1.37639
\(347\) 1.71049e6 0.762601 0.381300 0.924451i \(-0.375476\pi\)
0.381300 + 0.924451i \(0.375476\pi\)
\(348\) 33024.0 0.0146178
\(349\) −218822. −0.0961673 −0.0480836 0.998843i \(-0.515311\pi\)
−0.0480836 + 0.998843i \(0.515311\pi\)
\(350\) −347936. −0.151820
\(351\) 335620. 0.145405
\(352\) 0 0
\(353\) 3.68192e6 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(354\) −139956. −0.0593586
\(355\) −678861. −0.285897
\(356\) 2.00664e6 0.839159
\(357\) 122508. 0.0508738
\(358\) 1.21360e6 0.500457
\(359\) −1.88528e6 −0.772042 −0.386021 0.922490i \(-0.626151\pi\)
−0.386021 + 0.922490i \(0.626151\pi\)
\(360\) 789888. 0.321225
\(361\) −448323. −0.181060
\(362\) −1.14072e6 −0.457519
\(363\) 0 0
\(364\) −1.83795e6 −0.727078
\(365\) −2.71626e6 −1.06718
\(366\) 183760. 0.0717048
\(367\) −3.11666e6 −1.20788 −0.603940 0.797029i \(-0.706404\pi\)
−0.603940 + 0.797029i \(0.706404\pi\)
\(368\) −455424. −0.175306
\(369\) 1.28357e6 0.490742
\(370\) 3.01614e6 1.14537
\(371\) −5.10052e6 −1.92389
\(372\) 99920.0 0.0374365
\(373\) −1.39441e6 −0.518943 −0.259471 0.965751i \(-0.583548\pi\)
−0.259471 + 0.965751i \(0.583548\pi\)
\(374\) 0 0
\(375\) 186099. 0.0683386
\(376\) −1.09978e6 −0.401176
\(377\) −1.42829e6 −0.517562
\(378\) −322040. −0.115926
\(379\) −4.26036e6 −1.52352 −0.761759 0.647860i \(-0.775664\pi\)
−0.761759 + 0.647860i \(0.775664\pi\)
\(380\) 1.16198e6 0.412801
\(381\) 239416. 0.0844969
\(382\) 3.06827e6 1.07581
\(383\) 201765. 0.0702828 0.0351414 0.999382i \(-0.488812\pi\)
0.0351414 + 0.999382i \(0.488812\pi\)
\(384\) 16384.0 0.00567012
\(385\) 0 0
\(386\) −1.64667e6 −0.562521
\(387\) 4.30712e6 1.46187
\(388\) −1.42091e6 −0.479168
\(389\) 1.94882e6 0.652977 0.326489 0.945201i \(-0.394135\pi\)
0.326489 + 0.945201i \(0.394135\pi\)
\(390\) 141168. 0.0469975
\(391\) −1.31290e6 −0.434301
\(392\) 687936. 0.226117
\(393\) 98142.0 0.0320534
\(394\) 3.03703e6 0.985618
\(395\) 3.93893e6 1.27024
\(396\) 0 0
\(397\) −1.46826e6 −0.467548 −0.233774 0.972291i \(-0.575108\pi\)
−0.233774 + 0.972291i \(0.575108\pi\)
\(398\) −186400. −0.0589845
\(399\) −236384. −0.0743337
\(400\) −134144. −0.0419200
\(401\) 2.24618e6 0.697563 0.348781 0.937204i \(-0.386596\pi\)
0.348781 + 0.937204i \(0.386596\pi\)
\(402\) 101372. 0.0312862
\(403\) −4.32154e6 −1.32549
\(404\) −23712.0 −0.00722794
\(405\) −2.97437e6 −0.901068
\(406\) 1.37050e6 0.412632
\(407\) 0 0
\(408\) 47232.0 0.0140471
\(409\) 3.61488e6 1.06853 0.534263 0.845318i \(-0.320589\pi\)
0.534263 + 0.845318i \(0.320589\pi\)
\(410\) 1.08202e6 0.317888
\(411\) 400137. 0.116843
\(412\) −1.87994e6 −0.545632
\(413\) −5.80817e6 −1.67558
\(414\) 1.72207e6 0.493799
\(415\) 2.80571e6 0.799693
\(416\) −708608. −0.200758
\(417\) −205766. −0.0579473
\(418\) 0 0
\(419\) −3.81239e6 −1.06087 −0.530435 0.847726i \(-0.677971\pi\)
−0.530435 + 0.847726i \(0.677971\pi\)
\(420\) −135456. −0.0374693
\(421\) 1.97346e6 0.542655 0.271327 0.962487i \(-0.412537\pi\)
0.271327 + 0.962487i \(0.412537\pi\)
\(422\) 3.72971e6 1.01952
\(423\) 4.15853e6 1.13003
\(424\) −1.96646e6 −0.531216
\(425\) −386712. −0.103852
\(426\) 53244.0 0.0142150
\(427\) 7.62604e6 2.02409
\(428\) 1.26979e6 0.335060
\(429\) 0 0
\(430\) 3.63079e6 0.946957
\(431\) 2.08359e6 0.540280 0.270140 0.962821i \(-0.412930\pi\)
0.270140 + 0.962821i \(0.412930\pi\)
\(432\) −124160. −0.0320090
\(433\) −72691.0 −0.0186321 −0.00931603 0.999957i \(-0.502965\pi\)
−0.00931603 + 0.999957i \(0.502965\pi\)
\(434\) 4.14668e6 1.05676
\(435\) −105264. −0.0266721
\(436\) −1.40547e6 −0.354084
\(437\) 2.53330e6 0.634574
\(438\) 213040. 0.0530611
\(439\) −594392. −0.147201 −0.0736007 0.997288i \(-0.523449\pi\)
−0.0736007 + 0.997288i \(0.523449\pi\)
\(440\) 0 0
\(441\) −2.60126e6 −0.636923
\(442\) −2.04278e6 −0.497355
\(443\) 4.56651e6 1.10554 0.552770 0.833334i \(-0.313571\pi\)
0.552770 + 0.833334i \(0.313571\pi\)
\(444\) −236560. −0.0569487
\(445\) −6.39616e6 −1.53116
\(446\) 678980. 0.161629
\(447\) −87726.0 −0.0207663
\(448\) 679936. 0.160056
\(449\) −5.44382e6 −1.27435 −0.637174 0.770720i \(-0.719897\pi\)
−0.637174 + 0.770720i \(0.719897\pi\)
\(450\) 507232. 0.118080
\(451\) 0 0
\(452\) −755952. −0.174040
\(453\) 432778. 0.0990877
\(454\) −792312. −0.180408
\(455\) 5.85847e6 1.32665
\(456\) −91136.0 −0.0205247
\(457\) −6.70312e6 −1.50137 −0.750683 0.660662i \(-0.770276\pi\)
−0.750683 + 0.660662i \(0.770276\pi\)
\(458\) −3.39999e6 −0.757380
\(459\) −357930. −0.0792988
\(460\) 1.45166e6 0.319869
\(461\) 1.25994e6 0.276120 0.138060 0.990424i \(-0.455913\pi\)
0.138060 + 0.990424i \(0.455913\pi\)
\(462\) 0 0
\(463\) −5.02308e6 −1.08897 −0.544487 0.838769i \(-0.683276\pi\)
−0.544487 + 0.838769i \(0.683276\pi\)
\(464\) 528384. 0.113934
\(465\) −318495. −0.0683078
\(466\) 1.60733e6 0.342878
\(467\) −2.35660e6 −0.500028 −0.250014 0.968242i \(-0.580435\pi\)
−0.250014 + 0.968242i \(0.580435\pi\)
\(468\) 2.67942e6 0.565492
\(469\) 4.20694e6 0.883149
\(470\) 3.50554e6 0.731998
\(471\) −34075.0 −0.00707756
\(472\) −2.23930e6 −0.462654
\(473\) 0 0
\(474\) −308936. −0.0631572
\(475\) 746176. 0.151743
\(476\) 1.96013e6 0.396522
\(477\) 7.43569e6 1.49632
\(478\) −3.42070e6 −0.684770
\(479\) 6.72258e6 1.33874 0.669371 0.742928i \(-0.266563\pi\)
0.669371 + 0.742928i \(0.266563\pi\)
\(480\) −52224.0 −0.0103459
\(481\) 1.02312e7 2.01634
\(482\) −4.50186e6 −0.882620
\(483\) −295314. −0.0575992
\(484\) 0 0
\(485\) 4.52916e6 0.874305
\(486\) 704704. 0.135337
\(487\) 1.96001e6 0.374487 0.187243 0.982314i \(-0.440045\pi\)
0.187243 + 0.982314i \(0.440045\pi\)
\(488\) 2.94016e6 0.558884
\(489\) 45020.0 0.00851399
\(490\) −2.19280e6 −0.412580
\(491\) 579624. 0.108503 0.0542516 0.998527i \(-0.482723\pi\)
0.0542516 + 0.998527i \(0.482723\pi\)
\(492\) −84864.0 −0.0158056
\(493\) 1.52323e6 0.282260
\(494\) 3.94163e6 0.726706
\(495\) 0 0
\(496\) 1.59872e6 0.291789
\(497\) 2.20963e6 0.401262
\(498\) −220056. −0.0397612
\(499\) 1.36905e6 0.246132 0.123066 0.992398i \(-0.460727\pi\)
0.123066 + 0.992398i \(0.460727\pi\)
\(500\) 2.97758e6 0.532646
\(501\) −482556. −0.0858921
\(502\) −4.79005e6 −0.848361
\(503\) −1.83343e6 −0.323105 −0.161552 0.986864i \(-0.551650\pi\)
−0.161552 + 0.986864i \(0.551650\pi\)
\(504\) −2.57101e6 −0.450845
\(505\) 75582.0 0.0131883
\(506\) 0 0
\(507\) 107571. 0.0185855
\(508\) 3.83066e6 0.658588
\(509\) −1.71266e6 −0.293006 −0.146503 0.989210i \(-0.546802\pi\)
−0.146503 + 0.989210i \(0.546802\pi\)
\(510\) −150552. −0.0256307
\(511\) 8.84116e6 1.49781
\(512\) 262144. 0.0441942
\(513\) 690640. 0.115867
\(514\) 151032. 0.0252151
\(515\) 5.99230e6 0.995578
\(516\) −284768. −0.0470833
\(517\) 0 0
\(518\) −9.81724e6 −1.60755
\(519\) 766254. 0.124869
\(520\) 2.25869e6 0.366309
\(521\) −789435. −0.127415 −0.0637077 0.997969i \(-0.520293\pi\)
−0.0637077 + 0.997969i \(0.520293\pi\)
\(522\) −1.99795e6 −0.320929
\(523\) −627392. −0.100296 −0.0501481 0.998742i \(-0.515969\pi\)
−0.0501481 + 0.998742i \(0.515969\pi\)
\(524\) 1.57027e6 0.249831
\(525\) −86984.0 −0.0137734
\(526\) 2.52502e6 0.397924
\(527\) 4.60881e6 0.722873
\(528\) 0 0
\(529\) −3.27150e6 −0.508286
\(530\) 6.26810e6 0.969274
\(531\) 8.46734e6 1.30320
\(532\) −3.78214e6 −0.579374
\(533\) 3.67037e6 0.559618
\(534\) 501660. 0.0761301
\(535\) −4.04746e6 −0.611362
\(536\) 1.62195e6 0.243852
\(537\) 303399. 0.0454024
\(538\) −4.32137e6 −0.643673
\(539\) 0 0
\(540\) 395760. 0.0584047
\(541\) −3.20895e6 −0.471379 −0.235689 0.971828i \(-0.575735\pi\)
−0.235689 + 0.971828i \(0.575735\pi\)
\(542\) 3.26440e6 0.477315
\(543\) −285181. −0.0415070
\(544\) 755712. 0.109486
\(545\) 4.47994e6 0.646072
\(546\) −459488. −0.0659618
\(547\) −3.42658e6 −0.489658 −0.244829 0.969566i \(-0.578732\pi\)
−0.244829 + 0.969566i \(0.578732\pi\)
\(548\) 6.40219e6 0.910704
\(549\) −1.11175e7 −1.57426
\(550\) 0 0
\(551\) −2.93914e6 −0.412421
\(552\) −113856. −0.0159041
\(553\) −1.28208e7 −1.78280
\(554\) −6.75279e6 −0.934779
\(555\) 754035. 0.103910
\(556\) −3.29226e6 −0.451655
\(557\) −1.05198e7 −1.43672 −0.718358 0.695674i \(-0.755106\pi\)
−0.718358 + 0.695674i \(0.755106\pi\)
\(558\) −6.04516e6 −0.821906
\(559\) 1.23162e7 1.66705
\(560\) −2.16730e6 −0.292044
\(561\) 0 0
\(562\) 3.51617e6 0.469601
\(563\) −5.47288e6 −0.727687 −0.363844 0.931460i \(-0.618536\pi\)
−0.363844 + 0.931460i \(0.618536\pi\)
\(564\) −274944. −0.0363954
\(565\) 2.40960e6 0.317558
\(566\) −6.16107e6 −0.808379
\(567\) 9.68129e6 1.26466
\(568\) 851904. 0.110795
\(569\) 1.17787e7 1.52516 0.762580 0.646893i \(-0.223932\pi\)
0.762580 + 0.646893i \(0.223932\pi\)
\(570\) 290496. 0.0374501
\(571\) 8.35628e6 1.07256 0.536281 0.844039i \(-0.319829\pi\)
0.536281 + 0.844039i \(0.319829\pi\)
\(572\) 0 0
\(573\) 767067. 0.0975993
\(574\) −3.52186e6 −0.446161
\(575\) 932196. 0.117581
\(576\) −991232. −0.124486
\(577\) −1.37758e7 −1.72258 −0.861288 0.508117i \(-0.830342\pi\)
−0.861288 + 0.508117i \(0.830342\pi\)
\(578\) −3.50085e6 −0.435867
\(579\) −411668. −0.0510330
\(580\) −1.68422e6 −0.207888
\(581\) −9.13232e6 −1.12238
\(582\) −355228. −0.0434710
\(583\) 0 0
\(584\) 3.40864e6 0.413570
\(585\) −8.54066e6 −1.03182
\(586\) −2.88336e6 −0.346860
\(587\) −1.27093e7 −1.52239 −0.761196 0.648522i \(-0.775388\pi\)
−0.761196 + 0.648522i \(0.775388\pi\)
\(588\) 171984. 0.0205137
\(589\) −8.89288e6 −1.05622
\(590\) 7.13776e6 0.844173
\(591\) 759258. 0.0894171
\(592\) −3.78496e6 −0.443871
\(593\) −1.00825e6 −0.117742 −0.0588711 0.998266i \(-0.518750\pi\)
−0.0588711 + 0.998266i \(0.518750\pi\)
\(594\) 0 0
\(595\) −6.24791e6 −0.723506
\(596\) −1.40362e6 −0.161857
\(597\) −46600.0 −0.00535119
\(598\) 4.92427e6 0.563105
\(599\) 1.05100e7 1.19684 0.598421 0.801182i \(-0.295795\pi\)
0.598421 + 0.801182i \(0.295795\pi\)
\(600\) −33536.0 −0.00380306
\(601\) 199390. 0.0225173 0.0112587 0.999937i \(-0.496416\pi\)
0.0112587 + 0.999937i \(0.496416\pi\)
\(602\) −1.18179e7 −1.32907
\(603\) −6.13301e6 −0.686879
\(604\) 6.92445e6 0.772312
\(605\) 0 0
\(606\) −5928.00 −0.000655732 0
\(607\) −16190.0 −0.00178351 −0.000891754 1.00000i \(-0.500284\pi\)
−0.000891754 1.00000i \(0.500284\pi\)
\(608\) −1.45818e6 −0.159975
\(609\) 342624. 0.0374347
\(610\) −9.37176e6 −1.01976
\(611\) 1.18913e7 1.28863
\(612\) −2.85754e6 −0.308399
\(613\) 1.15253e7 1.23880 0.619402 0.785074i \(-0.287375\pi\)
0.619402 + 0.785074i \(0.287375\pi\)
\(614\) 4.15619e6 0.444913
\(615\) 270504. 0.0288394
\(616\) 0 0
\(617\) 1.69974e7 1.79750 0.898751 0.438459i \(-0.144476\pi\)
0.898751 + 0.438459i \(0.144476\pi\)
\(618\) −469984. −0.0495008
\(619\) −1.84875e7 −1.93933 −0.969663 0.244445i \(-0.921394\pi\)
−0.969663 + 0.244445i \(0.921394\pi\)
\(620\) −5.09592e6 −0.532407
\(621\) 862815. 0.0897819
\(622\) −5.00539e6 −0.518755
\(623\) 2.08189e7 2.14901
\(624\) −177152. −0.0182131
\(625\) −7.85355e6 −0.804203
\(626\) −5.77344e6 −0.588842
\(627\) 0 0
\(628\) −545200. −0.0551641
\(629\) −1.09113e7 −1.09964
\(630\) 8.19509e6 0.822626
\(631\) −4.54281e6 −0.454204 −0.227102 0.973871i \(-0.572925\pi\)
−0.227102 + 0.973871i \(0.572925\pi\)
\(632\) −4.94298e6 −0.492261
\(633\) 932428. 0.0924924
\(634\) −8.04832e6 −0.795210
\(635\) −1.22102e7 −1.20168
\(636\) −491616. −0.0481929
\(637\) −7.43831e6 −0.726316
\(638\) 0 0
\(639\) −3.22126e6 −0.312086
\(640\) −835584. −0.0806381
\(641\) 1.84286e7 1.77153 0.885764 0.464136i \(-0.153635\pi\)
0.885764 + 0.464136i \(0.153635\pi\)
\(642\) 317448. 0.0303973
\(643\) 9.66604e6 0.921979 0.460989 0.887406i \(-0.347495\pi\)
0.460989 + 0.887406i \(0.347495\pi\)
\(644\) −4.72502e6 −0.448941
\(645\) 907698. 0.0859097
\(646\) −4.20365e6 −0.396319
\(647\) −4.51430e6 −0.423965 −0.211982 0.977273i \(-0.567992\pi\)
−0.211982 + 0.977273i \(0.567992\pi\)
\(648\) 3.73254e6 0.349195
\(649\) 0 0
\(650\) 1.45043e6 0.134652
\(651\) 1.03667e6 0.0958712
\(652\) 720320. 0.0663600
\(653\) −5.37235e6 −0.493039 −0.246519 0.969138i \(-0.579287\pi\)
−0.246519 + 0.969138i \(0.579287\pi\)
\(654\) −351368. −0.0321231
\(655\) −5.00524e6 −0.455850
\(656\) −1.35782e6 −0.123192
\(657\) −1.28889e7 −1.16494
\(658\) −1.14102e7 −1.02737
\(659\) −9.87956e6 −0.886184 −0.443092 0.896476i \(-0.646119\pi\)
−0.443092 + 0.896476i \(0.646119\pi\)
\(660\) 0 0
\(661\) 1.08052e7 0.961898 0.480949 0.876748i \(-0.340292\pi\)
0.480949 + 0.876748i \(0.340292\pi\)
\(662\) 8.06935e6 0.715638
\(663\) −510696. −0.0451210
\(664\) −3.52090e6 −0.309908
\(665\) 1.20556e7 1.05714
\(666\) 1.43119e7 1.25029
\(667\) −3.67186e6 −0.319574
\(668\) −7.72090e6 −0.669463
\(669\) 169745. 0.0146633
\(670\) −5.16997e6 −0.444940
\(671\) 0 0
\(672\) 169984. 0.0145206
\(673\) −1.13275e7 −0.964042 −0.482021 0.876160i \(-0.660097\pi\)
−0.482021 + 0.876160i \(0.660097\pi\)
\(674\) −1.05649e6 −0.0895808
\(675\) 254140. 0.0214691
\(676\) 1.72114e6 0.144860
\(677\) 1.20595e7 1.01125 0.505624 0.862754i \(-0.331262\pi\)
0.505624 + 0.862754i \(0.331262\pi\)
\(678\) −188988. −0.0157892
\(679\) −1.47420e7 −1.22710
\(680\) −2.40883e6 −0.199772
\(681\) −198078. −0.0163670
\(682\) 0 0
\(683\) −5.14166e6 −0.421747 −0.210873 0.977513i \(-0.567631\pi\)
−0.210873 + 0.977513i \(0.567631\pi\)
\(684\) 5.51373e6 0.450614
\(685\) −2.04070e7 −1.66170
\(686\) −4.02251e6 −0.326353
\(687\) −849997. −0.0687109
\(688\) −4.55629e6 −0.366978
\(689\) 2.12624e7 1.70633
\(690\) 362916. 0.0290191
\(691\) 1.31243e7 1.04563 0.522817 0.852445i \(-0.324881\pi\)
0.522817 + 0.852445i \(0.324881\pi\)
\(692\) 1.22601e7 0.973257
\(693\) 0 0
\(694\) 6.84197e6 0.539240
\(695\) 1.04941e7 0.824104
\(696\) 132096. 0.0103363
\(697\) −3.91435e6 −0.305195
\(698\) −875288. −0.0680005
\(699\) 401832. 0.0311065
\(700\) −1.39174e6 −0.107353
\(701\) −3.65956e6 −0.281277 −0.140638 0.990061i \(-0.544916\pi\)
−0.140638 + 0.990061i \(0.544916\pi\)
\(702\) 1.34248e6 0.102817
\(703\) 2.10538e7 1.60673
\(704\) 0 0
\(705\) 876384. 0.0664082
\(706\) 1.47277e7 1.11204
\(707\) −246012. −0.0185101
\(708\) −559824. −0.0419728
\(709\) 1.02252e7 0.763935 0.381968 0.924176i \(-0.375247\pi\)
0.381968 + 0.924176i \(0.375247\pi\)
\(710\) −2.71544e6 −0.202160
\(711\) 1.86906e7 1.38660
\(712\) 8.02656e6 0.593375
\(713\) −1.11099e7 −0.818436
\(714\) 490032. 0.0359732
\(715\) 0 0
\(716\) 4.85438e6 0.353876
\(717\) −855174. −0.0621236
\(718\) −7.54114e6 −0.545916
\(719\) 2.41683e7 1.74351 0.871753 0.489945i \(-0.162983\pi\)
0.871753 + 0.489945i \(0.162983\pi\)
\(720\) 3.15955e6 0.227140
\(721\) −1.95043e7 −1.39731
\(722\) −1.79329e6 −0.128029
\(723\) −1.12546e6 −0.0800730
\(724\) −4.56290e6 −0.323515
\(725\) −1.08154e6 −0.0764181
\(726\) 0 0
\(727\) 1.68246e7 1.18062 0.590310 0.807177i \(-0.299006\pi\)
0.590310 + 0.807177i \(0.299006\pi\)
\(728\) −7.35181e6 −0.514121
\(729\) −1.39958e7 −0.975393
\(730\) −1.08650e7 −0.754613
\(731\) −1.31349e7 −0.909147
\(732\) 735040. 0.0507030
\(733\) 5.04168e6 0.346590 0.173295 0.984870i \(-0.444559\pi\)
0.173295 + 0.984870i \(0.444559\pi\)
\(734\) −1.24666e7 −0.854101
\(735\) −548199. −0.0374300
\(736\) −1.82170e6 −0.123960
\(737\) 0 0
\(738\) 5.13427e6 0.347007
\(739\) 6.26375e6 0.421913 0.210957 0.977495i \(-0.432342\pi\)
0.210957 + 0.977495i \(0.432342\pi\)
\(740\) 1.20646e7 0.809901
\(741\) 985408. 0.0659281
\(742\) −2.04021e7 −1.36039
\(743\) −3.63976e6 −0.241880 −0.120940 0.992660i \(-0.538591\pi\)
−0.120940 + 0.992660i \(0.538591\pi\)
\(744\) 399680. 0.0264716
\(745\) 4.47403e6 0.295330
\(746\) −5.57766e6 −0.366948
\(747\) 1.33134e7 0.872945
\(748\) 0 0
\(749\) 1.31741e7 0.858057
\(750\) 744396. 0.0483227
\(751\) −1.87370e7 −1.21227 −0.606135 0.795362i \(-0.707281\pi\)
−0.606135 + 0.795362i \(0.707281\pi\)
\(752\) −4.39910e6 −0.283674
\(753\) −1.19751e6 −0.0769649
\(754\) −5.71315e6 −0.365972
\(755\) −2.20717e7 −1.40918
\(756\) −1.28816e6 −0.0819720
\(757\) 489242. 0.0310302 0.0155151 0.999880i \(-0.495061\pi\)
0.0155151 + 0.999880i \(0.495061\pi\)
\(758\) −1.70414e7 −1.07729
\(759\) 0 0
\(760\) 4.64794e6 0.291895
\(761\) −1.46969e7 −0.919952 −0.459976 0.887931i \(-0.652142\pi\)
−0.459976 + 0.887931i \(0.652142\pi\)
\(762\) 957664. 0.0597483
\(763\) −1.45818e7 −0.906774
\(764\) 1.22731e7 0.760711
\(765\) 9.10840e6 0.562715
\(766\) 807060. 0.0496974
\(767\) 2.42124e7 1.48610
\(768\) 65536.0 0.00400938
\(769\) −2.42072e7 −1.47615 −0.738073 0.674721i \(-0.764264\pi\)
−0.738073 + 0.674721i \(0.764264\pi\)
\(770\) 0 0
\(771\) 37758.0 0.00228756
\(772\) −6.58669e6 −0.397763
\(773\) 1.35260e7 0.814181 0.407091 0.913388i \(-0.366543\pi\)
0.407091 + 0.913388i \(0.366543\pi\)
\(774\) 1.72285e7 1.03370
\(775\) −3.27238e6 −0.195708
\(776\) −5.68365e6 −0.338823
\(777\) −2.45431e6 −0.145840
\(778\) 7.79528e6 0.461725
\(779\) 7.55290e6 0.445933
\(780\) 564672. 0.0332323
\(781\) 0 0
\(782\) −5.25161e6 −0.307097
\(783\) −1.00104e6 −0.0583508
\(784\) 2.75174e6 0.159889
\(785\) 1.73782e6 0.100654
\(786\) 392568. 0.0226651
\(787\) −1.42094e7 −0.817786 −0.408893 0.912582i \(-0.634085\pi\)
−0.408893 + 0.912582i \(0.634085\pi\)
\(788\) 1.21481e7 0.696937
\(789\) 631254. 0.0361004
\(790\) 1.57557e7 0.898196
\(791\) −7.84300e6 −0.445698
\(792\) 0 0
\(793\) −3.17905e7 −1.79521
\(794\) −5.87303e6 −0.330606
\(795\) 1.56703e6 0.0879343
\(796\) −745600. −0.0417084
\(797\) −7.93333e6 −0.442395 −0.221197 0.975229i \(-0.570997\pi\)
−0.221197 + 0.975229i \(0.570997\pi\)
\(798\) −945536. −0.0525619
\(799\) −1.26818e7 −0.702771
\(800\) −536576. −0.0296419
\(801\) −3.03504e7 −1.67141
\(802\) 8.98471e6 0.493251
\(803\) 0 0
\(804\) 405488. 0.0221227
\(805\) 1.50610e7 0.819152
\(806\) −1.72862e7 −0.937262
\(807\) −1.08034e6 −0.0583952
\(808\) −94848.0 −0.00511093
\(809\) 1.04685e7 0.562359 0.281180 0.959655i \(-0.409274\pi\)
0.281180 + 0.959655i \(0.409274\pi\)
\(810\) −1.18975e7 −0.637151
\(811\) −1.19147e7 −0.636110 −0.318055 0.948072i \(-0.603030\pi\)
−0.318055 + 0.948072i \(0.603030\pi\)
\(812\) 5.48198e6 0.291775
\(813\) 816100. 0.0433029
\(814\) 0 0
\(815\) −2.29602e6 −0.121083
\(816\) 188928. 0.00993278
\(817\) 2.53444e7 1.32839
\(818\) 1.44595e7 0.755562
\(819\) 2.77990e7 1.44817
\(820\) 4.32806e6 0.224781
\(821\) 1.86112e6 0.0963645 0.0481822 0.998839i \(-0.484657\pi\)
0.0481822 + 0.998839i \(0.484657\pi\)
\(822\) 1.60055e6 0.0826208
\(823\) 2.30153e7 1.18445 0.592225 0.805773i \(-0.298250\pi\)
0.592225 + 0.805773i \(0.298250\pi\)
\(824\) −7.51974e6 −0.385820
\(825\) 0 0
\(826\) −2.32327e7 −1.18481
\(827\) 1.68351e7 0.855959 0.427980 0.903788i \(-0.359225\pi\)
0.427980 + 0.903788i \(0.359225\pi\)
\(828\) 6.88829e6 0.349169
\(829\) −2.35299e7 −1.18914 −0.594570 0.804044i \(-0.702678\pi\)
−0.594570 + 0.804044i \(0.702678\pi\)
\(830\) 1.12229e7 0.565468
\(831\) −1.68820e6 −0.0848049
\(832\) −2.83443e6 −0.141957
\(833\) 7.93276e6 0.396106
\(834\) −823064. −0.0409750
\(835\) 2.46104e7 1.22152
\(836\) 0 0
\(837\) −3.02882e6 −0.149438
\(838\) −1.52496e7 −0.750148
\(839\) 2.91549e7 1.42990 0.714952 0.699173i \(-0.246448\pi\)
0.714952 + 0.699173i \(0.246448\pi\)
\(840\) −541824. −0.0264948
\(841\) −1.62511e7 −0.792303
\(842\) 7.89385e6 0.383715
\(843\) 879042. 0.0426030
\(844\) 1.49188e7 0.720907
\(845\) −5.48612e6 −0.264316
\(846\) 1.66341e7 0.799050
\(847\) 0 0
\(848\) −7.86586e6 −0.375627
\(849\) −1.54027e6 −0.0733377
\(850\) −1.54685e6 −0.0734345
\(851\) 2.63025e7 1.24501
\(852\) 212976. 0.0100515
\(853\) 9.49052e6 0.446599 0.223299 0.974750i \(-0.428317\pi\)
0.223299 + 0.974750i \(0.428317\pi\)
\(854\) 3.05042e7 1.43125
\(855\) −1.75750e7 −0.822205
\(856\) 5.07917e6 0.236924
\(857\) 1.81553e6 0.0844405 0.0422203 0.999108i \(-0.486557\pi\)
0.0422203 + 0.999108i \(0.486557\pi\)
\(858\) 0 0
\(859\) −1.07812e7 −0.498522 −0.249261 0.968436i \(-0.580188\pi\)
−0.249261 + 0.968436i \(0.580188\pi\)
\(860\) 1.45232e7 0.669600
\(861\) −880464. −0.0404766
\(862\) 8.33436e6 0.382036
\(863\) −2.83355e7 −1.29510 −0.647550 0.762023i \(-0.724206\pi\)
−0.647550 + 0.762023i \(0.724206\pi\)
\(864\) −496640. −0.0226338
\(865\) −3.90790e7 −1.77584
\(866\) −290764. −0.0131749
\(867\) −875213. −0.0395427
\(868\) 1.65867e7 0.747242
\(869\) 0 0
\(870\) −421056. −0.0188600
\(871\) −1.75374e7 −0.783283
\(872\) −5.62189e6 −0.250375
\(873\) 2.14913e7 0.954392
\(874\) 1.01332e7 0.448712
\(875\) 3.08924e7 1.36406
\(876\) 852160. 0.0375198
\(877\) 2.68919e7 1.18065 0.590326 0.807165i \(-0.298999\pi\)
0.590326 + 0.807165i \(0.298999\pi\)
\(878\) −2.37757e6 −0.104087
\(879\) −720840. −0.0314678
\(880\) 0 0
\(881\) −1.92132e7 −0.833989 −0.416995 0.908909i \(-0.636917\pi\)
−0.416995 + 0.908909i \(0.636917\pi\)
\(882\) −1.04050e7 −0.450373
\(883\) 1.15931e7 0.500378 0.250189 0.968197i \(-0.419507\pi\)
0.250189 + 0.968197i \(0.419507\pi\)
\(884\) −8.17114e6 −0.351683
\(885\) 1.78444e6 0.0765850
\(886\) 1.82660e7 0.781735
\(887\) −1.31857e7 −0.562721 −0.281361 0.959602i \(-0.590786\pi\)
−0.281361 + 0.959602i \(0.590786\pi\)
\(888\) −946240. −0.0402688
\(889\) 3.97431e7 1.68658
\(890\) −2.55847e7 −1.08269
\(891\) 0 0
\(892\) 2.71592e6 0.114289
\(893\) 2.44700e7 1.02685
\(894\) −350904. −0.0146840
\(895\) −1.54733e7 −0.645694
\(896\) 2.71974e6 0.113177
\(897\) 1.23107e6 0.0510859
\(898\) −2.17753e7 −0.901100
\(899\) 1.28897e7 0.531916
\(900\) 2.02893e6 0.0834950
\(901\) −2.26758e7 −0.930573
\(902\) 0 0
\(903\) −2.95447e6 −0.120576
\(904\) −3.02381e6 −0.123065
\(905\) 1.45442e7 0.590295
\(906\) 1.73111e6 0.0700656
\(907\) 2.98195e6 0.120360 0.0601800 0.998188i \(-0.480833\pi\)
0.0601800 + 0.998188i \(0.480833\pi\)
\(908\) −3.16925e6 −0.127568
\(909\) 358644. 0.0143964
\(910\) 2.34339e7 0.938082
\(911\) 2.96579e7 1.18398 0.591989 0.805946i \(-0.298343\pi\)
0.591989 + 0.805946i \(0.298343\pi\)
\(912\) −364544. −0.0145132
\(913\) 0 0
\(914\) −2.68125e7 −1.06163
\(915\) −2.34294e6 −0.0925142
\(916\) −1.36000e7 −0.535548
\(917\) 1.62916e7 0.639793
\(918\) −1.43172e6 −0.0560727
\(919\) −3.18057e7 −1.24227 −0.621135 0.783704i \(-0.713328\pi\)
−0.621135 + 0.783704i \(0.713328\pi\)
\(920\) 5.80666e6 0.226181
\(921\) 1.03905e6 0.0403633
\(922\) 5.03976e6 0.195246
\(923\) −9.21121e6 −0.355887
\(924\) 0 0
\(925\) 7.74734e6 0.297713
\(926\) −2.00923e7 −0.770021
\(927\) 2.84340e7 1.08677
\(928\) 2.11354e6 0.0805638
\(929\) −2.33444e7 −0.887451 −0.443725 0.896163i \(-0.646343\pi\)
−0.443725 + 0.896163i \(0.646343\pi\)
\(930\) −1.27398e6 −0.0483009
\(931\) −1.53066e7 −0.578767
\(932\) 6.42931e6 0.242451
\(933\) −1.25135e6 −0.0470624
\(934\) −9.42642e6 −0.353573
\(935\) 0 0
\(936\) 1.07177e7 0.399864
\(937\) −2.07372e7 −0.771616 −0.385808 0.922579i \(-0.626077\pi\)
−0.385808 + 0.922579i \(0.626077\pi\)
\(938\) 1.68278e7 0.624481
\(939\) −1.44336e6 −0.0534209
\(940\) 1.40221e7 0.517601
\(941\) −2.69193e7 −0.991036 −0.495518 0.868598i \(-0.665022\pi\)
−0.495518 + 0.868598i \(0.665022\pi\)
\(942\) −136300. −0.00500459
\(943\) 9.43582e6 0.345542
\(944\) −8.95718e6 −0.327146
\(945\) 4.10601e6 0.149569
\(946\) 0 0
\(947\) 1.01896e7 0.369216 0.184608 0.982812i \(-0.440898\pi\)
0.184608 + 0.982812i \(0.440898\pi\)
\(948\) −1.23574e6 −0.0446589
\(949\) −3.68559e7 −1.32844
\(950\) 2.98470e6 0.107298
\(951\) −2.01208e6 −0.0721429
\(952\) 7.84051e6 0.280383
\(953\) −1.03924e7 −0.370665 −0.185333 0.982676i \(-0.559336\pi\)
−0.185333 + 0.982676i \(0.559336\pi\)
\(954\) 2.97428e7 1.05806
\(955\) −3.91204e7 −1.38802
\(956\) −1.36828e7 −0.484206
\(957\) 0 0
\(958\) 2.68903e7 0.946634
\(959\) 6.64227e7 2.33222
\(960\) −208896. −0.00731564
\(961\) 1.03709e7 0.362249
\(962\) 4.09249e7 1.42577
\(963\) −1.92056e7 −0.667363
\(964\) −1.80074e7 −0.624107
\(965\) 2.09951e7 0.725770
\(966\) −1.18126e6 −0.0407288
\(967\) 8.18877e6 0.281613 0.140806 0.990037i \(-0.455030\pi\)
0.140806 + 0.990037i \(0.455030\pi\)
\(968\) 0 0
\(969\) −1.05091e6 −0.0359548
\(970\) 1.81166e7 0.618227
\(971\) −1.73274e7 −0.589775 −0.294887 0.955532i \(-0.595282\pi\)
−0.294887 + 0.955532i \(0.595282\pi\)
\(972\) 2.81882e6 0.0956976
\(973\) −3.41572e7 −1.15664
\(974\) 7.84005e6 0.264802
\(975\) 362608. 0.0122159
\(976\) 1.17606e7 0.395190
\(977\) −438963. −0.0147127 −0.00735634 0.999973i \(-0.502342\pi\)
−0.00735634 + 0.999973i \(0.502342\pi\)
\(978\) 180080. 0.00602030
\(979\) 0 0
\(980\) −8.77118e6 −0.291738
\(981\) 2.12578e7 0.705253
\(982\) 2.31850e6 0.0767234
\(983\) −2.79124e7 −0.921326 −0.460663 0.887575i \(-0.652388\pi\)
−0.460663 + 0.887575i \(0.652388\pi\)
\(984\) −339456. −0.0111762
\(985\) −3.87222e7 −1.27165
\(986\) 6.09293e6 0.199588
\(987\) −2.85254e6 −0.0932051
\(988\) 1.57665e7 0.513859
\(989\) 3.16626e7 1.02933
\(990\) 0 0
\(991\) −4.26846e7 −1.38066 −0.690331 0.723494i \(-0.742535\pi\)
−0.690331 + 0.723494i \(0.742535\pi\)
\(992\) 6.39488e6 0.206326
\(993\) 2.01734e6 0.0649240
\(994\) 8.83850e6 0.283735
\(995\) 2.37660e6 0.0761024
\(996\) −880224. −0.0281154
\(997\) 2.21044e7 0.704273 0.352137 0.935949i \(-0.385455\pi\)
0.352137 + 0.935949i \(0.385455\pi\)
\(998\) 5.47621e6 0.174042
\(999\) 7.17072e6 0.227326
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 242.6.a.d.1.1 1
11.10 odd 2 22.6.a.b.1.1 1
33.32 even 2 198.6.a.i.1.1 1
44.43 even 2 176.6.a.b.1.1 1
55.32 even 4 550.6.b.f.199.1 2
55.43 even 4 550.6.b.f.199.2 2
55.54 odd 2 550.6.a.f.1.1 1
77.76 even 2 1078.6.a.a.1.1 1
88.21 odd 2 704.6.a.e.1.1 1
88.43 even 2 704.6.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.6.a.b.1.1 1 11.10 odd 2
176.6.a.b.1.1 1 44.43 even 2
198.6.a.i.1.1 1 33.32 even 2
242.6.a.d.1.1 1 1.1 even 1 trivial
550.6.a.f.1.1 1 55.54 odd 2
550.6.b.f.199.1 2 55.32 even 4
550.6.b.f.199.2 2 55.43 even 4
704.6.a.e.1.1 1 88.21 odd 2
704.6.a.f.1.1 1 88.43 even 2
1078.6.a.a.1.1 1 77.76 even 2