Properties

Label 114.6.a.g.1.1
Level $114$
Weight $6$
Character 114.1
Self dual yes
Analytic conductor $18.284$
Analytic rank $0$
Dimension $2$
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] = [114,6,Mod(1,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("114.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 114.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: \(18.2837554587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2441}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 610 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(25.2032\) of defining polynomial
Character \(\chi\) \(=\) 114.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -76.6097 q^{5} -36.0000 q^{6} -126.610 q^{7} +64.0000 q^{8} +81.0000 q^{9} -306.439 q^{10} +744.268 q^{11} -144.000 q^{12} +844.439 q^{13} -506.439 q^{14} +689.487 q^{15} +256.000 q^{16} +1306.17 q^{17} +324.000 q^{18} -361.000 q^{19} -1225.76 q^{20} +1139.49 q^{21} +2977.07 q^{22} -1823.41 q^{23} -576.000 q^{24} +2744.05 q^{25} +3377.76 q^{26} -729.000 q^{27} -2025.76 q^{28} -5539.75 q^{29} +2757.95 q^{30} +10200.7 q^{31} +1024.00 q^{32} -6698.41 q^{33} +5224.68 q^{34} +9699.53 q^{35} +1296.00 q^{36} +3328.10 q^{37} -1444.00 q^{38} -7599.95 q^{39} -4903.02 q^{40} +6050.53 q^{41} +4557.95 q^{42} +1922.46 q^{43} +11908.3 q^{44} -6205.39 q^{45} -7293.66 q^{46} +10370.8 q^{47} -2304.00 q^{48} -776.980 q^{49} +10976.2 q^{50} -11755.5 q^{51} +13511.0 q^{52} +3107.66 q^{53} -2916.00 q^{54} -57018.2 q^{55} -8103.02 q^{56} +3249.00 q^{57} -22159.0 q^{58} -3194.44 q^{59} +11031.8 q^{60} +17475.6 q^{61} +40802.9 q^{62} -10255.4 q^{63} +4096.00 q^{64} -64692.2 q^{65} -26793.6 q^{66} +13374.0 q^{67} +20898.7 q^{68} +16410.7 q^{69} +38798.1 q^{70} +6280.96 q^{71} +5184.00 q^{72} +78709.0 q^{73} +13312.4 q^{74} -24696.4 q^{75} -5776.00 q^{76} -94231.6 q^{77} -30399.8 q^{78} +41603.4 q^{79} -19612.1 q^{80} +6561.00 q^{81} +24202.1 q^{82} -9042.74 q^{83} +18231.8 q^{84} -100065. q^{85} +7689.85 q^{86} +49857.8 q^{87} +47633.2 q^{88} +3935.36 q^{89} -24821.5 q^{90} -106914. q^{91} -29174.6 q^{92} -91806.6 q^{93} +41483.4 q^{94} +27656.1 q^{95} -9216.00 q^{96} +144902. q^{97} -3107.92 q^{98} +60285.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 18 q^{3} + 32 q^{4} - 5 q^{5} - 72 q^{6} - 105 q^{7} + 128 q^{8} + 162 q^{9} - 20 q^{10} + 451 q^{11} - 288 q^{12} + 1096 q^{13} - 420 q^{14} + 45 q^{15} + 512 q^{16} + 3057 q^{17} + 648 q^{18}+ \cdots + 36531 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) −76.6097 −1.37044 −0.685218 0.728338i \(-0.740293\pi\)
−0.685218 + 0.728338i \(0.740293\pi\)
\(6\) −36.0000 −0.408248
\(7\) −126.610 −0.976612 −0.488306 0.872673i \(-0.662385\pi\)
−0.488306 + 0.872673i \(0.662385\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) −306.439 −0.969045
\(11\) 744.268 1.85459 0.927294 0.374333i \(-0.122128\pi\)
0.927294 + 0.374333i \(0.122128\pi\)
\(12\) −144.000 −0.288675
\(13\) 844.439 1.38583 0.692915 0.721019i \(-0.256326\pi\)
0.692915 + 0.721019i \(0.256326\pi\)
\(14\) −506.439 −0.690569
\(15\) 689.487 0.791222
\(16\) 256.000 0.250000
\(17\) 1306.17 1.09617 0.548085 0.836423i \(-0.315357\pi\)
0.548085 + 0.836423i \(0.315357\pi\)
\(18\) 324.000 0.235702
\(19\) −361.000 −0.229416
\(20\) −1225.76 −0.685218
\(21\) 1139.49 0.563847
\(22\) 2977.07 1.31139
\(23\) −1823.41 −0.718730 −0.359365 0.933197i \(-0.617007\pi\)
−0.359365 + 0.933197i \(0.617007\pi\)
\(24\) −576.000 −0.204124
\(25\) 2744.05 0.878096
\(26\) 3377.76 0.979930
\(27\) −729.000 −0.192450
\(28\) −2025.76 −0.488306
\(29\) −5539.75 −1.22319 −0.611597 0.791169i \(-0.709473\pi\)
−0.611597 + 0.791169i \(0.709473\pi\)
\(30\) 2757.95 0.559478
\(31\) 10200.7 1.90646 0.953229 0.302250i \(-0.0977379\pi\)
0.953229 + 0.302250i \(0.0977379\pi\)
\(32\) 1024.00 0.176777
\(33\) −6698.41 −1.07075
\(34\) 5224.68 0.775109
\(35\) 9699.53 1.33838
\(36\) 1296.00 0.166667
\(37\) 3328.10 0.399661 0.199830 0.979830i \(-0.435961\pi\)
0.199830 + 0.979830i \(0.435961\pi\)
\(38\) −1444.00 −0.162221
\(39\) −7599.95 −0.800109
\(40\) −4903.02 −0.484522
\(41\) 6050.53 0.562126 0.281063 0.959689i \(-0.409313\pi\)
0.281063 + 0.959689i \(0.409313\pi\)
\(42\) 4557.95 0.398700
\(43\) 1922.46 0.158557 0.0792787 0.996852i \(-0.474738\pi\)
0.0792787 + 0.996852i \(0.474738\pi\)
\(44\) 11908.3 0.927294
\(45\) −6205.39 −0.456812
\(46\) −7293.66 −0.508219
\(47\) 10370.8 0.684809 0.342405 0.939553i \(-0.388759\pi\)
0.342405 + 0.939553i \(0.388759\pi\)
\(48\) −2304.00 −0.144338
\(49\) −776.980 −0.0462295
\(50\) 10976.2 0.620907
\(51\) −11755.5 −0.632874
\(52\) 13511.0 0.692915
\(53\) 3107.66 0.151965 0.0759825 0.997109i \(-0.475791\pi\)
0.0759825 + 0.997109i \(0.475791\pi\)
\(54\) −2916.00 −0.136083
\(55\) −57018.2 −2.54160
\(56\) −8103.02 −0.345284
\(57\) 3249.00 0.132453
\(58\) −22159.0 −0.864929
\(59\) −3194.44 −0.119472 −0.0597358 0.998214i \(-0.519026\pi\)
−0.0597358 + 0.998214i \(0.519026\pi\)
\(60\) 11031.8 0.395611
\(61\) 17475.6 0.601323 0.300662 0.953731i \(-0.402792\pi\)
0.300662 + 0.953731i \(0.402792\pi\)
\(62\) 40802.9 1.34807
\(63\) −10255.4 −0.325537
\(64\) 4096.00 0.125000
\(65\) −64692.2 −1.89919
\(66\) −26793.6 −0.757133
\(67\) 13374.0 0.363976 0.181988 0.983301i \(-0.441747\pi\)
0.181988 + 0.983301i \(0.441747\pi\)
\(68\) 20898.7 0.548085
\(69\) 16410.7 0.414959
\(70\) 38798.1 0.946380
\(71\) 6280.96 0.147870 0.0739350 0.997263i \(-0.476444\pi\)
0.0739350 + 0.997263i \(0.476444\pi\)
\(72\) 5184.00 0.117851
\(73\) 78709.0 1.72869 0.864346 0.502898i \(-0.167733\pi\)
0.864346 + 0.502898i \(0.167733\pi\)
\(74\) 13312.4 0.282603
\(75\) −24696.4 −0.506969
\(76\) −5776.00 −0.114708
\(77\) −94231.6 −1.81121
\(78\) −30399.8 −0.565763
\(79\) 41603.4 0.750000 0.375000 0.927025i \(-0.377643\pi\)
0.375000 + 0.927025i \(0.377643\pi\)
\(80\) −19612.1 −0.342609
\(81\) 6561.00 0.111111
\(82\) 24202.1 0.397483
\(83\) −9042.74 −0.144080 −0.0720402 0.997402i \(-0.522951\pi\)
−0.0720402 + 0.997402i \(0.522951\pi\)
\(84\) 18231.8 0.281924
\(85\) −100065. −1.50223
\(86\) 7689.85 0.112117
\(87\) 49857.8 0.706211
\(88\) 47633.2 0.655696
\(89\) 3935.36 0.0526635 0.0263318 0.999653i \(-0.491617\pi\)
0.0263318 + 0.999653i \(0.491617\pi\)
\(90\) −24821.5 −0.323015
\(91\) −106914. −1.35342
\(92\) −29174.6 −0.359365
\(93\) −91806.6 −1.10069
\(94\) 41483.4 0.484233
\(95\) 27656.1 0.314400
\(96\) −9216.00 −0.102062
\(97\) 144902. 1.56367 0.781837 0.623483i \(-0.214283\pi\)
0.781837 + 0.623483i \(0.214283\pi\)
\(98\) −3107.92 −0.0326892
\(99\) 60285.7 0.618196
\(100\) 43904.8 0.439048
\(101\) −86084.3 −0.839692 −0.419846 0.907595i \(-0.637916\pi\)
−0.419846 + 0.907595i \(0.637916\pi\)
\(102\) −47022.2 −0.447509
\(103\) −194549. −1.80691 −0.903454 0.428686i \(-0.858977\pi\)
−0.903454 + 0.428686i \(0.858977\pi\)
\(104\) 54044.1 0.489965
\(105\) −87295.8 −0.772716
\(106\) 12430.6 0.107455
\(107\) −214997. −1.81540 −0.907702 0.419615i \(-0.862165\pi\)
−0.907702 + 0.419615i \(0.862165\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 143607. 1.15774 0.578870 0.815420i \(-0.303494\pi\)
0.578870 + 0.815420i \(0.303494\pi\)
\(110\) −228073. −1.79718
\(111\) −29952.9 −0.230744
\(112\) −32412.1 −0.244153
\(113\) −229554. −1.69117 −0.845586 0.533838i \(-0.820749\pi\)
−0.845586 + 0.533838i \(0.820749\pi\)
\(114\) 12996.0 0.0936586
\(115\) 139691. 0.984973
\(116\) −88636.0 −0.611597
\(117\) 68399.5 0.461943
\(118\) −12777.8 −0.0844792
\(119\) −165374. −1.07053
\(120\) 44127.2 0.279739
\(121\) 392884. 2.43950
\(122\) 69902.5 0.425200
\(123\) −54454.8 −0.324544
\(124\) 163212. 0.953229
\(125\) 29184.6 0.167062
\(126\) −41021.5 −0.230190
\(127\) −220694. −1.21418 −0.607088 0.794635i \(-0.707662\pi\)
−0.607088 + 0.794635i \(0.707662\pi\)
\(128\) 16384.0 0.0883883
\(129\) −17302.2 −0.0915432
\(130\) −258769. −1.34293
\(131\) 271930. 1.38445 0.692227 0.721680i \(-0.256630\pi\)
0.692227 + 0.721680i \(0.256630\pi\)
\(132\) −107175. −0.535374
\(133\) 45706.1 0.224050
\(134\) 53495.8 0.257370
\(135\) 55848.5 0.263741
\(136\) 83594.9 0.387554
\(137\) 302764. 1.37817 0.689085 0.724680i \(-0.258013\pi\)
0.689085 + 0.724680i \(0.258013\pi\)
\(138\) 65642.9 0.293420
\(139\) 405848. 1.78167 0.890833 0.454330i \(-0.150121\pi\)
0.890833 + 0.454330i \(0.150121\pi\)
\(140\) 155193. 0.669192
\(141\) −93337.6 −0.395375
\(142\) 25123.8 0.104560
\(143\) 628489. 2.57014
\(144\) 20736.0 0.0833333
\(145\) 424399. 1.67631
\(146\) 314836. 1.22237
\(147\) 6992.82 0.0266906
\(148\) 53249.6 0.199830
\(149\) −18679.2 −0.0689274 −0.0344637 0.999406i \(-0.510972\pi\)
−0.0344637 + 0.999406i \(0.510972\pi\)
\(150\) −98785.7 −0.358481
\(151\) −31638.8 −0.112922 −0.0564609 0.998405i \(-0.517982\pi\)
−0.0564609 + 0.998405i \(0.517982\pi\)
\(152\) −23104.0 −0.0811107
\(153\) 105800. 0.365390
\(154\) −376926. −1.28072
\(155\) −781475. −2.61268
\(156\) −121599. −0.400055
\(157\) −7817.53 −0.0253116 −0.0126558 0.999920i \(-0.504029\pi\)
−0.0126558 + 0.999920i \(0.504029\pi\)
\(158\) 166414. 0.530330
\(159\) −27968.9 −0.0877370
\(160\) −78448.3 −0.242261
\(161\) 230862. 0.701920
\(162\) 26244.0 0.0785674
\(163\) −245394. −0.723428 −0.361714 0.932289i \(-0.617808\pi\)
−0.361714 + 0.932289i \(0.617808\pi\)
\(164\) 96808.5 0.281063
\(165\) 513163. 1.46739
\(166\) −36171.0 −0.101880
\(167\) −687602. −1.90786 −0.953929 0.300034i \(-0.903002\pi\)
−0.953929 + 0.300034i \(0.903002\pi\)
\(168\) 72927.2 0.199350
\(169\) 341784. 0.920524
\(170\) −400262. −1.06224
\(171\) −29241.0 −0.0764719
\(172\) 30759.4 0.0792787
\(173\) 475094. 1.20688 0.603441 0.797408i \(-0.293796\pi\)
0.603441 + 0.797408i \(0.293796\pi\)
\(174\) 199431. 0.499367
\(175\) −347423. −0.857558
\(176\) 190533. 0.463647
\(177\) 28749.9 0.0689769
\(178\) 15741.5 0.0372387
\(179\) −151716. −0.353914 −0.176957 0.984219i \(-0.556625\pi\)
−0.176957 + 0.984219i \(0.556625\pi\)
\(180\) −99286.2 −0.228406
\(181\) 138508. 0.314251 0.157126 0.987579i \(-0.449777\pi\)
0.157126 + 0.987579i \(0.449777\pi\)
\(182\) −427657. −0.957011
\(183\) −157281. −0.347174
\(184\) −116698. −0.254109
\(185\) −254965. −0.547710
\(186\) −367226. −0.778308
\(187\) 972141. 2.03294
\(188\) 165934. 0.342405
\(189\) 92298.5 0.187949
\(190\) 110624. 0.222314
\(191\) −308867. −0.612616 −0.306308 0.951933i \(-0.599094\pi\)
−0.306308 + 0.951933i \(0.599094\pi\)
\(192\) −36864.0 −0.0721688
\(193\) 508873. 0.983367 0.491684 0.870774i \(-0.336382\pi\)
0.491684 + 0.870774i \(0.336382\pi\)
\(194\) 579610. 1.10568
\(195\) 582230. 1.09650
\(196\) −12431.7 −0.0231148
\(197\) −811685. −1.49012 −0.745061 0.666996i \(-0.767580\pi\)
−0.745061 + 0.666996i \(0.767580\pi\)
\(198\) 241143. 0.437131
\(199\) −707620. −1.26668 −0.633341 0.773873i \(-0.718317\pi\)
−0.633341 + 0.773873i \(0.718317\pi\)
\(200\) 175619. 0.310454
\(201\) −120366. −0.210142
\(202\) −344337. −0.593752
\(203\) 701386. 1.19459
\(204\) −188089. −0.316437
\(205\) −463530. −0.770358
\(206\) −778196. −1.27768
\(207\) −147697. −0.239577
\(208\) 216176. 0.346457
\(209\) −268681. −0.425472
\(210\) −349183. −0.546393
\(211\) 756138. 1.16922 0.584608 0.811316i \(-0.301248\pi\)
0.584608 + 0.811316i \(0.301248\pi\)
\(212\) 49722.5 0.0759825
\(213\) −56528.6 −0.0853728
\(214\) −859989. −1.28368
\(215\) −147279. −0.217293
\(216\) −46656.0 −0.0680414
\(217\) −1.29151e6 −1.86187
\(218\) 574430. 0.818645
\(219\) −708381. −0.998060
\(220\) −912291. −1.27080
\(221\) 1.10298e6 1.51910
\(222\) −119811. −0.163161
\(223\) −420229. −0.565880 −0.282940 0.959138i \(-0.591310\pi\)
−0.282940 + 0.959138i \(0.591310\pi\)
\(224\) −129648. −0.172642
\(225\) 222268. 0.292699
\(226\) −918214. −1.19584
\(227\) 918724. 1.18337 0.591684 0.806170i \(-0.298463\pi\)
0.591684 + 0.806170i \(0.298463\pi\)
\(228\) 51984.0 0.0662266
\(229\) 306139. 0.385772 0.192886 0.981221i \(-0.438215\pi\)
0.192886 + 0.981221i \(0.438215\pi\)
\(230\) 558765. 0.696481
\(231\) 848084. 1.04570
\(232\) −354544. −0.432464
\(233\) −394089. −0.475559 −0.237779 0.971319i \(-0.576420\pi\)
−0.237779 + 0.971319i \(0.576420\pi\)
\(234\) 273598. 0.326643
\(235\) −794508. −0.938487
\(236\) −51111.0 −0.0597358
\(237\) −374431. −0.433012
\(238\) −661496. −0.756980
\(239\) −1.44062e6 −1.63138 −0.815691 0.578488i \(-0.803643\pi\)
−0.815691 + 0.578488i \(0.803643\pi\)
\(240\) 176509. 0.197805
\(241\) −253673. −0.281340 −0.140670 0.990057i \(-0.544926\pi\)
−0.140670 + 0.990057i \(0.544926\pi\)
\(242\) 1.57154e6 1.72499
\(243\) −59049.0 −0.0641500
\(244\) 279610. 0.300662
\(245\) 59524.2 0.0633546
\(246\) −217819. −0.229487
\(247\) −304842. −0.317931
\(248\) 652847. 0.674034
\(249\) 81384.7 0.0831849
\(250\) 116738. 0.118131
\(251\) 1.07742e6 1.07945 0.539724 0.841842i \(-0.318528\pi\)
0.539724 + 0.841842i \(0.318528\pi\)
\(252\) −164086. −0.162769
\(253\) −1.35711e6 −1.33295
\(254\) −882777. −0.858552
\(255\) 900588. 0.867313
\(256\) 65536.0 0.0625000
\(257\) −1.65751e6 −1.56539 −0.782695 0.622405i \(-0.786156\pi\)
−0.782695 + 0.622405i \(0.786156\pi\)
\(258\) −69208.6 −0.0647308
\(259\) −421369. −0.390314
\(260\) −1.03508e6 −0.949596
\(261\) −448720. −0.407731
\(262\) 1.08772e6 0.978957
\(263\) −475083. −0.423526 −0.211763 0.977321i \(-0.567921\pi\)
−0.211763 + 0.977321i \(0.567921\pi\)
\(264\) −428698. −0.378566
\(265\) −238077. −0.208258
\(266\) 182824. 0.158427
\(267\) −35418.3 −0.0304053
\(268\) 213983. 0.181988
\(269\) 1.14894e6 0.968090 0.484045 0.875043i \(-0.339167\pi\)
0.484045 + 0.875043i \(0.339167\pi\)
\(270\) 223394. 0.186493
\(271\) −844947. −0.698886 −0.349443 0.936958i \(-0.613629\pi\)
−0.349443 + 0.936958i \(0.613629\pi\)
\(272\) 334380. 0.274042
\(273\) 962227. 0.781396
\(274\) 1.21106e6 0.974514
\(275\) 2.04231e6 1.62851
\(276\) 262572. 0.207479
\(277\) 1.25965e6 0.986391 0.493195 0.869919i \(-0.335829\pi\)
0.493195 + 0.869919i \(0.335829\pi\)
\(278\) 1.62339e6 1.25983
\(279\) 826259. 0.635486
\(280\) 620770. 0.473190
\(281\) 227822. 0.172119 0.0860595 0.996290i \(-0.472572\pi\)
0.0860595 + 0.996290i \(0.472572\pi\)
\(282\) −373351. −0.279572
\(283\) −2.47321e6 −1.83567 −0.917834 0.396965i \(-0.870064\pi\)
−0.917834 + 0.396965i \(0.870064\pi\)
\(284\) 100495. 0.0739350
\(285\) −248905. −0.181519
\(286\) 2.51396e6 1.81737
\(287\) −766056. −0.548979
\(288\) 82944.0 0.0589256
\(289\) 286225. 0.201587
\(290\) 1.69760e6 1.18533
\(291\) −1.30412e6 −0.902788
\(292\) 1.25934e6 0.864346
\(293\) −159716. −0.108687 −0.0543436 0.998522i \(-0.517307\pi\)
−0.0543436 + 0.998522i \(0.517307\pi\)
\(294\) 27971.3 0.0188731
\(295\) 244725. 0.163728
\(296\) 212998. 0.141301
\(297\) −542571. −0.356916
\(298\) −74716.7 −0.0487390
\(299\) −1.53976e6 −0.996037
\(300\) −395143. −0.253484
\(301\) −243402. −0.154849
\(302\) −126555. −0.0798477
\(303\) 774758. 0.484797
\(304\) −92416.0 −0.0573539
\(305\) −1.33880e6 −0.824075
\(306\) 423199. 0.258370
\(307\) 722474. 0.437498 0.218749 0.975781i \(-0.429802\pi\)
0.218749 + 0.975781i \(0.429802\pi\)
\(308\) −1.50770e6 −0.905607
\(309\) 1.75094e6 1.04322
\(310\) −3.12590e6 −1.84744
\(311\) 568481. 0.333284 0.166642 0.986017i \(-0.446708\pi\)
0.166642 + 0.986017i \(0.446708\pi\)
\(312\) −486397. −0.282881
\(313\) −39902.0 −0.0230215 −0.0115108 0.999934i \(-0.503664\pi\)
−0.0115108 + 0.999934i \(0.503664\pi\)
\(314\) −31270.1 −0.0178980
\(315\) 785662. 0.446128
\(316\) 665654. 0.375000
\(317\) 2.42559e6 1.35572 0.677859 0.735192i \(-0.262908\pi\)
0.677859 + 0.735192i \(0.262908\pi\)
\(318\) −111876. −0.0620395
\(319\) −4.12306e6 −2.26852
\(320\) −313793. −0.171305
\(321\) 1.93497e6 1.04812
\(322\) 923448. 0.496332
\(323\) −471528. −0.251479
\(324\) 104976. 0.0555556
\(325\) 2.31718e6 1.21689
\(326\) −981577. −0.511541
\(327\) −1.29247e6 −0.668421
\(328\) 387234. 0.198742
\(329\) −1.31305e6 −0.668793
\(330\) 2.05265e6 1.03760
\(331\) −87388.5 −0.0438414 −0.0219207 0.999760i \(-0.506978\pi\)
−0.0219207 + 0.999760i \(0.506978\pi\)
\(332\) −144684. −0.0720402
\(333\) 269576. 0.133220
\(334\) −2.75041e6 −1.34906
\(335\) −1.02457e6 −0.498806
\(336\) 291709. 0.140962
\(337\) 2.25546e6 1.08183 0.540917 0.841076i \(-0.318077\pi\)
0.540917 + 0.841076i \(0.318077\pi\)
\(338\) 1.36714e6 0.650909
\(339\) 2.06598e6 0.976399
\(340\) −1.60105e6 −0.751115
\(341\) 7.59208e6 3.53569
\(342\) −116964. −0.0540738
\(343\) 2.22630e6 1.02176
\(344\) 123038. 0.0560585
\(345\) −1.25722e6 −0.568675
\(346\) 1.90038e6 0.853394
\(347\) −2.30600e6 −1.02810 −0.514050 0.857760i \(-0.671855\pi\)
−0.514050 + 0.857760i \(0.671855\pi\)
\(348\) 797724. 0.353106
\(349\) 3.57261e6 1.57008 0.785041 0.619443i \(-0.212642\pi\)
0.785041 + 0.619443i \(0.212642\pi\)
\(350\) −1.38969e6 −0.606385
\(351\) −615596. −0.266703
\(352\) 762130. 0.327848
\(353\) −1.11293e6 −0.475371 −0.237685 0.971342i \(-0.576389\pi\)
−0.237685 + 0.971342i \(0.576389\pi\)
\(354\) 115000. 0.0487741
\(355\) −481182. −0.202646
\(356\) 62965.8 0.0263318
\(357\) 1.48837e6 0.618072
\(358\) −606863. −0.250255
\(359\) −3.44175e6 −1.40943 −0.704714 0.709492i \(-0.748925\pi\)
−0.704714 + 0.709492i \(0.748925\pi\)
\(360\) −397145. −0.161507
\(361\) 130321. 0.0526316
\(362\) 554030. 0.222209
\(363\) −3.53595e6 −1.40845
\(364\) −1.71063e6 −0.676709
\(365\) −6.02988e6 −2.36906
\(366\) −629122. −0.245489
\(367\) −1.46107e6 −0.566248 −0.283124 0.959083i \(-0.591371\pi\)
−0.283124 + 0.959083i \(0.591371\pi\)
\(368\) −466794. −0.179682
\(369\) 490093. 0.187375
\(370\) −1.01986e6 −0.387289
\(371\) −393460. −0.148411
\(372\) −1.46891e6 −0.550347
\(373\) −2.62654e6 −0.977488 −0.488744 0.872427i \(-0.662545\pi\)
−0.488744 + 0.872427i \(0.662545\pi\)
\(374\) 3.88856e6 1.43751
\(375\) −262661. −0.0964535
\(376\) 663734. 0.242117
\(377\) −4.67798e6 −1.69514
\(378\) 369194. 0.132900
\(379\) 302415. 0.108145 0.0540724 0.998537i \(-0.482780\pi\)
0.0540724 + 0.998537i \(0.482780\pi\)
\(380\) 442498. 0.157200
\(381\) 1.98625e6 0.701005
\(382\) −1.23547e6 −0.433185
\(383\) −161905. −0.0563981 −0.0281991 0.999602i \(-0.508977\pi\)
−0.0281991 + 0.999602i \(0.508977\pi\)
\(384\) −147456. −0.0510310
\(385\) 7.21905e6 2.48215
\(386\) 2.03549e6 0.695346
\(387\) 155719. 0.0528525
\(388\) 2.31844e6 0.781837
\(389\) 3.05129e6 1.02237 0.511187 0.859470i \(-0.329206\pi\)
0.511187 + 0.859470i \(0.329206\pi\)
\(390\) 2.32892e6 0.775342
\(391\) −2.38169e6 −0.787850
\(392\) −49726.7 −0.0163446
\(393\) −2.44737e6 −0.799315
\(394\) −3.24674e6 −1.05368
\(395\) −3.18722e6 −1.02783
\(396\) 964571. 0.309098
\(397\) 2.59147e6 0.825219 0.412610 0.910908i \(-0.364617\pi\)
0.412610 + 0.910908i \(0.364617\pi\)
\(398\) −2.83048e6 −0.895679
\(399\) −411355. −0.129355
\(400\) 702476. 0.219524
\(401\) 3.91055e6 1.21444 0.607221 0.794533i \(-0.292284\pi\)
0.607221 + 0.794533i \(0.292284\pi\)
\(402\) −481462. −0.148593
\(403\) 8.61389e6 2.64202
\(404\) −1.37735e6 −0.419846
\(405\) −502636. −0.152271
\(406\) 2.80555e6 0.844700
\(407\) 2.47700e6 0.741207
\(408\) −752354. −0.223755
\(409\) −3.71122e6 −1.09700 −0.548502 0.836149i \(-0.684802\pi\)
−0.548502 + 0.836149i \(0.684802\pi\)
\(410\) −1.85412e6 −0.544726
\(411\) −2.72488e6 −0.795687
\(412\) −3.11278e6 −0.903454
\(413\) 404447. 0.116677
\(414\) −590786. −0.169406
\(415\) 692762. 0.197453
\(416\) 864705. 0.244982
\(417\) −3.65263e6 −1.02865
\(418\) −1.07472e6 −0.300854
\(419\) −2.87932e6 −0.801225 −0.400613 0.916248i \(-0.631203\pi\)
−0.400613 + 0.916248i \(0.631203\pi\)
\(420\) −1.39673e6 −0.386358
\(421\) −6.67740e6 −1.83612 −0.918062 0.396436i \(-0.870247\pi\)
−0.918062 + 0.396436i \(0.870247\pi\)
\(422\) 3.02455e6 0.826761
\(423\) 840039. 0.228270
\(424\) 198890. 0.0537277
\(425\) 3.58420e6 0.962541
\(426\) −226115. −0.0603677
\(427\) −2.21258e6 −0.587260
\(428\) −3.43996e6 −0.907702
\(429\) −5.65640e6 −1.48387
\(430\) −589117. −0.153649
\(431\) 1.04017e6 0.269720 0.134860 0.990865i \(-0.456942\pi\)
0.134860 + 0.990865i \(0.456942\pi\)
\(432\) −186624. −0.0481125
\(433\) −924840. −0.237054 −0.118527 0.992951i \(-0.537817\pi\)
−0.118527 + 0.992951i \(0.537817\pi\)
\(434\) −5.16605e6 −1.31654
\(435\) −3.81959e6 −0.967818
\(436\) 2.29772e6 0.578870
\(437\) 658252. 0.164888
\(438\) −2.83353e6 −0.705735
\(439\) 689018. 0.170636 0.0853178 0.996354i \(-0.472809\pi\)
0.0853178 + 0.996354i \(0.472809\pi\)
\(440\) −3.64916e6 −0.898590
\(441\) −62935.4 −0.0154098
\(442\) 4.41193e6 1.07417
\(443\) 721377. 0.174644 0.0873219 0.996180i \(-0.472169\pi\)
0.0873219 + 0.996180i \(0.472169\pi\)
\(444\) −479246. −0.115372
\(445\) −301487. −0.0721720
\(446\) −1.68092e6 −0.400138
\(447\) 168113. 0.0397953
\(448\) −518593. −0.122076
\(449\) 6.19115e6 1.44929 0.724645 0.689123i \(-0.242004\pi\)
0.724645 + 0.689123i \(0.242004\pi\)
\(450\) 889072. 0.206969
\(451\) 4.50322e6 1.04251
\(452\) −3.67286e6 −0.845586
\(453\) 284749. 0.0651954
\(454\) 3.67489e6 0.836768
\(455\) 8.19066e6 1.85477
\(456\) 207936. 0.0468293
\(457\) −3.93674e6 −0.881753 −0.440876 0.897568i \(-0.645332\pi\)
−0.440876 + 0.897568i \(0.645332\pi\)
\(458\) 1.22456e6 0.272782
\(459\) −952199. −0.210958
\(460\) 2.23506e6 0.492487
\(461\) 1.63169e6 0.357589 0.178794 0.983886i \(-0.442780\pi\)
0.178794 + 0.983886i \(0.442780\pi\)
\(462\) 3.39234e6 0.739425
\(463\) 7.30809e6 1.58435 0.792176 0.610293i \(-0.208948\pi\)
0.792176 + 0.610293i \(0.208948\pi\)
\(464\) −1.41818e6 −0.305798
\(465\) 7.03328e6 1.50843
\(466\) −1.57636e6 −0.336271
\(467\) −567857. −0.120489 −0.0602444 0.998184i \(-0.519188\pi\)
−0.0602444 + 0.998184i \(0.519188\pi\)
\(468\) 1.09439e6 0.230972
\(469\) −1.69327e6 −0.355463
\(470\) −3.17803e6 −0.663611
\(471\) 70357.7 0.0146137
\(472\) −204444. −0.0422396
\(473\) 1.43083e6 0.294059
\(474\) −1.49772e6 −0.306186
\(475\) −990602. −0.201449
\(476\) −2.64598e6 −0.535266
\(477\) 251720. 0.0506550
\(478\) −5.76249e6 −1.15356
\(479\) −704030. −0.140201 −0.0701007 0.997540i \(-0.522332\pi\)
−0.0701007 + 0.997540i \(0.522332\pi\)
\(480\) 706035. 0.139870
\(481\) 2.81037e6 0.553862
\(482\) −1.01469e6 −0.198938
\(483\) −2.07776e6 −0.405254
\(484\) 6.28614e6 1.21975
\(485\) −1.11009e7 −2.14292
\(486\) −236196. −0.0453609
\(487\) 1.05439e6 0.201455 0.100727 0.994914i \(-0.467883\pi\)
0.100727 + 0.994914i \(0.467883\pi\)
\(488\) 1.11844e6 0.212600
\(489\) 2.20855e6 0.417671
\(490\) 238097. 0.0447985
\(491\) 4.58542e6 0.858371 0.429185 0.903216i \(-0.358801\pi\)
0.429185 + 0.903216i \(0.358801\pi\)
\(492\) −871277. −0.162272
\(493\) −7.23586e6 −1.34083
\(494\) −1.21937e6 −0.224811
\(495\) −4.61847e6 −0.847199
\(496\) 2.61139e6 0.476614
\(497\) −795230. −0.144412
\(498\) 325539. 0.0588206
\(499\) −4.71445e6 −0.847577 −0.423789 0.905761i \(-0.639300\pi\)
−0.423789 + 0.905761i \(0.639300\pi\)
\(500\) 466953. 0.0835311
\(501\) 6.18842e6 1.10150
\(502\) 4.30969e6 0.763285
\(503\) −8.80792e6 −1.55222 −0.776110 0.630597i \(-0.782810\pi\)
−0.776110 + 0.630597i \(0.782810\pi\)
\(504\) −656345. −0.115095
\(505\) 6.59489e6 1.15075
\(506\) −5.42843e6 −0.942537
\(507\) −3.07606e6 −0.531465
\(508\) −3.53111e6 −0.607088
\(509\) −2.92609e6 −0.500602 −0.250301 0.968168i \(-0.580530\pi\)
−0.250301 + 0.968168i \(0.580530\pi\)
\(510\) 3.60235e6 0.613283
\(511\) −9.96533e6 −1.68826
\(512\) 262144. 0.0441942
\(513\) 263169. 0.0441511
\(514\) −6.63003e6 −1.10690
\(515\) 1.49043e7 2.47625
\(516\) −276835. −0.0457716
\(517\) 7.71869e6 1.27004
\(518\) −1.68548e6 −0.275993
\(519\) −4.27585e6 −0.696793
\(520\) −4.14030e6 −0.671465
\(521\) −8.26613e6 −1.33416 −0.667080 0.744986i \(-0.732456\pi\)
−0.667080 + 0.744986i \(0.732456\pi\)
\(522\) −1.79488e6 −0.288310
\(523\) 1.64703e6 0.263298 0.131649 0.991296i \(-0.457973\pi\)
0.131649 + 0.991296i \(0.457973\pi\)
\(524\) 4.35088e6 0.692227
\(525\) 3.12681e6 0.495112
\(526\) −1.90033e6 −0.299478
\(527\) 1.33239e7 2.08980
\(528\) −1.71479e6 −0.267687
\(529\) −3.11151e6 −0.483428
\(530\) −952307. −0.147261
\(531\) −258750. −0.0398239
\(532\) 731298. 0.112025
\(533\) 5.10930e6 0.779011
\(534\) −141673. −0.0214998
\(535\) 1.64709e7 2.48790
\(536\) 855933. 0.128685
\(537\) 1.36544e6 0.204333
\(538\) 4.59575e6 0.684543
\(539\) −578281. −0.0857368
\(540\) 893576. 0.131870
\(541\) −9.24181e6 −1.35758 −0.678788 0.734335i \(-0.737494\pi\)
−0.678788 + 0.734335i \(0.737494\pi\)
\(542\) −3.37979e6 −0.494187
\(543\) −1.24657e6 −0.181433
\(544\) 1.33752e6 0.193777
\(545\) −1.10017e7 −1.58661
\(546\) 3.84891e6 0.552530
\(547\) −6.58733e6 −0.941329 −0.470664 0.882312i \(-0.655986\pi\)
−0.470664 + 0.882312i \(0.655986\pi\)
\(548\) 4.84423e6 0.689085
\(549\) 1.41553e6 0.200441
\(550\) 8.16923e6 1.15153
\(551\) 1.99985e6 0.280620
\(552\) 1.05029e6 0.146710
\(553\) −5.26739e6 −0.732458
\(554\) 5.03858e6 0.697484
\(555\) 2.29468e6 0.316220
\(556\) 6.49357e6 0.890833
\(557\) 7.38856e6 1.00907 0.504536 0.863391i \(-0.331664\pi\)
0.504536 + 0.863391i \(0.331664\pi\)
\(558\) 3.30504e6 0.449356
\(559\) 1.62340e6 0.219734
\(560\) 2.48308e6 0.334596
\(561\) −8.74927e6 −1.17372
\(562\) 911286. 0.121707
\(563\) 1.25576e7 1.66969 0.834843 0.550488i \(-0.185558\pi\)
0.834843 + 0.550488i \(0.185558\pi\)
\(564\) −1.49340e6 −0.197687
\(565\) 1.75860e7 2.31764
\(566\) −9.89282e6 −1.29801
\(567\) −830686. −0.108512
\(568\) 401981. 0.0522799
\(569\) 1.02834e7 1.33155 0.665775 0.746152i \(-0.268101\pi\)
0.665775 + 0.746152i \(0.268101\pi\)
\(570\) −995620. −0.128353
\(571\) −1.81737e6 −0.233267 −0.116633 0.993175i \(-0.537210\pi\)
−0.116633 + 0.993175i \(0.537210\pi\)
\(572\) 1.00558e7 1.28507
\(573\) 2.77980e6 0.353694
\(574\) −3.06422e6 −0.388187
\(575\) −5.00354e6 −0.631113
\(576\) 331776. 0.0416667
\(577\) −9.08710e6 −1.13628 −0.568140 0.822932i \(-0.692337\pi\)
−0.568140 + 0.822932i \(0.692337\pi\)
\(578\) 1.14490e6 0.142544
\(579\) −4.57985e6 −0.567747
\(580\) 6.79038e6 0.838155
\(581\) 1.14490e6 0.140711
\(582\) −5.21649e6 −0.638367
\(583\) 2.31293e6 0.281833
\(584\) 5.03738e6 0.611185
\(585\) −5.24007e6 −0.633064
\(586\) −638863. −0.0768535
\(587\) 1.14406e7 1.37042 0.685211 0.728344i \(-0.259710\pi\)
0.685211 + 0.728344i \(0.259710\pi\)
\(588\) 111885. 0.0133453
\(589\) −3.68246e6 −0.437371
\(590\) 978900. 0.115773
\(591\) 7.30516e6 0.860322
\(592\) 851993. 0.0999152
\(593\) −1.10567e7 −1.29119 −0.645596 0.763679i \(-0.723391\pi\)
−0.645596 + 0.763679i \(0.723391\pi\)
\(594\) −2.17029e6 −0.252378
\(595\) 1.26692e7 1.46710
\(596\) −298867. −0.0344637
\(597\) 6.36858e6 0.731319
\(598\) −6.15905e6 −0.704305
\(599\) 6.40039e6 0.728852 0.364426 0.931232i \(-0.381265\pi\)
0.364426 + 0.931232i \(0.381265\pi\)
\(600\) −1.58057e6 −0.179241
\(601\) 1.52981e7 1.72764 0.863818 0.503804i \(-0.168067\pi\)
0.863818 + 0.503804i \(0.168067\pi\)
\(602\) −973610. −0.109495
\(603\) 1.08329e6 0.121325
\(604\) −506221. −0.0564609
\(605\) −3.00987e7 −3.34318
\(606\) 3.09903e6 0.342803
\(607\) 241645. 0.0266199 0.0133100 0.999911i \(-0.495763\pi\)
0.0133100 + 0.999911i \(0.495763\pi\)
\(608\) −369664. −0.0405554
\(609\) −6.31248e6 −0.689694
\(610\) −5.35521e6 −0.582709
\(611\) 8.75755e6 0.949029
\(612\) 1.69280e6 0.182695
\(613\) −883791. −0.0949945 −0.0474973 0.998871i \(-0.515125\pi\)
−0.0474973 + 0.998871i \(0.515125\pi\)
\(614\) 2.88989e6 0.309358
\(615\) 4.17177e6 0.444767
\(616\) −6.03082e6 −0.640361
\(617\) −1.03688e7 −1.09652 −0.548260 0.836308i \(-0.684710\pi\)
−0.548260 + 0.836308i \(0.684710\pi\)
\(618\) 7.00376e6 0.737667
\(619\) −1.27847e7 −1.34111 −0.670553 0.741862i \(-0.733943\pi\)
−0.670553 + 0.741862i \(0.733943\pi\)
\(620\) −1.25036e7 −1.30634
\(621\) 1.32927e6 0.138320
\(622\) 2.27392e6 0.235667
\(623\) −498255. −0.0514318
\(624\) −1.94559e6 −0.200027
\(625\) −1.08110e7 −1.10704
\(626\) −159608. −0.0162787
\(627\) 2.41813e6 0.245646
\(628\) −125080. −0.0126558
\(629\) 4.34706e6 0.438096
\(630\) 3.14265e6 0.315460
\(631\) −1.69339e7 −1.69311 −0.846554 0.532304i \(-0.821326\pi\)
−0.846554 + 0.532304i \(0.821326\pi\)
\(632\) 2.66262e6 0.265165
\(633\) −6.80524e6 −0.675048
\(634\) 9.70237e6 0.958638
\(635\) 1.69073e7 1.66395
\(636\) −447503. −0.0438685
\(637\) −656112. −0.0640663
\(638\) −1.64922e7 −1.60409
\(639\) 508758. 0.0492900
\(640\) −1.25517e6 −0.121131
\(641\) 4.09564e6 0.393711 0.196855 0.980433i \(-0.436927\pi\)
0.196855 + 0.980433i \(0.436927\pi\)
\(642\) 7.73990e6 0.741136
\(643\) −6.06799e6 −0.578785 −0.289393 0.957211i \(-0.593453\pi\)
−0.289393 + 0.957211i \(0.593453\pi\)
\(644\) 3.69379e6 0.350960
\(645\) 1.32551e6 0.125454
\(646\) −1.88611e6 −0.177822
\(647\) −2.00355e7 −1.88165 −0.940825 0.338894i \(-0.889947\pi\)
−0.940825 + 0.338894i \(0.889947\pi\)
\(648\) 419904. 0.0392837
\(649\) −2.37752e6 −0.221571
\(650\) 9.26873e6 0.860472
\(651\) 1.16236e7 1.07495
\(652\) −3.92631e6 −0.361714
\(653\) −441926. −0.0405571 −0.0202785 0.999794i \(-0.506455\pi\)
−0.0202785 + 0.999794i \(0.506455\pi\)
\(654\) −5.16987e6 −0.472645
\(655\) −2.08325e7 −1.89731
\(656\) 1.54894e6 0.140532
\(657\) 6.37543e6 0.576230
\(658\) −5.25220e6 −0.472908
\(659\) −2.53352e6 −0.227253 −0.113627 0.993524i \(-0.536247\pi\)
−0.113627 + 0.993524i \(0.536247\pi\)
\(660\) 8.21062e6 0.733696
\(661\) −6.52457e6 −0.580829 −0.290414 0.956901i \(-0.593793\pi\)
−0.290414 + 0.956901i \(0.593793\pi\)
\(662\) −349554. −0.0310005
\(663\) −9.92683e6 −0.877055
\(664\) −578736. −0.0509401
\(665\) −3.50153e6 −0.307046
\(666\) 1.07830e6 0.0942010
\(667\) 1.01013e7 0.879146
\(668\) −1.10016e7 −0.953929
\(669\) 3.78206e6 0.326711
\(670\) −4.09830e6 −0.352709
\(671\) 1.30065e7 1.11521
\(672\) 1.16684e6 0.0996750
\(673\) 2.45884e6 0.209263 0.104631 0.994511i \(-0.466634\pi\)
0.104631 + 0.994511i \(0.466634\pi\)
\(674\) 9.02184e6 0.764972
\(675\) −2.00041e6 −0.168990
\(676\) 5.46854e6 0.460262
\(677\) −2.06428e7 −1.73100 −0.865500 0.500909i \(-0.832999\pi\)
−0.865500 + 0.500909i \(0.832999\pi\)
\(678\) 8.26393e6 0.690418
\(679\) −1.83461e7 −1.52710
\(680\) −6.40418e6 −0.531119
\(681\) −8.26851e6 −0.683218
\(682\) 3.03683e7 2.50011
\(683\) −1.33015e7 −1.09106 −0.545532 0.838090i \(-0.683672\pi\)
−0.545532 + 0.838090i \(0.683672\pi\)
\(684\) −467856. −0.0382360
\(685\) −2.31947e7 −1.88870
\(686\) 8.90521e6 0.722493
\(687\) −2.75525e6 −0.222725
\(688\) 492150. 0.0396394
\(689\) 2.62423e6 0.210598
\(690\) −5.02888e6 −0.402114
\(691\) 3.88585e6 0.309592 0.154796 0.987946i \(-0.450528\pi\)
0.154796 + 0.987946i \(0.450528\pi\)
\(692\) 7.60151e6 0.603441
\(693\) −7.63276e6 −0.603738
\(694\) −9.22400e6 −0.726977
\(695\) −3.10919e7 −2.44166
\(696\) 3.19090e6 0.249683
\(697\) 7.90303e6 0.616186
\(698\) 1.42905e7 1.11022
\(699\) 3.54680e6 0.274564
\(700\) −5.55877e6 −0.428779
\(701\) −6.00468e6 −0.461524 −0.230762 0.973010i \(-0.574122\pi\)
−0.230762 + 0.973010i \(0.574122\pi\)
\(702\) −2.46238e6 −0.188588
\(703\) −1.20144e6 −0.0916885
\(704\) 3.04852e6 0.231824
\(705\) 7.15057e6 0.541836
\(706\) −4.45174e6 −0.336138
\(707\) 1.08991e7 0.820054
\(708\) 459999. 0.0344885
\(709\) 1.88042e7 1.40488 0.702442 0.711741i \(-0.252093\pi\)
0.702442 + 0.711741i \(0.252093\pi\)
\(710\) −1.92473e6 −0.143293
\(711\) 3.36987e6 0.250000
\(712\) 251863. 0.0186194
\(713\) −1.86002e7 −1.37023
\(714\) 5.95346e6 0.437043
\(715\) −4.81484e7 −3.52222
\(716\) −2.42745e6 −0.176957
\(717\) 1.29656e7 0.941879
\(718\) −1.37670e7 −0.996616
\(719\) 2.03339e7 1.46689 0.733446 0.679748i \(-0.237911\pi\)
0.733446 + 0.679748i \(0.237911\pi\)
\(720\) −1.58858e6 −0.114203
\(721\) 2.46318e7 1.76465
\(722\) 521284. 0.0372161
\(723\) 2.28306e6 0.162432
\(724\) 2.21612e6 0.157126
\(725\) −1.52013e7 −1.07408
\(726\) −1.41438e7 −0.995922
\(727\) 2.53227e7 1.77694 0.888471 0.458933i \(-0.151768\pi\)
0.888471 + 0.458933i \(0.151768\pi\)
\(728\) −6.84251e6 −0.478505
\(729\) 531441. 0.0370370
\(730\) −2.41195e7 −1.67518
\(731\) 2.51106e6 0.173806
\(732\) −2.51649e6 −0.173587
\(733\) −2.71535e6 −0.186666 −0.0933331 0.995635i \(-0.529752\pi\)
−0.0933331 + 0.995635i \(0.529752\pi\)
\(734\) −5.84429e6 −0.400398
\(735\) −535718. −0.0365778
\(736\) −1.86718e6 −0.127055
\(737\) 9.95381e6 0.675026
\(738\) 1.96037e6 0.132494
\(739\) −2.19660e7 −1.47958 −0.739792 0.672835i \(-0.765076\pi\)
−0.739792 + 0.672835i \(0.765076\pi\)
\(740\) −4.07943e6 −0.273855
\(741\) 2.74358e6 0.183558
\(742\) −1.57384e6 −0.104942
\(743\) −3.22060e6 −0.214025 −0.107013 0.994258i \(-0.534129\pi\)
−0.107013 + 0.994258i \(0.534129\pi\)
\(744\) −5.87562e6 −0.389154
\(745\) 1.43101e6 0.0944606
\(746\) −1.05062e7 −0.691189
\(747\) −732462. −0.0480268
\(748\) 1.55543e7 1.01647
\(749\) 2.72207e7 1.77295
\(750\) −1.05064e6 −0.0682029
\(751\) 1.03346e7 0.668643 0.334322 0.942459i \(-0.391493\pi\)
0.334322 + 0.942459i \(0.391493\pi\)
\(752\) 2.65494e6 0.171202
\(753\) −9.69681e6 −0.623220
\(754\) −1.87119e7 −1.19864
\(755\) 2.42384e6 0.154752
\(756\) 1.47678e6 0.0939745
\(757\) −189369. −0.0120107 −0.00600536 0.999982i \(-0.501912\pi\)
−0.00600536 + 0.999982i \(0.501912\pi\)
\(758\) 1.20966e6 0.0764699
\(759\) 1.22140e7 0.769578
\(760\) 1.76999e6 0.111157
\(761\) 2.13628e7 1.33720 0.668601 0.743621i \(-0.266893\pi\)
0.668601 + 0.743621i \(0.266893\pi\)
\(762\) 7.94499e6 0.495685
\(763\) −1.81821e7 −1.13066
\(764\) −4.94187e6 −0.306308
\(765\) −8.10530e6 −0.500743
\(766\) −647622. −0.0398795
\(767\) −2.69751e6 −0.165567
\(768\) −589824. −0.0360844
\(769\) 4.12263e6 0.251396 0.125698 0.992069i \(-0.459883\pi\)
0.125698 + 0.992069i \(0.459883\pi\)
\(770\) 2.88762e7 1.75515
\(771\) 1.49176e7 0.903778
\(772\) 8.14196e6 0.491684
\(773\) 2.63212e6 0.158437 0.0792185 0.996857i \(-0.474758\pi\)
0.0792185 + 0.996857i \(0.474758\pi\)
\(774\) 622878. 0.0373724
\(775\) 2.79913e7 1.67405
\(776\) 9.27376e6 0.552842
\(777\) 3.79232e6 0.225348
\(778\) 1.22052e7 0.722927
\(779\) −2.18424e6 −0.128961
\(780\) 9.31568e6 0.548249
\(781\) 4.67472e6 0.274238
\(782\) −9.52676e6 −0.557094
\(783\) 4.03848e6 0.235404
\(784\) −198907. −0.0115574
\(785\) 598898. 0.0346880
\(786\) −9.78947e6 −0.565201
\(787\) −9.05359e6 −0.521055 −0.260528 0.965466i \(-0.583897\pi\)
−0.260528 + 0.965466i \(0.583897\pi\)
\(788\) −1.29870e7 −0.745061
\(789\) 4.27575e6 0.244523
\(790\) −1.27489e7 −0.726783
\(791\) 2.90637e7 1.65162
\(792\) 3.85829e6 0.218565
\(793\) 1.47571e7 0.833332
\(794\) 1.03659e7 0.583518
\(795\) 2.14269e6 0.120238
\(796\) −1.13219e7 −0.633341
\(797\) −1.36889e7 −0.763349 −0.381675 0.924297i \(-0.624653\pi\)
−0.381675 + 0.924297i \(0.624653\pi\)
\(798\) −1.64542e6 −0.0914681
\(799\) 1.35461e7 0.750667
\(800\) 2.80991e6 0.155227
\(801\) 318764. 0.0175545
\(802\) 1.56422e7 0.858740
\(803\) 5.85806e7 3.20601
\(804\) −1.92585e6 −0.105071
\(805\) −1.76863e7 −0.961937
\(806\) 3.44556e7 1.86819
\(807\) −1.03404e7 −0.558927
\(808\) −5.50939e6 −0.296876
\(809\) −2.60118e7 −1.39733 −0.698666 0.715448i \(-0.746222\pi\)
−0.698666 + 0.715448i \(0.746222\pi\)
\(810\) −2.01055e6 −0.107672
\(811\) 2.60995e7 1.39341 0.696706 0.717356i \(-0.254648\pi\)
0.696706 + 0.717356i \(0.254648\pi\)
\(812\) 1.12222e7 0.597293
\(813\) 7.60453e6 0.403502
\(814\) 9.90799e6 0.524112
\(815\) 1.87996e7 0.991412
\(816\) −3.00942e6 −0.158218
\(817\) −694009. −0.0363756
\(818\) −1.48449e7 −0.775699
\(819\) −8.66005e6 −0.451139
\(820\) −7.41647e6 −0.385179
\(821\) 1.26552e7 0.655255 0.327627 0.944807i \(-0.393751\pi\)
0.327627 + 0.944807i \(0.393751\pi\)
\(822\) −1.08995e7 −0.562636
\(823\) 8.61575e6 0.443398 0.221699 0.975115i \(-0.428840\pi\)
0.221699 + 0.975115i \(0.428840\pi\)
\(824\) −1.24511e7 −0.638838
\(825\) −1.83808e7 −0.940218
\(826\) 1.61779e6 0.0825033
\(827\) 2.37393e7 1.20699 0.603495 0.797367i \(-0.293774\pi\)
0.603495 + 0.797367i \(0.293774\pi\)
\(828\) −2.36314e6 −0.119788
\(829\) −1.01922e7 −0.515087 −0.257544 0.966267i \(-0.582913\pi\)
−0.257544 + 0.966267i \(0.582913\pi\)
\(830\) 2.77105e6 0.139620
\(831\) −1.13368e7 −0.569493
\(832\) 3.45882e6 0.173229
\(833\) −1.01487e6 −0.0506754
\(834\) −1.46105e7 −0.727362
\(835\) 5.26770e7 2.61460
\(836\) −4.29889e6 −0.212736
\(837\) −7.43633e6 −0.366898
\(838\) −1.15173e7 −0.566552
\(839\) 1.57496e7 0.772438 0.386219 0.922407i \(-0.373781\pi\)
0.386219 + 0.922407i \(0.373781\pi\)
\(840\) −5.58693e6 −0.273197
\(841\) 1.01777e7 0.496204
\(842\) −2.67096e7 −1.29834
\(843\) −2.05039e6 −0.0993730
\(844\) 1.20982e7 0.584608
\(845\) −2.61840e7 −1.26152
\(846\) 3.36015e6 0.161411
\(847\) −4.97429e7 −2.38244
\(848\) 795561. 0.0379913
\(849\) 2.22588e7 1.05982
\(850\) 1.43368e7 0.680620
\(851\) −6.06850e6 −0.287248
\(852\) −904458. −0.0426864
\(853\) −3.51443e7 −1.65380 −0.826899 0.562350i \(-0.809897\pi\)
−0.826899 + 0.562350i \(0.809897\pi\)
\(854\) −8.85034e6 −0.415255
\(855\) 2.24014e6 0.104800
\(856\) −1.37598e7 −0.641842
\(857\) −7.76419e6 −0.361114 −0.180557 0.983565i \(-0.557790\pi\)
−0.180557 + 0.983565i \(0.557790\pi\)
\(858\) −2.26256e7 −1.04926
\(859\) 3.50287e7 1.61973 0.809863 0.586619i \(-0.199541\pi\)
0.809863 + 0.586619i \(0.199541\pi\)
\(860\) −2.35647e6 −0.108646
\(861\) 6.89451e6 0.316953
\(862\) 4.16069e6 0.190721
\(863\) −1.31015e7 −0.598818 −0.299409 0.954125i \(-0.596790\pi\)
−0.299409 + 0.954125i \(0.596790\pi\)
\(864\) −746496. −0.0340207
\(865\) −3.63968e7 −1.65395
\(866\) −3.69936e6 −0.167622
\(867\) −2.57603e6 −0.116387
\(868\) −2.06642e7 −0.930934
\(869\) 3.09641e7 1.39094
\(870\) −1.52784e7 −0.684350
\(871\) 1.12935e7 0.504409
\(872\) 9.19088e6 0.409323
\(873\) 1.17371e7 0.521225
\(874\) 2.63301e6 0.116593
\(875\) −3.69505e6 −0.163155
\(876\) −1.13341e7 −0.499030
\(877\) −8.96391e6 −0.393549 −0.196774 0.980449i \(-0.563047\pi\)
−0.196774 + 0.980449i \(0.563047\pi\)
\(878\) 2.75607e6 0.120658
\(879\) 1.43744e6 0.0627506
\(880\) −1.45966e7 −0.635399
\(881\) −1.15038e7 −0.499347 −0.249673 0.968330i \(-0.580323\pi\)
−0.249673 + 0.968330i \(0.580323\pi\)
\(882\) −251741. −0.0108964
\(883\) 5.08324e6 0.219401 0.109701 0.993965i \(-0.465011\pi\)
0.109701 + 0.993965i \(0.465011\pi\)
\(884\) 1.76477e7 0.759552
\(885\) −2.20253e6 −0.0945285
\(886\) 2.88551e6 0.123492
\(887\) −2.73348e7 −1.16656 −0.583280 0.812271i \(-0.698231\pi\)
−0.583280 + 0.812271i \(0.698231\pi\)
\(888\) −1.91698e6 −0.0815804
\(889\) 2.79420e7 1.18578
\(890\) −1.20595e6 −0.0510333
\(891\) 4.88314e6 0.206065
\(892\) −6.72367e6 −0.282940
\(893\) −3.74388e6 −0.157106
\(894\) 672450. 0.0281395
\(895\) 1.16229e7 0.485017
\(896\) −2.07437e6 −0.0863211
\(897\) 1.38579e7 0.575062
\(898\) 2.47646e7 1.02480
\(899\) −5.65095e7 −2.33197
\(900\) 3.55629e6 0.146349
\(901\) 4.05913e6 0.166579
\(902\) 1.80129e7 0.737168
\(903\) 2.19062e6 0.0894022
\(904\) −1.46914e7 −0.597920
\(905\) −1.06110e7 −0.430661
\(906\) 1.13900e6 0.0461001
\(907\) 4.01778e7 1.62169 0.810846 0.585260i \(-0.199007\pi\)
0.810846 + 0.585260i \(0.199007\pi\)
\(908\) 1.46996e7 0.591684
\(909\) −6.97282e6 −0.279897
\(910\) 3.27627e7 1.31152
\(911\) 1.93885e7 0.774014 0.387007 0.922077i \(-0.373509\pi\)
0.387007 + 0.922077i \(0.373509\pi\)
\(912\) 831744. 0.0331133
\(913\) −6.73022e6 −0.267210
\(914\) −1.57470e7 −0.623493
\(915\) 1.20492e7 0.475780
\(916\) 4.89823e6 0.192886
\(917\) −3.44289e7 −1.35207
\(918\) −3.80879e6 −0.149170
\(919\) −3.38001e7 −1.32017 −0.660085 0.751191i \(-0.729480\pi\)
−0.660085 + 0.751191i \(0.729480\pi\)
\(920\) 8.94024e6 0.348241
\(921\) −6.50226e6 −0.252590
\(922\) 6.52674e6 0.252854
\(923\) 5.30389e6 0.204923
\(924\) 1.35693e7 0.522852
\(925\) 9.13246e6 0.350940
\(926\) 2.92324e7 1.12031
\(927\) −1.57585e7 −0.602302
\(928\) −5.67271e6 −0.216232
\(929\) −3.88357e6 −0.147636 −0.0738179 0.997272i \(-0.523518\pi\)
−0.0738179 + 0.997272i \(0.523518\pi\)
\(930\) 2.81331e7 1.06662
\(931\) 280490. 0.0106058
\(932\) −6.30542e6 −0.237779
\(933\) −5.11632e6 −0.192422
\(934\) −2.27143e6 −0.0851984
\(935\) −7.44755e7 −2.78602
\(936\) 4.37757e6 0.163322
\(937\) 2.61871e7 0.974402 0.487201 0.873290i \(-0.338018\pi\)
0.487201 + 0.873290i \(0.338018\pi\)
\(938\) −6.77309e6 −0.251351
\(939\) 359118. 0.0132915
\(940\) −1.27121e7 −0.469244
\(941\) −225612. −0.00830591 −0.00415296 0.999991i \(-0.501322\pi\)
−0.00415296 + 0.999991i \(0.501322\pi\)
\(942\) 281431. 0.0103334
\(943\) −1.10326e7 −0.404017
\(944\) −817776. −0.0298679
\(945\) −7.07096e6 −0.257572
\(946\) 5.72331e6 0.207931
\(947\) 3.64108e6 0.131934 0.0659669 0.997822i \(-0.478987\pi\)
0.0659669 + 0.997822i \(0.478987\pi\)
\(948\) −5.99089e6 −0.216506
\(949\) 6.64650e7 2.39567
\(950\) −3.96241e6 −0.142446
\(951\) −2.18303e7 −0.782724
\(952\) −1.05839e7 −0.378490
\(953\) 2.56251e7 0.913972 0.456986 0.889474i \(-0.348929\pi\)
0.456986 + 0.889474i \(0.348929\pi\)
\(954\) 1.00688e6 0.0358185
\(955\) 2.36622e7 0.839551
\(956\) −2.30500e7 −0.815691
\(957\) 3.71075e7 1.30973
\(958\) −2.81612e6 −0.0991373
\(959\) −3.83329e7 −1.34594
\(960\) 2.82414e6 0.0989027
\(961\) 7.54257e7 2.63458
\(962\) 1.12415e7 0.391640
\(963\) −1.74148e7 −0.605135
\(964\) −4.05877e6 −0.140670
\(965\) −3.89846e7 −1.34764
\(966\) −8.31103e6 −0.286558
\(967\) −1.66891e7 −0.573941 −0.286971 0.957939i \(-0.592648\pi\)
−0.286971 + 0.957939i \(0.592648\pi\)
\(968\) 2.51446e7 0.862493
\(969\) 4.24375e6 0.145191
\(970\) −4.44037e7 −1.51527
\(971\) −3.65408e6 −0.124374 −0.0621870 0.998065i \(-0.519808\pi\)
−0.0621870 + 0.998065i \(0.519808\pi\)
\(972\) −944784. −0.0320750
\(973\) −5.13843e7 −1.74000
\(974\) 4.21755e6 0.142450
\(975\) −2.08546e7 −0.702572
\(976\) 4.47376e6 0.150331
\(977\) 1.05458e6 0.0353463 0.0176732 0.999844i \(-0.494374\pi\)
0.0176732 + 0.999844i \(0.494374\pi\)
\(978\) 8.83419e6 0.295338
\(979\) 2.92897e6 0.0976692
\(980\) 952387. 0.0316773
\(981\) 1.16322e7 0.385913
\(982\) 1.83417e7 0.606960
\(983\) −3.53385e7 −1.16644 −0.583222 0.812313i \(-0.698208\pi\)
−0.583222 + 0.812313i \(0.698208\pi\)
\(984\) −3.48511e6 −0.114744
\(985\) 6.21829e7 2.04212
\(986\) −2.89435e7 −0.948108
\(987\) 1.18175e7 0.386128
\(988\) −4.87748e6 −0.158966
\(989\) −3.50544e6 −0.113960
\(990\) −1.84739e7 −0.599060
\(991\) −1.88975e7 −0.611252 −0.305626 0.952152i \(-0.598866\pi\)
−0.305626 + 0.952152i \(0.598866\pi\)
\(992\) 1.04455e7 0.337017
\(993\) 786496. 0.0253118
\(994\) −3.18092e6 −0.102114
\(995\) 5.42106e7 1.73591
\(996\) 1.30216e6 0.0415924
\(997\) −4.84709e6 −0.154434 −0.0772170 0.997014i \(-0.524603\pi\)
−0.0772170 + 0.997014i \(0.524603\pi\)
\(998\) −1.88578e7 −0.599328
\(999\) −2.42618e6 −0.0769148
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 114.6.a.g.1.1 2
3.2 odd 2 342.6.a.g.1.2 2
4.3 odd 2 912.6.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.6.a.g.1.1 2 1.1 even 1 trivial
342.6.a.g.1.2 2 3.2 odd 2
912.6.a.j.1.1 2 4.3 odd 2