Properties

Label 230.6.a.f.1.2
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $5$
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] = [230,6,Mod(1,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, 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("230.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.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: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 774x^{3} - 197x^{2} + 66287x + 154128 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.46823\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -8.46823 q^{3} +16.0000 q^{4} +25.0000 q^{5} +33.8729 q^{6} -118.977 q^{7} -64.0000 q^{8} -171.289 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -8.46823 q^{3} +16.0000 q^{4} +25.0000 q^{5} +33.8729 q^{6} -118.977 q^{7} -64.0000 q^{8} -171.289 q^{9} -100.000 q^{10} +358.711 q^{11} -135.492 q^{12} -515.740 q^{13} +475.909 q^{14} -211.706 q^{15} +256.000 q^{16} +2092.78 q^{17} +685.157 q^{18} -2775.19 q^{19} +400.000 q^{20} +1007.53 q^{21} -1434.84 q^{22} -529.000 q^{23} +541.967 q^{24} +625.000 q^{25} +2062.96 q^{26} +3508.29 q^{27} -1903.63 q^{28} -2344.09 q^{29} +846.823 q^{30} -9787.95 q^{31} -1024.00 q^{32} -3037.64 q^{33} -8371.12 q^{34} -2974.43 q^{35} -2740.63 q^{36} +4816.40 q^{37} +11100.8 q^{38} +4367.40 q^{39} -1600.00 q^{40} -24.0689 q^{41} -4030.10 q^{42} -1068.84 q^{43} +5739.37 q^{44} -4282.23 q^{45} +2116.00 q^{46} +17918.2 q^{47} -2167.87 q^{48} -2651.44 q^{49} -2500.00 q^{50} -17722.1 q^{51} -8251.83 q^{52} +15651.6 q^{53} -14033.2 q^{54} +8967.76 q^{55} +7614.54 q^{56} +23501.0 q^{57} +9376.37 q^{58} +45399.8 q^{59} -3387.29 q^{60} +18440.5 q^{61} +39151.8 q^{62} +20379.5 q^{63} +4096.00 q^{64} -12893.5 q^{65} +12150.6 q^{66} +48487.9 q^{67} +33484.5 q^{68} +4479.69 q^{69} +11897.7 q^{70} +3989.29 q^{71} +10962.5 q^{72} +5579.98 q^{73} -19265.6 q^{74} -5292.64 q^{75} -44403.1 q^{76} -42678.4 q^{77} -17469.6 q^{78} +38453.9 q^{79} +6400.00 q^{80} +11914.2 q^{81} +96.2757 q^{82} -29897.7 q^{83} +16120.4 q^{84} +52319.5 q^{85} +4275.37 q^{86} +19850.3 q^{87} -22957.5 q^{88} +50329.9 q^{89} +17128.9 q^{90} +61361.2 q^{91} -8464.00 q^{92} +82886.6 q^{93} -71672.7 q^{94} -69379.8 q^{95} +8671.46 q^{96} -71867.5 q^{97} +10605.8 q^{98} -61443.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 20 q^{2} + q^{3} + 80 q^{4} + 125 q^{5} - 4 q^{6} + 102 q^{7} - 320 q^{8} + 334 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 20 q^{2} + q^{3} + 80 q^{4} + 125 q^{5} - 4 q^{6} + 102 q^{7} - 320 q^{8} + 334 q^{9} - 500 q^{10} + 251 q^{11} + 16 q^{12} + 1743 q^{13} - 408 q^{14} + 25 q^{15} + 1280 q^{16} + 1944 q^{17} - 1336 q^{18} - 845 q^{19} + 2000 q^{20} + 4682 q^{21} - 1004 q^{22} - 2645 q^{23} - 64 q^{24} + 3125 q^{25} - 6972 q^{26} + 2428 q^{27} + 1632 q^{28} - 4021 q^{29} - 100 q^{30} - 15752 q^{31} - 5120 q^{32} + 2931 q^{33} - 7776 q^{34} + 2550 q^{35} + 5344 q^{36} - 3455 q^{37} + 3380 q^{38} - 16708 q^{39} - 8000 q^{40} - 11898 q^{41} - 18728 q^{42} + 6968 q^{43} + 4016 q^{44} + 8350 q^{45} + 10580 q^{46} + 13412 q^{47} + 256 q^{48} + 91041 q^{49} - 12500 q^{50} - 2115 q^{51} + 27888 q^{52} + 53029 q^{53} - 9712 q^{54} + 6275 q^{55} - 6528 q^{56} - 21730 q^{57} + 16084 q^{58} - 31223 q^{59} + 400 q^{60} + 71477 q^{61} + 63008 q^{62} + 262199 q^{63} + 20480 q^{64} + 43575 q^{65} - 11724 q^{66} + 76003 q^{67} + 31104 q^{68} - 529 q^{69} - 10200 q^{70} + 54418 q^{71} - 21376 q^{72} + 69418 q^{73} + 13820 q^{74} + 625 q^{75} - 13520 q^{76} + 283598 q^{77} + 66832 q^{78} + 105024 q^{79} + 32000 q^{80} + 102913 q^{81} + 47592 q^{82} + 89399 q^{83} + 74912 q^{84} + 48600 q^{85} - 27872 q^{86} + 276726 q^{87} - 16064 q^{88} + 96240 q^{89} - 33400 q^{90} + 59261 q^{91} - 42320 q^{92} + 84434 q^{93} - 53648 q^{94} - 21125 q^{95} - 1024 q^{96} + 216087 q^{97} - 364164 q^{98} + 386925 q^{99}+O(q^{100}) \) 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
\(3\) −8.46823 −0.543237 −0.271619 0.962405i \(-0.587559\pi\)
−0.271619 + 0.962405i \(0.587559\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) 33.8729 0.384127
\(7\) −118.977 −0.917737 −0.458869 0.888504i \(-0.651745\pi\)
−0.458869 + 0.888504i \(0.651745\pi\)
\(8\) −64.0000 −0.353553
\(9\) −171.289 −0.704894
\(10\) −100.000 −0.316228
\(11\) 358.711 0.893846 0.446923 0.894573i \(-0.352520\pi\)
0.446923 + 0.894573i \(0.352520\pi\)
\(12\) −135.492 −0.271619
\(13\) −515.740 −0.846393 −0.423197 0.906038i \(-0.639092\pi\)
−0.423197 + 0.906038i \(0.639092\pi\)
\(14\) 475.909 0.648938
\(15\) −211.706 −0.242943
\(16\) 256.000 0.250000
\(17\) 2092.78 1.75631 0.878155 0.478376i \(-0.158774\pi\)
0.878155 + 0.478376i \(0.158774\pi\)
\(18\) 685.157 0.498435
\(19\) −2775.19 −1.76364 −0.881818 0.471589i \(-0.843680\pi\)
−0.881818 + 0.471589i \(0.843680\pi\)
\(20\) 400.000 0.223607
\(21\) 1007.53 0.498549
\(22\) −1434.84 −0.632044
\(23\) −529.000 −0.208514
\(24\) 541.967 0.192063
\(25\) 625.000 0.200000
\(26\) 2062.96 0.598490
\(27\) 3508.29 0.926161
\(28\) −1903.63 −0.458869
\(29\) −2344.09 −0.517583 −0.258791 0.965933i \(-0.583324\pi\)
−0.258791 + 0.965933i \(0.583324\pi\)
\(30\) 846.823 0.171787
\(31\) −9787.95 −1.82931 −0.914655 0.404235i \(-0.867538\pi\)
−0.914655 + 0.404235i \(0.867538\pi\)
\(32\) −1024.00 −0.176777
\(33\) −3037.64 −0.485570
\(34\) −8371.12 −1.24190
\(35\) −2974.43 −0.410425
\(36\) −2740.63 −0.352447
\(37\) 4816.40 0.578387 0.289193 0.957271i \(-0.406613\pi\)
0.289193 + 0.957271i \(0.406613\pi\)
\(38\) 11100.8 1.24708
\(39\) 4367.40 0.459792
\(40\) −1600.00 −0.158114
\(41\) −24.0689 −0.00223613 −0.00111806 0.999999i \(-0.500356\pi\)
−0.00111806 + 0.999999i \(0.500356\pi\)
\(42\) −4030.10 −0.352527
\(43\) −1068.84 −0.0881541 −0.0440770 0.999028i \(-0.514035\pi\)
−0.0440770 + 0.999028i \(0.514035\pi\)
\(44\) 5739.37 0.446923
\(45\) −4282.23 −0.315238
\(46\) 2116.00 0.147442
\(47\) 17918.2 1.18317 0.591587 0.806241i \(-0.298501\pi\)
0.591587 + 0.806241i \(0.298501\pi\)
\(48\) −2167.87 −0.135809
\(49\) −2651.44 −0.157758
\(50\) −2500.00 −0.141421
\(51\) −17722.1 −0.954093
\(52\) −8251.83 −0.423197
\(53\) 15651.6 0.765365 0.382682 0.923880i \(-0.375000\pi\)
0.382682 + 0.923880i \(0.375000\pi\)
\(54\) −14033.2 −0.654895
\(55\) 8967.76 0.399740
\(56\) 7614.54 0.324469
\(57\) 23501.0 0.958073
\(58\) 9376.37 0.365986
\(59\) 45399.8 1.69795 0.848973 0.528436i \(-0.177221\pi\)
0.848973 + 0.528436i \(0.177221\pi\)
\(60\) −3387.29 −0.121471
\(61\) 18440.5 0.634525 0.317263 0.948338i \(-0.397236\pi\)
0.317263 + 0.948338i \(0.397236\pi\)
\(62\) 39151.8 1.29352
\(63\) 20379.5 0.646907
\(64\) 4096.00 0.125000
\(65\) −12893.5 −0.378519
\(66\) 12150.6 0.343350
\(67\) 48487.9 1.31961 0.659806 0.751436i \(-0.270639\pi\)
0.659806 + 0.751436i \(0.270639\pi\)
\(68\) 33484.5 0.878155
\(69\) 4479.69 0.113273
\(70\) 11897.7 0.290214
\(71\) 3989.29 0.0939182 0.0469591 0.998897i \(-0.485047\pi\)
0.0469591 + 0.998897i \(0.485047\pi\)
\(72\) 10962.5 0.249218
\(73\) 5579.98 0.122553 0.0612767 0.998121i \(-0.480483\pi\)
0.0612767 + 0.998121i \(0.480483\pi\)
\(74\) −19265.6 −0.408981
\(75\) −5292.64 −0.108647
\(76\) −44403.1 −0.881818
\(77\) −42678.4 −0.820316
\(78\) −17469.6 −0.325122
\(79\) 38453.9 0.693222 0.346611 0.938009i \(-0.387332\pi\)
0.346611 + 0.938009i \(0.387332\pi\)
\(80\) 6400.00 0.111803
\(81\) 11914.2 0.201768
\(82\) 96.2757 0.00158118
\(83\) −29897.7 −0.476368 −0.238184 0.971220i \(-0.576552\pi\)
−0.238184 + 0.971220i \(0.576552\pi\)
\(84\) 16120.4 0.249274
\(85\) 52319.5 0.785446
\(86\) 4275.37 0.0623343
\(87\) 19850.3 0.281170
\(88\) −22957.5 −0.316022
\(89\) 50329.9 0.673521 0.336760 0.941590i \(-0.390669\pi\)
0.336760 + 0.941590i \(0.390669\pi\)
\(90\) 17128.9 0.222907
\(91\) 61361.2 0.776767
\(92\) −8464.00 −0.104257
\(93\) 82886.6 0.993749
\(94\) −71672.7 −0.836631
\(95\) −69379.8 −0.788722
\(96\) 8671.46 0.0960316
\(97\) −71867.5 −0.775538 −0.387769 0.921757i \(-0.626754\pi\)
−0.387769 + 0.921757i \(0.626754\pi\)
\(98\) 10605.8 0.111552
\(99\) −61443.2 −0.630066
\(100\) 10000.0 0.100000
\(101\) 61080.4 0.595798 0.297899 0.954597i \(-0.403714\pi\)
0.297899 + 0.954597i \(0.403714\pi\)
\(102\) 70888.5 0.674645
\(103\) 181954. 1.68993 0.844964 0.534823i \(-0.179622\pi\)
0.844964 + 0.534823i \(0.179622\pi\)
\(104\) 33007.3 0.299245
\(105\) 25188.1 0.222958
\(106\) −62606.3 −0.541195
\(107\) −99429.8 −0.839571 −0.419785 0.907623i \(-0.637895\pi\)
−0.419785 + 0.907623i \(0.637895\pi\)
\(108\) 56132.7 0.463081
\(109\) 157698. 1.27134 0.635668 0.771963i \(-0.280725\pi\)
0.635668 + 0.771963i \(0.280725\pi\)
\(110\) −35871.1 −0.282659
\(111\) −40786.4 −0.314201
\(112\) −30458.2 −0.229434
\(113\) 178966. 1.31848 0.659242 0.751931i \(-0.270877\pi\)
0.659242 + 0.751931i \(0.270877\pi\)
\(114\) −94003.8 −0.677460
\(115\) −13225.0 −0.0932505
\(116\) −37505.5 −0.258791
\(117\) 88340.6 0.596617
\(118\) −181599. −1.20063
\(119\) −248993. −1.61183
\(120\) 13549.2 0.0858933
\(121\) −32377.7 −0.201040
\(122\) −73762.2 −0.448677
\(123\) 203.821 0.00121475
\(124\) −156607. −0.914655
\(125\) 15625.0 0.0894427
\(126\) −81518.0 −0.457432
\(127\) 161206. 0.886896 0.443448 0.896300i \(-0.353755\pi\)
0.443448 + 0.896300i \(0.353755\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 9051.19 0.0478885
\(130\) 51574.0 0.267653
\(131\) −137424. −0.699654 −0.349827 0.936814i \(-0.613760\pi\)
−0.349827 + 0.936814i \(0.613760\pi\)
\(132\) −48602.3 −0.242785
\(133\) 330184. 1.61856
\(134\) −193951. −0.933106
\(135\) 87707.4 0.414192
\(136\) −133938. −0.620949
\(137\) 305275. 1.38960 0.694800 0.719203i \(-0.255493\pi\)
0.694800 + 0.719203i \(0.255493\pi\)
\(138\) −17918.8 −0.0800959
\(139\) −143928. −0.631840 −0.315920 0.948786i \(-0.602313\pi\)
−0.315920 + 0.948786i \(0.602313\pi\)
\(140\) −47590.9 −0.205212
\(141\) −151735. −0.642744
\(142\) −15957.2 −0.0664102
\(143\) −185001. −0.756545
\(144\) −43850.0 −0.176223
\(145\) −58602.3 −0.231470
\(146\) −22319.9 −0.0866583
\(147\) 22453.0 0.0856999
\(148\) 77062.4 0.289193
\(149\) 355454. 1.31165 0.655824 0.754913i \(-0.272321\pi\)
0.655824 + 0.754913i \(0.272321\pi\)
\(150\) 21170.6 0.0768253
\(151\) −225364. −0.804344 −0.402172 0.915564i \(-0.631745\pi\)
−0.402172 + 0.915564i \(0.631745\pi\)
\(152\) 177612. 0.623540
\(153\) −358470. −1.23801
\(154\) 170713. 0.580051
\(155\) −244699. −0.818092
\(156\) 69878.4 0.229896
\(157\) 279940. 0.906391 0.453195 0.891411i \(-0.350284\pi\)
0.453195 + 0.891411i \(0.350284\pi\)
\(158\) −153816. −0.490182
\(159\) −132541. −0.415775
\(160\) −25600.0 −0.0790569
\(161\) 62938.9 0.191361
\(162\) −47656.9 −0.142672
\(163\) −642292. −1.89349 −0.946747 0.321980i \(-0.895652\pi\)
−0.946747 + 0.321980i \(0.895652\pi\)
\(164\) −385.103 −0.00111806
\(165\) −75941.1 −0.217153
\(166\) 119591. 0.336843
\(167\) −61147.1 −0.169662 −0.0848310 0.996395i \(-0.527035\pi\)
−0.0848310 + 0.996395i \(0.527035\pi\)
\(168\) −64481.6 −0.176264
\(169\) −105306. −0.283619
\(170\) −209278. −0.555394
\(171\) 475360. 1.24318
\(172\) −17101.5 −0.0440770
\(173\) −270177. −0.686331 −0.343165 0.939275i \(-0.611499\pi\)
−0.343165 + 0.939275i \(0.611499\pi\)
\(174\) −79401.2 −0.198817
\(175\) −74360.7 −0.183547
\(176\) 91829.9 0.223461
\(177\) −384456. −0.922387
\(178\) −201320. −0.476251
\(179\) −222698. −0.519499 −0.259749 0.965676i \(-0.583640\pi\)
−0.259749 + 0.965676i \(0.583640\pi\)
\(180\) −68515.7 −0.157619
\(181\) −277646. −0.629934 −0.314967 0.949103i \(-0.601994\pi\)
−0.314967 + 0.949103i \(0.601994\pi\)
\(182\) −245445. −0.549257
\(183\) −156159. −0.344698
\(184\) 33856.0 0.0737210
\(185\) 120410. 0.258662
\(186\) −331546. −0.702687
\(187\) 750702. 1.56987
\(188\) 286691. 0.591587
\(189\) −417407. −0.849973
\(190\) 277519. 0.557711
\(191\) 541372. 1.07377 0.536886 0.843655i \(-0.319600\pi\)
0.536886 + 0.843655i \(0.319600\pi\)
\(192\) −34685.9 −0.0679046
\(193\) −33487.9 −0.0647136 −0.0323568 0.999476i \(-0.510301\pi\)
−0.0323568 + 0.999476i \(0.510301\pi\)
\(194\) 287470. 0.548388
\(195\) 109185. 0.205625
\(196\) −42423.0 −0.0788790
\(197\) −149329. −0.274144 −0.137072 0.990561i \(-0.543769\pi\)
−0.137072 + 0.990561i \(0.543769\pi\)
\(198\) 245773. 0.445524
\(199\) 274716. 0.491759 0.245879 0.969300i \(-0.420923\pi\)
0.245879 + 0.969300i \(0.420923\pi\)
\(200\) −40000.0 −0.0707107
\(201\) −410606. −0.716862
\(202\) −244322. −0.421292
\(203\) 278893. 0.475005
\(204\) −283554. −0.477046
\(205\) −601.723 −0.00100003
\(206\) −727816. −1.19496
\(207\) 90611.9 0.146980
\(208\) −132029. −0.211598
\(209\) −995491. −1.57642
\(210\) −100753. −0.157655
\(211\) 353497. 0.546612 0.273306 0.961927i \(-0.411883\pi\)
0.273306 + 0.961927i \(0.411883\pi\)
\(212\) 250425. 0.382682
\(213\) −33782.2 −0.0510198
\(214\) 397719. 0.593666
\(215\) −26721.0 −0.0394237
\(216\) −224531. −0.327447
\(217\) 1.16454e6 1.67883
\(218\) −630792. −0.898970
\(219\) −47252.5 −0.0665755
\(220\) 143484. 0.199870
\(221\) −1.07933e6 −1.48653
\(222\) 163146. 0.222174
\(223\) −602989. −0.811984 −0.405992 0.913877i \(-0.633074\pi\)
−0.405992 + 0.913877i \(0.633074\pi\)
\(224\) 121833. 0.162235
\(225\) −107056. −0.140979
\(226\) −715865. −0.932309
\(227\) 543612. 0.700204 0.350102 0.936712i \(-0.386147\pi\)
0.350102 + 0.936712i \(0.386147\pi\)
\(228\) 376015. 0.479036
\(229\) 890736. 1.12243 0.561216 0.827669i \(-0.310334\pi\)
0.561216 + 0.827669i \(0.310334\pi\)
\(230\) 52900.0 0.0659380
\(231\) 361410. 0.445626
\(232\) 150022. 0.182993
\(233\) 1.05640e6 1.27478 0.637392 0.770539i \(-0.280013\pi\)
0.637392 + 0.770539i \(0.280013\pi\)
\(234\) −353362. −0.421872
\(235\) 447954. 0.529132
\(236\) 726397. 0.848973
\(237\) −325636. −0.376584
\(238\) 995972. 1.13974
\(239\) −805507. −0.912167 −0.456084 0.889937i \(-0.650748\pi\)
−0.456084 + 0.889937i \(0.650748\pi\)
\(240\) −54196.7 −0.0607357
\(241\) −848758. −0.941328 −0.470664 0.882312i \(-0.655986\pi\)
−0.470664 + 0.882312i \(0.655986\pi\)
\(242\) 129511. 0.142157
\(243\) −953408. −1.03577
\(244\) 295049. 0.317263
\(245\) −66285.9 −0.0705515
\(246\) −815.284 −0.000858957 0
\(247\) 1.43128e6 1.49273
\(248\) 626429. 0.646759
\(249\) 253180. 0.258781
\(250\) −62500.0 −0.0632456
\(251\) −1.61847e6 −1.62151 −0.810755 0.585386i \(-0.800943\pi\)
−0.810755 + 0.585386i \(0.800943\pi\)
\(252\) 326072. 0.323454
\(253\) −189758. −0.186380
\(254\) −644825. −0.627130
\(255\) −443053. −0.426683
\(256\) 65536.0 0.0625000
\(257\) 1.31624e6 1.24309 0.621544 0.783379i \(-0.286506\pi\)
0.621544 + 0.783379i \(0.286506\pi\)
\(258\) −36204.8 −0.0338623
\(259\) −573042. −0.530807
\(260\) −206296. −0.189259
\(261\) 401518. 0.364841
\(262\) 549695. 0.494730
\(263\) −605771. −0.540032 −0.270016 0.962856i \(-0.587029\pi\)
−0.270016 + 0.962856i \(0.587029\pi\)
\(264\) 194409. 0.171675
\(265\) 391290. 0.342282
\(266\) −1.32074e6 −1.14449
\(267\) −426205. −0.365881
\(268\) 775806. 0.659806
\(269\) 11040.1 0.00930233 0.00465116 0.999989i \(-0.498519\pi\)
0.00465116 + 0.999989i \(0.498519\pi\)
\(270\) −350829. −0.292878
\(271\) 1.30158e6 1.07658 0.538291 0.842759i \(-0.319070\pi\)
0.538291 + 0.842759i \(0.319070\pi\)
\(272\) 535752. 0.439078
\(273\) −519621. −0.421968
\(274\) −1.22110e6 −0.982596
\(275\) 224194. 0.178769
\(276\) 71675.1 0.0566364
\(277\) 2.07170e6 1.62229 0.811143 0.584848i \(-0.198846\pi\)
0.811143 + 0.584848i \(0.198846\pi\)
\(278\) 575710. 0.446778
\(279\) 1.67657e6 1.28947
\(280\) 190363. 0.145107
\(281\) 705629. 0.533102 0.266551 0.963821i \(-0.414116\pi\)
0.266551 + 0.963821i \(0.414116\pi\)
\(282\) 606940. 0.454489
\(283\) −1.85265e6 −1.37508 −0.687538 0.726148i \(-0.741309\pi\)
−0.687538 + 0.726148i \(0.741309\pi\)
\(284\) 63828.6 0.0469591
\(285\) 587524. 0.428463
\(286\) 740005. 0.534958
\(287\) 2863.65 0.00205218
\(288\) 175400. 0.124609
\(289\) 2.95987e6 2.08462
\(290\) 234409. 0.163674
\(291\) 608590. 0.421301
\(292\) 89279.6 0.0612767
\(293\) 1.06802e6 0.726790 0.363395 0.931635i \(-0.381618\pi\)
0.363395 + 0.931635i \(0.381618\pi\)
\(294\) −89811.9 −0.0605990
\(295\) 1.13500e6 0.759345
\(296\) −308250. −0.204491
\(297\) 1.25846e6 0.827845
\(298\) −1.42182e6 −0.927476
\(299\) 272826. 0.176485
\(300\) −84682.3 −0.0543237
\(301\) 127168. 0.0809023
\(302\) 901455. 0.568757
\(303\) −517243. −0.323659
\(304\) −710449. −0.440909
\(305\) 461013. 0.283768
\(306\) 1.43388e6 0.875406
\(307\) −34869.7 −0.0211156 −0.0105578 0.999944i \(-0.503361\pi\)
−0.0105578 + 0.999944i \(0.503361\pi\)
\(308\) −682854. −0.410158
\(309\) −1.54083e6 −0.918032
\(310\) 978795. 0.578479
\(311\) 2.27802e6 1.33554 0.667770 0.744368i \(-0.267249\pi\)
0.667770 + 0.744368i \(0.267249\pi\)
\(312\) −279514. −0.162561
\(313\) −1.31658e6 −0.759600 −0.379800 0.925069i \(-0.624007\pi\)
−0.379800 + 0.925069i \(0.624007\pi\)
\(314\) −1.11976e6 −0.640915
\(315\) 509487. 0.289306
\(316\) 615262. 0.346611
\(317\) −1.15067e6 −0.643135 −0.321568 0.946887i \(-0.604210\pi\)
−0.321568 + 0.946887i \(0.604210\pi\)
\(318\) 530165. 0.293997
\(319\) −840851. −0.462639
\(320\) 102400. 0.0559017
\(321\) 841995. 0.456086
\(322\) −251756. −0.135313
\(323\) −5.80787e6 −3.09749
\(324\) 190628. 0.100884
\(325\) −322337. −0.169279
\(326\) 2.56917e6 1.33890
\(327\) −1.33542e6 −0.690637
\(328\) 1540.41 0.000790591 0
\(329\) −2.13185e6 −1.08584
\(330\) 303764. 0.153551
\(331\) −3.62446e6 −1.81833 −0.909166 0.416434i \(-0.863280\pi\)
−0.909166 + 0.416434i \(0.863280\pi\)
\(332\) −478363. −0.238184
\(333\) −824997. −0.407701
\(334\) 244588. 0.119969
\(335\) 1.21220e6 0.590148
\(336\) 257927. 0.124637
\(337\) 308341. 0.147896 0.0739481 0.997262i \(-0.476440\pi\)
0.0739481 + 0.997262i \(0.476440\pi\)
\(338\) 421222. 0.200549
\(339\) −1.51553e6 −0.716249
\(340\) 837112. 0.392723
\(341\) −3.51104e6 −1.63512
\(342\) −1.90144e6 −0.879058
\(343\) 2.31511e6 1.06252
\(344\) 68405.9 0.0311672
\(345\) 111992. 0.0506571
\(346\) 1.08071e6 0.485309
\(347\) −850011. −0.378967 −0.189483 0.981884i \(-0.560681\pi\)
−0.189483 + 0.981884i \(0.560681\pi\)
\(348\) 317605. 0.140585
\(349\) −2.96385e6 −1.30255 −0.651273 0.758844i \(-0.725765\pi\)
−0.651273 + 0.758844i \(0.725765\pi\)
\(350\) 297443. 0.129788
\(351\) −1.80937e6 −0.783897
\(352\) −367320. −0.158011
\(353\) 2.14490e6 0.916157 0.458078 0.888912i \(-0.348538\pi\)
0.458078 + 0.888912i \(0.348538\pi\)
\(354\) 1.53782e6 0.652226
\(355\) 99732.2 0.0420015
\(356\) 805278. 0.336760
\(357\) 2.10853e6 0.875607
\(358\) 890793. 0.367341
\(359\) −438888. −0.179729 −0.0898643 0.995954i \(-0.528643\pi\)
−0.0898643 + 0.995954i \(0.528643\pi\)
\(360\) 274063. 0.111453
\(361\) 5.22559e6 2.11041
\(362\) 1.11058e6 0.445431
\(363\) 274182. 0.109212
\(364\) 981780. 0.388383
\(365\) 139499. 0.0548075
\(366\) 624635. 0.243738
\(367\) 1.38893e6 0.538289 0.269144 0.963100i \(-0.413259\pi\)
0.269144 + 0.963100i \(0.413259\pi\)
\(368\) −135424. −0.0521286
\(369\) 4122.74 0.00157623
\(370\) −481640. −0.182902
\(371\) −1.86218e6 −0.702404
\(372\) 1.32618e6 0.496875
\(373\) 2.14429e6 0.798017 0.399009 0.916947i \(-0.369354\pi\)
0.399009 + 0.916947i \(0.369354\pi\)
\(374\) −3.00281e6 −1.11007
\(375\) −132316. −0.0485886
\(376\) −1.14676e6 −0.418316
\(377\) 1.20894e6 0.438078
\(378\) 1.66963e6 0.601022
\(379\) 2.55976e6 0.915378 0.457689 0.889112i \(-0.348677\pi\)
0.457689 + 0.889112i \(0.348677\pi\)
\(380\) −1.11008e6 −0.394361
\(381\) −1.36513e6 −0.481795
\(382\) −2.16549e6 −0.759272
\(383\) −2.89975e6 −1.01010 −0.505049 0.863091i \(-0.668526\pi\)
−0.505049 + 0.863091i \(0.668526\pi\)
\(384\) 138743. 0.0480158
\(385\) −1.06696e6 −0.366856
\(386\) 133952. 0.0457594
\(387\) 183081. 0.0621392
\(388\) −1.14988e6 −0.387769
\(389\) 1.47101e6 0.492882 0.246441 0.969158i \(-0.420739\pi\)
0.246441 + 0.969158i \(0.420739\pi\)
\(390\) −436740. −0.145399
\(391\) −1.10708e6 −0.366216
\(392\) 169692. 0.0557759
\(393\) 1.16374e6 0.380078
\(394\) 597316. 0.193849
\(395\) 961347. 0.310018
\(396\) −983092. −0.315033
\(397\) −4.88528e6 −1.55565 −0.777827 0.628479i \(-0.783678\pi\)
−0.777827 + 0.628479i \(0.783678\pi\)
\(398\) −1.09887e6 −0.347726
\(399\) −2.79608e6 −0.879259
\(400\) 160000. 0.0500000
\(401\) −4.03714e6 −1.25376 −0.626878 0.779117i \(-0.715668\pi\)
−0.626878 + 0.779117i \(0.715668\pi\)
\(402\) 1.64243e6 0.506898
\(403\) 5.04803e6 1.54832
\(404\) 977287. 0.297899
\(405\) 297856. 0.0902336
\(406\) −1.11557e6 −0.335879
\(407\) 1.72769e6 0.516989
\(408\) 1.13422e6 0.337323
\(409\) −4.13883e6 −1.22340 −0.611702 0.791088i \(-0.709515\pi\)
−0.611702 + 0.791088i \(0.709515\pi\)
\(410\) 2406.89 0.000707126 0
\(411\) −2.58514e6 −0.754882
\(412\) 2.91126e6 0.844964
\(413\) −5.40154e6 −1.55827
\(414\) −362448. −0.103931
\(415\) −747442. −0.213038
\(416\) 528117. 0.149623
\(417\) 1.21881e6 0.343239
\(418\) 3.98196e6 1.11470
\(419\) 536280. 0.149230 0.0746150 0.997212i \(-0.476227\pi\)
0.0746150 + 0.997212i \(0.476227\pi\)
\(420\) 403010. 0.111479
\(421\) 6.59556e6 1.81362 0.906810 0.421539i \(-0.138510\pi\)
0.906810 + 0.421539i \(0.138510\pi\)
\(422\) −1.41399e6 −0.386513
\(423\) −3.06919e6 −0.834012
\(424\) −1.00170e6 −0.270597
\(425\) 1.30799e6 0.351262
\(426\) 135129. 0.0360765
\(427\) −2.19400e6 −0.582328
\(428\) −1.59088e6 −0.419785
\(429\) 1.56663e6 0.410983
\(430\) 106884. 0.0278768
\(431\) 3.65012e6 0.946486 0.473243 0.880932i \(-0.343083\pi\)
0.473243 + 0.880932i \(0.343083\pi\)
\(432\) 898123. 0.231540
\(433\) 5.50340e6 1.41062 0.705311 0.708898i \(-0.250807\pi\)
0.705311 + 0.708898i \(0.250807\pi\)
\(434\) −4.65817e6 −1.18711
\(435\) 496258. 0.125743
\(436\) 2.52317e6 0.635668
\(437\) 1.46808e6 0.367744
\(438\) 189010. 0.0470760
\(439\) −4.23173e6 −1.04799 −0.523994 0.851722i \(-0.675559\pi\)
−0.523994 + 0.851722i \(0.675559\pi\)
\(440\) −573937. −0.141329
\(441\) 454162. 0.111203
\(442\) 4.31732e6 1.05113
\(443\) −4.22418e6 −1.02266 −0.511332 0.859383i \(-0.670848\pi\)
−0.511332 + 0.859383i \(0.670848\pi\)
\(444\) −652582. −0.157101
\(445\) 1.25825e6 0.301208
\(446\) 2.41196e6 0.574159
\(447\) −3.01006e6 −0.712536
\(448\) −487330. −0.114717
\(449\) −6.45197e6 −1.51035 −0.755173 0.655526i \(-0.772447\pi\)
−0.755173 + 0.655526i \(0.772447\pi\)
\(450\) 428223. 0.0996870
\(451\) −8633.78 −0.00199875
\(452\) 2.86346e6 0.659242
\(453\) 1.90843e6 0.436949
\(454\) −2.17445e6 −0.495119
\(455\) 1.53403e6 0.347381
\(456\) −1.50406e6 −0.338730
\(457\) 1.98425e6 0.444432 0.222216 0.974997i \(-0.428671\pi\)
0.222216 + 0.974997i \(0.428671\pi\)
\(458\) −3.56294e6 −0.793679
\(459\) 7.34209e6 1.62663
\(460\) −211600. −0.0466252
\(461\) 5.78438e6 1.26767 0.633833 0.773470i \(-0.281481\pi\)
0.633833 + 0.773470i \(0.281481\pi\)
\(462\) −1.44564e6 −0.315105
\(463\) −7.76227e6 −1.68281 −0.841407 0.540402i \(-0.818272\pi\)
−0.841407 + 0.540402i \(0.818272\pi\)
\(464\) −600088. −0.129396
\(465\) 2.07216e6 0.444418
\(466\) −4.22558e6 −0.901409
\(467\) 7.51036e6 1.59356 0.796780 0.604269i \(-0.206535\pi\)
0.796780 + 0.604269i \(0.206535\pi\)
\(468\) 1.41345e6 0.298309
\(469\) −5.76895e6 −1.21106
\(470\) −1.79182e6 −0.374153
\(471\) −2.37059e6 −0.492385
\(472\) −2.90559e6 −0.600315
\(473\) −383405. −0.0787961
\(474\) 1.30255e6 0.266285
\(475\) −1.73450e6 −0.352727
\(476\) −3.98389e6 −0.805916
\(477\) −2.68095e6 −0.539501
\(478\) 3.22203e6 0.645000
\(479\) −7.38377e6 −1.47041 −0.735207 0.677843i \(-0.762915\pi\)
−0.735207 + 0.677843i \(0.762915\pi\)
\(480\) 216787. 0.0429467
\(481\) −2.48401e6 −0.489543
\(482\) 3.39503e6 0.665620
\(483\) −532981. −0.103955
\(484\) −518043. −0.100520
\(485\) −1.79669e6 −0.346831
\(486\) 3.81363e6 0.732400
\(487\) −9.26947e6 −1.77106 −0.885528 0.464586i \(-0.846203\pi\)
−0.885528 + 0.464586i \(0.846203\pi\)
\(488\) −1.18019e6 −0.224339
\(489\) 5.43908e6 1.02862
\(490\) 265144. 0.0498874
\(491\) −3.61323e6 −0.676381 −0.338190 0.941078i \(-0.609815\pi\)
−0.338190 + 0.941078i \(0.609815\pi\)
\(492\) 3261.14 0.000607374 0
\(493\) −4.90567e6 −0.909036
\(494\) −5.72511e6 −1.05552
\(495\) −1.53608e6 −0.281774
\(496\) −2.50571e6 −0.457328
\(497\) −474634. −0.0861922
\(498\) −1.01272e6 −0.182986
\(499\) 6.40637e6 1.15176 0.575878 0.817535i \(-0.304660\pi\)
0.575878 + 0.817535i \(0.304660\pi\)
\(500\) 250000. 0.0447214
\(501\) 517808. 0.0921667
\(502\) 6.47387e6 1.14658
\(503\) 1.04304e7 1.83815 0.919074 0.394086i \(-0.128939\pi\)
0.919074 + 0.394086i \(0.128939\pi\)
\(504\) −1.30429e6 −0.228716
\(505\) 1.52701e6 0.266449
\(506\) 759032. 0.131790
\(507\) 891752. 0.154072
\(508\) 2.57930e6 0.443448
\(509\) 44659.2 0.00764040 0.00382020 0.999993i \(-0.498784\pi\)
0.00382020 + 0.999993i \(0.498784\pi\)
\(510\) 1.77221e6 0.301711
\(511\) −663890. −0.112472
\(512\) −262144. −0.0441942
\(513\) −9.73619e6 −1.63341
\(514\) −5.26496e6 −0.878997
\(515\) 4.54885e6 0.755759
\(516\) 144819. 0.0239443
\(517\) 6.42744e6 1.05758
\(518\) 2.29217e6 0.375337
\(519\) 2.28792e6 0.372840
\(520\) 825183. 0.133827
\(521\) 6.03155e6 0.973497 0.486749 0.873542i \(-0.338183\pi\)
0.486749 + 0.873542i \(0.338183\pi\)
\(522\) −1.60607e6 −0.257981
\(523\) 1.12464e6 0.179788 0.0898939 0.995951i \(-0.471347\pi\)
0.0898939 + 0.995951i \(0.471347\pi\)
\(524\) −2.19878e6 −0.349827
\(525\) 629703. 0.0997098
\(526\) 2.42309e6 0.381860
\(527\) −2.04840e7 −3.21284
\(528\) −777637. −0.121392
\(529\) 279841. 0.0434783
\(530\) −1.56516e6 −0.242030
\(531\) −7.77649e6 −1.19687
\(532\) 5.28295e6 0.809278
\(533\) 12413.3 0.00189264
\(534\) 1.70482e6 0.258717
\(535\) −2.48575e6 −0.375467
\(536\) −3.10322e6 −0.466553
\(537\) 1.88586e6 0.282211
\(538\) −44160.3 −0.00657774
\(539\) −951099. −0.141011
\(540\) 1.40332e6 0.207096
\(541\) −8.07492e6 −1.18617 −0.593083 0.805142i \(-0.702089\pi\)
−0.593083 + 0.805142i \(0.702089\pi\)
\(542\) −5.20631e6 −0.761258
\(543\) 2.35117e6 0.342204
\(544\) −2.14301e6 −0.310475
\(545\) 3.94245e6 0.568559
\(546\) 2.07848e6 0.298377
\(547\) 5.64519e6 0.806697 0.403348 0.915047i \(-0.367846\pi\)
0.403348 + 0.915047i \(0.367846\pi\)
\(548\) 4.88440e6 0.694800
\(549\) −3.15866e6 −0.447273
\(550\) −896776. −0.126409
\(551\) 6.50531e6 0.912828
\(552\) −286700. −0.0400480
\(553\) −4.57513e6 −0.636196
\(554\) −8.28680e6 −1.14713
\(555\) −1.01966e6 −0.140515
\(556\) −2.30284e6 −0.315920
\(557\) 1.00532e7 1.37299 0.686496 0.727134i \(-0.259148\pi\)
0.686496 + 0.727134i \(0.259148\pi\)
\(558\) −6.70628e6 −0.911792
\(559\) 551244. 0.0746130
\(560\) −761454. −0.102606
\(561\) −6.35712e6 −0.852811
\(562\) −2.82251e6 −0.376960
\(563\) −7.02958e6 −0.934670 −0.467335 0.884080i \(-0.654786\pi\)
−0.467335 + 0.884080i \(0.654786\pi\)
\(564\) −2.42776e6 −0.321372
\(565\) 4.47415e6 0.589644
\(566\) 7.41059e6 0.972326
\(567\) −1.41752e6 −0.185170
\(568\) −255315. −0.0332051
\(569\) −6.22757e6 −0.806377 −0.403188 0.915117i \(-0.632098\pi\)
−0.403188 + 0.915117i \(0.632098\pi\)
\(570\) −2.35010e6 −0.302969
\(571\) 1.08461e7 1.39215 0.696074 0.717970i \(-0.254929\pi\)
0.696074 + 0.717970i \(0.254929\pi\)
\(572\) −2.96002e6 −0.378272
\(573\) −4.58446e6 −0.583313
\(574\) −11454.6 −0.00145111
\(575\) −330625. −0.0417029
\(576\) −701600. −0.0881117
\(577\) 1.08904e6 0.136178 0.0680889 0.997679i \(-0.478310\pi\)
0.0680889 + 0.997679i \(0.478310\pi\)
\(578\) −1.18395e7 −1.47405
\(579\) 283584. 0.0351548
\(580\) −937637. −0.115735
\(581\) 3.55714e6 0.437181
\(582\) −2.43436e6 −0.297905
\(583\) 5.61439e6 0.684118
\(584\) −357118. −0.0433292
\(585\) 2.20851e6 0.266815
\(586\) −4.27207e6 −0.513918
\(587\) 1.40443e7 1.68230 0.841150 0.540801i \(-0.181879\pi\)
0.841150 + 0.540801i \(0.181879\pi\)
\(588\) 359248. 0.0428500
\(589\) 2.71634e7 3.22624
\(590\) −4.53998e6 −0.536938
\(591\) 1.26455e6 0.148925
\(592\) 1.23300e6 0.144597
\(593\) 1.43036e7 1.67036 0.835180 0.549977i \(-0.185363\pi\)
0.835180 + 0.549977i \(0.185363\pi\)
\(594\) −5.03385e6 −0.585375
\(595\) −6.22482e6 −0.720833
\(596\) 5.68726e6 0.655824
\(597\) −2.32636e6 −0.267141
\(598\) −1.09131e6 −0.124794
\(599\) −4.51234e6 −0.513848 −0.256924 0.966432i \(-0.582709\pi\)
−0.256924 + 0.966432i \(0.582709\pi\)
\(600\) 338729. 0.0384127
\(601\) 1.00918e7 1.13968 0.569838 0.821757i \(-0.307006\pi\)
0.569838 + 0.821757i \(0.307006\pi\)
\(602\) −508671. −0.0572065
\(603\) −8.30544e6 −0.930186
\(604\) −3.60582e6 −0.402172
\(605\) −809443. −0.0899079
\(606\) 2.06897e6 0.228862
\(607\) 8.46363e6 0.932363 0.466182 0.884689i \(-0.345629\pi\)
0.466182 + 0.884689i \(0.345629\pi\)
\(608\) 2.84180e6 0.311770
\(609\) −2.36173e6 −0.258040
\(610\) −1.84405e6 −0.200655
\(611\) −9.24111e6 −1.00143
\(612\) −5.73553e6 −0.619006
\(613\) 453516. 0.0487463 0.0243732 0.999703i \(-0.492241\pi\)
0.0243732 + 0.999703i \(0.492241\pi\)
\(614\) 139479. 0.0149310
\(615\) 5095.53 0.000543252 0
\(616\) 2.73142e6 0.290025
\(617\) 6.89013e6 0.728643 0.364321 0.931273i \(-0.381301\pi\)
0.364321 + 0.931273i \(0.381301\pi\)
\(618\) 6.16331e6 0.649147
\(619\) 2.03233e6 0.213190 0.106595 0.994303i \(-0.466005\pi\)
0.106595 + 0.994303i \(0.466005\pi\)
\(620\) −3.91518e6 −0.409046
\(621\) −1.85589e6 −0.193118
\(622\) −9.11208e6 −0.944369
\(623\) −5.98811e6 −0.618115
\(624\) 1.11805e6 0.114948
\(625\) 390625. 0.0400000
\(626\) 5.26630e6 0.537118
\(627\) 8.43004e6 0.856369
\(628\) 4.47904e6 0.453195
\(629\) 1.00797e7 1.01583
\(630\) −2.03795e6 −0.204570
\(631\) 1.36301e7 1.36278 0.681392 0.731919i \(-0.261375\pi\)
0.681392 + 0.731919i \(0.261375\pi\)
\(632\) −2.46105e6 −0.245091
\(633\) −2.99349e6 −0.296940
\(634\) 4.60268e6 0.454765
\(635\) 4.03016e6 0.396632
\(636\) −2.12066e6 −0.207887
\(637\) 1.36745e6 0.133525
\(638\) 3.36340e6 0.327135
\(639\) −683322. −0.0662023
\(640\) −409600. −0.0395285
\(641\) −1.27425e7 −1.22492 −0.612462 0.790500i \(-0.709821\pi\)
−0.612462 + 0.790500i \(0.709821\pi\)
\(642\) −3.36798e6 −0.322501
\(643\) 1.23274e7 1.17583 0.587913 0.808924i \(-0.299950\pi\)
0.587913 + 0.808924i \(0.299950\pi\)
\(644\) 1.00702e6 0.0956807
\(645\) 226280. 0.0214164
\(646\) 2.32315e7 2.19026
\(647\) −1.16678e7 −1.09579 −0.547894 0.836548i \(-0.684570\pi\)
−0.547894 + 0.836548i \(0.684570\pi\)
\(648\) −762510. −0.0713359
\(649\) 1.62854e7 1.51770
\(650\) 1.28935e6 0.119698
\(651\) −9.86161e6 −0.912001
\(652\) −1.02767e7 −0.946747
\(653\) 1.07123e7 0.983108 0.491554 0.870847i \(-0.336429\pi\)
0.491554 + 0.870847i \(0.336429\pi\)
\(654\) 5.34169e6 0.488354
\(655\) −3.43559e6 −0.312895
\(656\) −6161.64 −0.000559032 0
\(657\) −955789. −0.0863871
\(658\) 8.52741e6 0.767808
\(659\) 1.03868e7 0.931686 0.465843 0.884867i \(-0.345751\pi\)
0.465843 + 0.884867i \(0.345751\pi\)
\(660\) −1.21506e6 −0.108577
\(661\) 1.32825e7 1.18243 0.591215 0.806514i \(-0.298649\pi\)
0.591215 + 0.806514i \(0.298649\pi\)
\(662\) 1.44978e7 1.28576
\(663\) 9.14001e6 0.807538
\(664\) 1.91345e6 0.168421
\(665\) 8.25461e6 0.723840
\(666\) 3.29999e6 0.288288
\(667\) 1.24002e6 0.107923
\(668\) −978354. −0.0848310
\(669\) 5.10625e6 0.441100
\(670\) −4.84879e6 −0.417298
\(671\) 6.61482e6 0.567168
\(672\) −1.03171e6 −0.0881318
\(673\) −1.21571e7 −1.03465 −0.517323 0.855790i \(-0.673071\pi\)
−0.517323 + 0.855790i \(0.673071\pi\)
\(674\) −1.23336e6 −0.104578
\(675\) 2.19268e6 0.185232
\(676\) −1.68489e6 −0.141809
\(677\) −1.63957e7 −1.37486 −0.687431 0.726250i \(-0.741262\pi\)
−0.687431 + 0.726250i \(0.741262\pi\)
\(678\) 6.06210e6 0.506465
\(679\) 8.55059e6 0.711740
\(680\) −3.34845e6 −0.277697
\(681\) −4.60343e6 −0.380377
\(682\) 1.40442e7 1.15621
\(683\) −5.50399e6 −0.451467 −0.225733 0.974189i \(-0.572478\pi\)
−0.225733 + 0.974189i \(0.572478\pi\)
\(684\) 7.60576e6 0.621588
\(685\) 7.63188e6 0.621448
\(686\) −9.26044e6 −0.751314
\(687\) −7.54295e6 −0.609747
\(688\) −273623. −0.0220385
\(689\) −8.07214e6 −0.647800
\(690\) −447969. −0.0358200
\(691\) 8.81682e6 0.702452 0.351226 0.936291i \(-0.385765\pi\)
0.351226 + 0.936291i \(0.385765\pi\)
\(692\) −4.32284e6 −0.343165
\(693\) 7.31034e6 0.578235
\(694\) 3.40004e6 0.267970
\(695\) −3.59819e6 −0.282567
\(696\) −1.27042e6 −0.0994086
\(697\) −50370.9 −0.00392734
\(698\) 1.18554e7 0.921039
\(699\) −8.94580e6 −0.692510
\(700\) −1.18977e6 −0.0917737
\(701\) 9.94017e6 0.764010 0.382005 0.924160i \(-0.375234\pi\)
0.382005 + 0.924160i \(0.375234\pi\)
\(702\) 7.23747e6 0.554299
\(703\) −1.33664e7 −1.02006
\(704\) 1.46928e6 0.111731
\(705\) −3.79338e6 −0.287444
\(706\) −8.57959e6 −0.647821
\(707\) −7.26718e6 −0.546786
\(708\) −6.15129e6 −0.461194
\(709\) 2.13658e7 1.59626 0.798128 0.602488i \(-0.205824\pi\)
0.798128 + 0.602488i \(0.205824\pi\)
\(710\) −398929. −0.0296995
\(711\) −6.58673e6 −0.488648
\(712\) −3.22111e6 −0.238126
\(713\) 5.17782e6 0.381438
\(714\) −8.43412e6 −0.619147
\(715\) −4.62503e6 −0.338337
\(716\) −3.56317e6 −0.259749
\(717\) 6.82121e6 0.495523
\(718\) 1.75555e6 0.127087
\(719\) 2.21914e6 0.160089 0.0800445 0.996791i \(-0.474494\pi\)
0.0800445 + 0.996791i \(0.474494\pi\)
\(720\) −1.09625e6 −0.0788095
\(721\) −2.16484e7 −1.55091
\(722\) −2.09024e7 −1.49229
\(723\) 7.18747e6 0.511364
\(724\) −4.44234e6 −0.314967
\(725\) −1.46506e6 −0.103517
\(726\) −1.09673e6 −0.0772249
\(727\) 1.66600e7 1.16906 0.584532 0.811371i \(-0.301278\pi\)
0.584532 + 0.811371i \(0.301278\pi\)
\(728\) −3.92712e6 −0.274629
\(729\) 5.17852e6 0.360900
\(730\) −557998. −0.0387548
\(731\) −2.23685e6 −0.154826
\(732\) −2.49854e6 −0.172349
\(733\) 6.84915e6 0.470844 0.235422 0.971893i \(-0.424353\pi\)
0.235422 + 0.971893i \(0.424353\pi\)
\(734\) −5.55572e6 −0.380627
\(735\) 561324. 0.0383262
\(736\) 541696. 0.0368605
\(737\) 1.73931e7 1.17953
\(738\) −16491.0 −0.00111457
\(739\) 4.66486e6 0.314216 0.157108 0.987581i \(-0.449783\pi\)
0.157108 + 0.987581i \(0.449783\pi\)
\(740\) 1.92656e6 0.129331
\(741\) −1.21204e7 −0.810906
\(742\) 7.44872e6 0.496675
\(743\) −7.85896e6 −0.522268 −0.261134 0.965303i \(-0.584096\pi\)
−0.261134 + 0.965303i \(0.584096\pi\)
\(744\) −5.30474e6 −0.351343
\(745\) 8.88635e6 0.586587
\(746\) −8.57718e6 −0.564284
\(747\) 5.12115e6 0.335789
\(748\) 1.20112e7 0.784935
\(749\) 1.18299e7 0.770506
\(750\) 529264. 0.0343573
\(751\) −1.85421e7 −1.19966 −0.599830 0.800128i \(-0.704765\pi\)
−0.599830 + 0.800128i \(0.704765\pi\)
\(752\) 4.58705e6 0.295794
\(753\) 1.37055e7 0.880864
\(754\) −4.83577e6 −0.309768
\(755\) −5.63409e6 −0.359714
\(756\) −6.67851e6 −0.424986
\(757\) −1.07470e7 −0.681629 −0.340815 0.940130i \(-0.610703\pi\)
−0.340815 + 0.940130i \(0.610703\pi\)
\(758\) −1.02390e7 −0.647270
\(759\) 1.60691e6 0.101248
\(760\) 4.44031e6 0.278855
\(761\) −6.27486e6 −0.392774 −0.196387 0.980526i \(-0.562921\pi\)
−0.196387 + 0.980526i \(0.562921\pi\)
\(762\) 5.46053e6 0.340680
\(763\) −1.87625e7 −1.16675
\(764\) 8.66195e6 0.536886
\(765\) −8.96176e6 −0.553656
\(766\) 1.15990e7 0.714247
\(767\) −2.34145e7 −1.43713
\(768\) −554974. −0.0339523
\(769\) −3.06263e7 −1.86758 −0.933788 0.357827i \(-0.883518\pi\)
−0.933788 + 0.357827i \(0.883518\pi\)
\(770\) 4.26784e6 0.259407
\(771\) −1.11462e7 −0.675292
\(772\) −535807. −0.0323568
\(773\) 2.73276e7 1.64495 0.822474 0.568803i \(-0.192593\pi\)
0.822474 + 0.568803i \(0.192593\pi\)
\(774\) −732324. −0.0439391
\(775\) −6.11747e6 −0.365862
\(776\) 4.59952e6 0.274194
\(777\) 4.85265e6 0.288354
\(778\) −5.88406e6 −0.348520
\(779\) 66795.9 0.00394372
\(780\) 1.74696e6 0.102813
\(781\) 1.43100e6 0.0839483
\(782\) 4.42832e6 0.258954
\(783\) −8.22377e6 −0.479365
\(784\) −678768. −0.0394395
\(785\) 6.99850e6 0.405350
\(786\) −4.65494e6 −0.268756
\(787\) −1.25785e7 −0.723925 −0.361963 0.932193i \(-0.617893\pi\)
−0.361963 + 0.932193i \(0.617893\pi\)
\(788\) −2.38927e6 −0.137072
\(789\) 5.12981e6 0.293365
\(790\) −3.84539e6 −0.219216
\(791\) −2.12929e7 −1.21002
\(792\) 3.93237e6 0.222762
\(793\) −9.51052e6 −0.537058
\(794\) 1.95411e7 1.10001
\(795\) −3.31353e6 −0.185940
\(796\) 4.39546e6 0.245879
\(797\) −2.82459e7 −1.57511 −0.787554 0.616246i \(-0.788653\pi\)
−0.787554 + 0.616246i \(0.788653\pi\)
\(798\) 1.11843e7 0.621730
\(799\) 3.74988e7 2.07802
\(800\) −640000. −0.0353553
\(801\) −8.62096e6 −0.474760
\(802\) 1.61486e7 0.886540
\(803\) 2.00160e6 0.109544
\(804\) −6.56970e6 −0.358431
\(805\) 1.57347e6 0.0855795
\(806\) −2.01921e7 −1.09482
\(807\) −93489.9 −0.00505337
\(808\) −3.90915e6 −0.210646
\(809\) 3.02669e7 1.62591 0.812956 0.582325i \(-0.197857\pi\)
0.812956 + 0.582325i \(0.197857\pi\)
\(810\) −1.19142e6 −0.0638048
\(811\) −2.24011e7 −1.19596 −0.597981 0.801510i \(-0.704030\pi\)
−0.597981 + 0.801510i \(0.704030\pi\)
\(812\) 4.46229e6 0.237503
\(813\) −1.10221e7 −0.584839
\(814\) −6.91078e6 −0.365566
\(815\) −1.60573e7 −0.846796
\(816\) −4.53687e6 −0.238523
\(817\) 2.96624e6 0.155472
\(818\) 1.65553e7 0.865077
\(819\) −1.05105e7 −0.547538
\(820\) −9627.57 −0.000500014 0
\(821\) 2.07746e7 1.07566 0.537830 0.843053i \(-0.319244\pi\)
0.537830 + 0.843053i \(0.319244\pi\)
\(822\) 1.03406e7 0.533782
\(823\) −1.77559e7 −0.913784 −0.456892 0.889522i \(-0.651037\pi\)
−0.456892 + 0.889522i \(0.651037\pi\)
\(824\) −1.16451e7 −0.597480
\(825\) −1.89853e6 −0.0971140
\(826\) 2.16062e7 1.10186
\(827\) −7.87984e6 −0.400639 −0.200320 0.979731i \(-0.564198\pi\)
−0.200320 + 0.979731i \(0.564198\pi\)
\(828\) 1.44979e6 0.0734902
\(829\) 2.16609e7 1.09469 0.547345 0.836907i \(-0.315639\pi\)
0.547345 + 0.836907i \(0.315639\pi\)
\(830\) 2.98977e6 0.150641
\(831\) −1.75436e7 −0.881286
\(832\) −2.11247e6 −0.105799
\(833\) −5.54887e6 −0.277072
\(834\) −4.87525e6 −0.242707
\(835\) −1.52868e6 −0.0758752
\(836\) −1.59279e7 −0.788209
\(837\) −3.43390e7 −1.69424
\(838\) −2.14512e6 −0.105522
\(839\) 2.69565e7 1.32208 0.661041 0.750350i \(-0.270115\pi\)
0.661041 + 0.750350i \(0.270115\pi\)
\(840\) −1.61204e6 −0.0788275
\(841\) −1.50164e7 −0.732108
\(842\) −2.63822e7 −1.28242
\(843\) −5.97542e6 −0.289601
\(844\) 5.65594e6 0.273306
\(845\) −2.63264e6 −0.126838
\(846\) 1.22767e7 0.589736
\(847\) 3.85221e6 0.184502
\(848\) 4.00680e6 0.191341
\(849\) 1.56886e7 0.746993
\(850\) −5.23195e6 −0.248380
\(851\) −2.54788e6 −0.120602
\(852\) −540515. −0.0255099
\(853\) −3.01511e7 −1.41883 −0.709415 0.704791i \(-0.751041\pi\)
−0.709415 + 0.704791i \(0.751041\pi\)
\(854\) 8.77601e6 0.411768
\(855\) 1.18840e7 0.555965
\(856\) 6.36351e6 0.296833
\(857\) 9.17229e6 0.426605 0.213302 0.976986i \(-0.431578\pi\)
0.213302 + 0.976986i \(0.431578\pi\)
\(858\) −6.26653e6 −0.290609
\(859\) −9.12169e6 −0.421786 −0.210893 0.977509i \(-0.567637\pi\)
−0.210893 + 0.977509i \(0.567637\pi\)
\(860\) −427537. −0.0197118
\(861\) −24250.1 −0.00111482
\(862\) −1.46005e7 −0.669267
\(863\) 1.54060e7 0.704145 0.352072 0.935973i \(-0.385477\pi\)
0.352072 + 0.935973i \(0.385477\pi\)
\(864\) −3.59249e6 −0.163724
\(865\) −6.75443e6 −0.306937
\(866\) −2.20136e7 −0.997461
\(867\) −2.50648e7 −1.13245
\(868\) 1.86327e7 0.839413
\(869\) 1.37938e7 0.619634
\(870\) −1.98503e6 −0.0889138
\(871\) −2.50071e7 −1.11691
\(872\) −1.00927e7 −0.449485
\(873\) 1.23101e7 0.546672
\(874\) −5.87231e6 −0.260034
\(875\) −1.85902e6 −0.0820849
\(876\) −756040. −0.0332878
\(877\) −1.74936e7 −0.768032 −0.384016 0.923326i \(-0.625459\pi\)
−0.384016 + 0.923326i \(0.625459\pi\)
\(878\) 1.69269e7 0.741040
\(879\) −9.04421e6 −0.394819
\(880\) 2.29575e6 0.0999350
\(881\) −1.82811e7 −0.793527 −0.396763 0.917921i \(-0.629867\pi\)
−0.396763 + 0.917921i \(0.629867\pi\)
\(882\) −1.81665e6 −0.0786321
\(883\) 1.89568e7 0.818206 0.409103 0.912488i \(-0.365842\pi\)
0.409103 + 0.912488i \(0.365842\pi\)
\(884\) −1.72693e7 −0.743264
\(885\) −9.61140e6 −0.412504
\(886\) 1.68967e7 0.723133
\(887\) 4.61134e6 0.196797 0.0983985 0.995147i \(-0.468628\pi\)
0.0983985 + 0.995147i \(0.468628\pi\)
\(888\) 2.61033e6 0.111087
\(889\) −1.91799e7 −0.813938
\(890\) −5.03299e6 −0.212986
\(891\) 4.27376e6 0.180350
\(892\) −9.64782e6 −0.405992
\(893\) −4.97264e7 −2.08669
\(894\) 1.20403e7 0.503839
\(895\) −5.56746e6 −0.232327
\(896\) 1.94932e6 0.0811173
\(897\) −2.31035e6 −0.0958733
\(898\) 2.58079e7 1.06798
\(899\) 2.29439e7 0.946819
\(900\) −1.71289e6 −0.0704894
\(901\) 3.27553e7 1.34422
\(902\) 34535.1 0.00141333
\(903\) −1.07689e6 −0.0439491
\(904\) −1.14538e7 −0.466154
\(905\) −6.94116e6 −0.281715
\(906\) −7.63373e6 −0.308970
\(907\) −1.05722e7 −0.426726 −0.213363 0.976973i \(-0.568442\pi\)
−0.213363 + 0.976973i \(0.568442\pi\)
\(908\) 8.69780e6 0.350102
\(909\) −1.04624e7 −0.419974
\(910\) −6.13612e6 −0.245635
\(911\) 1.06672e7 0.425849 0.212925 0.977069i \(-0.431701\pi\)
0.212925 + 0.977069i \(0.431701\pi\)
\(912\) 6.01625e6 0.239518
\(913\) −1.07246e7 −0.425799
\(914\) −7.93699e6 −0.314261
\(915\) −3.90397e6 −0.154153
\(916\) 1.42518e7 0.561216
\(917\) 1.63503e7 0.642099
\(918\) −2.93683e7 −1.15020
\(919\) 6018.39 0.000235067 0 0.000117534 1.00000i \(-0.499963\pi\)
0.000117534 1.00000i \(0.499963\pi\)
\(920\) 846400. 0.0329690
\(921\) 295285. 0.0114708
\(922\) −2.31375e7 −0.896375
\(923\) −2.05743e6 −0.0794917
\(924\) 5.78256e6 0.222813
\(925\) 3.01025e6 0.115677
\(926\) 3.10491e7 1.18993
\(927\) −3.11667e7 −1.19122
\(928\) 2.40035e6 0.0914966
\(929\) −4.69368e7 −1.78432 −0.892162 0.451715i \(-0.850812\pi\)
−0.892162 + 0.451715i \(0.850812\pi\)
\(930\) −8.28866e6 −0.314251
\(931\) 7.35825e6 0.278228
\(932\) 1.69023e7 0.637392
\(933\) −1.92908e7 −0.725514
\(934\) −3.00414e7 −1.12682
\(935\) 1.87676e7 0.702067
\(936\) −5.65380e6 −0.210936
\(937\) −4.09932e7 −1.52533 −0.762664 0.646795i \(-0.776109\pi\)
−0.762664 + 0.646795i \(0.776109\pi\)
\(938\) 2.30758e7 0.856347
\(939\) 1.11491e7 0.412643
\(940\) 7.16727e6 0.264566
\(941\) 2.29251e7 0.843988 0.421994 0.906599i \(-0.361330\pi\)
0.421994 + 0.906599i \(0.361330\pi\)
\(942\) 9.48238e6 0.348169
\(943\) 12732.5 0.000466265 0
\(944\) 1.16223e7 0.424487
\(945\) −1.04352e7 −0.380119
\(946\) 1.53362e6 0.0557173
\(947\) −1.17141e6 −0.0424456 −0.0212228 0.999775i \(-0.506756\pi\)
−0.0212228 + 0.999775i \(0.506756\pi\)
\(948\) −5.21018e6 −0.188292
\(949\) −2.87781e6 −0.103728
\(950\) 6.93798e6 0.249416
\(951\) 9.74413e6 0.349375
\(952\) 1.59355e7 0.569869
\(953\) −1.77064e7 −0.631537 −0.315768 0.948836i \(-0.602262\pi\)
−0.315768 + 0.948836i \(0.602262\pi\)
\(954\) 1.07238e7 0.381485
\(955\) 1.35343e7 0.480206
\(956\) −1.28881e7 −0.456084
\(957\) 7.12052e6 0.251323
\(958\) 2.95351e7 1.03974
\(959\) −3.63208e7 −1.27529
\(960\) −867146. −0.0303679
\(961\) 6.71748e7 2.34638
\(962\) 9.93604e6 0.346159
\(963\) 1.70313e7 0.591808
\(964\) −1.35801e7 −0.470664
\(965\) −837199. −0.0289408
\(966\) 2.13192e6 0.0735070
\(967\) −2.36387e7 −0.812937 −0.406468 0.913665i \(-0.633240\pi\)
−0.406468 + 0.913665i \(0.633240\pi\)
\(968\) 2.07217e6 0.0710784
\(969\) 4.91823e7 1.68267
\(970\) 7.18675e6 0.245247
\(971\) 2.33077e7 0.793324 0.396662 0.917965i \(-0.370168\pi\)
0.396662 + 0.917965i \(0.370168\pi\)
\(972\) −1.52545e7 −0.517885
\(973\) 1.71241e7 0.579863
\(974\) 3.70779e7 1.25233
\(975\) 2.72963e6 0.0919584
\(976\) 4.72078e6 0.158631
\(977\) 2.37183e7 0.794964 0.397482 0.917610i \(-0.369884\pi\)
0.397482 + 0.917610i \(0.369884\pi\)
\(978\) −2.17563e7 −0.727341
\(979\) 1.80539e7 0.602024
\(980\) −1.06058e6 −0.0352757
\(981\) −2.70120e7 −0.896156
\(982\) 1.44529e7 0.478274
\(983\) −2.54913e7 −0.841413 −0.420706 0.907197i \(-0.638218\pi\)
−0.420706 + 0.907197i \(0.638218\pi\)
\(984\) −13044.5 −0.000429478 0
\(985\) −3.73323e6 −0.122601
\(986\) 1.96227e7 0.642785
\(987\) 1.80530e7 0.589871
\(988\) 2.29004e7 0.746365
\(989\) 565417. 0.0183814
\(990\) 6.14432e6 0.199244
\(991\) −3.42847e7 −1.10896 −0.554481 0.832197i \(-0.687083\pi\)
−0.554481 + 0.832197i \(0.687083\pi\)
\(992\) 1.00229e7 0.323379
\(993\) 3.06927e7 0.987785
\(994\) 1.89854e6 0.0609471
\(995\) 6.86791e6 0.219921
\(996\) 4.05089e6 0.129390
\(997\) −2.64620e6 −0.0843111 −0.0421555 0.999111i \(-0.513423\pi\)
−0.0421555 + 0.999111i \(0.513423\pi\)
\(998\) −2.56255e7 −0.814415
\(999\) 1.68974e7 0.535680
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 230.6.a.f.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.f.1.2 5 1.1 even 1 trivial