Properties

Label 99.6.a.g.1.2
Level $99$
Weight $6$
Character 99.1
Self dual yes
Analytic conductor $15.878$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,6,Mod(1,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, 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("99.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(15.8779981615\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.04796\) of defining polynomial
Character \(\chi\) \(=\) 99.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.20859 q^{2} -27.1221 q^{4} -75.2230 q^{5} -225.525 q^{7} +130.577 q^{8} +O(q^{10})\) \(q-2.20859 q^{2} -27.1221 q^{4} -75.2230 q^{5} -225.525 q^{7} +130.577 q^{8} +166.137 q^{10} -121.000 q^{11} +455.465 q^{13} +498.092 q^{14} +579.518 q^{16} -190.657 q^{17} -135.393 q^{19} +2040.21 q^{20} +267.240 q^{22} -2796.65 q^{23} +2533.51 q^{25} -1005.94 q^{26} +6116.71 q^{28} +2608.58 q^{29} -1056.76 q^{31} -5458.37 q^{32} +421.082 q^{34} +16964.7 q^{35} +12536.8 q^{37} +299.028 q^{38} -9822.37 q^{40} -1130.09 q^{41} -14671.0 q^{43} +3281.78 q^{44} +6176.65 q^{46} +16882.2 q^{47} +34054.4 q^{49} -5595.48 q^{50} -12353.2 q^{52} -3313.02 q^{53} +9101.99 q^{55} -29448.3 q^{56} -5761.29 q^{58} -11454.0 q^{59} -28227.5 q^{61} +2333.95 q^{62} -6489.25 q^{64} -34261.4 q^{65} -51431.0 q^{67} +5171.01 q^{68} -37468.0 q^{70} +16218.0 q^{71} -10168.8 q^{73} -27688.7 q^{74} +3672.15 q^{76} +27288.5 q^{77} +60841.2 q^{79} -43593.1 q^{80} +2495.90 q^{82} -45770.6 q^{83} +14341.8 q^{85} +32402.3 q^{86} -15799.8 q^{88} +82267.9 q^{89} -102719. q^{91} +75851.0 q^{92} -37285.8 q^{94} +10184.7 q^{95} +53097.0 q^{97} -75212.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8} - 414 q^{10} - 363 q^{11} + 486 q^{13} + 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} + 23712 q^{28} + 3426 q^{29} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 17724 q^{37} + 9240 q^{38} - 15276 q^{40} - 5994 q^{41} - 26208 q^{43} - 10164 q^{44} - 16806 q^{46} + 17232 q^{47} + 48531 q^{49} - 41070 q^{50} - 35304 q^{52} - 50586 q^{53} + 2904 q^{55} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} + 18486 q^{61} + 19974 q^{62} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} - 123372 q^{70} - 39282 q^{71} + 15426 q^{73} - 153294 q^{74} + 103920 q^{76} - 10164 q^{77} + 125148 q^{79} - 118680 q^{80} - 255372 q^{82} + 143928 q^{83} - 104040 q^{85} - 243060 q^{86} - 68244 q^{88} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} - 74928 q^{94} + 22200 q^{95} + 9684 q^{97} - 3480 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.20859 −0.390428 −0.195214 0.980761i \(-0.562540\pi\)
−0.195214 + 0.980761i \(0.562540\pi\)
\(3\) 0 0
\(4\) −27.1221 −0.847566
\(5\) −75.2230 −1.34563 −0.672815 0.739810i \(-0.734915\pi\)
−0.672815 + 0.739810i \(0.734915\pi\)
\(6\) 0 0
\(7\) −225.525 −1.73960 −0.869799 0.493406i \(-0.835752\pi\)
−0.869799 + 0.493406i \(0.835752\pi\)
\(8\) 130.577 0.721341
\(9\) 0 0
\(10\) 166.137 0.525371
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) 455.465 0.747474 0.373737 0.927535i \(-0.378076\pi\)
0.373737 + 0.927535i \(0.378076\pi\)
\(14\) 498.092 0.679187
\(15\) 0 0
\(16\) 579.518 0.565935
\(17\) −190.657 −0.160003 −0.0800017 0.996795i \(-0.525493\pi\)
−0.0800017 + 0.996795i \(0.525493\pi\)
\(18\) 0 0
\(19\) −135.393 −0.0860424 −0.0430212 0.999074i \(-0.513698\pi\)
−0.0430212 + 0.999074i \(0.513698\pi\)
\(20\) 2040.21 1.14051
\(21\) 0 0
\(22\) 267.240 0.117718
\(23\) −2796.65 −1.10235 −0.551173 0.834391i \(-0.685820\pi\)
−0.551173 + 0.834391i \(0.685820\pi\)
\(24\) 0 0
\(25\) 2533.51 0.810722
\(26\) −1005.94 −0.291835
\(27\) 0 0
\(28\) 6116.71 1.47443
\(29\) 2608.58 0.575983 0.287991 0.957633i \(-0.407013\pi\)
0.287991 + 0.957633i \(0.407013\pi\)
\(30\) 0 0
\(31\) −1056.76 −0.197502 −0.0987510 0.995112i \(-0.531485\pi\)
−0.0987510 + 0.995112i \(0.531485\pi\)
\(32\) −5458.37 −0.942297
\(33\) 0 0
\(34\) 421.082 0.0624697
\(35\) 16964.7 2.34086
\(36\) 0 0
\(37\) 12536.8 1.50550 0.752752 0.658304i \(-0.228726\pi\)
0.752752 + 0.658304i \(0.228726\pi\)
\(38\) 299.028 0.0335933
\(39\) 0 0
\(40\) −9822.37 −0.970658
\(41\) −1130.09 −0.104991 −0.0524954 0.998621i \(-0.516718\pi\)
−0.0524954 + 0.998621i \(0.516718\pi\)
\(42\) 0 0
\(43\) −14671.0 −1.21001 −0.605005 0.796222i \(-0.706829\pi\)
−0.605005 + 0.796222i \(0.706829\pi\)
\(44\) 3281.78 0.255551
\(45\) 0 0
\(46\) 6176.65 0.430386
\(47\) 16882.2 1.11477 0.557383 0.830256i \(-0.311806\pi\)
0.557383 + 0.830256i \(0.311806\pi\)
\(48\) 0 0
\(49\) 34054.4 2.02620
\(50\) −5595.48 −0.316528
\(51\) 0 0
\(52\) −12353.2 −0.633534
\(53\) −3313.02 −0.162007 −0.0810035 0.996714i \(-0.525813\pi\)
−0.0810035 + 0.996714i \(0.525813\pi\)
\(54\) 0 0
\(55\) 9101.99 0.405723
\(56\) −29448.3 −1.25484
\(57\) 0 0
\(58\) −5761.29 −0.224880
\(59\) −11454.0 −0.428378 −0.214189 0.976792i \(-0.568711\pi\)
−0.214189 + 0.976792i \(0.568711\pi\)
\(60\) 0 0
\(61\) −28227.5 −0.971286 −0.485643 0.874157i \(-0.661415\pi\)
−0.485643 + 0.874157i \(0.661415\pi\)
\(62\) 2333.95 0.0771102
\(63\) 0 0
\(64\) −6489.25 −0.198036
\(65\) −34261.4 −1.00582
\(66\) 0 0
\(67\) −51431.0 −1.39971 −0.699855 0.714285i \(-0.746752\pi\)
−0.699855 + 0.714285i \(0.746752\pi\)
\(68\) 5171.01 0.135614
\(69\) 0 0
\(70\) −37468.0 −0.913935
\(71\) 16218.0 0.381814 0.190907 0.981608i \(-0.438857\pi\)
0.190907 + 0.981608i \(0.438857\pi\)
\(72\) 0 0
\(73\) −10168.8 −0.223337 −0.111669 0.993745i \(-0.535620\pi\)
−0.111669 + 0.993745i \(0.535620\pi\)
\(74\) −27688.7 −0.587791
\(75\) 0 0
\(76\) 3672.15 0.0729266
\(77\) 27288.5 0.524509
\(78\) 0 0
\(79\) 60841.2 1.09681 0.548404 0.836214i \(-0.315236\pi\)
0.548404 + 0.836214i \(0.315236\pi\)
\(80\) −43593.1 −0.761540
\(81\) 0 0
\(82\) 2495.90 0.0409913
\(83\) −45770.6 −0.729275 −0.364638 0.931150i \(-0.618807\pi\)
−0.364638 + 0.931150i \(0.618807\pi\)
\(84\) 0 0
\(85\) 14341.8 0.215306
\(86\) 32402.3 0.472421
\(87\) 0 0
\(88\) −15799.8 −0.217492
\(89\) 82267.9 1.10092 0.550460 0.834862i \(-0.314452\pi\)
0.550460 + 0.834862i \(0.314452\pi\)
\(90\) 0 0
\(91\) −102719. −1.30031
\(92\) 75851.0 0.934312
\(93\) 0 0
\(94\) −37285.8 −0.435235
\(95\) 10184.7 0.115781
\(96\) 0 0
\(97\) 53097.0 0.572981 0.286491 0.958083i \(-0.407511\pi\)
0.286491 + 0.958083i \(0.407511\pi\)
\(98\) −75212.2 −0.791085
\(99\) 0 0
\(100\) −68714.1 −0.687141
\(101\) −186821. −1.82231 −0.911153 0.412069i \(-0.864806\pi\)
−0.911153 + 0.412069i \(0.864806\pi\)
\(102\) 0 0
\(103\) 34290.5 0.318479 0.159240 0.987240i \(-0.449096\pi\)
0.159240 + 0.987240i \(0.449096\pi\)
\(104\) 59473.0 0.539184
\(105\) 0 0
\(106\) 7317.10 0.0632520
\(107\) 224117. 1.89241 0.946206 0.323565i \(-0.104881\pi\)
0.946206 + 0.323565i \(0.104881\pi\)
\(108\) 0 0
\(109\) 162229. 1.30786 0.653931 0.756554i \(-0.273118\pi\)
0.653931 + 0.756554i \(0.273118\pi\)
\(110\) −20102.6 −0.158405
\(111\) 0 0
\(112\) −130696. −0.984500
\(113\) −92225.0 −0.679442 −0.339721 0.940526i \(-0.610333\pi\)
−0.339721 + 0.940526i \(0.610333\pi\)
\(114\) 0 0
\(115\) 210372. 1.48335
\(116\) −70750.3 −0.488184
\(117\) 0 0
\(118\) 25297.2 0.167251
\(119\) 42997.8 0.278342
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 62342.9 0.379217
\(123\) 0 0
\(124\) 28661.5 0.167396
\(125\) 44493.9 0.254698
\(126\) 0 0
\(127\) 138299. 0.760868 0.380434 0.924808i \(-0.375775\pi\)
0.380434 + 0.924808i \(0.375775\pi\)
\(128\) 189000. 1.01962
\(129\) 0 0
\(130\) 75669.5 0.392702
\(131\) 54420.4 0.277066 0.138533 0.990358i \(-0.455761\pi\)
0.138533 + 0.990358i \(0.455761\pi\)
\(132\) 0 0
\(133\) 30534.5 0.149679
\(134\) 113590. 0.546485
\(135\) 0 0
\(136\) −24895.3 −0.115417
\(137\) −40555.1 −0.184605 −0.0923025 0.995731i \(-0.529423\pi\)
−0.0923025 + 0.995731i \(0.529423\pi\)
\(138\) 0 0
\(139\) 140537. 0.616955 0.308477 0.951232i \(-0.400181\pi\)
0.308477 + 0.951232i \(0.400181\pi\)
\(140\) −460117. −1.98403
\(141\) 0 0
\(142\) −35818.9 −0.149071
\(143\) −55111.2 −0.225372
\(144\) 0 0
\(145\) −196225. −0.775060
\(146\) 22458.7 0.0871971
\(147\) 0 0
\(148\) −340024. −1.27602
\(149\) −176073. −0.649722 −0.324861 0.945762i \(-0.605317\pi\)
−0.324861 + 0.945762i \(0.605317\pi\)
\(150\) 0 0
\(151\) 409241. 1.46062 0.730309 0.683117i \(-0.239376\pi\)
0.730309 + 0.683117i \(0.239376\pi\)
\(152\) −17679.2 −0.0620659
\(153\) 0 0
\(154\) −60269.1 −0.204783
\(155\) 79492.6 0.265765
\(156\) 0 0
\(157\) 14294.5 0.0462829 0.0231414 0.999732i \(-0.492633\pi\)
0.0231414 + 0.999732i \(0.492633\pi\)
\(158\) −134373. −0.428224
\(159\) 0 0
\(160\) 410595. 1.26798
\(161\) 630713. 1.91764
\(162\) 0 0
\(163\) −418474. −1.23367 −0.616836 0.787091i \(-0.711586\pi\)
−0.616836 + 0.787091i \(0.711586\pi\)
\(164\) 30650.3 0.0889868
\(165\) 0 0
\(166\) 101089. 0.284729
\(167\) 139747. 0.387749 0.193875 0.981026i \(-0.437894\pi\)
0.193875 + 0.981026i \(0.437894\pi\)
\(168\) 0 0
\(169\) −163845. −0.441282
\(170\) −31675.1 −0.0840612
\(171\) 0 0
\(172\) 397909. 1.02556
\(173\) 687104. 1.74545 0.872725 0.488213i \(-0.162351\pi\)
0.872725 + 0.488213i \(0.162351\pi\)
\(174\) 0 0
\(175\) −571368. −1.41033
\(176\) −70121.6 −0.170636
\(177\) 0 0
\(178\) −181696. −0.429829
\(179\) −35496.4 −0.0828042 −0.0414021 0.999143i \(-0.513182\pi\)
−0.0414021 + 0.999143i \(0.513182\pi\)
\(180\) 0 0
\(181\) 260469. 0.590963 0.295481 0.955349i \(-0.404520\pi\)
0.295481 + 0.955349i \(0.404520\pi\)
\(182\) 226863. 0.507675
\(183\) 0 0
\(184\) −365177. −0.795167
\(185\) −943056. −2.02585
\(186\) 0 0
\(187\) 23069.4 0.0482429
\(188\) −457880. −0.944837
\(189\) 0 0
\(190\) −22493.8 −0.0452042
\(191\) −392051. −0.777605 −0.388803 0.921321i \(-0.627111\pi\)
−0.388803 + 0.921321i \(0.627111\pi\)
\(192\) 0 0
\(193\) 15776.8 0.0304878 0.0152439 0.999884i \(-0.495148\pi\)
0.0152439 + 0.999884i \(0.495148\pi\)
\(194\) −117270. −0.223708
\(195\) 0 0
\(196\) −923627. −1.71734
\(197\) −545551. −1.00154 −0.500771 0.865580i \(-0.666950\pi\)
−0.500771 + 0.865580i \(0.666950\pi\)
\(198\) 0 0
\(199\) −546514. −0.978293 −0.489146 0.872202i \(-0.662692\pi\)
−0.489146 + 0.872202i \(0.662692\pi\)
\(200\) 330817. 0.584807
\(201\) 0 0
\(202\) 412610. 0.711478
\(203\) −588300. −1.00198
\(204\) 0 0
\(205\) 85008.5 0.141279
\(206\) −75733.7 −0.124343
\(207\) 0 0
\(208\) 263950. 0.423022
\(209\) 16382.6 0.0259428
\(210\) 0 0
\(211\) 537150. 0.830596 0.415298 0.909686i \(-0.363677\pi\)
0.415298 + 0.909686i \(0.363677\pi\)
\(212\) 89856.0 0.137312
\(213\) 0 0
\(214\) −494983. −0.738850
\(215\) 1.10360e6 1.62823
\(216\) 0 0
\(217\) 238325. 0.343574
\(218\) −358298. −0.510626
\(219\) 0 0
\(220\) −246865. −0.343877
\(221\) −86837.3 −0.119598
\(222\) 0 0
\(223\) −189640. −0.255368 −0.127684 0.991815i \(-0.540754\pi\)
−0.127684 + 0.991815i \(0.540754\pi\)
\(224\) 1.23100e6 1.63922
\(225\) 0 0
\(226\) 203687. 0.265273
\(227\) −363428. −0.468116 −0.234058 0.972223i \(-0.575201\pi\)
−0.234058 + 0.972223i \(0.575201\pi\)
\(228\) 0 0
\(229\) 504331. 0.635516 0.317758 0.948172i \(-0.397070\pi\)
0.317758 + 0.948172i \(0.397070\pi\)
\(230\) −464627. −0.579141
\(231\) 0 0
\(232\) 340620. 0.415480
\(233\) 1.20159e6 1.45000 0.724999 0.688750i \(-0.241840\pi\)
0.724999 + 0.688750i \(0.241840\pi\)
\(234\) 0 0
\(235\) −1.26993e6 −1.50006
\(236\) 310657. 0.363079
\(237\) 0 0
\(238\) −94964.5 −0.108672
\(239\) 185929. 0.210549 0.105275 0.994443i \(-0.466428\pi\)
0.105275 + 0.994443i \(0.466428\pi\)
\(240\) 0 0
\(241\) 174842. 0.193911 0.0969556 0.995289i \(-0.469090\pi\)
0.0969556 + 0.995289i \(0.469090\pi\)
\(242\) −32336.0 −0.0354934
\(243\) 0 0
\(244\) 765589. 0.823229
\(245\) −2.56168e6 −2.72652
\(246\) 0 0
\(247\) −61666.8 −0.0643145
\(248\) −137988. −0.142466
\(249\) 0 0
\(250\) −98269.0 −0.0994412
\(251\) −447906. −0.448748 −0.224374 0.974503i \(-0.572034\pi\)
−0.224374 + 0.974503i \(0.572034\pi\)
\(252\) 0 0
\(253\) 338394. 0.332370
\(254\) −305446. −0.297064
\(255\) 0 0
\(256\) −209768. −0.200050
\(257\) 1.14572e6 1.08204 0.541022 0.841009i \(-0.318038\pi\)
0.541022 + 0.841009i \(0.318038\pi\)
\(258\) 0 0
\(259\) −2.82736e6 −2.61897
\(260\) 929243. 0.852503
\(261\) 0 0
\(262\) −120192. −0.108174
\(263\) −443228. −0.395128 −0.197564 0.980290i \(-0.563303\pi\)
−0.197564 + 0.980290i \(0.563303\pi\)
\(264\) 0 0
\(265\) 249215. 0.218002
\(266\) −67438.2 −0.0584389
\(267\) 0 0
\(268\) 1.39492e6 1.18635
\(269\) 1.88722e6 1.59016 0.795082 0.606502i \(-0.207428\pi\)
0.795082 + 0.606502i \(0.207428\pi\)
\(270\) 0 0
\(271\) 2.24203e6 1.85446 0.927230 0.374491i \(-0.122183\pi\)
0.927230 + 0.374491i \(0.122183\pi\)
\(272\) −110489. −0.0905516
\(273\) 0 0
\(274\) 89569.6 0.0720749
\(275\) −306554. −0.244442
\(276\) 0 0
\(277\) 1.27824e6 1.00095 0.500474 0.865751i \(-0.333159\pi\)
0.500474 + 0.865751i \(0.333159\pi\)
\(278\) −310389. −0.240876
\(279\) 0 0
\(280\) 2.21519e6 1.68856
\(281\) −549325. −0.415015 −0.207508 0.978233i \(-0.566535\pi\)
−0.207508 + 0.978233i \(0.566535\pi\)
\(282\) 0 0
\(283\) −135813. −0.100803 −0.0504016 0.998729i \(-0.516050\pi\)
−0.0504016 + 0.998729i \(0.516050\pi\)
\(284\) −439867. −0.323612
\(285\) 0 0
\(286\) 121718. 0.0879914
\(287\) 254862. 0.182642
\(288\) 0 0
\(289\) −1.38351e6 −0.974399
\(290\) 433382. 0.302605
\(291\) 0 0
\(292\) 275799. 0.189293
\(293\) 1.76403e6 1.20043 0.600215 0.799839i \(-0.295082\pi\)
0.600215 + 0.799839i \(0.295082\pi\)
\(294\) 0 0
\(295\) 861606. 0.576439
\(296\) 1.63701e6 1.08598
\(297\) 0 0
\(298\) 388874. 0.253669
\(299\) −1.27377e6 −0.823976
\(300\) 0 0
\(301\) 3.30868e6 2.10493
\(302\) −903846. −0.570266
\(303\) 0 0
\(304\) −78462.7 −0.0486944
\(305\) 2.12336e6 1.30699
\(306\) 0 0
\(307\) −1.93533e6 −1.17195 −0.585975 0.810329i \(-0.699288\pi\)
−0.585975 + 0.810329i \(0.699288\pi\)
\(308\) −740122. −0.444556
\(309\) 0 0
\(310\) −175567. −0.103762
\(311\) −2.98327e6 −1.74901 −0.874504 0.485019i \(-0.838813\pi\)
−0.874504 + 0.485019i \(0.838813\pi\)
\(312\) 0 0
\(313\) −10701.5 −0.00617426 −0.00308713 0.999995i \(-0.500983\pi\)
−0.00308713 + 0.999995i \(0.500983\pi\)
\(314\) −31570.8 −0.0180701
\(315\) 0 0
\(316\) −1.65014e6 −0.929617
\(317\) 2.43658e6 1.36186 0.680929 0.732349i \(-0.261576\pi\)
0.680929 + 0.732349i \(0.261576\pi\)
\(318\) 0 0
\(319\) −315638. −0.173665
\(320\) 488141. 0.266484
\(321\) 0 0
\(322\) −1.39299e6 −0.748699
\(323\) 25813.6 0.0137671
\(324\) 0 0
\(325\) 1.15392e6 0.605994
\(326\) 924239. 0.481660
\(327\) 0 0
\(328\) −147563. −0.0757342
\(329\) −3.80734e6 −1.93924
\(330\) 0 0
\(331\) 119576. 0.0599894 0.0299947 0.999550i \(-0.490451\pi\)
0.0299947 + 0.999550i \(0.490451\pi\)
\(332\) 1.24140e6 0.618109
\(333\) 0 0
\(334\) −308644. −0.151388
\(335\) 3.86880e6 1.88349
\(336\) 0 0
\(337\) −2.02195e6 −0.969830 −0.484915 0.874561i \(-0.661149\pi\)
−0.484915 + 0.874561i \(0.661149\pi\)
\(338\) 361867. 0.172289
\(339\) 0 0
\(340\) −388979. −0.182486
\(341\) 127868. 0.0595491
\(342\) 0 0
\(343\) −3.88971e6 −1.78518
\(344\) −1.91569e6 −0.872829
\(345\) 0 0
\(346\) −1.51753e6 −0.681471
\(347\) −3.01864e6 −1.34582 −0.672912 0.739723i \(-0.734957\pi\)
−0.672912 + 0.739723i \(0.734957\pi\)
\(348\) 0 0
\(349\) 2.40399e6 1.05650 0.528250 0.849089i \(-0.322848\pi\)
0.528250 + 0.849089i \(0.322848\pi\)
\(350\) 1.26192e6 0.550632
\(351\) 0 0
\(352\) 660463. 0.284113
\(353\) −3.62981e6 −1.55041 −0.775206 0.631709i \(-0.782354\pi\)
−0.775206 + 0.631709i \(0.782354\pi\)
\(354\) 0 0
\(355\) −1.21997e6 −0.513780
\(356\) −2.23128e6 −0.933102
\(357\) 0 0
\(358\) 78397.1 0.0323290
\(359\) −939181. −0.384603 −0.192302 0.981336i \(-0.561595\pi\)
−0.192302 + 0.981336i \(0.561595\pi\)
\(360\) 0 0
\(361\) −2.45777e6 −0.992597
\(362\) −575270. −0.230728
\(363\) 0 0
\(364\) 2.78594e6 1.10210
\(365\) 764926. 0.300530
\(366\) 0 0
\(367\) −2.26697e6 −0.878577 −0.439288 0.898346i \(-0.644769\pi\)
−0.439288 + 0.898346i \(0.644769\pi\)
\(368\) −1.62071e6 −0.623857
\(369\) 0 0
\(370\) 2.08283e6 0.790949
\(371\) 747167. 0.281827
\(372\) 0 0
\(373\) −4.55029e6 −1.69343 −0.846714 0.532048i \(-0.821423\pi\)
−0.846714 + 0.532048i \(0.821423\pi\)
\(374\) −50951.0 −0.0188353
\(375\) 0 0
\(376\) 2.20442e6 0.804125
\(377\) 1.18812e6 0.430532
\(378\) 0 0
\(379\) 618788. 0.221281 0.110641 0.993860i \(-0.464710\pi\)
0.110641 + 0.993860i \(0.464710\pi\)
\(380\) −276230. −0.0981323
\(381\) 0 0
\(382\) 865881. 0.303599
\(383\) −2.23829e6 −0.779686 −0.389843 0.920881i \(-0.627471\pi\)
−0.389843 + 0.920881i \(0.627471\pi\)
\(384\) 0 0
\(385\) −2.05272e6 −0.705795
\(386\) −34844.5 −0.0119033
\(387\) 0 0
\(388\) −1.44010e6 −0.485640
\(389\) 4.60206e6 1.54198 0.770989 0.636848i \(-0.219762\pi\)
0.770989 + 0.636848i \(0.219762\pi\)
\(390\) 0 0
\(391\) 533199. 0.176379
\(392\) 4.44671e6 1.46158
\(393\) 0 0
\(394\) 1.20490e6 0.391030
\(395\) −4.57666e6 −1.47590
\(396\) 0 0
\(397\) −4.35532e6 −1.38690 −0.693448 0.720506i \(-0.743909\pi\)
−0.693448 + 0.720506i \(0.743909\pi\)
\(398\) 1.20703e6 0.381952
\(399\) 0 0
\(400\) 1.46821e6 0.458816
\(401\) −3.62515e6 −1.12581 −0.562905 0.826522i \(-0.690316\pi\)
−0.562905 + 0.826522i \(0.690316\pi\)
\(402\) 0 0
\(403\) −481316. −0.147628
\(404\) 5.06697e6 1.54453
\(405\) 0 0
\(406\) 1.29931e6 0.391200
\(407\) −1.51695e6 −0.453927
\(408\) 0 0
\(409\) 4.13585e6 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(410\) −187749. −0.0551592
\(411\) 0 0
\(412\) −930032. −0.269932
\(413\) 2.58316e6 0.745206
\(414\) 0 0
\(415\) 3.44300e6 0.981335
\(416\) −2.48609e6 −0.704343
\(417\) 0 0
\(418\) −36182.4 −0.0101288
\(419\) 2.46691e6 0.686464 0.343232 0.939251i \(-0.388478\pi\)
0.343232 + 0.939251i \(0.388478\pi\)
\(420\) 0 0
\(421\) 3.45258e6 0.949376 0.474688 0.880154i \(-0.342561\pi\)
0.474688 + 0.880154i \(0.342561\pi\)
\(422\) −1.18635e6 −0.324287
\(423\) 0 0
\(424\) −432602. −0.116862
\(425\) −483029. −0.129718
\(426\) 0 0
\(427\) 6.36599e6 1.68965
\(428\) −6.07853e6 −1.60394
\(429\) 0 0
\(430\) −2.43740e6 −0.635704
\(431\) 3.65893e6 0.948770 0.474385 0.880318i \(-0.342671\pi\)
0.474385 + 0.880318i \(0.342671\pi\)
\(432\) 0 0
\(433\) −1.59716e6 −0.409381 −0.204690 0.978827i \(-0.565619\pi\)
−0.204690 + 0.978827i \(0.565619\pi\)
\(434\) −526363. −0.134141
\(435\) 0 0
\(436\) −4.39999e6 −1.10850
\(437\) 378647. 0.0948485
\(438\) 0 0
\(439\) −1.58464e6 −0.392437 −0.196219 0.980560i \(-0.562866\pi\)
−0.196219 + 0.980560i \(0.562866\pi\)
\(440\) 1.18851e6 0.292664
\(441\) 0 0
\(442\) 191788. 0.0466945
\(443\) −2.29633e6 −0.555936 −0.277968 0.960590i \(-0.589661\pi\)
−0.277968 + 0.960590i \(0.589661\pi\)
\(444\) 0 0
\(445\) −6.18844e6 −1.48143
\(446\) 418837. 0.0997029
\(447\) 0 0
\(448\) 1.46349e6 0.344504
\(449\) −3.31569e6 −0.776171 −0.388086 0.921623i \(-0.626864\pi\)
−0.388086 + 0.921623i \(0.626864\pi\)
\(450\) 0 0
\(451\) 136740. 0.0316559
\(452\) 2.50134e6 0.575872
\(453\) 0 0
\(454\) 802664. 0.182765
\(455\) 7.72680e6 1.74973
\(456\) 0 0
\(457\) 2.20892e6 0.494754 0.247377 0.968919i \(-0.420431\pi\)
0.247377 + 0.968919i \(0.420431\pi\)
\(458\) −1.11386e6 −0.248123
\(459\) 0 0
\(460\) −5.70574e6 −1.25724
\(461\) 1.86064e6 0.407764 0.203882 0.978995i \(-0.434644\pi\)
0.203882 + 0.978995i \(0.434644\pi\)
\(462\) 0 0
\(463\) 1.20592e6 0.261437 0.130718 0.991420i \(-0.458272\pi\)
0.130718 + 0.991420i \(0.458272\pi\)
\(464\) 1.51172e6 0.325969
\(465\) 0 0
\(466\) −2.65383e6 −0.566119
\(467\) −2.29388e6 −0.486719 −0.243360 0.969936i \(-0.578250\pi\)
−0.243360 + 0.969936i \(0.578250\pi\)
\(468\) 0 0
\(469\) 1.15990e7 2.43493
\(470\) 2.80475e6 0.585666
\(471\) 0 0
\(472\) −1.49563e6 −0.309007
\(473\) 1.77519e6 0.364832
\(474\) 0 0
\(475\) −343019. −0.0697565
\(476\) −1.16619e6 −0.235913
\(477\) 0 0
\(478\) −410642. −0.0822041
\(479\) 7.90892e6 1.57499 0.787496 0.616320i \(-0.211377\pi\)
0.787496 + 0.616320i \(0.211377\pi\)
\(480\) 0 0
\(481\) 5.71007e6 1.12533
\(482\) −386154. −0.0757083
\(483\) 0 0
\(484\) −397095. −0.0770515
\(485\) −3.99412e6 −0.771021
\(486\) 0 0
\(487\) 3.48410e6 0.665684 0.332842 0.942983i \(-0.391992\pi\)
0.332842 + 0.942983i \(0.391992\pi\)
\(488\) −3.68585e6 −0.700628
\(489\) 0 0
\(490\) 5.65769e6 1.06451
\(491\) 8.98096e6 1.68120 0.840599 0.541658i \(-0.182203\pi\)
0.840599 + 0.541658i \(0.182203\pi\)
\(492\) 0 0
\(493\) −497343. −0.0921592
\(494\) 136197. 0.0251101
\(495\) 0 0
\(496\) −612410. −0.111773
\(497\) −3.65756e6 −0.664203
\(498\) 0 0
\(499\) 6.67736e6 1.20048 0.600238 0.799821i \(-0.295072\pi\)
0.600238 + 0.799821i \(0.295072\pi\)
\(500\) −1.20677e6 −0.215874
\(501\) 0 0
\(502\) 989242. 0.175204
\(503\) −7.58428e6 −1.33658 −0.668289 0.743902i \(-0.732973\pi\)
−0.668289 + 0.743902i \(0.732973\pi\)
\(504\) 0 0
\(505\) 1.40532e7 2.45215
\(506\) −747375. −0.129766
\(507\) 0 0
\(508\) −3.75096e6 −0.644886
\(509\) 6.02580e6 1.03091 0.515454 0.856917i \(-0.327623\pi\)
0.515454 + 0.856917i \(0.327623\pi\)
\(510\) 0 0
\(511\) 2.29331e6 0.388518
\(512\) −5.58471e6 −0.941511
\(513\) 0 0
\(514\) −2.53042e6 −0.422460
\(515\) −2.57944e6 −0.428555
\(516\) 0 0
\(517\) −2.04274e6 −0.336114
\(518\) 6.24448e6 1.02252
\(519\) 0 0
\(520\) −4.47374e6 −0.725542
\(521\) 4.58541e6 0.740088 0.370044 0.929014i \(-0.379343\pi\)
0.370044 + 0.929014i \(0.379343\pi\)
\(522\) 0 0
\(523\) −4.88145e6 −0.780359 −0.390179 0.920739i \(-0.627587\pi\)
−0.390179 + 0.920739i \(0.627587\pi\)
\(524\) −1.47600e6 −0.234832
\(525\) 0 0
\(526\) 978910. 0.154269
\(527\) 201478. 0.0316010
\(528\) 0 0
\(529\) 1.38489e6 0.215168
\(530\) −550414. −0.0851138
\(531\) 0 0
\(532\) −828160. −0.126863
\(533\) −514714. −0.0784780
\(534\) 0 0
\(535\) −1.68588e7 −2.54649
\(536\) −6.71568e6 −1.00967
\(537\) 0 0
\(538\) −4.16810e6 −0.620844
\(539\) −4.12058e6 −0.610923
\(540\) 0 0
\(541\) 6.21940e6 0.913598 0.456799 0.889570i \(-0.348996\pi\)
0.456799 + 0.889570i \(0.348996\pi\)
\(542\) −4.95172e6 −0.724033
\(543\) 0 0
\(544\) 1.04067e6 0.150771
\(545\) −1.22034e7 −1.75990
\(546\) 0 0
\(547\) −9.49047e6 −1.35619 −0.678093 0.734976i \(-0.737193\pi\)
−0.678093 + 0.734976i \(0.737193\pi\)
\(548\) 1.09994e6 0.156465
\(549\) 0 0
\(550\) 677053. 0.0954368
\(551\) −353184. −0.0495589
\(552\) 0 0
\(553\) −1.37212e7 −1.90800
\(554\) −2.82310e6 −0.390798
\(555\) 0 0
\(556\) −3.81166e6 −0.522910
\(557\) 2.92907e6 0.400029 0.200014 0.979793i \(-0.435901\pi\)
0.200014 + 0.979793i \(0.435901\pi\)
\(558\) 0 0
\(559\) −6.68213e6 −0.904451
\(560\) 9.83131e6 1.32477
\(561\) 0 0
\(562\) 1.21324e6 0.162033
\(563\) 455079. 0.0605084 0.0302542 0.999542i \(-0.490368\pi\)
0.0302542 + 0.999542i \(0.490368\pi\)
\(564\) 0 0
\(565\) 6.93744e6 0.914278
\(566\) 299955. 0.0393563
\(567\) 0 0
\(568\) 2.11769e6 0.275418
\(569\) 6.27664e6 0.812730 0.406365 0.913711i \(-0.366796\pi\)
0.406365 + 0.913711i \(0.366796\pi\)
\(570\) 0 0
\(571\) −621794. −0.0798098 −0.0399049 0.999203i \(-0.512706\pi\)
−0.0399049 + 0.999203i \(0.512706\pi\)
\(572\) 1.49473e6 0.191018
\(573\) 0 0
\(574\) −562886. −0.0713085
\(575\) −7.08532e6 −0.893697
\(576\) 0 0
\(577\) 1.28776e7 1.61026 0.805130 0.593098i \(-0.202095\pi\)
0.805130 + 0.593098i \(0.202095\pi\)
\(578\) 3.05560e6 0.380432
\(579\) 0 0
\(580\) 5.32205e6 0.656915
\(581\) 1.03224e7 1.26865
\(582\) 0 0
\(583\) 400875. 0.0488470
\(584\) −1.32780e6 −0.161102
\(585\) 0 0
\(586\) −3.89602e6 −0.468681
\(587\) −1.08775e7 −1.30296 −0.651482 0.758664i \(-0.725852\pi\)
−0.651482 + 0.758664i \(0.725852\pi\)
\(588\) 0 0
\(589\) 143078. 0.0169935
\(590\) −1.90293e6 −0.225058
\(591\) 0 0
\(592\) 7.26529e6 0.852018
\(593\) 7.50449e6 0.876364 0.438182 0.898886i \(-0.355622\pi\)
0.438182 + 0.898886i \(0.355622\pi\)
\(594\) 0 0
\(595\) −3.23442e6 −0.374545
\(596\) 4.77548e6 0.550682
\(597\) 0 0
\(598\) 2.81325e6 0.321703
\(599\) 7.69438e6 0.876207 0.438104 0.898925i \(-0.355650\pi\)
0.438104 + 0.898925i \(0.355650\pi\)
\(600\) 0 0
\(601\) 3.14770e6 0.355473 0.177737 0.984078i \(-0.443123\pi\)
0.177737 + 0.984078i \(0.443123\pi\)
\(602\) −7.30751e6 −0.821823
\(603\) 0 0
\(604\) −1.10995e7 −1.23797
\(605\) −1.10134e6 −0.122330
\(606\) 0 0
\(607\) −4.57397e6 −0.503874 −0.251937 0.967744i \(-0.581068\pi\)
−0.251937 + 0.967744i \(0.581068\pi\)
\(608\) 739025. 0.0810775
\(609\) 0 0
\(610\) −4.68962e6 −0.510286
\(611\) 7.68923e6 0.833258
\(612\) 0 0
\(613\) −1.56075e7 −1.67758 −0.838790 0.544455i \(-0.816736\pi\)
−0.838790 + 0.544455i \(0.816736\pi\)
\(614\) 4.27436e6 0.457562
\(615\) 0 0
\(616\) 3.56324e6 0.378349
\(617\) 1.18602e7 1.25424 0.627119 0.778924i \(-0.284234\pi\)
0.627119 + 0.778924i \(0.284234\pi\)
\(618\) 0 0
\(619\) 7.91821e6 0.830616 0.415308 0.909681i \(-0.363674\pi\)
0.415308 + 0.909681i \(0.363674\pi\)
\(620\) −2.15601e6 −0.225253
\(621\) 0 0
\(622\) 6.58883e6 0.682861
\(623\) −1.85534e7 −1.91516
\(624\) 0 0
\(625\) −1.12642e7 −1.15345
\(626\) 23635.3 0.00241060
\(627\) 0 0
\(628\) −387698. −0.0392278
\(629\) −2.39022e6 −0.240886
\(630\) 0 0
\(631\) −1.11561e7 −1.11542 −0.557709 0.830037i \(-0.688319\pi\)
−0.557709 + 0.830037i \(0.688319\pi\)
\(632\) 7.94444e6 0.791172
\(633\) 0 0
\(634\) −5.38140e6 −0.531707
\(635\) −1.04033e7 −1.02385
\(636\) 0 0
\(637\) 1.55106e7 1.51453
\(638\) 697116. 0.0678037
\(639\) 0 0
\(640\) −1.42172e7 −1.37203
\(641\) −7.17389e6 −0.689620 −0.344810 0.938672i \(-0.612057\pi\)
−0.344810 + 0.938672i \(0.612057\pi\)
\(642\) 0 0
\(643\) 7.14025e6 0.681061 0.340531 0.940233i \(-0.389393\pi\)
0.340531 + 0.940233i \(0.389393\pi\)
\(644\) −1.71063e7 −1.62533
\(645\) 0 0
\(646\) −57011.6 −0.00537505
\(647\) −1.56897e7 −1.47351 −0.736756 0.676159i \(-0.763643\pi\)
−0.736756 + 0.676159i \(0.763643\pi\)
\(648\) 0 0
\(649\) 1.38594e6 0.129161
\(650\) −2.54854e6 −0.236597
\(651\) 0 0
\(652\) 1.13499e7 1.04562
\(653\) 5.04236e6 0.462755 0.231378 0.972864i \(-0.425677\pi\)
0.231378 + 0.972864i \(0.425677\pi\)
\(654\) 0 0
\(655\) −4.09367e6 −0.372829
\(656\) −654904. −0.0594180
\(657\) 0 0
\(658\) 8.40887e6 0.757134
\(659\) 9.10902e6 0.817068 0.408534 0.912743i \(-0.366040\pi\)
0.408534 + 0.912743i \(0.366040\pi\)
\(660\) 0 0
\(661\) 1.31308e7 1.16893 0.584464 0.811420i \(-0.301305\pi\)
0.584464 + 0.811420i \(0.301305\pi\)
\(662\) −264095. −0.0234215
\(663\) 0 0
\(664\) −5.97657e6 −0.526056
\(665\) −2.29690e6 −0.201413
\(666\) 0 0
\(667\) −7.29528e6 −0.634933
\(668\) −3.79023e6 −0.328643
\(669\) 0 0
\(670\) −8.54459e6 −0.735367
\(671\) 3.41552e6 0.292854
\(672\) 0 0
\(673\) 1.55171e7 1.32061 0.660303 0.750999i \(-0.270428\pi\)
0.660303 + 0.750999i \(0.270428\pi\)
\(674\) 4.46566e6 0.378648
\(675\) 0 0
\(676\) 4.44382e6 0.374016
\(677\) 1.40356e7 1.17695 0.588476 0.808515i \(-0.299728\pi\)
0.588476 + 0.808515i \(0.299728\pi\)
\(678\) 0 0
\(679\) −1.19747e7 −0.996758
\(680\) 1.87270e6 0.155309
\(681\) 0 0
\(682\) −282408. −0.0232496
\(683\) −5.34969e6 −0.438810 −0.219405 0.975634i \(-0.570412\pi\)
−0.219405 + 0.975634i \(0.570412\pi\)
\(684\) 0 0
\(685\) 3.05068e6 0.248410
\(686\) 8.59079e6 0.696984
\(687\) 0 0
\(688\) −8.50211e6 −0.684787
\(689\) −1.50896e6 −0.121096
\(690\) 0 0
\(691\) 1.31390e7 1.04681 0.523404 0.852084i \(-0.324662\pi\)
0.523404 + 0.852084i \(0.324662\pi\)
\(692\) −1.86357e7 −1.47938
\(693\) 0 0
\(694\) 6.66695e6 0.525447
\(695\) −1.05716e7 −0.830194
\(696\) 0 0
\(697\) 215458. 0.0167989
\(698\) −5.30944e6 −0.412487
\(699\) 0 0
\(700\) 1.54967e7 1.19535
\(701\) 2.49888e7 1.92066 0.960330 0.278865i \(-0.0899581\pi\)
0.960330 + 0.278865i \(0.0899581\pi\)
\(702\) 0 0
\(703\) −1.69740e6 −0.129537
\(704\) 785200. 0.0597102
\(705\) 0 0
\(706\) 8.01676e6 0.605323
\(707\) 4.21327e7 3.17008
\(708\) 0 0
\(709\) −8.86200e6 −0.662089 −0.331044 0.943615i \(-0.607401\pi\)
−0.331044 + 0.943615i \(0.607401\pi\)
\(710\) 2.69441e6 0.200594
\(711\) 0 0
\(712\) 1.07423e7 0.794138
\(713\) 2.95538e6 0.217716
\(714\) 0 0
\(715\) 4.14563e6 0.303268
\(716\) 962739. 0.0701820
\(717\) 0 0
\(718\) 2.07427e6 0.150160
\(719\) −2.58635e7 −1.86580 −0.932901 0.360132i \(-0.882732\pi\)
−0.932901 + 0.360132i \(0.882732\pi\)
\(720\) 0 0
\(721\) −7.73336e6 −0.554026
\(722\) 5.42820e6 0.387537
\(723\) 0 0
\(724\) −7.06448e6 −0.500880
\(725\) 6.60886e6 0.466962
\(726\) 0 0
\(727\) −1.71871e7 −1.20605 −0.603026 0.797721i \(-0.706039\pi\)
−0.603026 + 0.797721i \(0.706039\pi\)
\(728\) −1.34126e7 −0.937963
\(729\) 0 0
\(730\) −1.68941e6 −0.117335
\(731\) 2.79712e6 0.193606
\(732\) 0 0
\(733\) 1.85650e7 1.27625 0.638125 0.769932i \(-0.279710\pi\)
0.638125 + 0.769932i \(0.279710\pi\)
\(734\) 5.00680e6 0.343021
\(735\) 0 0
\(736\) 1.52651e7 1.03874
\(737\) 6.22315e6 0.422028
\(738\) 0 0
\(739\) 5.94724e6 0.400594 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(740\) 2.55777e7 1.71705
\(741\) 0 0
\(742\) −1.65019e6 −0.110033
\(743\) 2.72654e7 1.81193 0.905963 0.423357i \(-0.139148\pi\)
0.905963 + 0.423357i \(0.139148\pi\)
\(744\) 0 0
\(745\) 1.32448e7 0.874285
\(746\) 1.00497e7 0.661161
\(747\) 0 0
\(748\) −625692. −0.0408890
\(749\) −5.05440e7 −3.29204
\(750\) 0 0
\(751\) 1.30069e7 0.841541 0.420770 0.907167i \(-0.361760\pi\)
0.420770 + 0.907167i \(0.361760\pi\)
\(752\) 9.78351e6 0.630885
\(753\) 0 0
\(754\) −2.62407e6 −0.168092
\(755\) −3.07844e7 −1.96545
\(756\) 0 0
\(757\) −9.22009e6 −0.584784 −0.292392 0.956299i \(-0.594451\pi\)
−0.292392 + 0.956299i \(0.594451\pi\)
\(758\) −1.36665e6 −0.0863942
\(759\) 0 0
\(760\) 1.32988e6 0.0835177
\(761\) −328083. −0.0205363 −0.0102682 0.999947i \(-0.503269\pi\)
−0.0102682 + 0.999947i \(0.503269\pi\)
\(762\) 0 0
\(763\) −3.65866e7 −2.27516
\(764\) 1.06333e7 0.659072
\(765\) 0 0
\(766\) 4.94347e6 0.304411
\(767\) −5.21690e6 −0.320202
\(768\) 0 0
\(769\) 2.19214e6 0.133676 0.0668380 0.997764i \(-0.478709\pi\)
0.0668380 + 0.997764i \(0.478709\pi\)
\(770\) 4.53363e6 0.275562
\(771\) 0 0
\(772\) −427900. −0.0258404
\(773\) −2.18539e7 −1.31547 −0.657735 0.753249i \(-0.728485\pi\)
−0.657735 + 0.753249i \(0.728485\pi\)
\(774\) 0 0
\(775\) −2.67730e6 −0.160119
\(776\) 6.93323e6 0.413315
\(777\) 0 0
\(778\) −1.01641e7 −0.602031
\(779\) 153006. 0.00903367
\(780\) 0 0
\(781\) −1.96238e6 −0.115121
\(782\) −1.17762e6 −0.0688633
\(783\) 0 0
\(784\) 1.97351e7 1.14670
\(785\) −1.07528e6 −0.0622797
\(786\) 0 0
\(787\) 2.61010e7 1.50217 0.751087 0.660203i \(-0.229530\pi\)
0.751087 + 0.660203i \(0.229530\pi\)
\(788\) 1.47965e7 0.848874
\(789\) 0 0
\(790\) 1.01080e7 0.576231
\(791\) 2.07990e7 1.18196
\(792\) 0 0
\(793\) −1.28566e7 −0.726011
\(794\) 9.61913e6 0.541483
\(795\) 0 0
\(796\) 1.48226e7 0.829168
\(797\) −1.39846e7 −0.779840 −0.389920 0.920849i \(-0.627497\pi\)
−0.389920 + 0.920849i \(0.627497\pi\)
\(798\) 0 0
\(799\) −3.21869e6 −0.178366
\(800\) −1.38288e7 −0.763941
\(801\) 0 0
\(802\) 8.00647e6 0.439547
\(803\) 1.23042e6 0.0673388
\(804\) 0 0
\(805\) −4.74442e7 −2.58044
\(806\) 1.06303e6 0.0576379
\(807\) 0 0
\(808\) −2.43944e7 −1.31450
\(809\) −2.70989e7 −1.45573 −0.727865 0.685721i \(-0.759487\pi\)
−0.727865 + 0.685721i \(0.759487\pi\)
\(810\) 0 0
\(811\) 1.99644e7 1.06587 0.532936 0.846156i \(-0.321089\pi\)
0.532936 + 0.846156i \(0.321089\pi\)
\(812\) 1.59559e7 0.849244
\(813\) 0 0
\(814\) 3.35033e6 0.177226
\(815\) 3.14789e7 1.66007
\(816\) 0 0
\(817\) 1.98635e6 0.104112
\(818\) −9.13439e6 −0.477306
\(819\) 0 0
\(820\) −2.30561e6 −0.119743
\(821\) −3.18829e7 −1.65082 −0.825410 0.564533i \(-0.809056\pi\)
−0.825410 + 0.564533i \(0.809056\pi\)
\(822\) 0 0
\(823\) −1.34203e7 −0.690655 −0.345328 0.938482i \(-0.612232\pi\)
−0.345328 + 0.938482i \(0.612232\pi\)
\(824\) 4.47754e6 0.229732
\(825\) 0 0
\(826\) −5.70515e6 −0.290949
\(827\) −1.19386e7 −0.607002 −0.303501 0.952831i \(-0.598156\pi\)
−0.303501 + 0.952831i \(0.598156\pi\)
\(828\) 0 0
\(829\) 2.59274e7 1.31031 0.655153 0.755497i \(-0.272604\pi\)
0.655153 + 0.755497i \(0.272604\pi\)
\(830\) −7.60419e6 −0.383140
\(831\) 0 0
\(832\) −2.95563e6 −0.148027
\(833\) −6.49269e6 −0.324199
\(834\) 0 0
\(835\) −1.05122e7 −0.521768
\(836\) −444330. −0.0219882
\(837\) 0 0
\(838\) −5.44839e6 −0.268015
\(839\) 3.21482e7 1.57671 0.788354 0.615222i \(-0.210933\pi\)
0.788354 + 0.615222i \(0.210933\pi\)
\(840\) 0 0
\(841\) −1.37064e7 −0.668244
\(842\) −7.62533e6 −0.370662
\(843\) 0 0
\(844\) −1.45687e7 −0.703985
\(845\) 1.23249e7 0.593803
\(846\) 0 0
\(847\) −3.30191e6 −0.158145
\(848\) −1.91995e6 −0.0916855
\(849\) 0 0
\(850\) 1.06681e6 0.0506456
\(851\) −3.50610e7 −1.65959
\(852\) 0 0
\(853\) −5.64308e6 −0.265548 −0.132774 0.991146i \(-0.542388\pi\)
−0.132774 + 0.991146i \(0.542388\pi\)
\(854\) −1.40599e7 −0.659685
\(855\) 0 0
\(856\) 2.92645e7 1.36507
\(857\) 1.77067e7 0.823543 0.411772 0.911287i \(-0.364910\pi\)
0.411772 + 0.911287i \(0.364910\pi\)
\(858\) 0 0
\(859\) 1.57119e7 0.726515 0.363258 0.931689i \(-0.381664\pi\)
0.363258 + 0.931689i \(0.381664\pi\)
\(860\) −2.99319e7 −1.38003
\(861\) 0 0
\(862\) −8.08108e6 −0.370426
\(863\) −245263. −0.0112100 −0.00560500 0.999984i \(-0.501784\pi\)
−0.00560500 + 0.999984i \(0.501784\pi\)
\(864\) 0 0
\(865\) −5.16861e7 −2.34873
\(866\) 3.52747e6 0.159834
\(867\) 0 0
\(868\) −6.46388e6 −0.291202
\(869\) −7.36179e6 −0.330700
\(870\) 0 0
\(871\) −2.34250e7 −1.04625
\(872\) 2.11833e7 0.943415
\(873\) 0 0
\(874\) −836276. −0.0370315
\(875\) −1.00345e7 −0.443073
\(876\) 0 0
\(877\) 1.04352e7 0.458142 0.229071 0.973410i \(-0.426431\pi\)
0.229071 + 0.973410i \(0.426431\pi\)
\(878\) 3.49983e6 0.153218
\(879\) 0 0
\(880\) 5.27476e6 0.229613
\(881\) 1.10430e7 0.479344 0.239672 0.970854i \(-0.422960\pi\)
0.239672 + 0.970854i \(0.422960\pi\)
\(882\) 0 0
\(883\) 5.41498e6 0.233720 0.116860 0.993148i \(-0.462717\pi\)
0.116860 + 0.993148i \(0.462717\pi\)
\(884\) 2.35521e6 0.101368
\(885\) 0 0
\(886\) 5.07166e6 0.217053
\(887\) 1.52663e7 0.651515 0.325757 0.945453i \(-0.394381\pi\)
0.325757 + 0.945453i \(0.394381\pi\)
\(888\) 0 0
\(889\) −3.11898e7 −1.32360
\(890\) 1.36677e7 0.578391
\(891\) 0 0
\(892\) 5.14343e6 0.216442
\(893\) −2.28573e6 −0.0959170
\(894\) 0 0
\(895\) 2.67015e6 0.111424
\(896\) −4.26242e7 −1.77372
\(897\) 0 0
\(898\) 7.32300e6 0.303039
\(899\) −2.75664e6 −0.113758
\(900\) 0 0
\(901\) 631648. 0.0259217
\(902\) −302004. −0.0123594
\(903\) 0 0
\(904\) −1.20424e7 −0.490109
\(905\) −1.95933e7 −0.795218
\(906\) 0 0
\(907\) 2.02437e7 0.817094 0.408547 0.912737i \(-0.366036\pi\)
0.408547 + 0.912737i \(0.366036\pi\)
\(908\) 9.85694e6 0.396759
\(909\) 0 0
\(910\) −1.70653e7 −0.683143
\(911\) 1.17158e7 0.467708 0.233854 0.972272i \(-0.424866\pi\)
0.233854 + 0.972272i \(0.424866\pi\)
\(912\) 0 0
\(913\) 5.53824e6 0.219885
\(914\) −4.87860e6 −0.193166
\(915\) 0 0
\(916\) −1.36785e7 −0.538642
\(917\) −1.22731e7 −0.481984
\(918\) 0 0
\(919\) −1.95296e7 −0.762788 −0.381394 0.924413i \(-0.624556\pi\)
−0.381394 + 0.924413i \(0.624556\pi\)
\(920\) 2.74697e7 1.07000
\(921\) 0 0
\(922\) −4.10939e6 −0.159202
\(923\) 7.38673e6 0.285396
\(924\) 0 0
\(925\) 3.17620e7 1.22055
\(926\) −2.66339e6 −0.102072
\(927\) 0 0
\(928\) −1.42386e7 −0.542747
\(929\) 4.29425e7 1.63248 0.816240 0.577713i \(-0.196055\pi\)
0.816240 + 0.577713i \(0.196055\pi\)
\(930\) 0 0
\(931\) −4.61073e6 −0.174339
\(932\) −3.25898e7 −1.22897
\(933\) 0 0
\(934\) 5.06625e6 0.190029
\(935\) −1.73535e6 −0.0649171
\(936\) 0 0
\(937\) 2.27191e7 0.845361 0.422681 0.906279i \(-0.361089\pi\)
0.422681 + 0.906279i \(0.361089\pi\)
\(938\) −2.56174e7 −0.950665
\(939\) 0 0
\(940\) 3.44431e7 1.27140
\(941\) −3.98095e6 −0.146559 −0.0732795 0.997311i \(-0.523347\pi\)
−0.0732795 + 0.997311i \(0.523347\pi\)
\(942\) 0 0
\(943\) 3.16045e6 0.115736
\(944\) −6.63780e6 −0.242434
\(945\) 0 0
\(946\) −3.92067e6 −0.142440
\(947\) 2.43639e7 0.882818 0.441409 0.897306i \(-0.354479\pi\)
0.441409 + 0.897306i \(0.354479\pi\)
\(948\) 0 0
\(949\) −4.63152e6 −0.166939
\(950\) 757589. 0.0272348
\(951\) 0 0
\(952\) 5.61450e6 0.200779
\(953\) −1.39017e7 −0.495833 −0.247916 0.968781i \(-0.579746\pi\)
−0.247916 + 0.968781i \(0.579746\pi\)
\(954\) 0 0
\(955\) 2.94913e7 1.04637
\(956\) −5.04280e6 −0.178454
\(957\) 0 0
\(958\) −1.74676e7 −0.614920
\(959\) 9.14617e6 0.321139
\(960\) 0 0
\(961\) −2.75124e7 −0.960993
\(962\) −1.26112e7 −0.439358
\(963\) 0 0
\(964\) −4.74208e6 −0.164353
\(965\) −1.18678e6 −0.0410253
\(966\) 0 0
\(967\) 5.16682e7 1.77688 0.888438 0.458998i \(-0.151791\pi\)
0.888438 + 0.458998i \(0.151791\pi\)
\(968\) 1.91177e6 0.0655764
\(969\) 0 0
\(970\) 8.82137e6 0.301028
\(971\) 1.45794e7 0.496240 0.248120 0.968729i \(-0.420187\pi\)
0.248120 + 0.968729i \(0.420187\pi\)
\(972\) 0 0
\(973\) −3.16945e7 −1.07325
\(974\) −7.69495e6 −0.259901
\(975\) 0 0
\(976\) −1.63583e7 −0.549685
\(977\) 3.09921e6 0.103876 0.0519379 0.998650i \(-0.483460\pi\)
0.0519379 + 0.998650i \(0.483460\pi\)
\(978\) 0 0
\(979\) −9.95442e6 −0.331940
\(980\) 6.94781e7 2.31091
\(981\) 0 0
\(982\) −1.98353e7 −0.656386
\(983\) −1.53445e7 −0.506489 −0.253245 0.967402i \(-0.581498\pi\)
−0.253245 + 0.967402i \(0.581498\pi\)
\(984\) 0 0
\(985\) 4.10380e7 1.34771
\(986\) 1.09843e6 0.0359815
\(987\) 0 0
\(988\) 1.67253e6 0.0545108
\(989\) 4.10296e7 1.33385
\(990\) 0 0
\(991\) 1.57747e7 0.510244 0.255122 0.966909i \(-0.417884\pi\)
0.255122 + 0.966909i \(0.417884\pi\)
\(992\) 5.76818e6 0.186106
\(993\) 0 0
\(994\) 8.07806e6 0.259323
\(995\) 4.11105e7 1.31642
\(996\) 0 0
\(997\) 1.85577e6 0.0591270 0.0295635 0.999563i \(-0.490588\pi\)
0.0295635 + 0.999563i \(0.490588\pi\)
\(998\) −1.47476e7 −0.468699
\(999\) 0 0
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 99.6.a.g.1.2 3
3.2 odd 2 11.6.a.b.1.2 3
11.10 odd 2 1089.6.a.r.1.2 3
12.11 even 2 176.6.a.i.1.2 3
15.2 even 4 275.6.b.b.199.4 6
15.8 even 4 275.6.b.b.199.3 6
15.14 odd 2 275.6.a.b.1.2 3
21.20 even 2 539.6.a.e.1.2 3
24.5 odd 2 704.6.a.q.1.2 3
24.11 even 2 704.6.a.t.1.2 3
33.32 even 2 121.6.a.d.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 3 3.2 odd 2
99.6.a.g.1.2 3 1.1 even 1 trivial
121.6.a.d.1.2 3 33.32 even 2
176.6.a.i.1.2 3 12.11 even 2
275.6.a.b.1.2 3 15.14 odd 2
275.6.b.b.199.3 6 15.8 even 4
275.6.b.b.199.4 6 15.2 even 4
539.6.a.e.1.2 3 21.20 even 2
704.6.a.q.1.2 3 24.5 odd 2
704.6.a.t.1.2 3 24.11 even 2
1089.6.a.r.1.2 3 11.10 odd 2