Properties

Label 1089.6.a.r.1.3
Level $1089$
Weight $6$
Character 1089.1
Self dual yes
Analytic conductor $174.658$
Analytic rank $1$
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] = [1089,6,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(174.657979776\)
Analytic rank: \(1\)
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 1.3
Root \(-6.29828\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.18772 q^{2} +35.0388 q^{4} +59.8722 q^{5} -145.071 q^{7} +24.8808 q^{8} +O(q^{10})\) \(q+8.18772 q^{2} +35.0388 q^{4} +59.8722 q^{5} -145.071 q^{7} +24.8808 q^{8} +490.217 q^{10} -615.772 q^{13} -1187.80 q^{14} -917.524 q^{16} +1840.68 q^{17} -366.633 q^{19} +2097.85 q^{20} +4516.38 q^{23} +459.685 q^{25} -5041.77 q^{26} -5083.12 q^{28} -1717.00 q^{29} -2650.54 q^{31} -8308.62 q^{32} +15070.9 q^{34} -8685.74 q^{35} +9660.61 q^{37} -3001.89 q^{38} +1489.67 q^{40} -11154.8 q^{41} -8368.48 q^{43} +36978.9 q^{46} +2221.22 q^{47} +4238.64 q^{49} +3763.77 q^{50} -21575.9 q^{52} -23707.9 q^{53} -3609.48 q^{56} -14058.3 q^{58} -19517.8 q^{59} -20937.3 q^{61} -21701.9 q^{62} -38667.9 q^{64} -36867.6 q^{65} -51707.7 q^{67} +64495.1 q^{68} -71116.4 q^{70} +1398.38 q^{71} -72466.6 q^{73} +79098.4 q^{74} -12846.4 q^{76} -64632.2 q^{79} -54934.2 q^{80} -91332.4 q^{82} -96790.3 q^{83} +110205. q^{85} -68518.8 q^{86} +47614.1 q^{89} +89330.7 q^{91} +158249. q^{92} +18186.7 q^{94} -21951.2 q^{95} -38399.6 q^{97} +34704.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{4} - 24 q^{5} - 84 q^{7} - 564 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{4} - 24 q^{5} - 84 q^{7} - 564 q^{8} + 414 q^{10} - 486 q^{13} + 1020 q^{14} + 1992 q^{16} + 1086 q^{17} - 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} - 12408 q^{32} + 25320 q^{34} - 24228 q^{35} + 17724 q^{37} + 9240 q^{38} + 15276 q^{40} + 5994 q^{41} + 26208 q^{43} + 16806 q^{46} + 17232 q^{47} + 48531 q^{49} + 41070 q^{50} + 35304 q^{52} - 50586 q^{53} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} - 18486 q^{61} - 19974 q^{62} - 20352 q^{64} - 7668 q^{65} - 47754 q^{67} - 12600 q^{68} - 123372 q^{70} - 39282 q^{71} - 15426 q^{73} + 153294 q^{74} - 103920 q^{76} - 125148 q^{79} - 118680 q^{80} - 255372 q^{82} - 143928 q^{83} + 104040 q^{85} - 243060 q^{86} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 74928 q^{94} - 22200 q^{95} + 9684 q^{97} + 3480 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.18772 1.44740 0.723699 0.690116i \(-0.242440\pi\)
0.723699 + 0.690116i \(0.242440\pi\)
\(3\) 0 0
\(4\) 35.0388 1.09496
\(5\) 59.8722 1.07103 0.535514 0.844527i \(-0.320118\pi\)
0.535514 + 0.844527i \(0.320118\pi\)
\(6\) 0 0
\(7\) −145.071 −1.11902 −0.559508 0.828825i \(-0.689010\pi\)
−0.559508 + 0.828825i \(0.689010\pi\)
\(8\) 24.8808 0.137448
\(9\) 0 0
\(10\) 490.217 1.55020
\(11\) 0 0
\(12\) 0 0
\(13\) −615.772 −1.01056 −0.505279 0.862956i \(-0.668610\pi\)
−0.505279 + 0.862956i \(0.668610\pi\)
\(14\) −1187.80 −1.61966
\(15\) 0 0
\(16\) −917.524 −0.896020
\(17\) 1840.68 1.54474 0.772369 0.635174i \(-0.219071\pi\)
0.772369 + 0.635174i \(0.219071\pi\)
\(18\) 0 0
\(19\) −366.633 −0.232996 −0.116498 0.993191i \(-0.537167\pi\)
−0.116498 + 0.993191i \(0.537167\pi\)
\(20\) 2097.85 1.17273
\(21\) 0 0
\(22\) 0 0
\(23\) 4516.38 1.78021 0.890104 0.455757i \(-0.150631\pi\)
0.890104 + 0.455757i \(0.150631\pi\)
\(24\) 0 0
\(25\) 459.685 0.147099
\(26\) −5041.77 −1.46268
\(27\) 0 0
\(28\) −5083.12 −1.22528
\(29\) −1717.00 −0.379119 −0.189560 0.981869i \(-0.560706\pi\)
−0.189560 + 0.981869i \(0.560706\pi\)
\(30\) 0 0
\(31\) −2650.54 −0.495371 −0.247685 0.968841i \(-0.579670\pi\)
−0.247685 + 0.968841i \(0.579670\pi\)
\(32\) −8308.62 −1.43435
\(33\) 0 0
\(34\) 15070.9 2.23585
\(35\) −8685.74 −1.19850
\(36\) 0 0
\(37\) 9660.61 1.16011 0.580057 0.814576i \(-0.303030\pi\)
0.580057 + 0.814576i \(0.303030\pi\)
\(38\) −3001.89 −0.337238
\(39\) 0 0
\(40\) 1489.67 0.147211
\(41\) −11154.8 −1.03634 −0.518170 0.855278i \(-0.673386\pi\)
−0.518170 + 0.855278i \(0.673386\pi\)
\(42\) 0 0
\(43\) −8368.48 −0.690201 −0.345100 0.938566i \(-0.612155\pi\)
−0.345100 + 0.938566i \(0.612155\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 36978.9 2.57667
\(47\) 2221.22 0.146672 0.0733360 0.997307i \(-0.476635\pi\)
0.0733360 + 0.997307i \(0.476635\pi\)
\(48\) 0 0
\(49\) 4238.64 0.252195
\(50\) 3763.77 0.212911
\(51\) 0 0
\(52\) −21575.9 −1.10652
\(53\) −23707.9 −1.15932 −0.579659 0.814859i \(-0.696814\pi\)
−0.579659 + 0.814859i \(0.696814\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3609.48 −0.153807
\(57\) 0 0
\(58\) −14058.3 −0.548737
\(59\) −19517.8 −0.729964 −0.364982 0.931015i \(-0.618925\pi\)
−0.364982 + 0.931015i \(0.618925\pi\)
\(60\) 0 0
\(61\) −20937.3 −0.720436 −0.360218 0.932868i \(-0.617298\pi\)
−0.360218 + 0.932868i \(0.617298\pi\)
\(62\) −21701.9 −0.716999
\(63\) 0 0
\(64\) −38667.9 −1.18005
\(65\) −36867.6 −1.08233
\(66\) 0 0
\(67\) −51707.7 −1.40724 −0.703619 0.710577i \(-0.748434\pi\)
−0.703619 + 0.710577i \(0.748434\pi\)
\(68\) 64495.1 1.69143
\(69\) 0 0
\(70\) −71116.4 −1.73470
\(71\) 1398.38 0.0329216 0.0164608 0.999865i \(-0.494760\pi\)
0.0164608 + 0.999865i \(0.494760\pi\)
\(72\) 0 0
\(73\) −72466.6 −1.59159 −0.795794 0.605567i \(-0.792946\pi\)
−0.795794 + 0.605567i \(0.792946\pi\)
\(74\) 79098.4 1.67915
\(75\) 0 0
\(76\) −12846.4 −0.255122
\(77\) 0 0
\(78\) 0 0
\(79\) −64632.2 −1.16515 −0.582574 0.812777i \(-0.697955\pi\)
−0.582574 + 0.812777i \(0.697955\pi\)
\(80\) −54934.2 −0.959662
\(81\) 0 0
\(82\) −91332.4 −1.50000
\(83\) −96790.3 −1.54219 −0.771093 0.636723i \(-0.780290\pi\)
−0.771093 + 0.636723i \(0.780290\pi\)
\(84\) 0 0
\(85\) 110205. 1.65446
\(86\) −68518.8 −0.998995
\(87\) 0 0
\(88\) 0 0
\(89\) 47614.1 0.637178 0.318589 0.947893i \(-0.396791\pi\)
0.318589 + 0.947893i \(0.396791\pi\)
\(90\) 0 0
\(91\) 89330.7 1.13083
\(92\) 158249. 1.94926
\(93\) 0 0
\(94\) 18186.7 0.212293
\(95\) −21951.2 −0.249545
\(96\) 0 0
\(97\) −38399.6 −0.414378 −0.207189 0.978301i \(-0.566432\pi\)
−0.207189 + 0.978301i \(0.566432\pi\)
\(98\) 34704.8 0.365027
\(99\) 0 0
\(100\) 16106.8 0.161068
\(101\) −41011.2 −0.400036 −0.200018 0.979792i \(-0.564100\pi\)
−0.200018 + 0.979792i \(0.564100\pi\)
\(102\) 0 0
\(103\) −49634.4 −0.460988 −0.230494 0.973074i \(-0.574034\pi\)
−0.230494 + 0.973074i \(0.574034\pi\)
\(104\) −15320.9 −0.138899
\(105\) 0 0
\(106\) −194113. −1.67800
\(107\) −6791.34 −0.0573450 −0.0286725 0.999589i \(-0.509128\pi\)
−0.0286725 + 0.999589i \(0.509128\pi\)
\(108\) 0 0
\(109\) −96780.7 −0.780230 −0.390115 0.920766i \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 133106. 1.00266
\(113\) 212938. 1.56876 0.784379 0.620281i \(-0.212982\pi\)
0.784379 + 0.620281i \(0.212982\pi\)
\(114\) 0 0
\(115\) 270406. 1.90665
\(116\) −60161.7 −0.415121
\(117\) 0 0
\(118\) −159806. −1.05655
\(119\) −267029. −1.72859
\(120\) 0 0
\(121\) 0 0
\(122\) −171428. −1.04276
\(123\) 0 0
\(124\) −92871.8 −0.542412
\(125\) −159578. −0.913480
\(126\) 0 0
\(127\) 90363.9 0.497148 0.248574 0.968613i \(-0.420038\pi\)
0.248574 + 0.968613i \(0.420038\pi\)
\(128\) −50726.1 −0.273657
\(129\) 0 0
\(130\) −301862. −1.56657
\(131\) 65299.5 0.332454 0.166227 0.986088i \(-0.446842\pi\)
0.166227 + 0.986088i \(0.446842\pi\)
\(132\) 0 0
\(133\) 53187.9 0.260726
\(134\) −423368. −2.03684
\(135\) 0 0
\(136\) 45797.4 0.212321
\(137\) −5322.74 −0.0242289 −0.0121145 0.999927i \(-0.503856\pi\)
−0.0121145 + 0.999927i \(0.503856\pi\)
\(138\) 0 0
\(139\) −89967.1 −0.394954 −0.197477 0.980308i \(-0.563275\pi\)
−0.197477 + 0.980308i \(0.563275\pi\)
\(140\) −304338. −1.31231
\(141\) 0 0
\(142\) 11449.6 0.0476506
\(143\) 0 0
\(144\) 0 0
\(145\) −102801. −0.406047
\(146\) −593336. −2.30366
\(147\) 0 0
\(148\) 338496. 1.27028
\(149\) 66489.8 0.245352 0.122676 0.992447i \(-0.460852\pi\)
0.122676 + 0.992447i \(0.460852\pi\)
\(150\) 0 0
\(151\) 130866. 0.467074 0.233537 0.972348i \(-0.424970\pi\)
0.233537 + 0.972348i \(0.424970\pi\)
\(152\) −9122.12 −0.0320248
\(153\) 0 0
\(154\) 0 0
\(155\) −158694. −0.530555
\(156\) 0 0
\(157\) −163297. −0.528723 −0.264362 0.964424i \(-0.585161\pi\)
−0.264362 + 0.964424i \(0.585161\pi\)
\(158\) −529191. −1.68643
\(159\) 0 0
\(160\) −497456. −1.53622
\(161\) −655197. −1.99208
\(162\) 0 0
\(163\) −535758. −1.57943 −0.789713 0.613477i \(-0.789771\pi\)
−0.789713 + 0.613477i \(0.789771\pi\)
\(164\) −390851. −1.13475
\(165\) 0 0
\(166\) −792492. −2.23216
\(167\) 553587. 1.53601 0.768005 0.640443i \(-0.221249\pi\)
0.768005 + 0.640443i \(0.221249\pi\)
\(168\) 0 0
\(169\) 7881.54 0.0212273
\(170\) 902331. 2.39466
\(171\) 0 0
\(172\) −293221. −0.755744
\(173\) 266973. 0.678190 0.339095 0.940752i \(-0.389879\pi\)
0.339095 + 0.940752i \(0.389879\pi\)
\(174\) 0 0
\(175\) −66687.0 −0.164606
\(176\) 0 0
\(177\) 0 0
\(178\) 389851. 0.922250
\(179\) −3030.33 −0.00706900 −0.00353450 0.999994i \(-0.501125\pi\)
−0.00353450 + 0.999994i \(0.501125\pi\)
\(180\) 0 0
\(181\) 761242. 1.72714 0.863568 0.504233i \(-0.168225\pi\)
0.863568 + 0.504233i \(0.168225\pi\)
\(182\) 731415. 1.63676
\(183\) 0 0
\(184\) 112371. 0.244686
\(185\) 578402. 1.24251
\(186\) 0 0
\(187\) 0 0
\(188\) 77828.9 0.160600
\(189\) 0 0
\(190\) −179730. −0.361191
\(191\) 430653. 0.854170 0.427085 0.904212i \(-0.359541\pi\)
0.427085 + 0.904212i \(0.359541\pi\)
\(192\) 0 0
\(193\) 272285. 0.526175 0.263088 0.964772i \(-0.415259\pi\)
0.263088 + 0.964772i \(0.415259\pi\)
\(194\) −314405. −0.599770
\(195\) 0 0
\(196\) 148517. 0.276144
\(197\) 574550. 1.05478 0.527390 0.849623i \(-0.323171\pi\)
0.527390 + 0.849623i \(0.323171\pi\)
\(198\) 0 0
\(199\) 269926. 0.483183 0.241592 0.970378i \(-0.422331\pi\)
0.241592 + 0.970378i \(0.422331\pi\)
\(200\) 11437.3 0.0202185
\(201\) 0 0
\(202\) −335788. −0.579011
\(203\) 249088. 0.424240
\(204\) 0 0
\(205\) −667863. −1.10995
\(206\) −406393. −0.667234
\(207\) 0 0
\(208\) 564985. 0.905480
\(209\) 0 0
\(210\) 0 0
\(211\) 753372. 1.16494 0.582470 0.812853i \(-0.302087\pi\)
0.582470 + 0.812853i \(0.302087\pi\)
\(212\) −830695. −1.26941
\(213\) 0 0
\(214\) −55605.6 −0.0830011
\(215\) −501040. −0.739224
\(216\) 0 0
\(217\) 384517. 0.554327
\(218\) −792414. −1.12930
\(219\) 0 0
\(220\) 0 0
\(221\) −1.13344e6 −1.56105
\(222\) 0 0
\(223\) −997692. −1.34349 −0.671745 0.740783i \(-0.734455\pi\)
−0.671745 + 0.740783i \(0.734455\pi\)
\(224\) 1.20534e6 1.60506
\(225\) 0 0
\(226\) 1.74347e6 2.27062
\(227\) −495214. −0.637864 −0.318932 0.947778i \(-0.603324\pi\)
−0.318932 + 0.947778i \(0.603324\pi\)
\(228\) 0 0
\(229\) −221893. −0.279611 −0.139806 0.990179i \(-0.544648\pi\)
−0.139806 + 0.990179i \(0.544648\pi\)
\(230\) 2.21401e6 2.75968
\(231\) 0 0
\(232\) −42720.4 −0.0521093
\(233\) 619425. 0.747479 0.373739 0.927534i \(-0.378075\pi\)
0.373739 + 0.927534i \(0.378075\pi\)
\(234\) 0 0
\(235\) 132989. 0.157090
\(236\) −683881. −0.799283
\(237\) 0 0
\(238\) −2.18636e6 −2.50195
\(239\) −295471. −0.334595 −0.167298 0.985906i \(-0.553504\pi\)
−0.167298 + 0.985906i \(0.553504\pi\)
\(240\) 0 0
\(241\) −693153. −0.768753 −0.384376 0.923176i \(-0.625584\pi\)
−0.384376 + 0.923176i \(0.625584\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −733616. −0.788850
\(245\) 253777. 0.270108
\(246\) 0 0
\(247\) 225762. 0.235456
\(248\) −65947.5 −0.0680878
\(249\) 0 0
\(250\) −1.30658e6 −1.32217
\(251\) −533816. −0.534820 −0.267410 0.963583i \(-0.586168\pi\)
−0.267410 + 0.963583i \(0.586168\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 739874. 0.719571
\(255\) 0 0
\(256\) 822041. 0.783960
\(257\) 652296. 0.616044 0.308022 0.951379i \(-0.400333\pi\)
0.308022 + 0.951379i \(0.400333\pi\)
\(258\) 0 0
\(259\) −1.40148e6 −1.29818
\(260\) −1.29180e6 −1.18512
\(261\) 0 0
\(262\) 534654. 0.481193
\(263\) −622045. −0.554540 −0.277270 0.960792i \(-0.589430\pi\)
−0.277270 + 0.960792i \(0.589430\pi\)
\(264\) 0 0
\(265\) −1.41944e6 −1.24166
\(266\) 435488. 0.377374
\(267\) 0 0
\(268\) −1.81177e6 −1.54087
\(269\) 482862. 0.406858 0.203429 0.979090i \(-0.434791\pi\)
0.203429 + 0.979090i \(0.434791\pi\)
\(270\) 0 0
\(271\) 1.10678e6 0.915460 0.457730 0.889091i \(-0.348663\pi\)
0.457730 + 0.889091i \(0.348663\pi\)
\(272\) −1.68887e6 −1.38412
\(273\) 0 0
\(274\) −43581.1 −0.0350689
\(275\) 0 0
\(276\) 0 0
\(277\) −639062. −0.500430 −0.250215 0.968190i \(-0.580501\pi\)
−0.250215 + 0.968190i \(0.580501\pi\)
\(278\) −736626. −0.571656
\(279\) 0 0
\(280\) −216108. −0.164731
\(281\) −257984. −0.194907 −0.0974534 0.995240i \(-0.531070\pi\)
−0.0974534 + 0.995240i \(0.531070\pi\)
\(282\) 0 0
\(283\) −1.02991e6 −0.764425 −0.382213 0.924074i \(-0.624838\pi\)
−0.382213 + 0.924074i \(0.624838\pi\)
\(284\) 48997.7 0.0360479
\(285\) 0 0
\(286\) 0 0
\(287\) 1.61824e6 1.15968
\(288\) 0 0
\(289\) 1.96823e6 1.38622
\(290\) −841704. −0.587712
\(291\) 0 0
\(292\) −2.53914e6 −1.74273
\(293\) 877712. 0.597287 0.298644 0.954365i \(-0.403466\pi\)
0.298644 + 0.954365i \(0.403466\pi\)
\(294\) 0 0
\(295\) −1.16858e6 −0.781811
\(296\) 240363. 0.159455
\(297\) 0 0
\(298\) 544400. 0.355122
\(299\) −2.78106e6 −1.79900
\(300\) 0 0
\(301\) 1.21403e6 0.772345
\(302\) 1.07150e6 0.676042
\(303\) 0 0
\(304\) 336395. 0.208769
\(305\) −1.25356e6 −0.771606
\(306\) 0 0
\(307\) 1.30925e6 0.792826 0.396413 0.918072i \(-0.370255\pi\)
0.396413 + 0.918072i \(0.370255\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.29934e6 −0.767925
\(311\) −3.35930e6 −1.96946 −0.984731 0.174083i \(-0.944304\pi\)
−0.984731 + 0.174083i \(0.944304\pi\)
\(312\) 0 0
\(313\) −3.00640e6 −1.73454 −0.867272 0.497835i \(-0.834129\pi\)
−0.867272 + 0.497835i \(0.834129\pi\)
\(314\) −1.33703e6 −0.765273
\(315\) 0 0
\(316\) −2.26464e6 −1.27579
\(317\) −2.10147e6 −1.17456 −0.587279 0.809385i \(-0.699801\pi\)
−0.587279 + 0.809385i \(0.699801\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.31513e6 −1.26387
\(321\) 0 0
\(322\) −5.36457e6 −2.88333
\(323\) −674853. −0.359918
\(324\) 0 0
\(325\) −283061. −0.148652
\(326\) −4.38663e6 −2.28606
\(327\) 0 0
\(328\) −277540. −0.142443
\(329\) −322235. −0.164128
\(330\) 0 0
\(331\) 1.23338e6 0.618766 0.309383 0.950938i \(-0.399878\pi\)
0.309383 + 0.950938i \(0.399878\pi\)
\(332\) −3.39142e6 −1.68864
\(333\) 0 0
\(334\) 4.53261e6 2.22322
\(335\) −3.09585e6 −1.50719
\(336\) 0 0
\(337\) −679319. −0.325836 −0.162918 0.986640i \(-0.552091\pi\)
−0.162918 + 0.986640i \(0.552091\pi\)
\(338\) 64531.9 0.0307243
\(339\) 0 0
\(340\) 3.86146e6 1.81157
\(341\) 0 0
\(342\) 0 0
\(343\) 1.82331e6 0.836805
\(344\) −208214. −0.0948668
\(345\) 0 0
\(346\) 2.18590e6 0.981611
\(347\) 2.67540e6 1.19279 0.596397 0.802690i \(-0.296598\pi\)
0.596397 + 0.802690i \(0.296598\pi\)
\(348\) 0 0
\(349\) 2.37636e6 1.04435 0.522177 0.852837i \(-0.325120\pi\)
0.522177 + 0.852837i \(0.325120\pi\)
\(350\) −546015. −0.238251
\(351\) 0 0
\(352\) 0 0
\(353\) −638696. −0.272808 −0.136404 0.990653i \(-0.543555\pi\)
−0.136404 + 0.990653i \(0.543555\pi\)
\(354\) 0 0
\(355\) 83724.4 0.0352599
\(356\) 1.66834e6 0.697686
\(357\) 0 0
\(358\) −24811.5 −0.0102317
\(359\) 1.50842e6 0.617712 0.308856 0.951109i \(-0.400054\pi\)
0.308856 + 0.951109i \(0.400054\pi\)
\(360\) 0 0
\(361\) −2.34168e6 −0.945713
\(362\) 6.23284e6 2.49985
\(363\) 0 0
\(364\) 3.13004e6 1.23822
\(365\) −4.33874e6 −1.70463
\(366\) 0 0
\(367\) 1.77368e6 0.687403 0.343701 0.939079i \(-0.388319\pi\)
0.343701 + 0.939079i \(0.388319\pi\)
\(368\) −4.14389e6 −1.59510
\(369\) 0 0
\(370\) 4.73580e6 1.79841
\(371\) 3.43933e6 1.29729
\(372\) 0 0
\(373\) −2.27176e6 −0.845456 −0.422728 0.906257i \(-0.638927\pi\)
−0.422728 + 0.906257i \(0.638927\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 55265.7 0.0201598
\(377\) 1.05728e6 0.383122
\(378\) 0 0
\(379\) −4.42409e6 −1.58207 −0.791035 0.611771i \(-0.790457\pi\)
−0.791035 + 0.611771i \(0.790457\pi\)
\(380\) −769142. −0.273242
\(381\) 0 0
\(382\) 3.52607e6 1.23632
\(383\) −2.37588e6 −0.827615 −0.413807 0.910364i \(-0.635801\pi\)
−0.413807 + 0.910364i \(0.635801\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.22939e6 0.761586
\(387\) 0 0
\(388\) −1.34547e6 −0.453728
\(389\) 2.65905e6 0.890949 0.445474 0.895295i \(-0.353035\pi\)
0.445474 + 0.895295i \(0.353035\pi\)
\(390\) 0 0
\(391\) 8.31319e6 2.74996
\(392\) 105461. 0.0346638
\(393\) 0 0
\(394\) 4.70425e6 1.52669
\(395\) −3.86968e6 −1.24791
\(396\) 0 0
\(397\) 2.15712e6 0.686907 0.343453 0.939170i \(-0.388403\pi\)
0.343453 + 0.939170i \(0.388403\pi\)
\(398\) 2.21008e6 0.699359
\(399\) 0 0
\(400\) −421772. −0.131804
\(401\) 2.43031e6 0.754744 0.377372 0.926062i \(-0.376828\pi\)
0.377372 + 0.926062i \(0.376828\pi\)
\(402\) 0 0
\(403\) 1.63213e6 0.500601
\(404\) −1.43698e6 −0.438024
\(405\) 0 0
\(406\) 2.03946e6 0.614045
\(407\) 0 0
\(408\) 0 0
\(409\) −6.12831e6 −1.81148 −0.905738 0.423839i \(-0.860682\pi\)
−0.905738 + 0.423839i \(0.860682\pi\)
\(410\) −5.46827e6 −1.60654
\(411\) 0 0
\(412\) −1.73913e6 −0.504765
\(413\) 2.83147e6 0.816841
\(414\) 0 0
\(415\) −5.79505e6 −1.65172
\(416\) 5.11621e6 1.44949
\(417\) 0 0
\(418\) 0 0
\(419\) 375626. 0.104525 0.0522626 0.998633i \(-0.483357\pi\)
0.0522626 + 0.998633i \(0.483357\pi\)
\(420\) 0 0
\(421\) 3.52333e6 0.968831 0.484416 0.874838i \(-0.339032\pi\)
0.484416 + 0.874838i \(0.339032\pi\)
\(422\) 6.16840e6 1.68613
\(423\) 0 0
\(424\) −589870. −0.159346
\(425\) 846131. 0.227230
\(426\) 0 0
\(427\) 3.03739e6 0.806178
\(428\) −237960. −0.0627906
\(429\) 0 0
\(430\) −4.10237e6 −1.06995
\(431\) 3.15287e6 0.817548 0.408774 0.912636i \(-0.365956\pi\)
0.408774 + 0.912636i \(0.365956\pi\)
\(432\) 0 0
\(433\) 1.62168e6 0.415667 0.207833 0.978164i \(-0.433359\pi\)
0.207833 + 0.978164i \(0.433359\pi\)
\(434\) 3.14832e6 0.802333
\(435\) 0 0
\(436\) −3.39108e6 −0.854322
\(437\) −1.65586e6 −0.414781
\(438\) 0 0
\(439\) 2.48145e6 0.614533 0.307266 0.951624i \(-0.400586\pi\)
0.307266 + 0.951624i \(0.400586\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −9.28026e6 −2.25946
\(443\) −3.75466e6 −0.908994 −0.454497 0.890748i \(-0.650181\pi\)
−0.454497 + 0.890748i \(0.650181\pi\)
\(444\) 0 0
\(445\) 2.85076e6 0.682435
\(446\) −8.16882e6 −1.94456
\(447\) 0 0
\(448\) 5.60960e6 1.32049
\(449\) 4.80916e6 1.12578 0.562890 0.826532i \(-0.309689\pi\)
0.562890 + 0.826532i \(0.309689\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.46108e6 1.71773
\(453\) 0 0
\(454\) −4.05467e6 −0.923244
\(455\) 5.34843e6 1.21115
\(456\) 0 0
\(457\) −7.09951e6 −1.59015 −0.795075 0.606512i \(-0.792568\pi\)
−0.795075 + 0.606512i \(0.792568\pi\)
\(458\) −1.81680e6 −0.404709
\(459\) 0 0
\(460\) 9.47469e6 2.08771
\(461\) 8.12745e6 1.78116 0.890578 0.454830i \(-0.150300\pi\)
0.890578 + 0.454830i \(0.150300\pi\)
\(462\) 0 0
\(463\) −2.67361e6 −0.579623 −0.289812 0.957084i \(-0.593593\pi\)
−0.289812 + 0.957084i \(0.593593\pi\)
\(464\) 1.57539e6 0.339699
\(465\) 0 0
\(466\) 5.07168e6 1.08190
\(467\) 4.32733e6 0.918180 0.459090 0.888390i \(-0.348175\pi\)
0.459090 + 0.888390i \(0.348175\pi\)
\(468\) 0 0
\(469\) 7.50129e6 1.57472
\(470\) 1.08888e6 0.227371
\(471\) 0 0
\(472\) −485618. −0.100332
\(473\) 0 0
\(474\) 0 0
\(475\) −168536. −0.0342735
\(476\) −9.35637e6 −1.89274
\(477\) 0 0
\(478\) −2.41923e6 −0.484293
\(479\) 1.55878e6 0.310417 0.155208 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(480\) 0 0
\(481\) −5.94873e6 −1.17236
\(482\) −5.67535e6 −1.11269
\(483\) 0 0
\(484\) 0 0
\(485\) −2.29907e6 −0.443810
\(486\) 0 0
\(487\) −7.63818e6 −1.45938 −0.729689 0.683779i \(-0.760335\pi\)
−0.729689 + 0.683779i \(0.760335\pi\)
\(488\) −520935. −0.0990225
\(489\) 0 0
\(490\) 2.07786e6 0.390954
\(491\) 3.60872e6 0.675537 0.337768 0.941229i \(-0.390328\pi\)
0.337768 + 0.941229i \(0.390328\pi\)
\(492\) 0 0
\(493\) −3.16045e6 −0.585640
\(494\) 1.84848e6 0.340798
\(495\) 0 0
\(496\) 2.43194e6 0.443862
\(497\) −202865. −0.0368397
\(498\) 0 0
\(499\) −8.46131e6 −1.52120 −0.760599 0.649221i \(-0.775095\pi\)
−0.760599 + 0.649221i \(0.775095\pi\)
\(500\) −5.59143e6 −1.00023
\(501\) 0 0
\(502\) −4.37074e6 −0.774098
\(503\) −8.28353e6 −1.45981 −0.729904 0.683550i \(-0.760435\pi\)
−0.729904 + 0.683550i \(0.760435\pi\)
\(504\) 0 0
\(505\) −2.45543e6 −0.428449
\(506\) 0 0
\(507\) 0 0
\(508\) 3.16624e6 0.544358
\(509\) 7.60138e6 1.30046 0.650232 0.759736i \(-0.274672\pi\)
0.650232 + 0.759736i \(0.274672\pi\)
\(510\) 0 0
\(511\) 1.05128e7 1.78101
\(512\) 8.35388e6 1.40836
\(513\) 0 0
\(514\) 5.34082e6 0.891661
\(515\) −2.97172e6 −0.493731
\(516\) 0 0
\(517\) 0 0
\(518\) −1.14749e7 −1.87899
\(519\) 0 0
\(520\) −917295. −0.148765
\(521\) −9.60432e6 −1.55015 −0.775073 0.631872i \(-0.782287\pi\)
−0.775073 + 0.631872i \(0.782287\pi\)
\(522\) 0 0
\(523\) 9.97831e6 1.59515 0.797577 0.603217i \(-0.206115\pi\)
0.797577 + 0.603217i \(0.206115\pi\)
\(524\) 2.28802e6 0.364025
\(525\) 0 0
\(526\) −5.09313e6 −0.802640
\(527\) −4.87879e6 −0.765218
\(528\) 0 0
\(529\) 1.39613e7 2.16914
\(530\) −1.16220e7 −1.79718
\(531\) 0 0
\(532\) 1.86364e6 0.285485
\(533\) 6.86881e6 1.04728
\(534\) 0 0
\(535\) −406612. −0.0614181
\(536\) −1.28653e6 −0.193422
\(537\) 0 0
\(538\) 3.95354e6 0.588885
\(539\) 0 0
\(540\) 0 0
\(541\) 4.34177e6 0.637784 0.318892 0.947791i \(-0.396689\pi\)
0.318892 + 0.947791i \(0.396689\pi\)
\(542\) 9.06203e6 1.32503
\(543\) 0 0
\(544\) −1.52935e7 −2.21569
\(545\) −5.79448e6 −0.835647
\(546\) 0 0
\(547\) −1.14668e7 −1.63860 −0.819302 0.573363i \(-0.805639\pi\)
−0.819302 + 0.573363i \(0.805639\pi\)
\(548\) −186503. −0.0265298
\(549\) 0 0
\(550\) 0 0
\(551\) 629511. 0.0883332
\(552\) 0 0
\(553\) 9.37627e6 1.30382
\(554\) −5.23246e6 −0.724322
\(555\) 0 0
\(556\) −3.15234e6 −0.432460
\(557\) −1.57100e6 −0.214555 −0.107278 0.994229i \(-0.534213\pi\)
−0.107278 + 0.994229i \(0.534213\pi\)
\(558\) 0 0
\(559\) 5.15307e6 0.697488
\(560\) 7.96938e6 1.07388
\(561\) 0 0
\(562\) −2.11230e6 −0.282108
\(563\) −850908. −0.113139 −0.0565694 0.998399i \(-0.518016\pi\)
−0.0565694 + 0.998399i \(0.518016\pi\)
\(564\) 0 0
\(565\) 1.27490e7 1.68018
\(566\) −8.43265e6 −1.10643
\(567\) 0 0
\(568\) 34792.9 0.00452501
\(569\) −1.19642e7 −1.54919 −0.774595 0.632458i \(-0.782046\pi\)
−0.774595 + 0.632458i \(0.782046\pi\)
\(570\) 0 0
\(571\) 7.97842e6 1.02406 0.512032 0.858967i \(-0.328893\pi\)
0.512032 + 0.858967i \(0.328893\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.32497e7 1.67852
\(575\) 2.07611e6 0.261867
\(576\) 0 0
\(577\) 5.90743e6 0.738685 0.369342 0.929293i \(-0.379583\pi\)
0.369342 + 0.929293i \(0.379583\pi\)
\(578\) 1.61153e7 2.00641
\(579\) 0 0
\(580\) −3.60202e6 −0.444606
\(581\) 1.40415e7 1.72573
\(582\) 0 0
\(583\) 0 0
\(584\) −1.80303e6 −0.218761
\(585\) 0 0
\(586\) 7.18646e6 0.864512
\(587\) −1.34766e6 −0.161430 −0.0807151 0.996737i \(-0.525720\pi\)
−0.0807151 + 0.996737i \(0.525720\pi\)
\(588\) 0 0
\(589\) 971777. 0.115419
\(590\) −9.56797e6 −1.13159
\(591\) 0 0
\(592\) −8.86385e6 −1.03948
\(593\) 1.05883e7 1.23649 0.618243 0.785987i \(-0.287845\pi\)
0.618243 + 0.785987i \(0.287845\pi\)
\(594\) 0 0
\(595\) −1.59876e7 −1.85136
\(596\) 2.32972e6 0.268651
\(597\) 0 0
\(598\) −2.27705e7 −2.60388
\(599\) −3.48377e6 −0.396718 −0.198359 0.980129i \(-0.563561\pi\)
−0.198359 + 0.980129i \(0.563561\pi\)
\(600\) 0 0
\(601\) 6.41433e6 0.724378 0.362189 0.932105i \(-0.382030\pi\)
0.362189 + 0.932105i \(0.382030\pi\)
\(602\) 9.94010e6 1.11789
\(603\) 0 0
\(604\) 4.58540e6 0.511428
\(605\) 0 0
\(606\) 0 0
\(607\) 700912. 0.0772132 0.0386066 0.999254i \(-0.487708\pi\)
0.0386066 + 0.999254i \(0.487708\pi\)
\(608\) 3.04622e6 0.334197
\(609\) 0 0
\(610\) −1.02638e7 −1.11682
\(611\) −1.36776e6 −0.148221
\(612\) 0 0
\(613\) 1.17591e7 1.26393 0.631966 0.774996i \(-0.282248\pi\)
0.631966 + 0.774996i \(0.282248\pi\)
\(614\) 1.07198e7 1.14753
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00683e6 0.106474 0.0532371 0.998582i \(-0.483046\pi\)
0.0532371 + 0.998582i \(0.483046\pi\)
\(618\) 0 0
\(619\) 1.27458e7 1.33703 0.668513 0.743700i \(-0.266931\pi\)
0.668513 + 0.743700i \(0.266931\pi\)
\(620\) −5.56044e6 −0.580938
\(621\) 0 0
\(622\) −2.75050e7 −2.85060
\(623\) −6.90744e6 −0.713012
\(624\) 0 0
\(625\) −1.09908e7 −1.12546
\(626\) −2.46155e7 −2.51058
\(627\) 0 0
\(628\) −5.72172e6 −0.578932
\(629\) 1.77821e7 1.79207
\(630\) 0 0
\(631\) 1.41284e7 1.41260 0.706299 0.707913i \(-0.250363\pi\)
0.706299 + 0.707913i \(0.250363\pi\)
\(632\) −1.60810e6 −0.160148
\(633\) 0 0
\(634\) −1.72062e7 −1.70005
\(635\) 5.41029e6 0.532459
\(636\) 0 0
\(637\) −2.61004e6 −0.254858
\(638\) 0 0
\(639\) 0 0
\(640\) −3.03708e6 −0.293094
\(641\) 4.36680e6 0.419777 0.209888 0.977725i \(-0.432690\pi\)
0.209888 + 0.977725i \(0.432690\pi\)
\(642\) 0 0
\(643\) 7.81597e6 0.745513 0.372757 0.927929i \(-0.378413\pi\)
0.372757 + 0.927929i \(0.378413\pi\)
\(644\) −2.29573e7 −2.18125
\(645\) 0 0
\(646\) −5.52551e6 −0.520944
\(647\) −2.01624e7 −1.89357 −0.946786 0.321863i \(-0.895691\pi\)
−0.946786 + 0.321863i \(0.895691\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −2.31762e6 −0.215159
\(651\) 0 0
\(652\) −1.87723e7 −1.72941
\(653\) −324619. −0.0297914 −0.0148957 0.999889i \(-0.504742\pi\)
−0.0148957 + 0.999889i \(0.504742\pi\)
\(654\) 0 0
\(655\) 3.90963e6 0.356067
\(656\) 1.02348e7 0.928581
\(657\) 0 0
\(658\) −2.63837e6 −0.237559
\(659\) −1.07107e7 −0.960740 −0.480370 0.877066i \(-0.659498\pi\)
−0.480370 + 0.877066i \(0.659498\pi\)
\(660\) 0 0
\(661\) −1.11064e7 −0.988712 −0.494356 0.869260i \(-0.664596\pi\)
−0.494356 + 0.869260i \(0.664596\pi\)
\(662\) 1.00986e7 0.895601
\(663\) 0 0
\(664\) −2.40822e6 −0.211971
\(665\) 3.18448e6 0.279244
\(666\) 0 0
\(667\) −7.75464e6 −0.674912
\(668\) 1.93970e7 1.68187
\(669\) 0 0
\(670\) −2.53480e7 −2.18151
\(671\) 0 0
\(672\) 0 0
\(673\) 1.38137e7 1.17564 0.587818 0.808993i \(-0.299987\pi\)
0.587818 + 0.808993i \(0.299987\pi\)
\(674\) −5.56208e6 −0.471615
\(675\) 0 0
\(676\) 276160. 0.0232431
\(677\) 2.29090e6 0.192103 0.0960514 0.995376i \(-0.469379\pi\)
0.0960514 + 0.995376i \(0.469379\pi\)
\(678\) 0 0
\(679\) 5.57067e6 0.463695
\(680\) 2.74200e6 0.227402
\(681\) 0 0
\(682\) 0 0
\(683\) −4.40512e6 −0.361332 −0.180666 0.983545i \(-0.557825\pi\)
−0.180666 + 0.983545i \(0.557825\pi\)
\(684\) 0 0
\(685\) −318685. −0.0259498
\(686\) 1.49287e7 1.21119
\(687\) 0 0
\(688\) 7.67828e6 0.618434
\(689\) 1.45986e7 1.17156
\(690\) 0 0
\(691\) −5.86199e6 −0.467035 −0.233518 0.972353i \(-0.575024\pi\)
−0.233518 + 0.972353i \(0.575024\pi\)
\(692\) 9.35440e6 0.742592
\(693\) 0 0
\(694\) 2.19054e7 1.72645
\(695\) −5.38653e6 −0.423007
\(696\) 0 0
\(697\) −2.05324e7 −1.60087
\(698\) 1.94569e7 1.51160
\(699\) 0 0
\(700\) −2.33663e6 −0.180238
\(701\) −8.02106e6 −0.616505 −0.308253 0.951305i \(-0.599744\pi\)
−0.308253 + 0.951305i \(0.599744\pi\)
\(702\) 0 0
\(703\) −3.54190e6 −0.270301
\(704\) 0 0
\(705\) 0 0
\(706\) −5.22946e6 −0.394862
\(707\) 5.94954e6 0.447646
\(708\) 0 0
\(709\) 2.17891e7 1.62788 0.813941 0.580948i \(-0.197318\pi\)
0.813941 + 0.580948i \(0.197318\pi\)
\(710\) 685512. 0.0510351
\(711\) 0 0
\(712\) 1.18468e6 0.0875789
\(713\) −1.19709e7 −0.881863
\(714\) 0 0
\(715\) 0 0
\(716\) −106179. −0.00774029
\(717\) 0 0
\(718\) 1.23505e7 0.894076
\(719\) −1.03483e7 −0.746531 −0.373266 0.927724i \(-0.621762\pi\)
−0.373266 + 0.927724i \(0.621762\pi\)
\(720\) 0 0
\(721\) 7.20052e6 0.515853
\(722\) −1.91730e7 −1.36882
\(723\) 0 0
\(724\) 2.66730e7 1.89115
\(725\) −789280. −0.0557682
\(726\) 0 0
\(727\) −2.03348e7 −1.42693 −0.713466 0.700690i \(-0.752876\pi\)
−0.713466 + 0.700690i \(0.752876\pi\)
\(728\) 2.22262e6 0.155430
\(729\) 0 0
\(730\) −3.55244e7 −2.46729
\(731\) −1.54037e7 −1.06618
\(732\) 0 0
\(733\) −4.78280e6 −0.328793 −0.164396 0.986394i \(-0.552568\pi\)
−0.164396 + 0.986394i \(0.552568\pi\)
\(734\) 1.45224e7 0.994946
\(735\) 0 0
\(736\) −3.75249e7 −2.55344
\(737\) 0 0
\(738\) 0 0
\(739\) 1.08737e7 0.732429 0.366215 0.930530i \(-0.380654\pi\)
0.366215 + 0.930530i \(0.380654\pi\)
\(740\) 2.02665e7 1.36050
\(741\) 0 0
\(742\) 2.81603e7 1.87770
\(743\) −1.01036e7 −0.671434 −0.335717 0.941963i \(-0.608979\pi\)
−0.335717 + 0.941963i \(0.608979\pi\)
\(744\) 0 0
\(745\) 3.98089e6 0.262778
\(746\) −1.86006e7 −1.22371
\(747\) 0 0
\(748\) 0 0
\(749\) 985227. 0.0641700
\(750\) 0 0
\(751\) 9.91947e6 0.641784 0.320892 0.947116i \(-0.396017\pi\)
0.320892 + 0.947116i \(0.396017\pi\)
\(752\) −2.03802e6 −0.131421
\(753\) 0 0
\(754\) 8.65673e6 0.554530
\(755\) 7.83526e6 0.500249
\(756\) 0 0
\(757\) −1.33506e7 −0.846759 −0.423380 0.905952i \(-0.639156\pi\)
−0.423380 + 0.905952i \(0.639156\pi\)
\(758\) −3.62232e7 −2.28988
\(759\) 0 0
\(760\) −546162. −0.0342995
\(761\) 1.28109e7 0.801898 0.400949 0.916100i \(-0.368680\pi\)
0.400949 + 0.916100i \(0.368680\pi\)
\(762\) 0 0
\(763\) 1.40401e7 0.873089
\(764\) 1.50896e7 0.935284
\(765\) 0 0
\(766\) −1.94531e7 −1.19789
\(767\) 1.20185e7 0.737671
\(768\) 0 0
\(769\) −1.90629e7 −1.16245 −0.581224 0.813743i \(-0.697426\pi\)
−0.581224 + 0.813743i \(0.697426\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.54054e6 0.576142
\(773\) 2.34434e6 0.141115 0.0705574 0.997508i \(-0.477522\pi\)
0.0705574 + 0.997508i \(0.477522\pi\)
\(774\) 0 0
\(775\) −1.21841e6 −0.0728686
\(776\) −955410. −0.0569555
\(777\) 0 0
\(778\) 2.17716e7 1.28956
\(779\) 4.08972e6 0.241463
\(780\) 0 0
\(781\) 0 0
\(782\) 6.80661e7 3.98028
\(783\) 0 0
\(784\) −3.88906e6 −0.225972
\(785\) −9.77694e6 −0.566277
\(786\) 0 0
\(787\) 1.52283e7 0.876425 0.438212 0.898871i \(-0.355612\pi\)
0.438212 + 0.898871i \(0.355612\pi\)
\(788\) 2.01315e7 1.15494
\(789\) 0 0
\(790\) −3.16838e7 −1.80622
\(791\) −3.08911e7 −1.75547
\(792\) 0 0
\(793\) 1.28926e7 0.728042
\(794\) 1.76619e7 0.994228
\(795\) 0 0
\(796\) 9.45788e6 0.529068
\(797\) 2.70618e7 1.50907 0.754537 0.656257i \(-0.227861\pi\)
0.754537 + 0.656257i \(0.227861\pi\)
\(798\) 0 0
\(799\) 4.08855e6 0.226570
\(800\) −3.81935e6 −0.210991
\(801\) 0 0
\(802\) 1.98987e7 1.09242
\(803\) 0 0
\(804\) 0 0
\(805\) −3.92281e7 −2.13357
\(806\) 1.33634e7 0.724569
\(807\) 0 0
\(808\) −1.02039e6 −0.0549842
\(809\) −2.25663e6 −0.121224 −0.0606121 0.998161i \(-0.519305\pi\)
−0.0606121 + 0.998161i \(0.519305\pi\)
\(810\) 0 0
\(811\) −4.49121e6 −0.239779 −0.119890 0.992787i \(-0.538254\pi\)
−0.119890 + 0.992787i \(0.538254\pi\)
\(812\) 8.72773e6 0.464527
\(813\) 0 0
\(814\) 0 0
\(815\) −3.20770e7 −1.69161
\(816\) 0 0
\(817\) 3.06816e6 0.160814
\(818\) −5.01769e7 −2.62193
\(819\) 0 0
\(820\) −2.34011e7 −1.21535
\(821\) 592581. 0.0306825 0.0153412 0.999882i \(-0.495117\pi\)
0.0153412 + 0.999882i \(0.495117\pi\)
\(822\) 0 0
\(823\) −1.14748e7 −0.590533 −0.295266 0.955415i \(-0.595408\pi\)
−0.295266 + 0.955415i \(0.595408\pi\)
\(824\) −1.23494e6 −0.0633620
\(825\) 0 0
\(826\) 2.31833e7 1.18229
\(827\) −8.47060e6 −0.430676 −0.215338 0.976540i \(-0.569085\pi\)
−0.215338 + 0.976540i \(0.569085\pi\)
\(828\) 0 0
\(829\) −1.58876e7 −0.802919 −0.401460 0.915877i \(-0.631497\pi\)
−0.401460 + 0.915877i \(0.631497\pi\)
\(830\) −4.74483e7 −2.39070
\(831\) 0 0
\(832\) 2.38106e7 1.19251
\(833\) 7.80197e6 0.389576
\(834\) 0 0
\(835\) 3.31445e7 1.64511
\(836\) 0 0
\(837\) 0 0
\(838\) 3.07552e6 0.151290
\(839\) −2.66963e7 −1.30932 −0.654659 0.755924i \(-0.727188\pi\)
−0.654659 + 0.755924i \(0.727188\pi\)
\(840\) 0 0
\(841\) −1.75631e7 −0.856268
\(842\) 2.88480e7 1.40228
\(843\) 0 0
\(844\) 2.63972e7 1.27556
\(845\) 471886. 0.0227350
\(846\) 0 0
\(847\) 0 0
\(848\) 2.17525e7 1.03877
\(849\) 0 0
\(850\) 6.92789e6 0.328892
\(851\) 4.36310e7 2.06524
\(852\) 0 0
\(853\) −3.58773e6 −0.168829 −0.0844146 0.996431i \(-0.526902\pi\)
−0.0844146 + 0.996431i \(0.526902\pi\)
\(854\) 2.48693e7 1.16686
\(855\) 0 0
\(856\) −168974. −0.00788197
\(857\) −6.00941e6 −0.279499 −0.139749 0.990187i \(-0.544630\pi\)
−0.139749 + 0.990187i \(0.544630\pi\)
\(858\) 0 0
\(859\) −1.74629e7 −0.807484 −0.403742 0.914873i \(-0.632291\pi\)
−0.403742 + 0.914873i \(0.632291\pi\)
\(860\) −1.75558e7 −0.809422
\(861\) 0 0
\(862\) 2.58149e7 1.18332
\(863\) −2.34431e6 −0.107149 −0.0535746 0.998564i \(-0.517061\pi\)
−0.0535746 + 0.998564i \(0.517061\pi\)
\(864\) 0 0
\(865\) 1.59842e7 0.726360
\(866\) 1.32779e7 0.601635
\(867\) 0 0
\(868\) 1.34730e7 0.606968
\(869\) 0 0
\(870\) 0 0
\(871\) 3.18401e7 1.42210
\(872\) −2.40798e6 −0.107241
\(873\) 0 0
\(874\) −1.35577e7 −0.600354
\(875\) 2.31502e7 1.02220
\(876\) 0 0
\(877\) −1.98979e7 −0.873591 −0.436796 0.899561i \(-0.643887\pi\)
−0.436796 + 0.899561i \(0.643887\pi\)
\(878\) 2.03175e7 0.889474
\(879\) 0 0
\(880\) 0 0
\(881\) −2.32718e7 −1.01016 −0.505081 0.863072i \(-0.668537\pi\)
−0.505081 + 0.863072i \(0.668537\pi\)
\(882\) 0 0
\(883\) −2.71777e7 −1.17304 −0.586518 0.809936i \(-0.699502\pi\)
−0.586518 + 0.809936i \(0.699502\pi\)
\(884\) −3.97142e7 −1.70929
\(885\) 0 0
\(886\) −3.07421e7 −1.31568
\(887\) −1.39671e7 −0.596069 −0.298034 0.954555i \(-0.596331\pi\)
−0.298034 + 0.954555i \(0.596331\pi\)
\(888\) 0 0
\(889\) −1.31092e7 −0.556316
\(890\) 2.33413e7 0.987755
\(891\) 0 0
\(892\) −3.49579e7 −1.47107
\(893\) −814374. −0.0341740
\(894\) 0 0
\(895\) −181433. −0.00757109
\(896\) 7.35889e6 0.306226
\(897\) 0 0
\(898\) 3.93761e7 1.62945
\(899\) 4.55099e6 0.187805
\(900\) 0 0
\(901\) −4.36385e7 −1.79084
\(902\) 0 0
\(903\) 0 0
\(904\) 5.29805e6 0.215623
\(905\) 4.55773e7 1.84981
\(906\) 0 0
\(907\) 1.77875e7 0.717954 0.358977 0.933346i \(-0.383126\pi\)
0.358977 + 0.933346i \(0.383126\pi\)
\(908\) −1.73517e7 −0.698437
\(909\) 0 0
\(910\) 4.37914e7 1.75302
\(911\) −30398.8 −0.00121356 −0.000606780 1.00000i \(-0.500193\pi\)
−0.000606780 1.00000i \(0.500193\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −5.81288e7 −2.30158
\(915\) 0 0
\(916\) −7.77486e6 −0.306164
\(917\) −9.47308e6 −0.372021
\(918\) 0 0
\(919\) 4.10055e6 0.160160 0.0800798 0.996788i \(-0.474482\pi\)
0.0800798 + 0.996788i \(0.474482\pi\)
\(920\) 6.72790e6 0.262066
\(921\) 0 0
\(922\) 6.65453e7 2.57804
\(923\) −861085. −0.0332692
\(924\) 0 0
\(925\) 4.44084e6 0.170652
\(926\) −2.18908e7 −0.838946
\(927\) 0 0
\(928\) 1.42659e7 0.543788
\(929\) 1.46532e7 0.557048 0.278524 0.960429i \(-0.410155\pi\)
0.278524 + 0.960429i \(0.410155\pi\)
\(930\) 0 0
\(931\) −1.55403e6 −0.0587604
\(932\) 2.17039e7 0.818461
\(933\) 0 0
\(934\) 3.54310e7 1.32897
\(935\) 0 0
\(936\) 0 0
\(937\) 3.97538e7 1.47921 0.739604 0.673042i \(-0.235013\pi\)
0.739604 + 0.673042i \(0.235013\pi\)
\(938\) 6.14185e7 2.27925
\(939\) 0 0
\(940\) 4.65979e6 0.172007
\(941\) 5.32850e6 0.196169 0.0980847 0.995178i \(-0.468728\pi\)
0.0980847 + 0.995178i \(0.468728\pi\)
\(942\) 0 0
\(943\) −5.03793e7 −1.84490
\(944\) 1.79081e7 0.654062
\(945\) 0 0
\(946\) 0 0
\(947\) −3.11430e7 −1.12846 −0.564230 0.825618i \(-0.690827\pi\)
−0.564230 + 0.825618i \(0.690827\pi\)
\(948\) 0 0
\(949\) 4.46229e7 1.60839
\(950\) −1.37993e6 −0.0496074
\(951\) 0 0
\(952\) −6.64389e6 −0.237591
\(953\) 4.87227e7 1.73780 0.868899 0.494990i \(-0.164828\pi\)
0.868899 + 0.494990i \(0.164828\pi\)
\(954\) 0 0
\(955\) 2.57842e7 0.914839
\(956\) −1.03529e7 −0.366369
\(957\) 0 0
\(958\) 1.27628e7 0.449297
\(959\) 772177. 0.0271125
\(960\) 0 0
\(961\) −2.16038e7 −0.754608
\(962\) −4.87065e7 −1.69687
\(963\) 0 0
\(964\) −2.42873e7 −0.841755
\(965\) 1.63023e7 0.563548
\(966\) 0 0
\(967\) −4.85436e7 −1.66942 −0.834711 0.550688i \(-0.814365\pi\)
−0.834711 + 0.550688i \(0.814365\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −1.88241e7 −0.642370
\(971\) 3.15035e7 1.07229 0.536143 0.844127i \(-0.319881\pi\)
0.536143 + 0.844127i \(0.319881\pi\)
\(972\) 0 0
\(973\) 1.30516e7 0.441960
\(974\) −6.25393e7 −2.11230
\(975\) 0 0
\(976\) 1.92104e7 0.645525
\(977\) 5.35354e7 1.79434 0.897170 0.441684i \(-0.145619\pi\)
0.897170 + 0.441684i \(0.145619\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 8.89204e6 0.295758
\(981\) 0 0
\(982\) 2.95472e7 0.977771
\(983\) 5.40925e7 1.78547 0.892736 0.450580i \(-0.148783\pi\)
0.892736 + 0.450580i \(0.148783\pi\)
\(984\) 0 0
\(985\) 3.43996e7 1.12970
\(986\) −2.58769e7 −0.847655
\(987\) 0 0
\(988\) 7.91044e6 0.257815
\(989\) −3.77952e7 −1.22870
\(990\) 0 0
\(991\) 2.31007e7 0.747208 0.373604 0.927588i \(-0.378122\pi\)
0.373604 + 0.927588i \(0.378122\pi\)
\(992\) 2.20223e7 0.710533
\(993\) 0 0
\(994\) −1.66100e6 −0.0533218
\(995\) 1.61611e7 0.517502
\(996\) 0 0
\(997\) 4.54061e7 1.44669 0.723346 0.690486i \(-0.242603\pi\)
0.723346 + 0.690486i \(0.242603\pi\)
\(998\) −6.92788e7 −2.20178
\(999\) 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 1089.6.a.r.1.3 3
3.2 odd 2 121.6.a.d.1.1 3
11.10 odd 2 99.6.a.g.1.1 3
33.32 even 2 11.6.a.b.1.3 3
132.131 odd 2 176.6.a.i.1.3 3
165.32 odd 4 275.6.b.b.199.5 6
165.98 odd 4 275.6.b.b.199.2 6
165.164 even 2 275.6.a.b.1.1 3
231.230 odd 2 539.6.a.e.1.3 3
264.131 odd 2 704.6.a.t.1.1 3
264.197 even 2 704.6.a.q.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 33.32 even 2
99.6.a.g.1.1 3 11.10 odd 2
121.6.a.d.1.1 3 3.2 odd 2
176.6.a.i.1.3 3 132.131 odd 2
275.6.a.b.1.1 3 165.164 even 2
275.6.b.b.199.2 6 165.98 odd 4
275.6.b.b.199.5 6 165.32 odd 4
539.6.a.e.1.3 3 231.230 odd 2
704.6.a.q.1.3 3 264.197 even 2
704.6.a.t.1.1 3 264.131 odd 2
1089.6.a.r.1.3 3 1.1 even 1 trivial