Properties

Label 99.6.a.g.1.1
Level $99$
Weight $6$
Character 99.1
Self dual yes
Analytic conductor $15.878$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.1
Root \(-6.29828\) 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)
 
\(f(q)\) \(=\) \(q-8.18772 q^{2} +35.0388 q^{4} +59.8722 q^{5} +145.071 q^{7} -24.8808 q^{8} +O(q^{10})\) \(q-8.18772 q^{2} +35.0388 q^{4} +59.8722 q^{5} +145.071 q^{7} -24.8808 q^{8} -490.217 q^{10} -121.000 q^{11} +615.772 q^{13} -1187.80 q^{14} -917.524 q^{16} -1840.68 q^{17} +366.633 q^{19} +2097.85 q^{20} +990.714 q^{22} +4516.38 q^{23} +459.685 q^{25} -5041.77 q^{26} +5083.12 q^{28} +1717.00 q^{29} -2650.54 q^{31} +8308.62 q^{32} +15070.9 q^{34} +8685.74 q^{35} +9660.61 q^{37} -3001.89 q^{38} -1489.67 q^{40} +11154.8 q^{41} +8368.48 q^{43} -4239.69 q^{44} -36978.9 q^{46} +2221.22 q^{47} +4238.64 q^{49} -3763.77 q^{50} +21575.9 q^{52} -23707.9 q^{53} -7244.54 q^{55} -3609.48 q^{56} -14058.3 q^{58} -19517.8 q^{59} +20937.3 q^{61} +21701.9 q^{62} -38667.9 q^{64} +36867.6 q^{65} -51707.7 q^{67} -64495.1 q^{68} -71116.4 q^{70} +1398.38 q^{71} +72466.6 q^{73} -79098.4 q^{74} +12846.4 q^{76} -17553.6 q^{77} +64632.2 q^{79} -54934.2 q^{80} -91332.4 q^{82} +96790.3 q^{83} -110205. q^{85} -68518.8 q^{86} +3010.57 q^{88} +47614.1 q^{89} +89330.7 q^{91} +158249. q^{92} -18186.7 q^{94} +21951.2 q^{95} -38399.6 q^{97} -34704.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 84 q^{4} - 24 q^{5} + 84 q^{7} + 564 q^{8} - 414 q^{10} - 363 q^{11} + 486 q^{13} + 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 1380 q^{19} + 3480 q^{20} + 3066 q^{23} - 57 q^{25} - 12132 q^{26} + 23712 q^{28} + 3426 q^{29} - 4098 q^{31} + 12408 q^{32} + 25320 q^{34} + 24228 q^{35} + 17724 q^{37} + 9240 q^{38} - 15276 q^{40} - 5994 q^{41} - 26208 q^{43} - 10164 q^{44} - 16806 q^{46} + 17232 q^{47} + 48531 q^{49} - 41070 q^{50} - 35304 q^{52} - 50586 q^{53} + 2904 q^{55} + 42312 q^{56} - 29172 q^{58} + 3738 q^{59} + 18486 q^{61} + 19974 q^{62} - 20352 q^{64} + 7668 q^{65} - 47754 q^{67} + 12600 q^{68} - 123372 q^{70} - 39282 q^{71} + 15426 q^{73} - 153294 q^{74} + 103920 q^{76} - 10164 q^{77} + 125148 q^{79} - 118680 q^{80} - 255372 q^{82} + 143928 q^{83} - 104040 q^{85} - 243060 q^{86} - 68244 q^{88} + 106824 q^{89} - 109632 q^{91} + 336528 q^{92} - 74928 q^{94} + 22200 q^{95} + 9684 q^{97} - 3480 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.18772 −1.44740 −0.723699 0.690116i \(-0.757560\pi\)
−0.723699 + 0.690116i \(0.757560\pi\)
\(3\) 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 \(-0.310990\pi\)
0.559508 + 0.828825i \(0.310990\pi\)
\(8\) −24.8808 −0.137448
\(9\) 0 0
\(10\) −490.217 −1.55020
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) 615.772 1.01056 0.505279 0.862956i \(-0.331390\pi\)
0.505279 + 0.862956i \(0.331390\pi\)
\(14\) −1187.80 −1.61966
\(15\) 0 0
\(16\) −917.524 −0.896020
\(17\) −1840.68 −1.54474 −0.772369 0.635174i \(-0.780929\pi\)
−0.772369 + 0.635174i \(0.780929\pi\)
\(18\) 0 0
\(19\) 366.633 0.232996 0.116498 0.993191i \(-0.462833\pi\)
0.116498 + 0.993191i \(0.462833\pi\)
\(20\) 2097.85 1.17273
\(21\) 0 0
\(22\) 990.714 0.436407
\(23\) 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 \(-0.439294\pi\)
0.189560 + 0.981869i \(0.439294\pi\)
\(30\) 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 \(-0.326614\pi\)
0.518170 + 0.855278i \(0.326614\pi\)
\(42\) 0 0
\(43\) 8368.48 0.690201 0.345100 0.938566i \(-0.387845\pi\)
0.345100 + 0.938566i \(0.387845\pi\)
\(44\) −4239.69 −0.330144
\(45\) 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
\(55\) −7244.54 −0.322927
\(56\) −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 \(-0.382702\pi\)
0.360218 + 0.932868i \(0.382702\pi\)
\(62\) 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 \(-0.207054\pi\)
0.795794 + 0.605567i \(0.207054\pi\)
\(74\) −79098.4 −1.67915
\(75\) 0 0
\(76\) 12846.4 0.255122
\(77\) −17553.6 −0.337396
\(78\) 0 0
\(79\) 64632.2 1.16515 0.582574 0.812777i \(-0.302045\pi\)
0.582574 + 0.812777i \(0.302045\pi\)
\(80\) −54934.2 −0.959662
\(81\) 0 0
\(82\) −91332.4 −1.50000
\(83\) 96790.3 1.54219 0.771093 0.636723i \(-0.219710\pi\)
0.771093 + 0.636723i \(0.219710\pi\)
\(84\) 0 0
\(85\) −110205. −1.65446
\(86\) −68518.8 −0.998995
\(87\) 0 0
\(88\) 3010.57 0.0414422
\(89\) 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 \(-0.435900\pi\)
0.200018 + 0.979792i \(0.435900\pi\)
\(102\) 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 \(-0.490872\pi\)
0.0286725 + 0.999589i \(0.490872\pi\)
\(108\) 0 0
\(109\) 96780.7 0.780230 0.390115 0.920766i \(-0.372435\pi\)
0.390115 + 0.920766i \(0.372435\pi\)
\(110\) 59316.3 0.467404
\(111\) 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
\(121\) 14641.0 0.0909091
\(122\) −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 \(-0.579962\pi\)
−0.248574 + 0.968613i \(0.579962\pi\)
\(128\) 50726.1 0.273657
\(129\) 0 0
\(130\) −301862. −1.56657
\(131\) −65299.5 −0.332454 −0.166227 0.986088i \(-0.553158\pi\)
−0.166227 + 0.986088i \(0.553158\pi\)
\(132\) 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 \(-0.436725\pi\)
0.197477 + 0.980308i \(0.436725\pi\)
\(140\) 304338. 1.31231
\(141\) 0 0
\(142\) −11449.6 −0.0476506
\(143\) −74508.4 −0.304695
\(144\) 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 \(-0.539148\pi\)
−0.122676 + 0.992447i \(0.539148\pi\)
\(150\) 0 0
\(151\) −130866. −0.467074 −0.233537 0.972348i \(-0.575030\pi\)
−0.233537 + 0.972348i \(0.575030\pi\)
\(152\) −9122.12 −0.0320248
\(153\) 0 0
\(154\) 143724. 0.488346
\(155\) −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 \(-0.778751\pi\)
−0.768005 + 0.640443i \(0.778751\pi\)
\(168\) 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 \(-0.610121\pi\)
−0.339095 + 0.940752i \(0.610121\pi\)
\(174\) 0 0
\(175\) 66687.0 0.164606
\(176\) 111020. 0.270160
\(177\) 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
\(187\) 222722. 0.465756
\(188\) 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 \(-0.584741\pi\)
−0.263088 + 0.964772i \(0.584741\pi\)
\(194\) 314405. 0.599770
\(195\) 0 0
\(196\) 148517. 0.276144
\(197\) −574550. −1.05478 −0.527390 0.849623i \(-0.676829\pi\)
−0.527390 + 0.849623i \(0.676829\pi\)
\(198\) 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
\(209\) −44362.6 −0.0702509
\(210\) 0 0
\(211\) −753372. −1.16494 −0.582470 0.812853i \(-0.697913\pi\)
−0.582470 + 0.812853i \(0.697913\pi\)
\(212\) −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
\(220\) −253840. −0.353593
\(221\) −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 \(-0.396676\pi\)
0.318932 + 0.947778i \(0.396676\pi\)
\(228\) 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 \(-0.621925\pi\)
−0.373739 + 0.927534i \(0.621925\pi\)
\(234\) 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 \(-0.446496\pi\)
0.167298 + 0.985906i \(0.446496\pi\)
\(240\) 0 0
\(241\) 693153. 0.768753 0.384376 0.923176i \(-0.374416\pi\)
0.384376 + 0.923176i \(0.374416\pi\)
\(242\) −119876. −0.131582
\(243\) 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
\(253\) −546482. −0.536753
\(254\) 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 \(-0.410570\pi\)
0.277270 + 0.960792i \(0.410570\pi\)
\(264\) 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 \(-0.651337\pi\)
−0.457730 + 0.889091i \(0.651337\pi\)
\(272\) 1.68887e6 1.38412
\(273\) 0 0
\(274\) 43581.1 0.0350689
\(275\) −55621.9 −0.0443521
\(276\) 0 0
\(277\) 639062. 0.500430 0.250215 0.968190i \(-0.419499\pi\)
0.250215 + 0.968190i \(0.419499\pi\)
\(278\) −736626. −0.571656
\(279\) 0 0
\(280\) −216108. −0.164731
\(281\) 257984. 0.194907 0.0974534 0.995240i \(-0.468930\pi\)
0.0974534 + 0.995240i \(0.468930\pi\)
\(282\) 0 0
\(283\) 1.02991e6 0.764425 0.382213 0.924074i \(-0.375162\pi\)
0.382213 + 0.924074i \(0.375162\pi\)
\(284\) 48997.7 0.0360479
\(285\) 0 0
\(286\) 610054. 0.441015
\(287\) 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 \(-0.596534\pi\)
−0.298644 + 0.954365i \(0.596534\pi\)
\(294\) 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 \(-0.629745\pi\)
−0.396413 + 0.918072i \(0.629745\pi\)
\(308\) −615057. −0.369436
\(309\) 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
\(319\) −207757. −0.114309
\(320\) −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 \(-0.447909\pi\)
0.162918 + 0.986640i \(0.447909\pi\)
\(338\) −64531.9 −0.0307243
\(339\) 0 0
\(340\) −3.86146e6 −1.81157
\(341\) 320715. 0.149360
\(342\) 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 \(-0.703402\pi\)
−0.596397 + 0.802690i \(0.703402\pi\)
\(348\) 0 0
\(349\) −2.37636e6 −1.04435 −0.522177 0.852837i \(-0.674880\pi\)
−0.522177 + 0.852837i \(0.674880\pi\)
\(350\) −546015. −0.238251
\(351\) 0 0
\(352\) −1.00534e6 −0.432472
\(353\) −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 \(-0.599946\pi\)
−0.308856 + 0.951109i \(0.599946\pi\)
\(360\) 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 \(-0.361073\pi\)
0.422728 + 0.906257i \(0.361073\pi\)
\(374\) −1.82358e6 −0.674135
\(375\) 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
\(385\) −1.05097e6 −0.361360
\(386\) 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
\(407\) −1.16893e6 −0.349787
\(408\) 0 0
\(409\) 6.12831e6 1.81148 0.905738 0.423839i \(-0.139318\pi\)
0.905738 + 0.423839i \(0.139318\pi\)
\(410\) −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
\(418\) 363229. 0.101681
\(419\) 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 \(-0.634044\pi\)
−0.408774 + 0.912636i \(0.634044\pi\)
\(432\) 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 \(-0.599414\pi\)
−0.307266 + 0.951624i \(0.599414\pi\)
\(440\) 180250. 0.0443857
\(441\) 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
\(451\) −1.34973e6 −0.312468
\(452\) 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 \(-0.207432\pi\)
0.795075 + 0.606512i \(0.207432\pi\)
\(458\) 1.81680e6 0.404709
\(459\) 0 0
\(460\) 9.47469e6 2.08771
\(461\) −8.12745e6 −1.78116 −0.890578 0.454830i \(-0.849700\pi\)
−0.890578 + 0.454830i \(0.849700\pi\)
\(462\) 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
\(473\) −1.01259e6 −0.208103
\(474\) 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 \(-0.549605\pi\)
−0.155208 + 0.987882i \(0.549605\pi\)
\(480\) 0 0
\(481\) 5.94873e6 1.17236
\(482\) −5.67535e6 −1.11269
\(483\) 0 0
\(484\) 513003. 0.0995420
\(485\) −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 \(-0.609672\pi\)
−0.337768 + 0.941229i \(0.609672\pi\)
\(492\) 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 \(-0.239565\pi\)
0.729904 + 0.683550i \(0.239565\pi\)
\(504\) 0 0
\(505\) 2.45543e6 0.428449
\(506\) 4.47444e6 0.776896
\(507\) 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
\(517\) −268768. −0.0442233
\(518\) −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 \(-0.793885\pi\)
−0.797577 + 0.603217i \(0.793885\pi\)
\(524\) −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
\(539\) −512876. −0.0760397
\(540\) 0 0
\(541\) −4.34177e6 −0.637784 −0.318892 0.947791i \(-0.603311\pi\)
−0.318892 + 0.947791i \(0.603311\pi\)
\(542\) 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 \(-0.194361\pi\)
0.819302 + 0.573363i \(0.194361\pi\)
\(548\) −186503. −0.0265298
\(549\) 0 0
\(550\) 455417. 0.0641951
\(551\) 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 \(-0.465787\pi\)
0.107278 + 0.994229i \(0.465787\pi\)
\(558\) 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 \(-0.481984\pi\)
0.0565694 + 0.998399i \(0.481984\pi\)
\(564\) 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 \(-0.217954\pi\)
0.774595 + 0.632458i \(0.217954\pi\)
\(570\) 0 0
\(571\) −7.97842e6 −1.02406 −0.512032 0.858967i \(-0.671107\pi\)
−0.512032 + 0.858967i \(0.671107\pi\)
\(572\) −2.61068e6 −0.333629
\(573\) 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
\(583\) 2.86865e6 0.349548
\(584\) −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 \(-0.712155\pi\)
−0.618243 + 0.785987i \(0.712155\pi\)
\(594\) 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 \(-0.617970\pi\)
−0.362189 + 0.932105i \(0.617970\pi\)
\(602\) −9.94010e6 −1.11789
\(603\) 0 0
\(604\) −4.58540e6 −0.511428
\(605\) 876589. 0.0973661
\(606\) 0 0
\(607\) −700912. −0.0772132 −0.0386066 0.999254i \(-0.512292\pi\)
−0.0386066 + 0.999254i \(0.512292\pi\)
\(608\) 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 \(-0.717752\pi\)
−0.631966 + 0.774996i \(0.717752\pi\)
\(614\) 1.07198e7 1.14753
\(615\) 0 0
\(616\) 436747. 0.0463744
\(617\) 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
\(638\) 1.70106e6 0.165450
\(639\) 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
\(649\) 2.36166e6 0.220092
\(650\) −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 \(-0.340502\pi\)
0.480370 + 0.877066i \(0.340502\pi\)
\(660\) 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
\(671\) −2.53341e6 −0.217219
\(672\) 0 0
\(673\) −1.38137e7 −1.17564 −0.587818 0.808993i \(-0.700013\pi\)
−0.587818 + 0.808993i \(0.700013\pi\)
\(674\) −5.56208e6 −0.471615
\(675\) 0 0
\(676\) 276160. 0.0232431
\(677\) −2.29090e6 −0.192103 −0.0960514 0.995376i \(-0.530621\pi\)
−0.0960514 + 0.995376i \(0.530621\pi\)
\(678\) 0 0
\(679\) −5.57067e6 −0.463695
\(680\) 2.74200e6 0.227402
\(681\) 0 0
\(682\) −2.62593e6 −0.216183
\(683\) −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 \(-0.400256\pi\)
0.308253 + 0.951305i \(0.400256\pi\)
\(702\) 0 0
\(703\) 3.54190e6 0.270301
\(704\) 4.67881e6 0.355799
\(705\) 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
\(715\) −4.46098e6 −0.326336
\(716\) −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 \(-0.447432\pi\)
0.164396 + 0.986394i \(0.447432\pi\)
\(734\) −1.45224e7 −0.994946
\(735\) 0 0
\(736\) 3.75249e7 2.55344
\(737\) 6.25663e6 0.424298
\(738\) 0 0
\(739\) −1.08737e7 −0.732429 −0.366215 0.930530i \(-0.619346\pi\)
−0.366215 + 0.930530i \(0.619346\pi\)
\(740\) 2.02665e7 1.36050
\(741\) 0 0
\(742\) 2.81603e7 1.87770
\(743\) 1.01036e7 0.671434 0.335717 0.941963i \(-0.391021\pi\)
0.335717 + 0.941963i \(0.391021\pi\)
\(744\) 0 0
\(745\) −3.98089e6 −0.262778
\(746\) −1.86006e7 −1.22371
\(747\) 0 0
\(748\) 7.80390e6 0.509986
\(749\) 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 \(-0.631320\pi\)
−0.400949 + 0.916100i \(0.631320\pi\)
\(762\) 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 \(-0.302574\pi\)
0.581224 + 0.813743i \(0.302574\pi\)
\(770\) 8.60508e6 0.523032
\(771\) 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
\(781\) −169204. −0.00992623
\(782\) 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 \(-0.644388\pi\)
−0.438212 + 0.898871i \(0.644388\pi\)
\(788\) −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
\(803\) −8.76846e6 −0.479882
\(804\) 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 \(-0.480695\pi\)
0.0606121 + 0.998161i \(0.480695\pi\)
\(810\) 0 0
\(811\) 4.49121e6 0.239779 0.119890 0.992787i \(-0.461746\pi\)
0.119890 + 0.992787i \(0.461746\pi\)
\(812\) 8.72773e6 0.464527
\(813\) 0 0
\(814\) 9.57091e6 0.506282
\(815\) −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 \(-0.504883\pi\)
−0.0153412 + 0.999882i \(0.504883\pi\)
\(822\) 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 \(-0.430915\pi\)
0.215338 + 0.976540i \(0.430915\pi\)
\(828\) 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
\(836\) −1.55441e6 −0.0769221
\(837\) 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
\(847\) 2.12399e6 0.101729
\(848\) 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 \(-0.473098\pi\)
0.0844146 + 0.996431i \(0.473098\pi\)
\(854\) −2.48693e7 −1.16686
\(855\) 0 0
\(856\) −168974. −0.00788197
\(857\) 6.00941e6 0.279499 0.139749 0.990187i \(-0.455370\pi\)
0.139749 + 0.990187i \(0.455370\pi\)
\(858\) 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
\(869\) −7.82050e6 −0.351306
\(870\) 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 \(-0.356113\pi\)
0.436796 + 0.899561i \(0.356113\pi\)
\(878\) 2.03175e7 0.889474
\(879\) 0 0
\(880\) 6.64704e6 0.289349
\(881\) −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 \(-0.403669\pi\)
0.298034 + 0.954555i \(0.403669\pi\)
\(888\) 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
\(902\) 1.10512e7 0.452266
\(903\) 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
\(913\) −1.17116e7 −0.464987
\(914\) −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 \(-0.525518\pi\)
−0.0800798 + 0.996788i \(0.525518\pi\)
\(920\) −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
\(935\) 1.33349e7 0.498838
\(936\) 0 0
\(937\) −3.97538e7 −1.47921 −0.739604 0.673042i \(-0.764987\pi\)
−0.739604 + 0.673042i \(0.764987\pi\)
\(938\) 6.14185e7 2.27925
\(939\) 0 0
\(940\) 4.65979e6 0.172007
\(941\) −5.32850e6 −0.196169 −0.0980847 0.995178i \(-0.531272\pi\)
−0.0980847 + 0.995178i \(0.531272\pi\)
\(942\) 0 0
\(943\) 5.03793e7 1.84490
\(944\) 1.79081e7 0.654062
\(945\) 0 0
\(946\) 8.29077e6 0.301208
\(947\) −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 \(-0.835172\pi\)
−0.868899 + 0.494990i \(0.835172\pi\)
\(954\) 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 \(-0.185635\pi\)
0.834711 + 0.550688i \(0.185635\pi\)
\(968\) −364279. −0.0124953
\(969\) 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
\(979\) −5.76131e6 −0.192116
\(980\) 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 \(-0.757397\pi\)
−0.723346 + 0.690486i \(0.757397\pi\)
\(998\) 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 99.6.a.g.1.1 3
3.2 odd 2 11.6.a.b.1.3 3
11.10 odd 2 1089.6.a.r.1.3 3
12.11 even 2 176.6.a.i.1.3 3
15.2 even 4 275.6.b.b.199.5 6
15.8 even 4 275.6.b.b.199.2 6
15.14 odd 2 275.6.a.b.1.1 3
21.20 even 2 539.6.a.e.1.3 3
24.5 odd 2 704.6.a.q.1.3 3
24.11 even 2 704.6.a.t.1.1 3
33.32 even 2 121.6.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 3.2 odd 2
99.6.a.g.1.1 3 1.1 even 1 trivial
121.6.a.d.1.1 3 33.32 even 2
176.6.a.i.1.3 3 12.11 even 2
275.6.a.b.1.1 3 15.14 odd 2
275.6.b.b.199.2 6 15.8 even 4
275.6.b.b.199.5 6 15.2 even 4
539.6.a.e.1.3 3 21.20 even 2
704.6.a.q.1.3 3 24.5 odd 2
704.6.a.t.1.1 3 24.11 even 2
1089.6.a.r.1.3 3 11.10 odd 2