Properties

Label 704.6.a.q.1.3
Level $704$
Weight $6$
Character 704.1
Self dual yes
Analytic conductor $112.910$
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] = [704,6,Mod(1,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("704.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 704.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: \(112.910209148\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(\chi\) \(=\) 704.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.48600 q^{3} +59.8722 q^{5} +145.071 q^{7} -230.848 q^{9} +O(q^{10})\) \(q+3.48600 q^{3} +59.8722 q^{5} +145.071 q^{7} -230.848 q^{9} -121.000 q^{11} -615.772 q^{13} +208.715 q^{15} +1840.68 q^{17} -366.633 q^{19} +505.718 q^{21} -4516.38 q^{23} +459.685 q^{25} -1651.83 q^{27} +1717.00 q^{29} -2650.54 q^{31} -421.806 q^{33} +8685.74 q^{35} -9660.61 q^{37} -2146.58 q^{39} -11154.8 q^{41} -8368.48 q^{43} -13821.4 q^{45} -2221.22 q^{47} +4238.64 q^{49} +6416.60 q^{51} -23707.9 q^{53} -7244.54 q^{55} -1278.08 q^{57} -19517.8 q^{59} -20937.3 q^{61} -33489.4 q^{63} -36867.6 q^{65} +51707.7 q^{67} -15744.1 q^{69} -1398.38 q^{71} +72466.6 q^{73} +1602.46 q^{75} -17553.6 q^{77} +64632.2 q^{79} +50337.7 q^{81} +96790.3 q^{83} +110205. q^{85} +5985.47 q^{87} -47614.1 q^{89} -89330.7 q^{91} -9239.79 q^{93} -21951.2 q^{95} -38399.6 q^{97} +27932.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} - 24 q^{5} + 84 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} - 24 q^{5} + 84 q^{7} - 7 q^{9} - 363 q^{11} - 486 q^{13} + 1654 q^{15} + 1086 q^{17} - 1380 q^{19} + 908 q^{21} - 3066 q^{23} - 57 q^{25} + 2990 q^{27} + 3426 q^{29} - 4098 q^{31} + 4114 q^{33} + 24228 q^{35} - 17724 q^{37} - 6560 q^{39} + 5994 q^{41} + 26208 q^{43} - 18458 q^{45} - 17232 q^{47} + 48531 q^{49} + 22724 q^{51} - 50586 q^{53} + 2904 q^{55} + 20160 q^{57} + 3738 q^{59} - 18486 q^{61} - 12496 q^{63} - 7668 q^{65} + 47754 q^{67} - 35042 q^{69} + 39282 q^{71} + 15426 q^{73} + 21916 q^{75} - 10164 q^{77} + 125148 q^{79} - 86917 q^{81} + 143928 q^{83} + 104040 q^{85} - 19368 q^{87} - 106824 q^{89} + 109632 q^{91} + 16622 q^{93} - 22200 q^{95} + 9684 q^{97} + 847 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.48600 0.223627 0.111814 0.993729i \(-0.464334\pi\)
0.111814 + 0.993729i \(0.464334\pi\)
\(4\) 0 0
\(5\) 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 \(0.310990\pi\)
\(8\) 0 0
\(9\) −230.848 −0.949991
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 0 0
\(13\) −615.772 −1.01056 −0.505279 0.862956i \(-0.668610\pi\)
−0.505279 + 0.862956i \(0.668610\pi\)
\(14\) 0 0
\(15\) 208.715 0.239511
\(16\) 0 0
\(17\) 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 \(0.537167\pi\)
\(20\) 0 0
\(21\) 505.718 0.250242
\(22\) 0 0
\(23\) −4516.38 −1.78021 −0.890104 0.455757i \(-0.849369\pi\)
−0.890104 + 0.455757i \(0.849369\pi\)
\(24\) 0 0
\(25\) 459.685 0.147099
\(26\) 0 0
\(27\) −1651.83 −0.436071
\(28\) 0 0
\(29\) 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 \(0.579670\pi\)
\(32\) 0 0
\(33\) −421.806 −0.0674261
\(34\) 0 0
\(35\) 8685.74 1.19850
\(36\) 0 0
\(37\) −9660.61 −1.16011 −0.580057 0.814576i \(-0.696970\pi\)
−0.580057 + 0.814576i \(0.696970\pi\)
\(38\) 0 0
\(39\) −2146.58 −0.225988
\(40\) 0 0
\(41\) −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
\(45\) −13821.4 −1.01747
\(46\) 0 0
\(47\) −2221.22 −0.146672 −0.0733360 0.997307i \(-0.523365\pi\)
−0.0733360 + 0.997307i \(0.523365\pi\)
\(48\) 0 0
\(49\) 4238.64 0.252195
\(50\) 0 0
\(51\) 6416.60 0.345445
\(52\) 0 0
\(53\) −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
\(56\) 0 0
\(57\) −1278.08 −0.0521042
\(58\) 0 0
\(59\) −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 \(0.617298\pi\)
\(62\) 0 0
\(63\) −33489.4 −1.06305
\(64\) 0 0
\(65\) −36867.6 −1.08233
\(66\) 0 0
\(67\) 51707.7 1.40724 0.703619 0.710577i \(-0.251566\pi\)
0.703619 + 0.710577i \(0.251566\pi\)
\(68\) 0 0
\(69\) −15744.1 −0.398103
\(70\) 0 0
\(71\) −1398.38 −0.0329216 −0.0164608 0.999865i \(-0.505240\pi\)
−0.0164608 + 0.999865i \(0.505240\pi\)
\(72\) 0 0
\(73\) 72466.6 1.59159 0.795794 0.605567i \(-0.207054\pi\)
0.795794 + 0.605567i \(0.207054\pi\)
\(74\) 0 0
\(75\) 1602.46 0.0328954
\(76\) 0 0
\(77\) −17553.6 −0.337396
\(78\) 0 0
\(79\) 64632.2 1.16515 0.582574 0.812777i \(-0.302045\pi\)
0.582574 + 0.812777i \(0.302045\pi\)
\(80\) 0 0
\(81\) 50337.7 0.852474
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 5985.47 0.0847814
\(88\) 0 0
\(89\) −47614.1 −0.637178 −0.318589 0.947893i \(-0.603209\pi\)
−0.318589 + 0.947893i \(0.603209\pi\)
\(90\) 0 0
\(91\) −89330.7 −1.13083
\(92\) 0 0
\(93\) −9239.79 −0.110778
\(94\) 0 0
\(95\) −21951.2 −0.249545
\(96\) 0 0
\(97\) −38399.6 −0.414378 −0.207189 0.978301i \(-0.566432\pi\)
−0.207189 + 0.978301i \(0.566432\pi\)
\(98\) 0 0
\(99\) 27932.6 0.286433
\(100\) 0 0
\(101\) 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 \(0.574034\pi\)
\(104\) 0 0
\(105\) 30278.5 0.268016
\(106\) 0 0
\(107\) 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 \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(110\) 0 0
\(111\) −33676.9 −0.259433
\(112\) 0 0
\(113\) −212938. −1.56876 −0.784379 0.620281i \(-0.787018\pi\)
−0.784379 + 0.620281i \(0.787018\pi\)
\(114\) 0 0
\(115\) −270406. −1.90665
\(116\) 0 0
\(117\) 142149. 0.960021
\(118\) 0 0
\(119\) 267029. 1.72859
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 0 0
\(123\) −38885.6 −0.231754
\(124\) 0 0
\(125\) −159578. −0.913480
\(126\) 0 0
\(127\) −90363.9 −0.497148 −0.248574 0.968613i \(-0.579962\pi\)
−0.248574 + 0.968613i \(0.579962\pi\)
\(128\) 0 0
\(129\) −29172.5 −0.154348
\(130\) 0 0
\(131\) −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
\(134\) 0 0
\(135\) −98899.0 −0.467044
\(136\) 0 0
\(137\) 5322.74 0.0242289 0.0121145 0.999927i \(-0.496144\pi\)
0.0121145 + 0.999927i \(0.496144\pi\)
\(138\) 0 0
\(139\) −89967.1 −0.394954 −0.197477 0.980308i \(-0.563275\pi\)
−0.197477 + 0.980308i \(0.563275\pi\)
\(140\) 0 0
\(141\) −7743.18 −0.0327998
\(142\) 0 0
\(143\) 74508.4 0.304695
\(144\) 0 0
\(145\) 102801. 0.406047
\(146\) 0 0
\(147\) 14775.9 0.0563977
\(148\) 0 0
\(149\) −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 \(0.575030\pi\)
\(152\) 0 0
\(153\) −424916. −1.46749
\(154\) 0 0
\(155\) −158694. −0.530555
\(156\) 0 0
\(157\) 163297. 0.528723 0.264362 0.964424i \(-0.414839\pi\)
0.264362 + 0.964424i \(0.414839\pi\)
\(158\) 0 0
\(159\) −82645.6 −0.259255
\(160\) 0 0
\(161\) −655197. −1.99208
\(162\) 0 0
\(163\) 535758. 1.57943 0.789713 0.613477i \(-0.210229\pi\)
0.789713 + 0.613477i \(0.210229\pi\)
\(164\) 0 0
\(165\) −25254.5 −0.0722152
\(166\) 0 0
\(167\) 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
\(170\) 0 0
\(171\) 84636.5 0.221344
\(172\) 0 0
\(173\) −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
\(176\) 0 0
\(177\) −68039.2 −0.163240
\(178\) 0 0
\(179\) −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 \(-0.831775\pi\)
−0.863568 + 0.504233i \(0.831775\pi\)
\(182\) 0 0
\(183\) −72987.3 −0.161109
\(184\) 0 0
\(185\) −578402. −1.24251
\(186\) 0 0
\(187\) −222722. −0.465756
\(188\) 0 0
\(189\) −239634. −0.487970
\(190\) 0 0
\(191\) −430653. −0.854170 −0.427085 0.904212i \(-0.640459\pi\)
−0.427085 + 0.904212i \(0.640459\pi\)
\(192\) 0 0
\(193\) −272285. −0.526175 −0.263088 0.964772i \(-0.584741\pi\)
−0.263088 + 0.964772i \(0.584741\pi\)
\(194\) 0 0
\(195\) −128521. −0.242039
\(196\) 0 0
\(197\) −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 \(0.422331\pi\)
\(200\) 0 0
\(201\) 180253. 0.314697
\(202\) 0 0
\(203\) 249088. 0.424240
\(204\) 0 0
\(205\) −667863. −1.10995
\(206\) 0 0
\(207\) 1.04260e6 1.69118
\(208\) 0 0
\(209\) 44362.6 0.0702509
\(210\) 0 0
\(211\) 753372. 1.16494 0.582470 0.812853i \(-0.302087\pi\)
0.582470 + 0.812853i \(0.302087\pi\)
\(212\) 0 0
\(213\) −4874.77 −0.00736216
\(214\) 0 0
\(215\) −501040. −0.739224
\(216\) 0 0
\(217\) −384517. −0.554327
\(218\) 0 0
\(219\) 252619. 0.355922
\(220\) 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 \(0.734455\pi\)
\(224\) 0 0
\(225\) −106117. −0.139743
\(226\) 0 0
\(227\) 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 \(-0.455352\pi\)
0.139806 + 0.990179i \(0.455352\pi\)
\(230\) 0 0
\(231\) −61191.9 −0.0754508
\(232\) 0 0
\(233\) 619425. 0.747479 0.373739 0.927534i \(-0.378075\pi\)
0.373739 + 0.927534i \(0.378075\pi\)
\(234\) 0 0
\(235\) −132989. −0.157090
\(236\) 0 0
\(237\) 225308. 0.260559
\(238\) 0 0
\(239\) −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 \(-0.374416\pi\)
0.384376 + 0.923176i \(0.374416\pi\)
\(242\) 0 0
\(243\) 576873. 0.626707
\(244\) 0 0
\(245\) 253777. 0.270108
\(246\) 0 0
\(247\) 225762. 0.235456
\(248\) 0 0
\(249\) 337411. 0.344875
\(250\) 0 0
\(251\) −533816. −0.534820 −0.267410 0.963583i \(-0.586168\pi\)
−0.267410 + 0.963583i \(0.586168\pi\)
\(252\) 0 0
\(253\) 546482. 0.536753
\(254\) 0 0
\(255\) 384176. 0.369981
\(256\) 0 0
\(257\) −652296. −0.616044 −0.308022 0.951379i \(-0.599667\pi\)
−0.308022 + 0.951379i \(0.599667\pi\)
\(258\) 0 0
\(259\) −1.40148e6 −1.29818
\(260\) 0 0
\(261\) −396366. −0.360160
\(262\) 0 0
\(263\) −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
\(266\) 0 0
\(267\) −165983. −0.142490
\(268\) 0 0
\(269\) 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 \(0.651337\pi\)
\(272\) 0 0
\(273\) −311407. −0.252884
\(274\) 0 0
\(275\) −55621.9 −0.0443521
\(276\) 0 0
\(277\) −639062. −0.500430 −0.250215 0.968190i \(-0.580501\pi\)
−0.250215 + 0.968190i \(0.580501\pi\)
\(278\) 0 0
\(279\) 611872. 0.470598
\(280\) 0 0
\(281\) −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 \(0.624838\pi\)
\(284\) 0 0
\(285\) −76521.8 −0.0558050
\(286\) 0 0
\(287\) −1.61824e6 −1.15968
\(288\) 0 0
\(289\) 1.96823e6 1.38622
\(290\) 0 0
\(291\) −133861. −0.0926662
\(292\) 0 0
\(293\) −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
\(296\) 0 0
\(297\) 199872. 0.131480
\(298\) 0 0
\(299\) 2.78106e6 1.79900
\(300\) 0 0
\(301\) −1.21403e6 −0.772345
\(302\) 0 0
\(303\) 142965. 0.0894589
\(304\) 0 0
\(305\) −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
\(309\) −173026. −0.103089
\(310\) 0 0
\(311\) 3.35930e6 1.96946 0.984731 0.174083i \(-0.0556962\pi\)
0.984731 + 0.174083i \(0.0556962\pi\)
\(312\) 0 0
\(313\) −3.00640e6 −1.73454 −0.867272 0.497835i \(-0.834129\pi\)
−0.867272 + 0.497835i \(0.834129\pi\)
\(314\) 0 0
\(315\) −2.00508e6 −1.13856
\(316\) 0 0
\(317\) −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
\(320\) 0 0
\(321\) 23674.6 0.0128239
\(322\) 0 0
\(323\) −674853. −0.359918
\(324\) 0 0
\(325\) −283061. −0.148652
\(326\) 0 0
\(327\) −337378. −0.174481
\(328\) 0 0
\(329\) −322235. −0.164128
\(330\) 0 0
\(331\) −1.23338e6 −0.618766 −0.309383 0.950938i \(-0.600122\pi\)
−0.309383 + 0.950938i \(0.600122\pi\)
\(332\) 0 0
\(333\) 2.23013e6 1.10210
\(334\) 0 0
\(335\) 3.09585e6 1.50719
\(336\) 0 0
\(337\) 679319. 0.325836 0.162918 0.986640i \(-0.447909\pi\)
0.162918 + 0.986640i \(0.447909\pi\)
\(338\) 0 0
\(339\) −742301. −0.350817
\(340\) 0 0
\(341\) 320715. 0.149360
\(342\) 0 0
\(343\) −1.82331e6 −0.836805
\(344\) 0 0
\(345\) −942635. −0.426379
\(346\) 0 0
\(347\) −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 \(-0.325120\pi\)
0.522177 + 0.852837i \(0.325120\pi\)
\(350\) 0 0
\(351\) 1.01715e6 0.440675
\(352\) 0 0
\(353\) 638696. 0.272808 0.136404 0.990653i \(-0.456445\pi\)
0.136404 + 0.990653i \(0.456445\pi\)
\(354\) 0 0
\(355\) −83724.4 −0.0352599
\(356\) 0 0
\(357\) 930864. 0.386559
\(358\) 0 0
\(359\) 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
\(362\) 0 0
\(363\) 51038.5 0.0203297
\(364\) 0 0
\(365\) 4.33874e6 1.70463
\(366\) 0 0
\(367\) 1.77368e6 0.687403 0.343701 0.939079i \(-0.388319\pi\)
0.343701 + 0.939079i \(0.388319\pi\)
\(368\) 0 0
\(369\) 2.57506e6 0.984513
\(370\) 0 0
\(371\) −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
\(375\) −556290. −0.204279
\(376\) 0 0
\(377\) −1.05728e6 −0.383122
\(378\) 0 0
\(379\) 4.42409e6 1.58207 0.791035 0.611771i \(-0.209543\pi\)
0.791035 + 0.611771i \(0.209543\pi\)
\(380\) 0 0
\(381\) −315009. −0.111176
\(382\) 0 0
\(383\) 2.37588e6 0.827615 0.413807 0.910364i \(-0.364199\pi\)
0.413807 + 0.910364i \(0.364199\pi\)
\(384\) 0 0
\(385\) −1.05097e6 −0.361360
\(386\) 0 0
\(387\) 1.93185e6 0.655684
\(388\) 0 0
\(389\) 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
\(392\) 0 0
\(393\) −227634. −0.0743457
\(394\) 0 0
\(395\) 3.86968e6 1.24791
\(396\) 0 0
\(397\) −2.15712e6 −0.686907 −0.343453 0.939170i \(-0.611597\pi\)
−0.343453 + 0.939170i \(0.611597\pi\)
\(398\) 0 0
\(399\) −185413. −0.0583054
\(400\) 0 0
\(401\) −2.43031e6 −0.754744 −0.377372 0.926062i \(-0.623172\pi\)
−0.377372 + 0.926062i \(0.623172\pi\)
\(402\) 0 0
\(403\) 1.63213e6 0.500601
\(404\) 0 0
\(405\) 3.01383e6 0.913022
\(406\) 0 0
\(407\) 1.16893e6 0.349787
\(408\) 0 0
\(409\) 6.12831e6 1.81148 0.905738 0.423839i \(-0.139318\pi\)
0.905738 + 0.423839i \(0.139318\pi\)
\(410\) 0 0
\(411\) 18555.1 0.00541824
\(412\) 0 0
\(413\) −2.83147e6 −0.816841
\(414\) 0 0
\(415\) 5.79505e6 1.65172
\(416\) 0 0
\(417\) −313625. −0.0883225
\(418\) 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 \(-0.660968\pi\)
−0.484416 + 0.874838i \(0.660968\pi\)
\(422\) 0 0
\(423\) 512764. 0.139337
\(424\) 0 0
\(425\) 846131. 0.227230
\(426\) 0 0
\(427\) −3.03739e6 −0.806178
\(428\) 0 0
\(429\) 259736. 0.0681380
\(430\) 0 0
\(431\) 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 \(0.433359\pi\)
\(434\) 0 0
\(435\) 358364. 0.0908032
\(436\) 0 0
\(437\) 1.65586e6 0.414781
\(438\) 0 0
\(439\) −2.48145e6 −0.614533 −0.307266 0.951624i \(-0.599414\pi\)
−0.307266 + 0.951624i \(0.599414\pi\)
\(440\) 0 0
\(441\) −978482. −0.239583
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) −231783. −0.0548673
\(448\) 0 0
\(449\) −4.80916e6 −1.12578 −0.562890 0.826532i \(-0.690311\pi\)
−0.562890 + 0.826532i \(0.690311\pi\)
\(450\) 0 0
\(451\) 1.34973e6 0.312468
\(452\) 0 0
\(453\) −456200. −0.104450
\(454\) 0 0
\(455\) −5.34843e6 −1.21115
\(456\) 0 0
\(457\) 7.09951e6 1.59015 0.795075 0.606512i \(-0.207432\pi\)
0.795075 + 0.606512i \(0.207432\pi\)
\(458\) 0 0
\(459\) −3.04049e6 −0.673616
\(460\) 0 0
\(461\) −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 \(0.593593\pi\)
\(464\) 0 0
\(465\) −553207. −0.118647
\(466\) 0 0
\(467\) 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
\(470\) 0 0
\(471\) 569253. 0.118237
\(472\) 0 0
\(473\) 1.01259e6 0.208103
\(474\) 0 0
\(475\) −168536. −0.0342735
\(476\) 0 0
\(477\) 5.47291e6 1.10134
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) −2.28402e6 −0.445483
\(484\) 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 \(0.760335\pi\)
\(488\) 0 0
\(489\) 1.86765e6 0.353202
\(490\) 0 0
\(491\) −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
\(494\) 0 0
\(495\) 1.67239e6 0.306778
\(496\) 0 0
\(497\) −202865. −0.0368397
\(498\) 0 0
\(499\) 8.46131e6 1.52120 0.760599 0.649221i \(-0.224905\pi\)
0.760599 + 0.649221i \(0.224905\pi\)
\(500\) 0 0
\(501\) 1.92980e6 0.343494
\(502\) 0 0
\(503\) −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
\(507\) 27475.1 0.00474700
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) 605618. 0.101603
\(514\) 0 0
\(515\) −2.97172e6 −0.493731
\(516\) 0 0
\(517\) 268768. 0.0442233
\(518\) 0 0
\(519\) −930667. −0.151662
\(520\) 0 0
\(521\) 9.60432e6 1.55015 0.775073 0.631872i \(-0.217713\pi\)
0.775073 + 0.631872i \(0.217713\pi\)
\(522\) 0 0
\(523\) 9.97831e6 1.59515 0.797577 0.603217i \(-0.206115\pi\)
0.797577 + 0.603217i \(0.206115\pi\)
\(524\) 0 0
\(525\) 232471. 0.0368104
\(526\) 0 0
\(527\) −4.87879e6 −0.765218
\(528\) 0 0
\(529\) 1.39613e7 2.16914
\(530\) 0 0
\(531\) 4.50565e6 0.693459
\(532\) 0 0
\(533\) 6.86881e6 1.04728
\(534\) 0 0
\(535\) 406612. 0.0614181
\(536\) 0 0
\(537\) −10563.8 −0.00158082
\(538\) 0 0
\(539\) −512876. −0.0760397
\(540\) 0 0
\(541\) 4.34177e6 0.637784 0.318892 0.947791i \(-0.396689\pi\)
0.318892 + 0.947791i \(0.396689\pi\)
\(542\) 0 0
\(543\) −2.65369e6 −0.386234
\(544\) 0 0
\(545\) −5.79448e6 −0.835647
\(546\) 0 0
\(547\) −1.14668e7 −1.63860 −0.819302 0.573363i \(-0.805639\pi\)
−0.819302 + 0.573363i \(0.805639\pi\)
\(548\) 0 0
\(549\) 4.83332e6 0.684407
\(550\) 0 0
\(551\) −629511. −0.0883332
\(552\) 0 0
\(553\) 9.37627e6 1.30382
\(554\) 0 0
\(555\) −2.01631e6 −0.277860
\(556\) 0 0
\(557\) 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
\(560\) 0 0
\(561\) −776409. −0.104156
\(562\) 0 0
\(563\) 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
\(566\) 0 0
\(567\) 7.30255e6 0.953931
\(568\) 0 0
\(569\) −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
\(573\) −1.50126e6 −0.191016
\(574\) 0 0
\(575\) −2.07611e6 −0.261867
\(576\) 0 0
\(577\) 5.90743e6 0.738685 0.369342 0.929293i \(-0.379583\pi\)
0.369342 + 0.929293i \(0.379583\pi\)
\(578\) 0 0
\(579\) −949186. −0.117667
\(580\) 0 0
\(581\) 1.40415e7 1.72573
\(582\) 0 0
\(583\) 2.86865e6 0.349548
\(584\) 0 0
\(585\) 8.51081e6 1.02821
\(586\) 0 0
\(587\) −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
\(590\) 0 0
\(591\) −2.00288e6 −0.235877
\(592\) 0 0
\(593\) 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
\(596\) 0 0
\(597\) 940962. 0.108053
\(598\) 0 0
\(599\) 3.48377e6 0.396718 0.198359 0.980129i \(-0.436439\pi\)
0.198359 + 0.980129i \(0.436439\pi\)
\(600\) 0 0
\(601\) −6.41433e6 −0.724378 −0.362189 0.932105i \(-0.617970\pi\)
−0.362189 + 0.932105i \(0.617970\pi\)
\(602\) 0 0
\(603\) −1.19366e7 −1.33686
\(604\) 0 0
\(605\) 876589. 0.0973661
\(606\) 0 0
\(607\) −700912. −0.0772132 −0.0386066 0.999254i \(-0.512292\pi\)
−0.0386066 + 0.999254i \(0.512292\pi\)
\(608\) 0 0
\(609\) 868320. 0.0948717
\(610\) 0 0
\(611\) 1.36776e6 0.148221
\(612\) 0 0
\(613\) 1.17591e7 1.26393 0.631966 0.774996i \(-0.282248\pi\)
0.631966 + 0.774996i \(0.282248\pi\)
\(614\) 0 0
\(615\) −2.32817e6 −0.248214
\(616\) 0 0
\(617\) −1.00683e6 −0.106474 −0.0532371 0.998582i \(-0.516954\pi\)
−0.0532371 + 0.998582i \(0.516954\pi\)
\(618\) 0 0
\(619\) −1.27458e7 −1.33703 −0.668513 0.743700i \(-0.733069\pi\)
−0.668513 + 0.743700i \(0.733069\pi\)
\(620\) 0 0
\(621\) 7.46031e6 0.776297
\(622\) 0 0
\(623\) −6.90744e6 −0.713012
\(624\) 0 0
\(625\) −1.09908e7 −1.12546
\(626\) 0 0
\(627\) 154648. 0.0157100
\(628\) 0 0
\(629\) −1.77821e7 −1.79207
\(630\) 0 0
\(631\) 1.41284e7 1.41260 0.706299 0.707913i \(-0.250363\pi\)
0.706299 + 0.707913i \(0.250363\pi\)
\(632\) 0 0
\(633\) 2.62626e6 0.260512
\(634\) 0 0
\(635\) −5.41029e6 −0.532459
\(636\) 0 0
\(637\) −2.61004e6 −0.254858
\(638\) 0 0
\(639\) 322814. 0.0312752
\(640\) 0 0
\(641\) −4.36680e6 −0.419777 −0.209888 0.977725i \(-0.567310\pi\)
−0.209888 + 0.977725i \(0.567310\pi\)
\(642\) 0 0
\(643\) −7.81597e6 −0.745513 −0.372757 0.927929i \(-0.621587\pi\)
−0.372757 + 0.927929i \(0.621587\pi\)
\(644\) 0 0
\(645\) −1.74662e6 −0.165310
\(646\) 0 0
\(647\) 2.01624e7 1.89357 0.946786 0.321863i \(-0.104309\pi\)
0.946786 + 0.321863i \(0.104309\pi\)
\(648\) 0 0
\(649\) 2.36166e6 0.220092
\(650\) 0 0
\(651\) −1.34043e6 −0.123963
\(652\) 0 0
\(653\) −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
\(656\) 0 0
\(657\) −1.67288e7 −1.51199
\(658\) 0 0
\(659\) 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 \(-0.335404\pi\)
0.494356 + 0.869260i \(0.335404\pi\)
\(662\) 0 0
\(663\) −3.95116e6 −0.349093
\(664\) 0 0
\(665\) −3.18448e6 −0.279244
\(666\) 0 0
\(667\) −7.75464e6 −0.674912
\(668\) 0 0
\(669\) −3.47795e6 −0.300441
\(670\) 0 0
\(671\) 2.53341e6 0.217219
\(672\) 0 0
\(673\) −1.38137e7 −1.17564 −0.587818 0.808993i \(-0.700013\pi\)
−0.587818 + 0.808993i \(0.700013\pi\)
\(674\) 0 0
\(675\) −759323. −0.0641457
\(676\) 0 0
\(677\) −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
\(680\) 0 0
\(681\) 1.72632e6 0.142644
\(682\) 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
\(686\) 0 0
\(687\) 773519. 0.0625287
\(688\) 0 0
\(689\) 1.45986e7 1.17156
\(690\) 0 0
\(691\) 5.86199e6 0.467035 0.233518 0.972353i \(-0.424976\pi\)
0.233518 + 0.972353i \(0.424976\pi\)
\(692\) 0 0
\(693\) 4.05221e6 0.320523
\(694\) 0 0
\(695\) −5.38653e6 −0.423007
\(696\) 0 0
\(697\) −2.05324e7 −1.60087
\(698\) 0 0
\(699\) 2.15932e6 0.167157
\(700\) 0 0
\(701\) 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
\(704\) 0 0
\(705\) −463602. −0.0351295
\(706\) 0 0
\(707\) 5.94954e6 0.447646
\(708\) 0 0
\(709\) −2.17891e7 −1.62788 −0.813941 0.580948i \(-0.802682\pi\)
−0.813941 + 0.580948i \(0.802682\pi\)
\(710\) 0 0
\(711\) −1.49202e7 −1.10688
\(712\) 0 0
\(713\) 1.19709e7 0.881863
\(714\) 0 0
\(715\) 4.46098e6 0.326336
\(716\) 0 0
\(717\) −1.03001e6 −0.0748246
\(718\) 0 0
\(719\) 1.03483e7 0.746531 0.373266 0.927724i \(-0.378238\pi\)
0.373266 + 0.927724i \(0.378238\pi\)
\(720\) 0 0
\(721\) −7.20052e6 −0.515853
\(722\) 0 0
\(723\) 2.41633e6 0.171914
\(724\) 0 0
\(725\) 789280. 0.0557682
\(726\) 0 0
\(727\) −2.03348e7 −1.42693 −0.713466 0.700690i \(-0.752876\pi\)
−0.713466 + 0.700690i \(0.752876\pi\)
\(728\) 0 0
\(729\) −1.02211e7 −0.712325
\(730\) 0 0
\(731\) −1.54037e7 −1.06618
\(732\) 0 0
\(733\) −4.78280e6 −0.328793 −0.164396 0.986394i \(-0.552568\pi\)
−0.164396 + 0.986394i \(0.552568\pi\)
\(734\) 0 0
\(735\) 884668. 0.0604034
\(736\) 0 0
\(737\) −6.25663e6 −0.424298
\(738\) 0 0
\(739\) 1.08737e7 0.732429 0.366215 0.930530i \(-0.380654\pi\)
0.366215 + 0.930530i \(0.380654\pi\)
\(740\) 0 0
\(741\) 787008. 0.0526543
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) −2.23438e7 −1.46506
\(748\) 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 \(0.396017\pi\)
\(752\) 0 0
\(753\) −1.86088e6 −0.119600
\(754\) 0 0
\(755\) −7.83526e6 −0.500249
\(756\) 0 0
\(757\) 1.33506e7 0.846759 0.423380 0.905952i \(-0.360844\pi\)
0.423380 + 0.905952i \(0.360844\pi\)
\(758\) 0 0
\(759\) 1.90504e6 0.120033
\(760\) 0 0
\(761\) 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
\(764\) 0 0
\(765\) −2.54407e7 −1.57172
\(766\) 0 0
\(767\) 1.20185e7 0.737671
\(768\) 0 0
\(769\) 1.90629e7 1.16245 0.581224 0.813743i \(-0.302574\pi\)
0.581224 + 0.813743i \(0.302574\pi\)
\(770\) 0 0
\(771\) −2.27390e6 −0.137764
\(772\) 0 0
\(773\) 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
\(776\) 0 0
\(777\) −4.88555e6 −0.290309
\(778\) 0 0
\(779\) 4.08972e6 0.241463
\(780\) 0 0
\(781\) 169204. 0.00992623
\(782\) 0 0
\(783\) −2.83620e6 −0.165323
\(784\) 0 0
\(785\) 9.77694e6 0.566277
\(786\) 0 0
\(787\) 1.52283e7 0.876425 0.438212 0.898871i \(-0.355612\pi\)
0.438212 + 0.898871i \(0.355612\pi\)
\(788\) 0 0
\(789\) −2.16845e6 −0.124010
\(790\) 0 0
\(791\) −3.08911e7 −1.75547
\(792\) 0 0
\(793\) 1.28926e7 0.728042
\(794\) 0 0
\(795\) −4.94818e6 −0.277669
\(796\) 0 0
\(797\) 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
\(800\) 0 0
\(801\) 1.09916e7 0.605313
\(802\) 0 0
\(803\) −8.76846e6 −0.479882
\(804\) 0 0
\(805\) −3.92281e7 −2.13357
\(806\) 0 0
\(807\) 1.68326e6 0.0909844
\(808\) 0 0
\(809\) −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 \(0.538254\pi\)
\(812\) 0 0
\(813\) −3.85825e6 −0.204722
\(814\) 0 0
\(815\) 3.20770e7 1.69161
\(816\) 0 0
\(817\) 3.06816e6 0.160814
\(818\) 0 0
\(819\) 2.06218e7 1.07428
\(820\) 0 0
\(821\) −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 \(0.595408\pi\)
\(824\) 0 0
\(825\) −193898. −0.00991833
\(826\) 0 0
\(827\) 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 \(-0.368503\pi\)
0.401460 + 0.915877i \(0.368503\pi\)
\(830\) 0 0
\(831\) −2.22777e6 −0.111910
\(832\) 0 0
\(833\) 7.80197e6 0.389576
\(834\) 0 0
\(835\) 3.31445e7 1.64511
\(836\) 0 0
\(837\) 4.37825e6 0.216017
\(838\) 0 0
\(839\) 2.66963e7 1.30932 0.654659 0.755924i \(-0.272812\pi\)
0.654659 + 0.755924i \(0.272812\pi\)
\(840\) 0 0
\(841\) −1.75631e7 −0.856268
\(842\) 0 0
\(843\) −899333. −0.0435864
\(844\) 0 0
\(845\) 471886. 0.0227350
\(846\) 0 0
\(847\) 2.12399e6 0.101729
\(848\) 0 0
\(849\) −3.59028e6 −0.170946
\(850\) 0 0
\(851\) 4.36310e7 2.06524
\(852\) 0 0
\(853\) −3.58773e6 −0.168829 −0.0844146 0.996431i \(-0.526902\pi\)
−0.0844146 + 0.996431i \(0.526902\pi\)
\(854\) 0 0
\(855\) 5.06738e6 0.237065
\(856\) 0 0
\(857\) −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 \(-0.367709\pi\)
0.403742 + 0.914873i \(0.367709\pi\)
\(860\) 0 0
\(861\) −5.64118e6 −0.259336
\(862\) 0 0
\(863\) 2.34431e6 0.107149 0.0535746 0.998564i \(-0.482939\pi\)
0.0535746 + 0.998564i \(0.482939\pi\)
\(864\) 0 0
\(865\) −1.59842e7 −0.726360
\(866\) 0 0
\(867\) 6.86126e6 0.309996
\(868\) 0 0
\(869\) −7.82050e6 −0.351306
\(870\) 0 0
\(871\) −3.18401e7 −1.42210
\(872\) 0 0
\(873\) 8.86445e6 0.393655
\(874\) 0 0
\(875\) −2.31502e7 −1.02220
\(876\) 0 0
\(877\) −1.98979e7 −0.873591 −0.436796 0.899561i \(-0.643887\pi\)
−0.436796 + 0.899561i \(0.643887\pi\)
\(878\) 0 0
\(879\) −3.05971e6 −0.133570
\(880\) 0 0
\(881\) 2.32718e7 1.01016 0.505081 0.863072i \(-0.331463\pi\)
0.505081 + 0.863072i \(0.331463\pi\)
\(882\) 0 0
\(883\) 2.71777e7 1.17304 0.586518 0.809936i \(-0.300498\pi\)
0.586518 + 0.809936i \(0.300498\pi\)
\(884\) 0 0
\(885\) −4.07366e6 −0.174834
\(886\) 0 0
\(887\) −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
\(890\) 0 0
\(891\) −6.09086e6 −0.257030
\(892\) 0 0
\(893\) 814374. 0.0341740
\(894\) 0 0
\(895\) −181433. −0.00757109
\(896\) 0 0
\(897\) 9.69477e6 0.402306
\(898\) 0 0
\(899\) −4.55099e6 −0.187805
\(900\) 0 0
\(901\) −4.36385e7 −1.79084
\(902\) 0 0
\(903\) −4.23209e6 −0.172717
\(904\) 0 0
\(905\) −4.55773e7 −1.84981
\(906\) 0 0
\(907\) −1.77875e7 −0.717954 −0.358977 0.933346i \(-0.616874\pi\)
−0.358977 + 0.933346i \(0.616874\pi\)
\(908\) 0 0
\(909\) −9.46734e6 −0.380031
\(910\) 0 0
\(911\) 30398.8 0.00121356 0.000606780 1.00000i \(-0.499807\pi\)
0.000606780 1.00000i \(0.499807\pi\)
\(912\) 0 0
\(913\) −1.17116e7 −0.464987
\(914\) 0 0
\(915\) −4.36991e6 −0.172552
\(916\) 0 0
\(917\) −9.47308e6 −0.372021
\(918\) 0 0
\(919\) −4.10055e6 −0.160160 −0.0800798 0.996788i \(-0.525518\pi\)
−0.0800798 + 0.996788i \(0.525518\pi\)
\(920\) 0 0
\(921\) 4.56406e6 0.177297
\(922\) 0 0
\(923\) 861085. 0.0332692
\(924\) 0 0
\(925\) −4.44084e6 −0.170652
\(926\) 0 0
\(927\) 1.14580e7 0.437935
\(928\) 0 0
\(929\) −1.46532e7 −0.557048 −0.278524 0.960429i \(-0.589845\pi\)
−0.278524 + 0.960429i \(0.589845\pi\)
\(930\) 0 0
\(931\) −1.55403e6 −0.0587604
\(932\) 0 0
\(933\) 1.17105e7 0.440425
\(934\) 0 0
\(935\) −1.33349e7 −0.498838
\(936\) 0 0
\(937\) −3.97538e7 −1.47921 −0.739604 0.673042i \(-0.764987\pi\)
−0.739604 + 0.673042i \(0.764987\pi\)
\(938\) 0 0
\(939\) −1.04803e7 −0.387891
\(940\) 0 0
\(941\) −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
\(944\) 0 0
\(945\) −1.43474e7 −0.522629
\(946\) 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
\(950\) 0 0
\(951\) −7.32571e6 −0.262663
\(952\) 0 0
\(953\) 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
\(956\) 0 0
\(957\) −724242. −0.0255625
\(958\) 0 0
\(959\) 772177. 0.0271125
\(960\) 0 0
\(961\) −2.16038e7 −0.754608
\(962\) 0 0
\(963\) −1.56776e6 −0.0544773
\(964\) 0 0
\(965\) −1.63023e7 −0.563548
\(966\) 0 0
\(967\) 4.85436e7 1.66942 0.834711 0.550688i \(-0.185635\pi\)
0.834711 + 0.550688i \(0.185635\pi\)
\(968\) 0 0
\(969\) −2.35254e6 −0.0804873
\(970\) 0 0
\(971\) 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
\(974\) 0 0
\(975\) −986751. −0.0332427
\(976\) 0 0
\(977\) −5.35354e7 −1.79434 −0.897170 0.441684i \(-0.854381\pi\)
−0.897170 + 0.441684i \(0.854381\pi\)
\(978\) 0 0
\(979\) 5.76131e6 0.192116
\(980\) 0 0
\(981\) 2.23416e7 0.741211
\(982\) 0 0
\(983\) −5.40925e7 −1.78547 −0.892736 0.450580i \(-0.851217\pi\)
−0.892736 + 0.450580i \(0.851217\pi\)
\(984\) 0 0
\(985\) −3.43996e7 −1.12970
\(986\) 0 0
\(987\) −1.12331e6 −0.0367035
\(988\) 0 0
\(989\) 3.77952e7 1.22870
\(990\) 0 0
\(991\) 2.31007e7 0.747208 0.373604 0.927588i \(-0.378122\pi\)
0.373604 + 0.927588i \(0.378122\pi\)
\(992\) 0 0
\(993\) −4.29956e6 −0.138373
\(994\) 0 0
\(995\) 1.61611e7 0.517502
\(996\) 0 0
\(997\) 4.54061e7 1.44669 0.723346 0.690486i \(-0.242603\pi\)
0.723346 + 0.690486i \(0.242603\pi\)
\(998\) 0 0
\(999\) 1.59577e7 0.505891
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 704.6.a.q.1.3 3
4.3 odd 2 704.6.a.t.1.1 3
8.3 odd 2 176.6.a.i.1.3 3
8.5 even 2 11.6.a.b.1.3 3
24.5 odd 2 99.6.a.g.1.1 3
40.13 odd 4 275.6.b.b.199.2 6
40.29 even 2 275.6.a.b.1.1 3
40.37 odd 4 275.6.b.b.199.5 6
56.13 odd 2 539.6.a.e.1.3 3
88.21 odd 2 121.6.a.d.1.1 3
264.197 even 2 1089.6.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 8.5 even 2
99.6.a.g.1.1 3 24.5 odd 2
121.6.a.d.1.1 3 88.21 odd 2
176.6.a.i.1.3 3 8.3 odd 2
275.6.a.b.1.1 3 40.29 even 2
275.6.b.b.199.2 6 40.13 odd 4
275.6.b.b.199.5 6 40.37 odd 4
539.6.a.e.1.3 3 56.13 odd 2
704.6.a.q.1.3 3 1.1 even 1 trivial
704.6.a.t.1.1 3 4.3 odd 2
1089.6.a.r.1.3 3 264.197 even 2