Properties

Label 1104.6.a.ba.1.5
Level $1104$
Weight $6$
Character 1104.1
Self dual yes
Analytic conductor $177.064$
Analytic rank $0$
Dimension $8$
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] = [1104,6,Mod(1,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1104.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1104.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: \(177.063737074\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4608x^{6} - 3161x^{5} + 6284039x^{4} - 8279002x^{3} - 2677454576x^{2} + 13573192447x + 77901299860 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(9.78333\) of defining polynomial
Character \(\chi\) \(=\) 1104.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.00000 q^{3} +21.5667 q^{5} +125.104 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +21.5667 q^{5} +125.104 q^{7} +81.0000 q^{9} +251.622 q^{11} -1049.36 q^{13} -194.100 q^{15} +1732.72 q^{17} +952.726 q^{19} -1125.93 q^{21} +529.000 q^{23} -2659.88 q^{25} -729.000 q^{27} +8873.99 q^{29} -5017.18 q^{31} -2264.60 q^{33} +2698.07 q^{35} -3179.77 q^{37} +9444.27 q^{39} +20516.0 q^{41} -8597.82 q^{43} +1746.90 q^{45} -5687.90 q^{47} -1156.09 q^{49} -15594.5 q^{51} +22168.2 q^{53} +5426.65 q^{55} -8574.54 q^{57} +541.135 q^{59} +18613.6 q^{61} +10133.4 q^{63} -22631.3 q^{65} -3031.98 q^{67} -4761.00 q^{69} +34389.7 q^{71} +23801.1 q^{73} +23938.9 q^{75} +31478.8 q^{77} +25487.8 q^{79} +6561.00 q^{81} -79525.8 q^{83} +37368.9 q^{85} -79865.9 q^{87} +91114.9 q^{89} -131279. q^{91} +45154.6 q^{93} +20547.1 q^{95} -66083.7 q^{97} +20381.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 72 q^{3} + 16 q^{5} + 36 q^{7} + 648 q^{9} + 148 q^{11} + 696 q^{13} - 144 q^{15} + 572 q^{17} - 2456 q^{19} - 324 q^{21} + 4232 q^{23} + 11896 q^{25} - 5832 q^{27} + 6504 q^{29} - 3128 q^{31} - 1332 q^{33} - 6960 q^{35} + 3844 q^{37} - 6264 q^{39} + 6440 q^{41} - 7048 q^{43} + 1296 q^{45} - 30464 q^{47} + 46984 q^{49} - 5148 q^{51} + 63696 q^{53} - 32688 q^{55} + 22104 q^{57} - 54872 q^{59} + 12108 q^{61} + 2916 q^{63} + 124088 q^{65} - 139216 q^{67} - 38088 q^{69} - 76216 q^{71} + 13632 q^{73} - 107064 q^{75} - 78248 q^{77} - 126380 q^{79} + 52488 q^{81} - 238196 q^{83} + 117536 q^{85} - 58536 q^{87} + 123668 q^{89} - 248176 q^{91} + 28152 q^{93} - 183408 q^{95} + 18576 q^{97} + 11988 q^{99}+O(q^{100}) \) 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
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) 21.5667 0.385796 0.192898 0.981219i \(-0.438211\pi\)
0.192898 + 0.981219i \(0.438211\pi\)
\(6\) 0 0
\(7\) 125.104 0.964994 0.482497 0.875898i \(-0.339730\pi\)
0.482497 + 0.875898i \(0.339730\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 251.622 0.626999 0.313500 0.949588i \(-0.398499\pi\)
0.313500 + 0.949588i \(0.398499\pi\)
\(12\) 0 0
\(13\) −1049.36 −1.72214 −0.861068 0.508489i \(-0.830204\pi\)
−0.861068 + 0.508489i \(0.830204\pi\)
\(14\) 0 0
\(15\) −194.100 −0.222739
\(16\) 0 0
\(17\) 1732.72 1.45414 0.727069 0.686564i \(-0.240882\pi\)
0.727069 + 0.686564i \(0.240882\pi\)
\(18\) 0 0
\(19\) 952.726 0.605458 0.302729 0.953077i \(-0.402102\pi\)
0.302729 + 0.953077i \(0.402102\pi\)
\(20\) 0 0
\(21\) −1125.93 −0.557140
\(22\) 0 0
\(23\) 529.000 0.208514
\(24\) 0 0
\(25\) −2659.88 −0.851161
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 8873.99 1.95940 0.979702 0.200459i \(-0.0642435\pi\)
0.979702 + 0.200459i \(0.0642435\pi\)
\(30\) 0 0
\(31\) −5017.18 −0.937681 −0.468841 0.883283i \(-0.655328\pi\)
−0.468841 + 0.883283i \(0.655328\pi\)
\(32\) 0 0
\(33\) −2264.60 −0.361998
\(34\) 0 0
\(35\) 2698.07 0.372291
\(36\) 0 0
\(37\) −3179.77 −0.381849 −0.190924 0.981605i \(-0.561149\pi\)
−0.190924 + 0.981605i \(0.561149\pi\)
\(38\) 0 0
\(39\) 9444.27 0.994276
\(40\) 0 0
\(41\) 20516.0 1.90604 0.953020 0.302906i \(-0.0979569\pi\)
0.953020 + 0.302906i \(0.0979569\pi\)
\(42\) 0 0
\(43\) −8597.82 −0.709116 −0.354558 0.935034i \(-0.615369\pi\)
−0.354558 + 0.935034i \(0.615369\pi\)
\(44\) 0 0
\(45\) 1746.90 0.128599
\(46\) 0 0
\(47\) −5687.90 −0.375584 −0.187792 0.982209i \(-0.560133\pi\)
−0.187792 + 0.982209i \(0.560133\pi\)
\(48\) 0 0
\(49\) −1156.09 −0.0687860
\(50\) 0 0
\(51\) −15594.5 −0.839547
\(52\) 0 0
\(53\) 22168.2 1.08403 0.542013 0.840370i \(-0.317662\pi\)
0.542013 + 0.840370i \(0.317662\pi\)
\(54\) 0 0
\(55\) 5426.65 0.241894
\(56\) 0 0
\(57\) −8574.54 −0.349561
\(58\) 0 0
\(59\) 541.135 0.0202384 0.0101192 0.999949i \(-0.496779\pi\)
0.0101192 + 0.999949i \(0.496779\pi\)
\(60\) 0 0
\(61\) 18613.6 0.640481 0.320241 0.947336i \(-0.396236\pi\)
0.320241 + 0.947336i \(0.396236\pi\)
\(62\) 0 0
\(63\) 10133.4 0.321665
\(64\) 0 0
\(65\) −22631.3 −0.664394
\(66\) 0 0
\(67\) −3031.98 −0.0825163 −0.0412581 0.999149i \(-0.513137\pi\)
−0.0412581 + 0.999149i \(0.513137\pi\)
\(68\) 0 0
\(69\) −4761.00 −0.120386
\(70\) 0 0
\(71\) 34389.7 0.809622 0.404811 0.914400i \(-0.367337\pi\)
0.404811 + 0.914400i \(0.367337\pi\)
\(72\) 0 0
\(73\) 23801.1 0.522744 0.261372 0.965238i \(-0.415825\pi\)
0.261372 + 0.965238i \(0.415825\pi\)
\(74\) 0 0
\(75\) 23938.9 0.491418
\(76\) 0 0
\(77\) 31478.8 0.605051
\(78\) 0 0
\(79\) 25487.8 0.459478 0.229739 0.973252i \(-0.426213\pi\)
0.229739 + 0.973252i \(0.426213\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −79525.8 −1.26711 −0.633553 0.773699i \(-0.718404\pi\)
−0.633553 + 0.773699i \(0.718404\pi\)
\(84\) 0 0
\(85\) 37368.9 0.561001
\(86\) 0 0
\(87\) −79865.9 −1.13126
\(88\) 0 0
\(89\) 91114.9 1.21931 0.609655 0.792667i \(-0.291308\pi\)
0.609655 + 0.792667i \(0.291308\pi\)
\(90\) 0 0
\(91\) −131279. −1.66185
\(92\) 0 0
\(93\) 45154.6 0.541371
\(94\) 0 0
\(95\) 20547.1 0.233583
\(96\) 0 0
\(97\) −66083.7 −0.713124 −0.356562 0.934272i \(-0.616051\pi\)
−0.356562 + 0.934272i \(0.616051\pi\)
\(98\) 0 0
\(99\) 20381.4 0.209000
\(100\) 0 0
\(101\) −47843.2 −0.466677 −0.233339 0.972396i \(-0.574965\pi\)
−0.233339 + 0.972396i \(0.574965\pi\)
\(102\) 0 0
\(103\) 26932.5 0.250140 0.125070 0.992148i \(-0.460084\pi\)
0.125070 + 0.992148i \(0.460084\pi\)
\(104\) 0 0
\(105\) −24282.6 −0.214942
\(106\) 0 0
\(107\) −137140. −1.15799 −0.578993 0.815332i \(-0.696554\pi\)
−0.578993 + 0.815332i \(0.696554\pi\)
\(108\) 0 0
\(109\) −151536. −1.22165 −0.610827 0.791764i \(-0.709163\pi\)
−0.610827 + 0.791764i \(0.709163\pi\)
\(110\) 0 0
\(111\) 28617.9 0.220460
\(112\) 0 0
\(113\) −84239.3 −0.620610 −0.310305 0.950637i \(-0.600431\pi\)
−0.310305 + 0.950637i \(0.600431\pi\)
\(114\) 0 0
\(115\) 11408.8 0.0804440
\(116\) 0 0
\(117\) −84998.5 −0.574046
\(118\) 0 0
\(119\) 216769. 1.40324
\(120\) 0 0
\(121\) −97737.3 −0.606872
\(122\) 0 0
\(123\) −184644. −1.10045
\(124\) 0 0
\(125\) −124761. −0.714171
\(126\) 0 0
\(127\) 7734.17 0.0425505 0.0212752 0.999774i \(-0.493227\pi\)
0.0212752 + 0.999774i \(0.493227\pi\)
\(128\) 0 0
\(129\) 77380.4 0.409408
\(130\) 0 0
\(131\) −158784. −0.808403 −0.404201 0.914670i \(-0.632450\pi\)
−0.404201 + 0.914670i \(0.632450\pi\)
\(132\) 0 0
\(133\) 119189. 0.584264
\(134\) 0 0
\(135\) −15722.1 −0.0742465
\(136\) 0 0
\(137\) 152990. 0.696405 0.348203 0.937419i \(-0.386792\pi\)
0.348203 + 0.937419i \(0.386792\pi\)
\(138\) 0 0
\(139\) −270171. −1.18605 −0.593023 0.805185i \(-0.702066\pi\)
−0.593023 + 0.805185i \(0.702066\pi\)
\(140\) 0 0
\(141\) 51191.1 0.216844
\(142\) 0 0
\(143\) −264043. −1.07978
\(144\) 0 0
\(145\) 191382. 0.755930
\(146\) 0 0
\(147\) 10404.8 0.0397136
\(148\) 0 0
\(149\) 220050. 0.812000 0.406000 0.913873i \(-0.366923\pi\)
0.406000 + 0.913873i \(0.366923\pi\)
\(150\) 0 0
\(151\) 432844. 1.54486 0.772430 0.635099i \(-0.219041\pi\)
0.772430 + 0.635099i \(0.219041\pi\)
\(152\) 0 0
\(153\) 140350. 0.484713
\(154\) 0 0
\(155\) −108204. −0.361754
\(156\) 0 0
\(157\) 8947.45 0.0289701 0.0144851 0.999895i \(-0.495389\pi\)
0.0144851 + 0.999895i \(0.495389\pi\)
\(158\) 0 0
\(159\) −199513. −0.625863
\(160\) 0 0
\(161\) 66179.8 0.201215
\(162\) 0 0
\(163\) −128349. −0.378375 −0.189187 0.981941i \(-0.560585\pi\)
−0.189187 + 0.981941i \(0.560585\pi\)
\(164\) 0 0
\(165\) −48839.8 −0.139657
\(166\) 0 0
\(167\) −84294.9 −0.233889 −0.116945 0.993138i \(-0.537310\pi\)
−0.116945 + 0.993138i \(0.537310\pi\)
\(168\) 0 0
\(169\) 729871. 1.96575
\(170\) 0 0
\(171\) 77170.8 0.201819
\(172\) 0 0
\(173\) 303801. 0.771744 0.385872 0.922552i \(-0.373901\pi\)
0.385872 + 0.922552i \(0.373901\pi\)
\(174\) 0 0
\(175\) −332761. −0.821366
\(176\) 0 0
\(177\) −4870.22 −0.0116846
\(178\) 0 0
\(179\) 365012. 0.851481 0.425740 0.904845i \(-0.360014\pi\)
0.425740 + 0.904845i \(0.360014\pi\)
\(180\) 0 0
\(181\) 561526. 1.27401 0.637006 0.770859i \(-0.280173\pi\)
0.637006 + 0.770859i \(0.280173\pi\)
\(182\) 0 0
\(183\) −167523. −0.369782
\(184\) 0 0
\(185\) −68577.0 −0.147316
\(186\) 0 0
\(187\) 435990. 0.911744
\(188\) 0 0
\(189\) −91200.5 −0.185713
\(190\) 0 0
\(191\) −337877. −0.670156 −0.335078 0.942190i \(-0.608763\pi\)
−0.335078 + 0.942190i \(0.608763\pi\)
\(192\) 0 0
\(193\) 795781. 1.53780 0.768900 0.639369i \(-0.220804\pi\)
0.768900 + 0.639369i \(0.220804\pi\)
\(194\) 0 0
\(195\) 203681. 0.383588
\(196\) 0 0
\(197\) 23009.5 0.0422417 0.0211208 0.999777i \(-0.493277\pi\)
0.0211208 + 0.999777i \(0.493277\pi\)
\(198\) 0 0
\(199\) 70554.9 0.126297 0.0631487 0.998004i \(-0.479886\pi\)
0.0631487 + 0.998004i \(0.479886\pi\)
\(200\) 0 0
\(201\) 27287.8 0.0476408
\(202\) 0 0
\(203\) 1.11017e6 1.89081
\(204\) 0 0
\(205\) 442461. 0.735343
\(206\) 0 0
\(207\) 42849.0 0.0695048
\(208\) 0 0
\(209\) 239727. 0.379622
\(210\) 0 0
\(211\) 1.07231e6 1.65812 0.829060 0.559159i \(-0.188876\pi\)
0.829060 + 0.559159i \(0.188876\pi\)
\(212\) 0 0
\(213\) −309507. −0.467436
\(214\) 0 0
\(215\) −185426. −0.273574
\(216\) 0 0
\(217\) −627667. −0.904857
\(218\) 0 0
\(219\) −214210. −0.301806
\(220\) 0 0
\(221\) −1.81825e6 −2.50423
\(222\) 0 0
\(223\) −1.43797e6 −1.93636 −0.968182 0.250246i \(-0.919489\pi\)
−0.968182 + 0.250246i \(0.919489\pi\)
\(224\) 0 0
\(225\) −215450. −0.283720
\(226\) 0 0
\(227\) −156122. −0.201095 −0.100547 0.994932i \(-0.532059\pi\)
−0.100547 + 0.994932i \(0.532059\pi\)
\(228\) 0 0
\(229\) 957388. 1.20642 0.603211 0.797582i \(-0.293888\pi\)
0.603211 + 0.797582i \(0.293888\pi\)
\(230\) 0 0
\(231\) −283309. −0.349326
\(232\) 0 0
\(233\) 395600. 0.477382 0.238691 0.971096i \(-0.423282\pi\)
0.238691 + 0.971096i \(0.423282\pi\)
\(234\) 0 0
\(235\) −122669. −0.144899
\(236\) 0 0
\(237\) −229390. −0.265280
\(238\) 0 0
\(239\) −1.10437e6 −1.25060 −0.625301 0.780384i \(-0.715024\pi\)
−0.625301 + 0.780384i \(0.715024\pi\)
\(240\) 0 0
\(241\) 1.10391e6 1.22431 0.612153 0.790739i \(-0.290304\pi\)
0.612153 + 0.790739i \(0.290304\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) −24932.9 −0.0265374
\(246\) 0 0
\(247\) −999756. −1.04268
\(248\) 0 0
\(249\) 715732. 0.731564
\(250\) 0 0
\(251\) 1.40907e6 1.41172 0.705858 0.708354i \(-0.250562\pi\)
0.705858 + 0.708354i \(0.250562\pi\)
\(252\) 0 0
\(253\) 133108. 0.130738
\(254\) 0 0
\(255\) −336321. −0.323894
\(256\) 0 0
\(257\) 251097. 0.237142 0.118571 0.992946i \(-0.462169\pi\)
0.118571 + 0.992946i \(0.462169\pi\)
\(258\) 0 0
\(259\) −397801. −0.368482
\(260\) 0 0
\(261\) 718793. 0.653135
\(262\) 0 0
\(263\) 1.46040e6 1.30191 0.650956 0.759115i \(-0.274368\pi\)
0.650956 + 0.759115i \(0.274368\pi\)
\(264\) 0 0
\(265\) 478093. 0.418213
\(266\) 0 0
\(267\) −820034. −0.703969
\(268\) 0 0
\(269\) 1.02652e6 0.864942 0.432471 0.901648i \(-0.357642\pi\)
0.432471 + 0.901648i \(0.357642\pi\)
\(270\) 0 0
\(271\) 2.34412e6 1.93891 0.969454 0.245273i \(-0.0788777\pi\)
0.969454 + 0.245273i \(0.0788777\pi\)
\(272\) 0 0
\(273\) 1.18151e6 0.959471
\(274\) 0 0
\(275\) −669284. −0.533678
\(276\) 0 0
\(277\) −858672. −0.672400 −0.336200 0.941791i \(-0.609142\pi\)
−0.336200 + 0.941791i \(0.609142\pi\)
\(278\) 0 0
\(279\) −406391. −0.312560
\(280\) 0 0
\(281\) 2.25896e6 1.70664 0.853322 0.521384i \(-0.174584\pi\)
0.853322 + 0.521384i \(0.174584\pi\)
\(282\) 0 0
\(283\) 1.35838e6 1.00822 0.504108 0.863640i \(-0.331821\pi\)
0.504108 + 0.863640i \(0.331821\pi\)
\(284\) 0 0
\(285\) −184924. −0.134859
\(286\) 0 0
\(287\) 2.56662e6 1.83932
\(288\) 0 0
\(289\) 1.58246e6 1.11452
\(290\) 0 0
\(291\) 594753. 0.411722
\(292\) 0 0
\(293\) 1.05384e6 0.717143 0.358571 0.933502i \(-0.383264\pi\)
0.358571 + 0.933502i \(0.383264\pi\)
\(294\) 0 0
\(295\) 11670.5 0.00780789
\(296\) 0 0
\(297\) −183433. −0.120666
\(298\) 0 0
\(299\) −555113. −0.359090
\(300\) 0 0
\(301\) −1.07562e6 −0.684293
\(302\) 0 0
\(303\) 430589. 0.269436
\(304\) 0 0
\(305\) 401434. 0.247095
\(306\) 0 0
\(307\) 144724. 0.0876385 0.0438193 0.999039i \(-0.486047\pi\)
0.0438193 + 0.999039i \(0.486047\pi\)
\(308\) 0 0
\(309\) −242392. −0.144418
\(310\) 0 0
\(311\) 1.10152e6 0.645791 0.322895 0.946435i \(-0.395344\pi\)
0.322895 + 0.946435i \(0.395344\pi\)
\(312\) 0 0
\(313\) −1.01260e6 −0.584218 −0.292109 0.956385i \(-0.594357\pi\)
−0.292109 + 0.956385i \(0.594357\pi\)
\(314\) 0 0
\(315\) 218543. 0.124097
\(316\) 0 0
\(317\) 597212. 0.333796 0.166898 0.985974i \(-0.446625\pi\)
0.166898 + 0.985974i \(0.446625\pi\)
\(318\) 0 0
\(319\) 2.23289e6 1.22854
\(320\) 0 0
\(321\) 1.23426e6 0.668564
\(322\) 0 0
\(323\) 1.65081e6 0.880420
\(324\) 0 0
\(325\) 2.79118e6 1.46582
\(326\) 0 0
\(327\) 1.36382e6 0.705323
\(328\) 0 0
\(329\) −711577. −0.362437
\(330\) 0 0
\(331\) −3.42336e6 −1.71744 −0.858722 0.512442i \(-0.828741\pi\)
−0.858722 + 0.512442i \(0.828741\pi\)
\(332\) 0 0
\(333\) −257561. −0.127283
\(334\) 0 0
\(335\) −65389.7 −0.0318345
\(336\) 0 0
\(337\) −2.22913e6 −1.06921 −0.534603 0.845103i \(-0.679539\pi\)
−0.534603 + 0.845103i \(0.679539\pi\)
\(338\) 0 0
\(339\) 758154. 0.358309
\(340\) 0 0
\(341\) −1.26243e6 −0.587926
\(342\) 0 0
\(343\) −2.24725e6 −1.03137
\(344\) 0 0
\(345\) −102679. −0.0464444
\(346\) 0 0
\(347\) −919683. −0.410029 −0.205014 0.978759i \(-0.565724\pi\)
−0.205014 + 0.978759i \(0.565724\pi\)
\(348\) 0 0
\(349\) −939403. −0.412846 −0.206423 0.978463i \(-0.566182\pi\)
−0.206423 + 0.978463i \(0.566182\pi\)
\(350\) 0 0
\(351\) 764986. 0.331425
\(352\) 0 0
\(353\) 3.95083e6 1.68753 0.843765 0.536712i \(-0.180334\pi\)
0.843765 + 0.536712i \(0.180334\pi\)
\(354\) 0 0
\(355\) 741671. 0.312349
\(356\) 0 0
\(357\) −1.95092e6 −0.810158
\(358\) 0 0
\(359\) 4.11247e6 1.68410 0.842048 0.539402i \(-0.181350\pi\)
0.842048 + 0.539402i \(0.181350\pi\)
\(360\) 0 0
\(361\) −1.56841e6 −0.633420
\(362\) 0 0
\(363\) 879636. 0.350378
\(364\) 0 0
\(365\) 513309. 0.201673
\(366\) 0 0
\(367\) −3.03521e6 −1.17631 −0.588157 0.808747i \(-0.700146\pi\)
−0.588157 + 0.808747i \(0.700146\pi\)
\(368\) 0 0
\(369\) 1.66179e6 0.635347
\(370\) 0 0
\(371\) 2.77332e6 1.04608
\(372\) 0 0
\(373\) −263081. −0.0979077 −0.0489538 0.998801i \(-0.515589\pi\)
−0.0489538 + 0.998801i \(0.515589\pi\)
\(374\) 0 0
\(375\) 1.12284e6 0.412327
\(376\) 0 0
\(377\) −9.31204e6 −3.37436
\(378\) 0 0
\(379\) 4.23360e6 1.51395 0.756976 0.653443i \(-0.226676\pi\)
0.756976 + 0.653443i \(0.226676\pi\)
\(380\) 0 0
\(381\) −69607.5 −0.0245665
\(382\) 0 0
\(383\) 4.88943e6 1.70318 0.851592 0.524206i \(-0.175638\pi\)
0.851592 + 0.524206i \(0.175638\pi\)
\(384\) 0 0
\(385\) 678893. 0.233426
\(386\) 0 0
\(387\) −696423. −0.236372
\(388\) 0 0
\(389\) −3.42103e6 −1.14626 −0.573130 0.819464i \(-0.694271\pi\)
−0.573130 + 0.819464i \(0.694271\pi\)
\(390\) 0 0
\(391\) 916608. 0.303209
\(392\) 0 0
\(393\) 1.42905e6 0.466732
\(394\) 0 0
\(395\) 549687. 0.177265
\(396\) 0 0
\(397\) 563878. 0.179560 0.0897798 0.995962i \(-0.471384\pi\)
0.0897798 + 0.995962i \(0.471384\pi\)
\(398\) 0 0
\(399\) −1.07271e6 −0.337325
\(400\) 0 0
\(401\) 590550. 0.183398 0.0916992 0.995787i \(-0.470770\pi\)
0.0916992 + 0.995787i \(0.470770\pi\)
\(402\) 0 0
\(403\) 5.26484e6 1.61482
\(404\) 0 0
\(405\) 141499. 0.0428662
\(406\) 0 0
\(407\) −800100. −0.239419
\(408\) 0 0
\(409\) −4.88465e6 −1.44386 −0.721929 0.691967i \(-0.756744\pi\)
−0.721929 + 0.691967i \(0.756744\pi\)
\(410\) 0 0
\(411\) −1.37691e6 −0.402070
\(412\) 0 0
\(413\) 67698.0 0.0195299
\(414\) 0 0
\(415\) −1.71511e6 −0.488844
\(416\) 0 0
\(417\) 2.43154e6 0.684764
\(418\) 0 0
\(419\) −3.32253e6 −0.924558 −0.462279 0.886735i \(-0.652968\pi\)
−0.462279 + 0.886735i \(0.652968\pi\)
\(420\) 0 0
\(421\) 338104. 0.0929706 0.0464853 0.998919i \(-0.485198\pi\)
0.0464853 + 0.998919i \(0.485198\pi\)
\(422\) 0 0
\(423\) −460720. −0.125195
\(424\) 0 0
\(425\) −4.60882e6 −1.23771
\(426\) 0 0
\(427\) 2.32863e6 0.618061
\(428\) 0 0
\(429\) 2.37639e6 0.623410
\(430\) 0 0
\(431\) 3.86095e6 1.00115 0.500577 0.865692i \(-0.333121\pi\)
0.500577 + 0.865692i \(0.333121\pi\)
\(432\) 0 0
\(433\) 2.91822e6 0.747995 0.373998 0.927430i \(-0.377987\pi\)
0.373998 + 0.927430i \(0.377987\pi\)
\(434\) 0 0
\(435\) −1.72244e6 −0.436437
\(436\) 0 0
\(437\) 503992. 0.126247
\(438\) 0 0
\(439\) 5405.96 0.00133879 0.000669394 1.00000i \(-0.499787\pi\)
0.000669394 1.00000i \(0.499787\pi\)
\(440\) 0 0
\(441\) −93642.9 −0.0229287
\(442\) 0 0
\(443\) 625169. 0.151352 0.0756760 0.997132i \(-0.475889\pi\)
0.0756760 + 0.997132i \(0.475889\pi\)
\(444\) 0 0
\(445\) 1.96504e6 0.470405
\(446\) 0 0
\(447\) −1.98045e6 −0.468808
\(448\) 0 0
\(449\) −3.52906e6 −0.826121 −0.413060 0.910704i \(-0.635540\pi\)
−0.413060 + 0.910704i \(0.635540\pi\)
\(450\) 0 0
\(451\) 5.16227e6 1.19509
\(452\) 0 0
\(453\) −3.89560e6 −0.891926
\(454\) 0 0
\(455\) −2.83125e6 −0.641136
\(456\) 0 0
\(457\) 5.99355e6 1.34244 0.671218 0.741260i \(-0.265771\pi\)
0.671218 + 0.741260i \(0.265771\pi\)
\(458\) 0 0
\(459\) −1.26315e6 −0.279849
\(460\) 0 0
\(461\) −1.60655e6 −0.352080 −0.176040 0.984383i \(-0.556329\pi\)
−0.176040 + 0.984383i \(0.556329\pi\)
\(462\) 0 0
\(463\) 803811. 0.174262 0.0871308 0.996197i \(-0.472230\pi\)
0.0871308 + 0.996197i \(0.472230\pi\)
\(464\) 0 0
\(465\) 973834. 0.208859
\(466\) 0 0
\(467\) 1.51941e6 0.322391 0.161195 0.986923i \(-0.448465\pi\)
0.161195 + 0.986923i \(0.448465\pi\)
\(468\) 0 0
\(469\) −379312. −0.0796277
\(470\) 0 0
\(471\) −80527.1 −0.0167259
\(472\) 0 0
\(473\) −2.16340e6 −0.444615
\(474\) 0 0
\(475\) −2.53414e6 −0.515343
\(476\) 0 0
\(477\) 1.79562e6 0.361342
\(478\) 0 0
\(479\) 7.26116e6 1.44600 0.722998 0.690850i \(-0.242764\pi\)
0.722998 + 0.690850i \(0.242764\pi\)
\(480\) 0 0
\(481\) 3.33673e6 0.657596
\(482\) 0 0
\(483\) −595618. −0.116172
\(484\) 0 0
\(485\) −1.42520e6 −0.275120
\(486\) 0 0
\(487\) 2.94840e6 0.563331 0.281666 0.959513i \(-0.409113\pi\)
0.281666 + 0.959513i \(0.409113\pi\)
\(488\) 0 0
\(489\) 1.15514e6 0.218455
\(490\) 0 0
\(491\) 3.80951e6 0.713125 0.356563 0.934271i \(-0.383949\pi\)
0.356563 + 0.934271i \(0.383949\pi\)
\(492\) 0 0
\(493\) 1.53761e7 2.84924
\(494\) 0 0
\(495\) 439558. 0.0806313
\(496\) 0 0
\(497\) 4.30227e6 0.781281
\(498\) 0 0
\(499\) −3.41466e6 −0.613898 −0.306949 0.951726i \(-0.599308\pi\)
−0.306949 + 0.951726i \(0.599308\pi\)
\(500\) 0 0
\(501\) 758655. 0.135036
\(502\) 0 0
\(503\) 8.54455e6 1.50581 0.752903 0.658132i \(-0.228653\pi\)
0.752903 + 0.658132i \(0.228653\pi\)
\(504\) 0 0
\(505\) −1.03182e6 −0.180042
\(506\) 0 0
\(507\) −6.56884e6 −1.13493
\(508\) 0 0
\(509\) −9.29142e6 −1.58960 −0.794800 0.606871i \(-0.792424\pi\)
−0.794800 + 0.606871i \(0.792424\pi\)
\(510\) 0 0
\(511\) 2.97760e6 0.504445
\(512\) 0 0
\(513\) −694537. −0.116520
\(514\) 0 0
\(515\) 580843. 0.0965030
\(516\) 0 0
\(517\) −1.43120e6 −0.235491
\(518\) 0 0
\(519\) −2.73421e6 −0.445567
\(520\) 0 0
\(521\) 9.43542e6 1.52288 0.761442 0.648233i \(-0.224492\pi\)
0.761442 + 0.648233i \(0.224492\pi\)
\(522\) 0 0
\(523\) −2.97282e6 −0.475241 −0.237620 0.971358i \(-0.576367\pi\)
−0.237620 + 0.971358i \(0.576367\pi\)
\(524\) 0 0
\(525\) 2.99484e6 0.474216
\(526\) 0 0
\(527\) −8.69336e6 −1.36352
\(528\) 0 0
\(529\) 279841. 0.0434783
\(530\) 0 0
\(531\) 43832.0 0.00674613
\(532\) 0 0
\(533\) −2.15287e7 −3.28246
\(534\) 0 0
\(535\) −2.95764e6 −0.446747
\(536\) 0 0
\(537\) −3.28511e6 −0.491603
\(538\) 0 0
\(539\) −290897. −0.0431288
\(540\) 0 0
\(541\) 6.75824e6 0.992751 0.496375 0.868108i \(-0.334664\pi\)
0.496375 + 0.868108i \(0.334664\pi\)
\(542\) 0 0
\(543\) −5.05373e6 −0.735551
\(544\) 0 0
\(545\) −3.26812e6 −0.471309
\(546\) 0 0
\(547\) 6.23996e6 0.891690 0.445845 0.895110i \(-0.352903\pi\)
0.445845 + 0.895110i \(0.352903\pi\)
\(548\) 0 0
\(549\) 1.50770e6 0.213494
\(550\) 0 0
\(551\) 8.45448e6 1.18634
\(552\) 0 0
\(553\) 3.18862e6 0.443394
\(554\) 0 0
\(555\) 617193. 0.0850528
\(556\) 0 0
\(557\) 1.06303e7 1.45180 0.725901 0.687799i \(-0.241423\pi\)
0.725901 + 0.687799i \(0.241423\pi\)
\(558\) 0 0
\(559\) 9.02224e6 1.22119
\(560\) 0 0
\(561\) −3.92391e6 −0.526396
\(562\) 0 0
\(563\) −9.89621e6 −1.31582 −0.657912 0.753095i \(-0.728560\pi\)
−0.657912 + 0.753095i \(0.728560\pi\)
\(564\) 0 0
\(565\) −1.81676e6 −0.239429
\(566\) 0 0
\(567\) 820805. 0.107222
\(568\) 0 0
\(569\) −1.02669e7 −1.32940 −0.664702 0.747108i \(-0.731442\pi\)
−0.664702 + 0.747108i \(0.731442\pi\)
\(570\) 0 0
\(571\) −6.72214e6 −0.862814 −0.431407 0.902157i \(-0.641983\pi\)
−0.431407 + 0.902157i \(0.641983\pi\)
\(572\) 0 0
\(573\) 3.04090e6 0.386915
\(574\) 0 0
\(575\) −1.40708e6 −0.177479
\(576\) 0 0
\(577\) −5.82589e6 −0.728489 −0.364244 0.931303i \(-0.618673\pi\)
−0.364244 + 0.931303i \(0.618673\pi\)
\(578\) 0 0
\(579\) −7.16203e6 −0.887850
\(580\) 0 0
\(581\) −9.94896e6 −1.22275
\(582\) 0 0
\(583\) 5.57800e6 0.679684
\(584\) 0 0
\(585\) −1.83313e6 −0.221465
\(586\) 0 0
\(587\) −6.28402e6 −0.752735 −0.376367 0.926471i \(-0.622827\pi\)
−0.376367 + 0.926471i \(0.622827\pi\)
\(588\) 0 0
\(589\) −4.78000e6 −0.567727
\(590\) 0 0
\(591\) −207085. −0.0243883
\(592\) 0 0
\(593\) 6.46432e6 0.754894 0.377447 0.926031i \(-0.376802\pi\)
0.377447 + 0.926031i \(0.376802\pi\)
\(594\) 0 0
\(595\) 4.67499e6 0.541363
\(596\) 0 0
\(597\) −634994. −0.0729179
\(598\) 0 0
\(599\) −9.39101e6 −1.06941 −0.534706 0.845038i \(-0.679578\pi\)
−0.534706 + 0.845038i \(0.679578\pi\)
\(600\) 0 0
\(601\) −5.92451e6 −0.669062 −0.334531 0.942385i \(-0.608578\pi\)
−0.334531 + 0.942385i \(0.608578\pi\)
\(602\) 0 0
\(603\) −245591. −0.0275054
\(604\) 0 0
\(605\) −2.10787e6 −0.234129
\(606\) 0 0
\(607\) −1.27590e7 −1.40555 −0.702774 0.711414i \(-0.748055\pi\)
−0.702774 + 0.711414i \(0.748055\pi\)
\(608\) 0 0
\(609\) −9.99152e6 −1.09166
\(610\) 0 0
\(611\) 5.96868e6 0.646807
\(612\) 0 0
\(613\) 9.84906e6 1.05863 0.529314 0.848426i \(-0.322449\pi\)
0.529314 + 0.848426i \(0.322449\pi\)
\(614\) 0 0
\(615\) −3.98215e6 −0.424550
\(616\) 0 0
\(617\) 1.75963e7 1.86084 0.930419 0.366498i \(-0.119443\pi\)
0.930419 + 0.366498i \(0.119443\pi\)
\(618\) 0 0
\(619\) −1.79122e7 −1.87898 −0.939491 0.342573i \(-0.888702\pi\)
−0.939491 + 0.342573i \(0.888702\pi\)
\(620\) 0 0
\(621\) −385641. −0.0401286
\(622\) 0 0
\(623\) 1.13988e7 1.17663
\(624\) 0 0
\(625\) 5.62146e6 0.575637
\(626\) 0 0
\(627\) −2.15754e6 −0.219175
\(628\) 0 0
\(629\) −5.50965e6 −0.555261
\(630\) 0 0
\(631\) 2.76600e6 0.276553 0.138277 0.990394i \(-0.455844\pi\)
0.138277 + 0.990394i \(0.455844\pi\)
\(632\) 0 0
\(633\) −9.65083e6 −0.957317
\(634\) 0 0
\(635\) 166800. 0.0164158
\(636\) 0 0
\(637\) 1.21315e6 0.118459
\(638\) 0 0
\(639\) 2.78556e6 0.269874
\(640\) 0 0
\(641\) 741083. 0.0712396 0.0356198 0.999365i \(-0.488659\pi\)
0.0356198 + 0.999365i \(0.488659\pi\)
\(642\) 0 0
\(643\) 8.91735e6 0.850567 0.425283 0.905060i \(-0.360174\pi\)
0.425283 + 0.905060i \(0.360174\pi\)
\(644\) 0 0
\(645\) 1.66884e6 0.157948
\(646\) 0 0
\(647\) −1.34913e7 −1.26705 −0.633523 0.773724i \(-0.718392\pi\)
−0.633523 + 0.773724i \(0.718392\pi\)
\(648\) 0 0
\(649\) 136162. 0.0126895
\(650\) 0 0
\(651\) 5.64900e6 0.522420
\(652\) 0 0
\(653\) −4.38594e6 −0.402513 −0.201256 0.979539i \(-0.564502\pi\)
−0.201256 + 0.979539i \(0.564502\pi\)
\(654\) 0 0
\(655\) −3.42443e6 −0.311879
\(656\) 0 0
\(657\) 1.92789e6 0.174248
\(658\) 0 0
\(659\) −9.23076e6 −0.827988 −0.413994 0.910280i \(-0.635866\pi\)
−0.413994 + 0.910280i \(0.635866\pi\)
\(660\) 0 0
\(661\) 880035. 0.0783423 0.0391711 0.999233i \(-0.487528\pi\)
0.0391711 + 0.999233i \(0.487528\pi\)
\(662\) 0 0
\(663\) 1.63643e7 1.44582
\(664\) 0 0
\(665\) 2.57052e6 0.225407
\(666\) 0 0
\(667\) 4.69434e6 0.408564
\(668\) 0 0
\(669\) 1.29417e7 1.11796
\(670\) 0 0
\(671\) 4.68360e6 0.401581
\(672\) 0 0
\(673\) 2.80265e6 0.238524 0.119262 0.992863i \(-0.461947\pi\)
0.119262 + 0.992863i \(0.461947\pi\)
\(674\) 0 0
\(675\) 1.93905e6 0.163806
\(676\) 0 0
\(677\) 1.05399e7 0.883822 0.441911 0.897059i \(-0.354301\pi\)
0.441911 + 0.897059i \(0.354301\pi\)
\(678\) 0 0
\(679\) −8.26731e6 −0.688161
\(680\) 0 0
\(681\) 1.40510e6 0.116102
\(682\) 0 0
\(683\) −1.50195e7 −1.23198 −0.615989 0.787755i \(-0.711243\pi\)
−0.615989 + 0.787755i \(0.711243\pi\)
\(684\) 0 0
\(685\) 3.29949e6 0.268670
\(686\) 0 0
\(687\) −8.61649e6 −0.696528
\(688\) 0 0
\(689\) −2.32625e7 −1.86684
\(690\) 0 0
\(691\) 1.82611e7 1.45489 0.727446 0.686165i \(-0.240707\pi\)
0.727446 + 0.686165i \(0.240707\pi\)
\(692\) 0 0
\(693\) 2.54979e6 0.201684
\(694\) 0 0
\(695\) −5.82669e6 −0.457572
\(696\) 0 0
\(697\) 3.55484e7 2.77165
\(698\) 0 0
\(699\) −3.56040e6 −0.275617
\(700\) 0 0
\(701\) 7.06422e6 0.542961 0.271481 0.962444i \(-0.412487\pi\)
0.271481 + 0.962444i \(0.412487\pi\)
\(702\) 0 0
\(703\) −3.02945e6 −0.231193
\(704\) 0 0
\(705\) 1.10402e6 0.0836574
\(706\) 0 0
\(707\) −5.98536e6 −0.450341
\(708\) 0 0
\(709\) 1.59888e7 1.19454 0.597269 0.802041i \(-0.296252\pi\)
0.597269 + 0.802041i \(0.296252\pi\)
\(710\) 0 0
\(711\) 2.06451e6 0.153159
\(712\) 0 0
\(713\) −2.65409e6 −0.195520
\(714\) 0 0
\(715\) −5.69453e6 −0.416574
\(716\) 0 0
\(717\) 9.93931e6 0.722035
\(718\) 0 0
\(719\) 1.77395e7 1.27974 0.639868 0.768485i \(-0.278989\pi\)
0.639868 + 0.768485i \(0.278989\pi\)
\(720\) 0 0
\(721\) 3.36935e6 0.241384
\(722\) 0 0
\(723\) −9.93517e6 −0.706854
\(724\) 0 0
\(725\) −2.36037e7 −1.66777
\(726\) 0 0
\(727\) 1.80420e7 1.26604 0.633020 0.774135i \(-0.281815\pi\)
0.633020 + 0.774135i \(0.281815\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.48976e7 −1.03115
\(732\) 0 0
\(733\) −1.81120e7 −1.24511 −0.622555 0.782576i \(-0.713905\pi\)
−0.622555 + 0.782576i \(0.713905\pi\)
\(734\) 0 0
\(735\) 224396. 0.0153213
\(736\) 0 0
\(737\) −762914. −0.0517376
\(738\) 0 0
\(739\) 1.22649e7 0.826142 0.413071 0.910699i \(-0.364456\pi\)
0.413071 + 0.910699i \(0.364456\pi\)
\(740\) 0 0
\(741\) 8.99781e6 0.601993
\(742\) 0 0
\(743\) 1.15977e6 0.0770727 0.0385363 0.999257i \(-0.487730\pi\)
0.0385363 + 0.999257i \(0.487730\pi\)
\(744\) 0 0
\(745\) 4.74575e6 0.313266
\(746\) 0 0
\(747\) −6.44159e6 −0.422369
\(748\) 0 0
\(749\) −1.71567e7 −1.11745
\(750\) 0 0
\(751\) −2.28675e7 −1.47952 −0.739758 0.672873i \(-0.765060\pi\)
−0.739758 + 0.672873i \(0.765060\pi\)
\(752\) 0 0
\(753\) −1.26816e7 −0.815054
\(754\) 0 0
\(755\) 9.33500e6 0.596001
\(756\) 0 0
\(757\) −2.04495e7 −1.29701 −0.648505 0.761210i \(-0.724606\pi\)
−0.648505 + 0.761210i \(0.724606\pi\)
\(758\) 0 0
\(759\) −1.19797e6 −0.0754818
\(760\) 0 0
\(761\) 4.13704e6 0.258957 0.129479 0.991582i \(-0.458670\pi\)
0.129479 + 0.991582i \(0.458670\pi\)
\(762\) 0 0
\(763\) −1.89576e7 −1.17889
\(764\) 0 0
\(765\) 3.02688e6 0.187000
\(766\) 0 0
\(767\) −567848. −0.0348533
\(768\) 0 0
\(769\) 2.34189e7 1.42807 0.714037 0.700108i \(-0.246865\pi\)
0.714037 + 0.700108i \(0.246865\pi\)
\(770\) 0 0
\(771\) −2.25987e6 −0.136914
\(772\) 0 0
\(773\) 2.26269e6 0.136200 0.0681000 0.997678i \(-0.478306\pi\)
0.0681000 + 0.997678i \(0.478306\pi\)
\(774\) 0 0
\(775\) 1.33451e7 0.798118
\(776\) 0 0
\(777\) 3.58021e6 0.212743
\(778\) 0 0
\(779\) 1.95461e7 1.15403
\(780\) 0 0
\(781\) 8.65320e6 0.507632
\(782\) 0 0
\(783\) −6.46914e6 −0.377087
\(784\) 0 0
\(785\) 192967. 0.0111766
\(786\) 0 0
\(787\) −3.10877e7 −1.78917 −0.894586 0.446896i \(-0.852529\pi\)
−0.894586 + 0.446896i \(0.852529\pi\)
\(788\) 0 0
\(789\) −1.31436e7 −0.751659
\(790\) 0 0
\(791\) −1.05386e7 −0.598885
\(792\) 0 0
\(793\) −1.95325e7 −1.10300
\(794\) 0 0
\(795\) −4.30284e6 −0.241456
\(796\) 0 0
\(797\) 4.36116e6 0.243196 0.121598 0.992579i \(-0.461198\pi\)
0.121598 + 0.992579i \(0.461198\pi\)
\(798\) 0 0
\(799\) −9.85553e6 −0.546152
\(800\) 0 0
\(801\) 7.38031e6 0.406437
\(802\) 0 0
\(803\) 5.98887e6 0.327760
\(804\) 0 0
\(805\) 1.42728e6 0.0776280
\(806\) 0 0
\(807\) −9.23869e6 −0.499375
\(808\) 0 0
\(809\) 1.01001e7 0.542568 0.271284 0.962499i \(-0.412552\pi\)
0.271284 + 0.962499i \(0.412552\pi\)
\(810\) 0 0
\(811\) −1.25188e7 −0.668360 −0.334180 0.942509i \(-0.608459\pi\)
−0.334180 + 0.942509i \(0.608459\pi\)
\(812\) 0 0
\(813\) −2.10971e7 −1.11943
\(814\) 0 0
\(815\) −2.76805e6 −0.145975
\(816\) 0 0
\(817\) −8.19137e6 −0.429340
\(818\) 0 0
\(819\) −1.06336e7 −0.553951
\(820\) 0 0
\(821\) 7.44647e6 0.385560 0.192780 0.981242i \(-0.438250\pi\)
0.192780 + 0.981242i \(0.438250\pi\)
\(822\) 0 0
\(823\) 2.59841e7 1.33724 0.668619 0.743605i \(-0.266886\pi\)
0.668619 + 0.743605i \(0.266886\pi\)
\(824\) 0 0
\(825\) 6.02356e6 0.308119
\(826\) 0 0
\(827\) −2.71328e6 −0.137953 −0.0689765 0.997618i \(-0.521973\pi\)
−0.0689765 + 0.997618i \(0.521973\pi\)
\(828\) 0 0
\(829\) −1.59063e7 −0.803867 −0.401933 0.915669i \(-0.631662\pi\)
−0.401933 + 0.915669i \(0.631662\pi\)
\(830\) 0 0
\(831\) 7.72805e6 0.388210
\(832\) 0 0
\(833\) −2.00317e6 −0.100024
\(834\) 0 0
\(835\) −1.81796e6 −0.0902336
\(836\) 0 0
\(837\) 3.65752e6 0.180457
\(838\) 0 0
\(839\) −1.05208e7 −0.515991 −0.257996 0.966146i \(-0.583062\pi\)
−0.257996 + 0.966146i \(0.583062\pi\)
\(840\) 0 0
\(841\) 5.82366e7 2.83926
\(842\) 0 0
\(843\) −2.03306e7 −0.985331
\(844\) 0 0
\(845\) 1.57409e7 0.758380
\(846\) 0 0
\(847\) −1.22273e7 −0.585628
\(848\) 0 0
\(849\) −1.22254e7 −0.582094
\(850\) 0 0
\(851\) −1.68210e6 −0.0796210
\(852\) 0 0
\(853\) −3.04557e6 −0.143316 −0.0716582 0.997429i \(-0.522829\pi\)
−0.0716582 + 0.997429i \(0.522829\pi\)
\(854\) 0 0
\(855\) 1.66432e6 0.0778611
\(856\) 0 0
\(857\) −2.54022e7 −1.18146 −0.590731 0.806868i \(-0.701161\pi\)
−0.590731 + 0.806868i \(0.701161\pi\)
\(858\) 0 0
\(859\) −3.29634e7 −1.52423 −0.762114 0.647443i \(-0.775838\pi\)
−0.762114 + 0.647443i \(0.775838\pi\)
\(860\) 0 0
\(861\) −2.30996e7 −1.06193
\(862\) 0 0
\(863\) 2.82024e7 1.28902 0.644510 0.764596i \(-0.277061\pi\)
0.644510 + 0.764596i \(0.277061\pi\)
\(864\) 0 0
\(865\) 6.55196e6 0.297736
\(866\) 0 0
\(867\) −1.42421e7 −0.643468
\(868\) 0 0
\(869\) 6.41329e6 0.288092
\(870\) 0 0
\(871\) 3.18165e6 0.142104
\(872\) 0 0
\(873\) −5.35278e6 −0.237708
\(874\) 0 0
\(875\) −1.56080e7 −0.689171
\(876\) 0 0
\(877\) −3.55923e7 −1.56263 −0.781316 0.624136i \(-0.785451\pi\)
−0.781316 + 0.624136i \(0.785451\pi\)
\(878\) 0 0
\(879\) −9.48456e6 −0.414042
\(880\) 0 0
\(881\) 2.51982e7 1.09378 0.546889 0.837205i \(-0.315812\pi\)
0.546889 + 0.837205i \(0.315812\pi\)
\(882\) 0 0
\(883\) 9.18247e6 0.396331 0.198165 0.980169i \(-0.436502\pi\)
0.198165 + 0.980169i \(0.436502\pi\)
\(884\) 0 0
\(885\) −105034. −0.00450789
\(886\) 0 0
\(887\) −2.18530e7 −0.932612 −0.466306 0.884623i \(-0.654415\pi\)
−0.466306 + 0.884623i \(0.654415\pi\)
\(888\) 0 0
\(889\) 967573. 0.0410610
\(890\) 0 0
\(891\) 1.65089e6 0.0696666
\(892\) 0 0
\(893\) −5.41901e6 −0.227401
\(894\) 0 0
\(895\) 7.87209e6 0.328498
\(896\) 0 0
\(897\) 4.99602e6 0.207321
\(898\) 0 0
\(899\) −4.45224e7 −1.83730
\(900\) 0 0
\(901\) 3.84112e7 1.57633
\(902\) 0 0
\(903\) 9.68056e6 0.395077
\(904\) 0 0
\(905\) 1.21102e7 0.491509
\(906\) 0 0
\(907\) 3.65912e7 1.47692 0.738462 0.674295i \(-0.235552\pi\)
0.738462 + 0.674295i \(0.235552\pi\)
\(908\) 0 0
\(909\) −3.87530e6 −0.155559
\(910\) 0 0
\(911\) −4.17284e6 −0.166585 −0.0832925 0.996525i \(-0.526544\pi\)
−0.0832925 + 0.996525i \(0.526544\pi\)
\(912\) 0 0
\(913\) −2.00104e7 −0.794474
\(914\) 0 0
\(915\) −3.61290e6 −0.142660
\(916\) 0 0
\(917\) −1.98644e7 −0.780104
\(918\) 0 0
\(919\) −8.44852e6 −0.329983 −0.164992 0.986295i \(-0.552760\pi\)
−0.164992 + 0.986295i \(0.552760\pi\)
\(920\) 0 0
\(921\) −1.30252e6 −0.0505981
\(922\) 0 0
\(923\) −3.60873e7 −1.39428
\(924\) 0 0
\(925\) 8.45780e6 0.325015
\(926\) 0 0
\(927\) 2.18153e6 0.0833800
\(928\) 0 0
\(929\) −4.57958e7 −1.74095 −0.870476 0.492211i \(-0.836189\pi\)
−0.870476 + 0.492211i \(0.836189\pi\)
\(930\) 0 0
\(931\) −1.10143e6 −0.0416470
\(932\) 0 0
\(933\) −9.91369e6 −0.372847
\(934\) 0 0
\(935\) 9.40285e6 0.351747
\(936\) 0 0
\(937\) −4.99626e7 −1.85907 −0.929536 0.368732i \(-0.879792\pi\)
−0.929536 + 0.368732i \(0.879792\pi\)
\(938\) 0 0
\(939\) 9.11336e6 0.337298
\(940\) 0 0
\(941\) 2.50119e6 0.0920814 0.0460407 0.998940i \(-0.485340\pi\)
0.0460407 + 0.998940i \(0.485340\pi\)
\(942\) 0 0
\(943\) 1.08529e7 0.397437
\(944\) 0 0
\(945\) −1.96689e6 −0.0716474
\(946\) 0 0
\(947\) −1.59210e6 −0.0576893 −0.0288446 0.999584i \(-0.509183\pi\)
−0.0288446 + 0.999584i \(0.509183\pi\)
\(948\) 0 0
\(949\) −2.49760e7 −0.900237
\(950\) 0 0
\(951\) −5.37491e6 −0.192717
\(952\) 0 0
\(953\) −1.66688e7 −0.594528 −0.297264 0.954795i \(-0.596074\pi\)
−0.297264 + 0.954795i \(0.596074\pi\)
\(954\) 0 0
\(955\) −7.28689e6 −0.258544
\(956\) 0 0
\(957\) −2.00960e7 −0.709301
\(958\) 0 0
\(959\) 1.91396e7 0.672027
\(960\) 0 0
\(961\) −3.45707e6 −0.120753
\(962\) 0 0
\(963\) −1.11083e7 −0.385996
\(964\) 0 0
\(965\) 1.71623e7 0.593278
\(966\) 0 0
\(967\) 3.17727e7 1.09267 0.546334 0.837567i \(-0.316023\pi\)
0.546334 + 0.837567i \(0.316023\pi\)
\(968\) 0 0
\(969\) −1.48573e7 −0.508311
\(970\) 0 0
\(971\) −1.25778e7 −0.428112 −0.214056 0.976821i \(-0.568668\pi\)
−0.214056 + 0.976821i \(0.568668\pi\)
\(972\) 0 0
\(973\) −3.37994e7 −1.14453
\(974\) 0 0
\(975\) −2.51206e7 −0.846289
\(976\) 0 0
\(977\) −1.21153e7 −0.406066 −0.203033 0.979172i \(-0.565080\pi\)
−0.203033 + 0.979172i \(0.565080\pi\)
\(978\) 0 0
\(979\) 2.29265e7 0.764507
\(980\) 0 0
\(981\) −1.22744e7 −0.407218
\(982\) 0 0
\(983\) −5.70328e7 −1.88253 −0.941263 0.337675i \(-0.890360\pi\)
−0.941263 + 0.337675i \(0.890360\pi\)
\(984\) 0 0
\(985\) 496238. 0.0162967
\(986\) 0 0
\(987\) 6.40419e6 0.209253
\(988\) 0 0
\(989\) −4.54825e6 −0.147861
\(990\) 0 0
\(991\) −7.45826e6 −0.241242 −0.120621 0.992699i \(-0.538489\pi\)
−0.120621 + 0.992699i \(0.538489\pi\)
\(992\) 0 0
\(993\) 3.08102e7 0.991567
\(994\) 0 0
\(995\) 1.52163e6 0.0487251
\(996\) 0 0
\(997\) −1.46400e7 −0.466447 −0.233224 0.972423i \(-0.574927\pi\)
−0.233224 + 0.972423i \(0.574927\pi\)
\(998\) 0 0
\(999\) 2.31805e6 0.0734868
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 1104.6.a.ba.1.5 8
4.3 odd 2 552.6.a.i.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
552.6.a.i.1.5 8 4.3 odd 2
1104.6.a.ba.1.5 8 1.1 even 1 trivial