Properties

Label 588.6.a.l.1.1
Level $588$
Weight $6$
Character 588.1
Self dual yes
Analytic conductor $94.306$
Analytic rank $1$
Dimension $2$
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] = [588,6,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, 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("588.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(94.3056860500\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7081}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1770 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(42.5743\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000 q^{3} -18.5743 q^{5} +81.0000 q^{9} -161.426 q^{11} +14.1283 q^{13} -167.169 q^{15} -765.703 q^{17} +1414.29 q^{19} +4185.43 q^{23} -2779.99 q^{25} +729.000 q^{27} -4202.60 q^{29} -2387.35 q^{31} -1452.83 q^{33} -672.939 q^{37} +127.155 q^{39} -4173.45 q^{41} -5430.94 q^{43} -1504.52 q^{45} +6302.68 q^{47} -6891.32 q^{51} +16416.7 q^{53} +2998.37 q^{55} +12728.6 q^{57} +3966.49 q^{59} -50338.7 q^{61} -262.424 q^{65} +13645.3 q^{67} +37668.9 q^{69} -83957.2 q^{71} -28578.3 q^{73} -25019.9 q^{75} -59955.5 q^{79} +6561.00 q^{81} -61583.0 q^{83} +14222.4 q^{85} -37823.4 q^{87} +42298.3 q^{89} -21486.2 q^{93} -26269.5 q^{95} +44638.1 q^{97} -13075.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 18 q^{3} + 47 q^{5} + 162 q^{9} - 407 q^{11} + 449 q^{13} + 423 q^{15} - 1868 q^{17} - 1463 q^{19} - 44 q^{23} - 1605 q^{25} + 1458 q^{27} + 767 q^{29} - 11170 q^{31} - 3663 q^{33} - 3113 q^{37} + 4041 q^{39}+ \cdots - 32967 q^{99}+O(q^{100}) \) 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
\(5\) −18.5743 −0.332268 −0.166134 0.986103i \(-0.553128\pi\)
−0.166134 + 0.986103i \(0.553128\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −161.426 −0.402245 −0.201123 0.979566i \(-0.564459\pi\)
−0.201123 + 0.979566i \(0.564459\pi\)
\(12\) 0 0
\(13\) 14.1283 0.0231863 0.0115932 0.999933i \(-0.496310\pi\)
0.0115932 + 0.999933i \(0.496310\pi\)
\(14\) 0 0
\(15\) −167.169 −0.191835
\(16\) 0 0
\(17\) −765.703 −0.642596 −0.321298 0.946978i \(-0.604119\pi\)
−0.321298 + 0.946978i \(0.604119\pi\)
\(18\) 0 0
\(19\) 1414.29 0.898783 0.449392 0.893335i \(-0.351641\pi\)
0.449392 + 0.893335i \(0.351641\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4185.43 1.64976 0.824880 0.565308i \(-0.191242\pi\)
0.824880 + 0.565308i \(0.191242\pi\)
\(24\) 0 0
\(25\) −2779.99 −0.889598
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −4202.60 −0.927947 −0.463974 0.885849i \(-0.653577\pi\)
−0.463974 + 0.885849i \(0.653577\pi\)
\(30\) 0 0
\(31\) −2387.35 −0.446182 −0.223091 0.974798i \(-0.571615\pi\)
−0.223091 + 0.974798i \(0.571615\pi\)
\(32\) 0 0
\(33\) −1452.83 −0.232236
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −672.939 −0.0808112 −0.0404056 0.999183i \(-0.512865\pi\)
−0.0404056 + 0.999183i \(0.512865\pi\)
\(38\) 0 0
\(39\) 127.155 0.0133866
\(40\) 0 0
\(41\) −4173.45 −0.387735 −0.193868 0.981028i \(-0.562103\pi\)
−0.193868 + 0.981028i \(0.562103\pi\)
\(42\) 0 0
\(43\) −5430.94 −0.447923 −0.223962 0.974598i \(-0.571899\pi\)
−0.223962 + 0.974598i \(0.571899\pi\)
\(44\) 0 0
\(45\) −1504.52 −0.110756
\(46\) 0 0
\(47\) 6302.68 0.416179 0.208090 0.978110i \(-0.433275\pi\)
0.208090 + 0.978110i \(0.433275\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6891.32 −0.371003
\(52\) 0 0
\(53\) 16416.7 0.802778 0.401389 0.915908i \(-0.368528\pi\)
0.401389 + 0.915908i \(0.368528\pi\)
\(54\) 0 0
\(55\) 2998.37 0.133653
\(56\) 0 0
\(57\) 12728.6 0.518913
\(58\) 0 0
\(59\) 3966.49 0.148346 0.0741731 0.997245i \(-0.476368\pi\)
0.0741731 + 0.997245i \(0.476368\pi\)
\(60\) 0 0
\(61\) −50338.7 −1.73212 −0.866059 0.499942i \(-0.833355\pi\)
−0.866059 + 0.499942i \(0.833355\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −262.424 −0.00770407
\(66\) 0 0
\(67\) 13645.3 0.371360 0.185680 0.982610i \(-0.440551\pi\)
0.185680 + 0.982610i \(0.440551\pi\)
\(68\) 0 0
\(69\) 37668.9 0.952490
\(70\) 0 0
\(71\) −83957.2 −1.97657 −0.988284 0.152624i \(-0.951228\pi\)
−0.988284 + 0.152624i \(0.951228\pi\)
\(72\) 0 0
\(73\) −28578.3 −0.627668 −0.313834 0.949478i \(-0.601613\pi\)
−0.313834 + 0.949478i \(0.601613\pi\)
\(74\) 0 0
\(75\) −25019.9 −0.513610
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −59955.5 −1.08084 −0.540420 0.841396i \(-0.681734\pi\)
−0.540420 + 0.841396i \(0.681734\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −61583.0 −0.981219 −0.490610 0.871380i \(-0.663226\pi\)
−0.490610 + 0.871380i \(0.663226\pi\)
\(84\) 0 0
\(85\) 14222.4 0.213514
\(86\) 0 0
\(87\) −37823.4 −0.535751
\(88\) 0 0
\(89\) 42298.3 0.566042 0.283021 0.959114i \(-0.408663\pi\)
0.283021 + 0.959114i \(0.408663\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −21486.2 −0.257603
\(94\) 0 0
\(95\) −26269.5 −0.298637
\(96\) 0 0
\(97\) 44638.1 0.481700 0.240850 0.970562i \(-0.422574\pi\)
0.240850 + 0.970562i \(0.422574\pi\)
\(98\) 0 0
\(99\) −13075.5 −0.134082
\(100\) 0 0
\(101\) −111924. −1.09175 −0.545873 0.837868i \(-0.683802\pi\)
−0.545873 + 0.837868i \(0.683802\pi\)
\(102\) 0 0
\(103\) 166949. 1.55057 0.775285 0.631612i \(-0.217606\pi\)
0.775285 + 0.631612i \(0.217606\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 102986. 0.869595 0.434798 0.900528i \(-0.356820\pi\)
0.434798 + 0.900528i \(0.356820\pi\)
\(108\) 0 0
\(109\) −137821. −1.11109 −0.555546 0.831486i \(-0.687491\pi\)
−0.555546 + 0.831486i \(0.687491\pi\)
\(110\) 0 0
\(111\) −6056.45 −0.0466563
\(112\) 0 0
\(113\) −129967. −0.957495 −0.478748 0.877953i \(-0.658909\pi\)
−0.478748 + 0.877953i \(0.658909\pi\)
\(114\) 0 0
\(115\) −77741.7 −0.548162
\(116\) 0 0
\(117\) 1144.39 0.00772877
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −134993. −0.838199
\(122\) 0 0
\(123\) −37561.0 −0.223859
\(124\) 0 0
\(125\) 109681. 0.627853
\(126\) 0 0
\(127\) −243265. −1.33835 −0.669177 0.743103i \(-0.733353\pi\)
−0.669177 + 0.743103i \(0.733353\pi\)
\(128\) 0 0
\(129\) −48878.4 −0.258609
\(130\) 0 0
\(131\) 194336. 0.989405 0.494703 0.869062i \(-0.335277\pi\)
0.494703 + 0.869062i \(0.335277\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −13540.7 −0.0639450
\(136\) 0 0
\(137\) −3093.70 −0.0140824 −0.00704121 0.999975i \(-0.502241\pi\)
−0.00704121 + 0.999975i \(0.502241\pi\)
\(138\) 0 0
\(139\) 22600.4 0.0992155 0.0496078 0.998769i \(-0.484203\pi\)
0.0496078 + 0.998769i \(0.484203\pi\)
\(140\) 0 0
\(141\) 56724.1 0.240281
\(142\) 0 0
\(143\) −2280.67 −0.00932658
\(144\) 0 0
\(145\) 78060.6 0.308327
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −352794. −1.30183 −0.650917 0.759149i \(-0.725616\pi\)
−0.650917 + 0.759149i \(0.725616\pi\)
\(150\) 0 0
\(151\) 145096. 0.517863 0.258931 0.965896i \(-0.416630\pi\)
0.258931 + 0.965896i \(0.416630\pi\)
\(152\) 0 0
\(153\) −62021.9 −0.214199
\(154\) 0 0
\(155\) 44343.5 0.148252
\(156\) 0 0
\(157\) −217895. −0.705503 −0.352751 0.935717i \(-0.614754\pi\)
−0.352751 + 0.935717i \(0.614754\pi\)
\(158\) 0 0
\(159\) 147750. 0.463484
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −322111. −0.949591 −0.474795 0.880096i \(-0.657478\pi\)
−0.474795 + 0.880096i \(0.657478\pi\)
\(164\) 0 0
\(165\) 26985.4 0.0771647
\(166\) 0 0
\(167\) 707392. 1.96277 0.981384 0.192056i \(-0.0615155\pi\)
0.981384 + 0.192056i \(0.0615155\pi\)
\(168\) 0 0
\(169\) −371093. −0.999462
\(170\) 0 0
\(171\) 114558. 0.299594
\(172\) 0 0
\(173\) −170507. −0.433140 −0.216570 0.976267i \(-0.569487\pi\)
−0.216570 + 0.976267i \(0.569487\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 35698.4 0.0856477
\(178\) 0 0
\(179\) −415406. −0.969037 −0.484519 0.874781i \(-0.661005\pi\)
−0.484519 + 0.874781i \(0.661005\pi\)
\(180\) 0 0
\(181\) −1162.38 −0.00263726 −0.00131863 0.999999i \(-0.500420\pi\)
−0.00131863 + 0.999999i \(0.500420\pi\)
\(182\) 0 0
\(183\) −453048. −1.00004
\(184\) 0 0
\(185\) 12499.4 0.0268510
\(186\) 0 0
\(187\) 123604. 0.258481
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −745614. −1.47887 −0.739436 0.673227i \(-0.764908\pi\)
−0.739436 + 0.673227i \(0.764908\pi\)
\(192\) 0 0
\(193\) −198373. −0.383344 −0.191672 0.981459i \(-0.561391\pi\)
−0.191672 + 0.981459i \(0.561391\pi\)
\(194\) 0 0
\(195\) −2361.82 −0.00444795
\(196\) 0 0
\(197\) −469368. −0.861684 −0.430842 0.902427i \(-0.641783\pi\)
−0.430842 + 0.902427i \(0.641783\pi\)
\(198\) 0 0
\(199\) −193593. −0.346544 −0.173272 0.984874i \(-0.555434\pi\)
−0.173272 + 0.984874i \(0.555434\pi\)
\(200\) 0 0
\(201\) 122807. 0.214405
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 77519.0 0.128832
\(206\) 0 0
\(207\) 339020. 0.549920
\(208\) 0 0
\(209\) −228303. −0.361531
\(210\) 0 0
\(211\) −298066. −0.460900 −0.230450 0.973084i \(-0.574020\pi\)
−0.230450 + 0.973084i \(0.574020\pi\)
\(212\) 0 0
\(213\) −755615. −1.14117
\(214\) 0 0
\(215\) 100876. 0.148831
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −257205. −0.362384
\(220\) 0 0
\(221\) −10818.1 −0.0148994
\(222\) 0 0
\(223\) −187215. −0.252103 −0.126051 0.992024i \(-0.540230\pi\)
−0.126051 + 0.992024i \(0.540230\pi\)
\(224\) 0 0
\(225\) −225180. −0.296533
\(226\) 0 0
\(227\) −1.33896e6 −1.72466 −0.862332 0.506344i \(-0.830997\pi\)
−0.862332 + 0.506344i \(0.830997\pi\)
\(228\) 0 0
\(229\) 951656. 1.19920 0.599600 0.800300i \(-0.295326\pi\)
0.599600 + 0.800300i \(0.295326\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 743419. 0.897106 0.448553 0.893756i \(-0.351940\pi\)
0.448553 + 0.893756i \(0.351940\pi\)
\(234\) 0 0
\(235\) −117068. −0.138283
\(236\) 0 0
\(237\) −539599. −0.624023
\(238\) 0 0
\(239\) −625847. −0.708718 −0.354359 0.935110i \(-0.615301\pi\)
−0.354359 + 0.935110i \(0.615301\pi\)
\(240\) 0 0
\(241\) −1.33282e6 −1.47819 −0.739094 0.673602i \(-0.764746\pi\)
−0.739094 + 0.673602i \(0.764746\pi\)
\(242\) 0 0
\(243\) 59049.0 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 19981.5 0.0208395
\(248\) 0 0
\(249\) −554247. −0.566507
\(250\) 0 0
\(251\) 1.78809e6 1.79145 0.895726 0.444606i \(-0.146656\pi\)
0.895726 + 0.444606i \(0.146656\pi\)
\(252\) 0 0
\(253\) −675636. −0.663608
\(254\) 0 0
\(255\) 128002. 0.123272
\(256\) 0 0
\(257\) 508924. 0.480640 0.240320 0.970694i \(-0.422748\pi\)
0.240320 + 0.970694i \(0.422748\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −340411. −0.309316
\(262\) 0 0
\(263\) −336493. −0.299976 −0.149988 0.988688i \(-0.547924\pi\)
−0.149988 + 0.988688i \(0.547924\pi\)
\(264\) 0 0
\(265\) −304929. −0.266737
\(266\) 0 0
\(267\) 380685. 0.326804
\(268\) 0 0
\(269\) 1.49553e6 1.26013 0.630064 0.776544i \(-0.283029\pi\)
0.630064 + 0.776544i \(0.283029\pi\)
\(270\) 0 0
\(271\) −1.77697e6 −1.46979 −0.734897 0.678179i \(-0.762770\pi\)
−0.734897 + 0.678179i \(0.762770\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 448762. 0.357837
\(276\) 0 0
\(277\) 881265. 0.690092 0.345046 0.938586i \(-0.387863\pi\)
0.345046 + 0.938586i \(0.387863\pi\)
\(278\) 0 0
\(279\) −193375. −0.148727
\(280\) 0 0
\(281\) −1.13932e6 −0.860756 −0.430378 0.902649i \(-0.641620\pi\)
−0.430378 + 0.902649i \(0.641620\pi\)
\(282\) 0 0
\(283\) −897466. −0.666119 −0.333059 0.942906i \(-0.608081\pi\)
−0.333059 + 0.942906i \(0.608081\pi\)
\(284\) 0 0
\(285\) −236426. −0.172418
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −833556. −0.587071
\(290\) 0 0
\(291\) 401743. 0.278110
\(292\) 0 0
\(293\) −1.84614e6 −1.25631 −0.628153 0.778090i \(-0.716189\pi\)
−0.628153 + 0.778090i \(0.716189\pi\)
\(294\) 0 0
\(295\) −73674.9 −0.0492907
\(296\) 0 0
\(297\) −117679. −0.0774121
\(298\) 0 0
\(299\) 59133.1 0.0382519
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.00732e6 −0.630320
\(304\) 0 0
\(305\) 935008. 0.575527
\(306\) 0 0
\(307\) 1.38565e6 0.839089 0.419544 0.907735i \(-0.362190\pi\)
0.419544 + 0.907735i \(0.362190\pi\)
\(308\) 0 0
\(309\) 1.50254e6 0.895221
\(310\) 0 0
\(311\) 2.31248e6 1.35574 0.677871 0.735181i \(-0.262903\pi\)
0.677871 + 0.735181i \(0.262903\pi\)
\(312\) 0 0
\(313\) −1.16880e6 −0.674341 −0.337171 0.941444i \(-0.609470\pi\)
−0.337171 + 0.941444i \(0.609470\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.43433e6 0.801681 0.400840 0.916148i \(-0.368718\pi\)
0.400840 + 0.916148i \(0.368718\pi\)
\(318\) 0 0
\(319\) 678408. 0.373262
\(320\) 0 0
\(321\) 926871. 0.502061
\(322\) 0 0
\(323\) −1.08293e6 −0.577554
\(324\) 0 0
\(325\) −39276.6 −0.0206265
\(326\) 0 0
\(327\) −1.24039e6 −0.641489
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.23418e6 −1.12085 −0.560426 0.828205i \(-0.689362\pi\)
−0.560426 + 0.828205i \(0.689362\pi\)
\(332\) 0 0
\(333\) −54508.0 −0.0269371
\(334\) 0 0
\(335\) −253452. −0.123391
\(336\) 0 0
\(337\) 3.08787e6 1.48110 0.740549 0.672002i \(-0.234565\pi\)
0.740549 + 0.672002i \(0.234565\pi\)
\(338\) 0 0
\(339\) −1.16970e6 −0.552810
\(340\) 0 0
\(341\) 385380. 0.179475
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −699675. −0.316482
\(346\) 0 0
\(347\) −3.11754e6 −1.38992 −0.694959 0.719050i \(-0.744577\pi\)
−0.694959 + 0.719050i \(0.744577\pi\)
\(348\) 0 0
\(349\) 613026. 0.269411 0.134706 0.990886i \(-0.456991\pi\)
0.134706 + 0.990886i \(0.456991\pi\)
\(350\) 0 0
\(351\) 10299.5 0.00446221
\(352\) 0 0
\(353\) 3.78370e6 1.61614 0.808071 0.589085i \(-0.200512\pi\)
0.808071 + 0.589085i \(0.200512\pi\)
\(354\) 0 0
\(355\) 1.55945e6 0.656750
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.72213e6 1.52425 0.762125 0.647430i \(-0.224156\pi\)
0.762125 + 0.647430i \(0.224156\pi\)
\(360\) 0 0
\(361\) −475879. −0.192189
\(362\) 0 0
\(363\) −1.21493e6 −0.483934
\(364\) 0 0
\(365\) 530824. 0.208554
\(366\) 0 0
\(367\) 4.13795e6 1.60369 0.801845 0.597532i \(-0.203852\pi\)
0.801845 + 0.597532i \(0.203852\pi\)
\(368\) 0 0
\(369\) −338049. −0.129245
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.94418e6 1.09570 0.547851 0.836576i \(-0.315446\pi\)
0.547851 + 0.836576i \(0.315446\pi\)
\(374\) 0 0
\(375\) 987132. 0.362491
\(376\) 0 0
\(377\) −59375.7 −0.0215157
\(378\) 0 0
\(379\) 2.97504e6 1.06388 0.531942 0.846781i \(-0.321462\pi\)
0.531942 + 0.846781i \(0.321462\pi\)
\(380\) 0 0
\(381\) −2.18939e6 −0.772699
\(382\) 0 0
\(383\) 2.06104e6 0.717942 0.358971 0.933349i \(-0.383128\pi\)
0.358971 + 0.933349i \(0.383128\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −439906. −0.149308
\(388\) 0 0
\(389\) 2.09084e6 0.700562 0.350281 0.936645i \(-0.386086\pi\)
0.350281 + 0.936645i \(0.386086\pi\)
\(390\) 0 0
\(391\) −3.20480e6 −1.06013
\(392\) 0 0
\(393\) 1.74902e6 0.571233
\(394\) 0 0
\(395\) 1.11363e6 0.359128
\(396\) 0 0
\(397\) 1.17129e6 0.372984 0.186492 0.982457i \(-0.440288\pi\)
0.186492 + 0.982457i \(0.440288\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.09149e6 0.960078 0.480039 0.877247i \(-0.340623\pi\)
0.480039 + 0.877247i \(0.340623\pi\)
\(402\) 0 0
\(403\) −33729.2 −0.0103453
\(404\) 0 0
\(405\) −121866. −0.0369187
\(406\) 0 0
\(407\) 108630. 0.0325059
\(408\) 0 0
\(409\) −4.34853e6 −1.28539 −0.642693 0.766124i \(-0.722183\pi\)
−0.642693 + 0.766124i \(0.722183\pi\)
\(410\) 0 0
\(411\) −27843.3 −0.00813048
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.14386e6 0.326028
\(416\) 0 0
\(417\) 203404. 0.0572821
\(418\) 0 0
\(419\) 3.13660e6 0.872818 0.436409 0.899748i \(-0.356250\pi\)
0.436409 + 0.899748i \(0.356250\pi\)
\(420\) 0 0
\(421\) 6.02560e6 1.65690 0.828448 0.560066i \(-0.189224\pi\)
0.828448 + 0.560066i \(0.189224\pi\)
\(422\) 0 0
\(423\) 510517. 0.138726
\(424\) 0 0
\(425\) 2.12865e6 0.571652
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −20526.0 −0.00538471
\(430\) 0 0
\(431\) −2.29863e6 −0.596040 −0.298020 0.954560i \(-0.596326\pi\)
−0.298020 + 0.954560i \(0.596326\pi\)
\(432\) 0 0
\(433\) −5.62982e6 −1.44303 −0.721515 0.692399i \(-0.756554\pi\)
−0.721515 + 0.692399i \(0.756554\pi\)
\(434\) 0 0
\(435\) 702545. 0.178013
\(436\) 0 0
\(437\) 5.91942e6 1.48278
\(438\) 0 0
\(439\) −5.44963e6 −1.34960 −0.674801 0.737999i \(-0.735771\pi\)
−0.674801 + 0.737999i \(0.735771\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.70933e6 1.38222 0.691108 0.722751i \(-0.257123\pi\)
0.691108 + 0.722751i \(0.257123\pi\)
\(444\) 0 0
\(445\) −785664. −0.188077
\(446\) 0 0
\(447\) −3.17515e6 −0.751615
\(448\) 0 0
\(449\) 5.38234e6 1.25996 0.629978 0.776613i \(-0.283064\pi\)
0.629978 + 0.776613i \(0.283064\pi\)
\(450\) 0 0
\(451\) 673701. 0.155965
\(452\) 0 0
\(453\) 1.30587e6 0.298988
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6.39051e6 −1.43135 −0.715673 0.698435i \(-0.753880\pi\)
−0.715673 + 0.698435i \(0.753880\pi\)
\(458\) 0 0
\(459\) −558197. −0.123668
\(460\) 0 0
\(461\) 7.34511e6 1.60970 0.804851 0.593476i \(-0.202245\pi\)
0.804851 + 0.593476i \(0.202245\pi\)
\(462\) 0 0
\(463\) −4.63416e6 −1.00466 −0.502329 0.864677i \(-0.667523\pi\)
−0.502329 + 0.864677i \(0.667523\pi\)
\(464\) 0 0
\(465\) 399091. 0.0855933
\(466\) 0 0
\(467\) −7.63195e6 −1.61936 −0.809680 0.586872i \(-0.800359\pi\)
−0.809680 + 0.586872i \(0.800359\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1.96106e6 −0.407322
\(472\) 0 0
\(473\) 876693. 0.180175
\(474\) 0 0
\(475\) −3.93172e6 −0.799556
\(476\) 0 0
\(477\) 1.32975e6 0.267593
\(478\) 0 0
\(479\) 9.61262e6 1.91427 0.957135 0.289644i \(-0.0935368\pi\)
0.957135 + 0.289644i \(0.0935368\pi\)
\(480\) 0 0
\(481\) −9507.49 −0.00187371
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −829124. −0.160053
\(486\) 0 0
\(487\) 2.80223e6 0.535404 0.267702 0.963502i \(-0.413736\pi\)
0.267702 + 0.963502i \(0.413736\pi\)
\(488\) 0 0
\(489\) −2.89900e6 −0.548246
\(490\) 0 0
\(491\) 4.82008e6 0.902299 0.451149 0.892448i \(-0.351014\pi\)
0.451149 + 0.892448i \(0.351014\pi\)
\(492\) 0 0
\(493\) 3.21794e6 0.596295
\(494\) 0 0
\(495\) 242868. 0.0445511
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.59658e6 0.826387 0.413194 0.910643i \(-0.364413\pi\)
0.413194 + 0.910643i \(0.364413\pi\)
\(500\) 0 0
\(501\) 6.36653e6 1.13320
\(502\) 0 0
\(503\) 1.80055e6 0.317311 0.158656 0.987334i \(-0.449284\pi\)
0.158656 + 0.987334i \(0.449284\pi\)
\(504\) 0 0
\(505\) 2.07892e6 0.362752
\(506\) 0 0
\(507\) −3.33984e6 −0.577040
\(508\) 0 0
\(509\) −4.83205e6 −0.826678 −0.413339 0.910577i \(-0.635638\pi\)
−0.413339 + 0.910577i \(0.635638\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.03102e6 0.172971
\(514\) 0 0
\(515\) −3.10097e6 −0.515204
\(516\) 0 0
\(517\) −1.01741e6 −0.167406
\(518\) 0 0
\(519\) −1.53457e6 −0.250073
\(520\) 0 0
\(521\) 2.81125e6 0.453738 0.226869 0.973925i \(-0.427151\pi\)
0.226869 + 0.973925i \(0.427151\pi\)
\(522\) 0 0
\(523\) −2.45429e6 −0.392349 −0.196174 0.980569i \(-0.562852\pi\)
−0.196174 + 0.980569i \(0.562852\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.82800e6 0.286715
\(528\) 0 0
\(529\) 1.10815e7 1.72171
\(530\) 0 0
\(531\) 321286. 0.0494487
\(532\) 0 0
\(533\) −58963.7 −0.00899015
\(534\) 0 0
\(535\) −1.91289e6 −0.288939
\(536\) 0 0
\(537\) −3.73866e6 −0.559474
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.15846e6 −0.170172 −0.0850862 0.996374i \(-0.527117\pi\)
−0.0850862 + 0.996374i \(0.527117\pi\)
\(542\) 0 0
\(543\) −10461.4 −0.00152262
\(544\) 0 0
\(545\) 2.55994e6 0.369180
\(546\) 0 0
\(547\) 4.63638e6 0.662538 0.331269 0.943536i \(-0.392523\pi\)
0.331269 + 0.943536i \(0.392523\pi\)
\(548\) 0 0
\(549\) −4.07744e6 −0.577373
\(550\) 0 0
\(551\) −5.94370e6 −0.834023
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 112495. 0.0155024
\(556\) 0 0
\(557\) 271832. 0.0371247 0.0185624 0.999828i \(-0.494091\pi\)
0.0185624 + 0.999828i \(0.494091\pi\)
\(558\) 0 0
\(559\) −76730.0 −0.0103857
\(560\) 0 0
\(561\) 1.11244e6 0.149234
\(562\) 0 0
\(563\) 3.38244e6 0.449738 0.224869 0.974389i \(-0.427805\pi\)
0.224869 + 0.974389i \(0.427805\pi\)
\(564\) 0 0
\(565\) 2.41405e6 0.318145
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.31226e7 1.69918 0.849590 0.527443i \(-0.176849\pi\)
0.849590 + 0.527443i \(0.176849\pi\)
\(570\) 0 0
\(571\) −1.47162e7 −1.88889 −0.944443 0.328674i \(-0.893398\pi\)
−0.944443 + 0.328674i \(0.893398\pi\)
\(572\) 0 0
\(573\) −6.71053e6 −0.853828
\(574\) 0 0
\(575\) −1.16355e7 −1.46762
\(576\) 0 0
\(577\) −3.88295e6 −0.485537 −0.242769 0.970084i \(-0.578056\pi\)
−0.242769 + 0.970084i \(0.578056\pi\)
\(578\) 0 0
\(579\) −1.78536e6 −0.221324
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.65007e6 −0.322914
\(584\) 0 0
\(585\) −21256.3 −0.00256802
\(586\) 0 0
\(587\) 4.97913e6 0.596428 0.298214 0.954499i \(-0.403609\pi\)
0.298214 + 0.954499i \(0.403609\pi\)
\(588\) 0 0
\(589\) −3.37641e6 −0.401021
\(590\) 0 0
\(591\) −4.22431e6 −0.497493
\(592\) 0 0
\(593\) 1.53269e7 1.78985 0.894926 0.446215i \(-0.147228\pi\)
0.894926 + 0.446215i \(0.147228\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.74234e6 −0.200077
\(598\) 0 0
\(599\) −6.62961e6 −0.754955 −0.377477 0.926019i \(-0.623208\pi\)
−0.377477 + 0.926019i \(0.623208\pi\)
\(600\) 0 0
\(601\) −1.45010e6 −0.163762 −0.0818808 0.996642i \(-0.526093\pi\)
−0.0818808 + 0.996642i \(0.526093\pi\)
\(602\) 0 0
\(603\) 1.10527e6 0.123787
\(604\) 0 0
\(605\) 2.50740e6 0.278507
\(606\) 0 0
\(607\) 3.91110e6 0.430851 0.215425 0.976520i \(-0.430886\pi\)
0.215425 + 0.976520i \(0.430886\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 89046.1 0.00964966
\(612\) 0 0
\(613\) 1.17753e7 1.26567 0.632835 0.774287i \(-0.281891\pi\)
0.632835 + 0.774287i \(0.281891\pi\)
\(614\) 0 0
\(615\) 697671. 0.0743812
\(616\) 0 0
\(617\) −4.61462e6 −0.488004 −0.244002 0.969775i \(-0.578460\pi\)
−0.244002 + 0.969775i \(0.578460\pi\)
\(618\) 0 0
\(619\) −5.67466e6 −0.595269 −0.297634 0.954680i \(-0.596198\pi\)
−0.297634 + 0.954680i \(0.596198\pi\)
\(620\) 0 0
\(621\) 3.05118e6 0.317497
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.65022e6 0.680983
\(626\) 0 0
\(627\) −2.05473e6 −0.208730
\(628\) 0 0
\(629\) 515271. 0.0519289
\(630\) 0 0
\(631\) 5.67894e6 0.567798 0.283899 0.958854i \(-0.408372\pi\)
0.283899 + 0.958854i \(0.408372\pi\)
\(632\) 0 0
\(633\) −2.68260e6 −0.266101
\(634\) 0 0
\(635\) 4.51849e6 0.444692
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6.80053e6 −0.658856
\(640\) 0 0
\(641\) −2.10737e6 −0.202580 −0.101290 0.994857i \(-0.532297\pi\)
−0.101290 + 0.994857i \(0.532297\pi\)
\(642\) 0 0
\(643\) −2.30987e6 −0.220323 −0.110162 0.993914i \(-0.535137\pi\)
−0.110162 + 0.993914i \(0.535137\pi\)
\(644\) 0 0
\(645\) 907885. 0.0859274
\(646\) 0 0
\(647\) 9.86066e6 0.926073 0.463036 0.886339i \(-0.346760\pi\)
0.463036 + 0.886339i \(0.346760\pi\)
\(648\) 0 0
\(649\) −640293. −0.0596715
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.00309e7 0.920570 0.460285 0.887771i \(-0.347747\pi\)
0.460285 + 0.887771i \(0.347747\pi\)
\(654\) 0 0
\(655\) −3.60966e6 −0.328748
\(656\) 0 0
\(657\) −2.31484e6 −0.209223
\(658\) 0 0
\(659\) −1.42828e7 −1.28115 −0.640575 0.767895i \(-0.721304\pi\)
−0.640575 + 0.767895i \(0.721304\pi\)
\(660\) 0 0
\(661\) −1.96419e7 −1.74856 −0.874280 0.485421i \(-0.838666\pi\)
−0.874280 + 0.485421i \(0.838666\pi\)
\(662\) 0 0
\(663\) −97362.7 −0.00860219
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.75897e7 −1.53089
\(668\) 0 0
\(669\) −1.68493e6 −0.145552
\(670\) 0 0
\(671\) 8.12596e6 0.696736
\(672\) 0 0
\(673\) −9.00150e6 −0.766086 −0.383043 0.923731i \(-0.625124\pi\)
−0.383043 + 0.923731i \(0.625124\pi\)
\(674\) 0 0
\(675\) −2.02662e6 −0.171203
\(676\) 0 0
\(677\) 1.34236e7 1.12564 0.562818 0.826581i \(-0.309717\pi\)
0.562818 + 0.826581i \(0.309717\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1.20507e7 −0.995735
\(682\) 0 0
\(683\) −8.01681e6 −0.657582 −0.328791 0.944403i \(-0.606641\pi\)
−0.328791 + 0.944403i \(0.606641\pi\)
\(684\) 0 0
\(685\) 57463.5 0.00467913
\(686\) 0 0
\(687\) 8.56491e6 0.692358
\(688\) 0 0
\(689\) 231940. 0.0186135
\(690\) 0 0
\(691\) −2.41973e7 −1.92785 −0.963923 0.266181i \(-0.914238\pi\)
−0.963923 + 0.266181i \(0.914238\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −419788. −0.0329661
\(696\) 0 0
\(697\) 3.19562e6 0.249157
\(698\) 0 0
\(699\) 6.69077e6 0.517944
\(700\) 0 0
\(701\) −1.56268e6 −0.120109 −0.0600543 0.998195i \(-0.519127\pi\)
−0.0600543 + 0.998195i \(0.519127\pi\)
\(702\) 0 0
\(703\) −951732. −0.0726317
\(704\) 0 0
\(705\) −1.05361e6 −0.0798377
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.95390e7 −1.45978 −0.729889 0.683566i \(-0.760428\pi\)
−0.729889 + 0.683566i \(0.760428\pi\)
\(710\) 0 0
\(711\) −4.85639e6 −0.360280
\(712\) 0 0
\(713\) −9.99210e6 −0.736093
\(714\) 0 0
\(715\) 42362.0 0.00309892
\(716\) 0 0
\(717\) −5.63262e6 −0.409178
\(718\) 0 0
\(719\) −8.10190e6 −0.584473 −0.292237 0.956346i \(-0.594399\pi\)
−0.292237 + 0.956346i \(0.594399\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.19954e7 −0.853432
\(724\) 0 0
\(725\) 1.16832e7 0.825500
\(726\) 0 0
\(727\) 2.52759e7 1.77366 0.886829 0.462098i \(-0.152903\pi\)
0.886829 + 0.462098i \(0.152903\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 4.15848e6 0.287834
\(732\) 0 0
\(733\) 2.74414e7 1.88646 0.943228 0.332145i \(-0.107772\pi\)
0.943228 + 0.332145i \(0.107772\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.20270e6 −0.149378
\(738\) 0 0
\(739\) −6.53569e6 −0.440231 −0.220115 0.975474i \(-0.570643\pi\)
−0.220115 + 0.975474i \(0.570643\pi\)
\(740\) 0 0
\(741\) 179834. 0.0120317
\(742\) 0 0
\(743\) −2.46784e7 −1.64000 −0.820001 0.572362i \(-0.806027\pi\)
−0.820001 + 0.572362i \(0.806027\pi\)
\(744\) 0 0
\(745\) 6.55292e6 0.432558
\(746\) 0 0
\(747\) −4.98823e6 −0.327073
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.27822e6 −0.212099 −0.106049 0.994361i \(-0.533820\pi\)
−0.106049 + 0.994361i \(0.533820\pi\)
\(752\) 0 0
\(753\) 1.60928e7 1.03430
\(754\) 0 0
\(755\) −2.69507e6 −0.172069
\(756\) 0 0
\(757\) 3.68090e6 0.233461 0.116731 0.993164i \(-0.462759\pi\)
0.116731 + 0.993164i \(0.462759\pi\)
\(758\) 0 0
\(759\) −6.08073e6 −0.383134
\(760\) 0 0
\(761\) 2.10198e7 1.31573 0.657866 0.753135i \(-0.271459\pi\)
0.657866 + 0.753135i \(0.271459\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.15202e6 0.0711713
\(766\) 0 0
\(767\) 56039.8 0.00343960
\(768\) 0 0
\(769\) −4.15418e6 −0.253320 −0.126660 0.991946i \(-0.540426\pi\)
−0.126660 + 0.991946i \(0.540426\pi\)
\(770\) 0 0
\(771\) 4.58032e6 0.277498
\(772\) 0 0
\(773\) 1.46348e6 0.0880924 0.0440462 0.999029i \(-0.485975\pi\)
0.0440462 + 0.999029i \(0.485975\pi\)
\(774\) 0 0
\(775\) 6.63682e6 0.396923
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.90247e6 −0.348490
\(780\) 0 0
\(781\) 1.35528e7 0.795065
\(782\) 0 0
\(783\) −3.06370e6 −0.178584
\(784\) 0 0
\(785\) 4.04726e6 0.234416
\(786\) 0 0
\(787\) −2.01711e7 −1.16089 −0.580446 0.814299i \(-0.697122\pi\)
−0.580446 + 0.814299i \(0.697122\pi\)
\(788\) 0 0
\(789\) −3.02844e6 −0.173191
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −711201. −0.0401614
\(794\) 0 0
\(795\) −2.74436e6 −0.154001
\(796\) 0 0
\(797\) 2.19496e7 1.22400 0.612000 0.790858i \(-0.290365\pi\)
0.612000 + 0.790858i \(0.290365\pi\)
\(798\) 0 0
\(799\) −4.82598e6 −0.267435
\(800\) 0 0
\(801\) 3.42617e6 0.188681
\(802\) 0 0
\(803\) 4.61328e6 0.252476
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.34598e7 0.727535
\(808\) 0 0
\(809\) −2.36276e7 −1.26926 −0.634628 0.772818i \(-0.718847\pi\)
−0.634628 + 0.772818i \(0.718847\pi\)
\(810\) 0 0
\(811\) −2.85032e6 −0.152174 −0.0760872 0.997101i \(-0.524243\pi\)
−0.0760872 + 0.997101i \(0.524243\pi\)
\(812\) 0 0
\(813\) −1.59927e7 −0.848586
\(814\) 0 0
\(815\) 5.98300e6 0.315518
\(816\) 0 0
\(817\) −7.68093e6 −0.402586
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.41313e7 −0.731685 −0.365843 0.930677i \(-0.619219\pi\)
−0.365843 + 0.930677i \(0.619219\pi\)
\(822\) 0 0
\(823\) −3.23188e7 −1.66324 −0.831620 0.555345i \(-0.812586\pi\)
−0.831620 + 0.555345i \(0.812586\pi\)
\(824\) 0 0
\(825\) 4.03886e6 0.206597
\(826\) 0 0
\(827\) −3.35291e7 −1.70474 −0.852369 0.522941i \(-0.824835\pi\)
−0.852369 + 0.522941i \(0.824835\pi\)
\(828\) 0 0
\(829\) −1.99517e7 −1.00831 −0.504154 0.863614i \(-0.668195\pi\)
−0.504154 + 0.863614i \(0.668195\pi\)
\(830\) 0 0
\(831\) 7.93138e6 0.398425
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.31393e7 −0.652165
\(836\) 0 0
\(837\) −1.74038e6 −0.0858677
\(838\) 0 0
\(839\) −2.34067e6 −0.114798 −0.0573990 0.998351i \(-0.518281\pi\)
−0.0573990 + 0.998351i \(0.518281\pi\)
\(840\) 0 0
\(841\) −2.84928e6 −0.138914
\(842\) 0 0
\(843\) −1.02539e7 −0.496958
\(844\) 0 0
\(845\) 6.89281e6 0.332089
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.07719e6 −0.384584
\(850\) 0 0
\(851\) −2.81654e6 −0.133319
\(852\) 0 0
\(853\) −1.79465e7 −0.844514 −0.422257 0.906476i \(-0.638762\pi\)
−0.422257 + 0.906476i \(0.638762\pi\)
\(854\) 0 0
\(855\) −2.12783e6 −0.0995456
\(856\) 0 0
\(857\) −3.68442e7 −1.71363 −0.856815 0.515624i \(-0.827560\pi\)
−0.856815 + 0.515624i \(0.827560\pi\)
\(858\) 0 0
\(859\) 2.40612e7 1.11259 0.556293 0.830986i \(-0.312223\pi\)
0.556293 + 0.830986i \(0.312223\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.64770e7 0.753098 0.376549 0.926397i \(-0.377111\pi\)
0.376549 + 0.926397i \(0.377111\pi\)
\(864\) 0 0
\(865\) 3.16706e6 0.143918
\(866\) 0 0
\(867\) −7.50201e6 −0.338945
\(868\) 0 0
\(869\) 9.67835e6 0.434762
\(870\) 0 0
\(871\) 192784. 0.00861046
\(872\) 0 0
\(873\) 3.61569e6 0.160567
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.04983e7 1.33899 0.669494 0.742817i \(-0.266511\pi\)
0.669494 + 0.742817i \(0.266511\pi\)
\(878\) 0 0
\(879\) −1.66153e7 −0.725329
\(880\) 0 0
\(881\) 2.35008e7 1.02010 0.510051 0.860144i \(-0.329627\pi\)
0.510051 + 0.860144i \(0.329627\pi\)
\(882\) 0 0
\(883\) −4.42812e7 −1.91125 −0.955625 0.294587i \(-0.904818\pi\)
−0.955625 + 0.294587i \(0.904818\pi\)
\(884\) 0 0
\(885\) −663074. −0.0284580
\(886\) 0 0
\(887\) −1.52662e7 −0.651510 −0.325755 0.945454i \(-0.605618\pi\)
−0.325755 + 0.945454i \(0.605618\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.05911e6 −0.0446939
\(892\) 0 0
\(893\) 8.91382e6 0.374055
\(894\) 0 0
\(895\) 7.71590e6 0.321980
\(896\) 0 0
\(897\) 532198. 0.0220847
\(898\) 0 0
\(899\) 1.00331e7 0.414033
\(900\) 0 0
\(901\) −1.25703e7 −0.515862
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21590.5 0.000876275 0
\(906\) 0 0
\(907\) 2.67694e7 1.08049 0.540245 0.841508i \(-0.318332\pi\)
0.540245 + 0.841508i \(0.318332\pi\)
\(908\) 0 0
\(909\) −9.06588e6 −0.363915
\(910\) 0 0
\(911\) 1.60537e7 0.640883 0.320442 0.947268i \(-0.396169\pi\)
0.320442 + 0.947268i \(0.396169\pi\)
\(912\) 0 0
\(913\) 9.94108e6 0.394691
\(914\) 0 0
\(915\) 8.41507e6 0.332281
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.08072e7 0.422107 0.211053 0.977475i \(-0.432311\pi\)
0.211053 + 0.977475i \(0.432311\pi\)
\(920\) 0 0
\(921\) 1.24709e7 0.484448
\(922\) 0 0
\(923\) −1.18617e6 −0.0458293
\(924\) 0 0
\(925\) 1.87077e6 0.0718894
\(926\) 0 0
\(927\) 1.35229e7 0.516856
\(928\) 0 0
\(929\) −2.16937e6 −0.0824695 −0.0412348 0.999149i \(-0.513129\pi\)
−0.0412348 + 0.999149i \(0.513129\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.08123e7 0.782738
\(934\) 0 0
\(935\) −2.29586e6 −0.0858849
\(936\) 0 0
\(937\) −3.51305e6 −0.130718 −0.0653589 0.997862i \(-0.520819\pi\)
−0.0653589 + 0.997862i \(0.520819\pi\)
\(938\) 0 0
\(939\) −1.05192e7 −0.389331
\(940\) 0 0
\(941\) −1.29297e7 −0.476008 −0.238004 0.971264i \(-0.576493\pi\)
−0.238004 + 0.971264i \(0.576493\pi\)
\(942\) 0 0
\(943\) −1.74677e7 −0.639670
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.48662e7 −0.901019 −0.450510 0.892772i \(-0.648758\pi\)
−0.450510 + 0.892772i \(0.648758\pi\)
\(948\) 0 0
\(949\) −403763. −0.0145533
\(950\) 0 0
\(951\) 1.29090e7 0.462850
\(952\) 0 0
\(953\) 3.33465e7 1.18937 0.594686 0.803958i \(-0.297276\pi\)
0.594686 + 0.803958i \(0.297276\pi\)
\(954\) 0 0
\(955\) 1.38493e7 0.491382
\(956\) 0 0
\(957\) 6.10567e6 0.215503
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.29297e7 −0.800922
\(962\) 0 0
\(963\) 8.34184e6 0.289865
\(964\) 0 0
\(965\) 3.68465e6 0.127373
\(966\) 0 0
\(967\) −2.48706e7 −0.855305 −0.427652 0.903943i \(-0.640659\pi\)
−0.427652 + 0.903943i \(0.640659\pi\)
\(968\) 0 0
\(969\) −9.74634e6 −0.333451
\(970\) 0 0
\(971\) −4.79491e7 −1.63205 −0.816024 0.578018i \(-0.803826\pi\)
−0.816024 + 0.578018i \(0.803826\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −353489. −0.0119087
\(976\) 0 0
\(977\) −7.85191e6 −0.263171 −0.131586 0.991305i \(-0.542007\pi\)
−0.131586 + 0.991305i \(0.542007\pi\)
\(978\) 0 0
\(979\) −6.82804e6 −0.227687
\(980\) 0 0
\(981\) −1.11635e7 −0.370364
\(982\) 0 0
\(983\) −2.84868e7 −0.940284 −0.470142 0.882591i \(-0.655797\pi\)
−0.470142 + 0.882591i \(0.655797\pi\)
\(984\) 0 0
\(985\) 8.71820e6 0.286310
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.27308e7 −0.738966
\(990\) 0 0
\(991\) −1.03795e7 −0.335733 −0.167867 0.985810i \(-0.553688\pi\)
−0.167867 + 0.985810i \(0.553688\pi\)
\(992\) 0 0
\(993\) −2.01076e7 −0.647124
\(994\) 0 0
\(995\) 3.59587e6 0.115145
\(996\) 0 0
\(997\) 1.68161e7 0.535781 0.267890 0.963449i \(-0.413674\pi\)
0.267890 + 0.963449i \(0.413674\pi\)
\(998\) 0 0
\(999\) −490572. −0.0155521
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 588.6.a.l.1.1 2
7.2 even 3 84.6.i.b.25.2 4
7.3 odd 6 588.6.i.m.373.1 4
7.4 even 3 84.6.i.b.37.2 yes 4
7.5 odd 6 588.6.i.m.361.1 4
7.6 odd 2 588.6.a.h.1.2 2
21.2 odd 6 252.6.k.e.109.1 4
21.11 odd 6 252.6.k.e.37.1 4
28.11 odd 6 336.6.q.g.289.2 4
28.23 odd 6 336.6.q.g.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.b.25.2 4 7.2 even 3
84.6.i.b.37.2 yes 4 7.4 even 3
252.6.k.e.37.1 4 21.11 odd 6
252.6.k.e.109.1 4 21.2 odd 6
336.6.q.g.193.2 4 28.23 odd 6
336.6.q.g.289.2 4 28.11 odd 6
588.6.a.h.1.2 2 7.6 odd 2
588.6.a.l.1.1 2 1.1 even 1 trivial
588.6.i.m.361.1 4 7.5 odd 6
588.6.i.m.373.1 4 7.3 odd 6