Properties

Label 400.6.c.n.49.1
Level $400$
Weight $6$
Character 400.49
Analytic conductor $64.154$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,6,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-7.26209i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.6.c.n.49.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-25.5242i q^{3} +131.048i q^{7} -408.483 q^{9} +O(q^{10})\) \(q-25.5242i q^{3} +131.048i q^{7} -408.483 q^{9} -290.104 q^{11} +68.3868i q^{13} +310.644i q^{17} -2133.35 q^{19} +3344.90 q^{21} -873.145i q^{23} +4223.83i q^{27} +2580.97 q^{29} +9086.30 q^{31} +7404.67i q^{33} -3990.64i q^{37} +1745.52 q^{39} +16981.8 q^{41} +18017.7i q^{43} +24864.7i q^{47} -366.670 q^{49} +7928.93 q^{51} -7652.91i q^{53} +54451.9i q^{57} -9233.69 q^{59} +3326.17 q^{61} -53531.1i q^{63} -32340.7i q^{67} -22286.3 q^{69} +35885.9 q^{71} -26513.6i q^{73} -38017.7i q^{77} +71705.7 q^{79} +8548.28 q^{81} -39630.1i q^{83} -65877.1i q^{87} +117441. q^{89} -8961.98 q^{91} -231920. i q^{93} -21878.3i q^{97} +118503. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 392 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 392 q^{9} + 392 q^{11} - 6360 q^{19} + 5928 q^{21} + 7840 q^{29} + 2192 q^{31} + 8224 q^{39} + 55508 q^{41} + 23372 q^{49} + 35752 q^{51} + 23920 q^{59} - 48792 q^{61} - 36984 q^{69} + 174592 q^{71} + 130960 q^{79} + 92564 q^{81} + 145620 q^{89} + 41152 q^{91} + 443584 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 25.5242i − 1.63738i −0.574238 0.818688i \(-0.694702\pi\)
0.574238 0.818688i \(-0.305298\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 131.048i 1.01085i 0.862871 + 0.505425i \(0.168664\pi\)
−0.862871 + 0.505425i \(0.831336\pi\)
\(8\) 0 0
\(9\) −408.483 −1.68100
\(10\) 0 0
\(11\) −290.104 −0.722891 −0.361445 0.932393i \(-0.617717\pi\)
−0.361445 + 0.932393i \(0.617717\pi\)
\(12\) 0 0
\(13\) 68.3868i 0.112231i 0.998424 + 0.0561156i \(0.0178715\pi\)
−0.998424 + 0.0561156i \(0.982128\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 310.644i 0.260700i 0.991468 + 0.130350i \(0.0416100\pi\)
−0.991468 + 0.130350i \(0.958390\pi\)
\(18\) 0 0
\(19\) −2133.35 −1.35574 −0.677871 0.735180i \(-0.737097\pi\)
−0.677871 + 0.735180i \(0.737097\pi\)
\(20\) 0 0
\(21\) 3344.90 1.65514
\(22\) 0 0
\(23\) − 873.145i − 0.344165i −0.985083 0.172083i \(-0.944950\pi\)
0.985083 0.172083i \(-0.0550496\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4223.83i 1.11506i
\(28\) 0 0
\(29\) 2580.97 0.569885 0.284943 0.958545i \(-0.408025\pi\)
0.284943 + 0.958545i \(0.408025\pi\)
\(30\) 0 0
\(31\) 9086.30 1.69818 0.849088 0.528252i \(-0.177152\pi\)
0.849088 + 0.528252i \(0.177152\pi\)
\(32\) 0 0
\(33\) 7404.67i 1.18364i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 3990.64i − 0.479224i −0.970869 0.239612i \(-0.922980\pi\)
0.970869 0.239612i \(-0.0770202\pi\)
\(38\) 0 0
\(39\) 1745.52 0.183765
\(40\) 0 0
\(41\) 16981.8 1.57770 0.788851 0.614584i \(-0.210676\pi\)
0.788851 + 0.614584i \(0.210676\pi\)
\(42\) 0 0
\(43\) 18017.7i 1.48603i 0.669273 + 0.743017i \(0.266606\pi\)
−0.669273 + 0.743017i \(0.733394\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 24864.7i 1.64187i 0.571024 + 0.820933i \(0.306546\pi\)
−0.571024 + 0.820933i \(0.693454\pi\)
\(48\) 0 0
\(49\) −366.670 −0.0218165
\(50\) 0 0
\(51\) 7928.93 0.426864
\(52\) 0 0
\(53\) − 7652.91i − 0.374229i −0.982338 0.187114i \(-0.940087\pi\)
0.982338 0.187114i \(-0.0599135\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 54451.9i 2.21986i
\(58\) 0 0
\(59\) −9233.69 −0.345339 −0.172669 0.984980i \(-0.555239\pi\)
−0.172669 + 0.984980i \(0.555239\pi\)
\(60\) 0 0
\(61\) 3326.17 0.114451 0.0572256 0.998361i \(-0.481775\pi\)
0.0572256 + 0.998361i \(0.481775\pi\)
\(62\) 0 0
\(63\) − 53531.1i − 1.69924i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 32340.7i − 0.880161i −0.897958 0.440080i \(-0.854950\pi\)
0.897958 0.440080i \(-0.145050\pi\)
\(68\) 0 0
\(69\) −22286.3 −0.563528
\(70\) 0 0
\(71\) 35885.9 0.844847 0.422424 0.906399i \(-0.361179\pi\)
0.422424 + 0.906399i \(0.361179\pi\)
\(72\) 0 0
\(73\) − 26513.6i − 0.582319i −0.956675 0.291159i \(-0.905959\pi\)
0.956675 0.291159i \(-0.0940410\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 38017.7i − 0.730733i
\(78\) 0 0
\(79\) 71705.7 1.29266 0.646332 0.763056i \(-0.276302\pi\)
0.646332 + 0.763056i \(0.276302\pi\)
\(80\) 0 0
\(81\) 8548.28 0.144766
\(82\) 0 0
\(83\) − 39630.1i − 0.631437i −0.948853 0.315719i \(-0.897754\pi\)
0.948853 0.315719i \(-0.102246\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 65877.1i − 0.933117i
\(88\) 0 0
\(89\) 117441. 1.57161 0.785806 0.618473i \(-0.212248\pi\)
0.785806 + 0.618473i \(0.212248\pi\)
\(90\) 0 0
\(91\) −8961.98 −0.113449
\(92\) 0 0
\(93\) − 231920.i − 2.78055i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 21878.3i − 0.236093i −0.993008 0.118047i \(-0.962337\pi\)
0.993008 0.118047i \(-0.0376633\pi\)
\(98\) 0 0
\(99\) 118503. 1.21518
\(100\) 0 0
\(101\) −75072.1 −0.732276 −0.366138 0.930561i \(-0.619320\pi\)
−0.366138 + 0.930561i \(0.619320\pi\)
\(102\) 0 0
\(103\) − 47928.6i − 0.445145i −0.974916 0.222573i \(-0.928555\pi\)
0.974916 0.222573i \(-0.0714455\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 92012.3i 0.776938i 0.921462 + 0.388469i \(0.126996\pi\)
−0.921462 + 0.388469i \(0.873004\pi\)
\(108\) 0 0
\(109\) 10647.5 0.0858387 0.0429194 0.999079i \(-0.486334\pi\)
0.0429194 + 0.999079i \(0.486334\pi\)
\(110\) 0 0
\(111\) −101858. −0.784670
\(112\) 0 0
\(113\) 87373.9i 0.643703i 0.946790 + 0.321852i \(0.104305\pi\)
−0.946790 + 0.321852i \(0.895695\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 27934.9i − 0.188661i
\(118\) 0 0
\(119\) −40709.4 −0.263528
\(120\) 0 0
\(121\) −76890.5 −0.477429
\(122\) 0 0
\(123\) − 433447.i − 2.58329i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 197379.i 1.08591i 0.839763 + 0.542953i \(0.182694\pi\)
−0.839763 + 0.542953i \(0.817306\pi\)
\(128\) 0 0
\(129\) 459887. 2.43320
\(130\) 0 0
\(131\) 118490. 0.603258 0.301629 0.953425i \(-0.402470\pi\)
0.301629 + 0.953425i \(0.402470\pi\)
\(132\) 0 0
\(133\) − 279571.i − 1.37045i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 302570.i 1.37728i 0.725101 + 0.688642i \(0.241793\pi\)
−0.725101 + 0.688642i \(0.758207\pi\)
\(138\) 0 0
\(139\) 157190. 0.690062 0.345031 0.938591i \(-0.387868\pi\)
0.345031 + 0.938591i \(0.387868\pi\)
\(140\) 0 0
\(141\) 634650. 2.68835
\(142\) 0 0
\(143\) − 19839.3i − 0.0811309i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9358.95i 0.0357218i
\(148\) 0 0
\(149\) −526340. −1.94223 −0.971115 0.238612i \(-0.923308\pi\)
−0.971115 + 0.238612i \(0.923308\pi\)
\(150\) 0 0
\(151\) −1849.08 −0.00659954 −0.00329977 0.999995i \(-0.501050\pi\)
−0.00329977 + 0.999995i \(0.501050\pi\)
\(152\) 0 0
\(153\) − 126893.i − 0.438237i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 343342.i − 1.11167i −0.831292 0.555837i \(-0.812398\pi\)
0.831292 0.555837i \(-0.187602\pi\)
\(158\) 0 0
\(159\) −195334. −0.612753
\(160\) 0 0
\(161\) 114424. 0.347899
\(162\) 0 0
\(163\) 267463.i 0.788487i 0.919006 + 0.394243i \(0.128993\pi\)
−0.919006 + 0.394243i \(0.871007\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 122968.i 0.341193i 0.985341 + 0.170596i \(0.0545694\pi\)
−0.985341 + 0.170596i \(0.945431\pi\)
\(168\) 0 0
\(169\) 366616. 0.987404
\(170\) 0 0
\(171\) 871437. 2.27901
\(172\) 0 0
\(173\) 288020.i 0.731657i 0.930682 + 0.365829i \(0.119214\pi\)
−0.930682 + 0.365829i \(0.880786\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 235682.i 0.565450i
\(178\) 0 0
\(179\) 246177. 0.574268 0.287134 0.957890i \(-0.407298\pi\)
0.287134 + 0.957890i \(0.407298\pi\)
\(180\) 0 0
\(181\) 433120. 0.982678 0.491339 0.870968i \(-0.336508\pi\)
0.491339 + 0.870968i \(0.336508\pi\)
\(182\) 0 0
\(183\) − 84897.9i − 0.187400i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 90119.1i − 0.188457i
\(188\) 0 0
\(189\) −553526. −1.12715
\(190\) 0 0
\(191\) 701011. 1.39040 0.695202 0.718814i \(-0.255315\pi\)
0.695202 + 0.718814i \(0.255315\pi\)
\(192\) 0 0
\(193\) 215730.i 0.416887i 0.978034 + 0.208443i \(0.0668397\pi\)
−0.978034 + 0.208443i \(0.933160\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 700484.i 1.28598i 0.765876 + 0.642988i \(0.222305\pi\)
−0.765876 + 0.642988i \(0.777695\pi\)
\(198\) 0 0
\(199\) 22097.5 0.0395558 0.0197779 0.999804i \(-0.493704\pi\)
0.0197779 + 0.999804i \(0.493704\pi\)
\(200\) 0 0
\(201\) −825469. −1.44115
\(202\) 0 0
\(203\) 338231.i 0.576068i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 356665.i 0.578542i
\(208\) 0 0
\(209\) 618893. 0.980054
\(210\) 0 0
\(211\) −910782. −1.40834 −0.704172 0.710030i \(-0.748681\pi\)
−0.704172 + 0.710030i \(0.748681\pi\)
\(212\) 0 0
\(213\) − 915958.i − 1.38333i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.19074e6i 1.71660i
\(218\) 0 0
\(219\) −676737. −0.953475
\(220\) 0 0
\(221\) −21243.9 −0.0292587
\(222\) 0 0
\(223\) − 132745.i − 0.178754i −0.995998 0.0893768i \(-0.971512\pi\)
0.995998 0.0893768i \(-0.0284875\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 354321.i − 0.456386i −0.973616 0.228193i \(-0.926718\pi\)
0.973616 0.228193i \(-0.0732817\pi\)
\(228\) 0 0
\(229\) −366643. −0.462013 −0.231007 0.972952i \(-0.574202\pi\)
−0.231007 + 0.972952i \(0.574202\pi\)
\(230\) 0 0
\(231\) −970370. −1.19649
\(232\) 0 0
\(233\) 1.02388e6i 1.23555i 0.786355 + 0.617776i \(0.211966\pi\)
−0.786355 + 0.617776i \(0.788034\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1.83023e6i − 2.11658i
\(238\) 0 0
\(239\) 1.19966e6 1.35852 0.679258 0.733899i \(-0.262302\pi\)
0.679258 + 0.733899i \(0.262302\pi\)
\(240\) 0 0
\(241\) −94967.5 −0.105325 −0.0526626 0.998612i \(-0.516771\pi\)
−0.0526626 + 0.998612i \(0.516771\pi\)
\(242\) 0 0
\(243\) 808203.i 0.878021i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 145893.i − 0.152157i
\(248\) 0 0
\(249\) −1.01153e6 −1.03390
\(250\) 0 0
\(251\) −418053. −0.418839 −0.209419 0.977826i \(-0.567157\pi\)
−0.209419 + 0.977826i \(0.567157\pi\)
\(252\) 0 0
\(253\) 253303.i 0.248794i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.04586e6i − 1.93216i −0.258246 0.966079i \(-0.583144\pi\)
0.258246 0.966079i \(-0.416856\pi\)
\(258\) 0 0
\(259\) 522967. 0.484423
\(260\) 0 0
\(261\) −1.05428e6 −0.957978
\(262\) 0 0
\(263\) − 1.64024e6i − 1.46224i −0.682250 0.731119i \(-0.738998\pi\)
0.682250 0.731119i \(-0.261002\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.99759e6i − 2.57332i
\(268\) 0 0
\(269\) 720582. 0.607160 0.303580 0.952806i \(-0.401818\pi\)
0.303580 + 0.952806i \(0.401818\pi\)
\(270\) 0 0
\(271\) −1.14186e6 −0.944477 −0.472238 0.881471i \(-0.656554\pi\)
−0.472238 + 0.881471i \(0.656554\pi\)
\(272\) 0 0
\(273\) 228747.i 0.185759i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 377028.i − 0.295239i −0.989044 0.147620i \(-0.952839\pi\)
0.989044 0.147620i \(-0.0471612\pi\)
\(278\) 0 0
\(279\) −3.71160e6 −2.85464
\(280\) 0 0
\(281\) −617249. −0.466331 −0.233166 0.972437i \(-0.574908\pi\)
−0.233166 + 0.972437i \(0.574908\pi\)
\(282\) 0 0
\(283\) 1.25311e6i 0.930087i 0.885288 + 0.465044i \(0.153961\pi\)
−0.885288 + 0.465044i \(0.846039\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.22544e6i 1.59482i
\(288\) 0 0
\(289\) 1.32336e6 0.932036
\(290\) 0 0
\(291\) −558425. −0.386574
\(292\) 0 0
\(293\) 818972.i 0.557314i 0.960391 + 0.278657i \(0.0898893\pi\)
−0.960391 + 0.278657i \(0.910111\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1.22535e6i − 0.806064i
\(298\) 0 0
\(299\) 59711.6 0.0386261
\(300\) 0 0
\(301\) −2.36119e6 −1.50216
\(302\) 0 0
\(303\) 1.91615e6i 1.19901i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 136224.i − 0.0824915i −0.999149 0.0412458i \(-0.986867\pi\)
0.999149 0.0412458i \(-0.0131327\pi\)
\(308\) 0 0
\(309\) −1.22334e6 −0.728871
\(310\) 0 0
\(311\) −2.62886e6 −1.54122 −0.770612 0.637304i \(-0.780050\pi\)
−0.770612 + 0.637304i \(0.780050\pi\)
\(312\) 0 0
\(313\) − 218161.i − 0.125868i −0.998018 0.0629341i \(-0.979954\pi\)
0.998018 0.0629341i \(-0.0200458\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.25865e6i 0.703491i 0.936096 + 0.351745i \(0.114412\pi\)
−0.936096 + 0.351745i \(0.885588\pi\)
\(318\) 0 0
\(319\) −748750. −0.411965
\(320\) 0 0
\(321\) 2.34854e6 1.27214
\(322\) 0 0
\(323\) − 662711.i − 0.353442i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 271770.i − 0.140550i
\(328\) 0 0
\(329\) −3.25847e6 −1.65968
\(330\) 0 0
\(331\) 3.21863e6 1.61473 0.807366 0.590051i \(-0.200892\pi\)
0.807366 + 0.590051i \(0.200892\pi\)
\(332\) 0 0
\(333\) 1.63011e6i 0.805576i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.63574e6i 0.784585i 0.919840 + 0.392293i \(0.128318\pi\)
−0.919840 + 0.392293i \(0.871682\pi\)
\(338\) 0 0
\(339\) 2.23015e6 1.05398
\(340\) 0 0
\(341\) −2.63597e6 −1.22760
\(342\) 0 0
\(343\) 2.15448e6i 0.988796i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.83815e6i 0.819514i 0.912195 + 0.409757i \(0.134386\pi\)
−0.912195 + 0.409757i \(0.865614\pi\)
\(348\) 0 0
\(349\) 2.53806e6 1.11542 0.557710 0.830036i \(-0.311680\pi\)
0.557710 + 0.830036i \(0.311680\pi\)
\(350\) 0 0
\(351\) −288854. −0.125144
\(352\) 0 0
\(353\) 1.88471e6i 0.805023i 0.915415 + 0.402511i \(0.131863\pi\)
−0.915415 + 0.402511i \(0.868137\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.03907e6i 0.431495i
\(358\) 0 0
\(359\) 305057. 0.124924 0.0624619 0.998047i \(-0.480105\pi\)
0.0624619 + 0.998047i \(0.480105\pi\)
\(360\) 0 0
\(361\) 2.07507e6 0.838039
\(362\) 0 0
\(363\) 1.96257e6i 0.781731i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 727834.i 0.282077i 0.990004 + 0.141038i \(0.0450441\pi\)
−0.990004 + 0.141038i \(0.954956\pi\)
\(368\) 0 0
\(369\) −6.93680e6 −2.65212
\(370\) 0 0
\(371\) 1.00290e6 0.378289
\(372\) 0 0
\(373\) 4.77676e6i 1.77771i 0.458188 + 0.888855i \(0.348499\pi\)
−0.458188 + 0.888855i \(0.651501\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 176504.i 0.0639590i
\(378\) 0 0
\(379\) −701558. −0.250880 −0.125440 0.992101i \(-0.540034\pi\)
−0.125440 + 0.992101i \(0.540034\pi\)
\(380\) 0 0
\(381\) 5.03794e6 1.77804
\(382\) 0 0
\(383\) 4.01069e6i 1.39708i 0.715570 + 0.698541i \(0.246167\pi\)
−0.715570 + 0.698541i \(0.753833\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 7.35994e6i − 2.49803i
\(388\) 0 0
\(389\) 4.45952e6 1.49422 0.747108 0.664702i \(-0.231442\pi\)
0.747108 + 0.664702i \(0.231442\pi\)
\(390\) 0 0
\(391\) 271237. 0.0897237
\(392\) 0 0
\(393\) − 3.02436e6i − 0.987761i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.36993e6i 1.07311i 0.843865 + 0.536555i \(0.180275\pi\)
−0.843865 + 0.536555i \(0.819725\pi\)
\(398\) 0 0
\(399\) −7.13583e6 −2.24395
\(400\) 0 0
\(401\) −3.00679e6 −0.933775 −0.466888 0.884317i \(-0.654625\pi\)
−0.466888 + 0.884317i \(0.654625\pi\)
\(402\) 0 0
\(403\) 621383.i 0.190588i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.15770e6i 0.346426i
\(408\) 0 0
\(409\) 998012. 0.295004 0.147502 0.989062i \(-0.452877\pi\)
0.147502 + 0.989062i \(0.452877\pi\)
\(410\) 0 0
\(411\) 7.72284e6 2.25513
\(412\) 0 0
\(413\) − 1.21006e6i − 0.349085i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.01215e6i − 1.12989i
\(418\) 0 0
\(419\) 5.53743e6 1.54090 0.770448 0.637503i \(-0.220033\pi\)
0.770448 + 0.637503i \(0.220033\pi\)
\(420\) 0 0
\(421\) 1.98635e6 0.546198 0.273099 0.961986i \(-0.411951\pi\)
0.273099 + 0.961986i \(0.411951\pi\)
\(422\) 0 0
\(423\) − 1.01568e7i − 2.75998i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 435890.i 0.115693i
\(428\) 0 0
\(429\) −506382. −0.132842
\(430\) 0 0
\(431\) −116512. −0.0302118 −0.0151059 0.999886i \(-0.504809\pi\)
−0.0151059 + 0.999886i \(0.504809\pi\)
\(432\) 0 0
\(433\) − 4.56166e6i − 1.16924i −0.811308 0.584619i \(-0.801244\pi\)
0.811308 0.584619i \(-0.198756\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.86272e6i 0.466599i
\(438\) 0 0
\(439\) 2.92172e6 0.723565 0.361782 0.932263i \(-0.382168\pi\)
0.361782 + 0.932263i \(0.382168\pi\)
\(440\) 0 0
\(441\) 149779. 0.0366736
\(442\) 0 0
\(443\) 1.59752e6i 0.386756i 0.981124 + 0.193378i \(0.0619444\pi\)
−0.981124 + 0.193378i \(0.938056\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.34344e7i 3.18016i
\(448\) 0 0
\(449\) −3.11073e6 −0.728193 −0.364096 0.931361i \(-0.618622\pi\)
−0.364096 + 0.931361i \(0.618622\pi\)
\(450\) 0 0
\(451\) −4.92650e6 −1.14051
\(452\) 0 0
\(453\) 47196.3i 0.0108059i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 6.47145e6i − 1.44948i −0.689025 0.724738i \(-0.741961\pi\)
0.689025 0.724738i \(-0.258039\pi\)
\(458\) 0 0
\(459\) −1.31211e6 −0.290695
\(460\) 0 0
\(461\) −5.47864e6 −1.20066 −0.600330 0.799752i \(-0.704964\pi\)
−0.600330 + 0.799752i \(0.704964\pi\)
\(462\) 0 0
\(463\) − 2.35489e6i − 0.510526i −0.966872 0.255263i \(-0.917838\pi\)
0.966872 0.255263i \(-0.0821621\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4.56027e6i − 0.967606i −0.875177 0.483803i \(-0.839255\pi\)
0.875177 0.483803i \(-0.160745\pi\)
\(468\) 0 0
\(469\) 4.23819e6 0.889710
\(470\) 0 0
\(471\) −8.76351e6 −1.82023
\(472\) 0 0
\(473\) − 5.22702e6i − 1.07424i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.12609e6i 0.629079i
\(478\) 0 0
\(479\) −1.88004e6 −0.374394 −0.187197 0.982322i \(-0.559940\pi\)
−0.187197 + 0.982322i \(0.559940\pi\)
\(480\) 0 0
\(481\) 272907. 0.0537839
\(482\) 0 0
\(483\) − 2.92058e6i − 0.569642i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.69396e6i 0.323654i 0.986819 + 0.161827i \(0.0517386\pi\)
−0.986819 + 0.161827i \(0.948261\pi\)
\(488\) 0 0
\(489\) 6.82677e6 1.29105
\(490\) 0 0
\(491\) −1.48645e6 −0.278258 −0.139129 0.990274i \(-0.544430\pi\)
−0.139129 + 0.990274i \(0.544430\pi\)
\(492\) 0 0
\(493\) 801762.i 0.148569i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.70279e6i 0.854013i
\(498\) 0 0
\(499\) 7.09934e6 1.27634 0.638170 0.769896i \(-0.279692\pi\)
0.638170 + 0.769896i \(0.279692\pi\)
\(500\) 0 0
\(501\) 3.13865e6 0.558661
\(502\) 0 0
\(503\) − 9.24224e6i − 1.62876i −0.580331 0.814381i \(-0.697077\pi\)
0.580331 0.814381i \(-0.302923\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.35758e6i − 1.61675i
\(508\) 0 0
\(509\) 8.12506e6 1.39006 0.695028 0.718983i \(-0.255392\pi\)
0.695028 + 0.718983i \(0.255392\pi\)
\(510\) 0 0
\(511\) 3.47456e6 0.588637
\(512\) 0 0
\(513\) − 9.01089e6i − 1.51173i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.21335e6i − 1.18689i
\(518\) 0 0
\(519\) 7.35148e6 1.19800
\(520\) 0 0
\(521\) 5.06245e6 0.817084 0.408542 0.912740i \(-0.366037\pi\)
0.408542 + 0.912740i \(0.366037\pi\)
\(522\) 0 0
\(523\) 4.76222e6i 0.761299i 0.924719 + 0.380649i \(0.124300\pi\)
−0.924719 + 0.380649i \(0.875700\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.82260e6i 0.442714i
\(528\) 0 0
\(529\) 5.67396e6 0.881550
\(530\) 0 0
\(531\) 3.77181e6 0.580515
\(532\) 0 0
\(533\) 1.16133e6i 0.177067i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 6.28346e6i − 0.940292i
\(538\) 0 0
\(539\) 106373. 0.0157709
\(540\) 0 0
\(541\) −2.89920e6 −0.425877 −0.212939 0.977066i \(-0.568303\pi\)
−0.212939 + 0.977066i \(0.568303\pi\)
\(542\) 0 0
\(543\) − 1.10550e7i − 1.60901i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 5.74434e6i − 0.820866i −0.911891 0.410433i \(-0.865378\pi\)
0.911891 0.410433i \(-0.134622\pi\)
\(548\) 0 0
\(549\) −1.35869e6 −0.192393
\(550\) 0 0
\(551\) −5.50610e6 −0.772618
\(552\) 0 0
\(553\) 9.39691e6i 1.30669i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.29174e6i 0.995848i 0.867221 + 0.497924i \(0.165904\pi\)
−0.867221 + 0.497924i \(0.834096\pi\)
\(558\) 0 0
\(559\) −1.23217e6 −0.166779
\(560\) 0 0
\(561\) −2.30022e6 −0.308576
\(562\) 0 0
\(563\) − 6.65348e6i − 0.884663i −0.896852 0.442331i \(-0.854151\pi\)
0.896852 0.442331i \(-0.145849\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.12024e6i 0.146336i
\(568\) 0 0
\(569\) −5.78715e6 −0.749349 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(570\) 0 0
\(571\) 1.22059e7 1.56667 0.783336 0.621599i \(-0.213517\pi\)
0.783336 + 0.621599i \(0.213517\pi\)
\(572\) 0 0
\(573\) − 1.78927e7i − 2.27662i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.02981e7i 1.28771i 0.765149 + 0.643853i \(0.222665\pi\)
−0.765149 + 0.643853i \(0.777335\pi\)
\(578\) 0 0
\(579\) 5.50634e6 0.682601
\(580\) 0 0
\(581\) 5.19346e6 0.638288
\(582\) 0 0
\(583\) 2.22014e6i 0.270526i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.30519e7i 1.56343i 0.623636 + 0.781715i \(0.285655\pi\)
−0.623636 + 0.781715i \(0.714345\pi\)
\(588\) 0 0
\(589\) −1.93842e7 −2.30229
\(590\) 0 0
\(591\) 1.78793e7 2.10563
\(592\) 0 0
\(593\) − 6.43920e6i − 0.751961i −0.926628 0.375980i \(-0.877306\pi\)
0.926628 0.375980i \(-0.122694\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 564020.i − 0.0647677i
\(598\) 0 0
\(599\) −1.00760e7 −1.14741 −0.573707 0.819061i \(-0.694495\pi\)
−0.573707 + 0.819061i \(0.694495\pi\)
\(600\) 0 0
\(601\) 1.57050e6 0.177358 0.0886791 0.996060i \(-0.471735\pi\)
0.0886791 + 0.996060i \(0.471735\pi\)
\(602\) 0 0
\(603\) 1.32106e7i 1.47955i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.31039e6i 0.805321i 0.915349 + 0.402660i \(0.131914\pi\)
−0.915349 + 0.402660i \(0.868086\pi\)
\(608\) 0 0
\(609\) 8.63308e6 0.943241
\(610\) 0 0
\(611\) −1.70041e6 −0.184269
\(612\) 0 0
\(613\) − 1.31997e7i − 1.41878i −0.704817 0.709389i \(-0.748971\pi\)
0.704817 0.709389i \(-0.251029\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.02423e7i 1.08314i 0.840655 + 0.541570i \(0.182170\pi\)
−0.840655 + 0.541570i \(0.817830\pi\)
\(618\) 0 0
\(619\) 1.05614e7 1.10788 0.553942 0.832555i \(-0.313123\pi\)
0.553942 + 0.832555i \(0.313123\pi\)
\(620\) 0 0
\(621\) 3.68802e6 0.383764
\(622\) 0 0
\(623\) 1.53905e7i 1.58866i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 1.57967e7i − 1.60472i
\(628\) 0 0
\(629\) 1.23967e6 0.124933
\(630\) 0 0
\(631\) −1.90535e7 −1.90503 −0.952513 0.304497i \(-0.901512\pi\)
−0.952513 + 0.304497i \(0.901512\pi\)
\(632\) 0 0
\(633\) 2.32470e7i 2.30599i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 25075.4i − 0.00244849i
\(638\) 0 0
\(639\) −1.46588e7 −1.42019
\(640\) 0 0
\(641\) 8.56937e6 0.823766 0.411883 0.911237i \(-0.364871\pi\)
0.411883 + 0.911237i \(0.364871\pi\)
\(642\) 0 0
\(643\) − 1.79513e7i − 1.71226i −0.516761 0.856130i \(-0.672862\pi\)
0.516761 0.856130i \(-0.327138\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.05470e7i 0.990534i 0.868741 + 0.495267i \(0.164930\pi\)
−0.868741 + 0.495267i \(0.835070\pi\)
\(648\) 0 0
\(649\) 2.67873e6 0.249642
\(650\) 0 0
\(651\) 3.03928e7 2.81072
\(652\) 0 0
\(653\) 1.00324e7i 0.920712i 0.887734 + 0.460356i \(0.152278\pi\)
−0.887734 + 0.460356i \(0.847722\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.08304e7i 0.978879i
\(658\) 0 0
\(659\) −8.99161e6 −0.806536 −0.403268 0.915082i \(-0.632126\pi\)
−0.403268 + 0.915082i \(0.632126\pi\)
\(660\) 0 0
\(661\) 2.39297e6 0.213027 0.106513 0.994311i \(-0.466031\pi\)
0.106513 + 0.994311i \(0.466031\pi\)
\(662\) 0 0
\(663\) 542234.i 0.0479074i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 2.25356e6i − 0.196135i
\(668\) 0 0
\(669\) −3.38820e6 −0.292687
\(670\) 0 0
\(671\) −964938. −0.0827357
\(672\) 0 0
\(673\) 1.53612e7i 1.30733i 0.756783 + 0.653666i \(0.226770\pi\)
−0.756783 + 0.653666i \(0.773230\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.16026e7i − 0.972934i −0.873699 0.486467i \(-0.838285\pi\)
0.873699 0.486467i \(-0.161715\pi\)
\(678\) 0 0
\(679\) 2.86711e6 0.238655
\(680\) 0 0
\(681\) −9.04375e6 −0.747275
\(682\) 0 0
\(683\) 1.20315e7i 0.986890i 0.869777 + 0.493445i \(0.164263\pi\)
−0.869777 + 0.493445i \(0.835737\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 9.35825e6i 0.756490i
\(688\) 0 0
\(689\) 523358. 0.0420001
\(690\) 0 0
\(691\) −5.18616e6 −0.413191 −0.206595 0.978426i \(-0.566238\pi\)
−0.206595 + 0.978426i \(0.566238\pi\)
\(692\) 0 0
\(693\) 1.55296e7i 1.22836i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.27530e6i 0.411306i
\(698\) 0 0
\(699\) 2.61338e7 2.02306
\(700\) 0 0
\(701\) 6.00859e6 0.461825 0.230913 0.972974i \(-0.425829\pi\)
0.230913 + 0.972974i \(0.425829\pi\)
\(702\) 0 0
\(703\) 8.51342e6i 0.649704i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.83807e6i − 0.740221i
\(708\) 0 0
\(709\) −5.90083e6 −0.440857 −0.220429 0.975403i \(-0.570746\pi\)
−0.220429 + 0.975403i \(0.570746\pi\)
\(710\) 0 0
\(711\) −2.92906e7 −2.17297
\(712\) 0 0
\(713\) − 7.93365e6i − 0.584453i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.06204e7i − 2.22440i
\(718\) 0 0
\(719\) −1.36592e7 −0.985382 −0.492691 0.870204i \(-0.663987\pi\)
−0.492691 + 0.870204i \(0.663987\pi\)
\(720\) 0 0
\(721\) 6.28097e6 0.449975
\(722\) 0 0
\(723\) 2.42397e6i 0.172457i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 1.11594e7i − 0.783079i −0.920161 0.391539i \(-0.871943\pi\)
0.920161 0.391539i \(-0.128057\pi\)
\(728\) 0 0
\(729\) 2.27059e7 1.58242
\(730\) 0 0
\(731\) −5.59710e6 −0.387409
\(732\) 0 0
\(733\) − 1.52510e7i − 1.04843i −0.851586 0.524215i \(-0.824359\pi\)
0.851586 0.524215i \(-0.175641\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.38217e6i 0.636260i
\(738\) 0 0
\(739\) −1.11820e7 −0.753196 −0.376598 0.926377i \(-0.622906\pi\)
−0.376598 + 0.926377i \(0.622906\pi\)
\(740\) 0 0
\(741\) −3.72379e6 −0.249138
\(742\) 0 0
\(743\) − 7.71450e6i − 0.512667i −0.966588 0.256334i \(-0.917485\pi\)
0.966588 0.256334i \(-0.0825146\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.61883e7i 1.06145i
\(748\) 0 0
\(749\) −1.20581e7 −0.785367
\(750\) 0 0
\(751\) 2.23973e7 1.44909 0.724545 0.689228i \(-0.242050\pi\)
0.724545 + 0.689228i \(0.242050\pi\)
\(752\) 0 0
\(753\) 1.06704e7i 0.685796i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2.57267e7i − 1.63171i −0.578254 0.815857i \(-0.696266\pi\)
0.578254 0.815857i \(-0.303734\pi\)
\(758\) 0 0
\(759\) 6.46535e6 0.407369
\(760\) 0 0
\(761\) −1.48340e7 −0.928533 −0.464267 0.885696i \(-0.653682\pi\)
−0.464267 + 0.885696i \(0.653682\pi\)
\(762\) 0 0
\(763\) 1.39534e6i 0.0867700i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 631463.i − 0.0387578i
\(768\) 0 0
\(769\) 5.57112e6 0.339724 0.169862 0.985468i \(-0.445668\pi\)
0.169862 + 0.985468i \(0.445668\pi\)
\(770\) 0 0
\(771\) −5.22188e7 −3.16367
\(772\) 0 0
\(773\) − 1.58230e7i − 0.952447i −0.879324 0.476224i \(-0.842005\pi\)
0.879324 0.476224i \(-0.157995\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 1.33483e7i − 0.793183i
\(778\) 0 0
\(779\) −3.62281e7 −2.13896
\(780\) 0 0
\(781\) −1.04107e7 −0.610732
\(782\) 0 0
\(783\) 1.09016e7i 0.635454i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 1.31529e7i − 0.756981i −0.925605 0.378491i \(-0.876443\pi\)
0.925605 0.378491i \(-0.123557\pi\)
\(788\) 0 0
\(789\) −4.18658e7 −2.39423
\(790\) 0 0
\(791\) −1.14502e7 −0.650687
\(792\) 0 0
\(793\) 227466.i 0.0128450i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.58443e7i 1.44118i 0.693361 + 0.720590i \(0.256129\pi\)
−0.693361 + 0.720590i \(0.743871\pi\)
\(798\) 0 0
\(799\) −7.72406e6 −0.428034
\(800\) 0 0
\(801\) −4.79728e7 −2.64188
\(802\) 0 0
\(803\) 7.69170e6i 0.420953i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.83923e7i − 0.994149i
\(808\) 0 0
\(809\) −1.78857e7 −0.960804 −0.480402 0.877048i \(-0.659509\pi\)
−0.480402 + 0.877048i \(0.659509\pi\)
\(810\) 0 0
\(811\) 1.41608e7 0.756026 0.378013 0.925800i \(-0.376607\pi\)
0.378013 + 0.925800i \(0.376607\pi\)
\(812\) 0 0
\(813\) 2.91451e7i 1.54646i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3.84380e7i − 2.01468i
\(818\) 0 0
\(819\) 3.66082e6 0.190708
\(820\) 0 0
\(821\) −3.46248e7 −1.79279 −0.896394 0.443258i \(-0.853823\pi\)
−0.896394 + 0.443258i \(0.853823\pi\)
\(822\) 0 0
\(823\) − 2.13360e7i − 1.09803i −0.835813 0.549015i \(-0.815003\pi\)
0.835813 0.549015i \(-0.184997\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.59813e6i − 0.0812548i −0.999174 0.0406274i \(-0.987064\pi\)
0.999174 0.0406274i \(-0.0129357\pi\)
\(828\) 0 0
\(829\) 2.53923e7 1.28327 0.641633 0.767012i \(-0.278257\pi\)
0.641633 + 0.767012i \(0.278257\pi\)
\(830\) 0 0
\(831\) −9.62333e6 −0.483418
\(832\) 0 0
\(833\) − 113904.i − 0.00568755i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.83790e7i 1.89356i
\(838\) 0 0
\(839\) 1.98528e7 0.973681 0.486841 0.873491i \(-0.338149\pi\)
0.486841 + 0.873491i \(0.338149\pi\)
\(840\) 0 0
\(841\) −1.38498e7 −0.675231
\(842\) 0 0
\(843\) 1.57548e7i 0.763560i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.00764e7i − 0.482609i
\(848\) 0 0
\(849\) 3.19846e7 1.52290
\(850\) 0 0
\(851\) −3.48441e6 −0.164932
\(852\) 0 0
\(853\) 1.59794e7i 0.751948i 0.926630 + 0.375974i \(0.122692\pi\)
−0.926630 + 0.375974i \(0.877308\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.00157e6i 0.325644i 0.986655 + 0.162822i \(0.0520597\pi\)
−0.986655 + 0.162822i \(0.947940\pi\)
\(858\) 0 0
\(859\) −7.28414e6 −0.336818 −0.168409 0.985717i \(-0.553863\pi\)
−0.168409 + 0.985717i \(0.553863\pi\)
\(860\) 0 0
\(861\) 5.68026e7 2.61132
\(862\) 0 0
\(863\) − 1.76361e7i − 0.806075i −0.915184 0.403037i \(-0.867954\pi\)
0.915184 0.403037i \(-0.132046\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 3.37776e7i − 1.52609i
\(868\) 0 0
\(869\) −2.08021e7 −0.934455
\(870\) 0 0
\(871\) 2.21167e6 0.0987816
\(872\) 0 0
\(873\) 8.93692e6i 0.396874i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 2.69004e7i − 1.18102i −0.807029 0.590512i \(-0.798926\pi\)
0.807029 0.590512i \(-0.201074\pi\)
\(878\) 0 0
\(879\) 2.09036e7 0.912533
\(880\) 0 0
\(881\) 2.51911e7 1.09347 0.546735 0.837306i \(-0.315870\pi\)
0.546735 + 0.837306i \(0.315870\pi\)
\(882\) 0 0
\(883\) 3.22126e7i 1.39035i 0.718840 + 0.695175i \(0.244673\pi\)
−0.718840 + 0.695175i \(0.755327\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.96139e6i 0.382443i 0.981547 + 0.191221i \(0.0612448\pi\)
−0.981547 + 0.191221i \(0.938755\pi\)
\(888\) 0 0
\(889\) −2.58662e7 −1.09769
\(890\) 0 0
\(891\) −2.47989e6 −0.104650
\(892\) 0 0
\(893\) − 5.30449e7i − 2.22595i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.52409e6i − 0.0632454i
\(898\) 0 0
\(899\) 2.34514e7 0.967765
\(900\) 0 0
\(901\) 2.37733e6 0.0975613
\(902\) 0 0
\(903\) 6.02675e7i 2.45960i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 5.81689e6i − 0.234786i −0.993086 0.117393i \(-0.962546\pi\)
0.993086 0.117393i \(-0.0374537\pi\)
\(908\) 0 0
\(909\) 3.06657e7 1.23096
\(910\) 0 0
\(911\) 1.96435e7 0.784192 0.392096 0.919924i \(-0.371750\pi\)
0.392096 + 0.919924i \(0.371750\pi\)
\(912\) 0 0
\(913\) 1.14969e7i 0.456460i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.55279e7i 0.609803i
\(918\) 0 0
\(919\) −89962.4 −0.00351376 −0.00175688 0.999998i \(-0.500559\pi\)
−0.00175688 + 0.999998i \(0.500559\pi\)
\(920\) 0 0
\(921\) −3.47702e6 −0.135070
\(922\) 0 0
\(923\) 2.45412e6i 0.0948183i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.95781e7i 0.748290i
\(928\) 0 0
\(929\) −3.65192e7 −1.38830 −0.694149 0.719832i \(-0.744219\pi\)
−0.694149 + 0.719832i \(0.744219\pi\)
\(930\) 0 0
\(931\) 782234. 0.0295776
\(932\) 0 0
\(933\) 6.70994e7i 2.52357i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.58659e7i 1.33454i 0.744814 + 0.667272i \(0.232538\pi\)
−0.744814 + 0.667272i \(0.767462\pi\)
\(938\) 0 0
\(939\) −5.56838e6 −0.206094
\(940\) 0 0
\(941\) −3.19693e7 −1.17695 −0.588476 0.808515i \(-0.700272\pi\)
−0.588476 + 0.808515i \(0.700272\pi\)
\(942\) 0 0
\(943\) − 1.48276e7i − 0.542990i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.71846e7i 1.70972i 0.518858 + 0.854861i \(0.326357\pi\)
−0.518858 + 0.854861i \(0.673643\pi\)
\(948\) 0 0
\(949\) 1.81318e6 0.0653544
\(950\) 0 0
\(951\) 3.21261e7 1.15188
\(952\) 0 0
\(953\) 1.65226e6i 0.0589315i 0.999566 + 0.0294657i \(0.00938059\pi\)
−0.999566 + 0.0294657i \(0.990619\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.91112e7i 0.674541i
\(958\) 0 0
\(959\) −3.96512e7 −1.39223
\(960\) 0 0
\(961\) 5.39316e7 1.88380
\(962\) 0 0
\(963\) − 3.75855e7i − 1.30603i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 3.23040e7i − 1.11094i −0.831537 0.555470i \(-0.812538\pi\)
0.831537 0.555470i \(-0.187462\pi\)
\(968\) 0 0
\(969\) −1.69151e7 −0.578717
\(970\) 0 0
\(971\) −1.15927e7 −0.394582 −0.197291 0.980345i \(-0.563214\pi\)
−0.197291 + 0.980345i \(0.563214\pi\)
\(972\) 0 0
\(973\) 2.05995e7i 0.697549i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 2.58947e7i − 0.867909i −0.900935 0.433954i \(-0.857118\pi\)
0.900935 0.433954i \(-0.142882\pi\)
\(978\) 0 0
\(979\) −3.40702e7 −1.13610
\(980\) 0 0
\(981\) −4.34935e6 −0.144295
\(982\) 0 0
\(983\) − 3.46040e7i − 1.14220i −0.820880 0.571101i \(-0.806517\pi\)
0.820880 0.571101i \(-0.193483\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.31698e7i 2.71752i
\(988\) 0 0
\(989\) 1.57321e7 0.511441
\(990\) 0 0
\(991\) 3.71464e7 1.20152 0.600762 0.799428i \(-0.294864\pi\)
0.600762 + 0.799428i \(0.294864\pi\)
\(992\) 0 0
\(993\) − 8.21528e7i − 2.64393i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.47350e7i 0.788086i 0.919092 + 0.394043i \(0.128924\pi\)
−0.919092 + 0.394043i \(0.871076\pi\)
\(998\) 0 0
\(999\) 1.68558e7 0.534362
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 400.6.c.n.49.1 4
4.3 odd 2 25.6.b.b.24.3 4
5.2 odd 4 400.6.a.o.1.1 2
5.3 odd 4 400.6.a.w.1.2 2
5.4 even 2 inner 400.6.c.n.49.4 4
12.11 even 2 225.6.b.i.199.2 4
20.3 even 4 25.6.a.b.1.2 2
20.7 even 4 25.6.a.d.1.1 yes 2
20.19 odd 2 25.6.b.b.24.2 4
60.23 odd 4 225.6.a.s.1.1 2
60.47 odd 4 225.6.a.l.1.2 2
60.59 even 2 225.6.b.i.199.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.6.a.b.1.2 2 20.3 even 4
25.6.a.d.1.1 yes 2 20.7 even 4
25.6.b.b.24.2 4 20.19 odd 2
25.6.b.b.24.3 4 4.3 odd 2
225.6.a.l.1.2 2 60.47 odd 4
225.6.a.s.1.1 2 60.23 odd 4
225.6.b.i.199.2 4 12.11 even 2
225.6.b.i.199.3 4 60.59 even 2
400.6.a.o.1.1 2 5.2 odd 4
400.6.a.w.1.2 2 5.3 odd 4
400.6.c.n.49.1 4 1.1 even 1 trivial
400.6.c.n.49.4 4 5.4 even 2 inner