Properties

Label 800.6.c.f.449.2
Level $800$
Weight $6$
Character 800.449
Analytic conductor $128.307$
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] = [800,6,Mod(449,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, 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("800.449");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{70})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(4.18330 + 4.18330i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.6.c.f.449.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-12.7332i q^{3} -68.7332i q^{7} +80.8656 q^{9} +O(q^{10})\) \(q-12.7332i q^{3} -68.7332i q^{7} +80.8656 q^{9} -327.332 q^{11} +719.328i q^{13} -379.328i q^{17} +1029.33 q^{19} -875.194 q^{21} -779.120i q^{23} -4123.85i q^{27} -1392.66 q^{29} -2744.68 q^{31} +4167.98i q^{33} +12640.6i q^{37} +9159.35 q^{39} +8210.43 q^{41} +22524.5i q^{43} -7739.18i q^{47} +12082.7 q^{49} -4830.06 q^{51} +2401.86i q^{53} -13106.6i q^{57} +15734.7 q^{59} +32082.1 q^{61} -5558.15i q^{63} -9009.07i q^{67} -9920.70 q^{69} -43832.3 q^{71} +65837.5i q^{73} +22498.6i q^{77} +39601.3 q^{79} -32859.4 q^{81} -63101.4i q^{83} +17733.0i q^{87} -34510.9 q^{89} +49441.7 q^{91} +34948.6i q^{93} -14081.3i q^{97} -26469.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 212 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 212 q^{9} - 640 q^{11} + 1440 q^{19} - 288 q^{21} - 216 q^{29} - 19680 q^{31} + 44000 q^{39} - 21240 q^{41} + 55292 q^{49} - 49440 q^{51} + 61600 q^{59} + 49080 q^{61} - 122144 q^{69} + 24800 q^{71} + 143680 q^{79} - 204796 q^{81} + 81496 q^{89} + 55200 q^{91} - 55680 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 12.7332i − 0.816835i −0.912795 0.408418i \(-0.866081\pi\)
0.912795 0.408418i \(-0.133919\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 68.7332i − 0.530178i −0.964224 0.265089i \(-0.914599\pi\)
0.964224 0.265089i \(-0.0854013\pi\)
\(8\) 0 0
\(9\) 80.8656 0.332780
\(10\) 0 0
\(11\) −327.332 −0.815655 −0.407828 0.913059i \(-0.633714\pi\)
−0.407828 + 0.913059i \(0.633714\pi\)
\(12\) 0 0
\(13\) 719.328i 1.18051i 0.807218 + 0.590254i \(0.200972\pi\)
−0.807218 + 0.590254i \(0.799028\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 379.328i − 0.318341i −0.987251 0.159171i \(-0.949118\pi\)
0.987251 0.159171i \(-0.0508820\pi\)
\(18\) 0 0
\(19\) 1029.33 0.654139 0.327069 0.945000i \(-0.393939\pi\)
0.327069 + 0.945000i \(0.393939\pi\)
\(20\) 0 0
\(21\) −875.194 −0.433068
\(22\) 0 0
\(23\) − 779.120i − 0.307104i −0.988141 0.153552i \(-0.950929\pi\)
0.988141 0.153552i \(-0.0490712\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4123.85i − 1.08866i
\(28\) 0 0
\(29\) −1392.66 −0.307503 −0.153751 0.988110i \(-0.549135\pi\)
−0.153751 + 0.988110i \(0.549135\pi\)
\(30\) 0 0
\(31\) −2744.68 −0.512965 −0.256483 0.966549i \(-0.582564\pi\)
−0.256483 + 0.966549i \(0.582564\pi\)
\(32\) 0 0
\(33\) 4167.98i 0.666256i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12640.6i 1.51797i 0.651108 + 0.758985i \(0.274304\pi\)
−0.651108 + 0.758985i \(0.725696\pi\)
\(38\) 0 0
\(39\) 9159.35 0.964280
\(40\) 0 0
\(41\) 8210.43 0.762792 0.381396 0.924412i \(-0.375443\pi\)
0.381396 + 0.924412i \(0.375443\pi\)
\(42\) 0 0
\(43\) 22524.5i 1.85774i 0.370408 + 0.928869i \(0.379218\pi\)
−0.370408 + 0.928869i \(0.620782\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 7739.18i − 0.511035i −0.966804 0.255517i \(-0.917754\pi\)
0.966804 0.255517i \(-0.0822458\pi\)
\(48\) 0 0
\(49\) 12082.7 0.718912
\(50\) 0 0
\(51\) −4830.06 −0.260032
\(52\) 0 0
\(53\) 2401.86i 0.117451i 0.998274 + 0.0587256i \(0.0187037\pi\)
−0.998274 + 0.0587256i \(0.981296\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 13106.6i − 0.534323i
\(58\) 0 0
\(59\) 15734.7 0.588474 0.294237 0.955732i \(-0.404934\pi\)
0.294237 + 0.955732i \(0.404934\pi\)
\(60\) 0 0
\(61\) 32082.1 1.10392 0.551961 0.833870i \(-0.313880\pi\)
0.551961 + 0.833870i \(0.313880\pi\)
\(62\) 0 0
\(63\) − 5558.15i − 0.176433i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 9009.07i − 0.245184i −0.992457 0.122592i \(-0.960879\pi\)
0.992457 0.122592i \(-0.0391207\pi\)
\(68\) 0 0
\(69\) −9920.70 −0.250853
\(70\) 0 0
\(71\) −43832.3 −1.03192 −0.515962 0.856611i \(-0.672566\pi\)
−0.515962 + 0.856611i \(0.672566\pi\)
\(72\) 0 0
\(73\) 65837.5i 1.44599i 0.690852 + 0.722997i \(0.257236\pi\)
−0.690852 + 0.722997i \(0.742764\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 22498.6i 0.432442i
\(78\) 0 0
\(79\) 39601.3 0.713907 0.356954 0.934122i \(-0.383815\pi\)
0.356954 + 0.934122i \(0.383815\pi\)
\(80\) 0 0
\(81\) −32859.4 −0.556477
\(82\) 0 0
\(83\) − 63101.4i − 1.00541i −0.864458 0.502706i \(-0.832338\pi\)
0.864458 0.502706i \(-0.167662\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 17733.0i 0.251179i
\(88\) 0 0
\(89\) −34510.9 −0.461829 −0.230915 0.972974i \(-0.574172\pi\)
−0.230915 + 0.972974i \(0.574172\pi\)
\(90\) 0 0
\(91\) 49441.7 0.625879
\(92\) 0 0
\(93\) 34948.6i 0.419008i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14081.3i − 0.151955i −0.997110 0.0759773i \(-0.975792\pi\)
0.997110 0.0759773i \(-0.0242076\pi\)
\(98\) 0 0
\(99\) −26469.9 −0.271434
\(100\) 0 0
\(101\) 184018. 1.79497 0.897485 0.441044i \(-0.145392\pi\)
0.897485 + 0.441044i \(0.145392\pi\)
\(102\) 0 0
\(103\) − 70168.0i − 0.651697i −0.945422 0.325849i \(-0.894350\pi\)
0.945422 0.325849i \(-0.105650\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7952.89i − 0.0671530i −0.999436 0.0335765i \(-0.989310\pi\)
0.999436 0.0335765i \(-0.0106897\pi\)
\(108\) 0 0
\(109\) −168681. −1.35988 −0.679939 0.733269i \(-0.737994\pi\)
−0.679939 + 0.733269i \(0.737994\pi\)
\(110\) 0 0
\(111\) 160955. 1.23993
\(112\) 0 0
\(113\) 61891.3i 0.455967i 0.973665 + 0.227984i \(0.0732133\pi\)
−0.973665 + 0.227984i \(0.926787\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 58168.9i 0.392849i
\(118\) 0 0
\(119\) −26072.4 −0.168777
\(120\) 0 0
\(121\) −53904.8 −0.334706
\(122\) 0 0
\(123\) − 104545.i − 0.623075i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 358695.i 1.97340i 0.162538 + 0.986702i \(0.448032\pi\)
−0.162538 + 0.986702i \(0.551968\pi\)
\(128\) 0 0
\(129\) 286809. 1.51747
\(130\) 0 0
\(131\) 312592. 1.59148 0.795738 0.605641i \(-0.207083\pi\)
0.795738 + 0.605641i \(0.207083\pi\)
\(132\) 0 0
\(133\) − 70749.0i − 0.346810i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 33573.9i − 0.152827i −0.997076 0.0764135i \(-0.975653\pi\)
0.997076 0.0764135i \(-0.0243469\pi\)
\(138\) 0 0
\(139\) 342175. 1.50214 0.751072 0.660220i \(-0.229537\pi\)
0.751072 + 0.660220i \(0.229537\pi\)
\(140\) 0 0
\(141\) −98544.6 −0.417431
\(142\) 0 0
\(143\) − 235459.i − 0.962887i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 153852.i − 0.587232i
\(148\) 0 0
\(149\) 239318. 0.883099 0.441549 0.897237i \(-0.354429\pi\)
0.441549 + 0.897237i \(0.354429\pi\)
\(150\) 0 0
\(151\) −169513. −0.605007 −0.302503 0.953148i \(-0.597822\pi\)
−0.302503 + 0.953148i \(0.597822\pi\)
\(152\) 0 0
\(153\) − 30674.6i − 0.105938i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 186382.i 0.603469i 0.953392 + 0.301735i \(0.0975656\pi\)
−0.953392 + 0.301735i \(0.902434\pi\)
\(158\) 0 0
\(159\) 30583.3 0.0959383
\(160\) 0 0
\(161\) −53551.4 −0.162820
\(162\) 0 0
\(163\) − 403058.i − 1.18822i −0.804382 0.594112i \(-0.797504\pi\)
0.804382 0.594112i \(-0.202496\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 167631.i − 0.465119i −0.972582 0.232560i \(-0.925290\pi\)
0.972582 0.232560i \(-0.0747101\pi\)
\(168\) 0 0
\(169\) −146140. −0.393597
\(170\) 0 0
\(171\) 83237.2 0.217684
\(172\) 0 0
\(173\) − 84080.4i − 0.213589i −0.994281 0.106795i \(-0.965941\pi\)
0.994281 0.106795i \(-0.0340588\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 200353.i − 0.480686i
\(178\) 0 0
\(179\) 741698. 1.73019 0.865096 0.501607i \(-0.167257\pi\)
0.865096 + 0.501607i \(0.167257\pi\)
\(180\) 0 0
\(181\) −343670. −0.779732 −0.389866 0.920872i \(-0.627479\pi\)
−0.389866 + 0.920872i \(0.627479\pi\)
\(182\) 0 0
\(183\) − 408508.i − 0.901722i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 124166.i 0.259657i
\(188\) 0 0
\(189\) −283445. −0.577184
\(190\) 0 0
\(191\) −631035. −1.25161 −0.625806 0.779978i \(-0.715230\pi\)
−0.625806 + 0.779978i \(0.715230\pi\)
\(192\) 0 0
\(193\) − 847791.i − 1.63831i −0.573574 0.819154i \(-0.694444\pi\)
0.573574 0.819154i \(-0.305556\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 397282.i − 0.729346i −0.931136 0.364673i \(-0.881181\pi\)
0.931136 0.364673i \(-0.118819\pi\)
\(198\) 0 0
\(199\) 896038. 1.60396 0.801980 0.597350i \(-0.203780\pi\)
0.801980 + 0.597350i \(0.203780\pi\)
\(200\) 0 0
\(201\) −114714. −0.200275
\(202\) 0 0
\(203\) 95721.7i 0.163031i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 63004.0i − 0.102198i
\(208\) 0 0
\(209\) −336932. −0.533552
\(210\) 0 0
\(211\) 876761. 1.35574 0.677868 0.735184i \(-0.262904\pi\)
0.677868 + 0.735184i \(0.262904\pi\)
\(212\) 0 0
\(213\) 558125.i 0.842913i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 188651.i 0.271963i
\(218\) 0 0
\(219\) 838322. 1.18114
\(220\) 0 0
\(221\) 272861. 0.375804
\(222\) 0 0
\(223\) 54891.1i 0.0739162i 0.999317 + 0.0369581i \(0.0117668\pi\)
−0.999317 + 0.0369581i \(0.988233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 803329.i − 1.03473i −0.855764 0.517367i \(-0.826912\pi\)
0.855764 0.517367i \(-0.173088\pi\)
\(228\) 0 0
\(229\) 589546. 0.742898 0.371449 0.928453i \(-0.378861\pi\)
0.371449 + 0.928453i \(0.378861\pi\)
\(230\) 0 0
\(231\) 286479. 0.353234
\(232\) 0 0
\(233\) − 1.02048e6i − 1.23144i −0.787965 0.615720i \(-0.788865\pi\)
0.787965 0.615720i \(-0.211135\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 504251.i − 0.583145i
\(238\) 0 0
\(239\) 1.65512e6 1.87428 0.937139 0.348956i \(-0.113464\pi\)
0.937139 + 0.348956i \(0.113464\pi\)
\(240\) 0 0
\(241\) −1.19028e6 −1.32010 −0.660049 0.751223i \(-0.729464\pi\)
−0.660049 + 0.751223i \(0.729464\pi\)
\(242\) 0 0
\(243\) − 583689.i − 0.634112i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 740424.i 0.772215i
\(248\) 0 0
\(249\) −803483. −0.821256
\(250\) 0 0
\(251\) 23776.0 0.0238207 0.0119104 0.999929i \(-0.496209\pi\)
0.0119104 + 0.999929i \(0.496209\pi\)
\(252\) 0 0
\(253\) 255031.i 0.250491i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 341681.i − 0.322692i −0.986898 0.161346i \(-0.948416\pi\)
0.986898 0.161346i \(-0.0515835\pi\)
\(258\) 0 0
\(259\) 868828. 0.804794
\(260\) 0 0
\(261\) −112618. −0.102331
\(262\) 0 0
\(263\) − 1.09120e6i − 0.972782i −0.873741 0.486391i \(-0.838313\pi\)
0.873741 0.486391i \(-0.161687\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 439434.i 0.377238i
\(268\) 0 0
\(269\) 922907. 0.777638 0.388819 0.921314i \(-0.372883\pi\)
0.388819 + 0.921314i \(0.372883\pi\)
\(270\) 0 0
\(271\) 1.34302e6 1.11086 0.555430 0.831564i \(-0.312554\pi\)
0.555430 + 0.831564i \(0.312554\pi\)
\(272\) 0 0
\(273\) − 629551.i − 0.511240i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 247543.i − 0.193843i −0.995292 0.0969217i \(-0.969100\pi\)
0.995292 0.0969217i \(-0.0308996\pi\)
\(278\) 0 0
\(279\) −221951. −0.170705
\(280\) 0 0
\(281\) 1.03588e6 0.782604 0.391302 0.920262i \(-0.372025\pi\)
0.391302 + 0.920262i \(0.372025\pi\)
\(282\) 0 0
\(283\) − 2.39427e6i − 1.77708i −0.458795 0.888542i \(-0.651719\pi\)
0.458795 0.888542i \(-0.348281\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 564329.i − 0.404415i
\(288\) 0 0
\(289\) 1.27597e6 0.898659
\(290\) 0 0
\(291\) −179300. −0.124122
\(292\) 0 0
\(293\) 2.44817e6i 1.66599i 0.553280 + 0.832995i \(0.313376\pi\)
−0.553280 + 0.832995i \(0.686624\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.34987e6i 0.887973i
\(298\) 0 0
\(299\) 560443. 0.362538
\(300\) 0 0
\(301\) 1.54818e6 0.984931
\(302\) 0 0
\(303\) − 2.34314e6i − 1.46620i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 939476.i 0.568905i 0.958690 + 0.284453i \(0.0918118\pi\)
−0.958690 + 0.284453i \(0.908188\pi\)
\(308\) 0 0
\(309\) −893463. −0.532329
\(310\) 0 0
\(311\) 1.13941e6 0.668004 0.334002 0.942572i \(-0.391601\pi\)
0.334002 + 0.942572i \(0.391601\pi\)
\(312\) 0 0
\(313\) − 1.51692e6i − 0.875191i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.73484e6i 1.52857i 0.644880 + 0.764284i \(0.276907\pi\)
−0.644880 + 0.764284i \(0.723093\pi\)
\(318\) 0 0
\(319\) 455861. 0.250816
\(320\) 0 0
\(321\) −101266. −0.0548530
\(322\) 0 0
\(323\) − 390453.i − 0.208239i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.14785e6i 1.11080i
\(328\) 0 0
\(329\) −531939. −0.270939
\(330\) 0 0
\(331\) −122807. −0.0616104 −0.0308052 0.999525i \(-0.509807\pi\)
−0.0308052 + 0.999525i \(0.509807\pi\)
\(332\) 0 0
\(333\) 1.02219e6i 0.505150i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.65582e6i − 0.794217i −0.917772 0.397108i \(-0.870014\pi\)
0.917772 0.397108i \(-0.129986\pi\)
\(338\) 0 0
\(339\) 788075. 0.372450
\(340\) 0 0
\(341\) 898423. 0.418403
\(342\) 0 0
\(343\) − 1.98568e6i − 0.911329i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.63896e6i 0.730711i 0.930868 + 0.365355i \(0.119052\pi\)
−0.930868 + 0.365355i \(0.880948\pi\)
\(348\) 0 0
\(349\) −2.07756e6 −0.913040 −0.456520 0.889713i \(-0.650904\pi\)
−0.456520 + 0.889713i \(0.650904\pi\)
\(350\) 0 0
\(351\) 2.96640e6 1.28517
\(352\) 0 0
\(353\) 3.87344e6i 1.65447i 0.561854 + 0.827236i \(0.310088\pi\)
−0.561854 + 0.827236i \(0.689912\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 331985.i 0.137863i
\(358\) 0 0
\(359\) 3.16016e6 1.29411 0.647057 0.762441i \(-0.275999\pi\)
0.647057 + 0.762441i \(0.275999\pi\)
\(360\) 0 0
\(361\) −1.41658e6 −0.572103
\(362\) 0 0
\(363\) 686380.i 0.273400i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.76875e6i − 0.685491i −0.939428 0.342745i \(-0.888643\pi\)
0.939428 0.342745i \(-0.111357\pi\)
\(368\) 0 0
\(369\) 663941. 0.253842
\(370\) 0 0
\(371\) 165087. 0.0622700
\(372\) 0 0
\(373\) 3.86744e6i 1.43930i 0.694336 + 0.719651i \(0.255698\pi\)
−0.694336 + 0.719651i \(0.744302\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.00178e6i − 0.363009i
\(378\) 0 0
\(379\) −5.06193e6 −1.81016 −0.905082 0.425236i \(-0.860191\pi\)
−0.905082 + 0.425236i \(0.860191\pi\)
\(380\) 0 0
\(381\) 4.56734e6 1.61195
\(382\) 0 0
\(383\) 4.23524e6i 1.47530i 0.675181 + 0.737652i \(0.264065\pi\)
−0.675181 + 0.737652i \(0.735935\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.82146e6i 0.618219i
\(388\) 0 0
\(389\) 390940. 0.130989 0.0654947 0.997853i \(-0.479137\pi\)
0.0654947 + 0.997853i \(0.479137\pi\)
\(390\) 0 0
\(391\) −295542. −0.0977637
\(392\) 0 0
\(393\) − 3.98030e6i − 1.29997i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.36908e6i 1.07284i 0.843951 + 0.536421i \(0.180224\pi\)
−0.843951 + 0.536421i \(0.819776\pi\)
\(398\) 0 0
\(399\) −900861. −0.283286
\(400\) 0 0
\(401\) −5.51542e6 −1.71284 −0.856422 0.516277i \(-0.827318\pi\)
−0.856422 + 0.516277i \(0.827318\pi\)
\(402\) 0 0
\(403\) − 1.97433e6i − 0.605559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.13767e6i − 1.23814i
\(408\) 0 0
\(409\) 3.29662e6 0.974453 0.487227 0.873276i \(-0.338009\pi\)
0.487227 + 0.873276i \(0.338009\pi\)
\(410\) 0 0
\(411\) −427503. −0.124834
\(412\) 0 0
\(413\) − 1.08149e6i − 0.311996i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.35699e6i − 1.22700i
\(418\) 0 0
\(419\) 6.88088e6 1.91474 0.957368 0.288870i \(-0.0932795\pi\)
0.957368 + 0.288870i \(0.0932795\pi\)
\(420\) 0 0
\(421\) 3.04971e6 0.838596 0.419298 0.907849i \(-0.362276\pi\)
0.419298 + 0.907849i \(0.362276\pi\)
\(422\) 0 0
\(423\) − 625834.i − 0.170062i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.20511e6i − 0.585275i
\(428\) 0 0
\(429\) −2.99815e6 −0.786520
\(430\) 0 0
\(431\) −6.20632e6 −1.60931 −0.804657 0.593740i \(-0.797651\pi\)
−0.804657 + 0.593740i \(0.797651\pi\)
\(432\) 0 0
\(433\) − 1.87723e6i − 0.481169i −0.970628 0.240585i \(-0.922661\pi\)
0.970628 0.240585i \(-0.0773392\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 801971.i − 0.200888i
\(438\) 0 0
\(439\) 1.85883e6 0.460341 0.230170 0.973150i \(-0.426072\pi\)
0.230170 + 0.973150i \(0.426072\pi\)
\(440\) 0 0
\(441\) 977079. 0.239240
\(442\) 0 0
\(443\) 4.39604e6i 1.06427i 0.846659 + 0.532136i \(0.178610\pi\)
−0.846659 + 0.532136i \(0.821390\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3.04728e6i − 0.721346i
\(448\) 0 0
\(449\) 984885. 0.230552 0.115276 0.993333i \(-0.463225\pi\)
0.115276 + 0.993333i \(0.463225\pi\)
\(450\) 0 0
\(451\) −2.68754e6 −0.622175
\(452\) 0 0
\(453\) 2.15844e6i 0.494191i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 696014.i − 0.155893i −0.996958 0.0779466i \(-0.975164\pi\)
0.996958 0.0779466i \(-0.0248364\pi\)
\(458\) 0 0
\(459\) −1.56429e6 −0.346566
\(460\) 0 0
\(461\) 6.03623e6 1.32286 0.661430 0.750007i \(-0.269950\pi\)
0.661430 + 0.750007i \(0.269950\pi\)
\(462\) 0 0
\(463\) 2.07793e6i 0.450483i 0.974303 + 0.225241i \(0.0723171\pi\)
−0.974303 + 0.225241i \(0.927683\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 1.81361e6i − 0.384816i −0.981315 0.192408i \(-0.938370\pi\)
0.981315 0.192408i \(-0.0616296\pi\)
\(468\) 0 0
\(469\) −619222. −0.129991
\(470\) 0 0
\(471\) 2.37324e6 0.492935
\(472\) 0 0
\(473\) − 7.37300e6i − 1.51527i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 194228.i 0.0390854i
\(478\) 0 0
\(479\) −6.10687e6 −1.21613 −0.608065 0.793887i \(-0.708054\pi\)
−0.608065 + 0.793887i \(0.708054\pi\)
\(480\) 0 0
\(481\) −9.09273e6 −1.79197
\(482\) 0 0
\(483\) 681881.i 0.132997i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.32848e6i 0.253823i 0.991914 + 0.126912i \(0.0405064\pi\)
−0.991914 + 0.126912i \(0.959494\pi\)
\(488\) 0 0
\(489\) −5.13222e6 −0.970583
\(490\) 0 0
\(491\) −8.72480e6 −1.63325 −0.816623 0.577171i \(-0.804157\pi\)
−0.816623 + 0.577171i \(0.804157\pi\)
\(492\) 0 0
\(493\) 528273.i 0.0978907i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.01273e6i 0.547104i
\(498\) 0 0
\(499\) −5.20143e6 −0.935129 −0.467564 0.883959i \(-0.654868\pi\)
−0.467564 + 0.883959i \(0.654868\pi\)
\(500\) 0 0
\(501\) −2.13448e6 −0.379926
\(502\) 0 0
\(503\) 6.44189e6i 1.13525i 0.823286 + 0.567627i \(0.192139\pi\)
−0.823286 + 0.567627i \(0.807861\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.86083e6i 0.321504i
\(508\) 0 0
\(509\) 2.31511e6 0.396075 0.198038 0.980194i \(-0.436543\pi\)
0.198038 + 0.980194i \(0.436543\pi\)
\(510\) 0 0
\(511\) 4.52522e6 0.766633
\(512\) 0 0
\(513\) − 4.24479e6i − 0.712136i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.53328e6i 0.416828i
\(518\) 0 0
\(519\) −1.07061e6 −0.174467
\(520\) 0 0
\(521\) −9.65617e6 −1.55851 −0.779257 0.626705i \(-0.784403\pi\)
−0.779257 + 0.626705i \(0.784403\pi\)
\(522\) 0 0
\(523\) 6.40583e6i 1.02405i 0.858970 + 0.512025i \(0.171105\pi\)
−0.858970 + 0.512025i \(0.828895\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.04114e6i 0.163298i
\(528\) 0 0
\(529\) 5.82931e6 0.905687
\(530\) 0 0
\(531\) 1.27239e6 0.195833
\(532\) 0 0
\(533\) 5.90599e6i 0.900481i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.44418e6i − 1.41328i
\(538\) 0 0
\(539\) −3.95507e6 −0.586384
\(540\) 0 0
\(541\) 1.32300e6 0.194342 0.0971709 0.995268i \(-0.469021\pi\)
0.0971709 + 0.995268i \(0.469021\pi\)
\(542\) 0 0
\(543\) 4.37602e6i 0.636912i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 4.68044e6i 0.668834i 0.942425 + 0.334417i \(0.108539\pi\)
−0.942425 + 0.334417i \(0.891461\pi\)
\(548\) 0 0
\(549\) 2.59434e6 0.367363
\(550\) 0 0
\(551\) −1.43350e6 −0.201149
\(552\) 0 0
\(553\) − 2.72192e6i − 0.378498i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.58860e6i 1.03639i 0.855262 + 0.518195i \(0.173396\pi\)
−0.855262 + 0.518195i \(0.826604\pi\)
\(558\) 0 0
\(559\) −1.62025e7 −2.19307
\(560\) 0 0
\(561\) 1.58103e6 0.212097
\(562\) 0 0
\(563\) 399946.i 0.0531777i 0.999646 + 0.0265889i \(0.00846450\pi\)
−0.999646 + 0.0265889i \(0.991536\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.25853e6i 0.295032i
\(568\) 0 0
\(569\) −1.57419e6 −0.203834 −0.101917 0.994793i \(-0.532498\pi\)
−0.101917 + 0.994793i \(0.532498\pi\)
\(570\) 0 0
\(571\) 3.11290e6 0.399554 0.199777 0.979841i \(-0.435978\pi\)
0.199777 + 0.979841i \(0.435978\pi\)
\(572\) 0 0
\(573\) 8.03509e6i 1.02236i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 5.57621e6i − 0.697267i −0.937259 0.348634i \(-0.886646\pi\)
0.937259 0.348634i \(-0.113354\pi\)
\(578\) 0 0
\(579\) −1.07951e7 −1.33823
\(580\) 0 0
\(581\) −4.33716e6 −0.533047
\(582\) 0 0
\(583\) − 786205.i − 0.0957997i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.11890e7i − 1.34028i −0.742236 0.670138i \(-0.766235\pi\)
0.742236 0.670138i \(-0.233765\pi\)
\(588\) 0 0
\(589\) −2.82518e6 −0.335551
\(590\) 0 0
\(591\) −5.05868e6 −0.595756
\(592\) 0 0
\(593\) − 1.44707e7i − 1.68987i −0.534871 0.844934i \(-0.679640\pi\)
0.534871 0.844934i \(-0.320360\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 1.14094e7i − 1.31017i
\(598\) 0 0
\(599\) −2.32734e6 −0.265028 −0.132514 0.991181i \(-0.542305\pi\)
−0.132514 + 0.991181i \(0.542305\pi\)
\(600\) 0 0
\(601\) 4.28568e6 0.483987 0.241993 0.970278i \(-0.422199\pi\)
0.241993 + 0.970278i \(0.422199\pi\)
\(602\) 0 0
\(603\) − 728524.i − 0.0815925i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.04310e7i 1.14909i 0.818471 + 0.574547i \(0.194822\pi\)
−0.818471 + 0.574547i \(0.805178\pi\)
\(608\) 0 0
\(609\) 1.21884e6 0.133170
\(610\) 0 0
\(611\) 5.56701e6 0.603280
\(612\) 0 0
\(613\) − 3.52495e6i − 0.378880i −0.981892 0.189440i \(-0.939333\pi\)
0.981892 0.189440i \(-0.0606673\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.16178e7i 1.22861i 0.789070 + 0.614303i \(0.210563\pi\)
−0.789070 + 0.614303i \(0.789437\pi\)
\(618\) 0 0
\(619\) −1.04132e7 −1.09234 −0.546171 0.837673i \(-0.683915\pi\)
−0.546171 + 0.837673i \(0.683915\pi\)
\(620\) 0 0
\(621\) −3.21297e6 −0.334332
\(622\) 0 0
\(623\) 2.37204e6i 0.244851i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 4.29022e6i 0.435824i
\(628\) 0 0
\(629\) 4.79493e6 0.483232
\(630\) 0 0
\(631\) −6.57325e6 −0.657214 −0.328607 0.944467i \(-0.606579\pi\)
−0.328607 + 0.944467i \(0.606579\pi\)
\(632\) 0 0
\(633\) − 1.11640e7i − 1.10741i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.69146e6i 0.848680i
\(638\) 0 0
\(639\) −3.54452e6 −0.343404
\(640\) 0 0
\(641\) 7.21768e6 0.693829 0.346914 0.937897i \(-0.387229\pi\)
0.346914 + 0.937897i \(0.387229\pi\)
\(642\) 0 0
\(643\) − 989729.i − 0.0944036i −0.998885 0.0472018i \(-0.984970\pi\)
0.998885 0.0472018i \(-0.0150304\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 4.31383e6i − 0.405138i −0.979268 0.202569i \(-0.935071\pi\)
0.979268 0.202569i \(-0.0649290\pi\)
\(648\) 0 0
\(649\) −5.15046e6 −0.479992
\(650\) 0 0
\(651\) 2.40213e6 0.222149
\(652\) 0 0
\(653\) 1.49637e7i 1.37327i 0.727004 + 0.686633i \(0.240912\pi\)
−0.727004 + 0.686633i \(0.759088\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.32399e6i 0.481198i
\(658\) 0 0
\(659\) −4.42210e6 −0.396657 −0.198328 0.980136i \(-0.563551\pi\)
−0.198328 + 0.980136i \(0.563551\pi\)
\(660\) 0 0
\(661\) 1.57925e7 1.40587 0.702937 0.711252i \(-0.251872\pi\)
0.702937 + 0.711252i \(0.251872\pi\)
\(662\) 0 0
\(663\) − 3.47440e6i − 0.306970i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.08505e6i 0.0944352i
\(668\) 0 0
\(669\) 698939. 0.0603773
\(670\) 0 0
\(671\) −1.05015e7 −0.900420
\(672\) 0 0
\(673\) 5.60799e6i 0.477276i 0.971109 + 0.238638i \(0.0767009\pi\)
−0.971109 + 0.238638i \(0.923299\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 7.01232e6i − 0.588017i −0.955803 0.294009i \(-0.905011\pi\)
0.955803 0.294009i \(-0.0949894\pi\)
\(678\) 0 0
\(679\) −967853. −0.0805629
\(680\) 0 0
\(681\) −1.02289e7 −0.845207
\(682\) 0 0
\(683\) 2.16662e7i 1.77718i 0.458703 + 0.888590i \(0.348314\pi\)
−0.458703 + 0.888590i \(0.651686\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 7.50681e6i − 0.606825i
\(688\) 0 0
\(689\) −1.72772e6 −0.138652
\(690\) 0 0
\(691\) −4.20276e6 −0.334842 −0.167421 0.985885i \(-0.553544\pi\)
−0.167421 + 0.985885i \(0.553544\pi\)
\(692\) 0 0
\(693\) 1.81936e6i 0.143908i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 3.11444e6i − 0.242828i
\(698\) 0 0
\(699\) −1.29939e7 −1.00588
\(700\) 0 0
\(701\) −1.50989e6 −0.116051 −0.0580256 0.998315i \(-0.518481\pi\)
−0.0580256 + 0.998315i \(0.518481\pi\)
\(702\) 0 0
\(703\) 1.30113e7i 0.992963i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.26482e7i − 0.951653i
\(708\) 0 0
\(709\) −2.08359e6 −0.155667 −0.0778336 0.996966i \(-0.524800\pi\)
−0.0778336 + 0.996966i \(0.524800\pi\)
\(710\) 0 0
\(711\) 3.20238e6 0.237574
\(712\) 0 0
\(713\) 2.13844e6i 0.157534i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 2.10749e7i − 1.53098i
\(718\) 0 0
\(719\) −1.03357e7 −0.745618 −0.372809 0.927908i \(-0.621605\pi\)
−0.372809 + 0.927908i \(0.621605\pi\)
\(720\) 0 0
\(721\) −4.82287e6 −0.345515
\(722\) 0 0
\(723\) 1.51561e7i 1.07830i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.39227e6i 0.238042i 0.992892 + 0.119021i \(0.0379756\pi\)
−0.992892 + 0.119021i \(0.962024\pi\)
\(728\) 0 0
\(729\) −1.54171e7 −1.07444
\(730\) 0 0
\(731\) 8.54418e6 0.591394
\(732\) 0 0
\(733\) 1.99922e7i 1.37436i 0.726487 + 0.687180i \(0.241152\pi\)
−0.726487 + 0.687180i \(0.758848\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.94896e6i 0.199986i
\(738\) 0 0
\(739\) 2.37515e7 1.59986 0.799928 0.600096i \(-0.204871\pi\)
0.799928 + 0.600096i \(0.204871\pi\)
\(740\) 0 0
\(741\) 9.42797e6 0.630773
\(742\) 0 0
\(743\) − 768079.i − 0.0510427i −0.999674 0.0255214i \(-0.991875\pi\)
0.999674 0.0255214i \(-0.00812458\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 5.10273e6i − 0.334581i
\(748\) 0 0
\(749\) −546628. −0.0356030
\(750\) 0 0
\(751\) −2.34656e7 −1.51821 −0.759105 0.650968i \(-0.774363\pi\)
−0.759105 + 0.650968i \(0.774363\pi\)
\(752\) 0 0
\(753\) − 302745.i − 0.0194576i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.58118e7i 1.63711i 0.574425 + 0.818557i \(0.305226\pi\)
−0.574425 + 0.818557i \(0.694774\pi\)
\(758\) 0 0
\(759\) 3.24736e6 0.204610
\(760\) 0 0
\(761\) 1.19501e7 0.748013 0.374006 0.927426i \(-0.377984\pi\)
0.374006 + 0.927426i \(0.377984\pi\)
\(762\) 0 0
\(763\) 1.15940e7i 0.720977i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.13184e7i 0.694698i
\(768\) 0 0
\(769\) 1.61907e7 0.987302 0.493651 0.869660i \(-0.335662\pi\)
0.493651 + 0.869660i \(0.335662\pi\)
\(770\) 0 0
\(771\) −4.35070e6 −0.263586
\(772\) 0 0
\(773\) 1.40818e7i 0.847637i 0.905747 + 0.423818i \(0.139311\pi\)
−0.905747 + 0.423818i \(0.860689\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 1.10630e7i − 0.657384i
\(778\) 0 0
\(779\) 8.45122e6 0.498972
\(780\) 0 0
\(781\) 1.43477e7 0.841695
\(782\) 0 0
\(783\) 5.74310e6i 0.334766i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.41819e6i − 0.254277i −0.991885 0.127139i \(-0.959421\pi\)
0.991885 0.127139i \(-0.0405793\pi\)
\(788\) 0 0
\(789\) −1.38945e7 −0.794602
\(790\) 0 0
\(791\) 4.25399e6 0.241744
\(792\) 0 0
\(793\) 2.30776e7i 1.30319i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.04472e6i 0.337078i 0.985695 + 0.168539i \(0.0539049\pi\)
−0.985695 + 0.168539i \(0.946095\pi\)
\(798\) 0 0
\(799\) −2.93569e6 −0.162683
\(800\) 0 0
\(801\) −2.79074e6 −0.153688
\(802\) 0 0
\(803\) − 2.15507e7i − 1.17943i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.17516e7i − 0.635202i
\(808\) 0 0
\(809\) 1.71556e7 0.921583 0.460791 0.887509i \(-0.347566\pi\)
0.460791 + 0.887509i \(0.347566\pi\)
\(810\) 0 0
\(811\) −1.83020e7 −0.977114 −0.488557 0.872532i \(-0.662477\pi\)
−0.488557 + 0.872532i \(0.662477\pi\)
\(812\) 0 0
\(813\) − 1.71009e7i − 0.907389i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.31851e7i 1.21522i
\(818\) 0 0
\(819\) 3.99813e6 0.208280
\(820\) 0 0
\(821\) −77887.9 −0.00403285 −0.00201643 0.999998i \(-0.500642\pi\)
−0.00201643 + 0.999998i \(0.500642\pi\)
\(822\) 0 0
\(823\) − 423648.i − 0.0218025i −0.999941 0.0109012i \(-0.996530\pi\)
0.999941 0.0109012i \(-0.00347004\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 5.18956e6i − 0.263856i −0.991259 0.131928i \(-0.957883\pi\)
0.991259 0.131928i \(-0.0421168\pi\)
\(828\) 0 0
\(829\) 2.08613e7 1.05428 0.527139 0.849779i \(-0.323265\pi\)
0.527139 + 0.849779i \(0.323265\pi\)
\(830\) 0 0
\(831\) −3.15201e6 −0.158338
\(832\) 0 0
\(833\) − 4.58332e6i − 0.228859i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.13187e7i 0.558446i
\(838\) 0 0
\(839\) −8.18798e6 −0.401580 −0.200790 0.979634i \(-0.564351\pi\)
−0.200790 + 0.979634i \(0.564351\pi\)
\(840\) 0 0
\(841\) −1.85717e7 −0.905442
\(842\) 0 0
\(843\) − 1.31900e7i − 0.639258i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.70505e6i 0.177454i
\(848\) 0 0
\(849\) −3.04868e7 −1.45158
\(850\) 0 0
\(851\) 9.84854e6 0.466174
\(852\) 0 0
\(853\) − 3.11924e7i − 1.46783i −0.679241 0.733915i \(-0.737691\pi\)
0.679241 0.733915i \(-0.262309\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1.88654e7i − 0.877433i −0.898626 0.438716i \(-0.855433\pi\)
0.898626 0.438716i \(-0.144567\pi\)
\(858\) 0 0
\(859\) 1.30703e7 0.604369 0.302184 0.953249i \(-0.402284\pi\)
0.302184 + 0.953249i \(0.402284\pi\)
\(860\) 0 0
\(861\) −7.18571e6 −0.330341
\(862\) 0 0
\(863\) 5.58115e6i 0.255092i 0.991833 + 0.127546i \(0.0407100\pi\)
−0.991833 + 0.127546i \(0.959290\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 1.62471e7i − 0.734056i
\(868\) 0 0
\(869\) −1.29628e7 −0.582302
\(870\) 0 0
\(871\) 6.48047e6 0.289442
\(872\) 0 0
\(873\) − 1.13869e6i − 0.0505675i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.17437e7i 0.954629i 0.878732 + 0.477315i \(0.158390\pi\)
−0.878732 + 0.477315i \(0.841610\pi\)
\(878\) 0 0
\(879\) 3.11730e7 1.36084
\(880\) 0 0
\(881\) 2.26789e7 0.984425 0.492212 0.870475i \(-0.336188\pi\)
0.492212 + 0.870475i \(0.336188\pi\)
\(882\) 0 0
\(883\) − 2.34145e7i − 1.01061i −0.862942 0.505304i \(-0.831381\pi\)
0.862942 0.505304i \(-0.168619\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.56731e7i 0.668875i 0.942418 + 0.334437i \(0.108546\pi\)
−0.942418 + 0.334437i \(0.891454\pi\)
\(888\) 0 0
\(889\) 2.46543e7 1.04626
\(890\) 0 0
\(891\) 1.07559e7 0.453894
\(892\) 0 0
\(893\) − 7.96616e6i − 0.334288i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 7.13624e6i − 0.296134i
\(898\) 0 0
\(899\) 3.82240e6 0.157738
\(900\) 0 0
\(901\) 911092. 0.0373895
\(902\) 0 0
\(903\) − 1.97133e7i − 0.804527i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 3.67553e7i 1.48355i 0.670649 + 0.741775i \(0.266016\pi\)
−0.670649 + 0.741775i \(0.733984\pi\)
\(908\) 0 0
\(909\) 1.48807e7 0.597331
\(910\) 0 0
\(911\) 2.11183e7 0.843067 0.421534 0.906813i \(-0.361492\pi\)
0.421534 + 0.906813i \(0.361492\pi\)
\(912\) 0 0
\(913\) 2.06551e7i 0.820070i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.14855e7i − 0.843765i
\(918\) 0 0
\(919\) 7.27922e6 0.284312 0.142156 0.989844i \(-0.454596\pi\)
0.142156 + 0.989844i \(0.454596\pi\)
\(920\) 0 0
\(921\) 1.19625e7 0.464702
\(922\) 0 0
\(923\) − 3.15298e7i − 1.21819i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 5.67418e6i − 0.216872i
\(928\) 0 0
\(929\) 3.41165e7 1.29696 0.648478 0.761234i \(-0.275406\pi\)
0.648478 + 0.761234i \(0.275406\pi\)
\(930\) 0 0
\(931\) 1.24371e7 0.470268
\(932\) 0 0
\(933\) − 1.45083e7i − 0.545649i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 4.73880e7i − 1.76327i −0.471929 0.881637i \(-0.656442\pi\)
0.471929 0.881637i \(-0.343558\pi\)
\(938\) 0 0
\(939\) −1.93153e7 −0.714887
\(940\) 0 0
\(941\) −3.97476e7 −1.46331 −0.731656 0.681674i \(-0.761252\pi\)
−0.731656 + 0.681674i \(0.761252\pi\)
\(942\) 0 0
\(943\) − 6.39691e6i − 0.234256i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.76410e7i 1.00156i 0.865573 + 0.500782i \(0.166954\pi\)
−0.865573 + 0.500782i \(0.833046\pi\)
\(948\) 0 0
\(949\) −4.73588e7 −1.70701
\(950\) 0 0
\(951\) 3.48233e7 1.24859
\(952\) 0 0
\(953\) 5.22977e6i 0.186531i 0.995641 + 0.0932654i \(0.0297305\pi\)
−0.995641 + 0.0932654i \(0.970269\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 5.80457e6i − 0.204876i
\(958\) 0 0
\(959\) −2.30764e6 −0.0810255
\(960\) 0 0
\(961\) −2.10959e7 −0.736866
\(962\) 0 0
\(963\) − 643115.i − 0.0223472i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 1.76477e7i − 0.606905i −0.952847 0.303453i \(-0.901861\pi\)
0.952847 0.303453i \(-0.0981394\pi\)
\(968\) 0 0
\(969\) −4.97172e6 −0.170097
\(970\) 0 0
\(971\) 3.66934e7 1.24893 0.624467 0.781051i \(-0.285316\pi\)
0.624467 + 0.781051i \(0.285316\pi\)
\(972\) 0 0
\(973\) − 2.35188e7i − 0.796404i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.16023e7i 1.05921i 0.848244 + 0.529605i \(0.177660\pi\)
−0.848244 + 0.529605i \(0.822340\pi\)
\(978\) 0 0
\(979\) 1.12965e7 0.376693
\(980\) 0 0
\(981\) −1.36405e7 −0.452540
\(982\) 0 0
\(983\) − 3.73829e7i − 1.23393i −0.786992 0.616963i \(-0.788363\pi\)
0.786992 0.616963i \(-0.211637\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.77328e6i 0.221313i
\(988\) 0 0
\(989\) 1.75493e7 0.570518
\(990\) 0 0
\(991\) 2.84243e7 0.919401 0.459701 0.888074i \(-0.347957\pi\)
0.459701 + 0.888074i \(0.347957\pi\)
\(992\) 0 0
\(993\) 1.56373e6i 0.0503256i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 1.70600e7i − 0.543553i −0.962360 0.271776i \(-0.912389\pi\)
0.962360 0.271776i \(-0.0876111\pi\)
\(998\) 0 0
\(999\) 5.21279e7 1.65256
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 800.6.c.f.449.2 4
4.3 odd 2 800.6.c.g.449.3 4
5.2 odd 4 800.6.a.l.1.1 2
5.3 odd 4 160.6.a.a.1.2 2
5.4 even 2 inner 800.6.c.f.449.3 4
20.3 even 4 160.6.a.e.1.1 yes 2
20.7 even 4 800.6.a.g.1.2 2
20.19 odd 2 800.6.c.g.449.2 4
40.3 even 4 320.6.a.r.1.2 2
40.13 odd 4 320.6.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.a.a.1.2 2 5.3 odd 4
160.6.a.e.1.1 yes 2 20.3 even 4
320.6.a.r.1.2 2 40.3 even 4
320.6.a.v.1.1 2 40.13 odd 4
800.6.a.g.1.2 2 20.7 even 4
800.6.a.l.1.1 2 5.2 odd 4
800.6.c.f.449.2 4 1.1 even 1 trivial
800.6.c.f.449.3 4 5.4 even 2 inner
800.6.c.g.449.2 4 20.19 odd 2
800.6.c.g.449.3 4 4.3 odd 2