Properties

Label 1100.6.b.a.749.2
Level $1100$
Weight $6$
Character 1100.749
Analytic conductor $176.422$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,6,Mod(749,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.749");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1100.b (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: \(176.422201794\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 749.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1100.749
Dual form 1100.6.b.a.749.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000i q^{3} +50.0000i q^{7} +194.000 q^{9} +O(q^{10})\) \(q+7.00000i q^{3} +50.0000i q^{7} +194.000 q^{9} +121.000 q^{11} -380.000i q^{13} +1154.00i q^{17} +1824.00 q^{19} -350.000 q^{21} +3591.00i q^{23} +3059.00i q^{27} -8032.00 q^{29} -2945.00 q^{31} +847.000i q^{33} -6979.00i q^{37} +2660.00 q^{39} -520.000 q^{41} -2486.00i q^{43} +6920.00i q^{47} +14307.0 q^{49} -8078.00 q^{51} -13718.0i q^{53} +12768.0i q^{57} +31779.0 q^{59} +34156.0 q^{61} +9700.00i q^{63} +61503.0i q^{67} -25137.0 q^{69} -14971.0 q^{71} -36444.0i q^{73} +6050.00i q^{77} +28538.0 q^{79} +25729.0 q^{81} +77482.0i q^{83} -56224.0i q^{87} -36271.0 q^{89} +19000.0 q^{91} -20615.0i q^{93} +49799.0i q^{97} +23474.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 388 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 388 q^{9} + 242 q^{11} + 3648 q^{19} - 700 q^{21} - 16064 q^{29} - 5890 q^{31} + 5320 q^{39} - 1040 q^{41} + 28614 q^{49} - 16156 q^{51} + 63558 q^{59} + 68312 q^{61} - 50274 q^{69} - 29942 q^{71} + 57076 q^{79} + 51458 q^{81} - 72542 q^{89} + 38000 q^{91} + 46948 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 7.00000i 0.449050i 0.974468 + 0.224525i \(0.0720831\pi\)
−0.974468 + 0.224525i \(0.927917\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 50.0000i 0.385678i 0.981230 + 0.192839i \(0.0617695\pi\)
−0.981230 + 0.192839i \(0.938230\pi\)
\(8\) 0 0
\(9\) 194.000 0.798354
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 0 0
\(13\) − 380.000i − 0.623627i −0.950143 0.311814i \(-0.899064\pi\)
0.950143 0.311814i \(-0.100936\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1154.00i 0.968464i 0.874940 + 0.484232i \(0.160901\pi\)
−0.874940 + 0.484232i \(0.839099\pi\)
\(18\) 0 0
\(19\) 1824.00 1.15915 0.579577 0.814918i \(-0.303218\pi\)
0.579577 + 0.814918i \(0.303218\pi\)
\(20\) 0 0
\(21\) −350.000 −0.173189
\(22\) 0 0
\(23\) 3591.00i 1.41545i 0.706486 + 0.707727i \(0.250279\pi\)
−0.706486 + 0.707727i \(0.749721\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3059.00i 0.807551i
\(28\) 0 0
\(29\) −8032.00 −1.77349 −0.886745 0.462259i \(-0.847039\pi\)
−0.886745 + 0.462259i \(0.847039\pi\)
\(30\) 0 0
\(31\) −2945.00 −0.550403 −0.275202 0.961387i \(-0.588745\pi\)
−0.275202 + 0.961387i \(0.588745\pi\)
\(32\) 0 0
\(33\) 847.000i 0.135394i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6979.00i − 0.838087i −0.907966 0.419043i \(-0.862366\pi\)
0.907966 0.419043i \(-0.137634\pi\)
\(38\) 0 0
\(39\) 2660.00 0.280040
\(40\) 0 0
\(41\) −520.000 −0.0483107 −0.0241554 0.999708i \(-0.507690\pi\)
−0.0241554 + 0.999708i \(0.507690\pi\)
\(42\) 0 0
\(43\) − 2486.00i − 0.205036i −0.994731 0.102518i \(-0.967310\pi\)
0.994731 0.102518i \(-0.0326899\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6920.00i 0.456942i 0.973551 + 0.228471i \(0.0733727\pi\)
−0.973551 + 0.228471i \(0.926627\pi\)
\(48\) 0 0
\(49\) 14307.0 0.851252
\(50\) 0 0
\(51\) −8078.00 −0.434889
\(52\) 0 0
\(53\) − 13718.0i − 0.670812i −0.942074 0.335406i \(-0.891126\pi\)
0.942074 0.335406i \(-0.108874\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12768.0i 0.520518i
\(58\) 0 0
\(59\) 31779.0 1.18853 0.594265 0.804269i \(-0.297443\pi\)
0.594265 + 0.804269i \(0.297443\pi\)
\(60\) 0 0
\(61\) 34156.0 1.17528 0.587641 0.809121i \(-0.300057\pi\)
0.587641 + 0.809121i \(0.300057\pi\)
\(62\) 0 0
\(63\) 9700.00i 0.307908i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 61503.0i 1.67382i 0.547339 + 0.836911i \(0.315641\pi\)
−0.547339 + 0.836911i \(0.684359\pi\)
\(68\) 0 0
\(69\) −25137.0 −0.635610
\(70\) 0 0
\(71\) −14971.0 −0.352456 −0.176228 0.984349i \(-0.556390\pi\)
−0.176228 + 0.984349i \(0.556390\pi\)
\(72\) 0 0
\(73\) − 36444.0i − 0.800422i −0.916423 0.400211i \(-0.868937\pi\)
0.916423 0.400211i \(-0.131063\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6050.00i 0.116286i
\(78\) 0 0
\(79\) 28538.0 0.514465 0.257232 0.966350i \(-0.417189\pi\)
0.257232 + 0.966350i \(0.417189\pi\)
\(80\) 0 0
\(81\) 25729.0 0.435723
\(82\) 0 0
\(83\) 77482.0i 1.23454i 0.786751 + 0.617271i \(0.211762\pi\)
−0.786751 + 0.617271i \(0.788238\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 56224.0i − 0.796386i
\(88\) 0 0
\(89\) −36271.0 −0.485383 −0.242691 0.970104i \(-0.578030\pi\)
−0.242691 + 0.970104i \(0.578030\pi\)
\(90\) 0 0
\(91\) 19000.0 0.240519
\(92\) 0 0
\(93\) − 20615.0i − 0.247159i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 49799.0i 0.537392i 0.963225 + 0.268696i \(0.0865927\pi\)
−0.963225 + 0.268696i \(0.913407\pi\)
\(98\) 0 0
\(99\) 23474.0 0.240713
\(100\) 0 0
\(101\) −153406. −1.49637 −0.748185 0.663490i \(-0.769074\pi\)
−0.748185 + 0.663490i \(0.769074\pi\)
\(102\) 0 0
\(103\) − 134720.i − 1.25124i −0.780130 0.625618i \(-0.784847\pi\)
0.780130 0.625618i \(-0.215153\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 169218.i − 1.42885i −0.699711 0.714426i \(-0.746688\pi\)
0.699711 0.714426i \(-0.253312\pi\)
\(108\) 0 0
\(109\) 233206. 1.88007 0.940034 0.341081i \(-0.110793\pi\)
0.940034 + 0.341081i \(0.110793\pi\)
\(110\) 0 0
\(111\) 48853.0 0.376343
\(112\) 0 0
\(113\) 94329.0i 0.694943i 0.937691 + 0.347471i \(0.112960\pi\)
−0.937691 + 0.347471i \(0.887040\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 73720.0i − 0.497875i
\(118\) 0 0
\(119\) −57700.0 −0.373515
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 0 0
\(123\) − 3640.00i − 0.0216939i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 259480.i 1.42756i 0.700370 + 0.713780i \(0.253019\pi\)
−0.700370 + 0.713780i \(0.746981\pi\)
\(128\) 0 0
\(129\) 17402.0 0.0920714
\(130\) 0 0
\(131\) −85410.0 −0.434841 −0.217420 0.976078i \(-0.569764\pi\)
−0.217420 + 0.976078i \(0.569764\pi\)
\(132\) 0 0
\(133\) 91200.0i 0.447060i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 427703.i 1.94689i 0.228926 + 0.973444i \(0.426479\pi\)
−0.228926 + 0.973444i \(0.573521\pi\)
\(138\) 0 0
\(139\) −309690. −1.35953 −0.679767 0.733428i \(-0.737919\pi\)
−0.679767 + 0.733428i \(0.737919\pi\)
\(140\) 0 0
\(141\) −48440.0 −0.205190
\(142\) 0 0
\(143\) − 45980.0i − 0.188031i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 100149.i 0.382255i
\(148\) 0 0
\(149\) −449846. −1.65996 −0.829981 0.557792i \(-0.811649\pi\)
−0.829981 + 0.557792i \(0.811649\pi\)
\(150\) 0 0
\(151\) 405074. 1.44575 0.722873 0.690981i \(-0.242821\pi\)
0.722873 + 0.690981i \(0.242821\pi\)
\(152\) 0 0
\(153\) 223876.i 0.773177i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 339321.i − 1.09866i −0.835607 0.549328i \(-0.814884\pi\)
0.835607 0.549328i \(-0.185116\pi\)
\(158\) 0 0
\(159\) 96026.0 0.301228
\(160\) 0 0
\(161\) −179550. −0.545910
\(162\) 0 0
\(163\) 271396.i 0.800082i 0.916497 + 0.400041i \(0.131004\pi\)
−0.916497 + 0.400041i \(0.868996\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 72468.0i − 0.201074i −0.994933 0.100537i \(-0.967944\pi\)
0.994933 0.100537i \(-0.0320560\pi\)
\(168\) 0 0
\(169\) 226893. 0.611089
\(170\) 0 0
\(171\) 353856. 0.925414
\(172\) 0 0
\(173\) 479226.i 1.21738i 0.793409 + 0.608689i \(0.208304\pi\)
−0.793409 + 0.608689i \(0.791696\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 222453.i 0.533710i
\(178\) 0 0
\(179\) 40935.0 0.0954910 0.0477455 0.998860i \(-0.484796\pi\)
0.0477455 + 0.998860i \(0.484796\pi\)
\(180\) 0 0
\(181\) −90169.0 −0.204579 −0.102289 0.994755i \(-0.532617\pi\)
−0.102289 + 0.994755i \(0.532617\pi\)
\(182\) 0 0
\(183\) 239092.i 0.527761i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 139634.i 0.292003i
\(188\) 0 0
\(189\) −152950. −0.311455
\(190\) 0 0
\(191\) −260375. −0.516435 −0.258218 0.966087i \(-0.583135\pi\)
−0.258218 + 0.966087i \(0.583135\pi\)
\(192\) 0 0
\(193\) 524324.i 1.01323i 0.862173 + 0.506613i \(0.169103\pi\)
−0.862173 + 0.506613i \(0.830897\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 759582.i − 1.39447i −0.716843 0.697235i \(-0.754413\pi\)
0.716843 0.697235i \(-0.245587\pi\)
\(198\) 0 0
\(199\) 882736. 1.58015 0.790075 0.613011i \(-0.210042\pi\)
0.790075 + 0.613011i \(0.210042\pi\)
\(200\) 0 0
\(201\) −430521. −0.751630
\(202\) 0 0
\(203\) − 401600.i − 0.683996i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 696654.i 1.13003i
\(208\) 0 0
\(209\) 220704. 0.349498
\(210\) 0 0
\(211\) −1.15285e6 −1.78266 −0.891328 0.453360i \(-0.850225\pi\)
−0.891328 + 0.453360i \(0.850225\pi\)
\(212\) 0 0
\(213\) − 104797.i − 0.158270i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 147250.i − 0.212278i
\(218\) 0 0
\(219\) 255108. 0.359430
\(220\) 0 0
\(221\) 438520. 0.603961
\(222\) 0 0
\(223\) − 65893.0i − 0.0887314i −0.999015 0.0443657i \(-0.985873\pi\)
0.999015 0.0443657i \(-0.0141267\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 314526.i 0.405128i 0.979269 + 0.202564i \(0.0649274\pi\)
−0.979269 + 0.202564i \(0.935073\pi\)
\(228\) 0 0
\(229\) −1.03846e6 −1.30859 −0.654293 0.756241i \(-0.727034\pi\)
−0.654293 + 0.756241i \(0.727034\pi\)
\(230\) 0 0
\(231\) −42350.0 −0.0522184
\(232\) 0 0
\(233\) 509976.i 0.615403i 0.951483 + 0.307702i \(0.0995599\pi\)
−0.951483 + 0.307702i \(0.900440\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 199766.i 0.231021i
\(238\) 0 0
\(239\) 444494. 0.503351 0.251676 0.967812i \(-0.419018\pi\)
0.251676 + 0.967812i \(0.419018\pi\)
\(240\) 0 0
\(241\) −283464. −0.314380 −0.157190 0.987568i \(-0.550244\pi\)
−0.157190 + 0.987568i \(0.550244\pi\)
\(242\) 0 0
\(243\) 923440.i 1.00321i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 693120.i − 0.722880i
\(248\) 0 0
\(249\) −542374. −0.554371
\(250\) 0 0
\(251\) −773807. −0.775262 −0.387631 0.921815i \(-0.626706\pi\)
−0.387631 + 0.921815i \(0.626706\pi\)
\(252\) 0 0
\(253\) 434511.i 0.426775i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 387714.i 0.366167i 0.983097 + 0.183083i \(0.0586078\pi\)
−0.983097 + 0.183083i \(0.941392\pi\)
\(258\) 0 0
\(259\) 348950. 0.323232
\(260\) 0 0
\(261\) −1.55821e6 −1.41587
\(262\) 0 0
\(263\) − 197602.i − 0.176158i −0.996113 0.0880789i \(-0.971927\pi\)
0.996113 0.0880789i \(-0.0280728\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 253897.i − 0.217961i
\(268\) 0 0
\(269\) 262694. 0.221345 0.110672 0.993857i \(-0.464700\pi\)
0.110672 + 0.993857i \(0.464700\pi\)
\(270\) 0 0
\(271\) −159068. −0.131571 −0.0657854 0.997834i \(-0.520955\pi\)
−0.0657854 + 0.997834i \(0.520955\pi\)
\(272\) 0 0
\(273\) 133000.i 0.108005i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.29385e6i − 1.01318i −0.862188 0.506589i \(-0.830906\pi\)
0.862188 0.506589i \(-0.169094\pi\)
\(278\) 0 0
\(279\) −571330. −0.439417
\(280\) 0 0
\(281\) −1.78114e6 −1.34565 −0.672824 0.739802i \(-0.734919\pi\)
−0.672824 + 0.739802i \(0.734919\pi\)
\(282\) 0 0
\(283\) 1.98279e6i 1.47167i 0.677161 + 0.735835i \(0.263210\pi\)
−0.677161 + 0.735835i \(0.736790\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 26000.0i − 0.0186324i
\(288\) 0 0
\(289\) 88141.0 0.0620774
\(290\) 0 0
\(291\) −348593. −0.241316
\(292\) 0 0
\(293\) 578360.i 0.393577i 0.980446 + 0.196788i \(0.0630512\pi\)
−0.980446 + 0.196788i \(0.936949\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 370139.i 0.243486i
\(298\) 0 0
\(299\) 1.36458e6 0.882716
\(300\) 0 0
\(301\) 124300. 0.0790779
\(302\) 0 0
\(303\) − 1.07384e6i − 0.671945i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.07602e6i 1.86270i 0.364120 + 0.931352i \(0.381370\pi\)
−0.364120 + 0.931352i \(0.618630\pi\)
\(308\) 0 0
\(309\) 943040. 0.561868
\(310\) 0 0
\(311\) −3.13757e6 −1.83947 −0.919735 0.392540i \(-0.871597\pi\)
−0.919735 + 0.392540i \(0.871597\pi\)
\(312\) 0 0
\(313\) 2.61784e6i 1.51037i 0.655514 + 0.755183i \(0.272452\pi\)
−0.655514 + 0.755183i \(0.727548\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.49220e6i 1.39294i 0.717584 + 0.696472i \(0.245248\pi\)
−0.717584 + 0.696472i \(0.754752\pi\)
\(318\) 0 0
\(319\) −971872. −0.534727
\(320\) 0 0
\(321\) 1.18453e6 0.641626
\(322\) 0 0
\(323\) 2.10490e6i 1.12260i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.63244e6i 0.844245i
\(328\) 0 0
\(329\) −346000. −0.176233
\(330\) 0 0
\(331\) −2.70125e6 −1.35517 −0.677586 0.735443i \(-0.736974\pi\)
−0.677586 + 0.735443i \(0.736974\pi\)
\(332\) 0 0
\(333\) − 1.35393e6i − 0.669090i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.42610e6i 0.684031i 0.939694 + 0.342016i \(0.111110\pi\)
−0.939694 + 0.342016i \(0.888890\pi\)
\(338\) 0 0
\(339\) −660303. −0.312064
\(340\) 0 0
\(341\) −356345. −0.165953
\(342\) 0 0
\(343\) 1.55570e6i 0.713987i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.86374e6i − 1.27676i −0.769721 0.638381i \(-0.779604\pi\)
0.769721 0.638381i \(-0.220396\pi\)
\(348\) 0 0
\(349\) −296350. −0.130239 −0.0651195 0.997877i \(-0.520743\pi\)
−0.0651195 + 0.997877i \(0.520743\pi\)
\(350\) 0 0
\(351\) 1.16242e6 0.503611
\(352\) 0 0
\(353\) − 2.12114e6i − 0.906010i −0.891508 0.453005i \(-0.850352\pi\)
0.891508 0.453005i \(-0.149648\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 403900.i − 0.167727i
\(358\) 0 0
\(359\) 3.47512e6 1.42310 0.711548 0.702638i \(-0.247994\pi\)
0.711548 + 0.702638i \(0.247994\pi\)
\(360\) 0 0
\(361\) 850877. 0.343636
\(362\) 0 0
\(363\) 102487.i 0.0408227i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.56190e6i − 0.605322i −0.953098 0.302661i \(-0.902125\pi\)
0.953098 0.302661i \(-0.0978750\pi\)
\(368\) 0 0
\(369\) −100880. −0.0385691
\(370\) 0 0
\(371\) 685900. 0.258718
\(372\) 0 0
\(373\) 1.93773e6i 0.721144i 0.932731 + 0.360572i \(0.117419\pi\)
−0.932731 + 0.360572i \(0.882581\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.05216e6i 1.10600i
\(378\) 0 0
\(379\) −3.07495e6 −1.09961 −0.549806 0.835292i \(-0.685298\pi\)
−0.549806 + 0.835292i \(0.685298\pi\)
\(380\) 0 0
\(381\) −1.81636e6 −0.641046
\(382\) 0 0
\(383\) 4.31553e6i 1.50327i 0.659579 + 0.751635i \(0.270734\pi\)
−0.659579 + 0.751635i \(0.729266\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 482284.i − 0.163691i
\(388\) 0 0
\(389\) −2.36251e6 −0.791590 −0.395795 0.918339i \(-0.629531\pi\)
−0.395795 + 0.918339i \(0.629531\pi\)
\(390\) 0 0
\(391\) −4.14401e6 −1.37082
\(392\) 0 0
\(393\) − 597870.i − 0.195265i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.77598e6i − 0.565539i −0.959188 0.282769i \(-0.908747\pi\)
0.959188 0.282769i \(-0.0912531\pi\)
\(398\) 0 0
\(399\) −638400. −0.200752
\(400\) 0 0
\(401\) 1.56967e6 0.487468 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(402\) 0 0
\(403\) 1.11910e6i 0.343247i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 844459.i − 0.252693i
\(408\) 0 0
\(409\) −1.29485e6 −0.382746 −0.191373 0.981517i \(-0.561294\pi\)
−0.191373 + 0.981517i \(0.561294\pi\)
\(410\) 0 0
\(411\) −2.99392e6 −0.874250
\(412\) 0 0
\(413\) 1.58895e6i 0.458390i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 2.16783e6i − 0.610499i
\(418\) 0 0
\(419\) −272916. −0.0759441 −0.0379720 0.999279i \(-0.512090\pi\)
−0.0379720 + 0.999279i \(0.512090\pi\)
\(420\) 0 0
\(421\) −2.61801e6 −0.719890 −0.359945 0.932974i \(-0.617205\pi\)
−0.359945 + 0.932974i \(0.617205\pi\)
\(422\) 0 0
\(423\) 1.34248e6i 0.364802i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.70780e6i 0.453281i
\(428\) 0 0
\(429\) 321860. 0.0844352
\(430\) 0 0
\(431\) 2.81037e6 0.728735 0.364368 0.931255i \(-0.381285\pi\)
0.364368 + 0.931255i \(0.381285\pi\)
\(432\) 0 0
\(433\) 5.98509e6i 1.53409i 0.641593 + 0.767046i \(0.278274\pi\)
−0.641593 + 0.767046i \(0.721726\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.54998e6i 1.64073i
\(438\) 0 0
\(439\) 7.50486e6 1.85858 0.929290 0.369352i \(-0.120420\pi\)
0.929290 + 0.369352i \(0.120420\pi\)
\(440\) 0 0
\(441\) 2.77556e6 0.679601
\(442\) 0 0
\(443\) 1.56806e6i 0.379624i 0.981820 + 0.189812i \(0.0607878\pi\)
−0.981820 + 0.189812i \(0.939212\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3.14892e6i − 0.745406i
\(448\) 0 0
\(449\) 4.04044e6 0.945831 0.472915 0.881108i \(-0.343202\pi\)
0.472915 + 0.881108i \(0.343202\pi\)
\(450\) 0 0
\(451\) −62920.0 −0.0145662
\(452\) 0 0
\(453\) 2.83552e6i 0.649213i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 2.21132e6i − 0.495291i −0.968851 0.247645i \(-0.920343\pi\)
0.968851 0.247645i \(-0.0796568\pi\)
\(458\) 0 0
\(459\) −3.53009e6 −0.782084
\(460\) 0 0
\(461\) −3.56735e6 −0.781795 −0.390898 0.920434i \(-0.627835\pi\)
−0.390898 + 0.920434i \(0.627835\pi\)
\(462\) 0 0
\(463\) 747757.i 0.162109i 0.996710 + 0.0810547i \(0.0258288\pi\)
−0.996710 + 0.0810547i \(0.974171\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.44511e6i 1.15535i 0.816266 + 0.577676i \(0.196040\pi\)
−0.816266 + 0.577676i \(0.803960\pi\)
\(468\) 0 0
\(469\) −3.07515e6 −0.645556
\(470\) 0 0
\(471\) 2.37525e6 0.493352
\(472\) 0 0
\(473\) − 300806.i − 0.0618207i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.66129e6i − 0.535546i
\(478\) 0 0
\(479\) 6.22046e6 1.23875 0.619375 0.785095i \(-0.287386\pi\)
0.619375 + 0.785095i \(0.287386\pi\)
\(480\) 0 0
\(481\) −2.65202e6 −0.522654
\(482\) 0 0
\(483\) − 1.25685e6i − 0.245141i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.34398e6i 0.638913i 0.947601 + 0.319457i \(0.103500\pi\)
−0.947601 + 0.319457i \(0.896500\pi\)
\(488\) 0 0
\(489\) −1.89977e6 −0.359277
\(490\) 0 0
\(491\) 5.58646e6 1.04576 0.522881 0.852406i \(-0.324857\pi\)
0.522881 + 0.852406i \(0.324857\pi\)
\(492\) 0 0
\(493\) − 9.26893e6i − 1.71756i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 748550.i − 0.135935i
\(498\) 0 0
\(499\) 8.29348e6 1.49103 0.745514 0.666490i \(-0.232204\pi\)
0.745514 + 0.666490i \(0.232204\pi\)
\(500\) 0 0
\(501\) 507276. 0.0902922
\(502\) 0 0
\(503\) 5.29951e6i 0.933933i 0.884275 + 0.466967i \(0.154653\pi\)
−0.884275 + 0.466967i \(0.845347\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.58825e6i 0.274410i
\(508\) 0 0
\(509\) −24415.0 −0.00417698 −0.00208849 0.999998i \(-0.500665\pi\)
−0.00208849 + 0.999998i \(0.500665\pi\)
\(510\) 0 0
\(511\) 1.82220e6 0.308705
\(512\) 0 0
\(513\) 5.57962e6i 0.936076i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 837320.i 0.137773i
\(518\) 0 0
\(519\) −3.35458e6 −0.546663
\(520\) 0 0
\(521\) 4.76275e6 0.768712 0.384356 0.923185i \(-0.374424\pi\)
0.384356 + 0.923185i \(0.374424\pi\)
\(522\) 0 0
\(523\) 735248.i 0.117538i 0.998272 + 0.0587692i \(0.0187176\pi\)
−0.998272 + 0.0587692i \(0.981282\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 3.39853e6i − 0.533046i
\(528\) 0 0
\(529\) −6.45894e6 −1.00351
\(530\) 0 0
\(531\) 6.16513e6 0.948868
\(532\) 0 0
\(533\) 197600.i 0.0301279i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 286545.i 0.0428802i
\(538\) 0 0
\(539\) 1.73115e6 0.256662
\(540\) 0 0
\(541\) −3.19649e6 −0.469548 −0.234774 0.972050i \(-0.575435\pi\)
−0.234774 + 0.972050i \(0.575435\pi\)
\(542\) 0 0
\(543\) − 631183.i − 0.0918662i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.85902e6i − 1.26595i −0.774171 0.632976i \(-0.781833\pi\)
0.774171 0.632976i \(-0.218167\pi\)
\(548\) 0 0
\(549\) 6.62626e6 0.938292
\(550\) 0 0
\(551\) −1.46504e7 −2.05575
\(552\) 0 0
\(553\) 1.42690e6i 0.198418i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.74512e6i 0.238335i 0.992874 + 0.119167i \(0.0380225\pi\)
−0.992874 + 0.119167i \(0.961977\pi\)
\(558\) 0 0
\(559\) −944680. −0.127866
\(560\) 0 0
\(561\) −977438. −0.131124
\(562\) 0 0
\(563\) − 1.32333e7i − 1.75953i −0.475410 0.879764i \(-0.657700\pi\)
0.475410 0.879764i \(-0.342300\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.28645e6i 0.168049i
\(568\) 0 0
\(569\) −1.04156e7 −1.34867 −0.674335 0.738426i \(-0.735570\pi\)
−0.674335 + 0.738426i \(0.735570\pi\)
\(570\) 0 0
\(571\) −2.48163e6 −0.318527 −0.159264 0.987236i \(-0.550912\pi\)
−0.159264 + 0.987236i \(0.550912\pi\)
\(572\) 0 0
\(573\) − 1.82262e6i − 0.231905i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.31244e7i 1.64112i 0.571557 + 0.820562i \(0.306339\pi\)
−0.571557 + 0.820562i \(0.693661\pi\)
\(578\) 0 0
\(579\) −3.67027e6 −0.454989
\(580\) 0 0
\(581\) −3.87410e6 −0.476135
\(582\) 0 0
\(583\) − 1.65988e6i − 0.202258i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.86010e6i 0.582170i 0.956697 + 0.291085i \(0.0940163\pi\)
−0.956697 + 0.291085i \(0.905984\pi\)
\(588\) 0 0
\(589\) −5.37168e6 −0.638002
\(590\) 0 0
\(591\) 5.31707e6 0.626187
\(592\) 0 0
\(593\) − 1.58559e6i − 0.185163i −0.995705 0.0925814i \(-0.970488\pi\)
0.995705 0.0925814i \(-0.0295118\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.17915e6i 0.709566i
\(598\) 0 0
\(599\) −9.04294e6 −1.02978 −0.514888 0.857258i \(-0.672166\pi\)
−0.514888 + 0.857258i \(0.672166\pi\)
\(600\) 0 0
\(601\) 729186. 0.0823478 0.0411739 0.999152i \(-0.486890\pi\)
0.0411739 + 0.999152i \(0.486890\pi\)
\(602\) 0 0
\(603\) 1.19316e7i 1.33630i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.91130e6i 0.430873i 0.976518 + 0.215437i \(0.0691175\pi\)
−0.976518 + 0.215437i \(0.930883\pi\)
\(608\) 0 0
\(609\) 2.81120e6 0.307149
\(610\) 0 0
\(611\) 2.62960e6 0.284962
\(612\) 0 0
\(613\) − 5.52184e6i − 0.593516i −0.954953 0.296758i \(-0.904094\pi\)
0.954953 0.296758i \(-0.0959055\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.88539e6i 0.516638i 0.966060 + 0.258319i \(0.0831686\pi\)
−0.966060 + 0.258319i \(0.916831\pi\)
\(618\) 0 0
\(619\) 4.11150e6 0.431295 0.215647 0.976471i \(-0.430814\pi\)
0.215647 + 0.976471i \(0.430814\pi\)
\(620\) 0 0
\(621\) −1.09849e7 −1.14305
\(622\) 0 0
\(623\) − 1.81355e6i − 0.187202i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.54493e6i 0.156942i
\(628\) 0 0
\(629\) 8.05377e6 0.811657
\(630\) 0 0
\(631\) 8.24910e6 0.824771 0.412385 0.911009i \(-0.364696\pi\)
0.412385 + 0.911009i \(0.364696\pi\)
\(632\) 0 0
\(633\) − 8.06996e6i − 0.800502i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 5.43666e6i − 0.530864i
\(638\) 0 0
\(639\) −2.90437e6 −0.281385
\(640\) 0 0
\(641\) −4.29330e6 −0.412711 −0.206355 0.978477i \(-0.566160\pi\)
−0.206355 + 0.978477i \(0.566160\pi\)
\(642\) 0 0
\(643\) − 1.63045e7i − 1.55518i −0.628774 0.777588i \(-0.716443\pi\)
0.628774 0.777588i \(-0.283557\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 4.42624e6i − 0.415695i −0.978161 0.207847i \(-0.933354\pi\)
0.978161 0.207847i \(-0.0666457\pi\)
\(648\) 0 0
\(649\) 3.84526e6 0.358355
\(650\) 0 0
\(651\) 1.03075e6 0.0953237
\(652\) 0 0
\(653\) − 6.27529e6i − 0.575905i −0.957645 0.287952i \(-0.907025\pi\)
0.957645 0.287952i \(-0.0929745\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 7.07014e6i − 0.639020i
\(658\) 0 0
\(659\) 1.09748e7 0.984422 0.492211 0.870476i \(-0.336189\pi\)
0.492211 + 0.870476i \(0.336189\pi\)
\(660\) 0 0
\(661\) 2.02025e7 1.79846 0.899229 0.437478i \(-0.144128\pi\)
0.899229 + 0.437478i \(0.144128\pi\)
\(662\) 0 0
\(663\) 3.06964e6i 0.271209i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 2.88429e7i − 2.51029i
\(668\) 0 0
\(669\) 461251. 0.0398448
\(670\) 0 0
\(671\) 4.13288e6 0.354361
\(672\) 0 0
\(673\) 1.14233e7i 0.972200i 0.873903 + 0.486100i \(0.161581\pi\)
−0.873903 + 0.486100i \(0.838419\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.43918e6i 0.204537i 0.994757 + 0.102268i \(0.0326101\pi\)
−0.994757 + 0.102268i \(0.967390\pi\)
\(678\) 0 0
\(679\) −2.48995e6 −0.207260
\(680\) 0 0
\(681\) −2.20168e6 −0.181923
\(682\) 0 0
\(683\) 1.01384e6i 0.0831606i 0.999135 + 0.0415803i \(0.0132392\pi\)
−0.999135 + 0.0415803i \(0.986761\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 7.26924e6i − 0.587621i
\(688\) 0 0
\(689\) −5.21284e6 −0.418337
\(690\) 0 0
\(691\) −8.03186e6 −0.639913 −0.319957 0.947432i \(-0.603668\pi\)
−0.319957 + 0.947432i \(0.603668\pi\)
\(692\) 0 0
\(693\) 1.17370e6i 0.0928376i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 600080.i − 0.0467872i
\(698\) 0 0
\(699\) −3.56983e6 −0.276347
\(700\) 0 0
\(701\) −259806. −0.0199689 −0.00998445 0.999950i \(-0.503178\pi\)
−0.00998445 + 0.999950i \(0.503178\pi\)
\(702\) 0 0
\(703\) − 1.27297e7i − 0.971471i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 7.67030e6i − 0.577117i
\(708\) 0 0
\(709\) 1.92848e7 1.44079 0.720393 0.693566i \(-0.243961\pi\)
0.720393 + 0.693566i \(0.243961\pi\)
\(710\) 0 0
\(711\) 5.53637e6 0.410725
\(712\) 0 0
\(713\) − 1.05755e7i − 0.779071i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.11146e6i 0.226030i
\(718\) 0 0
\(719\) −926119. −0.0668105 −0.0334052 0.999442i \(-0.510635\pi\)
−0.0334052 + 0.999442i \(0.510635\pi\)
\(720\) 0 0
\(721\) 6.73600e6 0.482574
\(722\) 0 0
\(723\) − 1.98425e6i − 0.141173i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 2.02599e7i − 1.42168i −0.703354 0.710840i \(-0.748315\pi\)
0.703354 0.710840i \(-0.251685\pi\)
\(728\) 0 0
\(729\) −211933. −0.0147700
\(730\) 0 0
\(731\) 2.86884e6 0.198570
\(732\) 0 0
\(733\) 1.10982e7i 0.762944i 0.924380 + 0.381472i \(0.124583\pi\)
−0.924380 + 0.381472i \(0.875417\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.44186e6i 0.504676i
\(738\) 0 0
\(739\) −624962. −0.0420962 −0.0210481 0.999778i \(-0.506700\pi\)
−0.0210481 + 0.999778i \(0.506700\pi\)
\(740\) 0 0
\(741\) 4.85184e6 0.324609
\(742\) 0 0
\(743\) 46436.0i 0.00308591i 0.999999 + 0.00154295i \(0.000491137\pi\)
−0.999999 + 0.00154295i \(0.999509\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.50315e7i 0.985601i
\(748\) 0 0
\(749\) 8.46090e6 0.551077
\(750\) 0 0
\(751\) −6.12144e6 −0.396053 −0.198027 0.980197i \(-0.563453\pi\)
−0.198027 + 0.980197i \(0.563453\pi\)
\(752\) 0 0
\(753\) − 5.41665e6i − 0.348131i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.26458e6i 0.207056i 0.994627 + 0.103528i \(0.0330131\pi\)
−0.994627 + 0.103528i \(0.966987\pi\)
\(758\) 0 0
\(759\) −3.04158e6 −0.191644
\(760\) 0 0
\(761\) 1.60311e7 1.00346 0.501732 0.865023i \(-0.332696\pi\)
0.501732 + 0.865023i \(0.332696\pi\)
\(762\) 0 0
\(763\) 1.16603e7i 0.725101i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.20760e7i − 0.741200i
\(768\) 0 0
\(769\) −2.64617e7 −1.61362 −0.806811 0.590810i \(-0.798808\pi\)
−0.806811 + 0.590810i \(0.798808\pi\)
\(770\) 0 0
\(771\) −2.71400e6 −0.164427
\(772\) 0 0
\(773\) − 2.63836e7i − 1.58813i −0.607836 0.794063i \(-0.707962\pi\)
0.607836 0.794063i \(-0.292038\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.44265e6i 0.145147i
\(778\) 0 0
\(779\) −948480. −0.0559995
\(780\) 0 0
\(781\) −1.81149e6 −0.106269
\(782\) 0 0
\(783\) − 2.45699e7i − 1.43218i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.68115e6i 0.326964i 0.986546 + 0.163482i \(0.0522725\pi\)
−0.986546 + 0.163482i \(0.947727\pi\)
\(788\) 0 0
\(789\) 1.38321e6 0.0791037
\(790\) 0 0
\(791\) −4.71645e6 −0.268024
\(792\) 0 0
\(793\) − 1.29793e7i − 0.732939i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9.99383e6i − 0.557296i −0.960393 0.278648i \(-0.910114\pi\)
0.960393 0.278648i \(-0.0898863\pi\)
\(798\) 0 0
\(799\) −7.98568e6 −0.442532
\(800\) 0 0
\(801\) −7.03657e6 −0.387507
\(802\) 0 0
\(803\) − 4.40972e6i − 0.241336i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.83886e6i 0.0993950i
\(808\) 0 0
\(809\) −2.32455e7 −1.24873 −0.624364 0.781134i \(-0.714642\pi\)
−0.624364 + 0.781134i \(0.714642\pi\)
\(810\) 0 0
\(811\) 1.27367e7 0.679991 0.339995 0.940427i \(-0.389574\pi\)
0.339995 + 0.940427i \(0.389574\pi\)
\(812\) 0 0
\(813\) − 1.11348e6i − 0.0590819i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 4.53446e6i − 0.237668i
\(818\) 0 0
\(819\) 3.68600e6 0.192020
\(820\) 0 0
\(821\) −7.85748e6 −0.406842 −0.203421 0.979091i \(-0.565206\pi\)
−0.203421 + 0.979091i \(0.565206\pi\)
\(822\) 0 0
\(823\) − 1.09499e7i − 0.563524i −0.959484 0.281762i \(-0.909081\pi\)
0.959484 0.281762i \(-0.0909188\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 2.20638e7i − 1.12180i −0.827883 0.560901i \(-0.810455\pi\)
0.827883 0.560901i \(-0.189545\pi\)
\(828\) 0 0
\(829\) −7.05255e6 −0.356418 −0.178209 0.983993i \(-0.557030\pi\)
−0.178209 + 0.983993i \(0.557030\pi\)
\(830\) 0 0
\(831\) 9.05698e6 0.454968
\(832\) 0 0
\(833\) 1.65103e7i 0.824407i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 9.00876e6i − 0.444479i
\(838\) 0 0
\(839\) 2.26195e7 1.10937 0.554686 0.832060i \(-0.312838\pi\)
0.554686 + 0.832060i \(0.312838\pi\)
\(840\) 0 0
\(841\) 4.40019e7 2.14527
\(842\) 0 0
\(843\) − 1.24680e7i − 0.604264i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 732050.i 0.0350616i
\(848\) 0 0
\(849\) −1.38795e7 −0.660853
\(850\) 0 0
\(851\) 2.50616e7 1.18627
\(852\) 0 0
\(853\) 9.46645e6i 0.445466i 0.974880 + 0.222733i \(0.0714978\pi\)
−0.974880 + 0.222733i \(0.928502\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 941480.i − 0.0437884i −0.999760 0.0218942i \(-0.993030\pi\)
0.999760 0.0218942i \(-0.00696970\pi\)
\(858\) 0 0
\(859\) 806423. 0.0372889 0.0186445 0.999826i \(-0.494065\pi\)
0.0186445 + 0.999826i \(0.494065\pi\)
\(860\) 0 0
\(861\) 182000. 0.00836688
\(862\) 0 0
\(863\) 1.19485e7i 0.546119i 0.961997 + 0.273059i \(0.0880356\pi\)
−0.961997 + 0.273059i \(0.911964\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 616987.i 0.0278759i
\(868\) 0 0
\(869\) 3.45310e6 0.155117
\(870\) 0 0
\(871\) 2.33711e7 1.04384
\(872\) 0 0
\(873\) 9.66101e6i 0.429029i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 7.84853e6i − 0.344580i −0.985046 0.172290i \(-0.944883\pi\)
0.985046 0.172290i \(-0.0551165\pi\)
\(878\) 0 0
\(879\) −4.04852e6 −0.176736
\(880\) 0 0
\(881\) −1.73933e7 −0.754991 −0.377496 0.926011i \(-0.623215\pi\)
−0.377496 + 0.926011i \(0.623215\pi\)
\(882\) 0 0
\(883\) − 4.31619e7i − 1.86294i −0.363818 0.931470i \(-0.618527\pi\)
0.363818 0.931470i \(-0.381473\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.25652e6i 0.395038i 0.980299 + 0.197519i \(0.0632884\pi\)
−0.980299 + 0.197519i \(0.936712\pi\)
\(888\) 0 0
\(889\) −1.29740e7 −0.550579
\(890\) 0 0
\(891\) 3.11321e6 0.131375
\(892\) 0 0
\(893\) 1.26221e7i 0.529666i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.55206e6i 0.396384i
\(898\) 0 0
\(899\) 2.36542e7 0.976135
\(900\) 0 0
\(901\) 1.58306e7 0.649658
\(902\) 0 0
\(903\) 870100.i 0.0355099i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 4.47293e7i − 1.80540i −0.430270 0.902700i \(-0.641582\pi\)
0.430270 0.902700i \(-0.358418\pi\)
\(908\) 0 0
\(909\) −2.97608e7 −1.19463
\(910\) 0 0
\(911\) 2.57577e7 1.02828 0.514139 0.857707i \(-0.328112\pi\)
0.514139 + 0.857707i \(0.328112\pi\)
\(912\) 0 0
\(913\) 9.37532e6i 0.372228i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.27050e6i − 0.167709i
\(918\) 0 0
\(919\) 3.63488e7 1.41972 0.709858 0.704344i \(-0.248759\pi\)
0.709858 + 0.704344i \(0.248759\pi\)
\(920\) 0 0
\(921\) −2.15322e7 −0.836447
\(922\) 0 0
\(923\) 5.68898e6i 0.219801i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2.61357e7i − 0.998929i
\(928\) 0 0
\(929\) 3.96617e6 0.150776 0.0753880 0.997154i \(-0.475980\pi\)
0.0753880 + 0.997154i \(0.475980\pi\)
\(930\) 0 0
\(931\) 2.60960e7 0.986732
\(932\) 0 0
\(933\) − 2.19630e7i − 0.826014i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.50528e7i 1.30429i 0.758094 + 0.652145i \(0.226131\pi\)
−0.758094 + 0.652145i \(0.773869\pi\)
\(938\) 0 0
\(939\) −1.83249e7 −0.678230
\(940\) 0 0
\(941\) −1.40738e7 −0.518130 −0.259065 0.965860i \(-0.583414\pi\)
−0.259065 + 0.965860i \(0.583414\pi\)
\(942\) 0 0
\(943\) − 1.86732e6i − 0.0683816i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.73759e7i 1.35431i 0.735842 + 0.677153i \(0.236786\pi\)
−0.735842 + 0.677153i \(0.763214\pi\)
\(948\) 0 0
\(949\) −1.38487e7 −0.499165
\(950\) 0 0
\(951\) −1.74454e7 −0.625502
\(952\) 0 0
\(953\) − 3.18424e7i − 1.13572i −0.823124 0.567862i \(-0.807771\pi\)
0.823124 0.567862i \(-0.192229\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 6.80310e6i − 0.240119i
\(958\) 0 0
\(959\) −2.13852e7 −0.750872
\(960\) 0 0
\(961\) −1.99561e7 −0.697056
\(962\) 0 0
\(963\) − 3.28283e7i − 1.14073i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.16276e7i 1.08768i 0.839190 + 0.543838i \(0.183029\pi\)
−0.839190 + 0.543838i \(0.816971\pi\)
\(968\) 0 0
\(969\) −1.47343e7 −0.504103
\(970\) 0 0
\(971\) −2.73412e7 −0.930614 −0.465307 0.885149i \(-0.654056\pi\)
−0.465307 + 0.885149i \(0.654056\pi\)
\(972\) 0 0
\(973\) − 1.54845e7i − 0.524343i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.81630e6i 0.194944i 0.995238 + 0.0974721i \(0.0310757\pi\)
−0.995238 + 0.0974721i \(0.968924\pi\)
\(978\) 0 0
\(979\) −4.38879e6 −0.146348
\(980\) 0 0
\(981\) 4.52420e7 1.50096
\(982\) 0 0
\(983\) − 3.81817e7i − 1.26029i −0.776476 0.630146i \(-0.782995\pi\)
0.776476 0.630146i \(-0.217005\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 2.42200e6i − 0.0791373i
\(988\) 0 0
\(989\) 8.92723e6 0.290219
\(990\) 0 0
\(991\) 5.44564e6 0.176143 0.0880714 0.996114i \(-0.471930\pi\)
0.0880714 + 0.996114i \(0.471930\pi\)
\(992\) 0 0
\(993\) − 1.89087e7i − 0.608541i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.77967e6i 0.120425i 0.998186 + 0.0602125i \(0.0191778\pi\)
−0.998186 + 0.0602125i \(0.980822\pi\)
\(998\) 0 0
\(999\) 2.13488e7 0.676798
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 1100.6.b.a.749.2 2
5.2 odd 4 44.6.a.a.1.1 1
5.3 odd 4 1100.6.a.a.1.1 1
5.4 even 2 inner 1100.6.b.a.749.1 2
15.2 even 4 396.6.a.e.1.1 1
20.7 even 4 176.6.a.a.1.1 1
40.27 even 4 704.6.a.g.1.1 1
40.37 odd 4 704.6.a.d.1.1 1
55.32 even 4 484.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.6.a.a.1.1 1 5.2 odd 4
176.6.a.a.1.1 1 20.7 even 4
396.6.a.e.1.1 1 15.2 even 4
484.6.a.b.1.1 1 55.32 even 4
704.6.a.d.1.1 1 40.37 odd 4
704.6.a.g.1.1 1 40.27 even 4
1100.6.a.a.1.1 1 5.3 odd 4
1100.6.b.a.749.1 2 5.4 even 2 inner
1100.6.b.a.749.2 2 1.1 even 1 trivial