Properties

Label 240.4.f.e.49.2
Level $240$
Weight $4$
Character 240.49
Analytic conductor $14.160$
Analytic rank $0$
Dimension $2$
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] = [240,4,Mod(49,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.f (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: \(14.1604584014\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 240.49
Dual form 240.4.f.e.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +(5.00000 - 10.0000i) q^{5} +4.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +(5.00000 - 10.0000i) q^{5} +4.00000i q^{7} -9.00000 q^{9} +28.0000 q^{11} -16.0000i q^{13} +(30.0000 + 15.0000i) q^{15} -108.000i q^{17} +32.0000 q^{19} -12.0000 q^{21} +28.0000i q^{23} +(-75.0000 - 100.000i) q^{25} -27.0000i q^{27} +238.000 q^{29} +180.000 q^{31} +84.0000i q^{33} +(40.0000 + 20.0000i) q^{35} +40.0000i q^{37} +48.0000 q^{39} +422.000 q^{41} -276.000i q^{43} +(-45.0000 + 90.0000i) q^{45} +60.0000i q^{47} +327.000 q^{49} +324.000 q^{51} +220.000i q^{53} +(140.000 - 280.000i) q^{55} +96.0000i q^{57} -804.000 q^{59} -358.000 q^{61} -36.0000i q^{63} +(-160.000 - 80.0000i) q^{65} -884.000i q^{67} -84.0000 q^{69} +64.0000 q^{71} -152.000i q^{73} +(300.000 - 225.000i) q^{75} +112.000i q^{77} -932.000 q^{79} +81.0000 q^{81} +1292.00i q^{83} +(-1080.00 - 540.000i) q^{85} +714.000i q^{87} +1146.00 q^{89} +64.0000 q^{91} +540.000i q^{93} +(160.000 - 320.000i) q^{95} -824.000i q^{97} -252.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} - 18 q^{9} + 56 q^{11} + 60 q^{15} + 64 q^{19} - 24 q^{21} - 150 q^{25} + 476 q^{29} + 360 q^{31} + 80 q^{35} + 96 q^{39} + 844 q^{41} - 90 q^{45} + 654 q^{49} + 648 q^{51} + 280 q^{55} - 1608 q^{59} - 716 q^{61} - 320 q^{65} - 168 q^{69} + 128 q^{71} + 600 q^{75} - 1864 q^{79} + 162 q^{81} - 2160 q^{85} + 2292 q^{89} + 128 q^{91} + 320 q^{95} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 5.00000 10.0000i 0.447214 0.894427i
\(6\) 0 0
\(7\) 4.00000i 0.215980i 0.994152 + 0.107990i \(0.0344414\pi\)
−0.994152 + 0.107990i \(0.965559\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) 16.0000i 0.341354i −0.985327 0.170677i \(-0.945405\pi\)
0.985327 0.170677i \(-0.0545955\pi\)
\(14\) 0 0
\(15\) 30.0000 + 15.0000i 0.516398 + 0.258199i
\(16\) 0 0
\(17\) 108.000i 1.54081i −0.637552 0.770407i \(-0.720053\pi\)
0.637552 0.770407i \(-0.279947\pi\)
\(18\) 0 0
\(19\) 32.0000 0.386384 0.193192 0.981161i \(-0.438116\pi\)
0.193192 + 0.981161i \(0.438116\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) 28.0000i 0.253844i 0.991913 + 0.126922i \(0.0405097\pi\)
−0.991913 + 0.126922i \(0.959490\pi\)
\(24\) 0 0
\(25\) −75.0000 100.000i −0.600000 0.800000i
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 238.000 1.52398 0.761991 0.647587i \(-0.224222\pi\)
0.761991 + 0.647587i \(0.224222\pi\)
\(30\) 0 0
\(31\) 180.000 1.04287 0.521435 0.853291i \(-0.325397\pi\)
0.521435 + 0.853291i \(0.325397\pi\)
\(32\) 0 0
\(33\) 84.0000i 0.443107i
\(34\) 0 0
\(35\) 40.0000 + 20.0000i 0.193178 + 0.0965891i
\(36\) 0 0
\(37\) 40.0000i 0.177729i 0.996044 + 0.0888643i \(0.0283238\pi\)
−0.996044 + 0.0888643i \(0.971676\pi\)
\(38\) 0 0
\(39\) 48.0000 0.197081
\(40\) 0 0
\(41\) 422.000 1.60745 0.803724 0.595003i \(-0.202849\pi\)
0.803724 + 0.595003i \(0.202849\pi\)
\(42\) 0 0
\(43\) 276.000i 0.978828i −0.872052 0.489414i \(-0.837211\pi\)
0.872052 0.489414i \(-0.162789\pi\)
\(44\) 0 0
\(45\) −45.0000 + 90.0000i −0.149071 + 0.298142i
\(46\) 0 0
\(47\) 60.0000i 0.186211i 0.995656 + 0.0931053i \(0.0296793\pi\)
−0.995656 + 0.0931053i \(0.970321\pi\)
\(48\) 0 0
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) 324.000 0.889590
\(52\) 0 0
\(53\) 220.000i 0.570176i 0.958501 + 0.285088i \(0.0920228\pi\)
−0.958501 + 0.285088i \(0.907977\pi\)
\(54\) 0 0
\(55\) 140.000 280.000i 0.343229 0.686458i
\(56\) 0 0
\(57\) 96.0000i 0.223079i
\(58\) 0 0
\(59\) −804.000 −1.77410 −0.887050 0.461674i \(-0.847249\pi\)
−0.887050 + 0.461674i \(0.847249\pi\)
\(60\) 0 0
\(61\) −358.000 −0.751430 −0.375715 0.926735i \(-0.622603\pi\)
−0.375715 + 0.926735i \(0.622603\pi\)
\(62\) 0 0
\(63\) 36.0000i 0.0719932i
\(64\) 0 0
\(65\) −160.000 80.0000i −0.305316 0.152658i
\(66\) 0 0
\(67\) 884.000i 1.61191i −0.591979 0.805954i \(-0.701653\pi\)
0.591979 0.805954i \(-0.298347\pi\)
\(68\) 0 0
\(69\) −84.0000 −0.146557
\(70\) 0 0
\(71\) 64.0000 0.106978 0.0534888 0.998568i \(-0.482966\pi\)
0.0534888 + 0.998568i \(0.482966\pi\)
\(72\) 0 0
\(73\) 152.000i 0.243702i −0.992548 0.121851i \(-0.961117\pi\)
0.992548 0.121851i \(-0.0388830\pi\)
\(74\) 0 0
\(75\) 300.000 225.000i 0.461880 0.346410i
\(76\) 0 0
\(77\) 112.000i 0.165761i
\(78\) 0 0
\(79\) −932.000 −1.32732 −0.663659 0.748035i \(-0.730998\pi\)
−0.663659 + 0.748035i \(0.730998\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1292.00i 1.70862i 0.519764 + 0.854310i \(0.326020\pi\)
−0.519764 + 0.854310i \(0.673980\pi\)
\(84\) 0 0
\(85\) −1080.00 540.000i −1.37815 0.689073i
\(86\) 0 0
\(87\) 714.000i 0.879872i
\(88\) 0 0
\(89\) 1146.00 1.36490 0.682448 0.730934i \(-0.260915\pi\)
0.682448 + 0.730934i \(0.260915\pi\)
\(90\) 0 0
\(91\) 64.0000 0.0737255
\(92\) 0 0
\(93\) 540.000i 0.602101i
\(94\) 0 0
\(95\) 160.000 320.000i 0.172796 0.345593i
\(96\) 0 0
\(97\) 824.000i 0.862521i −0.902227 0.431260i \(-0.858069\pi\)
0.902227 0.431260i \(-0.141931\pi\)
\(98\) 0 0
\(99\) −252.000 −0.255828
\(100\) 0 0
\(101\) −1290.00 −1.27089 −0.635445 0.772147i \(-0.719183\pi\)
−0.635445 + 0.772147i \(0.719183\pi\)
\(102\) 0 0
\(103\) 1604.00i 1.53444i 0.641387 + 0.767218i \(0.278359\pi\)
−0.641387 + 0.767218i \(0.721641\pi\)
\(104\) 0 0
\(105\) −60.0000 + 120.000i −0.0557657 + 0.111531i
\(106\) 0 0
\(107\) 892.000i 0.805915i 0.915219 + 0.402957i \(0.132018\pi\)
−0.915219 + 0.402957i \(0.867982\pi\)
\(108\) 0 0
\(109\) −966.000 −0.848863 −0.424431 0.905460i \(-0.639526\pi\)
−0.424431 + 0.905460i \(0.639526\pi\)
\(110\) 0 0
\(111\) −120.000 −0.102612
\(112\) 0 0
\(113\) 1124.00i 0.935726i 0.883801 + 0.467863i \(0.154976\pi\)
−0.883801 + 0.467863i \(0.845024\pi\)
\(114\) 0 0
\(115\) 280.000 + 140.000i 0.227045 + 0.113522i
\(116\) 0 0
\(117\) 144.000i 0.113785i
\(118\) 0 0
\(119\) 432.000 0.332785
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) 1266.00i 0.928060i
\(124\) 0 0
\(125\) −1375.00 + 250.000i −0.983870 + 0.178885i
\(126\) 0 0
\(127\) 1884.00i 1.31636i −0.752860 0.658181i \(-0.771326\pi\)
0.752860 0.658181i \(-0.228674\pi\)
\(128\) 0 0
\(129\) 828.000 0.565127
\(130\) 0 0
\(131\) 588.000 0.392166 0.196083 0.980587i \(-0.437178\pi\)
0.196083 + 0.980587i \(0.437178\pi\)
\(132\) 0 0
\(133\) 128.000i 0.0834512i
\(134\) 0 0
\(135\) −270.000 135.000i −0.172133 0.0860663i
\(136\) 0 0
\(137\) 1060.00i 0.661036i −0.943800 0.330518i \(-0.892777\pi\)
0.943800 0.330518i \(-0.107223\pi\)
\(138\) 0 0
\(139\) −2864.00 −1.74764 −0.873818 0.486254i \(-0.838363\pi\)
−0.873818 + 0.486254i \(0.838363\pi\)
\(140\) 0 0
\(141\) −180.000 −0.107509
\(142\) 0 0
\(143\) 448.000i 0.261984i
\(144\) 0 0
\(145\) 1190.00 2380.00i 0.681546 1.36309i
\(146\) 0 0
\(147\) 981.000i 0.550418i
\(148\) 0 0
\(149\) −342.000 −0.188038 −0.0940192 0.995570i \(-0.529972\pi\)
−0.0940192 + 0.995570i \(0.529972\pi\)
\(150\) 0 0
\(151\) −1636.00 −0.881694 −0.440847 0.897582i \(-0.645322\pi\)
−0.440847 + 0.897582i \(0.645322\pi\)
\(152\) 0 0
\(153\) 972.000i 0.513605i
\(154\) 0 0
\(155\) 900.000 1800.00i 0.466385 0.932771i
\(156\) 0 0
\(157\) 2072.00i 1.05327i 0.850091 + 0.526636i \(0.176547\pi\)
−0.850091 + 0.526636i \(0.823453\pi\)
\(158\) 0 0
\(159\) −660.000 −0.329191
\(160\) 0 0
\(161\) −112.000 −0.0548251
\(162\) 0 0
\(163\) 772.000i 0.370968i −0.982647 0.185484i \(-0.940615\pi\)
0.982647 0.185484i \(-0.0593852\pi\)
\(164\) 0 0
\(165\) 840.000 + 420.000i 0.396327 + 0.198163i
\(166\) 0 0
\(167\) 1044.00i 0.483755i −0.970307 0.241878i \(-0.922237\pi\)
0.970307 0.241878i \(-0.0777633\pi\)
\(168\) 0 0
\(169\) 1941.00 0.883477
\(170\) 0 0
\(171\) −288.000 −0.128795
\(172\) 0 0
\(173\) 4404.00i 1.93543i 0.252041 + 0.967717i \(0.418898\pi\)
−0.252041 + 0.967717i \(0.581102\pi\)
\(174\) 0 0
\(175\) 400.000 300.000i 0.172784 0.129588i
\(176\) 0 0
\(177\) 2412.00i 1.02428i
\(178\) 0 0
\(179\) 3452.00 1.44142 0.720711 0.693235i \(-0.243815\pi\)
0.720711 + 0.693235i \(0.243815\pi\)
\(180\) 0 0
\(181\) 526.000 0.216007 0.108004 0.994151i \(-0.465554\pi\)
0.108004 + 0.994151i \(0.465554\pi\)
\(182\) 0 0
\(183\) 1074.00i 0.433838i
\(184\) 0 0
\(185\) 400.000 + 200.000i 0.158965 + 0.0794827i
\(186\) 0 0
\(187\) 3024.00i 1.18255i
\(188\) 0 0
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) −72.0000 −0.0272761 −0.0136381 0.999907i \(-0.504341\pi\)
−0.0136381 + 0.999907i \(0.504341\pi\)
\(192\) 0 0
\(193\) 208.000i 0.0775760i 0.999247 + 0.0387880i \(0.0123497\pi\)
−0.999247 + 0.0387880i \(0.987650\pi\)
\(194\) 0 0
\(195\) 240.000 480.000i 0.0881372 0.176274i
\(196\) 0 0
\(197\) 372.000i 0.134538i −0.997735 0.0672688i \(-0.978571\pi\)
0.997735 0.0672688i \(-0.0214285\pi\)
\(198\) 0 0
\(199\) 4348.00 1.54885 0.774426 0.632665i \(-0.218039\pi\)
0.774426 + 0.632665i \(0.218039\pi\)
\(200\) 0 0
\(201\) 2652.00 0.930635
\(202\) 0 0
\(203\) 952.000i 0.329149i
\(204\) 0 0
\(205\) 2110.00 4220.00i 0.718872 1.43774i
\(206\) 0 0
\(207\) 252.000i 0.0846145i
\(208\) 0 0
\(209\) 896.000 0.296544
\(210\) 0 0
\(211\) 416.000 0.135728 0.0678640 0.997695i \(-0.478382\pi\)
0.0678640 + 0.997695i \(0.478382\pi\)
\(212\) 0 0
\(213\) 192.000i 0.0617635i
\(214\) 0 0
\(215\) −2760.00 1380.00i −0.875490 0.437745i
\(216\) 0 0
\(217\) 720.000i 0.225239i
\(218\) 0 0
\(219\) 456.000 0.140701
\(220\) 0 0
\(221\) −1728.00 −0.525963
\(222\) 0 0
\(223\) 5748.00i 1.72607i 0.505141 + 0.863037i \(0.331441\pi\)
−0.505141 + 0.863037i \(0.668559\pi\)
\(224\) 0 0
\(225\) 675.000 + 900.000i 0.200000 + 0.266667i
\(226\) 0 0
\(227\) 1148.00i 0.335663i 0.985816 + 0.167831i \(0.0536764\pi\)
−0.985816 + 0.167831i \(0.946324\pi\)
\(228\) 0 0
\(229\) −3234.00 −0.933226 −0.466613 0.884462i \(-0.654526\pi\)
−0.466613 + 0.884462i \(0.654526\pi\)
\(230\) 0 0
\(231\) −336.000 −0.0957021
\(232\) 0 0
\(233\) 228.000i 0.0641063i −0.999486 0.0320532i \(-0.989795\pi\)
0.999486 0.0320532i \(-0.0102046\pi\)
\(234\) 0 0
\(235\) 600.000 + 300.000i 0.166552 + 0.0832759i
\(236\) 0 0
\(237\) 2796.00i 0.766328i
\(238\) 0 0
\(239\) 4760.00 1.28828 0.644140 0.764908i \(-0.277216\pi\)
0.644140 + 0.764908i \(0.277216\pi\)
\(240\) 0 0
\(241\) 3230.00 0.863330 0.431665 0.902034i \(-0.357926\pi\)
0.431665 + 0.902034i \(0.357926\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 1635.00 3270.00i 0.426352 0.852705i
\(246\) 0 0
\(247\) 512.000i 0.131894i
\(248\) 0 0
\(249\) −3876.00 −0.986472
\(250\) 0 0
\(251\) −1708.00 −0.429514 −0.214757 0.976668i \(-0.568896\pi\)
−0.214757 + 0.976668i \(0.568896\pi\)
\(252\) 0 0
\(253\) 784.000i 0.194821i
\(254\) 0 0
\(255\) 1620.00 3240.00i 0.397837 0.795673i
\(256\) 0 0
\(257\) 6372.00i 1.54659i −0.634044 0.773297i \(-0.718606\pi\)
0.634044 0.773297i \(-0.281394\pi\)
\(258\) 0 0
\(259\) −160.000 −0.0383858
\(260\) 0 0
\(261\) −2142.00 −0.507994
\(262\) 0 0
\(263\) 3036.00i 0.711817i 0.934521 + 0.355908i \(0.115828\pi\)
−0.934521 + 0.355908i \(0.884172\pi\)
\(264\) 0 0
\(265\) 2200.00 + 1100.00i 0.509981 + 0.254990i
\(266\) 0 0
\(267\) 3438.00i 0.788023i
\(268\) 0 0
\(269\) 114.000 0.0258390 0.0129195 0.999917i \(-0.495887\pi\)
0.0129195 + 0.999917i \(0.495887\pi\)
\(270\) 0 0
\(271\) 5236.00 1.17367 0.586835 0.809707i \(-0.300374\pi\)
0.586835 + 0.809707i \(0.300374\pi\)
\(272\) 0 0
\(273\) 192.000i 0.0425655i
\(274\) 0 0
\(275\) −2100.00 2800.00i −0.460490 0.613987i
\(276\) 0 0
\(277\) 5712.00i 1.23899i 0.785000 + 0.619496i \(0.212663\pi\)
−0.785000 + 0.619496i \(0.787337\pi\)
\(278\) 0 0
\(279\) −1620.00 −0.347623
\(280\) 0 0
\(281\) 3222.00 0.684016 0.342008 0.939697i \(-0.388893\pi\)
0.342008 + 0.939697i \(0.388893\pi\)
\(282\) 0 0
\(283\) 4620.00i 0.970426i 0.874396 + 0.485213i \(0.161258\pi\)
−0.874396 + 0.485213i \(0.838742\pi\)
\(284\) 0 0
\(285\) 960.000 + 480.000i 0.199528 + 0.0997640i
\(286\) 0 0
\(287\) 1688.00i 0.347176i
\(288\) 0 0
\(289\) −6751.00 −1.37411
\(290\) 0 0
\(291\) 2472.00 0.497977
\(292\) 0 0
\(293\) 5404.00i 1.07749i −0.842468 0.538746i \(-0.818898\pi\)
0.842468 0.538746i \(-0.181102\pi\)
\(294\) 0 0
\(295\) −4020.00 + 8040.00i −0.793402 + 1.58680i
\(296\) 0 0
\(297\) 756.000i 0.147702i
\(298\) 0 0
\(299\) 448.000 0.0866505
\(300\) 0 0
\(301\) 1104.00 0.211407
\(302\) 0 0
\(303\) 3870.00i 0.733748i
\(304\) 0 0
\(305\) −1790.00 + 3580.00i −0.336050 + 0.672099i
\(306\) 0 0
\(307\) 9700.00i 1.80328i −0.432483 0.901642i \(-0.642362\pi\)
0.432483 0.901642i \(-0.357638\pi\)
\(308\) 0 0
\(309\) −4812.00 −0.885907
\(310\) 0 0
\(311\) −9672.00 −1.76350 −0.881750 0.471716i \(-0.843635\pi\)
−0.881750 + 0.471716i \(0.843635\pi\)
\(312\) 0 0
\(313\) 4048.00i 0.731011i 0.930809 + 0.365506i \(0.119104\pi\)
−0.930809 + 0.365506i \(0.880896\pi\)
\(314\) 0 0
\(315\) −360.000 180.000i −0.0643927 0.0321964i
\(316\) 0 0
\(317\) 84.0000i 0.0148830i 0.999972 + 0.00744150i \(0.00236872\pi\)
−0.999972 + 0.00744150i \(0.997631\pi\)
\(318\) 0 0
\(319\) 6664.00 1.16963
\(320\) 0 0
\(321\) −2676.00 −0.465295
\(322\) 0 0
\(323\) 3456.00i 0.595347i
\(324\) 0 0
\(325\) −1600.00 + 1200.00i −0.273083 + 0.204812i
\(326\) 0 0
\(327\) 2898.00i 0.490091i
\(328\) 0 0
\(329\) −240.000 −0.0402177
\(330\) 0 0
\(331\) −5416.00 −0.899366 −0.449683 0.893188i \(-0.648463\pi\)
−0.449683 + 0.893188i \(0.648463\pi\)
\(332\) 0 0
\(333\) 360.000i 0.0592429i
\(334\) 0 0
\(335\) −8840.00 4420.00i −1.44173 0.720867i
\(336\) 0 0
\(337\) 8216.00i 1.32805i 0.747709 + 0.664027i \(0.231154\pi\)
−0.747709 + 0.664027i \(0.768846\pi\)
\(338\) 0 0
\(339\) −3372.00 −0.540242
\(340\) 0 0
\(341\) 5040.00 0.800385
\(342\) 0 0
\(343\) 2680.00i 0.421885i
\(344\) 0 0
\(345\) −420.000 + 840.000i −0.0655421 + 0.131084i
\(346\) 0 0
\(347\) 3836.00i 0.593450i 0.954963 + 0.296725i \(0.0958945\pi\)
−0.954963 + 0.296725i \(0.904105\pi\)
\(348\) 0 0
\(349\) 2038.00 0.312583 0.156292 0.987711i \(-0.450046\pi\)
0.156292 + 0.987711i \(0.450046\pi\)
\(350\) 0 0
\(351\) −432.000 −0.0656936
\(352\) 0 0
\(353\) 5292.00i 0.797917i 0.916969 + 0.398959i \(0.130628\pi\)
−0.916969 + 0.398959i \(0.869372\pi\)
\(354\) 0 0
\(355\) 320.000 640.000i 0.0478418 0.0956836i
\(356\) 0 0
\(357\) 1296.00i 0.192133i
\(358\) 0 0
\(359\) −3896.00 −0.572766 −0.286383 0.958115i \(-0.592453\pi\)
−0.286383 + 0.958115i \(0.592453\pi\)
\(360\) 0 0
\(361\) −5835.00 −0.850707
\(362\) 0 0
\(363\) 1641.00i 0.237273i
\(364\) 0 0
\(365\) −1520.00 760.000i −0.217974 0.108987i
\(366\) 0 0
\(367\) 7652.00i 1.08837i 0.838966 + 0.544184i \(0.183161\pi\)
−0.838966 + 0.544184i \(0.816839\pi\)
\(368\) 0 0
\(369\) −3798.00 −0.535816
\(370\) 0 0
\(371\) −880.000 −0.123146
\(372\) 0 0
\(373\) 1576.00i 0.218773i −0.993999 0.109386i \(-0.965111\pi\)
0.993999 0.109386i \(-0.0348886\pi\)
\(374\) 0 0
\(375\) −750.000 4125.00i −0.103280 0.568038i
\(376\) 0 0
\(377\) 3808.00i 0.520217i
\(378\) 0 0
\(379\) −5416.00 −0.734040 −0.367020 0.930213i \(-0.619622\pi\)
−0.367020 + 0.930213i \(0.619622\pi\)
\(380\) 0 0
\(381\) 5652.00 0.760002
\(382\) 0 0
\(383\) 8292.00i 1.10627i −0.833092 0.553135i \(-0.813431\pi\)
0.833092 0.553135i \(-0.186569\pi\)
\(384\) 0 0
\(385\) 1120.00 + 560.000i 0.148261 + 0.0741305i
\(386\) 0 0
\(387\) 2484.00i 0.326276i
\(388\) 0 0
\(389\) 9642.00 1.25673 0.628366 0.777918i \(-0.283724\pi\)
0.628366 + 0.777918i \(0.283724\pi\)
\(390\) 0 0
\(391\) 3024.00 0.391126
\(392\) 0 0
\(393\) 1764.00i 0.226417i
\(394\) 0 0
\(395\) −4660.00 + 9320.00i −0.593595 + 1.18719i
\(396\) 0 0
\(397\) 13032.0i 1.64750i 0.566954 + 0.823750i \(0.308122\pi\)
−0.566954 + 0.823750i \(0.691878\pi\)
\(398\) 0 0
\(399\) −384.000 −0.0481806
\(400\) 0 0
\(401\) −13358.0 −1.66351 −0.831754 0.555144i \(-0.812663\pi\)
−0.831754 + 0.555144i \(0.812663\pi\)
\(402\) 0 0
\(403\) 2880.00i 0.355988i
\(404\) 0 0
\(405\) 405.000 810.000i 0.0496904 0.0993808i
\(406\) 0 0
\(407\) 1120.00i 0.136404i
\(408\) 0 0
\(409\) −6410.00 −0.774949 −0.387474 0.921880i \(-0.626652\pi\)
−0.387474 + 0.921880i \(0.626652\pi\)
\(410\) 0 0
\(411\) 3180.00 0.381649
\(412\) 0 0
\(413\) 3216.00i 0.383170i
\(414\) 0 0
\(415\) 12920.0 + 6460.00i 1.52824 + 0.764118i
\(416\) 0 0
\(417\) 8592.00i 1.00900i
\(418\) 0 0
\(419\) 7644.00 0.891250 0.445625 0.895220i \(-0.352981\pi\)
0.445625 + 0.895220i \(0.352981\pi\)
\(420\) 0 0
\(421\) 14674.0 1.69873 0.849367 0.527803i \(-0.176984\pi\)
0.849367 + 0.527803i \(0.176984\pi\)
\(422\) 0 0
\(423\) 540.000i 0.0620702i
\(424\) 0 0
\(425\) −10800.0 + 8100.00i −1.23265 + 0.924489i
\(426\) 0 0
\(427\) 1432.00i 0.162294i
\(428\) 0 0
\(429\) 1344.00 0.151256
\(430\) 0 0
\(431\) 9704.00 1.08451 0.542257 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(432\) 0 0
\(433\) 1296.00i 0.143838i 0.997410 + 0.0719189i \(0.0229123\pi\)
−0.997410 + 0.0719189i \(0.977088\pi\)
\(434\) 0 0
\(435\) 7140.00 + 3570.00i 0.786981 + 0.393491i
\(436\) 0 0
\(437\) 896.000i 0.0980812i
\(438\) 0 0
\(439\) −15684.0 −1.70514 −0.852570 0.522613i \(-0.824957\pi\)
−0.852570 + 0.522613i \(0.824957\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) 5772.00i 0.619043i 0.950892 + 0.309521i \(0.100169\pi\)
−0.950892 + 0.309521i \(0.899831\pi\)
\(444\) 0 0
\(445\) 5730.00 11460.0i 0.610400 1.22080i
\(446\) 0 0
\(447\) 1026.00i 0.108564i
\(448\) 0 0
\(449\) 4782.00 0.502620 0.251310 0.967907i \(-0.419139\pi\)
0.251310 + 0.967907i \(0.419139\pi\)
\(450\) 0 0
\(451\) 11816.0 1.23369
\(452\) 0 0
\(453\) 4908.00i 0.509046i
\(454\) 0 0
\(455\) 320.000 640.000i 0.0329711 0.0659421i
\(456\) 0 0
\(457\) 15000.0i 1.53538i −0.640819 0.767692i \(-0.721405\pi\)
0.640819 0.767692i \(-0.278595\pi\)
\(458\) 0 0
\(459\) −2916.00 −0.296530
\(460\) 0 0
\(461\) −3762.00 −0.380073 −0.190037 0.981777i \(-0.560861\pi\)
−0.190037 + 0.981777i \(0.560861\pi\)
\(462\) 0 0
\(463\) 5036.00i 0.505492i −0.967533 0.252746i \(-0.918666\pi\)
0.967533 0.252746i \(-0.0813337\pi\)
\(464\) 0 0
\(465\) 5400.00 + 2700.00i 0.538535 + 0.269268i
\(466\) 0 0
\(467\) 2268.00i 0.224733i 0.993667 + 0.112367i \(0.0358431\pi\)
−0.993667 + 0.112367i \(0.964157\pi\)
\(468\) 0 0
\(469\) 3536.00 0.348139
\(470\) 0 0
\(471\) −6216.00 −0.608106
\(472\) 0 0
\(473\) 7728.00i 0.751234i
\(474\) 0 0
\(475\) −2400.00 3200.00i −0.231831 0.309108i
\(476\) 0 0
\(477\) 1980.00i 0.190059i
\(478\) 0 0
\(479\) 16208.0 1.54606 0.773030 0.634370i \(-0.218740\pi\)
0.773030 + 0.634370i \(0.218740\pi\)
\(480\) 0 0
\(481\) 640.000 0.0606684
\(482\) 0 0
\(483\) 336.000i 0.0316533i
\(484\) 0 0
\(485\) −8240.00 4120.00i −0.771462 0.385731i
\(486\) 0 0
\(487\) 11572.0i 1.07675i 0.842705 + 0.538375i \(0.180962\pi\)
−0.842705 + 0.538375i \(0.819038\pi\)
\(488\) 0 0
\(489\) 2316.00 0.214178
\(490\) 0 0
\(491\) −5636.00 −0.518023 −0.259011 0.965874i \(-0.583397\pi\)
−0.259011 + 0.965874i \(0.583397\pi\)
\(492\) 0 0
\(493\) 25704.0i 2.34817i
\(494\) 0 0
\(495\) −1260.00 + 2520.00i −0.114410 + 0.228819i
\(496\) 0 0
\(497\) 256.000i 0.0231050i
\(498\) 0 0
\(499\) −5560.00 −0.498797 −0.249399 0.968401i \(-0.580233\pi\)
−0.249399 + 0.968401i \(0.580233\pi\)
\(500\) 0 0
\(501\) 3132.00 0.279296
\(502\) 0 0
\(503\) 15172.0i 1.34490i −0.740141 0.672451i \(-0.765241\pi\)
0.740141 0.672451i \(-0.234759\pi\)
\(504\) 0 0
\(505\) −6450.00 + 12900.0i −0.568359 + 1.13672i
\(506\) 0 0
\(507\) 5823.00i 0.510076i
\(508\) 0 0
\(509\) 17342.0 1.51016 0.755079 0.655634i \(-0.227598\pi\)
0.755079 + 0.655634i \(0.227598\pi\)
\(510\) 0 0
\(511\) 608.000 0.0526347
\(512\) 0 0
\(513\) 864.000i 0.0743597i
\(514\) 0 0
\(515\) 16040.0 + 8020.00i 1.37244 + 0.686220i
\(516\) 0 0
\(517\) 1680.00i 0.142914i
\(518\) 0 0
\(519\) −13212.0 −1.11742
\(520\) 0 0
\(521\) −4886.00 −0.410863 −0.205431 0.978672i \(-0.565860\pi\)
−0.205431 + 0.978672i \(0.565860\pi\)
\(522\) 0 0
\(523\) 18548.0i 1.55076i −0.631495 0.775380i \(-0.717558\pi\)
0.631495 0.775380i \(-0.282442\pi\)
\(524\) 0 0
\(525\) 900.000 + 1200.00i 0.0748176 + 0.0997567i
\(526\) 0 0
\(527\) 19440.0i 1.60687i
\(528\) 0 0
\(529\) 11383.0 0.935563
\(530\) 0 0
\(531\) 7236.00 0.591367
\(532\) 0 0
\(533\) 6752.00i 0.548708i
\(534\) 0 0
\(535\) 8920.00 + 4460.00i 0.720832 + 0.360416i
\(536\) 0 0
\(537\) 10356.0i 0.832206i
\(538\) 0 0
\(539\) 9156.00 0.731682
\(540\) 0 0
\(541\) −15770.0 −1.25324 −0.626622 0.779323i \(-0.715563\pi\)
−0.626622 + 0.779323i \(0.715563\pi\)
\(542\) 0 0
\(543\) 1578.00i 0.124712i
\(544\) 0 0
\(545\) −4830.00 + 9660.00i −0.379623 + 0.759246i
\(546\) 0 0
\(547\) 7700.00i 0.601880i −0.953643 0.300940i \(-0.902700\pi\)
0.953643 0.300940i \(-0.0973004\pi\)
\(548\) 0 0
\(549\) 3222.00 0.250477
\(550\) 0 0
\(551\) 7616.00 0.588843
\(552\) 0 0
\(553\) 3728.00i 0.286674i
\(554\) 0 0
\(555\) −600.000 + 1200.00i −0.0458893 + 0.0917787i
\(556\) 0 0
\(557\) 19236.0i 1.46330i 0.681683 + 0.731648i \(0.261248\pi\)
−0.681683 + 0.731648i \(0.738752\pi\)
\(558\) 0 0
\(559\) −4416.00 −0.334127
\(560\) 0 0
\(561\) 9072.00 0.682745
\(562\) 0 0
\(563\) 8388.00i 0.627908i −0.949438 0.313954i \(-0.898346\pi\)
0.949438 0.313954i \(-0.101654\pi\)
\(564\) 0 0
\(565\) 11240.0 + 5620.00i 0.836939 + 0.418469i
\(566\) 0 0
\(567\) 324.000i 0.0239977i
\(568\) 0 0
\(569\) 16758.0 1.23468 0.617339 0.786697i \(-0.288211\pi\)
0.617339 + 0.786697i \(0.288211\pi\)
\(570\) 0 0
\(571\) −8056.00 −0.590426 −0.295213 0.955432i \(-0.595391\pi\)
−0.295213 + 0.955432i \(0.595391\pi\)
\(572\) 0 0
\(573\) 216.000i 0.0157479i
\(574\) 0 0
\(575\) 2800.00 2100.00i 0.203075 0.152306i
\(576\) 0 0
\(577\) 5728.00i 0.413275i −0.978418 0.206638i \(-0.933748\pi\)
0.978418 0.206638i \(-0.0662521\pi\)
\(578\) 0 0
\(579\) −624.000 −0.0447885
\(580\) 0 0
\(581\) −5168.00 −0.369027
\(582\) 0 0
\(583\) 6160.00i 0.437601i
\(584\) 0 0
\(585\) 1440.00 + 720.000i 0.101772 + 0.0508860i
\(586\) 0 0
\(587\) 12172.0i 0.855864i 0.903811 + 0.427932i \(0.140758\pi\)
−0.903811 + 0.427932i \(0.859242\pi\)
\(588\) 0 0
\(589\) 5760.00 0.402948
\(590\) 0 0
\(591\) 1116.00 0.0776753
\(592\) 0 0
\(593\) 10708.0i 0.741526i 0.928728 + 0.370763i \(0.120904\pi\)
−0.928728 + 0.370763i \(0.879096\pi\)
\(594\) 0 0
\(595\) 2160.00 4320.00i 0.148826 0.297652i
\(596\) 0 0
\(597\) 13044.0i 0.894230i
\(598\) 0 0
\(599\) −9416.00 −0.642283 −0.321141 0.947031i \(-0.604066\pi\)
−0.321141 + 0.947031i \(0.604066\pi\)
\(600\) 0 0
\(601\) 9270.00 0.629170 0.314585 0.949229i \(-0.398135\pi\)
0.314585 + 0.949229i \(0.398135\pi\)
\(602\) 0 0
\(603\) 7956.00i 0.537302i
\(604\) 0 0
\(605\) −2735.00 + 5470.00i −0.183791 + 0.367582i
\(606\) 0 0
\(607\) 7996.00i 0.534675i −0.963603 0.267337i \(-0.913856\pi\)
0.963603 0.267337i \(-0.0861438\pi\)
\(608\) 0 0
\(609\) −2856.00 −0.190034
\(610\) 0 0
\(611\) 960.000 0.0635637
\(612\) 0 0
\(613\) 232.000i 0.0152861i −0.999971 0.00764306i \(-0.997567\pi\)
0.999971 0.00764306i \(-0.00243289\pi\)
\(614\) 0 0
\(615\) 12660.0 + 6330.00i 0.830082 + 0.415041i
\(616\) 0 0
\(617\) 3740.00i 0.244030i −0.992528 0.122015i \(-0.961064\pi\)
0.992528 0.122015i \(-0.0389357\pi\)
\(618\) 0 0
\(619\) −26000.0 −1.68825 −0.844126 0.536145i \(-0.819880\pi\)
−0.844126 + 0.536145i \(0.819880\pi\)
\(620\) 0 0
\(621\) 756.000 0.0488522
\(622\) 0 0
\(623\) 4584.00i 0.294790i
\(624\) 0 0
\(625\) −4375.00 + 15000.0i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 2688.00i 0.171210i
\(628\) 0 0
\(629\) 4320.00 0.273847
\(630\) 0 0
\(631\) −11660.0 −0.735622 −0.367811 0.929901i \(-0.619893\pi\)
−0.367811 + 0.929901i \(0.619893\pi\)
\(632\) 0 0
\(633\) 1248.00i 0.0783626i
\(634\) 0 0
\(635\) −18840.0 9420.00i −1.17739 0.588695i
\(636\) 0 0
\(637\) 5232.00i 0.325431i
\(638\) 0 0
\(639\) −576.000 −0.0356592
\(640\) 0 0
\(641\) −7602.00 −0.468426 −0.234213 0.972185i \(-0.575251\pi\)
−0.234213 + 0.972185i \(0.575251\pi\)
\(642\) 0 0
\(643\) 29268.0i 1.79505i −0.440963 0.897525i \(-0.645363\pi\)
0.440963 0.897525i \(-0.354637\pi\)
\(644\) 0 0
\(645\) 4140.00 8280.00i 0.252732 0.505465i
\(646\) 0 0
\(647\) 17836.0i 1.08378i 0.840449 + 0.541890i \(0.182291\pi\)
−0.840449 + 0.541890i \(0.817709\pi\)
\(648\) 0 0
\(649\) −22512.0 −1.36159
\(650\) 0 0
\(651\) −2160.00 −0.130042
\(652\) 0 0
\(653\) 19188.0i 1.14990i −0.818189 0.574950i \(-0.805022\pi\)
0.818189 0.574950i \(-0.194978\pi\)
\(654\) 0 0
\(655\) 2940.00 5880.00i 0.175382 0.350764i
\(656\) 0 0
\(657\) 1368.00i 0.0812340i
\(658\) 0 0
\(659\) 13860.0 0.819285 0.409643 0.912246i \(-0.365653\pi\)
0.409643 + 0.912246i \(0.365653\pi\)
\(660\) 0 0
\(661\) −16558.0 −0.974329 −0.487165 0.873310i \(-0.661969\pi\)
−0.487165 + 0.873310i \(0.661969\pi\)
\(662\) 0 0
\(663\) 5184.00i 0.303665i
\(664\) 0 0
\(665\) 1280.00 + 640.000i 0.0746410 + 0.0373205i
\(666\) 0 0
\(667\) 6664.00i 0.386853i
\(668\) 0 0
\(669\) −17244.0 −0.996549
\(670\) 0 0
\(671\) −10024.0 −0.576710
\(672\) 0 0
\(673\) 4640.00i 0.265764i −0.991132 0.132882i \(-0.957577\pi\)
0.991132 0.132882i \(-0.0424231\pi\)
\(674\) 0 0
\(675\) −2700.00 + 2025.00i −0.153960 + 0.115470i
\(676\) 0 0
\(677\) 34260.0i 1.94493i 0.233045 + 0.972466i \(0.425131\pi\)
−0.233045 + 0.972466i \(0.574869\pi\)
\(678\) 0 0
\(679\) 3296.00 0.186287
\(680\) 0 0
\(681\) −3444.00 −0.193795
\(682\) 0 0
\(683\) 19420.0i 1.08797i 0.839094 + 0.543987i \(0.183086\pi\)
−0.839094 + 0.543987i \(0.816914\pi\)
\(684\) 0 0
\(685\) −10600.0 5300.00i −0.591248 0.295624i
\(686\) 0 0
\(687\) 9702.00i 0.538798i
\(688\) 0 0
\(689\) 3520.00 0.194632
\(690\) 0 0
\(691\) −4608.00 −0.253685 −0.126843 0.991923i \(-0.540484\pi\)
−0.126843 + 0.991923i \(0.540484\pi\)
\(692\) 0 0
\(693\) 1008.00i 0.0552536i
\(694\) 0 0
\(695\) −14320.0 + 28640.0i −0.781566 + 1.56313i
\(696\) 0 0
\(697\) 45576.0i 2.47678i
\(698\) 0 0
\(699\) 684.000 0.0370118
\(700\) 0 0
\(701\) −2318.00 −0.124893 −0.0624463 0.998048i \(-0.519890\pi\)
−0.0624463 + 0.998048i \(0.519890\pi\)
\(702\) 0 0
\(703\) 1280.00i 0.0686716i
\(704\) 0 0
\(705\) −900.000 + 1800.00i −0.0480794 + 0.0961588i
\(706\) 0 0
\(707\) 5160.00i 0.274486i
\(708\) 0 0
\(709\) −16834.0 −0.891698 −0.445849 0.895108i \(-0.647098\pi\)
−0.445849 + 0.895108i \(0.647098\pi\)
\(710\) 0 0
\(711\) 8388.00 0.442440
\(712\) 0 0
\(713\) 5040.00i 0.264726i
\(714\) 0 0
\(715\) −4480.00 2240.00i −0.234325 0.117163i
\(716\) 0 0
\(717\) 14280.0i 0.743789i
\(718\) 0 0
\(719\) −7400.00 −0.383830 −0.191915 0.981412i \(-0.561470\pi\)
−0.191915 + 0.981412i \(0.561470\pi\)
\(720\) 0 0
\(721\) −6416.00 −0.331407
\(722\) 0 0
\(723\) 9690.00i 0.498444i
\(724\) 0 0
\(725\) −17850.0 23800.0i −0.914389 1.21919i
\(726\) 0 0
\(727\) 20340.0i 1.03765i 0.854882 + 0.518823i \(0.173630\pi\)
−0.854882 + 0.518823i \(0.826370\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −29808.0 −1.50819
\(732\) 0 0
\(733\) 4896.00i 0.246709i 0.992363 + 0.123355i \(0.0393653\pi\)
−0.992363 + 0.123355i \(0.960635\pi\)
\(734\) 0 0
\(735\) 9810.00 + 4905.00i 0.492309 + 0.246155i
\(736\) 0 0
\(737\) 24752.0i 1.23711i
\(738\) 0 0
\(739\) 26040.0 1.29621 0.648103 0.761552i \(-0.275562\pi\)
0.648103 + 0.761552i \(0.275562\pi\)
\(740\) 0 0
\(741\) 1536.00 0.0761489
\(742\) 0 0
\(743\) 6780.00i 0.334770i 0.985892 + 0.167385i \(0.0535323\pi\)
−0.985892 + 0.167385i \(0.946468\pi\)
\(744\) 0 0
\(745\) −1710.00 + 3420.00i −0.0840934 + 0.168187i
\(746\) 0 0
\(747\) 11628.0i 0.569540i
\(748\) 0 0
\(749\) −3568.00 −0.174061
\(750\) 0 0
\(751\) 20692.0 1.00541 0.502704 0.864458i \(-0.332338\pi\)
0.502704 + 0.864458i \(0.332338\pi\)
\(752\) 0 0
\(753\) 5124.00i 0.247980i
\(754\) 0 0
\(755\) −8180.00 + 16360.0i −0.394306 + 0.788611i
\(756\) 0 0
\(757\) 10816.0i 0.519305i 0.965702 + 0.259653i \(0.0836081\pi\)
−0.965702 + 0.259653i \(0.916392\pi\)
\(758\) 0 0
\(759\) −2352.00 −0.112480
\(760\) 0 0
\(761\) 13978.0 0.665837 0.332919 0.942956i \(-0.391967\pi\)
0.332919 + 0.942956i \(0.391967\pi\)
\(762\) 0 0
\(763\) 3864.00i 0.183337i
\(764\) 0 0
\(765\) 9720.00 + 4860.00i 0.459382 + 0.229691i
\(766\) 0 0
\(767\) 12864.0i 0.605596i
\(768\) 0 0
\(769\) −2926.00 −0.137210 −0.0686048 0.997644i \(-0.521855\pi\)
−0.0686048 + 0.997644i \(0.521855\pi\)
\(770\) 0 0
\(771\) 19116.0 0.892926
\(772\) 0 0
\(773\) 13916.0i 0.647508i 0.946141 + 0.323754i \(0.104945\pi\)
−0.946141 + 0.323754i \(0.895055\pi\)
\(774\) 0 0
\(775\) −13500.0 18000.0i −0.625722 0.834296i
\(776\) 0 0
\(777\) 480.000i 0.0221620i
\(778\) 0 0
\(779\) 13504.0 0.621092
\(780\) 0 0
\(781\) 1792.00 0.0821035
\(782\) 0 0
\(783\) 6426.00i 0.293291i
\(784\) 0 0
\(785\) 20720.0 + 10360.0i 0.942075 + 0.471037i
\(786\) 0 0
\(787\) 29996.0i 1.35863i 0.733847 + 0.679315i \(0.237723\pi\)
−0.733847 + 0.679315i \(0.762277\pi\)
\(788\) 0 0
\(789\) −9108.00 −0.410968
\(790\) 0 0
\(791\) −4496.00 −0.202098
\(792\) 0 0
\(793\) 5728.00i 0.256503i
\(794\) 0 0
\(795\) −3300.00 + 6600.00i −0.147219 + 0.294438i
\(796\) 0 0
\(797\) 8940.00i 0.397329i 0.980068 + 0.198664i \(0.0636604\pi\)
−0.980068 + 0.198664i \(0.936340\pi\)
\(798\) 0 0
\(799\) 6480.00 0.286916
\(800\) 0 0
\(801\) −10314.0 −0.454965
\(802\) 0 0
\(803\) 4256.00i 0.187037i
\(804\) 0 0
\(805\) −560.000 + 1120.00i −0.0245185 + 0.0490370i
\(806\) 0 0
\(807\) 342.000i 0.0149182i
\(808\) 0 0
\(809\) 10698.0 0.464922 0.232461 0.972606i \(-0.425322\pi\)
0.232461 + 0.972606i \(0.425322\pi\)
\(810\) 0 0
\(811\) 6408.00 0.277454 0.138727 0.990331i \(-0.455699\pi\)
0.138727 + 0.990331i \(0.455699\pi\)
\(812\) 0 0
\(813\) 15708.0i 0.677618i
\(814\) 0 0
\(815\) −7720.00 3860.00i −0.331803 0.165902i
\(816\) 0 0
\(817\) 8832.00i 0.378204i
\(818\) 0 0
\(819\) −576.000 −0.0245752
\(820\) 0 0
\(821\) 23130.0 0.983243 0.491622 0.870809i \(-0.336404\pi\)
0.491622 + 0.870809i \(0.336404\pi\)
\(822\) 0 0
\(823\) 11852.0i 0.501986i −0.967989 0.250993i \(-0.919243\pi\)
0.967989 0.250993i \(-0.0807572\pi\)
\(824\) 0 0
\(825\) 8400.00 6300.00i 0.354485 0.265864i
\(826\) 0 0
\(827\) 32628.0i 1.37193i −0.727634 0.685965i \(-0.759380\pi\)
0.727634 0.685965i \(-0.240620\pi\)
\(828\) 0 0
\(829\) −36694.0 −1.53732 −0.768658 0.639660i \(-0.779075\pi\)
−0.768658 + 0.639660i \(0.779075\pi\)
\(830\) 0 0
\(831\) −17136.0 −0.715332
\(832\) 0 0
\(833\) 35316.0i 1.46894i
\(834\) 0 0
\(835\) −10440.0 5220.00i −0.432684 0.216342i
\(836\) 0 0
\(837\) 4860.00i 0.200700i
\(838\) 0 0
\(839\) −1704.00 −0.0701175 −0.0350588 0.999385i \(-0.511162\pi\)
−0.0350588 + 0.999385i \(0.511162\pi\)
\(840\) 0 0
\(841\) 32255.0 1.32252
\(842\) 0 0
\(843\) 9666.00i 0.394917i
\(844\) 0 0
\(845\) 9705.00 19410.0i 0.395103 0.790206i
\(846\) 0 0
\(847\) 2188.00i 0.0887610i
\(848\) 0 0
\(849\) −13860.0 −0.560276
\(850\) 0 0
\(851\) −1120.00 −0.0451153
\(852\) 0 0
\(853\) 31880.0i 1.27966i 0.768516 + 0.639830i \(0.220995\pi\)
−0.768516 + 0.639830i \(0.779005\pi\)
\(854\) 0 0
\(855\) −1440.00 + 2880.00i −0.0575988 + 0.115198i
\(856\) 0 0
\(857\) 7972.00i 0.317758i 0.987298 + 0.158879i \(0.0507879\pi\)
−0.987298 + 0.158879i \(0.949212\pi\)
\(858\) 0 0
\(859\) −6008.00 −0.238638 −0.119319 0.992856i \(-0.538071\pi\)
−0.119319 + 0.992856i \(0.538071\pi\)
\(860\) 0 0
\(861\) −5064.00 −0.200442
\(862\) 0 0
\(863\) 1716.00i 0.0676863i −0.999427 0.0338432i \(-0.989225\pi\)
0.999427 0.0338432i \(-0.0107747\pi\)
\(864\) 0 0
\(865\) 44040.0 + 22020.0i 1.73110 + 0.865552i
\(866\) 0 0
\(867\) 20253.0i 0.793342i
\(868\) 0 0
\(869\) −26096.0 −1.01870
\(870\) 0 0
\(871\) −14144.0 −0.550231
\(872\) 0 0
\(873\) 7416.00i 0.287507i
\(874\) 0 0
\(875\) −1000.00 5500.00i −0.0386356 0.212496i
\(876\) 0 0
\(877\) 9032.00i 0.347764i 0.984767 + 0.173882i \(0.0556311\pi\)
−0.984767 + 0.173882i \(0.944369\pi\)
\(878\) 0 0
\(879\) 16212.0 0.622090
\(880\) 0 0
\(881\) 27838.0 1.06457 0.532285 0.846565i \(-0.321334\pi\)
0.532285 + 0.846565i \(0.321334\pi\)
\(882\) 0 0
\(883\) 4316.00i 0.164490i 0.996612 + 0.0822452i \(0.0262091\pi\)
−0.996612 + 0.0822452i \(0.973791\pi\)
\(884\) 0 0
\(885\) −24120.0 12060.0i −0.916141 0.458071i
\(886\) 0 0
\(887\) 43524.0i 1.64757i −0.566904 0.823784i \(-0.691859\pi\)
0.566904 0.823784i \(-0.308141\pi\)
\(888\) 0 0
\(889\) 7536.00 0.284307
\(890\) 0 0
\(891\) 2268.00 0.0852759
\(892\) 0 0
\(893\) 1920.00i 0.0719489i
\(894\) 0 0
\(895\) 17260.0 34520.0i 0.644624 1.28925i
\(896\) 0 0
\(897\) 1344.00i 0.0500277i
\(898\) 0 0
\(899\) 42840.0 1.58931
\(900\) 0 0
\(901\) 23760.0 0.878535
\(902\) 0 0
\(903\) 3312.00i 0.122056i
\(904\) 0 0
\(905\) 2630.00 5260.00i 0.0966013 0.193203i
\(906\) 0 0
\(907\) 10556.0i 0.386446i 0.981155 + 0.193223i \(0.0618940\pi\)
−0.981155 + 0.193223i \(0.938106\pi\)
\(908\) 0 0
\(909\) 11610.0 0.423630
\(910\) 0 0
\(911\) 47472.0 1.72647 0.863237 0.504799i \(-0.168433\pi\)
0.863237 + 0.504799i \(0.168433\pi\)
\(912\) 0 0
\(913\) 36176.0i 1.31134i
\(914\) 0 0
\(915\) −10740.0 5370.00i −0.388037 0.194018i
\(916\) 0 0
\(917\) 2352.00i 0.0847000i
\(918\) 0 0
\(919\) 11964.0 0.429441 0.214720 0.976676i \(-0.431116\pi\)
0.214720 + 0.976676i \(0.431116\pi\)
\(920\) 0 0
\(921\) 29100.0 1.04113
\(922\) 0 0
\(923\) 1024.00i 0.0365172i
\(924\) 0 0
\(925\) 4000.00 3000.00i 0.142183 0.106637i
\(926\) 0 0
\(927\) 14436.0i 0.511478i
\(928\) 0 0
\(929\) 15214.0 0.537304 0.268652 0.963237i \(-0.413422\pi\)
0.268652 + 0.963237i \(0.413422\pi\)
\(930\) 0 0
\(931\) 10464.0 0.368361
\(932\) 0 0
\(933\) 29016.0i 1.01816i
\(934\) 0 0
\(935\) −30240.0 15120.0i −1.05770 0.528852i
\(936\) 0 0
\(937\) 39712.0i 1.38456i −0.721628 0.692281i \(-0.756606\pi\)
0.721628 0.692281i \(-0.243394\pi\)
\(938\) 0 0
\(939\) −12144.0 −0.422049
\(940\) 0 0
\(941\) −36034.0 −1.24833 −0.624163 0.781294i \(-0.714560\pi\)
−0.624163 + 0.781294i \(0.714560\pi\)
\(942\) 0 0
\(943\) 11816.0i 0.408040i
\(944\) 0 0
\(945\) 540.000 1080.00i 0.0185886 0.0371771i
\(946\) 0 0
\(947\) 2532.00i 0.0868838i −0.999056 0.0434419i \(-0.986168\pi\)
0.999056 0.0434419i \(-0.0138323\pi\)
\(948\) 0 0
\(949\) −2432.00 −0.0831887
\(950\) 0 0
\(951\) −252.000 −0.00859270
\(952\) 0 0
\(953\) 55284.0i 1.87914i 0.342351 + 0.939572i \(0.388777\pi\)
−0.342351 + 0.939572i \(0.611223\pi\)
\(954\) 0 0
\(955\) −360.000 + 720.000i −0.0121982 + 0.0243965i
\(956\) 0 0
\(957\) 19992.0i 0.675287i
\(958\) 0 0
\(959\) 4240.00 0.142770
\(960\) 0 0
\(961\) 2609.00 0.0875768
\(962\) 0 0
\(963\) 8028.00i 0.268638i
\(964\) 0 0
\(965\) 2080.00 + 1040.00i 0.0693861 + 0.0346930i
\(966\) 0 0
\(967\) 4372.00i 0.145392i 0.997354 + 0.0726960i \(0.0231603\pi\)
−0.997354 + 0.0726960i \(0.976840\pi\)
\(968\) 0 0
\(969\) 10368.0 0.343724
\(970\) 0 0
\(971\) −24300.0 −0.803114 −0.401557 0.915834i \(-0.631531\pi\)
−0.401557 + 0.915834i \(0.631531\pi\)
\(972\) 0 0
\(973\) 11456.0i 0.377454i
\(974\) 0 0
\(975\) −3600.00 4800.00i −0.118248 0.157665i
\(976\) 0 0
\(977\) 46204.0i 1.51300i −0.653996 0.756498i \(-0.726909\pi\)
0.653996 0.756498i \(-0.273091\pi\)
\(978\) 0 0
\(979\) 32088.0 1.04754
\(980\) 0 0
\(981\) 8694.00 0.282954
\(982\) 0 0
\(983\) 25468.0i 0.826351i 0.910651 + 0.413176i \(0.135580\pi\)
−0.910651 + 0.413176i \(0.864420\pi\)
\(984\) 0 0
\(985\) −3720.00 1860.00i −0.120334 0.0601670i
\(986\) 0 0
\(987\) 720.000i 0.0232197i
\(988\) 0 0
\(989\) 7728.00 0.248469
\(990\) 0 0
\(991\) −11668.0 −0.374012 −0.187006 0.982359i \(-0.559878\pi\)
−0.187006 + 0.982359i \(0.559878\pi\)
\(992\) 0 0
\(993\) 16248.0i 0.519249i
\(994\) 0 0
\(995\) 21740.0 43480.0i 0.692668 1.38534i
\(996\) 0 0
\(997\) 7224.00i 0.229475i 0.993396 + 0.114737i \(0.0366027\pi\)
−0.993396 + 0.114737i \(0.963397\pi\)
\(998\) 0 0
\(999\) 1080.00 0.0342039
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 240.4.f.e.49.2 2
3.2 odd 2 720.4.f.b.289.2 2
4.3 odd 2 120.4.f.c.49.1 2
5.2 odd 4 1200.4.a.z.1.1 1
5.3 odd 4 1200.4.a.l.1.1 1
5.4 even 2 inner 240.4.f.e.49.1 2
8.3 odd 2 960.4.f.b.769.2 2
8.5 even 2 960.4.f.a.769.1 2
12.11 even 2 360.4.f.a.289.2 2
15.14 odd 2 720.4.f.b.289.1 2
20.3 even 4 600.4.a.k.1.1 1
20.7 even 4 600.4.a.f.1.1 1
20.19 odd 2 120.4.f.c.49.2 yes 2
40.19 odd 2 960.4.f.b.769.1 2
40.29 even 2 960.4.f.a.769.2 2
60.23 odd 4 1800.4.a.o.1.1 1
60.47 odd 4 1800.4.a.u.1.1 1
60.59 even 2 360.4.f.a.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.c.49.1 2 4.3 odd 2
120.4.f.c.49.2 yes 2 20.19 odd 2
240.4.f.e.49.1 2 5.4 even 2 inner
240.4.f.e.49.2 2 1.1 even 1 trivial
360.4.f.a.289.1 2 60.59 even 2
360.4.f.a.289.2 2 12.11 even 2
600.4.a.f.1.1 1 20.7 even 4
600.4.a.k.1.1 1 20.3 even 4
720.4.f.b.289.1 2 15.14 odd 2
720.4.f.b.289.2 2 3.2 odd 2
960.4.f.a.769.1 2 8.5 even 2
960.4.f.a.769.2 2 40.29 even 2
960.4.f.b.769.1 2 40.19 odd 2
960.4.f.b.769.2 2 8.3 odd 2
1200.4.a.l.1.1 1 5.3 odd 4
1200.4.a.z.1.1 1 5.2 odd 4
1800.4.a.o.1.1 1 60.23 odd 4
1800.4.a.u.1.1 1 60.47 odd 4