Properties

Label 600.4.f.i.49.2
Level $600$
Weight $4$
Character 600.49
Analytic conductor $35.401$
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] = [600,4,Mod(49,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, 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("600.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.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: \(35.4011460034\)
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 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 600.49
Dual form 600.4.f.i.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +4.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +4.00000i q^{7} -9.00000 q^{9} +72.0000 q^{11} +6.00000i q^{13} +38.0000i q^{17} -52.0000 q^{19} -12.0000 q^{21} -152.000i q^{23} -27.0000i q^{27} +78.0000 q^{29} +120.000 q^{31} +216.000i q^{33} -150.000i q^{37} -18.0000 q^{39} +362.000 q^{41} +484.000i q^{43} +280.000i q^{47} +327.000 q^{49} -114.000 q^{51} +670.000i q^{53} -156.000i q^{57} -696.000 q^{59} +222.000 q^{61} -36.0000i q^{63} -4.00000i q^{67} +456.000 q^{69} +96.0000 q^{71} -178.000i q^{73} +288.000i q^{77} +632.000 q^{79} +81.0000 q^{81} +612.000i q^{83} +234.000i q^{87} -994.000 q^{89} -24.0000 q^{91} +360.000i q^{93} +1634.00i q^{97} -648.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 144 q^{11} - 104 q^{19} - 24 q^{21} + 156 q^{29} + 240 q^{31} - 36 q^{39} + 724 q^{41} + 654 q^{49} - 228 q^{51} - 1392 q^{59} + 444 q^{61} + 912 q^{69} + 192 q^{71} + 1264 q^{79} + 162 q^{81} - 1988 q^{89} - 48 q^{91} - 1296 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 0 0
\(6\) 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
\(11\) 72.0000 1.97353 0.986764 0.162160i \(-0.0518462\pi\)
0.986764 + 0.162160i \(0.0518462\pi\)
\(12\) 0 0
\(13\) 6.00000i 0.128008i 0.997950 + 0.0640039i \(0.0203870\pi\)
−0.997950 + 0.0640039i \(0.979613\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 38.0000i 0.542138i 0.962560 + 0.271069i \(0.0873772\pi\)
−0.962560 + 0.271069i \(0.912623\pi\)
\(18\) 0 0
\(19\) −52.0000 −0.627875 −0.313937 0.949444i \(-0.601648\pi\)
−0.313937 + 0.949444i \(0.601648\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) − 152.000i − 1.37801i −0.724757 0.689004i \(-0.758048\pi\)
0.724757 0.689004i \(-0.241952\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 78.0000 0.499456 0.249728 0.968316i \(-0.419659\pi\)
0.249728 + 0.968316i \(0.419659\pi\)
\(30\) 0 0
\(31\) 120.000 0.695246 0.347623 0.937634i \(-0.386989\pi\)
0.347623 + 0.937634i \(0.386989\pi\)
\(32\) 0 0
\(33\) 216.000i 1.13942i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 150.000i − 0.666482i −0.942842 0.333241i \(-0.891858\pi\)
0.942842 0.333241i \(-0.108142\pi\)
\(38\) 0 0
\(39\) −18.0000 −0.0739053
\(40\) 0 0
\(41\) 362.000 1.37890 0.689450 0.724333i \(-0.257852\pi\)
0.689450 + 0.724333i \(0.257852\pi\)
\(42\) 0 0
\(43\) 484.000i 1.71650i 0.513236 + 0.858248i \(0.328447\pi\)
−0.513236 + 0.858248i \(0.671553\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 280.000i 0.868983i 0.900676 + 0.434491i \(0.143072\pi\)
−0.900676 + 0.434491i \(0.856928\pi\)
\(48\) 0 0
\(49\) 327.000 0.953353
\(50\) 0 0
\(51\) −114.000 −0.313004
\(52\) 0 0
\(53\) 670.000i 1.73644i 0.496175 + 0.868222i \(0.334737\pi\)
−0.496175 + 0.868222i \(0.665263\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 156.000i − 0.362504i
\(58\) 0 0
\(59\) −696.000 −1.53579 −0.767894 0.640577i \(-0.778695\pi\)
−0.767894 + 0.640577i \(0.778695\pi\)
\(60\) 0 0
\(61\) 222.000 0.465970 0.232985 0.972480i \(-0.425151\pi\)
0.232985 + 0.972480i \(0.425151\pi\)
\(62\) 0 0
\(63\) − 36.0000i − 0.0719932i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.00729370i −0.999993 0.00364685i \(-0.998839\pi\)
0.999993 0.00364685i \(-0.00116083\pi\)
\(68\) 0 0
\(69\) 456.000 0.795593
\(70\) 0 0
\(71\) 96.0000 0.160466 0.0802331 0.996776i \(-0.474434\pi\)
0.0802331 + 0.996776i \(0.474434\pi\)
\(72\) 0 0
\(73\) − 178.000i − 0.285388i −0.989767 0.142694i \(-0.954424\pi\)
0.989767 0.142694i \(-0.0455765\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 288.000i 0.426242i
\(78\) 0 0
\(79\) 632.000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 612.000i 0.809346i 0.914461 + 0.404673i \(0.132615\pi\)
−0.914461 + 0.404673i \(0.867385\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 234.000i 0.288361i
\(88\) 0 0
\(89\) −994.000 −1.18386 −0.591931 0.805988i \(-0.701634\pi\)
−0.591931 + 0.805988i \(0.701634\pi\)
\(90\) 0 0
\(91\) −24.0000 −0.0276471
\(92\) 0 0
\(93\) 360.000i 0.401401i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1634.00i 1.71039i 0.518309 + 0.855194i \(0.326562\pi\)
−0.518309 + 0.855194i \(0.673438\pi\)
\(98\) 0 0
\(99\) −648.000 −0.657843
\(100\) 0 0
\(101\) 890.000 0.876815 0.438407 0.898776i \(-0.355543\pi\)
0.438407 + 0.898776i \(0.355543\pi\)
\(102\) 0 0
\(103\) 524.000i 0.501274i 0.968081 + 0.250637i \(0.0806401\pi\)
−0.968081 + 0.250637i \(0.919360\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 932.000i 0.842055i 0.907048 + 0.421027i \(0.138330\pi\)
−0.907048 + 0.421027i \(0.861670\pi\)
\(108\) 0 0
\(109\) −446.000 −0.391918 −0.195959 0.980612i \(-0.562782\pi\)
−0.195959 + 0.980612i \(0.562782\pi\)
\(110\) 0 0
\(111\) 450.000 0.384794
\(112\) 0 0
\(113\) 786.000i 0.654342i 0.944965 + 0.327171i \(0.106095\pi\)
−0.944965 + 0.327171i \(0.893905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 54.0000i − 0.0426692i
\(118\) 0 0
\(119\) −152.000 −0.117091
\(120\) 0 0
\(121\) 3853.00 2.89482
\(122\) 0 0
\(123\) 1086.00i 0.796108i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 716.000i 0.500273i 0.968211 + 0.250137i \(0.0804756\pi\)
−0.968211 + 0.250137i \(0.919524\pi\)
\(128\) 0 0
\(129\) −1452.00 −0.991019
\(130\) 0 0
\(131\) −808.000 −0.538895 −0.269448 0.963015i \(-0.586841\pi\)
−0.269448 + 0.963015i \(0.586841\pi\)
\(132\) 0 0
\(133\) − 208.000i − 0.135608i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1770.00i − 1.10381i −0.833909 0.551903i \(-0.813902\pi\)
0.833909 0.551903i \(-0.186098\pi\)
\(138\) 0 0
\(139\) 924.000 0.563832 0.281916 0.959439i \(-0.409030\pi\)
0.281916 + 0.959439i \(0.409030\pi\)
\(140\) 0 0
\(141\) −840.000 −0.501708
\(142\) 0 0
\(143\) 432.000i 0.252627i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 981.000i 0.550418i
\(148\) 0 0
\(149\) 3198.00 1.75832 0.879162 0.476522i \(-0.158103\pi\)
0.879162 + 0.476522i \(0.158103\pi\)
\(150\) 0 0
\(151\) −3384.00 −1.82375 −0.911874 0.410470i \(-0.865365\pi\)
−0.911874 + 0.410470i \(0.865365\pi\)
\(152\) 0 0
\(153\) − 342.000i − 0.180713i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 3302.00i − 1.67852i −0.543727 0.839262i \(-0.682987\pi\)
0.543727 0.839262i \(-0.317013\pi\)
\(158\) 0 0
\(159\) −2010.00 −1.00254
\(160\) 0 0
\(161\) 608.000 0.297622
\(162\) 0 0
\(163\) − 2252.00i − 1.08215i −0.840975 0.541074i \(-0.818018\pi\)
0.840975 0.541074i \(-0.181982\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 184.000i − 0.0852596i −0.999091 0.0426298i \(-0.986426\pi\)
0.999091 0.0426298i \(-0.0135736\pi\)
\(168\) 0 0
\(169\) 2161.00 0.983614
\(170\) 0 0
\(171\) 468.000 0.209292
\(172\) 0 0
\(173\) 2646.00i 1.16284i 0.813603 + 0.581421i \(0.197503\pi\)
−0.813603 + 0.581421i \(0.802497\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 2088.00i − 0.886688i
\(178\) 0 0
\(179\) 608.000 0.253877 0.126939 0.991911i \(-0.459485\pi\)
0.126939 + 0.991911i \(0.459485\pi\)
\(180\) 0 0
\(181\) 2246.00 0.922342 0.461171 0.887311i \(-0.347430\pi\)
0.461171 + 0.887311i \(0.347430\pi\)
\(182\) 0 0
\(183\) 666.000i 0.269028i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2736.00i 1.06993i
\(188\) 0 0
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) −3848.00 −1.45776 −0.728878 0.684643i \(-0.759958\pi\)
−0.728878 + 0.684643i \(0.759958\pi\)
\(192\) 0 0
\(193\) − 2058.00i − 0.767555i −0.923426 0.383777i \(-0.874623\pi\)
0.923426 0.383777i \(-0.125377\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3838.00i − 1.38805i −0.719950 0.694026i \(-0.755835\pi\)
0.719950 0.694026i \(-0.244165\pi\)
\(198\) 0 0
\(199\) 1992.00 0.709594 0.354797 0.934943i \(-0.384550\pi\)
0.354797 + 0.934943i \(0.384550\pi\)
\(200\) 0 0
\(201\) 12.0000 0.00421102
\(202\) 0 0
\(203\) 312.000i 0.107872i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1368.00i 0.459336i
\(208\) 0 0
\(209\) −3744.00 −1.23913
\(210\) 0 0
\(211\) 4764.00 1.55435 0.777174 0.629286i \(-0.216653\pi\)
0.777174 + 0.629286i \(0.216653\pi\)
\(212\) 0 0
\(213\) 288.000i 0.0926452i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 480.000i 0.150159i
\(218\) 0 0
\(219\) 534.000 0.164769
\(220\) 0 0
\(221\) −228.000 −0.0693979
\(222\) 0 0
\(223\) − 4092.00i − 1.22879i −0.788998 0.614396i \(-0.789400\pi\)
0.788998 0.614396i \(-0.210600\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 468.000i 0.136838i 0.997657 + 0.0684191i \(0.0217955\pi\)
−0.997657 + 0.0684191i \(0.978205\pi\)
\(228\) 0 0
\(229\) 5586.00 1.61194 0.805968 0.591959i \(-0.201645\pi\)
0.805968 + 0.591959i \(0.201645\pi\)
\(230\) 0 0
\(231\) −864.000 −0.246091
\(232\) 0 0
\(233\) 1058.00i 0.297476i 0.988877 + 0.148738i \(0.0475211\pi\)
−0.988877 + 0.148738i \(0.952479\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1896.00i 0.519656i
\(238\) 0 0
\(239\) −6840.00 −1.85123 −0.925613 0.378472i \(-0.876450\pi\)
−0.925613 + 0.378472i \(0.876450\pi\)
\(240\) 0 0
\(241\) −6430.00 −1.71864 −0.859321 0.511437i \(-0.829113\pi\)
−0.859321 + 0.511437i \(0.829113\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 312.000i − 0.0803728i
\(248\) 0 0
\(249\) −1836.00 −0.467276
\(250\) 0 0
\(251\) −6352.00 −1.59735 −0.798675 0.601763i \(-0.794465\pi\)
−0.798675 + 0.601763i \(0.794465\pi\)
\(252\) 0 0
\(253\) − 10944.0i − 2.71954i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1422.00i 0.345144i 0.984997 + 0.172572i \(0.0552077\pi\)
−0.984997 + 0.172572i \(0.944792\pi\)
\(258\) 0 0
\(259\) 600.000 0.143947
\(260\) 0 0
\(261\) −702.000 −0.166485
\(262\) 0 0
\(263\) − 7224.00i − 1.69373i −0.531808 0.846865i \(-0.678487\pi\)
0.531808 0.846865i \(-0.321513\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2982.00i − 0.683504i
\(268\) 0 0
\(269\) −3186.00 −0.722133 −0.361067 0.932540i \(-0.617587\pi\)
−0.361067 + 0.932540i \(0.617587\pi\)
\(270\) 0 0
\(271\) −256.000 −0.0573834 −0.0286917 0.999588i \(-0.509134\pi\)
−0.0286917 + 0.999588i \(0.509134\pi\)
\(272\) 0 0
\(273\) − 72.0000i − 0.0159620i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 5942.00i − 1.28888i −0.764654 0.644441i \(-0.777090\pi\)
0.764654 0.644441i \(-0.222910\pi\)
\(278\) 0 0
\(279\) −1080.00 −0.231749
\(280\) 0 0
\(281\) 3202.00 0.679770 0.339885 0.940467i \(-0.389612\pi\)
0.339885 + 0.940467i \(0.389612\pi\)
\(282\) 0 0
\(283\) − 3940.00i − 0.827593i −0.910370 0.413796i \(-0.864203\pi\)
0.910370 0.413796i \(-0.135797\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1448.00i 0.297814i
\(288\) 0 0
\(289\) 3469.00 0.706086
\(290\) 0 0
\(291\) −4902.00 −0.987493
\(292\) 0 0
\(293\) − 1826.00i − 0.364082i −0.983291 0.182041i \(-0.941730\pi\)
0.983291 0.182041i \(-0.0582704\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1944.00i − 0.379806i
\(298\) 0 0
\(299\) 912.000 0.176396
\(300\) 0 0
\(301\) −1936.00 −0.370728
\(302\) 0 0
\(303\) 2670.00i 0.506229i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6580.00i 1.22326i 0.791144 + 0.611629i \(0.209486\pi\)
−0.791144 + 0.611629i \(0.790514\pi\)
\(308\) 0 0
\(309\) −1572.00 −0.289411
\(310\) 0 0
\(311\) −5728.00 −1.04439 −0.522195 0.852826i \(-0.674887\pi\)
−0.522195 + 0.852826i \(0.674887\pi\)
\(312\) 0 0
\(313\) 1742.00i 0.314580i 0.987552 + 0.157290i \(0.0502758\pi\)
−0.987552 + 0.157290i \(0.949724\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8746.00i 1.54960i 0.632204 + 0.774802i \(0.282150\pi\)
−0.632204 + 0.774802i \(0.717850\pi\)
\(318\) 0 0
\(319\) 5616.00 0.985692
\(320\) 0 0
\(321\) −2796.00 −0.486160
\(322\) 0 0
\(323\) − 1976.00i − 0.340395i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 1338.00i − 0.226274i
\(328\) 0 0
\(329\) −1120.00 −0.187683
\(330\) 0 0
\(331\) −2564.00 −0.425771 −0.212885 0.977077i \(-0.568286\pi\)
−0.212885 + 0.977077i \(0.568286\pi\)
\(332\) 0 0
\(333\) 1350.00i 0.222161i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 4166.00i − 0.673402i −0.941612 0.336701i \(-0.890689\pi\)
0.941612 0.336701i \(-0.109311\pi\)
\(338\) 0 0
\(339\) −2358.00 −0.377785
\(340\) 0 0
\(341\) 8640.00 1.37209
\(342\) 0 0
\(343\) 2680.00i 0.421885i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9444.00i − 1.46104i −0.682892 0.730519i \(-0.739278\pi\)
0.682892 0.730519i \(-0.260722\pi\)
\(348\) 0 0
\(349\) 9218.00 1.41383 0.706917 0.707296i \(-0.250085\pi\)
0.706917 + 0.707296i \(0.250085\pi\)
\(350\) 0 0
\(351\) 162.000 0.0246351
\(352\) 0 0
\(353\) 4698.00i 0.708355i 0.935178 + 0.354177i \(0.115239\pi\)
−0.935178 + 0.354177i \(0.884761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 456.000i − 0.0676025i
\(358\) 0 0
\(359\) 6056.00 0.890316 0.445158 0.895452i \(-0.353148\pi\)
0.445158 + 0.895452i \(0.353148\pi\)
\(360\) 0 0
\(361\) −4155.00 −0.605773
\(362\) 0 0
\(363\) 11559.0i 1.67132i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8228.00i − 1.17029i −0.810927 0.585147i \(-0.801037\pi\)
0.810927 0.585147i \(-0.198963\pi\)
\(368\) 0 0
\(369\) −3258.00 −0.459633
\(370\) 0 0
\(371\) −2680.00 −0.375037
\(372\) 0 0
\(373\) − 5954.00i − 0.826505i −0.910616 0.413253i \(-0.864393\pi\)
0.910616 0.413253i \(-0.135607\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 468.000i 0.0639343i
\(378\) 0 0
\(379\) −5284.00 −0.716150 −0.358075 0.933693i \(-0.616567\pi\)
−0.358075 + 0.933693i \(0.616567\pi\)
\(380\) 0 0
\(381\) −2148.00 −0.288833
\(382\) 0 0
\(383\) − 9832.00i − 1.31173i −0.754879 0.655864i \(-0.772305\pi\)
0.754879 0.655864i \(-0.227695\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4356.00i − 0.572165i
\(388\) 0 0
\(389\) 222.000 0.0289353 0.0144677 0.999895i \(-0.495395\pi\)
0.0144677 + 0.999895i \(0.495395\pi\)
\(390\) 0 0
\(391\) 5776.00 0.747071
\(392\) 0 0
\(393\) − 2424.00i − 0.311131i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 12098.0i 1.52942i 0.644372 + 0.764712i \(0.277119\pi\)
−0.644372 + 0.764712i \(0.722881\pi\)
\(398\) 0 0
\(399\) 624.000 0.0782934
\(400\) 0 0
\(401\) −5958.00 −0.741966 −0.370983 0.928640i \(-0.620979\pi\)
−0.370983 + 0.928640i \(0.620979\pi\)
\(402\) 0 0
\(403\) 720.000i 0.0889969i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 10800.0i − 1.31532i
\(408\) 0 0
\(409\) −1930.00 −0.233331 −0.116665 0.993171i \(-0.537221\pi\)
−0.116665 + 0.993171i \(0.537221\pi\)
\(410\) 0 0
\(411\) 5310.00 0.637282
\(412\) 0 0
\(413\) − 2784.00i − 0.331699i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2772.00i 0.325529i
\(418\) 0 0
\(419\) −4744.00 −0.553125 −0.276563 0.960996i \(-0.589195\pi\)
−0.276563 + 0.960996i \(0.589195\pi\)
\(420\) 0 0
\(421\) 1614.00 0.186845 0.0934223 0.995627i \(-0.470219\pi\)
0.0934223 + 0.995627i \(0.470219\pi\)
\(422\) 0 0
\(423\) − 2520.00i − 0.289661i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 888.000i 0.100640i
\(428\) 0 0
\(429\) −1296.00 −0.145854
\(430\) 0 0
\(431\) 9296.00 1.03892 0.519458 0.854496i \(-0.326134\pi\)
0.519458 + 0.854496i \(0.326134\pi\)
\(432\) 0 0
\(433\) 3494.00i 0.387785i 0.981023 + 0.193893i \(0.0621113\pi\)
−0.981023 + 0.193893i \(0.937889\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7904.00i 0.865216i
\(438\) 0 0
\(439\) 12584.0 1.36811 0.684056 0.729429i \(-0.260214\pi\)
0.684056 + 0.729429i \(0.260214\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) 12852.0i 1.37837i 0.724586 + 0.689184i \(0.242031\pi\)
−0.724586 + 0.689184i \(0.757969\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9594.00i 1.01517i
\(448\) 0 0
\(449\) −14458.0 −1.51963 −0.759816 0.650138i \(-0.774711\pi\)
−0.759816 + 0.650138i \(0.774711\pi\)
\(450\) 0 0
\(451\) 26064.0 2.72130
\(452\) 0 0
\(453\) − 10152.0i − 1.05294i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 4310.00i − 0.441167i −0.975368 0.220583i \(-0.929204\pi\)
0.975368 0.220583i \(-0.0707961\pi\)
\(458\) 0 0
\(459\) 1026.00 0.104335
\(460\) 0 0
\(461\) 5338.00 0.539296 0.269648 0.962959i \(-0.413093\pi\)
0.269648 + 0.962959i \(0.413093\pi\)
\(462\) 0 0
\(463\) − 1156.00i − 0.116034i −0.998316 0.0580171i \(-0.981522\pi\)
0.998316 0.0580171i \(-0.0184778\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5948.00i 0.589380i 0.955593 + 0.294690i \(0.0952164\pi\)
−0.955593 + 0.294690i \(0.904784\pi\)
\(468\) 0 0
\(469\) 16.0000 0.00157529
\(470\) 0 0
\(471\) 9906.00 0.969096
\(472\) 0 0
\(473\) 34848.0i 3.38755i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6030.00i − 0.578815i
\(478\) 0 0
\(479\) −6888.00 −0.657037 −0.328519 0.944498i \(-0.606549\pi\)
−0.328519 + 0.944498i \(0.606549\pi\)
\(480\) 0 0
\(481\) 900.000 0.0853149
\(482\) 0 0
\(483\) 1824.00i 0.171832i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2892.00i 0.269095i 0.990907 + 0.134547i \(0.0429580\pi\)
−0.990907 + 0.134547i \(0.957042\pi\)
\(488\) 0 0
\(489\) 6756.00 0.624779
\(490\) 0 0
\(491\) 4096.00 0.376476 0.188238 0.982123i \(-0.439722\pi\)
0.188238 + 0.982123i \(0.439722\pi\)
\(492\) 0 0
\(493\) 2964.00i 0.270775i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 384.000i 0.0346575i
\(498\) 0 0
\(499\) −11060.0 −0.992212 −0.496106 0.868262i \(-0.665237\pi\)
−0.496106 + 0.868262i \(0.665237\pi\)
\(500\) 0 0
\(501\) 552.000 0.0492246
\(502\) 0 0
\(503\) 9648.00i 0.855235i 0.903960 + 0.427617i \(0.140647\pi\)
−0.903960 + 0.427617i \(0.859353\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6483.00i 0.567890i
\(508\) 0 0
\(509\) 10062.0 0.876209 0.438104 0.898924i \(-0.355650\pi\)
0.438104 + 0.898924i \(0.355650\pi\)
\(510\) 0 0
\(511\) 712.000 0.0616380
\(512\) 0 0
\(513\) 1404.00i 0.120835i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 20160.0i 1.71496i
\(518\) 0 0
\(519\) −7938.00 −0.671367
\(520\) 0 0
\(521\) −7966.00 −0.669859 −0.334930 0.942243i \(-0.608713\pi\)
−0.334930 + 0.942243i \(0.608713\pi\)
\(522\) 0 0
\(523\) − 7668.00i − 0.641106i −0.947231 0.320553i \(-0.896131\pi\)
0.947231 0.320553i \(-0.103869\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4560.00i 0.376920i
\(528\) 0 0
\(529\) −10937.0 −0.898907
\(530\) 0 0
\(531\) 6264.00 0.511929
\(532\) 0 0
\(533\) 2172.00i 0.176510i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1824.00i 0.146576i
\(538\) 0 0
\(539\) 23544.0 1.88147
\(540\) 0 0
\(541\) 6590.00 0.523708 0.261854 0.965107i \(-0.415666\pi\)
0.261854 + 0.965107i \(0.415666\pi\)
\(542\) 0 0
\(543\) 6738.00i 0.532514i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 4700.00i − 0.367381i −0.982984 0.183691i \(-0.941196\pi\)
0.982984 0.183691i \(-0.0588044\pi\)
\(548\) 0 0
\(549\) −1998.00 −0.155323
\(550\) 0 0
\(551\) −4056.00 −0.313596
\(552\) 0 0
\(553\) 2528.00i 0.194397i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 15766.0i − 1.19933i −0.800251 0.599665i \(-0.795300\pi\)
0.800251 0.599665i \(-0.204700\pi\)
\(558\) 0 0
\(559\) −2904.00 −0.219725
\(560\) 0 0
\(561\) −8208.00 −0.617722
\(562\) 0 0
\(563\) − 22788.0i − 1.70586i −0.522025 0.852930i \(-0.674823\pi\)
0.522025 0.852930i \(-0.325177\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 324.000i 0.0239977i
\(568\) 0 0
\(569\) 3358.00 0.247407 0.123704 0.992319i \(-0.460523\pi\)
0.123704 + 0.992319i \(0.460523\pi\)
\(570\) 0 0
\(571\) −11444.0 −0.838733 −0.419366 0.907817i \(-0.637748\pi\)
−0.419366 + 0.907817i \(0.637748\pi\)
\(572\) 0 0
\(573\) − 11544.0i − 0.841636i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 10622.0i − 0.766377i −0.923670 0.383189i \(-0.874826\pi\)
0.923670 0.383189i \(-0.125174\pi\)
\(578\) 0 0
\(579\) 6174.00 0.443148
\(580\) 0 0
\(581\) −2448.00 −0.174802
\(582\) 0 0
\(583\) 48240.0i 3.42692i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 6588.00i − 0.463230i −0.972808 0.231615i \(-0.925599\pi\)
0.972808 0.231615i \(-0.0744009\pi\)
\(588\) 0 0
\(589\) −6240.00 −0.436528
\(590\) 0 0
\(591\) 11514.0 0.801392
\(592\) 0 0
\(593\) 11362.0i 0.786815i 0.919364 + 0.393408i \(0.128704\pi\)
−0.919364 + 0.393408i \(0.871296\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5976.00i 0.409684i
\(598\) 0 0
\(599\) −1624.00 −0.110776 −0.0553880 0.998465i \(-0.517640\pi\)
−0.0553880 + 0.998465i \(0.517640\pi\)
\(600\) 0 0
\(601\) −14950.0 −1.01468 −0.507340 0.861746i \(-0.669371\pi\)
−0.507340 + 0.861746i \(0.669371\pi\)
\(602\) 0 0
\(603\) 36.0000i 0.00243123i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8244.00i 0.551258i 0.961264 + 0.275629i \(0.0888861\pi\)
−0.961264 + 0.275629i \(0.911114\pi\)
\(608\) 0 0
\(609\) −936.000 −0.0622802
\(610\) 0 0
\(611\) −1680.00 −0.111237
\(612\) 0 0
\(613\) − 6698.00i − 0.441321i −0.975351 0.220660i \(-0.929179\pi\)
0.975351 0.220660i \(-0.0708213\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22670.0i 1.47919i 0.673053 + 0.739595i \(0.264983\pi\)
−0.673053 + 0.739595i \(0.735017\pi\)
\(618\) 0 0
\(619\) 10060.0 0.653224 0.326612 0.945159i \(-0.394093\pi\)
0.326612 + 0.945159i \(0.394093\pi\)
\(620\) 0 0
\(621\) −4104.00 −0.265198
\(622\) 0 0
\(623\) − 3976.00i − 0.255690i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 11232.0i − 0.715411i
\(628\) 0 0
\(629\) 5700.00 0.361326
\(630\) 0 0
\(631\) 10240.0 0.646035 0.323017 0.946393i \(-0.395303\pi\)
0.323017 + 0.946393i \(0.395303\pi\)
\(632\) 0 0
\(633\) 14292.0i 0.897403i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1962.00i 0.122037i
\(638\) 0 0
\(639\) −864.000 −0.0534888
\(640\) 0 0
\(641\) 13218.0 0.814477 0.407238 0.913322i \(-0.366492\pi\)
0.407238 + 0.913322i \(0.366492\pi\)
\(642\) 0 0
\(643\) 23412.0i 1.43589i 0.696098 + 0.717946i \(0.254918\pi\)
−0.696098 + 0.717946i \(0.745082\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 15264.0i − 0.927496i −0.885967 0.463748i \(-0.846504\pi\)
0.885967 0.463748i \(-0.153496\pi\)
\(648\) 0 0
\(649\) −50112.0 −3.03092
\(650\) 0 0
\(651\) −1440.00 −0.0866944
\(652\) 0 0
\(653\) − 1482.00i − 0.0888134i −0.999014 0.0444067i \(-0.985860\pi\)
0.999014 0.0444067i \(-0.0141397\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1602.00i 0.0951293i
\(658\) 0 0
\(659\) 18920.0 1.11839 0.559195 0.829036i \(-0.311110\pi\)
0.559195 + 0.829036i \(0.311110\pi\)
\(660\) 0 0
\(661\) −24218.0 −1.42507 −0.712535 0.701637i \(-0.752453\pi\)
−0.712535 + 0.701637i \(0.752453\pi\)
\(662\) 0 0
\(663\) − 684.000i − 0.0400669i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 11856.0i − 0.688255i
\(668\) 0 0
\(669\) 12276.0 0.709443
\(670\) 0 0
\(671\) 15984.0 0.919606
\(672\) 0 0
\(673\) − 890.000i − 0.0509762i −0.999675 0.0254881i \(-0.991886\pi\)
0.999675 0.0254881i \(-0.00811399\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29250.0i 1.66052i 0.557380 + 0.830258i \(0.311807\pi\)
−0.557380 + 0.830258i \(0.688193\pi\)
\(678\) 0 0
\(679\) −6536.00 −0.369409
\(680\) 0 0
\(681\) −1404.00 −0.0790035
\(682\) 0 0
\(683\) − 14580.0i − 0.816820i −0.912799 0.408410i \(-0.866083\pi\)
0.912799 0.408410i \(-0.133917\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16758.0i 0.930651i
\(688\) 0 0
\(689\) −4020.00 −0.222278
\(690\) 0 0
\(691\) 23668.0 1.30300 0.651500 0.758649i \(-0.274140\pi\)
0.651500 + 0.758649i \(0.274140\pi\)
\(692\) 0 0
\(693\) − 2592.00i − 0.142081i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13756.0i 0.747555i
\(698\) 0 0
\(699\) −3174.00 −0.171748
\(700\) 0 0
\(701\) 32402.0 1.74580 0.872901 0.487898i \(-0.162236\pi\)
0.872901 + 0.487898i \(0.162236\pi\)
\(702\) 0 0
\(703\) 7800.00i 0.418467i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3560.00i 0.189374i
\(708\) 0 0
\(709\) 30626.0 1.62226 0.811131 0.584865i \(-0.198852\pi\)
0.811131 + 0.584865i \(0.198852\pi\)
\(710\) 0 0
\(711\) −5688.00 −0.300023
\(712\) 0 0
\(713\) − 18240.0i − 0.958055i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 20520.0i − 1.06881i
\(718\) 0 0
\(719\) −13440.0 −0.697117 −0.348559 0.937287i \(-0.613329\pi\)
−0.348559 + 0.937287i \(0.613329\pi\)
\(720\) 0 0
\(721\) −2096.00 −0.108265
\(722\) 0 0
\(723\) − 19290.0i − 0.992258i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 24820.0i − 1.26619i −0.774073 0.633097i \(-0.781783\pi\)
0.774073 0.633097i \(-0.218217\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −18392.0 −0.930578
\(732\) 0 0
\(733\) − 21986.0i − 1.10787i −0.832559 0.553937i \(-0.813125\pi\)
0.832559 0.553937i \(-0.186875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 288.000i − 0.0143943i
\(738\) 0 0
\(739\) −4420.00 −0.220017 −0.110008 0.993931i \(-0.535088\pi\)
−0.110008 + 0.993931i \(0.535088\pi\)
\(740\) 0 0
\(741\) 936.000 0.0464033
\(742\) 0 0
\(743\) − 34560.0i − 1.70644i −0.521553 0.853219i \(-0.674647\pi\)
0.521553 0.853219i \(-0.325353\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 5508.00i − 0.269782i
\(748\) 0 0
\(749\) −3728.00 −0.181867
\(750\) 0 0
\(751\) −24792.0 −1.20462 −0.602312 0.798261i \(-0.705754\pi\)
−0.602312 + 0.798261i \(0.705754\pi\)
\(752\) 0 0
\(753\) − 19056.0i − 0.922230i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 2166.00i − 0.103996i −0.998647 0.0519978i \(-0.983441\pi\)
0.998647 0.0519978i \(-0.0165589\pi\)
\(758\) 0 0
\(759\) 32832.0 1.57013
\(760\) 0 0
\(761\) −10622.0 −0.505975 −0.252988 0.967470i \(-0.581413\pi\)
−0.252988 + 0.967470i \(0.581413\pi\)
\(762\) 0 0
\(763\) − 1784.00i − 0.0846463i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 4176.00i − 0.196593i
\(768\) 0 0
\(769\) −29826.0 −1.39864 −0.699319 0.714809i \(-0.746513\pi\)
−0.699319 + 0.714809i \(0.746513\pi\)
\(770\) 0 0
\(771\) −4266.00 −0.199269
\(772\) 0 0
\(773\) − 6386.00i − 0.297139i −0.988902 0.148570i \(-0.952533\pi\)
0.988902 0.148570i \(-0.0474669\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1800.00i 0.0831076i
\(778\) 0 0
\(779\) −18824.0 −0.865776
\(780\) 0 0
\(781\) 6912.00 0.316685
\(782\) 0 0
\(783\) − 2106.00i − 0.0961204i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 3516.00i 0.159253i 0.996825 + 0.0796263i \(0.0253727\pi\)
−0.996825 + 0.0796263i \(0.974627\pi\)
\(788\) 0 0
\(789\) 21672.0 0.977875
\(790\) 0 0
\(791\) −3144.00 −0.141325
\(792\) 0 0
\(793\) 1332.00i 0.0596478i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 25030.0i − 1.11243i −0.831038 0.556216i \(-0.812253\pi\)
0.831038 0.556216i \(-0.187747\pi\)
\(798\) 0 0
\(799\) −10640.0 −0.471109
\(800\) 0 0
\(801\) 8946.00 0.394621
\(802\) 0 0
\(803\) − 12816.0i − 0.563221i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 9558.00i − 0.416924i
\(808\) 0 0
\(809\) −7962.00 −0.346019 −0.173009 0.984920i \(-0.555349\pi\)
−0.173009 + 0.984920i \(0.555349\pi\)
\(810\) 0 0
\(811\) −34668.0 −1.50106 −0.750529 0.660837i \(-0.770201\pi\)
−0.750529 + 0.660837i \(0.770201\pi\)
\(812\) 0 0
\(813\) − 768.000i − 0.0331303i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 25168.0i − 1.07774i
\(818\) 0 0
\(819\) 216.000 0.00921569
\(820\) 0 0
\(821\) 250.000 0.0106274 0.00531368 0.999986i \(-0.498309\pi\)
0.00531368 + 0.999986i \(0.498309\pi\)
\(822\) 0 0
\(823\) 6388.00i 0.270561i 0.990807 + 0.135280i \(0.0431936\pi\)
−0.990807 + 0.135280i \(0.956806\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3932.00i 0.165331i 0.996577 + 0.0826657i \(0.0263434\pi\)
−0.996577 + 0.0826657i \(0.973657\pi\)
\(828\) 0 0
\(829\) 25906.0 1.08535 0.542673 0.839944i \(-0.317412\pi\)
0.542673 + 0.839944i \(0.317412\pi\)
\(830\) 0 0
\(831\) 17826.0 0.744136
\(832\) 0 0
\(833\) 12426.0i 0.516849i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 3240.00i − 0.133800i
\(838\) 0 0
\(839\) 9944.00 0.409184 0.204592 0.978847i \(-0.434413\pi\)
0.204592 + 0.978847i \(0.434413\pi\)
\(840\) 0 0
\(841\) −18305.0 −0.750543
\(842\) 0 0
\(843\) 9606.00i 0.392465i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 15412.0i 0.625221i
\(848\) 0 0
\(849\) 11820.0 0.477811
\(850\) 0 0
\(851\) −22800.0 −0.918418
\(852\) 0 0
\(853\) 14630.0i 0.587247i 0.955921 + 0.293623i \(0.0948612\pi\)
−0.955921 + 0.293623i \(0.905139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 478.000i 0.0190527i 0.999955 + 0.00952635i \(0.00303238\pi\)
−0.999955 + 0.00952635i \(0.996968\pi\)
\(858\) 0 0
\(859\) −24132.0 −0.958525 −0.479263 0.877672i \(-0.659096\pi\)
−0.479263 + 0.877672i \(0.659096\pi\)
\(860\) 0 0
\(861\) −4344.00 −0.171943
\(862\) 0 0
\(863\) − 15776.0i − 0.622273i −0.950365 0.311136i \(-0.899290\pi\)
0.950365 0.311136i \(-0.100710\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 10407.0i 0.407659i
\(868\) 0 0
\(869\) 45504.0 1.77631
\(870\) 0 0
\(871\) 24.0000 0.000933650 0
\(872\) 0 0
\(873\) − 14706.0i − 0.570129i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 33542.0i − 1.29149i −0.763555 0.645743i \(-0.776548\pi\)
0.763555 0.645743i \(-0.223452\pi\)
\(878\) 0 0
\(879\) 5478.00 0.210203
\(880\) 0 0
\(881\) 22858.0 0.874127 0.437063 0.899431i \(-0.356019\pi\)
0.437063 + 0.899431i \(0.356019\pi\)
\(882\) 0 0
\(883\) − 2764.00i − 0.105341i −0.998612 0.0526704i \(-0.983227\pi\)
0.998612 0.0526704i \(-0.0167733\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6216.00i 0.235302i 0.993055 + 0.117651i \(0.0375364\pi\)
−0.993055 + 0.117651i \(0.962464\pi\)
\(888\) 0 0
\(889\) −2864.00 −0.108049
\(890\) 0 0
\(891\) 5832.00 0.219281
\(892\) 0 0
\(893\) − 14560.0i − 0.545612i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2736.00i 0.101842i
\(898\) 0 0
\(899\) 9360.00 0.347245
\(900\) 0 0
\(901\) −25460.0 −0.941394
\(902\) 0 0
\(903\) − 5808.00i − 0.214040i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 18884.0i − 0.691326i −0.938359 0.345663i \(-0.887654\pi\)
0.938359 0.345663i \(-0.112346\pi\)
\(908\) 0 0
\(909\) −8010.00 −0.292272
\(910\) 0 0
\(911\) −15232.0 −0.553961 −0.276981 0.960876i \(-0.589334\pi\)
−0.276981 + 0.960876i \(0.589334\pi\)
\(912\) 0 0
\(913\) 44064.0i 1.59727i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3232.00i − 0.116390i
\(918\) 0 0
\(919\) −7744.00 −0.277966 −0.138983 0.990295i \(-0.544383\pi\)
−0.138983 + 0.990295i \(0.544383\pi\)
\(920\) 0 0
\(921\) −19740.0 −0.706249
\(922\) 0 0
\(923\) 576.000i 0.0205409i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4716.00i − 0.167091i
\(928\) 0 0
\(929\) −22266.0 −0.786355 −0.393177 0.919463i \(-0.628624\pi\)
−0.393177 + 0.919463i \(0.628624\pi\)
\(930\) 0 0
\(931\) −17004.0 −0.598586
\(932\) 0 0
\(933\) − 17184.0i − 0.602978i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16202.0i 0.564884i 0.959284 + 0.282442i \(0.0911445\pi\)
−0.959284 + 0.282442i \(0.908856\pi\)
\(938\) 0 0
\(939\) −5226.00 −0.181623
\(940\) 0 0
\(941\) −53494.0 −1.85319 −0.926596 0.376057i \(-0.877280\pi\)
−0.926596 + 0.376057i \(0.877280\pi\)
\(942\) 0 0
\(943\) − 55024.0i − 1.90014i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2332.00i − 0.0800209i −0.999199 0.0400105i \(-0.987261\pi\)
0.999199 0.0400105i \(-0.0127391\pi\)
\(948\) 0 0
\(949\) 1068.00 0.0365319
\(950\) 0 0
\(951\) −26238.0 −0.894664
\(952\) 0 0
\(953\) − 15414.0i − 0.523933i −0.965077 0.261967i \(-0.915629\pi\)
0.965077 0.261967i \(-0.0843710\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 16848.0i 0.569089i
\(958\) 0 0
\(959\) 7080.00 0.238400
\(960\) 0 0
\(961\) −15391.0 −0.516633
\(962\) 0 0
\(963\) − 8388.00i − 0.280685i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 35012.0i 1.16433i 0.813070 + 0.582167i \(0.197795\pi\)
−0.813070 + 0.582167i \(0.802205\pi\)
\(968\) 0 0
\(969\) 5928.00 0.196527
\(970\) 0 0
\(971\) 11360.0 0.375448 0.187724 0.982222i \(-0.439889\pi\)
0.187724 + 0.982222i \(0.439889\pi\)
\(972\) 0 0
\(973\) 3696.00i 0.121776i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 24586.0i − 0.805093i −0.915400 0.402546i \(-0.868125\pi\)
0.915400 0.402546i \(-0.131875\pi\)
\(978\) 0 0
\(979\) −71568.0 −2.33639
\(980\) 0 0
\(981\) 4014.00 0.130639
\(982\) 0 0
\(983\) − 8832.00i − 0.286569i −0.989682 0.143284i \(-0.954234\pi\)
0.989682 0.143284i \(-0.0457663\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 3360.00i − 0.108359i
\(988\) 0 0
\(989\) 73568.0 2.36535
\(990\) 0 0
\(991\) −22912.0 −0.734434 −0.367217 0.930135i \(-0.619689\pi\)
−0.367217 + 0.930135i \(0.619689\pi\)
\(992\) 0 0
\(993\) − 7692.00i − 0.245819i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 10974.0i − 0.348596i −0.984693 0.174298i \(-0.944234\pi\)
0.984693 0.174298i \(-0.0557656\pi\)
\(998\) 0 0
\(999\) −4050.00 −0.128265
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 600.4.f.i.49.2 2
3.2 odd 2 1800.4.f.a.649.2 2
4.3 odd 2 1200.4.f.a.49.1 2
5.2 odd 4 600.4.a.l.1.1 1
5.3 odd 4 120.4.a.a.1.1 1
5.4 even 2 inner 600.4.f.i.49.1 2
15.2 even 4 1800.4.a.n.1.1 1
15.8 even 4 360.4.a.l.1.1 1
15.14 odd 2 1800.4.f.a.649.1 2
20.3 even 4 240.4.a.h.1.1 1
20.7 even 4 1200.4.a.k.1.1 1
20.19 odd 2 1200.4.f.a.49.2 2
40.3 even 4 960.4.a.o.1.1 1
40.13 odd 4 960.4.a.bf.1.1 1
60.23 odd 4 720.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.a.1.1 1 5.3 odd 4
240.4.a.h.1.1 1 20.3 even 4
360.4.a.l.1.1 1 15.8 even 4
600.4.a.l.1.1 1 5.2 odd 4
600.4.f.i.49.1 2 5.4 even 2 inner
600.4.f.i.49.2 2 1.1 even 1 trivial
720.4.a.v.1.1 1 60.23 odd 4
960.4.a.o.1.1 1 40.3 even 4
960.4.a.bf.1.1 1 40.13 odd 4
1200.4.a.k.1.1 1 20.7 even 4
1200.4.f.a.49.1 2 4.3 odd 2
1200.4.f.a.49.2 2 20.19 odd 2
1800.4.a.n.1.1 1 15.2 even 4
1800.4.f.a.649.1 2 15.14 odd 2
1800.4.f.a.649.2 2 3.2 odd 2