Properties

Label 1200.4.f.q.49.1
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
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] = [1200,4,Mod(49,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, 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("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
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 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.q.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -23.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -23.0000i q^{7} -9.00000 q^{9} +30.0000 q^{11} -29.0000i q^{13} +78.0000i q^{17} +149.000 q^{19} -69.0000 q^{21} +150.000i q^{23} +27.0000i q^{27} +234.000 q^{29} +217.000 q^{31} -90.0000i q^{33} +146.000i q^{37} -87.0000 q^{39} -156.000 q^{41} -433.000i q^{43} -30.0000i q^{47} -186.000 q^{49} +234.000 q^{51} +552.000i q^{53} -447.000i q^{57} -270.000 q^{59} +275.000 q^{61} +207.000i q^{63} -803.000i q^{67} +450.000 q^{69} -660.000 q^{71} +646.000i q^{73} -690.000i q^{77} +992.000 q^{79} +81.0000 q^{81} -846.000i q^{83} -702.000i q^{87} +1488.00 q^{89} -667.000 q^{91} -651.000i q^{93} -319.000i q^{97} -270.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 60 q^{11} + 298 q^{19} - 138 q^{21} + 468 q^{29} + 434 q^{31} - 174 q^{39} - 312 q^{41} - 372 q^{49} + 468 q^{51} - 540 q^{59} + 550 q^{61} + 900 q^{69} - 1320 q^{71} + 1984 q^{79} + 162 q^{81} + 2976 q^{89} - 1334 q^{91} - 540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(7\) − 23.0000i − 1.24188i −0.783857 0.620942i \(-0.786750\pi\)
0.783857 0.620942i \(-0.213250\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 30.0000 0.822304 0.411152 0.911567i \(-0.365127\pi\)
0.411152 + 0.911567i \(0.365127\pi\)
\(12\) 0 0
\(13\) − 29.0000i − 0.618704i −0.950948 0.309352i \(-0.899888\pi\)
0.950948 0.309352i \(-0.100112\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 78.0000i 1.11281i 0.830911 + 0.556405i \(0.187820\pi\)
−0.830911 + 0.556405i \(0.812180\pi\)
\(18\) 0 0
\(19\) 149.000 1.79910 0.899551 0.436815i \(-0.143894\pi\)
0.899551 + 0.436815i \(0.143894\pi\)
\(20\) 0 0
\(21\) −69.0000 −0.717002
\(22\) 0 0
\(23\) 150.000i 1.35988i 0.733269 + 0.679938i \(0.237993\pi\)
−0.733269 + 0.679938i \(0.762007\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 234.000 1.49837 0.749185 0.662361i \(-0.230446\pi\)
0.749185 + 0.662361i \(0.230446\pi\)
\(30\) 0 0
\(31\) 217.000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) − 90.0000i − 0.474757i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 146.000i 0.648710i 0.945936 + 0.324355i \(0.105147\pi\)
−0.945936 + 0.324355i \(0.894853\pi\)
\(38\) 0 0
\(39\) −87.0000 −0.357209
\(40\) 0 0
\(41\) −156.000 −0.594222 −0.297111 0.954843i \(-0.596023\pi\)
−0.297111 + 0.954843i \(0.596023\pi\)
\(42\) 0 0
\(43\) − 433.000i − 1.53563i −0.640675 0.767813i \(-0.721345\pi\)
0.640675 0.767813i \(-0.278655\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 30.0000i − 0.0931053i −0.998916 0.0465527i \(-0.985176\pi\)
0.998916 0.0465527i \(-0.0148235\pi\)
\(48\) 0 0
\(49\) −186.000 −0.542274
\(50\) 0 0
\(51\) 234.000 0.642481
\(52\) 0 0
\(53\) 552.000i 1.43062i 0.698806 + 0.715312i \(0.253715\pi\)
−0.698806 + 0.715312i \(0.746285\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 447.000i − 1.03871i
\(58\) 0 0
\(59\) −270.000 −0.595780 −0.297890 0.954600i \(-0.596283\pi\)
−0.297890 + 0.954600i \(0.596283\pi\)
\(60\) 0 0
\(61\) 275.000 0.577215 0.288608 0.957447i \(-0.406808\pi\)
0.288608 + 0.957447i \(0.406808\pi\)
\(62\) 0 0
\(63\) 207.000i 0.413961i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 803.000i − 1.46421i −0.681192 0.732105i \(-0.738538\pi\)
0.681192 0.732105i \(-0.261462\pi\)
\(68\) 0 0
\(69\) 450.000 0.785125
\(70\) 0 0
\(71\) −660.000 −1.10321 −0.551603 0.834107i \(-0.685984\pi\)
−0.551603 + 0.834107i \(0.685984\pi\)
\(72\) 0 0
\(73\) 646.000i 1.03573i 0.855461 + 0.517867i \(0.173274\pi\)
−0.855461 + 0.517867i \(0.826726\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 690.000i − 1.02121i
\(78\) 0 0
\(79\) 992.000 1.41277 0.706384 0.707829i \(-0.250325\pi\)
0.706384 + 0.707829i \(0.250325\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 846.000i − 1.11880i −0.828897 0.559401i \(-0.811031\pi\)
0.828897 0.559401i \(-0.188969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 702.000i − 0.865084i
\(88\) 0 0
\(89\) 1488.00 1.77222 0.886111 0.463474i \(-0.153397\pi\)
0.886111 + 0.463474i \(0.153397\pi\)
\(90\) 0 0
\(91\) −667.000 −0.768358
\(92\) 0 0
\(93\) − 651.000i − 0.725866i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 319.000i − 0.333913i −0.985964 0.166956i \(-0.946606\pi\)
0.985964 0.166956i \(-0.0533939\pi\)
\(98\) 0 0
\(99\) −270.000 −0.274101
\(100\) 0 0
\(101\) −792.000 −0.780267 −0.390133 0.920758i \(-0.627571\pi\)
−0.390133 + 0.920758i \(0.627571\pi\)
\(102\) 0 0
\(103\) 812.000i 0.776784i 0.921494 + 0.388392i \(0.126969\pi\)
−0.921494 + 0.388392i \(0.873031\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1416.00i 1.27934i 0.768648 + 0.639672i \(0.220930\pi\)
−0.768648 + 0.639672i \(0.779070\pi\)
\(108\) 0 0
\(109\) 55.0000 0.0483307 0.0241653 0.999708i \(-0.492307\pi\)
0.0241653 + 0.999708i \(0.492307\pi\)
\(110\) 0 0
\(111\) 438.000 0.374533
\(112\) 0 0
\(113\) − 1404.00i − 1.16882i −0.811457 0.584412i \(-0.801325\pi\)
0.811457 0.584412i \(-0.198675\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 261.000i 0.206235i
\(118\) 0 0
\(119\) 1794.00 1.38198
\(120\) 0 0
\(121\) −431.000 −0.323817
\(122\) 0 0
\(123\) 468.000i 0.343074i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1280.00i − 0.894344i −0.894448 0.447172i \(-0.852431\pi\)
0.894448 0.447172i \(-0.147569\pi\)
\(128\) 0 0
\(129\) −1299.00 −0.886594
\(130\) 0 0
\(131\) −480.000 −0.320136 −0.160068 0.987106i \(-0.551171\pi\)
−0.160068 + 0.987106i \(0.551171\pi\)
\(132\) 0 0
\(133\) − 3427.00i − 2.23428i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 282.000i − 0.175860i −0.996127 0.0879302i \(-0.971975\pi\)
0.996127 0.0879302i \(-0.0280253\pi\)
\(138\) 0 0
\(139\) 1604.00 0.978773 0.489387 0.872067i \(-0.337221\pi\)
0.489387 + 0.872067i \(0.337221\pi\)
\(140\) 0 0
\(141\) −90.0000 −0.0537544
\(142\) 0 0
\(143\) − 870.000i − 0.508763i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 558.000i 0.313082i
\(148\) 0 0
\(149\) 774.000 0.425561 0.212780 0.977100i \(-0.431748\pi\)
0.212780 + 0.977100i \(0.431748\pi\)
\(150\) 0 0
\(151\) −293.000 −0.157907 −0.0789536 0.996878i \(-0.525158\pi\)
−0.0789536 + 0.996878i \(0.525158\pi\)
\(152\) 0 0
\(153\) − 702.000i − 0.370937i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1729.00i − 0.878912i −0.898264 0.439456i \(-0.855171\pi\)
0.898264 0.439456i \(-0.144829\pi\)
\(158\) 0 0
\(159\) 1656.00 0.825971
\(160\) 0 0
\(161\) 3450.00 1.68881
\(162\) 0 0
\(163\) − 1123.00i − 0.539633i −0.962912 0.269816i \(-0.913037\pi\)
0.962912 0.269816i \(-0.0869630\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1200.00i − 0.556041i −0.960575 0.278020i \(-0.910322\pi\)
0.960575 0.278020i \(-0.0896783\pi\)
\(168\) 0 0
\(169\) 1356.00 0.617205
\(170\) 0 0
\(171\) −1341.00 −0.599701
\(172\) 0 0
\(173\) 1734.00i 0.762044i 0.924566 + 0.381022i \(0.124428\pi\)
−0.924566 + 0.381022i \(0.875572\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 810.000i 0.343974i
\(178\) 0 0
\(179\) 2586.00 1.07981 0.539907 0.841725i \(-0.318459\pi\)
0.539907 + 0.841725i \(0.318459\pi\)
\(180\) 0 0
\(181\) −3931.00 −1.61430 −0.807152 0.590344i \(-0.798992\pi\)
−0.807152 + 0.590344i \(0.798992\pi\)
\(182\) 0 0
\(183\) − 825.000i − 0.333255i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2340.00i 0.915068i
\(188\) 0 0
\(189\) 621.000 0.239001
\(190\) 0 0
\(191\) −1566.00 −0.593255 −0.296628 0.954993i \(-0.595862\pi\)
−0.296628 + 0.954993i \(0.595862\pi\)
\(192\) 0 0
\(193\) − 2291.00i − 0.854455i −0.904144 0.427227i \(-0.859490\pi\)
0.904144 0.427227i \(-0.140510\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2142.00i 0.774676i 0.921938 + 0.387338i \(0.126605\pi\)
−0.921938 + 0.387338i \(0.873395\pi\)
\(198\) 0 0
\(199\) −4903.00 −1.74656 −0.873278 0.487223i \(-0.838010\pi\)
−0.873278 + 0.487223i \(0.838010\pi\)
\(200\) 0 0
\(201\) −2409.00 −0.845362
\(202\) 0 0
\(203\) − 5382.00i − 1.86080i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1350.00i − 0.453292i
\(208\) 0 0
\(209\) 4470.00 1.47941
\(210\) 0 0
\(211\) −605.000 −0.197393 −0.0986965 0.995118i \(-0.531467\pi\)
−0.0986965 + 0.995118i \(0.531467\pi\)
\(212\) 0 0
\(213\) 1980.00i 0.636936i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4991.00i − 1.56134i
\(218\) 0 0
\(219\) 1938.00 0.597981
\(220\) 0 0
\(221\) 2262.00 0.688500
\(222\) 0 0
\(223\) − 145.000i − 0.0435422i −0.999763 0.0217711i \(-0.993069\pi\)
0.999763 0.0217711i \(-0.00693051\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2964.00i − 0.866641i −0.901240 0.433321i \(-0.857342\pi\)
0.901240 0.433321i \(-0.142658\pi\)
\(228\) 0 0
\(229\) 5635.00 1.62608 0.813038 0.582211i \(-0.197812\pi\)
0.813038 + 0.582211i \(0.197812\pi\)
\(230\) 0 0
\(231\) −2070.00 −0.589593
\(232\) 0 0
\(233\) − 4164.00i − 1.17078i −0.810750 0.585392i \(-0.800941\pi\)
0.810750 0.585392i \(-0.199059\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 2976.00i − 0.815662i
\(238\) 0 0
\(239\) −1944.00 −0.526138 −0.263069 0.964777i \(-0.584735\pi\)
−0.263069 + 0.964777i \(0.584735\pi\)
\(240\) 0 0
\(241\) 857.000 0.229063 0.114532 0.993420i \(-0.463463\pi\)
0.114532 + 0.993420i \(0.463463\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4321.00i − 1.11311i
\(248\) 0 0
\(249\) −2538.00 −0.645941
\(250\) 0 0
\(251\) 3924.00 0.986776 0.493388 0.869809i \(-0.335758\pi\)
0.493388 + 0.869809i \(0.335758\pi\)
\(252\) 0 0
\(253\) 4500.00i 1.11823i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2844.00i − 0.690287i −0.938550 0.345144i \(-0.887830\pi\)
0.938550 0.345144i \(-0.112170\pi\)
\(258\) 0 0
\(259\) 3358.00 0.805621
\(260\) 0 0
\(261\) −2106.00 −0.499456
\(262\) 0 0
\(263\) − 6060.00i − 1.42082i −0.703788 0.710410i \(-0.748510\pi\)
0.703788 0.710410i \(-0.251490\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4464.00i − 1.02319i
\(268\) 0 0
\(269\) −3906.00 −0.885327 −0.442664 0.896688i \(-0.645966\pi\)
−0.442664 + 0.896688i \(0.645966\pi\)
\(270\) 0 0
\(271\) −2144.00 −0.480586 −0.240293 0.970700i \(-0.577243\pi\)
−0.240293 + 0.970700i \(0.577243\pi\)
\(272\) 0 0
\(273\) 2001.00i 0.443612i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2321.00i 0.503449i 0.967799 + 0.251725i \(0.0809977\pi\)
−0.967799 + 0.251725i \(0.919002\pi\)
\(278\) 0 0
\(279\) −1953.00 −0.419079
\(280\) 0 0
\(281\) −6822.00 −1.44828 −0.724140 0.689654i \(-0.757763\pi\)
−0.724140 + 0.689654i \(0.757763\pi\)
\(282\) 0 0
\(283\) 4049.00i 0.850488i 0.905079 + 0.425244i \(0.139812\pi\)
−0.905079 + 0.425244i \(0.860188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3588.00i 0.737955i
\(288\) 0 0
\(289\) −1171.00 −0.238347
\(290\) 0 0
\(291\) −957.000 −0.192785
\(292\) 0 0
\(293\) − 2238.00i − 0.446230i −0.974792 0.223115i \(-0.928377\pi\)
0.974792 0.223115i \(-0.0716225\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 810.000i 0.158252i
\(298\) 0 0
\(299\) 4350.00 0.841361
\(300\) 0 0
\(301\) −9959.00 −1.90707
\(302\) 0 0
\(303\) 2376.00i 0.450487i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1385.00i − 0.257479i −0.991678 0.128740i \(-0.958907\pi\)
0.991678 0.128740i \(-0.0410931\pi\)
\(308\) 0 0
\(309\) 2436.00 0.448476
\(310\) 0 0
\(311\) 5670.00 1.03381 0.516907 0.856042i \(-0.327083\pi\)
0.516907 + 0.856042i \(0.327083\pi\)
\(312\) 0 0
\(313\) 421.000i 0.0760266i 0.999277 + 0.0380133i \(0.0121029\pi\)
−0.999277 + 0.0380133i \(0.987897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9984.00i 1.76895i 0.466587 + 0.884475i \(0.345483\pi\)
−0.466587 + 0.884475i \(0.654517\pi\)
\(318\) 0 0
\(319\) 7020.00 1.23211
\(320\) 0 0
\(321\) 4248.00 0.738630
\(322\) 0 0
\(323\) 11622.0i 2.00206i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 165.000i − 0.0279037i
\(328\) 0 0
\(329\) −690.000 −0.115626
\(330\) 0 0
\(331\) 4228.00 0.702090 0.351045 0.936359i \(-0.385826\pi\)
0.351045 + 0.936359i \(0.385826\pi\)
\(332\) 0 0
\(333\) − 1314.00i − 0.216237i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5393.00i 0.871737i 0.900010 + 0.435869i \(0.143559\pi\)
−0.900010 + 0.435869i \(0.856441\pi\)
\(338\) 0 0
\(339\) −4212.00 −0.674821
\(340\) 0 0
\(341\) 6510.00 1.03383
\(342\) 0 0
\(343\) − 3611.00i − 0.568442i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7914.00i 1.22434i 0.790726 + 0.612170i \(0.209703\pi\)
−0.790726 + 0.612170i \(0.790297\pi\)
\(348\) 0 0
\(349\) −1010.00 −0.154911 −0.0774557 0.996996i \(-0.524680\pi\)
−0.0774557 + 0.996996i \(0.524680\pi\)
\(350\) 0 0
\(351\) 783.000 0.119070
\(352\) 0 0
\(353\) − 4722.00i − 0.711974i −0.934491 0.355987i \(-0.884145\pi\)
0.934491 0.355987i \(-0.115855\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 5382.00i − 0.797887i
\(358\) 0 0
\(359\) 6204.00 0.912074 0.456037 0.889961i \(-0.349268\pi\)
0.456037 + 0.889961i \(0.349268\pi\)
\(360\) 0 0
\(361\) 15342.0 2.23677
\(362\) 0 0
\(363\) 1293.00i 0.186956i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 1361.00i − 0.193579i −0.995305 0.0967897i \(-0.969143\pi\)
0.995305 0.0967897i \(-0.0308574\pi\)
\(368\) 0 0
\(369\) 1404.00 0.198074
\(370\) 0 0
\(371\) 12696.0 1.77667
\(372\) 0 0
\(373\) 913.000i 0.126738i 0.997990 + 0.0633691i \(0.0201845\pi\)
−0.997990 + 0.0633691i \(0.979815\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 6786.00i − 0.927047i
\(378\) 0 0
\(379\) −8881.00 −1.20366 −0.601829 0.798625i \(-0.705561\pi\)
−0.601829 + 0.798625i \(0.705561\pi\)
\(380\) 0 0
\(381\) −3840.00 −0.516350
\(382\) 0 0
\(383\) 5460.00i 0.728441i 0.931313 + 0.364221i \(0.118665\pi\)
−0.931313 + 0.364221i \(0.881335\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3897.00i 0.511875i
\(388\) 0 0
\(389\) 13884.0 1.80963 0.904816 0.425803i \(-0.140008\pi\)
0.904816 + 0.425803i \(0.140008\pi\)
\(390\) 0 0
\(391\) −11700.0 −1.51328
\(392\) 0 0
\(393\) 1440.00i 0.184831i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 3781.00i − 0.477992i −0.971021 0.238996i \(-0.923182\pi\)
0.971021 0.238996i \(-0.0768183\pi\)
\(398\) 0 0
\(399\) −10281.0 −1.28996
\(400\) 0 0
\(401\) 9024.00 1.12378 0.561892 0.827211i \(-0.310074\pi\)
0.561892 + 0.827211i \(0.310074\pi\)
\(402\) 0 0
\(403\) − 6293.00i − 0.777858i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4380.00i 0.533436i
\(408\) 0 0
\(409\) −14789.0 −1.78794 −0.893972 0.448123i \(-0.852093\pi\)
−0.893972 + 0.448123i \(0.852093\pi\)
\(410\) 0 0
\(411\) −846.000 −0.101533
\(412\) 0 0
\(413\) 6210.00i 0.739889i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4812.00i − 0.565095i
\(418\) 0 0
\(419\) 9840.00 1.14729 0.573646 0.819103i \(-0.305528\pi\)
0.573646 + 0.819103i \(0.305528\pi\)
\(420\) 0 0
\(421\) 5510.00 0.637865 0.318932 0.947778i \(-0.396676\pi\)
0.318932 + 0.947778i \(0.396676\pi\)
\(422\) 0 0
\(423\) 270.000i 0.0310351i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6325.00i − 0.716834i
\(428\) 0 0
\(429\) −2610.00 −0.293734
\(430\) 0 0
\(431\) −11070.0 −1.23718 −0.618588 0.785715i \(-0.712295\pi\)
−0.618588 + 0.785715i \(0.712295\pi\)
\(432\) 0 0
\(433\) 12133.0i 1.34659i 0.739373 + 0.673297i \(0.235122\pi\)
−0.739373 + 0.673297i \(0.764878\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22350.0i 2.44656i
\(438\) 0 0
\(439\) −1873.00 −0.203630 −0.101815 0.994803i \(-0.532465\pi\)
−0.101815 + 0.994803i \(0.532465\pi\)
\(440\) 0 0
\(441\) 1674.00 0.180758
\(442\) 0 0
\(443\) − 576.000i − 0.0617756i −0.999523 0.0308878i \(-0.990167\pi\)
0.999523 0.0308878i \(-0.00983345\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 2322.00i − 0.245698i
\(448\) 0 0
\(449\) 4884.00 0.513341 0.256671 0.966499i \(-0.417374\pi\)
0.256671 + 0.966499i \(0.417374\pi\)
\(450\) 0 0
\(451\) −4680.00 −0.488631
\(452\) 0 0
\(453\) 879.000i 0.0911678i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 15802.0i − 1.61748i −0.588169 0.808738i \(-0.700151\pi\)
0.588169 0.808738i \(-0.299849\pi\)
\(458\) 0 0
\(459\) −2106.00 −0.214160
\(460\) 0 0
\(461\) −15360.0 −1.55181 −0.775907 0.630847i \(-0.782708\pi\)
−0.775907 + 0.630847i \(0.782708\pi\)
\(462\) 0 0
\(463\) 1712.00i 0.171843i 0.996302 + 0.0859216i \(0.0273835\pi\)
−0.996302 + 0.0859216i \(0.972617\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16278.0i 1.61297i 0.591256 + 0.806484i \(0.298632\pi\)
−0.591256 + 0.806484i \(0.701368\pi\)
\(468\) 0 0
\(469\) −18469.0 −1.81838
\(470\) 0 0
\(471\) −5187.00 −0.507440
\(472\) 0 0
\(473\) − 12990.0i − 1.26275i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4968.00i − 0.476874i
\(478\) 0 0
\(479\) −14766.0 −1.40851 −0.704254 0.709948i \(-0.748719\pi\)
−0.704254 + 0.709948i \(0.748719\pi\)
\(480\) 0 0
\(481\) 4234.00 0.401359
\(482\) 0 0
\(483\) − 10350.0i − 0.975034i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3319.00i 0.308826i 0.988006 + 0.154413i \(0.0493486\pi\)
−0.988006 + 0.154413i \(0.950651\pi\)
\(488\) 0 0
\(489\) −3369.00 −0.311557
\(490\) 0 0
\(491\) −11064.0 −1.01693 −0.508464 0.861083i \(-0.669786\pi\)
−0.508464 + 0.861083i \(0.669786\pi\)
\(492\) 0 0
\(493\) 18252.0i 1.66740i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15180.0i 1.37005i
\(498\) 0 0
\(499\) −14131.0 −1.26772 −0.633858 0.773449i \(-0.718530\pi\)
−0.633858 + 0.773449i \(0.718530\pi\)
\(500\) 0 0
\(501\) −3600.00 −0.321030
\(502\) 0 0
\(503\) 11988.0i 1.06266i 0.847165 + 0.531331i \(0.178308\pi\)
−0.847165 + 0.531331i \(0.821692\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 4068.00i − 0.356344i
\(508\) 0 0
\(509\) −10806.0 −0.940997 −0.470499 0.882401i \(-0.655926\pi\)
−0.470499 + 0.882401i \(0.655926\pi\)
\(510\) 0 0
\(511\) 14858.0 1.28626
\(512\) 0 0
\(513\) 4023.00i 0.346237i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 900.000i − 0.0765608i
\(518\) 0 0
\(519\) 5202.00 0.439966
\(520\) 0 0
\(521\) 22578.0 1.89858 0.949290 0.314402i \(-0.101804\pi\)
0.949290 + 0.314402i \(0.101804\pi\)
\(522\) 0 0
\(523\) 12065.0i 1.00873i 0.863491 + 0.504365i \(0.168273\pi\)
−0.863491 + 0.504365i \(0.831727\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16926.0i 1.39907i
\(528\) 0 0
\(529\) −10333.0 −0.849264
\(530\) 0 0
\(531\) 2430.00 0.198593
\(532\) 0 0
\(533\) 4524.00i 0.367648i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 7758.00i − 0.623431i
\(538\) 0 0
\(539\) −5580.00 −0.445914
\(540\) 0 0
\(541\) −12055.0 −0.958013 −0.479006 0.877811i \(-0.659003\pi\)
−0.479006 + 0.877811i \(0.659003\pi\)
\(542\) 0 0
\(543\) 11793.0i 0.932019i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 6176.00i − 0.482754i −0.970431 0.241377i \(-0.922401\pi\)
0.970431 0.241377i \(-0.0775991\pi\)
\(548\) 0 0
\(549\) −2475.00 −0.192405
\(550\) 0 0
\(551\) 34866.0 2.69572
\(552\) 0 0
\(553\) − 22816.0i − 1.75449i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8274.00i 0.629409i 0.949190 + 0.314704i \(0.101905\pi\)
−0.949190 + 0.314704i \(0.898095\pi\)
\(558\) 0 0
\(559\) −12557.0 −0.950098
\(560\) 0 0
\(561\) 7020.00 0.528315
\(562\) 0 0
\(563\) 966.000i 0.0723127i 0.999346 + 0.0361563i \(0.0115114\pi\)
−0.999346 + 0.0361563i \(0.988489\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1863.00i − 0.137987i
\(568\) 0 0
\(569\) −19002.0 −1.40001 −0.700005 0.714138i \(-0.746819\pi\)
−0.700005 + 0.714138i \(0.746819\pi\)
\(570\) 0 0
\(571\) −8645.00 −0.633594 −0.316797 0.948493i \(-0.602607\pi\)
−0.316797 + 0.948493i \(0.602607\pi\)
\(572\) 0 0
\(573\) 4698.00i 0.342516i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10931.0i 0.788672i 0.918966 + 0.394336i \(0.129025\pi\)
−0.918966 + 0.394336i \(0.870975\pi\)
\(578\) 0 0
\(579\) −6873.00 −0.493320
\(580\) 0 0
\(581\) −19458.0 −1.38942
\(582\) 0 0
\(583\) 16560.0i 1.17641i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 8904.00i − 0.626077i −0.949740 0.313039i \(-0.898653\pi\)
0.949740 0.313039i \(-0.101347\pi\)
\(588\) 0 0
\(589\) 32333.0 2.26190
\(590\) 0 0
\(591\) 6426.00 0.447259
\(592\) 0 0
\(593\) − 8820.00i − 0.610782i −0.952227 0.305391i \(-0.901213\pi\)
0.952227 0.305391i \(-0.0987872\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 14709.0i 1.00837i
\(598\) 0 0
\(599\) 9804.00 0.668749 0.334374 0.942440i \(-0.391475\pi\)
0.334374 + 0.942440i \(0.391475\pi\)
\(600\) 0 0
\(601\) −23437.0 −1.59071 −0.795354 0.606146i \(-0.792715\pi\)
−0.795354 + 0.606146i \(0.792715\pi\)
\(602\) 0 0
\(603\) 7227.00i 0.488070i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 2648.00i − 0.177066i −0.996073 0.0885330i \(-0.971782\pi\)
0.996073 0.0885330i \(-0.0282179\pi\)
\(608\) 0 0
\(609\) −16146.0 −1.07433
\(610\) 0 0
\(611\) −870.000 −0.0576046
\(612\) 0 0
\(613\) − 794.000i − 0.0523154i −0.999658 0.0261577i \(-0.991673\pi\)
0.999658 0.0261577i \(-0.00832721\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18720.0i 1.22146i 0.791840 + 0.610728i \(0.209123\pi\)
−0.791840 + 0.610728i \(0.790877\pi\)
\(618\) 0 0
\(619\) −8959.00 −0.581733 −0.290866 0.956764i \(-0.593944\pi\)
−0.290866 + 0.956764i \(0.593944\pi\)
\(620\) 0 0
\(621\) −4050.00 −0.261708
\(622\) 0 0
\(623\) − 34224.0i − 2.20089i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 13410.0i − 0.854137i
\(628\) 0 0
\(629\) −11388.0 −0.721891
\(630\) 0 0
\(631\) 12373.0 0.780604 0.390302 0.920687i \(-0.372371\pi\)
0.390302 + 0.920687i \(0.372371\pi\)
\(632\) 0 0
\(633\) 1815.00i 0.113965i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5394.00i 0.335507i
\(638\) 0 0
\(639\) 5940.00 0.367735
\(640\) 0 0
\(641\) 24900.0 1.53431 0.767154 0.641463i \(-0.221672\pi\)
0.767154 + 0.641463i \(0.221672\pi\)
\(642\) 0 0
\(643\) − 14668.0i − 0.899610i −0.893127 0.449805i \(-0.851493\pi\)
0.893127 0.449805i \(-0.148507\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10788.0i 0.655518i 0.944761 + 0.327759i \(0.106293\pi\)
−0.944761 + 0.327759i \(0.893707\pi\)
\(648\) 0 0
\(649\) −8100.00 −0.489912
\(650\) 0 0
\(651\) −14973.0 −0.901441
\(652\) 0 0
\(653\) 14214.0i 0.851817i 0.904766 + 0.425909i \(0.140046\pi\)
−0.904766 + 0.425909i \(0.859954\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 5814.00i − 0.345245i
\(658\) 0 0
\(659\) −588.000 −0.0347576 −0.0173788 0.999849i \(-0.505532\pi\)
−0.0173788 + 0.999849i \(0.505532\pi\)
\(660\) 0 0
\(661\) −3166.00 −0.186298 −0.0931491 0.995652i \(-0.529693\pi\)
−0.0931491 + 0.995652i \(0.529693\pi\)
\(662\) 0 0
\(663\) − 6786.00i − 0.397506i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35100.0i 2.03760i
\(668\) 0 0
\(669\) −435.000 −0.0251391
\(670\) 0 0
\(671\) 8250.00 0.474646
\(672\) 0 0
\(673\) − 9182.00i − 0.525914i −0.964808 0.262957i \(-0.915302\pi\)
0.964808 0.262957i \(-0.0846977\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11742.0i 0.666590i 0.942823 + 0.333295i \(0.108161\pi\)
−0.942823 + 0.333295i \(0.891839\pi\)
\(678\) 0 0
\(679\) −7337.00 −0.414681
\(680\) 0 0
\(681\) −8892.00 −0.500356
\(682\) 0 0
\(683\) − 6024.00i − 0.337485i −0.985660 0.168742i \(-0.946029\pi\)
0.985660 0.168742i \(-0.0539706\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 16905.0i − 0.938815i
\(688\) 0 0
\(689\) 16008.0 0.885132
\(690\) 0 0
\(691\) −9344.00 −0.514418 −0.257209 0.966356i \(-0.582803\pi\)
−0.257209 + 0.966356i \(0.582803\pi\)
\(692\) 0 0
\(693\) 6210.00i 0.340402i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 12168.0i − 0.661257i
\(698\) 0 0
\(699\) −12492.0 −0.675953
\(700\) 0 0
\(701\) 21234.0 1.14408 0.572038 0.820227i \(-0.306153\pi\)
0.572038 + 0.820227i \(0.306153\pi\)
\(702\) 0 0
\(703\) 21754.0i 1.16709i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18216.0i 0.969000i
\(708\) 0 0
\(709\) 1723.00 0.0912675 0.0456337 0.998958i \(-0.485469\pi\)
0.0456337 + 0.998958i \(0.485469\pi\)
\(710\) 0 0
\(711\) −8928.00 −0.470923
\(712\) 0 0
\(713\) 32550.0i 1.70969i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5832.00i 0.303766i
\(718\) 0 0
\(719\) 18510.0 0.960093 0.480046 0.877243i \(-0.340620\pi\)
0.480046 + 0.877243i \(0.340620\pi\)
\(720\) 0 0
\(721\) 18676.0 0.964675
\(722\) 0 0
\(723\) − 2571.00i − 0.132250i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1009.00i 0.0514742i 0.999669 + 0.0257371i \(0.00819328\pi\)
−0.999669 + 0.0257371i \(0.991807\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 33774.0 1.70886
\(732\) 0 0
\(733\) 21994.0i 1.10828i 0.832425 + 0.554138i \(0.186952\pi\)
−0.832425 + 0.554138i \(0.813048\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 24090.0i − 1.20403i
\(738\) 0 0
\(739\) −13948.0 −0.694297 −0.347148 0.937810i \(-0.612850\pi\)
−0.347148 + 0.937810i \(0.612850\pi\)
\(740\) 0 0
\(741\) −12963.0 −0.642655
\(742\) 0 0
\(743\) − 26508.0i − 1.30886i −0.756122 0.654431i \(-0.772908\pi\)
0.756122 0.654431i \(-0.227092\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 7614.00i 0.372934i
\(748\) 0 0
\(749\) 32568.0 1.58880
\(750\) 0 0
\(751\) 1600.00 0.0777428 0.0388714 0.999244i \(-0.487624\pi\)
0.0388714 + 0.999244i \(0.487624\pi\)
\(752\) 0 0
\(753\) − 11772.0i − 0.569715i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30101.0i 1.44523i 0.691250 + 0.722615i \(0.257060\pi\)
−0.691250 + 0.722615i \(0.742940\pi\)
\(758\) 0 0
\(759\) 13500.0 0.645611
\(760\) 0 0
\(761\) 35628.0 1.69713 0.848564 0.529093i \(-0.177468\pi\)
0.848564 + 0.529093i \(0.177468\pi\)
\(762\) 0 0
\(763\) − 1265.00i − 0.0600211i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7830.00i 0.368611i
\(768\) 0 0
\(769\) 12517.0 0.586963 0.293482 0.955965i \(-0.405186\pi\)
0.293482 + 0.955965i \(0.405186\pi\)
\(770\) 0 0
\(771\) −8532.00 −0.398538
\(772\) 0 0
\(773\) − 14124.0i − 0.657186i −0.944472 0.328593i \(-0.893426\pi\)
0.944472 0.328593i \(-0.106574\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 10074.0i − 0.465126i
\(778\) 0 0
\(779\) −23244.0 −1.06907
\(780\) 0 0
\(781\) −19800.0 −0.907170
\(782\) 0 0
\(783\) 6318.00i 0.288361i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 40433.0i − 1.83136i −0.401907 0.915680i \(-0.631653\pi\)
0.401907 0.915680i \(-0.368347\pi\)
\(788\) 0 0
\(789\) −18180.0 −0.820311
\(790\) 0 0
\(791\) −32292.0 −1.45154
\(792\) 0 0
\(793\) − 7975.00i − 0.357126i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 27300.0i − 1.21332i −0.794962 0.606660i \(-0.792509\pi\)
0.794962 0.606660i \(-0.207491\pi\)
\(798\) 0 0
\(799\) 2340.00 0.103609
\(800\) 0 0
\(801\) −13392.0 −0.590740
\(802\) 0 0
\(803\) 19380.0i 0.851688i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 11718.0i 0.511144i
\(808\) 0 0
\(809\) 2856.00 0.124118 0.0620591 0.998072i \(-0.480233\pi\)
0.0620591 + 0.998072i \(0.480233\pi\)
\(810\) 0 0
\(811\) 12619.0 0.546379 0.273189 0.961960i \(-0.411921\pi\)
0.273189 + 0.961960i \(0.411921\pi\)
\(812\) 0 0
\(813\) 6432.00i 0.277466i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 64517.0i − 2.76275i
\(818\) 0 0
\(819\) 6003.00 0.256119
\(820\) 0 0
\(821\) −29082.0 −1.23626 −0.618130 0.786076i \(-0.712109\pi\)
−0.618130 + 0.786076i \(0.712109\pi\)
\(822\) 0 0
\(823\) 10235.0i 0.433499i 0.976227 + 0.216749i \(0.0695455\pi\)
−0.976227 + 0.216749i \(0.930455\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 26976.0i − 1.13428i −0.823622 0.567139i \(-0.808050\pi\)
0.823622 0.567139i \(-0.191950\pi\)
\(828\) 0 0
\(829\) −37802.0 −1.58374 −0.791868 0.610692i \(-0.790891\pi\)
−0.791868 + 0.610692i \(0.790891\pi\)
\(830\) 0 0
\(831\) 6963.00 0.290666
\(832\) 0 0
\(833\) − 14508.0i − 0.603448i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5859.00i 0.241955i
\(838\) 0 0
\(839\) −16974.0 −0.698460 −0.349230 0.937037i \(-0.613557\pi\)
−0.349230 + 0.937037i \(0.613557\pi\)
\(840\) 0 0
\(841\) 30367.0 1.24511
\(842\) 0 0
\(843\) 20466.0i 0.836164i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9913.00i 0.402143i
\(848\) 0 0
\(849\) 12147.0 0.491029
\(850\) 0 0
\(851\) −21900.0 −0.882165
\(852\) 0 0
\(853\) 24937.0i 1.00097i 0.865745 + 0.500485i \(0.166845\pi\)
−0.865745 + 0.500485i \(0.833155\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15756.0i 0.628022i 0.949419 + 0.314011i \(0.101673\pi\)
−0.949419 + 0.314011i \(0.898327\pi\)
\(858\) 0 0
\(859\) 38144.0 1.51508 0.757542 0.652787i \(-0.226400\pi\)
0.757542 + 0.652787i \(0.226400\pi\)
\(860\) 0 0
\(861\) 10764.0 0.426058
\(862\) 0 0
\(863\) 5448.00i 0.214892i 0.994211 + 0.107446i \(0.0342673\pi\)
−0.994211 + 0.107446i \(0.965733\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3513.00i 0.137610i
\(868\) 0 0
\(869\) 29760.0 1.16172
\(870\) 0 0
\(871\) −23287.0 −0.905913
\(872\) 0 0
\(873\) 2871.00i 0.111304i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 21191.0i 0.815928i 0.912998 + 0.407964i \(0.133761\pi\)
−0.912998 + 0.407964i \(0.866239\pi\)
\(878\) 0 0
\(879\) −6714.00 −0.257631
\(880\) 0 0
\(881\) 18216.0 0.696609 0.348305 0.937381i \(-0.386758\pi\)
0.348305 + 0.937381i \(0.386758\pi\)
\(882\) 0 0
\(883\) 12767.0i 0.486573i 0.969955 + 0.243286i \(0.0782255\pi\)
−0.969955 + 0.243286i \(0.921775\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 11010.0i − 0.416775i −0.978046 0.208388i \(-0.933178\pi\)
0.978046 0.208388i \(-0.0668215\pi\)
\(888\) 0 0
\(889\) −29440.0 −1.11067
\(890\) 0 0
\(891\) 2430.00 0.0913671
\(892\) 0 0
\(893\) − 4470.00i − 0.167506i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 13050.0i − 0.485760i
\(898\) 0 0
\(899\) 50778.0 1.88381
\(900\) 0 0
\(901\) −43056.0 −1.59201
\(902\) 0 0
\(903\) 29877.0i 1.10105i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 22772.0i − 0.833662i −0.908984 0.416831i \(-0.863141\pi\)
0.908984 0.416831i \(-0.136859\pi\)
\(908\) 0 0
\(909\) 7128.00 0.260089
\(910\) 0 0
\(911\) −29802.0 −1.08385 −0.541923 0.840428i \(-0.682304\pi\)
−0.541923 + 0.840428i \(0.682304\pi\)
\(912\) 0 0
\(913\) − 25380.0i − 0.919995i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11040.0i 0.397571i
\(918\) 0 0
\(919\) 48941.0 1.75671 0.878354 0.478011i \(-0.158642\pi\)
0.878354 + 0.478011i \(0.158642\pi\)
\(920\) 0 0
\(921\) −4155.00 −0.148656
\(922\) 0 0
\(923\) 19140.0i 0.682558i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 7308.00i − 0.258928i
\(928\) 0 0
\(929\) −31026.0 −1.09573 −0.547863 0.836568i \(-0.684559\pi\)
−0.547863 + 0.836568i \(0.684559\pi\)
\(930\) 0 0
\(931\) −27714.0 −0.975607
\(932\) 0 0
\(933\) − 17010.0i − 0.596873i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11183.0i 0.389896i 0.980814 + 0.194948i \(0.0624538\pi\)
−0.980814 + 0.194948i \(0.937546\pi\)
\(938\) 0 0
\(939\) 1263.00 0.0438940
\(940\) 0 0
\(941\) −2562.00 −0.0887554 −0.0443777 0.999015i \(-0.514130\pi\)
−0.0443777 + 0.999015i \(0.514130\pi\)
\(942\) 0 0
\(943\) − 23400.0i − 0.808069i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7638.00i − 0.262093i −0.991376 0.131046i \(-0.958166\pi\)
0.991376 0.131046i \(-0.0418336\pi\)
\(948\) 0 0
\(949\) 18734.0 0.640813
\(950\) 0 0
\(951\) 29952.0 1.02130
\(952\) 0 0
\(953\) − 51432.0i − 1.74821i −0.485735 0.874106i \(-0.661448\pi\)
0.485735 0.874106i \(-0.338552\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 21060.0i − 0.711362i
\(958\) 0 0
\(959\) −6486.00 −0.218398
\(960\) 0 0
\(961\) 17298.0 0.580645
\(962\) 0 0
\(963\) − 12744.0i − 0.426448i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 39728.0i − 1.32116i −0.750754 0.660582i \(-0.770309\pi\)
0.750754 0.660582i \(-0.229691\pi\)
\(968\) 0 0
\(969\) 34866.0 1.15589
\(970\) 0 0
\(971\) 47946.0 1.58461 0.792307 0.610123i \(-0.208880\pi\)
0.792307 + 0.610123i \(0.208880\pi\)
\(972\) 0 0
\(973\) − 36892.0i − 1.21552i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 22326.0i − 0.731087i −0.930794 0.365544i \(-0.880883\pi\)
0.930794 0.365544i \(-0.119117\pi\)
\(978\) 0 0
\(979\) 44640.0 1.45730
\(980\) 0 0
\(981\) −495.000 −0.0161102
\(982\) 0 0
\(983\) − 48468.0i − 1.57262i −0.617830 0.786312i \(-0.711988\pi\)
0.617830 0.786312i \(-0.288012\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2070.00i 0.0667567i
\(988\) 0 0
\(989\) 64950.0 2.08826
\(990\) 0 0
\(991\) 25141.0 0.805883 0.402942 0.915226i \(-0.367988\pi\)
0.402942 + 0.915226i \(0.367988\pi\)
\(992\) 0 0
\(993\) − 12684.0i − 0.405352i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 35422.0i − 1.12520i −0.826729 0.562601i \(-0.809801\pi\)
0.826729 0.562601i \(-0.190199\pi\)
\(998\) 0 0
\(999\) −3942.00 −0.124844
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 1200.4.f.q.49.1 2
4.3 odd 2 150.4.c.b.49.2 2
5.2 odd 4 1200.4.a.r.1.1 1
5.3 odd 4 1200.4.a.v.1.1 1
5.4 even 2 inner 1200.4.f.q.49.2 2
12.11 even 2 450.4.c.h.199.1 2
20.3 even 4 150.4.a.g.1.1 yes 1
20.7 even 4 150.4.a.c.1.1 1
20.19 odd 2 150.4.c.b.49.1 2
60.23 odd 4 450.4.a.i.1.1 1
60.47 odd 4 450.4.a.l.1.1 1
60.59 even 2 450.4.c.h.199.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.4.a.c.1.1 1 20.7 even 4
150.4.a.g.1.1 yes 1 20.3 even 4
150.4.c.b.49.1 2 20.19 odd 2
150.4.c.b.49.2 2 4.3 odd 2
450.4.a.i.1.1 1 60.23 odd 4
450.4.a.l.1.1 1 60.47 odd 4
450.4.c.h.199.1 2 12.11 even 2
450.4.c.h.199.2 2 60.59 even 2
1200.4.a.r.1.1 1 5.2 odd 4
1200.4.a.v.1.1 1 5.3 odd 4
1200.4.f.q.49.1 2 1.1 even 1 trivial
1200.4.f.q.49.2 2 5.4 even 2 inner