Properties

Label 4050.2.c.h.649.2
Level $4050$
Weight $2$
Character 4050.649
Analytic conductor $32.339$
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] = [4050,2,Mod(649,4050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4050, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4050.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.3394128186\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4050.649
Dual form 4050.2.c.h.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +4.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +7.00000 q^{19} -4.00000 q^{26} -2.00000i q^{28} +6.00000 q^{29} -10.0000 q^{31} +1.00000i q^{32} +6.00000 q^{34} +2.00000i q^{37} +7.00000i q^{38} +9.00000 q^{41} +1.00000i q^{43} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{52} +12.0000i q^{53} +2.00000 q^{56} +6.00000i q^{58} +9.00000 q^{59} -4.00000 q^{61} -10.0000i q^{62} -1.00000 q^{64} -13.0000i q^{67} +6.00000i q^{68} +6.00000 q^{71} +1.00000i q^{73} -2.00000 q^{74} -7.00000 q^{76} -2.00000 q^{79} +9.00000i q^{82} -9.00000i q^{83} -1.00000 q^{86} -15.0000 q^{89} -8.00000 q^{91} -6.00000 q^{94} +17.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{14} + 2 q^{16} + 14 q^{19} - 8 q^{26} + 12 q^{29} - 20 q^{31} + 12 q^{34} + 18 q^{41} + 6 q^{49} + 4 q^{56} + 18 q^{59} - 8 q^{61} - 2 q^{64} + 12 q^{71} - 4 q^{74} - 14 q^{76} - 4 q^{79} - 2 q^{86} - 30 q^{89} - 16 q^{91} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times\).

\(n\) \(2351\) \(3727\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 7.00000i 1.13555i
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 4.00000i − 0.554700i
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.0000i − 1.58820i −0.607785 0.794101i \(-0.707942\pi\)
0.607785 0.794101i \(-0.292058\pi\)
\(68\) 6.00000i 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 9.00000i 0.993884i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000i 1.72609i 0.505128 + 0.863044i \(0.331445\pi\)
−0.505128 + 0.863044i \(0.668555\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 9.00000i 0.870063i 0.900415 + 0.435031i \(0.143263\pi\)
−0.900415 + 0.435031i \(0.856737\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000i 0.188982i
\(113\) − 3.00000i − 0.282216i −0.989994 0.141108i \(-0.954933\pi\)
0.989994 0.141108i \(-0.0450665\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 9.00000i 0.828517i
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 4.00000i − 0.362143i
\(123\) 0 0
\(124\) 10.0000 0.898027
\(125\) 0 0
\(126\) 0 0
\(127\) 20.0000i 1.77471i 0.461084 + 0.887357i \(0.347461\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 14.0000i 1.21395i
\(134\) 13.0000 1.12303
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 3.00000i − 0.256307i −0.991754 0.128154i \(-0.959095\pi\)
0.991754 0.128154i \(-0.0409051\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000i 0.503509i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −1.00000 −0.0827606
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) − 7.00000i − 0.567775i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) − 2.00000i − 0.159111i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 7.00000i 0.548282i 0.961689 + 0.274141i \(0.0883936\pi\)
−0.961689 + 0.274141i \(0.911606\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) − 1.00000i − 0.0762493i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) − 15.0000i − 1.12430i
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) − 8.00000i − 0.592999i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 12.0000i 0.844317i
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) − 12.0000i − 0.824163i
\(213\) 0 0
\(214\) −9.00000 −0.615227
\(215\) 0 0
\(216\) 0 0
\(217\) − 20.0000i − 1.35769i
\(218\) − 2.00000i − 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 3.00000 0.199557
\(227\) − 15.0000i − 0.995585i −0.867296 0.497792i \(-0.834144\pi\)
0.867296 0.497792i \(-0.165856\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 6.00000i − 0.393919i
\(233\) − 3.00000i − 0.196537i −0.995160 0.0982683i \(-0.968670\pi\)
0.995160 0.0982683i \(-0.0313303\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.00000 −0.585850
\(237\) 0 0
\(238\) 12.0000i 0.777844i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −19.0000 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(242\) − 11.0000i − 0.707107i
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 28.0000i 1.78160i
\(248\) 10.0000i 0.635001i
\(249\) 0 0
\(250\) 0 0
\(251\) 15.0000 0.946792 0.473396 0.880850i \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.00000i 0.187135i 0.995613 + 0.0935674i \(0.0298271\pi\)
−0.995613 + 0.0935674i \(0.970173\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) − 12.0000i − 0.741362i
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −14.0000 −0.858395
\(267\) 0 0
\(268\) 13.0000i 0.794101i
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 19.0000i 1.12943i 0.825285 + 0.564716i \(0.191014\pi\)
−0.825285 + 0.564716i \(0.808986\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) 18.0000i 1.06251i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) − 1.00000i − 0.0585206i
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 6.00000i 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) − 4.00000i − 0.230174i
\(303\) 0 0
\(304\) 7.00000 0.401478
\(305\) 0 0
\(306\) 0 0
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 19.0000i 1.07394i 0.843600 + 0.536972i \(0.180432\pi\)
−0.843600 + 0.536972i \(0.819568\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 30.0000i 1.68497i 0.538721 + 0.842484i \(0.318908\pi\)
−0.538721 + 0.842484i \(0.681092\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 42.0000i − 2.33694i
\(324\) 0 0
\(325\) 0 0
\(326\) −7.00000 −0.387694
\(327\) 0 0
\(328\) − 9.00000i − 0.496942i
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 11.0000 0.604615 0.302307 0.953211i \(-0.402243\pi\)
0.302307 + 0.953211i \(0.402243\pi\)
\(332\) 9.00000i 0.493939i
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) − 3.00000i − 0.163178i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 15.0000i − 0.798369i −0.916871 0.399185i \(-0.869293\pi\)
0.916871 0.399185i \(-0.130707\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) 9.00000i 0.475665i
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) − 16.0000i − 0.840941i
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) − 10.0000i − 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −24.0000 −1.24602
\(372\) 0 0
\(373\) 4.00000i 0.207112i 0.994624 + 0.103556i \(0.0330221\pi\)
−0.994624 + 0.103556i \(0.966978\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 6.00000i 0.306987i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) − 17.0000i − 0.863044i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 3.00000i − 0.151523i
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) − 40.0000i − 1.99254i
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) 31.0000 1.53285 0.766426 0.642333i \(-0.222033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.00000i 0.0985329i
\(413\) 18.0000i 0.885722i
\(414\) 0 0
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 5.00000i 0.243396i
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.00000i − 0.387147i
\(428\) − 9.00000i − 0.435031i
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 20.0000 0.960031
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.0000i 1.14156i
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 3.00000i 0.141108i
\(453\) 0 0
\(454\) 15.0000 0.703985
\(455\) 0 0
\(456\) 0 0
\(457\) − 19.0000i − 0.888783i −0.895833 0.444391i \(-0.853420\pi\)
0.895833 0.444391i \(-0.146580\pi\)
\(458\) 10.0000i 0.467269i
\(459\) 0 0
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) − 26.0000i − 1.20832i −0.796862 0.604161i \(-0.793508\pi\)
0.796862 0.604161i \(-0.206492\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 3.00000 0.138972
\(467\) − 15.0000i − 0.694117i −0.937843 0.347059i \(-0.887180\pi\)
0.937843 0.347059i \(-0.112820\pi\)
\(468\) 0 0
\(469\) 26.0000 1.20057
\(470\) 0 0
\(471\) 0 0
\(472\) − 9.00000i − 0.414259i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 0 0
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) − 19.0000i − 0.865426i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 4.00000i 0.181071i
\(489\) 0 0
\(490\) 0 0
\(491\) −3.00000 −0.135388 −0.0676941 0.997706i \(-0.521564\pi\)
−0.0676941 + 0.997706i \(0.521564\pi\)
\(492\) 0 0
\(493\) − 36.0000i − 1.62136i
\(494\) −28.0000 −1.25978
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) 12.0000i 0.538274i
\(498\) 0 0
\(499\) −11.0000 −0.492428 −0.246214 0.969216i \(-0.579187\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 15.0000i 0.669483i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) − 20.0000i − 0.887357i
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −3.00000 −0.132324
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) − 4.00000i − 0.175750i
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 1.00000i 0.0437269i 0.999761 + 0.0218635i \(0.00695991\pi\)
−0.999761 + 0.0218635i \(0.993040\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 60.0000i 2.61364i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) − 14.0000i − 0.606977i
\(533\) 36.0000i 1.55933i
\(534\) 0 0
\(535\) 0 0
\(536\) −13.0000 −0.561514
\(537\) 0 0
\(538\) 30.0000i 1.29339i
\(539\) 0 0
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) − 28.0000i − 1.20270i
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 0 0
\(547\) 5.00000i 0.213785i 0.994271 + 0.106892i \(0.0340900\pi\)
−0.994271 + 0.106892i \(0.965910\pi\)
\(548\) 3.00000i 0.128154i
\(549\) 0 0
\(550\) 0 0
\(551\) 42.0000 1.78926
\(552\) 0 0
\(553\) − 4.00000i − 0.170097i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) − 12.0000i − 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) − 18.0000i − 0.759284i
\(563\) − 9.00000i − 0.379305i −0.981851 0.189652i \(-0.939264\pi\)
0.981851 0.189652i \(-0.0607361\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −19.0000 −0.798630
\(567\) 0 0
\(568\) − 6.00000i − 0.251754i
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 35.0000 1.46470 0.732352 0.680926i \(-0.238422\pi\)
0.732352 + 0.680926i \(0.238422\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −18.0000 −0.751305
\(575\) 0 0
\(576\) 0 0
\(577\) 35.0000i 1.45707i 0.685009 + 0.728535i \(0.259798\pi\)
−0.685009 + 0.728535i \(0.740202\pi\)
\(578\) − 19.0000i − 0.790296i
\(579\) 0 0
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) 0 0
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) −70.0000 −2.88430
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000i 0.0821995i
\(593\) 9.00000i 0.369586i 0.982777 + 0.184793i \(0.0591614\pi\)
−0.982777 + 0.184793i \(0.940839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 0 0
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) − 2.00000i − 0.0815139i
\(603\) 0 0
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 7.00000i 0.283887i
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) − 44.0000i − 1.77714i −0.458738 0.888572i \(-0.651698\pi\)
0.458738 0.888572i \(-0.348302\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) − 9.00000i − 0.362326i −0.983453 0.181163i \(-0.942014\pi\)
0.983453 0.181163i \(-0.0579862\pi\)
\(618\) 0 0
\(619\) −23.0000 −0.924448 −0.462224 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 30.0000i − 1.20289i
\(623\) − 30.0000i − 1.20192i
\(624\) 0 0
\(625\) 0 0
\(626\) −19.0000 −0.759393
\(627\) 0 0
\(628\) − 14.0000i − 0.558661i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 2.00000i 0.0795557i
\(633\) 0 0
\(634\) −30.0000 −1.19145
\(635\) 0 0
\(636\) 0 0
\(637\) 12.0000i 0.475457i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) 13.0000i 0.512670i 0.966588 + 0.256335i \(0.0825150\pi\)
−0.966588 + 0.256335i \(0.917485\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 42.0000 1.65247
\(647\) − 48.0000i − 1.88707i −0.331266 0.943537i \(-0.607476\pi\)
0.331266 0.943537i \(-0.392524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) − 7.00000i − 0.274141i
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) 0 0
\(658\) − 12.0000i − 0.467809i
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 11.0000i 0.427527i
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 12.0000i − 0.464294i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 12.0000i − 0.461197i −0.973049 0.230599i \(-0.925932\pi\)
0.973049 0.230599i \(-0.0740685\pi\)
\(678\) 0 0
\(679\) −34.0000 −1.30480
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.00000i 0.114792i 0.998351 + 0.0573959i \(0.0182797\pi\)
−0.998351 + 0.0573959i \(0.981720\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 1.00000i 0.0381246i
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) − 54.0000i − 2.04540i
\(698\) 10.0000i 0.378506i
\(699\) 0 0
\(700\) 0 0
\(701\) −48.0000 −1.81293 −0.906467 0.422276i \(-0.861231\pi\)
−0.906467 + 0.422276i \(0.861231\pi\)
\(702\) 0 0
\(703\) 14.0000i 0.528020i
\(704\) 0 0
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 15.0000i 0.562149i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −9.00000 −0.336346
\(717\) 0 0
\(718\) − 12.0000i − 0.447836i
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 30.0000i 1.11648i
\(723\) 0 0
\(724\) 16.0000 0.594635
\(725\) 0 0
\(726\) 0 0
\(727\) − 40.0000i − 1.48352i −0.670667 0.741759i \(-0.733992\pi\)
0.670667 0.741759i \(-0.266008\pi\)
\(728\) 8.00000i 0.296500i
\(729\) 0 0
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 24.0000i − 0.881068i
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) 0 0
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 0 0
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 0 0
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) − 4.00000i − 0.144810i
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 36.0000i 1.29988i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000i 0.0719816i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 17.0000 0.610264
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) 63.0000 2.25721
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) 20.0000i 0.712923i 0.934310 + 0.356462i \(0.116017\pi\)
−0.934310 + 0.356462i \(0.883983\pi\)
\(788\) − 18.0000i − 0.641223i
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) − 16.0000i − 0.568177i
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 36.0000i − 1.27519i −0.770374 0.637593i \(-0.779930\pi\)
0.770374 0.637593i \(-0.220070\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 0 0
\(802\) 18.0000i 0.635602i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 40.0000 1.40894
\(807\) 0 0
\(808\) − 12.0000i − 0.422159i
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 5.00000 0.175574 0.0877869 0.996139i \(-0.472021\pi\)
0.0877869 + 0.996139i \(0.472021\pi\)
\(812\) − 12.0000i − 0.421117i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.00000i 0.244899i
\(818\) 31.0000i 1.08389i
\(819\) 0 0
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 0 0
\(826\) −18.0000 −0.626300
\(827\) − 27.0000i − 0.938882i −0.882964 0.469441i \(-0.844455\pi\)
0.882964 0.469441i \(-0.155545\pi\)
\(828\) 0 0
\(829\) −32.0000 −1.11141 −0.555703 0.831381i \(-0.687551\pi\)
−0.555703 + 0.831381i \(0.687551\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 4.00000i − 0.138675i
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 3.00000i − 0.103633i
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 10.0000i − 0.344623i
\(843\) 0 0
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) − 22.0000i − 0.755929i
\(848\) 12.0000i 0.412082i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) − 39.0000i − 1.33221i −0.745856 0.666107i \(-0.767959\pi\)
0.745856 0.666107i \(-0.232041\pi\)
\(858\) 0 0
\(859\) 31.0000 1.05771 0.528853 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.00000i 0.204361i
\(863\) − 12.0000i − 0.408485i −0.978920 0.204242i \(-0.934527\pi\)
0.978920 0.204242i \(-0.0654731\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 0 0
\(868\) 20.0000i 0.678844i
\(869\) 0 0
\(870\) 0 0
\(871\) 52.0000 1.76195
\(872\) 2.00000i 0.0677285i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 4.00000i 0.134993i
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) − 20.0000i − 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 6.00000i 0.201460i 0.994914 + 0.100730i \(0.0321179\pi\)
−0.994914 + 0.100730i \(0.967882\pi\)
\(888\) 0 0
\(889\) −40.0000 −1.34156
\(890\) 0 0
\(891\) 0 0
\(892\) − 16.0000i − 0.535720i
\(893\) 42.0000i 1.40548i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 15.0000i 0.500556i
\(899\) −60.0000 −2.00111
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) 0 0
\(904\) −3.00000 −0.0997785
\(905\) 0 0
\(906\) 0 0
\(907\) − 7.00000i − 0.232431i −0.993224 0.116216i \(-0.962924\pi\)
0.993224 0.116216i \(-0.0370764\pi\)
\(908\) 15.0000i 0.497792i
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 19.0000 0.628464
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) − 24.0000i − 0.792550i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 6.00000i 0.197599i
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) 0 0
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) 6.00000i 0.196960i
\(929\) 54.0000 1.77168 0.885841 0.463988i \(-0.153582\pi\)
0.885841 + 0.463988i \(0.153582\pi\)
\(930\) 0 0
\(931\) 21.0000 0.688247
\(932\) 3.00000i 0.0982683i
\(933\) 0 0
\(934\) 15.0000 0.490815
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0000i 0.751377i 0.926746 + 0.375689i \(0.122594\pi\)
−0.926746 + 0.375689i \(0.877406\pi\)
\(938\) 26.0000i 0.848930i
\(939\) 0 0
\(940\) 0 0
\(941\) −54.0000 −1.76035 −0.880175 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) 0 0
\(947\) − 21.0000i − 0.682408i −0.939989 0.341204i \(-0.889165\pi\)
0.939989 0.341204i \(-0.110835\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) − 12.0000i − 0.388922i
\(953\) 30.0000i 0.971795i 0.874016 + 0.485898i \(0.161507\pi\)
−0.874016 + 0.485898i \(0.838493\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) − 36.0000i − 1.16311i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) − 8.00000i − 0.257930i
\(963\) 0 0
\(964\) 19.0000 0.611949
\(965\) 0 0
\(966\) 0 0
\(967\) 20.0000i 0.643157i 0.946883 + 0.321578i \(0.104213\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) 11.0000i 0.353553i
\(969\) 0 0
\(970\) 0 0
\(971\) −21.0000 −0.673922 −0.336961 0.941519i \(-0.609399\pi\)
−0.336961 + 0.941519i \(0.609399\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) − 15.0000i − 0.479893i −0.970786 0.239946i \(-0.922870\pi\)
0.970786 0.239946i \(-0.0771298\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) − 3.00000i − 0.0957338i
\(983\) 30.0000i 0.956851i 0.878128 + 0.478426i \(0.158792\pi\)
−0.878128 + 0.478426i \(0.841208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) − 28.0000i − 0.890799i
\(989\) 0 0
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) − 10.0000i − 0.317500i
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) − 28.0000i − 0.886769i −0.896332 0.443384i \(-0.853778\pi\)
0.896332 0.443384i \(-0.146222\pi\)
\(998\) − 11.0000i − 0.348199i
\(999\) 0 0
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 4050.2.c.h.649.2 2
3.2 odd 2 4050.2.c.m.649.1 2
5.2 odd 4 4050.2.a.d.1.1 1
5.3 odd 4 4050.2.a.bg.1.1 1
5.4 even 2 inner 4050.2.c.h.649.1 2
9.2 odd 6 1350.2.j.b.199.1 4
9.4 even 3 450.2.j.b.349.1 4
9.5 odd 6 1350.2.j.b.1099.2 4
9.7 even 3 450.2.j.b.49.2 4
15.2 even 4 4050.2.a.u.1.1 1
15.8 even 4 4050.2.a.o.1.1 1
15.14 odd 2 4050.2.c.m.649.2 2
45.2 even 12 1350.2.e.d.901.1 2
45.4 even 6 450.2.j.b.349.2 4
45.7 odd 12 450.2.e.f.301.1 yes 2
45.13 odd 12 450.2.e.c.151.1 2
45.14 odd 6 1350.2.j.b.1099.1 4
45.22 odd 12 450.2.e.f.151.1 yes 2
45.23 even 12 1350.2.e.h.451.1 2
45.29 odd 6 1350.2.j.b.199.2 4
45.32 even 12 1350.2.e.d.451.1 2
45.34 even 6 450.2.j.b.49.1 4
45.38 even 12 1350.2.e.h.901.1 2
45.43 odd 12 450.2.e.c.301.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.2.e.c.151.1 2 45.13 odd 12
450.2.e.c.301.1 yes 2 45.43 odd 12
450.2.e.f.151.1 yes 2 45.22 odd 12
450.2.e.f.301.1 yes 2 45.7 odd 12
450.2.j.b.49.1 4 45.34 even 6
450.2.j.b.49.2 4 9.7 even 3
450.2.j.b.349.1 4 9.4 even 3
450.2.j.b.349.2 4 45.4 even 6
1350.2.e.d.451.1 2 45.32 even 12
1350.2.e.d.901.1 2 45.2 even 12
1350.2.e.h.451.1 2 45.23 even 12
1350.2.e.h.901.1 2 45.38 even 12
1350.2.j.b.199.1 4 9.2 odd 6
1350.2.j.b.199.2 4 45.29 odd 6
1350.2.j.b.1099.1 4 45.14 odd 6
1350.2.j.b.1099.2 4 9.5 odd 6
4050.2.a.d.1.1 1 5.2 odd 4
4050.2.a.o.1.1 1 15.8 even 4
4050.2.a.u.1.1 1 15.2 even 4
4050.2.a.bg.1.1 1 5.3 odd 4
4050.2.c.h.649.1 2 5.4 even 2 inner
4050.2.c.h.649.2 2 1.1 even 1 trivial
4050.2.c.m.649.1 2 3.2 odd 2
4050.2.c.m.649.2 2 15.14 odd 2