Properties

Label 150.4.c.e.49.2
Level $150$
Weight $4$
Character 150.49
Analytic conductor $8.850$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,4,Mod(49,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("150.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(8.85028650086\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.4.c.e.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} +1.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} +6.00000 q^{6} +1.00000i q^{7} -8.00000i q^{8} -9.00000 q^{9} +42.0000 q^{11} +12.0000i q^{12} -67.0000i q^{13} -2.00000 q^{14} +16.0000 q^{16} -54.0000i q^{17} -18.0000i q^{18} +115.000 q^{19} +3.00000 q^{21} +84.0000i q^{22} -162.000i q^{23} -24.0000 q^{24} +134.000 q^{26} +27.0000i q^{27} -4.00000i q^{28} +210.000 q^{29} -193.000 q^{31} +32.0000i q^{32} -126.000i q^{33} +108.000 q^{34} +36.0000 q^{36} +286.000i q^{37} +230.000i q^{38} -201.000 q^{39} +12.0000 q^{41} +6.00000i q^{42} +263.000i q^{43} -168.000 q^{44} +324.000 q^{46} -414.000i q^{47} -48.0000i q^{48} +342.000 q^{49} -162.000 q^{51} +268.000i q^{52} -192.000i q^{53} -54.0000 q^{54} +8.00000 q^{56} -345.000i q^{57} +420.000i q^{58} -690.000 q^{59} -733.000 q^{61} -386.000i q^{62} -9.00000i q^{63} -64.0000 q^{64} +252.000 q^{66} -299.000i q^{67} +216.000i q^{68} -486.000 q^{69} -228.000 q^{71} +72.0000i q^{72} +938.000i q^{73} -572.000 q^{74} -460.000 q^{76} +42.0000i q^{77} -402.000i q^{78} +160.000 q^{79} +81.0000 q^{81} +24.0000i q^{82} -462.000i q^{83} -12.0000 q^{84} -526.000 q^{86} -630.000i q^{87} -336.000i q^{88} +240.000 q^{89} +67.0000 q^{91} +648.000i q^{92} +579.000i q^{93} +828.000 q^{94} +96.0000 q^{96} +511.000i q^{97} +684.000i q^{98} -378.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} + 84 q^{11} - 4 q^{14} + 32 q^{16} + 230 q^{19} + 6 q^{21} - 48 q^{24} + 268 q^{26} + 420 q^{29} - 386 q^{31} + 216 q^{34} + 72 q^{36} - 402 q^{39} + 24 q^{41} - 336 q^{44} + 648 q^{46} + 684 q^{49} - 324 q^{51} - 108 q^{54} + 16 q^{56} - 1380 q^{59} - 1466 q^{61} - 128 q^{64} + 504 q^{66} - 972 q^{69} - 456 q^{71} - 1144 q^{74} - 920 q^{76} + 320 q^{79} + 162 q^{81} - 24 q^{84} - 1052 q^{86} + 480 q^{89} + 134 q^{91} + 1656 q^{94} + 192 q^{96} - 756 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 0.707107i
\(3\) − 3.00000i − 0.577350i
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 6.00000 0.408248
\(7\) 1.00000i 0.0539949i 0.999636 + 0.0269975i \(0.00859460\pi\)
−0.999636 + 0.0269975i \(0.991405\pi\)
\(8\) − 8.00000i − 0.353553i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 42.0000 1.15123 0.575613 0.817723i \(-0.304764\pi\)
0.575613 + 0.817723i \(0.304764\pi\)
\(12\) 12.0000i 0.288675i
\(13\) − 67.0000i − 1.42942i −0.699421 0.714710i \(-0.746559\pi\)
0.699421 0.714710i \(-0.253441\pi\)
\(14\) −2.00000 −0.0381802
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 54.0000i − 0.770407i −0.922832 0.385204i \(-0.874131\pi\)
0.922832 0.385204i \(-0.125869\pi\)
\(18\) − 18.0000i − 0.235702i
\(19\) 115.000 1.38857 0.694284 0.719701i \(-0.255721\pi\)
0.694284 + 0.719701i \(0.255721\pi\)
\(20\) 0 0
\(21\) 3.00000 0.0311740
\(22\) 84.0000i 0.814039i
\(23\) − 162.000i − 1.46867i −0.678789 0.734333i \(-0.737495\pi\)
0.678789 0.734333i \(-0.262505\pi\)
\(24\) −24.0000 −0.204124
\(25\) 0 0
\(26\) 134.000 1.01075
\(27\) 27.0000i 0.192450i
\(28\) − 4.00000i − 0.0269975i
\(29\) 210.000 1.34469 0.672345 0.740238i \(-0.265287\pi\)
0.672345 + 0.740238i \(0.265287\pi\)
\(30\) 0 0
\(31\) −193.000 −1.11819 −0.559094 0.829104i \(-0.688851\pi\)
−0.559094 + 0.829104i \(0.688851\pi\)
\(32\) 32.0000i 0.176777i
\(33\) − 126.000i − 0.664660i
\(34\) 108.000 0.544760
\(35\) 0 0
\(36\) 36.0000 0.166667
\(37\) 286.000i 1.27076i 0.772200 + 0.635380i \(0.219156\pi\)
−0.772200 + 0.635380i \(0.780844\pi\)
\(38\) 230.000i 0.981866i
\(39\) −201.000 −0.825276
\(40\) 0 0
\(41\) 12.0000 0.0457094 0.0228547 0.999739i \(-0.492724\pi\)
0.0228547 + 0.999739i \(0.492724\pi\)
\(42\) 6.00000i 0.0220433i
\(43\) 263.000i 0.932724i 0.884594 + 0.466362i \(0.154436\pi\)
−0.884594 + 0.466362i \(0.845564\pi\)
\(44\) −168.000 −0.575613
\(45\) 0 0
\(46\) 324.000 1.03850
\(47\) − 414.000i − 1.28485i −0.766347 0.642427i \(-0.777928\pi\)
0.766347 0.642427i \(-0.222072\pi\)
\(48\) − 48.0000i − 0.144338i
\(49\) 342.000 0.997085
\(50\) 0 0
\(51\) −162.000 −0.444795
\(52\) 268.000i 0.714710i
\(53\) − 192.000i − 0.497608i −0.968554 0.248804i \(-0.919962\pi\)
0.968554 0.248804i \(-0.0800375\pi\)
\(54\) −54.0000 −0.136083
\(55\) 0 0
\(56\) 8.00000 0.0190901
\(57\) − 345.000i − 0.801691i
\(58\) 420.000i 0.950840i
\(59\) −690.000 −1.52255 −0.761274 0.648430i \(-0.775426\pi\)
−0.761274 + 0.648430i \(0.775426\pi\)
\(60\) 0 0
\(61\) −733.000 −1.53854 −0.769271 0.638923i \(-0.779380\pi\)
−0.769271 + 0.638923i \(0.779380\pi\)
\(62\) − 386.000i − 0.790678i
\(63\) − 9.00000i − 0.0179983i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 252.000 0.469986
\(67\) − 299.000i − 0.545204i −0.962127 0.272602i \(-0.912116\pi\)
0.962127 0.272602i \(-0.0878842\pi\)
\(68\) 216.000i 0.385204i
\(69\) −486.000 −0.847935
\(70\) 0 0
\(71\) −228.000 −0.381107 −0.190554 0.981677i \(-0.561028\pi\)
−0.190554 + 0.981677i \(0.561028\pi\)
\(72\) 72.0000i 0.117851i
\(73\) 938.000i 1.50390i 0.659221 + 0.751949i \(0.270886\pi\)
−0.659221 + 0.751949i \(0.729114\pi\)
\(74\) −572.000 −0.898563
\(75\) 0 0
\(76\) −460.000 −0.694284
\(77\) 42.0000i 0.0621603i
\(78\) − 402.000i − 0.583558i
\(79\) 160.000 0.227866 0.113933 0.993488i \(-0.463655\pi\)
0.113933 + 0.993488i \(0.463655\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 24.0000i 0.0323214i
\(83\) − 462.000i − 0.610977i −0.952196 0.305488i \(-0.901180\pi\)
0.952196 0.305488i \(-0.0988197\pi\)
\(84\) −12.0000 −0.0155870
\(85\) 0 0
\(86\) −526.000 −0.659535
\(87\) − 630.000i − 0.776357i
\(88\) − 336.000i − 0.407020i
\(89\) 240.000 0.285842 0.142921 0.989734i \(-0.454350\pi\)
0.142921 + 0.989734i \(0.454350\pi\)
\(90\) 0 0
\(91\) 67.0000 0.0771814
\(92\) 648.000i 0.734333i
\(93\) 579.000i 0.645586i
\(94\) 828.000 0.908529
\(95\) 0 0
\(96\) 96.0000 0.102062
\(97\) 511.000i 0.534889i 0.963573 + 0.267444i \(0.0861791\pi\)
−0.963573 + 0.267444i \(0.913821\pi\)
\(98\) 684.000i 0.705045i
\(99\) −378.000 −0.383742
\(100\) 0 0
\(101\) 912.000 0.898489 0.449245 0.893409i \(-0.351693\pi\)
0.449245 + 0.893409i \(0.351693\pi\)
\(102\) − 324.000i − 0.314517i
\(103\) 668.000i 0.639029i 0.947581 + 0.319515i \(0.103520\pi\)
−0.947581 + 0.319515i \(0.896480\pi\)
\(104\) −536.000 −0.505376
\(105\) 0 0
\(106\) 384.000 0.351862
\(107\) 1296.00i 1.17093i 0.810699 + 0.585463i \(0.199087\pi\)
−0.810699 + 0.585463i \(0.800913\pi\)
\(108\) − 108.000i − 0.0962250i
\(109\) 1735.00 1.52461 0.762307 0.647216i \(-0.224067\pi\)
0.762307 + 0.647216i \(0.224067\pi\)
\(110\) 0 0
\(111\) 858.000 0.733673
\(112\) 16.0000i 0.0134987i
\(113\) − 1092.00i − 0.909086i −0.890725 0.454543i \(-0.849803\pi\)
0.890725 0.454543i \(-0.150197\pi\)
\(114\) 690.000 0.566881
\(115\) 0 0
\(116\) −840.000 −0.672345
\(117\) 603.000i 0.476473i
\(118\) − 1380.00i − 1.07660i
\(119\) 54.0000 0.0415981
\(120\) 0 0
\(121\) 433.000 0.325319
\(122\) − 1466.00i − 1.08791i
\(123\) − 36.0000i − 0.0263903i
\(124\) 772.000 0.559094
\(125\) 0 0
\(126\) 18.0000 0.0127267
\(127\) 16.0000i 0.0111793i 0.999984 + 0.00558965i \(0.00177925\pi\)
−0.999984 + 0.00558965i \(0.998221\pi\)
\(128\) − 128.000i − 0.0883883i
\(129\) 789.000 0.538508
\(130\) 0 0
\(131\) 1992.00 1.32856 0.664282 0.747482i \(-0.268737\pi\)
0.664282 + 0.747482i \(0.268737\pi\)
\(132\) 504.000i 0.332330i
\(133\) 115.000i 0.0749757i
\(134\) 598.000 0.385517
\(135\) 0 0
\(136\) −432.000 −0.272380
\(137\) 2346.00i 1.46301i 0.681836 + 0.731505i \(0.261182\pi\)
−0.681836 + 0.731505i \(0.738818\pi\)
\(138\) − 972.000i − 0.599581i
\(139\) −2900.00 −1.76960 −0.884801 0.465968i \(-0.845706\pi\)
−0.884801 + 0.465968i \(0.845706\pi\)
\(140\) 0 0
\(141\) −1242.00 −0.741810
\(142\) − 456.000i − 0.269484i
\(143\) − 2814.00i − 1.64558i
\(144\) −144.000 −0.0833333
\(145\) 0 0
\(146\) −1876.00 −1.06342
\(147\) − 1026.00i − 0.575667i
\(148\) − 1144.00i − 0.635380i
\(149\) 2070.00 1.13813 0.569064 0.822293i \(-0.307306\pi\)
0.569064 + 0.822293i \(0.307306\pi\)
\(150\) 0 0
\(151\) 2237.00 1.20559 0.602796 0.797895i \(-0.294053\pi\)
0.602796 + 0.797895i \(0.294053\pi\)
\(152\) − 920.000i − 0.490933i
\(153\) 486.000i 0.256802i
\(154\) −84.0000 −0.0439540
\(155\) 0 0
\(156\) 804.000 0.412638
\(157\) 241.000i 0.122509i 0.998122 + 0.0612544i \(0.0195101\pi\)
−0.998122 + 0.0612544i \(0.980490\pi\)
\(158\) 320.000i 0.161126i
\(159\) −576.000 −0.287294
\(160\) 0 0
\(161\) 162.000 0.0793006
\(162\) 162.000i 0.0785674i
\(163\) − 3547.00i − 1.70443i −0.523190 0.852216i \(-0.675258\pi\)
0.523190 0.852216i \(-0.324742\pi\)
\(164\) −48.0000 −0.0228547
\(165\) 0 0
\(166\) 924.000 0.432026
\(167\) − 984.000i − 0.455953i −0.973667 0.227977i \(-0.926789\pi\)
0.973667 0.227977i \(-0.0732110\pi\)
\(168\) − 24.0000i − 0.0110217i
\(169\) −2292.00 −1.04324
\(170\) 0 0
\(171\) −1035.00 −0.462856
\(172\) − 1052.00i − 0.466362i
\(173\) 3618.00i 1.59001i 0.606604 + 0.795004i \(0.292531\pi\)
−0.606604 + 0.795004i \(0.707469\pi\)
\(174\) 1260.00 0.548968
\(175\) 0 0
\(176\) 672.000 0.287806
\(177\) 2070.00i 0.879044i
\(178\) 480.000i 0.202121i
\(179\) 150.000 0.0626342 0.0313171 0.999509i \(-0.490030\pi\)
0.0313171 + 0.999509i \(0.490030\pi\)
\(180\) 0 0
\(181\) 197.000 0.0809000 0.0404500 0.999182i \(-0.487121\pi\)
0.0404500 + 0.999182i \(0.487121\pi\)
\(182\) 134.000i 0.0545755i
\(183\) 2199.00i 0.888277i
\(184\) −1296.00 −0.519252
\(185\) 0 0
\(186\) −1158.00 −0.456498
\(187\) − 2268.00i − 0.886912i
\(188\) 1656.00i 0.642427i
\(189\) −27.0000 −0.0103913
\(190\) 0 0
\(191\) 1302.00 0.493243 0.246622 0.969112i \(-0.420680\pi\)
0.246622 + 0.969112i \(0.420680\pi\)
\(192\) 192.000i 0.0721688i
\(193\) 4163.00i 1.55264i 0.630340 + 0.776319i \(0.282916\pi\)
−0.630340 + 0.776319i \(0.717084\pi\)
\(194\) −1022.00 −0.378223
\(195\) 0 0
\(196\) −1368.00 −0.498542
\(197\) − 3054.00i − 1.10451i −0.833675 0.552255i \(-0.813767\pi\)
0.833675 0.552255i \(-0.186233\pi\)
\(198\) − 756.000i − 0.271346i
\(199\) −3425.00 −1.22006 −0.610030 0.792379i \(-0.708842\pi\)
−0.610030 + 0.792379i \(0.708842\pi\)
\(200\) 0 0
\(201\) −897.000 −0.314774
\(202\) 1824.00i 0.635328i
\(203\) 210.000i 0.0726065i
\(204\) 648.000 0.222397
\(205\) 0 0
\(206\) −1336.00 −0.451862
\(207\) 1458.00i 0.489556i
\(208\) − 1072.00i − 0.357355i
\(209\) 4830.00 1.59856
\(210\) 0 0
\(211\) −2443.00 −0.797076 −0.398538 0.917152i \(-0.630482\pi\)
−0.398538 + 0.917152i \(0.630482\pi\)
\(212\) 768.000i 0.248804i
\(213\) 684.000i 0.220032i
\(214\) −2592.00 −0.827969
\(215\) 0 0
\(216\) 216.000 0.0680414
\(217\) − 193.000i − 0.0603765i
\(218\) 3470.00i 1.07806i
\(219\) 2814.00 0.868276
\(220\) 0 0
\(221\) −3618.00 −1.10124
\(222\) 1716.00i 0.518785i
\(223\) 23.0000i 0.00690670i 0.999994 + 0.00345335i \(0.00109924\pi\)
−0.999994 + 0.00345335i \(0.998901\pi\)
\(224\) −32.0000 −0.00954504
\(225\) 0 0
\(226\) 2184.00 0.642821
\(227\) 1956.00i 0.571913i 0.958243 + 0.285957i \(0.0923113\pi\)
−0.958243 + 0.285957i \(0.907689\pi\)
\(228\) 1380.00i 0.400845i
\(229\) −1805.00 −0.520864 −0.260432 0.965492i \(-0.583865\pi\)
−0.260432 + 0.965492i \(0.583865\pi\)
\(230\) 0 0
\(231\) 126.000 0.0358883
\(232\) − 1680.00i − 0.475420i
\(233\) 3468.00i 0.975091i 0.873098 + 0.487546i \(0.162108\pi\)
−0.873098 + 0.487546i \(0.837892\pi\)
\(234\) −1206.00 −0.336917
\(235\) 0 0
\(236\) 2760.00 0.761274
\(237\) − 480.000i − 0.131558i
\(238\) 108.000i 0.0294143i
\(239\) −2640.00 −0.714508 −0.357254 0.934007i \(-0.616287\pi\)
−0.357254 + 0.934007i \(0.616287\pi\)
\(240\) 0 0
\(241\) −5383.00 −1.43879 −0.719397 0.694599i \(-0.755582\pi\)
−0.719397 + 0.694599i \(0.755582\pi\)
\(242\) 866.000i 0.230035i
\(243\) − 243.000i − 0.0641500i
\(244\) 2932.00 0.769271
\(245\) 0 0
\(246\) 72.0000 0.0186608
\(247\) − 7705.00i − 1.98485i
\(248\) 1544.00i 0.395339i
\(249\) −1386.00 −0.352748
\(250\) 0 0
\(251\) −5028.00 −1.26440 −0.632200 0.774805i \(-0.717848\pi\)
−0.632200 + 0.774805i \(0.717848\pi\)
\(252\) 36.0000i 0.00899915i
\(253\) − 6804.00i − 1.69077i
\(254\) −32.0000 −0.00790496
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) − 564.000i − 0.136892i −0.997655 0.0684462i \(-0.978196\pi\)
0.997655 0.0684462i \(-0.0218042\pi\)
\(258\) 1578.00i 0.380783i
\(259\) −286.000 −0.0686146
\(260\) 0 0
\(261\) −1890.00 −0.448230
\(262\) 3984.00i 0.939436i
\(263\) − 1812.00i − 0.424839i −0.977179 0.212420i \(-0.931866\pi\)
0.977179 0.212420i \(-0.0681344\pi\)
\(264\) −1008.00 −0.234993
\(265\) 0 0
\(266\) −230.000 −0.0530158
\(267\) − 720.000i − 0.165031i
\(268\) 1196.00i 0.272602i
\(269\) 5190.00 1.17636 0.588178 0.808731i \(-0.299845\pi\)
0.588178 + 0.808731i \(0.299845\pi\)
\(270\) 0 0
\(271\) 4592.00 1.02931 0.514657 0.857396i \(-0.327919\pi\)
0.514657 + 0.857396i \(0.327919\pi\)
\(272\) − 864.000i − 0.192602i
\(273\) − 201.000i − 0.0445607i
\(274\) −4692.00 −1.03450
\(275\) 0 0
\(276\) 1944.00 0.423968
\(277\) 2191.00i 0.475251i 0.971357 + 0.237625i \(0.0763690\pi\)
−0.971357 + 0.237625i \(0.923631\pi\)
\(278\) − 5800.00i − 1.25130i
\(279\) 1737.00 0.372729
\(280\) 0 0
\(281\) 7842.00 1.66482 0.832410 0.554160i \(-0.186960\pi\)
0.832410 + 0.554160i \(0.186960\pi\)
\(282\) − 2484.00i − 0.524539i
\(283\) − 247.000i − 0.0518821i −0.999663 0.0259410i \(-0.991742\pi\)
0.999663 0.0259410i \(-0.00825821\pi\)
\(284\) 912.000 0.190554
\(285\) 0 0
\(286\) 5628.00 1.16360
\(287\) 12.0000i 0.00246808i
\(288\) − 288.000i − 0.0589256i
\(289\) 1997.00 0.406473
\(290\) 0 0
\(291\) 1533.00 0.308818
\(292\) − 3752.00i − 0.751949i
\(293\) − 5442.00i − 1.08507i −0.840034 0.542534i \(-0.817465\pi\)
0.840034 0.542534i \(-0.182535\pi\)
\(294\) 2052.00 0.407058
\(295\) 0 0
\(296\) 2288.00 0.449281
\(297\) 1134.00i 0.221553i
\(298\) 4140.00i 0.804778i
\(299\) −10854.0 −2.09934
\(300\) 0 0
\(301\) −263.000 −0.0503624
\(302\) 4474.00i 0.852483i
\(303\) − 2736.00i − 0.518743i
\(304\) 1840.00 0.347142
\(305\) 0 0
\(306\) −972.000 −0.181587
\(307\) 3871.00i 0.719641i 0.933022 + 0.359820i \(0.117162\pi\)
−0.933022 + 0.359820i \(0.882838\pi\)
\(308\) − 168.000i − 0.0310802i
\(309\) 2004.00 0.368944
\(310\) 0 0
\(311\) −5718.00 −1.04257 −0.521283 0.853384i \(-0.674546\pi\)
−0.521283 + 0.853384i \(0.674546\pi\)
\(312\) 1608.00i 0.291779i
\(313\) − 3637.00i − 0.656790i −0.944540 0.328395i \(-0.893492\pi\)
0.944540 0.328395i \(-0.106508\pi\)
\(314\) −482.000 −0.0866269
\(315\) 0 0
\(316\) −640.000 −0.113933
\(317\) 1296.00i 0.229623i 0.993387 + 0.114812i \(0.0366265\pi\)
−0.993387 + 0.114812i \(0.963374\pi\)
\(318\) − 1152.00i − 0.203148i
\(319\) 8820.00 1.54804
\(320\) 0 0
\(321\) 3888.00 0.676034
\(322\) 324.000i 0.0560740i
\(323\) − 6210.00i − 1.06976i
\(324\) −324.000 −0.0555556
\(325\) 0 0
\(326\) 7094.00 1.20522
\(327\) − 5205.00i − 0.880236i
\(328\) − 96.0000i − 0.0161607i
\(329\) 414.000 0.0693756
\(330\) 0 0
\(331\) 5132.00 0.852206 0.426103 0.904675i \(-0.359886\pi\)
0.426103 + 0.904675i \(0.359886\pi\)
\(332\) 1848.00i 0.305488i
\(333\) − 2574.00i − 0.423587i
\(334\) 1968.00 0.322408
\(335\) 0 0
\(336\) 48.0000 0.00779350
\(337\) 6751.00i 1.09125i 0.838030 + 0.545624i \(0.183707\pi\)
−0.838030 + 0.545624i \(0.816293\pi\)
\(338\) − 4584.00i − 0.737683i
\(339\) −3276.00 −0.524861
\(340\) 0 0
\(341\) −8106.00 −1.28729
\(342\) − 2070.00i − 0.327289i
\(343\) 685.000i 0.107832i
\(344\) 2104.00 0.329768
\(345\) 0 0
\(346\) −7236.00 −1.12431
\(347\) 5226.00i 0.808491i 0.914651 + 0.404246i \(0.132466\pi\)
−0.914651 + 0.404246i \(0.867534\pi\)
\(348\) 2520.00i 0.388179i
\(349\) 6190.00 0.949407 0.474704 0.880146i \(-0.342555\pi\)
0.474704 + 0.880146i \(0.342555\pi\)
\(350\) 0 0
\(351\) 1809.00 0.275092
\(352\) 1344.00i 0.203510i
\(353\) 6618.00i 0.997849i 0.866646 + 0.498924i \(0.166271\pi\)
−0.866646 + 0.498924i \(0.833729\pi\)
\(354\) −4140.00 −0.621578
\(355\) 0 0
\(356\) −960.000 −0.142921
\(357\) − 162.000i − 0.0240167i
\(358\) 300.000i 0.0442891i
\(359\) 3420.00 0.502787 0.251394 0.967885i \(-0.419111\pi\)
0.251394 + 0.967885i \(0.419111\pi\)
\(360\) 0 0
\(361\) 6366.00 0.928124
\(362\) 394.000i 0.0572049i
\(363\) − 1299.00i − 0.187823i
\(364\) −268.000 −0.0385907
\(365\) 0 0
\(366\) −4398.00 −0.628107
\(367\) 871.000i 0.123885i 0.998080 + 0.0619425i \(0.0197296\pi\)
−0.998080 + 0.0619425i \(0.980270\pi\)
\(368\) − 2592.00i − 0.367167i
\(369\) −108.000 −0.0152365
\(370\) 0 0
\(371\) 192.000 0.0268683
\(372\) − 2316.00i − 0.322793i
\(373\) 6383.00i 0.886057i 0.896508 + 0.443028i \(0.146096\pi\)
−0.896508 + 0.443028i \(0.853904\pi\)
\(374\) 4536.00 0.627142
\(375\) 0 0
\(376\) −3312.00 −0.454264
\(377\) − 14070.0i − 1.92213i
\(378\) − 54.0000i − 0.00734778i
\(379\) 9865.00 1.33702 0.668511 0.743703i \(-0.266932\pi\)
0.668511 + 0.743703i \(0.266932\pi\)
\(380\) 0 0
\(381\) 48.0000 0.00645437
\(382\) 2604.00i 0.348775i
\(383\) 9828.00i 1.31119i 0.755111 + 0.655597i \(0.227583\pi\)
−0.755111 + 0.655597i \(0.772417\pi\)
\(384\) −384.000 −0.0510310
\(385\) 0 0
\(386\) −8326.00 −1.09788
\(387\) − 2367.00i − 0.310908i
\(388\) − 2044.00i − 0.267444i
\(389\) −12540.0 −1.63446 −0.817228 0.576315i \(-0.804490\pi\)
−0.817228 + 0.576315i \(0.804490\pi\)
\(390\) 0 0
\(391\) −8748.00 −1.13147
\(392\) − 2736.00i − 0.352523i
\(393\) − 5976.00i − 0.767047i
\(394\) 6108.00 0.781007
\(395\) 0 0
\(396\) 1512.00 0.191871
\(397\) 1381.00i 0.174585i 0.996183 + 0.0872927i \(0.0278215\pi\)
−0.996183 + 0.0872927i \(0.972178\pi\)
\(398\) − 6850.00i − 0.862712i
\(399\) 345.000 0.0432872
\(400\) 0 0
\(401\) 14232.0 1.77235 0.886175 0.463351i \(-0.153353\pi\)
0.886175 + 0.463351i \(0.153353\pi\)
\(402\) − 1794.00i − 0.222579i
\(403\) 12931.0i 1.59836i
\(404\) −3648.00 −0.449245
\(405\) 0 0
\(406\) −420.000 −0.0513405
\(407\) 12012.0i 1.46293i
\(408\) 1296.00i 0.157259i
\(409\) −2645.00 −0.319772 −0.159886 0.987135i \(-0.551113\pi\)
−0.159886 + 0.987135i \(0.551113\pi\)
\(410\) 0 0
\(411\) 7038.00 0.844669
\(412\) − 2672.00i − 0.319515i
\(413\) − 690.000i − 0.0822099i
\(414\) −2916.00 −0.346168
\(415\) 0 0
\(416\) 2144.00 0.252688
\(417\) 8700.00i 1.02168i
\(418\) 9660.00i 1.13035i
\(419\) −3000.00 −0.349784 −0.174892 0.984588i \(-0.555958\pi\)
−0.174892 + 0.984588i \(0.555958\pi\)
\(420\) 0 0
\(421\) −11338.0 −1.31254 −0.656271 0.754525i \(-0.727867\pi\)
−0.656271 + 0.754525i \(0.727867\pi\)
\(422\) − 4886.00i − 0.563618i
\(423\) 3726.00i 0.428284i
\(424\) −1536.00 −0.175931
\(425\) 0 0
\(426\) −1368.00 −0.155586
\(427\) − 733.000i − 0.0830734i
\(428\) − 5184.00i − 0.585463i
\(429\) −8442.00 −0.950078
\(430\) 0 0
\(431\) −3258.00 −0.364112 −0.182056 0.983288i \(-0.558275\pi\)
−0.182056 + 0.983288i \(0.558275\pi\)
\(432\) 432.000i 0.0481125i
\(433\) 1163.00i 0.129077i 0.997915 + 0.0645384i \(0.0205575\pi\)
−0.997915 + 0.0645384i \(0.979443\pi\)
\(434\) 386.000 0.0426926
\(435\) 0 0
\(436\) −6940.00 −0.762307
\(437\) − 18630.0i − 2.03934i
\(438\) 5628.00i 0.613964i
\(439\) −6695.00 −0.727870 −0.363935 0.931424i \(-0.618567\pi\)
−0.363935 + 0.931424i \(0.618567\pi\)
\(440\) 0 0
\(441\) −3078.00 −0.332362
\(442\) − 7236.00i − 0.778691i
\(443\) 16368.0i 1.75546i 0.479159 + 0.877728i \(0.340942\pi\)
−0.479159 + 0.877728i \(0.659058\pi\)
\(444\) −3432.00 −0.366837
\(445\) 0 0
\(446\) −46.0000 −0.00488377
\(447\) − 6210.00i − 0.657098i
\(448\) − 64.0000i − 0.00674937i
\(449\) −16380.0 −1.72165 −0.860824 0.508903i \(-0.830051\pi\)
−0.860824 + 0.508903i \(0.830051\pi\)
\(450\) 0 0
\(451\) 504.000 0.0526218
\(452\) 4368.00i 0.454543i
\(453\) − 6711.00i − 0.696049i
\(454\) −3912.00 −0.404404
\(455\) 0 0
\(456\) −2760.00 −0.283440
\(457\) 13786.0i 1.41112i 0.708650 + 0.705560i \(0.249304\pi\)
−0.708650 + 0.705560i \(0.750696\pi\)
\(458\) − 3610.00i − 0.368306i
\(459\) 1458.00 0.148265
\(460\) 0 0
\(461\) 11832.0 1.19538 0.597691 0.801726i \(-0.296085\pi\)
0.597691 + 0.801726i \(0.296085\pi\)
\(462\) 252.000i 0.0253768i
\(463\) 3008.00i 0.301930i 0.988539 + 0.150965i \(0.0482381\pi\)
−0.988539 + 0.150965i \(0.951762\pi\)
\(464\) 3360.00 0.336173
\(465\) 0 0
\(466\) −6936.00 −0.689494
\(467\) − 4434.00i − 0.439360i −0.975572 0.219680i \(-0.929499\pi\)
0.975572 0.219680i \(-0.0705013\pi\)
\(468\) − 2412.00i − 0.238237i
\(469\) 299.000 0.0294382
\(470\) 0 0
\(471\) 723.000 0.0707305
\(472\) 5520.00i 0.538302i
\(473\) 11046.0i 1.07378i
\(474\) 960.000 0.0930259
\(475\) 0 0
\(476\) −216.000 −0.0207990
\(477\) 1728.00i 0.165869i
\(478\) − 5280.00i − 0.505233i
\(479\) −7410.00 −0.706830 −0.353415 0.935467i \(-0.614980\pi\)
−0.353415 + 0.935467i \(0.614980\pi\)
\(480\) 0 0
\(481\) 19162.0 1.81645
\(482\) − 10766.0i − 1.01738i
\(483\) − 486.000i − 0.0457842i
\(484\) −1732.00 −0.162660
\(485\) 0 0
\(486\) 486.000 0.0453609
\(487\) 8671.00i 0.806818i 0.915020 + 0.403409i \(0.132175\pi\)
−0.915020 + 0.403409i \(0.867825\pi\)
\(488\) 5864.00i 0.543957i
\(489\) −10641.0 −0.984055
\(490\) 0 0
\(491\) −19368.0 −1.78017 −0.890087 0.455790i \(-0.849357\pi\)
−0.890087 + 0.455790i \(0.849357\pi\)
\(492\) 144.000i 0.0131952i
\(493\) − 11340.0i − 1.03596i
\(494\) 15410.0 1.40350
\(495\) 0 0
\(496\) −3088.00 −0.279547
\(497\) − 228.000i − 0.0205779i
\(498\) − 2772.00i − 0.249430i
\(499\) 8875.00 0.796192 0.398096 0.917344i \(-0.369671\pi\)
0.398096 + 0.917344i \(0.369671\pi\)
\(500\) 0 0
\(501\) −2952.00 −0.263245
\(502\) − 10056.0i − 0.894066i
\(503\) − 10452.0i − 0.926504i −0.886227 0.463252i \(-0.846682\pi\)
0.886227 0.463252i \(-0.153318\pi\)
\(504\) −72.0000 −0.00636336
\(505\) 0 0
\(506\) 13608.0 1.19555
\(507\) 6876.00i 0.602315i
\(508\) − 64.0000i − 0.00558965i
\(509\) 19770.0 1.72159 0.860796 0.508951i \(-0.169967\pi\)
0.860796 + 0.508951i \(0.169967\pi\)
\(510\) 0 0
\(511\) −938.000 −0.0812029
\(512\) 512.000i 0.0441942i
\(513\) 3105.00i 0.267230i
\(514\) 1128.00 0.0967976
\(515\) 0 0
\(516\) −3156.00 −0.269254
\(517\) − 17388.0i − 1.47916i
\(518\) − 572.000i − 0.0485178i
\(519\) 10854.0 0.917992
\(520\) 0 0
\(521\) −11238.0 −0.945001 −0.472501 0.881330i \(-0.656649\pi\)
−0.472501 + 0.881330i \(0.656649\pi\)
\(522\) − 3780.00i − 0.316947i
\(523\) − 7447.00i − 0.622628i −0.950307 0.311314i \(-0.899231\pi\)
0.950307 0.311314i \(-0.100769\pi\)
\(524\) −7968.00 −0.664282
\(525\) 0 0
\(526\) 3624.00 0.300407
\(527\) 10422.0i 0.861460i
\(528\) − 2016.00i − 0.166165i
\(529\) −14077.0 −1.15698
\(530\) 0 0
\(531\) 6210.00 0.507516
\(532\) − 460.000i − 0.0374878i
\(533\) − 804.000i − 0.0653379i
\(534\) 1440.00 0.116695
\(535\) 0 0
\(536\) −2392.00 −0.192759
\(537\) − 450.000i − 0.0361619i
\(538\) 10380.0i 0.831810i
\(539\) 14364.0 1.14787
\(540\) 0 0
\(541\) −17623.0 −1.40050 −0.700251 0.713896i \(-0.746929\pi\)
−0.700251 + 0.713896i \(0.746929\pi\)
\(542\) 9184.00i 0.727835i
\(543\) − 591.000i − 0.0467076i
\(544\) 1728.00 0.136190
\(545\) 0 0
\(546\) 402.000 0.0315092
\(547\) 10096.0i 0.789166i 0.918860 + 0.394583i \(0.129111\pi\)
−0.918860 + 0.394583i \(0.870889\pi\)
\(548\) − 9384.00i − 0.731505i
\(549\) 6597.00 0.512847
\(550\) 0 0
\(551\) 24150.0 1.86720
\(552\) 3888.00i 0.299790i
\(553\) 160.000i 0.0123036i
\(554\) −4382.00 −0.336053
\(555\) 0 0
\(556\) 11600.0 0.884801
\(557\) − 14514.0i − 1.10409i −0.833814 0.552045i \(-0.813848\pi\)
0.833814 0.552045i \(-0.186152\pi\)
\(558\) 3474.00i 0.263559i
\(559\) 17621.0 1.33325
\(560\) 0 0
\(561\) −6804.00 −0.512059
\(562\) 15684.0i 1.17721i
\(563\) − 10242.0i − 0.766694i −0.923604 0.383347i \(-0.874771\pi\)
0.923604 0.383347i \(-0.125229\pi\)
\(564\) 4968.00 0.370905
\(565\) 0 0
\(566\) 494.000 0.0366862
\(567\) 81.0000i 0.00599944i
\(568\) 1824.00i 0.134742i
\(569\) 6750.00 0.497319 0.248660 0.968591i \(-0.420010\pi\)
0.248660 + 0.968591i \(0.420010\pi\)
\(570\) 0 0
\(571\) 17117.0 1.25451 0.627254 0.778815i \(-0.284179\pi\)
0.627254 + 0.778815i \(0.284179\pi\)
\(572\) 11256.0i 0.822792i
\(573\) − 3906.00i − 0.284774i
\(574\) −24.0000 −0.00174519
\(575\) 0 0
\(576\) 576.000 0.0416667
\(577\) 301.000i 0.0217171i 0.999941 + 0.0108586i \(0.00345646\pi\)
−0.999941 + 0.0108586i \(0.996544\pi\)
\(578\) 3994.00i 0.287420i
\(579\) 12489.0 0.896416
\(580\) 0 0
\(581\) 462.000 0.0329897
\(582\) 3066.00i 0.218367i
\(583\) − 8064.00i − 0.572859i
\(584\) 7504.00 0.531708
\(585\) 0 0
\(586\) 10884.0 0.767259
\(587\) 15456.0i 1.08678i 0.839482 + 0.543388i \(0.182859\pi\)
−0.839482 + 0.543388i \(0.817141\pi\)
\(588\) 4104.00i 0.287834i
\(589\) −22195.0 −1.55268
\(590\) 0 0
\(591\) −9162.00 −0.637689
\(592\) 4576.00i 0.317690i
\(593\) − 9492.00i − 0.657318i −0.944449 0.328659i \(-0.893403\pi\)
0.944449 0.328659i \(-0.106597\pi\)
\(594\) −2268.00 −0.156662
\(595\) 0 0
\(596\) −8280.00 −0.569064
\(597\) 10275.0i 0.704402i
\(598\) − 21708.0i − 1.48446i
\(599\) −1500.00 −0.102318 −0.0511589 0.998691i \(-0.516291\pi\)
−0.0511589 + 0.998691i \(0.516291\pi\)
\(600\) 0 0
\(601\) 14627.0 0.992758 0.496379 0.868106i \(-0.334663\pi\)
0.496379 + 0.868106i \(0.334663\pi\)
\(602\) − 526.000i − 0.0356116i
\(603\) 2691.00i 0.181735i
\(604\) −8948.00 −0.602796
\(605\) 0 0
\(606\) 5472.00 0.366807
\(607\) − 16184.0i − 1.08219i −0.840962 0.541094i \(-0.818010\pi\)
0.840962 0.541094i \(-0.181990\pi\)
\(608\) 3680.00i 0.245467i
\(609\) 630.000 0.0419194
\(610\) 0 0
\(611\) −27738.0 −1.83659
\(612\) − 1944.00i − 0.128401i
\(613\) − 18502.0i − 1.21907i −0.792760 0.609534i \(-0.791357\pi\)
0.792760 0.609534i \(-0.208643\pi\)
\(614\) −7742.00 −0.508863
\(615\) 0 0
\(616\) 336.000 0.0219770
\(617\) 13896.0i 0.906697i 0.891333 + 0.453348i \(0.149771\pi\)
−0.891333 + 0.453348i \(0.850229\pi\)
\(618\) 4008.00i 0.260883i
\(619\) 9895.00 0.642510 0.321255 0.946993i \(-0.395895\pi\)
0.321255 + 0.946993i \(0.395895\pi\)
\(620\) 0 0
\(621\) 4374.00 0.282645
\(622\) − 11436.0i − 0.737206i
\(623\) 240.000i 0.0154340i
\(624\) −3216.00 −0.206319
\(625\) 0 0
\(626\) 7274.00 0.464421
\(627\) − 14490.0i − 0.922926i
\(628\) − 964.000i − 0.0612544i
\(629\) 15444.0 0.979003
\(630\) 0 0
\(631\) 467.000 0.0294627 0.0147314 0.999891i \(-0.495311\pi\)
0.0147314 + 0.999891i \(0.495311\pi\)
\(632\) − 1280.00i − 0.0805628i
\(633\) 7329.00i 0.460192i
\(634\) −2592.00 −0.162368
\(635\) 0 0
\(636\) 2304.00 0.143647
\(637\) − 22914.0i − 1.42525i
\(638\) 17640.0i 1.09463i
\(639\) 2052.00 0.127036
\(640\) 0 0
\(641\) 30612.0 1.88627 0.943137 0.332405i \(-0.107860\pi\)
0.943137 + 0.332405i \(0.107860\pi\)
\(642\) 7776.00i 0.478028i
\(643\) − 1852.00i − 0.113586i −0.998386 0.0567930i \(-0.981913\pi\)
0.998386 0.0567930i \(-0.0180875\pi\)
\(644\) −648.000 −0.0396503
\(645\) 0 0
\(646\) 12420.0 0.756437
\(647\) 21156.0i 1.28551i 0.766070 + 0.642757i \(0.222210\pi\)
−0.766070 + 0.642757i \(0.777790\pi\)
\(648\) − 648.000i − 0.0392837i
\(649\) −28980.0 −1.75280
\(650\) 0 0
\(651\) −579.000 −0.0348584
\(652\) 14188.0i 0.852216i
\(653\) − 9702.00i − 0.581422i −0.956811 0.290711i \(-0.906108\pi\)
0.956811 0.290711i \(-0.0938918\pi\)
\(654\) 10410.0 0.622421
\(655\) 0 0
\(656\) 192.000 0.0114273
\(657\) − 8442.00i − 0.501300i
\(658\) 828.000i 0.0490559i
\(659\) −1980.00 −0.117041 −0.0585204 0.998286i \(-0.518638\pi\)
−0.0585204 + 0.998286i \(0.518638\pi\)
\(660\) 0 0
\(661\) −20158.0 −1.18617 −0.593083 0.805142i \(-0.702089\pi\)
−0.593083 + 0.805142i \(0.702089\pi\)
\(662\) 10264.0i 0.602601i
\(663\) 10854.0i 0.635799i
\(664\) −3696.00 −0.216013
\(665\) 0 0
\(666\) 5148.00 0.299521
\(667\) − 34020.0i − 1.97490i
\(668\) 3936.00i 0.227977i
\(669\) 69.0000 0.00398758
\(670\) 0 0
\(671\) −30786.0 −1.77121
\(672\) 96.0000i 0.00551083i
\(673\) − 16882.0i − 0.966944i −0.875360 0.483472i \(-0.839376\pi\)
0.875360 0.483472i \(-0.160624\pi\)
\(674\) −13502.0 −0.771628
\(675\) 0 0
\(676\) 9168.00 0.521620
\(677\) − 20934.0i − 1.18842i −0.804311 0.594209i \(-0.797465\pi\)
0.804311 0.594209i \(-0.202535\pi\)
\(678\) − 6552.00i − 0.371133i
\(679\) −511.000 −0.0288813
\(680\) 0 0
\(681\) 5868.00 0.330194
\(682\) − 16212.0i − 0.910249i
\(683\) − 8712.00i − 0.488075i −0.969766 0.244038i \(-0.921528\pi\)
0.969766 0.244038i \(-0.0784720\pi\)
\(684\) 4140.00 0.231428
\(685\) 0 0
\(686\) −1370.00 −0.0762490
\(687\) 5415.00i 0.300721i
\(688\) 4208.00i 0.233181i
\(689\) −12864.0 −0.711291
\(690\) 0 0
\(691\) −14128.0 −0.777792 −0.388896 0.921282i \(-0.627144\pi\)
−0.388896 + 0.921282i \(0.627144\pi\)
\(692\) − 14472.0i − 0.795004i
\(693\) − 378.000i − 0.0207201i
\(694\) −10452.0 −0.571689
\(695\) 0 0
\(696\) −5040.00 −0.274484
\(697\) − 648.000i − 0.0352148i
\(698\) 12380.0i 0.671332i
\(699\) 10404.0 0.562969
\(700\) 0 0
\(701\) −28278.0 −1.52360 −0.761801 0.647811i \(-0.775685\pi\)
−0.761801 + 0.647811i \(0.775685\pi\)
\(702\) 3618.00i 0.194519i
\(703\) 32890.0i 1.76454i
\(704\) −2688.00 −0.143903
\(705\) 0 0
\(706\) −13236.0 −0.705586
\(707\) 912.000i 0.0485138i
\(708\) − 8280.00i − 0.439522i
\(709\) −8885.00 −0.470639 −0.235320 0.971918i \(-0.575614\pi\)
−0.235320 + 0.971918i \(0.575614\pi\)
\(710\) 0 0
\(711\) −1440.00 −0.0759553
\(712\) − 1920.00i − 0.101060i
\(713\) 31266.0i 1.64225i
\(714\) 324.000 0.0169823
\(715\) 0 0
\(716\) −600.000 −0.0313171
\(717\) 7920.00i 0.412521i
\(718\) 6840.00i 0.355524i
\(719\) 7530.00 0.390572 0.195286 0.980746i \(-0.437436\pi\)
0.195286 + 0.980746i \(0.437436\pi\)
\(720\) 0 0
\(721\) −668.000 −0.0345043
\(722\) 12732.0i 0.656283i
\(723\) 16149.0i 0.830688i
\(724\) −788.000 −0.0404500
\(725\) 0 0
\(726\) 2598.00 0.132811
\(727\) 1801.00i 0.0918781i 0.998944 + 0.0459391i \(0.0146280\pi\)
−0.998944 + 0.0459391i \(0.985372\pi\)
\(728\) − 536.000i − 0.0272877i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 14202.0 0.718577
\(732\) − 8796.00i − 0.444139i
\(733\) − 7882.00i − 0.397174i −0.980083 0.198587i \(-0.936365\pi\)
0.980083 0.198587i \(-0.0636352\pi\)
\(734\) −1742.00 −0.0876000
\(735\) 0 0
\(736\) 5184.00 0.259626
\(737\) − 12558.0i − 0.627652i
\(738\) − 216.000i − 0.0107738i
\(739\) −33860.0 −1.68547 −0.842734 0.538331i \(-0.819055\pi\)
−0.842734 + 0.538331i \(0.819055\pi\)
\(740\) 0 0
\(741\) −23115.0 −1.14595
\(742\) 384.000i 0.0189988i
\(743\) − 20652.0i − 1.01972i −0.860259 0.509858i \(-0.829698\pi\)
0.860259 0.509858i \(-0.170302\pi\)
\(744\) 4632.00 0.228249
\(745\) 0 0
\(746\) −12766.0 −0.626537
\(747\) 4158.00i 0.203659i
\(748\) 9072.00i 0.443456i
\(749\) −1296.00 −0.0632240
\(750\) 0 0
\(751\) 7472.00 0.363059 0.181529 0.983386i \(-0.441895\pi\)
0.181529 + 0.983386i \(0.441895\pi\)
\(752\) − 6624.00i − 0.321213i
\(753\) 15084.0i 0.730002i
\(754\) 28140.0 1.35915
\(755\) 0 0
\(756\) 108.000 0.00519566
\(757\) 32251.0i 1.54846i 0.632906 + 0.774229i \(0.281862\pi\)
−0.632906 + 0.774229i \(0.718138\pi\)
\(758\) 19730.0i 0.945417i
\(759\) −20412.0 −0.976164
\(760\) 0 0
\(761\) 16812.0 0.800834 0.400417 0.916333i \(-0.368865\pi\)
0.400417 + 0.916333i \(0.368865\pi\)
\(762\) 96.0000i 0.00456393i
\(763\) 1735.00i 0.0823214i
\(764\) −5208.00 −0.246622
\(765\) 0 0
\(766\) −19656.0 −0.927154
\(767\) 46230.0i 2.17636i
\(768\) − 768.000i − 0.0360844i
\(769\) 34645.0 1.62462 0.812309 0.583228i \(-0.198211\pi\)
0.812309 + 0.583228i \(0.198211\pi\)
\(770\) 0 0
\(771\) −1692.00 −0.0790349
\(772\) − 16652.0i − 0.776319i
\(773\) − 8412.00i − 0.391408i −0.980663 0.195704i \(-0.937301\pi\)
0.980663 0.195704i \(-0.0626992\pi\)
\(774\) 4734.00 0.219845
\(775\) 0 0
\(776\) 4088.00 0.189112
\(777\) 858.000i 0.0396146i
\(778\) − 25080.0i − 1.15573i
\(779\) 1380.00 0.0634706
\(780\) 0 0
\(781\) −9576.00 −0.438740
\(782\) − 17496.0i − 0.800071i
\(783\) 5670.00i 0.258786i
\(784\) 5472.00 0.249271
\(785\) 0 0
\(786\) 11952.0 0.542384
\(787\) − 18329.0i − 0.830188i −0.909778 0.415094i \(-0.863749\pi\)
0.909778 0.415094i \(-0.136251\pi\)
\(788\) 12216.0i 0.552255i
\(789\) −5436.00 −0.245281
\(790\) 0 0
\(791\) 1092.00 0.0490860
\(792\) 3024.00i 0.135673i
\(793\) 49111.0i 2.19922i
\(794\) −2762.00 −0.123451
\(795\) 0 0
\(796\) 13700.0 0.610030
\(797\) − 16044.0i − 0.713059i −0.934284 0.356529i \(-0.883960\pi\)
0.934284 0.356529i \(-0.116040\pi\)
\(798\) 690.000i 0.0306087i
\(799\) −22356.0 −0.989860
\(800\) 0 0
\(801\) −2160.00 −0.0952807
\(802\) 28464.0i 1.25324i
\(803\) 39396.0i 1.73133i
\(804\) 3588.00 0.157387
\(805\) 0 0
\(806\) −25862.0 −1.13021
\(807\) − 15570.0i − 0.679170i
\(808\) − 7296.00i − 0.317664i
\(809\) 24000.0 1.04301 0.521505 0.853248i \(-0.325371\pi\)
0.521505 + 0.853248i \(0.325371\pi\)
\(810\) 0 0
\(811\) 5117.00 0.221556 0.110778 0.993845i \(-0.464666\pi\)
0.110778 + 0.993845i \(0.464666\pi\)
\(812\) − 840.000i − 0.0363032i
\(813\) − 13776.0i − 0.594275i
\(814\) −24024.0 −1.03445
\(815\) 0 0
\(816\) −2592.00 −0.111199
\(817\) 30245.0i 1.29515i
\(818\) − 5290.00i − 0.226113i
\(819\) −603.000 −0.0257271
\(820\) 0 0
\(821\) 13542.0 0.575663 0.287831 0.957681i \(-0.407066\pi\)
0.287831 + 0.957681i \(0.407066\pi\)
\(822\) 14076.0i 0.597271i
\(823\) 1283.00i 0.0543409i 0.999631 + 0.0271705i \(0.00864969\pi\)
−0.999631 + 0.0271705i \(0.991350\pi\)
\(824\) 5344.00 0.225931
\(825\) 0 0
\(826\) 1380.00 0.0581312
\(827\) − 16344.0i − 0.687227i −0.939111 0.343613i \(-0.888349\pi\)
0.939111 0.343613i \(-0.111651\pi\)
\(828\) − 5832.00i − 0.244778i
\(829\) 790.000 0.0330975 0.0165488 0.999863i \(-0.494732\pi\)
0.0165488 + 0.999863i \(0.494732\pi\)
\(830\) 0 0
\(831\) 6573.00 0.274386
\(832\) 4288.00i 0.178677i
\(833\) − 18468.0i − 0.768161i
\(834\) −17400.0 −0.722437
\(835\) 0 0
\(836\) −19320.0 −0.799278
\(837\) − 5211.00i − 0.215195i
\(838\) − 6000.00i − 0.247335i
\(839\) 9990.00 0.411076 0.205538 0.978649i \(-0.434106\pi\)
0.205538 + 0.978649i \(0.434106\pi\)
\(840\) 0 0
\(841\) 19711.0 0.808192
\(842\) − 22676.0i − 0.928108i
\(843\) − 23526.0i − 0.961184i
\(844\) 9772.00 0.398538
\(845\) 0 0
\(846\) −7452.00 −0.302843
\(847\) 433.000i 0.0175656i
\(848\) − 3072.00i − 0.124402i
\(849\) −741.000 −0.0299541
\(850\) 0 0
\(851\) 46332.0 1.86632
\(852\) − 2736.00i − 0.110016i
\(853\) 24743.0i 0.993182i 0.867985 + 0.496591i \(0.165415\pi\)
−0.867985 + 0.496591i \(0.834585\pi\)
\(854\) 1466.00 0.0587418
\(855\) 0 0
\(856\) 10368.0 0.413985
\(857\) 23556.0i 0.938924i 0.882953 + 0.469462i \(0.155552\pi\)
−0.882953 + 0.469462i \(0.844448\pi\)
\(858\) − 16884.0i − 0.671807i
\(859\) 34000.0 1.35048 0.675242 0.737597i \(-0.264039\pi\)
0.675242 + 0.737597i \(0.264039\pi\)
\(860\) 0 0
\(861\) 36.0000 0.00142494
\(862\) − 6516.00i − 0.257466i
\(863\) − 37032.0i − 1.46070i −0.683073 0.730350i \(-0.739357\pi\)
0.683073 0.730350i \(-0.260643\pi\)
\(864\) −864.000 −0.0340207
\(865\) 0 0
\(866\) −2326.00 −0.0912710
\(867\) − 5991.00i − 0.234677i
\(868\) 772.000i 0.0301882i
\(869\) 6720.00 0.262325
\(870\) 0 0
\(871\) −20033.0 −0.779325
\(872\) − 13880.0i − 0.539032i
\(873\) − 4599.00i − 0.178296i
\(874\) 37260.0 1.44203
\(875\) 0 0
\(876\) −11256.0 −0.434138
\(877\) − 2519.00i − 0.0969904i −0.998823 0.0484952i \(-0.984557\pi\)
0.998823 0.0484952i \(-0.0154426\pi\)
\(878\) − 13390.0i − 0.514682i
\(879\) −16326.0 −0.626465
\(880\) 0 0
\(881\) 43992.0 1.68232 0.841162 0.540783i \(-0.181872\pi\)
0.841162 + 0.540783i \(0.181872\pi\)
\(882\) − 6156.00i − 0.235015i
\(883\) − 19177.0i − 0.730869i −0.930837 0.365435i \(-0.880920\pi\)
0.930837 0.365435i \(-0.119080\pi\)
\(884\) 14472.0 0.550618
\(885\) 0 0
\(886\) −32736.0 −1.24130
\(887\) − 44994.0i − 1.70321i −0.524181 0.851607i \(-0.675628\pi\)
0.524181 0.851607i \(-0.324372\pi\)
\(888\) − 6864.00i − 0.259393i
\(889\) −16.0000 −0.000603625 0
\(890\) 0 0
\(891\) 3402.00 0.127914
\(892\) − 92.0000i − 0.00345335i
\(893\) − 47610.0i − 1.78411i
\(894\) 12420.0 0.464639
\(895\) 0 0
\(896\) 128.000 0.00477252
\(897\) 32562.0i 1.21206i
\(898\) − 32760.0i − 1.21739i
\(899\) −40530.0 −1.50362
\(900\) 0 0
\(901\) −10368.0 −0.383361
\(902\) 1008.00i 0.0372092i
\(903\) 789.000i 0.0290767i
\(904\) −8736.00 −0.321410
\(905\) 0 0
\(906\) 13422.0 0.492181
\(907\) 52396.0i 1.91817i 0.283117 + 0.959085i \(0.408631\pi\)
−0.283117 + 0.959085i \(0.591369\pi\)
\(908\) − 7824.00i − 0.285957i
\(909\) −8208.00 −0.299496
\(910\) 0 0
\(911\) 7242.00 0.263379 0.131689 0.991291i \(-0.457960\pi\)
0.131689 + 0.991291i \(0.457960\pi\)
\(912\) − 5520.00i − 0.200423i
\(913\) − 19404.0i − 0.703372i
\(914\) −27572.0 −0.997813
\(915\) 0 0
\(916\) 7220.00 0.260432
\(917\) 1992.00i 0.0717357i
\(918\) 2916.00i 0.104839i
\(919\) −4085.00 −0.146629 −0.0733143 0.997309i \(-0.523358\pi\)
−0.0733143 + 0.997309i \(0.523358\pi\)
\(920\) 0 0
\(921\) 11613.0 0.415485
\(922\) 23664.0i 0.845263i
\(923\) 15276.0i 0.544762i
\(924\) −504.000 −0.0179441
\(925\) 0 0
\(926\) −6016.00 −0.213497
\(927\) − 6012.00i − 0.213010i
\(928\) 6720.00i 0.237710i
\(929\) 3030.00 0.107009 0.0535043 0.998568i \(-0.482961\pi\)
0.0535043 + 0.998568i \(0.482961\pi\)
\(930\) 0 0
\(931\) 39330.0 1.38452
\(932\) − 13872.0i − 0.487546i
\(933\) 17154.0i 0.601926i
\(934\) 8868.00 0.310674
\(935\) 0 0
\(936\) 4824.00 0.168459
\(937\) − 5759.00i − 0.200788i −0.994948 0.100394i \(-0.967990\pi\)
0.994948 0.100394i \(-0.0320103\pi\)
\(938\) 598.000i 0.0208160i
\(939\) −10911.0 −0.379198
\(940\) 0 0
\(941\) −258.000 −0.00893790 −0.00446895 0.999990i \(-0.501423\pi\)
−0.00446895 + 0.999990i \(0.501423\pi\)
\(942\) 1446.00i 0.0500140i
\(943\) − 1944.00i − 0.0671319i
\(944\) −11040.0 −0.380637
\(945\) 0 0
\(946\) −22092.0 −0.759274
\(947\) − 1374.00i − 0.0471478i −0.999722 0.0235739i \(-0.992495\pi\)
0.999722 0.0235739i \(-0.00750451\pi\)
\(948\) 1920.00i 0.0657792i
\(949\) 62846.0 2.14970
\(950\) 0 0
\(951\) 3888.00 0.132573
\(952\) − 432.000i − 0.0147071i
\(953\) 9288.00i 0.315706i 0.987463 + 0.157853i \(0.0504572\pi\)
−0.987463 + 0.157853i \(0.949543\pi\)
\(954\) −3456.00 −0.117287
\(955\) 0 0
\(956\) 10560.0 0.357254
\(957\) − 26460.0i − 0.893762i
\(958\) − 14820.0i − 0.499804i
\(959\) −2346.00 −0.0789951
\(960\) 0 0
\(961\) 7458.00 0.250344
\(962\) 38324.0i 1.28442i
\(963\) − 11664.0i − 0.390309i
\(964\) 21532.0 0.719397
\(965\) 0 0
\(966\) 972.000 0.0323743
\(967\) 21616.0i 0.718846i 0.933175 + 0.359423i \(0.117026\pi\)
−0.933175 + 0.359423i \(0.882974\pi\)
\(968\) − 3464.00i − 0.115018i
\(969\) −18630.0 −0.617628
\(970\) 0 0
\(971\) −19098.0 −0.631188 −0.315594 0.948894i \(-0.602204\pi\)
−0.315594 + 0.948894i \(0.602204\pi\)
\(972\) 972.000i 0.0320750i
\(973\) − 2900.00i − 0.0955496i
\(974\) −17342.0 −0.570507
\(975\) 0 0
\(976\) −11728.0 −0.384635
\(977\) 18246.0i 0.597483i 0.954334 + 0.298742i \(0.0965669\pi\)
−0.954334 + 0.298742i \(0.903433\pi\)
\(978\) − 21282.0i − 0.695832i
\(979\) 10080.0 0.329069
\(980\) 0 0
\(981\) −15615.0 −0.508204
\(982\) − 38736.0i − 1.25877i
\(983\) − 38772.0i − 1.25802i −0.777397 0.629011i \(-0.783460\pi\)
0.777397 0.629011i \(-0.216540\pi\)
\(984\) −288.000 −0.00933039
\(985\) 0 0
\(986\) 22680.0 0.732534
\(987\) − 1242.00i − 0.0400540i
\(988\) 30820.0i 0.992424i
\(989\) 42606.0 1.36986
\(990\) 0 0
\(991\) −23053.0 −0.738953 −0.369477 0.929240i \(-0.620463\pi\)
−0.369477 + 0.929240i \(0.620463\pi\)
\(992\) − 6176.00i − 0.197670i
\(993\) − 15396.0i − 0.492021i
\(994\) 456.000 0.0145507
\(995\) 0 0
\(996\) 5544.00 0.176374
\(997\) 10366.0i 0.329282i 0.986354 + 0.164641i \(0.0526466\pi\)
−0.986354 + 0.164641i \(0.947353\pi\)
\(998\) 17750.0i 0.562992i
\(999\) −7722.00 −0.244558
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 150.4.c.e.49.2 2
3.2 odd 2 450.4.c.a.199.1 2
4.3 odd 2 1200.4.f.c.49.2 2
5.2 odd 4 150.4.a.a.1.1 1
5.3 odd 4 150.4.a.h.1.1 yes 1
5.4 even 2 inner 150.4.c.e.49.1 2
15.2 even 4 450.4.a.o.1.1 1
15.8 even 4 450.4.a.f.1.1 1
15.14 odd 2 450.4.c.a.199.2 2
20.3 even 4 1200.4.a.i.1.1 1
20.7 even 4 1200.4.a.bb.1.1 1
20.19 odd 2 1200.4.f.c.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.4.a.a.1.1 1 5.2 odd 4
150.4.a.h.1.1 yes 1 5.3 odd 4
150.4.c.e.49.1 2 5.4 even 2 inner
150.4.c.e.49.2 2 1.1 even 1 trivial
450.4.a.f.1.1 1 15.8 even 4
450.4.a.o.1.1 1 15.2 even 4
450.4.c.a.199.1 2 3.2 odd 2
450.4.c.a.199.2 2 15.14 odd 2
1200.4.a.i.1.1 1 20.3 even 4
1200.4.a.bb.1.1 1 20.7 even 4
1200.4.f.c.49.1 2 20.19 odd 2
1200.4.f.c.49.2 2 4.3 odd 2