Properties

Label 825.2.a.h.1.2
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,2,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

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: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} -1.53919 q^{6} -0.290725 q^{7} +2.51026 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.53919 q^{2} +1.00000 q^{3} +0.369102 q^{4} -1.53919 q^{6} -0.290725 q^{7} +2.51026 q^{8} +1.00000 q^{9} -1.00000 q^{11} +0.369102 q^{12} -6.97107 q^{13} +0.447480 q^{14} -4.60197 q^{16} -4.78765 q^{17} -1.53919 q^{18} +7.75872 q^{19} -0.290725 q^{21} +1.53919 q^{22} -4.00000 q^{23} +2.51026 q^{24} +10.7298 q^{26} +1.00000 q^{27} -0.107307 q^{28} -7.41855 q^{29} +6.34017 q^{31} +2.06278 q^{32} -1.00000 q^{33} +7.36910 q^{34} +0.369102 q^{36} +3.41855 q^{37} -11.9421 q^{38} -6.97107 q^{39} -7.41855 q^{41} +0.447480 q^{42} -0.290725 q^{43} -0.369102 q^{44} +6.15676 q^{46} -5.26180 q^{47} -4.60197 q^{48} -6.91548 q^{49} -4.78765 q^{51} -2.57304 q^{52} -5.75872 q^{53} -1.53919 q^{54} -0.729794 q^{56} +7.75872 q^{57} +11.4186 q^{58} +3.60197 q^{59} -6.68035 q^{61} -9.75872 q^{62} -0.290725 q^{63} +6.02893 q^{64} +1.53919 q^{66} -6.15676 q^{67} -1.76713 q^{68} -4.00000 q^{69} -5.07838 q^{71} +2.51026 q^{72} +1.12783 q^{73} -5.26180 q^{74} +2.86376 q^{76} +0.290725 q^{77} +10.7298 q^{78} -0.921622 q^{79} +1.00000 q^{81} +11.4186 q^{82} -1.70928 q^{83} -0.107307 q^{84} +0.447480 q^{86} -7.41855 q^{87} -2.51026 q^{88} +4.34017 q^{89} +2.02666 q^{91} -1.47641 q^{92} +6.34017 q^{93} +8.09890 q^{94} +2.06278 q^{96} -4.68035 q^{97} +10.6442 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{6} - 8 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{3} + 5 q^{4} - 3 q^{6} - 8 q^{7} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 5 q^{12} - 6 q^{13} + 2 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 2 q^{19} - 8 q^{21} + 3 q^{22} - 12 q^{23} - 9 q^{24} - 8 q^{26} + 3 q^{27} - 12 q^{28} - 8 q^{29} + 8 q^{31} - 11 q^{32} - 3 q^{33} + 26 q^{34} + 5 q^{36} - 4 q^{37} - 6 q^{38} - 6 q^{39} - 8 q^{41} + 2 q^{42} - 8 q^{43} - 5 q^{44} + 12 q^{46} - 8 q^{47} + 5 q^{48} + 11 q^{49} - 4 q^{51} + 26 q^{52} + 8 q^{53} - 3 q^{54} + 38 q^{56} - 2 q^{57} + 20 q^{58} - 8 q^{59} + 2 q^{61} - 4 q^{62} - 8 q^{63} + 33 q^{64} + 3 q^{66} - 12 q^{67} - 28 q^{68} - 12 q^{69} - 12 q^{71} - 9 q^{72} - 18 q^{73} - 8 q^{74} - 18 q^{76} + 8 q^{77} - 8 q^{78} - 6 q^{79} + 3 q^{81} + 20 q^{82} + 2 q^{83} - 12 q^{84} + 2 q^{86} - 8 q^{87} + 9 q^{88} + 2 q^{89} + 8 q^{91} - 20 q^{92} + 8 q^{93} - 12 q^{94} - 11 q^{96} + 8 q^{97} + 29 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53919 −1.08837 −0.544185 0.838965i \(-0.683161\pi\)
−0.544185 + 0.838965i \(0.683161\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.369102 0.184551
\(5\) 0 0
\(6\) −1.53919 −0.628371
\(7\) −0.290725 −0.109884 −0.0549418 0.998490i \(-0.517497\pi\)
−0.0549418 + 0.998490i \(0.517497\pi\)
\(8\) 2.51026 0.887511
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0.369102 0.106551
\(13\) −6.97107 −1.93343 −0.966714 0.255861i \(-0.917641\pi\)
−0.966714 + 0.255861i \(0.917641\pi\)
\(14\) 0.447480 0.119594
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) −4.78765 −1.16118 −0.580588 0.814197i \(-0.697177\pi\)
−0.580588 + 0.814197i \(0.697177\pi\)
\(18\) −1.53919 −0.362790
\(19\) 7.75872 1.77997 0.889987 0.455987i \(-0.150714\pi\)
0.889987 + 0.455987i \(0.150714\pi\)
\(20\) 0 0
\(21\) −0.290725 −0.0634413
\(22\) 1.53919 0.328156
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 2.51026 0.512405
\(25\) 0 0
\(26\) 10.7298 2.10429
\(27\) 1.00000 0.192450
\(28\) −0.107307 −0.0202791
\(29\) −7.41855 −1.37759 −0.688795 0.724956i \(-0.741860\pi\)
−0.688795 + 0.724956i \(0.741860\pi\)
\(30\) 0 0
\(31\) 6.34017 1.13873 0.569364 0.822085i \(-0.307189\pi\)
0.569364 + 0.822085i \(0.307189\pi\)
\(32\) 2.06278 0.364651
\(33\) −1.00000 −0.174078
\(34\) 7.36910 1.26379
\(35\) 0 0
\(36\) 0.369102 0.0615171
\(37\) 3.41855 0.562006 0.281003 0.959707i \(-0.409333\pi\)
0.281003 + 0.959707i \(0.409333\pi\)
\(38\) −11.9421 −1.93727
\(39\) −6.97107 −1.11626
\(40\) 0 0
\(41\) −7.41855 −1.15858 −0.579291 0.815120i \(-0.696671\pi\)
−0.579291 + 0.815120i \(0.696671\pi\)
\(42\) 0.447480 0.0690477
\(43\) −0.290725 −0.0443351 −0.0221675 0.999754i \(-0.507057\pi\)
−0.0221675 + 0.999754i \(0.507057\pi\)
\(44\) −0.369102 −0.0556443
\(45\) 0 0
\(46\) 6.15676 0.907764
\(47\) −5.26180 −0.767512 −0.383756 0.923435i \(-0.625370\pi\)
−0.383756 + 0.923435i \(0.625370\pi\)
\(48\) −4.60197 −0.664237
\(49\) −6.91548 −0.987926
\(50\) 0 0
\(51\) −4.78765 −0.670406
\(52\) −2.57304 −0.356816
\(53\) −5.75872 −0.791022 −0.395511 0.918461i \(-0.629432\pi\)
−0.395511 + 0.918461i \(0.629432\pi\)
\(54\) −1.53919 −0.209457
\(55\) 0 0
\(56\) −0.729794 −0.0975229
\(57\) 7.75872 1.02767
\(58\) 11.4186 1.49933
\(59\) 3.60197 0.468936 0.234468 0.972124i \(-0.424665\pi\)
0.234468 + 0.972124i \(0.424665\pi\)
\(60\) 0 0
\(61\) −6.68035 −0.855331 −0.427665 0.903937i \(-0.640664\pi\)
−0.427665 + 0.903937i \(0.640664\pi\)
\(62\) −9.75872 −1.23936
\(63\) −0.290725 −0.0366279
\(64\) 6.02893 0.753616
\(65\) 0 0
\(66\) 1.53919 0.189461
\(67\) −6.15676 −0.752167 −0.376084 0.926586i \(-0.622729\pi\)
−0.376084 + 0.926586i \(0.622729\pi\)
\(68\) −1.76713 −0.214296
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −5.07838 −0.602693 −0.301346 0.953515i \(-0.597436\pi\)
−0.301346 + 0.953515i \(0.597436\pi\)
\(72\) 2.51026 0.295837
\(73\) 1.12783 0.132002 0.0660010 0.997820i \(-0.478976\pi\)
0.0660010 + 0.997820i \(0.478976\pi\)
\(74\) −5.26180 −0.611671
\(75\) 0 0
\(76\) 2.86376 0.328496
\(77\) 0.290725 0.0331311
\(78\) 10.7298 1.21491
\(79\) −0.921622 −0.103691 −0.0518453 0.998655i \(-0.516510\pi\)
−0.0518453 + 0.998655i \(0.516510\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 11.4186 1.26097
\(83\) −1.70928 −0.187617 −0.0938087 0.995590i \(-0.529904\pi\)
−0.0938087 + 0.995590i \(0.529904\pi\)
\(84\) −0.107307 −0.0117082
\(85\) 0 0
\(86\) 0.447480 0.0482530
\(87\) −7.41855 −0.795352
\(88\) −2.51026 −0.267595
\(89\) 4.34017 0.460057 0.230029 0.973184i \(-0.426118\pi\)
0.230029 + 0.973184i \(0.426118\pi\)
\(90\) 0 0
\(91\) 2.02666 0.212452
\(92\) −1.47641 −0.153926
\(93\) 6.34017 0.657445
\(94\) 8.09890 0.835337
\(95\) 0 0
\(96\) 2.06278 0.210532
\(97\) −4.68035 −0.475217 −0.237609 0.971361i \(-0.576364\pi\)
−0.237609 + 0.971361i \(0.576364\pi\)
\(98\) 10.6442 1.07523
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −8.58145 −0.853886 −0.426943 0.904279i \(-0.640410\pi\)
−0.426943 + 0.904279i \(0.640410\pi\)
\(102\) 7.36910 0.729650
\(103\) −6.73820 −0.663935 −0.331968 0.943291i \(-0.607712\pi\)
−0.331968 + 0.943291i \(0.607712\pi\)
\(104\) −17.4992 −1.71594
\(105\) 0 0
\(106\) 8.86376 0.860925
\(107\) −12.2329 −1.18260 −0.591298 0.806453i \(-0.701384\pi\)
−0.591298 + 0.806453i \(0.701384\pi\)
\(108\) 0.369102 0.0355169
\(109\) −6.31351 −0.604725 −0.302362 0.953193i \(-0.597775\pi\)
−0.302362 + 0.953193i \(0.597775\pi\)
\(110\) 0 0
\(111\) 3.41855 0.324474
\(112\) 1.33791 0.126420
\(113\) 16.4969 1.55190 0.775950 0.630794i \(-0.217271\pi\)
0.775950 + 0.630794i \(0.217271\pi\)
\(114\) −11.9421 −1.11848
\(115\) 0 0
\(116\) −2.73820 −0.254236
\(117\) −6.97107 −0.644476
\(118\) −5.54411 −0.510377
\(119\) 1.39189 0.127594
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 10.2823 0.930917
\(123\) −7.41855 −0.668908
\(124\) 2.34017 0.210154
\(125\) 0 0
\(126\) 0.447480 0.0398647
\(127\) −4.87217 −0.432336 −0.216168 0.976356i \(-0.569356\pi\)
−0.216168 + 0.976356i \(0.569356\pi\)
\(128\) −13.4052 −1.18487
\(129\) −0.290725 −0.0255969
\(130\) 0 0
\(131\) −8.68035 −0.758405 −0.379203 0.925314i \(-0.623802\pi\)
−0.379203 + 0.925314i \(0.623802\pi\)
\(132\) −0.369102 −0.0321262
\(133\) −2.25565 −0.195590
\(134\) 9.47641 0.818637
\(135\) 0 0
\(136\) −12.0183 −1.03056
\(137\) 12.5958 1.07613 0.538067 0.842902i \(-0.319155\pi\)
0.538067 + 0.842902i \(0.319155\pi\)
\(138\) 6.15676 0.524098
\(139\) −9.91548 −0.841020 −0.420510 0.907288i \(-0.638149\pi\)
−0.420510 + 0.907288i \(0.638149\pi\)
\(140\) 0 0
\(141\) −5.26180 −0.443123
\(142\) 7.81658 0.655953
\(143\) 6.97107 0.582950
\(144\) −4.60197 −0.383497
\(145\) 0 0
\(146\) −1.73594 −0.143667
\(147\) −6.91548 −0.570379
\(148\) 1.26180 0.103719
\(149\) −1.26180 −0.103370 −0.0516851 0.998663i \(-0.516459\pi\)
−0.0516851 + 0.998663i \(0.516459\pi\)
\(150\) 0 0
\(151\) 1.60197 0.130366 0.0651832 0.997873i \(-0.479237\pi\)
0.0651832 + 0.997873i \(0.479237\pi\)
\(152\) 19.4764 1.57975
\(153\) −4.78765 −0.387059
\(154\) −0.447480 −0.0360590
\(155\) 0 0
\(156\) −2.57304 −0.206008
\(157\) 7.10504 0.567044 0.283522 0.958966i \(-0.408497\pi\)
0.283522 + 0.958966i \(0.408497\pi\)
\(158\) 1.41855 0.112854
\(159\) −5.75872 −0.456696
\(160\) 0 0
\(161\) 1.16290 0.0916492
\(162\) −1.53919 −0.120930
\(163\) 22.9360 1.79649 0.898243 0.439499i \(-0.144844\pi\)
0.898243 + 0.439499i \(0.144844\pi\)
\(164\) −2.73820 −0.213818
\(165\) 0 0
\(166\) 2.63090 0.204197
\(167\) −4.81432 −0.372543 −0.186271 0.982498i \(-0.559640\pi\)
−0.186271 + 0.982498i \(0.559640\pi\)
\(168\) −0.729794 −0.0563049
\(169\) 35.5958 2.73814
\(170\) 0 0
\(171\) 7.75872 0.593324
\(172\) −0.107307 −0.00818209
\(173\) 12.8865 0.979746 0.489873 0.871794i \(-0.337043\pi\)
0.489873 + 0.871794i \(0.337043\pi\)
\(174\) 11.4186 0.865638
\(175\) 0 0
\(176\) 4.60197 0.346886
\(177\) 3.60197 0.270741
\(178\) −6.68035 −0.500713
\(179\) 1.84324 0.137771 0.0688853 0.997625i \(-0.478056\pi\)
0.0688853 + 0.997625i \(0.478056\pi\)
\(180\) 0 0
\(181\) 10.2823 0.764278 0.382139 0.924105i \(-0.375187\pi\)
0.382139 + 0.924105i \(0.375187\pi\)
\(182\) −3.11942 −0.231226
\(183\) −6.68035 −0.493825
\(184\) −10.0410 −0.740235
\(185\) 0 0
\(186\) −9.75872 −0.715544
\(187\) 4.78765 0.350108
\(188\) −1.94214 −0.141645
\(189\) −0.290725 −0.0211471
\(190\) 0 0
\(191\) −11.5174 −0.833373 −0.416687 0.909050i \(-0.636809\pi\)
−0.416687 + 0.909050i \(0.636809\pi\)
\(192\) 6.02893 0.435101
\(193\) −3.86603 −0.278283 −0.139141 0.990273i \(-0.544434\pi\)
−0.139141 + 0.990273i \(0.544434\pi\)
\(194\) 7.20394 0.517212
\(195\) 0 0
\(196\) −2.55252 −0.182323
\(197\) 8.57304 0.610804 0.305402 0.952224i \(-0.401209\pi\)
0.305402 + 0.952224i \(0.401209\pi\)
\(198\) 1.53919 0.109385
\(199\) 8.31351 0.589329 0.294665 0.955601i \(-0.404792\pi\)
0.294665 + 0.955601i \(0.404792\pi\)
\(200\) 0 0
\(201\) −6.15676 −0.434264
\(202\) 13.2085 0.929345
\(203\) 2.15676 0.151375
\(204\) −1.76713 −0.123724
\(205\) 0 0
\(206\) 10.3714 0.722608
\(207\) −4.00000 −0.278019
\(208\) 32.0806 2.22439
\(209\) −7.75872 −0.536682
\(210\) 0 0
\(211\) 25.9155 1.78410 0.892048 0.451941i \(-0.149268\pi\)
0.892048 + 0.451941i \(0.149268\pi\)
\(212\) −2.12556 −0.145984
\(213\) −5.07838 −0.347965
\(214\) 18.8287 1.28710
\(215\) 0 0
\(216\) 2.51026 0.170802
\(217\) −1.84324 −0.125128
\(218\) 9.71769 0.658165
\(219\) 1.12783 0.0762114
\(220\) 0 0
\(221\) 33.3751 2.24505
\(222\) −5.26180 −0.353149
\(223\) 9.62863 0.644781 0.322390 0.946607i \(-0.395514\pi\)
0.322390 + 0.946607i \(0.395514\pi\)
\(224\) −0.599701 −0.0400692
\(225\) 0 0
\(226\) −25.3919 −1.68904
\(227\) 18.3896 1.22056 0.610281 0.792185i \(-0.291057\pi\)
0.610281 + 0.792185i \(0.291057\pi\)
\(228\) 2.86376 0.189657
\(229\) 17.1506 1.13334 0.566672 0.823943i \(-0.308231\pi\)
0.566672 + 0.823943i \(0.308231\pi\)
\(230\) 0 0
\(231\) 0.290725 0.0191283
\(232\) −18.6225 −1.22263
\(233\) 4.10731 0.269079 0.134539 0.990908i \(-0.457045\pi\)
0.134539 + 0.990908i \(0.457045\pi\)
\(234\) 10.7298 0.701429
\(235\) 0 0
\(236\) 1.32950 0.0865428
\(237\) −0.921622 −0.0598658
\(238\) −2.14238 −0.138870
\(239\) −4.36683 −0.282467 −0.141234 0.989976i \(-0.545107\pi\)
−0.141234 + 0.989976i \(0.545107\pi\)
\(240\) 0 0
\(241\) −5.20394 −0.335215 −0.167608 0.985854i \(-0.553604\pi\)
−0.167608 + 0.985854i \(0.553604\pi\)
\(242\) −1.53919 −0.0989428
\(243\) 1.00000 0.0641500
\(244\) −2.46573 −0.157852
\(245\) 0 0
\(246\) 11.4186 0.728020
\(247\) −54.0866 −3.44145
\(248\) 15.9155 1.01063
\(249\) −1.70928 −0.108321
\(250\) 0 0
\(251\) −8.28231 −0.522775 −0.261388 0.965234i \(-0.584180\pi\)
−0.261388 + 0.965234i \(0.584180\pi\)
\(252\) −0.107307 −0.00675972
\(253\) 4.00000 0.251478
\(254\) 7.49920 0.470541
\(255\) 0 0
\(256\) 8.57531 0.535957
\(257\) −11.0205 −0.687441 −0.343721 0.939072i \(-0.611687\pi\)
−0.343721 + 0.939072i \(0.611687\pi\)
\(258\) 0.447480 0.0278589
\(259\) −0.993857 −0.0617553
\(260\) 0 0
\(261\) −7.41855 −0.459197
\(262\) 13.3607 0.825426
\(263\) 2.97107 0.183204 0.0916020 0.995796i \(-0.470801\pi\)
0.0916020 + 0.995796i \(0.470801\pi\)
\(264\) −2.51026 −0.154496
\(265\) 0 0
\(266\) 3.47187 0.212874
\(267\) 4.34017 0.265614
\(268\) −2.27247 −0.138813
\(269\) −15.8432 −0.965980 −0.482990 0.875626i \(-0.660449\pi\)
−0.482990 + 0.875626i \(0.660449\pi\)
\(270\) 0 0
\(271\) −6.28231 −0.381623 −0.190812 0.981627i \(-0.561112\pi\)
−0.190812 + 0.981627i \(0.561112\pi\)
\(272\) 22.0326 1.33592
\(273\) 2.02666 0.122659
\(274\) −19.3874 −1.17123
\(275\) 0 0
\(276\) −1.47641 −0.0888694
\(277\) −5.12783 −0.308101 −0.154051 0.988063i \(-0.549232\pi\)
−0.154051 + 0.988063i \(0.549232\pi\)
\(278\) 15.2618 0.915342
\(279\) 6.34017 0.379576
\(280\) 0 0
\(281\) 21.8888 1.30578 0.652889 0.757454i \(-0.273557\pi\)
0.652889 + 0.757454i \(0.273557\pi\)
\(282\) 8.09890 0.482282
\(283\) −25.9649 −1.54345 −0.771727 0.635954i \(-0.780607\pi\)
−0.771727 + 0.635954i \(0.780607\pi\)
\(284\) −1.87444 −0.111228
\(285\) 0 0
\(286\) −10.7298 −0.634466
\(287\) 2.15676 0.127309
\(288\) 2.06278 0.121550
\(289\) 5.92162 0.348331
\(290\) 0 0
\(291\) −4.68035 −0.274367
\(292\) 0.416283 0.0243611
\(293\) −10.4163 −0.608526 −0.304263 0.952588i \(-0.598410\pi\)
−0.304263 + 0.952588i \(0.598410\pi\)
\(294\) 10.6442 0.620784
\(295\) 0 0
\(296\) 8.58145 0.498787
\(297\) −1.00000 −0.0580259
\(298\) 1.94214 0.112505
\(299\) 27.8843 1.61259
\(300\) 0 0
\(301\) 0.0845208 0.00487170
\(302\) −2.46573 −0.141887
\(303\) −8.58145 −0.492991
\(304\) −35.7054 −2.04785
\(305\) 0 0
\(306\) 7.36910 0.421264
\(307\) 29.0700 1.65911 0.829555 0.558425i \(-0.188594\pi\)
0.829555 + 0.558425i \(0.188594\pi\)
\(308\) 0.107307 0.00611439
\(309\) −6.73820 −0.383323
\(310\) 0 0
\(311\) 5.44521 0.308770 0.154385 0.988011i \(-0.450660\pi\)
0.154385 + 0.988011i \(0.450660\pi\)
\(312\) −17.4992 −0.990697
\(313\) −25.0928 −1.41833 −0.709163 0.705044i \(-0.750927\pi\)
−0.709163 + 0.705044i \(0.750927\pi\)
\(314\) −10.9360 −0.617154
\(315\) 0 0
\(316\) −0.340173 −0.0191362
\(317\) −21.1773 −1.18943 −0.594717 0.803935i \(-0.702736\pi\)
−0.594717 + 0.803935i \(0.702736\pi\)
\(318\) 8.86376 0.497055
\(319\) 7.41855 0.415359
\(320\) 0 0
\(321\) −12.2329 −0.682772
\(322\) −1.78992 −0.0997484
\(323\) −37.1461 −2.06686
\(324\) 0.369102 0.0205057
\(325\) 0 0
\(326\) −35.3028 −1.95524
\(327\) −6.31351 −0.349138
\(328\) −18.6225 −1.02825
\(329\) 1.52973 0.0843369
\(330\) 0 0
\(331\) −6.70701 −0.368650 −0.184325 0.982865i \(-0.559010\pi\)
−0.184325 + 0.982865i \(0.559010\pi\)
\(332\) −0.630898 −0.0346250
\(333\) 3.41855 0.187335
\(334\) 7.41014 0.405465
\(335\) 0 0
\(336\) 1.33791 0.0729887
\(337\) −23.7503 −1.29376 −0.646881 0.762591i \(-0.723927\pi\)
−0.646881 + 0.762591i \(0.723927\pi\)
\(338\) −54.7887 −2.98011
\(339\) 16.4969 0.895990
\(340\) 0 0
\(341\) −6.34017 −0.343340
\(342\) −11.9421 −0.645757
\(343\) 4.04557 0.218440
\(344\) −0.729794 −0.0393479
\(345\) 0 0
\(346\) −19.8348 −1.06633
\(347\) 31.1689 1.67323 0.836616 0.547790i \(-0.184531\pi\)
0.836616 + 0.547790i \(0.184531\pi\)
\(348\) −2.73820 −0.146783
\(349\) 11.0472 0.591342 0.295671 0.955290i \(-0.404457\pi\)
0.295671 + 0.955290i \(0.404457\pi\)
\(350\) 0 0
\(351\) −6.97107 −0.372088
\(352\) −2.06278 −0.109947
\(353\) 27.4329 1.46011 0.730054 0.683390i \(-0.239495\pi\)
0.730054 + 0.683390i \(0.239495\pi\)
\(354\) −5.54411 −0.294666
\(355\) 0 0
\(356\) 1.60197 0.0849041
\(357\) 1.39189 0.0736666
\(358\) −2.83710 −0.149945
\(359\) −19.1506 −1.01073 −0.505365 0.862905i \(-0.668642\pi\)
−0.505365 + 0.862905i \(0.668642\pi\)
\(360\) 0 0
\(361\) 41.1978 2.16830
\(362\) −15.8264 −0.831818
\(363\) 1.00000 0.0524864
\(364\) 0.748046 0.0392083
\(365\) 0 0
\(366\) 10.2823 0.537465
\(367\) 14.5692 0.760504 0.380252 0.924883i \(-0.375837\pi\)
0.380252 + 0.924883i \(0.375837\pi\)
\(368\) 18.4079 0.959577
\(369\) −7.41855 −0.386194
\(370\) 0 0
\(371\) 1.67420 0.0869203
\(372\) 2.34017 0.121332
\(373\) 8.81432 0.456388 0.228194 0.973616i \(-0.426718\pi\)
0.228194 + 0.973616i \(0.426718\pi\)
\(374\) −7.36910 −0.381047
\(375\) 0 0
\(376\) −13.2085 −0.681175
\(377\) 51.7152 2.66347
\(378\) 0.447480 0.0230159
\(379\) −23.5174 −1.20801 −0.604005 0.796980i \(-0.706429\pi\)
−0.604005 + 0.796980i \(0.706429\pi\)
\(380\) 0 0
\(381\) −4.87217 −0.249609
\(382\) 17.7275 0.907019
\(383\) 19.3028 0.986329 0.493164 0.869936i \(-0.335840\pi\)
0.493164 + 0.869936i \(0.335840\pi\)
\(384\) −13.4052 −0.684082
\(385\) 0 0
\(386\) 5.95055 0.302875
\(387\) −0.290725 −0.0147784
\(388\) −1.72753 −0.0877019
\(389\) −20.0410 −1.01612 −0.508060 0.861321i \(-0.669637\pi\)
−0.508060 + 0.861321i \(0.669637\pi\)
\(390\) 0 0
\(391\) 19.1506 0.968488
\(392\) −17.3596 −0.876795
\(393\) −8.68035 −0.437866
\(394\) −13.1955 −0.664781
\(395\) 0 0
\(396\) −0.369102 −0.0185481
\(397\) 33.3607 1.67433 0.837163 0.546954i \(-0.184213\pi\)
0.837163 + 0.546954i \(0.184213\pi\)
\(398\) −12.7961 −0.641409
\(399\) −2.25565 −0.112924
\(400\) 0 0
\(401\) −37.6475 −1.88003 −0.940014 0.341135i \(-0.889189\pi\)
−0.940014 + 0.341135i \(0.889189\pi\)
\(402\) 9.47641 0.472640
\(403\) −44.1978 −2.20165
\(404\) −3.16743 −0.157586
\(405\) 0 0
\(406\) −3.31965 −0.164752
\(407\) −3.41855 −0.169451
\(408\) −12.0183 −0.594992
\(409\) −25.2039 −1.24625 −0.623127 0.782120i \(-0.714138\pi\)
−0.623127 + 0.782120i \(0.714138\pi\)
\(410\) 0 0
\(411\) 12.5958 0.621306
\(412\) −2.48709 −0.122530
\(413\) −1.04718 −0.0515284
\(414\) 6.15676 0.302588
\(415\) 0 0
\(416\) −14.3798 −0.705027
\(417\) −9.91548 −0.485563
\(418\) 11.9421 0.584109
\(419\) 17.8432 0.871700 0.435850 0.900019i \(-0.356448\pi\)
0.435850 + 0.900019i \(0.356448\pi\)
\(420\) 0 0
\(421\) −11.8120 −0.575684 −0.287842 0.957678i \(-0.592938\pi\)
−0.287842 + 0.957678i \(0.592938\pi\)
\(422\) −39.8888 −1.94176
\(423\) −5.26180 −0.255837
\(424\) −14.4559 −0.702040
\(425\) 0 0
\(426\) 7.81658 0.378715
\(427\) 1.94214 0.0939868
\(428\) −4.51518 −0.218249
\(429\) 6.97107 0.336566
\(430\) 0 0
\(431\) 16.6803 0.803464 0.401732 0.915757i \(-0.368408\pi\)
0.401732 + 0.915757i \(0.368408\pi\)
\(432\) −4.60197 −0.221412
\(433\) 28.3135 1.36066 0.680330 0.732906i \(-0.261836\pi\)
0.680330 + 0.732906i \(0.261836\pi\)
\(434\) 2.83710 0.136185
\(435\) 0 0
\(436\) −2.33033 −0.111603
\(437\) −31.0349 −1.48460
\(438\) −1.73594 −0.0829463
\(439\) −27.4452 −1.30989 −0.654944 0.755677i \(-0.727308\pi\)
−0.654944 + 0.755677i \(0.727308\pi\)
\(440\) 0 0
\(441\) −6.91548 −0.329309
\(442\) −51.3705 −2.44345
\(443\) −0.412408 −0.0195941 −0.00979704 0.999952i \(-0.503119\pi\)
−0.00979704 + 0.999952i \(0.503119\pi\)
\(444\) 1.26180 0.0598822
\(445\) 0 0
\(446\) −14.8203 −0.701761
\(447\) −1.26180 −0.0596809
\(448\) −1.75276 −0.0828100
\(449\) 1.33403 0.0629568 0.0314784 0.999504i \(-0.489978\pi\)
0.0314784 + 0.999504i \(0.489978\pi\)
\(450\) 0 0
\(451\) 7.41855 0.349326
\(452\) 6.08906 0.286405
\(453\) 1.60197 0.0752670
\(454\) −28.3051 −1.32842
\(455\) 0 0
\(456\) 19.4764 0.912066
\(457\) −11.6514 −0.545030 −0.272515 0.962151i \(-0.587855\pi\)
−0.272515 + 0.962151i \(0.587855\pi\)
\(458\) −26.3980 −1.23350
\(459\) −4.78765 −0.223469
\(460\) 0 0
\(461\) −2.05786 −0.0958440 −0.0479220 0.998851i \(-0.515260\pi\)
−0.0479220 + 0.998851i \(0.515260\pi\)
\(462\) −0.447480 −0.0208187
\(463\) −28.7792 −1.33748 −0.668742 0.743494i \(-0.733167\pi\)
−0.668742 + 0.743494i \(0.733167\pi\)
\(464\) 34.1399 1.58491
\(465\) 0 0
\(466\) −6.32192 −0.292857
\(467\) −1.84324 −0.0852952 −0.0426476 0.999090i \(-0.513579\pi\)
−0.0426476 + 0.999090i \(0.513579\pi\)
\(468\) −2.57304 −0.118939
\(469\) 1.78992 0.0826509
\(470\) 0 0
\(471\) 7.10504 0.327383
\(472\) 9.04187 0.416186
\(473\) 0.290725 0.0133675
\(474\) 1.41855 0.0651562
\(475\) 0 0
\(476\) 0.513749 0.0235477
\(477\) −5.75872 −0.263674
\(478\) 6.72138 0.307429
\(479\) 26.8371 1.22622 0.613109 0.789998i \(-0.289919\pi\)
0.613109 + 0.789998i \(0.289919\pi\)
\(480\) 0 0
\(481\) −23.8310 −1.08660
\(482\) 8.00984 0.364838
\(483\) 1.16290 0.0529137
\(484\) 0.369102 0.0167774
\(485\) 0 0
\(486\) −1.53919 −0.0698190
\(487\) −28.5646 −1.29439 −0.647193 0.762326i \(-0.724057\pi\)
−0.647193 + 0.762326i \(0.724057\pi\)
\(488\) −16.7694 −0.759115
\(489\) 22.9360 1.03720
\(490\) 0 0
\(491\) 25.9877 1.17281 0.586405 0.810018i \(-0.300543\pi\)
0.586405 + 0.810018i \(0.300543\pi\)
\(492\) −2.73820 −0.123448
\(493\) 35.5174 1.59963
\(494\) 83.2495 3.74557
\(495\) 0 0
\(496\) −29.1773 −1.31010
\(497\) 1.47641 0.0662260
\(498\) 2.63090 0.117893
\(499\) −27.5174 −1.23185 −0.615925 0.787805i \(-0.711218\pi\)
−0.615925 + 0.787805i \(0.711218\pi\)
\(500\) 0 0
\(501\) −4.81432 −0.215088
\(502\) 12.7480 0.568973
\(503\) −22.6576 −1.01025 −0.505125 0.863046i \(-0.668554\pi\)
−0.505125 + 0.863046i \(0.668554\pi\)
\(504\) −0.729794 −0.0325076
\(505\) 0 0
\(506\) −6.15676 −0.273701
\(507\) 35.5958 1.58087
\(508\) −1.79833 −0.0797880
\(509\) −27.8432 −1.23413 −0.617065 0.786912i \(-0.711678\pi\)
−0.617065 + 0.786912i \(0.711678\pi\)
\(510\) 0 0
\(511\) −0.327887 −0.0145049
\(512\) 13.6114 0.601546
\(513\) 7.75872 0.342556
\(514\) 16.9627 0.748191
\(515\) 0 0
\(516\) −0.107307 −0.00472393
\(517\) 5.26180 0.231413
\(518\) 1.52973 0.0672126
\(519\) 12.8865 0.565657
\(520\) 0 0
\(521\) 29.7009 1.30122 0.650609 0.759413i \(-0.274514\pi\)
0.650609 + 0.759413i \(0.274514\pi\)
\(522\) 11.4186 0.499776
\(523\) −24.7565 −1.08252 −0.541262 0.840854i \(-0.682053\pi\)
−0.541262 + 0.840854i \(0.682053\pi\)
\(524\) −3.20394 −0.139965
\(525\) 0 0
\(526\) −4.57304 −0.199394
\(527\) −30.3545 −1.32226
\(528\) 4.60197 0.200275
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 3.60197 0.156312
\(532\) −0.832567 −0.0360963
\(533\) 51.7152 2.24004
\(534\) −6.68035 −0.289087
\(535\) 0 0
\(536\) −15.4551 −0.667557
\(537\) 1.84324 0.0795419
\(538\) 24.3857 1.05134
\(539\) 6.91548 0.297871
\(540\) 0 0
\(541\) −12.5236 −0.538431 −0.269216 0.963080i \(-0.586764\pi\)
−0.269216 + 0.963080i \(0.586764\pi\)
\(542\) 9.66967 0.415348
\(543\) 10.2823 0.441256
\(544\) −9.87587 −0.423425
\(545\) 0 0
\(546\) −3.11942 −0.133499
\(547\) 14.2784 0.610502 0.305251 0.952272i \(-0.401260\pi\)
0.305251 + 0.952272i \(0.401260\pi\)
\(548\) 4.64915 0.198602
\(549\) −6.68035 −0.285110
\(550\) 0 0
\(551\) −57.5585 −2.45207
\(552\) −10.0410 −0.427375
\(553\) 0.267938 0.0113939
\(554\) 7.89269 0.335328
\(555\) 0 0
\(556\) −3.65983 −0.155211
\(557\) 3.57918 0.151655 0.0758274 0.997121i \(-0.475840\pi\)
0.0758274 + 0.997121i \(0.475840\pi\)
\(558\) −9.75872 −0.413120
\(559\) 2.02666 0.0857187
\(560\) 0 0
\(561\) 4.78765 0.202135
\(562\) −33.6910 −1.42117
\(563\) −28.9588 −1.22047 −0.610234 0.792222i \(-0.708924\pi\)
−0.610234 + 0.792222i \(0.708924\pi\)
\(564\) −1.94214 −0.0817789
\(565\) 0 0
\(566\) 39.9649 1.67985
\(567\) −0.290725 −0.0122093
\(568\) −12.7480 −0.534896
\(569\) −25.8264 −1.08270 −0.541350 0.840797i \(-0.682087\pi\)
−0.541350 + 0.840797i \(0.682087\pi\)
\(570\) 0 0
\(571\) 17.1194 0.716425 0.358213 0.933640i \(-0.383386\pi\)
0.358213 + 0.933640i \(0.383386\pi\)
\(572\) 2.57304 0.107584
\(573\) −11.5174 −0.481148
\(574\) −3.31965 −0.138560
\(575\) 0 0
\(576\) 6.02893 0.251205
\(577\) −22.5692 −0.939567 −0.469783 0.882782i \(-0.655668\pi\)
−0.469783 + 0.882782i \(0.655668\pi\)
\(578\) −9.11450 −0.379113
\(579\) −3.86603 −0.160667
\(580\) 0 0
\(581\) 0.496928 0.0206161
\(582\) 7.20394 0.298613
\(583\) 5.75872 0.238502
\(584\) 2.83114 0.117153
\(585\) 0 0
\(586\) 16.0326 0.662302
\(587\) −3.63317 −0.149957 −0.0749784 0.997185i \(-0.523889\pi\)
−0.0749784 + 0.997185i \(0.523889\pi\)
\(588\) −2.55252 −0.105264
\(589\) 49.1917 2.02691
\(590\) 0 0
\(591\) 8.57304 0.352648
\(592\) −15.7321 −0.646584
\(593\) −12.3051 −0.505310 −0.252655 0.967556i \(-0.581304\pi\)
−0.252655 + 0.967556i \(0.581304\pi\)
\(594\) 1.53919 0.0631537
\(595\) 0 0
\(596\) −0.465732 −0.0190771
\(597\) 8.31351 0.340249
\(598\) −42.9192 −1.75510
\(599\) 19.6865 0.804368 0.402184 0.915559i \(-0.368251\pi\)
0.402184 + 0.915559i \(0.368251\pi\)
\(600\) 0 0
\(601\) 25.8843 1.05584 0.527921 0.849294i \(-0.322972\pi\)
0.527921 + 0.849294i \(0.322972\pi\)
\(602\) −0.130094 −0.00530222
\(603\) −6.15676 −0.250722
\(604\) 0.591290 0.0240593
\(605\) 0 0
\(606\) 13.2085 0.536557
\(607\) 41.5357 1.68588 0.842941 0.538006i \(-0.180822\pi\)
0.842941 + 0.538006i \(0.180822\pi\)
\(608\) 16.0045 0.649070
\(609\) 2.15676 0.0873961
\(610\) 0 0
\(611\) 36.6803 1.48393
\(612\) −1.76713 −0.0714322
\(613\) 20.6453 0.833855 0.416927 0.908940i \(-0.363107\pi\)
0.416927 + 0.908940i \(0.363107\pi\)
\(614\) −44.7442 −1.80573
\(615\) 0 0
\(616\) 0.729794 0.0294042
\(617\) −8.69472 −0.350036 −0.175018 0.984565i \(-0.555998\pi\)
−0.175018 + 0.984565i \(0.555998\pi\)
\(618\) 10.3714 0.417198
\(619\) 2.65368 0.106661 0.0533303 0.998577i \(-0.483016\pi\)
0.0533303 + 0.998577i \(0.483016\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −8.38121 −0.336056
\(623\) −1.26180 −0.0505528
\(624\) 32.0806 1.28425
\(625\) 0 0
\(626\) 38.6225 1.54367
\(627\) −7.75872 −0.309854
\(628\) 2.62249 0.104649
\(629\) −16.3668 −0.652588
\(630\) 0 0
\(631\) −10.6393 −0.423544 −0.211772 0.977319i \(-0.567923\pi\)
−0.211772 + 0.977319i \(0.567923\pi\)
\(632\) −2.31351 −0.0920265
\(633\) 25.9155 1.03005
\(634\) 32.5958 1.29455
\(635\) 0 0
\(636\) −2.12556 −0.0842839
\(637\) 48.2083 1.91008
\(638\) −11.4186 −0.452065
\(639\) −5.07838 −0.200898
\(640\) 0 0
\(641\) −35.8576 −1.41629 −0.708145 0.706067i \(-0.750468\pi\)
−0.708145 + 0.706067i \(0.750468\pi\)
\(642\) 18.8287 0.743109
\(643\) −12.2146 −0.481697 −0.240849 0.970563i \(-0.577426\pi\)
−0.240849 + 0.970563i \(0.577426\pi\)
\(644\) 0.429229 0.0169140
\(645\) 0 0
\(646\) 57.1748 2.24951
\(647\) −25.9421 −1.01989 −0.509945 0.860207i \(-0.670334\pi\)
−0.509945 + 0.860207i \(0.670334\pi\)
\(648\) 2.51026 0.0986123
\(649\) −3.60197 −0.141390
\(650\) 0 0
\(651\) −1.84324 −0.0722424
\(652\) 8.46573 0.331544
\(653\) 14.3402 0.561174 0.280587 0.959829i \(-0.409471\pi\)
0.280587 + 0.959829i \(0.409471\pi\)
\(654\) 9.71769 0.379992
\(655\) 0 0
\(656\) 34.1399 1.33294
\(657\) 1.12783 0.0440007
\(658\) −2.35455 −0.0917899
\(659\) 6.52359 0.254123 0.127062 0.991895i \(-0.459445\pi\)
0.127062 + 0.991895i \(0.459445\pi\)
\(660\) 0 0
\(661\) 3.16290 0.123022 0.0615112 0.998106i \(-0.480408\pi\)
0.0615112 + 0.998106i \(0.480408\pi\)
\(662\) 10.3234 0.401228
\(663\) 33.3751 1.29618
\(664\) −4.29072 −0.166512
\(665\) 0 0
\(666\) −5.26180 −0.203890
\(667\) 29.6742 1.14899
\(668\) −1.77698 −0.0687532
\(669\) 9.62863 0.372264
\(670\) 0 0
\(671\) 6.68035 0.257892
\(672\) −0.599701 −0.0231340
\(673\) 18.4885 0.712680 0.356340 0.934356i \(-0.384024\pi\)
0.356340 + 0.934356i \(0.384024\pi\)
\(674\) 36.5562 1.40809
\(675\) 0 0
\(676\) 13.1385 0.505327
\(677\) −15.0966 −0.580211 −0.290105 0.956995i \(-0.593690\pi\)
−0.290105 + 0.956995i \(0.593690\pi\)
\(678\) −25.3919 −0.975170
\(679\) 1.36069 0.0522186
\(680\) 0 0
\(681\) 18.3896 0.704692
\(682\) 9.75872 0.373681
\(683\) 12.4657 0.476988 0.238494 0.971144i \(-0.423346\pi\)
0.238494 + 0.971144i \(0.423346\pi\)
\(684\) 2.86376 0.109499
\(685\) 0 0
\(686\) −6.22690 −0.237744
\(687\) 17.1506 0.654337
\(688\) 1.33791 0.0510072
\(689\) 40.1445 1.52938
\(690\) 0 0
\(691\) −30.7214 −1.16870 −0.584348 0.811503i \(-0.698650\pi\)
−0.584348 + 0.811503i \(0.698650\pi\)
\(692\) 4.75646 0.180813
\(693\) 0.290725 0.0110437
\(694\) −47.9748 −1.82110
\(695\) 0 0
\(696\) −18.6225 −0.705884
\(697\) 35.5174 1.34532
\(698\) −17.0037 −0.643599
\(699\) 4.10731 0.155353
\(700\) 0 0
\(701\) 17.9955 0.679679 0.339840 0.940483i \(-0.389627\pi\)
0.339840 + 0.940483i \(0.389627\pi\)
\(702\) 10.7298 0.404970
\(703\) 26.5236 1.00036
\(704\) −6.02893 −0.227224
\(705\) 0 0
\(706\) −42.2245 −1.58914
\(707\) 2.49484 0.0938281
\(708\) 1.32950 0.0499655
\(709\) 23.5897 0.885929 0.442965 0.896539i \(-0.353927\pi\)
0.442965 + 0.896539i \(0.353927\pi\)
\(710\) 0 0
\(711\) −0.921622 −0.0345635
\(712\) 10.8950 0.408306
\(713\) −25.3607 −0.949765
\(714\) −2.14238 −0.0801765
\(715\) 0 0
\(716\) 0.680346 0.0254257
\(717\) −4.36683 −0.163082
\(718\) 29.4764 1.10005
\(719\) 24.5646 0.916106 0.458053 0.888925i \(-0.348547\pi\)
0.458053 + 0.888925i \(0.348547\pi\)
\(720\) 0 0
\(721\) 1.95896 0.0729556
\(722\) −63.4112 −2.35992
\(723\) −5.20394 −0.193536
\(724\) 3.79523 0.141048
\(725\) 0 0
\(726\) −1.53919 −0.0571247
\(727\) −8.51130 −0.315667 −0.157833 0.987466i \(-0.550451\pi\)
−0.157833 + 0.987466i \(0.550451\pi\)
\(728\) 5.08745 0.188553
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.39189 0.0514809
\(732\) −2.46573 −0.0911361
\(733\) 29.8615 1.10296 0.551480 0.834188i \(-0.314063\pi\)
0.551480 + 0.834188i \(0.314063\pi\)
\(734\) −22.4247 −0.827711
\(735\) 0 0
\(736\) −8.25112 −0.304140
\(737\) 6.15676 0.226787
\(738\) 11.4186 0.420323
\(739\) −36.7526 −1.35197 −0.675983 0.736917i \(-0.736281\pi\)
−0.675983 + 0.736917i \(0.736281\pi\)
\(740\) 0 0
\(741\) −54.0866 −1.98692
\(742\) −2.57691 −0.0946015
\(743\) 2.17501 0.0797933 0.0398966 0.999204i \(-0.487297\pi\)
0.0398966 + 0.999204i \(0.487297\pi\)
\(744\) 15.9155 0.583490
\(745\) 0 0
\(746\) −13.5669 −0.496719
\(747\) −1.70928 −0.0625391
\(748\) 1.76713 0.0646128
\(749\) 3.55640 0.129948
\(750\) 0 0
\(751\) −44.4580 −1.62229 −0.811147 0.584842i \(-0.801157\pi\)
−0.811147 + 0.584842i \(0.801157\pi\)
\(752\) 24.2146 0.883016
\(753\) −8.28231 −0.301824
\(754\) −79.5995 −2.89884
\(755\) 0 0
\(756\) −0.107307 −0.00390272
\(757\) 26.4247 0.960422 0.480211 0.877153i \(-0.340560\pi\)
0.480211 + 0.877153i \(0.340560\pi\)
\(758\) 36.1978 1.31476
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −27.6163 −1.00109 −0.500546 0.865710i \(-0.666867\pi\)
−0.500546 + 0.865710i \(0.666867\pi\)
\(762\) 7.49920 0.271667
\(763\) 1.83549 0.0664493
\(764\) −4.25112 −0.153800
\(765\) 0 0
\(766\) −29.7107 −1.07349
\(767\) −25.1096 −0.906654
\(768\) 8.57531 0.309435
\(769\) 27.8432 1.00405 0.502027 0.864852i \(-0.332588\pi\)
0.502027 + 0.864852i \(0.332588\pi\)
\(770\) 0 0
\(771\) −11.0205 −0.396894
\(772\) −1.42696 −0.0513575
\(773\) 18.0267 0.648374 0.324187 0.945993i \(-0.394909\pi\)
0.324187 + 0.945993i \(0.394909\pi\)
\(774\) 0.447480 0.0160843
\(775\) 0 0
\(776\) −11.7489 −0.421760
\(777\) −0.993857 −0.0356544
\(778\) 30.8469 1.10592
\(779\) −57.5585 −2.06225
\(780\) 0 0
\(781\) 5.07838 0.181719
\(782\) −29.4764 −1.05407
\(783\) −7.41855 −0.265117
\(784\) 31.8248 1.13660
\(785\) 0 0
\(786\) 13.3607 0.476560
\(787\) −35.6391 −1.27040 −0.635199 0.772349i \(-0.719082\pi\)
−0.635199 + 0.772349i \(0.719082\pi\)
\(788\) 3.16433 0.112725
\(789\) 2.97107 0.105773
\(790\) 0 0
\(791\) −4.79606 −0.170528
\(792\) −2.51026 −0.0891982
\(793\) 46.5692 1.65372
\(794\) −51.3484 −1.82229
\(795\) 0 0
\(796\) 3.06854 0.108761
\(797\) 4.97948 0.176382 0.0881911 0.996104i \(-0.471891\pi\)
0.0881911 + 0.996104i \(0.471891\pi\)
\(798\) 3.47187 0.122903
\(799\) 25.1917 0.891217
\(800\) 0 0
\(801\) 4.34017 0.153352
\(802\) 57.9467 2.04617
\(803\) −1.12783 −0.0398001
\(804\) −2.27247 −0.0801439
\(805\) 0 0
\(806\) 68.0288 2.39621
\(807\) −15.8432 −0.557709
\(808\) −21.5417 −0.757833
\(809\) 31.6697 1.11345 0.556723 0.830698i \(-0.312058\pi\)
0.556723 + 0.830698i \(0.312058\pi\)
\(810\) 0 0
\(811\) −21.7998 −0.765493 −0.382747 0.923853i \(-0.625022\pi\)
−0.382747 + 0.923853i \(0.625022\pi\)
\(812\) 0.796064 0.0279364
\(813\) −6.28231 −0.220330
\(814\) 5.26180 0.184426
\(815\) 0 0
\(816\) 22.0326 0.771296
\(817\) −2.25565 −0.0789153
\(818\) 38.7936 1.35639
\(819\) 2.02666 0.0708173
\(820\) 0 0
\(821\) −10.9360 −0.381669 −0.190834 0.981622i \(-0.561119\pi\)
−0.190834 + 0.981622i \(0.561119\pi\)
\(822\) −19.3874 −0.676212
\(823\) −45.9709 −1.60244 −0.801222 0.598367i \(-0.795817\pi\)
−0.801222 + 0.598367i \(0.795817\pi\)
\(824\) −16.9146 −0.589249
\(825\) 0 0
\(826\) 1.61181 0.0560820
\(827\) 22.9711 0.798782 0.399391 0.916781i \(-0.369222\pi\)
0.399391 + 0.916781i \(0.369222\pi\)
\(828\) −1.47641 −0.0513088
\(829\) −48.1543 −1.67247 −0.836234 0.548373i \(-0.815248\pi\)
−0.836234 + 0.548373i \(0.815248\pi\)
\(830\) 0 0
\(831\) −5.12783 −0.177882
\(832\) −42.0281 −1.45706
\(833\) 33.1089 1.14716
\(834\) 15.2618 0.528473
\(835\) 0 0
\(836\) −2.86376 −0.0990453
\(837\) 6.34017 0.219148
\(838\) −27.4641 −0.948732
\(839\) −17.9278 −0.618935 −0.309468 0.950910i \(-0.600151\pi\)
−0.309468 + 0.950910i \(0.600151\pi\)
\(840\) 0 0
\(841\) 26.0349 0.897755
\(842\) 18.1810 0.626558
\(843\) 21.8888 0.753891
\(844\) 9.56547 0.329257
\(845\) 0 0
\(846\) 8.09890 0.278446
\(847\) −0.290725 −0.00998942
\(848\) 26.5015 0.910064
\(849\) −25.9649 −0.891114
\(850\) 0 0
\(851\) −13.6742 −0.468746
\(852\) −1.87444 −0.0642173
\(853\) 9.54799 0.326917 0.163458 0.986550i \(-0.447735\pi\)
0.163458 + 0.986550i \(0.447735\pi\)
\(854\) −2.98932 −0.102292
\(855\) 0 0
\(856\) −30.7077 −1.04957
\(857\) −10.5776 −0.361323 −0.180662 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(858\) −10.7298 −0.366309
\(859\) −31.3340 −1.06910 −0.534552 0.845136i \(-0.679520\pi\)
−0.534552 + 0.845136i \(0.679520\pi\)
\(860\) 0 0
\(861\) 2.15676 0.0735020
\(862\) −25.6742 −0.874467
\(863\) −56.7792 −1.93279 −0.966394 0.257066i \(-0.917244\pi\)
−0.966394 + 0.257066i \(0.917244\pi\)
\(864\) 2.06278 0.0701772
\(865\) 0 0
\(866\) −43.5798 −1.48090
\(867\) 5.92162 0.201109
\(868\) −0.680346 −0.0230924
\(869\) 0.921622 0.0312639
\(870\) 0 0
\(871\) 42.9192 1.45426
\(872\) −15.8486 −0.536700
\(873\) −4.68035 −0.158406
\(874\) 47.7686 1.61580
\(875\) 0 0
\(876\) 0.416283 0.0140649
\(877\) −5.87832 −0.198497 −0.0992483 0.995063i \(-0.531644\pi\)
−0.0992483 + 0.995063i \(0.531644\pi\)
\(878\) 42.2434 1.42564
\(879\) −10.4163 −0.351333
\(880\) 0 0
\(881\) 8.21008 0.276605 0.138302 0.990390i \(-0.455835\pi\)
0.138302 + 0.990390i \(0.455835\pi\)
\(882\) 10.6442 0.358410
\(883\) 27.0349 0.909797 0.454898 0.890543i \(-0.349676\pi\)
0.454898 + 0.890543i \(0.349676\pi\)
\(884\) 12.3188 0.414327
\(885\) 0 0
\(886\) 0.634773 0.0213256
\(887\) 48.7442 1.63667 0.818335 0.574742i \(-0.194898\pi\)
0.818335 + 0.574742i \(0.194898\pi\)
\(888\) 8.58145 0.287975
\(889\) 1.41646 0.0475066
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 3.55395 0.118995
\(893\) −40.8248 −1.36615
\(894\) 1.94214 0.0649549
\(895\) 0 0
\(896\) 3.89723 0.130197
\(897\) 27.8843 0.931029
\(898\) −2.05332 −0.0685203
\(899\) −47.0349 −1.56870
\(900\) 0 0
\(901\) 27.5708 0.918516
\(902\) −11.4186 −0.380196
\(903\) 0.0845208 0.00281268
\(904\) 41.4116 1.37733
\(905\) 0 0
\(906\) −2.46573 −0.0819184
\(907\) 32.4657 1.07801 0.539003 0.842304i \(-0.318801\pi\)
0.539003 + 0.842304i \(0.318801\pi\)
\(908\) 6.78765 0.225256
\(909\) −8.58145 −0.284629
\(910\) 0 0
\(911\) −4.84939 −0.160667 −0.0803337 0.996768i \(-0.525599\pi\)
−0.0803337 + 0.996768i \(0.525599\pi\)
\(912\) −35.7054 −1.18232
\(913\) 1.70928 0.0565688
\(914\) 17.9337 0.593195
\(915\) 0 0
\(916\) 6.33033 0.209160
\(917\) 2.52359 0.0833363
\(918\) 7.36910 0.243217
\(919\) −14.3980 −0.474947 −0.237474 0.971394i \(-0.576319\pi\)
−0.237474 + 0.971394i \(0.576319\pi\)
\(920\) 0 0
\(921\) 29.0700 0.957888
\(922\) 3.16743 0.104314
\(923\) 35.4017 1.16526
\(924\) 0.107307 0.00353015
\(925\) 0 0
\(926\) 44.2967 1.45568
\(927\) −6.73820 −0.221312
\(928\) −15.3028 −0.502340
\(929\) −11.1629 −0.366243 −0.183121 0.983090i \(-0.558620\pi\)
−0.183121 + 0.983090i \(0.558620\pi\)
\(930\) 0 0
\(931\) −53.6553 −1.75848
\(932\) 1.51602 0.0496588
\(933\) 5.44521 0.178268
\(934\) 2.83710 0.0928328
\(935\) 0 0
\(936\) −17.4992 −0.571979
\(937\) −48.6453 −1.58917 −0.794586 0.607152i \(-0.792312\pi\)
−0.794586 + 0.607152i \(0.792312\pi\)
\(938\) −2.75503 −0.0899548
\(939\) −25.0928 −0.818871
\(940\) 0 0
\(941\) 16.4124 0.535029 0.267515 0.963554i \(-0.413798\pi\)
0.267515 + 0.963554i \(0.413798\pi\)
\(942\) −10.9360 −0.356314
\(943\) 29.6742 0.966325
\(944\) −16.5761 −0.539508
\(945\) 0 0
\(946\) −0.447480 −0.0145488
\(947\) 45.7198 1.48569 0.742847 0.669462i \(-0.233475\pi\)
0.742847 + 0.669462i \(0.233475\pi\)
\(948\) −0.340173 −0.0110483
\(949\) −7.86216 −0.255216
\(950\) 0 0
\(951\) −21.1773 −0.686720
\(952\) 3.49400 0.113241
\(953\) 26.4619 0.857184 0.428592 0.903498i \(-0.359010\pi\)
0.428592 + 0.903498i \(0.359010\pi\)
\(954\) 8.86376 0.286975
\(955\) 0 0
\(956\) −1.61181 −0.0521296
\(957\) 7.41855 0.239808
\(958\) −41.3074 −1.33458
\(959\) −3.66192 −0.118249
\(960\) 0 0
\(961\) 9.19779 0.296703
\(962\) 36.6803 1.18262
\(963\) −12.2329 −0.394199
\(964\) −1.92079 −0.0618643
\(965\) 0 0
\(966\) −1.78992 −0.0575897
\(967\) −50.5464 −1.62546 −0.812731 0.582639i \(-0.802020\pi\)
−0.812731 + 0.582639i \(0.802020\pi\)
\(968\) 2.51026 0.0806828
\(969\) −37.1461 −1.19330
\(970\) 0 0
\(971\) −57.3074 −1.83908 −0.919540 0.392995i \(-0.871439\pi\)
−0.919540 + 0.392995i \(0.871439\pi\)
\(972\) 0.369102 0.0118390
\(973\) 2.88267 0.0924143
\(974\) 43.9664 1.40877
\(975\) 0 0
\(976\) 30.7427 0.984051
\(977\) 45.7875 1.46487 0.732436 0.680836i \(-0.238384\pi\)
0.732436 + 0.680836i \(0.238384\pi\)
\(978\) −35.3028 −1.12886
\(979\) −4.34017 −0.138713
\(980\) 0 0
\(981\) −6.31351 −0.201575
\(982\) −40.0000 −1.27645
\(983\) −55.0805 −1.75679 −0.878397 0.477932i \(-0.841387\pi\)
−0.878397 + 0.477932i \(0.841387\pi\)
\(984\) −18.6225 −0.593663
\(985\) 0 0
\(986\) −54.6681 −1.74099
\(987\) 1.52973 0.0486920
\(988\) −19.9635 −0.635123
\(989\) 1.16290 0.0369780
\(990\) 0 0
\(991\) 40.6947 1.29271 0.646355 0.763037i \(-0.276292\pi\)
0.646355 + 0.763037i \(0.276292\pi\)
\(992\) 13.0784 0.415239
\(993\) −6.70701 −0.212840
\(994\) −2.27247 −0.0720785
\(995\) 0 0
\(996\) −0.630898 −0.0199908
\(997\) 32.7610 1.03755 0.518775 0.854911i \(-0.326388\pi\)
0.518775 + 0.854911i \(0.326388\pi\)
\(998\) 42.3545 1.34071
\(999\) 3.41855 0.108158
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 825.2.a.h.1.2 3
3.2 odd 2 2475.2.a.be.1.2 3
5.2 odd 4 165.2.c.a.34.2 6
5.3 odd 4 165.2.c.a.34.5 yes 6
5.4 even 2 825.2.a.n.1.2 3
11.10 odd 2 9075.2.a.ck.1.2 3
15.2 even 4 495.2.c.d.199.5 6
15.8 even 4 495.2.c.d.199.2 6
15.14 odd 2 2475.2.a.y.1.2 3
20.3 even 4 2640.2.d.i.529.3 6
20.7 even 4 2640.2.d.i.529.6 6
55.32 even 4 1815.2.c.d.364.5 6
55.43 even 4 1815.2.c.d.364.2 6
55.54 odd 2 9075.2.a.cc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.c.a.34.2 6 5.2 odd 4
165.2.c.a.34.5 yes 6 5.3 odd 4
495.2.c.d.199.2 6 15.8 even 4
495.2.c.d.199.5 6 15.2 even 4
825.2.a.h.1.2 3 1.1 even 1 trivial
825.2.a.n.1.2 3 5.4 even 2
1815.2.c.d.364.2 6 55.43 even 4
1815.2.c.d.364.5 6 55.32 even 4
2475.2.a.y.1.2 3 15.14 odd 2
2475.2.a.be.1.2 3 3.2 odd 2
2640.2.d.i.529.3 6 20.3 even 4
2640.2.d.i.529.6 6 20.7 even 4
9075.2.a.cc.1.2 3 55.54 odd 2
9075.2.a.ck.1.2 3 11.10 odd 2