Properties

Label 1925.2.a.u.1.2
Level $1925$
Weight $2$
Character 1925.1
Self dual yes
Analytic conductor $15.371$
Analytic rank $0$
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] = [1925,2,Mod(1,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, 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("1925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.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: \(15.3712023891\)
Analytic rank: \(0\)
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 385)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 1925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.311108 q^{2} +2.21432 q^{3} -1.90321 q^{4} -0.688892 q^{6} -1.00000 q^{7} +1.21432 q^{8} +1.90321 q^{9} +O(q^{10})\) \(q-0.311108 q^{2} +2.21432 q^{3} -1.90321 q^{4} -0.688892 q^{6} -1.00000 q^{7} +1.21432 q^{8} +1.90321 q^{9} +1.00000 q^{11} -4.21432 q^{12} +2.21432 q^{13} +0.311108 q^{14} +3.42864 q^{16} -4.21432 q^{17} -0.592104 q^{18} +7.80642 q^{19} -2.21432 q^{21} -0.311108 q^{22} -2.90321 q^{23} +2.68889 q^{24} -0.688892 q^{26} -2.42864 q^{27} +1.90321 q^{28} +0.755569 q^{29} +7.11753 q^{31} -3.49532 q^{32} +2.21432 q^{33} +1.31111 q^{34} -3.62222 q^{36} +6.28100 q^{37} -2.42864 q^{38} +4.90321 q^{39} -5.54617 q^{41} +0.688892 q^{42} +4.14764 q^{43} -1.90321 q^{44} +0.903212 q^{46} -1.03011 q^{47} +7.59210 q^{48} +1.00000 q^{49} -9.33185 q^{51} -4.21432 q^{52} +6.57628 q^{53} +0.755569 q^{54} -1.21432 q^{56} +17.2859 q^{57} -0.235063 q^{58} +2.68889 q^{59} +8.79060 q^{61} -2.21432 q^{62} -1.90321 q^{63} -5.76986 q^{64} -0.688892 q^{66} -1.52543 q^{67} +8.02074 q^{68} -6.42864 q^{69} +7.61285 q^{71} +2.31111 q^{72} +12.2143 q^{73} -1.95407 q^{74} -14.8573 q^{76} -1.00000 q^{77} -1.52543 q^{78} +3.52543 q^{79} -11.0874 q^{81} +1.72546 q^{82} +2.13335 q^{83} +4.21432 q^{84} -1.29036 q^{86} +1.67307 q^{87} +1.21432 q^{88} +17.2859 q^{89} -2.21432 q^{91} +5.52543 q^{92} +15.7605 q^{93} +0.320476 q^{94} -7.73975 q^{96} -5.28592 q^{97} -0.311108 q^{98} +1.90321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + q^{4} - 2 q^{6} - 3 q^{7} - 3 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + q^{4} - 2 q^{6} - 3 q^{7} - 3 q^{8} - q^{9} + 3 q^{11} - 6 q^{12} + q^{14} - 3 q^{16} - 6 q^{17} + 5 q^{18} + 10 q^{19} - q^{22} - 2 q^{23} + 8 q^{24} - 2 q^{26} + 6 q^{27} - q^{28} + 2 q^{29} + 8 q^{31} + 3 q^{32} + 4 q^{34} - 11 q^{36} + 12 q^{37} + 6 q^{38} + 8 q^{39} + 10 q^{41} + 2 q^{42} + 6 q^{43} + q^{44} - 4 q^{46} - 10 q^{47} + 16 q^{48} + 3 q^{49} - 8 q^{51} - 6 q^{52} + 2 q^{54} + 3 q^{56} + 12 q^{57} + 26 q^{58} + 8 q^{59} + q^{63} - 11 q^{64} - 2 q^{66} + 2 q^{67} + 4 q^{68} - 6 q^{69} - 4 q^{71} + 7 q^{72} + 30 q^{73} + 14 q^{74} - 18 q^{76} - 3 q^{77} + 2 q^{78} + 4 q^{79} - 13 q^{81} - 2 q^{82} + 6 q^{83} + 6 q^{84} - 24 q^{86} - 8 q^{87} - 3 q^{88} + 12 q^{89} + 10 q^{92} + 14 q^{93} + 28 q^{94} - 10 q^{96} + 24 q^{97} - q^{98} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.311108 −0.219986 −0.109993 0.993932i \(-0.535083\pi\)
−0.109993 + 0.993932i \(0.535083\pi\)
\(3\) 2.21432 1.27844 0.639219 0.769025i \(-0.279258\pi\)
0.639219 + 0.769025i \(0.279258\pi\)
\(4\) −1.90321 −0.951606
\(5\) 0 0
\(6\) −0.688892 −0.281239
\(7\) −1.00000 −0.377964
\(8\) 1.21432 0.429327
\(9\) 1.90321 0.634404
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −4.21432 −1.21657
\(13\) 2.21432 0.614142 0.307071 0.951687i \(-0.400651\pi\)
0.307071 + 0.951687i \(0.400651\pi\)
\(14\) 0.311108 0.0831471
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) −4.21432 −1.02212 −0.511061 0.859544i \(-0.670748\pi\)
−0.511061 + 0.859544i \(0.670748\pi\)
\(18\) −0.592104 −0.139560
\(19\) 7.80642 1.79092 0.895458 0.445146i \(-0.146848\pi\)
0.895458 + 0.445146i \(0.146848\pi\)
\(20\) 0 0
\(21\) −2.21432 −0.483204
\(22\) −0.311108 −0.0663284
\(23\) −2.90321 −0.605362 −0.302681 0.953092i \(-0.597882\pi\)
−0.302681 + 0.953092i \(0.597882\pi\)
\(24\) 2.68889 0.548868
\(25\) 0 0
\(26\) −0.688892 −0.135103
\(27\) −2.42864 −0.467392
\(28\) 1.90321 0.359673
\(29\) 0.755569 0.140306 0.0701528 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) 7.11753 1.27835 0.639173 0.769063i \(-0.279277\pi\)
0.639173 + 0.769063i \(0.279277\pi\)
\(32\) −3.49532 −0.617890
\(33\) 2.21432 0.385464
\(34\) 1.31111 0.224853
\(35\) 0 0
\(36\) −3.62222 −0.603703
\(37\) 6.28100 1.03259 0.516295 0.856411i \(-0.327311\pi\)
0.516295 + 0.856411i \(0.327311\pi\)
\(38\) −2.42864 −0.393977
\(39\) 4.90321 0.785142
\(40\) 0 0
\(41\) −5.54617 −0.866166 −0.433083 0.901354i \(-0.642574\pi\)
−0.433083 + 0.901354i \(0.642574\pi\)
\(42\) 0.688892 0.106298
\(43\) 4.14764 0.632510 0.316255 0.948674i \(-0.397575\pi\)
0.316255 + 0.948674i \(0.397575\pi\)
\(44\) −1.90321 −0.286920
\(45\) 0 0
\(46\) 0.903212 0.133171
\(47\) −1.03011 −0.150257 −0.0751286 0.997174i \(-0.523937\pi\)
−0.0751286 + 0.997174i \(0.523937\pi\)
\(48\) 7.59210 1.09583
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −9.33185 −1.30672
\(52\) −4.21432 −0.584421
\(53\) 6.57628 0.903322 0.451661 0.892190i \(-0.350832\pi\)
0.451661 + 0.892190i \(0.350832\pi\)
\(54\) 0.755569 0.102820
\(55\) 0 0
\(56\) −1.21432 −0.162270
\(57\) 17.2859 2.28958
\(58\) −0.235063 −0.0308653
\(59\) 2.68889 0.350064 0.175032 0.984563i \(-0.443997\pi\)
0.175032 + 0.984563i \(0.443997\pi\)
\(60\) 0 0
\(61\) 8.79060 1.12552 0.562761 0.826620i \(-0.309739\pi\)
0.562761 + 0.826620i \(0.309739\pi\)
\(62\) −2.21432 −0.281219
\(63\) −1.90321 −0.239782
\(64\) −5.76986 −0.721232
\(65\) 0 0
\(66\) −0.688892 −0.0847968
\(67\) −1.52543 −0.186361 −0.0931803 0.995649i \(-0.529703\pi\)
−0.0931803 + 0.995649i \(0.529703\pi\)
\(68\) 8.02074 0.972658
\(69\) −6.42864 −0.773917
\(70\) 0 0
\(71\) 7.61285 0.903479 0.451739 0.892150i \(-0.350804\pi\)
0.451739 + 0.892150i \(0.350804\pi\)
\(72\) 2.31111 0.272367
\(73\) 12.2143 1.42958 0.714789 0.699340i \(-0.246523\pi\)
0.714789 + 0.699340i \(0.246523\pi\)
\(74\) −1.95407 −0.227156
\(75\) 0 0
\(76\) −14.8573 −1.70425
\(77\) −1.00000 −0.113961
\(78\) −1.52543 −0.172721
\(79\) 3.52543 0.396642 0.198321 0.980137i \(-0.436451\pi\)
0.198321 + 0.980137i \(0.436451\pi\)
\(80\) 0 0
\(81\) −11.0874 −1.23194
\(82\) 1.72546 0.190545
\(83\) 2.13335 0.234166 0.117083 0.993122i \(-0.462646\pi\)
0.117083 + 0.993122i \(0.462646\pi\)
\(84\) 4.21432 0.459820
\(85\) 0 0
\(86\) −1.29036 −0.139144
\(87\) 1.67307 0.179372
\(88\) 1.21432 0.129447
\(89\) 17.2859 1.83230 0.916152 0.400831i \(-0.131279\pi\)
0.916152 + 0.400831i \(0.131279\pi\)
\(90\) 0 0
\(91\) −2.21432 −0.232124
\(92\) 5.52543 0.576066
\(93\) 15.7605 1.63429
\(94\) 0.320476 0.0330545
\(95\) 0 0
\(96\) −7.73975 −0.789935
\(97\) −5.28592 −0.536704 −0.268352 0.963321i \(-0.586479\pi\)
−0.268352 + 0.963321i \(0.586479\pi\)
\(98\) −0.311108 −0.0314266
\(99\) 1.90321 0.191280
\(100\) 0 0
\(101\) 18.1082 1.80183 0.900915 0.433996i \(-0.142897\pi\)
0.900915 + 0.433996i \(0.142897\pi\)
\(102\) 2.90321 0.287461
\(103\) −16.7447 −1.64990 −0.824951 0.565205i \(-0.808797\pi\)
−0.824951 + 0.565205i \(0.808797\pi\)
\(104\) 2.68889 0.263668
\(105\) 0 0
\(106\) −2.04593 −0.198719
\(107\) 0.561993 0.0543299 0.0271649 0.999631i \(-0.491352\pi\)
0.0271649 + 0.999631i \(0.491352\pi\)
\(108\) 4.62222 0.444773
\(109\) −3.93978 −0.377362 −0.188681 0.982038i \(-0.560421\pi\)
−0.188681 + 0.982038i \(0.560421\pi\)
\(110\) 0 0
\(111\) 13.9081 1.32010
\(112\) −3.42864 −0.323976
\(113\) −11.2859 −1.06169 −0.530845 0.847469i \(-0.678125\pi\)
−0.530845 + 0.847469i \(0.678125\pi\)
\(114\) −5.37778 −0.503676
\(115\) 0 0
\(116\) −1.43801 −0.133516
\(117\) 4.21432 0.389614
\(118\) −0.836535 −0.0770093
\(119\) 4.21432 0.386326
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.73483 −0.247599
\(123\) −12.2810 −1.10734
\(124\) −13.5462 −1.21648
\(125\) 0 0
\(126\) 0.592104 0.0527488
\(127\) 12.9906 1.15273 0.576366 0.817192i \(-0.304470\pi\)
0.576366 + 0.817192i \(0.304470\pi\)
\(128\) 8.78568 0.776552
\(129\) 9.18421 0.808624
\(130\) 0 0
\(131\) −8.47013 −0.740038 −0.370019 0.929024i \(-0.620649\pi\)
−0.370019 + 0.929024i \(0.620649\pi\)
\(132\) −4.21432 −0.366809
\(133\) −7.80642 −0.676903
\(134\) 0.474572 0.0409968
\(135\) 0 0
\(136\) −5.11753 −0.438825
\(137\) 11.1383 0.951607 0.475804 0.879552i \(-0.342157\pi\)
0.475804 + 0.879552i \(0.342157\pi\)
\(138\) 2.00000 0.170251
\(139\) −8.04149 −0.682070 −0.341035 0.940051i \(-0.610777\pi\)
−0.341035 + 0.940051i \(0.610777\pi\)
\(140\) 0 0
\(141\) −2.28100 −0.192095
\(142\) −2.36842 −0.198753
\(143\) 2.21432 0.185171
\(144\) 6.52543 0.543786
\(145\) 0 0
\(146\) −3.79997 −0.314488
\(147\) 2.21432 0.182634
\(148\) −11.9541 −0.982618
\(149\) −6.13335 −0.502464 −0.251232 0.967927i \(-0.580836\pi\)
−0.251232 + 0.967927i \(0.580836\pi\)
\(150\) 0 0
\(151\) 10.1476 0.825803 0.412902 0.910776i \(-0.364515\pi\)
0.412902 + 0.910776i \(0.364515\pi\)
\(152\) 9.47949 0.768889
\(153\) −8.02074 −0.648439
\(154\) 0.311108 0.0250698
\(155\) 0 0
\(156\) −9.33185 −0.747146
\(157\) 21.7146 1.73301 0.866505 0.499168i \(-0.166361\pi\)
0.866505 + 0.499168i \(0.166361\pi\)
\(158\) −1.09679 −0.0872558
\(159\) 14.5620 1.15484
\(160\) 0 0
\(161\) 2.90321 0.228805
\(162\) 3.44938 0.271009
\(163\) 8.01429 0.627728 0.313864 0.949468i \(-0.398376\pi\)
0.313864 + 0.949468i \(0.398376\pi\)
\(164\) 10.5555 0.824249
\(165\) 0 0
\(166\) −0.663703 −0.0515133
\(167\) −15.1240 −1.17033 −0.585165 0.810915i \(-0.698970\pi\)
−0.585165 + 0.810915i \(0.698970\pi\)
\(168\) −2.68889 −0.207453
\(169\) −8.09679 −0.622830
\(170\) 0 0
\(171\) 14.8573 1.13616
\(172\) −7.89384 −0.601900
\(173\) −19.6938 −1.49729 −0.748646 0.662969i \(-0.769296\pi\)
−0.748646 + 0.662969i \(0.769296\pi\)
\(174\) −0.520505 −0.0394594
\(175\) 0 0
\(176\) 3.42864 0.258443
\(177\) 5.95407 0.447535
\(178\) −5.37778 −0.403082
\(179\) 5.53972 0.414058 0.207029 0.978335i \(-0.433621\pi\)
0.207029 + 0.978335i \(0.433621\pi\)
\(180\) 0 0
\(181\) −19.9081 −1.47976 −0.739880 0.672739i \(-0.765118\pi\)
−0.739880 + 0.672739i \(0.765118\pi\)
\(182\) 0.688892 0.0510641
\(183\) 19.4652 1.43891
\(184\) −3.52543 −0.259898
\(185\) 0 0
\(186\) −4.90321 −0.359521
\(187\) −4.21432 −0.308182
\(188\) 1.96052 0.142986
\(189\) 2.42864 0.176658
\(190\) 0 0
\(191\) −20.8988 −1.51218 −0.756091 0.654467i \(-0.772893\pi\)
−0.756091 + 0.654467i \(0.772893\pi\)
\(192\) −12.7763 −0.922051
\(193\) 13.6271 0.980903 0.490451 0.871469i \(-0.336832\pi\)
0.490451 + 0.871469i \(0.336832\pi\)
\(194\) 1.64449 0.118068
\(195\) 0 0
\(196\) −1.90321 −0.135944
\(197\) −15.7003 −1.11860 −0.559299 0.828966i \(-0.688930\pi\)
−0.559299 + 0.828966i \(0.688930\pi\)
\(198\) −0.592104 −0.0420790
\(199\) −23.9146 −1.69526 −0.847630 0.530588i \(-0.821971\pi\)
−0.847630 + 0.530588i \(0.821971\pi\)
\(200\) 0 0
\(201\) −3.37778 −0.238251
\(202\) −5.63359 −0.396378
\(203\) −0.755569 −0.0530305
\(204\) 17.7605 1.24348
\(205\) 0 0
\(206\) 5.20940 0.362956
\(207\) −5.52543 −0.384044
\(208\) 7.59210 0.526418
\(209\) 7.80642 0.539982
\(210\) 0 0
\(211\) 11.4795 0.790281 0.395141 0.918621i \(-0.370696\pi\)
0.395141 + 0.918621i \(0.370696\pi\)
\(212\) −12.5161 −0.859607
\(213\) 16.8573 1.15504
\(214\) −0.174840 −0.0119518
\(215\) 0 0
\(216\) −2.94914 −0.200664
\(217\) −7.11753 −0.483170
\(218\) 1.22570 0.0830146
\(219\) 27.0464 1.82763
\(220\) 0 0
\(221\) −9.33185 −0.627728
\(222\) −4.32693 −0.290404
\(223\) −19.7669 −1.32369 −0.661846 0.749640i \(-0.730227\pi\)
−0.661846 + 0.749640i \(0.730227\pi\)
\(224\) 3.49532 0.233541
\(225\) 0 0
\(226\) 3.51114 0.233557
\(227\) −3.61285 −0.239793 −0.119897 0.992786i \(-0.538256\pi\)
−0.119897 + 0.992786i \(0.538256\pi\)
\(228\) −32.8988 −2.17877
\(229\) −18.7239 −1.23731 −0.618656 0.785662i \(-0.712322\pi\)
−0.618656 + 0.785662i \(0.712322\pi\)
\(230\) 0 0
\(231\) −2.21432 −0.145692
\(232\) 0.917502 0.0602370
\(233\) −7.00492 −0.458908 −0.229454 0.973320i \(-0.573694\pi\)
−0.229454 + 0.973320i \(0.573694\pi\)
\(234\) −1.31111 −0.0857098
\(235\) 0 0
\(236\) −5.11753 −0.333123
\(237\) 7.80642 0.507082
\(238\) −1.31111 −0.0849865
\(239\) 20.9906 1.35777 0.678886 0.734244i \(-0.262463\pi\)
0.678886 + 0.734244i \(0.262463\pi\)
\(240\) 0 0
\(241\) 10.7491 0.692411 0.346206 0.938159i \(-0.387470\pi\)
0.346206 + 0.938159i \(0.387470\pi\)
\(242\) −0.311108 −0.0199988
\(243\) −17.2652 −1.10756
\(244\) −16.7304 −1.07105
\(245\) 0 0
\(246\) 3.82071 0.243600
\(247\) 17.2859 1.09988
\(248\) 8.64296 0.548828
\(249\) 4.72393 0.299367
\(250\) 0 0
\(251\) −6.30174 −0.397762 −0.198881 0.980024i \(-0.563731\pi\)
−0.198881 + 0.980024i \(0.563731\pi\)
\(252\) 3.62222 0.228178
\(253\) −2.90321 −0.182523
\(254\) −4.04149 −0.253585
\(255\) 0 0
\(256\) 8.80642 0.550401
\(257\) −26.2766 −1.63909 −0.819543 0.573018i \(-0.805773\pi\)
−0.819543 + 0.573018i \(0.805773\pi\)
\(258\) −2.85728 −0.177886
\(259\) −6.28100 −0.390282
\(260\) 0 0
\(261\) 1.43801 0.0890104
\(262\) 2.63512 0.162798
\(263\) −12.7239 −0.784591 −0.392295 0.919839i \(-0.628319\pi\)
−0.392295 + 0.919839i \(0.628319\pi\)
\(264\) 2.68889 0.165490
\(265\) 0 0
\(266\) 2.42864 0.148909
\(267\) 38.2766 2.34249
\(268\) 2.90321 0.177342
\(269\) −14.1017 −0.859796 −0.429898 0.902877i \(-0.641451\pi\)
−0.429898 + 0.902877i \(0.641451\pi\)
\(270\) 0 0
\(271\) −5.28592 −0.321097 −0.160548 0.987028i \(-0.551326\pi\)
−0.160548 + 0.987028i \(0.551326\pi\)
\(272\) −14.4494 −0.876123
\(273\) −4.90321 −0.296756
\(274\) −3.46520 −0.209341
\(275\) 0 0
\(276\) 12.2351 0.736464
\(277\) −15.1798 −0.912064 −0.456032 0.889963i \(-0.650730\pi\)
−0.456032 + 0.889963i \(0.650730\pi\)
\(278\) 2.50177 0.150046
\(279\) 13.5462 0.810988
\(280\) 0 0
\(281\) 32.8671 1.96069 0.980344 0.197295i \(-0.0632158\pi\)
0.980344 + 0.197295i \(0.0632158\pi\)
\(282\) 0.709636 0.0422582
\(283\) 8.94914 0.531971 0.265986 0.963977i \(-0.414303\pi\)
0.265986 + 0.963977i \(0.414303\pi\)
\(284\) −14.4889 −0.859756
\(285\) 0 0
\(286\) −0.688892 −0.0407350
\(287\) 5.54617 0.327380
\(288\) −6.65233 −0.391992
\(289\) 0.760491 0.0447348
\(290\) 0 0
\(291\) −11.7047 −0.686142
\(292\) −23.2464 −1.36039
\(293\) 22.9195 1.33897 0.669486 0.742825i \(-0.266514\pi\)
0.669486 + 0.742825i \(0.266514\pi\)
\(294\) −0.688892 −0.0401770
\(295\) 0 0
\(296\) 7.62714 0.443318
\(297\) −2.42864 −0.140924
\(298\) 1.90813 0.110535
\(299\) −6.42864 −0.371778
\(300\) 0 0
\(301\) −4.14764 −0.239066
\(302\) −3.15701 −0.181666
\(303\) 40.0973 2.30353
\(304\) 26.7654 1.53510
\(305\) 0 0
\(306\) 2.49532 0.142648
\(307\) 18.2953 1.04417 0.522084 0.852894i \(-0.325155\pi\)
0.522084 + 0.852894i \(0.325155\pi\)
\(308\) 1.90321 0.108446
\(309\) −37.0781 −2.10930
\(310\) 0 0
\(311\) −5.54617 −0.314495 −0.157247 0.987559i \(-0.550262\pi\)
−0.157247 + 0.987559i \(0.550262\pi\)
\(312\) 5.95407 0.337083
\(313\) 23.8064 1.34562 0.672809 0.739816i \(-0.265087\pi\)
0.672809 + 0.739816i \(0.265087\pi\)
\(314\) −6.75557 −0.381239
\(315\) 0 0
\(316\) −6.70964 −0.377447
\(317\) −10.5906 −0.594826 −0.297413 0.954749i \(-0.596124\pi\)
−0.297413 + 0.954749i \(0.596124\pi\)
\(318\) −4.53035 −0.254049
\(319\) 0.755569 0.0423037
\(320\) 0 0
\(321\) 1.24443 0.0694574
\(322\) −0.903212 −0.0503340
\(323\) −32.8988 −1.83054
\(324\) 21.1017 1.17232
\(325\) 0 0
\(326\) −2.49331 −0.138092
\(327\) −8.72393 −0.482434
\(328\) −6.73483 −0.371869
\(329\) 1.03011 0.0567919
\(330\) 0 0
\(331\) 30.1146 1.65525 0.827625 0.561282i \(-0.189692\pi\)
0.827625 + 0.561282i \(0.189692\pi\)
\(332\) −4.06022 −0.222834
\(333\) 11.9541 0.655079
\(334\) 4.70519 0.257457
\(335\) 0 0
\(336\) −7.59210 −0.414183
\(337\) 30.4558 1.65904 0.829518 0.558481i \(-0.188615\pi\)
0.829518 + 0.558481i \(0.188615\pi\)
\(338\) 2.51897 0.137014
\(339\) −24.9906 −1.35730
\(340\) 0 0
\(341\) 7.11753 0.385436
\(342\) −4.62222 −0.249941
\(343\) −1.00000 −0.0539949
\(344\) 5.03657 0.271553
\(345\) 0 0
\(346\) 6.12690 0.329384
\(347\) 28.3511 1.52196 0.760982 0.648772i \(-0.224717\pi\)
0.760982 + 0.648772i \(0.224717\pi\)
\(348\) −3.18421 −0.170691
\(349\) −32.5654 −1.74319 −0.871593 0.490231i \(-0.836912\pi\)
−0.871593 + 0.490231i \(0.836912\pi\)
\(350\) 0 0
\(351\) −5.37778 −0.287045
\(352\) −3.49532 −0.186301
\(353\) 24.2766 1.29211 0.646055 0.763291i \(-0.276418\pi\)
0.646055 + 0.763291i \(0.276418\pi\)
\(354\) −1.85236 −0.0984517
\(355\) 0 0
\(356\) −32.8988 −1.74363
\(357\) 9.33185 0.493894
\(358\) −1.72345 −0.0910871
\(359\) 16.4746 0.869495 0.434747 0.900552i \(-0.356838\pi\)
0.434747 + 0.900552i \(0.356838\pi\)
\(360\) 0 0
\(361\) 41.9403 2.20738
\(362\) 6.19358 0.325527
\(363\) 2.21432 0.116222
\(364\) 4.21432 0.220890
\(365\) 0 0
\(366\) −6.05578 −0.316541
\(367\) −6.71609 −0.350577 −0.175289 0.984517i \(-0.556086\pi\)
−0.175289 + 0.984517i \(0.556086\pi\)
\(368\) −9.95407 −0.518892
\(369\) −10.5555 −0.549499
\(370\) 0 0
\(371\) −6.57628 −0.341424
\(372\) −29.9956 −1.55520
\(373\) −10.5477 −0.546139 −0.273070 0.961994i \(-0.588039\pi\)
−0.273070 + 0.961994i \(0.588039\pi\)
\(374\) 1.31111 0.0677958
\(375\) 0 0
\(376\) −1.25088 −0.0645095
\(377\) 1.67307 0.0861675
\(378\) −0.755569 −0.0388623
\(379\) −33.0607 −1.69821 −0.849107 0.528221i \(-0.822859\pi\)
−0.849107 + 0.528221i \(0.822859\pi\)
\(380\) 0 0
\(381\) 28.7654 1.47370
\(382\) 6.50177 0.332659
\(383\) 17.3067 0.884329 0.442165 0.896934i \(-0.354211\pi\)
0.442165 + 0.896934i \(0.354211\pi\)
\(384\) 19.4543 0.992773
\(385\) 0 0
\(386\) −4.23951 −0.215785
\(387\) 7.89384 0.401267
\(388\) 10.0602 0.510730
\(389\) −33.1842 −1.68251 −0.841253 0.540642i \(-0.818182\pi\)
−0.841253 + 0.540642i \(0.818182\pi\)
\(390\) 0 0
\(391\) 12.2351 0.618754
\(392\) 1.21432 0.0613324
\(393\) −18.7556 −0.946093
\(394\) 4.88448 0.246076
\(395\) 0 0
\(396\) −3.62222 −0.182023
\(397\) −10.6637 −0.535196 −0.267598 0.963531i \(-0.586230\pi\)
−0.267598 + 0.963531i \(0.586230\pi\)
\(398\) 7.44002 0.372934
\(399\) −17.2859 −0.865378
\(400\) 0 0
\(401\) −10.5906 −0.528868 −0.264434 0.964404i \(-0.585185\pi\)
−0.264434 + 0.964404i \(0.585185\pi\)
\(402\) 1.05086 0.0524119
\(403\) 15.7605 0.785086
\(404\) −34.4637 −1.71463
\(405\) 0 0
\(406\) 0.235063 0.0116660
\(407\) 6.28100 0.311337
\(408\) −11.3319 −0.561010
\(409\) 11.8829 0.587574 0.293787 0.955871i \(-0.405084\pi\)
0.293787 + 0.955871i \(0.405084\pi\)
\(410\) 0 0
\(411\) 24.6637 1.21657
\(412\) 31.8687 1.57006
\(413\) −2.68889 −0.132312
\(414\) 1.71900 0.0844844
\(415\) 0 0
\(416\) −7.73975 −0.379472
\(417\) −17.8064 −0.871984
\(418\) −2.42864 −0.118789
\(419\) −2.61576 −0.127788 −0.0638942 0.997957i \(-0.520352\pi\)
−0.0638942 + 0.997957i \(0.520352\pi\)
\(420\) 0 0
\(421\) −21.5254 −1.04909 −0.524543 0.851384i \(-0.675764\pi\)
−0.524543 + 0.851384i \(0.675764\pi\)
\(422\) −3.57136 −0.173851
\(423\) −1.96052 −0.0953238
\(424\) 7.98571 0.387820
\(425\) 0 0
\(426\) −5.24443 −0.254094
\(427\) −8.79060 −0.425407
\(428\) −1.06959 −0.0517006
\(429\) 4.90321 0.236729
\(430\) 0 0
\(431\) −19.0464 −0.917433 −0.458717 0.888583i \(-0.651691\pi\)
−0.458717 + 0.888583i \(0.651691\pi\)
\(432\) −8.32693 −0.400630
\(433\) −18.6953 −0.898441 −0.449220 0.893421i \(-0.648298\pi\)
−0.449220 + 0.893421i \(0.648298\pi\)
\(434\) 2.21432 0.106291
\(435\) 0 0
\(436\) 7.49823 0.359100
\(437\) −22.6637 −1.08415
\(438\) −8.41435 −0.402053
\(439\) −0.990632 −0.0472803 −0.0236401 0.999721i \(-0.507526\pi\)
−0.0236401 + 0.999721i \(0.507526\pi\)
\(440\) 0 0
\(441\) 1.90321 0.0906291
\(442\) 2.90321 0.138092
\(443\) −20.3511 −0.966908 −0.483454 0.875370i \(-0.660618\pi\)
−0.483454 + 0.875370i \(0.660618\pi\)
\(444\) −26.4701 −1.25622
\(445\) 0 0
\(446\) 6.14965 0.291194
\(447\) −13.5812 −0.642369
\(448\) 5.76986 0.272600
\(449\) −29.1383 −1.37512 −0.687560 0.726127i \(-0.741318\pi\)
−0.687560 + 0.726127i \(0.741318\pi\)
\(450\) 0 0
\(451\) −5.54617 −0.261159
\(452\) 21.4795 1.01031
\(453\) 22.4701 1.05574
\(454\) 1.12399 0.0527512
\(455\) 0 0
\(456\) 20.9906 0.982976
\(457\) −31.3131 −1.46477 −0.732383 0.680893i \(-0.761592\pi\)
−0.732383 + 0.680893i \(0.761592\pi\)
\(458\) 5.82516 0.272192
\(459\) 10.2351 0.477732
\(460\) 0 0
\(461\) 11.9017 0.554317 0.277158 0.960824i \(-0.410607\pi\)
0.277158 + 0.960824i \(0.410607\pi\)
\(462\) 0.688892 0.0320502
\(463\) −3.89384 −0.180962 −0.0904811 0.995898i \(-0.528840\pi\)
−0.0904811 + 0.995898i \(0.528840\pi\)
\(464\) 2.59057 0.120264
\(465\) 0 0
\(466\) 2.17929 0.100953
\(467\) 21.3985 0.990206 0.495103 0.868834i \(-0.335130\pi\)
0.495103 + 0.868834i \(0.335130\pi\)
\(468\) −8.02074 −0.370759
\(469\) 1.52543 0.0704377
\(470\) 0 0
\(471\) 48.0830 2.21555
\(472\) 3.26517 0.150292
\(473\) 4.14764 0.190709
\(474\) −2.42864 −0.111551
\(475\) 0 0
\(476\) −8.02074 −0.367630
\(477\) 12.5161 0.573071
\(478\) −6.53035 −0.298691
\(479\) −11.9813 −0.547438 −0.273719 0.961810i \(-0.588254\pi\)
−0.273719 + 0.961810i \(0.588254\pi\)
\(480\) 0 0
\(481\) 13.9081 0.634156
\(482\) −3.34413 −0.152321
\(483\) 6.42864 0.292513
\(484\) −1.90321 −0.0865096
\(485\) 0 0
\(486\) 5.37133 0.243649
\(487\) 16.5892 0.751728 0.375864 0.926675i \(-0.377346\pi\)
0.375864 + 0.926675i \(0.377346\pi\)
\(488\) 10.6746 0.483217
\(489\) 17.7462 0.802511
\(490\) 0 0
\(491\) −41.2400 −1.86113 −0.930567 0.366121i \(-0.880686\pi\)
−0.930567 + 0.366121i \(0.880686\pi\)
\(492\) 23.3733 1.05375
\(493\) −3.18421 −0.143410
\(494\) −5.37778 −0.241958
\(495\) 0 0
\(496\) 24.4035 1.09575
\(497\) −7.61285 −0.341483
\(498\) −1.46965 −0.0658566
\(499\) −32.3783 −1.44945 −0.724725 0.689038i \(-0.758033\pi\)
−0.724725 + 0.689038i \(0.758033\pi\)
\(500\) 0 0
\(501\) −33.4893 −1.49619
\(502\) 1.96052 0.0875023
\(503\) −33.4795 −1.49278 −0.746388 0.665511i \(-0.768214\pi\)
−0.746388 + 0.665511i \(0.768214\pi\)
\(504\) −2.31111 −0.102945
\(505\) 0 0
\(506\) 0.903212 0.0401527
\(507\) −17.9289 −0.796249
\(508\) −24.7239 −1.09695
\(509\) −8.79706 −0.389923 −0.194961 0.980811i \(-0.562458\pi\)
−0.194961 + 0.980811i \(0.562458\pi\)
\(510\) 0 0
\(511\) −12.2143 −0.540330
\(512\) −20.3111 −0.897633
\(513\) −18.9590 −0.837060
\(514\) 8.17484 0.360577
\(515\) 0 0
\(516\) −17.4795 −0.769492
\(517\) −1.03011 −0.0453043
\(518\) 1.95407 0.0858568
\(519\) −43.6084 −1.91420
\(520\) 0 0
\(521\) 28.4415 1.24605 0.623023 0.782203i \(-0.285904\pi\)
0.623023 + 0.782203i \(0.285904\pi\)
\(522\) −0.447375 −0.0195811
\(523\) 3.80642 0.166443 0.0832216 0.996531i \(-0.473479\pi\)
0.0832216 + 0.996531i \(0.473479\pi\)
\(524\) 16.1204 0.704225
\(525\) 0 0
\(526\) 3.95851 0.172599
\(527\) −29.9956 −1.30663
\(528\) 7.59210 0.330404
\(529\) −14.5714 −0.633537
\(530\) 0 0
\(531\) 5.11753 0.222082
\(532\) 14.8573 0.644145
\(533\) −12.2810 −0.531949
\(534\) −11.9081 −0.515315
\(535\) 0 0
\(536\) −1.85236 −0.0800096
\(537\) 12.2667 0.529347
\(538\) 4.38715 0.189144
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 32.2351 1.38589 0.692947 0.720989i \(-0.256312\pi\)
0.692947 + 0.720989i \(0.256312\pi\)
\(542\) 1.64449 0.0706369
\(543\) −44.0830 −1.89178
\(544\) 14.7304 0.631560
\(545\) 0 0
\(546\) 1.52543 0.0652823
\(547\) −19.3876 −0.828955 −0.414478 0.910060i \(-0.636036\pi\)
−0.414478 + 0.910060i \(0.636036\pi\)
\(548\) −21.1985 −0.905555
\(549\) 16.7304 0.714035
\(550\) 0 0
\(551\) 5.89829 0.251276
\(552\) −7.80642 −0.332263
\(553\) −3.52543 −0.149916
\(554\) 4.72254 0.200642
\(555\) 0 0
\(556\) 15.3047 0.649062
\(557\) 1.58565 0.0671862 0.0335931 0.999436i \(-0.489305\pi\)
0.0335931 + 0.999436i \(0.489305\pi\)
\(558\) −4.21432 −0.178406
\(559\) 9.18421 0.388451
\(560\) 0 0
\(561\) −9.33185 −0.393991
\(562\) −10.2252 −0.431325
\(563\) −28.8069 −1.21407 −0.607033 0.794677i \(-0.707640\pi\)
−0.607033 + 0.794677i \(0.707640\pi\)
\(564\) 4.34122 0.182798
\(565\) 0 0
\(566\) −2.78415 −0.117027
\(567\) 11.0874 0.465628
\(568\) 9.24443 0.387888
\(569\) −14.6953 −0.616061 −0.308030 0.951376i \(-0.599670\pi\)
−0.308030 + 0.951376i \(0.599670\pi\)
\(570\) 0 0
\(571\) −12.3368 −0.516278 −0.258139 0.966108i \(-0.583109\pi\)
−0.258139 + 0.966108i \(0.583109\pi\)
\(572\) −4.21432 −0.176210
\(573\) −46.2766 −1.93323
\(574\) −1.72546 −0.0720192
\(575\) 0 0
\(576\) −10.9813 −0.457553
\(577\) 8.13335 0.338596 0.169298 0.985565i \(-0.445850\pi\)
0.169298 + 0.985565i \(0.445850\pi\)
\(578\) −0.236595 −0.00984104
\(579\) 30.1748 1.25402
\(580\) 0 0
\(581\) −2.13335 −0.0885064
\(582\) 3.64143 0.150942
\(583\) 6.57628 0.272362
\(584\) 14.8321 0.613756
\(585\) 0 0
\(586\) −7.13044 −0.294556
\(587\) −2.30619 −0.0951865 −0.0475932 0.998867i \(-0.515155\pi\)
−0.0475932 + 0.998867i \(0.515155\pi\)
\(588\) −4.21432 −0.173796
\(589\) 55.5625 2.28941
\(590\) 0 0
\(591\) −34.7654 −1.43006
\(592\) 21.5353 0.885094
\(593\) 25.7067 1.05565 0.527824 0.849354i \(-0.323008\pi\)
0.527824 + 0.849354i \(0.323008\pi\)
\(594\) 0.755569 0.0310014
\(595\) 0 0
\(596\) 11.6731 0.478148
\(597\) −52.9545 −2.16729
\(598\) 2.00000 0.0817861
\(599\) 27.5210 1.12448 0.562238 0.826975i \(-0.309940\pi\)
0.562238 + 0.826975i \(0.309940\pi\)
\(600\) 0 0
\(601\) 17.1175 0.698239 0.349119 0.937078i \(-0.386481\pi\)
0.349119 + 0.937078i \(0.386481\pi\)
\(602\) 1.29036 0.0525913
\(603\) −2.90321 −0.118228
\(604\) −19.3131 −0.785840
\(605\) 0 0
\(606\) −12.4746 −0.506745
\(607\) 21.4291 0.869781 0.434890 0.900483i \(-0.356787\pi\)
0.434890 + 0.900483i \(0.356787\pi\)
\(608\) −27.2859 −1.10659
\(609\) −1.67307 −0.0677962
\(610\) 0 0
\(611\) −2.28100 −0.0922792
\(612\) 15.2652 0.617058
\(613\) 40.3037 1.62785 0.813927 0.580968i \(-0.197326\pi\)
0.813927 + 0.580968i \(0.197326\pi\)
\(614\) −5.69181 −0.229703
\(615\) 0 0
\(616\) −1.21432 −0.0489263
\(617\) 15.0553 0.606104 0.303052 0.952974i \(-0.401994\pi\)
0.303052 + 0.952974i \(0.401994\pi\)
\(618\) 11.5353 0.464017
\(619\) 14.6702 0.589643 0.294822 0.955552i \(-0.404740\pi\)
0.294822 + 0.955552i \(0.404740\pi\)
\(620\) 0 0
\(621\) 7.05086 0.282941
\(622\) 1.72546 0.0691845
\(623\) −17.2859 −0.692546
\(624\) 16.8113 0.672992
\(625\) 0 0
\(626\) −7.40636 −0.296018
\(627\) 17.2859 0.690333
\(628\) −41.3274 −1.64914
\(629\) −26.4701 −1.05543
\(630\) 0 0
\(631\) −16.2034 −0.645048 −0.322524 0.946561i \(-0.604531\pi\)
−0.322524 + 0.946561i \(0.604531\pi\)
\(632\) 4.28100 0.170289
\(633\) 25.4193 1.01033
\(634\) 3.29481 0.130854
\(635\) 0 0
\(636\) −27.7146 −1.09895
\(637\) 2.21432 0.0877345
\(638\) −0.235063 −0.00930625
\(639\) 14.4889 0.573171
\(640\) 0 0
\(641\) −28.7225 −1.13447 −0.567236 0.823555i \(-0.691987\pi\)
−0.567236 + 0.823555i \(0.691987\pi\)
\(642\) −0.387152 −0.0152797
\(643\) 7.87755 0.310660 0.155330 0.987863i \(-0.450356\pi\)
0.155330 + 0.987863i \(0.450356\pi\)
\(644\) −5.52543 −0.217732
\(645\) 0 0
\(646\) 10.2351 0.402693
\(647\) −29.7540 −1.16975 −0.584876 0.811123i \(-0.698857\pi\)
−0.584876 + 0.811123i \(0.698857\pi\)
\(648\) −13.4637 −0.528903
\(649\) 2.68889 0.105548
\(650\) 0 0
\(651\) −15.7605 −0.617702
\(652\) −15.2529 −0.597349
\(653\) −16.7141 −0.654073 −0.327036 0.945012i \(-0.606050\pi\)
−0.327036 + 0.945012i \(0.606050\pi\)
\(654\) 2.71408 0.106129
\(655\) 0 0
\(656\) −19.0158 −0.742443
\(657\) 23.2464 0.906930
\(658\) −0.320476 −0.0124934
\(659\) 1.38223 0.0538440 0.0269220 0.999638i \(-0.491429\pi\)
0.0269220 + 0.999638i \(0.491429\pi\)
\(660\) 0 0
\(661\) 0.723926 0.0281575 0.0140787 0.999901i \(-0.495518\pi\)
0.0140787 + 0.999901i \(0.495518\pi\)
\(662\) −9.36889 −0.364132
\(663\) −20.6637 −0.802512
\(664\) 2.59057 0.100534
\(665\) 0 0
\(666\) −3.71900 −0.144108
\(667\) −2.19358 −0.0849356
\(668\) 28.7841 1.11369
\(669\) −43.7703 −1.69226
\(670\) 0 0
\(671\) 8.79060 0.339357
\(672\) 7.73975 0.298567
\(673\) −27.5254 −1.06103 −0.530514 0.847676i \(-0.678001\pi\)
−0.530514 + 0.847676i \(0.678001\pi\)
\(674\) −9.47505 −0.364965
\(675\) 0 0
\(676\) 15.4099 0.592689
\(677\) 20.4079 0.784339 0.392170 0.919893i \(-0.371725\pi\)
0.392170 + 0.919893i \(0.371725\pi\)
\(678\) 7.77478 0.298589
\(679\) 5.28592 0.202855
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) −2.21432 −0.0847907
\(683\) 38.5575 1.47536 0.737682 0.675149i \(-0.235920\pi\)
0.737682 + 0.675149i \(0.235920\pi\)
\(684\) −28.2766 −1.08118
\(685\) 0 0
\(686\) 0.311108 0.0118782
\(687\) −41.4608 −1.58183
\(688\) 14.2208 0.542162
\(689\) 14.5620 0.554768
\(690\) 0 0
\(691\) 9.59655 0.365070 0.182535 0.983199i \(-0.441570\pi\)
0.182535 + 0.983199i \(0.441570\pi\)
\(692\) 37.4815 1.42483
\(693\) −1.90321 −0.0722970
\(694\) −8.82024 −0.334812
\(695\) 0 0
\(696\) 2.03164 0.0770092
\(697\) 23.3733 0.885328
\(698\) 10.1313 0.383477
\(699\) −15.5111 −0.586685
\(700\) 0 0
\(701\) −30.0701 −1.13573 −0.567865 0.823121i \(-0.692231\pi\)
−0.567865 + 0.823121i \(0.692231\pi\)
\(702\) 1.67307 0.0631460
\(703\) 49.0321 1.84928
\(704\) −5.76986 −0.217460
\(705\) 0 0
\(706\) −7.55262 −0.284247
\(707\) −18.1082 −0.681028
\(708\) −11.3319 −0.425877
\(709\) −4.10171 −0.154043 −0.0770215 0.997029i \(-0.524541\pi\)
−0.0770215 + 0.997029i \(0.524541\pi\)
\(710\) 0 0
\(711\) 6.70964 0.251631
\(712\) 20.9906 0.786657
\(713\) −20.6637 −0.773862
\(714\) −2.90321 −0.108650
\(715\) 0 0
\(716\) −10.5433 −0.394020
\(717\) 46.4800 1.73583
\(718\) −5.12537 −0.191277
\(719\) 19.2696 0.718636 0.359318 0.933215i \(-0.383009\pi\)
0.359318 + 0.933215i \(0.383009\pi\)
\(720\) 0 0
\(721\) 16.7447 0.623604
\(722\) −13.0479 −0.485594
\(723\) 23.8020 0.885205
\(724\) 37.8894 1.40815
\(725\) 0 0
\(726\) −0.688892 −0.0255672
\(727\) 37.4084 1.38740 0.693700 0.720264i \(-0.255979\pi\)
0.693700 + 0.720264i \(0.255979\pi\)
\(728\) −2.68889 −0.0996570
\(729\) −4.96836 −0.184013
\(730\) 0 0
\(731\) −17.4795 −0.646502
\(732\) −37.0464 −1.36927
\(733\) −4.60147 −0.169959 −0.0849796 0.996383i \(-0.527083\pi\)
−0.0849796 + 0.996383i \(0.527083\pi\)
\(734\) 2.08943 0.0771222
\(735\) 0 0
\(736\) 10.1476 0.374047
\(737\) −1.52543 −0.0561898
\(738\) 3.28391 0.120882
\(739\) 35.0420 1.28904 0.644520 0.764588i \(-0.277057\pi\)
0.644520 + 0.764588i \(0.277057\pi\)
\(740\) 0 0
\(741\) 38.2766 1.40612
\(742\) 2.04593 0.0751086
\(743\) −40.7654 −1.49554 −0.747769 0.663959i \(-0.768875\pi\)
−0.747769 + 0.663959i \(0.768875\pi\)
\(744\) 19.1383 0.701643
\(745\) 0 0
\(746\) 3.28147 0.120143
\(747\) 4.06022 0.148556
\(748\) 8.02074 0.293267
\(749\) −0.561993 −0.0205348
\(750\) 0 0
\(751\) 24.4286 0.891414 0.445707 0.895179i \(-0.352952\pi\)
0.445707 + 0.895179i \(0.352952\pi\)
\(752\) −3.53188 −0.128794
\(753\) −13.9541 −0.508514
\(754\) −0.520505 −0.0189557
\(755\) 0 0
\(756\) −4.62222 −0.168108
\(757\) −36.9086 −1.34147 −0.670733 0.741699i \(-0.734020\pi\)
−0.670733 + 0.741699i \(0.734020\pi\)
\(758\) 10.2854 0.373584
\(759\) −6.42864 −0.233345
\(760\) 0 0
\(761\) −3.96190 −0.143619 −0.0718094 0.997418i \(-0.522877\pi\)
−0.0718094 + 0.997418i \(0.522877\pi\)
\(762\) −8.94914 −0.324193
\(763\) 3.93978 0.142630
\(764\) 39.7748 1.43900
\(765\) 0 0
\(766\) −5.38424 −0.194540
\(767\) 5.95407 0.214989
\(768\) 19.5002 0.703654
\(769\) 2.29190 0.0826479 0.0413239 0.999146i \(-0.486842\pi\)
0.0413239 + 0.999146i \(0.486842\pi\)
\(770\) 0 0
\(771\) −58.1847 −2.09547
\(772\) −25.9353 −0.933433
\(773\) 11.3047 0.406600 0.203300 0.979116i \(-0.434833\pi\)
0.203300 + 0.979116i \(0.434833\pi\)
\(774\) −2.45584 −0.0882732
\(775\) 0 0
\(776\) −6.41880 −0.230421
\(777\) −13.9081 −0.498952
\(778\) 10.3239 0.370129
\(779\) −43.2958 −1.55123
\(780\) 0 0
\(781\) 7.61285 0.272409
\(782\) −3.80642 −0.136117
\(783\) −1.83500 −0.0655777
\(784\) 3.42864 0.122451
\(785\) 0 0
\(786\) 5.83500 0.208128
\(787\) −20.2163 −0.720634 −0.360317 0.932830i \(-0.617332\pi\)
−0.360317 + 0.932830i \(0.617332\pi\)
\(788\) 29.8809 1.06446
\(789\) −28.1748 −1.00305
\(790\) 0 0
\(791\) 11.2859 0.401281
\(792\) 2.31111 0.0821216
\(793\) 19.4652 0.691230
\(794\) 3.31756 0.117736
\(795\) 0 0
\(796\) 45.5145 1.61322
\(797\) −50.4514 −1.78708 −0.893540 0.448984i \(-0.851786\pi\)
−0.893540 + 0.448984i \(0.851786\pi\)
\(798\) 5.37778 0.190372
\(799\) 4.34122 0.153581
\(800\) 0 0
\(801\) 32.8988 1.16242
\(802\) 3.29481 0.116344
\(803\) 12.2143 0.431034
\(804\) 6.42864 0.226721
\(805\) 0 0
\(806\) −4.90321 −0.172708
\(807\) −31.2257 −1.09920
\(808\) 21.9891 0.773574
\(809\) 17.9911 0.632534 0.316267 0.948670i \(-0.397571\pi\)
0.316267 + 0.948670i \(0.397571\pi\)
\(810\) 0 0
\(811\) −38.3082 −1.34518 −0.672591 0.740014i \(-0.734819\pi\)
−0.672591 + 0.740014i \(0.734819\pi\)
\(812\) 1.43801 0.0504642
\(813\) −11.7047 −0.410502
\(814\) −1.95407 −0.0684900
\(815\) 0 0
\(816\) −31.9956 −1.12007
\(817\) 32.3783 1.13277
\(818\) −3.69688 −0.129258
\(819\) −4.21432 −0.147260
\(820\) 0 0
\(821\) 22.3082 0.778561 0.389281 0.921119i \(-0.372724\pi\)
0.389281 + 0.921119i \(0.372724\pi\)
\(822\) −7.67307 −0.267629
\(823\) −38.3541 −1.33694 −0.668470 0.743739i \(-0.733051\pi\)
−0.668470 + 0.743739i \(0.733051\pi\)
\(824\) −20.3334 −0.708347
\(825\) 0 0
\(826\) 0.836535 0.0291068
\(827\) 30.5446 1.06214 0.531071 0.847328i \(-0.321790\pi\)
0.531071 + 0.847328i \(0.321790\pi\)
\(828\) 10.5161 0.365458
\(829\) −52.9215 −1.83804 −0.919020 0.394211i \(-0.871018\pi\)
−0.919020 + 0.394211i \(0.871018\pi\)
\(830\) 0 0
\(831\) −33.6128 −1.16602
\(832\) −12.7763 −0.442939
\(833\) −4.21432 −0.146018
\(834\) 5.53972 0.191825
\(835\) 0 0
\(836\) −14.8573 −0.513850
\(837\) −17.2859 −0.597489
\(838\) 0.813784 0.0281117
\(839\) 38.6987 1.33603 0.668014 0.744148i \(-0.267144\pi\)
0.668014 + 0.744148i \(0.267144\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 6.69673 0.230785
\(843\) 72.7783 2.50662
\(844\) −21.8479 −0.752036
\(845\) 0 0
\(846\) 0.609933 0.0209699
\(847\) −1.00000 −0.0343604
\(848\) 22.5477 0.774291
\(849\) 19.8163 0.680093
\(850\) 0 0
\(851\) −18.2351 −0.625090
\(852\) −32.0830 −1.09914
\(853\) −29.3352 −1.00442 −0.502210 0.864746i \(-0.667479\pi\)
−0.502210 + 0.864746i \(0.667479\pi\)
\(854\) 2.73483 0.0935838
\(855\) 0 0
\(856\) 0.682439 0.0233253
\(857\) 21.0627 0.719488 0.359744 0.933051i \(-0.382864\pi\)
0.359744 + 0.933051i \(0.382864\pi\)
\(858\) −1.52543 −0.0520772
\(859\) 16.9525 0.578413 0.289207 0.957267i \(-0.406609\pi\)
0.289207 + 0.957267i \(0.406609\pi\)
\(860\) 0 0
\(861\) 12.2810 0.418535
\(862\) 5.92549 0.201823
\(863\) 49.8336 1.69636 0.848178 0.529711i \(-0.177700\pi\)
0.848178 + 0.529711i \(0.177700\pi\)
\(864\) 8.48886 0.288797
\(865\) 0 0
\(866\) 5.81627 0.197645
\(867\) 1.68397 0.0571906
\(868\) 13.5462 0.459787
\(869\) 3.52543 0.119592
\(870\) 0 0
\(871\) −3.37778 −0.114452
\(872\) −4.78415 −0.162012
\(873\) −10.0602 −0.340487
\(874\) 7.05086 0.238499
\(875\) 0 0
\(876\) −51.4750 −1.73918
\(877\) −1.29036 −0.0435725 −0.0217863 0.999763i \(-0.506935\pi\)
−0.0217863 + 0.999763i \(0.506935\pi\)
\(878\) 0.308193 0.0104010
\(879\) 50.7511 1.71179
\(880\) 0 0
\(881\) 23.4380 0.789647 0.394823 0.918757i \(-0.370806\pi\)
0.394823 + 0.918757i \(0.370806\pi\)
\(882\) −0.592104 −0.0199372
\(883\) 31.0366 1.04446 0.522232 0.852804i \(-0.325100\pi\)
0.522232 + 0.852804i \(0.325100\pi\)
\(884\) 17.7605 0.597350
\(885\) 0 0
\(886\) 6.33138 0.212707
\(887\) 1.40636 0.0472211 0.0236106 0.999721i \(-0.492484\pi\)
0.0236106 + 0.999721i \(0.492484\pi\)
\(888\) 16.8889 0.566755
\(889\) −12.9906 −0.435692
\(890\) 0 0
\(891\) −11.0874 −0.371443
\(892\) 37.6207 1.25963
\(893\) −8.04149 −0.269098
\(894\) 4.22522 0.141312
\(895\) 0 0
\(896\) −8.78568 −0.293509
\(897\) −14.2351 −0.475295
\(898\) 9.06515 0.302508
\(899\) 5.37778 0.179359
\(900\) 0 0
\(901\) −27.7146 −0.923306
\(902\) 1.72546 0.0574514
\(903\) −9.18421 −0.305631
\(904\) −13.7047 −0.455812
\(905\) 0 0
\(906\) −6.99063 −0.232248
\(907\) −5.86220 −0.194651 −0.0973256 0.995253i \(-0.531029\pi\)
−0.0973256 + 0.995253i \(0.531029\pi\)
\(908\) 6.87601 0.228189
\(909\) 34.4637 1.14309
\(910\) 0 0
\(911\) −19.2257 −0.636976 −0.318488 0.947927i \(-0.603175\pi\)
−0.318488 + 0.947927i \(0.603175\pi\)
\(912\) 59.2672 1.96253
\(913\) 2.13335 0.0706037
\(914\) 9.74176 0.322229
\(915\) 0 0
\(916\) 35.6356 1.17743
\(917\) 8.47013 0.279708
\(918\) −3.18421 −0.105095
\(919\) 0.520505 0.0171699 0.00858494 0.999963i \(-0.497267\pi\)
0.00858494 + 0.999963i \(0.497267\pi\)
\(920\) 0 0
\(921\) 40.5116 1.33490
\(922\) −3.70271 −0.121942
\(923\) 16.8573 0.554864
\(924\) 4.21432 0.138641
\(925\) 0 0
\(926\) 1.21141 0.0398092
\(927\) −31.8687 −1.04670
\(928\) −2.64095 −0.0866935
\(929\) 19.6414 0.644414 0.322207 0.946669i \(-0.395575\pi\)
0.322207 + 0.946669i \(0.395575\pi\)
\(930\) 0 0
\(931\) 7.80642 0.255845
\(932\) 13.3319 0.436699
\(933\) −12.2810 −0.402062
\(934\) −6.65725 −0.217832
\(935\) 0 0
\(936\) 5.11753 0.167272
\(937\) 54.2973 1.77382 0.886908 0.461947i \(-0.152849\pi\)
0.886908 + 0.461947i \(0.152849\pi\)
\(938\) −0.474572 −0.0154953
\(939\) 52.7150 1.72029
\(940\) 0 0
\(941\) 1.95206 0.0636353 0.0318177 0.999494i \(-0.489870\pi\)
0.0318177 + 0.999494i \(0.489870\pi\)
\(942\) −14.9590 −0.487390
\(943\) 16.1017 0.524344
\(944\) 9.21924 0.300061
\(945\) 0 0
\(946\) −1.29036 −0.0419534
\(947\) 9.39207 0.305201 0.152601 0.988288i \(-0.451235\pi\)
0.152601 + 0.988288i \(0.451235\pi\)
\(948\) −14.8573 −0.482542
\(949\) 27.0464 0.877964
\(950\) 0 0
\(951\) −23.4509 −0.760448
\(952\) 5.11753 0.165860
\(953\) 11.2716 0.365124 0.182562 0.983194i \(-0.441561\pi\)
0.182562 + 0.983194i \(0.441561\pi\)
\(954\) −3.89384 −0.126068
\(955\) 0 0
\(956\) −39.9496 −1.29206
\(957\) 1.67307 0.0540827
\(958\) 3.72746 0.120429
\(959\) −11.1383 −0.359674
\(960\) 0 0
\(961\) 19.6593 0.634170
\(962\) −4.32693 −0.139506
\(963\) 1.06959 0.0344671
\(964\) −20.4578 −0.658903
\(965\) 0 0
\(966\) −2.00000 −0.0643489
\(967\) −17.3002 −0.556337 −0.278169 0.960532i \(-0.589727\pi\)
−0.278169 + 0.960532i \(0.589727\pi\)
\(968\) 1.21432 0.0390297
\(969\) −72.8484 −2.34023
\(970\) 0 0
\(971\) 28.2928 0.907961 0.453980 0.891012i \(-0.350004\pi\)
0.453980 + 0.891012i \(0.350004\pi\)
\(972\) 32.8593 1.05396
\(973\) 8.04149 0.257798
\(974\) −5.16103 −0.165370
\(975\) 0 0
\(976\) 30.1398 0.964752
\(977\) 45.9782 1.47097 0.735486 0.677539i \(-0.236954\pi\)
0.735486 + 0.677539i \(0.236954\pi\)
\(978\) −5.52098 −0.176542
\(979\) 17.2859 0.552460
\(980\) 0 0
\(981\) −7.49823 −0.239400
\(982\) 12.8301 0.409424
\(983\) −1.41726 −0.0452037 −0.0226018 0.999745i \(-0.507195\pi\)
−0.0226018 + 0.999745i \(0.507195\pi\)
\(984\) −14.9131 −0.475411
\(985\) 0 0
\(986\) 0.990632 0.0315482
\(987\) 2.28100 0.0726049
\(988\) −32.8988 −1.04665
\(989\) −12.0415 −0.382897
\(990\) 0 0
\(991\) −28.9906 −0.920918 −0.460459 0.887681i \(-0.652315\pi\)
−0.460459 + 0.887681i \(0.652315\pi\)
\(992\) −24.8780 −0.789878
\(993\) 66.6834 2.11613
\(994\) 2.36842 0.0751216
\(995\) 0 0
\(996\) −8.99063 −0.284879
\(997\) 5.80843 0.183955 0.0919774 0.995761i \(-0.470681\pi\)
0.0919774 + 0.995761i \(0.470681\pi\)
\(998\) 10.0731 0.318859
\(999\) −15.2543 −0.482624
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 1925.2.a.u.1.2 3
5.2 odd 4 1925.2.b.o.1849.3 6
5.3 odd 4 1925.2.b.o.1849.4 6
5.4 even 2 385.2.a.g.1.2 3
15.14 odd 2 3465.2.a.ba.1.2 3
20.19 odd 2 6160.2.a.bj.1.3 3
35.34 odd 2 2695.2.a.i.1.2 3
55.54 odd 2 4235.2.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.a.g.1.2 3 5.4 even 2
1925.2.a.u.1.2 3 1.1 even 1 trivial
1925.2.b.o.1849.3 6 5.2 odd 4
1925.2.b.o.1849.4 6 5.3 odd 4
2695.2.a.i.1.2 3 35.34 odd 2
3465.2.a.ba.1.2 3 15.14 odd 2
4235.2.a.o.1.2 3 55.54 odd 2
6160.2.a.bj.1.3 3 20.19 odd 2