Properties

Label 1075.2.a.o.1.1
Level $1075$
Weight $2$
Character 1075.1
Self dual yes
Analytic conductor $8.584$
Analytic rank $1$
Dimension $5$
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] = [1075,2,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.24217.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.96003\) of defining polynomial
Character \(\chi\) \(=\) 1075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.96003 q^{2} +0.199933 q^{3} +1.84170 q^{4} -0.391875 q^{6} +3.80173 q^{7} +0.310264 q^{8} -2.96003 q^{9} +O(q^{10})\) \(q-1.96003 q^{2} +0.199933 q^{3} +1.84170 q^{4} -0.391875 q^{6} +3.80173 q^{7} +0.310264 q^{8} -2.96003 q^{9} -1.77967 q^{11} +0.368218 q^{12} -1.73545 q^{13} -7.45149 q^{14} -4.29153 q^{16} +0.964104 q^{17} +5.80173 q^{18} -4.34358 q^{19} +0.760093 q^{21} +3.48820 q^{22} -4.20160 q^{23} +0.0620321 q^{24} +3.40153 q^{26} -1.19161 q^{27} +7.00166 q^{28} -9.77008 q^{29} +5.61064 q^{31} +7.79099 q^{32} -0.355816 q^{33} -1.88967 q^{34} -5.45149 q^{36} +0.837627 q^{37} +8.51353 q^{38} -0.346975 q^{39} +7.13997 q^{41} -1.48980 q^{42} +1.00000 q^{43} -3.27763 q^{44} +8.23524 q^{46} -7.86053 q^{47} -0.858021 q^{48} +7.45316 q^{49} +0.192757 q^{51} -3.19619 q^{52} +1.84463 q^{53} +2.33558 q^{54} +1.17954 q^{56} -0.868426 q^{57} +19.1496 q^{58} -8.35323 q^{59} -11.1011 q^{61} -10.9970 q^{62} -11.2532 q^{63} -6.68749 q^{64} +0.697409 q^{66} +2.64062 q^{67} +1.77559 q^{68} -0.840040 q^{69} -10.4442 q^{71} -0.918389 q^{72} +4.21367 q^{73} -1.64177 q^{74} -7.99958 q^{76} -6.76584 q^{77} +0.680080 q^{78} +16.4139 q^{79} +8.64184 q^{81} -13.9945 q^{82} -17.1653 q^{83} +1.39987 q^{84} -1.96003 q^{86} -1.95337 q^{87} -0.552168 q^{88} -11.0755 q^{89} -6.59772 q^{91} -7.73810 q^{92} +1.12175 q^{93} +15.4069 q^{94} +1.55768 q^{96} +0.955297 q^{97} -14.6084 q^{98} +5.26788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{6} + 3 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{6} + 3 q^{8} - 5 q^{9} - 9 q^{11} - 8 q^{12} + q^{13} - 7 q^{14} - 2 q^{16} - 3 q^{17} + 10 q^{18} - 11 q^{19} - 5 q^{21} - 13 q^{22} - 9 q^{24} - 5 q^{26} - 3 q^{27} + 15 q^{28} - 22 q^{29} - 5 q^{31} - 4 q^{32} + 12 q^{33} - 7 q^{34} + 3 q^{36} - 7 q^{37} + 3 q^{38} - 3 q^{39} - 21 q^{41} - 7 q^{42} + 5 q^{43} - 20 q^{44} + 13 q^{46} + 3 q^{47} + 6 q^{48} - 13 q^{49} - 5 q^{51} - 18 q^{52} + 5 q^{53} + 4 q^{54} + 4 q^{56} - 13 q^{57} + 16 q^{58} - 8 q^{59} - 16 q^{61} + 2 q^{62} - 7 q^{63} - 17 q^{64} + 19 q^{66} + 8 q^{67} + 7 q^{68} - 15 q^{69} - 10 q^{71} - 5 q^{72} - q^{73} - 3 q^{76} - 7 q^{77} + 25 q^{78} + 12 q^{79} - 15 q^{81} - 41 q^{82} - 20 q^{83} + 5 q^{84} + 22 q^{87} + 2 q^{88} - 36 q^{89} - 13 q^{91} + q^{92} + 13 q^{93} + 28 q^{96} + 8 q^{97} - 25 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.96003 −1.38595 −0.692974 0.720963i \(-0.743700\pi\)
−0.692974 + 0.720963i \(0.743700\pi\)
\(3\) 0.199933 0.115432 0.0577158 0.998333i \(-0.481618\pi\)
0.0577158 + 0.998333i \(0.481618\pi\)
\(4\) 1.84170 0.920852
\(5\) 0 0
\(6\) −0.391875 −0.159982
\(7\) 3.80173 1.43692 0.718460 0.695569i \(-0.244848\pi\)
0.718460 + 0.695569i \(0.244848\pi\)
\(8\) 0.310264 0.109695
\(9\) −2.96003 −0.986676
\(10\) 0 0
\(11\) −1.77967 −0.536591 −0.268296 0.963337i \(-0.586460\pi\)
−0.268296 + 0.963337i \(0.586460\pi\)
\(12\) 0.368218 0.106295
\(13\) −1.73545 −0.481328 −0.240664 0.970609i \(-0.577365\pi\)
−0.240664 + 0.970609i \(0.577365\pi\)
\(14\) −7.45149 −1.99150
\(15\) 0 0
\(16\) −4.29153 −1.07288
\(17\) 0.964104 0.233830 0.116915 0.993142i \(-0.462700\pi\)
0.116915 + 0.993142i \(0.462700\pi\)
\(18\) 5.80173 1.36748
\(19\) −4.34358 −0.996485 −0.498242 0.867038i \(-0.666021\pi\)
−0.498242 + 0.867038i \(0.666021\pi\)
\(20\) 0 0
\(21\) 0.760093 0.165866
\(22\) 3.48820 0.743688
\(23\) −4.20160 −0.876094 −0.438047 0.898952i \(-0.644330\pi\)
−0.438047 + 0.898952i \(0.644330\pi\)
\(24\) 0.0620321 0.0126622
\(25\) 0 0
\(26\) 3.40153 0.667095
\(27\) −1.19161 −0.229325
\(28\) 7.00166 1.32319
\(29\) −9.77008 −1.81426 −0.907129 0.420852i \(-0.861731\pi\)
−0.907129 + 0.420852i \(0.861731\pi\)
\(30\) 0 0
\(31\) 5.61064 1.00770 0.503850 0.863791i \(-0.331916\pi\)
0.503850 + 0.863791i \(0.331916\pi\)
\(32\) 7.79099 1.37727
\(33\) −0.355816 −0.0619396
\(34\) −1.88967 −0.324076
\(35\) 0 0
\(36\) −5.45149 −0.908582
\(37\) 0.837627 0.137705 0.0688525 0.997627i \(-0.478066\pi\)
0.0688525 + 0.997627i \(0.478066\pi\)
\(38\) 8.51353 1.38108
\(39\) −0.346975 −0.0555604
\(40\) 0 0
\(41\) 7.13997 1.11508 0.557538 0.830152i \(-0.311746\pi\)
0.557538 + 0.830152i \(0.311746\pi\)
\(42\) −1.48980 −0.229882
\(43\) 1.00000 0.152499
\(44\) −3.27763 −0.494121
\(45\) 0 0
\(46\) 8.23524 1.21422
\(47\) −7.86053 −1.14658 −0.573288 0.819354i \(-0.694333\pi\)
−0.573288 + 0.819354i \(0.694333\pi\)
\(48\) −0.858021 −0.123845
\(49\) 7.45316 1.06474
\(50\) 0 0
\(51\) 0.192757 0.0269913
\(52\) −3.19619 −0.443232
\(53\) 1.84463 0.253380 0.126690 0.991942i \(-0.459565\pi\)
0.126690 + 0.991942i \(0.459565\pi\)
\(54\) 2.33558 0.317833
\(55\) 0 0
\(56\) 1.17954 0.157623
\(57\) −0.868426 −0.115026
\(58\) 19.1496 2.51447
\(59\) −8.35323 −1.08750 −0.543749 0.839248i \(-0.682996\pi\)
−0.543749 + 0.839248i \(0.682996\pi\)
\(60\) 0 0
\(61\) −11.1011 −1.42135 −0.710674 0.703521i \(-0.751610\pi\)
−0.710674 + 0.703521i \(0.751610\pi\)
\(62\) −10.9970 −1.39662
\(63\) −11.2532 −1.41777
\(64\) −6.68749 −0.835936
\(65\) 0 0
\(66\) 0.697409 0.0858451
\(67\) 2.64062 0.322603 0.161302 0.986905i \(-0.448431\pi\)
0.161302 + 0.986905i \(0.448431\pi\)
\(68\) 1.77559 0.215322
\(69\) −0.840040 −0.101129
\(70\) 0 0
\(71\) −10.4442 −1.23949 −0.619747 0.784802i \(-0.712765\pi\)
−0.619747 + 0.784802i \(0.712765\pi\)
\(72\) −0.918389 −0.108233
\(73\) 4.21367 0.493172 0.246586 0.969121i \(-0.420691\pi\)
0.246586 + 0.969121i \(0.420691\pi\)
\(74\) −1.64177 −0.190852
\(75\) 0 0
\(76\) −7.99958 −0.917615
\(77\) −6.76584 −0.771038
\(78\) 0.680080 0.0770039
\(79\) 16.4139 1.84670 0.923352 0.383955i \(-0.125438\pi\)
0.923352 + 0.383955i \(0.125438\pi\)
\(80\) 0 0
\(81\) 8.64184 0.960204
\(82\) −13.9945 −1.54544
\(83\) −17.1653 −1.88413 −0.942067 0.335423i \(-0.891121\pi\)
−0.942067 + 0.335423i \(0.891121\pi\)
\(84\) 1.39987 0.152738
\(85\) 0 0
\(86\) −1.96003 −0.211355
\(87\) −1.95337 −0.209423
\(88\) −0.552168 −0.0588613
\(89\) −11.0755 −1.17400 −0.586998 0.809588i \(-0.699690\pi\)
−0.586998 + 0.809588i \(0.699690\pi\)
\(90\) 0 0
\(91\) −6.59772 −0.691629
\(92\) −7.73810 −0.806753
\(93\) 1.12175 0.116321
\(94\) 15.4069 1.58910
\(95\) 0 0
\(96\) 1.55768 0.158980
\(97\) 0.955297 0.0969957 0.0484979 0.998823i \(-0.484557\pi\)
0.0484979 + 0.998823i \(0.484557\pi\)
\(98\) −14.6084 −1.47567
\(99\) 5.26788 0.529442
\(100\) 0 0
\(101\) 2.32158 0.231006 0.115503 0.993307i \(-0.463152\pi\)
0.115503 + 0.993307i \(0.463152\pi\)
\(102\) −0.377808 −0.0374086
\(103\) −0.607140 −0.0598233 −0.0299116 0.999553i \(-0.509523\pi\)
−0.0299116 + 0.999553i \(0.509523\pi\)
\(104\) −0.538448 −0.0527991
\(105\) 0 0
\(106\) −3.61553 −0.351171
\(107\) −0.107747 −0.0104163 −0.00520817 0.999986i \(-0.501658\pi\)
−0.00520817 + 0.999986i \(0.501658\pi\)
\(108\) −2.19459 −0.211175
\(109\) −9.60330 −0.919829 −0.459915 0.887963i \(-0.652120\pi\)
−0.459915 + 0.887963i \(0.652120\pi\)
\(110\) 0 0
\(111\) 0.167470 0.0158955
\(112\) −16.3153 −1.54165
\(113\) 5.32060 0.500520 0.250260 0.968179i \(-0.419484\pi\)
0.250260 + 0.968179i \(0.419484\pi\)
\(114\) 1.70214 0.159420
\(115\) 0 0
\(116\) −17.9936 −1.67066
\(117\) 5.13698 0.474914
\(118\) 16.3726 1.50722
\(119\) 3.66526 0.335994
\(120\) 0 0
\(121\) −7.83277 −0.712070
\(122\) 21.7584 1.96992
\(123\) 1.42752 0.128715
\(124\) 10.3331 0.927943
\(125\) 0 0
\(126\) 22.0566 1.96496
\(127\) −6.70870 −0.595301 −0.297651 0.954675i \(-0.596203\pi\)
−0.297651 + 0.954675i \(0.596203\pi\)
\(128\) −2.47434 −0.218702
\(129\) 0.199933 0.0176032
\(130\) 0 0
\(131\) −10.6812 −0.933217 −0.466609 0.884464i \(-0.654524\pi\)
−0.466609 + 0.884464i \(0.654524\pi\)
\(132\) −0.655308 −0.0570372
\(133\) −16.5131 −1.43187
\(134\) −5.17569 −0.447112
\(135\) 0 0
\(136\) 0.299127 0.0256499
\(137\) −1.64034 −0.140144 −0.0700720 0.997542i \(-0.522323\pi\)
−0.0700720 + 0.997542i \(0.522323\pi\)
\(138\) 1.64650 0.140159
\(139\) −5.01948 −0.425746 −0.212873 0.977080i \(-0.568282\pi\)
−0.212873 + 0.977080i \(0.568282\pi\)
\(140\) 0 0
\(141\) −1.57158 −0.132351
\(142\) 20.4708 1.71787
\(143\) 3.08854 0.258276
\(144\) 12.7031 1.05859
\(145\) 0 0
\(146\) −8.25890 −0.683511
\(147\) 1.49014 0.122904
\(148\) 1.54266 0.126806
\(149\) −8.48755 −0.695327 −0.347664 0.937619i \(-0.613025\pi\)
−0.347664 + 0.937619i \(0.613025\pi\)
\(150\) 0 0
\(151\) −20.2094 −1.64462 −0.822309 0.569041i \(-0.807314\pi\)
−0.822309 + 0.569041i \(0.807314\pi\)
\(152\) −1.34765 −0.109309
\(153\) −2.85377 −0.230714
\(154\) 13.2612 1.06862
\(155\) 0 0
\(156\) −0.639025 −0.0511629
\(157\) 14.9980 1.19697 0.598485 0.801134i \(-0.295770\pi\)
0.598485 + 0.801134i \(0.295770\pi\)
\(158\) −32.1716 −2.55944
\(159\) 0.368804 0.0292481
\(160\) 0 0
\(161\) −15.9733 −1.25888
\(162\) −16.9382 −1.33079
\(163\) −8.48233 −0.664387 −0.332194 0.943211i \(-0.607789\pi\)
−0.332194 + 0.943211i \(0.607789\pi\)
\(164\) 13.1497 1.02682
\(165\) 0 0
\(166\) 33.6444 2.61131
\(167\) −3.27586 −0.253494 −0.126747 0.991935i \(-0.540454\pi\)
−0.126747 + 0.991935i \(0.540454\pi\)
\(168\) 0.235829 0.0181946
\(169\) −9.98821 −0.768324
\(170\) 0 0
\(171\) 12.8571 0.983207
\(172\) 1.84170 0.140429
\(173\) 18.7232 1.42350 0.711750 0.702432i \(-0.247903\pi\)
0.711750 + 0.702432i \(0.247903\pi\)
\(174\) 3.82865 0.290249
\(175\) 0 0
\(176\) 7.63752 0.575700
\(177\) −1.67009 −0.125532
\(178\) 21.7082 1.62710
\(179\) −8.90541 −0.665621 −0.332811 0.942994i \(-0.607997\pi\)
−0.332811 + 0.942994i \(0.607997\pi\)
\(180\) 0 0
\(181\) −6.61322 −0.491557 −0.245778 0.969326i \(-0.579044\pi\)
−0.245778 + 0.969326i \(0.579044\pi\)
\(182\) 12.9317 0.958562
\(183\) −2.21948 −0.164069
\(184\) −1.30360 −0.0961029
\(185\) 0 0
\(186\) −2.19867 −0.161214
\(187\) −1.71579 −0.125471
\(188\) −14.4768 −1.05583
\(189\) −4.53018 −0.329522
\(190\) 0 0
\(191\) 24.6849 1.78613 0.893067 0.449924i \(-0.148549\pi\)
0.893067 + 0.449924i \(0.148549\pi\)
\(192\) −1.33705 −0.0964934
\(193\) −11.8102 −0.850114 −0.425057 0.905167i \(-0.639746\pi\)
−0.425057 + 0.905167i \(0.639746\pi\)
\(194\) −1.87241 −0.134431
\(195\) 0 0
\(196\) 13.7265 0.980465
\(197\) −15.0039 −1.06899 −0.534493 0.845173i \(-0.679497\pi\)
−0.534493 + 0.845173i \(0.679497\pi\)
\(198\) −10.3252 −0.733779
\(199\) 14.4597 1.02502 0.512511 0.858681i \(-0.328716\pi\)
0.512511 + 0.858681i \(0.328716\pi\)
\(200\) 0 0
\(201\) 0.527949 0.0372386
\(202\) −4.55037 −0.320163
\(203\) −37.1432 −2.60694
\(204\) 0.355001 0.0248550
\(205\) 0 0
\(206\) 1.19001 0.0829120
\(207\) 12.4368 0.864420
\(208\) 7.44775 0.516408
\(209\) 7.73014 0.534705
\(210\) 0 0
\(211\) 10.2369 0.704738 0.352369 0.935861i \(-0.385376\pi\)
0.352369 + 0.935861i \(0.385376\pi\)
\(212\) 3.39727 0.233325
\(213\) −2.08814 −0.143077
\(214\) 0.211188 0.0144365
\(215\) 0 0
\(216\) −0.369713 −0.0251558
\(217\) 21.3301 1.44798
\(218\) 18.8227 1.27484
\(219\) 0.842453 0.0569277
\(220\) 0 0
\(221\) −1.67316 −0.112549
\(222\) −0.328245 −0.0220304
\(223\) 28.7779 1.92711 0.963554 0.267515i \(-0.0862024\pi\)
0.963554 + 0.267515i \(0.0862024\pi\)
\(224\) 29.6193 1.97902
\(225\) 0 0
\(226\) −10.4285 −0.693695
\(227\) 4.24838 0.281975 0.140987 0.990011i \(-0.454972\pi\)
0.140987 + 0.990011i \(0.454972\pi\)
\(228\) −1.59938 −0.105922
\(229\) 29.2506 1.93293 0.966466 0.256793i \(-0.0826660\pi\)
0.966466 + 0.256793i \(0.0826660\pi\)
\(230\) 0 0
\(231\) −1.35272 −0.0890022
\(232\) −3.03130 −0.199015
\(233\) 15.7785 1.03369 0.516843 0.856080i \(-0.327107\pi\)
0.516843 + 0.856080i \(0.327107\pi\)
\(234\) −10.0686 −0.658206
\(235\) 0 0
\(236\) −15.3842 −1.00143
\(237\) 3.28168 0.213168
\(238\) −7.18402 −0.465671
\(239\) 10.3196 0.667520 0.333760 0.942658i \(-0.391682\pi\)
0.333760 + 0.942658i \(0.391682\pi\)
\(240\) 0 0
\(241\) 2.87891 0.185447 0.0927235 0.995692i \(-0.470443\pi\)
0.0927235 + 0.995692i \(0.470443\pi\)
\(242\) 15.3524 0.986892
\(243\) 5.30262 0.340163
\(244\) −20.4449 −1.30885
\(245\) 0 0
\(246\) −2.79797 −0.178392
\(247\) 7.53807 0.479636
\(248\) 1.74078 0.110539
\(249\) −3.43191 −0.217489
\(250\) 0 0
\(251\) −11.4846 −0.724903 −0.362451 0.932003i \(-0.618060\pi\)
−0.362451 + 0.932003i \(0.618060\pi\)
\(252\) −20.7251 −1.30556
\(253\) 7.47747 0.470104
\(254\) 13.1492 0.825056
\(255\) 0 0
\(256\) 18.2247 1.13905
\(257\) −9.57691 −0.597391 −0.298696 0.954348i \(-0.596552\pi\)
−0.298696 + 0.954348i \(0.596552\pi\)
\(258\) −0.391875 −0.0243971
\(259\) 3.18443 0.197871
\(260\) 0 0
\(261\) 28.9197 1.79008
\(262\) 20.9354 1.29339
\(263\) −20.8514 −1.28575 −0.642875 0.765971i \(-0.722259\pi\)
−0.642875 + 0.765971i \(0.722259\pi\)
\(264\) −0.110397 −0.00679445
\(265\) 0 0
\(266\) 32.3661 1.98450
\(267\) −2.21435 −0.135516
\(268\) 4.86325 0.297070
\(269\) 2.09034 0.127450 0.0637250 0.997967i \(-0.479702\pi\)
0.0637250 + 0.997967i \(0.479702\pi\)
\(270\) 0 0
\(271\) −8.13823 −0.494362 −0.247181 0.968969i \(-0.579504\pi\)
−0.247181 + 0.968969i \(0.579504\pi\)
\(272\) −4.13749 −0.250872
\(273\) −1.31910 −0.0798359
\(274\) 3.21512 0.194232
\(275\) 0 0
\(276\) −1.54711 −0.0931248
\(277\) −12.1111 −0.727688 −0.363844 0.931460i \(-0.618536\pi\)
−0.363844 + 0.931460i \(0.618536\pi\)
\(278\) 9.83831 0.590062
\(279\) −16.6076 −0.994273
\(280\) 0 0
\(281\) 13.1976 0.787306 0.393653 0.919259i \(-0.371211\pi\)
0.393653 + 0.919259i \(0.371211\pi\)
\(282\) 3.08035 0.183432
\(283\) −5.09205 −0.302691 −0.151345 0.988481i \(-0.548361\pi\)
−0.151345 + 0.988481i \(0.548361\pi\)
\(284\) −19.2350 −1.14139
\(285\) 0 0
\(286\) −6.05361 −0.357958
\(287\) 27.1442 1.60227
\(288\) −23.0615 −1.35891
\(289\) −16.0705 −0.945324
\(290\) 0 0
\(291\) 0.190996 0.0111964
\(292\) 7.76033 0.454139
\(293\) −11.1081 −0.648943 −0.324471 0.945895i \(-0.605186\pi\)
−0.324471 + 0.945895i \(0.605186\pi\)
\(294\) −2.92071 −0.170339
\(295\) 0 0
\(296\) 0.259885 0.0151055
\(297\) 2.12067 0.123054
\(298\) 16.6358 0.963688
\(299\) 7.29167 0.421688
\(300\) 0 0
\(301\) 3.80173 0.219128
\(302\) 39.6110 2.27936
\(303\) 0.464162 0.0266654
\(304\) 18.6406 1.06911
\(305\) 0 0
\(306\) 5.59347 0.319758
\(307\) 27.6299 1.57692 0.788462 0.615084i \(-0.210878\pi\)
0.788462 + 0.615084i \(0.210878\pi\)
\(308\) −12.4607 −0.710012
\(309\) −0.121388 −0.00690550
\(310\) 0 0
\(311\) −12.6190 −0.715556 −0.357778 0.933807i \(-0.616466\pi\)
−0.357778 + 0.933807i \(0.616466\pi\)
\(312\) −0.107654 −0.00609469
\(313\) −9.95430 −0.562650 −0.281325 0.959613i \(-0.590774\pi\)
−0.281325 + 0.959613i \(0.590774\pi\)
\(314\) −29.3965 −1.65894
\(315\) 0 0
\(316\) 30.2295 1.70054
\(317\) 27.6663 1.55389 0.776947 0.629566i \(-0.216767\pi\)
0.776947 + 0.629566i \(0.216767\pi\)
\(318\) −0.722866 −0.0405363
\(319\) 17.3875 0.973516
\(320\) 0 0
\(321\) −0.0215423 −0.00120237
\(322\) 31.3082 1.74474
\(323\) −4.18766 −0.233008
\(324\) 15.9157 0.884206
\(325\) 0 0
\(326\) 16.6256 0.920806
\(327\) −1.92002 −0.106177
\(328\) 2.21527 0.122318
\(329\) −29.8836 −1.64754
\(330\) 0 0
\(331\) 11.0556 0.607668 0.303834 0.952725i \(-0.401733\pi\)
0.303834 + 0.952725i \(0.401733\pi\)
\(332\) −31.6134 −1.73501
\(333\) −2.47940 −0.135870
\(334\) 6.42078 0.351329
\(335\) 0 0
\(336\) −3.26197 −0.177955
\(337\) −11.6752 −0.635991 −0.317996 0.948092i \(-0.603010\pi\)
−0.317996 + 0.948092i \(0.603010\pi\)
\(338\) 19.5772 1.06486
\(339\) 1.06377 0.0577758
\(340\) 0 0
\(341\) −9.98510 −0.540723
\(342\) −25.2003 −1.36267
\(343\) 1.72279 0.0930217
\(344\) 0.310264 0.0167283
\(345\) 0 0
\(346\) −36.6980 −1.97290
\(347\) 23.5556 1.26453 0.632265 0.774752i \(-0.282125\pi\)
0.632265 + 0.774752i \(0.282125\pi\)
\(348\) −3.59752 −0.192847
\(349\) 19.9469 1.06773 0.533865 0.845570i \(-0.320739\pi\)
0.533865 + 0.845570i \(0.320739\pi\)
\(350\) 0 0
\(351\) 2.06798 0.110381
\(352\) −13.8654 −0.739029
\(353\) 30.3607 1.61594 0.807968 0.589226i \(-0.200567\pi\)
0.807968 + 0.589226i \(0.200567\pi\)
\(354\) 3.27342 0.173980
\(355\) 0 0
\(356\) −20.3977 −1.08108
\(357\) 0.732809 0.0387844
\(358\) 17.4548 0.922517
\(359\) 14.6239 0.771822 0.385911 0.922536i \(-0.373887\pi\)
0.385911 + 0.922536i \(0.373887\pi\)
\(360\) 0 0
\(361\) −0.133341 −0.00701797
\(362\) 12.9621 0.681272
\(363\) −1.56603 −0.0821954
\(364\) −12.1510 −0.636888
\(365\) 0 0
\(366\) 4.35024 0.227391
\(367\) 3.79697 0.198200 0.0991000 0.995077i \(-0.468404\pi\)
0.0991000 + 0.995077i \(0.468404\pi\)
\(368\) 18.0313 0.939946
\(369\) −21.1345 −1.10022
\(370\) 0 0
\(371\) 7.01280 0.364087
\(372\) 2.06594 0.107114
\(373\) −13.1551 −0.681147 −0.340573 0.940218i \(-0.610621\pi\)
−0.340573 + 0.940218i \(0.610621\pi\)
\(374\) 3.36299 0.173896
\(375\) 0 0
\(376\) −2.43884 −0.125773
\(377\) 16.9555 0.873253
\(378\) 8.87926 0.456700
\(379\) −26.2492 −1.34833 −0.674164 0.738581i \(-0.735496\pi\)
−0.674164 + 0.738581i \(0.735496\pi\)
\(380\) 0 0
\(381\) −1.34129 −0.0687166
\(382\) −48.3830 −2.47549
\(383\) 25.4582 1.30085 0.650426 0.759569i \(-0.274590\pi\)
0.650426 + 0.759569i \(0.274590\pi\)
\(384\) −0.494702 −0.0252452
\(385\) 0 0
\(386\) 23.1482 1.17821
\(387\) −2.96003 −0.150467
\(388\) 1.75937 0.0893187
\(389\) −6.70189 −0.339799 −0.169900 0.985461i \(-0.554344\pi\)
−0.169900 + 0.985461i \(0.554344\pi\)
\(390\) 0 0
\(391\) −4.05078 −0.204857
\(392\) 2.31244 0.116796
\(393\) −2.13552 −0.107723
\(394\) 29.4081 1.48156
\(395\) 0 0
\(396\) 9.70187 0.487537
\(397\) −19.5353 −0.980450 −0.490225 0.871596i \(-0.663085\pi\)
−0.490225 + 0.871596i \(0.663085\pi\)
\(398\) −28.3414 −1.42063
\(399\) −3.30152 −0.165283
\(400\) 0 0
\(401\) −36.4161 −1.81853 −0.909267 0.416214i \(-0.863357\pi\)
−0.909267 + 0.416214i \(0.863357\pi\)
\(402\) −1.03479 −0.0516108
\(403\) −9.73699 −0.485034
\(404\) 4.27567 0.212723
\(405\) 0 0
\(406\) 72.8017 3.61309
\(407\) −1.49070 −0.0738913
\(408\) 0.0598054 0.00296081
\(409\) −3.61297 −0.178650 −0.0893250 0.996003i \(-0.528471\pi\)
−0.0893250 + 0.996003i \(0.528471\pi\)
\(410\) 0 0
\(411\) −0.327959 −0.0161770
\(412\) −1.11817 −0.0550884
\(413\) −31.7567 −1.56265
\(414\) −24.3765 −1.19804
\(415\) 0 0
\(416\) −13.5209 −0.662916
\(417\) −1.00356 −0.0491446
\(418\) −15.1513 −0.741074
\(419\) 8.05241 0.393386 0.196693 0.980465i \(-0.436980\pi\)
0.196693 + 0.980465i \(0.436980\pi\)
\(420\) 0 0
\(421\) −37.1449 −1.81033 −0.905165 0.425060i \(-0.860253\pi\)
−0.905165 + 0.425060i \(0.860253\pi\)
\(422\) −20.0646 −0.976730
\(423\) 23.2674 1.13130
\(424\) 0.572323 0.0277945
\(425\) 0 0
\(426\) 4.09280 0.198297
\(427\) −42.2033 −2.04236
\(428\) −0.198439 −0.00959190
\(429\) 0.617501 0.0298133
\(430\) 0 0
\(431\) −10.3223 −0.497209 −0.248604 0.968605i \(-0.579972\pi\)
−0.248604 + 0.968605i \(0.579972\pi\)
\(432\) 5.11383 0.246039
\(433\) 17.4592 0.839036 0.419518 0.907747i \(-0.362199\pi\)
0.419518 + 0.907747i \(0.362199\pi\)
\(434\) −41.8076 −2.00683
\(435\) 0 0
\(436\) −17.6864 −0.847027
\(437\) 18.2500 0.873014
\(438\) −1.65123 −0.0788988
\(439\) −7.85325 −0.374815 −0.187408 0.982282i \(-0.560009\pi\)
−0.187408 + 0.982282i \(0.560009\pi\)
\(440\) 0 0
\(441\) −22.0615 −1.05055
\(442\) 3.27943 0.155987
\(443\) 6.36246 0.302289 0.151145 0.988512i \(-0.451704\pi\)
0.151145 + 0.988512i \(0.451704\pi\)
\(444\) 0.308430 0.0146374
\(445\) 0 0
\(446\) −56.4054 −2.67087
\(447\) −1.69695 −0.0802628
\(448\) −25.4240 −1.20117
\(449\) 14.6676 0.692208 0.346104 0.938196i \(-0.387504\pi\)
0.346104 + 0.938196i \(0.387504\pi\)
\(450\) 0 0
\(451\) −12.7068 −0.598340
\(452\) 9.79897 0.460905
\(453\) −4.04054 −0.189841
\(454\) −8.32694 −0.390803
\(455\) 0 0
\(456\) −0.269441 −0.0126177
\(457\) −37.7674 −1.76668 −0.883342 0.468728i \(-0.844712\pi\)
−0.883342 + 0.468728i \(0.844712\pi\)
\(458\) −57.3319 −2.67894
\(459\) −1.14883 −0.0536230
\(460\) 0 0
\(461\) −31.2202 −1.45407 −0.727035 0.686601i \(-0.759102\pi\)
−0.727035 + 0.686601i \(0.759102\pi\)
\(462\) 2.65136 0.123352
\(463\) 34.5627 1.60627 0.803133 0.595799i \(-0.203165\pi\)
0.803133 + 0.595799i \(0.203165\pi\)
\(464\) 41.9286 1.94649
\(465\) 0 0
\(466\) −30.9263 −1.43263
\(467\) −34.8475 −1.61255 −0.806274 0.591542i \(-0.798519\pi\)
−0.806274 + 0.591542i \(0.798519\pi\)
\(468\) 9.46080 0.437326
\(469\) 10.0389 0.463555
\(470\) 0 0
\(471\) 2.99860 0.138168
\(472\) −2.59171 −0.119293
\(473\) −1.77967 −0.0818294
\(474\) −6.43218 −0.295440
\(475\) 0 0
\(476\) 6.75033 0.309401
\(477\) −5.46017 −0.250004
\(478\) −20.2267 −0.925149
\(479\) 27.1580 1.24088 0.620441 0.784253i \(-0.286954\pi\)
0.620441 + 0.784253i \(0.286954\pi\)
\(480\) 0 0
\(481\) −1.45366 −0.0662812
\(482\) −5.64274 −0.257020
\(483\) −3.19361 −0.145314
\(484\) −14.4256 −0.655711
\(485\) 0 0
\(486\) −10.3933 −0.471448
\(487\) 30.2897 1.37256 0.686279 0.727338i \(-0.259243\pi\)
0.686279 + 0.727338i \(0.259243\pi\)
\(488\) −3.44426 −0.155915
\(489\) −1.69590 −0.0766913
\(490\) 0 0
\(491\) 9.93544 0.448380 0.224190 0.974545i \(-0.428026\pi\)
0.224190 + 0.974545i \(0.428026\pi\)
\(492\) 2.62907 0.118527
\(493\) −9.41938 −0.424227
\(494\) −14.7748 −0.664750
\(495\) 0 0
\(496\) −24.0782 −1.08115
\(497\) −39.7059 −1.78105
\(498\) 6.72664 0.301428
\(499\) 28.0062 1.25373 0.626865 0.779128i \(-0.284338\pi\)
0.626865 + 0.779128i \(0.284338\pi\)
\(500\) 0 0
\(501\) −0.654954 −0.0292612
\(502\) 22.5102 1.00468
\(503\) −5.77091 −0.257312 −0.128656 0.991689i \(-0.541066\pi\)
−0.128656 + 0.991689i \(0.541066\pi\)
\(504\) −3.49147 −0.155522
\(505\) 0 0
\(506\) −14.6560 −0.651540
\(507\) −1.99698 −0.0886889
\(508\) −12.3554 −0.548184
\(509\) −37.3060 −1.65356 −0.826780 0.562525i \(-0.809830\pi\)
−0.826780 + 0.562525i \(0.809830\pi\)
\(510\) 0 0
\(511\) 16.0192 0.708649
\(512\) −30.7723 −1.35996
\(513\) 5.17584 0.228519
\(514\) 18.7710 0.827953
\(515\) 0 0
\(516\) 0.368218 0.0162099
\(517\) 13.9892 0.615243
\(518\) −6.24157 −0.274239
\(519\) 3.74340 0.164317
\(520\) 0 0
\(521\) −37.0968 −1.62524 −0.812620 0.582794i \(-0.801959\pi\)
−0.812620 + 0.582794i \(0.801959\pi\)
\(522\) −56.6834 −2.48096
\(523\) 27.8814 1.21917 0.609585 0.792721i \(-0.291336\pi\)
0.609585 + 0.792721i \(0.291336\pi\)
\(524\) −19.6715 −0.859355
\(525\) 0 0
\(526\) 40.8692 1.78198
\(527\) 5.40924 0.235630
\(528\) 1.52700 0.0664540
\(529\) −5.34658 −0.232460
\(530\) 0 0
\(531\) 24.7258 1.07301
\(532\) −30.4123 −1.31854
\(533\) −12.3911 −0.536717
\(534\) 4.34019 0.187818
\(535\) 0 0
\(536\) 0.819289 0.0353879
\(537\) −1.78049 −0.0768338
\(538\) −4.09711 −0.176639
\(539\) −13.2642 −0.571329
\(540\) 0 0
\(541\) 17.5536 0.754689 0.377345 0.926073i \(-0.376837\pi\)
0.377345 + 0.926073i \(0.376837\pi\)
\(542\) 15.9512 0.685161
\(543\) −1.32220 −0.0567412
\(544\) 7.51133 0.322046
\(545\) 0 0
\(546\) 2.58548 0.110648
\(547\) 10.0280 0.428768 0.214384 0.976749i \(-0.431226\pi\)
0.214384 + 0.976749i \(0.431226\pi\)
\(548\) −3.02103 −0.129052
\(549\) 32.8595 1.40241
\(550\) 0 0
\(551\) 42.4371 1.80788
\(552\) −0.260634 −0.0110933
\(553\) 62.4011 2.65356
\(554\) 23.7382 1.00854
\(555\) 0 0
\(556\) −9.24439 −0.392049
\(557\) −20.8856 −0.884951 −0.442475 0.896781i \(-0.645900\pi\)
−0.442475 + 0.896781i \(0.645900\pi\)
\(558\) 32.5514 1.37801
\(559\) −1.73545 −0.0734018
\(560\) 0 0
\(561\) −0.343044 −0.0144833
\(562\) −25.8677 −1.09116
\(563\) 22.3621 0.942452 0.471226 0.882013i \(-0.343812\pi\)
0.471226 + 0.882013i \(0.343812\pi\)
\(564\) −2.89439 −0.121876
\(565\) 0 0
\(566\) 9.98055 0.419514
\(567\) 32.8539 1.37974
\(568\) −3.24044 −0.135966
\(569\) −0.514833 −0.0215829 −0.0107915 0.999942i \(-0.503435\pi\)
−0.0107915 + 0.999942i \(0.503435\pi\)
\(570\) 0 0
\(571\) 30.4847 1.27574 0.637872 0.770142i \(-0.279815\pi\)
0.637872 + 0.770142i \(0.279815\pi\)
\(572\) 5.68817 0.237834
\(573\) 4.93533 0.206176
\(574\) −53.2034 −2.22067
\(575\) 0 0
\(576\) 19.7951 0.824797
\(577\) −40.5019 −1.68612 −0.843059 0.537821i \(-0.819248\pi\)
−0.843059 + 0.537821i \(0.819248\pi\)
\(578\) 31.4986 1.31017
\(579\) −2.36125 −0.0981300
\(580\) 0 0
\(581\) −65.2578 −2.70735
\(582\) −0.374357 −0.0155176
\(583\) −3.28284 −0.135961
\(584\) 1.30735 0.0540984
\(585\) 0 0
\(586\) 21.7722 0.899401
\(587\) −4.75388 −0.196214 −0.0981068 0.995176i \(-0.531279\pi\)
−0.0981068 + 0.995176i \(0.531279\pi\)
\(588\) 2.74439 0.113177
\(589\) −24.3702 −1.00416
\(590\) 0 0
\(591\) −2.99979 −0.123395
\(592\) −3.59470 −0.147741
\(593\) 44.3233 1.82014 0.910069 0.414456i \(-0.136028\pi\)
0.910069 + 0.414456i \(0.136028\pi\)
\(594\) −4.15658 −0.170546
\(595\) 0 0
\(596\) −15.6316 −0.640294
\(597\) 2.89098 0.118320
\(598\) −14.2919 −0.584438
\(599\) 16.2752 0.664986 0.332493 0.943106i \(-0.392110\pi\)
0.332493 + 0.943106i \(0.392110\pi\)
\(600\) 0 0
\(601\) 22.1970 0.905433 0.452716 0.891655i \(-0.350455\pi\)
0.452716 + 0.891655i \(0.350455\pi\)
\(602\) −7.45149 −0.303700
\(603\) −7.81631 −0.318305
\(604\) −37.2197 −1.51445
\(605\) 0 0
\(606\) −0.909771 −0.0369569
\(607\) −44.7910 −1.81801 −0.909005 0.416785i \(-0.863157\pi\)
−0.909005 + 0.416785i \(0.863157\pi\)
\(608\) −33.8408 −1.37242
\(609\) −7.42617 −0.300924
\(610\) 0 0
\(611\) 13.6416 0.551879
\(612\) −5.25581 −0.212453
\(613\) 43.4227 1.75383 0.876914 0.480647i \(-0.159598\pi\)
0.876914 + 0.480647i \(0.159598\pi\)
\(614\) −54.1554 −2.18553
\(615\) 0 0
\(616\) −2.09919 −0.0845789
\(617\) −12.2954 −0.494994 −0.247497 0.968889i \(-0.579608\pi\)
−0.247497 + 0.968889i \(0.579608\pi\)
\(618\) 0.237923 0.00957066
\(619\) 39.3457 1.58144 0.790719 0.612179i \(-0.209707\pi\)
0.790719 + 0.612179i \(0.209707\pi\)
\(620\) 0 0
\(621\) 5.00666 0.200910
\(622\) 24.7335 0.991723
\(623\) −42.1059 −1.68694
\(624\) 1.48905 0.0596099
\(625\) 0 0
\(626\) 19.5107 0.779804
\(627\) 1.54551 0.0617219
\(628\) 27.6219 1.10223
\(629\) 0.807559 0.0321995
\(630\) 0 0
\(631\) −27.4877 −1.09427 −0.547134 0.837045i \(-0.684281\pi\)
−0.547134 + 0.837045i \(0.684281\pi\)
\(632\) 5.09263 0.202574
\(633\) 2.04670 0.0813490
\(634\) −54.2267 −2.15362
\(635\) 0 0
\(636\) 0.679228 0.0269331
\(637\) −12.9346 −0.512487
\(638\) −34.0800 −1.34924
\(639\) 30.9150 1.22298
\(640\) 0 0
\(641\) 28.9559 1.14369 0.571845 0.820362i \(-0.306228\pi\)
0.571845 + 0.820362i \(0.306228\pi\)
\(642\) 0.0422235 0.00166643
\(643\) 20.9097 0.824597 0.412299 0.911049i \(-0.364726\pi\)
0.412299 + 0.911049i \(0.364726\pi\)
\(644\) −29.4182 −1.15924
\(645\) 0 0
\(646\) 8.20793 0.322937
\(647\) 37.6568 1.48044 0.740221 0.672363i \(-0.234721\pi\)
0.740221 + 0.672363i \(0.234721\pi\)
\(648\) 2.68125 0.105329
\(649\) 14.8660 0.583542
\(650\) 0 0
\(651\) 4.26461 0.167143
\(652\) −15.6219 −0.611802
\(653\) 15.3216 0.599579 0.299790 0.954005i \(-0.403084\pi\)
0.299790 + 0.954005i \(0.403084\pi\)
\(654\) 3.76329 0.147156
\(655\) 0 0
\(656\) −30.6414 −1.19635
\(657\) −12.4726 −0.486601
\(658\) 58.5727 2.28340
\(659\) 21.7590 0.847612 0.423806 0.905753i \(-0.360694\pi\)
0.423806 + 0.905753i \(0.360694\pi\)
\(660\) 0 0
\(661\) −36.7921 −1.43105 −0.715524 0.698589i \(-0.753812\pi\)
−0.715524 + 0.698589i \(0.753812\pi\)
\(662\) −21.6692 −0.842197
\(663\) −0.334520 −0.0129917
\(664\) −5.32577 −0.206680
\(665\) 0 0
\(666\) 4.85969 0.188309
\(667\) 41.0500 1.58946
\(668\) −6.03317 −0.233430
\(669\) 5.75366 0.222449
\(670\) 0 0
\(671\) 19.7563 0.762683
\(672\) 5.92188 0.228442
\(673\) −38.7475 −1.49361 −0.746804 0.665044i \(-0.768413\pi\)
−0.746804 + 0.665044i \(0.768413\pi\)
\(674\) 22.8838 0.881451
\(675\) 0 0
\(676\) −18.3953 −0.707513
\(677\) 44.4839 1.70966 0.854828 0.518912i \(-0.173663\pi\)
0.854828 + 0.518912i \(0.173663\pi\)
\(678\) −2.08501 −0.0800743
\(679\) 3.63178 0.139375
\(680\) 0 0
\(681\) 0.849394 0.0325488
\(682\) 19.5711 0.749415
\(683\) −40.9070 −1.56526 −0.782631 0.622485i \(-0.786123\pi\)
−0.782631 + 0.622485i \(0.786123\pi\)
\(684\) 23.6790 0.905388
\(685\) 0 0
\(686\) −3.37671 −0.128923
\(687\) 5.84817 0.223122
\(688\) −4.29153 −0.163613
\(689\) −3.20127 −0.121959
\(690\) 0 0
\(691\) 24.0512 0.914949 0.457475 0.889223i \(-0.348754\pi\)
0.457475 + 0.889223i \(0.348754\pi\)
\(692\) 34.4827 1.31083
\(693\) 20.0271 0.760765
\(694\) −46.1696 −1.75257
\(695\) 0 0
\(696\) −0.606059 −0.0229726
\(697\) 6.88367 0.260738
\(698\) −39.0964 −1.47982
\(699\) 3.15465 0.119320
\(700\) 0 0
\(701\) 42.7283 1.61383 0.806914 0.590670i \(-0.201136\pi\)
0.806914 + 0.590670i \(0.201136\pi\)
\(702\) −4.05329 −0.152982
\(703\) −3.63830 −0.137221
\(704\) 11.9015 0.448556
\(705\) 0 0
\(706\) −59.5078 −2.23960
\(707\) 8.82604 0.331937
\(708\) −3.07581 −0.115596
\(709\) 4.99747 0.187684 0.0938420 0.995587i \(-0.470085\pi\)
0.0938420 + 0.995587i \(0.470085\pi\)
\(710\) 0 0
\(711\) −48.5855 −1.82210
\(712\) −3.43631 −0.128781
\(713\) −23.5736 −0.882840
\(714\) −1.43633 −0.0537531
\(715\) 0 0
\(716\) −16.4011 −0.612939
\(717\) 2.06324 0.0770530
\(718\) −28.6633 −1.06971
\(719\) −28.1008 −1.04798 −0.523991 0.851724i \(-0.675558\pi\)
−0.523991 + 0.851724i \(0.675558\pi\)
\(720\) 0 0
\(721\) −2.30818 −0.0859612
\(722\) 0.261353 0.00972654
\(723\) 0.575591 0.0214064
\(724\) −12.1796 −0.452651
\(725\) 0 0
\(726\) 3.06946 0.113919
\(727\) −32.7169 −1.21340 −0.606701 0.794930i \(-0.707508\pi\)
−0.606701 + 0.794930i \(0.707508\pi\)
\(728\) −2.04703 −0.0758681
\(729\) −24.8653 −0.920939
\(730\) 0 0
\(731\) 0.964104 0.0356587
\(732\) −4.08762 −0.151083
\(733\) 7.67000 0.283298 0.141649 0.989917i \(-0.454760\pi\)
0.141649 + 0.989917i \(0.454760\pi\)
\(734\) −7.44216 −0.274695
\(735\) 0 0
\(736\) −32.7346 −1.20661
\(737\) −4.69944 −0.173106
\(738\) 41.4242 1.52484
\(739\) −3.07210 −0.113009 −0.0565045 0.998402i \(-0.517996\pi\)
−0.0565045 + 0.998402i \(0.517996\pi\)
\(740\) 0 0
\(741\) 1.50711 0.0553651
\(742\) −13.7453 −0.504605
\(743\) −1.63867 −0.0601169 −0.0300585 0.999548i \(-0.509569\pi\)
−0.0300585 + 0.999548i \(0.509569\pi\)
\(744\) 0.348040 0.0127598
\(745\) 0 0
\(746\) 25.7844 0.944034
\(747\) 50.8097 1.85903
\(748\) −3.15998 −0.115540
\(749\) −0.409627 −0.0149674
\(750\) 0 0
\(751\) −1.29060 −0.0470948 −0.0235474 0.999723i \(-0.507496\pi\)
−0.0235474 + 0.999723i \(0.507496\pi\)
\(752\) 33.7338 1.23014
\(753\) −2.29616 −0.0836767
\(754\) −33.2332 −1.21028
\(755\) 0 0
\(756\) −8.34324 −0.303441
\(757\) −25.9057 −0.941559 −0.470780 0.882251i \(-0.656027\pi\)
−0.470780 + 0.882251i \(0.656027\pi\)
\(758\) 51.4490 1.86871
\(759\) 1.49500 0.0542649
\(760\) 0 0
\(761\) 23.4619 0.850495 0.425247 0.905077i \(-0.360187\pi\)
0.425247 + 0.905077i \(0.360187\pi\)
\(762\) 2.62897 0.0952376
\(763\) −36.5092 −1.32172
\(764\) 45.4622 1.64477
\(765\) 0 0
\(766\) −49.8987 −1.80291
\(767\) 14.4966 0.523443
\(768\) 3.64373 0.131482
\(769\) 42.1622 1.52041 0.760205 0.649684i \(-0.225099\pi\)
0.760205 + 0.649684i \(0.225099\pi\)
\(770\) 0 0
\(771\) −1.91474 −0.0689578
\(772\) −21.7508 −0.782829
\(773\) 13.0799 0.470452 0.235226 0.971941i \(-0.424417\pi\)
0.235226 + 0.971941i \(0.424417\pi\)
\(774\) 5.80173 0.208539
\(775\) 0 0
\(776\) 0.296394 0.0106399
\(777\) 0.636674 0.0228406
\(778\) 13.1359 0.470944
\(779\) −31.0130 −1.11116
\(780\) 0 0
\(781\) 18.5872 0.665101
\(782\) 7.93963 0.283921
\(783\) 11.6421 0.416055
\(784\) −31.9855 −1.14234
\(785\) 0 0
\(786\) 4.18568 0.149298
\(787\) 18.6616 0.665215 0.332607 0.943065i \(-0.392072\pi\)
0.332607 + 0.943065i \(0.392072\pi\)
\(788\) −27.6328 −0.984378
\(789\) −4.16888 −0.148416
\(790\) 0 0
\(791\) 20.2275 0.719207
\(792\) 1.63443 0.0580770
\(793\) 19.2654 0.684134
\(794\) 38.2897 1.35885
\(795\) 0 0
\(796\) 26.6305 0.943893
\(797\) 17.6220 0.624203 0.312102 0.950049i \(-0.398967\pi\)
0.312102 + 0.950049i \(0.398967\pi\)
\(798\) 6.47107 0.229073
\(799\) −7.57837 −0.268104
\(800\) 0 0
\(801\) 32.7836 1.15835
\(802\) 71.3765 2.52039
\(803\) −7.49895 −0.264632
\(804\) 0.972326 0.0342913
\(805\) 0 0
\(806\) 19.0848 0.672232
\(807\) 0.417928 0.0147118
\(808\) 0.720303 0.0253402
\(809\) 3.90127 0.137161 0.0685807 0.997646i \(-0.478153\pi\)
0.0685807 + 0.997646i \(0.478153\pi\)
\(810\) 0 0
\(811\) 21.7960 0.765361 0.382681 0.923881i \(-0.375001\pi\)
0.382681 + 0.923881i \(0.375001\pi\)
\(812\) −68.4068 −2.40061
\(813\) −1.62710 −0.0570651
\(814\) 2.92181 0.102410
\(815\) 0 0
\(816\) −0.827222 −0.0289586
\(817\) −4.34358 −0.151963
\(818\) 7.08152 0.247600
\(819\) 19.5294 0.682413
\(820\) 0 0
\(821\) −42.5054 −1.48345 −0.741724 0.670705i \(-0.765991\pi\)
−0.741724 + 0.670705i \(0.765991\pi\)
\(822\) 0.642809 0.0224205
\(823\) −27.8169 −0.969637 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(824\) −0.188374 −0.00656230
\(825\) 0 0
\(826\) 62.2441 2.16575
\(827\) −34.9291 −1.21460 −0.607302 0.794471i \(-0.707748\pi\)
−0.607302 + 0.794471i \(0.707748\pi\)
\(828\) 22.9050 0.796003
\(829\) −31.8932 −1.10769 −0.553847 0.832618i \(-0.686841\pi\)
−0.553847 + 0.832618i \(0.686841\pi\)
\(830\) 0 0
\(831\) −2.42142 −0.0839982
\(832\) 11.6058 0.402359
\(833\) 7.18562 0.248967
\(834\) 1.96701 0.0681119
\(835\) 0 0
\(836\) 14.2366 0.492384
\(837\) −6.68569 −0.231091
\(838\) −15.7829 −0.545212
\(839\) −34.5556 −1.19299 −0.596497 0.802616i \(-0.703441\pi\)
−0.596497 + 0.802616i \(0.703441\pi\)
\(840\) 0 0
\(841\) 66.4545 2.29153
\(842\) 72.8050 2.50902
\(843\) 2.63865 0.0908800
\(844\) 18.8534 0.648959
\(845\) 0 0
\(846\) −45.6047 −1.56792
\(847\) −29.7781 −1.02319
\(848\) −7.91631 −0.271847
\(849\) −1.01807 −0.0349401
\(850\) 0 0
\(851\) −3.51937 −0.120642
\(852\) −3.84573 −0.131752
\(853\) −49.1182 −1.68178 −0.840888 0.541209i \(-0.817967\pi\)
−0.840888 + 0.541209i \(0.817967\pi\)
\(854\) 82.7197 2.83061
\(855\) 0 0
\(856\) −0.0334301 −0.00114262
\(857\) 32.0392 1.09444 0.547220 0.836989i \(-0.315686\pi\)
0.547220 + 0.836989i \(0.315686\pi\)
\(858\) −1.21032 −0.0413196
\(859\) 33.2548 1.13464 0.567320 0.823498i \(-0.307980\pi\)
0.567320 + 0.823498i \(0.307980\pi\)
\(860\) 0 0
\(861\) 5.42704 0.184953
\(862\) 20.2320 0.689106
\(863\) 37.0582 1.26147 0.630737 0.775996i \(-0.282753\pi\)
0.630737 + 0.775996i \(0.282753\pi\)
\(864\) −9.28381 −0.315842
\(865\) 0 0
\(866\) −34.2205 −1.16286
\(867\) −3.21303 −0.109120
\(868\) 39.2838 1.33338
\(869\) −29.2113 −0.990925
\(870\) 0 0
\(871\) −4.58267 −0.155278
\(872\) −2.97956 −0.100900
\(873\) −2.82770 −0.0957033
\(874\) −35.7704 −1.20995
\(875\) 0 0
\(876\) 1.55155 0.0524220
\(877\) −25.6636 −0.866597 −0.433298 0.901250i \(-0.642650\pi\)
−0.433298 + 0.901250i \(0.642650\pi\)
\(878\) 15.3926 0.519475
\(879\) −2.22088 −0.0749085
\(880\) 0 0
\(881\) 2.09265 0.0705031 0.0352515 0.999378i \(-0.488777\pi\)
0.0352515 + 0.999378i \(0.488777\pi\)
\(882\) 43.2412 1.45601
\(883\) −55.5019 −1.86779 −0.933894 0.357551i \(-0.883612\pi\)
−0.933894 + 0.357551i \(0.883612\pi\)
\(884\) −3.08146 −0.103641
\(885\) 0 0
\(886\) −12.4706 −0.418957
\(887\) 4.21678 0.141586 0.0707928 0.997491i \(-0.477447\pi\)
0.0707928 + 0.997491i \(0.477447\pi\)
\(888\) 0.0519597 0.00174365
\(889\) −25.5047 −0.855400
\(890\) 0 0
\(891\) −15.3796 −0.515237
\(892\) 53.0003 1.77458
\(893\) 34.1428 1.14255
\(894\) 3.32606 0.111240
\(895\) 0 0
\(896\) −9.40676 −0.314258
\(897\) 1.45785 0.0486761
\(898\) −28.7489 −0.959364
\(899\) −54.8164 −1.82823
\(900\) 0 0
\(901\) 1.77842 0.0592477
\(902\) 24.9057 0.829268
\(903\) 0.760093 0.0252943
\(904\) 1.65079 0.0549044
\(905\) 0 0
\(906\) 7.91956 0.263110
\(907\) 35.4222 1.17617 0.588087 0.808797i \(-0.299881\pi\)
0.588087 + 0.808797i \(0.299881\pi\)
\(908\) 7.82426 0.259657
\(909\) −6.87195 −0.227928
\(910\) 0 0
\(911\) 10.6877 0.354100 0.177050 0.984202i \(-0.443345\pi\)
0.177050 + 0.984202i \(0.443345\pi\)
\(912\) 3.72688 0.123409
\(913\) 30.5486 1.01101
\(914\) 74.0251 2.44853
\(915\) 0 0
\(916\) 53.8709 1.77995
\(917\) −40.6069 −1.34096
\(918\) 2.25175 0.0743187
\(919\) −1.25334 −0.0413437 −0.0206719 0.999786i \(-0.506581\pi\)
−0.0206719 + 0.999786i \(0.506581\pi\)
\(920\) 0 0
\(921\) 5.52415 0.182027
\(922\) 61.1924 2.01526
\(923\) 18.1253 0.596602
\(924\) −2.49130 −0.0819579
\(925\) 0 0
\(926\) −67.7439 −2.22620
\(927\) 1.79715 0.0590262
\(928\) −76.1186 −2.49872
\(929\) −8.43820 −0.276848 −0.138424 0.990373i \(-0.544204\pi\)
−0.138424 + 0.990373i \(0.544204\pi\)
\(930\) 0 0
\(931\) −32.3734 −1.06099
\(932\) 29.0594 0.951871
\(933\) −2.52295 −0.0825978
\(934\) 68.3020 2.23491
\(935\) 0 0
\(936\) 1.59382 0.0520956
\(937\) −41.1596 −1.34463 −0.672313 0.740267i \(-0.734699\pi\)
−0.672313 + 0.740267i \(0.734699\pi\)
\(938\) −19.6766 −0.642463
\(939\) −1.99020 −0.0649476
\(940\) 0 0
\(941\) 39.7706 1.29649 0.648243 0.761434i \(-0.275504\pi\)
0.648243 + 0.761434i \(0.275504\pi\)
\(942\) −5.87733 −0.191494
\(943\) −29.9993 −0.976910
\(944\) 35.8482 1.16676
\(945\) 0 0
\(946\) 3.48820 0.113411
\(947\) −5.84267 −0.189861 −0.0949306 0.995484i \(-0.530263\pi\)
−0.0949306 + 0.995484i \(0.530263\pi\)
\(948\) 6.04389 0.196296
\(949\) −7.31262 −0.237378
\(950\) 0 0
\(951\) 5.53142 0.179368
\(952\) 1.13720 0.0368568
\(953\) 28.3236 0.917492 0.458746 0.888567i \(-0.348299\pi\)
0.458746 + 0.888567i \(0.348299\pi\)
\(954\) 10.7021 0.346492
\(955\) 0 0
\(956\) 19.0057 0.614688
\(957\) 3.47635 0.112374
\(958\) −53.2305 −1.71980
\(959\) −6.23614 −0.201376
\(960\) 0 0
\(961\) 0.479267 0.0154602
\(962\) 2.84921 0.0918623
\(963\) 0.318935 0.0102775
\(964\) 5.30210 0.170769
\(965\) 0 0
\(966\) 6.25955 0.201398
\(967\) −8.21946 −0.264320 −0.132160 0.991228i \(-0.542191\pi\)
−0.132160 + 0.991228i \(0.542191\pi\)
\(968\) −2.43022 −0.0781103
\(969\) −0.837253 −0.0268965
\(970\) 0 0
\(971\) −33.9899 −1.09079 −0.545394 0.838180i \(-0.683620\pi\)
−0.545394 + 0.838180i \(0.683620\pi\)
\(972\) 9.76586 0.313240
\(973\) −19.0827 −0.611763
\(974\) −59.3686 −1.90229
\(975\) 0 0
\(976\) 47.6407 1.52494
\(977\) 27.6406 0.884302 0.442151 0.896941i \(-0.354215\pi\)
0.442151 + 0.896941i \(0.354215\pi\)
\(978\) 3.32401 0.106290
\(979\) 19.7107 0.629956
\(980\) 0 0
\(981\) 28.4260 0.907573
\(982\) −19.4737 −0.621432
\(983\) 33.7780 1.07735 0.538675 0.842513i \(-0.318925\pi\)
0.538675 + 0.842513i \(0.318925\pi\)
\(984\) 0.442907 0.0141194
\(985\) 0 0
\(986\) 18.4622 0.587957
\(987\) −5.97474 −0.190178
\(988\) 13.8829 0.441674
\(989\) −4.20160 −0.133603
\(990\) 0 0
\(991\) 32.6864 1.03832 0.519160 0.854677i \(-0.326245\pi\)
0.519160 + 0.854677i \(0.326245\pi\)
\(992\) 43.7124 1.38787
\(993\) 2.21038 0.0701441
\(994\) 77.8246 2.46844
\(995\) 0 0
\(996\) −6.32057 −0.200275
\(997\) 24.8137 0.785859 0.392930 0.919569i \(-0.371462\pi\)
0.392930 + 0.919569i \(0.371462\pi\)
\(998\) −54.8929 −1.73760
\(999\) −0.998123 −0.0315792
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 1075.2.a.o.1.1 yes 5
3.2 odd 2 9675.2.a.cb.1.5 5
5.2 odd 4 1075.2.b.i.474.2 10
5.3 odd 4 1075.2.b.i.474.9 10
5.4 even 2 1075.2.a.n.1.5 5
15.14 odd 2 9675.2.a.cc.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1075.2.a.n.1.5 5 5.4 even 2
1075.2.a.o.1.1 yes 5 1.1 even 1 trivial
1075.2.b.i.474.2 10 5.2 odd 4
1075.2.b.i.474.9 10 5.3 odd 4
9675.2.a.cb.1.5 5 3.2 odd 2
9675.2.a.cc.1.1 5 15.14 odd 2