Properties

Label 3525.2.a.u.1.2
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $0$
Dimension $4$
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] = [3525,2,Mod(1,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, 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("3525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.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: \(28.1472667125\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.14656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 705)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.385537\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.38554 q^{2} +1.00000 q^{3} -0.0802864 q^{4} -1.38554 q^{6} -4.18757 q^{7} +2.88231 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.38554 q^{2} +1.00000 q^{3} -0.0802864 q^{4} -1.38554 q^{6} -4.18757 q^{7} +2.88231 q^{8} +1.00000 q^{9} +0.919714 q^{11} -0.0802864 q^{12} +0.614463 q^{13} +5.80203 q^{14} -3.83298 q^{16} -6.98959 q^{17} -1.38554 q^{18} +1.12962 q^{19} -4.18757 q^{21} -1.27430 q^{22} -8.34814 q^{23} +2.88231 q^{24} -0.851361 q^{26} +1.00000 q^{27} +0.336204 q^{28} +5.00000 q^{29} -3.59544 q^{31} -0.453890 q^{32} +0.919714 q^{33} +9.68434 q^{34} -0.0802864 q^{36} +3.75270 q^{37} -1.56513 q^{38} +0.614463 q^{39} +6.08029 q^{41} +5.80203 q^{42} +8.21456 q^{43} -0.0738405 q^{44} +11.5667 q^{46} -1.00000 q^{47} -3.83298 q^{48} +10.5357 q^{49} -6.98959 q^{51} -0.0493330 q^{52} +1.49282 q^{53} -1.38554 q^{54} -12.0699 q^{56} +1.12962 q^{57} -6.92769 q^{58} -2.57310 q^{59} -5.86974 q^{61} +4.98162 q^{62} -4.18757 q^{63} +8.29484 q^{64} -1.27430 q^{66} +0.824368 q^{67} +0.561169 q^{68} -8.34814 q^{69} +12.1232 q^{71} +2.88231 q^{72} -11.7732 q^{73} -5.19950 q^{74} -0.0906930 q^{76} -3.85136 q^{77} -0.851361 q^{78} -4.06592 q^{79} +1.00000 q^{81} -8.42446 q^{82} +14.4738 q^{83} +0.336204 q^{84} -11.3816 q^{86} +5.00000 q^{87} +2.65090 q^{88} -16.7049 q^{89} -2.57310 q^{91} +0.670242 q^{92} -3.59544 q^{93} +1.38554 q^{94} -0.453890 q^{96} +1.93809 q^{97} -14.5976 q^{98} +0.919714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{3} + 4 q^{4} - 2 q^{6} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{3} + 4 q^{4} - 2 q^{6} - 6 q^{8} + 4 q^{9} + 8 q^{11} + 4 q^{12} + 6 q^{13} + 10 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{18} + 2 q^{19} - 12 q^{22} - 8 q^{23} - 6 q^{24} + 8 q^{26} + 4 q^{27} - 4 q^{28} + 20 q^{29} - 4 q^{31} - 14 q^{32} + 8 q^{33} + 8 q^{34} + 4 q^{36} - 8 q^{38} + 6 q^{39} + 20 q^{41} + 10 q^{42} + 8 q^{43} + 32 q^{44} - 2 q^{46} - 4 q^{47} + 4 q^{48} + 2 q^{51} - 2 q^{52} - 10 q^{53} - 2 q^{54} - 14 q^{56} + 2 q^{57} - 10 q^{58} + 10 q^{59} - 20 q^{61} + 12 q^{62} + 4 q^{64} - 12 q^{66} + 2 q^{68} - 8 q^{69} + 18 q^{71} - 6 q^{72} + 4 q^{73} + 16 q^{74} - 26 q^{76} - 4 q^{77} + 8 q^{78} + 20 q^{79} + 4 q^{81} - 2 q^{82} + 28 q^{83} - 4 q^{84} - 40 q^{86} + 20 q^{87} - 40 q^{88} + 10 q^{91} + 36 q^{92} - 4 q^{93} + 2 q^{94} - 14 q^{96} + 20 q^{97} - 4 q^{98} + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38554 −0.979723 −0.489861 0.871800i \(-0.662953\pi\)
−0.489861 + 0.871800i \(0.662953\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.0802864 −0.0401432
\(5\) 0 0
\(6\) −1.38554 −0.565643
\(7\) −4.18757 −1.58275 −0.791375 0.611330i \(-0.790635\pi\)
−0.791375 + 0.611330i \(0.790635\pi\)
\(8\) 2.88231 1.01905
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.919714 0.277304 0.138652 0.990341i \(-0.455723\pi\)
0.138652 + 0.990341i \(0.455723\pi\)
\(12\) −0.0802864 −0.0231767
\(13\) 0.614463 0.170421 0.0852106 0.996363i \(-0.472844\pi\)
0.0852106 + 0.996363i \(0.472844\pi\)
\(14\) 5.80203 1.55066
\(15\) 0 0
\(16\) −3.83298 −0.958245
\(17\) −6.98959 −1.69523 −0.847613 0.530615i \(-0.821961\pi\)
−0.847613 + 0.530615i \(0.821961\pi\)
\(18\) −1.38554 −0.326574
\(19\) 1.12962 0.259152 0.129576 0.991569i \(-0.458638\pi\)
0.129576 + 0.991569i \(0.458638\pi\)
\(20\) 0 0
\(21\) −4.18757 −0.913802
\(22\) −1.27430 −0.271681
\(23\) −8.34814 −1.74071 −0.870354 0.492427i \(-0.836110\pi\)
−0.870354 + 0.492427i \(0.836110\pi\)
\(24\) 2.88231 0.588350
\(25\) 0 0
\(26\) −0.851361 −0.166966
\(27\) 1.00000 0.192450
\(28\) 0.336204 0.0635366
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −3.59544 −0.645761 −0.322880 0.946440i \(-0.604651\pi\)
−0.322880 + 0.946440i \(0.604651\pi\)
\(32\) −0.453890 −0.0802372
\(33\) 0.919714 0.160102
\(34\) 9.68434 1.66085
\(35\) 0 0
\(36\) −0.0802864 −0.0133811
\(37\) 3.75270 0.616939 0.308470 0.951234i \(-0.400183\pi\)
0.308470 + 0.951234i \(0.400183\pi\)
\(38\) −1.56513 −0.253898
\(39\) 0.614463 0.0983928
\(40\) 0 0
\(41\) 6.08029 0.949581 0.474791 0.880099i \(-0.342524\pi\)
0.474791 + 0.880099i \(0.342524\pi\)
\(42\) 5.80203 0.895272
\(43\) 8.21456 1.25271 0.626354 0.779539i \(-0.284546\pi\)
0.626354 + 0.779539i \(0.284546\pi\)
\(44\) −0.0738405 −0.0111319
\(45\) 0 0
\(46\) 11.5667 1.70541
\(47\) −1.00000 −0.145865
\(48\) −3.83298 −0.553243
\(49\) 10.5357 1.50510
\(50\) 0 0
\(51\) −6.98959 −0.978739
\(52\) −0.0493330 −0.00684125
\(53\) 1.49282 0.205054 0.102527 0.994730i \(-0.467307\pi\)
0.102527 + 0.994730i \(0.467307\pi\)
\(54\) −1.38554 −0.188548
\(55\) 0 0
\(56\) −12.0699 −1.61291
\(57\) 1.12962 0.149622
\(58\) −6.92769 −0.909650
\(59\) −2.57310 −0.334989 −0.167495 0.985873i \(-0.553568\pi\)
−0.167495 + 0.985873i \(0.553568\pi\)
\(60\) 0 0
\(61\) −5.86974 −0.751543 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(62\) 4.98162 0.632666
\(63\) −4.18757 −0.527584
\(64\) 8.29484 1.03686
\(65\) 0 0
\(66\) −1.27430 −0.156855
\(67\) 0.824368 0.100713 0.0503563 0.998731i \(-0.483964\pi\)
0.0503563 + 0.998731i \(0.483964\pi\)
\(68\) 0.561169 0.0680517
\(69\) −8.34814 −1.00500
\(70\) 0 0
\(71\) 12.1232 1.43876 0.719378 0.694619i \(-0.244427\pi\)
0.719378 + 0.694619i \(0.244427\pi\)
\(72\) 2.88231 0.339684
\(73\) −11.7732 −1.37795 −0.688977 0.724783i \(-0.741940\pi\)
−0.688977 + 0.724783i \(0.741940\pi\)
\(74\) −5.19950 −0.604429
\(75\) 0 0
\(76\) −0.0906930 −0.0104032
\(77\) −3.85136 −0.438903
\(78\) −0.851361 −0.0963976
\(79\) −4.06592 −0.457452 −0.228726 0.973491i \(-0.573456\pi\)
−0.228726 + 0.973491i \(0.573456\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −8.42446 −0.930326
\(83\) 14.4738 1.58871 0.794353 0.607457i \(-0.207810\pi\)
0.794353 + 0.607457i \(0.207810\pi\)
\(84\) 0.336204 0.0366829
\(85\) 0 0
\(86\) −11.3816 −1.22731
\(87\) 5.00000 0.536056
\(88\) 2.65090 0.282587
\(89\) −16.7049 −1.77071 −0.885357 0.464911i \(-0.846086\pi\)
−0.885357 + 0.464911i \(0.846086\pi\)
\(90\) 0 0
\(91\) −2.57310 −0.269734
\(92\) 0.670242 0.0698775
\(93\) −3.59544 −0.372830
\(94\) 1.38554 0.142907
\(95\) 0 0
\(96\) −0.453890 −0.0463250
\(97\) 1.93809 0.196784 0.0983918 0.995148i \(-0.468630\pi\)
0.0983918 + 0.995148i \(0.468630\pi\)
\(98\) −14.5976 −1.47458
\(99\) 0.919714 0.0924347
\(100\) 0 0
\(101\) −4.94026 −0.491574 −0.245787 0.969324i \(-0.579046\pi\)
−0.245787 + 0.969324i \(0.579046\pi\)
\(102\) 9.68434 0.958893
\(103\) 8.87191 0.874175 0.437088 0.899419i \(-0.356010\pi\)
0.437088 + 0.899419i \(0.356010\pi\)
\(104\) 1.77107 0.173668
\(105\) 0 0
\(106\) −2.06835 −0.200896
\(107\) −8.21456 −0.794131 −0.397066 0.917790i \(-0.629971\pi\)
−0.397066 + 0.917790i \(0.629971\pi\)
\(108\) −0.0802864 −0.00772556
\(109\) −10.1692 −0.974031 −0.487016 0.873393i \(-0.661915\pi\)
−0.487016 + 0.873393i \(0.661915\pi\)
\(110\) 0 0
\(111\) 3.75270 0.356190
\(112\) 16.0509 1.51666
\(113\) 6.01193 0.565555 0.282778 0.959186i \(-0.408744\pi\)
0.282778 + 0.959186i \(0.408744\pi\)
\(114\) −1.56513 −0.146588
\(115\) 0 0
\(116\) −0.401432 −0.0372720
\(117\) 0.614463 0.0568071
\(118\) 3.56513 0.328197
\(119\) 29.2694 2.68312
\(120\) 0 0
\(121\) −10.1541 −0.923102
\(122\) 8.13274 0.736304
\(123\) 6.08029 0.548241
\(124\) 0.288665 0.0259229
\(125\) 0 0
\(126\) 5.80203 0.516886
\(127\) 8.69628 0.771670 0.385835 0.922568i \(-0.373913\pi\)
0.385835 + 0.922568i \(0.373913\pi\)
\(128\) −10.5850 −0.935594
\(129\) 8.21456 0.723251
\(130\) 0 0
\(131\) 12.1073 1.05782 0.528909 0.848679i \(-0.322601\pi\)
0.528909 + 0.848679i \(0.322601\pi\)
\(132\) −0.0738405 −0.00642699
\(133\) −4.73035 −0.410174
\(134\) −1.14219 −0.0986705
\(135\) 0 0
\(136\) −20.1462 −1.72752
\(137\) 17.9792 1.53607 0.768033 0.640411i \(-0.221236\pi\)
0.768033 + 0.640411i \(0.221236\pi\)
\(138\) 11.5667 0.984619
\(139\) 9.65671 0.819071 0.409536 0.912294i \(-0.365691\pi\)
0.409536 + 0.912294i \(0.365691\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −16.7971 −1.40958
\(143\) 0.565130 0.0472585
\(144\) −3.83298 −0.319415
\(145\) 0 0
\(146\) 16.3123 1.35001
\(147\) 10.5357 0.868970
\(148\) −0.301290 −0.0247659
\(149\) 20.5529 1.68376 0.841881 0.539664i \(-0.181449\pi\)
0.841881 + 0.539664i \(0.181449\pi\)
\(150\) 0 0
\(151\) 9.64146 0.784611 0.392305 0.919835i \(-0.371678\pi\)
0.392305 + 0.919835i \(0.371678\pi\)
\(152\) 3.25592 0.264090
\(153\) −6.98959 −0.565075
\(154\) 5.33620 0.430004
\(155\) 0 0
\(156\) −0.0493330 −0.00394980
\(157\) 12.6724 1.01137 0.505684 0.862719i \(-0.331240\pi\)
0.505684 + 0.862719i \(0.331240\pi\)
\(158\) 5.63348 0.448176
\(159\) 1.49282 0.118388
\(160\) 0 0
\(161\) 34.9584 2.75511
\(162\) −1.38554 −0.108858
\(163\) −6.16057 −0.482533 −0.241267 0.970459i \(-0.577563\pi\)
−0.241267 + 0.970459i \(0.577563\pi\)
\(164\) −0.488164 −0.0381192
\(165\) 0 0
\(166\) −20.0540 −1.55649
\(167\) 3.84159 0.297272 0.148636 0.988892i \(-0.452512\pi\)
0.148636 + 0.988892i \(0.452512\pi\)
\(168\) −12.0699 −0.931211
\(169\) −12.6224 −0.970957
\(170\) 0 0
\(171\) 1.12962 0.0863841
\(172\) −0.659517 −0.0502877
\(173\) 14.2964 1.08693 0.543466 0.839431i \(-0.317111\pi\)
0.543466 + 0.839431i \(0.317111\pi\)
\(174\) −6.92769 −0.525187
\(175\) 0 0
\(176\) −3.52525 −0.265725
\(177\) −2.57310 −0.193406
\(178\) 23.1452 1.73481
\(179\) 15.8633 1.18568 0.592839 0.805321i \(-0.298007\pi\)
0.592839 + 0.805321i \(0.298007\pi\)
\(180\) 0 0
\(181\) −0.639928 −0.0475655 −0.0237827 0.999717i \(-0.507571\pi\)
−0.0237827 + 0.999717i \(0.507571\pi\)
\(182\) 3.56513 0.264265
\(183\) −5.86974 −0.433904
\(184\) −24.0620 −1.77387
\(185\) 0 0
\(186\) 4.98162 0.365270
\(187\) −6.42842 −0.470093
\(188\) 0.0802864 0.00585548
\(189\) −4.18757 −0.304601
\(190\) 0 0
\(191\) −7.31539 −0.529323 −0.264662 0.964341i \(-0.585260\pi\)
−0.264662 + 0.964341i \(0.585260\pi\)
\(192\) 8.29484 0.598629
\(193\) −9.10083 −0.655092 −0.327546 0.944835i \(-0.606222\pi\)
−0.327546 + 0.944835i \(0.606222\pi\)
\(194\) −2.68530 −0.192793
\(195\) 0 0
\(196\) −0.845873 −0.0604195
\(197\) −18.3950 −1.31059 −0.655296 0.755372i \(-0.727456\pi\)
−0.655296 + 0.755372i \(0.727456\pi\)
\(198\) −1.27430 −0.0905604
\(199\) 17.9798 1.27456 0.637278 0.770634i \(-0.280060\pi\)
0.637278 + 0.770634i \(0.280060\pi\)
\(200\) 0 0
\(201\) 0.824368 0.0581465
\(202\) 6.84492 0.481607
\(203\) −20.9378 −1.46955
\(204\) 0.561169 0.0392897
\(205\) 0 0
\(206\) −12.2924 −0.856449
\(207\) −8.34814 −0.580236
\(208\) −2.35522 −0.163305
\(209\) 1.03893 0.0718640
\(210\) 0 0
\(211\) 13.2375 0.911310 0.455655 0.890157i \(-0.349405\pi\)
0.455655 + 0.890157i \(0.349405\pi\)
\(212\) −0.119853 −0.00823152
\(213\) 12.1232 0.830666
\(214\) 11.3816 0.778029
\(215\) 0 0
\(216\) 2.88231 0.196117
\(217\) 15.0562 1.02208
\(218\) 14.0898 0.954281
\(219\) −11.7732 −0.795562
\(220\) 0 0
\(221\) −4.29484 −0.288902
\(222\) −5.19950 −0.348968
\(223\) −7.05974 −0.472755 −0.236378 0.971661i \(-0.575960\pi\)
−0.236378 + 0.971661i \(0.575960\pi\)
\(224\) 1.90069 0.126995
\(225\) 0 0
\(226\) −8.32976 −0.554087
\(227\) 24.3720 1.61763 0.808813 0.588065i \(-0.200110\pi\)
0.808813 + 0.588065i \(0.200110\pi\)
\(228\) −0.0906930 −0.00600629
\(229\) 19.8640 1.31265 0.656325 0.754479i \(-0.272110\pi\)
0.656325 + 0.754479i \(0.272110\pi\)
\(230\) 0 0
\(231\) −3.85136 −0.253401
\(232\) 14.4116 0.946166
\(233\) 0.923726 0.0605153 0.0302576 0.999542i \(-0.490367\pi\)
0.0302576 + 0.999542i \(0.490367\pi\)
\(234\) −0.851361 −0.0556552
\(235\) 0 0
\(236\) 0.206585 0.0134475
\(237\) −4.06592 −0.264110
\(238\) −40.5538 −2.62871
\(239\) 12.1073 0.783155 0.391577 0.920145i \(-0.371930\pi\)
0.391577 + 0.920145i \(0.371930\pi\)
\(240\) 0 0
\(241\) −11.8870 −0.765707 −0.382853 0.923809i \(-0.625059\pi\)
−0.382853 + 0.923809i \(0.625059\pi\)
\(242\) 14.0689 0.904385
\(243\) 1.00000 0.0641500
\(244\) 0.471260 0.0301693
\(245\) 0 0
\(246\) −8.42446 −0.537124
\(247\) 0.694109 0.0441651
\(248\) −10.3632 −0.658064
\(249\) 14.4738 0.917240
\(250\) 0 0
\(251\) 22.0858 1.39404 0.697022 0.717049i \(-0.254508\pi\)
0.697022 + 0.717049i \(0.254508\pi\)
\(252\) 0.336204 0.0211789
\(253\) −7.67790 −0.482705
\(254\) −12.0490 −0.756022
\(255\) 0 0
\(256\) −1.92373 −0.120233
\(257\) 5.60004 0.349321 0.174661 0.984629i \(-0.444117\pi\)
0.174661 + 0.984629i \(0.444117\pi\)
\(258\) −11.3816 −0.708586
\(259\) −15.7147 −0.976461
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) −16.7751 −1.03637
\(263\) 13.7892 0.850278 0.425139 0.905128i \(-0.360225\pi\)
0.425139 + 0.905128i \(0.360225\pi\)
\(264\) 2.65090 0.163152
\(265\) 0 0
\(266\) 6.55408 0.401857
\(267\) −16.7049 −1.02232
\(268\) −0.0661855 −0.00404292
\(269\) −4.57374 −0.278866 −0.139433 0.990232i \(-0.544528\pi\)
−0.139433 + 0.990232i \(0.544528\pi\)
\(270\) 0 0
\(271\) −29.8057 −1.81057 −0.905284 0.424806i \(-0.860342\pi\)
−0.905284 + 0.424806i \(0.860342\pi\)
\(272\) 26.7910 1.62444
\(273\) −2.57310 −0.155731
\(274\) −24.9108 −1.50492
\(275\) 0 0
\(276\) 0.670242 0.0403438
\(277\) −19.4187 −1.16675 −0.583377 0.812201i \(-0.698269\pi\)
−0.583377 + 0.812201i \(0.698269\pi\)
\(278\) −13.3797 −0.802463
\(279\) −3.59544 −0.215254
\(280\) 0 0
\(281\) 17.0239 1.01556 0.507779 0.861487i \(-0.330467\pi\)
0.507779 + 0.861487i \(0.330467\pi\)
\(282\) 1.38554 0.0825075
\(283\) 11.0303 0.655684 0.327842 0.944733i \(-0.393679\pi\)
0.327842 + 0.944733i \(0.393679\pi\)
\(284\) −0.973325 −0.0577562
\(285\) 0 0
\(286\) −0.783008 −0.0463003
\(287\) −25.4616 −1.50295
\(288\) −0.453890 −0.0267457
\(289\) 31.8544 1.87379
\(290\) 0 0
\(291\) 1.93809 0.113613
\(292\) 0.945231 0.0553154
\(293\) 33.5084 1.95758 0.978792 0.204856i \(-0.0656726\pi\)
0.978792 + 0.204856i \(0.0656726\pi\)
\(294\) −14.5976 −0.851350
\(295\) 0 0
\(296\) 10.8164 0.628693
\(297\) 0.919714 0.0533672
\(298\) −28.4769 −1.64962
\(299\) −5.12962 −0.296654
\(300\) 0 0
\(301\) −34.3990 −1.98273
\(302\) −13.3586 −0.768701
\(303\) −4.94026 −0.283811
\(304\) −4.32981 −0.248332
\(305\) 0 0
\(306\) 9.68434 0.553617
\(307\) −9.98494 −0.569871 −0.284935 0.958547i \(-0.591972\pi\)
−0.284935 + 0.958547i \(0.591972\pi\)
\(308\) 0.309212 0.0176190
\(309\) 8.87191 0.504705
\(310\) 0 0
\(311\) 12.9816 0.736120 0.368060 0.929802i \(-0.380022\pi\)
0.368060 + 0.929802i \(0.380022\pi\)
\(312\) 1.77107 0.100267
\(313\) 21.9378 1.24000 0.619998 0.784603i \(-0.287133\pi\)
0.619998 + 0.784603i \(0.287133\pi\)
\(314\) −17.5581 −0.990861
\(315\) 0 0
\(316\) 0.326438 0.0183636
\(317\) −11.0856 −0.622628 −0.311314 0.950307i \(-0.600769\pi\)
−0.311314 + 0.950307i \(0.600769\pi\)
\(318\) −2.06835 −0.115987
\(319\) 4.59857 0.257470
\(320\) 0 0
\(321\) −8.21456 −0.458492
\(322\) −48.4361 −2.69924
\(323\) −7.89558 −0.439322
\(324\) −0.0802864 −0.00446035
\(325\) 0 0
\(326\) 8.53570 0.472749
\(327\) −10.1692 −0.562357
\(328\) 17.5253 0.967673
\(329\) 4.18757 0.230868
\(330\) 0 0
\(331\) 7.88716 0.433518 0.216759 0.976225i \(-0.430451\pi\)
0.216759 + 0.976225i \(0.430451\pi\)
\(332\) −1.16205 −0.0637757
\(333\) 3.75270 0.205646
\(334\) −5.32267 −0.291244
\(335\) 0 0
\(336\) 16.0509 0.875646
\(337\) 22.0819 1.20288 0.601438 0.798920i \(-0.294595\pi\)
0.601438 + 0.798920i \(0.294595\pi\)
\(338\) 17.4889 0.951268
\(339\) 6.01193 0.326523
\(340\) 0 0
\(341\) −3.30678 −0.179072
\(342\) −1.56513 −0.0846325
\(343\) −14.8060 −0.799448
\(344\) 23.6769 1.27657
\(345\) 0 0
\(346\) −19.8082 −1.06489
\(347\) 24.5149 1.31603 0.658014 0.753006i \(-0.271397\pi\)
0.658014 + 0.753006i \(0.271397\pi\)
\(348\) −0.401432 −0.0215190
\(349\) −28.2103 −1.51006 −0.755031 0.655689i \(-0.772378\pi\)
−0.755031 + 0.655689i \(0.772378\pi\)
\(350\) 0 0
\(351\) 0.614463 0.0327976
\(352\) −0.417449 −0.0222501
\(353\) 32.9115 1.75170 0.875852 0.482580i \(-0.160300\pi\)
0.875852 + 0.482580i \(0.160300\pi\)
\(354\) 3.56513 0.189485
\(355\) 0 0
\(356\) 1.34117 0.0710821
\(357\) 29.2694 1.54910
\(358\) −21.9792 −1.16164
\(359\) −17.7819 −0.938490 −0.469245 0.883068i \(-0.655474\pi\)
−0.469245 + 0.883068i \(0.655474\pi\)
\(360\) 0 0
\(361\) −17.7240 −0.932840
\(362\) 0.886645 0.0466010
\(363\) −10.1541 −0.532953
\(364\) 0.206585 0.0108280
\(365\) 0 0
\(366\) 8.13274 0.425105
\(367\) −14.0079 −0.731208 −0.365604 0.930771i \(-0.619137\pi\)
−0.365604 + 0.930771i \(0.619137\pi\)
\(368\) 31.9983 1.66802
\(369\) 6.08029 0.316527
\(370\) 0 0
\(371\) −6.25127 −0.324550
\(372\) 0.288665 0.0149666
\(373\) −21.9577 −1.13693 −0.568463 0.822709i \(-0.692462\pi\)
−0.568463 + 0.822709i \(0.692462\pi\)
\(374\) 8.90682 0.460561
\(375\) 0 0
\(376\) −2.88231 −0.148644
\(377\) 3.07231 0.158232
\(378\) 5.80203 0.298424
\(379\) −5.10415 −0.262183 −0.131091 0.991370i \(-0.541848\pi\)
−0.131091 + 0.991370i \(0.541848\pi\)
\(380\) 0 0
\(381\) 8.69628 0.445524
\(382\) 10.1357 0.518590
\(383\) −6.43635 −0.328882 −0.164441 0.986387i \(-0.552582\pi\)
−0.164441 + 0.986387i \(0.552582\pi\)
\(384\) −10.5850 −0.540165
\(385\) 0 0
\(386\) 12.6095 0.641809
\(387\) 8.21456 0.417569
\(388\) −0.155602 −0.00789952
\(389\) −36.7518 −1.86339 −0.931696 0.363239i \(-0.881671\pi\)
−0.931696 + 0.363239i \(0.881671\pi\)
\(390\) 0 0
\(391\) 58.3501 2.95089
\(392\) 30.3672 1.53378
\(393\) 12.1073 0.610731
\(394\) 25.4870 1.28402
\(395\) 0 0
\(396\) −0.0738405 −0.00371062
\(397\) 20.7423 1.04103 0.520514 0.853853i \(-0.325740\pi\)
0.520514 + 0.853853i \(0.325740\pi\)
\(398\) −24.9117 −1.24871
\(399\) −4.73035 −0.236814
\(400\) 0 0
\(401\) 37.7899 1.88714 0.943568 0.331178i \(-0.107446\pi\)
0.943568 + 0.331178i \(0.107446\pi\)
\(402\) −1.14219 −0.0569674
\(403\) −2.20927 −0.110051
\(404\) 0.396635 0.0197334
\(405\) 0 0
\(406\) 29.0101 1.43975
\(407\) 3.45140 0.171080
\(408\) −20.1462 −0.997386
\(409\) −1.49678 −0.0740109 −0.0370054 0.999315i \(-0.511782\pi\)
−0.0370054 + 0.999315i \(0.511782\pi\)
\(410\) 0 0
\(411\) 17.9792 0.886848
\(412\) −0.712293 −0.0350922
\(413\) 10.7750 0.530205
\(414\) 11.5667 0.568470
\(415\) 0 0
\(416\) −0.278898 −0.0136741
\(417\) 9.65671 0.472891
\(418\) −1.43947 −0.0704068
\(419\) −2.85379 −0.139417 −0.0697085 0.997567i \(-0.522207\pi\)
−0.0697085 + 0.997567i \(0.522207\pi\)
\(420\) 0 0
\(421\) −27.0334 −1.31753 −0.658763 0.752351i \(-0.728920\pi\)
−0.658763 + 0.752351i \(0.728920\pi\)
\(422\) −18.3411 −0.892831
\(423\) −1.00000 −0.0486217
\(424\) 4.30277 0.208961
\(425\) 0 0
\(426\) −16.7971 −0.813823
\(427\) 24.5799 1.18951
\(428\) 0.659517 0.0318790
\(429\) 0.565130 0.0272847
\(430\) 0 0
\(431\) −6.47444 −0.311863 −0.155931 0.987768i \(-0.549838\pi\)
−0.155931 + 0.987768i \(0.549838\pi\)
\(432\) −3.83298 −0.184414
\(433\) 37.4885 1.80158 0.900792 0.434251i \(-0.142987\pi\)
0.900792 + 0.434251i \(0.142987\pi\)
\(434\) −20.8609 −1.00135
\(435\) 0 0
\(436\) 0.816447 0.0391007
\(437\) −9.43022 −0.451109
\(438\) 16.3123 0.779430
\(439\) 31.0135 1.48019 0.740097 0.672500i \(-0.234780\pi\)
0.740097 + 0.672500i \(0.234780\pi\)
\(440\) 0 0
\(441\) 10.5357 0.501700
\(442\) 5.95067 0.283044
\(443\) −7.04971 −0.334942 −0.167471 0.985877i \(-0.553560\pi\)
−0.167471 + 0.985877i \(0.553560\pi\)
\(444\) −0.301290 −0.0142986
\(445\) 0 0
\(446\) 9.78153 0.463169
\(447\) 20.5529 0.972120
\(448\) −34.7352 −1.64108
\(449\) −5.58166 −0.263415 −0.131708 0.991289i \(-0.542046\pi\)
−0.131708 + 0.991289i \(0.542046\pi\)
\(450\) 0 0
\(451\) 5.59212 0.263323
\(452\) −0.482676 −0.0227032
\(453\) 9.64146 0.452995
\(454\) −33.7683 −1.58483
\(455\) 0 0
\(456\) 3.25592 0.152472
\(457\) 1.07141 0.0501183 0.0250591 0.999686i \(-0.492023\pi\)
0.0250591 + 0.999686i \(0.492023\pi\)
\(458\) −27.5223 −1.28603
\(459\) −6.98959 −0.326246
\(460\) 0 0
\(461\) 5.04754 0.235087 0.117544 0.993068i \(-0.462498\pi\)
0.117544 + 0.993068i \(0.462498\pi\)
\(462\) 5.33620 0.248263
\(463\) −6.75454 −0.313910 −0.156955 0.987606i \(-0.550168\pi\)
−0.156955 + 0.987606i \(0.550168\pi\)
\(464\) −19.1649 −0.889708
\(465\) 0 0
\(466\) −1.27986 −0.0592882
\(467\) −19.5324 −0.903851 −0.451925 0.892056i \(-0.649263\pi\)
−0.451925 + 0.892056i \(0.649263\pi\)
\(468\) −0.0493330 −0.00228042
\(469\) −3.45210 −0.159403
\(470\) 0 0
\(471\) 12.6724 0.583914
\(472\) −7.41649 −0.341372
\(473\) 7.55504 0.347381
\(474\) 5.63348 0.258754
\(475\) 0 0
\(476\) −2.34993 −0.107709
\(477\) 1.49282 0.0683514
\(478\) −16.7751 −0.767275
\(479\) −10.9843 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(480\) 0 0
\(481\) 2.30589 0.105140
\(482\) 16.4698 0.750180
\(483\) 34.9584 1.59066
\(484\) 0.815238 0.0370563
\(485\) 0 0
\(486\) −1.38554 −0.0628492
\(487\) 25.9801 1.17727 0.588635 0.808399i \(-0.299666\pi\)
0.588635 + 0.808399i \(0.299666\pi\)
\(488\) −16.9184 −0.765862
\(489\) −6.16057 −0.278591
\(490\) 0 0
\(491\) 29.1879 1.31723 0.658617 0.752479i \(-0.271142\pi\)
0.658617 + 0.752479i \(0.271142\pi\)
\(492\) −0.488164 −0.0220081
\(493\) −34.9480 −1.57398
\(494\) −0.961714 −0.0432696
\(495\) 0 0
\(496\) 13.7813 0.618797
\(497\) −50.7666 −2.27719
\(498\) −20.0540 −0.898641
\(499\) −2.73795 −0.122568 −0.0612838 0.998120i \(-0.519519\pi\)
−0.0612838 + 0.998120i \(0.519519\pi\)
\(500\) 0 0
\(501\) 3.84159 0.171630
\(502\) −30.6007 −1.36578
\(503\) −6.50871 −0.290209 −0.145105 0.989416i \(-0.546352\pi\)
−0.145105 + 0.989416i \(0.546352\pi\)
\(504\) −12.0699 −0.537635
\(505\) 0 0
\(506\) 10.6380 0.472917
\(507\) −12.6224 −0.560582
\(508\) −0.698192 −0.0309773
\(509\) −10.4054 −0.461213 −0.230607 0.973047i \(-0.574071\pi\)
−0.230607 + 0.973047i \(0.574071\pi\)
\(510\) 0 0
\(511\) 49.3012 2.18096
\(512\) 23.8355 1.05339
\(513\) 1.12962 0.0498739
\(514\) −7.75907 −0.342238
\(515\) 0 0
\(516\) −0.659517 −0.0290336
\(517\) −0.919714 −0.0404490
\(518\) 21.7732 0.956661
\(519\) 14.2964 0.627541
\(520\) 0 0
\(521\) −10.2017 −0.446943 −0.223472 0.974710i \(-0.571739\pi\)
−0.223472 + 0.974710i \(0.571739\pi\)
\(522\) −6.92769 −0.303217
\(523\) −10.8838 −0.475917 −0.237959 0.971275i \(-0.576478\pi\)
−0.237959 + 0.971275i \(0.576478\pi\)
\(524\) −0.972049 −0.0424642
\(525\) 0 0
\(526\) −19.1054 −0.833037
\(527\) 25.1307 1.09471
\(528\) −3.52525 −0.153417
\(529\) 46.6914 2.03006
\(530\) 0 0
\(531\) −2.57310 −0.111663
\(532\) 0.379783 0.0164657
\(533\) 3.73611 0.161829
\(534\) 23.1452 1.00159
\(535\) 0 0
\(536\) 2.37609 0.102631
\(537\) 15.8633 0.684552
\(538\) 6.33709 0.273211
\(539\) 9.68983 0.417371
\(540\) 0 0
\(541\) −4.50138 −0.193529 −0.0967647 0.995307i \(-0.530849\pi\)
−0.0967647 + 0.995307i \(0.530849\pi\)
\(542\) 41.2969 1.77386
\(543\) −0.639928 −0.0274620
\(544\) 3.17251 0.136020
\(545\) 0 0
\(546\) 3.56513 0.152573
\(547\) −42.9758 −1.83751 −0.918756 0.394825i \(-0.870805\pi\)
−0.918756 + 0.394825i \(0.870805\pi\)
\(548\) −1.44348 −0.0616626
\(549\) −5.86974 −0.250514
\(550\) 0 0
\(551\) 5.64810 0.240617
\(552\) −24.0620 −1.02414
\(553\) 17.0263 0.724032
\(554\) 26.9053 1.14310
\(555\) 0 0
\(556\) −0.775302 −0.0328801
\(557\) −27.3479 −1.15877 −0.579383 0.815055i \(-0.696707\pi\)
−0.579383 + 0.815055i \(0.696707\pi\)
\(558\) 4.98162 0.210889
\(559\) 5.04754 0.213488
\(560\) 0 0
\(561\) −6.42842 −0.271408
\(562\) −23.5872 −0.994966
\(563\) 21.5293 0.907350 0.453675 0.891167i \(-0.350113\pi\)
0.453675 + 0.891167i \(0.350113\pi\)
\(564\) 0.0802864 0.00338067
\(565\) 0 0
\(566\) −15.2829 −0.642389
\(567\) −4.18757 −0.175861
\(568\) 34.9428 1.46617
\(569\) 9.20811 0.386024 0.193012 0.981196i \(-0.438174\pi\)
0.193012 + 0.981196i \(0.438174\pi\)
\(570\) 0 0
\(571\) 29.9531 1.25350 0.626749 0.779221i \(-0.284385\pi\)
0.626749 + 0.779221i \(0.284385\pi\)
\(572\) −0.0453722 −0.00189711
\(573\) −7.31539 −0.305605
\(574\) 35.2780 1.47247
\(575\) 0 0
\(576\) 8.29484 0.345619
\(577\) −41.6419 −1.73357 −0.866787 0.498678i \(-0.833819\pi\)
−0.866787 + 0.498678i \(0.833819\pi\)
\(578\) −44.1355 −1.83579
\(579\) −9.10083 −0.378218
\(580\) 0 0
\(581\) −60.6100 −2.51453
\(582\) −2.68530 −0.111309
\(583\) 1.37296 0.0568623
\(584\) −33.9342 −1.40421
\(585\) 0 0
\(586\) −46.4272 −1.91789
\(587\) 36.9519 1.52517 0.762585 0.646888i \(-0.223930\pi\)
0.762585 + 0.646888i \(0.223930\pi\)
\(588\) −0.845873 −0.0348832
\(589\) −4.06148 −0.167350
\(590\) 0 0
\(591\) −18.3950 −0.756671
\(592\) −14.3840 −0.591179
\(593\) 27.6951 1.13730 0.568651 0.822579i \(-0.307465\pi\)
0.568651 + 0.822579i \(0.307465\pi\)
\(594\) −1.27430 −0.0522851
\(595\) 0 0
\(596\) −1.65012 −0.0675915
\(597\) 17.9798 0.735865
\(598\) 7.10728 0.290638
\(599\) −6.30033 −0.257425 −0.128712 0.991682i \(-0.541084\pi\)
−0.128712 + 0.991682i \(0.541084\pi\)
\(600\) 0 0
\(601\) −21.1106 −0.861119 −0.430560 0.902562i \(-0.641684\pi\)
−0.430560 + 0.902562i \(0.641684\pi\)
\(602\) 47.6611 1.94252
\(603\) 0.824368 0.0335709
\(604\) −0.774077 −0.0314968
\(605\) 0 0
\(606\) 6.84492 0.278056
\(607\) 40.9648 1.66271 0.831355 0.555741i \(-0.187565\pi\)
0.831355 + 0.555741i \(0.187565\pi\)
\(608\) −0.512723 −0.0207937
\(609\) −20.9378 −0.848444
\(610\) 0 0
\(611\) −0.614463 −0.0248585
\(612\) 0.561169 0.0226839
\(613\) 48.1149 1.94334 0.971672 0.236334i \(-0.0759459\pi\)
0.971672 + 0.236334i \(0.0759459\pi\)
\(614\) 13.8345 0.558315
\(615\) 0 0
\(616\) −11.1008 −0.447265
\(617\) 3.74809 0.150893 0.0754463 0.997150i \(-0.475962\pi\)
0.0754463 + 0.997150i \(0.475962\pi\)
\(618\) −12.2924 −0.494471
\(619\) −15.6789 −0.630186 −0.315093 0.949061i \(-0.602036\pi\)
−0.315093 + 0.949061i \(0.602036\pi\)
\(620\) 0 0
\(621\) −8.34814 −0.334999
\(622\) −17.9865 −0.721194
\(623\) 69.9528 2.80260
\(624\) −2.35522 −0.0942844
\(625\) 0 0
\(626\) −30.3956 −1.21485
\(627\) 1.03893 0.0414907
\(628\) −1.01742 −0.0405995
\(629\) −26.2298 −1.04585
\(630\) 0 0
\(631\) −30.2501 −1.20424 −0.602119 0.798407i \(-0.705677\pi\)
−0.602119 + 0.798407i \(0.705677\pi\)
\(632\) −11.7193 −0.466167
\(633\) 13.2375 0.526145
\(634\) 15.3595 0.610003
\(635\) 0 0
\(636\) −0.119853 −0.00475247
\(637\) 6.47380 0.256501
\(638\) −6.37149 −0.252250
\(639\) 12.1232 0.479585
\(640\) 0 0
\(641\) −23.5134 −0.928724 −0.464362 0.885645i \(-0.653716\pi\)
−0.464362 + 0.885645i \(0.653716\pi\)
\(642\) 11.3816 0.449195
\(643\) 24.5259 0.967209 0.483604 0.875287i \(-0.339327\pi\)
0.483604 + 0.875287i \(0.339327\pi\)
\(644\) −2.80668 −0.110599
\(645\) 0 0
\(646\) 10.9396 0.430414
\(647\) −10.2318 −0.402253 −0.201126 0.979565i \(-0.564460\pi\)
−0.201126 + 0.979565i \(0.564460\pi\)
\(648\) 2.88231 0.113228
\(649\) −2.36652 −0.0928939
\(650\) 0 0
\(651\) 15.0562 0.590097
\(652\) 0.494610 0.0193704
\(653\) −15.4266 −0.603691 −0.301845 0.953357i \(-0.597603\pi\)
−0.301845 + 0.953357i \(0.597603\pi\)
\(654\) 14.0898 0.550954
\(655\) 0 0
\(656\) −23.3056 −0.909932
\(657\) −11.7732 −0.459318
\(658\) −5.80203 −0.226187
\(659\) −40.8238 −1.59027 −0.795135 0.606432i \(-0.792600\pi\)
−0.795135 + 0.606432i \(0.792600\pi\)
\(660\) 0 0
\(661\) −28.2581 −1.09911 −0.549556 0.835457i \(-0.685203\pi\)
−0.549556 + 0.835457i \(0.685203\pi\)
\(662\) −10.9280 −0.424727
\(663\) −4.29484 −0.166798
\(664\) 41.7180 1.61897
\(665\) 0 0
\(666\) −5.19950 −0.201476
\(667\) −41.7407 −1.61621
\(668\) −0.308428 −0.0119334
\(669\) −7.05974 −0.272945
\(670\) 0 0
\(671\) −5.39848 −0.208406
\(672\) 1.90069 0.0733209
\(673\) 13.6620 0.526631 0.263316 0.964710i \(-0.415184\pi\)
0.263316 + 0.964710i \(0.415184\pi\)
\(674\) −30.5952 −1.17848
\(675\) 0 0
\(676\) 1.01341 0.0389773
\(677\) −50.5626 −1.94328 −0.971640 0.236466i \(-0.924011\pi\)
−0.971640 + 0.236466i \(0.924011\pi\)
\(678\) −8.32976 −0.319902
\(679\) −8.11589 −0.311459
\(680\) 0 0
\(681\) 24.3720 0.933937
\(682\) 4.58166 0.175441
\(683\) −20.6092 −0.788590 −0.394295 0.918984i \(-0.629011\pi\)
−0.394295 + 0.918984i \(0.629011\pi\)
\(684\) −0.0906930 −0.00346773
\(685\) 0 0
\(686\) 20.5142 0.783238
\(687\) 19.8640 0.757858
\(688\) −31.4862 −1.20040
\(689\) 0.917280 0.0349456
\(690\) 0 0
\(691\) 25.9719 0.988018 0.494009 0.869457i \(-0.335531\pi\)
0.494009 + 0.869457i \(0.335531\pi\)
\(692\) −1.14780 −0.0436329
\(693\) −3.85136 −0.146301
\(694\) −33.9663 −1.28934
\(695\) 0 0
\(696\) 14.4116 0.546269
\(697\) −42.4987 −1.60975
\(698\) 39.0864 1.47944
\(699\) 0.923726 0.0349385
\(700\) 0 0
\(701\) −37.7609 −1.42621 −0.713105 0.701057i \(-0.752712\pi\)
−0.713105 + 0.701057i \(0.752712\pi\)
\(702\) −0.851361 −0.0321325
\(703\) 4.23912 0.159881
\(704\) 7.62888 0.287524
\(705\) 0 0
\(706\) −45.6001 −1.71618
\(707\) 20.6877 0.778040
\(708\) 0.206585 0.00776394
\(709\) 27.1125 1.01823 0.509116 0.860698i \(-0.329973\pi\)
0.509116 + 0.860698i \(0.329973\pi\)
\(710\) 0 0
\(711\) −4.06592 −0.152484
\(712\) −48.1487 −1.80445
\(713\) 30.0153 1.12408
\(714\) −40.5538 −1.51769
\(715\) 0 0
\(716\) −1.27361 −0.0475969
\(717\) 12.1073 0.452155
\(718\) 24.6374 0.919460
\(719\) 5.44412 0.203032 0.101516 0.994834i \(-0.467631\pi\)
0.101516 + 0.994834i \(0.467631\pi\)
\(720\) 0 0
\(721\) −37.1517 −1.38360
\(722\) 24.5572 0.913925
\(723\) −11.8870 −0.442081
\(724\) 0.0513775 0.00190943
\(725\) 0 0
\(726\) 14.0689 0.522147
\(727\) −32.3623 −1.20025 −0.600126 0.799905i \(-0.704883\pi\)
−0.600126 + 0.799905i \(0.704883\pi\)
\(728\) −7.41649 −0.274873
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −57.4164 −2.12362
\(732\) 0.471260 0.0174183
\(733\) 22.6016 0.834810 0.417405 0.908721i \(-0.362940\pi\)
0.417405 + 0.908721i \(0.362940\pi\)
\(734\) 19.4085 0.716381
\(735\) 0 0
\(736\) 3.78914 0.139669
\(737\) 0.758183 0.0279280
\(738\) −8.42446 −0.310109
\(739\) −10.6215 −0.390717 −0.195359 0.980732i \(-0.562587\pi\)
−0.195359 + 0.980732i \(0.562587\pi\)
\(740\) 0 0
\(741\) 0.694109 0.0254987
\(742\) 8.66136 0.317969
\(743\) −10.2836 −0.377269 −0.188634 0.982047i \(-0.560406\pi\)
−0.188634 + 0.982047i \(0.560406\pi\)
\(744\) −10.3632 −0.379933
\(745\) 0 0
\(746\) 30.4232 1.11387
\(747\) 14.4738 0.529569
\(748\) 0.516115 0.0188710
\(749\) 34.3990 1.25691
\(750\) 0 0
\(751\) 15.6206 0.570005 0.285003 0.958527i \(-0.408006\pi\)
0.285003 + 0.958527i \(0.408006\pi\)
\(752\) 3.83298 0.139774
\(753\) 22.0858 0.804852
\(754\) −4.25680 −0.155024
\(755\) 0 0
\(756\) 0.336204 0.0122276
\(757\) −30.0073 −1.09064 −0.545318 0.838230i \(-0.683591\pi\)
−0.545318 + 0.838230i \(0.683591\pi\)
\(758\) 7.07200 0.256866
\(759\) −7.67790 −0.278690
\(760\) 0 0
\(761\) −5.88716 −0.213409 −0.106705 0.994291i \(-0.534030\pi\)
−0.106705 + 0.994291i \(0.534030\pi\)
\(762\) −12.0490 −0.436490
\(763\) 42.5841 1.54165
\(764\) 0.587326 0.0212487
\(765\) 0 0
\(766\) 8.91780 0.322213
\(767\) −1.58108 −0.0570893
\(768\) −1.92373 −0.0694165
\(769\) −28.8029 −1.03866 −0.519329 0.854574i \(-0.673818\pi\)
−0.519329 + 0.854574i \(0.673818\pi\)
\(770\) 0 0
\(771\) 5.60004 0.201681
\(772\) 0.730673 0.0262975
\(773\) 24.5651 0.883546 0.441773 0.897127i \(-0.354350\pi\)
0.441773 + 0.897127i \(0.354350\pi\)
\(774\) −11.3816 −0.409102
\(775\) 0 0
\(776\) 5.58619 0.200533
\(777\) −15.7147 −0.563760
\(778\) 50.9210 1.82561
\(779\) 6.86841 0.246086
\(780\) 0 0
\(781\) 11.1498 0.398973
\(782\) −80.8462 −2.89106
\(783\) 5.00000 0.178685
\(784\) −40.3832 −1.44226
\(785\) 0 0
\(786\) −16.7751 −0.598348
\(787\) 51.2953 1.82848 0.914239 0.405174i \(-0.132789\pi\)
0.914239 + 0.405174i \(0.132789\pi\)
\(788\) 1.47687 0.0526113
\(789\) 13.7892 0.490908
\(790\) 0 0
\(791\) −25.1754 −0.895133
\(792\) 2.65090 0.0941958
\(793\) −3.60674 −0.128079
\(794\) −28.7393 −1.01992
\(795\) 0 0
\(796\) −1.44353 −0.0511647
\(797\) −32.4937 −1.15098 −0.575492 0.817807i \(-0.695189\pi\)
−0.575492 + 0.817807i \(0.695189\pi\)
\(798\) 6.55408 0.232012
\(799\) 6.98959 0.247274
\(800\) 0 0
\(801\) −16.7049 −0.590238
\(802\) −52.3593 −1.84887
\(803\) −10.8280 −0.382112
\(804\) −0.0661855 −0.00233418
\(805\) 0 0
\(806\) 3.06102 0.107820
\(807\) −4.57374 −0.161003
\(808\) −14.2394 −0.500940
\(809\) 49.2124 1.73022 0.865109 0.501584i \(-0.167249\pi\)
0.865109 + 0.501584i \(0.167249\pi\)
\(810\) 0 0
\(811\) 6.70917 0.235591 0.117795 0.993038i \(-0.462417\pi\)
0.117795 + 0.993038i \(0.462417\pi\)
\(812\) 1.68102 0.0589923
\(813\) −29.8057 −1.04533
\(814\) −4.78205 −0.167611
\(815\) 0 0
\(816\) 26.7910 0.937872
\(817\) 9.27932 0.324642
\(818\) 2.07384 0.0725101
\(819\) −2.57310 −0.0899115
\(820\) 0 0
\(821\) 45.5348 1.58918 0.794589 0.607148i \(-0.207687\pi\)
0.794589 + 0.607148i \(0.207687\pi\)
\(822\) −24.9108 −0.868865
\(823\) 8.76979 0.305696 0.152848 0.988250i \(-0.451156\pi\)
0.152848 + 0.988250i \(0.451156\pi\)
\(824\) 25.5716 0.890830
\(825\) 0 0
\(826\) −14.9292 −0.519454
\(827\) −49.4171 −1.71840 −0.859201 0.511639i \(-0.829039\pi\)
−0.859201 + 0.511639i \(0.829039\pi\)
\(828\) 0.670242 0.0232925
\(829\) −56.2875 −1.95495 −0.977474 0.211058i \(-0.932309\pi\)
−0.977474 + 0.211058i \(0.932309\pi\)
\(830\) 0 0
\(831\) −19.4187 −0.673626
\(832\) 5.09687 0.176702
\(833\) −73.6403 −2.55148
\(834\) −13.3797 −0.463302
\(835\) 0 0
\(836\) −0.0834116 −0.00288485
\(837\) −3.59544 −0.124277
\(838\) 3.95404 0.136590
\(839\) 22.8330 0.788282 0.394141 0.919050i \(-0.371042\pi\)
0.394141 + 0.919050i \(0.371042\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 37.4557 1.29081
\(843\) 17.0239 0.586333
\(844\) −1.06279 −0.0365829
\(845\) 0 0
\(846\) 1.38554 0.0476358
\(847\) 42.5211 1.46104
\(848\) −5.72194 −0.196492
\(849\) 11.0303 0.378559
\(850\) 0 0
\(851\) −31.3280 −1.07391
\(852\) −0.973325 −0.0333456
\(853\) 12.4069 0.424805 0.212402 0.977182i \(-0.431871\pi\)
0.212402 + 0.977182i \(0.431871\pi\)
\(854\) −34.0564 −1.16539
\(855\) 0 0
\(856\) −23.6769 −0.809261
\(857\) 28.5268 0.974458 0.487229 0.873274i \(-0.338008\pi\)
0.487229 + 0.873274i \(0.338008\pi\)
\(858\) −0.783008 −0.0267315
\(859\) −27.2313 −0.929121 −0.464561 0.885541i \(-0.653788\pi\)
−0.464561 + 0.885541i \(0.653788\pi\)
\(860\) 0 0
\(861\) −25.4616 −0.867729
\(862\) 8.97057 0.305539
\(863\) −8.61981 −0.293422 −0.146711 0.989179i \(-0.546869\pi\)
−0.146711 + 0.989179i \(0.546869\pi\)
\(864\) −0.453890 −0.0154417
\(865\) 0 0
\(866\) −51.9418 −1.76505
\(867\) 31.8544 1.08183
\(868\) −1.20880 −0.0410295
\(869\) −3.73948 −0.126853
\(870\) 0 0
\(871\) 0.506544 0.0171636
\(872\) −29.3108 −0.992589
\(873\) 1.93809 0.0655945
\(874\) 13.0659 0.441961
\(875\) 0 0
\(876\) 0.945231 0.0319364
\(877\) −28.3424 −0.957056 −0.478528 0.878072i \(-0.658829\pi\)
−0.478528 + 0.878072i \(0.658829\pi\)
\(878\) −42.9704 −1.45018
\(879\) 33.5084 1.13021
\(880\) 0 0
\(881\) 12.8920 0.434344 0.217172 0.976133i \(-0.430317\pi\)
0.217172 + 0.976133i \(0.430317\pi\)
\(882\) −14.5976 −0.491527
\(883\) −48.1627 −1.62081 −0.810403 0.585873i \(-0.800752\pi\)
−0.810403 + 0.585873i \(0.800752\pi\)
\(884\) 0.344817 0.0115975
\(885\) 0 0
\(886\) 9.76763 0.328150
\(887\) 17.0254 0.571659 0.285829 0.958281i \(-0.407731\pi\)
0.285829 + 0.958281i \(0.407731\pi\)
\(888\) 10.8164 0.362976
\(889\) −36.4162 −1.22136
\(890\) 0 0
\(891\) 0.919714 0.0308116
\(892\) 0.566801 0.0189779
\(893\) −1.12962 −0.0378013
\(894\) −28.4769 −0.952408
\(895\) 0 0
\(896\) 44.3255 1.48081
\(897\) −5.12962 −0.171273
\(898\) 7.73360 0.258074
\(899\) −17.9772 −0.599574
\(900\) 0 0
\(901\) −10.4342 −0.347613
\(902\) −7.74809 −0.257983
\(903\) −34.3990 −1.14473
\(904\) 17.3283 0.576330
\(905\) 0 0
\(906\) −13.3586 −0.443810
\(907\) 19.3378 0.642101 0.321050 0.947062i \(-0.395964\pi\)
0.321050 + 0.947062i \(0.395964\pi\)
\(908\) −1.95674 −0.0649367
\(909\) −4.94026 −0.163858
\(910\) 0 0
\(911\) −46.8527 −1.55230 −0.776149 0.630549i \(-0.782830\pi\)
−0.776149 + 0.630549i \(0.782830\pi\)
\(912\) −4.32981 −0.143374
\(913\) 13.3117 0.440555
\(914\) −1.48447 −0.0491020
\(915\) 0 0
\(916\) −1.59481 −0.0526939
\(917\) −50.7000 −1.67426
\(918\) 9.68434 0.319631
\(919\) 26.0103 0.858002 0.429001 0.903304i \(-0.358866\pi\)
0.429001 + 0.903304i \(0.358866\pi\)
\(920\) 0 0
\(921\) −9.98494 −0.329015
\(922\) −6.99355 −0.230320
\(923\) 7.44924 0.245195
\(924\) 0.309212 0.0101723
\(925\) 0 0
\(926\) 9.35867 0.307545
\(927\) 8.87191 0.291392
\(928\) −2.26945 −0.0744983
\(929\) 57.6243 1.89059 0.945297 0.326212i \(-0.105772\pi\)
0.945297 + 0.326212i \(0.105772\pi\)
\(930\) 0 0
\(931\) 11.9013 0.390050
\(932\) −0.0741626 −0.00242928
\(933\) 12.9816 0.424999
\(934\) 27.0628 0.885523
\(935\) 0 0
\(936\) 1.77107 0.0578894
\(937\) −61.0181 −1.99337 −0.996687 0.0813275i \(-0.974084\pi\)
−0.996687 + 0.0813275i \(0.974084\pi\)
\(938\) 4.78301 0.156171
\(939\) 21.9378 0.715912
\(940\) 0 0
\(941\) 43.9597 1.43304 0.716522 0.697565i \(-0.245733\pi\)
0.716522 + 0.697565i \(0.245733\pi\)
\(942\) −17.5581 −0.572074
\(943\) −50.7591 −1.65294
\(944\) 9.86265 0.321002
\(945\) 0 0
\(946\) −10.4678 −0.340337
\(947\) −44.4156 −1.44331 −0.721657 0.692251i \(-0.756619\pi\)
−0.721657 + 0.692251i \(0.756619\pi\)
\(948\) 0.326438 0.0106022
\(949\) −7.23422 −0.234833
\(950\) 0 0
\(951\) −11.0856 −0.359475
\(952\) 84.3636 2.73424
\(953\) 56.6835 1.83616 0.918079 0.396397i \(-0.129740\pi\)
0.918079 + 0.396397i \(0.129740\pi\)
\(954\) −2.06835 −0.0669654
\(955\) 0 0
\(956\) −0.972049 −0.0314383
\(957\) 4.59857 0.148651
\(958\) 15.2192 0.491709
\(959\) −75.2890 −2.43121
\(960\) 0 0
\(961\) −18.0728 −0.582993
\(962\) −3.19490 −0.103008
\(963\) −8.21456 −0.264710
\(964\) 0.954361 0.0307379
\(965\) 0 0
\(966\) −48.4361 −1.55841
\(967\) −9.38822 −0.301905 −0.150952 0.988541i \(-0.548234\pi\)
−0.150952 + 0.988541i \(0.548234\pi\)
\(968\) −29.2674 −0.940689
\(969\) −7.89558 −0.253643
\(970\) 0 0
\(971\) −1.85360 −0.0594848 −0.0297424 0.999558i \(-0.509469\pi\)
−0.0297424 + 0.999558i \(0.509469\pi\)
\(972\) −0.0802864 −0.00257519
\(973\) −40.4381 −1.29639
\(974\) −35.9964 −1.15340
\(975\) 0 0
\(976\) 22.4986 0.720163
\(977\) −39.6717 −1.26921 −0.634605 0.772837i \(-0.718837\pi\)
−0.634605 + 0.772837i \(0.718837\pi\)
\(978\) 8.53570 0.272942
\(979\) −15.3637 −0.491026
\(980\) 0 0
\(981\) −10.1692 −0.324677
\(982\) −40.4410 −1.29052
\(983\) −3.32976 −0.106203 −0.0531014 0.998589i \(-0.516911\pi\)
−0.0531014 + 0.998589i \(0.516911\pi\)
\(984\) 17.5253 0.558686
\(985\) 0 0
\(986\) 48.4217 1.54206
\(987\) 4.18757 0.133292
\(988\) −0.0557275 −0.00177293
\(989\) −68.5763 −2.18060
\(990\) 0 0
\(991\) 23.6961 0.752731 0.376365 0.926471i \(-0.377174\pi\)
0.376365 + 0.926471i \(0.377174\pi\)
\(992\) 1.63194 0.0518140
\(993\) 7.88716 0.250292
\(994\) 70.3390 2.23102
\(995\) 0 0
\(996\) −1.16205 −0.0368209
\(997\) 12.5238 0.396633 0.198317 0.980138i \(-0.436453\pi\)
0.198317 + 0.980138i \(0.436453\pi\)
\(998\) 3.79354 0.120082
\(999\) 3.75270 0.118730
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 3525.2.a.u.1.2 4
5.4 even 2 705.2.a.j.1.3 4
15.14 odd 2 2115.2.a.p.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
705.2.a.j.1.3 4 5.4 even 2
2115.2.a.p.1.2 4 15.14 odd 2
3525.2.a.u.1.2 4 1.1 even 1 trivial