Properties

Label 475.2.a.g.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
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] = [475,2,Mod(1,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, 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("475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) 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 \(2.65109\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.65109 q^{2} +2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} -0.377203 q^{7} +2.10331 q^{8} +2.65109 q^{9} +O(q^{10})\) \(q-1.65109 q^{2} +2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} -0.377203 q^{7} +2.10331 q^{8} +2.65109 q^{9} -1.37720 q^{11} +1.72611 q^{12} +2.82167 q^{13} +0.622797 q^{14} -4.92498 q^{16} +6.37720 q^{17} -4.37720 q^{18} +1.00000 q^{19} -0.896688 q^{21} +2.27389 q^{22} +6.19887 q^{23} +5.00000 q^{24} -4.65884 q^{26} -0.829422 q^{27} -0.273891 q^{28} -3.37720 q^{29} +2.48052 q^{31} +3.92498 q^{32} -3.27389 q^{33} -10.5294 q^{34} +1.92498 q^{36} +5.58383 q^{37} -1.65109 q^{38} +6.70769 q^{39} +8.50106 q^{41} +1.48052 q^{42} -12.1522 q^{43} -1.00000 q^{44} -10.2349 q^{46} -6.87826 q^{47} -11.7077 q^{48} -6.85772 q^{49} +15.1599 q^{51} +2.04884 q^{52} +11.5478 q^{53} +1.36945 q^{54} -0.793375 q^{56} +2.37720 q^{57} +5.57608 q^{58} +6.05659 q^{59} +5.02830 q^{61} -4.09556 q^{62} -1.00000 q^{63} +3.36945 q^{64} +5.40550 q^{66} +3.22717 q^{67} +4.63055 q^{68} +14.7360 q^{69} -2.30219 q^{71} +5.57608 q^{72} -3.19887 q^{73} -9.21942 q^{74} +0.726109 q^{76} +0.519485 q^{77} -11.0750 q^{78} -6.71836 q^{79} -9.92498 q^{81} -14.0360 q^{82} +18.2165 q^{83} -0.651093 q^{84} +20.0643 q^{86} -8.02830 q^{87} -2.89669 q^{88} +1.50106 q^{89} -1.06434 q^{91} +4.50106 q^{92} +5.89669 q^{93} +11.3567 q^{94} +9.33048 q^{96} -11.7827 q^{97} +11.3227 q^{98} -3.65109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} + 7 q^{14} - 6 q^{16} + 14 q^{17} - 8 q^{18} + 3 q^{19} - 6 q^{21} + 5 q^{22} + 8 q^{23} + 15 q^{24} - 11 q^{26} - q^{27} + q^{28} - 5 q^{29} - q^{31} + 3 q^{32} - 8 q^{33} + 5 q^{34} - 3 q^{36} + 5 q^{37} + 2 q^{38} - 11 q^{39} + q^{41} - 4 q^{42} - 5 q^{43} - 3 q^{44} - 12 q^{46} + 9 q^{47} - 4 q^{48} - 7 q^{49} + 18 q^{51} - 22 q^{52} + 31 q^{53} - 5 q^{54} - 9 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} + 3 q^{61} - 5 q^{62} - 3 q^{63} + q^{64} - q^{66} - 13 q^{67} + 23 q^{68} + q^{69} + 7 q^{71} + q^{72} + q^{73} - q^{74} + 4 q^{76} + 10 q^{77} - 42 q^{78} - 18 q^{79} - 21 q^{81} - 34 q^{82} + 3 q^{83} + 5 q^{84} + 40 q^{86} - 12 q^{87} - 12 q^{88} - 20 q^{89} + 17 q^{91} - 11 q^{92} + 21 q^{93} + 45 q^{94} + 2 q^{96} - 13 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.65109 −1.16750 −0.583750 0.811934i \(-0.698415\pi\)
−0.583750 + 0.811934i \(0.698415\pi\)
\(3\) 2.37720 1.37248 0.686239 0.727376i \(-0.259260\pi\)
0.686239 + 0.727376i \(0.259260\pi\)
\(4\) 0.726109 0.363055
\(5\) 0 0
\(6\) −3.92498 −1.60237
\(7\) −0.377203 −0.142569 −0.0712846 0.997456i \(-0.522710\pi\)
−0.0712846 + 0.997456i \(0.522710\pi\)
\(8\) 2.10331 0.743633
\(9\) 2.65109 0.883698
\(10\) 0 0
\(11\) −1.37720 −0.415242 −0.207621 0.978209i \(-0.566572\pi\)
−0.207621 + 0.978209i \(0.566572\pi\)
\(12\) 1.72611 0.498285
\(13\) 2.82167 0.782591 0.391295 0.920265i \(-0.372027\pi\)
0.391295 + 0.920265i \(0.372027\pi\)
\(14\) 0.622797 0.166450
\(15\) 0 0
\(16\) −4.92498 −1.23125
\(17\) 6.37720 1.54670 0.773349 0.633980i \(-0.218580\pi\)
0.773349 + 0.633980i \(0.218580\pi\)
\(18\) −4.37720 −1.03172
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −0.896688 −0.195673
\(22\) 2.27389 0.484795
\(23\) 6.19887 1.29255 0.646277 0.763103i \(-0.276325\pi\)
0.646277 + 0.763103i \(0.276325\pi\)
\(24\) 5.00000 1.02062
\(25\) 0 0
\(26\) −4.65884 −0.913674
\(27\) −0.829422 −0.159622
\(28\) −0.273891 −0.0517604
\(29\) −3.37720 −0.627131 −0.313565 0.949567i \(-0.601524\pi\)
−0.313565 + 0.949567i \(0.601524\pi\)
\(30\) 0 0
\(31\) 2.48052 0.445514 0.222757 0.974874i \(-0.428494\pi\)
0.222757 + 0.974874i \(0.428494\pi\)
\(32\) 3.92498 0.693846
\(33\) −3.27389 −0.569911
\(34\) −10.5294 −1.80577
\(35\) 0 0
\(36\) 1.92498 0.320831
\(37\) 5.58383 0.917976 0.458988 0.888443i \(-0.348212\pi\)
0.458988 + 0.888443i \(0.348212\pi\)
\(38\) −1.65109 −0.267843
\(39\) 6.70769 1.07409
\(40\) 0 0
\(41\) 8.50106 1.32764 0.663821 0.747891i \(-0.268934\pi\)
0.663821 + 0.747891i \(0.268934\pi\)
\(42\) 1.48052 0.228448
\(43\) −12.1522 −1.85319 −0.926593 0.376065i \(-0.877277\pi\)
−0.926593 + 0.376065i \(0.877277\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −10.2349 −1.50906
\(47\) −6.87826 −1.00330 −0.501649 0.865071i \(-0.667273\pi\)
−0.501649 + 0.865071i \(0.667273\pi\)
\(48\) −11.7077 −1.68986
\(49\) −6.85772 −0.979674
\(50\) 0 0
\(51\) 15.1599 2.12281
\(52\) 2.04884 0.284123
\(53\) 11.5478 1.58621 0.793105 0.609085i \(-0.208463\pi\)
0.793105 + 0.609085i \(0.208463\pi\)
\(54\) 1.36945 0.186359
\(55\) 0 0
\(56\) −0.793375 −0.106019
\(57\) 2.37720 0.314868
\(58\) 5.57608 0.732175
\(59\) 6.05659 0.788501 0.394251 0.919003i \(-0.371004\pi\)
0.394251 + 0.919003i \(0.371004\pi\)
\(60\) 0 0
\(61\) 5.02830 0.643807 0.321904 0.946772i \(-0.395677\pi\)
0.321904 + 0.946772i \(0.395677\pi\)
\(62\) −4.09556 −0.520137
\(63\) −1.00000 −0.125988
\(64\) 3.36945 0.421182
\(65\) 0 0
\(66\) 5.40550 0.665371
\(67\) 3.22717 0.394262 0.197131 0.980377i \(-0.436838\pi\)
0.197131 + 0.980377i \(0.436838\pi\)
\(68\) 4.63055 0.561536
\(69\) 14.7360 1.77400
\(70\) 0 0
\(71\) −2.30219 −0.273219 −0.136610 0.990625i \(-0.543621\pi\)
−0.136610 + 0.990625i \(0.543621\pi\)
\(72\) 5.57608 0.657147
\(73\) −3.19887 −0.374400 −0.187200 0.982322i \(-0.559941\pi\)
−0.187200 + 0.982322i \(0.559941\pi\)
\(74\) −9.21942 −1.07174
\(75\) 0 0
\(76\) 0.726109 0.0832905
\(77\) 0.519485 0.0592008
\(78\) −11.0750 −1.25400
\(79\) −6.71836 −0.755874 −0.377937 0.925831i \(-0.623366\pi\)
−0.377937 + 0.925831i \(0.623366\pi\)
\(80\) 0 0
\(81\) −9.92498 −1.10278
\(82\) −14.0360 −1.55002
\(83\) 18.2165 1.99952 0.999760 0.0218996i \(-0.00697141\pi\)
0.999760 + 0.0218996i \(0.00697141\pi\)
\(84\) −0.651093 −0.0710401
\(85\) 0 0
\(86\) 20.0643 2.16359
\(87\) −8.02830 −0.860724
\(88\) −2.89669 −0.308788
\(89\) 1.50106 0.159112 0.0795561 0.996830i \(-0.474650\pi\)
0.0795561 + 0.996830i \(0.474650\pi\)
\(90\) 0 0
\(91\) −1.06434 −0.111573
\(92\) 4.50106 0.469268
\(93\) 5.89669 0.611458
\(94\) 11.3567 1.17135
\(95\) 0 0
\(96\) 9.33048 0.952288
\(97\) −11.7827 −1.19635 −0.598176 0.801365i \(-0.704108\pi\)
−0.598176 + 0.801365i \(0.704108\pi\)
\(98\) 11.3227 1.14377
\(99\) −3.65109 −0.366949
\(100\) 0 0
\(101\) −9.18820 −0.914260 −0.457130 0.889400i \(-0.651123\pi\)
−0.457130 + 0.889400i \(0.651123\pi\)
\(102\) −25.0304 −2.47838
\(103\) −9.47277 −0.933379 −0.466690 0.884421i \(-0.654553\pi\)
−0.466690 + 0.884421i \(0.654553\pi\)
\(104\) 5.93486 0.581961
\(105\) 0 0
\(106\) −19.0665 −1.85190
\(107\) −15.3510 −1.48404 −0.742020 0.670378i \(-0.766132\pi\)
−0.742020 + 0.670378i \(0.766132\pi\)
\(108\) −0.602251 −0.0579516
\(109\) −16.7643 −1.60573 −0.802863 0.596163i \(-0.796691\pi\)
−0.802863 + 0.596163i \(0.796691\pi\)
\(110\) 0 0
\(111\) 13.2739 1.25990
\(112\) 1.85772 0.175538
\(113\) −13.3305 −1.25403 −0.627013 0.779009i \(-0.715723\pi\)
−0.627013 + 0.779009i \(0.715723\pi\)
\(114\) −3.92498 −0.367608
\(115\) 0 0
\(116\) −2.45222 −0.227683
\(117\) 7.48052 0.691574
\(118\) −10.0000 −0.920575
\(119\) −2.40550 −0.220512
\(120\) 0 0
\(121\) −9.10331 −0.827574
\(122\) −8.30219 −0.751645
\(123\) 20.2087 1.82216
\(124\) 1.80113 0.161746
\(125\) 0 0
\(126\) 1.65109 0.147091
\(127\) −3.04672 −0.270353 −0.135176 0.990822i \(-0.543160\pi\)
−0.135176 + 0.990822i \(0.543160\pi\)
\(128\) −13.4132 −1.18557
\(129\) −28.8881 −2.54346
\(130\) 0 0
\(131\) −8.70769 −0.760794 −0.380397 0.924823i \(-0.624213\pi\)
−0.380397 + 0.924823i \(0.624213\pi\)
\(132\) −2.37720 −0.206909
\(133\) −0.377203 −0.0327076
\(134\) −5.32836 −0.460300
\(135\) 0 0
\(136\) 13.4132 1.15018
\(137\) 6.59450 0.563406 0.281703 0.959502i \(-0.409101\pi\)
0.281703 + 0.959502i \(0.409101\pi\)
\(138\) −24.3305 −2.07115
\(139\) −10.1677 −0.862409 −0.431205 0.902254i \(-0.641911\pi\)
−0.431205 + 0.902254i \(0.641911\pi\)
\(140\) 0 0
\(141\) −16.3510 −1.37701
\(142\) 3.80113 0.318983
\(143\) −3.88601 −0.324965
\(144\) −13.0566 −1.08805
\(145\) 0 0
\(146\) 5.28164 0.437112
\(147\) −16.3022 −1.34458
\(148\) 4.05447 0.333275
\(149\) −5.54003 −0.453857 −0.226929 0.973911i \(-0.572868\pi\)
−0.226929 + 0.973911i \(0.572868\pi\)
\(150\) 0 0
\(151\) 5.12386 0.416974 0.208487 0.978025i \(-0.433146\pi\)
0.208487 + 0.978025i \(0.433146\pi\)
\(152\) 2.10331 0.170601
\(153\) 16.9066 1.36681
\(154\) −0.857718 −0.0691169
\(155\) 0 0
\(156\) 4.87051 0.389953
\(157\) 19.7643 1.57736 0.788681 0.614803i \(-0.210765\pi\)
0.788681 + 0.614803i \(0.210765\pi\)
\(158\) 11.0926 0.882483
\(159\) 27.4514 2.17704
\(160\) 0 0
\(161\) −2.33823 −0.184279
\(162\) 16.3871 1.28749
\(163\) 13.5195 1.05893 0.529464 0.848332i \(-0.322393\pi\)
0.529464 + 0.848332i \(0.322393\pi\)
\(164\) 6.17270 0.482007
\(165\) 0 0
\(166\) −30.0771 −2.33444
\(167\) 13.1054 1.01413 0.507065 0.861908i \(-0.330731\pi\)
0.507065 + 0.861908i \(0.330731\pi\)
\(168\) −1.88601 −0.145509
\(169\) −5.03817 −0.387551
\(170\) 0 0
\(171\) 2.65109 0.202734
\(172\) −8.82379 −0.672808
\(173\) 1.48827 0.113151 0.0565754 0.998398i \(-0.481982\pi\)
0.0565754 + 0.998398i \(0.481982\pi\)
\(174\) 13.2555 1.00489
\(175\) 0 0
\(176\) 6.78270 0.511265
\(177\) 14.3977 1.08220
\(178\) −2.47839 −0.185763
\(179\) −17.1132 −1.27910 −0.639550 0.768750i \(-0.720879\pi\)
−0.639550 + 0.768750i \(0.720879\pi\)
\(180\) 0 0
\(181\) −5.73891 −0.426569 −0.213285 0.976990i \(-0.568416\pi\)
−0.213285 + 0.976990i \(0.568416\pi\)
\(182\) 1.75733 0.130262
\(183\) 11.9533 0.883612
\(184\) 13.0382 0.961187
\(185\) 0 0
\(186\) −9.73598 −0.713877
\(187\) −8.78270 −0.642255
\(188\) −4.99437 −0.364252
\(189\) 0.312860 0.0227572
\(190\) 0 0
\(191\) 22.3948 1.62043 0.810216 0.586131i \(-0.199350\pi\)
0.810216 + 0.586131i \(0.199350\pi\)
\(192\) 8.00987 0.578063
\(193\) −21.3687 −1.53815 −0.769075 0.639159i \(-0.779283\pi\)
−0.769075 + 0.639159i \(0.779283\pi\)
\(194\) 19.4543 1.39674
\(195\) 0 0
\(196\) −4.97945 −0.355675
\(197\) 1.11399 0.0793682 0.0396841 0.999212i \(-0.487365\pi\)
0.0396841 + 0.999212i \(0.487365\pi\)
\(198\) 6.02830 0.428412
\(199\) −2.22505 −0.157729 −0.0788647 0.996885i \(-0.525130\pi\)
−0.0788647 + 0.996885i \(0.525130\pi\)
\(200\) 0 0
\(201\) 7.67164 0.541116
\(202\) 15.1706 1.06740
\(203\) 1.27389 0.0894096
\(204\) 11.0078 0.770697
\(205\) 0 0
\(206\) 15.6404 1.08972
\(207\) 16.4338 1.14223
\(208\) −13.8967 −0.963562
\(209\) −1.37720 −0.0952631
\(210\) 0 0
\(211\) −9.75441 −0.671521 −0.335760 0.941947i \(-0.608993\pi\)
−0.335760 + 0.941947i \(0.608993\pi\)
\(212\) 8.38495 0.575881
\(213\) −5.47277 −0.374988
\(214\) 25.3460 1.73262
\(215\) 0 0
\(216\) −1.74453 −0.118700
\(217\) −0.935657 −0.0635166
\(218\) 27.6794 1.87468
\(219\) −7.60437 −0.513856
\(220\) 0 0
\(221\) 17.9944 1.21043
\(222\) −21.9164 −1.47093
\(223\) −18.6433 −1.24845 −0.624225 0.781244i \(-0.714585\pi\)
−0.624225 + 0.781244i \(0.714585\pi\)
\(224\) −1.48052 −0.0989211
\(225\) 0 0
\(226\) 22.0099 1.46407
\(227\) −8.04672 −0.534080 −0.267040 0.963686i \(-0.586046\pi\)
−0.267040 + 0.963686i \(0.586046\pi\)
\(228\) 1.72611 0.114314
\(229\) 20.1415 1.33099 0.665493 0.746404i \(-0.268221\pi\)
0.665493 + 0.746404i \(0.268221\pi\)
\(230\) 0 0
\(231\) 1.23492 0.0812518
\(232\) −7.10331 −0.466355
\(233\) −1.73386 −0.113589 −0.0567945 0.998386i \(-0.518088\pi\)
−0.0567945 + 0.998386i \(0.518088\pi\)
\(234\) −12.3510 −0.807412
\(235\) 0 0
\(236\) 4.39775 0.286269
\(237\) −15.9709 −1.03742
\(238\) 3.97170 0.257447
\(239\) −18.3150 −1.18470 −0.592349 0.805682i \(-0.701799\pi\)
−0.592349 + 0.805682i \(0.701799\pi\)
\(240\) 0 0
\(241\) −2.19675 −0.141505 −0.0707526 0.997494i \(-0.522540\pi\)
−0.0707526 + 0.997494i \(0.522540\pi\)
\(242\) 15.0304 0.966192
\(243\) −21.1054 −1.35391
\(244\) 3.65109 0.233737
\(245\) 0 0
\(246\) −33.3665 −2.12737
\(247\) 2.82167 0.179539
\(248\) 5.21730 0.331299
\(249\) 43.3043 2.74430
\(250\) 0 0
\(251\) −13.6716 −0.862946 −0.431473 0.902126i \(-0.642006\pi\)
−0.431473 + 0.902126i \(0.642006\pi\)
\(252\) −0.726109 −0.0457406
\(253\) −8.53711 −0.536723
\(254\) 5.03042 0.315637
\(255\) 0 0
\(256\) 15.4076 0.962976
\(257\) 28.4904 1.77718 0.888591 0.458701i \(-0.151685\pi\)
0.888591 + 0.458701i \(0.151685\pi\)
\(258\) 47.6970 2.96949
\(259\) −2.10624 −0.130875
\(260\) 0 0
\(261\) −8.95328 −0.554194
\(262\) 14.3772 0.888227
\(263\) 5.36945 0.331095 0.165547 0.986202i \(-0.447061\pi\)
0.165547 + 0.986202i \(0.447061\pi\)
\(264\) −6.88601 −0.423805
\(265\) 0 0
\(266\) 0.622797 0.0381861
\(267\) 3.56833 0.218378
\(268\) 2.34328 0.143139
\(269\) 7.06727 0.430899 0.215449 0.976515i \(-0.430878\pi\)
0.215449 + 0.976515i \(0.430878\pi\)
\(270\) 0 0
\(271\) 21.6970 1.31800 0.659000 0.752143i \(-0.270980\pi\)
0.659000 + 0.752143i \(0.270980\pi\)
\(272\) −31.4076 −1.90437
\(273\) −2.53016 −0.153132
\(274\) −10.8881 −0.657776
\(275\) 0 0
\(276\) 10.6999 0.644060
\(277\) 25.1132 1.50891 0.754453 0.656355i \(-0.227902\pi\)
0.754453 + 0.656355i \(0.227902\pi\)
\(278\) 16.7877 1.00686
\(279\) 6.57608 0.393699
\(280\) 0 0
\(281\) −10.0411 −0.599001 −0.299501 0.954096i \(-0.596820\pi\)
−0.299501 + 0.954096i \(0.596820\pi\)
\(282\) 26.9971 1.60765
\(283\) −20.9143 −1.24323 −0.621613 0.783324i \(-0.713522\pi\)
−0.621613 + 0.783324i \(0.713522\pi\)
\(284\) −1.67164 −0.0991936
\(285\) 0 0
\(286\) 6.41617 0.379396
\(287\) −3.20662 −0.189281
\(288\) 10.4055 0.613150
\(289\) 23.6687 1.39228
\(290\) 0 0
\(291\) −28.0099 −1.64197
\(292\) −2.32273 −0.135928
\(293\) −0.818748 −0.0478318 −0.0239159 0.999714i \(-0.507613\pi\)
−0.0239159 + 0.999714i \(0.507613\pi\)
\(294\) 26.9164 1.56980
\(295\) 0 0
\(296\) 11.7445 0.682637
\(297\) 1.14228 0.0662819
\(298\) 9.14711 0.529878
\(299\) 17.4912 1.01154
\(300\) 0 0
\(301\) 4.58383 0.264207
\(302\) −8.45997 −0.486817
\(303\) −21.8422 −1.25480
\(304\) −4.92498 −0.282467
\(305\) 0 0
\(306\) −27.9143 −1.59575
\(307\) −7.44447 −0.424878 −0.212439 0.977174i \(-0.568141\pi\)
−0.212439 + 0.977174i \(0.568141\pi\)
\(308\) 0.377203 0.0214931
\(309\) −22.5187 −1.28104
\(310\) 0 0
\(311\) −0.956204 −0.0542213 −0.0271107 0.999632i \(-0.508631\pi\)
−0.0271107 + 0.999632i \(0.508631\pi\)
\(312\) 14.1084 0.798729
\(313\) 5.98158 0.338099 0.169049 0.985608i \(-0.445930\pi\)
0.169049 + 0.985608i \(0.445930\pi\)
\(314\) −32.6327 −1.84157
\(315\) 0 0
\(316\) −4.87826 −0.274424
\(317\) 22.6511 1.27221 0.636106 0.771602i \(-0.280544\pi\)
0.636106 + 0.771602i \(0.280544\pi\)
\(318\) −45.3249 −2.54169
\(319\) 4.65109 0.260411
\(320\) 0 0
\(321\) −36.4925 −2.03681
\(322\) 3.86064 0.215145
\(323\) 6.37720 0.354837
\(324\) −7.20662 −0.400368
\(325\) 0 0
\(326\) −22.3219 −1.23630
\(327\) −39.8521 −2.20383
\(328\) 17.8804 0.987279
\(329\) 2.59450 0.143039
\(330\) 0 0
\(331\) 4.16283 0.228810 0.114405 0.993434i \(-0.463504\pi\)
0.114405 + 0.993434i \(0.463504\pi\)
\(332\) 13.2272 0.725935
\(333\) 14.8032 0.811213
\(334\) −21.6383 −1.18399
\(335\) 0 0
\(336\) 4.41617 0.240922
\(337\) −18.0304 −0.982180 −0.491090 0.871109i \(-0.663401\pi\)
−0.491090 + 0.871109i \(0.663401\pi\)
\(338\) 8.31849 0.452466
\(339\) −31.6893 −1.72112
\(340\) 0 0
\(341\) −3.41617 −0.184996
\(342\) −4.37720 −0.236692
\(343\) 5.22717 0.282241
\(344\) −25.5598 −1.37809
\(345\) 0 0
\(346\) −2.45726 −0.132103
\(347\) 7.43380 0.399067 0.199534 0.979891i \(-0.436057\pi\)
0.199534 + 0.979891i \(0.436057\pi\)
\(348\) −5.82942 −0.312490
\(349\) 21.9194 1.17332 0.586658 0.809835i \(-0.300443\pi\)
0.586658 + 0.809835i \(0.300443\pi\)
\(350\) 0 0
\(351\) −2.34036 −0.124919
\(352\) −5.40550 −0.288114
\(353\) 10.3695 0.551910 0.275955 0.961171i \(-0.411006\pi\)
0.275955 + 0.961171i \(0.411006\pi\)
\(354\) −23.7720 −1.26347
\(355\) 0 0
\(356\) 1.08993 0.0577664
\(357\) −5.71836 −0.302648
\(358\) 28.2555 1.49335
\(359\) −23.9893 −1.26611 −0.633054 0.774108i \(-0.718199\pi\)
−0.633054 + 0.774108i \(0.718199\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 9.47547 0.498020
\(363\) −21.6404 −1.13583
\(364\) −0.772829 −0.0405073
\(365\) 0 0
\(366\) −19.7360 −1.03162
\(367\) 35.0275 1.82842 0.914210 0.405240i \(-0.132812\pi\)
0.914210 + 0.405240i \(0.132812\pi\)
\(368\) −30.5294 −1.59145
\(369\) 22.5371 1.17323
\(370\) 0 0
\(371\) −4.35586 −0.226145
\(372\) 4.28164 0.221993
\(373\) 30.8003 1.59478 0.797390 0.603464i \(-0.206213\pi\)
0.797390 + 0.603464i \(0.206213\pi\)
\(374\) 14.5011 0.749832
\(375\) 0 0
\(376\) −14.4671 −0.746086
\(377\) −9.52936 −0.490787
\(378\) −0.516561 −0.0265691
\(379\) 0.671640 0.0344998 0.0172499 0.999851i \(-0.494509\pi\)
0.0172499 + 0.999851i \(0.494509\pi\)
\(380\) 0 0
\(381\) −7.24267 −0.371053
\(382\) −36.9759 −1.89185
\(383\) −30.1805 −1.54215 −0.771075 0.636745i \(-0.780280\pi\)
−0.771075 + 0.636745i \(0.780280\pi\)
\(384\) −31.8860 −1.62718
\(385\) 0 0
\(386\) 35.2816 1.79579
\(387\) −32.2165 −1.63766
\(388\) −8.55553 −0.434341
\(389\) −31.5109 −1.59767 −0.798834 0.601552i \(-0.794549\pi\)
−0.798834 + 0.601552i \(0.794549\pi\)
\(390\) 0 0
\(391\) 39.5315 1.99919
\(392\) −14.4239 −0.728518
\(393\) −20.6999 −1.04417
\(394\) −1.83929 −0.0926623
\(395\) 0 0
\(396\) −2.65109 −0.133222
\(397\) −12.5761 −0.631175 −0.315588 0.948896i \(-0.602202\pi\)
−0.315588 + 0.948896i \(0.602202\pi\)
\(398\) 3.67376 0.184149
\(399\) −0.896688 −0.0448905
\(400\) 0 0
\(401\) 24.6036 1.22864 0.614322 0.789056i \(-0.289430\pi\)
0.614322 + 0.789056i \(0.289430\pi\)
\(402\) −12.6666 −0.631752
\(403\) 6.99920 0.348655
\(404\) −6.67164 −0.331926
\(405\) 0 0
\(406\) −2.10331 −0.104386
\(407\) −7.69006 −0.381182
\(408\) 31.8860 1.57859
\(409\) 13.6666 0.675770 0.337885 0.941187i \(-0.390289\pi\)
0.337885 + 0.941187i \(0.390289\pi\)
\(410\) 0 0
\(411\) 15.6765 0.773263
\(412\) −6.87826 −0.338868
\(413\) −2.28456 −0.112416
\(414\) −27.1337 −1.33355
\(415\) 0 0
\(416\) 11.0750 0.542997
\(417\) −24.1706 −1.18364
\(418\) 2.27389 0.111220
\(419\) 12.6893 0.619911 0.309956 0.950751i \(-0.399686\pi\)
0.309956 + 0.950751i \(0.399686\pi\)
\(420\) 0 0
\(421\) 29.1826 1.42227 0.711136 0.703055i \(-0.248181\pi\)
0.711136 + 0.703055i \(0.248181\pi\)
\(422\) 16.1054 0.784000
\(423\) −18.2349 −0.886612
\(424\) 24.2886 1.17956
\(425\) 0 0
\(426\) 9.03605 0.437798
\(427\) −1.89669 −0.0917872
\(428\) −11.1465 −0.538788
\(429\) −9.23784 −0.446007
\(430\) 0 0
\(431\) −29.9816 −1.44416 −0.722081 0.691809i \(-0.756814\pi\)
−0.722081 + 0.691809i \(0.756814\pi\)
\(432\) 4.08489 0.196534
\(433\) −4.76216 −0.228855 −0.114427 0.993432i \(-0.536503\pi\)
−0.114427 + 0.993432i \(0.536503\pi\)
\(434\) 1.54486 0.0741555
\(435\) 0 0
\(436\) −12.1727 −0.582967
\(437\) 6.19887 0.296532
\(438\) 12.5555 0.599926
\(439\) 6.88601 0.328652 0.164326 0.986406i \(-0.447455\pi\)
0.164326 + 0.986406i \(0.447455\pi\)
\(440\) 0 0
\(441\) −18.1805 −0.865736
\(442\) −29.7104 −1.41318
\(443\) 8.65109 0.411026 0.205513 0.978654i \(-0.434114\pi\)
0.205513 + 0.978654i \(0.434114\pi\)
\(444\) 9.63830 0.457413
\(445\) 0 0
\(446\) 30.7819 1.45757
\(447\) −13.1698 −0.622909
\(448\) −1.27097 −0.0600476
\(449\) 8.63055 0.407301 0.203650 0.979044i \(-0.434719\pi\)
0.203650 + 0.979044i \(0.434719\pi\)
\(450\) 0 0
\(451\) −11.7077 −0.551293
\(452\) −9.67939 −0.455280
\(453\) 12.1805 0.572288
\(454\) 13.2859 0.623538
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) −28.6092 −1.33828 −0.669141 0.743135i \(-0.733338\pi\)
−0.669141 + 0.743135i \(0.733338\pi\)
\(458\) −33.2555 −1.55393
\(459\) −5.28939 −0.246888
\(460\) 0 0
\(461\) −39.1025 −1.82119 −0.910593 0.413305i \(-0.864374\pi\)
−0.910593 + 0.413305i \(0.864374\pi\)
\(462\) −2.03897 −0.0948615
\(463\) −8.04380 −0.373827 −0.186913 0.982376i \(-0.559848\pi\)
−0.186913 + 0.982376i \(0.559848\pi\)
\(464\) 16.6327 0.772152
\(465\) 0 0
\(466\) 2.86276 0.132615
\(467\) 17.0488 0.788926 0.394463 0.918912i \(-0.370931\pi\)
0.394463 + 0.918912i \(0.370931\pi\)
\(468\) 5.43167 0.251079
\(469\) −1.21730 −0.0562096
\(470\) 0 0
\(471\) 46.9837 2.16489
\(472\) 12.7389 0.586356
\(473\) 16.7360 0.769521
\(474\) 26.3695 1.21119
\(475\) 0 0
\(476\) −1.74666 −0.0800578
\(477\) 30.6142 1.40173
\(478\) 30.2397 1.38313
\(479\) 22.5478 1.03023 0.515117 0.857120i \(-0.327748\pi\)
0.515117 + 0.857120i \(0.327748\pi\)
\(480\) 0 0
\(481\) 15.7557 0.718399
\(482\) 3.62704 0.165207
\(483\) −5.55845 −0.252918
\(484\) −6.61000 −0.300455
\(485\) 0 0
\(486\) 34.8470 1.58069
\(487\) −24.5577 −1.11281 −0.556407 0.830910i \(-0.687820\pi\)
−0.556407 + 0.830910i \(0.687820\pi\)
\(488\) 10.5761 0.478757
\(489\) 32.1386 1.45336
\(490\) 0 0
\(491\) −16.9298 −0.764032 −0.382016 0.924156i \(-0.624770\pi\)
−0.382016 + 0.924156i \(0.624770\pi\)
\(492\) 14.6738 0.661544
\(493\) −21.5371 −0.969983
\(494\) −4.65884 −0.209611
\(495\) 0 0
\(496\) −12.2165 −0.548537
\(497\) 0.868391 0.0389527
\(498\) −71.4995 −3.20397
\(499\) 0.418295 0.0187255 0.00936273 0.999956i \(-0.497020\pi\)
0.00936273 + 0.999956i \(0.497020\pi\)
\(500\) 0 0
\(501\) 31.1543 1.39187
\(502\) 22.5732 1.00749
\(503\) −12.5993 −0.561776 −0.280888 0.959741i \(-0.590629\pi\)
−0.280888 + 0.959741i \(0.590629\pi\)
\(504\) −2.10331 −0.0936890
\(505\) 0 0
\(506\) 14.0956 0.626624
\(507\) −11.9767 −0.531906
\(508\) −2.21225 −0.0981528
\(509\) 12.8209 0.568275 0.284138 0.958784i \(-0.408293\pi\)
0.284138 + 0.958784i \(0.408293\pi\)
\(510\) 0 0
\(511\) 1.20662 0.0533779
\(512\) 1.38708 0.0613007
\(513\) −0.829422 −0.0366199
\(514\) −47.0403 −2.07486
\(515\) 0 0
\(516\) −20.9759 −0.923415
\(517\) 9.47277 0.416612
\(518\) 3.47759 0.152797
\(519\) 3.53791 0.155297
\(520\) 0 0
\(521\) 11.3743 0.498316 0.249158 0.968463i \(-0.419846\pi\)
0.249158 + 0.968463i \(0.419846\pi\)
\(522\) 14.7827 0.647021
\(523\) 33.0948 1.44713 0.723566 0.690255i \(-0.242502\pi\)
0.723566 + 0.690255i \(0.242502\pi\)
\(524\) −6.32273 −0.276210
\(525\) 0 0
\(526\) −8.86547 −0.386553
\(527\) 15.8187 0.689075
\(528\) 16.1239 0.701701
\(529\) 15.4260 0.670698
\(530\) 0 0
\(531\) 16.0566 0.696797
\(532\) −0.273891 −0.0118747
\(533\) 23.9872 1.03900
\(534\) −5.89164 −0.254956
\(535\) 0 0
\(536\) 6.78775 0.293186
\(537\) −40.6815 −1.75554
\(538\) −11.6687 −0.503074
\(539\) 9.44447 0.406802
\(540\) 0 0
\(541\) 31.4338 1.35144 0.675722 0.737156i \(-0.263832\pi\)
0.675722 + 0.737156i \(0.263832\pi\)
\(542\) −35.8238 −1.53876
\(543\) −13.6425 −0.585458
\(544\) 25.0304 1.07317
\(545\) 0 0
\(546\) 4.17753 0.178782
\(547\) −23.3460 −0.998202 −0.499101 0.866544i \(-0.666336\pi\)
−0.499101 + 0.866544i \(0.666336\pi\)
\(548\) 4.78833 0.204547
\(549\) 13.3305 0.568931
\(550\) 0 0
\(551\) −3.37720 −0.143874
\(552\) 30.9944 1.31921
\(553\) 2.53418 0.107764
\(554\) −41.4642 −1.76165
\(555\) 0 0
\(556\) −7.38283 −0.313102
\(557\) 15.9229 0.674673 0.337337 0.941384i \(-0.390474\pi\)
0.337337 + 0.941384i \(0.390474\pi\)
\(558\) −10.8577 −0.459644
\(559\) −34.2894 −1.45029
\(560\) 0 0
\(561\) −20.8783 −0.881481
\(562\) 16.5788 0.699334
\(563\) −29.6404 −1.24919 −0.624597 0.780947i \(-0.714737\pi\)
−0.624597 + 0.780947i \(0.714737\pi\)
\(564\) −11.8726 −0.499928
\(565\) 0 0
\(566\) 34.5315 1.45147
\(567\) 3.74373 0.157222
\(568\) −4.84222 −0.203175
\(569\) −7.64334 −0.320426 −0.160213 0.987082i \(-0.551218\pi\)
−0.160213 + 0.987082i \(0.551218\pi\)
\(570\) 0 0
\(571\) −10.5577 −0.441824 −0.220912 0.975294i \(-0.570903\pi\)
−0.220912 + 0.975294i \(0.570903\pi\)
\(572\) −2.82167 −0.117980
\(573\) 53.2370 2.22401
\(574\) 5.29444 0.220986
\(575\) 0 0
\(576\) 8.93273 0.372197
\(577\) 38.9447 1.62129 0.810645 0.585538i \(-0.199117\pi\)
0.810645 + 0.585538i \(0.199117\pi\)
\(578\) −39.0793 −1.62548
\(579\) −50.7976 −2.11108
\(580\) 0 0
\(581\) −6.87131 −0.285070
\(582\) 46.2469 1.91700
\(583\) −15.9036 −0.658661
\(584\) −6.72823 −0.278416
\(585\) 0 0
\(586\) 1.35183 0.0558436
\(587\) −10.9992 −0.453986 −0.226993 0.973896i \(-0.572889\pi\)
−0.226993 + 0.973896i \(0.572889\pi\)
\(588\) −11.8372 −0.488157
\(589\) 2.48052 0.102208
\(590\) 0 0
\(591\) 2.64817 0.108931
\(592\) −27.5003 −1.13025
\(593\) 26.7848 1.09992 0.549960 0.835191i \(-0.314643\pi\)
0.549960 + 0.835191i \(0.314643\pi\)
\(594\) −1.88601 −0.0773841
\(595\) 0 0
\(596\) −4.02267 −0.164775
\(597\) −5.28939 −0.216480
\(598\) −28.8796 −1.18097
\(599\) −3.44930 −0.140934 −0.0704672 0.997514i \(-0.522449\pi\)
−0.0704672 + 0.997514i \(0.522449\pi\)
\(600\) 0 0
\(601\) −37.0686 −1.51206 −0.756030 0.654537i \(-0.772863\pi\)
−0.756030 + 0.654537i \(0.772863\pi\)
\(602\) −7.56833 −0.308462
\(603\) 8.55553 0.348408
\(604\) 3.72048 0.151384
\(605\) 0 0
\(606\) 36.0635 1.46498
\(607\) −14.8732 −0.603685 −0.301843 0.953358i \(-0.597602\pi\)
−0.301843 + 0.953358i \(0.597602\pi\)
\(608\) 3.92498 0.159179
\(609\) 3.02830 0.122713
\(610\) 0 0
\(611\) −19.4082 −0.785172
\(612\) 12.2760 0.496228
\(613\) −40.4671 −1.63445 −0.817226 0.576317i \(-0.804489\pi\)
−0.817226 + 0.576317i \(0.804489\pi\)
\(614\) 12.2915 0.496045
\(615\) 0 0
\(616\) 1.09264 0.0440237
\(617\) 8.76991 0.353063 0.176532 0.984295i \(-0.443512\pi\)
0.176532 + 0.984295i \(0.443512\pi\)
\(618\) 37.1805 1.49562
\(619\) 32.0510 1.28824 0.644119 0.764926i \(-0.277224\pi\)
0.644119 + 0.764926i \(0.277224\pi\)
\(620\) 0 0
\(621\) −5.14148 −0.206321
\(622\) 1.57878 0.0633034
\(623\) −0.566205 −0.0226845
\(624\) −33.0352 −1.32247
\(625\) 0 0
\(626\) −9.87614 −0.394730
\(627\) −3.27389 −0.130747
\(628\) 14.3510 0.572668
\(629\) 35.6092 1.41983
\(630\) 0 0
\(631\) −18.9709 −0.755220 −0.377610 0.925965i \(-0.623254\pi\)
−0.377610 + 0.925965i \(0.623254\pi\)
\(632\) −14.1308 −0.562093
\(633\) −23.1882 −0.921648
\(634\) −37.3991 −1.48531
\(635\) 0 0
\(636\) 19.9327 0.790384
\(637\) −19.3502 −0.766684
\(638\) −7.67939 −0.304030
\(639\) −6.10331 −0.241443
\(640\) 0 0
\(641\) −44.3433 −1.75145 −0.875727 0.482806i \(-0.839617\pi\)
−0.875727 + 0.482806i \(0.839617\pi\)
\(642\) 60.2525 2.37798
\(643\) 42.9397 1.69338 0.846688 0.532090i \(-0.178593\pi\)
0.846688 + 0.532090i \(0.178593\pi\)
\(644\) −1.69781 −0.0669032
\(645\) 0 0
\(646\) −10.5294 −0.414272
\(647\) 17.9864 0.707118 0.353559 0.935412i \(-0.384971\pi\)
0.353559 + 0.935412i \(0.384971\pi\)
\(648\) −20.8753 −0.820061
\(649\) −8.34116 −0.327419
\(650\) 0 0
\(651\) −2.22425 −0.0871751
\(652\) 9.81663 0.384449
\(653\) −21.7274 −0.850260 −0.425130 0.905132i \(-0.639772\pi\)
−0.425130 + 0.905132i \(0.639772\pi\)
\(654\) 65.7995 2.57297
\(655\) 0 0
\(656\) −41.8676 −1.63465
\(657\) −8.48052 −0.330856
\(658\) −4.28376 −0.166998
\(659\) −40.8590 −1.59164 −0.795821 0.605532i \(-0.792960\pi\)
−0.795821 + 0.605532i \(0.792960\pi\)
\(660\) 0 0
\(661\) 33.7282 1.31188 0.655938 0.754815i \(-0.272273\pi\)
0.655938 + 0.754815i \(0.272273\pi\)
\(662\) −6.87322 −0.267135
\(663\) 42.7763 1.66129
\(664\) 38.3150 1.48691
\(665\) 0 0
\(666\) −24.4415 −0.947091
\(667\) −20.9349 −0.810601
\(668\) 9.51598 0.368184
\(669\) −44.3190 −1.71347
\(670\) 0 0
\(671\) −6.92498 −0.267336
\(672\) −3.51948 −0.135767
\(673\) −11.0622 −0.426417 −0.213209 0.977007i \(-0.568391\pi\)
−0.213209 + 0.977007i \(0.568391\pi\)
\(674\) 29.7699 1.14669
\(675\) 0 0
\(676\) −3.65826 −0.140702
\(677\) −6.14711 −0.236253 −0.118126 0.992999i \(-0.537689\pi\)
−0.118126 + 0.992999i \(0.537689\pi\)
\(678\) 52.3219 2.00941
\(679\) 4.44447 0.170563
\(680\) 0 0
\(681\) −19.1287 −0.733013
\(682\) 5.64042 0.215983
\(683\) −13.6073 −0.520669 −0.260334 0.965519i \(-0.583833\pi\)
−0.260334 + 0.965519i \(0.583833\pi\)
\(684\) 1.92498 0.0736036
\(685\) 0 0
\(686\) −8.63055 −0.329516
\(687\) 47.8804 1.82675
\(688\) 59.8492 2.28173
\(689\) 32.5840 1.24135
\(690\) 0 0
\(691\) −40.3043 −1.53325 −0.766624 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(692\) 1.08064 0.0410799
\(693\) 1.37720 0.0523156
\(694\) −12.2739 −0.465911
\(695\) 0 0
\(696\) −16.8860 −0.640063
\(697\) 54.2130 2.05346
\(698\) −36.1909 −1.36985
\(699\) −4.12174 −0.155898
\(700\) 0 0
\(701\) −3.23704 −0.122261 −0.0611307 0.998130i \(-0.519471\pi\)
−0.0611307 + 0.998130i \(0.519471\pi\)
\(702\) 3.86415 0.145843
\(703\) 5.58383 0.210598
\(704\) −4.64042 −0.174892
\(705\) 0 0
\(706\) −17.1209 −0.644355
\(707\) 3.46582 0.130345
\(708\) 10.4543 0.392898
\(709\) 20.2370 0.760018 0.380009 0.924983i \(-0.375921\pi\)
0.380009 + 0.924983i \(0.375921\pi\)
\(710\) 0 0
\(711\) −17.8110 −0.667965
\(712\) 3.15720 0.118321
\(713\) 15.3764 0.575851
\(714\) 9.44155 0.353341
\(715\) 0 0
\(716\) −12.4260 −0.464383
\(717\) −43.5384 −1.62597
\(718\) 39.6086 1.47818
\(719\) 47.8804 1.78564 0.892819 0.450416i \(-0.148724\pi\)
0.892819 + 0.450416i \(0.148724\pi\)
\(720\) 0 0
\(721\) 3.57315 0.133071
\(722\) −1.65109 −0.0614473
\(723\) −5.22212 −0.194213
\(724\) −4.16707 −0.154868
\(725\) 0 0
\(726\) 35.7304 1.32608
\(727\) 17.7926 0.659890 0.329945 0.944000i \(-0.392970\pi\)
0.329945 + 0.944000i \(0.392970\pi\)
\(728\) −2.23864 −0.0829697
\(729\) −20.3969 −0.755443
\(730\) 0 0
\(731\) −77.4968 −2.86632
\(732\) 8.67939 0.320799
\(733\) −5.66097 −0.209093 −0.104546 0.994520i \(-0.533339\pi\)
−0.104546 + 0.994520i \(0.533339\pi\)
\(734\) −57.8337 −2.13468
\(735\) 0 0
\(736\) 24.3305 0.896834
\(737\) −4.44447 −0.163714
\(738\) −37.2109 −1.36975
\(739\) −5.59955 −0.205983 −0.102991 0.994682i \(-0.532841\pi\)
−0.102991 + 0.994682i \(0.532841\pi\)
\(740\) 0 0
\(741\) 6.70769 0.246413
\(742\) 7.19193 0.264024
\(743\) 5.81663 0.213391 0.106696 0.994292i \(-0.465973\pi\)
0.106696 + 0.994292i \(0.465973\pi\)
\(744\) 12.4026 0.454700
\(745\) 0 0
\(746\) −50.8542 −1.86191
\(747\) 48.2936 1.76697
\(748\) −6.37720 −0.233174
\(749\) 5.79045 0.211579
\(750\) 0 0
\(751\) −41.1797 −1.50267 −0.751333 0.659923i \(-0.770589\pi\)
−0.751333 + 0.659923i \(0.770589\pi\)
\(752\) 33.8753 1.23531
\(753\) −32.5003 −1.18438
\(754\) 15.7339 0.572993
\(755\) 0 0
\(756\) 0.227171 0.00826212
\(757\) −33.5208 −1.21833 −0.609167 0.793042i \(-0.708496\pi\)
−0.609167 + 0.793042i \(0.708496\pi\)
\(758\) −1.10894 −0.0402785
\(759\) −20.2944 −0.736641
\(760\) 0 0
\(761\) 31.5032 1.14199 0.570995 0.820954i \(-0.306558\pi\)
0.570995 + 0.820954i \(0.306558\pi\)
\(762\) 11.9583 0.433204
\(763\) 6.32353 0.228927
\(764\) 16.2611 0.588306
\(765\) 0 0
\(766\) 49.8307 1.80046
\(767\) 17.0897 0.617074
\(768\) 36.6270 1.32166
\(769\) −1.22425 −0.0441475 −0.0220737 0.999756i \(-0.507027\pi\)
−0.0220737 + 0.999756i \(0.507027\pi\)
\(770\) 0 0
\(771\) 67.7274 2.43914
\(772\) −15.5160 −0.558432
\(773\) 40.6687 1.46275 0.731376 0.681974i \(-0.238878\pi\)
0.731376 + 0.681974i \(0.238878\pi\)
\(774\) 53.1924 1.91196
\(775\) 0 0
\(776\) −24.7827 −0.889647
\(777\) −5.00695 −0.179623
\(778\) 52.0275 1.86528
\(779\) 8.50106 0.304582
\(780\) 0 0
\(781\) 3.17058 0.113452
\(782\) −65.2702 −2.33406
\(783\) 2.80113 0.100104
\(784\) 33.7742 1.20622
\(785\) 0 0
\(786\) 34.1775 1.21907
\(787\) −39.9525 −1.42415 −0.712076 0.702102i \(-0.752245\pi\)
−0.712076 + 0.702102i \(0.752245\pi\)
\(788\) 0.808876 0.0288150
\(789\) 12.7643 0.454420
\(790\) 0 0
\(791\) 5.02830 0.178786
\(792\) −7.67939 −0.272875
\(793\) 14.1882 0.503838
\(794\) 20.7643 0.736897
\(795\) 0 0
\(796\) −1.61563 −0.0572644
\(797\) −4.53791 −0.160741 −0.0803705 0.996765i \(-0.525610\pi\)
−0.0803705 + 0.996765i \(0.525610\pi\)
\(798\) 1.48052 0.0524097
\(799\) −43.8641 −1.55180
\(800\) 0 0
\(801\) 3.97945 0.140607
\(802\) −40.6228 −1.43444
\(803\) 4.40550 0.155467
\(804\) 5.57045 0.196455
\(805\) 0 0
\(806\) −11.5563 −0.407054
\(807\) 16.8003 0.591399
\(808\) −19.3257 −0.679874
\(809\) 55.5958 1.95465 0.977323 0.211756i \(-0.0679183\pi\)
0.977323 + 0.211756i \(0.0679183\pi\)
\(810\) 0 0
\(811\) 42.3326 1.48650 0.743249 0.669014i \(-0.233284\pi\)
0.743249 + 0.669014i \(0.233284\pi\)
\(812\) 0.924984 0.0324606
\(813\) 51.5782 1.80893
\(814\) 12.6970 0.445030
\(815\) 0 0
\(816\) −74.6623 −2.61370
\(817\) −12.1522 −0.425150
\(818\) −22.5648 −0.788961
\(819\) −2.82167 −0.0985972
\(820\) 0 0
\(821\) −3.10543 −0.108380 −0.0541902 0.998531i \(-0.517258\pi\)
−0.0541902 + 0.998531i \(0.517258\pi\)
\(822\) −25.8833 −0.902784
\(823\) −45.2002 −1.57558 −0.787790 0.615944i \(-0.788775\pi\)
−0.787790 + 0.615944i \(0.788775\pi\)
\(824\) −19.9242 −0.694092
\(825\) 0 0
\(826\) 3.77203 0.131246
\(827\) −35.6503 −1.23968 −0.619841 0.784727i \(-0.712803\pi\)
−0.619841 + 0.784727i \(0.712803\pi\)
\(828\) 11.9327 0.414691
\(829\) −18.4330 −0.640204 −0.320102 0.947383i \(-0.603717\pi\)
−0.320102 + 0.947383i \(0.603717\pi\)
\(830\) 0 0
\(831\) 59.6991 2.07094
\(832\) 9.50749 0.329613
\(833\) −43.7331 −1.51526
\(834\) 39.9079 1.38190
\(835\) 0 0
\(836\) −1.00000 −0.0345857
\(837\) −2.05739 −0.0711139
\(838\) −20.9512 −0.723746
\(839\) −14.1805 −0.489564 −0.244782 0.969578i \(-0.578716\pi\)
−0.244782 + 0.969578i \(0.578716\pi\)
\(840\) 0 0
\(841\) −17.5945 −0.606707
\(842\) −48.1832 −1.66050
\(843\) −23.8697 −0.822117
\(844\) −7.08277 −0.243799
\(845\) 0 0
\(846\) 30.1076 1.03512
\(847\) 3.43380 0.117987
\(848\) −56.8726 −1.95301
\(849\) −49.7176 −1.70630
\(850\) 0 0
\(851\) 34.6134 1.18653
\(852\) −3.97383 −0.136141
\(853\) 42.6639 1.46078 0.730392 0.683028i \(-0.239337\pi\)
0.730392 + 0.683028i \(0.239337\pi\)
\(854\) 3.13161 0.107161
\(855\) 0 0
\(856\) −32.2880 −1.10358
\(857\) −1.42392 −0.0486403 −0.0243201 0.999704i \(-0.507742\pi\)
−0.0243201 + 0.999704i \(0.507742\pi\)
\(858\) 15.2525 0.520713
\(859\) −8.27894 −0.282474 −0.141237 0.989976i \(-0.545108\pi\)
−0.141237 + 0.989976i \(0.545108\pi\)
\(860\) 0 0
\(861\) −7.62280 −0.259784
\(862\) 49.5024 1.68606
\(863\) 30.4154 1.03535 0.517676 0.855577i \(-0.326797\pi\)
0.517676 + 0.855577i \(0.326797\pi\)
\(864\) −3.25547 −0.110753
\(865\) 0 0
\(866\) 7.86276 0.267188
\(867\) 56.2653 1.91087
\(868\) −0.679390 −0.0230600
\(869\) 9.25254 0.313871
\(870\) 0 0
\(871\) 9.10602 0.308546
\(872\) −35.2605 −1.19407
\(873\) −31.2370 −1.05721
\(874\) −10.2349 −0.346201
\(875\) 0 0
\(876\) −5.52161 −0.186558
\(877\) 57.8484 1.95340 0.976700 0.214608i \(-0.0688474\pi\)
0.976700 + 0.214608i \(0.0688474\pi\)
\(878\) −11.3695 −0.383700
\(879\) −1.94633 −0.0656481
\(880\) 0 0
\(881\) 18.4055 0.620097 0.310049 0.950721i \(-0.399655\pi\)
0.310049 + 0.950721i \(0.399655\pi\)
\(882\) 30.0176 1.01075
\(883\) −18.7947 −0.632492 −0.316246 0.948677i \(-0.602422\pi\)
−0.316246 + 0.948677i \(0.602422\pi\)
\(884\) 13.0659 0.439453
\(885\) 0 0
\(886\) −14.2838 −0.479872
\(887\) −11.1471 −0.374283 −0.187142 0.982333i \(-0.559922\pi\)
−0.187142 + 0.982333i \(0.559922\pi\)
\(888\) 27.9191 0.936905
\(889\) 1.14923 0.0385440
\(890\) 0 0
\(891\) 13.6687 0.457919
\(892\) −13.5371 −0.453256
\(893\) −6.87826 −0.230172
\(894\) 21.7445 0.727246
\(895\) 0 0
\(896\) 5.05952 0.169027
\(897\) 41.5801 1.38832
\(898\) −14.2498 −0.475523
\(899\) −8.37720 −0.279395
\(900\) 0 0
\(901\) 73.6425 2.45339
\(902\) 19.3305 0.643635
\(903\) 10.8967 0.362619
\(904\) −28.0382 −0.932536
\(905\) 0 0
\(906\) −20.1111 −0.668145
\(907\) 38.5598 1.28036 0.640178 0.768226i \(-0.278861\pi\)
0.640178 + 0.768226i \(0.278861\pi\)
\(908\) −5.84280 −0.193900
\(909\) −24.3588 −0.807930
\(910\) 0 0
\(911\) −1.79820 −0.0595771 −0.0297885 0.999556i \(-0.509483\pi\)
−0.0297885 + 0.999556i \(0.509483\pi\)
\(912\) −11.7077 −0.387680
\(913\) −25.0878 −0.830285
\(914\) 47.2365 1.56244
\(915\) 0 0
\(916\) 14.6249 0.483221
\(917\) 3.28456 0.108466
\(918\) 8.73328 0.288241
\(919\) −32.4458 −1.07029 −0.535144 0.844761i \(-0.679743\pi\)
−0.535144 + 0.844761i \(0.679743\pi\)
\(920\) 0 0
\(921\) −17.6970 −0.583136
\(922\) 64.5619 2.12623
\(923\) −6.49602 −0.213819
\(924\) 0.896688 0.0294989
\(925\) 0 0
\(926\) 13.2811 0.436443
\(927\) −25.1132 −0.824825
\(928\) −13.2555 −0.435132
\(929\) −18.4231 −0.604443 −0.302222 0.953238i \(-0.597728\pi\)
−0.302222 + 0.953238i \(0.597728\pi\)
\(930\) 0 0
\(931\) −6.85772 −0.224753
\(932\) −1.25897 −0.0412390
\(933\) −2.27309 −0.0744176
\(934\) −28.1492 −0.921071
\(935\) 0 0
\(936\) 15.7339 0.514277
\(937\) 8.29926 0.271125 0.135563 0.990769i \(-0.456716\pi\)
0.135563 + 0.990769i \(0.456716\pi\)
\(938\) 2.00987 0.0656247
\(939\) 14.2194 0.464033
\(940\) 0 0
\(941\) 33.6687 1.09757 0.548784 0.835964i \(-0.315091\pi\)
0.548784 + 0.835964i \(0.315091\pi\)
\(942\) −77.5745 −2.52751
\(943\) 52.6970 1.71605
\(944\) −29.8286 −0.970839
\(945\) 0 0
\(946\) −27.6327 −0.898416
\(947\) 1.35103 0.0439026 0.0219513 0.999759i \(-0.493012\pi\)
0.0219513 + 0.999759i \(0.493012\pi\)
\(948\) −11.5966 −0.376641
\(949\) −9.02617 −0.293002
\(950\) 0 0
\(951\) 53.8462 1.74608
\(952\) −5.05952 −0.163980
\(953\) −6.70769 −0.217283 −0.108642 0.994081i \(-0.534650\pi\)
−0.108642 + 0.994081i \(0.534650\pi\)
\(954\) −50.5470 −1.63652
\(955\) 0 0
\(956\) −13.2987 −0.430110
\(957\) 11.0566 0.357409
\(958\) −37.2285 −1.20280
\(959\) −2.48746 −0.0803244
\(960\) 0 0
\(961\) −24.8470 −0.801518
\(962\) −26.0142 −0.838731
\(963\) −40.6970 −1.31144
\(964\) −1.59508 −0.0513741
\(965\) 0 0
\(966\) 9.17753 0.295282
\(967\) 19.5174 0.627636 0.313818 0.949483i \(-0.398392\pi\)
0.313818 + 0.949483i \(0.398392\pi\)
\(968\) −19.1471 −0.615411
\(969\) 15.1599 0.487006
\(970\) 0 0
\(971\) −8.50669 −0.272993 −0.136496 0.990641i \(-0.543584\pi\)
−0.136496 + 0.990641i \(0.543584\pi\)
\(972\) −15.3249 −0.491545
\(973\) 3.83527 0.122953
\(974\) 40.5470 1.29921
\(975\) 0 0
\(976\) −24.7643 −0.792685
\(977\) 19.7048 0.630411 0.315206 0.949023i \(-0.397927\pi\)
0.315206 + 0.949023i \(0.397927\pi\)
\(978\) −53.0638 −1.69679
\(979\) −2.06727 −0.0660701
\(980\) 0 0
\(981\) −44.4437 −1.41898
\(982\) 27.9527 0.892006
\(983\) −12.4677 −0.397658 −0.198829 0.980034i \(-0.563714\pi\)
−0.198829 + 0.980034i \(0.563714\pi\)
\(984\) 42.5053 1.35502
\(985\) 0 0
\(986\) 35.5598 1.13245
\(987\) 6.16765 0.196319
\(988\) 2.04884 0.0651824
\(989\) −75.3297 −2.39534
\(990\) 0 0
\(991\) 25.0841 0.796822 0.398411 0.917207i \(-0.369562\pi\)
0.398411 + 0.917207i \(0.369562\pi\)
\(992\) 9.73598 0.309118
\(993\) 9.89589 0.314036
\(994\) −1.43380 −0.0454772
\(995\) 0 0
\(996\) 31.4437 0.996331
\(997\) 11.8775 0.376163 0.188082 0.982153i \(-0.439773\pi\)
0.188082 + 0.982153i \(0.439773\pi\)
\(998\) −0.690644 −0.0218620
\(999\) −4.63135 −0.146529
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 475.2.a.g.1.1 yes 3
3.2 odd 2 4275.2.a.ba.1.3 3
4.3 odd 2 7600.2.a.bh.1.1 3
5.2 odd 4 475.2.b.b.324.2 6
5.3 odd 4 475.2.b.b.324.5 6
5.4 even 2 475.2.a.e.1.3 3
15.14 odd 2 4275.2.a.bm.1.1 3
19.18 odd 2 9025.2.a.y.1.3 3
20.19 odd 2 7600.2.a.cc.1.3 3
95.94 odd 2 9025.2.a.bc.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.3 3 5.4 even 2
475.2.a.g.1.1 yes 3 1.1 even 1 trivial
475.2.b.b.324.2 6 5.2 odd 4
475.2.b.b.324.5 6 5.3 odd 4
4275.2.a.ba.1.3 3 3.2 odd 2
4275.2.a.bm.1.1 3 15.14 odd 2
7600.2.a.bh.1.1 3 4.3 odd 2
7600.2.a.cc.1.3 3 20.19 odd 2
9025.2.a.y.1.3 3 19.18 odd 2
9025.2.a.bc.1.1 3 95.94 odd 2