Properties

Label 2275.2.a.m.1.1
Level $2275$
Weight $2$
Character 2275.1
Self dual yes
Analytic conductor $18.166$
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] = [2275,2,Mod(1,2275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2275, 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("2275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2275.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: \(18.1659664598\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 2275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.34292 q^{2} +1.14637 q^{3} +3.48929 q^{4} -2.68585 q^{6} +1.00000 q^{7} -3.48929 q^{8} -1.68585 q^{9} +O(q^{10})\) \(q-2.34292 q^{2} +1.14637 q^{3} +3.48929 q^{4} -2.68585 q^{6} +1.00000 q^{7} -3.48929 q^{8} -1.68585 q^{9} +1.14637 q^{11} +4.00000 q^{12} -1.00000 q^{13} -2.34292 q^{14} +1.19656 q^{16} -5.83221 q^{17} +3.94981 q^{18} -3.34292 q^{19} +1.14637 q^{21} -2.68585 q^{22} +3.17513 q^{23} -4.00000 q^{24} +2.34292 q^{26} -5.37169 q^{27} +3.48929 q^{28} +10.4893 q^{29} +1.63565 q^{31} +4.17513 q^{32} +1.31415 q^{33} +13.6644 q^{34} -5.88240 q^{36} -8.51806 q^{37} +7.83221 q^{38} -1.14637 q^{39} -0.292731 q^{41} -2.68585 q^{42} +8.15371 q^{43} +4.00000 q^{44} -7.43910 q^{46} +10.6142 q^{47} +1.37169 q^{48} +1.00000 q^{49} -6.68585 q^{51} -3.48929 q^{52} +0.782020 q^{53} +12.5855 q^{54} -3.48929 q^{56} -3.83221 q^{57} -24.5756 q^{58} +12.6430 q^{59} -2.00000 q^{61} -3.83221 q^{62} -1.68585 q^{63} -12.1751 q^{64} -3.07896 q^{66} +6.10038 q^{67} -20.3503 q^{68} +3.63986 q^{69} +1.53948 q^{71} +5.88240 q^{72} +15.3001 q^{73} +19.9572 q^{74} -11.6644 q^{76} +1.14637 q^{77} +2.68585 q^{78} +0.882404 q^{79} -1.10038 q^{81} +0.685846 q^{82} +12.1292 q^{83} +4.00000 q^{84} -19.1035 q^{86} +12.0246 q^{87} -4.00000 q^{88} +5.73604 q^{89} -1.00000 q^{91} +11.0790 q^{92} +1.87506 q^{93} -24.8683 q^{94} +4.78623 q^{96} +5.34292 q^{97} -2.34292 q^{98} -1.93260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 2 q^{3} + 3 q^{4} + 4 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 2 q^{3} + 3 q^{4} + 4 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} + 2 q^{11} + 12 q^{12} - 3 q^{13} - q^{14} - q^{16} - 4 q^{17} + 15 q^{18} - 4 q^{19} + 2 q^{21} + 4 q^{22} - 10 q^{23} - 12 q^{24} + q^{26} + 8 q^{27} + 3 q^{28} + 24 q^{29} - 4 q^{31} - 7 q^{32} + 16 q^{33} + 14 q^{34} - q^{36} + 10 q^{38} - 2 q^{39} + 2 q^{41} + 4 q^{42} - 10 q^{43} + 12 q^{44} - 18 q^{46} + 8 q^{47} - 20 q^{48} + 3 q^{49} - 8 q^{51} - 3 q^{52} - 8 q^{53} + 32 q^{54} - 3 q^{56} + 2 q^{57} - 12 q^{58} - 4 q^{59} - 6 q^{61} + 2 q^{62} + 7 q^{63} - 17 q^{64} + 12 q^{66} + 12 q^{67} - 22 q^{68} - 6 q^{69} - 6 q^{71} + q^{72} + 10 q^{73} + 30 q^{74} - 8 q^{76} + 2 q^{77} - 4 q^{78} - 14 q^{79} + 3 q^{81} - 10 q^{82} + 12 q^{83} + 12 q^{84} - 26 q^{86} + 26 q^{87} - 12 q^{88} + 2 q^{89} - 3 q^{91} + 12 q^{92} + 22 q^{93} - 10 q^{94} - 4 q^{96} + 10 q^{97} - q^{98} + 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.34292 −1.65670 −0.828348 0.560213i \(-0.810719\pi\)
−0.828348 + 0.560213i \(0.810719\pi\)
\(3\) 1.14637 0.661854 0.330927 0.943656i \(-0.392639\pi\)
0.330927 + 0.943656i \(0.392639\pi\)
\(4\) 3.48929 1.74464
\(5\) 0 0
\(6\) −2.68585 −1.09649
\(7\) 1.00000 0.377964
\(8\) −3.48929 −1.23365
\(9\) −1.68585 −0.561949
\(10\) 0 0
\(11\) 1.14637 0.345642 0.172821 0.984953i \(-0.444712\pi\)
0.172821 + 0.984953i \(0.444712\pi\)
\(12\) 4.00000 1.15470
\(13\) −1.00000 −0.277350
\(14\) −2.34292 −0.626173
\(15\) 0 0
\(16\) 1.19656 0.299139
\(17\) −5.83221 −1.41452 −0.707260 0.706954i \(-0.750069\pi\)
−0.707260 + 0.706954i \(0.750069\pi\)
\(18\) 3.94981 0.930979
\(19\) −3.34292 −0.766919 −0.383460 0.923558i \(-0.625267\pi\)
−0.383460 + 0.923558i \(0.625267\pi\)
\(20\) 0 0
\(21\) 1.14637 0.250157
\(22\) −2.68585 −0.572624
\(23\) 3.17513 0.662061 0.331031 0.943620i \(-0.392604\pi\)
0.331031 + 0.943620i \(0.392604\pi\)
\(24\) −4.00000 −0.816497
\(25\) 0 0
\(26\) 2.34292 0.459485
\(27\) −5.37169 −1.03378
\(28\) 3.48929 0.659414
\(29\) 10.4893 1.94781 0.973906 0.226952i \(-0.0728760\pi\)
0.973906 + 0.226952i \(0.0728760\pi\)
\(30\) 0 0
\(31\) 1.63565 0.293772 0.146886 0.989153i \(-0.453075\pi\)
0.146886 + 0.989153i \(0.453075\pi\)
\(32\) 4.17513 0.738067
\(33\) 1.31415 0.228765
\(34\) 13.6644 2.34343
\(35\) 0 0
\(36\) −5.88240 −0.980401
\(37\) −8.51806 −1.40036 −0.700180 0.713966i \(-0.746897\pi\)
−0.700180 + 0.713966i \(0.746897\pi\)
\(38\) 7.83221 1.27055
\(39\) −1.14637 −0.183565
\(40\) 0 0
\(41\) −0.292731 −0.0457169 −0.0228584 0.999739i \(-0.507277\pi\)
−0.0228584 + 0.999739i \(0.507277\pi\)
\(42\) −2.68585 −0.414435
\(43\) 8.15371 1.24343 0.621715 0.783244i \(-0.286436\pi\)
0.621715 + 0.783244i \(0.286436\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −7.43910 −1.09683
\(47\) 10.6142 1.54824 0.774122 0.633036i \(-0.218191\pi\)
0.774122 + 0.633036i \(0.218191\pi\)
\(48\) 1.37169 0.197987
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.68585 −0.936206
\(52\) −3.48929 −0.483877
\(53\) 0.782020 0.107419 0.0537093 0.998557i \(-0.482896\pi\)
0.0537093 + 0.998557i \(0.482896\pi\)
\(54\) 12.5855 1.71266
\(55\) 0 0
\(56\) −3.48929 −0.466276
\(57\) −3.83221 −0.507589
\(58\) −24.5756 −3.22693
\(59\) 12.6430 1.64598 0.822989 0.568057i \(-0.192305\pi\)
0.822989 + 0.568057i \(0.192305\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −3.83221 −0.486691
\(63\) −1.68585 −0.212397
\(64\) −12.1751 −1.52189
\(65\) 0 0
\(66\) −3.07896 −0.378994
\(67\) 6.10038 0.745281 0.372640 0.927976i \(-0.378453\pi\)
0.372640 + 0.927976i \(0.378453\pi\)
\(68\) −20.3503 −2.46783
\(69\) 3.63986 0.438188
\(70\) 0 0
\(71\) 1.53948 0.182703 0.0913514 0.995819i \(-0.470881\pi\)
0.0913514 + 0.995819i \(0.470881\pi\)
\(72\) 5.88240 0.693248
\(73\) 15.3001 1.79074 0.895369 0.445324i \(-0.146912\pi\)
0.895369 + 0.445324i \(0.146912\pi\)
\(74\) 19.9572 2.31997
\(75\) 0 0
\(76\) −11.6644 −1.33800
\(77\) 1.14637 0.130640
\(78\) 2.68585 0.304112
\(79\) 0.882404 0.0992782 0.0496391 0.998767i \(-0.484193\pi\)
0.0496391 + 0.998767i \(0.484193\pi\)
\(80\) 0 0
\(81\) −1.10038 −0.122265
\(82\) 0.685846 0.0757390
\(83\) 12.1292 1.33135 0.665674 0.746243i \(-0.268144\pi\)
0.665674 + 0.746243i \(0.268144\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −19.1035 −2.05999
\(87\) 12.0246 1.28917
\(88\) −4.00000 −0.426401
\(89\) 5.73604 0.608019 0.304009 0.952669i \(-0.401675\pi\)
0.304009 + 0.952669i \(0.401675\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 11.0790 1.15506
\(93\) 1.87506 0.194434
\(94\) −24.8683 −2.56497
\(95\) 0 0
\(96\) 4.78623 0.488493
\(97\) 5.34292 0.542492 0.271246 0.962510i \(-0.412564\pi\)
0.271246 + 0.962510i \(0.412564\pi\)
\(98\) −2.34292 −0.236671
\(99\) −1.93260 −0.194233
\(100\) 0 0
\(101\) 11.1464 1.10910 0.554552 0.832149i \(-0.312889\pi\)
0.554552 + 0.832149i \(0.312889\pi\)
\(102\) 15.6644 1.55101
\(103\) 3.41454 0.336444 0.168222 0.985749i \(-0.446197\pi\)
0.168222 + 0.985749i \(0.446197\pi\)
\(104\) 3.48929 0.342153
\(105\) 0 0
\(106\) −1.83221 −0.177960
\(107\) −4.97858 −0.481297 −0.240649 0.970612i \(-0.577360\pi\)
−0.240649 + 0.970612i \(0.577360\pi\)
\(108\) −18.7434 −1.80358
\(109\) −13.4966 −1.29274 −0.646372 0.763023i \(-0.723714\pi\)
−0.646372 + 0.763023i \(0.723714\pi\)
\(110\) 0 0
\(111\) −9.76481 −0.926835
\(112\) 1.19656 0.113064
\(113\) −16.4464 −1.54715 −0.773576 0.633704i \(-0.781534\pi\)
−0.773576 + 0.633704i \(0.781534\pi\)
\(114\) 8.97858 0.840921
\(115\) 0 0
\(116\) 36.6002 3.39824
\(117\) 1.68585 0.155857
\(118\) −29.6216 −2.72689
\(119\) −5.83221 −0.534638
\(120\) 0 0
\(121\) −9.68585 −0.880531
\(122\) 4.68585 0.424237
\(123\) −0.335577 −0.0302579
\(124\) 5.70727 0.512528
\(125\) 0 0
\(126\) 3.94981 0.351877
\(127\) −12.0575 −1.06993 −0.534967 0.844873i \(-0.679676\pi\)
−0.534967 + 0.844873i \(0.679676\pi\)
\(128\) 20.1751 1.78325
\(129\) 9.34713 0.822969
\(130\) 0 0
\(131\) −3.66442 −0.320162 −0.160081 0.987104i \(-0.551176\pi\)
−0.160081 + 0.987104i \(0.551176\pi\)
\(132\) 4.58546 0.399113
\(133\) −3.34292 −0.289868
\(134\) −14.2927 −1.23470
\(135\) 0 0
\(136\) 20.3503 1.74502
\(137\) 13.1035 1.11951 0.559755 0.828658i \(-0.310895\pi\)
0.559755 + 0.828658i \(0.310895\pi\)
\(138\) −8.52792 −0.725945
\(139\) 7.49663 0.635856 0.317928 0.948115i \(-0.397013\pi\)
0.317928 + 0.948115i \(0.397013\pi\)
\(140\) 0 0
\(141\) 12.1678 1.02471
\(142\) −3.60688 −0.302683
\(143\) −1.14637 −0.0958639
\(144\) −2.01721 −0.168101
\(145\) 0 0
\(146\) −35.8469 −2.96671
\(147\) 1.14637 0.0945506
\(148\) −29.7220 −2.44313
\(149\) 2.16779 0.177592 0.0887961 0.996050i \(-0.471698\pi\)
0.0887961 + 0.996050i \(0.471698\pi\)
\(150\) 0 0
\(151\) 14.9112 1.21345 0.606727 0.794910i \(-0.292482\pi\)
0.606727 + 0.794910i \(0.292482\pi\)
\(152\) 11.6644 0.946110
\(153\) 9.83221 0.794887
\(154\) −2.68585 −0.216432
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −22.8683 −1.82509 −0.912546 0.408975i \(-0.865886\pi\)
−0.912546 + 0.408975i \(0.865886\pi\)
\(158\) −2.06740 −0.164474
\(159\) 0.896480 0.0710955
\(160\) 0 0
\(161\) 3.17513 0.250236
\(162\) 2.57812 0.202556
\(163\) −7.07896 −0.554467 −0.277234 0.960803i \(-0.589418\pi\)
−0.277234 + 0.960803i \(0.589418\pi\)
\(164\) −1.02142 −0.0797597
\(165\) 0 0
\(166\) −28.4177 −2.20564
\(167\) −2.61423 −0.202295 −0.101148 0.994871i \(-0.532251\pi\)
−0.101148 + 0.994871i \(0.532251\pi\)
\(168\) −4.00000 −0.308607
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 5.63565 0.430969
\(172\) 28.4507 2.16934
\(173\) −11.0031 −0.836553 −0.418276 0.908320i \(-0.637366\pi\)
−0.418276 + 0.908320i \(0.637366\pi\)
\(174\) −28.1726 −2.13576
\(175\) 0 0
\(176\) 1.37169 0.103395
\(177\) 14.4935 1.08940
\(178\) −13.4391 −1.00730
\(179\) 23.9614 1.79096 0.895478 0.445105i \(-0.146834\pi\)
0.895478 + 0.445105i \(0.146834\pi\)
\(180\) 0 0
\(181\) 6.56090 0.487668 0.243834 0.969817i \(-0.421595\pi\)
0.243834 + 0.969817i \(0.421595\pi\)
\(182\) 2.34292 0.173669
\(183\) −2.29273 −0.169484
\(184\) −11.0790 −0.816752
\(185\) 0 0
\(186\) −4.39312 −0.322119
\(187\) −6.68585 −0.488917
\(188\) 37.0361 2.70114
\(189\) −5.37169 −0.390733
\(190\) 0 0
\(191\) −4.39312 −0.317875 −0.158937 0.987289i \(-0.550807\pi\)
−0.158937 + 0.987289i \(0.550807\pi\)
\(192\) −13.9572 −1.00727
\(193\) 8.29273 0.596924 0.298462 0.954422i \(-0.403526\pi\)
0.298462 + 0.954422i \(0.403526\pi\)
\(194\) −12.5181 −0.898744
\(195\) 0 0
\(196\) 3.48929 0.249235
\(197\) 3.17092 0.225919 0.112959 0.993600i \(-0.463967\pi\)
0.112959 + 0.993600i \(0.463967\pi\)
\(198\) 4.52792 0.321786
\(199\) −13.5970 −0.963867 −0.481934 0.876208i \(-0.660065\pi\)
−0.481934 + 0.876208i \(0.660065\pi\)
\(200\) 0 0
\(201\) 6.99327 0.493267
\(202\) −26.1151 −1.83745
\(203\) 10.4893 0.736204
\(204\) −23.3288 −1.63335
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −5.35279 −0.372045
\(208\) −1.19656 −0.0829663
\(209\) −3.83221 −0.265080
\(210\) 0 0
\(211\) 9.27552 0.638553 0.319277 0.947662i \(-0.396560\pi\)
0.319277 + 0.947662i \(0.396560\pi\)
\(212\) 2.72869 0.187407
\(213\) 1.76481 0.120923
\(214\) 11.6644 0.797364
\(215\) 0 0
\(216\) 18.7434 1.27533
\(217\) 1.63565 0.111035
\(218\) 31.6216 2.14168
\(219\) 17.5395 1.18521
\(220\) 0 0
\(221\) 5.83221 0.392317
\(222\) 22.8782 1.53548
\(223\) 19.5928 1.31203 0.656016 0.754747i \(-0.272240\pi\)
0.656016 + 0.754747i \(0.272240\pi\)
\(224\) 4.17513 0.278963
\(225\) 0 0
\(226\) 38.5328 2.56316
\(227\) 19.6644 1.30517 0.652587 0.757714i \(-0.273684\pi\)
0.652587 + 0.757714i \(0.273684\pi\)
\(228\) −13.3717 −0.885562
\(229\) 7.76481 0.513113 0.256556 0.966529i \(-0.417412\pi\)
0.256556 + 0.966529i \(0.417412\pi\)
\(230\) 0 0
\(231\) 1.31415 0.0864650
\(232\) −36.6002 −2.40292
\(233\) −12.1966 −0.799023 −0.399512 0.916728i \(-0.630820\pi\)
−0.399512 + 0.916728i \(0.630820\pi\)
\(234\) −3.94981 −0.258207
\(235\) 0 0
\(236\) 44.1151 2.87165
\(237\) 1.01156 0.0657077
\(238\) 13.6644 0.885733
\(239\) −10.2927 −0.665781 −0.332891 0.942965i \(-0.608024\pi\)
−0.332891 + 0.942965i \(0.608024\pi\)
\(240\) 0 0
\(241\) −4.02877 −0.259516 −0.129758 0.991546i \(-0.541420\pi\)
−0.129758 + 0.991546i \(0.541420\pi\)
\(242\) 22.6932 1.45877
\(243\) 14.8536 0.952861
\(244\) −6.97858 −0.446758
\(245\) 0 0
\(246\) 0.786230 0.0501282
\(247\) 3.34292 0.212705
\(248\) −5.70727 −0.362412
\(249\) 13.9044 0.881158
\(250\) 0 0
\(251\) 2.91117 0.183752 0.0918758 0.995770i \(-0.470714\pi\)
0.0918758 + 0.995770i \(0.470714\pi\)
\(252\) −5.88240 −0.370557
\(253\) 3.63986 0.228836
\(254\) 28.2499 1.77256
\(255\) 0 0
\(256\) −22.9185 −1.43241
\(257\) 19.5970 1.22243 0.611214 0.791465i \(-0.290681\pi\)
0.611214 + 0.791465i \(0.290681\pi\)
\(258\) −21.8996 −1.36341
\(259\) −8.51806 −0.529286
\(260\) 0 0
\(261\) −17.6833 −1.09457
\(262\) 8.58546 0.530412
\(263\) −7.56825 −0.466678 −0.233339 0.972395i \(-0.574965\pi\)
−0.233339 + 0.972395i \(0.574965\pi\)
\(264\) −4.58546 −0.282216
\(265\) 0 0
\(266\) 7.83221 0.480224
\(267\) 6.57560 0.402420
\(268\) 21.2860 1.30025
\(269\) −9.47208 −0.577523 −0.288761 0.957401i \(-0.593243\pi\)
−0.288761 + 0.957401i \(0.593243\pi\)
\(270\) 0 0
\(271\) 29.3717 1.78420 0.892102 0.451835i \(-0.149230\pi\)
0.892102 + 0.451835i \(0.149230\pi\)
\(272\) −6.97858 −0.423138
\(273\) −1.14637 −0.0693812
\(274\) −30.7005 −1.85469
\(275\) 0 0
\(276\) 12.7005 0.764483
\(277\) 1.90383 0.114390 0.0571949 0.998363i \(-0.481784\pi\)
0.0571949 + 0.998363i \(0.481784\pi\)
\(278\) −17.5640 −1.05342
\(279\) −2.75746 −0.165085
\(280\) 0 0
\(281\) −20.5756 −1.22744 −0.613719 0.789525i \(-0.710327\pi\)
−0.613719 + 0.789525i \(0.710327\pi\)
\(282\) −28.5082 −1.69764
\(283\) 26.9933 1.60458 0.802292 0.596932i \(-0.203614\pi\)
0.802292 + 0.596932i \(0.203614\pi\)
\(284\) 5.37169 0.318751
\(285\) 0 0
\(286\) 2.68585 0.158817
\(287\) −0.292731 −0.0172794
\(288\) −7.03863 −0.414756
\(289\) 17.0147 1.00086
\(290\) 0 0
\(291\) 6.12494 0.359050
\(292\) 53.3864 3.12420
\(293\) 14.9070 0.870874 0.435437 0.900219i \(-0.356594\pi\)
0.435437 + 0.900219i \(0.356594\pi\)
\(294\) −2.68585 −0.156642
\(295\) 0 0
\(296\) 29.7220 1.72755
\(297\) −6.15792 −0.357319
\(298\) −5.07896 −0.294216
\(299\) −3.17513 −0.183623
\(300\) 0 0
\(301\) 8.15371 0.469972
\(302\) −34.9357 −2.01033
\(303\) 12.7778 0.734066
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −23.0361 −1.31689
\(307\) −26.0288 −1.48554 −0.742770 0.669546i \(-0.766488\pi\)
−0.742770 + 0.669546i \(0.766488\pi\)
\(308\) 4.00000 0.227921
\(309\) 3.91431 0.222677
\(310\) 0 0
\(311\) 19.4966 1.10555 0.552776 0.833330i \(-0.313568\pi\)
0.552776 + 0.833330i \(0.313568\pi\)
\(312\) 4.00000 0.226455
\(313\) 3.48194 0.196811 0.0984055 0.995146i \(-0.468626\pi\)
0.0984055 + 0.995146i \(0.468626\pi\)
\(314\) 53.5787 3.02362
\(315\) 0 0
\(316\) 3.07896 0.173205
\(317\) −5.02142 −0.282031 −0.141016 0.990007i \(-0.545037\pi\)
−0.141016 + 0.990007i \(0.545037\pi\)
\(318\) −2.10038 −0.117784
\(319\) 12.0246 0.673246
\(320\) 0 0
\(321\) −5.70727 −0.318549
\(322\) −7.43910 −0.414565
\(323\) 19.4966 1.08482
\(324\) −3.83956 −0.213309
\(325\) 0 0
\(326\) 16.5855 0.918584
\(327\) −15.4721 −0.855608
\(328\) 1.02142 0.0563986
\(329\) 10.6142 0.585182
\(330\) 0 0
\(331\) −6.14950 −0.338007 −0.169004 0.985615i \(-0.554055\pi\)
−0.169004 + 0.985615i \(0.554055\pi\)
\(332\) 42.3221 2.32273
\(333\) 14.3601 0.786931
\(334\) 6.12494 0.335142
\(335\) 0 0
\(336\) 1.37169 0.0748320
\(337\) 25.6258 1.39593 0.697963 0.716134i \(-0.254090\pi\)
0.697963 + 0.716134i \(0.254090\pi\)
\(338\) −2.34292 −0.127438
\(339\) −18.8536 −1.02399
\(340\) 0 0
\(341\) 1.87506 0.101540
\(342\) −13.2039 −0.713985
\(343\) 1.00000 0.0539949
\(344\) −28.4507 −1.53396
\(345\) 0 0
\(346\) 25.7795 1.38591
\(347\) 16.7005 0.896532 0.448266 0.893900i \(-0.352042\pi\)
0.448266 + 0.893900i \(0.352042\pi\)
\(348\) 41.9572 2.24914
\(349\) −23.5500 −1.26060 −0.630300 0.776351i \(-0.717068\pi\)
−0.630300 + 0.776351i \(0.717068\pi\)
\(350\) 0 0
\(351\) 5.37169 0.286720
\(352\) 4.78623 0.255107
\(353\) 7.64973 0.407154 0.203577 0.979059i \(-0.434743\pi\)
0.203577 + 0.979059i \(0.434743\pi\)
\(354\) −33.9572 −1.80480
\(355\) 0 0
\(356\) 20.0147 1.06078
\(357\) −6.68585 −0.353853
\(358\) −56.1396 −2.96707
\(359\) 18.3748 0.969786 0.484893 0.874573i \(-0.338858\pi\)
0.484893 + 0.874573i \(0.338858\pi\)
\(360\) 0 0
\(361\) −7.82487 −0.411835
\(362\) −15.3717 −0.807918
\(363\) −11.1035 −0.582784
\(364\) −3.48929 −0.182888
\(365\) 0 0
\(366\) 5.37169 0.280783
\(367\) −5.33871 −0.278679 −0.139339 0.990245i \(-0.544498\pi\)
−0.139339 + 0.990245i \(0.544498\pi\)
\(368\) 3.79923 0.198049
\(369\) 0.493499 0.0256906
\(370\) 0 0
\(371\) 0.782020 0.0406004
\(372\) 6.54262 0.339219
\(373\) −21.5212 −1.11433 −0.557163 0.830403i \(-0.688110\pi\)
−0.557163 + 0.830403i \(0.688110\pi\)
\(374\) 15.6644 0.809988
\(375\) 0 0
\(376\) −37.0361 −1.90999
\(377\) −10.4893 −0.540226
\(378\) 12.5855 0.647326
\(379\) 4.61002 0.236801 0.118400 0.992966i \(-0.462223\pi\)
0.118400 + 0.992966i \(0.462223\pi\)
\(380\) 0 0
\(381\) −13.8223 −0.708140
\(382\) 10.2927 0.526622
\(383\) 8.33558 0.425928 0.212964 0.977060i \(-0.431688\pi\)
0.212964 + 0.977060i \(0.431688\pi\)
\(384\) 23.1281 1.18025
\(385\) 0 0
\(386\) −19.4292 −0.988922
\(387\) −13.7459 −0.698744
\(388\) 18.6430 0.946455
\(389\) −6.44223 −0.326634 −0.163317 0.986574i \(-0.552219\pi\)
−0.163317 + 0.986574i \(0.552219\pi\)
\(390\) 0 0
\(391\) −18.5181 −0.936498
\(392\) −3.48929 −0.176236
\(393\) −4.20077 −0.211901
\(394\) −7.42923 −0.374279
\(395\) 0 0
\(396\) −6.74338 −0.338868
\(397\) −1.40046 −0.0702872 −0.0351436 0.999382i \(-0.511189\pi\)
−0.0351436 + 0.999382i \(0.511189\pi\)
\(398\) 31.8568 1.59684
\(399\) −3.83221 −0.191851
\(400\) 0 0
\(401\) −6.97858 −0.348494 −0.174247 0.984702i \(-0.555749\pi\)
−0.174247 + 0.984702i \(0.555749\pi\)
\(402\) −16.3847 −0.817194
\(403\) −1.63565 −0.0814777
\(404\) 38.8929 1.93499
\(405\) 0 0
\(406\) −24.5756 −1.21967
\(407\) −9.76481 −0.484024
\(408\) 23.3288 1.15495
\(409\) −18.3790 −0.908785 −0.454392 0.890802i \(-0.650144\pi\)
−0.454392 + 0.890802i \(0.650144\pi\)
\(410\) 0 0
\(411\) 15.0214 0.740952
\(412\) 11.9143 0.586976
\(413\) 12.6430 0.622121
\(414\) 12.5412 0.616365
\(415\) 0 0
\(416\) −4.17513 −0.204703
\(417\) 8.59388 0.420844
\(418\) 8.97858 0.439157
\(419\) −30.0393 −1.46751 −0.733757 0.679412i \(-0.762235\pi\)
−0.733757 + 0.679412i \(0.762235\pi\)
\(420\) 0 0
\(421\) −8.31729 −0.405360 −0.202680 0.979245i \(-0.564965\pi\)
−0.202680 + 0.979245i \(0.564965\pi\)
\(422\) −21.7318 −1.05789
\(423\) −17.8940 −0.870034
\(424\) −2.72869 −0.132517
\(425\) 0 0
\(426\) −4.13481 −0.200332
\(427\) −2.00000 −0.0967868
\(428\) −17.3717 −0.839692
\(429\) −1.31415 −0.0634479
\(430\) 0 0
\(431\) 9.64973 0.464811 0.232406 0.972619i \(-0.425340\pi\)
0.232406 + 0.972619i \(0.425340\pi\)
\(432\) −6.42754 −0.309245
\(433\) −26.3074 −1.26425 −0.632127 0.774865i \(-0.717818\pi\)
−0.632127 + 0.774865i \(0.717818\pi\)
\(434\) −3.83221 −0.183952
\(435\) 0 0
\(436\) −47.0937 −2.25538
\(437\) −10.6142 −0.507748
\(438\) −41.0937 −1.96353
\(439\) 33.8139 1.61385 0.806925 0.590654i \(-0.201130\pi\)
0.806925 + 0.590654i \(0.201130\pi\)
\(440\) 0 0
\(441\) −1.68585 −0.0802784
\(442\) −13.6644 −0.649950
\(443\) −26.4464 −1.25651 −0.628254 0.778008i \(-0.716230\pi\)
−0.628254 + 0.778008i \(0.716230\pi\)
\(444\) −34.0722 −1.61700
\(445\) 0 0
\(446\) −45.9044 −2.17364
\(447\) 2.48508 0.117540
\(448\) −12.1751 −0.575221
\(449\) 2.64300 0.124731 0.0623655 0.998053i \(-0.480136\pi\)
0.0623655 + 0.998053i \(0.480136\pi\)
\(450\) 0 0
\(451\) −0.335577 −0.0158017
\(452\) −57.3864 −2.69923
\(453\) 17.0937 0.803130
\(454\) −46.0722 −2.16228
\(455\) 0 0
\(456\) 13.3717 0.626187
\(457\) 33.6890 1.57590 0.787952 0.615737i \(-0.211141\pi\)
0.787952 + 0.615737i \(0.211141\pi\)
\(458\) −18.1923 −0.850073
\(459\) 31.3288 1.46231
\(460\) 0 0
\(461\) 33.0790 1.54064 0.770320 0.637657i \(-0.220096\pi\)
0.770320 + 0.637657i \(0.220096\pi\)
\(462\) −3.07896 −0.143246
\(463\) 2.51806 0.117024 0.0585120 0.998287i \(-0.481364\pi\)
0.0585120 + 0.998287i \(0.481364\pi\)
\(464\) 12.5510 0.582667
\(465\) 0 0
\(466\) 28.5756 1.32374
\(467\) −2.57560 −0.119184 −0.0595922 0.998223i \(-0.518980\pi\)
−0.0595922 + 0.998223i \(0.518980\pi\)
\(468\) 5.88240 0.271914
\(469\) 6.10038 0.281690
\(470\) 0 0
\(471\) −26.2155 −1.20794
\(472\) −44.1151 −2.03056
\(473\) 9.34713 0.429782
\(474\) −2.37000 −0.108858
\(475\) 0 0
\(476\) −20.3503 −0.932753
\(477\) −1.31836 −0.0603638
\(478\) 24.1151 1.10300
\(479\) 0.513847 0.0234783 0.0117391 0.999931i \(-0.496263\pi\)
0.0117391 + 0.999931i \(0.496263\pi\)
\(480\) 0 0
\(481\) 8.51806 0.388390
\(482\) 9.43910 0.429939
\(483\) 3.63986 0.165620
\(484\) −33.7967 −1.53621
\(485\) 0 0
\(486\) −34.8009 −1.57860
\(487\) −36.0575 −1.63392 −0.816962 0.576692i \(-0.804343\pi\)
−0.816962 + 0.576692i \(0.804343\pi\)
\(488\) 6.97858 0.315905
\(489\) −8.11508 −0.366976
\(490\) 0 0
\(491\) −9.22846 −0.416475 −0.208237 0.978078i \(-0.566773\pi\)
−0.208237 + 0.978078i \(0.566773\pi\)
\(492\) −1.17092 −0.0527893
\(493\) −61.1758 −2.75522
\(494\) −7.83221 −0.352388
\(495\) 0 0
\(496\) 1.95715 0.0878788
\(497\) 1.53948 0.0690551
\(498\) −32.5770 −1.45981
\(499\) 1.00314 0.0449065 0.0224533 0.999748i \(-0.492852\pi\)
0.0224533 + 0.999748i \(0.492852\pi\)
\(500\) 0 0
\(501\) −2.99686 −0.133890
\(502\) −6.82065 −0.304421
\(503\) 30.3503 1.35325 0.676626 0.736327i \(-0.263441\pi\)
0.676626 + 0.736327i \(0.263441\pi\)
\(504\) 5.88240 0.262023
\(505\) 0 0
\(506\) −8.52792 −0.379112
\(507\) 1.14637 0.0509119
\(508\) −42.0722 −1.86665
\(509\) 10.5995 0.469816 0.234908 0.972018i \(-0.424521\pi\)
0.234908 + 0.972018i \(0.424521\pi\)
\(510\) 0 0
\(511\) 15.3001 0.676836
\(512\) 13.3461 0.589818
\(513\) 17.9572 0.792828
\(514\) −45.9143 −2.02519
\(515\) 0 0
\(516\) 32.6148 1.43579
\(517\) 12.1678 0.535139
\(518\) 19.9572 0.876867
\(519\) −12.6136 −0.553676
\(520\) 0 0
\(521\) −16.2646 −0.712564 −0.356282 0.934378i \(-0.615956\pi\)
−0.356282 + 0.934378i \(0.615956\pi\)
\(522\) 41.4307 1.81337
\(523\) −7.22219 −0.315804 −0.157902 0.987455i \(-0.550473\pi\)
−0.157902 + 0.987455i \(0.550473\pi\)
\(524\) −12.7862 −0.558569
\(525\) 0 0
\(526\) 17.7318 0.773144
\(527\) −9.53948 −0.415546
\(528\) 1.57246 0.0684326
\(529\) −12.9185 −0.561675
\(530\) 0 0
\(531\) −21.3142 −0.924955
\(532\) −11.6644 −0.505717
\(533\) 0.292731 0.0126796
\(534\) −15.4061 −0.666688
\(535\) 0 0
\(536\) −21.2860 −0.919415
\(537\) 27.4685 1.18535
\(538\) 22.1923 0.956780
\(539\) 1.14637 0.0493775
\(540\) 0 0
\(541\) 17.3534 0.746081 0.373041 0.927815i \(-0.378315\pi\)
0.373041 + 0.927815i \(0.378315\pi\)
\(542\) −68.8156 −2.95588
\(543\) 7.52119 0.322765
\(544\) −24.3503 −1.04401
\(545\) 0 0
\(546\) 2.68585 0.114944
\(547\) 34.1109 1.45848 0.729238 0.684261i \(-0.239875\pi\)
0.729238 + 0.684261i \(0.239875\pi\)
\(548\) 45.7220 1.95315
\(549\) 3.37169 0.143900
\(550\) 0 0
\(551\) −35.0649 −1.49381
\(552\) −12.7005 −0.540571
\(553\) 0.882404 0.0375236
\(554\) −4.46052 −0.189509
\(555\) 0 0
\(556\) 26.1579 1.10934
\(557\) 13.2222 0.560242 0.280121 0.959965i \(-0.409625\pi\)
0.280121 + 0.959965i \(0.409625\pi\)
\(558\) 6.46052 0.273496
\(559\) −8.15371 −0.344865
\(560\) 0 0
\(561\) −7.66442 −0.323592
\(562\) 48.2070 2.03349
\(563\) 41.3717 1.74361 0.871804 0.489854i \(-0.162950\pi\)
0.871804 + 0.489854i \(0.162950\pi\)
\(564\) 42.4569 1.78776
\(565\) 0 0
\(566\) −63.2432 −2.65831
\(567\) −1.10038 −0.0462118
\(568\) −5.37169 −0.225391
\(569\) −2.68164 −0.112420 −0.0562100 0.998419i \(-0.517902\pi\)
−0.0562100 + 0.998419i \(0.517902\pi\)
\(570\) 0 0
\(571\) −28.9315 −1.21075 −0.605373 0.795942i \(-0.706976\pi\)
−0.605373 + 0.795942i \(0.706976\pi\)
\(572\) −4.00000 −0.167248
\(573\) −5.03612 −0.210387
\(574\) 0.685846 0.0286267
\(575\) 0 0
\(576\) 20.5254 0.855225
\(577\) −37.5296 −1.56238 −0.781189 0.624294i \(-0.785387\pi\)
−0.781189 + 0.624294i \(0.785387\pi\)
\(578\) −39.8641 −1.65813
\(579\) 9.50650 0.395077
\(580\) 0 0
\(581\) 12.1292 0.503202
\(582\) −14.3503 −0.594838
\(583\) 0.896480 0.0371284
\(584\) −53.3864 −2.20914
\(585\) 0 0
\(586\) −34.9259 −1.44277
\(587\) −23.0649 −0.951990 −0.475995 0.879448i \(-0.657912\pi\)
−0.475995 + 0.879448i \(0.657912\pi\)
\(588\) 4.00000 0.164957
\(589\) −5.46787 −0.225299
\(590\) 0 0
\(591\) 3.63504 0.149525
\(592\) −10.1923 −0.418903
\(593\) 11.0502 0.453777 0.226889 0.973921i \(-0.427145\pi\)
0.226889 + 0.973921i \(0.427145\pi\)
\(594\) 14.4275 0.591969
\(595\) 0 0
\(596\) 7.56404 0.309835
\(597\) −15.5872 −0.637940
\(598\) 7.43910 0.304207
\(599\) −44.6044 −1.82248 −0.911242 0.411870i \(-0.864876\pi\)
−0.911242 + 0.411870i \(0.864876\pi\)
\(600\) 0 0
\(601\) 47.8715 1.95272 0.976359 0.216156i \(-0.0693520\pi\)
0.976359 + 0.216156i \(0.0693520\pi\)
\(602\) −19.1035 −0.778601
\(603\) −10.2843 −0.418809
\(604\) 52.0294 2.11705
\(605\) 0 0
\(606\) −29.9374 −1.21612
\(607\) −45.0691 −1.82930 −0.914649 0.404249i \(-0.867533\pi\)
−0.914649 + 0.404249i \(0.867533\pi\)
\(608\) −13.9572 −0.566037
\(609\) 12.0246 0.487260
\(610\) 0 0
\(611\) −10.6142 −0.429406
\(612\) 34.3074 1.38680
\(613\) 25.5212 1.03079 0.515396 0.856952i \(-0.327645\pi\)
0.515396 + 0.856952i \(0.327645\pi\)
\(614\) 60.9834 2.46109
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) −29.2432 −1.17729 −0.588643 0.808393i \(-0.700337\pi\)
−0.588643 + 0.808393i \(0.700337\pi\)
\(618\) −9.17092 −0.368909
\(619\) 4.78623 0.192375 0.0961874 0.995363i \(-0.469335\pi\)
0.0961874 + 0.995363i \(0.469335\pi\)
\(620\) 0 0
\(621\) −17.0558 −0.684428
\(622\) −45.6791 −1.83157
\(623\) 5.73604 0.229810
\(624\) −1.37169 −0.0549116
\(625\) 0 0
\(626\) −8.15792 −0.326056
\(627\) −4.39312 −0.175444
\(628\) −79.7942 −3.18413
\(629\) 49.6791 1.98084
\(630\) 0 0
\(631\) −28.3931 −1.13031 −0.565156 0.824984i \(-0.691184\pi\)
−0.565156 + 0.824984i \(0.691184\pi\)
\(632\) −3.07896 −0.122475
\(633\) 10.6331 0.422629
\(634\) 11.7648 0.467240
\(635\) 0 0
\(636\) 3.12808 0.124036
\(637\) −1.00000 −0.0396214
\(638\) −28.1726 −1.11536
\(639\) −2.59533 −0.102670
\(640\) 0 0
\(641\) 5.96137 0.235460 0.117730 0.993046i \(-0.462438\pi\)
0.117730 + 0.993046i \(0.462438\pi\)
\(642\) 13.3717 0.527739
\(643\) −31.1940 −1.23017 −0.615086 0.788460i \(-0.710879\pi\)
−0.615086 + 0.788460i \(0.710879\pi\)
\(644\) 11.0790 0.436572
\(645\) 0 0
\(646\) −45.6791 −1.79722
\(647\) −14.9112 −0.586219 −0.293109 0.956079i \(-0.594690\pi\)
−0.293109 + 0.956079i \(0.594690\pi\)
\(648\) 3.83956 0.150832
\(649\) 14.4935 0.568920
\(650\) 0 0
\(651\) 1.87506 0.0734893
\(652\) −24.7005 −0.967348
\(653\) 3.57246 0.139801 0.0699006 0.997554i \(-0.477732\pi\)
0.0699006 + 0.997554i \(0.477732\pi\)
\(654\) 36.2499 1.41748
\(655\) 0 0
\(656\) −0.350269 −0.0136757
\(657\) −25.7936 −1.00630
\(658\) −24.8683 −0.969468
\(659\) −3.90383 −0.152071 −0.0760357 0.997105i \(-0.524226\pi\)
−0.0760357 + 0.997105i \(0.524226\pi\)
\(660\) 0 0
\(661\) 13.7936 0.536508 0.268254 0.963348i \(-0.413553\pi\)
0.268254 + 0.963348i \(0.413553\pi\)
\(662\) 14.4078 0.559975
\(663\) 6.68585 0.259657
\(664\) −42.3221 −1.64242
\(665\) 0 0
\(666\) −33.6447 −1.30371
\(667\) 33.3049 1.28957
\(668\) −9.12181 −0.352933
\(669\) 22.4605 0.868374
\(670\) 0 0
\(671\) −2.29273 −0.0885099
\(672\) 4.78623 0.184633
\(673\) −5.70306 −0.219837 −0.109918 0.993941i \(-0.535059\pi\)
−0.109918 + 0.993941i \(0.535059\pi\)
\(674\) −60.0393 −2.31263
\(675\) 0 0
\(676\) 3.48929 0.134203
\(677\) −35.2614 −1.35521 −0.677604 0.735427i \(-0.736981\pi\)
−0.677604 + 0.735427i \(0.736981\pi\)
\(678\) 44.1726 1.69644
\(679\) 5.34292 0.205043
\(680\) 0 0
\(681\) 22.5426 0.863835
\(682\) −4.39312 −0.168221
\(683\) −1.03612 −0.0396459 −0.0198229 0.999804i \(-0.506310\pi\)
−0.0198229 + 0.999804i \(0.506310\pi\)
\(684\) 19.6644 0.751888
\(685\) 0 0
\(686\) −2.34292 −0.0894532
\(687\) 8.90131 0.339606
\(688\) 9.75639 0.371959
\(689\) −0.782020 −0.0297926
\(690\) 0 0
\(691\) 3.67850 0.139937 0.0699684 0.997549i \(-0.477710\pi\)
0.0699684 + 0.997549i \(0.477710\pi\)
\(692\) −38.3931 −1.45949
\(693\) −1.93260 −0.0734132
\(694\) −39.1281 −1.48528
\(695\) 0 0
\(696\) −41.9572 −1.59038
\(697\) 1.70727 0.0646674
\(698\) 55.1758 2.08843
\(699\) −13.9817 −0.528837
\(700\) 0 0
\(701\) −0.0617493 −0.00233224 −0.00116612 0.999999i \(-0.500371\pi\)
−0.00116612 + 0.999999i \(0.500371\pi\)
\(702\) −12.5855 −0.475008
\(703\) 28.4752 1.07396
\(704\) −13.9572 −0.526030
\(705\) 0 0
\(706\) −17.9227 −0.674531
\(707\) 11.1464 0.419202
\(708\) 50.5720 1.90061
\(709\) −42.9834 −1.61428 −0.807138 0.590363i \(-0.798985\pi\)
−0.807138 + 0.590363i \(0.798985\pi\)
\(710\) 0 0
\(711\) −1.48760 −0.0557892
\(712\) −20.0147 −0.750082
\(713\) 5.19342 0.194495
\(714\) 15.6644 0.586226
\(715\) 0 0
\(716\) 83.6081 3.12458
\(717\) −11.7992 −0.440650
\(718\) −43.0508 −1.60664
\(719\) 17.6546 0.658404 0.329202 0.944260i \(-0.393220\pi\)
0.329202 + 0.944260i \(0.393220\pi\)
\(720\) 0 0
\(721\) 3.41454 0.127164
\(722\) 18.3331 0.682286
\(723\) −4.61844 −0.171762
\(724\) 22.8929 0.850807
\(725\) 0 0
\(726\) 26.0147 0.965496
\(727\) 23.8077 0.882977 0.441488 0.897267i \(-0.354451\pi\)
0.441488 + 0.897267i \(0.354451\pi\)
\(728\) 3.48929 0.129322
\(729\) 20.3288 0.752920
\(730\) 0 0
\(731\) −47.5542 −1.75885
\(732\) −8.00000 −0.295689
\(733\) −31.3492 −1.15791 −0.578954 0.815360i \(-0.696539\pi\)
−0.578954 + 0.815360i \(0.696539\pi\)
\(734\) 12.5082 0.461686
\(735\) 0 0
\(736\) 13.2566 0.488645
\(737\) 6.99327 0.257600
\(738\) −1.15623 −0.0425615
\(739\) −24.8108 −0.912680 −0.456340 0.889806i \(-0.650840\pi\)
−0.456340 + 0.889806i \(0.650840\pi\)
\(740\) 0 0
\(741\) 3.83221 0.140780
\(742\) −1.83221 −0.0672626
\(743\) −21.3717 −0.784051 −0.392026 0.919954i \(-0.628226\pi\)
−0.392026 + 0.919954i \(0.628226\pi\)
\(744\) −6.54262 −0.239864
\(745\) 0 0
\(746\) 50.4225 1.84610
\(747\) −20.4479 −0.748149
\(748\) −23.3288 −0.852987
\(749\) −4.97858 −0.181913
\(750\) 0 0
\(751\) 19.2243 0.701503 0.350751 0.936469i \(-0.385926\pi\)
0.350751 + 0.936469i \(0.385926\pi\)
\(752\) 12.7005 0.463141
\(753\) 3.33727 0.121617
\(754\) 24.5756 0.894990
\(755\) 0 0
\(756\) −18.7434 −0.681690
\(757\) 19.8610 0.721860 0.360930 0.932593i \(-0.382459\pi\)
0.360930 + 0.932593i \(0.382459\pi\)
\(758\) −10.8009 −0.392307
\(759\) 4.17262 0.151456
\(760\) 0 0
\(761\) 6.12073 0.221876 0.110938 0.993827i \(-0.464614\pi\)
0.110938 + 0.993827i \(0.464614\pi\)
\(762\) 32.3847 1.17317
\(763\) −13.4966 −0.488611
\(764\) −15.3288 −0.554578
\(765\) 0 0
\(766\) −19.5296 −0.705634
\(767\) −12.6430 −0.456512
\(768\) −26.2730 −0.948045
\(769\) 3.82800 0.138041 0.0690206 0.997615i \(-0.478013\pi\)
0.0690206 + 0.997615i \(0.478013\pi\)
\(770\) 0 0
\(771\) 22.4653 0.809070
\(772\) 28.9357 1.04142
\(773\) 6.53635 0.235096 0.117548 0.993067i \(-0.462497\pi\)
0.117548 + 0.993067i \(0.462497\pi\)
\(774\) 32.2056 1.15761
\(775\) 0 0
\(776\) −18.6430 −0.669245
\(777\) −9.76481 −0.350311
\(778\) 15.0937 0.541134
\(779\) 0.978577 0.0350612
\(780\) 0 0
\(781\) 1.76481 0.0631498
\(782\) 43.3864 1.55149
\(783\) −56.3452 −2.01361
\(784\) 1.19656 0.0427342
\(785\) 0 0
\(786\) 9.84208 0.351055
\(787\) −30.8066 −1.09814 −0.549068 0.835778i \(-0.685017\pi\)
−0.549068 + 0.835778i \(0.685017\pi\)
\(788\) 11.0643 0.394148
\(789\) −8.67598 −0.308873
\(790\) 0 0
\(791\) −16.4464 −0.584768
\(792\) 6.74338 0.239616
\(793\) 2.00000 0.0710221
\(794\) 3.28117 0.116444
\(795\) 0 0
\(796\) −47.4439 −1.68161
\(797\) 38.8156 1.37492 0.687460 0.726222i \(-0.258726\pi\)
0.687460 + 0.726222i \(0.258726\pi\)
\(798\) 8.97858 0.317838
\(799\) −61.9044 −2.19002
\(800\) 0 0
\(801\) −9.67008 −0.341675
\(802\) 16.3503 0.577348
\(803\) 17.5395 0.618955
\(804\) 24.4015 0.860576
\(805\) 0 0
\(806\) 3.83221 0.134984
\(807\) −10.8585 −0.382236
\(808\) −38.8929 −1.36825
\(809\) 1.04033 0.0365759 0.0182880 0.999833i \(-0.494178\pi\)
0.0182880 + 0.999833i \(0.494178\pi\)
\(810\) 0 0
\(811\) 12.6712 0.444944 0.222472 0.974939i \(-0.428587\pi\)
0.222472 + 0.974939i \(0.428587\pi\)
\(812\) 36.6002 1.28441
\(813\) 33.6707 1.18088
\(814\) 22.8782 0.801880
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) −27.2572 −0.953610
\(818\) 43.0607 1.50558
\(819\) 1.68585 0.0589082
\(820\) 0 0
\(821\) 27.0361 0.943567 0.471783 0.881714i \(-0.343610\pi\)
0.471783 + 0.881714i \(0.343610\pi\)
\(822\) −35.1940 −1.22753
\(823\) 31.6363 1.10277 0.551386 0.834251i \(-0.314099\pi\)
0.551386 + 0.834251i \(0.314099\pi\)
\(824\) −11.9143 −0.415055
\(825\) 0 0
\(826\) −29.6216 −1.03067
\(827\) −56.4800 −1.96400 −0.982002 0.188872i \(-0.939517\pi\)
−0.982002 + 0.188872i \(0.939517\pi\)
\(828\) −18.6774 −0.649085
\(829\) −42.6760 −1.48220 −0.741099 0.671396i \(-0.765695\pi\)
−0.741099 + 0.671396i \(0.765695\pi\)
\(830\) 0 0
\(831\) 2.18248 0.0757094
\(832\) 12.1751 0.422097
\(833\) −5.83221 −0.202074
\(834\) −20.1348 −0.697211
\(835\) 0 0
\(836\) −13.3717 −0.462470
\(837\) −8.78623 −0.303697
\(838\) 70.3797 2.43122
\(839\) −40.1642 −1.38662 −0.693311 0.720639i \(-0.743849\pi\)
−0.693311 + 0.720639i \(0.743849\pi\)
\(840\) 0 0
\(841\) 81.0252 2.79397
\(842\) 19.4868 0.671558
\(843\) −23.5872 −0.812385
\(844\) 32.3650 1.11405
\(845\) 0 0
\(846\) 41.9242 1.44138
\(847\) −9.68585 −0.332810
\(848\) 0.935731 0.0321331
\(849\) 30.9442 1.06200
\(850\) 0 0
\(851\) −27.0460 −0.927124
\(852\) 6.15792 0.210967
\(853\) −19.6932 −0.674282 −0.337141 0.941454i \(-0.609460\pi\)
−0.337141 + 0.941454i \(0.609460\pi\)
\(854\) 4.68585 0.160346
\(855\) 0 0
\(856\) 17.3717 0.593752
\(857\) −1.66442 −0.0568556 −0.0284278 0.999596i \(-0.509050\pi\)
−0.0284278 + 0.999596i \(0.509050\pi\)
\(858\) 3.07896 0.105114
\(859\) −41.2944 −1.40895 −0.704474 0.709730i \(-0.748817\pi\)
−0.704474 + 0.709730i \(0.748817\pi\)
\(860\) 0 0
\(861\) −0.335577 −0.0114364
\(862\) −22.6086 −0.770051
\(863\) 31.3288 1.06645 0.533223 0.845975i \(-0.320981\pi\)
0.533223 + 0.845975i \(0.320981\pi\)
\(864\) −22.4275 −0.763000
\(865\) 0 0
\(866\) 61.6363 2.09449
\(867\) 19.5051 0.662426
\(868\) 5.70727 0.193717
\(869\) 1.01156 0.0343147
\(870\) 0 0
\(871\) −6.10038 −0.206704
\(872\) 47.0937 1.59479
\(873\) −9.00735 −0.304852
\(874\) 24.8683 0.841184
\(875\) 0 0
\(876\) 61.2003 2.06777
\(877\) 53.8041 1.81683 0.908417 0.418065i \(-0.137292\pi\)
0.908417 + 0.418065i \(0.137292\pi\)
\(878\) −79.2234 −2.67366
\(879\) 17.0888 0.576392
\(880\) 0 0
\(881\) 8.09196 0.272625 0.136313 0.990666i \(-0.456475\pi\)
0.136313 + 0.990666i \(0.456475\pi\)
\(882\) 3.94981 0.132997
\(883\) 27.0705 0.910996 0.455498 0.890237i \(-0.349461\pi\)
0.455498 + 0.890237i \(0.349461\pi\)
\(884\) 20.3503 0.684454
\(885\) 0 0
\(886\) 61.9620 2.08165
\(887\) 38.4935 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(888\) 34.0722 1.14339
\(889\) −12.0575 −0.404397
\(890\) 0 0
\(891\) −1.26144 −0.0422599
\(892\) 68.3650 2.28903
\(893\) −35.4826 −1.18738
\(894\) −5.82235 −0.194728
\(895\) 0 0
\(896\) 20.1751 0.674004
\(897\) −3.63986 −0.121532
\(898\) −6.19235 −0.206641
\(899\) 17.1568 0.572213
\(900\) 0 0
\(901\) −4.56090 −0.151946
\(902\) 0.786230 0.0261786
\(903\) 9.34713 0.311053
\(904\) 57.3864 1.90864
\(905\) 0 0
\(906\) −40.0491 −1.33054
\(907\) −15.7031 −0.521411 −0.260706 0.965418i \(-0.583955\pi\)
−0.260706 + 0.965418i \(0.583955\pi\)
\(908\) 68.6148 2.27706
\(909\) −18.7911 −0.623260
\(910\) 0 0
\(911\) 44.9399 1.48893 0.744463 0.667663i \(-0.232705\pi\)
0.744463 + 0.667663i \(0.232705\pi\)
\(912\) −4.58546 −0.151840
\(913\) 13.9044 0.460170
\(914\) −78.9307 −2.61080
\(915\) 0 0
\(916\) 27.0937 0.895200
\(917\) −3.66442 −0.121010
\(918\) −73.4011 −2.42260
\(919\) −27.2432 −0.898669 −0.449334 0.893364i \(-0.648339\pi\)
−0.449334 + 0.893364i \(0.648339\pi\)
\(920\) 0 0
\(921\) −29.8385 −0.983211
\(922\) −77.5015 −2.55237
\(923\) −1.53948 −0.0506726
\(924\) 4.58546 0.150851
\(925\) 0 0
\(926\) −5.89962 −0.193873
\(927\) −5.75639 −0.189065
\(928\) 43.7942 1.43761
\(929\) −17.1422 −0.562416 −0.281208 0.959647i \(-0.590735\pi\)
−0.281208 + 0.959647i \(0.590735\pi\)
\(930\) 0 0
\(931\) −3.34292 −0.109560
\(932\) −42.5573 −1.39401
\(933\) 22.3503 0.731715
\(934\) 6.03442 0.197452
\(935\) 0 0
\(936\) −5.88240 −0.192272
\(937\) 51.5197 1.68308 0.841538 0.540197i \(-0.181650\pi\)
0.841538 + 0.540197i \(0.181650\pi\)
\(938\) −14.2927 −0.466674
\(939\) 3.99158 0.130260
\(940\) 0 0
\(941\) 32.7575 1.06786 0.533931 0.845528i \(-0.320714\pi\)
0.533931 + 0.845528i \(0.320714\pi\)
\(942\) 61.4208 2.00120
\(943\) −0.929460 −0.0302674
\(944\) 15.1281 0.492377
\(945\) 0 0
\(946\) −21.8996 −0.712018
\(947\) −20.9295 −0.680116 −0.340058 0.940404i \(-0.610447\pi\)
−0.340058 + 0.940404i \(0.610447\pi\)
\(948\) 3.52962 0.114637
\(949\) −15.3001 −0.496662
\(950\) 0 0
\(951\) −5.75639 −0.186664
\(952\) 20.3503 0.659556
\(953\) −46.4120 −1.50343 −0.751716 0.659487i \(-0.770774\pi\)
−0.751716 + 0.659487i \(0.770774\pi\)
\(954\) 3.08883 0.100004
\(955\) 0 0
\(956\) −35.9143 −1.16155
\(957\) 13.7845 0.445591
\(958\) −1.20390 −0.0388964
\(959\) 13.1035 0.423135
\(960\) 0 0
\(961\) −28.3246 −0.913698
\(962\) −19.9572 −0.643444
\(963\) 8.39312 0.270464
\(964\) −14.0575 −0.452763
\(965\) 0 0
\(966\) −8.52792 −0.274381
\(967\) 23.2186 0.746660 0.373330 0.927699i \(-0.378216\pi\)
0.373330 + 0.927699i \(0.378216\pi\)
\(968\) 33.7967 1.08627
\(969\) 22.3503 0.717994
\(970\) 0 0
\(971\) −14.1004 −0.452503 −0.226251 0.974069i \(-0.572647\pi\)
−0.226251 + 0.974069i \(0.572647\pi\)
\(972\) 51.8286 1.66240
\(973\) 7.49663 0.240331
\(974\) 84.4800 2.70692
\(975\) 0 0
\(976\) −2.39312 −0.0766018
\(977\) 2.95402 0.0945074 0.0472537 0.998883i \(-0.484953\pi\)
0.0472537 + 0.998883i \(0.484953\pi\)
\(978\) 19.0130 0.607969
\(979\) 6.57560 0.210157
\(980\) 0 0
\(981\) 22.7533 0.726455
\(982\) 21.6216 0.689972
\(983\) 35.0367 1.11750 0.558749 0.829337i \(-0.311281\pi\)
0.558749 + 0.829337i \(0.311281\pi\)
\(984\) 1.17092 0.0373277
\(985\) 0 0
\(986\) 143.330 4.56456
\(987\) 12.1678 0.387305
\(988\) 11.6644 0.371095
\(989\) 25.8891 0.823227
\(990\) 0 0
\(991\) 47.9718 1.52388 0.761938 0.647650i \(-0.224248\pi\)
0.761938 + 0.647650i \(0.224248\pi\)
\(992\) 6.82908 0.216823
\(993\) −7.04958 −0.223712
\(994\) −3.60688 −0.114403
\(995\) 0 0
\(996\) 48.5166 1.53731
\(997\) 38.4422 1.21748 0.608739 0.793371i \(-0.291676\pi\)
0.608739 + 0.793371i \(0.291676\pi\)
\(998\) −2.35027 −0.0743965
\(999\) 45.7564 1.44767
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 2275.2.a.m.1.1 3
5.4 even 2 91.2.a.d.1.3 3
15.14 odd 2 819.2.a.i.1.1 3
20.19 odd 2 1456.2.a.t.1.2 3
35.4 even 6 637.2.e.j.79.1 6
35.9 even 6 637.2.e.j.508.1 6
35.19 odd 6 637.2.e.i.508.1 6
35.24 odd 6 637.2.e.i.79.1 6
35.34 odd 2 637.2.a.j.1.3 3
40.19 odd 2 5824.2.a.bs.1.2 3
40.29 even 2 5824.2.a.by.1.2 3
65.34 odd 4 1183.2.c.f.337.6 6
65.44 odd 4 1183.2.c.f.337.1 6
65.64 even 2 1183.2.a.i.1.1 3
105.104 even 2 5733.2.a.x.1.1 3
455.454 odd 2 8281.2.a.bg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 5.4 even 2
637.2.a.j.1.3 3 35.34 odd 2
637.2.e.i.79.1 6 35.24 odd 6
637.2.e.i.508.1 6 35.19 odd 6
637.2.e.j.79.1 6 35.4 even 6
637.2.e.j.508.1 6 35.9 even 6
819.2.a.i.1.1 3 15.14 odd 2
1183.2.a.i.1.1 3 65.64 even 2
1183.2.c.f.337.1 6 65.44 odd 4
1183.2.c.f.337.6 6 65.34 odd 4
1456.2.a.t.1.2 3 20.19 odd 2
2275.2.a.m.1.1 3 1.1 even 1 trivial
5733.2.a.x.1.1 3 105.104 even 2
5824.2.a.bs.1.2 3 40.19 odd 2
5824.2.a.by.1.2 3 40.29 even 2
8281.2.a.bg.1.1 3 455.454 odd 2