Properties

Label 2013.2.a.g.1.1
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $13$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} - 640 x^{4} + 274 x^{3} + 256 x^{2} - 74 x - 35 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.33467\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.33467 q^{2} +1.00000 q^{3} +3.45069 q^{4} +0.0293558 q^{5} -2.33467 q^{6} -2.70286 q^{7} -3.38688 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.33467 q^{2} +1.00000 q^{3} +3.45069 q^{4} +0.0293558 q^{5} -2.33467 q^{6} -2.70286 q^{7} -3.38688 q^{8} +1.00000 q^{9} -0.0685361 q^{10} +1.00000 q^{11} +3.45069 q^{12} +2.89225 q^{13} +6.31030 q^{14} +0.0293558 q^{15} +1.00587 q^{16} +7.17621 q^{17} -2.33467 q^{18} +4.17563 q^{19} +0.101298 q^{20} -2.70286 q^{21} -2.33467 q^{22} -2.06467 q^{23} -3.38688 q^{24} -4.99914 q^{25} -6.75246 q^{26} +1.00000 q^{27} -9.32674 q^{28} +4.39436 q^{29} -0.0685361 q^{30} -4.71288 q^{31} +4.42539 q^{32} +1.00000 q^{33} -16.7541 q^{34} -0.0793448 q^{35} +3.45069 q^{36} -2.29994 q^{37} -9.74873 q^{38} +2.89225 q^{39} -0.0994245 q^{40} -2.67070 q^{41} +6.31030 q^{42} -8.92207 q^{43} +3.45069 q^{44} +0.0293558 q^{45} +4.82032 q^{46} +10.7872 q^{47} +1.00587 q^{48} +0.305477 q^{49} +11.6713 q^{50} +7.17621 q^{51} +9.98027 q^{52} -4.86692 q^{53} -2.33467 q^{54} +0.0293558 q^{55} +9.15427 q^{56} +4.17563 q^{57} -10.2594 q^{58} -0.181028 q^{59} +0.101298 q^{60} +1.00000 q^{61} +11.0030 q^{62} -2.70286 q^{63} -12.3436 q^{64} +0.0849045 q^{65} -2.33467 q^{66} +14.4055 q^{67} +24.7628 q^{68} -2.06467 q^{69} +0.185244 q^{70} +4.73226 q^{71} -3.38688 q^{72} +7.51871 q^{73} +5.36960 q^{74} -4.99914 q^{75} +14.4088 q^{76} -2.70286 q^{77} -6.75246 q^{78} +11.1725 q^{79} +0.0295280 q^{80} +1.00000 q^{81} +6.23522 q^{82} +12.9734 q^{83} -9.32674 q^{84} +0.210663 q^{85} +20.8301 q^{86} +4.39436 q^{87} -3.38688 q^{88} -10.1884 q^{89} -0.0685361 q^{90} -7.81737 q^{91} -7.12452 q^{92} -4.71288 q^{93} -25.1846 q^{94} +0.122579 q^{95} +4.42539 q^{96} -16.7207 q^{97} -0.713188 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 7 q^{7} + 9 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 7 q^{7} + 9 q^{8} + 13 q^{9} + 2 q^{10} + 13 q^{11} + 12 q^{12} + 9 q^{13} + 7 q^{14} + 7 q^{15} + 2 q^{16} + 19 q^{17} + 4 q^{18} + 14 q^{19} + 19 q^{20} + 7 q^{21} + 4 q^{22} + 5 q^{23} + 9 q^{24} + 2 q^{25} - 4 q^{26} + 13 q^{27} + 7 q^{28} + 10 q^{29} + 2 q^{30} - q^{31} + 7 q^{32} + 13 q^{33} - 2 q^{34} + 16 q^{35} + 12 q^{36} - 8 q^{37} - 10 q^{38} + 9 q^{39} + 14 q^{40} + 21 q^{41} + 7 q^{42} + 11 q^{43} + 12 q^{44} + 7 q^{45} - 8 q^{46} + 22 q^{47} + 2 q^{48} + 19 q^{50} + 19 q^{51} - q^{52} + 16 q^{53} + 4 q^{54} + 7 q^{55} + 14 q^{57} - 13 q^{58} + 19 q^{59} + 19 q^{60} + 13 q^{61} + 3 q^{62} + 7 q^{63} - 13 q^{64} + 13 q^{65} + 4 q^{66} + 12 q^{67} + 36 q^{68} + 5 q^{69} - 20 q^{70} + 5 q^{71} + 9 q^{72} + 18 q^{73} + 6 q^{74} + 2 q^{75} - 5 q^{76} + 7 q^{77} - 4 q^{78} - q^{79} + 6 q^{80} + 13 q^{81} - 22 q^{82} + 48 q^{83} + 7 q^{84} - 2 q^{85} + 26 q^{86} + 10 q^{87} + 9 q^{88} + 15 q^{89} + 2 q^{90} - 11 q^{91} - 24 q^{92} - q^{93} - 23 q^{94} + 17 q^{95} + 7 q^{96} - 17 q^{97} - 15 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.33467 −1.65086 −0.825431 0.564503i \(-0.809068\pi\)
−0.825431 + 0.564503i \(0.809068\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.45069 1.72534
\(5\) 0.0293558 0.0131283 0.00656416 0.999978i \(-0.497911\pi\)
0.00656416 + 0.999978i \(0.497911\pi\)
\(6\) −2.33467 −0.953125
\(7\) −2.70286 −1.02159 −0.510793 0.859703i \(-0.670648\pi\)
−0.510793 + 0.859703i \(0.670648\pi\)
\(8\) −3.38688 −1.19744
\(9\) 1.00000 0.333333
\(10\) −0.0685361 −0.0216730
\(11\) 1.00000 0.301511
\(12\) 3.45069 0.996128
\(13\) 2.89225 0.802167 0.401084 0.916041i \(-0.368634\pi\)
0.401084 + 0.916041i \(0.368634\pi\)
\(14\) 6.31030 1.68650
\(15\) 0.0293558 0.00757963
\(16\) 1.00587 0.251467
\(17\) 7.17621 1.74049 0.870243 0.492623i \(-0.163962\pi\)
0.870243 + 0.492623i \(0.163962\pi\)
\(18\) −2.33467 −0.550287
\(19\) 4.17563 0.957956 0.478978 0.877827i \(-0.341007\pi\)
0.478978 + 0.877827i \(0.341007\pi\)
\(20\) 0.101298 0.0226508
\(21\) −2.70286 −0.589813
\(22\) −2.33467 −0.497753
\(23\) −2.06467 −0.430513 −0.215256 0.976558i \(-0.569059\pi\)
−0.215256 + 0.976558i \(0.569059\pi\)
\(24\) −3.38688 −0.691343
\(25\) −4.99914 −0.999828
\(26\) −6.75246 −1.32427
\(27\) 1.00000 0.192450
\(28\) −9.32674 −1.76259
\(29\) 4.39436 0.816012 0.408006 0.912979i \(-0.366224\pi\)
0.408006 + 0.912979i \(0.366224\pi\)
\(30\) −0.0685361 −0.0125129
\(31\) −4.71288 −0.846457 −0.423229 0.906023i \(-0.639103\pi\)
−0.423229 + 0.906023i \(0.639103\pi\)
\(32\) 4.42539 0.782305
\(33\) 1.00000 0.174078
\(34\) −16.7541 −2.87330
\(35\) −0.0793448 −0.0134117
\(36\) 3.45069 0.575114
\(37\) −2.29994 −0.378108 −0.189054 0.981967i \(-0.560542\pi\)
−0.189054 + 0.981967i \(0.560542\pi\)
\(38\) −9.74873 −1.58145
\(39\) 2.89225 0.463131
\(40\) −0.0994245 −0.0157204
\(41\) −2.67070 −0.417094 −0.208547 0.978012i \(-0.566873\pi\)
−0.208547 + 0.978012i \(0.566873\pi\)
\(42\) 6.31030 0.973700
\(43\) −8.92207 −1.36060 −0.680301 0.732932i \(-0.738151\pi\)
−0.680301 + 0.732932i \(0.738151\pi\)
\(44\) 3.45069 0.520211
\(45\) 0.0293558 0.00437610
\(46\) 4.82032 0.710717
\(47\) 10.7872 1.57348 0.786738 0.617287i \(-0.211768\pi\)
0.786738 + 0.617287i \(0.211768\pi\)
\(48\) 1.00587 0.145184
\(49\) 0.305477 0.0436396
\(50\) 11.6713 1.65058
\(51\) 7.17621 1.00487
\(52\) 9.98027 1.38401
\(53\) −4.86692 −0.668523 −0.334261 0.942480i \(-0.608487\pi\)
−0.334261 + 0.942480i \(0.608487\pi\)
\(54\) −2.33467 −0.317708
\(55\) 0.0293558 0.00395834
\(56\) 9.15427 1.22329
\(57\) 4.17563 0.553076
\(58\) −10.2594 −1.34712
\(59\) −0.181028 −0.0235678 −0.0117839 0.999931i \(-0.503751\pi\)
−0.0117839 + 0.999931i \(0.503751\pi\)
\(60\) 0.101298 0.0130775
\(61\) 1.00000 0.128037
\(62\) 11.0030 1.39738
\(63\) −2.70286 −0.340529
\(64\) −12.3436 −1.54294
\(65\) 0.0849045 0.0105311
\(66\) −2.33467 −0.287378
\(67\) 14.4055 1.75991 0.879955 0.475057i \(-0.157573\pi\)
0.879955 + 0.475057i \(0.157573\pi\)
\(68\) 24.7628 3.00294
\(69\) −2.06467 −0.248557
\(70\) 0.185244 0.0221409
\(71\) 4.73226 0.561616 0.280808 0.959764i \(-0.409398\pi\)
0.280808 + 0.959764i \(0.409398\pi\)
\(72\) −3.38688 −0.399147
\(73\) 7.51871 0.879999 0.439999 0.897998i \(-0.354979\pi\)
0.439999 + 0.897998i \(0.354979\pi\)
\(74\) 5.36960 0.624204
\(75\) −4.99914 −0.577251
\(76\) 14.4088 1.65280
\(77\) −2.70286 −0.308020
\(78\) −6.75246 −0.764566
\(79\) 11.1725 1.25701 0.628503 0.777807i \(-0.283668\pi\)
0.628503 + 0.777807i \(0.283668\pi\)
\(80\) 0.0295280 0.00330133
\(81\) 1.00000 0.111111
\(82\) 6.23522 0.688564
\(83\) 12.9734 1.42402 0.712008 0.702171i \(-0.247786\pi\)
0.712008 + 0.702171i \(0.247786\pi\)
\(84\) −9.32674 −1.01763
\(85\) 0.210663 0.0228496
\(86\) 20.8301 2.24617
\(87\) 4.39436 0.471125
\(88\) −3.38688 −0.361042
\(89\) −10.1884 −1.07997 −0.539983 0.841676i \(-0.681569\pi\)
−0.539983 + 0.841676i \(0.681569\pi\)
\(90\) −0.0685361 −0.00722434
\(91\) −7.81737 −0.819483
\(92\) −7.12452 −0.742783
\(93\) −4.71288 −0.488702
\(94\) −25.1846 −2.59759
\(95\) 0.122579 0.0125763
\(96\) 4.42539 0.451664
\(97\) −16.7207 −1.69773 −0.848867 0.528606i \(-0.822715\pi\)
−0.848867 + 0.528606i \(0.822715\pi\)
\(98\) −0.713188 −0.0720429
\(99\) 1.00000 0.100504
\(100\) −17.2505 −1.72505
\(101\) 1.97502 0.196522 0.0982610 0.995161i \(-0.468672\pi\)
0.0982610 + 0.995161i \(0.468672\pi\)
\(102\) −16.7541 −1.65890
\(103\) 8.68750 0.856005 0.428003 0.903777i \(-0.359217\pi\)
0.428003 + 0.903777i \(0.359217\pi\)
\(104\) −9.79571 −0.960548
\(105\) −0.0793448 −0.00774325
\(106\) 11.3627 1.10364
\(107\) −6.12215 −0.591851 −0.295925 0.955211i \(-0.595628\pi\)
−0.295925 + 0.955211i \(0.595628\pi\)
\(108\) 3.45069 0.332043
\(109\) −3.59659 −0.344491 −0.172245 0.985054i \(-0.555102\pi\)
−0.172245 + 0.985054i \(0.555102\pi\)
\(110\) −0.0685361 −0.00653466
\(111\) −2.29994 −0.218301
\(112\) −2.71872 −0.256895
\(113\) 6.81766 0.641352 0.320676 0.947189i \(-0.396090\pi\)
0.320676 + 0.947189i \(0.396090\pi\)
\(114\) −9.74873 −0.913052
\(115\) −0.0606100 −0.00565191
\(116\) 15.1636 1.40790
\(117\) 2.89225 0.267389
\(118\) 0.422640 0.0389072
\(119\) −19.3963 −1.77806
\(120\) −0.0994245 −0.00907617
\(121\) 1.00000 0.0909091
\(122\) −2.33467 −0.211371
\(123\) −2.67070 −0.240809
\(124\) −16.2627 −1.46043
\(125\) −0.293533 −0.0262544
\(126\) 6.31030 0.562166
\(127\) 5.40526 0.479639 0.239819 0.970817i \(-0.422912\pi\)
0.239819 + 0.970817i \(0.422912\pi\)
\(128\) 19.9674 1.76488
\(129\) −8.92207 −0.785544
\(130\) −0.198224 −0.0173854
\(131\) 19.3818 1.69340 0.846698 0.532075i \(-0.178587\pi\)
0.846698 + 0.532075i \(0.178587\pi\)
\(132\) 3.45069 0.300344
\(133\) −11.2862 −0.978635
\(134\) −33.6321 −2.90537
\(135\) 0.0293558 0.00252654
\(136\) −24.3049 −2.08413
\(137\) 15.7164 1.34275 0.671373 0.741119i \(-0.265705\pi\)
0.671373 + 0.741119i \(0.265705\pi\)
\(138\) 4.82032 0.410333
\(139\) 1.57642 0.133711 0.0668553 0.997763i \(-0.478703\pi\)
0.0668553 + 0.997763i \(0.478703\pi\)
\(140\) −0.273794 −0.0231398
\(141\) 10.7872 0.908447
\(142\) −11.0483 −0.927150
\(143\) 2.89225 0.241862
\(144\) 1.00587 0.0838222
\(145\) 0.129000 0.0107129
\(146\) −17.5537 −1.45276
\(147\) 0.305477 0.0251953
\(148\) −7.93637 −0.652366
\(149\) 1.10490 0.0905172 0.0452586 0.998975i \(-0.485589\pi\)
0.0452586 + 0.998975i \(0.485589\pi\)
\(150\) 11.6713 0.952961
\(151\) −15.0032 −1.22095 −0.610473 0.792037i \(-0.709021\pi\)
−0.610473 + 0.792037i \(0.709021\pi\)
\(152\) −14.1424 −1.14710
\(153\) 7.17621 0.580162
\(154\) 6.31030 0.508498
\(155\) −0.138350 −0.0111126
\(156\) 9.98027 0.799061
\(157\) −9.75971 −0.778910 −0.389455 0.921046i \(-0.627337\pi\)
−0.389455 + 0.921046i \(0.627337\pi\)
\(158\) −26.0842 −2.07514
\(159\) −4.86692 −0.385972
\(160\) 0.129911 0.0102703
\(161\) 5.58052 0.439806
\(162\) −2.33467 −0.183429
\(163\) 21.3224 1.67010 0.835050 0.550174i \(-0.185439\pi\)
0.835050 + 0.550174i \(0.185439\pi\)
\(164\) −9.21577 −0.719630
\(165\) 0.0293558 0.00228535
\(166\) −30.2886 −2.35085
\(167\) 19.0985 1.47789 0.738943 0.673768i \(-0.235325\pi\)
0.738943 + 0.673768i \(0.235325\pi\)
\(168\) 9.15427 0.706267
\(169\) −4.63486 −0.356528
\(170\) −0.491829 −0.0377216
\(171\) 4.17563 0.319319
\(172\) −30.7873 −2.34751
\(173\) −17.4746 −1.32857 −0.664285 0.747479i \(-0.731264\pi\)
−0.664285 + 0.747479i \(0.731264\pi\)
\(174\) −10.2594 −0.777762
\(175\) 13.5120 1.02141
\(176\) 1.00587 0.0758200
\(177\) −0.181028 −0.0136069
\(178\) 23.7865 1.78288
\(179\) 15.9382 1.19127 0.595637 0.803253i \(-0.296900\pi\)
0.595637 + 0.803253i \(0.296900\pi\)
\(180\) 0.101298 0.00755028
\(181\) 7.41273 0.550984 0.275492 0.961303i \(-0.411159\pi\)
0.275492 + 0.961303i \(0.411159\pi\)
\(182\) 18.2510 1.35285
\(183\) 1.00000 0.0739221
\(184\) 6.99277 0.515514
\(185\) −0.0675166 −0.00496392
\(186\) 11.0030 0.806780
\(187\) 7.17621 0.524776
\(188\) 37.2233 2.71479
\(189\) −2.70286 −0.196604
\(190\) −0.286182 −0.0207618
\(191\) 11.6075 0.839886 0.419943 0.907550i \(-0.362050\pi\)
0.419943 + 0.907550i \(0.362050\pi\)
\(192\) −12.3436 −0.890819
\(193\) 20.4885 1.47479 0.737396 0.675460i \(-0.236055\pi\)
0.737396 + 0.675460i \(0.236055\pi\)
\(194\) 39.0374 2.80272
\(195\) 0.0849045 0.00608013
\(196\) 1.05411 0.0752933
\(197\) 7.27132 0.518060 0.259030 0.965869i \(-0.416597\pi\)
0.259030 + 0.965869i \(0.416597\pi\)
\(198\) −2.33467 −0.165918
\(199\) −10.0401 −0.711723 −0.355862 0.934539i \(-0.615813\pi\)
−0.355862 + 0.934539i \(0.615813\pi\)
\(200\) 16.9315 1.19724
\(201\) 14.4055 1.01608
\(202\) −4.61103 −0.324431
\(203\) −11.8774 −0.833627
\(204\) 24.7628 1.73375
\(205\) −0.0784007 −0.00547574
\(206\) −20.2825 −1.41315
\(207\) −2.06467 −0.143504
\(208\) 2.90922 0.201718
\(209\) 4.17563 0.288835
\(210\) 0.185244 0.0127830
\(211\) −7.36422 −0.506974 −0.253487 0.967339i \(-0.581577\pi\)
−0.253487 + 0.967339i \(0.581577\pi\)
\(212\) −16.7942 −1.15343
\(213\) 4.73226 0.324249
\(214\) 14.2932 0.977063
\(215\) −0.261915 −0.0178624
\(216\) −3.38688 −0.230448
\(217\) 12.7383 0.864730
\(218\) 8.39685 0.568706
\(219\) 7.51871 0.508068
\(220\) 0.101298 0.00682949
\(221\) 20.7554 1.39616
\(222\) 5.36960 0.360384
\(223\) 1.02663 0.0687482 0.0343741 0.999409i \(-0.489056\pi\)
0.0343741 + 0.999409i \(0.489056\pi\)
\(224\) −11.9612 −0.799193
\(225\) −4.99914 −0.333276
\(226\) −15.9170 −1.05878
\(227\) 21.6433 1.43651 0.718257 0.695778i \(-0.244940\pi\)
0.718257 + 0.695778i \(0.244940\pi\)
\(228\) 14.4088 0.954246
\(229\) 9.88816 0.653428 0.326714 0.945123i \(-0.394059\pi\)
0.326714 + 0.945123i \(0.394059\pi\)
\(230\) 0.141504 0.00933052
\(231\) −2.70286 −0.177835
\(232\) −14.8832 −0.977127
\(233\) −24.8039 −1.62496 −0.812480 0.582989i \(-0.801883\pi\)
−0.812480 + 0.582989i \(0.801883\pi\)
\(234\) −6.75246 −0.441422
\(235\) 0.316667 0.0206571
\(236\) −0.624670 −0.0406626
\(237\) 11.1725 0.725733
\(238\) 45.2840 2.93533
\(239\) 15.9365 1.03084 0.515422 0.856936i \(-0.327635\pi\)
0.515422 + 0.856936i \(0.327635\pi\)
\(240\) 0.0295280 0.00190602
\(241\) −10.6415 −0.685481 −0.342740 0.939430i \(-0.611355\pi\)
−0.342740 + 0.939430i \(0.611355\pi\)
\(242\) −2.33467 −0.150078
\(243\) 1.00000 0.0641500
\(244\) 3.45069 0.220908
\(245\) 0.00896753 0.000572914 0
\(246\) 6.23522 0.397543
\(247\) 12.0770 0.768441
\(248\) 15.9619 1.01358
\(249\) 12.9734 0.822156
\(250\) 0.685302 0.0433423
\(251\) −24.6375 −1.55510 −0.777552 0.628819i \(-0.783539\pi\)
−0.777552 + 0.628819i \(0.783539\pi\)
\(252\) −9.32674 −0.587529
\(253\) −2.06467 −0.129805
\(254\) −12.6195 −0.791818
\(255\) 0.210663 0.0131922
\(256\) −21.9301 −1.37063
\(257\) 8.56747 0.534424 0.267212 0.963638i \(-0.413898\pi\)
0.267212 + 0.963638i \(0.413898\pi\)
\(258\) 20.8301 1.29683
\(259\) 6.21643 0.386270
\(260\) 0.292979 0.0181698
\(261\) 4.39436 0.272004
\(262\) −45.2501 −2.79556
\(263\) −7.12601 −0.439408 −0.219704 0.975567i \(-0.570509\pi\)
−0.219704 + 0.975567i \(0.570509\pi\)
\(264\) −3.38688 −0.208448
\(265\) −0.142872 −0.00877658
\(266\) 26.3495 1.61559
\(267\) −10.1884 −0.623519
\(268\) 49.7088 3.03645
\(269\) 14.1919 0.865295 0.432648 0.901563i \(-0.357579\pi\)
0.432648 + 0.901563i \(0.357579\pi\)
\(270\) −0.0685361 −0.00417098
\(271\) 29.2260 1.77535 0.887677 0.460466i \(-0.152318\pi\)
0.887677 + 0.460466i \(0.152318\pi\)
\(272\) 7.21830 0.437674
\(273\) −7.81737 −0.473129
\(274\) −36.6927 −2.21669
\(275\) −4.99914 −0.301459
\(276\) −7.12452 −0.428846
\(277\) −13.9600 −0.838774 −0.419387 0.907808i \(-0.637755\pi\)
−0.419387 + 0.907808i \(0.637755\pi\)
\(278\) −3.68043 −0.220738
\(279\) −4.71288 −0.282152
\(280\) 0.268731 0.0160597
\(281\) −5.84690 −0.348797 −0.174398 0.984675i \(-0.555798\pi\)
−0.174398 + 0.984675i \(0.555798\pi\)
\(282\) −25.1846 −1.49972
\(283\) 25.6344 1.52380 0.761902 0.647692i \(-0.224266\pi\)
0.761902 + 0.647692i \(0.224266\pi\)
\(284\) 16.3296 0.968981
\(285\) 0.122579 0.00726096
\(286\) −6.75246 −0.399281
\(287\) 7.21855 0.426098
\(288\) 4.42539 0.260768
\(289\) 34.4979 2.02929
\(290\) −0.301172 −0.0176855
\(291\) −16.7207 −0.980187
\(292\) 25.9447 1.51830
\(293\) 1.56499 0.0914279 0.0457140 0.998955i \(-0.485444\pi\)
0.0457140 + 0.998955i \(0.485444\pi\)
\(294\) −0.713188 −0.0415940
\(295\) −0.00531422 −0.000309406 0
\(296\) 7.78961 0.452762
\(297\) 1.00000 0.0580259
\(298\) −2.57959 −0.149431
\(299\) −5.97154 −0.345343
\(300\) −17.2505 −0.995956
\(301\) 24.1151 1.38997
\(302\) 35.0276 2.01561
\(303\) 1.97502 0.113462
\(304\) 4.20013 0.240894
\(305\) 0.0293558 0.00168091
\(306\) −16.7541 −0.957767
\(307\) 8.60524 0.491127 0.245563 0.969381i \(-0.421027\pi\)
0.245563 + 0.969381i \(0.421027\pi\)
\(308\) −9.32674 −0.531440
\(309\) 8.68750 0.494215
\(310\) 0.323002 0.0183453
\(311\) −31.7733 −1.80170 −0.900850 0.434130i \(-0.857056\pi\)
−0.900850 + 0.434130i \(0.857056\pi\)
\(312\) −9.79571 −0.554573
\(313\) 26.8892 1.51987 0.759934 0.650000i \(-0.225231\pi\)
0.759934 + 0.650000i \(0.225231\pi\)
\(314\) 22.7857 1.28587
\(315\) −0.0793448 −0.00447057
\(316\) 38.5529 2.16877
\(317\) −21.6911 −1.21830 −0.609148 0.793057i \(-0.708488\pi\)
−0.609148 + 0.793057i \(0.708488\pi\)
\(318\) 11.3627 0.637186
\(319\) 4.39436 0.246037
\(320\) −0.362355 −0.0202562
\(321\) −6.12215 −0.341705
\(322\) −13.0287 −0.726059
\(323\) 29.9652 1.66731
\(324\) 3.45069 0.191705
\(325\) −14.4588 −0.802029
\(326\) −49.7808 −2.75710
\(327\) −3.59659 −0.198892
\(328\) 9.04535 0.499446
\(329\) −29.1564 −1.60744
\(330\) −0.0685361 −0.00377279
\(331\) −3.16662 −0.174053 −0.0870265 0.996206i \(-0.527736\pi\)
−0.0870265 + 0.996206i \(0.527736\pi\)
\(332\) 44.7672 2.45692
\(333\) −2.29994 −0.126036
\(334\) −44.5887 −2.43979
\(335\) 0.422884 0.0231046
\(336\) −2.71872 −0.148318
\(337\) 29.0756 1.58385 0.791924 0.610619i \(-0.209079\pi\)
0.791924 + 0.610619i \(0.209079\pi\)
\(338\) 10.8209 0.588578
\(339\) 6.81766 0.370285
\(340\) 0.726933 0.0394235
\(341\) −4.71288 −0.255217
\(342\) −9.74873 −0.527151
\(343\) 18.0944 0.977005
\(344\) 30.2179 1.62924
\(345\) −0.0606100 −0.00326313
\(346\) 40.7975 2.19329
\(347\) −5.76588 −0.309529 −0.154764 0.987951i \(-0.549462\pi\)
−0.154764 + 0.987951i \(0.549462\pi\)
\(348\) 15.1636 0.812852
\(349\) −29.8833 −1.59961 −0.799807 0.600257i \(-0.795065\pi\)
−0.799807 + 0.600257i \(0.795065\pi\)
\(350\) −31.5461 −1.68621
\(351\) 2.89225 0.154377
\(352\) 4.42539 0.235874
\(353\) 8.68646 0.462334 0.231167 0.972914i \(-0.425746\pi\)
0.231167 + 0.972914i \(0.425746\pi\)
\(354\) 0.422640 0.0224631
\(355\) 0.138919 0.00737307
\(356\) −35.1569 −1.86331
\(357\) −19.3963 −1.02656
\(358\) −37.2104 −1.96663
\(359\) −0.320645 −0.0169230 −0.00846150 0.999964i \(-0.502693\pi\)
−0.00846150 + 0.999964i \(0.502693\pi\)
\(360\) −0.0994245 −0.00524013
\(361\) −1.56409 −0.0823205
\(362\) −17.3063 −0.909598
\(363\) 1.00000 0.0524864
\(364\) −26.9753 −1.41389
\(365\) 0.220718 0.0115529
\(366\) −2.33467 −0.122035
\(367\) −14.7522 −0.770060 −0.385030 0.922904i \(-0.625809\pi\)
−0.385030 + 0.922904i \(0.625809\pi\)
\(368\) −2.07678 −0.108260
\(369\) −2.67070 −0.139031
\(370\) 0.157629 0.00819474
\(371\) 13.1546 0.682954
\(372\) −16.2627 −0.843180
\(373\) −7.54533 −0.390683 −0.195341 0.980735i \(-0.562581\pi\)
−0.195341 + 0.980735i \(0.562581\pi\)
\(374\) −16.7541 −0.866333
\(375\) −0.293533 −0.0151580
\(376\) −36.5349 −1.88415
\(377\) 12.7096 0.654578
\(378\) 6.31030 0.324567
\(379\) −5.90833 −0.303491 −0.151745 0.988420i \(-0.548489\pi\)
−0.151745 + 0.988420i \(0.548489\pi\)
\(380\) 0.422982 0.0216985
\(381\) 5.40526 0.276920
\(382\) −27.0996 −1.38654
\(383\) 28.3128 1.44672 0.723358 0.690473i \(-0.242598\pi\)
0.723358 + 0.690473i \(0.242598\pi\)
\(384\) 19.9674 1.01895
\(385\) −0.0793448 −0.00404378
\(386\) −47.8338 −2.43468
\(387\) −8.92207 −0.453534
\(388\) −57.6981 −2.92918
\(389\) −16.1105 −0.816833 −0.408417 0.912796i \(-0.633919\pi\)
−0.408417 + 0.912796i \(0.633919\pi\)
\(390\) −0.198224 −0.0100375
\(391\) −14.8165 −0.749302
\(392\) −1.03461 −0.0522559
\(393\) 19.3818 0.977682
\(394\) −16.9761 −0.855245
\(395\) 0.327978 0.0165024
\(396\) 3.45069 0.173404
\(397\) −12.9139 −0.648128 −0.324064 0.946035i \(-0.605049\pi\)
−0.324064 + 0.946035i \(0.605049\pi\)
\(398\) 23.4403 1.17496
\(399\) −11.2862 −0.565015
\(400\) −5.02846 −0.251423
\(401\) 21.6181 1.07956 0.539778 0.841808i \(-0.318508\pi\)
0.539778 + 0.841808i \(0.318508\pi\)
\(402\) −33.6321 −1.67741
\(403\) −13.6308 −0.679000
\(404\) 6.81518 0.339068
\(405\) 0.0293558 0.00145870
\(406\) 27.7297 1.37620
\(407\) −2.29994 −0.114004
\(408\) −24.3049 −1.20327
\(409\) −10.5185 −0.520108 −0.260054 0.965594i \(-0.583740\pi\)
−0.260054 + 0.965594i \(0.583740\pi\)
\(410\) 0.183040 0.00903969
\(411\) 15.7164 0.775235
\(412\) 29.9779 1.47690
\(413\) 0.489294 0.0240766
\(414\) 4.82032 0.236906
\(415\) 0.380845 0.0186949
\(416\) 12.7993 0.627539
\(417\) 1.57642 0.0771978
\(418\) −9.74873 −0.476826
\(419\) 5.19468 0.253777 0.126888 0.991917i \(-0.459501\pi\)
0.126888 + 0.991917i \(0.459501\pi\)
\(420\) −0.273794 −0.0133598
\(421\) −22.0265 −1.07351 −0.536753 0.843740i \(-0.680349\pi\)
−0.536753 + 0.843740i \(0.680349\pi\)
\(422\) 17.1930 0.836943
\(423\) 10.7872 0.524492
\(424\) 16.4837 0.800517
\(425\) −35.8748 −1.74019
\(426\) −11.0483 −0.535290
\(427\) −2.70286 −0.130801
\(428\) −21.1256 −1.02115
\(429\) 2.89225 0.139639
\(430\) 0.611484 0.0294884
\(431\) −0.844284 −0.0406677 −0.0203339 0.999793i \(-0.506473\pi\)
−0.0203339 + 0.999793i \(0.506473\pi\)
\(432\) 1.00587 0.0483948
\(433\) 28.8544 1.38665 0.693326 0.720624i \(-0.256144\pi\)
0.693326 + 0.720624i \(0.256144\pi\)
\(434\) −29.7397 −1.42755
\(435\) 0.129000 0.00618508
\(436\) −12.4107 −0.594365
\(437\) −8.62129 −0.412412
\(438\) −17.5537 −0.838749
\(439\) −21.6203 −1.03188 −0.515941 0.856624i \(-0.672557\pi\)
−0.515941 + 0.856624i \(0.672557\pi\)
\(440\) −0.0994245 −0.00473987
\(441\) 0.305477 0.0145465
\(442\) −48.4571 −2.30487
\(443\) −3.90144 −0.185363 −0.0926815 0.995696i \(-0.529544\pi\)
−0.0926815 + 0.995696i \(0.529544\pi\)
\(444\) −7.93637 −0.376644
\(445\) −0.299088 −0.0141781
\(446\) −2.39684 −0.113494
\(447\) 1.10490 0.0522601
\(448\) 33.3629 1.57625
\(449\) −7.16531 −0.338152 −0.169076 0.985603i \(-0.554078\pi\)
−0.169076 + 0.985603i \(0.554078\pi\)
\(450\) 11.6713 0.550192
\(451\) −2.67070 −0.125759
\(452\) 23.5256 1.10655
\(453\) −15.0032 −0.704914
\(454\) −50.5299 −2.37149
\(455\) −0.229485 −0.0107584
\(456\) −14.1424 −0.662276
\(457\) 3.35821 0.157091 0.0785453 0.996911i \(-0.474972\pi\)
0.0785453 + 0.996911i \(0.474972\pi\)
\(458\) −23.0856 −1.07872
\(459\) 7.17621 0.334957
\(460\) −0.209146 −0.00975148
\(461\) −41.2484 −1.92113 −0.960565 0.278055i \(-0.910310\pi\)
−0.960565 + 0.278055i \(0.910310\pi\)
\(462\) 6.31030 0.293582
\(463\) 18.7387 0.870862 0.435431 0.900222i \(-0.356596\pi\)
0.435431 + 0.900222i \(0.356596\pi\)
\(464\) 4.42014 0.205200
\(465\) −0.138350 −0.00641584
\(466\) 57.9091 2.68259
\(467\) −24.4499 −1.13141 −0.565704 0.824608i \(-0.691396\pi\)
−0.565704 + 0.824608i \(0.691396\pi\)
\(468\) 9.98027 0.461338
\(469\) −38.9361 −1.79790
\(470\) −0.739314 −0.0341020
\(471\) −9.75971 −0.449704
\(472\) 0.613119 0.0282211
\(473\) −8.92207 −0.410237
\(474\) −26.0842 −1.19809
\(475\) −20.8746 −0.957791
\(476\) −66.9306 −3.06776
\(477\) −4.86692 −0.222841
\(478\) −37.2064 −1.70178
\(479\) −34.9779 −1.59818 −0.799089 0.601212i \(-0.794685\pi\)
−0.799089 + 0.601212i \(0.794685\pi\)
\(480\) 0.129911 0.00592959
\(481\) −6.65201 −0.303306
\(482\) 24.8445 1.13163
\(483\) 5.58052 0.253922
\(484\) 3.45069 0.156849
\(485\) −0.490851 −0.0222884
\(486\) −2.33467 −0.105903
\(487\) 11.8692 0.537847 0.268923 0.963162i \(-0.413332\pi\)
0.268923 + 0.963162i \(0.413332\pi\)
\(488\) −3.38688 −0.153317
\(489\) 21.3224 0.964233
\(490\) −0.0209362 −0.000945802 0
\(491\) 34.2782 1.54695 0.773477 0.633824i \(-0.218516\pi\)
0.773477 + 0.633824i \(0.218516\pi\)
\(492\) −9.21577 −0.415479
\(493\) 31.5348 1.42026
\(494\) −28.1958 −1.26859
\(495\) 0.0293558 0.00131945
\(496\) −4.74052 −0.212856
\(497\) −12.7907 −0.573740
\(498\) −30.2886 −1.35727
\(499\) 22.9767 1.02858 0.514289 0.857617i \(-0.328056\pi\)
0.514289 + 0.857617i \(0.328056\pi\)
\(500\) −1.01289 −0.0452978
\(501\) 19.0985 0.853258
\(502\) 57.5204 2.56726
\(503\) −31.8586 −1.42051 −0.710253 0.703946i \(-0.751420\pi\)
−0.710253 + 0.703946i \(0.751420\pi\)
\(504\) 9.15427 0.407763
\(505\) 0.0579784 0.00258000
\(506\) 4.82032 0.214289
\(507\) −4.63486 −0.205841
\(508\) 18.6518 0.827542
\(509\) −14.4168 −0.639012 −0.319506 0.947584i \(-0.603517\pi\)
−0.319506 + 0.947584i \(0.603517\pi\)
\(510\) −0.491829 −0.0217786
\(511\) −20.3221 −0.898995
\(512\) 11.2648 0.497840
\(513\) 4.17563 0.184359
\(514\) −20.0022 −0.882261
\(515\) 0.255029 0.0112379
\(516\) −30.7873 −1.35533
\(517\) 10.7872 0.474421
\(518\) −14.5133 −0.637678
\(519\) −17.4746 −0.767050
\(520\) −0.287561 −0.0126104
\(521\) 5.61958 0.246198 0.123099 0.992394i \(-0.460717\pi\)
0.123099 + 0.992394i \(0.460717\pi\)
\(522\) −10.2594 −0.449041
\(523\) 13.6790 0.598142 0.299071 0.954231i \(-0.403323\pi\)
0.299071 + 0.954231i \(0.403323\pi\)
\(524\) 66.8805 2.92169
\(525\) 13.5120 0.589712
\(526\) 16.6369 0.725402
\(527\) −33.8206 −1.47325
\(528\) 1.00587 0.0437747
\(529\) −18.7371 −0.814659
\(530\) 0.333560 0.0144889
\(531\) −0.181028 −0.00785594
\(532\) −38.9450 −1.68848
\(533\) −7.72436 −0.334579
\(534\) 23.7865 1.02934
\(535\) −0.179721 −0.00777000
\(536\) −48.7896 −2.10739
\(537\) 15.9382 0.687783
\(538\) −33.1334 −1.42848
\(539\) 0.305477 0.0131578
\(540\) 0.101298 0.00435916
\(541\) 7.06159 0.303602 0.151801 0.988411i \(-0.451493\pi\)
0.151801 + 0.988411i \(0.451493\pi\)
\(542\) −68.2331 −2.93086
\(543\) 7.41273 0.318111
\(544\) 31.7575 1.36159
\(545\) −0.105581 −0.00452258
\(546\) 18.2510 0.781070
\(547\) −21.3505 −0.912881 −0.456441 0.889754i \(-0.650876\pi\)
−0.456441 + 0.889754i \(0.650876\pi\)
\(548\) 54.2325 2.31670
\(549\) 1.00000 0.0426790
\(550\) 11.6713 0.497668
\(551\) 18.3492 0.781704
\(552\) 6.99277 0.297632
\(553\) −30.1978 −1.28414
\(554\) 32.5919 1.38470
\(555\) −0.0675166 −0.00286592
\(556\) 5.43975 0.230697
\(557\) 12.8162 0.543039 0.271520 0.962433i \(-0.412474\pi\)
0.271520 + 0.962433i \(0.412474\pi\)
\(558\) 11.0030 0.465795
\(559\) −25.8049 −1.09143
\(560\) −0.0798102 −0.00337260
\(561\) 7.17621 0.302980
\(562\) 13.6506 0.575815
\(563\) −30.0977 −1.26846 −0.634232 0.773142i \(-0.718684\pi\)
−0.634232 + 0.773142i \(0.718684\pi\)
\(564\) 37.2233 1.56738
\(565\) 0.200138 0.00841986
\(566\) −59.8478 −2.51559
\(567\) −2.70286 −0.113510
\(568\) −16.0276 −0.672502
\(569\) 33.1992 1.39178 0.695891 0.718148i \(-0.255010\pi\)
0.695891 + 0.718148i \(0.255010\pi\)
\(570\) −0.286182 −0.0119868
\(571\) −21.8687 −0.915177 −0.457589 0.889164i \(-0.651287\pi\)
−0.457589 + 0.889164i \(0.651287\pi\)
\(572\) 9.98027 0.417296
\(573\) 11.6075 0.484908
\(574\) −16.8529 −0.703428
\(575\) 10.3216 0.430439
\(576\) −12.3436 −0.514315
\(577\) −10.2833 −0.428101 −0.214050 0.976823i \(-0.568666\pi\)
−0.214050 + 0.976823i \(0.568666\pi\)
\(578\) −80.5413 −3.35008
\(579\) 20.4885 0.851472
\(580\) 0.445139 0.0184834
\(581\) −35.0654 −1.45476
\(582\) 39.0374 1.61815
\(583\) −4.86692 −0.201567
\(584\) −25.4650 −1.05375
\(585\) 0.0849045 0.00351037
\(586\) −3.65374 −0.150935
\(587\) −21.6507 −0.893622 −0.446811 0.894628i \(-0.647440\pi\)
−0.446811 + 0.894628i \(0.647440\pi\)
\(588\) 1.05411 0.0434706
\(589\) −19.6792 −0.810869
\(590\) 0.0124069 0.000510786 0
\(591\) 7.27132 0.299102
\(592\) −2.31343 −0.0950815
\(593\) −12.8896 −0.529312 −0.264656 0.964343i \(-0.585258\pi\)
−0.264656 + 0.964343i \(0.585258\pi\)
\(594\) −2.33467 −0.0957927
\(595\) −0.569394 −0.0233429
\(596\) 3.81268 0.156173
\(597\) −10.0401 −0.410914
\(598\) 13.9416 0.570114
\(599\) −22.2972 −0.911039 −0.455520 0.890226i \(-0.650547\pi\)
−0.455520 + 0.890226i \(0.650547\pi\)
\(600\) 16.9315 0.691224
\(601\) −7.06663 −0.288254 −0.144127 0.989559i \(-0.546037\pi\)
−0.144127 + 0.989559i \(0.546037\pi\)
\(602\) −56.3009 −2.29465
\(603\) 14.4055 0.586637
\(604\) −51.7715 −2.10655
\(605\) 0.0293558 0.00119348
\(606\) −4.61103 −0.187310
\(607\) −10.6898 −0.433886 −0.216943 0.976184i \(-0.569609\pi\)
−0.216943 + 0.976184i \(0.569609\pi\)
\(608\) 18.4788 0.749414
\(609\) −11.8774 −0.481295
\(610\) −0.0685361 −0.00277495
\(611\) 31.1994 1.26219
\(612\) 24.7628 1.00098
\(613\) 24.7225 0.998533 0.499267 0.866448i \(-0.333603\pi\)
0.499267 + 0.866448i \(0.333603\pi\)
\(614\) −20.0904 −0.810782
\(615\) −0.0784007 −0.00316142
\(616\) 9.15427 0.368836
\(617\) 47.5636 1.91484 0.957420 0.288700i \(-0.0932230\pi\)
0.957420 + 0.288700i \(0.0932230\pi\)
\(618\) −20.2825 −0.815880
\(619\) 19.1635 0.770246 0.385123 0.922865i \(-0.374159\pi\)
0.385123 + 0.922865i \(0.374159\pi\)
\(620\) −0.477403 −0.0191730
\(621\) −2.06467 −0.0828523
\(622\) 74.1803 2.97436
\(623\) 27.5378 1.10328
\(624\) 2.90922 0.116462
\(625\) 24.9871 0.999483
\(626\) −62.7775 −2.50909
\(627\) 4.17563 0.166759
\(628\) −33.6777 −1.34389
\(629\) −16.5048 −0.658091
\(630\) 0.185244 0.00738029
\(631\) 4.32941 0.172351 0.0861755 0.996280i \(-0.472535\pi\)
0.0861755 + 0.996280i \(0.472535\pi\)
\(632\) −37.8399 −1.50519
\(633\) −7.36422 −0.292701
\(634\) 50.6417 2.01124
\(635\) 0.158676 0.00629685
\(636\) −16.7942 −0.665934
\(637\) 0.883518 0.0350062
\(638\) −10.2594 −0.406173
\(639\) 4.73226 0.187205
\(640\) 0.586158 0.0231699
\(641\) 28.9219 1.14234 0.571172 0.820830i \(-0.306489\pi\)
0.571172 + 0.820830i \(0.306489\pi\)
\(642\) 14.2932 0.564108
\(643\) −33.7472 −1.33086 −0.665430 0.746460i \(-0.731752\pi\)
−0.665430 + 0.746460i \(0.731752\pi\)
\(644\) 19.2566 0.758817
\(645\) −0.261915 −0.0103129
\(646\) −69.9589 −2.75250
\(647\) −43.6251 −1.71508 −0.857540 0.514418i \(-0.828008\pi\)
−0.857540 + 0.514418i \(0.828008\pi\)
\(648\) −3.38688 −0.133049
\(649\) −0.181028 −0.00710597
\(650\) 33.7565 1.32404
\(651\) 12.7383 0.499252
\(652\) 73.5770 2.88150
\(653\) 13.5296 0.529455 0.264727 0.964323i \(-0.414718\pi\)
0.264727 + 0.964323i \(0.414718\pi\)
\(654\) 8.39685 0.328343
\(655\) 0.568968 0.0222314
\(656\) −2.68637 −0.104885
\(657\) 7.51871 0.293333
\(658\) 68.0705 2.65367
\(659\) −3.51759 −0.137026 −0.0685129 0.997650i \(-0.521825\pi\)
−0.0685129 + 0.997650i \(0.521825\pi\)
\(660\) 0.101298 0.00394301
\(661\) −43.5404 −1.69353 −0.846763 0.531971i \(-0.821452\pi\)
−0.846763 + 0.531971i \(0.821452\pi\)
\(662\) 7.39301 0.287337
\(663\) 20.7554 0.806074
\(664\) −43.9393 −1.70518
\(665\) −0.331315 −0.0128478
\(666\) 5.36960 0.208068
\(667\) −9.07289 −0.351304
\(668\) 65.9030 2.54986
\(669\) 1.02663 0.0396918
\(670\) −0.987296 −0.0381426
\(671\) 1.00000 0.0386046
\(672\) −11.9612 −0.461414
\(673\) 31.6788 1.22113 0.610563 0.791968i \(-0.290943\pi\)
0.610563 + 0.791968i \(0.290943\pi\)
\(674\) −67.8819 −2.61471
\(675\) −4.99914 −0.192417
\(676\) −15.9935 −0.615133
\(677\) 0.472792 0.0181709 0.00908543 0.999959i \(-0.497108\pi\)
0.00908543 + 0.999959i \(0.497108\pi\)
\(678\) −15.9170 −0.611288
\(679\) 45.1939 1.73438
\(680\) −0.713490 −0.0273611
\(681\) 21.6433 0.829372
\(682\) 11.0030 0.421327
\(683\) 1.89167 0.0723828 0.0361914 0.999345i \(-0.488477\pi\)
0.0361914 + 0.999345i \(0.488477\pi\)
\(684\) 14.4088 0.550934
\(685\) 0.461369 0.0176280
\(686\) −42.2444 −1.61290
\(687\) 9.88816 0.377257
\(688\) −8.97441 −0.342146
\(689\) −14.0764 −0.536267
\(690\) 0.141504 0.00538698
\(691\) 47.6079 1.81109 0.905545 0.424250i \(-0.139462\pi\)
0.905545 + 0.424250i \(0.139462\pi\)
\(692\) −60.2994 −2.29224
\(693\) −2.70286 −0.102673
\(694\) 13.4614 0.510989
\(695\) 0.0462772 0.00175539
\(696\) −14.8832 −0.564145
\(697\) −19.1655 −0.725946
\(698\) 69.7676 2.64074
\(699\) −24.8039 −0.938172
\(700\) 46.6257 1.76228
\(701\) 29.5010 1.11424 0.557119 0.830433i \(-0.311907\pi\)
0.557119 + 0.830433i \(0.311907\pi\)
\(702\) −6.75246 −0.254855
\(703\) −9.60370 −0.362211
\(704\) −12.3436 −0.465215
\(705\) 0.316667 0.0119264
\(706\) −20.2800 −0.763249
\(707\) −5.33822 −0.200764
\(708\) −0.624670 −0.0234766
\(709\) −24.1831 −0.908215 −0.454107 0.890947i \(-0.650042\pi\)
−0.454107 + 0.890947i \(0.650042\pi\)
\(710\) −0.324331 −0.0121719
\(711\) 11.1725 0.419002
\(712\) 34.5068 1.29320
\(713\) 9.73052 0.364411
\(714\) 45.2840 1.69471
\(715\) 0.0849045 0.00317525
\(716\) 54.9976 2.05536
\(717\) 15.9365 0.595158
\(718\) 0.748601 0.0279375
\(719\) 5.78255 0.215653 0.107826 0.994170i \(-0.465611\pi\)
0.107826 + 0.994170i \(0.465611\pi\)
\(720\) 0.0295280 0.00110044
\(721\) −23.4811 −0.874484
\(722\) 3.65163 0.135900
\(723\) −10.6415 −0.395763
\(724\) 25.5790 0.950636
\(725\) −21.9680 −0.815872
\(726\) −2.33467 −0.0866478
\(727\) 11.9924 0.444774 0.222387 0.974958i \(-0.428615\pi\)
0.222387 + 0.974958i \(0.428615\pi\)
\(728\) 26.4765 0.981283
\(729\) 1.00000 0.0370370
\(730\) −0.515304 −0.0190722
\(731\) −64.0266 −2.36811
\(732\) 3.45069 0.127541
\(733\) 6.07851 0.224515 0.112257 0.993679i \(-0.464192\pi\)
0.112257 + 0.993679i \(0.464192\pi\)
\(734\) 34.4416 1.27126
\(735\) 0.00896753 0.000330772 0
\(736\) −9.13695 −0.336792
\(737\) 14.4055 0.530633
\(738\) 6.23522 0.229521
\(739\) 27.1397 0.998350 0.499175 0.866501i \(-0.333636\pi\)
0.499175 + 0.866501i \(0.333636\pi\)
\(740\) −0.232979 −0.00856446
\(741\) 12.0770 0.443659
\(742\) −30.7117 −1.12746
\(743\) 1.32191 0.0484963 0.0242482 0.999706i \(-0.492281\pi\)
0.0242482 + 0.999706i \(0.492281\pi\)
\(744\) 15.9619 0.585193
\(745\) 0.0324353 0.00118834
\(746\) 17.6159 0.644963
\(747\) 12.9734 0.474672
\(748\) 24.7628 0.905419
\(749\) 16.5473 0.604627
\(750\) 0.685302 0.0250237
\(751\) 5.66385 0.206677 0.103338 0.994646i \(-0.467048\pi\)
0.103338 + 0.994646i \(0.467048\pi\)
\(752\) 10.8505 0.395677
\(753\) −24.6375 −0.897840
\(754\) −29.6728 −1.08062
\(755\) −0.440432 −0.0160290
\(756\) −9.32674 −0.339210
\(757\) −43.1109 −1.56689 −0.783447 0.621459i \(-0.786540\pi\)
−0.783447 + 0.621459i \(0.786540\pi\)
\(758\) 13.7940 0.501021
\(759\) −2.06467 −0.0749427
\(760\) −0.415160 −0.0150594
\(761\) 15.5334 0.563086 0.281543 0.959549i \(-0.409154\pi\)
0.281543 + 0.959549i \(0.409154\pi\)
\(762\) −12.6195 −0.457156
\(763\) 9.72109 0.351927
\(764\) 40.0537 1.44909
\(765\) 0.210663 0.00761655
\(766\) −66.1011 −2.38833
\(767\) −0.523579 −0.0189053
\(768\) −21.9301 −0.791334
\(769\) −44.5062 −1.60493 −0.802467 0.596697i \(-0.796479\pi\)
−0.802467 + 0.596697i \(0.796479\pi\)
\(770\) 0.185244 0.00667573
\(771\) 8.56747 0.308550
\(772\) 70.6993 2.54452
\(773\) −3.75321 −0.134993 −0.0674967 0.997719i \(-0.521501\pi\)
−0.0674967 + 0.997719i \(0.521501\pi\)
\(774\) 20.8301 0.748722
\(775\) 23.5603 0.846312
\(776\) 56.6311 2.03294
\(777\) 6.21643 0.223013
\(778\) 37.6126 1.34848
\(779\) −11.1519 −0.399558
\(780\) 0.292979 0.0104903
\(781\) 4.73226 0.169334
\(782\) 34.5916 1.23699
\(783\) 4.39436 0.157042
\(784\) 0.307269 0.0109739
\(785\) −0.286504 −0.0102258
\(786\) −45.2501 −1.61402
\(787\) −44.9918 −1.60379 −0.801893 0.597468i \(-0.796174\pi\)
−0.801893 + 0.597468i \(0.796174\pi\)
\(788\) 25.0910 0.893832
\(789\) −7.12601 −0.253693
\(790\) −0.765721 −0.0272431
\(791\) −18.4272 −0.655196
\(792\) −3.38688 −0.120347
\(793\) 2.89225 0.102707
\(794\) 30.1496 1.06997
\(795\) −0.142872 −0.00506716
\(796\) −34.6452 −1.22797
\(797\) 43.6737 1.54700 0.773501 0.633795i \(-0.218504\pi\)
0.773501 + 0.633795i \(0.218504\pi\)
\(798\) 26.3495 0.932762
\(799\) 77.4112 2.73861
\(800\) −22.1231 −0.782170
\(801\) −10.1884 −0.359989
\(802\) −50.4711 −1.78220
\(803\) 7.51871 0.265330
\(804\) 49.7088 1.75309
\(805\) 0.163821 0.00577392
\(806\) 31.8235 1.12094
\(807\) 14.1919 0.499578
\(808\) −6.68916 −0.235324
\(809\) −21.0958 −0.741690 −0.370845 0.928695i \(-0.620932\pi\)
−0.370845 + 0.928695i \(0.620932\pi\)
\(810\) −0.0685361 −0.00240811
\(811\) 47.7913 1.67818 0.839090 0.543992i \(-0.183088\pi\)
0.839090 + 0.543992i \(0.183088\pi\)
\(812\) −40.9851 −1.43829
\(813\) 29.2260 1.02500
\(814\) 5.36960 0.188204
\(815\) 0.625936 0.0219256
\(816\) 7.21830 0.252691
\(817\) −37.2553 −1.30340
\(818\) 24.5573 0.858626
\(819\) −7.81737 −0.273161
\(820\) −0.270536 −0.00944753
\(821\) 8.42000 0.293860 0.146930 0.989147i \(-0.453061\pi\)
0.146930 + 0.989147i \(0.453061\pi\)
\(822\) −36.6927 −1.27981
\(823\) 21.4926 0.749186 0.374593 0.927189i \(-0.377782\pi\)
0.374593 + 0.927189i \(0.377782\pi\)
\(824\) −29.4235 −1.02502
\(825\) −4.99914 −0.174048
\(826\) −1.14234 −0.0397471
\(827\) 32.0863 1.11575 0.557875 0.829925i \(-0.311617\pi\)
0.557875 + 0.829925i \(0.311617\pi\)
\(828\) −7.12452 −0.247594
\(829\) −43.5722 −1.51332 −0.756662 0.653806i \(-0.773171\pi\)
−0.756662 + 0.653806i \(0.773171\pi\)
\(830\) −0.889147 −0.0308627
\(831\) −13.9600 −0.484266
\(832\) −35.7007 −1.23770
\(833\) 2.19217 0.0759541
\(834\) −3.68043 −0.127443
\(835\) 0.560652 0.0194022
\(836\) 14.4088 0.498339
\(837\) −4.71288 −0.162901
\(838\) −12.1279 −0.418951
\(839\) −9.71778 −0.335495 −0.167748 0.985830i \(-0.553649\pi\)
−0.167748 + 0.985830i \(0.553649\pi\)
\(840\) 0.268731 0.00927209
\(841\) −9.68959 −0.334124
\(842\) 51.4246 1.77221
\(843\) −5.84690 −0.201378
\(844\) −25.4116 −0.874704
\(845\) −0.136060 −0.00468061
\(846\) −25.1846 −0.865864
\(847\) −2.70286 −0.0928715
\(848\) −4.89547 −0.168111
\(849\) 25.6344 0.879769
\(850\) 83.7559 2.87281
\(851\) 4.74861 0.162780
\(852\) 16.3296 0.559441
\(853\) 10.1506 0.347551 0.173776 0.984785i \(-0.444403\pi\)
0.173776 + 0.984785i \(0.444403\pi\)
\(854\) 6.31030 0.215934
\(855\) 0.122579 0.00419211
\(856\) 20.7350 0.708707
\(857\) 1.00340 0.0342754 0.0171377 0.999853i \(-0.494545\pi\)
0.0171377 + 0.999853i \(0.494545\pi\)
\(858\) −6.75246 −0.230525
\(859\) −30.9914 −1.05741 −0.528706 0.848805i \(-0.677323\pi\)
−0.528706 + 0.848805i \(0.677323\pi\)
\(860\) −0.903785 −0.0308188
\(861\) 7.21855 0.246008
\(862\) 1.97113 0.0671368
\(863\) −36.5669 −1.24475 −0.622377 0.782718i \(-0.713833\pi\)
−0.622377 + 0.782718i \(0.713833\pi\)
\(864\) 4.42539 0.150555
\(865\) −0.512981 −0.0174419
\(866\) −67.3655 −2.28917
\(867\) 34.4979 1.17161
\(868\) 43.9558 1.49196
\(869\) 11.1725 0.379002
\(870\) −0.301172 −0.0102107
\(871\) 41.6643 1.41174
\(872\) 12.1812 0.412507
\(873\) −16.7207 −0.565911
\(874\) 20.1279 0.680836
\(875\) 0.793379 0.0268211
\(876\) 25.9447 0.876591
\(877\) 4.64834 0.156963 0.0784816 0.996916i \(-0.474993\pi\)
0.0784816 + 0.996916i \(0.474993\pi\)
\(878\) 50.4763 1.70349
\(879\) 1.56499 0.0527859
\(880\) 0.0295280 0.000995389 0
\(881\) −29.7857 −1.00351 −0.501753 0.865011i \(-0.667311\pi\)
−0.501753 + 0.865011i \(0.667311\pi\)
\(882\) −0.713188 −0.0240143
\(883\) −43.9469 −1.47893 −0.739466 0.673194i \(-0.764922\pi\)
−0.739466 + 0.673194i \(0.764922\pi\)
\(884\) 71.6204 2.40886
\(885\) −0.00531422 −0.000178635 0
\(886\) 9.10858 0.306009
\(887\) −31.7216 −1.06511 −0.532554 0.846396i \(-0.678767\pi\)
−0.532554 + 0.846396i \(0.678767\pi\)
\(888\) 7.78961 0.261402
\(889\) −14.6097 −0.489993
\(890\) 0.698272 0.0234061
\(891\) 1.00000 0.0335013
\(892\) 3.54258 0.118614
\(893\) 45.0434 1.50732
\(894\) −2.57959 −0.0862742
\(895\) 0.467878 0.0156394
\(896\) −53.9691 −1.80298
\(897\) −5.97154 −0.199384
\(898\) 16.7286 0.558242
\(899\) −20.7101 −0.690720
\(900\) −17.2505 −0.575015
\(901\) −34.9260 −1.16355
\(902\) 6.23522 0.207610
\(903\) 24.1151 0.802502
\(904\) −23.0906 −0.767981
\(905\) 0.217607 0.00723349
\(906\) 35.0276 1.16372
\(907\) −43.3090 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(908\) 74.6841 2.47848
\(909\) 1.97502 0.0655074
\(910\) 0.535772 0.0177607
\(911\) −39.7332 −1.31642 −0.658210 0.752834i \(-0.728686\pi\)
−0.658210 + 0.752834i \(0.728686\pi\)
\(912\) 4.20013 0.139080
\(913\) 12.9734 0.429357
\(914\) −7.84032 −0.259335
\(915\) 0.0293558 0.000970473 0
\(916\) 34.1209 1.12739
\(917\) −52.3864 −1.72995
\(918\) −16.7541 −0.552967
\(919\) −47.8215 −1.57749 −0.788744 0.614722i \(-0.789268\pi\)
−0.788744 + 0.614722i \(0.789268\pi\)
\(920\) 0.205278 0.00676783
\(921\) 8.60524 0.283552
\(922\) 96.3015 3.17152
\(923\) 13.6869 0.450510
\(924\) −9.32674 −0.306827
\(925\) 11.4977 0.378043
\(926\) −43.7487 −1.43767
\(927\) 8.68750 0.285335
\(928\) 19.4467 0.638371
\(929\) 33.7455 1.10715 0.553576 0.832798i \(-0.313263\pi\)
0.553576 + 0.832798i \(0.313263\pi\)
\(930\) 0.323002 0.0105917
\(931\) 1.27556 0.0418048
\(932\) −85.5907 −2.80362
\(933\) −31.7733 −1.04021
\(934\) 57.0826 1.86780
\(935\) 0.210663 0.00688943
\(936\) −9.79571 −0.320183
\(937\) 28.0948 0.917817 0.458908 0.888484i \(-0.348241\pi\)
0.458908 + 0.888484i \(0.348241\pi\)
\(938\) 90.9029 2.96808
\(939\) 26.8892 0.877497
\(940\) 1.09272 0.0356406
\(941\) −2.34219 −0.0763531 −0.0381765 0.999271i \(-0.512155\pi\)
−0.0381765 + 0.999271i \(0.512155\pi\)
\(942\) 22.7857 0.742398
\(943\) 5.51412 0.179564
\(944\) −0.182090 −0.00592652
\(945\) −0.0793448 −0.00258108
\(946\) 20.8301 0.677245
\(947\) 0.361908 0.0117604 0.00588021 0.999983i \(-0.498128\pi\)
0.00588021 + 0.999983i \(0.498128\pi\)
\(948\) 38.5529 1.25214
\(949\) 21.7460 0.705906
\(950\) 48.7352 1.58118
\(951\) −21.6911 −0.703383
\(952\) 65.6929 2.12912
\(953\) 20.2966 0.657473 0.328736 0.944422i \(-0.393377\pi\)
0.328736 + 0.944422i \(0.393377\pi\)
\(954\) 11.3627 0.367879
\(955\) 0.340746 0.0110263
\(956\) 54.9918 1.77856
\(957\) 4.39436 0.142050
\(958\) 81.6618 2.63837
\(959\) −42.4794 −1.37173
\(960\) −0.362355 −0.0116950
\(961\) −8.78880 −0.283510
\(962\) 15.5303 0.500716
\(963\) −6.12215 −0.197284
\(964\) −36.7206 −1.18269
\(965\) 0.601455 0.0193615
\(966\) −13.0287 −0.419191
\(967\) −18.7607 −0.603303 −0.301651 0.953418i \(-0.597538\pi\)
−0.301651 + 0.953418i \(0.597538\pi\)
\(968\) −3.38688 −0.108858
\(969\) 29.9652 0.962621
\(970\) 1.14598 0.0367950
\(971\) 56.5296 1.81412 0.907061 0.420999i \(-0.138320\pi\)
0.907061 + 0.420999i \(0.138320\pi\)
\(972\) 3.45069 0.110681
\(973\) −4.26086 −0.136597
\(974\) −27.7108 −0.887911
\(975\) −14.4588 −0.463052
\(976\) 1.00587 0.0321970
\(977\) 3.17054 0.101435 0.0507173 0.998713i \(-0.483849\pi\)
0.0507173 + 0.998713i \(0.483849\pi\)
\(978\) −49.7808 −1.59181
\(979\) −10.1884 −0.325622
\(980\) 0.0309441 0.000988474 0
\(981\) −3.59659 −0.114830
\(982\) −80.0284 −2.55381
\(983\) −40.8147 −1.30179 −0.650893 0.759169i \(-0.725606\pi\)
−0.650893 + 0.759169i \(0.725606\pi\)
\(984\) 9.04535 0.288355
\(985\) 0.213455 0.00680125
\(986\) −73.6235 −2.34465
\(987\) −29.1564 −0.928058
\(988\) 41.6739 1.32582
\(989\) 18.4211 0.585757
\(990\) −0.0685361 −0.00217822
\(991\) −56.3255 −1.78924 −0.894619 0.446829i \(-0.852553\pi\)
−0.894619 + 0.446829i \(0.852553\pi\)
\(992\) −20.8563 −0.662188
\(993\) −3.16662 −0.100490
\(994\) 29.8620 0.947164
\(995\) −0.294735 −0.00934372
\(996\) 44.7672 1.41850
\(997\) −44.4996 −1.40932 −0.704658 0.709547i \(-0.748900\pi\)
−0.704658 + 0.709547i \(0.748900\pi\)
\(998\) −53.6430 −1.69804
\(999\) −2.29994 −0.0727669
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 2013.2.a.g.1.1 13
3.2 odd 2 6039.2.a.h.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.g.1.1 13 1.1 even 1 trivial
6039.2.a.h.1.13 13 3.2 odd 2