Properties

Label 2013.2.a.f.1.12
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: \( x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} - 404 x^{4} + 98 x^{3} + 118 x^{2} - 16 x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(2.60074\) 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)
 
\(f(q)\) \(=\) \(q+2.60074 q^{2} +1.00000 q^{3} +4.76385 q^{4} -0.540623 q^{5} +2.60074 q^{6} +4.31275 q^{7} +7.18805 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.60074 q^{2} +1.00000 q^{3} +4.76385 q^{4} -0.540623 q^{5} +2.60074 q^{6} +4.31275 q^{7} +7.18805 q^{8} +1.00000 q^{9} -1.40602 q^{10} -1.00000 q^{11} +4.76385 q^{12} -6.04924 q^{13} +11.2163 q^{14} -0.540623 q^{15} +9.16656 q^{16} -1.53220 q^{17} +2.60074 q^{18} +5.71304 q^{19} -2.57545 q^{20} +4.31275 q^{21} -2.60074 q^{22} +2.05269 q^{23} +7.18805 q^{24} -4.70773 q^{25} -15.7325 q^{26} +1.00000 q^{27} +20.5453 q^{28} -0.0227019 q^{29} -1.40602 q^{30} -7.86398 q^{31} +9.46374 q^{32} -1.00000 q^{33} -3.98485 q^{34} -2.33157 q^{35} +4.76385 q^{36} -6.22862 q^{37} +14.8581 q^{38} -6.04924 q^{39} -3.88603 q^{40} +7.86011 q^{41} +11.2163 q^{42} -2.59570 q^{43} -4.76385 q^{44} -0.540623 q^{45} +5.33851 q^{46} -6.08525 q^{47} +9.16656 q^{48} +11.5998 q^{49} -12.2436 q^{50} -1.53220 q^{51} -28.8177 q^{52} -5.04706 q^{53} +2.60074 q^{54} +0.540623 q^{55} +31.0003 q^{56} +5.71304 q^{57} -0.0590417 q^{58} +5.62146 q^{59} -2.57545 q^{60} -1.00000 q^{61} -20.4522 q^{62} +4.31275 q^{63} +6.27960 q^{64} +3.27036 q^{65} -2.60074 q^{66} +8.08014 q^{67} -7.29916 q^{68} +2.05269 q^{69} -6.06382 q^{70} -8.56121 q^{71} +7.18805 q^{72} +13.6253 q^{73} -16.1990 q^{74} -4.70773 q^{75} +27.2161 q^{76} -4.31275 q^{77} -15.7325 q^{78} -8.84218 q^{79} -4.95566 q^{80} +1.00000 q^{81} +20.4421 q^{82} +9.24421 q^{83} +20.5453 q^{84} +0.828342 q^{85} -6.75073 q^{86} -0.0227019 q^{87} -7.18805 q^{88} +11.7669 q^{89} -1.40602 q^{90} -26.0889 q^{91} +9.77869 q^{92} -7.86398 q^{93} -15.8262 q^{94} -3.08860 q^{95} +9.46374 q^{96} -9.60720 q^{97} +30.1681 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 5 q^{7} + 15 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 4 q^{2} + 13 q^{3} + 12 q^{4} + 7 q^{5} + 4 q^{6} + 5 q^{7} + 15 q^{8} + 13 q^{9} + 8 q^{10} - 13 q^{11} + 12 q^{12} - 9 q^{13} + 19 q^{14} + 7 q^{15} + 18 q^{16} + 7 q^{17} + 4 q^{18} + 2 q^{19} + 15 q^{20} + 5 q^{21} - 4 q^{22} + 23 q^{23} + 15 q^{24} + 10 q^{25} + 8 q^{26} + 13 q^{27} + 9 q^{28} + 16 q^{29} + 8 q^{30} + 9 q^{31} + 29 q^{32} - 13 q^{33} + 2 q^{34} + 16 q^{35} + 12 q^{36} + 14 q^{37} + 8 q^{38} - 9 q^{39} + 16 q^{40} + 19 q^{41} + 19 q^{42} + 7 q^{43} - 12 q^{44} + 7 q^{45} + 4 q^{46} + 26 q^{47} + 18 q^{48} + 8 q^{49} - 15 q^{50} + 7 q^{51} - 17 q^{52} + 18 q^{53} + 4 q^{54} - 7 q^{55} + 44 q^{56} + 2 q^{57} - q^{58} + 31 q^{59} + 15 q^{60} - 13 q^{61} - 5 q^{62} + 5 q^{63} - 17 q^{64} + 31 q^{65} - 4 q^{66} + 14 q^{67} - 32 q^{68} + 23 q^{69} - 20 q^{70} + 37 q^{71} + 15 q^{72} - 16 q^{73} - 6 q^{74} + 10 q^{75} - 7 q^{76} - 5 q^{77} + 8 q^{78} - 17 q^{79} - 2 q^{80} + 13 q^{81} - 2 q^{82} + 30 q^{83} + 9 q^{84} - 16 q^{85} - 22 q^{86} + 16 q^{87} - 15 q^{88} + 35 q^{89} + 8 q^{90} - q^{91} + 24 q^{92} + 9 q^{93} - 11 q^{94} + 13 q^{95} + 29 q^{96} - q^{97} - q^{98} - 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.60074 1.83900 0.919500 0.393089i \(-0.128594\pi\)
0.919500 + 0.393089i \(0.128594\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.76385 2.38192
\(5\) −0.540623 −0.241774 −0.120887 0.992666i \(-0.538574\pi\)
−0.120887 + 0.992666i \(0.538574\pi\)
\(6\) 2.60074 1.06175
\(7\) 4.31275 1.63007 0.815033 0.579414i \(-0.196719\pi\)
0.815033 + 0.579414i \(0.196719\pi\)
\(8\) 7.18805 2.54136
\(9\) 1.00000 0.333333
\(10\) −1.40602 −0.444623
\(11\) −1.00000 −0.301511
\(12\) 4.76385 1.37520
\(13\) −6.04924 −1.67776 −0.838879 0.544319i \(-0.816788\pi\)
−0.838879 + 0.544319i \(0.816788\pi\)
\(14\) 11.2163 2.99769
\(15\) −0.540623 −0.139588
\(16\) 9.16656 2.29164
\(17\) −1.53220 −0.371613 −0.185806 0.982586i \(-0.559490\pi\)
−0.185806 + 0.982586i \(0.559490\pi\)
\(18\) 2.60074 0.613000
\(19\) 5.71304 1.31066 0.655331 0.755342i \(-0.272529\pi\)
0.655331 + 0.755342i \(0.272529\pi\)
\(20\) −2.57545 −0.575888
\(21\) 4.31275 0.941119
\(22\) −2.60074 −0.554480
\(23\) 2.05269 0.428015 0.214007 0.976832i \(-0.431348\pi\)
0.214007 + 0.976832i \(0.431348\pi\)
\(24\) 7.18805 1.46726
\(25\) −4.70773 −0.941545
\(26\) −15.7325 −3.08540
\(27\) 1.00000 0.192450
\(28\) 20.5453 3.88270
\(29\) −0.0227019 −0.00421563 −0.00210782 0.999998i \(-0.500671\pi\)
−0.00210782 + 0.999998i \(0.500671\pi\)
\(30\) −1.40602 −0.256703
\(31\) −7.86398 −1.41241 −0.706206 0.708006i \(-0.749595\pi\)
−0.706206 + 0.708006i \(0.749595\pi\)
\(32\) 9.46374 1.67297
\(33\) −1.00000 −0.174078
\(34\) −3.98485 −0.683396
\(35\) −2.33157 −0.394108
\(36\) 4.76385 0.793975
\(37\) −6.22862 −1.02398 −0.511990 0.858992i \(-0.671091\pi\)
−0.511990 + 0.858992i \(0.671091\pi\)
\(38\) 14.8581 2.41031
\(39\) −6.04924 −0.968654
\(40\) −3.88603 −0.614435
\(41\) 7.86011 1.22754 0.613772 0.789483i \(-0.289651\pi\)
0.613772 + 0.789483i \(0.289651\pi\)
\(42\) 11.2163 1.73072
\(43\) −2.59570 −0.395840 −0.197920 0.980218i \(-0.563419\pi\)
−0.197920 + 0.980218i \(0.563419\pi\)
\(44\) −4.76385 −0.718177
\(45\) −0.540623 −0.0805914
\(46\) 5.33851 0.787120
\(47\) −6.08525 −0.887625 −0.443813 0.896120i \(-0.646374\pi\)
−0.443813 + 0.896120i \(0.646374\pi\)
\(48\) 9.16656 1.32308
\(49\) 11.5998 1.65712
\(50\) −12.2436 −1.73150
\(51\) −1.53220 −0.214551
\(52\) −28.8177 −3.99629
\(53\) −5.04706 −0.693268 −0.346634 0.938001i \(-0.612675\pi\)
−0.346634 + 0.938001i \(0.612675\pi\)
\(54\) 2.60074 0.353916
\(55\) 0.540623 0.0728976
\(56\) 31.0003 4.14259
\(57\) 5.71304 0.756711
\(58\) −0.0590417 −0.00775255
\(59\) 5.62146 0.731852 0.365926 0.930644i \(-0.380752\pi\)
0.365926 + 0.930644i \(0.380752\pi\)
\(60\) −2.57545 −0.332489
\(61\) −1.00000 −0.128037
\(62\) −20.4522 −2.59743
\(63\) 4.31275 0.543355
\(64\) 6.27960 0.784950
\(65\) 3.27036 0.405638
\(66\) −2.60074 −0.320129
\(67\) 8.08014 0.987146 0.493573 0.869704i \(-0.335691\pi\)
0.493573 + 0.869704i \(0.335691\pi\)
\(68\) −7.29916 −0.885153
\(69\) 2.05269 0.247115
\(70\) −6.06382 −0.724765
\(71\) −8.56121 −1.01603 −0.508014 0.861349i \(-0.669620\pi\)
−0.508014 + 0.861349i \(0.669620\pi\)
\(72\) 7.18805 0.847120
\(73\) 13.6253 1.59472 0.797358 0.603506i \(-0.206230\pi\)
0.797358 + 0.603506i \(0.206230\pi\)
\(74\) −16.1990 −1.88310
\(75\) −4.70773 −0.543601
\(76\) 27.2161 3.12190
\(77\) −4.31275 −0.491483
\(78\) −15.7325 −1.78135
\(79\) −8.84218 −0.994823 −0.497411 0.867515i \(-0.665716\pi\)
−0.497411 + 0.867515i \(0.665716\pi\)
\(80\) −4.95566 −0.554059
\(81\) 1.00000 0.111111
\(82\) 20.4421 2.25745
\(83\) 9.24421 1.01468 0.507342 0.861745i \(-0.330628\pi\)
0.507342 + 0.861745i \(0.330628\pi\)
\(84\) 20.5453 2.24168
\(85\) 0.828342 0.0898463
\(86\) −6.75073 −0.727950
\(87\) −0.0227019 −0.00243390
\(88\) −7.18805 −0.766249
\(89\) 11.7669 1.24729 0.623644 0.781708i \(-0.285651\pi\)
0.623644 + 0.781708i \(0.285651\pi\)
\(90\) −1.40602 −0.148208
\(91\) −26.0889 −2.73486
\(92\) 9.77869 1.01950
\(93\) −7.86398 −0.815457
\(94\) −15.8262 −1.63234
\(95\) −3.08860 −0.316884
\(96\) 9.46374 0.965889
\(97\) −9.60720 −0.975463 −0.487732 0.872994i \(-0.662175\pi\)
−0.487732 + 0.872994i \(0.662175\pi\)
\(98\) 30.1681 3.04744
\(99\) −1.00000 −0.100504
\(100\) −22.4269 −2.24269
\(101\) 15.3839 1.53075 0.765377 0.643582i \(-0.222552\pi\)
0.765377 + 0.643582i \(0.222552\pi\)
\(102\) −3.98485 −0.394559
\(103\) −10.3446 −1.01928 −0.509642 0.860386i \(-0.670222\pi\)
−0.509642 + 0.860386i \(0.670222\pi\)
\(104\) −43.4823 −4.26379
\(105\) −2.33157 −0.227538
\(106\) −13.1261 −1.27492
\(107\) 5.66352 0.547514 0.273757 0.961799i \(-0.411734\pi\)
0.273757 + 0.961799i \(0.411734\pi\)
\(108\) 4.76385 0.458402
\(109\) −14.4612 −1.38513 −0.692564 0.721357i \(-0.743519\pi\)
−0.692564 + 0.721357i \(0.743519\pi\)
\(110\) 1.40602 0.134059
\(111\) −6.22862 −0.591195
\(112\) 39.5331 3.73553
\(113\) 3.56615 0.335475 0.167738 0.985832i \(-0.446354\pi\)
0.167738 + 0.985832i \(0.446354\pi\)
\(114\) 14.8581 1.39159
\(115\) −1.10973 −0.103483
\(116\) −0.108148 −0.0100413
\(117\) −6.04924 −0.559252
\(118\) 14.6200 1.34588
\(119\) −6.60799 −0.605753
\(120\) −3.88603 −0.354744
\(121\) 1.00000 0.0909091
\(122\) −2.60074 −0.235460
\(123\) 7.86011 0.708723
\(124\) −37.4628 −3.36426
\(125\) 5.24822 0.469415
\(126\) 11.2163 0.999231
\(127\) −15.5050 −1.37585 −0.687925 0.725782i \(-0.741478\pi\)
−0.687925 + 0.725782i \(0.741478\pi\)
\(128\) −2.59587 −0.229445
\(129\) −2.59570 −0.228538
\(130\) 8.50535 0.745969
\(131\) 18.8326 1.64541 0.822707 0.568465i \(-0.192463\pi\)
0.822707 + 0.568465i \(0.192463\pi\)
\(132\) −4.76385 −0.414640
\(133\) 24.6389 2.13646
\(134\) 21.0143 1.81536
\(135\) −0.540623 −0.0465294
\(136\) −11.0135 −0.944402
\(137\) −16.6626 −1.42359 −0.711793 0.702390i \(-0.752116\pi\)
−0.711793 + 0.702390i \(0.752116\pi\)
\(138\) 5.33851 0.454444
\(139\) −17.2228 −1.46082 −0.730411 0.683008i \(-0.760671\pi\)
−0.730411 + 0.683008i \(0.760671\pi\)
\(140\) −11.1073 −0.938735
\(141\) −6.08525 −0.512471
\(142\) −22.2655 −1.86848
\(143\) 6.04924 0.505863
\(144\) 9.16656 0.763880
\(145\) 0.0122732 0.00101923
\(146\) 35.4358 2.93269
\(147\) 11.5998 0.956737
\(148\) −29.6722 −2.43904
\(149\) −8.95498 −0.733620 −0.366810 0.930296i \(-0.619550\pi\)
−0.366810 + 0.930296i \(0.619550\pi\)
\(150\) −12.2436 −0.999684
\(151\) −12.6384 −1.02849 −0.514247 0.857642i \(-0.671929\pi\)
−0.514247 + 0.857642i \(0.671929\pi\)
\(152\) 41.0656 3.33086
\(153\) −1.53220 −0.123871
\(154\) −11.2163 −0.903839
\(155\) 4.25145 0.341485
\(156\) −28.8177 −2.30726
\(157\) 16.8355 1.34362 0.671810 0.740723i \(-0.265517\pi\)
0.671810 + 0.740723i \(0.265517\pi\)
\(158\) −22.9962 −1.82948
\(159\) −5.04706 −0.400258
\(160\) −5.11632 −0.404480
\(161\) 8.85273 0.697693
\(162\) 2.60074 0.204333
\(163\) 6.53716 0.512030 0.256015 0.966673i \(-0.417590\pi\)
0.256015 + 0.966673i \(0.417590\pi\)
\(164\) 37.4444 2.92392
\(165\) 0.540623 0.0420875
\(166\) 24.0418 1.86600
\(167\) −1.74070 −0.134699 −0.0673497 0.997729i \(-0.521454\pi\)
−0.0673497 + 0.997729i \(0.521454\pi\)
\(168\) 31.0003 2.39172
\(169\) 23.5933 1.81487
\(170\) 2.15430 0.165227
\(171\) 5.71304 0.436887
\(172\) −12.3655 −0.942861
\(173\) 8.99614 0.683964 0.341982 0.939707i \(-0.388902\pi\)
0.341982 + 0.939707i \(0.388902\pi\)
\(174\) −0.0590417 −0.00447594
\(175\) −20.3032 −1.53478
\(176\) −9.16656 −0.690956
\(177\) 5.62146 0.422535
\(178\) 30.6026 2.29376
\(179\) −12.0429 −0.900129 −0.450064 0.892996i \(-0.648599\pi\)
−0.450064 + 0.892996i \(0.648599\pi\)
\(180\) −2.57545 −0.191963
\(181\) 22.4118 1.66585 0.832926 0.553384i \(-0.186664\pi\)
0.832926 + 0.553384i \(0.186664\pi\)
\(182\) −67.8503 −5.02940
\(183\) −1.00000 −0.0739221
\(184\) 14.7548 1.08774
\(185\) 3.36734 0.247572
\(186\) −20.4522 −1.49963
\(187\) 1.53220 0.112045
\(188\) −28.9892 −2.11426
\(189\) 4.31275 0.313706
\(190\) −8.03265 −0.582750
\(191\) 19.2565 1.39335 0.696677 0.717385i \(-0.254661\pi\)
0.696677 + 0.717385i \(0.254661\pi\)
\(192\) 6.27960 0.453191
\(193\) 11.2997 0.813370 0.406685 0.913569i \(-0.366685\pi\)
0.406685 + 0.913569i \(0.366685\pi\)
\(194\) −24.9858 −1.79388
\(195\) 3.27036 0.234195
\(196\) 55.2598 3.94713
\(197\) 11.8065 0.841176 0.420588 0.907252i \(-0.361824\pi\)
0.420588 + 0.907252i \(0.361824\pi\)
\(198\) −2.60074 −0.184827
\(199\) 5.62741 0.398916 0.199458 0.979906i \(-0.436082\pi\)
0.199458 + 0.979906i \(0.436082\pi\)
\(200\) −33.8394 −2.39281
\(201\) 8.08014 0.569929
\(202\) 40.0095 2.81506
\(203\) −0.0979075 −0.00687176
\(204\) −7.29916 −0.511043
\(205\) −4.24936 −0.296788
\(206\) −26.9036 −1.87446
\(207\) 2.05269 0.142672
\(208\) −55.4507 −3.84482
\(209\) −5.71304 −0.395179
\(210\) −6.06382 −0.418443
\(211\) −7.91724 −0.545045 −0.272523 0.962149i \(-0.587858\pi\)
−0.272523 + 0.962149i \(0.587858\pi\)
\(212\) −24.0435 −1.65131
\(213\) −8.56121 −0.586605
\(214\) 14.7294 1.00688
\(215\) 1.40329 0.0957038
\(216\) 7.18805 0.489085
\(217\) −33.9154 −2.30233
\(218\) −37.6097 −2.54725
\(219\) 13.6253 0.920710
\(220\) 2.57545 0.173637
\(221\) 9.26863 0.623476
\(222\) −16.1990 −1.08721
\(223\) 14.3852 0.963302 0.481651 0.876363i \(-0.340037\pi\)
0.481651 + 0.876363i \(0.340037\pi\)
\(224\) 40.8147 2.72705
\(225\) −4.70773 −0.313848
\(226\) 9.27463 0.616940
\(227\) 1.89780 0.125961 0.0629806 0.998015i \(-0.479939\pi\)
0.0629806 + 0.998015i \(0.479939\pi\)
\(228\) 27.2161 1.80243
\(229\) −15.0613 −0.995276 −0.497638 0.867385i \(-0.665799\pi\)
−0.497638 + 0.867385i \(0.665799\pi\)
\(230\) −2.88612 −0.190305
\(231\) −4.31275 −0.283758
\(232\) −0.163182 −0.0107134
\(233\) 11.0715 0.725318 0.362659 0.931922i \(-0.381869\pi\)
0.362659 + 0.931922i \(0.381869\pi\)
\(234\) −15.7325 −1.02847
\(235\) 3.28983 0.214605
\(236\) 26.7798 1.74322
\(237\) −8.84218 −0.574361
\(238\) −17.1857 −1.11398
\(239\) 29.2019 1.88891 0.944457 0.328635i \(-0.106589\pi\)
0.944457 + 0.328635i \(0.106589\pi\)
\(240\) −4.95566 −0.319886
\(241\) −13.1945 −0.849932 −0.424966 0.905209i \(-0.639714\pi\)
−0.424966 + 0.905209i \(0.639714\pi\)
\(242\) 2.60074 0.167182
\(243\) 1.00000 0.0641500
\(244\) −4.76385 −0.304974
\(245\) −6.27113 −0.400648
\(246\) 20.4421 1.30334
\(247\) −34.5595 −2.19897
\(248\) −56.5267 −3.58945
\(249\) 9.24421 0.585828
\(250\) 13.6493 0.863255
\(251\) −21.4744 −1.35545 −0.677727 0.735314i \(-0.737035\pi\)
−0.677727 + 0.735314i \(0.737035\pi\)
\(252\) 20.5453 1.29423
\(253\) −2.05269 −0.129051
\(254\) −40.3246 −2.53019
\(255\) 0.828342 0.0518728
\(256\) −19.3104 −1.20690
\(257\) 17.8382 1.11271 0.556357 0.830943i \(-0.312199\pi\)
0.556357 + 0.830943i \(0.312199\pi\)
\(258\) −6.75073 −0.420282
\(259\) −26.8625 −1.66915
\(260\) 15.5795 0.966200
\(261\) −0.0227019 −0.00140521
\(262\) 48.9788 3.02592
\(263\) −29.8044 −1.83782 −0.918908 0.394472i \(-0.870928\pi\)
−0.918908 + 0.394472i \(0.870928\pi\)
\(264\) −7.18805 −0.442394
\(265\) 2.72856 0.167614
\(266\) 64.0794 3.92896
\(267\) 11.7669 0.720122
\(268\) 38.4926 2.35131
\(269\) −25.4442 −1.55136 −0.775681 0.631125i \(-0.782593\pi\)
−0.775681 + 0.631125i \(0.782593\pi\)
\(270\) −1.40602 −0.0855677
\(271\) −4.56237 −0.277144 −0.138572 0.990352i \(-0.544251\pi\)
−0.138572 + 0.990352i \(0.544251\pi\)
\(272\) −14.0450 −0.851602
\(273\) −26.0889 −1.57897
\(274\) −43.3352 −2.61797
\(275\) 4.70773 0.283887
\(276\) 9.77869 0.588608
\(277\) 17.4846 1.05055 0.525273 0.850934i \(-0.323963\pi\)
0.525273 + 0.850934i \(0.323963\pi\)
\(278\) −44.7921 −2.68645
\(279\) −7.86398 −0.470804
\(280\) −16.7595 −1.00157
\(281\) −23.6079 −1.40833 −0.704165 0.710036i \(-0.748679\pi\)
−0.704165 + 0.710036i \(0.748679\pi\)
\(282\) −15.8262 −0.942434
\(283\) −2.59000 −0.153960 −0.0769799 0.997033i \(-0.524528\pi\)
−0.0769799 + 0.997033i \(0.524528\pi\)
\(284\) −40.7843 −2.42010
\(285\) −3.08860 −0.182953
\(286\) 15.7325 0.930282
\(287\) 33.8987 2.00098
\(288\) 9.46374 0.557656
\(289\) −14.6524 −0.861904
\(290\) 0.0319193 0.00187437
\(291\) −9.60720 −0.563184
\(292\) 64.9087 3.79849
\(293\) −30.1955 −1.76404 −0.882020 0.471213i \(-0.843816\pi\)
−0.882020 + 0.471213i \(0.843816\pi\)
\(294\) 30.1681 1.75944
\(295\) −3.03909 −0.176943
\(296\) −44.7717 −2.60230
\(297\) −1.00000 −0.0580259
\(298\) −23.2896 −1.34913
\(299\) −12.4172 −0.718105
\(300\) −22.4269 −1.29482
\(301\) −11.1946 −0.645245
\(302\) −32.8691 −1.89140
\(303\) 15.3839 0.883782
\(304\) 52.3689 3.00356
\(305\) 0.540623 0.0309560
\(306\) −3.98485 −0.227799
\(307\) 15.3667 0.877022 0.438511 0.898726i \(-0.355506\pi\)
0.438511 + 0.898726i \(0.355506\pi\)
\(308\) −20.5453 −1.17068
\(309\) −10.3446 −0.588484
\(310\) 11.0569 0.627991
\(311\) −22.5782 −1.28029 −0.640145 0.768254i \(-0.721126\pi\)
−0.640145 + 0.768254i \(0.721126\pi\)
\(312\) −43.4823 −2.46170
\(313\) −1.93752 −0.109515 −0.0547576 0.998500i \(-0.517439\pi\)
−0.0547576 + 0.998500i \(0.517439\pi\)
\(314\) 43.7848 2.47092
\(315\) −2.33157 −0.131369
\(316\) −42.1228 −2.36959
\(317\) 17.4823 0.981901 0.490950 0.871187i \(-0.336650\pi\)
0.490950 + 0.871187i \(0.336650\pi\)
\(318\) −13.1261 −0.736075
\(319\) 0.0227019 0.00127106
\(320\) −3.39490 −0.189781
\(321\) 5.66352 0.316107
\(322\) 23.0236 1.28306
\(323\) −8.75351 −0.487058
\(324\) 4.76385 0.264658
\(325\) 28.4782 1.57968
\(326\) 17.0014 0.941623
\(327\) −14.4612 −0.799704
\(328\) 56.4989 3.11963
\(329\) −26.2442 −1.44689
\(330\) 1.40602 0.0773989
\(331\) −3.64398 −0.200291 −0.100146 0.994973i \(-0.531931\pi\)
−0.100146 + 0.994973i \(0.531931\pi\)
\(332\) 44.0380 2.41690
\(333\) −6.22862 −0.341326
\(334\) −4.52711 −0.247712
\(335\) −4.36831 −0.238666
\(336\) 39.5331 2.15671
\(337\) −25.1928 −1.37234 −0.686169 0.727442i \(-0.740709\pi\)
−0.686169 + 0.727442i \(0.740709\pi\)
\(338\) 61.3600 3.33755
\(339\) 3.56615 0.193687
\(340\) 3.94610 0.214007
\(341\) 7.86398 0.425858
\(342\) 14.8581 0.803436
\(343\) 19.8378 1.07114
\(344\) −18.6580 −1.00597
\(345\) −1.10973 −0.0597459
\(346\) 23.3966 1.25781
\(347\) 15.0795 0.809510 0.404755 0.914425i \(-0.367357\pi\)
0.404755 + 0.914425i \(0.367357\pi\)
\(348\) −0.108148 −0.00579736
\(349\) 18.4363 0.986875 0.493437 0.869781i \(-0.335740\pi\)
0.493437 + 0.869781i \(0.335740\pi\)
\(350\) −52.8035 −2.82246
\(351\) −6.04924 −0.322885
\(352\) −9.46374 −0.504419
\(353\) 0.328084 0.0174622 0.00873109 0.999962i \(-0.497221\pi\)
0.00873109 + 0.999962i \(0.497221\pi\)
\(354\) 14.6200 0.777042
\(355\) 4.62839 0.245649
\(356\) 56.0557 2.97095
\(357\) −6.60799 −0.349732
\(358\) −31.3205 −1.65534
\(359\) 9.73592 0.513842 0.256921 0.966432i \(-0.417292\pi\)
0.256921 + 0.966432i \(0.417292\pi\)
\(360\) −3.88603 −0.204812
\(361\) 13.6388 0.717833
\(362\) 58.2872 3.06351
\(363\) 1.00000 0.0524864
\(364\) −124.283 −6.51422
\(365\) −7.36613 −0.385561
\(366\) −2.60074 −0.135943
\(367\) −7.31107 −0.381635 −0.190817 0.981626i \(-0.561114\pi\)
−0.190817 + 0.981626i \(0.561114\pi\)
\(368\) 18.8161 0.980856
\(369\) 7.86011 0.409181
\(370\) 8.75757 0.455284
\(371\) −21.7667 −1.13007
\(372\) −37.4628 −1.94236
\(373\) 17.6033 0.911464 0.455732 0.890117i \(-0.349377\pi\)
0.455732 + 0.890117i \(0.349377\pi\)
\(374\) 3.98485 0.206052
\(375\) 5.24822 0.271017
\(376\) −43.7411 −2.25578
\(377\) 0.137329 0.00707280
\(378\) 11.2163 0.576906
\(379\) 23.1208 1.18763 0.593817 0.804600i \(-0.297620\pi\)
0.593817 + 0.804600i \(0.297620\pi\)
\(380\) −14.7136 −0.754794
\(381\) −15.5050 −0.794348
\(382\) 50.0812 2.56238
\(383\) 10.0134 0.511663 0.255831 0.966721i \(-0.417651\pi\)
0.255831 + 0.966721i \(0.417651\pi\)
\(384\) −2.59587 −0.132470
\(385\) 2.33157 0.118828
\(386\) 29.3876 1.49579
\(387\) −2.59570 −0.131947
\(388\) −45.7672 −2.32348
\(389\) 2.12966 0.107978 0.0539889 0.998542i \(-0.482806\pi\)
0.0539889 + 0.998542i \(0.482806\pi\)
\(390\) 8.50535 0.430685
\(391\) −3.14512 −0.159056
\(392\) 83.3801 4.21133
\(393\) 18.8326 0.949981
\(394\) 30.7056 1.54692
\(395\) 4.78029 0.240522
\(396\) −4.76385 −0.239392
\(397\) 0.991316 0.0497527 0.0248764 0.999691i \(-0.492081\pi\)
0.0248764 + 0.999691i \(0.492081\pi\)
\(398\) 14.6354 0.733608
\(399\) 24.6389 1.23349
\(400\) −43.1537 −2.15768
\(401\) 4.94978 0.247180 0.123590 0.992333i \(-0.460559\pi\)
0.123590 + 0.992333i \(0.460559\pi\)
\(402\) 21.0143 1.04810
\(403\) 47.5711 2.36968
\(404\) 73.2865 3.64614
\(405\) −0.540623 −0.0268638
\(406\) −0.254632 −0.0126372
\(407\) 6.22862 0.308741
\(408\) −11.0135 −0.545251
\(409\) −18.5646 −0.917963 −0.458981 0.888446i \(-0.651785\pi\)
−0.458981 + 0.888446i \(0.651785\pi\)
\(410\) −11.0515 −0.545794
\(411\) −16.6626 −0.821907
\(412\) −49.2801 −2.42786
\(413\) 24.2439 1.19297
\(414\) 5.33851 0.262373
\(415\) −4.99763 −0.245324
\(416\) −57.2484 −2.80683
\(417\) −17.2228 −0.843406
\(418\) −14.8581 −0.726735
\(419\) −22.7594 −1.11187 −0.555935 0.831225i \(-0.687640\pi\)
−0.555935 + 0.831225i \(0.687640\pi\)
\(420\) −11.1073 −0.541979
\(421\) −10.7741 −0.525096 −0.262548 0.964919i \(-0.584563\pi\)
−0.262548 + 0.964919i \(0.584563\pi\)
\(422\) −20.5907 −1.00234
\(423\) −6.08525 −0.295875
\(424\) −36.2786 −1.76184
\(425\) 7.21317 0.349890
\(426\) −22.2655 −1.07877
\(427\) −4.31275 −0.208709
\(428\) 26.9802 1.30414
\(429\) 6.04924 0.292060
\(430\) 3.64960 0.175999
\(431\) 7.60900 0.366512 0.183256 0.983065i \(-0.441336\pi\)
0.183256 + 0.983065i \(0.441336\pi\)
\(432\) 9.16656 0.441026
\(433\) 2.06386 0.0991829 0.0495914 0.998770i \(-0.484208\pi\)
0.0495914 + 0.998770i \(0.484208\pi\)
\(434\) −88.2051 −4.23398
\(435\) 0.0122732 0.000588453 0
\(436\) −68.8908 −3.29927
\(437\) 11.7271 0.560983
\(438\) 35.4358 1.69319
\(439\) 7.60854 0.363136 0.181568 0.983378i \(-0.441883\pi\)
0.181568 + 0.983378i \(0.441883\pi\)
\(440\) 3.88603 0.185259
\(441\) 11.5998 0.552372
\(442\) 24.1053 1.14657
\(443\) 9.27130 0.440493 0.220246 0.975444i \(-0.429314\pi\)
0.220246 + 0.975444i \(0.429314\pi\)
\(444\) −29.6722 −1.40818
\(445\) −6.36146 −0.301562
\(446\) 37.4121 1.77151
\(447\) −8.95498 −0.423556
\(448\) 27.0823 1.27952
\(449\) 7.28337 0.343724 0.171862 0.985121i \(-0.445022\pi\)
0.171862 + 0.985121i \(0.445022\pi\)
\(450\) −12.2436 −0.577168
\(451\) −7.86011 −0.370118
\(452\) 16.9886 0.799077
\(453\) −12.6384 −0.593802
\(454\) 4.93568 0.231643
\(455\) 14.1042 0.661217
\(456\) 41.0656 1.92307
\(457\) 9.93757 0.464860 0.232430 0.972613i \(-0.425332\pi\)
0.232430 + 0.972613i \(0.425332\pi\)
\(458\) −39.1704 −1.83031
\(459\) −1.53220 −0.0715169
\(460\) −5.28659 −0.246488
\(461\) −11.3804 −0.530038 −0.265019 0.964243i \(-0.585378\pi\)
−0.265019 + 0.964243i \(0.585378\pi\)
\(462\) −11.2163 −0.521831
\(463\) 31.7317 1.47470 0.737349 0.675512i \(-0.236077\pi\)
0.737349 + 0.675512i \(0.236077\pi\)
\(464\) −0.208098 −0.00966071
\(465\) 4.25145 0.197156
\(466\) 28.7941 1.33386
\(467\) 34.2226 1.58363 0.791815 0.610760i \(-0.209136\pi\)
0.791815 + 0.610760i \(0.209136\pi\)
\(468\) −28.8177 −1.33210
\(469\) 34.8476 1.60911
\(470\) 8.55599 0.394658
\(471\) 16.8355 0.775740
\(472\) 40.4073 1.85990
\(473\) 2.59570 0.119350
\(474\) −22.9962 −1.05625
\(475\) −26.8954 −1.23405
\(476\) −31.4795 −1.44286
\(477\) −5.04706 −0.231089
\(478\) 75.9465 3.47371
\(479\) −0.0317724 −0.00145172 −0.000725859 1.00000i \(-0.500231\pi\)
−0.000725859 1.00000i \(0.500231\pi\)
\(480\) −5.11632 −0.233527
\(481\) 37.6784 1.71799
\(482\) −34.3154 −1.56302
\(483\) 8.85273 0.402813
\(484\) 4.76385 0.216539
\(485\) 5.19387 0.235842
\(486\) 2.60074 0.117972
\(487\) −6.71443 −0.304260 −0.152130 0.988360i \(-0.548613\pi\)
−0.152130 + 0.988360i \(0.548613\pi\)
\(488\) −7.18805 −0.325388
\(489\) 6.53716 0.295620
\(490\) −16.3096 −0.736792
\(491\) 15.6320 0.705464 0.352732 0.935724i \(-0.385253\pi\)
0.352732 + 0.935724i \(0.385253\pi\)
\(492\) 37.4444 1.68812
\(493\) 0.0347838 0.00156658
\(494\) −89.8804 −4.04391
\(495\) 0.540623 0.0242992
\(496\) −72.0856 −3.23674
\(497\) −36.9224 −1.65619
\(498\) 24.0418 1.07734
\(499\) −42.6023 −1.90714 −0.953570 0.301170i \(-0.902623\pi\)
−0.953570 + 0.301170i \(0.902623\pi\)
\(500\) 25.0017 1.11811
\(501\) −1.74070 −0.0777687
\(502\) −55.8494 −2.49268
\(503\) 20.7317 0.924380 0.462190 0.886781i \(-0.347064\pi\)
0.462190 + 0.886781i \(0.347064\pi\)
\(504\) 31.0003 1.38086
\(505\) −8.31689 −0.370097
\(506\) −5.33851 −0.237326
\(507\) 23.5933 1.04782
\(508\) −73.8637 −3.27717
\(509\) −10.5993 −0.469807 −0.234904 0.972019i \(-0.575477\pi\)
−0.234904 + 0.972019i \(0.575477\pi\)
\(510\) 2.15430 0.0953941
\(511\) 58.7624 2.59949
\(512\) −45.0295 −1.99004
\(513\) 5.71304 0.252237
\(514\) 46.3925 2.04628
\(515\) 5.59253 0.246437
\(516\) −12.3655 −0.544361
\(517\) 6.08525 0.267629
\(518\) −69.8624 −3.06958
\(519\) 8.99614 0.394887
\(520\) 23.5075 1.03087
\(521\) 17.7172 0.776203 0.388101 0.921617i \(-0.373131\pi\)
0.388101 + 0.921617i \(0.373131\pi\)
\(522\) −0.0590417 −0.00258418
\(523\) 33.2492 1.45389 0.726944 0.686697i \(-0.240940\pi\)
0.726944 + 0.686697i \(0.240940\pi\)
\(524\) 89.7158 3.91925
\(525\) −20.3032 −0.886106
\(526\) −77.5134 −3.37975
\(527\) 12.0492 0.524870
\(528\) −9.16656 −0.398923
\(529\) −18.7865 −0.816803
\(530\) 7.09628 0.308243
\(531\) 5.62146 0.243951
\(532\) 117.376 5.08890
\(533\) −47.5477 −2.05952
\(534\) 30.6026 1.32431
\(535\) −3.06183 −0.132375
\(536\) 58.0805 2.50869
\(537\) −12.0429 −0.519690
\(538\) −66.1738 −2.85296
\(539\) −11.5998 −0.499639
\(540\) −2.57545 −0.110830
\(541\) 0.0519933 0.00223537 0.00111768 0.999999i \(-0.499644\pi\)
0.00111768 + 0.999999i \(0.499644\pi\)
\(542\) −11.8655 −0.509668
\(543\) 22.4118 0.961781
\(544\) −14.5003 −0.621696
\(545\) 7.81804 0.334888
\(546\) −67.8503 −2.90373
\(547\) −41.7142 −1.78357 −0.891785 0.452458i \(-0.850547\pi\)
−0.891785 + 0.452458i \(0.850547\pi\)
\(548\) −79.3783 −3.39087
\(549\) −1.00000 −0.0426790
\(550\) 12.2436 0.522068
\(551\) −0.129697 −0.00552526
\(552\) 14.7548 0.628007
\(553\) −38.1341 −1.62163
\(554\) 45.4728 1.93195
\(555\) 3.36734 0.142936
\(556\) −82.0470 −3.47957
\(557\) 7.84719 0.332496 0.166248 0.986084i \(-0.446835\pi\)
0.166248 + 0.986084i \(0.446835\pi\)
\(558\) −20.4522 −0.865809
\(559\) 15.7020 0.664123
\(560\) −21.3725 −0.903153
\(561\) 1.53220 0.0646894
\(562\) −61.3981 −2.58992
\(563\) 11.3050 0.476450 0.238225 0.971210i \(-0.423434\pi\)
0.238225 + 0.971210i \(0.423434\pi\)
\(564\) −28.9892 −1.22067
\(565\) −1.92794 −0.0811092
\(566\) −6.73593 −0.283132
\(567\) 4.31275 0.181118
\(568\) −61.5385 −2.58210
\(569\) 24.2745 1.01764 0.508820 0.860873i \(-0.330082\pi\)
0.508820 + 0.860873i \(0.330082\pi\)
\(570\) −8.03265 −0.336451
\(571\) −3.48083 −0.145668 −0.0728342 0.997344i \(-0.523204\pi\)
−0.0728342 + 0.997344i \(0.523204\pi\)
\(572\) 28.8177 1.20493
\(573\) 19.2565 0.804453
\(574\) 88.1617 3.67980
\(575\) −9.66349 −0.402995
\(576\) 6.27960 0.261650
\(577\) 4.02370 0.167509 0.0837545 0.996486i \(-0.473309\pi\)
0.0837545 + 0.996486i \(0.473309\pi\)
\(578\) −38.1070 −1.58504
\(579\) 11.2997 0.469599
\(580\) 0.0584675 0.00242773
\(581\) 39.8680 1.65400
\(582\) −24.9858 −1.03570
\(583\) 5.04706 0.209028
\(584\) 97.9391 4.05275
\(585\) 3.27036 0.135213
\(586\) −78.5306 −3.24407
\(587\) 23.1317 0.954746 0.477373 0.878701i \(-0.341589\pi\)
0.477373 + 0.878701i \(0.341589\pi\)
\(588\) 55.2598 2.27887
\(589\) −44.9272 −1.85119
\(590\) −7.90389 −0.325398
\(591\) 11.8065 0.485653
\(592\) −57.0951 −2.34659
\(593\) −26.1833 −1.07522 −0.537610 0.843194i \(-0.680673\pi\)
−0.537610 + 0.843194i \(0.680673\pi\)
\(594\) −2.60074 −0.106710
\(595\) 3.57243 0.146455
\(596\) −42.6602 −1.74743
\(597\) 5.62741 0.230314
\(598\) −32.2939 −1.32060
\(599\) −22.7916 −0.931240 −0.465620 0.884985i \(-0.654169\pi\)
−0.465620 + 0.884985i \(0.654169\pi\)
\(600\) −33.8394 −1.38149
\(601\) 37.8468 1.54380 0.771902 0.635741i \(-0.219306\pi\)
0.771902 + 0.635741i \(0.219306\pi\)
\(602\) −29.1142 −1.18661
\(603\) 8.08014 0.329049
\(604\) −60.2072 −2.44980
\(605\) −0.540623 −0.0219795
\(606\) 40.0095 1.62528
\(607\) 16.0621 0.651939 0.325969 0.945380i \(-0.394309\pi\)
0.325969 + 0.945380i \(0.394309\pi\)
\(608\) 54.0667 2.19269
\(609\) −0.0979075 −0.00396741
\(610\) 1.40602 0.0569281
\(611\) 36.8111 1.48922
\(612\) −7.29916 −0.295051
\(613\) 36.9475 1.49229 0.746147 0.665781i \(-0.231901\pi\)
0.746147 + 0.665781i \(0.231901\pi\)
\(614\) 39.9647 1.61284
\(615\) −4.24936 −0.171351
\(616\) −31.0003 −1.24904
\(617\) −13.0177 −0.524072 −0.262036 0.965058i \(-0.584394\pi\)
−0.262036 + 0.965058i \(0.584394\pi\)
\(618\) −26.9036 −1.08222
\(619\) 22.0427 0.885971 0.442985 0.896529i \(-0.353919\pi\)
0.442985 + 0.896529i \(0.353919\pi\)
\(620\) 20.2533 0.813391
\(621\) 2.05269 0.0823715
\(622\) −58.7199 −2.35445
\(623\) 50.7477 2.03316
\(624\) −55.4507 −2.21981
\(625\) 20.7013 0.828053
\(626\) −5.03899 −0.201399
\(627\) −5.71304 −0.228157
\(628\) 80.2019 3.20040
\(629\) 9.54348 0.380524
\(630\) −6.06382 −0.241588
\(631\) −21.1563 −0.842220 −0.421110 0.907010i \(-0.638359\pi\)
−0.421110 + 0.907010i \(0.638359\pi\)
\(632\) −63.5581 −2.52820
\(633\) −7.91724 −0.314682
\(634\) 45.4668 1.80572
\(635\) 8.38239 0.332645
\(636\) −24.0435 −0.953385
\(637\) −70.1700 −2.78024
\(638\) 0.0590417 0.00233748
\(639\) −8.56121 −0.338676
\(640\) 1.40339 0.0554738
\(641\) 25.2664 0.997965 0.498982 0.866612i \(-0.333707\pi\)
0.498982 + 0.866612i \(0.333707\pi\)
\(642\) 14.7294 0.581321
\(643\) −20.5251 −0.809430 −0.404715 0.914443i \(-0.632629\pi\)
−0.404715 + 0.914443i \(0.632629\pi\)
\(644\) 42.1731 1.66185
\(645\) 1.40329 0.0552546
\(646\) −22.7656 −0.895701
\(647\) −3.31668 −0.130392 −0.0651960 0.997872i \(-0.520767\pi\)
−0.0651960 + 0.997872i \(0.520767\pi\)
\(648\) 7.18805 0.282373
\(649\) −5.62146 −0.220662
\(650\) 74.0643 2.90504
\(651\) −33.9154 −1.32925
\(652\) 31.1420 1.21962
\(653\) 40.2518 1.57517 0.787587 0.616204i \(-0.211330\pi\)
0.787587 + 0.616204i \(0.211330\pi\)
\(654\) −37.6097 −1.47066
\(655\) −10.1814 −0.397819
\(656\) 72.0502 2.81309
\(657\) 13.6253 0.531572
\(658\) −68.2543 −2.66083
\(659\) −11.2229 −0.437183 −0.218591 0.975816i \(-0.570146\pi\)
−0.218591 + 0.975816i \(0.570146\pi\)
\(660\) 2.57545 0.100249
\(661\) 18.4223 0.716546 0.358273 0.933617i \(-0.383366\pi\)
0.358273 + 0.933617i \(0.383366\pi\)
\(662\) −9.47704 −0.368336
\(663\) 9.26863 0.359964
\(664\) 66.4479 2.57868
\(665\) −13.3204 −0.516542
\(666\) −16.1990 −0.627700
\(667\) −0.0465998 −0.00180435
\(668\) −8.29243 −0.320844
\(669\) 14.3852 0.556162
\(670\) −11.3608 −0.438907
\(671\) 1.00000 0.0386046
\(672\) 40.8147 1.57446
\(673\) −32.2878 −1.24460 −0.622301 0.782778i \(-0.713802\pi\)
−0.622301 + 0.782778i \(0.713802\pi\)
\(674\) −65.5199 −2.52373
\(675\) −4.70773 −0.181200
\(676\) 112.395 4.32288
\(677\) −45.6192 −1.75329 −0.876644 0.481139i \(-0.840223\pi\)
−0.876644 + 0.481139i \(0.840223\pi\)
\(678\) 9.27463 0.356190
\(679\) −41.4334 −1.59007
\(680\) 5.95417 0.228332
\(681\) 1.89780 0.0727237
\(682\) 20.4522 0.783154
\(683\) 6.70817 0.256681 0.128340 0.991730i \(-0.459035\pi\)
0.128340 + 0.991730i \(0.459035\pi\)
\(684\) 27.2161 1.04063
\(685\) 9.00821 0.344186
\(686\) 51.5931 1.96983
\(687\) −15.0613 −0.574623
\(688\) −23.7936 −0.907123
\(689\) 30.5309 1.16313
\(690\) −2.88612 −0.109873
\(691\) 6.61809 0.251764 0.125882 0.992045i \(-0.459824\pi\)
0.125882 + 0.992045i \(0.459824\pi\)
\(692\) 42.8562 1.62915
\(693\) −4.31275 −0.163828
\(694\) 39.2179 1.48869
\(695\) 9.31106 0.353189
\(696\) −0.163182 −0.00618541
\(697\) −12.0433 −0.456171
\(698\) 47.9481 1.81486
\(699\) 11.0715 0.418762
\(700\) −96.7216 −3.65573
\(701\) 36.1896 1.36686 0.683432 0.730014i \(-0.260487\pi\)
0.683432 + 0.730014i \(0.260487\pi\)
\(702\) −15.7325 −0.593785
\(703\) −35.5844 −1.34209
\(704\) −6.27960 −0.236671
\(705\) 3.28983 0.123902
\(706\) 0.853263 0.0321130
\(707\) 66.3469 2.49523
\(708\) 26.7798 1.00645
\(709\) 42.6357 1.60122 0.800609 0.599187i \(-0.204510\pi\)
0.800609 + 0.599187i \(0.204510\pi\)
\(710\) 12.0372 0.451750
\(711\) −8.84218 −0.331608
\(712\) 84.5811 3.16981
\(713\) −16.1423 −0.604533
\(714\) −17.1857 −0.643157
\(715\) −3.27036 −0.122305
\(716\) −57.3706 −2.14404
\(717\) 29.2019 1.09056
\(718\) 25.3206 0.944957
\(719\) −18.9095 −0.705206 −0.352603 0.935773i \(-0.614703\pi\)
−0.352603 + 0.935773i \(0.614703\pi\)
\(720\) −4.95566 −0.184686
\(721\) −44.6137 −1.66150
\(722\) 35.4710 1.32010
\(723\) −13.1945 −0.490708
\(724\) 106.766 3.96794
\(725\) 0.106874 0.00396921
\(726\) 2.60074 0.0965225
\(727\) 7.37299 0.273449 0.136725 0.990609i \(-0.456342\pi\)
0.136725 + 0.990609i \(0.456342\pi\)
\(728\) −187.528 −6.95025
\(729\) 1.00000 0.0370370
\(730\) −19.1574 −0.709047
\(731\) 3.97712 0.147099
\(732\) −4.76385 −0.176077
\(733\) 12.6913 0.468764 0.234382 0.972145i \(-0.424693\pi\)
0.234382 + 0.972145i \(0.424693\pi\)
\(734\) −19.0142 −0.701827
\(735\) −6.27113 −0.231314
\(736\) 19.4261 0.716055
\(737\) −8.08014 −0.297636
\(738\) 20.4421 0.752485
\(739\) −9.50571 −0.349673 −0.174837 0.984597i \(-0.555940\pi\)
−0.174837 + 0.984597i \(0.555940\pi\)
\(740\) 16.0415 0.589697
\(741\) −34.5595 −1.26958
\(742\) −56.6096 −2.07820
\(743\) 1.76760 0.0648470 0.0324235 0.999474i \(-0.489677\pi\)
0.0324235 + 0.999474i \(0.489677\pi\)
\(744\) −56.5267 −2.07237
\(745\) 4.84127 0.177370
\(746\) 45.7816 1.67618
\(747\) 9.24421 0.338228
\(748\) 7.29916 0.266884
\(749\) 24.4254 0.892483
\(750\) 13.6493 0.498401
\(751\) −21.8139 −0.796001 −0.398000 0.917385i \(-0.630296\pi\)
−0.398000 + 0.917385i \(0.630296\pi\)
\(752\) −55.7808 −2.03412
\(753\) −21.4744 −0.782571
\(754\) 0.357157 0.0130069
\(755\) 6.83259 0.248663
\(756\) 20.5453 0.747225
\(757\) 20.5403 0.746552 0.373276 0.927720i \(-0.378235\pi\)
0.373276 + 0.927720i \(0.378235\pi\)
\(758\) 60.1311 2.18406
\(759\) −2.05269 −0.0745078
\(760\) −22.2010 −0.805316
\(761\) 42.2493 1.53154 0.765768 0.643116i \(-0.222359\pi\)
0.765768 + 0.643116i \(0.222359\pi\)
\(762\) −40.3246 −1.46081
\(763\) −62.3674 −2.25785
\(764\) 91.7352 3.31886
\(765\) 0.828342 0.0299488
\(766\) 26.0423 0.940948
\(767\) −34.0055 −1.22787
\(768\) −19.3104 −0.696803
\(769\) 12.3073 0.443814 0.221907 0.975068i \(-0.428772\pi\)
0.221907 + 0.975068i \(0.428772\pi\)
\(770\) 6.06382 0.218525
\(771\) 17.8382 0.642426
\(772\) 53.8300 1.93739
\(773\) 41.7428 1.50138 0.750692 0.660652i \(-0.229720\pi\)
0.750692 + 0.660652i \(0.229720\pi\)
\(774\) −6.75073 −0.242650
\(775\) 37.0215 1.32985
\(776\) −69.0570 −2.47900
\(777\) −26.8625 −0.963687
\(778\) 5.53868 0.198571
\(779\) 44.9051 1.60889
\(780\) 15.5795 0.557836
\(781\) 8.56121 0.306344
\(782\) −8.17965 −0.292504
\(783\) −0.0227019 −0.000811299 0
\(784\) 106.330 3.79751
\(785\) −9.10167 −0.324853
\(786\) 48.9788 1.74702
\(787\) −20.4654 −0.729512 −0.364756 0.931103i \(-0.618848\pi\)
−0.364756 + 0.931103i \(0.618848\pi\)
\(788\) 56.2442 2.00362
\(789\) −29.8044 −1.06106
\(790\) 12.4323 0.442321
\(791\) 15.3799 0.546847
\(792\) −7.18805 −0.255416
\(793\) 6.04924 0.214815
\(794\) 2.57816 0.0914953
\(795\) 2.72856 0.0967721
\(796\) 26.8081 0.950189
\(797\) 17.3900 0.615986 0.307993 0.951389i \(-0.400343\pi\)
0.307993 + 0.951389i \(0.400343\pi\)
\(798\) 64.0794 2.26839
\(799\) 9.32381 0.329853
\(800\) −44.5527 −1.57518
\(801\) 11.7669 0.415763
\(802\) 12.8731 0.454565
\(803\) −13.6253 −0.480825
\(804\) 38.4926 1.35753
\(805\) −4.78599 −0.168684
\(806\) 123.720 4.35785
\(807\) −25.4442 −0.895679
\(808\) 110.580 3.89020
\(809\) 28.4398 0.999891 0.499946 0.866057i \(-0.333353\pi\)
0.499946 + 0.866057i \(0.333353\pi\)
\(810\) −1.40602 −0.0494025
\(811\) −12.1286 −0.425892 −0.212946 0.977064i \(-0.568306\pi\)
−0.212946 + 0.977064i \(0.568306\pi\)
\(812\) −0.466416 −0.0163680
\(813\) −4.56237 −0.160009
\(814\) 16.1990 0.567776
\(815\) −3.53414 −0.123795
\(816\) −14.0450 −0.491673
\(817\) −14.8293 −0.518812
\(818\) −48.2818 −1.68813
\(819\) −26.0889 −0.911618
\(820\) −20.2433 −0.706927
\(821\) −2.72354 −0.0950521 −0.0475261 0.998870i \(-0.515134\pi\)
−0.0475261 + 0.998870i \(0.515134\pi\)
\(822\) −43.3352 −1.51149
\(823\) −10.1098 −0.352404 −0.176202 0.984354i \(-0.556381\pi\)
−0.176202 + 0.984354i \(0.556381\pi\)
\(824\) −74.3576 −2.59037
\(825\) 4.70773 0.163902
\(826\) 63.0522 2.19387
\(827\) 27.8472 0.968341 0.484170 0.874974i \(-0.339122\pi\)
0.484170 + 0.874974i \(0.339122\pi\)
\(828\) 9.77869 0.339833
\(829\) 32.8263 1.14010 0.570052 0.821608i \(-0.306923\pi\)
0.570052 + 0.821608i \(0.306923\pi\)
\(830\) −12.9975 −0.451151
\(831\) 17.4846 0.606533
\(832\) −37.9868 −1.31696
\(833\) −17.7732 −0.615805
\(834\) −44.7921 −1.55102
\(835\) 0.941063 0.0325668
\(836\) −27.2161 −0.941287
\(837\) −7.86398 −0.271819
\(838\) −59.1914 −2.04473
\(839\) −38.5543 −1.33104 −0.665521 0.746379i \(-0.731791\pi\)
−0.665521 + 0.746379i \(0.731791\pi\)
\(840\) −16.7595 −0.578257
\(841\) −28.9995 −0.999982
\(842\) −28.0205 −0.965652
\(843\) −23.6079 −0.813100
\(844\) −37.7165 −1.29826
\(845\) −12.7551 −0.438788
\(846\) −15.8262 −0.544114
\(847\) 4.31275 0.148188
\(848\) −46.2642 −1.58872
\(849\) −2.59000 −0.0888887
\(850\) 18.7596 0.643448
\(851\) −12.7854 −0.438278
\(852\) −40.7843 −1.39725
\(853\) −14.1974 −0.486111 −0.243055 0.970012i \(-0.578150\pi\)
−0.243055 + 0.970012i \(0.578150\pi\)
\(854\) −11.2163 −0.383815
\(855\) −3.08860 −0.105628
\(856\) 40.7097 1.39143
\(857\) −46.1146 −1.57524 −0.787622 0.616159i \(-0.788688\pi\)
−0.787622 + 0.616159i \(0.788688\pi\)
\(858\) 15.7325 0.537099
\(859\) 57.2683 1.95397 0.976985 0.213306i \(-0.0684231\pi\)
0.976985 + 0.213306i \(0.0684231\pi\)
\(860\) 6.68508 0.227959
\(861\) 33.8987 1.15526
\(862\) 19.7890 0.674017
\(863\) −21.7311 −0.739737 −0.369868 0.929084i \(-0.620597\pi\)
−0.369868 + 0.929084i \(0.620597\pi\)
\(864\) 9.46374 0.321963
\(865\) −4.86352 −0.165365
\(866\) 5.36757 0.182397
\(867\) −14.6524 −0.497621
\(868\) −161.568 −5.48397
\(869\) 8.84218 0.299950
\(870\) 0.0319193 0.00108217
\(871\) −48.8787 −1.65619
\(872\) −103.948 −3.52011
\(873\) −9.60720 −0.325154
\(874\) 30.4991 1.03165
\(875\) 22.6343 0.765178
\(876\) 64.9087 2.19306
\(877\) −44.3014 −1.49595 −0.747976 0.663726i \(-0.768974\pi\)
−0.747976 + 0.663726i \(0.768974\pi\)
\(878\) 19.7878 0.667807
\(879\) −30.1955 −1.01847
\(880\) 4.95566 0.167055
\(881\) 20.2667 0.682802 0.341401 0.939918i \(-0.389099\pi\)
0.341401 + 0.939918i \(0.389099\pi\)
\(882\) 30.1681 1.01581
\(883\) 49.6889 1.67216 0.836081 0.548605i \(-0.184841\pi\)
0.836081 + 0.548605i \(0.184841\pi\)
\(884\) 44.1544 1.48507
\(885\) −3.03909 −0.102158
\(886\) 24.1123 0.810067
\(887\) −21.0928 −0.708228 −0.354114 0.935202i \(-0.615218\pi\)
−0.354114 + 0.935202i \(0.615218\pi\)
\(888\) −44.7717 −1.50244
\(889\) −66.8694 −2.24273
\(890\) −16.5445 −0.554573
\(891\) −1.00000 −0.0335013
\(892\) 68.5287 2.29451
\(893\) −34.7653 −1.16338
\(894\) −23.2896 −0.778920
\(895\) 6.51067 0.217628
\(896\) −11.1953 −0.374010
\(897\) −12.4172 −0.414598
\(898\) 18.9422 0.632108
\(899\) 0.178527 0.00595421
\(900\) −22.4269 −0.747563
\(901\) 7.73310 0.257627
\(902\) −20.4421 −0.680648
\(903\) −11.1946 −0.372533
\(904\) 25.6337 0.852564
\(905\) −12.1163 −0.402760
\(906\) −32.8691 −1.09200
\(907\) 24.8398 0.824794 0.412397 0.911004i \(-0.364692\pi\)
0.412397 + 0.911004i \(0.364692\pi\)
\(908\) 9.04082 0.300030
\(909\) 15.3839 0.510252
\(910\) 36.6815 1.21598
\(911\) 23.9432 0.793275 0.396637 0.917975i \(-0.370177\pi\)
0.396637 + 0.917975i \(0.370177\pi\)
\(912\) 52.3689 1.73411
\(913\) −9.24421 −0.305939
\(914\) 25.8450 0.854878
\(915\) 0.540623 0.0178725
\(916\) −71.7495 −2.37067
\(917\) 81.2204 2.68214
\(918\) −3.98485 −0.131520
\(919\) 14.3672 0.473931 0.236965 0.971518i \(-0.423847\pi\)
0.236965 + 0.971518i \(0.423847\pi\)
\(920\) −7.97680 −0.262987
\(921\) 15.3667 0.506349
\(922\) −29.5974 −0.974740
\(923\) 51.7888 1.70465
\(924\) −20.5453 −0.675890
\(925\) 29.3227 0.964123
\(926\) 82.5259 2.71197
\(927\) −10.3446 −0.339761
\(928\) −0.214845 −0.00705262
\(929\) −39.5612 −1.29796 −0.648980 0.760806i \(-0.724804\pi\)
−0.648980 + 0.760806i \(0.724804\pi\)
\(930\) 11.0569 0.362571
\(931\) 66.2702 2.17192
\(932\) 52.7429 1.72765
\(933\) −22.5782 −0.739176
\(934\) 89.0040 2.91230
\(935\) −0.828342 −0.0270897
\(936\) −43.4823 −1.42126
\(937\) −48.6379 −1.58893 −0.794465 0.607310i \(-0.792249\pi\)
−0.794465 + 0.607310i \(0.792249\pi\)
\(938\) 90.6296 2.95916
\(939\) −1.93752 −0.0632287
\(940\) 15.6722 0.511172
\(941\) 28.6981 0.935530 0.467765 0.883853i \(-0.345059\pi\)
0.467765 + 0.883853i \(0.345059\pi\)
\(942\) 43.7848 1.42659
\(943\) 16.1344 0.525407
\(944\) 51.5294 1.67714
\(945\) −2.33157 −0.0758461
\(946\) 6.75073 0.219485
\(947\) −9.54462 −0.310158 −0.155079 0.987902i \(-0.549563\pi\)
−0.155079 + 0.987902i \(0.549563\pi\)
\(948\) −42.1228 −1.36809
\(949\) −82.4225 −2.67555
\(950\) −69.9480 −2.26941
\(951\) 17.4823 0.566901
\(952\) −47.4986 −1.53944
\(953\) 31.2116 1.01104 0.505521 0.862814i \(-0.331300\pi\)
0.505521 + 0.862814i \(0.331300\pi\)
\(954\) −13.1261 −0.424973
\(955\) −10.4105 −0.336877
\(956\) 139.113 4.49925
\(957\) 0.0227019 0.000733847 0
\(958\) −0.0826318 −0.00266971
\(959\) −71.8618 −2.32054
\(960\) −3.39490 −0.109570
\(961\) 30.8422 0.994908
\(962\) 97.9918 3.15938
\(963\) 5.66352 0.182505
\(964\) −62.8565 −2.02447
\(965\) −6.10888 −0.196652
\(966\) 23.0236 0.740774
\(967\) 49.0672 1.57790 0.788948 0.614460i \(-0.210626\pi\)
0.788948 + 0.614460i \(0.210626\pi\)
\(968\) 7.18805 0.231033
\(969\) −8.75351 −0.281203
\(970\) 13.5079 0.433713
\(971\) 30.9733 0.993980 0.496990 0.867756i \(-0.334439\pi\)
0.496990 + 0.867756i \(0.334439\pi\)
\(972\) 4.76385 0.152801
\(973\) −74.2778 −2.38124
\(974\) −17.4625 −0.559534
\(975\) 28.4782 0.912031
\(976\) −9.16656 −0.293414
\(977\) −2.27911 −0.0729153 −0.0364577 0.999335i \(-0.511607\pi\)
−0.0364577 + 0.999335i \(0.511607\pi\)
\(978\) 17.0014 0.543646
\(979\) −11.7669 −0.376072
\(980\) −29.8747 −0.954313
\(981\) −14.4612 −0.461709
\(982\) 40.6549 1.29735
\(983\) 24.0060 0.765671 0.382835 0.923817i \(-0.374948\pi\)
0.382835 + 0.923817i \(0.374948\pi\)
\(984\) 56.4989 1.80112
\(985\) −6.38285 −0.203375
\(986\) 0.0904635 0.00288094
\(987\) −26.2442 −0.835361
\(988\) −164.636 −5.23778
\(989\) −5.32815 −0.169425
\(990\) 1.40602 0.0446863
\(991\) −15.1701 −0.481894 −0.240947 0.970538i \(-0.577458\pi\)
−0.240947 + 0.970538i \(0.577458\pi\)
\(992\) −74.4226 −2.36292
\(993\) −3.64398 −0.115638
\(994\) −96.0255 −3.04574
\(995\) −3.04231 −0.0964476
\(996\) 44.0380 1.39540
\(997\) −39.1763 −1.24072 −0.620362 0.784315i \(-0.713014\pi\)
−0.620362 + 0.784315i \(0.713014\pi\)
\(998\) −110.798 −3.50723
\(999\) −6.22862 −0.197065
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.f.1.12 13
3.2 odd 2 6039.2.a.g.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.f.1.12 13 1.1 even 1 trivial
6039.2.a.g.1.2 13 3.2 odd 2