Properties

Label 8015.2.a.m.1.11
Level $8015$
Weight $2$
Character 8015.1
Self dual yes
Analytic conductor $64.000$
Analytic rank $0$
Dimension $67$
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] = [8015,2,Mod(1,8015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, 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("8015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8015.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: \(64.0000972201\)
Analytic rank: \(0\)
Dimension: \(67\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 8015.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.94503 q^{2} +2.18285 q^{3} +1.78313 q^{4} +1.00000 q^{5} -4.24571 q^{6} -1.00000 q^{7} +0.421811 q^{8} +1.76484 q^{9} +O(q^{10})\) \(q-1.94503 q^{2} +2.18285 q^{3} +1.78313 q^{4} +1.00000 q^{5} -4.24571 q^{6} -1.00000 q^{7} +0.421811 q^{8} +1.76484 q^{9} -1.94503 q^{10} +6.09451 q^{11} +3.89231 q^{12} +3.26846 q^{13} +1.94503 q^{14} +2.18285 q^{15} -4.38670 q^{16} -5.66347 q^{17} -3.43266 q^{18} +7.89079 q^{19} +1.78313 q^{20} -2.18285 q^{21} -11.8540 q^{22} -3.46719 q^{23} +0.920750 q^{24} +1.00000 q^{25} -6.35724 q^{26} -2.69618 q^{27} -1.78313 q^{28} +6.35413 q^{29} -4.24571 q^{30} -4.38922 q^{31} +7.68864 q^{32} +13.3034 q^{33} +11.0156 q^{34} -1.00000 q^{35} +3.14694 q^{36} -5.29552 q^{37} -15.3478 q^{38} +7.13455 q^{39} +0.421811 q^{40} -1.38738 q^{41} +4.24571 q^{42} +6.06975 q^{43} +10.8673 q^{44} +1.76484 q^{45} +6.74379 q^{46} -5.49249 q^{47} -9.57551 q^{48} +1.00000 q^{49} -1.94503 q^{50} -12.3625 q^{51} +5.82810 q^{52} -11.7735 q^{53} +5.24414 q^{54} +6.09451 q^{55} -0.421811 q^{56} +17.2244 q^{57} -12.3590 q^{58} +11.9506 q^{59} +3.89231 q^{60} +7.00385 q^{61} +8.53715 q^{62} -1.76484 q^{63} -6.18121 q^{64} +3.26846 q^{65} -25.8755 q^{66} +8.63416 q^{67} -10.0987 q^{68} -7.56836 q^{69} +1.94503 q^{70} +14.7022 q^{71} +0.744427 q^{72} +4.02309 q^{73} +10.2999 q^{74} +2.18285 q^{75} +14.0703 q^{76} -6.09451 q^{77} -13.8769 q^{78} +4.96540 q^{79} -4.38670 q^{80} -11.1799 q^{81} +2.69848 q^{82} +0.709802 q^{83} -3.89231 q^{84} -5.66347 q^{85} -11.8058 q^{86} +13.8701 q^{87} +2.57073 q^{88} +14.7952 q^{89} -3.43266 q^{90} -3.26846 q^{91} -6.18247 q^{92} -9.58101 q^{93} +10.6830 q^{94} +7.89079 q^{95} +16.7831 q^{96} -7.08866 q^{97} -1.94503 q^{98} +10.7558 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 67 q + 3 q^{2} + 73 q^{4} + 67 q^{5} + 17 q^{6} - 67 q^{7} + 12 q^{8} + 97 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 67 q + 3 q^{2} + 73 q^{4} + 67 q^{5} + 17 q^{6} - 67 q^{7} + 12 q^{8} + 97 q^{9} + 3 q^{10} + 11 q^{11} + 9 q^{12} + 19 q^{13} - 3 q^{14} + 93 q^{16} + 7 q^{17} + 9 q^{18} + 36 q^{19} + 73 q^{20} - 8 q^{22} - 2 q^{23} + 37 q^{24} + 67 q^{25} + 39 q^{26} + 6 q^{27} - 73 q^{28} + 56 q^{29} + 17 q^{30} + 63 q^{31} + 27 q^{32} + 51 q^{33} + 55 q^{34} - 67 q^{35} + 148 q^{36} + 28 q^{37} + 10 q^{38} + 7 q^{39} + 12 q^{40} + 84 q^{41} - 17 q^{42} - 11 q^{43} + 41 q^{44} + 97 q^{45} + 17 q^{46} - 10 q^{47} + 10 q^{48} + 67 q^{49} + 3 q^{50} - q^{51} + 49 q^{52} + 3 q^{53} + 20 q^{54} + 11 q^{55} - 12 q^{56} + 45 q^{57} + 22 q^{58} + 80 q^{59} + 9 q^{60} + 64 q^{61} - 37 q^{62} - 97 q^{63} + 110 q^{64} + 19 q^{65} + 75 q^{66} + 22 q^{67} + 7 q^{68} + 107 q^{69} - 3 q^{70} + 24 q^{71} + 72 q^{72} + 83 q^{73} + 52 q^{74} + 115 q^{76} - 11 q^{77} - 70 q^{78} - 32 q^{79} + 93 q^{80} + 183 q^{81} + 56 q^{82} - 58 q^{83} - 9 q^{84} + 7 q^{85} + 51 q^{86} + 20 q^{87} - 5 q^{88} + 129 q^{89} + 9 q^{90} - 19 q^{91} - 37 q^{92} + 33 q^{93} + 89 q^{94} + 36 q^{95} + 129 q^{96} + 126 q^{97} + 3 q^{98} - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.94503 −1.37534 −0.687671 0.726022i \(-0.741367\pi\)
−0.687671 + 0.726022i \(0.741367\pi\)
\(3\) 2.18285 1.26027 0.630135 0.776486i \(-0.283000\pi\)
0.630135 + 0.776486i \(0.283000\pi\)
\(4\) 1.78313 0.891567
\(5\) 1.00000 0.447214
\(6\) −4.24571 −1.73330
\(7\) −1.00000 −0.377964
\(8\) 0.421811 0.149133
\(9\) 1.76484 0.588279
\(10\) −1.94503 −0.615072
\(11\) 6.09451 1.83756 0.918782 0.394766i \(-0.129174\pi\)
0.918782 + 0.394766i \(0.129174\pi\)
\(12\) 3.89231 1.12361
\(13\) 3.26846 0.906507 0.453253 0.891382i \(-0.350263\pi\)
0.453253 + 0.891382i \(0.350263\pi\)
\(14\) 1.94503 0.519831
\(15\) 2.18285 0.563610
\(16\) −4.38670 −1.09668
\(17\) −5.66347 −1.37359 −0.686797 0.726850i \(-0.740984\pi\)
−0.686797 + 0.726850i \(0.740984\pi\)
\(18\) −3.43266 −0.809085
\(19\) 7.89079 1.81027 0.905135 0.425124i \(-0.139769\pi\)
0.905135 + 0.425124i \(0.139769\pi\)
\(20\) 1.78313 0.398721
\(21\) −2.18285 −0.476337
\(22\) −11.8540 −2.52728
\(23\) −3.46719 −0.722960 −0.361480 0.932380i \(-0.617728\pi\)
−0.361480 + 0.932380i \(0.617728\pi\)
\(24\) 0.920750 0.187947
\(25\) 1.00000 0.200000
\(26\) −6.35724 −1.24676
\(27\) −2.69618 −0.518880
\(28\) −1.78313 −0.336981
\(29\) 6.35413 1.17993 0.589966 0.807428i \(-0.299141\pi\)
0.589966 + 0.807428i \(0.299141\pi\)
\(30\) −4.24571 −0.775156
\(31\) −4.38922 −0.788327 −0.394163 0.919040i \(-0.628966\pi\)
−0.394163 + 0.919040i \(0.628966\pi\)
\(32\) 7.68864 1.35917
\(33\) 13.3034 2.31582
\(34\) 11.0156 1.88916
\(35\) −1.00000 −0.169031
\(36\) 3.14694 0.524490
\(37\) −5.29552 −0.870579 −0.435289 0.900291i \(-0.643354\pi\)
−0.435289 + 0.900291i \(0.643354\pi\)
\(38\) −15.3478 −2.48974
\(39\) 7.13455 1.14244
\(40\) 0.421811 0.0666942
\(41\) −1.38738 −0.216672 −0.108336 0.994114i \(-0.534552\pi\)
−0.108336 + 0.994114i \(0.534552\pi\)
\(42\) 4.24571 0.655127
\(43\) 6.06975 0.925629 0.462814 0.886455i \(-0.346840\pi\)
0.462814 + 0.886455i \(0.346840\pi\)
\(44\) 10.8673 1.63831
\(45\) 1.76484 0.263086
\(46\) 6.74379 0.994317
\(47\) −5.49249 −0.801162 −0.400581 0.916261i \(-0.631192\pi\)
−0.400581 + 0.916261i \(0.631192\pi\)
\(48\) −9.57551 −1.38211
\(49\) 1.00000 0.142857
\(50\) −1.94503 −0.275068
\(51\) −12.3625 −1.73110
\(52\) 5.82810 0.808211
\(53\) −11.7735 −1.61722 −0.808611 0.588344i \(-0.799780\pi\)
−0.808611 + 0.588344i \(0.799780\pi\)
\(54\) 5.24414 0.713637
\(55\) 6.09451 0.821783
\(56\) −0.421811 −0.0563669
\(57\) 17.2244 2.28143
\(58\) −12.3590 −1.62281
\(59\) 11.9506 1.55584 0.777919 0.628365i \(-0.216276\pi\)
0.777919 + 0.628365i \(0.216276\pi\)
\(60\) 3.89231 0.502496
\(61\) 7.00385 0.896751 0.448375 0.893845i \(-0.352003\pi\)
0.448375 + 0.893845i \(0.352003\pi\)
\(62\) 8.53715 1.08422
\(63\) −1.76484 −0.222348
\(64\) −6.18121 −0.772651
\(65\) 3.26846 0.405402
\(66\) −25.8755 −3.18505
\(67\) 8.63416 1.05483 0.527415 0.849608i \(-0.323161\pi\)
0.527415 + 0.849608i \(0.323161\pi\)
\(68\) −10.0987 −1.22465
\(69\) −7.56836 −0.911124
\(70\) 1.94503 0.232475
\(71\) 14.7022 1.74483 0.872414 0.488767i \(-0.162553\pi\)
0.872414 + 0.488767i \(0.162553\pi\)
\(72\) 0.744427 0.0877316
\(73\) 4.02309 0.470867 0.235434 0.971890i \(-0.424349\pi\)
0.235434 + 0.971890i \(0.424349\pi\)
\(74\) 10.2999 1.19734
\(75\) 2.18285 0.252054
\(76\) 14.0703 1.61398
\(77\) −6.09451 −0.694534
\(78\) −13.8769 −1.57125
\(79\) 4.96540 0.558651 0.279326 0.960196i \(-0.409889\pi\)
0.279326 + 0.960196i \(0.409889\pi\)
\(80\) −4.38670 −0.490448
\(81\) −11.1799 −1.24221
\(82\) 2.69848 0.297998
\(83\) 0.709802 0.0779109 0.0389554 0.999241i \(-0.487597\pi\)
0.0389554 + 0.999241i \(0.487597\pi\)
\(84\) −3.89231 −0.424686
\(85\) −5.66347 −0.614290
\(86\) −11.8058 −1.27306
\(87\) 13.8701 1.48703
\(88\) 2.57073 0.274041
\(89\) 14.7952 1.56829 0.784145 0.620577i \(-0.213102\pi\)
0.784145 + 0.620577i \(0.213102\pi\)
\(90\) −3.43266 −0.361834
\(91\) −3.26846 −0.342627
\(92\) −6.18247 −0.644567
\(93\) −9.58101 −0.993504
\(94\) 10.6830 1.10187
\(95\) 7.89079 0.809578
\(96\) 16.7831 1.71292
\(97\) −7.08866 −0.719744 −0.359872 0.933002i \(-0.617180\pi\)
−0.359872 + 0.933002i \(0.617180\pi\)
\(98\) −1.94503 −0.196477
\(99\) 10.7558 1.08100
\(100\) 1.78313 0.178313
\(101\) −5.16388 −0.513826 −0.256913 0.966435i \(-0.582705\pi\)
−0.256913 + 0.966435i \(0.582705\pi\)
\(102\) 24.0454 2.38085
\(103\) 6.94975 0.684779 0.342390 0.939558i \(-0.388764\pi\)
0.342390 + 0.939558i \(0.388764\pi\)
\(104\) 1.37867 0.135190
\(105\) −2.18285 −0.213024
\(106\) 22.8999 2.22423
\(107\) 18.0290 1.74293 0.871464 0.490460i \(-0.163171\pi\)
0.871464 + 0.490460i \(0.163171\pi\)
\(108\) −4.80765 −0.462616
\(109\) −13.0278 −1.24784 −0.623918 0.781490i \(-0.714460\pi\)
−0.623918 + 0.781490i \(0.714460\pi\)
\(110\) −11.8540 −1.13023
\(111\) −11.5593 −1.09716
\(112\) 4.38670 0.414504
\(113\) −6.68878 −0.629228 −0.314614 0.949220i \(-0.601875\pi\)
−0.314614 + 0.949220i \(0.601875\pi\)
\(114\) −33.5020 −3.13775
\(115\) −3.46719 −0.323317
\(116\) 11.3303 1.05199
\(117\) 5.76829 0.533279
\(118\) −23.2443 −2.13981
\(119\) 5.66347 0.519169
\(120\) 0.920750 0.0840526
\(121\) 26.1430 2.37664
\(122\) −13.6227 −1.23334
\(123\) −3.02843 −0.273065
\(124\) −7.82656 −0.702846
\(125\) 1.00000 0.0894427
\(126\) 3.43266 0.305805
\(127\) 15.7252 1.39539 0.697695 0.716395i \(-0.254209\pi\)
0.697695 + 0.716395i \(0.254209\pi\)
\(128\) −3.35465 −0.296512
\(129\) 13.2494 1.16654
\(130\) −6.35724 −0.557567
\(131\) 9.24702 0.807916 0.403958 0.914777i \(-0.367634\pi\)
0.403958 + 0.914777i \(0.367634\pi\)
\(132\) 23.7217 2.06471
\(133\) −7.89079 −0.684218
\(134\) −16.7937 −1.45075
\(135\) −2.69618 −0.232050
\(136\) −2.38891 −0.204848
\(137\) −6.76591 −0.578051 −0.289025 0.957321i \(-0.593331\pi\)
−0.289025 + 0.957321i \(0.593331\pi\)
\(138\) 14.7207 1.25311
\(139\) 3.08504 0.261670 0.130835 0.991404i \(-0.458234\pi\)
0.130835 + 0.991404i \(0.458234\pi\)
\(140\) −1.78313 −0.150702
\(141\) −11.9893 −1.00968
\(142\) −28.5962 −2.39974
\(143\) 19.9196 1.66576
\(144\) −7.74181 −0.645151
\(145\) 6.35413 0.527681
\(146\) −7.82503 −0.647604
\(147\) 2.18285 0.180038
\(148\) −9.44263 −0.776179
\(149\) −8.33305 −0.682670 −0.341335 0.939942i \(-0.610879\pi\)
−0.341335 + 0.939942i \(0.610879\pi\)
\(150\) −4.24571 −0.346660
\(151\) 2.27792 0.185374 0.0926872 0.995695i \(-0.470454\pi\)
0.0926872 + 0.995695i \(0.470454\pi\)
\(152\) 3.32842 0.269971
\(153\) −9.99509 −0.808055
\(154\) 11.8540 0.955222
\(155\) −4.38922 −0.352550
\(156\) 12.7219 1.01856
\(157\) −22.9011 −1.82771 −0.913853 0.406044i \(-0.866908\pi\)
−0.913853 + 0.406044i \(0.866908\pi\)
\(158\) −9.65785 −0.768337
\(159\) −25.6999 −2.03813
\(160\) 7.68864 0.607840
\(161\) 3.46719 0.273253
\(162\) 21.7451 1.70846
\(163\) −3.83777 −0.300597 −0.150299 0.988641i \(-0.548024\pi\)
−0.150299 + 0.988641i \(0.548024\pi\)
\(164\) −2.47388 −0.193177
\(165\) 13.3034 1.03567
\(166\) −1.38058 −0.107154
\(167\) −5.22631 −0.404424 −0.202212 0.979342i \(-0.564813\pi\)
−0.202212 + 0.979342i \(0.564813\pi\)
\(168\) −0.920750 −0.0710374
\(169\) −2.31719 −0.178245
\(170\) 11.0156 0.844858
\(171\) 13.9259 1.06494
\(172\) 10.8232 0.825260
\(173\) 4.83749 0.367788 0.183894 0.982946i \(-0.441130\pi\)
0.183894 + 0.982946i \(0.441130\pi\)
\(174\) −26.9777 −2.04518
\(175\) −1.00000 −0.0755929
\(176\) −26.7348 −2.01521
\(177\) 26.0864 1.96077
\(178\) −28.7771 −2.15694
\(179\) −6.15653 −0.460161 −0.230080 0.973172i \(-0.573899\pi\)
−0.230080 + 0.973172i \(0.573899\pi\)
\(180\) 3.14694 0.234559
\(181\) −14.0167 −1.04185 −0.520926 0.853602i \(-0.674413\pi\)
−0.520926 + 0.853602i \(0.674413\pi\)
\(182\) 6.35724 0.471230
\(183\) 15.2884 1.13015
\(184\) −1.46250 −0.107817
\(185\) −5.29552 −0.389335
\(186\) 18.6353 1.36641
\(187\) −34.5161 −2.52406
\(188\) −9.79385 −0.714290
\(189\) 2.69618 0.196118
\(190\) −15.3478 −1.11345
\(191\) 0.0806972 0.00583905 0.00291952 0.999996i \(-0.499071\pi\)
0.00291952 + 0.999996i \(0.499071\pi\)
\(192\) −13.4927 −0.973748
\(193\) −2.01661 −0.145159 −0.0725794 0.997363i \(-0.523123\pi\)
−0.0725794 + 0.997363i \(0.523123\pi\)
\(194\) 13.7876 0.989895
\(195\) 7.13455 0.510916
\(196\) 1.78313 0.127367
\(197\) −1.09518 −0.0780286 −0.0390143 0.999239i \(-0.512422\pi\)
−0.0390143 + 0.999239i \(0.512422\pi\)
\(198\) −20.9203 −1.48674
\(199\) −10.5428 −0.747358 −0.373679 0.927558i \(-0.621904\pi\)
−0.373679 + 0.927558i \(0.621904\pi\)
\(200\) 0.421811 0.0298265
\(201\) 18.8471 1.32937
\(202\) 10.0439 0.706686
\(203\) −6.35413 −0.445972
\(204\) −22.0440 −1.54339
\(205\) −1.38738 −0.0968985
\(206\) −13.5175 −0.941806
\(207\) −6.11903 −0.425302
\(208\) −14.3377 −0.994144
\(209\) 48.0905 3.32649
\(210\) 4.24571 0.292981
\(211\) −13.0622 −0.899237 −0.449618 0.893221i \(-0.648440\pi\)
−0.449618 + 0.893221i \(0.648440\pi\)
\(212\) −20.9938 −1.44186
\(213\) 32.0927 2.19895
\(214\) −35.0669 −2.39712
\(215\) 6.06975 0.413954
\(216\) −1.13728 −0.0773819
\(217\) 4.38922 0.297960
\(218\) 25.3394 1.71620
\(219\) 8.78181 0.593420
\(220\) 10.8673 0.732675
\(221\) −18.5108 −1.24517
\(222\) 22.4832 1.50898
\(223\) −2.70168 −0.180918 −0.0904589 0.995900i \(-0.528833\pi\)
−0.0904589 + 0.995900i \(0.528833\pi\)
\(224\) −7.68864 −0.513719
\(225\) 1.76484 0.117656
\(226\) 13.0099 0.865403
\(227\) −20.0411 −1.33018 −0.665088 0.746765i \(-0.731606\pi\)
−0.665088 + 0.746765i \(0.731606\pi\)
\(228\) 30.7134 2.03405
\(229\) −1.00000 −0.0660819
\(230\) 6.74379 0.444672
\(231\) −13.3034 −0.875299
\(232\) 2.68024 0.175966
\(233\) 24.1462 1.58187 0.790936 0.611899i \(-0.209594\pi\)
0.790936 + 0.611899i \(0.209594\pi\)
\(234\) −11.2195 −0.733441
\(235\) −5.49249 −0.358291
\(236\) 21.3095 1.38713
\(237\) 10.8387 0.704051
\(238\) −11.0156 −0.714036
\(239\) 8.77862 0.567841 0.283921 0.958848i \(-0.408365\pi\)
0.283921 + 0.958848i \(0.408365\pi\)
\(240\) −9.57551 −0.618097
\(241\) 5.29756 0.341246 0.170623 0.985336i \(-0.445422\pi\)
0.170623 + 0.985336i \(0.445422\pi\)
\(242\) −50.8489 −3.26869
\(243\) −16.3154 −1.04664
\(244\) 12.4888 0.799513
\(245\) 1.00000 0.0638877
\(246\) 5.89039 0.375557
\(247\) 25.7907 1.64102
\(248\) −1.85142 −0.117565
\(249\) 1.54939 0.0981887
\(250\) −1.94503 −0.123014
\(251\) −8.70336 −0.549351 −0.274676 0.961537i \(-0.588570\pi\)
−0.274676 + 0.961537i \(0.588570\pi\)
\(252\) −3.14694 −0.198238
\(253\) −21.1308 −1.32848
\(254\) −30.5860 −1.91914
\(255\) −12.3625 −0.774170
\(256\) 18.8873 1.18046
\(257\) 29.3730 1.83224 0.916120 0.400905i \(-0.131304\pi\)
0.916120 + 0.400905i \(0.131304\pi\)
\(258\) −25.7704 −1.60439
\(259\) 5.29552 0.329048
\(260\) 5.82810 0.361443
\(261\) 11.2140 0.694128
\(262\) −17.9857 −1.11116
\(263\) 14.1841 0.874631 0.437315 0.899308i \(-0.355929\pi\)
0.437315 + 0.899308i \(0.355929\pi\)
\(264\) 5.61152 0.345365
\(265\) −11.7735 −0.723243
\(266\) 15.3478 0.941034
\(267\) 32.2958 1.97647
\(268\) 15.3959 0.940452
\(269\) 13.3952 0.816719 0.408359 0.912821i \(-0.366101\pi\)
0.408359 + 0.912821i \(0.366101\pi\)
\(270\) 5.24414 0.319148
\(271\) 5.97771 0.363120 0.181560 0.983380i \(-0.441885\pi\)
0.181560 + 0.983380i \(0.441885\pi\)
\(272\) 24.8440 1.50639
\(273\) −7.13455 −0.431803
\(274\) 13.1599 0.795018
\(275\) 6.09451 0.367513
\(276\) −13.4954 −0.812328
\(277\) 8.87570 0.533289 0.266645 0.963795i \(-0.414085\pi\)
0.266645 + 0.963795i \(0.414085\pi\)
\(278\) −6.00050 −0.359886
\(279\) −7.74625 −0.463756
\(280\) −0.421811 −0.0252080
\(281\) −22.1107 −1.31902 −0.659508 0.751697i \(-0.729235\pi\)
−0.659508 + 0.751697i \(0.729235\pi\)
\(282\) 23.3195 1.38866
\(283\) 2.35505 0.139993 0.0699966 0.997547i \(-0.477701\pi\)
0.0699966 + 0.997547i \(0.477701\pi\)
\(284\) 26.2160 1.55563
\(285\) 17.2244 1.02029
\(286\) −38.7442 −2.29100
\(287\) 1.38738 0.0818942
\(288\) 13.5692 0.799572
\(289\) 15.0749 0.886758
\(290\) −12.3590 −0.725743
\(291\) −15.4735 −0.907071
\(292\) 7.17371 0.419810
\(293\) −3.36842 −0.196785 −0.0983926 0.995148i \(-0.531370\pi\)
−0.0983926 + 0.995148i \(0.531370\pi\)
\(294\) −4.24571 −0.247615
\(295\) 11.9506 0.695791
\(296\) −2.23371 −0.129832
\(297\) −16.4319 −0.953474
\(298\) 16.2080 0.938905
\(299\) −11.3324 −0.655368
\(300\) 3.89231 0.224723
\(301\) −6.06975 −0.349855
\(302\) −4.43061 −0.254953
\(303\) −11.2720 −0.647559
\(304\) −34.6145 −1.98528
\(305\) 7.00385 0.401039
\(306\) 19.4407 1.11135
\(307\) 17.5637 1.00241 0.501206 0.865328i \(-0.332890\pi\)
0.501206 + 0.865328i \(0.332890\pi\)
\(308\) −10.8673 −0.619223
\(309\) 15.1703 0.863006
\(310\) 8.53715 0.484878
\(311\) 1.86503 0.105756 0.0528780 0.998601i \(-0.483161\pi\)
0.0528780 + 0.998601i \(0.483161\pi\)
\(312\) 3.00943 0.170376
\(313\) 25.7982 1.45820 0.729099 0.684408i \(-0.239939\pi\)
0.729099 + 0.684408i \(0.239939\pi\)
\(314\) 44.5433 2.51372
\(315\) −1.76484 −0.0994372
\(316\) 8.85398 0.498075
\(317\) 7.33811 0.412149 0.206075 0.978536i \(-0.433931\pi\)
0.206075 + 0.978536i \(0.433931\pi\)
\(318\) 49.9870 2.80313
\(319\) 38.7253 2.16820
\(320\) −6.18121 −0.345540
\(321\) 39.3546 2.19656
\(322\) −6.74379 −0.375817
\(323\) −44.6892 −2.48658
\(324\) −19.9352 −1.10751
\(325\) 3.26846 0.181301
\(326\) 7.46457 0.413424
\(327\) −28.4377 −1.57261
\(328\) −0.585210 −0.0323128
\(329\) 5.49249 0.302811
\(330\) −25.8755 −1.42440
\(331\) 11.3771 0.625343 0.312672 0.949861i \(-0.398776\pi\)
0.312672 + 0.949861i \(0.398776\pi\)
\(332\) 1.26567 0.0694627
\(333\) −9.34573 −0.512143
\(334\) 10.1653 0.556222
\(335\) 8.63416 0.471735
\(336\) 9.57551 0.522387
\(337\) −31.0498 −1.69139 −0.845696 0.533664i \(-0.820815\pi\)
−0.845696 + 0.533664i \(0.820815\pi\)
\(338\) 4.50700 0.245149
\(339\) −14.6006 −0.792996
\(340\) −10.0987 −0.547680
\(341\) −26.7501 −1.44860
\(342\) −27.0864 −1.46466
\(343\) −1.00000 −0.0539949
\(344\) 2.56029 0.138041
\(345\) −7.56836 −0.407467
\(346\) −9.40905 −0.505834
\(347\) 9.88075 0.530426 0.265213 0.964190i \(-0.414558\pi\)
0.265213 + 0.964190i \(0.414558\pi\)
\(348\) 24.7323 1.32579
\(349\) −17.6303 −0.943726 −0.471863 0.881672i \(-0.656418\pi\)
−0.471863 + 0.881672i \(0.656418\pi\)
\(350\) 1.94503 0.103966
\(351\) −8.81234 −0.470368
\(352\) 46.8584 2.49756
\(353\) −21.1391 −1.12512 −0.562560 0.826757i \(-0.690183\pi\)
−0.562560 + 0.826757i \(0.690183\pi\)
\(354\) −50.7388 −2.69674
\(355\) 14.7022 0.780311
\(356\) 26.3819 1.39824
\(357\) 12.3625 0.654293
\(358\) 11.9746 0.632879
\(359\) −27.9344 −1.47432 −0.737161 0.675717i \(-0.763834\pi\)
−0.737161 + 0.675717i \(0.763834\pi\)
\(360\) 0.744427 0.0392348
\(361\) 43.2645 2.27708
\(362\) 27.2629 1.43290
\(363\) 57.0663 2.99520
\(364\) −5.82810 −0.305475
\(365\) 4.02309 0.210578
\(366\) −29.7363 −1.55434
\(367\) 2.46148 0.128488 0.0642440 0.997934i \(-0.479536\pi\)
0.0642440 + 0.997934i \(0.479536\pi\)
\(368\) 15.2095 0.792852
\(369\) −2.44849 −0.127463
\(370\) 10.2999 0.535468
\(371\) 11.7735 0.611252
\(372\) −17.0842 −0.885775
\(373\) 31.3663 1.62409 0.812044 0.583597i \(-0.198355\pi\)
0.812044 + 0.583597i \(0.198355\pi\)
\(374\) 67.1347 3.47145
\(375\) 2.18285 0.112722
\(376\) −2.31679 −0.119479
\(377\) 20.7682 1.06962
\(378\) −5.24414 −0.269730
\(379\) 34.5706 1.77577 0.887886 0.460063i \(-0.152173\pi\)
0.887886 + 0.460063i \(0.152173\pi\)
\(380\) 14.0703 0.721793
\(381\) 34.3259 1.75857
\(382\) −0.156958 −0.00803069
\(383\) 19.7372 1.00852 0.504261 0.863551i \(-0.331765\pi\)
0.504261 + 0.863551i \(0.331765\pi\)
\(384\) −7.32270 −0.373685
\(385\) −6.09451 −0.310605
\(386\) 3.92236 0.199643
\(387\) 10.7121 0.544528
\(388\) −12.6400 −0.641700
\(389\) −28.9690 −1.46879 −0.734394 0.678723i \(-0.762533\pi\)
−0.734394 + 0.678723i \(0.762533\pi\)
\(390\) −13.8769 −0.702684
\(391\) 19.6363 0.993052
\(392\) 0.421811 0.0213047
\(393\) 20.1849 1.01819
\(394\) 2.13016 0.107316
\(395\) 4.96540 0.249836
\(396\) 19.1790 0.963783
\(397\) 26.0902 1.30943 0.654716 0.755875i \(-0.272788\pi\)
0.654716 + 0.755875i \(0.272788\pi\)
\(398\) 20.5060 1.02787
\(399\) −17.2244 −0.862299
\(400\) −4.38670 −0.219335
\(401\) −21.7214 −1.08472 −0.542359 0.840147i \(-0.682469\pi\)
−0.542359 + 0.840147i \(0.682469\pi\)
\(402\) −36.6581 −1.82834
\(403\) −14.3460 −0.714624
\(404\) −9.20790 −0.458110
\(405\) −11.1799 −0.555532
\(406\) 12.3590 0.613364
\(407\) −32.2736 −1.59974
\(408\) −5.21464 −0.258163
\(409\) 17.8012 0.880212 0.440106 0.897946i \(-0.354941\pi\)
0.440106 + 0.897946i \(0.354941\pi\)
\(410\) 2.69848 0.133269
\(411\) −14.7690 −0.728500
\(412\) 12.3923 0.610527
\(413\) −11.9506 −0.588051
\(414\) 11.9017 0.584935
\(415\) 0.709802 0.0348428
\(416\) 25.1300 1.23210
\(417\) 6.73419 0.329775
\(418\) −93.5373 −4.57506
\(419\) 8.72315 0.426154 0.213077 0.977035i \(-0.431652\pi\)
0.213077 + 0.977035i \(0.431652\pi\)
\(420\) −3.89231 −0.189925
\(421\) 4.72652 0.230356 0.115178 0.993345i \(-0.463256\pi\)
0.115178 + 0.993345i \(0.463256\pi\)
\(422\) 25.4063 1.23676
\(423\) −9.69335 −0.471307
\(424\) −4.96621 −0.241181
\(425\) −5.66347 −0.274719
\(426\) −62.4212 −3.02431
\(427\) −7.00385 −0.338940
\(428\) 32.1481 1.55394
\(429\) 43.4816 2.09931
\(430\) −11.8058 −0.569328
\(431\) −24.2116 −1.16623 −0.583116 0.812389i \(-0.698167\pi\)
−0.583116 + 0.812389i \(0.698167\pi\)
\(432\) 11.8273 0.569043
\(433\) −17.8543 −0.858022 −0.429011 0.903299i \(-0.641138\pi\)
−0.429011 + 0.903299i \(0.641138\pi\)
\(434\) −8.53715 −0.409796
\(435\) 13.8701 0.665021
\(436\) −23.2303 −1.11253
\(437\) −27.3589 −1.30875
\(438\) −17.0809 −0.816155
\(439\) 27.3246 1.30413 0.652065 0.758163i \(-0.273903\pi\)
0.652065 + 0.758163i \(0.273903\pi\)
\(440\) 2.57073 0.122555
\(441\) 1.76484 0.0840398
\(442\) 36.0040 1.71254
\(443\) 22.0902 1.04954 0.524768 0.851245i \(-0.324152\pi\)
0.524768 + 0.851245i \(0.324152\pi\)
\(444\) −20.6118 −0.978195
\(445\) 14.7952 0.701361
\(446\) 5.25484 0.248824
\(447\) −18.1898 −0.860348
\(448\) 6.18121 0.292035
\(449\) −33.0520 −1.55982 −0.779909 0.625893i \(-0.784735\pi\)
−0.779909 + 0.625893i \(0.784735\pi\)
\(450\) −3.43266 −0.161817
\(451\) −8.45537 −0.398148
\(452\) −11.9270 −0.560998
\(453\) 4.97235 0.233622
\(454\) 38.9805 1.82945
\(455\) −3.26846 −0.153228
\(456\) 7.26544 0.340236
\(457\) 13.5275 0.632789 0.316394 0.948628i \(-0.397528\pi\)
0.316394 + 0.948628i \(0.397528\pi\)
\(458\) 1.94503 0.0908852
\(459\) 15.2697 0.712730
\(460\) −6.18247 −0.288259
\(461\) 27.8589 1.29752 0.648760 0.760993i \(-0.275288\pi\)
0.648760 + 0.760993i \(0.275288\pi\)
\(462\) 25.8755 1.20384
\(463\) −5.80127 −0.269608 −0.134804 0.990872i \(-0.543040\pi\)
−0.134804 + 0.990872i \(0.543040\pi\)
\(464\) −27.8737 −1.29400
\(465\) −9.58101 −0.444308
\(466\) −46.9651 −2.17562
\(467\) −27.0355 −1.25106 −0.625528 0.780202i \(-0.715116\pi\)
−0.625528 + 0.780202i \(0.715116\pi\)
\(468\) 10.2856 0.475453
\(469\) −8.63416 −0.398689
\(470\) 10.6830 0.492772
\(471\) −49.9897 −2.30340
\(472\) 5.04090 0.232026
\(473\) 36.9922 1.70090
\(474\) −21.0816 −0.968311
\(475\) 7.89079 0.362054
\(476\) 10.0987 0.462874
\(477\) −20.7784 −0.951377
\(478\) −17.0747 −0.780976
\(479\) −11.8206 −0.540097 −0.270048 0.962847i \(-0.587040\pi\)
−0.270048 + 0.962847i \(0.587040\pi\)
\(480\) 16.7831 0.766042
\(481\) −17.3082 −0.789185
\(482\) −10.3039 −0.469330
\(483\) 7.56836 0.344372
\(484\) 46.6165 2.11893
\(485\) −7.08866 −0.321879
\(486\) 31.7340 1.43948
\(487\) −39.4987 −1.78986 −0.894929 0.446208i \(-0.852774\pi\)
−0.894929 + 0.446208i \(0.852774\pi\)
\(488\) 2.95430 0.133735
\(489\) −8.37728 −0.378833
\(490\) −1.94503 −0.0878674
\(491\) −17.4209 −0.786192 −0.393096 0.919497i \(-0.628596\pi\)
−0.393096 + 0.919497i \(0.628596\pi\)
\(492\) −5.40010 −0.243455
\(493\) −35.9864 −1.62075
\(494\) −50.1636 −2.25697
\(495\) 10.7558 0.483438
\(496\) 19.2542 0.864539
\(497\) −14.7022 −0.659483
\(498\) −3.01361 −0.135043
\(499\) 32.2937 1.44567 0.722833 0.691022i \(-0.242839\pi\)
0.722833 + 0.691022i \(0.242839\pi\)
\(500\) 1.78313 0.0797442
\(501\) −11.4083 −0.509684
\(502\) 16.9283 0.755546
\(503\) −9.08506 −0.405083 −0.202541 0.979274i \(-0.564920\pi\)
−0.202541 + 0.979274i \(0.564920\pi\)
\(504\) −0.744427 −0.0331594
\(505\) −5.16388 −0.229790
\(506\) 41.1001 1.82712
\(507\) −5.05808 −0.224637
\(508\) 28.0402 1.24408
\(509\) −36.6965 −1.62654 −0.813272 0.581884i \(-0.802316\pi\)
−0.813272 + 0.581884i \(0.802316\pi\)
\(510\) 24.0454 1.06475
\(511\) −4.02309 −0.177971
\(512\) −30.0270 −1.32702
\(513\) −21.2750 −0.939313
\(514\) −57.1314 −2.51996
\(515\) 6.94975 0.306243
\(516\) 23.6254 1.04005
\(517\) −33.4740 −1.47219
\(518\) −10.2999 −0.452553
\(519\) 10.5595 0.463511
\(520\) 1.37867 0.0604587
\(521\) −9.09670 −0.398534 −0.199267 0.979945i \(-0.563856\pi\)
−0.199267 + 0.979945i \(0.563856\pi\)
\(522\) −21.8115 −0.954664
\(523\) −9.79116 −0.428138 −0.214069 0.976819i \(-0.568672\pi\)
−0.214069 + 0.976819i \(0.568672\pi\)
\(524\) 16.4887 0.720311
\(525\) −2.18285 −0.0952674
\(526\) −27.5885 −1.20292
\(527\) 24.8582 1.08284
\(528\) −58.3580 −2.53971
\(529\) −10.9786 −0.477329
\(530\) 22.8999 0.994707
\(531\) 21.0909 0.915266
\(532\) −14.0703 −0.610026
\(533\) −4.53458 −0.196414
\(534\) −62.8162 −2.71832
\(535\) 18.0290 0.779461
\(536\) 3.64198 0.157310
\(537\) −13.4388 −0.579926
\(538\) −26.0540 −1.12327
\(539\) 6.09451 0.262509
\(540\) −4.80765 −0.206888
\(541\) 24.4772 1.05236 0.526178 0.850375i \(-0.323625\pi\)
0.526178 + 0.850375i \(0.323625\pi\)
\(542\) −11.6268 −0.499415
\(543\) −30.5964 −1.31302
\(544\) −43.5444 −1.86695
\(545\) −13.0278 −0.558049
\(546\) 13.8769 0.593877
\(547\) 2.77529 0.118663 0.0593315 0.998238i \(-0.481103\pi\)
0.0593315 + 0.998238i \(0.481103\pi\)
\(548\) −12.0645 −0.515371
\(549\) 12.3606 0.527539
\(550\) −11.8540 −0.505456
\(551\) 50.1390 2.13600
\(552\) −3.19242 −0.135878
\(553\) −4.96540 −0.211150
\(554\) −17.2635 −0.733456
\(555\) −11.5593 −0.490666
\(556\) 5.50104 0.233296
\(557\) −20.8517 −0.883517 −0.441758 0.897134i \(-0.645645\pi\)
−0.441758 + 0.897134i \(0.645645\pi\)
\(558\) 15.0667 0.637823
\(559\) 19.8387 0.839089
\(560\) 4.38670 0.185372
\(561\) −75.3434 −3.18100
\(562\) 43.0060 1.81410
\(563\) 20.5705 0.866943 0.433472 0.901167i \(-0.357288\pi\)
0.433472 + 0.901167i \(0.357288\pi\)
\(564\) −21.3785 −0.900197
\(565\) −6.68878 −0.281399
\(566\) −4.58064 −0.192539
\(567\) 11.1799 0.469510
\(568\) 6.20154 0.260211
\(569\) −25.6461 −1.07514 −0.537571 0.843219i \(-0.680658\pi\)
−0.537571 + 0.843219i \(0.680658\pi\)
\(570\) −33.5020 −1.40324
\(571\) 11.2058 0.468949 0.234474 0.972122i \(-0.424663\pi\)
0.234474 + 0.972122i \(0.424663\pi\)
\(572\) 35.5194 1.48514
\(573\) 0.176150 0.00735877
\(574\) −2.69848 −0.112633
\(575\) −3.46719 −0.144592
\(576\) −10.9088 −0.454534
\(577\) 43.5874 1.81457 0.907283 0.420521i \(-0.138153\pi\)
0.907283 + 0.420521i \(0.138153\pi\)
\(578\) −29.3211 −1.21960
\(579\) −4.40196 −0.182939
\(580\) 11.3303 0.470463
\(581\) −0.709802 −0.0294475
\(582\) 30.0963 1.24753
\(583\) −71.7540 −2.97175
\(584\) 1.69698 0.0702217
\(585\) 5.76829 0.238489
\(586\) 6.55167 0.270647
\(587\) 5.94833 0.245514 0.122757 0.992437i \(-0.460826\pi\)
0.122757 + 0.992437i \(0.460826\pi\)
\(588\) 3.89231 0.160516
\(589\) −34.6344 −1.42708
\(590\) −23.2443 −0.956952
\(591\) −2.39062 −0.0983370
\(592\) 23.2299 0.954742
\(593\) 4.79044 0.196720 0.0983600 0.995151i \(-0.468640\pi\)
0.0983600 + 0.995151i \(0.468640\pi\)
\(594\) 31.9605 1.31135
\(595\) 5.66347 0.232180
\(596\) −14.8589 −0.608646
\(597\) −23.0133 −0.941872
\(598\) 22.0418 0.901355
\(599\) 26.2777 1.07368 0.536840 0.843684i \(-0.319618\pi\)
0.536840 + 0.843684i \(0.319618\pi\)
\(600\) 0.920750 0.0375895
\(601\) 46.1625 1.88301 0.941505 0.337000i \(-0.109412\pi\)
0.941505 + 0.337000i \(0.109412\pi\)
\(602\) 11.8058 0.481170
\(603\) 15.2379 0.620535
\(604\) 4.06183 0.165274
\(605\) 26.1430 1.06287
\(606\) 21.9243 0.890615
\(607\) −21.0505 −0.854415 −0.427207 0.904154i \(-0.640503\pi\)
−0.427207 + 0.904154i \(0.640503\pi\)
\(608\) 60.6694 2.46047
\(609\) −13.8701 −0.562045
\(610\) −13.6227 −0.551566
\(611\) −17.9520 −0.726259
\(612\) −17.8226 −0.720435
\(613\) −44.7882 −1.80898 −0.904489 0.426496i \(-0.859748\pi\)
−0.904489 + 0.426496i \(0.859748\pi\)
\(614\) −34.1618 −1.37866
\(615\) −3.02843 −0.122118
\(616\) −2.57073 −0.103578
\(617\) 18.4952 0.744589 0.372294 0.928115i \(-0.378571\pi\)
0.372294 + 0.928115i \(0.378571\pi\)
\(618\) −29.5066 −1.18693
\(619\) 17.7844 0.714814 0.357407 0.933949i \(-0.383661\pi\)
0.357407 + 0.933949i \(0.383661\pi\)
\(620\) −7.82656 −0.314322
\(621\) 9.34817 0.375129
\(622\) −3.62753 −0.145451
\(623\) −14.7952 −0.592758
\(624\) −31.2971 −1.25289
\(625\) 1.00000 0.0400000
\(626\) −50.1781 −2.00552
\(627\) 104.974 4.19227
\(628\) −40.8357 −1.62952
\(629\) 29.9910 1.19582
\(630\) 3.43266 0.136760
\(631\) 40.9276 1.62930 0.814651 0.579952i \(-0.196929\pi\)
0.814651 + 0.579952i \(0.196929\pi\)
\(632\) 2.09446 0.0833132
\(633\) −28.5128 −1.13328
\(634\) −14.2728 −0.566846
\(635\) 15.7252 0.624037
\(636\) −45.8263 −1.81713
\(637\) 3.26846 0.129501
\(638\) −75.3217 −2.98202
\(639\) 25.9470 1.02645
\(640\) −3.35465 −0.132604
\(641\) −47.7144 −1.88460 −0.942302 0.334763i \(-0.891344\pi\)
−0.942302 + 0.334763i \(0.891344\pi\)
\(642\) −76.5457 −3.02102
\(643\) 31.0126 1.22302 0.611508 0.791238i \(-0.290563\pi\)
0.611508 + 0.791238i \(0.290563\pi\)
\(644\) 6.18247 0.243623
\(645\) 13.2494 0.521693
\(646\) 86.9218 3.41989
\(647\) −2.51737 −0.0989681 −0.0494840 0.998775i \(-0.515758\pi\)
−0.0494840 + 0.998775i \(0.515758\pi\)
\(648\) −4.71579 −0.185254
\(649\) 72.8331 2.85895
\(650\) −6.35724 −0.249351
\(651\) 9.58101 0.375509
\(652\) −6.84326 −0.268003
\(653\) −5.10081 −0.199610 −0.0998050 0.995007i \(-0.531822\pi\)
−0.0998050 + 0.995007i \(0.531822\pi\)
\(654\) 55.3122 2.16288
\(655\) 9.24702 0.361311
\(656\) 6.08600 0.237619
\(657\) 7.10010 0.277001
\(658\) −10.6830 −0.416469
\(659\) 23.5714 0.918213 0.459106 0.888381i \(-0.348170\pi\)
0.459106 + 0.888381i \(0.348170\pi\)
\(660\) 23.7217 0.923367
\(661\) −23.1544 −0.900601 −0.450301 0.892877i \(-0.648683\pi\)
−0.450301 + 0.892877i \(0.648683\pi\)
\(662\) −22.1288 −0.860061
\(663\) −40.4063 −1.56925
\(664\) 0.299402 0.0116191
\(665\) −7.89079 −0.305992
\(666\) 18.1777 0.704372
\(667\) −22.0310 −0.853043
\(668\) −9.31922 −0.360571
\(669\) −5.89736 −0.228005
\(670\) −16.7937 −0.648797
\(671\) 42.6850 1.64784
\(672\) −16.7831 −0.647424
\(673\) −21.3514 −0.823035 −0.411517 0.911402i \(-0.635001\pi\)
−0.411517 + 0.911402i \(0.635001\pi\)
\(674\) 60.3928 2.32624
\(675\) −2.69618 −0.103776
\(676\) −4.13186 −0.158918
\(677\) 4.28442 0.164664 0.0823318 0.996605i \(-0.473763\pi\)
0.0823318 + 0.996605i \(0.473763\pi\)
\(678\) 28.3986 1.09064
\(679\) 7.08866 0.272038
\(680\) −2.38891 −0.0916107
\(681\) −43.7468 −1.67638
\(682\) 52.0297 1.99232
\(683\) 14.4763 0.553921 0.276960 0.960881i \(-0.410673\pi\)
0.276960 + 0.960881i \(0.410673\pi\)
\(684\) 24.8318 0.949468
\(685\) −6.76591 −0.258512
\(686\) 1.94503 0.0742615
\(687\) −2.18285 −0.0832809
\(688\) −26.6262 −1.01511
\(689\) −38.4813 −1.46602
\(690\) 14.7207 0.560407
\(691\) −17.0017 −0.646775 −0.323387 0.946267i \(-0.604822\pi\)
−0.323387 + 0.946267i \(0.604822\pi\)
\(692\) 8.62589 0.327907
\(693\) −10.7558 −0.408579
\(694\) −19.2183 −0.729518
\(695\) 3.08504 0.117022
\(696\) 5.85056 0.221765
\(697\) 7.85736 0.297619
\(698\) 34.2913 1.29795
\(699\) 52.7076 1.99359
\(700\) −1.78313 −0.0673961
\(701\) −15.8705 −0.599421 −0.299710 0.954030i \(-0.596890\pi\)
−0.299710 + 0.954030i \(0.596890\pi\)
\(702\) 17.1403 0.646917
\(703\) −41.7858 −1.57598
\(704\) −37.6714 −1.41979
\(705\) −11.9893 −0.451543
\(706\) 41.1161 1.54743
\(707\) 5.16388 0.194208
\(708\) 46.5155 1.74816
\(709\) −4.60502 −0.172945 −0.0864726 0.996254i \(-0.527559\pi\)
−0.0864726 + 0.996254i \(0.527559\pi\)
\(710\) −28.5962 −1.07320
\(711\) 8.76312 0.328643
\(712\) 6.24079 0.233883
\(713\) 15.2183 0.569928
\(714\) −24.0454 −0.899877
\(715\) 19.9196 0.744952
\(716\) −10.9779 −0.410264
\(717\) 19.1624 0.715633
\(718\) 54.3332 2.02770
\(719\) 43.7672 1.63224 0.816121 0.577881i \(-0.196120\pi\)
0.816121 + 0.577881i \(0.196120\pi\)
\(720\) −7.74181 −0.288520
\(721\) −6.94975 −0.258822
\(722\) −84.1507 −3.13176
\(723\) 11.5638 0.430062
\(724\) −24.9936 −0.928881
\(725\) 6.35413 0.235986
\(726\) −110.996 −4.11943
\(727\) 37.0322 1.37345 0.686725 0.726918i \(-0.259048\pi\)
0.686725 + 0.726918i \(0.259048\pi\)
\(728\) −1.37867 −0.0510969
\(729\) −2.07456 −0.0768355
\(730\) −7.82503 −0.289617
\(731\) −34.3759 −1.27144
\(732\) 27.2612 1.00760
\(733\) −11.5213 −0.425548 −0.212774 0.977101i \(-0.568250\pi\)
−0.212774 + 0.977101i \(0.568250\pi\)
\(734\) −4.78764 −0.176715
\(735\) 2.18285 0.0805157
\(736\) −26.6580 −0.982626
\(737\) 52.6210 1.93832
\(738\) 4.76238 0.175306
\(739\) −3.09239 −0.113755 −0.0568777 0.998381i \(-0.518115\pi\)
−0.0568777 + 0.998381i \(0.518115\pi\)
\(740\) −9.44263 −0.347118
\(741\) 56.2972 2.06813
\(742\) −22.8999 −0.840681
\(743\) −40.5707 −1.48839 −0.744197 0.667960i \(-0.767168\pi\)
−0.744197 + 0.667960i \(0.767168\pi\)
\(744\) −4.04137 −0.148164
\(745\) −8.33305 −0.305299
\(746\) −61.0084 −2.23368
\(747\) 1.25268 0.0458333
\(748\) −61.5468 −2.25037
\(749\) −18.0290 −0.658765
\(750\) −4.24571 −0.155031
\(751\) 19.8437 0.724106 0.362053 0.932157i \(-0.382076\pi\)
0.362053 + 0.932157i \(0.382076\pi\)
\(752\) 24.0939 0.878615
\(753\) −18.9981 −0.692330
\(754\) −40.3947 −1.47109
\(755\) 2.27792 0.0829019
\(756\) 4.80765 0.174852
\(757\) 5.05572 0.183753 0.0918766 0.995770i \(-0.470713\pi\)
0.0918766 + 0.995770i \(0.470713\pi\)
\(758\) −67.2408 −2.44230
\(759\) −46.1254 −1.67425
\(760\) 3.32842 0.120734
\(761\) 3.88152 0.140705 0.0703525 0.997522i \(-0.477588\pi\)
0.0703525 + 0.997522i \(0.477588\pi\)
\(762\) −66.7648 −2.41863
\(763\) 13.0278 0.471638
\(764\) 0.143894 0.00520590
\(765\) −9.99509 −0.361373
\(766\) −38.3893 −1.38706
\(767\) 39.0601 1.41038
\(768\) 41.2282 1.48769
\(769\) −23.3927 −0.843562 −0.421781 0.906698i \(-0.638595\pi\)
−0.421781 + 0.906698i \(0.638595\pi\)
\(770\) 11.8540 0.427188
\(771\) 64.1169 2.30912
\(772\) −3.59589 −0.129419
\(773\) −24.6255 −0.885719 −0.442859 0.896591i \(-0.646036\pi\)
−0.442859 + 0.896591i \(0.646036\pi\)
\(774\) −20.8354 −0.748912
\(775\) −4.38922 −0.157665
\(776\) −2.99007 −0.107337
\(777\) 11.5593 0.414689
\(778\) 56.3456 2.02009
\(779\) −10.9475 −0.392234
\(780\) 12.7219 0.455516
\(781\) 89.6026 3.20623
\(782\) −38.1932 −1.36579
\(783\) −17.1319 −0.612243
\(784\) −4.38670 −0.156668
\(785\) −22.9011 −0.817375
\(786\) −39.2601 −1.40036
\(787\) 35.5427 1.26696 0.633481 0.773759i \(-0.281626\pi\)
0.633481 + 0.773759i \(0.281626\pi\)
\(788\) −1.95286 −0.0695677
\(789\) 30.9618 1.10227
\(790\) −9.65785 −0.343611
\(791\) 6.68878 0.237826
\(792\) 4.53692 0.161212
\(793\) 22.8918 0.812911
\(794\) −50.7463 −1.80092
\(795\) −25.6999 −0.911481
\(796\) −18.7992 −0.666319
\(797\) −7.86184 −0.278481 −0.139240 0.990259i \(-0.544466\pi\)
−0.139240 + 0.990259i \(0.544466\pi\)
\(798\) 33.5020 1.18596
\(799\) 31.1066 1.10047
\(800\) 7.68864 0.271834
\(801\) 26.1111 0.922592
\(802\) 42.2488 1.49186
\(803\) 24.5188 0.865248
\(804\) 33.6069 1.18522
\(805\) 3.46719 0.122202
\(806\) 27.9033 0.982852
\(807\) 29.2397 1.02929
\(808\) −2.17818 −0.0766282
\(809\) 4.72254 0.166036 0.0830178 0.996548i \(-0.473544\pi\)
0.0830178 + 0.996548i \(0.473544\pi\)
\(810\) 21.7451 0.764046
\(811\) 11.1467 0.391412 0.195706 0.980663i \(-0.437300\pi\)
0.195706 + 0.980663i \(0.437300\pi\)
\(812\) −11.3303 −0.397614
\(813\) 13.0485 0.457629
\(814\) 62.7731 2.20019
\(815\) −3.83777 −0.134431
\(816\) 54.2306 1.89845
\(817\) 47.8951 1.67564
\(818\) −34.6238 −1.21059
\(819\) −5.76829 −0.201560
\(820\) −2.47388 −0.0863915
\(821\) 15.6043 0.544593 0.272297 0.962213i \(-0.412217\pi\)
0.272297 + 0.962213i \(0.412217\pi\)
\(822\) 28.7261 1.00194
\(823\) −25.9271 −0.903763 −0.451881 0.892078i \(-0.649247\pi\)
−0.451881 + 0.892078i \(0.649247\pi\)
\(824\) 2.93148 0.102123
\(825\) 13.3034 0.463165
\(826\) 23.2443 0.808772
\(827\) −4.83166 −0.168013 −0.0840066 0.996465i \(-0.526772\pi\)
−0.0840066 + 0.996465i \(0.526772\pi\)
\(828\) −10.9110 −0.379185
\(829\) 28.6631 0.995509 0.497754 0.867318i \(-0.334158\pi\)
0.497754 + 0.867318i \(0.334158\pi\)
\(830\) −1.38058 −0.0479208
\(831\) 19.3743 0.672088
\(832\) −20.2030 −0.700413
\(833\) −5.66347 −0.196228
\(834\) −13.0982 −0.453553
\(835\) −5.22631 −0.180864
\(836\) 85.7517 2.96579
\(837\) 11.8341 0.409047
\(838\) −16.9668 −0.586107
\(839\) 19.8815 0.686385 0.343193 0.939265i \(-0.388492\pi\)
0.343193 + 0.939265i \(0.388492\pi\)
\(840\) −0.920750 −0.0317689
\(841\) 11.3749 0.392238
\(842\) −9.19321 −0.316819
\(843\) −48.2644 −1.66232
\(844\) −23.2916 −0.801730
\(845\) −2.31719 −0.0797138
\(846\) 18.8538 0.648208
\(847\) −26.1430 −0.898285
\(848\) 51.6470 1.77357
\(849\) 5.14072 0.176429
\(850\) 11.0156 0.377832
\(851\) 18.3606 0.629393
\(852\) 57.2255 1.96051
\(853\) 3.69666 0.126571 0.0632856 0.997995i \(-0.479842\pi\)
0.0632856 + 0.997995i \(0.479842\pi\)
\(854\) 13.6227 0.466158
\(855\) 13.9259 0.476257
\(856\) 7.60482 0.259927
\(857\) −12.2556 −0.418642 −0.209321 0.977847i \(-0.567125\pi\)
−0.209321 + 0.977847i \(0.567125\pi\)
\(858\) −84.5729 −2.88727
\(859\) −28.2914 −0.965291 −0.482645 0.875816i \(-0.660324\pi\)
−0.482645 + 0.875816i \(0.660324\pi\)
\(860\) 10.8232 0.369067
\(861\) 3.02843 0.103209
\(862\) 47.0923 1.60397
\(863\) 20.0807 0.683555 0.341777 0.939781i \(-0.388971\pi\)
0.341777 + 0.939781i \(0.388971\pi\)
\(864\) −20.7299 −0.705247
\(865\) 4.83749 0.164480
\(866\) 34.7271 1.18007
\(867\) 32.9062 1.11755
\(868\) 7.82656 0.265651
\(869\) 30.2617 1.02656
\(870\) −26.9777 −0.914631
\(871\) 28.2204 0.956211
\(872\) −5.49527 −0.186093
\(873\) −12.5103 −0.423410
\(874\) 53.2138 1.79998
\(875\) −1.00000 −0.0338062
\(876\) 15.6591 0.529073
\(877\) 39.5867 1.33675 0.668374 0.743825i \(-0.266991\pi\)
0.668374 + 0.743825i \(0.266991\pi\)
\(878\) −53.1470 −1.79363
\(879\) −7.35276 −0.248002
\(880\) −26.7348 −0.901230
\(881\) 18.1508 0.611516 0.305758 0.952109i \(-0.401090\pi\)
0.305758 + 0.952109i \(0.401090\pi\)
\(882\) −3.43266 −0.115584
\(883\) −34.6179 −1.16499 −0.582493 0.812836i \(-0.697923\pi\)
−0.582493 + 0.812836i \(0.697923\pi\)
\(884\) −33.0072 −1.11015
\(885\) 26.0864 0.876885
\(886\) −42.9661 −1.44347
\(887\) 39.6897 1.33265 0.666325 0.745662i \(-0.267866\pi\)
0.666325 + 0.745662i \(0.267866\pi\)
\(888\) −4.87585 −0.163623
\(889\) −15.7252 −0.527408
\(890\) −28.7771 −0.964612
\(891\) −68.1358 −2.28263
\(892\) −4.81746 −0.161300
\(893\) −43.3401 −1.45032
\(894\) 35.3797 1.18327
\(895\) −6.15653 −0.205790
\(896\) 3.35465 0.112071
\(897\) −24.7369 −0.825940
\(898\) 64.2870 2.14528
\(899\) −27.8896 −0.930172
\(900\) 3.14694 0.104898
\(901\) 66.6791 2.22140
\(902\) 16.4459 0.547590
\(903\) −13.2494 −0.440911
\(904\) −2.82140 −0.0938384
\(905\) −14.0167 −0.465931
\(906\) −9.67137 −0.321310
\(907\) −50.2224 −1.66761 −0.833804 0.552060i \(-0.813842\pi\)
−0.833804 + 0.552060i \(0.813842\pi\)
\(908\) −35.7360 −1.18594
\(909\) −9.11341 −0.302273
\(910\) 6.35724 0.210740
\(911\) −45.3281 −1.50179 −0.750894 0.660423i \(-0.770377\pi\)
−0.750894 + 0.660423i \(0.770377\pi\)
\(912\) −75.5583 −2.50199
\(913\) 4.32589 0.143166
\(914\) −26.3113 −0.870301
\(915\) 15.2884 0.505417
\(916\) −1.78313 −0.0589164
\(917\) −9.24702 −0.305364
\(918\) −29.7000 −0.980247
\(919\) −41.1704 −1.35809 −0.679043 0.734099i \(-0.737605\pi\)
−0.679043 + 0.734099i \(0.737605\pi\)
\(920\) −1.46250 −0.0482172
\(921\) 38.3389 1.26331
\(922\) −54.1864 −1.78454
\(923\) 48.0535 1.58170
\(924\) −23.7217 −0.780388
\(925\) −5.29552 −0.174116
\(926\) 11.2836 0.370803
\(927\) 12.2652 0.402841
\(928\) 48.8546 1.60373
\(929\) −58.9729 −1.93484 −0.967419 0.253179i \(-0.918524\pi\)
−0.967419 + 0.253179i \(0.918524\pi\)
\(930\) 18.6353 0.611076
\(931\) 7.89079 0.258610
\(932\) 43.0560 1.41035
\(933\) 4.07107 0.133281
\(934\) 52.5849 1.72063
\(935\) −34.5161 −1.12880
\(936\) 2.43313 0.0795293
\(937\) −46.3749 −1.51500 −0.757501 0.652834i \(-0.773580\pi\)
−0.757501 + 0.652834i \(0.773580\pi\)
\(938\) 16.7937 0.548333
\(939\) 56.3135 1.83772
\(940\) −9.79385 −0.319440
\(941\) −35.2353 −1.14864 −0.574319 0.818632i \(-0.694733\pi\)
−0.574319 + 0.818632i \(0.694733\pi\)
\(942\) 97.2313 3.16797
\(943\) 4.81030 0.156645
\(944\) −52.4238 −1.70625
\(945\) 2.69618 0.0877067
\(946\) −71.9508 −2.33932
\(947\) 31.5509 1.02527 0.512633 0.858608i \(-0.328670\pi\)
0.512633 + 0.858608i \(0.328670\pi\)
\(948\) 19.3269 0.627709
\(949\) 13.1493 0.426844
\(950\) −15.3478 −0.497948
\(951\) 16.0180 0.519419
\(952\) 2.38891 0.0774251
\(953\) 22.4196 0.726242 0.363121 0.931742i \(-0.381711\pi\)
0.363121 + 0.931742i \(0.381711\pi\)
\(954\) 40.4145 1.30847
\(955\) 0.0806972 0.00261130
\(956\) 15.6534 0.506269
\(957\) 84.5315 2.73251
\(958\) 22.9914 0.742818
\(959\) 6.76591 0.218483
\(960\) −13.4927 −0.435473
\(961\) −11.7348 −0.378541
\(962\) 33.6649 1.08540
\(963\) 31.8182 1.02533
\(964\) 9.44627 0.304244
\(965\) −2.01661 −0.0649170
\(966\) −14.7207 −0.473630
\(967\) −25.0441 −0.805365 −0.402683 0.915340i \(-0.631922\pi\)
−0.402683 + 0.915340i \(0.631922\pi\)
\(968\) 11.0274 0.354435
\(969\) −97.5499 −3.13375
\(970\) 13.7876 0.442694
\(971\) 46.5107 1.49260 0.746299 0.665610i \(-0.231829\pi\)
0.746299 + 0.665610i \(0.231829\pi\)
\(972\) −29.0926 −0.933145
\(973\) −3.08504 −0.0989019
\(974\) 76.8261 2.46167
\(975\) 7.13455 0.228489
\(976\) −30.7238 −0.983444
\(977\) −37.4191 −1.19714 −0.598572 0.801069i \(-0.704265\pi\)
−0.598572 + 0.801069i \(0.704265\pi\)
\(978\) 16.2940 0.521026
\(979\) 90.1696 2.88183
\(980\) 1.78313 0.0569601
\(981\) −22.9919 −0.734075
\(982\) 33.8840 1.08128
\(983\) −32.5613 −1.03854 −0.519272 0.854609i \(-0.673797\pi\)
−0.519272 + 0.854609i \(0.673797\pi\)
\(984\) −1.27743 −0.0407229
\(985\) −1.09518 −0.0348954
\(986\) 69.9945 2.22908
\(987\) 11.9893 0.381623
\(988\) 45.9883 1.46308
\(989\) −21.0450 −0.669192
\(990\) −20.9203 −0.664892
\(991\) 0.547631 0.0173961 0.00869804 0.999962i \(-0.497231\pi\)
0.00869804 + 0.999962i \(0.497231\pi\)
\(992\) −33.7471 −1.07147
\(993\) 24.8345 0.788101
\(994\) 28.5962 0.907015
\(995\) −10.5428 −0.334229
\(996\) 2.76277 0.0875418
\(997\) 44.2506 1.40143 0.700716 0.713441i \(-0.252864\pi\)
0.700716 + 0.713441i \(0.252864\pi\)
\(998\) −62.8122 −1.98829
\(999\) 14.2777 0.451726
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 8015.2.a.m.1.11 67
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8015.2.a.m.1.11 67 1.1 even 1 trivial