Properties

Label 8047.2.a.c.1.11
Level $8047$
Weight $2$
Character 8047.1
Self dual yes
Analytic conductor $64.256$
Analytic rank $1$
Dimension $151$
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] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.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: \(64.2556185065\)
Analytic rank: \(1\)
Dimension: \(151\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 8047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.59872 q^{2} +0.688709 q^{3} +4.75335 q^{4} -1.04737 q^{5} -1.78976 q^{6} +2.76372 q^{7} -7.15519 q^{8} -2.52568 q^{9} +O(q^{10})\) \(q-2.59872 q^{2} +0.688709 q^{3} +4.75335 q^{4} -1.04737 q^{5} -1.78976 q^{6} +2.76372 q^{7} -7.15519 q^{8} -2.52568 q^{9} +2.72181 q^{10} +2.17885 q^{11} +3.27368 q^{12} -1.00000 q^{13} -7.18213 q^{14} -0.721331 q^{15} +9.08764 q^{16} -2.98651 q^{17} +6.56354 q^{18} +7.89798 q^{19} -4.97850 q^{20} +1.90340 q^{21} -5.66223 q^{22} +9.29513 q^{23} -4.92785 q^{24} -3.90302 q^{25} +2.59872 q^{26} -3.80559 q^{27} +13.1369 q^{28} -4.12598 q^{29} +1.87454 q^{30} +6.39056 q^{31} -9.30587 q^{32} +1.50060 q^{33} +7.76110 q^{34} -2.89462 q^{35} -12.0054 q^{36} -6.47592 q^{37} -20.5247 q^{38} -0.688709 q^{39} +7.49410 q^{40} -5.96707 q^{41} -4.94640 q^{42} -6.98240 q^{43} +10.3569 q^{44} +2.64531 q^{45} -24.1555 q^{46} -8.32173 q^{47} +6.25875 q^{48} +0.638136 q^{49} +10.1429 q^{50} -2.05684 q^{51} -4.75335 q^{52} -11.0940 q^{53} +9.88966 q^{54} -2.28206 q^{55} -19.7749 q^{56} +5.43942 q^{57} +10.7223 q^{58} +5.14294 q^{59} -3.42874 q^{60} +9.66893 q^{61} -16.6073 q^{62} -6.98026 q^{63} +6.00807 q^{64} +1.04737 q^{65} -3.89963 q^{66} -4.15316 q^{67} -14.1959 q^{68} +6.40164 q^{69} +7.52232 q^{70} -1.40495 q^{71} +18.0717 q^{72} -8.94604 q^{73} +16.8291 q^{74} -2.68805 q^{75} +37.5419 q^{76} +6.02174 q^{77} +1.78976 q^{78} +15.8910 q^{79} -9.51809 q^{80} +4.95609 q^{81} +15.5067 q^{82} -10.6444 q^{83} +9.04752 q^{84} +3.12797 q^{85} +18.1453 q^{86} -2.84160 q^{87} -15.5901 q^{88} -7.65560 q^{89} -6.87443 q^{90} -2.76372 q^{91} +44.1830 q^{92} +4.40124 q^{93} +21.6258 q^{94} -8.27208 q^{95} -6.40904 q^{96} +5.80198 q^{97} -1.65834 q^{98} -5.50309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.59872 −1.83757 −0.918787 0.394755i \(-0.870830\pi\)
−0.918787 + 0.394755i \(0.870830\pi\)
\(3\) 0.688709 0.397627 0.198813 0.980037i \(-0.436291\pi\)
0.198813 + 0.980037i \(0.436291\pi\)
\(4\) 4.75335 2.37668
\(5\) −1.04737 −0.468396 −0.234198 0.972189i \(-0.575246\pi\)
−0.234198 + 0.972189i \(0.575246\pi\)
\(6\) −1.78976 −0.730668
\(7\) 2.76372 1.04459 0.522294 0.852766i \(-0.325077\pi\)
0.522294 + 0.852766i \(0.325077\pi\)
\(8\) −7.15519 −2.52974
\(9\) −2.52568 −0.841893
\(10\) 2.72181 0.860713
\(11\) 2.17885 0.656949 0.328475 0.944513i \(-0.393465\pi\)
0.328475 + 0.944513i \(0.393465\pi\)
\(12\) 3.27368 0.945029
\(13\) −1.00000 −0.277350
\(14\) −7.18213 −1.91951
\(15\) −0.721331 −0.186247
\(16\) 9.08764 2.27191
\(17\) −2.98651 −0.724335 −0.362167 0.932113i \(-0.617963\pi\)
−0.362167 + 0.932113i \(0.617963\pi\)
\(18\) 6.56354 1.54704
\(19\) 7.89798 1.81192 0.905961 0.423361i \(-0.139150\pi\)
0.905961 + 0.423361i \(0.139150\pi\)
\(20\) −4.97850 −1.11323
\(21\) 1.90340 0.415356
\(22\) −5.66223 −1.20719
\(23\) 9.29513 1.93817 0.969084 0.246729i \(-0.0793558\pi\)
0.969084 + 0.246729i \(0.0793558\pi\)
\(24\) −4.92785 −1.00589
\(25\) −3.90302 −0.780605
\(26\) 2.59872 0.509651
\(27\) −3.80559 −0.732386
\(28\) 13.1369 2.48264
\(29\) −4.12598 −0.766175 −0.383087 0.923712i \(-0.625139\pi\)
−0.383087 + 0.923712i \(0.625139\pi\)
\(30\) 1.87454 0.342242
\(31\) 6.39056 1.14778 0.573889 0.818933i \(-0.305434\pi\)
0.573889 + 0.818933i \(0.305434\pi\)
\(32\) −9.30587 −1.64506
\(33\) 1.50060 0.261220
\(34\) 7.76110 1.33102
\(35\) −2.89462 −0.489281
\(36\) −12.0054 −2.00091
\(37\) −6.47592 −1.06463 −0.532317 0.846545i \(-0.678679\pi\)
−0.532317 + 0.846545i \(0.678679\pi\)
\(38\) −20.5247 −3.32954
\(39\) −0.688709 −0.110282
\(40\) 7.49410 1.18492
\(41\) −5.96707 −0.931899 −0.465950 0.884811i \(-0.654287\pi\)
−0.465950 + 0.884811i \(0.654287\pi\)
\(42\) −4.94640 −0.763246
\(43\) −6.98240 −1.06481 −0.532403 0.846491i \(-0.678711\pi\)
−0.532403 + 0.846491i \(0.678711\pi\)
\(44\) 10.3569 1.56136
\(45\) 2.64531 0.394340
\(46\) −24.1555 −3.56153
\(47\) −8.32173 −1.21385 −0.606924 0.794760i \(-0.707597\pi\)
−0.606924 + 0.794760i \(0.707597\pi\)
\(48\) 6.25875 0.903372
\(49\) 0.638136 0.0911623
\(50\) 10.1429 1.43442
\(51\) −2.05684 −0.288015
\(52\) −4.75335 −0.659171
\(53\) −11.0940 −1.52388 −0.761940 0.647648i \(-0.775753\pi\)
−0.761940 + 0.647648i \(0.775753\pi\)
\(54\) 9.88966 1.34581
\(55\) −2.28206 −0.307713
\(56\) −19.7749 −2.64254
\(57\) 5.43942 0.720468
\(58\) 10.7223 1.40790
\(59\) 5.14294 0.669554 0.334777 0.942297i \(-0.391339\pi\)
0.334777 + 0.942297i \(0.391339\pi\)
\(60\) −3.42874 −0.442648
\(61\) 9.66893 1.23798 0.618990 0.785399i \(-0.287542\pi\)
0.618990 + 0.785399i \(0.287542\pi\)
\(62\) −16.6073 −2.10913
\(63\) −6.98026 −0.879431
\(64\) 6.00807 0.751008
\(65\) 1.04737 0.129910
\(66\) −3.89963 −0.480012
\(67\) −4.15316 −0.507389 −0.253694 0.967284i \(-0.581646\pi\)
−0.253694 + 0.967284i \(0.581646\pi\)
\(68\) −14.1959 −1.72151
\(69\) 6.40164 0.770667
\(70\) 7.52232 0.899089
\(71\) −1.40495 −0.166737 −0.0833683 0.996519i \(-0.526568\pi\)
−0.0833683 + 0.996519i \(0.526568\pi\)
\(72\) 18.0717 2.12977
\(73\) −8.94604 −1.04706 −0.523528 0.852009i \(-0.675384\pi\)
−0.523528 + 0.852009i \(0.675384\pi\)
\(74\) 16.8291 1.95634
\(75\) −2.68805 −0.310389
\(76\) 37.5419 4.30635
\(77\) 6.02174 0.686241
\(78\) 1.78976 0.202651
\(79\) 15.8910 1.78787 0.893937 0.448194i \(-0.147932\pi\)
0.893937 + 0.448194i \(0.147932\pi\)
\(80\) −9.51809 −1.06415
\(81\) 4.95609 0.550677
\(82\) 15.5067 1.71243
\(83\) −10.6444 −1.16838 −0.584188 0.811619i \(-0.698587\pi\)
−0.584188 + 0.811619i \(0.698587\pi\)
\(84\) 9.04752 0.987165
\(85\) 3.12797 0.339276
\(86\) 18.1453 1.95666
\(87\) −2.84160 −0.304651
\(88\) −15.5901 −1.66191
\(89\) −7.65560 −0.811492 −0.405746 0.913986i \(-0.632988\pi\)
−0.405746 + 0.913986i \(0.632988\pi\)
\(90\) −6.87443 −0.724628
\(91\) −2.76372 −0.289716
\(92\) 44.1830 4.60640
\(93\) 4.40124 0.456387
\(94\) 21.6258 2.23054
\(95\) −8.27208 −0.848698
\(96\) −6.40904 −0.654120
\(97\) 5.80198 0.589102 0.294551 0.955636i \(-0.404830\pi\)
0.294551 + 0.955636i \(0.404830\pi\)
\(98\) −1.65834 −0.167517
\(99\) −5.50309 −0.553081
\(100\) −18.5524 −1.85524
\(101\) 1.08317 0.107780 0.0538899 0.998547i \(-0.482838\pi\)
0.0538899 + 0.998547i \(0.482838\pi\)
\(102\) 5.34514 0.529248
\(103\) 3.95448 0.389646 0.194823 0.980838i \(-0.437587\pi\)
0.194823 + 0.980838i \(0.437587\pi\)
\(104\) 7.15519 0.701624
\(105\) −1.99356 −0.194551
\(106\) 28.8302 2.80024
\(107\) −14.5159 −1.40331 −0.701654 0.712518i \(-0.747555\pi\)
−0.701654 + 0.712518i \(0.747555\pi\)
\(108\) −18.0893 −1.74064
\(109\) −19.0446 −1.82414 −0.912071 0.410031i \(-0.865518\pi\)
−0.912071 + 0.410031i \(0.865518\pi\)
\(110\) 5.93043 0.565445
\(111\) −4.46003 −0.423327
\(112\) 25.1157 2.37321
\(113\) −2.85441 −0.268520 −0.134260 0.990946i \(-0.542866\pi\)
−0.134260 + 0.990946i \(0.542866\pi\)
\(114\) −14.1355 −1.32391
\(115\) −9.73541 −0.907831
\(116\) −19.6122 −1.82095
\(117\) 2.52568 0.233499
\(118\) −13.3651 −1.23035
\(119\) −8.25387 −0.756631
\(120\) 5.16126 0.471156
\(121\) −6.25259 −0.568418
\(122\) −25.1268 −2.27488
\(123\) −4.10958 −0.370548
\(124\) 30.3766 2.72790
\(125\) 9.32473 0.834029
\(126\) 18.1398 1.61602
\(127\) 6.28579 0.557773 0.278887 0.960324i \(-0.410035\pi\)
0.278887 + 0.960324i \(0.410035\pi\)
\(128\) 2.99845 0.265028
\(129\) −4.80884 −0.423395
\(130\) −2.72181 −0.238719
\(131\) 11.2745 0.985055 0.492528 0.870297i \(-0.336073\pi\)
0.492528 + 0.870297i \(0.336073\pi\)
\(132\) 7.13287 0.620836
\(133\) 21.8278 1.89271
\(134\) 10.7929 0.932364
\(135\) 3.98584 0.343047
\(136\) 21.3690 1.83238
\(137\) 9.19000 0.785155 0.392577 0.919719i \(-0.371583\pi\)
0.392577 + 0.919719i \(0.371583\pi\)
\(138\) −16.6361 −1.41616
\(139\) 3.59163 0.304638 0.152319 0.988331i \(-0.451326\pi\)
0.152319 + 0.988331i \(0.451326\pi\)
\(140\) −13.7592 −1.16286
\(141\) −5.73125 −0.482659
\(142\) 3.65107 0.306391
\(143\) −2.17885 −0.182205
\(144\) −22.9525 −1.91271
\(145\) 4.32141 0.358873
\(146\) 23.2483 1.92404
\(147\) 0.439490 0.0362486
\(148\) −30.7823 −2.53029
\(149\) −9.10469 −0.745885 −0.372943 0.927854i \(-0.621651\pi\)
−0.372943 + 0.927854i \(0.621651\pi\)
\(150\) 6.98549 0.570363
\(151\) −0.944906 −0.0768953 −0.0384477 0.999261i \(-0.512241\pi\)
−0.0384477 + 0.999261i \(0.512241\pi\)
\(152\) −56.5116 −4.58370
\(153\) 7.54296 0.609812
\(154\) −15.6488 −1.26102
\(155\) −6.69326 −0.537616
\(156\) −3.27368 −0.262104
\(157\) 0.0201883 0.00161120 0.000805600 1.00000i \(-0.499744\pi\)
0.000805600 1.00000i \(0.499744\pi\)
\(158\) −41.2962 −3.28535
\(159\) −7.64055 −0.605935
\(160\) 9.74665 0.770540
\(161\) 25.6891 2.02459
\(162\) −12.8795 −1.01191
\(163\) 18.3239 1.43524 0.717619 0.696436i \(-0.245232\pi\)
0.717619 + 0.696436i \(0.245232\pi\)
\(164\) −28.3636 −2.21482
\(165\) −1.57167 −0.122355
\(166\) 27.6618 2.14697
\(167\) −17.3408 −1.34187 −0.670934 0.741517i \(-0.734107\pi\)
−0.670934 + 0.741517i \(0.734107\pi\)
\(168\) −13.6192 −1.05074
\(169\) 1.00000 0.0769231
\(170\) −8.12872 −0.623444
\(171\) −19.9478 −1.52544
\(172\) −33.1898 −2.53070
\(173\) 19.4028 1.47517 0.737585 0.675254i \(-0.235966\pi\)
0.737585 + 0.675254i \(0.235966\pi\)
\(174\) 7.38452 0.559819
\(175\) −10.7869 −0.815410
\(176\) 19.8006 1.49253
\(177\) 3.54199 0.266232
\(178\) 19.8948 1.49118
\(179\) −18.8738 −1.41069 −0.705345 0.708864i \(-0.749208\pi\)
−0.705345 + 0.708864i \(0.749208\pi\)
\(180\) 12.5741 0.937217
\(181\) −9.17970 −0.682322 −0.341161 0.940005i \(-0.610820\pi\)
−0.341161 + 0.940005i \(0.610820\pi\)
\(182\) 7.18213 0.532375
\(183\) 6.65908 0.492254
\(184\) −66.5084 −4.90307
\(185\) 6.78266 0.498671
\(186\) −11.4376 −0.838645
\(187\) −6.50717 −0.475851
\(188\) −39.5561 −2.88492
\(189\) −10.5176 −0.765041
\(190\) 21.4968 1.55954
\(191\) 1.23686 0.0894960 0.0447480 0.998998i \(-0.485752\pi\)
0.0447480 + 0.998998i \(0.485752\pi\)
\(192\) 4.13781 0.298621
\(193\) 16.9145 1.21753 0.608765 0.793350i \(-0.291665\pi\)
0.608765 + 0.793350i \(0.291665\pi\)
\(194\) −15.0777 −1.08252
\(195\) 0.721331 0.0516556
\(196\) 3.03329 0.216663
\(197\) −21.4795 −1.53035 −0.765175 0.643823i \(-0.777347\pi\)
−0.765175 + 0.643823i \(0.777347\pi\)
\(198\) 14.3010 1.01633
\(199\) −16.3287 −1.15751 −0.578754 0.815502i \(-0.696461\pi\)
−0.578754 + 0.815502i \(0.696461\pi\)
\(200\) 27.9269 1.97473
\(201\) −2.86032 −0.201751
\(202\) −2.81486 −0.198053
\(203\) −11.4030 −0.800336
\(204\) −9.77687 −0.684518
\(205\) 6.24970 0.436498
\(206\) −10.2766 −0.716003
\(207\) −23.4765 −1.63173
\(208\) −9.08764 −0.630115
\(209\) 17.2086 1.19034
\(210\) 5.18069 0.357502
\(211\) 11.8390 0.815031 0.407516 0.913198i \(-0.366395\pi\)
0.407516 + 0.913198i \(0.366395\pi\)
\(212\) −52.7337 −3.62177
\(213\) −0.967601 −0.0662989
\(214\) 37.7228 2.57868
\(215\) 7.31313 0.498751
\(216\) 27.2297 1.85275
\(217\) 17.6617 1.19896
\(218\) 49.4916 3.35200
\(219\) −6.16122 −0.416337
\(220\) −10.8474 −0.731333
\(221\) 2.98651 0.200894
\(222\) 11.5904 0.777894
\(223\) −13.2083 −0.884496 −0.442248 0.896893i \(-0.645819\pi\)
−0.442248 + 0.896893i \(0.645819\pi\)
\(224\) −25.7188 −1.71841
\(225\) 9.85779 0.657186
\(226\) 7.41781 0.493425
\(227\) 14.9772 0.994069 0.497034 0.867731i \(-0.334422\pi\)
0.497034 + 0.867731i \(0.334422\pi\)
\(228\) 25.8555 1.71232
\(229\) −0.684727 −0.0452481 −0.0226240 0.999744i \(-0.507202\pi\)
−0.0226240 + 0.999744i \(0.507202\pi\)
\(230\) 25.2996 1.66821
\(231\) 4.14723 0.272868
\(232\) 29.5222 1.93822
\(233\) −6.24751 −0.409288 −0.204644 0.978836i \(-0.565604\pi\)
−0.204644 + 0.978836i \(0.565604\pi\)
\(234\) −6.56354 −0.429072
\(235\) 8.71590 0.568562
\(236\) 24.4462 1.59131
\(237\) 10.9443 0.710906
\(238\) 21.4495 1.39036
\(239\) −12.3898 −0.801427 −0.400713 0.916203i \(-0.631238\pi\)
−0.400713 + 0.916203i \(0.631238\pi\)
\(240\) −6.55520 −0.423136
\(241\) −19.5597 −1.25995 −0.629977 0.776614i \(-0.716936\pi\)
−0.629977 + 0.776614i \(0.716936\pi\)
\(242\) 16.2487 1.04451
\(243\) 14.8301 0.951349
\(244\) 45.9598 2.94228
\(245\) −0.668362 −0.0427001
\(246\) 10.6796 0.680909
\(247\) −7.89798 −0.502537
\(248\) −45.7257 −2.90358
\(249\) −7.33090 −0.464577
\(250\) −24.2324 −1.53259
\(251\) 20.6154 1.30123 0.650617 0.759406i \(-0.274510\pi\)
0.650617 + 0.759406i \(0.274510\pi\)
\(252\) −33.1796 −2.09012
\(253\) 20.2527 1.27328
\(254\) −16.3350 −1.02495
\(255\) 2.15426 0.134905
\(256\) −19.8083 −1.23802
\(257\) 17.9214 1.11791 0.558953 0.829199i \(-0.311203\pi\)
0.558953 + 0.829199i \(0.311203\pi\)
\(258\) 12.4968 0.778020
\(259\) −17.8976 −1.11210
\(260\) 4.97850 0.308753
\(261\) 10.4209 0.645037
\(262\) −29.2992 −1.81011
\(263\) −24.2219 −1.49359 −0.746793 0.665056i \(-0.768408\pi\)
−0.746793 + 0.665056i \(0.768408\pi\)
\(264\) −10.7371 −0.660820
\(265\) 11.6195 0.713780
\(266\) −56.7244 −3.47799
\(267\) −5.27248 −0.322671
\(268\) −19.7414 −1.20590
\(269\) 11.2016 0.682972 0.341486 0.939887i \(-0.389070\pi\)
0.341486 + 0.939887i \(0.389070\pi\)
\(270\) −10.3581 −0.630374
\(271\) −4.33678 −0.263440 −0.131720 0.991287i \(-0.542050\pi\)
−0.131720 + 0.991287i \(0.542050\pi\)
\(272\) −27.1403 −1.64562
\(273\) −1.90340 −0.115199
\(274\) −23.8823 −1.44278
\(275\) −8.50412 −0.512818
\(276\) 30.4293 1.83163
\(277\) 6.25708 0.375951 0.187976 0.982174i \(-0.439807\pi\)
0.187976 + 0.982174i \(0.439807\pi\)
\(278\) −9.33366 −0.559796
\(279\) −16.1405 −0.966307
\(280\) 20.7116 1.23775
\(281\) −19.4833 −1.16227 −0.581137 0.813806i \(-0.697392\pi\)
−0.581137 + 0.813806i \(0.697392\pi\)
\(282\) 14.8939 0.886920
\(283\) 2.43171 0.144550 0.0722751 0.997385i \(-0.476974\pi\)
0.0722751 + 0.997385i \(0.476974\pi\)
\(284\) −6.67821 −0.396279
\(285\) −5.69706 −0.337465
\(286\) 5.66223 0.334815
\(287\) −16.4913 −0.973450
\(288\) 23.5036 1.38497
\(289\) −8.08077 −0.475339
\(290\) −11.2301 −0.659456
\(291\) 3.99588 0.234242
\(292\) −42.5237 −2.48851
\(293\) 23.0117 1.34436 0.672179 0.740389i \(-0.265359\pi\)
0.672179 + 0.740389i \(0.265359\pi\)
\(294\) −1.14211 −0.0666094
\(295\) −5.38654 −0.313616
\(296\) 46.3364 2.69325
\(297\) −8.29182 −0.481140
\(298\) 23.6605 1.37062
\(299\) −9.29513 −0.537551
\(300\) −12.7772 −0.737694
\(301\) −19.2974 −1.11228
\(302\) 2.45555 0.141301
\(303\) 0.745992 0.0428561
\(304\) 71.7741 4.11652
\(305\) −10.1269 −0.579865
\(306\) −19.6021 −1.12058
\(307\) 24.3955 1.39233 0.696164 0.717883i \(-0.254889\pi\)
0.696164 + 0.717883i \(0.254889\pi\)
\(308\) 28.6234 1.63097
\(309\) 2.72349 0.154934
\(310\) 17.3939 0.987908
\(311\) −26.9813 −1.52997 −0.764984 0.644049i \(-0.777253\pi\)
−0.764984 + 0.644049i \(0.777253\pi\)
\(312\) 4.92785 0.278984
\(313\) 19.3728 1.09502 0.547509 0.836800i \(-0.315576\pi\)
0.547509 + 0.836800i \(0.315576\pi\)
\(314\) −0.0524637 −0.00296070
\(315\) 7.31089 0.411922
\(316\) 75.5353 4.24919
\(317\) −9.00074 −0.505532 −0.252766 0.967527i \(-0.581340\pi\)
−0.252766 + 0.967527i \(0.581340\pi\)
\(318\) 19.8557 1.11345
\(319\) −8.98990 −0.503338
\(320\) −6.29265 −0.351770
\(321\) −9.99726 −0.557992
\(322\) −66.7589 −3.72033
\(323\) −23.5874 −1.31244
\(324\) 23.5581 1.30878
\(325\) 3.90302 0.216501
\(326\) −47.6187 −2.63736
\(327\) −13.1162 −0.725328
\(328\) 42.6955 2.35746
\(329\) −22.9989 −1.26797
\(330\) 4.08434 0.224836
\(331\) 27.5757 1.51570 0.757849 0.652430i \(-0.226250\pi\)
0.757849 + 0.652430i \(0.226250\pi\)
\(332\) −50.5966 −2.77685
\(333\) 16.3561 0.896309
\(334\) 45.0638 2.46578
\(335\) 4.34988 0.237659
\(336\) 17.2974 0.943651
\(337\) −31.1307 −1.69580 −0.847898 0.530159i \(-0.822132\pi\)
−0.847898 + 0.530159i \(0.822132\pi\)
\(338\) −2.59872 −0.141352
\(339\) −1.96586 −0.106771
\(340\) 14.8683 0.806348
\(341\) 13.9241 0.754033
\(342\) 51.8387 2.80312
\(343\) −17.5824 −0.949360
\(344\) 49.9604 2.69368
\(345\) −6.70487 −0.360978
\(346\) −50.4226 −2.71073
\(347\) −29.6154 −1.58984 −0.794920 0.606715i \(-0.792487\pi\)
−0.794920 + 0.606715i \(0.792487\pi\)
\(348\) −13.5071 −0.724058
\(349\) −28.9624 −1.55032 −0.775162 0.631762i \(-0.782332\pi\)
−0.775162 + 0.631762i \(0.782332\pi\)
\(350\) 28.0320 1.49838
\(351\) 3.80559 0.203127
\(352\) −20.2761 −1.08072
\(353\) −22.3910 −1.19175 −0.595877 0.803075i \(-0.703196\pi\)
−0.595877 + 0.803075i \(0.703196\pi\)
\(354\) −9.20464 −0.489221
\(355\) 1.47150 0.0780989
\(356\) −36.3897 −1.92865
\(357\) −5.68452 −0.300857
\(358\) 49.0476 2.59225
\(359\) 1.15725 0.0610772 0.0305386 0.999534i \(-0.490278\pi\)
0.0305386 + 0.999534i \(0.490278\pi\)
\(360\) −18.9277 −0.997578
\(361\) 43.3782 2.28306
\(362\) 23.8555 1.25382
\(363\) −4.30622 −0.226018
\(364\) −13.1369 −0.688562
\(365\) 9.36978 0.490437
\(366\) −17.3051 −0.904552
\(367\) 20.8216 1.08688 0.543438 0.839449i \(-0.317122\pi\)
0.543438 + 0.839449i \(0.317122\pi\)
\(368\) 84.4708 4.40335
\(369\) 15.0709 0.784560
\(370\) −17.6262 −0.916344
\(371\) −30.6607 −1.59183
\(372\) 20.9206 1.08468
\(373\) −22.7516 −1.17803 −0.589017 0.808121i \(-0.700485\pi\)
−0.589017 + 0.808121i \(0.700485\pi\)
\(374\) 16.9103 0.874411
\(375\) 6.42203 0.331632
\(376\) 59.5436 3.07072
\(377\) 4.12598 0.212499
\(378\) 27.3322 1.40582
\(379\) −10.4205 −0.535264 −0.267632 0.963521i \(-0.586241\pi\)
−0.267632 + 0.963521i \(0.586241\pi\)
\(380\) −39.3201 −2.01708
\(381\) 4.32908 0.221785
\(382\) −3.21425 −0.164455
\(383\) −19.8913 −1.01640 −0.508199 0.861240i \(-0.669689\pi\)
−0.508199 + 0.861240i \(0.669689\pi\)
\(384\) 2.06506 0.105382
\(385\) −6.30696 −0.321433
\(386\) −43.9560 −2.23730
\(387\) 17.6353 0.896453
\(388\) 27.5788 1.40010
\(389\) 8.25398 0.418493 0.209247 0.977863i \(-0.432899\pi\)
0.209247 + 0.977863i \(0.432899\pi\)
\(390\) −1.87454 −0.0949209
\(391\) −27.7600 −1.40388
\(392\) −4.56599 −0.230617
\(393\) 7.76484 0.391684
\(394\) 55.8192 2.81213
\(395\) −16.6437 −0.837433
\(396\) −26.1581 −1.31449
\(397\) −30.0695 −1.50914 −0.754571 0.656218i \(-0.772155\pi\)
−0.754571 + 0.656218i \(0.772155\pi\)
\(398\) 42.4336 2.12701
\(399\) 15.0330 0.752592
\(400\) −35.4693 −1.77346
\(401\) −5.42244 −0.270783 −0.135392 0.990792i \(-0.543229\pi\)
−0.135392 + 0.990792i \(0.543229\pi\)
\(402\) 7.43317 0.370733
\(403\) −6.39056 −0.318337
\(404\) 5.14870 0.256158
\(405\) −5.19085 −0.257935
\(406\) 29.6333 1.47068
\(407\) −14.1101 −0.699411
\(408\) 14.7171 0.728603
\(409\) 36.7863 1.81896 0.909482 0.415744i \(-0.136479\pi\)
0.909482 + 0.415744i \(0.136479\pi\)
\(410\) −16.2412 −0.802098
\(411\) 6.32924 0.312198
\(412\) 18.7970 0.926063
\(413\) 14.2136 0.699407
\(414\) 61.0089 2.99843
\(415\) 11.1486 0.547263
\(416\) 9.30587 0.456258
\(417\) 2.47359 0.121132
\(418\) −44.7202 −2.18734
\(419\) 13.9172 0.679899 0.339950 0.940444i \(-0.389590\pi\)
0.339950 + 0.940444i \(0.389590\pi\)
\(420\) −9.47607 −0.462385
\(421\) −8.08024 −0.393807 −0.196903 0.980423i \(-0.563089\pi\)
−0.196903 + 0.980423i \(0.563089\pi\)
\(422\) −30.7663 −1.49768
\(423\) 21.0180 1.02193
\(424\) 79.3798 3.85502
\(425\) 11.6564 0.565419
\(426\) 2.51453 0.121829
\(427\) 26.7222 1.29318
\(428\) −68.9993 −3.33521
\(429\) −1.50060 −0.0724495
\(430\) −19.0048 −0.916492
\(431\) 15.1622 0.730340 0.365170 0.930941i \(-0.381011\pi\)
0.365170 + 0.930941i \(0.381011\pi\)
\(432\) −34.5838 −1.66391
\(433\) 7.09373 0.340903 0.170451 0.985366i \(-0.445477\pi\)
0.170451 + 0.985366i \(0.445477\pi\)
\(434\) −45.8979 −2.20317
\(435\) 2.97619 0.142698
\(436\) −90.5257 −4.33540
\(437\) 73.4128 3.51181
\(438\) 16.0113 0.765050
\(439\) 40.8912 1.95163 0.975815 0.218597i \(-0.0701479\pi\)
0.975815 + 0.218597i \(0.0701479\pi\)
\(440\) 16.3286 0.778434
\(441\) −1.61173 −0.0767489
\(442\) −7.76110 −0.369158
\(443\) 17.6417 0.838180 0.419090 0.907945i \(-0.362349\pi\)
0.419090 + 0.907945i \(0.362349\pi\)
\(444\) −21.2001 −1.00611
\(445\) 8.01821 0.380100
\(446\) 34.3248 1.62533
\(447\) −6.27049 −0.296584
\(448\) 16.6046 0.784494
\(449\) −4.66444 −0.220129 −0.110064 0.993924i \(-0.535106\pi\)
−0.110064 + 0.993924i \(0.535106\pi\)
\(450\) −25.6176 −1.20763
\(451\) −13.0014 −0.612211
\(452\) −13.5680 −0.638185
\(453\) −0.650765 −0.0305756
\(454\) −38.9215 −1.82667
\(455\) 2.89462 0.135702
\(456\) −38.9201 −1.82260
\(457\) 40.8629 1.91149 0.955743 0.294203i \(-0.0950542\pi\)
0.955743 + 0.294203i \(0.0950542\pi\)
\(458\) 1.77942 0.0831466
\(459\) 11.3654 0.530492
\(460\) −46.2758 −2.15762
\(461\) 17.0262 0.792991 0.396495 0.918037i \(-0.370226\pi\)
0.396495 + 0.918037i \(0.370226\pi\)
\(462\) −10.7775 −0.501414
\(463\) −37.5228 −1.74383 −0.871917 0.489654i \(-0.837123\pi\)
−0.871917 + 0.489654i \(0.837123\pi\)
\(464\) −37.4954 −1.74068
\(465\) −4.60971 −0.213770
\(466\) 16.2355 0.752097
\(467\) −26.7958 −1.23996 −0.619980 0.784618i \(-0.712859\pi\)
−0.619980 + 0.784618i \(0.712859\pi\)
\(468\) 12.0054 0.554952
\(469\) −11.4782 −0.530012
\(470\) −22.6502 −1.04477
\(471\) 0.0139039 0.000640656 0
\(472\) −36.7987 −1.69380
\(473\) −15.2136 −0.699523
\(474\) −28.4411 −1.30634
\(475\) −30.8260 −1.41440
\(476\) −39.2335 −1.79827
\(477\) 28.0199 1.28294
\(478\) 32.1975 1.47268
\(479\) −30.7475 −1.40489 −0.702445 0.711738i \(-0.747908\pi\)
−0.702445 + 0.711738i \(0.747908\pi\)
\(480\) 6.71261 0.306387
\(481\) 6.47592 0.295277
\(482\) 50.8303 2.31526
\(483\) 17.6923 0.805029
\(484\) −29.7208 −1.35094
\(485\) −6.07680 −0.275933
\(486\) −38.5392 −1.74817
\(487\) 16.4350 0.744743 0.372371 0.928084i \(-0.378545\pi\)
0.372371 + 0.928084i \(0.378545\pi\)
\(488\) −69.1830 −3.13177
\(489\) 12.6198 0.570689
\(490\) 1.73689 0.0784646
\(491\) 35.1988 1.58850 0.794250 0.607592i \(-0.207864\pi\)
0.794250 + 0.607592i \(0.207864\pi\)
\(492\) −19.5343 −0.880672
\(493\) 12.3223 0.554967
\(494\) 20.5247 0.923448
\(495\) 5.76375 0.259061
\(496\) 58.0752 2.60765
\(497\) −3.88288 −0.174171
\(498\) 19.0510 0.853694
\(499\) −17.4592 −0.781582 −0.390791 0.920479i \(-0.627798\pi\)
−0.390791 + 0.920479i \(0.627798\pi\)
\(500\) 44.3237 1.98222
\(501\) −11.9427 −0.533563
\(502\) −53.5737 −2.39111
\(503\) −20.8128 −0.927998 −0.463999 0.885836i \(-0.653586\pi\)
−0.463999 + 0.885836i \(0.653586\pi\)
\(504\) 49.9451 2.22473
\(505\) −1.13448 −0.0504836
\(506\) −52.6312 −2.33974
\(507\) 0.688709 0.0305867
\(508\) 29.8785 1.32565
\(509\) −24.1331 −1.06968 −0.534840 0.844953i \(-0.679628\pi\)
−0.534840 + 0.844953i \(0.679628\pi\)
\(510\) −5.59832 −0.247898
\(511\) −24.7243 −1.09374
\(512\) 45.4792 2.00992
\(513\) −30.0565 −1.32703
\(514\) −46.5727 −2.05423
\(515\) −4.14179 −0.182509
\(516\) −22.8581 −1.00627
\(517\) −18.1318 −0.797437
\(518\) 46.5109 2.04357
\(519\) 13.3629 0.586567
\(520\) −7.49410 −0.328638
\(521\) 24.1945 1.05998 0.529989 0.848004i \(-0.322196\pi\)
0.529989 + 0.848004i \(0.322196\pi\)
\(522\) −27.0810 −1.18530
\(523\) 9.60899 0.420172 0.210086 0.977683i \(-0.432626\pi\)
0.210086 + 0.977683i \(0.432626\pi\)
\(524\) 53.5915 2.34116
\(525\) −7.42901 −0.324229
\(526\) 62.9460 2.74457
\(527\) −19.0855 −0.831376
\(528\) 13.6369 0.593470
\(529\) 63.3995 2.75650
\(530\) −30.1958 −1.31162
\(531\) −12.9894 −0.563693
\(532\) 103.755 4.49836
\(533\) 5.96707 0.258462
\(534\) 13.7017 0.592931
\(535\) 15.2035 0.657304
\(536\) 29.7166 1.28356
\(537\) −12.9985 −0.560928
\(538\) −29.1098 −1.25501
\(539\) 1.39041 0.0598890
\(540\) 18.9461 0.815311
\(541\) −19.1507 −0.823353 −0.411676 0.911330i \(-0.635057\pi\)
−0.411676 + 0.911330i \(0.635057\pi\)
\(542\) 11.2701 0.484091
\(543\) −6.32215 −0.271309
\(544\) 27.7921 1.19157
\(545\) 19.9467 0.854422
\(546\) 4.94640 0.211686
\(547\) −25.7697 −1.10183 −0.550916 0.834561i \(-0.685721\pi\)
−0.550916 + 0.834561i \(0.685721\pi\)
\(548\) 43.6833 1.86606
\(549\) −24.4206 −1.04225
\(550\) 22.0998 0.942340
\(551\) −32.5869 −1.38825
\(552\) −45.8050 −1.94959
\(553\) 43.9181 1.86759
\(554\) −16.2604 −0.690838
\(555\) 4.67128 0.198285
\(556\) 17.0723 0.724027
\(557\) 43.2228 1.83141 0.915704 0.401853i \(-0.131634\pi\)
0.915704 + 0.401853i \(0.131634\pi\)
\(558\) 41.9447 1.77566
\(559\) 6.98240 0.295324
\(560\) −26.3053 −1.11160
\(561\) −4.48155 −0.189211
\(562\) 50.6316 2.13576
\(563\) 16.5646 0.698116 0.349058 0.937101i \(-0.386502\pi\)
0.349058 + 0.937101i \(0.386502\pi\)
\(564\) −27.2427 −1.14712
\(565\) 2.98961 0.125774
\(566\) −6.31934 −0.265622
\(567\) 13.6972 0.575230
\(568\) 10.0527 0.421801
\(569\) 11.4237 0.478904 0.239452 0.970908i \(-0.423032\pi\)
0.239452 + 0.970908i \(0.423032\pi\)
\(570\) 14.8051 0.620116
\(571\) −7.28710 −0.304956 −0.152478 0.988307i \(-0.548725\pi\)
−0.152478 + 0.988307i \(0.548725\pi\)
\(572\) −10.3569 −0.433042
\(573\) 0.851836 0.0355860
\(574\) 42.8563 1.78879
\(575\) −36.2791 −1.51294
\(576\) −15.1745 −0.632269
\(577\) 9.90803 0.412477 0.206238 0.978502i \(-0.433878\pi\)
0.206238 + 0.978502i \(0.433878\pi\)
\(578\) 20.9997 0.873470
\(579\) 11.6492 0.484122
\(580\) 20.5412 0.852926
\(581\) −29.4181 −1.22047
\(582\) −10.3842 −0.430438
\(583\) −24.1722 −1.00111
\(584\) 64.0106 2.64878
\(585\) −2.64531 −0.109370
\(586\) −59.8010 −2.47036
\(587\) 24.8505 1.02569 0.512845 0.858481i \(-0.328591\pi\)
0.512845 + 0.858481i \(0.328591\pi\)
\(588\) 2.08905 0.0861511
\(589\) 50.4726 2.07969
\(590\) 13.9981 0.576293
\(591\) −14.7931 −0.608508
\(592\) −58.8508 −2.41875
\(593\) −38.7850 −1.59271 −0.796354 0.604830i \(-0.793241\pi\)
−0.796354 + 0.604830i \(0.793241\pi\)
\(594\) 21.5481 0.884130
\(595\) 8.64482 0.354403
\(596\) −43.2778 −1.77273
\(597\) −11.2457 −0.460256
\(598\) 24.1555 0.987790
\(599\) 17.0945 0.698462 0.349231 0.937037i \(-0.386443\pi\)
0.349231 + 0.937037i \(0.386443\pi\)
\(600\) 19.2335 0.785205
\(601\) −21.2362 −0.866244 −0.433122 0.901335i \(-0.642588\pi\)
−0.433122 + 0.901335i \(0.642588\pi\)
\(602\) 50.1485 2.04390
\(603\) 10.4895 0.427167
\(604\) −4.49147 −0.182755
\(605\) 6.54876 0.266245
\(606\) −1.93862 −0.0787512
\(607\) −48.5361 −1.97002 −0.985010 0.172498i \(-0.944816\pi\)
−0.985010 + 0.172498i \(0.944816\pi\)
\(608\) −73.4976 −2.98072
\(609\) −7.85338 −0.318235
\(610\) 26.3170 1.06554
\(611\) 8.32173 0.336661
\(612\) 35.8544 1.44933
\(613\) −4.34559 −0.175517 −0.0877584 0.996142i \(-0.527970\pi\)
−0.0877584 + 0.996142i \(0.527970\pi\)
\(614\) −63.3972 −2.55850
\(615\) 4.30423 0.173563
\(616\) −43.0867 −1.73601
\(617\) −47.3895 −1.90783 −0.953914 0.300080i \(-0.902986\pi\)
−0.953914 + 0.300080i \(0.902986\pi\)
\(618\) −7.07758 −0.284702
\(619\) −1.00000 −0.0401934
\(620\) −31.8154 −1.27774
\(621\) −35.3734 −1.41949
\(622\) 70.1168 2.81143
\(623\) −21.1579 −0.847674
\(624\) −6.25875 −0.250550
\(625\) 9.74872 0.389949
\(626\) −50.3446 −2.01217
\(627\) 11.8517 0.473311
\(628\) 0.0959620 0.00382930
\(629\) 19.3404 0.771152
\(630\) −18.9990 −0.756937
\(631\) 10.0256 0.399113 0.199557 0.979886i \(-0.436050\pi\)
0.199557 + 0.979886i \(0.436050\pi\)
\(632\) −113.703 −4.52286
\(633\) 8.15364 0.324078
\(634\) 23.3904 0.928952
\(635\) −6.58352 −0.261259
\(636\) −36.3182 −1.44011
\(637\) −0.638136 −0.0252839
\(638\) 23.3622 0.924920
\(639\) 3.54845 0.140374
\(640\) −3.14047 −0.124138
\(641\) −27.2086 −1.07468 −0.537338 0.843367i \(-0.680570\pi\)
−0.537338 + 0.843367i \(0.680570\pi\)
\(642\) 25.9801 1.02535
\(643\) −32.6845 −1.28895 −0.644476 0.764625i \(-0.722924\pi\)
−0.644476 + 0.764625i \(0.722924\pi\)
\(644\) 122.109 4.81178
\(645\) 5.03662 0.198317
\(646\) 61.2971 2.41170
\(647\) 7.71902 0.303466 0.151733 0.988422i \(-0.451515\pi\)
0.151733 + 0.988422i \(0.451515\pi\)
\(648\) −35.4618 −1.39307
\(649\) 11.2057 0.439863
\(650\) −10.1429 −0.397836
\(651\) 12.1638 0.476736
\(652\) 87.0999 3.41110
\(653\) 8.19855 0.320834 0.160417 0.987049i \(-0.448716\pi\)
0.160417 + 0.987049i \(0.448716\pi\)
\(654\) 34.0854 1.33284
\(655\) −11.8085 −0.461396
\(656\) −54.2266 −2.11719
\(657\) 22.5948 0.881508
\(658\) 59.7677 2.32999
\(659\) −1.86454 −0.0726322 −0.0363161 0.999340i \(-0.511562\pi\)
−0.0363161 + 0.999340i \(0.511562\pi\)
\(660\) −7.47072 −0.290797
\(661\) −18.7343 −0.728680 −0.364340 0.931266i \(-0.618705\pi\)
−0.364340 + 0.931266i \(0.618705\pi\)
\(662\) −71.6616 −2.78521
\(663\) 2.05684 0.0798809
\(664\) 76.1627 2.95569
\(665\) −22.8617 −0.886539
\(666\) −42.5049 −1.64703
\(667\) −38.3515 −1.48498
\(668\) −82.4268 −3.18919
\(669\) −9.09671 −0.351699
\(670\) −11.3041 −0.436716
\(671\) 21.0672 0.813290
\(672\) −17.7128 −0.683285
\(673\) −11.3377 −0.437036 −0.218518 0.975833i \(-0.570122\pi\)
−0.218518 + 0.975833i \(0.570122\pi\)
\(674\) 80.8999 3.11615
\(675\) 14.8533 0.571704
\(676\) 4.75335 0.182821
\(677\) −1.04186 −0.0400418 −0.0200209 0.999800i \(-0.506373\pi\)
−0.0200209 + 0.999800i \(0.506373\pi\)
\(678\) 5.10872 0.196199
\(679\) 16.0350 0.615368
\(680\) −22.3812 −0.858280
\(681\) 10.3149 0.395268
\(682\) −36.1849 −1.38559
\(683\) 10.1832 0.389647 0.194824 0.980838i \(-0.437587\pi\)
0.194824 + 0.980838i \(0.437587\pi\)
\(684\) −94.8188 −3.62549
\(685\) −9.62530 −0.367764
\(686\) 45.6917 1.74452
\(687\) −0.471578 −0.0179918
\(688\) −63.4536 −2.41914
\(689\) 11.0940 0.422648
\(690\) 17.4241 0.663323
\(691\) −25.5194 −0.970805 −0.485402 0.874291i \(-0.661327\pi\)
−0.485402 + 0.874291i \(0.661327\pi\)
\(692\) 92.2285 3.50600
\(693\) −15.2090 −0.577741
\(694\) 76.9622 2.92145
\(695\) −3.76176 −0.142692
\(696\) 20.3322 0.770689
\(697\) 17.8207 0.675007
\(698\) 75.2653 2.84883
\(699\) −4.30272 −0.162744
\(700\) −51.2737 −1.93796
\(701\) 40.8717 1.54370 0.771851 0.635803i \(-0.219331\pi\)
0.771851 + 0.635803i \(0.219331\pi\)
\(702\) −9.88966 −0.373261
\(703\) −51.1467 −1.92903
\(704\) 13.0907 0.493374
\(705\) 6.00272 0.226075
\(706\) 58.1881 2.18994
\(707\) 2.99359 0.112585
\(708\) 16.8363 0.632748
\(709\) −27.5022 −1.03287 −0.516433 0.856328i \(-0.672740\pi\)
−0.516433 + 0.856328i \(0.672740\pi\)
\(710\) −3.82401 −0.143512
\(711\) −40.1355 −1.50520
\(712\) 54.7773 2.05286
\(713\) 59.4011 2.22459
\(714\) 14.7725 0.552846
\(715\) 2.28206 0.0853441
\(716\) −89.7136 −3.35275
\(717\) −8.53294 −0.318669
\(718\) −3.00736 −0.112234
\(719\) −27.9410 −1.04202 −0.521012 0.853549i \(-0.674445\pi\)
−0.521012 + 0.853549i \(0.674445\pi\)
\(720\) 24.0396 0.895905
\(721\) 10.9291 0.407019
\(722\) −112.728 −4.19529
\(723\) −13.4710 −0.500991
\(724\) −43.6343 −1.62166
\(725\) 16.1038 0.598080
\(726\) 11.1907 0.415325
\(727\) 30.5808 1.13418 0.567090 0.823656i \(-0.308069\pi\)
0.567090 + 0.823656i \(0.308069\pi\)
\(728\) 19.7749 0.732908
\(729\) −4.65467 −0.172395
\(730\) −24.3494 −0.901214
\(731\) 20.8530 0.771276
\(732\) 31.6530 1.16993
\(733\) 26.1846 0.967150 0.483575 0.875303i \(-0.339338\pi\)
0.483575 + 0.875303i \(0.339338\pi\)
\(734\) −54.1094 −1.99722
\(735\) −0.460307 −0.0169787
\(736\) −86.4993 −3.18841
\(737\) −9.04912 −0.333329
\(738\) −39.1651 −1.44169
\(739\) −39.5814 −1.45603 −0.728013 0.685563i \(-0.759556\pi\)
−0.728013 + 0.685563i \(0.759556\pi\)
\(740\) 32.2404 1.18518
\(741\) −5.43942 −0.199822
\(742\) 79.6787 2.92510
\(743\) 11.4306 0.419349 0.209675 0.977771i \(-0.432759\pi\)
0.209675 + 0.977771i \(0.432759\pi\)
\(744\) −31.4917 −1.15454
\(745\) 9.53594 0.349370
\(746\) 59.1251 2.16472
\(747\) 26.8843 0.983647
\(748\) −30.9308 −1.13094
\(749\) −40.1179 −1.46588
\(750\) −16.6891 −0.609398
\(751\) 32.6042 1.18974 0.594872 0.803821i \(-0.297203\pi\)
0.594872 + 0.803821i \(0.297203\pi\)
\(752\) −75.6249 −2.75776
\(753\) 14.1980 0.517405
\(754\) −10.7223 −0.390482
\(755\) 0.989662 0.0360175
\(756\) −49.9937 −1.81825
\(757\) −43.1269 −1.56747 −0.783737 0.621093i \(-0.786689\pi\)
−0.783737 + 0.621093i \(0.786689\pi\)
\(758\) 27.0799 0.983586
\(759\) 13.9483 0.506289
\(760\) 59.1883 2.14699
\(761\) −38.3672 −1.39081 −0.695405 0.718618i \(-0.744775\pi\)
−0.695405 + 0.718618i \(0.744775\pi\)
\(762\) −11.2501 −0.407547
\(763\) −52.6339 −1.90548
\(764\) 5.87922 0.212703
\(765\) −7.90024 −0.285634
\(766\) 51.6920 1.86771
\(767\) −5.14294 −0.185701
\(768\) −13.6421 −0.492268
\(769\) −33.8874 −1.22201 −0.611006 0.791626i \(-0.709235\pi\)
−0.611006 + 0.791626i \(0.709235\pi\)
\(770\) 16.3900 0.590656
\(771\) 12.3426 0.444509
\(772\) 80.4004 2.89367
\(773\) −29.0798 −1.04593 −0.522963 0.852355i \(-0.675174\pi\)
−0.522963 + 0.852355i \(0.675174\pi\)
\(774\) −45.8292 −1.64730
\(775\) −24.9425 −0.895962
\(776\) −41.5143 −1.49028
\(777\) −12.3263 −0.442202
\(778\) −21.4498 −0.769012
\(779\) −47.1278 −1.68853
\(780\) 3.42874 0.122769
\(781\) −3.06118 −0.109538
\(782\) 72.1405 2.57974
\(783\) 15.7018 0.561135
\(784\) 5.79915 0.207113
\(785\) −0.0211445 −0.000754681 0
\(786\) −20.1786 −0.719748
\(787\) −6.19685 −0.220894 −0.110447 0.993882i \(-0.535228\pi\)
−0.110447 + 0.993882i \(0.535228\pi\)
\(788\) −102.099 −3.63714
\(789\) −16.6819 −0.593890
\(790\) 43.2522 1.53884
\(791\) −7.88878 −0.280493
\(792\) 39.3756 1.39915
\(793\) −9.66893 −0.343354
\(794\) 78.1421 2.77316
\(795\) 8.00245 0.283818
\(796\) −77.6158 −2.75102
\(797\) 31.0023 1.09816 0.549079 0.835771i \(-0.314979\pi\)
0.549079 + 0.835771i \(0.314979\pi\)
\(798\) −39.0666 −1.38294
\(799\) 24.8529 0.879233
\(800\) 36.3210 1.28414
\(801\) 19.3356 0.683189
\(802\) 14.0914 0.497585
\(803\) −19.4921 −0.687862
\(804\) −13.5961 −0.479497
\(805\) −26.9059 −0.948309
\(806\) 16.6073 0.584967
\(807\) 7.71463 0.271568
\(808\) −7.75031 −0.272655
\(809\) 6.11111 0.214855 0.107428 0.994213i \(-0.465739\pi\)
0.107428 + 0.994213i \(0.465739\pi\)
\(810\) 13.4896 0.473975
\(811\) 22.4734 0.789147 0.394573 0.918864i \(-0.370892\pi\)
0.394573 + 0.918864i \(0.370892\pi\)
\(812\) −54.2026 −1.90214
\(813\) −2.98678 −0.104751
\(814\) 36.6682 1.28522
\(815\) −19.1918 −0.672261
\(816\) −18.6918 −0.654344
\(817\) −55.1469 −1.92935
\(818\) −95.5972 −3.34248
\(819\) 6.98026 0.243910
\(820\) 29.7070 1.03741
\(821\) −38.2799 −1.33598 −0.667989 0.744171i \(-0.732845\pi\)
−0.667989 + 0.744171i \(0.732845\pi\)
\(822\) −16.4479 −0.573688
\(823\) −48.1451 −1.67823 −0.839116 0.543952i \(-0.816927\pi\)
−0.839116 + 0.543952i \(0.816927\pi\)
\(824\) −28.2950 −0.985704
\(825\) −5.85687 −0.203910
\(826\) −36.9373 −1.28521
\(827\) −37.7005 −1.31098 −0.655488 0.755206i \(-0.727537\pi\)
−0.655488 + 0.755206i \(0.727537\pi\)
\(828\) −111.592 −3.87810
\(829\) −11.8709 −0.412292 −0.206146 0.978521i \(-0.566092\pi\)
−0.206146 + 0.978521i \(0.566092\pi\)
\(830\) −28.9721 −1.00564
\(831\) 4.30931 0.149488
\(832\) −6.00807 −0.208292
\(833\) −1.90580 −0.0660320
\(834\) −6.42818 −0.222590
\(835\) 18.1621 0.628527
\(836\) 81.7983 2.82905
\(837\) −24.3198 −0.840617
\(838\) −36.1669 −1.24936
\(839\) 39.7211 1.37132 0.685662 0.727920i \(-0.259513\pi\)
0.685662 + 0.727920i \(0.259513\pi\)
\(840\) 14.2643 0.492164
\(841\) −11.9763 −0.412976
\(842\) 20.9983 0.723649
\(843\) −13.4183 −0.462151
\(844\) 56.2750 1.93706
\(845\) −1.04737 −0.0360305
\(846\) −54.6200 −1.87787
\(847\) −17.2804 −0.593762
\(848\) −100.818 −3.46212
\(849\) 1.67474 0.0574770
\(850\) −30.2918 −1.03900
\(851\) −60.1945 −2.06344
\(852\) −4.59935 −0.157571
\(853\) 36.9809 1.26620 0.633101 0.774069i \(-0.281782\pi\)
0.633101 + 0.774069i \(0.281782\pi\)
\(854\) −69.4435 −2.37631
\(855\) 20.8926 0.714513
\(856\) 103.864 3.55001
\(857\) −21.2559 −0.726088 −0.363044 0.931772i \(-0.618263\pi\)
−0.363044 + 0.931772i \(0.618263\pi\)
\(858\) 3.89963 0.133131
\(859\) 29.1910 0.995983 0.497992 0.867182i \(-0.334071\pi\)
0.497992 + 0.867182i \(0.334071\pi\)
\(860\) 34.7619 1.18537
\(861\) −11.3577 −0.387070
\(862\) −39.4024 −1.34205
\(863\) 10.8183 0.368260 0.184130 0.982902i \(-0.441053\pi\)
0.184130 + 0.982902i \(0.441053\pi\)
\(864\) 35.4143 1.20482
\(865\) −20.3219 −0.690965
\(866\) −18.4346 −0.626434
\(867\) −5.56530 −0.189007
\(868\) 83.9523 2.84953
\(869\) 34.6241 1.17454
\(870\) −7.73430 −0.262217
\(871\) 4.15316 0.140724
\(872\) 136.268 4.61461
\(873\) −14.6539 −0.495961
\(874\) −190.779 −6.45321
\(875\) 25.7709 0.871216
\(876\) −29.2865 −0.989498
\(877\) −38.4298 −1.29768 −0.648841 0.760924i \(-0.724746\pi\)
−0.648841 + 0.760924i \(0.724746\pi\)
\(878\) −106.265 −3.58626
\(879\) 15.8484 0.534552
\(880\) −20.7385 −0.699096
\(881\) 2.72589 0.0918375 0.0459187 0.998945i \(-0.485378\pi\)
0.0459187 + 0.998945i \(0.485378\pi\)
\(882\) 4.18843 0.141032
\(883\) 18.0096 0.606071 0.303035 0.952979i \(-0.402000\pi\)
0.303035 + 0.952979i \(0.402000\pi\)
\(884\) 14.1959 0.477461
\(885\) −3.70976 −0.124702
\(886\) −45.8457 −1.54022
\(887\) −8.47061 −0.284415 −0.142208 0.989837i \(-0.545420\pi\)
−0.142208 + 0.989837i \(0.545420\pi\)
\(888\) 31.9123 1.07091
\(889\) 17.3721 0.582643
\(890\) −20.8371 −0.698461
\(891\) 10.7986 0.361767
\(892\) −62.7839 −2.10216
\(893\) −65.7249 −2.19940
\(894\) 16.2952 0.544994
\(895\) 19.7677 0.660762
\(896\) 8.28686 0.276844
\(897\) −6.40164 −0.213745
\(898\) 12.1216 0.404503
\(899\) −26.3673 −0.879399
\(900\) 46.8575 1.56192
\(901\) 33.1324 1.10380
\(902\) 33.7869 1.12498
\(903\) −13.2903 −0.442273
\(904\) 20.4238 0.679287
\(905\) 9.61451 0.319597
\(906\) 1.69116 0.0561850
\(907\) −19.7512 −0.655826 −0.327913 0.944708i \(-0.606345\pi\)
−0.327913 + 0.944708i \(0.606345\pi\)
\(908\) 71.1917 2.36258
\(909\) −2.73575 −0.0907390
\(910\) −7.52232 −0.249363
\(911\) 4.90474 0.162501 0.0812507 0.996694i \(-0.474109\pi\)
0.0812507 + 0.996694i \(0.474109\pi\)
\(912\) 49.4315 1.63684
\(913\) −23.1926 −0.767563
\(914\) −106.191 −3.51250
\(915\) −6.97450 −0.230570
\(916\) −3.25475 −0.107540
\(917\) 31.1595 1.02898
\(918\) −29.5356 −0.974819
\(919\) 29.1191 0.960551 0.480276 0.877118i \(-0.340537\pi\)
0.480276 + 0.877118i \(0.340537\pi\)
\(920\) 69.6587 2.29658
\(921\) 16.8014 0.553626
\(922\) −44.2464 −1.45718
\(923\) 1.40495 0.0462444
\(924\) 19.7132 0.648518
\(925\) 25.2757 0.831059
\(926\) 97.5113 3.20442
\(927\) −9.98774 −0.328040
\(928\) 38.3958 1.26040
\(929\) −54.8629 −1.79999 −0.899997 0.435896i \(-0.856431\pi\)
−0.899997 + 0.435896i \(0.856431\pi\)
\(930\) 11.9794 0.392818
\(931\) 5.03999 0.165179
\(932\) −29.6966 −0.972745
\(933\) −18.5823 −0.608356
\(934\) 69.6347 2.27852
\(935\) 6.81539 0.222887
\(936\) −18.0717 −0.590693
\(937\) 3.01907 0.0986286 0.0493143 0.998783i \(-0.484296\pi\)
0.0493143 + 0.998783i \(0.484296\pi\)
\(938\) 29.8285 0.973936
\(939\) 13.3423 0.435408
\(940\) 41.4297 1.35129
\(941\) 12.6616 0.412758 0.206379 0.978472i \(-0.433832\pi\)
0.206379 + 0.978472i \(0.433832\pi\)
\(942\) −0.0361323 −0.00117725
\(943\) −55.4647 −1.80618
\(944\) 46.7372 1.52117
\(945\) 11.0157 0.358342
\(946\) 39.5360 1.28543
\(947\) −11.8605 −0.385415 −0.192708 0.981256i \(-0.561727\pi\)
−0.192708 + 0.981256i \(0.561727\pi\)
\(948\) 52.0219 1.68959
\(949\) 8.94604 0.290401
\(950\) 80.1082 2.59905
\(951\) −6.19890 −0.201013
\(952\) 59.0580 1.91408
\(953\) −49.5246 −1.60426 −0.802129 0.597151i \(-0.796299\pi\)
−0.802129 + 0.597151i \(0.796299\pi\)
\(954\) −72.8160 −2.35750
\(955\) −1.29544 −0.0419196
\(956\) −58.8929 −1.90473
\(957\) −6.19143 −0.200141
\(958\) 79.9042 2.58159
\(959\) 25.3986 0.820163
\(960\) −4.33380 −0.139873
\(961\) 9.83930 0.317397
\(962\) −16.8291 −0.542592
\(963\) 36.6626 1.18144
\(964\) −92.9743 −2.99450
\(965\) −17.7156 −0.570287
\(966\) −45.9775 −1.47930
\(967\) −50.4159 −1.62127 −0.810634 0.585554i \(-0.800877\pi\)
−0.810634 + 0.585554i \(0.800877\pi\)
\(968\) 44.7385 1.43795
\(969\) −16.2449 −0.521860
\(970\) 15.7919 0.507047
\(971\) −8.01922 −0.257349 −0.128675 0.991687i \(-0.541072\pi\)
−0.128675 + 0.991687i \(0.541072\pi\)
\(972\) 70.4925 2.26105
\(973\) 9.92627 0.318221
\(974\) −42.7101 −1.36852
\(975\) 2.68805 0.0860865
\(976\) 87.8678 2.81258
\(977\) −8.72755 −0.279219 −0.139610 0.990207i \(-0.544585\pi\)
−0.139610 + 0.990207i \(0.544585\pi\)
\(978\) −32.7955 −1.04868
\(979\) −16.6804 −0.533109
\(980\) −3.17696 −0.101484
\(981\) 48.1006 1.53573
\(982\) −91.4718 −2.91898
\(983\) 48.4105 1.54406 0.772028 0.635589i \(-0.219243\pi\)
0.772028 + 0.635589i \(0.219243\pi\)
\(984\) 29.4048 0.937391
\(985\) 22.4969 0.716810
\(986\) −32.0221 −1.01979
\(987\) −15.8396 −0.504179
\(988\) −37.5419 −1.19437
\(989\) −64.9023 −2.06377
\(990\) −14.9784 −0.476044
\(991\) 9.26982 0.294466 0.147233 0.989102i \(-0.452963\pi\)
0.147233 + 0.989102i \(0.452963\pi\)
\(992\) −59.4697 −1.88817
\(993\) 18.9916 0.602682
\(994\) 10.0905 0.320052
\(995\) 17.1021 0.542172
\(996\) −34.8463 −1.10415
\(997\) −29.5072 −0.934503 −0.467251 0.884125i \(-0.654756\pi\)
−0.467251 + 0.884125i \(0.654756\pi\)
\(998\) 45.3716 1.43621
\(999\) 24.6447 0.779723
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 8047.2.a.c.1.11 151
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8047.2.a.c.1.11 151 1.1 even 1 trivial