Properties

Label 1502.2.a.g.1.16
Level $1502$
Weight $2$
Character 1502.1
Self dual yes
Analytic conductor $11.994$
Analytic rank $0$
Dimension $16$
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] = [1502,2,Mod(1,1502)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1502, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1502.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1502 = 2 \cdot 751 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1502.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: \(11.9935303836\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} - 25 x^{14} + 59 x^{13} + 273 x^{12} - 443 x^{11} - 1620 x^{10} + 1595 x^{9} + 5490 x^{8} - 2787 x^{7} - 10540 x^{6} + 1919 x^{5} + 10822 x^{4} + 132 x^{3} + \cdots + 864 \) 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 1.16
Root \(-2.17558\) of defining polynomial
Character \(\chi\) \(=\) 1502.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.17558 q^{3} +1.00000 q^{4} +2.74937 q^{5} +3.17558 q^{6} -3.80515 q^{7} +1.00000 q^{8} +7.08430 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.17558 q^{3} +1.00000 q^{4} +2.74937 q^{5} +3.17558 q^{6} -3.80515 q^{7} +1.00000 q^{8} +7.08430 q^{9} +2.74937 q^{10} -0.994384 q^{11} +3.17558 q^{12} -1.61008 q^{13} -3.80515 q^{14} +8.73083 q^{15} +1.00000 q^{16} -3.55443 q^{17} +7.08430 q^{18} +1.37594 q^{19} +2.74937 q^{20} -12.0836 q^{21} -0.994384 q^{22} +4.07706 q^{23} +3.17558 q^{24} +2.55902 q^{25} -1.61008 q^{26} +12.9700 q^{27} -3.80515 q^{28} +7.53219 q^{29} +8.73083 q^{30} -3.04814 q^{31} +1.00000 q^{32} -3.15774 q^{33} -3.55443 q^{34} -10.4618 q^{35} +7.08430 q^{36} +4.67498 q^{37} +1.37594 q^{38} -5.11293 q^{39} +2.74937 q^{40} -7.29785 q^{41} -12.0836 q^{42} -7.79973 q^{43} -0.994384 q^{44} +19.4773 q^{45} +4.07706 q^{46} +5.89187 q^{47} +3.17558 q^{48} +7.47920 q^{49} +2.55902 q^{50} -11.2874 q^{51} -1.61008 q^{52} -8.19172 q^{53} +12.9700 q^{54} -2.73393 q^{55} -3.80515 q^{56} +4.36940 q^{57} +7.53219 q^{58} +2.50465 q^{59} +8.73083 q^{60} -15.2149 q^{61} -3.04814 q^{62} -26.9568 q^{63} +1.00000 q^{64} -4.42669 q^{65} -3.15774 q^{66} +0.0467268 q^{67} -3.55443 q^{68} +12.9470 q^{69} -10.4618 q^{70} +10.0920 q^{71} +7.08430 q^{72} -6.04356 q^{73} +4.67498 q^{74} +8.12638 q^{75} +1.37594 q^{76} +3.78378 q^{77} -5.11293 q^{78} -5.29564 q^{79} +2.74937 q^{80} +19.9344 q^{81} -7.29785 q^{82} +11.2022 q^{83} -12.0836 q^{84} -9.77243 q^{85} -7.79973 q^{86} +23.9191 q^{87} -0.994384 q^{88} -14.6766 q^{89} +19.4773 q^{90} +6.12659 q^{91} +4.07706 q^{92} -9.67962 q^{93} +5.89187 q^{94} +3.78296 q^{95} +3.17558 q^{96} -6.54687 q^{97} +7.47920 q^{98} -7.04451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 13 q^{3} + 16 q^{4} + 4 q^{5} + 13 q^{6} + 7 q^{7} + 16 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 13 q^{3} + 16 q^{4} + 4 q^{5} + 13 q^{6} + 7 q^{7} + 16 q^{8} + 21 q^{9} + 4 q^{10} + 4 q^{11} + 13 q^{12} + 17 q^{13} + 7 q^{14} + 8 q^{15} + 16 q^{16} - q^{17} + 21 q^{18} + 23 q^{19} + 4 q^{20} + 9 q^{21} + 4 q^{22} + 15 q^{23} + 13 q^{24} + 24 q^{25} + 17 q^{26} + 31 q^{27} + 7 q^{28} + 4 q^{29} + 8 q^{30} + 42 q^{31} + 16 q^{32} + 3 q^{33} - q^{34} - 13 q^{35} + 21 q^{36} + 31 q^{37} + 23 q^{38} - 2 q^{39} + 4 q^{40} - 9 q^{41} + 9 q^{42} + 13 q^{43} + 4 q^{44} - 2 q^{45} + 15 q^{46} + 18 q^{47} + 13 q^{48} - 9 q^{49} + 24 q^{50} - 2 q^{51} + 17 q^{52} - 14 q^{53} + 31 q^{54} - 2 q^{55} + 7 q^{56} - 18 q^{57} + 4 q^{58} + 4 q^{59} + 8 q^{60} + q^{61} + 42 q^{62} + 17 q^{63} + 16 q^{64} - 32 q^{65} + 3 q^{66} + 5 q^{67} - q^{68} + 6 q^{69} - 13 q^{70} + 9 q^{71} + 21 q^{72} + 28 q^{73} + 31 q^{74} + 16 q^{75} + 23 q^{76} - 30 q^{77} - 2 q^{78} + 10 q^{79} + 4 q^{80} + 12 q^{81} - 9 q^{82} + 3 q^{83} + 9 q^{84} - 7 q^{85} + 13 q^{86} - 22 q^{87} + 4 q^{88} - 17 q^{89} - 2 q^{90} + 12 q^{91} + 15 q^{92} - q^{93} + 18 q^{94} - 4 q^{95} + 13 q^{96} - 17 q^{97} - 9 q^{98} - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 3.17558 1.83342 0.916710 0.399552i \(-0.130834\pi\)
0.916710 + 0.399552i \(0.130834\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.74937 1.22955 0.614777 0.788701i \(-0.289246\pi\)
0.614777 + 0.788701i \(0.289246\pi\)
\(6\) 3.17558 1.29642
\(7\) −3.80515 −1.43821 −0.719107 0.694900i \(-0.755449\pi\)
−0.719107 + 0.694900i \(0.755449\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.08430 2.36143
\(10\) 2.74937 0.869426
\(11\) −0.994384 −0.299818 −0.149909 0.988700i \(-0.547898\pi\)
−0.149909 + 0.988700i \(0.547898\pi\)
\(12\) 3.17558 0.916710
\(13\) −1.61008 −0.446555 −0.223278 0.974755i \(-0.571676\pi\)
−0.223278 + 0.974755i \(0.571676\pi\)
\(14\) −3.80515 −1.01697
\(15\) 8.73083 2.25429
\(16\) 1.00000 0.250000
\(17\) −3.55443 −0.862076 −0.431038 0.902334i \(-0.641852\pi\)
−0.431038 + 0.902334i \(0.641852\pi\)
\(18\) 7.08430 1.66978
\(19\) 1.37594 0.315662 0.157831 0.987466i \(-0.449550\pi\)
0.157831 + 0.987466i \(0.449550\pi\)
\(20\) 2.74937 0.614777
\(21\) −12.0836 −2.63685
\(22\) −0.994384 −0.212003
\(23\) 4.07706 0.850126 0.425063 0.905164i \(-0.360252\pi\)
0.425063 + 0.905164i \(0.360252\pi\)
\(24\) 3.17558 0.648212
\(25\) 2.55902 0.511805
\(26\) −1.61008 −0.315762
\(27\) 12.9700 2.49608
\(28\) −3.80515 −0.719107
\(29\) 7.53219 1.39869 0.699346 0.714783i \(-0.253475\pi\)
0.699346 + 0.714783i \(0.253475\pi\)
\(30\) 8.73083 1.59402
\(31\) −3.04814 −0.547463 −0.273731 0.961806i \(-0.588258\pi\)
−0.273731 + 0.961806i \(0.588258\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.15774 −0.549693
\(34\) −3.55443 −0.609580
\(35\) −10.4618 −1.76836
\(36\) 7.08430 1.18072
\(37\) 4.67498 0.768562 0.384281 0.923216i \(-0.374449\pi\)
0.384281 + 0.923216i \(0.374449\pi\)
\(38\) 1.37594 0.223207
\(39\) −5.11293 −0.818723
\(40\) 2.74937 0.434713
\(41\) −7.29785 −1.13973 −0.569866 0.821738i \(-0.693005\pi\)
−0.569866 + 0.821738i \(0.693005\pi\)
\(42\) −12.0836 −1.86453
\(43\) −7.79973 −1.18945 −0.594724 0.803930i \(-0.702739\pi\)
−0.594724 + 0.803930i \(0.702739\pi\)
\(44\) −0.994384 −0.149909
\(45\) 19.4773 2.90351
\(46\) 4.07706 0.601130
\(47\) 5.89187 0.859418 0.429709 0.902967i \(-0.358616\pi\)
0.429709 + 0.902967i \(0.358616\pi\)
\(48\) 3.17558 0.458355
\(49\) 7.47920 1.06846
\(50\) 2.55902 0.361900
\(51\) −11.2874 −1.58055
\(52\) −1.61008 −0.223278
\(53\) −8.19172 −1.12522 −0.562609 0.826723i \(-0.690202\pi\)
−0.562609 + 0.826723i \(0.690202\pi\)
\(54\) 12.9700 1.76499
\(55\) −2.73393 −0.368643
\(56\) −3.80515 −0.508485
\(57\) 4.36940 0.578742
\(58\) 7.53219 0.989025
\(59\) 2.50465 0.326078 0.163039 0.986620i \(-0.447870\pi\)
0.163039 + 0.986620i \(0.447870\pi\)
\(60\) 8.73083 1.12715
\(61\) −15.2149 −1.94807 −0.974037 0.226389i \(-0.927308\pi\)
−0.974037 + 0.226389i \(0.927308\pi\)
\(62\) −3.04814 −0.387115
\(63\) −26.9568 −3.39624
\(64\) 1.00000 0.125000
\(65\) −4.42669 −0.549064
\(66\) −3.15774 −0.388691
\(67\) 0.0467268 0.00570859 0.00285430 0.999996i \(-0.499091\pi\)
0.00285430 + 0.999996i \(0.499091\pi\)
\(68\) −3.55443 −0.431038
\(69\) 12.9470 1.55864
\(70\) −10.4618 −1.25042
\(71\) 10.0920 1.19770 0.598848 0.800863i \(-0.295625\pi\)
0.598848 + 0.800863i \(0.295625\pi\)
\(72\) 7.08430 0.834892
\(73\) −6.04356 −0.707345 −0.353673 0.935369i \(-0.615067\pi\)
−0.353673 + 0.935369i \(0.615067\pi\)
\(74\) 4.67498 0.543456
\(75\) 8.12638 0.938353
\(76\) 1.37594 0.157831
\(77\) 3.78378 0.431202
\(78\) −5.11293 −0.578925
\(79\) −5.29564 −0.595806 −0.297903 0.954596i \(-0.596287\pi\)
−0.297903 + 0.954596i \(0.596287\pi\)
\(80\) 2.74937 0.307389
\(81\) 19.9344 2.21493
\(82\) −7.29785 −0.805912
\(83\) 11.2022 1.22960 0.614802 0.788681i \(-0.289236\pi\)
0.614802 + 0.788681i \(0.289236\pi\)
\(84\) −12.0836 −1.31842
\(85\) −9.77243 −1.05997
\(86\) −7.79973 −0.841067
\(87\) 23.9191 2.56439
\(88\) −0.994384 −0.106002
\(89\) −14.6766 −1.55571 −0.777857 0.628442i \(-0.783693\pi\)
−0.777857 + 0.628442i \(0.783693\pi\)
\(90\) 19.4773 2.05309
\(91\) 6.12659 0.642241
\(92\) 4.07706 0.425063
\(93\) −9.67962 −1.00373
\(94\) 5.89187 0.607700
\(95\) 3.78296 0.388124
\(96\) 3.17558 0.324106
\(97\) −6.54687 −0.664734 −0.332367 0.943150i \(-0.607847\pi\)
−0.332367 + 0.943150i \(0.607847\pi\)
\(98\) 7.47920 0.755513
\(99\) −7.04451 −0.708000
\(100\) 2.55902 0.255902
\(101\) −4.04332 −0.402325 −0.201163 0.979558i \(-0.564472\pi\)
−0.201163 + 0.979558i \(0.564472\pi\)
\(102\) −11.2874 −1.11762
\(103\) 5.06385 0.498956 0.249478 0.968380i \(-0.419741\pi\)
0.249478 + 0.968380i \(0.419741\pi\)
\(104\) −1.61008 −0.157881
\(105\) −33.2222 −3.24215
\(106\) −8.19172 −0.795650
\(107\) −9.95084 −0.961984 −0.480992 0.876725i \(-0.659723\pi\)
−0.480992 + 0.876725i \(0.659723\pi\)
\(108\) 12.9700 1.24804
\(109\) −5.88020 −0.563221 −0.281610 0.959529i \(-0.590869\pi\)
−0.281610 + 0.959529i \(0.590869\pi\)
\(110\) −2.73393 −0.260670
\(111\) 14.8458 1.40910
\(112\) −3.80515 −0.359553
\(113\) −16.1497 −1.51924 −0.759619 0.650369i \(-0.774614\pi\)
−0.759619 + 0.650369i \(0.774614\pi\)
\(114\) 4.36940 0.409232
\(115\) 11.2093 1.04528
\(116\) 7.53219 0.699346
\(117\) −11.4063 −1.05451
\(118\) 2.50465 0.230572
\(119\) 13.5252 1.23985
\(120\) 8.73083 0.797012
\(121\) −10.0112 −0.910109
\(122\) −15.2149 −1.37750
\(123\) −23.1749 −2.08961
\(124\) −3.04814 −0.273731
\(125\) −6.71114 −0.600263
\(126\) −26.9568 −2.40151
\(127\) −16.6680 −1.47905 −0.739524 0.673130i \(-0.764949\pi\)
−0.739524 + 0.673130i \(0.764949\pi\)
\(128\) 1.00000 0.0883883
\(129\) −24.7687 −2.18076
\(130\) −4.42669 −0.388247
\(131\) 18.6452 1.62904 0.814520 0.580136i \(-0.197001\pi\)
0.814520 + 0.580136i \(0.197001\pi\)
\(132\) −3.15774 −0.274846
\(133\) −5.23566 −0.453989
\(134\) 0.0467268 0.00403658
\(135\) 35.6593 3.06906
\(136\) −3.55443 −0.304790
\(137\) 1.77649 0.151776 0.0758880 0.997116i \(-0.475821\pi\)
0.0758880 + 0.997116i \(0.475821\pi\)
\(138\) 12.9470 1.10212
\(139\) 2.27140 0.192657 0.0963287 0.995350i \(-0.469290\pi\)
0.0963287 + 0.995350i \(0.469290\pi\)
\(140\) −10.4618 −0.884181
\(141\) 18.7101 1.57568
\(142\) 10.0920 0.846899
\(143\) 1.60103 0.133885
\(144\) 7.08430 0.590358
\(145\) 20.7088 1.71977
\(146\) −6.04356 −0.500169
\(147\) 23.7508 1.95893
\(148\) 4.67498 0.384281
\(149\) 12.1540 0.995695 0.497848 0.867264i \(-0.334124\pi\)
0.497848 + 0.867264i \(0.334124\pi\)
\(150\) 8.12638 0.663516
\(151\) −2.34401 −0.190753 −0.0953765 0.995441i \(-0.530406\pi\)
−0.0953765 + 0.995441i \(0.530406\pi\)
\(152\) 1.37594 0.111603
\(153\) −25.1806 −2.03573
\(154\) 3.78378 0.304906
\(155\) −8.38047 −0.673135
\(156\) −5.11293 −0.409362
\(157\) 17.7604 1.41743 0.708715 0.705495i \(-0.249275\pi\)
0.708715 + 0.705495i \(0.249275\pi\)
\(158\) −5.29564 −0.421299
\(159\) −26.0134 −2.06300
\(160\) 2.74937 0.217357
\(161\) −15.5138 −1.22266
\(162\) 19.9344 1.56619
\(163\) −3.80818 −0.298280 −0.149140 0.988816i \(-0.547650\pi\)
−0.149140 + 0.988816i \(0.547650\pi\)
\(164\) −7.29785 −0.569866
\(165\) −8.68180 −0.675877
\(166\) 11.2022 0.869461
\(167\) 13.7656 1.06522 0.532608 0.846362i \(-0.321212\pi\)
0.532608 + 0.846362i \(0.321212\pi\)
\(168\) −12.0836 −0.932267
\(169\) −10.4077 −0.800589
\(170\) −9.77243 −0.749511
\(171\) 9.74756 0.745415
\(172\) −7.79973 −0.594724
\(173\) 19.3413 1.47049 0.735246 0.677800i \(-0.237067\pi\)
0.735246 + 0.677800i \(0.237067\pi\)
\(174\) 23.9191 1.81330
\(175\) −9.73748 −0.736084
\(176\) −0.994384 −0.0749545
\(177\) 7.95371 0.597837
\(178\) −14.6766 −1.10006
\(179\) −0.585208 −0.0437405 −0.0218702 0.999761i \(-0.506962\pi\)
−0.0218702 + 0.999761i \(0.506962\pi\)
\(180\) 19.4773 1.45175
\(181\) −1.34299 −0.0998240 −0.0499120 0.998754i \(-0.515894\pi\)
−0.0499120 + 0.998754i \(0.515894\pi\)
\(182\) 6.12659 0.454133
\(183\) −48.3162 −3.57164
\(184\) 4.07706 0.300565
\(185\) 12.8532 0.944989
\(186\) −9.67962 −0.709744
\(187\) 3.53447 0.258466
\(188\) 5.89187 0.429709
\(189\) −49.3529 −3.58989
\(190\) 3.78296 0.274445
\(191\) 24.4599 1.76985 0.884927 0.465730i \(-0.154208\pi\)
0.884927 + 0.465730i \(0.154208\pi\)
\(192\) 3.17558 0.229178
\(193\) 20.2681 1.45893 0.729464 0.684019i \(-0.239770\pi\)
0.729464 + 0.684019i \(0.239770\pi\)
\(194\) −6.54687 −0.470038
\(195\) −14.0573 −1.00667
\(196\) 7.47920 0.534228
\(197\) 11.6804 0.832197 0.416098 0.909320i \(-0.363397\pi\)
0.416098 + 0.909320i \(0.363397\pi\)
\(198\) −7.04451 −0.500632
\(199\) 11.4681 0.812955 0.406477 0.913661i \(-0.366757\pi\)
0.406477 + 0.913661i \(0.366757\pi\)
\(200\) 2.55902 0.180950
\(201\) 0.148385 0.0104662
\(202\) −4.04332 −0.284487
\(203\) −28.6611 −2.01162
\(204\) −11.2874 −0.790274
\(205\) −20.0645 −1.40136
\(206\) 5.06385 0.352815
\(207\) 28.8831 2.00751
\(208\) −1.61008 −0.111639
\(209\) −1.36821 −0.0946412
\(210\) −33.2222 −2.29255
\(211\) −3.56683 −0.245551 −0.122775 0.992434i \(-0.539179\pi\)
−0.122775 + 0.992434i \(0.539179\pi\)
\(212\) −8.19172 −0.562609
\(213\) 32.0478 2.19588
\(214\) −9.95084 −0.680226
\(215\) −21.4443 −1.46249
\(216\) 12.9700 0.882497
\(217\) 11.5987 0.787368
\(218\) −5.88020 −0.398257
\(219\) −19.1918 −1.29686
\(220\) −2.73393 −0.184321
\(221\) 5.72291 0.384964
\(222\) 14.8458 0.996383
\(223\) 20.0646 1.34362 0.671811 0.740723i \(-0.265517\pi\)
0.671811 + 0.740723i \(0.265517\pi\)
\(224\) −3.80515 −0.254243
\(225\) 18.1289 1.20859
\(226\) −16.1497 −1.07426
\(227\) 16.1415 1.07135 0.535675 0.844424i \(-0.320057\pi\)
0.535675 + 0.844424i \(0.320057\pi\)
\(228\) 4.36940 0.289371
\(229\) −21.5723 −1.42554 −0.712768 0.701399i \(-0.752559\pi\)
−0.712768 + 0.701399i \(0.752559\pi\)
\(230\) 11.2093 0.739122
\(231\) 12.0157 0.790575
\(232\) 7.53219 0.494513
\(233\) 24.3039 1.59220 0.796099 0.605166i \(-0.206893\pi\)
0.796099 + 0.605166i \(0.206893\pi\)
\(234\) −11.4063 −0.745651
\(235\) 16.1989 1.05670
\(236\) 2.50465 0.163039
\(237\) −16.8167 −1.09236
\(238\) 13.5252 0.876705
\(239\) 17.5860 1.13754 0.568771 0.822496i \(-0.307419\pi\)
0.568771 + 0.822496i \(0.307419\pi\)
\(240\) 8.73083 0.563573
\(241\) 9.83124 0.633286 0.316643 0.948545i \(-0.397444\pi\)
0.316643 + 0.948545i \(0.397444\pi\)
\(242\) −10.0112 −0.643544
\(243\) 24.3931 1.56482
\(244\) −15.2149 −0.974037
\(245\) 20.5631 1.31373
\(246\) −23.1749 −1.47758
\(247\) −2.21537 −0.140960
\(248\) −3.04814 −0.193557
\(249\) 35.5735 2.25438
\(250\) −6.71114 −0.424450
\(251\) 16.1118 1.01697 0.508485 0.861071i \(-0.330206\pi\)
0.508485 + 0.861071i \(0.330206\pi\)
\(252\) −26.9568 −1.69812
\(253\) −4.05416 −0.254883
\(254\) −16.6680 −1.04584
\(255\) −31.0331 −1.94337
\(256\) 1.00000 0.0625000
\(257\) 3.93245 0.245299 0.122650 0.992450i \(-0.460861\pi\)
0.122650 + 0.992450i \(0.460861\pi\)
\(258\) −24.7687 −1.54203
\(259\) −17.7890 −1.10536
\(260\) −4.42669 −0.274532
\(261\) 53.3603 3.30292
\(262\) 18.6452 1.15190
\(263\) −12.7883 −0.788560 −0.394280 0.918990i \(-0.629006\pi\)
−0.394280 + 0.918990i \(0.629006\pi\)
\(264\) −3.15774 −0.194346
\(265\) −22.5220 −1.38352
\(266\) −5.23566 −0.321019
\(267\) −46.6066 −2.85228
\(268\) 0.0467268 0.00285430
\(269\) 27.2182 1.65952 0.829761 0.558119i \(-0.188477\pi\)
0.829761 + 0.558119i \(0.188477\pi\)
\(270\) 35.6593 2.17016
\(271\) −15.6191 −0.948792 −0.474396 0.880311i \(-0.657334\pi\)
−0.474396 + 0.880311i \(0.657334\pi\)
\(272\) −3.55443 −0.215519
\(273\) 19.4555 1.17750
\(274\) 1.77649 0.107322
\(275\) −2.54465 −0.153448
\(276\) 12.9470 0.779319
\(277\) 15.0948 0.906958 0.453479 0.891267i \(-0.350183\pi\)
0.453479 + 0.891267i \(0.350183\pi\)
\(278\) 2.27140 0.136229
\(279\) −21.5940 −1.29280
\(280\) −10.4618 −0.625210
\(281\) −13.1122 −0.782210 −0.391105 0.920346i \(-0.627907\pi\)
−0.391105 + 0.920346i \(0.627907\pi\)
\(282\) 18.7101 1.11417
\(283\) 15.6931 0.932859 0.466429 0.884558i \(-0.345540\pi\)
0.466429 + 0.884558i \(0.345540\pi\)
\(284\) 10.0920 0.598848
\(285\) 12.0131 0.711594
\(286\) 1.60103 0.0946712
\(287\) 27.7694 1.63918
\(288\) 7.08430 0.417446
\(289\) −4.36603 −0.256825
\(290\) 20.7088 1.21606
\(291\) −20.7901 −1.21874
\(292\) −6.04356 −0.353673
\(293\) 7.06624 0.412814 0.206407 0.978466i \(-0.433823\pi\)
0.206407 + 0.978466i \(0.433823\pi\)
\(294\) 23.7508 1.38517
\(295\) 6.88620 0.400930
\(296\) 4.67498 0.271728
\(297\) −12.8972 −0.748369
\(298\) 12.1540 0.704063
\(299\) −6.56438 −0.379628
\(300\) 8.12638 0.469177
\(301\) 29.6792 1.71068
\(302\) −2.34401 −0.134883
\(303\) −12.8399 −0.737631
\(304\) 1.37594 0.0789155
\(305\) −41.8315 −2.39526
\(306\) −25.1806 −1.43948
\(307\) −3.77208 −0.215284 −0.107642 0.994190i \(-0.534330\pi\)
−0.107642 + 0.994190i \(0.534330\pi\)
\(308\) 3.78378 0.215601
\(309\) 16.0807 0.914796
\(310\) −8.38047 −0.475979
\(311\) −19.8737 −1.12693 −0.563467 0.826138i \(-0.690533\pi\)
−0.563467 + 0.826138i \(0.690533\pi\)
\(312\) −5.11293 −0.289462
\(313\) −25.6759 −1.45129 −0.725644 0.688070i \(-0.758458\pi\)
−0.725644 + 0.688070i \(0.758458\pi\)
\(314\) 17.7604 1.00227
\(315\) −74.1143 −4.17587
\(316\) −5.29564 −0.297903
\(317\) −13.4370 −0.754699 −0.377350 0.926071i \(-0.623165\pi\)
−0.377350 + 0.926071i \(0.623165\pi\)
\(318\) −26.0134 −1.45876
\(319\) −7.48989 −0.419353
\(320\) 2.74937 0.153694
\(321\) −31.5997 −1.76372
\(322\) −15.5138 −0.864552
\(323\) −4.89068 −0.272125
\(324\) 19.9344 1.10747
\(325\) −4.12022 −0.228549
\(326\) −3.80818 −0.210916
\(327\) −18.6730 −1.03262
\(328\) −7.29785 −0.402956
\(329\) −22.4195 −1.23603
\(330\) −8.68180 −0.477917
\(331\) 4.05100 0.222663 0.111332 0.993783i \(-0.464488\pi\)
0.111332 + 0.993783i \(0.464488\pi\)
\(332\) 11.2022 0.614802
\(333\) 33.1190 1.81491
\(334\) 13.7656 0.753221
\(335\) 0.128469 0.00701902
\(336\) −12.0836 −0.659212
\(337\) −4.81379 −0.262224 −0.131112 0.991368i \(-0.541855\pi\)
−0.131112 + 0.991368i \(0.541855\pi\)
\(338\) −10.4077 −0.566102
\(339\) −51.2847 −2.78540
\(340\) −9.77243 −0.529985
\(341\) 3.03103 0.164139
\(342\) 9.74756 0.527088
\(343\) −1.82342 −0.0984555
\(344\) −7.79973 −0.420533
\(345\) 35.5961 1.91643
\(346\) 19.3413 1.03979
\(347\) −28.0304 −1.50475 −0.752377 0.658733i \(-0.771093\pi\)
−0.752377 + 0.658733i \(0.771093\pi\)
\(348\) 23.9191 1.28220
\(349\) −14.5076 −0.776573 −0.388286 0.921539i \(-0.626933\pi\)
−0.388286 + 0.921539i \(0.626933\pi\)
\(350\) −9.73748 −0.520490
\(351\) −20.8827 −1.11464
\(352\) −0.994384 −0.0530009
\(353\) 18.4431 0.981626 0.490813 0.871265i \(-0.336700\pi\)
0.490813 + 0.871265i \(0.336700\pi\)
\(354\) 7.95371 0.422735
\(355\) 27.7465 1.47263
\(356\) −14.6766 −0.777857
\(357\) 42.9502 2.27316
\(358\) −0.585208 −0.0309292
\(359\) 18.4513 0.973821 0.486910 0.873452i \(-0.338124\pi\)
0.486910 + 0.873452i \(0.338124\pi\)
\(360\) 19.4773 1.02655
\(361\) −17.1068 −0.900357
\(362\) −1.34299 −0.0705862
\(363\) −31.7913 −1.66861
\(364\) 6.12659 0.321121
\(365\) −16.6160 −0.869719
\(366\) −48.3162 −2.52553
\(367\) 29.7137 1.55104 0.775521 0.631322i \(-0.217487\pi\)
0.775521 + 0.631322i \(0.217487\pi\)
\(368\) 4.07706 0.212531
\(369\) −51.7001 −2.69140
\(370\) 12.8532 0.668208
\(371\) 31.1707 1.61830
\(372\) −9.67962 −0.501865
\(373\) 28.2684 1.46368 0.731840 0.681476i \(-0.238662\pi\)
0.731840 + 0.681476i \(0.238662\pi\)
\(374\) 3.53447 0.182763
\(375\) −21.3118 −1.10053
\(376\) 5.89187 0.303850
\(377\) −12.1274 −0.624593
\(378\) −49.3529 −2.53844
\(379\) −3.52360 −0.180995 −0.0904975 0.995897i \(-0.528846\pi\)
−0.0904975 + 0.995897i \(0.528846\pi\)
\(380\) 3.78296 0.194062
\(381\) −52.9306 −2.71172
\(382\) 24.4599 1.25148
\(383\) 30.7104 1.56923 0.784613 0.619986i \(-0.212862\pi\)
0.784613 + 0.619986i \(0.212862\pi\)
\(384\) 3.17558 0.162053
\(385\) 10.4030 0.530187
\(386\) 20.2681 1.03162
\(387\) −55.2556 −2.80880
\(388\) −6.54687 −0.332367
\(389\) −14.8158 −0.751193 −0.375597 0.926783i \(-0.622562\pi\)
−0.375597 + 0.926783i \(0.622562\pi\)
\(390\) −14.0573 −0.711820
\(391\) −14.4916 −0.732873
\(392\) 7.47920 0.377757
\(393\) 59.2093 2.98672
\(394\) 11.6804 0.588452
\(395\) −14.5597 −0.732577
\(396\) −7.04451 −0.354000
\(397\) 13.4129 0.673174 0.336587 0.941652i \(-0.390727\pi\)
0.336587 + 0.941652i \(0.390727\pi\)
\(398\) 11.4681 0.574846
\(399\) −16.6263 −0.832354
\(400\) 2.55902 0.127951
\(401\) −37.5566 −1.87549 −0.937744 0.347327i \(-0.887089\pi\)
−0.937744 + 0.347327i \(0.887089\pi\)
\(402\) 0.148385 0.00740076
\(403\) 4.90775 0.244472
\(404\) −4.04332 −0.201163
\(405\) 54.8069 2.72338
\(406\) −28.6611 −1.42243
\(407\) −4.64873 −0.230429
\(408\) −11.2874 −0.558808
\(409\) 24.0263 1.18802 0.594012 0.804456i \(-0.297543\pi\)
0.594012 + 0.804456i \(0.297543\pi\)
\(410\) −20.0645 −0.990913
\(411\) 5.64139 0.278269
\(412\) 5.06385 0.249478
\(413\) −9.53058 −0.468969
\(414\) 28.8831 1.41953
\(415\) 30.7990 1.51187
\(416\) −1.61008 −0.0789405
\(417\) 7.21300 0.353222
\(418\) −1.36821 −0.0669214
\(419\) −32.4579 −1.58567 −0.792837 0.609434i \(-0.791397\pi\)
−0.792837 + 0.609434i \(0.791397\pi\)
\(420\) −33.2222 −1.62108
\(421\) 14.0767 0.686055 0.343027 0.939325i \(-0.388548\pi\)
0.343027 + 0.939325i \(0.388548\pi\)
\(422\) −3.56683 −0.173631
\(423\) 41.7398 2.02946
\(424\) −8.19172 −0.397825
\(425\) −9.09587 −0.441214
\(426\) 32.0478 1.55272
\(427\) 57.8952 2.80175
\(428\) −9.95084 −0.480992
\(429\) 5.08421 0.245468
\(430\) −21.4443 −1.03414
\(431\) −9.86060 −0.474969 −0.237484 0.971391i \(-0.576323\pi\)
−0.237484 + 0.971391i \(0.576323\pi\)
\(432\) 12.9700 0.624020
\(433\) 29.3909 1.41244 0.706218 0.707994i \(-0.250400\pi\)
0.706218 + 0.707994i \(0.250400\pi\)
\(434\) 11.5987 0.556753
\(435\) 65.7623 3.15306
\(436\) −5.88020 −0.281610
\(437\) 5.60979 0.268352
\(438\) −19.1918 −0.917019
\(439\) −4.57873 −0.218531 −0.109266 0.994013i \(-0.534850\pi\)
−0.109266 + 0.994013i \(0.534850\pi\)
\(440\) −2.73393 −0.130335
\(441\) 52.9849 2.52309
\(442\) 5.72291 0.272211
\(443\) −36.6705 −1.74227 −0.871134 0.491045i \(-0.836615\pi\)
−0.871134 + 0.491045i \(0.836615\pi\)
\(444\) 14.8458 0.704549
\(445\) −40.3513 −1.91283
\(446\) 20.0646 0.950084
\(447\) 38.5960 1.82553
\(448\) −3.80515 −0.179777
\(449\) −10.6630 −0.503216 −0.251608 0.967829i \(-0.580959\pi\)
−0.251608 + 0.967829i \(0.580959\pi\)
\(450\) 18.1289 0.854603
\(451\) 7.25686 0.341712
\(452\) −16.1497 −0.759619
\(453\) −7.44359 −0.349731
\(454\) 16.1415 0.757558
\(455\) 16.8443 0.789671
\(456\) 4.36940 0.204616
\(457\) 39.1144 1.82969 0.914847 0.403801i \(-0.132311\pi\)
0.914847 + 0.403801i \(0.132311\pi\)
\(458\) −21.5723 −1.00801
\(459\) −46.1010 −2.15181
\(460\) 11.2093 0.522638
\(461\) −32.6190 −1.51922 −0.759609 0.650380i \(-0.774610\pi\)
−0.759609 + 0.650380i \(0.774610\pi\)
\(462\) 12.0157 0.559021
\(463\) −7.87682 −0.366067 −0.183033 0.983107i \(-0.558592\pi\)
−0.183033 + 0.983107i \(0.558592\pi\)
\(464\) 7.53219 0.349673
\(465\) −26.6128 −1.23414
\(466\) 24.3039 1.12585
\(467\) 29.1726 1.34995 0.674973 0.737842i \(-0.264155\pi\)
0.674973 + 0.737842i \(0.264155\pi\)
\(468\) −11.4063 −0.527255
\(469\) −0.177803 −0.00821017
\(470\) 16.1989 0.747201
\(471\) 56.3994 2.59875
\(472\) 2.50465 0.115286
\(473\) 7.75593 0.356618
\(474\) −16.8167 −0.772418
\(475\) 3.52106 0.161557
\(476\) 13.5252 0.619924
\(477\) −58.0326 −2.65713
\(478\) 17.5860 0.804363
\(479\) −10.5245 −0.480877 −0.240439 0.970664i \(-0.577291\pi\)
−0.240439 + 0.970664i \(0.577291\pi\)
\(480\) 8.73083 0.398506
\(481\) −7.52708 −0.343205
\(482\) 9.83124 0.447801
\(483\) −49.2654 −2.24165
\(484\) −10.0112 −0.455055
\(485\) −17.9998 −0.817327
\(486\) 24.3931 1.10650
\(487\) 40.4170 1.83147 0.915734 0.401786i \(-0.131610\pi\)
0.915734 + 0.401786i \(0.131610\pi\)
\(488\) −15.2149 −0.688748
\(489\) −12.0932 −0.546872
\(490\) 20.5631 0.928945
\(491\) 12.7430 0.575085 0.287543 0.957768i \(-0.407162\pi\)
0.287543 + 0.957768i \(0.407162\pi\)
\(492\) −23.1749 −1.04480
\(493\) −26.7726 −1.20578
\(494\) −2.21537 −0.0996741
\(495\) −19.3680 −0.870525
\(496\) −3.04814 −0.136866
\(497\) −38.4015 −1.72254
\(498\) 35.5735 1.59409
\(499\) −11.4162 −0.511061 −0.255531 0.966801i \(-0.582250\pi\)
−0.255531 + 0.966801i \(0.582250\pi\)
\(500\) −6.71114 −0.300131
\(501\) 43.7138 1.95299
\(502\) 16.1118 0.719106
\(503\) −33.8483 −1.50922 −0.754610 0.656174i \(-0.772174\pi\)
−0.754610 + 0.656174i \(0.772174\pi\)
\(504\) −26.9568 −1.20075
\(505\) −11.1166 −0.494681
\(506\) −4.05416 −0.180230
\(507\) −33.0503 −1.46782
\(508\) −16.6680 −0.739524
\(509\) −1.42384 −0.0631106 −0.0315553 0.999502i \(-0.510046\pi\)
−0.0315553 + 0.999502i \(0.510046\pi\)
\(510\) −31.0331 −1.37417
\(511\) 22.9967 1.01731
\(512\) 1.00000 0.0441942
\(513\) 17.8459 0.787917
\(514\) 3.93245 0.173453
\(515\) 13.9224 0.613494
\(516\) −24.7687 −1.09038
\(517\) −5.85879 −0.257669
\(518\) −17.7890 −0.781605
\(519\) 61.4198 2.69603
\(520\) −4.42669 −0.194123
\(521\) 14.4693 0.633913 0.316957 0.948440i \(-0.397339\pi\)
0.316957 + 0.948440i \(0.397339\pi\)
\(522\) 53.3603 2.33552
\(523\) −41.3027 −1.80604 −0.903021 0.429595i \(-0.858656\pi\)
−0.903021 + 0.429595i \(0.858656\pi\)
\(524\) 18.6452 0.814520
\(525\) −30.9221 −1.34955
\(526\) −12.7883 −0.557596
\(527\) 10.8344 0.471954
\(528\) −3.15774 −0.137423
\(529\) −6.37759 −0.277286
\(530\) −22.5220 −0.978295
\(531\) 17.7437 0.770010
\(532\) −5.23566 −0.226995
\(533\) 11.7501 0.508953
\(534\) −46.6066 −2.01687
\(535\) −27.3585 −1.18281
\(536\) 0.0467268 0.00201829
\(537\) −1.85837 −0.0801947
\(538\) 27.2182 1.17346
\(539\) −7.43720 −0.320343
\(540\) 35.6593 1.53453
\(541\) −31.6641 −1.36135 −0.680673 0.732588i \(-0.738312\pi\)
−0.680673 + 0.732588i \(0.738312\pi\)
\(542\) −15.6191 −0.670897
\(543\) −4.26478 −0.183019
\(544\) −3.55443 −0.152395
\(545\) −16.1668 −0.692511
\(546\) 19.4555 0.832617
\(547\) −11.9782 −0.512150 −0.256075 0.966657i \(-0.582429\pi\)
−0.256075 + 0.966657i \(0.582429\pi\)
\(548\) 1.77649 0.0758880
\(549\) −107.787 −4.60024
\(550\) −2.54465 −0.108504
\(551\) 10.3638 0.441514
\(552\) 12.9470 0.551062
\(553\) 20.1507 0.856897
\(554\) 15.0948 0.641316
\(555\) 40.8165 1.73256
\(556\) 2.27140 0.0963287
\(557\) −6.47782 −0.274474 −0.137237 0.990538i \(-0.543822\pi\)
−0.137237 + 0.990538i \(0.543822\pi\)
\(558\) −21.5940 −0.914145
\(559\) 12.5582 0.531154
\(560\) −10.4618 −0.442090
\(561\) 11.2240 0.473877
\(562\) −13.1122 −0.553106
\(563\) 13.1123 0.552618 0.276309 0.961069i \(-0.410889\pi\)
0.276309 + 0.961069i \(0.410889\pi\)
\(564\) 18.7101 0.787838
\(565\) −44.4015 −1.86799
\(566\) 15.6931 0.659631
\(567\) −75.8534 −3.18554
\(568\) 10.0920 0.423450
\(569\) 40.7627 1.70886 0.854431 0.519565i \(-0.173906\pi\)
0.854431 + 0.519565i \(0.173906\pi\)
\(570\) 12.0131 0.503173
\(571\) −2.15286 −0.0900942 −0.0450471 0.998985i \(-0.514344\pi\)
−0.0450471 + 0.998985i \(0.514344\pi\)
\(572\) 1.60103 0.0669426
\(573\) 77.6742 3.24489
\(574\) 27.7694 1.15907
\(575\) 10.4333 0.435098
\(576\) 7.08430 0.295179
\(577\) 21.6408 0.900919 0.450459 0.892797i \(-0.351260\pi\)
0.450459 + 0.892797i \(0.351260\pi\)
\(578\) −4.36603 −0.181603
\(579\) 64.3628 2.67483
\(580\) 20.7088 0.859885
\(581\) −42.6262 −1.76843
\(582\) −20.7901 −0.861778
\(583\) 8.14571 0.337361
\(584\) −6.04356 −0.250084
\(585\) −31.3600 −1.29658
\(586\) 7.06624 0.291904
\(587\) −22.8445 −0.942894 −0.471447 0.881894i \(-0.656268\pi\)
−0.471447 + 0.881894i \(0.656268\pi\)
\(588\) 23.7508 0.979466
\(589\) −4.19406 −0.172813
\(590\) 6.88620 0.283500
\(591\) 37.0921 1.52577
\(592\) 4.67498 0.192141
\(593\) 16.3335 0.670735 0.335368 0.942087i \(-0.391139\pi\)
0.335368 + 0.942087i \(0.391139\pi\)
\(594\) −12.8972 −0.529177
\(595\) 37.1856 1.52446
\(596\) 12.1540 0.497848
\(597\) 36.4180 1.49049
\(598\) −6.56438 −0.268437
\(599\) −30.5769 −1.24934 −0.624668 0.780890i \(-0.714766\pi\)
−0.624668 + 0.780890i \(0.714766\pi\)
\(600\) 8.12638 0.331758
\(601\) 20.5192 0.836997 0.418499 0.908217i \(-0.362556\pi\)
0.418499 + 0.908217i \(0.362556\pi\)
\(602\) 29.6792 1.20963
\(603\) 0.331027 0.0134805
\(604\) −2.34401 −0.0953765
\(605\) −27.5245 −1.11903
\(606\) −12.8399 −0.521584
\(607\) −18.3804 −0.746039 −0.373019 0.927824i \(-0.621678\pi\)
−0.373019 + 0.927824i \(0.621678\pi\)
\(608\) 1.37594 0.0558017
\(609\) −91.0157 −3.68814
\(610\) −41.8315 −1.69371
\(611\) −9.48637 −0.383778
\(612\) −25.1806 −1.01787
\(613\) −0.655604 −0.0264796 −0.0132398 0.999912i \(-0.504214\pi\)
−0.0132398 + 0.999912i \(0.504214\pi\)
\(614\) −3.77208 −0.152229
\(615\) −63.7163 −2.56929
\(616\) 3.78378 0.152453
\(617\) −23.5073 −0.946367 −0.473184 0.880964i \(-0.656895\pi\)
−0.473184 + 0.880964i \(0.656895\pi\)
\(618\) 16.0807 0.646859
\(619\) 9.69469 0.389663 0.194831 0.980837i \(-0.437584\pi\)
0.194831 + 0.980837i \(0.437584\pi\)
\(620\) −8.38047 −0.336568
\(621\) 52.8795 2.12198
\(622\) −19.8737 −0.796863
\(623\) 55.8466 2.23745
\(624\) −5.11293 −0.204681
\(625\) −31.2465 −1.24986
\(626\) −25.6759 −1.02622
\(627\) −4.34486 −0.173517
\(628\) 17.7604 0.708715
\(629\) −16.6169 −0.662559
\(630\) −74.1143 −2.95278
\(631\) 31.8474 1.26783 0.633913 0.773404i \(-0.281448\pi\)
0.633913 + 0.773404i \(0.281448\pi\)
\(632\) −5.29564 −0.210649
\(633\) −11.3268 −0.450198
\(634\) −13.4370 −0.533653
\(635\) −45.8265 −1.81857
\(636\) −26.0134 −1.03150
\(637\) −12.0421 −0.477125
\(638\) −7.48989 −0.296528
\(639\) 71.4945 2.82828
\(640\) 2.74937 0.108678
\(641\) 2.87122 0.113406 0.0567032 0.998391i \(-0.481941\pi\)
0.0567032 + 0.998391i \(0.481941\pi\)
\(642\) −31.5997 −1.24714
\(643\) −7.13336 −0.281312 −0.140656 0.990058i \(-0.544921\pi\)
−0.140656 + 0.990058i \(0.544921\pi\)
\(644\) −15.5138 −0.611331
\(645\) −68.0981 −2.68136
\(646\) −4.89068 −0.192421
\(647\) 7.48721 0.294353 0.147176 0.989110i \(-0.452982\pi\)
0.147176 + 0.989110i \(0.452982\pi\)
\(648\) 19.9344 0.783096
\(649\) −2.49058 −0.0977640
\(650\) −4.12022 −0.161608
\(651\) 36.8324 1.44358
\(652\) −3.80818 −0.149140
\(653\) −43.3869 −1.69786 −0.848929 0.528506i \(-0.822752\pi\)
−0.848929 + 0.528506i \(0.822752\pi\)
\(654\) −18.6730 −0.730173
\(655\) 51.2625 2.00299
\(656\) −7.29785 −0.284933
\(657\) −42.8144 −1.67035
\(658\) −22.4195 −0.874003
\(659\) −44.7861 −1.74462 −0.872309 0.488955i \(-0.837378\pi\)
−0.872309 + 0.488955i \(0.837378\pi\)
\(660\) −8.68180 −0.337939
\(661\) −10.0915 −0.392514 −0.196257 0.980552i \(-0.562879\pi\)
−0.196257 + 0.980552i \(0.562879\pi\)
\(662\) 4.05100 0.157447
\(663\) 18.1735 0.705802
\(664\) 11.2022 0.434731
\(665\) −14.3948 −0.558205
\(666\) 33.1190 1.28333
\(667\) 30.7092 1.18906
\(668\) 13.7656 0.532608
\(669\) 63.7166 2.46342
\(670\) 0.128469 0.00496320
\(671\) 15.1295 0.584068
\(672\) −12.0836 −0.466134
\(673\) 6.51802 0.251251 0.125626 0.992078i \(-0.459906\pi\)
0.125626 + 0.992078i \(0.459906\pi\)
\(674\) −4.81379 −0.185420
\(675\) 33.1905 1.27750
\(676\) −10.4077 −0.400294
\(677\) −0.510876 −0.0196346 −0.00981728 0.999952i \(-0.503125\pi\)
−0.00981728 + 0.999952i \(0.503125\pi\)
\(678\) −51.2847 −1.96958
\(679\) 24.9119 0.956030
\(680\) −9.77243 −0.374756
\(681\) 51.2586 1.96423
\(682\) 3.03103 0.116064
\(683\) −11.2651 −0.431047 −0.215524 0.976499i \(-0.569146\pi\)
−0.215524 + 0.976499i \(0.569146\pi\)
\(684\) 9.74756 0.372707
\(685\) 4.88423 0.186617
\(686\) −1.82342 −0.0696186
\(687\) −68.5045 −2.61361
\(688\) −7.79973 −0.297362
\(689\) 13.1893 0.502472
\(690\) 35.5961 1.35512
\(691\) −1.08585 −0.0413077 −0.0206539 0.999787i \(-0.506575\pi\)
−0.0206539 + 0.999787i \(0.506575\pi\)
\(692\) 19.3413 0.735246
\(693\) 26.8055 1.01826
\(694\) −28.0304 −1.06402
\(695\) 6.24491 0.236883
\(696\) 23.9191 0.906650
\(697\) 25.9397 0.982536
\(698\) −14.5076 −0.549120
\(699\) 77.1788 2.91917
\(700\) −9.73748 −0.368042
\(701\) −4.11280 −0.155338 −0.0776691 0.996979i \(-0.524748\pi\)
−0.0776691 + 0.996979i \(0.524748\pi\)
\(702\) −20.8827 −0.788167
\(703\) 6.43249 0.242606
\(704\) −0.994384 −0.0374773
\(705\) 51.4410 1.93738
\(706\) 18.4431 0.694114
\(707\) 15.3854 0.578629
\(708\) 7.95371 0.298919
\(709\) 43.8863 1.64819 0.824093 0.566455i \(-0.191686\pi\)
0.824093 + 0.566455i \(0.191686\pi\)
\(710\) 27.7465 1.04131
\(711\) −37.5159 −1.40696
\(712\) −14.6766 −0.550028
\(713\) −12.4275 −0.465412
\(714\) 42.9502 1.60737
\(715\) 4.40183 0.164619
\(716\) −0.585208 −0.0218702
\(717\) 55.8456 2.08559
\(718\) 18.4513 0.688595
\(719\) −17.6201 −0.657120 −0.328560 0.944483i \(-0.606563\pi\)
−0.328560 + 0.944483i \(0.606563\pi\)
\(720\) 19.4773 0.725877
\(721\) −19.2687 −0.717605
\(722\) −17.1068 −0.636649
\(723\) 31.2199 1.16108
\(724\) −1.34299 −0.0499120
\(725\) 19.2750 0.715857
\(726\) −31.7913 −1.17989
\(727\) −3.49375 −0.129576 −0.0647881 0.997899i \(-0.520637\pi\)
−0.0647881 + 0.997899i \(0.520637\pi\)
\(728\) 6.12659 0.227067
\(729\) 17.6592 0.654045
\(730\) −16.6160 −0.614984
\(731\) 27.7236 1.02539
\(732\) −48.3162 −1.78582
\(733\) −28.8437 −1.06537 −0.532684 0.846314i \(-0.678817\pi\)
−0.532684 + 0.846314i \(0.678817\pi\)
\(734\) 29.7137 1.09675
\(735\) 65.2996 2.40861
\(736\) 4.07706 0.150282
\(737\) −0.0464644 −0.00171154
\(738\) −51.7001 −1.90311
\(739\) 20.2013 0.743118 0.371559 0.928409i \(-0.378823\pi\)
0.371559 + 0.928409i \(0.378823\pi\)
\(740\) 12.8532 0.472495
\(741\) −7.03507 −0.258440
\(742\) 31.1707 1.14431
\(743\) 6.51897 0.239158 0.119579 0.992825i \(-0.461846\pi\)
0.119579 + 0.992825i \(0.461846\pi\)
\(744\) −9.67962 −0.354872
\(745\) 33.4158 1.22426
\(746\) 28.2684 1.03498
\(747\) 79.3599 2.90363
\(748\) 3.53447 0.129233
\(749\) 37.8645 1.38354
\(750\) −21.3118 −0.778196
\(751\) −1.00000 −0.0364905
\(752\) 5.89187 0.214855
\(753\) 51.1643 1.86453
\(754\) −12.1274 −0.441654
\(755\) −6.44455 −0.234541
\(756\) −49.3529 −1.79495
\(757\) −46.8268 −1.70195 −0.850975 0.525206i \(-0.823988\pi\)
−0.850975 + 0.525206i \(0.823988\pi\)
\(758\) −3.52360 −0.127983
\(759\) −12.8743 −0.467308
\(760\) 3.78296 0.137222
\(761\) 41.4119 1.50118 0.750590 0.660768i \(-0.229769\pi\)
0.750590 + 0.660768i \(0.229769\pi\)
\(762\) −52.9306 −1.91747
\(763\) 22.3751 0.810032
\(764\) 24.4599 0.884927
\(765\) −69.2308 −2.50305
\(766\) 30.7104 1.10961
\(767\) −4.03268 −0.145612
\(768\) 3.17558 0.114589
\(769\) −9.69312 −0.349543 −0.174771 0.984609i \(-0.555919\pi\)
−0.174771 + 0.984609i \(0.555919\pi\)
\(770\) 10.4030 0.374899
\(771\) 12.4878 0.449737
\(772\) 20.2681 0.729464
\(773\) −50.4613 −1.81497 −0.907484 0.420087i \(-0.861999\pi\)
−0.907484 + 0.420087i \(0.861999\pi\)
\(774\) −55.2556 −1.98612
\(775\) −7.80027 −0.280194
\(776\) −6.54687 −0.235019
\(777\) −56.4904 −2.02658
\(778\) −14.8158 −0.531174
\(779\) −10.0414 −0.359770
\(780\) −14.0573 −0.503333
\(781\) −10.0353 −0.359091
\(782\) −14.4916 −0.518219
\(783\) 97.6925 3.49125
\(784\) 7.47920 0.267114
\(785\) 48.8297 1.74281
\(786\) 59.2093 2.11193
\(787\) 37.8382 1.34878 0.674392 0.738373i \(-0.264406\pi\)
0.674392 + 0.738373i \(0.264406\pi\)
\(788\) 11.6804 0.416098
\(789\) −40.6102 −1.44576
\(790\) −14.5597 −0.518010
\(791\) 61.4522 2.18499
\(792\) −7.04451 −0.250316
\(793\) 24.4972 0.869922
\(794\) 13.4129 0.476006
\(795\) −71.5205 −2.53657
\(796\) 11.4681 0.406477
\(797\) 30.8500 1.09276 0.546381 0.837537i \(-0.316005\pi\)
0.546381 + 0.837537i \(0.316005\pi\)
\(798\) −16.6263 −0.588563
\(799\) −20.9423 −0.740884
\(800\) 2.55902 0.0904751
\(801\) −103.973 −3.67371
\(802\) −37.5566 −1.32617
\(803\) 6.00962 0.212075
\(804\) 0.148385 0.00523312
\(805\) −42.6532 −1.50333
\(806\) 4.90775 0.172868
\(807\) 86.4334 3.04260
\(808\) −4.04332 −0.142243
\(809\) 23.9700 0.842740 0.421370 0.906889i \(-0.361549\pi\)
0.421370 + 0.906889i \(0.361549\pi\)
\(810\) 54.8069 1.92572
\(811\) −49.0754 −1.72327 −0.861636 0.507527i \(-0.830560\pi\)
−0.861636 + 0.507527i \(0.830560\pi\)
\(812\) −28.6611 −1.00581
\(813\) −49.5997 −1.73954
\(814\) −4.64873 −0.162938
\(815\) −10.4701 −0.366751
\(816\) −11.2874 −0.395137
\(817\) −10.7320 −0.375464
\(818\) 24.0263 0.840060
\(819\) 43.4026 1.51661
\(820\) −20.0645 −0.700682
\(821\) 49.4463 1.72569 0.862843 0.505471i \(-0.168681\pi\)
0.862843 + 0.505471i \(0.168681\pi\)
\(822\) 5.64139 0.196766
\(823\) 46.6706 1.62683 0.813417 0.581681i \(-0.197605\pi\)
0.813417 + 0.581681i \(0.197605\pi\)
\(824\) 5.06385 0.176408
\(825\) −8.08074 −0.281335
\(826\) −9.53058 −0.331611
\(827\) −2.64911 −0.0921185 −0.0460593 0.998939i \(-0.514666\pi\)
−0.0460593 + 0.998939i \(0.514666\pi\)
\(828\) 28.8831 1.00376
\(829\) −0.533637 −0.0185340 −0.00926699 0.999957i \(-0.502950\pi\)
−0.00926699 + 0.999957i \(0.502950\pi\)
\(830\) 30.7990 1.06905
\(831\) 47.9347 1.66284
\(832\) −1.61008 −0.0558194
\(833\) −26.5843 −0.921091
\(834\) 7.21300 0.249766
\(835\) 37.8468 1.30974
\(836\) −1.36821 −0.0473206
\(837\) −39.5344 −1.36651
\(838\) −32.4579 −1.12124
\(839\) 3.26426 0.112695 0.0563474 0.998411i \(-0.482055\pi\)
0.0563474 + 0.998411i \(0.482055\pi\)
\(840\) −33.2222 −1.14627
\(841\) 27.7339 0.956341
\(842\) 14.0767 0.485114
\(843\) −41.6389 −1.43412
\(844\) −3.56683 −0.122775
\(845\) −28.6145 −0.984367
\(846\) 41.7398 1.43504
\(847\) 38.0942 1.30893
\(848\) −8.19172 −0.281305
\(849\) 49.8347 1.71032
\(850\) −9.09587 −0.311986
\(851\) 19.0602 0.653374
\(852\) 32.0478 1.09794
\(853\) 49.6838 1.70114 0.850571 0.525860i \(-0.176256\pi\)
0.850571 + 0.525860i \(0.176256\pi\)
\(854\) 57.8952 1.98113
\(855\) 26.7996 0.916528
\(856\) −9.95084 −0.340113
\(857\) 16.8291 0.574870 0.287435 0.957800i \(-0.407198\pi\)
0.287435 + 0.957800i \(0.407198\pi\)
\(858\) 5.08421 0.173572
\(859\) 2.57724 0.0879343 0.0439671 0.999033i \(-0.486000\pi\)
0.0439671 + 0.999033i \(0.486000\pi\)
\(860\) −21.4443 −0.731246
\(861\) 88.1840 3.00530
\(862\) −9.86060 −0.335853
\(863\) 24.7556 0.842690 0.421345 0.906901i \(-0.361558\pi\)
0.421345 + 0.906901i \(0.361558\pi\)
\(864\) 12.9700 0.441248
\(865\) 53.1764 1.80805
\(866\) 29.3909 0.998743
\(867\) −13.8647 −0.470869
\(868\) 11.5987 0.393684
\(869\) 5.26590 0.178634
\(870\) 65.7623 2.22955
\(871\) −0.0752338 −0.00254920
\(872\) −5.88020 −0.199129
\(873\) −46.3800 −1.56973
\(874\) 5.60979 0.189754
\(875\) 25.5369 0.863306
\(876\) −19.1918 −0.648431
\(877\) 27.8969 0.942012 0.471006 0.882130i \(-0.343891\pi\)
0.471006 + 0.882130i \(0.343891\pi\)
\(878\) −4.57873 −0.154525
\(879\) 22.4394 0.756863
\(880\) −2.73393 −0.0921607
\(881\) −55.8653 −1.88215 −0.941074 0.338200i \(-0.890182\pi\)
−0.941074 + 0.338200i \(0.890182\pi\)
\(882\) 52.9849 1.78409
\(883\) −44.8709 −1.51003 −0.755014 0.655709i \(-0.772370\pi\)
−0.755014 + 0.655709i \(0.772370\pi\)
\(884\) 5.72291 0.192482
\(885\) 21.8677 0.735074
\(886\) −36.6705 −1.23197
\(887\) −7.01672 −0.235598 −0.117799 0.993037i \(-0.537584\pi\)
−0.117799 + 0.993037i \(0.537584\pi\)
\(888\) 14.8458 0.498191
\(889\) 63.4244 2.12719
\(890\) −40.3513 −1.35258
\(891\) −19.8224 −0.664076
\(892\) 20.0646 0.671811
\(893\) 8.10686 0.271286
\(894\) 38.5960 1.29084
\(895\) −1.60895 −0.0537813
\(896\) −3.80515 −0.127121
\(897\) −20.8457 −0.696018
\(898\) −10.6630 −0.355827
\(899\) −22.9592 −0.765732
\(900\) 18.1289 0.604296
\(901\) 29.1169 0.970024
\(902\) 7.25686 0.241627
\(903\) 94.2486 3.13640
\(904\) −16.1497 −0.537132
\(905\) −3.69239 −0.122739
\(906\) −7.44359 −0.247297
\(907\) −4.56437 −0.151557 −0.0757787 0.997125i \(-0.524144\pi\)
−0.0757787 + 0.997125i \(0.524144\pi\)
\(908\) 16.1415 0.535675
\(909\) −28.6441 −0.950063
\(910\) 16.8443 0.558382
\(911\) 24.8789 0.824275 0.412138 0.911122i \(-0.364782\pi\)
0.412138 + 0.911122i \(0.364782\pi\)
\(912\) 4.36940 0.144685
\(913\) −11.1393 −0.368658
\(914\) 39.1144 1.29379
\(915\) −132.839 −4.39153
\(916\) −21.5723 −0.712768
\(917\) −70.9479 −2.34291
\(918\) −46.1010 −1.52156
\(919\) 21.3410 0.703976 0.351988 0.936005i \(-0.385506\pi\)
0.351988 + 0.936005i \(0.385506\pi\)
\(920\) 11.2093 0.369561
\(921\) −11.9785 −0.394706
\(922\) −32.6190 −1.07425
\(923\) −16.2488 −0.534837
\(924\) 12.0157 0.395288
\(925\) 11.9634 0.393354
\(926\) −7.87682 −0.258848
\(927\) 35.8738 1.17825
\(928\) 7.53219 0.247256
\(929\) −5.14653 −0.168852 −0.0844261 0.996430i \(-0.526906\pi\)
−0.0844261 + 0.996430i \(0.526906\pi\)
\(930\) −26.6128 −0.872669
\(931\) 10.2909 0.337271
\(932\) 24.3039 0.796099
\(933\) −63.1105 −2.06614
\(934\) 29.1726 0.954556
\(935\) 9.71755 0.317798
\(936\) −11.4063 −0.372825
\(937\) 22.1701 0.724267 0.362134 0.932126i \(-0.382048\pi\)
0.362134 + 0.932126i \(0.382048\pi\)
\(938\) −0.177803 −0.00580547
\(939\) −81.5359 −2.66082
\(940\) 16.1989 0.528351
\(941\) −13.8102 −0.450199 −0.225100 0.974336i \(-0.572271\pi\)
−0.225100 + 0.974336i \(0.572271\pi\)
\(942\) 56.3994 1.83759
\(943\) −29.7538 −0.968916
\(944\) 2.50465 0.0815194
\(945\) −135.689 −4.41397
\(946\) 7.75593 0.252167
\(947\) 2.78480 0.0904938 0.0452469 0.998976i \(-0.485593\pi\)
0.0452469 + 0.998976i \(0.485593\pi\)
\(948\) −16.8167 −0.546182
\(949\) 9.73060 0.315868
\(950\) 3.52106 0.114238
\(951\) −42.6704 −1.38368
\(952\) 13.5252 0.438353
\(953\) 10.7432 0.348006 0.174003 0.984745i \(-0.444330\pi\)
0.174003 + 0.984745i \(0.444330\pi\)
\(954\) −58.0326 −1.87887
\(955\) 67.2492 2.17613
\(956\) 17.5860 0.568771
\(957\) −23.7847 −0.768851
\(958\) −10.5245 −0.340032
\(959\) −6.75983 −0.218286
\(960\) 8.73083 0.281786
\(961\) −21.7088 −0.700285
\(962\) −7.52708 −0.242683
\(963\) −70.4947 −2.27166
\(964\) 9.83124 0.316643
\(965\) 55.7244 1.79383
\(966\) −49.2654 −1.58509
\(967\) 28.4197 0.913918 0.456959 0.889488i \(-0.348939\pi\)
0.456959 + 0.889488i \(0.348939\pi\)
\(968\) −10.0112 −0.321772
\(969\) −15.5307 −0.498919
\(970\) −17.9998 −0.577938
\(971\) −26.3117 −0.844382 −0.422191 0.906507i \(-0.638739\pi\)
−0.422191 + 0.906507i \(0.638739\pi\)
\(972\) 24.3931 0.782410
\(973\) −8.64302 −0.277083
\(974\) 40.4170 1.29504
\(975\) −13.0841 −0.419026
\(976\) −15.2149 −0.487019
\(977\) 33.3912 1.06828 0.534139 0.845396i \(-0.320636\pi\)
0.534139 + 0.845396i \(0.320636\pi\)
\(978\) −12.0932 −0.386697
\(979\) 14.5942 0.466431
\(980\) 20.5631 0.656863
\(981\) −41.6571 −1.33001
\(982\) 12.7430 0.406647
\(983\) 14.4612 0.461239 0.230620 0.973044i \(-0.425925\pi\)
0.230620 + 0.973044i \(0.425925\pi\)
\(984\) −23.1749 −0.738788
\(985\) 32.1138 1.02323
\(986\) −26.7726 −0.852615
\(987\) −71.1948 −2.26616
\(988\) −2.21537 −0.0704802
\(989\) −31.8000 −1.01118
\(990\) −19.3680 −0.615554
\(991\) −22.3146 −0.708848 −0.354424 0.935085i \(-0.615323\pi\)
−0.354424 + 0.935085i \(0.615323\pi\)
\(992\) −3.04814 −0.0967786
\(993\) 12.8643 0.408235
\(994\) −38.4015 −1.21802
\(995\) 31.5301 0.999573
\(996\) 35.5735 1.12719
\(997\) −34.8376 −1.10332 −0.551659 0.834070i \(-0.686005\pi\)
−0.551659 + 0.834070i \(0.686005\pi\)
\(998\) −11.4162 −0.361375
\(999\) 60.6345 1.91839
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 1502.2.a.g.1.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1502.2.a.g.1.16 16 1.1 even 1 trivial