Properties

Label 2013.2.a.h.1.3
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} - 745 x^{5} - 802 x^{4} + 386 x^{3} + 74 x^{2} - 15 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.93923\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.93923 q^{2} -1.00000 q^{3} +1.76061 q^{4} +3.07794 q^{5} +1.93923 q^{6} -3.15900 q^{7} +0.464237 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.93923 q^{2} -1.00000 q^{3} +1.76061 q^{4} +3.07794 q^{5} +1.93923 q^{6} -3.15900 q^{7} +0.464237 q^{8} +1.00000 q^{9} -5.96883 q^{10} -1.00000 q^{11} -1.76061 q^{12} -4.31016 q^{13} +6.12601 q^{14} -3.07794 q^{15} -4.42148 q^{16} -3.81369 q^{17} -1.93923 q^{18} -4.23281 q^{19} +5.41904 q^{20} +3.15900 q^{21} +1.93923 q^{22} +6.48918 q^{23} -0.464237 q^{24} +4.47371 q^{25} +8.35838 q^{26} -1.00000 q^{27} -5.56175 q^{28} -3.27479 q^{29} +5.96883 q^{30} +10.0842 q^{31} +7.64578 q^{32} +1.00000 q^{33} +7.39562 q^{34} -9.72319 q^{35} +1.76061 q^{36} -1.85262 q^{37} +8.20839 q^{38} +4.31016 q^{39} +1.42889 q^{40} -10.7788 q^{41} -6.12601 q^{42} +10.4003 q^{43} -1.76061 q^{44} +3.07794 q^{45} -12.5840 q^{46} +10.3730 q^{47} +4.42148 q^{48} +2.97925 q^{49} -8.67555 q^{50} +3.81369 q^{51} -7.58850 q^{52} +8.28262 q^{53} +1.93923 q^{54} -3.07794 q^{55} -1.46652 q^{56} +4.23281 q^{57} +6.35056 q^{58} -10.0151 q^{59} -5.41904 q^{60} +1.00000 q^{61} -19.5555 q^{62} -3.15900 q^{63} -5.98396 q^{64} -13.2664 q^{65} -1.93923 q^{66} -5.80576 q^{67} -6.71441 q^{68} -6.48918 q^{69} +18.8555 q^{70} -3.40168 q^{71} +0.464237 q^{72} +11.2828 q^{73} +3.59265 q^{74} -4.47371 q^{75} -7.45232 q^{76} +3.15900 q^{77} -8.35838 q^{78} +4.26886 q^{79} -13.6090 q^{80} +1.00000 q^{81} +20.9026 q^{82} +17.4095 q^{83} +5.56175 q^{84} -11.7383 q^{85} -20.1685 q^{86} +3.27479 q^{87} -0.464237 q^{88} -9.75884 q^{89} -5.96883 q^{90} +13.6158 q^{91} +11.4249 q^{92} -10.0842 q^{93} -20.1156 q^{94} -13.0283 q^{95} -7.64578 q^{96} +0.814992 q^{97} -5.77745 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9} + 6 q^{10} - 14 q^{11} - 15 q^{12} + q^{13} - 7 q^{14} - q^{15} + 17 q^{16} - 9 q^{17} - q^{18} + 22 q^{19} + 23 q^{20} - 9 q^{21} + q^{22} + q^{23} + 25 q^{25} + 4 q^{26} - 14 q^{27} + 37 q^{28} - 6 q^{29} - 6 q^{30} + 9 q^{31} + 4 q^{32} + 14 q^{33} + 8 q^{34} + 18 q^{35} + 15 q^{36} + 18 q^{37} + 8 q^{38} - q^{39} + 16 q^{40} - 25 q^{41} + 7 q^{42} + 25 q^{43} - 15 q^{44} + q^{45} + 20 q^{46} + 36 q^{47} - 17 q^{48} + 25 q^{49} + 2 q^{50} + 9 q^{51} - 13 q^{52} + q^{54} - q^{55} - 40 q^{56} - 22 q^{57} + 33 q^{58} + 17 q^{59} - 23 q^{60} + 14 q^{61} - 13 q^{62} + 9 q^{63} - 6 q^{64} - 61 q^{65} - q^{66} + 22 q^{67} + 66 q^{68} - q^{69} + 44 q^{70} - 13 q^{71} + 20 q^{73} - 12 q^{74} - 25 q^{75} + 49 q^{76} - 9 q^{77} - 4 q^{78} + 31 q^{79} + 88 q^{80} + 14 q^{81} + 2 q^{82} + 32 q^{83} - 37 q^{84} + 2 q^{85} - 14 q^{86} + 6 q^{87} - 21 q^{89} + 6 q^{90} + 45 q^{91} - 14 q^{92} - 9 q^{93} - 31 q^{94} + 23 q^{95} - 4 q^{96} + 37 q^{97} - 38 q^{98} - 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93923 −1.37124 −0.685621 0.727959i \(-0.740469\pi\)
−0.685621 + 0.727959i \(0.740469\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.76061 0.880304
\(5\) 3.07794 1.37650 0.688248 0.725475i \(-0.258380\pi\)
0.688248 + 0.725475i \(0.258380\pi\)
\(6\) 1.93923 0.791687
\(7\) −3.15900 −1.19399 −0.596994 0.802246i \(-0.703638\pi\)
−0.596994 + 0.802246i \(0.703638\pi\)
\(8\) 0.464237 0.164132
\(9\) 1.00000 0.333333
\(10\) −5.96883 −1.88751
\(11\) −1.00000 −0.301511
\(12\) −1.76061 −0.508244
\(13\) −4.31016 −1.19542 −0.597711 0.801711i \(-0.703923\pi\)
−0.597711 + 0.801711i \(0.703923\pi\)
\(14\) 6.12601 1.63725
\(15\) −3.07794 −0.794720
\(16\) −4.42148 −1.10537
\(17\) −3.81369 −0.924956 −0.462478 0.886631i \(-0.653040\pi\)
−0.462478 + 0.886631i \(0.653040\pi\)
\(18\) −1.93923 −0.457081
\(19\) −4.23281 −0.971073 −0.485537 0.874216i \(-0.661376\pi\)
−0.485537 + 0.874216i \(0.661376\pi\)
\(20\) 5.41904 1.21173
\(21\) 3.15900 0.689349
\(22\) 1.93923 0.413445
\(23\) 6.48918 1.35309 0.676544 0.736402i \(-0.263477\pi\)
0.676544 + 0.736402i \(0.263477\pi\)
\(24\) −0.464237 −0.0947619
\(25\) 4.47371 0.894742
\(26\) 8.35838 1.63921
\(27\) −1.00000 −0.192450
\(28\) −5.56175 −1.05107
\(29\) −3.27479 −0.608113 −0.304056 0.952654i \(-0.598341\pi\)
−0.304056 + 0.952654i \(0.598341\pi\)
\(30\) 5.96883 1.08975
\(31\) 10.0842 1.81117 0.905585 0.424164i \(-0.139432\pi\)
0.905585 + 0.424164i \(0.139432\pi\)
\(32\) 7.64578 1.35160
\(33\) 1.00000 0.174078
\(34\) 7.39562 1.26834
\(35\) −9.72319 −1.64352
\(36\) 1.76061 0.293435
\(37\) −1.85262 −0.304569 −0.152284 0.988337i \(-0.548663\pi\)
−0.152284 + 0.988337i \(0.548663\pi\)
\(38\) 8.20839 1.33158
\(39\) 4.31016 0.690178
\(40\) 1.42889 0.225928
\(41\) −10.7788 −1.68337 −0.841685 0.539969i \(-0.818436\pi\)
−0.841685 + 0.539969i \(0.818436\pi\)
\(42\) −6.12601 −0.945264
\(43\) 10.4003 1.58603 0.793013 0.609205i \(-0.208511\pi\)
0.793013 + 0.609205i \(0.208511\pi\)
\(44\) −1.76061 −0.265422
\(45\) 3.07794 0.458832
\(46\) −12.5840 −1.85541
\(47\) 10.3730 1.51305 0.756527 0.653962i \(-0.226894\pi\)
0.756527 + 0.653962i \(0.226894\pi\)
\(48\) 4.42148 0.638185
\(49\) 2.97925 0.425607
\(50\) −8.67555 −1.22691
\(51\) 3.81369 0.534024
\(52\) −7.58850 −1.05234
\(53\) 8.28262 1.13771 0.568853 0.822439i \(-0.307387\pi\)
0.568853 + 0.822439i \(0.307387\pi\)
\(54\) 1.93923 0.263896
\(55\) −3.07794 −0.415029
\(56\) −1.46652 −0.195972
\(57\) 4.23281 0.560649
\(58\) 6.35056 0.833869
\(59\) −10.0151 −1.30385 −0.651925 0.758283i \(-0.726038\pi\)
−0.651925 + 0.758283i \(0.726038\pi\)
\(60\) −5.41904 −0.699595
\(61\) 1.00000 0.128037
\(62\) −19.5555 −2.48355
\(63\) −3.15900 −0.397996
\(64\) −5.98396 −0.747995
\(65\) −13.2664 −1.64550
\(66\) −1.93923 −0.238703
\(67\) −5.80576 −0.709287 −0.354643 0.935002i \(-0.615398\pi\)
−0.354643 + 0.935002i \(0.615398\pi\)
\(68\) −6.71441 −0.814242
\(69\) −6.48918 −0.781205
\(70\) 18.8555 2.25366
\(71\) −3.40168 −0.403705 −0.201853 0.979416i \(-0.564696\pi\)
−0.201853 + 0.979416i \(0.564696\pi\)
\(72\) 0.464237 0.0547108
\(73\) 11.2828 1.32055 0.660276 0.751023i \(-0.270439\pi\)
0.660276 + 0.751023i \(0.270439\pi\)
\(74\) 3.59265 0.417638
\(75\) −4.47371 −0.516580
\(76\) −7.45232 −0.854840
\(77\) 3.15900 0.360001
\(78\) −8.35838 −0.946400
\(79\) 4.26886 0.480284 0.240142 0.970738i \(-0.422806\pi\)
0.240142 + 0.970738i \(0.422806\pi\)
\(80\) −13.6090 −1.52154
\(81\) 1.00000 0.111111
\(82\) 20.9026 2.30831
\(83\) 17.4095 1.91094 0.955469 0.295092i \(-0.0953505\pi\)
0.955469 + 0.295092i \(0.0953505\pi\)
\(84\) 5.56175 0.606837
\(85\) −11.7383 −1.27320
\(86\) −20.1685 −2.17483
\(87\) 3.27479 0.351094
\(88\) −0.464237 −0.0494878
\(89\) −9.75884 −1.03444 −0.517218 0.855854i \(-0.673032\pi\)
−0.517218 + 0.855854i \(0.673032\pi\)
\(90\) −5.96883 −0.629170
\(91\) 13.6158 1.42732
\(92\) 11.4249 1.19113
\(93\) −10.0842 −1.04568
\(94\) −20.1156 −2.07476
\(95\) −13.0283 −1.33668
\(96\) −7.64578 −0.780344
\(97\) 0.814992 0.0827499 0.0413749 0.999144i \(-0.486826\pi\)
0.0413749 + 0.999144i \(0.486826\pi\)
\(98\) −5.77745 −0.583610
\(99\) −1.00000 −0.100504
\(100\) 7.87645 0.787645
\(101\) 8.64218 0.859929 0.429965 0.902846i \(-0.358526\pi\)
0.429965 + 0.902846i \(0.358526\pi\)
\(102\) −7.39562 −0.732275
\(103\) 1.53886 0.151629 0.0758143 0.997122i \(-0.475844\pi\)
0.0758143 + 0.997122i \(0.475844\pi\)
\(104\) −2.00093 −0.196208
\(105\) 9.72319 0.948887
\(106\) −16.0619 −1.56007
\(107\) −1.27301 −0.123067 −0.0615333 0.998105i \(-0.519599\pi\)
−0.0615333 + 0.998105i \(0.519599\pi\)
\(108\) −1.76061 −0.169415
\(109\) 10.8977 1.04381 0.521905 0.853004i \(-0.325222\pi\)
0.521905 + 0.853004i \(0.325222\pi\)
\(110\) 5.96883 0.569105
\(111\) 1.85262 0.175843
\(112\) 13.9674 1.31980
\(113\) −12.6330 −1.18841 −0.594206 0.804313i \(-0.702534\pi\)
−0.594206 + 0.804313i \(0.702534\pi\)
\(114\) −8.20839 −0.768786
\(115\) 19.9733 1.86252
\(116\) −5.76561 −0.535324
\(117\) −4.31016 −0.398474
\(118\) 19.4215 1.78789
\(119\) 12.0474 1.10439
\(120\) −1.42889 −0.130439
\(121\) 1.00000 0.0909091
\(122\) −1.93923 −0.175570
\(123\) 10.7788 0.971894
\(124\) 17.7543 1.59438
\(125\) −1.61989 −0.144887
\(126\) 6.12601 0.545749
\(127\) 12.4848 1.10784 0.553921 0.832569i \(-0.313131\pi\)
0.553921 + 0.832569i \(0.313131\pi\)
\(128\) −3.68729 −0.325913
\(129\) −10.4003 −0.915693
\(130\) 25.7266 2.25637
\(131\) 0.0498953 0.00435938 0.00217969 0.999998i \(-0.499306\pi\)
0.00217969 + 0.999998i \(0.499306\pi\)
\(132\) 1.76061 0.153241
\(133\) 13.3714 1.15945
\(134\) 11.2587 0.972604
\(135\) −3.07794 −0.264907
\(136\) −1.77045 −0.151815
\(137\) 0.243742 0.0208243 0.0104121 0.999946i \(-0.496686\pi\)
0.0104121 + 0.999946i \(0.496686\pi\)
\(138\) 12.5840 1.07122
\(139\) −21.1871 −1.79706 −0.898531 0.438910i \(-0.855365\pi\)
−0.898531 + 0.438910i \(0.855365\pi\)
\(140\) −17.1187 −1.44680
\(141\) −10.3730 −0.873562
\(142\) 6.59664 0.553578
\(143\) 4.31016 0.360434
\(144\) −4.42148 −0.368456
\(145\) −10.0796 −0.837065
\(146\) −21.8799 −1.81080
\(147\) −2.97925 −0.245724
\(148\) −3.26174 −0.268113
\(149\) 13.4964 1.10567 0.552834 0.833291i \(-0.313546\pi\)
0.552834 + 0.833291i \(0.313546\pi\)
\(150\) 8.67555 0.708355
\(151\) 2.14503 0.174560 0.0872801 0.996184i \(-0.472182\pi\)
0.0872801 + 0.996184i \(0.472182\pi\)
\(152\) −1.96503 −0.159385
\(153\) −3.81369 −0.308319
\(154\) −6.12601 −0.493648
\(155\) 31.0385 2.49307
\(156\) 7.58850 0.607566
\(157\) −2.78006 −0.221873 −0.110936 0.993828i \(-0.535385\pi\)
−0.110936 + 0.993828i \(0.535385\pi\)
\(158\) −8.27830 −0.658586
\(159\) −8.28262 −0.656855
\(160\) 23.5332 1.86047
\(161\) −20.4993 −1.61557
\(162\) −1.93923 −0.152360
\(163\) 16.8129 1.31689 0.658444 0.752630i \(-0.271215\pi\)
0.658444 + 0.752630i \(0.271215\pi\)
\(164\) −18.9773 −1.48188
\(165\) 3.07794 0.239617
\(166\) −33.7609 −2.62036
\(167\) 14.1771 1.09706 0.548529 0.836132i \(-0.315188\pi\)
0.548529 + 0.836132i \(0.315188\pi\)
\(168\) 1.46652 0.113145
\(169\) 5.57746 0.429036
\(170\) 22.7633 1.74586
\(171\) −4.23281 −0.323691
\(172\) 18.3108 1.39618
\(173\) 19.2019 1.45989 0.729945 0.683505i \(-0.239545\pi\)
0.729945 + 0.683505i \(0.239545\pi\)
\(174\) −6.35056 −0.481435
\(175\) −14.1324 −1.06831
\(176\) 4.42148 0.333281
\(177\) 10.0151 0.752778
\(178\) 18.9246 1.41846
\(179\) −6.27478 −0.468999 −0.234500 0.972116i \(-0.575345\pi\)
−0.234500 + 0.972116i \(0.575345\pi\)
\(180\) 5.41904 0.403912
\(181\) 14.9701 1.11272 0.556359 0.830942i \(-0.312198\pi\)
0.556359 + 0.830942i \(0.312198\pi\)
\(182\) −26.4041 −1.95720
\(183\) −1.00000 −0.0739221
\(184\) 3.01251 0.222085
\(185\) −5.70225 −0.419238
\(186\) 19.5555 1.43388
\(187\) 3.81369 0.278885
\(188\) 18.2627 1.33195
\(189\) 3.15900 0.229783
\(190\) 25.2649 1.83291
\(191\) −1.11274 −0.0805152 −0.0402576 0.999189i \(-0.512818\pi\)
−0.0402576 + 0.999189i \(0.512818\pi\)
\(192\) 5.98396 0.431855
\(193\) −16.4747 −1.18587 −0.592937 0.805249i \(-0.702032\pi\)
−0.592937 + 0.805249i \(0.702032\pi\)
\(194\) −1.58046 −0.113470
\(195\) 13.2664 0.950027
\(196\) 5.24529 0.374664
\(197\) 25.4239 1.81138 0.905690 0.423941i \(-0.139354\pi\)
0.905690 + 0.423941i \(0.139354\pi\)
\(198\) 1.93923 0.137815
\(199\) −8.05688 −0.571137 −0.285569 0.958358i \(-0.592182\pi\)
−0.285569 + 0.958358i \(0.592182\pi\)
\(200\) 2.07686 0.146856
\(201\) 5.80576 0.409507
\(202\) −16.7592 −1.17917
\(203\) 10.3450 0.726079
\(204\) 6.71441 0.470103
\(205\) −33.1766 −2.31715
\(206\) −2.98420 −0.207919
\(207\) 6.48918 0.451029
\(208\) 19.0573 1.32138
\(209\) 4.23281 0.292790
\(210\) −18.8555 −1.30115
\(211\) 21.8405 1.50356 0.751780 0.659414i \(-0.229196\pi\)
0.751780 + 0.659414i \(0.229196\pi\)
\(212\) 14.5825 1.00153
\(213\) 3.40168 0.233079
\(214\) 2.46866 0.168754
\(215\) 32.0114 2.18316
\(216\) −0.464237 −0.0315873
\(217\) −31.8559 −2.16252
\(218\) −21.1331 −1.43132
\(219\) −11.2828 −0.762422
\(220\) −5.41904 −0.365352
\(221\) 16.4376 1.10571
\(222\) −3.59265 −0.241123
\(223\) 12.6761 0.848852 0.424426 0.905463i \(-0.360476\pi\)
0.424426 + 0.905463i \(0.360476\pi\)
\(224\) −24.1530 −1.61379
\(225\) 4.47371 0.298247
\(226\) 24.4983 1.62960
\(227\) 18.0164 1.19579 0.597897 0.801573i \(-0.296003\pi\)
0.597897 + 0.801573i \(0.296003\pi\)
\(228\) 7.45232 0.493542
\(229\) 21.1854 1.39997 0.699984 0.714159i \(-0.253191\pi\)
0.699984 + 0.714159i \(0.253191\pi\)
\(230\) −38.7328 −2.55397
\(231\) −3.15900 −0.207847
\(232\) −1.52028 −0.0998110
\(233\) −28.3068 −1.85444 −0.927221 0.374516i \(-0.877809\pi\)
−0.927221 + 0.374516i \(0.877809\pi\)
\(234\) 8.35838 0.546405
\(235\) 31.9274 2.08271
\(236\) −17.6326 −1.14778
\(237\) −4.26886 −0.277292
\(238\) −23.3627 −1.51438
\(239\) 4.89918 0.316902 0.158451 0.987367i \(-0.449350\pi\)
0.158451 + 0.987367i \(0.449350\pi\)
\(240\) 13.6090 0.878459
\(241\) 27.3485 1.76167 0.880836 0.473422i \(-0.156981\pi\)
0.880836 + 0.473422i \(0.156981\pi\)
\(242\) −1.93923 −0.124658
\(243\) −1.00000 −0.0641500
\(244\) 1.76061 0.112711
\(245\) 9.16995 0.585847
\(246\) −20.9026 −1.33270
\(247\) 18.2441 1.16084
\(248\) 4.68144 0.297272
\(249\) −17.4095 −1.10328
\(250\) 3.14134 0.198676
\(251\) 2.95590 0.186575 0.0932874 0.995639i \(-0.470262\pi\)
0.0932874 + 0.995639i \(0.470262\pi\)
\(252\) −5.56175 −0.350357
\(253\) −6.48918 −0.407971
\(254\) −24.2108 −1.51912
\(255\) 11.7383 0.735082
\(256\) 19.1184 1.19490
\(257\) 8.57140 0.534669 0.267335 0.963604i \(-0.413857\pi\)
0.267335 + 0.963604i \(0.413857\pi\)
\(258\) 20.1685 1.25564
\(259\) 5.85242 0.363652
\(260\) −23.3569 −1.44854
\(261\) −3.27479 −0.202704
\(262\) −0.0967585 −0.00597776
\(263\) −1.84001 −0.113460 −0.0567300 0.998390i \(-0.518067\pi\)
−0.0567300 + 0.998390i \(0.518067\pi\)
\(264\) 0.464237 0.0285718
\(265\) 25.4934 1.56605
\(266\) −25.9303 −1.58989
\(267\) 9.75884 0.597232
\(268\) −10.2217 −0.624388
\(269\) −8.65480 −0.527692 −0.263846 0.964565i \(-0.584991\pi\)
−0.263846 + 0.964565i \(0.584991\pi\)
\(270\) 5.96883 0.363251
\(271\) 18.4447 1.12044 0.560219 0.828344i \(-0.310717\pi\)
0.560219 + 0.828344i \(0.310717\pi\)
\(272\) 16.8621 1.02242
\(273\) −13.6158 −0.824064
\(274\) −0.472671 −0.0285551
\(275\) −4.47371 −0.269775
\(276\) −11.4249 −0.687698
\(277\) 8.09606 0.486445 0.243223 0.969971i \(-0.421795\pi\)
0.243223 + 0.969971i \(0.421795\pi\)
\(278\) 41.0865 2.46421
\(279\) 10.0842 0.603724
\(280\) −4.51386 −0.269755
\(281\) −17.8764 −1.06642 −0.533208 0.845984i \(-0.679014\pi\)
−0.533208 + 0.845984i \(0.679014\pi\)
\(282\) 20.1156 1.19787
\(283\) 17.7034 1.05236 0.526179 0.850374i \(-0.323624\pi\)
0.526179 + 0.850374i \(0.323624\pi\)
\(284\) −5.98903 −0.355383
\(285\) 13.0283 0.771732
\(286\) −8.35838 −0.494241
\(287\) 34.0503 2.00992
\(288\) 7.64578 0.450532
\(289\) −2.45576 −0.144456
\(290\) 19.5466 1.14782
\(291\) −0.814992 −0.0477757
\(292\) 19.8646 1.16249
\(293\) −13.3362 −0.779109 −0.389554 0.921003i \(-0.627371\pi\)
−0.389554 + 0.921003i \(0.627371\pi\)
\(294\) 5.77745 0.336948
\(295\) −30.8258 −1.79474
\(296\) −0.860054 −0.0499896
\(297\) 1.00000 0.0580259
\(298\) −26.1726 −1.51614
\(299\) −27.9694 −1.61751
\(300\) −7.87645 −0.454747
\(301\) −32.8544 −1.89370
\(302\) −4.15971 −0.239364
\(303\) −8.64218 −0.496480
\(304\) 18.7153 1.07339
\(305\) 3.07794 0.176242
\(306\) 7.39562 0.422779
\(307\) −32.1020 −1.83216 −0.916080 0.400996i \(-0.868664\pi\)
−0.916080 + 0.400996i \(0.868664\pi\)
\(308\) 5.56175 0.316910
\(309\) −1.53886 −0.0875428
\(310\) −60.1907 −3.41860
\(311\) 19.1427 1.08548 0.542742 0.839899i \(-0.317386\pi\)
0.542742 + 0.839899i \(0.317386\pi\)
\(312\) 2.00093 0.113281
\(313\) 7.12592 0.402781 0.201390 0.979511i \(-0.435454\pi\)
0.201390 + 0.979511i \(0.435454\pi\)
\(314\) 5.39117 0.304241
\(315\) −9.72319 −0.547840
\(316\) 7.51579 0.422796
\(317\) −30.7643 −1.72789 −0.863947 0.503584i \(-0.832015\pi\)
−0.863947 + 0.503584i \(0.832015\pi\)
\(318\) 16.0619 0.900707
\(319\) 3.27479 0.183353
\(320\) −18.4183 −1.02961
\(321\) 1.27301 0.0710525
\(322\) 39.7528 2.21534
\(323\) 16.1426 0.898200
\(324\) 1.76061 0.0978115
\(325\) −19.2824 −1.06959
\(326\) −32.6040 −1.80577
\(327\) −10.8977 −0.602644
\(328\) −5.00393 −0.276296
\(329\) −32.7682 −1.80657
\(330\) −5.96883 −0.328573
\(331\) −1.20244 −0.0660920 −0.0330460 0.999454i \(-0.510521\pi\)
−0.0330460 + 0.999454i \(0.510521\pi\)
\(332\) 30.6512 1.68221
\(333\) −1.85262 −0.101523
\(334\) −27.4926 −1.50433
\(335\) −17.8698 −0.976331
\(336\) −13.9674 −0.761985
\(337\) −27.8849 −1.51898 −0.759492 0.650516i \(-0.774553\pi\)
−0.759492 + 0.650516i \(0.774553\pi\)
\(338\) −10.8160 −0.588312
\(339\) 12.6330 0.686131
\(340\) −20.6666 −1.12080
\(341\) −10.0842 −0.546089
\(342\) 8.20839 0.443859
\(343\) 12.7015 0.685818
\(344\) 4.82818 0.260318
\(345\) −19.9733 −1.07533
\(346\) −37.2368 −2.00186
\(347\) 8.66808 0.465327 0.232663 0.972557i \(-0.425256\pi\)
0.232663 + 0.972557i \(0.425256\pi\)
\(348\) 5.76561 0.309069
\(349\) −3.27364 −0.175234 −0.0876170 0.996154i \(-0.527925\pi\)
−0.0876170 + 0.996154i \(0.527925\pi\)
\(350\) 27.4060 1.46491
\(351\) 4.31016 0.230059
\(352\) −7.64578 −0.407521
\(353\) 23.4498 1.24811 0.624054 0.781381i \(-0.285485\pi\)
0.624054 + 0.781381i \(0.285485\pi\)
\(354\) −19.4215 −1.03224
\(355\) −10.4702 −0.555699
\(356\) −17.1815 −0.910617
\(357\) −12.0474 −0.637618
\(358\) 12.1682 0.643111
\(359\) −26.9502 −1.42238 −0.711188 0.703001i \(-0.751843\pi\)
−0.711188 + 0.703001i \(0.751843\pi\)
\(360\) 1.42889 0.0753092
\(361\) −1.08332 −0.0570167
\(362\) −29.0304 −1.52580
\(363\) −1.00000 −0.0524864
\(364\) 23.9720 1.25648
\(365\) 34.7278 1.81774
\(366\) 1.93923 0.101365
\(367\) −6.45575 −0.336987 −0.168494 0.985703i \(-0.553890\pi\)
−0.168494 + 0.985703i \(0.553890\pi\)
\(368\) −28.6918 −1.49566
\(369\) −10.7788 −0.561123
\(370\) 11.0580 0.574877
\(371\) −26.1648 −1.35841
\(372\) −17.7543 −0.920516
\(373\) 9.95114 0.515250 0.257625 0.966245i \(-0.417060\pi\)
0.257625 + 0.966245i \(0.417060\pi\)
\(374\) −7.39562 −0.382418
\(375\) 1.61989 0.0836507
\(376\) 4.81552 0.248341
\(377\) 14.1148 0.726952
\(378\) −6.12601 −0.315088
\(379\) −5.99448 −0.307916 −0.153958 0.988077i \(-0.549202\pi\)
−0.153958 + 0.988077i \(0.549202\pi\)
\(380\) −22.9378 −1.17668
\(381\) −12.4848 −0.639613
\(382\) 2.15786 0.110406
\(383\) 2.43976 0.124666 0.0623329 0.998055i \(-0.480146\pi\)
0.0623329 + 0.998055i \(0.480146\pi\)
\(384\) 3.68729 0.188166
\(385\) 9.72319 0.495540
\(386\) 31.9482 1.62612
\(387\) 10.4003 0.528675
\(388\) 1.43488 0.0728450
\(389\) −1.63009 −0.0826491 −0.0413245 0.999146i \(-0.513158\pi\)
−0.0413245 + 0.999146i \(0.513158\pi\)
\(390\) −25.7266 −1.30272
\(391\) −24.7477 −1.25155
\(392\) 1.38308 0.0698559
\(393\) −0.0498953 −0.00251689
\(394\) −49.3028 −2.48384
\(395\) 13.1393 0.661110
\(396\) −1.76061 −0.0884739
\(397\) −29.1550 −1.46325 −0.731624 0.681709i \(-0.761237\pi\)
−0.731624 + 0.681709i \(0.761237\pi\)
\(398\) 15.6241 0.783167
\(399\) −13.3714 −0.669409
\(400\) −19.7804 −0.989020
\(401\) −28.8479 −1.44059 −0.720297 0.693666i \(-0.755995\pi\)
−0.720297 + 0.693666i \(0.755995\pi\)
\(402\) −11.2587 −0.561533
\(403\) −43.4644 −2.16512
\(404\) 15.2155 0.756999
\(405\) 3.07794 0.152944
\(406\) −20.0614 −0.995630
\(407\) 1.85262 0.0918310
\(408\) 1.77045 0.0876506
\(409\) 19.7104 0.974619 0.487309 0.873229i \(-0.337978\pi\)
0.487309 + 0.873229i \(0.337978\pi\)
\(410\) 64.3370 3.17738
\(411\) −0.243742 −0.0120229
\(412\) 2.70933 0.133479
\(413\) 31.6375 1.55678
\(414\) −12.5840 −0.618470
\(415\) 53.5853 2.63040
\(416\) −32.9545 −1.61573
\(417\) 21.1871 1.03753
\(418\) −8.20839 −0.401485
\(419\) −6.33286 −0.309381 −0.154690 0.987963i \(-0.549438\pi\)
−0.154690 + 0.987963i \(0.549438\pi\)
\(420\) 17.1187 0.835309
\(421\) 30.9731 1.50954 0.754768 0.655992i \(-0.227750\pi\)
0.754768 + 0.655992i \(0.227750\pi\)
\(422\) −42.3537 −2.06174
\(423\) 10.3730 0.504351
\(424\) 3.84510 0.186734
\(425\) −17.0613 −0.827597
\(426\) −6.59664 −0.319608
\(427\) −3.15900 −0.152874
\(428\) −2.24127 −0.108336
\(429\) −4.31016 −0.208096
\(430\) −62.0774 −2.99364
\(431\) −0.711680 −0.0342804 −0.0171402 0.999853i \(-0.505456\pi\)
−0.0171402 + 0.999853i \(0.505456\pi\)
\(432\) 4.42148 0.212728
\(433\) 38.6317 1.85652 0.928259 0.371934i \(-0.121305\pi\)
0.928259 + 0.371934i \(0.121305\pi\)
\(434\) 61.7758 2.96533
\(435\) 10.0796 0.483280
\(436\) 19.1866 0.918870
\(437\) −27.4675 −1.31395
\(438\) 21.8799 1.04546
\(439\) 10.1209 0.483043 0.241522 0.970395i \(-0.422353\pi\)
0.241522 + 0.970395i \(0.422353\pi\)
\(440\) −1.42889 −0.0681197
\(441\) 2.97925 0.141869
\(442\) −31.8763 −1.51620
\(443\) −27.6987 −1.31600 −0.658002 0.753016i \(-0.728598\pi\)
−0.658002 + 0.753016i \(0.728598\pi\)
\(444\) 3.26174 0.154795
\(445\) −30.0371 −1.42390
\(446\) −24.5818 −1.16398
\(447\) −13.4964 −0.638358
\(448\) 18.9033 0.893097
\(449\) 20.3574 0.960727 0.480364 0.877069i \(-0.340505\pi\)
0.480364 + 0.877069i \(0.340505\pi\)
\(450\) −8.67555 −0.408969
\(451\) 10.7788 0.507555
\(452\) −22.2418 −1.04616
\(453\) −2.14503 −0.100782
\(454\) −34.9380 −1.63972
\(455\) 41.9085 1.96470
\(456\) 1.96503 0.0920207
\(457\) −29.6278 −1.38593 −0.692964 0.720972i \(-0.743696\pi\)
−0.692964 + 0.720972i \(0.743696\pi\)
\(458\) −41.0832 −1.91969
\(459\) 3.81369 0.178008
\(460\) 35.1651 1.63958
\(461\) 13.6161 0.634163 0.317081 0.948398i \(-0.397297\pi\)
0.317081 + 0.948398i \(0.397297\pi\)
\(462\) 6.12601 0.285008
\(463\) −16.1578 −0.750914 −0.375457 0.926840i \(-0.622514\pi\)
−0.375457 + 0.926840i \(0.622514\pi\)
\(464\) 14.4794 0.672189
\(465\) −31.0385 −1.43937
\(466\) 54.8934 2.54289
\(467\) −12.4292 −0.575155 −0.287577 0.957757i \(-0.592850\pi\)
−0.287577 + 0.957757i \(0.592850\pi\)
\(468\) −7.58850 −0.350778
\(469\) 18.3404 0.846880
\(470\) −61.9145 −2.85590
\(471\) 2.78006 0.128098
\(472\) −4.64936 −0.214004
\(473\) −10.4003 −0.478205
\(474\) 8.27830 0.380235
\(475\) −18.9364 −0.868860
\(476\) 21.2108 0.972196
\(477\) 8.28262 0.379235
\(478\) −9.50063 −0.434549
\(479\) 29.1644 1.33255 0.666277 0.745704i \(-0.267887\pi\)
0.666277 + 0.745704i \(0.267887\pi\)
\(480\) −23.5332 −1.07414
\(481\) 7.98509 0.364089
\(482\) −53.0350 −2.41568
\(483\) 20.4993 0.932750
\(484\) 1.76061 0.0800276
\(485\) 2.50850 0.113905
\(486\) 1.93923 0.0879652
\(487\) 36.8143 1.66822 0.834108 0.551601i \(-0.185983\pi\)
0.834108 + 0.551601i \(0.185983\pi\)
\(488\) 0.464237 0.0210150
\(489\) −16.8129 −0.760305
\(490\) −17.7826 −0.803337
\(491\) 38.5259 1.73865 0.869326 0.494239i \(-0.164553\pi\)
0.869326 + 0.494239i \(0.164553\pi\)
\(492\) 18.9773 0.855562
\(493\) 12.4890 0.562477
\(494\) −35.3794 −1.59180
\(495\) −3.07794 −0.138343
\(496\) −44.5869 −2.00201
\(497\) 10.7459 0.482019
\(498\) 33.7609 1.51286
\(499\) 1.68198 0.0752959 0.0376480 0.999291i \(-0.488013\pi\)
0.0376480 + 0.999291i \(0.488013\pi\)
\(500\) −2.85199 −0.127545
\(501\) −14.1771 −0.633386
\(502\) −5.73217 −0.255839
\(503\) 20.1033 0.896360 0.448180 0.893943i \(-0.352072\pi\)
0.448180 + 0.893943i \(0.352072\pi\)
\(504\) −1.46652 −0.0653240
\(505\) 26.6001 1.18369
\(506\) 12.5840 0.559427
\(507\) −5.57746 −0.247704
\(508\) 21.9807 0.975238
\(509\) 32.9375 1.45993 0.729964 0.683485i \(-0.239537\pi\)
0.729964 + 0.683485i \(0.239537\pi\)
\(510\) −22.7633 −1.00797
\(511\) −35.6423 −1.57672
\(512\) −29.7004 −1.31259
\(513\) 4.23281 0.186883
\(514\) −16.6219 −0.733161
\(515\) 4.73652 0.208716
\(516\) −18.3108 −0.806088
\(517\) −10.3730 −0.456203
\(518\) −11.3492 −0.498654
\(519\) −19.2019 −0.842868
\(520\) −6.15875 −0.270079
\(521\) 6.18719 0.271066 0.135533 0.990773i \(-0.456725\pi\)
0.135533 + 0.990773i \(0.456725\pi\)
\(522\) 6.35056 0.277956
\(523\) 19.2567 0.842035 0.421017 0.907053i \(-0.361673\pi\)
0.421017 + 0.907053i \(0.361673\pi\)
\(524\) 0.0878461 0.00383757
\(525\) 14.1324 0.616790
\(526\) 3.56820 0.155581
\(527\) −38.4579 −1.67525
\(528\) −4.42148 −0.192420
\(529\) 19.1095 0.830846
\(530\) −49.4376 −2.14743
\(531\) −10.0151 −0.434617
\(532\) 23.5418 1.02067
\(533\) 46.4585 2.01234
\(534\) −18.9246 −0.818949
\(535\) −3.91825 −0.169401
\(536\) −2.69525 −0.116417
\(537\) 6.27478 0.270777
\(538\) 16.7836 0.723594
\(539\) −2.97925 −0.128325
\(540\) −5.41904 −0.233198
\(541\) −32.4081 −1.39333 −0.696666 0.717395i \(-0.745334\pi\)
−0.696666 + 0.717395i \(0.745334\pi\)
\(542\) −35.7686 −1.53639
\(543\) −14.9701 −0.642427
\(544\) −29.1586 −1.25017
\(545\) 33.5425 1.43680
\(546\) 26.4041 1.12999
\(547\) −10.6214 −0.454140 −0.227070 0.973878i \(-0.572915\pi\)
−0.227070 + 0.973878i \(0.572915\pi\)
\(548\) 0.429133 0.0183317
\(549\) 1.00000 0.0426790
\(550\) 8.67555 0.369927
\(551\) 13.8615 0.590522
\(552\) −3.01251 −0.128221
\(553\) −13.4853 −0.573454
\(554\) −15.7001 −0.667034
\(555\) 5.70225 0.242047
\(556\) −37.3021 −1.58196
\(557\) −33.3894 −1.41475 −0.707376 0.706837i \(-0.750121\pi\)
−0.707376 + 0.706837i \(0.750121\pi\)
\(558\) −19.5555 −0.827851
\(559\) −44.8268 −1.89597
\(560\) 42.9909 1.81670
\(561\) −3.81369 −0.161014
\(562\) 34.6664 1.46231
\(563\) 8.93867 0.376720 0.188360 0.982100i \(-0.439683\pi\)
0.188360 + 0.982100i \(0.439683\pi\)
\(564\) −18.2627 −0.769000
\(565\) −38.8836 −1.63585
\(566\) −34.3309 −1.44304
\(567\) −3.15900 −0.132665
\(568\) −1.57918 −0.0662611
\(569\) −2.35475 −0.0987163 −0.0493582 0.998781i \(-0.515718\pi\)
−0.0493582 + 0.998781i \(0.515718\pi\)
\(570\) −25.2649 −1.05823
\(571\) 4.24507 0.177651 0.0888253 0.996047i \(-0.471689\pi\)
0.0888253 + 0.996047i \(0.471689\pi\)
\(572\) 7.58850 0.317291
\(573\) 1.11274 0.0464855
\(574\) −66.0313 −2.75609
\(575\) 29.0307 1.21066
\(576\) −5.98396 −0.249332
\(577\) −8.98406 −0.374011 −0.187006 0.982359i \(-0.559878\pi\)
−0.187006 + 0.982359i \(0.559878\pi\)
\(578\) 4.76227 0.198084
\(579\) 16.4747 0.684664
\(580\) −17.7462 −0.736871
\(581\) −54.9964 −2.28164
\(582\) 1.58046 0.0655120
\(583\) −8.28262 −0.343031
\(584\) 5.23789 0.216745
\(585\) −13.2664 −0.548498
\(586\) 25.8619 1.06835
\(587\) 0.0674569 0.00278424 0.00139212 0.999999i \(-0.499557\pi\)
0.00139212 + 0.999999i \(0.499557\pi\)
\(588\) −5.24529 −0.216312
\(589\) −42.6844 −1.75878
\(590\) 59.7782 2.46103
\(591\) −25.4239 −1.04580
\(592\) 8.19132 0.336661
\(593\) −19.4214 −0.797540 −0.398770 0.917051i \(-0.630563\pi\)
−0.398770 + 0.917051i \(0.630563\pi\)
\(594\) −1.93923 −0.0795675
\(595\) 37.0813 1.52018
\(596\) 23.7619 0.973324
\(597\) 8.05688 0.329746
\(598\) 54.2390 2.21800
\(599\) −19.6058 −0.801070 −0.400535 0.916282i \(-0.631176\pi\)
−0.400535 + 0.916282i \(0.631176\pi\)
\(600\) −2.07686 −0.0847874
\(601\) 32.7244 1.33485 0.667427 0.744675i \(-0.267395\pi\)
0.667427 + 0.744675i \(0.267395\pi\)
\(602\) 63.7122 2.59672
\(603\) −5.80576 −0.236429
\(604\) 3.77656 0.153666
\(605\) 3.07794 0.125136
\(606\) 16.7592 0.680795
\(607\) −6.92562 −0.281102 −0.140551 0.990073i \(-0.544887\pi\)
−0.140551 + 0.990073i \(0.544887\pi\)
\(608\) −32.3631 −1.31250
\(609\) −10.3450 −0.419202
\(610\) −5.96883 −0.241671
\(611\) −44.7092 −1.80874
\(612\) −6.71441 −0.271414
\(613\) 10.1296 0.409131 0.204566 0.978853i \(-0.434422\pi\)
0.204566 + 0.978853i \(0.434422\pi\)
\(614\) 62.2532 2.51233
\(615\) 33.1766 1.33781
\(616\) 1.46652 0.0590878
\(617\) 3.90198 0.157088 0.0785439 0.996911i \(-0.474973\pi\)
0.0785439 + 0.996911i \(0.474973\pi\)
\(618\) 2.98420 0.120042
\(619\) 8.08612 0.325009 0.162504 0.986708i \(-0.448043\pi\)
0.162504 + 0.986708i \(0.448043\pi\)
\(620\) 54.6466 2.19466
\(621\) −6.48918 −0.260402
\(622\) −37.1221 −1.48846
\(623\) 30.8281 1.23510
\(624\) −19.0573 −0.762901
\(625\) −27.3545 −1.09418
\(626\) −13.8188 −0.552310
\(627\) −4.23281 −0.169042
\(628\) −4.89459 −0.195315
\(629\) 7.06532 0.281713
\(630\) 18.8555 0.751221
\(631\) 14.8852 0.592569 0.296284 0.955100i \(-0.404252\pi\)
0.296284 + 0.955100i \(0.404252\pi\)
\(632\) 1.98176 0.0788302
\(633\) −21.8405 −0.868081
\(634\) 59.6589 2.36936
\(635\) 38.4273 1.52494
\(636\) −14.5825 −0.578232
\(637\) −12.8410 −0.508780
\(638\) −6.35056 −0.251421
\(639\) −3.40168 −0.134568
\(640\) −11.3492 −0.448618
\(641\) 6.49332 0.256471 0.128235 0.991744i \(-0.459069\pi\)
0.128235 + 0.991744i \(0.459069\pi\)
\(642\) −2.46866 −0.0974301
\(643\) −0.679815 −0.0268093 −0.0134046 0.999910i \(-0.504267\pi\)
−0.0134046 + 0.999910i \(0.504267\pi\)
\(644\) −36.0912 −1.42219
\(645\) −32.0114 −1.26045
\(646\) −31.3043 −1.23165
\(647\) 9.53160 0.374726 0.187363 0.982291i \(-0.440006\pi\)
0.187363 + 0.982291i \(0.440006\pi\)
\(648\) 0.464237 0.0182369
\(649\) 10.0151 0.393126
\(650\) 37.3930 1.46667
\(651\) 31.8559 1.24853
\(652\) 29.6009 1.15926
\(653\) −15.7563 −0.616591 −0.308296 0.951291i \(-0.599759\pi\)
−0.308296 + 0.951291i \(0.599759\pi\)
\(654\) 21.1331 0.826371
\(655\) 0.153575 0.00600066
\(656\) 47.6583 1.86075
\(657\) 11.2828 0.440184
\(658\) 63.5450 2.47724
\(659\) 22.5633 0.878941 0.439470 0.898257i \(-0.355166\pi\)
0.439470 + 0.898257i \(0.355166\pi\)
\(660\) 5.41904 0.210936
\(661\) −47.5006 −1.84756 −0.923779 0.382927i \(-0.874916\pi\)
−0.923779 + 0.382927i \(0.874916\pi\)
\(662\) 2.33180 0.0906281
\(663\) −16.4376 −0.638384
\(664\) 8.08211 0.313647
\(665\) 41.1564 1.59598
\(666\) 3.59265 0.139213
\(667\) −21.2507 −0.822830
\(668\) 24.9603 0.965744
\(669\) −12.6761 −0.490085
\(670\) 34.6536 1.33879
\(671\) −1.00000 −0.0386046
\(672\) 24.1530 0.931721
\(673\) 6.74671 0.260067 0.130033 0.991510i \(-0.458492\pi\)
0.130033 + 0.991510i \(0.458492\pi\)
\(674\) 54.0751 2.08290
\(675\) −4.47371 −0.172193
\(676\) 9.81973 0.377682
\(677\) 31.7778 1.22132 0.610660 0.791893i \(-0.290904\pi\)
0.610660 + 0.791893i \(0.290904\pi\)
\(678\) −24.4983 −0.940851
\(679\) −2.57456 −0.0988024
\(680\) −5.44935 −0.208973
\(681\) −18.0164 −0.690392
\(682\) 19.5555 0.748819
\(683\) −4.24491 −0.162427 −0.0812135 0.996697i \(-0.525880\pi\)
−0.0812135 + 0.996697i \(0.525880\pi\)
\(684\) −7.45232 −0.284947
\(685\) 0.750222 0.0286645
\(686\) −24.6312 −0.940422
\(687\) −21.1854 −0.808272
\(688\) −45.9845 −1.75314
\(689\) −35.6994 −1.36004
\(690\) 38.7328 1.47453
\(691\) 39.5060 1.50288 0.751440 0.659801i \(-0.229359\pi\)
0.751440 + 0.659801i \(0.229359\pi\)
\(692\) 33.8069 1.28515
\(693\) 3.15900 0.120000
\(694\) −16.8094 −0.638076
\(695\) −65.2125 −2.47365
\(696\) 1.52028 0.0576259
\(697\) 41.1071 1.55704
\(698\) 6.34834 0.240288
\(699\) 28.3068 1.07066
\(700\) −24.8817 −0.940438
\(701\) −2.69413 −0.101756 −0.0508779 0.998705i \(-0.516202\pi\)
−0.0508779 + 0.998705i \(0.516202\pi\)
\(702\) −8.35838 −0.315467
\(703\) 7.84179 0.295759
\(704\) 5.98396 0.225529
\(705\) −31.9274 −1.20246
\(706\) −45.4746 −1.71146
\(707\) −27.3006 −1.02675
\(708\) 17.6326 0.662673
\(709\) 44.7755 1.68158 0.840790 0.541362i \(-0.182091\pi\)
0.840790 + 0.541362i \(0.182091\pi\)
\(710\) 20.3041 0.761997
\(711\) 4.26886 0.160095
\(712\) −4.53041 −0.169784
\(713\) 65.4380 2.45067
\(714\) 23.3627 0.874328
\(715\) 13.2664 0.496135
\(716\) −11.0474 −0.412862
\(717\) −4.89918 −0.182963
\(718\) 52.2626 1.95042
\(719\) −28.5746 −1.06565 −0.532827 0.846224i \(-0.678870\pi\)
−0.532827 + 0.846224i \(0.678870\pi\)
\(720\) −13.6090 −0.507179
\(721\) −4.86126 −0.181043
\(722\) 2.10080 0.0781836
\(723\) −27.3485 −1.01710
\(724\) 26.3564 0.979529
\(725\) −14.6504 −0.544104
\(726\) 1.93923 0.0719715
\(727\) −25.0251 −0.928130 −0.464065 0.885801i \(-0.653610\pi\)
−0.464065 + 0.885801i \(0.653610\pi\)
\(728\) 6.32094 0.234270
\(729\) 1.00000 0.0370370
\(730\) −67.3451 −2.49256
\(731\) −39.6634 −1.46700
\(732\) −1.76061 −0.0650739
\(733\) −0.528034 −0.0195034 −0.00975169 0.999952i \(-0.503104\pi\)
−0.00975169 + 0.999952i \(0.503104\pi\)
\(734\) 12.5192 0.462091
\(735\) −9.16995 −0.338239
\(736\) 49.6148 1.82883
\(737\) 5.80576 0.213858
\(738\) 20.9026 0.769436
\(739\) 20.9456 0.770496 0.385248 0.922813i \(-0.374116\pi\)
0.385248 + 0.922813i \(0.374116\pi\)
\(740\) −10.0394 −0.369057
\(741\) −18.2441 −0.670213
\(742\) 50.7395 1.86270
\(743\) −17.0344 −0.624932 −0.312466 0.949929i \(-0.601155\pi\)
−0.312466 + 0.949929i \(0.601155\pi\)
\(744\) −4.68144 −0.171630
\(745\) 41.5411 1.52195
\(746\) −19.2975 −0.706533
\(747\) 17.4095 0.636979
\(748\) 6.71441 0.245503
\(749\) 4.02143 0.146940
\(750\) −3.14134 −0.114705
\(751\) −10.2079 −0.372491 −0.186246 0.982503i \(-0.559632\pi\)
−0.186246 + 0.982503i \(0.559632\pi\)
\(752\) −45.8639 −1.67248
\(753\) −2.95590 −0.107719
\(754\) −27.3719 −0.996826
\(755\) 6.60228 0.240281
\(756\) 5.56175 0.202279
\(757\) 18.2567 0.663549 0.331775 0.943359i \(-0.392353\pi\)
0.331775 + 0.943359i \(0.392353\pi\)
\(758\) 11.6247 0.422227
\(759\) 6.48918 0.235542
\(760\) −6.04823 −0.219392
\(761\) 5.09224 0.184594 0.0922968 0.995732i \(-0.470579\pi\)
0.0922968 + 0.995732i \(0.470579\pi\)
\(762\) 24.2108 0.877064
\(763\) −34.4258 −1.24630
\(764\) −1.95910 −0.0708779
\(765\) −11.7383 −0.424400
\(766\) −4.73125 −0.170947
\(767\) 43.1665 1.55865
\(768\) −19.1184 −0.689877
\(769\) −9.02962 −0.325617 −0.162808 0.986658i \(-0.552055\pi\)
−0.162808 + 0.986658i \(0.552055\pi\)
\(770\) −18.8555 −0.679505
\(771\) −8.57140 −0.308692
\(772\) −29.0054 −1.04393
\(773\) −9.13122 −0.328427 −0.164214 0.986425i \(-0.552509\pi\)
−0.164214 + 0.986425i \(0.552509\pi\)
\(774\) −20.1685 −0.724942
\(775\) 45.1137 1.62053
\(776\) 0.378349 0.0135819
\(777\) −5.85242 −0.209954
\(778\) 3.16113 0.113332
\(779\) 45.6247 1.63468
\(780\) 23.3569 0.836312
\(781\) 3.40168 0.121722
\(782\) 47.9915 1.71617
\(783\) 3.27479 0.117031
\(784\) −13.1727 −0.470453
\(785\) −8.55685 −0.305407
\(786\) 0.0967585 0.00345126
\(787\) 28.3647 1.01109 0.505546 0.862800i \(-0.331291\pi\)
0.505546 + 0.862800i \(0.331291\pi\)
\(788\) 44.7615 1.59456
\(789\) 1.84001 0.0655062
\(790\) −25.4801 −0.906541
\(791\) 39.9076 1.41895
\(792\) −0.464237 −0.0164959
\(793\) −4.31016 −0.153058
\(794\) 56.5382 2.00647
\(795\) −25.4934 −0.904158
\(796\) −14.1850 −0.502774
\(797\) 36.2460 1.28390 0.641950 0.766747i \(-0.278126\pi\)
0.641950 + 0.766747i \(0.278126\pi\)
\(798\) 25.9303 0.917921
\(799\) −39.5593 −1.39951
\(800\) 34.2050 1.20933
\(801\) −9.75884 −0.344812
\(802\) 55.9426 1.97540
\(803\) −11.2828 −0.398162
\(804\) 10.2217 0.360491
\(805\) −63.0956 −2.22383
\(806\) 84.2874 2.96890
\(807\) 8.65480 0.304663
\(808\) 4.01202 0.141142
\(809\) −53.3004 −1.87394 −0.936971 0.349408i \(-0.886383\pi\)
−0.936971 + 0.349408i \(0.886383\pi\)
\(810\) −5.96883 −0.209723
\(811\) 13.8664 0.486914 0.243457 0.969912i \(-0.421719\pi\)
0.243457 + 0.969912i \(0.421719\pi\)
\(812\) 18.2135 0.639170
\(813\) −18.4447 −0.646885
\(814\) −3.59265 −0.125922
\(815\) 51.7491 1.81269
\(816\) −16.8621 −0.590293
\(817\) −44.0224 −1.54015
\(818\) −38.2231 −1.33644
\(819\) 13.6158 0.475773
\(820\) −58.4109 −2.03980
\(821\) −42.4160 −1.48033 −0.740164 0.672427i \(-0.765252\pi\)
−0.740164 + 0.672427i \(0.765252\pi\)
\(822\) 0.472671 0.0164863
\(823\) 35.8679 1.25028 0.625138 0.780514i \(-0.285043\pi\)
0.625138 + 0.780514i \(0.285043\pi\)
\(824\) 0.714396 0.0248871
\(825\) 4.47371 0.155755
\(826\) −61.3524 −2.13472
\(827\) −7.27629 −0.253022 −0.126511 0.991965i \(-0.540378\pi\)
−0.126511 + 0.991965i \(0.540378\pi\)
\(828\) 11.4249 0.397043
\(829\) −21.2488 −0.738000 −0.369000 0.929429i \(-0.620300\pi\)
−0.369000 + 0.929429i \(0.620300\pi\)
\(830\) −103.914 −3.60691
\(831\) −8.09606 −0.280849
\(832\) 25.7918 0.894171
\(833\) −11.3619 −0.393668
\(834\) −41.0865 −1.42271
\(835\) 43.6363 1.51010
\(836\) 7.45232 0.257744
\(837\) −10.0842 −0.348560
\(838\) 12.2809 0.424236
\(839\) 27.7675 0.958641 0.479320 0.877640i \(-0.340883\pi\)
0.479320 + 0.877640i \(0.340883\pi\)
\(840\) 4.51386 0.155743
\(841\) −18.2758 −0.630199
\(842\) −60.0639 −2.06994
\(843\) 17.8764 0.615695
\(844\) 38.4525 1.32359
\(845\) 17.1671 0.590566
\(846\) −20.1156 −0.691588
\(847\) −3.15900 −0.108544
\(848\) −36.6214 −1.25758
\(849\) −17.7034 −0.607579
\(850\) 33.0859 1.13484
\(851\) −12.0220 −0.412108
\(852\) 5.98903 0.205181
\(853\) −7.41186 −0.253777 −0.126889 0.991917i \(-0.540499\pi\)
−0.126889 + 0.991917i \(0.540499\pi\)
\(854\) 6.12601 0.209628
\(855\) −13.0283 −0.445560
\(856\) −0.590978 −0.0201992
\(857\) −13.6447 −0.466095 −0.233047 0.972465i \(-0.574870\pi\)
−0.233047 + 0.972465i \(0.574870\pi\)
\(858\) 8.35838 0.285350
\(859\) −39.3829 −1.34373 −0.671864 0.740675i \(-0.734506\pi\)
−0.671864 + 0.740675i \(0.734506\pi\)
\(860\) 56.3595 1.92184
\(861\) −34.0503 −1.16043
\(862\) 1.38011 0.0470068
\(863\) −48.1420 −1.63877 −0.819387 0.573241i \(-0.805686\pi\)
−0.819387 + 0.573241i \(0.805686\pi\)
\(864\) −7.64578 −0.260115
\(865\) 59.1022 2.00953
\(866\) −74.9156 −2.54574
\(867\) 2.45576 0.0834019
\(868\) −56.0857 −1.90367
\(869\) −4.26886 −0.144811
\(870\) −19.5466 −0.662693
\(871\) 25.0238 0.847898
\(872\) 5.05911 0.171323
\(873\) 0.814992 0.0275833
\(874\) 53.2657 1.80174
\(875\) 5.11722 0.172994
\(876\) −19.8646 −0.671163
\(877\) 37.6587 1.27164 0.635822 0.771836i \(-0.280661\pi\)
0.635822 + 0.771836i \(0.280661\pi\)
\(878\) −19.6267 −0.662369
\(879\) 13.3362 0.449819
\(880\) 13.6090 0.458760
\(881\) 32.6783 1.10096 0.550480 0.834848i \(-0.314445\pi\)
0.550480 + 0.834848i \(0.314445\pi\)
\(882\) −5.77745 −0.194537
\(883\) −2.47331 −0.0832335 −0.0416167 0.999134i \(-0.513251\pi\)
−0.0416167 + 0.999134i \(0.513251\pi\)
\(884\) 28.9402 0.973364
\(885\) 30.8258 1.03620
\(886\) 53.7141 1.80456
\(887\) 2.57792 0.0865580 0.0432790 0.999063i \(-0.486220\pi\)
0.0432790 + 0.999063i \(0.486220\pi\)
\(888\) 0.860054 0.0288615
\(889\) −39.4393 −1.32275
\(890\) 58.2489 1.95251
\(891\) −1.00000 −0.0335013
\(892\) 22.3176 0.747248
\(893\) −43.9068 −1.46929
\(894\) 26.1726 0.875343
\(895\) −19.3134 −0.645576
\(896\) 11.6481 0.389136
\(897\) 27.9694 0.933871
\(898\) −39.4777 −1.31739
\(899\) −33.0235 −1.10140
\(900\) 7.87645 0.262548
\(901\) −31.5874 −1.05233
\(902\) −20.9026 −0.695981
\(903\) 32.8544 1.09333
\(904\) −5.86470 −0.195057
\(905\) 46.0770 1.53165
\(906\) 4.15971 0.138197
\(907\) 8.34904 0.277225 0.138613 0.990347i \(-0.455736\pi\)
0.138613 + 0.990347i \(0.455736\pi\)
\(908\) 31.7199 1.05266
\(909\) 8.64218 0.286643
\(910\) −81.2702 −2.69408
\(911\) −8.66605 −0.287119 −0.143560 0.989642i \(-0.545855\pi\)
−0.143560 + 0.989642i \(0.545855\pi\)
\(912\) −18.7153 −0.619724
\(913\) −17.4095 −0.576169
\(914\) 57.4550 1.90044
\(915\) −3.07794 −0.101754
\(916\) 37.2991 1.23240
\(917\) −0.157619 −0.00520504
\(918\) −7.39562 −0.244092
\(919\) 17.4019 0.574035 0.287017 0.957925i \(-0.407336\pi\)
0.287017 + 0.957925i \(0.407336\pi\)
\(920\) 9.27234 0.305700
\(921\) 32.1020 1.05780
\(922\) −26.4046 −0.869591
\(923\) 14.6618 0.482599
\(924\) −5.56175 −0.182968
\(925\) −8.28809 −0.272511
\(926\) 31.3336 1.02969
\(927\) 1.53886 0.0505428
\(928\) −25.0383 −0.821922
\(929\) 7.38998 0.242458 0.121229 0.992625i \(-0.461317\pi\)
0.121229 + 0.992625i \(0.461317\pi\)
\(930\) 60.1907 1.97373
\(931\) −12.6106 −0.413296
\(932\) −49.8372 −1.63247
\(933\) −19.1427 −0.626705
\(934\) 24.1031 0.788676
\(935\) 11.7383 0.383884
\(936\) −2.00093 −0.0654025
\(937\) 41.9703 1.37111 0.685555 0.728021i \(-0.259560\pi\)
0.685555 + 0.728021i \(0.259560\pi\)
\(938\) −35.5662 −1.16128
\(939\) −7.12592 −0.232545
\(940\) 56.2116 1.83342
\(941\) −51.7763 −1.68786 −0.843929 0.536454i \(-0.819763\pi\)
−0.843929 + 0.536454i \(0.819763\pi\)
\(942\) −5.39117 −0.175654
\(943\) −69.9458 −2.27775
\(944\) 44.2814 1.44124
\(945\) 9.72319 0.316296
\(946\) 20.1685 0.655734
\(947\) 22.0964 0.718037 0.359019 0.933330i \(-0.383111\pi\)
0.359019 + 0.933330i \(0.383111\pi\)
\(948\) −7.51579 −0.244101
\(949\) −48.6307 −1.57862
\(950\) 36.7219 1.19142
\(951\) 30.7643 0.997600
\(952\) 5.59286 0.181266
\(953\) −36.4561 −1.18093 −0.590464 0.807064i \(-0.701055\pi\)
−0.590464 + 0.807064i \(0.701055\pi\)
\(954\) −16.0619 −0.520023
\(955\) −3.42495 −0.110829
\(956\) 8.62553 0.278970
\(957\) −3.27479 −0.105859
\(958\) −56.5564 −1.82725
\(959\) −0.769979 −0.0248639
\(960\) 18.4183 0.594447
\(961\) 70.6905 2.28034
\(962\) −15.4849 −0.499253
\(963\) −1.27301 −0.0410222
\(964\) 48.1500 1.55081
\(965\) −50.7081 −1.63235
\(966\) −39.7528 −1.27903
\(967\) 1.95027 0.0627164 0.0313582 0.999508i \(-0.490017\pi\)
0.0313582 + 0.999508i \(0.490017\pi\)
\(968\) 0.464237 0.0149211
\(969\) −16.1426 −0.518576
\(970\) −4.86455 −0.156191
\(971\) −40.8271 −1.31020 −0.655102 0.755540i \(-0.727374\pi\)
−0.655102 + 0.755540i \(0.727374\pi\)
\(972\) −1.76061 −0.0564715
\(973\) 66.9298 2.14567
\(974\) −71.3914 −2.28753
\(975\) 19.2824 0.617531
\(976\) −4.42148 −0.141528
\(977\) −29.4296 −0.941535 −0.470767 0.882257i \(-0.656023\pi\)
−0.470767 + 0.882257i \(0.656023\pi\)
\(978\) 32.6040 1.04256
\(979\) 9.75884 0.311894
\(980\) 16.1447 0.515723
\(981\) 10.8977 0.347937
\(982\) −74.7106 −2.38411
\(983\) 12.2766 0.391562 0.195781 0.980648i \(-0.437276\pi\)
0.195781 + 0.980648i \(0.437276\pi\)
\(984\) 5.00393 0.159519
\(985\) 78.2533 2.49336
\(986\) −24.2191 −0.771292
\(987\) 32.7682 1.04302
\(988\) 32.1207 1.02189
\(989\) 67.4892 2.14603
\(990\) 5.96883 0.189702
\(991\) −30.0821 −0.955588 −0.477794 0.878472i \(-0.658563\pi\)
−0.477794 + 0.878472i \(0.658563\pi\)
\(992\) 77.1014 2.44797
\(993\) 1.20244 0.0381582
\(994\) −20.8387 −0.660965
\(995\) −24.7986 −0.786168
\(996\) −30.6512 −0.971222
\(997\) 28.9187 0.915866 0.457933 0.888987i \(-0.348590\pi\)
0.457933 + 0.888987i \(0.348590\pi\)
\(998\) −3.26175 −0.103249
\(999\) 1.85262 0.0586143
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2013.2.a.h.1.3 14
3.2 odd 2 6039.2.a.j.1.12 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.h.1.3 14 1.1 even 1 trivial
6039.2.a.j.1.12 14 3.2 odd 2