Properties

Label 4017.2.a.e.1.7
Level $4017$
Weight $2$
Character 4017.1
Self dual yes
Analytic conductor $32.076$
Analytic rank $1$
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] = [4017,2,Mod(1,4017)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, 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("4017.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4017.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: \(32.0759064919\)
Analytic rank: \(1\)
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: \( x^{16} - 21 x^{14} - 3 x^{13} + 177 x^{12} + 45 x^{11} - 763 x^{10} - 251 x^{9} + 1771 x^{8} + 639 x^{7} - 2118 x^{6} - 710 x^{5} + 1113 x^{4} + 243 x^{3} - 183 x^{2} - 10 x + 7 \) 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 1.7
Root \(0.291985\) of defining polynomial
Character \(\chi\) \(=\) 4017.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.291985 q^{2} +1.00000 q^{3} -1.91474 q^{4} -3.42004 q^{5} -0.291985 q^{6} -3.90865 q^{7} +1.14305 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.291985 q^{2} +1.00000 q^{3} -1.91474 q^{4} -3.42004 q^{5} -0.291985 q^{6} -3.90865 q^{7} +1.14305 q^{8} +1.00000 q^{9} +0.998600 q^{10} +4.06263 q^{11} -1.91474 q^{12} -1.00000 q^{13} +1.14127 q^{14} -3.42004 q^{15} +3.49574 q^{16} -0.751286 q^{17} -0.291985 q^{18} +5.37839 q^{19} +6.54851 q^{20} -3.90865 q^{21} -1.18623 q^{22} -8.24911 q^{23} +1.14305 q^{24} +6.69670 q^{25} +0.291985 q^{26} +1.00000 q^{27} +7.48407 q^{28} +8.16240 q^{29} +0.998600 q^{30} +1.89795 q^{31} -3.30679 q^{32} +4.06263 q^{33} +0.219364 q^{34} +13.3678 q^{35} -1.91474 q^{36} -4.14408 q^{37} -1.57041 q^{38} -1.00000 q^{39} -3.90927 q^{40} +0.461243 q^{41} +1.14127 q^{42} +0.847965 q^{43} -7.77891 q^{44} -3.42004 q^{45} +2.40861 q^{46} +8.10038 q^{47} +3.49574 q^{48} +8.27757 q^{49} -1.95533 q^{50} -0.751286 q^{51} +1.91474 q^{52} +2.55597 q^{53} -0.291985 q^{54} -13.8944 q^{55} -4.46777 q^{56} +5.37839 q^{57} -2.38330 q^{58} -5.39458 q^{59} +6.54851 q^{60} -1.43944 q^{61} -0.554172 q^{62} -3.90865 q^{63} -6.02594 q^{64} +3.42004 q^{65} -1.18623 q^{66} +5.33432 q^{67} +1.43852 q^{68} -8.24911 q^{69} -3.90318 q^{70} -1.28042 q^{71} +1.14305 q^{72} +5.97616 q^{73} +1.21001 q^{74} +6.69670 q^{75} -10.2982 q^{76} -15.8794 q^{77} +0.291985 q^{78} -8.83363 q^{79} -11.9556 q^{80} +1.00000 q^{81} -0.134676 q^{82} +11.3247 q^{83} +7.48407 q^{84} +2.56943 q^{85} -0.247593 q^{86} +8.16240 q^{87} +4.64378 q^{88} -12.6221 q^{89} +0.998600 q^{90} +3.90865 q^{91} +15.7949 q^{92} +1.89795 q^{93} -2.36519 q^{94} -18.3943 q^{95} -3.30679 q^{96} -4.26288 q^{97} -2.41692 q^{98} +4.06263 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{3} + 10 q^{4} - 6 q^{5} - 13 q^{7} - 9 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{3} + 10 q^{4} - 6 q^{5} - 13 q^{7} - 9 q^{8} + 16 q^{9} - 8 q^{10} - 5 q^{11} + 10 q^{12} - 16 q^{13} - 8 q^{14} - 6 q^{15} - 14 q^{16} - q^{17} + 6 q^{19} - 4 q^{20} - 13 q^{21} - 11 q^{22} - 21 q^{23} - 9 q^{24} - 10 q^{25} + 16 q^{27} - 10 q^{28} - 17 q^{29} - 8 q^{30} - 33 q^{31} - 18 q^{32} - 5 q^{33} - 5 q^{34} - 4 q^{35} + 10 q^{36} - 23 q^{37} - 28 q^{38} - 16 q^{39} - 12 q^{40} + 7 q^{41} - 8 q^{42} - 33 q^{43} + 11 q^{44} - 6 q^{45} - 15 q^{46} - 13 q^{47} - 14 q^{48} - 17 q^{49} + 35 q^{50} - q^{51} - 10 q^{52} - 20 q^{53} - 54 q^{55} + 12 q^{56} + 6 q^{57} - 33 q^{58} + 6 q^{59} - 4 q^{60} - 49 q^{61} - 13 q^{62} - 13 q^{63} - 35 q^{64} + 6 q^{65} - 11 q^{66} - 4 q^{67} - 14 q^{68} - 21 q^{69} - 33 q^{70} - 29 q^{71} - 9 q^{72} - 21 q^{73} + 22 q^{74} - 10 q^{75} + 10 q^{76} - 21 q^{77} - 70 q^{79} - 8 q^{80} + 16 q^{81} - 10 q^{82} + 5 q^{83} - 10 q^{84} + 14 q^{85} + 29 q^{86} - 17 q^{87} - 45 q^{88} - 8 q^{89} - 8 q^{90} + 13 q^{91} - 29 q^{92} - 33 q^{93} + 12 q^{94} - 45 q^{95} - 18 q^{96} - 30 q^{97} + 15 q^{98} - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.291985 −0.206464 −0.103232 0.994657i \(-0.532918\pi\)
−0.103232 + 0.994657i \(0.532918\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.91474 −0.957372
\(5\) −3.42004 −1.52949 −0.764745 0.644333i \(-0.777135\pi\)
−0.764745 + 0.644333i \(0.777135\pi\)
\(6\) −0.291985 −0.119202
\(7\) −3.90865 −1.47733 −0.738666 0.674072i \(-0.764544\pi\)
−0.738666 + 0.674072i \(0.764544\pi\)
\(8\) 1.14305 0.404128
\(9\) 1.00000 0.333333
\(10\) 0.998600 0.315785
\(11\) 4.06263 1.22493 0.612465 0.790498i \(-0.290178\pi\)
0.612465 + 0.790498i \(0.290178\pi\)
\(12\) −1.91474 −0.552739
\(13\) −1.00000 −0.277350
\(14\) 1.14127 0.305016
\(15\) −3.42004 −0.883051
\(16\) 3.49574 0.873935
\(17\) −0.751286 −0.182214 −0.0911068 0.995841i \(-0.529040\pi\)
−0.0911068 + 0.995841i \(0.529040\pi\)
\(18\) −0.291985 −0.0688215
\(19\) 5.37839 1.23389 0.616944 0.787007i \(-0.288371\pi\)
0.616944 + 0.787007i \(0.288371\pi\)
\(20\) 6.54851 1.46429
\(21\) −3.90865 −0.852938
\(22\) −1.18623 −0.252904
\(23\) −8.24911 −1.72006 −0.860029 0.510245i \(-0.829555\pi\)
−0.860029 + 0.510245i \(0.829555\pi\)
\(24\) 1.14305 0.233323
\(25\) 6.69670 1.33934
\(26\) 0.291985 0.0572629
\(27\) 1.00000 0.192450
\(28\) 7.48407 1.41436
\(29\) 8.16240 1.51572 0.757860 0.652417i \(-0.226245\pi\)
0.757860 + 0.652417i \(0.226245\pi\)
\(30\) 0.998600 0.182319
\(31\) 1.89795 0.340881 0.170441 0.985368i \(-0.445481\pi\)
0.170441 + 0.985368i \(0.445481\pi\)
\(32\) −3.30679 −0.584564
\(33\) 4.06263 0.707214
\(34\) 0.219364 0.0376206
\(35\) 13.3678 2.25956
\(36\) −1.91474 −0.319124
\(37\) −4.14408 −0.681283 −0.340641 0.940193i \(-0.610644\pi\)
−0.340641 + 0.940193i \(0.610644\pi\)
\(38\) −1.57041 −0.254754
\(39\) −1.00000 −0.160128
\(40\) −3.90927 −0.618109
\(41\) 0.461243 0.0720341 0.0360170 0.999351i \(-0.488533\pi\)
0.0360170 + 0.999351i \(0.488533\pi\)
\(42\) 1.14127 0.176101
\(43\) 0.847965 0.129314 0.0646568 0.997908i \(-0.479405\pi\)
0.0646568 + 0.997908i \(0.479405\pi\)
\(44\) −7.77891 −1.17271
\(45\) −3.42004 −0.509830
\(46\) 2.40861 0.355131
\(47\) 8.10038 1.18156 0.590781 0.806832i \(-0.298820\pi\)
0.590781 + 0.806832i \(0.298820\pi\)
\(48\) 3.49574 0.504566
\(49\) 8.27757 1.18251
\(50\) −1.95533 −0.276526
\(51\) −0.751286 −0.105201
\(52\) 1.91474 0.265527
\(53\) 2.55597 0.351089 0.175545 0.984471i \(-0.443831\pi\)
0.175545 + 0.984471i \(0.443831\pi\)
\(54\) −0.291985 −0.0397341
\(55\) −13.8944 −1.87352
\(56\) −4.46777 −0.597031
\(57\) 5.37839 0.712385
\(58\) −2.38330 −0.312942
\(59\) −5.39458 −0.702315 −0.351157 0.936316i \(-0.614212\pi\)
−0.351157 + 0.936316i \(0.614212\pi\)
\(60\) 6.54851 0.845409
\(61\) −1.43944 −0.184302 −0.0921509 0.995745i \(-0.529374\pi\)
−0.0921509 + 0.995745i \(0.529374\pi\)
\(62\) −0.554172 −0.0703799
\(63\) −3.90865 −0.492444
\(64\) −6.02594 −0.753243
\(65\) 3.42004 0.424204
\(66\) −1.18623 −0.146014
\(67\) 5.33432 0.651690 0.325845 0.945423i \(-0.394351\pi\)
0.325845 + 0.945423i \(0.394351\pi\)
\(68\) 1.43852 0.174446
\(69\) −8.24911 −0.993076
\(70\) −3.90318 −0.466519
\(71\) −1.28042 −0.151958 −0.0759790 0.997109i \(-0.524208\pi\)
−0.0759790 + 0.997109i \(0.524208\pi\)
\(72\) 1.14305 0.134709
\(73\) 5.97616 0.699456 0.349728 0.936851i \(-0.386274\pi\)
0.349728 + 0.936851i \(0.386274\pi\)
\(74\) 1.21001 0.140661
\(75\) 6.69670 0.773268
\(76\) −10.2982 −1.18129
\(77\) −15.8794 −1.80963
\(78\) 0.291985 0.0330608
\(79\) −8.83363 −0.993861 −0.496930 0.867790i \(-0.665540\pi\)
−0.496930 + 0.867790i \(0.665540\pi\)
\(80\) −11.9556 −1.33667
\(81\) 1.00000 0.111111
\(82\) −0.134676 −0.0148725
\(83\) 11.3247 1.24305 0.621524 0.783395i \(-0.286514\pi\)
0.621524 + 0.783395i \(0.286514\pi\)
\(84\) 7.48407 0.816579
\(85\) 2.56943 0.278694
\(86\) −0.247593 −0.0266986
\(87\) 8.16240 0.875101
\(88\) 4.64378 0.495028
\(89\) −12.6221 −1.33794 −0.668972 0.743288i \(-0.733265\pi\)
−0.668972 + 0.743288i \(0.733265\pi\)
\(90\) 0.998600 0.105262
\(91\) 3.90865 0.409738
\(92\) 15.7949 1.64674
\(93\) 1.89795 0.196808
\(94\) −2.36519 −0.243950
\(95\) −18.3943 −1.88722
\(96\) −3.30679 −0.337498
\(97\) −4.26288 −0.432829 −0.216415 0.976302i \(-0.569436\pi\)
−0.216415 + 0.976302i \(0.569436\pi\)
\(98\) −2.41692 −0.244146
\(99\) 4.06263 0.408310
\(100\) −12.8225 −1.28225
\(101\) −3.27285 −0.325661 −0.162830 0.986654i \(-0.552062\pi\)
−0.162830 + 0.986654i \(0.552062\pi\)
\(102\) 0.219364 0.0217203
\(103\) 1.00000 0.0985329
\(104\) −1.14305 −0.112085
\(105\) 13.3678 1.30456
\(106\) −0.746304 −0.0724874
\(107\) −18.1309 −1.75278 −0.876392 0.481599i \(-0.840056\pi\)
−0.876392 + 0.481599i \(0.840056\pi\)
\(108\) −1.91474 −0.184246
\(109\) 3.57733 0.342646 0.171323 0.985215i \(-0.445196\pi\)
0.171323 + 0.985215i \(0.445196\pi\)
\(110\) 4.05695 0.386815
\(111\) −4.14408 −0.393339
\(112\) −13.6636 −1.29109
\(113\) 7.41181 0.697244 0.348622 0.937263i \(-0.386650\pi\)
0.348622 + 0.937263i \(0.386650\pi\)
\(114\) −1.57041 −0.147082
\(115\) 28.2123 2.63081
\(116\) −15.6289 −1.45111
\(117\) −1.00000 −0.0924500
\(118\) 1.57514 0.145003
\(119\) 2.93652 0.269190
\(120\) −3.90927 −0.356865
\(121\) 5.50499 0.500454
\(122\) 0.420295 0.0380517
\(123\) 0.461243 0.0415889
\(124\) −3.63409 −0.326350
\(125\) −5.80277 −0.519016
\(126\) 1.14127 0.101672
\(127\) −11.5345 −1.02352 −0.511761 0.859128i \(-0.671007\pi\)
−0.511761 + 0.859128i \(0.671007\pi\)
\(128\) 8.37307 0.740082
\(129\) 0.847965 0.0746592
\(130\) −0.998600 −0.0875830
\(131\) −10.3087 −0.900677 −0.450339 0.892858i \(-0.648697\pi\)
−0.450339 + 0.892858i \(0.648697\pi\)
\(132\) −7.77891 −0.677067
\(133\) −21.0223 −1.82286
\(134\) −1.55754 −0.134551
\(135\) −3.42004 −0.294350
\(136\) −0.858755 −0.0736376
\(137\) −0.280481 −0.0239631 −0.0119815 0.999928i \(-0.503814\pi\)
−0.0119815 + 0.999928i \(0.503814\pi\)
\(138\) 2.40861 0.205035
\(139\) 16.5220 1.40137 0.700687 0.713469i \(-0.252877\pi\)
0.700687 + 0.713469i \(0.252877\pi\)
\(140\) −25.5959 −2.16324
\(141\) 8.10038 0.682175
\(142\) 0.373863 0.0313739
\(143\) −4.06263 −0.339735
\(144\) 3.49574 0.291312
\(145\) −27.9158 −2.31828
\(146\) −1.74495 −0.144413
\(147\) 8.27757 0.682722
\(148\) 7.93486 0.652241
\(149\) 3.22493 0.264196 0.132098 0.991237i \(-0.457829\pi\)
0.132098 + 0.991237i \(0.457829\pi\)
\(150\) −1.95533 −0.159652
\(151\) −12.1172 −0.986085 −0.493043 0.870005i \(-0.664115\pi\)
−0.493043 + 0.870005i \(0.664115\pi\)
\(152\) 6.14775 0.498648
\(153\) −0.751286 −0.0607379
\(154\) 4.63655 0.373624
\(155\) −6.49106 −0.521375
\(156\) 1.91474 0.153302
\(157\) −6.90402 −0.551001 −0.275501 0.961301i \(-0.588844\pi\)
−0.275501 + 0.961301i \(0.588844\pi\)
\(158\) 2.57928 0.205197
\(159\) 2.55597 0.202702
\(160\) 11.3094 0.894085
\(161\) 32.2429 2.54110
\(162\) −0.291985 −0.0229405
\(163\) −10.0169 −0.784585 −0.392293 0.919840i \(-0.628318\pi\)
−0.392293 + 0.919840i \(0.628318\pi\)
\(164\) −0.883163 −0.0689635
\(165\) −13.8944 −1.08168
\(166\) −3.30664 −0.256645
\(167\) −3.84284 −0.297368 −0.148684 0.988885i \(-0.547504\pi\)
−0.148684 + 0.988885i \(0.547504\pi\)
\(168\) −4.46777 −0.344696
\(169\) 1.00000 0.0769231
\(170\) −0.750235 −0.0575404
\(171\) 5.37839 0.411296
\(172\) −1.62364 −0.123801
\(173\) −18.6286 −1.41631 −0.708154 0.706058i \(-0.750472\pi\)
−0.708154 + 0.706058i \(0.750472\pi\)
\(174\) −2.38330 −0.180677
\(175\) −26.1751 −1.97865
\(176\) 14.2019 1.07051
\(177\) −5.39458 −0.405482
\(178\) 3.68547 0.276238
\(179\) −20.6618 −1.54433 −0.772166 0.635421i \(-0.780827\pi\)
−0.772166 + 0.635421i \(0.780827\pi\)
\(180\) 6.54851 0.488097
\(181\) −21.6202 −1.60701 −0.803507 0.595295i \(-0.797035\pi\)
−0.803507 + 0.595295i \(0.797035\pi\)
\(182\) −1.14127 −0.0845963
\(183\) −1.43944 −0.106407
\(184\) −9.42911 −0.695123
\(185\) 14.1729 1.04202
\(186\) −0.554172 −0.0406338
\(187\) −3.05220 −0.223199
\(188\) −15.5102 −1.13119
\(189\) −3.90865 −0.284313
\(190\) 5.37086 0.389643
\(191\) 16.0629 1.16227 0.581134 0.813808i \(-0.302609\pi\)
0.581134 + 0.813808i \(0.302609\pi\)
\(192\) −6.02594 −0.434885
\(193\) −1.18317 −0.0851665 −0.0425832 0.999093i \(-0.513559\pi\)
−0.0425832 + 0.999093i \(0.513559\pi\)
\(194\) 1.24469 0.0893638
\(195\) 3.42004 0.244914
\(196\) −15.8494 −1.13210
\(197\) −21.1485 −1.50677 −0.753383 0.657582i \(-0.771579\pi\)
−0.753383 + 0.657582i \(0.771579\pi\)
\(198\) −1.18623 −0.0843015
\(199\) −10.8945 −0.772287 −0.386144 0.922439i \(-0.626193\pi\)
−0.386144 + 0.922439i \(0.626193\pi\)
\(200\) 7.65463 0.541264
\(201\) 5.33432 0.376254
\(202\) 0.955622 0.0672374
\(203\) −31.9040 −2.23922
\(204\) 1.43852 0.100717
\(205\) −1.57747 −0.110175
\(206\) −0.291985 −0.0203435
\(207\) −8.24911 −0.573353
\(208\) −3.49574 −0.242386
\(209\) 21.8504 1.51143
\(210\) −3.90318 −0.269345
\(211\) 1.53355 0.105574 0.0527869 0.998606i \(-0.483190\pi\)
0.0527869 + 0.998606i \(0.483190\pi\)
\(212\) −4.89403 −0.336123
\(213\) −1.28042 −0.0877330
\(214\) 5.29396 0.361887
\(215\) −2.90008 −0.197784
\(216\) 1.14305 0.0777744
\(217\) −7.41842 −0.503595
\(218\) −1.04452 −0.0707441
\(219\) 5.97616 0.403831
\(220\) 26.6042 1.79365
\(221\) 0.751286 0.0505370
\(222\) 1.21001 0.0812104
\(223\) −7.27176 −0.486953 −0.243476 0.969907i \(-0.578288\pi\)
−0.243476 + 0.969907i \(0.578288\pi\)
\(224\) 12.9251 0.863595
\(225\) 6.69670 0.446446
\(226\) −2.16413 −0.143956
\(227\) −19.6771 −1.30602 −0.653008 0.757351i \(-0.726493\pi\)
−0.653008 + 0.757351i \(0.726493\pi\)
\(228\) −10.2982 −0.682018
\(229\) −12.7100 −0.839899 −0.419949 0.907548i \(-0.637952\pi\)
−0.419949 + 0.907548i \(0.637952\pi\)
\(230\) −8.23757 −0.543169
\(231\) −15.8794 −1.04479
\(232\) 9.33000 0.612544
\(233\) 11.1281 0.729029 0.364515 0.931198i \(-0.381235\pi\)
0.364515 + 0.931198i \(0.381235\pi\)
\(234\) 0.291985 0.0190876
\(235\) −27.7037 −1.80719
\(236\) 10.3292 0.672377
\(237\) −8.83363 −0.573806
\(238\) −0.857418 −0.0555782
\(239\) 23.2690 1.50514 0.752572 0.658510i \(-0.228813\pi\)
0.752572 + 0.658510i \(0.228813\pi\)
\(240\) −11.9556 −0.771729
\(241\) −7.36801 −0.474616 −0.237308 0.971435i \(-0.576265\pi\)
−0.237308 + 0.971435i \(0.576265\pi\)
\(242\) −1.60737 −0.103326
\(243\) 1.00000 0.0641500
\(244\) 2.75617 0.176445
\(245\) −28.3096 −1.80864
\(246\) −0.134676 −0.00858663
\(247\) −5.37839 −0.342219
\(248\) 2.16944 0.137760
\(249\) 11.3247 0.717674
\(250\) 1.69432 0.107158
\(251\) 7.87972 0.497364 0.248682 0.968585i \(-0.420003\pi\)
0.248682 + 0.968585i \(0.420003\pi\)
\(252\) 7.48407 0.471452
\(253\) −33.5131 −2.10695
\(254\) 3.36790 0.211321
\(255\) 2.56943 0.160904
\(256\) 9.60708 0.600442
\(257\) 6.16495 0.384559 0.192280 0.981340i \(-0.438412\pi\)
0.192280 + 0.981340i \(0.438412\pi\)
\(258\) −0.247593 −0.0154145
\(259\) 16.1978 1.00648
\(260\) −6.54851 −0.406121
\(261\) 8.16240 0.505240
\(262\) 3.00999 0.185958
\(263\) 10.4331 0.643330 0.321665 0.946853i \(-0.395757\pi\)
0.321665 + 0.946853i \(0.395757\pi\)
\(264\) 4.64378 0.285805
\(265\) −8.74152 −0.536988
\(266\) 6.13818 0.376356
\(267\) −12.6221 −0.772462
\(268\) −10.2139 −0.623910
\(269\) 2.25633 0.137571 0.0687853 0.997631i \(-0.478088\pi\)
0.0687853 + 0.997631i \(0.478088\pi\)
\(270\) 0.998600 0.0607729
\(271\) −7.93979 −0.482308 −0.241154 0.970487i \(-0.577526\pi\)
−0.241154 + 0.970487i \(0.577526\pi\)
\(272\) −2.62630 −0.159243
\(273\) 3.90865 0.236562
\(274\) 0.0818961 0.00494752
\(275\) 27.2062 1.64060
\(276\) 15.7949 0.950744
\(277\) −24.1539 −1.45127 −0.725635 0.688080i \(-0.758454\pi\)
−0.725635 + 0.688080i \(0.758454\pi\)
\(278\) −4.82416 −0.289334
\(279\) 1.89795 0.113627
\(280\) 15.2800 0.913152
\(281\) 26.4669 1.57888 0.789441 0.613827i \(-0.210371\pi\)
0.789441 + 0.613827i \(0.210371\pi\)
\(282\) −2.36519 −0.140845
\(283\) −5.24325 −0.311679 −0.155839 0.987782i \(-0.549808\pi\)
−0.155839 + 0.987782i \(0.549808\pi\)
\(284\) 2.45168 0.145480
\(285\) −18.3943 −1.08959
\(286\) 1.18623 0.0701431
\(287\) −1.80284 −0.106418
\(288\) −3.30679 −0.194855
\(289\) −16.4356 −0.966798
\(290\) 8.15097 0.478642
\(291\) −4.26288 −0.249894
\(292\) −11.4428 −0.669640
\(293\) 27.1084 1.58369 0.791845 0.610722i \(-0.209121\pi\)
0.791845 + 0.610722i \(0.209121\pi\)
\(294\) −2.41692 −0.140958
\(295\) 18.4497 1.07418
\(296\) −4.73687 −0.275325
\(297\) 4.06263 0.235738
\(298\) −0.941629 −0.0545471
\(299\) 8.24911 0.477058
\(300\) −12.8225 −0.740305
\(301\) −3.31440 −0.191039
\(302\) 3.53804 0.203591
\(303\) −3.27285 −0.188020
\(304\) 18.8014 1.07834
\(305\) 4.92296 0.281888
\(306\) 0.219364 0.0125402
\(307\) 29.6254 1.69081 0.845404 0.534127i \(-0.179359\pi\)
0.845404 + 0.534127i \(0.179359\pi\)
\(308\) 30.4050 1.73249
\(309\) 1.00000 0.0568880
\(310\) 1.89529 0.107645
\(311\) 27.3840 1.55280 0.776402 0.630238i \(-0.217043\pi\)
0.776402 + 0.630238i \(0.217043\pi\)
\(312\) −1.14305 −0.0647122
\(313\) −3.74737 −0.211814 −0.105907 0.994376i \(-0.533775\pi\)
−0.105907 + 0.994376i \(0.533775\pi\)
\(314\) 2.01587 0.113762
\(315\) 13.3678 0.753188
\(316\) 16.9141 0.951495
\(317\) 27.9021 1.56714 0.783569 0.621304i \(-0.213397\pi\)
0.783569 + 0.621304i \(0.213397\pi\)
\(318\) −0.746304 −0.0418506
\(319\) 33.1608 1.85665
\(320\) 20.6090 1.15208
\(321\) −18.1309 −1.01197
\(322\) −9.41444 −0.524646
\(323\) −4.04071 −0.224831
\(324\) −1.91474 −0.106375
\(325\) −6.69670 −0.371466
\(326\) 2.92479 0.161989
\(327\) 3.57733 0.197827
\(328\) 0.527222 0.0291110
\(329\) −31.6616 −1.74556
\(330\) 4.05695 0.223328
\(331\) −16.3140 −0.896699 −0.448350 0.893858i \(-0.647988\pi\)
−0.448350 + 0.893858i \(0.647988\pi\)
\(332\) −21.6839 −1.19006
\(333\) −4.14408 −0.227094
\(334\) 1.12205 0.0613958
\(335\) −18.2436 −0.996754
\(336\) −13.6636 −0.745412
\(337\) −9.27981 −0.505504 −0.252752 0.967531i \(-0.581336\pi\)
−0.252752 + 0.967531i \(0.581336\pi\)
\(338\) −0.291985 −0.0158819
\(339\) 7.41181 0.402554
\(340\) −4.91981 −0.266814
\(341\) 7.71066 0.417556
\(342\) −1.57041 −0.0849179
\(343\) −4.99356 −0.269627
\(344\) 0.969263 0.0522592
\(345\) 28.2123 1.51890
\(346\) 5.43927 0.292417
\(347\) −24.4834 −1.31434 −0.657169 0.753743i \(-0.728246\pi\)
−0.657169 + 0.753743i \(0.728246\pi\)
\(348\) −15.6289 −0.837798
\(349\) −20.5689 −1.10103 −0.550514 0.834826i \(-0.685568\pi\)
−0.550514 + 0.834826i \(0.685568\pi\)
\(350\) 7.64272 0.408520
\(351\) −1.00000 −0.0533761
\(352\) −13.4343 −0.716050
\(353\) 9.75708 0.519317 0.259659 0.965700i \(-0.416390\pi\)
0.259659 + 0.965700i \(0.416390\pi\)
\(354\) 1.57514 0.0837175
\(355\) 4.37909 0.232418
\(356\) 24.1682 1.28091
\(357\) 2.93652 0.155417
\(358\) 6.03292 0.318850
\(359\) 28.4144 1.49965 0.749827 0.661634i \(-0.230137\pi\)
0.749827 + 0.661634i \(0.230137\pi\)
\(360\) −3.90927 −0.206036
\(361\) 9.92708 0.522478
\(362\) 6.31276 0.331791
\(363\) 5.50499 0.288937
\(364\) −7.48407 −0.392272
\(365\) −20.4387 −1.06981
\(366\) 0.420295 0.0219692
\(367\) −28.0983 −1.46672 −0.733359 0.679841i \(-0.762049\pi\)
−0.733359 + 0.679841i \(0.762049\pi\)
\(368\) −28.8367 −1.50322
\(369\) 0.461243 0.0240114
\(370\) −4.13828 −0.215139
\(371\) −9.99039 −0.518675
\(372\) −3.63409 −0.188419
\(373\) −19.8663 −1.02864 −0.514319 0.857599i \(-0.671955\pi\)
−0.514319 + 0.857599i \(0.671955\pi\)
\(374\) 0.891196 0.0460826
\(375\) −5.80277 −0.299654
\(376\) 9.25911 0.477502
\(377\) −8.16240 −0.420385
\(378\) 1.14127 0.0587004
\(379\) 10.7347 0.551406 0.275703 0.961243i \(-0.411089\pi\)
0.275703 + 0.961243i \(0.411089\pi\)
\(380\) 35.2204 1.80677
\(381\) −11.5345 −0.590931
\(382\) −4.69011 −0.239967
\(383\) 14.5351 0.742710 0.371355 0.928491i \(-0.378893\pi\)
0.371355 + 0.928491i \(0.378893\pi\)
\(384\) 8.37307 0.427286
\(385\) 54.3083 2.76781
\(386\) 0.345468 0.0175838
\(387\) 0.847965 0.0431045
\(388\) 8.16232 0.414379
\(389\) 37.4501 1.89879 0.949397 0.314080i \(-0.101696\pi\)
0.949397 + 0.314080i \(0.101696\pi\)
\(390\) −0.998600 −0.0505661
\(391\) 6.19744 0.313418
\(392\) 9.46164 0.477885
\(393\) −10.3087 −0.520006
\(394\) 6.17503 0.311093
\(395\) 30.2114 1.52010
\(396\) −7.77891 −0.390905
\(397\) −16.5911 −0.832683 −0.416342 0.909208i \(-0.636688\pi\)
−0.416342 + 0.909208i \(0.636688\pi\)
\(398\) 3.18101 0.159450
\(399\) −21.0223 −1.05243
\(400\) 23.4099 1.17049
\(401\) −34.1489 −1.70531 −0.852656 0.522472i \(-0.825010\pi\)
−0.852656 + 0.522472i \(0.825010\pi\)
\(402\) −1.55754 −0.0776830
\(403\) −1.89795 −0.0945435
\(404\) 6.26667 0.311779
\(405\) −3.42004 −0.169943
\(406\) 9.31548 0.462319
\(407\) −16.8359 −0.834524
\(408\) −0.858755 −0.0425147
\(409\) −12.3348 −0.609917 −0.304958 0.952366i \(-0.598643\pi\)
−0.304958 + 0.952366i \(0.598643\pi\)
\(410\) 0.460598 0.0227473
\(411\) −0.280481 −0.0138351
\(412\) −1.91474 −0.0943327
\(413\) 21.0855 1.03755
\(414\) 2.40861 0.118377
\(415\) −38.7310 −1.90123
\(416\) 3.30679 0.162129
\(417\) 16.5220 0.809084
\(418\) −6.37999 −0.312056
\(419\) 22.7240 1.11014 0.555070 0.831804i \(-0.312692\pi\)
0.555070 + 0.831804i \(0.312692\pi\)
\(420\) −25.5959 −1.24895
\(421\) 30.5654 1.48967 0.744833 0.667251i \(-0.232529\pi\)
0.744833 + 0.667251i \(0.232529\pi\)
\(422\) −0.447773 −0.0217972
\(423\) 8.10038 0.393854
\(424\) 2.92159 0.141885
\(425\) −5.03114 −0.244046
\(426\) 0.373863 0.0181137
\(427\) 5.62628 0.272275
\(428\) 34.7161 1.67807
\(429\) −4.06263 −0.196146
\(430\) 0.846778 0.0408353
\(431\) −15.5146 −0.747314 −0.373657 0.927567i \(-0.621896\pi\)
−0.373657 + 0.927567i \(0.621896\pi\)
\(432\) 3.49574 0.168189
\(433\) −17.7167 −0.851409 −0.425704 0.904862i \(-0.639974\pi\)
−0.425704 + 0.904862i \(0.639974\pi\)
\(434\) 2.16606 0.103974
\(435\) −27.9158 −1.33846
\(436\) −6.84967 −0.328039
\(437\) −44.3669 −2.12236
\(438\) −1.74495 −0.0833767
\(439\) 24.4918 1.16893 0.584464 0.811419i \(-0.301305\pi\)
0.584464 + 0.811419i \(0.301305\pi\)
\(440\) −15.8819 −0.757141
\(441\) 8.27757 0.394170
\(442\) −0.219364 −0.0104341
\(443\) 14.0806 0.668988 0.334494 0.942398i \(-0.391435\pi\)
0.334494 + 0.942398i \(0.391435\pi\)
\(444\) 7.93486 0.376572
\(445\) 43.1683 2.04637
\(446\) 2.12324 0.100538
\(447\) 3.22493 0.152534
\(448\) 23.5533 1.11279
\(449\) 26.2006 1.23648 0.618241 0.785989i \(-0.287846\pi\)
0.618241 + 0.785989i \(0.287846\pi\)
\(450\) −1.95533 −0.0921753
\(451\) 1.87386 0.0882367
\(452\) −14.1917 −0.667522
\(453\) −12.1172 −0.569317
\(454\) 5.74542 0.269646
\(455\) −13.3678 −0.626690
\(456\) 6.14775 0.287895
\(457\) 24.4187 1.14226 0.571129 0.820860i \(-0.306505\pi\)
0.571129 + 0.820860i \(0.306505\pi\)
\(458\) 3.71112 0.173409
\(459\) −0.751286 −0.0350670
\(460\) −54.0194 −2.51867
\(461\) −2.64630 −0.123250 −0.0616252 0.998099i \(-0.519628\pi\)
−0.0616252 + 0.998099i \(0.519628\pi\)
\(462\) 4.63655 0.215712
\(463\) −8.27785 −0.384704 −0.192352 0.981326i \(-0.561612\pi\)
−0.192352 + 0.981326i \(0.561612\pi\)
\(464\) 28.5336 1.32464
\(465\) −6.49106 −0.301016
\(466\) −3.24925 −0.150519
\(467\) 6.41694 0.296941 0.148470 0.988917i \(-0.452565\pi\)
0.148470 + 0.988917i \(0.452565\pi\)
\(468\) 1.91474 0.0885091
\(469\) −20.8500 −0.962763
\(470\) 8.08904 0.373120
\(471\) −6.90402 −0.318121
\(472\) −6.16625 −0.283825
\(473\) 3.44497 0.158400
\(474\) 2.57928 0.118470
\(475\) 36.0174 1.65259
\(476\) −5.62268 −0.257715
\(477\) 2.55597 0.117030
\(478\) −6.79418 −0.310759
\(479\) −18.2673 −0.834653 −0.417327 0.908757i \(-0.637033\pi\)
−0.417327 + 0.908757i \(0.637033\pi\)
\(480\) 11.3094 0.516200
\(481\) 4.14408 0.188954
\(482\) 2.15135 0.0979912
\(483\) 32.2429 1.46710
\(484\) −10.5407 −0.479121
\(485\) 14.5792 0.662008
\(486\) −0.291985 −0.0132447
\(487\) 20.5854 0.932814 0.466407 0.884570i \(-0.345548\pi\)
0.466407 + 0.884570i \(0.345548\pi\)
\(488\) −1.64535 −0.0744814
\(489\) −10.0169 −0.452981
\(490\) 8.26598 0.373419
\(491\) −35.2613 −1.59132 −0.795660 0.605743i \(-0.792876\pi\)
−0.795660 + 0.605743i \(0.792876\pi\)
\(492\) −0.883163 −0.0398161
\(493\) −6.13230 −0.276185
\(494\) 1.57041 0.0706560
\(495\) −13.8944 −0.624506
\(496\) 6.63473 0.297908
\(497\) 5.00472 0.224492
\(498\) −3.30664 −0.148174
\(499\) 12.3590 0.553267 0.276633 0.960976i \(-0.410781\pi\)
0.276633 + 0.960976i \(0.410781\pi\)
\(500\) 11.1108 0.496891
\(501\) −3.84284 −0.171685
\(502\) −2.30076 −0.102688
\(503\) 8.11879 0.361999 0.180999 0.983483i \(-0.442067\pi\)
0.180999 + 0.983483i \(0.442067\pi\)
\(504\) −4.46777 −0.199010
\(505\) 11.1933 0.498095
\(506\) 9.78532 0.435010
\(507\) 1.00000 0.0444116
\(508\) 22.0856 0.979892
\(509\) −1.23756 −0.0548540 −0.0274270 0.999624i \(-0.508731\pi\)
−0.0274270 + 0.999624i \(0.508731\pi\)
\(510\) −0.750235 −0.0332210
\(511\) −23.3587 −1.03333
\(512\) −19.5513 −0.864052
\(513\) 5.37839 0.237462
\(514\) −1.80007 −0.0793978
\(515\) −3.42004 −0.150705
\(516\) −1.62364 −0.0714767
\(517\) 32.9089 1.44733
\(518\) −4.72950 −0.207802
\(519\) −18.6286 −0.817705
\(520\) 3.90927 0.171433
\(521\) 25.9861 1.13847 0.569236 0.822174i \(-0.307239\pi\)
0.569236 + 0.822174i \(0.307239\pi\)
\(522\) −2.38330 −0.104314
\(523\) 8.85525 0.387213 0.193606 0.981079i \(-0.437982\pi\)
0.193606 + 0.981079i \(0.437982\pi\)
\(524\) 19.7386 0.862283
\(525\) −26.1751 −1.14237
\(526\) −3.04630 −0.132825
\(527\) −1.42590 −0.0621133
\(528\) 14.2019 0.618059
\(529\) 45.0478 1.95860
\(530\) 2.55239 0.110869
\(531\) −5.39458 −0.234105
\(532\) 40.2523 1.74516
\(533\) −0.461243 −0.0199787
\(534\) 3.68547 0.159486
\(535\) 62.0086 2.68086
\(536\) 6.09737 0.263366
\(537\) −20.6618 −0.891620
\(538\) −0.658813 −0.0284034
\(539\) 33.6287 1.44849
\(540\) 6.54851 0.281803
\(541\) −1.98118 −0.0851774 −0.0425887 0.999093i \(-0.513561\pi\)
−0.0425887 + 0.999093i \(0.513561\pi\)
\(542\) 2.31830 0.0995795
\(543\) −21.6202 −0.927810
\(544\) 2.48435 0.106516
\(545\) −12.2346 −0.524073
\(546\) −1.14127 −0.0488417
\(547\) −38.2448 −1.63523 −0.817615 0.575765i \(-0.804704\pi\)
−0.817615 + 0.575765i \(0.804704\pi\)
\(548\) 0.537049 0.0229416
\(549\) −1.43944 −0.0614339
\(550\) −7.94380 −0.338725
\(551\) 43.9006 1.87023
\(552\) −9.42911 −0.401330
\(553\) 34.5276 1.46826
\(554\) 7.05258 0.299636
\(555\) 14.1729 0.601608
\(556\) −31.6353 −1.34164
\(557\) 18.2115 0.771646 0.385823 0.922573i \(-0.373918\pi\)
0.385823 + 0.922573i \(0.373918\pi\)
\(558\) −0.554172 −0.0234600
\(559\) −0.847965 −0.0358651
\(560\) 46.7302 1.97471
\(561\) −3.05220 −0.128864
\(562\) −7.72792 −0.325983
\(563\) −28.3927 −1.19661 −0.598305 0.801269i \(-0.704159\pi\)
−0.598305 + 0.801269i \(0.704159\pi\)
\(564\) −15.5102 −0.653096
\(565\) −25.3487 −1.06643
\(566\) 1.53095 0.0643505
\(567\) −3.90865 −0.164148
\(568\) −1.46358 −0.0614104
\(569\) −12.0751 −0.506214 −0.253107 0.967438i \(-0.581452\pi\)
−0.253107 + 0.967438i \(0.581452\pi\)
\(570\) 5.37086 0.224961
\(571\) −4.87460 −0.203996 −0.101998 0.994785i \(-0.532523\pi\)
−0.101998 + 0.994785i \(0.532523\pi\)
\(572\) 7.77891 0.325252
\(573\) 16.0629 0.671036
\(574\) 0.526402 0.0219716
\(575\) −55.2418 −2.30374
\(576\) −6.02594 −0.251081
\(577\) −3.94288 −0.164144 −0.0820721 0.996626i \(-0.526154\pi\)
−0.0820721 + 0.996626i \(0.526154\pi\)
\(578\) 4.79893 0.199609
\(579\) −1.18317 −0.0491709
\(580\) 53.4516 2.21946
\(581\) −44.2644 −1.83639
\(582\) 1.24469 0.0515942
\(583\) 10.3840 0.430060
\(584\) 6.83102 0.282670
\(585\) 3.42004 0.141401
\(586\) −7.91524 −0.326975
\(587\) −6.69981 −0.276531 −0.138265 0.990395i \(-0.544153\pi\)
−0.138265 + 0.990395i \(0.544153\pi\)
\(588\) −15.8494 −0.653619
\(589\) 10.2079 0.420609
\(590\) −5.38703 −0.221781
\(591\) −21.1485 −0.869932
\(592\) −14.4866 −0.595397
\(593\) −35.9666 −1.47697 −0.738486 0.674269i \(-0.764459\pi\)
−0.738486 + 0.674269i \(0.764459\pi\)
\(594\) −1.18623 −0.0486715
\(595\) −10.0430 −0.411723
\(596\) −6.17491 −0.252934
\(597\) −10.8945 −0.445880
\(598\) −2.40861 −0.0984956
\(599\) −44.9990 −1.83861 −0.919304 0.393547i \(-0.871248\pi\)
−0.919304 + 0.393547i \(0.871248\pi\)
\(600\) 7.65463 0.312499
\(601\) −41.4206 −1.68958 −0.844791 0.535097i \(-0.820275\pi\)
−0.844791 + 0.535097i \(0.820275\pi\)
\(602\) 0.967755 0.0394427
\(603\) 5.33432 0.217230
\(604\) 23.2014 0.944051
\(605\) −18.8273 −0.765439
\(606\) 0.955622 0.0388195
\(607\) −34.4093 −1.39663 −0.698316 0.715789i \(-0.746067\pi\)
−0.698316 + 0.715789i \(0.746067\pi\)
\(608\) −17.7852 −0.721286
\(609\) −31.9040 −1.29281
\(610\) −1.43743 −0.0581998
\(611\) −8.10038 −0.327706
\(612\) 1.43852 0.0581488
\(613\) −8.32488 −0.336239 −0.168119 0.985767i \(-0.553769\pi\)
−0.168119 + 0.985767i \(0.553769\pi\)
\(614\) −8.65015 −0.349092
\(615\) −1.57747 −0.0636098
\(616\) −18.1509 −0.731321
\(617\) −5.27308 −0.212286 −0.106143 0.994351i \(-0.533850\pi\)
−0.106143 + 0.994351i \(0.533850\pi\)
\(618\) −0.291985 −0.0117453
\(619\) 1.41263 0.0567785 0.0283892 0.999597i \(-0.490962\pi\)
0.0283892 + 0.999597i \(0.490962\pi\)
\(620\) 12.4287 0.499150
\(621\) −8.24911 −0.331025
\(622\) −7.99571 −0.320599
\(623\) 49.3356 1.97659
\(624\) −3.49574 −0.139942
\(625\) −13.6377 −0.545510
\(626\) 1.09417 0.0437320
\(627\) 21.8504 0.872622
\(628\) 13.2194 0.527513
\(629\) 3.11339 0.124139
\(630\) −3.90318 −0.155506
\(631\) −3.73469 −0.148675 −0.0743377 0.997233i \(-0.523684\pi\)
−0.0743377 + 0.997233i \(0.523684\pi\)
\(632\) −10.0972 −0.401647
\(633\) 1.53355 0.0609531
\(634\) −8.14699 −0.323558
\(635\) 39.4485 1.56547
\(636\) −4.89403 −0.194061
\(637\) −8.27757 −0.327969
\(638\) −9.68246 −0.383332
\(639\) −1.28042 −0.0506527
\(640\) −28.6363 −1.13195
\(641\) 7.40940 0.292654 0.146327 0.989236i \(-0.453255\pi\)
0.146327 + 0.989236i \(0.453255\pi\)
\(642\) 5.29396 0.208936
\(643\) −25.2640 −0.996316 −0.498158 0.867086i \(-0.665990\pi\)
−0.498158 + 0.867086i \(0.665990\pi\)
\(644\) −61.7370 −2.43278
\(645\) −2.90008 −0.114190
\(646\) 1.17983 0.0464196
\(647\) −14.8010 −0.581887 −0.290944 0.956740i \(-0.593969\pi\)
−0.290944 + 0.956740i \(0.593969\pi\)
\(648\) 1.14305 0.0449031
\(649\) −21.9162 −0.860286
\(650\) 1.95533 0.0766945
\(651\) −7.41842 −0.290751
\(652\) 19.1798 0.751140
\(653\) −19.8418 −0.776469 −0.388234 0.921561i \(-0.626915\pi\)
−0.388234 + 0.921561i \(0.626915\pi\)
\(654\) −1.04452 −0.0408441
\(655\) 35.2563 1.37758
\(656\) 1.61239 0.0629531
\(657\) 5.97616 0.233152
\(658\) 9.24470 0.360396
\(659\) −26.6577 −1.03844 −0.519219 0.854641i \(-0.673777\pi\)
−0.519219 + 0.854641i \(0.673777\pi\)
\(660\) 26.6042 1.03557
\(661\) −39.6098 −1.54064 −0.770321 0.637656i \(-0.779904\pi\)
−0.770321 + 0.637656i \(0.779904\pi\)
\(662\) 4.76344 0.185136
\(663\) 0.751286 0.0291775
\(664\) 12.9447 0.502350
\(665\) 71.8970 2.78805
\(666\) 1.21001 0.0468869
\(667\) −67.3325 −2.60713
\(668\) 7.35805 0.284692
\(669\) −7.27176 −0.281142
\(670\) 5.32685 0.205794
\(671\) −5.84793 −0.225757
\(672\) 12.9251 0.498597
\(673\) −47.7062 −1.83894 −0.919469 0.393163i \(-0.871381\pi\)
−0.919469 + 0.393163i \(0.871381\pi\)
\(674\) 2.70956 0.104368
\(675\) 6.69670 0.257756
\(676\) −1.91474 −0.0736440
\(677\) −26.2074 −1.00723 −0.503615 0.863928i \(-0.667997\pi\)
−0.503615 + 0.863928i \(0.667997\pi\)
\(678\) −2.16413 −0.0831131
\(679\) 16.6621 0.639433
\(680\) 2.93698 0.112628
\(681\) −19.6771 −0.754029
\(682\) −2.25140 −0.0862104
\(683\) 4.31379 0.165063 0.0825314 0.996588i \(-0.473700\pi\)
0.0825314 + 0.996588i \(0.473700\pi\)
\(684\) −10.2982 −0.393763
\(685\) 0.959256 0.0366513
\(686\) 1.45804 0.0556683
\(687\) −12.7100 −0.484916
\(688\) 2.96426 0.113012
\(689\) −2.55597 −0.0973747
\(690\) −8.23757 −0.313599
\(691\) −5.79495 −0.220450 −0.110225 0.993907i \(-0.535157\pi\)
−0.110225 + 0.993907i \(0.535157\pi\)
\(692\) 35.6690 1.35593
\(693\) −15.8794 −0.603209
\(694\) 7.14878 0.271364
\(695\) −56.5058 −2.14339
\(696\) 9.33000 0.353653
\(697\) −0.346526 −0.0131256
\(698\) 6.00581 0.227323
\(699\) 11.1281 0.420905
\(700\) 50.1186 1.89430
\(701\) −10.8920 −0.411386 −0.205693 0.978617i \(-0.565945\pi\)
−0.205693 + 0.978617i \(0.565945\pi\)
\(702\) 0.291985 0.0110203
\(703\) −22.2885 −0.840626
\(704\) −24.4812 −0.922670
\(705\) −27.7037 −1.04338
\(706\) −2.84892 −0.107220
\(707\) 12.7924 0.481109
\(708\) 10.3292 0.388197
\(709\) 22.8349 0.857582 0.428791 0.903404i \(-0.358940\pi\)
0.428791 + 0.903404i \(0.358940\pi\)
\(710\) −1.27863 −0.0479861
\(711\) −8.83363 −0.331287
\(712\) −14.4277 −0.540700
\(713\) −15.6564 −0.586336
\(714\) −0.857418 −0.0320881
\(715\) 13.8944 0.519620
\(716\) 39.5620 1.47850
\(717\) 23.2690 0.868996
\(718\) −8.29657 −0.309625
\(719\) 18.6716 0.696332 0.348166 0.937433i \(-0.386805\pi\)
0.348166 + 0.937433i \(0.386805\pi\)
\(720\) −11.9556 −0.445558
\(721\) −3.90865 −0.145566
\(722\) −2.89855 −0.107873
\(723\) −7.36801 −0.274019
\(724\) 41.3971 1.53851
\(725\) 54.6611 2.03006
\(726\) −1.60737 −0.0596552
\(727\) −9.24966 −0.343051 −0.171525 0.985180i \(-0.554870\pi\)
−0.171525 + 0.985180i \(0.554870\pi\)
\(728\) 4.46777 0.165587
\(729\) 1.00000 0.0370370
\(730\) 5.96779 0.220878
\(731\) −0.637065 −0.0235627
\(732\) 2.75617 0.101871
\(733\) 7.64093 0.282224 0.141112 0.989994i \(-0.454932\pi\)
0.141112 + 0.989994i \(0.454932\pi\)
\(734\) 8.20427 0.302825
\(735\) −28.3096 −1.04422
\(736\) 27.2781 1.00548
\(737\) 21.6714 0.798275
\(738\) −0.134676 −0.00495749
\(739\) 52.5423 1.93280 0.966399 0.257045i \(-0.0827488\pi\)
0.966399 + 0.257045i \(0.0827488\pi\)
\(740\) −27.1376 −0.997596
\(741\) −5.37839 −0.197580
\(742\) 2.91704 0.107088
\(743\) 14.8996 0.546613 0.273306 0.961927i \(-0.411883\pi\)
0.273306 + 0.961927i \(0.411883\pi\)
\(744\) 2.16944 0.0795355
\(745\) −11.0294 −0.404086
\(746\) 5.80066 0.212377
\(747\) 11.3247 0.414349
\(748\) 5.84419 0.213685
\(749\) 70.8675 2.58944
\(750\) 1.69432 0.0618678
\(751\) −50.0771 −1.82734 −0.913669 0.406458i \(-0.866764\pi\)
−0.913669 + 0.406458i \(0.866764\pi\)
\(752\) 28.3168 1.03261
\(753\) 7.87972 0.287153
\(754\) 2.38330 0.0867945
\(755\) 41.4414 1.50821
\(756\) 7.48407 0.272193
\(757\) −12.5890 −0.457555 −0.228777 0.973479i \(-0.573473\pi\)
−0.228777 + 0.973479i \(0.573473\pi\)
\(758\) −3.13437 −0.113846
\(759\) −33.5131 −1.21645
\(760\) −21.0256 −0.762677
\(761\) −10.7172 −0.388497 −0.194248 0.980952i \(-0.562227\pi\)
−0.194248 + 0.980952i \(0.562227\pi\)
\(762\) 3.36790 0.122006
\(763\) −13.9825 −0.506201
\(764\) −30.7563 −1.11272
\(765\) 2.56943 0.0928980
\(766\) −4.24404 −0.153343
\(767\) 5.39458 0.194787
\(768\) 9.60708 0.346666
\(769\) 3.80655 0.137268 0.0686338 0.997642i \(-0.478136\pi\)
0.0686338 + 0.997642i \(0.478136\pi\)
\(770\) −15.8572 −0.571454
\(771\) 6.16495 0.222025
\(772\) 2.26547 0.0815360
\(773\) −35.4403 −1.27470 −0.637349 0.770575i \(-0.719969\pi\)
−0.637349 + 0.770575i \(0.719969\pi\)
\(774\) −0.247593 −0.00889954
\(775\) 12.7100 0.456556
\(776\) −4.87266 −0.174918
\(777\) 16.1978 0.581092
\(778\) −10.9348 −0.392033
\(779\) 2.48075 0.0888819
\(780\) −6.54851 −0.234474
\(781\) −5.20188 −0.186138
\(782\) −1.80956 −0.0647097
\(783\) 8.16240 0.291700
\(784\) 28.9362 1.03344
\(785\) 23.6121 0.842751
\(786\) 3.00999 0.107363
\(787\) 6.93573 0.247232 0.123616 0.992330i \(-0.460551\pi\)
0.123616 + 0.992330i \(0.460551\pi\)
\(788\) 40.4939 1.44254
\(789\) 10.4331 0.371427
\(790\) −8.82126 −0.313846
\(791\) −28.9702 −1.03006
\(792\) 4.64378 0.165009
\(793\) 1.43944 0.0511161
\(794\) 4.84435 0.171919
\(795\) −8.74152 −0.310030
\(796\) 20.8601 0.739366
\(797\) 13.4366 0.475950 0.237975 0.971271i \(-0.423516\pi\)
0.237975 + 0.971271i \(0.423516\pi\)
\(798\) 6.13818 0.217289
\(799\) −6.08571 −0.215297
\(800\) −22.1446 −0.782929
\(801\) −12.6221 −0.445981
\(802\) 9.97095 0.352086
\(803\) 24.2789 0.856785
\(804\) −10.2139 −0.360215
\(805\) −110.272 −3.88658
\(806\) 0.554172 0.0195199
\(807\) 2.25633 0.0794264
\(808\) −3.74102 −0.131609
\(809\) −23.5074 −0.826475 −0.413237 0.910623i \(-0.635602\pi\)
−0.413237 + 0.910623i \(0.635602\pi\)
\(810\) 0.998600 0.0350872
\(811\) −25.7091 −0.902768 −0.451384 0.892330i \(-0.649070\pi\)
−0.451384 + 0.892330i \(0.649070\pi\)
\(812\) 61.0880 2.14377
\(813\) −7.93979 −0.278461
\(814\) 4.91582 0.172299
\(815\) 34.2583 1.20002
\(816\) −2.62630 −0.0919389
\(817\) 4.56069 0.159558
\(818\) 3.60157 0.125926
\(819\) 3.90865 0.136579
\(820\) 3.02046 0.105479
\(821\) −41.0294 −1.43194 −0.715968 0.698133i \(-0.754015\pi\)
−0.715968 + 0.698133i \(0.754015\pi\)
\(822\) 0.0818961 0.00285645
\(823\) −2.30228 −0.0802525 −0.0401262 0.999195i \(-0.512776\pi\)
−0.0401262 + 0.999195i \(0.512776\pi\)
\(824\) 1.14305 0.0398199
\(825\) 27.2062 0.947199
\(826\) −6.15666 −0.214217
\(827\) 27.3361 0.950571 0.475285 0.879832i \(-0.342345\pi\)
0.475285 + 0.879832i \(0.342345\pi\)
\(828\) 15.7949 0.548912
\(829\) 44.5572 1.54754 0.773768 0.633469i \(-0.218370\pi\)
0.773768 + 0.633469i \(0.218370\pi\)
\(830\) 11.3089 0.392536
\(831\) −24.1539 −0.837891
\(832\) 6.02594 0.208912
\(833\) −6.21882 −0.215469
\(834\) −4.82416 −0.167047
\(835\) 13.1427 0.454821
\(836\) −41.8380 −1.44700
\(837\) 1.89795 0.0656027
\(838\) −6.63506 −0.229204
\(839\) 7.92724 0.273679 0.136839 0.990593i \(-0.456306\pi\)
0.136839 + 0.990593i \(0.456306\pi\)
\(840\) 15.2800 0.527209
\(841\) 37.6248 1.29741
\(842\) −8.92463 −0.307563
\(843\) 26.4669 0.911567
\(844\) −2.93635 −0.101073
\(845\) −3.42004 −0.117653
\(846\) −2.36519 −0.0813168
\(847\) −21.5171 −0.739337
\(848\) 8.93500 0.306829
\(849\) −5.24325 −0.179948
\(850\) 1.46901 0.0503868
\(851\) 34.1850 1.17185
\(852\) 2.45168 0.0839931
\(853\) −18.8584 −0.645700 −0.322850 0.946450i \(-0.604641\pi\)
−0.322850 + 0.946450i \(0.604641\pi\)
\(854\) −1.64279 −0.0562151
\(855\) −18.3943 −0.629073
\(856\) −20.7245 −0.708348
\(857\) −7.48466 −0.255671 −0.127836 0.991795i \(-0.540803\pi\)
−0.127836 + 0.991795i \(0.540803\pi\)
\(858\) 1.18623 0.0404971
\(859\) −0.276798 −0.00944423 −0.00472211 0.999989i \(-0.501503\pi\)
−0.00472211 + 0.999989i \(0.501503\pi\)
\(860\) 5.55291 0.189353
\(861\) −1.80284 −0.0614406
\(862\) 4.53004 0.154294
\(863\) −25.5255 −0.868897 −0.434448 0.900697i \(-0.643057\pi\)
−0.434448 + 0.900697i \(0.643057\pi\)
\(864\) −3.30679 −0.112499
\(865\) 63.7107 2.16623
\(866\) 5.17300 0.175786
\(867\) −16.4356 −0.558181
\(868\) 14.2044 0.482128
\(869\) −35.8878 −1.21741
\(870\) 8.15097 0.276344
\(871\) −5.33432 −0.180746
\(872\) 4.08905 0.138473
\(873\) −4.26288 −0.144276
\(874\) 12.9545 0.438191
\(875\) 22.6810 0.766758
\(876\) −11.4428 −0.386617
\(877\) −20.1987 −0.682063 −0.341032 0.940052i \(-0.610776\pi\)
−0.341032 + 0.940052i \(0.610776\pi\)
\(878\) −7.15122 −0.241342
\(879\) 27.1084 0.914344
\(880\) −48.5711 −1.63733
\(881\) −10.8204 −0.364550 −0.182275 0.983248i \(-0.558346\pi\)
−0.182275 + 0.983248i \(0.558346\pi\)
\(882\) −2.41692 −0.0813820
\(883\) −41.9002 −1.41005 −0.705027 0.709181i \(-0.749065\pi\)
−0.705027 + 0.709181i \(0.749065\pi\)
\(884\) −1.43852 −0.0483827
\(885\) 18.4497 0.620180
\(886\) −4.11131 −0.138122
\(887\) 12.0341 0.404066 0.202033 0.979379i \(-0.435245\pi\)
0.202033 + 0.979379i \(0.435245\pi\)
\(888\) −4.73687 −0.158959
\(889\) 45.0844 1.51208
\(890\) −12.6045 −0.422503
\(891\) 4.06263 0.136103
\(892\) 13.9236 0.466195
\(893\) 43.5670 1.45791
\(894\) −0.941629 −0.0314928
\(895\) 70.6641 2.36204
\(896\) −32.7274 −1.09335
\(897\) 8.24911 0.275430
\(898\) −7.65017 −0.255289
\(899\) 15.4918 0.516681
\(900\) −12.8225 −0.427415
\(901\) −1.92026 −0.0639733
\(902\) −0.547139 −0.0182177
\(903\) −3.31440 −0.110296
\(904\) 8.47203 0.281776
\(905\) 73.9419 2.45791
\(906\) 3.53804 0.117544
\(907\) 8.55232 0.283975 0.141987 0.989868i \(-0.454651\pi\)
0.141987 + 0.989868i \(0.454651\pi\)
\(908\) 37.6767 1.25034
\(909\) −3.27285 −0.108554
\(910\) 3.90318 0.129389
\(911\) −15.7307 −0.521182 −0.260591 0.965449i \(-0.583917\pi\)
−0.260591 + 0.965449i \(0.583917\pi\)
\(912\) 18.8014 0.622578
\(913\) 46.0081 1.52265
\(914\) −7.12989 −0.235836
\(915\) 4.92296 0.162748
\(916\) 24.3364 0.804096
\(917\) 40.2932 1.33060
\(918\) 0.219364 0.00724009
\(919\) −40.0281 −1.32040 −0.660202 0.751088i \(-0.729529\pi\)
−0.660202 + 0.751088i \(0.729529\pi\)
\(920\) 32.2480 1.06318
\(921\) 29.6254 0.976189
\(922\) 0.772678 0.0254468
\(923\) 1.28042 0.0421456
\(924\) 30.4050 1.00025
\(925\) −27.7516 −0.912469
\(926\) 2.41701 0.0794278
\(927\) 1.00000 0.0328443
\(928\) −26.9914 −0.886035
\(929\) 22.5677 0.740422 0.370211 0.928948i \(-0.379285\pi\)
0.370211 + 0.928948i \(0.379285\pi\)
\(930\) 1.89529 0.0621490
\(931\) 44.5200 1.45908
\(932\) −21.3076 −0.697952
\(933\) 27.3840 0.896512
\(934\) −1.87365 −0.0613077
\(935\) 10.4387 0.341381
\(936\) −1.14305 −0.0373616
\(937\) 58.2171 1.90187 0.950935 0.309389i \(-0.100125\pi\)
0.950935 + 0.309389i \(0.100125\pi\)
\(938\) 6.08788 0.198776
\(939\) −3.74737 −0.122291
\(940\) 53.0454 1.73015
\(941\) −55.5396 −1.81054 −0.905270 0.424837i \(-0.860332\pi\)
−0.905270 + 0.424837i \(0.860332\pi\)
\(942\) 2.01587 0.0656806
\(943\) −3.80485 −0.123903
\(944\) −18.8580 −0.613777
\(945\) 13.3678 0.434853
\(946\) −1.00588 −0.0327040
\(947\) −45.9699 −1.49382 −0.746910 0.664925i \(-0.768463\pi\)
−0.746910 + 0.664925i \(0.768463\pi\)
\(948\) 16.9141 0.549346
\(949\) −5.97616 −0.193994
\(950\) −10.5165 −0.341202
\(951\) 27.9021 0.904788
\(952\) 3.35657 0.108787
\(953\) −11.0045 −0.356471 −0.178235 0.983988i \(-0.557039\pi\)
−0.178235 + 0.983988i \(0.557039\pi\)
\(954\) −0.746304 −0.0241625
\(955\) −54.9357 −1.77768
\(956\) −44.5541 −1.44098
\(957\) 33.1608 1.07194
\(958\) 5.33377 0.172326
\(959\) 1.09630 0.0354014
\(960\) 20.6090 0.665152
\(961\) −27.3978 −0.883800
\(962\) −1.21001 −0.0390122
\(963\) −18.1309 −0.584261
\(964\) 14.1079 0.454384
\(965\) 4.04649 0.130261
\(966\) −9.41444 −0.302905
\(967\) 20.3983 0.655966 0.327983 0.944684i \(-0.393631\pi\)
0.327983 + 0.944684i \(0.393631\pi\)
\(968\) 6.29246 0.202247
\(969\) −4.04071 −0.129806
\(970\) −4.25691 −0.136681
\(971\) −39.5917 −1.27056 −0.635278 0.772283i \(-0.719115\pi\)
−0.635278 + 0.772283i \(0.719115\pi\)
\(972\) −1.91474 −0.0614155
\(973\) −64.5786 −2.07029
\(974\) −6.01063 −0.192593
\(975\) −6.69670 −0.214466
\(976\) −5.03191 −0.161068
\(977\) −45.1151 −1.44336 −0.721679 0.692227i \(-0.756630\pi\)
−0.721679 + 0.692227i \(0.756630\pi\)
\(978\) 2.92479 0.0935244
\(979\) −51.2791 −1.63889
\(980\) 54.2057 1.73154
\(981\) 3.57733 0.114215
\(982\) 10.2958 0.328551
\(983\) −43.7929 −1.39678 −0.698388 0.715720i \(-0.746099\pi\)
−0.698388 + 0.715720i \(0.746099\pi\)
\(984\) 0.527222 0.0168072
\(985\) 72.3287 2.30458
\(986\) 1.79054 0.0570223
\(987\) −31.6616 −1.00780
\(988\) 10.2982 0.327631
\(989\) −6.99496 −0.222427
\(990\) 4.05695 0.128938
\(991\) 23.1871 0.736562 0.368281 0.929714i \(-0.379946\pi\)
0.368281 + 0.929714i \(0.379946\pi\)
\(992\) −6.27612 −0.199267
\(993\) −16.3140 −0.517710
\(994\) −1.46130 −0.0463497
\(995\) 37.2595 1.18121
\(996\) −21.6839 −0.687082
\(997\) −1.26380 −0.0400248 −0.0200124 0.999800i \(-0.506371\pi\)
−0.0200124 + 0.999800i \(0.506371\pi\)
\(998\) −3.60865 −0.114230
\(999\) −4.14408 −0.131113
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 4017.2.a.e.1.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4017.2.a.e.1.7 16 1.1 even 1 trivial