Properties

Label 1502.2.a.h.1.18
Level $1502$
Weight $2$
Character 1502.1
Self dual yes
Analytic conductor $11.994$
Analytic rank $0$
Dimension $19$
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: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} - 11338 x^{11} - 56744 x^{10} + 50183 x^{9} + 120237 x^{8} - 102992 x^{7} + \cdots + 4222 \) 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 1.18
Root \(2.91512\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.91512 q^{3} +1.00000 q^{4} -1.49286 q^{5} -2.91512 q^{6} +3.25503 q^{7} -1.00000 q^{8} +5.49794 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.91512 q^{3} +1.00000 q^{4} -1.49286 q^{5} -2.91512 q^{6} +3.25503 q^{7} -1.00000 q^{8} +5.49794 q^{9} +1.49286 q^{10} +0.928887 q^{11} +2.91512 q^{12} +4.49624 q^{13} -3.25503 q^{14} -4.35188 q^{15} +1.00000 q^{16} -2.77875 q^{17} -5.49794 q^{18} +7.13629 q^{19} -1.49286 q^{20} +9.48881 q^{21} -0.928887 q^{22} -4.26032 q^{23} -2.91512 q^{24} -2.77136 q^{25} -4.49624 q^{26} +7.28181 q^{27} +3.25503 q^{28} +5.01897 q^{29} +4.35188 q^{30} -8.01202 q^{31} -1.00000 q^{32} +2.70782 q^{33} +2.77875 q^{34} -4.85931 q^{35} +5.49794 q^{36} -5.12886 q^{37} -7.13629 q^{38} +13.1071 q^{39} +1.49286 q^{40} -5.10790 q^{41} -9.48881 q^{42} +4.93752 q^{43} +0.928887 q^{44} -8.20768 q^{45} +4.26032 q^{46} -3.97539 q^{47} +2.91512 q^{48} +3.59521 q^{49} +2.77136 q^{50} -8.10041 q^{51} +4.49624 q^{52} +4.97697 q^{53} -7.28181 q^{54} -1.38670 q^{55} -3.25503 q^{56} +20.8032 q^{57} -5.01897 q^{58} -3.30780 q^{59} -4.35188 q^{60} +2.18885 q^{61} +8.01202 q^{62} +17.8960 q^{63} +1.00000 q^{64} -6.71227 q^{65} -2.70782 q^{66} +6.27089 q^{67} -2.77875 q^{68} -12.4193 q^{69} +4.85931 q^{70} +0.214802 q^{71} -5.49794 q^{72} +15.0530 q^{73} +5.12886 q^{74} -8.07885 q^{75} +7.13629 q^{76} +3.02355 q^{77} -13.1071 q^{78} +5.14629 q^{79} -1.49286 q^{80} +4.73355 q^{81} +5.10790 q^{82} +12.4793 q^{83} +9.48881 q^{84} +4.14830 q^{85} -4.93752 q^{86} +14.6309 q^{87} -0.928887 q^{88} -3.90016 q^{89} +8.20768 q^{90} +14.6354 q^{91} -4.26032 q^{92} -23.3560 q^{93} +3.97539 q^{94} -10.6535 q^{95} -2.91512 q^{96} -7.22768 q^{97} -3.59521 q^{98} +5.10697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 2.91512 1.68305 0.841524 0.540220i \(-0.181659\pi\)
0.841524 + 0.540220i \(0.181659\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.49286 −0.667629 −0.333814 0.942639i \(-0.608336\pi\)
−0.333814 + 0.942639i \(0.608336\pi\)
\(6\) −2.91512 −1.19009
\(7\) 3.25503 1.23028 0.615142 0.788416i \(-0.289098\pi\)
0.615142 + 0.788416i \(0.289098\pi\)
\(8\) −1.00000 −0.353553
\(9\) 5.49794 1.83265
\(10\) 1.49286 0.472085
\(11\) 0.928887 0.280070 0.140035 0.990147i \(-0.455279\pi\)
0.140035 + 0.990147i \(0.455279\pi\)
\(12\) 2.91512 0.841524
\(13\) 4.49624 1.24703 0.623516 0.781810i \(-0.285704\pi\)
0.623516 + 0.781810i \(0.285704\pi\)
\(14\) −3.25503 −0.869943
\(15\) −4.35188 −1.12365
\(16\) 1.00000 0.250000
\(17\) −2.77875 −0.673947 −0.336973 0.941514i \(-0.609403\pi\)
−0.336973 + 0.941514i \(0.609403\pi\)
\(18\) −5.49794 −1.29588
\(19\) 7.13629 1.63718 0.818589 0.574380i \(-0.194757\pi\)
0.818589 + 0.574380i \(0.194757\pi\)
\(20\) −1.49286 −0.333814
\(21\) 9.48881 2.07063
\(22\) −0.928887 −0.198039
\(23\) −4.26032 −0.888337 −0.444169 0.895943i \(-0.646501\pi\)
−0.444169 + 0.895943i \(0.646501\pi\)
\(24\) −2.91512 −0.595047
\(25\) −2.77136 −0.554272
\(26\) −4.49624 −0.881785
\(27\) 7.28181 1.40139
\(28\) 3.25503 0.615142
\(29\) 5.01897 0.931999 0.465999 0.884785i \(-0.345695\pi\)
0.465999 + 0.884785i \(0.345695\pi\)
\(30\) 4.35188 0.794541
\(31\) −8.01202 −1.43900 −0.719501 0.694492i \(-0.755629\pi\)
−0.719501 + 0.694492i \(0.755629\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.70782 0.471371
\(34\) 2.77875 0.476552
\(35\) −4.85931 −0.821374
\(36\) 5.49794 0.916324
\(37\) −5.12886 −0.843180 −0.421590 0.906787i \(-0.638528\pi\)
−0.421590 + 0.906787i \(0.638528\pi\)
\(38\) −7.13629 −1.15766
\(39\) 13.1071 2.09881
\(40\) 1.49286 0.236042
\(41\) −5.10790 −0.797720 −0.398860 0.917012i \(-0.630594\pi\)
−0.398860 + 0.917012i \(0.630594\pi\)
\(42\) −9.48881 −1.46415
\(43\) 4.93752 0.752965 0.376483 0.926424i \(-0.377133\pi\)
0.376483 + 0.926424i \(0.377133\pi\)
\(44\) 0.928887 0.140035
\(45\) −8.20768 −1.22353
\(46\) 4.26032 0.628149
\(47\) −3.97539 −0.579871 −0.289935 0.957046i \(-0.593634\pi\)
−0.289935 + 0.957046i \(0.593634\pi\)
\(48\) 2.91512 0.420762
\(49\) 3.59521 0.513601
\(50\) 2.77136 0.391929
\(51\) −8.10041 −1.13428
\(52\) 4.49624 0.623516
\(53\) 4.97697 0.683640 0.341820 0.939766i \(-0.388957\pi\)
0.341820 + 0.939766i \(0.388957\pi\)
\(54\) −7.28181 −0.990929
\(55\) −1.38670 −0.186983
\(56\) −3.25503 −0.434971
\(57\) 20.8032 2.75545
\(58\) −5.01897 −0.659023
\(59\) −3.30780 −0.430639 −0.215320 0.976544i \(-0.569079\pi\)
−0.215320 + 0.976544i \(0.569079\pi\)
\(60\) −4.35188 −0.561826
\(61\) 2.18885 0.280253 0.140127 0.990134i \(-0.455249\pi\)
0.140127 + 0.990134i \(0.455249\pi\)
\(62\) 8.01202 1.01753
\(63\) 17.8960 2.25468
\(64\) 1.00000 0.125000
\(65\) −6.71227 −0.832555
\(66\) −2.70782 −0.333310
\(67\) 6.27089 0.766111 0.383055 0.923725i \(-0.374872\pi\)
0.383055 + 0.923725i \(0.374872\pi\)
\(68\) −2.77875 −0.336973
\(69\) −12.4193 −1.49511
\(70\) 4.85931 0.580799
\(71\) 0.214802 0.0254923 0.0127461 0.999919i \(-0.495943\pi\)
0.0127461 + 0.999919i \(0.495943\pi\)
\(72\) −5.49794 −0.647939
\(73\) 15.0530 1.76182 0.880912 0.473280i \(-0.156930\pi\)
0.880912 + 0.473280i \(0.156930\pi\)
\(74\) 5.12886 0.596218
\(75\) −8.07885 −0.932865
\(76\) 7.13629 0.818589
\(77\) 3.02355 0.344566
\(78\) −13.1071 −1.48409
\(79\) 5.14629 0.579003 0.289502 0.957178i \(-0.406510\pi\)
0.289502 + 0.957178i \(0.406510\pi\)
\(80\) −1.49286 −0.166907
\(81\) 4.73355 0.525950
\(82\) 5.10790 0.564074
\(83\) 12.4793 1.36978 0.684889 0.728647i \(-0.259850\pi\)
0.684889 + 0.728647i \(0.259850\pi\)
\(84\) 9.48881 1.03531
\(85\) 4.14830 0.449946
\(86\) −4.93752 −0.532427
\(87\) 14.6309 1.56860
\(88\) −0.928887 −0.0990197
\(89\) −3.90016 −0.413416 −0.206708 0.978403i \(-0.566275\pi\)
−0.206708 + 0.978403i \(0.566275\pi\)
\(90\) 8.20768 0.865166
\(91\) 14.6354 1.53421
\(92\) −4.26032 −0.444169
\(93\) −23.3560 −2.42191
\(94\) 3.97539 0.410031
\(95\) −10.6535 −1.09303
\(96\) −2.91512 −0.297524
\(97\) −7.22768 −0.733860 −0.366930 0.930249i \(-0.619591\pi\)
−0.366930 + 0.930249i \(0.619591\pi\)
\(98\) −3.59521 −0.363171
\(99\) 5.10697 0.513270
\(100\) −2.77136 −0.277136
\(101\) −3.30748 −0.329107 −0.164553 0.986368i \(-0.552618\pi\)
−0.164553 + 0.986368i \(0.552618\pi\)
\(102\) 8.10041 0.802060
\(103\) 0.832862 0.0820643 0.0410322 0.999158i \(-0.486935\pi\)
0.0410322 + 0.999158i \(0.486935\pi\)
\(104\) −4.49624 −0.440893
\(105\) −14.1655 −1.38241
\(106\) −4.97697 −0.483406
\(107\) 13.8805 1.34188 0.670941 0.741511i \(-0.265890\pi\)
0.670941 + 0.741511i \(0.265890\pi\)
\(108\) 7.28181 0.700693
\(109\) 4.18477 0.400828 0.200414 0.979711i \(-0.435771\pi\)
0.200414 + 0.979711i \(0.435771\pi\)
\(110\) 1.38670 0.132217
\(111\) −14.9513 −1.41911
\(112\) 3.25503 0.307571
\(113\) −16.7698 −1.57757 −0.788783 0.614672i \(-0.789288\pi\)
−0.788783 + 0.614672i \(0.789288\pi\)
\(114\) −20.8032 −1.94840
\(115\) 6.36007 0.593080
\(116\) 5.01897 0.465999
\(117\) 24.7201 2.28537
\(118\) 3.30780 0.304508
\(119\) −9.04492 −0.829147
\(120\) 4.35188 0.397271
\(121\) −10.1372 −0.921561
\(122\) −2.18885 −0.198169
\(123\) −14.8902 −1.34260
\(124\) −8.01202 −0.719501
\(125\) 11.6016 1.03768
\(126\) −17.8960 −1.59430
\(127\) −20.1589 −1.78882 −0.894408 0.447253i \(-0.852402\pi\)
−0.894408 + 0.447253i \(0.852402\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 14.3935 1.26728
\(130\) 6.71227 0.588705
\(131\) 5.52709 0.482904 0.241452 0.970413i \(-0.422376\pi\)
0.241452 + 0.970413i \(0.422376\pi\)
\(132\) 2.70782 0.235685
\(133\) 23.2288 2.01420
\(134\) −6.27089 −0.541722
\(135\) −10.8708 −0.935606
\(136\) 2.77875 0.238276
\(137\) −4.83457 −0.413045 −0.206522 0.978442i \(-0.566215\pi\)
−0.206522 + 0.978442i \(0.566215\pi\)
\(138\) 12.4193 1.05720
\(139\) −9.72387 −0.824768 −0.412384 0.911010i \(-0.635304\pi\)
−0.412384 + 0.911010i \(0.635304\pi\)
\(140\) −4.85931 −0.410687
\(141\) −11.5888 −0.975950
\(142\) −0.214802 −0.0180258
\(143\) 4.17650 0.349256
\(144\) 5.49794 0.458162
\(145\) −7.49264 −0.622229
\(146\) −15.0530 −1.24580
\(147\) 10.4805 0.864415
\(148\) −5.12886 −0.421590
\(149\) −0.478897 −0.0392328 −0.0196164 0.999808i \(-0.506244\pi\)
−0.0196164 + 0.999808i \(0.506244\pi\)
\(150\) 8.07885 0.659635
\(151\) −19.0949 −1.55392 −0.776959 0.629551i \(-0.783239\pi\)
−0.776959 + 0.629551i \(0.783239\pi\)
\(152\) −7.13629 −0.578830
\(153\) −15.2774 −1.23511
\(154\) −3.02355 −0.243645
\(155\) 11.9609 0.960719
\(156\) 13.1071 1.04941
\(157\) −23.4859 −1.87438 −0.937189 0.348821i \(-0.886582\pi\)
−0.937189 + 0.348821i \(0.886582\pi\)
\(158\) −5.14629 −0.409417
\(159\) 14.5085 1.15060
\(160\) 1.49286 0.118021
\(161\) −13.8674 −1.09291
\(162\) −4.73355 −0.371903
\(163\) 8.95881 0.701708 0.350854 0.936430i \(-0.385891\pi\)
0.350854 + 0.936430i \(0.385891\pi\)
\(164\) −5.10790 −0.398860
\(165\) −4.04241 −0.314701
\(166\) −12.4793 −0.968580
\(167\) 4.96670 0.384335 0.192167 0.981362i \(-0.438448\pi\)
0.192167 + 0.981362i \(0.438448\pi\)
\(168\) −9.48881 −0.732077
\(169\) 7.21617 0.555090
\(170\) −4.14830 −0.318160
\(171\) 39.2349 3.00037
\(172\) 4.93752 0.376483
\(173\) −3.09836 −0.235564 −0.117782 0.993039i \(-0.537578\pi\)
−0.117782 + 0.993039i \(0.537578\pi\)
\(174\) −14.6309 −1.10917
\(175\) −9.02085 −0.681912
\(176\) 0.928887 0.0700175
\(177\) −9.64266 −0.724787
\(178\) 3.90016 0.292329
\(179\) 4.95495 0.370350 0.185175 0.982706i \(-0.440715\pi\)
0.185175 + 0.982706i \(0.440715\pi\)
\(180\) −8.20768 −0.611764
\(181\) 0.0380714 0.00282982 0.00141491 0.999999i \(-0.499550\pi\)
0.00141491 + 0.999999i \(0.499550\pi\)
\(182\) −14.6354 −1.08485
\(183\) 6.38076 0.471679
\(184\) 4.26032 0.314075
\(185\) 7.65669 0.562931
\(186\) 23.3560 1.71255
\(187\) −2.58115 −0.188752
\(188\) −3.97539 −0.289935
\(189\) 23.7025 1.72410
\(190\) 10.6535 0.772887
\(191\) 26.7411 1.93492 0.967459 0.253028i \(-0.0814265\pi\)
0.967459 + 0.253028i \(0.0814265\pi\)
\(192\) 2.91512 0.210381
\(193\) 3.83897 0.276335 0.138168 0.990409i \(-0.455879\pi\)
0.138168 + 0.990409i \(0.455879\pi\)
\(194\) 7.22768 0.518917
\(195\) −19.5671 −1.40123
\(196\) 3.59521 0.256801
\(197\) −27.7419 −1.97653 −0.988263 0.152762i \(-0.951183\pi\)
−0.988263 + 0.152762i \(0.951183\pi\)
\(198\) −5.10697 −0.362936
\(199\) 15.2322 1.07978 0.539891 0.841735i \(-0.318466\pi\)
0.539891 + 0.841735i \(0.318466\pi\)
\(200\) 2.77136 0.195965
\(201\) 18.2804 1.28940
\(202\) 3.30748 0.232713
\(203\) 16.3369 1.14662
\(204\) −8.10041 −0.567142
\(205\) 7.62540 0.532581
\(206\) −0.832862 −0.0580282
\(207\) −23.4230 −1.62801
\(208\) 4.49624 0.311758
\(209\) 6.62881 0.458524
\(210\) 14.1655 0.977512
\(211\) 8.31911 0.572711 0.286356 0.958123i \(-0.407556\pi\)
0.286356 + 0.958123i \(0.407556\pi\)
\(212\) 4.97697 0.341820
\(213\) 0.626173 0.0429047
\(214\) −13.8805 −0.948854
\(215\) −7.37105 −0.502701
\(216\) −7.28181 −0.495465
\(217\) −26.0793 −1.77038
\(218\) −4.18477 −0.283429
\(219\) 43.8814 2.96523
\(220\) −1.38670 −0.0934914
\(221\) −12.4939 −0.840434
\(222\) 14.9513 1.00346
\(223\) 4.08943 0.273848 0.136924 0.990582i \(-0.456278\pi\)
0.136924 + 0.990582i \(0.456278\pi\)
\(224\) −3.25503 −0.217486
\(225\) −15.2368 −1.01578
\(226\) 16.7698 1.11551
\(227\) 11.9712 0.794558 0.397279 0.917698i \(-0.369955\pi\)
0.397279 + 0.917698i \(0.369955\pi\)
\(228\) 20.8032 1.37772
\(229\) 26.2492 1.73460 0.867299 0.497788i \(-0.165854\pi\)
0.867299 + 0.497788i \(0.165854\pi\)
\(230\) −6.36007 −0.419371
\(231\) 8.81403 0.579920
\(232\) −5.01897 −0.329511
\(233\) 0.168495 0.0110384 0.00551922 0.999985i \(-0.498243\pi\)
0.00551922 + 0.999985i \(0.498243\pi\)
\(234\) −24.7201 −1.61600
\(235\) 5.93472 0.387139
\(236\) −3.30780 −0.215320
\(237\) 15.0021 0.974490
\(238\) 9.04492 0.586295
\(239\) −5.00385 −0.323672 −0.161836 0.986818i \(-0.551742\pi\)
−0.161836 + 0.986818i \(0.551742\pi\)
\(240\) −4.35188 −0.280913
\(241\) −26.3551 −1.69768 −0.848841 0.528648i \(-0.822699\pi\)
−0.848841 + 0.528648i \(0.822699\pi\)
\(242\) 10.1372 0.651642
\(243\) −8.04655 −0.516186
\(244\) 2.18885 0.140127
\(245\) −5.36715 −0.342895
\(246\) 14.8902 0.949362
\(247\) 32.0865 2.04161
\(248\) 8.01202 0.508764
\(249\) 36.3786 2.30540
\(250\) −11.6016 −0.733748
\(251\) 14.6780 0.926465 0.463233 0.886237i \(-0.346689\pi\)
0.463233 + 0.886237i \(0.346689\pi\)
\(252\) 17.8960 1.12734
\(253\) −3.95735 −0.248797
\(254\) 20.1589 1.26488
\(255\) 12.0928 0.757281
\(256\) 1.00000 0.0625000
\(257\) 9.00926 0.561983 0.280991 0.959710i \(-0.409337\pi\)
0.280991 + 0.959710i \(0.409337\pi\)
\(258\) −14.3935 −0.896099
\(259\) −16.6946 −1.03735
\(260\) −6.71227 −0.416277
\(261\) 27.5940 1.70803
\(262\) −5.52709 −0.341464
\(263\) −29.0832 −1.79335 −0.896674 0.442691i \(-0.854024\pi\)
−0.896674 + 0.442691i \(0.854024\pi\)
\(264\) −2.70782 −0.166655
\(265\) −7.42994 −0.456418
\(266\) −23.2288 −1.42425
\(267\) −11.3694 −0.695798
\(268\) 6.27089 0.383055
\(269\) −11.7279 −0.715063 −0.357532 0.933901i \(-0.616382\pi\)
−0.357532 + 0.933901i \(0.616382\pi\)
\(270\) 10.8708 0.661573
\(271\) −24.2213 −1.47134 −0.735671 0.677339i \(-0.763133\pi\)
−0.735671 + 0.677339i \(0.763133\pi\)
\(272\) −2.77875 −0.168487
\(273\) 42.6640 2.58214
\(274\) 4.83457 0.292067
\(275\) −2.57428 −0.155235
\(276\) −12.4193 −0.747557
\(277\) −6.71572 −0.403509 −0.201754 0.979436i \(-0.564664\pi\)
−0.201754 + 0.979436i \(0.564664\pi\)
\(278\) 9.72387 0.583199
\(279\) −44.0496 −2.63718
\(280\) 4.85931 0.290400
\(281\) 15.6567 0.934003 0.467001 0.884257i \(-0.345334\pi\)
0.467001 + 0.884257i \(0.345334\pi\)
\(282\) 11.5888 0.690101
\(283\) −23.0211 −1.36846 −0.684232 0.729264i \(-0.739862\pi\)
−0.684232 + 0.729264i \(0.739862\pi\)
\(284\) 0.214802 0.0127461
\(285\) −31.0563 −1.83962
\(286\) −4.17650 −0.246961
\(287\) −16.6264 −0.981423
\(288\) −5.49794 −0.323969
\(289\) −9.27853 −0.545796
\(290\) 7.49264 0.439983
\(291\) −21.0696 −1.23512
\(292\) 15.0530 0.880912
\(293\) −11.3369 −0.662306 −0.331153 0.943577i \(-0.607438\pi\)
−0.331153 + 0.943577i \(0.607438\pi\)
\(294\) −10.4805 −0.611234
\(295\) 4.93810 0.287507
\(296\) 5.12886 0.298109
\(297\) 6.76398 0.392486
\(298\) 0.478897 0.0277418
\(299\) −19.1554 −1.10779
\(300\) −8.07885 −0.466433
\(301\) 16.0718 0.926362
\(302\) 19.0949 1.09879
\(303\) −9.64171 −0.553902
\(304\) 7.13629 0.409295
\(305\) −3.26765 −0.187105
\(306\) 15.2774 0.873353
\(307\) 17.3822 0.992052 0.496026 0.868308i \(-0.334792\pi\)
0.496026 + 0.868308i \(0.334792\pi\)
\(308\) 3.02355 0.172283
\(309\) 2.42789 0.138118
\(310\) −11.9609 −0.679331
\(311\) −22.4588 −1.27352 −0.636761 0.771061i \(-0.719726\pi\)
−0.636761 + 0.771061i \(0.719726\pi\)
\(312\) −13.1071 −0.742043
\(313\) −23.6953 −1.33934 −0.669669 0.742660i \(-0.733564\pi\)
−0.669669 + 0.742660i \(0.733564\pi\)
\(314\) 23.4859 1.32539
\(315\) −26.7162 −1.50529
\(316\) 5.14629 0.289502
\(317\) −7.48626 −0.420470 −0.210235 0.977651i \(-0.567423\pi\)
−0.210235 + 0.977651i \(0.567423\pi\)
\(318\) −14.5085 −0.813595
\(319\) 4.66205 0.261025
\(320\) −1.49286 −0.0834536
\(321\) 40.4635 2.25845
\(322\) 13.8674 0.772803
\(323\) −19.8300 −1.10337
\(324\) 4.73355 0.262975
\(325\) −12.4607 −0.691195
\(326\) −8.95881 −0.496183
\(327\) 12.1991 0.674613
\(328\) 5.10790 0.282037
\(329\) −12.9400 −0.713406
\(330\) 4.04241 0.222527
\(331\) 3.52357 0.193673 0.0968364 0.995300i \(-0.469128\pi\)
0.0968364 + 0.995300i \(0.469128\pi\)
\(332\) 12.4793 0.684889
\(333\) −28.1982 −1.54525
\(334\) −4.96670 −0.271766
\(335\) −9.36158 −0.511478
\(336\) 9.48881 0.517657
\(337\) −7.72046 −0.420560 −0.210280 0.977641i \(-0.567438\pi\)
−0.210280 + 0.977641i \(0.567438\pi\)
\(338\) −7.21617 −0.392508
\(339\) −48.8859 −2.65512
\(340\) 4.14830 0.224973
\(341\) −7.44226 −0.403021
\(342\) −39.2349 −2.12158
\(343\) −11.0827 −0.598409
\(344\) −4.93752 −0.266213
\(345\) 18.5404 0.998181
\(346\) 3.09836 0.166569
\(347\) 6.82229 0.366240 0.183120 0.983091i \(-0.441380\pi\)
0.183120 + 0.983091i \(0.441380\pi\)
\(348\) 14.6309 0.784299
\(349\) −23.4882 −1.25729 −0.628647 0.777691i \(-0.716391\pi\)
−0.628647 + 0.777691i \(0.716391\pi\)
\(350\) 9.02085 0.482185
\(351\) 32.7408 1.74757
\(352\) −0.928887 −0.0495098
\(353\) 20.6971 1.10160 0.550798 0.834639i \(-0.314323\pi\)
0.550798 + 0.834639i \(0.314323\pi\)
\(354\) 9.64266 0.512501
\(355\) −0.320670 −0.0170194
\(356\) −3.90016 −0.206708
\(357\) −26.3671 −1.39549
\(358\) −4.95495 −0.261877
\(359\) 20.9481 1.10560 0.552799 0.833315i \(-0.313560\pi\)
0.552799 + 0.833315i \(0.313560\pi\)
\(360\) 8.20768 0.432583
\(361\) 31.9267 1.68035
\(362\) −0.0380714 −0.00200099
\(363\) −29.5511 −1.55103
\(364\) 14.6354 0.767103
\(365\) −22.4721 −1.17624
\(366\) −6.38076 −0.333528
\(367\) 12.8420 0.670348 0.335174 0.942156i \(-0.391205\pi\)
0.335174 + 0.942156i \(0.391205\pi\)
\(368\) −4.26032 −0.222084
\(369\) −28.0830 −1.46194
\(370\) −7.65669 −0.398053
\(371\) 16.2002 0.841071
\(372\) −23.3560 −1.21095
\(373\) 13.3270 0.690048 0.345024 0.938594i \(-0.387871\pi\)
0.345024 + 0.938594i \(0.387871\pi\)
\(374\) 2.58115 0.133468
\(375\) 33.8200 1.74646
\(376\) 3.97539 0.205015
\(377\) 22.5665 1.16223
\(378\) −23.7025 −1.21913
\(379\) 6.33684 0.325502 0.162751 0.986667i \(-0.447963\pi\)
0.162751 + 0.986667i \(0.447963\pi\)
\(380\) −10.6535 −0.546514
\(381\) −58.7658 −3.01066
\(382\) −26.7411 −1.36819
\(383\) −24.1211 −1.23253 −0.616266 0.787538i \(-0.711356\pi\)
−0.616266 + 0.787538i \(0.711356\pi\)
\(384\) −2.91512 −0.148762
\(385\) −4.51375 −0.230042
\(386\) −3.83897 −0.195398
\(387\) 27.1462 1.37992
\(388\) −7.22768 −0.366930
\(389\) −22.4082 −1.13614 −0.568069 0.822981i \(-0.692309\pi\)
−0.568069 + 0.822981i \(0.692309\pi\)
\(390\) 19.5671 0.990819
\(391\) 11.8384 0.598692
\(392\) −3.59521 −0.181585
\(393\) 16.1121 0.812750
\(394\) 27.7419 1.39761
\(395\) −7.68271 −0.386559
\(396\) 5.10697 0.256635
\(397\) 7.85601 0.394282 0.197141 0.980375i \(-0.436834\pi\)
0.197141 + 0.980375i \(0.436834\pi\)
\(398\) −15.2322 −0.763521
\(399\) 67.7149 3.38999
\(400\) −2.77136 −0.138568
\(401\) 16.1596 0.806971 0.403486 0.914986i \(-0.367799\pi\)
0.403486 + 0.914986i \(0.367799\pi\)
\(402\) −18.2804 −0.911744
\(403\) −36.0240 −1.79448
\(404\) −3.30748 −0.164553
\(405\) −7.06655 −0.351140
\(406\) −16.3369 −0.810786
\(407\) −4.76413 −0.236149
\(408\) 8.10041 0.401030
\(409\) 21.9292 1.08433 0.542165 0.840272i \(-0.317605\pi\)
0.542165 + 0.840272i \(0.317605\pi\)
\(410\) −7.62540 −0.376592
\(411\) −14.0934 −0.695174
\(412\) 0.832862 0.0410322
\(413\) −10.7670 −0.529809
\(414\) 23.4230 1.15118
\(415\) −18.6299 −0.914504
\(416\) −4.49624 −0.220446
\(417\) −28.3463 −1.38812
\(418\) −6.62881 −0.324226
\(419\) −15.8461 −0.774135 −0.387067 0.922051i \(-0.626512\pi\)
−0.387067 + 0.922051i \(0.626512\pi\)
\(420\) −14.1655 −0.691205
\(421\) −14.2514 −0.694570 −0.347285 0.937760i \(-0.612896\pi\)
−0.347285 + 0.937760i \(0.612896\pi\)
\(422\) −8.31911 −0.404968
\(423\) −21.8565 −1.06270
\(424\) −4.97697 −0.241703
\(425\) 7.70092 0.373550
\(426\) −0.626173 −0.0303382
\(427\) 7.12476 0.344791
\(428\) 13.8805 0.670941
\(429\) 12.1750 0.587815
\(430\) 7.37105 0.355464
\(431\) −26.6546 −1.28391 −0.641953 0.766744i \(-0.721876\pi\)
−0.641953 + 0.766744i \(0.721876\pi\)
\(432\) 7.28181 0.350346
\(433\) −11.0415 −0.530622 −0.265311 0.964163i \(-0.585475\pi\)
−0.265311 + 0.964163i \(0.585475\pi\)
\(434\) 26.0793 1.25185
\(435\) −21.8420 −1.04724
\(436\) 4.18477 0.200414
\(437\) −30.4029 −1.45437
\(438\) −43.8814 −2.09674
\(439\) −8.58467 −0.409724 −0.204862 0.978791i \(-0.565675\pi\)
−0.204862 + 0.978791i \(0.565675\pi\)
\(440\) 1.38670 0.0661084
\(441\) 19.7662 0.941250
\(442\) 12.4939 0.594276
\(443\) −27.8370 −1.32258 −0.661288 0.750132i \(-0.729990\pi\)
−0.661288 + 0.750132i \(0.729990\pi\)
\(444\) −14.9513 −0.709556
\(445\) 5.82240 0.276008
\(446\) −4.08943 −0.193640
\(447\) −1.39604 −0.0660306
\(448\) 3.25503 0.153786
\(449\) 18.9956 0.896457 0.448228 0.893919i \(-0.352055\pi\)
0.448228 + 0.893919i \(0.352055\pi\)
\(450\) 15.2368 0.718268
\(451\) −4.74466 −0.223417
\(452\) −16.7698 −0.788783
\(453\) −55.6639 −2.61532
\(454\) −11.9712 −0.561837
\(455\) −21.8486 −1.02428
\(456\) −20.8032 −0.974198
\(457\) 20.7789 0.971994 0.485997 0.873960i \(-0.338457\pi\)
0.485997 + 0.873960i \(0.338457\pi\)
\(458\) −26.2492 −1.22655
\(459\) −20.2344 −0.944460
\(460\) 6.36007 0.296540
\(461\) −0.601112 −0.0279966 −0.0139983 0.999902i \(-0.504456\pi\)
−0.0139983 + 0.999902i \(0.504456\pi\)
\(462\) −8.81403 −0.410066
\(463\) 6.18736 0.287551 0.143775 0.989610i \(-0.454076\pi\)
0.143775 + 0.989610i \(0.454076\pi\)
\(464\) 5.01897 0.233000
\(465\) 34.8674 1.61693
\(466\) −0.168495 −0.00780536
\(467\) −18.7722 −0.868673 −0.434337 0.900751i \(-0.643017\pi\)
−0.434337 + 0.900751i \(0.643017\pi\)
\(468\) 24.7201 1.14269
\(469\) 20.4119 0.942535
\(470\) −5.93472 −0.273748
\(471\) −68.4643 −3.15467
\(472\) 3.30780 0.152254
\(473\) 4.58640 0.210883
\(474\) −15.0021 −0.689068
\(475\) −19.7772 −0.907441
\(476\) −9.04492 −0.414573
\(477\) 27.3631 1.25287
\(478\) 5.00385 0.228871
\(479\) −22.8394 −1.04356 −0.521780 0.853080i \(-0.674732\pi\)
−0.521780 + 0.853080i \(0.674732\pi\)
\(480\) 4.35188 0.198635
\(481\) −23.0606 −1.05147
\(482\) 26.3551 1.20044
\(483\) −40.4253 −1.83942
\(484\) −10.1372 −0.460780
\(485\) 10.7899 0.489946
\(486\) 8.04655 0.364999
\(487\) 35.9728 1.63008 0.815042 0.579402i \(-0.196714\pi\)
0.815042 + 0.579402i \(0.196714\pi\)
\(488\) −2.18885 −0.0990845
\(489\) 26.1160 1.18101
\(490\) 5.36715 0.242463
\(491\) −21.8992 −0.988299 −0.494149 0.869377i \(-0.664520\pi\)
−0.494149 + 0.869377i \(0.664520\pi\)
\(492\) −14.8902 −0.671301
\(493\) −13.9465 −0.628118
\(494\) −32.0865 −1.44364
\(495\) −7.62401 −0.342674
\(496\) −8.01202 −0.359750
\(497\) 0.699185 0.0313627
\(498\) −36.3786 −1.63017
\(499\) −2.85591 −0.127848 −0.0639240 0.997955i \(-0.520362\pi\)
−0.0639240 + 0.997955i \(0.520362\pi\)
\(500\) 11.6016 0.518838
\(501\) 14.4785 0.646854
\(502\) −14.6780 −0.655110
\(503\) 25.3371 1.12972 0.564862 0.825185i \(-0.308929\pi\)
0.564862 + 0.825185i \(0.308929\pi\)
\(504\) −17.8960 −0.797149
\(505\) 4.93762 0.219721
\(506\) 3.95735 0.175926
\(507\) 21.0360 0.934242
\(508\) −20.1589 −0.894408
\(509\) −8.66996 −0.384289 −0.192145 0.981367i \(-0.561544\pi\)
−0.192145 + 0.981367i \(0.561544\pi\)
\(510\) −12.0928 −0.535479
\(511\) 48.9980 2.16755
\(512\) −1.00000 −0.0441942
\(513\) 51.9652 2.29432
\(514\) −9.00926 −0.397382
\(515\) −1.24335 −0.0547885
\(516\) 14.3935 0.633638
\(517\) −3.69269 −0.162404
\(518\) 16.6946 0.733518
\(519\) −9.03211 −0.396465
\(520\) 6.71227 0.294353
\(521\) 25.4825 1.11641 0.558204 0.829704i \(-0.311491\pi\)
0.558204 + 0.829704i \(0.311491\pi\)
\(522\) −27.5940 −1.20776
\(523\) 25.8411 1.12995 0.564975 0.825108i \(-0.308886\pi\)
0.564975 + 0.825108i \(0.308886\pi\)
\(524\) 5.52709 0.241452
\(525\) −26.2969 −1.14769
\(526\) 29.0832 1.26809
\(527\) 22.2634 0.969810
\(528\) 2.70782 0.117843
\(529\) −4.84971 −0.210857
\(530\) 7.42994 0.322736
\(531\) −18.1861 −0.789210
\(532\) 23.2288 1.00710
\(533\) −22.9664 −0.994783
\(534\) 11.3694 0.492004
\(535\) −20.7217 −0.895879
\(536\) −6.27089 −0.270861
\(537\) 14.4443 0.623317
\(538\) 11.7279 0.505626
\(539\) 3.33954 0.143844
\(540\) −10.8708 −0.467803
\(541\) 15.8467 0.681304 0.340652 0.940190i \(-0.389352\pi\)
0.340652 + 0.940190i \(0.389352\pi\)
\(542\) 24.2213 1.04040
\(543\) 0.110983 0.00476273
\(544\) 2.77875 0.119138
\(545\) −6.24729 −0.267605
\(546\) −42.6640 −1.82585
\(547\) 35.8623 1.53336 0.766681 0.642028i \(-0.221907\pi\)
0.766681 + 0.642028i \(0.221907\pi\)
\(548\) −4.83457 −0.206522
\(549\) 12.0342 0.513605
\(550\) 2.57428 0.109768
\(551\) 35.8168 1.52585
\(552\) 12.4193 0.528602
\(553\) 16.7513 0.712339
\(554\) 6.71572 0.285324
\(555\) 22.3202 0.947440
\(556\) −9.72387 −0.412384
\(557\) 14.4638 0.612852 0.306426 0.951895i \(-0.400867\pi\)
0.306426 + 0.951895i \(0.400867\pi\)
\(558\) 44.0496 1.86477
\(559\) 22.2003 0.938972
\(560\) −4.85931 −0.205343
\(561\) −7.52436 −0.317679
\(562\) −15.6567 −0.660440
\(563\) −12.4977 −0.526717 −0.263358 0.964698i \(-0.584830\pi\)
−0.263358 + 0.964698i \(0.584830\pi\)
\(564\) −11.5888 −0.487975
\(565\) 25.0350 1.05323
\(566\) 23.0211 0.967650
\(567\) 15.4079 0.647069
\(568\) −0.214802 −0.00901288
\(569\) −8.90081 −0.373141 −0.186571 0.982442i \(-0.559737\pi\)
−0.186571 + 0.982442i \(0.559737\pi\)
\(570\) 31.0563 1.30081
\(571\) −4.72569 −0.197764 −0.0988819 0.995099i \(-0.531527\pi\)
−0.0988819 + 0.995099i \(0.531527\pi\)
\(572\) 4.17650 0.174628
\(573\) 77.9536 3.25656
\(574\) 16.6264 0.693971
\(575\) 11.8069 0.492380
\(576\) 5.49794 0.229081
\(577\) 4.78401 0.199161 0.0995805 0.995030i \(-0.468250\pi\)
0.0995805 + 0.995030i \(0.468250\pi\)
\(578\) 9.27853 0.385936
\(579\) 11.1911 0.465085
\(580\) −7.49264 −0.311115
\(581\) 40.6204 1.68522
\(582\) 21.0696 0.873362
\(583\) 4.62304 0.191467
\(584\) −15.0530 −0.622899
\(585\) −36.9037 −1.52578
\(586\) 11.3369 0.468321
\(587\) 38.9819 1.60896 0.804479 0.593982i \(-0.202445\pi\)
0.804479 + 0.593982i \(0.202445\pi\)
\(588\) 10.4805 0.432207
\(589\) −57.1761 −2.35590
\(590\) −4.93810 −0.203298
\(591\) −80.8710 −3.32659
\(592\) −5.12886 −0.210795
\(593\) −16.9640 −0.696629 −0.348314 0.937378i \(-0.613246\pi\)
−0.348314 + 0.937378i \(0.613246\pi\)
\(594\) −6.76398 −0.277529
\(595\) 13.5028 0.553562
\(596\) −0.478897 −0.0196164
\(597\) 44.4037 1.81732
\(598\) 19.1554 0.783323
\(599\) −44.8188 −1.83125 −0.915624 0.402036i \(-0.868303\pi\)
−0.915624 + 0.402036i \(0.868303\pi\)
\(600\) 8.07885 0.329818
\(601\) 24.3965 0.995156 0.497578 0.867419i \(-0.334223\pi\)
0.497578 + 0.867419i \(0.334223\pi\)
\(602\) −16.0718 −0.655037
\(603\) 34.4770 1.40401
\(604\) −19.0949 −0.776959
\(605\) 15.1334 0.615261
\(606\) 9.64171 0.391668
\(607\) −45.3298 −1.83988 −0.919940 0.392059i \(-0.871763\pi\)
−0.919940 + 0.392059i \(0.871763\pi\)
\(608\) −7.13629 −0.289415
\(609\) 47.6240 1.92982
\(610\) 3.26765 0.132303
\(611\) −17.8743 −0.723118
\(612\) −15.2774 −0.617554
\(613\) −4.79941 −0.193847 −0.0969233 0.995292i \(-0.530900\pi\)
−0.0969233 + 0.995292i \(0.530900\pi\)
\(614\) −17.3822 −0.701487
\(615\) 22.2290 0.896359
\(616\) −3.02355 −0.121822
\(617\) 2.06921 0.0833033 0.0416516 0.999132i \(-0.486738\pi\)
0.0416516 + 0.999132i \(0.486738\pi\)
\(618\) −2.42789 −0.0976642
\(619\) −41.1248 −1.65295 −0.826473 0.562976i \(-0.809656\pi\)
−0.826473 + 0.562976i \(0.809656\pi\)
\(620\) 11.9609 0.480359
\(621\) −31.0228 −1.24490
\(622\) 22.4588 0.900516
\(623\) −12.6951 −0.508619
\(624\) 13.1071 0.524704
\(625\) −3.46279 −0.138511
\(626\) 23.6953 0.947055
\(627\) 19.3238 0.771718
\(628\) −23.4859 −0.937189
\(629\) 14.2518 0.568258
\(630\) 26.7162 1.06440
\(631\) −29.4760 −1.17342 −0.586710 0.809797i \(-0.699577\pi\)
−0.586710 + 0.809797i \(0.699577\pi\)
\(632\) −5.14629 −0.204709
\(633\) 24.2512 0.963900
\(634\) 7.48626 0.297317
\(635\) 30.0945 1.19426
\(636\) 14.5085 0.575299
\(637\) 16.1649 0.640477
\(638\) −4.66205 −0.184572
\(639\) 1.18097 0.0467183
\(640\) 1.49286 0.0590106
\(641\) −27.1851 −1.07375 −0.536873 0.843663i \(-0.680395\pi\)
−0.536873 + 0.843663i \(0.680395\pi\)
\(642\) −40.4635 −1.59697
\(643\) −31.4818 −1.24152 −0.620761 0.784000i \(-0.713176\pi\)
−0.620761 + 0.784000i \(0.713176\pi\)
\(644\) −13.8674 −0.546454
\(645\) −21.4875 −0.846070
\(646\) 19.8300 0.780201
\(647\) −7.84978 −0.308607 −0.154303 0.988024i \(-0.549313\pi\)
−0.154303 + 0.988024i \(0.549313\pi\)
\(648\) −4.73355 −0.185952
\(649\) −3.07258 −0.120609
\(650\) 12.4607 0.488748
\(651\) −76.0245 −2.97964
\(652\) 8.95881 0.350854
\(653\) 23.3571 0.914032 0.457016 0.889458i \(-0.348918\pi\)
0.457016 + 0.889458i \(0.348918\pi\)
\(654\) −12.1991 −0.477024
\(655\) −8.25118 −0.322400
\(656\) −5.10790 −0.199430
\(657\) 82.7607 3.22880
\(658\) 12.9400 0.504454
\(659\) 10.7555 0.418976 0.209488 0.977811i \(-0.432820\pi\)
0.209488 + 0.977811i \(0.432820\pi\)
\(660\) −4.04241 −0.157350
\(661\) 50.1851 1.95197 0.975987 0.217830i \(-0.0698977\pi\)
0.975987 + 0.217830i \(0.0698977\pi\)
\(662\) −3.52357 −0.136947
\(663\) −36.4214 −1.41449
\(664\) −12.4793 −0.484290
\(665\) −34.6775 −1.34474
\(666\) 28.1982 1.09266
\(667\) −21.3824 −0.827929
\(668\) 4.96670 0.192167
\(669\) 11.9212 0.460900
\(670\) 9.36158 0.361669
\(671\) 2.03319 0.0784905
\(672\) −9.48881 −0.366039
\(673\) 12.9753 0.500161 0.250081 0.968225i \(-0.419543\pi\)
0.250081 + 0.968225i \(0.419543\pi\)
\(674\) 7.72046 0.297381
\(675\) −20.1805 −0.776748
\(676\) 7.21617 0.277545
\(677\) 49.2178 1.89159 0.945796 0.324760i \(-0.105284\pi\)
0.945796 + 0.324760i \(0.105284\pi\)
\(678\) 48.8859 1.87745
\(679\) −23.5263 −0.902857
\(680\) −4.14830 −0.159080
\(681\) 34.8976 1.33728
\(682\) 7.44226 0.284979
\(683\) 23.7176 0.907527 0.453764 0.891122i \(-0.350081\pi\)
0.453764 + 0.891122i \(0.350081\pi\)
\(684\) 39.2349 1.50019
\(685\) 7.21735 0.275761
\(686\) 11.0827 0.423139
\(687\) 76.5197 2.91941
\(688\) 4.93752 0.188241
\(689\) 22.3777 0.852521
\(690\) −18.5404 −0.705821
\(691\) −9.28834 −0.353345 −0.176672 0.984270i \(-0.556533\pi\)
−0.176672 + 0.984270i \(0.556533\pi\)
\(692\) −3.09836 −0.117782
\(693\) 16.6233 0.631468
\(694\) −6.82229 −0.258971
\(695\) 14.5164 0.550639
\(696\) −14.6309 −0.554583
\(697\) 14.1936 0.537621
\(698\) 23.4882 0.889041
\(699\) 0.491182 0.0185782
\(700\) −9.02085 −0.340956
\(701\) −27.3674 −1.03365 −0.516826 0.856090i \(-0.672887\pi\)
−0.516826 + 0.856090i \(0.672887\pi\)
\(702\) −32.7408 −1.23572
\(703\) −36.6011 −1.38044
\(704\) 0.928887 0.0350087
\(705\) 17.3004 0.651572
\(706\) −20.6971 −0.778946
\(707\) −10.7659 −0.404895
\(708\) −9.64266 −0.362393
\(709\) −24.0369 −0.902723 −0.451362 0.892341i \(-0.649062\pi\)
−0.451362 + 0.892341i \(0.649062\pi\)
\(710\) 0.320670 0.0120345
\(711\) 28.2940 1.06111
\(712\) 3.90016 0.146165
\(713\) 34.1337 1.27832
\(714\) 26.3671 0.986763
\(715\) −6.23494 −0.233174
\(716\) 4.95495 0.185175
\(717\) −14.5868 −0.544755
\(718\) −20.9481 −0.781776
\(719\) −21.6220 −0.806364 −0.403182 0.915120i \(-0.632096\pi\)
−0.403182 + 0.915120i \(0.632096\pi\)
\(720\) −8.20768 −0.305882
\(721\) 2.71099 0.100962
\(722\) −31.9267 −1.18819
\(723\) −76.8284 −2.85728
\(724\) 0.0380714 0.00141491
\(725\) −13.9094 −0.516581
\(726\) 29.5511 1.09674
\(727\) −3.66114 −0.135784 −0.0678920 0.997693i \(-0.521627\pi\)
−0.0678920 + 0.997693i \(0.521627\pi\)
\(728\) −14.6354 −0.542423
\(729\) −37.6573 −1.39472
\(730\) 22.4721 0.831731
\(731\) −13.7202 −0.507459
\(732\) 6.38076 0.235840
\(733\) −36.2282 −1.33812 −0.669060 0.743209i \(-0.733303\pi\)
−0.669060 + 0.743209i \(0.733303\pi\)
\(734\) −12.8420 −0.474007
\(735\) −15.6459 −0.577108
\(736\) 4.26032 0.157037
\(737\) 5.82495 0.214565
\(738\) 28.0830 1.03375
\(739\) 46.2968 1.70306 0.851528 0.524310i \(-0.175677\pi\)
0.851528 + 0.524310i \(0.175677\pi\)
\(740\) 7.65669 0.281466
\(741\) 93.5361 3.43613
\(742\) −16.2002 −0.594727
\(743\) 46.5840 1.70900 0.854500 0.519451i \(-0.173864\pi\)
0.854500 + 0.519451i \(0.173864\pi\)
\(744\) 23.3560 0.856273
\(745\) 0.714928 0.0261929
\(746\) −13.3270 −0.487937
\(747\) 68.6104 2.51032
\(748\) −2.58115 −0.0943761
\(749\) 45.1815 1.65090
\(750\) −33.8200 −1.23493
\(751\) 1.00000 0.0364905
\(752\) −3.97539 −0.144968
\(753\) 42.7881 1.55928
\(754\) −22.5665 −0.821823
\(755\) 28.5060 1.03744
\(756\) 23.7025 0.862052
\(757\) −24.8281 −0.902392 −0.451196 0.892425i \(-0.649002\pi\)
−0.451196 + 0.892425i \(0.649002\pi\)
\(758\) −6.33684 −0.230164
\(759\) −11.5362 −0.418736
\(760\) 10.6535 0.386444
\(761\) 16.0722 0.582618 0.291309 0.956629i \(-0.405909\pi\)
0.291309 + 0.956629i \(0.405909\pi\)
\(762\) 58.7658 2.12886
\(763\) 13.6215 0.493133
\(764\) 26.7411 0.967459
\(765\) 22.8071 0.824593
\(766\) 24.1211 0.871532
\(767\) −14.8727 −0.537021
\(768\) 2.91512 0.105190
\(769\) 14.3686 0.518147 0.259073 0.965858i \(-0.416583\pi\)
0.259073 + 0.965858i \(0.416583\pi\)
\(770\) 4.51375 0.162664
\(771\) 26.2631 0.945843
\(772\) 3.83897 0.138168
\(773\) 7.91260 0.284597 0.142298 0.989824i \(-0.454551\pi\)
0.142298 + 0.989824i \(0.454551\pi\)
\(774\) −27.1462 −0.975751
\(775\) 22.2042 0.797597
\(776\) 7.22768 0.259459
\(777\) −48.6668 −1.74591
\(778\) 22.4082 0.803371
\(779\) −36.4515 −1.30601
\(780\) −19.5671 −0.700615
\(781\) 0.199526 0.00713962
\(782\) −11.8384 −0.423339
\(783\) 36.5472 1.30609
\(784\) 3.59521 0.128400
\(785\) 35.0612 1.25139
\(786\) −16.1121 −0.574701
\(787\) 33.6860 1.20078 0.600388 0.799709i \(-0.295013\pi\)
0.600388 + 0.799709i \(0.295013\pi\)
\(788\) −27.7419 −0.988263
\(789\) −84.7812 −3.01829
\(790\) 7.68271 0.273339
\(791\) −54.5860 −1.94085
\(792\) −5.10697 −0.181468
\(793\) 9.84158 0.349485
\(794\) −7.85601 −0.278799
\(795\) −21.6592 −0.768172
\(796\) 15.2322 0.539891
\(797\) 43.7146 1.54845 0.774225 0.632910i \(-0.218140\pi\)
0.774225 + 0.632910i \(0.218140\pi\)
\(798\) −67.7149 −2.39708
\(799\) 11.0466 0.390802
\(800\) 2.77136 0.0979823
\(801\) −21.4428 −0.757646
\(802\) −16.1596 −0.570615
\(803\) 13.9826 0.493434
\(804\) 18.2804 0.644700
\(805\) 20.7022 0.729657
\(806\) 36.0240 1.26889
\(807\) −34.1883 −1.20348
\(808\) 3.30748 0.116357
\(809\) 36.9956 1.30070 0.650348 0.759637i \(-0.274623\pi\)
0.650348 + 0.759637i \(0.274623\pi\)
\(810\) 7.06655 0.248293
\(811\) 40.4822 1.42152 0.710761 0.703434i \(-0.248351\pi\)
0.710761 + 0.703434i \(0.248351\pi\)
\(812\) 16.3369 0.573312
\(813\) −70.6082 −2.47634
\(814\) 4.76413 0.166983
\(815\) −13.3743 −0.468481
\(816\) −8.10041 −0.283571
\(817\) 35.2356 1.23274
\(818\) −21.9292 −0.766738
\(819\) 80.4645 2.81166
\(820\) 7.62540 0.266291
\(821\) 52.1358 1.81955 0.909776 0.415100i \(-0.136253\pi\)
0.909776 + 0.415100i \(0.136253\pi\)
\(822\) 14.0934 0.491562
\(823\) −10.9477 −0.381611 −0.190806 0.981628i \(-0.561110\pi\)
−0.190806 + 0.981628i \(0.561110\pi\)
\(824\) −0.832862 −0.0290141
\(825\) −7.50434 −0.261267
\(826\) 10.7670 0.374632
\(827\) −42.5251 −1.47874 −0.739371 0.673298i \(-0.764877\pi\)
−0.739371 + 0.673298i \(0.764877\pi\)
\(828\) −23.4230 −0.814005
\(829\) −53.8318 −1.86965 −0.934827 0.355103i \(-0.884446\pi\)
−0.934827 + 0.355103i \(0.884446\pi\)
\(830\) 18.6299 0.646652
\(831\) −19.5772 −0.679124
\(832\) 4.49624 0.155879
\(833\) −9.99020 −0.346140
\(834\) 28.3463 0.981552
\(835\) −7.41461 −0.256593
\(836\) 6.62881 0.229262
\(837\) −58.3420 −2.01660
\(838\) 15.8461 0.547396
\(839\) −42.1427 −1.45493 −0.727463 0.686147i \(-0.759301\pi\)
−0.727463 + 0.686147i \(0.759301\pi\)
\(840\) 14.1655 0.488756
\(841\) −3.80996 −0.131378
\(842\) 14.2514 0.491135
\(843\) 45.6413 1.57197
\(844\) 8.31911 0.286356
\(845\) −10.7728 −0.370594
\(846\) 21.8565 0.751442
\(847\) −32.9968 −1.13378
\(848\) 4.97697 0.170910
\(849\) −67.1094 −2.30319
\(850\) −7.70092 −0.264139
\(851\) 21.8506 0.749028
\(852\) 0.626173 0.0214523
\(853\) 29.8307 1.02138 0.510692 0.859764i \(-0.329389\pi\)
0.510692 + 0.859764i \(0.329389\pi\)
\(854\) −7.12476 −0.243804
\(855\) −58.5724 −2.00313
\(856\) −13.8805 −0.474427
\(857\) −4.55580 −0.155623 −0.0778116 0.996968i \(-0.524793\pi\)
−0.0778116 + 0.996968i \(0.524793\pi\)
\(858\) −12.1750 −0.415648
\(859\) 33.6547 1.14828 0.574141 0.818756i \(-0.305336\pi\)
0.574141 + 0.818756i \(0.305336\pi\)
\(860\) −7.37105 −0.251351
\(861\) −48.4679 −1.65178
\(862\) 26.6546 0.907859
\(863\) 28.1850 0.959429 0.479714 0.877425i \(-0.340740\pi\)
0.479714 + 0.877425i \(0.340740\pi\)
\(864\) −7.28181 −0.247732
\(865\) 4.62543 0.157269
\(866\) 11.0415 0.375207
\(867\) −27.0480 −0.918600
\(868\) −26.0793 −0.885191
\(869\) 4.78032 0.162161
\(870\) 21.8420 0.740512
\(871\) 28.1954 0.955365
\(872\) −4.18477 −0.141714
\(873\) −39.7374 −1.34491
\(874\) 30.4029 1.02839
\(875\) 37.7635 1.27664
\(876\) 43.8814 1.48262
\(877\) −46.0368 −1.55455 −0.777276 0.629160i \(-0.783399\pi\)
−0.777276 + 0.629160i \(0.783399\pi\)
\(878\) 8.58467 0.289719
\(879\) −33.0483 −1.11469
\(880\) −1.38670 −0.0467457
\(881\) 5.88384 0.198231 0.0991157 0.995076i \(-0.468399\pi\)
0.0991157 + 0.995076i \(0.468399\pi\)
\(882\) −19.7662 −0.665564
\(883\) 24.4465 0.822691 0.411346 0.911479i \(-0.365059\pi\)
0.411346 + 0.911479i \(0.365059\pi\)
\(884\) −12.4939 −0.420217
\(885\) 14.3952 0.483888
\(886\) 27.8370 0.935203
\(887\) 50.0875 1.68177 0.840887 0.541211i \(-0.182034\pi\)
0.840887 + 0.541211i \(0.182034\pi\)
\(888\) 14.9513 0.501732
\(889\) −65.6179 −2.20075
\(890\) −5.82240 −0.195167
\(891\) 4.39694 0.147303
\(892\) 4.08943 0.136924
\(893\) −28.3696 −0.949352
\(894\) 1.39604 0.0466907
\(895\) −7.39706 −0.247257
\(896\) −3.25503 −0.108743
\(897\) −55.8404 −1.86446
\(898\) −18.9956 −0.633891
\(899\) −40.2121 −1.34115
\(900\) −15.2368 −0.507892
\(901\) −13.8298 −0.460737
\(902\) 4.74466 0.157980
\(903\) 46.8512 1.55911
\(904\) 16.7698 0.557754
\(905\) −0.0568354 −0.00188927
\(906\) 55.6639 1.84931
\(907\) −55.4951 −1.84268 −0.921342 0.388753i \(-0.872906\pi\)
−0.921342 + 0.388753i \(0.872906\pi\)
\(908\) 11.9712 0.397279
\(909\) −18.1843 −0.603136
\(910\) 21.8486 0.724275
\(911\) −42.1133 −1.39528 −0.697638 0.716451i \(-0.745766\pi\)
−0.697638 + 0.716451i \(0.745766\pi\)
\(912\) 20.8032 0.688862
\(913\) 11.5918 0.383634
\(914\) −20.7789 −0.687303
\(915\) −9.52560 −0.314907
\(916\) 26.2492 0.867299
\(917\) 17.9908 0.594109
\(918\) 20.2344 0.667834
\(919\) 48.3699 1.59558 0.797788 0.602938i \(-0.206003\pi\)
0.797788 + 0.602938i \(0.206003\pi\)
\(920\) −6.36007 −0.209685
\(921\) 50.6711 1.66967
\(922\) 0.601112 0.0197966
\(923\) 0.965799 0.0317897
\(924\) 8.81403 0.289960
\(925\) 14.2139 0.467351
\(926\) −6.18736 −0.203329
\(927\) 4.57903 0.150395
\(928\) −5.01897 −0.164756
\(929\) 51.9047 1.70294 0.851470 0.524404i \(-0.175712\pi\)
0.851470 + 0.524404i \(0.175712\pi\)
\(930\) −34.8674 −1.14335
\(931\) 25.6565 0.840856
\(932\) 0.168495 0.00551922
\(933\) −65.4702 −2.14340
\(934\) 18.7722 0.614245
\(935\) 3.85330 0.126016
\(936\) −24.7201 −0.808001
\(937\) −35.1507 −1.14832 −0.574161 0.818743i \(-0.694672\pi\)
−0.574161 + 0.818743i \(0.694672\pi\)
\(938\) −20.4119 −0.666473
\(939\) −69.0747 −2.25417
\(940\) 5.93472 0.193569
\(941\) −45.3754 −1.47920 −0.739598 0.673049i \(-0.764984\pi\)
−0.739598 + 0.673049i \(0.764984\pi\)
\(942\) 68.4643 2.23069
\(943\) 21.7613 0.708645
\(944\) −3.30780 −0.107660
\(945\) −35.3846 −1.15106
\(946\) −4.58640 −0.149117
\(947\) 28.8082 0.936141 0.468070 0.883691i \(-0.344949\pi\)
0.468070 + 0.883691i \(0.344949\pi\)
\(948\) 15.0021 0.487245
\(949\) 67.6820 2.19705
\(950\) 19.7772 0.641658
\(951\) −21.8234 −0.707671
\(952\) 9.04492 0.293148
\(953\) −25.9657 −0.841111 −0.420555 0.907267i \(-0.638165\pi\)
−0.420555 + 0.907267i \(0.638165\pi\)
\(954\) −27.3631 −0.885913
\(955\) −39.9208 −1.29181
\(956\) −5.00385 −0.161836
\(957\) 13.5905 0.439317
\(958\) 22.8394 0.737909
\(959\) −15.7366 −0.508163
\(960\) −4.35188 −0.140456
\(961\) 33.1924 1.07072
\(962\) 23.0606 0.743503
\(963\) 76.3144 2.45920
\(964\) −26.3551 −0.848841
\(965\) −5.73106 −0.184489
\(966\) 40.4253 1.30066
\(967\) 14.0626 0.452224 0.226112 0.974101i \(-0.427398\pi\)
0.226112 + 0.974101i \(0.427398\pi\)
\(968\) 10.1372 0.325821
\(969\) −57.8069 −1.85703
\(970\) −10.7899 −0.346444
\(971\) 12.6707 0.406622 0.203311 0.979114i \(-0.434830\pi\)
0.203311 + 0.979114i \(0.434830\pi\)
\(972\) −8.04655 −0.258093
\(973\) −31.6515 −1.01470
\(974\) −35.9728 −1.15264
\(975\) −36.3244 −1.16331
\(976\) 2.18885 0.0700633
\(977\) 5.59021 0.178847 0.0894234 0.995994i \(-0.471498\pi\)
0.0894234 + 0.995994i \(0.471498\pi\)
\(978\) −26.1160 −0.835099
\(979\) −3.62280 −0.115785
\(980\) −5.36715 −0.171447
\(981\) 23.0076 0.734577
\(982\) 21.8992 0.698833
\(983\) −21.6558 −0.690714 −0.345357 0.938471i \(-0.612242\pi\)
−0.345357 + 0.938471i \(0.612242\pi\)
\(984\) 14.8902 0.474681
\(985\) 41.4148 1.31959
\(986\) 13.9465 0.444146
\(987\) −37.7218 −1.20070
\(988\) 32.0865 1.02081
\(989\) −21.0354 −0.668887
\(990\) 7.62401 0.242307
\(991\) 36.3543 1.15483 0.577417 0.816450i \(-0.304061\pi\)
0.577417 + 0.816450i \(0.304061\pi\)
\(992\) 8.01202 0.254382
\(993\) 10.2716 0.325961
\(994\) −0.699185 −0.0221768
\(995\) −22.7396 −0.720893
\(996\) 36.3786 1.15270
\(997\) 43.8215 1.38784 0.693921 0.720051i \(-0.255882\pi\)
0.693921 + 0.720051i \(0.255882\pi\)
\(998\) 2.85591 0.0904022
\(999\) −37.3474 −1.18162
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.h.1.18 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1502.2.a.h.1.18 19 1.1 even 1 trivial