Properties

Label 1849.2.a.j.1.1
Level $1849$
Weight $2$
Character 1849.1
Self dual yes
Analytic conductor $14.764$
Analytic rank $0$
Dimension $3$
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] = [1849,2,Mod(1,1849)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1849, 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("1849.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1849.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: \(14.7643393337\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.80194 q^{2} +1.24698 q^{3} +1.24698 q^{4} -2.24698 q^{5} -2.24698 q^{6} -1.69202 q^{7} +1.35690 q^{8} -1.44504 q^{9} +O(q^{10})\) \(q-1.80194 q^{2} +1.24698 q^{3} +1.24698 q^{4} -2.24698 q^{5} -2.24698 q^{6} -1.69202 q^{7} +1.35690 q^{8} -1.44504 q^{9} +4.04892 q^{10} -1.13706 q^{11} +1.55496 q^{12} -6.54288 q^{13} +3.04892 q^{14} -2.80194 q^{15} -4.93900 q^{16} -3.60388 q^{17} +2.60388 q^{18} +4.44504 q^{19} -2.80194 q^{20} -2.10992 q^{21} +2.04892 q^{22} -3.71379 q^{23} +1.69202 q^{24} +0.0489173 q^{25} +11.7899 q^{26} -5.54288 q^{27} -2.10992 q^{28} +6.00000 q^{29} +5.04892 q^{30} +10.3230 q^{31} +6.18598 q^{32} -1.41789 q^{33} +6.49396 q^{34} +3.80194 q^{35} -1.80194 q^{36} -3.65279 q^{37} -8.00969 q^{38} -8.15883 q^{39} -3.04892 q^{40} +4.39612 q^{41} +3.80194 q^{42} -1.41789 q^{44} +3.24698 q^{45} +6.69202 q^{46} -3.14914 q^{47} -6.15883 q^{48} -4.13706 q^{49} -0.0881460 q^{50} -4.49396 q^{51} -8.15883 q^{52} +11.1468 q^{53} +9.98792 q^{54} +2.55496 q^{55} -2.29590 q^{56} +5.54288 q^{57} -10.8116 q^{58} +0.158834 q^{59} -3.49396 q^{60} +0.713792 q^{61} -18.6015 q^{62} +2.44504 q^{63} -1.26875 q^{64} +14.7017 q^{65} +2.55496 q^{66} -2.09783 q^{67} -4.49396 q^{68} -4.63102 q^{69} -6.85086 q^{70} +3.72587 q^{71} -1.96077 q^{72} -1.43296 q^{73} +6.58211 q^{74} +0.0609989 q^{75} +5.54288 q^{76} +1.92394 q^{77} +14.7017 q^{78} -0.0609989 q^{79} +11.0978 q^{80} -2.57673 q^{81} -7.92154 q^{82} -12.2838 q^{83} -2.63102 q^{84} +8.09783 q^{85} +7.48188 q^{87} -1.54288 q^{88} +10.6189 q^{89} -5.85086 q^{90} +11.0707 q^{91} -4.63102 q^{92} +12.8726 q^{93} +5.67456 q^{94} -9.98792 q^{95} +7.71379 q^{96} +3.50365 q^{97} +7.45473 q^{98} +1.64310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - q^{3} - q^{4} - 2 q^{5} - 2 q^{6} - 4 q^{9} + 3 q^{10} + 2 q^{11} + 5 q^{12} - q^{13} - 4 q^{15} - 5 q^{16} - 2 q^{17} - q^{18} + 13 q^{19} - 4 q^{20} - 7 q^{21} - 3 q^{22} - 3 q^{23} - 9 q^{25} + 12 q^{26} + 2 q^{27} - 7 q^{28} + 18 q^{29} + 6 q^{30} + 11 q^{31} + 4 q^{32} - 10 q^{33} + 10 q^{34} + 7 q^{35} - q^{36} + 7 q^{37} - 2 q^{38} - 16 q^{39} + 22 q^{41} + 7 q^{42} - 10 q^{44} + 5 q^{45} + 15 q^{46} - 23 q^{47} - 10 q^{48} - 7 q^{49} - 4 q^{50} - 4 q^{51} - 16 q^{52} + 6 q^{53} + 11 q^{54} + 8 q^{55} + 7 q^{56} - 2 q^{57} - 6 q^{58} - 8 q^{59} - q^{60} - 6 q^{61} - 6 q^{62} + 7 q^{63} + 4 q^{64} + 17 q^{65} + 8 q^{66} + 12 q^{67} - 4 q^{68} + q^{69} - 7 q^{70} + 22 q^{71} + 7 q^{72} + 15 q^{73} + 14 q^{74} + 10 q^{75} - 2 q^{76} + 21 q^{77} + 17 q^{78} - 10 q^{79} + 15 q^{80} - 5 q^{81} + 2 q^{82} - 4 q^{83} + 7 q^{84} + 6 q^{85} - 6 q^{87} + 14 q^{88} - 2 q^{89} - 4 q^{90} + 21 q^{91} + q^{92} + 22 q^{93} - 4 q^{94} - 11 q^{95} + 15 q^{96} - 21 q^{97} + 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80194 −1.27416 −0.637081 0.770797i \(-0.719858\pi\)
−0.637081 + 0.770797i \(0.719858\pi\)
\(3\) 1.24698 0.719944 0.359972 0.932963i \(-0.382786\pi\)
0.359972 + 0.932963i \(0.382786\pi\)
\(4\) 1.24698 0.623490
\(5\) −2.24698 −1.00488 −0.502440 0.864612i \(-0.667564\pi\)
−0.502440 + 0.864612i \(0.667564\pi\)
\(6\) −2.24698 −0.917326
\(7\) −1.69202 −0.639524 −0.319762 0.947498i \(-0.603603\pi\)
−0.319762 + 0.947498i \(0.603603\pi\)
\(8\) 1.35690 0.479735
\(9\) −1.44504 −0.481681
\(10\) 4.04892 1.28038
\(11\) −1.13706 −0.342837 −0.171419 0.985198i \(-0.554835\pi\)
−0.171419 + 0.985198i \(0.554835\pi\)
\(12\) 1.55496 0.448878
\(13\) −6.54288 −1.81467 −0.907334 0.420411i \(-0.861886\pi\)
−0.907334 + 0.420411i \(0.861886\pi\)
\(14\) 3.04892 0.814857
\(15\) −2.80194 −0.723457
\(16\) −4.93900 −1.23475
\(17\) −3.60388 −0.874068 −0.437034 0.899445i \(-0.643971\pi\)
−0.437034 + 0.899445i \(0.643971\pi\)
\(18\) 2.60388 0.613739
\(19\) 4.44504 1.01976 0.509881 0.860245i \(-0.329689\pi\)
0.509881 + 0.860245i \(0.329689\pi\)
\(20\) −2.80194 −0.626532
\(21\) −2.10992 −0.460421
\(22\) 2.04892 0.436831
\(23\) −3.71379 −0.774379 −0.387190 0.922000i \(-0.626554\pi\)
−0.387190 + 0.922000i \(0.626554\pi\)
\(24\) 1.69202 0.345382
\(25\) 0.0489173 0.00978347
\(26\) 11.7899 2.31218
\(27\) −5.54288 −1.06673
\(28\) −2.10992 −0.398737
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 5.04892 0.921802
\(31\) 10.3230 1.85407 0.927037 0.374971i \(-0.122347\pi\)
0.927037 + 0.374971i \(0.122347\pi\)
\(32\) 6.18598 1.09354
\(33\) −1.41789 −0.246824
\(34\) 6.49396 1.11370
\(35\) 3.80194 0.642645
\(36\) −1.80194 −0.300323
\(37\) −3.65279 −0.600515 −0.300258 0.953858i \(-0.597073\pi\)
−0.300258 + 0.953858i \(0.597073\pi\)
\(38\) −8.00969 −1.29934
\(39\) −8.15883 −1.30646
\(40\) −3.04892 −0.482076
\(41\) 4.39612 0.686559 0.343280 0.939233i \(-0.388462\pi\)
0.343280 + 0.939233i \(0.388462\pi\)
\(42\) 3.80194 0.586652
\(43\) 0 0
\(44\) −1.41789 −0.213756
\(45\) 3.24698 0.484031
\(46\) 6.69202 0.986685
\(47\) −3.14914 −0.459350 −0.229675 0.973267i \(-0.573766\pi\)
−0.229675 + 0.973267i \(0.573766\pi\)
\(48\) −6.15883 −0.888951
\(49\) −4.13706 −0.591009
\(50\) −0.0881460 −0.0124657
\(51\) −4.49396 −0.629280
\(52\) −8.15883 −1.13143
\(53\) 11.1468 1.53112 0.765562 0.643362i \(-0.222461\pi\)
0.765562 + 0.643362i \(0.222461\pi\)
\(54\) 9.98792 1.35918
\(55\) 2.55496 0.344510
\(56\) −2.29590 −0.306802
\(57\) 5.54288 0.734172
\(58\) −10.8116 −1.41964
\(59\) 0.158834 0.0206784 0.0103392 0.999947i \(-0.496709\pi\)
0.0103392 + 0.999947i \(0.496709\pi\)
\(60\) −3.49396 −0.451068
\(61\) 0.713792 0.0913917 0.0456958 0.998955i \(-0.485449\pi\)
0.0456958 + 0.998955i \(0.485449\pi\)
\(62\) −18.6015 −2.36239
\(63\) 2.44504 0.308046
\(64\) −1.26875 −0.158594
\(65\) 14.7017 1.82352
\(66\) 2.55496 0.314494
\(67\) −2.09783 −0.256291 −0.128146 0.991755i \(-0.540902\pi\)
−0.128146 + 0.991755i \(0.540902\pi\)
\(68\) −4.49396 −0.544973
\(69\) −4.63102 −0.557510
\(70\) −6.85086 −0.818834
\(71\) 3.72587 0.442180 0.221090 0.975253i \(-0.429039\pi\)
0.221090 + 0.975253i \(0.429039\pi\)
\(72\) −1.96077 −0.231079
\(73\) −1.43296 −0.167715 −0.0838577 0.996478i \(-0.526724\pi\)
−0.0838577 + 0.996478i \(0.526724\pi\)
\(74\) 6.58211 0.765154
\(75\) 0.0609989 0.00704355
\(76\) 5.54288 0.635812
\(77\) 1.92394 0.219253
\(78\) 14.7017 1.66464
\(79\) −0.0609989 −0.00686292 −0.00343146 0.999994i \(-0.501092\pi\)
−0.00343146 + 0.999994i \(0.501092\pi\)
\(80\) 11.0978 1.24078
\(81\) −2.57673 −0.286303
\(82\) −7.92154 −0.874788
\(83\) −12.2838 −1.34832 −0.674162 0.738584i \(-0.735495\pi\)
−0.674162 + 0.738584i \(0.735495\pi\)
\(84\) −2.63102 −0.287068
\(85\) 8.09783 0.878333
\(86\) 0 0
\(87\) 7.48188 0.802141
\(88\) −1.54288 −0.164471
\(89\) 10.6189 1.12561 0.562803 0.826591i \(-0.309723\pi\)
0.562803 + 0.826591i \(0.309723\pi\)
\(90\) −5.85086 −0.616734
\(91\) 11.0707 1.16052
\(92\) −4.63102 −0.482817
\(93\) 12.8726 1.33483
\(94\) 5.67456 0.585286
\(95\) −9.98792 −1.02474
\(96\) 7.71379 0.787286
\(97\) 3.50365 0.355742 0.177871 0.984054i \(-0.443079\pi\)
0.177871 + 0.984054i \(0.443079\pi\)
\(98\) 7.45473 0.753042
\(99\) 1.64310 0.165138
\(100\) 0.0609989 0.00609989
\(101\) 12.8726 1.28087 0.640437 0.768011i \(-0.278753\pi\)
0.640437 + 0.768011i \(0.278753\pi\)
\(102\) 8.09783 0.801805
\(103\) 3.71379 0.365931 0.182965 0.983119i \(-0.441430\pi\)
0.182965 + 0.983119i \(0.441430\pi\)
\(104\) −8.87800 −0.870560
\(105\) 4.74094 0.462668
\(106\) −20.0858 −1.95090
\(107\) 5.04892 0.488097 0.244049 0.969763i \(-0.421524\pi\)
0.244049 + 0.969763i \(0.421524\pi\)
\(108\) −6.91185 −0.665093
\(109\) 2.17629 0.208451 0.104225 0.994554i \(-0.466764\pi\)
0.104225 + 0.994554i \(0.466764\pi\)
\(110\) −4.60388 −0.438962
\(111\) −4.55496 −0.432337
\(112\) 8.35690 0.789652
\(113\) 11.7463 1.10500 0.552500 0.833513i \(-0.313674\pi\)
0.552500 + 0.833513i \(0.313674\pi\)
\(114\) −9.98792 −0.935454
\(115\) 8.34481 0.778158
\(116\) 7.48188 0.694675
\(117\) 9.45473 0.874090
\(118\) −0.286208 −0.0263476
\(119\) 6.09783 0.558988
\(120\) −3.80194 −0.347068
\(121\) −9.70709 −0.882462
\(122\) −1.28621 −0.116448
\(123\) 5.48188 0.494284
\(124\) 12.8726 1.15600
\(125\) 11.1250 0.995049
\(126\) −4.40581 −0.392501
\(127\) −20.2228 −1.79448 −0.897242 0.441538i \(-0.854433\pi\)
−0.897242 + 0.441538i \(0.854433\pi\)
\(128\) −10.0858 −0.891463
\(129\) 0 0
\(130\) −26.4916 −2.32346
\(131\) −2.05861 −0.179861 −0.0899306 0.995948i \(-0.528665\pi\)
−0.0899306 + 0.995948i \(0.528665\pi\)
\(132\) −1.76809 −0.153892
\(133\) −7.52111 −0.652163
\(134\) 3.78017 0.326557
\(135\) 12.4547 1.07193
\(136\) −4.89008 −0.419321
\(137\) −15.6015 −1.33292 −0.666462 0.745539i \(-0.732192\pi\)
−0.666462 + 0.745539i \(0.732192\pi\)
\(138\) 8.34481 0.710358
\(139\) −7.07606 −0.600184 −0.300092 0.953910i \(-0.597017\pi\)
−0.300092 + 0.953910i \(0.597017\pi\)
\(140\) 4.74094 0.400682
\(141\) −3.92692 −0.330706
\(142\) −6.71379 −0.563409
\(143\) 7.43967 0.622136
\(144\) 7.13706 0.594755
\(145\) −13.4819 −1.11961
\(146\) 2.58211 0.213697
\(147\) −5.15883 −0.425493
\(148\) −4.55496 −0.374415
\(149\) 5.80194 0.475313 0.237657 0.971349i \(-0.423621\pi\)
0.237657 + 0.971349i \(0.423621\pi\)
\(150\) −0.109916 −0.00897463
\(151\) −7.44935 −0.606220 −0.303110 0.952956i \(-0.598025\pi\)
−0.303110 + 0.952956i \(0.598025\pi\)
\(152\) 6.03146 0.489216
\(153\) 5.20775 0.421022
\(154\) −3.46681 −0.279364
\(155\) −23.1957 −1.86312
\(156\) −10.1739 −0.814564
\(157\) −13.2131 −1.05452 −0.527261 0.849703i \(-0.676781\pi\)
−0.527261 + 0.849703i \(0.676781\pi\)
\(158\) 0.109916 0.00874447
\(159\) 13.8998 1.10232
\(160\) −13.8998 −1.09887
\(161\) 6.28382 0.495234
\(162\) 4.64310 0.364797
\(163\) 14.1860 1.11113 0.555566 0.831473i \(-0.312502\pi\)
0.555566 + 0.831473i \(0.312502\pi\)
\(164\) 5.48188 0.428063
\(165\) 3.18598 0.248028
\(166\) 22.1347 1.71798
\(167\) 9.47219 0.732980 0.366490 0.930422i \(-0.380559\pi\)
0.366490 + 0.930422i \(0.380559\pi\)
\(168\) −2.86294 −0.220880
\(169\) 29.8092 2.29302
\(170\) −14.5918 −1.11914
\(171\) −6.42327 −0.491200
\(172\) 0 0
\(173\) 4.76271 0.362102 0.181051 0.983474i \(-0.442050\pi\)
0.181051 + 0.983474i \(0.442050\pi\)
\(174\) −13.4819 −1.02206
\(175\) −0.0827692 −0.00625676
\(176\) 5.61596 0.423319
\(177\) 0.198062 0.0148873
\(178\) −19.1347 −1.43420
\(179\) −1.64742 −0.123134 −0.0615668 0.998103i \(-0.519610\pi\)
−0.0615668 + 0.998103i \(0.519610\pi\)
\(180\) 4.04892 0.301788
\(181\) 2.27652 0.169212 0.0846062 0.996414i \(-0.473037\pi\)
0.0846062 + 0.996414i \(0.473037\pi\)
\(182\) −19.9487 −1.47870
\(183\) 0.890084 0.0657969
\(184\) −5.03923 −0.371497
\(185\) 8.20775 0.603446
\(186\) −23.1957 −1.70079
\(187\) 4.09783 0.299663
\(188\) −3.92692 −0.286400
\(189\) 9.37867 0.682198
\(190\) 17.9976 1.30568
\(191\) 15.0315 1.08764 0.543819 0.839202i \(-0.316978\pi\)
0.543819 + 0.839202i \(0.316978\pi\)
\(192\) −1.58211 −0.114179
\(193\) 9.85086 0.709080 0.354540 0.935041i \(-0.384637\pi\)
0.354540 + 0.935041i \(0.384637\pi\)
\(194\) −6.31336 −0.453272
\(195\) 18.3327 1.31283
\(196\) −5.15883 −0.368488
\(197\) 10.9608 0.780923 0.390461 0.920619i \(-0.372316\pi\)
0.390461 + 0.920619i \(0.372316\pi\)
\(198\) −2.96077 −0.210413
\(199\) 24.3545 1.72644 0.863222 0.504824i \(-0.168442\pi\)
0.863222 + 0.504824i \(0.168442\pi\)
\(200\) 0.0663757 0.00469347
\(201\) −2.61596 −0.184515
\(202\) −23.1957 −1.63204
\(203\) −10.1521 −0.712540
\(204\) −5.60388 −0.392350
\(205\) −9.87800 −0.689910
\(206\) −6.69202 −0.466255
\(207\) 5.36658 0.373003
\(208\) 32.3153 2.24066
\(209\) −5.05429 −0.349613
\(210\) −8.54288 −0.589514
\(211\) −10.3110 −0.709836 −0.354918 0.934897i \(-0.615491\pi\)
−0.354918 + 0.934897i \(0.615491\pi\)
\(212\) 13.8998 0.954640
\(213\) 4.64609 0.318345
\(214\) −9.09783 −0.621915
\(215\) 0 0
\(216\) −7.52111 −0.511746
\(217\) −17.4668 −1.18572
\(218\) −3.92154 −0.265600
\(219\) −1.78687 −0.120746
\(220\) 3.18598 0.214799
\(221\) 23.5797 1.58614
\(222\) 8.20775 0.550868
\(223\) −2.83340 −0.189738 −0.0948691 0.995490i \(-0.530243\pi\)
−0.0948691 + 0.995490i \(0.530243\pi\)
\(224\) −10.4668 −0.699343
\(225\) −0.0706876 −0.00471251
\(226\) −21.1661 −1.40795
\(227\) 13.0640 0.867087 0.433543 0.901133i \(-0.357263\pi\)
0.433543 + 0.901133i \(0.357263\pi\)
\(228\) 6.91185 0.457749
\(229\) −17.9162 −1.18393 −0.591967 0.805962i \(-0.701648\pi\)
−0.591967 + 0.805962i \(0.701648\pi\)
\(230\) −15.0368 −0.991500
\(231\) 2.39911 0.157850
\(232\) 8.14138 0.534507
\(233\) 9.34050 0.611917 0.305958 0.952045i \(-0.401023\pi\)
0.305958 + 0.952045i \(0.401023\pi\)
\(234\) −17.0368 −1.11373
\(235\) 7.07606 0.461592
\(236\) 0.198062 0.0128928
\(237\) −0.0760644 −0.00494091
\(238\) −10.9879 −0.712241
\(239\) 7.24160 0.468420 0.234210 0.972186i \(-0.424750\pi\)
0.234210 + 0.972186i \(0.424750\pi\)
\(240\) 13.8388 0.893289
\(241\) 0.109916 0.00708033 0.00354016 0.999994i \(-0.498873\pi\)
0.00354016 + 0.999994i \(0.498873\pi\)
\(242\) 17.4916 1.12440
\(243\) 13.4155 0.860605
\(244\) 0.890084 0.0569818
\(245\) 9.29590 0.593893
\(246\) −9.87800 −0.629798
\(247\) −29.0834 −1.85053
\(248\) 14.0073 0.889464
\(249\) −15.3177 −0.970718
\(250\) −20.0465 −1.26785
\(251\) −27.8974 −1.76087 −0.880433 0.474170i \(-0.842748\pi\)
−0.880433 + 0.474170i \(0.842748\pi\)
\(252\) 3.04892 0.192064
\(253\) 4.22282 0.265486
\(254\) 36.4403 2.28647
\(255\) 10.0978 0.632351
\(256\) 20.7114 1.29446
\(257\) 2.66487 0.166230 0.0831151 0.996540i \(-0.473513\pi\)
0.0831151 + 0.996540i \(0.473513\pi\)
\(258\) 0 0
\(259\) 6.18060 0.384044
\(260\) 18.3327 1.13695
\(261\) −8.67025 −0.536675
\(262\) 3.70948 0.229172
\(263\) 24.1347 1.48821 0.744104 0.668064i \(-0.232877\pi\)
0.744104 + 0.668064i \(0.232877\pi\)
\(264\) −1.92394 −0.118410
\(265\) −25.0465 −1.53860
\(266\) 13.5526 0.830961
\(267\) 13.2416 0.810373
\(268\) −2.61596 −0.159795
\(269\) 30.1511 1.83834 0.919171 0.393858i \(-0.128860\pi\)
0.919171 + 0.393858i \(0.128860\pi\)
\(270\) −22.4426 −1.36582
\(271\) −13.6920 −0.831731 −0.415866 0.909426i \(-0.636521\pi\)
−0.415866 + 0.909426i \(0.636521\pi\)
\(272\) 17.7995 1.07926
\(273\) 13.8049 0.835512
\(274\) 28.1129 1.69836
\(275\) −0.0556221 −0.00335414
\(276\) −5.77479 −0.347602
\(277\) 18.5743 1.11602 0.558012 0.829833i \(-0.311564\pi\)
0.558012 + 0.829833i \(0.311564\pi\)
\(278\) 12.7506 0.764732
\(279\) −14.9172 −0.893071
\(280\) 5.15883 0.308299
\(281\) −2.71618 −0.162034 −0.0810170 0.996713i \(-0.525817\pi\)
−0.0810170 + 0.996713i \(0.525817\pi\)
\(282\) 7.07606 0.421374
\(283\) 17.0121 1.01126 0.505632 0.862749i \(-0.331259\pi\)
0.505632 + 0.862749i \(0.331259\pi\)
\(284\) 4.64609 0.275695
\(285\) −12.4547 −0.737755
\(286\) −13.4058 −0.792702
\(287\) −7.43834 −0.439071
\(288\) −8.93900 −0.526736
\(289\) −4.01208 −0.236005
\(290\) 24.2935 1.42656
\(291\) 4.36898 0.256114
\(292\) −1.78687 −0.104569
\(293\) −11.0315 −0.644465 −0.322232 0.946661i \(-0.604433\pi\)
−0.322232 + 0.946661i \(0.604433\pi\)
\(294\) 9.29590 0.542148
\(295\) −0.356896 −0.0207793
\(296\) −4.95646 −0.288088
\(297\) 6.30260 0.365714
\(298\) −10.4547 −0.605626
\(299\) 24.2989 1.40524
\(300\) 0.0760644 0.00439158
\(301\) 0 0
\(302\) 13.4233 0.772422
\(303\) 16.0519 0.922158
\(304\) −21.9541 −1.25915
\(305\) −1.60388 −0.0918376
\(306\) −9.38404 −0.536450
\(307\) 15.7952 0.901482 0.450741 0.892655i \(-0.351160\pi\)
0.450741 + 0.892655i \(0.351160\pi\)
\(308\) 2.39911 0.136702
\(309\) 4.63102 0.263450
\(310\) 41.7972 2.37392
\(311\) −12.3666 −0.701245 −0.350622 0.936517i \(-0.614030\pi\)
−0.350622 + 0.936517i \(0.614030\pi\)
\(312\) −11.0707 −0.626754
\(313\) 5.83446 0.329783 0.164892 0.986312i \(-0.447273\pi\)
0.164892 + 0.986312i \(0.447273\pi\)
\(314\) 23.8092 1.34363
\(315\) −5.49396 −0.309550
\(316\) −0.0760644 −0.00427896
\(317\) 26.1685 1.46977 0.734885 0.678191i \(-0.237236\pi\)
0.734885 + 0.678191i \(0.237236\pi\)
\(318\) −25.0465 −1.40454
\(319\) −6.82238 −0.381980
\(320\) 2.85086 0.159368
\(321\) 6.29590 0.351403
\(322\) −11.3230 −0.631009
\(323\) −16.0194 −0.891342
\(324\) −3.21313 −0.178507
\(325\) −0.320060 −0.0177537
\(326\) −25.5623 −1.41576
\(327\) 2.71379 0.150073
\(328\) 5.96508 0.329367
\(329\) 5.32842 0.293765
\(330\) −5.74094 −0.316028
\(331\) −18.2664 −1.00401 −0.502005 0.864865i \(-0.667404\pi\)
−0.502005 + 0.864865i \(0.667404\pi\)
\(332\) −15.3177 −0.840666
\(333\) 5.27844 0.289257
\(334\) −17.0683 −0.933936
\(335\) 4.71379 0.257542
\(336\) 10.4209 0.568506
\(337\) −7.09113 −0.386278 −0.193139 0.981171i \(-0.561867\pi\)
−0.193139 + 0.981171i \(0.561867\pi\)
\(338\) −53.7144 −2.92168
\(339\) 14.6474 0.795538
\(340\) 10.0978 0.547632
\(341\) −11.7380 −0.635646
\(342\) 11.5743 0.625868
\(343\) 18.8442 1.01749
\(344\) 0 0
\(345\) 10.4058 0.560230
\(346\) −8.58211 −0.461377
\(347\) 3.21446 0.172561 0.0862805 0.996271i \(-0.472502\pi\)
0.0862805 + 0.996271i \(0.472502\pi\)
\(348\) 9.32975 0.500127
\(349\) −27.3502 −1.46402 −0.732011 0.681293i \(-0.761418\pi\)
−0.732011 + 0.681293i \(0.761418\pi\)
\(350\) 0.149145 0.00797213
\(351\) 36.2664 1.93575
\(352\) −7.03385 −0.374906
\(353\) −23.0978 −1.22937 −0.614687 0.788771i \(-0.710718\pi\)
−0.614687 + 0.788771i \(0.710718\pi\)
\(354\) −0.356896 −0.0189688
\(355\) −8.37196 −0.444338
\(356\) 13.2416 0.701804
\(357\) 7.60388 0.402440
\(358\) 2.96854 0.156892
\(359\) −0.254749 −0.0134452 −0.00672258 0.999977i \(-0.502140\pi\)
−0.00672258 + 0.999977i \(0.502140\pi\)
\(360\) 4.40581 0.232207
\(361\) 0.758397 0.0399156
\(362\) −4.10215 −0.215604
\(363\) −12.1045 −0.635324
\(364\) 13.8049 0.723575
\(365\) 3.21983 0.168534
\(366\) −1.60388 −0.0838359
\(367\) −14.8092 −0.773036 −0.386518 0.922282i \(-0.626322\pi\)
−0.386518 + 0.922282i \(0.626322\pi\)
\(368\) 18.3424 0.956165
\(369\) −6.35258 −0.330702
\(370\) −14.7899 −0.768888
\(371\) −18.8605 −0.979191
\(372\) 16.0519 0.832252
\(373\) −22.4722 −1.16357 −0.581783 0.813344i \(-0.697645\pi\)
−0.581783 + 0.813344i \(0.697645\pi\)
\(374\) −7.38404 −0.381820
\(375\) 13.8726 0.716379
\(376\) −4.27306 −0.220366
\(377\) −39.2573 −2.02185
\(378\) −16.8998 −0.869230
\(379\) −26.0508 −1.33814 −0.669071 0.743199i \(-0.733308\pi\)
−0.669071 + 0.743199i \(0.733308\pi\)
\(380\) −12.4547 −0.638914
\(381\) −25.2174 −1.29193
\(382\) −27.0858 −1.38583
\(383\) −12.3405 −0.630570 −0.315285 0.948997i \(-0.602100\pi\)
−0.315285 + 0.948997i \(0.602100\pi\)
\(384\) −12.5767 −0.641803
\(385\) −4.32304 −0.220323
\(386\) −17.7506 −0.903483
\(387\) 0 0
\(388\) 4.36898 0.221801
\(389\) 19.9138 1.00967 0.504834 0.863216i \(-0.331554\pi\)
0.504834 + 0.863216i \(0.331554\pi\)
\(390\) −33.0344 −1.67276
\(391\) 13.3840 0.676860
\(392\) −5.61356 −0.283528
\(393\) −2.56704 −0.129490
\(394\) −19.7506 −0.995022
\(395\) 0.137063 0.00689641
\(396\) 2.04892 0.102962
\(397\) 9.12498 0.457970 0.228985 0.973430i \(-0.426459\pi\)
0.228985 + 0.973430i \(0.426459\pi\)
\(398\) −43.8853 −2.19977
\(399\) −9.37867 −0.469521
\(400\) −0.241603 −0.0120801
\(401\) 0.0435405 0.00217431 0.00108716 0.999999i \(-0.499654\pi\)
0.00108716 + 0.999999i \(0.499654\pi\)
\(402\) 4.71379 0.235103
\(403\) −67.5424 −3.36453
\(404\) 16.0519 0.798612
\(405\) 5.78986 0.287700
\(406\) 18.2935 0.907891
\(407\) 4.15346 0.205879
\(408\) −6.09783 −0.301888
\(409\) 10.6866 0.528421 0.264210 0.964465i \(-0.414889\pi\)
0.264210 + 0.964465i \(0.414889\pi\)
\(410\) 17.7995 0.879057
\(411\) −19.4547 −0.959631
\(412\) 4.63102 0.228154
\(413\) −0.268750 −0.0132243
\(414\) −9.67025 −0.475267
\(415\) 27.6015 1.35490
\(416\) −40.4741 −1.98441
\(417\) −8.82371 −0.432099
\(418\) 9.10752 0.445464
\(419\) 7.38106 0.360588 0.180294 0.983613i \(-0.442295\pi\)
0.180294 + 0.983613i \(0.442295\pi\)
\(420\) 5.91185 0.288469
\(421\) −9.79225 −0.477245 −0.238623 0.971112i \(-0.576696\pi\)
−0.238623 + 0.971112i \(0.576696\pi\)
\(422\) 18.5797 0.904446
\(423\) 4.55065 0.221260
\(424\) 15.1250 0.734534
\(425\) −0.176292 −0.00855142
\(426\) −8.37196 −0.405623
\(427\) −1.20775 −0.0584472
\(428\) 6.29590 0.304324
\(429\) 9.27711 0.447903
\(430\) 0 0
\(431\) 27.3297 1.31643 0.658214 0.752831i \(-0.271312\pi\)
0.658214 + 0.752831i \(0.271312\pi\)
\(432\) 27.3763 1.31714
\(433\) 2.77777 0.133491 0.0667457 0.997770i \(-0.478738\pi\)
0.0667457 + 0.997770i \(0.478738\pi\)
\(434\) 31.4741 1.51081
\(435\) −16.8116 −0.806056
\(436\) 2.71379 0.129967
\(437\) −16.5080 −0.789683
\(438\) 3.21983 0.153850
\(439\) 9.58211 0.457329 0.228664 0.973505i \(-0.426564\pi\)
0.228664 + 0.973505i \(0.426564\pi\)
\(440\) 3.46681 0.165274
\(441\) 5.97823 0.284678
\(442\) −42.4892 −2.02100
\(443\) 23.7754 1.12960 0.564801 0.825227i \(-0.308953\pi\)
0.564801 + 0.825227i \(0.308953\pi\)
\(444\) −5.67994 −0.269558
\(445\) −23.8605 −1.13110
\(446\) 5.10560 0.241757
\(447\) 7.23490 0.342199
\(448\) 2.14675 0.101424
\(449\) 18.3502 0.865999 0.433000 0.901394i \(-0.357455\pi\)
0.433000 + 0.901394i \(0.357455\pi\)
\(450\) 0.127375 0.00600450
\(451\) −4.99867 −0.235378
\(452\) 14.6474 0.688956
\(453\) −9.28919 −0.436444
\(454\) −23.5405 −1.10481
\(455\) −24.8756 −1.16619
\(456\) 7.52111 0.352208
\(457\) −39.8116 −1.86231 −0.931155 0.364624i \(-0.881197\pi\)
−0.931155 + 0.364624i \(0.881197\pi\)
\(458\) 32.2838 1.50852
\(459\) 19.9758 0.932392
\(460\) 10.4058 0.485174
\(461\) −19.2301 −0.895636 −0.447818 0.894125i \(-0.647799\pi\)
−0.447818 + 0.894125i \(0.647799\pi\)
\(462\) −4.32304 −0.201126
\(463\) −14.9584 −0.695175 −0.347587 0.937648i \(-0.612999\pi\)
−0.347587 + 0.937648i \(0.612999\pi\)
\(464\) −29.6340 −1.37572
\(465\) −28.9245 −1.34134
\(466\) −16.8310 −0.779681
\(467\) −25.7904 −1.19344 −0.596720 0.802450i \(-0.703530\pi\)
−0.596720 + 0.802450i \(0.703530\pi\)
\(468\) 11.7899 0.544986
\(469\) 3.54958 0.163904
\(470\) −12.7506 −0.588143
\(471\) −16.4765 −0.759197
\(472\) 0.215521 0.00992014
\(473\) 0 0
\(474\) 0.137063 0.00629553
\(475\) 0.217440 0.00997681
\(476\) 7.60388 0.348523
\(477\) −16.1075 −0.737513
\(478\) −13.0489 −0.596844
\(479\) −29.7157 −1.35775 −0.678873 0.734256i \(-0.737531\pi\)
−0.678873 + 0.734256i \(0.737531\pi\)
\(480\) −17.3327 −0.791127
\(481\) 23.8998 1.08974
\(482\) −0.198062 −0.00902149
\(483\) 7.83579 0.356541
\(484\) −12.1045 −0.550206
\(485\) −7.87263 −0.357478
\(486\) −24.1739 −1.09655
\(487\) 1.52111 0.0689279 0.0344640 0.999406i \(-0.489028\pi\)
0.0344640 + 0.999406i \(0.489028\pi\)
\(488\) 0.968541 0.0438438
\(489\) 17.6896 0.799953
\(490\) −16.7506 −0.756716
\(491\) −28.6631 −1.29355 −0.646773 0.762682i \(-0.723882\pi\)
−0.646773 + 0.762682i \(0.723882\pi\)
\(492\) 6.83579 0.308181
\(493\) −21.6233 −0.973862
\(494\) 52.4064 2.35788
\(495\) −3.69202 −0.165944
\(496\) −50.9855 −2.28932
\(497\) −6.30426 −0.282785
\(498\) 27.6015 1.23685
\(499\) 20.8159 0.931849 0.465925 0.884824i \(-0.345722\pi\)
0.465925 + 0.884824i \(0.345722\pi\)
\(500\) 13.8726 0.620403
\(501\) 11.8116 0.527705
\(502\) 50.2693 2.24363
\(503\) −3.65087 −0.162784 −0.0813922 0.996682i \(-0.525937\pi\)
−0.0813922 + 0.996682i \(0.525937\pi\)
\(504\) 3.31767 0.147781
\(505\) −28.9245 −1.28712
\(506\) −7.60925 −0.338273
\(507\) 37.1715 1.65084
\(508\) −25.2174 −1.11884
\(509\) −6.27114 −0.277964 −0.138982 0.990295i \(-0.544383\pi\)
−0.138982 + 0.990295i \(0.544383\pi\)
\(510\) −18.1957 −0.805718
\(511\) 2.42460 0.107258
\(512\) −17.1491 −0.757892
\(513\) −24.6383 −1.08781
\(514\) −4.80194 −0.211804
\(515\) −8.34481 −0.367716
\(516\) 0 0
\(517\) 3.58078 0.157482
\(518\) −11.1371 −0.489334
\(519\) 5.93900 0.260693
\(520\) 19.9487 0.874808
\(521\) 0.128703 0.00563856 0.00281928 0.999996i \(-0.499103\pi\)
0.00281928 + 0.999996i \(0.499103\pi\)
\(522\) 15.6233 0.683811
\(523\) 21.6209 0.945414 0.472707 0.881220i \(-0.343277\pi\)
0.472707 + 0.881220i \(0.343277\pi\)
\(524\) −2.56704 −0.112142
\(525\) −0.103211 −0.00450452
\(526\) −43.4892 −1.89622
\(527\) −37.2030 −1.62059
\(528\) 7.00298 0.304766
\(529\) −9.20775 −0.400337
\(530\) 45.1323 1.96042
\(531\) −0.229521 −0.00996037
\(532\) −9.37867 −0.406617
\(533\) −28.7633 −1.24588
\(534\) −23.8605 −1.03255
\(535\) −11.3448 −0.490479
\(536\) −2.84654 −0.122952
\(537\) −2.05429 −0.0886493
\(538\) −54.3303 −2.34235
\(539\) 4.70410 0.202620
\(540\) 15.5308 0.668339
\(541\) 14.0519 0.604138 0.302069 0.953286i \(-0.402323\pi\)
0.302069 + 0.953286i \(0.402323\pi\)
\(542\) 24.6722 1.05976
\(543\) 2.83877 0.121823
\(544\) −22.2935 −0.955826
\(545\) −4.89008 −0.209468
\(546\) −24.8756 −1.06458
\(547\) −33.7318 −1.44227 −0.721135 0.692795i \(-0.756379\pi\)
−0.721135 + 0.692795i \(0.756379\pi\)
\(548\) −19.4547 −0.831065
\(549\) −1.03146 −0.0440216
\(550\) 0.100228 0.00427372
\(551\) 26.6703 1.13619
\(552\) −6.28382 −0.267457
\(553\) 0.103211 0.00438900
\(554\) −33.4698 −1.42200
\(555\) 10.2349 0.434447
\(556\) −8.82371 −0.374209
\(557\) 25.4588 1.07872 0.539362 0.842074i \(-0.318666\pi\)
0.539362 + 0.842074i \(0.318666\pi\)
\(558\) 26.8799 1.13792
\(559\) 0 0
\(560\) −18.7778 −0.793506
\(561\) 5.10992 0.215741
\(562\) 4.89440 0.206458
\(563\) −6.42327 −0.270709 −0.135354 0.990797i \(-0.543217\pi\)
−0.135354 + 0.990797i \(0.543217\pi\)
\(564\) −4.89679 −0.206192
\(565\) −26.3937 −1.11039
\(566\) −30.6547 −1.28851
\(567\) 4.35988 0.183098
\(568\) 5.05562 0.212129
\(569\) −1.22414 −0.0513188 −0.0256594 0.999671i \(-0.508169\pi\)
−0.0256594 + 0.999671i \(0.508169\pi\)
\(570\) 22.4426 0.940019
\(571\) 36.9221 1.54514 0.772572 0.634928i \(-0.218970\pi\)
0.772572 + 0.634928i \(0.218970\pi\)
\(572\) 9.27711 0.387895
\(573\) 18.7439 0.783039
\(574\) 13.4034 0.559448
\(575\) −0.181669 −0.00757611
\(576\) 1.83340 0.0763915
\(577\) 25.7904 1.07367 0.536835 0.843687i \(-0.319620\pi\)
0.536835 + 0.843687i \(0.319620\pi\)
\(578\) 7.22952 0.300708
\(579\) 12.2838 0.510498
\(580\) −16.8116 −0.698065
\(581\) 20.7845 0.862286
\(582\) −7.87263 −0.326331
\(583\) −12.6746 −0.524927
\(584\) −1.94438 −0.0804589
\(585\) −21.2446 −0.878356
\(586\) 19.8780 0.821153
\(587\) −9.42519 −0.389019 −0.194510 0.980901i \(-0.562312\pi\)
−0.194510 + 0.980901i \(0.562312\pi\)
\(588\) −6.43296 −0.265291
\(589\) 45.8864 1.89071
\(590\) 0.643104 0.0264762
\(591\) 13.6679 0.562221
\(592\) 18.0411 0.741487
\(593\) 8.71810 0.358010 0.179005 0.983848i \(-0.442712\pi\)
0.179005 + 0.983848i \(0.442712\pi\)
\(594\) −11.3569 −0.465979
\(595\) −13.7017 −0.561715
\(596\) 7.23490 0.296353
\(597\) 30.3696 1.24294
\(598\) −43.7851 −1.79050
\(599\) 18.7791 0.767293 0.383647 0.923480i \(-0.374668\pi\)
0.383647 + 0.923480i \(0.374668\pi\)
\(600\) 0.0827692 0.00337904
\(601\) 13.5646 0.553313 0.276657 0.960969i \(-0.410774\pi\)
0.276657 + 0.960969i \(0.410774\pi\)
\(602\) 0 0
\(603\) 3.03146 0.123451
\(604\) −9.28919 −0.377972
\(605\) 21.8116 0.886769
\(606\) −28.9245 −1.17498
\(607\) 30.1866 1.22524 0.612618 0.790379i \(-0.290117\pi\)
0.612618 + 0.790379i \(0.290117\pi\)
\(608\) 27.4969 1.11515
\(609\) −12.6595 −0.512989
\(610\) 2.89008 0.117016
\(611\) 20.6045 0.833567
\(612\) 6.49396 0.262503
\(613\) 4.49694 0.181630 0.0908149 0.995868i \(-0.471053\pi\)
0.0908149 + 0.995868i \(0.471053\pi\)
\(614\) −28.4620 −1.14863
\(615\) −12.3177 −0.496696
\(616\) 2.61058 0.105183
\(617\) −28.5579 −1.14970 −0.574850 0.818259i \(-0.694939\pi\)
−0.574850 + 0.818259i \(0.694939\pi\)
\(618\) −8.34481 −0.335678
\(619\) −2.90515 −0.116768 −0.0583839 0.998294i \(-0.518595\pi\)
−0.0583839 + 0.998294i \(0.518595\pi\)
\(620\) −28.9245 −1.16164
\(621\) 20.5851 0.826051
\(622\) 22.2838 0.893500
\(623\) −17.9675 −0.719852
\(624\) 40.2965 1.61315
\(625\) −25.2422 −1.00969
\(626\) −10.5133 −0.420197
\(627\) −6.30260 −0.251702
\(628\) −16.4765 −0.657484
\(629\) 13.1642 0.524891
\(630\) 9.89977 0.394416
\(631\) −15.1196 −0.601902 −0.300951 0.953640i \(-0.597304\pi\)
−0.300951 + 0.953640i \(0.597304\pi\)
\(632\) −0.0827692 −0.00329238
\(633\) −12.8576 −0.511042
\(634\) −47.1540 −1.87273
\(635\) 45.4403 1.80324
\(636\) 17.3327 0.687288
\(637\) 27.0683 1.07248
\(638\) 12.2935 0.486704
\(639\) −5.38404 −0.212989
\(640\) 22.6625 0.895813
\(641\) −3.40714 −0.134574 −0.0672870 0.997734i \(-0.521434\pi\)
−0.0672870 + 0.997734i \(0.521434\pi\)
\(642\) −11.3448 −0.447744
\(643\) 0.977165 0.0385356 0.0192678 0.999814i \(-0.493866\pi\)
0.0192678 + 0.999814i \(0.493866\pi\)
\(644\) 7.83579 0.308773
\(645\) 0 0
\(646\) 28.8659 1.13571
\(647\) 16.4558 0.646944 0.323472 0.946238i \(-0.395150\pi\)
0.323472 + 0.946238i \(0.395150\pi\)
\(648\) −3.49635 −0.137350
\(649\) −0.180604 −0.00708932
\(650\) 0.576728 0.0226211
\(651\) −21.7808 −0.853655
\(652\) 17.6896 0.692779
\(653\) −33.7077 −1.31908 −0.659542 0.751668i \(-0.729250\pi\)
−0.659542 + 0.751668i \(0.729250\pi\)
\(654\) −4.89008 −0.191217
\(655\) 4.62565 0.180739
\(656\) −21.7125 −0.847729
\(657\) 2.07069 0.0807852
\(658\) −9.60148 −0.374305
\(659\) −35.1008 −1.36733 −0.683667 0.729794i \(-0.739616\pi\)
−0.683667 + 0.729794i \(0.739616\pi\)
\(660\) 3.97285 0.154643
\(661\) 2.10693 0.0819502 0.0409751 0.999160i \(-0.486954\pi\)
0.0409751 + 0.999160i \(0.486954\pi\)
\(662\) 32.9148 1.27927
\(663\) 29.4034 1.14193
\(664\) −16.6679 −0.646838
\(665\) 16.8998 0.655345
\(666\) −9.51142 −0.368560
\(667\) −22.2828 −0.862792
\(668\) 11.8116 0.457006
\(669\) −3.53319 −0.136601
\(670\) −8.49396 −0.328150
\(671\) −0.811626 −0.0313325
\(672\) −13.0519 −0.503488
\(673\) −17.4964 −0.674435 −0.337217 0.941427i \(-0.609486\pi\)
−0.337217 + 0.941427i \(0.609486\pi\)
\(674\) 12.7778 0.492181
\(675\) −0.271143 −0.0104363
\(676\) 37.1715 1.42967
\(677\) 29.4252 1.13090 0.565451 0.824782i \(-0.308702\pi\)
0.565451 + 0.824782i \(0.308702\pi\)
\(678\) −26.3937 −1.01365
\(679\) −5.92825 −0.227505
\(680\) 10.9879 0.421367
\(681\) 16.2905 0.624254
\(682\) 21.1511 0.809916
\(683\) −51.8504 −1.98400 −0.992000 0.126239i \(-0.959709\pi\)
−0.992000 + 0.126239i \(0.959709\pi\)
\(684\) −8.00969 −0.306258
\(685\) 35.0562 1.33943
\(686\) −33.9560 −1.29645
\(687\) −22.3411 −0.852366
\(688\) 0 0
\(689\) −72.9318 −2.77848
\(690\) −18.7506 −0.713824
\(691\) −1.26337 −0.0480610 −0.0240305 0.999711i \(-0.507650\pi\)
−0.0240305 + 0.999711i \(0.507650\pi\)
\(692\) 5.93900 0.225767
\(693\) −2.78017 −0.105610
\(694\) −5.79225 −0.219871
\(695\) 15.8998 0.603113
\(696\) 10.1521 0.384815
\(697\) −15.8431 −0.600100
\(698\) 49.2833 1.86540
\(699\) 11.6474 0.440546
\(700\) −0.103211 −0.00390103
\(701\) 33.9922 1.28387 0.641934 0.766760i \(-0.278132\pi\)
0.641934 + 0.766760i \(0.278132\pi\)
\(702\) −65.3497 −2.46647
\(703\) −16.2368 −0.612383
\(704\) 1.44265 0.0543719
\(705\) 8.82371 0.332320
\(706\) 41.6209 1.56642
\(707\) −21.7808 −0.819150
\(708\) 0.246980 0.00928206
\(709\) 35.1618 1.32053 0.660265 0.751033i \(-0.270444\pi\)
0.660265 + 0.751033i \(0.270444\pi\)
\(710\) 15.0858 0.566158
\(711\) 0.0881460 0.00330573
\(712\) 14.4088 0.539993
\(713\) −38.3376 −1.43576
\(714\) −13.7017 −0.512774
\(715\) −16.7168 −0.625172
\(716\) −2.05429 −0.0767726
\(717\) 9.03013 0.337236
\(718\) 0.459042 0.0171313
\(719\) 24.4983 0.913631 0.456816 0.889561i \(-0.348990\pi\)
0.456816 + 0.889561i \(0.348990\pi\)
\(720\) −16.0368 −0.597658
\(721\) −6.28382 −0.234022
\(722\) −1.36658 −0.0508590
\(723\) 0.137063 0.00509744
\(724\) 2.83877 0.105502
\(725\) 0.293504 0.0109005
\(726\) 21.8116 0.809505
\(727\) 38.9855 1.44589 0.722947 0.690904i \(-0.242787\pi\)
0.722947 + 0.690904i \(0.242787\pi\)
\(728\) 15.0218 0.556744
\(729\) 24.4590 0.905890
\(730\) −5.80194 −0.214739
\(731\) 0 0
\(732\) 1.10992 0.0410237
\(733\) 5.86294 0.216553 0.108276 0.994121i \(-0.465467\pi\)
0.108276 + 0.994121i \(0.465467\pi\)
\(734\) 26.6853 0.984973
\(735\) 11.5918 0.427570
\(736\) −22.9734 −0.846812
\(737\) 2.38537 0.0878663
\(738\) 11.4470 0.421368
\(739\) 22.5840 0.830767 0.415383 0.909646i \(-0.363647\pi\)
0.415383 + 0.909646i \(0.363647\pi\)
\(740\) 10.2349 0.376242
\(741\) −36.2664 −1.33228
\(742\) 33.9855 1.24765
\(743\) 44.8877 1.64677 0.823385 0.567483i \(-0.192083\pi\)
0.823385 + 0.567483i \(0.192083\pi\)
\(744\) 17.4668 0.640364
\(745\) −13.0368 −0.477633
\(746\) 40.4935 1.48257
\(747\) 17.7506 0.649461
\(748\) 5.10992 0.186837
\(749\) −8.54288 −0.312150
\(750\) −24.9976 −0.912784
\(751\) 0.476501 0.0173878 0.00869388 0.999962i \(-0.497233\pi\)
0.00869388 + 0.999962i \(0.497233\pi\)
\(752\) 15.5536 0.567183
\(753\) −34.7875 −1.26773
\(754\) 70.7391 2.57617
\(755\) 16.7385 0.609178
\(756\) 11.6950 0.425343
\(757\) 10.7855 0.392007 0.196004 0.980603i \(-0.437204\pi\)
0.196004 + 0.980603i \(0.437204\pi\)
\(758\) 46.9420 1.70501
\(759\) 5.26577 0.191135
\(760\) −13.5526 −0.491603
\(761\) −8.87023 −0.321546 −0.160773 0.986991i \(-0.551399\pi\)
−0.160773 + 0.986991i \(0.551399\pi\)
\(762\) 45.4403 1.64613
\(763\) −3.68233 −0.133309
\(764\) 18.7439 0.678131
\(765\) −11.7017 −0.423076
\(766\) 22.2368 0.803449
\(767\) −1.03923 −0.0375244
\(768\) 25.8267 0.931940
\(769\) 38.7928 1.39891 0.699453 0.714679i \(-0.253427\pi\)
0.699453 + 0.714679i \(0.253427\pi\)
\(770\) 7.78986 0.280727
\(771\) 3.32304 0.119677
\(772\) 12.2838 0.442104
\(773\) −44.4730 −1.59958 −0.799792 0.600277i \(-0.795057\pi\)
−0.799792 + 0.600277i \(0.795057\pi\)
\(774\) 0 0
\(775\) 0.504976 0.0181393
\(776\) 4.75409 0.170662
\(777\) 7.70709 0.276490
\(778\) −35.8834 −1.28648
\(779\) 19.5410 0.700127
\(780\) 22.8605 0.818539
\(781\) −4.23655 −0.151596
\(782\) −24.1172 −0.862430
\(783\) −33.2573 −1.18852
\(784\) 20.4330 0.729749
\(785\) 29.6896 1.05967
\(786\) 4.62565 0.164991
\(787\) −10.6243 −0.378716 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(788\) 13.6679 0.486897
\(789\) 30.0954 1.07143
\(790\) −0.246980 −0.00878714
\(791\) −19.8750 −0.706674
\(792\) 2.22952 0.0792226
\(793\) −4.67025 −0.165845
\(794\) −16.4426 −0.583528
\(795\) −31.2325 −1.10770
\(796\) 30.3696 1.07642
\(797\) 0.528402 0.0187170 0.00935848 0.999956i \(-0.497021\pi\)
0.00935848 + 0.999956i \(0.497021\pi\)
\(798\) 16.8998 0.598245
\(799\) 11.3491 0.401503
\(800\) 0.302602 0.0106986
\(801\) −15.3448 −0.542182
\(802\) −0.0784573 −0.00277042
\(803\) 1.62937 0.0574991
\(804\) −3.26205 −0.115043
\(805\) −14.1196 −0.497651
\(806\) 121.707 4.28695
\(807\) 37.5978 1.32350
\(808\) 17.4668 0.614480
\(809\) 10.7127 0.376639 0.188320 0.982108i \(-0.439696\pi\)
0.188320 + 0.982108i \(0.439696\pi\)
\(810\) −10.4330 −0.366577
\(811\) 52.4782 1.84276 0.921379 0.388666i \(-0.127064\pi\)
0.921379 + 0.388666i \(0.127064\pi\)
\(812\) −12.6595 −0.444261
\(813\) −17.0737 −0.598800
\(814\) −7.48427 −0.262324
\(815\) −31.8756 −1.11655
\(816\) 22.1957 0.777004
\(817\) 0 0
\(818\) −19.2567 −0.673294
\(819\) −15.9976 −0.559002
\(820\) −12.3177 −0.430152
\(821\) 14.1032 0.492205 0.246103 0.969244i \(-0.420850\pi\)
0.246103 + 0.969244i \(0.420850\pi\)
\(822\) 35.0562 1.22273
\(823\) 45.8146 1.59700 0.798498 0.601997i \(-0.205628\pi\)
0.798498 + 0.601997i \(0.205628\pi\)
\(824\) 5.03923 0.175550
\(825\) −0.0693596 −0.00241479
\(826\) 0.484271 0.0168499
\(827\) −38.4655 −1.33758 −0.668788 0.743453i \(-0.733187\pi\)
−0.668788 + 0.743453i \(0.733187\pi\)
\(828\) 6.69202 0.232564
\(829\) −19.2849 −0.669792 −0.334896 0.942255i \(-0.608701\pi\)
−0.334896 + 0.942255i \(0.608701\pi\)
\(830\) −49.7362 −1.72637
\(831\) 23.1618 0.803475
\(832\) 8.30127 0.287795
\(833\) 14.9095 0.516582
\(834\) 15.8998 0.550564
\(835\) −21.2838 −0.736557
\(836\) −6.30260 −0.217980
\(837\) −57.2194 −1.97779
\(838\) −13.3002 −0.459448
\(839\) 5.90946 0.204017 0.102009 0.994784i \(-0.467473\pi\)
0.102009 + 0.994784i \(0.467473\pi\)
\(840\) 6.43296 0.221958
\(841\) 7.00000 0.241379
\(842\) 17.6450 0.608088
\(843\) −3.38703 −0.116655
\(844\) −12.8576 −0.442575
\(845\) −66.9807 −2.30421
\(846\) −8.19998 −0.281921
\(847\) 16.4246 0.564356
\(848\) −55.0538 −1.89056
\(849\) 21.2137 0.728053
\(850\) 0.317667 0.0108959
\(851\) 13.5657 0.465027
\(852\) 5.79358 0.198485
\(853\) 27.6021 0.945077 0.472538 0.881310i \(-0.343338\pi\)
0.472538 + 0.881310i \(0.343338\pi\)
\(854\) 2.17629 0.0744712
\(855\) 14.4330 0.493597
\(856\) 6.85086 0.234157
\(857\) 17.0683 0.583042 0.291521 0.956564i \(-0.405839\pi\)
0.291521 + 0.956564i \(0.405839\pi\)
\(858\) −16.7168 −0.570701
\(859\) −4.80838 −0.164060 −0.0820299 0.996630i \(-0.526140\pi\)
−0.0820299 + 0.996630i \(0.526140\pi\)
\(860\) 0 0
\(861\) −9.27545 −0.316107
\(862\) −49.2465 −1.67734
\(863\) −54.3909 −1.85149 −0.925743 0.378153i \(-0.876559\pi\)
−0.925743 + 0.378153i \(0.876559\pi\)
\(864\) −34.2881 −1.16651
\(865\) −10.7017 −0.363869
\(866\) −5.00538 −0.170090
\(867\) −5.00298 −0.169910
\(868\) −21.7808 −0.739287
\(869\) 0.0693596 0.00235286
\(870\) 30.2935 1.02705
\(871\) 13.7259 0.465083
\(872\) 2.95300 0.100001
\(873\) −5.06292 −0.171354
\(874\) 29.7463 1.00618
\(875\) −18.8237 −0.636357
\(876\) −2.22819 −0.0752837
\(877\) −13.4940 −0.455659 −0.227829 0.973701i \(-0.573163\pi\)
−0.227829 + 0.973701i \(0.573163\pi\)
\(878\) −17.2664 −0.582711
\(879\) −13.7560 −0.463979
\(880\) −12.6189 −0.425384
\(881\) −35.3564 −1.19119 −0.595594 0.803286i \(-0.703083\pi\)
−0.595594 + 0.803286i \(0.703083\pi\)
\(882\) −10.7724 −0.362726
\(883\) 53.6902 1.80682 0.903410 0.428778i \(-0.141056\pi\)
0.903410 + 0.428778i \(0.141056\pi\)
\(884\) 29.4034 0.988944
\(885\) −0.445042 −0.0149599
\(886\) −42.8418 −1.43930
\(887\) 9.72156 0.326418 0.163209 0.986592i \(-0.447816\pi\)
0.163209 + 0.986592i \(0.447816\pi\)
\(888\) −6.18060 −0.207407
\(889\) 34.2174 1.14762
\(890\) 42.9952 1.44120
\(891\) 2.92990 0.0981555
\(892\) −3.53319 −0.118300
\(893\) −13.9981 −0.468428
\(894\) −13.0368 −0.436017
\(895\) 3.70171 0.123735
\(896\) 17.0653 0.570112
\(897\) 30.3002 1.01169
\(898\) −33.0659 −1.10342
\(899\) 61.9383 2.06576
\(900\) −0.0881460 −0.00293820
\(901\) −40.1715 −1.33831
\(902\) 9.00730 0.299910
\(903\) 0 0
\(904\) 15.9385 0.530108
\(905\) −5.11529 −0.170038
\(906\) 16.7385 0.556101
\(907\) −28.8455 −0.957798 −0.478899 0.877870i \(-0.658964\pi\)
−0.478899 + 0.877870i \(0.658964\pi\)
\(908\) 16.2905 0.540620
\(909\) −18.6015 −0.616972
\(910\) 44.8243 1.48591
\(911\) −14.3274 −0.474686 −0.237343 0.971426i \(-0.576277\pi\)
−0.237343 + 0.971426i \(0.576277\pi\)
\(912\) −27.3763 −0.906519
\(913\) 13.9675 0.462256
\(914\) 71.7381 2.37288
\(915\) −2.00000 −0.0661180
\(916\) −22.3411 −0.738171
\(917\) 3.48321 0.115026
\(918\) −35.9952 −1.18802
\(919\) 42.9077 1.41539 0.707697 0.706517i \(-0.249734\pi\)
0.707697 + 0.706517i \(0.249734\pi\)
\(920\) 11.3230 0.373310
\(921\) 19.6963 0.649016
\(922\) 34.6515 1.14119
\(923\) −24.3779 −0.802409
\(924\) 2.99164 0.0984177
\(925\) −0.178685 −0.00587512
\(926\) 26.9541 0.885766
\(927\) −5.36658 −0.176262
\(928\) 37.1159 1.21839
\(929\) 19.1919 0.629667 0.314834 0.949147i \(-0.398051\pi\)
0.314834 + 0.949147i \(0.398051\pi\)
\(930\) 52.1202 1.70909
\(931\) −18.3894 −0.602689
\(932\) 11.6474 0.381524
\(933\) −15.4209 −0.504857
\(934\) 46.4728 1.52064
\(935\) −9.20775 −0.301126
\(936\) 12.8291 0.419332
\(937\) −35.0767 −1.14590 −0.572952 0.819589i \(-0.694202\pi\)
−0.572952 + 0.819589i \(0.694202\pi\)
\(938\) −6.39612 −0.208841
\(939\) 7.27545 0.237425
\(940\) 8.82371 0.287798
\(941\) −8.92048 −0.290799 −0.145400 0.989373i \(-0.546447\pi\)
−0.145400 + 0.989373i \(0.546447\pi\)
\(942\) 29.6896 0.967340
\(943\) −16.3263 −0.531657
\(944\) −0.784479 −0.0255326
\(945\) −21.0737 −0.685527
\(946\) 0 0
\(947\) 27.3672 0.889314 0.444657 0.895701i \(-0.353326\pi\)
0.444657 + 0.895701i \(0.353326\pi\)
\(948\) −0.0948508 −0.00308061
\(949\) 9.37568 0.304348
\(950\) −0.391813 −0.0127121
\(951\) 32.6316 1.05815
\(952\) 8.27413 0.268166
\(953\) 52.1221 1.68840 0.844200 0.536028i \(-0.180076\pi\)
0.844200 + 0.536028i \(0.180076\pi\)
\(954\) 29.0248 0.939711
\(955\) −33.7754 −1.09295
\(956\) 9.03013 0.292055
\(957\) −8.50737 −0.275004
\(958\) 53.5459 1.72999
\(959\) 26.3980 0.852437
\(960\) 3.55496 0.114736
\(961\) 75.5652 2.43759
\(962\) −43.0659 −1.38850
\(963\) −7.29590 −0.235107
\(964\) 0.137063 0.00441451
\(965\) −22.1347 −0.712540
\(966\) −14.1196 −0.454291
\(967\) 10.1618 0.326782 0.163391 0.986561i \(-0.447757\pi\)
0.163391 + 0.986561i \(0.447757\pi\)
\(968\) −13.1715 −0.423348
\(969\) −19.9758 −0.641716
\(970\) 14.1860 0.455484
\(971\) 7.46250 0.239483 0.119742 0.992805i \(-0.461793\pi\)
0.119742 + 0.992805i \(0.461793\pi\)
\(972\) 16.7289 0.536578
\(973\) 11.9729 0.383832
\(974\) −2.74094 −0.0878253
\(975\) −0.399108 −0.0127817
\(976\) −3.52542 −0.112846
\(977\) −2.22414 −0.0711567 −0.0355783 0.999367i \(-0.511327\pi\)
−0.0355783 + 0.999367i \(0.511327\pi\)
\(978\) −31.8756 −1.01927
\(979\) −12.0744 −0.385900
\(980\) 11.5918 0.370286
\(981\) −3.14483 −0.100407
\(982\) 51.6491 1.64819
\(983\) 21.5050 0.685902 0.342951 0.939353i \(-0.388573\pi\)
0.342951 + 0.939353i \(0.388573\pi\)
\(984\) 7.43834 0.237126
\(985\) −24.6286 −0.784733
\(986\) 38.9638 1.24086
\(987\) 6.64443 0.211495
\(988\) −36.2664 −1.15379
\(989\) 0 0
\(990\) 6.65279 0.211440
\(991\) −50.7900 −1.61340 −0.806698 0.590963i \(-0.798748\pi\)
−0.806698 + 0.590963i \(0.798748\pi\)
\(992\) 63.8582 2.02750
\(993\) −22.7778 −0.722831
\(994\) 11.3599 0.360314
\(995\) −54.7241 −1.73487
\(996\) −19.1008 −0.605233
\(997\) 1.00059 0.0316890 0.0158445 0.999874i \(-0.494956\pi\)
0.0158445 + 0.999874i \(0.494956\pi\)
\(998\) −37.5090 −1.18733
\(999\) 20.2470 0.640586
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 1849.2.a.j.1.1 3
43.8 odd 14 43.2.e.a.21.1 6
43.27 odd 14 43.2.e.a.41.1 yes 6
43.42 odd 2 1849.2.a.k.1.3 3
129.8 even 14 387.2.u.c.64.1 6
129.113 even 14 387.2.u.c.127.1 6
172.27 even 14 688.2.u.b.385.1 6
172.51 even 14 688.2.u.b.193.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.2.e.a.21.1 6 43.8 odd 14
43.2.e.a.41.1 yes 6 43.27 odd 14
387.2.u.c.64.1 6 129.8 even 14
387.2.u.c.127.1 6 129.113 even 14
688.2.u.b.193.1 6 172.51 even 14
688.2.u.b.385.1 6 172.27 even 14
1849.2.a.j.1.1 3 1.1 even 1 trivial
1849.2.a.k.1.3 3 43.42 odd 2