Properties

Label 1849.2.a.k.1.3
Level $1849$
Weight $2$
Character 1849.1
Self dual yes
Analytic conductor $14.764$
Analytic rank $1$
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: \(1\)
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 1.3
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 \(-0.280142\pi\)
0.637081 + 0.770797i \(0.280142\pi\)
\(3\) −1.24698 −0.719944 −0.359972 0.932963i \(-0.617214\pi\)
−0.359972 + 0.932963i \(0.617214\pi\)
\(4\) 1.24698 0.623490
\(5\) 2.24698 1.00488 0.502440 0.864612i \(-0.332436\pi\)
0.502440 + 0.864612i \(0.332436\pi\)
\(6\) −2.24698 −0.917326
\(7\) 1.69202 0.639524 0.319762 0.947498i \(-0.396397\pi\)
0.319762 + 0.947498i \(0.396397\pi\)
\(8\) −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 \(-0.670311\pi\)
−0.509881 + 0.860245i \(0.670311\pi\)
\(20\) 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 \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −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 \(-0.402927\pi\)
0.300258 + 0.953858i \(0.402927\pi\)
\(38\) −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 \(-0.514551\pi\)
−0.0456958 + 0.998955i \(0.514551\pi\)
\(62\) 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 \(-0.570961\pi\)
−0.221090 + 0.975253i \(0.570961\pi\)
\(72\) 1.96077 0.231079
\(73\) 1.43296 0.167715 0.0838577 0.996478i \(-0.473276\pi\)
0.0838577 + 0.996478i \(0.473276\pi\)
\(74\) 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 \(-0.690277\pi\)
−0.562803 + 0.826591i \(0.690277\pi\)
\(90\) −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 \(-0.686326\pi\)
−0.552500 + 0.833513i \(0.686326\pi\)
\(114\) 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 \(-0.471335\pi\)
0.0899306 + 0.995948i \(0.471335\pi\)
\(132\) 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 \(-0.267808\pi\)
0.666462 + 0.745539i \(0.267808\pi\)
\(138\) 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 \(-0.576379\pi\)
−0.237657 + 0.971349i \(0.576379\pi\)
\(150\) −0.109916 −0.00897463
\(151\) 7.44935 0.606220 0.303110 0.952956i \(-0.401975\pi\)
0.303110 + 0.952956i \(0.401975\pi\)
\(152\) 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 \(-0.323219\pi\)
0.527261 + 0.849703i \(0.323219\pi\)
\(158\) −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 \(-0.687498\pi\)
−0.555566 + 0.831473i \(0.687498\pi\)
\(164\) 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 \(-0.480390\pi\)
0.0615668 + 0.998103i \(0.480390\pi\)
\(180\) −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 \(-0.683022\pi\)
−0.543819 + 0.839202i \(0.683022\pi\)
\(192\) 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 \(-0.831558\pi\)
−0.863222 + 0.504824i \(0.831558\pi\)
\(200\) −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 \(-0.384509\pi\)
0.354918 + 0.934897i \(0.384509\pi\)
\(212\) 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 \(-0.469757\pi\)
0.0948691 + 0.995490i \(0.469757\pi\)
\(224\) −10.4668 −0.699343
\(225\) −0.0706876 −0.00471251
\(226\) −21.1661 −1.40795
\(227\) −13.0640 −0.867087 −0.433543 0.901133i \(-0.642737\pi\)
−0.433543 + 0.901133i \(0.642737\pi\)
\(228\) 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 \(-0.598977\pi\)
−0.305958 + 0.952045i \(0.598977\pi\)
\(234\) 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 \(-0.501127\pi\)
−0.00354016 + 0.999994i \(0.501127\pi\)
\(242\) −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 \(-0.526487\pi\)
−0.0831151 + 0.996540i \(0.526487\pi\)
\(258\) 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 \(-0.767123\pi\)
−0.744104 + 0.668064i \(0.767123\pi\)
\(264\) −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 \(-0.688436\pi\)
−0.558012 + 0.829833i \(0.688436\pi\)
\(278\) −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 \(-0.552727\pi\)
−0.164892 + 0.986312i \(0.552727\pi\)
\(314\) 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 \(-0.332596\pi\)
0.502005 + 0.864865i \(0.332596\pi\)
\(332\) −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 \(-0.527498\pi\)
−0.0862805 + 0.996271i \(0.527498\pi\)
\(348\) 9.32975 0.500127
\(349\) 27.3502 1.46402 0.732011 0.681293i \(-0.238582\pi\)
0.732011 + 0.681293i \(0.238582\pi\)
\(350\) 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 \(-0.302355\pi\)
0.581783 + 0.813344i \(0.302355\pi\)
\(374\) 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 \(-0.397900\pi\)
0.315285 + 0.948997i \(0.397900\pi\)
\(384\) −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 \(-0.668446\pi\)
−0.504834 + 0.863216i \(0.668446\pi\)
\(390\) 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 \(-0.585111\pi\)
−0.264210 + 0.964465i \(0.585111\pi\)
\(410\) 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 \(-0.557705\pi\)
−0.180294 + 0.983613i \(0.557705\pi\)
\(420\) −5.91185 −0.288469
\(421\) 9.79225 0.477245 0.238623 0.971112i \(-0.423304\pi\)
0.238623 + 0.971112i \(0.423304\pi\)
\(422\) 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 \(-0.521262\pi\)
−0.0667457 + 0.997770i \(0.521262\pi\)
\(434\) 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 \(-0.642545\pi\)
−0.433000 + 0.901394i \(0.642545\pi\)
\(450\) −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 \(-0.118803\pi\)
0.931155 + 0.364624i \(0.118803\pi\)
\(458\) −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 \(-0.387001\pi\)
0.347587 + 0.937648i \(0.387001\pi\)
\(464\) 29.6340 1.37572
\(465\) −28.9245 −1.34134
\(466\) −16.8310 −0.779681
\(467\) 25.7904 1.19344 0.596720 0.802450i \(-0.296470\pi\)
0.596720 + 0.802450i \(0.296470\pi\)
\(468\) 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 \(-0.276118\pi\)
0.646773 + 0.762682i \(0.276118\pi\)
\(492\) −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 \(-0.654278\pi\)
−0.465925 + 0.884824i \(0.654278\pi\)
\(500\) −13.8726 −0.620403
\(501\) −11.8116 −0.527705
\(502\) −50.2693 −2.24363
\(503\) 3.65087 0.162784 0.0813922 0.996682i \(-0.474063\pi\)
0.0813922 + 0.996682i \(0.474063\pi\)
\(504\) 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 \(-0.500897\pi\)
−0.00281928 + 0.999996i \(0.500897\pi\)
\(522\) 15.6233 0.683811
\(523\) −21.6209 −0.945414 −0.472707 0.881220i \(-0.656723\pi\)
−0.472707 + 0.881220i \(0.656723\pi\)
\(524\) 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 \(-0.781030\pi\)
−0.772572 + 0.634928i \(0.781030\pi\)
\(572\) 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 \(-0.680380\pi\)
−0.536835 + 0.843687i \(0.680380\pi\)
\(578\) −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 \(-0.437688\pi\)
0.194510 + 0.980901i \(0.437688\pi\)
\(588\) 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 \(-0.557288\pi\)
−0.179005 + 0.983848i \(0.557288\pi\)
\(594\) −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 \(-0.589226\pi\)
−0.276657 + 0.960969i \(0.589226\pi\)
\(602\) 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 \(-0.709883\pi\)
−0.612618 + 0.790379i \(0.709883\pi\)
\(608\) 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 \(-0.402696\pi\)
0.300951 + 0.953640i \(0.402696\pi\)
\(632\) 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 \(-0.478566\pi\)
0.0672870 + 0.997734i \(0.478566\pi\)
\(642\) −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 \(-0.604850\pi\)
−0.323472 + 0.946238i \(0.604850\pi\)
\(648\) 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 \(-0.270750\pi\)
0.659542 + 0.751668i \(0.270750\pi\)
\(654\) −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 \(-0.390514\pi\)
0.337217 + 0.941427i \(0.390514\pi\)
\(674\) −12.7778 −0.492181
\(675\) 0.271143 0.0104363
\(676\) 37.1715 1.42967
\(677\) −29.4252 −1.13090 −0.565451 0.824782i \(-0.691298\pi\)
−0.565451 + 0.824782i \(0.691298\pi\)
\(678\) 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 \(-0.492350\pi\)
0.0240305 + 0.999711i \(0.492350\pi\)
\(692\) 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 \(-0.757213\pi\)
−0.722947 + 0.690904i \(0.757213\pi\)
\(728\) 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 \(-0.534533\pi\)
−0.108276 + 0.994121i \(0.534533\pi\)
\(734\) −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 \(-0.636353\pi\)
−0.415383 + 0.909646i \(0.636353\pi\)
\(740\) 10.2349 0.376242
\(741\) −36.2664 −1.33228
\(742\) 33.9855 1.24765
\(743\) −44.8877 −1.64677 −0.823385 0.567483i \(-0.807917\pi\)
−0.823385 + 0.567483i \(0.807917\pi\)
\(744\) 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 \(-0.502767\pi\)
−0.00869388 + 0.999962i \(0.502767\pi\)
\(752\) 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 \(-0.562796\pi\)
−0.196004 + 0.980603i \(0.562796\pi\)
\(758\) −46.9420 −1.70501
\(759\) −5.26577 −0.191135
\(760\) 13.5526 0.491603
\(761\) 8.87023 0.321546 0.160773 0.986991i \(-0.448601\pi\)
0.160773 + 0.986991i \(0.448601\pi\)
\(762\) 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 \(-0.204943\pi\)
0.799792 + 0.600277i \(0.204943\pi\)
\(774\) 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 \(-0.872936\pi\)
−0.921379 + 0.388666i \(0.872936\pi\)
\(812\) −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 \(-0.391299\pi\)
0.334896 + 0.942255i \(0.391299\pi\)
\(830\) −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 \(-0.532527\pi\)
−0.102009 + 0.994784i \(0.532527\pi\)
\(840\) 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 \(-0.473860\pi\)
0.0820299 + 0.996630i \(0.473860\pi\)
\(860\) 0 0
\(861\) −9.27545 −0.316107
\(862\) 49.2465 1.67734
\(863\) 54.3909 1.85149 0.925743 0.378153i \(-0.123441\pi\)
0.925743 + 0.378153i \(0.123441\pi\)
\(864\) −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 \(-0.552184\pi\)
−0.163209 + 0.986592i \(0.552184\pi\)
\(888\) 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 \(-0.423723\pi\)
0.237343 + 0.971426i \(0.423723\pi\)
\(912\) −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 \(-0.601949\pi\)
−0.314834 + 0.949147i \(0.601949\pi\)
\(930\) −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 \(-0.305798\pi\)
0.572952 + 0.819589i \(0.305798\pi\)
\(938\) −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 \(-0.819924\pi\)
−0.844200 + 0.536028i \(0.819924\pi\)
\(954\) −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 \(-0.611427\pi\)
−0.342951 + 0.939353i \(0.611427\pi\)
\(984\) 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 \(-0.201252\pi\)
0.806698 + 0.590963i \(0.201252\pi\)
\(992\) −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 \(-0.505044\pi\)
−0.0158445 + 0.999874i \(0.505044\pi\)
\(998\) −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.k.1.3 3
43.16 even 7 43.2.e.a.41.1 yes 6
43.35 even 7 43.2.e.a.21.1 6
43.42 odd 2 1849.2.a.j.1.1 3
129.35 odd 14 387.2.u.c.64.1 6
129.59 odd 14 387.2.u.c.127.1 6
172.35 odd 14 688.2.u.b.193.1 6
172.59 odd 14 688.2.u.b.385.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.2.e.a.21.1 6 43.35 even 7
43.2.e.a.41.1 yes 6 43.16 even 7
387.2.u.c.64.1 6 129.35 odd 14
387.2.u.c.127.1 6 129.59 odd 14
688.2.u.b.193.1 6 172.35 odd 14
688.2.u.b.385.1 6 172.59 odd 14
1849.2.a.j.1.1 3 43.42 odd 2
1849.2.a.k.1.3 3 1.1 even 1 trivial