Properties

Label 1183.2.a.j.1.2
Level $1183$
Weight $2$
Character 1183.1
Self dual yes
Analytic conductor $9.446$
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] = [1183,2,Mod(1,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 1183.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.688892 q^{2} -2.21432 q^{3} -1.52543 q^{4} +3.21432 q^{5} -1.52543 q^{6} -1.00000 q^{7} -2.42864 q^{8} +1.90321 q^{9} +O(q^{10})\) \(q+0.688892 q^{2} -2.21432 q^{3} -1.52543 q^{4} +3.21432 q^{5} -1.52543 q^{6} -1.00000 q^{7} -2.42864 q^{8} +1.90321 q^{9} +2.21432 q^{10} +2.68889 q^{11} +3.37778 q^{12} -0.688892 q^{14} -7.11753 q^{15} +1.37778 q^{16} +3.59210 q^{17} +1.31111 q^{18} -8.54617 q^{19} -4.90321 q^{20} +2.21432 q^{21} +1.85236 q^{22} -3.28100 q^{23} +5.37778 q^{24} +5.33185 q^{25} +2.42864 q^{27} +1.52543 q^{28} +2.05086 q^{29} -4.90321 q^{30} +5.83654 q^{31} +5.80642 q^{32} -5.95407 q^{33} +2.47457 q^{34} -3.21432 q^{35} -2.90321 q^{36} +3.93332 q^{37} -5.88739 q^{38} -7.80642 q^{40} +0.755569 q^{41} +1.52543 q^{42} +8.80642 q^{43} -4.10171 q^{44} +6.11753 q^{45} -2.26025 q^{46} +1.88247 q^{47} -3.05086 q^{48} +1.00000 q^{49} +3.67307 q^{50} -7.95407 q^{51} +2.52543 q^{53} +1.67307 q^{54} +8.64296 q^{55} +2.42864 q^{56} +18.9240 q^{57} +1.41282 q^{58} +7.33185 q^{59} +10.8573 q^{60} +9.05086 q^{61} +4.02074 q^{62} -1.90321 q^{63} +1.24443 q^{64} -4.10171 q^{66} +0.428639 q^{67} -5.47949 q^{68} +7.26517 q^{69} -2.21432 q^{70} +8.98418 q^{71} -4.62222 q^{72} +5.79060 q^{73} +2.70964 q^{74} -11.8064 q^{75} +13.0366 q^{76} -2.68889 q^{77} -4.47949 q^{79} +4.42864 q^{80} -11.0874 q^{81} +0.520505 q^{82} -10.8272 q^{83} -3.37778 q^{84} +11.5462 q^{85} +6.06668 q^{86} -4.54125 q^{87} -6.53035 q^{88} +5.36196 q^{89} +4.21432 q^{90} +5.00492 q^{92} -12.9240 q^{93} +1.29682 q^{94} -27.4701 q^{95} -12.8573 q^{96} +9.62867 q^{97} +0.688892 q^{98} +5.11753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 3 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 3 q^{7} + 6 q^{8} - q^{9} + 8 q^{11} + 10 q^{12} - 2 q^{14} - 8 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + q^{19} - 8 q^{20} + 12 q^{22} - 3 q^{23} + 16 q^{24} - 4 q^{25} - 6 q^{27} - 2 q^{28} - 7 q^{29} - 8 q^{30} + 11 q^{31} + 4 q^{32} + 2 q^{33} + 14 q^{34} - 3 q^{35} - 2 q^{36} + 12 q^{37} + 2 q^{38} - 10 q^{40} + 2 q^{41} - 2 q^{42} + 13 q^{43} + 14 q^{44} + 5 q^{45} - 20 q^{46} + 19 q^{47} + 4 q^{48} + 3 q^{49} - 2 q^{50} - 4 q^{51} + q^{53} - 8 q^{54} + 6 q^{55} - 6 q^{56} + 30 q^{57} - 22 q^{58} + 2 q^{59} + 6 q^{60} + 14 q^{61} - 8 q^{62} + q^{63} + 4 q^{64} + 14 q^{66} - 12 q^{67} + 10 q^{68} + 2 q^{69} + 14 q^{71} - 14 q^{72} - 9 q^{73} - 12 q^{74} - 22 q^{75} + 32 q^{76} - 8 q^{77} + 13 q^{79} - 13 q^{81} + 28 q^{82} + q^{83} - 10 q^{84} + 8 q^{85} + 18 q^{86} - 20 q^{87} + 20 q^{88} + 3 q^{89} + 6 q^{90} - 18 q^{92} - 12 q^{93} + 10 q^{94} - 29 q^{95} - 12 q^{96} + 15 q^{97} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.688892 0.487120 0.243560 0.969886i \(-0.421685\pi\)
0.243560 + 0.969886i \(0.421685\pi\)
\(3\) −2.21432 −1.27844 −0.639219 0.769025i \(-0.720742\pi\)
−0.639219 + 0.769025i \(0.720742\pi\)
\(4\) −1.52543 −0.762714
\(5\) 3.21432 1.43749 0.718744 0.695275i \(-0.244717\pi\)
0.718744 + 0.695275i \(0.244717\pi\)
\(6\) −1.52543 −0.622753
\(7\) −1.00000 −0.377964
\(8\) −2.42864 −0.858654
\(9\) 1.90321 0.634404
\(10\) 2.21432 0.700229
\(11\) 2.68889 0.810731 0.405366 0.914155i \(-0.367144\pi\)
0.405366 + 0.914155i \(0.367144\pi\)
\(12\) 3.37778 0.975082
\(13\) 0 0
\(14\) −0.688892 −0.184114
\(15\) −7.11753 −1.83774
\(16\) 1.37778 0.344446
\(17\) 3.59210 0.871213 0.435607 0.900137i \(-0.356534\pi\)
0.435607 + 0.900137i \(0.356534\pi\)
\(18\) 1.31111 0.309031
\(19\) −8.54617 −1.96063 −0.980313 0.197449i \(-0.936734\pi\)
−0.980313 + 0.197449i \(0.936734\pi\)
\(20\) −4.90321 −1.09639
\(21\) 2.21432 0.483204
\(22\) 1.85236 0.394924
\(23\) −3.28100 −0.684135 −0.342068 0.939675i \(-0.611127\pi\)
−0.342068 + 0.939675i \(0.611127\pi\)
\(24\) 5.37778 1.09774
\(25\) 5.33185 1.06637
\(26\) 0 0
\(27\) 2.42864 0.467392
\(28\) 1.52543 0.288279
\(29\) 2.05086 0.380834 0.190417 0.981703i \(-0.439016\pi\)
0.190417 + 0.981703i \(0.439016\pi\)
\(30\) −4.90321 −0.895200
\(31\) 5.83654 1.04827 0.524136 0.851634i \(-0.324388\pi\)
0.524136 + 0.851634i \(0.324388\pi\)
\(32\) 5.80642 1.02644
\(33\) −5.95407 −1.03647
\(34\) 2.47457 0.424386
\(35\) −3.21432 −0.543319
\(36\) −2.90321 −0.483869
\(37\) 3.93332 0.646634 0.323317 0.946291i \(-0.395202\pi\)
0.323317 + 0.946291i \(0.395202\pi\)
\(38\) −5.88739 −0.955061
\(39\) 0 0
\(40\) −7.80642 −1.23430
\(41\) 0.755569 0.118000 0.0590000 0.998258i \(-0.481209\pi\)
0.0590000 + 0.998258i \(0.481209\pi\)
\(42\) 1.52543 0.235379
\(43\) 8.80642 1.34297 0.671484 0.741019i \(-0.265657\pi\)
0.671484 + 0.741019i \(0.265657\pi\)
\(44\) −4.10171 −0.618356
\(45\) 6.11753 0.911948
\(46\) −2.26025 −0.333256
\(47\) 1.88247 0.274586 0.137293 0.990530i \(-0.456160\pi\)
0.137293 + 0.990530i \(0.456160\pi\)
\(48\) −3.05086 −0.440353
\(49\) 1.00000 0.142857
\(50\) 3.67307 0.519451
\(51\) −7.95407 −1.11379
\(52\) 0 0
\(53\) 2.52543 0.346894 0.173447 0.984843i \(-0.444509\pi\)
0.173447 + 0.984843i \(0.444509\pi\)
\(54\) 1.67307 0.227676
\(55\) 8.64296 1.16542
\(56\) 2.42864 0.324541
\(57\) 18.9240 2.50654
\(58\) 1.41282 0.185512
\(59\) 7.33185 0.954526 0.477263 0.878761i \(-0.341629\pi\)
0.477263 + 0.878761i \(0.341629\pi\)
\(60\) 10.8573 1.40167
\(61\) 9.05086 1.15884 0.579422 0.815028i \(-0.303278\pi\)
0.579422 + 0.815028i \(0.303278\pi\)
\(62\) 4.02074 0.510635
\(63\) −1.90321 −0.239782
\(64\) 1.24443 0.155554
\(65\) 0 0
\(66\) −4.10171 −0.504886
\(67\) 0.428639 0.0523666 0.0261833 0.999657i \(-0.491665\pi\)
0.0261833 + 0.999657i \(0.491665\pi\)
\(68\) −5.47949 −0.664486
\(69\) 7.26517 0.874624
\(70\) −2.21432 −0.264662
\(71\) 8.98418 1.06623 0.533113 0.846044i \(-0.321022\pi\)
0.533113 + 0.846044i \(0.321022\pi\)
\(72\) −4.62222 −0.544733
\(73\) 5.79060 0.677739 0.338869 0.940833i \(-0.389956\pi\)
0.338869 + 0.940833i \(0.389956\pi\)
\(74\) 2.70964 0.314989
\(75\) −11.8064 −1.36329
\(76\) 13.0366 1.49540
\(77\) −2.68889 −0.306428
\(78\) 0 0
\(79\) −4.47949 −0.503983 −0.251991 0.967730i \(-0.581085\pi\)
−0.251991 + 0.967730i \(0.581085\pi\)
\(80\) 4.42864 0.495137
\(81\) −11.0874 −1.23194
\(82\) 0.520505 0.0574802
\(83\) −10.8272 −1.18844 −0.594218 0.804304i \(-0.702538\pi\)
−0.594218 + 0.804304i \(0.702538\pi\)
\(84\) −3.37778 −0.368546
\(85\) 11.5462 1.25236
\(86\) 6.06668 0.654187
\(87\) −4.54125 −0.486873
\(88\) −6.53035 −0.696138
\(89\) 5.36196 0.568367 0.284183 0.958770i \(-0.408278\pi\)
0.284183 + 0.958770i \(0.408278\pi\)
\(90\) 4.21432 0.444228
\(91\) 0 0
\(92\) 5.00492 0.521799
\(93\) −12.9240 −1.34015
\(94\) 1.29682 0.133757
\(95\) −27.4701 −2.81838
\(96\) −12.8573 −1.31224
\(97\) 9.62867 0.977643 0.488822 0.872384i \(-0.337427\pi\)
0.488822 + 0.872384i \(0.337427\pi\)
\(98\) 0.688892 0.0695886
\(99\) 5.11753 0.514331
\(100\) −8.13335 −0.813335
\(101\) 13.6938 1.36259 0.681293 0.732011i \(-0.261418\pi\)
0.681293 + 0.732011i \(0.261418\pi\)
\(102\) −5.47949 −0.542551
\(103\) 12.2953 1.21149 0.605745 0.795659i \(-0.292875\pi\)
0.605745 + 0.795659i \(0.292875\pi\)
\(104\) 0 0
\(105\) 7.11753 0.694600
\(106\) 1.73975 0.168979
\(107\) −18.1891 −1.75841 −0.879205 0.476444i \(-0.841925\pi\)
−0.879205 + 0.476444i \(0.841925\pi\)
\(108\) −3.70471 −0.356486
\(109\) 8.36196 0.800931 0.400465 0.916312i \(-0.368848\pi\)
0.400465 + 0.916312i \(0.368848\pi\)
\(110\) 5.95407 0.567698
\(111\) −8.70964 −0.826682
\(112\) −1.37778 −0.130188
\(113\) −8.46520 −0.796339 −0.398170 0.917312i \(-0.630354\pi\)
−0.398170 + 0.917312i \(0.630354\pi\)
\(114\) 13.0366 1.22099
\(115\) −10.5462 −0.983436
\(116\) −3.12843 −0.290468
\(117\) 0 0
\(118\) 5.05086 0.464969
\(119\) −3.59210 −0.329288
\(120\) 17.2859 1.57798
\(121\) −3.76986 −0.342714
\(122\) 6.23506 0.564496
\(123\) −1.67307 −0.150856
\(124\) −8.90321 −0.799532
\(125\) 1.06668 0.0954065
\(126\) −1.31111 −0.116803
\(127\) 4.08742 0.362700 0.181350 0.983419i \(-0.441953\pi\)
0.181350 + 0.983419i \(0.441953\pi\)
\(128\) −10.7556 −0.950667
\(129\) −19.5002 −1.71690
\(130\) 0 0
\(131\) 5.93978 0.518961 0.259480 0.965748i \(-0.416449\pi\)
0.259480 + 0.965748i \(0.416449\pi\)
\(132\) 9.08250 0.790530
\(133\) 8.54617 0.741047
\(134\) 0.295286 0.0255089
\(135\) 7.80642 0.671870
\(136\) −8.72393 −0.748070
\(137\) 16.3620 1.39790 0.698948 0.715172i \(-0.253652\pi\)
0.698948 + 0.715172i \(0.253652\pi\)
\(138\) 5.00492 0.426047
\(139\) 3.03011 0.257011 0.128505 0.991709i \(-0.458982\pi\)
0.128505 + 0.991709i \(0.458982\pi\)
\(140\) 4.90321 0.414397
\(141\) −4.16839 −0.351041
\(142\) 6.18913 0.519380
\(143\) 0 0
\(144\) 2.62222 0.218518
\(145\) 6.59210 0.547444
\(146\) 3.98910 0.330140
\(147\) −2.21432 −0.182634
\(148\) −6.00000 −0.493197
\(149\) −14.5368 −1.19090 −0.595451 0.803392i \(-0.703026\pi\)
−0.595451 + 0.803392i \(0.703026\pi\)
\(150\) −8.13335 −0.664085
\(151\) −19.9748 −1.62553 −0.812764 0.582594i \(-0.802038\pi\)
−0.812764 + 0.582594i \(0.802038\pi\)
\(152\) 20.7556 1.68350
\(153\) 6.83654 0.552701
\(154\) −1.85236 −0.149267
\(155\) 18.7605 1.50688
\(156\) 0 0
\(157\) 7.39853 0.590467 0.295233 0.955425i \(-0.404603\pi\)
0.295233 + 0.955425i \(0.404603\pi\)
\(158\) −3.08589 −0.245500
\(159\) −5.59210 −0.443483
\(160\) 18.6637 1.47550
\(161\) 3.28100 0.258579
\(162\) −7.63804 −0.600101
\(163\) 2.32693 0.182259 0.0911296 0.995839i \(-0.470952\pi\)
0.0911296 + 0.995839i \(0.470952\pi\)
\(164\) −1.15257 −0.0900002
\(165\) −19.1383 −1.48991
\(166\) −7.45875 −0.578911
\(167\) −3.42219 −0.264817 −0.132408 0.991195i \(-0.542271\pi\)
−0.132408 + 0.991195i \(0.542271\pi\)
\(168\) −5.37778 −0.414905
\(169\) 0 0
\(170\) 7.95407 0.610049
\(171\) −16.2652 −1.24383
\(172\) −13.4336 −1.02430
\(173\) 8.27454 0.629102 0.314551 0.949241i \(-0.398146\pi\)
0.314551 + 0.949241i \(0.398146\pi\)
\(174\) −3.12843 −0.237166
\(175\) −5.33185 −0.403050
\(176\) 3.70471 0.279253
\(177\) −16.2351 −1.22030
\(178\) 3.69381 0.276863
\(179\) −16.2257 −1.21277 −0.606383 0.795173i \(-0.707380\pi\)
−0.606383 + 0.795173i \(0.707380\pi\)
\(180\) −9.33185 −0.695555
\(181\) −9.20495 −0.684199 −0.342099 0.939664i \(-0.611138\pi\)
−0.342099 + 0.939664i \(0.611138\pi\)
\(182\) 0 0
\(183\) −20.0415 −1.48151
\(184\) 7.96836 0.587435
\(185\) 12.6430 0.929529
\(186\) −8.90321 −0.652815
\(187\) 9.65878 0.706320
\(188\) −2.87157 −0.209431
\(189\) −2.42864 −0.176658
\(190\) −18.9240 −1.37289
\(191\) −6.66815 −0.482490 −0.241245 0.970464i \(-0.577556\pi\)
−0.241245 + 0.970464i \(0.577556\pi\)
\(192\) −2.75557 −0.198866
\(193\) 20.6035 1.48307 0.741535 0.670914i \(-0.234098\pi\)
0.741535 + 0.670914i \(0.234098\pi\)
\(194\) 6.63311 0.476230
\(195\) 0 0
\(196\) −1.52543 −0.108959
\(197\) −8.36842 −0.596225 −0.298112 0.954531i \(-0.596357\pi\)
−0.298112 + 0.954531i \(0.596357\pi\)
\(198\) 3.52543 0.250541
\(199\) −0.601472 −0.0426372 −0.0213186 0.999773i \(-0.506786\pi\)
−0.0213186 + 0.999773i \(0.506786\pi\)
\(200\) −12.9491 −0.915643
\(201\) −0.949145 −0.0669475
\(202\) 9.43356 0.663743
\(203\) −2.05086 −0.143942
\(204\) 12.1334 0.849505
\(205\) 2.42864 0.169624
\(206\) 8.47013 0.590142
\(207\) −6.24443 −0.434018
\(208\) 0 0
\(209\) −22.9797 −1.58954
\(210\) 4.90321 0.338354
\(211\) −1.90321 −0.131023 −0.0655113 0.997852i \(-0.520868\pi\)
−0.0655113 + 0.997852i \(0.520868\pi\)
\(212\) −3.85236 −0.264581
\(213\) −19.8938 −1.36310
\(214\) −12.5303 −0.856557
\(215\) 28.3067 1.93050
\(216\) −5.89829 −0.401328
\(217\) −5.83654 −0.396210
\(218\) 5.76049 0.390150
\(219\) −12.8222 −0.866447
\(220\) −13.1842 −0.888879
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 10.6336 0.712078 0.356039 0.934471i \(-0.384127\pi\)
0.356039 + 0.934471i \(0.384127\pi\)
\(224\) −5.80642 −0.387958
\(225\) 10.1476 0.676510
\(226\) −5.83161 −0.387913
\(227\) 15.0509 0.998960 0.499480 0.866325i \(-0.333524\pi\)
0.499480 + 0.866325i \(0.333524\pi\)
\(228\) −28.8671 −1.91177
\(229\) −6.53480 −0.431831 −0.215916 0.976412i \(-0.569274\pi\)
−0.215916 + 0.976412i \(0.569274\pi\)
\(230\) −7.26517 −0.479051
\(231\) 5.95407 0.391749
\(232\) −4.98079 −0.327005
\(233\) −27.9590 −1.83165 −0.915827 0.401573i \(-0.868464\pi\)
−0.915827 + 0.401573i \(0.868464\pi\)
\(234\) 0 0
\(235\) 6.05086 0.394714
\(236\) −11.1842 −0.728030
\(237\) 9.91903 0.644310
\(238\) −2.47457 −0.160403
\(239\) −19.5812 −1.26660 −0.633301 0.773905i \(-0.718301\pi\)
−0.633301 + 0.773905i \(0.718301\pi\)
\(240\) −9.80642 −0.633002
\(241\) −13.3575 −0.860433 −0.430217 0.902726i \(-0.641563\pi\)
−0.430217 + 0.902726i \(0.641563\pi\)
\(242\) −2.59703 −0.166943
\(243\) 17.2652 1.10756
\(244\) −13.8064 −0.883866
\(245\) 3.21432 0.205355
\(246\) −1.15257 −0.0734849
\(247\) 0 0
\(248\) −14.1748 −0.900103
\(249\) 23.9748 1.51934
\(250\) 0.734825 0.0464744
\(251\) 3.29682 0.208093 0.104047 0.994572i \(-0.466821\pi\)
0.104047 + 0.994572i \(0.466821\pi\)
\(252\) 2.90321 0.182885
\(253\) −8.82225 −0.554650
\(254\) 2.81579 0.176678
\(255\) −25.5669 −1.60106
\(256\) −9.89829 −0.618643
\(257\) −23.6938 −1.47798 −0.738990 0.673717i \(-0.764697\pi\)
−0.738990 + 0.673717i \(0.764697\pi\)
\(258\) −13.4336 −0.836337
\(259\) −3.93332 −0.244405
\(260\) 0 0
\(261\) 3.90321 0.241603
\(262\) 4.09187 0.252796
\(263\) 9.99063 0.616049 0.308024 0.951378i \(-0.400332\pi\)
0.308024 + 0.951378i \(0.400332\pi\)
\(264\) 14.4603 0.889969
\(265\) 8.11753 0.498656
\(266\) 5.88739 0.360979
\(267\) −11.8731 −0.726622
\(268\) −0.653858 −0.0399408
\(269\) −18.2034 −1.10988 −0.554941 0.831890i \(-0.687259\pi\)
−0.554941 + 0.831890i \(0.687259\pi\)
\(270\) 5.37778 0.327282
\(271\) −24.1748 −1.46852 −0.734258 0.678870i \(-0.762470\pi\)
−0.734258 + 0.678870i \(0.762470\pi\)
\(272\) 4.94914 0.300086
\(273\) 0 0
\(274\) 11.2716 0.680944
\(275\) 14.3368 0.864540
\(276\) −11.0825 −0.667088
\(277\) 1.69535 0.101863 0.0509317 0.998702i \(-0.483781\pi\)
0.0509317 + 0.998702i \(0.483781\pi\)
\(278\) 2.08742 0.125195
\(279\) 11.1082 0.665028
\(280\) 7.80642 0.466523
\(281\) 11.6479 0.694854 0.347427 0.937707i \(-0.387055\pi\)
0.347427 + 0.937707i \(0.387055\pi\)
\(282\) −2.87157 −0.170999
\(283\) 12.1334 0.721253 0.360626 0.932710i \(-0.382563\pi\)
0.360626 + 0.932710i \(0.382563\pi\)
\(284\) −13.7047 −0.813225
\(285\) 60.8276 3.60312
\(286\) 0 0
\(287\) −0.755569 −0.0445998
\(288\) 11.0509 0.651178
\(289\) −4.09679 −0.240988
\(290\) 4.54125 0.266671
\(291\) −21.3210 −1.24986
\(292\) −8.83314 −0.516921
\(293\) 11.4538 0.669140 0.334570 0.942371i \(-0.391409\pi\)
0.334570 + 0.942371i \(0.391409\pi\)
\(294\) −1.52543 −0.0889647
\(295\) 23.5669 1.37212
\(296\) −9.55262 −0.555235
\(297\) 6.53035 0.378929
\(298\) −10.0143 −0.580112
\(299\) 0 0
\(300\) 18.0098 1.03980
\(301\) −8.80642 −0.507594
\(302\) −13.7605 −0.791827
\(303\) −30.3225 −1.74198
\(304\) −11.7748 −0.675330
\(305\) 29.0923 1.66582
\(306\) 4.70964 0.269232
\(307\) 3.96989 0.226574 0.113287 0.993562i \(-0.463862\pi\)
0.113287 + 0.993562i \(0.463862\pi\)
\(308\) 4.10171 0.233717
\(309\) −27.2257 −1.54882
\(310\) 12.9240 0.734031
\(311\) 27.3481 1.55077 0.775386 0.631488i \(-0.217556\pi\)
0.775386 + 0.631488i \(0.217556\pi\)
\(312\) 0 0
\(313\) −19.1032 −1.07978 −0.539890 0.841736i \(-0.681534\pi\)
−0.539890 + 0.841736i \(0.681534\pi\)
\(314\) 5.09679 0.287628
\(315\) −6.11753 −0.344684
\(316\) 6.83314 0.384394
\(317\) 9.90813 0.556496 0.278248 0.960509i \(-0.410246\pi\)
0.278248 + 0.960509i \(0.410246\pi\)
\(318\) −3.85236 −0.216029
\(319\) 5.51453 0.308754
\(320\) 4.00000 0.223607
\(321\) 40.2766 2.24802
\(322\) 2.26025 0.125959
\(323\) −30.6987 −1.70812
\(324\) 16.9131 0.939614
\(325\) 0 0
\(326\) 1.60300 0.0887821
\(327\) −18.5161 −1.02394
\(328\) −1.83500 −0.101321
\(329\) −1.88247 −0.103784
\(330\) −13.1842 −0.725767
\(331\) 23.4193 1.28724 0.643620 0.765345i \(-0.277432\pi\)
0.643620 + 0.765345i \(0.277432\pi\)
\(332\) 16.5161 0.906437
\(333\) 7.48595 0.410227
\(334\) −2.35752 −0.128998
\(335\) 1.37778 0.0752764
\(336\) 3.05086 0.166438
\(337\) −7.51606 −0.409426 −0.204713 0.978822i \(-0.565626\pi\)
−0.204713 + 0.978822i \(0.565626\pi\)
\(338\) 0 0
\(339\) 18.7447 1.01807
\(340\) −17.6128 −0.955191
\(341\) 15.6938 0.849868
\(342\) −11.2050 −0.605894
\(343\) −1.00000 −0.0539949
\(344\) −21.3876 −1.15314
\(345\) 23.3526 1.25726
\(346\) 5.70027 0.306448
\(347\) 5.64449 0.303012 0.151506 0.988456i \(-0.451588\pi\)
0.151506 + 0.988456i \(0.451588\pi\)
\(348\) 6.92735 0.371345
\(349\) 24.1590 1.29320 0.646601 0.762828i \(-0.276190\pi\)
0.646601 + 0.762828i \(0.276190\pi\)
\(350\) −3.67307 −0.196334
\(351\) 0 0
\(352\) 15.6128 0.832168
\(353\) 36.1289 1.92295 0.961474 0.274896i \(-0.0886436\pi\)
0.961474 + 0.274896i \(0.0886436\pi\)
\(354\) −11.1842 −0.594434
\(355\) 28.8780 1.53269
\(356\) −8.17929 −0.433501
\(357\) 7.95407 0.420974
\(358\) −11.1778 −0.590763
\(359\) 17.4128 0.919013 0.459507 0.888174i \(-0.348026\pi\)
0.459507 + 0.888174i \(0.348026\pi\)
\(360\) −14.8573 −0.783047
\(361\) 54.0370 2.84405
\(362\) −6.34122 −0.333287
\(363\) 8.34767 0.438139
\(364\) 0 0
\(365\) 18.6128 0.974241
\(366\) −13.8064 −0.721673
\(367\) 2.93825 0.153375 0.0766876 0.997055i \(-0.475566\pi\)
0.0766876 + 0.997055i \(0.475566\pi\)
\(368\) −4.52051 −0.235648
\(369\) 1.43801 0.0748597
\(370\) 8.70964 0.452792
\(371\) −2.52543 −0.131114
\(372\) 19.7146 1.02215
\(373\) −18.7699 −0.971866 −0.485933 0.873996i \(-0.661520\pi\)
−0.485933 + 0.873996i \(0.661520\pi\)
\(374\) 6.65386 0.344063
\(375\) −2.36196 −0.121971
\(376\) −4.57184 −0.235774
\(377\) 0 0
\(378\) −1.67307 −0.0860535
\(379\) 23.6894 1.21684 0.608421 0.793615i \(-0.291803\pi\)
0.608421 + 0.793615i \(0.291803\pi\)
\(380\) 41.9037 2.14961
\(381\) −9.05086 −0.463689
\(382\) −4.59364 −0.235031
\(383\) 31.6128 1.61534 0.807671 0.589634i \(-0.200728\pi\)
0.807671 + 0.589634i \(0.200728\pi\)
\(384\) 23.8163 1.21537
\(385\) −8.64296 −0.440486
\(386\) 14.1936 0.722434
\(387\) 16.7605 0.851984
\(388\) −14.6878 −0.745662
\(389\) −20.0558 −1.01687 −0.508434 0.861101i \(-0.669775\pi\)
−0.508434 + 0.861101i \(0.669775\pi\)
\(390\) 0 0
\(391\) −11.7857 −0.596027
\(392\) −2.42864 −0.122665
\(393\) −13.1526 −0.663459
\(394\) −5.76494 −0.290433
\(395\) −14.3985 −0.724469
\(396\) −7.80642 −0.392288
\(397\) 22.8731 1.14797 0.573984 0.818867i \(-0.305397\pi\)
0.573984 + 0.818867i \(0.305397\pi\)
\(398\) −0.414349 −0.0207695
\(399\) −18.9240 −0.947383
\(400\) 7.34614 0.367307
\(401\) −5.61285 −0.280292 −0.140146 0.990131i \(-0.544757\pi\)
−0.140146 + 0.990131i \(0.544757\pi\)
\(402\) −0.653858 −0.0326115
\(403\) 0 0
\(404\) −20.8889 −1.03926
\(405\) −35.6385 −1.77089
\(406\) −1.41282 −0.0701170
\(407\) 10.5763 0.524247
\(408\) 19.3176 0.956362
\(409\) −26.1175 −1.29143 −0.645714 0.763579i \(-0.723440\pi\)
−0.645714 + 0.763579i \(0.723440\pi\)
\(410\) 1.67307 0.0826271
\(411\) −36.2306 −1.78712
\(412\) −18.7556 −0.924021
\(413\) −7.33185 −0.360777
\(414\) −4.30174 −0.211419
\(415\) −34.8020 −1.70836
\(416\) 0 0
\(417\) −6.70964 −0.328572
\(418\) −15.8306 −0.774298
\(419\) −25.2464 −1.23337 −0.616685 0.787210i \(-0.711525\pi\)
−0.616685 + 0.787210i \(0.711525\pi\)
\(420\) −10.8573 −0.529781
\(421\) −8.22861 −0.401038 −0.200519 0.979690i \(-0.564263\pi\)
−0.200519 + 0.979690i \(0.564263\pi\)
\(422\) −1.31111 −0.0638237
\(423\) 3.58274 0.174199
\(424\) −6.13335 −0.297862
\(425\) 19.1526 0.929036
\(426\) −13.7047 −0.663996
\(427\) −9.05086 −0.438002
\(428\) 27.7462 1.34116
\(429\) 0 0
\(430\) 19.5002 0.940385
\(431\) −1.43801 −0.0692664 −0.0346332 0.999400i \(-0.511026\pi\)
−0.0346332 + 0.999400i \(0.511026\pi\)
\(432\) 3.34614 0.160991
\(433\) −16.5018 −0.793024 −0.396512 0.918029i \(-0.629780\pi\)
−0.396512 + 0.918029i \(0.629780\pi\)
\(434\) −4.02074 −0.193002
\(435\) −14.5970 −0.699874
\(436\) −12.7556 −0.610881
\(437\) 28.0400 1.34133
\(438\) −8.83314 −0.422064
\(439\) −10.1619 −0.485003 −0.242501 0.970151i \(-0.577968\pi\)
−0.242501 + 0.970151i \(0.577968\pi\)
\(440\) −20.9906 −1.00069
\(441\) 1.90321 0.0906291
\(442\) 0 0
\(443\) 3.20787 0.152410 0.0762052 0.997092i \(-0.475720\pi\)
0.0762052 + 0.997092i \(0.475720\pi\)
\(444\) 13.2859 0.630522
\(445\) 17.2351 0.817020
\(446\) 7.32540 0.346868
\(447\) 32.1891 1.52249
\(448\) −1.24443 −0.0587939
\(449\) −18.4099 −0.868817 −0.434409 0.900716i \(-0.643043\pi\)
−0.434409 + 0.900716i \(0.643043\pi\)
\(450\) 6.99063 0.329542
\(451\) 2.03164 0.0956663
\(452\) 12.9131 0.607379
\(453\) 44.2306 2.07814
\(454\) 10.3684 0.486614
\(455\) 0 0
\(456\) −45.9595 −2.15225
\(457\) 3.40297 0.159184 0.0795922 0.996828i \(-0.474638\pi\)
0.0795922 + 0.996828i \(0.474638\pi\)
\(458\) −4.50177 −0.210354
\(459\) 8.72393 0.407198
\(460\) 16.0874 0.750080
\(461\) 17.5714 0.818380 0.409190 0.912449i \(-0.365811\pi\)
0.409190 + 0.912449i \(0.365811\pi\)
\(462\) 4.10171 0.190829
\(463\) 15.7714 0.732959 0.366479 0.930426i \(-0.380563\pi\)
0.366479 + 0.930426i \(0.380563\pi\)
\(464\) 2.82564 0.131177
\(465\) −41.5417 −1.92645
\(466\) −19.2607 −0.892236
\(467\) −3.76694 −0.174313 −0.0871567 0.996195i \(-0.527778\pi\)
−0.0871567 + 0.996195i \(0.527778\pi\)
\(468\) 0 0
\(469\) −0.428639 −0.0197927
\(470\) 4.16839 0.192273
\(471\) −16.3827 −0.754875
\(472\) −17.8064 −0.819607
\(473\) 23.6795 1.08879
\(474\) 6.83314 0.313857
\(475\) −45.5669 −2.09075
\(476\) 5.47949 0.251152
\(477\) 4.80642 0.220071
\(478\) −13.4893 −0.616988
\(479\) −12.1032 −0.553011 −0.276506 0.961012i \(-0.589176\pi\)
−0.276506 + 0.961012i \(0.589176\pi\)
\(480\) −41.3274 −1.88633
\(481\) 0 0
\(482\) −9.20189 −0.419135
\(483\) −7.26517 −0.330577
\(484\) 5.75065 0.261393
\(485\) 30.9496 1.40535
\(486\) 11.8938 0.539516
\(487\) −17.3778 −0.787463 −0.393731 0.919226i \(-0.628816\pi\)
−0.393731 + 0.919226i \(0.628816\pi\)
\(488\) −21.9813 −0.995045
\(489\) −5.15257 −0.233007
\(490\) 2.21432 0.100033
\(491\) −19.3921 −0.875152 −0.437576 0.899181i \(-0.644163\pi\)
−0.437576 + 0.899181i \(0.644163\pi\)
\(492\) 2.55215 0.115060
\(493\) 7.36689 0.331788
\(494\) 0 0
\(495\) 16.4494 0.739345
\(496\) 8.04149 0.361073
\(497\) −8.98418 −0.402995
\(498\) 16.5161 0.740102
\(499\) −10.6702 −0.477662 −0.238831 0.971061i \(-0.576764\pi\)
−0.238831 + 0.971061i \(0.576764\pi\)
\(500\) −1.62714 −0.0727678
\(501\) 7.57781 0.338552
\(502\) 2.27115 0.101366
\(503\) −4.27655 −0.190682 −0.0953410 0.995445i \(-0.530394\pi\)
−0.0953410 + 0.995445i \(0.530394\pi\)
\(504\) 4.62222 0.205890
\(505\) 44.0163 1.95870
\(506\) −6.07758 −0.270181
\(507\) 0 0
\(508\) −6.23506 −0.276636
\(509\) 33.0765 1.46609 0.733046 0.680180i \(-0.238098\pi\)
0.733046 + 0.680180i \(0.238098\pi\)
\(510\) −17.6128 −0.779910
\(511\) −5.79060 −0.256161
\(512\) 14.6923 0.649313
\(513\) −20.7556 −0.916381
\(514\) −16.3225 −0.719954
\(515\) 39.5210 1.74150
\(516\) 29.7462 1.30950
\(517\) 5.06175 0.222616
\(518\) −2.70964 −0.119055
\(519\) −18.3225 −0.804268
\(520\) 0 0
\(521\) 24.3783 1.06803 0.534015 0.845475i \(-0.320682\pi\)
0.534015 + 0.845475i \(0.320682\pi\)
\(522\) 2.68889 0.117690
\(523\) 34.9403 1.52783 0.763915 0.645317i \(-0.223275\pi\)
0.763915 + 0.645317i \(0.223275\pi\)
\(524\) −9.06070 −0.395818
\(525\) 11.8064 0.515275
\(526\) 6.88247 0.300090
\(527\) 20.9654 0.913269
\(528\) −8.20342 −0.357008
\(529\) −12.2351 −0.531959
\(530\) 5.59210 0.242905
\(531\) 13.9541 0.605555
\(532\) −13.0366 −0.565207
\(533\) 0 0
\(534\) −8.17929 −0.353952
\(535\) −58.4657 −2.52769
\(536\) −1.04101 −0.0449648
\(537\) 35.9289 1.55045
\(538\) −12.5402 −0.540646
\(539\) 2.68889 0.115819
\(540\) −11.9081 −0.512445
\(541\) −30.4953 −1.31110 −0.655548 0.755153i \(-0.727562\pi\)
−0.655548 + 0.755153i \(0.727562\pi\)
\(542\) −16.6539 −0.715344
\(543\) 20.3827 0.874706
\(544\) 20.8573 0.894248
\(545\) 26.8780 1.15133
\(546\) 0 0
\(547\) −10.0049 −0.427780 −0.213890 0.976858i \(-0.568613\pi\)
−0.213890 + 0.976858i \(0.568613\pi\)
\(548\) −24.9590 −1.06620
\(549\) 17.2257 0.735175
\(550\) 9.87649 0.421135
\(551\) −17.5270 −0.746674
\(552\) −17.6445 −0.750999
\(553\) 4.47949 0.190487
\(554\) 1.16791 0.0496198
\(555\) −27.9956 −1.18835
\(556\) −4.62222 −0.196026
\(557\) 14.1936 0.601401 0.300701 0.953719i \(-0.402780\pi\)
0.300701 + 0.953719i \(0.402780\pi\)
\(558\) 7.65233 0.323949
\(559\) 0 0
\(560\) −4.42864 −0.187144
\(561\) −21.3876 −0.902986
\(562\) 8.02413 0.338478
\(563\) −21.3590 −0.900177 −0.450088 0.892984i \(-0.648607\pi\)
−0.450088 + 0.892984i \(0.648607\pi\)
\(564\) 6.35857 0.267744
\(565\) −27.2099 −1.14473
\(566\) 8.35857 0.351337
\(567\) 11.0874 0.465628
\(568\) −21.8193 −0.915519
\(569\) 34.2672 1.43656 0.718278 0.695757i \(-0.244931\pi\)
0.718278 + 0.695757i \(0.244931\pi\)
\(570\) 41.9037 1.75515
\(571\) −37.6494 −1.57558 −0.787789 0.615945i \(-0.788774\pi\)
−0.787789 + 0.615945i \(0.788774\pi\)
\(572\) 0 0
\(573\) 14.7654 0.616834
\(574\) −0.520505 −0.0217255
\(575\) −17.4938 −0.729541
\(576\) 2.36842 0.0986840
\(577\) 28.3970 1.18218 0.591091 0.806605i \(-0.298697\pi\)
0.591091 + 0.806605i \(0.298697\pi\)
\(578\) −2.82225 −0.117390
\(579\) −45.6227 −1.89601
\(580\) −10.0558 −0.417543
\(581\) 10.8272 0.449187
\(582\) −14.6878 −0.608830
\(583\) 6.79060 0.281238
\(584\) −14.0633 −0.581943
\(585\) 0 0
\(586\) 7.89045 0.325952
\(587\) −6.23659 −0.257412 −0.128706 0.991683i \(-0.541082\pi\)
−0.128706 + 0.991683i \(0.541082\pi\)
\(588\) 3.37778 0.139297
\(589\) −49.8800 −2.05527
\(590\) 16.2351 0.668387
\(591\) 18.5303 0.762237
\(592\) 5.41927 0.222731
\(593\) 17.5698 0.721506 0.360753 0.932661i \(-0.382520\pi\)
0.360753 + 0.932661i \(0.382520\pi\)
\(594\) 4.49871 0.184584
\(595\) −11.5462 −0.473347
\(596\) 22.1748 0.908317
\(597\) 1.33185 0.0545090
\(598\) 0 0
\(599\) 16.9813 0.693836 0.346918 0.937896i \(-0.387228\pi\)
0.346918 + 0.937896i \(0.387228\pi\)
\(600\) 28.6735 1.17059
\(601\) −18.7052 −0.763001 −0.381500 0.924369i \(-0.624592\pi\)
−0.381500 + 0.924369i \(0.624592\pi\)
\(602\) −6.06668 −0.247259
\(603\) 0.815792 0.0332216
\(604\) 30.4701 1.23981
\(605\) −12.1175 −0.492648
\(606\) −20.8889 −0.848554
\(607\) −10.3575 −0.420399 −0.210199 0.977659i \(-0.567411\pi\)
−0.210199 + 0.977659i \(0.567411\pi\)
\(608\) −49.6227 −2.01247
\(609\) 4.54125 0.184021
\(610\) 20.0415 0.811456
\(611\) 0 0
\(612\) −10.4286 −0.421553
\(613\) −7.02227 −0.283627 −0.141814 0.989893i \(-0.545293\pi\)
−0.141814 + 0.989893i \(0.545293\pi\)
\(614\) 2.73483 0.110369
\(615\) −5.37778 −0.216853
\(616\) 6.53035 0.263115
\(617\) 29.9813 1.20700 0.603500 0.797363i \(-0.293772\pi\)
0.603500 + 0.797363i \(0.293772\pi\)
\(618\) −18.7556 −0.754460
\(619\) 0.0285802 0.00114874 0.000574368 1.00000i \(-0.499817\pi\)
0.000574368 1.00000i \(0.499817\pi\)
\(620\) −28.6178 −1.14932
\(621\) −7.96836 −0.319759
\(622\) 18.8399 0.755412
\(623\) −5.36196 −0.214823
\(624\) 0 0
\(625\) −23.2306 −0.929225
\(626\) −13.1601 −0.525982
\(627\) 50.8845 2.03213
\(628\) −11.2859 −0.450357
\(629\) 14.1289 0.563356
\(630\) −4.21432 −0.167903
\(631\) −12.1936 −0.485419 −0.242709 0.970099i \(-0.578036\pi\)
−0.242709 + 0.970099i \(0.578036\pi\)
\(632\) 10.8791 0.432746
\(633\) 4.21432 0.167504
\(634\) 6.82564 0.271081
\(635\) 13.1383 0.521377
\(636\) 8.53035 0.338250
\(637\) 0 0
\(638\) 3.79892 0.150401
\(639\) 17.0988 0.676418
\(640\) −34.5718 −1.36657
\(641\) −2.82516 −0.111587 −0.0557935 0.998442i \(-0.517769\pi\)
−0.0557935 + 0.998442i \(0.517769\pi\)
\(642\) 27.7462 1.09506
\(643\) −37.7275 −1.48783 −0.743913 0.668276i \(-0.767032\pi\)
−0.743913 + 0.668276i \(0.767032\pi\)
\(644\) −5.00492 −0.197222
\(645\) −62.6800 −2.46802
\(646\) −21.1481 −0.832062
\(647\) 12.3664 0.486174 0.243087 0.970005i \(-0.421840\pi\)
0.243087 + 0.970005i \(0.421840\pi\)
\(648\) 26.9273 1.05781
\(649\) 19.7146 0.773864
\(650\) 0 0
\(651\) 12.9240 0.506530
\(652\) −3.54956 −0.139012
\(653\) −33.0005 −1.29141 −0.645704 0.763588i \(-0.723436\pi\)
−0.645704 + 0.763588i \(0.723436\pi\)
\(654\) −12.7556 −0.498782
\(655\) 19.0923 0.746000
\(656\) 1.04101 0.0406446
\(657\) 11.0207 0.429960
\(658\) −1.29682 −0.0505552
\(659\) 32.8118 1.27817 0.639084 0.769137i \(-0.279314\pi\)
0.639084 + 0.769137i \(0.279314\pi\)
\(660\) 29.1941 1.13638
\(661\) −14.7067 −0.572025 −0.286013 0.958226i \(-0.592330\pi\)
−0.286013 + 0.958226i \(0.592330\pi\)
\(662\) 16.1334 0.627041
\(663\) 0 0
\(664\) 26.2953 1.02046
\(665\) 27.4701 1.06525
\(666\) 5.15701 0.199830
\(667\) −6.72885 −0.260542
\(668\) 5.22030 0.201979
\(669\) −23.5462 −0.910348
\(670\) 0.949145 0.0366687
\(671\) 24.3368 0.939511
\(672\) 12.8573 0.495980
\(673\) 21.2908 0.820702 0.410351 0.911928i \(-0.365406\pi\)
0.410351 + 0.911928i \(0.365406\pi\)
\(674\) −5.17775 −0.199440
\(675\) 12.9491 0.498413
\(676\) 0 0
\(677\) −3.07160 −0.118051 −0.0590256 0.998256i \(-0.518799\pi\)
−0.0590256 + 0.998256i \(0.518799\pi\)
\(678\) 12.9131 0.495923
\(679\) −9.62867 −0.369514
\(680\) −28.0415 −1.07534
\(681\) −33.3274 −1.27711
\(682\) 10.8113 0.413988
\(683\) −24.7971 −0.948833 −0.474416 0.880301i \(-0.657341\pi\)
−0.474416 + 0.880301i \(0.657341\pi\)
\(684\) 24.8113 0.948686
\(685\) 52.5926 2.00946
\(686\) −0.688892 −0.0263020
\(687\) 14.4701 0.552070
\(688\) 12.1334 0.462580
\(689\) 0 0
\(690\) 16.0874 0.612438
\(691\) −27.0953 −1.03075 −0.515376 0.856964i \(-0.672348\pi\)
−0.515376 + 0.856964i \(0.672348\pi\)
\(692\) −12.6222 −0.479825
\(693\) −5.11753 −0.194399
\(694\) 3.88845 0.147603
\(695\) 9.73975 0.369450
\(696\) 11.0291 0.418055
\(697\) 2.71408 0.102803
\(698\) 16.6430 0.629945
\(699\) 61.9101 2.34166
\(700\) 8.13335 0.307412
\(701\) 13.5205 0.510662 0.255331 0.966854i \(-0.417815\pi\)
0.255331 + 0.966854i \(0.417815\pi\)
\(702\) 0 0
\(703\) −33.6149 −1.26781
\(704\) 3.34614 0.126112
\(705\) −13.3985 −0.504618
\(706\) 24.8889 0.936707
\(707\) −13.6938 −0.515009
\(708\) 24.7654 0.930741
\(709\) −50.9753 −1.91442 −0.957209 0.289399i \(-0.906545\pi\)
−0.957209 + 0.289399i \(0.906545\pi\)
\(710\) 19.8938 0.746603
\(711\) −8.52543 −0.319729
\(712\) −13.0223 −0.488030
\(713\) −19.1497 −0.717160
\(714\) 5.47949 0.205065
\(715\) 0 0
\(716\) 24.7511 0.924993
\(717\) 43.3590 1.61927
\(718\) 11.9956 0.447670
\(719\) 41.5417 1.54924 0.774622 0.632424i \(-0.217940\pi\)
0.774622 + 0.632424i \(0.217940\pi\)
\(720\) 8.42864 0.314117
\(721\) −12.2953 −0.457900
\(722\) 37.2257 1.38540
\(723\) 29.5778 1.10001
\(724\) 14.0415 0.521848
\(725\) 10.9349 0.406110
\(726\) 5.75065 0.213427
\(727\) −20.8988 −0.775092 −0.387546 0.921850i \(-0.626677\pi\)
−0.387546 + 0.921850i \(0.626677\pi\)
\(728\) 0 0
\(729\) −4.96836 −0.184013
\(730\) 12.8222 0.474573
\(731\) 31.6336 1.17001
\(732\) 30.5718 1.12997
\(733\) 19.3575 0.714986 0.357493 0.933916i \(-0.383632\pi\)
0.357493 + 0.933916i \(0.383632\pi\)
\(734\) 2.02413 0.0747122
\(735\) −7.11753 −0.262534
\(736\) −19.0509 −0.702224
\(737\) 1.15257 0.0424553
\(738\) 0.990632 0.0364657
\(739\) −1.31111 −0.0482299 −0.0241149 0.999709i \(-0.507677\pi\)
−0.0241149 + 0.999709i \(0.507677\pi\)
\(740\) −19.2859 −0.708964
\(741\) 0 0
\(742\) −1.73975 −0.0638681
\(743\) −21.5210 −0.789528 −0.394764 0.918783i \(-0.629174\pi\)
−0.394764 + 0.918783i \(0.629174\pi\)
\(744\) 31.3876 1.15073
\(745\) −46.7259 −1.71191
\(746\) −12.9304 −0.473416
\(747\) −20.6064 −0.753949
\(748\) −14.7338 −0.538720
\(749\) 18.1891 0.664616
\(750\) −1.62714 −0.0594147
\(751\) −16.3176 −0.595436 −0.297718 0.954654i \(-0.596226\pi\)
−0.297718 + 0.954654i \(0.596226\pi\)
\(752\) 2.59364 0.0945802
\(753\) −7.30021 −0.266034
\(754\) 0 0
\(755\) −64.2054 −2.33667
\(756\) 3.70471 0.134739
\(757\) −18.7462 −0.681342 −0.340671 0.940183i \(-0.610654\pi\)
−0.340671 + 0.940183i \(0.610654\pi\)
\(758\) 16.3194 0.592748
\(759\) 19.5353 0.709085
\(760\) 66.7150 2.42001
\(761\) −10.2968 −0.373259 −0.186630 0.982430i \(-0.559756\pi\)
−0.186630 + 0.982430i \(0.559756\pi\)
\(762\) −6.23506 −0.225873
\(763\) −8.36196 −0.302723
\(764\) 10.1718 0.368002
\(765\) 21.9748 0.794501
\(766\) 21.7778 0.786865
\(767\) 0 0
\(768\) 21.9180 0.790897
\(769\) 9.36641 0.337761 0.168881 0.985637i \(-0.445985\pi\)
0.168881 + 0.985637i \(0.445985\pi\)
\(770\) −5.95407 −0.214570
\(771\) 52.4657 1.88951
\(772\) −31.4291 −1.13116
\(773\) −3.71456 −0.133603 −0.0668017 0.997766i \(-0.521279\pi\)
−0.0668017 + 0.997766i \(0.521279\pi\)
\(774\) 11.5462 0.415019
\(775\) 31.1195 1.11785
\(776\) −23.3846 −0.839457
\(777\) 8.70964 0.312456
\(778\) −13.8163 −0.495337
\(779\) −6.45722 −0.231354
\(780\) 0 0
\(781\) 24.1575 0.864423
\(782\) −8.11906 −0.290337
\(783\) 4.98079 0.177999
\(784\) 1.37778 0.0492066
\(785\) 23.7812 0.848789
\(786\) −9.06070 −0.323184
\(787\) 6.23659 0.222311 0.111155 0.993803i \(-0.464545\pi\)
0.111155 + 0.993803i \(0.464545\pi\)
\(788\) 12.7654 0.454749
\(789\) −22.1225 −0.787580
\(790\) −9.91903 −0.352903
\(791\) 8.46520 0.300988
\(792\) −12.4286 −0.441632
\(793\) 0 0
\(794\) 15.7571 0.559199
\(795\) −17.9748 −0.637501
\(796\) 0.917502 0.0325200
\(797\) −40.0701 −1.41935 −0.709677 0.704527i \(-0.751159\pi\)
−0.709677 + 0.704527i \(0.751159\pi\)
\(798\) −13.0366 −0.461489
\(799\) 6.76202 0.239223
\(800\) 30.9590 1.09457
\(801\) 10.2050 0.360574
\(802\) −3.86665 −0.136536
\(803\) 15.5703 0.549464
\(804\) 1.44785 0.0510618
\(805\) 10.5462 0.371704
\(806\) 0 0
\(807\) 40.3082 1.41892
\(808\) −33.2573 −1.16999
\(809\) −2.57136 −0.0904042 −0.0452021 0.998978i \(-0.514393\pi\)
−0.0452021 + 0.998978i \(0.514393\pi\)
\(810\) −24.5511 −0.862637
\(811\) −28.3654 −0.996042 −0.498021 0.867165i \(-0.665940\pi\)
−0.498021 + 0.867165i \(0.665940\pi\)
\(812\) 3.12843 0.109786
\(813\) 53.5308 1.87741
\(814\) 7.28592 0.255371
\(815\) 7.47949 0.261995
\(816\) −10.9590 −0.383641
\(817\) −75.2612 −2.63306
\(818\) −17.9922 −0.629081
\(819\) 0 0
\(820\) −3.70471 −0.129374
\(821\) −31.7846 −1.10929 −0.554646 0.832087i \(-0.687146\pi\)
−0.554646 + 0.832087i \(0.687146\pi\)
\(822\) −24.9590 −0.870545
\(823\) 30.9131 1.07756 0.538781 0.842446i \(-0.318885\pi\)
0.538781 + 0.842446i \(0.318885\pi\)
\(824\) −29.8608 −1.04025
\(825\) −31.7462 −1.10526
\(826\) −5.05086 −0.175742
\(827\) 49.7560 1.73019 0.865094 0.501610i \(-0.167259\pi\)
0.865094 + 0.501610i \(0.167259\pi\)
\(828\) 9.52543 0.331031
\(829\) −19.9190 −0.691817 −0.345908 0.938268i \(-0.612429\pi\)
−0.345908 + 0.938268i \(0.612429\pi\)
\(830\) −23.9748 −0.832178
\(831\) −3.75404 −0.130226
\(832\) 0 0
\(833\) 3.59210 0.124459
\(834\) −4.62222 −0.160054
\(835\) −11.0000 −0.380671
\(836\) 35.0539 1.21237
\(837\) 14.1748 0.489954
\(838\) −17.3921 −0.600799
\(839\) 11.5625 0.399181 0.199590 0.979879i \(-0.436039\pi\)
0.199590 + 0.979879i \(0.436039\pi\)
\(840\) −17.2859 −0.596421
\(841\) −24.7940 −0.854965
\(842\) −5.66862 −0.195354
\(843\) −25.7921 −0.888328
\(844\) 2.90321 0.0999327
\(845\) 0 0
\(846\) 2.46812 0.0848557
\(847\) 3.76986 0.129534
\(848\) 3.47949 0.119486
\(849\) −26.8671 −0.922077
\(850\) 13.1941 0.452552
\(851\) −12.9052 −0.442385
\(852\) 30.3466 1.03966
\(853\) 1.74467 0.0597363 0.0298682 0.999554i \(-0.490491\pi\)
0.0298682 + 0.999554i \(0.490491\pi\)
\(854\) −6.23506 −0.213359
\(855\) −52.2815 −1.78799
\(856\) 44.1748 1.50986
\(857\) −42.3368 −1.44620 −0.723098 0.690745i \(-0.757283\pi\)
−0.723098 + 0.690745i \(0.757283\pi\)
\(858\) 0 0
\(859\) −5.37778 −0.183488 −0.0917438 0.995783i \(-0.529244\pi\)
−0.0917438 + 0.995783i \(0.529244\pi\)
\(860\) −43.1798 −1.47242
\(861\) 1.67307 0.0570181
\(862\) −0.990632 −0.0337411
\(863\) 15.4291 0.525213 0.262607 0.964903i \(-0.415418\pi\)
0.262607 + 0.964903i \(0.415418\pi\)
\(864\) 14.1017 0.479750
\(865\) 26.5970 0.904326
\(866\) −11.3679 −0.386298
\(867\) 9.07160 0.308088
\(868\) 8.90321 0.302195
\(869\) −12.0449 −0.408595
\(870\) −10.0558 −0.340923
\(871\) 0 0
\(872\) −20.3082 −0.687722
\(873\) 18.3254 0.620221
\(874\) 19.3165 0.653391
\(875\) −1.06668 −0.0360602
\(876\) 19.5594 0.660851
\(877\) 38.5181 1.30066 0.650331 0.759651i \(-0.274630\pi\)
0.650331 + 0.759651i \(0.274630\pi\)
\(878\) −7.00048 −0.236255
\(879\) −25.3624 −0.855454
\(880\) 11.9081 0.401423
\(881\) −41.7373 −1.40617 −0.703083 0.711108i \(-0.748194\pi\)
−0.703083 + 0.711108i \(0.748194\pi\)
\(882\) 1.31111 0.0441473
\(883\) 52.8439 1.77834 0.889170 0.457577i \(-0.151283\pi\)
0.889170 + 0.457577i \(0.151283\pi\)
\(884\) 0 0
\(885\) −52.1847 −1.75417
\(886\) 2.20987 0.0742422
\(887\) 8.99063 0.301876 0.150938 0.988543i \(-0.451771\pi\)
0.150938 + 0.988543i \(0.451771\pi\)
\(888\) 21.1526 0.709834
\(889\) −4.08742 −0.137088
\(890\) 11.8731 0.397987
\(891\) −29.8129 −0.998769
\(892\) −16.2208 −0.543112
\(893\) −16.0879 −0.538361
\(894\) 22.1748 0.741638
\(895\) −52.1546 −1.74334
\(896\) 10.7556 0.359318
\(897\) 0 0
\(898\) −12.6824 −0.423218
\(899\) 11.9699 0.399218
\(900\) −15.4795 −0.515983
\(901\) 9.07160 0.302219
\(902\) 1.39958 0.0466010
\(903\) 19.5002 0.648927
\(904\) 20.5589 0.683780
\(905\) −29.5877 −0.983527
\(906\) 30.4701 1.01230
\(907\) −20.7003 −0.687341 −0.343671 0.939090i \(-0.611670\pi\)
−0.343671 + 0.939090i \(0.611670\pi\)
\(908\) −22.9590 −0.761921
\(909\) 26.0622 0.864430
\(910\) 0 0
\(911\) 29.8988 0.990590 0.495295 0.868725i \(-0.335060\pi\)
0.495295 + 0.868725i \(0.335060\pi\)
\(912\) 26.0731 0.863368
\(913\) −29.1131 −0.963503
\(914\) 2.34428 0.0775420
\(915\) −64.4197 −2.12965
\(916\) 9.96836 0.329364
\(917\) −5.93978 −0.196149
\(918\) 6.00984 0.198354
\(919\) 38.5847 1.27279 0.636397 0.771362i \(-0.280424\pi\)
0.636397 + 0.771362i \(0.280424\pi\)
\(920\) 25.6128 0.844431
\(921\) −8.79060 −0.289660
\(922\) 12.1048 0.398649
\(923\) 0 0
\(924\) −9.08250 −0.298792
\(925\) 20.9719 0.689552
\(926\) 10.8648 0.357039
\(927\) 23.4005 0.768574
\(928\) 11.9081 0.390904
\(929\) 3.25581 0.106820 0.0534098 0.998573i \(-0.482991\pi\)
0.0534098 + 0.998573i \(0.482991\pi\)
\(930\) −28.6178 −0.938414
\(931\) −8.54617 −0.280089
\(932\) 42.6494 1.39703
\(933\) −60.5575 −1.98257
\(934\) −2.59502 −0.0849116
\(935\) 31.0464 1.01533
\(936\) 0 0
\(937\) −11.6840 −0.381699 −0.190849 0.981619i \(-0.561124\pi\)
−0.190849 + 0.981619i \(0.561124\pi\)
\(938\) −0.295286 −0.00964144
\(939\) 42.3007 1.38043
\(940\) −9.23014 −0.301054
\(941\) −41.9699 −1.36818 −0.684090 0.729398i \(-0.739800\pi\)
−0.684090 + 0.729398i \(0.739800\pi\)
\(942\) −11.2859 −0.367715
\(943\) −2.47902 −0.0807279
\(944\) 10.1017 0.328783
\(945\) −7.80642 −0.253943
\(946\) 16.3126 0.530370
\(947\) −20.3555 −0.661465 −0.330733 0.943725i \(-0.607296\pi\)
−0.330733 + 0.943725i \(0.607296\pi\)
\(948\) −15.1308 −0.491424
\(949\) 0 0
\(950\) −31.3907 −1.01845
\(951\) −21.9398 −0.711446
\(952\) 8.72393 0.282744
\(953\) −30.5496 −0.989597 −0.494799 0.869008i \(-0.664758\pi\)
−0.494799 + 0.869008i \(0.664758\pi\)
\(954\) 3.31111 0.107201
\(955\) −21.4336 −0.693574
\(956\) 29.8697 0.966055
\(957\) −12.2109 −0.394723
\(958\) −8.33783 −0.269383
\(959\) −16.3620 −0.528355
\(960\) −8.85728 −0.285867
\(961\) 3.06515 0.0988757
\(962\) 0 0
\(963\) −34.6178 −1.11554
\(964\) 20.3759 0.656264
\(965\) 66.2262 2.13190
\(966\) −5.00492 −0.161031
\(967\) 54.4548 1.75115 0.875574 0.483084i \(-0.160484\pi\)
0.875574 + 0.483084i \(0.160484\pi\)
\(968\) 9.15563 0.294273
\(969\) 67.9768 2.18373
\(970\) 21.3210 0.684575
\(971\) −54.6035 −1.75231 −0.876155 0.482030i \(-0.839899\pi\)
−0.876155 + 0.482030i \(0.839899\pi\)
\(972\) −26.3368 −0.844752
\(973\) −3.03011 −0.0971409
\(974\) −11.9714 −0.383589
\(975\) 0 0
\(976\) 12.4701 0.399159
\(977\) 32.6474 1.04448 0.522242 0.852798i \(-0.325096\pi\)
0.522242 + 0.852798i \(0.325096\pi\)
\(978\) −3.54956 −0.113502
\(979\) 14.4177 0.460793
\(980\) −4.90321 −0.156627
\(981\) 15.9146 0.508114
\(982\) −13.3590 −0.426304
\(983\) −27.5052 −0.877278 −0.438639 0.898663i \(-0.644539\pi\)
−0.438639 + 0.898663i \(0.644539\pi\)
\(984\) 4.06329 0.129533
\(985\) −26.8988 −0.857066
\(986\) 5.07499 0.161621
\(987\) 4.16839 0.132681
\(988\) 0 0
\(989\) −28.8938 −0.918771
\(990\) 11.3319 0.360150
\(991\) −6.29390 −0.199932 −0.0999662 0.994991i \(-0.531873\pi\)
−0.0999662 + 0.994991i \(0.531873\pi\)
\(992\) 33.8894 1.07599
\(993\) −51.8578 −1.64566
\(994\) −6.18913 −0.196307
\(995\) −1.93332 −0.0612905
\(996\) −36.5718 −1.15882
\(997\) −20.8702 −0.660965 −0.330483 0.943812i \(-0.607212\pi\)
−0.330483 + 0.943812i \(0.607212\pi\)
\(998\) −7.35059 −0.232679
\(999\) 9.55262 0.302232
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 1183.2.a.j.1.2 3
7.6 odd 2 8281.2.a.bi.1.2 3
13.5 odd 4 91.2.c.a.64.3 6
13.8 odd 4 91.2.c.a.64.4 yes 6
13.12 even 2 1183.2.a.h.1.2 3
39.5 even 4 819.2.c.b.64.4 6
39.8 even 4 819.2.c.b.64.3 6
52.31 even 4 1456.2.k.c.337.5 6
52.47 even 4 1456.2.k.c.337.6 6
91.5 even 12 637.2.r.d.116.3 12
91.18 odd 12 637.2.r.e.324.4 12
91.31 even 12 637.2.r.d.324.4 12
91.34 even 4 637.2.c.d.246.4 6
91.44 odd 12 637.2.r.e.116.3 12
91.47 even 12 637.2.r.d.116.4 12
91.60 odd 12 637.2.r.e.324.3 12
91.73 even 12 637.2.r.d.324.3 12
91.83 even 4 637.2.c.d.246.3 6
91.86 odd 12 637.2.r.e.116.4 12
91.90 odd 2 8281.2.a.be.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.c.a.64.3 6 13.5 odd 4
91.2.c.a.64.4 yes 6 13.8 odd 4
637.2.c.d.246.3 6 91.83 even 4
637.2.c.d.246.4 6 91.34 even 4
637.2.r.d.116.3 12 91.5 even 12
637.2.r.d.116.4 12 91.47 even 12
637.2.r.d.324.3 12 91.73 even 12
637.2.r.d.324.4 12 91.31 even 12
637.2.r.e.116.3 12 91.44 odd 12
637.2.r.e.116.4 12 91.86 odd 12
637.2.r.e.324.3 12 91.60 odd 12
637.2.r.e.324.4 12 91.18 odd 12
819.2.c.b.64.3 6 39.8 even 4
819.2.c.b.64.4 6 39.5 even 4
1183.2.a.h.1.2 3 13.12 even 2
1183.2.a.j.1.2 3 1.1 even 1 trivial
1456.2.k.c.337.5 6 52.31 even 4
1456.2.k.c.337.6 6 52.47 even 4
8281.2.a.be.1.2 3 91.90 odd 2
8281.2.a.bi.1.2 3 7.6 odd 2