Properties

Label 4205.2.a.f.1.3
Level $4205$
Weight $2$
Character 4205.1
Self dual yes
Analytic conductor $33.577$
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] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, 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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
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 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 4205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} +1.00000 q^{5} -1.19394 q^{6} +1.19394 q^{7} -2.67513 q^{8} -2.35026 q^{9} +O(q^{10})\) \(q+1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} +1.00000 q^{5} -1.19394 q^{6} +1.19394 q^{7} -2.67513 q^{8} -2.35026 q^{9} +1.48119 q^{10} -4.15633 q^{11} -0.156325 q^{12} +2.96239 q^{13} +1.76845 q^{14} -0.806063 q^{15} -4.35026 q^{16} -5.50659 q^{17} -3.48119 q^{18} +3.19394 q^{19} +0.193937 q^{20} -0.962389 q^{21} -6.15633 q^{22} +1.84367 q^{23} +2.15633 q^{24} +1.00000 q^{25} +4.38787 q^{26} +4.31265 q^{27} +0.231548 q^{28} -1.19394 q^{30} +4.80606 q^{31} -1.09332 q^{32} +3.35026 q^{33} -8.15633 q^{34} +1.19394 q^{35} -0.455802 q^{36} +9.50659 q^{37} +4.73084 q^{38} -2.38787 q^{39} -2.67513 q^{40} +11.2750 q^{41} -1.42548 q^{42} +0.0303172 q^{43} -0.806063 q^{44} -2.35026 q^{45} +2.73084 q^{46} -4.80606 q^{47} +3.50659 q^{48} -5.57452 q^{49} +1.48119 q^{50} +4.43866 q^{51} +0.574515 q^{52} -1.35026 q^{53} +6.38787 q^{54} -4.15633 q^{55} -3.19394 q^{56} -2.57452 q^{57} +13.2750 q^{59} -0.156325 q^{60} -8.88717 q^{61} +7.11871 q^{62} -2.80606 q^{63} +7.08110 q^{64} +2.96239 q^{65} +4.96239 q^{66} +5.84367 q^{67} -1.06793 q^{68} -1.48612 q^{69} +1.76845 q^{70} -1.27504 q^{71} +6.28726 q^{72} +15.2447 q^{73} +14.0811 q^{74} -0.806063 q^{75} +0.619421 q^{76} -4.96239 q^{77} -3.53690 q^{78} +4.93207 q^{79} -4.35026 q^{80} +3.57452 q^{81} +16.7005 q^{82} +4.41819 q^{83} -0.186642 q^{84} -5.50659 q^{85} +0.0449056 q^{86} +11.1187 q^{88} +3.61213 q^{89} -3.48119 q^{90} +3.53690 q^{91} +0.357556 q^{92} -3.87399 q^{93} -7.11871 q^{94} +3.19394 q^{95} +0.881286 q^{96} +1.38058 q^{97} -8.25694 q^{98} +9.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 2 q^{3} + q^{4} + 3 q^{5} - 4 q^{6} + 4 q^{7} - 3 q^{8} + 3 q^{9} - q^{10} - 2 q^{11} + 10 q^{12} - 2 q^{13} - 6 q^{14} - 2 q^{15} - 3 q^{16} + 4 q^{17} - 5 q^{18} + 10 q^{19} + q^{20} + 8 q^{21} - 8 q^{22} + 16 q^{23} - 4 q^{24} + 3 q^{25} + 14 q^{26} - 8 q^{27} + 12 q^{28} - 4 q^{30} + 14 q^{31} + 3 q^{32} - 14 q^{34} + 4 q^{35} - 11 q^{36} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 3 q^{40} + 2 q^{41} - 16 q^{42} - 2 q^{43} - 2 q^{44} + 3 q^{45} - 14 q^{46} - 14 q^{47} - 10 q^{48} - 5 q^{49} - q^{50} - 16 q^{51} - 10 q^{52} + 6 q^{53} + 20 q^{54} - 2 q^{55} - 10 q^{56} + 4 q^{57} + 8 q^{59} + 10 q^{60} + 6 q^{61} - 8 q^{63} - 11 q^{64} - 2 q^{65} + 4 q^{66} + 28 q^{67} - 12 q^{68} - 12 q^{69} - 6 q^{70} + 28 q^{71} + 13 q^{72} + 16 q^{73} + 10 q^{74} - 2 q^{75} + 14 q^{76} - 4 q^{77} + 12 q^{78} + 6 q^{79} - 3 q^{80} - q^{81} + 30 q^{82} + 12 q^{83} + 12 q^{84} + 4 q^{85} + 24 q^{86} + 12 q^{88} + 10 q^{89} - 5 q^{90} - 12 q^{91} + 4 q^{92} - 20 q^{93} + 10 q^{95} + 24 q^{96} - 8 q^{97} - 21 q^{98} + 18 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.48119 1.04736 0.523681 0.851914i \(-0.324558\pi\)
0.523681 + 0.851914i \(0.324558\pi\)
\(3\) −0.806063 −0.465381 −0.232690 0.972551i \(-0.574753\pi\)
−0.232690 + 0.972551i \(0.574753\pi\)
\(4\) 0.193937 0.0969683
\(5\) 1.00000 0.447214
\(6\) −1.19394 −0.487423
\(7\) 1.19394 0.451266 0.225633 0.974212i \(-0.427555\pi\)
0.225633 + 0.974212i \(0.427555\pi\)
\(8\) −2.67513 −0.945802
\(9\) −2.35026 −0.783421
\(10\) 1.48119 0.468395
\(11\) −4.15633 −1.25318 −0.626590 0.779349i \(-0.715550\pi\)
−0.626590 + 0.779349i \(0.715550\pi\)
\(12\) −0.156325 −0.0451272
\(13\) 2.96239 0.821619 0.410809 0.911721i \(-0.365246\pi\)
0.410809 + 0.911721i \(0.365246\pi\)
\(14\) 1.76845 0.472639
\(15\) −0.806063 −0.208125
\(16\) −4.35026 −1.08757
\(17\) −5.50659 −1.33554 −0.667772 0.744366i \(-0.732752\pi\)
−0.667772 + 0.744366i \(0.732752\pi\)
\(18\) −3.48119 −0.820525
\(19\) 3.19394 0.732739 0.366370 0.930469i \(-0.380601\pi\)
0.366370 + 0.930469i \(0.380601\pi\)
\(20\) 0.193937 0.0433655
\(21\) −0.962389 −0.210010
\(22\) −6.15633 −1.31253
\(23\) 1.84367 0.384433 0.192216 0.981353i \(-0.438432\pi\)
0.192216 + 0.981353i \(0.438432\pi\)
\(24\) 2.15633 0.440158
\(25\) 1.00000 0.200000
\(26\) 4.38787 0.860533
\(27\) 4.31265 0.829970
\(28\) 0.231548 0.0437585
\(29\) 0 0
\(30\) −1.19394 −0.217982
\(31\) 4.80606 0.863194 0.431597 0.902066i \(-0.357950\pi\)
0.431597 + 0.902066i \(0.357950\pi\)
\(32\) −1.09332 −0.193274
\(33\) 3.35026 0.583206
\(34\) −8.15633 −1.39880
\(35\) 1.19394 0.201812
\(36\) −0.455802 −0.0759669
\(37\) 9.50659 1.56287 0.781437 0.623985i \(-0.214487\pi\)
0.781437 + 0.623985i \(0.214487\pi\)
\(38\) 4.73084 0.767444
\(39\) −2.38787 −0.382366
\(40\) −2.67513 −0.422975
\(41\) 11.2750 1.76087 0.880433 0.474171i \(-0.157252\pi\)
0.880433 + 0.474171i \(0.157252\pi\)
\(42\) −1.42548 −0.219957
\(43\) 0.0303172 0.00462332 0.00231166 0.999997i \(-0.499264\pi\)
0.00231166 + 0.999997i \(0.499264\pi\)
\(44\) −0.806063 −0.121519
\(45\) −2.35026 −0.350356
\(46\) 2.73084 0.402640
\(47\) −4.80606 −0.701036 −0.350518 0.936556i \(-0.613995\pi\)
−0.350518 + 0.936556i \(0.613995\pi\)
\(48\) 3.50659 0.506132
\(49\) −5.57452 −0.796359
\(50\) 1.48119 0.209473
\(51\) 4.43866 0.621536
\(52\) 0.574515 0.0796710
\(53\) −1.35026 −0.185473 −0.0927364 0.995691i \(-0.529561\pi\)
−0.0927364 + 0.995691i \(0.529561\pi\)
\(54\) 6.38787 0.869279
\(55\) −4.15633 −0.560439
\(56\) −3.19394 −0.426808
\(57\) −2.57452 −0.341003
\(58\) 0 0
\(59\) 13.2750 1.72826 0.864131 0.503266i \(-0.167868\pi\)
0.864131 + 0.503266i \(0.167868\pi\)
\(60\) −0.156325 −0.0201815
\(61\) −8.88717 −1.13788 −0.568942 0.822377i \(-0.692647\pi\)
−0.568942 + 0.822377i \(0.692647\pi\)
\(62\) 7.11871 0.904078
\(63\) −2.80606 −0.353531
\(64\) 7.08110 0.885138
\(65\) 2.96239 0.367439
\(66\) 4.96239 0.610828
\(67\) 5.84367 0.713919 0.356959 0.934120i \(-0.383813\pi\)
0.356959 + 0.934120i \(0.383813\pi\)
\(68\) −1.06793 −0.129505
\(69\) −1.48612 −0.178908
\(70\) 1.76845 0.211370
\(71\) −1.27504 −0.151319 −0.0756596 0.997134i \(-0.524106\pi\)
−0.0756596 + 0.997134i \(0.524106\pi\)
\(72\) 6.28726 0.740960
\(73\) 15.2447 1.78426 0.892130 0.451779i \(-0.149210\pi\)
0.892130 + 0.451779i \(0.149210\pi\)
\(74\) 14.0811 1.63689
\(75\) −0.806063 −0.0930762
\(76\) 0.619421 0.0710525
\(77\) −4.96239 −0.565517
\(78\) −3.53690 −0.400476
\(79\) 4.93207 0.554901 0.277451 0.960740i \(-0.410510\pi\)
0.277451 + 0.960740i \(0.410510\pi\)
\(80\) −4.35026 −0.486374
\(81\) 3.57452 0.397168
\(82\) 16.7005 1.84426
\(83\) 4.41819 0.484959 0.242480 0.970156i \(-0.422039\pi\)
0.242480 + 0.970156i \(0.422039\pi\)
\(84\) −0.186642 −0.0203643
\(85\) −5.50659 −0.597273
\(86\) 0.0449056 0.00484230
\(87\) 0 0
\(88\) 11.1187 1.18526
\(89\) 3.61213 0.382885 0.191442 0.981504i \(-0.438684\pi\)
0.191442 + 0.981504i \(0.438684\pi\)
\(90\) −3.48119 −0.366950
\(91\) 3.53690 0.370768
\(92\) 0.357556 0.0372778
\(93\) −3.87399 −0.401714
\(94\) −7.11871 −0.734239
\(95\) 3.19394 0.327691
\(96\) 0.881286 0.0899459
\(97\) 1.38058 0.140177 0.0700883 0.997541i \(-0.477672\pi\)
0.0700883 + 0.997541i \(0.477672\pi\)
\(98\) −8.25694 −0.834077
\(99\) 9.76845 0.981766
\(100\) 0.193937 0.0193937
\(101\) 13.0132 1.29486 0.647430 0.762125i \(-0.275844\pi\)
0.647430 + 0.762125i \(0.275844\pi\)
\(102\) 6.57452 0.650974
\(103\) 5.31994 0.524190 0.262095 0.965042i \(-0.415587\pi\)
0.262095 + 0.965042i \(0.415587\pi\)
\(104\) −7.92478 −0.777088
\(105\) −0.962389 −0.0939195
\(106\) −2.00000 −0.194257
\(107\) −13.8192 −1.33596 −0.667978 0.744181i \(-0.732840\pi\)
−0.667978 + 0.744181i \(0.732840\pi\)
\(108\) 0.836381 0.0804808
\(109\) −1.87399 −0.179496 −0.0897479 0.995965i \(-0.528606\pi\)
−0.0897479 + 0.995965i \(0.528606\pi\)
\(110\) −6.15633 −0.586983
\(111\) −7.66291 −0.727331
\(112\) −5.19394 −0.490781
\(113\) 11.7685 1.10708 0.553541 0.832822i \(-0.313276\pi\)
0.553541 + 0.832822i \(0.313276\pi\)
\(114\) −3.81336 −0.357154
\(115\) 1.84367 0.171924
\(116\) 0 0
\(117\) −6.96239 −0.643673
\(118\) 19.6629 1.81012
\(119\) −6.57452 −0.602685
\(120\) 2.15633 0.196845
\(121\) 6.27504 0.570458
\(122\) −13.1636 −1.19178
\(123\) −9.08840 −0.819473
\(124\) 0.932071 0.0837025
\(125\) 1.00000 0.0894427
\(126\) −4.15633 −0.370275
\(127\) −14.2677 −1.26606 −0.633029 0.774128i \(-0.718189\pi\)
−0.633029 + 0.774128i \(0.718189\pi\)
\(128\) 12.6751 1.12033
\(129\) −0.0244376 −0.00215161
\(130\) 4.38787 0.384842
\(131\) −5.89446 −0.515001 −0.257501 0.966278i \(-0.582899\pi\)
−0.257501 + 0.966278i \(0.582899\pi\)
\(132\) 0.649738 0.0565525
\(133\) 3.81336 0.330660
\(134\) 8.65562 0.747731
\(135\) 4.31265 0.371174
\(136\) 14.7308 1.26316
\(137\) −18.2823 −1.56197 −0.780983 0.624553i \(-0.785281\pi\)
−0.780983 + 0.624553i \(0.785281\pi\)
\(138\) −2.20123 −0.187381
\(139\) −11.5369 −0.978547 −0.489274 0.872130i \(-0.662738\pi\)
−0.489274 + 0.872130i \(0.662738\pi\)
\(140\) 0.231548 0.0195694
\(141\) 3.87399 0.326249
\(142\) −1.88858 −0.158486
\(143\) −12.3127 −1.02964
\(144\) 10.2243 0.852021
\(145\) 0 0
\(146\) 22.5804 1.86877
\(147\) 4.49341 0.370610
\(148\) 1.84367 0.151549
\(149\) 2.77575 0.227398 0.113699 0.993515i \(-0.463730\pi\)
0.113699 + 0.993515i \(0.463730\pi\)
\(150\) −1.19394 −0.0974845
\(151\) 1.79877 0.146382 0.0731909 0.997318i \(-0.476682\pi\)
0.0731909 + 0.997318i \(0.476682\pi\)
\(152\) −8.54420 −0.693026
\(153\) 12.9419 1.04629
\(154\) −7.35026 −0.592301
\(155\) 4.80606 0.386032
\(156\) −0.463096 −0.0370773
\(157\) −3.76845 −0.300755 −0.150378 0.988629i \(-0.548049\pi\)
−0.150378 + 0.988629i \(0.548049\pi\)
\(158\) 7.30536 0.581183
\(159\) 1.08840 0.0863155
\(160\) −1.09332 −0.0864346
\(161\) 2.20123 0.173481
\(162\) 5.29455 0.415979
\(163\) −1.64244 −0.128646 −0.0643231 0.997929i \(-0.520489\pi\)
−0.0643231 + 0.997929i \(0.520489\pi\)
\(164\) 2.18664 0.170748
\(165\) 3.35026 0.260818
\(166\) 6.54420 0.507928
\(167\) 8.08110 0.625334 0.312667 0.949863i \(-0.398778\pi\)
0.312667 + 0.949863i \(0.398778\pi\)
\(168\) 2.57452 0.198628
\(169\) −4.22425 −0.324943
\(170\) −8.15633 −0.625562
\(171\) −7.50659 −0.574043
\(172\) 0.00587961 0.000448316 0
\(173\) −7.73813 −0.588320 −0.294160 0.955756i \(-0.595040\pi\)
−0.294160 + 0.955756i \(0.595040\pi\)
\(174\) 0 0
\(175\) 1.19394 0.0902531
\(176\) 18.0811 1.36291
\(177\) −10.7005 −0.804301
\(178\) 5.35026 0.401019
\(179\) −21.4010 −1.59959 −0.799795 0.600274i \(-0.795058\pi\)
−0.799795 + 0.600274i \(0.795058\pi\)
\(180\) −0.455802 −0.0339735
\(181\) 15.2750 1.13538 0.567692 0.823241i \(-0.307836\pi\)
0.567692 + 0.823241i \(0.307836\pi\)
\(182\) 5.23884 0.388329
\(183\) 7.16362 0.529550
\(184\) −4.93207 −0.363597
\(185\) 9.50659 0.698938
\(186\) −5.73813 −0.420740
\(187\) 22.8872 1.67368
\(188\) −0.932071 −0.0679783
\(189\) 5.14903 0.374537
\(190\) 4.73084 0.343211
\(191\) −3.31994 −0.240223 −0.120111 0.992760i \(-0.538325\pi\)
−0.120111 + 0.992760i \(0.538325\pi\)
\(192\) −5.70782 −0.411926
\(193\) 4.88129 0.351363 0.175681 0.984447i \(-0.443787\pi\)
0.175681 + 0.984447i \(0.443787\pi\)
\(194\) 2.04491 0.146816
\(195\) −2.38787 −0.170999
\(196\) −1.08110 −0.0772216
\(197\) −24.2374 −1.72685 −0.863423 0.504481i \(-0.831684\pi\)
−0.863423 + 0.504481i \(0.831684\pi\)
\(198\) 14.4690 1.02827
\(199\) 16.7513 1.18747 0.593734 0.804661i \(-0.297653\pi\)
0.593734 + 0.804661i \(0.297653\pi\)
\(200\) −2.67513 −0.189160
\(201\) −4.71037 −0.332244
\(202\) 19.2750 1.35619
\(203\) 0 0
\(204\) 0.860818 0.0602693
\(205\) 11.2750 0.787483
\(206\) 7.87987 0.549017
\(207\) −4.33312 −0.301173
\(208\) −12.8872 −0.893564
\(209\) −13.2750 −0.918254
\(210\) −1.42548 −0.0983678
\(211\) 25.3054 1.74209 0.871046 0.491201i \(-0.163442\pi\)
0.871046 + 0.491201i \(0.163442\pi\)
\(212\) −0.261865 −0.0179850
\(213\) 1.02776 0.0704211
\(214\) −20.4690 −1.39923
\(215\) 0.0303172 0.00206761
\(216\) −11.5369 −0.784987
\(217\) 5.73813 0.389530
\(218\) −2.77575 −0.187997
\(219\) −12.2882 −0.830360
\(220\) −0.806063 −0.0543448
\(221\) −16.3127 −1.09731
\(222\) −11.3503 −0.761780
\(223\) 17.6932 1.18483 0.592413 0.805634i \(-0.298175\pi\)
0.592413 + 0.805634i \(0.298175\pi\)
\(224\) −1.30536 −0.0872178
\(225\) −2.35026 −0.156684
\(226\) 17.4314 1.15952
\(227\) 26.8423 1.78158 0.890792 0.454412i \(-0.150151\pi\)
0.890792 + 0.454412i \(0.150151\pi\)
\(228\) −0.499293 −0.0330665
\(229\) 17.2243 1.13821 0.569105 0.822265i \(-0.307290\pi\)
0.569105 + 0.822265i \(0.307290\pi\)
\(230\) 2.73084 0.180066
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −9.07381 −0.594445 −0.297222 0.954808i \(-0.596060\pi\)
−0.297222 + 0.954808i \(0.596060\pi\)
\(234\) −10.3127 −0.674159
\(235\) −4.80606 −0.313513
\(236\) 2.57452 0.167587
\(237\) −3.97556 −0.258241
\(238\) −9.73813 −0.631230
\(239\) 20.4993 1.32599 0.662995 0.748624i \(-0.269285\pi\)
0.662995 + 0.748624i \(0.269285\pi\)
\(240\) 3.50659 0.226349
\(241\) 5.47627 0.352758 0.176379 0.984322i \(-0.443562\pi\)
0.176379 + 0.984322i \(0.443562\pi\)
\(242\) 9.29455 0.597476
\(243\) −15.8192 −1.01480
\(244\) −1.72355 −0.110339
\(245\) −5.57452 −0.356143
\(246\) −13.4617 −0.858285
\(247\) 9.46168 0.602032
\(248\) −12.8568 −0.816411
\(249\) −3.56134 −0.225691
\(250\) 1.48119 0.0936790
\(251\) 29.6180 1.86947 0.934736 0.355343i \(-0.115636\pi\)
0.934736 + 0.355343i \(0.115636\pi\)
\(252\) −0.544198 −0.0342813
\(253\) −7.66291 −0.481763
\(254\) −21.1333 −1.32602
\(255\) 4.43866 0.277960
\(256\) 4.61213 0.288258
\(257\) 17.6629 1.10178 0.550891 0.834577i \(-0.314288\pi\)
0.550891 + 0.834577i \(0.314288\pi\)
\(258\) −0.0361968 −0.00225351
\(259\) 11.3503 0.705271
\(260\) 0.574515 0.0356299
\(261\) 0 0
\(262\) −8.73084 −0.539393
\(263\) −27.3561 −1.68685 −0.843426 0.537245i \(-0.819465\pi\)
−0.843426 + 0.537245i \(0.819465\pi\)
\(264\) −8.96239 −0.551597
\(265\) −1.35026 −0.0829459
\(266\) 5.64832 0.346321
\(267\) −2.91160 −0.178187
\(268\) 1.13330 0.0692275
\(269\) −10.4993 −0.640153 −0.320077 0.947392i \(-0.603709\pi\)
−0.320077 + 0.947392i \(0.603709\pi\)
\(270\) 6.38787 0.388754
\(271\) 9.61801 0.584252 0.292126 0.956380i \(-0.405637\pi\)
0.292126 + 0.956380i \(0.405637\pi\)
\(272\) 23.9551 1.45249
\(273\) −2.85097 −0.172548
\(274\) −27.0797 −1.63594
\(275\) −4.15633 −0.250636
\(276\) −0.288213 −0.0173484
\(277\) 13.3503 0.802139 0.401070 0.916048i \(-0.368638\pi\)
0.401070 + 0.916048i \(0.368638\pi\)
\(278\) −17.0884 −1.02489
\(279\) −11.2955 −0.676244
\(280\) −3.19394 −0.190874
\(281\) 20.4241 1.21840 0.609199 0.793017i \(-0.291491\pi\)
0.609199 + 0.793017i \(0.291491\pi\)
\(282\) 5.73813 0.341701
\(283\) −8.02047 −0.476767 −0.238384 0.971171i \(-0.576618\pi\)
−0.238384 + 0.971171i \(0.576618\pi\)
\(284\) −0.247277 −0.0146732
\(285\) −2.57452 −0.152501
\(286\) −18.2374 −1.07840
\(287\) 13.4617 0.794618
\(288\) 2.56959 0.151415
\(289\) 13.3225 0.783676
\(290\) 0 0
\(291\) −1.11283 −0.0652355
\(292\) 2.95651 0.173017
\(293\) 23.3054 1.36151 0.680757 0.732510i \(-0.261651\pi\)
0.680757 + 0.732510i \(0.261651\pi\)
\(294\) 6.65562 0.388164
\(295\) 13.2750 0.772903
\(296\) −25.4314 −1.47817
\(297\) −17.9248 −1.04010
\(298\) 4.11142 0.238168
\(299\) 5.46168 0.315857
\(300\) −0.156325 −0.00902544
\(301\) 0.0361968 0.00208635
\(302\) 2.66433 0.153315
\(303\) −10.4894 −0.602603
\(304\) −13.8945 −0.796902
\(305\) −8.88717 −0.508878
\(306\) 19.1695 1.09585
\(307\) 6.73084 0.384149 0.192075 0.981380i \(-0.438478\pi\)
0.192075 + 0.981380i \(0.438478\pi\)
\(308\) −0.962389 −0.0548372
\(309\) −4.28821 −0.243948
\(310\) 7.11871 0.404316
\(311\) 22.0567 1.25072 0.625359 0.780337i \(-0.284952\pi\)
0.625359 + 0.780337i \(0.284952\pi\)
\(312\) 6.38787 0.361642
\(313\) 5.03761 0.284743 0.142371 0.989813i \(-0.454527\pi\)
0.142371 + 0.989813i \(0.454527\pi\)
\(314\) −5.58181 −0.315000
\(315\) −2.80606 −0.158104
\(316\) 0.956509 0.0538078
\(317\) −34.2941 −1.92615 −0.963074 0.269237i \(-0.913229\pi\)
−0.963074 + 0.269237i \(0.913229\pi\)
\(318\) 1.61213 0.0904036
\(319\) 0 0
\(320\) 7.08110 0.395846
\(321\) 11.1392 0.621729
\(322\) 3.26045 0.181698
\(323\) −17.5877 −0.978605
\(324\) 0.693229 0.0385127
\(325\) 2.96239 0.164324
\(326\) −2.43278 −0.134739
\(327\) 1.51056 0.0835340
\(328\) −30.1622 −1.66543
\(329\) −5.73813 −0.316354
\(330\) 4.96239 0.273171
\(331\) 34.8324 1.91456 0.957281 0.289159i \(-0.0933755\pi\)
0.957281 + 0.289159i \(0.0933755\pi\)
\(332\) 0.856849 0.0470257
\(333\) −22.3430 −1.22439
\(334\) 11.9697 0.654952
\(335\) 5.84367 0.319274
\(336\) 4.18664 0.228400
\(337\) 17.6326 0.960509 0.480254 0.877129i \(-0.340544\pi\)
0.480254 + 0.877129i \(0.340544\pi\)
\(338\) −6.25694 −0.340333
\(339\) −9.48612 −0.515215
\(340\) −1.06793 −0.0579166
\(341\) −19.9756 −1.08174
\(342\) −11.1187 −0.601231
\(343\) −15.0132 −0.810635
\(344\) −0.0811024 −0.00437275
\(345\) −1.48612 −0.0800099
\(346\) −11.4617 −0.616184
\(347\) −3.11871 −0.167421 −0.0837107 0.996490i \(-0.526677\pi\)
−0.0837107 + 0.996490i \(0.526677\pi\)
\(348\) 0 0
\(349\) −13.0738 −0.699825 −0.349912 0.936782i \(-0.613789\pi\)
−0.349912 + 0.936782i \(0.613789\pi\)
\(350\) 1.76845 0.0945277
\(351\) 12.7757 0.681919
\(352\) 4.54420 0.242207
\(353\) 5.19982 0.276758 0.138379 0.990379i \(-0.455811\pi\)
0.138379 + 0.990379i \(0.455811\pi\)
\(354\) −15.8496 −0.842394
\(355\) −1.27504 −0.0676720
\(356\) 0.700523 0.0371277
\(357\) 5.29948 0.280478
\(358\) −31.6991 −1.67535
\(359\) −30.4182 −1.60541 −0.802705 0.596376i \(-0.796607\pi\)
−0.802705 + 0.596376i \(0.796607\pi\)
\(360\) 6.28726 0.331368
\(361\) −8.79877 −0.463093
\(362\) 22.6253 1.18916
\(363\) −5.05808 −0.265480
\(364\) 0.685935 0.0359528
\(365\) 15.2447 0.797945
\(366\) 10.6107 0.554631
\(367\) −20.6556 −1.07821 −0.539107 0.842237i \(-0.681238\pi\)
−0.539107 + 0.842237i \(0.681238\pi\)
\(368\) −8.02047 −0.418096
\(369\) −26.4993 −1.37950
\(370\) 14.0811 0.732042
\(371\) −1.61213 −0.0836975
\(372\) −0.751309 −0.0389535
\(373\) 11.0884 0.574135 0.287068 0.957910i \(-0.407320\pi\)
0.287068 + 0.957910i \(0.407320\pi\)
\(374\) 33.9003 1.75294
\(375\) −0.806063 −0.0416249
\(376\) 12.8568 0.663041
\(377\) 0 0
\(378\) 7.62672 0.392276
\(379\) −10.0811 −0.517831 −0.258916 0.965900i \(-0.583365\pi\)
−0.258916 + 0.965900i \(0.583365\pi\)
\(380\) 0.619421 0.0317756
\(381\) 11.5007 0.589199
\(382\) −4.91748 −0.251600
\(383\) 16.3576 0.835832 0.417916 0.908486i \(-0.362761\pi\)
0.417916 + 0.908486i \(0.362761\pi\)
\(384\) −10.2170 −0.521382
\(385\) −4.96239 −0.252907
\(386\) 7.23013 0.368004
\(387\) −0.0712533 −0.00362201
\(388\) 0.267745 0.0135927
\(389\) −31.9003 −1.61741 −0.808706 0.588213i \(-0.799831\pi\)
−0.808706 + 0.588213i \(0.799831\pi\)
\(390\) −3.53690 −0.179098
\(391\) −10.1524 −0.513427
\(392\) 14.9126 0.753198
\(393\) 4.75131 0.239672
\(394\) −35.9003 −1.80863
\(395\) 4.93207 0.248159
\(396\) 1.89446 0.0952002
\(397\) 2.98683 0.149905 0.0749523 0.997187i \(-0.476120\pi\)
0.0749523 + 0.997187i \(0.476120\pi\)
\(398\) 24.8119 1.24371
\(399\) −3.07381 −0.153883
\(400\) −4.35026 −0.217513
\(401\) −21.9756 −1.09741 −0.548704 0.836017i \(-0.684878\pi\)
−0.548704 + 0.836017i \(0.684878\pi\)
\(402\) −6.97698 −0.347980
\(403\) 14.2374 0.709217
\(404\) 2.52373 0.125560
\(405\) 3.57452 0.177619
\(406\) 0 0
\(407\) −39.5125 −1.95856
\(408\) −11.8740 −0.587850
\(409\) 22.4387 1.10952 0.554760 0.832010i \(-0.312810\pi\)
0.554760 + 0.832010i \(0.312810\pi\)
\(410\) 16.7005 0.824780
\(411\) 14.7367 0.726909
\(412\) 1.03173 0.0508298
\(413\) 15.8496 0.779906
\(414\) −6.41819 −0.315437
\(415\) 4.41819 0.216880
\(416\) −3.23884 −0.158797
\(417\) 9.29948 0.455397
\(418\) −19.6629 −0.961744
\(419\) 10.3634 0.506287 0.253143 0.967429i \(-0.418536\pi\)
0.253143 + 0.967429i \(0.418536\pi\)
\(420\) −0.186642 −0.00910721
\(421\) −34.0362 −1.65882 −0.829411 0.558638i \(-0.811324\pi\)
−0.829411 + 0.558638i \(0.811324\pi\)
\(422\) 37.4821 1.82460
\(423\) 11.2955 0.549206
\(424\) 3.61213 0.175420
\(425\) −5.50659 −0.267109
\(426\) 1.52232 0.0737564
\(427\) −10.6107 −0.513488
\(428\) −2.68006 −0.129545
\(429\) 9.92478 0.479173
\(430\) 0.0449056 0.00216554
\(431\) 25.7743 1.24151 0.620753 0.784006i \(-0.286827\pi\)
0.620753 + 0.784006i \(0.286827\pi\)
\(432\) −18.7612 −0.902647
\(433\) −2.18076 −0.104801 −0.0524004 0.998626i \(-0.516687\pi\)
−0.0524004 + 0.998626i \(0.516687\pi\)
\(434\) 8.49929 0.407979
\(435\) 0 0
\(436\) −0.363436 −0.0174054
\(437\) 5.88858 0.281689
\(438\) −18.2012 −0.869688
\(439\) −35.5125 −1.69492 −0.847459 0.530861i \(-0.821869\pi\)
−0.847459 + 0.530861i \(0.821869\pi\)
\(440\) 11.1187 0.530064
\(441\) 13.1016 0.623884
\(442\) −24.1622 −1.14928
\(443\) 4.34297 0.206341 0.103170 0.994664i \(-0.467101\pi\)
0.103170 + 0.994664i \(0.467101\pi\)
\(444\) −1.48612 −0.0705281
\(445\) 3.61213 0.171231
\(446\) 26.2071 1.24094
\(447\) −2.23743 −0.105827
\(448\) 8.45439 0.399432
\(449\) −31.3357 −1.47882 −0.739411 0.673254i \(-0.764896\pi\)
−0.739411 + 0.673254i \(0.764896\pi\)
\(450\) −3.48119 −0.164105
\(451\) −46.8627 −2.20668
\(452\) 2.28233 0.107352
\(453\) −1.44992 −0.0681233
\(454\) 39.7586 1.86596
\(455\) 3.53690 0.165813
\(456\) 6.88717 0.322521
\(457\) 34.3488 1.60677 0.803386 0.595459i \(-0.203030\pi\)
0.803386 + 0.595459i \(0.203030\pi\)
\(458\) 25.5125 1.19212
\(459\) −23.7480 −1.10846
\(460\) 0.357556 0.0166711
\(461\) −11.8641 −0.552568 −0.276284 0.961076i \(-0.589103\pi\)
−0.276284 + 0.961076i \(0.589103\pi\)
\(462\) 5.92478 0.275646
\(463\) 40.4953 1.88198 0.940989 0.338438i \(-0.109899\pi\)
0.940989 + 0.338438i \(0.109899\pi\)
\(464\) 0 0
\(465\) −3.87399 −0.179652
\(466\) −13.4401 −0.622599
\(467\) 30.2071 1.39782 0.698909 0.715210i \(-0.253669\pi\)
0.698909 + 0.715210i \(0.253669\pi\)
\(468\) −1.35026 −0.0624159
\(469\) 6.97698 0.322167
\(470\) −7.11871 −0.328362
\(471\) 3.03761 0.139966
\(472\) −35.5125 −1.63459
\(473\) −0.126008 −0.00579385
\(474\) −5.88858 −0.270471
\(475\) 3.19394 0.146548
\(476\) −1.27504 −0.0584413
\(477\) 3.17347 0.145303
\(478\) 30.3634 1.38879
\(479\) −0.0547547 −0.00250181 −0.00125090 0.999999i \(-0.500398\pi\)
−0.00125090 + 0.999999i \(0.500398\pi\)
\(480\) 0.881286 0.0402250
\(481\) 28.1622 1.28409
\(482\) 8.11142 0.369465
\(483\) −1.77433 −0.0807349
\(484\) 1.21696 0.0553163
\(485\) 1.38058 0.0626889
\(486\) −23.4314 −1.06287
\(487\) −0.881286 −0.0399349 −0.0199674 0.999801i \(-0.506356\pi\)
−0.0199674 + 0.999801i \(0.506356\pi\)
\(488\) 23.7743 1.07621
\(489\) 1.32391 0.0598695
\(490\) −8.25694 −0.373011
\(491\) −41.0698 −1.85346 −0.926728 0.375733i \(-0.877391\pi\)
−0.926728 + 0.375733i \(0.877391\pi\)
\(492\) −1.76257 −0.0794629
\(493\) 0 0
\(494\) 14.0146 0.630546
\(495\) 9.76845 0.439059
\(496\) −20.9076 −0.938780
\(497\) −1.52232 −0.0682852
\(498\) −5.27504 −0.236380
\(499\) 12.3733 0.553904 0.276952 0.960884i \(-0.410676\pi\)
0.276952 + 0.960884i \(0.410676\pi\)
\(500\) 0.193937 0.00867311
\(501\) −6.51388 −0.291019
\(502\) 43.8700 1.95801
\(503\) 2.26774 0.101114 0.0505569 0.998721i \(-0.483900\pi\)
0.0505569 + 0.998721i \(0.483900\pi\)
\(504\) 7.50659 0.334370
\(505\) 13.0132 0.579079
\(506\) −11.3503 −0.504581
\(507\) 3.40502 0.151222
\(508\) −2.76704 −0.122767
\(509\) −10.9018 −0.483212 −0.241606 0.970374i \(-0.577674\pi\)
−0.241606 + 0.970374i \(0.577674\pi\)
\(510\) 6.57452 0.291124
\(511\) 18.2012 0.805175
\(512\) −18.5188 −0.818423
\(513\) 13.7743 0.608152
\(514\) 26.1622 1.15397
\(515\) 5.31994 0.234425
\(516\) −0.00473934 −0.000208638 0
\(517\) 19.9756 0.878524
\(518\) 16.8119 0.738674
\(519\) 6.23743 0.273793
\(520\) −7.92478 −0.347524
\(521\) −4.72496 −0.207004 −0.103502 0.994629i \(-0.533005\pi\)
−0.103502 + 0.994629i \(0.533005\pi\)
\(522\) 0 0
\(523\) −1.06793 −0.0466973 −0.0233486 0.999727i \(-0.507433\pi\)
−0.0233486 + 0.999727i \(0.507433\pi\)
\(524\) −1.14315 −0.0499388
\(525\) −0.962389 −0.0420021
\(526\) −40.5198 −1.76675
\(527\) −26.4650 −1.15283
\(528\) −14.5745 −0.634274
\(529\) −19.6009 −0.852211
\(530\) −2.00000 −0.0868744
\(531\) −31.1998 −1.35396
\(532\) 0.739549 0.0320635
\(533\) 33.4010 1.44676
\(534\) −4.31265 −0.186627
\(535\) −13.8192 −0.597458
\(536\) −15.6326 −0.675225
\(537\) 17.2506 0.744418
\(538\) −15.5515 −0.670472
\(539\) 23.1695 0.997981
\(540\) 0.836381 0.0359921
\(541\) 7.46168 0.320803 0.160401 0.987052i \(-0.448721\pi\)
0.160401 + 0.987052i \(0.448721\pi\)
\(542\) 14.2461 0.611924
\(543\) −12.3127 −0.528386
\(544\) 6.02047 0.258125
\(545\) −1.87399 −0.0802730
\(546\) −4.22284 −0.180721
\(547\) −38.9683 −1.66616 −0.833081 0.553150i \(-0.813425\pi\)
−0.833081 + 0.553150i \(0.813425\pi\)
\(548\) −3.54561 −0.151461
\(549\) 20.8872 0.891443
\(550\) −6.15633 −0.262507
\(551\) 0 0
\(552\) 3.97556 0.169211
\(553\) 5.88858 0.250408
\(554\) 19.7743 0.840131
\(555\) −7.66291 −0.325273
\(556\) −2.23743 −0.0948881
\(557\) −22.9986 −0.974481 −0.487241 0.873268i \(-0.661997\pi\)
−0.487241 + 0.873268i \(0.661997\pi\)
\(558\) −16.7308 −0.708273
\(559\) 0.0898112 0.00379861
\(560\) −5.19394 −0.219484
\(561\) −18.4485 −0.778897
\(562\) 30.2520 1.27610
\(563\) −11.6688 −0.491781 −0.245890 0.969298i \(-0.579080\pi\)
−0.245890 + 0.969298i \(0.579080\pi\)
\(564\) 0.751309 0.0316358
\(565\) 11.7685 0.495102
\(566\) −11.8799 −0.499348
\(567\) 4.26774 0.179228
\(568\) 3.41090 0.143118
\(569\) −11.3357 −0.475216 −0.237608 0.971361i \(-0.576363\pi\)
−0.237608 + 0.971361i \(0.576363\pi\)
\(570\) −3.81336 −0.159724
\(571\) 27.1754 1.13725 0.568627 0.822595i \(-0.307475\pi\)
0.568627 + 0.822595i \(0.307475\pi\)
\(572\) −2.38787 −0.0998420
\(573\) 2.67609 0.111795
\(574\) 19.9394 0.832253
\(575\) 1.84367 0.0768866
\(576\) −16.6424 −0.693435
\(577\) −22.5950 −0.940641 −0.470321 0.882496i \(-0.655862\pi\)
−0.470321 + 0.882496i \(0.655862\pi\)
\(578\) 19.7332 0.820793
\(579\) −3.93463 −0.163517
\(580\) 0 0
\(581\) 5.27504 0.218845
\(582\) −1.64832 −0.0683252
\(583\) 5.61213 0.232431
\(584\) −40.7816 −1.68756
\(585\) −6.96239 −0.287859
\(586\) 34.5198 1.42600
\(587\) 9.31994 0.384675 0.192338 0.981329i \(-0.438393\pi\)
0.192338 + 0.981329i \(0.438393\pi\)
\(588\) 0.871437 0.0359375
\(589\) 15.3503 0.632497
\(590\) 19.6629 0.809509
\(591\) 19.5369 0.803641
\(592\) −41.3561 −1.69973
\(593\) −15.1246 −0.621093 −0.310546 0.950558i \(-0.600512\pi\)
−0.310546 + 0.950558i \(0.600512\pi\)
\(594\) −26.5501 −1.08936
\(595\) −6.57452 −0.269529
\(596\) 0.538319 0.0220504
\(597\) −13.5026 −0.552625
\(598\) 8.08981 0.330817
\(599\) −4.09569 −0.167345 −0.0836727 0.996493i \(-0.526665\pi\)
−0.0836727 + 0.996493i \(0.526665\pi\)
\(600\) 2.15633 0.0880316
\(601\) −22.2276 −0.906682 −0.453341 0.891337i \(-0.649768\pi\)
−0.453341 + 0.891337i \(0.649768\pi\)
\(602\) 0.0536145 0.00218516
\(603\) −13.7342 −0.559298
\(604\) 0.348847 0.0141944
\(605\) 6.27504 0.255117
\(606\) −15.5369 −0.631144
\(607\) 48.2941 1.96020 0.980098 0.198512i \(-0.0636110\pi\)
0.980098 + 0.198512i \(0.0636110\pi\)
\(608\) −3.49200 −0.141619
\(609\) 0 0
\(610\) −13.1636 −0.532979
\(611\) −14.2374 −0.575985
\(612\) 2.50991 0.101457
\(613\) −9.74798 −0.393717 −0.196859 0.980432i \(-0.563074\pi\)
−0.196859 + 0.980432i \(0.563074\pi\)
\(614\) 9.96968 0.402344
\(615\) −9.08840 −0.366480
\(616\) 13.2750 0.534867
\(617\) −18.2170 −0.733387 −0.366694 0.930342i \(-0.619510\pi\)
−0.366694 + 0.930342i \(0.619510\pi\)
\(618\) −6.35168 −0.255502
\(619\) −25.0943 −1.00862 −0.504312 0.863521i \(-0.668254\pi\)
−0.504312 + 0.863521i \(0.668254\pi\)
\(620\) 0.932071 0.0374329
\(621\) 7.95112 0.319068
\(622\) 32.6702 1.30996
\(623\) 4.31265 0.172783
\(624\) 10.3879 0.415848
\(625\) 1.00000 0.0400000
\(626\) 7.46168 0.298229
\(627\) 10.7005 0.427338
\(628\) −0.730841 −0.0291637
\(629\) −52.3488 −2.08729
\(630\) −4.15633 −0.165592
\(631\) 21.4617 0.854376 0.427188 0.904163i \(-0.359504\pi\)
0.427188 + 0.904163i \(0.359504\pi\)
\(632\) −13.1939 −0.524827
\(633\) −20.3977 −0.810737
\(634\) −50.7962 −2.01738
\(635\) −14.2677 −0.566198
\(636\) 0.211080 0.00836986
\(637\) −16.5139 −0.654304
\(638\) 0 0
\(639\) 2.99668 0.118547
\(640\) 12.6751 0.501029
\(641\) −3.17347 −0.125344 −0.0626722 0.998034i \(-0.519962\pi\)
−0.0626722 + 0.998034i \(0.519962\pi\)
\(642\) 16.4993 0.651175
\(643\) −2.74069 −0.108082 −0.0540411 0.998539i \(-0.517210\pi\)
−0.0540411 + 0.998539i \(0.517210\pi\)
\(644\) 0.426899 0.0168222
\(645\) −0.0244376 −0.000962228 0
\(646\) −26.0508 −1.02495
\(647\) 6.34297 0.249368 0.124684 0.992197i \(-0.460208\pi\)
0.124684 + 0.992197i \(0.460208\pi\)
\(648\) −9.56230 −0.375642
\(649\) −55.1754 −2.16582
\(650\) 4.38787 0.172107
\(651\) −4.62530 −0.181280
\(652\) −0.318530 −0.0124746
\(653\) −4.08110 −0.159706 −0.0798529 0.996807i \(-0.525445\pi\)
−0.0798529 + 0.996807i \(0.525445\pi\)
\(654\) 2.23743 0.0874903
\(655\) −5.89446 −0.230316
\(656\) −49.0494 −1.91506
\(657\) −35.8291 −1.39783
\(658\) −8.49929 −0.331337
\(659\) −9.58181 −0.373254 −0.186627 0.982431i \(-0.559756\pi\)
−0.186627 + 0.982431i \(0.559756\pi\)
\(660\) 0.649738 0.0252910
\(661\) −27.5271 −1.07068 −0.535339 0.844637i \(-0.679816\pi\)
−0.535339 + 0.844637i \(0.679816\pi\)
\(662\) 51.5936 2.00524
\(663\) 13.1490 0.510666
\(664\) −11.8192 −0.458675
\(665\) 3.81336 0.147876
\(666\) −33.0943 −1.28238
\(667\) 0 0
\(668\) 1.56722 0.0606376
\(669\) −14.2619 −0.551396
\(670\) 8.65562 0.334396
\(671\) 36.9380 1.42597
\(672\) 1.05220 0.0405895
\(673\) 3.13727 0.120933 0.0604665 0.998170i \(-0.480741\pi\)
0.0604665 + 0.998170i \(0.480741\pi\)
\(674\) 26.1173 1.00600
\(675\) 4.31265 0.165994
\(676\) −0.819237 −0.0315091
\(677\) 46.2579 1.77784 0.888918 0.458067i \(-0.151458\pi\)
0.888918 + 0.458067i \(0.151458\pi\)
\(678\) −14.0508 −0.539617
\(679\) 1.64832 0.0632569
\(680\) 14.7308 0.564902
\(681\) −21.6366 −0.829115
\(682\) −29.5877 −1.13297
\(683\) −9.01905 −0.345104 −0.172552 0.985000i \(-0.555201\pi\)
−0.172552 + 0.985000i \(0.555201\pi\)
\(684\) −1.45580 −0.0556640
\(685\) −18.2823 −0.698532
\(686\) −22.2374 −0.849029
\(687\) −13.8838 −0.529702
\(688\) −0.131888 −0.00502817
\(689\) −4.00000 −0.152388
\(690\) −2.20123 −0.0837994
\(691\) −50.0625 −1.90447 −0.952234 0.305368i \(-0.901221\pi\)
−0.952234 + 0.305368i \(0.901221\pi\)
\(692\) −1.50071 −0.0570483
\(693\) 11.6629 0.443037
\(694\) −4.61942 −0.175351
\(695\) −11.5369 −0.437620
\(696\) 0 0
\(697\) −62.0870 −2.35171
\(698\) −19.3649 −0.732970
\(699\) 7.31406 0.276643
\(700\) 0.231548 0.00875169
\(701\) 45.3014 1.71101 0.855505 0.517795i \(-0.173247\pi\)
0.855505 + 0.517795i \(0.173247\pi\)
\(702\) 18.9234 0.714216
\(703\) 30.3634 1.14518
\(704\) −29.4314 −1.10924
\(705\) 3.87399 0.145903
\(706\) 7.70194 0.289866
\(707\) 15.5369 0.584325
\(708\) −2.07522 −0.0779916
\(709\) −3.27504 −0.122997 −0.0614983 0.998107i \(-0.519588\pi\)
−0.0614983 + 0.998107i \(0.519588\pi\)
\(710\) −1.88858 −0.0708772
\(711\) −11.5917 −0.434721
\(712\) −9.66291 −0.362133
\(713\) 8.86082 0.331840
\(714\) 7.84955 0.293762
\(715\) −12.3127 −0.460467
\(716\) −4.15045 −0.155109
\(717\) −16.5237 −0.617090
\(718\) −45.0553 −1.68145
\(719\) 27.7235 1.03391 0.516957 0.856011i \(-0.327065\pi\)
0.516957 + 0.856011i \(0.327065\pi\)
\(720\) 10.2243 0.381035
\(721\) 6.35168 0.236549
\(722\) −13.0327 −0.485026
\(723\) −4.41422 −0.164167
\(724\) 2.96239 0.110096
\(725\) 0 0
\(726\) −7.49200 −0.278054
\(727\) 26.8930 0.997408 0.498704 0.866772i \(-0.333810\pi\)
0.498704 + 0.866772i \(0.333810\pi\)
\(728\) −9.46168 −0.350673
\(729\) 2.02776 0.0751023
\(730\) 22.5804 0.835738
\(731\) −0.166944 −0.00617465
\(732\) 1.38929 0.0513496
\(733\) −3.17935 −0.117432 −0.0587160 0.998275i \(-0.518701\pi\)
−0.0587160 + 0.998275i \(0.518701\pi\)
\(734\) −30.5950 −1.12928
\(735\) 4.49341 0.165742
\(736\) −2.01573 −0.0743007
\(737\) −24.2882 −0.894668
\(738\) −39.2506 −1.44483
\(739\) 29.7440 1.09415 0.547076 0.837083i \(-0.315741\pi\)
0.547076 + 0.837083i \(0.315741\pi\)
\(740\) 1.84367 0.0677748
\(741\) −7.62672 −0.280174
\(742\) −2.38787 −0.0876616
\(743\) 4.34297 0.159328 0.0796640 0.996822i \(-0.474615\pi\)
0.0796640 + 0.996822i \(0.474615\pi\)
\(744\) 10.3634 0.379942
\(745\) 2.77575 0.101695
\(746\) 16.4241 0.601328
\(747\) −10.3839 −0.379927
\(748\) 4.43866 0.162293
\(749\) −16.4993 −0.602871
\(750\) −1.19394 −0.0435964
\(751\) −22.5804 −0.823970 −0.411985 0.911191i \(-0.635164\pi\)
−0.411985 + 0.911191i \(0.635164\pi\)
\(752\) 20.9076 0.762423
\(753\) −23.8740 −0.870017
\(754\) 0 0
\(755\) 1.79877 0.0654639
\(756\) 0.998585 0.0363182
\(757\) −9.88461 −0.359262 −0.179631 0.983734i \(-0.557490\pi\)
−0.179631 + 0.983734i \(0.557490\pi\)
\(758\) −14.9321 −0.542357
\(759\) 6.17679 0.224203
\(760\) −8.54420 −0.309931
\(761\) 13.6991 0.496592 0.248296 0.968684i \(-0.420129\pi\)
0.248296 + 0.968684i \(0.420129\pi\)
\(762\) 17.0348 0.617105
\(763\) −2.23743 −0.0810003
\(764\) −0.643859 −0.0232940
\(765\) 12.9419 0.467916
\(766\) 24.2287 0.875419
\(767\) 39.3258 1.41997
\(768\) −3.71767 −0.134150
\(769\) −25.0132 −0.901998 −0.450999 0.892524i \(-0.648932\pi\)
−0.450999 + 0.892524i \(0.648932\pi\)
\(770\) −7.35026 −0.264885
\(771\) −14.2374 −0.512748
\(772\) 0.946660 0.0340710
\(773\) 35.9062 1.29146 0.645728 0.763567i \(-0.276554\pi\)
0.645728 + 0.763567i \(0.276554\pi\)
\(774\) −0.105540 −0.00379356
\(775\) 4.80606 0.172639
\(776\) −3.69323 −0.132579
\(777\) −9.14903 −0.328220
\(778\) −47.2506 −1.69402
\(779\) 36.0118 1.29026
\(780\) −0.463096 −0.0165815
\(781\) 5.29948 0.189630
\(782\) −15.0376 −0.537744
\(783\) 0 0
\(784\) 24.2506 0.866093
\(785\) −3.76845 −0.134502
\(786\) 7.03761 0.251023
\(787\) 50.3839 1.79599 0.897996 0.440003i \(-0.145023\pi\)
0.897996 + 0.440003i \(0.145023\pi\)
\(788\) −4.70052 −0.167449
\(789\) 22.0508 0.785029
\(790\) 7.30536 0.259913
\(791\) 14.0508 0.499588
\(792\) −26.1319 −0.928556
\(793\) −26.3272 −0.934908
\(794\) 4.42407 0.157004
\(795\) 1.08840 0.0386014
\(796\) 3.24869 0.115147
\(797\) 5.69323 0.201665 0.100832 0.994903i \(-0.467849\pi\)
0.100832 + 0.994903i \(0.467849\pi\)
\(798\) −4.55291 −0.161171
\(799\) 26.4650 0.936265
\(800\) −1.09332 −0.0386547
\(801\) −8.48944 −0.299960
\(802\) −32.5501 −1.14938
\(803\) −63.3620 −2.23600
\(804\) −0.913513 −0.0322171
\(805\) 2.20123 0.0775832
\(806\) 21.0884 0.742807
\(807\) 8.46310 0.297915
\(808\) −34.8119 −1.22468
\(809\) 7.76257 0.272918 0.136459 0.990646i \(-0.456428\pi\)
0.136459 + 0.990646i \(0.456428\pi\)
\(810\) 5.29455 0.186032
\(811\) −26.4894 −0.930170 −0.465085 0.885266i \(-0.653976\pi\)
−0.465085 + 0.885266i \(0.653976\pi\)
\(812\) 0 0
\(813\) −7.75272 −0.271900
\(814\) −58.5256 −2.05132
\(815\) −1.64244 −0.0575323
\(816\) −19.3093 −0.675962
\(817\) 0.0968311 0.00338769
\(818\) 33.2360 1.16207
\(819\) −8.31265 −0.290468
\(820\) 2.18664 0.0763609
\(821\) 25.4763 0.889128 0.444564 0.895747i \(-0.353359\pi\)
0.444564 + 0.895747i \(0.353359\pi\)
\(822\) 21.8279 0.761337
\(823\) 9.22028 0.321399 0.160699 0.987003i \(-0.448625\pi\)
0.160699 + 0.987003i \(0.448625\pi\)
\(824\) −14.2315 −0.495779
\(825\) 3.35026 0.116641
\(826\) 23.4763 0.816844
\(827\) −24.5343 −0.853143 −0.426571 0.904454i \(-0.640279\pi\)
−0.426571 + 0.904454i \(0.640279\pi\)
\(828\) −0.840350 −0.0292042
\(829\) −0.201231 −0.00698903 −0.00349452 0.999994i \(-0.501112\pi\)
−0.00349452 + 0.999994i \(0.501112\pi\)
\(830\) 6.54420 0.227152
\(831\) −10.7612 −0.373300
\(832\) 20.9770 0.727246
\(833\) 30.6966 1.06357
\(834\) 13.7743 0.476966
\(835\) 8.08110 0.279658
\(836\) −2.57452 −0.0890415
\(837\) 20.7269 0.716425
\(838\) 15.3503 0.530266
\(839\) −1.45580 −0.0502599 −0.0251299 0.999684i \(-0.508000\pi\)
−0.0251299 + 0.999684i \(0.508000\pi\)
\(840\) 2.57452 0.0888292
\(841\) 0 0
\(842\) −50.4142 −1.73739
\(843\) −16.4631 −0.567019
\(844\) 4.90763 0.168928
\(845\) −4.22425 −0.145319
\(846\) 16.7308 0.575218
\(847\) 7.49200 0.257428
\(848\) 5.87399 0.201714
\(849\) 6.46501 0.221878
\(850\) −8.15633 −0.279760
\(851\) 17.5271 0.600820
\(852\) 0.199321 0.00682861
\(853\) −43.1793 −1.47843 −0.739216 0.673468i \(-0.764804\pi\)
−0.739216 + 0.673468i \(0.764804\pi\)
\(854\) −15.7165 −0.537808
\(855\) −7.50659 −0.256720
\(856\) 36.9683 1.26355
\(857\) −20.9887 −0.716962 −0.358481 0.933537i \(-0.616705\pi\)
−0.358481 + 0.933537i \(0.616705\pi\)
\(858\) 14.7005 0.501868
\(859\) 49.4069 1.68574 0.842871 0.538115i \(-0.180863\pi\)
0.842871 + 0.538115i \(0.180863\pi\)
\(860\) 0.00587961 0.000200493 0
\(861\) −10.8510 −0.369800
\(862\) 38.1768 1.30031
\(863\) 56.6820 1.92948 0.964738 0.263211i \(-0.0847816\pi\)
0.964738 + 0.263211i \(0.0847816\pi\)
\(864\) −4.71511 −0.160411
\(865\) −7.73813 −0.263104
\(866\) −3.23013 −0.109764
\(867\) −10.7388 −0.364708
\(868\) 1.11283 0.0377721
\(869\) −20.4993 −0.695391
\(870\) 0 0
\(871\) 17.3112 0.586569
\(872\) 5.01317 0.169767
\(873\) −3.24472 −0.109817
\(874\) 8.72213 0.295031
\(875\) 1.19394 0.0403624
\(876\) −2.38313 −0.0805186
\(877\) −13.1998 −0.445726 −0.222863 0.974850i \(-0.571540\pi\)
−0.222863 + 0.974850i \(0.571540\pi\)
\(878\) −52.6009 −1.77519
\(879\) −18.7856 −0.633622
\(880\) 18.0811 0.609514
\(881\) −6.37802 −0.214881 −0.107441 0.994212i \(-0.534266\pi\)
−0.107441 + 0.994212i \(0.534266\pi\)
\(882\) 19.4060 0.653433
\(883\) 48.6213 1.63624 0.818119 0.575049i \(-0.195017\pi\)
0.818119 + 0.575049i \(0.195017\pi\)
\(884\) −3.16362 −0.106404
\(885\) −10.7005 −0.359694
\(886\) 6.43278 0.216113
\(887\) 15.0317 0.504716 0.252358 0.967634i \(-0.418794\pi\)
0.252358 + 0.967634i \(0.418794\pi\)
\(888\) 20.4993 0.687911
\(889\) −17.0348 −0.571328
\(890\) 5.35026 0.179341
\(891\) −14.8568 −0.497723
\(892\) 3.43136 0.114891
\(893\) −15.3503 −0.513677
\(894\) −3.31406 −0.110839
\(895\) −21.4010 −0.715358
\(896\) 15.1333 0.505568
\(897\) −4.40246 −0.146994
\(898\) −46.4142 −1.54886
\(899\) 0 0
\(900\) −0.455802 −0.0151934
\(901\) 7.43533 0.247707
\(902\) −69.4128 −2.31119
\(903\) −0.0291769 −0.000970946 0
\(904\) −31.4821 −1.04708
\(905\) 15.2750 0.507759
\(906\) −2.14762 −0.0713498
\(907\) 0.342968 0.0113880 0.00569402 0.999984i \(-0.498188\pi\)
0.00569402 + 0.999984i \(0.498188\pi\)
\(908\) 5.20570 0.172757
\(909\) −30.5844 −1.01442
\(910\) 5.23884 0.173666
\(911\) −20.9076 −0.692701 −0.346350 0.938105i \(-0.612579\pi\)
−0.346350 + 0.938105i \(0.612579\pi\)
\(912\) 11.1998 0.370863
\(913\) −18.3634 −0.607741
\(914\) 50.8773 1.68287
\(915\) 7.16362 0.236822
\(916\) 3.34041 0.110370
\(917\) −7.03761 −0.232402
\(918\) −35.1754 −1.16096
\(919\) −1.90034 −0.0626864 −0.0313432 0.999509i \(-0.509978\pi\)
−0.0313432 + 0.999509i \(0.509978\pi\)
\(920\) −4.93207 −0.162606
\(921\) −5.42548 −0.178776
\(922\) −17.5731 −0.578739
\(923\) −3.77716 −0.124327
\(924\) 0.775746 0.0255202
\(925\) 9.50659 0.312575
\(926\) 59.9814 1.97111
\(927\) −12.5033 −0.410661
\(928\) 0 0
\(929\) 39.3522 1.29110 0.645551 0.763717i \(-0.276628\pi\)
0.645551 + 0.763717i \(0.276628\pi\)
\(930\) −5.73813 −0.188161
\(931\) −17.8046 −0.583524
\(932\) −1.75974 −0.0576423
\(933\) −17.7791 −0.582061
\(934\) 44.7426 1.46402
\(935\) 22.8872 0.748490
\(936\) 18.6253 0.608787
\(937\) −6.37802 −0.208361 −0.104180 0.994558i \(-0.533222\pi\)
−0.104180 + 0.994558i \(0.533222\pi\)
\(938\) 10.3343 0.337426
\(939\) −4.06063 −0.132514
\(940\) −0.932071 −0.0304008
\(941\) −26.6253 −0.867960 −0.433980 0.900923i \(-0.642891\pi\)
−0.433980 + 0.900923i \(0.642891\pi\)
\(942\) 4.49929 0.146595
\(943\) 20.7875 0.676934
\(944\) −57.7499 −1.87960
\(945\) 5.14903 0.167498
\(946\) −0.186642 −0.00606827
\(947\) 12.2823 0.399122 0.199561 0.979885i \(-0.436048\pi\)
0.199561 + 0.979885i \(0.436048\pi\)
\(948\) −0.771007 −0.0250411
\(949\) 45.1608 1.46598
\(950\) 4.73084 0.153489
\(951\) 27.6432 0.896393
\(952\) 17.5877 0.570020
\(953\) 0.821792 0.0266205 0.0133102 0.999911i \(-0.495763\pi\)
0.0133102 + 0.999911i \(0.495763\pi\)
\(954\) 4.70052 0.152185
\(955\) −3.31994 −0.107431
\(956\) 3.97556 0.128579
\(957\) 0 0
\(958\) −0.0811024 −0.00262030
\(959\) −21.8279 −0.704861
\(960\) −5.70782 −0.184219
\(961\) −7.90175 −0.254895
\(962\) 41.7137 1.34490
\(963\) 32.4788 1.04662
\(964\) 1.06205 0.0342063
\(965\) 4.88129 0.157134
\(966\) −2.62813 −0.0845587
\(967\) 37.4314 1.20371 0.601856 0.798605i \(-0.294428\pi\)
0.601856 + 0.798605i \(0.294428\pi\)
\(968\) −16.7866 −0.539540
\(969\) 14.1768 0.455424
\(970\) 2.04491 0.0656580
\(971\) 8.71625 0.279718 0.139859 0.990171i \(-0.455335\pi\)
0.139859 + 0.990171i \(0.455335\pi\)
\(972\) −3.06793 −0.0984039
\(973\) −13.7743 −0.441585
\(974\) −1.30536 −0.0418263
\(975\) −2.38787 −0.0764731
\(976\) 38.6615 1.23752
\(977\) −33.7645 −1.08022 −0.540111 0.841594i \(-0.681618\pi\)
−0.540111 + 0.841594i \(0.681618\pi\)
\(978\) 1.96097 0.0627050
\(979\) −15.0132 −0.479823
\(980\) −1.08110 −0.0345345
\(981\) 4.40437 0.140621
\(982\) −60.8324 −1.94124
\(983\) −43.6082 −1.39088 −0.695442 0.718582i \(-0.744791\pi\)
−0.695442 + 0.718582i \(0.744791\pi\)
\(984\) 24.3127 0.775059
\(985\) −24.2374 −0.772269
\(986\) 0 0
\(987\) 4.62530 0.147225
\(988\) 1.83497 0.0583780
\(989\) 0.0558950 0.00177736
\(990\) 14.4690 0.459854
\(991\) −52.9741 −1.68278 −0.841390 0.540429i \(-0.818262\pi\)
−0.841390 + 0.540429i \(0.818262\pi\)
\(992\) −5.25457 −0.166833
\(993\) −28.0771 −0.891001
\(994\) −2.25485 −0.0715193
\(995\) 16.7513 0.531052
\(996\) −0.690674 −0.0218849
\(997\) −13.6326 −0.431749 −0.215874 0.976421i \(-0.569260\pi\)
−0.215874 + 0.976421i \(0.569260\pi\)
\(998\) 18.3272 0.580139
\(999\) 40.9986 1.29714
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 4205.2.a.f.1.3 3
29.28 even 2 145.2.a.c.1.1 3
87.86 odd 2 1305.2.a.p.1.3 3
116.115 odd 2 2320.2.a.n.1.2 3
145.28 odd 4 725.2.b.e.349.5 6
145.57 odd 4 725.2.b.e.349.2 6
145.144 even 2 725.2.a.e.1.3 3
203.202 odd 2 7105.2.a.o.1.1 3
232.115 odd 2 9280.2.a.br.1.2 3
232.173 even 2 9280.2.a.bj.1.2 3
435.434 odd 2 6525.2.a.be.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.1 3 29.28 even 2
725.2.a.e.1.3 3 145.144 even 2
725.2.b.e.349.2 6 145.57 odd 4
725.2.b.e.349.5 6 145.28 odd 4
1305.2.a.p.1.3 3 87.86 odd 2
2320.2.a.n.1.2 3 116.115 odd 2
4205.2.a.f.1.3 3 1.1 even 1 trivial
6525.2.a.be.1.1 3 435.434 odd 2
7105.2.a.o.1.1 3 203.202 odd 2
9280.2.a.bj.1.2 3 232.173 even 2
9280.2.a.br.1.2 3 232.115 odd 2