Properties

Label 4205.2.a.e.1.3
Level $4205$
Weight $2$
Character 4205.1
Self dual yes
Analytic conductor $33.577$
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] = [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: \(1\)
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 \(0.311108\) 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)
 
\(f(q)\) \(=\) \(q+1.21432 q^{2} +2.90321 q^{3} -0.525428 q^{4} -1.00000 q^{5} +3.52543 q^{6} +1.52543 q^{7} -3.06668 q^{8} +5.42864 q^{9} +O(q^{10})\) \(q+1.21432 q^{2} +2.90321 q^{3} -0.525428 q^{4} -1.00000 q^{5} +3.52543 q^{6} +1.52543 q^{7} -3.06668 q^{8} +5.42864 q^{9} -1.21432 q^{10} -4.90321 q^{11} -1.52543 q^{12} -6.42864 q^{13} +1.85236 q^{14} -2.90321 q^{15} -2.67307 q^{16} -2.14764 q^{17} +6.59210 q^{18} -2.28100 q^{19} +0.525428 q^{20} +4.42864 q^{21} -5.95407 q^{22} +6.90321 q^{23} -8.90321 q^{24} +1.00000 q^{25} -7.80642 q^{26} +7.05086 q^{27} -0.801502 q^{28} -3.52543 q^{30} -1.71900 q^{31} +2.88739 q^{32} -14.2351 q^{33} -2.60793 q^{34} -1.52543 q^{35} -2.85236 q^{36} -7.95407 q^{37} -2.76986 q^{38} -18.6637 q^{39} +3.06668 q^{40} +3.37778 q^{41} +5.37778 q^{42} +1.09679 q^{43} +2.57628 q^{44} -5.42864 q^{45} +8.38271 q^{46} -12.7096 q^{47} -7.76049 q^{48} -4.67307 q^{49} +1.21432 q^{50} -6.23506 q^{51} +3.37778 q^{52} +3.37778 q^{53} +8.56199 q^{54} +4.90321 q^{55} -4.67799 q^{56} -6.62222 q^{57} -3.18421 q^{59} +1.52543 q^{60} +2.42864 q^{61} -2.08742 q^{62} +8.28100 q^{63} +8.85236 q^{64} +6.42864 q^{65} -17.2859 q^{66} -1.09679 q^{67} +1.12843 q^{68} +20.0415 q^{69} -1.85236 q^{70} +3.57136 q^{71} -16.6479 q^{72} -14.1891 q^{73} -9.65878 q^{74} +2.90321 q^{75} +1.19850 q^{76} -7.47949 q^{77} -22.6637 q^{78} -0.341219 q^{79} +2.67307 q^{80} +4.18421 q^{81} +4.10171 q^{82} -7.33185 q^{83} -2.32693 q^{84} +2.14764 q^{85} +1.33185 q^{86} +15.0366 q^{88} -2.94914 q^{89} -6.59210 q^{90} -9.80642 q^{91} -3.62714 q^{92} -4.99063 q^{93} -15.4336 q^{94} +2.28100 q^{95} +8.38271 q^{96} +18.5763 q^{97} -5.67460 q^{98} -26.6178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 3 q^{10} - 8 q^{11} + 2 q^{12} - 6 q^{13} + 12 q^{14} - 2 q^{15} + 5 q^{16} + 13 q^{18} - 5 q^{20} + 2 q^{22} + 14 q^{23} - 20 q^{24} + 3 q^{25} - 10 q^{26} + 8 q^{27} - 22 q^{28} - 4 q^{30} - 12 q^{31} - 11 q^{32} - 16 q^{33} - 14 q^{34} + 2 q^{35} - 15 q^{36} - 4 q^{37} - 2 q^{38} - 16 q^{39} + 9 q^{40} + 10 q^{41} + 16 q^{42} + 10 q^{43} - 12 q^{44} - 3 q^{45} - 8 q^{46} - 18 q^{47} + 10 q^{48} - q^{49} - 3 q^{50} + 8 q^{51} + 10 q^{52} + 10 q^{53} + 12 q^{54} + 8 q^{55} + 32 q^{56} - 20 q^{57} + 4 q^{59} - 2 q^{60} - 6 q^{61} + 14 q^{62} + 18 q^{63} + 33 q^{64} + 6 q^{65} - 12 q^{66} - 10 q^{67} + 36 q^{68} + 20 q^{69} - 12 q^{70} + 24 q^{71} + 3 q^{72} + 4 q^{73} - 22 q^{74} + 2 q^{75} - 16 q^{76} + 4 q^{77} - 28 q^{78} - 8 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 2 q^{83} - 20 q^{84} - 16 q^{86} + 38 q^{88} - 22 q^{89} - 13 q^{90} - 16 q^{91} + 22 q^{92} + 12 q^{93} - 8 q^{96} + 36 q^{97} - 23 q^{98} - 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21432 0.858654 0.429327 0.903149i \(-0.358751\pi\)
0.429327 + 0.903149i \(0.358751\pi\)
\(3\) 2.90321 1.67617 0.838085 0.545540i \(-0.183675\pi\)
0.838085 + 0.545540i \(0.183675\pi\)
\(4\) −0.525428 −0.262714
\(5\) −1.00000 −0.447214
\(6\) 3.52543 1.43925
\(7\) 1.52543 0.576557 0.288279 0.957547i \(-0.406917\pi\)
0.288279 + 0.957547i \(0.406917\pi\)
\(8\) −3.06668 −1.08423
\(9\) 5.42864 1.80955
\(10\) −1.21432 −0.384002
\(11\) −4.90321 −1.47837 −0.739187 0.673500i \(-0.764790\pi\)
−0.739187 + 0.673500i \(0.764790\pi\)
\(12\) −1.52543 −0.440353
\(13\) −6.42864 −1.78298 −0.891492 0.453037i \(-0.850341\pi\)
−0.891492 + 0.453037i \(0.850341\pi\)
\(14\) 1.85236 0.495063
\(15\) −2.90321 −0.749606
\(16\) −2.67307 −0.668268
\(17\) −2.14764 −0.520880 −0.260440 0.965490i \(-0.583868\pi\)
−0.260440 + 0.965490i \(0.583868\pi\)
\(18\) 6.59210 1.55377
\(19\) −2.28100 −0.523296 −0.261648 0.965163i \(-0.584266\pi\)
−0.261648 + 0.965163i \(0.584266\pi\)
\(20\) 0.525428 0.117489
\(21\) 4.42864 0.966408
\(22\) −5.95407 −1.26941
\(23\) 6.90321 1.43942 0.719710 0.694275i \(-0.244275\pi\)
0.719710 + 0.694275i \(0.244275\pi\)
\(24\) −8.90321 −1.81736
\(25\) 1.00000 0.200000
\(26\) −7.80642 −1.53097
\(27\) 7.05086 1.35694
\(28\) −0.801502 −0.151470
\(29\) 0 0
\(30\) −3.52543 −0.643652
\(31\) −1.71900 −0.308742 −0.154371 0.988013i \(-0.549335\pi\)
−0.154371 + 0.988013i \(0.549335\pi\)
\(32\) 2.88739 0.510423
\(33\) −14.2351 −2.47801
\(34\) −2.60793 −0.447256
\(35\) −1.52543 −0.257844
\(36\) −2.85236 −0.475393
\(37\) −7.95407 −1.30764 −0.653820 0.756650i \(-0.726835\pi\)
−0.653820 + 0.756650i \(0.726835\pi\)
\(38\) −2.76986 −0.449330
\(39\) −18.6637 −2.98858
\(40\) 3.06668 0.484884
\(41\) 3.37778 0.527521 0.263761 0.964588i \(-0.415037\pi\)
0.263761 + 0.964588i \(0.415037\pi\)
\(42\) 5.37778 0.829810
\(43\) 1.09679 0.167259 0.0836293 0.996497i \(-0.473349\pi\)
0.0836293 + 0.996497i \(0.473349\pi\)
\(44\) 2.57628 0.388389
\(45\) −5.42864 −0.809254
\(46\) 8.38271 1.23596
\(47\) −12.7096 −1.85389 −0.926945 0.375196i \(-0.877575\pi\)
−0.926945 + 0.375196i \(0.877575\pi\)
\(48\) −7.76049 −1.12013
\(49\) −4.67307 −0.667582
\(50\) 1.21432 0.171731
\(51\) −6.23506 −0.873084
\(52\) 3.37778 0.468414
\(53\) 3.37778 0.463974 0.231987 0.972719i \(-0.425477\pi\)
0.231987 + 0.972719i \(0.425477\pi\)
\(54\) 8.56199 1.16514
\(55\) 4.90321 0.661149
\(56\) −4.67799 −0.625123
\(57\) −6.62222 −0.877134
\(58\) 0 0
\(59\) −3.18421 −0.414549 −0.207274 0.978283i \(-0.566459\pi\)
−0.207274 + 0.978283i \(0.566459\pi\)
\(60\) 1.52543 0.196932
\(61\) 2.42864 0.310955 0.155478 0.987839i \(-0.450308\pi\)
0.155478 + 0.987839i \(0.450308\pi\)
\(62\) −2.08742 −0.265103
\(63\) 8.28100 1.04331
\(64\) 8.85236 1.10654
\(65\) 6.42864 0.797375
\(66\) −17.2859 −2.12775
\(67\) −1.09679 −0.133994 −0.0669970 0.997753i \(-0.521342\pi\)
−0.0669970 + 0.997753i \(0.521342\pi\)
\(68\) 1.12843 0.136842
\(69\) 20.0415 2.41271
\(70\) −1.85236 −0.221399
\(71\) 3.57136 0.423843 0.211921 0.977287i \(-0.432028\pi\)
0.211921 + 0.977287i \(0.432028\pi\)
\(72\) −16.6479 −1.96197
\(73\) −14.1891 −1.66071 −0.830356 0.557233i \(-0.811863\pi\)
−0.830356 + 0.557233i \(0.811863\pi\)
\(74\) −9.65878 −1.12281
\(75\) 2.90321 0.335234
\(76\) 1.19850 0.137477
\(77\) −7.47949 −0.852368
\(78\) −22.6637 −2.56616
\(79\) −0.341219 −0.0383902 −0.0191951 0.999816i \(-0.506110\pi\)
−0.0191951 + 0.999816i \(0.506110\pi\)
\(80\) 2.67307 0.298858
\(81\) 4.18421 0.464912
\(82\) 4.10171 0.452958
\(83\) −7.33185 −0.804775 −0.402388 0.915469i \(-0.631820\pi\)
−0.402388 + 0.915469i \(0.631820\pi\)
\(84\) −2.32693 −0.253889
\(85\) 2.14764 0.232945
\(86\) 1.33185 0.143617
\(87\) 0 0
\(88\) 15.0366 1.60290
\(89\) −2.94914 −0.312609 −0.156304 0.987709i \(-0.549958\pi\)
−0.156304 + 0.987709i \(0.549958\pi\)
\(90\) −6.59210 −0.694869
\(91\) −9.80642 −1.02799
\(92\) −3.62714 −0.378155
\(93\) −4.99063 −0.517504
\(94\) −15.4336 −1.59185
\(95\) 2.28100 0.234025
\(96\) 8.38271 0.855556
\(97\) 18.5763 1.88614 0.943068 0.332600i \(-0.107926\pi\)
0.943068 + 0.332600i \(0.107926\pi\)
\(98\) −5.67460 −0.573221
\(99\) −26.6178 −2.67519
\(100\) −0.525428 −0.0525428
\(101\) −15.4193 −1.53427 −0.767137 0.641483i \(-0.778320\pi\)
−0.767137 + 0.641483i \(0.778320\pi\)
\(102\) −7.57136 −0.749676
\(103\) 7.76049 0.764664 0.382332 0.924025i \(-0.375121\pi\)
0.382332 + 0.924025i \(0.375121\pi\)
\(104\) 19.7146 1.93317
\(105\) −4.42864 −0.432191
\(106\) 4.10171 0.398393
\(107\) −3.03657 −0.293556 −0.146778 0.989169i \(-0.546890\pi\)
−0.146778 + 0.989169i \(0.546890\pi\)
\(108\) −3.70471 −0.356486
\(109\) −7.93978 −0.760493 −0.380246 0.924885i \(-0.624161\pi\)
−0.380246 + 0.924885i \(0.624161\pi\)
\(110\) 5.95407 0.567698
\(111\) −23.0923 −2.19183
\(112\) −4.07758 −0.385295
\(113\) −7.82071 −0.735711 −0.367855 0.929883i \(-0.619908\pi\)
−0.367855 + 0.929883i \(0.619908\pi\)
\(114\) −8.04149 −0.753154
\(115\) −6.90321 −0.643728
\(116\) 0 0
\(117\) −34.8988 −3.22639
\(118\) −3.86665 −0.355954
\(119\) −3.27607 −0.300317
\(120\) 8.90321 0.812748
\(121\) 13.0415 1.18559
\(122\) 2.94914 0.267003
\(123\) 9.80642 0.884215
\(124\) 0.903212 0.0811108
\(125\) −1.00000 −0.0894427
\(126\) 10.0558 0.895840
\(127\) 12.4429 1.10413 0.552066 0.833801i \(-0.313840\pi\)
0.552066 + 0.833801i \(0.313840\pi\)
\(128\) 4.97481 0.439715
\(129\) 3.18421 0.280354
\(130\) 7.80642 0.684669
\(131\) −4.08742 −0.357120 −0.178560 0.983929i \(-0.557144\pi\)
−0.178560 + 0.983929i \(0.557144\pi\)
\(132\) 7.47949 0.651006
\(133\) −3.47949 −0.301710
\(134\) −1.33185 −0.115054
\(135\) −7.05086 −0.606841
\(136\) 6.58613 0.564756
\(137\) 19.9541 1.70479 0.852395 0.522898i \(-0.175149\pi\)
0.852395 + 0.522898i \(0.175149\pi\)
\(138\) 24.3368 2.07168
\(139\) 7.90813 0.670759 0.335380 0.942083i \(-0.391135\pi\)
0.335380 + 0.942083i \(0.391135\pi\)
\(140\) 0.801502 0.0677393
\(141\) −36.8988 −3.10744
\(142\) 4.33677 0.363934
\(143\) 31.5210 2.63592
\(144\) −14.5111 −1.20926
\(145\) 0 0
\(146\) −17.2301 −1.42598
\(147\) −13.5669 −1.11898
\(148\) 4.17929 0.343535
\(149\) −16.1017 −1.31910 −0.659552 0.751659i \(-0.729254\pi\)
−0.659552 + 0.751659i \(0.729254\pi\)
\(150\) 3.52543 0.287850
\(151\) 5.67307 0.461668 0.230834 0.972993i \(-0.425855\pi\)
0.230834 + 0.972993i \(0.425855\pi\)
\(152\) 6.99508 0.567376
\(153\) −11.6588 −0.942557
\(154\) −9.08250 −0.731889
\(155\) 1.71900 0.138074
\(156\) 9.80642 0.785142
\(157\) −1.89384 −0.151145 −0.0755726 0.997140i \(-0.524078\pi\)
−0.0755726 + 0.997140i \(0.524078\pi\)
\(158\) −0.414349 −0.0329639
\(159\) 9.80642 0.777700
\(160\) −2.88739 −0.228268
\(161\) 10.5303 0.829908
\(162\) 5.08097 0.399198
\(163\) 5.95407 0.466359 0.233179 0.972434i \(-0.425087\pi\)
0.233179 + 0.972434i \(0.425087\pi\)
\(164\) −1.77478 −0.138587
\(165\) 14.2351 1.10820
\(166\) −8.90321 −0.691023
\(167\) 4.23951 0.328063 0.164032 0.986455i \(-0.447550\pi\)
0.164032 + 0.986455i \(0.447550\pi\)
\(168\) −13.5812 −1.04781
\(169\) 28.3274 2.17903
\(170\) 2.60793 0.200019
\(171\) −12.3827 −0.946929
\(172\) −0.576283 −0.0439411
\(173\) 24.4099 1.85585 0.927925 0.372766i \(-0.121591\pi\)
0.927925 + 0.372766i \(0.121591\pi\)
\(174\) 0 0
\(175\) 1.52543 0.115311
\(176\) 13.1066 0.987950
\(177\) −9.24443 −0.694854
\(178\) −3.58120 −0.268423
\(179\) −3.61285 −0.270037 −0.135018 0.990843i \(-0.543109\pi\)
−0.135018 + 0.990843i \(0.543109\pi\)
\(180\) 2.85236 0.212602
\(181\) 18.0415 1.34101 0.670507 0.741904i \(-0.266077\pi\)
0.670507 + 0.741904i \(0.266077\pi\)
\(182\) −11.9081 −0.882690
\(183\) 7.05086 0.521214
\(184\) −21.1699 −1.56067
\(185\) 7.95407 0.584795
\(186\) −6.06022 −0.444357
\(187\) 10.5303 0.770055
\(188\) 6.67799 0.487043
\(189\) 10.7556 0.782353
\(190\) 2.76986 0.200947
\(191\) −9.85236 −0.712892 −0.356446 0.934316i \(-0.616012\pi\)
−0.356446 + 0.934316i \(0.616012\pi\)
\(192\) 25.7003 1.85476
\(193\) −2.23951 −0.161203 −0.0806017 0.996746i \(-0.525684\pi\)
−0.0806017 + 0.996746i \(0.525684\pi\)
\(194\) 22.5575 1.61954
\(195\) 18.6637 1.33654
\(196\) 2.45536 0.175383
\(197\) −10.5620 −0.752511 −0.376255 0.926516i \(-0.622788\pi\)
−0.376255 + 0.926516i \(0.622788\pi\)
\(198\) −32.3225 −2.29706
\(199\) 3.18421 0.225723 0.112861 0.993611i \(-0.463998\pi\)
0.112861 + 0.993611i \(0.463998\pi\)
\(200\) −3.06668 −0.216847
\(201\) −3.18421 −0.224597
\(202\) −18.7239 −1.31741
\(203\) 0 0
\(204\) 3.27607 0.229371
\(205\) −3.37778 −0.235915
\(206\) 9.42372 0.656581
\(207\) 37.4750 2.60470
\(208\) 17.1842 1.19151
\(209\) 11.1842 0.773628
\(210\) −5.37778 −0.371102
\(211\) 5.76049 0.396569 0.198284 0.980145i \(-0.436463\pi\)
0.198284 + 0.980145i \(0.436463\pi\)
\(212\) −1.77478 −0.121892
\(213\) 10.3684 0.710432
\(214\) −3.68736 −0.252063
\(215\) −1.09679 −0.0748003
\(216\) −21.6227 −1.47124
\(217\) −2.62222 −0.178008
\(218\) −9.64143 −0.653000
\(219\) −41.1941 −2.78364
\(220\) −2.57628 −0.173693
\(221\) 13.8064 0.928721
\(222\) −28.0415 −1.88202
\(223\) 8.14764 0.545607 0.272803 0.962070i \(-0.412049\pi\)
0.272803 + 0.962070i \(0.412049\pi\)
\(224\) 4.40451 0.294288
\(225\) 5.42864 0.361909
\(226\) −9.49685 −0.631721
\(227\) 10.5161 0.697975 0.348988 0.937127i \(-0.386525\pi\)
0.348988 + 0.937127i \(0.386525\pi\)
\(228\) 3.47949 0.230435
\(229\) −8.48886 −0.560960 −0.280480 0.959860i \(-0.590494\pi\)
−0.280480 + 0.959860i \(0.590494\pi\)
\(230\) −8.38271 −0.552739
\(231\) −21.7146 −1.42871
\(232\) 0 0
\(233\) −20.5718 −1.34771 −0.673853 0.738866i \(-0.735362\pi\)
−0.673853 + 0.738866i \(0.735362\pi\)
\(234\) −42.3783 −2.77035
\(235\) 12.7096 0.829085
\(236\) 1.67307 0.108908
\(237\) −0.990632 −0.0643485
\(238\) −3.97820 −0.257869
\(239\) 0.815792 0.0527692 0.0263846 0.999652i \(-0.491601\pi\)
0.0263846 + 0.999652i \(0.491601\pi\)
\(240\) 7.76049 0.500938
\(241\) −7.24443 −0.466655 −0.233327 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(242\) 15.8365 1.01801
\(243\) −9.00492 −0.577666
\(244\) −1.27607 −0.0816923
\(245\) 4.67307 0.298552
\(246\) 11.9081 0.759235
\(247\) 14.6637 0.933029
\(248\) 5.27163 0.334749
\(249\) −21.2859 −1.34894
\(250\) −1.21432 −0.0768003
\(251\) 20.4242 1.28916 0.644582 0.764535i \(-0.277031\pi\)
0.644582 + 0.764535i \(0.277031\pi\)
\(252\) −4.35106 −0.274091
\(253\) −33.8479 −2.12800
\(254\) 15.1097 0.948067
\(255\) 6.23506 0.390455
\(256\) −11.6637 −0.728981
\(257\) 3.08250 0.192281 0.0961405 0.995368i \(-0.469350\pi\)
0.0961405 + 0.995368i \(0.469350\pi\)
\(258\) 3.86665 0.240727
\(259\) −12.1334 −0.753930
\(260\) −3.37778 −0.209481
\(261\) 0 0
\(262\) −4.96343 −0.306642
\(263\) −20.0558 −1.23669 −0.618346 0.785906i \(-0.712197\pi\)
−0.618346 + 0.785906i \(0.712197\pi\)
\(264\) 43.6543 2.68674
\(265\) −3.37778 −0.207496
\(266\) −4.22522 −0.259065
\(267\) −8.56199 −0.523985
\(268\) 0.576283 0.0352021
\(269\) 23.4608 1.43043 0.715214 0.698906i \(-0.246329\pi\)
0.715214 + 0.698906i \(0.246329\pi\)
\(270\) −8.56199 −0.521066
\(271\) 21.9353 1.33248 0.666238 0.745739i \(-0.267903\pi\)
0.666238 + 0.745739i \(0.267903\pi\)
\(272\) 5.74080 0.348087
\(273\) −28.4701 −1.72309
\(274\) 24.2306 1.46383
\(275\) −4.90321 −0.295675
\(276\) −10.5303 −0.633853
\(277\) 18.3368 1.10175 0.550875 0.834588i \(-0.314294\pi\)
0.550875 + 0.834588i \(0.314294\pi\)
\(278\) 9.60300 0.575950
\(279\) −9.33185 −0.558683
\(280\) 4.67799 0.279564
\(281\) 6.89877 0.411546 0.205773 0.978600i \(-0.434029\pi\)
0.205773 + 0.978600i \(0.434029\pi\)
\(282\) −44.8069 −2.66821
\(283\) −29.0049 −1.72416 −0.862082 0.506769i \(-0.830840\pi\)
−0.862082 + 0.506769i \(0.830840\pi\)
\(284\) −1.87649 −0.111349
\(285\) 6.62222 0.392266
\(286\) 38.2766 2.26334
\(287\) 5.15257 0.304146
\(288\) 15.6746 0.923635
\(289\) −12.3876 −0.728684
\(290\) 0 0
\(291\) 53.9309 3.16148
\(292\) 7.45536 0.436292
\(293\) −7.79213 −0.455221 −0.227611 0.973752i \(-0.573091\pi\)
−0.227611 + 0.973752i \(0.573091\pi\)
\(294\) −16.4746 −0.960817
\(295\) 3.18421 0.185392
\(296\) 24.3926 1.41779
\(297\) −34.5718 −2.00606
\(298\) −19.5526 −1.13265
\(299\) −44.3783 −2.56646
\(300\) −1.52543 −0.0880706
\(301\) 1.67307 0.0964342
\(302\) 6.88892 0.396413
\(303\) −44.7654 −2.57171
\(304\) 6.09726 0.349702
\(305\) −2.42864 −0.139063
\(306\) −14.1575 −0.809330
\(307\) −5.92549 −0.338185 −0.169093 0.985600i \(-0.554084\pi\)
−0.169093 + 0.985600i \(0.554084\pi\)
\(308\) 3.92993 0.223929
\(309\) 22.5303 1.28171
\(310\) 2.08742 0.118557
\(311\) −8.94470 −0.507207 −0.253604 0.967308i \(-0.581616\pi\)
−0.253604 + 0.967308i \(0.581616\pi\)
\(312\) 57.2355 3.24032
\(313\) −30.5116 −1.72462 −0.862309 0.506382i \(-0.830983\pi\)
−0.862309 + 0.506382i \(0.830983\pi\)
\(314\) −2.29973 −0.129781
\(315\) −8.28100 −0.466581
\(316\) 0.179286 0.0100856
\(317\) 22.2306 1.24860 0.624298 0.781186i \(-0.285385\pi\)
0.624298 + 0.781186i \(0.285385\pi\)
\(318\) 11.9081 0.667775
\(319\) 0 0
\(320\) −8.85236 −0.494862
\(321\) −8.81579 −0.492050
\(322\) 12.7872 0.712603
\(323\) 4.89877 0.272575
\(324\) −2.19850 −0.122139
\(325\) −6.42864 −0.356597
\(326\) 7.23014 0.400440
\(327\) −23.0509 −1.27472
\(328\) −10.3586 −0.571956
\(329\) −19.3876 −1.06887
\(330\) 17.2859 0.951558
\(331\) 6.54770 0.359894 0.179947 0.983676i \(-0.442407\pi\)
0.179947 + 0.983676i \(0.442407\pi\)
\(332\) 3.85236 0.211426
\(333\) −43.1798 −2.36624
\(334\) 5.14812 0.281693
\(335\) 1.09679 0.0599239
\(336\) −11.8381 −0.645819
\(337\) −2.66815 −0.145343 −0.0726717 0.997356i \(-0.523153\pi\)
−0.0726717 + 0.997356i \(0.523153\pi\)
\(338\) 34.3985 1.87103
\(339\) −22.7052 −1.23318
\(340\) −1.12843 −0.0611978
\(341\) 8.42864 0.456436
\(342\) −15.0366 −0.813084
\(343\) −17.8064 −0.961457
\(344\) −3.36349 −0.181347
\(345\) −20.0415 −1.07900
\(346\) 29.6414 1.59353
\(347\) 14.6780 0.787956 0.393978 0.919120i \(-0.371099\pi\)
0.393978 + 0.919120i \(0.371099\pi\)
\(348\) 0 0
\(349\) −11.1240 −0.595453 −0.297727 0.954651i \(-0.596228\pi\)
−0.297727 + 0.954651i \(0.596228\pi\)
\(350\) 1.85236 0.0990126
\(351\) −45.3274 −2.41940
\(352\) −14.1575 −0.754597
\(353\) −13.4795 −0.717441 −0.358721 0.933445i \(-0.616787\pi\)
−0.358721 + 0.933445i \(0.616787\pi\)
\(354\) −11.2257 −0.596639
\(355\) −3.57136 −0.189548
\(356\) 1.54956 0.0821266
\(357\) −9.51114 −0.503383
\(358\) −4.38715 −0.231868
\(359\) −26.1891 −1.38221 −0.691105 0.722755i \(-0.742876\pi\)
−0.691105 + 0.722755i \(0.742876\pi\)
\(360\) 16.6479 0.877420
\(361\) −13.7971 −0.726161
\(362\) 21.9081 1.15147
\(363\) 37.8622 1.98725
\(364\) 5.15257 0.270068
\(365\) 14.1891 0.742693
\(366\) 8.56199 0.447543
\(367\) 22.9862 1.19987 0.599935 0.800049i \(-0.295193\pi\)
0.599935 + 0.800049i \(0.295193\pi\)
\(368\) −18.4528 −0.961917
\(369\) 18.3368 0.954574
\(370\) 9.65878 0.502136
\(371\) 5.15257 0.267508
\(372\) 2.62222 0.135956
\(373\) −24.2766 −1.25699 −0.628496 0.777813i \(-0.716329\pi\)
−0.628496 + 0.777813i \(0.716329\pi\)
\(374\) 12.7872 0.661211
\(375\) −2.90321 −0.149921
\(376\) 38.9763 2.01005
\(377\) 0 0
\(378\) 13.0607 0.671770
\(379\) −29.7605 −1.52869 −0.764347 0.644805i \(-0.776938\pi\)
−0.764347 + 0.644805i \(0.776938\pi\)
\(380\) −1.19850 −0.0614817
\(381\) 36.1245 1.85071
\(382\) −11.9639 −0.612127
\(383\) 23.8020 1.21622 0.608112 0.793851i \(-0.291927\pi\)
0.608112 + 0.793851i \(0.291927\pi\)
\(384\) 14.4429 0.737038
\(385\) 7.47949 0.381190
\(386\) −2.71948 −0.138418
\(387\) 5.95407 0.302662
\(388\) −9.76049 −0.495514
\(389\) 6.52051 0.330603 0.165301 0.986243i \(-0.447140\pi\)
0.165301 + 0.986243i \(0.447140\pi\)
\(390\) 22.6637 1.14762
\(391\) −14.8256 −0.749765
\(392\) 14.3308 0.723815
\(393\) −11.8666 −0.598593
\(394\) −12.8256 −0.646146
\(395\) 0.341219 0.0171686
\(396\) 13.9857 0.702808
\(397\) −14.7654 −0.741055 −0.370527 0.928822i \(-0.620823\pi\)
−0.370527 + 0.928822i \(0.620823\pi\)
\(398\) 3.86665 0.193817
\(399\) −10.1017 −0.505718
\(400\) −2.67307 −0.133654
\(401\) 6.81579 0.340364 0.170182 0.985413i \(-0.445564\pi\)
0.170182 + 0.985413i \(0.445564\pi\)
\(402\) −3.86665 −0.192851
\(403\) 11.0509 0.550482
\(404\) 8.10171 0.403075
\(405\) −4.18421 −0.207915
\(406\) 0 0
\(407\) 39.0005 1.93318
\(408\) 19.1209 0.946627
\(409\) 11.0825 0.547994 0.273997 0.961731i \(-0.411654\pi\)
0.273997 + 0.961731i \(0.411654\pi\)
\(410\) −4.10171 −0.202569
\(411\) 57.9309 2.85752
\(412\) −4.07758 −0.200888
\(413\) −4.85728 −0.239011
\(414\) 45.5067 2.23653
\(415\) 7.33185 0.359906
\(416\) −18.5620 −0.910077
\(417\) 22.9590 1.12431
\(418\) 13.5812 0.664278
\(419\) 30.9719 1.51308 0.756538 0.653950i \(-0.226889\pi\)
0.756538 + 0.653950i \(0.226889\pi\)
\(420\) 2.32693 0.113543
\(421\) −22.8988 −1.11602 −0.558009 0.829835i \(-0.688434\pi\)
−0.558009 + 0.829835i \(0.688434\pi\)
\(422\) 6.99508 0.340515
\(423\) −68.9960 −3.35470
\(424\) −10.3586 −0.503057
\(425\) −2.14764 −0.104176
\(426\) 12.5906 0.610015
\(427\) 3.70471 0.179284
\(428\) 1.59549 0.0771212
\(429\) 91.5121 4.41825
\(430\) −1.33185 −0.0642276
\(431\) −28.0830 −1.35271 −0.676355 0.736576i \(-0.736441\pi\)
−0.676355 + 0.736576i \(0.736441\pi\)
\(432\) −18.8474 −0.906798
\(433\) −23.0049 −1.10555 −0.552773 0.833332i \(-0.686430\pi\)
−0.552773 + 0.833332i \(0.686430\pi\)
\(434\) −3.18421 −0.152847
\(435\) 0 0
\(436\) 4.17178 0.199792
\(437\) −15.7462 −0.753243
\(438\) −50.0228 −2.39018
\(439\) 11.7462 0.560616 0.280308 0.959910i \(-0.409563\pi\)
0.280308 + 0.959910i \(0.409563\pi\)
\(440\) −15.0366 −0.716840
\(441\) −25.3684 −1.20802
\(442\) 16.7654 0.797449
\(443\) −17.6874 −0.840352 −0.420176 0.907443i \(-0.638032\pi\)
−0.420176 + 0.907443i \(0.638032\pi\)
\(444\) 12.1334 0.575823
\(445\) 2.94914 0.139803
\(446\) 9.89384 0.468487
\(447\) −46.7467 −2.21104
\(448\) 13.5036 0.637987
\(449\) −1.57136 −0.0741571 −0.0370785 0.999312i \(-0.511805\pi\)
−0.0370785 + 0.999312i \(0.511805\pi\)
\(450\) 6.59210 0.310755
\(451\) −16.5620 −0.779874
\(452\) 4.10922 0.193281
\(453\) 16.4701 0.773834
\(454\) 12.7699 0.599319
\(455\) 9.80642 0.459732
\(456\) 20.3082 0.951018
\(457\) 1.47949 0.0692078 0.0346039 0.999401i \(-0.488983\pi\)
0.0346039 + 0.999401i \(0.488983\pi\)
\(458\) −10.3082 −0.481670
\(459\) −15.1427 −0.706802
\(460\) 3.62714 0.169116
\(461\) 41.2543 1.92140 0.960702 0.277583i \(-0.0895334\pi\)
0.960702 + 0.277583i \(0.0895334\pi\)
\(462\) −26.3684 −1.22677
\(463\) −34.4242 −1.59983 −0.799914 0.600115i \(-0.795122\pi\)
−0.799914 + 0.600115i \(0.795122\pi\)
\(464\) 0 0
\(465\) 4.99063 0.231435
\(466\) −24.9808 −1.15721
\(467\) −15.1699 −0.701980 −0.350990 0.936379i \(-0.614155\pi\)
−0.350990 + 0.936379i \(0.614155\pi\)
\(468\) 18.3368 0.847618
\(469\) −1.67307 −0.0772552
\(470\) 15.4336 0.711897
\(471\) −5.49823 −0.253345
\(472\) 9.76494 0.449468
\(473\) −5.37778 −0.247271
\(474\) −1.20294 −0.0552531
\(475\) −2.28100 −0.104659
\(476\) 1.72134 0.0788975
\(477\) 18.3368 0.839583
\(478\) 0.990632 0.0453105
\(479\) 18.9763 0.867051 0.433526 0.901141i \(-0.357269\pi\)
0.433526 + 0.901141i \(0.357269\pi\)
\(480\) −8.38271 −0.382616
\(481\) 51.1338 2.33150
\(482\) −8.79706 −0.400695
\(483\) 30.5718 1.39107
\(484\) −6.85236 −0.311471
\(485\) −18.5763 −0.843506
\(486\) −10.9349 −0.496015
\(487\) −32.3926 −1.46785 −0.733923 0.679232i \(-0.762313\pi\)
−0.733923 + 0.679232i \(0.762313\pi\)
\(488\) −7.44785 −0.337148
\(489\) 17.2859 0.781696
\(490\) 5.67460 0.256352
\(491\) −2.69673 −0.121702 −0.0608508 0.998147i \(-0.519381\pi\)
−0.0608508 + 0.998147i \(0.519381\pi\)
\(492\) −5.15257 −0.232296
\(493\) 0 0
\(494\) 17.8064 0.801149
\(495\) 26.6178 1.19638
\(496\) 4.59502 0.206322
\(497\) 5.44785 0.244370
\(498\) −25.8479 −1.15827
\(499\) 14.5718 0.652325 0.326163 0.945314i \(-0.394244\pi\)
0.326163 + 0.945314i \(0.394244\pi\)
\(500\) 0.525428 0.0234978
\(501\) 12.3082 0.549890
\(502\) 24.8015 1.10695
\(503\) 22.2494 0.992050 0.496025 0.868308i \(-0.334792\pi\)
0.496025 + 0.868308i \(0.334792\pi\)
\(504\) −25.3951 −1.13119
\(505\) 15.4193 0.686149
\(506\) −41.1022 −1.82722
\(507\) 82.2405 3.65243
\(508\) −6.53786 −0.290071
\(509\) −9.18421 −0.407083 −0.203541 0.979066i \(-0.565245\pi\)
−0.203541 + 0.979066i \(0.565245\pi\)
\(510\) 7.57136 0.335265
\(511\) −21.6445 −0.957496
\(512\) −24.1131 −1.06566
\(513\) −16.0830 −0.710081
\(514\) 3.74314 0.165103
\(515\) −7.76049 −0.341968
\(516\) −1.67307 −0.0736528
\(517\) 62.3180 2.74074
\(518\) −14.7338 −0.647365
\(519\) 70.8671 3.11072
\(520\) −19.7146 −0.864541
\(521\) 7.01921 0.307517 0.153759 0.988108i \(-0.450862\pi\)
0.153759 + 0.988108i \(0.450862\pi\)
\(522\) 0 0
\(523\) −7.29036 −0.318785 −0.159393 0.987215i \(-0.550954\pi\)
−0.159393 + 0.987215i \(0.550954\pi\)
\(524\) 2.14764 0.0938202
\(525\) 4.42864 0.193282
\(526\) −24.3541 −1.06189
\(527\) 3.69181 0.160818
\(528\) 38.0513 1.65597
\(529\) 24.6543 1.07193
\(530\) −4.10171 −0.178167
\(531\) −17.2859 −0.750145
\(532\) 1.82822 0.0792635
\(533\) −21.7146 −0.940562
\(534\) −10.3970 −0.449922
\(535\) 3.03657 0.131282
\(536\) 3.36349 0.145281
\(537\) −10.4889 −0.452628
\(538\) 28.4889 1.22824
\(539\) 22.9131 0.986935
\(540\) 3.70471 0.159425
\(541\) −30.9491 −1.33061 −0.665304 0.746573i \(-0.731698\pi\)
−0.665304 + 0.746573i \(0.731698\pi\)
\(542\) 26.6365 1.14414
\(543\) 52.3783 2.24777
\(544\) −6.20108 −0.265869
\(545\) 7.93978 0.340103
\(546\) −34.5718 −1.47954
\(547\) −19.4237 −0.830498 −0.415249 0.909708i \(-0.636306\pi\)
−0.415249 + 0.909708i \(0.636306\pi\)
\(548\) −10.4844 −0.447872
\(549\) 13.1842 0.562688
\(550\) −5.95407 −0.253882
\(551\) 0 0
\(552\) −61.4608 −2.61594
\(553\) −0.520505 −0.0221341
\(554\) 22.2667 0.946022
\(555\) 23.0923 0.980215
\(556\) −4.15515 −0.176218
\(557\) 30.3497 1.28596 0.642979 0.765884i \(-0.277698\pi\)
0.642979 + 0.765884i \(0.277698\pi\)
\(558\) −11.3319 −0.479716
\(559\) −7.05086 −0.298219
\(560\) 4.07758 0.172309
\(561\) 30.5718 1.29074
\(562\) 8.37731 0.353375
\(563\) 33.1798 1.39836 0.699180 0.714946i \(-0.253549\pi\)
0.699180 + 0.714946i \(0.253549\pi\)
\(564\) 19.3876 0.816366
\(565\) 7.82071 0.329020
\(566\) −35.2212 −1.48046
\(567\) 6.38271 0.268048
\(568\) −10.9522 −0.459544
\(569\) 4.06022 0.170213 0.0851067 0.996372i \(-0.472877\pi\)
0.0851067 + 0.996372i \(0.472877\pi\)
\(570\) 8.04149 0.336821
\(571\) −31.5496 −1.32031 −0.660154 0.751130i \(-0.729509\pi\)
−0.660154 + 0.751130i \(0.729509\pi\)
\(572\) −16.5620 −0.692492
\(573\) −28.6035 −1.19493
\(574\) 6.25686 0.261156
\(575\) 6.90321 0.287884
\(576\) 48.0563 2.00234
\(577\) −33.7891 −1.40666 −0.703329 0.710865i \(-0.748304\pi\)
−0.703329 + 0.710865i \(0.748304\pi\)
\(578\) −15.0425 −0.625687
\(579\) −6.50177 −0.270204
\(580\) 0 0
\(581\) −11.1842 −0.463999
\(582\) 65.4893 2.71462
\(583\) −16.5620 −0.685928
\(584\) 43.5135 1.80060
\(585\) 34.8988 1.44289
\(586\) −9.46214 −0.390877
\(587\) −25.5669 −1.05526 −0.527630 0.849474i \(-0.676919\pi\)
−0.527630 + 0.849474i \(0.676919\pi\)
\(588\) 7.12843 0.293972
\(589\) 3.92104 0.161564
\(590\) 3.86665 0.159187
\(591\) −30.6637 −1.26134
\(592\) 21.2618 0.873854
\(593\) 7.96836 0.327221 0.163611 0.986525i \(-0.447686\pi\)
0.163611 + 0.986525i \(0.447686\pi\)
\(594\) −41.9813 −1.72251
\(595\) 3.27607 0.134306
\(596\) 8.46028 0.346547
\(597\) 9.24443 0.378349
\(598\) −53.8894 −2.20370
\(599\) −37.2815 −1.52328 −0.761640 0.648001i \(-0.775605\pi\)
−0.761640 + 0.648001i \(0.775605\pi\)
\(600\) −8.90321 −0.363472
\(601\) 29.9496 1.22167 0.610835 0.791758i \(-0.290834\pi\)
0.610835 + 0.791758i \(0.290834\pi\)
\(602\) 2.03164 0.0828036
\(603\) −5.95407 −0.242468
\(604\) −2.98079 −0.121287
\(605\) −13.0415 −0.530212
\(606\) −54.3595 −2.20820
\(607\) 31.8435 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(608\) −6.58613 −0.267103
\(609\) 0 0
\(610\) −2.94914 −0.119407
\(611\) 81.7057 3.30546
\(612\) 6.12584 0.247623
\(613\) −2.65386 −0.107188 −0.0535942 0.998563i \(-0.517068\pi\)
−0.0535942 + 0.998563i \(0.517068\pi\)
\(614\) −7.19544 −0.290384
\(615\) −9.80642 −0.395433
\(616\) 22.9372 0.924166
\(617\) 18.3096 0.737116 0.368558 0.929605i \(-0.379852\pi\)
0.368558 + 0.929605i \(0.379852\pi\)
\(618\) 27.3590 1.10054
\(619\) −12.8113 −0.514931 −0.257466 0.966287i \(-0.582887\pi\)
−0.257466 + 0.966287i \(0.582887\pi\)
\(620\) −0.903212 −0.0362739
\(621\) 48.6735 1.95320
\(622\) −10.8617 −0.435515
\(623\) −4.49871 −0.180237
\(624\) 49.8894 1.99717
\(625\) 1.00000 0.0400000
\(626\) −37.0509 −1.48085
\(627\) 32.4701 1.29673
\(628\) 0.995078 0.0397079
\(629\) 17.0825 0.681124
\(630\) −10.0558 −0.400632
\(631\) 30.2766 1.20529 0.602645 0.798009i \(-0.294113\pi\)
0.602645 + 0.798009i \(0.294113\pi\)
\(632\) 1.04641 0.0416239
\(633\) 16.7239 0.664716
\(634\) 26.9951 1.07211
\(635\) −12.4429 −0.493783
\(636\) −5.15257 −0.204313
\(637\) 30.0415 1.19029
\(638\) 0 0
\(639\) 19.3876 0.766963
\(640\) −4.97481 −0.196647
\(641\) 4.50177 0.177809 0.0889046 0.996040i \(-0.471663\pi\)
0.0889046 + 0.996040i \(0.471663\pi\)
\(642\) −10.7052 −0.422500
\(643\) −40.0272 −1.57852 −0.789259 0.614060i \(-0.789535\pi\)
−0.789259 + 0.614060i \(0.789535\pi\)
\(644\) −5.53294 −0.218028
\(645\) −3.18421 −0.125378
\(646\) 5.94867 0.234047
\(647\) 27.3604 1.07565 0.537825 0.843057i \(-0.319246\pi\)
0.537825 + 0.843057i \(0.319246\pi\)
\(648\) −12.8316 −0.504073
\(649\) 15.6128 0.612858
\(650\) −7.80642 −0.306193
\(651\) −7.61285 −0.298371
\(652\) −3.12843 −0.122519
\(653\) −22.8430 −0.893915 −0.446958 0.894555i \(-0.647493\pi\)
−0.446958 + 0.894555i \(0.647493\pi\)
\(654\) −27.9911 −1.09454
\(655\) 4.08742 0.159709
\(656\) −9.02906 −0.352525
\(657\) −77.0277 −3.00514
\(658\) −23.5428 −0.917793
\(659\) −15.0178 −0.585012 −0.292506 0.956264i \(-0.594489\pi\)
−0.292506 + 0.956264i \(0.594489\pi\)
\(660\) −7.47949 −0.291139
\(661\) −4.65080 −0.180895 −0.0904475 0.995901i \(-0.528830\pi\)
−0.0904475 + 0.995901i \(0.528830\pi\)
\(662\) 7.95100 0.309025
\(663\) 40.0830 1.55669
\(664\) 22.4844 0.872565
\(665\) 3.47949 0.134929
\(666\) −52.4340 −2.03178
\(667\) 0 0
\(668\) −2.22755 −0.0861867
\(669\) 23.6543 0.914529
\(670\) 1.33185 0.0514539
\(671\) −11.9081 −0.459708
\(672\) 12.7872 0.493277
\(673\) −44.8671 −1.72950 −0.864750 0.502202i \(-0.832523\pi\)
−0.864750 + 0.502202i \(0.832523\pi\)
\(674\) −3.23999 −0.124800
\(675\) 7.05086 0.271388
\(676\) −14.8840 −0.572462
\(677\) −27.2212 −1.04620 −0.523099 0.852272i \(-0.675224\pi\)
−0.523099 + 0.852272i \(0.675224\pi\)
\(678\) −27.5714 −1.05887
\(679\) 28.3368 1.08747
\(680\) −6.58613 −0.252566
\(681\) 30.5303 1.16993
\(682\) 10.2351 0.391921
\(683\) 12.0558 0.461301 0.230651 0.973037i \(-0.425915\pi\)
0.230651 + 0.973037i \(0.425915\pi\)
\(684\) 6.50622 0.248771
\(685\) −19.9541 −0.762406
\(686\) −21.6227 −0.825558
\(687\) −24.6450 −0.940264
\(688\) −2.93179 −0.111774
\(689\) −21.7146 −0.827259
\(690\) −24.3368 −0.926485
\(691\) 37.5812 1.42966 0.714828 0.699300i \(-0.246505\pi\)
0.714828 + 0.699300i \(0.246505\pi\)
\(692\) −12.8256 −0.487558
\(693\) −40.6035 −1.54240
\(694\) 17.8238 0.676581
\(695\) −7.90813 −0.299973
\(696\) 0 0
\(697\) −7.25428 −0.274775
\(698\) −13.5081 −0.511288
\(699\) −59.7244 −2.25898
\(700\) −0.801502 −0.0302939
\(701\) 2.04149 0.0771059 0.0385530 0.999257i \(-0.487725\pi\)
0.0385530 + 0.999257i \(0.487725\pi\)
\(702\) −55.0420 −2.07743
\(703\) 18.1432 0.684284
\(704\) −43.4050 −1.63589
\(705\) 36.8988 1.38969
\(706\) −16.3684 −0.616033
\(707\) −23.5210 −0.884598
\(708\) 4.85728 0.182548
\(709\) −32.1432 −1.20716 −0.603582 0.797301i \(-0.706260\pi\)
−0.603582 + 0.797301i \(0.706260\pi\)
\(710\) −4.33677 −0.162756
\(711\) −1.85236 −0.0694688
\(712\) 9.04407 0.338941
\(713\) −11.8666 −0.444409
\(714\) −11.5496 −0.432231
\(715\) −31.5210 −1.17882
\(716\) 1.89829 0.0709424
\(717\) 2.36842 0.0884501
\(718\) −31.8020 −1.18684
\(719\) 1.01921 0.0380102 0.0190051 0.999819i \(-0.493950\pi\)
0.0190051 + 0.999819i \(0.493950\pi\)
\(720\) 14.5111 0.540798
\(721\) 11.8381 0.440873
\(722\) −16.7540 −0.623521
\(723\) −21.0321 −0.782193
\(724\) −9.47949 −0.352303
\(725\) 0 0
\(726\) 45.9768 1.70636
\(727\) 24.1476 0.895587 0.447793 0.894137i \(-0.352210\pi\)
0.447793 + 0.894137i \(0.352210\pi\)
\(728\) 30.0731 1.11458
\(729\) −38.6958 −1.43318
\(730\) 17.2301 0.637716
\(731\) −2.35551 −0.0871217
\(732\) −3.70471 −0.136930
\(733\) −18.8845 −0.697514 −0.348757 0.937213i \(-0.613396\pi\)
−0.348757 + 0.937213i \(0.613396\pi\)
\(734\) 27.9126 1.03027
\(735\) 13.5669 0.500423
\(736\) 19.9323 0.734713
\(737\) 5.37778 0.198093
\(738\) 22.2667 0.819649
\(739\) 3.31312 0.121875 0.0609375 0.998142i \(-0.480591\pi\)
0.0609375 + 0.998142i \(0.480591\pi\)
\(740\) −4.17929 −0.153634
\(741\) 42.5718 1.56392
\(742\) 6.25686 0.229697
\(743\) −2.99508 −0.109879 −0.0549394 0.998490i \(-0.517497\pi\)
−0.0549394 + 0.998490i \(0.517497\pi\)
\(744\) 15.3047 0.561096
\(745\) 16.1017 0.589921
\(746\) −29.4795 −1.07932
\(747\) −39.8020 −1.45628
\(748\) −5.53294 −0.202304
\(749\) −4.63206 −0.169252
\(750\) −3.52543 −0.128730
\(751\) −11.9956 −0.437724 −0.218862 0.975756i \(-0.570234\pi\)
−0.218862 + 0.975756i \(0.570234\pi\)
\(752\) 33.9738 1.23890
\(753\) 59.2958 2.16086
\(754\) 0 0
\(755\) −5.67307 −0.206464
\(756\) −5.65127 −0.205535
\(757\) −18.5763 −0.675166 −0.337583 0.941296i \(-0.609609\pi\)
−0.337583 + 0.941296i \(0.609609\pi\)
\(758\) −36.1388 −1.31262
\(759\) −98.2677 −3.56689
\(760\) −6.99508 −0.253738
\(761\) 2.59057 0.0939082 0.0469541 0.998897i \(-0.485049\pi\)
0.0469541 + 0.998897i \(0.485049\pi\)
\(762\) 43.8666 1.58912
\(763\) −12.1116 −0.438468
\(764\) 5.17670 0.187286
\(765\) 11.6588 0.421524
\(766\) 28.9032 1.04432
\(767\) 20.4701 0.739133
\(768\) −33.8622 −1.22190
\(769\) −32.7467 −1.18088 −0.590438 0.807083i \(-0.701045\pi\)
−0.590438 + 0.807083i \(0.701045\pi\)
\(770\) 9.08250 0.327311
\(771\) 8.94914 0.322296
\(772\) 1.17670 0.0423504
\(773\) −28.2208 −1.01503 −0.507515 0.861643i \(-0.669436\pi\)
−0.507515 + 0.861643i \(0.669436\pi\)
\(774\) 7.23014 0.259882
\(775\) −1.71900 −0.0617484
\(776\) −56.9675 −2.04501
\(777\) −35.2257 −1.26371
\(778\) 7.91798 0.283873
\(779\) −7.70471 −0.276050
\(780\) −9.80642 −0.351126
\(781\) −17.5111 −0.626598
\(782\) −18.0031 −0.643788
\(783\) 0 0
\(784\) 12.4914 0.446123
\(785\) 1.89384 0.0675942
\(786\) −14.4099 −0.513984
\(787\) 11.9857 0.427244 0.213622 0.976916i \(-0.431474\pi\)
0.213622 + 0.976916i \(0.431474\pi\)
\(788\) 5.54956 0.197695
\(789\) −58.2262 −2.07291
\(790\) 0.414349 0.0147419
\(791\) −11.9299 −0.424180
\(792\) 81.6281 2.90053
\(793\) −15.6128 −0.554428
\(794\) −17.9299 −0.636309
\(795\) −9.80642 −0.347798
\(796\) −1.67307 −0.0593004
\(797\) 33.0366 1.17022 0.585108 0.810956i \(-0.301052\pi\)
0.585108 + 0.810956i \(0.301052\pi\)
\(798\) −12.2667 −0.434237
\(799\) 27.2958 0.965655
\(800\) 2.88739 0.102085
\(801\) −16.0098 −0.565680
\(802\) 8.27655 0.292255
\(803\) 69.5723 2.45515
\(804\) 1.67307 0.0590047
\(805\) −10.5303 −0.371146
\(806\) 13.4193 0.472674
\(807\) 68.1116 2.39764
\(808\) 47.2859 1.66351
\(809\) 32.1303 1.12964 0.564820 0.825214i \(-0.308945\pi\)
0.564820 + 0.825214i \(0.308945\pi\)
\(810\) −5.08097 −0.178527
\(811\) 15.3176 0.537872 0.268936 0.963158i \(-0.413328\pi\)
0.268936 + 0.963158i \(0.413328\pi\)
\(812\) 0 0
\(813\) 63.6829 2.23346
\(814\) 47.3590 1.65993
\(815\) −5.95407 −0.208562
\(816\) 16.6668 0.583453
\(817\) −2.50177 −0.0875258
\(818\) 13.4577 0.470537
\(819\) −53.2355 −1.86020
\(820\) 1.77478 0.0619780
\(821\) −17.1427 −0.598285 −0.299143 0.954208i \(-0.596701\pi\)
−0.299143 + 0.954208i \(0.596701\pi\)
\(822\) 70.3466 2.45362
\(823\) 35.6400 1.24233 0.621167 0.783678i \(-0.286659\pi\)
0.621167 + 0.783678i \(0.286659\pi\)
\(824\) −23.7989 −0.829075
\(825\) −14.2351 −0.495601
\(826\) −5.89829 −0.205228
\(827\) −4.70964 −0.163770 −0.0818850 0.996642i \(-0.526094\pi\)
−0.0818850 + 0.996642i \(0.526094\pi\)
\(828\) −19.6904 −0.684290
\(829\) −2.25380 −0.0782777 −0.0391388 0.999234i \(-0.512461\pi\)
−0.0391388 + 0.999234i \(0.512461\pi\)
\(830\) 8.90321 0.309035
\(831\) 53.2355 1.84672
\(832\) −56.9086 −1.97295
\(833\) 10.0361 0.347730
\(834\) 27.8796 0.965390
\(835\) −4.23951 −0.146714
\(836\) −5.87649 −0.203243
\(837\) −12.1204 −0.418944
\(838\) 37.6098 1.29921
\(839\) 8.42419 0.290835 0.145418 0.989370i \(-0.453547\pi\)
0.145418 + 0.989370i \(0.453547\pi\)
\(840\) 13.5812 0.468596
\(841\) 0 0
\(842\) −27.8064 −0.958273
\(843\) 20.0286 0.689821
\(844\) −3.02672 −0.104184
\(845\) −28.3274 −0.974492
\(846\) −83.7832 −2.88053
\(847\) 19.8938 0.683561
\(848\) −9.02906 −0.310059
\(849\) −84.2074 −2.88999
\(850\) −2.60793 −0.0894511
\(851\) −54.9086 −1.88224
\(852\) −5.44785 −0.186640
\(853\) −51.8247 −1.77444 −0.887222 0.461342i \(-0.847368\pi\)
−0.887222 + 0.461342i \(0.847368\pi\)
\(854\) 4.49871 0.153943
\(855\) 12.3827 0.423480
\(856\) 9.31216 0.318283
\(857\) −30.0415 −1.02620 −0.513099 0.858330i \(-0.671503\pi\)
−0.513099 + 0.858330i \(0.671503\pi\)
\(858\) 111.125 3.79374
\(859\) −19.6874 −0.671724 −0.335862 0.941911i \(-0.609028\pi\)
−0.335862 + 0.941911i \(0.609028\pi\)
\(860\) 0.576283 0.0196511
\(861\) 14.9590 0.509801
\(862\) −34.1017 −1.16151
\(863\) 19.5986 0.667143 0.333571 0.942725i \(-0.391746\pi\)
0.333571 + 0.942725i \(0.391746\pi\)
\(864\) 20.3586 0.692613
\(865\) −24.4099 −0.829962
\(866\) −27.9353 −0.949281
\(867\) −35.9639 −1.22140
\(868\) 1.37778 0.0467650
\(869\) 1.67307 0.0567550
\(870\) 0 0
\(871\) 7.05086 0.238909
\(872\) 24.3487 0.824552
\(873\) 100.844 3.41305
\(874\) −19.1209 −0.646775
\(875\) −1.52543 −0.0515689
\(876\) 21.6445 0.731300
\(877\) −16.2351 −0.548219 −0.274110 0.961698i \(-0.588383\pi\)
−0.274110 + 0.961698i \(0.588383\pi\)
\(878\) 14.2636 0.481375
\(879\) −22.6222 −0.763028
\(880\) −13.1066 −0.441824
\(881\) 20.7052 0.697576 0.348788 0.937202i \(-0.386593\pi\)
0.348788 + 0.937202i \(0.386593\pi\)
\(882\) −30.8054 −1.03727
\(883\) 8.75112 0.294499 0.147249 0.989099i \(-0.452958\pi\)
0.147249 + 0.989099i \(0.452958\pi\)
\(884\) −7.25428 −0.243988
\(885\) 9.24443 0.310748
\(886\) −21.4781 −0.721571
\(887\) −0.414349 −0.0139125 −0.00695625 0.999976i \(-0.502214\pi\)
−0.00695625 + 0.999976i \(0.502214\pi\)
\(888\) 70.8167 2.37645
\(889\) 18.9808 0.636595
\(890\) 3.58120 0.120042
\(891\) −20.5161 −0.687314
\(892\) −4.28100 −0.143338
\(893\) 28.9906 0.970135
\(894\) −56.7654 −1.89852
\(895\) 3.61285 0.120764
\(896\) 7.58871 0.253521
\(897\) −128.839 −4.30183
\(898\) −1.90813 −0.0636753
\(899\) 0 0
\(900\) −2.85236 −0.0950786
\(901\) −7.25428 −0.241675
\(902\) −20.1116 −0.669642
\(903\) 4.85728 0.161640
\(904\) 23.9836 0.797683
\(905\) −18.0415 −0.599719
\(906\) 20.0000 0.664455
\(907\) 46.9862 1.56015 0.780075 0.625686i \(-0.215181\pi\)
0.780075 + 0.625686i \(0.215181\pi\)
\(908\) −5.52543 −0.183368
\(909\) −83.7057 −2.77634
\(910\) 11.9081 0.394751
\(911\) 12.1704 0.403223 0.201612 0.979466i \(-0.435382\pi\)
0.201612 + 0.979466i \(0.435382\pi\)
\(912\) 17.7017 0.586160
\(913\) 35.9496 1.18976
\(914\) 1.79658 0.0594256
\(915\) −7.05086 −0.233094
\(916\) 4.46028 0.147372
\(917\) −6.23506 −0.205900
\(918\) −18.3881 −0.606898
\(919\) −23.2672 −0.767514 −0.383757 0.923434i \(-0.625370\pi\)
−0.383757 + 0.923434i \(0.625370\pi\)
\(920\) 21.1699 0.697952
\(921\) −17.2029 −0.566856
\(922\) 50.0959 1.64982
\(923\) −22.9590 −0.755704
\(924\) 11.4094 0.375343
\(925\) −7.95407 −0.261528
\(926\) −41.8020 −1.37370
\(927\) 42.1289 1.38369
\(928\) 0 0
\(929\) 44.7556 1.46838 0.734191 0.678943i \(-0.237562\pi\)
0.734191 + 0.678943i \(0.237562\pi\)
\(930\) 6.06022 0.198723
\(931\) 10.6593 0.349343
\(932\) 10.8090 0.354061
\(933\) −25.9684 −0.850166
\(934\) −18.4211 −0.602758
\(935\) −10.5303 −0.344379
\(936\) 107.023 3.49816
\(937\) −22.7239 −0.742358 −0.371179 0.928561i \(-0.621046\pi\)
−0.371179 + 0.928561i \(0.621046\pi\)
\(938\) −2.03164 −0.0663355
\(939\) −88.5817 −2.89075
\(940\) −6.67799 −0.217812
\(941\) 4.10171 0.133712 0.0668560 0.997763i \(-0.478703\pi\)
0.0668560 + 0.997763i \(0.478703\pi\)
\(942\) −6.67661 −0.217536
\(943\) 23.3176 0.759324
\(944\) 8.51161 0.277029
\(945\) −10.7556 −0.349879
\(946\) −6.53035 −0.212320
\(947\) −16.6178 −0.540005 −0.270002 0.962860i \(-0.587025\pi\)
−0.270002 + 0.962860i \(0.587025\pi\)
\(948\) 0.520505 0.0169052
\(949\) 91.2168 2.96102
\(950\) −2.76986 −0.0898661
\(951\) 64.5402 2.09286
\(952\) 10.0467 0.325614
\(953\) 2.85728 0.0925563 0.0462782 0.998929i \(-0.485264\pi\)
0.0462782 + 0.998929i \(0.485264\pi\)
\(954\) 22.2667 0.720911
\(955\) 9.85236 0.318815
\(956\) −0.428639 −0.0138632
\(957\) 0 0
\(958\) 23.0433 0.744497
\(959\) 30.4385 0.982910
\(960\) −25.7003 −0.829473
\(961\) −28.0450 −0.904678
\(962\) 62.0928 2.00195
\(963\) −16.4844 −0.531203
\(964\) 3.80642 0.122597
\(965\) 2.23951 0.0720923
\(966\) 37.1240 1.19444
\(967\) −16.8015 −0.540300 −0.270150 0.962818i \(-0.587073\pi\)
−0.270150 + 0.962818i \(0.587073\pi\)
\(968\) −39.9940 −1.28546
\(969\) 14.2222 0.456881
\(970\) −22.5575 −0.724279
\(971\) −38.3640 −1.23116 −0.615579 0.788075i \(-0.711078\pi\)
−0.615579 + 0.788075i \(0.711078\pi\)
\(972\) 4.73143 0.151761
\(973\) 12.0633 0.386731
\(974\) −39.3349 −1.26037
\(975\) −18.6637 −0.597717
\(976\) −6.49193 −0.207801
\(977\) 31.6356 1.01211 0.506056 0.862500i \(-0.331103\pi\)
0.506056 + 0.862500i \(0.331103\pi\)
\(978\) 20.9906 0.671206
\(979\) 14.4603 0.462153
\(980\) −2.45536 −0.0784336
\(981\) −43.1022 −1.37615
\(982\) −3.27469 −0.104500
\(983\) 35.3733 1.12823 0.564117 0.825695i \(-0.309217\pi\)
0.564117 + 0.825695i \(0.309217\pi\)
\(984\) −30.0731 −0.958696
\(985\) 10.5620 0.336533
\(986\) 0 0
\(987\) −56.2864 −1.79162
\(988\) −7.70471 −0.245120
\(989\) 7.57136 0.240755
\(990\) 32.3225 1.02728
\(991\) 24.6953 0.784474 0.392237 0.919864i \(-0.371701\pi\)
0.392237 + 0.919864i \(0.371701\pi\)
\(992\) −4.96343 −0.157589
\(993\) 19.0094 0.603244
\(994\) 6.61543 0.209829
\(995\) −3.18421 −0.100946
\(996\) 11.1842 0.354385
\(997\) 11.4050 0.361199 0.180600 0.983557i \(-0.442196\pi\)
0.180600 + 0.983557i \(0.442196\pi\)
\(998\) 17.6949 0.560121
\(999\) −56.0830 −1.77439
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.e.1.3 3
29.28 even 2 145.2.a.d.1.1 3
87.86 odd 2 1305.2.a.o.1.3 3
116.115 odd 2 2320.2.a.s.1.3 3
145.28 odd 4 725.2.b.d.349.4 6
145.57 odd 4 725.2.b.d.349.3 6
145.144 even 2 725.2.a.d.1.3 3
203.202 odd 2 7105.2.a.p.1.1 3
232.115 odd 2 9280.2.a.bm.1.1 3
232.173 even 2 9280.2.a.bu.1.3 3
435.434 odd 2 6525.2.a.bh.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.1 3 29.28 even 2
725.2.a.d.1.3 3 145.144 even 2
725.2.b.d.349.3 6 145.57 odd 4
725.2.b.d.349.4 6 145.28 odd 4
1305.2.a.o.1.3 3 87.86 odd 2
2320.2.a.s.1.3 3 116.115 odd 2
4205.2.a.e.1.3 3 1.1 even 1 trivial
6525.2.a.bh.1.1 3 435.434 odd 2
7105.2.a.p.1.1 3 203.202 odd 2
9280.2.a.bm.1.1 3 232.115 odd 2
9280.2.a.bu.1.3 3 232.173 even 2