Properties

Label 4205.2.a.e.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-2.67513 q^{2} +0.806063 q^{3} +5.15633 q^{4} -1.00000 q^{5} -2.15633 q^{6} -4.15633 q^{7} -8.44358 q^{8} -2.35026 q^{9} +O(q^{10})\) \(q-2.67513 q^{2} +0.806063 q^{3} +5.15633 q^{4} -1.00000 q^{5} -2.15633 q^{6} -4.15633 q^{7} -8.44358 q^{8} -2.35026 q^{9} +2.67513 q^{10} -2.80606 q^{11} +4.15633 q^{12} +1.35026 q^{13} +11.1187 q^{14} -0.806063 q^{15} +12.2750 q^{16} +7.11871 q^{17} +6.28726 q^{18} -3.76845 q^{19} -5.15633 q^{20} -3.35026 q^{21} +7.50659 q^{22} +4.80606 q^{23} -6.80606 q^{24} +1.00000 q^{25} -3.61213 q^{26} -4.31265 q^{27} -21.4314 q^{28} +2.15633 q^{30} -0.231548 q^{31} -15.9502 q^{32} -2.26187 q^{33} -19.0435 q^{34} +4.15633 q^{35} -12.1187 q^{36} +5.50659 q^{37} +10.0811 q^{38} +1.08840 q^{39} +8.44358 q^{40} +6.96239 q^{41} +8.96239 q^{42} +3.19394 q^{43} -14.4690 q^{44} +2.35026 q^{45} -12.8568 q^{46} -6.41819 q^{47} +9.89446 q^{48} +10.2750 q^{49} -2.67513 q^{50} +5.73813 q^{51} +6.96239 q^{52} +6.96239 q^{53} +11.5369 q^{54} +2.80606 q^{55} +35.0943 q^{56} -3.03761 q^{57} -2.57452 q^{59} -4.15633 q^{60} -5.35026 q^{61} +0.619421 q^{62} +9.76845 q^{63} +18.1187 q^{64} -1.35026 q^{65} +6.05079 q^{66} -3.19394 q^{67} +36.7064 q^{68} +3.87399 q^{69} -11.1187 q^{70} +11.3503 q^{71} +19.8446 q^{72} +11.2447 q^{73} -14.7308 q^{74} +0.806063 q^{75} -19.4314 q^{76} +11.6629 q^{77} -2.91160 q^{78} +4.73084 q^{79} -12.2750 q^{80} +3.57452 q^{81} -18.6253 q^{82} +2.54420 q^{83} -17.2750 q^{84} -7.11871 q^{85} -8.54420 q^{86} +23.6932 q^{88} -14.3127 q^{89} -6.28726 q^{90} -5.61213 q^{91} +24.7816 q^{92} -0.186642 q^{93} +17.1695 q^{94} +3.76845 q^{95} -12.8568 q^{96} +1.53102 q^{97} -27.4871 q^{98} +6.59498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.67513 −1.89160 −0.945802 0.324745i \(-0.894721\pi\)
−0.945802 + 0.324745i \(0.894721\pi\)
\(3\) 0.806063 0.465381 0.232690 0.972551i \(-0.425247\pi\)
0.232690 + 0.972551i \(0.425247\pi\)
\(4\) 5.15633 2.57816
\(5\) −1.00000 −0.447214
\(6\) −2.15633 −0.880316
\(7\) −4.15633 −1.57094 −0.785472 0.618898i \(-0.787580\pi\)
−0.785472 + 0.618898i \(0.787580\pi\)
\(8\) −8.44358 −2.98526
\(9\) −2.35026 −0.783421
\(10\) 2.67513 0.845951
\(11\) −2.80606 −0.846060 −0.423030 0.906116i \(-0.639034\pi\)
−0.423030 + 0.906116i \(0.639034\pi\)
\(12\) 4.15633 1.19983
\(13\) 1.35026 0.374495 0.187248 0.982313i \(-0.440043\pi\)
0.187248 + 0.982313i \(0.440043\pi\)
\(14\) 11.1187 2.97160
\(15\) −0.806063 −0.208125
\(16\) 12.2750 3.06876
\(17\) 7.11871 1.72654 0.863271 0.504741i \(-0.168412\pi\)
0.863271 + 0.504741i \(0.168412\pi\)
\(18\) 6.28726 1.48192
\(19\) −3.76845 −0.864542 −0.432271 0.901744i \(-0.642288\pi\)
−0.432271 + 0.901744i \(0.642288\pi\)
\(20\) −5.15633 −1.15299
\(21\) −3.35026 −0.731087
\(22\) 7.50659 1.60041
\(23\) 4.80606 1.00213 0.501067 0.865409i \(-0.332941\pi\)
0.501067 + 0.865409i \(0.332941\pi\)
\(24\) −6.80606 −1.38928
\(25\) 1.00000 0.200000
\(26\) −3.61213 −0.708396
\(27\) −4.31265 −0.829970
\(28\) −21.4314 −4.05015
\(29\) 0 0
\(30\) 2.15633 0.393689
\(31\) −0.231548 −0.0415872 −0.0207936 0.999784i \(-0.506619\pi\)
−0.0207936 + 0.999784i \(0.506619\pi\)
\(32\) −15.9502 −2.81962
\(33\) −2.26187 −0.393740
\(34\) −19.0435 −3.26593
\(35\) 4.15633 0.702547
\(36\) −12.1187 −2.01979
\(37\) 5.50659 0.905277 0.452639 0.891694i \(-0.350483\pi\)
0.452639 + 0.891694i \(0.350483\pi\)
\(38\) 10.0811 1.63537
\(39\) 1.08840 0.174283
\(40\) 8.44358 1.33505
\(41\) 6.96239 1.08734 0.543671 0.839298i \(-0.317034\pi\)
0.543671 + 0.839298i \(0.317034\pi\)
\(42\) 8.96239 1.38293
\(43\) 3.19394 0.487071 0.243535 0.969892i \(-0.421693\pi\)
0.243535 + 0.969892i \(0.421693\pi\)
\(44\) −14.4690 −2.18128
\(45\) 2.35026 0.350356
\(46\) −12.8568 −1.89564
\(47\) −6.41819 −0.936189 −0.468095 0.883678i \(-0.655059\pi\)
−0.468095 + 0.883678i \(0.655059\pi\)
\(48\) 9.89446 1.42814
\(49\) 10.2750 1.46786
\(50\) −2.67513 −0.378321
\(51\) 5.73813 0.803500
\(52\) 6.96239 0.965510
\(53\) 6.96239 0.956358 0.478179 0.878263i \(-0.341297\pi\)
0.478179 + 0.878263i \(0.341297\pi\)
\(54\) 11.5369 1.56997
\(55\) 2.80606 0.378370
\(56\) 35.0943 4.68967
\(57\) −3.03761 −0.402341
\(58\) 0 0
\(59\) −2.57452 −0.335173 −0.167587 0.985857i \(-0.553597\pi\)
−0.167587 + 0.985857i \(0.553597\pi\)
\(60\) −4.15633 −0.536579
\(61\) −5.35026 −0.685031 −0.342515 0.939512i \(-0.611279\pi\)
−0.342515 + 0.939512i \(0.611279\pi\)
\(62\) 0.619421 0.0786666
\(63\) 9.76845 1.23071
\(64\) 18.1187 2.26484
\(65\) −1.35026 −0.167479
\(66\) 6.05079 0.744800
\(67\) −3.19394 −0.390201 −0.195101 0.980783i \(-0.562503\pi\)
−0.195101 + 0.980783i \(0.562503\pi\)
\(68\) 36.7064 4.45131
\(69\) 3.87399 0.466374
\(70\) −11.1187 −1.32894
\(71\) 11.3503 1.34703 0.673514 0.739174i \(-0.264784\pi\)
0.673514 + 0.739174i \(0.264784\pi\)
\(72\) 19.8446 2.33871
\(73\) 11.2447 1.31610 0.658048 0.752976i \(-0.271383\pi\)
0.658048 + 0.752976i \(0.271383\pi\)
\(74\) −14.7308 −1.71243
\(75\) 0.806063 0.0930762
\(76\) −19.4314 −2.22893
\(77\) 11.6629 1.32911
\(78\) −2.91160 −0.329674
\(79\) 4.73084 0.532261 0.266131 0.963937i \(-0.414255\pi\)
0.266131 + 0.963937i \(0.414255\pi\)
\(80\) −12.2750 −1.37239
\(81\) 3.57452 0.397168
\(82\) −18.6253 −2.05682
\(83\) 2.54420 0.279262 0.139631 0.990204i \(-0.455408\pi\)
0.139631 + 0.990204i \(0.455408\pi\)
\(84\) −17.2750 −1.88486
\(85\) −7.11871 −0.772133
\(86\) −8.54420 −0.921345
\(87\) 0 0
\(88\) 23.6932 2.52571
\(89\) −14.3127 −1.51714 −0.758569 0.651593i \(-0.774101\pi\)
−0.758569 + 0.651593i \(0.774101\pi\)
\(90\) −6.28726 −0.662735
\(91\) −5.61213 −0.588311
\(92\) 24.7816 2.58366
\(93\) −0.186642 −0.0193539
\(94\) 17.1695 1.77090
\(95\) 3.76845 0.386635
\(96\) −12.8568 −1.31220
\(97\) 1.53102 0.155452 0.0777260 0.996975i \(-0.475234\pi\)
0.0777260 + 0.996975i \(0.475234\pi\)
\(98\) −27.4871 −2.77661
\(99\) 6.59498 0.662821
\(100\) 5.15633 0.515633
\(101\) −2.83638 −0.282230 −0.141115 0.989993i \(-0.545069\pi\)
−0.141115 + 0.989993i \(0.545069\pi\)
\(102\) −15.3503 −1.51990
\(103\) −9.89446 −0.974930 −0.487465 0.873142i \(-0.662078\pi\)
−0.487465 + 0.873142i \(0.662078\pi\)
\(104\) −11.4010 −1.11796
\(105\) 3.35026 0.326952
\(106\) −18.6253 −1.80905
\(107\) −11.6932 −1.13043 −0.565214 0.824945i \(-0.691206\pi\)
−0.565214 + 0.824945i \(0.691206\pi\)
\(108\) −22.2374 −2.13980
\(109\) −14.4993 −1.38878 −0.694390 0.719599i \(-0.744326\pi\)
−0.694390 + 0.719599i \(0.744326\pi\)
\(110\) −7.50659 −0.715725
\(111\) 4.43866 0.421299
\(112\) −51.0191 −4.82085
\(113\) 16.3938 1.54219 0.771097 0.636717i \(-0.219708\pi\)
0.771097 + 0.636717i \(0.219708\pi\)
\(114\) 8.12601 0.761070
\(115\) −4.80606 −0.448168
\(116\) 0 0
\(117\) −3.17347 −0.293387
\(118\) 6.88717 0.634015
\(119\) −29.5877 −2.71230
\(120\) 6.80606 0.621306
\(121\) −3.12601 −0.284183
\(122\) 14.3127 1.29581
\(123\) 5.61213 0.506028
\(124\) −1.19394 −0.107219
\(125\) −1.00000 −0.0894427
\(126\) −26.1319 −2.32801
\(127\) −15.3561 −1.36264 −0.681319 0.731987i \(-0.738593\pi\)
−0.681319 + 0.731987i \(0.738593\pi\)
\(128\) −16.5696 −1.46456
\(129\) 2.57452 0.226673
\(130\) 3.61213 0.316804
\(131\) −1.38058 −0.120622 −0.0603109 0.998180i \(-0.519209\pi\)
−0.0603109 + 0.998180i \(0.519209\pi\)
\(132\) −11.6629 −1.01513
\(133\) 15.6629 1.35815
\(134\) 8.54420 0.738106
\(135\) 4.31265 0.371174
\(136\) −60.1075 −5.15417
\(137\) 6.49341 0.554770 0.277385 0.960759i \(-0.410532\pi\)
0.277385 + 0.960759i \(0.410532\pi\)
\(138\) −10.3634 −0.882194
\(139\) −19.0132 −1.61268 −0.806338 0.591455i \(-0.798554\pi\)
−0.806338 + 0.591455i \(0.798554\pi\)
\(140\) 21.4314 1.81128
\(141\) −5.17347 −0.435685
\(142\) −30.3634 −2.54804
\(143\) −3.78892 −0.316845
\(144\) −28.8496 −2.40413
\(145\) 0 0
\(146\) −30.0811 −2.48953
\(147\) 8.28233 0.683115
\(148\) 28.3938 2.33395
\(149\) 6.62530 0.542766 0.271383 0.962471i \(-0.412519\pi\)
0.271383 + 0.962471i \(0.412519\pi\)
\(150\) −2.15633 −0.176063
\(151\) −9.27504 −0.754792 −0.377396 0.926052i \(-0.623180\pi\)
−0.377396 + 0.926052i \(0.623180\pi\)
\(152\) 31.8192 2.58088
\(153\) −16.7308 −1.35261
\(154\) −31.1998 −2.51415
\(155\) 0.231548 0.0185984
\(156\) 5.61213 0.449330
\(157\) 5.00729 0.399626 0.199813 0.979834i \(-0.435967\pi\)
0.199813 + 0.979834i \(0.435967\pi\)
\(158\) −12.6556 −1.00683
\(159\) 5.61213 0.445071
\(160\) 15.9502 1.26097
\(161\) −19.9756 −1.57429
\(162\) −9.56230 −0.751285
\(163\) −7.50659 −0.587961 −0.293981 0.955811i \(-0.594980\pi\)
−0.293981 + 0.955811i \(0.594980\pi\)
\(164\) 35.9003 2.80335
\(165\) 2.26187 0.176086
\(166\) −6.80606 −0.528253
\(167\) 21.8945 1.69424 0.847122 0.531398i \(-0.178333\pi\)
0.847122 + 0.531398i \(0.178333\pi\)
\(168\) 28.2882 2.18248
\(169\) −11.1768 −0.859753
\(170\) 19.0435 1.46057
\(171\) 8.85685 0.677300
\(172\) 16.4690 1.25575
\(173\) 7.02302 0.533951 0.266975 0.963703i \(-0.413976\pi\)
0.266975 + 0.963703i \(0.413976\pi\)
\(174\) 0 0
\(175\) −4.15633 −0.314189
\(176\) −34.4445 −2.59635
\(177\) −2.07522 −0.155983
\(178\) 38.2882 2.86982
\(179\) 4.77575 0.356956 0.178478 0.983944i \(-0.442883\pi\)
0.178478 + 0.983944i \(0.442883\pi\)
\(180\) 12.1187 0.903276
\(181\) 1.87399 0.139293 0.0696464 0.997572i \(-0.477813\pi\)
0.0696464 + 0.997572i \(0.477813\pi\)
\(182\) 15.0132 1.11285
\(183\) −4.31265 −0.318800
\(184\) −40.5804 −2.99163
\(185\) −5.50659 −0.404852
\(186\) 0.499293 0.0366099
\(187\) −19.9756 −1.46076
\(188\) −33.0943 −2.41365
\(189\) 17.9248 1.30384
\(190\) −10.0811 −0.731360
\(191\) −19.1187 −1.38338 −0.691691 0.722194i \(-0.743134\pi\)
−0.691691 + 0.722194i \(0.743134\pi\)
\(192\) 14.6048 1.05401
\(193\) −19.8945 −1.43203 −0.716017 0.698083i \(-0.754037\pi\)
−0.716017 + 0.698083i \(0.754037\pi\)
\(194\) −4.09569 −0.294053
\(195\) −1.08840 −0.0779417
\(196\) 52.9814 3.78439
\(197\) −13.5369 −0.964464 −0.482232 0.876043i \(-0.660174\pi\)
−0.482232 + 0.876043i \(0.660174\pi\)
\(198\) −17.6424 −1.25379
\(199\) 2.57452 0.182503 0.0912513 0.995828i \(-0.470913\pi\)
0.0912513 + 0.995828i \(0.470913\pi\)
\(200\) −8.44358 −0.597051
\(201\) −2.57452 −0.181592
\(202\) 7.58769 0.533868
\(203\) 0 0
\(204\) 29.5877 2.07155
\(205\) −6.96239 −0.486274
\(206\) 26.4690 1.84418
\(207\) −11.2955 −0.785092
\(208\) 16.5745 1.14924
\(209\) 10.5745 0.731455
\(210\) −8.96239 −0.618464
\(211\) −11.8945 −0.818848 −0.409424 0.912344i \(-0.634270\pi\)
−0.409424 + 0.912344i \(0.634270\pi\)
\(212\) 35.9003 2.46565
\(213\) 9.14903 0.626881
\(214\) 31.2809 2.13832
\(215\) −3.19394 −0.217825
\(216\) 36.4142 2.47767
\(217\) 0.962389 0.0653312
\(218\) 38.7875 2.62702
\(219\) 9.06396 0.612486
\(220\) 14.4690 0.975498
\(221\) 9.61213 0.646582
\(222\) −11.8740 −0.796930
\(223\) −1.11871 −0.0749146 −0.0374573 0.999298i \(-0.511926\pi\)
−0.0374573 + 0.999298i \(0.511926\pi\)
\(224\) 66.2941 4.42946
\(225\) −2.35026 −0.156684
\(226\) −43.8554 −2.91722
\(227\) 0.0303172 0.00201222 0.00100611 0.999999i \(-0.499680\pi\)
0.00100611 + 0.999999i \(0.499680\pi\)
\(228\) −15.6629 −1.03730
\(229\) 5.84955 0.386549 0.193275 0.981145i \(-0.438089\pi\)
0.193275 + 0.981145i \(0.438089\pi\)
\(230\) 12.8568 0.847755
\(231\) 9.40105 0.618543
\(232\) 0 0
\(233\) 26.1016 1.70997 0.854985 0.518652i \(-0.173566\pi\)
0.854985 + 0.518652i \(0.173566\pi\)
\(234\) 8.48944 0.554972
\(235\) 6.41819 0.418677
\(236\) −13.2750 −0.864131
\(237\) 3.81336 0.247704
\(238\) 79.1509 5.13059
\(239\) 1.42548 0.0922069 0.0461035 0.998937i \(-0.485320\pi\)
0.0461035 + 0.998937i \(0.485320\pi\)
\(240\) −9.89446 −0.638685
\(241\) −0.0752228 −0.00484553 −0.00242276 0.999997i \(-0.500771\pi\)
−0.00242276 + 0.999997i \(0.500771\pi\)
\(242\) 8.36248 0.537561
\(243\) 15.8192 1.01480
\(244\) −27.5877 −1.76612
\(245\) −10.2750 −0.656448
\(246\) −15.0132 −0.957205
\(247\) −5.08840 −0.323767
\(248\) 1.95509 0.124149
\(249\) 2.05079 0.129963
\(250\) 2.67513 0.169190
\(251\) −16.9829 −1.07195 −0.535974 0.844234i \(-0.680056\pi\)
−0.535974 + 0.844234i \(0.680056\pi\)
\(252\) 50.3693 3.17297
\(253\) −13.4861 −0.847865
\(254\) 41.0797 2.57757
\(255\) −5.73813 −0.359336
\(256\) 8.08840 0.505525
\(257\) 25.1998 1.57192 0.785961 0.618276i \(-0.212169\pi\)
0.785961 + 0.618276i \(0.212169\pi\)
\(258\) −6.88717 −0.428776
\(259\) −22.8872 −1.42214
\(260\) −6.96239 −0.431789
\(261\) 0 0
\(262\) 3.69323 0.228168
\(263\) 16.1319 0.994735 0.497367 0.867540i \(-0.334300\pi\)
0.497367 + 0.867540i \(0.334300\pi\)
\(264\) 19.0982 1.17542
\(265\) −6.96239 −0.427696
\(266\) −41.9003 −2.56907
\(267\) −11.5369 −0.706047
\(268\) −16.4690 −1.00600
\(269\) −5.28963 −0.322514 −0.161257 0.986912i \(-0.551555\pi\)
−0.161257 + 0.986912i \(0.551555\pi\)
\(270\) −11.5369 −0.702114
\(271\) −1.13330 −0.0688432 −0.0344216 0.999407i \(-0.510959\pi\)
−0.0344216 + 0.999407i \(0.510959\pi\)
\(272\) 87.3825 5.29834
\(273\) −4.52373 −0.273789
\(274\) −17.3707 −1.04940
\(275\) −2.80606 −0.169212
\(276\) 19.9756 1.20239
\(277\) −16.3634 −0.983184 −0.491592 0.870826i \(-0.663585\pi\)
−0.491592 + 0.870826i \(0.663585\pi\)
\(278\) 50.8627 3.05054
\(279\) 0.544198 0.0325803
\(280\) −35.0943 −2.09728
\(281\) −24.8265 −1.48103 −0.740513 0.672042i \(-0.765418\pi\)
−0.740513 + 0.672042i \(0.765418\pi\)
\(282\) 13.8397 0.824142
\(283\) −4.18076 −0.248521 −0.124260 0.992250i \(-0.539656\pi\)
−0.124260 + 0.992250i \(0.539656\pi\)
\(284\) 58.5256 3.47286
\(285\) 3.03761 0.179933
\(286\) 10.1359 0.599346
\(287\) −28.9380 −1.70815
\(288\) 37.4871 2.20895
\(289\) 33.6761 1.98095
\(290\) 0 0
\(291\) 1.23410 0.0723444
\(292\) 57.9814 3.39311
\(293\) −23.6180 −1.37978 −0.689889 0.723915i \(-0.742341\pi\)
−0.689889 + 0.723915i \(0.742341\pi\)
\(294\) −22.1563 −1.29218
\(295\) 2.57452 0.149894
\(296\) −46.4953 −2.70249
\(297\) 12.1016 0.702204
\(298\) −17.7235 −1.02670
\(299\) 6.48944 0.375294
\(300\) 4.15633 0.239966
\(301\) −13.2750 −0.765161
\(302\) 24.8119 1.42777
\(303\) −2.28630 −0.131345
\(304\) −46.2579 −2.65307
\(305\) 5.35026 0.306355
\(306\) 44.7572 2.55860
\(307\) −32.5052 −1.85517 −0.927584 0.373614i \(-0.878118\pi\)
−0.927584 + 0.373614i \(0.878118\pi\)
\(308\) 60.1378 3.42667
\(309\) −7.97556 −0.453714
\(310\) −0.619421 −0.0351808
\(311\) 9.31994 0.528486 0.264243 0.964456i \(-0.414878\pi\)
0.264243 + 0.964456i \(0.414878\pi\)
\(312\) −9.18997 −0.520279
\(313\) 9.60228 0.542753 0.271376 0.962473i \(-0.412521\pi\)
0.271376 + 0.962473i \(0.412521\pi\)
\(314\) −13.3952 −0.755933
\(315\) −9.76845 −0.550390
\(316\) 24.3938 1.37226
\(317\) −19.3707 −1.08797 −0.543984 0.839095i \(-0.683085\pi\)
−0.543984 + 0.839095i \(0.683085\pi\)
\(318\) −15.0132 −0.841897
\(319\) 0 0
\(320\) −18.1187 −1.01287
\(321\) −9.42548 −0.526079
\(322\) 53.4372 2.97794
\(323\) −26.8265 −1.49267
\(324\) 18.4314 1.02396
\(325\) 1.35026 0.0748990
\(326\) 20.0811 1.11219
\(327\) −11.6873 −0.646312
\(328\) −58.7875 −3.24600
\(329\) 26.6761 1.47070
\(330\) −6.05079 −0.333085
\(331\) 29.5428 1.62382 0.811909 0.583784i \(-0.198428\pi\)
0.811909 + 0.583784i \(0.198428\pi\)
\(332\) 13.1187 0.719983
\(333\) −12.9419 −0.709213
\(334\) −58.5705 −3.20484
\(335\) 3.19394 0.174503
\(336\) −41.1246 −2.24353
\(337\) −12.5442 −0.683326 −0.341663 0.939823i \(-0.610990\pi\)
−0.341663 + 0.939823i \(0.610990\pi\)
\(338\) 29.8994 1.62631
\(339\) 13.2144 0.717708
\(340\) −36.7064 −1.99068
\(341\) 0.649738 0.0351853
\(342\) −23.6932 −1.28118
\(343\) −13.6121 −0.734986
\(344\) −26.9683 −1.45403
\(345\) −3.87399 −0.208569
\(346\) −18.7875 −1.01002
\(347\) −25.0943 −1.34713 −0.673566 0.739127i \(-0.735238\pi\)
−0.673566 + 0.739127i \(0.735238\pi\)
\(348\) 0 0
\(349\) −17.0738 −0.913940 −0.456970 0.889482i \(-0.651065\pi\)
−0.456970 + 0.889482i \(0.651065\pi\)
\(350\) 11.1187 0.594320
\(351\) −5.82321 −0.310820
\(352\) 44.7572 2.38557
\(353\) 5.66291 0.301406 0.150703 0.988579i \(-0.451846\pi\)
0.150703 + 0.988579i \(0.451846\pi\)
\(354\) 5.55149 0.295058
\(355\) −11.3503 −0.602409
\(356\) −73.8007 −3.91143
\(357\) −23.8496 −1.26225
\(358\) −12.7757 −0.675219
\(359\) −0.755278 −0.0398621 −0.0199310 0.999801i \(-0.506345\pi\)
−0.0199310 + 0.999801i \(0.506345\pi\)
\(360\) −19.8446 −1.04590
\(361\) −4.79877 −0.252567
\(362\) −5.01317 −0.263487
\(363\) −2.51976 −0.132253
\(364\) −28.9380 −1.51676
\(365\) −11.2447 −0.588576
\(366\) 11.5369 0.603044
\(367\) −11.4460 −0.597474 −0.298737 0.954335i \(-0.596565\pi\)
−0.298737 + 0.954335i \(0.596565\pi\)
\(368\) 58.9946 3.07531
\(369\) −16.3634 −0.851846
\(370\) 14.7308 0.765820
\(371\) −28.9380 −1.50238
\(372\) −0.962389 −0.0498975
\(373\) 3.86414 0.200078 0.100039 0.994984i \(-0.468103\pi\)
0.100039 + 0.994984i \(0.468103\pi\)
\(374\) 53.4372 2.76317
\(375\) −0.806063 −0.0416249
\(376\) 54.1925 2.79477
\(377\) 0 0
\(378\) −47.9511 −2.46634
\(379\) −12.1055 −0.621820 −0.310910 0.950439i \(-0.600634\pi\)
−0.310910 + 0.950439i \(0.600634\pi\)
\(380\) 19.4314 0.996808
\(381\) −12.3780 −0.634145
\(382\) 51.1451 2.61681
\(383\) −10.0205 −0.512022 −0.256011 0.966674i \(-0.582408\pi\)
−0.256011 + 0.966674i \(0.582408\pi\)
\(384\) −13.3561 −0.681578
\(385\) −11.6629 −0.594397
\(386\) 53.2203 2.70884
\(387\) −7.50659 −0.381581
\(388\) 7.89446 0.400780
\(389\) 25.6629 1.30116 0.650581 0.759437i \(-0.274526\pi\)
0.650581 + 0.759437i \(0.274526\pi\)
\(390\) 2.91160 0.147435
\(391\) 34.2130 1.73023
\(392\) −86.7581 −4.38195
\(393\) −1.11283 −0.0561351
\(394\) 36.2130 1.82438
\(395\) −4.73084 −0.238034
\(396\) 34.0059 1.70886
\(397\) 27.7137 1.39091 0.695455 0.718569i \(-0.255203\pi\)
0.695455 + 0.718569i \(0.255203\pi\)
\(398\) −6.88717 −0.345222
\(399\) 12.6253 0.632056
\(400\) 12.2750 0.613752
\(401\) 7.42548 0.370811 0.185406 0.982662i \(-0.440640\pi\)
0.185406 + 0.982662i \(0.440640\pi\)
\(402\) 6.88717 0.343501
\(403\) −0.312650 −0.0155742
\(404\) −14.6253 −0.727636
\(405\) −3.57452 −0.177619
\(406\) 0 0
\(407\) −15.4518 −0.765919
\(408\) −48.4504 −2.39865
\(409\) 33.1998 1.64163 0.820813 0.571198i \(-0.193521\pi\)
0.820813 + 0.571198i \(0.193521\pi\)
\(410\) 18.6253 0.919838
\(411\) 5.23410 0.258179
\(412\) −51.0191 −2.51353
\(413\) 10.7005 0.526538
\(414\) 30.2170 1.48508
\(415\) −2.54420 −0.124890
\(416\) −21.5369 −1.05593
\(417\) −15.3258 −0.750509
\(418\) −28.2882 −1.38362
\(419\) 16.5599 0.809005 0.404503 0.914537i \(-0.367445\pi\)
0.404503 + 0.914537i \(0.367445\pi\)
\(420\) 17.2750 0.842936
\(421\) 8.82653 0.430179 0.215089 0.976594i \(-0.430996\pi\)
0.215089 + 0.976594i \(0.430996\pi\)
\(422\) 31.8192 1.54894
\(423\) 15.0844 0.733430
\(424\) −58.7875 −2.85497
\(425\) 7.11871 0.345308
\(426\) −24.4749 −1.18581
\(427\) 22.2374 1.07614
\(428\) −60.2941 −2.91442
\(429\) −3.05411 −0.147454
\(430\) 8.54420 0.412038
\(431\) 4.25202 0.204812 0.102406 0.994743i \(-0.467346\pi\)
0.102406 + 0.994743i \(0.467346\pi\)
\(432\) −52.9380 −2.54698
\(433\) 1.81924 0.0874270 0.0437135 0.999044i \(-0.486081\pi\)
0.0437135 + 0.999044i \(0.486081\pi\)
\(434\) −2.57452 −0.123581
\(435\) 0 0
\(436\) −74.7631 −3.58050
\(437\) −18.1114 −0.866387
\(438\) −24.2473 −1.15858
\(439\) 14.1114 0.673501 0.336751 0.941594i \(-0.390672\pi\)
0.336751 + 0.941594i \(0.390672\pi\)
\(440\) −23.6932 −1.12953
\(441\) −24.1490 −1.14995
\(442\) −25.7137 −1.22308
\(443\) 17.2809 0.821041 0.410521 0.911851i \(-0.365347\pi\)
0.410521 + 0.911851i \(0.365347\pi\)
\(444\) 22.8872 1.08618
\(445\) 14.3127 0.678485
\(446\) 2.99271 0.141709
\(447\) 5.34041 0.252593
\(448\) −75.3073 −3.55793
\(449\) −9.35026 −0.441266 −0.220633 0.975357i \(-0.570812\pi\)
−0.220633 + 0.975357i \(0.570812\pi\)
\(450\) 6.28726 0.296384
\(451\) −19.5369 −0.919957
\(452\) 84.5315 3.97603
\(453\) −7.47627 −0.351266
\(454\) −0.0811024 −0.00380632
\(455\) 5.61213 0.263101
\(456\) 25.6483 1.20109
\(457\) −17.6629 −0.826236 −0.413118 0.910677i \(-0.635560\pi\)
−0.413118 + 0.910677i \(0.635560\pi\)
\(458\) −15.6483 −0.731198
\(459\) −30.7005 −1.43298
\(460\) −24.7816 −1.15545
\(461\) −15.5633 −0.724853 −0.362426 0.932012i \(-0.618052\pi\)
−0.362426 + 0.932012i \(0.618052\pi\)
\(462\) −25.1490 −1.17004
\(463\) 2.98286 0.138625 0.0693126 0.997595i \(-0.477919\pi\)
0.0693126 + 0.997595i \(0.477919\pi\)
\(464\) 0 0
\(465\) 0.186642 0.00865533
\(466\) −69.8251 −3.23459
\(467\) −34.5804 −1.60019 −0.800095 0.599873i \(-0.795218\pi\)
−0.800095 + 0.599873i \(0.795218\pi\)
\(468\) −16.3634 −0.756400
\(469\) 13.2750 0.612984
\(470\) −17.1695 −0.791970
\(471\) 4.03620 0.185978
\(472\) 21.7381 1.00058
\(473\) −8.96239 −0.412091
\(474\) −10.2012 −0.468558
\(475\) −3.76845 −0.172908
\(476\) −152.564 −6.99275
\(477\) −16.3634 −0.749230
\(478\) −3.81336 −0.174419
\(479\) 34.1925 1.56230 0.781148 0.624346i \(-0.214634\pi\)
0.781148 + 0.624346i \(0.214634\pi\)
\(480\) 12.8568 0.586832
\(481\) 7.43533 0.339022
\(482\) 0.201231 0.00916581
\(483\) −16.1016 −0.732647
\(484\) −16.1187 −0.732669
\(485\) −1.53102 −0.0695202
\(486\) −42.3185 −1.91961
\(487\) 38.4953 1.74439 0.872195 0.489159i \(-0.162696\pi\)
0.872195 + 0.489159i \(0.162696\pi\)
\(488\) 45.1754 2.04499
\(489\) −6.05079 −0.273626
\(490\) 27.4871 1.24174
\(491\) 27.4676 1.23959 0.619797 0.784762i \(-0.287215\pi\)
0.619797 + 0.784762i \(0.287215\pi\)
\(492\) 28.9380 1.30462
\(493\) 0 0
\(494\) 13.6121 0.612439
\(495\) −6.59498 −0.296422
\(496\) −2.84226 −0.127621
\(497\) −47.1754 −2.11610
\(498\) −5.48612 −0.245839
\(499\) −32.1016 −1.43706 −0.718532 0.695494i \(-0.755186\pi\)
−0.718532 + 0.695494i \(0.755186\pi\)
\(500\) −5.15633 −0.230598
\(501\) 17.6483 0.788469
\(502\) 45.4314 2.02770
\(503\) −9.74401 −0.434464 −0.217232 0.976120i \(-0.569703\pi\)
−0.217232 + 0.976120i \(0.569703\pi\)
\(504\) −82.4807 −3.67398
\(505\) 2.83638 0.126217
\(506\) 36.0771 1.60382
\(507\) −9.00920 −0.400113
\(508\) −79.1813 −3.51310
\(509\) −8.57452 −0.380059 −0.190029 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(510\) 15.3503 0.679721
\(511\) −46.7367 −2.06751
\(512\) 11.5017 0.508306
\(513\) 16.2520 0.717544
\(514\) −67.4128 −2.97345
\(515\) 9.89446 0.436002
\(516\) 13.2750 0.584401
\(517\) 18.0098 0.792072
\(518\) 61.2262 2.69012
\(519\) 5.66100 0.248490
\(520\) 11.4010 0.499969
\(521\) −37.8251 −1.65715 −0.828574 0.559879i \(-0.810848\pi\)
−0.828574 + 0.559879i \(0.810848\pi\)
\(522\) 0 0
\(523\) −13.5818 −0.593891 −0.296946 0.954894i \(-0.595968\pi\)
−0.296946 + 0.954894i \(0.595968\pi\)
\(524\) −7.11871 −0.310982
\(525\) −3.35026 −0.146217
\(526\) −43.1549 −1.88164
\(527\) −1.64832 −0.0718021
\(528\) −27.7645 −1.20829
\(529\) 0.0982457 0.00427155
\(530\) 18.6253 0.809031
\(531\) 6.05079 0.262582
\(532\) 80.7631 3.50152
\(533\) 9.40105 0.407205
\(534\) 30.8627 1.33556
\(535\) 11.6932 0.505542
\(536\) 26.9683 1.16485
\(537\) 3.84955 0.166121
\(538\) 14.1504 0.610069
\(539\) −28.8324 −1.24190
\(540\) 22.2374 0.956947
\(541\) −42.3127 −1.81916 −0.909581 0.415526i \(-0.863598\pi\)
−0.909581 + 0.415526i \(0.863598\pi\)
\(542\) 3.03173 0.130224
\(543\) 1.51056 0.0648242
\(544\) −113.545 −4.86819
\(545\) 14.4993 0.621081
\(546\) 12.1016 0.517899
\(547\) −36.4690 −1.55930 −0.779650 0.626215i \(-0.784603\pi\)
−0.779650 + 0.626215i \(0.784603\pi\)
\(548\) 33.4821 1.43029
\(549\) 12.5745 0.536667
\(550\) 7.50659 0.320082
\(551\) 0 0
\(552\) −32.7104 −1.39225
\(553\) −19.6629 −0.836152
\(554\) 43.7743 1.85979
\(555\) −4.43866 −0.188411
\(556\) −98.0381 −4.15774
\(557\) 19.5223 0.827187 0.413594 0.910462i \(-0.364273\pi\)
0.413594 + 0.910462i \(0.364273\pi\)
\(558\) −1.45580 −0.0616290
\(559\) 4.31265 0.182406
\(560\) 51.0191 2.15595
\(561\) −16.1016 −0.679809
\(562\) 66.4142 2.80151
\(563\) 2.94192 0.123987 0.0619936 0.998077i \(-0.480254\pi\)
0.0619936 + 0.998077i \(0.480254\pi\)
\(564\) −26.6761 −1.12327
\(565\) −16.3938 −0.689690
\(566\) 11.1841 0.470102
\(567\) −14.8568 −0.623929
\(568\) −95.8369 −4.02123
\(569\) −2.49929 −0.104776 −0.0523879 0.998627i \(-0.516683\pi\)
−0.0523879 + 0.998627i \(0.516683\pi\)
\(570\) −8.12601 −0.340361
\(571\) 43.8007 1.83300 0.916501 0.400033i \(-0.131001\pi\)
0.916501 + 0.400033i \(0.131001\pi\)
\(572\) −19.5369 −0.816879
\(573\) −15.4109 −0.643799
\(574\) 77.4128 3.23115
\(575\) 4.80606 0.200427
\(576\) −42.5837 −1.77432
\(577\) 23.9062 0.995229 0.497614 0.867398i \(-0.334209\pi\)
0.497614 + 0.867398i \(0.334209\pi\)
\(578\) −90.0879 −3.74716
\(579\) −16.0362 −0.666442
\(580\) 0 0
\(581\) −10.5745 −0.438705
\(582\) −3.30139 −0.136847
\(583\) −19.5369 −0.809136
\(584\) −94.9457 −3.92888
\(585\) 3.17347 0.131207
\(586\) 63.1813 2.60999
\(587\) −3.71767 −0.153445 −0.0767223 0.997053i \(-0.524445\pi\)
−0.0767223 + 0.997053i \(0.524445\pi\)
\(588\) 42.7064 1.76118
\(589\) 0.872577 0.0359539
\(590\) −6.88717 −0.283540
\(591\) −10.9116 −0.448843
\(592\) 67.5936 2.77808
\(593\) −25.5125 −1.04767 −0.523836 0.851819i \(-0.675499\pi\)
−0.523836 + 0.851819i \(0.675499\pi\)
\(594\) −32.3733 −1.32829
\(595\) 29.5877 1.21298
\(596\) 34.1622 1.39934
\(597\) 2.07522 0.0849332
\(598\) −17.3601 −0.709908
\(599\) 15.6834 0.640806 0.320403 0.947281i \(-0.396182\pi\)
0.320403 + 0.947281i \(0.396182\pi\)
\(600\) −6.80606 −0.277856
\(601\) −13.1392 −0.535958 −0.267979 0.963425i \(-0.586356\pi\)
−0.267979 + 0.963425i \(0.586356\pi\)
\(602\) 35.5125 1.44738
\(603\) 7.50659 0.305692
\(604\) −47.8251 −1.94598
\(605\) 3.12601 0.127090
\(606\) 6.11616 0.248452
\(607\) −18.1465 −0.736543 −0.368271 0.929718i \(-0.620050\pi\)
−0.368271 + 0.929718i \(0.620050\pi\)
\(608\) 60.1075 2.43768
\(609\) 0 0
\(610\) −14.3127 −0.579502
\(611\) −8.66624 −0.350598
\(612\) −86.2697 −3.48724
\(613\) −32.5501 −1.31469 −0.657343 0.753592i \(-0.728320\pi\)
−0.657343 + 0.753592i \(0.728320\pi\)
\(614\) 86.9556 3.50924
\(615\) −5.61213 −0.226303
\(616\) −98.4768 −3.96774
\(617\) −20.2433 −0.814965 −0.407482 0.913213i \(-0.633593\pi\)
−0.407482 + 0.913213i \(0.633593\pi\)
\(618\) 21.3357 0.858247
\(619\) 16.2071 0.651419 0.325709 0.945470i \(-0.394397\pi\)
0.325709 + 0.945470i \(0.394397\pi\)
\(620\) 1.19394 0.0479496
\(621\) −20.7269 −0.831741
\(622\) −24.9321 −0.999685
\(623\) 59.4880 2.38334
\(624\) 13.3601 0.534832
\(625\) 1.00000 0.0400000
\(626\) −25.6873 −1.02667
\(627\) 8.52373 0.340405
\(628\) 25.8192 1.03030
\(629\) 39.1998 1.56300
\(630\) 26.1319 1.04112
\(631\) 2.13586 0.0850271 0.0425136 0.999096i \(-0.486463\pi\)
0.0425136 + 0.999096i \(0.486463\pi\)
\(632\) −39.9452 −1.58894
\(633\) −9.58769 −0.381076
\(634\) 51.8192 2.05800
\(635\) 15.3561 0.609390
\(636\) 28.9380 1.14746
\(637\) 13.8740 0.549708
\(638\) 0 0
\(639\) −26.6761 −1.05529
\(640\) 16.5696 0.654971
\(641\) 14.0362 0.554396 0.277198 0.960813i \(-0.410594\pi\)
0.277198 + 0.960813i \(0.410594\pi\)
\(642\) 25.2144 0.995133
\(643\) −43.8799 −1.73045 −0.865227 0.501381i \(-0.832826\pi\)
−0.865227 + 0.501381i \(0.832826\pi\)
\(644\) −103.000 −4.05879
\(645\) −2.57452 −0.101371
\(646\) 71.7645 2.82354
\(647\) −22.5560 −0.886766 −0.443383 0.896332i \(-0.646222\pi\)
−0.443383 + 0.896332i \(0.646222\pi\)
\(648\) −30.1817 −1.18565
\(649\) 7.22425 0.283577
\(650\) −3.61213 −0.141679
\(651\) 0.775746 0.0304039
\(652\) −38.7064 −1.51586
\(653\) −27.3054 −1.06854 −0.534271 0.845314i \(-0.679414\pi\)
−0.534271 + 0.845314i \(0.679414\pi\)
\(654\) 31.2652 1.22257
\(655\) 1.38058 0.0539437
\(656\) 85.4636 3.33679
\(657\) −26.4280 −1.03106
\(658\) −71.3620 −2.78198
\(659\) −14.0665 −0.547954 −0.273977 0.961736i \(-0.588339\pi\)
−0.273977 + 0.961736i \(0.588339\pi\)
\(660\) 11.6629 0.453978
\(661\) 38.9741 1.51592 0.757959 0.652302i \(-0.226197\pi\)
0.757959 + 0.652302i \(0.226197\pi\)
\(662\) −79.0308 −3.07162
\(663\) 7.74798 0.300907
\(664\) −21.4821 −0.833669
\(665\) −15.6629 −0.607382
\(666\) 34.6213 1.34155
\(667\) 0 0
\(668\) 112.895 4.36804
\(669\) −0.901754 −0.0348638
\(670\) −8.54420 −0.330091
\(671\) 15.0132 0.579577
\(672\) 53.4372 2.06139
\(673\) 20.3390 0.784011 0.392005 0.919963i \(-0.371781\pi\)
0.392005 + 0.919963i \(0.371781\pi\)
\(674\) 33.5574 1.29258
\(675\) −4.31265 −0.165994
\(676\) −57.6312 −2.21658
\(677\) 19.1841 0.737304 0.368652 0.929567i \(-0.379819\pi\)
0.368652 + 0.929567i \(0.379819\pi\)
\(678\) −35.3503 −1.35762
\(679\) −6.36344 −0.244206
\(680\) 60.1075 2.30502
\(681\) 0.0244376 0.000936449 0
\(682\) −1.73813 −0.0665566
\(683\) −24.1319 −0.923381 −0.461691 0.887041i \(-0.652757\pi\)
−0.461691 + 0.887041i \(0.652757\pi\)
\(684\) 45.6688 1.74619
\(685\) −6.49341 −0.248101
\(686\) 36.4142 1.39030
\(687\) 4.71511 0.179893
\(688\) 39.2057 1.49470
\(689\) 9.40105 0.358151
\(690\) 10.3634 0.394529
\(691\) −4.28821 −0.163131 −0.0815657 0.996668i \(-0.525992\pi\)
−0.0815657 + 0.996668i \(0.525992\pi\)
\(692\) 36.2130 1.37661
\(693\) −27.4109 −1.04125
\(694\) 67.1305 2.54824
\(695\) 19.0132 0.721211
\(696\) 0 0
\(697\) 49.5633 1.87734
\(698\) 45.6747 1.72881
\(699\) 21.0395 0.795788
\(700\) −21.4314 −0.810029
\(701\) −14.1260 −0.533532 −0.266766 0.963761i \(-0.585955\pi\)
−0.266766 + 0.963761i \(0.585955\pi\)
\(702\) 15.5778 0.587948
\(703\) −20.7513 −0.782650
\(704\) −50.8423 −1.91619
\(705\) 5.17347 0.194844
\(706\) −15.1490 −0.570141
\(707\) 11.7889 0.443368
\(708\) −10.7005 −0.402150
\(709\) 6.75131 0.253551 0.126775 0.991931i \(-0.459537\pi\)
0.126775 + 0.991931i \(0.459537\pi\)
\(710\) 30.3634 1.13952
\(711\) −11.1187 −0.416984
\(712\) 120.850 4.52905
\(713\) −1.11283 −0.0416760
\(714\) 63.8007 2.38768
\(715\) 3.78892 0.141698
\(716\) 24.6253 0.920291
\(717\) 1.14903 0.0429113
\(718\) 2.02047 0.0754032
\(719\) −43.8251 −1.63440 −0.817201 0.576353i \(-0.804475\pi\)
−0.817201 + 0.576353i \(0.804475\pi\)
\(720\) 28.8496 1.07516
\(721\) 41.1246 1.53156
\(722\) 12.8373 0.477756
\(723\) −0.0606343 −0.00225502
\(724\) 9.66291 0.359119
\(725\) 0 0
\(726\) 6.74069 0.250170
\(727\) 14.8813 0.551916 0.275958 0.961170i \(-0.411005\pi\)
0.275958 + 0.961170i \(0.411005\pi\)
\(728\) 47.3865 1.75626
\(729\) 2.02776 0.0751023
\(730\) 30.0811 1.11335
\(731\) 22.7367 0.840948
\(732\) −22.2374 −0.821919
\(733\) −7.17935 −0.265175 −0.132588 0.991171i \(-0.542329\pi\)
−0.132588 + 0.991171i \(0.542329\pi\)
\(734\) 30.6194 1.13018
\(735\) −8.28233 −0.305498
\(736\) −76.6575 −2.82563
\(737\) 8.96239 0.330134
\(738\) 43.7743 1.61136
\(739\) −16.1709 −0.594857 −0.297428 0.954744i \(-0.596129\pi\)
−0.297428 + 0.954744i \(0.596129\pi\)
\(740\) −28.3938 −1.04378
\(741\) −4.10157 −0.150675
\(742\) 77.4128 2.84191
\(743\) −27.8192 −1.02059 −0.510294 0.860000i \(-0.670464\pi\)
−0.510294 + 0.860000i \(0.670464\pi\)
\(744\) 1.57593 0.0577764
\(745\) −6.62530 −0.242732
\(746\) −10.3371 −0.378468
\(747\) −5.97953 −0.218780
\(748\) −103.000 −3.76607
\(749\) 48.6009 1.77584
\(750\) 2.15633 0.0787379
\(751\) 17.6326 0.643423 0.321711 0.946838i \(-0.395742\pi\)
0.321711 + 0.946838i \(0.395742\pi\)
\(752\) −78.7835 −2.87294
\(753\) −13.6893 −0.498864
\(754\) 0 0
\(755\) 9.27504 0.337553
\(756\) 92.4260 3.36150
\(757\) −1.53102 −0.0556460 −0.0278230 0.999613i \(-0.508857\pi\)
−0.0278230 + 0.999613i \(0.508857\pi\)
\(758\) 32.3839 1.17624
\(759\) −10.8707 −0.394580
\(760\) −31.8192 −1.15421
\(761\) −34.4749 −1.24971 −0.624856 0.780740i \(-0.714842\pi\)
−0.624856 + 0.780740i \(0.714842\pi\)
\(762\) 33.1128 1.19955
\(763\) 60.2638 2.18170
\(764\) −98.5823 −3.56658
\(765\) 16.7308 0.604905
\(766\) 26.8061 0.968542
\(767\) −3.47627 −0.125521
\(768\) 6.51976 0.235262
\(769\) 19.3404 0.697433 0.348717 0.937228i \(-0.386618\pi\)
0.348717 + 0.937228i \(0.386618\pi\)
\(770\) 31.1998 1.12436
\(771\) 20.3127 0.731542
\(772\) −102.582 −3.69202
\(773\) −36.2677 −1.30446 −0.652230 0.758021i \(-0.726166\pi\)
−0.652230 + 0.758021i \(0.726166\pi\)
\(774\) 20.0811 0.721800
\(775\) −0.231548 −0.00831745
\(776\) −12.9273 −0.464064
\(777\) −18.4485 −0.661837
\(778\) −68.6516 −2.46128
\(779\) −26.2374 −0.940053
\(780\) −5.61213 −0.200946
\(781\) −31.8496 −1.13967
\(782\) −91.5242 −3.27290
\(783\) 0 0
\(784\) 126.127 4.50452
\(785\) −5.00729 −0.178718
\(786\) 2.97698 0.106185
\(787\) 32.0059 1.14089 0.570443 0.821337i \(-0.306771\pi\)
0.570443 + 0.821337i \(0.306771\pi\)
\(788\) −69.8007 −2.48655
\(789\) 13.0033 0.462931
\(790\) 12.6556 0.450267
\(791\) −68.1378 −2.42270
\(792\) −55.6853 −1.97869
\(793\) −7.22425 −0.256541
\(794\) −74.1378 −2.63105
\(795\) −5.61213 −0.199042
\(796\) 13.2750 0.470521
\(797\) 41.6932 1.47685 0.738425 0.674336i \(-0.235570\pi\)
0.738425 + 0.674336i \(0.235570\pi\)
\(798\) −33.7743 −1.19560
\(799\) −45.6893 −1.61637
\(800\) −15.9502 −0.563924
\(801\) 33.6385 1.18856
\(802\) −19.8641 −0.701427
\(803\) −31.5534 −1.11350
\(804\) −13.2750 −0.468175
\(805\) 19.9756 0.704046
\(806\) 0.836381 0.0294603
\(807\) −4.26378 −0.150092
\(808\) 23.9492 0.842530
\(809\) −30.6371 −1.07714 −0.538571 0.842580i \(-0.681036\pi\)
−0.538571 + 0.842580i \(0.681036\pi\)
\(810\) 9.56230 0.335985
\(811\) 25.4617 0.894081 0.447040 0.894514i \(-0.352478\pi\)
0.447040 + 0.894514i \(0.352478\pi\)
\(812\) 0 0
\(813\) −0.913513 −0.0320383
\(814\) 41.3357 1.44881
\(815\) 7.50659 0.262944
\(816\) 70.4358 2.46575
\(817\) −12.0362 −0.421093
\(818\) −88.8139 −3.10530
\(819\) 13.1900 0.460895
\(820\) −35.9003 −1.25369
\(821\) −32.7005 −1.14126 −0.570628 0.821209i \(-0.693300\pi\)
−0.570628 + 0.821209i \(0.693300\pi\)
\(822\) −14.0019 −0.488373
\(823\) 31.1041 1.08422 0.542111 0.840307i \(-0.317625\pi\)
0.542111 + 0.840307i \(0.317625\pi\)
\(824\) 83.5447 2.91042
\(825\) −2.26187 −0.0787480
\(826\) −28.6253 −0.996002
\(827\) 1.58181 0.0550049 0.0275025 0.999622i \(-0.491245\pi\)
0.0275025 + 0.999622i \(0.491245\pi\)
\(828\) −58.2433 −2.02409
\(829\) 0.111420 0.00386976 0.00193488 0.999998i \(-0.499384\pi\)
0.00193488 + 0.999998i \(0.499384\pi\)
\(830\) 6.80606 0.236242
\(831\) −13.1900 −0.457555
\(832\) 24.4650 0.848171
\(833\) 73.1451 2.53433
\(834\) 40.9986 1.41966
\(835\) −21.8945 −0.757689
\(836\) 54.5256 1.88581
\(837\) 0.998585 0.0345162
\(838\) −44.3000 −1.53032
\(839\) −28.9829 −1.00060 −0.500300 0.865852i \(-0.666777\pi\)
−0.500300 + 0.865852i \(0.666777\pi\)
\(840\) −28.2882 −0.976036
\(841\) 0 0
\(842\) −23.6121 −0.813728
\(843\) −20.0118 −0.689242
\(844\) −61.3317 −2.11112
\(845\) 11.1768 0.384493
\(846\) −40.3528 −1.38736
\(847\) 12.9927 0.446435
\(848\) 85.4636 2.93483
\(849\) −3.36996 −0.115657
\(850\) −19.0435 −0.653186
\(851\) 26.4650 0.907209
\(852\) 47.1754 1.61620
\(853\) 7.77319 0.266149 0.133075 0.991106i \(-0.457515\pi\)
0.133075 + 0.991106i \(0.457515\pi\)
\(854\) −59.4880 −2.03564
\(855\) −8.85685 −0.302898
\(856\) 98.7328 3.37462
\(857\) −13.8740 −0.473927 −0.236963 0.971519i \(-0.576152\pi\)
−0.236963 + 0.971519i \(0.576152\pi\)
\(858\) 8.17014 0.278924
\(859\) 15.2809 0.521378 0.260689 0.965423i \(-0.416050\pi\)
0.260689 + 0.965423i \(0.416050\pi\)
\(860\) −16.4690 −0.561587
\(861\) −23.3258 −0.794942
\(862\) −11.3747 −0.387424
\(863\) 31.2301 1.06309 0.531543 0.847031i \(-0.321612\pi\)
0.531543 + 0.847031i \(0.321612\pi\)
\(864\) 68.7875 2.34020
\(865\) −7.02302 −0.238790
\(866\) −4.86670 −0.165377
\(867\) 27.1451 0.921895
\(868\) 4.96239 0.168434
\(869\) −13.2750 −0.450325
\(870\) 0 0
\(871\) −4.31265 −0.146129
\(872\) 122.426 4.14587
\(873\) −3.59831 −0.121784
\(874\) 48.4504 1.63886
\(875\) 4.15633 0.140509
\(876\) 46.7367 1.57909
\(877\) −4.26187 −0.143913 −0.0719565 0.997408i \(-0.522924\pi\)
−0.0719565 + 0.997408i \(0.522924\pi\)
\(878\) −37.7499 −1.27400
\(879\) −19.0376 −0.642123
\(880\) 34.4445 1.16113
\(881\) −15.2144 −0.512586 −0.256293 0.966599i \(-0.582501\pi\)
−0.256293 + 0.966599i \(0.582501\pi\)
\(882\) 64.6018 2.17526
\(883\) −13.7078 −0.461305 −0.230652 0.973036i \(-0.574086\pi\)
−0.230652 + 0.973036i \(0.574086\pi\)
\(884\) 49.5633 1.66699
\(885\) 2.07522 0.0697579
\(886\) −46.2287 −1.55308
\(887\) −12.6556 −0.424934 −0.212467 0.977168i \(-0.568150\pi\)
−0.212467 + 0.977168i \(0.568150\pi\)
\(888\) −37.4782 −1.25769
\(889\) 63.8251 2.14063
\(890\) −38.2882 −1.28342
\(891\) −10.0303 −0.336028
\(892\) −5.76845 −0.193142
\(893\) 24.1866 0.809375
\(894\) −14.2863 −0.477805
\(895\) −4.77575 −0.159636
\(896\) 68.8686 2.30074
\(897\) 5.23090 0.174655
\(898\) 25.0132 0.834700
\(899\) 0 0
\(900\) −12.1187 −0.403957
\(901\) 49.5633 1.65119
\(902\) 52.2638 1.74019
\(903\) −10.7005 −0.356091
\(904\) −138.422 −4.60385
\(905\) −1.87399 −0.0622936
\(906\) 20.0000 0.664455
\(907\) 12.5540 0.416850 0.208425 0.978038i \(-0.433166\pi\)
0.208425 + 0.978038i \(0.433166\pi\)
\(908\) 0.156325 0.00518783
\(909\) 6.66624 0.221105
\(910\) −15.0132 −0.497682
\(911\) −22.8714 −0.757765 −0.378882 0.925445i \(-0.623691\pi\)
−0.378882 + 0.925445i \(0.623691\pi\)
\(912\) −37.2868 −1.23469
\(913\) −7.13918 −0.236272
\(914\) 47.2506 1.56291
\(915\) 4.31265 0.142572
\(916\) 30.1622 0.996587
\(917\) 5.73813 0.189490
\(918\) 82.1279 2.71063
\(919\) 9.67750 0.319231 0.159616 0.987179i \(-0.448975\pi\)
0.159616 + 0.987179i \(0.448975\pi\)
\(920\) 40.5804 1.33790
\(921\) −26.2012 −0.863360
\(922\) 41.6337 1.37113
\(923\) 15.3258 0.504456
\(924\) 48.4749 1.59471
\(925\) 5.50659 0.181055
\(926\) −7.97953 −0.262224
\(927\) 23.2546 0.763780
\(928\) 0 0
\(929\) 51.9248 1.70360 0.851798 0.523870i \(-0.175512\pi\)
0.851798 + 0.523870i \(0.175512\pi\)
\(930\) −0.499293 −0.0163725
\(931\) −38.7210 −1.26903
\(932\) 134.588 4.40858
\(933\) 7.51247 0.245947
\(934\) 92.5071 3.02692
\(935\) 19.9756 0.653271
\(936\) 26.7954 0.875837
\(937\) 3.58769 0.117205 0.0586024 0.998281i \(-0.481336\pi\)
0.0586024 + 0.998281i \(0.481336\pi\)
\(938\) −35.5125 −1.15952
\(939\) 7.74004 0.252587
\(940\) 33.0943 1.07942
\(941\) −18.6253 −0.607167 −0.303584 0.952805i \(-0.598183\pi\)
−0.303584 + 0.952805i \(0.598183\pi\)
\(942\) −10.7974 −0.351797
\(943\) 33.4617 1.08966
\(944\) −31.6023 −1.02857
\(945\) −17.9248 −0.583093
\(946\) 23.9756 0.779513
\(947\) 16.5950 0.539265 0.269632 0.962963i \(-0.413098\pi\)
0.269632 + 0.962963i \(0.413098\pi\)
\(948\) 19.6629 0.638622
\(949\) 15.1833 0.492871
\(950\) 10.0811 0.327074
\(951\) −15.6140 −0.506320
\(952\) 249.826 8.09691
\(953\) −12.7005 −0.411410 −0.205705 0.978614i \(-0.565949\pi\)
−0.205705 + 0.978614i \(0.565949\pi\)
\(954\) 43.7743 1.41725
\(955\) 19.1187 0.618667
\(956\) 7.35026 0.237724
\(957\) 0 0
\(958\) −91.4695 −2.95524
\(959\) −26.9887 −0.871512
\(960\) −14.6048 −0.471369
\(961\) −30.9464 −0.998271
\(962\) −19.8905 −0.641295
\(963\) 27.4821 0.885600
\(964\) −0.387873 −0.0124926
\(965\) 19.8945 0.640425
\(966\) 43.0738 1.38588
\(967\) −37.4314 −1.20371 −0.601856 0.798605i \(-0.705572\pi\)
−0.601856 + 0.798605i \(0.705572\pi\)
\(968\) 26.3947 0.848358
\(969\) −21.6239 −0.694659
\(970\) 4.09569 0.131505
\(971\) −7.51644 −0.241214 −0.120607 0.992700i \(-0.538484\pi\)
−0.120607 + 0.992700i \(0.538484\pi\)
\(972\) 81.5691 2.61633
\(973\) 79.0249 2.53342
\(974\) −102.980 −3.29969
\(975\) 1.08840 0.0348566
\(976\) −65.6747 −2.10220
\(977\) −2.52847 −0.0808929 −0.0404465 0.999182i \(-0.512878\pi\)
−0.0404465 + 0.999182i \(0.512878\pi\)
\(978\) 16.1866 0.517592
\(979\) 40.1622 1.28359
\(980\) −52.9814 −1.69243
\(981\) 34.0771 1.08800
\(982\) −73.4793 −2.34482
\(983\) 9.32979 0.297574 0.148787 0.988869i \(-0.452463\pi\)
0.148787 + 0.988869i \(0.452463\pi\)
\(984\) −47.3865 −1.51063
\(985\) 13.5369 0.431322
\(986\) 0 0
\(987\) 21.5026 0.684436
\(988\) −26.2374 −0.834724
\(989\) 15.3503 0.488110
\(990\) 17.6424 0.560714
\(991\) 38.4241 1.22058 0.610290 0.792178i \(-0.291053\pi\)
0.610290 + 0.792178i \(0.291053\pi\)
\(992\) 3.69323 0.117260
\(993\) 23.8134 0.755694
\(994\) 126.200 4.00283
\(995\) −2.57452 −0.0816176
\(996\) 10.5745 0.335066
\(997\) 18.8423 0.596740 0.298370 0.954450i \(-0.403557\pi\)
0.298370 + 0.954450i \(0.403557\pi\)
\(998\) 85.8759 2.71835
\(999\) −23.7480 −0.751353
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.1.1 3
29.28 even 2 145.2.a.d.1.3 3
87.86 odd 2 1305.2.a.o.1.1 3
116.115 odd 2 2320.2.a.s.1.2 3
145.28 odd 4 725.2.b.d.349.1 6
145.57 odd 4 725.2.b.d.349.6 6
145.144 even 2 725.2.a.d.1.1 3
203.202 odd 2 7105.2.a.p.1.3 3
232.115 odd 2 9280.2.a.bm.1.2 3
232.173 even 2 9280.2.a.bu.1.2 3
435.434 odd 2 6525.2.a.bh.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.3 3 29.28 even 2
725.2.a.d.1.1 3 145.144 even 2
725.2.b.d.349.1 6 145.28 odd 4
725.2.b.d.349.6 6 145.57 odd 4
1305.2.a.o.1.1 3 87.86 odd 2
2320.2.a.s.1.2 3 116.115 odd 2
4205.2.a.e.1.1 3 1.1 even 1 trivial
6525.2.a.bh.1.3 3 435.434 odd 2
7105.2.a.p.1.3 3 203.202 odd 2
9280.2.a.bm.1.2 3 232.115 odd 2
9280.2.a.bu.1.2 3 232.173 even 2