Properties

Label 6034.2.a.k.1.16
Level $6034$
Weight $2$
Character 6034.1
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + 2170 x^{12} - 42069 x^{11} + 19553 x^{10} + 84697 x^{9} - 82713 x^{8} + \cdots - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(1.69438\) of defining polynomial
Character \(\chi\) \(=\) 6034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.69438 q^{3} +1.00000 q^{4} -4.04996 q^{5} -1.69438 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.129071 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.69438 q^{3} +1.00000 q^{4} -4.04996 q^{5} -1.69438 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.129071 q^{9} +4.04996 q^{10} -1.22767 q^{11} +1.69438 q^{12} -1.01270 q^{13} -1.00000 q^{14} -6.86217 q^{15} +1.00000 q^{16} +0.455224 q^{17} +0.129071 q^{18} +3.36763 q^{19} -4.04996 q^{20} +1.69438 q^{21} +1.22767 q^{22} -0.370189 q^{23} -1.69438 q^{24} +11.4022 q^{25} +1.01270 q^{26} -5.30184 q^{27} +1.00000 q^{28} +3.63036 q^{29} +6.86217 q^{30} +4.99252 q^{31} -1.00000 q^{32} -2.08013 q^{33} -0.455224 q^{34} -4.04996 q^{35} -0.129071 q^{36} -3.73589 q^{37} -3.36763 q^{38} -1.71590 q^{39} +4.04996 q^{40} +3.64381 q^{41} -1.69438 q^{42} +7.02184 q^{43} -1.22767 q^{44} +0.522731 q^{45} +0.370189 q^{46} -9.70825 q^{47} +1.69438 q^{48} +1.00000 q^{49} -11.4022 q^{50} +0.771323 q^{51} -1.01270 q^{52} +5.61269 q^{53} +5.30184 q^{54} +4.97199 q^{55} -1.00000 q^{56} +5.70605 q^{57} -3.63036 q^{58} -3.93901 q^{59} -6.86217 q^{60} +5.28219 q^{61} -4.99252 q^{62} -0.129071 q^{63} +1.00000 q^{64} +4.10138 q^{65} +2.08013 q^{66} +11.6016 q^{67} +0.455224 q^{68} -0.627241 q^{69} +4.04996 q^{70} -16.0530 q^{71} +0.129071 q^{72} +3.51325 q^{73} +3.73589 q^{74} +19.3196 q^{75} +3.36763 q^{76} -1.22767 q^{77} +1.71590 q^{78} -13.8367 q^{79} -4.04996 q^{80} -8.59613 q^{81} -3.64381 q^{82} -12.9247 q^{83} +1.69438 q^{84} -1.84364 q^{85} -7.02184 q^{86} +6.15121 q^{87} +1.22767 q^{88} +2.53987 q^{89} -0.522731 q^{90} -1.01270 q^{91} -0.370189 q^{92} +8.45924 q^{93} +9.70825 q^{94} -13.6388 q^{95} -1.69438 q^{96} +10.8779 q^{97} -1.00000 q^{98} +0.158456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{2} + 3 q^{3} + 20 q^{4} - 3 q^{5} - 3 q^{6} + 20 q^{7} - 20 q^{8} + 13 q^{9} + 3 q^{10} - 8 q^{11} + 3 q^{12} - 4 q^{13} - 20 q^{14} - 25 q^{15} + 20 q^{16} + 9 q^{17} - 13 q^{18} - 14 q^{19} - 3 q^{20} + 3 q^{21} + 8 q^{22} - 23 q^{23} - 3 q^{24} + 31 q^{25} + 4 q^{26} - 21 q^{27} + 20 q^{28} - 48 q^{29} + 25 q^{30} - q^{31} - 20 q^{32} - 29 q^{33} - 9 q^{34} - 3 q^{35} + 13 q^{36} - q^{37} + 14 q^{38} - q^{39} + 3 q^{40} - 27 q^{41} - 3 q^{42} - 3 q^{43} - 8 q^{44} - 12 q^{45} + 23 q^{46} - 26 q^{47} + 3 q^{48} + 20 q^{49} - 31 q^{50} - 17 q^{51} - 4 q^{52} - 43 q^{53} + 21 q^{54} - 16 q^{55} - 20 q^{56} - 25 q^{57} + 48 q^{58} - 19 q^{59} - 25 q^{60} + 9 q^{61} + q^{62} + 13 q^{63} + 20 q^{64} - 87 q^{65} + 29 q^{66} + 32 q^{67} + 9 q^{68} - 23 q^{69} + 3 q^{70} - 63 q^{71} - 13 q^{72} + 2 q^{73} + q^{74} - 8 q^{75} - 14 q^{76} - 8 q^{77} + q^{78} - 51 q^{79} - 3 q^{80} + 4 q^{81} + 27 q^{82} - 24 q^{83} + 3 q^{84} + 31 q^{85} + 3 q^{86} - 33 q^{87} + 8 q^{88} - 35 q^{89} + 12 q^{90} - 4 q^{91} - 23 q^{92} + 17 q^{93} + 26 q^{94} - 30 q^{95} - 3 q^{96} + 5 q^{97} - 20 q^{98} - 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.69438 0.978252 0.489126 0.872213i \(-0.337316\pi\)
0.489126 + 0.872213i \(0.337316\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.04996 −1.81120 −0.905598 0.424137i \(-0.860577\pi\)
−0.905598 + 0.424137i \(0.860577\pi\)
\(6\) −1.69438 −0.691728
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −0.129071 −0.0430236
\(10\) 4.04996 1.28071
\(11\) −1.22767 −0.370155 −0.185078 0.982724i \(-0.559254\pi\)
−0.185078 + 0.982724i \(0.559254\pi\)
\(12\) 1.69438 0.489126
\(13\) −1.01270 −0.280872 −0.140436 0.990090i \(-0.544850\pi\)
−0.140436 + 0.990090i \(0.544850\pi\)
\(14\) −1.00000 −0.267261
\(15\) −6.86217 −1.77181
\(16\) 1.00000 0.250000
\(17\) 0.455224 0.110408 0.0552040 0.998475i \(-0.482419\pi\)
0.0552040 + 0.998475i \(0.482419\pi\)
\(18\) 0.129071 0.0304223
\(19\) 3.36763 0.772588 0.386294 0.922376i \(-0.373755\pi\)
0.386294 + 0.922376i \(0.373755\pi\)
\(20\) −4.04996 −0.905598
\(21\) 1.69438 0.369744
\(22\) 1.22767 0.261739
\(23\) −0.370189 −0.0771897 −0.0385948 0.999255i \(-0.512288\pi\)
−0.0385948 + 0.999255i \(0.512288\pi\)
\(24\) −1.69438 −0.345864
\(25\) 11.4022 2.28043
\(26\) 1.01270 0.198606
\(27\) −5.30184 −1.02034
\(28\) 1.00000 0.188982
\(29\) 3.63036 0.674140 0.337070 0.941480i \(-0.390564\pi\)
0.337070 + 0.941480i \(0.390564\pi\)
\(30\) 6.86217 1.25286
\(31\) 4.99252 0.896684 0.448342 0.893862i \(-0.352015\pi\)
0.448342 + 0.893862i \(0.352015\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.08013 −0.362105
\(34\) −0.455224 −0.0780702
\(35\) −4.04996 −0.684568
\(36\) −0.129071 −0.0215118
\(37\) −3.73589 −0.614176 −0.307088 0.951681i \(-0.599355\pi\)
−0.307088 + 0.951681i \(0.599355\pi\)
\(38\) −3.36763 −0.546302
\(39\) −1.71590 −0.274763
\(40\) 4.04996 0.640355
\(41\) 3.64381 0.569068 0.284534 0.958666i \(-0.408161\pi\)
0.284534 + 0.958666i \(0.408161\pi\)
\(42\) −1.69438 −0.261449
\(43\) 7.02184 1.07082 0.535410 0.844592i \(-0.320157\pi\)
0.535410 + 0.844592i \(0.320157\pi\)
\(44\) −1.22767 −0.185078
\(45\) 0.522731 0.0779242
\(46\) 0.370189 0.0545813
\(47\) −9.70825 −1.41609 −0.708047 0.706166i \(-0.750423\pi\)
−0.708047 + 0.706166i \(0.750423\pi\)
\(48\) 1.69438 0.244563
\(49\) 1.00000 0.142857
\(50\) −11.4022 −1.61251
\(51\) 0.771323 0.108007
\(52\) −1.01270 −0.140436
\(53\) 5.61269 0.770962 0.385481 0.922716i \(-0.374036\pi\)
0.385481 + 0.922716i \(0.374036\pi\)
\(54\) 5.30184 0.721489
\(55\) 4.97199 0.670424
\(56\) −1.00000 −0.133631
\(57\) 5.70605 0.755785
\(58\) −3.63036 −0.476689
\(59\) −3.93901 −0.512816 −0.256408 0.966569i \(-0.582539\pi\)
−0.256408 + 0.966569i \(0.582539\pi\)
\(60\) −6.86217 −0.885903
\(61\) 5.28219 0.676316 0.338158 0.941089i \(-0.390196\pi\)
0.338158 + 0.941089i \(0.390196\pi\)
\(62\) −4.99252 −0.634051
\(63\) −0.129071 −0.0162614
\(64\) 1.00000 0.125000
\(65\) 4.10138 0.508714
\(66\) 2.08013 0.256047
\(67\) 11.6016 1.41736 0.708678 0.705532i \(-0.249292\pi\)
0.708678 + 0.705532i \(0.249292\pi\)
\(68\) 0.455224 0.0552040
\(69\) −0.627241 −0.0755109
\(70\) 4.04996 0.484063
\(71\) −16.0530 −1.90514 −0.952572 0.304313i \(-0.901573\pi\)
−0.952572 + 0.304313i \(0.901573\pi\)
\(72\) 0.129071 0.0152111
\(73\) 3.51325 0.411195 0.205597 0.978637i \(-0.434086\pi\)
0.205597 + 0.978637i \(0.434086\pi\)
\(74\) 3.73589 0.434288
\(75\) 19.3196 2.23084
\(76\) 3.36763 0.386294
\(77\) −1.22767 −0.139905
\(78\) 1.71590 0.194287
\(79\) −13.8367 −1.55675 −0.778377 0.627797i \(-0.783957\pi\)
−0.778377 + 0.627797i \(0.783957\pi\)
\(80\) −4.04996 −0.452799
\(81\) −8.59613 −0.955125
\(82\) −3.64381 −0.402392
\(83\) −12.9247 −1.41867 −0.709333 0.704874i \(-0.751004\pi\)
−0.709333 + 0.704874i \(0.751004\pi\)
\(84\) 1.69438 0.184872
\(85\) −1.84364 −0.199971
\(86\) −7.02184 −0.757185
\(87\) 6.15121 0.659479
\(88\) 1.22767 0.130870
\(89\) 2.53987 0.269226 0.134613 0.990898i \(-0.457021\pi\)
0.134613 + 0.990898i \(0.457021\pi\)
\(90\) −0.522731 −0.0551007
\(91\) −1.01270 −0.106160
\(92\) −0.370189 −0.0385948
\(93\) 8.45924 0.877182
\(94\) 9.70825 1.00133
\(95\) −13.6388 −1.39931
\(96\) −1.69438 −0.172932
\(97\) 10.8779 1.10448 0.552242 0.833684i \(-0.313772\pi\)
0.552242 + 0.833684i \(0.313772\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0.158456 0.0159254
\(100\) 11.4022 1.14022
\(101\) 9.17481 0.912928 0.456464 0.889742i \(-0.349116\pi\)
0.456464 + 0.889742i \(0.349116\pi\)
\(102\) −0.771323 −0.0763723
\(103\) −13.6620 −1.34616 −0.673079 0.739571i \(-0.735029\pi\)
−0.673079 + 0.739571i \(0.735029\pi\)
\(104\) 1.01270 0.0993031
\(105\) −6.86217 −0.669680
\(106\) −5.61269 −0.545152
\(107\) −1.17574 −0.113663 −0.0568316 0.998384i \(-0.518100\pi\)
−0.0568316 + 0.998384i \(0.518100\pi\)
\(108\) −5.30184 −0.510170
\(109\) −15.0842 −1.44480 −0.722401 0.691474i \(-0.756961\pi\)
−0.722401 + 0.691474i \(0.756961\pi\)
\(110\) −4.97199 −0.474061
\(111\) −6.33002 −0.600819
\(112\) 1.00000 0.0944911
\(113\) −18.0915 −1.70190 −0.850951 0.525244i \(-0.823974\pi\)
−0.850951 + 0.525244i \(0.823974\pi\)
\(114\) −5.70605 −0.534421
\(115\) 1.49925 0.139806
\(116\) 3.63036 0.337070
\(117\) 0.130710 0.0120841
\(118\) 3.93901 0.362615
\(119\) 0.455224 0.0417303
\(120\) 6.86217 0.626428
\(121\) −9.49284 −0.862985
\(122\) −5.28219 −0.478227
\(123\) 6.17401 0.556692
\(124\) 4.99252 0.448342
\(125\) −25.9285 −2.31911
\(126\) 0.129071 0.0114985
\(127\) 9.26295 0.821954 0.410977 0.911646i \(-0.365188\pi\)
0.410977 + 0.911646i \(0.365188\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.8977 1.04753
\(130\) −4.10138 −0.359715
\(131\) 12.8101 1.11923 0.559613 0.828754i \(-0.310950\pi\)
0.559613 + 0.828754i \(0.310950\pi\)
\(132\) −2.08013 −0.181052
\(133\) 3.36763 0.292011
\(134\) −11.6016 −1.00222
\(135\) 21.4722 1.84804
\(136\) −0.455224 −0.0390351
\(137\) 14.6909 1.25513 0.627564 0.778565i \(-0.284052\pi\)
0.627564 + 0.778565i \(0.284052\pi\)
\(138\) 0.627241 0.0533943
\(139\) −17.0680 −1.44769 −0.723845 0.689962i \(-0.757627\pi\)
−0.723845 + 0.689962i \(0.757627\pi\)
\(140\) −4.04996 −0.342284
\(141\) −16.4495 −1.38530
\(142\) 16.0530 1.34714
\(143\) 1.24325 0.103966
\(144\) −0.129071 −0.0107559
\(145\) −14.7028 −1.22100
\(146\) −3.51325 −0.290759
\(147\) 1.69438 0.139750
\(148\) −3.73589 −0.307088
\(149\) −14.4653 −1.18505 −0.592523 0.805554i \(-0.701868\pi\)
−0.592523 + 0.805554i \(0.701868\pi\)
\(150\) −19.3196 −1.57744
\(151\) 0.765320 0.0622809 0.0311404 0.999515i \(-0.490086\pi\)
0.0311404 + 0.999515i \(0.490086\pi\)
\(152\) −3.36763 −0.273151
\(153\) −0.0587561 −0.00475015
\(154\) 1.22767 0.0989281
\(155\) −20.2195 −1.62407
\(156\) −1.71590 −0.137382
\(157\) −20.2064 −1.61264 −0.806321 0.591478i \(-0.798545\pi\)
−0.806321 + 0.591478i \(0.798545\pi\)
\(158\) 13.8367 1.10079
\(159\) 9.51003 0.754195
\(160\) 4.04996 0.320177
\(161\) −0.370189 −0.0291749
\(162\) 8.59613 0.675376
\(163\) −19.0857 −1.49491 −0.747454 0.664314i \(-0.768724\pi\)
−0.747454 + 0.664314i \(0.768724\pi\)
\(164\) 3.64381 0.284534
\(165\) 8.42446 0.655843
\(166\) 12.9247 1.00315
\(167\) −1.80685 −0.139818 −0.0699090 0.997553i \(-0.522271\pi\)
−0.0699090 + 0.997553i \(0.522271\pi\)
\(168\) −1.69438 −0.130724
\(169\) −11.9744 −0.921111
\(170\) 1.84364 0.141400
\(171\) −0.434663 −0.0332395
\(172\) 7.02184 0.535410
\(173\) 13.1842 1.00238 0.501189 0.865338i \(-0.332896\pi\)
0.501189 + 0.865338i \(0.332896\pi\)
\(174\) −6.15121 −0.466322
\(175\) 11.4022 0.861922
\(176\) −1.22767 −0.0925388
\(177\) −6.67419 −0.501663
\(178\) −2.53987 −0.190371
\(179\) 21.9446 1.64022 0.820109 0.572208i \(-0.193913\pi\)
0.820109 + 0.572208i \(0.193913\pi\)
\(180\) 0.522731 0.0389621
\(181\) 17.2828 1.28462 0.642311 0.766444i \(-0.277976\pi\)
0.642311 + 0.766444i \(0.277976\pi\)
\(182\) 1.01270 0.0750661
\(183\) 8.95005 0.661607
\(184\) 0.370189 0.0272907
\(185\) 15.1302 1.11239
\(186\) −8.45924 −0.620261
\(187\) −0.558863 −0.0408681
\(188\) −9.70825 −0.708047
\(189\) −5.30184 −0.385652
\(190\) 13.6388 0.989460
\(191\) −20.5812 −1.48921 −0.744603 0.667507i \(-0.767361\pi\)
−0.744603 + 0.667507i \(0.767361\pi\)
\(192\) 1.69438 0.122281
\(193\) 7.07694 0.509409 0.254705 0.967019i \(-0.418022\pi\)
0.254705 + 0.967019i \(0.418022\pi\)
\(194\) −10.8779 −0.780988
\(195\) 6.94930 0.497650
\(196\) 1.00000 0.0714286
\(197\) −24.8708 −1.77197 −0.885987 0.463710i \(-0.846518\pi\)
−0.885987 + 0.463710i \(0.846518\pi\)
\(198\) −0.158456 −0.0112610
\(199\) 10.9476 0.776058 0.388029 0.921647i \(-0.373156\pi\)
0.388029 + 0.921647i \(0.373156\pi\)
\(200\) −11.4022 −0.806254
\(201\) 19.6575 1.38653
\(202\) −9.17481 −0.645538
\(203\) 3.63036 0.254801
\(204\) 0.771323 0.0540034
\(205\) −14.7573 −1.03069
\(206\) 13.6620 0.951877
\(207\) 0.0477805 0.00332098
\(208\) −1.01270 −0.0702179
\(209\) −4.13433 −0.285977
\(210\) 6.86217 0.473535
\(211\) −21.3568 −1.47026 −0.735130 0.677926i \(-0.762879\pi\)
−0.735130 + 0.677926i \(0.762879\pi\)
\(212\) 5.61269 0.385481
\(213\) −27.2000 −1.86371
\(214\) 1.17574 0.0803721
\(215\) −28.4382 −1.93947
\(216\) 5.30184 0.360745
\(217\) 4.99252 0.338915
\(218\) 15.0842 1.02163
\(219\) 5.95279 0.402252
\(220\) 4.97199 0.335212
\(221\) −0.461004 −0.0310105
\(222\) 6.33002 0.424843
\(223\) 19.2587 1.28966 0.644830 0.764326i \(-0.276928\pi\)
0.644830 + 0.764326i \(0.276928\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.47169 −0.0981123
\(226\) 18.0915 1.20343
\(227\) −27.8002 −1.84516 −0.922581 0.385803i \(-0.873924\pi\)
−0.922581 + 0.385803i \(0.873924\pi\)
\(228\) 5.70605 0.377893
\(229\) 11.0885 0.732747 0.366373 0.930468i \(-0.380599\pi\)
0.366373 + 0.930468i \(0.380599\pi\)
\(230\) −1.49925 −0.0988575
\(231\) −2.08013 −0.136863
\(232\) −3.63036 −0.238345
\(233\) 26.3989 1.72945 0.864724 0.502248i \(-0.167494\pi\)
0.864724 + 0.502248i \(0.167494\pi\)
\(234\) −0.130710 −0.00854476
\(235\) 39.3180 2.56482
\(236\) −3.93901 −0.256408
\(237\) −23.4447 −1.52290
\(238\) −0.455224 −0.0295078
\(239\) −1.44825 −0.0936793 −0.0468397 0.998902i \(-0.514915\pi\)
−0.0468397 + 0.998902i \(0.514915\pi\)
\(240\) −6.86217 −0.442951
\(241\) −18.0067 −1.15992 −0.579958 0.814647i \(-0.696931\pi\)
−0.579958 + 0.814647i \(0.696931\pi\)
\(242\) 9.49284 0.610223
\(243\) 1.34040 0.0859866
\(244\) 5.28219 0.338158
\(245\) −4.04996 −0.258742
\(246\) −6.17401 −0.393641
\(247\) −3.41039 −0.216998
\(248\) −4.99252 −0.317026
\(249\) −21.8993 −1.38781
\(250\) 25.9285 1.63986
\(251\) 3.57879 0.225891 0.112946 0.993601i \(-0.463971\pi\)
0.112946 + 0.993601i \(0.463971\pi\)
\(252\) −0.129071 −0.00813070
\(253\) 0.454468 0.0285721
\(254\) −9.26295 −0.581209
\(255\) −3.12382 −0.195621
\(256\) 1.00000 0.0625000
\(257\) 0.128614 0.00802275 0.00401138 0.999992i \(-0.498723\pi\)
0.00401138 + 0.999992i \(0.498723\pi\)
\(258\) −11.8977 −0.740717
\(259\) −3.73589 −0.232137
\(260\) 4.10138 0.254357
\(261\) −0.468573 −0.0290039
\(262\) −12.8101 −0.791412
\(263\) 6.36157 0.392271 0.196136 0.980577i \(-0.437161\pi\)
0.196136 + 0.980577i \(0.437161\pi\)
\(264\) 2.08013 0.128023
\(265\) −22.7311 −1.39636
\(266\) −3.36763 −0.206483
\(267\) 4.30351 0.263370
\(268\) 11.6016 0.708678
\(269\) −10.9848 −0.669756 −0.334878 0.942261i \(-0.608695\pi\)
−0.334878 + 0.942261i \(0.608695\pi\)
\(270\) −21.4722 −1.30676
\(271\) −27.0448 −1.64285 −0.821427 0.570314i \(-0.806822\pi\)
−0.821427 + 0.570314i \(0.806822\pi\)
\(272\) 0.455224 0.0276020
\(273\) −1.71590 −0.103851
\(274\) −14.6909 −0.887509
\(275\) −13.9980 −0.844113
\(276\) −0.627241 −0.0377555
\(277\) −8.73221 −0.524667 −0.262334 0.964977i \(-0.584492\pi\)
−0.262334 + 0.964977i \(0.584492\pi\)
\(278\) 17.0680 1.02367
\(279\) −0.644389 −0.0385786
\(280\) 4.04996 0.242031
\(281\) 11.2372 0.670357 0.335179 0.942155i \(-0.391203\pi\)
0.335179 + 0.942155i \(0.391203\pi\)
\(282\) 16.4495 0.979552
\(283\) −21.5351 −1.28013 −0.640065 0.768321i \(-0.721093\pi\)
−0.640065 + 0.768321i \(0.721093\pi\)
\(284\) −16.0530 −0.952572
\(285\) −23.1093 −1.36888
\(286\) −1.24325 −0.0735151
\(287\) 3.64381 0.215088
\(288\) 0.129071 0.00760557
\(289\) −16.7928 −0.987810
\(290\) 14.7028 0.863377
\(291\) 18.4313 1.08046
\(292\) 3.51325 0.205597
\(293\) −17.9971 −1.05140 −0.525700 0.850670i \(-0.676197\pi\)
−0.525700 + 0.850670i \(0.676197\pi\)
\(294\) −1.69438 −0.0988183
\(295\) 15.9528 0.928810
\(296\) 3.73589 0.217144
\(297\) 6.50889 0.377684
\(298\) 14.4653 0.837954
\(299\) 0.374889 0.0216804
\(300\) 19.3196 1.11542
\(301\) 7.02184 0.404732
\(302\) −0.765320 −0.0440392
\(303\) 15.5456 0.893073
\(304\) 3.36763 0.193147
\(305\) −21.3927 −1.22494
\(306\) 0.0587561 0.00335886
\(307\) −6.09863 −0.348067 −0.174034 0.984740i \(-0.555680\pi\)
−0.174034 + 0.984740i \(0.555680\pi\)
\(308\) −1.22767 −0.0699527
\(309\) −23.1487 −1.31688
\(310\) 20.2195 1.14839
\(311\) 0.136369 0.00773276 0.00386638 0.999993i \(-0.498769\pi\)
0.00386638 + 0.999993i \(0.498769\pi\)
\(312\) 1.71590 0.0971435
\(313\) 14.6617 0.828728 0.414364 0.910111i \(-0.364004\pi\)
0.414364 + 0.910111i \(0.364004\pi\)
\(314\) 20.2064 1.14031
\(315\) 0.522731 0.0294526
\(316\) −13.8367 −0.778377
\(317\) 17.8736 1.00388 0.501940 0.864902i \(-0.332620\pi\)
0.501940 + 0.864902i \(0.332620\pi\)
\(318\) −9.51003 −0.533296
\(319\) −4.45686 −0.249536
\(320\) −4.04996 −0.226400
\(321\) −1.99216 −0.111191
\(322\) 0.370189 0.0206298
\(323\) 1.53303 0.0852998
\(324\) −8.59613 −0.477563
\(325\) −11.5469 −0.640509
\(326\) 19.0857 1.05706
\(327\) −25.5583 −1.41338
\(328\) −3.64381 −0.201196
\(329\) −9.70825 −0.535233
\(330\) −8.42446 −0.463751
\(331\) −34.9839 −1.92289 −0.961443 0.275003i \(-0.911321\pi\)
−0.961443 + 0.275003i \(0.911321\pi\)
\(332\) −12.9247 −0.709333
\(333\) 0.482194 0.0264241
\(334\) 1.80685 0.0988663
\(335\) −46.9858 −2.56711
\(336\) 1.69438 0.0924361
\(337\) 8.29791 0.452016 0.226008 0.974125i \(-0.427432\pi\)
0.226008 + 0.974125i \(0.427432\pi\)
\(338\) 11.9744 0.651324
\(339\) −30.6539 −1.66489
\(340\) −1.84364 −0.0999853
\(341\) −6.12915 −0.331912
\(342\) 0.434663 0.0235039
\(343\) 1.00000 0.0539949
\(344\) −7.02184 −0.378592
\(345\) 2.54030 0.136765
\(346\) −13.1842 −0.708789
\(347\) −9.70924 −0.521219 −0.260610 0.965444i \(-0.583924\pi\)
−0.260610 + 0.965444i \(0.583924\pi\)
\(348\) 6.15121 0.329739
\(349\) −6.21256 −0.332551 −0.166275 0.986079i \(-0.553174\pi\)
−0.166275 + 0.986079i \(0.553174\pi\)
\(350\) −11.4022 −0.609471
\(351\) 5.36916 0.286585
\(352\) 1.22767 0.0654348
\(353\) −24.0250 −1.27872 −0.639362 0.768906i \(-0.720801\pi\)
−0.639362 + 0.768906i \(0.720801\pi\)
\(354\) 6.67419 0.354729
\(355\) 65.0141 3.45059
\(356\) 2.53987 0.134613
\(357\) 0.771323 0.0408227
\(358\) −21.9446 −1.15981
\(359\) −10.2462 −0.540774 −0.270387 0.962752i \(-0.587152\pi\)
−0.270387 + 0.962752i \(0.587152\pi\)
\(360\) −0.522731 −0.0275504
\(361\) −7.65906 −0.403108
\(362\) −17.2828 −0.908365
\(363\) −16.0845 −0.844217
\(364\) −1.01270 −0.0530798
\(365\) −14.2285 −0.744754
\(366\) −8.95005 −0.467827
\(367\) 14.8716 0.776290 0.388145 0.921598i \(-0.373116\pi\)
0.388145 + 0.921598i \(0.373116\pi\)
\(368\) −0.370189 −0.0192974
\(369\) −0.470310 −0.0244834
\(370\) −15.1302 −0.786580
\(371\) 5.61269 0.291396
\(372\) 8.45924 0.438591
\(373\) 17.1193 0.886403 0.443202 0.896422i \(-0.353843\pi\)
0.443202 + 0.896422i \(0.353843\pi\)
\(374\) 0.558863 0.0288981
\(375\) −43.9327 −2.26868
\(376\) 9.70825 0.500665
\(377\) −3.67645 −0.189347
\(378\) 5.30184 0.272697
\(379\) 26.1615 1.34383 0.671914 0.740629i \(-0.265473\pi\)
0.671914 + 0.740629i \(0.265473\pi\)
\(380\) −13.6388 −0.699654
\(381\) 15.6950 0.804078
\(382\) 20.5812 1.05303
\(383\) −25.4018 −1.29797 −0.648986 0.760801i \(-0.724807\pi\)
−0.648986 + 0.760801i \(0.724807\pi\)
\(384\) −1.69438 −0.0864661
\(385\) 4.97199 0.253396
\(386\) −7.07694 −0.360207
\(387\) −0.906315 −0.0460706
\(388\) 10.8779 0.552242
\(389\) −26.6661 −1.35202 −0.676011 0.736891i \(-0.736293\pi\)
−0.676011 + 0.736891i \(0.736293\pi\)
\(390\) −6.94930 −0.351892
\(391\) −0.168519 −0.00852235
\(392\) −1.00000 −0.0505076
\(393\) 21.7052 1.09488
\(394\) 24.8708 1.25298
\(395\) 56.0382 2.81959
\(396\) 0.158456 0.00796270
\(397\) 18.6176 0.934391 0.467196 0.884154i \(-0.345264\pi\)
0.467196 + 0.884154i \(0.345264\pi\)
\(398\) −10.9476 −0.548756
\(399\) 5.70605 0.285660
\(400\) 11.4022 0.570108
\(401\) 11.3576 0.567170 0.283585 0.958947i \(-0.408476\pi\)
0.283585 + 0.958947i \(0.408476\pi\)
\(402\) −19.6575 −0.980425
\(403\) −5.05591 −0.251853
\(404\) 9.17481 0.456464
\(405\) 34.8140 1.72992
\(406\) −3.63036 −0.180172
\(407\) 4.58642 0.227340
\(408\) −0.771323 −0.0381862
\(409\) −14.6733 −0.725550 −0.362775 0.931877i \(-0.618171\pi\)
−0.362775 + 0.931877i \(0.618171\pi\)
\(410\) 14.7573 0.728811
\(411\) 24.8920 1.22783
\(412\) −13.6620 −0.673079
\(413\) −3.93901 −0.193826
\(414\) −0.0477805 −0.00234828
\(415\) 52.3443 2.56948
\(416\) 1.01270 0.0496516
\(417\) −28.9197 −1.41621
\(418\) 4.13433 0.202216
\(419\) −25.1561 −1.22896 −0.614478 0.788934i \(-0.710633\pi\)
−0.614478 + 0.788934i \(0.710633\pi\)
\(420\) −6.86217 −0.334840
\(421\) 7.60205 0.370501 0.185251 0.982691i \(-0.440690\pi\)
0.185251 + 0.982691i \(0.440690\pi\)
\(422\) 21.3568 1.03963
\(423\) 1.25305 0.0609254
\(424\) −5.61269 −0.272576
\(425\) 5.19053 0.251778
\(426\) 27.2000 1.31784
\(427\) 5.28219 0.255623
\(428\) −1.17574 −0.0568316
\(429\) 2.10655 0.101705
\(430\) 28.4382 1.37141
\(431\) 1.00000 0.0481683
\(432\) −5.30184 −0.255085
\(433\) 5.77007 0.277292 0.138646 0.990342i \(-0.455725\pi\)
0.138646 + 0.990342i \(0.455725\pi\)
\(434\) −4.99252 −0.239649
\(435\) −24.9121 −1.19445
\(436\) −15.0842 −0.722401
\(437\) −1.24666 −0.0596358
\(438\) −5.95279 −0.284435
\(439\) −21.9324 −1.04677 −0.523387 0.852095i \(-0.675332\pi\)
−0.523387 + 0.852095i \(0.675332\pi\)
\(440\) −4.97199 −0.237031
\(441\) −0.129071 −0.00614623
\(442\) 0.461004 0.0219277
\(443\) 11.3476 0.539143 0.269571 0.962980i \(-0.413118\pi\)
0.269571 + 0.962980i \(0.413118\pi\)
\(444\) −6.33002 −0.300409
\(445\) −10.2864 −0.487621
\(446\) −19.2587 −0.911928
\(447\) −24.5098 −1.15927
\(448\) 1.00000 0.0472456
\(449\) −9.13298 −0.431012 −0.215506 0.976503i \(-0.569140\pi\)
−0.215506 + 0.976503i \(0.569140\pi\)
\(450\) 1.47169 0.0693759
\(451\) −4.47339 −0.210644
\(452\) −18.0915 −0.850951
\(453\) 1.29674 0.0609264
\(454\) 27.8002 1.30473
\(455\) 4.10138 0.192276
\(456\) −5.70605 −0.267210
\(457\) −11.0013 −0.514621 −0.257311 0.966329i \(-0.582836\pi\)
−0.257311 + 0.966329i \(0.582836\pi\)
\(458\) −11.0885 −0.518130
\(459\) −2.41352 −0.112654
\(460\) 1.49925 0.0699028
\(461\) 22.9284 1.06788 0.533941 0.845522i \(-0.320710\pi\)
0.533941 + 0.845522i \(0.320710\pi\)
\(462\) 2.08013 0.0967766
\(463\) −20.1512 −0.936507 −0.468254 0.883594i \(-0.655117\pi\)
−0.468254 + 0.883594i \(0.655117\pi\)
\(464\) 3.63036 0.168535
\(465\) −34.2596 −1.58875
\(466\) −26.3989 −1.22290
\(467\) 33.3189 1.54181 0.770907 0.636948i \(-0.219804\pi\)
0.770907 + 0.636948i \(0.219804\pi\)
\(468\) 0.130710 0.00604205
\(469\) 11.6016 0.535710
\(470\) −39.3180 −1.81360
\(471\) −34.2373 −1.57757
\(472\) 3.93901 0.181308
\(473\) −8.62047 −0.396370
\(474\) 23.4447 1.07685
\(475\) 38.3983 1.76183
\(476\) 0.455224 0.0208651
\(477\) −0.724434 −0.0331695
\(478\) 1.44825 0.0662413
\(479\) −2.45506 −0.112175 −0.0560873 0.998426i \(-0.517863\pi\)
−0.0560873 + 0.998426i \(0.517863\pi\)
\(480\) 6.86217 0.313214
\(481\) 3.78332 0.172505
\(482\) 18.0067 0.820184
\(483\) −0.627241 −0.0285404
\(484\) −9.49284 −0.431493
\(485\) −44.0551 −2.00044
\(486\) −1.34040 −0.0608017
\(487\) 37.8017 1.71296 0.856478 0.516183i \(-0.172648\pi\)
0.856478 + 0.516183i \(0.172648\pi\)
\(488\) −5.28219 −0.239114
\(489\) −32.3385 −1.46240
\(490\) 4.04996 0.182958
\(491\) 26.4129 1.19200 0.596000 0.802985i \(-0.296756\pi\)
0.596000 + 0.802985i \(0.296756\pi\)
\(492\) 6.17401 0.278346
\(493\) 1.65262 0.0744305
\(494\) 3.41039 0.153441
\(495\) −0.641739 −0.0288440
\(496\) 4.99252 0.224171
\(497\) −16.0530 −0.720077
\(498\) 21.8993 0.981332
\(499\) 31.6075 1.41495 0.707473 0.706740i \(-0.249835\pi\)
0.707473 + 0.706740i \(0.249835\pi\)
\(500\) −25.9285 −1.15956
\(501\) −3.06149 −0.136777
\(502\) −3.57879 −0.159729
\(503\) 35.8375 1.59792 0.798958 0.601387i \(-0.205385\pi\)
0.798958 + 0.601387i \(0.205385\pi\)
\(504\) 0.129071 0.00574927
\(505\) −37.1576 −1.65349
\(506\) −0.454468 −0.0202036
\(507\) −20.2893 −0.901079
\(508\) 9.26295 0.410977
\(509\) 28.4515 1.26109 0.630546 0.776152i \(-0.282831\pi\)
0.630546 + 0.776152i \(0.282831\pi\)
\(510\) 3.12382 0.138325
\(511\) 3.51325 0.155417
\(512\) −1.00000 −0.0441942
\(513\) −17.8546 −0.788302
\(514\) −0.128614 −0.00567294
\(515\) 55.3306 2.43816
\(516\) 11.8977 0.523766
\(517\) 11.9185 0.524174
\(518\) 3.73589 0.164145
\(519\) 22.3391 0.980578
\(520\) −4.10138 −0.179857
\(521\) −5.09817 −0.223355 −0.111678 0.993744i \(-0.535622\pi\)
−0.111678 + 0.993744i \(0.535622\pi\)
\(522\) 0.468573 0.0205089
\(523\) 33.2179 1.45252 0.726259 0.687421i \(-0.241257\pi\)
0.726259 + 0.687421i \(0.241257\pi\)
\(524\) 12.8101 0.559613
\(525\) 19.3196 0.843177
\(526\) −6.36157 −0.277378
\(527\) 2.27272 0.0990010
\(528\) −2.08013 −0.0905262
\(529\) −22.8630 −0.994042
\(530\) 22.7311 0.987378
\(531\) 0.508411 0.0220632
\(532\) 3.36763 0.146005
\(533\) −3.69008 −0.159835
\(534\) −4.30351 −0.186231
\(535\) 4.76171 0.205866
\(536\) −11.6016 −0.501111
\(537\) 37.1825 1.60455
\(538\) 10.9848 0.473589
\(539\) −1.22767 −0.0528793
\(540\) 21.4722 0.924018
\(541\) −25.4103 −1.09248 −0.546238 0.837630i \(-0.683940\pi\)
−0.546238 + 0.837630i \(0.683940\pi\)
\(542\) 27.0448 1.16167
\(543\) 29.2837 1.25668
\(544\) −0.455224 −0.0195176
\(545\) 61.0903 2.61682
\(546\) 1.71590 0.0734336
\(547\) −4.47439 −0.191311 −0.0956555 0.995414i \(-0.530495\pi\)
−0.0956555 + 0.995414i \(0.530495\pi\)
\(548\) 14.6909 0.627564
\(549\) −0.681777 −0.0290975
\(550\) 13.9980 0.596878
\(551\) 12.2257 0.520832
\(552\) 0.627241 0.0266971
\(553\) −13.8367 −0.588398
\(554\) 8.73221 0.370996
\(555\) 25.6363 1.08820
\(556\) −17.0680 −0.723845
\(557\) −39.9598 −1.69315 −0.846576 0.532268i \(-0.821340\pi\)
−0.846576 + 0.532268i \(0.821340\pi\)
\(558\) 0.644389 0.0272792
\(559\) −7.11100 −0.300763
\(560\) −4.04996 −0.171142
\(561\) −0.946927 −0.0399793
\(562\) −11.2372 −0.474014
\(563\) −16.8827 −0.711520 −0.355760 0.934577i \(-0.615778\pi\)
−0.355760 + 0.934577i \(0.615778\pi\)
\(564\) −16.4495 −0.692648
\(565\) 73.2697 3.08248
\(566\) 21.5351 0.905189
\(567\) −8.59613 −0.361003
\(568\) 16.0530 0.673570
\(569\) 25.8643 1.08429 0.542145 0.840285i \(-0.317612\pi\)
0.542145 + 0.840285i \(0.317612\pi\)
\(570\) 23.1093 0.967941
\(571\) 23.5807 0.986823 0.493412 0.869796i \(-0.335750\pi\)
0.493412 + 0.869796i \(0.335750\pi\)
\(572\) 1.24325 0.0519830
\(573\) −34.8725 −1.45682
\(574\) −3.64381 −0.152090
\(575\) −4.22095 −0.176026
\(576\) −0.129071 −0.00537795
\(577\) −10.5502 −0.439211 −0.219605 0.975589i \(-0.570477\pi\)
−0.219605 + 0.975589i \(0.570477\pi\)
\(578\) 16.7928 0.698487
\(579\) 11.9910 0.498331
\(580\) −14.7028 −0.610500
\(581\) −12.9247 −0.536205
\(582\) −18.4313 −0.764003
\(583\) −6.89050 −0.285375
\(584\) −3.51325 −0.145379
\(585\) −0.529368 −0.0218867
\(586\) 17.9971 0.743453
\(587\) 28.6982 1.18450 0.592251 0.805753i \(-0.298239\pi\)
0.592251 + 0.805753i \(0.298239\pi\)
\(588\) 1.69438 0.0698751
\(589\) 16.8130 0.692767
\(590\) −15.9528 −0.656768
\(591\) −42.1407 −1.73344
\(592\) −3.73589 −0.153544
\(593\) −19.5840 −0.804217 −0.402108 0.915592i \(-0.631723\pi\)
−0.402108 + 0.915592i \(0.631723\pi\)
\(594\) −6.50889 −0.267063
\(595\) −1.84364 −0.0755817
\(596\) −14.4653 −0.592523
\(597\) 18.5495 0.759180
\(598\) −0.374889 −0.0153303
\(599\) 17.1604 0.701157 0.350578 0.936533i \(-0.385985\pi\)
0.350578 + 0.936533i \(0.385985\pi\)
\(600\) −19.3196 −0.788720
\(601\) −16.2600 −0.663260 −0.331630 0.943410i \(-0.607598\pi\)
−0.331630 + 0.943410i \(0.607598\pi\)
\(602\) −7.02184 −0.286189
\(603\) −1.49742 −0.0609797
\(604\) 0.765320 0.0311404
\(605\) 38.4456 1.56304
\(606\) −15.5456 −0.631498
\(607\) 33.8465 1.37379 0.686894 0.726757i \(-0.258973\pi\)
0.686894 + 0.726757i \(0.258973\pi\)
\(608\) −3.36763 −0.136575
\(609\) 6.15121 0.249260
\(610\) 21.3927 0.866164
\(611\) 9.83151 0.397740
\(612\) −0.0587561 −0.00237507
\(613\) −15.9246 −0.643189 −0.321594 0.946878i \(-0.604219\pi\)
−0.321594 + 0.946878i \(0.604219\pi\)
\(614\) 6.09863 0.246121
\(615\) −25.0045 −1.00828
\(616\) 1.22767 0.0494641
\(617\) −1.27042 −0.0511452 −0.0255726 0.999673i \(-0.508141\pi\)
−0.0255726 + 0.999673i \(0.508141\pi\)
\(618\) 23.1487 0.931176
\(619\) 11.6977 0.470169 0.235085 0.971975i \(-0.424463\pi\)
0.235085 + 0.971975i \(0.424463\pi\)
\(620\) −20.2195 −0.812035
\(621\) 1.96268 0.0787597
\(622\) −0.136369 −0.00546789
\(623\) 2.53987 0.101758
\(624\) −1.71590 −0.0686908
\(625\) 47.9984 1.91994
\(626\) −14.6617 −0.585999
\(627\) −7.00513 −0.279758
\(628\) −20.2064 −0.806321
\(629\) −1.70066 −0.0678099
\(630\) −0.522731 −0.0208261
\(631\) 35.6124 1.41771 0.708854 0.705355i \(-0.249212\pi\)
0.708854 + 0.705355i \(0.249212\pi\)
\(632\) 13.8367 0.550396
\(633\) −36.1865 −1.43829
\(634\) −17.8736 −0.709851
\(635\) −37.5145 −1.48872
\(636\) 9.51003 0.377097
\(637\) −1.01270 −0.0401245
\(638\) 4.45686 0.176449
\(639\) 2.07198 0.0819661
\(640\) 4.04996 0.160089
\(641\) −42.5386 −1.68018 −0.840088 0.542451i \(-0.817496\pi\)
−0.840088 + 0.542451i \(0.817496\pi\)
\(642\) 1.99216 0.0786241
\(643\) −33.9921 −1.34052 −0.670259 0.742127i \(-0.733817\pi\)
−0.670259 + 0.742127i \(0.733817\pi\)
\(644\) −0.370189 −0.0145875
\(645\) −48.1851 −1.89729
\(646\) −1.53303 −0.0603161
\(647\) −37.3199 −1.46720 −0.733598 0.679584i \(-0.762160\pi\)
−0.733598 + 0.679584i \(0.762160\pi\)
\(648\) 8.59613 0.337688
\(649\) 4.83579 0.189821
\(650\) 11.5469 0.452908
\(651\) 8.45924 0.331544
\(652\) −19.0857 −0.747454
\(653\) −43.1135 −1.68716 −0.843581 0.537002i \(-0.819557\pi\)
−0.843581 + 0.537002i \(0.819557\pi\)
\(654\) 25.5583 0.999410
\(655\) −51.8805 −2.02714
\(656\) 3.64381 0.142267
\(657\) −0.453458 −0.0176911
\(658\) 9.70825 0.378467
\(659\) −13.1760 −0.513264 −0.256632 0.966509i \(-0.582613\pi\)
−0.256632 + 0.966509i \(0.582613\pi\)
\(660\) 8.42446 0.327921
\(661\) −15.9695 −0.621141 −0.310570 0.950550i \(-0.600520\pi\)
−0.310570 + 0.950550i \(0.600520\pi\)
\(662\) 34.9839 1.35969
\(663\) −0.781116 −0.0303361
\(664\) 12.9247 0.501574
\(665\) −13.6388 −0.528889
\(666\) −0.482194 −0.0186846
\(667\) −1.34392 −0.0520366
\(668\) −1.80685 −0.0699090
\(669\) 32.6317 1.26161
\(670\) 46.9858 1.81522
\(671\) −6.48477 −0.250342
\(672\) −1.69438 −0.0653622
\(673\) −24.6479 −0.950106 −0.475053 0.879957i \(-0.657571\pi\)
−0.475053 + 0.879957i \(0.657571\pi\)
\(674\) −8.29791 −0.319624
\(675\) −60.4524 −2.32681
\(676\) −11.9744 −0.460556
\(677\) 6.98276 0.268369 0.134185 0.990956i \(-0.457158\pi\)
0.134185 + 0.990956i \(0.457158\pi\)
\(678\) 30.6539 1.17725
\(679\) 10.8779 0.417456
\(680\) 1.84364 0.0707002
\(681\) −47.1041 −1.80503
\(682\) 6.12915 0.234697
\(683\) −46.3513 −1.77358 −0.886792 0.462169i \(-0.847071\pi\)
−0.886792 + 0.462169i \(0.847071\pi\)
\(684\) −0.434663 −0.0166197
\(685\) −59.4975 −2.27328
\(686\) −1.00000 −0.0381802
\(687\) 18.7881 0.716811
\(688\) 7.02184 0.267705
\(689\) −5.68395 −0.216541
\(690\) −2.54030 −0.0967075
\(691\) 39.4119 1.49930 0.749650 0.661835i \(-0.230222\pi\)
0.749650 + 0.661835i \(0.230222\pi\)
\(692\) 13.1842 0.501189
\(693\) 0.158456 0.00601924
\(694\) 9.70924 0.368558
\(695\) 69.1248 2.62205
\(696\) −6.15121 −0.233161
\(697\) 1.65875 0.0628297
\(698\) 6.21256 0.235149
\(699\) 44.7298 1.69183
\(700\) 11.4022 0.430961
\(701\) 9.42552 0.355997 0.177998 0.984031i \(-0.443038\pi\)
0.177998 + 0.984031i \(0.443038\pi\)
\(702\) −5.36916 −0.202646
\(703\) −12.5811 −0.474505
\(704\) −1.22767 −0.0462694
\(705\) 66.6197 2.50904
\(706\) 24.0250 0.904194
\(707\) 9.17481 0.345054
\(708\) −6.67419 −0.250831
\(709\) 19.7652 0.742299 0.371149 0.928573i \(-0.378964\pi\)
0.371149 + 0.928573i \(0.378964\pi\)
\(710\) −65.0141 −2.43993
\(711\) 1.78592 0.0669772
\(712\) −2.53987 −0.0951857
\(713\) −1.84817 −0.0692147
\(714\) −0.771323 −0.0288660
\(715\) −5.03512 −0.188303
\(716\) 21.9446 0.820109
\(717\) −2.45388 −0.0916420
\(718\) 10.2462 0.382385
\(719\) −31.6490 −1.18031 −0.590154 0.807291i \(-0.700933\pi\)
−0.590154 + 0.807291i \(0.700933\pi\)
\(720\) 0.522731 0.0194810
\(721\) −13.6620 −0.508800
\(722\) 7.65906 0.285041
\(723\) −30.5103 −1.13469
\(724\) 17.2828 0.642311
\(725\) 41.3939 1.53733
\(726\) 16.0845 0.596951
\(727\) −14.9907 −0.555973 −0.277986 0.960585i \(-0.589667\pi\)
−0.277986 + 0.960585i \(0.589667\pi\)
\(728\) 1.01270 0.0375331
\(729\) 28.0595 1.03924
\(730\) 14.2285 0.526621
\(731\) 3.19651 0.118227
\(732\) 8.95005 0.330803
\(733\) 3.95892 0.146226 0.0731130 0.997324i \(-0.476707\pi\)
0.0731130 + 0.997324i \(0.476707\pi\)
\(734\) −14.8716 −0.548920
\(735\) −6.86217 −0.253115
\(736\) 0.370189 0.0136453
\(737\) −14.2428 −0.524641
\(738\) 0.470310 0.0173124
\(739\) 25.9678 0.955239 0.477620 0.878567i \(-0.341500\pi\)
0.477620 + 0.878567i \(0.341500\pi\)
\(740\) 15.1302 0.556196
\(741\) −5.77850 −0.212279
\(742\) −5.61269 −0.206048
\(743\) −40.6421 −1.49101 −0.745507 0.666498i \(-0.767793\pi\)
−0.745507 + 0.666498i \(0.767793\pi\)
\(744\) −8.45924 −0.310131
\(745\) 58.5839 2.14635
\(746\) −17.1193 −0.626782
\(747\) 1.66820 0.0610361
\(748\) −0.558863 −0.0204340
\(749\) −1.17574 −0.0429607
\(750\) 43.9327 1.60420
\(751\) −35.0716 −1.27978 −0.639890 0.768466i \(-0.721020\pi\)
−0.639890 + 0.768466i \(0.721020\pi\)
\(752\) −9.70825 −0.354023
\(753\) 6.06383 0.220978
\(754\) 3.67645 0.133888
\(755\) −3.09951 −0.112803
\(756\) −5.30184 −0.192826
\(757\) −29.5195 −1.07290 −0.536452 0.843931i \(-0.680236\pi\)
−0.536452 + 0.843931i \(0.680236\pi\)
\(758\) −26.1615 −0.950230
\(759\) 0.770042 0.0279508
\(760\) 13.6388 0.494730
\(761\) −35.0810 −1.27168 −0.635842 0.771819i \(-0.719347\pi\)
−0.635842 + 0.771819i \(0.719347\pi\)
\(762\) −15.6950 −0.568569
\(763\) −15.0842 −0.546084
\(764\) −20.5812 −0.744603
\(765\) 0.237960 0.00860345
\(766\) 25.4018 0.917804
\(767\) 3.98903 0.144035
\(768\) 1.69438 0.0611407
\(769\) −8.63816 −0.311500 −0.155750 0.987796i \(-0.549779\pi\)
−0.155750 + 0.987796i \(0.549779\pi\)
\(770\) −4.97199 −0.179178
\(771\) 0.217922 0.00784827
\(772\) 7.07694 0.254705
\(773\) −6.80172 −0.244641 −0.122320 0.992491i \(-0.539034\pi\)
−0.122320 + 0.992491i \(0.539034\pi\)
\(774\) 0.906315 0.0325768
\(775\) 56.9255 2.04483
\(776\) −10.8779 −0.390494
\(777\) −6.33002 −0.227088
\(778\) 26.6661 0.956024
\(779\) 12.2710 0.439655
\(780\) 6.94930 0.248825
\(781\) 19.7078 0.705199
\(782\) 0.168519 0.00602621
\(783\) −19.2476 −0.687852
\(784\) 1.00000 0.0357143
\(785\) 81.8349 2.92081
\(786\) −21.7052 −0.774201
\(787\) 30.6448 1.09237 0.546184 0.837665i \(-0.316080\pi\)
0.546184 + 0.837665i \(0.316080\pi\)
\(788\) −24.8708 −0.885987
\(789\) 10.7789 0.383740
\(790\) −56.0382 −1.99375
\(791\) −18.0915 −0.643259
\(792\) −0.158456 −0.00563048
\(793\) −5.34926 −0.189958
\(794\) −18.6176 −0.660714
\(795\) −38.5152 −1.36599
\(796\) 10.9476 0.388029
\(797\) 6.16844 0.218497 0.109249 0.994014i \(-0.465156\pi\)
0.109249 + 0.994014i \(0.465156\pi\)
\(798\) −5.70605 −0.201992
\(799\) −4.41942 −0.156348
\(800\) −11.4022 −0.403127
\(801\) −0.327823 −0.0115831
\(802\) −11.3576 −0.401050
\(803\) −4.31310 −0.152206
\(804\) 19.6575 0.693265
\(805\) 1.49925 0.0528415
\(806\) 5.05591 0.178087
\(807\) −18.6125 −0.655190
\(808\) −9.17481 −0.322769
\(809\) −19.2956 −0.678396 −0.339198 0.940715i \(-0.610156\pi\)
−0.339198 + 0.940715i \(0.610156\pi\)
\(810\) −34.8140 −1.22324
\(811\) 28.1345 0.987934 0.493967 0.869481i \(-0.335546\pi\)
0.493967 + 0.869481i \(0.335546\pi\)
\(812\) 3.63036 0.127401
\(813\) −45.8242 −1.60712
\(814\) −4.58642 −0.160754
\(815\) 77.2963 2.70757
\(816\) 0.771323 0.0270017
\(817\) 23.6470 0.827303
\(818\) 14.6733 0.513041
\(819\) 0.130710 0.00456736
\(820\) −14.7573 −0.515347
\(821\) 21.3065 0.743601 0.371800 0.928313i \(-0.378741\pi\)
0.371800 + 0.928313i \(0.378741\pi\)
\(822\) −24.8920 −0.868207
\(823\) 0.638428 0.0222542 0.0111271 0.999938i \(-0.496458\pi\)
0.0111271 + 0.999938i \(0.496458\pi\)
\(824\) 13.6620 0.475939
\(825\) −23.7180 −0.825755
\(826\) 3.93901 0.137056
\(827\) −20.2058 −0.702624 −0.351312 0.936258i \(-0.614264\pi\)
−0.351312 + 0.936258i \(0.614264\pi\)
\(828\) 0.0477805 0.00166049
\(829\) 35.8559 1.24533 0.622663 0.782490i \(-0.286051\pi\)
0.622663 + 0.782490i \(0.286051\pi\)
\(830\) −52.3443 −1.81690
\(831\) −14.7957 −0.513257
\(832\) −1.01270 −0.0351090
\(833\) 0.455224 0.0157726
\(834\) 28.9197 1.00141
\(835\) 7.31765 0.253238
\(836\) −4.13433 −0.142989
\(837\) −26.4696 −0.914922
\(838\) 25.1561 0.869003
\(839\) −30.0836 −1.03860 −0.519300 0.854592i \(-0.673807\pi\)
−0.519300 + 0.854592i \(0.673807\pi\)
\(840\) 6.86217 0.236767
\(841\) −15.8205 −0.545535
\(842\) −7.60205 −0.261984
\(843\) 19.0402 0.655778
\(844\) −21.3568 −0.735130
\(845\) 48.4960 1.66831
\(846\) −1.25305 −0.0430808
\(847\) −9.49284 −0.326178
\(848\) 5.61269 0.192740
\(849\) −36.4887 −1.25229
\(850\) −5.19053 −0.178034
\(851\) 1.38298 0.0474080
\(852\) −27.2000 −0.931855
\(853\) 7.66644 0.262494 0.131247 0.991350i \(-0.458102\pi\)
0.131247 + 0.991350i \(0.458102\pi\)
\(854\) −5.28219 −0.180753
\(855\) 1.76037 0.0602033
\(856\) 1.17574 0.0401860
\(857\) 42.9555 1.46733 0.733666 0.679510i \(-0.237808\pi\)
0.733666 + 0.679510i \(0.237808\pi\)
\(858\) −2.10655 −0.0719163
\(859\) −32.0705 −1.09423 −0.547115 0.837058i \(-0.684274\pi\)
−0.547115 + 0.837058i \(0.684274\pi\)
\(860\) −28.4382 −0.969733
\(861\) 6.17401 0.210410
\(862\) −1.00000 −0.0340601
\(863\) −40.8641 −1.39103 −0.695516 0.718511i \(-0.744824\pi\)
−0.695516 + 0.718511i \(0.744824\pi\)
\(864\) 5.30184 0.180372
\(865\) −53.3956 −1.81550
\(866\) −5.77007 −0.196075
\(867\) −28.4534 −0.966327
\(868\) 4.99252 0.169457
\(869\) 16.9869 0.576241
\(870\) 24.9121 0.844600
\(871\) −11.7489 −0.398095
\(872\) 15.0842 0.510815
\(873\) −1.40402 −0.0475189
\(874\) 1.24666 0.0421689
\(875\) −25.9285 −0.876542
\(876\) 5.95279 0.201126
\(877\) −33.7627 −1.14009 −0.570043 0.821615i \(-0.693074\pi\)
−0.570043 + 0.821615i \(0.693074\pi\)
\(878\) 21.9324 0.740181
\(879\) −30.4939 −1.02853
\(880\) 4.97199 0.167606
\(881\) 18.6504 0.628346 0.314173 0.949366i \(-0.398273\pi\)
0.314173 + 0.949366i \(0.398273\pi\)
\(882\) 0.129071 0.00434604
\(883\) −18.4592 −0.621203 −0.310602 0.950540i \(-0.600530\pi\)
−0.310602 + 0.950540i \(0.600530\pi\)
\(884\) −0.461004 −0.0155052
\(885\) 27.0302 0.908610
\(886\) −11.3476 −0.381231
\(887\) −38.6016 −1.29611 −0.648057 0.761592i \(-0.724418\pi\)
−0.648057 + 0.761592i \(0.724418\pi\)
\(888\) 6.33002 0.212421
\(889\) 9.26295 0.310669
\(890\) 10.2864 0.344800
\(891\) 10.5532 0.353545
\(892\) 19.2587 0.644830
\(893\) −32.6938 −1.09406
\(894\) 24.5098 0.819729
\(895\) −88.8747 −2.97075
\(896\) −1.00000 −0.0334077
\(897\) 0.635205 0.0212089
\(898\) 9.13298 0.304771
\(899\) 18.1246 0.604490
\(900\) −1.47169 −0.0490562
\(901\) 2.55503 0.0851203
\(902\) 4.47339 0.148947
\(903\) 11.8977 0.395930
\(904\) 18.0915 0.601713
\(905\) −69.9947 −2.32670
\(906\) −1.29674 −0.0430814
\(907\) 10.2804 0.341355 0.170677 0.985327i \(-0.445404\pi\)
0.170677 + 0.985327i \(0.445404\pi\)
\(908\) −27.8002 −0.922581
\(909\) −1.18420 −0.0392774
\(910\) −4.10138 −0.135959
\(911\) 27.4015 0.907852 0.453926 0.891039i \(-0.350023\pi\)
0.453926 + 0.891039i \(0.350023\pi\)
\(912\) 5.70605 0.188946
\(913\) 15.8672 0.525127
\(914\) 11.0013 0.363892
\(915\) −36.2473 −1.19830
\(916\) 11.0885 0.366373
\(917\) 12.8101 0.423028
\(918\) 2.41352 0.0796582
\(919\) −53.2178 −1.75549 −0.877747 0.479125i \(-0.840954\pi\)
−0.877747 + 0.479125i \(0.840954\pi\)
\(920\) −1.49925 −0.0494287
\(921\) −10.3334 −0.340497
\(922\) −22.9284 −0.755107
\(923\) 16.2569 0.535101
\(924\) −2.08013 −0.0684314
\(925\) −42.5971 −1.40059
\(926\) 20.1512 0.662211
\(927\) 1.76337 0.0579165
\(928\) −3.63036 −0.119172
\(929\) −24.6008 −0.807127 −0.403563 0.914952i \(-0.632229\pi\)
−0.403563 + 0.914952i \(0.632229\pi\)
\(930\) 34.2596 1.12342
\(931\) 3.36763 0.110370
\(932\) 26.3989 0.864724
\(933\) 0.231061 0.00756458
\(934\) −33.3189 −1.09023
\(935\) 2.26337 0.0740201
\(936\) −0.130710 −0.00427238
\(937\) −32.7632 −1.07033 −0.535164 0.844748i \(-0.679750\pi\)
−0.535164 + 0.844748i \(0.679750\pi\)
\(938\) −11.6016 −0.378804
\(939\) 24.8425 0.810705
\(940\) 39.3180 1.28241
\(941\) 19.4389 0.633690 0.316845 0.948477i \(-0.397376\pi\)
0.316845 + 0.948477i \(0.397376\pi\)
\(942\) 34.2373 1.11551
\(943\) −1.34890 −0.0439262
\(944\) −3.93901 −0.128204
\(945\) 21.4722 0.698492
\(946\) 8.62047 0.280276
\(947\) 15.7947 0.513258 0.256629 0.966510i \(-0.417388\pi\)
0.256629 + 0.966510i \(0.417388\pi\)
\(948\) −23.4447 −0.761449
\(949\) −3.55786 −0.115493
\(950\) −38.3983 −1.24580
\(951\) 30.2847 0.982048
\(952\) −0.455224 −0.0147539
\(953\) −44.7353 −1.44912 −0.724559 0.689213i \(-0.757957\pi\)
−0.724559 + 0.689213i \(0.757957\pi\)
\(954\) 0.724434 0.0234544
\(955\) 83.3532 2.69724
\(956\) −1.44825 −0.0468397
\(957\) −7.55163 −0.244109
\(958\) 2.45506 0.0793194
\(959\) 14.6909 0.474393
\(960\) −6.86217 −0.221476
\(961\) −6.07472 −0.195959
\(962\) −3.78332 −0.121979
\(963\) 0.151754 0.00489020
\(964\) −18.0067 −0.579958
\(965\) −28.6613 −0.922640
\(966\) 0.627241 0.0201811
\(967\) −28.1299 −0.904597 −0.452299 0.891867i \(-0.649396\pi\)
−0.452299 + 0.891867i \(0.649396\pi\)
\(968\) 9.49284 0.305111
\(969\) 2.59753 0.0834447
\(970\) 44.0551 1.41452
\(971\) 37.4754 1.20264 0.601322 0.799007i \(-0.294641\pi\)
0.601322 + 0.799007i \(0.294641\pi\)
\(972\) 1.34040 0.0429933
\(973\) −17.0680 −0.547176
\(974\) −37.8017 −1.21124
\(975\) −19.5649 −0.626579
\(976\) 5.28219 0.169079
\(977\) 24.9367 0.797795 0.398898 0.916995i \(-0.369393\pi\)
0.398898 + 0.916995i \(0.369393\pi\)
\(978\) 32.3385 1.03407
\(979\) −3.11811 −0.0996553
\(980\) −4.04996 −0.129371
\(981\) 1.94693 0.0621606
\(982\) −26.4129 −0.842871
\(983\) 16.4300 0.524035 0.262018 0.965063i \(-0.415612\pi\)
0.262018 + 0.965063i \(0.415612\pi\)
\(984\) −6.17401 −0.196820
\(985\) 100.726 3.20939
\(986\) −1.65262 −0.0526303
\(987\) −16.4495 −0.523593
\(988\) −3.41039 −0.108499
\(989\) −2.59941 −0.0826563
\(990\) 0.641739 0.0203958
\(991\) 11.3800 0.361496 0.180748 0.983529i \(-0.442148\pi\)
0.180748 + 0.983529i \(0.442148\pi\)
\(992\) −4.99252 −0.158513
\(993\) −59.2760 −1.88107
\(994\) 16.0530 0.509171
\(995\) −44.3375 −1.40559
\(996\) −21.8993 −0.693906
\(997\) 21.4949 0.680751 0.340376 0.940290i \(-0.389446\pi\)
0.340376 + 0.940290i \(0.389446\pi\)
\(998\) −31.6075 −1.00052
\(999\) 19.8071 0.626668
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 6034.2.a.k.1.16 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6034.2.a.k.1.16 20 1.1 even 1 trivial