Properties

Label 6034.2.a.l.1.10
Level $6034$
Weight $2$
Character 6034.1
Self dual yes
Analytic conductor $48.182$
Analytic rank $1$
Dimension $20$
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] = [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: \( x^{20} - 3 x^{19} - 36 x^{18} + 97 x^{17} + 573 x^{16} - 1292 x^{15} - 5329 x^{14} + 9121 x^{13} + 31784 x^{12} - 36075 x^{11} - 124276 x^{10} + 74594 x^{9} + 312410 x^{8} + \cdots - 21776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-0.395900\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.395900 q^{3} +1.00000 q^{4} +0.151677 q^{5} +0.395900 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.84326 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.395900 q^{3} +1.00000 q^{4} +0.151677 q^{5} +0.395900 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.84326 q^{9} +0.151677 q^{10} +2.85030 q^{11} +0.395900 q^{12} -2.70854 q^{13} -1.00000 q^{14} +0.0600489 q^{15} +1.00000 q^{16} -1.00368 q^{17} -2.84326 q^{18} +2.34557 q^{19} +0.151677 q^{20} -0.395900 q^{21} +2.85030 q^{22} -0.830286 q^{23} +0.395900 q^{24} -4.97699 q^{25} -2.70854 q^{26} -2.31335 q^{27} -1.00000 q^{28} +3.14114 q^{29} +0.0600489 q^{30} -4.76935 q^{31} +1.00000 q^{32} +1.12843 q^{33} -1.00368 q^{34} -0.151677 q^{35} -2.84326 q^{36} +2.32662 q^{37} +2.34557 q^{38} -1.07231 q^{39} +0.151677 q^{40} -3.35096 q^{41} -0.395900 q^{42} -11.1914 q^{43} +2.85030 q^{44} -0.431257 q^{45} -0.830286 q^{46} -9.19342 q^{47} +0.395900 q^{48} +1.00000 q^{49} -4.97699 q^{50} -0.397357 q^{51} -2.70854 q^{52} +0.902309 q^{53} -2.31335 q^{54} +0.432324 q^{55} -1.00000 q^{56} +0.928613 q^{57} +3.14114 q^{58} -0.539748 q^{59} +0.0600489 q^{60} -3.28025 q^{61} -4.76935 q^{62} +2.84326 q^{63} +1.00000 q^{64} -0.410823 q^{65} +1.12843 q^{66} -8.88542 q^{67} -1.00368 q^{68} -0.328710 q^{69} -0.151677 q^{70} +9.68947 q^{71} -2.84326 q^{72} +1.66771 q^{73} +2.32662 q^{74} -1.97039 q^{75} +2.34557 q^{76} -2.85030 q^{77} -1.07231 q^{78} +4.67827 q^{79} +0.151677 q^{80} +7.61393 q^{81} -3.35096 q^{82} +13.9915 q^{83} -0.395900 q^{84} -0.152235 q^{85} -11.1914 q^{86} +1.24358 q^{87} +2.85030 q^{88} -1.06964 q^{89} -0.431257 q^{90} +2.70854 q^{91} -0.830286 q^{92} -1.88819 q^{93} -9.19342 q^{94} +0.355769 q^{95} +0.395900 q^{96} -6.33555 q^{97} +1.00000 q^{98} -8.10415 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{2} - 3 q^{3} + 20 q^{4} - 10 q^{5} - 3 q^{6} - 20 q^{7} + 20 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{2} - 3 q^{3} + 20 q^{4} - 10 q^{5} - 3 q^{6} - 20 q^{7} + 20 q^{8} + 21 q^{9} - 10 q^{10} - 17 q^{11} - 3 q^{12} - 23 q^{13} - 20 q^{14} - 3 q^{15} + 20 q^{16} - 21 q^{17} + 21 q^{18} - 22 q^{19} - 10 q^{20} + 3 q^{21} - 17 q^{22} + 15 q^{23} - 3 q^{24} - 23 q^{26} - 42 q^{27} - 20 q^{28} - 3 q^{29} - 3 q^{30} - 3 q^{31} + 20 q^{32} - 12 q^{33} - 21 q^{34} + 10 q^{35} + 21 q^{36} - 14 q^{37} - 22 q^{38} + q^{39} - 10 q^{40} - 37 q^{41} + 3 q^{42} - 5 q^{43} - 17 q^{44} - 55 q^{45} + 15 q^{46} - 29 q^{47} - 3 q^{48} + 20 q^{49} - 7 q^{51} - 23 q^{52} - 28 q^{53} - 42 q^{54} + 4 q^{55} - 20 q^{56} - 23 q^{57} - 3 q^{58} - 47 q^{59} - 3 q^{60} - 13 q^{61} - 3 q^{62} - 21 q^{63} + 20 q^{64} - 26 q^{65} - 12 q^{66} - 24 q^{67} - 21 q^{68} - 76 q^{69} + 10 q^{70} - 22 q^{71} + 21 q^{72} - 37 q^{73} - 14 q^{74} - 39 q^{75} - 22 q^{76} + 17 q^{77} + q^{78} + 25 q^{79} - 10 q^{80} - 36 q^{81} - 37 q^{82} - 33 q^{83} + 3 q^{84} - 2 q^{85} - 5 q^{86} - 26 q^{87} - 17 q^{88} - 71 q^{89} - 55 q^{90} + 23 q^{91} + 15 q^{92} - 49 q^{93} - 29 q^{94} - 14 q^{95} - 3 q^{96} - 51 q^{97} + 20 q^{98} - 10 q^{99}+O(q^{100}) \) 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
\(3\) 0.395900 0.228573 0.114287 0.993448i \(-0.463542\pi\)
0.114287 + 0.993448i \(0.463542\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.151677 0.0678319 0.0339160 0.999425i \(-0.489202\pi\)
0.0339160 + 0.999425i \(0.489202\pi\)
\(6\) 0.395900 0.161626
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.84326 −0.947754
\(10\) 0.151677 0.0479644
\(11\) 2.85030 0.859398 0.429699 0.902972i \(-0.358620\pi\)
0.429699 + 0.902972i \(0.358620\pi\)
\(12\) 0.395900 0.114287
\(13\) −2.70854 −0.751214 −0.375607 0.926779i \(-0.622566\pi\)
−0.375607 + 0.926779i \(0.622566\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.0600489 0.0155046
\(16\) 1.00000 0.250000
\(17\) −1.00368 −0.243428 −0.121714 0.992565i \(-0.538839\pi\)
−0.121714 + 0.992565i \(0.538839\pi\)
\(18\) −2.84326 −0.670164
\(19\) 2.34557 0.538112 0.269056 0.963125i \(-0.413288\pi\)
0.269056 + 0.963125i \(0.413288\pi\)
\(20\) 0.151677 0.0339160
\(21\) −0.395900 −0.0863925
\(22\) 2.85030 0.607686
\(23\) −0.830286 −0.173127 −0.0865633 0.996246i \(-0.527588\pi\)
−0.0865633 + 0.996246i \(0.527588\pi\)
\(24\) 0.395900 0.0808128
\(25\) −4.97699 −0.995399
\(26\) −2.70854 −0.531188
\(27\) −2.31335 −0.445204
\(28\) −1.00000 −0.188982
\(29\) 3.14114 0.583296 0.291648 0.956526i \(-0.405796\pi\)
0.291648 + 0.956526i \(0.405796\pi\)
\(30\) 0.0600489 0.0109634
\(31\) −4.76935 −0.856601 −0.428301 0.903636i \(-0.640888\pi\)
−0.428301 + 0.903636i \(0.640888\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.12843 0.196435
\(34\) −1.00368 −0.172130
\(35\) −0.151677 −0.0256381
\(36\) −2.84326 −0.473877
\(37\) 2.32662 0.382494 0.191247 0.981542i \(-0.438747\pi\)
0.191247 + 0.981542i \(0.438747\pi\)
\(38\) 2.34557 0.380502
\(39\) −1.07231 −0.171707
\(40\) 0.151677 0.0239822
\(41\) −3.35096 −0.523332 −0.261666 0.965158i \(-0.584272\pi\)
−0.261666 + 0.965158i \(0.584272\pi\)
\(42\) −0.395900 −0.0610887
\(43\) −11.1914 −1.70668 −0.853339 0.521356i \(-0.825426\pi\)
−0.853339 + 0.521356i \(0.825426\pi\)
\(44\) 2.85030 0.429699
\(45\) −0.431257 −0.0642880
\(46\) −0.830286 −0.122419
\(47\) −9.19342 −1.34100 −0.670499 0.741910i \(-0.733920\pi\)
−0.670499 + 0.741910i \(0.733920\pi\)
\(48\) 0.395900 0.0571433
\(49\) 1.00000 0.142857
\(50\) −4.97699 −0.703853
\(51\) −0.397357 −0.0556411
\(52\) −2.70854 −0.375607
\(53\) 0.902309 0.123942 0.0619708 0.998078i \(-0.480261\pi\)
0.0619708 + 0.998078i \(0.480261\pi\)
\(54\) −2.31335 −0.314807
\(55\) 0.432324 0.0582946
\(56\) −1.00000 −0.133631
\(57\) 0.928613 0.122998
\(58\) 3.14114 0.412453
\(59\) −0.539748 −0.0702692 −0.0351346 0.999383i \(-0.511186\pi\)
−0.0351346 + 0.999383i \(0.511186\pi\)
\(60\) 0.0600489 0.00775228
\(61\) −3.28025 −0.419992 −0.209996 0.977702i \(-0.567345\pi\)
−0.209996 + 0.977702i \(0.567345\pi\)
\(62\) −4.76935 −0.605708
\(63\) 2.84326 0.358217
\(64\) 1.00000 0.125000
\(65\) −0.410823 −0.0509563
\(66\) 1.12843 0.138901
\(67\) −8.88542 −1.08553 −0.542764 0.839885i \(-0.682622\pi\)
−0.542764 + 0.839885i \(0.682622\pi\)
\(68\) −1.00368 −0.121714
\(69\) −0.328710 −0.0395721
\(70\) −0.151677 −0.0181288
\(71\) 9.68947 1.14993 0.574964 0.818178i \(-0.305016\pi\)
0.574964 + 0.818178i \(0.305016\pi\)
\(72\) −2.84326 −0.335082
\(73\) 1.66771 0.195191 0.0975953 0.995226i \(-0.468885\pi\)
0.0975953 + 0.995226i \(0.468885\pi\)
\(74\) 2.32662 0.270464
\(75\) −1.97039 −0.227521
\(76\) 2.34557 0.269056
\(77\) −2.85030 −0.324822
\(78\) −1.07231 −0.121415
\(79\) 4.67827 0.526346 0.263173 0.964749i \(-0.415231\pi\)
0.263173 + 0.964749i \(0.415231\pi\)
\(80\) 0.151677 0.0169580
\(81\) 7.61393 0.845993
\(82\) −3.35096 −0.370052
\(83\) 13.9915 1.53577 0.767886 0.640587i \(-0.221309\pi\)
0.767886 + 0.640587i \(0.221309\pi\)
\(84\) −0.395900 −0.0431962
\(85\) −0.152235 −0.0165122
\(86\) −11.1914 −1.20680
\(87\) 1.24358 0.133326
\(88\) 2.85030 0.303843
\(89\) −1.06964 −0.113382 −0.0566910 0.998392i \(-0.518055\pi\)
−0.0566910 + 0.998392i \(0.518055\pi\)
\(90\) −0.431257 −0.0454585
\(91\) 2.70854 0.283932
\(92\) −0.830286 −0.0865633
\(93\) −1.88819 −0.195796
\(94\) −9.19342 −0.948229
\(95\) 0.355769 0.0365011
\(96\) 0.395900 0.0404064
\(97\) −6.33555 −0.643277 −0.321639 0.946863i \(-0.604234\pi\)
−0.321639 + 0.946863i \(0.604234\pi\)
\(98\) 1.00000 0.101015
\(99\) −8.10415 −0.814498
\(100\) −4.97699 −0.497699
\(101\) −2.64274 −0.262963 −0.131481 0.991319i \(-0.541973\pi\)
−0.131481 + 0.991319i \(0.541973\pi\)
\(102\) −0.397357 −0.0393442
\(103\) −7.36597 −0.725791 −0.362895 0.931830i \(-0.618212\pi\)
−0.362895 + 0.931830i \(0.618212\pi\)
\(104\) −2.70854 −0.265594
\(105\) −0.0600489 −0.00586017
\(106\) 0.902309 0.0876400
\(107\) 2.57808 0.249232 0.124616 0.992205i \(-0.460230\pi\)
0.124616 + 0.992205i \(0.460230\pi\)
\(108\) −2.31335 −0.222602
\(109\) −17.7365 −1.69885 −0.849425 0.527710i \(-0.823051\pi\)
−0.849425 + 0.527710i \(0.823051\pi\)
\(110\) 0.432324 0.0412205
\(111\) 0.921109 0.0874278
\(112\) −1.00000 −0.0944911
\(113\) −17.6543 −1.66077 −0.830387 0.557187i \(-0.811881\pi\)
−0.830387 + 0.557187i \(0.811881\pi\)
\(114\) 0.928613 0.0869726
\(115\) −0.125935 −0.0117435
\(116\) 3.14114 0.291648
\(117\) 7.70109 0.711966
\(118\) −0.539748 −0.0496878
\(119\) 1.00368 0.0920071
\(120\) 0.0600489 0.00548169
\(121\) −2.87579 −0.261435
\(122\) −3.28025 −0.296979
\(123\) −1.32665 −0.119620
\(124\) −4.76935 −0.428301
\(125\) −1.51328 −0.135352
\(126\) 2.84326 0.253298
\(127\) −1.89718 −0.168348 −0.0841738 0.996451i \(-0.526825\pi\)
−0.0841738 + 0.996451i \(0.526825\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.43069 −0.390101
\(130\) −0.410823 −0.0360315
\(131\) −18.4798 −1.61459 −0.807295 0.590148i \(-0.799069\pi\)
−0.807295 + 0.590148i \(0.799069\pi\)
\(132\) 1.12843 0.0982176
\(133\) −2.34557 −0.203387
\(134\) −8.88542 −0.767584
\(135\) −0.350881 −0.0301991
\(136\) −1.00368 −0.0860648
\(137\) 7.37889 0.630421 0.315211 0.949022i \(-0.397925\pi\)
0.315211 + 0.949022i \(0.397925\pi\)
\(138\) −0.328710 −0.0279817
\(139\) −12.3861 −1.05058 −0.525288 0.850924i \(-0.676042\pi\)
−0.525288 + 0.850924i \(0.676042\pi\)
\(140\) −0.151677 −0.0128190
\(141\) −3.63968 −0.306516
\(142\) 9.68947 0.813122
\(143\) −7.72015 −0.645592
\(144\) −2.84326 −0.236939
\(145\) 0.476439 0.0395661
\(146\) 1.66771 0.138021
\(147\) 0.395900 0.0326533
\(148\) 2.32662 0.191247
\(149\) 15.5998 1.27798 0.638992 0.769214i \(-0.279352\pi\)
0.638992 + 0.769214i \(0.279352\pi\)
\(150\) −1.97039 −0.160882
\(151\) −7.65766 −0.623171 −0.311586 0.950218i \(-0.600860\pi\)
−0.311586 + 0.950218i \(0.600860\pi\)
\(152\) 2.34557 0.190251
\(153\) 2.85372 0.230710
\(154\) −2.85030 −0.229684
\(155\) −0.723400 −0.0581049
\(156\) −1.07231 −0.0858536
\(157\) 3.84490 0.306856 0.153428 0.988160i \(-0.450969\pi\)
0.153428 + 0.988160i \(0.450969\pi\)
\(158\) 4.67827 0.372183
\(159\) 0.357224 0.0283297
\(160\) 0.151677 0.0119911
\(161\) 0.830286 0.0654357
\(162\) 7.61393 0.598207
\(163\) 21.9243 1.71725 0.858623 0.512607i \(-0.171320\pi\)
0.858623 + 0.512607i \(0.171320\pi\)
\(164\) −3.35096 −0.261666
\(165\) 0.171157 0.0133246
\(166\) 13.9915 1.08595
\(167\) −9.30474 −0.720022 −0.360011 0.932948i \(-0.617227\pi\)
−0.360011 + 0.932948i \(0.617227\pi\)
\(168\) −0.395900 −0.0305444
\(169\) −5.66381 −0.435678
\(170\) −0.152235 −0.0116759
\(171\) −6.66908 −0.509998
\(172\) −11.1914 −0.853339
\(173\) −13.3146 −1.01229 −0.506144 0.862449i \(-0.668930\pi\)
−0.506144 + 0.862449i \(0.668930\pi\)
\(174\) 1.24358 0.0942755
\(175\) 4.97699 0.376225
\(176\) 2.85030 0.214849
\(177\) −0.213686 −0.0160617
\(178\) −1.06964 −0.0801732
\(179\) 2.54428 0.190169 0.0950843 0.995469i \(-0.469688\pi\)
0.0950843 + 0.995469i \(0.469688\pi\)
\(180\) −0.431257 −0.0321440
\(181\) 4.36483 0.324435 0.162218 0.986755i \(-0.448135\pi\)
0.162218 + 0.986755i \(0.448135\pi\)
\(182\) 2.70854 0.200770
\(183\) −1.29865 −0.0959989
\(184\) −0.830286 −0.0612095
\(185\) 0.352894 0.0259453
\(186\) −1.88819 −0.138449
\(187\) −2.86079 −0.209202
\(188\) −9.19342 −0.670499
\(189\) 2.31335 0.168271
\(190\) 0.355769 0.0258102
\(191\) −20.2581 −1.46582 −0.732911 0.680324i \(-0.761839\pi\)
−0.732911 + 0.680324i \(0.761839\pi\)
\(192\) 0.395900 0.0285716
\(193\) 25.3832 1.82712 0.913562 0.406699i \(-0.133320\pi\)
0.913562 + 0.406699i \(0.133320\pi\)
\(194\) −6.33555 −0.454866
\(195\) −0.162645 −0.0116472
\(196\) 1.00000 0.0714286
\(197\) 5.58473 0.397896 0.198948 0.980010i \(-0.436248\pi\)
0.198948 + 0.980010i \(0.436248\pi\)
\(198\) −8.10415 −0.575937
\(199\) −19.5254 −1.38412 −0.692058 0.721842i \(-0.743296\pi\)
−0.692058 + 0.721842i \(0.743296\pi\)
\(200\) −4.97699 −0.351927
\(201\) −3.51774 −0.248122
\(202\) −2.64274 −0.185943
\(203\) −3.14114 −0.220465
\(204\) −0.397357 −0.0278205
\(205\) −0.508263 −0.0354986
\(206\) −7.36597 −0.513212
\(207\) 2.36072 0.164081
\(208\) −2.70854 −0.187803
\(209\) 6.68559 0.462452
\(210\) −0.0600489 −0.00414377
\(211\) −12.8303 −0.883273 −0.441637 0.897194i \(-0.645602\pi\)
−0.441637 + 0.897194i \(0.645602\pi\)
\(212\) 0.902309 0.0619708
\(213\) 3.83606 0.262843
\(214\) 2.57808 0.176234
\(215\) −1.69748 −0.115767
\(216\) −2.31335 −0.157403
\(217\) 4.76935 0.323765
\(218\) −17.7365 −1.20127
\(219\) 0.660247 0.0446153
\(220\) 0.432324 0.0291473
\(221\) 2.71851 0.182866
\(222\) 0.921109 0.0618208
\(223\) 21.9733 1.47144 0.735722 0.677284i \(-0.236843\pi\)
0.735722 + 0.677284i \(0.236843\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 14.1509 0.943394
\(226\) −17.6543 −1.17434
\(227\) 28.8717 1.91628 0.958140 0.286302i \(-0.0924259\pi\)
0.958140 + 0.286302i \(0.0924259\pi\)
\(228\) 0.928613 0.0614989
\(229\) −25.3652 −1.67618 −0.838089 0.545534i \(-0.816327\pi\)
−0.838089 + 0.545534i \(0.816327\pi\)
\(230\) −0.125935 −0.00830391
\(231\) −1.12843 −0.0742455
\(232\) 3.14114 0.206226
\(233\) −5.57673 −0.365344 −0.182672 0.983174i \(-0.558475\pi\)
−0.182672 + 0.983174i \(0.558475\pi\)
\(234\) 7.70109 0.503436
\(235\) −1.39443 −0.0909625
\(236\) −0.539748 −0.0351346
\(237\) 1.85213 0.120309
\(238\) 1.00368 0.0650589
\(239\) −29.2351 −1.89106 −0.945530 0.325535i \(-0.894456\pi\)
−0.945530 + 0.325535i \(0.894456\pi\)
\(240\) 0.0600489 0.00387614
\(241\) 0.626558 0.0403602 0.0201801 0.999796i \(-0.493576\pi\)
0.0201801 + 0.999796i \(0.493576\pi\)
\(242\) −2.87579 −0.184863
\(243\) 9.95440 0.638575
\(244\) −3.28025 −0.209996
\(245\) 0.151677 0.00969028
\(246\) −1.32665 −0.0845839
\(247\) −6.35308 −0.404237
\(248\) −4.76935 −0.302854
\(249\) 5.53925 0.351036
\(250\) −1.51328 −0.0957081
\(251\) 10.7106 0.676050 0.338025 0.941137i \(-0.390241\pi\)
0.338025 + 0.941137i \(0.390241\pi\)
\(252\) 2.84326 0.179109
\(253\) −2.36656 −0.148785
\(254\) −1.89718 −0.119040
\(255\) −0.0602698 −0.00377424
\(256\) 1.00000 0.0625000
\(257\) 12.7103 0.792845 0.396422 0.918068i \(-0.370252\pi\)
0.396422 + 0.918068i \(0.370252\pi\)
\(258\) −4.43069 −0.275843
\(259\) −2.32662 −0.144569
\(260\) −0.410823 −0.0254781
\(261\) −8.93110 −0.552821
\(262\) −18.4798 −1.14169
\(263\) −5.36770 −0.330986 −0.165493 0.986211i \(-0.552922\pi\)
−0.165493 + 0.986211i \(0.552922\pi\)
\(264\) 1.12843 0.0694503
\(265\) 0.136859 0.00840720
\(266\) −2.34557 −0.143816
\(267\) −0.423472 −0.0259161
\(268\) −8.88542 −0.542764
\(269\) 26.7652 1.63190 0.815952 0.578119i \(-0.196213\pi\)
0.815952 + 0.578119i \(0.196213\pi\)
\(270\) −0.350881 −0.0213540
\(271\) −26.7417 −1.62444 −0.812222 0.583349i \(-0.801742\pi\)
−0.812222 + 0.583349i \(0.801742\pi\)
\(272\) −1.00368 −0.0608570
\(273\) 1.07231 0.0648992
\(274\) 7.37889 0.445775
\(275\) −14.1859 −0.855444
\(276\) −0.328710 −0.0197860
\(277\) 0.0718655 0.00431798 0.00215899 0.999998i \(-0.499313\pi\)
0.00215899 + 0.999998i \(0.499313\pi\)
\(278\) −12.3861 −0.742869
\(279\) 13.5605 0.811847
\(280\) −0.151677 −0.00906442
\(281\) 16.4509 0.981378 0.490689 0.871335i \(-0.336745\pi\)
0.490689 + 0.871335i \(0.336745\pi\)
\(282\) −3.63968 −0.216740
\(283\) −19.2773 −1.14592 −0.572959 0.819584i \(-0.694205\pi\)
−0.572959 + 0.819584i \(0.694205\pi\)
\(284\) 9.68947 0.574964
\(285\) 0.140849 0.00834318
\(286\) −7.72015 −0.456502
\(287\) 3.35096 0.197801
\(288\) −2.84326 −0.167541
\(289\) −15.9926 −0.940743
\(290\) 0.476439 0.0279774
\(291\) −2.50824 −0.147036
\(292\) 1.66771 0.0975953
\(293\) 31.6786 1.85068 0.925340 0.379137i \(-0.123779\pi\)
0.925340 + 0.379137i \(0.123779\pi\)
\(294\) 0.395900 0.0230894
\(295\) −0.0818673 −0.00476650
\(296\) 2.32662 0.135232
\(297\) −6.59374 −0.382608
\(298\) 15.5998 0.903671
\(299\) 2.24886 0.130055
\(300\) −1.97039 −0.113761
\(301\) 11.1914 0.645064
\(302\) −7.65766 −0.440649
\(303\) −1.04626 −0.0601061
\(304\) 2.34557 0.134528
\(305\) −0.497537 −0.0284889
\(306\) 2.85372 0.163137
\(307\) −12.1393 −0.692829 −0.346415 0.938082i \(-0.612601\pi\)
−0.346415 + 0.938082i \(0.612601\pi\)
\(308\) −2.85030 −0.162411
\(309\) −2.91619 −0.165896
\(310\) −0.723400 −0.0410864
\(311\) −1.46226 −0.0829174 −0.0414587 0.999140i \(-0.513200\pi\)
−0.0414587 + 0.999140i \(0.513200\pi\)
\(312\) −1.07231 −0.0607077
\(313\) −7.45357 −0.421300 −0.210650 0.977562i \(-0.567558\pi\)
−0.210650 + 0.977562i \(0.567558\pi\)
\(314\) 3.84490 0.216980
\(315\) 0.431257 0.0242986
\(316\) 4.67827 0.263173
\(317\) 8.92427 0.501237 0.250619 0.968086i \(-0.419366\pi\)
0.250619 + 0.968086i \(0.419366\pi\)
\(318\) 0.357224 0.0200321
\(319\) 8.95321 0.501283
\(320\) 0.151677 0.00847899
\(321\) 1.02066 0.0569677
\(322\) 0.830286 0.0462700
\(323\) −2.35420 −0.130991
\(324\) 7.61393 0.422996
\(325\) 13.4804 0.747757
\(326\) 21.9243 1.21428
\(327\) −7.02189 −0.388311
\(328\) −3.35096 −0.185026
\(329\) 9.19342 0.506850
\(330\) 0.171157 0.00942190
\(331\) 26.5579 1.45975 0.729877 0.683578i \(-0.239577\pi\)
0.729877 + 0.683578i \(0.239577\pi\)
\(332\) 13.9915 0.767886
\(333\) −6.61519 −0.362510
\(334\) −9.30474 −0.509133
\(335\) −1.34771 −0.0736334
\(336\) −0.395900 −0.0215981
\(337\) 22.4767 1.22439 0.612193 0.790708i \(-0.290288\pi\)
0.612193 + 0.790708i \(0.290288\pi\)
\(338\) −5.66381 −0.308071
\(339\) −6.98933 −0.379608
\(340\) −0.152235 −0.00825610
\(341\) −13.5941 −0.736161
\(342\) −6.66908 −0.360623
\(343\) −1.00000 −0.0539949
\(344\) −11.1914 −0.603402
\(345\) −0.0498577 −0.00268425
\(346\) −13.3146 −0.715796
\(347\) −8.23114 −0.441871 −0.220935 0.975288i \(-0.570911\pi\)
−0.220935 + 0.975288i \(0.570911\pi\)
\(348\) 1.24358 0.0666629
\(349\) −16.0146 −0.857240 −0.428620 0.903485i \(-0.641000\pi\)
−0.428620 + 0.903485i \(0.641000\pi\)
\(350\) 4.97699 0.266032
\(351\) 6.26580 0.334443
\(352\) 2.85030 0.151922
\(353\) −17.9707 −0.956486 −0.478243 0.878227i \(-0.658726\pi\)
−0.478243 + 0.878227i \(0.658726\pi\)
\(354\) −0.213686 −0.0113573
\(355\) 1.46967 0.0780019
\(356\) −1.06964 −0.0566910
\(357\) 0.397357 0.0210304
\(358\) 2.54428 0.134470
\(359\) −7.87372 −0.415559 −0.207780 0.978176i \(-0.566624\pi\)
−0.207780 + 0.978176i \(0.566624\pi\)
\(360\) −0.431257 −0.0227292
\(361\) −13.4983 −0.710436
\(362\) 4.36483 0.229410
\(363\) −1.13852 −0.0597570
\(364\) 2.70854 0.141966
\(365\) 0.252953 0.0132402
\(366\) −1.29865 −0.0678815
\(367\) 6.21171 0.324249 0.162124 0.986770i \(-0.448165\pi\)
0.162124 + 0.986770i \(0.448165\pi\)
\(368\) −0.830286 −0.0432816
\(369\) 9.52766 0.495990
\(370\) 0.352894 0.0183461
\(371\) −0.902309 −0.0468456
\(372\) −1.88819 −0.0978980
\(373\) 23.3144 1.20718 0.603588 0.797297i \(-0.293737\pi\)
0.603588 + 0.797297i \(0.293737\pi\)
\(374\) −2.86079 −0.147928
\(375\) −0.599107 −0.0309378
\(376\) −9.19342 −0.474114
\(377\) −8.50791 −0.438180
\(378\) 2.31335 0.118986
\(379\) 13.6025 0.698713 0.349357 0.936990i \(-0.386400\pi\)
0.349357 + 0.936990i \(0.386400\pi\)
\(380\) 0.355769 0.0182506
\(381\) −0.751094 −0.0384797
\(382\) −20.2581 −1.03649
\(383\) −27.4489 −1.40257 −0.701287 0.712879i \(-0.747391\pi\)
−0.701287 + 0.712879i \(0.747391\pi\)
\(384\) 0.395900 0.0202032
\(385\) −0.432324 −0.0220333
\(386\) 25.3832 1.29197
\(387\) 31.8202 1.61751
\(388\) −6.33555 −0.321639
\(389\) 29.7462 1.50819 0.754096 0.656764i \(-0.228075\pi\)
0.754096 + 0.656764i \(0.228075\pi\)
\(390\) −0.162645 −0.00823584
\(391\) 0.833341 0.0421439
\(392\) 1.00000 0.0505076
\(393\) −7.31617 −0.369052
\(394\) 5.58473 0.281355
\(395\) 0.709585 0.0357031
\(396\) −8.10415 −0.407249
\(397\) 8.96900 0.450141 0.225071 0.974342i \(-0.427739\pi\)
0.225071 + 0.974342i \(0.427739\pi\)
\(398\) −19.5254 −0.978718
\(399\) −0.928613 −0.0464888
\(400\) −4.97699 −0.248850
\(401\) −31.8397 −1.59000 −0.795000 0.606609i \(-0.792529\pi\)
−0.795000 + 0.606609i \(0.792529\pi\)
\(402\) −3.51774 −0.175449
\(403\) 12.9180 0.643490
\(404\) −2.64274 −0.131481
\(405\) 1.15486 0.0573853
\(406\) −3.14114 −0.155892
\(407\) 6.63156 0.328714
\(408\) −0.397357 −0.0196721
\(409\) −10.4532 −0.516879 −0.258440 0.966027i \(-0.583208\pi\)
−0.258440 + 0.966027i \(0.583208\pi\)
\(410\) −0.508263 −0.0251013
\(411\) 2.92130 0.144097
\(412\) −7.36597 −0.362895
\(413\) 0.539748 0.0265593
\(414\) 2.36072 0.116023
\(415\) 2.12219 0.104174
\(416\) −2.70854 −0.132797
\(417\) −4.90366 −0.240133
\(418\) 6.68559 0.327003
\(419\) −17.1744 −0.839022 −0.419511 0.907750i \(-0.637799\pi\)
−0.419511 + 0.907750i \(0.637799\pi\)
\(420\) −0.0600489 −0.00293008
\(421\) 29.5853 1.44190 0.720949 0.692988i \(-0.243706\pi\)
0.720949 + 0.692988i \(0.243706\pi\)
\(422\) −12.8303 −0.624569
\(423\) 26.1393 1.27094
\(424\) 0.902309 0.0438200
\(425\) 4.99531 0.242308
\(426\) 3.83606 0.185858
\(427\) 3.28025 0.158742
\(428\) 2.57808 0.124616
\(429\) −3.05641 −0.147565
\(430\) −1.69748 −0.0818598
\(431\) 1.00000 0.0481683
\(432\) −2.31335 −0.111301
\(433\) −26.4582 −1.27150 −0.635749 0.771896i \(-0.719309\pi\)
−0.635749 + 0.771896i \(0.719309\pi\)
\(434\) 4.76935 0.228936
\(435\) 0.188622 0.00904374
\(436\) −17.7365 −0.849425
\(437\) −1.94750 −0.0931614
\(438\) 0.660247 0.0315478
\(439\) 34.8667 1.66410 0.832049 0.554702i \(-0.187168\pi\)
0.832049 + 0.554702i \(0.187168\pi\)
\(440\) 0.432324 0.0206103
\(441\) −2.84326 −0.135393
\(442\) 2.71851 0.129306
\(443\) 21.3817 1.01588 0.507938 0.861394i \(-0.330408\pi\)
0.507938 + 0.861394i \(0.330408\pi\)
\(444\) 0.921109 0.0437139
\(445\) −0.162240 −0.00769093
\(446\) 21.9733 1.04047
\(447\) 6.17595 0.292113
\(448\) −1.00000 −0.0472456
\(449\) −1.49850 −0.0707186 −0.0353593 0.999375i \(-0.511258\pi\)
−0.0353593 + 0.999375i \(0.511258\pi\)
\(450\) 14.1509 0.667080
\(451\) −9.55125 −0.449751
\(452\) −17.6543 −0.830387
\(453\) −3.03167 −0.142440
\(454\) 28.8717 1.35501
\(455\) 0.410823 0.0192597
\(456\) 0.928613 0.0434863
\(457\) 7.11181 0.332677 0.166338 0.986069i \(-0.446806\pi\)
0.166338 + 0.986069i \(0.446806\pi\)
\(458\) −25.3652 −1.18524
\(459\) 2.32186 0.108375
\(460\) −0.125935 −0.00587175
\(461\) 5.48316 0.255376 0.127688 0.991814i \(-0.459244\pi\)
0.127688 + 0.991814i \(0.459244\pi\)
\(462\) −1.12843 −0.0524995
\(463\) 12.5409 0.582824 0.291412 0.956598i \(-0.405875\pi\)
0.291412 + 0.956598i \(0.405875\pi\)
\(464\) 3.14114 0.145824
\(465\) −0.286394 −0.0132812
\(466\) −5.57673 −0.258337
\(467\) −12.2467 −0.566708 −0.283354 0.959015i \(-0.591447\pi\)
−0.283354 + 0.959015i \(0.591447\pi\)
\(468\) 7.70109 0.355983
\(469\) 8.88542 0.410291
\(470\) −1.39443 −0.0643202
\(471\) 1.52220 0.0701391
\(472\) −0.539748 −0.0248439
\(473\) −31.8990 −1.46672
\(474\) 1.85213 0.0850710
\(475\) −11.6739 −0.535636
\(476\) 1.00368 0.0460036
\(477\) −2.56550 −0.117466
\(478\) −29.2351 −1.33718
\(479\) 30.2923 1.38409 0.692045 0.721854i \(-0.256710\pi\)
0.692045 + 0.721854i \(0.256710\pi\)
\(480\) 0.0600489 0.00274084
\(481\) −6.30174 −0.287335
\(482\) 0.626558 0.0285389
\(483\) 0.328710 0.0149568
\(484\) −2.87579 −0.130718
\(485\) −0.960955 −0.0436347
\(486\) 9.95440 0.451541
\(487\) 28.9776 1.31310 0.656549 0.754283i \(-0.272015\pi\)
0.656549 + 0.754283i \(0.272015\pi\)
\(488\) −3.28025 −0.148490
\(489\) 8.67985 0.392516
\(490\) 0.151677 0.00685206
\(491\) −0.484813 −0.0218793 −0.0109396 0.999940i \(-0.503482\pi\)
−0.0109396 + 0.999940i \(0.503482\pi\)
\(492\) −1.32665 −0.0598098
\(493\) −3.15270 −0.141991
\(494\) −6.35308 −0.285839
\(495\) −1.22921 −0.0552490
\(496\) −4.76935 −0.214150
\(497\) −9.68947 −0.434632
\(498\) 5.53925 0.248220
\(499\) 9.87310 0.441981 0.220990 0.975276i \(-0.429071\pi\)
0.220990 + 0.975276i \(0.429071\pi\)
\(500\) −1.51328 −0.0676759
\(501\) −3.68375 −0.164578
\(502\) 10.7106 0.478040
\(503\) 2.07393 0.0924719 0.0462360 0.998931i \(-0.485277\pi\)
0.0462360 + 0.998931i \(0.485277\pi\)
\(504\) 2.84326 0.126649
\(505\) −0.400842 −0.0178373
\(506\) −2.36656 −0.105207
\(507\) −2.24230 −0.0995842
\(508\) −1.89718 −0.0841738
\(509\) −3.03420 −0.134489 −0.0672443 0.997737i \(-0.521421\pi\)
−0.0672443 + 0.997737i \(0.521421\pi\)
\(510\) −0.0602698 −0.00266879
\(511\) −1.66771 −0.0737751
\(512\) 1.00000 0.0441942
\(513\) −5.42613 −0.239570
\(514\) 12.7103 0.560626
\(515\) −1.11725 −0.0492318
\(516\) −4.43069 −0.195050
\(517\) −26.2040 −1.15245
\(518\) −2.32662 −0.102226
\(519\) −5.27124 −0.231382
\(520\) −0.410823 −0.0180158
\(521\) −30.2969 −1.32733 −0.663667 0.748028i \(-0.731001\pi\)
−0.663667 + 0.748028i \(0.731001\pi\)
\(522\) −8.93110 −0.390904
\(523\) −18.2395 −0.797557 −0.398778 0.917047i \(-0.630566\pi\)
−0.398778 + 0.917047i \(0.630566\pi\)
\(524\) −18.4798 −0.807295
\(525\) 1.97039 0.0859950
\(526\) −5.36770 −0.234043
\(527\) 4.78690 0.208521
\(528\) 1.12843 0.0491088
\(529\) −22.3106 −0.970027
\(530\) 0.136859 0.00594479
\(531\) 1.53465 0.0665980
\(532\) −2.34557 −0.101694
\(533\) 9.07621 0.393134
\(534\) −0.423472 −0.0183254
\(535\) 0.391034 0.0169059
\(536\) −8.88542 −0.383792
\(537\) 1.00728 0.0434674
\(538\) 26.7652 1.15393
\(539\) 2.85030 0.122771
\(540\) −0.350881 −0.0150995
\(541\) −22.6644 −0.974419 −0.487210 0.873285i \(-0.661985\pi\)
−0.487210 + 0.873285i \(0.661985\pi\)
\(542\) −26.7417 −1.14866
\(543\) 1.72804 0.0741571
\(544\) −1.00368 −0.0430324
\(545\) −2.69022 −0.115236
\(546\) 1.07231 0.0458907
\(547\) 12.6896 0.542569 0.271285 0.962499i \(-0.412552\pi\)
0.271285 + 0.962499i \(0.412552\pi\)
\(548\) 7.37889 0.315211
\(549\) 9.32660 0.398050
\(550\) −14.1859 −0.604890
\(551\) 7.36779 0.313878
\(552\) −0.328710 −0.0139908
\(553\) −4.67827 −0.198940
\(554\) 0.0718655 0.00305327
\(555\) 0.139711 0.00593039
\(556\) −12.3861 −0.525288
\(557\) 37.7129 1.59795 0.798974 0.601366i \(-0.205377\pi\)
0.798974 + 0.601366i \(0.205377\pi\)
\(558\) 13.5605 0.574063
\(559\) 30.3125 1.28208
\(560\) −0.151677 −0.00640951
\(561\) −1.13259 −0.0478178
\(562\) 16.4509 0.693939
\(563\) 29.4977 1.24318 0.621591 0.783342i \(-0.286487\pi\)
0.621591 + 0.783342i \(0.286487\pi\)
\(564\) −3.63968 −0.153258
\(565\) −2.67774 −0.112654
\(566\) −19.2773 −0.810286
\(567\) −7.61393 −0.319755
\(568\) 9.68947 0.406561
\(569\) −10.3836 −0.435301 −0.217651 0.976027i \(-0.569839\pi\)
−0.217651 + 0.976027i \(0.569839\pi\)
\(570\) 0.140849 0.00589952
\(571\) −15.1532 −0.634142 −0.317071 0.948402i \(-0.602699\pi\)
−0.317071 + 0.948402i \(0.602699\pi\)
\(572\) −7.72015 −0.322796
\(573\) −8.02017 −0.335047
\(574\) 3.35096 0.139866
\(575\) 4.13233 0.172330
\(576\) −2.84326 −0.118469
\(577\) −26.9099 −1.12027 −0.560137 0.828400i \(-0.689252\pi\)
−0.560137 + 0.828400i \(0.689252\pi\)
\(578\) −15.9926 −0.665206
\(579\) 10.0492 0.417632
\(580\) 0.476439 0.0197830
\(581\) −13.9915 −0.580467
\(582\) −2.50824 −0.103970
\(583\) 2.57185 0.106515
\(584\) 1.66771 0.0690103
\(585\) 1.16808 0.0482940
\(586\) 31.6786 1.30863
\(587\) −24.0191 −0.991375 −0.495687 0.868501i \(-0.665084\pi\)
−0.495687 + 0.868501i \(0.665084\pi\)
\(588\) 0.395900 0.0163266
\(589\) −11.1869 −0.460947
\(590\) −0.0818673 −0.00337042
\(591\) 2.21100 0.0909482
\(592\) 2.32662 0.0956234
\(593\) 46.5065 1.90979 0.954897 0.296938i \(-0.0959656\pi\)
0.954897 + 0.296938i \(0.0959656\pi\)
\(594\) −6.59374 −0.270544
\(595\) 0.152235 0.00624102
\(596\) 15.5998 0.638992
\(597\) −7.73009 −0.316372
\(598\) 2.24886 0.0919628
\(599\) 28.6066 1.16883 0.584417 0.811453i \(-0.301323\pi\)
0.584417 + 0.811453i \(0.301323\pi\)
\(600\) −1.97039 −0.0804409
\(601\) 1.96721 0.0802441 0.0401220 0.999195i \(-0.487225\pi\)
0.0401220 + 0.999195i \(0.487225\pi\)
\(602\) 11.1914 0.456129
\(603\) 25.2636 1.02881
\(604\) −7.65766 −0.311586
\(605\) −0.436190 −0.0177336
\(606\) −1.04626 −0.0425015
\(607\) −37.0758 −1.50486 −0.752430 0.658673i \(-0.771118\pi\)
−0.752430 + 0.658673i \(0.771118\pi\)
\(608\) 2.34557 0.0951256
\(609\) −1.24358 −0.0503924
\(610\) −0.497537 −0.0201447
\(611\) 24.9007 1.00738
\(612\) 2.85372 0.115355
\(613\) 5.14413 0.207770 0.103885 0.994589i \(-0.466873\pi\)
0.103885 + 0.994589i \(0.466873\pi\)
\(614\) −12.1393 −0.489904
\(615\) −0.201221 −0.00811403
\(616\) −2.85030 −0.114842
\(617\) 17.9122 0.721119 0.360559 0.932736i \(-0.382586\pi\)
0.360559 + 0.932736i \(0.382586\pi\)
\(618\) −2.91619 −0.117306
\(619\) 32.9535 1.32451 0.662256 0.749278i \(-0.269599\pi\)
0.662256 + 0.749278i \(0.269599\pi\)
\(620\) −0.723400 −0.0290525
\(621\) 1.92074 0.0770767
\(622\) −1.46226 −0.0586314
\(623\) 1.06964 0.0428544
\(624\) −1.07231 −0.0429268
\(625\) 24.6554 0.986218
\(626\) −7.45357 −0.297904
\(627\) 2.64683 0.105704
\(628\) 3.84490 0.153428
\(629\) −2.33518 −0.0931097
\(630\) 0.431257 0.0171817
\(631\) −40.1058 −1.59659 −0.798293 0.602270i \(-0.794263\pi\)
−0.798293 + 0.602270i \(0.794263\pi\)
\(632\) 4.67827 0.186092
\(633\) −5.07951 −0.201892
\(634\) 8.92427 0.354428
\(635\) −0.287758 −0.0114193
\(636\) 0.357224 0.0141649
\(637\) −2.70854 −0.107316
\(638\) 8.95321 0.354461
\(639\) −27.5497 −1.08985
\(640\) 0.151677 0.00599555
\(641\) −7.94400 −0.313769 −0.156885 0.987617i \(-0.550145\pi\)
−0.156885 + 0.987617i \(0.550145\pi\)
\(642\) 1.02066 0.0402823
\(643\) 31.7676 1.25279 0.626397 0.779504i \(-0.284529\pi\)
0.626397 + 0.779504i \(0.284529\pi\)
\(644\) 0.830286 0.0327178
\(645\) −0.672033 −0.0264613
\(646\) −2.35420 −0.0926249
\(647\) 34.6495 1.36221 0.681106 0.732185i \(-0.261499\pi\)
0.681106 + 0.732185i \(0.261499\pi\)
\(648\) 7.61393 0.299104
\(649\) −1.53844 −0.0603892
\(650\) 13.4804 0.528744
\(651\) 1.88819 0.0740039
\(652\) 21.9243 0.858623
\(653\) −15.9056 −0.622433 −0.311216 0.950339i \(-0.600736\pi\)
−0.311216 + 0.950339i \(0.600736\pi\)
\(654\) −7.02189 −0.274578
\(655\) −2.80296 −0.109521
\(656\) −3.35096 −0.130833
\(657\) −4.74174 −0.184993
\(658\) 9.19342 0.358397
\(659\) 36.1413 1.40787 0.703933 0.710266i \(-0.251425\pi\)
0.703933 + 0.710266i \(0.251425\pi\)
\(660\) 0.171157 0.00666229
\(661\) 5.86031 0.227940 0.113970 0.993484i \(-0.463643\pi\)
0.113970 + 0.993484i \(0.463643\pi\)
\(662\) 26.5579 1.03220
\(663\) 1.07626 0.0417983
\(664\) 13.9915 0.542977
\(665\) −0.355769 −0.0137961
\(666\) −6.61519 −0.256333
\(667\) −2.60805 −0.100984
\(668\) −9.30474 −0.360011
\(669\) 8.69925 0.336332
\(670\) −1.34771 −0.0520667
\(671\) −9.34969 −0.360941
\(672\) −0.395900 −0.0152722
\(673\) 0.806263 0.0310791 0.0155396 0.999879i \(-0.495053\pi\)
0.0155396 + 0.999879i \(0.495053\pi\)
\(674\) 22.4767 0.865772
\(675\) 11.5135 0.443156
\(676\) −5.66381 −0.217839
\(677\) 18.4218 0.708009 0.354005 0.935244i \(-0.384820\pi\)
0.354005 + 0.935244i \(0.384820\pi\)
\(678\) −6.98933 −0.268424
\(679\) 6.33555 0.243136
\(680\) −0.152235 −0.00583794
\(681\) 11.4303 0.438010
\(682\) −13.5941 −0.520545
\(683\) 26.7634 1.02407 0.512036 0.858964i \(-0.328892\pi\)
0.512036 + 0.858964i \(0.328892\pi\)
\(684\) −6.66908 −0.254999
\(685\) 1.11921 0.0427627
\(686\) −1.00000 −0.0381802
\(687\) −10.0421 −0.383129
\(688\) −11.1914 −0.426670
\(689\) −2.44394 −0.0931067
\(690\) −0.0498577 −0.00189805
\(691\) 1.30138 0.0495068 0.0247534 0.999694i \(-0.492120\pi\)
0.0247534 + 0.999694i \(0.492120\pi\)
\(692\) −13.3146 −0.506144
\(693\) 8.10415 0.307851
\(694\) −8.23114 −0.312450
\(695\) −1.87869 −0.0712626
\(696\) 1.24358 0.0471378
\(697\) 3.36329 0.127394
\(698\) −16.0146 −0.606161
\(699\) −2.20783 −0.0835078
\(700\) 4.97699 0.188113
\(701\) −37.9923 −1.43495 −0.717474 0.696585i \(-0.754702\pi\)
−0.717474 + 0.696585i \(0.754702\pi\)
\(702\) 6.26580 0.236487
\(703\) 5.45726 0.205824
\(704\) 2.85030 0.107425
\(705\) −0.552055 −0.0207916
\(706\) −17.9707 −0.676338
\(707\) 2.64274 0.0993905
\(708\) −0.213686 −0.00803083
\(709\) 34.7336 1.30445 0.652224 0.758026i \(-0.273836\pi\)
0.652224 + 0.758026i \(0.273836\pi\)
\(710\) 1.46967 0.0551557
\(711\) −13.3015 −0.498847
\(712\) −1.06964 −0.0400866
\(713\) 3.95993 0.148300
\(714\) 0.397357 0.0148707
\(715\) −1.17097 −0.0437917
\(716\) 2.54428 0.0950843
\(717\) −11.5742 −0.432245
\(718\) −7.87372 −0.293845
\(719\) 2.02151 0.0753896 0.0376948 0.999289i \(-0.487999\pi\)
0.0376948 + 0.999289i \(0.487999\pi\)
\(720\) −0.431257 −0.0160720
\(721\) 7.36597 0.274323
\(722\) −13.4983 −0.502354
\(723\) 0.248054 0.00922525
\(724\) 4.36483 0.162218
\(725\) −15.6335 −0.580612
\(726\) −1.13852 −0.0422546
\(727\) −23.2705 −0.863054 −0.431527 0.902100i \(-0.642025\pi\)
−0.431527 + 0.902100i \(0.642025\pi\)
\(728\) 2.70854 0.100385
\(729\) −18.9009 −0.700032
\(730\) 0.252953 0.00936221
\(731\) 11.2326 0.415453
\(732\) −1.29865 −0.0479995
\(733\) −7.55748 −0.279142 −0.139571 0.990212i \(-0.544572\pi\)
−0.139571 + 0.990212i \(0.544572\pi\)
\(734\) 6.21171 0.229278
\(735\) 0.0600489 0.00221494
\(736\) −0.830286 −0.0306047
\(737\) −25.3261 −0.932900
\(738\) 9.52766 0.350718
\(739\) 27.3338 1.00549 0.502745 0.864435i \(-0.332323\pi\)
0.502745 + 0.864435i \(0.332323\pi\)
\(740\) 0.352894 0.0129726
\(741\) −2.51519 −0.0923976
\(742\) −0.902309 −0.0331248
\(743\) −27.0621 −0.992812 −0.496406 0.868091i \(-0.665347\pi\)
−0.496406 + 0.868091i \(0.665347\pi\)
\(744\) −1.88819 −0.0692243
\(745\) 2.36612 0.0866881
\(746\) 23.3144 0.853602
\(747\) −39.7816 −1.45553
\(748\) −2.86079 −0.104601
\(749\) −2.57808 −0.0942008
\(750\) −0.599107 −0.0218763
\(751\) 37.4171 1.36537 0.682684 0.730713i \(-0.260812\pi\)
0.682684 + 0.730713i \(0.260812\pi\)
\(752\) −9.19342 −0.335250
\(753\) 4.24035 0.154527
\(754\) −8.50791 −0.309840
\(755\) −1.16149 −0.0422709
\(756\) 2.31335 0.0841357
\(757\) −38.6902 −1.40622 −0.703109 0.711082i \(-0.748205\pi\)
−0.703109 + 0.711082i \(0.748205\pi\)
\(758\) 13.6025 0.494065
\(759\) −0.936923 −0.0340082
\(760\) 0.355769 0.0129051
\(761\) −18.6624 −0.676510 −0.338255 0.941055i \(-0.609837\pi\)
−0.338255 + 0.941055i \(0.609837\pi\)
\(762\) −0.751094 −0.0272093
\(763\) 17.7365 0.642105
\(764\) −20.2581 −0.732911
\(765\) 0.432844 0.0156495
\(766\) −27.4489 −0.991770
\(767\) 1.46193 0.0527872
\(768\) 0.395900 0.0142858
\(769\) −3.03393 −0.109406 −0.0547032 0.998503i \(-0.517421\pi\)
−0.0547032 + 0.998503i \(0.517421\pi\)
\(770\) −0.432324 −0.0155799
\(771\) 5.03200 0.181223
\(772\) 25.3832 0.913562
\(773\) 27.8610 1.00209 0.501046 0.865421i \(-0.332949\pi\)
0.501046 + 0.865421i \(0.332949\pi\)
\(774\) 31.8202 1.14375
\(775\) 23.7370 0.852660
\(776\) −6.33555 −0.227433
\(777\) −0.921109 −0.0330446
\(778\) 29.7462 1.06645
\(779\) −7.85993 −0.281611
\(780\) −0.162645 −0.00582362
\(781\) 27.6179 0.988246
\(782\) 0.833341 0.0298002
\(783\) −7.26656 −0.259686
\(784\) 1.00000 0.0357143
\(785\) 0.583182 0.0208147
\(786\) −7.31617 −0.260959
\(787\) 6.10093 0.217475 0.108737 0.994071i \(-0.465319\pi\)
0.108737 + 0.994071i \(0.465319\pi\)
\(788\) 5.58473 0.198948
\(789\) −2.12507 −0.0756545
\(790\) 0.709585 0.0252459
\(791\) 17.6543 0.627714
\(792\) −8.10415 −0.287969
\(793\) 8.88467 0.315504
\(794\) 8.96900 0.318298
\(795\) 0.0541826 0.00192166
\(796\) −19.5254 −0.692058
\(797\) 24.3675 0.863140 0.431570 0.902079i \(-0.357960\pi\)
0.431570 + 0.902079i \(0.357960\pi\)
\(798\) −0.928613 −0.0328725
\(799\) 9.22725 0.326437
\(800\) −4.97699 −0.175963
\(801\) 3.04128 0.107458
\(802\) −31.8397 −1.12430
\(803\) 4.75347 0.167746
\(804\) −3.51774 −0.124061
\(805\) 0.125935 0.00443863
\(806\) 12.9180 0.455016
\(807\) 10.5964 0.373009
\(808\) −2.64274 −0.0929713
\(809\) −49.8699 −1.75333 −0.876665 0.481100i \(-0.840237\pi\)
−0.876665 + 0.481100i \(0.840237\pi\)
\(810\) 1.15486 0.0405775
\(811\) −39.7231 −1.39487 −0.697434 0.716649i \(-0.745675\pi\)
−0.697434 + 0.716649i \(0.745675\pi\)
\(812\) −3.14114 −0.110233
\(813\) −10.5871 −0.371304
\(814\) 6.63156 0.232436
\(815\) 3.32541 0.116484
\(816\) −0.397357 −0.0139103
\(817\) −26.2503 −0.918383
\(818\) −10.4532 −0.365489
\(819\) −7.70109 −0.269098
\(820\) −0.508263 −0.0177493
\(821\) −5.74872 −0.200632 −0.100316 0.994956i \(-0.531985\pi\)
−0.100316 + 0.994956i \(0.531985\pi\)
\(822\) 2.92130 0.101892
\(823\) −10.5288 −0.367012 −0.183506 0.983019i \(-0.558745\pi\)
−0.183506 + 0.983019i \(0.558745\pi\)
\(824\) −7.36597 −0.256606
\(825\) −5.61621 −0.195531
\(826\) 0.539748 0.0187802
\(827\) 2.75411 0.0957699 0.0478849 0.998853i \(-0.484752\pi\)
0.0478849 + 0.998853i \(0.484752\pi\)
\(828\) 2.36072 0.0820407
\(829\) 13.2129 0.458903 0.229451 0.973320i \(-0.426307\pi\)
0.229451 + 0.973320i \(0.426307\pi\)
\(830\) 2.12219 0.0736624
\(831\) 0.0284516 0.000986974 0
\(832\) −2.70854 −0.0939017
\(833\) −1.00368 −0.0347754
\(834\) −4.90366 −0.169800
\(835\) −1.41131 −0.0488405
\(836\) 6.68559 0.231226
\(837\) 11.0332 0.381362
\(838\) −17.1744 −0.593278
\(839\) −35.9953 −1.24270 −0.621348 0.783535i \(-0.713415\pi\)
−0.621348 + 0.783535i \(0.713415\pi\)
\(840\) −0.0600489 −0.00207188
\(841\) −19.1332 −0.659766
\(842\) 29.5853 1.01958
\(843\) 6.51291 0.224317
\(844\) −12.8303 −0.441637
\(845\) −0.859069 −0.0295529
\(846\) 26.1393 0.898688
\(847\) 2.87579 0.0988132
\(848\) 0.902309 0.0309854
\(849\) −7.63190 −0.261926
\(850\) 4.99531 0.171338
\(851\) −1.93176 −0.0662198
\(852\) 3.83606 0.131421
\(853\) −24.2029 −0.828691 −0.414346 0.910120i \(-0.635990\pi\)
−0.414346 + 0.910120i \(0.635990\pi\)
\(854\) 3.28025 0.112248
\(855\) −1.01155 −0.0345941
\(856\) 2.57808 0.0881168
\(857\) −31.6458 −1.08100 −0.540501 0.841344i \(-0.681765\pi\)
−0.540501 + 0.841344i \(0.681765\pi\)
\(858\) −3.05641 −0.104344
\(859\) 23.9937 0.818656 0.409328 0.912387i \(-0.365763\pi\)
0.409328 + 0.912387i \(0.365763\pi\)
\(860\) −1.69748 −0.0578836
\(861\) 1.32665 0.0452120
\(862\) 1.00000 0.0340601
\(863\) 38.5603 1.31261 0.656304 0.754496i \(-0.272119\pi\)
0.656304 + 0.754496i \(0.272119\pi\)
\(864\) −2.31335 −0.0787017
\(865\) −2.01951 −0.0686655
\(866\) −26.4582 −0.899085
\(867\) −6.33148 −0.215028
\(868\) 4.76935 0.161882
\(869\) 13.3345 0.452341
\(870\) 0.188622 0.00639489
\(871\) 24.0665 0.815463
\(872\) −17.7365 −0.600634
\(873\) 18.0136 0.609669
\(874\) −1.94750 −0.0658751
\(875\) 1.51328 0.0511582
\(876\) 0.660247 0.0223077
\(877\) −17.9125 −0.604864 −0.302432 0.953171i \(-0.597798\pi\)
−0.302432 + 0.953171i \(0.597798\pi\)
\(878\) 34.8667 1.17670
\(879\) 12.5415 0.423016
\(880\) 0.432324 0.0145737
\(881\) −39.4006 −1.32744 −0.663720 0.747981i \(-0.731023\pi\)
−0.663720 + 0.747981i \(0.731023\pi\)
\(882\) −2.84326 −0.0957376
\(883\) −7.44940 −0.250692 −0.125346 0.992113i \(-0.540004\pi\)
−0.125346 + 0.992113i \(0.540004\pi\)
\(884\) 2.71851 0.0914332
\(885\) −0.0324113 −0.00108949
\(886\) 21.3817 0.718333
\(887\) −56.6982 −1.90374 −0.951869 0.306505i \(-0.900840\pi\)
−0.951869 + 0.306505i \(0.900840\pi\)
\(888\) 0.921109 0.0309104
\(889\) 1.89718 0.0636294
\(890\) −0.162240 −0.00543831
\(891\) 21.7020 0.727044
\(892\) 21.9733 0.735722
\(893\) −21.5638 −0.721607
\(894\) 6.17595 0.206555
\(895\) 0.385909 0.0128995
\(896\) −1.00000 −0.0334077
\(897\) 0.890325 0.0297271
\(898\) −1.49850 −0.0500056
\(899\) −14.9812 −0.499652
\(900\) 14.1509 0.471697
\(901\) −0.905629 −0.0301709
\(902\) −9.55125 −0.318022
\(903\) 4.43069 0.147444
\(904\) −17.6543 −0.587172
\(905\) 0.662043 0.0220071
\(906\) −3.03167 −0.100720
\(907\) −14.2665 −0.473711 −0.236856 0.971545i \(-0.576117\pi\)
−0.236856 + 0.971545i \(0.576117\pi\)
\(908\) 28.8717 0.958140
\(909\) 7.51401 0.249224
\(910\) 0.410823 0.0136186
\(911\) 15.5405 0.514880 0.257440 0.966294i \(-0.417121\pi\)
0.257440 + 0.966294i \(0.417121\pi\)
\(912\) 0.928613 0.0307495
\(913\) 39.8801 1.31984
\(914\) 7.11181 0.235238
\(915\) −0.196975 −0.00651179
\(916\) −25.3652 −0.838089
\(917\) 18.4798 0.610258
\(918\) 2.32186 0.0766328
\(919\) 0.128723 0.00424617 0.00212308 0.999998i \(-0.499324\pi\)
0.00212308 + 0.999998i \(0.499324\pi\)
\(920\) −0.125935 −0.00415196
\(921\) −4.80597 −0.158362
\(922\) 5.48316 0.180578
\(923\) −26.2443 −0.863842
\(924\) −1.12843 −0.0371228
\(925\) −11.5796 −0.380734
\(926\) 12.5409 0.412119
\(927\) 20.9434 0.687871
\(928\) 3.14114 0.103113
\(929\) −50.4325 −1.65464 −0.827319 0.561732i \(-0.810135\pi\)
−0.827319 + 0.561732i \(0.810135\pi\)
\(930\) −0.286394 −0.00939124
\(931\) 2.34557 0.0768731
\(932\) −5.57673 −0.182672
\(933\) −0.578911 −0.0189527
\(934\) −12.2467 −0.400723
\(935\) −0.433915 −0.0141905
\(936\) 7.70109 0.251718
\(937\) −5.71367 −0.186657 −0.0933287 0.995635i \(-0.529751\pi\)
−0.0933287 + 0.995635i \(0.529751\pi\)
\(938\) 8.88542 0.290119
\(939\) −2.95087 −0.0962979
\(940\) −1.39443 −0.0454812
\(941\) 0.0891990 0.00290780 0.00145390 0.999999i \(-0.499537\pi\)
0.00145390 + 0.999999i \(0.499537\pi\)
\(942\) 1.52220 0.0495958
\(943\) 2.78226 0.0906027
\(944\) −0.539748 −0.0175673
\(945\) 0.350881 0.0114142
\(946\) −31.8990 −1.03712
\(947\) 29.5625 0.960652 0.480326 0.877090i \(-0.340518\pi\)
0.480326 + 0.877090i \(0.340518\pi\)
\(948\) 1.85213 0.0601543
\(949\) −4.51706 −0.146630
\(950\) −11.6739 −0.378752
\(951\) 3.53312 0.114569
\(952\) 1.00368 0.0325294
\(953\) 24.1278 0.781577 0.390788 0.920481i \(-0.372202\pi\)
0.390788 + 0.920481i \(0.372202\pi\)
\(954\) −2.56550 −0.0830612
\(955\) −3.07268 −0.0994295
\(956\) −29.2351 −0.945530
\(957\) 3.54458 0.114580
\(958\) 30.2923 0.978700
\(959\) −7.37889 −0.238277
\(960\) 0.0600489 0.00193807
\(961\) −8.25327 −0.266235
\(962\) −6.30174 −0.203176
\(963\) −7.33015 −0.236211
\(964\) 0.626558 0.0201801
\(965\) 3.85005 0.123937
\(966\) 0.328710 0.0105761
\(967\) −10.9786 −0.353048 −0.176524 0.984296i \(-0.556485\pi\)
−0.176524 + 0.984296i \(0.556485\pi\)
\(968\) −2.87579 −0.0924313
\(969\) −0.932030 −0.0299411
\(970\) −0.960955 −0.0308544
\(971\) −0.692288 −0.0222166 −0.0111083 0.999938i \(-0.503536\pi\)
−0.0111083 + 0.999938i \(0.503536\pi\)
\(972\) 9.95440 0.319288
\(973\) 12.3861 0.397080
\(974\) 28.9776 0.928501
\(975\) 5.33689 0.170917
\(976\) −3.28025 −0.104998
\(977\) 22.0437 0.705240 0.352620 0.935767i \(-0.385291\pi\)
0.352620 + 0.935767i \(0.385291\pi\)
\(978\) 8.67985 0.277551
\(979\) −3.04881 −0.0974403
\(980\) 0.151677 0.00484514
\(981\) 50.4296 1.61009
\(982\) −0.484813 −0.0154710
\(983\) 53.9526 1.72082 0.860410 0.509603i \(-0.170208\pi\)
0.860410 + 0.509603i \(0.170208\pi\)
\(984\) −1.32665 −0.0422919
\(985\) 0.847074 0.0269900
\(986\) −3.15270 −0.100402
\(987\) 3.63968 0.115852
\(988\) −6.35308 −0.202118
\(989\) 9.29209 0.295471
\(990\) −1.22921 −0.0390669
\(991\) 19.2823 0.612522 0.306261 0.951948i \(-0.400922\pi\)
0.306261 + 0.951948i \(0.400922\pi\)
\(992\) −4.76935 −0.151427
\(993\) 10.5143 0.333661
\(994\) −9.68947 −0.307331
\(995\) −2.96154 −0.0938872
\(996\) 5.53925 0.175518
\(997\) −17.2621 −0.546696 −0.273348 0.961915i \(-0.588131\pi\)
−0.273348 + 0.961915i \(0.588131\pi\)
\(998\) 9.87310 0.312527
\(999\) −5.38228 −0.170288
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.l.1.10 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6034.2.a.l.1.10 20 1.1 even 1 trivial