Properties

Label 273.2.c.b.64.5
Level $273$
Weight $2$
Character 273.64
Analytic conductor $2.180$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(64,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 64.5
Root \(0.403032 + 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 273.64
Dual form 273.2.c.b.64.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.48119i q^{2} +1.00000 q^{3} -0.193937 q^{4} -4.15633i q^{5} +1.48119i q^{6} -1.00000i q^{7} +2.67513i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.48119i q^{2} +1.00000 q^{3} -0.193937 q^{4} -4.15633i q^{5} +1.48119i q^{6} -1.00000i q^{7} +2.67513i q^{8} +1.00000 q^{9} +6.15633 q^{10} -3.19394i q^{11} -0.193937 q^{12} +(1.48119 + 3.28726i) q^{13} +1.48119 q^{14} -4.15633i q^{15} -4.35026 q^{16} +3.35026 q^{17} +1.48119i q^{18} +2.38787i q^{19} +0.806063i q^{20} -1.00000i q^{21} +4.73084 q^{22} +0.387873 q^{23} +2.67513i q^{24} -12.2750 q^{25} +(-4.86907 + 2.19394i) q^{26} +1.00000 q^{27} +0.193937i q^{28} -7.92478 q^{29} +6.15633 q^{30} +10.7005i q^{31} -1.09332i q^{32} -3.19394i q^{33} +4.96239i q^{34} -4.15633 q^{35} -0.193937 q^{36} -1.61213i q^{37} -3.53690 q^{38} +(1.48119 + 3.28726i) q^{39} +11.1187 q^{40} +1.45580i q^{41} +1.48119 q^{42} +1.92478 q^{43} +0.619421i q^{44} -4.15633i q^{45} +0.574515i q^{46} -3.76845i q^{47} -4.35026 q^{48} -1.00000 q^{49} -18.1817i q^{50} +3.35026 q^{51} +(-0.287258 - 0.637519i) q^{52} -6.00000 q^{53} +1.48119i q^{54} -13.2750 q^{55} +2.67513 q^{56} +2.38787i q^{57} -11.7381i q^{58} -6.15633i q^{59} +0.806063i q^{60} +14.4387 q^{61} -15.8496 q^{62} -1.00000i q^{63} -7.08110 q^{64} +(13.6629 - 6.15633i) q^{65} +4.73084 q^{66} +5.61213i q^{67} -0.649738 q^{68} +0.387873 q^{69} -6.15633i q^{70} +11.8192i q^{71} +2.67513i q^{72} -15.6629i q^{73} +2.38787 q^{74} -12.2750 q^{75} -0.463096i q^{76} -3.19394 q^{77} +(-4.86907 + 2.19394i) q^{78} -8.96239 q^{79} +18.0811i q^{80} +1.00000 q^{81} -2.15633 q^{82} +6.99271i q^{83} +0.193937i q^{84} -13.9248i q^{85} +2.85097i q^{86} -7.92478 q^{87} +8.54420 q^{88} +0.932071i q^{89} +6.15633 q^{90} +(3.28726 - 1.48119i) q^{91} -0.0752228 q^{92} +10.7005i q^{93} +5.58181 q^{94} +9.92478 q^{95} -1.09332i q^{96} -3.35026i q^{97} -1.48119i q^{98} -3.19394i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} - 2 q^{4} + 6 q^{9} + 16 q^{10} - 2 q^{12} - 2 q^{13} - 2 q^{14} - 6 q^{16} - 16 q^{22} + 4 q^{23} - 10 q^{25} - 20 q^{26} + 6 q^{27} - 4 q^{29} + 16 q^{30} - 4 q^{35} - 2 q^{36} + 24 q^{38} - 2 q^{39} + 24 q^{40} - 2 q^{42} - 32 q^{43} - 6 q^{48} - 6 q^{49} + 10 q^{52} - 36 q^{53} - 16 q^{55} + 6 q^{56} + 28 q^{61} - 8 q^{62} + 22 q^{64} + 20 q^{65} - 16 q^{66} - 24 q^{68} + 4 q^{69} + 16 q^{74} - 10 q^{75} - 20 q^{77} - 20 q^{78} - 32 q^{79} + 6 q^{81} + 8 q^{82} - 4 q^{87} + 32 q^{88} + 16 q^{90} + 8 q^{91} - 44 q^{92} + 36 q^{94} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.48119i 1.04736i 0.851914 + 0.523681i \(0.175442\pi\)
−0.851914 + 0.523681i \(0.824558\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.193937 −0.0969683
\(5\) 4.15633i 1.85877i −0.369118 0.929383i \(-0.620340\pi\)
0.369118 0.929383i \(-0.379660\pi\)
\(6\) 1.48119i 0.604695i
\(7\) 1.00000i 0.377964i
\(8\) 2.67513i 0.945802i
\(9\) 1.00000 0.333333
\(10\) 6.15633 1.94680
\(11\) 3.19394i 0.963008i −0.876444 0.481504i \(-0.840091\pi\)
0.876444 0.481504i \(-0.159909\pi\)
\(12\) −0.193937 −0.0559847
\(13\) 1.48119 + 3.28726i 0.410809 + 0.911721i
\(14\) 1.48119 0.395866
\(15\) 4.15633i 1.07316i
\(16\) −4.35026 −1.08757
\(17\) 3.35026 0.812558 0.406279 0.913749i \(-0.366826\pi\)
0.406279 + 0.913749i \(0.366826\pi\)
\(18\) 1.48119i 0.349121i
\(19\) 2.38787i 0.547816i 0.961756 + 0.273908i \(0.0883163\pi\)
−0.961756 + 0.273908i \(0.911684\pi\)
\(20\) 0.806063i 0.180241i
\(21\) 1.00000i 0.218218i
\(22\) 4.73084 1.00862
\(23\) 0.387873 0.0808771 0.0404386 0.999182i \(-0.487124\pi\)
0.0404386 + 0.999182i \(0.487124\pi\)
\(24\) 2.67513i 0.546059i
\(25\) −12.2750 −2.45501
\(26\) −4.86907 + 2.19394i −0.954903 + 0.430266i
\(27\) 1.00000 0.192450
\(28\) 0.193937i 0.0366506i
\(29\) −7.92478 −1.47159 −0.735797 0.677202i \(-0.763192\pi\)
−0.735797 + 0.677202i \(0.763192\pi\)
\(30\) 6.15633 1.12399
\(31\) 10.7005i 1.92187i 0.276773 + 0.960935i \(0.410735\pi\)
−0.276773 + 0.960935i \(0.589265\pi\)
\(32\) 1.09332i 0.193274i
\(33\) 3.19394i 0.555993i
\(34\) 4.96239i 0.851043i
\(35\) −4.15633 −0.702547
\(36\) −0.193937 −0.0323228
\(37\) 1.61213i 0.265032i −0.991181 0.132516i \(-0.957694\pi\)
0.991181 0.132516i \(-0.0423056\pi\)
\(38\) −3.53690 −0.573762
\(39\) 1.48119 + 3.28726i 0.237181 + 0.526383i
\(40\) 11.1187 1.75802
\(41\) 1.45580i 0.227358i 0.993518 + 0.113679i \(0.0362635\pi\)
−0.993518 + 0.113679i \(0.963736\pi\)
\(42\) 1.48119 0.228553
\(43\) 1.92478 0.293526 0.146763 0.989172i \(-0.453115\pi\)
0.146763 + 0.989172i \(0.453115\pi\)
\(44\) 0.619421i 0.0933812i
\(45\) 4.15633i 0.619588i
\(46\) 0.574515i 0.0847077i
\(47\) 3.76845i 0.549685i −0.961489 0.274843i \(-0.911374\pi\)
0.961489 0.274843i \(-0.0886258\pi\)
\(48\) −4.35026 −0.627906
\(49\) −1.00000 −0.142857
\(50\) 18.1817i 2.57128i
\(51\) 3.35026 0.469130
\(52\) −0.287258 0.637519i −0.0398355 0.0884080i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.48119i 0.201565i
\(55\) −13.2750 −1.79001
\(56\) 2.67513 0.357479
\(57\) 2.38787i 0.316282i
\(58\) 11.7381i 1.54129i
\(59\) 6.15633i 0.801485i −0.916191 0.400743i \(-0.868752\pi\)
0.916191 0.400743i \(-0.131248\pi\)
\(60\) 0.806063i 0.104062i
\(61\) 14.4387 1.84868 0.924340 0.381569i \(-0.124616\pi\)
0.924340 + 0.381569i \(0.124616\pi\)
\(62\) −15.8496 −2.01290
\(63\) 1.00000i 0.125988i
\(64\) −7.08110 −0.885138
\(65\) 13.6629 6.15633i 1.69468 0.763598i
\(66\) 4.73084 0.582326
\(67\) 5.61213i 0.685630i 0.939403 + 0.342815i \(0.111380\pi\)
−0.939403 + 0.342815i \(0.888620\pi\)
\(68\) −0.649738 −0.0787923
\(69\) 0.387873 0.0466944
\(70\) 6.15633i 0.735822i
\(71\) 11.8192i 1.40269i 0.712824 + 0.701343i \(0.247416\pi\)
−0.712824 + 0.701343i \(0.752584\pi\)
\(72\) 2.67513i 0.315267i
\(73\) 15.6629i 1.83321i −0.399800 0.916603i \(-0.630920\pi\)
0.399800 0.916603i \(-0.369080\pi\)
\(74\) 2.38787 0.277585
\(75\) −12.2750 −1.41740
\(76\) 0.463096i 0.0531207i
\(77\) −3.19394 −0.363983
\(78\) −4.86907 + 2.19394i −0.551313 + 0.248414i
\(79\) −8.96239 −1.00835 −0.504174 0.863602i \(-0.668203\pi\)
−0.504174 + 0.863602i \(0.668203\pi\)
\(80\) 18.0811i 2.02153i
\(81\) 1.00000 0.111111
\(82\) −2.15633 −0.238126
\(83\) 6.99271i 0.767549i 0.923427 + 0.383775i \(0.125376\pi\)
−0.923427 + 0.383775i \(0.874624\pi\)
\(84\) 0.193937i 0.0211602i
\(85\) 13.9248i 1.51035i
\(86\) 2.85097i 0.307428i
\(87\) −7.92478 −0.849625
\(88\) 8.54420 0.910815
\(89\) 0.932071i 0.0987994i 0.998779 + 0.0493997i \(0.0157308\pi\)
−0.998779 + 0.0493997i \(0.984269\pi\)
\(90\) 6.15633 0.648934
\(91\) 3.28726 1.48119i 0.344598 0.155271i
\(92\) −0.0752228 −0.00784252
\(93\) 10.7005i 1.10959i
\(94\) 5.58181 0.575720
\(95\) 9.92478 1.01826
\(96\) 1.09332i 0.111587i
\(97\) 3.35026i 0.340168i −0.985430 0.170084i \(-0.945596\pi\)
0.985430 0.170084i \(-0.0544038\pi\)
\(98\) 1.48119i 0.149623i
\(99\) 3.19394i 0.321003i
\(100\) 2.38058 0.238058
\(101\) −14.0508 −1.39811 −0.699053 0.715070i \(-0.746395\pi\)
−0.699053 + 0.715070i \(0.746395\pi\)
\(102\) 4.96239i 0.491350i
\(103\) 1.42548 0.140457 0.0702286 0.997531i \(-0.477627\pi\)
0.0702286 + 0.997531i \(0.477627\pi\)
\(104\) −8.79384 + 3.96239i −0.862307 + 0.388544i
\(105\) −4.15633 −0.405616
\(106\) 8.88717i 0.863198i
\(107\) 9.53690 0.921967 0.460984 0.887409i \(-0.347497\pi\)
0.460984 + 0.887409i \(0.347497\pi\)
\(108\) −0.193937 −0.0186616
\(109\) 6.23743i 0.597437i −0.954341 0.298719i \(-0.903441\pi\)
0.954341 0.298719i \(-0.0965592\pi\)
\(110\) 19.6629i 1.87479i
\(111\) 1.61213i 0.153016i
\(112\) 4.35026i 0.411061i
\(113\) −3.92478 −0.369212 −0.184606 0.982813i \(-0.559101\pi\)
−0.184606 + 0.982813i \(0.559101\pi\)
\(114\) −3.53690 −0.331261
\(115\) 1.61213i 0.150332i
\(116\) 1.53690 0.142698
\(117\) 1.48119 + 3.28726i 0.136936 + 0.303907i
\(118\) 9.11871 0.839446
\(119\) 3.35026i 0.307118i
\(120\) 11.1187 1.01500
\(121\) 0.798769 0.0726154
\(122\) 21.3865i 1.93624i
\(123\) 1.45580i 0.131265i
\(124\) 2.07522i 0.186361i
\(125\) 30.2374i 2.70452i
\(126\) 1.48119 0.131955
\(127\) −0.962389 −0.0853982 −0.0426991 0.999088i \(-0.513596\pi\)
−0.0426991 + 0.999088i \(0.513596\pi\)
\(128\) 12.6751i 1.12033i
\(129\) 1.92478 0.169467
\(130\) 9.11871 + 20.2374i 0.799764 + 1.77494i
\(131\) 17.1490 1.49832 0.749159 0.662390i \(-0.230458\pi\)
0.749159 + 0.662390i \(0.230458\pi\)
\(132\) 0.619421i 0.0539137i
\(133\) 2.38787 0.207055
\(134\) −8.31265 −0.718104
\(135\) 4.15633i 0.357720i
\(136\) 8.96239i 0.768518i
\(137\) 4.85685i 0.414949i 0.978240 + 0.207474i \(0.0665243\pi\)
−0.978240 + 0.207474i \(0.933476\pi\)
\(138\) 0.574515i 0.0489060i
\(139\) 2.57452 0.218368 0.109184 0.994022i \(-0.465176\pi\)
0.109184 + 0.994022i \(0.465176\pi\)
\(140\) 0.806063 0.0681248
\(141\) 3.76845i 0.317361i
\(142\) −17.5066 −1.46912
\(143\) 10.4993 4.73084i 0.877995 0.395613i
\(144\) −4.35026 −0.362522
\(145\) 32.9380i 2.73535i
\(146\) 23.1998 1.92003
\(147\) −1.00000 −0.0824786
\(148\) 0.312650i 0.0256997i
\(149\) 13.3806i 1.09618i −0.836419 0.548090i \(-0.815355\pi\)
0.836419 0.548090i \(-0.184645\pi\)
\(150\) 18.1817i 1.48453i
\(151\) 15.8496i 1.28982i 0.764259 + 0.644909i \(0.223105\pi\)
−0.764259 + 0.644909i \(0.776895\pi\)
\(152\) −6.38787 −0.518125
\(153\) 3.35026 0.270853
\(154\) 4.73084i 0.381222i
\(155\) 44.4749 3.57231
\(156\) −0.287258 0.637519i −0.0229990 0.0510424i
\(157\) −12.7005 −1.01361 −0.506806 0.862060i \(-0.669174\pi\)
−0.506806 + 0.862060i \(0.669174\pi\)
\(158\) 13.2750i 1.05611i
\(159\) −6.00000 −0.475831
\(160\) −4.54420 −0.359250
\(161\) 0.387873i 0.0305687i
\(162\) 1.48119i 0.116374i
\(163\) 9.61213i 0.752880i −0.926441 0.376440i \(-0.877148\pi\)
0.926441 0.376440i \(-0.122852\pi\)
\(164\) 0.282333i 0.0220465i
\(165\) −13.2750 −1.03346
\(166\) −10.3576 −0.803902
\(167\) 12.0811i 0.934864i 0.884029 + 0.467432i \(0.154821\pi\)
−0.884029 + 0.467432i \(0.845179\pi\)
\(168\) 2.67513 0.206391
\(169\) −8.61213 + 9.73813i −0.662471 + 0.749087i
\(170\) 20.6253 1.58189
\(171\) 2.38787i 0.182605i
\(172\) −0.373285 −0.0284627
\(173\) −9.27504 −0.705168 −0.352584 0.935780i \(-0.614697\pi\)
−0.352584 + 0.935780i \(0.614697\pi\)
\(174\) 11.7381i 0.889866i
\(175\) 12.2750i 0.927906i
\(176\) 13.8945i 1.04733i
\(177\) 6.15633i 0.462738i
\(178\) −1.38058 −0.103479
\(179\) −5.01317 −0.374702 −0.187351 0.982293i \(-0.559990\pi\)
−0.187351 + 0.982293i \(0.559990\pi\)
\(180\) 0.806063i 0.0600804i
\(181\) −13.8496 −1.02943 −0.514715 0.857362i \(-0.672102\pi\)
−0.514715 + 0.857362i \(0.672102\pi\)
\(182\) 2.19394 + 4.86907i 0.162625 + 0.360919i
\(183\) 14.4387 1.06734
\(184\) 1.03761i 0.0764937i
\(185\) −6.70052 −0.492632
\(186\) −15.8496 −1.16215
\(187\) 10.7005i 0.782500i
\(188\) 0.730841i 0.0533020i
\(189\) 1.00000i 0.0727393i
\(190\) 14.7005i 1.06649i
\(191\) −18.5647 −1.34329 −0.671646 0.740872i \(-0.734412\pi\)
−0.671646 + 0.740872i \(0.734412\pi\)
\(192\) −7.08110 −0.511035
\(193\) 13.0884i 0.942123i −0.882100 0.471062i \(-0.843871\pi\)
0.882100 0.471062i \(-0.156129\pi\)
\(194\) 4.96239 0.356279
\(195\) 13.6629 6.15633i 0.978421 0.440864i
\(196\) 0.193937 0.0138526
\(197\) 7.24472i 0.516165i −0.966123 0.258083i \(-0.916909\pi\)
0.966123 0.258083i \(-0.0830906\pi\)
\(198\) 4.73084 0.336206
\(199\) 8.62530 0.611431 0.305716 0.952123i \(-0.401104\pi\)
0.305716 + 0.952123i \(0.401104\pi\)
\(200\) 32.8373i 2.32195i
\(201\) 5.61213i 0.395849i
\(202\) 20.8119i 1.46432i
\(203\) 7.92478i 0.556210i
\(204\) −0.649738 −0.0454908
\(205\) 6.05079 0.422605
\(206\) 2.11142i 0.147110i
\(207\) 0.387873 0.0269590
\(208\) −6.44358 14.3004i −0.446782 0.991557i
\(209\) 7.62672 0.527551
\(210\) 6.15633i 0.424827i
\(211\) 18.8872 1.30025 0.650123 0.759829i \(-0.274717\pi\)
0.650123 + 0.759829i \(0.274717\pi\)
\(212\) 1.16362 0.0799177
\(213\) 11.8192i 0.809841i
\(214\) 14.1260i 0.965634i
\(215\) 8.00000i 0.545595i
\(216\) 2.67513i 0.182020i
\(217\) 10.7005 0.726399
\(218\) 9.23884 0.625733
\(219\) 15.6629i 1.05840i
\(220\) 2.57452 0.173574
\(221\) 4.96239 + 11.0132i 0.333806 + 0.740826i
\(222\) 2.38787 0.160264
\(223\) 16.1622i 1.08230i −0.840926 0.541151i \(-0.817989\pi\)
0.840926 0.541151i \(-0.182011\pi\)
\(224\) −1.09332 −0.0730506
\(225\) −12.2750 −0.818336
\(226\) 5.81336i 0.386699i
\(227\) 27.2447i 1.80830i −0.427220 0.904148i \(-0.640507\pi\)
0.427220 0.904148i \(-0.359493\pi\)
\(228\) 0.463096i 0.0306693i
\(229\) 19.5125i 1.28942i −0.764427 0.644710i \(-0.776978\pi\)
0.764427 0.644710i \(-0.223022\pi\)
\(230\) 2.38787 0.157452
\(231\) −3.19394 −0.210146
\(232\) 21.1998i 1.39184i
\(233\) −21.8496 −1.43141 −0.715706 0.698402i \(-0.753895\pi\)
−0.715706 + 0.698402i \(0.753895\pi\)
\(234\) −4.86907 + 2.19394i −0.318301 + 0.143422i
\(235\) −15.6629 −1.02174
\(236\) 1.19394i 0.0777187i
\(237\) −8.96239 −0.582170
\(238\) 4.96239 0.321664
\(239\) 8.49341i 0.549393i 0.961531 + 0.274697i \(0.0885774\pi\)
−0.961531 + 0.274697i \(0.911423\pi\)
\(240\) 18.0811i 1.16713i
\(241\) 5.48612i 0.353392i −0.984265 0.176696i \(-0.943459\pi\)
0.984265 0.176696i \(-0.0565409\pi\)
\(242\) 1.18313i 0.0760546i
\(243\) 1.00000 0.0641500
\(244\) −2.80018 −0.179263
\(245\) 4.15633i 0.265538i
\(246\) −2.15633 −0.137482
\(247\) −7.84955 + 3.53690i −0.499455 + 0.225048i
\(248\) −28.6253 −1.81771
\(249\) 6.99271i 0.443145i
\(250\) −44.7875 −2.83261
\(251\) 6.44851 0.407026 0.203513 0.979072i \(-0.434764\pi\)
0.203513 + 0.979072i \(0.434764\pi\)
\(252\) 0.193937i 0.0122169i
\(253\) 1.23884i 0.0778853i
\(254\) 1.42548i 0.0894429i
\(255\) 13.9248i 0.872003i
\(256\) 4.61213 0.288258
\(257\) 9.90034 0.617566 0.308783 0.951132i \(-0.400078\pi\)
0.308783 + 0.951132i \(0.400078\pi\)
\(258\) 2.85097i 0.177494i
\(259\) −1.61213 −0.100173
\(260\) −2.64974 + 1.19394i −0.164330 + 0.0740448i
\(261\) −7.92478 −0.490531
\(262\) 25.4010i 1.56928i
\(263\) 3.46168 0.213456 0.106728 0.994288i \(-0.465963\pi\)
0.106728 + 0.994288i \(0.465963\pi\)
\(264\) 8.54420 0.525859
\(265\) 24.9380i 1.53193i
\(266\) 3.53690i 0.216862i
\(267\) 0.932071i 0.0570418i
\(268\) 1.08840i 0.0664844i
\(269\) 16.7513 1.02135 0.510673 0.859775i \(-0.329396\pi\)
0.510673 + 0.859775i \(0.329396\pi\)
\(270\) 6.15633 0.374662
\(271\) 6.38787i 0.388036i 0.980998 + 0.194018i \(0.0621520\pi\)
−0.980998 + 0.194018i \(0.937848\pi\)
\(272\) −14.5745 −0.883710
\(273\) 3.28726 1.48119i 0.198954 0.0896460i
\(274\) −7.19394 −0.434602
\(275\) 39.2057i 2.36419i
\(276\) −0.0752228 −0.00452788
\(277\) 12.1768 0.731633 0.365816 0.930687i \(-0.380790\pi\)
0.365816 + 0.930687i \(0.380790\pi\)
\(278\) 3.81336i 0.228710i
\(279\) 10.7005i 0.640624i
\(280\) 11.1187i 0.664470i
\(281\) 27.3054i 1.62890i 0.580233 + 0.814450i \(0.302961\pi\)
−0.580233 + 0.814450i \(0.697039\pi\)
\(282\) 5.58181 0.332392
\(283\) 27.0738 1.60937 0.804685 0.593701i \(-0.202334\pi\)
0.804685 + 0.593701i \(0.202334\pi\)
\(284\) 2.29218i 0.136016i
\(285\) 9.92478 0.587893
\(286\) 7.00729 + 15.5515i 0.414350 + 0.919579i
\(287\) 1.45580 0.0859333
\(288\) 1.09332i 0.0644246i
\(289\) −5.77575 −0.339750
\(290\) −48.7875 −2.86490
\(291\) 3.35026i 0.196396i
\(292\) 3.03761i 0.177763i
\(293\) 16.7816i 0.980393i −0.871612 0.490197i \(-0.836925\pi\)
0.871612 0.490197i \(-0.163075\pi\)
\(294\) 1.48119i 0.0863850i
\(295\) −25.5877 −1.48977
\(296\) 4.31265 0.250668
\(297\) 3.19394i 0.185331i
\(298\) 19.8192 1.14810
\(299\) 0.574515 + 1.27504i 0.0332251 + 0.0737374i
\(300\) 2.38058 0.137443
\(301\) 1.92478i 0.110942i
\(302\) −23.4763 −1.35091
\(303\) −14.0508 −0.807197
\(304\) 10.3879i 0.595785i
\(305\) 60.0118i 3.43626i
\(306\) 4.96239i 0.283681i
\(307\) 17.2995i 0.987333i −0.869651 0.493667i \(-0.835656\pi\)
0.869651 0.493667i \(-0.164344\pi\)
\(308\) 0.619421 0.0352948
\(309\) 1.42548 0.0810930
\(310\) 65.8759i 3.74150i
\(311\) 17.1490 0.972432 0.486216 0.873839i \(-0.338377\pi\)
0.486216 + 0.873839i \(0.338377\pi\)
\(312\) −8.79384 + 3.96239i −0.497853 + 0.224326i
\(313\) −10.8119 −0.611127 −0.305564 0.952172i \(-0.598845\pi\)
−0.305564 + 0.952172i \(0.598845\pi\)
\(314\) 18.8119i 1.06162i
\(315\) −4.15633 −0.234182
\(316\) 1.73813 0.0977777
\(317\) 14.1563i 0.795098i 0.917581 + 0.397549i \(0.130139\pi\)
−0.917581 + 0.397549i \(0.869861\pi\)
\(318\) 8.88717i 0.498368i
\(319\) 25.3112i 1.41716i
\(320\) 29.4314i 1.64526i
\(321\) 9.53690 0.532298
\(322\) 0.574515 0.0320165
\(323\) 8.00000i 0.445132i
\(324\) −0.193937 −0.0107743
\(325\) −18.1817 40.3512i −1.00854 2.23828i
\(326\) 14.2374 0.788538
\(327\) 6.23743i 0.344931i
\(328\) −3.89446 −0.215036
\(329\) −3.76845 −0.207761
\(330\) 19.6629i 1.08241i
\(331\) 17.2506i 0.948179i −0.880477 0.474089i \(-0.842777\pi\)
0.880477 0.474089i \(-0.157223\pi\)
\(332\) 1.35614i 0.0744279i
\(333\) 1.61213i 0.0883440i
\(334\) −17.8945 −0.979141
\(335\) 23.3258 1.27443
\(336\) 4.35026i 0.237326i
\(337\) −5.97556 −0.325510 −0.162755 0.986667i \(-0.552038\pi\)
−0.162755 + 0.986667i \(0.552038\pi\)
\(338\) −14.4241 12.7562i −0.784566 0.693848i
\(339\) −3.92478 −0.213165
\(340\) 2.70052i 0.146456i
\(341\) 34.1768 1.85078
\(342\) −3.53690 −0.191254
\(343\) 1.00000i 0.0539949i
\(344\) 5.14903i 0.277617i
\(345\) 1.61213i 0.0867940i
\(346\) 13.7381i 0.738567i
\(347\) 25.7889 1.38442 0.692211 0.721695i \(-0.256637\pi\)
0.692211 + 0.721695i \(0.256637\pi\)
\(348\) 1.53690 0.0823867
\(349\) 12.9624i 0.693861i −0.937891 0.346930i \(-0.887224\pi\)
0.937891 0.346930i \(-0.112776\pi\)
\(350\) −18.1817 −0.971854
\(351\) 1.48119 + 3.28726i 0.0790603 + 0.175461i
\(352\) −3.49200 −0.186124
\(353\) 3.78304i 0.201351i 0.994919 + 0.100675i \(0.0321004\pi\)
−0.994919 + 0.100675i \(0.967900\pi\)
\(354\) 9.11871 0.484654
\(355\) 49.1246 2.60726
\(356\) 0.180763i 0.00958041i
\(357\) 3.35026i 0.177315i
\(358\) 7.42548i 0.392449i
\(359\) 10.6702i 0.563152i −0.959539 0.281576i \(-0.909143\pi\)
0.959539 0.281576i \(-0.0908571\pi\)
\(360\) 11.1187 0.586008
\(361\) 13.2981 0.699898
\(362\) 20.5139i 1.07819i
\(363\) 0.798769 0.0419245
\(364\) −0.637519 + 0.287258i −0.0334151 + 0.0150564i
\(365\) −65.1002 −3.40750
\(366\) 21.3865i 1.11789i
\(367\) 6.82653 0.356342 0.178171 0.984000i \(-0.442982\pi\)
0.178171 + 0.984000i \(0.442982\pi\)
\(368\) −1.68735 −0.0879592
\(369\) 1.45580i 0.0757860i
\(370\) 9.92478i 0.515965i
\(371\) 6.00000i 0.311504i
\(372\) 2.07522i 0.107595i
\(373\) 16.5745 0.858196 0.429098 0.903258i \(-0.358832\pi\)
0.429098 + 0.903258i \(0.358832\pi\)
\(374\) 15.8496 0.819561
\(375\) 30.2374i 1.56145i
\(376\) 10.0811 0.519893
\(377\) −11.7381 26.0508i −0.604545 1.34168i
\(378\) 1.48119 0.0761844
\(379\) 19.0738i 0.979756i 0.871791 + 0.489878i \(0.162959\pi\)
−0.871791 + 0.489878i \(0.837041\pi\)
\(380\) −1.92478 −0.0987390
\(381\) −0.962389 −0.0493047
\(382\) 27.4979i 1.40691i
\(383\) 8.29218i 0.423711i −0.977301 0.211855i \(-0.932049\pi\)
0.977301 0.211855i \(-0.0679506\pi\)
\(384\) 12.6751i 0.646825i
\(385\) 13.2750i 0.676559i
\(386\) 19.3865 0.986745
\(387\) 1.92478 0.0978419
\(388\) 0.649738i 0.0329855i
\(389\) 0.700523 0.0355180 0.0177590 0.999842i \(-0.494347\pi\)
0.0177590 + 0.999842i \(0.494347\pi\)
\(390\) 9.11871 + 20.2374i 0.461744 + 1.02476i
\(391\) 1.29948 0.0657174
\(392\) 2.67513i 0.135115i
\(393\) 17.1490 0.865054
\(394\) 10.7308 0.540612
\(395\) 37.2506i 1.87428i
\(396\) 0.619421i 0.0311271i
\(397\) 12.4993i 0.627322i 0.949535 + 0.313661i \(0.101555\pi\)
−0.949535 + 0.313661i \(0.898445\pi\)
\(398\) 12.7757i 0.640390i
\(399\) 2.38787 0.119543
\(400\) 53.3996 2.66998
\(401\) 28.7064i 1.43353i −0.697315 0.716765i \(-0.745622\pi\)
0.697315 0.716765i \(-0.254378\pi\)
\(402\) −8.31265 −0.414597
\(403\) −35.1754 + 15.8496i −1.75221 + 0.789523i
\(404\) 2.72496 0.135572
\(405\) 4.15633i 0.206529i
\(406\) −11.7381 −0.582554
\(407\) −5.14903 −0.255228
\(408\) 8.96239i 0.443704i
\(409\) 23.4518i 1.15962i 0.814752 + 0.579809i \(0.196873\pi\)
−0.814752 + 0.579809i \(0.803127\pi\)
\(410\) 8.96239i 0.442621i
\(411\) 4.85685i 0.239571i
\(412\) −0.276454 −0.0136199
\(413\) −6.15633 −0.302933
\(414\) 0.574515i 0.0282359i
\(415\) 29.0640 1.42669
\(416\) 3.59403 1.61942i 0.176212 0.0793987i
\(417\) 2.57452 0.126075
\(418\) 11.2966i 0.552537i
\(419\) −34.1768 −1.66965 −0.834823 0.550519i \(-0.814430\pi\)
−0.834823 + 0.550519i \(0.814430\pi\)
\(420\) 0.806063 0.0393319
\(421\) 5.55149i 0.270563i −0.990807 0.135282i \(-0.956806\pi\)
0.990807 0.135282i \(-0.0431939\pi\)
\(422\) 27.9756i 1.36183i
\(423\) 3.76845i 0.183228i
\(424\) 16.0508i 0.779495i
\(425\) −41.1246 −1.99484
\(426\) −17.5066 −0.848197
\(427\) 14.4387i 0.698736i
\(428\) −1.84955 −0.0894016
\(429\) 10.4993 4.73084i 0.506911 0.228407i
\(430\) 11.8496 0.571436
\(431\) 5.43136i 0.261620i 0.991407 + 0.130810i \(0.0417577\pi\)
−0.991407 + 0.130810i \(0.958242\pi\)
\(432\) −4.35026 −0.209302
\(433\) 40.2130 1.93251 0.966256 0.257582i \(-0.0829257\pi\)
0.966256 + 0.257582i \(0.0829257\pi\)
\(434\) 15.8496i 0.760803i
\(435\) 32.9380i 1.57925i
\(436\) 1.20967i 0.0579325i
\(437\) 0.926192i 0.0443058i
\(438\) 23.1998 1.10853
\(439\) −22.5745 −1.07742 −0.538711 0.842490i \(-0.681089\pi\)
−0.538711 + 0.842490i \(0.681089\pi\)
\(440\) 35.5125i 1.69299i
\(441\) −1.00000 −0.0476190
\(442\) −16.3127 + 7.35026i −0.775914 + 0.349616i
\(443\) −8.23743 −0.391372 −0.195686 0.980667i \(-0.562693\pi\)
−0.195686 + 0.980667i \(0.562693\pi\)
\(444\) 0.312650i 0.0148377i
\(445\) 3.87399 0.183645
\(446\) 23.9394 1.13356
\(447\) 13.3806i 0.632880i
\(448\) 7.08110i 0.334551i
\(449\) 2.40834i 0.113657i −0.998384 0.0568283i \(-0.981901\pi\)
0.998384 0.0568283i \(-0.0180988\pi\)
\(450\) 18.1817i 0.857094i
\(451\) 4.64974 0.218948
\(452\) 0.761158 0.0358019
\(453\) 15.8496i 0.744677i
\(454\) 40.3547 1.89394
\(455\) −6.15633 13.6629i −0.288613 0.640527i
\(456\) −6.38787 −0.299140
\(457\) 29.5633i 1.38291i −0.722419 0.691455i \(-0.756970\pi\)
0.722419 0.691455i \(-0.243030\pi\)
\(458\) 28.9018 1.35049
\(459\) 3.35026 0.156377
\(460\) 0.312650i 0.0145774i
\(461\) 33.7196i 1.57048i 0.619193 + 0.785239i \(0.287460\pi\)
−0.619193 + 0.785239i \(0.712540\pi\)
\(462\) 4.73084i 0.220099i
\(463\) 16.8364i 0.782453i −0.920294 0.391226i \(-0.872051\pi\)
0.920294 0.391226i \(-0.127949\pi\)
\(464\) 34.4749 1.60045
\(465\) 44.4749 2.06247
\(466\) 32.3634i 1.49921i
\(467\) −22.7005 −1.05045 −0.525227 0.850962i \(-0.676020\pi\)
−0.525227 + 0.850962i \(0.676020\pi\)
\(468\) −0.287258 0.637519i −0.0132785 0.0294693i
\(469\) 5.61213 0.259144
\(470\) 23.1998i 1.07013i
\(471\) −12.7005 −0.585209
\(472\) 16.4690 0.758046
\(473\) 6.14762i 0.282668i
\(474\) 13.2750i 0.609743i
\(475\) 29.3112i 1.34489i
\(476\) 0.649738i 0.0297807i
\(477\) −6.00000 −0.274721
\(478\) −12.5804 −0.575414
\(479\) 10.0957i 0.461284i 0.973039 + 0.230642i \(0.0740826\pi\)
−0.973039 + 0.230642i \(0.925917\pi\)
\(480\) −4.54420 −0.207413
\(481\) 5.29948 2.38787i 0.241635 0.108878i
\(482\) 8.12601 0.370130
\(483\) 0.387873i 0.0176488i
\(484\) −0.154911 −0.00704139
\(485\) −13.9248 −0.632292
\(486\) 1.48119i 0.0671883i
\(487\) 9.40105i 0.426002i −0.977052 0.213001i \(-0.931676\pi\)
0.977052 0.213001i \(-0.0683238\pi\)
\(488\) 38.6253i 1.74849i
\(489\) 9.61213i 0.434675i
\(490\) −6.15633 −0.278114
\(491\) 20.7612 0.936938 0.468469 0.883480i \(-0.344806\pi\)
0.468469 + 0.883480i \(0.344806\pi\)
\(492\) 0.282333i 0.0127286i
\(493\) −26.5501 −1.19576
\(494\) −5.23884 11.6267i −0.235707 0.523111i
\(495\) −13.2750 −0.596669
\(496\) 46.5501i 2.09016i
\(497\) 11.8192 0.530165
\(498\) −10.3576 −0.464133
\(499\) 22.9525i 1.02750i 0.857941 + 0.513748i \(0.171744\pi\)
−0.857941 + 0.513748i \(0.828256\pi\)
\(500\) 5.86414i 0.262252i
\(501\) 12.0811i 0.539744i
\(502\) 9.55149i 0.426304i
\(503\) −34.9234 −1.55716 −0.778578 0.627548i \(-0.784059\pi\)
−0.778578 + 0.627548i \(0.784059\pi\)
\(504\) 2.67513 0.119160
\(505\) 58.3996i 2.59875i
\(506\) 1.83497 0.0815742
\(507\) −8.61213 + 9.73813i −0.382478 + 0.432486i
\(508\) 0.186642 0.00828091
\(509\) 19.2301i 0.852361i −0.904638 0.426180i \(-0.859859\pi\)
0.904638 0.426180i \(-0.140141\pi\)
\(510\) 20.6253 0.913304
\(511\) −15.6629 −0.692886
\(512\) 18.5188i 0.818423i
\(513\) 2.38787i 0.105427i
\(514\) 14.6643i 0.646816i
\(515\) 5.92478i 0.261077i
\(516\) −0.373285 −0.0164329
\(517\) −12.0362 −0.529351
\(518\) 2.38787i 0.104917i
\(519\) −9.27504 −0.407129
\(520\) 16.4690 + 36.5501i 0.722212 + 1.60283i
\(521\) 21.5271 0.943117 0.471559 0.881835i \(-0.343692\pi\)
0.471559 + 0.881835i \(0.343692\pi\)
\(522\) 11.7381i 0.513764i
\(523\) 39.6747 1.73485 0.867426 0.497566i \(-0.165773\pi\)
0.867426 + 0.497566i \(0.165773\pi\)
\(524\) −3.32582 −0.145289
\(525\) 12.2750i 0.535727i
\(526\) 5.12742i 0.223566i
\(527\) 35.8496i 1.56163i
\(528\) 13.8945i 0.604679i
\(529\) −22.8496 −0.993459
\(530\) −36.9380 −1.60448
\(531\) 6.15633i 0.267162i
\(532\) −0.463096 −0.0200778
\(533\) −4.78560 + 2.15633i −0.207287 + 0.0934008i
\(534\) −1.38058 −0.0597435
\(535\) 39.6385i 1.71372i
\(536\) −15.0132 −0.648470
\(537\) −5.01317 −0.216334
\(538\) 24.8119i 1.06972i
\(539\) 3.19394i 0.137573i
\(540\) 0.806063i 0.0346874i
\(541\) 10.0263i 0.431066i 0.976497 + 0.215533i \(0.0691489\pi\)
−0.976497 + 0.215533i \(0.930851\pi\)
\(542\) −9.46168 −0.406414
\(543\) −13.8496 −0.594341
\(544\) 3.66291i 0.157046i
\(545\) −25.9248 −1.11050
\(546\) 2.19394 + 4.86907i 0.0938918 + 0.208377i
\(547\) −26.4485 −1.13086 −0.565428 0.824797i \(-0.691289\pi\)
−0.565428 + 0.824797i \(0.691289\pi\)
\(548\) 0.941921i 0.0402369i
\(549\) 14.4387 0.616227
\(550\) −58.0713 −2.47617
\(551\) 18.9234i 0.806162i
\(552\) 1.03761i 0.0441637i
\(553\) 8.96239i 0.381120i
\(554\) 18.0362i 0.766285i
\(555\) −6.70052 −0.284421
\(556\) −0.499293 −0.0211747
\(557\) 5.84367i 0.247604i −0.992307 0.123802i \(-0.960491\pi\)
0.992307 0.123802i \(-0.0395088\pi\)
\(558\) −15.8496 −0.670965
\(559\) 2.85097 + 6.32724i 0.120583 + 0.267614i
\(560\) 18.0811 0.764066
\(561\) 10.7005i 0.451776i
\(562\) −40.4445 −1.70605
\(563\) 1.25060 0.0527066 0.0263533 0.999653i \(-0.491611\pi\)
0.0263533 + 0.999653i \(0.491611\pi\)
\(564\) 0.730841i 0.0307739i
\(565\) 16.3127i 0.686278i
\(566\) 40.1016i 1.68559i
\(567\) 1.00000i 0.0419961i
\(568\) −31.6180 −1.32666
\(569\) −38.2228 −1.60238 −0.801192 0.598407i \(-0.795801\pi\)
−0.801192 + 0.598407i \(0.795801\pi\)
\(570\) 14.7005i 0.615737i
\(571\) −9.73813 −0.407528 −0.203764 0.979020i \(-0.565318\pi\)
−0.203764 + 0.979020i \(0.565318\pi\)
\(572\) −2.03620 + 0.917483i −0.0851377 + 0.0383619i
\(573\) −18.5647 −0.775550
\(574\) 2.15633i 0.0900033i
\(575\) −4.76116 −0.198554
\(576\) −7.08110 −0.295046
\(577\) 3.87399i 0.161276i 0.996743 + 0.0806382i \(0.0256958\pi\)
−0.996743 + 0.0806382i \(0.974304\pi\)
\(578\) 8.55500i 0.355841i
\(579\) 13.0884i 0.543935i
\(580\) 6.38787i 0.265242i
\(581\) 6.99271 0.290106
\(582\) 4.96239 0.205698
\(583\) 19.1636i 0.793676i
\(584\) 41.9003 1.73385
\(585\) 13.6629 6.15633i 0.564892 0.254533i
\(586\) 24.8568 1.02683
\(587\) 11.2447i 0.464119i 0.972702 + 0.232060i \(0.0745465\pi\)
−0.972702 + 0.232060i \(0.925454\pi\)
\(588\) 0.193937 0.00799781
\(589\) −25.5515 −1.05283
\(590\) 37.9003i 1.56033i
\(591\) 7.24472i 0.298008i
\(592\) 7.01317i 0.288240i
\(593\) 28.7210i 1.17943i −0.807612 0.589715i \(-0.799240\pi\)
0.807612 0.589715i \(-0.200760\pi\)
\(594\) 4.73084 0.194109
\(595\) −13.9248 −0.570860
\(596\) 2.59498i 0.106295i
\(597\) 8.62530 0.353010
\(598\) −1.88858 + 0.850969i −0.0772298 + 0.0347987i
\(599\) −21.1344 −0.863530 −0.431765 0.901986i \(-0.642109\pi\)
−0.431765 + 0.901986i \(0.642109\pi\)
\(600\) 32.8373i 1.34058i
\(601\) 18.9135 0.771498 0.385749 0.922604i \(-0.373943\pi\)
0.385749 + 0.922604i \(0.373943\pi\)
\(602\) 2.85097 0.116197
\(603\) 5.61213i 0.228543i
\(604\) 3.07381i 0.125071i
\(605\) 3.31994i 0.134975i
\(606\) 20.8119i 0.845427i
\(607\) −41.1246 −1.66920 −0.834598 0.550860i \(-0.814300\pi\)
−0.834598 + 0.550860i \(0.814300\pi\)
\(608\) 2.61071 0.105878
\(609\) 7.92478i 0.321128i
\(610\) 88.8891 3.59901
\(611\) 12.3879 5.58181i 0.501160 0.225816i
\(612\) −0.649738 −0.0262641
\(613\) 23.3258i 0.942121i 0.882101 + 0.471061i \(0.156129\pi\)
−0.882101 + 0.471061i \(0.843871\pi\)
\(614\) 25.6239 1.03410
\(615\) 6.05079 0.243991
\(616\) 8.54420i 0.344256i
\(617\) 17.7830i 0.715918i −0.933737 0.357959i \(-0.883473\pi\)
0.933737 0.357959i \(-0.116527\pi\)
\(618\) 2.11142i 0.0849338i
\(619\) 8.93795i 0.359247i 0.983735 + 0.179623i \(0.0574879\pi\)
−0.983735 + 0.179623i \(0.942512\pi\)
\(620\) −8.62530 −0.346400
\(621\) 0.387873 0.0155648
\(622\) 25.4010i 1.01849i
\(623\) 0.932071 0.0373427
\(624\) −6.44358 14.3004i −0.257950 0.572475i
\(625\) 64.3014 2.57206
\(626\) 16.0146i 0.640072i
\(627\) 7.62672 0.304582
\(628\) 2.46310 0.0982882
\(629\) 5.40105i 0.215354i
\(630\) 6.15633i 0.245274i
\(631\) 32.2638i 1.28440i −0.766537 0.642200i \(-0.778022\pi\)
0.766537 0.642200i \(-0.221978\pi\)
\(632\) 23.9756i 0.953697i
\(633\) 18.8872 0.750697
\(634\) −20.9683 −0.832756
\(635\) 4.00000i 0.158735i
\(636\) 1.16362 0.0461405
\(637\) −1.48119 3.28726i −0.0586871 0.130246i
\(638\) −37.4909 −1.48428
\(639\) 11.8192i 0.467562i
\(640\) −52.6820 −2.08244
\(641\) −49.4274 −1.95226 −0.976132 0.217176i \(-0.930315\pi\)
−0.976132 + 0.217176i \(0.930315\pi\)
\(642\) 14.1260i 0.557509i
\(643\) 6.07522i 0.239583i 0.992799 + 0.119792i \(0.0382227\pi\)
−0.992799 + 0.119792i \(0.961777\pi\)
\(644\) 0.0752228i 0.00296419i
\(645\) 8.00000i 0.315000i
\(646\) −11.8496 −0.466214
\(647\) −12.9986 −0.511027 −0.255514 0.966805i \(-0.582245\pi\)
−0.255514 + 0.966805i \(0.582245\pi\)
\(648\) 2.67513i 0.105089i
\(649\) −19.6629 −0.771837
\(650\) 59.7680 26.9307i 2.34429 1.05631i
\(651\) 10.7005 0.419387
\(652\) 1.86414i 0.0730055i
\(653\) 15.4010 0.602690 0.301345 0.953515i \(-0.402564\pi\)
0.301345 + 0.953515i \(0.402564\pi\)
\(654\) 9.23884 0.361267
\(655\) 71.2769i 2.78502i
\(656\) 6.33312i 0.247267i
\(657\) 15.6629i 0.611068i
\(658\) 5.58181i 0.217602i
\(659\) 8.76116 0.341286 0.170643 0.985333i \(-0.445415\pi\)
0.170643 + 0.985333i \(0.445415\pi\)
\(660\) 2.57452 0.100213
\(661\) 33.4372i 1.30056i −0.759695 0.650279i \(-0.774652\pi\)
0.759695 0.650279i \(-0.225348\pi\)
\(662\) 25.5515 0.993087
\(663\) 4.96239 + 11.0132i 0.192723 + 0.427716i
\(664\) −18.7064 −0.725949
\(665\) 9.92478i 0.384866i
\(666\) 2.38787 0.0925282
\(667\) −3.07381 −0.119018
\(668\) 2.34297i 0.0906521i
\(669\) 16.1622i 0.624867i
\(670\) 34.5501i 1.33479i
\(671\) 46.1162i 1.78029i
\(672\) −1.09332 −0.0421758
\(673\) −0.176793 −0.00681488 −0.00340744 0.999994i \(-0.501085\pi\)
−0.00340744 + 0.999994i \(0.501085\pi\)
\(674\) 8.85097i 0.340927i
\(675\) −12.2750 −0.472466
\(676\) 1.67021 1.88858i 0.0642387 0.0726377i
\(677\) −5.67750 −0.218204 −0.109102 0.994031i \(-0.534798\pi\)
−0.109102 + 0.994031i \(0.534798\pi\)
\(678\) 5.81336i 0.223261i
\(679\) −3.35026 −0.128571
\(680\) 37.2506 1.42850
\(681\) 27.2447i 1.04402i
\(682\) 50.6225i 1.93843i
\(683\) 50.4299i 1.92965i 0.262896 + 0.964824i \(0.415322\pi\)
−0.262896 + 0.964824i \(0.584678\pi\)
\(684\) 0.463096i 0.0177069i
\(685\) 20.1866 0.771292
\(686\) −1.48119 −0.0565523
\(687\) 19.5125i 0.744447i
\(688\) −8.37328 −0.319228
\(689\) −8.88717 19.7235i −0.338574 0.751407i
\(690\) 2.38787 0.0909048
\(691\) 41.5633i 1.58114i 0.612371 + 0.790570i \(0.290216\pi\)
−0.612371 + 0.790570i \(0.709784\pi\)
\(692\) 1.79877 0.0683789
\(693\) −3.19394 −0.121328
\(694\) 38.1984i 1.44999i
\(695\) 10.7005i 0.405894i
\(696\) 21.1998i 0.803577i
\(697\) 4.87732i 0.184742i
\(698\) 19.1998 0.726724
\(699\) −21.8496 −0.826426
\(700\) 2.38058i 0.0899774i
\(701\) −2.47486 −0.0934740 −0.0467370 0.998907i \(-0.514882\pi\)
−0.0467370 + 0.998907i \(0.514882\pi\)
\(702\) −4.86907 + 2.19394i −0.183771 + 0.0828048i
\(703\) 3.84955 0.145189
\(704\) 22.6166i 0.852395i
\(705\) −15.6629 −0.589899
\(706\) −5.60342 −0.210887
\(707\) 14.0508i 0.528434i
\(708\) 1.19394i 0.0448709i
\(709\) 7.91019i 0.297073i −0.988907 0.148537i \(-0.952544\pi\)
0.988907 0.148537i \(-0.0474563\pi\)
\(710\) 72.7631i 2.73075i
\(711\) −8.96239 −0.336116
\(712\) −2.49341 −0.0934446
\(713\) 4.15045i 0.155435i
\(714\) 4.96239 0.185713
\(715\) −19.6629 43.6385i −0.735351 1.63199i
\(716\) 0.972238 0.0363342
\(717\) 8.49341i 0.317192i
\(718\) 15.8046 0.589824
\(719\) 26.5501 0.990151 0.495075 0.868850i \(-0.335140\pi\)
0.495075 + 0.868850i \(0.335140\pi\)
\(720\) 18.0811i 0.673843i
\(721\) 1.42548i 0.0530878i
\(722\) 19.6970i 0.733047i
\(723\) 5.48612i 0.204031i
\(724\) 2.68594 0.0998220
\(725\) 97.2769 3.61278
\(726\) 1.18313i 0.0439102i
\(727\) −6.44851 −0.239162 −0.119581 0.992824i \(-0.538155\pi\)
−0.119581 + 0.992824i \(0.538155\pi\)
\(728\) 3.96239 + 8.79384i 0.146856 + 0.325922i
\(729\) 1.00000 0.0370370
\(730\) 96.4260i 3.56889i
\(731\) 6.44851 0.238507
\(732\) −2.80018 −0.103498
\(733\) 28.8119i 1.06419i 0.846684 + 0.532097i \(0.178596\pi\)
−0.846684 + 0.532097i \(0.821404\pi\)
\(734\) 10.1114i 0.373219i
\(735\) 4.15633i 0.153308i
\(736\) 0.424070i 0.0156314i
\(737\) 17.9248 0.660268
\(738\) −2.15633 −0.0793754
\(739\) 29.6121i 1.08930i 0.838664 + 0.544650i \(0.183337\pi\)
−0.838664 + 0.544650i \(0.816663\pi\)
\(740\) 1.29948 0.0477697
\(741\) −7.84955 + 3.53690i −0.288361 + 0.129931i
\(742\) −8.88717 −0.326258
\(743\) 3.77830i 0.138612i −0.997595 0.0693062i \(-0.977921\pi\)
0.997595 0.0693062i \(-0.0220786\pi\)
\(744\) −28.6253 −1.04945
\(745\) −55.6140 −2.03754
\(746\) 24.5501i 0.898842i
\(747\) 6.99271i 0.255850i
\(748\) 2.07522i 0.0758777i
\(749\) 9.53690i 0.348471i
\(750\) −44.7875 −1.63541
\(751\) −10.0752 −0.367650 −0.183825 0.982959i \(-0.558848\pi\)
−0.183825 + 0.982959i \(0.558848\pi\)
\(752\) 16.3938i 0.597819i
\(753\) 6.44851 0.234997
\(754\) 38.5863 17.3865i 1.40523 0.633177i
\(755\) 65.8759 2.39747
\(756\) 0.193937i 0.00705340i
\(757\) 39.4979 1.43557 0.717787 0.696262i \(-0.245155\pi\)
0.717787 + 0.696262i \(0.245155\pi\)
\(758\) −28.2520 −1.02616
\(759\) 1.23884i 0.0449671i
\(760\) 26.5501i 0.963073i
\(761\) 25.7685i 0.934106i 0.884229 + 0.467053i \(0.154684\pi\)
−0.884229 + 0.467053i \(0.845316\pi\)
\(762\) 1.42548i 0.0516399i
\(763\) −6.23743 −0.225810
\(764\) 3.60037 0.130257
\(765\) 13.9248i 0.503451i
\(766\) 12.2823 0.443779
\(767\) 20.2374 9.11871i 0.730731 0.329258i
\(768\) 4.61213 0.166426
\(769\) 49.4372i 1.78275i 0.453264 + 0.891376i \(0.350259\pi\)
−0.453264 + 0.891376i \(0.649741\pi\)
\(770\) −19.6629 −0.708602
\(771\) 9.90034 0.356552
\(772\) 2.53832i 0.0913561i
\(773\) 15.2184i 0.547367i −0.961820 0.273683i \(-0.911758\pi\)
0.961820 0.273683i \(-0.0882421\pi\)
\(774\) 2.85097i 0.102476i
\(775\) 131.349i 4.71821i
\(776\) 8.96239 0.321731
\(777\) −1.61213 −0.0578347
\(778\) 1.03761i 0.0372002i
\(779\) −3.47627 −0.124550
\(780\) −2.64974 + 1.19394i −0.0948758 + 0.0427498i
\(781\) 37.7499 1.35080
\(782\) 1.92478i 0.0688299i
\(783\) −7.92478 −0.283208
\(784\) 4.35026 0.155366
\(785\) 52.7875i 1.88407i
\(786\) 25.4010i 0.906025i
\(787\) 26.6107i 0.948569i 0.880372 + 0.474285i \(0.157293\pi\)
−0.880372 + 0.474285i \(0.842707\pi\)
\(788\) 1.40502i 0.0500516i
\(789\) 3.46168 0.123239
\(790\) −55.1754 −1.96305
\(791\) 3.92478i 0.139549i
\(792\) 8.54420 0.303605
\(793\) 21.3865 + 47.4636i 0.759455 + 1.68548i
\(794\) −18.5139 −0.657033
\(795\) 24.9380i 0.884458i
\(796\) −1.67276 −0.0592894
\(797\) 12.2473 0.433821 0.216910 0.976192i \(-0.430402\pi\)
0.216910 + 0.976192i \(0.430402\pi\)
\(798\) 3.53690i 0.125205i
\(799\) 12.6253i 0.446651i
\(800\) 13.4206i 0.474488i
\(801\) 0.932071i 0.0329331i
\(802\) 42.5198 1.50142
\(803\) −50.0263 −1.76539
\(804\) 1.08840i 0.0383848i
\(805\) −1.61213 −0.0568200
\(806\) −23.4763 52.1016i −0.826916 1.83520i
\(807\) 16.7513 0.589674
\(808\) 37.5877i 1.32233i
\(809\) −28.9234 −1.01689 −0.508446 0.861094i \(-0.669780\pi\)
−0.508446 + 0.861094i \(0.669780\pi\)
\(810\) 6.15633 0.216311
\(811\) 33.1002i 1.16230i 0.813795 + 0.581152i \(0.197398\pi\)
−0.813795 + 0.581152i \(0.802602\pi\)
\(812\) 1.53690i 0.0539348i
\(813\) 6.38787i 0.224032i
\(814\) 7.62672i 0.267316i
\(815\) −39.9511 −1.39943
\(816\) −14.5745 −0.510210
\(817\) 4.59612i 0.160798i
\(818\) −34.7367 −1.21454
\(819\) 3.28726 1.48119i 0.114866 0.0517571i
\(820\) −1.17347 −0.0409793
\(821\) 6.46898i 0.225769i 0.993608 + 0.112884i \(0.0360090\pi\)
−0.993608 + 0.112884i \(0.963991\pi\)
\(822\) −7.19394 −0.250917
\(823\) 9.55149 0.332944 0.166472 0.986046i \(-0.446762\pi\)
0.166472 + 0.986046i \(0.446762\pi\)
\(824\) 3.81336i 0.132845i
\(825\) 39.2057i 1.36497i
\(826\) 9.11871i 0.317281i
\(827\) 36.3430i 1.26377i −0.775063 0.631884i \(-0.782282\pi\)
0.775063 0.631884i \(-0.217718\pi\)
\(828\) −0.0752228 −0.00261417
\(829\) −21.7645 −0.755912 −0.377956 0.925824i \(-0.623373\pi\)
−0.377956 + 0.925824i \(0.623373\pi\)
\(830\) 43.0494i 1.49427i
\(831\) 12.1768 0.422408
\(832\) −10.4885 23.2774i −0.363623 0.806999i
\(833\) −3.35026 −0.116080
\(834\) 3.81336i 0.132046i
\(835\) 50.2130 1.73769
\(836\) −1.47910 −0.0511557
\(837\) 10.7005i 0.369864i
\(838\) 50.6225i 1.74872i
\(839\) 0.0693432i 0.00239399i 0.999999 + 0.00119700i \(0.000381016\pi\)
−0.999999 + 0.00119700i \(0.999619\pi\)
\(840\) 11.1187i 0.383632i
\(841\) 33.8021 1.16559
\(842\) 8.22284 0.283378
\(843\) 27.3054i 0.940446i
\(844\) −3.66291 −0.126083
\(845\) 40.4749 + 35.7948i 1.39238 + 1.23138i
\(846\) 5.58181 0.191907
\(847\) 0.798769i 0.0274460i
\(848\) 26.1016 0.896332
\(849\) 27.0738 0.929171
\(850\) 60.9135i 2.08932i
\(851\) 0.625301i 0.0214350i
\(852\) 2.29218i 0.0785289i
\(853\) 5.17347i 0.177136i 0.996070 + 0.0885681i \(0.0282291\pi\)
−0.996070 + 0.0885681i \(0.971771\pi\)
\(854\) 21.3865 0.731830
\(855\) 9.92478 0.339420
\(856\) 25.5125i 0.871998i
\(857\) 41.2750 1.40993 0.704964 0.709243i \(-0.250963\pi\)
0.704964 + 0.709243i \(0.250963\pi\)
\(858\) 7.00729 + 15.5515i 0.239225 + 0.530919i
\(859\) −14.1768 −0.483706 −0.241853 0.970313i \(-0.577755\pi\)
−0.241853 + 0.970313i \(0.577755\pi\)
\(860\) 1.55149i 0.0529055i
\(861\) 1.45580 0.0496136
\(862\) −8.04491 −0.274011
\(863\) 7.62786i 0.259655i 0.991537 + 0.129828i \(0.0414424\pi\)
−0.991537 + 0.129828i \(0.958558\pi\)
\(864\) 1.09332i 0.0371955i
\(865\) 38.5501i 1.31074i
\(866\) 59.5633i 2.02404i
\(867\) −5.77575 −0.196155
\(868\) −2.07522 −0.0704377
\(869\) 28.6253i 0.971047i
\(870\) −48.7875 −1.65405
\(871\) −18.4485 + 8.31265i −0.625104 + 0.281663i
\(872\) 16.6859 0.565057
\(873\) 3.35026i 0.113389i
\(874\) −1.37187 −0.0464042
\(875\) 30.2374 1.02221
\(876\) 3.03761i 0.102631i
\(877\) 43.1754i 1.45793i −0.684552 0.728964i \(-0.740002\pi\)
0.684552 0.728964i \(-0.259998\pi\)
\(878\) 33.4372i 1.12845i
\(879\) 16.7816i 0.566030i
\(880\) 57.7499 1.94675
\(881\) −29.9003 −1.00737 −0.503684 0.863888i \(-0.668022\pi\)
−0.503684 + 0.863888i \(0.668022\pi\)
\(882\) 1.48119i 0.0498744i
\(883\) −9.83971 −0.331132 −0.165566 0.986199i \(-0.552945\pi\)
−0.165566 + 0.986199i \(0.552945\pi\)
\(884\) −0.962389 2.13586i −0.0323686 0.0718366i
\(885\) −25.5877 −0.860121
\(886\) 12.2012i 0.409908i
\(887\) −43.0249 −1.44464 −0.722318 0.691561i \(-0.756923\pi\)
−0.722318 + 0.691561i \(0.756923\pi\)
\(888\) 4.31265 0.144723
\(889\) 0.962389i 0.0322775i
\(890\) 5.73813i 0.192343i
\(891\) 3.19394i 0.107001i
\(892\) 3.13444i 0.104949i
\(893\) 8.99859 0.301126
\(894\) 19.8192 0.662854
\(895\) 20.8364i 0.696483i
\(896\) −12.6751 −0.423446
\(897\) 0.574515 + 1.27504i 0.0191825 + 0.0425723i
\(898\) 3.56722 0.119040
\(899\) 84.7993i 2.82821i
\(900\) 2.38058 0.0793526
\(901\) −20.1016 −0.669680
\(902\) 6.88717i 0.229318i
\(903\) 1.92478i 0.0640526i
\(904\) 10.4993i 0.349201i
\(905\) 57.5633i 1.91347i
\(906\) −23.4763 −0.779947
\(907\) −11.2605 −0.373897 −0.186949 0.982370i \(-0.559860\pi\)
−0.186949 + 0.982370i \(0.559860\pi\)
\(908\) 5.28375i 0.175347i
\(909\) −14.0508 −0.466035
\(910\) 20.2374 9.11871i 0.670864 0.302282i
\(911\) 23.5633 0.780685 0.390343 0.920670i \(-0.372357\pi\)
0.390343 + 0.920670i \(0.372357\pi\)
\(912\) 10.3879i 0.343977i
\(913\) 22.3343 0.739156
\(914\) 43.7889 1.44841
\(915\) 60.0118i 1.98393i
\(916\) 3.78418i 0.125033i
\(917\) 17.1490i 0.566311i
\(918\) 4.96239i 0.163783i
\(919\) 12.8411 0.423589 0.211795 0.977314i \(-0.432069\pi\)
0.211795 + 0.977314i \(0.432069\pi\)
\(920\) 4.31265 0.142184
\(921\) 17.2995i 0.570037i
\(922\) −49.9452 −1.64486
\(923\) −38.8529 + 17.5066i −1.27886 + 0.576236i
\(924\) 0.619421 0.0203775
\(925\) 19.7889i 0.650656i
\(926\) 24.9380 0.819512
\(927\) 1.42548 0.0468191
\(928\) 8.66433i 0.284420i
\(929\) 24.6194i 0.807737i −0.914817 0.403869i \(-0.867665\pi\)
0.914817 0.403869i \(-0.132335\pi\)
\(930\) 65.8759i 2.16016i
\(931\) 2.38787i 0.0782594i
\(932\) 4.23743 0.138802
\(933\) 17.1490 0.561434
\(934\) 33.6239i 1.10021i
\(935\) −44.4749 −1.45448
\(936\) −8.79384 + 3.96239i −0.287436 + 0.129515i
\(937\) 44.4847 1.45325 0.726626 0.687033i \(-0.241087\pi\)
0.726626 + 0.687033i \(0.241087\pi\)
\(938\) 8.31265i 0.271418i
\(939\) −10.8119 −0.352834
\(940\) 3.03761 0.0990760
\(941\) 2.69464i 0.0878429i −0.999035 0.0439214i \(-0.986015\pi\)
0.999035 0.0439214i \(-0.0139851\pi\)
\(942\) 18.8119i 0.612926i
\(943\) 0.564666i 0.0183881i
\(944\) 26.7816i 0.871668i
\(945\) −4.15633 −0.135205
\(946\) 9.10581 0.296056
\(947\) 3.09237i 0.100488i 0.998737 + 0.0502442i \(0.0160000\pi\)
−0.998737 + 0.0502442i \(0.984000\pi\)
\(948\) 1.73813 0.0564520
\(949\) 51.4880 23.1998i 1.67137 0.753098i
\(950\) 43.4156 1.40859
\(951\) 14.1563i 0.459050i
\(952\) 8.96239 0.290473
\(953\) 8.57925 0.277909 0.138955 0.990299i \(-0.455626\pi\)
0.138955 + 0.990299i \(0.455626\pi\)
\(954\) 8.88717i 0.287733i
\(955\) 77.1608i 2.49686i
\(956\) 1.64718i 0.0532737i
\(957\) 25.3112i 0.818196i
\(958\) −14.9537 −0.483131
\(959\) 4.85685 0.156836
\(960\) 29.4314i 0.949893i
\(961\) −83.5012 −2.69359
\(962\) 3.53690 + 7.84955i 0.114034 + 0.253080i
\(963\) 9.53690 0.307322
\(964\) 1.06396i 0.0342678i
\(965\) −54.3996 −1.75119
\(966\) 0.574515 0.0184847
\(967\) 7.43533i 0.239104i −0.992828 0.119552i \(-0.961854\pi\)
0.992828 0.119552i \(-0.0381459\pi\)
\(968\) 2.13681i 0.0686797i
\(969\) 8.00000i 0.256997i
\(970\) 20.6253i 0.662238i
\(971\) 52.3244 1.67917 0.839585 0.543228i \(-0.182798\pi\)
0.839585 + 0.543228i \(0.182798\pi\)
\(972\) −0.193937 −0.00622052
\(973\) 2.57452i 0.0825352i
\(974\) 13.9248 0.446179
\(975\) −18.1817 40.3512i −0.582281 1.29227i
\(976\) −62.8119 −2.01056
\(977\) 12.0811i 0.386509i 0.981149 + 0.193254i \(0.0619043\pi\)
−0.981149 + 0.193254i \(0.938096\pi\)
\(978\) 14.2374 0.455263
\(979\) 2.97698 0.0951446
\(980\) 0.806063i 0.0257488i
\(981\) 6.23743i 0.199146i
\(982\) 30.7513i 0.981314i
\(983\) 16.7553i 0.534410i −0.963640 0.267205i \(-0.913900\pi\)
0.963640 0.267205i \(-0.0861001\pi\)
\(984\) −3.89446 −0.124151
\(985\) −30.1114 −0.959430
\(986\) 39.3258i 1.25239i
\(987\) −3.76845 −0.119951
\(988\) 1.52232 0.685935i 0.0484313 0.0218225i
\(989\) 0.746569 0.0237395
\(990\) 19.6629i 0.624928i
\(991\) −1.77433 −0.0563635 −0.0281818 0.999603i \(-0.508972\pi\)
−0.0281818 + 0.999603i \(0.508972\pi\)
\(992\) 11.6991 0.371447
\(993\) 17.2506i 0.547431i
\(994\) 17.5066i 0.555275i
\(995\) 35.8496i 1.13651i
\(996\) 1.35614i 0.0429710i
\(997\) 24.5501 0.777509 0.388754 0.921341i \(-0.372905\pi\)
0.388754 + 0.921341i \(0.372905\pi\)
\(998\) −33.9972 −1.07616
\(999\) 1.61213i 0.0510054i
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 273.2.c.b.64.5 yes 6
3.2 odd 2 819.2.c.c.64.2 6
4.3 odd 2 4368.2.h.o.337.1 6
7.6 odd 2 1911.2.c.h.883.5 6
13.5 odd 4 3549.2.a.k.1.3 3
13.8 odd 4 3549.2.a.q.1.1 3
13.12 even 2 inner 273.2.c.b.64.2 6
39.38 odd 2 819.2.c.c.64.5 6
52.51 odd 2 4368.2.h.o.337.6 6
91.90 odd 2 1911.2.c.h.883.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.c.b.64.2 6 13.12 even 2 inner
273.2.c.b.64.5 yes 6 1.1 even 1 trivial
819.2.c.c.64.2 6 3.2 odd 2
819.2.c.c.64.5 6 39.38 odd 2
1911.2.c.h.883.2 6 91.90 odd 2
1911.2.c.h.883.5 6 7.6 odd 2
3549.2.a.k.1.3 3 13.5 odd 4
3549.2.a.q.1.1 3 13.8 odd 4
4368.2.h.o.337.1 6 4.3 odd 2
4368.2.h.o.337.6 6 52.51 odd 2