Properties

Label 273.2.c.c.64.1
Level $273$
Weight $2$
Character 273.64
Analytic conductor $2.180$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 \) 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 64.1
Root \(-2.60520i\) of defining polynomial
Character \(\chi\) \(=\) 273.64
Dual form 273.2.c.c.64.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.60520i q^{2} -1.00000 q^{3} -4.78706 q^{4} +3.78706i q^{5} +2.60520i q^{6} +1.00000i q^{7} +7.26084i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.60520i q^{2} -1.00000 q^{3} -4.78706 q^{4} +3.78706i q^{5} +2.60520i q^{6} +1.00000i q^{7} +7.26084i q^{8} +1.00000 q^{9} +9.86604 q^{10} -2.55475i q^{11} +4.78706 q^{12} +(0.0504439 + 3.60520i) q^{13} +2.60520 q^{14} -3.78706i q^{15} +9.34181 q^{16} -3.21040 q^{17} -2.60520i q^{18} +8.44270i q^{19} -18.1289i q^{20} -1.00000i q^{21} -6.65564 q^{22} +3.97809 q^{23} -7.26084i q^{24} -9.34181 q^{25} +(9.39226 - 0.131416i) q^{26} -1.00000 q^{27} -4.78706i q^{28} -2.86858 q^{29} -9.86604 q^{30} -0.868584i q^{31} -9.81560i q^{32} +2.55475i q^{33} +8.36372i q^{34} -3.78706 q^{35} -4.78706 q^{36} +5.10951i q^{37} +21.9949 q^{38} +(-0.0504439 - 3.60520i) q^{39} -27.4972 q^{40} +3.34436i q^{41} -2.60520 q^{42} +4.86858 q^{43} +12.2298i q^{44} +3.78706i q^{45} -10.3637i q^{46} -11.0983i q^{47} -9.34181 q^{48} -1.00000 q^{49} +24.3373i q^{50} +3.21040 q^{51} +(-0.241478 - 17.2583i) q^{52} +1.33319 q^{53} +2.60520i q^{54} +9.67501 q^{55} -7.26084 q^{56} -8.44270i q^{57} +7.47323i q^{58} -10.6556i q^{59} +18.1289i q^{60} -4.10089 q^{61} -2.26283 q^{62} +1.00000i q^{63} -6.88795 q^{64} +(-13.6531 + 0.191034i) q^{65} +6.65564 q^{66} +10.8467i q^{67} +15.3684 q^{68} -3.97809 q^{69} +9.86604i q^{70} +2.55475i q^{71} +7.26084i q^{72} +6.76770i q^{73} +13.3113 q^{74} +9.34181 q^{75} -40.4157i q^{76} +2.55475 q^{77} +(-9.39226 + 0.131416i) q^{78} -1.23230 q^{79} +35.3780i q^{80} +1.00000 q^{81} +8.71272 q^{82} +11.0983i q^{83} +4.78706i q^{84} -12.1580i q^{85} -12.6836i q^{86} +2.86858 q^{87} +18.5497 q^{88} -5.52423i q^{89} +9.86604 q^{90} +(-3.60520 + 0.0504439i) q^{91} -19.0434 q^{92} +0.868584i q^{93} -28.9134 q^{94} -31.9730 q^{95} +9.81560i q^{96} -5.65819i q^{97} +2.60520i q^{98} -2.55475i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.60520i 1.84215i −0.389381 0.921077i \(-0.627311\pi\)
0.389381 0.921077i \(-0.372689\pi\)
\(3\) −1.00000 −0.577350
\(4\) −4.78706 −2.39353
\(5\) 3.78706i 1.69362i 0.531892 + 0.846812i \(0.321481\pi\)
−0.531892 + 0.846812i \(0.678519\pi\)
\(6\) 2.60520i 1.06357i
\(7\) 1.00000i 0.377964i
\(8\) 7.26084i 2.56709i
\(9\) 1.00000 0.333333
\(10\) 9.86604 3.11992
\(11\) 2.55475i 0.770287i −0.922857 0.385144i \(-0.874152\pi\)
0.922857 0.385144i \(-0.125848\pi\)
\(12\) 4.78706 1.38190
\(13\) 0.0504439 + 3.60520i 0.0139906 + 0.999902i
\(14\) 2.60520 0.696269
\(15\) 3.78706i 0.977814i
\(16\) 9.34181 2.33545
\(17\) −3.21040 −0.778636 −0.389318 0.921103i \(-0.627289\pi\)
−0.389318 + 0.921103i \(0.627289\pi\)
\(18\) 2.60520i 0.614051i
\(19\) 8.44270i 1.93689i 0.249230 + 0.968444i \(0.419822\pi\)
−0.249230 + 0.968444i \(0.580178\pi\)
\(20\) 18.1289i 4.05374i
\(21\) 1.00000i 0.218218i
\(22\) −6.65564 −1.41899
\(23\) 3.97809 0.829490 0.414745 0.909938i \(-0.363871\pi\)
0.414745 + 0.909938i \(0.363871\pi\)
\(24\) 7.26084i 1.48211i
\(25\) −9.34181 −1.86836
\(26\) 9.39226 0.131416i 1.84197 0.0257729i
\(27\) −1.00000 −0.192450
\(28\) 4.78706i 0.904669i
\(29\) −2.86858 −0.532683 −0.266341 0.963879i \(-0.585815\pi\)
−0.266341 + 0.963879i \(0.585815\pi\)
\(30\) −9.86604 −1.80128
\(31\) 0.868584i 0.156002i −0.996953 0.0780011i \(-0.975146\pi\)
0.996953 0.0780011i \(-0.0248538\pi\)
\(32\) 9.81560i 1.73517i
\(33\) 2.55475i 0.444726i
\(34\) 8.36372i 1.43437i
\(35\) −3.78706 −0.640130
\(36\) −4.78706 −0.797843
\(37\) 5.10951i 0.839998i 0.907525 + 0.419999i \(0.137970\pi\)
−0.907525 + 0.419999i \(0.862030\pi\)
\(38\) 21.9949 3.56805
\(39\) −0.0504439 3.60520i −0.00807749 0.577294i
\(40\) −27.4972 −4.34769
\(41\) 3.34436i 0.522301i 0.965298 + 0.261150i \(0.0841019\pi\)
−0.965298 + 0.261150i \(0.915898\pi\)
\(42\) −2.60520 −0.401991
\(43\) 4.86858 0.742452 0.371226 0.928543i \(-0.378938\pi\)
0.371226 + 0.928543i \(0.378938\pi\)
\(44\) 12.2298i 1.84371i
\(45\) 3.78706i 0.564541i
\(46\) 10.3637i 1.52805i
\(47\) 11.0983i 1.61886i −0.587217 0.809430i \(-0.699776\pi\)
0.587217 0.809430i \(-0.300224\pi\)
\(48\) −9.34181 −1.34837
\(49\) −1.00000 −0.142857
\(50\) 24.3373i 3.44181i
\(51\) 3.21040 0.449545
\(52\) −0.241478 17.2583i −0.0334870 2.39330i
\(53\) 1.33319 0.183128 0.0915640 0.995799i \(-0.470813\pi\)
0.0915640 + 0.995799i \(0.470813\pi\)
\(54\) 2.60520i 0.354523i
\(55\) 9.67501 1.30458
\(56\) −7.26084 −0.970271
\(57\) 8.44270i 1.11826i
\(58\) 7.47323i 0.981283i
\(59\) 10.6556i 1.38725i −0.720338 0.693623i \(-0.756013\pi\)
0.720338 0.693623i \(-0.243987\pi\)
\(60\) 18.1289i 2.34043i
\(61\) −4.10089 −0.525065 −0.262532 0.964923i \(-0.584558\pi\)
−0.262532 + 0.964923i \(0.584558\pi\)
\(62\) −2.26283 −0.287380
\(63\) 1.00000i 0.125988i
\(64\) −6.88795 −0.860993
\(65\) −13.6531 + 0.191034i −1.69346 + 0.0236949i
\(66\) 6.65564 0.819253
\(67\) 10.8467i 1.32513i 0.749003 + 0.662566i \(0.230533\pi\)
−0.749003 + 0.662566i \(0.769467\pi\)
\(68\) 15.3684 1.86369
\(69\) −3.97809 −0.478906
\(70\) 9.86604i 1.17922i
\(71\) 2.55475i 0.303194i 0.988442 + 0.151597i \(0.0484415\pi\)
−0.988442 + 0.151597i \(0.951558\pi\)
\(72\) 7.26084i 0.855698i
\(73\) 6.76770i 0.792099i 0.918229 + 0.396049i \(0.129619\pi\)
−0.918229 + 0.396049i \(0.870381\pi\)
\(74\) 13.3113 1.54741
\(75\) 9.34181 1.07870
\(76\) 40.4157i 4.63600i
\(77\) 2.55475 0.291141
\(78\) −9.39226 + 0.131416i −1.06346 + 0.0148800i
\(79\) −1.23230 −0.138645 −0.0693225 0.997594i \(-0.522084\pi\)
−0.0693225 + 0.997594i \(0.522084\pi\)
\(80\) 35.3780i 3.95538i
\(81\) 1.00000 0.111111
\(82\) 8.71272 0.962158
\(83\) 11.0983i 1.21820i 0.793093 + 0.609101i \(0.208469\pi\)
−0.793093 + 0.609101i \(0.791531\pi\)
\(84\) 4.78706i 0.522311i
\(85\) 12.1580i 1.31872i
\(86\) 12.6836i 1.36771i
\(87\) 2.86858 0.307544
\(88\) 18.5497 1.97740
\(89\) 5.52423i 0.585567i −0.956179 0.292783i \(-0.905419\pi\)
0.956179 0.292783i \(-0.0945815\pi\)
\(90\) 9.86604 1.03997
\(91\) −3.60520 + 0.0504439i −0.377927 + 0.00528796i
\(92\) −19.0434 −1.98541
\(93\) 0.868584i 0.0900679i
\(94\) −28.9134 −2.98219
\(95\) −31.9730 −3.28036
\(96\) 9.81560i 1.00180i
\(97\) 5.65819i 0.574502i −0.957855 0.287251i \(-0.907259\pi\)
0.957855 0.287251i \(-0.0927414\pi\)
\(98\) 2.60520i 0.263165i
\(99\) 2.55475i 0.256762i
\(100\) 44.7198 4.47198
\(101\) −8.94756 −0.890316 −0.445158 0.895452i \(-0.646852\pi\)
−0.445158 + 0.895452i \(0.646852\pi\)
\(102\) 8.36372i 0.828132i
\(103\) −1.67501 −0.165043 −0.0825216 0.996589i \(-0.526297\pi\)
−0.0825216 + 0.996589i \(0.526297\pi\)
\(104\) −26.1768 + 0.366265i −2.56684 + 0.0359152i
\(105\) 3.78706 0.369579
\(106\) 3.47323i 0.337350i
\(107\) −0.846676 −0.0818513 −0.0409256 0.999162i \(-0.513031\pi\)
−0.0409256 + 0.999162i \(0.513031\pi\)
\(108\) 4.78706 0.460635
\(109\) 17.9949i 1.72360i −0.507248 0.861800i \(-0.669337\pi\)
0.507248 0.861800i \(-0.330663\pi\)
\(110\) 25.2053i 2.40323i
\(111\) 5.10951i 0.484973i
\(112\) 9.34181i 0.882718i
\(113\) 2.86858 0.269854 0.134927 0.990856i \(-0.456920\pi\)
0.134927 + 0.990856i \(0.456920\pi\)
\(114\) −21.9949 −2.06001
\(115\) 15.0653i 1.40484i
\(116\) 13.7321 1.27499
\(117\) 0.0504439 + 3.60520i 0.00466354 + 0.333301i
\(118\) −27.7601 −2.55552
\(119\) 3.21040i 0.294297i
\(120\) 27.4972 2.51014
\(121\) 4.47323 0.406657
\(122\) 10.6836i 0.967250i
\(123\) 3.34436i 0.301551i
\(124\) 4.15796i 0.373396i
\(125\) 16.4427i 1.47068i
\(126\) 2.60520 0.232090
\(127\) 17.2053 1.52672 0.763362 0.645971i \(-0.223547\pi\)
0.763362 + 0.645971i \(0.223547\pi\)
\(128\) 1.68672i 0.149087i
\(129\) −4.86858 −0.428655
\(130\) 0.497681 + 35.5690i 0.0436496 + 3.11961i
\(131\) 16.6836 1.45766 0.728828 0.684697i \(-0.240066\pi\)
0.728828 + 0.684697i \(0.240066\pi\)
\(132\) 12.2298i 1.06446i
\(133\) −8.44270 −0.732075
\(134\) 28.2577 2.44110
\(135\) 3.78706i 0.325938i
\(136\) 23.3102i 1.99883i
\(137\) 18.9134i 1.61588i 0.589265 + 0.807940i \(0.299417\pi\)
−0.589265 + 0.807940i \(0.700583\pi\)
\(138\) 10.3637i 0.882218i
\(139\) 5.67501 0.481348 0.240674 0.970606i \(-0.422632\pi\)
0.240674 + 0.970606i \(0.422632\pi\)
\(140\) 18.1289 1.53217
\(141\) 11.0983i 0.934649i
\(142\) 6.65564 0.558529
\(143\) 9.21040 0.128872i 0.770212 0.0107768i
\(144\) 9.34181 0.778484
\(145\) 10.8635i 0.902164i
\(146\) 17.6312 1.45917
\(147\) 1.00000 0.0824786
\(148\) 24.4595i 2.01056i
\(149\) 8.49259i 0.695740i −0.937543 0.347870i \(-0.886905\pi\)
0.937543 0.347870i \(-0.113095\pi\)
\(150\) 24.3373i 1.98713i
\(151\) 14.2190i 1.15713i −0.815637 0.578564i \(-0.803613\pi\)
0.815637 0.578564i \(-0.196387\pi\)
\(152\) −61.3011 −4.97218
\(153\) −3.21040 −0.259545
\(154\) 6.65564i 0.536327i
\(155\) 3.28938 0.264209
\(156\) 0.241478 + 17.2583i 0.0193337 + 1.38177i
\(157\) 8.42079 0.672052 0.336026 0.941853i \(-0.390917\pi\)
0.336026 + 0.941853i \(0.390917\pi\)
\(158\) 3.21040i 0.255405i
\(159\) −1.33319 −0.105729
\(160\) 37.1722 2.93872
\(161\) 3.97809i 0.313518i
\(162\) 2.60520i 0.204684i
\(163\) 3.77589i 0.295751i 0.989006 + 0.147875i \(0.0472435\pi\)
−0.989006 + 0.147875i \(0.952757\pi\)
\(164\) 16.0096i 1.25014i
\(165\) −9.67501 −0.753198
\(166\) 28.9134 2.24411
\(167\) 8.83551i 0.683712i −0.939752 0.341856i \(-0.888944\pi\)
0.939752 0.341856i \(-0.111056\pi\)
\(168\) 7.26084 0.560186
\(169\) −12.9949 + 0.363721i −0.999609 + 0.0279785i
\(170\) −31.6739 −2.42928
\(171\) 8.44270i 0.645629i
\(172\) −23.3062 −1.77708
\(173\) 17.6750 1.34381 0.671903 0.740639i \(-0.265477\pi\)
0.671903 + 0.740639i \(0.265477\pi\)
\(174\) 7.47323i 0.566544i
\(175\) 9.34181i 0.706175i
\(176\) 23.8660i 1.79897i
\(177\) 10.6556i 0.800927i
\(178\) −14.3917 −1.07870
\(179\) 16.9073 1.26371 0.631856 0.775086i \(-0.282293\pi\)
0.631856 + 0.775086i \(0.282293\pi\)
\(180\) 18.1289i 1.35125i
\(181\) 7.73717 0.575099 0.287550 0.957766i \(-0.407159\pi\)
0.287550 + 0.957766i \(0.407159\pi\)
\(182\) 0.131416 + 9.39226i 0.00974123 + 0.696200i
\(183\) 4.10089 0.303146
\(184\) 28.8843i 2.12938i
\(185\) −19.3500 −1.42264
\(186\) 2.26283 0.165919
\(187\) 8.20178i 0.599773i
\(188\) 53.1284i 3.87479i
\(189\) 1.00000i 0.0727393i
\(190\) 83.2960i 6.04293i
\(191\) 5.57412 0.403329 0.201664 0.979455i \(-0.435365\pi\)
0.201664 + 0.979455i \(0.435365\pi\)
\(192\) 6.88795 0.497095
\(193\) 7.57412i 0.545197i 0.962128 + 0.272598i \(0.0878830\pi\)
−0.962128 + 0.272598i \(0.912117\pi\)
\(194\) −14.7407 −1.05832
\(195\) 13.6531 0.191034i 0.977719 0.0136802i
\(196\) 4.78706 0.341933
\(197\) 13.3444i 0.950746i 0.879784 + 0.475373i \(0.157687\pi\)
−0.879784 + 0.475373i \(0.842313\pi\)
\(198\) −6.65564 −0.472996
\(199\) −16.8854 −1.19697 −0.598487 0.801132i \(-0.704231\pi\)
−0.598487 + 0.801132i \(0.704231\pi\)
\(200\) 67.8294i 4.79626i
\(201\) 10.8467i 0.765066i
\(202\) 23.3102i 1.64010i
\(203\) 2.86858i 0.201335i
\(204\) −15.3684 −1.07600
\(205\) −12.6653 −0.884581
\(206\) 4.36372i 0.304035i
\(207\) 3.97809 0.276497
\(208\) 0.471237 + 33.6791i 0.0326744 + 2.33522i
\(209\) 21.5690 1.49196
\(210\) 9.86604i 0.680821i
\(211\) 11.6969 0.805249 0.402624 0.915365i \(-0.368098\pi\)
0.402624 + 0.915365i \(0.368098\pi\)
\(212\) −6.38207 −0.438322
\(213\) 2.55475i 0.175049i
\(214\) 2.20576i 0.150783i
\(215\) 18.4376i 1.25743i
\(216\) 7.26084i 0.494038i
\(217\) 0.868584 0.0589633
\(218\) −46.8803 −3.17514
\(219\) 6.76770i 0.457319i
\(220\) −46.3148 −3.12254
\(221\) −0.161945 11.5741i −0.0108936 0.778559i
\(222\) −13.3113 −0.893395
\(223\) 4.24093i 0.283993i −0.989867 0.141997i \(-0.954648\pi\)
0.989867 0.141997i \(-0.0453522\pi\)
\(224\) 9.81560 0.655832
\(225\) −9.34181 −0.622788
\(226\) 7.47323i 0.497112i
\(227\) 1.72643i 0.114587i 0.998357 + 0.0572934i \(0.0182471\pi\)
−0.998357 + 0.0572934i \(0.981753\pi\)
\(228\) 40.4157i 2.67660i
\(229\) 6.82833i 0.451229i −0.974217 0.225614i \(-0.927561\pi\)
0.974217 0.225614i \(-0.0724389\pi\)
\(230\) 39.2480 2.58794
\(231\) −2.55475 −0.168091
\(232\) 20.8283i 1.36745i
\(233\) −1.33319 −0.0873403 −0.0436702 0.999046i \(-0.513905\pi\)
−0.0436702 + 0.999046i \(0.513905\pi\)
\(234\) 9.39226 0.131416i 0.613991 0.00859096i
\(235\) 42.0301 2.74174
\(236\) 51.0092i 3.32042i
\(237\) 1.23230 0.0800468
\(238\) −8.36372 −0.542139
\(239\) 3.90985i 0.252907i −0.991973 0.126454i \(-0.959640\pi\)
0.991973 0.126454i \(-0.0403595\pi\)
\(240\) 35.3780i 2.28364i
\(241\) 25.3903i 1.63553i 0.575552 + 0.817765i \(0.304787\pi\)
−0.575552 + 0.817765i \(0.695213\pi\)
\(242\) 11.6537i 0.749125i
\(243\) −1.00000 −0.0641500
\(244\) 19.6312 1.25676
\(245\) 3.78706i 0.241946i
\(246\) −8.71272 −0.555502
\(247\) −30.4376 + 0.425883i −1.93670 + 0.0270983i
\(248\) 6.30665 0.400473
\(249\) 11.0983i 0.703329i
\(250\) −42.8365 −2.70922
\(251\) −25.7708 −1.62664 −0.813319 0.581818i \(-0.802342\pi\)
−0.813319 + 0.581818i \(0.802342\pi\)
\(252\) 4.78706i 0.301556i
\(253\) 10.1631i 0.638945i
\(254\) 44.8232i 2.81246i
\(255\) 12.1580i 0.761361i
\(256\) −18.1701 −1.13563
\(257\) 5.87678 0.366584 0.183292 0.983059i \(-0.441325\pi\)
0.183292 + 0.983059i \(0.441325\pi\)
\(258\) 12.6836i 0.789648i
\(259\) −5.10951 −0.317489
\(260\) 65.3582 0.914491i 4.05334 0.0567143i
\(261\) −2.86858 −0.177561
\(262\) 43.4642i 2.68522i
\(263\) −10.2409 −0.631483 −0.315741 0.948845i \(-0.602253\pi\)
−0.315741 + 0.948845i \(0.602253\pi\)
\(264\) −18.5497 −1.14165
\(265\) 5.04888i 0.310150i
\(266\) 21.9949i 1.34859i
\(267\) 5.52423i 0.338077i
\(268\) 51.9237i 3.17174i
\(269\) −16.1396 −0.984050 −0.492025 0.870581i \(-0.663743\pi\)
−0.492025 + 0.870581i \(0.663743\pi\)
\(270\) −9.86604 −0.600428
\(271\) 12.2577i 0.744605i 0.928111 + 0.372302i \(0.121432\pi\)
−0.928111 + 0.372302i \(0.878568\pi\)
\(272\) −29.9909 −1.81847
\(273\) 3.60520 0.0504439i 0.218197 0.00305300i
\(274\) 49.2731 2.97670
\(275\) 23.8660i 1.43918i
\(276\) 19.0434 1.14628
\(277\) 9.81504 0.589729 0.294864 0.955539i \(-0.404726\pi\)
0.294864 + 0.955539i \(0.404726\pi\)
\(278\) 14.7845i 0.886716i
\(279\) 0.868584i 0.0520007i
\(280\) 27.4972i 1.64327i
\(281\) 16.1910i 0.965876i −0.875655 0.482938i \(-0.839570\pi\)
0.875655 0.482938i \(-0.160430\pi\)
\(282\) 28.9134 1.72177
\(283\) 19.1482 1.13824 0.569122 0.822253i \(-0.307283\pi\)
0.569122 + 0.822253i \(0.307283\pi\)
\(284\) 12.2298i 0.725703i
\(285\) 31.9730 1.89392
\(286\) −0.335737 23.9949i −0.0198525 1.41885i
\(287\) −3.34436 −0.197411
\(288\) 9.81560i 0.578389i
\(289\) −6.69335 −0.393727
\(290\) −28.3016 −1.66192
\(291\) 5.65819i 0.331689i
\(292\) 32.3974i 1.89591i
\(293\) 24.6725i 1.44138i 0.693257 + 0.720690i \(0.256175\pi\)
−0.693257 + 0.720690i \(0.743825\pi\)
\(294\) 2.60520i 0.151938i
\(295\) 40.3535 2.34947
\(296\) −37.0993 −2.15635
\(297\) 2.55475i 0.148242i
\(298\) −22.1249 −1.28166
\(299\) 0.200670 + 14.3418i 0.0116051 + 0.829408i
\(300\) −44.7198 −2.58190
\(301\) 4.86858i 0.280620i
\(302\) −37.0434 −2.13161
\(303\) 8.94756 0.514024
\(304\) 78.8701i 4.52351i
\(305\) 15.5303i 0.889263i
\(306\) 8.36372i 0.478122i
\(307\) 13.7540i 0.784981i 0.919756 + 0.392491i \(0.128386\pi\)
−0.919756 + 0.392491i \(0.871614\pi\)
\(308\) −12.2298 −0.696855
\(309\) 1.67501 0.0952877
\(310\) 8.56948i 0.486714i
\(311\) 12.1580 0.689415 0.344707 0.938710i \(-0.387978\pi\)
0.344707 + 0.938710i \(0.387978\pi\)
\(312\) 26.1768 0.366265i 1.48197 0.0207357i
\(313\) 21.6699 1.22486 0.612428 0.790526i \(-0.290193\pi\)
0.612428 + 0.790526i \(0.290193\pi\)
\(314\) 21.9378i 1.23802i
\(315\) −3.78706 −0.213377
\(316\) 5.89911 0.331851
\(317\) 29.3780i 1.65003i 0.565109 + 0.825016i \(0.308834\pi\)
−0.565109 + 0.825016i \(0.691166\pi\)
\(318\) 3.47323i 0.194769i
\(319\) 7.32853i 0.410319i
\(320\) 26.0851i 1.45820i
\(321\) 0.846676 0.0472569
\(322\) 10.3637 0.577548
\(323\) 27.1044i 1.50813i
\(324\) −4.78706 −0.265948
\(325\) −0.471237 33.6791i −0.0261396 1.86818i
\(326\) 9.83695 0.544818
\(327\) 17.9949i 0.995121i
\(328\) −24.2828 −1.34080
\(329\) 11.0983 0.611871
\(330\) 25.2053i 1.38751i
\(331\) 23.9124i 1.31434i −0.753741 0.657171i \(-0.771753\pi\)
0.753741 0.657171i \(-0.228247\pi\)
\(332\) 53.1284i 2.91580i
\(333\) 5.10951i 0.279999i
\(334\) −23.0183 −1.25950
\(335\) −41.0770 −2.24428
\(336\) 9.34181i 0.509638i
\(337\) −15.5706 −0.848182 −0.424091 0.905620i \(-0.639406\pi\)
−0.424091 + 0.905620i \(0.639406\pi\)
\(338\) 0.947564 + 33.8543i 0.0515407 + 1.84143i
\(339\) −2.86858 −0.155800
\(340\) 58.2009i 3.15639i
\(341\) −2.21902 −0.120167
\(342\) 21.9949 1.18935
\(343\) 1.00000i 0.0539949i
\(344\) 35.3500i 1.90594i
\(345\) 15.0653i 0.811087i
\(346\) 46.0469i 2.47550i
\(347\) 1.04845 0.0562838 0.0281419 0.999604i \(-0.491041\pi\)
0.0281419 + 0.999604i \(0.491041\pi\)
\(348\) −13.7321 −0.736117
\(349\) 13.8721i 0.742557i −0.928521 0.371279i \(-0.878919\pi\)
0.928521 0.371279i \(-0.121081\pi\)
\(350\) −24.3373 −1.30088
\(351\) −0.0504439 3.60520i −0.00269250 0.192431i
\(352\) −25.0764 −1.33658
\(353\) 9.96693i 0.530486i 0.964182 + 0.265243i \(0.0854522\pi\)
−0.964182 + 0.265243i \(0.914548\pi\)
\(354\) 27.7601 1.47543
\(355\) −9.67501 −0.513496
\(356\) 26.4448i 1.40157i
\(357\) 3.21040i 0.169912i
\(358\) 44.0469i 2.32795i
\(359\) 18.9705i 1.00122i 0.865672 + 0.500611i \(0.166891\pi\)
−0.865672 + 0.500611i \(0.833109\pi\)
\(360\) −27.4972 −1.44923
\(361\) −52.2792 −2.75154
\(362\) 20.1569i 1.05942i
\(363\) −4.47323 −0.234784
\(364\) 17.2583 0.241478i 0.904581 0.0126569i
\(365\) −25.6297 −1.34152
\(366\) 10.6836i 0.558442i
\(367\) 22.3586 1.16711 0.583556 0.812073i \(-0.301661\pi\)
0.583556 + 0.812073i \(0.301661\pi\)
\(368\) 37.1626 1.93723
\(369\) 3.34436i 0.174100i
\(370\) 50.4106i 2.62072i
\(371\) 1.33319i 0.0692159i
\(372\) 4.15796i 0.215580i
\(373\) 26.0958 1.35119 0.675595 0.737273i \(-0.263887\pi\)
0.675595 + 0.737273i \(0.263887\pi\)
\(374\) 21.3673 1.10487
\(375\) 16.4427i 0.849097i
\(376\) 80.5833 4.15577
\(377\) −0.144703 10.3418i −0.00745256 0.532630i
\(378\) −2.60520 −0.133997
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 153.057 7.85164
\(381\) −17.2053 −0.881455
\(382\) 14.5217i 0.742994i
\(383\) 4.96229i 0.253561i 0.991931 + 0.126781i \(0.0404644\pi\)
−0.991931 + 0.126781i \(0.959536\pi\)
\(384\) 1.68672i 0.0860752i
\(385\) 9.67501i 0.493084i
\(386\) 19.7321 1.00434
\(387\) 4.86858 0.247484
\(388\) 27.0861i 1.37509i
\(389\) 17.8757 0.906333 0.453166 0.891426i \(-0.350294\pi\)
0.453166 + 0.891426i \(0.350294\pi\)
\(390\) −0.497681 35.5690i −0.0252011 1.80111i
\(391\) −12.7713 −0.645870
\(392\) 7.26084i 0.366728i
\(393\) −16.6836 −0.841578
\(394\) 34.7647 1.75142
\(395\) 4.66681i 0.234813i
\(396\) 12.2298i 0.614569i
\(397\) 7.06925i 0.354796i −0.984139 0.177398i \(-0.943232\pi\)
0.984139 0.177398i \(-0.0567679\pi\)
\(398\) 43.9898i 2.20501i
\(399\) 8.44270 0.422664
\(400\) −87.2695 −4.36347
\(401\) 9.12025i 0.455444i 0.973726 + 0.227722i \(0.0731277\pi\)
−0.973726 + 0.227722i \(0.926872\pi\)
\(402\) −28.2577 −1.40937
\(403\) 3.13142 0.0438147i 0.155987 0.00218257i
\(404\) 42.8325 2.13100
\(405\) 3.78706i 0.188180i
\(406\) −7.47323 −0.370890
\(407\) 13.0535 0.647040
\(408\) 23.3102i 1.15403i
\(409\) 18.2542i 0.902611i 0.892369 + 0.451306i \(0.149042\pi\)
−0.892369 + 0.451306i \(0.850958\pi\)
\(410\) 32.9956i 1.62953i
\(411\) 18.9134i 0.932929i
\(412\) 8.01835 0.395036
\(413\) 10.6556 0.524330
\(414\) 10.3637i 0.509349i
\(415\) −42.0301 −2.06318
\(416\) 35.3872 0.495137i 1.73500 0.0242761i
\(417\) −5.67501 −0.277906
\(418\) 56.1916i 2.74842i
\(419\) 13.6934 0.668964 0.334482 0.942402i \(-0.391439\pi\)
0.334482 + 0.942402i \(0.391439\pi\)
\(420\) −18.1289 −0.884598
\(421\) 33.8146i 1.64802i 0.566573 + 0.824012i \(0.308269\pi\)
−0.566573 + 0.824012i \(0.691731\pi\)
\(422\) 30.4728i 1.48339i
\(423\) 11.0983i 0.539620i
\(424\) 9.68009i 0.470107i
\(425\) 29.9909 1.45477
\(426\) −6.65564 −0.322467
\(427\) 4.10089i 0.198456i
\(428\) 4.05309 0.195913
\(429\) −9.21040 + 0.128872i −0.444682 + 0.00622199i
\(430\) 48.0336 2.31639
\(431\) 20.3256i 0.979048i −0.871990 0.489524i \(-0.837171\pi\)
0.871990 0.489524i \(-0.162829\pi\)
\(432\) −9.34181 −0.449458
\(433\) −17.1106 −0.822284 −0.411142 0.911571i \(-0.634870\pi\)
−0.411142 + 0.911571i \(0.634870\pi\)
\(434\) 2.26283i 0.108619i
\(435\) 10.8635i 0.520865i
\(436\) 86.1427i 4.12549i
\(437\) 33.5858i 1.60663i
\(438\) −17.6312 −0.842451
\(439\) −1.67501 −0.0799436 −0.0399718 0.999201i \(-0.512727\pi\)
−0.0399718 + 0.999201i \(0.512727\pi\)
\(440\) 70.2487i 3.34897i
\(441\) −1.00000 −0.0476190
\(442\) −30.1529 + 0.421899i −1.43423 + 0.0200677i
\(443\) −12.9073 −0.613245 −0.306622 0.951831i \(-0.599199\pi\)
−0.306622 + 0.951831i \(0.599199\pi\)
\(444\) 24.4595i 1.16080i
\(445\) 20.9206 0.991730
\(446\) −11.0485 −0.523159
\(447\) 8.49259i 0.401686i
\(448\) 6.88795i 0.325425i
\(449\) 21.1762i 0.999368i −0.866208 0.499684i \(-0.833450\pi\)
0.866208 0.499684i \(-0.166550\pi\)
\(450\) 24.3373i 1.14727i
\(451\) 8.54401 0.402322
\(452\) −13.7321 −0.645903
\(453\) 14.2190i 0.668068i
\(454\) 4.49768 0.211087
\(455\) −0.191034 13.6531i −0.00895581 0.640067i
\(456\) 61.3011 2.87069
\(457\) 8.41570i 0.393670i 0.980437 + 0.196835i \(0.0630663\pi\)
−0.980437 + 0.196835i \(0.936934\pi\)
\(458\) −17.7892 −0.831232
\(459\) 3.21040 0.149848
\(460\) 72.1183i 3.36253i
\(461\) 1.76515i 0.0822113i −0.999155 0.0411056i \(-0.986912\pi\)
0.999155 0.0411056i \(-0.0130880\pi\)
\(462\) 6.65564i 0.309649i
\(463\) 38.9241i 1.80896i −0.426519 0.904479i \(-0.640260\pi\)
0.426519 0.904479i \(-0.359740\pi\)
\(464\) −26.7978 −1.24406
\(465\) −3.28938 −0.152541
\(466\) 3.47323i 0.160894i
\(467\) −30.4544 −1.40926 −0.704631 0.709573i \(-0.748888\pi\)
−0.704631 + 0.709573i \(0.748888\pi\)
\(468\) −0.241478 17.2583i −0.0111623 0.797765i
\(469\) −10.8467 −0.500853
\(470\) 109.497i 5.05071i
\(471\) −8.42079 −0.388010
\(472\) 77.3689 3.56119
\(473\) 12.4380i 0.571902i
\(474\) 3.21040i 0.147458i
\(475\) 78.8701i 3.61881i
\(476\) 15.3684i 0.704408i
\(477\) 1.33319 0.0610427
\(478\) −10.1859 −0.465894
\(479\) 3.54656i 0.162046i 0.996712 + 0.0810232i \(0.0258188\pi\)
−0.996712 + 0.0810232i \(0.974181\pi\)
\(480\) −37.1722 −1.69667
\(481\) −18.4208 + 0.257744i −0.839916 + 0.0117521i
\(482\) 66.1467 3.01290
\(483\) 3.97809i 0.181009i
\(484\) −21.4136 −0.973346
\(485\) 21.4279 0.972990
\(486\) 2.60520i 0.118174i
\(487\) 39.9898i 1.81211i −0.423158 0.906056i \(-0.639078\pi\)
0.423158 0.906056i \(-0.360922\pi\)
\(488\) 29.7759i 1.34789i
\(489\) 3.77589i 0.170752i
\(490\) −9.86604 −0.445702
\(491\) −9.69844 −0.437685 −0.218842 0.975760i \(-0.570228\pi\)
−0.218842 + 0.975760i \(0.570228\pi\)
\(492\) 16.0096i 0.721770i
\(493\) 9.20929 0.414766
\(494\) 1.10951 + 79.2960i 0.0499192 + 3.56770i
\(495\) 9.67501 0.434859
\(496\) 8.11415i 0.364336i
\(497\) −2.55475 −0.114596
\(498\) −28.9134 −1.29564
\(499\) 24.0336i 1.07589i −0.842979 0.537947i \(-0.819200\pi\)
0.842979 0.537947i \(-0.180800\pi\)
\(500\) 78.7122i 3.52012i
\(501\) 8.83551i 0.394741i
\(502\) 67.1381i 2.99652i
\(503\) 3.67390 0.163811 0.0819056 0.996640i \(-0.473899\pi\)
0.0819056 + 0.996640i \(0.473899\pi\)
\(504\) −7.26084 −0.323424
\(505\) 33.8849i 1.50786i
\(506\) −26.4768 −1.17704
\(507\) 12.9949 0.363721i 0.577124 0.0161534i
\(508\) −82.3628 −3.65426
\(509\) 19.3612i 0.858169i −0.903264 0.429085i \(-0.858836\pi\)
0.903264 0.429085i \(-0.141164\pi\)
\(510\) 31.6739 1.40254
\(511\) −6.76770 −0.299385
\(512\) 43.9634i 1.94293i
\(513\) 8.44270i 0.372754i
\(514\) 15.3102i 0.675303i
\(515\) 6.34334i 0.279521i
\(516\) 23.3062 1.02600
\(517\) −28.3535 −1.24699
\(518\) 13.3113i 0.584864i
\(519\) −17.6750 −0.775847
\(520\) −1.38707 99.1330i −0.0608269 4.34727i
\(521\) 34.1130 1.49452 0.747260 0.664532i \(-0.231369\pi\)
0.747260 + 0.664532i \(0.231369\pi\)
\(522\) 7.47323i 0.327094i
\(523\) −33.0250 −1.44408 −0.722042 0.691850i \(-0.756796\pi\)
−0.722042 + 0.691850i \(0.756796\pi\)
\(524\) −79.8655 −3.48894
\(525\) 9.34181i 0.407710i
\(526\) 26.6796i 1.16329i
\(527\) 2.78850i 0.121469i
\(528\) 23.8660i 1.03864i
\(529\) −7.17478 −0.311947
\(530\) 13.1533 0.571344
\(531\) 10.6556i 0.462415i
\(532\) 40.4157 1.75224
\(533\) −12.0571 + 0.168702i −0.522250 + 0.00730731i
\(534\) 14.3917 0.622790
\(535\) 3.20641i 0.138625i
\(536\) −78.7560 −3.40174
\(537\) −16.9073 −0.729604
\(538\) 42.0469i 1.81277i
\(539\) 2.55475i 0.110041i
\(540\) 18.1289i 0.780142i
\(541\) 24.8416i 1.06802i 0.845477 + 0.534012i \(0.179316\pi\)
−0.845477 + 0.534012i \(0.820684\pi\)
\(542\) 31.9339 1.37168
\(543\) −7.73717 −0.332034
\(544\) 31.5120i 1.35106i
\(545\) 68.1478 2.91913
\(546\) −0.131416 9.39226i −0.00562410 0.401951i
\(547\) 8.14114 0.348090 0.174045 0.984738i \(-0.444316\pi\)
0.174045 + 0.984738i \(0.444316\pi\)
\(548\) 90.5395i 3.86766i
\(549\) −4.10089 −0.175022
\(550\) 62.1758 2.65118
\(551\) 24.2186i 1.03175i
\(552\) 28.8843i 1.22940i
\(553\) 1.23230i 0.0524029i
\(554\) 25.5701i 1.08637i
\(555\) 19.3500 0.821362
\(556\) −27.1666 −1.15212
\(557\) 8.11052i 0.343654i −0.985127 0.171827i \(-0.945033\pi\)
0.985127 0.171827i \(-0.0549670\pi\)
\(558\) −2.26283 −0.0957933
\(559\) 0.245590 + 17.5522i 0.0103874 + 0.742379i
\(560\) −35.3780 −1.49499
\(561\) 8.20178i 0.346279i
\(562\) −42.1809 −1.77929
\(563\) 11.1044 0.467995 0.233998 0.972237i \(-0.424819\pi\)
0.233998 + 0.972237i \(0.424819\pi\)
\(564\) 53.1284i 2.23711i
\(565\) 10.8635i 0.457031i
\(566\) 49.8849i 2.09682i
\(567\) 1.00000i 0.0419961i
\(568\) −18.5497 −0.778327
\(569\) 2.86858 0.120257 0.0601286 0.998191i \(-0.480849\pi\)
0.0601286 + 0.998191i \(0.480849\pi\)
\(570\) 83.2960i 3.48889i
\(571\) 32.4947 1.35986 0.679930 0.733277i \(-0.262010\pi\)
0.679930 + 0.733277i \(0.262010\pi\)
\(572\) −44.0907 + 0.616917i −1.84353 + 0.0257946i
\(573\) −5.57412 −0.232862
\(574\) 8.71272i 0.363662i
\(575\) −37.1626 −1.54979
\(576\) −6.88795 −0.286998
\(577\) 24.3759i 1.01478i 0.861716 + 0.507390i \(0.169390\pi\)
−0.861716 + 0.507390i \(0.830610\pi\)
\(578\) 17.4375i 0.725305i
\(579\) 7.57412i 0.314770i
\(580\) 52.0042i 2.15936i
\(581\) −11.0983 −0.460437
\(582\) 14.7407 0.611022
\(583\) 3.40598i 0.141061i
\(584\) −49.1392 −2.03339
\(585\) −13.6531 + 0.191034i −0.564486 + 0.00789828i
\(586\) 64.2767 2.65524
\(587\) 11.7483i 0.484906i 0.970163 + 0.242453i \(0.0779520\pi\)
−0.970163 + 0.242453i \(0.922048\pi\)
\(588\) −4.78706 −0.197415
\(589\) 7.33319 0.302159
\(590\) 105.129i 4.32809i
\(591\) 13.3444i 0.548914i
\(592\) 47.7321i 1.96178i
\(593\) 46.6459i 1.91552i −0.287574 0.957759i \(-0.592849\pi\)
0.287574 0.957759i \(-0.407151\pi\)
\(594\) 6.65564 0.273084
\(595\) 12.1580 0.498428
\(596\) 40.6545i 1.66527i
\(597\) 16.8854 0.691073
\(598\) 37.3633 0.522786i 1.52790 0.0213783i
\(599\) −20.8197 −0.850669 −0.425335 0.905036i \(-0.639844\pi\)
−0.425335 + 0.905036i \(0.639844\pi\)
\(600\) 67.8294i 2.76912i
\(601\) 24.2588 0.989539 0.494770 0.869024i \(-0.335252\pi\)
0.494770 + 0.869024i \(0.335252\pi\)
\(602\) 12.6836 0.516946
\(603\) 10.8467i 0.441711i
\(604\) 68.0673i 2.76962i
\(605\) 16.9404i 0.688724i
\(606\) 23.3102i 0.946911i
\(607\) 6.20067 0.251677 0.125839 0.992051i \(-0.459838\pi\)
0.125839 + 0.992051i \(0.459838\pi\)
\(608\) 82.8701 3.36083
\(609\) 2.86858i 0.116241i
\(610\) −40.4595 −1.63816
\(611\) 40.0117 0.559844i 1.61870 0.0226489i
\(612\) 15.3684 0.621229
\(613\) 14.9026i 0.601912i 0.953638 + 0.300956i \(0.0973058\pi\)
−0.953638 + 0.300956i \(0.902694\pi\)
\(614\) 35.8319 1.44606
\(615\) 12.6653 0.510713
\(616\) 18.5497i 0.747387i
\(617\) 24.1472i 0.972130i 0.873923 + 0.486065i \(0.161568\pi\)
−0.873923 + 0.486065i \(0.838432\pi\)
\(618\) 4.36372i 0.175535i
\(619\) 43.9339i 1.76585i −0.469513 0.882925i \(-0.655571\pi\)
0.469513 0.882925i \(-0.344429\pi\)
\(620\) −15.7464 −0.632392
\(621\) −3.97809 −0.159635
\(622\) 31.6739i 1.27001i
\(623\) 5.52423 0.221323
\(624\) −0.471237 33.6791i −0.0188646 1.34824i
\(625\) 15.5604 0.622416
\(626\) 56.4544i 2.25637i
\(627\) −21.5690 −0.861384
\(628\) −40.3108 −1.60858
\(629\) 16.4036i 0.654052i
\(630\) 9.86604i 0.393072i
\(631\) 25.9511i 1.03310i 0.856258 + 0.516548i \(0.172783\pi\)
−0.856258 + 0.516548i \(0.827217\pi\)
\(632\) 8.94756i 0.355915i
\(633\) −11.6969 −0.464911
\(634\) 76.5355 3.03961
\(635\) 65.1575i 2.58570i
\(636\) 6.38207 0.253065
\(637\) −0.0504439 3.60520i −0.00199866 0.142843i
\(638\) 19.0923 0.755870
\(639\) 2.55475i 0.101065i
\(640\) 6.38772 0.252497
\(641\) −39.5858 −1.56355 −0.781773 0.623563i \(-0.785685\pi\)
−0.781773 + 0.623563i \(0.785685\pi\)
\(642\) 2.20576i 0.0870544i
\(643\) 17.0434i 0.672125i 0.941840 + 0.336062i \(0.109095\pi\)
−0.941840 + 0.336062i \(0.890905\pi\)
\(644\) 19.0434i 0.750414i
\(645\) 18.4376i 0.725980i
\(646\) −70.6124 −2.77821
\(647\) −27.7544 −1.09114 −0.545569 0.838066i \(-0.683687\pi\)
−0.545569 + 0.838066i \(0.683687\pi\)
\(648\) 7.26084i 0.285233i
\(649\) −27.2225 −1.06858
\(650\) −87.7407 + 1.22767i −3.44147 + 0.0481531i
\(651\) −0.868584 −0.0340425
\(652\) 18.0754i 0.707888i
\(653\) −7.77080 −0.304095 −0.152048 0.988373i \(-0.548587\pi\)
−0.152048 + 0.988373i \(0.548587\pi\)
\(654\) 46.8803 1.83317
\(655\) 63.1819i 2.46872i
\(656\) 31.2424i 1.21981i
\(657\) 6.76770i 0.264033i
\(658\) 28.9134i 1.12716i
\(659\) −4.74935 −0.185008 −0.0925042 0.995712i \(-0.529487\pi\)
−0.0925042 + 0.995712i \(0.529487\pi\)
\(660\) 46.3148 1.80280
\(661\) 38.0301i 1.47920i 0.673047 + 0.739599i \(0.264985\pi\)
−0.673047 + 0.739599i \(0.735015\pi\)
\(662\) −62.2965 −2.42122
\(663\) 0.161945 + 11.5741i 0.00628942 + 0.449501i
\(664\) −80.5833 −3.12724
\(665\) 31.9730i 1.23986i
\(666\) 13.3113 0.515802
\(667\) −11.4115 −0.441855
\(668\) 42.2961i 1.63649i
\(669\) 4.24093i 0.163964i
\(670\) 107.014i 4.13430i
\(671\) 10.4768i 0.404451i
\(672\) −9.81560 −0.378645
\(673\) 23.0602 0.888905 0.444452 0.895803i \(-0.353398\pi\)
0.444452 + 0.895803i \(0.353398\pi\)
\(674\) 40.5644i 1.56248i
\(675\) 9.34181 0.359567
\(676\) 62.2074 1.74115i 2.39259 0.0669674i
\(677\) −44.3759 −1.70550 −0.852752 0.522317i \(-0.825068\pi\)
−0.852752 + 0.522317i \(0.825068\pi\)
\(678\) 7.47323i 0.287008i
\(679\) 5.65819 0.217141
\(680\) 88.2770 3.38527
\(681\) 1.72643i 0.0661568i
\(682\) 5.78098i 0.221365i
\(683\) 5.91494i 0.226329i −0.993576 0.113165i \(-0.963901\pi\)
0.993576 0.113165i \(-0.0360987\pi\)
\(684\) 40.4157i 1.54533i
\(685\) −71.6261 −2.73669
\(686\) −2.60520 −0.0994669
\(687\) 6.82833i 0.260517i
\(688\) 45.4814 1.73396
\(689\) 0.0672514 + 4.80642i 0.00256207 + 0.183110i
\(690\) −39.2480 −1.49415
\(691\) 12.6883i 0.482685i 0.970440 + 0.241343i \(0.0775878\pi\)
−0.970440 + 0.241343i \(0.922412\pi\)
\(692\) −84.6113 −3.21644
\(693\) 2.55475 0.0970471
\(694\) 2.73143i 0.103683i
\(695\) 21.4916i 0.815222i
\(696\) 20.8283i 0.789496i
\(697\) 10.7367i 0.406682i
\(698\) −36.1396 −1.36790
\(699\) 1.33319 0.0504260
\(700\) 44.7198i 1.69025i
\(701\) 25.8487 0.976291 0.488146 0.872762i \(-0.337674\pi\)
0.488146 + 0.872762i \(0.337674\pi\)
\(702\) −9.39226 + 0.131416i −0.354488 + 0.00495999i
\(703\) −43.1381 −1.62698
\(704\) 17.5970i 0.663212i
\(705\) −42.0301 −1.58294
\(706\) 25.9658 0.977237
\(707\) 8.94756i 0.336508i
\(708\) 51.0092i 1.91704i
\(709\) 18.8803i 0.709065i −0.935044 0.354533i \(-0.884640\pi\)
0.935044 0.354533i \(-0.115360\pi\)
\(710\) 25.2053i 0.945938i
\(711\) −1.23230 −0.0462150
\(712\) 40.1105 1.50321
\(713\) 3.45531i 0.129402i
\(714\) 8.36372 0.313004
\(715\) 0.488045 + 34.8803i 0.0182518 + 1.30445i
\(716\) −80.9363 −3.02473
\(717\) 3.90985i 0.146016i
\(718\) 49.4218 1.84441
\(719\) −40.1580 −1.49764 −0.748820 0.662774i \(-0.769379\pi\)
−0.748820 + 0.662774i \(0.769379\pi\)
\(720\) 35.3780i 1.31846i
\(721\) 1.67501i 0.0623804i
\(722\) 136.198i 5.06875i
\(723\) 25.3903i 0.944274i
\(724\) −37.0383 −1.37652
\(725\) 26.7978 0.995244
\(726\) 11.6537i 0.432508i
\(727\) −32.0336 −1.18806 −0.594031 0.804442i \(-0.702464\pi\)
−0.594031 + 0.804442i \(0.702464\pi\)
\(728\) −0.366265 26.1768i −0.0135747 0.970176i
\(729\) 1.00000 0.0370370
\(730\) 66.7704i 2.47128i
\(731\) −15.6301 −0.578100
\(732\) −19.6312 −0.725590
\(733\) 48.3265i 1.78498i −0.451066 0.892491i \(-0.648956\pi\)
0.451066 0.892491i \(-0.351044\pi\)
\(734\) 58.2487i 2.15000i
\(735\) 3.78706i 0.139688i
\(736\) 39.0473i 1.43930i
\(737\) 27.7106 1.02073
\(738\) 8.71272 0.320719
\(739\) 35.7321i 1.31443i −0.753705 0.657213i \(-0.771735\pi\)
0.753705 0.657213i \(-0.228265\pi\)
\(740\) 92.6296 3.40513
\(741\) 30.4376 0.425883i 1.11815 0.0156452i
\(742\) 3.47323 0.127506
\(743\) 45.0092i 1.65123i 0.564236 + 0.825613i \(0.309171\pi\)
−0.564236 + 0.825613i \(0.690829\pi\)
\(744\) −6.30665 −0.231213
\(745\) 32.1619 1.17832
\(746\) 67.9847i 2.48910i
\(747\) 11.0983i 0.406067i
\(748\) 39.2624i 1.43557i
\(749\) 0.846676i 0.0309369i
\(750\) 42.8365 1.56417
\(751\) 5.39425 0.196839 0.0984195 0.995145i \(-0.468621\pi\)
0.0984195 + 0.995145i \(0.468621\pi\)
\(752\) 103.679i 3.78077i
\(753\) 25.7708 0.939140
\(754\) −26.9425 + 0.376979i −0.981187 + 0.0137288i
\(755\) 53.8483 1.95974
\(756\) 4.78706i 0.174104i
\(757\) 13.7896 0.501191 0.250595 0.968092i \(-0.419374\pi\)
0.250595 + 0.968092i \(0.419374\pi\)
\(758\) 31.2624 1.13550
\(759\) 10.1631i 0.368895i
\(760\) 232.151i 8.42100i
\(761\) 30.6674i 1.11169i −0.831286 0.555846i \(-0.812395\pi\)
0.831286 0.555846i \(-0.187605\pi\)
\(762\) 44.8232i 1.62377i
\(763\) 17.9949 0.651460
\(764\) −26.6836 −0.965380
\(765\) 12.1580i 0.439572i
\(766\) 12.9277 0.467099
\(767\) 38.4157 0.537512i 1.38711 0.0194084i
\(768\) 18.1701 0.655659
\(769\) 48.6160i 1.75314i 0.481278 + 0.876568i \(0.340173\pi\)
−0.481278 + 0.876568i \(0.659827\pi\)
\(770\) 25.2053 0.908336
\(771\) −5.87678 −0.211647
\(772\) 36.2577i 1.30494i
\(773\) 18.6729i 0.671617i −0.941930 0.335808i \(-0.890991\pi\)
0.941930 0.335808i \(-0.109009\pi\)
\(774\) 12.6836i 0.455904i
\(775\) 8.11415i 0.291469i
\(776\) 41.0832 1.47480
\(777\) 5.10951 0.183303
\(778\) 46.5697i 1.66960i
\(779\) −28.2354 −1.01164
\(780\) −65.3582 + 0.914491i −2.34020 + 0.0327440i
\(781\) 6.52677 0.233546
\(782\) 33.2717i 1.18979i
\(783\) 2.86858 0.102515
\(784\) −9.34181 −0.333636
\(785\) 31.8900i 1.13820i
\(786\) 43.4642i 1.55032i
\(787\) 9.61751i 0.342827i 0.985199 + 0.171414i \(0.0548334\pi\)
−0.985199 + 0.171414i \(0.945167\pi\)
\(788\) 63.8802i 2.27564i
\(789\) 10.2409 0.364587
\(790\) −12.1580 −0.432561
\(791\) 2.86858i 0.101995i
\(792\) 18.5497 0.659134
\(793\) −0.206865 14.7845i −0.00734598 0.525013i
\(794\) −18.4168 −0.653588
\(795\) 5.04888i 0.179065i
\(796\) 80.8314 2.86499
\(797\) −3.73606 −0.132338 −0.0661691 0.997808i \(-0.521078\pi\)
−0.0661691 + 0.997808i \(0.521078\pi\)
\(798\) 21.9949i 0.778611i
\(799\) 35.6301i 1.26050i
\(800\) 91.6955i 3.24192i
\(801\) 5.52423i 0.195189i
\(802\) 23.7601 0.838997
\(803\) 17.2898 0.610144
\(804\) 51.9237i 1.83121i
\(805\) −15.0653 −0.530981
\(806\) −0.114146 8.15796i −0.00402062 0.287352i
\(807\) 16.1396 0.568141
\(808\) 64.9668i 2.28553i
\(809\) −10.1850 −0.358084 −0.179042 0.983841i \(-0.557300\pi\)
−0.179042 + 0.983841i \(0.557300\pi\)
\(810\) 9.86604 0.346657
\(811\) 6.21902i 0.218379i −0.994021 0.109190i \(-0.965174\pi\)
0.994021 0.109190i \(-0.0348256\pi\)
\(812\) 13.7321i 0.481901i
\(813\) 12.2577i 0.429898i
\(814\) 34.0071i 1.19195i
\(815\) −14.2995 −0.500891
\(816\) 29.9909 1.04989
\(817\) 41.1040i 1.43805i
\(818\) 47.5558 1.66275
\(819\) −3.60520 + 0.0504439i −0.125976 + 0.00176265i
\(820\) 60.6294 2.11727
\(821\) 55.4534i 1.93534i 0.252224 + 0.967669i \(0.418838\pi\)
−0.252224 + 0.967669i \(0.581162\pi\)
\(822\) −49.2731 −1.71860
\(823\) −20.4380 −0.712425 −0.356213 0.934405i \(-0.615932\pi\)
−0.356213 + 0.934405i \(0.615932\pi\)
\(824\) 12.1619i 0.423681i
\(825\) 23.8660i 0.830909i
\(826\) 27.7601i 0.965896i
\(827\) 14.1289i 0.491309i 0.969357 + 0.245655i \(0.0790029\pi\)
−0.969357 + 0.245655i \(0.920997\pi\)
\(828\) −19.0434 −0.661803
\(829\) 34.9159 1.21268 0.606340 0.795206i \(-0.292637\pi\)
0.606340 + 0.795206i \(0.292637\pi\)
\(830\) 109.497i 3.80069i
\(831\) −9.81504 −0.340480
\(832\) −0.347455 24.8324i −0.0120458 0.860909i
\(833\) 3.21040 0.111234
\(834\) 14.7845i 0.511946i
\(835\) 33.4606 1.15795
\(836\) −103.252 −3.57105
\(837\) 0.868584i 0.0300226i
\(838\) 35.6739i 1.23233i
\(839\) 46.4437i 1.60341i −0.597717 0.801707i \(-0.703925\pi\)
0.597717 0.801707i \(-0.296075\pi\)
\(840\) 27.4972i 0.948745i
\(841\) −20.7712 −0.716249
\(842\) 88.0938 3.03591
\(843\) 16.1910i 0.557649i
\(844\) −55.9938 −1.92739
\(845\) −1.37743 49.2125i −0.0473851 1.69296i
\(846\) −28.9134 −0.994063
\(847\) 4.47323i 0.153702i
\(848\) 12.4544 0.427687
\(849\) −19.1482 −0.657166
\(850\) 78.1323i 2.67992i
\(851\) 20.3261i 0.696770i
\(852\) 12.2298i 0.418985i
\(853\) 41.4462i 1.41909i −0.704659 0.709546i \(-0.748900\pi\)
0.704659 0.709546i \(-0.251100\pi\)
\(854\) −10.6836 −0.365586
\(855\) −31.9730 −1.09345
\(856\) 6.14758i 0.210120i
\(857\) 33.8666 1.15686 0.578431 0.815732i \(-0.303665\pi\)
0.578431 + 0.815732i \(0.303665\pi\)
\(858\) 0.335737 + 23.9949i 0.0114619 + 0.819173i
\(859\) −37.2890 −1.27228 −0.636141 0.771573i \(-0.719470\pi\)
−0.636141 + 0.771573i \(0.719470\pi\)
\(860\) 88.2619i 3.00971i
\(861\) 3.34436 0.113975
\(862\) −52.9521 −1.80356
\(863\) 38.7464i 1.31894i −0.751730 0.659471i \(-0.770781\pi\)
0.751730 0.659471i \(-0.229219\pi\)
\(864\) 9.81560i 0.333933i
\(865\) 66.9363i 2.27590i
\(866\) 44.5765i 1.51477i
\(867\) 6.69335 0.227318
\(868\) −4.15796 −0.141130
\(869\) 3.14823i 0.106797i
\(870\) 28.3016 0.959513
\(871\) −39.1044 + 0.547149i −1.32500 + 0.0185394i
\(872\) 130.658 4.42464
\(873\) 5.65819i 0.191501i
\(874\) 87.4978 2.95966
\(875\) 16.4427 0.555865
\(876\) 32.3974i 1.09461i
\(877\) 51.8319i 1.75024i −0.483908 0.875119i \(-0.660783\pi\)
0.483908 0.875119i \(-0.339217\pi\)
\(878\) 4.36372i 0.147268i
\(879\) 24.6725i 0.832181i
\(880\) 90.3821 3.04678
\(881\) 20.7896 0.700420 0.350210 0.936671i \(-0.386110\pi\)
0.350210 + 0.936671i \(0.386110\pi\)
\(882\) 2.60520i 0.0877216i
\(883\) 17.5292 0.589904 0.294952 0.955512i \(-0.404696\pi\)
0.294952 + 0.955512i \(0.404696\pi\)
\(884\) 0.775240 + 55.4060i 0.0260741 + 1.86350i
\(885\) −40.3535 −1.35647
\(886\) 33.6261i 1.12969i
\(887\) 9.53539 0.320167 0.160084 0.987103i \(-0.448824\pi\)
0.160084 + 0.987103i \(0.448824\pi\)
\(888\) 37.0993 1.24497
\(889\) 17.2053i 0.577047i
\(890\) 54.5022i 1.82692i
\(891\) 2.55475i 0.0855875i
\(892\) 20.3016i 0.679746i
\(893\) 93.7000 3.13555
\(894\) 22.1249 0.739967
\(895\) 64.0290i 2.14025i
\(896\) 1.68672 0.0563495
\(897\) −0.200670 14.3418i −0.00670019 0.478859i
\(898\) −55.1683 −1.84099
\(899\) 2.49160i 0.0830997i
\(900\) 44.7198 1.49066
\(901\) −4.28008 −0.142590
\(902\) 22.2588i 0.741139i
\(903\) 4.86858i 0.162016i
\(904\) 20.8283i 0.692740i
\(905\) 29.3011i 0.974002i
\(906\) 37.0434 1.23068
\(907\) 27.3270 0.907378 0.453689 0.891160i \(-0.350108\pi\)
0.453689 + 0.891160i \(0.350108\pi\)
\(908\) 8.26450i 0.274267i
\(909\) −8.94756 −0.296772
\(910\) −35.5690 + 0.497681i −1.17910 + 0.0164980i
\(911\) −38.9582 −1.29074 −0.645371 0.763869i \(-0.723297\pi\)
−0.645371 + 0.763869i \(0.723297\pi\)
\(912\) 78.8701i 2.61165i
\(913\) 28.3535 0.938365
\(914\) 21.9246 0.725201
\(915\) 15.5303i 0.513416i
\(916\) 32.6876i 1.08003i
\(917\) 16.6836i 0.550942i
\(918\) 8.36372i 0.276044i
\(919\) 9.94293 0.327987 0.163993 0.986461i \(-0.447562\pi\)
0.163993 + 0.986461i \(0.447562\pi\)
\(920\) −109.387 −3.60637
\(921\) 13.7540i 0.453209i
\(922\) −4.59857 −0.151446
\(923\) −9.21040 + 0.128872i −0.303164 + 0.00424187i
\(924\) 12.2298 0.402330
\(925\) 47.7321i 1.56942i
\(926\) −101.405 −3.33238
\(927\) −1.67501 −0.0550144
\(928\) 28.1569i 0.924294i
\(929\) 7.80854i 0.256190i 0.991762 + 0.128095i \(0.0408862\pi\)
−0.991762 + 0.128095i \(0.959114\pi\)
\(930\) 8.56948i 0.281004i
\(931\) 8.44270i 0.276698i
\(932\) 6.38207 0.209052
\(933\) −12.1580 −0.398034
\(934\) 79.3398i 2.59608i
\(935\) −31.0606 −1.01579
\(936\) −26.1768 + 0.366265i −0.855615 + 0.0119717i
\(937\) −9.59246 −0.313372 −0.156686 0.987648i \(-0.550081\pi\)
−0.156686 + 0.987648i \(0.550081\pi\)
\(938\) 28.2577i 0.922648i
\(939\) −21.6699 −0.707171
\(940\) −201.200 −6.56243
\(941\) 28.5043i 0.929214i −0.885517 0.464607i \(-0.846196\pi\)
0.885517 0.464607i \(-0.153804\pi\)
\(942\) 21.9378i 0.714773i
\(943\) 13.3042i 0.433243i
\(944\) 99.5430i 3.23985i
\(945\) 3.78706 0.123193
\(946\) −32.4036 −1.05353
\(947\) 47.8997i 1.55653i −0.627936 0.778265i \(-0.716100\pi\)
0.627936 0.778265i \(-0.283900\pi\)
\(948\) −5.89911 −0.191594
\(949\) −24.3989 + 0.341389i −0.792021 + 0.0110820i
\(950\) −205.472 −6.66640
\(951\) 29.3780i 0.952647i
\(952\) 23.3102 0.755487
\(953\) 32.5181 1.05337 0.526683 0.850062i \(-0.323436\pi\)
0.526683 + 0.850062i \(0.323436\pi\)
\(954\) 3.47323i 0.112450i
\(955\) 21.1095i 0.683088i
\(956\) 18.7167i 0.605341i
\(957\) 7.32853i 0.236898i
\(958\) 9.23949 0.298514
\(959\) −18.9134 −0.610745
\(960\) 26.0851i 0.841892i
\(961\) 30.2456 0.975663
\(962\) 0.671473 + 47.9898i 0.0216492 + 1.54725i
\(963\) −0.846676 −0.0272838
\(964\) 121.545i 3.91469i
\(965\) −28.6836 −0.923359
\(966\) −10.3637 −0.333447
\(967\) 18.0387i 0.580086i −0.957014 0.290043i \(-0.906330\pi\)
0.957014 0.290043i \(-0.0936697\pi\)
\(968\) 32.4794i 1.04393i
\(969\) 27.1044i 0.870719i
\(970\) 55.8239i 1.79240i
\(971\) 30.2190 0.969774 0.484887 0.874577i \(-0.338861\pi\)
0.484887 + 0.874577i \(0.338861\pi\)
\(972\) 4.78706 0.153545
\(973\) 5.67501i 0.181932i
\(974\) −104.181 −3.33819
\(975\) 0.471237 + 33.6791i 0.0150917 + 1.07859i
\(976\) −38.3097 −1.22626
\(977\) 38.4652i 1.23061i 0.788289 + 0.615305i \(0.210967\pi\)
−0.788289 + 0.615305i \(0.789033\pi\)
\(978\) −9.83695 −0.314551
\(979\) −14.1130 −0.451055
\(980\) 18.1289i 0.579106i
\(981\) 17.9949i 0.574533i
\(982\) 25.2664i 0.806282i
\(983\) 12.1742i 0.388297i −0.980972 0.194149i \(-0.937806\pi\)
0.980972 0.194149i \(-0.0621944\pi\)
\(984\) 24.2828 0.774109
\(985\) −50.5359 −1.61021
\(986\) 23.9920i 0.764062i
\(987\) −11.0983 −0.353264
\(988\) 145.707 2.03873i 4.63555 0.0648605i
\(989\) 19.3677 0.615856
\(990\) 25.2053i 0.801077i
\(991\) 14.4982 0.460552 0.230276 0.973125i \(-0.426037\pi\)
0.230276 + 0.973125i \(0.426037\pi\)
\(992\) −8.52567 −0.270690
\(993\) 23.9124i 0.758836i
\(994\) 6.65564i 0.211104i
\(995\) 63.9460i 2.02722i
\(996\) 53.1284i 1.68344i
\(997\) −9.35086 −0.296145 −0.148072 0.988977i \(-0.547307\pi\)
−0.148072 + 0.988977i \(0.547307\pi\)
\(998\) −62.6124 −1.98196
\(999\) 5.10951i 0.161658i
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.c.64.1 8
3.2 odd 2 819.2.c.d.64.8 8
4.3 odd 2 4368.2.h.q.337.8 8
7.6 odd 2 1911.2.c.l.883.1 8
13.5 odd 4 3549.2.a.x.1.1 4
13.8 odd 4 3549.2.a.v.1.4 4
13.12 even 2 inner 273.2.c.c.64.8 yes 8
39.38 odd 2 819.2.c.d.64.1 8
52.51 odd 2 4368.2.h.q.337.1 8
91.90 odd 2 1911.2.c.l.883.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.c.c.64.1 8 1.1 even 1 trivial
273.2.c.c.64.8 yes 8 13.12 even 2 inner
819.2.c.d.64.1 8 39.38 odd 2
819.2.c.d.64.8 8 3.2 odd 2
1911.2.c.l.883.1 8 7.6 odd 2
1911.2.c.l.883.8 8 91.90 odd 2
3549.2.a.v.1.4 4 13.8 odd 4
3549.2.a.x.1.1 4 13.5 odd 4
4368.2.h.q.337.1 8 52.51 odd 2
4368.2.h.q.337.8 8 4.3 odd 2