Properties

Label 230.2.b.a.139.2
Level $230$
Weight $2$
Character 230.139
Analytic conductor $1.837$
Analytic rank $0$
Dimension $4$
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] = [230,2,Mod(139,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.b (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: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 139.2
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 230.139
Dual form 230.2.b.a.139.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.618034i q^{3} -1.00000 q^{4} +2.23607 q^{5} +0.618034 q^{6} +4.85410i q^{7} +1.00000i q^{8} +2.61803 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.618034i q^{3} -1.00000 q^{4} +2.23607 q^{5} +0.618034 q^{6} +4.85410i q^{7} +1.00000i q^{8} +2.61803 q^{9} -2.23607i q^{10} -3.38197 q^{11} -0.618034i q^{12} -0.381966i q^{13} +4.85410 q^{14} +1.38197i q^{15} +1.00000 q^{16} -5.85410i q^{17} -2.61803i q^{18} +6.85410 q^{19} -2.23607 q^{20} -3.00000 q^{21} +3.38197i q^{22} -1.00000i q^{23} -0.618034 q^{24} +5.00000 q^{25} -0.381966 q^{26} +3.47214i q^{27} -4.85410i q^{28} -3.70820 q^{29} +1.38197 q^{30} -8.85410 q^{31} -1.00000i q^{32} -2.09017i q^{33} -5.85410 q^{34} +10.8541i q^{35} -2.61803 q^{36} +3.70820i q^{37} -6.85410i q^{38} +0.236068 q^{39} +2.23607i q^{40} -3.38197 q^{41} +3.00000i q^{42} -6.76393i q^{43} +3.38197 q^{44} +5.85410 q^{45} -1.00000 q^{46} -11.7082i q^{47} +0.618034i q^{48} -16.5623 q^{49} -5.00000i q^{50} +3.61803 q^{51} +0.381966i q^{52} +2.00000i q^{53} +3.47214 q^{54} -7.56231 q^{55} -4.85410 q^{56} +4.23607i q^{57} +3.70820i q^{58} +6.00000 q^{59} -1.38197i q^{60} -3.85410 q^{61} +8.85410i q^{62} +12.7082i q^{63} -1.00000 q^{64} -0.854102i q^{65} -2.09017 q^{66} +0.763932i q^{67} +5.85410i q^{68} +0.618034 q^{69} +10.8541 q^{70} +2.61803 q^{71} +2.61803i q^{72} +7.52786i q^{73} +3.70820 q^{74} +3.09017i q^{75} -6.85410 q^{76} -16.4164i q^{77} -0.236068i q^{78} -5.70820 q^{79} +2.23607 q^{80} +5.70820 q^{81} +3.38197i q^{82} +5.70820i q^{83} +3.00000 q^{84} -13.0902i q^{85} -6.76393 q^{86} -2.29180i q^{87} -3.38197i q^{88} +9.70820 q^{89} -5.85410i q^{90} +1.85410 q^{91} +1.00000i q^{92} -5.47214i q^{93} -11.7082 q^{94} +15.3262 q^{95} +0.618034 q^{96} -16.0344i q^{97} +16.5623i q^{98} -8.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 2 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 2 q^{6} + 6 q^{9} - 18 q^{11} + 6 q^{14} + 4 q^{16} + 14 q^{19} - 12 q^{21} + 2 q^{24} + 20 q^{25} - 6 q^{26} + 12 q^{29} + 10 q^{30} - 22 q^{31} - 10 q^{34} - 6 q^{36} - 8 q^{39} - 18 q^{41} + 18 q^{44} + 10 q^{45} - 4 q^{46} - 26 q^{49} + 10 q^{51} - 4 q^{54} + 10 q^{55} - 6 q^{56} + 24 q^{59} - 2 q^{61} - 4 q^{64} + 14 q^{66} - 2 q^{69} + 30 q^{70} + 6 q^{71} - 12 q^{74} - 14 q^{76} + 4 q^{79} - 4 q^{81} + 12 q^{84} - 36 q^{86} + 12 q^{89} - 6 q^{91} - 20 q^{94} + 30 q^{95} - 2 q^{96} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.618034i 0.356822i 0.983956 + 0.178411i \(0.0570957\pi\)
−0.983956 + 0.178411i \(0.942904\pi\)
\(4\) −1.00000 −0.500000
\(5\) 2.23607 1.00000
\(6\) 0.618034 0.252311
\(7\) 4.85410i 1.83468i 0.398107 + 0.917339i \(0.369667\pi\)
−0.398107 + 0.917339i \(0.630333\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.61803 0.872678
\(10\) − 2.23607i − 0.707107i
\(11\) −3.38197 −1.01970 −0.509851 0.860263i \(-0.670299\pi\)
−0.509851 + 0.860263i \(0.670299\pi\)
\(12\) − 0.618034i − 0.178411i
\(13\) − 0.381966i − 0.105938i −0.998596 0.0529692i \(-0.983131\pi\)
0.998596 0.0529692i \(-0.0168685\pi\)
\(14\) 4.85410 1.29731
\(15\) 1.38197i 0.356822i
\(16\) 1.00000 0.250000
\(17\) − 5.85410i − 1.41983i −0.704288 0.709914i \(-0.748734\pi\)
0.704288 0.709914i \(-0.251266\pi\)
\(18\) − 2.61803i − 0.617077i
\(19\) 6.85410 1.57244 0.786219 0.617947i \(-0.212036\pi\)
0.786219 + 0.617947i \(0.212036\pi\)
\(20\) −2.23607 −0.500000
\(21\) −3.00000 −0.654654
\(22\) 3.38197i 0.721038i
\(23\) − 1.00000i − 0.208514i
\(24\) −0.618034 −0.126156
\(25\) 5.00000 1.00000
\(26\) −0.381966 −0.0749097
\(27\) 3.47214i 0.668213i
\(28\) − 4.85410i − 0.917339i
\(29\) −3.70820 −0.688596 −0.344298 0.938860i \(-0.611883\pi\)
−0.344298 + 0.938860i \(0.611883\pi\)
\(30\) 1.38197 0.252311
\(31\) −8.85410 −1.59024 −0.795122 0.606450i \(-0.792593\pi\)
−0.795122 + 0.606450i \(0.792593\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 2.09017i − 0.363852i
\(34\) −5.85410 −1.00397
\(35\) 10.8541i 1.83468i
\(36\) −2.61803 −0.436339
\(37\) 3.70820i 0.609625i 0.952412 + 0.304812i \(0.0985938\pi\)
−0.952412 + 0.304812i \(0.901406\pi\)
\(38\) − 6.85410i − 1.11188i
\(39\) 0.236068 0.0378011
\(40\) 2.23607i 0.353553i
\(41\) −3.38197 −0.528174 −0.264087 0.964499i \(-0.585071\pi\)
−0.264087 + 0.964499i \(0.585071\pi\)
\(42\) 3.00000i 0.462910i
\(43\) − 6.76393i − 1.03149i −0.856742 0.515745i \(-0.827515\pi\)
0.856742 0.515745i \(-0.172485\pi\)
\(44\) 3.38197 0.509851
\(45\) 5.85410 0.872678
\(46\) −1.00000 −0.147442
\(47\) − 11.7082i − 1.70782i −0.520423 0.853909i \(-0.674226\pi\)
0.520423 0.853909i \(-0.325774\pi\)
\(48\) 0.618034i 0.0892055i
\(49\) −16.5623 −2.36604
\(50\) − 5.00000i − 0.707107i
\(51\) 3.61803 0.506626
\(52\) 0.381966i 0.0529692i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 3.47214 0.472498
\(55\) −7.56231 −1.01970
\(56\) −4.85410 −0.648657
\(57\) 4.23607i 0.561081i
\(58\) 3.70820i 0.486911i
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) − 1.38197i − 0.178411i
\(61\) −3.85410 −0.493467 −0.246734 0.969083i \(-0.579357\pi\)
−0.246734 + 0.969083i \(0.579357\pi\)
\(62\) 8.85410i 1.12447i
\(63\) 12.7082i 1.60108i
\(64\) −1.00000 −0.125000
\(65\) − 0.854102i − 0.105938i
\(66\) −2.09017 −0.257282
\(67\) 0.763932i 0.0933292i 0.998911 + 0.0466646i \(0.0148592\pi\)
−0.998911 + 0.0466646i \(0.985141\pi\)
\(68\) 5.85410i 0.709914i
\(69\) 0.618034 0.0744025
\(70\) 10.8541 1.29731
\(71\) 2.61803 0.310703 0.155352 0.987859i \(-0.450349\pi\)
0.155352 + 0.987859i \(0.450349\pi\)
\(72\) 2.61803i 0.308538i
\(73\) 7.52786i 0.881070i 0.897735 + 0.440535i \(0.145211\pi\)
−0.897735 + 0.440535i \(0.854789\pi\)
\(74\) 3.70820 0.431070
\(75\) 3.09017i 0.356822i
\(76\) −6.85410 −0.786219
\(77\) − 16.4164i − 1.87082i
\(78\) − 0.236068i − 0.0267294i
\(79\) −5.70820 −0.642223 −0.321112 0.947041i \(-0.604056\pi\)
−0.321112 + 0.947041i \(0.604056\pi\)
\(80\) 2.23607 0.250000
\(81\) 5.70820 0.634245
\(82\) 3.38197i 0.373476i
\(83\) 5.70820i 0.626557i 0.949661 + 0.313278i \(0.101427\pi\)
−0.949661 + 0.313278i \(0.898573\pi\)
\(84\) 3.00000 0.327327
\(85\) − 13.0902i − 1.41983i
\(86\) −6.76393 −0.729374
\(87\) − 2.29180i − 0.245706i
\(88\) − 3.38197i − 0.360519i
\(89\) 9.70820 1.02907 0.514534 0.857470i \(-0.327965\pi\)
0.514534 + 0.857470i \(0.327965\pi\)
\(90\) − 5.85410i − 0.617077i
\(91\) 1.85410 0.194363
\(92\) 1.00000i 0.104257i
\(93\) − 5.47214i − 0.567434i
\(94\) −11.7082 −1.20761
\(95\) 15.3262 1.57244
\(96\) 0.618034 0.0630778
\(97\) − 16.0344i − 1.62805i −0.580829 0.814025i \(-0.697272\pi\)
0.580829 0.814025i \(-0.302728\pi\)
\(98\) 16.5623i 1.67305i
\(99\) −8.85410 −0.889871
\(100\) −5.00000 −0.500000
\(101\) −10.4721 −1.04202 −0.521008 0.853552i \(-0.674444\pi\)
−0.521008 + 0.853552i \(0.674444\pi\)
\(102\) − 3.61803i − 0.358239i
\(103\) − 4.14590i − 0.408507i −0.978918 0.204254i \(-0.934523\pi\)
0.978918 0.204254i \(-0.0654768\pi\)
\(104\) 0.381966 0.0374548
\(105\) −6.70820 −0.654654
\(106\) 2.00000 0.194257
\(107\) − 1.70820i − 0.165138i −0.996585 0.0825692i \(-0.973687\pi\)
0.996585 0.0825692i \(-0.0263125\pi\)
\(108\) − 3.47214i − 0.334106i
\(109\) 12.5623 1.20325 0.601625 0.798778i \(-0.294520\pi\)
0.601625 + 0.798778i \(0.294520\pi\)
\(110\) 7.56231i 0.721038i
\(111\) −2.29180 −0.217528
\(112\) 4.85410i 0.458670i
\(113\) − 13.4164i − 1.26211i −0.775738 0.631055i \(-0.782622\pi\)
0.775738 0.631055i \(-0.217378\pi\)
\(114\) 4.23607 0.396744
\(115\) − 2.23607i − 0.208514i
\(116\) 3.70820 0.344298
\(117\) − 1.00000i − 0.0924500i
\(118\) − 6.00000i − 0.552345i
\(119\) 28.4164 2.60493
\(120\) −1.38197 −0.126156
\(121\) 0.437694 0.0397904
\(122\) 3.85410i 0.348934i
\(123\) − 2.09017i − 0.188464i
\(124\) 8.85410 0.795122
\(125\) 11.1803 1.00000
\(126\) 12.7082 1.13214
\(127\) 3.70820i 0.329050i 0.986373 + 0.164525i \(0.0526091\pi\)
−0.986373 + 0.164525i \(0.947391\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.18034 0.368058
\(130\) −0.854102 −0.0749097
\(131\) 8.18034 0.714720 0.357360 0.933967i \(-0.383677\pi\)
0.357360 + 0.933967i \(0.383677\pi\)
\(132\) 2.09017i 0.181926i
\(133\) 33.2705i 2.88492i
\(134\) 0.763932 0.0659937
\(135\) 7.76393i 0.668213i
\(136\) 5.85410 0.501985
\(137\) − 7.14590i − 0.610515i −0.952270 0.305258i \(-0.901257\pi\)
0.952270 0.305258i \(-0.0987426\pi\)
\(138\) − 0.618034i − 0.0526105i
\(139\) −17.7082 −1.50199 −0.750995 0.660308i \(-0.770426\pi\)
−0.750995 + 0.660308i \(0.770426\pi\)
\(140\) − 10.8541i − 0.917339i
\(141\) 7.23607 0.609387
\(142\) − 2.61803i − 0.219701i
\(143\) 1.29180i 0.108025i
\(144\) 2.61803 0.218169
\(145\) −8.29180 −0.688596
\(146\) 7.52786 0.623010
\(147\) − 10.2361i − 0.844257i
\(148\) − 3.70820i − 0.304812i
\(149\) 0.381966 0.0312919 0.0156459 0.999878i \(-0.495020\pi\)
0.0156459 + 0.999878i \(0.495020\pi\)
\(150\) 3.09017 0.252311
\(151\) −19.2705 −1.56821 −0.784106 0.620627i \(-0.786878\pi\)
−0.784106 + 0.620627i \(0.786878\pi\)
\(152\) 6.85410i 0.555941i
\(153\) − 15.3262i − 1.23905i
\(154\) −16.4164 −1.32287
\(155\) −19.7984 −1.59024
\(156\) −0.236068 −0.0189006
\(157\) 15.7082i 1.25365i 0.779160 + 0.626826i \(0.215646\pi\)
−0.779160 + 0.626826i \(0.784354\pi\)
\(158\) 5.70820i 0.454120i
\(159\) −1.23607 −0.0980266
\(160\) − 2.23607i − 0.176777i
\(161\) 4.85410 0.382557
\(162\) − 5.70820i − 0.448479i
\(163\) − 7.03444i − 0.550980i −0.961304 0.275490i \(-0.911160\pi\)
0.961304 0.275490i \(-0.0888401\pi\)
\(164\) 3.38197 0.264087
\(165\) − 4.67376i − 0.363852i
\(166\) 5.70820 0.443043
\(167\) 3.70820i 0.286949i 0.989654 + 0.143475i \(0.0458276\pi\)
−0.989654 + 0.143475i \(0.954172\pi\)
\(168\) − 3.00000i − 0.231455i
\(169\) 12.8541 0.988777
\(170\) −13.0902 −1.00397
\(171\) 17.9443 1.37223
\(172\) 6.76393i 0.515745i
\(173\) 3.56231i 0.270837i 0.990788 + 0.135419i \(0.0432379\pi\)
−0.990788 + 0.135419i \(0.956762\pi\)
\(174\) −2.29180 −0.173741
\(175\) 24.2705i 1.83468i
\(176\) −3.38197 −0.254925
\(177\) 3.70820i 0.278726i
\(178\) − 9.70820i − 0.727661i
\(179\) −16.4721 −1.23119 −0.615593 0.788065i \(-0.711083\pi\)
−0.615593 + 0.788065i \(0.711083\pi\)
\(180\) −5.85410 −0.436339
\(181\) −4.56231 −0.339114 −0.169557 0.985520i \(-0.554234\pi\)
−0.169557 + 0.985520i \(0.554234\pi\)
\(182\) − 1.85410i − 0.137435i
\(183\) − 2.38197i − 0.176080i
\(184\) 1.00000 0.0737210
\(185\) 8.29180i 0.609625i
\(186\) −5.47214 −0.401236
\(187\) 19.7984i 1.44780i
\(188\) 11.7082i 0.853909i
\(189\) −16.8541 −1.22596
\(190\) − 15.3262i − 1.11188i
\(191\) −1.41641 −0.102488 −0.0512438 0.998686i \(-0.516319\pi\)
−0.0512438 + 0.998686i \(0.516319\pi\)
\(192\) − 0.618034i − 0.0446028i
\(193\) 2.29180i 0.164967i 0.996592 + 0.0824835i \(0.0262852\pi\)
−0.996592 + 0.0824835i \(0.973715\pi\)
\(194\) −16.0344 −1.15121
\(195\) 0.527864 0.0378011
\(196\) 16.5623 1.18302
\(197\) − 0.437694i − 0.0311844i −0.999878 0.0155922i \(-0.995037\pi\)
0.999878 0.0155922i \(-0.00496335\pi\)
\(198\) 8.85410i 0.629234i
\(199\) 15.4164 1.09284 0.546420 0.837511i \(-0.315990\pi\)
0.546420 + 0.837511i \(0.315990\pi\)
\(200\) 5.00000i 0.353553i
\(201\) −0.472136 −0.0333019
\(202\) 10.4721i 0.736817i
\(203\) − 18.0000i − 1.26335i
\(204\) −3.61803 −0.253313
\(205\) −7.56231 −0.528174
\(206\) −4.14590 −0.288858
\(207\) − 2.61803i − 0.181966i
\(208\) − 0.381966i − 0.0264846i
\(209\) −23.1803 −1.60342
\(210\) 6.70820i 0.462910i
\(211\) 5.70820 0.392969 0.196484 0.980507i \(-0.437047\pi\)
0.196484 + 0.980507i \(0.437047\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 1.61803i 0.110866i
\(214\) −1.70820 −0.116770
\(215\) − 15.1246i − 1.03149i
\(216\) −3.47214 −0.236249
\(217\) − 42.9787i − 2.91759i
\(218\) − 12.5623i − 0.850827i
\(219\) −4.65248 −0.314385
\(220\) 7.56231 0.509851
\(221\) −2.23607 −0.150414
\(222\) 2.29180i 0.153815i
\(223\) 22.3607i 1.49738i 0.662919 + 0.748691i \(0.269317\pi\)
−0.662919 + 0.748691i \(0.730683\pi\)
\(224\) 4.85410 0.324328
\(225\) 13.0902 0.872678
\(226\) −13.4164 −0.892446
\(227\) 23.7082i 1.57357i 0.617228 + 0.786784i \(0.288256\pi\)
−0.617228 + 0.786784i \(0.711744\pi\)
\(228\) − 4.23607i − 0.280540i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −2.23607 −0.147442
\(231\) 10.1459 0.667551
\(232\) − 3.70820i − 0.243456i
\(233\) 19.1246i 1.25289i 0.779464 + 0.626447i \(0.215492\pi\)
−0.779464 + 0.626447i \(0.784508\pi\)
\(234\) −1.00000 −0.0653720
\(235\) − 26.1803i − 1.70782i
\(236\) −6.00000 −0.390567
\(237\) − 3.52786i − 0.229159i
\(238\) − 28.4164i − 1.84196i
\(239\) 1.52786 0.0988293 0.0494147 0.998778i \(-0.484264\pi\)
0.0494147 + 0.998778i \(0.484264\pi\)
\(240\) 1.38197i 0.0892055i
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) − 0.437694i − 0.0281360i
\(243\) 13.9443i 0.894525i
\(244\) 3.85410 0.246734
\(245\) −37.0344 −2.36604
\(246\) −2.09017 −0.133264
\(247\) − 2.61803i − 0.166582i
\(248\) − 8.85410i − 0.562236i
\(249\) −3.52786 −0.223569
\(250\) − 11.1803i − 0.707107i
\(251\) 7.79837 0.492229 0.246114 0.969241i \(-0.420846\pi\)
0.246114 + 0.969241i \(0.420846\pi\)
\(252\) − 12.7082i − 0.800542i
\(253\) 3.38197i 0.212622i
\(254\) 3.70820 0.232673
\(255\) 8.09017 0.506626
\(256\) 1.00000 0.0625000
\(257\) 15.4164i 0.961649i 0.876817 + 0.480825i \(0.159663\pi\)
−0.876817 + 0.480825i \(0.840337\pi\)
\(258\) − 4.18034i − 0.260257i
\(259\) −18.0000 −1.11847
\(260\) 0.854102i 0.0529692i
\(261\) −9.70820 −0.600923
\(262\) − 8.18034i − 0.505383i
\(263\) − 15.2705i − 0.941620i −0.882235 0.470810i \(-0.843962\pi\)
0.882235 0.470810i \(-0.156038\pi\)
\(264\) 2.09017 0.128641
\(265\) 4.47214i 0.274721i
\(266\) 33.2705 2.03995
\(267\) 6.00000i 0.367194i
\(268\) − 0.763932i − 0.0466646i
\(269\) 10.4721 0.638497 0.319249 0.947671i \(-0.396569\pi\)
0.319249 + 0.947671i \(0.396569\pi\)
\(270\) 7.76393 0.472498
\(271\) 19.1459 1.16303 0.581515 0.813536i \(-0.302460\pi\)
0.581515 + 0.813536i \(0.302460\pi\)
\(272\) − 5.85410i − 0.354957i
\(273\) 1.14590i 0.0693529i
\(274\) −7.14590 −0.431699
\(275\) −16.9098 −1.01970
\(276\) −0.618034 −0.0372013
\(277\) 16.4721i 0.989715i 0.868974 + 0.494857i \(0.164780\pi\)
−0.868974 + 0.494857i \(0.835220\pi\)
\(278\) 17.7082i 1.06207i
\(279\) −23.1803 −1.38777
\(280\) −10.8541 −0.648657
\(281\) −0.652476 −0.0389234 −0.0194617 0.999811i \(-0.506195\pi\)
−0.0194617 + 0.999811i \(0.506195\pi\)
\(282\) − 7.23607i − 0.430902i
\(283\) − 3.05573i − 0.181644i −0.995867 0.0908221i \(-0.971051\pi\)
0.995867 0.0908221i \(-0.0289495\pi\)
\(284\) −2.61803 −0.155352
\(285\) 9.47214i 0.561081i
\(286\) 1.29180 0.0763855
\(287\) − 16.4164i − 0.969030i
\(288\) − 2.61803i − 0.154269i
\(289\) −17.2705 −1.01591
\(290\) 8.29180i 0.486911i
\(291\) 9.90983 0.580925
\(292\) − 7.52786i − 0.440535i
\(293\) 27.7082i 1.61873i 0.587306 + 0.809365i \(0.300189\pi\)
−0.587306 + 0.809365i \(0.699811\pi\)
\(294\) −10.2361 −0.596980
\(295\) 13.4164 0.781133
\(296\) −3.70820 −0.215535
\(297\) − 11.7426i − 0.681377i
\(298\) − 0.381966i − 0.0221267i
\(299\) −0.381966 −0.0220897
\(300\) − 3.09017i − 0.178411i
\(301\) 32.8328 1.89245
\(302\) 19.2705i 1.10889i
\(303\) − 6.47214i − 0.371814i
\(304\) 6.85410 0.393110
\(305\) −8.61803 −0.493467
\(306\) −15.3262 −0.876143
\(307\) 10.1459i 0.579057i 0.957169 + 0.289528i \(0.0934985\pi\)
−0.957169 + 0.289528i \(0.906502\pi\)
\(308\) 16.4164i 0.935412i
\(309\) 2.56231 0.145764
\(310\) 19.7984i 1.12447i
\(311\) −13.5279 −0.767095 −0.383547 0.923521i \(-0.625298\pi\)
−0.383547 + 0.923521i \(0.625298\pi\)
\(312\) 0.236068i 0.0133647i
\(313\) − 19.1459i − 1.08219i −0.840961 0.541095i \(-0.818010\pi\)
0.840961 0.541095i \(-0.181990\pi\)
\(314\) 15.7082 0.886465
\(315\) 28.4164i 1.60108i
\(316\) 5.70820 0.321112
\(317\) 26.2705i 1.47550i 0.675075 + 0.737749i \(0.264111\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(318\) 1.23607i 0.0693153i
\(319\) 12.5410 0.702162
\(320\) −2.23607 −0.125000
\(321\) 1.05573 0.0589250
\(322\) − 4.85410i − 0.270509i
\(323\) − 40.1246i − 2.23259i
\(324\) −5.70820 −0.317122
\(325\) − 1.90983i − 0.105938i
\(326\) −7.03444 −0.389602
\(327\) 7.76393i 0.429346i
\(328\) − 3.38197i − 0.186738i
\(329\) 56.8328 3.13329
\(330\) −4.67376 −0.257282
\(331\) 15.1246 0.831324 0.415662 0.909519i \(-0.363550\pi\)
0.415662 + 0.909519i \(0.363550\pi\)
\(332\) − 5.70820i − 0.313278i
\(333\) 9.70820i 0.532006i
\(334\) 3.70820 0.202904
\(335\) 1.70820i 0.0933292i
\(336\) −3.00000 −0.163663
\(337\) − 5.61803i − 0.306034i −0.988224 0.153017i \(-0.951101\pi\)
0.988224 0.153017i \(-0.0488989\pi\)
\(338\) − 12.8541i − 0.699171i
\(339\) 8.29180 0.450349
\(340\) 13.0902i 0.709914i
\(341\) 29.9443 1.62157
\(342\) − 17.9443i − 0.970315i
\(343\) − 46.4164i − 2.50625i
\(344\) 6.76393 0.364687
\(345\) 1.38197 0.0744025
\(346\) 3.56231 0.191511
\(347\) − 5.56231i − 0.298600i −0.988792 0.149300i \(-0.952298\pi\)
0.988792 0.149300i \(-0.0477021\pi\)
\(348\) 2.29180i 0.122853i
\(349\) −14.2918 −0.765022 −0.382511 0.923951i \(-0.624941\pi\)
−0.382511 + 0.923951i \(0.624941\pi\)
\(350\) 24.2705 1.29731
\(351\) 1.32624 0.0707893
\(352\) 3.38197i 0.180259i
\(353\) − 8.00000i − 0.425797i −0.977074 0.212899i \(-0.931710\pi\)
0.977074 0.212899i \(-0.0682904\pi\)
\(354\) 3.70820 0.197089
\(355\) 5.85410 0.310703
\(356\) −9.70820 −0.514534
\(357\) 17.5623i 0.929496i
\(358\) 16.4721i 0.870579i
\(359\) −13.4164 −0.708091 −0.354045 0.935228i \(-0.615194\pi\)
−0.354045 + 0.935228i \(0.615194\pi\)
\(360\) 5.85410i 0.308538i
\(361\) 27.9787 1.47256
\(362\) 4.56231i 0.239789i
\(363\) 0.270510i 0.0141981i
\(364\) −1.85410 −0.0971813
\(365\) 16.8328i 0.881070i
\(366\) −2.38197 −0.124507
\(367\) 12.0000i 0.626395i 0.949688 + 0.313197i \(0.101400\pi\)
−0.949688 + 0.313197i \(0.898600\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) −8.85410 −0.460926
\(370\) 8.29180 0.431070
\(371\) −9.70820 −0.504025
\(372\) 5.47214i 0.283717i
\(373\) 3.05573i 0.158220i 0.996866 + 0.0791098i \(0.0252078\pi\)
−0.996866 + 0.0791098i \(0.974792\pi\)
\(374\) 19.7984 1.02375
\(375\) 6.90983i 0.356822i
\(376\) 11.7082 0.603805
\(377\) 1.41641i 0.0729487i
\(378\) 16.8541i 0.866881i
\(379\) −9.27051 −0.476194 −0.238097 0.971241i \(-0.576524\pi\)
−0.238097 + 0.971241i \(0.576524\pi\)
\(380\) −15.3262 −0.786219
\(381\) −2.29180 −0.117412
\(382\) 1.41641i 0.0724697i
\(383\) − 27.4164i − 1.40091i −0.713695 0.700456i \(-0.752980\pi\)
0.713695 0.700456i \(-0.247020\pi\)
\(384\) −0.618034 −0.0315389
\(385\) − 36.7082i − 1.87082i
\(386\) 2.29180 0.116649
\(387\) − 17.7082i − 0.900159i
\(388\) 16.0344i 0.814025i
\(389\) 5.67376 0.287671 0.143836 0.989602i \(-0.454056\pi\)
0.143836 + 0.989602i \(0.454056\pi\)
\(390\) − 0.527864i − 0.0267294i
\(391\) −5.85410 −0.296055
\(392\) − 16.5623i − 0.836523i
\(393\) 5.05573i 0.255028i
\(394\) −0.437694 −0.0220507
\(395\) −12.7639 −0.642223
\(396\) 8.85410 0.444935
\(397\) 35.5623i 1.78482i 0.451224 + 0.892410i \(0.350987\pi\)
−0.451224 + 0.892410i \(0.649013\pi\)
\(398\) − 15.4164i − 0.772755i
\(399\) −20.5623 −1.02940
\(400\) 5.00000 0.250000
\(401\) −34.4721 −1.72146 −0.860728 0.509065i \(-0.829991\pi\)
−0.860728 + 0.509065i \(0.829991\pi\)
\(402\) 0.472136i 0.0235480i
\(403\) 3.38197i 0.168468i
\(404\) 10.4721 0.521008
\(405\) 12.7639 0.634245
\(406\) −18.0000 −0.893325
\(407\) − 12.5410i − 0.621635i
\(408\) 3.61803i 0.179119i
\(409\) 6.56231 0.324485 0.162243 0.986751i \(-0.448127\pi\)
0.162243 + 0.986751i \(0.448127\pi\)
\(410\) 7.56231i 0.373476i
\(411\) 4.41641 0.217845
\(412\) 4.14590i 0.204254i
\(413\) 29.1246i 1.43313i
\(414\) −2.61803 −0.128669
\(415\) 12.7639i 0.626557i
\(416\) −0.381966 −0.0187274
\(417\) − 10.9443i − 0.535943i
\(418\) 23.1803i 1.13379i
\(419\) −5.88854 −0.287674 −0.143837 0.989601i \(-0.545944\pi\)
−0.143837 + 0.989601i \(0.545944\pi\)
\(420\) 6.70820 0.327327
\(421\) 6.85410 0.334048 0.167024 0.985953i \(-0.446584\pi\)
0.167024 + 0.985953i \(0.446584\pi\)
\(422\) − 5.70820i − 0.277871i
\(423\) − 30.6525i − 1.49037i
\(424\) −2.00000 −0.0971286
\(425\) − 29.2705i − 1.41983i
\(426\) 1.61803 0.0783940
\(427\) − 18.7082i − 0.905353i
\(428\) 1.70820i 0.0825692i
\(429\) −0.798374 −0.0385459
\(430\) −15.1246 −0.729374
\(431\) −6.76393 −0.325807 −0.162904 0.986642i \(-0.552086\pi\)
−0.162904 + 0.986642i \(0.552086\pi\)
\(432\) 3.47214i 0.167053i
\(433\) 29.6180i 1.42335i 0.702508 + 0.711676i \(0.252064\pi\)
−0.702508 + 0.711676i \(0.747936\pi\)
\(434\) −42.9787 −2.06304
\(435\) − 5.12461i − 0.245706i
\(436\) −12.5623 −0.601625
\(437\) − 6.85410i − 0.327876i
\(438\) 4.65248i 0.222304i
\(439\) 0.270510 0.0129107 0.00645536 0.999979i \(-0.497945\pi\)
0.00645536 + 0.999979i \(0.497945\pi\)
\(440\) − 7.56231i − 0.360519i
\(441\) −43.3607 −2.06479
\(442\) 2.23607i 0.106359i
\(443\) − 6.85410i − 0.325648i −0.986655 0.162824i \(-0.947940\pi\)
0.986655 0.162824i \(-0.0520603\pi\)
\(444\) 2.29180 0.108764
\(445\) 21.7082 1.02907
\(446\) 22.3607 1.05881
\(447\) 0.236068i 0.0111656i
\(448\) − 4.85410i − 0.229335i
\(449\) −2.56231 −0.120923 −0.0604613 0.998171i \(-0.519257\pi\)
−0.0604613 + 0.998171i \(0.519257\pi\)
\(450\) − 13.0902i − 0.617077i
\(451\) 11.4377 0.538580
\(452\) 13.4164i 0.631055i
\(453\) − 11.9098i − 0.559573i
\(454\) 23.7082 1.11268
\(455\) 4.14590 0.194363
\(456\) −4.23607 −0.198372
\(457\) − 25.4164i − 1.18893i −0.804122 0.594465i \(-0.797364\pi\)
0.804122 0.594465i \(-0.202636\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 20.3262 0.948748
\(460\) 2.23607i 0.104257i
\(461\) −25.3050 −1.17857 −0.589285 0.807926i \(-0.700590\pi\)
−0.589285 + 0.807926i \(0.700590\pi\)
\(462\) − 10.1459i − 0.472030i
\(463\) 2.29180i 0.106509i 0.998581 + 0.0532544i \(0.0169594\pi\)
−0.998581 + 0.0532544i \(0.983041\pi\)
\(464\) −3.70820 −0.172149
\(465\) − 12.2361i − 0.567434i
\(466\) 19.1246 0.885931
\(467\) − 17.1246i − 0.792433i −0.918157 0.396216i \(-0.870323\pi\)
0.918157 0.396216i \(-0.129677\pi\)
\(468\) 1.00000i 0.0462250i
\(469\) −3.70820 −0.171229
\(470\) −26.1803 −1.20761
\(471\) −9.70820 −0.447330
\(472\) 6.00000i 0.276172i
\(473\) 22.8754i 1.05181i
\(474\) −3.52786 −0.162040
\(475\) 34.2705 1.57244
\(476\) −28.4164 −1.30246
\(477\) 5.23607i 0.239743i
\(478\) − 1.52786i − 0.0698829i
\(479\) 19.5279 0.892251 0.446125 0.894970i \(-0.352804\pi\)
0.446125 + 0.894970i \(0.352804\pi\)
\(480\) 1.38197 0.0630778
\(481\) 1.41641 0.0645826
\(482\) 2.00000i 0.0910975i
\(483\) 3.00000i 0.136505i
\(484\) −0.437694 −0.0198952
\(485\) − 35.8541i − 1.62805i
\(486\) 13.9443 0.632525
\(487\) − 21.7082i − 0.983693i −0.870682 0.491846i \(-0.836322\pi\)
0.870682 0.491846i \(-0.163678\pi\)
\(488\) − 3.85410i − 0.174467i
\(489\) 4.34752 0.196602
\(490\) 37.0344i 1.67305i
\(491\) −26.8328 −1.21095 −0.605474 0.795865i \(-0.707016\pi\)
−0.605474 + 0.795865i \(0.707016\pi\)
\(492\) 2.09017i 0.0942321i
\(493\) 21.7082i 0.977688i
\(494\) −2.61803 −0.117791
\(495\) −19.7984 −0.889871
\(496\) −8.85410 −0.397561
\(497\) 12.7082i 0.570041i
\(498\) 3.52786i 0.158087i
\(499\) −11.4164 −0.511069 −0.255534 0.966800i \(-0.582251\pi\)
−0.255534 + 0.966800i \(0.582251\pi\)
\(500\) −11.1803 −0.500000
\(501\) −2.29180 −0.102390
\(502\) − 7.79837i − 0.348058i
\(503\) − 16.8541i − 0.751487i −0.926724 0.375744i \(-0.877387\pi\)
0.926724 0.375744i \(-0.122613\pi\)
\(504\) −12.7082 −0.566068
\(505\) −23.4164 −1.04202
\(506\) 3.38197 0.150347
\(507\) 7.94427i 0.352818i
\(508\) − 3.70820i − 0.164525i
\(509\) 24.6525 1.09270 0.546351 0.837556i \(-0.316017\pi\)
0.546351 + 0.837556i \(0.316017\pi\)
\(510\) − 8.09017i − 0.358239i
\(511\) −36.5410 −1.61648
\(512\) − 1.00000i − 0.0441942i
\(513\) 23.7984i 1.05072i
\(514\) 15.4164 0.679989
\(515\) − 9.27051i − 0.408507i
\(516\) −4.18034 −0.184029
\(517\) 39.5967i 1.74146i
\(518\) 18.0000i 0.790875i
\(519\) −2.20163 −0.0966407
\(520\) 0.854102 0.0374548
\(521\) 20.0689 0.879234 0.439617 0.898185i \(-0.355114\pi\)
0.439617 + 0.898185i \(0.355114\pi\)
\(522\) 9.70820i 0.424917i
\(523\) 40.3607i 1.76485i 0.470454 + 0.882425i \(0.344090\pi\)
−0.470454 + 0.882425i \(0.655910\pi\)
\(524\) −8.18034 −0.357360
\(525\) −15.0000 −0.654654
\(526\) −15.2705 −0.665826
\(527\) 51.8328i 2.25787i
\(528\) − 2.09017i − 0.0909630i
\(529\) −1.00000 −0.0434783
\(530\) 4.47214 0.194257
\(531\) 15.7082 0.681678
\(532\) − 33.2705i − 1.44246i
\(533\) 1.29180i 0.0559539i
\(534\) 6.00000 0.259645
\(535\) − 3.81966i − 0.165138i
\(536\) −0.763932 −0.0329968
\(537\) − 10.1803i − 0.439314i
\(538\) − 10.4721i − 0.451486i
\(539\) 56.0132 2.41266
\(540\) − 7.76393i − 0.334106i
\(541\) −27.4164 −1.17872 −0.589362 0.807869i \(-0.700621\pi\)
−0.589362 + 0.807869i \(0.700621\pi\)
\(542\) − 19.1459i − 0.822387i
\(543\) − 2.81966i − 0.121003i
\(544\) −5.85410 −0.250993
\(545\) 28.0902 1.20325
\(546\) 1.14590 0.0490399
\(547\) − 1.14590i − 0.0489951i −0.999700 0.0244975i \(-0.992201\pi\)
0.999700 0.0244975i \(-0.00779859\pi\)
\(548\) 7.14590i 0.305258i
\(549\) −10.0902 −0.430638
\(550\) 16.9098i 0.721038i
\(551\) −25.4164 −1.08278
\(552\) 0.618034i 0.0263053i
\(553\) − 27.7082i − 1.17827i
\(554\) 16.4721 0.699834
\(555\) −5.12461 −0.217528
\(556\) 17.7082 0.750995
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) 23.1803i 0.981302i
\(559\) −2.58359 −0.109274
\(560\) 10.8541i 0.458670i
\(561\) −12.2361 −0.516607
\(562\) 0.652476i 0.0275230i
\(563\) 2.29180i 0.0965877i 0.998833 + 0.0482938i \(0.0153784\pi\)
−0.998833 + 0.0482938i \(0.984622\pi\)
\(564\) −7.23607 −0.304693
\(565\) − 30.0000i − 1.26211i
\(566\) −3.05573 −0.128442
\(567\) 27.7082i 1.16364i
\(568\) 2.61803i 0.109850i
\(569\) −28.3607 −1.18894 −0.594471 0.804117i \(-0.702638\pi\)
−0.594471 + 0.804117i \(0.702638\pi\)
\(570\) 9.47214 0.396744
\(571\) −25.2705 −1.05754 −0.528769 0.848766i \(-0.677346\pi\)
−0.528769 + 0.848766i \(0.677346\pi\)
\(572\) − 1.29180i − 0.0540127i
\(573\) − 0.875388i − 0.0365699i
\(574\) −16.4164 −0.685208
\(575\) − 5.00000i − 0.208514i
\(576\) −2.61803 −0.109085
\(577\) − 26.2918i − 1.09454i −0.836956 0.547271i \(-0.815667\pi\)
0.836956 0.547271i \(-0.184333\pi\)
\(578\) 17.2705i 0.718359i
\(579\) −1.41641 −0.0588639
\(580\) 8.29180 0.344298
\(581\) −27.7082 −1.14953
\(582\) − 9.90983i − 0.410776i
\(583\) − 6.76393i − 0.280133i
\(584\) −7.52786 −0.311505
\(585\) − 2.23607i − 0.0924500i
\(586\) 27.7082 1.14462
\(587\) − 2.14590i − 0.0885707i −0.999019 0.0442853i \(-0.985899\pi\)
0.999019 0.0442853i \(-0.0141011\pi\)
\(588\) 10.2361i 0.422128i
\(589\) −60.6869 −2.50056
\(590\) − 13.4164i − 0.552345i
\(591\) 0.270510 0.0111273
\(592\) 3.70820i 0.152406i
\(593\) − 33.4164i − 1.37225i −0.727485 0.686124i \(-0.759311\pi\)
0.727485 0.686124i \(-0.240689\pi\)
\(594\) −11.7426 −0.481807
\(595\) 63.5410 2.60493
\(596\) −0.381966 −0.0156459
\(597\) 9.52786i 0.389950i
\(598\) 0.381966i 0.0156198i
\(599\) −21.3820 −0.873643 −0.436822 0.899548i \(-0.643896\pi\)
−0.436822 + 0.899548i \(0.643896\pi\)
\(600\) −3.09017 −0.126156
\(601\) −3.14590 −0.128324 −0.0641619 0.997940i \(-0.520437\pi\)
−0.0641619 + 0.997940i \(0.520437\pi\)
\(602\) − 32.8328i − 1.33817i
\(603\) 2.00000i 0.0814463i
\(604\) 19.2705 0.784106
\(605\) 0.978714 0.0397904
\(606\) −6.47214 −0.262913
\(607\) 38.9443i 1.58070i 0.612656 + 0.790350i \(0.290101\pi\)
−0.612656 + 0.790350i \(0.709899\pi\)
\(608\) − 6.85410i − 0.277971i
\(609\) 11.1246 0.450792
\(610\) 8.61803i 0.348934i
\(611\) −4.47214 −0.180923
\(612\) 15.3262i 0.619526i
\(613\) 5.23607i 0.211483i 0.994394 + 0.105741i \(0.0337216\pi\)
−0.994394 + 0.105741i \(0.966278\pi\)
\(614\) 10.1459 0.409455
\(615\) − 4.67376i − 0.188464i
\(616\) 16.4164 0.661436
\(617\) 17.5623i 0.707032i 0.935429 + 0.353516i \(0.115014\pi\)
−0.935429 + 0.353516i \(0.884986\pi\)
\(618\) − 2.56231i − 0.103071i
\(619\) 21.1459 0.849925 0.424963 0.905211i \(-0.360287\pi\)
0.424963 + 0.905211i \(0.360287\pi\)
\(620\) 19.7984 0.795122
\(621\) 3.47214 0.139332
\(622\) 13.5279i 0.542418i
\(623\) 47.1246i 1.88801i
\(624\) 0.236068 0.00945028
\(625\) 25.0000 1.00000
\(626\) −19.1459 −0.765224
\(627\) − 14.3262i − 0.572135i
\(628\) − 15.7082i − 0.626826i
\(629\) 21.7082 0.865563
\(630\) 28.4164 1.13214
\(631\) 3.41641 0.136005 0.0680025 0.997685i \(-0.478337\pi\)
0.0680025 + 0.997685i \(0.478337\pi\)
\(632\) − 5.70820i − 0.227060i
\(633\) 3.52786i 0.140220i
\(634\) 26.2705 1.04334
\(635\) 8.29180i 0.329050i
\(636\) 1.23607 0.0490133
\(637\) 6.32624i 0.250655i
\(638\) − 12.5410i − 0.496504i
\(639\) 6.85410 0.271144
\(640\) 2.23607i 0.0883883i
\(641\) 30.6525 1.21070 0.605350 0.795959i \(-0.293033\pi\)
0.605350 + 0.795959i \(0.293033\pi\)
\(642\) − 1.05573i − 0.0416663i
\(643\) − 35.0132i − 1.38078i −0.723435 0.690392i \(-0.757438\pi\)
0.723435 0.690392i \(-0.242562\pi\)
\(644\) −4.85410 −0.191278
\(645\) 9.34752 0.368058
\(646\) −40.1246 −1.57868
\(647\) − 0.291796i − 0.0114717i −0.999984 0.00573584i \(-0.998174\pi\)
0.999984 0.00573584i \(-0.00182579\pi\)
\(648\) 5.70820i 0.224239i
\(649\) −20.2918 −0.796523
\(650\) −1.90983 −0.0749097
\(651\) 26.5623 1.04106
\(652\) 7.03444i 0.275490i
\(653\) 11.9787i 0.468763i 0.972145 + 0.234382i \(0.0753065\pi\)
−0.972145 + 0.234382i \(0.924693\pi\)
\(654\) 7.76393 0.303594
\(655\) 18.2918 0.714720
\(656\) −3.38197 −0.132044
\(657\) 19.7082i 0.768890i
\(658\) − 56.8328i − 2.21557i
\(659\) −17.8885 −0.696839 −0.348419 0.937339i \(-0.613281\pi\)
−0.348419 + 0.937339i \(0.613281\pi\)
\(660\) 4.67376i 0.181926i
\(661\) 34.3951 1.33782 0.668908 0.743346i \(-0.266762\pi\)
0.668908 + 0.743346i \(0.266762\pi\)
\(662\) − 15.1246i − 0.587835i
\(663\) − 1.38197i − 0.0536711i
\(664\) −5.70820 −0.221521
\(665\) 74.3951i 2.88492i
\(666\) 9.70820 0.376185
\(667\) 3.70820i 0.143582i
\(668\) − 3.70820i − 0.143475i
\(669\) −13.8197 −0.534299
\(670\) 1.70820 0.0659937
\(671\) 13.0344 0.503189
\(672\) 3.00000i 0.115728i
\(673\) 14.2918i 0.550908i 0.961314 + 0.275454i \(0.0888282\pi\)
−0.961314 + 0.275454i \(0.911172\pi\)
\(674\) −5.61803 −0.216399
\(675\) 17.3607i 0.668213i
\(676\) −12.8541 −0.494389
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) − 8.29180i − 0.318445i
\(679\) 77.8328 2.98695
\(680\) 13.0902 0.501985
\(681\) −14.6525 −0.561484
\(682\) − 29.9443i − 1.14663i
\(683\) − 35.9787i − 1.37669i −0.725385 0.688344i \(-0.758338\pi\)
0.725385 0.688344i \(-0.241662\pi\)
\(684\) −17.9443 −0.686116
\(685\) − 15.9787i − 0.610515i
\(686\) −46.4164 −1.77219
\(687\) 6.18034i 0.235795i
\(688\) − 6.76393i − 0.257872i
\(689\) 0.763932 0.0291035
\(690\) − 1.38197i − 0.0526105i
\(691\) 19.1246 0.727535 0.363767 0.931490i \(-0.381490\pi\)
0.363767 + 0.931490i \(0.381490\pi\)
\(692\) − 3.56231i − 0.135419i
\(693\) − 42.9787i − 1.63263i
\(694\) −5.56231 −0.211142
\(695\) −39.5967 −1.50199
\(696\) 2.29180 0.0868703
\(697\) 19.7984i 0.749917i
\(698\) 14.2918i 0.540952i
\(699\) −11.8197 −0.447061
\(700\) − 24.2705i − 0.917339i
\(701\) 36.9230 1.39456 0.697281 0.716798i \(-0.254393\pi\)
0.697281 + 0.716798i \(0.254393\pi\)
\(702\) − 1.32624i − 0.0500556i
\(703\) 25.4164i 0.958598i
\(704\) 3.38197 0.127463
\(705\) 16.1803 0.609387
\(706\) −8.00000 −0.301084
\(707\) − 50.8328i − 1.91176i
\(708\) − 3.70820i − 0.139363i
\(709\) 11.5623 0.434232 0.217116 0.976146i \(-0.430335\pi\)
0.217116 + 0.976146i \(0.430335\pi\)
\(710\) − 5.85410i − 0.219701i
\(711\) −14.9443 −0.560454
\(712\) 9.70820i 0.363830i
\(713\) 8.85410i 0.331589i
\(714\) 17.5623 0.657253
\(715\) 2.88854i 0.108025i
\(716\) 16.4721 0.615593
\(717\) 0.944272i 0.0352645i
\(718\) 13.4164i 0.500696i
\(719\) 32.4508 1.21021 0.605106 0.796145i \(-0.293131\pi\)
0.605106 + 0.796145i \(0.293131\pi\)
\(720\) 5.85410 0.218169
\(721\) 20.1246 0.749480
\(722\) − 27.9787i − 1.04126i
\(723\) − 1.23607i − 0.0459699i
\(724\) 4.56231 0.169557
\(725\) −18.5410 −0.688596
\(726\) 0.270510 0.0100396
\(727\) − 1.74265i − 0.0646312i −0.999478 0.0323156i \(-0.989712\pi\)
0.999478 0.0323156i \(-0.0102882\pi\)
\(728\) 1.85410i 0.0687176i
\(729\) 8.50658 0.315058
\(730\) 16.8328 0.623010
\(731\) −39.5967 −1.46454
\(732\) 2.38197i 0.0880400i
\(733\) − 35.8885i − 1.32557i −0.748808 0.662787i \(-0.769374\pi\)
0.748808 0.662787i \(-0.230626\pi\)
\(734\) 12.0000 0.442928
\(735\) − 22.8885i − 0.844257i
\(736\) −1.00000 −0.0368605
\(737\) − 2.58359i − 0.0951678i
\(738\) 8.85410i 0.325924i
\(739\) 32.5410 1.19704 0.598520 0.801108i \(-0.295756\pi\)
0.598520 + 0.801108i \(0.295756\pi\)
\(740\) − 8.29180i − 0.304812i
\(741\) 1.61803 0.0594400
\(742\) 9.70820i 0.356399i
\(743\) 9.97871i 0.366084i 0.983105 + 0.183042i \(0.0585944\pi\)
−0.983105 + 0.183042i \(0.941406\pi\)
\(744\) 5.47214 0.200618
\(745\) 0.854102 0.0312919
\(746\) 3.05573 0.111878
\(747\) 14.9443i 0.546782i
\(748\) − 19.7984i − 0.723900i
\(749\) 8.29180 0.302976
\(750\) 6.90983 0.252311
\(751\) −41.1246 −1.50066 −0.750329 0.661064i \(-0.770105\pi\)
−0.750329 + 0.661064i \(0.770105\pi\)
\(752\) − 11.7082i − 0.426954i
\(753\) 4.81966i 0.175638i
\(754\) 1.41641 0.0515825
\(755\) −43.0902 −1.56821
\(756\) 16.8541 0.612978
\(757\) − 47.0132i − 1.70872i −0.519680 0.854361i \(-0.673949\pi\)
0.519680 0.854361i \(-0.326051\pi\)
\(758\) 9.27051i 0.336720i
\(759\) −2.09017 −0.0758684
\(760\) 15.3262i 0.555941i
\(761\) −11.5066 −0.417113 −0.208557 0.978010i \(-0.566877\pi\)
−0.208557 + 0.978010i \(0.566877\pi\)
\(762\) 2.29180i 0.0830230i
\(763\) 60.9787i 2.20758i
\(764\) 1.41641 0.0512438
\(765\) − 34.2705i − 1.23905i
\(766\) −27.4164 −0.990595
\(767\) − 2.29180i − 0.0827520i
\(768\) 0.618034i 0.0223014i
\(769\) 34.2918 1.23659 0.618297 0.785945i \(-0.287823\pi\)
0.618297 + 0.785945i \(0.287823\pi\)
\(770\) −36.7082 −1.32287
\(771\) −9.52786 −0.343138
\(772\) − 2.29180i − 0.0824835i
\(773\) − 14.0000i − 0.503545i −0.967786 0.251773i \(-0.918987\pi\)
0.967786 0.251773i \(-0.0810135\pi\)
\(774\) −17.7082 −0.636508
\(775\) −44.2705 −1.59024
\(776\) 16.0344 0.575603
\(777\) − 11.1246i − 0.399093i
\(778\) − 5.67376i − 0.203414i
\(779\) −23.1803 −0.830522
\(780\) −0.527864 −0.0189006
\(781\) −8.85410 −0.316825
\(782\) 5.85410i 0.209342i
\(783\) − 12.8754i − 0.460129i
\(784\) −16.5623 −0.591511
\(785\) 35.1246i 1.25365i
\(786\) 5.05573 0.180332
\(787\) 31.4164i 1.11987i 0.828535 + 0.559937i \(0.189175\pi\)
−0.828535 + 0.559937i \(0.810825\pi\)
\(788\) 0.437694i 0.0155922i
\(789\) 9.43769 0.335991
\(790\) 12.7639i 0.454120i
\(791\) 65.1246 2.31556
\(792\) − 8.85410i − 0.314617i
\(793\) 1.47214i 0.0522771i
\(794\) 35.5623 1.26206
\(795\) −2.76393 −0.0980266
\(796\) −15.4164 −0.546420
\(797\) 18.5836i 0.658265i 0.944284 + 0.329132i \(0.106756\pi\)
−0.944284 + 0.329132i \(0.893244\pi\)
\(798\) 20.5623i 0.727898i
\(799\) −68.5410 −2.42481
\(800\) − 5.00000i − 0.176777i
\(801\) 25.4164 0.898045
\(802\) 34.4721i 1.21725i
\(803\) − 25.4590i − 0.898428i
\(804\) 0.472136 0.0166510
\(805\) 10.8541 0.382557
\(806\) 3.38197 0.119125
\(807\) 6.47214i 0.227830i
\(808\) − 10.4721i − 0.368408i
\(809\) −27.1591 −0.954861 −0.477431 0.878669i \(-0.658432\pi\)
−0.477431 + 0.878669i \(0.658432\pi\)
\(810\) − 12.7639i − 0.448479i
\(811\) −7.70820 −0.270672 −0.135336 0.990800i \(-0.543211\pi\)
−0.135336 + 0.990800i \(0.543211\pi\)
\(812\) 18.0000i 0.631676i
\(813\) 11.8328i 0.414995i
\(814\) −12.5410 −0.439563
\(815\) − 15.7295i − 0.550980i
\(816\) 3.61803 0.126657
\(817\) − 46.3607i − 1.62195i
\(818\) − 6.56231i − 0.229446i
\(819\) 4.85410 0.169616
\(820\) 7.56231 0.264087
\(821\) −2.94427 −0.102756 −0.0513779 0.998679i \(-0.516361\pi\)
−0.0513779 + 0.998679i \(0.516361\pi\)
\(822\) − 4.41641i − 0.154040i
\(823\) − 20.8328i − 0.726186i −0.931753 0.363093i \(-0.881721\pi\)
0.931753 0.363093i \(-0.118279\pi\)
\(824\) 4.14590 0.144429
\(825\) − 10.4508i − 0.363852i
\(826\) 29.1246 1.01337
\(827\) 48.2492i 1.67779i 0.544293 + 0.838895i \(0.316798\pi\)
−0.544293 + 0.838895i \(0.683202\pi\)
\(828\) 2.61803i 0.0909830i
\(829\) −32.5410 −1.13020 −0.565098 0.825024i \(-0.691162\pi\)
−0.565098 + 0.825024i \(0.691162\pi\)
\(830\) 12.7639 0.443043
\(831\) −10.1803 −0.353152
\(832\) 0.381966i 0.0132423i
\(833\) 96.9574i 3.35938i
\(834\) −10.9443 −0.378969
\(835\) 8.29180i 0.286949i
\(836\) 23.1803 0.801709
\(837\) − 30.7426i − 1.06262i
\(838\) 5.88854i 0.203416i
\(839\) 6.76393 0.233517 0.116758 0.993160i \(-0.462750\pi\)
0.116758 + 0.993160i \(0.462750\pi\)
\(840\) − 6.70820i − 0.231455i
\(841\) −15.2492 −0.525835
\(842\) − 6.85410i − 0.236208i
\(843\) − 0.403252i − 0.0138887i
\(844\) −5.70820 −0.196484
\(845\) 28.7426 0.988777
\(846\) −30.6525 −1.05385
\(847\) 2.12461i 0.0730025i
\(848\) 2.00000i 0.0686803i
\(849\) 1.88854 0.0648147
\(850\) −29.2705 −1.00397
\(851\) 3.70820 0.127116
\(852\) − 1.61803i − 0.0554329i
\(853\) 51.2148i 1.75356i 0.480891 + 0.876780i \(0.340313\pi\)
−0.480891 + 0.876780i \(0.659687\pi\)
\(854\) −18.7082 −0.640182
\(855\) 40.1246 1.37223
\(856\) 1.70820 0.0583852
\(857\) − 36.0000i − 1.22974i −0.788630 0.614868i \(-0.789209\pi\)
0.788630 0.614868i \(-0.210791\pi\)
\(858\) 0.798374i 0.0272560i
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 15.1246i 0.515745i
\(861\) 10.1459 0.345771
\(862\) 6.76393i 0.230380i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 3.47214 0.118124
\(865\) 7.96556i 0.270837i
\(866\) 29.6180 1.00646
\(867\) − 10.6738i − 0.362500i
\(868\) 42.9787i 1.45879i
\(869\) 19.3050 0.654876
\(870\) −5.12461 −0.173741
\(871\) 0.291796 0.00988713
\(872\) 12.5623i 0.425413i
\(873\) − 41.9787i − 1.42076i
\(874\) −6.85410 −0.231843
\(875\) 54.2705i 1.83468i
\(876\) 4.65248 0.157193
\(877\) − 34.6869i − 1.17129i −0.810566 0.585647i \(-0.800840\pi\)
0.810566 0.585647i \(-0.199160\pi\)
\(878\) − 0.270510i − 0.00912926i
\(879\) −17.1246 −0.577599
\(880\) −7.56231 −0.254925
\(881\) −38.2918 −1.29008 −0.645042 0.764147i \(-0.723160\pi\)
−0.645042 + 0.764147i \(0.723160\pi\)
\(882\) 43.3607i 1.46003i
\(883\) 17.7295i 0.596645i 0.954465 + 0.298322i \(0.0964271\pi\)
−0.954465 + 0.298322i \(0.903573\pi\)
\(884\) 2.23607 0.0752071
\(885\) 8.29180i 0.278726i
\(886\) −6.85410 −0.230268
\(887\) − 36.5410i − 1.22693i −0.789723 0.613464i \(-0.789776\pi\)
0.789723 0.613464i \(-0.210224\pi\)
\(888\) − 2.29180i − 0.0769076i
\(889\) −18.0000 −0.603701
\(890\) − 21.7082i − 0.727661i
\(891\) −19.3050 −0.646740
\(892\) − 22.3607i − 0.748691i
\(893\) − 80.2492i − 2.68544i
\(894\) 0.236068 0.00789529
\(895\) −36.8328 −1.23119
\(896\) −4.85410 −0.162164
\(897\) − 0.236068i − 0.00788208i
\(898\) 2.56231i 0.0855053i
\(899\) 32.8328 1.09504
\(900\) −13.0902 −0.436339
\(901\) 11.7082 0.390057
\(902\) − 11.4377i − 0.380834i
\(903\) 20.2918i 0.675269i
\(904\) 13.4164 0.446223
\(905\) −10.2016 −0.339114
\(906\) −11.9098 −0.395678
\(907\) 30.5410i 1.01410i 0.861917 + 0.507049i \(0.169264\pi\)
−0.861917 + 0.507049i \(0.830736\pi\)
\(908\) − 23.7082i − 0.786784i
\(909\) −27.4164 −0.909345
\(910\) − 4.14590i − 0.137435i
\(911\) 22.3607 0.740842 0.370421 0.928864i \(-0.379213\pi\)
0.370421 + 0.928864i \(0.379213\pi\)
\(912\) 4.23607i 0.140270i
\(913\) − 19.3050i − 0.638901i
\(914\) −25.4164 −0.840700
\(915\) − 5.32624i − 0.176080i
\(916\) −10.0000 −0.330409
\(917\) 39.7082i 1.31128i
\(918\) − 20.3262i − 0.670866i
\(919\) 14.5836 0.481068 0.240534 0.970641i \(-0.422677\pi\)
0.240534 + 0.970641i \(0.422677\pi\)
\(920\) 2.23607 0.0737210
\(921\) −6.27051 −0.206620
\(922\) 25.3050i 0.833374i
\(923\) − 1.00000i − 0.0329154i
\(924\) −10.1459 −0.333776
\(925\) 18.5410i 0.609625i
\(926\) 2.29180 0.0753131
\(927\) − 10.8541i − 0.356495i
\(928\) 3.70820i 0.121728i
\(929\) −19.3050 −0.633375 −0.316687 0.948530i \(-0.602571\pi\)
−0.316687 + 0.948530i \(0.602571\pi\)
\(930\) −12.2361 −0.401236
\(931\) −113.520 −3.72046
\(932\) − 19.1246i − 0.626447i
\(933\) − 8.36068i − 0.273716i
\(934\) −17.1246 −0.560334
\(935\) 44.2705i 1.44780i
\(936\) 1.00000 0.0326860
\(937\) − 8.67376i − 0.283359i −0.989913 0.141680i \(-0.954750\pi\)
0.989913 0.141680i \(-0.0452503\pi\)
\(938\) 3.70820i 0.121077i
\(939\) 11.8328 0.386149
\(940\) 26.1803i 0.853909i
\(941\) −24.2148 −0.789379 −0.394690 0.918814i \(-0.629148\pi\)
−0.394690 + 0.918814i \(0.629148\pi\)
\(942\) 9.70820i 0.316310i
\(943\) 3.38197i 0.110132i
\(944\) 6.00000 0.195283
\(945\) −37.6869 −1.22596
\(946\) 22.8754 0.743743
\(947\) − 47.3951i − 1.54013i −0.637963 0.770067i \(-0.720223\pi\)
0.637963 0.770067i \(-0.279777\pi\)
\(948\) 3.52786i 0.114580i
\(949\) 2.87539 0.0933391
\(950\) − 34.2705i − 1.11188i
\(951\) −16.2361 −0.526491
\(952\) 28.4164i 0.920981i
\(953\) 36.3951i 1.17895i 0.807785 + 0.589477i \(0.200666\pi\)
−0.807785 + 0.589477i \(0.799334\pi\)
\(954\) 5.23607 0.169524
\(955\) −3.16718 −0.102488
\(956\) −1.52786 −0.0494147
\(957\) 7.75078i 0.250547i
\(958\) − 19.5279i − 0.630917i
\(959\) 34.6869 1.12010
\(960\) − 1.38197i − 0.0446028i
\(961\) 47.3951 1.52887
\(962\) − 1.41641i − 0.0456668i
\(963\) − 4.47214i − 0.144113i
\(964\) 2.00000 0.0644157
\(965\) 5.12461i 0.164967i
\(966\) 3.00000 0.0965234
\(967\) 24.7639i 0.796354i 0.917309 + 0.398177i \(0.130357\pi\)
−0.917309 + 0.398177i \(0.869643\pi\)
\(968\) 0.437694i 0.0140680i
\(969\) 24.7984 0.796639
\(970\) −35.8541 −1.15121
\(971\) 25.8541 0.829698 0.414849 0.909890i \(-0.363834\pi\)
0.414849 + 0.909890i \(0.363834\pi\)
\(972\) − 13.9443i − 0.447263i
\(973\) − 85.9574i − 2.75567i
\(974\) −21.7082 −0.695576
\(975\) 1.18034 0.0378011
\(976\) −3.85410 −0.123367
\(977\) − 32.2705i − 1.03243i −0.856461 0.516213i \(-0.827341\pi\)
0.856461 0.516213i \(-0.172659\pi\)
\(978\) − 4.34752i − 0.139018i
\(979\) −32.8328 −1.04934
\(980\) 37.0344 1.18302
\(981\) 32.8885 1.05005
\(982\) 26.8328i 0.856270i
\(983\) − 26.3951i − 0.841874i −0.907090 0.420937i \(-0.861701\pi\)
0.907090 0.420937i \(-0.138299\pi\)
\(984\) 2.09017 0.0666322
\(985\) − 0.978714i − 0.0311844i
\(986\) 21.7082 0.691330
\(987\) 35.1246i 1.11803i
\(988\) 2.61803i 0.0832908i
\(989\) −6.76393 −0.215081
\(990\) 19.7984i 0.629234i
\(991\) 51.2705 1.62866 0.814331 0.580401i \(-0.197104\pi\)
0.814331 + 0.580401i \(0.197104\pi\)
\(992\) 8.85410i 0.281118i
\(993\) 9.34752i 0.296635i
\(994\) 12.7082 0.403080
\(995\) 34.4721 1.09284
\(996\) 3.52786 0.111785
\(997\) 26.7214i 0.846274i 0.906066 + 0.423137i \(0.139071\pi\)
−0.906066 + 0.423137i \(0.860929\pi\)
\(998\) 11.4164i 0.361380i
\(999\) −12.8754 −0.407359
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 230.2.b.a.139.2 4
3.2 odd 2 2070.2.d.c.829.3 4
4.3 odd 2 1840.2.e.c.369.2 4
5.2 odd 4 1150.2.a.n.1.2 2
5.3 odd 4 1150.2.a.l.1.1 2
5.4 even 2 inner 230.2.b.a.139.3 yes 4
15.14 odd 2 2070.2.d.c.829.1 4
20.3 even 4 9200.2.a.bo.1.2 2
20.7 even 4 9200.2.a.by.1.1 2
20.19 odd 2 1840.2.e.c.369.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.b.a.139.2 4 1.1 even 1 trivial
230.2.b.a.139.3 yes 4 5.4 even 2 inner
1150.2.a.l.1.1 2 5.3 odd 4
1150.2.a.n.1.2 2 5.2 odd 4
1840.2.e.c.369.2 4 4.3 odd 2
1840.2.e.c.369.3 4 20.19 odd 2
2070.2.d.c.829.1 4 15.14 odd 2
2070.2.d.c.829.3 4 3.2 odd 2
9200.2.a.bo.1.2 2 20.3 even 4
9200.2.a.by.1.1 2 20.7 even 4