Properties

Label 5054.2.a.i.1.1
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5054,2,Mod(1,5054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -4.23607 q^{5} -0.381966 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.381966 q^{3} +1.00000 q^{4} -4.23607 q^{5} -0.381966 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.85410 q^{9} +4.23607 q^{10} +3.00000 q^{11} +0.381966 q^{12} -3.85410 q^{13} +1.00000 q^{14} -1.61803 q^{15} +1.00000 q^{16} +1.85410 q^{17} +2.85410 q^{18} -4.23607 q^{20} -0.381966 q^{21} -3.00000 q^{22} -6.70820 q^{23} -0.381966 q^{24} +12.9443 q^{25} +3.85410 q^{26} -2.23607 q^{27} -1.00000 q^{28} -8.23607 q^{29} +1.61803 q^{30} +5.00000 q^{31} -1.00000 q^{32} +1.14590 q^{33} -1.85410 q^{34} +4.23607 q^{35} -2.85410 q^{36} -5.00000 q^{37} -1.47214 q^{39} +4.23607 q^{40} -4.09017 q^{41} +0.381966 q^{42} -11.5623 q^{43} +3.00000 q^{44} +12.0902 q^{45} +6.70820 q^{46} -6.23607 q^{47} +0.381966 q^{48} +1.00000 q^{49} -12.9443 q^{50} +0.708204 q^{51} -3.85410 q^{52} -12.7082 q^{53} +2.23607 q^{54} -12.7082 q^{55} +1.00000 q^{56} +8.23607 q^{58} +10.2361 q^{59} -1.61803 q^{60} -6.70820 q^{61} -5.00000 q^{62} +2.85410 q^{63} +1.00000 q^{64} +16.3262 q^{65} -1.14590 q^{66} -4.56231 q^{67} +1.85410 q^{68} -2.56231 q^{69} -4.23607 q^{70} +3.52786 q^{71} +2.85410 q^{72} -9.09017 q^{73} +5.00000 q^{74} +4.94427 q^{75} -3.00000 q^{77} +1.47214 q^{78} +10.1459 q^{79} -4.23607 q^{80} +7.70820 q^{81} +4.09017 q^{82} -6.47214 q^{83} -0.381966 q^{84} -7.85410 q^{85} +11.5623 q^{86} -3.14590 q^{87} -3.00000 q^{88} -1.38197 q^{89} -12.0902 q^{90} +3.85410 q^{91} -6.70820 q^{92} +1.90983 q^{93} +6.23607 q^{94} -0.381966 q^{96} -5.29180 q^{97} -1.00000 q^{98} -8.56231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 4 q^{5} - 3 q^{6} - 2 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 4 q^{5} - 3 q^{6} - 2 q^{7} - 2 q^{8} + q^{9} + 4 q^{10} + 6 q^{11} + 3 q^{12} - q^{13} + 2 q^{14} - q^{15} + 2 q^{16} - 3 q^{17} - q^{18} - 4 q^{20} - 3 q^{21} - 6 q^{22} - 3 q^{24} + 8 q^{25} + q^{26} - 2 q^{28} - 12 q^{29} + q^{30} + 10 q^{31} - 2 q^{32} + 9 q^{33} + 3 q^{34} + 4 q^{35} + q^{36} - 10 q^{37} + 6 q^{39} + 4 q^{40} + 3 q^{41} + 3 q^{42} - 3 q^{43} + 6 q^{44} + 13 q^{45} - 8 q^{47} + 3 q^{48} + 2 q^{49} - 8 q^{50} - 12 q^{51} - q^{52} - 12 q^{53} - 12 q^{55} + 2 q^{56} + 12 q^{58} + 16 q^{59} - q^{60} - 10 q^{62} - q^{63} + 2 q^{64} + 17 q^{65} - 9 q^{66} + 11 q^{67} - 3 q^{68} + 15 q^{69} - 4 q^{70} + 16 q^{71} - q^{72} - 7 q^{73} + 10 q^{74} - 8 q^{75} - 6 q^{77} - 6 q^{78} + 27 q^{79} - 4 q^{80} + 2 q^{81} - 3 q^{82} - 4 q^{83} - 3 q^{84} - 9 q^{85} + 3 q^{86} - 13 q^{87} - 6 q^{88} - 5 q^{89} - 13 q^{90} + q^{91} + 15 q^{93} + 8 q^{94} - 3 q^{96} - 24 q^{97} - 2 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.381966 0.220528 0.110264 0.993902i \(-0.464830\pi\)
0.110264 + 0.993902i \(0.464830\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.23607 −1.89443 −0.947214 0.320603i \(-0.896114\pi\)
−0.947214 + 0.320603i \(0.896114\pi\)
\(6\) −0.381966 −0.155937
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.85410 −0.951367
\(10\) 4.23607 1.33956
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0.381966 0.110264
\(13\) −3.85410 −1.06894 −0.534468 0.845189i \(-0.679488\pi\)
−0.534468 + 0.845189i \(0.679488\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.61803 −0.417775
\(16\) 1.00000 0.250000
\(17\) 1.85410 0.449686 0.224843 0.974395i \(-0.427813\pi\)
0.224843 + 0.974395i \(0.427813\pi\)
\(18\) 2.85410 0.672718
\(19\) 0 0
\(20\) −4.23607 −0.947214
\(21\) −0.381966 −0.0833518
\(22\) −3.00000 −0.639602
\(23\) −6.70820 −1.39876 −0.699379 0.714751i \(-0.746540\pi\)
−0.699379 + 0.714751i \(0.746540\pi\)
\(24\) −0.381966 −0.0779685
\(25\) 12.9443 2.58885
\(26\) 3.85410 0.755852
\(27\) −2.23607 −0.430331
\(28\) −1.00000 −0.188982
\(29\) −8.23607 −1.52940 −0.764700 0.644387i \(-0.777113\pi\)
−0.764700 + 0.644387i \(0.777113\pi\)
\(30\) 1.61803 0.295411
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.14590 0.199475
\(34\) −1.85410 −0.317976
\(35\) 4.23607 0.716026
\(36\) −2.85410 −0.475684
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) −1.47214 −0.235730
\(40\) 4.23607 0.669781
\(41\) −4.09017 −0.638777 −0.319389 0.947624i \(-0.603478\pi\)
−0.319389 + 0.947624i \(0.603478\pi\)
\(42\) 0.381966 0.0589386
\(43\) −11.5623 −1.76324 −0.881618 0.471964i \(-0.843545\pi\)
−0.881618 + 0.471964i \(0.843545\pi\)
\(44\) 3.00000 0.452267
\(45\) 12.0902 1.80230
\(46\) 6.70820 0.989071
\(47\) −6.23607 −0.909624 −0.454812 0.890587i \(-0.650294\pi\)
−0.454812 + 0.890587i \(0.650294\pi\)
\(48\) 0.381966 0.0551320
\(49\) 1.00000 0.142857
\(50\) −12.9443 −1.83060
\(51\) 0.708204 0.0991684
\(52\) −3.85410 −0.534468
\(53\) −12.7082 −1.74561 −0.872803 0.488073i \(-0.837700\pi\)
−0.872803 + 0.488073i \(0.837700\pi\)
\(54\) 2.23607 0.304290
\(55\) −12.7082 −1.71357
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 8.23607 1.08145
\(59\) 10.2361 1.33262 0.666311 0.745674i \(-0.267872\pi\)
0.666311 + 0.745674i \(0.267872\pi\)
\(60\) −1.61803 −0.208887
\(61\) −6.70820 −0.858898 −0.429449 0.903091i \(-0.641292\pi\)
−0.429449 + 0.903091i \(0.641292\pi\)
\(62\) −5.00000 −0.635001
\(63\) 2.85410 0.359583
\(64\) 1.00000 0.125000
\(65\) 16.3262 2.02502
\(66\) −1.14590 −0.141050
\(67\) −4.56231 −0.557374 −0.278687 0.960382i \(-0.589899\pi\)
−0.278687 + 0.960382i \(0.589899\pi\)
\(68\) 1.85410 0.224843
\(69\) −2.56231 −0.308465
\(70\) −4.23607 −0.506307
\(71\) 3.52786 0.418680 0.209340 0.977843i \(-0.432868\pi\)
0.209340 + 0.977843i \(0.432868\pi\)
\(72\) 2.85410 0.336359
\(73\) −9.09017 −1.06392 −0.531962 0.846768i \(-0.678545\pi\)
−0.531962 + 0.846768i \(0.678545\pi\)
\(74\) 5.00000 0.581238
\(75\) 4.94427 0.570915
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 1.47214 0.166687
\(79\) 10.1459 1.14150 0.570751 0.821123i \(-0.306652\pi\)
0.570751 + 0.821123i \(0.306652\pi\)
\(80\) −4.23607 −0.473607
\(81\) 7.70820 0.856467
\(82\) 4.09017 0.451684
\(83\) −6.47214 −0.710409 −0.355205 0.934789i \(-0.615589\pi\)
−0.355205 + 0.934789i \(0.615589\pi\)
\(84\) −0.381966 −0.0416759
\(85\) −7.85410 −0.851897
\(86\) 11.5623 1.24680
\(87\) −3.14590 −0.337276
\(88\) −3.00000 −0.319801
\(89\) −1.38197 −0.146488 −0.0732441 0.997314i \(-0.523335\pi\)
−0.0732441 + 0.997314i \(0.523335\pi\)
\(90\) −12.0902 −1.27442
\(91\) 3.85410 0.404020
\(92\) −6.70820 −0.699379
\(93\) 1.90983 0.198040
\(94\) 6.23607 0.643201
\(95\) 0 0
\(96\) −0.381966 −0.0389842
\(97\) −5.29180 −0.537300 −0.268650 0.963238i \(-0.586578\pi\)
−0.268650 + 0.963238i \(0.586578\pi\)
\(98\) −1.00000 −0.101015
\(99\) −8.56231 −0.860544
\(100\) 12.9443 1.29443
\(101\) −8.09017 −0.805002 −0.402501 0.915420i \(-0.631859\pi\)
−0.402501 + 0.915420i \(0.631859\pi\)
\(102\) −0.708204 −0.0701226
\(103\) 6.85410 0.675355 0.337677 0.941262i \(-0.390359\pi\)
0.337677 + 0.941262i \(0.390359\pi\)
\(104\) 3.85410 0.377926
\(105\) 1.61803 0.157904
\(106\) 12.7082 1.23433
\(107\) −13.4721 −1.30240 −0.651200 0.758906i \(-0.725734\pi\)
−0.651200 + 0.758906i \(0.725734\pi\)
\(108\) −2.23607 −0.215166
\(109\) 4.85410 0.464939 0.232469 0.972604i \(-0.425319\pi\)
0.232469 + 0.972604i \(0.425319\pi\)
\(110\) 12.7082 1.21168
\(111\) −1.90983 −0.181273
\(112\) −1.00000 −0.0944911
\(113\) 11.7984 1.10990 0.554949 0.831884i \(-0.312738\pi\)
0.554949 + 0.831884i \(0.312738\pi\)
\(114\) 0 0
\(115\) 28.4164 2.64984
\(116\) −8.23607 −0.764700
\(117\) 11.0000 1.01695
\(118\) −10.2361 −0.942306
\(119\) −1.85410 −0.169965
\(120\) 1.61803 0.147706
\(121\) −2.00000 −0.181818
\(122\) 6.70820 0.607332
\(123\) −1.56231 −0.140868
\(124\) 5.00000 0.449013
\(125\) −33.6525 −3.00997
\(126\) −2.85410 −0.254264
\(127\) 20.0344 1.77777 0.888885 0.458131i \(-0.151481\pi\)
0.888885 + 0.458131i \(0.151481\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.41641 −0.388843
\(130\) −16.3262 −1.43191
\(131\) −3.38197 −0.295484 −0.147742 0.989026i \(-0.547200\pi\)
−0.147742 + 0.989026i \(0.547200\pi\)
\(132\) 1.14590 0.0997376
\(133\) 0 0
\(134\) 4.56231 0.394123
\(135\) 9.47214 0.815232
\(136\) −1.85410 −0.158988
\(137\) 9.23607 0.789091 0.394545 0.918877i \(-0.370902\pi\)
0.394545 + 0.918877i \(0.370902\pi\)
\(138\) 2.56231 0.218118
\(139\) 19.0000 1.61156 0.805779 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 4.23607 0.358013
\(141\) −2.38197 −0.200598
\(142\) −3.52786 −0.296052
\(143\) −11.5623 −0.966889
\(144\) −2.85410 −0.237842
\(145\) 34.8885 2.89734
\(146\) 9.09017 0.752308
\(147\) 0.381966 0.0315040
\(148\) −5.00000 −0.410997
\(149\) −1.90983 −0.156459 −0.0782297 0.996935i \(-0.524927\pi\)
−0.0782297 + 0.996935i \(0.524927\pi\)
\(150\) −4.94427 −0.403698
\(151\) 0.236068 0.0192109 0.00960547 0.999954i \(-0.496942\pi\)
0.00960547 + 0.999954i \(0.496942\pi\)
\(152\) 0 0
\(153\) −5.29180 −0.427816
\(154\) 3.00000 0.241747
\(155\) −21.1803 −1.70125
\(156\) −1.47214 −0.117865
\(157\) −10.3820 −0.828571 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(158\) −10.1459 −0.807164
\(159\) −4.85410 −0.384955
\(160\) 4.23607 0.334891
\(161\) 6.70820 0.528681
\(162\) −7.70820 −0.605614
\(163\) −3.52786 −0.276324 −0.138162 0.990410i \(-0.544119\pi\)
−0.138162 + 0.990410i \(0.544119\pi\)
\(164\) −4.09017 −0.319389
\(165\) −4.85410 −0.377891
\(166\) 6.47214 0.502335
\(167\) 12.7639 0.987703 0.493851 0.869546i \(-0.335589\pi\)
0.493851 + 0.869546i \(0.335589\pi\)
\(168\) 0.381966 0.0294693
\(169\) 1.85410 0.142623
\(170\) 7.85410 0.602382
\(171\) 0 0
\(172\) −11.5623 −0.881618
\(173\) −4.41641 −0.335773 −0.167887 0.985806i \(-0.553694\pi\)
−0.167887 + 0.985806i \(0.553694\pi\)
\(174\) 3.14590 0.238490
\(175\) −12.9443 −0.978495
\(176\) 3.00000 0.226134
\(177\) 3.90983 0.293881
\(178\) 1.38197 0.103583
\(179\) 3.67376 0.274590 0.137295 0.990530i \(-0.456159\pi\)
0.137295 + 0.990530i \(0.456159\pi\)
\(180\) 12.0902 0.901148
\(181\) 15.9443 1.18513 0.592564 0.805523i \(-0.298116\pi\)
0.592564 + 0.805523i \(0.298116\pi\)
\(182\) −3.85410 −0.285685
\(183\) −2.56231 −0.189411
\(184\) 6.70820 0.494535
\(185\) 21.1803 1.55721
\(186\) −1.90983 −0.140036
\(187\) 5.56231 0.406756
\(188\) −6.23607 −0.454812
\(189\) 2.23607 0.162650
\(190\) 0 0
\(191\) −7.76393 −0.561778 −0.280889 0.959740i \(-0.590629\pi\)
−0.280889 + 0.959740i \(0.590629\pi\)
\(192\) 0.381966 0.0275660
\(193\) −4.18034 −0.300907 −0.150454 0.988617i \(-0.548073\pi\)
−0.150454 + 0.988617i \(0.548073\pi\)
\(194\) 5.29180 0.379929
\(195\) 6.23607 0.446574
\(196\) 1.00000 0.0714286
\(197\) 19.0344 1.35615 0.678074 0.734994i \(-0.262815\pi\)
0.678074 + 0.734994i \(0.262815\pi\)
\(198\) 8.56231 0.608497
\(199\) 15.7984 1.11992 0.559959 0.828521i \(-0.310817\pi\)
0.559959 + 0.828521i \(0.310817\pi\)
\(200\) −12.9443 −0.915298
\(201\) −1.74265 −0.122917
\(202\) 8.09017 0.569222
\(203\) 8.23607 0.578059
\(204\) 0.708204 0.0495842
\(205\) 17.3262 1.21012
\(206\) −6.85410 −0.477548
\(207\) 19.1459 1.33073
\(208\) −3.85410 −0.267234
\(209\) 0 0
\(210\) −1.61803 −0.111655
\(211\) 21.0000 1.44570 0.722850 0.691005i \(-0.242832\pi\)
0.722850 + 0.691005i \(0.242832\pi\)
\(212\) −12.7082 −0.872803
\(213\) 1.34752 0.0923308
\(214\) 13.4721 0.920936
\(215\) 48.9787 3.34032
\(216\) 2.23607 0.152145
\(217\) −5.00000 −0.339422
\(218\) −4.85410 −0.328761
\(219\) −3.47214 −0.234625
\(220\) −12.7082 −0.856787
\(221\) −7.14590 −0.480685
\(222\) 1.90983 0.128179
\(223\) 9.00000 0.602685 0.301342 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 1.00000 0.0668153
\(225\) −36.9443 −2.46295
\(226\) −11.7984 −0.784816
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 0 0
\(229\) −17.5623 −1.16055 −0.580275 0.814421i \(-0.697055\pi\)
−0.580275 + 0.814421i \(0.697055\pi\)
\(230\) −28.4164 −1.87372
\(231\) −1.14590 −0.0753946
\(232\) 8.23607 0.540724
\(233\) −16.7984 −1.10050 −0.550249 0.835001i \(-0.685467\pi\)
−0.550249 + 0.835001i \(0.685467\pi\)
\(234\) −11.0000 −0.719092
\(235\) 26.4164 1.72322
\(236\) 10.2361 0.666311
\(237\) 3.87539 0.251734
\(238\) 1.85410 0.120184
\(239\) 4.03444 0.260966 0.130483 0.991451i \(-0.458347\pi\)
0.130483 + 0.991451i \(0.458347\pi\)
\(240\) −1.61803 −0.104444
\(241\) 20.1459 1.29771 0.648856 0.760911i \(-0.275248\pi\)
0.648856 + 0.760911i \(0.275248\pi\)
\(242\) 2.00000 0.128565
\(243\) 9.65248 0.619207
\(244\) −6.70820 −0.429449
\(245\) −4.23607 −0.270632
\(246\) 1.56231 0.0996090
\(247\) 0 0
\(248\) −5.00000 −0.317500
\(249\) −2.47214 −0.156665
\(250\) 33.6525 2.12837
\(251\) 16.3607 1.03268 0.516338 0.856385i \(-0.327295\pi\)
0.516338 + 0.856385i \(0.327295\pi\)
\(252\) 2.85410 0.179792
\(253\) −20.1246 −1.26522
\(254\) −20.0344 −1.25707
\(255\) −3.00000 −0.187867
\(256\) 1.00000 0.0625000
\(257\) −0.708204 −0.0441765 −0.0220883 0.999756i \(-0.507031\pi\)
−0.0220883 + 0.999756i \(0.507031\pi\)
\(258\) 4.41641 0.274954
\(259\) 5.00000 0.310685
\(260\) 16.3262 1.01251
\(261\) 23.5066 1.45502
\(262\) 3.38197 0.208939
\(263\) −5.29180 −0.326306 −0.163153 0.986601i \(-0.552166\pi\)
−0.163153 + 0.986601i \(0.552166\pi\)
\(264\) −1.14590 −0.0705251
\(265\) 53.8328 3.30692
\(266\) 0 0
\(267\) −0.527864 −0.0323048
\(268\) −4.56231 −0.278687
\(269\) −31.4721 −1.91889 −0.959445 0.281896i \(-0.909037\pi\)
−0.959445 + 0.281896i \(0.909037\pi\)
\(270\) −9.47214 −0.576456
\(271\) 3.58359 0.217688 0.108844 0.994059i \(-0.465285\pi\)
0.108844 + 0.994059i \(0.465285\pi\)
\(272\) 1.85410 0.112421
\(273\) 1.47214 0.0890977
\(274\) −9.23607 −0.557971
\(275\) 38.8328 2.34171
\(276\) −2.56231 −0.154233
\(277\) −18.4721 −1.10988 −0.554942 0.831889i \(-0.687259\pi\)
−0.554942 + 0.831889i \(0.687259\pi\)
\(278\) −19.0000 −1.13954
\(279\) −14.2705 −0.854353
\(280\) −4.23607 −0.253153
\(281\) −15.6525 −0.933748 −0.466874 0.884324i \(-0.654620\pi\)
−0.466874 + 0.884324i \(0.654620\pi\)
\(282\) 2.38197 0.141844
\(283\) 15.0344 0.893705 0.446852 0.894608i \(-0.352545\pi\)
0.446852 + 0.894608i \(0.352545\pi\)
\(284\) 3.52786 0.209340
\(285\) 0 0
\(286\) 11.5623 0.683693
\(287\) 4.09017 0.241435
\(288\) 2.85410 0.168180
\(289\) −13.5623 −0.797783
\(290\) −34.8885 −2.04873
\(291\) −2.02129 −0.118490
\(292\) −9.09017 −0.531962
\(293\) −1.58359 −0.0925144 −0.0462572 0.998930i \(-0.514729\pi\)
−0.0462572 + 0.998930i \(0.514729\pi\)
\(294\) −0.381966 −0.0222767
\(295\) −43.3607 −2.52456
\(296\) 5.00000 0.290619
\(297\) −6.70820 −0.389249
\(298\) 1.90983 0.110633
\(299\) 25.8541 1.49518
\(300\) 4.94427 0.285458
\(301\) 11.5623 0.666440
\(302\) −0.236068 −0.0135842
\(303\) −3.09017 −0.177526
\(304\) 0 0
\(305\) 28.4164 1.62712
\(306\) 5.29180 0.302512
\(307\) 26.8328 1.53143 0.765715 0.643180i \(-0.222385\pi\)
0.765715 + 0.643180i \(0.222385\pi\)
\(308\) −3.00000 −0.170941
\(309\) 2.61803 0.148935
\(310\) 21.1803 1.20296
\(311\) −33.9443 −1.92480 −0.962402 0.271631i \(-0.912437\pi\)
−0.962402 + 0.271631i \(0.912437\pi\)
\(312\) 1.47214 0.0833433
\(313\) −4.27051 −0.241383 −0.120692 0.992690i \(-0.538511\pi\)
−0.120692 + 0.992690i \(0.538511\pi\)
\(314\) 10.3820 0.585888
\(315\) −12.0902 −0.681204
\(316\) 10.1459 0.570751
\(317\) −18.5279 −1.04063 −0.520314 0.853975i \(-0.674185\pi\)
−0.520314 + 0.853975i \(0.674185\pi\)
\(318\) 4.85410 0.272205
\(319\) −24.7082 −1.38339
\(320\) −4.23607 −0.236803
\(321\) −5.14590 −0.287216
\(322\) −6.70820 −0.373834
\(323\) 0 0
\(324\) 7.70820 0.428234
\(325\) −49.8885 −2.76732
\(326\) 3.52786 0.195390
\(327\) 1.85410 0.102532
\(328\) 4.09017 0.225842
\(329\) 6.23607 0.343806
\(330\) 4.85410 0.267210
\(331\) −0.472136 −0.0259509 −0.0129755 0.999916i \(-0.504130\pi\)
−0.0129755 + 0.999916i \(0.504130\pi\)
\(332\) −6.47214 −0.355205
\(333\) 14.2705 0.782019
\(334\) −12.7639 −0.698411
\(335\) 19.3262 1.05591
\(336\) −0.381966 −0.0208380
\(337\) −8.56231 −0.466419 −0.233209 0.972427i \(-0.574923\pi\)
−0.233209 + 0.972427i \(0.574923\pi\)
\(338\) −1.85410 −0.100850
\(339\) 4.50658 0.244764
\(340\) −7.85410 −0.425948
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 11.5623 0.623398
\(345\) 10.8541 0.584365
\(346\) 4.41641 0.237428
\(347\) 4.47214 0.240077 0.120038 0.992769i \(-0.461698\pi\)
0.120038 + 0.992769i \(0.461698\pi\)
\(348\) −3.14590 −0.168638
\(349\) 21.8328 1.16868 0.584342 0.811508i \(-0.301353\pi\)
0.584342 + 0.811508i \(0.301353\pi\)
\(350\) 12.9443 0.691900
\(351\) 8.61803 0.459997
\(352\) −3.00000 −0.159901
\(353\) 25.7984 1.37311 0.686554 0.727078i \(-0.259122\pi\)
0.686554 + 0.727078i \(0.259122\pi\)
\(354\) −3.90983 −0.207805
\(355\) −14.9443 −0.793160
\(356\) −1.38197 −0.0732441
\(357\) −0.708204 −0.0374821
\(358\) −3.67376 −0.194164
\(359\) −1.94427 −0.102615 −0.0513074 0.998683i \(-0.516339\pi\)
−0.0513074 + 0.998683i \(0.516339\pi\)
\(360\) −12.0902 −0.637208
\(361\) 0 0
\(362\) −15.9443 −0.838012
\(363\) −0.763932 −0.0400960
\(364\) 3.85410 0.202010
\(365\) 38.5066 2.01553
\(366\) 2.56231 0.133934
\(367\) 1.43769 0.0750470 0.0375235 0.999296i \(-0.488053\pi\)
0.0375235 + 0.999296i \(0.488053\pi\)
\(368\) −6.70820 −0.349689
\(369\) 11.6738 0.607712
\(370\) −21.1803 −1.10111
\(371\) 12.7082 0.659777
\(372\) 1.90983 0.0990201
\(373\) −11.8541 −0.613782 −0.306891 0.951745i \(-0.599289\pi\)
−0.306891 + 0.951745i \(0.599289\pi\)
\(374\) −5.56231 −0.287620
\(375\) −12.8541 −0.663783
\(376\) 6.23607 0.321601
\(377\) 31.7426 1.63483
\(378\) −2.23607 −0.115011
\(379\) −12.2705 −0.630294 −0.315147 0.949043i \(-0.602054\pi\)
−0.315147 + 0.949043i \(0.602054\pi\)
\(380\) 0 0
\(381\) 7.65248 0.392048
\(382\) 7.76393 0.397237
\(383\) −0.708204 −0.0361875 −0.0180938 0.999836i \(-0.505760\pi\)
−0.0180938 + 0.999836i \(0.505760\pi\)
\(384\) −0.381966 −0.0194921
\(385\) 12.7082 0.647670
\(386\) 4.18034 0.212774
\(387\) 33.0000 1.67748
\(388\) −5.29180 −0.268650
\(389\) 10.2016 0.517243 0.258621 0.965979i \(-0.416732\pi\)
0.258621 + 0.965979i \(0.416732\pi\)
\(390\) −6.23607 −0.315776
\(391\) −12.4377 −0.629001
\(392\) −1.00000 −0.0505076
\(393\) −1.29180 −0.0651625
\(394\) −19.0344 −0.958941
\(395\) −42.9787 −2.16249
\(396\) −8.56231 −0.430272
\(397\) −14.3262 −0.719013 −0.359507 0.933143i \(-0.617055\pi\)
−0.359507 + 0.933143i \(0.617055\pi\)
\(398\) −15.7984 −0.791901
\(399\) 0 0
\(400\) 12.9443 0.647214
\(401\) −24.4721 −1.22208 −0.611040 0.791600i \(-0.709249\pi\)
−0.611040 + 0.791600i \(0.709249\pi\)
\(402\) 1.74265 0.0869153
\(403\) −19.2705 −0.959932
\(404\) −8.09017 −0.402501
\(405\) −32.6525 −1.62251
\(406\) −8.23607 −0.408749
\(407\) −15.0000 −0.743522
\(408\) −0.708204 −0.0350613
\(409\) 16.1459 0.798363 0.399182 0.916872i \(-0.369294\pi\)
0.399182 + 0.916872i \(0.369294\pi\)
\(410\) −17.3262 −0.855682
\(411\) 3.52786 0.174017
\(412\) 6.85410 0.337677
\(413\) −10.2361 −0.503684
\(414\) −19.1459 −0.940970
\(415\) 27.4164 1.34582
\(416\) 3.85410 0.188963
\(417\) 7.25735 0.355394
\(418\) 0 0
\(419\) −28.0902 −1.37229 −0.686147 0.727463i \(-0.740699\pi\)
−0.686147 + 0.727463i \(0.740699\pi\)
\(420\) 1.61803 0.0789520
\(421\) −16.8885 −0.823097 −0.411549 0.911388i \(-0.635012\pi\)
−0.411549 + 0.911388i \(0.635012\pi\)
\(422\) −21.0000 −1.02226
\(423\) 17.7984 0.865387
\(424\) 12.7082 0.617165
\(425\) 24.0000 1.16417
\(426\) −1.34752 −0.0652878
\(427\) 6.70820 0.324633
\(428\) −13.4721 −0.651200
\(429\) −4.41641 −0.213226
\(430\) −48.9787 −2.36196
\(431\) 22.3607 1.07708 0.538538 0.842601i \(-0.318977\pi\)
0.538538 + 0.842601i \(0.318977\pi\)
\(432\) −2.23607 −0.107583
\(433\) −25.8541 −1.24247 −0.621234 0.783625i \(-0.713369\pi\)
−0.621234 + 0.783625i \(0.713369\pi\)
\(434\) 5.00000 0.240008
\(435\) 13.3262 0.638944
\(436\) 4.85410 0.232469
\(437\) 0 0
\(438\) 3.47214 0.165905
\(439\) 13.7639 0.656917 0.328458 0.944518i \(-0.393471\pi\)
0.328458 + 0.944518i \(0.393471\pi\)
\(440\) 12.7082 0.605840
\(441\) −2.85410 −0.135910
\(442\) 7.14590 0.339896
\(443\) 26.5066 1.25937 0.629683 0.776852i \(-0.283185\pi\)
0.629683 + 0.776852i \(0.283185\pi\)
\(444\) −1.90983 −0.0906365
\(445\) 5.85410 0.277511
\(446\) −9.00000 −0.426162
\(447\) −0.729490 −0.0345037
\(448\) −1.00000 −0.0472456
\(449\) −28.7639 −1.35745 −0.678727 0.734391i \(-0.737468\pi\)
−0.678727 + 0.734391i \(0.737468\pi\)
\(450\) 36.9443 1.74157
\(451\) −12.2705 −0.577796
\(452\) 11.7984 0.554949
\(453\) 0.0901699 0.00423655
\(454\) −27.0000 −1.26717
\(455\) −16.3262 −0.765386
\(456\) 0 0
\(457\) −37.8885 −1.77235 −0.886176 0.463349i \(-0.846647\pi\)
−0.886176 + 0.463349i \(0.846647\pi\)
\(458\) 17.5623 0.820633
\(459\) −4.14590 −0.193514
\(460\) 28.4164 1.32492
\(461\) −21.7082 −1.01105 −0.505526 0.862811i \(-0.668702\pi\)
−0.505526 + 0.862811i \(0.668702\pi\)
\(462\) 1.14590 0.0533120
\(463\) 33.9787 1.57912 0.789562 0.613670i \(-0.210308\pi\)
0.789562 + 0.613670i \(0.210308\pi\)
\(464\) −8.23607 −0.382350
\(465\) −8.09017 −0.375173
\(466\) 16.7984 0.778170
\(467\) 38.1246 1.76420 0.882098 0.471065i \(-0.156130\pi\)
0.882098 + 0.471065i \(0.156130\pi\)
\(468\) 11.0000 0.508475
\(469\) 4.56231 0.210668
\(470\) −26.4164 −1.21850
\(471\) −3.96556 −0.182723
\(472\) −10.2361 −0.471153
\(473\) −34.6869 −1.59491
\(474\) −3.87539 −0.178002
\(475\) 0 0
\(476\) −1.85410 −0.0849826
\(477\) 36.2705 1.66071
\(478\) −4.03444 −0.184531
\(479\) −27.7082 −1.26602 −0.633010 0.774144i \(-0.718181\pi\)
−0.633010 + 0.774144i \(0.718181\pi\)
\(480\) 1.61803 0.0738528
\(481\) 19.2705 0.878660
\(482\) −20.1459 −0.917621
\(483\) 2.56231 0.116589
\(484\) −2.00000 −0.0909091
\(485\) 22.4164 1.01788
\(486\) −9.65248 −0.437845
\(487\) −10.3475 −0.468891 −0.234446 0.972129i \(-0.575327\pi\)
−0.234446 + 0.972129i \(0.575327\pi\)
\(488\) 6.70820 0.303666
\(489\) −1.34752 −0.0609371
\(490\) 4.23607 0.191366
\(491\) −24.0689 −1.08621 −0.543107 0.839664i \(-0.682752\pi\)
−0.543107 + 0.839664i \(0.682752\pi\)
\(492\) −1.56231 −0.0704342
\(493\) −15.2705 −0.687749
\(494\) 0 0
\(495\) 36.2705 1.63024
\(496\) 5.00000 0.224507
\(497\) −3.52786 −0.158246
\(498\) 2.47214 0.110779
\(499\) −3.00000 −0.134298 −0.0671492 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(500\) −33.6525 −1.50498
\(501\) 4.87539 0.217816
\(502\) −16.3607 −0.730213
\(503\) −12.9443 −0.577157 −0.288578 0.957456i \(-0.593183\pi\)
−0.288578 + 0.957456i \(0.593183\pi\)
\(504\) −2.85410 −0.127132
\(505\) 34.2705 1.52502
\(506\) 20.1246 0.894648
\(507\) 0.708204 0.0314524
\(508\) 20.0344 0.888885
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 3.00000 0.132842
\(511\) 9.09017 0.402125
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 0.708204 0.0312375
\(515\) −29.0344 −1.27941
\(516\) −4.41641 −0.194422
\(517\) −18.7082 −0.822786
\(518\) −5.00000 −0.219687
\(519\) −1.68692 −0.0740475
\(520\) −16.3262 −0.715953
\(521\) 10.8885 0.477036 0.238518 0.971138i \(-0.423338\pi\)
0.238518 + 0.971138i \(0.423338\pi\)
\(522\) −23.5066 −1.02885
\(523\) 30.0902 1.31575 0.657876 0.753127i \(-0.271455\pi\)
0.657876 + 0.753127i \(0.271455\pi\)
\(524\) −3.38197 −0.147742
\(525\) −4.94427 −0.215786
\(526\) 5.29180 0.230733
\(527\) 9.27051 0.403830
\(528\) 1.14590 0.0498688
\(529\) 22.0000 0.956522
\(530\) −53.8328 −2.33835
\(531\) −29.2148 −1.26781
\(532\) 0 0
\(533\) 15.7639 0.682812
\(534\) 0.527864 0.0228429
\(535\) 57.0689 2.46730
\(536\) 4.56231 0.197062
\(537\) 1.40325 0.0605548
\(538\) 31.4721 1.35686
\(539\) 3.00000 0.129219
\(540\) 9.47214 0.407616
\(541\) −3.47214 −0.149279 −0.0746394 0.997211i \(-0.523781\pi\)
−0.0746394 + 0.997211i \(0.523781\pi\)
\(542\) −3.58359 −0.153928
\(543\) 6.09017 0.261354
\(544\) −1.85410 −0.0794940
\(545\) −20.5623 −0.880792
\(546\) −1.47214 −0.0630016
\(547\) 21.1803 0.905606 0.452803 0.891611i \(-0.350424\pi\)
0.452803 + 0.891611i \(0.350424\pi\)
\(548\) 9.23607 0.394545
\(549\) 19.1459 0.817127
\(550\) −38.8328 −1.65584
\(551\) 0 0
\(552\) 2.56231 0.109059
\(553\) −10.1459 −0.431447
\(554\) 18.4721 0.784806
\(555\) 8.09017 0.343409
\(556\) 19.0000 0.805779
\(557\) −37.7639 −1.60011 −0.800055 0.599927i \(-0.795196\pi\)
−0.800055 + 0.599927i \(0.795196\pi\)
\(558\) 14.2705 0.604119
\(559\) 44.5623 1.88478
\(560\) 4.23607 0.179007
\(561\) 2.12461 0.0897012
\(562\) 15.6525 0.660260
\(563\) −12.1803 −0.513340 −0.256670 0.966499i \(-0.582625\pi\)
−0.256670 + 0.966499i \(0.582625\pi\)
\(564\) −2.38197 −0.100299
\(565\) −49.9787 −2.10262
\(566\) −15.0344 −0.631945
\(567\) −7.70820 −0.323714
\(568\) −3.52786 −0.148026
\(569\) −21.3820 −0.896379 −0.448189 0.893939i \(-0.647931\pi\)
−0.448189 + 0.893939i \(0.647931\pi\)
\(570\) 0 0
\(571\) 7.34752 0.307484 0.153742 0.988111i \(-0.450867\pi\)
0.153742 + 0.988111i \(0.450867\pi\)
\(572\) −11.5623 −0.483444
\(573\) −2.96556 −0.123888
\(574\) −4.09017 −0.170720
\(575\) −86.8328 −3.62118
\(576\) −2.85410 −0.118921
\(577\) 38.4164 1.59930 0.799648 0.600469i \(-0.205019\pi\)
0.799648 + 0.600469i \(0.205019\pi\)
\(578\) 13.5623 0.564118
\(579\) −1.59675 −0.0663586
\(580\) 34.8885 1.44867
\(581\) 6.47214 0.268509
\(582\) 2.02129 0.0837850
\(583\) −38.1246 −1.57896
\(584\) 9.09017 0.376154
\(585\) −46.5967 −1.92654
\(586\) 1.58359 0.0654176
\(587\) −5.65248 −0.233303 −0.116651 0.993173i \(-0.537216\pi\)
−0.116651 + 0.993173i \(0.537216\pi\)
\(588\) 0.381966 0.0157520
\(589\) 0 0
\(590\) 43.3607 1.78513
\(591\) 7.27051 0.299069
\(592\) −5.00000 −0.205499
\(593\) −5.67376 −0.232993 −0.116497 0.993191i \(-0.537166\pi\)
−0.116497 + 0.993191i \(0.537166\pi\)
\(594\) 6.70820 0.275241
\(595\) 7.85410 0.321987
\(596\) −1.90983 −0.0782297
\(597\) 6.03444 0.246973
\(598\) −25.8541 −1.05725
\(599\) 13.4721 0.550457 0.275228 0.961379i \(-0.411247\pi\)
0.275228 + 0.961379i \(0.411247\pi\)
\(600\) −4.94427 −0.201849
\(601\) 37.8328 1.54323 0.771616 0.636088i \(-0.219449\pi\)
0.771616 + 0.636088i \(0.219449\pi\)
\(602\) −11.5623 −0.471244
\(603\) 13.0213 0.530268
\(604\) 0.236068 0.00960547
\(605\) 8.47214 0.344441
\(606\) 3.09017 0.125530
\(607\) 30.8328 1.25147 0.625733 0.780038i \(-0.284800\pi\)
0.625733 + 0.780038i \(0.284800\pi\)
\(608\) 0 0
\(609\) 3.14590 0.127478
\(610\) −28.4164 −1.15055
\(611\) 24.0344 0.972329
\(612\) −5.29180 −0.213908
\(613\) −18.1803 −0.734297 −0.367149 0.930162i \(-0.619666\pi\)
−0.367149 + 0.930162i \(0.619666\pi\)
\(614\) −26.8328 −1.08288
\(615\) 6.61803 0.266865
\(616\) 3.00000 0.120873
\(617\) 13.7984 0.555502 0.277751 0.960653i \(-0.410411\pi\)
0.277751 + 0.960653i \(0.410411\pi\)
\(618\) −2.61803 −0.105313
\(619\) 6.85410 0.275490 0.137745 0.990468i \(-0.456015\pi\)
0.137745 + 0.990468i \(0.456015\pi\)
\(620\) −21.1803 −0.850623
\(621\) 15.0000 0.601929
\(622\) 33.9443 1.36104
\(623\) 1.38197 0.0553673
\(624\) −1.47214 −0.0589326
\(625\) 77.8328 3.11331
\(626\) 4.27051 0.170684
\(627\) 0 0
\(628\) −10.3820 −0.414286
\(629\) −9.27051 −0.369639
\(630\) 12.0902 0.481684
\(631\) 14.8328 0.590485 0.295243 0.955422i \(-0.404600\pi\)
0.295243 + 0.955422i \(0.404600\pi\)
\(632\) −10.1459 −0.403582
\(633\) 8.02129 0.318818
\(634\) 18.5279 0.735835
\(635\) −84.8673 −3.36785
\(636\) −4.85410 −0.192478
\(637\) −3.85410 −0.152705
\(638\) 24.7082 0.978207
\(639\) −10.0689 −0.398319
\(640\) 4.23607 0.167445
\(641\) 23.7984 0.939979 0.469990 0.882672i \(-0.344258\pi\)
0.469990 + 0.882672i \(0.344258\pi\)
\(642\) 5.14590 0.203092
\(643\) 5.59675 0.220714 0.110357 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(644\) 6.70820 0.264340
\(645\) 18.7082 0.736635
\(646\) 0 0
\(647\) −6.79837 −0.267272 −0.133636 0.991031i \(-0.542665\pi\)
−0.133636 + 0.991031i \(0.542665\pi\)
\(648\) −7.70820 −0.302807
\(649\) 30.7082 1.20540
\(650\) 49.8885 1.95679
\(651\) −1.90983 −0.0748521
\(652\) −3.52786 −0.138162
\(653\) 7.67376 0.300298 0.150149 0.988663i \(-0.452025\pi\)
0.150149 + 0.988663i \(0.452025\pi\)
\(654\) −1.85410 −0.0725011
\(655\) 14.3262 0.559772
\(656\) −4.09017 −0.159694
\(657\) 25.9443 1.01218
\(658\) −6.23607 −0.243107
\(659\) 2.34752 0.0914466 0.0457233 0.998954i \(-0.485441\pi\)
0.0457233 + 0.998954i \(0.485441\pi\)
\(660\) −4.85410 −0.188946
\(661\) −11.2918 −0.439200 −0.219600 0.975590i \(-0.570475\pi\)
−0.219600 + 0.975590i \(0.570475\pi\)
\(662\) 0.472136 0.0183501
\(663\) −2.72949 −0.106005
\(664\) 6.47214 0.251168
\(665\) 0 0
\(666\) −14.2705 −0.552971
\(667\) 55.2492 2.13926
\(668\) 12.7639 0.493851
\(669\) 3.43769 0.132909
\(670\) −19.3262 −0.746638
\(671\) −20.1246 −0.776902
\(672\) 0.381966 0.0147347
\(673\) −9.61803 −0.370748 −0.185374 0.982668i \(-0.559350\pi\)
−0.185374 + 0.982668i \(0.559350\pi\)
\(674\) 8.56231 0.329808
\(675\) −28.9443 −1.11407
\(676\) 1.85410 0.0713116
\(677\) 34.0689 1.30937 0.654687 0.755900i \(-0.272801\pi\)
0.654687 + 0.755900i \(0.272801\pi\)
\(678\) −4.50658 −0.173074
\(679\) 5.29180 0.203080
\(680\) 7.85410 0.301191
\(681\) 10.3131 0.395198
\(682\) −15.0000 −0.574380
\(683\) −43.7426 −1.67377 −0.836883 0.547382i \(-0.815625\pi\)
−0.836883 + 0.547382i \(0.815625\pi\)
\(684\) 0 0
\(685\) −39.1246 −1.49487
\(686\) 1.00000 0.0381802
\(687\) −6.70820 −0.255934
\(688\) −11.5623 −0.440809
\(689\) 48.9787 1.86594
\(690\) −10.8541 −0.413209
\(691\) −32.1246 −1.22208 −0.611039 0.791601i \(-0.709248\pi\)
−0.611039 + 0.791601i \(0.709248\pi\)
\(692\) −4.41641 −0.167887
\(693\) 8.56231 0.325255
\(694\) −4.47214 −0.169760
\(695\) −80.4853 −3.05298
\(696\) 3.14590 0.119245
\(697\) −7.58359 −0.287249
\(698\) −21.8328 −0.826384
\(699\) −6.41641 −0.242691
\(700\) −12.9443 −0.489247
\(701\) 8.21478 0.310268 0.155134 0.987893i \(-0.450419\pi\)
0.155134 + 0.987893i \(0.450419\pi\)
\(702\) −8.61803 −0.325267
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) 10.0902 0.380018
\(706\) −25.7984 −0.970935
\(707\) 8.09017 0.304262
\(708\) 3.90983 0.146940
\(709\) −36.2361 −1.36087 −0.680437 0.732807i \(-0.738210\pi\)
−0.680437 + 0.732807i \(0.738210\pi\)
\(710\) 14.9443 0.560849
\(711\) −28.9574 −1.08599
\(712\) 1.38197 0.0517914
\(713\) −33.5410 −1.25612
\(714\) 0.708204 0.0265039
\(715\) 48.9787 1.83170
\(716\) 3.67376 0.137295
\(717\) 1.54102 0.0575504
\(718\) 1.94427 0.0725596
\(719\) 41.0689 1.53161 0.765805 0.643072i \(-0.222341\pi\)
0.765805 + 0.643072i \(0.222341\pi\)
\(720\) 12.0902 0.450574
\(721\) −6.85410 −0.255260
\(722\) 0 0
\(723\) 7.69505 0.286182
\(724\) 15.9443 0.592564
\(725\) −106.610 −3.95939
\(726\) 0.763932 0.0283522
\(727\) 0.639320 0.0237111 0.0118555 0.999930i \(-0.496226\pi\)
0.0118555 + 0.999930i \(0.496226\pi\)
\(728\) −3.85410 −0.142843
\(729\) −19.4377 −0.719915
\(730\) −38.5066 −1.42519
\(731\) −21.4377 −0.792902
\(732\) −2.56231 −0.0947056
\(733\) 31.0000 1.14501 0.572506 0.819901i \(-0.305971\pi\)
0.572506 + 0.819901i \(0.305971\pi\)
\(734\) −1.43769 −0.0530663
\(735\) −1.61803 −0.0596821
\(736\) 6.70820 0.247268
\(737\) −13.6869 −0.504164
\(738\) −11.6738 −0.429717
\(739\) −14.8328 −0.545634 −0.272817 0.962066i \(-0.587955\pi\)
−0.272817 + 0.962066i \(0.587955\pi\)
\(740\) 21.1803 0.778605
\(741\) 0 0
\(742\) −12.7082 −0.466533
\(743\) −7.14590 −0.262158 −0.131079 0.991372i \(-0.541844\pi\)
−0.131079 + 0.991372i \(0.541844\pi\)
\(744\) −1.90983 −0.0700178
\(745\) 8.09017 0.296401
\(746\) 11.8541 0.434010
\(747\) 18.4721 0.675860
\(748\) 5.56231 0.203378
\(749\) 13.4721 0.492261
\(750\) 12.8541 0.469365
\(751\) −15.1246 −0.551905 −0.275952 0.961171i \(-0.588993\pi\)
−0.275952 + 0.961171i \(0.588993\pi\)
\(752\) −6.23607 −0.227406
\(753\) 6.24922 0.227734
\(754\) −31.7426 −1.15600
\(755\) −1.00000 −0.0363937
\(756\) 2.23607 0.0813250
\(757\) 27.4164 0.996466 0.498233 0.867043i \(-0.333982\pi\)
0.498233 + 0.867043i \(0.333982\pi\)
\(758\) 12.2705 0.445685
\(759\) −7.68692 −0.279017
\(760\) 0 0
\(761\) −20.1246 −0.729517 −0.364758 0.931102i \(-0.618848\pi\)
−0.364758 + 0.931102i \(0.618848\pi\)
\(762\) −7.65248 −0.277220
\(763\) −4.85410 −0.175730
\(764\) −7.76393 −0.280889
\(765\) 22.4164 0.810467
\(766\) 0.708204 0.0255884
\(767\) −39.4508 −1.42449
\(768\) 0.381966 0.0137830
\(769\) 30.2492 1.09081 0.545407 0.838171i \(-0.316375\pi\)
0.545407 + 0.838171i \(0.316375\pi\)
\(770\) −12.7082 −0.457972
\(771\) −0.270510 −0.00974217
\(772\) −4.18034 −0.150454
\(773\) 17.2148 0.619173 0.309586 0.950871i \(-0.399809\pi\)
0.309586 + 0.950871i \(0.399809\pi\)
\(774\) −33.0000 −1.18616
\(775\) 64.7214 2.32486
\(776\) 5.29180 0.189964
\(777\) 1.90983 0.0685148
\(778\) −10.2016 −0.365746
\(779\) 0 0
\(780\) 6.23607 0.223287
\(781\) 10.5836 0.378711
\(782\) 12.4377 0.444771
\(783\) 18.4164 0.658149
\(784\) 1.00000 0.0357143
\(785\) 43.9787 1.56967
\(786\) 1.29180 0.0460768
\(787\) 39.0902 1.39341 0.696707 0.717356i \(-0.254648\pi\)
0.696707 + 0.717356i \(0.254648\pi\)
\(788\) 19.0344 0.678074
\(789\) −2.02129 −0.0719597
\(790\) 42.9787 1.52911
\(791\) −11.7984 −0.419502
\(792\) 8.56231 0.304248
\(793\) 25.8541 0.918106
\(794\) 14.3262 0.508419
\(795\) 20.5623 0.729270
\(796\) 15.7984 0.559959
\(797\) 13.7984 0.488763 0.244382 0.969679i \(-0.421415\pi\)
0.244382 + 0.969679i \(0.421415\pi\)
\(798\) 0 0
\(799\) −11.5623 −0.409045
\(800\) −12.9443 −0.457649
\(801\) 3.94427 0.139364
\(802\) 24.4721 0.864141
\(803\) −27.2705 −0.962355
\(804\) −1.74265 −0.0614584
\(805\) −28.4164 −1.00155
\(806\) 19.2705 0.678775
\(807\) −12.0213 −0.423169
\(808\) 8.09017 0.284611
\(809\) 11.7295 0.412387 0.206193 0.978511i \(-0.433892\pi\)
0.206193 + 0.978511i \(0.433892\pi\)
\(810\) 32.6525 1.14729
\(811\) 1.23607 0.0434042 0.0217021 0.999764i \(-0.493091\pi\)
0.0217021 + 0.999764i \(0.493091\pi\)
\(812\) 8.23607 0.289029
\(813\) 1.36881 0.0480063
\(814\) 15.0000 0.525750
\(815\) 14.9443 0.523475
\(816\) 0.708204 0.0247921
\(817\) 0 0
\(818\) −16.1459 −0.564528
\(819\) −11.0000 −0.384371
\(820\) 17.3262 0.605058
\(821\) −13.1803 −0.459997 −0.229999 0.973191i \(-0.573872\pi\)
−0.229999 + 0.973191i \(0.573872\pi\)
\(822\) −3.52786 −0.123048
\(823\) −33.0689 −1.15271 −0.576354 0.817200i \(-0.695525\pi\)
−0.576354 + 0.817200i \(0.695525\pi\)
\(824\) −6.85410 −0.238774
\(825\) 14.8328 0.516412
\(826\) 10.2361 0.356158
\(827\) −7.61803 −0.264905 −0.132452 0.991189i \(-0.542285\pi\)
−0.132452 + 0.991189i \(0.542285\pi\)
\(828\) 19.1459 0.665366
\(829\) 54.1246 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(830\) −27.4164 −0.951637
\(831\) −7.05573 −0.244760
\(832\) −3.85410 −0.133617
\(833\) 1.85410 0.0642408
\(834\) −7.25735 −0.251302
\(835\) −54.0689 −1.87113
\(836\) 0 0
\(837\) −11.1803 −0.386449
\(838\) 28.0902 0.970359
\(839\) −51.4721 −1.77702 −0.888508 0.458862i \(-0.848257\pi\)
−0.888508 + 0.458862i \(0.848257\pi\)
\(840\) −1.61803 −0.0558275
\(841\) 38.8328 1.33906
\(842\) 16.8885 0.582018
\(843\) −5.97871 −0.205918
\(844\) 21.0000 0.722850
\(845\) −7.85410 −0.270189
\(846\) −17.7984 −0.611921
\(847\) 2.00000 0.0687208
\(848\) −12.7082 −0.436402
\(849\) 5.74265 0.197087
\(850\) −24.0000 −0.823193
\(851\) 33.5410 1.14977
\(852\) 1.34752 0.0461654
\(853\) −41.0000 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(854\) −6.70820 −0.229550
\(855\) 0 0
\(856\) 13.4721 0.460468
\(857\) −10.2016 −0.348481 −0.174240 0.984703i \(-0.555747\pi\)
−0.174240 + 0.984703i \(0.555747\pi\)
\(858\) 4.41641 0.150774
\(859\) 54.3050 1.85286 0.926431 0.376466i \(-0.122861\pi\)
0.926431 + 0.376466i \(0.122861\pi\)
\(860\) 48.9787 1.67016
\(861\) 1.56231 0.0532432
\(862\) −22.3607 −0.761608
\(863\) −51.6525 −1.75827 −0.879135 0.476572i \(-0.841879\pi\)
−0.879135 + 0.476572i \(0.841879\pi\)
\(864\) 2.23607 0.0760726
\(865\) 18.7082 0.636098
\(866\) 25.8541 0.878558
\(867\) −5.18034 −0.175934
\(868\) −5.00000 −0.169711
\(869\) 30.4377 1.03253
\(870\) −13.3262 −0.451802
\(871\) 17.5836 0.595797
\(872\) −4.85410 −0.164381
\(873\) 15.1033 0.511170
\(874\) 0 0
\(875\) 33.6525 1.13766
\(876\) −3.47214 −0.117313
\(877\) 2.32624 0.0785515 0.0392757 0.999228i \(-0.487495\pi\)
0.0392757 + 0.999228i \(0.487495\pi\)
\(878\) −13.7639 −0.464510
\(879\) −0.604878 −0.0204020
\(880\) −12.7082 −0.428393
\(881\) 4.74265 0.159784 0.0798919 0.996804i \(-0.474542\pi\)
0.0798919 + 0.996804i \(0.474542\pi\)
\(882\) 2.85410 0.0961026
\(883\) 43.1033 1.45054 0.725271 0.688463i \(-0.241714\pi\)
0.725271 + 0.688463i \(0.241714\pi\)
\(884\) −7.14590 −0.240343
\(885\) −16.5623 −0.556736
\(886\) −26.5066 −0.890506
\(887\) 41.7771 1.40274 0.701369 0.712799i \(-0.252573\pi\)
0.701369 + 0.712799i \(0.252573\pi\)
\(888\) 1.90983 0.0640897
\(889\) −20.0344 −0.671934
\(890\) −5.85410 −0.196230
\(891\) 23.1246 0.774704
\(892\) 9.00000 0.301342
\(893\) 0 0
\(894\) 0.729490 0.0243978
\(895\) −15.5623 −0.520191
\(896\) 1.00000 0.0334077
\(897\) 9.87539 0.329730
\(898\) 28.7639 0.959865
\(899\) −41.1803 −1.37344
\(900\) −36.9443 −1.23148
\(901\) −23.5623 −0.784974
\(902\) 12.2705 0.408563
\(903\) 4.41641 0.146969
\(904\) −11.7984 −0.392408
\(905\) −67.5410 −2.24514
\(906\) −0.0901699 −0.00299570
\(907\) 22.1459 0.735342 0.367671 0.929956i \(-0.380155\pi\)
0.367671 + 0.929956i \(0.380155\pi\)
\(908\) 27.0000 0.896026
\(909\) 23.0902 0.765853
\(910\) 16.3262 0.541210
\(911\) 7.74265 0.256525 0.128263 0.991740i \(-0.459060\pi\)
0.128263 + 0.991740i \(0.459060\pi\)
\(912\) 0 0
\(913\) −19.4164 −0.642589
\(914\) 37.8885 1.25324
\(915\) 10.8541 0.358826
\(916\) −17.5623 −0.580275
\(917\) 3.38197 0.111682
\(918\) 4.14590 0.136835
\(919\) −35.3951 −1.16758 −0.583789 0.811906i \(-0.698430\pi\)
−0.583789 + 0.811906i \(0.698430\pi\)
\(920\) −28.4164 −0.936861
\(921\) 10.2492 0.337723
\(922\) 21.7082 0.714922
\(923\) −13.5967 −0.447542
\(924\) −1.14590 −0.0376973
\(925\) −64.7214 −2.12803
\(926\) −33.9787 −1.11661
\(927\) −19.5623 −0.642510
\(928\) 8.23607 0.270362
\(929\) 31.0689 1.01934 0.509669 0.860371i \(-0.329768\pi\)
0.509669 + 0.860371i \(0.329768\pi\)
\(930\) 8.09017 0.265287
\(931\) 0 0
\(932\) −16.7984 −0.550249
\(933\) −12.9656 −0.424473
\(934\) −38.1246 −1.24748
\(935\) −23.5623 −0.770570
\(936\) −11.0000 −0.359546
\(937\) −39.0344 −1.27520 −0.637600 0.770368i \(-0.720073\pi\)
−0.637600 + 0.770368i \(0.720073\pi\)
\(938\) −4.56231 −0.148965
\(939\) −1.63119 −0.0532319
\(940\) 26.4164 0.861608
\(941\) 8.61803 0.280940 0.140470 0.990085i \(-0.455139\pi\)
0.140470 + 0.990085i \(0.455139\pi\)
\(942\) 3.96556 0.129205
\(943\) 27.4377 0.893494
\(944\) 10.2361 0.333156
\(945\) −9.47214 −0.308129
\(946\) 34.6869 1.12777
\(947\) −37.5279 −1.21949 −0.609746 0.792597i \(-0.708728\pi\)
−0.609746 + 0.792597i \(0.708728\pi\)
\(948\) 3.87539 0.125867
\(949\) 35.0344 1.13727
\(950\) 0 0
\(951\) −7.07701 −0.229488
\(952\) 1.85410 0.0600918
\(953\) 34.4164 1.11486 0.557428 0.830225i \(-0.311788\pi\)
0.557428 + 0.830225i \(0.311788\pi\)
\(954\) −36.2705 −1.17430
\(955\) 32.8885 1.06425
\(956\) 4.03444 0.130483
\(957\) −9.43769 −0.305077
\(958\) 27.7082 0.895211
\(959\) −9.23607 −0.298248
\(960\) −1.61803 −0.0522218
\(961\) −6.00000 −0.193548
\(962\) −19.2705 −0.621306
\(963\) 38.4508 1.23906
\(964\) 20.1459 0.648856
\(965\) 17.7082 0.570047
\(966\) −2.56231 −0.0824408
\(967\) −8.56231 −0.275345 −0.137673 0.990478i \(-0.543962\pi\)
−0.137673 + 0.990478i \(0.543962\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) −22.4164 −0.719747
\(971\) 20.2361 0.649406 0.324703 0.945816i \(-0.394736\pi\)
0.324703 + 0.945816i \(0.394736\pi\)
\(972\) 9.65248 0.309603
\(973\) −19.0000 −0.609112
\(974\) 10.3475 0.331556
\(975\) −19.0557 −0.610272
\(976\) −6.70820 −0.214724
\(977\) −32.5279 −1.04066 −0.520329 0.853966i \(-0.674191\pi\)
−0.520329 + 0.853966i \(0.674191\pi\)
\(978\) 1.34752 0.0430891
\(979\) −4.14590 −0.132503
\(980\) −4.23607 −0.135316
\(981\) −13.8541 −0.442327
\(982\) 24.0689 0.768069
\(983\) 23.7984 0.759050 0.379525 0.925181i \(-0.376087\pi\)
0.379525 + 0.925181i \(0.376087\pi\)
\(984\) 1.56231 0.0498045
\(985\) −80.6312 −2.56912
\(986\) 15.2705 0.486312
\(987\) 2.38197 0.0758188
\(988\) 0 0
\(989\) 77.5623 2.46634
\(990\) −36.2705 −1.15275
\(991\) 13.4164 0.426186 0.213093 0.977032i \(-0.431646\pi\)
0.213093 + 0.977032i \(0.431646\pi\)
\(992\) −5.00000 −0.158750
\(993\) −0.180340 −0.00572291
\(994\) 3.52786 0.111897
\(995\) −66.9230 −2.12160
\(996\) −2.47214 −0.0783326
\(997\) −15.9656 −0.505634 −0.252817 0.967514i \(-0.581357\pi\)
−0.252817 + 0.967514i \(0.581357\pi\)
\(998\) 3.00000 0.0949633
\(999\) 11.1803 0.353730
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 5054.2.a.i.1.1 2
19.18 odd 2 5054.2.a.j.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.i.1.1 2 1.1 even 1 trivial
5054.2.a.j.1.2 yes 2 19.18 odd 2