Properties

Label 4830.2.a.cb.1.2
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
Dimension $3$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.925197 q^{11} -1.00000 q^{12} -4.64681 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -3.72161 q^{17} +1.00000 q^{18} -1.72161 q^{19} +1.00000 q^{20} -1.00000 q^{21} -0.925197 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.64681 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.92520 q^{29} -1.00000 q^{30} +5.72161 q^{31} +1.00000 q^{32} +0.925197 q^{33} -3.72161 q^{34} +1.00000 q^{35} +1.00000 q^{36} +9.44322 q^{37} -1.72161 q^{38} +4.64681 q^{39} +1.00000 q^{40} +5.44322 q^{41} -1.00000 q^{42} +8.92520 q^{43} -0.925197 q^{44} +1.00000 q^{45} -1.00000 q^{46} +5.72161 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.72161 q^{51} -4.64681 q^{52} +9.44322 q^{53} -1.00000 q^{54} -0.925197 q^{55} +1.00000 q^{56} +1.72161 q^{57} +2.92520 q^{58} -7.44322 q^{59} -1.00000 q^{60} -2.00000 q^{61} +5.72161 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.64681 q^{65} +0.925197 q^{66} +0.925197 q^{67} -3.72161 q^{68} +1.00000 q^{69} +1.00000 q^{70} +6.77559 q^{71} +1.00000 q^{72} +15.2936 q^{73} +9.44322 q^{74} -1.00000 q^{75} -1.72161 q^{76} -0.925197 q^{77} +4.64681 q^{78} +1.00000 q^{80} +1.00000 q^{81} +5.44322 q^{82} +15.0152 q^{83} -1.00000 q^{84} -3.72161 q^{85} +8.92520 q^{86} -2.92520 q^{87} -0.925197 q^{88} -6.79641 q^{89} +1.00000 q^{90} -4.64681 q^{91} -1.00000 q^{92} -5.72161 q^{93} +5.72161 q^{94} -1.72161 q^{95} -1.00000 q^{96} -10.2396 q^{97} +1.00000 q^{98} -0.925197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 2 q^{11} - 3 q^{12} + 2 q^{13} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{18} + 6 q^{19} + 3 q^{20} - 3 q^{21} + 2 q^{22} - 3 q^{23} - 3 q^{24} + 3 q^{25} + 2 q^{26} - 3 q^{27} + 3 q^{28} + 4 q^{29} - 3 q^{30} + 6 q^{31} + 3 q^{32} - 2 q^{33} + 3 q^{35} + 3 q^{36} + 6 q^{37} + 6 q^{38} - 2 q^{39} + 3 q^{40} - 6 q^{41} - 3 q^{42} + 22 q^{43} + 2 q^{44} + 3 q^{45} - 3 q^{46} + 6 q^{47} - 3 q^{48} + 3 q^{49} + 3 q^{50} + 2 q^{52} + 6 q^{53} - 3 q^{54} + 2 q^{55} + 3 q^{56} - 6 q^{57} + 4 q^{58} - 3 q^{60} - 6 q^{61} + 6 q^{62} + 3 q^{63} + 3 q^{64} + 2 q^{65} - 2 q^{66} - 2 q^{67} + 3 q^{69} + 3 q^{70} + 6 q^{71} + 3 q^{72} + 14 q^{73} + 6 q^{74} - 3 q^{75} + 6 q^{76} + 2 q^{77} - 2 q^{78} + 3 q^{80} + 3 q^{81} - 6 q^{82} + 2 q^{83} - 3 q^{84} + 22 q^{86} - 4 q^{87} + 2 q^{88} - 14 q^{89} + 3 q^{90} + 2 q^{91} - 3 q^{92} - 6 q^{93} + 6 q^{94} + 6 q^{95} - 3 q^{96} - 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −0.925197 −0.278957 −0.139479 0.990225i \(-0.544543\pi\)
−0.139479 + 0.990225i \(0.544543\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.64681 −1.28879 −0.644396 0.764692i \(-0.722891\pi\)
−0.644396 + 0.764692i \(0.722891\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −3.72161 −0.902623 −0.451312 0.892366i \(-0.649044\pi\)
−0.451312 + 0.892366i \(0.649044\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.72161 −0.394965 −0.197482 0.980306i \(-0.563277\pi\)
−0.197482 + 0.980306i \(0.563277\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −0.925197 −0.197253
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −4.64681 −0.911314
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 2.92520 0.543195 0.271598 0.962411i \(-0.412448\pi\)
0.271598 + 0.962411i \(0.412448\pi\)
\(30\) −1.00000 −0.182574
\(31\) 5.72161 1.02763 0.513816 0.857900i \(-0.328231\pi\)
0.513816 + 0.857900i \(0.328231\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.925197 0.161056
\(34\) −3.72161 −0.638251
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 9.44322 1.55246 0.776228 0.630452i \(-0.217130\pi\)
0.776228 + 0.630452i \(0.217130\pi\)
\(38\) −1.72161 −0.279282
\(39\) 4.64681 0.744085
\(40\) 1.00000 0.158114
\(41\) 5.44322 0.850089 0.425044 0.905173i \(-0.360259\pi\)
0.425044 + 0.905173i \(0.360259\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.92520 1.36108 0.680540 0.732711i \(-0.261745\pi\)
0.680540 + 0.732711i \(0.261745\pi\)
\(44\) −0.925197 −0.139479
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) 5.72161 0.834583 0.417291 0.908773i \(-0.362979\pi\)
0.417291 + 0.908773i \(0.362979\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 3.72161 0.521130
\(52\) −4.64681 −0.644396
\(53\) 9.44322 1.29713 0.648563 0.761161i \(-0.275370\pi\)
0.648563 + 0.761161i \(0.275370\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.925197 −0.124754
\(56\) 1.00000 0.133631
\(57\) 1.72161 0.228033
\(58\) 2.92520 0.384097
\(59\) −7.44322 −0.969025 −0.484513 0.874784i \(-0.661003\pi\)
−0.484513 + 0.874784i \(0.661003\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 5.72161 0.726645
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −4.64681 −0.576366
\(66\) 0.925197 0.113884
\(67\) 0.925197 0.113031 0.0565154 0.998402i \(-0.482001\pi\)
0.0565154 + 0.998402i \(0.482001\pi\)
\(68\) −3.72161 −0.451312
\(69\) 1.00000 0.120386
\(70\) 1.00000 0.119523
\(71\) 6.77559 0.804115 0.402057 0.915614i \(-0.368295\pi\)
0.402057 + 0.915614i \(0.368295\pi\)
\(72\) 1.00000 0.117851
\(73\) 15.2936 1.78998 0.894991 0.446084i \(-0.147182\pi\)
0.894991 + 0.446084i \(0.147182\pi\)
\(74\) 9.44322 1.09775
\(75\) −1.00000 −0.115470
\(76\) −1.72161 −0.197482
\(77\) −0.925197 −0.105436
\(78\) 4.64681 0.526147
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 5.44322 0.601103
\(83\) 15.0152 1.64814 0.824068 0.566491i \(-0.191700\pi\)
0.824068 + 0.566491i \(0.191700\pi\)
\(84\) −1.00000 −0.109109
\(85\) −3.72161 −0.403665
\(86\) 8.92520 0.962429
\(87\) −2.92520 −0.313614
\(88\) −0.925197 −0.0986263
\(89\) −6.79641 −0.720419 −0.360209 0.932872i \(-0.617295\pi\)
−0.360209 + 0.932872i \(0.617295\pi\)
\(90\) 1.00000 0.105409
\(91\) −4.64681 −0.487118
\(92\) −1.00000 −0.104257
\(93\) −5.72161 −0.593303
\(94\) 5.72161 0.590139
\(95\) −1.72161 −0.176634
\(96\) −1.00000 −0.102062
\(97\) −10.2396 −1.03968 −0.519839 0.854264i \(-0.674008\pi\)
−0.519839 + 0.854264i \(0.674008\pi\)
\(98\) 1.00000 0.101015
\(99\) −0.925197 −0.0929858
\(100\) 1.00000 0.100000
\(101\) −2.79641 −0.278254 −0.139127 0.990275i \(-0.544430\pi\)
−0.139127 + 0.990275i \(0.544430\pi\)
\(102\) 3.72161 0.368494
\(103\) 1.85039 0.182325 0.0911624 0.995836i \(-0.470942\pi\)
0.0911624 + 0.995836i \(0.470942\pi\)
\(104\) −4.64681 −0.455657
\(105\) −1.00000 −0.0975900
\(106\) 9.44322 0.917207
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −0.149606 −0.0143297 −0.00716484 0.999974i \(-0.502281\pi\)
−0.00716484 + 0.999974i \(0.502281\pi\)
\(110\) −0.925197 −0.0882141
\(111\) −9.44322 −0.896311
\(112\) 1.00000 0.0944911
\(113\) 13.4432 1.26463 0.632316 0.774711i \(-0.282104\pi\)
0.632316 + 0.774711i \(0.282104\pi\)
\(114\) 1.72161 0.161244
\(115\) −1.00000 −0.0932505
\(116\) 2.92520 0.271598
\(117\) −4.64681 −0.429598
\(118\) −7.44322 −0.685204
\(119\) −3.72161 −0.341160
\(120\) −1.00000 −0.0912871
\(121\) −10.1440 −0.922183
\(122\) −2.00000 −0.181071
\(123\) −5.44322 −0.490799
\(124\) 5.72161 0.513816
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) −0.368420 −0.0326920 −0.0163460 0.999866i \(-0.505203\pi\)
−0.0163460 + 0.999866i \(0.505203\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.92520 −0.785820
\(130\) −4.64681 −0.407552
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0.925197 0.0805280
\(133\) −1.72161 −0.149283
\(134\) 0.925197 0.0799248
\(135\) −1.00000 −0.0860663
\(136\) −3.72161 −0.319126
\(137\) −9.70079 −0.828794 −0.414397 0.910096i \(-0.636008\pi\)
−0.414397 + 0.910096i \(0.636008\pi\)
\(138\) 1.00000 0.0851257
\(139\) 9.29362 0.788274 0.394137 0.919052i \(-0.371043\pi\)
0.394137 + 0.919052i \(0.371043\pi\)
\(140\) 1.00000 0.0845154
\(141\) −5.72161 −0.481847
\(142\) 6.77559 0.568595
\(143\) 4.29921 0.359518
\(144\) 1.00000 0.0833333
\(145\) 2.92520 0.242924
\(146\) 15.2936 1.26571
\(147\) −1.00000 −0.0824786
\(148\) 9.44322 0.776228
\(149\) −17.1440 −1.40449 −0.702246 0.711934i \(-0.747819\pi\)
−0.702246 + 0.711934i \(0.747819\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −13.2936 −1.08182 −0.540910 0.841081i \(-0.681920\pi\)
−0.540910 + 0.841081i \(0.681920\pi\)
\(152\) −1.72161 −0.139641
\(153\) −3.72161 −0.300874
\(154\) −0.925197 −0.0745545
\(155\) 5.72161 0.459571
\(156\) 4.64681 0.372042
\(157\) 14.7368 1.17613 0.588064 0.808814i \(-0.299890\pi\)
0.588064 + 0.808814i \(0.299890\pi\)
\(158\) 0 0
\(159\) −9.44322 −0.748896
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) −1.29362 −0.101324 −0.0506620 0.998716i \(-0.516133\pi\)
−0.0506620 + 0.998716i \(0.516133\pi\)
\(164\) 5.44322 0.425044
\(165\) 0.925197 0.0720265
\(166\) 15.0152 1.16541
\(167\) −14.4585 −1.11883 −0.559414 0.828888i \(-0.688974\pi\)
−0.559414 + 0.828888i \(0.688974\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 8.59283 0.660987
\(170\) −3.72161 −0.285435
\(171\) −1.72161 −0.131655
\(172\) 8.92520 0.680540
\(173\) 7.16484 0.544732 0.272366 0.962194i \(-0.412194\pi\)
0.272366 + 0.962194i \(0.412194\pi\)
\(174\) −2.92520 −0.221759
\(175\) 1.00000 0.0755929
\(176\) −0.925197 −0.0697393
\(177\) 7.44322 0.559467
\(178\) −6.79641 −0.509413
\(179\) 20.7368 1.54994 0.774972 0.631995i \(-0.217764\pi\)
0.774972 + 0.631995i \(0.217764\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) −4.64681 −0.344444
\(183\) 2.00000 0.147844
\(184\) −1.00000 −0.0737210
\(185\) 9.44322 0.694280
\(186\) −5.72161 −0.419529
\(187\) 3.44322 0.251793
\(188\) 5.72161 0.417291
\(189\) −1.00000 −0.0727393
\(190\) −1.72161 −0.124899
\(191\) 4.55678 0.329717 0.164858 0.986317i \(-0.447283\pi\)
0.164858 + 0.986317i \(0.447283\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.55678 0.471967 0.235984 0.971757i \(-0.424169\pi\)
0.235984 + 0.971757i \(0.424169\pi\)
\(194\) −10.2396 −0.735163
\(195\) 4.64681 0.332765
\(196\) 1.00000 0.0714286
\(197\) −1.74244 −0.124143 −0.0620717 0.998072i \(-0.519771\pi\)
−0.0620717 + 0.998072i \(0.519771\pi\)
\(198\) −0.925197 −0.0657509
\(199\) −3.44322 −0.244084 −0.122042 0.992525i \(-0.538944\pi\)
−0.122042 + 0.992525i \(0.538944\pi\)
\(200\) 1.00000 0.0707107
\(201\) −0.925197 −0.0652584
\(202\) −2.79641 −0.196755
\(203\) 2.92520 0.205309
\(204\) 3.72161 0.260565
\(205\) 5.44322 0.380171
\(206\) 1.85039 0.128923
\(207\) −1.00000 −0.0695048
\(208\) −4.64681 −0.322198
\(209\) 1.59283 0.110178
\(210\) −1.00000 −0.0690066
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 9.44322 0.648563
\(213\) −6.77559 −0.464256
\(214\) 4.00000 0.273434
\(215\) 8.92520 0.608693
\(216\) −1.00000 −0.0680414
\(217\) 5.72161 0.388408
\(218\) −0.149606 −0.0101326
\(219\) −15.2936 −1.03345
\(220\) −0.925197 −0.0623768
\(221\) 17.2936 1.16329
\(222\) −9.44322 −0.633788
\(223\) −4.49720 −0.301155 −0.150577 0.988598i \(-0.548113\pi\)
−0.150577 + 0.988598i \(0.548113\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 13.4432 0.894230
\(227\) −6.27839 −0.416711 −0.208356 0.978053i \(-0.566811\pi\)
−0.208356 + 0.978053i \(0.566811\pi\)
\(228\) 1.72161 0.114017
\(229\) −0.149606 −0.00988626 −0.00494313 0.999988i \(-0.501573\pi\)
−0.00494313 + 0.999988i \(0.501573\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0.925197 0.0608735
\(232\) 2.92520 0.192049
\(233\) 5.44322 0.356597 0.178299 0.983976i \(-0.442941\pi\)
0.178299 + 0.983976i \(0.442941\pi\)
\(234\) −4.64681 −0.303771
\(235\) 5.72161 0.373237
\(236\) −7.44322 −0.484513
\(237\) 0 0
\(238\) −3.72161 −0.241236
\(239\) 14.7756 0.955753 0.477877 0.878427i \(-0.341407\pi\)
0.477877 + 0.878427i \(0.341407\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 3.16484 0.203865 0.101932 0.994791i \(-0.467497\pi\)
0.101932 + 0.994791i \(0.467497\pi\)
\(242\) −10.1440 −0.652082
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) −5.44322 −0.347047
\(247\) 8.00000 0.509028
\(248\) 5.72161 0.363323
\(249\) −15.0152 −0.951551
\(250\) 1.00000 0.0632456
\(251\) −4.79641 −0.302747 −0.151374 0.988477i \(-0.548370\pi\)
−0.151374 + 0.988477i \(0.548370\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0.925197 0.0581666
\(254\) −0.368420 −0.0231167
\(255\) 3.72161 0.233056
\(256\) 1.00000 0.0625000
\(257\) −29.1440 −1.81795 −0.908977 0.416847i \(-0.863135\pi\)
−0.908977 + 0.416847i \(0.863135\pi\)
\(258\) −8.92520 −0.555658
\(259\) 9.44322 0.586773
\(260\) −4.64681 −0.288183
\(261\) 2.92520 0.181065
\(262\) −4.00000 −0.247121
\(263\) −20.1801 −1.24436 −0.622178 0.782876i \(-0.713752\pi\)
−0.622178 + 0.782876i \(0.713752\pi\)
\(264\) 0.925197 0.0569419
\(265\) 9.44322 0.580093
\(266\) −1.72161 −0.105559
\(267\) 6.79641 0.415934
\(268\) 0.925197 0.0565154
\(269\) −27.5333 −1.67873 −0.839366 0.543567i \(-0.817074\pi\)
−0.839366 + 0.543567i \(0.817074\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 2.02082 0.122756 0.0613782 0.998115i \(-0.480450\pi\)
0.0613782 + 0.998115i \(0.480450\pi\)
\(272\) −3.72161 −0.225656
\(273\) 4.64681 0.281238
\(274\) −9.70079 −0.586046
\(275\) −0.925197 −0.0557915
\(276\) 1.00000 0.0601929
\(277\) 6.36842 0.382641 0.191321 0.981528i \(-0.438723\pi\)
0.191321 + 0.981528i \(0.438723\pi\)
\(278\) 9.29362 0.557394
\(279\) 5.72161 0.342544
\(280\) 1.00000 0.0597614
\(281\) 16.5180 0.985383 0.492691 0.870204i \(-0.336013\pi\)
0.492691 + 0.870204i \(0.336013\pi\)
\(282\) −5.72161 −0.340717
\(283\) 29.5333 1.75557 0.877785 0.479055i \(-0.159021\pi\)
0.877785 + 0.479055i \(0.159021\pi\)
\(284\) 6.77559 0.402057
\(285\) 1.72161 0.101979
\(286\) 4.29921 0.254218
\(287\) 5.44322 0.321303
\(288\) 1.00000 0.0589256
\(289\) −3.14961 −0.185271
\(290\) 2.92520 0.171773
\(291\) 10.2396 0.600258
\(292\) 15.2936 0.894991
\(293\) −3.85039 −0.224942 −0.112471 0.993655i \(-0.535877\pi\)
−0.112471 + 0.993655i \(0.535877\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −7.44322 −0.433361
\(296\) 9.44322 0.548876
\(297\) 0.925197 0.0536854
\(298\) −17.1440 −0.993126
\(299\) 4.64681 0.268732
\(300\) −1.00000 −0.0577350
\(301\) 8.92520 0.514440
\(302\) −13.2936 −0.764962
\(303\) 2.79641 0.160650
\(304\) −1.72161 −0.0987412
\(305\) −2.00000 −0.114520
\(306\) −3.72161 −0.212750
\(307\) 25.0361 1.42888 0.714442 0.699695i \(-0.246681\pi\)
0.714442 + 0.699695i \(0.246681\pi\)
\(308\) −0.925197 −0.0527180
\(309\) −1.85039 −0.105265
\(310\) 5.72161 0.324966
\(311\) 14.0900 0.798972 0.399486 0.916739i \(-0.369189\pi\)
0.399486 + 0.916739i \(0.369189\pi\)
\(312\) 4.64681 0.263074
\(313\) 3.05398 0.172621 0.0863105 0.996268i \(-0.472492\pi\)
0.0863105 + 0.996268i \(0.472492\pi\)
\(314\) 14.7368 0.831648
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 3.29362 0.184988 0.0924940 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(318\) −9.44322 −0.529550
\(319\) −2.70638 −0.151528
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) −1.00000 −0.0557278
\(323\) 6.40717 0.356504
\(324\) 1.00000 0.0555556
\(325\) −4.64681 −0.257759
\(326\) −1.29362 −0.0716468
\(327\) 0.149606 0.00827325
\(328\) 5.44322 0.300552
\(329\) 5.72161 0.315443
\(330\) 0.925197 0.0509304
\(331\) 10.4072 0.572030 0.286015 0.958225i \(-0.407669\pi\)
0.286015 + 0.958225i \(0.407669\pi\)
\(332\) 15.0152 0.824068
\(333\) 9.44322 0.517485
\(334\) −14.4585 −0.791131
\(335\) 0.925197 0.0505489
\(336\) −1.00000 −0.0545545
\(337\) 5.33237 0.290473 0.145236 0.989397i \(-0.453606\pi\)
0.145236 + 0.989397i \(0.453606\pi\)
\(338\) 8.59283 0.467388
\(339\) −13.4432 −0.730136
\(340\) −3.72161 −0.201833
\(341\) −5.29362 −0.286665
\(342\) −1.72161 −0.0930941
\(343\) 1.00000 0.0539949
\(344\) 8.92520 0.481214
\(345\) 1.00000 0.0538382
\(346\) 7.16484 0.385184
\(347\) −5.59283 −0.300239 −0.150119 0.988668i \(-0.547966\pi\)
−0.150119 + 0.988668i \(0.547966\pi\)
\(348\) −2.92520 −0.156807
\(349\) −12.0208 −0.643460 −0.321730 0.946831i \(-0.604264\pi\)
−0.321730 + 0.946831i \(0.604264\pi\)
\(350\) 1.00000 0.0534522
\(351\) 4.64681 0.248028
\(352\) −0.925197 −0.0493132
\(353\) −11.0361 −0.587390 −0.293695 0.955899i \(-0.594885\pi\)
−0.293695 + 0.955899i \(0.594885\pi\)
\(354\) 7.44322 0.395603
\(355\) 6.77559 0.359611
\(356\) −6.79641 −0.360209
\(357\) 3.72161 0.196969
\(358\) 20.7368 1.09598
\(359\) 7.14401 0.377046 0.188523 0.982069i \(-0.439630\pi\)
0.188523 + 0.982069i \(0.439630\pi\)
\(360\) 1.00000 0.0527046
\(361\) −16.0361 −0.844003
\(362\) 6.00000 0.315353
\(363\) 10.1440 0.532422
\(364\) −4.64681 −0.243559
\(365\) 15.2936 0.800505
\(366\) 2.00000 0.104542
\(367\) −20.4376 −1.06684 −0.533418 0.845852i \(-0.679093\pi\)
−0.533418 + 0.845852i \(0.679093\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 5.44322 0.283363
\(370\) 9.44322 0.490930
\(371\) 9.44322 0.490268
\(372\) −5.72161 −0.296652
\(373\) −1.14401 −0.0592346 −0.0296173 0.999561i \(-0.509429\pi\)
−0.0296173 + 0.999561i \(0.509429\pi\)
\(374\) 3.44322 0.178045
\(375\) −1.00000 −0.0516398
\(376\) 5.72161 0.295070
\(377\) −13.5928 −0.700066
\(378\) −1.00000 −0.0514344
\(379\) 0.299213 0.0153695 0.00768476 0.999970i \(-0.497554\pi\)
0.00768476 + 0.999970i \(0.497554\pi\)
\(380\) −1.72161 −0.0883168
\(381\) 0.368420 0.0188747
\(382\) 4.55678 0.233145
\(383\) 0.736841 0.0376508 0.0188254 0.999823i \(-0.494007\pi\)
0.0188254 + 0.999823i \(0.494007\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.925197 −0.0471524
\(386\) 6.55678 0.333731
\(387\) 8.92520 0.453693
\(388\) −10.2396 −0.519839
\(389\) −31.0361 −1.57359 −0.786795 0.617214i \(-0.788261\pi\)
−0.786795 + 0.617214i \(0.788261\pi\)
\(390\) 4.64681 0.235300
\(391\) 3.72161 0.188210
\(392\) 1.00000 0.0505076
\(393\) 4.00000 0.201773
\(394\) −1.74244 −0.0877827
\(395\) 0 0
\(396\) −0.925197 −0.0464929
\(397\) −4.64681 −0.233217 −0.116608 0.993178i \(-0.537202\pi\)
−0.116608 + 0.993178i \(0.537202\pi\)
\(398\) −3.44322 −0.172593
\(399\) 1.72161 0.0861884
\(400\) 1.00000 0.0500000
\(401\) −8.21881 −0.410428 −0.205214 0.978717i \(-0.565789\pi\)
−0.205214 + 0.978717i \(0.565789\pi\)
\(402\) −0.925197 −0.0461446
\(403\) −26.5872 −1.32440
\(404\) −2.79641 −0.139127
\(405\) 1.00000 0.0496904
\(406\) 2.92520 0.145175
\(407\) −8.73684 −0.433069
\(408\) 3.72161 0.184247
\(409\) −12.1496 −0.600759 −0.300380 0.953820i \(-0.597113\pi\)
−0.300380 + 0.953820i \(0.597113\pi\)
\(410\) 5.44322 0.268822
\(411\) 9.70079 0.478505
\(412\) 1.85039 0.0911624
\(413\) −7.44322 −0.366257
\(414\) −1.00000 −0.0491473
\(415\) 15.0152 0.737069
\(416\) −4.64681 −0.227829
\(417\) −9.29362 −0.455110
\(418\) 1.59283 0.0779078
\(419\) −26.3476 −1.28716 −0.643582 0.765377i \(-0.722553\pi\)
−0.643582 + 0.765377i \(0.722553\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 8.70638 0.424323 0.212162 0.977235i \(-0.431950\pi\)
0.212162 + 0.977235i \(0.431950\pi\)
\(422\) 4.00000 0.194717
\(423\) 5.72161 0.278194
\(424\) 9.44322 0.458603
\(425\) −3.72161 −0.180525
\(426\) −6.77559 −0.328278
\(427\) −2.00000 −0.0967868
\(428\) 4.00000 0.193347
\(429\) −4.29921 −0.207568
\(430\) 8.92520 0.430411
\(431\) 22.0305 1.06117 0.530585 0.847632i \(-0.321972\pi\)
0.530585 + 0.847632i \(0.321972\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −3.35319 −0.161144 −0.0805720 0.996749i \(-0.525675\pi\)
−0.0805720 + 0.996749i \(0.525675\pi\)
\(434\) 5.72161 0.274646
\(435\) −2.92520 −0.140252
\(436\) −0.149606 −0.00716484
\(437\) 1.72161 0.0823559
\(438\) −15.2936 −0.730757
\(439\) −23.5720 −1.12503 −0.562515 0.826787i \(-0.690166\pi\)
−0.562515 + 0.826787i \(0.690166\pi\)
\(440\) −0.925197 −0.0441070
\(441\) 1.00000 0.0476190
\(442\) 17.2936 0.822573
\(443\) −12.4793 −0.592908 −0.296454 0.955047i \(-0.595804\pi\)
−0.296454 + 0.955047i \(0.595804\pi\)
\(444\) −9.44322 −0.448156
\(445\) −6.79641 −0.322181
\(446\) −4.49720 −0.212949
\(447\) 17.1440 0.810884
\(448\) 1.00000 0.0472456
\(449\) 30.1801 1.42429 0.712143 0.702035i \(-0.247725\pi\)
0.712143 + 0.702035i \(0.247725\pi\)
\(450\) 1.00000 0.0471405
\(451\) −5.03605 −0.237138
\(452\) 13.4432 0.632316
\(453\) 13.2936 0.624589
\(454\) −6.27839 −0.294659
\(455\) −4.64681 −0.217846
\(456\) 1.72161 0.0806219
\(457\) 22.3268 1.04440 0.522201 0.852822i \(-0.325111\pi\)
0.522201 + 0.852822i \(0.325111\pi\)
\(458\) −0.149606 −0.00699064
\(459\) 3.72161 0.173710
\(460\) −1.00000 −0.0466252
\(461\) 8.38924 0.390726 0.195363 0.980731i \(-0.437411\pi\)
0.195363 + 0.980731i \(0.437411\pi\)
\(462\) 0.925197 0.0430441
\(463\) 24.1109 1.12053 0.560263 0.828315i \(-0.310700\pi\)
0.560263 + 0.828315i \(0.310700\pi\)
\(464\) 2.92520 0.135799
\(465\) −5.72161 −0.265333
\(466\) 5.44322 0.252152
\(467\) 3.57201 0.165293 0.0826463 0.996579i \(-0.473663\pi\)
0.0826463 + 0.996579i \(0.473663\pi\)
\(468\) −4.64681 −0.214799
\(469\) 0.925197 0.0427216
\(470\) 5.72161 0.263918
\(471\) −14.7368 −0.679038
\(472\) −7.44322 −0.342602
\(473\) −8.25756 −0.379683
\(474\) 0 0
\(475\) −1.72161 −0.0789930
\(476\) −3.72161 −0.170580
\(477\) 9.44322 0.432375
\(478\) 14.7756 0.675820
\(479\) 23.6233 1.07938 0.539688 0.841865i \(-0.318542\pi\)
0.539688 + 0.841865i \(0.318542\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −43.8809 −2.00079
\(482\) 3.16484 0.144154
\(483\) 1.00000 0.0455016
\(484\) −10.1440 −0.461091
\(485\) −10.2396 −0.464958
\(486\) −1.00000 −0.0453609
\(487\) −3.07480 −0.139333 −0.0696663 0.997570i \(-0.522193\pi\)
−0.0696663 + 0.997570i \(0.522193\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 1.29362 0.0584994
\(490\) 1.00000 0.0451754
\(491\) 18.8864 0.852333 0.426167 0.904645i \(-0.359864\pi\)
0.426167 + 0.904645i \(0.359864\pi\)
\(492\) −5.44322 −0.245399
\(493\) −10.8864 −0.490301
\(494\) 8.00000 0.359937
\(495\) −0.925197 −0.0415845
\(496\) 5.72161 0.256908
\(497\) 6.77559 0.303927
\(498\) −15.0152 −0.672848
\(499\) 33.7729 1.51188 0.755941 0.654640i \(-0.227180\pi\)
0.755941 + 0.654640i \(0.227180\pi\)
\(500\) 1.00000 0.0447214
\(501\) 14.4585 0.645956
\(502\) −4.79641 −0.214074
\(503\) −21.0361 −0.937951 −0.468976 0.883211i \(-0.655377\pi\)
−0.468976 + 0.883211i \(0.655377\pi\)
\(504\) 1.00000 0.0445435
\(505\) −2.79641 −0.124439
\(506\) 0.925197 0.0411300
\(507\) −8.59283 −0.381621
\(508\) −0.368420 −0.0163460
\(509\) −30.4972 −1.35176 −0.675882 0.737010i \(-0.736237\pi\)
−0.675882 + 0.737010i \(0.736237\pi\)
\(510\) 3.72161 0.164796
\(511\) 15.2936 0.676550
\(512\) 1.00000 0.0441942
\(513\) 1.72161 0.0760110
\(514\) −29.1440 −1.28549
\(515\) 1.85039 0.0815381
\(516\) −8.92520 −0.392910
\(517\) −5.29362 −0.232813
\(518\) 9.44322 0.414911
\(519\) −7.16484 −0.314501
\(520\) −4.64681 −0.203776
\(521\) −33.3836 −1.46256 −0.731282 0.682075i \(-0.761078\pi\)
−0.731282 + 0.682075i \(0.761078\pi\)
\(522\) 2.92520 0.128032
\(523\) 29.5333 1.29140 0.645700 0.763592i \(-0.276566\pi\)
0.645700 + 0.763592i \(0.276566\pi\)
\(524\) −4.00000 −0.174741
\(525\) −1.00000 −0.0436436
\(526\) −20.1801 −0.879893
\(527\) −21.2936 −0.927565
\(528\) 0.925197 0.0402640
\(529\) 1.00000 0.0434783
\(530\) 9.44322 0.410187
\(531\) −7.44322 −0.323008
\(532\) −1.72161 −0.0746413
\(533\) −25.2936 −1.09559
\(534\) 6.79641 0.294110
\(535\) 4.00000 0.172935
\(536\) 0.925197 0.0399624
\(537\) −20.7368 −0.894861
\(538\) −27.5333 −1.18704
\(539\) −0.925197 −0.0398510
\(540\) −1.00000 −0.0430331
\(541\) 4.14961 0.178406 0.0892028 0.996013i \(-0.471568\pi\)
0.0892028 + 0.996013i \(0.471568\pi\)
\(542\) 2.02082 0.0868018
\(543\) −6.00000 −0.257485
\(544\) −3.72161 −0.159563
\(545\) −0.149606 −0.00640843
\(546\) 4.64681 0.198865
\(547\) −34.0305 −1.45504 −0.727519 0.686088i \(-0.759327\pi\)
−0.727519 + 0.686088i \(0.759327\pi\)
\(548\) −9.70079 −0.414397
\(549\) −2.00000 −0.0853579
\(550\) −0.925197 −0.0394505
\(551\) −5.03605 −0.214543
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 6.36842 0.270568
\(555\) −9.44322 −0.400843
\(556\) 9.29362 0.394137
\(557\) −3.33527 −0.141320 −0.0706599 0.997500i \(-0.522510\pi\)
−0.0706599 + 0.997500i \(0.522510\pi\)
\(558\) 5.72161 0.242215
\(559\) −41.4737 −1.75415
\(560\) 1.00000 0.0422577
\(561\) −3.44322 −0.145373
\(562\) 16.5180 0.696771
\(563\) 15.0152 0.632816 0.316408 0.948623i \(-0.397523\pi\)
0.316408 + 0.948623i \(0.397523\pi\)
\(564\) −5.72161 −0.240923
\(565\) 13.4432 0.565561
\(566\) 29.5333 1.24138
\(567\) 1.00000 0.0419961
\(568\) 6.77559 0.284297
\(569\) −32.2188 −1.35068 −0.675341 0.737505i \(-0.736004\pi\)
−0.675341 + 0.737505i \(0.736004\pi\)
\(570\) 1.72161 0.0721104
\(571\) −0.556777 −0.0233004 −0.0116502 0.999932i \(-0.503708\pi\)
−0.0116502 + 0.999932i \(0.503708\pi\)
\(572\) 4.29921 0.179759
\(573\) −4.55678 −0.190362
\(574\) 5.44322 0.227196
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −7.11355 −0.296141 −0.148071 0.988977i \(-0.547306\pi\)
−0.148071 + 0.988977i \(0.547306\pi\)
\(578\) −3.14961 −0.131006
\(579\) −6.55678 −0.272490
\(580\) 2.92520 0.121462
\(581\) 15.0152 0.622937
\(582\) 10.2396 0.424447
\(583\) −8.73684 −0.361843
\(584\) 15.2936 0.632854
\(585\) −4.64681 −0.192122
\(586\) −3.85039 −0.159058
\(587\) −11.1440 −0.459963 −0.229981 0.973195i \(-0.573866\pi\)
−0.229981 + 0.973195i \(0.573866\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −9.85039 −0.405878
\(590\) −7.44322 −0.306433
\(591\) 1.74244 0.0716742
\(592\) 9.44322 0.388114
\(593\) −22.2576 −0.914009 −0.457004 0.889464i \(-0.651078\pi\)
−0.457004 + 0.889464i \(0.651078\pi\)
\(594\) 0.925197 0.0379613
\(595\) −3.72161 −0.152571
\(596\) −17.1440 −0.702246
\(597\) 3.44322 0.140922
\(598\) 4.64681 0.190022
\(599\) −11.3324 −0.463028 −0.231514 0.972832i \(-0.574368\pi\)
−0.231514 + 0.972832i \(0.574368\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 6.18006 0.252090 0.126045 0.992025i \(-0.459772\pi\)
0.126045 + 0.992025i \(0.459772\pi\)
\(602\) 8.92520 0.363764
\(603\) 0.925197 0.0376769
\(604\) −13.2936 −0.540910
\(605\) −10.1440 −0.412413
\(606\) 2.79641 0.113597
\(607\) 37.2340 1.51128 0.755642 0.654985i \(-0.227325\pi\)
0.755642 + 0.654985i \(0.227325\pi\)
\(608\) −1.72161 −0.0698206
\(609\) −2.92520 −0.118535
\(610\) −2.00000 −0.0809776
\(611\) −26.5872 −1.07560
\(612\) −3.72161 −0.150437
\(613\) −22.4376 −0.906247 −0.453124 0.891448i \(-0.649690\pi\)
−0.453124 + 0.891448i \(0.649690\pi\)
\(614\) 25.0361 1.01037
\(615\) −5.44322 −0.219492
\(616\) −0.925197 −0.0372772
\(617\) 7.03605 0.283261 0.141630 0.989920i \(-0.454766\pi\)
0.141630 + 0.989920i \(0.454766\pi\)
\(618\) −1.85039 −0.0744338
\(619\) 19.1953 0.771524 0.385762 0.922598i \(-0.373939\pi\)
0.385762 + 0.922598i \(0.373939\pi\)
\(620\) 5.72161 0.229785
\(621\) 1.00000 0.0401286
\(622\) 14.0900 0.564959
\(623\) −6.79641 −0.272293
\(624\) 4.64681 0.186021
\(625\) 1.00000 0.0400000
\(626\) 3.05398 0.122062
\(627\) −1.59283 −0.0636115
\(628\) 14.7368 0.588064
\(629\) −35.1440 −1.40128
\(630\) 1.00000 0.0398410
\(631\) 4.29921 0.171149 0.0855745 0.996332i \(-0.472727\pi\)
0.0855745 + 0.996332i \(0.472727\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 3.29362 0.130806
\(635\) −0.368420 −0.0146203
\(636\) −9.44322 −0.374448
\(637\) −4.64681 −0.184113
\(638\) −2.70638 −0.107147
\(639\) 6.77559 0.268038
\(640\) 1.00000 0.0395285
\(641\) −14.3684 −0.567518 −0.283759 0.958896i \(-0.591582\pi\)
−0.283759 + 0.958896i \(0.591582\pi\)
\(642\) −4.00000 −0.157867
\(643\) 29.7908 1.17484 0.587418 0.809284i \(-0.300145\pi\)
0.587418 + 0.809284i \(0.300145\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −8.92520 −0.351429
\(646\) 6.40717 0.252087
\(647\) 16.0513 0.631041 0.315521 0.948919i \(-0.397821\pi\)
0.315521 + 0.948919i \(0.397821\pi\)
\(648\) 1.00000 0.0392837
\(649\) 6.88645 0.270317
\(650\) −4.64681 −0.182263
\(651\) −5.72161 −0.224248
\(652\) −1.29362 −0.0506620
\(653\) 7.85039 0.307210 0.153605 0.988132i \(-0.450912\pi\)
0.153605 + 0.988132i \(0.450912\pi\)
\(654\) 0.149606 0.00585007
\(655\) −4.00000 −0.156293
\(656\) 5.44322 0.212522
\(657\) 15.2936 0.596661
\(658\) 5.72161 0.223052
\(659\) −20.3684 −0.793441 −0.396721 0.917939i \(-0.629852\pi\)
−0.396721 + 0.917939i \(0.629852\pi\)
\(660\) 0.925197 0.0360132
\(661\) −19.8504 −0.772091 −0.386045 0.922480i \(-0.626159\pi\)
−0.386045 + 0.922480i \(0.626159\pi\)
\(662\) 10.4072 0.404486
\(663\) −17.2936 −0.671628
\(664\) 15.0152 0.582704
\(665\) −1.72161 −0.0667612
\(666\) 9.44322 0.365917
\(667\) −2.92520 −0.113264
\(668\) −14.4585 −0.559414
\(669\) 4.49720 0.173872
\(670\) 0.925197 0.0357435
\(671\) 1.85039 0.0714337
\(672\) −1.00000 −0.0385758
\(673\) −3.29362 −0.126960 −0.0634798 0.997983i \(-0.520220\pi\)
−0.0634798 + 0.997983i \(0.520220\pi\)
\(674\) 5.33237 0.205395
\(675\) −1.00000 −0.0384900
\(676\) 8.59283 0.330493
\(677\) 12.8864 0.495266 0.247633 0.968854i \(-0.420347\pi\)
0.247633 + 0.968854i \(0.420347\pi\)
\(678\) −13.4432 −0.516284
\(679\) −10.2396 −0.392961
\(680\) −3.72161 −0.142717
\(681\) 6.27839 0.240588
\(682\) −5.29362 −0.202703
\(683\) 16.7785 0.642011 0.321006 0.947077i \(-0.395979\pi\)
0.321006 + 0.947077i \(0.395979\pi\)
\(684\) −1.72161 −0.0658275
\(685\) −9.70079 −0.370648
\(686\) 1.00000 0.0381802
\(687\) 0.149606 0.00570784
\(688\) 8.92520 0.340270
\(689\) −43.8809 −1.67173
\(690\) 1.00000 0.0380693
\(691\) −1.89204 −0.0719767 −0.0359883 0.999352i \(-0.511458\pi\)
−0.0359883 + 0.999352i \(0.511458\pi\)
\(692\) 7.16484 0.272366
\(693\) −0.925197 −0.0351453
\(694\) −5.59283 −0.212301
\(695\) 9.29362 0.352527
\(696\) −2.92520 −0.110879
\(697\) −20.2576 −0.767310
\(698\) −12.0208 −0.454995
\(699\) −5.44322 −0.205882
\(700\) 1.00000 0.0377964
\(701\) −22.1801 −0.837729 −0.418865 0.908049i \(-0.637572\pi\)
−0.418865 + 0.908049i \(0.637572\pi\)
\(702\) 4.64681 0.175382
\(703\) −16.2576 −0.613166
\(704\) −0.925197 −0.0348697
\(705\) −5.72161 −0.215488
\(706\) −11.0361 −0.415347
\(707\) −2.79641 −0.105170
\(708\) 7.44322 0.279733
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 6.77559 0.254283
\(711\) 0 0
\(712\) −6.79641 −0.254706
\(713\) −5.72161 −0.214276
\(714\) 3.72161 0.139278
\(715\) 4.29921 0.160781
\(716\) 20.7368 0.774972
\(717\) −14.7756 −0.551805
\(718\) 7.14401 0.266612
\(719\) 34.2701 1.27806 0.639029 0.769182i \(-0.279336\pi\)
0.639029 + 0.769182i \(0.279336\pi\)
\(720\) 1.00000 0.0372678
\(721\) 1.85039 0.0689123
\(722\) −16.0361 −0.596800
\(723\) −3.16484 −0.117701
\(724\) 6.00000 0.222988
\(725\) 2.92520 0.108639
\(726\) 10.1440 0.376480
\(727\) −4.18006 −0.155030 −0.0775150 0.996991i \(-0.524699\pi\)
−0.0775150 + 0.996991i \(0.524699\pi\)
\(728\) −4.64681 −0.172222
\(729\) 1.00000 0.0370370
\(730\) 15.2936 0.566042
\(731\) −33.2161 −1.22854
\(732\) 2.00000 0.0739221
\(733\) 25.4432 0.939767 0.469883 0.882728i \(-0.344296\pi\)
0.469883 + 0.882728i \(0.344296\pi\)
\(734\) −20.4376 −0.754367
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −0.855989 −0.0315308
\(738\) 5.44322 0.200368
\(739\) −29.5928 −1.08859 −0.544295 0.838894i \(-0.683203\pi\)
−0.544295 + 0.838894i \(0.683203\pi\)
\(740\) 9.44322 0.347140
\(741\) −8.00000 −0.293887
\(742\) 9.44322 0.346672
\(743\) −48.3601 −1.77416 −0.887081 0.461615i \(-0.847270\pi\)
−0.887081 + 0.461615i \(0.847270\pi\)
\(744\) −5.72161 −0.209764
\(745\) −17.1440 −0.628108
\(746\) −1.14401 −0.0418852
\(747\) 15.0152 0.549378
\(748\) 3.44322 0.125897
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) 29.7729 1.08643 0.543214 0.839594i \(-0.317207\pi\)
0.543214 + 0.839594i \(0.317207\pi\)
\(752\) 5.72161 0.208646
\(753\) 4.79641 0.174791
\(754\) −13.5928 −0.495022
\(755\) −13.2936 −0.483804
\(756\) −1.00000 −0.0363696
\(757\) 34.4376 1.25166 0.625828 0.779961i \(-0.284761\pi\)
0.625828 + 0.779961i \(0.284761\pi\)
\(758\) 0.299213 0.0108679
\(759\) −0.925197 −0.0335825
\(760\) −1.72161 −0.0624494
\(761\) −24.8448 −0.900623 −0.450312 0.892871i \(-0.648687\pi\)
−0.450312 + 0.892871i \(0.648687\pi\)
\(762\) 0.368420 0.0133465
\(763\) −0.149606 −0.00541611
\(764\) 4.55678 0.164858
\(765\) −3.72161 −0.134555
\(766\) 0.736841 0.0266231
\(767\) 34.5872 1.24887
\(768\) −1.00000 −0.0360844
\(769\) −1.27279 −0.0458981 −0.0229490 0.999737i \(-0.507306\pi\)
−0.0229490 + 0.999737i \(0.507306\pi\)
\(770\) −0.925197 −0.0333418
\(771\) 29.1440 1.04960
\(772\) 6.55678 0.235984
\(773\) 8.10796 0.291623 0.145811 0.989312i \(-0.453421\pi\)
0.145811 + 0.989312i \(0.453421\pi\)
\(774\) 8.92520 0.320810
\(775\) 5.72161 0.205526
\(776\) −10.2396 −0.367582
\(777\) −9.44322 −0.338774
\(778\) −31.0361 −1.11270
\(779\) −9.37112 −0.335755
\(780\) 4.64681 0.166382
\(781\) −6.26875 −0.224314
\(782\) 3.72161 0.133085
\(783\) −2.92520 −0.104538
\(784\) 1.00000 0.0357143
\(785\) 14.7368 0.525980
\(786\) 4.00000 0.142675
\(787\) 19.2036 0.684534 0.342267 0.939603i \(-0.388805\pi\)
0.342267 + 0.939603i \(0.388805\pi\)
\(788\) −1.74244 −0.0620717
\(789\) 20.1801 0.718429
\(790\) 0 0
\(791\) 13.4432 0.477986
\(792\) −0.925197 −0.0328754
\(793\) 9.29362 0.330026
\(794\) −4.64681 −0.164909
\(795\) −9.44322 −0.334917
\(796\) −3.44322 −0.122042
\(797\) 20.0305 0.709515 0.354758 0.934958i \(-0.384563\pi\)
0.354758 + 0.934958i \(0.384563\pi\)
\(798\) 1.72161 0.0609444
\(799\) −21.2936 −0.753314
\(800\) 1.00000 0.0353553
\(801\) −6.79641 −0.240140
\(802\) −8.21881 −0.290216
\(803\) −14.1496 −0.499329
\(804\) −0.925197 −0.0326292
\(805\) −1.00000 −0.0352454
\(806\) −26.5872 −0.936495
\(807\) 27.5333 0.969216
\(808\) −2.79641 −0.0983775
\(809\) −29.7424 −1.04569 −0.522844 0.852428i \(-0.675129\pi\)
−0.522844 + 0.852428i \(0.675129\pi\)
\(810\) 1.00000 0.0351364
\(811\) 33.5512 1.17814 0.589071 0.808082i \(-0.299494\pi\)
0.589071 + 0.808082i \(0.299494\pi\)
\(812\) 2.92520 0.102654
\(813\) −2.02082 −0.0708734
\(814\) −8.73684 −0.306226
\(815\) −1.29362 −0.0453134
\(816\) 3.72161 0.130282
\(817\) −15.3657 −0.537579
\(818\) −12.1496 −0.424801
\(819\) −4.64681 −0.162373
\(820\) 5.44322 0.190086
\(821\) 46.5068 1.62310 0.811550 0.584283i \(-0.198624\pi\)
0.811550 + 0.584283i \(0.198624\pi\)
\(822\) 9.70079 0.338354
\(823\) 5.40447 0.188388 0.0941940 0.995554i \(-0.469973\pi\)
0.0941940 + 0.995554i \(0.469973\pi\)
\(824\) 1.85039 0.0644615
\(825\) 0.925197 0.0322112
\(826\) −7.44322 −0.258983
\(827\) 10.8864 0.378559 0.189279 0.981923i \(-0.439385\pi\)
0.189279 + 0.981923i \(0.439385\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −30.3505 −1.05412 −0.527058 0.849829i \(-0.676705\pi\)
−0.527058 + 0.849829i \(0.676705\pi\)
\(830\) 15.0152 0.521186
\(831\) −6.36842 −0.220918
\(832\) −4.64681 −0.161099
\(833\) −3.72161 −0.128946
\(834\) −9.29362 −0.321812
\(835\) −14.4585 −0.500355
\(836\) 1.59283 0.0550892
\(837\) −5.72161 −0.197768
\(838\) −26.3476 −0.910163
\(839\) −12.5568 −0.433508 −0.216754 0.976226i \(-0.569547\pi\)
−0.216754 + 0.976226i \(0.569547\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −20.4432 −0.704939
\(842\) 8.70638 0.300042
\(843\) −16.5180 −0.568911
\(844\) 4.00000 0.137686
\(845\) 8.59283 0.295602
\(846\) 5.72161 0.196713
\(847\) −10.1440 −0.348552
\(848\) 9.44322 0.324282
\(849\) −29.5333 −1.01358
\(850\) −3.72161 −0.127650
\(851\) −9.44322 −0.323710
\(852\) −6.77559 −0.232128
\(853\) 6.53885 0.223886 0.111943 0.993715i \(-0.464293\pi\)
0.111943 + 0.993715i \(0.464293\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −1.72161 −0.0588779
\(856\) 4.00000 0.136717
\(857\) 21.3241 0.728417 0.364208 0.931318i \(-0.381340\pi\)
0.364208 + 0.931318i \(0.381340\pi\)
\(858\) −4.29921 −0.146773
\(859\) −11.8809 −0.405369 −0.202685 0.979244i \(-0.564967\pi\)
−0.202685 + 0.979244i \(0.564967\pi\)
\(860\) 8.92520 0.304347
\(861\) −5.44322 −0.185505
\(862\) 22.0305 0.750360
\(863\) 35.0665 1.19368 0.596839 0.802361i \(-0.296423\pi\)
0.596839 + 0.802361i \(0.296423\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 7.16484 0.243612
\(866\) −3.35319 −0.113946
\(867\) 3.14961 0.106966
\(868\) 5.72161 0.194204
\(869\) 0 0
\(870\) −2.92520 −0.0991735
\(871\) −4.29921 −0.145673
\(872\) −0.149606 −0.00506631
\(873\) −10.2396 −0.346559
\(874\) 1.72161 0.0582344
\(875\) 1.00000 0.0338062
\(876\) −15.2936 −0.516723
\(877\) −4.47638 −0.151157 −0.0755783 0.997140i \(-0.524080\pi\)
−0.0755783 + 0.997140i \(0.524080\pi\)
\(878\) −23.5720 −0.795517
\(879\) 3.85039 0.129871
\(880\) −0.925197 −0.0311884
\(881\) −0.167532 −0.00564430 −0.00282215 0.999996i \(-0.500898\pi\)
−0.00282215 + 0.999996i \(0.500898\pi\)
\(882\) 1.00000 0.0336718
\(883\) 48.1801 1.62139 0.810694 0.585470i \(-0.199090\pi\)
0.810694 + 0.585470i \(0.199090\pi\)
\(884\) 17.2936 0.581647
\(885\) 7.44322 0.250201
\(886\) −12.4793 −0.419249
\(887\) −28.0096 −0.940471 −0.470236 0.882541i \(-0.655831\pi\)
−0.470236 + 0.882541i \(0.655831\pi\)
\(888\) −9.44322 −0.316894
\(889\) −0.368420 −0.0123564
\(890\) −6.79641 −0.227816
\(891\) −0.925197 −0.0309953
\(892\) −4.49720 −0.150577
\(893\) −9.85039 −0.329631
\(894\) 17.1440 0.573381
\(895\) 20.7368 0.693156
\(896\) 1.00000 0.0334077
\(897\) −4.64681 −0.155152
\(898\) 30.1801 1.00712
\(899\) 16.7368 0.558205
\(900\) 1.00000 0.0333333
\(901\) −35.1440 −1.17082
\(902\) −5.03605 −0.167682
\(903\) −8.92520 −0.297012
\(904\) 13.4432 0.447115
\(905\) 6.00000 0.199447
\(906\) 13.2936 0.441651
\(907\) 36.3684 1.20759 0.603797 0.797138i \(-0.293654\pi\)
0.603797 + 0.797138i \(0.293654\pi\)
\(908\) −6.27839 −0.208356
\(909\) −2.79641 −0.0927512
\(910\) −4.64681 −0.154040
\(911\) 28.1801 0.933647 0.466824 0.884350i \(-0.345398\pi\)
0.466824 + 0.884350i \(0.345398\pi\)
\(912\) 1.72161 0.0570083
\(913\) −13.8920 −0.459759
\(914\) 22.3268 0.738504
\(915\) 2.00000 0.0661180
\(916\) −0.149606 −0.00494313
\(917\) −4.00000 −0.132092
\(918\) 3.72161 0.122831
\(919\) −29.2936 −0.966307 −0.483154 0.875536i \(-0.660509\pi\)
−0.483154 + 0.875536i \(0.660509\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −25.0361 −0.824966
\(922\) 8.38924 0.276285
\(923\) −31.4849 −1.03634
\(924\) 0.925197 0.0304367
\(925\) 9.44322 0.310491
\(926\) 24.1109 0.792332
\(927\) 1.85039 0.0607749
\(928\) 2.92520 0.0960243
\(929\) −30.7368 −1.00844 −0.504222 0.863574i \(-0.668220\pi\)
−0.504222 + 0.863574i \(0.668220\pi\)
\(930\) −5.72161 −0.187619
\(931\) −1.72161 −0.0564235
\(932\) 5.44322 0.178299
\(933\) −14.0900 −0.461287
\(934\) 3.57201 0.116880
\(935\) 3.44322 0.112605
\(936\) −4.64681 −0.151886
\(937\) 6.01793 0.196597 0.0982985 0.995157i \(-0.468660\pi\)
0.0982985 + 0.995157i \(0.468660\pi\)
\(938\) 0.925197 0.0302087
\(939\) −3.05398 −0.0996628
\(940\) 5.72161 0.186618
\(941\) −30.2992 −0.987726 −0.493863 0.869540i \(-0.664416\pi\)
−0.493863 + 0.869540i \(0.664416\pi\)
\(942\) −14.7368 −0.480152
\(943\) −5.44322 −0.177256
\(944\) −7.44322 −0.242256
\(945\) −1.00000 −0.0325300
\(946\) −8.25756 −0.268477
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) −71.0665 −2.30692
\(950\) −1.72161 −0.0558565
\(951\) −3.29362 −0.106803
\(952\) −3.72161 −0.120618
\(953\) 8.62888 0.279517 0.139758 0.990186i \(-0.455367\pi\)
0.139758 + 0.990186i \(0.455367\pi\)
\(954\) 9.44322 0.305736
\(955\) 4.55678 0.147454
\(956\) 14.7756 0.477877
\(957\) 2.70638 0.0874849
\(958\) 23.6233 0.763234
\(959\) −9.70079 −0.313255
\(960\) −1.00000 −0.0322749
\(961\) 1.73684 0.0560271
\(962\) −43.8809 −1.41478
\(963\) 4.00000 0.128898
\(964\) 3.16484 0.101932
\(965\) 6.55678 0.211070
\(966\) 1.00000 0.0321745
\(967\) −17.3628 −0.558351 −0.279175 0.960240i \(-0.590061\pi\)
−0.279175 + 0.960240i \(0.590061\pi\)
\(968\) −10.1440 −0.326041
\(969\) −6.40717 −0.205828
\(970\) −10.2396 −0.328775
\(971\) 14.3892 0.461773 0.230886 0.972981i \(-0.425837\pi\)
0.230886 + 0.972981i \(0.425837\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 9.29362 0.297940
\(974\) −3.07480 −0.0985230
\(975\) 4.64681 0.148817
\(976\) −2.00000 −0.0640184
\(977\) −30.2217 −0.966878 −0.483439 0.875378i \(-0.660613\pi\)
−0.483439 + 0.875378i \(0.660613\pi\)
\(978\) 1.29362 0.0413653
\(979\) 6.28802 0.200966
\(980\) 1.00000 0.0319438
\(981\) −0.149606 −0.00477656
\(982\) 18.8864 0.602691
\(983\) −55.0249 −1.75502 −0.877510 0.479558i \(-0.840797\pi\)
−0.877510 + 0.479558i \(0.840797\pi\)
\(984\) −5.44322 −0.173524
\(985\) −1.74244 −0.0555186
\(986\) −10.8864 −0.346695
\(987\) −5.72161 −0.182121
\(988\) 8.00000 0.254514
\(989\) −8.92520 −0.283805
\(990\) −0.925197 −0.0294047
\(991\) 24.4793 0.777610 0.388805 0.921320i \(-0.372888\pi\)
0.388805 + 0.921320i \(0.372888\pi\)
\(992\) 5.72161 0.181661
\(993\) −10.4072 −0.330262
\(994\) 6.77559 0.214909
\(995\) −3.44322 −0.109158
\(996\) −15.0152 −0.475776
\(997\) 51.1149 1.61882 0.809412 0.587241i \(-0.199786\pi\)
0.809412 + 0.587241i \(0.199786\pi\)
\(998\) 33.7729 1.06906
\(999\) −9.44322 −0.298770
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 4830.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.cb.1.2 3 1.1 even 1 trivial