Properties

Label 29.2.b.a.28.1
Level $29$
Weight $2$
Character 29.28
Analytic conductor $0.232$
Analytic rank $0$
Dimension $2$
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] = [29,2,Mod(28,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.28");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 29.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: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 28.1
Root \(-2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 29.28
Dual form 29.2.b.a.28.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{2} +2.23607i q^{3} -3.00000 q^{4} -3.00000 q^{5} +5.00000 q^{6} +2.00000 q^{7} +2.23607i q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-2.23607i q^{2} +2.23607i q^{3} -3.00000 q^{4} -3.00000 q^{5} +5.00000 q^{6} +2.00000 q^{7} +2.23607i q^{8} -2.00000 q^{9} +6.70820i q^{10} -2.23607i q^{11} -6.70820i q^{12} -1.00000 q^{13} -4.47214i q^{14} -6.70820i q^{15} -1.00000 q^{16} +4.47214i q^{17} +4.47214i q^{18} +9.00000 q^{20} +4.47214i q^{21} -5.00000 q^{22} +6.00000 q^{23} -5.00000 q^{24} +4.00000 q^{25} +2.23607i q^{26} +2.23607i q^{27} -6.00000 q^{28} +(-3.00000 - 4.47214i) q^{29} -15.0000 q^{30} -6.70820i q^{31} +6.70820i q^{32} +5.00000 q^{33} +10.0000 q^{34} -6.00000 q^{35} +6.00000 q^{36} -2.23607i q^{39} -6.70820i q^{40} +4.47214i q^{41} +10.0000 q^{42} +6.70820i q^{43} +6.70820i q^{44} +6.00000 q^{45} -13.4164i q^{46} -2.23607i q^{47} -2.23607i q^{48} -3.00000 q^{49} -8.94427i q^{50} -10.0000 q^{51} +3.00000 q^{52} -9.00000 q^{53} +5.00000 q^{54} +6.70820i q^{55} +4.47214i q^{56} +(-10.0000 + 6.70820i) q^{58} +6.00000 q^{59} +20.1246i q^{60} -13.4164i q^{61} -15.0000 q^{62} -4.00000 q^{63} +13.0000 q^{64} +3.00000 q^{65} -11.1803i q^{66} +8.00000 q^{67} -13.4164i q^{68} +13.4164i q^{69} +13.4164i q^{70} -4.47214i q^{72} +8.94427i q^{75} -4.47214i q^{77} -5.00000 q^{78} +6.70820i q^{79} +3.00000 q^{80} -11.0000 q^{81} +10.0000 q^{82} -6.00000 q^{83} -13.4164i q^{84} -13.4164i q^{85} +15.0000 q^{86} +(10.0000 - 6.70820i) q^{87} +5.00000 q^{88} +4.47214i q^{89} -13.4164i q^{90} -2.00000 q^{91} -18.0000 q^{92} +15.0000 q^{93} -5.00000 q^{94} -15.0000 q^{96} -13.4164i q^{97} +6.70820i q^{98} +4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{4} - 6 q^{5} + 10 q^{6} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{4} - 6 q^{5} + 10 q^{6} + 4 q^{7} - 4 q^{9} - 2 q^{13} - 2 q^{16} + 18 q^{20} - 10 q^{22} + 12 q^{23} - 10 q^{24} + 8 q^{25} - 12 q^{28} - 6 q^{29} - 30 q^{30} + 10 q^{33} + 20 q^{34} - 12 q^{35} + 12 q^{36} + 20 q^{42} + 12 q^{45} - 6 q^{49} - 20 q^{51} + 6 q^{52} - 18 q^{53} + 10 q^{54} - 20 q^{58} + 12 q^{59} - 30 q^{62} - 8 q^{63} + 26 q^{64} + 6 q^{65} + 16 q^{67} - 10 q^{78} + 6 q^{80} - 22 q^{81} + 20 q^{82} - 12 q^{83} + 30 q^{86} + 20 q^{87} + 10 q^{88} - 4 q^{91} - 36 q^{92} + 30 q^{93} - 10 q^{94} - 30 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i −0.612372 0.790569i \(-0.709785\pi\)
0.612372 0.790569i \(-0.290215\pi\)
\(3\) 2.23607i 1.29099i 0.763763 + 0.645497i \(0.223350\pi\)
−0.763763 + 0.645497i \(0.776650\pi\)
\(4\) −3.00000 −1.50000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 5.00000 2.04124
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 2.23607i 0.790569i
\(9\) −2.00000 −0.666667
\(10\) 6.70820i 2.12132i
\(11\) 2.23607i 0.674200i −0.941469 0.337100i \(-0.890554\pi\)
0.941469 0.337100i \(-0.109446\pi\)
\(12\) 6.70820i 1.93649i
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 4.47214i 1.19523i
\(15\) 6.70820i 1.73205i
\(16\) −1.00000 −0.250000
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 4.47214i 1.05409i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 9.00000 2.01246
\(21\) 4.47214i 0.975900i
\(22\) −5.00000 −1.06600
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −5.00000 −1.02062
\(25\) 4.00000 0.800000
\(26\) 2.23607i 0.438529i
\(27\) 2.23607i 0.430331i
\(28\) −6.00000 −1.13389
\(29\) −3.00000 4.47214i −0.557086 0.830455i
\(30\) −15.0000 −2.73861
\(31\) 6.70820i 1.20483i −0.798183 0.602414i \(-0.794205\pi\)
0.798183 0.602414i \(-0.205795\pi\)
\(32\) 6.70820i 1.18585i
\(33\) 5.00000 0.870388
\(34\) 10.0000 1.71499
\(35\) −6.00000 −1.01419
\(36\) 6.00000 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 2.23607i 0.358057i
\(40\) 6.70820i 1.06066i
\(41\) 4.47214i 0.698430i 0.937043 + 0.349215i \(0.113552\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 10.0000 1.54303
\(43\) 6.70820i 1.02299i 0.859286 + 0.511496i \(0.170908\pi\)
−0.859286 + 0.511496i \(0.829092\pi\)
\(44\) 6.70820i 1.01130i
\(45\) 6.00000 0.894427
\(46\) 13.4164i 1.97814i
\(47\) 2.23607i 0.326164i −0.986613 0.163082i \(-0.947856\pi\)
0.986613 0.163082i \(-0.0521435\pi\)
\(48\) 2.23607i 0.322749i
\(49\) −3.00000 −0.428571
\(50\) 8.94427i 1.26491i
\(51\) −10.0000 −1.40028
\(52\) 3.00000 0.416025
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 5.00000 0.680414
\(55\) 6.70820i 0.904534i
\(56\) 4.47214i 0.597614i
\(57\) 0 0
\(58\) −10.0000 + 6.70820i −1.31306 + 0.880830i
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 20.1246i 2.59808i
\(61\) 13.4164i 1.71780i −0.512148 0.858898i \(-0.671150\pi\)
0.512148 0.858898i \(-0.328850\pi\)
\(62\) −15.0000 −1.90500
\(63\) −4.00000 −0.503953
\(64\) 13.0000 1.62500
\(65\) 3.00000 0.372104
\(66\) 11.1803i 1.37620i
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 13.4164i 1.62698i
\(69\) 13.4164i 1.61515i
\(70\) 13.4164i 1.60357i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 4.47214i 0.527046i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 8.94427i 1.03280i
\(76\) 0 0
\(77\) 4.47214i 0.509647i
\(78\) −5.00000 −0.566139
\(79\) 6.70820i 0.754732i 0.926064 + 0.377366i \(0.123170\pi\)
−0.926064 + 0.377366i \(0.876830\pi\)
\(80\) 3.00000 0.335410
\(81\) −11.0000 −1.22222
\(82\) 10.0000 1.10432
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 13.4164i 1.46385i
\(85\) 13.4164i 1.45521i
\(86\) 15.0000 1.61749
\(87\) 10.0000 6.70820i 1.07211 0.719195i
\(88\) 5.00000 0.533002
\(89\) 4.47214i 0.474045i 0.971504 + 0.237023i \(0.0761716\pi\)
−0.971504 + 0.237023i \(0.923828\pi\)
\(90\) 13.4164i 1.41421i
\(91\) −2.00000 −0.209657
\(92\) −18.0000 −1.87663
\(93\) 15.0000 1.55543
\(94\) −5.00000 −0.515711
\(95\) 0 0
\(96\) −15.0000 −1.53093
\(97\) 13.4164i 1.36223i −0.732177 0.681115i \(-0.761495\pi\)
0.732177 0.681115i \(-0.238505\pi\)
\(98\) 6.70820i 0.677631i
\(99\) 4.47214i 0.449467i
\(100\) −12.0000 −1.20000
\(101\) 17.8885i 1.77998i 0.455983 + 0.889988i \(0.349288\pi\)
−0.455983 + 0.889988i \(0.650712\pi\)
\(102\) 22.3607i 2.21404i
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.23607i 0.219265i
\(105\) 13.4164i 1.30931i
\(106\) 20.1246i 1.95468i
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 6.70820i 0.645497i
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 15.0000 1.43019
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) 8.94427i 0.841406i −0.907198 0.420703i \(-0.861783\pi\)
0.907198 0.420703i \(-0.138217\pi\)
\(114\) 0 0
\(115\) −18.0000 −1.67851
\(116\) 9.00000 + 13.4164i 0.835629 + 1.24568i
\(117\) 2.00000 0.184900
\(118\) 13.4164i 1.23508i
\(119\) 8.94427i 0.819920i
\(120\) 15.0000 1.36931
\(121\) 6.00000 0.545455
\(122\) −30.0000 −2.71607
\(123\) −10.0000 −0.901670
\(124\) 20.1246i 1.80724i
\(125\) 3.00000 0.268328
\(126\) 8.94427i 0.796819i
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 15.6525i 1.38350i
\(129\) −15.0000 −1.32068
\(130\) 6.70820i 0.588348i
\(131\) 8.94427i 0.781465i −0.920504 0.390732i \(-0.872222\pi\)
0.920504 0.390732i \(-0.127778\pi\)
\(132\) −15.0000 −1.30558
\(133\) 0 0
\(134\) 17.8885i 1.54533i
\(135\) 6.70820i 0.577350i
\(136\) −10.0000 −0.857493
\(137\) 8.94427i 0.764161i −0.924129 0.382080i \(-0.875208\pi\)
0.924129 0.382080i \(-0.124792\pi\)
\(138\) 30.0000 2.55377
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 18.0000 1.52128
\(141\) 5.00000 0.421076
\(142\) 0 0
\(143\) 2.23607i 0.186989i
\(144\) 2.00000 0.166667
\(145\) 9.00000 + 13.4164i 0.747409 + 1.11417i
\(146\) 0 0
\(147\) 6.70820i 0.553283i
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 20.0000 1.63299
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 8.94427i 0.723102i
\(154\) −10.0000 −0.805823
\(155\) 20.1246i 1.61645i
\(156\) 6.70820i 0.537086i
\(157\) 13.4164i 1.07075i 0.844616 + 0.535373i \(0.179829\pi\)
−0.844616 + 0.535373i \(0.820171\pi\)
\(158\) 15.0000 1.19334
\(159\) 20.1246i 1.59599i
\(160\) 20.1246i 1.59099i
\(161\) 12.0000 0.945732
\(162\) 24.5967i 1.93250i
\(163\) 20.1246i 1.57628i 0.615495 + 0.788141i \(0.288956\pi\)
−0.615495 + 0.788141i \(0.711044\pi\)
\(164\) 13.4164i 1.04765i
\(165\) −15.0000 −1.16775
\(166\) 13.4164i 1.04132i
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) −10.0000 −0.771517
\(169\) −12.0000 −0.923077
\(170\) −30.0000 −2.30089
\(171\) 0 0
\(172\) 20.1246i 1.53449i
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −15.0000 22.3607i −1.13715 1.69516i
\(175\) 8.00000 0.604743
\(176\) 2.23607i 0.168550i
\(177\) 13.4164i 1.00844i
\(178\) 10.0000 0.749532
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −18.0000 −1.34164
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 4.47214i 0.331497i
\(183\) 30.0000 2.21766
\(184\) 13.4164i 0.989071i
\(185\) 0 0
\(186\) 33.5410i 2.45935i
\(187\) 10.0000 0.731272
\(188\) 6.70820i 0.489246i
\(189\) 4.47214i 0.325300i
\(190\) 0 0
\(191\) 8.94427i 0.647185i −0.946197 0.323592i \(-0.895109\pi\)
0.946197 0.323592i \(-0.104891\pi\)
\(192\) 29.0689i 2.09787i
\(193\) 13.4164i 0.965734i 0.875694 + 0.482867i \(0.160405\pi\)
−0.875694 + 0.482867i \(0.839595\pi\)
\(194\) −30.0000 −2.15387
\(195\) 6.70820i 0.480384i
\(196\) 9.00000 0.642857
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 10.0000 0.710669
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 8.94427i 0.632456i
\(201\) 17.8885i 1.26176i
\(202\) 40.0000 2.81439
\(203\) −6.00000 8.94427i −0.421117 0.627765i
\(204\) 30.0000 2.10042
\(205\) 13.4164i 0.937043i
\(206\) 8.94427i 0.623177i
\(207\) −12.0000 −0.834058
\(208\) 1.00000 0.0693375
\(209\) 0 0
\(210\) −30.0000 −2.07020
\(211\) 6.70820i 0.461812i −0.972976 0.230906i \(-0.925831\pi\)
0.972976 0.230906i \(-0.0741690\pi\)
\(212\) 27.0000 1.85437
\(213\) 0 0
\(214\) 40.2492i 2.75138i
\(215\) 20.1246i 1.37249i
\(216\) −5.00000 −0.340207
\(217\) 13.4164i 0.910765i
\(218\) 11.1803i 0.757228i
\(219\) 0 0
\(220\) 20.1246i 1.35680i
\(221\) 4.47214i 0.300828i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 13.4164i 0.896421i
\(225\) −8.00000 −0.533333
\(226\) −20.0000 −1.33038
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 13.4164i 0.886581i 0.896378 + 0.443291i \(0.146189\pi\)
−0.896378 + 0.443291i \(0.853811\pi\)
\(230\) 40.2492i 2.65396i
\(231\) 10.0000 0.657952
\(232\) 10.0000 6.70820i 0.656532 0.440415i
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 4.47214i 0.292353i
\(235\) 6.70820i 0.437595i
\(236\) −18.0000 −1.17170
\(237\) −15.0000 −0.974355
\(238\) 20.0000 1.29641
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 6.70820i 0.433013i
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 13.4164i 0.862439i
\(243\) 17.8885i 1.14755i
\(244\) 40.2492i 2.57669i
\(245\) 9.00000 0.574989
\(246\) 22.3607i 1.42566i
\(247\) 0 0
\(248\) 15.0000 0.952501
\(249\) 13.4164i 0.850230i
\(250\) 6.70820i 0.424264i
\(251\) 15.6525i 0.987976i −0.869469 0.493988i \(-0.835539\pi\)
0.869469 0.493988i \(-0.164461\pi\)
\(252\) 12.0000 0.755929
\(253\) 13.4164i 0.843482i
\(254\) 0 0
\(255\) 30.0000 1.87867
\(256\) −9.00000 −0.562500
\(257\) −3.00000 −0.187135 −0.0935674 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) 33.5410i 2.08817i
\(259\) 0 0
\(260\) −9.00000 −0.558156
\(261\) 6.00000 + 8.94427i 0.371391 + 0.553637i
\(262\) −20.0000 −1.23560
\(263\) 11.1803i 0.689409i 0.938711 + 0.344705i \(0.112021\pi\)
−0.938711 + 0.344705i \(0.887979\pi\)
\(264\) 11.1803i 0.688102i
\(265\) 27.0000 1.65860
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) −24.0000 −1.46603
\(269\) 4.47214i 0.272671i 0.990663 + 0.136335i \(0.0435325\pi\)
−0.990663 + 0.136335i \(0.956467\pi\)
\(270\) −15.0000 −0.912871
\(271\) 20.1246i 1.22248i 0.791444 + 0.611242i \(0.209330\pi\)
−0.791444 + 0.611242i \(0.790670\pi\)
\(272\) 4.47214i 0.271163i
\(273\) 4.47214i 0.270666i
\(274\) −20.0000 −1.20824
\(275\) 8.94427i 0.539360i
\(276\) 40.2492i 2.42272i
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 22.3607i 1.34110i
\(279\) 13.4164i 0.803219i
\(280\) 13.4164i 0.801784i
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 11.1803i 0.665780i
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 5.00000 0.295656
\(287\) 8.94427i 0.527964i
\(288\) 13.4164i 0.790569i
\(289\) −3.00000 −0.176471
\(290\) 30.0000 20.1246i 1.76166 1.18176i
\(291\) 30.0000 1.75863
\(292\) 0 0
\(293\) 8.94427i 0.522530i −0.965267 0.261265i \(-0.915860\pi\)
0.965267 0.261265i \(-0.0841396\pi\)
\(294\) −15.0000 −0.874818
\(295\) −18.0000 −1.04800
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 33.5410i 1.94298i
\(299\) −6.00000 −0.346989
\(300\) 26.8328i 1.54919i
\(301\) 13.4164i 0.773309i
\(302\) 22.3607i 1.28671i
\(303\) −40.0000 −2.29794
\(304\) 0 0
\(305\) 40.2492i 2.30466i
\(306\) −20.0000 −1.14332
\(307\) 20.1246i 1.14857i −0.818655 0.574286i \(-0.805280\pi\)
0.818655 0.574286i \(-0.194720\pi\)
\(308\) 13.4164i 0.764471i
\(309\) 8.94427i 0.508822i
\(310\) 45.0000 2.55583
\(311\) 8.94427i 0.507183i −0.967311 0.253592i \(-0.918388\pi\)
0.967311 0.253592i \(-0.0816119\pi\)
\(312\) 5.00000 0.283069
\(313\) 29.0000 1.63918 0.819588 0.572953i \(-0.194202\pi\)
0.819588 + 0.572953i \(0.194202\pi\)
\(314\) 30.0000 1.69300
\(315\) 12.0000 0.676123
\(316\) 20.1246i 1.13210i
\(317\) 22.3607i 1.25590i −0.778253 0.627950i \(-0.783894\pi\)
0.778253 0.627950i \(-0.216106\pi\)
\(318\) −45.0000 −2.52347
\(319\) −10.0000 + 6.70820i −0.559893 + 0.375587i
\(320\) −39.0000 −2.18017
\(321\) 40.2492i 2.24649i
\(322\) 26.8328i 1.49533i
\(323\) 0 0
\(324\) 33.0000 1.83333
\(325\) −4.00000 −0.221880
\(326\) 45.0000 2.49232
\(327\) 11.1803i 0.618274i
\(328\) −10.0000 −0.552158
\(329\) 4.47214i 0.246557i
\(330\) 33.5410i 1.84637i
\(331\) 33.5410i 1.84358i −0.387688 0.921791i \(-0.626726\pi\)
0.387688 0.921791i \(-0.373274\pi\)
\(332\) 18.0000 0.987878
\(333\) 0 0
\(334\) 26.8328i 1.46823i
\(335\) −24.0000 −1.31126
\(336\) 4.47214i 0.243975i
\(337\) 13.4164i 0.730838i 0.930843 + 0.365419i \(0.119074\pi\)
−0.930843 + 0.365419i \(0.880926\pi\)
\(338\) 26.8328i 1.45951i
\(339\) 20.0000 1.08625
\(340\) 40.2492i 2.18282i
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) −15.0000 −0.808746
\(345\) 40.2492i 2.16695i
\(346\) 13.4164i 0.721271i
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −30.0000 + 20.1246i −1.60817 + 1.07879i
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 17.8885i 0.956183i
\(351\) 2.23607i 0.119352i
\(352\) 15.0000 0.799503
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 30.0000 1.59448
\(355\) 0 0
\(356\) 13.4164i 0.711068i
\(357\) −20.0000 −1.05851
\(358\) 0 0
\(359\) 2.23607i 0.118015i −0.998258 0.0590076i \(-0.981206\pi\)
0.998258 0.0590076i \(-0.0187936\pi\)
\(360\) 13.4164i 0.707107i
\(361\) 19.0000 1.00000
\(362\) 11.1803i 0.587626i
\(363\) 13.4164i 0.704179i
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 67.0820i 3.50643i
\(367\) 26.8328i 1.40066i 0.713818 + 0.700331i \(0.246964\pi\)
−0.713818 + 0.700331i \(0.753036\pi\)
\(368\) −6.00000 −0.312772
\(369\) 8.94427i 0.465620i
\(370\) 0 0
\(371\) −18.0000 −0.934513
\(372\) −45.0000 −2.33314
\(373\) −31.0000 −1.60512 −0.802560 0.596572i \(-0.796529\pi\)
−0.802560 + 0.596572i \(0.796529\pi\)
\(374\) 22.3607i 1.15624i
\(375\) 6.70820i 0.346410i
\(376\) 5.00000 0.257855
\(377\) 3.00000 + 4.47214i 0.154508 + 0.230327i
\(378\) 10.0000 0.514344
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.0000 −1.02329
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 35.0000 1.78609
\(385\) 13.4164i 0.683763i
\(386\) 30.0000 1.52696
\(387\) 13.4164i 0.681994i
\(388\) 40.2492i 2.04334i
\(389\) 17.8885i 0.906985i 0.891260 + 0.453493i \(0.149822\pi\)
−0.891260 + 0.453493i \(0.850178\pi\)
\(390\) 15.0000 0.759555
\(391\) 26.8328i 1.35699i
\(392\) 6.70820i 0.338815i
\(393\) 20.0000 1.00887
\(394\) 40.2492i 2.02773i
\(395\) 20.1246i 1.01258i
\(396\) 13.4164i 0.674200i
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 31.3050i 1.56918i
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 15.0000 0.749064 0.374532 0.927214i \(-0.377803\pi\)
0.374532 + 0.927214i \(0.377803\pi\)
\(402\) 40.0000 1.99502
\(403\) 6.70820i 0.334159i
\(404\) 53.6656i 2.66996i
\(405\) 33.0000 1.63978
\(406\) −20.0000 + 13.4164i −0.992583 + 0.665845i
\(407\) 0 0
\(408\) 22.3607i 1.10702i
\(409\) 26.8328i 1.32680i −0.748266 0.663399i \(-0.769113\pi\)
0.748266 0.663399i \(-0.230887\pi\)
\(410\) −30.0000 −1.48159
\(411\) 20.0000 0.986527
\(412\) 12.0000 0.591198
\(413\) 12.0000 0.590481
\(414\) 26.8328i 1.31876i
\(415\) 18.0000 0.883585
\(416\) 6.70820i 0.328897i
\(417\) 22.3607i 1.09501i
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 40.2492i 1.96396i
\(421\) 13.4164i 0.653876i −0.945046 0.326938i \(-0.893983\pi\)
0.945046 0.326938i \(-0.106017\pi\)
\(422\) −15.0000 −0.730189
\(423\) 4.47214i 0.217443i
\(424\) 20.1246i 0.977338i
\(425\) 17.8885i 0.867722i
\(426\) 0 0
\(427\) 26.8328i 1.29853i
\(428\) −54.0000 −2.61019
\(429\) −5.00000 −0.241402
\(430\) −45.0000 −2.17009
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 2.23607i 0.107583i
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −30.0000 −1.44005
\(435\) −30.0000 + 20.1246i −1.43839 + 0.964901i
\(436\) −15.0000 −0.718370
\(437\) 0 0
\(438\) 0 0
\(439\) −34.0000 −1.62273 −0.811366 0.584539i \(-0.801275\pi\)
−0.811366 + 0.584539i \(0.801275\pi\)
\(440\) −15.0000 −0.715097
\(441\) 6.00000 0.285714
\(442\) −10.0000 −0.475651
\(443\) 17.8885i 0.849910i 0.905214 + 0.424955i \(0.139710\pi\)
−0.905214 + 0.424955i \(0.860290\pi\)
\(444\) 0 0
\(445\) 13.4164i 0.635999i
\(446\) 35.7771i 1.69409i
\(447\) 33.5410i 1.58644i
\(448\) 26.0000 1.22838
\(449\) 35.7771i 1.68843i −0.536009 0.844213i \(-0.680069\pi\)
0.536009 0.844213i \(-0.319931\pi\)
\(450\) 17.8885i 0.843274i
\(451\) 10.0000 0.470882
\(452\) 26.8328i 1.26211i
\(453\) 22.3607i 1.05060i
\(454\) 26.8328i 1.25933i
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 30.0000 1.40181
\(459\) −10.0000 −0.466760
\(460\) 54.0000 2.51776
\(461\) 8.94427i 0.416576i −0.978068 0.208288i \(-0.933211\pi\)
0.978068 0.208288i \(-0.0667892\pi\)
\(462\) 22.3607i 1.04031i
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) 3.00000 + 4.47214i 0.139272 + 0.207614i
\(465\) −45.0000 −2.08683
\(466\) 20.1246i 0.932255i
\(467\) 2.23607i 0.103473i −0.998661 0.0517364i \(-0.983524\pi\)
0.998661 0.0517364i \(-0.0164756\pi\)
\(468\) −6.00000 −0.277350
\(469\) 16.0000 0.738811
\(470\) 15.0000 0.691898
\(471\) −30.0000 −1.38233
\(472\) 13.4164i 0.617540i
\(473\) 15.0000 0.689701
\(474\) 33.5410i 1.54059i
\(475\) 0 0
\(476\) 26.8328i 1.22988i
\(477\) 18.0000 0.824163
\(478\) 13.4164i 0.613652i
\(479\) 11.1803i 0.510843i 0.966830 + 0.255421i \(0.0822142\pi\)
−0.966830 + 0.255421i \(0.917786\pi\)
\(480\) 45.0000 2.05396
\(481\) 0 0
\(482\) 55.9017i 2.54625i
\(483\) 26.8328i 1.22094i
\(484\) −18.0000 −0.818182
\(485\) 40.2492i 1.82762i
\(486\) −40.0000 −1.81444
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 30.0000 1.35804
\(489\) −45.0000 −2.03497
\(490\) 20.1246i 0.909137i
\(491\) 24.5967i 1.11004i 0.831838 + 0.555018i \(0.187289\pi\)
−0.831838 + 0.555018i \(0.812711\pi\)
\(492\) 30.0000 1.35250
\(493\) 20.0000 13.4164i 0.900755 0.604245i
\(494\) 0 0
\(495\) 13.4164i 0.603023i
\(496\) 6.70820i 0.301207i
\(497\) 0 0
\(498\) −30.0000 −1.34433
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) −9.00000 −0.402492
\(501\) 26.8328i 1.19880i
\(502\) −35.0000 −1.56213
\(503\) 15.6525i 0.697909i −0.937140 0.348955i \(-0.886537\pi\)
0.937140 0.348955i \(-0.113463\pi\)
\(504\) 8.94427i 0.398410i
\(505\) 53.6656i 2.38809i
\(506\) −30.0000 −1.33366
\(507\) 26.8328i 1.19169i
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 67.0820i 2.97044i
\(511\) 0 0
\(512\) 11.1803i 0.494106i
\(513\) 0 0
\(514\) 6.70820i 0.295886i
\(515\) 12.0000 0.528783
\(516\) 45.0000 1.98101
\(517\) −5.00000 −0.219900
\(518\) 0 0
\(519\) 13.4164i 0.588915i
\(520\) 6.70820i 0.294174i
\(521\) −27.0000 −1.18289 −0.591446 0.806345i \(-0.701443\pi\)
−0.591446 + 0.806345i \(0.701443\pi\)
\(522\) 20.0000 13.4164i 0.875376 0.587220i
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 26.8328i 1.17220i
\(525\) 17.8885i 0.780720i
\(526\) 25.0000 1.09005
\(527\) 30.0000 1.30682
\(528\) −5.00000 −0.217597
\(529\) 13.0000 0.565217
\(530\) 60.3738i 2.62247i
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 4.47214i 0.193710i
\(534\) 22.3607i 0.967641i
\(535\) −54.0000 −2.33462
\(536\) 17.8885i 0.772667i
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 6.70820i 0.288943i
\(540\) 20.1246i 0.866025i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 45.0000 1.93292
\(543\) 11.1803i 0.479794i
\(544\) −30.0000 −1.28624
\(545\) −15.0000 −0.642529
\(546\) −10.0000 −0.427960
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) 26.8328i 1.14624i
\(549\) 26.8328i 1.14520i
\(550\) −20.0000 −0.852803
\(551\) 0 0
\(552\) −30.0000 −1.27688
\(553\) 13.4164i 0.570524i
\(554\) 4.47214i 0.190003i
\(555\) 0 0
\(556\) 30.0000 1.27228
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 30.0000 1.27000
\(559\) 6.70820i 0.283727i
\(560\) 6.00000 0.253546
\(561\) 22.3607i 0.944069i
\(562\) 6.70820i 0.282969i
\(563\) 11.1803i 0.471195i 0.971851 + 0.235598i \(0.0757047\pi\)
−0.971851 + 0.235598i \(0.924295\pi\)
\(564\) −15.0000 −0.631614
\(565\) 26.8328i 1.12887i
\(566\) 31.3050i 1.31585i
\(567\) −22.0000 −0.923913
\(568\) 0 0
\(569\) 17.8885i 0.749927i 0.927040 + 0.374963i \(0.122345\pi\)
−0.927040 + 0.374963i \(0.877655\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 6.70820i 0.280484i
\(573\) 20.0000 0.835512
\(574\) 20.0000 0.834784
\(575\) 24.0000 1.00087
\(576\) −26.0000 −1.08333
\(577\) 40.2492i 1.67560i 0.545979 + 0.837799i \(0.316158\pi\)
−0.545979 + 0.837799i \(0.683842\pi\)
\(578\) 6.70820i 0.279024i
\(579\) −30.0000 −1.24676
\(580\) −27.0000 40.2492i −1.12111 1.67126i
\(581\) −12.0000 −0.497844
\(582\) 67.0820i 2.78064i
\(583\) 20.1246i 0.833476i
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) −20.0000 −0.826192
\(587\) −42.0000 −1.73353 −0.866763 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 20.1246i 0.829925i
\(589\) 0 0
\(590\) 40.2492i 1.65703i
\(591\) 40.2492i 1.65563i
\(592\) 0 0
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 11.1803i 0.458735i
\(595\) 26.8328i 1.10004i
\(596\) −45.0000 −1.84327
\(597\) 31.3050i 1.28123i
\(598\) 13.4164i 0.548638i
\(599\) 29.0689i 1.18772i −0.804568 0.593861i \(-0.797603\pi\)
0.804568 0.593861i \(-0.202397\pi\)
\(600\) −20.0000 −0.816497
\(601\) 13.4164i 0.547267i −0.961834 0.273633i \(-0.911775\pi\)
0.961834 0.273633i \(-0.0882255\pi\)
\(602\) 30.0000 1.22271
\(603\) −16.0000 −0.651570
\(604\) 30.0000 1.22068
\(605\) −18.0000 −0.731804
\(606\) 89.4427i 3.63336i
\(607\) 6.70820i 0.272278i −0.990690 0.136139i \(-0.956531\pi\)
0.990690 0.136139i \(-0.0434693\pi\)
\(608\) 0 0
\(609\) 20.0000 13.4164i 0.810441 0.543660i
\(610\) 90.0000 3.64399
\(611\) 2.23607i 0.0904616i
\(612\) 26.8328i 1.08465i
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) −45.0000 −1.81605
\(615\) 30.0000 1.20972
\(616\) 10.0000 0.402911
\(617\) 44.7214i 1.80041i 0.435462 + 0.900207i \(0.356585\pi\)
−0.435462 + 0.900207i \(0.643415\pi\)
\(618\) −20.0000 −0.804518
\(619\) 33.5410i 1.34813i −0.738673 0.674064i \(-0.764547\pi\)
0.738673 0.674064i \(-0.235453\pi\)
\(620\) 60.3738i 2.42467i
\(621\) 13.4164i 0.538382i
\(622\) −20.0000 −0.801927
\(623\) 8.94427i 0.358345i
\(624\) 2.23607i 0.0895144i
\(625\) −29.0000 −1.16000
\(626\) 64.8460i 2.59177i
\(627\) 0 0
\(628\) 40.2492i 1.60612i
\(629\) 0 0
\(630\) 26.8328i 1.06904i
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −15.0000 −0.596668
\(633\) 15.0000 0.596196
\(634\) −50.0000 −1.98575
\(635\) 0 0
\(636\) 60.3738i 2.39398i
\(637\) 3.00000 0.118864
\(638\) 15.0000 + 22.3607i 0.593856 + 0.885268i
\(639\) 0 0
\(640\) 46.9574i 1.85616i
\(641\) 22.3607i 0.883194i −0.897214 0.441597i \(-0.854412\pi\)
0.897214 0.441597i \(-0.145588\pi\)
\(642\) 90.0000 3.55202
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) −36.0000 −1.41860
\(645\) 45.0000 1.77187
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 24.5967i 0.966252i
\(649\) 13.4164i 0.526640i
\(650\) 8.94427i 0.350823i
\(651\) 30.0000 1.17579
\(652\) 60.3738i 2.36442i
\(653\) 49.1935i 1.92509i −0.271122 0.962545i \(-0.587395\pi\)
0.271122 0.962545i \(-0.412605\pi\)
\(654\) 25.0000 0.977577
\(655\) 26.8328i 1.04844i
\(656\) 4.47214i 0.174608i
\(657\) 0 0
\(658\) −10.0000 −0.389841
\(659\) 38.0132i 1.48078i 0.672176 + 0.740391i \(0.265360\pi\)
−0.672176 + 0.740391i \(0.734640\pi\)
\(660\) 45.0000 1.75162
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −75.0000 −2.91496
\(663\) 10.0000 0.388368
\(664\) 13.4164i 0.520658i
\(665\) 0 0
\(666\) 0 0
\(667\) −18.0000 26.8328i −0.696963 1.03897i
\(668\) 36.0000 1.39288
\(669\) 35.7771i 1.38322i
\(670\) 53.6656i 2.07328i
\(671\) −30.0000 −1.15814
\(672\) −30.0000 −1.15728
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 30.0000 1.15556
\(675\) 8.94427i 0.344265i
\(676\) 36.0000 1.38462
\(677\) 17.8885i 0.687513i 0.939059 + 0.343756i \(0.111699\pi\)
−0.939059 + 0.343756i \(0.888301\pi\)
\(678\) 44.7214i 1.71751i
\(679\) 26.8328i 1.02975i
\(680\) 30.0000 1.15045
\(681\) 26.8328i 1.02824i
\(682\) 33.5410i 1.28435i
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 26.8328i 1.02523i
\(686\) 44.7214i 1.70747i
\(687\) −30.0000 −1.14457
\(688\) 6.70820i 0.255748i
\(689\) 9.00000 0.342873
\(690\) −90.0000 −3.42624
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 18.0000 0.684257
\(693\) 8.94427i 0.339765i
\(694\) 26.8328i 1.01856i
\(695\) 30.0000 1.13796
\(696\) 15.0000 + 22.3607i 0.568574 + 0.847579i
\(697\) −20.0000 −0.757554
\(698\) 2.23607i 0.0846364i
\(699\) 20.1246i 0.761183i
\(700\) −24.0000 −0.907115
\(701\) −45.0000 −1.69963 −0.849813 0.527084i \(-0.823285\pi\)
−0.849813 + 0.527084i \(0.823285\pi\)
\(702\) −5.00000 −0.188713
\(703\) 0 0
\(704\) 29.0689i 1.09557i
\(705\) −15.0000 −0.564933
\(706\) 13.4164i 0.504933i
\(707\) 35.7771i 1.34554i
\(708\) 40.2492i 1.51266i
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 13.4164i 0.503155i
\(712\) −10.0000 −0.374766
\(713\) 40.2492i 1.50735i
\(714\) 44.7214i 1.67365i
\(715\) 6.70820i 0.250873i
\(716\) 0 0
\(717\) 13.4164i 0.501045i
\(718\) −5.00000 −0.186598
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −6.00000 −0.223607
\(721\) −8.00000 −0.297936
\(722\) 42.4853i 1.58114i
\(723\) 55.9017i 2.07901i
\(724\) −15.0000 −0.557471
\(725\) −12.0000 17.8885i −0.445669 0.664364i
\(726\) 30.0000 1.11340
\(727\) 26.8328i 0.995174i 0.867414 + 0.497587i \(0.165780\pi\)
−0.867414 + 0.497587i \(0.834220\pi\)
\(728\) 4.47214i 0.165748i
\(729\) 7.00000 0.259259
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) −90.0000 −3.32650
\(733\) 26.8328i 0.991093i −0.868582 0.495546i \(-0.834968\pi\)
0.868582 0.495546i \(-0.165032\pi\)
\(734\) 60.0000 2.21464
\(735\) 20.1246i 0.742307i
\(736\) 40.2492i 1.48361i
\(737\) 17.8885i 0.658933i
\(738\) −20.0000 −0.736210
\(739\) 20.1246i 0.740296i 0.928973 + 0.370148i \(0.120693\pi\)
−0.928973 + 0.370148i \(0.879307\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 40.2492i 1.47760i
\(743\) 8.94427i 0.328134i −0.986449 0.164067i \(-0.947539\pi\)
0.986449 0.164067i \(-0.0524612\pi\)
\(744\) 33.5410i 1.22967i
\(745\) −45.0000 −1.64867
\(746\) 69.3181i 2.53792i
\(747\) 12.0000 0.439057
\(748\) −30.0000 −1.09691
\(749\) 36.0000 1.31541
\(750\) 15.0000 0.547723
\(751\) 26.8328i 0.979143i −0.871963 0.489572i \(-0.837153\pi\)
0.871963 0.489572i \(-0.162847\pi\)
\(752\) 2.23607i 0.0815410i
\(753\) 35.0000 1.27547
\(754\) 10.0000 6.70820i 0.364179 0.244298i
\(755\) 30.0000 1.09181
\(756\) 13.4164i 0.487950i
\(757\) 40.2492i 1.46288i 0.681904 + 0.731441i \(0.261152\pi\)
−0.681904 + 0.731441i \(0.738848\pi\)
\(758\) 0 0
\(759\) 30.0000 1.08893
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) 26.8328i 0.970777i
\(765\) 26.8328i 0.970143i
\(766\) 13.4164i 0.484755i
\(767\) −6.00000 −0.216647
\(768\) 20.1246i 0.726184i
\(769\) 13.4164i 0.483808i 0.970300 + 0.241904i \(0.0777719\pi\)
−0.970300 + 0.241904i \(0.922228\pi\)
\(770\) 30.0000 1.08112
\(771\) 6.70820i 0.241590i
\(772\) 40.2492i 1.44860i
\(773\) 22.3607i 0.804258i −0.915583 0.402129i \(-0.868270\pi\)
0.915583 0.402129i \(-0.131730\pi\)
\(774\) −30.0000 −1.07833
\(775\) 26.8328i 0.963863i
\(776\) 30.0000 1.07694
\(777\) 0 0
\(778\) 40.0000 1.43407
\(779\) 0 0
\(780\) 20.1246i 0.720577i
\(781\) 0 0
\(782\) 60.0000 2.14560
\(783\) 10.0000 6.70820i 0.357371 0.239732i
\(784\) 3.00000 0.107143
\(785\) 40.2492i 1.43656i
\(786\) 44.7214i 1.59516i
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 54.0000 1.92367
\(789\) −25.0000 −0.890024
\(790\) −45.0000 −1.60103
\(791\) 17.8885i 0.636043i
\(792\) −10.0000 −0.355335
\(793\) 13.4164i 0.476431i
\(794\) 15.6525i 0.555486i
\(795\) 60.3738i 2.14124i
\(796\) −42.0000 −1.48865
\(797\) 4.47214i 0.158411i 0.996858 + 0.0792056i \(0.0252384\pi\)
−0.996858 + 0.0792056i \(0.974762\pi\)
\(798\) 0 0
\(799\) 10.0000 0.353775
\(800\) 26.8328i 0.948683i
\(801\) 8.94427i 0.316030i
\(802\) 33.5410i 1.18437i
\(803\) 0 0
\(804\) 53.6656i 1.89264i
\(805\) −36.0000 −1.26883
\(806\) 15.0000 0.528352
\(807\) −10.0000 −0.352017
\(808\) −40.0000 −1.40720
\(809\) 44.7214i 1.57232i 0.618023 + 0.786160i \(0.287934\pi\)
−0.618023 + 0.786160i \(0.712066\pi\)
\(810\) 73.7902i 2.59272i
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 18.0000 + 26.8328i 0.631676 + 0.941647i
\(813\) −45.0000 −1.57822
\(814\) 0 0
\(815\) 60.3738i 2.11480i
\(816\) 10.0000 0.350070
\(817\) 0 0
\(818\) −60.0000 −2.09785
\(819\) 4.00000 0.139771
\(820\) 40.2492i 1.40556i
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 44.7214i 1.55984i
\(823\) 53.6656i 1.87067i 0.353768 + 0.935333i \(0.384900\pi\)
−0.353768 + 0.935333i \(0.615100\pi\)
\(824\) 8.94427i 0.311588i
\(825\) 20.0000 0.696311
\(826\) 26.8328i 0.933633i
\(827\) 38.0132i 1.32185i 0.750453 + 0.660923i \(0.229835\pi\)
−0.750453 + 0.660923i \(0.770165\pi\)
\(828\) 36.0000 1.25109
\(829\) 40.2492i 1.39791i 0.715164 + 0.698957i \(0.246352\pi\)
−0.715164 + 0.698957i \(0.753648\pi\)
\(830\) 40.2492i 1.39707i
\(831\) 4.47214i 0.155137i
\(832\) −13.0000 −0.450694
\(833\) 13.4164i 0.464851i
\(834\) −50.0000 −1.73136
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) 53.6656i 1.85385i
\(839\) 2.23607i 0.0771976i −0.999255 0.0385988i \(-0.987711\pi\)
0.999255 0.0385988i \(-0.0122894\pi\)
\(840\) 30.0000 1.03510
\(841\) −11.0000 + 26.8328i −0.379310 + 0.925270i
\(842\) −30.0000 −1.03387
\(843\) 6.70820i 0.231043i
\(844\) 20.1246i 0.692718i
\(845\) 36.0000 1.23844
\(846\) 10.0000 0.343807
\(847\) 12.0000 0.412325
\(848\) 9.00000 0.309061
\(849\) 31.3050i 1.07438i
\(850\) 40.0000 1.37199
\(851\) 0 0
\(852\) 0 0
\(853\) 26.8328i 0.918738i 0.888246 + 0.459369i \(0.151924\pi\)
−0.888246 + 0.459369i \(0.848076\pi\)
\(854\) −60.0000 −2.05316
\(855\) 0 0
\(856\) 40.2492i 1.37569i
\(857\) 57.0000 1.94708 0.973541 0.228510i \(-0.0733855\pi\)
0.973541 + 0.228510i \(0.0733855\pi\)
\(858\) 11.1803i 0.381691i
\(859\) 46.9574i 1.60217i −0.598553 0.801083i \(-0.704257\pi\)
0.598553 0.801083i \(-0.295743\pi\)
\(860\) 60.3738i 2.05873i
\(861\) −20.0000 −0.681598
\(862\) 40.2492i 1.37089i
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) −15.0000 −0.510310
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) 6.70820i 0.227823i
\(868\) 40.2492i 1.36615i
\(869\) 15.0000 0.508840
\(870\) 45.0000 + 67.0820i 1.52564 + 2.27429i
\(871\) −8.00000 −0.271070
\(872\) 11.1803i 0.378614i
\(873\) 26.8328i 0.908153i
\(874\) 0 0
\(875\) 6.00000 0.202837
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 76.0263i 2.56576i
\(879\) 20.0000 0.674583
\(880\) 6.70820i 0.226134i
\(881\) 35.7771i 1.20536i −0.797983 0.602680i \(-0.794099\pi\)
0.797983 0.602680i \(-0.205901\pi\)
\(882\) 13.4164i 0.451754i
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 13.4164i 0.451243i
\(885\) 40.2492i 1.35296i
\(886\) 40.0000 1.34383
\(887\) 11.1803i 0.375399i 0.982226 + 0.187700i \(0.0601031\pi\)
−0.982226 + 0.187700i \(0.939897\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −30.0000 −1.00560
\(891\) 24.5967i 0.824022i
\(892\) 48.0000 1.60716
\(893\) 0 0
\(894\) 75.0000 2.50838
\(895\) 0 0
\(896\) 31.3050i 1.04583i
\(897\) 13.4164i 0.447961i
\(898\) −80.0000 −2.66963
\(899\) −30.0000 + 20.1246i −1.00056 + 0.671193i
\(900\) 24.0000 0.800000
\(901\) 40.2492i 1.34090i
\(902\) 22.3607i 0.744529i
\(903\) −30.0000 −0.998337
\(904\) 20.0000 0.665190
\(905\) −15.0000 −0.498617
\(906\) −50.0000 −1.66114
\(907\) 53.6656i 1.78194i −0.454064 0.890969i \(-0.650026\pi\)
0.454064 0.890969i \(-0.349974\pi\)
\(908\) −36.0000 −1.19470
\(909\) 35.7771i 1.18665i
\(910\) 13.4164i 0.444750i
\(911\) 38.0132i 1.25943i 0.776825 + 0.629716i \(0.216829\pi\)
−0.776825 + 0.629716i \(0.783171\pi\)
\(912\) 0 0
\(913\) 13.4164i 0.444018i
\(914\) 4.47214i 0.147925i
\(915\) −90.0000 −2.97531
\(916\) 40.2492i 1.32987i
\(917\) 17.8885i 0.590732i
\(918\) 22.3607i 0.738012i
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 40.2492i 1.32698i
\(921\) 45.0000 1.48280
\(922\) −20.0000 −0.658665
\(923\) 0 0
\(924\) −30.0000 −0.986928
\(925\) 0 0
\(926\) 58.1378i 1.91053i
\(927\) 8.00000 0.262754
\(928\) 30.0000 20.1246i 0.984798 0.660623i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 100.623i 3.29956i
\(931\) 0 0
\(932\) 27.0000 0.884414
\(933\) 20.0000 0.654771
\(934\) −5.00000 −0.163605
\(935\) −30.0000 −0.981105
\(936\) 4.47214i 0.146176i
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 35.7771i 1.16816i
\(939\) 64.8460i 2.11617i
\(940\) 20.1246i 0.656392i
\(941\) −3.00000 −0.0977972 −0.0488986 0.998804i \(-0.515571\pi\)
−0.0488986 + 0.998804i \(0.515571\pi\)
\(942\) 67.0820i 2.18565i
\(943\) 26.8328i 0.873797i
\(944\) −6.00000 −0.195283
\(945\) 13.4164i 0.436436i
\(946\) 33.5410i 1.09051i
\(947\) 2.23607i 0.0726624i −0.999340 0.0363312i \(-0.988433\pi\)
0.999340 0.0363312i \(-0.0115671\pi\)
\(948\) 45.0000 1.46153
\(949\) 0 0
\(950\) 0 0
\(951\) 50.0000 1.62136
\(952\) −20.0000 −0.648204
\(953\) −9.00000 −0.291539 −0.145769 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 40.2492i 1.30312i
\(955\) 26.8328i 0.868290i
\(956\) −18.0000 −0.582162
\(957\) −15.0000 22.3607i −0.484881 0.722818i
\(958\) 25.0000 0.807713
\(959\) 17.8885i 0.577651i
\(960\) 87.2067i 2.81458i
\(961\) −14.0000 −0.451613
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 75.0000 2.41559
\(965\) 40.2492i 1.29567i
\(966\) 60.0000 1.93047
\(967\) 46.9574i 1.51005i −0.655697 0.755025i \(-0.727625\pi\)
0.655697 0.755025i \(-0.272375\pi\)
\(968\) 13.4164i 0.431220i
\(969\) 0 0
\(970\) 90.0000 2.88973
\(971\) 17.8885i 0.574071i 0.957920 + 0.287035i \(0.0926697\pi\)
−0.957920 + 0.287035i \(0.907330\pi\)
\(972\) 53.6656i 1.72133i
\(973\) −20.0000 −0.641171
\(974\) 4.47214i 0.143296i
\(975\) 8.94427i 0.286446i
\(976\) 13.4164i 0.429449i
\(977\) 33.0000 1.05576 0.527882 0.849318i \(-0.322986\pi\)
0.527882 + 0.849318i \(0.322986\pi\)
\(978\) 100.623i 3.21757i
\(979\) 10.0000 0.319601
\(980\) −27.0000 −0.862483
\(981\) −10.0000 −0.319275
\(982\) 55.0000 1.75512
\(983\) 29.0689i 0.927153i −0.886057 0.463577i \(-0.846566\pi\)
0.886057 0.463577i \(-0.153434\pi\)
\(984\) 22.3607i 0.712832i
\(985\) 54.0000 1.72058
\(986\) −30.0000 44.7214i −0.955395 1.42422i
\(987\) 10.0000 0.318304
\(988\) 0 0
\(989\) 40.2492i 1.27985i
\(990\) −30.0000 −0.953463
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 45.0000 1.42875
\(993\) 75.0000 2.38005
\(994\) 0 0
\(995\) −42.0000 −1.33149
\(996\) 40.2492i 1.27535i
\(997\) 26.8328i 0.849804i 0.905239 + 0.424902i \(0.139691\pi\)
−0.905239 + 0.424902i \(0.860309\pi\)
\(998\) 22.3607i 0.707815i
\(999\) 0 0
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 29.2.b.a.28.1 2
3.2 odd 2 261.2.c.a.28.2 2
4.3 odd 2 464.2.e.a.289.1 2
5.2 odd 4 725.2.d.a.724.4 4
5.3 odd 4 725.2.d.a.724.1 4
5.4 even 2 725.2.c.c.376.2 2
7.6 odd 2 1421.2.b.b.1275.1 2
8.3 odd 2 1856.2.e.f.1217.2 2
8.5 even 2 1856.2.e.g.1217.1 2
12.11 even 2 4176.2.o.k.289.1 2
29.2 odd 28 841.2.d.h.605.2 12
29.3 odd 28 841.2.d.h.571.2 12
29.4 even 14 841.2.e.g.651.2 12
29.5 even 14 841.2.e.g.236.1 12
29.6 even 14 841.2.e.g.196.2 12
29.7 even 7 841.2.e.g.270.2 12
29.8 odd 28 841.2.d.h.574.2 12
29.9 even 14 841.2.e.g.267.2 12
29.10 odd 28 841.2.d.h.190.1 12
29.11 odd 28 841.2.d.h.778.1 12
29.12 odd 4 841.2.a.b.1.2 2
29.13 even 14 841.2.e.g.63.1 12
29.14 odd 28 841.2.d.h.645.2 12
29.15 odd 28 841.2.d.h.645.1 12
29.16 even 7 841.2.e.g.63.2 12
29.17 odd 4 841.2.a.b.1.1 2
29.18 odd 28 841.2.d.h.778.2 12
29.19 odd 28 841.2.d.h.190.2 12
29.20 even 7 841.2.e.g.267.1 12
29.21 odd 28 841.2.d.h.574.1 12
29.22 even 14 841.2.e.g.270.1 12
29.23 even 7 841.2.e.g.196.1 12
29.24 even 7 841.2.e.g.236.2 12
29.25 even 7 841.2.e.g.651.1 12
29.26 odd 28 841.2.d.h.571.1 12
29.27 odd 28 841.2.d.h.605.1 12
29.28 even 2 inner 29.2.b.a.28.2 yes 2
87.17 even 4 7569.2.a.i.1.2 2
87.41 even 4 7569.2.a.i.1.1 2
87.86 odd 2 261.2.c.a.28.1 2
116.115 odd 2 464.2.e.a.289.2 2
145.28 odd 4 725.2.d.a.724.3 4
145.57 odd 4 725.2.d.a.724.2 4
145.144 even 2 725.2.c.c.376.1 2
203.202 odd 2 1421.2.b.b.1275.2 2
232.115 odd 2 1856.2.e.f.1217.1 2
232.173 even 2 1856.2.e.g.1217.2 2
348.347 even 2 4176.2.o.k.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.2.b.a.28.1 2 1.1 even 1 trivial
29.2.b.a.28.2 yes 2 29.28 even 2 inner
261.2.c.a.28.1 2 87.86 odd 2
261.2.c.a.28.2 2 3.2 odd 2
464.2.e.a.289.1 2 4.3 odd 2
464.2.e.a.289.2 2 116.115 odd 2
725.2.c.c.376.1 2 145.144 even 2
725.2.c.c.376.2 2 5.4 even 2
725.2.d.a.724.1 4 5.3 odd 4
725.2.d.a.724.2 4 145.57 odd 4
725.2.d.a.724.3 4 145.28 odd 4
725.2.d.a.724.4 4 5.2 odd 4
841.2.a.b.1.1 2 29.17 odd 4
841.2.a.b.1.2 2 29.12 odd 4
841.2.d.h.190.1 12 29.10 odd 28
841.2.d.h.190.2 12 29.19 odd 28
841.2.d.h.571.1 12 29.26 odd 28
841.2.d.h.571.2 12 29.3 odd 28
841.2.d.h.574.1 12 29.21 odd 28
841.2.d.h.574.2 12 29.8 odd 28
841.2.d.h.605.1 12 29.27 odd 28
841.2.d.h.605.2 12 29.2 odd 28
841.2.d.h.645.1 12 29.15 odd 28
841.2.d.h.645.2 12 29.14 odd 28
841.2.d.h.778.1 12 29.11 odd 28
841.2.d.h.778.2 12 29.18 odd 28
841.2.e.g.63.1 12 29.13 even 14
841.2.e.g.63.2 12 29.16 even 7
841.2.e.g.196.1 12 29.23 even 7
841.2.e.g.196.2 12 29.6 even 14
841.2.e.g.236.1 12 29.5 even 14
841.2.e.g.236.2 12 29.24 even 7
841.2.e.g.267.1 12 29.20 even 7
841.2.e.g.267.2 12 29.9 even 14
841.2.e.g.270.1 12 29.22 even 14
841.2.e.g.270.2 12 29.7 even 7
841.2.e.g.651.1 12 29.25 even 7
841.2.e.g.651.2 12 29.4 even 14
1421.2.b.b.1275.1 2 7.6 odd 2
1421.2.b.b.1275.2 2 203.202 odd 2
1856.2.e.f.1217.1 2 232.115 odd 2
1856.2.e.f.1217.2 2 8.3 odd 2
1856.2.e.g.1217.1 2 8.5 even 2
1856.2.e.g.1217.2 2 232.173 even 2
4176.2.o.k.289.1 2 12.11 even 2
4176.2.o.k.289.2 2 348.347 even 2
7569.2.a.i.1.1 2 87.41 even 4
7569.2.a.i.1.2 2 87.17 even 4