Properties

Label 77.2.b.b.76.3
Level $77$
Weight $2$
Character 77.76
Analytic conductor $0.615$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [77,2,Mod(76,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.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: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 76.3
Root \(1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 77.76
Dual form 77.2.b.b.76.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.23607i q^{3} +2.23607i q^{5} +3.16228 q^{6} +(1.58114 - 2.12132i) q^{7} +2.82843i q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} -2.23607i q^{3} +2.23607i q^{5} +3.16228 q^{6} +(1.58114 - 2.12132i) q^{7} +2.82843i q^{8} -2.00000 q^{9} -3.16228 q^{10} +(-3.00000 - 1.41421i) q^{11} -6.32456 q^{13} +(3.00000 + 2.23607i) q^{14} +5.00000 q^{15} -4.00000 q^{16} -2.82843i q^{18} +3.16228 q^{19} +(-4.74342 - 3.53553i) q^{21} +(2.00000 - 4.24264i) q^{22} -3.00000 q^{23} +6.32456 q^{24} -8.94427i q^{26} -2.23607i q^{27} +1.41421i q^{29} +7.07107i q^{30} +6.70820i q^{31} +(-3.16228 + 6.70820i) q^{33} +(4.74342 + 3.53553i) q^{35} -1.00000 q^{37} +4.47214i q^{38} +14.1421i q^{39} -6.32456 q^{40} +9.48683 q^{41} +(5.00000 - 6.70820i) q^{42} -4.24264i q^{43} -4.47214i q^{45} -4.24264i q^{46} -4.47214i q^{47} +8.94427i q^{48} +(-2.00000 - 6.70820i) q^{49} +3.16228 q^{54} +(3.16228 - 6.70820i) q^{55} +(6.00000 + 4.47214i) q^{56} -7.07107i q^{57} -2.00000 q^{58} +2.23607i q^{59} +3.16228 q^{61} -9.48683 q^{62} +(-3.16228 + 4.24264i) q^{63} -8.00000 q^{64} -14.1421i q^{65} +(-9.48683 - 4.47214i) q^{66} +11.0000 q^{67} +6.70820i q^{69} +(-5.00000 + 6.70820i) q^{70} +9.00000 q^{71} -5.65685i q^{72} +3.16228 q^{73} -1.41421i q^{74} +(-7.74342 + 4.12789i) q^{77} -20.0000 q^{78} +8.48528i q^{79} -8.94427i q^{80} -11.0000 q^{81} +13.4164i q^{82} -9.48683 q^{83} +6.00000 q^{86} +3.16228 q^{87} +(4.00000 - 8.48528i) q^{88} +2.23607i q^{89} +6.32456 q^{90} +(-10.0000 + 13.4164i) q^{91} +15.0000 q^{93} +6.32456 q^{94} +7.07107i q^{95} -6.70820i q^{97} +(9.48683 - 2.82843i) q^{98} +(6.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} - 12 q^{11} + 12 q^{14} + 20 q^{15} - 16 q^{16} + 8 q^{22} - 12 q^{23} - 4 q^{37} + 20 q^{42} - 8 q^{49} + 24 q^{56} - 8 q^{58} - 32 q^{64} + 44 q^{67} - 20 q^{70} + 36 q^{71} - 12 q^{77} - 80 q^{78} - 44 q^{81} + 24 q^{86} + 16 q^{88} - 40 q^{91} + 60 q^{93} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 2.23607i 1.29099i −0.763763 0.645497i \(-0.776650\pi\)
0.763763 0.645497i \(-0.223350\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 3.16228 1.29099
\(7\) 1.58114 2.12132i 0.597614 0.801784i
\(8\) 2.82843i 1.00000i
\(9\) −2.00000 −0.666667
\(10\) −3.16228 −1.00000
\(11\) −3.00000 1.41421i −0.904534 0.426401i
\(12\) 0 0
\(13\) −6.32456 −1.75412 −0.877058 0.480384i \(-0.840497\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 3.00000 + 2.23607i 0.801784 + 0.597614i
\(15\) 5.00000 1.29099
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 2.82843i 0.666667i
\(19\) 3.16228 0.725476 0.362738 0.931891i \(-0.381842\pi\)
0.362738 + 0.931891i \(0.381842\pi\)
\(20\) 0 0
\(21\) −4.74342 3.53553i −1.03510 0.771517i
\(22\) 2.00000 4.24264i 0.426401 0.904534i
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 6.32456 1.29099
\(25\) 0 0
\(26\) 8.94427i 1.75412i
\(27\) 2.23607i 0.430331i
\(28\) 0 0
\(29\) 1.41421i 0.262613i 0.991342 + 0.131306i \(0.0419172\pi\)
−0.991342 + 0.131306i \(0.958083\pi\)
\(30\) 7.07107i 1.29099i
\(31\) 6.70820i 1.20483i 0.798183 + 0.602414i \(0.205795\pi\)
−0.798183 + 0.602414i \(0.794205\pi\)
\(32\) 0 0
\(33\) −3.16228 + 6.70820i −0.550482 + 1.16775i
\(34\) 0 0
\(35\) 4.74342 + 3.53553i 0.801784 + 0.597614i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 4.47214i 0.725476i
\(39\) 14.1421i 2.26455i
\(40\) −6.32456 −1.00000
\(41\) 9.48683 1.48159 0.740797 0.671729i \(-0.234448\pi\)
0.740797 + 0.671729i \(0.234448\pi\)
\(42\) 5.00000 6.70820i 0.771517 1.03510i
\(43\) 4.24264i 0.646997i −0.946229 0.323498i \(-0.895141\pi\)
0.946229 0.323498i \(-0.104859\pi\)
\(44\) 0 0
\(45\) 4.47214i 0.666667i
\(46\) 4.24264i 0.625543i
\(47\) 4.47214i 0.652328i −0.945313 0.326164i \(-0.894244\pi\)
0.945313 0.326164i \(-0.105756\pi\)
\(48\) 8.94427i 1.29099i
\(49\) −2.00000 6.70820i −0.285714 0.958315i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 3.16228 0.430331
\(55\) 3.16228 6.70820i 0.426401 0.904534i
\(56\) 6.00000 + 4.47214i 0.801784 + 0.597614i
\(57\) 7.07107i 0.936586i
\(58\) −2.00000 −0.262613
\(59\) 2.23607i 0.291111i 0.989350 + 0.145556i \(0.0464970\pi\)
−0.989350 + 0.145556i \(0.953503\pi\)
\(60\) 0 0
\(61\) 3.16228 0.404888 0.202444 0.979294i \(-0.435112\pi\)
0.202444 + 0.979294i \(0.435112\pi\)
\(62\) −9.48683 −1.20483
\(63\) −3.16228 + 4.24264i −0.398410 + 0.534522i
\(64\) −8.00000 −1.00000
\(65\) 14.1421i 1.75412i
\(66\) −9.48683 4.47214i −1.16775 0.550482i
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 6.70820i 0.807573i
\(70\) −5.00000 + 6.70820i −0.597614 + 0.801784i
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 5.65685i 0.666667i
\(73\) 3.16228 0.370117 0.185058 0.982728i \(-0.440753\pi\)
0.185058 + 0.982728i \(0.440753\pi\)
\(74\) 1.41421i 0.164399i
\(75\) 0 0
\(76\) 0 0
\(77\) −7.74342 + 4.12789i −0.882444 + 0.470417i
\(78\) −20.0000 −2.26455
\(79\) 8.48528i 0.954669i 0.878722 + 0.477334i \(0.158397\pi\)
−0.878722 + 0.477334i \(0.841603\pi\)
\(80\) 8.94427i 1.00000i
\(81\) −11.0000 −1.22222
\(82\) 13.4164i 1.48159i
\(83\) −9.48683 −1.04132 −0.520658 0.853766i \(-0.674313\pi\)
−0.520658 + 0.853766i \(0.674313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 3.16228 0.339032
\(88\) 4.00000 8.48528i 0.426401 0.904534i
\(89\) 2.23607i 0.237023i 0.992953 + 0.118511i \(0.0378122\pi\)
−0.992953 + 0.118511i \(0.962188\pi\)
\(90\) 6.32456 0.666667
\(91\) −10.0000 + 13.4164i −1.04828 + 1.40642i
\(92\) 0 0
\(93\) 15.0000 1.55543
\(94\) 6.32456 0.652328
\(95\) 7.07107i 0.725476i
\(96\) 0 0
\(97\) 6.70820i 0.681115i −0.940224 0.340557i \(-0.889384\pi\)
0.940224 0.340557i \(-0.110616\pi\)
\(98\) 9.48683 2.82843i 0.958315 0.285714i
\(99\) 6.00000 + 2.82843i 0.603023 + 0.284268i
\(100\) 0 0
\(101\) −9.48683 −0.943975 −0.471988 0.881605i \(-0.656463\pi\)
−0.471988 + 0.881605i \(0.656463\pi\)
\(102\) 0 0
\(103\) 13.4164i 1.32196i −0.750404 0.660979i \(-0.770141\pi\)
0.750404 0.660979i \(-0.229859\pi\)
\(104\) 17.8885i 1.75412i
\(105\) 7.90569 10.6066i 0.771517 1.03510i
\(106\) 0 0
\(107\) 2.82843i 0.273434i −0.990610 0.136717i \(-0.956345\pi\)
0.990610 0.136717i \(-0.0436552\pi\)
\(108\) 0 0
\(109\) 12.7279i 1.21911i 0.792742 + 0.609557i \(0.208653\pi\)
−0.792742 + 0.609557i \(0.791347\pi\)
\(110\) 9.48683 + 4.47214i 0.904534 + 0.426401i
\(111\) 2.23607i 0.212238i
\(112\) −6.32456 + 8.48528i −0.597614 + 0.801784i
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 10.0000 0.936586
\(115\) 6.70820i 0.625543i
\(116\) 0 0
\(117\) 12.6491 1.16941
\(118\) −3.16228 −0.291111
\(119\) 0 0
\(120\) 14.1421i 1.29099i
\(121\) 7.00000 + 8.48528i 0.636364 + 0.771389i
\(122\) 4.47214i 0.404888i
\(123\) 21.2132i 1.91273i
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) −6.00000 4.47214i −0.534522 0.398410i
\(127\) 12.7279i 1.12942i 0.825289 + 0.564710i \(0.191012\pi\)
−0.825289 + 0.564710i \(0.808988\pi\)
\(128\) 11.3137i 1.00000i
\(129\) −9.48683 −0.835269
\(130\) 20.0000 1.75412
\(131\) −18.9737 −1.65774 −0.828868 0.559444i \(-0.811015\pi\)
−0.828868 + 0.559444i \(0.811015\pi\)
\(132\) 0 0
\(133\) 5.00000 6.70820i 0.433555 0.581675i
\(134\) 15.5563i 1.34386i
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) −15.0000 −1.28154 −0.640768 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) −9.48683 −0.807573
\(139\) 12.6491 1.07288 0.536442 0.843937i \(-0.319768\pi\)
0.536442 + 0.843937i \(0.319768\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 12.7279i 1.06810i
\(143\) 18.9737 + 8.94427i 1.58666 + 0.747958i
\(144\) 8.00000 0.666667
\(145\) −3.16228 −0.262613
\(146\) 4.47214i 0.370117i
\(147\) −15.0000 + 4.47214i −1.23718 + 0.368856i
\(148\) 0 0
\(149\) 19.7990i 1.62200i −0.585049 0.810998i \(-0.698925\pi\)
0.585049 0.810998i \(-0.301075\pi\)
\(150\) 0 0
\(151\) 16.9706i 1.38104i −0.723311 0.690522i \(-0.757381\pi\)
0.723311 0.690522i \(-0.242619\pi\)
\(152\) 8.94427i 0.725476i
\(153\) 0 0
\(154\) −5.83772 10.9508i −0.470417 0.882444i
\(155\) −15.0000 −1.20483
\(156\) 0 0
\(157\) 6.70820i 0.535373i 0.963506 + 0.267686i \(0.0862591\pi\)
−0.963506 + 0.267686i \(0.913741\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) 0 0
\(161\) −4.74342 + 6.36396i −0.373834 + 0.501550i
\(162\) 15.5563i 1.22222i
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) −15.0000 7.07107i −1.16775 0.550482i
\(166\) 13.4164i 1.04132i
\(167\) 9.48683 0.734113 0.367057 0.930199i \(-0.380366\pi\)
0.367057 + 0.930199i \(0.380366\pi\)
\(168\) 10.0000 13.4164i 0.771517 1.03510i
\(169\) 27.0000 2.07692
\(170\) 0 0
\(171\) −6.32456 −0.483651
\(172\) 0 0
\(173\) −9.48683 −0.721271 −0.360635 0.932707i \(-0.617440\pi\)
−0.360635 + 0.932707i \(0.617440\pi\)
\(174\) 4.47214i 0.339032i
\(175\) 0 0
\(176\) 12.0000 + 5.65685i 0.904534 + 0.426401i
\(177\) 5.00000 0.375823
\(178\) −3.16228 −0.237023
\(179\) 9.00000 0.672692 0.336346 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) 20.1246i 1.49585i 0.663783 + 0.747925i \(0.268950\pi\)
−0.663783 + 0.747925i \(0.731050\pi\)
\(182\) −18.9737 14.1421i −1.40642 1.04828i
\(183\) 7.07107i 0.522708i
\(184\) 8.48528i 0.625543i
\(185\) 2.23607i 0.164399i
\(186\) 21.2132i 1.55543i
\(187\) 0 0
\(188\) 0 0
\(189\) −4.74342 3.53553i −0.345033 0.257172i
\(190\) −10.0000 −0.725476
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 17.8885i 1.29099i
\(193\) 4.24264i 0.305392i 0.988273 + 0.152696i \(0.0487955\pi\)
−0.988273 + 0.152696i \(0.951204\pi\)
\(194\) 9.48683 0.681115
\(195\) −31.6228 −2.26455
\(196\) 0 0
\(197\) 5.65685i 0.403034i 0.979485 + 0.201517i \(0.0645872\pi\)
−0.979485 + 0.201517i \(0.935413\pi\)
\(198\) −4.00000 + 8.48528i −0.284268 + 0.603023i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 24.5967i 1.73492i
\(202\) 13.4164i 0.943975i
\(203\) 3.00000 + 2.23607i 0.210559 + 0.156941i
\(204\) 0 0
\(205\) 21.2132i 1.48159i
\(206\) 18.9737 1.32196
\(207\) 6.00000 0.417029
\(208\) 25.2982 1.75412
\(209\) −9.48683 4.47214i −0.656218 0.309344i
\(210\) 15.0000 + 11.1803i 1.03510 + 0.771517i
\(211\) 21.2132i 1.46038i −0.683246 0.730189i \(-0.739432\pi\)
0.683246 0.730189i \(-0.260568\pi\)
\(212\) 0 0
\(213\) 20.1246i 1.37892i
\(214\) 4.00000 0.273434
\(215\) 9.48683 0.646997
\(216\) 6.32456 0.430331
\(217\) 14.2302 + 10.6066i 0.966012 + 0.720023i
\(218\) −18.0000 −1.21911
\(219\) 7.07107i 0.477818i
\(220\) 0 0
\(221\) 0 0
\(222\) −3.16228 −0.212238
\(223\) 6.70820i 0.449215i −0.974449 0.224607i \(-0.927890\pi\)
0.974449 0.224607i \(-0.0721099\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4.24264i 0.282216i
\(227\) −9.48683 −0.629663 −0.314832 0.949148i \(-0.601948\pi\)
−0.314832 + 0.949148i \(0.601948\pi\)
\(228\) 0 0
\(229\) 6.70820i 0.443291i 0.975127 + 0.221645i \(0.0711427\pi\)
−0.975127 + 0.221645i \(0.928857\pi\)
\(230\) 9.48683 0.625543
\(231\) 9.23025 + 17.3148i 0.607306 + 1.13923i
\(232\) −4.00000 −0.262613
\(233\) 18.3848i 1.20443i 0.798335 + 0.602213i \(0.205714\pi\)
−0.798335 + 0.602213i \(0.794286\pi\)
\(234\) 17.8885i 1.16941i
\(235\) 10.0000 0.652328
\(236\) 0 0
\(237\) 18.9737 1.23247
\(238\) 0 0
\(239\) 7.07107i 0.457389i −0.973498 0.228695i \(-0.926554\pi\)
0.973498 0.228695i \(-0.0734457\pi\)
\(240\) −20.0000 −1.29099
\(241\) 12.6491 0.814801 0.407400 0.913250i \(-0.366435\pi\)
0.407400 + 0.913250i \(0.366435\pi\)
\(242\) −12.0000 + 9.89949i −0.771389 + 0.636364i
\(243\) 17.8885i 1.14755i
\(244\) 0 0
\(245\) 15.0000 4.47214i 0.958315 0.285714i
\(246\) 30.0000 1.91273
\(247\) −20.0000 −1.27257
\(248\) −18.9737 −1.20483
\(249\) 21.2132i 1.34433i
\(250\) −15.8114 −1.00000
\(251\) 11.1803i 0.705697i −0.935681 0.352848i \(-0.885213\pi\)
0.935681 0.352848i \(-0.114787\pi\)
\(252\) 0 0
\(253\) 9.00000 + 4.24264i 0.565825 + 0.266733i
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 0 0
\(257\) 22.3607i 1.39482i 0.716672 + 0.697410i \(0.245665\pi\)
−0.716672 + 0.697410i \(0.754335\pi\)
\(258\) 13.4164i 0.835269i
\(259\) −1.58114 + 2.12132i −0.0982472 + 0.131812i
\(260\) 0 0
\(261\) 2.82843i 0.175075i
\(262\) 26.8328i 1.65774i
\(263\) 11.3137i 0.697633i −0.937191 0.348817i \(-0.886584\pi\)
0.937191 0.348817i \(-0.113416\pi\)
\(264\) −18.9737 8.94427i −1.16775 0.550482i
\(265\) 0 0
\(266\) 9.48683 + 7.07107i 0.581675 + 0.433555i
\(267\) 5.00000 0.305995
\(268\) 0 0
\(269\) 17.8885i 1.09068i −0.838214 0.545342i \(-0.816400\pi\)
0.838214 0.545342i \(-0.183600\pi\)
\(270\) 7.07107i 0.430331i
\(271\) −25.2982 −1.53676 −0.768379 0.639995i \(-0.778936\pi\)
−0.768379 + 0.639995i \(0.778936\pi\)
\(272\) 0 0
\(273\) 30.0000 + 22.3607i 1.81568 + 1.35333i
\(274\) 21.2132i 1.28154i
\(275\) 0 0
\(276\) 0 0
\(277\) 4.24264i 0.254916i −0.991844 0.127458i \(-0.959318\pi\)
0.991844 0.127458i \(-0.0406817\pi\)
\(278\) 17.8885i 1.07288i
\(279\) 13.4164i 0.803219i
\(280\) −10.0000 + 13.4164i −0.597614 + 0.801784i
\(281\) 24.0416i 1.43420i −0.696969 0.717102i \(-0.745468\pi\)
0.696969 0.717102i \(-0.254532\pi\)
\(282\) 14.1421i 0.842152i
\(283\) −6.32456 −0.375956 −0.187978 0.982173i \(-0.560193\pi\)
−0.187978 + 0.982173i \(0.560193\pi\)
\(284\) 0 0
\(285\) 15.8114 0.936586
\(286\) −12.6491 + 26.8328i −0.747958 + 1.58666i
\(287\) 15.0000 20.1246i 0.885422 1.18792i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 4.47214i 0.262613i
\(291\) −15.0000 −0.879316
\(292\) 0 0
\(293\) 9.48683 0.554227 0.277113 0.960837i \(-0.410622\pi\)
0.277113 + 0.960837i \(0.410622\pi\)
\(294\) −6.32456 21.2132i −0.368856 1.23718i
\(295\) −5.00000 −0.291111
\(296\) 2.82843i 0.164399i
\(297\) −3.16228 + 6.70820i −0.183494 + 0.389249i
\(298\) 28.0000 1.62200
\(299\) 18.9737 1.09728
\(300\) 0 0
\(301\) −9.00000 6.70820i −0.518751 0.386654i
\(302\) 24.0000 1.38104
\(303\) 21.2132i 1.21867i
\(304\) −12.6491 −0.725476
\(305\) 7.07107i 0.404888i
\(306\) 0 0
\(307\) −6.32456 −0.360961 −0.180481 0.983579i \(-0.557765\pi\)
−0.180481 + 0.983579i \(0.557765\pi\)
\(308\) 0 0
\(309\) −30.0000 −1.70664
\(310\) 21.2132i 1.20483i
\(311\) 4.47214i 0.253592i −0.991929 0.126796i \(-0.959531\pi\)
0.991929 0.126796i \(-0.0404693\pi\)
\(312\) −40.0000 −2.26455
\(313\) 6.70820i 0.379170i −0.981864 0.189585i \(-0.939286\pi\)
0.981864 0.189585i \(-0.0607143\pi\)
\(314\) −9.48683 −0.535373
\(315\) −9.48683 7.07107i −0.534522 0.398410i
\(316\) 0 0
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) 0 0
\(319\) 2.00000 4.24264i 0.111979 0.237542i
\(320\) 17.8885i 1.00000i
\(321\) −6.32456 −0.353002
\(322\) −9.00000 6.70820i −0.501550 0.373834i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 14.1421i 0.783260i
\(327\) 28.4605 1.57387
\(328\) 26.8328i 1.48159i
\(329\) −9.48683 7.07107i −0.523026 0.389841i
\(330\) 10.0000 21.2132i 0.550482 1.16775i
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 13.4164i 0.734113i
\(335\) 24.5967i 1.34386i
\(336\) 18.9737 + 14.1421i 1.03510 + 0.771517i
\(337\) 16.9706i 0.924445i 0.886764 + 0.462223i \(0.152948\pi\)
−0.886764 + 0.462223i \(0.847052\pi\)
\(338\) 38.1838i 2.07692i
\(339\) 6.70820i 0.364340i
\(340\) 0 0
\(341\) 9.48683 20.1246i 0.513741 1.08981i
\(342\) 8.94427i 0.483651i
\(343\) −17.3925 6.36396i −0.939108 0.343622i
\(344\) 12.0000 0.646997
\(345\) −15.0000 −0.807573
\(346\) 13.4164i 0.721271i
\(347\) 1.41421i 0.0759190i 0.999279 + 0.0379595i \(0.0120858\pi\)
−0.999279 + 0.0379595i \(0.987914\pi\)
\(348\) 0 0
\(349\) 12.6491 0.677091 0.338546 0.940950i \(-0.390065\pi\)
0.338546 + 0.940950i \(0.390065\pi\)
\(350\) 0 0
\(351\) 14.1421i 0.754851i
\(352\) 0 0
\(353\) 15.6525i 0.833097i 0.909113 + 0.416549i \(0.136760\pi\)
−0.909113 + 0.416549i \(0.863240\pi\)
\(354\) 7.07107i 0.375823i
\(355\) 20.1246i 1.06810i
\(356\) 0 0
\(357\) 0 0
\(358\) 12.7279i 0.672692i
\(359\) 5.65685i 0.298557i 0.988795 + 0.149279i \(0.0476951\pi\)
−0.988795 + 0.149279i \(0.952305\pi\)
\(360\) 12.6491 0.666667
\(361\) −9.00000 −0.473684
\(362\) −28.4605 −1.49585
\(363\) 18.9737 15.6525i 0.995859 0.821542i
\(364\) 0 0
\(365\) 7.07107i 0.370117i
\(366\) 10.0000 0.522708
\(367\) 33.5410i 1.75083i 0.483375 + 0.875413i \(0.339411\pi\)
−0.483375 + 0.875413i \(0.660589\pi\)
\(368\) 12.0000 0.625543
\(369\) −18.9737 −0.987730
\(370\) 3.16228 0.164399
\(371\) 0 0
\(372\) 0 0
\(373\) 4.24264i 0.219676i 0.993950 + 0.109838i \(0.0350331\pi\)
−0.993950 + 0.109838i \(0.964967\pi\)
\(374\) 0 0
\(375\) 25.0000 1.29099
\(376\) 12.6491 0.652328
\(377\) 8.94427i 0.460653i
\(378\) 5.00000 6.70820i 0.257172 0.345033i
\(379\) 17.0000 0.873231 0.436616 0.899648i \(-0.356177\pi\)
0.436616 + 0.899648i \(0.356177\pi\)
\(380\) 0 0
\(381\) 28.4605 1.45808
\(382\) 4.24264i 0.217072i
\(383\) 24.5967i 1.25684i −0.777876 0.628418i \(-0.783703\pi\)
0.777876 0.628418i \(-0.216297\pi\)
\(384\) −25.2982 −1.29099
\(385\) −9.23025 17.3148i −0.470417 0.882444i
\(386\) −6.00000 −0.305392
\(387\) 8.48528i 0.431331i
\(388\) 0 0
\(389\) 3.00000 0.152106 0.0760530 0.997104i \(-0.475768\pi\)
0.0760530 + 0.997104i \(0.475768\pi\)
\(390\) 44.7214i 2.26455i
\(391\) 0 0
\(392\) 18.9737 5.65685i 0.958315 0.285714i
\(393\) 42.4264i 2.14013i
\(394\) −8.00000 −0.403034
\(395\) −18.9737 −0.954669
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) −15.0000 11.1803i −0.750939 0.559717i
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 34.7851 1.73492
\(403\) 42.4264i 2.11341i
\(404\) 0 0
\(405\) 24.5967i 1.22222i
\(406\) −3.16228 + 4.24264i −0.156941 + 0.210559i
\(407\) 3.00000 + 1.41421i 0.148704 + 0.0701000i
\(408\) 0 0
\(409\) 12.6491 0.625458 0.312729 0.949842i \(-0.398757\pi\)
0.312729 + 0.949842i \(0.398757\pi\)
\(410\) −30.0000 −1.48159
\(411\) 33.5410i 1.65446i
\(412\) 0 0
\(413\) 4.74342 + 3.53553i 0.233408 + 0.173972i
\(414\) 8.48528i 0.417029i
\(415\) 21.2132i 1.04132i
\(416\) 0 0
\(417\) 28.2843i 1.38509i
\(418\) 6.32456 13.4164i 0.309344 0.656218i
\(419\) 22.3607i 1.09239i 0.837658 + 0.546195i \(0.183924\pi\)
−0.837658 + 0.546195i \(0.816076\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 30.0000 1.46038
\(423\) 8.94427i 0.434885i
\(424\) 0 0
\(425\) 0 0
\(426\) 28.4605 1.37892
\(427\) 5.00000 6.70820i 0.241967 0.324633i
\(428\) 0 0
\(429\) 20.0000 42.4264i 0.965609 2.04837i
\(430\) 13.4164i 0.646997i
\(431\) 35.3553i 1.70301i 0.524349 + 0.851503i \(0.324309\pi\)
−0.524349 + 0.851503i \(0.675691\pi\)
\(432\) 8.94427i 0.430331i
\(433\) 20.1246i 0.967127i −0.875309 0.483564i \(-0.839342\pi\)
0.875309 0.483564i \(-0.160658\pi\)
\(434\) −15.0000 + 20.1246i −0.720023 + 0.966012i
\(435\) 7.07107i 0.339032i
\(436\) 0 0
\(437\) −9.48683 −0.453817
\(438\) 10.0000 0.477818
\(439\) −15.8114 −0.754636 −0.377318 0.926084i \(-0.623154\pi\)
−0.377318 + 0.926084i \(0.623154\pi\)
\(440\) 18.9737 + 8.94427i 0.904534 + 0.426401i
\(441\) 4.00000 + 13.4164i 0.190476 + 0.638877i
\(442\) 0 0
\(443\) −15.0000 −0.712672 −0.356336 0.934358i \(-0.615974\pi\)
−0.356336 + 0.934358i \(0.615974\pi\)
\(444\) 0 0
\(445\) −5.00000 −0.237023
\(446\) 9.48683 0.449215
\(447\) −44.2719 −2.09399
\(448\) −12.6491 + 16.9706i −0.597614 + 0.801784i
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −28.4605 13.4164i −1.34015 0.631754i
\(452\) 0 0
\(453\) −37.9473 −1.78292
\(454\) 13.4164i 0.629663i
\(455\) −30.0000 22.3607i −1.40642 1.04828i
\(456\) 20.0000 0.936586
\(457\) 8.48528i 0.396925i 0.980109 + 0.198462i \(0.0635948\pi\)
−0.980109 + 0.198462i \(0.936405\pi\)
\(458\) −9.48683 −0.443291
\(459\) 0 0
\(460\) 0 0
\(461\) −9.48683 −0.441846 −0.220923 0.975291i \(-0.570907\pi\)
−0.220923 + 0.975291i \(0.570907\pi\)
\(462\) −24.4868 + 13.0535i −1.13923 + 0.607306i
\(463\) −25.0000 −1.16185 −0.580924 0.813958i \(-0.697309\pi\)
−0.580924 + 0.813958i \(0.697309\pi\)
\(464\) 5.65685i 0.262613i
\(465\) 33.5410i 1.55543i
\(466\) −26.0000 −1.20443
\(467\) 11.1803i 0.517364i −0.965962 0.258682i \(-0.916712\pi\)
0.965962 0.258682i \(-0.0832882\pi\)
\(468\) 0 0
\(469\) 17.3925 23.3345i 0.803112 1.07749i
\(470\) 14.1421i 0.652328i
\(471\) 15.0000 0.691164
\(472\) −6.32456 −0.291111
\(473\) −6.00000 + 12.7279i −0.275880 + 0.585230i
\(474\) 26.8328i 1.23247i
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 10.0000 0.457389
\(479\) 18.9737 0.866929 0.433464 0.901171i \(-0.357291\pi\)
0.433464 + 0.901171i \(0.357291\pi\)
\(480\) 0 0
\(481\) 6.32456 0.288375
\(482\) 17.8885i 0.814801i
\(483\) 14.2302 + 10.6066i 0.647499 + 0.482617i
\(484\) 0 0
\(485\) 15.0000 0.681115
\(486\) −25.2982 −1.14755
\(487\) 29.0000 1.31412 0.657058 0.753840i \(-0.271801\pi\)
0.657058 + 0.753840i \(0.271801\pi\)
\(488\) 8.94427i 0.404888i
\(489\) 22.3607i 1.01118i
\(490\) 6.32456 + 21.2132i 0.285714 + 0.958315i
\(491\) 32.5269i 1.46792i −0.679193 0.733959i \(-0.737670\pi\)
0.679193 0.733959i \(-0.262330\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 28.2843i 1.27257i
\(495\) −6.32456 + 13.4164i −0.284268 + 0.603023i
\(496\) 26.8328i 1.20483i
\(497\) 14.2302 19.0919i 0.638314 0.856388i
\(498\) −30.0000 −1.34433
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 21.2132i 0.947736i
\(502\) 15.8114 0.705697
\(503\) 28.4605 1.26899 0.634495 0.772927i \(-0.281208\pi\)
0.634495 + 0.772927i \(0.281208\pi\)
\(504\) −12.0000 8.94427i −0.534522 0.398410i
\(505\) 21.2132i 0.943975i
\(506\) −6.00000 + 12.7279i −0.266733 + 0.565825i
\(507\) 60.3738i 2.68130i
\(508\) 0 0
\(509\) 15.6525i 0.693784i 0.937905 + 0.346892i \(0.112763\pi\)
−0.937905 + 0.346892i \(0.887237\pi\)
\(510\) 0 0
\(511\) 5.00000 6.70820i 0.221187 0.296753i
\(512\) 22.6274i 1.00000i
\(513\) 7.07107i 0.312195i
\(514\) −31.6228 −1.39482
\(515\) 30.0000 1.32196
\(516\) 0 0
\(517\) −6.32456 + 13.4164i −0.278154 + 0.590053i
\(518\) −3.00000 2.23607i −0.131812 0.0982472i
\(519\) 21.2132i 0.931156i
\(520\) 40.0000 1.75412
\(521\) 11.1803i 0.489820i −0.969546 0.244910i \(-0.921242\pi\)
0.969546 0.244910i \(-0.0787583\pi\)
\(522\) 4.00000 0.175075
\(523\) 31.6228 1.38277 0.691384 0.722488i \(-0.257001\pi\)
0.691384 + 0.722488i \(0.257001\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 12.6491 26.8328i 0.550482 1.16775i
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 4.47214i 0.194074i
\(532\) 0 0
\(533\) −60.0000 −2.59889
\(534\) 7.07107i 0.305995i
\(535\) 6.32456 0.273434
\(536\) 31.1127i 1.34386i
\(537\) 20.1246i 0.868441i
\(538\) 25.2982 1.09068
\(539\) −3.48683 + 22.9530i −0.150189 + 0.988657i
\(540\) 0 0
\(541\) 25.4558i 1.09443i −0.836991 0.547216i \(-0.815688\pi\)
0.836991 0.547216i \(-0.184312\pi\)
\(542\) 35.7771i 1.53676i
\(543\) 45.0000 1.93113
\(544\) 0 0
\(545\) −28.4605 −1.21911
\(546\) −31.6228 + 42.4264i −1.35333 + 1.81568i
\(547\) 8.48528i 0.362804i 0.983409 + 0.181402i \(0.0580636\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 0 0
\(549\) −6.32456 −0.269925
\(550\) 0 0
\(551\) 4.47214i 0.190519i
\(552\) −18.9737 −0.807573
\(553\) 18.0000 + 13.4164i 0.765438 + 0.570524i
\(554\) 6.00000 0.254916
\(555\) −5.00000 −0.212238
\(556\) 0 0
\(557\) 22.6274i 0.958754i 0.877609 + 0.479377i \(0.159137\pi\)
−0.877609 + 0.479377i \(0.840863\pi\)
\(558\) 18.9737 0.803219
\(559\) 26.8328i 1.13491i
\(560\) −18.9737 14.1421i −0.801784 0.597614i
\(561\) 0 0
\(562\) 34.0000 1.43420
\(563\) 37.9473 1.59929 0.799645 0.600473i \(-0.205021\pi\)
0.799645 + 0.600473i \(0.205021\pi\)
\(564\) 0 0
\(565\) 6.70820i 0.282216i
\(566\) 8.94427i 0.375956i
\(567\) −17.3925 + 23.3345i −0.730417 + 0.979958i
\(568\) 25.4558i 1.06810i
\(569\) 15.5563i 0.652156i −0.945343 0.326078i \(-0.894273\pi\)
0.945343 0.326078i \(-0.105727\pi\)
\(570\) 22.3607i 0.936586i
\(571\) 16.9706i 0.710196i 0.934829 + 0.355098i \(0.115552\pi\)
−0.934829 + 0.355098i \(0.884448\pi\)
\(572\) 0 0
\(573\) 6.70820i 0.280239i
\(574\) 28.4605 + 21.2132i 1.18792 + 0.885422i
\(575\) 0 0
\(576\) 16.0000 0.666667
\(577\) 20.1246i 0.837799i −0.908033 0.418899i \(-0.862416\pi\)
0.908033 0.418899i \(-0.137584\pi\)
\(578\) 24.0416i 1.00000i
\(579\) 9.48683 0.394259
\(580\) 0 0
\(581\) −15.0000 + 20.1246i −0.622305 + 0.834910i
\(582\) 21.2132i 0.879316i
\(583\) 0 0
\(584\) 8.94427i 0.370117i
\(585\) 28.2843i 1.16941i
\(586\) 13.4164i 0.554227i
\(587\) 4.47214i 0.184585i −0.995732 0.0922924i \(-0.970581\pi\)
0.995732 0.0922924i \(-0.0294195\pi\)
\(588\) 0 0
\(589\) 21.2132i 0.874075i
\(590\) 7.07107i 0.291111i
\(591\) 12.6491 0.520315
\(592\) 4.00000 0.164399
\(593\) 28.4605 1.16873 0.584366 0.811490i \(-0.301343\pi\)
0.584366 + 0.811490i \(0.301343\pi\)
\(594\) −9.48683 4.47214i −0.389249 0.183494i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 26.8328i 1.09728i
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) −15.8114 −0.644960 −0.322480 0.946576i \(-0.604517\pi\)
−0.322480 + 0.946576i \(0.604517\pi\)
\(602\) 9.48683 12.7279i 0.386654 0.518751i
\(603\) −22.0000 −0.895909
\(604\) 0 0
\(605\) −18.9737 + 15.6525i −0.771389 + 0.636364i
\(606\) −30.0000 −1.21867
\(607\) −6.32456 −0.256706 −0.128353 0.991729i \(-0.540969\pi\)
−0.128353 + 0.991729i \(0.540969\pi\)
\(608\) 0 0
\(609\) 5.00000 6.70820i 0.202610 0.271830i
\(610\) −10.0000 −0.404888
\(611\) 28.2843i 1.14426i
\(612\) 0 0
\(613\) 25.4558i 1.02815i 0.857745 + 0.514076i \(0.171865\pi\)
−0.857745 + 0.514076i \(0.828135\pi\)
\(614\) 8.94427i 0.360961i
\(615\) 47.4342 1.91273
\(616\) −11.6754 21.9017i −0.470417 0.882444i
\(617\) 24.0000 0.966204 0.483102 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) 42.4264i 1.70664i
\(619\) 46.9574i 1.88738i −0.330833 0.943689i \(-0.607330\pi\)
0.330833 0.943689i \(-0.392670\pi\)
\(620\) 0 0
\(621\) 6.70820i 0.269191i
\(622\) 6.32456 0.253592
\(623\) 4.74342 + 3.53553i 0.190041 + 0.141648i
\(624\) 56.5685i 2.26455i
\(625\) −25.0000 −1.00000
\(626\) 9.48683 0.379170
\(627\) −10.0000 + 21.2132i −0.399362 + 0.847174i
\(628\) 0 0
\(629\) 0 0
\(630\) 10.0000 13.4164i 0.398410 0.534522i
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) −24.0000 −0.954669
\(633\) −47.4342 −1.88534
\(634\) 29.6985i 1.17948i
\(635\) −28.4605 −1.12942
\(636\) 0 0
\(637\) 12.6491 + 42.4264i 0.501176 + 1.68100i
\(638\) 6.00000 + 2.82843i 0.237542 + 0.111979i
\(639\) −18.0000 −0.712069
\(640\) 25.2982 1.00000
\(641\) 21.0000 0.829450 0.414725 0.909947i \(-0.363878\pi\)
0.414725 + 0.909947i \(0.363878\pi\)
\(642\) 8.94427i 0.353002i
\(643\) 6.70820i 0.264546i 0.991213 + 0.132273i \(0.0422275\pi\)
−0.991213 + 0.132273i \(0.957772\pi\)
\(644\) 0 0
\(645\) 21.2132i 0.835269i
\(646\) 0 0
\(647\) 29.0689i 1.14282i 0.820666 + 0.571408i \(0.193603\pi\)
−0.820666 + 0.571408i \(0.806397\pi\)
\(648\) 31.1127i 1.22222i
\(649\) 3.16228 6.70820i 0.124130 0.263320i
\(650\) 0 0
\(651\) 23.7171 31.8198i 0.929546 1.24712i
\(652\) 0 0
\(653\) 15.0000 0.586995 0.293498 0.955960i \(-0.405181\pi\)
0.293498 + 0.955960i \(0.405181\pi\)
\(654\) 40.2492i 1.57387i
\(655\) 42.4264i 1.65774i
\(656\) −37.9473 −1.48159
\(657\) −6.32456 −0.246744
\(658\) 10.0000 13.4164i 0.389841 0.523026i
\(659\) 14.1421i 0.550899i 0.961315 + 0.275450i \(0.0888267\pi\)
−0.961315 + 0.275450i \(0.911173\pi\)
\(660\) 0 0
\(661\) 46.9574i 1.82643i 0.407476 + 0.913216i \(0.366409\pi\)
−0.407476 + 0.913216i \(0.633591\pi\)
\(662\) 26.8701i 1.04433i
\(663\) 0 0
\(664\) 26.8328i 1.04132i
\(665\) 15.0000 + 11.1803i 0.581675 + 0.433555i
\(666\) 2.82843i 0.109599i
\(667\) 4.24264i 0.164276i
\(668\) 0 0
\(669\) −15.0000 −0.579934
\(670\) −34.7851 −1.34386
\(671\) −9.48683 4.47214i −0.366235 0.172645i
\(672\) 0 0
\(673\) 29.6985i 1.14479i −0.819977 0.572396i \(-0.806014\pi\)
0.819977 0.572396i \(-0.193986\pi\)
\(674\) −24.0000 −0.924445
\(675\) 0 0
\(676\) 0 0
\(677\) −37.9473 −1.45843 −0.729217 0.684282i \(-0.760116\pi\)
−0.729217 + 0.684282i \(0.760116\pi\)
\(678\) −9.48683 −0.364340
\(679\) −14.2302 10.6066i −0.546107 0.407044i
\(680\) 0 0
\(681\) 21.2132i 0.812892i
\(682\) 28.4605 + 13.4164i 1.08981 + 0.513741i
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 0 0
\(685\) 33.5410i 1.28154i
\(686\) 9.00000 24.5967i 0.343622 0.939108i
\(687\) 15.0000 0.572286
\(688\) 16.9706i 0.646997i
\(689\) 0 0
\(690\) 21.2132i 0.807573i
\(691\) 6.70820i 0.255192i −0.991826 0.127596i \(-0.959274\pi\)
0.991826 0.127596i \(-0.0407261\pi\)
\(692\) 0 0
\(693\) 15.4868 8.25579i 0.588296 0.313611i
\(694\) −2.00000 −0.0759190
\(695\) 28.2843i 1.07288i
\(696\) 8.94427i 0.339032i
\(697\) 0 0
\(698\) 17.8885i 0.677091i
\(699\) 41.1096 1.55491
\(700\) 0 0
\(701\) 7.07107i 0.267071i −0.991044 0.133535i \(-0.957367\pi\)
0.991044 0.133535i \(-0.0426329\pi\)
\(702\) −20.0000 −0.754851
\(703\) −3.16228 −0.119268
\(704\) 24.0000 + 11.3137i 0.904534 + 0.426401i
\(705\) 22.3607i 0.842152i
\(706\) −22.1359 −0.833097
\(707\) −15.0000 + 20.1246i −0.564133 + 0.756864i
\(708\) 0 0
\(709\) 41.0000 1.53979 0.769894 0.638172i \(-0.220309\pi\)
0.769894 + 0.638172i \(0.220309\pi\)
\(710\) −28.4605 −1.06810
\(711\) 16.9706i 0.636446i
\(712\) −6.32456 −0.237023
\(713\) 20.1246i 0.753673i
\(714\) 0 0
\(715\) −20.0000 + 42.4264i −0.747958 + 1.58666i
\(716\) 0 0
\(717\) −15.8114 −0.590487
\(718\) −8.00000 −0.298557
\(719\) 51.4296i 1.91800i −0.283408 0.959000i \(-0.591465\pi\)
0.283408 0.959000i \(-0.408535\pi\)
\(720\) 17.8885i 0.666667i
\(721\) −28.4605 21.2132i −1.05992 0.790021i
\(722\) 12.7279i 0.473684i
\(723\) 28.2843i 1.05190i
\(724\) 0 0
\(725\) 0 0
\(726\) 22.1359 + 26.8328i 0.821542 + 0.995859i
\(727\) 33.5410i 1.24397i 0.783030 + 0.621984i \(0.213673\pi\)
−0.783030 + 0.621984i \(0.786327\pi\)
\(728\) −37.9473 28.2843i −1.40642 1.04828i
\(729\) 7.00000 0.259259
\(730\) −10.0000 −0.370117
\(731\) 0 0
\(732\) 0 0
\(733\) 12.6491 0.467206 0.233603 0.972332i \(-0.424948\pi\)
0.233603 + 0.972332i \(0.424948\pi\)
\(734\) −47.4342 −1.75083
\(735\) −10.0000 33.5410i −0.368856 1.23718i
\(736\) 0 0
\(737\) −33.0000 15.5563i −1.21557 0.573025i
\(738\) 26.8328i 0.987730i
\(739\) 50.9117i 1.87282i 0.350912 + 0.936408i \(0.385872\pi\)
−0.350912 + 0.936408i \(0.614128\pi\)
\(740\) 0 0
\(741\) 44.7214i 1.64288i
\(742\) 0 0
\(743\) 19.7990i 0.726354i −0.931720 0.363177i \(-0.881692\pi\)
0.931720 0.363177i \(-0.118308\pi\)
\(744\) 42.4264i 1.55543i
\(745\) 44.2719 1.62200
\(746\) −6.00000 −0.219676
\(747\) 18.9737 0.694210
\(748\) 0 0
\(749\) −6.00000 4.47214i −0.219235 0.163408i
\(750\) 35.3553i 1.29099i
\(751\) −7.00000 −0.255434 −0.127717 0.991811i \(-0.540765\pi\)
−0.127717 + 0.991811i \(0.540765\pi\)
\(752\) 17.8885i 0.652328i
\(753\) −25.0000 −0.911051
\(754\) 12.6491 0.460653
\(755\) 37.9473 1.38104
\(756\) 0 0
\(757\) 44.0000 1.59921 0.799604 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) 24.0416i 0.873231i
\(759\) 9.48683 20.1246i 0.344350 0.730477i
\(760\) −20.0000 −0.725476
\(761\) 37.9473 1.37559 0.687795 0.725905i \(-0.258579\pi\)
0.687795 + 0.725905i \(0.258579\pi\)
\(762\) 40.2492i 1.45808i
\(763\) 27.0000 + 20.1246i 0.977466 + 0.728560i
\(764\) 0 0
\(765\) 0 0
\(766\) 34.7851 1.25684
\(767\) 14.1421i 0.510643i
\(768\) 0 0
\(769\) −6.32456 −0.228069 −0.114035 0.993477i \(-0.536377\pi\)
−0.114035 + 0.993477i \(0.536377\pi\)
\(770\) 24.4868 13.0535i 0.882444 0.470417i
\(771\) 50.0000 1.80071
\(772\) 0 0
\(773\) 8.94427i 0.321703i 0.986979 + 0.160852i \(0.0514240\pi\)
−0.986979 + 0.160852i \(0.948576\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 18.9737 0.681115
\(777\) 4.74342 + 3.53553i 0.170169 + 0.126837i
\(778\) 4.24264i 0.152106i
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) −27.0000 12.7279i −0.966136 0.455441i
\(782\) 0 0
\(783\) 3.16228 0.113011
\(784\) 8.00000 + 26.8328i 0.285714 + 0.958315i
\(785\) −15.0000 −0.535373
\(786\) −60.0000 −2.14013
\(787\) 22.1359 0.789061 0.394531 0.918883i \(-0.370907\pi\)
0.394531 + 0.918883i \(0.370907\pi\)
\(788\) 0 0
\(789\) −25.2982 −0.900641
\(790\) 26.8328i 0.954669i
\(791\) −4.74342 + 6.36396i −0.168656 + 0.226276i
\(792\) −8.00000 + 16.9706i −0.284268 + 0.603023i
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.0132i 1.34650i −0.739417 0.673248i \(-0.764899\pi\)
0.739417 0.673248i \(-0.235101\pi\)
\(798\) 15.8114 21.2132i 0.559717 0.750939i
\(799\) 0 0
\(800\) 0 0
\(801\) 4.47214i 0.158015i
\(802\) 33.9411i 1.19850i
\(803\) −9.48683 4.47214i −0.334783 0.157818i
\(804\) 0 0
\(805\) −14.2302 10.6066i −0.501550 0.373834i
\(806\) 60.0000 2.11341
\(807\) −40.0000 −1.40807
\(808\) 26.8328i 0.943975i
\(809\) 41.0122i 1.44191i −0.692981 0.720956i \(-0.743703\pi\)
0.692981 0.720956i \(-0.256297\pi\)
\(810\) 34.7851 1.22222
\(811\) −25.2982 −0.888341 −0.444170 0.895942i \(-0.646502\pi\)
−0.444170 + 0.895942i \(0.646502\pi\)
\(812\) 0 0
\(813\) 56.5685i 1.98395i
\(814\) −2.00000 + 4.24264i −0.0701000 + 0.148704i
\(815\) 22.3607i 0.783260i
\(816\) 0 0
\(817\) 13.4164i 0.469381i
\(818\) 17.8885i 0.625458i
\(819\) 20.0000 26.8328i 0.698857 0.937614i
\(820\) 0 0
\(821\) 14.1421i 0.493564i 0.969071 + 0.246782i \(0.0793731\pi\)
−0.969071 + 0.246782i \(0.920627\pi\)
\(822\) −47.4342 −1.65446
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) 37.9473 1.32196
\(825\) 0 0
\(826\) −5.00000 + 6.70820i −0.173972 + 0.233408i
\(827\) 22.6274i 0.786832i 0.919360 + 0.393416i \(0.128707\pi\)
−0.919360 + 0.393416i \(0.871293\pi\)
\(828\) 0 0
\(829\) 20.1246i 0.698957i −0.936944 0.349478i \(-0.886359\pi\)
0.936944 0.349478i \(-0.113641\pi\)
\(830\) 30.0000 1.04132
\(831\) −9.48683 −0.329095
\(832\) 50.5964 1.75412
\(833\) 0 0
\(834\) 40.0000 1.38509
\(835\) 21.2132i 0.734113i
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) −31.6228 −1.09239
\(839\) 15.6525i 0.540383i 0.962807 + 0.270192i \(0.0870871\pi\)
−0.962807 + 0.270192i \(0.912913\pi\)
\(840\) 30.0000 + 22.3607i 1.03510 + 0.771517i
\(841\) 27.0000 0.931034
\(842\) 5.65685i 0.194948i
\(843\) −53.7587 −1.85155
\(844\) 0 0
\(845\) 60.3738i 2.07692i
\(846\) −12.6491 −0.434885
\(847\) 29.0680 1.43283i 0.998787 0.0492328i
\(848\) 0 0
\(849\) 14.1421i 0.485357i
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) −25.2982 −0.866195 −0.433097 0.901347i \(-0.642579\pi\)
−0.433097 + 0.901347i \(0.642579\pi\)
\(854\) 9.48683 + 7.07107i 0.324633 + 0.241967i
\(855\) 14.1421i 0.483651i
\(856\) 8.00000 0.273434
\(857\) 18.9737 0.648128 0.324064 0.946035i \(-0.394951\pi\)
0.324064 + 0.946035i \(0.394951\pi\)
\(858\) 60.0000 + 28.2843i 2.04837 + 0.965609i
\(859\) 6.70820i 0.228881i 0.993430 + 0.114440i \(0.0365075\pi\)
−0.993430 + 0.114440i \(0.963492\pi\)
\(860\) 0 0
\(861\) −45.0000 33.5410i −1.53360 1.14307i
\(862\) −50.0000 −1.70301
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 0 0
\(865\) 21.2132i 0.721271i
\(866\) 28.4605 0.967127
\(867\) 38.0132i 1.29099i
\(868\) 0 0
\(869\) 12.0000 25.4558i 0.407072 0.863530i
\(870\) −10.0000 −0.339032
\(871\) −69.5701 −2.35729
\(872\) −36.0000 −1.21911
\(873\) 13.4164i 0.454077i
\(874\) 13.4164i 0.453817i
\(875\) 23.7171 + 17.6777i 0.801784 + 0.597614i
\(876\) 0 0
\(877\) 4.24264i 0.143264i 0.997431 + 0.0716319i \(0.0228207\pi\)
−0.997431 + 0.0716319i \(0.977179\pi\)
\(878\) 22.3607i 0.754636i
\(879\) 21.2132i 0.715504i
\(880\) −12.6491 + 26.8328i −0.426401 + 0.904534i
\(881\) 51.4296i 1.73271i −0.499432 0.866353i \(-0.666458\pi\)
0.499432 0.866353i \(-0.333542\pi\)
\(882\) −18.9737 + 5.65685i −0.638877 + 0.190476i
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 11.1803i 0.375823i
\(886\) 21.2132i 0.712672i
\(887\) −18.9737 −0.637073 −0.318537 0.947911i \(-0.603191\pi\)
−0.318537 + 0.947911i \(0.603191\pi\)
\(888\) −6.32456 −0.212238
\(889\) 27.0000 + 20.1246i 0.905551 + 0.674958i
\(890\) 7.07107i 0.237023i
\(891\) 33.0000 + 15.5563i 1.10554 + 0.521157i
\(892\) 0 0
\(893\) 14.1421i 0.473249i
\(894\) 62.6099i 2.09399i
\(895\) 20.1246i 0.672692i
\(896\) −24.0000 17.8885i −0.801784 0.597614i
\(897\) 42.4264i 1.41658i
\(898\) 38.1838i 1.27421i
\(899\) −9.48683 −0.316404
\(900\) 0 0
\(901\) 0 0
\(902\) 18.9737 40.2492i 0.631754 1.34015i
\(903\) −15.0000 + 20.1246i −0.499169 + 0.669705i
\(904\) 8.48528i 0.282216i
\(905\) −45.0000 −1.49585
\(906\) 53.6656i 1.78292i
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 0 0
\(909\) 18.9737 0.629317
\(910\) 31.6228 42.4264i 1.04828 1.40642i
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 28.2843i 0.936586i
\(913\) 28.4605 + 13.4164i 0.941905 + 0.444018i
\(914\) −12.0000 −0.396925
\(915\) 15.8114 0.522708
\(916\) 0 0
\(917\) −30.0000 + 40.2492i −0.990687 + 1.32915i
\(918\) 0 0
\(919\) 12.7279i 0.419855i −0.977717 0.209928i \(-0.932677\pi\)
0.977717 0.209928i \(-0.0673229\pi\)
\(920\) 18.9737 0.625543
\(921\) 14.1421i 0.465999i
\(922\) 13.4164i 0.441846i
\(923\) −56.9210 −1.87358
\(924\) 0 0
\(925\) 0 0
\(926\) 35.3553i 1.16185i
\(927\) 26.8328i 0.881305i
\(928\) 0 0
\(929\) 35.7771i 1.17381i 0.809656 + 0.586904i \(0.199653\pi\)
−0.809656 + 0.586904i \(0.800347\pi\)
\(930\) −47.4342 −1.55543
\(931\) −6.32456 21.2132i −0.207279 0.695235i
\(932\) 0 0
\(933\) −10.0000 −0.327385
\(934\) 15.8114 0.517364
\(935\) 0 0
\(936\) 35.7771i 1.16941i
\(937\) 50.5964 1.65291 0.826457 0.563000i \(-0.190353\pi\)
0.826457 + 0.563000i \(0.190353\pi\)
\(938\) 33.0000 + 24.5967i 1.07749 + 0.803112i
\(939\) −15.0000 −0.489506
\(940\) 0 0
\(941\) 37.9473 1.23705 0.618524 0.785766i \(-0.287731\pi\)
0.618524 + 0.785766i \(0.287731\pi\)
\(942\) 21.2132i 0.691164i
\(943\) −28.4605 −0.926801
\(944\) 8.94427i 0.291111i
\(945\) 7.90569 10.6066i 0.257172 0.345033i
\(946\) −18.0000 8.48528i −0.585230 0.275880i
\(947\) −15.0000 −0.487435 −0.243717 0.969846i \(-0.578367\pi\)
−0.243717 + 0.969846i \(0.578367\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 46.9574i 1.52270i
\(952\) 0 0
\(953\) 9.89949i 0.320676i 0.987062 + 0.160338i \(0.0512584\pi\)
−0.987062 + 0.160338i \(0.948742\pi\)
\(954\) 0 0
\(955\) 6.70820i 0.217072i
\(956\) 0 0
\(957\) −9.48683 4.47214i −0.306666 0.144564i
\(958\) 26.8328i 0.866929i
\(959\) −23.7171 + 31.8198i −0.765865 + 1.02752i
\(960\) −40.0000 −1.29099
\(961\) −14.0000 −0.451613
\(962\) 8.94427i 0.288375i
\(963\) 5.65685i 0.182290i
\(964\) 0 0
\(965\) −9.48683 −0.305392
\(966\) −15.0000 + 20.1246i −0.482617 + 0.647499i
\(967\) 46.6690i 1.50078i −0.660998 0.750388i \(-0.729867\pi\)
0.660998 0.750388i \(-0.270133\pi\)
\(968\) −24.0000 + 19.7990i −0.771389 + 0.636364i
\(969\) 0 0
\(970\) 21.2132i 0.681115i
\(971\) 29.0689i 0.932865i 0.884557 + 0.466432i \(0.154461\pi\)
−0.884557 + 0.466432i \(0.845539\pi\)
\(972\) 0 0
\(973\) 20.0000 26.8328i 0.641171 0.860221i
\(974\) 41.0122i 1.31412i
\(975\) 0 0
\(976\) −12.6491 −0.404888
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) −31.6228 −1.01118
\(979\) 3.16228 6.70820i 0.101067 0.214395i
\(980\) 0 0
\(981\) 25.4558i 0.812743i
\(982\) 46.0000 1.46792
\(983\) 29.0689i 0.927153i 0.886057 + 0.463577i \(0.153434\pi\)
−0.886057 + 0.463577i \(0.846566\pi\)
\(984\) 60.0000 1.91273
\(985\) −12.6491 −0.403034
\(986\) 0 0
\(987\) −15.8114 + 21.2132i −0.503282 + 0.675224i
\(988\) 0 0
\(989\) 12.7279i 0.404724i
\(990\) −18.9737 8.94427i −0.603023 0.284268i
\(991\) −46.0000 −1.46124 −0.730619 0.682785i \(-0.760768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(992\) 0 0
\(993\) 42.4853i 1.34823i
\(994\) 27.0000 + 20.1246i 0.856388 + 0.638314i
\(995\) 0 0
\(996\) 0 0
\(997\) −53.7587 −1.70256 −0.851278 0.524715i \(-0.824172\pi\)
−0.851278 + 0.524715i \(0.824172\pi\)
\(998\) 22.6274i 0.716258i
\(999\) 2.23607i 0.0707461i
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 77.2.b.b.76.3 yes 4
3.2 odd 2 693.2.c.b.307.1 4
4.3 odd 2 1232.2.e.c.769.3 4
7.2 even 3 539.2.i.b.472.4 8
7.3 odd 6 539.2.i.b.362.2 8
7.4 even 3 539.2.i.b.362.1 8
7.5 odd 6 539.2.i.b.472.3 8
7.6 odd 2 inner 77.2.b.b.76.4 yes 4
11.2 odd 10 847.2.l.g.524.1 16
11.3 even 5 847.2.l.g.475.4 16
11.4 even 5 847.2.l.g.699.1 16
11.5 even 5 847.2.l.g.118.2 16
11.6 odd 10 847.2.l.g.118.4 16
11.7 odd 10 847.2.l.g.699.3 16
11.8 odd 10 847.2.l.g.475.2 16
11.9 even 5 847.2.l.g.524.3 16
11.10 odd 2 inner 77.2.b.b.76.1 4
21.20 even 2 693.2.c.b.307.2 4
28.27 even 2 1232.2.e.c.769.2 4
33.32 even 2 693.2.c.b.307.3 4
44.43 even 2 1232.2.e.c.769.4 4
77.6 even 10 847.2.l.g.118.3 16
77.10 even 6 539.2.i.b.362.4 8
77.13 even 10 847.2.l.g.524.2 16
77.20 odd 10 847.2.l.g.524.4 16
77.27 odd 10 847.2.l.g.118.1 16
77.32 odd 6 539.2.i.b.362.3 8
77.41 even 10 847.2.l.g.475.1 16
77.48 odd 10 847.2.l.g.699.2 16
77.54 even 6 539.2.i.b.472.1 8
77.62 even 10 847.2.l.g.699.4 16
77.65 odd 6 539.2.i.b.472.2 8
77.69 odd 10 847.2.l.g.475.3 16
77.76 even 2 inner 77.2.b.b.76.2 yes 4
231.230 odd 2 693.2.c.b.307.4 4
308.307 odd 2 1232.2.e.c.769.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.b.b.76.1 4 11.10 odd 2 inner
77.2.b.b.76.2 yes 4 77.76 even 2 inner
77.2.b.b.76.3 yes 4 1.1 even 1 trivial
77.2.b.b.76.4 yes 4 7.6 odd 2 inner
539.2.i.b.362.1 8 7.4 even 3
539.2.i.b.362.2 8 7.3 odd 6
539.2.i.b.362.3 8 77.32 odd 6
539.2.i.b.362.4 8 77.10 even 6
539.2.i.b.472.1 8 77.54 even 6
539.2.i.b.472.2 8 77.65 odd 6
539.2.i.b.472.3 8 7.5 odd 6
539.2.i.b.472.4 8 7.2 even 3
693.2.c.b.307.1 4 3.2 odd 2
693.2.c.b.307.2 4 21.20 even 2
693.2.c.b.307.3 4 33.32 even 2
693.2.c.b.307.4 4 231.230 odd 2
847.2.l.g.118.1 16 77.27 odd 10
847.2.l.g.118.2 16 11.5 even 5
847.2.l.g.118.3 16 77.6 even 10
847.2.l.g.118.4 16 11.6 odd 10
847.2.l.g.475.1 16 77.41 even 10
847.2.l.g.475.2 16 11.8 odd 10
847.2.l.g.475.3 16 77.69 odd 10
847.2.l.g.475.4 16 11.3 even 5
847.2.l.g.524.1 16 11.2 odd 10
847.2.l.g.524.2 16 77.13 even 10
847.2.l.g.524.3 16 11.9 even 5
847.2.l.g.524.4 16 77.20 odd 10
847.2.l.g.699.1 16 11.4 even 5
847.2.l.g.699.2 16 77.48 odd 10
847.2.l.g.699.3 16 11.7 odd 10
847.2.l.g.699.4 16 77.62 even 10
1232.2.e.c.769.1 4 308.307 odd 2
1232.2.e.c.769.2 4 28.27 even 2
1232.2.e.c.769.3 4 4.3 odd 2
1232.2.e.c.769.4 4 44.43 even 2