Properties

Label 1560.2.w.c.781.3
Level $1560$
Weight $2$
Character 1560.781
Analytic conductor $12.457$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,2,Mod(781,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1560.781");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.w (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: \(12.4566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 781.3
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1560.781
Dual form 1560.2.w.c.781.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{5} -1.41421i q^{6} -0.585786 q^{7} +2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} -1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{5} -1.41421i q^{6} -0.585786 q^{7} +2.82843 q^{8} -1.00000 q^{9} -1.41421i q^{10} -4.82843i q^{11} -2.00000i q^{12} +1.00000i q^{13} -0.828427 q^{14} -1.00000 q^{15} +4.00000 q^{16} -1.41421 q^{17} -1.41421 q^{18} +0.242641i q^{19} -2.00000i q^{20} +0.585786i q^{21} -6.82843i q^{22} +4.24264 q^{23} -2.82843i q^{24} -1.00000 q^{25} +1.41421i q^{26} +1.00000i q^{27} -1.17157 q^{28} -10.2426i q^{29} -1.41421 q^{30} +1.17157 q^{31} +5.65685 q^{32} -4.82843 q^{33} -2.00000 q^{34} +0.585786i q^{35} -2.00000 q^{36} -6.48528i q^{37} +0.343146i q^{38} +1.00000 q^{39} -2.82843i q^{40} -5.17157 q^{41} +0.828427i q^{42} +4.82843i q^{43} -9.65685i q^{44} +1.00000i q^{45} +6.00000 q^{46} +1.65685 q^{47} -4.00000i q^{48} -6.65685 q^{49} -1.41421 q^{50} +1.41421i q^{51} +2.00000i q^{52} +0.343146i q^{53} +1.41421i q^{54} -4.82843 q^{55} -1.65685 q^{56} +0.242641 q^{57} -14.4853i q^{58} +3.17157i q^{59} -2.00000 q^{60} +8.00000i q^{61} +1.65685 q^{62} +0.585786 q^{63} +8.00000 q^{64} +1.00000 q^{65} -6.82843 q^{66} +1.17157i q^{67} -2.82843 q^{68} -4.24264i q^{69} +0.828427i q^{70} +4.82843 q^{71} -2.82843 q^{72} +8.58579 q^{73} -9.17157i q^{74} +1.00000i q^{75} +0.485281i q^{76} +2.82843i q^{77} +1.41421 q^{78} -1.65685 q^{79} -4.00000i q^{80} +1.00000 q^{81} -7.31371 q^{82} +7.65685i q^{83} +1.17157i q^{84} +1.41421i q^{85} +6.82843i q^{86} -10.2426 q^{87} -13.6569i q^{88} +7.65685 q^{89} +1.41421i q^{90} -0.585786i q^{91} +8.48528 q^{92} -1.17157i q^{93} +2.34315 q^{94} +0.242641 q^{95} -5.65685i q^{96} +6.72792 q^{97} -9.41421 q^{98} +4.82843i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 8 q^{7} - 4 q^{9} + 8 q^{14} - 4 q^{15} + 16 q^{16} - 4 q^{25} - 16 q^{28} + 16 q^{31} - 8 q^{33} - 8 q^{34} - 8 q^{36} + 4 q^{39} - 32 q^{41} + 24 q^{46} - 16 q^{47} - 4 q^{49} - 8 q^{55} + 16 q^{56} - 16 q^{57} - 8 q^{60} - 16 q^{62} + 8 q^{63} + 32 q^{64} + 4 q^{65} - 16 q^{66} + 8 q^{71} + 40 q^{73} + 16 q^{79} + 4 q^{81} + 16 q^{82} - 24 q^{87} + 8 q^{89} + 32 q^{94} - 16 q^{95} - 24 q^{97} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000
\(3\) − 1.00000i − 0.577350i
\(4\) 2.00000 1.00000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.41421i − 0.577350i
\(7\) −0.585786 −0.221406 −0.110703 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) 2.82843 1.00000
\(9\) −1.00000 −0.333333
\(10\) − 1.41421i − 0.447214i
\(11\) − 4.82843i − 1.45583i −0.685670 0.727913i \(-0.740491\pi\)
0.685670 0.727913i \(-0.259509\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) 1.00000i 0.277350i
\(14\) −0.828427 −0.221406
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) −1.41421 −0.333333
\(19\) 0.242641i 0.0556656i 0.999613 + 0.0278328i \(0.00886060\pi\)
−0.999613 + 0.0278328i \(0.991139\pi\)
\(20\) − 2.00000i − 0.447214i
\(21\) 0.585786i 0.127829i
\(22\) − 6.82843i − 1.45583i
\(23\) 4.24264 0.884652 0.442326 0.896854i \(-0.354153\pi\)
0.442326 + 0.896854i \(0.354153\pi\)
\(24\) − 2.82843i − 0.577350i
\(25\) −1.00000 −0.200000
\(26\) 1.41421i 0.277350i
\(27\) 1.00000i 0.192450i
\(28\) −1.17157 −0.221406
\(29\) − 10.2426i − 1.90201i −0.309175 0.951005i \(-0.600053\pi\)
0.309175 0.951005i \(-0.399947\pi\)
\(30\) −1.41421 −0.258199
\(31\) 1.17157 0.210421 0.105210 0.994450i \(-0.466448\pi\)
0.105210 + 0.994450i \(0.466448\pi\)
\(32\) 5.65685 1.00000
\(33\) −4.82843 −0.840521
\(34\) −2.00000 −0.342997
\(35\) 0.585786i 0.0990160i
\(36\) −2.00000 −0.333333
\(37\) − 6.48528i − 1.06617i −0.846061 0.533087i \(-0.821032\pi\)
0.846061 0.533087i \(-0.178968\pi\)
\(38\) 0.343146i 0.0556656i
\(39\) 1.00000 0.160128
\(40\) − 2.82843i − 0.447214i
\(41\) −5.17157 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(42\) 0.828427i 0.127829i
\(43\) 4.82843i 0.736328i 0.929761 + 0.368164i \(0.120014\pi\)
−0.929761 + 0.368164i \(0.879986\pi\)
\(44\) − 9.65685i − 1.45583i
\(45\) 1.00000i 0.149071i
\(46\) 6.00000 0.884652
\(47\) 1.65685 0.241677 0.120839 0.992672i \(-0.461442\pi\)
0.120839 + 0.992672i \(0.461442\pi\)
\(48\) − 4.00000i − 0.577350i
\(49\) −6.65685 −0.950979
\(50\) −1.41421 −0.200000
\(51\) 1.41421i 0.198030i
\(52\) 2.00000i 0.277350i
\(53\) 0.343146i 0.0471347i 0.999722 + 0.0235673i \(0.00750241\pi\)
−0.999722 + 0.0235673i \(0.992498\pi\)
\(54\) 1.41421i 0.192450i
\(55\) −4.82843 −0.651065
\(56\) −1.65685 −0.221406
\(57\) 0.242641 0.0321385
\(58\) − 14.4853i − 1.90201i
\(59\) 3.17157i 0.412904i 0.978457 + 0.206452i \(0.0661917\pi\)
−0.978457 + 0.206452i \(0.933808\pi\)
\(60\) −2.00000 −0.258199
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 1.65685 0.210421
\(63\) 0.585786 0.0738022
\(64\) 8.00000 1.00000
\(65\) 1.00000 0.124035
\(66\) −6.82843 −0.840521
\(67\) 1.17157i 0.143130i 0.997436 + 0.0715652i \(0.0227994\pi\)
−0.997436 + 0.0715652i \(0.977201\pi\)
\(68\) −2.82843 −0.342997
\(69\) − 4.24264i − 0.510754i
\(70\) 0.828427i 0.0990160i
\(71\) 4.82843 0.573029 0.286514 0.958076i \(-0.407503\pi\)
0.286514 + 0.958076i \(0.407503\pi\)
\(72\) −2.82843 −0.333333
\(73\) 8.58579 1.00489 0.502445 0.864609i \(-0.332434\pi\)
0.502445 + 0.864609i \(0.332434\pi\)
\(74\) − 9.17157i − 1.06617i
\(75\) 1.00000i 0.115470i
\(76\) 0.485281i 0.0556656i
\(77\) 2.82843i 0.322329i
\(78\) 1.41421 0.160128
\(79\) −1.65685 −0.186411 −0.0932053 0.995647i \(-0.529711\pi\)
−0.0932053 + 0.995647i \(0.529711\pi\)
\(80\) − 4.00000i − 0.447214i
\(81\) 1.00000 0.111111
\(82\) −7.31371 −0.807664
\(83\) 7.65685i 0.840449i 0.907420 + 0.420224i \(0.138049\pi\)
−0.907420 + 0.420224i \(0.861951\pi\)
\(84\) 1.17157i 0.127829i
\(85\) 1.41421i 0.153393i
\(86\) 6.82843i 0.736328i
\(87\) −10.2426 −1.09813
\(88\) − 13.6569i − 1.45583i
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 1.41421i 0.149071i
\(91\) − 0.585786i − 0.0614071i
\(92\) 8.48528 0.884652
\(93\) − 1.17157i − 0.121486i
\(94\) 2.34315 0.241677
\(95\) 0.242641 0.0248944
\(96\) − 5.65685i − 0.577350i
\(97\) 6.72792 0.683117 0.341558 0.939861i \(-0.389045\pi\)
0.341558 + 0.939861i \(0.389045\pi\)
\(98\) −9.41421 −0.950979
\(99\) 4.82843i 0.485275i
\(100\) −2.00000 −0.200000
\(101\) 0.585786i 0.0582879i 0.999575 + 0.0291440i \(0.00927813\pi\)
−0.999575 + 0.0291440i \(0.990722\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 7.65685 0.754452 0.377226 0.926121i \(-0.376878\pi\)
0.377226 + 0.926121i \(0.376878\pi\)
\(104\) 2.82843i 0.277350i
\(105\) 0.585786 0.0571669
\(106\) 0.485281i 0.0471347i
\(107\) − 1.17157i − 0.113260i −0.998395 0.0566301i \(-0.981964\pi\)
0.998395 0.0566301i \(-0.0180356\pi\)
\(108\) 2.00000i 0.192450i
\(109\) − 4.24264i − 0.406371i −0.979140 0.203186i \(-0.934871\pi\)
0.979140 0.203186i \(-0.0651295\pi\)
\(110\) −6.82843 −0.651065
\(111\) −6.48528 −0.615556
\(112\) −2.34315 −0.221406
\(113\) 10.5858 0.995827 0.497914 0.867227i \(-0.334100\pi\)
0.497914 + 0.867227i \(0.334100\pi\)
\(114\) 0.343146 0.0321385
\(115\) − 4.24264i − 0.395628i
\(116\) − 20.4853i − 1.90201i
\(117\) − 1.00000i − 0.0924500i
\(118\) 4.48528i 0.412904i
\(119\) 0.828427 0.0759418
\(120\) −2.82843 −0.258199
\(121\) −12.3137 −1.11943
\(122\) 11.3137i 1.02430i
\(123\) 5.17157i 0.466305i
\(124\) 2.34315 0.210421
\(125\) 1.00000i 0.0894427i
\(126\) 0.828427 0.0738022
\(127\) −18.9706 −1.68337 −0.841683 0.539973i \(-0.818435\pi\)
−0.841683 + 0.539973i \(0.818435\pi\)
\(128\) 11.3137 1.00000
\(129\) 4.82843 0.425119
\(130\) 1.41421 0.124035
\(131\) 10.2426i 0.894904i 0.894308 + 0.447452i \(0.147668\pi\)
−0.894308 + 0.447452i \(0.852332\pi\)
\(132\) −9.65685 −0.840521
\(133\) − 0.142136i − 0.0123247i
\(134\) 1.65685i 0.143130i
\(135\) 1.00000 0.0860663
\(136\) −4.00000 −0.342997
\(137\) −8.48528 −0.724947 −0.362473 0.931994i \(-0.618068\pi\)
−0.362473 + 0.931994i \(0.618068\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) 17.1716i 1.45647i 0.685325 + 0.728237i \(0.259660\pi\)
−0.685325 + 0.728237i \(0.740340\pi\)
\(140\) 1.17157i 0.0990160i
\(141\) − 1.65685i − 0.139532i
\(142\) 6.82843 0.573029
\(143\) 4.82843 0.403773
\(144\) −4.00000 −0.333333
\(145\) −10.2426 −0.850605
\(146\) 12.1421 1.00489
\(147\) 6.65685i 0.549048i
\(148\) − 12.9706i − 1.06617i
\(149\) 11.1716i 0.915211i 0.889155 + 0.457605i \(0.151293\pi\)
−0.889155 + 0.457605i \(0.848707\pi\)
\(150\) 1.41421i 0.115470i
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0.686292i 0.0556656i
\(153\) 1.41421 0.114332
\(154\) 4.00000i 0.322329i
\(155\) − 1.17157i − 0.0941030i
\(156\) 2.00000 0.160128
\(157\) − 6.82843i − 0.544968i −0.962160 0.272484i \(-0.912155\pi\)
0.962160 0.272484i \(-0.0878452\pi\)
\(158\) −2.34315 −0.186411
\(159\) 0.343146 0.0272132
\(160\) − 5.65685i − 0.447214i
\(161\) −2.48528 −0.195868
\(162\) 1.41421 0.111111
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −10.3431 −0.807664
\(165\) 4.82843i 0.375893i
\(166\) 10.8284i 0.840449i
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 1.65685i 0.127829i
\(169\) −1.00000 −0.0769231
\(170\) 2.00000i 0.153393i
\(171\) − 0.242641i − 0.0185552i
\(172\) 9.65685i 0.736328i
\(173\) − 12.1421i − 0.923149i −0.887101 0.461575i \(-0.847285\pi\)
0.887101 0.461575i \(-0.152715\pi\)
\(174\) −14.4853 −1.09813
\(175\) 0.585786 0.0442813
\(176\) − 19.3137i − 1.45583i
\(177\) 3.17157 0.238390
\(178\) 10.8284 0.811625
\(179\) 15.8995i 1.18838i 0.804323 + 0.594192i \(0.202528\pi\)
−0.804323 + 0.594192i \(0.797472\pi\)
\(180\) 2.00000i 0.149071i
\(181\) 6.82843i 0.507553i 0.967263 + 0.253776i \(0.0816728\pi\)
−0.967263 + 0.253776i \(0.918327\pi\)
\(182\) − 0.828427i − 0.0614071i
\(183\) 8.00000 0.591377
\(184\) 12.0000 0.884652
\(185\) −6.48528 −0.476807
\(186\) − 1.65685i − 0.121486i
\(187\) 6.82843i 0.499344i
\(188\) 3.31371 0.241677
\(189\) − 0.585786i − 0.0426097i
\(190\) 0.343146 0.0248944
\(191\) 15.7990 1.14317 0.571587 0.820541i \(-0.306328\pi\)
0.571587 + 0.820541i \(0.306328\pi\)
\(192\) − 8.00000i − 0.577350i
\(193\) 3.41421 0.245760 0.122880 0.992422i \(-0.460787\pi\)
0.122880 + 0.992422i \(0.460787\pi\)
\(194\) 9.51472 0.683117
\(195\) − 1.00000i − 0.0716115i
\(196\) −13.3137 −0.950979
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 6.82843i 0.485275i
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −2.82843 −0.200000
\(201\) 1.17157 0.0826364
\(202\) 0.828427i 0.0582879i
\(203\) 6.00000i 0.421117i
\(204\) 2.82843i 0.198030i
\(205\) 5.17157i 0.361198i
\(206\) 10.8284 0.754452
\(207\) −4.24264 −0.294884
\(208\) 4.00000i 0.277350i
\(209\) 1.17157 0.0810394
\(210\) 0.828427 0.0571669
\(211\) 24.9706i 1.71904i 0.511098 + 0.859522i \(0.329239\pi\)
−0.511098 + 0.859522i \(0.670761\pi\)
\(212\) 0.686292i 0.0471347i
\(213\) − 4.82843i − 0.330838i
\(214\) − 1.65685i − 0.113260i
\(215\) 4.82843 0.329296
\(216\) 2.82843i 0.192450i
\(217\) −0.686292 −0.0465885
\(218\) − 6.00000i − 0.406371i
\(219\) − 8.58579i − 0.580174i
\(220\) −9.65685 −0.651065
\(221\) − 1.41421i − 0.0951303i
\(222\) −9.17157 −0.615556
\(223\) 27.2132 1.82233 0.911165 0.412041i \(-0.135184\pi\)
0.911165 + 0.412041i \(0.135184\pi\)
\(224\) −3.31371 −0.221406
\(225\) 1.00000 0.0666667
\(226\) 14.9706 0.995827
\(227\) 15.3137i 1.01641i 0.861237 + 0.508203i \(0.169690\pi\)
−0.861237 + 0.508203i \(0.830310\pi\)
\(228\) 0.485281 0.0321385
\(229\) 5.89949i 0.389850i 0.980818 + 0.194925i \(0.0624463\pi\)
−0.980818 + 0.194925i \(0.937554\pi\)
\(230\) − 6.00000i − 0.395628i
\(231\) 2.82843 0.186097
\(232\) − 28.9706i − 1.90201i
\(233\) 22.5858 1.47964 0.739822 0.672803i \(-0.234910\pi\)
0.739822 + 0.672803i \(0.234910\pi\)
\(234\) − 1.41421i − 0.0924500i
\(235\) − 1.65685i − 0.108081i
\(236\) 6.34315i 0.412904i
\(237\) 1.65685i 0.107624i
\(238\) 1.17157 0.0759418
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) −4.00000 −0.258199
\(241\) −1.51472 −0.0975716 −0.0487858 0.998809i \(-0.515535\pi\)
−0.0487858 + 0.998809i \(0.515535\pi\)
\(242\) −17.4142 −1.11943
\(243\) − 1.00000i − 0.0641500i
\(244\) 16.0000i 1.02430i
\(245\) 6.65685i 0.425291i
\(246\) 7.31371i 0.466305i
\(247\) −0.242641 −0.0154389
\(248\) 3.31371 0.210421
\(249\) 7.65685 0.485233
\(250\) 1.41421i 0.0894427i
\(251\) 2.92893i 0.184873i 0.995719 + 0.0924363i \(0.0294654\pi\)
−0.995719 + 0.0924363i \(0.970535\pi\)
\(252\) 1.17157 0.0738022
\(253\) − 20.4853i − 1.28790i
\(254\) −26.8284 −1.68337
\(255\) 1.41421 0.0885615
\(256\) 16.0000 1.00000
\(257\) −10.1005 −0.630052 −0.315026 0.949083i \(-0.602013\pi\)
−0.315026 + 0.949083i \(0.602013\pi\)
\(258\) 6.82843 0.425119
\(259\) 3.79899i 0.236058i
\(260\) 2.00000 0.124035
\(261\) 10.2426i 0.634004i
\(262\) 14.4853i 0.894904i
\(263\) 4.24264 0.261612 0.130806 0.991408i \(-0.458243\pi\)
0.130806 + 0.991408i \(0.458243\pi\)
\(264\) −13.6569 −0.840521
\(265\) 0.343146 0.0210793
\(266\) − 0.201010i − 0.0123247i
\(267\) − 7.65685i − 0.468592i
\(268\) 2.34315i 0.143130i
\(269\) − 15.2132i − 0.927565i −0.885949 0.463783i \(-0.846492\pi\)
0.885949 0.463783i \(-0.153508\pi\)
\(270\) 1.41421 0.0860663
\(271\) 1.65685 0.100647 0.0503234 0.998733i \(-0.483975\pi\)
0.0503234 + 0.998733i \(0.483975\pi\)
\(272\) −5.65685 −0.342997
\(273\) −0.585786 −0.0354534
\(274\) −12.0000 −0.724947
\(275\) 4.82843i 0.291165i
\(276\) − 8.48528i − 0.510754i
\(277\) − 13.1716i − 0.791403i −0.918379 0.395702i \(-0.870501\pi\)
0.918379 0.395702i \(-0.129499\pi\)
\(278\) 24.2843i 1.45647i
\(279\) −1.17157 −0.0701402
\(280\) 1.65685i 0.0990160i
\(281\) −27.6569 −1.64987 −0.824935 0.565228i \(-0.808788\pi\)
−0.824935 + 0.565228i \(0.808788\pi\)
\(282\) − 2.34315i − 0.139532i
\(283\) 16.1421i 0.959550i 0.877391 + 0.479775i \(0.159282\pi\)
−0.877391 + 0.479775i \(0.840718\pi\)
\(284\) 9.65685 0.573029
\(285\) − 0.242641i − 0.0143728i
\(286\) 6.82843 0.403773
\(287\) 3.02944 0.178822
\(288\) −5.65685 −0.333333
\(289\) −15.0000 −0.882353
\(290\) −14.4853 −0.850605
\(291\) − 6.72792i − 0.394398i
\(292\) 17.1716 1.00489
\(293\) 7.31371i 0.427271i 0.976913 + 0.213636i \(0.0685306\pi\)
−0.976913 + 0.213636i \(0.931469\pi\)
\(294\) 9.41421i 0.549048i
\(295\) 3.17157 0.184656
\(296\) − 18.3431i − 1.06617i
\(297\) 4.82843 0.280174
\(298\) 15.7990i 0.915211i
\(299\) 4.24264i 0.245358i
\(300\) 2.00000i 0.115470i
\(301\) − 2.82843i − 0.163028i
\(302\) −16.9706 −0.976546
\(303\) 0.585786 0.0336526
\(304\) 0.970563i 0.0556656i
\(305\) 8.00000 0.458079
\(306\) 2.00000 0.114332
\(307\) − 3.31371i − 0.189123i −0.995519 0.0945617i \(-0.969855\pi\)
0.995519 0.0945617i \(-0.0301450\pi\)
\(308\) 5.65685i 0.322329i
\(309\) − 7.65685i − 0.435583i
\(310\) − 1.65685i − 0.0941030i
\(311\) 14.1421 0.801927 0.400963 0.916094i \(-0.368675\pi\)
0.400963 + 0.916094i \(0.368675\pi\)
\(312\) 2.82843 0.160128
\(313\) 16.8284 0.951199 0.475599 0.879662i \(-0.342231\pi\)
0.475599 + 0.879662i \(0.342231\pi\)
\(314\) − 9.65685i − 0.544968i
\(315\) − 0.585786i − 0.0330053i
\(316\) −3.31371 −0.186411
\(317\) 20.6274i 1.15855i 0.815132 + 0.579276i \(0.196664\pi\)
−0.815132 + 0.579276i \(0.803336\pi\)
\(318\) 0.485281 0.0272132
\(319\) −49.4558 −2.76900
\(320\) − 8.00000i − 0.447214i
\(321\) −1.17157 −0.0653908
\(322\) −3.51472 −0.195868
\(323\) − 0.343146i − 0.0190931i
\(324\) 2.00000 0.111111
\(325\) − 1.00000i − 0.0554700i
\(326\) − 11.3137i − 0.626608i
\(327\) −4.24264 −0.234619
\(328\) −14.6274 −0.807664
\(329\) −0.970563 −0.0535089
\(330\) 6.82843i 0.375893i
\(331\) 33.6985i 1.85224i 0.377234 + 0.926118i \(0.376875\pi\)
−0.377234 + 0.926118i \(0.623125\pi\)
\(332\) 15.3137i 0.840449i
\(333\) 6.48528i 0.355391i
\(334\) 28.0000 1.53209
\(335\) 1.17157 0.0640099
\(336\) 2.34315i 0.127829i
\(337\) −34.4853 −1.87853 −0.939266 0.343189i \(-0.888493\pi\)
−0.939266 + 0.343189i \(0.888493\pi\)
\(338\) −1.41421 −0.0769231
\(339\) − 10.5858i − 0.574941i
\(340\) 2.82843i 0.153393i
\(341\) − 5.65685i − 0.306336i
\(342\) − 0.343146i − 0.0185552i
\(343\) 8.00000 0.431959
\(344\) 13.6569i 0.736328i
\(345\) −4.24264 −0.228416
\(346\) − 17.1716i − 0.923149i
\(347\) 11.3137i 0.607352i 0.952775 + 0.303676i \(0.0982140\pi\)
−0.952775 + 0.303676i \(0.901786\pi\)
\(348\) −20.4853 −1.09813
\(349\) − 20.7279i − 1.10954i −0.832004 0.554770i \(-0.812806\pi\)
0.832004 0.554770i \(-0.187194\pi\)
\(350\) 0.828427 0.0442813
\(351\) −1.00000 −0.0533761
\(352\) − 27.3137i − 1.45583i
\(353\) 6.14214 0.326913 0.163456 0.986551i \(-0.447736\pi\)
0.163456 + 0.986551i \(0.447736\pi\)
\(354\) 4.48528 0.238390
\(355\) − 4.82843i − 0.256266i
\(356\) 15.3137 0.811625
\(357\) − 0.828427i − 0.0438450i
\(358\) 22.4853i 1.18838i
\(359\) 23.1716 1.22295 0.611474 0.791264i \(-0.290577\pi\)
0.611474 + 0.791264i \(0.290577\pi\)
\(360\) 2.82843i 0.149071i
\(361\) 18.9411 0.996901
\(362\) 9.65685i 0.507553i
\(363\) 12.3137i 0.646302i
\(364\) − 1.17157i − 0.0614071i
\(365\) − 8.58579i − 0.449401i
\(366\) 11.3137 0.591377
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 16.9706 0.884652
\(369\) 5.17157 0.269221
\(370\) −9.17157 −0.476807
\(371\) − 0.201010i − 0.0104359i
\(372\) − 2.34315i − 0.121486i
\(373\) − 22.8284i − 1.18201i −0.806667 0.591006i \(-0.798731\pi\)
0.806667 0.591006i \(-0.201269\pi\)
\(374\) 9.65685i 0.499344i
\(375\) 1.00000 0.0516398
\(376\) 4.68629 0.241677
\(377\) 10.2426 0.527523
\(378\) − 0.828427i − 0.0426097i
\(379\) 1.89949i 0.0975705i 0.998809 + 0.0487853i \(0.0155350\pi\)
−0.998809 + 0.0487853i \(0.984465\pi\)
\(380\) 0.485281 0.0248944
\(381\) 18.9706i 0.971891i
\(382\) 22.3431 1.14317
\(383\) 35.7990 1.82924 0.914621 0.404311i \(-0.132489\pi\)
0.914621 + 0.404311i \(0.132489\pi\)
\(384\) − 11.3137i − 0.577350i
\(385\) 2.82843 0.144150
\(386\) 4.82843 0.245760
\(387\) − 4.82843i − 0.245443i
\(388\) 13.4558 0.683117
\(389\) 27.4142i 1.38996i 0.719031 + 0.694978i \(0.244586\pi\)
−0.719031 + 0.694978i \(0.755414\pi\)
\(390\) − 1.41421i − 0.0716115i
\(391\) −6.00000 −0.303433
\(392\) −18.8284 −0.950979
\(393\) 10.2426 0.516673
\(394\) 0 0
\(395\) 1.65685i 0.0833654i
\(396\) 9.65685i 0.485275i
\(397\) 10.6863i 0.536330i 0.963373 + 0.268165i \(0.0864172\pi\)
−0.963373 + 0.268165i \(0.913583\pi\)
\(398\) 0 0
\(399\) −0.142136 −0.00711568
\(400\) −4.00000 −0.200000
\(401\) −10.1421 −0.506474 −0.253237 0.967404i \(-0.581495\pi\)
−0.253237 + 0.967404i \(0.581495\pi\)
\(402\) 1.65685 0.0826364
\(403\) 1.17157i 0.0583602i
\(404\) 1.17157i 0.0582879i
\(405\) − 1.00000i − 0.0496904i
\(406\) 8.48528i 0.421117i
\(407\) −31.3137 −1.55216
\(408\) 4.00000i 0.198030i
\(409\) −5.79899 −0.286742 −0.143371 0.989669i \(-0.545794\pi\)
−0.143371 + 0.989669i \(0.545794\pi\)
\(410\) 7.31371i 0.361198i
\(411\) 8.48528i 0.418548i
\(412\) 15.3137 0.754452
\(413\) − 1.85786i − 0.0914195i
\(414\) −6.00000 −0.294884
\(415\) 7.65685 0.375860
\(416\) 5.65685i 0.277350i
\(417\) 17.1716 0.840896
\(418\) 1.65685 0.0810394
\(419\) − 0.100505i − 0.00490999i −0.999997 0.00245500i \(-0.999219\pi\)
0.999997 0.00245500i \(-0.000781451\pi\)
\(420\) 1.17157 0.0571669
\(421\) 7.27208i 0.354419i 0.984173 + 0.177210i \(0.0567071\pi\)
−0.984173 + 0.177210i \(0.943293\pi\)
\(422\) 35.3137i 1.71904i
\(423\) −1.65685 −0.0805590
\(424\) 0.970563i 0.0471347i
\(425\) 1.41421 0.0685994
\(426\) − 6.82843i − 0.330838i
\(427\) − 4.68629i − 0.226786i
\(428\) − 2.34315i − 0.113260i
\(429\) − 4.82843i − 0.233119i
\(430\) 6.82843 0.329296
\(431\) 19.3137 0.930309 0.465154 0.885230i \(-0.345999\pi\)
0.465154 + 0.885230i \(0.345999\pi\)
\(432\) 4.00000i 0.192450i
\(433\) −34.4853 −1.65726 −0.828628 0.559799i \(-0.810878\pi\)
−0.828628 + 0.559799i \(0.810878\pi\)
\(434\) −0.970563 −0.0465885
\(435\) 10.2426i 0.491097i
\(436\) − 8.48528i − 0.406371i
\(437\) 1.02944i 0.0492447i
\(438\) − 12.1421i − 0.580174i
\(439\) −31.6569 −1.51090 −0.755450 0.655207i \(-0.772582\pi\)
−0.755450 + 0.655207i \(0.772582\pi\)
\(440\) −13.6569 −0.651065
\(441\) 6.65685 0.316993
\(442\) − 2.00000i − 0.0951303i
\(443\) − 13.6569i − 0.648857i −0.945910 0.324428i \(-0.894828\pi\)
0.945910 0.324428i \(-0.105172\pi\)
\(444\) −12.9706 −0.615556
\(445\) − 7.65685i − 0.362970i
\(446\) 38.4853 1.82233
\(447\) 11.1716 0.528397
\(448\) −4.68629 −0.221406
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.41421 0.0666667
\(451\) 24.9706i 1.17582i
\(452\) 21.1716 0.995827
\(453\) 12.0000i 0.563809i
\(454\) 21.6569i 1.01641i
\(455\) −0.585786 −0.0274621
\(456\) 0.686292 0.0321385
\(457\) 5.75736 0.269318 0.134659 0.990892i \(-0.457006\pi\)
0.134659 + 0.990892i \(0.457006\pi\)
\(458\) 8.34315i 0.389850i
\(459\) − 1.41421i − 0.0660098i
\(460\) − 8.48528i − 0.395628i
\(461\) − 35.6569i − 1.66071i −0.557238 0.830353i \(-0.688139\pi\)
0.557238 0.830353i \(-0.311861\pi\)
\(462\) 4.00000 0.186097
\(463\) −34.7279 −1.61394 −0.806972 0.590590i \(-0.798895\pi\)
−0.806972 + 0.590590i \(0.798895\pi\)
\(464\) − 40.9706i − 1.90201i
\(465\) −1.17157 −0.0543304
\(466\) 31.9411 1.47964
\(467\) − 35.1127i − 1.62482i −0.583085 0.812411i \(-0.698155\pi\)
0.583085 0.812411i \(-0.301845\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) − 0.686292i − 0.0316900i
\(470\) − 2.34315i − 0.108081i
\(471\) −6.82843 −0.314637
\(472\) 8.97056i 0.412904i
\(473\) 23.3137 1.07197
\(474\) 2.34315i 0.107624i
\(475\) − 0.242641i − 0.0111331i
\(476\) 1.65685 0.0759418
\(477\) − 0.343146i − 0.0157116i
\(478\) −16.0000 −0.731823
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) −5.65685 −0.258199
\(481\) 6.48528 0.295703
\(482\) −2.14214 −0.0975716
\(483\) 2.48528i 0.113084i
\(484\) −24.6274 −1.11943
\(485\) − 6.72792i − 0.305499i
\(486\) − 1.41421i − 0.0641500i
\(487\) −28.3848 −1.28624 −0.643118 0.765767i \(-0.722360\pi\)
−0.643118 + 0.765767i \(0.722360\pi\)
\(488\) 22.6274i 1.02430i
\(489\) −8.00000 −0.361773
\(490\) 9.41421i 0.425291i
\(491\) − 40.8701i − 1.84444i −0.386666 0.922220i \(-0.626373\pi\)
0.386666 0.922220i \(-0.373627\pi\)
\(492\) 10.3431i 0.466305i
\(493\) 14.4853i 0.652384i
\(494\) −0.343146 −0.0154389
\(495\) 4.82843 0.217022
\(496\) 4.68629 0.210421
\(497\) −2.82843 −0.126872
\(498\) 10.8284 0.485233
\(499\) 2.58579i 0.115756i 0.998324 + 0.0578778i \(0.0184334\pi\)
−0.998324 + 0.0578778i \(0.981567\pi\)
\(500\) 2.00000i 0.0894427i
\(501\) − 19.7990i − 0.884554i
\(502\) 4.14214i 0.184873i
\(503\) −10.5858 −0.471997 −0.235998 0.971753i \(-0.575836\pi\)
−0.235998 + 0.971753i \(0.575836\pi\)
\(504\) 1.65685 0.0738022
\(505\) 0.585786 0.0260672
\(506\) − 28.9706i − 1.28790i
\(507\) 1.00000i 0.0444116i
\(508\) −37.9411 −1.68337
\(509\) − 24.6274i − 1.09159i −0.837918 0.545796i \(-0.816228\pi\)
0.837918 0.545796i \(-0.183772\pi\)
\(510\) 2.00000 0.0885615
\(511\) −5.02944 −0.222489
\(512\) 22.6274 1.00000
\(513\) −0.242641 −0.0107128
\(514\) −14.2843 −0.630052
\(515\) − 7.65685i − 0.337401i
\(516\) 9.65685 0.425119
\(517\) − 8.00000i − 0.351840i
\(518\) 5.37258i 0.236058i
\(519\) −12.1421 −0.532981
\(520\) 2.82843 0.124035
\(521\) 3.17157 0.138949 0.0694746 0.997584i \(-0.477868\pi\)
0.0694746 + 0.997584i \(0.477868\pi\)
\(522\) 14.4853i 0.634004i
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 20.4853i 0.894904i
\(525\) − 0.585786i − 0.0255658i
\(526\) 6.00000 0.261612
\(527\) −1.65685 −0.0721737
\(528\) −19.3137 −0.840521
\(529\) −5.00000 −0.217391
\(530\) 0.485281 0.0210793
\(531\) − 3.17157i − 0.137635i
\(532\) − 0.284271i − 0.0123247i
\(533\) − 5.17157i − 0.224006i
\(534\) − 10.8284i − 0.468592i
\(535\) −1.17157 −0.0506515
\(536\) 3.31371i 0.143130i
\(537\) 15.8995 0.686114
\(538\) − 21.5147i − 0.927565i
\(539\) 32.1421i 1.38446i
\(540\) 2.00000 0.0860663
\(541\) − 28.7279i − 1.23511i −0.786528 0.617555i \(-0.788123\pi\)
0.786528 0.617555i \(-0.211877\pi\)
\(542\) 2.34315 0.100647
\(543\) 6.82843 0.293036
\(544\) −8.00000 −0.342997
\(545\) −4.24264 −0.181735
\(546\) −0.828427 −0.0354534
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) −16.9706 −0.724947
\(549\) − 8.00000i − 0.341432i
\(550\) 6.82843i 0.291165i
\(551\) 2.48528 0.105877
\(552\) − 12.0000i − 0.510754i
\(553\) 0.970563 0.0412725
\(554\) − 18.6274i − 0.791403i
\(555\) 6.48528i 0.275285i
\(556\) 34.3431i 1.45647i
\(557\) − 29.3137i − 1.24206i −0.783786 0.621031i \(-0.786714\pi\)
0.783786 0.621031i \(-0.213286\pi\)
\(558\) −1.65685 −0.0701402
\(559\) −4.82843 −0.204221
\(560\) 2.34315i 0.0990160i
\(561\) 6.82843 0.288296
\(562\) −39.1127 −1.64987
\(563\) − 6.34315i − 0.267332i −0.991026 0.133666i \(-0.957325\pi\)
0.991026 0.133666i \(-0.0426749\pi\)
\(564\) − 3.31371i − 0.139532i
\(565\) − 10.5858i − 0.445347i
\(566\) 22.8284i 0.959550i
\(567\) −0.585786 −0.0246007
\(568\) 13.6569 0.573029
\(569\) −24.8284 −1.04086 −0.520431 0.853904i \(-0.674229\pi\)
−0.520431 + 0.853904i \(0.674229\pi\)
\(570\) − 0.343146i − 0.0143728i
\(571\) − 30.1421i − 1.26141i −0.776023 0.630705i \(-0.782766\pi\)
0.776023 0.630705i \(-0.217234\pi\)
\(572\) 9.65685 0.403773
\(573\) − 15.7990i − 0.660012i
\(574\) 4.28427 0.178822
\(575\) −4.24264 −0.176930
\(576\) −8.00000 −0.333333
\(577\) 36.8701 1.53492 0.767460 0.641096i \(-0.221520\pi\)
0.767460 + 0.641096i \(0.221520\pi\)
\(578\) −21.2132 −0.882353
\(579\) − 3.41421i − 0.141890i
\(580\) −20.4853 −0.850605
\(581\) − 4.48528i − 0.186081i
\(582\) − 9.51472i − 0.394398i
\(583\) 1.65685 0.0686199
\(584\) 24.2843 1.00489
\(585\) −1.00000 −0.0413449
\(586\) 10.3431i 0.427271i
\(587\) 5.65685i 0.233483i 0.993162 + 0.116742i \(0.0372450\pi\)
−0.993162 + 0.116742i \(0.962755\pi\)
\(588\) 13.3137i 0.549048i
\(589\) 0.284271i 0.0117132i
\(590\) 4.48528 0.184656
\(591\) 0 0
\(592\) − 25.9411i − 1.06617i
\(593\) 18.3431 0.753263 0.376631 0.926363i \(-0.377082\pi\)
0.376631 + 0.926363i \(0.377082\pi\)
\(594\) 6.82843 0.280174
\(595\) − 0.828427i − 0.0339622i
\(596\) 22.3431i 0.915211i
\(597\) 0 0
\(598\) 6.00000i 0.245358i
\(599\) −21.9411 −0.896490 −0.448245 0.893911i \(-0.647951\pi\)
−0.448245 + 0.893911i \(0.647951\pi\)
\(600\) 2.82843i 0.115470i
\(601\) −44.2843 −1.80639 −0.903197 0.429227i \(-0.858786\pi\)
−0.903197 + 0.429227i \(0.858786\pi\)
\(602\) − 4.00000i − 0.163028i
\(603\) − 1.17157i − 0.0477101i
\(604\) −24.0000 −0.976546
\(605\) 12.3137i 0.500623i
\(606\) 0.828427 0.0336526
\(607\) −14.9706 −0.607636 −0.303818 0.952730i \(-0.598262\pi\)
−0.303818 + 0.952730i \(0.598262\pi\)
\(608\) 1.37258i 0.0556656i
\(609\) 6.00000 0.243132
\(610\) 11.3137 0.458079
\(611\) 1.65685i 0.0670291i
\(612\) 2.82843 0.114332
\(613\) 31.1716i 1.25901i 0.776997 + 0.629504i \(0.216742\pi\)
−0.776997 + 0.629504i \(0.783258\pi\)
\(614\) − 4.68629i − 0.189123i
\(615\) 5.17157 0.208538
\(616\) 8.00000i 0.322329i
\(617\) 37.4558 1.50792 0.753958 0.656923i \(-0.228142\pi\)
0.753958 + 0.656923i \(0.228142\pi\)
\(618\) − 10.8284i − 0.435583i
\(619\) − 11.2721i − 0.453063i −0.974004 0.226532i \(-0.927261\pi\)
0.974004 0.226532i \(-0.0727386\pi\)
\(620\) − 2.34315i − 0.0941030i
\(621\) 4.24264i 0.170251i
\(622\) 20.0000 0.801927
\(623\) −4.48528 −0.179699
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) 23.7990 0.951199
\(627\) − 1.17157i − 0.0467881i
\(628\) − 13.6569i − 0.544968i
\(629\) 9.17157i 0.365695i
\(630\) − 0.828427i − 0.0330053i
\(631\) −36.4853 −1.45246 −0.726228 0.687454i \(-0.758728\pi\)
−0.726228 + 0.687454i \(0.758728\pi\)
\(632\) −4.68629 −0.186411
\(633\) 24.9706 0.992491
\(634\) 29.1716i 1.15855i
\(635\) 18.9706i 0.752824i
\(636\) 0.686292 0.0272132
\(637\) − 6.65685i − 0.263754i
\(638\) −69.9411 −2.76900
\(639\) −4.82843 −0.191010
\(640\) − 11.3137i − 0.447214i
\(641\) −37.7990 −1.49297 −0.746485 0.665402i \(-0.768260\pi\)
−0.746485 + 0.665402i \(0.768260\pi\)
\(642\) −1.65685 −0.0653908
\(643\) − 42.1421i − 1.66192i −0.556329 0.830962i \(-0.687791\pi\)
0.556329 0.830962i \(-0.312209\pi\)
\(644\) −4.97056 −0.195868
\(645\) − 4.82843i − 0.190119i
\(646\) − 0.485281i − 0.0190931i
\(647\) −10.1005 −0.397092 −0.198546 0.980092i \(-0.563622\pi\)
−0.198546 + 0.980092i \(0.563622\pi\)
\(648\) 2.82843 0.111111
\(649\) 15.3137 0.601116
\(650\) − 1.41421i − 0.0554700i
\(651\) 0.686292i 0.0268979i
\(652\) − 16.0000i − 0.626608i
\(653\) − 40.1421i − 1.57088i −0.618936 0.785442i \(-0.712436\pi\)
0.618936 0.785442i \(-0.287564\pi\)
\(654\) −6.00000 −0.234619
\(655\) 10.2426 0.400213
\(656\) −20.6863 −0.807664
\(657\) −8.58579 −0.334963
\(658\) −1.37258 −0.0535089
\(659\) 6.92893i 0.269913i 0.990852 + 0.134956i \(0.0430894\pi\)
−0.990852 + 0.134956i \(0.956911\pi\)
\(660\) 9.65685i 0.375893i
\(661\) − 4.24264i − 0.165020i −0.996590 0.0825098i \(-0.973706\pi\)
0.996590 0.0825098i \(-0.0262936\pi\)
\(662\) 47.6569i 1.85224i
\(663\) −1.41421 −0.0549235
\(664\) 21.6569i 0.840449i
\(665\) −0.142136 −0.00551178
\(666\) 9.17157i 0.355391i
\(667\) − 43.4558i − 1.68262i
\(668\) 39.5980 1.53209
\(669\) − 27.2132i − 1.05212i
\(670\) 1.65685 0.0640099
\(671\) 38.6274 1.49119
\(672\) 3.31371i 0.127829i
\(673\) 35.9411 1.38543 0.692714 0.721212i \(-0.256415\pi\)
0.692714 + 0.721212i \(0.256415\pi\)
\(674\) −48.7696 −1.87853
\(675\) − 1.00000i − 0.0384900i
\(676\) −2.00000 −0.0769231
\(677\) 37.1127i 1.42636i 0.700983 + 0.713178i \(0.252745\pi\)
−0.700983 + 0.713178i \(0.747255\pi\)
\(678\) − 14.9706i − 0.574941i
\(679\) −3.94113 −0.151247
\(680\) 4.00000i 0.153393i
\(681\) 15.3137 0.586823
\(682\) − 8.00000i − 0.306336i
\(683\) 22.9706i 0.878944i 0.898256 + 0.439472i \(0.144834\pi\)
−0.898256 + 0.439472i \(0.855166\pi\)
\(684\) − 0.485281i − 0.0185552i
\(685\) 8.48528i 0.324206i
\(686\) 11.3137 0.431959
\(687\) 5.89949 0.225080
\(688\) 19.3137i 0.736328i
\(689\) −0.343146 −0.0130728
\(690\) −6.00000 −0.228416
\(691\) 23.2721i 0.885312i 0.896692 + 0.442656i \(0.145964\pi\)
−0.896692 + 0.442656i \(0.854036\pi\)
\(692\) − 24.2843i − 0.923149i
\(693\) − 2.82843i − 0.107443i
\(694\) 16.0000i 0.607352i
\(695\) 17.1716 0.651355
\(696\) −28.9706 −1.09813
\(697\) 7.31371 0.277026
\(698\) − 29.3137i − 1.10954i
\(699\) − 22.5858i − 0.854273i
\(700\) 1.17157 0.0442813
\(701\) − 46.2426i − 1.74656i −0.487218 0.873280i \(-0.661988\pi\)
0.487218 0.873280i \(-0.338012\pi\)
\(702\) −1.41421 −0.0533761
\(703\) 1.57359 0.0593492
\(704\) − 38.6274i − 1.45583i
\(705\) −1.65685 −0.0624007
\(706\) 8.68629 0.326913
\(707\) − 0.343146i − 0.0129053i
\(708\) 6.34315 0.238390
\(709\) − 13.8995i − 0.522006i −0.965338 0.261003i \(-0.915947\pi\)
0.965338 0.261003i \(-0.0840533\pi\)
\(710\) − 6.82843i − 0.256266i
\(711\) 1.65685 0.0621369
\(712\) 21.6569 0.811625
\(713\) 4.97056 0.186149
\(714\) − 1.17157i − 0.0438450i
\(715\) − 4.82843i − 0.180573i
\(716\) 31.7990i 1.18838i
\(717\) 11.3137i 0.422518i
\(718\) 32.7696 1.22295
\(719\) 42.4264 1.58224 0.791119 0.611662i \(-0.209499\pi\)
0.791119 + 0.611662i \(0.209499\pi\)
\(720\) 4.00000i 0.149071i
\(721\) −4.48528 −0.167041
\(722\) 26.7868 0.996901
\(723\) 1.51472i 0.0563330i
\(724\) 13.6569i 0.507553i
\(725\) 10.2426i 0.380402i
\(726\) 17.4142i 0.646302i
\(727\) 23.1716 0.859386 0.429693 0.902975i \(-0.358622\pi\)
0.429693 + 0.902975i \(0.358622\pi\)
\(728\) − 1.65685i − 0.0614071i
\(729\) −1.00000 −0.0370370
\(730\) − 12.1421i − 0.449401i
\(731\) − 6.82843i − 0.252559i
\(732\) 16.0000 0.591377
\(733\) 42.9706i 1.58715i 0.608470 + 0.793577i \(0.291784\pi\)
−0.608470 + 0.793577i \(0.708216\pi\)
\(734\) −36.7696 −1.35719
\(735\) 6.65685 0.245542
\(736\) 24.0000 0.884652
\(737\) 5.65685 0.208373
\(738\) 7.31371 0.269221
\(739\) 0.242641i 0.00892568i 0.999990 + 0.00446284i \(0.00142057\pi\)
−0.999990 + 0.00446284i \(0.998579\pi\)
\(740\) −12.9706 −0.476807
\(741\) 0.242641i 0.00891363i
\(742\) − 0.284271i − 0.0104359i
\(743\) 33.1716 1.21695 0.608473 0.793574i \(-0.291782\pi\)
0.608473 + 0.793574i \(0.291782\pi\)
\(744\) − 3.31371i − 0.121486i
\(745\) 11.1716 0.409295
\(746\) − 32.2843i − 1.18201i
\(747\) − 7.65685i − 0.280150i
\(748\) 13.6569i 0.499344i
\(749\) 0.686292i 0.0250765i
\(750\) 1.41421 0.0516398
\(751\) −1.02944 −0.0375647 −0.0187823 0.999824i \(-0.505979\pi\)
−0.0187823 + 0.999824i \(0.505979\pi\)
\(752\) 6.62742 0.241677
\(753\) 2.92893 0.106736
\(754\) 14.4853 0.527523
\(755\) 12.0000i 0.436725i
\(756\) − 1.17157i − 0.0426097i
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 2.68629i 0.0975705i
\(759\) −20.4853 −0.743569
\(760\) 0.686292 0.0248944
\(761\) −25.4558 −0.922774 −0.461387 0.887199i \(-0.652648\pi\)
−0.461387 + 0.887199i \(0.652648\pi\)
\(762\) 26.8284i 0.971891i
\(763\) 2.48528i 0.0899732i
\(764\) 31.5980 1.14317
\(765\) − 1.41421i − 0.0511310i
\(766\) 50.6274 1.82924
\(767\) −3.17157 −0.114519
\(768\) − 16.0000i − 0.577350i
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 4.00000 0.144150
\(771\) 10.1005i 0.363761i
\(772\) 6.82843 0.245760
\(773\) − 13.6569i − 0.491203i −0.969371 0.245601i \(-0.921015\pi\)
0.969371 0.245601i \(-0.0789854\pi\)
\(774\) − 6.82843i − 0.245443i
\(775\) −1.17157 −0.0420841
\(776\) 19.0294 0.683117
\(777\) 3.79899 0.136288
\(778\) 38.7696i 1.38996i
\(779\) − 1.25483i − 0.0449591i
\(780\) − 2.00000i − 0.0716115i
\(781\) − 23.3137i − 0.834230i
\(782\) −8.48528 −0.303433
\(783\) 10.2426 0.366042
\(784\) −26.6274 −0.950979
\(785\) −6.82843 −0.243717
\(786\) 14.4853 0.516673
\(787\) 13.6569i 0.486814i 0.969924 + 0.243407i \(0.0782651\pi\)
−0.969924 + 0.243407i \(0.921735\pi\)
\(788\) 0 0
\(789\) − 4.24264i − 0.151042i
\(790\) 2.34315i 0.0833654i
\(791\) −6.20101 −0.220483
\(792\) 13.6569i 0.485275i
\(793\) −8.00000 −0.284088
\(794\) 15.1127i 0.536330i
\(795\) − 0.343146i − 0.0121701i
\(796\) 0 0
\(797\) 38.7696i 1.37329i 0.726994 + 0.686644i \(0.240917\pi\)
−0.726994 + 0.686644i \(0.759083\pi\)
\(798\) −0.201010 −0.00711568
\(799\) −2.34315 −0.0828945
\(800\) −5.65685 −0.200000
\(801\) −7.65685 −0.270542
\(802\) −14.3431 −0.506474
\(803\) − 41.4558i − 1.46294i
\(804\) 2.34315 0.0826364
\(805\) 2.48528i 0.0875947i
\(806\) 1.65685i 0.0583602i
\(807\) −15.2132 −0.535530
\(808\) 1.65685i 0.0582879i
\(809\) −49.1127 −1.72671 −0.863355 0.504597i \(-0.831641\pi\)
−0.863355 + 0.504597i \(0.831641\pi\)
\(810\) − 1.41421i − 0.0496904i
\(811\) − 45.6985i − 1.60469i −0.596860 0.802345i \(-0.703585\pi\)
0.596860 0.802345i \(-0.296415\pi\)
\(812\) 12.0000i 0.421117i
\(813\) − 1.65685i − 0.0581084i
\(814\) −44.2843 −1.55216
\(815\) −8.00000 −0.280228
\(816\) 5.65685i 0.198030i
\(817\) −1.17157 −0.0409881
\(818\) −8.20101 −0.286742
\(819\) 0.585786i 0.0204690i
\(820\) 10.3431i 0.361198i
\(821\) 20.1421i 0.702965i 0.936194 + 0.351483i \(0.114322\pi\)
−0.936194 + 0.351483i \(0.885678\pi\)
\(822\) 12.0000i 0.418548i
\(823\) 21.3137 0.742949 0.371475 0.928443i \(-0.378852\pi\)
0.371475 + 0.928443i \(0.378852\pi\)
\(824\) 21.6569 0.754452
\(825\) 4.82843 0.168104
\(826\) − 2.62742i − 0.0914195i
\(827\) 37.3137i 1.29752i 0.760991 + 0.648762i \(0.224713\pi\)
−0.760991 + 0.648762i \(0.775287\pi\)
\(828\) −8.48528 −0.294884
\(829\) − 7.02944i − 0.244142i −0.992521 0.122071i \(-0.961046\pi\)
0.992521 0.122071i \(-0.0389536\pi\)
\(830\) 10.8284 0.375860
\(831\) −13.1716 −0.456917
\(832\) 8.00000i 0.277350i
\(833\) 9.41421 0.326183
\(834\) 24.2843 0.840896
\(835\) − 19.7990i − 0.685172i
\(836\) 2.34315 0.0810394
\(837\) 1.17157i 0.0404955i
\(838\) − 0.142136i − 0.00490999i
\(839\) −45.6569 −1.57625 −0.788125 0.615515i \(-0.788948\pi\)
−0.788125 + 0.615515i \(0.788948\pi\)
\(840\) 1.65685 0.0571669
\(841\) −75.9117 −2.61764
\(842\) 10.2843i 0.354419i
\(843\) 27.6569i 0.952553i
\(844\) 49.9411i 1.71904i
\(845\) 1.00000i 0.0344010i
\(846\) −2.34315 −0.0805590
\(847\) 7.21320 0.247849
\(848\) 1.37258i 0.0471347i
\(849\) 16.1421 0.553997
\(850\) 2.00000 0.0685994
\(851\) − 27.5147i − 0.943192i
\(852\) − 9.65685i − 0.330838i
\(853\) 47.9411i 1.64147i 0.571307 + 0.820736i \(0.306437\pi\)
−0.571307 + 0.820736i \(0.693563\pi\)
\(854\) − 6.62742i − 0.226786i
\(855\) −0.242641 −0.00829814
\(856\) − 3.31371i − 0.113260i
\(857\) −23.5563 −0.804670 −0.402335 0.915493i \(-0.631801\pi\)
−0.402335 + 0.915493i \(0.631801\pi\)
\(858\) − 6.82843i − 0.233119i
\(859\) − 13.9411i − 0.475665i −0.971306 0.237833i \(-0.923563\pi\)
0.971306 0.237833i \(-0.0764369\pi\)
\(860\) 9.65685 0.329296
\(861\) − 3.02944i − 0.103243i
\(862\) 27.3137 0.930309
\(863\) 26.1421 0.889889 0.444944 0.895558i \(-0.353223\pi\)
0.444944 + 0.895558i \(0.353223\pi\)
\(864\) 5.65685i 0.192450i
\(865\) −12.1421 −0.412845
\(866\) −48.7696 −1.65726
\(867\) 15.0000i 0.509427i
\(868\) −1.37258 −0.0465885
\(869\) 8.00000i 0.271381i
\(870\) 14.4853i 0.491097i
\(871\) −1.17157 −0.0396972
\(872\) − 12.0000i − 0.406371i
\(873\) −6.72792 −0.227706
\(874\) 1.45584i 0.0492447i
\(875\) − 0.585786i − 0.0198032i
\(876\) − 17.1716i − 0.580174i
\(877\) − 36.4264i − 1.23003i −0.788514 0.615016i \(-0.789149\pi\)
0.788514 0.615016i \(-0.210851\pi\)
\(878\) −44.7696 −1.51090
\(879\) 7.31371 0.246685
\(880\) −19.3137 −0.651065
\(881\) −51.9411 −1.74994 −0.874970 0.484176i \(-0.839119\pi\)
−0.874970 + 0.484176i \(0.839119\pi\)
\(882\) 9.41421 0.316993
\(883\) 4.97056i 0.167273i 0.996496 + 0.0836364i \(0.0266534\pi\)
−0.996496 + 0.0836364i \(0.973347\pi\)
\(884\) − 2.82843i − 0.0951303i
\(885\) − 3.17157i − 0.106611i
\(886\) − 19.3137i − 0.648857i
\(887\) 0.928932 0.0311905 0.0155952 0.999878i \(-0.495036\pi\)
0.0155952 + 0.999878i \(0.495036\pi\)
\(888\) −18.3431 −0.615556
\(889\) 11.1127 0.372708
\(890\) − 10.8284i − 0.362970i
\(891\) − 4.82843i − 0.161758i
\(892\) 54.4264 1.82233
\(893\) 0.402020i 0.0134531i
\(894\) 15.7990 0.528397
\(895\) 15.8995 0.531462
\(896\) −6.62742 −0.221406
\(897\) 4.24264 0.141658
\(898\) −8.48528 −0.283158
\(899\) − 12.0000i − 0.400222i
\(900\) 2.00000 0.0666667
\(901\) − 0.485281i − 0.0161671i
\(902\) 35.3137i 1.17582i
\(903\) −2.82843 −0.0941242
\(904\) 29.9411 0.995827
\(905\) 6.82843 0.226985
\(906\) 16.9706i 0.563809i
\(907\) − 32.4264i − 1.07670i −0.842721 0.538351i \(-0.819048\pi\)
0.842721 0.538351i \(-0.180952\pi\)
\(908\) 30.6274i 1.01641i
\(909\) − 0.585786i − 0.0194293i
\(910\) −0.828427 −0.0274621
\(911\) 32.4853 1.07629 0.538143 0.842854i \(-0.319126\pi\)
0.538143 + 0.842854i \(0.319126\pi\)
\(912\) 0.970563 0.0321385
\(913\) 36.9706 1.22355
\(914\) 8.14214 0.269318
\(915\) − 8.00000i − 0.264472i
\(916\) 11.7990i 0.389850i
\(917\) − 6.00000i − 0.198137i
\(918\) − 2.00000i − 0.0660098i
\(919\) 27.9411 0.921693 0.460846 0.887480i \(-0.347546\pi\)
0.460846 + 0.887480i \(0.347546\pi\)
\(920\) − 12.0000i − 0.395628i
\(921\) −3.31371 −0.109190
\(922\) − 50.4264i − 1.66071i
\(923\) 4.82843i 0.158930i
\(924\) 5.65685 0.186097
\(925\) 6.48528i 0.213235i
\(926\) −49.1127 −1.61394
\(927\) −7.65685 −0.251484
\(928\) − 57.9411i − 1.90201i
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −1.65685 −0.0543304
\(931\) − 1.61522i − 0.0529368i
\(932\) 45.1716 1.47964
\(933\) − 14.1421i − 0.462993i
\(934\) − 49.6569i − 1.62482i
\(935\) 6.82843 0.223313
\(936\) − 2.82843i − 0.0924500i
\(937\) 30.9706 1.01176 0.505882 0.862603i \(-0.331167\pi\)
0.505882 + 0.862603i \(0.331167\pi\)
\(938\) − 0.970563i − 0.0316900i
\(939\) − 16.8284i − 0.549175i
\(940\) − 3.31371i − 0.108081i
\(941\) − 10.2010i − 0.332543i −0.986080 0.166272i \(-0.946827\pi\)
0.986080 0.166272i \(-0.0531729\pi\)
\(942\) −9.65685 −0.314637
\(943\) −21.9411 −0.714501
\(944\) 12.6863i 0.412904i
\(945\) −0.585786 −0.0190556
\(946\) 32.9706 1.07197
\(947\) − 37.5980i − 1.22177i −0.791719 0.610885i \(-0.790814\pi\)
0.791719 0.610885i \(-0.209186\pi\)
\(948\) 3.31371i 0.107624i
\(949\) 8.58579i 0.278706i
\(950\) − 0.343146i − 0.0111331i
\(951\) 20.6274 0.668890
\(952\) 2.34315 0.0759418
\(953\) −48.3259 −1.56543 −0.782715 0.622381i \(-0.786166\pi\)
−0.782715 + 0.622381i \(0.786166\pi\)
\(954\) − 0.485281i − 0.0157116i
\(955\) − 15.7990i − 0.511243i
\(956\) −22.6274 −0.731823
\(957\) 49.4558i 1.59868i
\(958\) 16.0000 0.516937
\(959\) 4.97056 0.160508
\(960\) −8.00000 −0.258199
\(961\) −29.6274 −0.955723
\(962\) 9.17157 0.295703
\(963\) 1.17157i 0.0377534i
\(964\) −3.02944 −0.0975716
\(965\) − 3.41421i − 0.109907i
\(966\) 3.51472i 0.113084i
\(967\) 8.38478 0.269636 0.134818 0.990870i \(-0.456955\pi\)
0.134818 + 0.990870i \(0.456955\pi\)
\(968\) −34.8284 −1.11943
\(969\) −0.343146 −0.0110234
\(970\) − 9.51472i − 0.305499i
\(971\) − 24.5858i − 0.788995i −0.918897 0.394498i \(-0.870919\pi\)
0.918897 0.394498i \(-0.129081\pi\)
\(972\) − 2.00000i − 0.0641500i
\(973\) − 10.0589i − 0.322473i
\(974\) −40.1421 −1.28624
\(975\) −1.00000 −0.0320256
\(976\) 32.0000i 1.02430i
\(977\) 18.3431 0.586849 0.293425 0.955982i \(-0.405205\pi\)
0.293425 + 0.955982i \(0.405205\pi\)
\(978\) −11.3137 −0.361773
\(979\) − 36.9706i − 1.18158i
\(980\) 13.3137i 0.425291i
\(981\) 4.24264i 0.135457i
\(982\) − 57.7990i − 1.84444i
\(983\) −39.7990 −1.26939 −0.634695 0.772762i \(-0.718874\pi\)
−0.634695 + 0.772762i \(0.718874\pi\)
\(984\) 14.6274i 0.466305i
\(985\) 0 0
\(986\) 20.4853i 0.652384i
\(987\) 0.970563i 0.0308934i
\(988\) −0.485281 −0.0154389
\(989\) 20.4853i 0.651394i
\(990\) 6.82843 0.217022
\(991\) 48.9706 1.55560 0.777801 0.628511i \(-0.216335\pi\)
0.777801 + 0.628511i \(0.216335\pi\)
\(992\) 6.62742 0.210421
\(993\) 33.6985 1.06939
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 15.3137 0.485233
\(997\) − 8.48528i − 0.268732i −0.990932 0.134366i \(-0.957100\pi\)
0.990932 0.134366i \(-0.0428997\pi\)
\(998\) 3.65685i 0.115756i
\(999\) 6.48528 0.205185
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 1560.2.w.c.781.3 4
4.3 odd 2 6240.2.w.d.3121.3 4
8.3 odd 2 6240.2.w.d.3121.1 4
8.5 even 2 inner 1560.2.w.c.781.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.2.w.c.781.3 4 1.1 even 1 trivial
1560.2.w.c.781.4 yes 4 8.5 even 2 inner
6240.2.w.d.3121.1 4 8.3 odd 2
6240.2.w.d.3121.3 4 4.3 odd 2