Properties

Label 725.2.b.c.349.3
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(349,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.b (of order \(2\), degree \(1\), not 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: \(5.78915414654\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.c.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} -4.82843i q^{7} +1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} -4.82843i q^{7} +1.58579i q^{8} -1.00000 q^{9} +0.828427 q^{11} +3.65685i q^{12} +2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843i q^{17} -0.414214i q^{18} +4.82843 q^{19} +9.65685 q^{21} +0.343146i q^{22} +3.17157i q^{23} -3.17157 q^{24} -0.828427 q^{26} +4.00000i q^{27} -8.82843i q^{28} -1.00000 q^{29} +6.48528 q^{31} +4.41421i q^{32} +1.65685i q^{33} -1.17157 q^{34} -1.82843 q^{36} -8.48528i q^{37} +2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{42} +6.00000i q^{43} +1.51472 q^{44} -1.31371 q^{46} -11.6569i q^{47} +6.00000i q^{48} -16.3137 q^{49} -5.65685 q^{51} +3.65685i q^{52} +3.65685i q^{53} -1.65685 q^{54} +7.65685 q^{56} +9.65685i q^{57} -0.414214i q^{58} -3.65685 q^{61} +2.68629i q^{62} +4.82843i q^{63} +4.17157 q^{64} -0.686292 q^{66} +6.48528i q^{67} +5.17157i q^{68} -6.34315 q^{69} -15.3137 q^{71} -1.58579i q^{72} -8.48528i q^{73} +3.51472 q^{74} +8.82843 q^{76} -4.00000i q^{77} -1.65685i q^{78} +2.48528 q^{79} -11.0000 q^{81} -2.48528i q^{82} -7.17157i q^{83} +17.6569 q^{84} -2.48528 q^{86} -2.00000i q^{87} +1.31371i q^{88} +7.65685 q^{89} +9.65685 q^{91} +5.79899i q^{92} +12.9706i q^{93} +4.82843 q^{94} -8.82843 q^{96} -12.4853i q^{97} -6.75736i q^{98} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46} - 20 q^{49} + 16 q^{54} + 8 q^{56} + 8 q^{61} + 28 q^{64} - 48 q^{66} - 48 q^{69} - 16 q^{71} + 48 q^{74} + 24 q^{76} - 24 q^{79} - 44 q^{81} + 48 q^{84} + 24 q^{86} + 8 q^{89} + 16 q^{91} + 8 q^{94} - 24 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214i 0.292893i 0.989219 + 0.146447i \(0.0467837\pi\)
−0.989219 + 0.146447i \(0.953216\pi\)
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) −0.828427 −0.338204
\(7\) − 4.82843i − 1.82497i −0.409106 0.912487i \(-0.634159\pi\)
0.409106 0.912487i \(-0.365841\pi\)
\(8\) 1.58579i 0.560660i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 3.65685i 1.05564i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) − 0.414214i − 0.0976311i
\(19\) 4.82843 1.10772 0.553859 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 0 0
\(21\) 9.65685 2.10730
\(22\) 0.343146i 0.0731589i
\(23\) 3.17157i 0.661319i 0.943750 + 0.330659i \(0.107271\pi\)
−0.943750 + 0.330659i \(0.892729\pi\)
\(24\) −3.17157 −0.647395
\(25\) 0 0
\(26\) −0.828427 −0.162468
\(27\) 4.00000i 0.769800i
\(28\) − 8.82843i − 1.66842i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) 4.41421i 0.780330i
\(33\) 1.65685i 0.288421i
\(34\) −1.17157 −0.200923
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) − 8.48528i − 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) 2.00000i 0.324443i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 1.51472 0.228352
\(45\) 0 0
\(46\) −1.31371 −0.193696
\(47\) − 11.6569i − 1.70033i −0.526519 0.850163i \(-0.676503\pi\)
0.526519 0.850163i \(-0.323497\pi\)
\(48\) 6.00000i 0.866025i
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) −5.65685 −0.792118
\(52\) 3.65685i 0.507114i
\(53\) 3.65685i 0.502308i 0.967947 + 0.251154i \(0.0808100\pi\)
−0.967947 + 0.251154i \(0.919190\pi\)
\(54\) −1.65685 −0.225469
\(55\) 0 0
\(56\) 7.65685 1.02319
\(57\) 9.65685i 1.27908i
\(58\) − 0.414214i − 0.0543889i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 2.68629i 0.341159i
\(63\) 4.82843i 0.608325i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) −0.686292 −0.0844766
\(67\) 6.48528i 0.792303i 0.918185 + 0.396152i \(0.129655\pi\)
−0.918185 + 0.396152i \(0.870345\pi\)
\(68\) 5.17157i 0.627145i
\(69\) −6.34315 −0.763625
\(70\) 0 0
\(71\) −15.3137 −1.81740 −0.908701 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) − 1.58579i − 0.186887i
\(73\) − 8.48528i − 0.993127i −0.868000 0.496564i \(-0.834595\pi\)
0.868000 0.496564i \(-0.165405\pi\)
\(74\) 3.51472 0.408578
\(75\) 0 0
\(76\) 8.82843 1.01269
\(77\) − 4.00000i − 0.455842i
\(78\) − 1.65685i − 0.187602i
\(79\) 2.48528 0.279616 0.139808 0.990179i \(-0.455351\pi\)
0.139808 + 0.990179i \(0.455351\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 2.48528i − 0.274453i
\(83\) − 7.17157i − 0.787182i −0.919286 0.393591i \(-0.871233\pi\)
0.919286 0.393591i \(-0.128767\pi\)
\(84\) 17.6569 1.92652
\(85\) 0 0
\(86\) −2.48528 −0.267995
\(87\) − 2.00000i − 0.214423i
\(88\) 1.31371i 0.140042i
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) 5.79899i 0.604586i
\(93\) 12.9706i 1.34498i
\(94\) 4.82843 0.498014
\(95\) 0 0
\(96\) −8.82843 −0.901048
\(97\) − 12.4853i − 1.26769i −0.773461 0.633844i \(-0.781476\pi\)
0.773461 0.633844i \(-0.218524\pi\)
\(98\) − 6.75736i − 0.682596i
\(99\) −0.828427 −0.0832601
\(100\) 0 0
\(101\) 15.6569 1.55792 0.778958 0.627077i \(-0.215749\pi\)
0.778958 + 0.627077i \(0.215749\pi\)
\(102\) − 2.34315i − 0.232006i
\(103\) − 16.1421i − 1.59053i −0.606261 0.795266i \(-0.707331\pi\)
0.606261 0.795266i \(-0.292669\pi\)
\(104\) −3.17157 −0.310998
\(105\) 0 0
\(106\) −1.51472 −0.147122
\(107\) 20.1421i 1.94721i 0.228232 + 0.973607i \(0.426706\pi\)
−0.228232 + 0.973607i \(0.573294\pi\)
\(108\) 7.31371i 0.703762i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 16.9706 1.61077
\(112\) − 14.4853i − 1.36873i
\(113\) 2.82843i 0.266076i 0.991111 + 0.133038i \(0.0424732\pi\)
−0.991111 + 0.133038i \(0.957527\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −1.82843 −0.169765
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) − 1.51472i − 0.137136i
\(123\) − 12.0000i − 1.08200i
\(124\) 11.8579 1.06487
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 10.5563i 0.933058i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −12.1421 −1.06086 −0.530432 0.847728i \(-0.677970\pi\)
−0.530432 + 0.847728i \(0.677970\pi\)
\(132\) 3.02944i 0.263679i
\(133\) − 23.3137i − 2.02155i
\(134\) −2.68629 −0.232060
\(135\) 0 0
\(136\) −4.48528 −0.384610
\(137\) − 5.17157i − 0.441837i −0.975292 0.220919i \(-0.929094\pi\)
0.975292 0.220919i \(-0.0709055\pi\)
\(138\) − 2.62742i − 0.223661i
\(139\) −21.6569 −1.83691 −0.918455 0.395525i \(-0.870563\pi\)
−0.918455 + 0.395525i \(0.870563\pi\)
\(140\) 0 0
\(141\) 23.3137 1.96337
\(142\) − 6.34315i − 0.532305i
\(143\) 1.65685i 0.138553i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 3.51472 0.290880
\(147\) − 32.6274i − 2.69106i
\(148\) − 15.5147i − 1.27530i
\(149\) −9.31371 −0.763009 −0.381504 0.924367i \(-0.624594\pi\)
−0.381504 + 0.924367i \(0.624594\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 7.65685i 0.621053i
\(153\) − 2.82843i − 0.228665i
\(154\) 1.65685 0.133513
\(155\) 0 0
\(156\) −7.31371 −0.585565
\(157\) 0.485281i 0.0387297i 0.999812 + 0.0193648i \(0.00616440\pi\)
−0.999812 + 0.0193648i \(0.993836\pi\)
\(158\) 1.02944i 0.0818976i
\(159\) −7.31371 −0.580015
\(160\) 0 0
\(161\) 15.3137 1.20689
\(162\) − 4.55635i − 0.357981i
\(163\) 8.34315i 0.653486i 0.945113 + 0.326743i \(0.105951\pi\)
−0.945113 + 0.326743i \(0.894049\pi\)
\(164\) −10.9706 −0.856657
\(165\) 0 0
\(166\) 2.97056 0.230560
\(167\) − 2.48528i − 0.192317i −0.995366 0.0961584i \(-0.969344\pi\)
0.995366 0.0961584i \(-0.0306555\pi\)
\(168\) 15.3137i 1.18148i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.82843 −0.369239
\(172\) 10.9706i 0.836498i
\(173\) − 17.3137i − 1.31634i −0.752871 0.658168i \(-0.771331\pi\)
0.752871 0.658168i \(-0.228669\pi\)
\(174\) 0.828427 0.0628029
\(175\) 0 0
\(176\) 2.48528 0.187335
\(177\) 0 0
\(178\) 3.17157i 0.237719i
\(179\) 23.3137 1.74255 0.871274 0.490797i \(-0.163294\pi\)
0.871274 + 0.490797i \(0.163294\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 4.00000i 0.296500i
\(183\) − 7.31371i − 0.540645i
\(184\) −5.02944 −0.370775
\(185\) 0 0
\(186\) −5.37258 −0.393937
\(187\) 2.34315i 0.171348i
\(188\) − 21.3137i − 1.55446i
\(189\) 19.3137 1.40487
\(190\) 0 0
\(191\) −20.8284 −1.50709 −0.753546 0.657395i \(-0.771658\pi\)
−0.753546 + 0.657395i \(0.771658\pi\)
\(192\) 8.34315i 0.602115i
\(193\) − 4.48528i − 0.322858i −0.986884 0.161429i \(-0.948390\pi\)
0.986884 0.161429i \(-0.0516102\pi\)
\(194\) 5.17157 0.371297
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) − 19.6569i − 1.40049i −0.713901 0.700246i \(-0.753073\pi\)
0.713901 0.700246i \(-0.246927\pi\)
\(198\) − 0.343146i − 0.0243863i
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) −12.9706 −0.914873
\(202\) 6.48528i 0.456303i
\(203\) 4.82843i 0.338889i
\(204\) −10.3431 −0.724165
\(205\) 0 0
\(206\) 6.68629 0.465856
\(207\) − 3.17157i − 0.220440i
\(208\) 6.00000i 0.416025i
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −0.828427 −0.0570313 −0.0285156 0.999593i \(-0.509078\pi\)
−0.0285156 + 0.999593i \(0.509078\pi\)
\(212\) 6.68629i 0.459216i
\(213\) − 30.6274i − 2.09856i
\(214\) −8.34315 −0.570326
\(215\) 0 0
\(216\) −6.34315 −0.431596
\(217\) − 31.3137i − 2.12571i
\(218\) − 0.828427i − 0.0561082i
\(219\) 16.9706 1.14676
\(220\) 0 0
\(221\) −5.65685 −0.380521
\(222\) 7.02944i 0.471785i
\(223\) 17.7990i 1.19191i 0.803018 + 0.595954i \(0.203226\pi\)
−0.803018 + 0.595954i \(0.796774\pi\)
\(224\) 21.3137 1.42408
\(225\) 0 0
\(226\) −1.17157 −0.0779319
\(227\) 20.1421i 1.33688i 0.743766 + 0.668440i \(0.233038\pi\)
−0.743766 + 0.668440i \(0.766962\pi\)
\(228\) 17.6569i 1.16935i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) − 1.58579i − 0.104112i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0.828427 0.0541560
\(235\) 0 0
\(236\) 0 0
\(237\) 4.97056i 0.322873i
\(238\) 5.65685i 0.366679i
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 4.27208i − 0.274620i
\(243\) − 10.0000i − 0.641500i
\(244\) −6.68629 −0.428046
\(245\) 0 0
\(246\) 4.97056 0.316912
\(247\) 9.65685i 0.614451i
\(248\) 10.2843i 0.653052i
\(249\) 14.3431 0.908960
\(250\) 0 0
\(251\) 8.82843 0.557245 0.278623 0.960401i \(-0.410122\pi\)
0.278623 + 0.960401i \(0.410122\pi\)
\(252\) 8.82843i 0.556139i
\(253\) 2.62742i 0.165184i
\(254\) −2.48528 −0.155940
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 6.68629i 0.417079i 0.978014 + 0.208540i \(0.0668710\pi\)
−0.978014 + 0.208540i \(0.933129\pi\)
\(258\) − 4.97056i − 0.309454i
\(259\) −40.9706 −2.54579
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) − 5.02944i − 0.310720i
\(263\) 19.6569i 1.21209i 0.795429 + 0.606047i \(0.207246\pi\)
−0.795429 + 0.606047i \(0.792754\pi\)
\(264\) −2.62742 −0.161706
\(265\) 0 0
\(266\) 9.65685 0.592100
\(267\) 15.3137i 0.937184i
\(268\) 11.8579i 0.724334i
\(269\) 21.3137 1.29952 0.649760 0.760140i \(-0.274869\pi\)
0.649760 + 0.760140i \(0.274869\pi\)
\(270\) 0 0
\(271\) −9.79899 −0.595246 −0.297623 0.954683i \(-0.596194\pi\)
−0.297623 + 0.954683i \(0.596194\pi\)
\(272\) 8.48528i 0.514496i
\(273\) 19.3137i 1.16892i
\(274\) 2.14214 0.129411
\(275\) 0 0
\(276\) −11.5980 −0.698116
\(277\) − 3.65685i − 0.219719i −0.993947 0.109860i \(-0.964960\pi\)
0.993947 0.109860i \(-0.0350401\pi\)
\(278\) − 8.97056i − 0.538019i
\(279\) −6.48528 −0.388264
\(280\) 0 0
\(281\) −29.3137 −1.74871 −0.874355 0.485288i \(-0.838715\pi\)
−0.874355 + 0.485288i \(0.838715\pi\)
\(282\) 9.65685i 0.575057i
\(283\) − 4.82843i − 0.287020i −0.989649 0.143510i \(-0.954161\pi\)
0.989649 0.143510i \(-0.0458390\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) −0.686292 −0.0405813
\(287\) 28.9706i 1.71008i
\(288\) − 4.41421i − 0.260110i
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) 24.9706 1.46380
\(292\) − 15.5147i − 0.907930i
\(293\) − 8.48528i − 0.495715i −0.968796 0.247858i \(-0.920273\pi\)
0.968796 0.247858i \(-0.0797265\pi\)
\(294\) 13.5147 0.788194
\(295\) 0 0
\(296\) 13.4558 0.782105
\(297\) 3.31371i 0.192281i
\(298\) − 3.85786i − 0.223480i
\(299\) −6.34315 −0.366834
\(300\) 0 0
\(301\) 28.9706 1.66984
\(302\) − 4.97056i − 0.286024i
\(303\) 31.3137i 1.79893i
\(304\) 14.4853 0.830788
\(305\) 0 0
\(306\) 1.17157 0.0669744
\(307\) − 22.9706i − 1.31100i −0.755195 0.655500i \(-0.772458\pi\)
0.755195 0.655500i \(-0.227542\pi\)
\(308\) − 7.31371i − 0.416737i
\(309\) 32.2843 1.83659
\(310\) 0 0
\(311\) 14.4853 0.821385 0.410692 0.911774i \(-0.365287\pi\)
0.410692 + 0.911774i \(0.365287\pi\)
\(312\) − 6.34315i − 0.359110i
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) −0.201010 −0.0113437
\(315\) 0 0
\(316\) 4.54416 0.255629
\(317\) 2.82843i 0.158860i 0.996840 + 0.0794301i \(0.0253101\pi\)
−0.996840 + 0.0794301i \(0.974690\pi\)
\(318\) − 3.02944i − 0.169882i
\(319\) −0.828427 −0.0463830
\(320\) 0 0
\(321\) −40.2843 −2.24845
\(322\) 6.34315i 0.353490i
\(323\) 13.6569i 0.759888i
\(324\) −20.1127 −1.11737
\(325\) 0 0
\(326\) −3.45584 −0.191402
\(327\) − 4.00000i − 0.221201i
\(328\) − 9.51472i − 0.525362i
\(329\) −56.2843 −3.10305
\(330\) 0 0
\(331\) 21.7990 1.19818 0.599090 0.800681i \(-0.295529\pi\)
0.599090 + 0.800681i \(0.295529\pi\)
\(332\) − 13.1127i − 0.719653i
\(333\) 8.48528i 0.464991i
\(334\) 1.02944 0.0563283
\(335\) 0 0
\(336\) 28.9706 1.58047
\(337\) − 1.17157i − 0.0638196i −0.999491 0.0319098i \(-0.989841\pi\)
0.999491 0.0319098i \(-0.0101589\pi\)
\(338\) 3.72792i 0.202772i
\(339\) −5.65685 −0.307238
\(340\) 0 0
\(341\) 5.37258 0.290942
\(342\) − 2.00000i − 0.108148i
\(343\) 44.9706i 2.42818i
\(344\) −9.51472 −0.512999
\(345\) 0 0
\(346\) 7.17157 0.385546
\(347\) − 8.14214i − 0.437093i −0.975827 0.218546i \(-0.929869\pi\)
0.975827 0.218546i \(-0.0701315\pi\)
\(348\) − 3.65685i − 0.196028i
\(349\) −20.6274 −1.10416 −0.552080 0.833791i \(-0.686166\pi\)
−0.552080 + 0.833791i \(0.686166\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) 3.65685i 0.194911i
\(353\) 4.34315i 0.231162i 0.993298 + 0.115581i \(0.0368730\pi\)
−0.993298 + 0.115581i \(0.963127\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 27.3137i 1.44559i
\(358\) 9.65685i 0.510381i
\(359\) 3.85786 0.203610 0.101805 0.994804i \(-0.467538\pi\)
0.101805 + 0.994804i \(0.467538\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) − 2.48528i − 0.130623i
\(363\) − 20.6274i − 1.08266i
\(364\) 17.6569 0.925471
\(365\) 0 0
\(366\) 3.02944 0.158351
\(367\) 18.0000i 0.939592i 0.882775 + 0.469796i \(0.155673\pi\)
−0.882775 + 0.469796i \(0.844327\pi\)
\(368\) 9.51472i 0.495989i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 17.6569 0.916698
\(372\) 23.7157i 1.22960i
\(373\) 6.97056i 0.360922i 0.983582 + 0.180461i \(0.0577590\pi\)
−0.983582 + 0.180461i \(0.942241\pi\)
\(374\) −0.970563 −0.0501866
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) − 2.00000i − 0.103005i
\(378\) 8.00000i 0.411476i
\(379\) −22.4853 −1.15499 −0.577496 0.816394i \(-0.695970\pi\)
−0.577496 + 0.816394i \(0.695970\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) − 8.62742i − 0.441417i
\(383\) − 2.48528i − 0.126992i −0.997982 0.0634960i \(-0.979775\pi\)
0.997982 0.0634960i \(-0.0202250\pi\)
\(384\) −21.1127 −1.07740
\(385\) 0 0
\(386\) 1.85786 0.0945628
\(387\) − 6.00000i − 0.304997i
\(388\) − 22.8284i − 1.15894i
\(389\) 29.3137 1.48626 0.743132 0.669145i \(-0.233339\pi\)
0.743132 + 0.669145i \(0.233339\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) − 25.8701i − 1.30664i
\(393\) − 24.2843i − 1.22498i
\(394\) 8.14214 0.410195
\(395\) 0 0
\(396\) −1.51472 −0.0761175
\(397\) − 19.6569i − 0.986549i −0.869874 0.493275i \(-0.835800\pi\)
0.869874 0.493275i \(-0.164200\pi\)
\(398\) − 4.97056i − 0.249152i
\(399\) 46.6274 2.33429
\(400\) 0 0
\(401\) −6.68629 −0.333897 −0.166949 0.985966i \(-0.553391\pi\)
−0.166949 + 0.985966i \(0.553391\pi\)
\(402\) − 5.37258i − 0.267960i
\(403\) 12.9706i 0.646110i
\(404\) 28.6274 1.42427
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) − 7.02944i − 0.348436i
\(408\) − 8.97056i − 0.444109i
\(409\) 2.97056 0.146885 0.0734424 0.997299i \(-0.476601\pi\)
0.0734424 + 0.997299i \(0.476601\pi\)
\(410\) 0 0
\(411\) 10.3431 0.510190
\(412\) − 29.5147i − 1.45409i
\(413\) 0 0
\(414\) 1.31371 0.0645653
\(415\) 0 0
\(416\) −8.82843 −0.432849
\(417\) − 43.3137i − 2.12108i
\(418\) 1.65685i 0.0810394i
\(419\) −28.9706 −1.41530 −0.707652 0.706561i \(-0.750246\pi\)
−0.707652 + 0.706561i \(0.750246\pi\)
\(420\) 0 0
\(421\) 18.9706 0.924569 0.462284 0.886732i \(-0.347030\pi\)
0.462284 + 0.886732i \(0.347030\pi\)
\(422\) − 0.343146i − 0.0167041i
\(423\) 11.6569i 0.566776i
\(424\) −5.79899 −0.281624
\(425\) 0 0
\(426\) 12.6863 0.614653
\(427\) 17.6569i 0.854475i
\(428\) 36.8284i 1.78017i
\(429\) −3.31371 −0.159987
\(430\) 0 0
\(431\) 3.31371 0.159616 0.0798079 0.996810i \(-0.474569\pi\)
0.0798079 + 0.996810i \(0.474569\pi\)
\(432\) 12.0000i 0.577350i
\(433\) 29.1716i 1.40190i 0.713212 + 0.700948i \(0.247240\pi\)
−0.713212 + 0.700948i \(0.752760\pi\)
\(434\) 12.9706 0.622607
\(435\) 0 0
\(436\) −3.65685 −0.175132
\(437\) 15.3137i 0.732554i
\(438\) 7.02944i 0.335880i
\(439\) 10.3431 0.493651 0.246826 0.969060i \(-0.420612\pi\)
0.246826 + 0.969060i \(0.420612\pi\)
\(440\) 0 0
\(441\) 16.3137 0.776843
\(442\) − 2.34315i − 0.111452i
\(443\) 7.65685i 0.363788i 0.983318 + 0.181894i \(0.0582228\pi\)
−0.983318 + 0.181894i \(0.941777\pi\)
\(444\) 31.0294 1.47259
\(445\) 0 0
\(446\) −7.37258 −0.349102
\(447\) − 18.6274i − 0.881047i
\(448\) − 20.1421i − 0.951626i
\(449\) −11.6569 −0.550121 −0.275060 0.961427i \(-0.588698\pi\)
−0.275060 + 0.961427i \(0.588698\pi\)
\(450\) 0 0
\(451\) −4.97056 −0.234055
\(452\) 5.17157i 0.243250i
\(453\) − 24.0000i − 1.12762i
\(454\) −8.34315 −0.391563
\(455\) 0 0
\(456\) −15.3137 −0.717130
\(457\) 19.6569i 0.919509i 0.888046 + 0.459754i \(0.152063\pi\)
−0.888046 + 0.459754i \(0.847937\pi\)
\(458\) 0.828427i 0.0387099i
\(459\) −11.3137 −0.528079
\(460\) 0 0
\(461\) −35.6569 −1.66071 −0.830353 0.557238i \(-0.811861\pi\)
−0.830353 + 0.557238i \(0.811861\pi\)
\(462\) 3.31371i 0.154168i
\(463\) − 21.7990i − 1.01308i −0.862215 0.506542i \(-0.830923\pi\)
0.862215 0.506542i \(-0.169077\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 7.45584 0.345385
\(467\) 10.9706i 0.507657i 0.967249 + 0.253829i \(0.0816899\pi\)
−0.967249 + 0.253829i \(0.918310\pi\)
\(468\) − 3.65685i − 0.169038i
\(469\) 31.3137 1.44593
\(470\) 0 0
\(471\) −0.970563 −0.0447212
\(472\) 0 0
\(473\) 4.97056i 0.228547i
\(474\) −2.05887 −0.0945672
\(475\) 0 0
\(476\) 24.9706 1.14452
\(477\) − 3.65685i − 0.167436i
\(478\) 0.284271i 0.0130023i
\(479\) −7.17157 −0.327678 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(480\) 0 0
\(481\) 16.9706 0.773791
\(482\) 4.14214i 0.188669i
\(483\) 30.6274i 1.39360i
\(484\) −18.8579 −0.857176
\(485\) 0 0
\(486\) 4.14214 0.187891
\(487\) 9.79899i 0.444035i 0.975043 + 0.222017i \(0.0712641\pi\)
−0.975043 + 0.222017i \(0.928736\pi\)
\(488\) − 5.79899i − 0.262508i
\(489\) −16.6863 −0.754580
\(490\) 0 0
\(491\) −7.45584 −0.336478 −0.168239 0.985746i \(-0.553808\pi\)
−0.168239 + 0.985746i \(0.553808\pi\)
\(492\) − 21.9411i − 0.989182i
\(493\) − 2.82843i − 0.127386i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 19.4558 0.873593
\(497\) 73.9411i 3.31671i
\(498\) 5.94113i 0.266228i
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 4.97056 0.222068
\(502\) 3.65685i 0.163213i
\(503\) 30.0000i 1.33763i 0.743427 + 0.668817i \(0.233199\pi\)
−0.743427 + 0.668817i \(0.766801\pi\)
\(504\) −7.65685 −0.341063
\(505\) 0 0
\(506\) −1.08831 −0.0483814
\(507\) 18.0000i 0.799408i
\(508\) 10.9706i 0.486740i
\(509\) −0.627417 −0.0278098 −0.0139049 0.999903i \(-0.504426\pi\)
−0.0139049 + 0.999903i \(0.504426\pi\)
\(510\) 0 0
\(511\) −40.9706 −1.81243
\(512\) 22.7574i 1.00574i
\(513\) 19.3137i 0.852721i
\(514\) −2.76955 −0.122160
\(515\) 0 0
\(516\) −21.9411 −0.965904
\(517\) − 9.65685i − 0.424708i
\(518\) − 16.9706i − 0.745644i
\(519\) 34.6274 1.51997
\(520\) 0 0
\(521\) −21.3137 −0.933771 −0.466885 0.884318i \(-0.654624\pi\)
−0.466885 + 0.884318i \(0.654624\pi\)
\(522\) 0.414214i 0.0181296i
\(523\) − 2.48528i − 0.108674i −0.998523 0.0543369i \(-0.982696\pi\)
0.998523 0.0543369i \(-0.0173045\pi\)
\(524\) −22.2010 −0.969856
\(525\) 0 0
\(526\) −8.14214 −0.355014
\(527\) 18.3431i 0.799040i
\(528\) 4.97056i 0.216316i
\(529\) 12.9411 0.562658
\(530\) 0 0
\(531\) 0 0
\(532\) − 42.6274i − 1.84813i
\(533\) − 12.0000i − 0.519778i
\(534\) −6.34315 −0.274495
\(535\) 0 0
\(536\) −10.2843 −0.444213
\(537\) 46.6274i 2.01212i
\(538\) 8.82843i 0.380621i
\(539\) −13.5147 −0.582120
\(540\) 0 0
\(541\) −5.02944 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(542\) − 4.05887i − 0.174344i
\(543\) − 12.0000i − 0.514969i
\(544\) −12.4853 −0.535302
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) − 2.48528i − 0.106263i −0.998588 0.0531315i \(-0.983080\pi\)
0.998588 0.0531315i \(-0.0169202\pi\)
\(548\) − 9.45584i − 0.403934i
\(549\) 3.65685 0.156071
\(550\) 0 0
\(551\) −4.82843 −0.205698
\(552\) − 10.0589i − 0.428134i
\(553\) − 12.0000i − 0.510292i
\(554\) 1.51472 0.0643542
\(555\) 0 0
\(556\) −39.5980 −1.67933
\(557\) − 27.9411i − 1.18390i −0.805973 0.591952i \(-0.798358\pi\)
0.805973 0.591952i \(-0.201642\pi\)
\(558\) − 2.68629i − 0.113720i
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −4.68629 −0.197855
\(562\) − 12.1421i − 0.512185i
\(563\) − 7.65685i − 0.322698i −0.986897 0.161349i \(-0.948416\pi\)
0.986897 0.161349i \(-0.0515845\pi\)
\(564\) 42.6274 1.79494
\(565\) 0 0
\(566\) 2.00000 0.0840663
\(567\) 53.1127i 2.23052i
\(568\) − 24.2843i − 1.01895i
\(569\) −27.6569 −1.15944 −0.579718 0.814817i \(-0.696837\pi\)
−0.579718 + 0.814817i \(0.696837\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 3.02944i 0.126667i
\(573\) − 41.6569i − 1.74024i
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) 23.7990i 0.990765i 0.868675 + 0.495382i \(0.164972\pi\)
−0.868675 + 0.495382i \(0.835028\pi\)
\(578\) 3.72792i 0.155061i
\(579\) 8.97056 0.372804
\(580\) 0 0
\(581\) −34.6274 −1.43659
\(582\) 10.3431i 0.428737i
\(583\) 3.02944i 0.125466i
\(584\) 13.4558 0.556807
\(585\) 0 0
\(586\) 3.51472 0.145192
\(587\) − 29.7990i − 1.22994i −0.788552 0.614968i \(-0.789169\pi\)
0.788552 0.614968i \(-0.210831\pi\)
\(588\) − 59.6569i − 2.46021i
\(589\) 31.3137 1.29026
\(590\) 0 0
\(591\) 39.3137 1.61715
\(592\) − 25.4558i − 1.04623i
\(593\) 7.65685i 0.314429i 0.987564 + 0.157215i \(0.0502515\pi\)
−0.987564 + 0.157215i \(0.949749\pi\)
\(594\) −1.37258 −0.0563178
\(595\) 0 0
\(596\) −17.0294 −0.697553
\(597\) − 24.0000i − 0.982255i
\(598\) − 2.62742i − 0.107443i
\(599\) 37.7990 1.54442 0.772212 0.635364i \(-0.219150\pi\)
0.772212 + 0.635364i \(0.219150\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 12.0000i 0.489083i
\(603\) − 6.48528i − 0.264101i
\(604\) −21.9411 −0.892772
\(605\) 0 0
\(606\) −12.9706 −0.526893
\(607\) 9.02944i 0.366494i 0.983067 + 0.183247i \(0.0586607\pi\)
−0.983067 + 0.183247i \(0.941339\pi\)
\(608\) 21.3137i 0.864385i
\(609\) −9.65685 −0.391315
\(610\) 0 0
\(611\) 23.3137 0.943172
\(612\) − 5.17157i − 0.209048i
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 9.51472 0.383983
\(615\) 0 0
\(616\) 6.34315 0.255573
\(617\) − 9.17157i − 0.369234i −0.982811 0.184617i \(-0.940896\pi\)
0.982811 0.184617i \(-0.0591044\pi\)
\(618\) 13.3726i 0.537924i
\(619\) 9.79899 0.393855 0.196927 0.980418i \(-0.436904\pi\)
0.196927 + 0.980418i \(0.436904\pi\)
\(620\) 0 0
\(621\) −12.6863 −0.509083
\(622\) 6.00000i 0.240578i
\(623\) − 36.9706i − 1.48119i
\(624\) −12.0000 −0.480384
\(625\) 0 0
\(626\) −2.48528 −0.0993318
\(627\) 8.00000i 0.319489i
\(628\) 0.887302i 0.0354072i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 36.9706 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(632\) 3.94113i 0.156770i
\(633\) − 1.65685i − 0.0658540i
\(634\) −1.17157 −0.0465291
\(635\) 0 0
\(636\) −13.3726 −0.530257
\(637\) − 32.6274i − 1.29275i
\(638\) − 0.343146i − 0.0135853i
\(639\) 15.3137 0.605801
\(640\) 0 0
\(641\) 0.627417 0.0247815 0.0123907 0.999923i \(-0.496056\pi\)
0.0123907 + 0.999923i \(0.496056\pi\)
\(642\) − 16.6863i − 0.658555i
\(643\) − 19.4558i − 0.767264i −0.923486 0.383632i \(-0.874673\pi\)
0.923486 0.383632i \(-0.125327\pi\)
\(644\) 28.0000 1.10335
\(645\) 0 0
\(646\) −5.65685 −0.222566
\(647\) − 41.1127i − 1.61631i −0.588972 0.808153i \(-0.700467\pi\)
0.588972 0.808153i \(-0.299533\pi\)
\(648\) − 17.4437i − 0.685251i
\(649\) 0 0
\(650\) 0 0
\(651\) 62.6274 2.45456
\(652\) 15.2548i 0.597425i
\(653\) 17.1716i 0.671976i 0.941866 + 0.335988i \(0.109070\pi\)
−0.941866 + 0.335988i \(0.890930\pi\)
\(654\) 1.65685 0.0647881
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) 8.48528i 0.331042i
\(658\) − 23.3137i − 0.908863i
\(659\) −1.79899 −0.0700787 −0.0350393 0.999386i \(-0.511156\pi\)
−0.0350393 + 0.999386i \(0.511156\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 9.02944i 0.350939i
\(663\) − 11.3137i − 0.439388i
\(664\) 11.3726 0.441342
\(665\) 0 0
\(666\) −3.51472 −0.136193
\(667\) − 3.17157i − 0.122804i
\(668\) − 4.54416i − 0.175819i
\(669\) −35.5980 −1.37630
\(670\) 0 0
\(671\) −3.02944 −0.116950
\(672\) 42.6274i 1.64439i
\(673\) − 22.9706i − 0.885450i −0.896657 0.442725i \(-0.854012\pi\)
0.896657 0.442725i \(-0.145988\pi\)
\(674\) 0.485281 0.0186923
\(675\) 0 0
\(676\) 16.4558 0.632917
\(677\) − 36.7696i − 1.41317i −0.707629 0.706584i \(-0.750235\pi\)
0.707629 0.706584i \(-0.249765\pi\)
\(678\) − 2.34315i − 0.0899880i
\(679\) −60.2843 −2.31350
\(680\) 0 0
\(681\) −40.2843 −1.54370
\(682\) 2.22540i 0.0852148i
\(683\) − 11.8579i − 0.453729i −0.973926 0.226864i \(-0.927153\pi\)
0.973926 0.226864i \(-0.0728474\pi\)
\(684\) −8.82843 −0.337563
\(685\) 0 0
\(686\) −18.6274 −0.711198
\(687\) 4.00000i 0.152610i
\(688\) 18.0000i 0.686244i
\(689\) −7.31371 −0.278630
\(690\) 0 0
\(691\) −44.9706 −1.71076 −0.855380 0.518000i \(-0.826677\pi\)
−0.855380 + 0.518000i \(0.826677\pi\)
\(692\) − 31.6569i − 1.20341i
\(693\) 4.00000i 0.151947i
\(694\) 3.37258 0.128022
\(695\) 0 0
\(696\) 3.17157 0.120218
\(697\) − 16.9706i − 0.642806i
\(698\) − 8.54416i − 0.323401i
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 6.68629 0.252538 0.126269 0.991996i \(-0.459700\pi\)
0.126269 + 0.991996i \(0.459700\pi\)
\(702\) − 3.31371i − 0.125068i
\(703\) − 40.9706i − 1.54523i
\(704\) 3.45584 0.130247
\(705\) 0 0
\(706\) −1.79899 −0.0677059
\(707\) − 75.5980i − 2.84315i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −2.48528 −0.0932053
\(712\) 12.1421i 0.455046i
\(713\) 20.5685i 0.770298i
\(714\) −11.3137 −0.423405
\(715\) 0 0
\(716\) 42.6274 1.59306
\(717\) 1.37258i 0.0512601i
\(718\) 1.59798i 0.0596361i
\(719\) 34.6274 1.29138 0.645692 0.763598i \(-0.276569\pi\)
0.645692 + 0.763598i \(0.276569\pi\)
\(720\) 0 0
\(721\) −77.9411 −2.90268
\(722\) 1.78680i 0.0664977i
\(723\) 20.0000i 0.743808i
\(724\) −10.9706 −0.407718
\(725\) 0 0
\(726\) 8.54416 0.317103
\(727\) − 23.9411i − 0.887927i −0.896045 0.443964i \(-0.853572\pi\)
0.896045 0.443964i \(-0.146428\pi\)
\(728\) 15.3137i 0.567564i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −16.9706 −0.627679
\(732\) − 13.3726i − 0.494265i
\(733\) 22.8284i 0.843187i 0.906785 + 0.421594i \(0.138529\pi\)
−0.906785 + 0.421594i \(0.861471\pi\)
\(734\) −7.45584 −0.275200
\(735\) 0 0
\(736\) −14.0000 −0.516047
\(737\) 5.37258i 0.197902i
\(738\) 2.48528i 0.0914845i
\(739\) −14.4853 −0.532850 −0.266425 0.963856i \(-0.585842\pi\)
−0.266425 + 0.963856i \(0.585842\pi\)
\(740\) 0 0
\(741\) −19.3137 −0.709507
\(742\) 7.31371i 0.268495i
\(743\) 52.6274i 1.93071i 0.260935 + 0.965356i \(0.415969\pi\)
−0.260935 + 0.965356i \(0.584031\pi\)
\(744\) −20.5685 −0.754079
\(745\) 0 0
\(746\) −2.88730 −0.105712
\(747\) 7.17157i 0.262394i
\(748\) 4.28427i 0.156648i
\(749\) 97.2548 3.55361
\(750\) 0 0
\(751\) 16.1421 0.589035 0.294517 0.955646i \(-0.404841\pi\)
0.294517 + 0.955646i \(0.404841\pi\)
\(752\) − 34.9706i − 1.27525i
\(753\) 17.6569i 0.643452i
\(754\) 0.828427 0.0301695
\(755\) 0 0
\(756\) 35.3137 1.28435
\(757\) 19.5147i 0.709275i 0.935004 + 0.354637i \(0.115396\pi\)
−0.935004 + 0.354637i \(0.884604\pi\)
\(758\) − 9.31371i − 0.338289i
\(759\) −5.25483 −0.190738
\(760\) 0 0
\(761\) 8.62742 0.312744 0.156372 0.987698i \(-0.450020\pi\)
0.156372 + 0.987698i \(0.450020\pi\)
\(762\) − 4.97056i − 0.180064i
\(763\) 9.65685i 0.349602i
\(764\) −38.0833 −1.37780
\(765\) 0 0
\(766\) 1.02944 0.0371951
\(767\) 0 0
\(768\) 7.94113i 0.286551i
\(769\) 15.6569 0.564601 0.282300 0.959326i \(-0.408903\pi\)
0.282300 + 0.959326i \(0.408903\pi\)
\(770\) 0 0
\(771\) −13.3726 −0.481602
\(772\) − 8.20101i − 0.295161i
\(773\) 8.48528i 0.305194i 0.988288 + 0.152597i \(0.0487637\pi\)
−0.988288 + 0.152597i \(0.951236\pi\)
\(774\) 2.48528 0.0893316
\(775\) 0 0
\(776\) 19.7990 0.710742
\(777\) − 81.9411i − 2.93962i
\(778\) 12.1421i 0.435317i
\(779\) −28.9706 −1.03798
\(780\) 0 0
\(781\) −12.6863 −0.453951
\(782\) − 3.71573i − 0.132874i
\(783\) − 4.00000i − 0.142948i
\(784\) −48.9411 −1.74790
\(785\) 0 0
\(786\) 10.0589 0.358788
\(787\) 17.7990i 0.634465i 0.948348 + 0.317233i \(0.102754\pi\)
−0.948348 + 0.317233i \(0.897246\pi\)
\(788\) − 35.9411i − 1.28035i
\(789\) −39.3137 −1.39961
\(790\) 0 0
\(791\) 13.6569 0.485582
\(792\) − 1.31371i − 0.0466806i
\(793\) − 7.31371i − 0.259717i
\(794\) 8.14214 0.288954
\(795\) 0 0
\(796\) −21.9411 −0.777683
\(797\) − 5.85786i − 0.207496i −0.994604 0.103748i \(-0.966916\pi\)
0.994604 0.103748i \(-0.0330836\pi\)
\(798\) 19.3137i 0.683698i
\(799\) 32.9706 1.16641
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) − 2.76955i − 0.0977963i
\(803\) − 7.02944i − 0.248063i
\(804\) −23.7157 −0.836389
\(805\) 0 0
\(806\) −5.37258 −0.189241
\(807\) 42.6274i 1.50056i
\(808\) 24.8284i 0.873461i
\(809\) −42.2843 −1.48664 −0.743318 0.668938i \(-0.766749\pi\)
−0.743318 + 0.668938i \(0.766749\pi\)
\(810\) 0 0
\(811\) 37.6569 1.32231 0.661155 0.750249i \(-0.270066\pi\)
0.661155 + 0.750249i \(0.270066\pi\)
\(812\) 8.82843i 0.309817i
\(813\) − 19.5980i − 0.687331i
\(814\) 2.91169 0.102055
\(815\) 0 0
\(816\) −16.9706 −0.594089
\(817\) 28.9706i 1.01355i
\(818\) 1.23045i 0.0430216i
\(819\) −9.65685 −0.337438
\(820\) 0 0
\(821\) −22.6863 −0.791757 −0.395879 0.918303i \(-0.629560\pi\)
−0.395879 + 0.918303i \(0.629560\pi\)
\(822\) 4.28427i 0.149431i
\(823\) 30.9706i 1.07957i 0.841804 + 0.539783i \(0.181494\pi\)
−0.841804 + 0.539783i \(0.818506\pi\)
\(824\) 25.5980 0.891748
\(825\) 0 0
\(826\) 0 0
\(827\) 17.3137i 0.602057i 0.953615 + 0.301028i \(0.0973299\pi\)
−0.953615 + 0.301028i \(0.902670\pi\)
\(828\) − 5.79899i − 0.201529i
\(829\) −20.6274 −0.716420 −0.358210 0.933641i \(-0.616613\pi\)
−0.358210 + 0.933641i \(0.616613\pi\)
\(830\) 0 0
\(831\) 7.31371 0.253710
\(832\) 8.34315i 0.289247i
\(833\) − 46.1421i − 1.59873i
\(834\) 17.9411 0.621250
\(835\) 0 0
\(836\) 7.31371 0.252950
\(837\) 25.9411i 0.896656i
\(838\) − 12.0000i − 0.414533i
\(839\) −2.48528 −0.0858014 −0.0429007 0.999079i \(-0.513660\pi\)
−0.0429007 + 0.999079i \(0.513660\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 7.85786i 0.270800i
\(843\) − 58.6274i − 2.01924i
\(844\) −1.51472 −0.0521388
\(845\) 0 0
\(846\) −4.82843 −0.166005
\(847\) 49.7990i 1.71111i
\(848\) 10.9706i 0.376731i
\(849\) 9.65685 0.331422
\(850\) 0 0
\(851\) 26.9117 0.922521
\(852\) − 56.0000i − 1.91853i
\(853\) − 51.1127i − 1.75007i −0.484064 0.875033i \(-0.660840\pi\)
0.484064 0.875033i \(-0.339160\pi\)
\(854\) −7.31371 −0.250270
\(855\) 0 0
\(856\) −31.9411 −1.09173
\(857\) 3.37258i 0.115205i 0.998340 + 0.0576026i \(0.0183456\pi\)
−0.998340 + 0.0576026i \(0.981654\pi\)
\(858\) − 1.37258i − 0.0468592i
\(859\) 56.4264 1.92524 0.962622 0.270848i \(-0.0873041\pi\)
0.962622 + 0.270848i \(0.0873041\pi\)
\(860\) 0 0
\(861\) −57.9411 −1.97463
\(862\) 1.37258i 0.0467504i
\(863\) 36.1421i 1.23029i 0.788413 + 0.615146i \(0.210903\pi\)
−0.788413 + 0.615146i \(0.789097\pi\)
\(864\) −17.6569 −0.600698
\(865\) 0 0
\(866\) −12.0833 −0.410606
\(867\) 18.0000i 0.611312i
\(868\) − 57.2548i − 1.94336i
\(869\) 2.05887 0.0698425
\(870\) 0 0
\(871\) −12.9706 −0.439491
\(872\) − 3.17157i − 0.107403i
\(873\) 12.4853i 0.422563i
\(874\) −6.34315 −0.214560
\(875\) 0 0
\(876\) 31.0294 1.04839
\(877\) 38.2843i 1.29277i 0.763012 + 0.646384i \(0.223720\pi\)
−0.763012 + 0.646384i \(0.776280\pi\)
\(878\) 4.28427i 0.144587i
\(879\) 16.9706 0.572403
\(880\) 0 0
\(881\) 29.3137 0.987604 0.493802 0.869574i \(-0.335607\pi\)
0.493802 + 0.869574i \(0.335607\pi\)
\(882\) 6.75736i 0.227532i
\(883\) 14.4853i 0.487469i 0.969842 + 0.243734i \(0.0783725\pi\)
−0.969842 + 0.243734i \(0.921628\pi\)
\(884\) −10.3431 −0.347878
\(885\) 0 0
\(886\) −3.17157 −0.106551
\(887\) − 6.68629i − 0.224504i −0.993680 0.112252i \(-0.964194\pi\)
0.993680 0.112252i \(-0.0358063\pi\)
\(888\) 26.9117i 0.903097i
\(889\) 28.9706 0.971641
\(890\) 0 0
\(891\) −9.11270 −0.305287
\(892\) 32.5442i 1.08966i
\(893\) − 56.2843i − 1.88348i
\(894\) 7.71573 0.258053
\(895\) 0 0
\(896\) 50.9706 1.70281
\(897\) − 12.6863i − 0.423583i
\(898\) − 4.82843i − 0.161127i
\(899\) −6.48528 −0.216296
\(900\) 0 0
\(901\) −10.3431 −0.344580
\(902\) − 2.05887i − 0.0685530i
\(903\) 57.9411i 1.92816i
\(904\) −4.48528 −0.149178
\(905\) 0 0
\(906\) 9.94113 0.330272
\(907\) − 10.0000i − 0.332045i −0.986122 0.166022i \(-0.946908\pi\)
0.986122 0.166022i \(-0.0530924\pi\)
\(908\) 36.8284i 1.22219i
\(909\) −15.6569 −0.519305
\(910\) 0 0
\(911\) 32.1421 1.06492 0.532458 0.846456i \(-0.321268\pi\)
0.532458 + 0.846456i \(0.321268\pi\)
\(912\) 28.9706i 0.959311i
\(913\) − 5.94113i − 0.196623i
\(914\) −8.14214 −0.269318
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) 58.6274i 1.93605i
\(918\) − 4.68629i − 0.154671i
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) 45.9411 1.51381
\(922\) − 14.7696i − 0.486409i
\(923\) − 30.6274i − 1.00811i
\(924\) 14.6274 0.481207
\(925\) 0 0
\(926\) 9.02944 0.296726
\(927\) 16.1421i 0.530177i
\(928\) − 4.41421i − 0.144904i
\(929\) 4.62742 0.151821 0.0759103 0.997115i \(-0.475814\pi\)
0.0759103 + 0.997115i \(0.475814\pi\)
\(930\) 0 0
\(931\) −78.7696 −2.58157
\(932\) − 32.9117i − 1.07806i
\(933\) 28.9706i 0.948454i
\(934\) −4.54416 −0.148689
\(935\) 0 0
\(936\) 3.17157 0.103666
\(937\) 19.6569i 0.642161i 0.947052 + 0.321081i \(0.104046\pi\)
−0.947052 + 0.321081i \(0.895954\pi\)
\(938\) 12.9706i 0.423504i
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) −27.9411 −0.910855 −0.455427 0.890273i \(-0.650514\pi\)
−0.455427 + 0.890273i \(0.650514\pi\)
\(942\) − 0.402020i − 0.0130985i
\(943\) − 19.0294i − 0.619684i
\(944\) 0 0
\(945\) 0 0
\(946\) −2.05887 −0.0669398
\(947\) 44.9117i 1.45943i 0.683749 + 0.729717i \(0.260348\pi\)
−0.683749 + 0.729717i \(0.739652\pi\)
\(948\) 9.08831i 0.295175i
\(949\) 16.9706 0.550888
\(950\) 0 0
\(951\) −5.65685 −0.183436
\(952\) 21.6569i 0.701903i
\(953\) − 29.3137i − 0.949564i −0.880103 0.474782i \(-0.842527\pi\)
0.880103 0.474782i \(-0.157473\pi\)
\(954\) 1.51472 0.0490408
\(955\) 0 0
\(956\) 1.25483 0.0405842
\(957\) − 1.65685i − 0.0535585i
\(958\) − 2.97056i − 0.0959745i
\(959\) −24.9706 −0.806342
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) 7.02944i 0.226638i
\(963\) − 20.1421i − 0.649071i
\(964\) 18.2843 0.588897
\(965\) 0 0
\(966\) −12.6863 −0.408175
\(967\) 14.9706i 0.481421i 0.970597 + 0.240710i \(0.0773804\pi\)
−0.970597 + 0.240710i \(0.922620\pi\)
\(968\) − 16.3553i − 0.525681i
\(969\) −27.3137 −0.877443
\(970\) 0 0
\(971\) 28.1421 0.903124 0.451562 0.892240i \(-0.350867\pi\)
0.451562 + 0.892240i \(0.350867\pi\)
\(972\) − 18.2843i − 0.586468i
\(973\) 104.569i 3.35231i
\(974\) −4.05887 −0.130055
\(975\) 0 0
\(976\) −10.9706 −0.351159
\(977\) − 2.68629i − 0.0859421i −0.999076 0.0429710i \(-0.986318\pi\)
0.999076 0.0429710i \(-0.0136823\pi\)
\(978\) − 6.91169i − 0.221011i
\(979\) 6.34315 0.202728
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) − 3.08831i − 0.0985520i
\(983\) 9.31371i 0.297061i 0.988908 + 0.148531i \(0.0474543\pi\)
−0.988908 + 0.148531i \(0.952546\pi\)
\(984\) 19.0294 0.606636
\(985\) 0 0
\(986\) 1.17157 0.0373105
\(987\) − 112.569i − 3.58310i
\(988\) 17.6569i 0.561739i
\(989\) −19.0294 −0.605101
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 28.6274i 0.908921i
\(993\) 43.5980i 1.38354i
\(994\) −30.6274 −0.971443
\(995\) 0 0
\(996\) 26.2254 0.830983
\(997\) 6.82843i 0.216258i 0.994137 + 0.108129i \(0.0344860\pi\)
−0.994137 + 0.108129i \(0.965514\pi\)
\(998\) − 14.9117i − 0.472021i
\(999\) 33.9411 1.07385
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 725.2.b.c.349.3 4
5.2 odd 4 725.2.a.c.1.1 2
5.3 odd 4 145.2.a.b.1.2 2
5.4 even 2 inner 725.2.b.c.349.2 4
15.2 even 4 6525.2.a.p.1.2 2
15.8 even 4 1305.2.a.n.1.1 2
20.3 even 4 2320.2.a.k.1.2 2
35.13 even 4 7105.2.a.e.1.2 2
40.3 even 4 9280.2.a.w.1.2 2
40.13 odd 4 9280.2.a.be.1.1 2
145.28 odd 4 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 5.3 odd 4
725.2.a.c.1.1 2 5.2 odd 4
725.2.b.c.349.2 4 5.4 even 2 inner
725.2.b.c.349.3 4 1.1 even 1 trivial
1305.2.a.n.1.1 2 15.8 even 4
2320.2.a.k.1.2 2 20.3 even 4
4205.2.a.d.1.1 2 145.28 odd 4
6525.2.a.p.1.2 2 15.2 even 4
7105.2.a.e.1.2 2 35.13 even 4
9280.2.a.w.1.2 2 40.3 even 4
9280.2.a.be.1.1 2 40.13 odd 4