Properties

Label 1305.2.a.n.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -4.82843 q^{7} +1.58579 q^{8} +O(q^{10})\) \(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -4.82843 q^{7} +1.58579 q^{8} +0.414214 q^{10} -0.828427 q^{11} -2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -4.82843 q^{19} +1.82843 q^{20} +0.343146 q^{22} +3.17157 q^{23} +1.00000 q^{25} +0.828427 q^{26} +8.82843 q^{28} -1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.17157 q^{34} +4.82843 q^{35} -8.48528 q^{37} +2.00000 q^{38} -1.58579 q^{40} +6.00000 q^{41} -6.00000 q^{43} +1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +16.3137 q^{49} -0.414214 q^{50} +3.65685 q^{52} +3.65685 q^{53} +0.828427 q^{55} -7.65685 q^{56} +0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} -4.17157 q^{64} +2.00000 q^{65} +6.48528 q^{67} +5.17157 q^{68} -2.00000 q^{70} +15.3137 q^{71} +8.48528 q^{73} +3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -2.48528 q^{79} -3.00000 q^{80} -2.48528 q^{82} -7.17157 q^{83} +2.82843 q^{85} +2.48528 q^{86} -1.31371 q^{88} +7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} -4.82843 q^{94} +4.82843 q^{95} -12.4853 q^{97} -6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 2 q^{20} + 12 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} - 2 q^{29} - 4 q^{31} - 6 q^{32} + 8 q^{34} + 4 q^{35} + 4 q^{38} - 6 q^{40} + 12 q^{41} - 12 q^{43} + 20 q^{44} + 20 q^{46} + 12 q^{47} + 10 q^{49} + 2 q^{50} - 4 q^{52} - 4 q^{53} - 4 q^{55} - 4 q^{56} - 2 q^{58} + 4 q^{61} - 28 q^{62} - 14 q^{64} + 4 q^{65} - 4 q^{67} + 16 q^{68} - 4 q^{70} + 8 q^{71} + 24 q^{74} + 12 q^{76} + 8 q^{77} + 12 q^{79} - 6 q^{80} + 12 q^{82} - 20 q^{83} - 12 q^{86} + 20 q^{88} + 4 q^{89} + 8 q^{91} + 28 q^{92} - 4 q^{94} + 4 q^{95} - 8 q^{97} - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.82843 −1.82497 −0.912487 0.409106i \(-0.865841\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(8\) 1.58579 0.560660
\(9\) 0 0
\(10\) 0.414214 0.130986
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) −4.82843 −1.10772 −0.553859 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) 1.82843 0.408849
\(21\) 0 0
\(22\) 0.343146 0.0731589
\(23\) 3.17157 0.661319 0.330659 0.943750i \(-0.392729\pi\)
0.330659 + 0.943750i \(0.392729\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.828427 0.162468
\(27\) 0 0
\(28\) 8.82843 1.66842
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) −4.41421 −0.780330
\(33\) 0 0
\(34\) 1.17157 0.200923
\(35\) 4.82843 0.816153
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.58579 −0.250735
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 1.51472 0.228352
\(45\) 0 0
\(46\) −1.31371 −0.193696
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) −0.414214 −0.0585786
\(51\) 0 0
\(52\) 3.65685 0.507114
\(53\) 3.65685 0.502308 0.251154 0.967947i \(-0.419190\pi\)
0.251154 + 0.967947i \(0.419190\pi\)
\(54\) 0 0
\(55\) 0.828427 0.111705
\(56\) −7.65685 −1.02319
\(57\) 0 0
\(58\) 0.414214 0.0543889
\(59\) 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 \(0.575216\pi\)
\(62\) −2.68629 −0.341159
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 6.48528 0.792303 0.396152 0.918185i \(-0.370345\pi\)
0.396152 + 0.918185i \(0.370345\pi\)
\(68\) 5.17157 0.627145
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 15.3137 1.81740 0.908701 0.417447i \(-0.137075\pi\)
0.908701 + 0.417447i \(0.137075\pi\)
\(72\) 0 0
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 3.51472 0.408578
\(75\) 0 0
\(76\) 8.82843 1.01269
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −2.48528 −0.279616 −0.139808 0.990179i \(-0.544649\pi\)
−0.139808 + 0.990179i \(0.544649\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) −2.48528 −0.274453
\(83\) −7.17157 −0.787182 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 2.48528 0.267995
\(87\) 0 0
\(88\) −1.31371 −0.140042
\(89\) 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
\(92\) −5.79899 −0.604586
\(93\) 0 0
\(94\) −4.82843 −0.498014
\(95\) 4.82843 0.495386
\(96\) 0 0
\(97\) −12.4853 −1.26769 −0.633844 0.773461i \(-0.718524\pi\)
−0.633844 + 0.773461i \(0.718524\pi\)
\(98\) −6.75736 −0.682596
\(99\) 0 0
\(100\) −1.82843 −0.182843
\(101\) −15.6569 −1.55792 −0.778958 0.627077i \(-0.784251\pi\)
−0.778958 + 0.627077i \(0.784251\pi\)
\(102\) 0 0
\(103\) 16.1421 1.59053 0.795266 0.606261i \(-0.207331\pi\)
0.795266 + 0.606261i \(0.207331\pi\)
\(104\) −3.17157 −0.310998
\(105\) 0 0
\(106\) −1.51472 −0.147122
\(107\) −20.1421 −1.94721 −0.973607 0.228232i \(-0.926706\pi\)
−0.973607 + 0.228232i \(0.926706\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −0.343146 −0.0327177
\(111\) 0 0
\(112\) −14.4853 −1.36873
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) 0 0
\(115\) −3.17157 −0.295751
\(116\) 1.82843 0.169765
\(117\) 0 0
\(118\) 0 0
\(119\) 13.6569 1.25192
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 1.51472 0.137136
\(123\) 0 0
\(124\) −11.8579 −1.06487
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 10.5563 0.933058
\(129\) 0 0
\(130\) −0.828427 −0.0726579
\(131\) 12.1421 1.06086 0.530432 0.847728i \(-0.322030\pi\)
0.530432 + 0.847728i \(0.322030\pi\)
\(132\) 0 0
\(133\) 23.3137 2.02155
\(134\) −2.68629 −0.232060
\(135\) 0 0
\(136\) −4.48528 −0.384610
\(137\) 5.17157 0.441837 0.220919 0.975292i \(-0.429094\pi\)
0.220919 + 0.975292i \(0.429094\pi\)
\(138\) 0 0
\(139\) 21.6569 1.83691 0.918455 0.395525i \(-0.129437\pi\)
0.918455 + 0.395525i \(0.129437\pi\)
\(140\) −8.82843 −0.746138
\(141\) 0 0
\(142\) −6.34315 −0.532305
\(143\) 1.65685 0.138553
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) −3.51472 −0.290880
\(147\) 0 0
\(148\) 15.5147 1.27530
\(149\) −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 \(0.662373\pi\)
\(152\) −7.65685 −0.621053
\(153\) 0 0
\(154\) −1.65685 −0.133513
\(155\) −6.48528 −0.520910
\(156\) 0 0
\(157\) 0.485281 0.0387297 0.0193648 0.999812i \(-0.493836\pi\)
0.0193648 + 0.999812i \(0.493836\pi\)
\(158\) 1.02944 0.0818976
\(159\) 0 0
\(160\) 4.41421 0.348974
\(161\) −15.3137 −1.20689
\(162\) 0 0
\(163\) −8.34315 −0.653486 −0.326743 0.945113i \(-0.605951\pi\)
−0.326743 + 0.945113i \(0.605951\pi\)
\(164\) −10.9706 −0.856657
\(165\) 0 0
\(166\) 2.97056 0.230560
\(167\) 2.48528 0.192317 0.0961584 0.995366i \(-0.469344\pi\)
0.0961584 + 0.995366i \(0.469344\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −1.17157 −0.0898555
\(171\) 0 0
\(172\) 10.9706 0.836498
\(173\) −17.3137 −1.31634 −0.658168 0.752871i \(-0.728669\pi\)
−0.658168 + 0.752871i \(0.728669\pi\)
\(174\) 0 0
\(175\) −4.82843 −0.364995
\(176\) −2.48528 −0.187335
\(177\) 0 0
\(178\) −3.17157 −0.237719
\(179\) 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 \(0.571581\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 5.02944 0.370775
\(185\) 8.48528 0.623850
\(186\) 0 0
\(187\) 2.34315 0.171348
\(188\) −21.3137 −1.55446
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 20.8284 1.50709 0.753546 0.657395i \(-0.228342\pi\)
0.753546 + 0.657395i \(0.228342\pi\)
\(192\) 0 0
\(193\) 4.48528 0.322858 0.161429 0.986884i \(-0.448390\pi\)
0.161429 + 0.986884i \(0.448390\pi\)
\(194\) 5.17157 0.371297
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) 19.6569 1.40049 0.700246 0.713901i \(-0.253073\pi\)
0.700246 + 0.713901i \(0.253073\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 1.58579 0.112132
\(201\) 0 0
\(202\) 6.48528 0.456303
\(203\) 4.82843 0.338889
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −6.68629 −0.465856
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −0.828427 −0.0570313 −0.0285156 0.999593i \(-0.509078\pi\)
−0.0285156 + 0.999593i \(0.509078\pi\)
\(212\) −6.68629 −0.459216
\(213\) 0 0
\(214\) 8.34315 0.570326
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) −31.3137 −2.12571
\(218\) −0.828427 −0.0561082
\(219\) 0 0
\(220\) −1.51472 −0.102122
\(221\) 5.65685 0.380521
\(222\) 0 0
\(223\) −17.7990 −1.19191 −0.595954 0.803018i \(-0.703226\pi\)
−0.595954 + 0.803018i \(0.703226\pi\)
\(224\) 21.3137 1.42408
\(225\) 0 0
\(226\) −1.17157 −0.0779319
\(227\) −20.1421 −1.33688 −0.668440 0.743766i \(-0.733038\pi\)
−0.668440 + 0.743766i \(0.733038\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 1.31371 0.0866234
\(231\) 0 0
\(232\) −1.58579 −0.104112
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −11.6569 −0.760409
\(236\) 0 0
\(237\) 0 0
\(238\) −5.65685 −0.366679
\(239\) 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 \(0.395619\pi\)
\(242\) 4.27208 0.274620
\(243\) 0 0
\(244\) 6.68629 0.428046
\(245\) −16.3137 −1.04224
\(246\) 0 0
\(247\) 9.65685 0.614451
\(248\) 10.2843 0.653052
\(249\) 0 0
\(250\) 0.414214 0.0261972
\(251\) −8.82843 −0.557245 −0.278623 0.960401i \(-0.589878\pi\)
−0.278623 + 0.960401i \(0.589878\pi\)
\(252\) 0 0
\(253\) −2.62742 −0.165184
\(254\) −2.48528 −0.155940
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −6.68629 −0.417079 −0.208540 0.978014i \(-0.566871\pi\)
−0.208540 + 0.978014i \(0.566871\pi\)
\(258\) 0 0
\(259\) 40.9706 2.54579
\(260\) −3.65685 −0.226788
\(261\) 0 0
\(262\) −5.02944 −0.310720
\(263\) 19.6569 1.21209 0.606047 0.795429i \(-0.292754\pi\)
0.606047 + 0.795429i \(0.292754\pi\)
\(264\) 0 0
\(265\) −3.65685 −0.224639
\(266\) −9.65685 −0.592100
\(267\) 0 0
\(268\) −11.8579 −0.724334
\(269\) 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 \(0.596194\pi\)
\(272\) −8.48528 −0.514496
\(273\) 0 0
\(274\) −2.14214 −0.129411
\(275\) −0.828427 −0.0499560
\(276\) 0 0
\(277\) −3.65685 −0.219719 −0.109860 0.993947i \(-0.535040\pi\)
−0.109860 + 0.993947i \(0.535040\pi\)
\(278\) −8.97056 −0.538019
\(279\) 0 0
\(280\) 7.65685 0.457585
\(281\) 29.3137 1.74871 0.874355 0.485288i \(-0.161285\pi\)
0.874355 + 0.485288i \(0.161285\pi\)
\(282\) 0 0
\(283\) 4.82843 0.287020 0.143510 0.989649i \(-0.454161\pi\)
0.143510 + 0.989649i \(0.454161\pi\)
\(284\) −28.0000 −1.66149
\(285\) 0 0
\(286\) −0.686292 −0.0405813
\(287\) −28.9706 −1.71008
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) −0.414214 −0.0243235
\(291\) 0 0
\(292\) −15.5147 −0.907930
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −13.4558 −0.782105
\(297\) 0 0
\(298\) 3.85786 0.223480
\(299\) −6.34315 −0.366834
\(300\) 0 0
\(301\) 28.9706 1.66984
\(302\) 4.97056 0.286024
\(303\) 0 0
\(304\) −14.4853 −0.830788
\(305\) 3.65685 0.209391
\(306\) 0 0
\(307\) −22.9706 −1.31100 −0.655500 0.755195i \(-0.727542\pi\)
−0.655500 + 0.755195i \(0.727542\pi\)
\(308\) −7.31371 −0.416737
\(309\) 0 0
\(310\) 2.68629 0.152571
\(311\) −14.4853 −0.821385 −0.410692 0.911774i \(-0.634713\pi\)
−0.410692 + 0.911774i \(0.634713\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −0.201010 −0.0113437
\(315\) 0 0
\(316\) 4.54416 0.255629
\(317\) −2.82843 −0.158860 −0.0794301 0.996840i \(-0.525310\pi\)
−0.0794301 + 0.996840i \(0.525310\pi\)
\(318\) 0 0
\(319\) 0.828427 0.0463830
\(320\) 4.17157 0.233198
\(321\) 0 0
\(322\) 6.34315 0.353490
\(323\) 13.6569 0.759888
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 3.45584 0.191402
\(327\) 0 0
\(328\) 9.51472 0.525362
\(329\) −56.2843 −3.10305
\(330\) 0 0
\(331\) 21.7990 1.19818 0.599090 0.800681i \(-0.295529\pi\)
0.599090 + 0.800681i \(0.295529\pi\)
\(332\) 13.1127 0.719653
\(333\) 0 0
\(334\) −1.02944 −0.0563283
\(335\) −6.48528 −0.354329
\(336\) 0 0
\(337\) −1.17157 −0.0638196 −0.0319098 0.999491i \(-0.510159\pi\)
−0.0319098 + 0.999491i \(0.510159\pi\)
\(338\) 3.72792 0.202772
\(339\) 0 0
\(340\) −5.17157 −0.280468
\(341\) −5.37258 −0.290942
\(342\) 0 0
\(343\) −44.9706 −2.42818
\(344\) −9.51472 −0.512999
\(345\) 0 0
\(346\) 7.17157 0.385546
\(347\) 8.14214 0.437093 0.218546 0.975827i \(-0.429869\pi\)
0.218546 + 0.975827i \(0.429869\pi\)
\(348\) 0 0
\(349\) 20.6274 1.10416 0.552080 0.833791i \(-0.313834\pi\)
0.552080 + 0.833791i \(0.313834\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) 3.65685 0.194911
\(353\) 4.34315 0.231162 0.115581 0.993298i \(-0.463127\pi\)
0.115581 + 0.993298i \(0.463127\pi\)
\(354\) 0 0
\(355\) −15.3137 −0.812767
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −9.65685 −0.510381
\(359\) 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
\(362\) 2.48528 0.130623
\(363\) 0 0
\(364\) −17.6569 −0.925471
\(365\) −8.48528 −0.444140
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 9.51472 0.495989
\(369\) 0 0
\(370\) −3.51472 −0.182722
\(371\) −17.6569 −0.916698
\(372\) 0 0
\(373\) −6.97056 −0.360922 −0.180461 0.983582i \(-0.557759\pi\)
−0.180461 + 0.983582i \(0.557759\pi\)
\(374\) −0.970563 −0.0501866
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 22.4853 1.15499 0.577496 0.816394i \(-0.304030\pi\)
0.577496 + 0.816394i \(0.304030\pi\)
\(380\) −8.82843 −0.452889
\(381\) 0 0
\(382\) −8.62742 −0.441417
\(383\) −2.48528 −0.126992 −0.0634960 0.997982i \(-0.520225\pi\)
−0.0634960 + 0.997982i \(0.520225\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) −1.85786 −0.0945628
\(387\) 0 0
\(388\) 22.8284 1.15894
\(389\) 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
\(392\) 25.8701 1.30664
\(393\) 0 0
\(394\) −8.14214 −0.410195
\(395\) 2.48528 0.125048
\(396\) 0 0
\(397\) −19.6569 −0.986549 −0.493275 0.869874i \(-0.664200\pi\)
−0.493275 + 0.869874i \(0.664200\pi\)
\(398\) −4.97056 −0.249152
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) 6.68629 0.333897 0.166949 0.985966i \(-0.446609\pi\)
0.166949 + 0.985966i \(0.446609\pi\)
\(402\) 0 0
\(403\) −12.9706 −0.646110
\(404\) 28.6274 1.42427
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 7.02944 0.348436
\(408\) 0 0
\(409\) −2.97056 −0.146885 −0.0734424 0.997299i \(-0.523399\pi\)
−0.0734424 + 0.997299i \(0.523399\pi\)
\(410\) 2.48528 0.122739
\(411\) 0 0
\(412\) −29.5147 −1.45409
\(413\) 0 0
\(414\) 0 0
\(415\) 7.17157 0.352039
\(416\) 8.82843 0.432849
\(417\) 0 0
\(418\) −1.65685 −0.0810394
\(419\) −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 \(0.347030\pi\)
\(422\) 0.343146 0.0167041
\(423\) 0 0
\(424\) 5.79899 0.281624
\(425\) −2.82843 −0.137199
\(426\) 0 0
\(427\) 17.6569 0.854475
\(428\) 36.8284 1.78017
\(429\) 0 0
\(430\) −2.48528 −0.119851
\(431\) −3.31371 −0.159616 −0.0798079 0.996810i \(-0.525431\pi\)
−0.0798079 + 0.996810i \(0.525431\pi\)
\(432\) 0 0
\(433\) −29.1716 −1.40190 −0.700948 0.713212i \(-0.747240\pi\)
−0.700948 + 0.713212i \(0.747240\pi\)
\(434\) 12.9706 0.622607
\(435\) 0 0
\(436\) −3.65685 −0.175132
\(437\) −15.3137 −0.732554
\(438\) 0 0
\(439\) −10.3431 −0.493651 −0.246826 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(440\) 1.31371 0.0626286
\(441\) 0 0
\(442\) −2.34315 −0.111452
\(443\) 7.65685 0.363788 0.181894 0.983318i \(-0.441777\pi\)
0.181894 + 0.983318i \(0.441777\pi\)
\(444\) 0 0
\(445\) −7.65685 −0.362970
\(446\) 7.37258 0.349102
\(447\) 0 0
\(448\) 20.1421 0.951626
\(449\) −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
\(452\) −5.17157 −0.243250
\(453\) 0 0
\(454\) 8.34315 0.391563
\(455\) −9.65685 −0.452720
\(456\) 0 0
\(457\) 19.6569 0.919509 0.459754 0.888046i \(-0.347937\pi\)
0.459754 + 0.888046i \(0.347937\pi\)
\(458\) 0.828427 0.0387099
\(459\) 0 0
\(460\) 5.79899 0.270379
\(461\) 35.6569 1.66071 0.830353 0.557238i \(-0.188139\pi\)
0.830353 + 0.557238i \(0.188139\pi\)
\(462\) 0 0
\(463\) 21.7990 1.01308 0.506542 0.862215i \(-0.330923\pi\)
0.506542 + 0.862215i \(0.330923\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 7.45584 0.345385
\(467\) −10.9706 −0.507657 −0.253829 0.967249i \(-0.581690\pi\)
−0.253829 + 0.967249i \(0.581690\pi\)
\(468\) 0 0
\(469\) −31.3137 −1.44593
\(470\) 4.82843 0.222719
\(471\) 0 0
\(472\) 0 0
\(473\) 4.97056 0.228547
\(474\) 0 0
\(475\) −4.82843 −0.221543
\(476\) −24.9706 −1.14452
\(477\) 0 0
\(478\) −0.284271 −0.0130023
\(479\) −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
\(482\) −4.14214 −0.188669
\(483\) 0 0
\(484\) 18.8579 0.857176
\(485\) 12.4853 0.566927
\(486\) 0 0
\(487\) 9.79899 0.444035 0.222017 0.975043i \(-0.428736\pi\)
0.222017 + 0.975043i \(0.428736\pi\)
\(488\) −5.79899 −0.262508
\(489\) 0 0
\(490\) 6.75736 0.305266
\(491\) 7.45584 0.336478 0.168239 0.985746i \(-0.446192\pi\)
0.168239 + 0.985746i \(0.446192\pi\)
\(492\) 0 0
\(493\) 2.82843 0.127386
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 19.4558 0.873593
\(497\) −73.9411 −3.31671
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 1.82843 0.0817697
\(501\) 0 0
\(502\) 3.65685 0.163213
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 15.6569 0.696721
\(506\) 1.08831 0.0483814
\(507\) 0 0
\(508\) −10.9706 −0.486740
\(509\) −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
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) 2.76955 0.122160
\(515\) −16.1421 −0.711307
\(516\) 0 0
\(517\) −9.65685 −0.424708
\(518\) −16.9706 −0.745644
\(519\) 0 0
\(520\) 3.17157 0.139083
\(521\) 21.3137 0.933771 0.466885 0.884318i \(-0.345376\pi\)
0.466885 + 0.884318i \(0.345376\pi\)
\(522\) 0 0
\(523\) 2.48528 0.108674 0.0543369 0.998523i \(-0.482696\pi\)
0.0543369 + 0.998523i \(0.482696\pi\)
\(524\) −22.2010 −0.969856
\(525\) 0 0
\(526\) −8.14214 −0.355014
\(527\) −18.3431 −0.799040
\(528\) 0 0
\(529\) −12.9411 −0.562658
\(530\) 1.51472 0.0657952
\(531\) 0 0
\(532\) −42.6274 −1.84813
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 20.1421 0.870820
\(536\) 10.2843 0.444213
\(537\) 0 0
\(538\) −8.82843 −0.380621
\(539\) −13.5147 −0.582120
\(540\) 0 0
\(541\) −5.02944 −0.216232 −0.108116 0.994138i \(-0.534482\pi\)
−0.108116 + 0.994138i \(0.534482\pi\)
\(542\) 4.05887 0.174344
\(543\) 0 0
\(544\) 12.4853 0.535302
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) −2.48528 −0.106263 −0.0531315 0.998588i \(-0.516920\pi\)
−0.0531315 + 0.998588i \(0.516920\pi\)
\(548\) −9.45584 −0.403934
\(549\) 0 0
\(550\) 0.343146 0.0146318
\(551\) 4.82843 0.205698
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 1.51472 0.0643542
\(555\) 0 0
\(556\) −39.5980 −1.67933
\(557\) 27.9411 1.18390 0.591952 0.805973i \(-0.298358\pi\)
0.591952 + 0.805973i \(0.298358\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 14.4853 0.612115
\(561\) 0 0
\(562\) −12.1421 −0.512185
\(563\) −7.65685 −0.322698 −0.161349 0.986897i \(-0.551584\pi\)
−0.161349 + 0.986897i \(0.551584\pi\)
\(564\) 0 0
\(565\) −2.82843 −0.118993
\(566\) −2.00000 −0.0840663
\(567\) 0 0
\(568\) 24.2843 1.01895
\(569\) −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 \(0.300748\pi\)
\(572\) −3.02944 −0.126667
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 3.17157 0.132264
\(576\) 0 0
\(577\) 23.7990 0.990765 0.495382 0.868675i \(-0.335028\pi\)
0.495382 + 0.868675i \(0.335028\pi\)
\(578\) 3.72792 0.155061
\(579\) 0 0
\(580\) −1.82843 −0.0759213
\(581\) 34.6274 1.43659
\(582\) 0 0
\(583\) −3.02944 −0.125466
\(584\) 13.4558 0.556807
\(585\) 0 0
\(586\) 3.51472 0.145192
\(587\) 29.7990 1.22994 0.614968 0.788552i \(-0.289169\pi\)
0.614968 + 0.788552i \(0.289169\pi\)
\(588\) 0 0
\(589\) −31.3137 −1.29026
\(590\) 0 0
\(591\) 0 0
\(592\) −25.4558 −1.04623
\(593\) 7.65685 0.314429 0.157215 0.987564i \(-0.449749\pi\)
0.157215 + 0.987564i \(0.449749\pi\)
\(594\) 0 0
\(595\) −13.6569 −0.559876
\(596\) 17.0294 0.697553
\(597\) 0 0
\(598\) 2.62742 0.107443
\(599\) 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 \(0.487012\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) 21.9411 0.892772
\(605\) 10.3137 0.419312
\(606\) 0 0
\(607\) 9.02944 0.366494 0.183247 0.983067i \(-0.441339\pi\)
0.183247 + 0.983067i \(0.441339\pi\)
\(608\) 21.3137 0.864385
\(609\) 0 0
\(610\) −1.51472 −0.0613292
\(611\) −23.3137 −0.943172
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 9.51472 0.383983
\(615\) 0 0
\(616\) 6.34315 0.255573
\(617\) 9.17157 0.369234 0.184617 0.982811i \(-0.440896\pi\)
0.184617 + 0.982811i \(0.440896\pi\)
\(618\) 0 0
\(619\) −9.79899 −0.393855 −0.196927 0.980418i \(-0.563096\pi\)
−0.196927 + 0.980418i \(0.563096\pi\)
\(620\) 11.8579 0.476223
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) −36.9706 −1.48119
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.48528 0.0993318
\(627\) 0 0
\(628\) −0.887302 −0.0354072
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 36.9706 1.47177 0.735887 0.677104i \(-0.236765\pi\)
0.735887 + 0.677104i \(0.236765\pi\)
\(632\) −3.94113 −0.156770
\(633\) 0 0
\(634\) 1.17157 0.0465291
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) −32.6274 −1.29275
\(638\) −0.343146 −0.0135853
\(639\) 0 0
\(640\) −10.5563 −0.417276
\(641\) −0.627417 −0.0247815 −0.0123907 0.999923i \(-0.503944\pi\)
−0.0123907 + 0.999923i \(0.503944\pi\)
\(642\) 0 0
\(643\) 19.4558 0.767264 0.383632 0.923486i \(-0.374673\pi\)
0.383632 + 0.923486i \(0.374673\pi\)
\(644\) 28.0000 1.10335
\(645\) 0 0
\(646\) −5.65685 −0.222566
\(647\) 41.1127 1.61631 0.808153 0.588972i \(-0.200467\pi\)
0.808153 + 0.588972i \(0.200467\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0.828427 0.0324936
\(651\) 0 0
\(652\) 15.2548 0.597425
\(653\) 17.1716 0.671976 0.335988 0.941866i \(-0.390930\pi\)
0.335988 + 0.941866i \(0.390930\pi\)
\(654\) 0 0
\(655\) −12.1421 −0.474432
\(656\) 18.0000 0.702782
\(657\) 0 0
\(658\) 23.3137 0.908863
\(659\) −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 \(0.331256\pi\)
\(662\) −9.02944 −0.350939
\(663\) 0 0
\(664\) −11.3726 −0.441342
\(665\) −23.3137 −0.904067
\(666\) 0 0
\(667\) −3.17157 −0.122804
\(668\) −4.54416 −0.175819
\(669\) 0 0
\(670\) 2.68629 0.103780
\(671\) 3.02944 0.116950
\(672\) 0 0
\(673\) 22.9706 0.885450 0.442725 0.896657i \(-0.354012\pi\)
0.442725 + 0.896657i \(0.354012\pi\)
\(674\) 0.485281 0.0186923
\(675\) 0 0
\(676\) 16.4558 0.632917
\(677\) 36.7696 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(678\) 0 0
\(679\) 60.2843 2.31350
\(680\) 4.48528 0.172003
\(681\) 0 0
\(682\) 2.22540 0.0852148
\(683\) −11.8579 −0.453729 −0.226864 0.973926i \(-0.572847\pi\)
−0.226864 + 0.973926i \(0.572847\pi\)
\(684\) 0 0
\(685\) −5.17157 −0.197596
\(686\) 18.6274 0.711198
\(687\) 0 0
\(688\) −18.0000 −0.686244
\(689\) −7.31371 −0.278630
\(690\) 0 0
\(691\) −44.9706 −1.71076 −0.855380 0.518000i \(-0.826677\pi\)
−0.855380 + 0.518000i \(0.826677\pi\)
\(692\) 31.6569 1.20341
\(693\) 0 0
\(694\) −3.37258 −0.128022
\(695\) −21.6569 −0.821491
\(696\) 0 0
\(697\) −16.9706 −0.642806
\(698\) −8.54416 −0.323401
\(699\) 0 0
\(700\) 8.82843 0.333683
\(701\) −6.68629 −0.252538 −0.126269 0.991996i \(-0.540300\pi\)
−0.126269 + 0.991996i \(0.540300\pi\)
\(702\) 0 0
\(703\) 40.9706 1.54523
\(704\) 3.45584 0.130247
\(705\) 0 0
\(706\) −1.79899 −0.0677059
\(707\) 75.5980 2.84315
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 6.34315 0.238054
\(711\) 0 0
\(712\) 12.1421 0.455046
\(713\) 20.5685 0.770298
\(714\) 0 0
\(715\) −1.65685 −0.0619628
\(716\) −42.6274 −1.59306
\(717\) 0 0
\(718\) −1.59798 −0.0596361
\(719\) 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
\(722\) −1.78680 −0.0664977
\(723\) 0 0
\(724\) 10.9706 0.407718
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −23.9411 −0.887927 −0.443964 0.896045i \(-0.646428\pi\)
−0.443964 + 0.896045i \(0.646428\pi\)
\(728\) 15.3137 0.567564
\(729\) 0 0
\(730\) 3.51472 0.130086
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) −22.8284 −0.843187 −0.421594 0.906785i \(-0.638529\pi\)
−0.421594 + 0.906785i \(0.638529\pi\)
\(734\) −7.45584 −0.275200
\(735\) 0 0
\(736\) −14.0000 −0.516047
\(737\) −5.37258 −0.197902
\(738\) 0 0
\(739\) 14.4853 0.532850 0.266425 0.963856i \(-0.414158\pi\)
0.266425 + 0.963856i \(0.414158\pi\)
\(740\) −15.5147 −0.570332
\(741\) 0 0
\(742\) 7.31371 0.268495
\(743\) 52.6274 1.93071 0.965356 0.260935i \(-0.0840309\pi\)
0.965356 + 0.260935i \(0.0840309\pi\)
\(744\) 0 0
\(745\) 9.31371 0.341228
\(746\) 2.88730 0.105712
\(747\) 0 0
\(748\) −4.28427 −0.156648
\(749\) 97.2548 3.55361
\(750\) 0 0
\(751\) 16.1421 0.589035 0.294517 0.955646i \(-0.404841\pi\)
0.294517 + 0.955646i \(0.404841\pi\)
\(752\) 34.9706 1.27525
\(753\) 0 0
\(754\) −0.828427 −0.0301695
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 19.5147 0.709275 0.354637 0.935004i \(-0.384604\pi\)
0.354637 + 0.935004i \(0.384604\pi\)
\(758\) −9.31371 −0.338289
\(759\) 0 0
\(760\) 7.65685 0.277743
\(761\) −8.62742 −0.312744 −0.156372 0.987698i \(-0.549980\pi\)
−0.156372 + 0.987698i \(0.549980\pi\)
\(762\) 0 0
\(763\) −9.65685 −0.349602
\(764\) −38.0833 −1.37780
\(765\) 0 0
\(766\) 1.02944 0.0371951
\(767\) 0 0
\(768\) 0 0
\(769\) −15.6569 −0.564601 −0.282300 0.959326i \(-0.591097\pi\)
−0.282300 + 0.959326i \(0.591097\pi\)
\(770\) 1.65685 0.0597089
\(771\) 0 0
\(772\) −8.20101 −0.295161
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) 0 0
\(775\) 6.48528 0.232958
\(776\) −19.7990 −0.710742
\(777\) 0 0
\(778\) −12.1421 −0.435317
\(779\) −28.9706 −1.03798
\(780\) 0 0
\(781\) −12.6863 −0.453951
\(782\) 3.71573 0.132874
\(783\) 0 0
\(784\) 48.9411 1.74790
\(785\) −0.485281 −0.0173204
\(786\) 0 0
\(787\) 17.7990 0.634465 0.317233 0.948348i \(-0.397246\pi\)
0.317233 + 0.948348i \(0.397246\pi\)
\(788\) −35.9411 −1.28035
\(789\) 0 0
\(790\) −1.02944 −0.0366257
\(791\) −13.6569 −0.485582
\(792\) 0 0
\(793\) 7.31371 0.259717
\(794\) 8.14214 0.288954
\(795\) 0 0
\(796\) −21.9411 −0.777683
\(797\) 5.85786 0.207496 0.103748 0.994604i \(-0.466916\pi\)
0.103748 + 0.994604i \(0.466916\pi\)
\(798\) 0 0
\(799\) −32.9706 −1.16641
\(800\) −4.41421 −0.156066
\(801\) 0 0
\(802\) −2.76955 −0.0977963
\(803\) −7.02944 −0.248063
\(804\) 0 0
\(805\) 15.3137 0.539737
\(806\) 5.37258 0.189241
\(807\) 0 0
\(808\) −24.8284 −0.873461
\(809\) −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 \(0.270066\pi\)
\(812\) −8.82843 −0.309817
\(813\) 0 0
\(814\) −2.91169 −0.102055
\(815\) 8.34315 0.292248
\(816\) 0 0
\(817\) 28.9706 1.01355
\(818\) 1.23045 0.0430216
\(819\) 0 0
\(820\) 10.9706 0.383109
\(821\) 22.6863 0.791757 0.395879 0.918303i \(-0.370440\pi\)
0.395879 + 0.918303i \(0.370440\pi\)
\(822\) 0 0
\(823\) −30.9706 −1.07957 −0.539783 0.841804i \(-0.681494\pi\)
−0.539783 + 0.841804i \(0.681494\pi\)
\(824\) 25.5980 0.891748
\(825\) 0 0
\(826\) 0 0
\(827\) −17.3137 −0.602057 −0.301028 0.953615i \(-0.597330\pi\)
−0.301028 + 0.953615i \(0.597330\pi\)
\(828\) 0 0
\(829\) 20.6274 0.716420 0.358210 0.933641i \(-0.383387\pi\)
0.358210 + 0.933641i \(0.383387\pi\)
\(830\) −2.97056 −0.103110
\(831\) 0 0
\(832\) 8.34315 0.289247
\(833\) −46.1421 −1.59873
\(834\) 0 0
\(835\) −2.48528 −0.0860067
\(836\) −7.31371 −0.252950
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −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
\(842\) −7.85786 −0.270800
\(843\) 0 0
\(844\) 1.51472 0.0521388
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 49.7990 1.71111
\(848\) 10.9706 0.376731
\(849\) 0 0
\(850\) 1.17157 0.0401846
\(851\) −26.9117 −0.922521
\(852\) 0 0
\(853\) 51.1127 1.75007 0.875033 0.484064i \(-0.160840\pi\)
0.875033 + 0.484064i \(0.160840\pi\)
\(854\) −7.31371 −0.250270
\(855\) 0 0
\(856\) −31.9411 −1.09173
\(857\) −3.37258 −0.115205 −0.0576026 0.998340i \(-0.518346\pi\)
−0.0576026 + 0.998340i \(0.518346\pi\)
\(858\) 0 0
\(859\) −56.4264 −1.92524 −0.962622 0.270848i \(-0.912696\pi\)
−0.962622 + 0.270848i \(0.912696\pi\)
\(860\) −10.9706 −0.374093
\(861\) 0 0
\(862\) 1.37258 0.0467504
\(863\) 36.1421 1.23029 0.615146 0.788413i \(-0.289097\pi\)
0.615146 + 0.788413i \(0.289097\pi\)
\(864\) 0 0
\(865\) 17.3137 0.588684
\(866\) 12.0833 0.410606
\(867\) 0 0
\(868\) 57.2548 1.94336
\(869\) 2.05887 0.0698425
\(870\) 0 0
\(871\) −12.9706 −0.439491
\(872\) 3.17157 0.107403
\(873\) 0 0
\(874\) 6.34315 0.214560
\(875\) 4.82843 0.163231
\(876\) 0 0
\(877\) 38.2843 1.29277 0.646384 0.763012i \(-0.276280\pi\)
0.646384 + 0.763012i \(0.276280\pi\)
\(878\) 4.28427 0.144587
\(879\) 0 0
\(880\) 2.48528 0.0837788
\(881\) −29.3137 −0.987604 −0.493802 0.869574i \(-0.664393\pi\)
−0.493802 + 0.869574i \(0.664393\pi\)
\(882\) 0 0
\(883\) −14.4853 −0.487469 −0.243734 0.969842i \(-0.578372\pi\)
−0.243734 + 0.969842i \(0.578372\pi\)
\(884\) −10.3431 −0.347878
\(885\) 0 0
\(886\) −3.17157 −0.106551
\(887\) 6.68629 0.224504 0.112252 0.993680i \(-0.464194\pi\)
0.112252 + 0.993680i \(0.464194\pi\)
\(888\) 0 0
\(889\) −28.9706 −0.971641
\(890\) 3.17157 0.106311
\(891\) 0 0
\(892\) 32.5442 1.08966
\(893\) −56.2843 −1.88348
\(894\) 0 0
\(895\) −23.3137 −0.779291
\(896\) −50.9706 −1.70281
\(897\) 0 0
\(898\) 4.82843 0.161127
\(899\) −6.48528 −0.216296
\(900\) 0 0
\(901\) −10.3431 −0.344580
\(902\) 2.05887 0.0685530
\(903\) 0 0
\(904\) 4.48528 0.149178
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 36.8284 1.22219
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) −32.1421 −1.06492 −0.532458 0.846456i \(-0.678732\pi\)
−0.532458 + 0.846456i \(0.678732\pi\)
\(912\) 0 0
\(913\) 5.94113 0.196623
\(914\) −8.14214 −0.269318
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) −58.6274 −1.93605
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) −5.02944 −0.165816
\(921\) 0 0
\(922\) −14.7696 −0.486409
\(923\) −30.6274 −1.00811
\(924\) 0 0
\(925\) −8.48528 −0.278994
\(926\) −9.02944 −0.296726
\(927\) 0 0
\(928\) 4.41421 0.144904
\(929\) 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
\(932\) 32.9117 1.07806
\(933\) 0 0
\(934\) 4.54416 0.148689
\(935\) −2.34315 −0.0766291
\(936\) 0 0
\(937\) 19.6569 0.642161 0.321081 0.947052i \(-0.395954\pi\)
0.321081 + 0.947052i \(0.395954\pi\)
\(938\) 12.9706 0.423504
\(939\) 0 0
\(940\) 21.3137 0.695177
\(941\) 27.9411 0.910855 0.455427 0.890273i \(-0.349486\pi\)
0.455427 + 0.890273i \(0.349486\pi\)
\(942\) 0 0
\(943\) 19.0294 0.619684
\(944\) 0 0
\(945\) 0 0
\(946\) −2.05887 −0.0669398
\(947\) −44.9117 −1.45943 −0.729717 0.683749i \(-0.760348\pi\)
−0.729717 + 0.683749i \(0.760348\pi\)
\(948\) 0 0
\(949\) −16.9706 −0.550888
\(950\) 2.00000 0.0648886
\(951\) 0 0
\(952\) 21.6569 0.701903
\(953\) −29.3137 −0.949564 −0.474782 0.880103i \(-0.657473\pi\)
−0.474782 + 0.880103i \(0.657473\pi\)
\(954\) 0 0
\(955\) −20.8284 −0.673992
\(956\) −1.25483 −0.0405842
\(957\) 0 0
\(958\) 2.97056 0.0959745
\(959\) −24.9706 −0.806342
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) −7.02944 −0.226638
\(963\) 0 0
\(964\) −18.2843 −0.588897
\(965\) −4.48528 −0.144386
\(966\) 0 0
\(967\) 14.9706 0.481421 0.240710 0.970597i \(-0.422620\pi\)
0.240710 + 0.970597i \(0.422620\pi\)
\(968\) −16.3553 −0.525681
\(969\) 0 0
\(970\) −5.17157 −0.166049
\(971\) −28.1421 −0.903124 −0.451562 0.892240i \(-0.649133\pi\)
−0.451562 + 0.892240i \(0.649133\pi\)
\(972\) 0 0
\(973\) −104.569 −3.35231
\(974\) −4.05887 −0.130055
\(975\) 0 0
\(976\) −10.9706 −0.351159
\(977\) 2.68629 0.0859421 0.0429710 0.999076i \(-0.486318\pi\)
0.0429710 + 0.999076i \(0.486318\pi\)
\(978\) 0 0
\(979\) −6.34315 −0.202728
\(980\) 29.8284 0.952834
\(981\) 0 0
\(982\) −3.08831 −0.0985520
\(983\) 9.31371 0.297061 0.148531 0.988908i \(-0.452546\pi\)
0.148531 + 0.988908i \(0.452546\pi\)
\(984\) 0 0
\(985\) −19.6569 −0.626319
\(986\) −1.17157 −0.0373105
\(987\) 0 0
\(988\) −17.6569 −0.561739
\(989\) −19.0294 −0.605101
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) −28.6274 −0.908921
\(993\) 0 0
\(994\) 30.6274 0.971443
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 6.82843 0.216258 0.108129 0.994137i \(-0.465514\pi\)
0.108129 + 0.994137i \(0.465514\pi\)
\(998\) −14.9117 −0.472021
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1305.2.a.n.1.1 2
3.2 odd 2 145.2.a.b.1.2 2
5.4 even 2 6525.2.a.p.1.2 2
12.11 even 2 2320.2.a.k.1.2 2
15.2 even 4 725.2.b.c.349.3 4
15.8 even 4 725.2.b.c.349.2 4
15.14 odd 2 725.2.a.c.1.1 2
21.20 even 2 7105.2.a.e.1.2 2
24.5 odd 2 9280.2.a.be.1.1 2
24.11 even 2 9280.2.a.w.1.2 2
87.86 odd 2 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 2 3.2 odd 2
725.2.a.c.1.1 2 15.14 odd 2
725.2.b.c.349.2 4 15.8 even 4
725.2.b.c.349.3 4 15.2 even 4
1305.2.a.n.1.1 2 1.1 even 1 trivial
2320.2.a.k.1.2 2 12.11 even 2
4205.2.a.d.1.1 2 87.86 odd 2
6525.2.a.p.1.2 2 5.4 even 2
7105.2.a.e.1.2 2 21.20 even 2
9280.2.a.w.1.2 2 24.11 even 2
9280.2.a.be.1.1 2 24.5 odd 2