Properties

Label 4508.2.a.f.1.1
Level $4508$
Weight $2$
Character 4508.1
Self dual yes
Analytic conductor $35.997$
Analytic rank $1$
Dimension $5$
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] = [4508,2,Mod(1,4508)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4508, 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("4508.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4508 = 2^{2} \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4508.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: \(35.9965612312\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.6963152.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 644)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.25688\) of defining polynomial
Character \(\chi\) \(=\) 4508.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.25688 q^{3} +3.04771 q^{5} +7.60729 q^{9} +O(q^{10})\) \(q-3.25688 q^{3} +3.04771 q^{5} +7.60729 q^{9} +5.22523 q^{11} -3.38206 q^{13} -9.92604 q^{15} +2.63895 q^{17} -3.39461 q^{19} -1.00000 q^{23} +4.28854 q^{25} -15.0054 q^{27} -4.00190 q^{29} -6.97185 q^{31} -17.0180 q^{33} -5.11916 q^{37} +11.0150 q^{39} +5.89583 q^{41} -7.90838 q^{43} +23.1848 q^{45} -9.54893 q^{47} -8.59475 q^{51} -11.8200 q^{53} +15.9250 q^{55} +11.0558 q^{57} -1.51979 q^{59} -1.13934 q^{61} -10.3076 q^{65} -8.80688 q^{67} +3.25688 q^{69} -14.5333 q^{71} +13.2271 q^{73} -13.9673 q^{75} +14.4398 q^{79} +26.0490 q^{81} +0.693795 q^{83} +8.04275 q^{85} +13.0337 q^{87} -10.8511 q^{89} +22.7065 q^{93} -10.3458 q^{95} +3.23476 q^{97} +39.7498 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{3} - 2 q^{5} + 10 q^{9} + 2 q^{11} - 13 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} - 5 q^{23} + 19 q^{25} - 15 q^{27} + 13 q^{29} + 3 q^{31} - 24 q^{33} - 4 q^{37} + 3 q^{39} - q^{41} - 8 q^{43} + 16 q^{45} - 5 q^{47} - 16 q^{51} - 8 q^{53} + 2 q^{55} + 12 q^{57} - 12 q^{59} - 20 q^{61} - 12 q^{65} - 12 q^{67} + 3 q^{69} + 9 q^{71} + 9 q^{73} - 35 q^{75} - 8 q^{79} - 11 q^{81} + 28 q^{83} + 16 q^{85} + 15 q^{87} - 32 q^{89} - 15 q^{93} - 36 q^{95} - 4 q^{97} + 28 q^{99}+O(q^{100}) \) 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 0
\(3\) −3.25688 −1.88036 −0.940181 0.340675i \(-0.889345\pi\)
−0.940181 + 0.340675i \(0.889345\pi\)
\(4\) 0 0
\(5\) 3.04771 1.36298 0.681489 0.731828i \(-0.261333\pi\)
0.681489 + 0.731828i \(0.261333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 7.60729 2.53576
\(10\) 0 0
\(11\) 5.22523 1.57546 0.787732 0.616018i \(-0.211255\pi\)
0.787732 + 0.616018i \(0.211255\pi\)
\(12\) 0 0
\(13\) −3.38206 −0.938016 −0.469008 0.883194i \(-0.655388\pi\)
−0.469008 + 0.883194i \(0.655388\pi\)
\(14\) 0 0
\(15\) −9.92604 −2.56289
\(16\) 0 0
\(17\) 2.63895 0.640039 0.320019 0.947411i \(-0.396311\pi\)
0.320019 + 0.947411i \(0.396311\pi\)
\(18\) 0 0
\(19\) −3.39461 −0.778777 −0.389388 0.921074i \(-0.627314\pi\)
−0.389388 + 0.921074i \(0.627314\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 4.28854 0.857708
\(26\) 0 0
\(27\) −15.0054 −2.88779
\(28\) 0 0
\(29\) −4.00190 −0.743134 −0.371567 0.928406i \(-0.621179\pi\)
−0.371567 + 0.928406i \(0.621179\pi\)
\(30\) 0 0
\(31\) −6.97185 −1.25218 −0.626091 0.779750i \(-0.715346\pi\)
−0.626091 + 0.779750i \(0.715346\pi\)
\(32\) 0 0
\(33\) −17.0180 −2.96245
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.11916 −0.841585 −0.420792 0.907157i \(-0.638248\pi\)
−0.420792 + 0.907157i \(0.638248\pi\)
\(38\) 0 0
\(39\) 11.0150 1.76381
\(40\) 0 0
\(41\) 5.89583 0.920774 0.460387 0.887718i \(-0.347711\pi\)
0.460387 + 0.887718i \(0.347711\pi\)
\(42\) 0 0
\(43\) −7.90838 −1.20602 −0.603008 0.797735i \(-0.706031\pi\)
−0.603008 + 0.797735i \(0.706031\pi\)
\(44\) 0 0
\(45\) 23.1848 3.45619
\(46\) 0 0
\(47\) −9.54893 −1.39286 −0.696428 0.717627i \(-0.745228\pi\)
−0.696428 + 0.717627i \(0.745228\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.59475 −1.20351
\(52\) 0 0
\(53\) −11.8200 −1.62360 −0.811799 0.583937i \(-0.801512\pi\)
−0.811799 + 0.583937i \(0.801512\pi\)
\(54\) 0 0
\(55\) 15.9250 2.14732
\(56\) 0 0
\(57\) 11.0558 1.46438
\(58\) 0 0
\(59\) −1.51979 −0.197860 −0.0989299 0.995094i \(-0.531542\pi\)
−0.0989299 + 0.995094i \(0.531542\pi\)
\(60\) 0 0
\(61\) −1.13934 −0.145877 −0.0729385 0.997336i \(-0.523238\pi\)
−0.0729385 + 0.997336i \(0.523238\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.3076 −1.27849
\(66\) 0 0
\(67\) −8.80688 −1.07593 −0.537966 0.842967i \(-0.680807\pi\)
−0.537966 + 0.842967i \(0.680807\pi\)
\(68\) 0 0
\(69\) 3.25688 0.392083
\(70\) 0 0
\(71\) −14.5333 −1.72479 −0.862394 0.506237i \(-0.831036\pi\)
−0.862394 + 0.506237i \(0.831036\pi\)
\(72\) 0 0
\(73\) 13.2271 1.54812 0.774059 0.633114i \(-0.218223\pi\)
0.774059 + 0.633114i \(0.218223\pi\)
\(74\) 0 0
\(75\) −13.9673 −1.61280
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 14.4398 1.62461 0.812303 0.583236i \(-0.198214\pi\)
0.812303 + 0.583236i \(0.198214\pi\)
\(80\) 0 0
\(81\) 26.0490 2.89433
\(82\) 0 0
\(83\) 0.693795 0.0761538 0.0380769 0.999275i \(-0.487877\pi\)
0.0380769 + 0.999275i \(0.487877\pi\)
\(84\) 0 0
\(85\) 8.04275 0.872359
\(86\) 0 0
\(87\) 13.0337 1.39736
\(88\) 0 0
\(89\) −10.8511 −1.15021 −0.575106 0.818079i \(-0.695039\pi\)
−0.575106 + 0.818079i \(0.695039\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 22.7065 2.35456
\(94\) 0 0
\(95\) −10.3458 −1.06146
\(96\) 0 0
\(97\) 3.23476 0.328440 0.164220 0.986424i \(-0.447489\pi\)
0.164220 + 0.986424i \(0.447489\pi\)
\(98\) 0 0
\(99\) 39.7498 3.99501
\(100\) 0 0
\(101\) 8.32672 0.828540 0.414270 0.910154i \(-0.364037\pi\)
0.414270 + 0.910154i \(0.364037\pi\)
\(102\) 0 0
\(103\) −0.418345 −0.0412208 −0.0206104 0.999788i \(-0.506561\pi\)
−0.0206104 + 0.999788i \(0.506561\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.9084 −1.15123 −0.575613 0.817722i \(-0.695237\pi\)
−0.575613 + 0.817722i \(0.695237\pi\)
\(108\) 0 0
\(109\) 17.9154 1.71598 0.857992 0.513663i \(-0.171712\pi\)
0.857992 + 0.513663i \(0.171712\pi\)
\(110\) 0 0
\(111\) 16.6725 1.58248
\(112\) 0 0
\(113\) −12.6375 −1.18884 −0.594418 0.804156i \(-0.702617\pi\)
−0.594418 + 0.804156i \(0.702617\pi\)
\(114\) 0 0
\(115\) −3.04771 −0.284201
\(116\) 0 0
\(117\) −25.7283 −2.37859
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 16.3030 1.48209
\(122\) 0 0
\(123\) −19.2020 −1.73139
\(124\) 0 0
\(125\) −2.16832 −0.193940
\(126\) 0 0
\(127\) −3.22333 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(128\) 0 0
\(129\) 25.7567 2.26775
\(130\) 0 0
\(131\) 14.0910 1.23114 0.615569 0.788083i \(-0.288926\pi\)
0.615569 + 0.788083i \(0.288926\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −45.7321 −3.93600
\(136\) 0 0
\(137\) 2.67250 0.228327 0.114164 0.993462i \(-0.463581\pi\)
0.114164 + 0.993462i \(0.463581\pi\)
\(138\) 0 0
\(139\) −6.23315 −0.528689 −0.264344 0.964428i \(-0.585156\pi\)
−0.264344 + 0.964428i \(0.585156\pi\)
\(140\) 0 0
\(141\) 31.0998 2.61907
\(142\) 0 0
\(143\) −17.6721 −1.47781
\(144\) 0 0
\(145\) −12.1966 −1.01287
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.58165 −0.457267 −0.228633 0.973513i \(-0.573426\pi\)
−0.228633 + 0.973513i \(0.573426\pi\)
\(150\) 0 0
\(151\) −11.8325 −0.962916 −0.481458 0.876469i \(-0.659893\pi\)
−0.481458 + 0.876469i \(0.659893\pi\)
\(152\) 0 0
\(153\) 20.0752 1.62299
\(154\) 0 0
\(155\) −21.2482 −1.70670
\(156\) 0 0
\(157\) −17.3015 −1.38081 −0.690406 0.723422i \(-0.742568\pi\)
−0.690406 + 0.723422i \(0.742568\pi\)
\(158\) 0 0
\(159\) 38.4963 3.05295
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 14.3212 1.12172 0.560861 0.827910i \(-0.310470\pi\)
0.560861 + 0.827910i \(0.310470\pi\)
\(164\) 0 0
\(165\) −51.8658 −4.03775
\(166\) 0 0
\(167\) −6.28358 −0.486238 −0.243119 0.969996i \(-0.578171\pi\)
−0.243119 + 0.969996i \(0.578171\pi\)
\(168\) 0 0
\(169\) −1.56164 −0.120126
\(170\) 0 0
\(171\) −25.8238 −1.97479
\(172\) 0 0
\(173\) 2.60539 0.198084 0.0990421 0.995083i \(-0.468422\pi\)
0.0990421 + 0.995083i \(0.468422\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.94978 0.372048
\(178\) 0 0
\(179\) 25.2341 1.88609 0.943044 0.332667i \(-0.107949\pi\)
0.943044 + 0.332667i \(0.107949\pi\)
\(180\) 0 0
\(181\) 1.36485 0.101448 0.0507242 0.998713i \(-0.483847\pi\)
0.0507242 + 0.998713i \(0.483847\pi\)
\(182\) 0 0
\(183\) 3.71068 0.274302
\(184\) 0 0
\(185\) −15.6017 −1.14706
\(186\) 0 0
\(187\) 13.7891 1.00836
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −22.1467 −1.60248 −0.801239 0.598344i \(-0.795826\pi\)
−0.801239 + 0.598344i \(0.795826\pi\)
\(192\) 0 0
\(193\) 18.3653 1.32197 0.660983 0.750401i \(-0.270139\pi\)
0.660983 + 0.750401i \(0.270139\pi\)
\(194\) 0 0
\(195\) 33.5705 2.40403
\(196\) 0 0
\(197\) −1.07831 −0.0768261 −0.0384131 0.999262i \(-0.512230\pi\)
−0.0384131 + 0.999262i \(0.512230\pi\)
\(198\) 0 0
\(199\) 5.55335 0.393666 0.196833 0.980437i \(-0.436934\pi\)
0.196833 + 0.980437i \(0.436934\pi\)
\(200\) 0 0
\(201\) 28.6830 2.02314
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 17.9688 1.25499
\(206\) 0 0
\(207\) −7.60729 −0.528743
\(208\) 0 0
\(209\) −17.7376 −1.22693
\(210\) 0 0
\(211\) −19.3543 −1.33240 −0.666201 0.745772i \(-0.732081\pi\)
−0.666201 + 0.745772i \(0.732081\pi\)
\(212\) 0 0
\(213\) 47.3334 3.24323
\(214\) 0 0
\(215\) −24.1024 −1.64377
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −43.0792 −2.91102
\(220\) 0 0
\(221\) −8.92509 −0.600367
\(222\) 0 0
\(223\) 2.69234 0.180293 0.0901464 0.995929i \(-0.471267\pi\)
0.0901464 + 0.995929i \(0.471267\pi\)
\(224\) 0 0
\(225\) 32.6242 2.17495
\(226\) 0 0
\(227\) 8.26663 0.548675 0.274338 0.961633i \(-0.411541\pi\)
0.274338 + 0.961633i \(0.411541\pi\)
\(228\) 0 0
\(229\) −26.2369 −1.73378 −0.866891 0.498499i \(-0.833885\pi\)
−0.866891 + 0.498499i \(0.833885\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.17735 0.0771310 0.0385655 0.999256i \(-0.487721\pi\)
0.0385655 + 0.999256i \(0.487721\pi\)
\(234\) 0 0
\(235\) −29.1024 −1.89843
\(236\) 0 0
\(237\) −47.0288 −3.05485
\(238\) 0 0
\(239\) 5.56911 0.360236 0.180118 0.983645i \(-0.442352\pi\)
0.180118 + 0.983645i \(0.442352\pi\)
\(240\) 0 0
\(241\) −8.12518 −0.523389 −0.261694 0.965151i \(-0.584281\pi\)
−0.261694 + 0.965151i \(0.584281\pi\)
\(242\) 0 0
\(243\) −39.8223 −2.55460
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 11.4808 0.730505
\(248\) 0 0
\(249\) −2.25961 −0.143197
\(250\) 0 0
\(251\) −2.78542 −0.175814 −0.0879071 0.996129i \(-0.528018\pi\)
−0.0879071 + 0.996129i \(0.528018\pi\)
\(252\) 0 0
\(253\) −5.22523 −0.328507
\(254\) 0 0
\(255\) −26.1943 −1.64035
\(256\) 0 0
\(257\) 24.1150 1.50425 0.752126 0.659020i \(-0.229029\pi\)
0.752126 + 0.659020i \(0.229029\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −30.4436 −1.88441
\(262\) 0 0
\(263\) 3.40905 0.210211 0.105106 0.994461i \(-0.466482\pi\)
0.105106 + 0.994461i \(0.466482\pi\)
\(264\) 0 0
\(265\) −36.0239 −2.21293
\(266\) 0 0
\(267\) 35.3407 2.16282
\(268\) 0 0
\(269\) −5.29044 −0.322564 −0.161282 0.986908i \(-0.551563\pi\)
−0.161282 + 0.986908i \(0.551563\pi\)
\(270\) 0 0
\(271\) −16.2206 −0.985331 −0.492666 0.870219i \(-0.663977\pi\)
−0.492666 + 0.870219i \(0.663977\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.4086 1.35129
\(276\) 0 0
\(277\) −22.2211 −1.33513 −0.667567 0.744550i \(-0.732664\pi\)
−0.667567 + 0.744550i \(0.732664\pi\)
\(278\) 0 0
\(279\) −53.0369 −3.17524
\(280\) 0 0
\(281\) −5.74964 −0.342995 −0.171497 0.985185i \(-0.554860\pi\)
−0.171497 + 0.985185i \(0.554860\pi\)
\(282\) 0 0
\(283\) 14.0108 0.832857 0.416428 0.909169i \(-0.363282\pi\)
0.416428 + 0.909169i \(0.363282\pi\)
\(284\) 0 0
\(285\) 33.6950 1.99592
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.0360 −0.590350
\(290\) 0 0
\(291\) −10.5352 −0.617586
\(292\) 0 0
\(293\) −15.7616 −0.920801 −0.460400 0.887711i \(-0.652294\pi\)
−0.460400 + 0.887711i \(0.652294\pi\)
\(294\) 0 0
\(295\) −4.63188 −0.269678
\(296\) 0 0
\(297\) −78.4066 −4.54961
\(298\) 0 0
\(299\) 3.38206 0.195590
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −27.1192 −1.55795
\(304\) 0 0
\(305\) −3.47237 −0.198827
\(306\) 0 0
\(307\) −11.4944 −0.656018 −0.328009 0.944675i \(-0.606378\pi\)
−0.328009 + 0.944675i \(0.606378\pi\)
\(308\) 0 0
\(309\) 1.36250 0.0775100
\(310\) 0 0
\(311\) −4.41105 −0.250128 −0.125064 0.992149i \(-0.539914\pi\)
−0.125064 + 0.992149i \(0.539914\pi\)
\(312\) 0 0
\(313\) 6.20154 0.350532 0.175266 0.984521i \(-0.443921\pi\)
0.175266 + 0.984521i \(0.443921\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.50769 −0.534005 −0.267003 0.963696i \(-0.586033\pi\)
−0.267003 + 0.963696i \(0.586033\pi\)
\(318\) 0 0
\(319\) −20.9108 −1.17078
\(320\) 0 0
\(321\) 38.7842 2.16472
\(322\) 0 0
\(323\) −8.95819 −0.498447
\(324\) 0 0
\(325\) −14.5041 −0.804544
\(326\) 0 0
\(327\) −58.3484 −3.22667
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0087 0.550130 0.275065 0.961426i \(-0.411301\pi\)
0.275065 + 0.961426i \(0.411301\pi\)
\(332\) 0 0
\(333\) −38.9429 −2.13406
\(334\) 0 0
\(335\) −26.8408 −1.46647
\(336\) 0 0
\(337\) 16.5389 0.900929 0.450464 0.892794i \(-0.351258\pi\)
0.450464 + 0.892794i \(0.351258\pi\)
\(338\) 0 0
\(339\) 41.1589 2.23544
\(340\) 0 0
\(341\) −36.4295 −1.97277
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 9.92604 0.534400
\(346\) 0 0
\(347\) 34.8917 1.87308 0.936541 0.350558i \(-0.114008\pi\)
0.936541 + 0.350558i \(0.114008\pi\)
\(348\) 0 0
\(349\) −6.92337 −0.370599 −0.185300 0.982682i \(-0.559326\pi\)
−0.185300 + 0.982682i \(0.559326\pi\)
\(350\) 0 0
\(351\) 50.7493 2.70880
\(352\) 0 0
\(353\) −21.6050 −1.14992 −0.574959 0.818182i \(-0.694982\pi\)
−0.574959 + 0.818182i \(0.694982\pi\)
\(354\) 0 0
\(355\) −44.2934 −2.35085
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.2085 −1.38323 −0.691616 0.722265i \(-0.743101\pi\)
−0.691616 + 0.722265i \(0.743101\pi\)
\(360\) 0 0
\(361\) −7.47664 −0.393507
\(362\) 0 0
\(363\) −53.0969 −2.78687
\(364\) 0 0
\(365\) 40.3124 2.11005
\(366\) 0 0
\(367\) −16.5650 −0.864688 −0.432344 0.901709i \(-0.642313\pi\)
−0.432344 + 0.901709i \(0.642313\pi\)
\(368\) 0 0
\(369\) 44.8513 2.33487
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 15.6116 0.808340 0.404170 0.914684i \(-0.367560\pi\)
0.404170 + 0.914684i \(0.367560\pi\)
\(374\) 0 0
\(375\) 7.06197 0.364678
\(376\) 0 0
\(377\) 13.5347 0.697071
\(378\) 0 0
\(379\) −9.80933 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(380\) 0 0
\(381\) 10.4980 0.537829
\(382\) 0 0
\(383\) 18.4925 0.944921 0.472461 0.881352i \(-0.343366\pi\)
0.472461 + 0.881352i \(0.343366\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −60.1613 −3.05817
\(388\) 0 0
\(389\) 11.3030 0.573084 0.286542 0.958068i \(-0.407494\pi\)
0.286542 + 0.958068i \(0.407494\pi\)
\(390\) 0 0
\(391\) −2.63895 −0.133457
\(392\) 0 0
\(393\) −45.8928 −2.31498
\(394\) 0 0
\(395\) 44.0084 2.21430
\(396\) 0 0
\(397\) 1.20204 0.0603285 0.0301643 0.999545i \(-0.490397\pi\)
0.0301643 + 0.999545i \(0.490397\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.01127 −0.150376 −0.0751878 0.997169i \(-0.523956\pi\)
−0.0751878 + 0.997169i \(0.523956\pi\)
\(402\) 0 0
\(403\) 23.5793 1.17457
\(404\) 0 0
\(405\) 79.3898 3.94491
\(406\) 0 0
\(407\) −26.7488 −1.32589
\(408\) 0 0
\(409\) 33.4725 1.65511 0.827553 0.561387i \(-0.189732\pi\)
0.827553 + 0.561387i \(0.189732\pi\)
\(410\) 0 0
\(411\) −8.70404 −0.429338
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2.11449 0.103796
\(416\) 0 0
\(417\) 20.3006 0.994126
\(418\) 0 0
\(419\) 6.40028 0.312674 0.156337 0.987704i \(-0.450031\pi\)
0.156337 + 0.987704i \(0.450031\pi\)
\(420\) 0 0
\(421\) −8.01082 −0.390423 −0.195212 0.980761i \(-0.562539\pi\)
−0.195212 + 0.980761i \(0.562539\pi\)
\(422\) 0 0
\(423\) −72.6415 −3.53195
\(424\) 0 0
\(425\) 11.3172 0.548967
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 57.5558 2.77882
\(430\) 0 0
\(431\) −27.6084 −1.32985 −0.664925 0.746910i \(-0.731537\pi\)
−0.664925 + 0.746910i \(0.731537\pi\)
\(432\) 0 0
\(433\) 14.8797 0.715074 0.357537 0.933899i \(-0.383617\pi\)
0.357537 + 0.933899i \(0.383617\pi\)
\(434\) 0 0
\(435\) 39.7230 1.90457
\(436\) 0 0
\(437\) 3.39461 0.162386
\(438\) 0 0
\(439\) 36.1931 1.72740 0.863702 0.504004i \(-0.168140\pi\)
0.863702 + 0.504004i \(0.168140\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5546 −0.501465 −0.250733 0.968056i \(-0.580671\pi\)
−0.250733 + 0.968056i \(0.580671\pi\)
\(444\) 0 0
\(445\) −33.0710 −1.56771
\(446\) 0 0
\(447\) 18.1788 0.859828
\(448\) 0 0
\(449\) 39.9560 1.88564 0.942821 0.333301i \(-0.108162\pi\)
0.942821 + 0.333301i \(0.108162\pi\)
\(450\) 0 0
\(451\) 30.8071 1.45065
\(452\) 0 0
\(453\) 38.5371 1.81063
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.1870 0.476530 0.238265 0.971200i \(-0.423421\pi\)
0.238265 + 0.971200i \(0.423421\pi\)
\(458\) 0 0
\(459\) −39.5985 −1.84830
\(460\) 0 0
\(461\) 6.70956 0.312495 0.156248 0.987718i \(-0.450060\pi\)
0.156248 + 0.987718i \(0.450060\pi\)
\(462\) 0 0
\(463\) 16.6512 0.773848 0.386924 0.922112i \(-0.373538\pi\)
0.386924 + 0.922112i \(0.373538\pi\)
\(464\) 0 0
\(465\) 69.2029 3.20921
\(466\) 0 0
\(467\) −39.1863 −1.81332 −0.906662 0.421857i \(-0.861378\pi\)
−0.906662 + 0.421857i \(0.861378\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 56.3491 2.59643
\(472\) 0 0
\(473\) −41.3230 −1.90004
\(474\) 0 0
\(475\) −14.5579 −0.667963
\(476\) 0 0
\(477\) −89.9180 −4.11706
\(478\) 0 0
\(479\) 36.2704 1.65724 0.828619 0.559813i \(-0.189127\pi\)
0.828619 + 0.559813i \(0.189127\pi\)
\(480\) 0 0
\(481\) 17.3133 0.789420
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.85861 0.447656
\(486\) 0 0
\(487\) 23.7517 1.07629 0.538146 0.842851i \(-0.319125\pi\)
0.538146 + 0.842851i \(0.319125\pi\)
\(488\) 0 0
\(489\) −46.6425 −2.10925
\(490\) 0 0
\(491\) −22.3752 −1.00978 −0.504889 0.863185i \(-0.668466\pi\)
−0.504889 + 0.863185i \(0.668466\pi\)
\(492\) 0 0
\(493\) −10.5608 −0.475635
\(494\) 0 0
\(495\) 121.146 5.44510
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 16.2296 0.726535 0.363268 0.931685i \(-0.381661\pi\)
0.363268 + 0.931685i \(0.381661\pi\)
\(500\) 0 0
\(501\) 20.4649 0.914304
\(502\) 0 0
\(503\) 2.57708 0.114906 0.0574532 0.998348i \(-0.481702\pi\)
0.0574532 + 0.998348i \(0.481702\pi\)
\(504\) 0 0
\(505\) 25.3774 1.12928
\(506\) 0 0
\(507\) 5.08608 0.225881
\(508\) 0 0
\(509\) −33.4091 −1.48083 −0.740417 0.672148i \(-0.765372\pi\)
−0.740417 + 0.672148i \(0.765372\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 50.9375 2.24894
\(514\) 0 0
\(515\) −1.27500 −0.0561830
\(516\) 0 0
\(517\) −49.8953 −2.19439
\(518\) 0 0
\(519\) −8.48546 −0.372470
\(520\) 0 0
\(521\) −12.4303 −0.544580 −0.272290 0.962215i \(-0.587781\pi\)
−0.272290 + 0.962215i \(0.587781\pi\)
\(522\) 0 0
\(523\) −1.05262 −0.0460279 −0.0230140 0.999735i \(-0.507326\pi\)
−0.0230140 + 0.999735i \(0.507326\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.3984 −0.801445
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.5615 −0.501725
\(532\) 0 0
\(533\) −19.9401 −0.863701
\(534\) 0 0
\(535\) −36.2933 −1.56910
\(536\) 0 0
\(537\) −82.1847 −3.54653
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 29.4265 1.26514 0.632572 0.774502i \(-0.281999\pi\)
0.632572 + 0.774502i \(0.281999\pi\)
\(542\) 0 0
\(543\) −4.44515 −0.190760
\(544\) 0 0
\(545\) 54.6009 2.33885
\(546\) 0 0
\(547\) −36.3851 −1.55571 −0.777857 0.628441i \(-0.783693\pi\)
−0.777857 + 0.628441i \(0.783693\pi\)
\(548\) 0 0
\(549\) −8.66726 −0.369910
\(550\) 0 0
\(551\) 13.5849 0.578735
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 50.8130 2.15689
\(556\) 0 0
\(557\) −22.2666 −0.943467 −0.471734 0.881741i \(-0.656372\pi\)
−0.471734 + 0.881741i \(0.656372\pi\)
\(558\) 0 0
\(559\) 26.7466 1.13126
\(560\) 0 0
\(561\) −44.9095 −1.89608
\(562\) 0 0
\(563\) −22.0300 −0.928453 −0.464227 0.885717i \(-0.653668\pi\)
−0.464227 + 0.885717i \(0.653668\pi\)
\(564\) 0 0
\(565\) −38.5154 −1.62036
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.9918 1.17348 0.586738 0.809777i \(-0.300412\pi\)
0.586738 + 0.809777i \(0.300412\pi\)
\(570\) 0 0
\(571\) 19.7000 0.824421 0.412210 0.911089i \(-0.364757\pi\)
0.412210 + 0.911089i \(0.364757\pi\)
\(572\) 0 0
\(573\) 72.1292 3.01324
\(574\) 0 0
\(575\) −4.28854 −0.178845
\(576\) 0 0
\(577\) −11.9624 −0.498000 −0.249000 0.968503i \(-0.580102\pi\)
−0.249000 + 0.968503i \(0.580102\pi\)
\(578\) 0 0
\(579\) −59.8138 −2.48578
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −61.7620 −2.55792
\(584\) 0 0
\(585\) −78.4126 −3.24196
\(586\) 0 0
\(587\) 19.3278 0.797743 0.398872 0.917007i \(-0.369402\pi\)
0.398872 + 0.917007i \(0.369402\pi\)
\(588\) 0 0
\(589\) 23.6667 0.975170
\(590\) 0 0
\(591\) 3.51192 0.144461
\(592\) 0 0
\(593\) −8.45045 −0.347018 −0.173509 0.984832i \(-0.555511\pi\)
−0.173509 + 0.984832i \(0.555511\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.0866 −0.740235
\(598\) 0 0
\(599\) 5.56581 0.227413 0.113706 0.993514i \(-0.463728\pi\)
0.113706 + 0.993514i \(0.463728\pi\)
\(600\) 0 0
\(601\) 18.2258 0.743445 0.371722 0.928344i \(-0.378767\pi\)
0.371722 + 0.928344i \(0.378767\pi\)
\(602\) 0 0
\(603\) −66.9965 −2.72831
\(604\) 0 0
\(605\) 49.6868 2.02005
\(606\) 0 0
\(607\) −28.3740 −1.15166 −0.575832 0.817568i \(-0.695322\pi\)
−0.575832 + 0.817568i \(0.695322\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.2951 1.30652
\(612\) 0 0
\(613\) 15.8620 0.640660 0.320330 0.947306i \(-0.396206\pi\)
0.320330 + 0.947306i \(0.396206\pi\)
\(614\) 0 0
\(615\) −58.5223 −2.35985
\(616\) 0 0
\(617\) −14.9140 −0.600417 −0.300208 0.953874i \(-0.597056\pi\)
−0.300208 + 0.953874i \(0.597056\pi\)
\(618\) 0 0
\(619\) −32.3397 −1.29984 −0.649920 0.760002i \(-0.725198\pi\)
−0.649920 + 0.760002i \(0.725198\pi\)
\(620\) 0 0
\(621\) 15.0054 0.602146
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −28.0511 −1.12204
\(626\) 0 0
\(627\) 57.7693 2.30708
\(628\) 0 0
\(629\) −13.5092 −0.538647
\(630\) 0 0
\(631\) −15.3840 −0.612426 −0.306213 0.951963i \(-0.599062\pi\)
−0.306213 + 0.951963i \(0.599062\pi\)
\(632\) 0 0
\(633\) 63.0346 2.50540
\(634\) 0 0
\(635\) −9.82377 −0.389844
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −110.559 −4.37366
\(640\) 0 0
\(641\) 37.0193 1.46217 0.731087 0.682284i \(-0.239013\pi\)
0.731087 + 0.682284i \(0.239013\pi\)
\(642\) 0 0
\(643\) −24.7183 −0.974796 −0.487398 0.873180i \(-0.662054\pi\)
−0.487398 + 0.873180i \(0.662054\pi\)
\(644\) 0 0
\(645\) 78.4988 3.09089
\(646\) 0 0
\(647\) 20.3207 0.798887 0.399444 0.916758i \(-0.369203\pi\)
0.399444 + 0.916758i \(0.369203\pi\)
\(648\) 0 0
\(649\) −7.94124 −0.311721
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.7950 −0.539839 −0.269919 0.962883i \(-0.586997\pi\)
−0.269919 + 0.962883i \(0.586997\pi\)
\(654\) 0 0
\(655\) 42.9453 1.67801
\(656\) 0 0
\(657\) 100.623 3.92566
\(658\) 0 0
\(659\) 39.4316 1.53604 0.768019 0.640427i \(-0.221243\pi\)
0.768019 + 0.640427i \(0.221243\pi\)
\(660\) 0 0
\(661\) −38.9211 −1.51385 −0.756927 0.653499i \(-0.773300\pi\)
−0.756927 + 0.653499i \(0.773300\pi\)
\(662\) 0 0
\(663\) 29.0680 1.12891
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00190 0.154954
\(668\) 0 0
\(669\) −8.76865 −0.339016
\(670\) 0 0
\(671\) −5.95329 −0.229824
\(672\) 0 0
\(673\) −43.7150 −1.68509 −0.842544 0.538627i \(-0.818943\pi\)
−0.842544 + 0.538627i \(0.818943\pi\)
\(674\) 0 0
\(675\) −64.3513 −2.47688
\(676\) 0 0
\(677\) −16.6777 −0.640977 −0.320489 0.947252i \(-0.603847\pi\)
−0.320489 + 0.947252i \(0.603847\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −26.9234 −1.03171
\(682\) 0 0
\(683\) 48.7312 1.86465 0.932324 0.361625i \(-0.117778\pi\)
0.932324 + 0.361625i \(0.117778\pi\)
\(684\) 0 0
\(685\) 8.14502 0.311205
\(686\) 0 0
\(687\) 85.4504 3.26014
\(688\) 0 0
\(689\) 39.9759 1.52296
\(690\) 0 0
\(691\) 41.7882 1.58970 0.794849 0.606807i \(-0.207550\pi\)
0.794849 + 0.606807i \(0.207550\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.9968 −0.720591
\(696\) 0 0
\(697\) 15.5588 0.589331
\(698\) 0 0
\(699\) −3.83450 −0.145034
\(700\) 0 0
\(701\) −28.1499 −1.06321 −0.531604 0.846993i \(-0.678410\pi\)
−0.531604 + 0.846993i \(0.678410\pi\)
\(702\) 0 0
\(703\) 17.3775 0.655406
\(704\) 0 0
\(705\) 94.7831 3.56974
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22.1680 0.832536 0.416268 0.909242i \(-0.363338\pi\)
0.416268 + 0.909242i \(0.363338\pi\)
\(710\) 0 0
\(711\) 109.848 4.11961
\(712\) 0 0
\(713\) 6.97185 0.261098
\(714\) 0 0
\(715\) −53.8593 −2.01422
\(716\) 0 0
\(717\) −18.1379 −0.677374
\(718\) 0 0
\(719\) 22.2623 0.830243 0.415122 0.909766i \(-0.363739\pi\)
0.415122 + 0.909766i \(0.363739\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 26.4628 0.984161
\(724\) 0 0
\(725\) −17.1623 −0.637392
\(726\) 0 0
\(727\) 21.6618 0.803392 0.401696 0.915773i \(-0.368421\pi\)
0.401696 + 0.915773i \(0.368421\pi\)
\(728\) 0 0
\(729\) 51.5497 1.90925
\(730\) 0 0
\(731\) −20.8698 −0.771897
\(732\) 0 0
\(733\) 4.50596 0.166431 0.0832156 0.996532i \(-0.473481\pi\)
0.0832156 + 0.996532i \(0.473481\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −46.0179 −1.69509
\(738\) 0 0
\(739\) −15.7692 −0.580079 −0.290040 0.957015i \(-0.593669\pi\)
−0.290040 + 0.957015i \(0.593669\pi\)
\(740\) 0 0
\(741\) −37.3916 −1.37361
\(742\) 0 0
\(743\) 30.8619 1.13222 0.566108 0.824331i \(-0.308449\pi\)
0.566108 + 0.824331i \(0.308449\pi\)
\(744\) 0 0
\(745\) −17.0113 −0.623245
\(746\) 0 0
\(747\) 5.27790 0.193108
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 38.1808 1.39324 0.696618 0.717442i \(-0.254687\pi\)
0.696618 + 0.717442i \(0.254687\pi\)
\(752\) 0 0
\(753\) 9.07179 0.330594
\(754\) 0 0
\(755\) −36.0621 −1.31243
\(756\) 0 0
\(757\) −15.6733 −0.569655 −0.284828 0.958579i \(-0.591936\pi\)
−0.284828 + 0.958579i \(0.591936\pi\)
\(758\) 0 0
\(759\) 17.0180 0.617713
\(760\) 0 0
\(761\) −41.8363 −1.51657 −0.758283 0.651926i \(-0.773961\pi\)
−0.758283 + 0.651926i \(0.773961\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 61.1835 2.21210
\(766\) 0 0
\(767\) 5.14003 0.185596
\(768\) 0 0
\(769\) −5.39928 −0.194703 −0.0973515 0.995250i \(-0.531037\pi\)
−0.0973515 + 0.995250i \(0.531037\pi\)
\(770\) 0 0
\(771\) −78.5397 −2.82854
\(772\) 0 0
\(773\) −37.1956 −1.33783 −0.668917 0.743337i \(-0.733242\pi\)
−0.668917 + 0.743337i \(0.733242\pi\)
\(774\) 0 0
\(775\) −29.8991 −1.07401
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0140 −0.717077
\(780\) 0 0
\(781\) −75.9399 −2.71734
\(782\) 0 0
\(783\) 60.0501 2.14602
\(784\) 0 0
\(785\) −52.7301 −1.88202
\(786\) 0 0
\(787\) 38.9942 1.38999 0.694997 0.719013i \(-0.255406\pi\)
0.694997 + 0.719013i \(0.255406\pi\)
\(788\) 0 0
\(789\) −11.1029 −0.395273
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.85331 0.136835
\(794\) 0 0
\(795\) 117.326 4.16111
\(796\) 0 0
\(797\) −25.8311 −0.914986 −0.457493 0.889213i \(-0.651253\pi\)
−0.457493 + 0.889213i \(0.651253\pi\)
\(798\) 0 0
\(799\) −25.1991 −0.891482
\(800\) 0 0
\(801\) −82.5473 −2.91667
\(802\) 0 0
\(803\) 69.1147 2.43901
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 17.2303 0.606537
\(808\) 0 0
\(809\) 26.7612 0.940875 0.470437 0.882433i \(-0.344096\pi\)
0.470437 + 0.882433i \(0.344096\pi\)
\(810\) 0 0
\(811\) −30.3856 −1.06698 −0.533492 0.845805i \(-0.679120\pi\)
−0.533492 + 0.845805i \(0.679120\pi\)
\(812\) 0 0
\(813\) 52.8286 1.85278
\(814\) 0 0
\(815\) 43.6469 1.52888
\(816\) 0 0
\(817\) 26.8458 0.939217
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16.5069 −0.576095 −0.288048 0.957616i \(-0.593006\pi\)
−0.288048 + 0.957616i \(0.593006\pi\)
\(822\) 0 0
\(823\) −19.9562 −0.695631 −0.347816 0.937563i \(-0.613076\pi\)
−0.347816 + 0.937563i \(0.613076\pi\)
\(824\) 0 0
\(825\) −72.9822 −2.54091
\(826\) 0 0
\(827\) −36.8443 −1.28120 −0.640601 0.767874i \(-0.721315\pi\)
−0.640601 + 0.767874i \(0.721315\pi\)
\(828\) 0 0
\(829\) −5.80916 −0.201760 −0.100880 0.994899i \(-0.532166\pi\)
−0.100880 + 0.994899i \(0.532166\pi\)
\(830\) 0 0
\(831\) 72.3714 2.51054
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −19.1505 −0.662732
\(836\) 0 0
\(837\) 104.615 3.61604
\(838\) 0 0
\(839\) 12.2733 0.423722 0.211861 0.977300i \(-0.432048\pi\)
0.211861 + 0.977300i \(0.432048\pi\)
\(840\) 0 0
\(841\) −12.9848 −0.447752
\(842\) 0 0
\(843\) 18.7259 0.644954
\(844\) 0 0
\(845\) −4.75942 −0.163729
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −45.6316 −1.56607
\(850\) 0 0
\(851\) 5.11916 0.175483
\(852\) 0 0
\(853\) 13.2560 0.453878 0.226939 0.973909i \(-0.427128\pi\)
0.226939 + 0.973909i \(0.427128\pi\)
\(854\) 0 0
\(855\) −78.7034 −2.69160
\(856\) 0 0
\(857\) 19.1066 0.652670 0.326335 0.945254i \(-0.394186\pi\)
0.326335 + 0.945254i \(0.394186\pi\)
\(858\) 0 0
\(859\) 46.1878 1.57591 0.787953 0.615735i \(-0.211141\pi\)
0.787953 + 0.615735i \(0.211141\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11.5341 0.392625 0.196313 0.980541i \(-0.437103\pi\)
0.196313 + 0.980541i \(0.437103\pi\)
\(864\) 0 0
\(865\) 7.94048 0.269984
\(866\) 0 0
\(867\) 32.6859 1.11007
\(868\) 0 0
\(869\) 75.4512 2.55951
\(870\) 0 0
\(871\) 29.7854 1.00924
\(872\) 0 0
\(873\) 24.6077 0.832846
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.9695 −0.573020 −0.286510 0.958077i \(-0.592495\pi\)
−0.286510 + 0.958077i \(0.592495\pi\)
\(878\) 0 0
\(879\) 51.3336 1.73144
\(880\) 0 0
\(881\) −11.4505 −0.385775 −0.192888 0.981221i \(-0.561785\pi\)
−0.192888 + 0.981221i \(0.561785\pi\)
\(882\) 0 0
\(883\) 17.3634 0.584325 0.292162 0.956369i \(-0.405625\pi\)
0.292162 + 0.956369i \(0.405625\pi\)
\(884\) 0 0
\(885\) 15.0855 0.507093
\(886\) 0 0
\(887\) −22.7378 −0.763460 −0.381730 0.924274i \(-0.624672\pi\)
−0.381730 + 0.924274i \(0.624672\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 136.112 4.55992
\(892\) 0 0
\(893\) 32.4149 1.08472
\(894\) 0 0
\(895\) 76.9064 2.57070
\(896\) 0 0
\(897\) −11.0150 −0.367780
\(898\) 0 0
\(899\) 27.9006 0.930539
\(900\) 0 0
\(901\) −31.1923 −1.03917
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.15966 0.138272
\(906\) 0 0
\(907\) −41.5359 −1.37918 −0.689588 0.724202i \(-0.742208\pi\)
−0.689588 + 0.724202i \(0.742208\pi\)
\(908\) 0 0
\(909\) 63.3438 2.10098
\(910\) 0 0
\(911\) 2.92749 0.0969921 0.0484961 0.998823i \(-0.484557\pi\)
0.0484961 + 0.998823i \(0.484557\pi\)
\(912\) 0 0
\(913\) 3.62523 0.119978
\(914\) 0 0
\(915\) 11.3091 0.373867
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 13.9648 0.460658 0.230329 0.973113i \(-0.426020\pi\)
0.230329 + 0.973113i \(0.426020\pi\)
\(920\) 0 0
\(921\) 37.4358 1.23355
\(922\) 0 0
\(923\) 49.1527 1.61788
\(924\) 0 0
\(925\) −21.9537 −0.721834
\(926\) 0 0
\(927\) −3.18247 −0.104526
\(928\) 0 0
\(929\) 23.3541 0.766222 0.383111 0.923702i \(-0.374853\pi\)
0.383111 + 0.923702i \(0.374853\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 14.3663 0.470331
\(934\) 0 0
\(935\) 42.0252 1.37437
\(936\) 0 0
\(937\) −11.1156 −0.363130 −0.181565 0.983379i \(-0.558116\pi\)
−0.181565 + 0.983379i \(0.558116\pi\)
\(938\) 0 0
\(939\) −20.1977 −0.659127
\(940\) 0 0
\(941\) −5.13868 −0.167516 −0.0837580 0.996486i \(-0.526692\pi\)
−0.0837580 + 0.996486i \(0.526692\pi\)
\(942\) 0 0
\(943\) −5.89583 −0.191995
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.4147 0.955851 0.477925 0.878400i \(-0.341389\pi\)
0.477925 + 0.878400i \(0.341389\pi\)
\(948\) 0 0
\(949\) −44.7350 −1.45216
\(950\) 0 0
\(951\) 30.9655 1.00412
\(952\) 0 0
\(953\) −5.67396 −0.183797 −0.0918987 0.995768i \(-0.529294\pi\)
−0.0918987 + 0.995768i \(0.529294\pi\)
\(954\) 0 0
\(955\) −67.4967 −2.18414
\(956\) 0 0
\(957\) 68.1041 2.20149
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 17.6067 0.567959
\(962\) 0 0
\(963\) −90.5905 −2.91924
\(964\) 0 0
\(965\) 55.9723 1.80181
\(966\) 0 0
\(967\) −0.898731 −0.0289013 −0.0144506 0.999896i \(-0.504600\pi\)
−0.0144506 + 0.999896i \(0.504600\pi\)
\(968\) 0 0
\(969\) 29.1758 0.937262
\(970\) 0 0
\(971\) −34.2084 −1.09780 −0.548899 0.835889i \(-0.684953\pi\)
−0.548899 + 0.835889i \(0.684953\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 47.2382 1.51283
\(976\) 0 0
\(977\) 8.19697 0.262244 0.131122 0.991366i \(-0.458142\pi\)
0.131122 + 0.991366i \(0.458142\pi\)
\(978\) 0 0
\(979\) −56.6994 −1.81212
\(980\) 0 0
\(981\) 136.288 4.35133
\(982\) 0 0
\(983\) 1.12997 0.0360406 0.0180203 0.999838i \(-0.494264\pi\)
0.0180203 + 0.999838i \(0.494264\pi\)
\(984\) 0 0
\(985\) −3.28637 −0.104712
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.90838 0.251472
\(990\) 0 0
\(991\) 14.6766 0.466218 0.233109 0.972451i \(-0.425110\pi\)
0.233109 + 0.972451i \(0.425110\pi\)
\(992\) 0 0
\(993\) −32.5973 −1.03444
\(994\) 0 0
\(995\) 16.9250 0.536558
\(996\) 0 0
\(997\) 1.64207 0.0520049 0.0260024 0.999662i \(-0.491722\pi\)
0.0260024 + 0.999662i \(0.491722\pi\)
\(998\) 0 0
\(999\) 76.8151 2.43032
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 4508.2.a.f.1.1 5
7.6 odd 2 644.2.a.d.1.5 5
21.20 even 2 5796.2.a.t.1.4 5
28.27 even 2 2576.2.a.bb.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.d.1.5 5 7.6 odd 2
2576.2.a.bb.1.1 5 28.27 even 2
4508.2.a.f.1.1 5 1.1 even 1 trivial
5796.2.a.t.1.4 5 21.20 even 2