Properties

Label 4275.2.a.x.1.1
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
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] = [4275,2,Mod(1,4275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4275, 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("4275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.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: \(34.1360468641\)
Analytic rank: \(1\)
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 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} -1.82843 q^{4} -0.585786 q^{7} +1.58579 q^{8} +O(q^{10})\) \(q-0.414214 q^{2} -1.82843 q^{4} -0.585786 q^{7} +1.58579 q^{8} -1.41421 q^{11} -5.41421 q^{13} +0.242641 q^{14} +3.00000 q^{16} -1.17157 q^{17} +1.00000 q^{19} +0.585786 q^{22} +7.65685 q^{23} +2.24264 q^{26} +1.07107 q^{28} +9.07107 q^{29} +6.48528 q^{31} -4.41421 q^{32} +0.485281 q^{34} -11.0711 q^{37} -0.414214 q^{38} +7.41421 q^{41} -0.585786 q^{43} +2.58579 q^{44} -3.17157 q^{46} +0.343146 q^{47} -6.65685 q^{49} +9.89949 q^{52} +4.00000 q^{53} -0.928932 q^{56} -3.75736 q^{58} -8.48528 q^{59} +5.65685 q^{61} -2.68629 q^{62} -4.17157 q^{64} -12.0000 q^{67} +2.14214 q^{68} +4.48528 q^{71} +2.00000 q^{73} +4.58579 q^{74} -1.82843 q^{76} +0.828427 q^{77} -11.3137 q^{79} -3.07107 q^{82} -10.4853 q^{83} +0.242641 q^{86} -2.24264 q^{88} -10.7279 q^{89} +3.17157 q^{91} -14.0000 q^{92} -0.142136 q^{94} +4.24264 q^{97} +2.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8} - 8 q^{13} - 8 q^{14} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} - 4 q^{26} - 12 q^{28} + 4 q^{29} - 4 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 2 q^{38} + 12 q^{41} - 4 q^{43} + 8 q^{44} - 12 q^{46} + 12 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{56} - 16 q^{58} - 28 q^{62} - 14 q^{64} - 24 q^{67} - 24 q^{68} - 8 q^{71} + 4 q^{73} + 12 q^{74} + 2 q^{76} - 4 q^{77} + 8 q^{82} - 4 q^{83} - 8 q^{86} + 4 q^{88} + 4 q^{89} + 12 q^{91} - 28 q^{92} + 28 q^{94} + 14 q^{98}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 0 0
\(7\) −0.585786 −0.221406 −0.110703 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) 1.58579 0.560660
\(9\) 0 0
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) −5.41421 −1.50163 −0.750816 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0.242641 0.0648485
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −1.17157 −0.284148 −0.142074 0.989856i \(-0.545377\pi\)
−0.142074 + 0.989856i \(0.545377\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0.585786 0.124890
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.24264 0.439818
\(27\) 0 0
\(28\) 1.07107 0.202413
\(29\) 9.07107 1.68446 0.842228 0.539122i \(-0.181244\pi\)
0.842228 + 0.539122i \(0.181244\pi\)
\(30\) 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
\(34\) 0.485281 0.0832251
\(35\) 0 0
\(36\) 0 0
\(37\) −11.0711 −1.82007 −0.910036 0.414529i \(-0.863946\pi\)
−0.910036 + 0.414529i \(0.863946\pi\)
\(38\) −0.414214 −0.0671943
\(39\) 0 0
\(40\) 0 0
\(41\) 7.41421 1.15791 0.578953 0.815361i \(-0.303462\pi\)
0.578953 + 0.815361i \(0.303462\pi\)
\(42\) 0 0
\(43\) −0.585786 −0.0893316 −0.0446658 0.999002i \(-0.514222\pi\)
−0.0446658 + 0.999002i \(0.514222\pi\)
\(44\) 2.58579 0.389822
\(45\) 0 0
\(46\) −3.17157 −0.467623
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 9.89949 1.37281
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.928932 −0.124134
\(57\) 0 0
\(58\) −3.75736 −0.493365
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) −2.68629 −0.341159
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.14214 0.259772
\(69\) 0 0
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 4.58579 0.533087
\(75\) 0 0
\(76\) −1.82843 −0.209735
\(77\) 0.828427 0.0944080
\(78\) 0 0
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.07107 −0.339143
\(83\) −10.4853 −1.15091 −0.575455 0.817834i \(-0.695175\pi\)
−0.575455 + 0.817834i \(0.695175\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.242641 0.0261646
\(87\) 0 0
\(88\) −2.24264 −0.239066
\(89\) −10.7279 −1.13716 −0.568579 0.822629i \(-0.692507\pi\)
−0.568579 + 0.822629i \(0.692507\pi\)
\(90\) 0 0
\(91\) 3.17157 0.332471
\(92\) −14.0000 −1.45960
\(93\) 0 0
\(94\) −0.142136 −0.0146602
\(95\) 0 0
\(96\) 0 0
\(97\) 4.24264 0.430775 0.215387 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) 2.75736 0.278535
\(99\) 0 0
\(100\) 0 0
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) 0 0
\(103\) 1.65685 0.163255 0.0816274 0.996663i \(-0.473988\pi\)
0.0816274 + 0.996663i \(0.473988\pi\)
\(104\) −8.58579 −0.841906
\(105\) 0 0
\(106\) −1.65685 −0.160928
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 3.17157 0.303782 0.151891 0.988397i \(-0.451464\pi\)
0.151891 + 0.988397i \(0.451464\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.75736 −0.166055
\(113\) 12.4853 1.17452 0.587258 0.809400i \(-0.300207\pi\)
0.587258 + 0.809400i \(0.300207\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −16.5858 −1.53995
\(117\) 0 0
\(118\) 3.51472 0.323556
\(119\) 0.686292 0.0629122
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) −2.34315 −0.212138
\(123\) 0 0
\(124\) −11.8579 −1.06487
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 10.5563 0.933058
\(129\) 0 0
\(130\) 0 0
\(131\) −16.7279 −1.46153 −0.730763 0.682632i \(-0.760835\pi\)
−0.730763 + 0.682632i \(0.760835\pi\)
\(132\) 0 0
\(133\) −0.585786 −0.0507941
\(134\) 4.97056 0.429391
\(135\) 0 0
\(136\) −1.85786 −0.159311
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 1.17157 0.0993715 0.0496858 0.998765i \(-0.484178\pi\)
0.0496858 + 0.998765i \(0.484178\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.85786 −0.155909
\(143\) 7.65685 0.640298
\(144\) 0 0
\(145\) 0 0
\(146\) −0.828427 −0.0685611
\(147\) 0 0
\(148\) 20.2426 1.66393
\(149\) 3.65685 0.299581 0.149791 0.988718i \(-0.452140\pi\)
0.149791 + 0.988718i \(0.452140\pi\)
\(150\) 0 0
\(151\) 17.7990 1.44846 0.724231 0.689558i \(-0.242195\pi\)
0.724231 + 0.689558i \(0.242195\pi\)
\(152\) 1.58579 0.128624
\(153\) 0 0
\(154\) −0.343146 −0.0276515
\(155\) 0 0
\(156\) 0 0
\(157\) 21.7990 1.73975 0.869874 0.493273i \(-0.164200\pi\)
0.869874 + 0.493273i \(0.164200\pi\)
\(158\) 4.68629 0.372821
\(159\) 0 0
\(160\) 0 0
\(161\) −4.48528 −0.353490
\(162\) 0 0
\(163\) −7.89949 −0.618736 −0.309368 0.950942i \(-0.600118\pi\)
−0.309368 + 0.950942i \(0.600118\pi\)
\(164\) −13.5563 −1.05857
\(165\) 0 0
\(166\) 4.34315 0.337093
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) 0 0
\(171\) 0 0
\(172\) 1.07107 0.0816682
\(173\) −6.14214 −0.466978 −0.233489 0.972359i \(-0.575014\pi\)
−0.233489 + 0.972359i \(0.575014\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.24264 −0.319801
\(177\) 0 0
\(178\) 4.44365 0.333066
\(179\) −17.1716 −1.28346 −0.641732 0.766929i \(-0.721784\pi\)
−0.641732 + 0.766929i \(0.721784\pi\)
\(180\) 0 0
\(181\) −19.1716 −1.42501 −0.712506 0.701666i \(-0.752440\pi\)
−0.712506 + 0.701666i \(0.752440\pi\)
\(182\) −1.31371 −0.0973786
\(183\) 0 0
\(184\) 12.1421 0.895130
\(185\) 0 0
\(186\) 0 0
\(187\) 1.65685 0.121161
\(188\) −0.627417 −0.0457591
\(189\) 0 0
\(190\) 0 0
\(191\) −1.89949 −0.137443 −0.0687213 0.997636i \(-0.521892\pi\)
−0.0687213 + 0.997636i \(0.521892\pi\)
\(192\) 0 0
\(193\) 15.0711 1.08484 0.542420 0.840108i \(-0.317508\pi\)
0.542420 + 0.840108i \(0.317508\pi\)
\(194\) −1.75736 −0.126171
\(195\) 0 0
\(196\) 12.1716 0.869398
\(197\) −14.8284 −1.05648 −0.528241 0.849095i \(-0.677148\pi\)
−0.528241 + 0.849095i \(0.677148\pi\)
\(198\) 0 0
\(199\) −16.4853 −1.16861 −0.584305 0.811534i \(-0.698633\pi\)
−0.584305 + 0.811534i \(0.698633\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −5.31371 −0.372949
\(204\) 0 0
\(205\) 0 0
\(206\) −0.686292 −0.0478162
\(207\) 0 0
\(208\) −16.2426 −1.12622
\(209\) −1.41421 −0.0978232
\(210\) 0 0
\(211\) −15.3137 −1.05424 −0.527120 0.849791i \(-0.676728\pi\)
−0.527120 + 0.849791i \(0.676728\pi\)
\(212\) −7.31371 −0.502308
\(213\) 0 0
\(214\) −3.31371 −0.226520
\(215\) 0 0
\(216\) 0 0
\(217\) −3.79899 −0.257892
\(218\) −1.31371 −0.0889756
\(219\) 0 0
\(220\) 0 0
\(221\) 6.34315 0.426686
\(222\) 0 0
\(223\) −6.34315 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(224\) 2.58579 0.172770
\(225\) 0 0
\(226\) −5.17157 −0.344008
\(227\) −18.9706 −1.25912 −0.629560 0.776952i \(-0.716765\pi\)
−0.629560 + 0.776952i \(0.716765\pi\)
\(228\) 0 0
\(229\) −1.65685 −0.109488 −0.0547440 0.998500i \(-0.517434\pi\)
−0.0547440 + 0.998500i \(0.517434\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 14.3848 0.944407
\(233\) −8.34315 −0.546578 −0.273289 0.961932i \(-0.588111\pi\)
−0.273289 + 0.961932i \(0.588111\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 15.5147 1.00992
\(237\) 0 0
\(238\) −0.284271 −0.0184266
\(239\) −2.58579 −0.167261 −0.0836303 0.996497i \(-0.526651\pi\)
−0.0836303 + 0.996497i \(0.526651\pi\)
\(240\) 0 0
\(241\) −14.9706 −0.964339 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(242\) 3.72792 0.239640
\(243\) 0 0
\(244\) −10.3431 −0.662152
\(245\) 0 0
\(246\) 0 0
\(247\) −5.41421 −0.344498
\(248\) 10.2843 0.653052
\(249\) 0 0
\(250\) 0 0
\(251\) −12.9289 −0.816067 −0.408033 0.912967i \(-0.633785\pi\)
−0.408033 + 0.912967i \(0.633785\pi\)
\(252\) 0 0
\(253\) −10.8284 −0.680777
\(254\) 3.31371 0.207921
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 1.17157 0.0730807 0.0365404 0.999332i \(-0.488366\pi\)
0.0365404 + 0.999332i \(0.488366\pi\)
\(258\) 0 0
\(259\) 6.48528 0.402976
\(260\) 0 0
\(261\) 0 0
\(262\) 6.92893 0.428071
\(263\) −32.1421 −1.98197 −0.990984 0.133977i \(-0.957225\pi\)
−0.990984 + 0.133977i \(0.957225\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.242641 0.0148773
\(267\) 0 0
\(268\) 21.9411 1.34027
\(269\) −8.38478 −0.511229 −0.255614 0.966779i \(-0.582278\pi\)
−0.255614 + 0.966779i \(0.582278\pi\)
\(270\) 0 0
\(271\) −30.8284 −1.87269 −0.936347 0.351076i \(-0.885816\pi\)
−0.936347 + 0.351076i \(0.885816\pi\)
\(272\) −3.51472 −0.213111
\(273\) 0 0
\(274\) 5.79899 0.350330
\(275\) 0 0
\(276\) 0 0
\(277\) −10.9706 −0.659157 −0.329579 0.944128i \(-0.606907\pi\)
−0.329579 + 0.944128i \(0.606907\pi\)
\(278\) −0.485281 −0.0291052
\(279\) 0 0
\(280\) 0 0
\(281\) 14.7279 0.878594 0.439297 0.898342i \(-0.355228\pi\)
0.439297 + 0.898342i \(0.355228\pi\)
\(282\) 0 0
\(283\) −6.24264 −0.371086 −0.185543 0.982636i \(-0.559404\pi\)
−0.185543 + 0.982636i \(0.559404\pi\)
\(284\) −8.20101 −0.486640
\(285\) 0 0
\(286\) −3.17157 −0.187539
\(287\) −4.34315 −0.256368
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 0 0
\(292\) −3.65685 −0.214001
\(293\) −31.7990 −1.85772 −0.928858 0.370435i \(-0.879209\pi\)
−0.928858 + 0.370435i \(0.879209\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −17.5563 −1.02044
\(297\) 0 0
\(298\) −1.51472 −0.0877453
\(299\) −41.4558 −2.39745
\(300\) 0 0
\(301\) 0.343146 0.0197786
\(302\) −7.37258 −0.424244
\(303\) 0 0
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) 0 0
\(307\) −7.79899 −0.445112 −0.222556 0.974920i \(-0.571440\pi\)
−0.222556 + 0.974920i \(0.571440\pi\)
\(308\) −1.51472 −0.0863091
\(309\) 0 0
\(310\) 0 0
\(311\) 32.2426 1.82831 0.914156 0.405362i \(-0.132855\pi\)
0.914156 + 0.405362i \(0.132855\pi\)
\(312\) 0 0
\(313\) 9.51472 0.537804 0.268902 0.963168i \(-0.413339\pi\)
0.268902 + 0.963168i \(0.413339\pi\)
\(314\) −9.02944 −0.509561
\(315\) 0 0
\(316\) 20.6863 1.16369
\(317\) 11.3137 0.635441 0.317721 0.948184i \(-0.397083\pi\)
0.317721 + 0.948184i \(0.397083\pi\)
\(318\) 0 0
\(319\) −12.8284 −0.718254
\(320\) 0 0
\(321\) 0 0
\(322\) 1.85786 0.103535
\(323\) −1.17157 −0.0651881
\(324\) 0 0
\(325\) 0 0
\(326\) 3.27208 0.181224
\(327\) 0 0
\(328\) 11.7574 0.649192
\(329\) −0.201010 −0.0110820
\(330\) 0 0
\(331\) 7.17157 0.394185 0.197093 0.980385i \(-0.436850\pi\)
0.197093 + 0.980385i \(0.436850\pi\)
\(332\) 19.1716 1.05218
\(333\) 0 0
\(334\) −4.14214 −0.226648
\(335\) 0 0
\(336\) 0 0
\(337\) −28.2426 −1.53847 −0.769237 0.638963i \(-0.779364\pi\)
−0.769237 + 0.638963i \(0.779364\pi\)
\(338\) −6.75736 −0.367552
\(339\) 0 0
\(340\) 0 0
\(341\) −9.17157 −0.496669
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −0.928932 −0.0500847
\(345\) 0 0
\(346\) 2.54416 0.136775
\(347\) 10.4853 0.562879 0.281440 0.959579i \(-0.409188\pi\)
0.281440 + 0.959579i \(0.409188\pi\)
\(348\) 0 0
\(349\) 29.3137 1.56913 0.784563 0.620049i \(-0.212887\pi\)
0.784563 + 0.620049i \(0.212887\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.24264 0.332734
\(353\) 3.65685 0.194635 0.0973174 0.995253i \(-0.468974\pi\)
0.0973174 + 0.995253i \(0.468974\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 19.6152 1.03960
\(357\) 0 0
\(358\) 7.11270 0.375918
\(359\) −9.89949 −0.522475 −0.261238 0.965275i \(-0.584131\pi\)
−0.261238 + 0.965275i \(0.584131\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 7.94113 0.417376
\(363\) 0 0
\(364\) −5.79899 −0.303950
\(365\) 0 0
\(366\) 0 0
\(367\) 19.4142 1.01341 0.506707 0.862118i \(-0.330863\pi\)
0.506707 + 0.862118i \(0.330863\pi\)
\(368\) 22.9706 1.19742
\(369\) 0 0
\(370\) 0 0
\(371\) −2.34315 −0.121650
\(372\) 0 0
\(373\) 9.89949 0.512576 0.256288 0.966600i \(-0.417500\pi\)
0.256288 + 0.966600i \(0.417500\pi\)
\(374\) −0.686292 −0.0354873
\(375\) 0 0
\(376\) 0.544156 0.0280627
\(377\) −49.1127 −2.52943
\(378\) 0 0
\(379\) 24.1421 1.24010 0.620049 0.784563i \(-0.287113\pi\)
0.620049 + 0.784563i \(0.287113\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.786797 0.0402560
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.24264 −0.317742
\(387\) 0 0
\(388\) −7.75736 −0.393820
\(389\) −14.9706 −0.759038 −0.379519 0.925184i \(-0.623910\pi\)
−0.379519 + 0.925184i \(0.623910\pi\)
\(390\) 0 0
\(391\) −8.97056 −0.453661
\(392\) −10.5563 −0.533176
\(393\) 0 0
\(394\) 6.14214 0.309436
\(395\) 0 0
\(396\) 0 0
\(397\) −35.6569 −1.78957 −0.894783 0.446501i \(-0.852670\pi\)
−0.894783 + 0.446501i \(0.852670\pi\)
\(398\) 6.82843 0.342278
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0416 −1.50021 −0.750104 0.661320i \(-0.769996\pi\)
−0.750104 + 0.661320i \(0.769996\pi\)
\(402\) 0 0
\(403\) −35.1127 −1.74909
\(404\) −8.82843 −0.439231
\(405\) 0 0
\(406\) 2.20101 0.109234
\(407\) 15.6569 0.776081
\(408\) 0 0
\(409\) 9.51472 0.470473 0.235236 0.971938i \(-0.424414\pi\)
0.235236 + 0.971938i \(0.424414\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.02944 −0.149250
\(413\) 4.97056 0.244585
\(414\) 0 0
\(415\) 0 0
\(416\) 23.8995 1.17177
\(417\) 0 0
\(418\) 0.585786 0.0286518
\(419\) 34.8701 1.70351 0.851757 0.523937i \(-0.175537\pi\)
0.851757 + 0.523937i \(0.175537\pi\)
\(420\) 0 0
\(421\) −14.6863 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(422\) 6.34315 0.308780
\(423\) 0 0
\(424\) 6.34315 0.308050
\(425\) 0 0
\(426\) 0 0
\(427\) −3.31371 −0.160362
\(428\) −14.6274 −0.707043
\(429\) 0 0
\(430\) 0 0
\(431\) −3.51472 −0.169298 −0.0846490 0.996411i \(-0.526977\pi\)
−0.0846490 + 0.996411i \(0.526977\pi\)
\(432\) 0 0
\(433\) −0.928932 −0.0446416 −0.0223208 0.999751i \(-0.507106\pi\)
−0.0223208 + 0.999751i \(0.507106\pi\)
\(434\) 1.57359 0.0755349
\(435\) 0 0
\(436\) −5.79899 −0.277721
\(437\) 7.65685 0.366277
\(438\) 0 0
\(439\) 0.970563 0.0463224 0.0231612 0.999732i \(-0.492627\pi\)
0.0231612 + 0.999732i \(0.492627\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.62742 −0.124973
\(443\) −1.31371 −0.0624162 −0.0312081 0.999513i \(-0.509935\pi\)
−0.0312081 + 0.999513i \(0.509935\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.62742 0.124412
\(447\) 0 0
\(448\) 2.44365 0.115452
\(449\) −7.89949 −0.372800 −0.186400 0.982474i \(-0.559682\pi\)
−0.186400 + 0.982474i \(0.559682\pi\)
\(450\) 0 0
\(451\) −10.4853 −0.493733
\(452\) −22.8284 −1.07376
\(453\) 0 0
\(454\) 7.85786 0.368788
\(455\) 0 0
\(456\) 0 0
\(457\) −28.8284 −1.34854 −0.674268 0.738486i \(-0.735541\pi\)
−0.674268 + 0.738486i \(0.735541\pi\)
\(458\) 0.686292 0.0320683
\(459\) 0 0
\(460\) 0 0
\(461\) −10.6863 −0.497710 −0.248855 0.968541i \(-0.580054\pi\)
−0.248855 + 0.968541i \(0.580054\pi\)
\(462\) 0 0
\(463\) 8.10051 0.376462 0.188231 0.982125i \(-0.439725\pi\)
0.188231 + 0.982125i \(0.439725\pi\)
\(464\) 27.2132 1.26334
\(465\) 0 0
\(466\) 3.45584 0.160089
\(467\) 16.3431 0.756271 0.378135 0.925750i \(-0.376565\pi\)
0.378135 + 0.925750i \(0.376565\pi\)
\(468\) 0 0
\(469\) 7.02944 0.324589
\(470\) 0 0
\(471\) 0 0
\(472\) −13.4558 −0.619355
\(473\) 0.828427 0.0380911
\(474\) 0 0
\(475\) 0 0
\(476\) −1.25483 −0.0575152
\(477\) 0 0
\(478\) 1.07107 0.0489895
\(479\) 38.1838 1.74466 0.872330 0.488917i \(-0.162608\pi\)
0.872330 + 0.488917i \(0.162608\pi\)
\(480\) 0 0
\(481\) 59.9411 2.73308
\(482\) 6.20101 0.282448
\(483\) 0 0
\(484\) 16.4558 0.747993
\(485\) 0 0
\(486\) 0 0
\(487\) 2.82843 0.128168 0.0640841 0.997944i \(-0.479587\pi\)
0.0640841 + 0.997944i \(0.479587\pi\)
\(488\) 8.97056 0.406078
\(489\) 0 0
\(490\) 0 0
\(491\) −21.8995 −0.988310 −0.494155 0.869374i \(-0.664523\pi\)
−0.494155 + 0.869374i \(0.664523\pi\)
\(492\) 0 0
\(493\) −10.6274 −0.478635
\(494\) 2.24264 0.100901
\(495\) 0 0
\(496\) 19.4558 0.873593
\(497\) −2.62742 −0.117856
\(498\) 0 0
\(499\) −23.7990 −1.06539 −0.532695 0.846308i \(-0.678821\pi\)
−0.532695 + 0.846308i \(0.678821\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.35534 0.239020
\(503\) 0.828427 0.0369377 0.0184689 0.999829i \(-0.494121\pi\)
0.0184689 + 0.999829i \(0.494121\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 4.48528 0.199395
\(507\) 0 0
\(508\) 14.6274 0.648987
\(509\) −32.3848 −1.43543 −0.717715 0.696337i \(-0.754812\pi\)
−0.717715 + 0.696337i \(0.754812\pi\)
\(510\) 0 0
\(511\) −1.17157 −0.0518273
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) −0.485281 −0.0214048
\(515\) 0 0
\(516\) 0 0
\(517\) −0.485281 −0.0213427
\(518\) −2.68629 −0.118029
\(519\) 0 0
\(520\) 0 0
\(521\) 24.3848 1.06832 0.534158 0.845385i \(-0.320629\pi\)
0.534158 + 0.845385i \(0.320629\pi\)
\(522\) 0 0
\(523\) 15.7990 0.690842 0.345421 0.938448i \(-0.387736\pi\)
0.345421 + 0.938448i \(0.387736\pi\)
\(524\) 30.5858 1.33615
\(525\) 0 0
\(526\) 13.3137 0.580505
\(527\) −7.59798 −0.330973
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) 0 0
\(532\) 1.07107 0.0464367
\(533\) −40.1421 −1.73875
\(534\) 0 0
\(535\) 0 0
\(536\) −19.0294 −0.821946
\(537\) 0 0
\(538\) 3.47309 0.149735
\(539\) 9.41421 0.405499
\(540\) 0 0
\(541\) −32.6274 −1.40276 −0.701381 0.712786i \(-0.747433\pi\)
−0.701381 + 0.712786i \(0.747433\pi\)
\(542\) 12.7696 0.548499
\(543\) 0 0
\(544\) 5.17157 0.221729
\(545\) 0 0
\(546\) 0 0
\(547\) 34.1421 1.45981 0.729906 0.683547i \(-0.239564\pi\)
0.729906 + 0.683547i \(0.239564\pi\)
\(548\) 25.5980 1.09349
\(549\) 0 0
\(550\) 0 0
\(551\) 9.07107 0.386440
\(552\) 0 0
\(553\) 6.62742 0.281826
\(554\) 4.54416 0.193063
\(555\) 0 0
\(556\) −2.14214 −0.0908468
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) 3.17157 0.134143
\(560\) 0 0
\(561\) 0 0
\(562\) −6.10051 −0.257334
\(563\) 42.2843 1.78207 0.891035 0.453935i \(-0.149980\pi\)
0.891035 + 0.453935i \(0.149980\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.58579 0.108689
\(567\) 0 0
\(568\) 7.11270 0.298442
\(569\) 6.72792 0.282049 0.141025 0.990006i \(-0.454960\pi\)
0.141025 + 0.990006i \(0.454960\pi\)
\(570\) 0 0
\(571\) 19.7990 0.828562 0.414281 0.910149i \(-0.364033\pi\)
0.414281 + 0.910149i \(0.364033\pi\)
\(572\) −14.0000 −0.585369
\(573\) 0 0
\(574\) 1.79899 0.0750884
\(575\) 0 0
\(576\) 0 0
\(577\) 37.7990 1.57359 0.786796 0.617213i \(-0.211738\pi\)
0.786796 + 0.617213i \(0.211738\pi\)
\(578\) 6.47309 0.269245
\(579\) 0 0
\(580\) 0 0
\(581\) 6.14214 0.254819
\(582\) 0 0
\(583\) −5.65685 −0.234283
\(584\) 3.17157 0.131241
\(585\) 0 0
\(586\) 13.1716 0.544113
\(587\) 11.6569 0.481130 0.240565 0.970633i \(-0.422667\pi\)
0.240565 + 0.970633i \(0.422667\pi\)
\(588\) 0 0
\(589\) 6.48528 0.267221
\(590\) 0 0
\(591\) 0 0
\(592\) −33.2132 −1.36505
\(593\) −29.3137 −1.20377 −0.601885 0.798583i \(-0.705583\pi\)
−0.601885 + 0.798583i \(0.705583\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.68629 −0.273881
\(597\) 0 0
\(598\) 17.1716 0.702198
\(599\) 21.9411 0.896490 0.448245 0.893911i \(-0.352049\pi\)
0.448245 + 0.893911i \(0.352049\pi\)
\(600\) 0 0
\(601\) −28.8284 −1.17594 −0.587968 0.808884i \(-0.700072\pi\)
−0.587968 + 0.808884i \(0.700072\pi\)
\(602\) −0.142136 −0.00579302
\(603\) 0 0
\(604\) −32.5442 −1.32420
\(605\) 0 0
\(606\) 0 0
\(607\) 2.14214 0.0869466 0.0434733 0.999055i \(-0.486158\pi\)
0.0434733 + 0.999055i \(0.486158\pi\)
\(608\) −4.41421 −0.179020
\(609\) 0 0
\(610\) 0 0
\(611\) −1.85786 −0.0751611
\(612\) 0 0
\(613\) −25.1127 −1.01429 −0.507146 0.861860i \(-0.669300\pi\)
−0.507146 + 0.861860i \(0.669300\pi\)
\(614\) 3.23045 0.130370
\(615\) 0 0
\(616\) 1.31371 0.0529308
\(617\) 10.1421 0.408307 0.204154 0.978939i \(-0.434556\pi\)
0.204154 + 0.978939i \(0.434556\pi\)
\(618\) 0 0
\(619\) −43.7990 −1.76043 −0.880215 0.474575i \(-0.842602\pi\)
−0.880215 + 0.474575i \(0.842602\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −13.3553 −0.535500
\(623\) 6.28427 0.251774
\(624\) 0 0
\(625\) 0 0
\(626\) −3.94113 −0.157519
\(627\) 0 0
\(628\) −39.8579 −1.59050
\(629\) 12.9706 0.517170
\(630\) 0 0
\(631\) 22.6274 0.900783 0.450392 0.892831i \(-0.351284\pi\)
0.450392 + 0.892831i \(0.351284\pi\)
\(632\) −17.9411 −0.713660
\(633\) 0 0
\(634\) −4.68629 −0.186116
\(635\) 0 0
\(636\) 0 0
\(637\) 36.0416 1.42802
\(638\) 5.31371 0.210372
\(639\) 0 0
\(640\) 0 0
\(641\) 8.58579 0.339118 0.169559 0.985520i \(-0.445766\pi\)
0.169559 + 0.985520i \(0.445766\pi\)
\(642\) 0 0
\(643\) −6.04163 −0.238259 −0.119129 0.992879i \(-0.538010\pi\)
−0.119129 + 0.992879i \(0.538010\pi\)
\(644\) 8.20101 0.323165
\(645\) 0 0
\(646\) 0.485281 0.0190931
\(647\) 45.1127 1.77356 0.886782 0.462189i \(-0.152936\pi\)
0.886782 + 0.462189i \(0.152936\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 14.4437 0.565657
\(653\) 42.4264 1.66027 0.830137 0.557560i \(-0.188262\pi\)
0.830137 + 0.557560i \(0.188262\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 22.2426 0.868429
\(657\) 0 0
\(658\) 0.0832611 0.00324586
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 18.4853 0.718994 0.359497 0.933146i \(-0.382948\pi\)
0.359497 + 0.933146i \(0.382948\pi\)
\(662\) −2.97056 −0.115454
\(663\) 0 0
\(664\) −16.6274 −0.645269
\(665\) 0 0
\(666\) 0 0
\(667\) 69.4558 2.68934
\(668\) −18.2843 −0.707440
\(669\) 0 0
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −21.8995 −0.844163 −0.422082 0.906558i \(-0.638700\pi\)
−0.422082 + 0.906558i \(0.638700\pi\)
\(674\) 11.6985 0.450609
\(675\) 0 0
\(676\) −29.8284 −1.14725
\(677\) −44.9706 −1.72836 −0.864180 0.503184i \(-0.832162\pi\)
−0.864180 + 0.503184i \(0.832162\pi\)
\(678\) 0 0
\(679\) −2.48528 −0.0953763
\(680\) 0 0
\(681\) 0 0
\(682\) 3.79899 0.145471
\(683\) 5.65685 0.216454 0.108227 0.994126i \(-0.465483\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −3.31371 −0.126518
\(687\) 0 0
\(688\) −1.75736 −0.0669987
\(689\) −21.6569 −0.825060
\(690\) 0 0
\(691\) −23.1127 −0.879248 −0.439624 0.898182i \(-0.644888\pi\)
−0.439624 + 0.898182i \(0.644888\pi\)
\(692\) 11.2304 0.426918
\(693\) 0 0
\(694\) −4.34315 −0.164864
\(695\) 0 0
\(696\) 0 0
\(697\) −8.68629 −0.329017
\(698\) −12.1421 −0.459587
\(699\) 0 0
\(700\) 0 0
\(701\) 0.343146 0.0129604 0.00648022 0.999979i \(-0.497937\pi\)
0.00648022 + 0.999979i \(0.497937\pi\)
\(702\) 0 0
\(703\) −11.0711 −0.417553
\(704\) 5.89949 0.222346
\(705\) 0 0
\(706\) −1.51472 −0.0570072
\(707\) −2.82843 −0.106374
\(708\) 0 0
\(709\) −35.3137 −1.32623 −0.663117 0.748516i \(-0.730767\pi\)
−0.663117 + 0.748516i \(0.730767\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −17.0122 −0.637559
\(713\) 49.6569 1.85966
\(714\) 0 0
\(715\) 0 0
\(716\) 31.3970 1.17336
\(717\) 0 0
\(718\) 4.10051 0.153029
\(719\) 16.4437 0.613245 0.306622 0.951831i \(-0.400801\pi\)
0.306622 + 0.951831i \(0.400801\pi\)
\(720\) 0 0
\(721\) −0.970563 −0.0361456
\(722\) −0.414214 −0.0154154
\(723\) 0 0
\(724\) 35.0538 1.30277
\(725\) 0 0
\(726\) 0 0
\(727\) 4.58579 0.170077 0.0850387 0.996378i \(-0.472899\pi\)
0.0850387 + 0.996378i \(0.472899\pi\)
\(728\) 5.02944 0.186403
\(729\) 0 0
\(730\) 0 0
\(731\) 0.686292 0.0253834
\(732\) 0 0
\(733\) 1.31371 0.0485229 0.0242615 0.999706i \(-0.492277\pi\)
0.0242615 + 0.999706i \(0.492277\pi\)
\(734\) −8.04163 −0.296822
\(735\) 0 0
\(736\) −33.7990 −1.24585
\(737\) 16.9706 0.625119
\(738\) 0 0
\(739\) −9.65685 −0.355233 −0.177617 0.984100i \(-0.556839\pi\)
−0.177617 + 0.984100i \(0.556839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.970563 0.0356305
\(743\) −15.3137 −0.561805 −0.280903 0.959736i \(-0.590634\pi\)
−0.280903 + 0.959736i \(0.590634\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.10051 −0.150130
\(747\) 0 0
\(748\) −3.02944 −0.110767
\(749\) −4.68629 −0.171233
\(750\) 0 0
\(751\) −37.1127 −1.35426 −0.677131 0.735863i \(-0.736777\pi\)
−0.677131 + 0.735863i \(0.736777\pi\)
\(752\) 1.02944 0.0375397
\(753\) 0 0
\(754\) 20.3431 0.740854
\(755\) 0 0
\(756\) 0 0
\(757\) −16.8284 −0.611640 −0.305820 0.952089i \(-0.598931\pi\)
−0.305820 + 0.952089i \(0.598931\pi\)
\(758\) −10.0000 −0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) 10.2843 0.372805 0.186402 0.982474i \(-0.440317\pi\)
0.186402 + 0.982474i \(0.440317\pi\)
\(762\) 0 0
\(763\) −1.85786 −0.0672592
\(764\) 3.47309 0.125652
\(765\) 0 0
\(766\) −4.97056 −0.179594
\(767\) 45.9411 1.65884
\(768\) 0 0
\(769\) −35.6569 −1.28582 −0.642910 0.765942i \(-0.722273\pi\)
−0.642910 + 0.765942i \(0.722273\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −27.5563 −0.991775
\(773\) −32.9706 −1.18587 −0.592934 0.805251i \(-0.702031\pi\)
−0.592934 + 0.805251i \(0.702031\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.72792 0.241518
\(777\) 0 0
\(778\) 6.20101 0.222317
\(779\) 7.41421 0.265642
\(780\) 0 0
\(781\) −6.34315 −0.226976
\(782\) 3.71573 0.132874
\(783\) 0 0
\(784\) −19.9706 −0.713234
\(785\) 0 0
\(786\) 0 0
\(787\) −13.4558 −0.479649 −0.239825 0.970816i \(-0.577090\pi\)
−0.239825 + 0.970816i \(0.577090\pi\)
\(788\) 27.1127 0.965850
\(789\) 0 0
\(790\) 0 0
\(791\) −7.31371 −0.260046
\(792\) 0 0
\(793\) −30.6274 −1.08761
\(794\) 14.7696 0.524152
\(795\) 0 0
\(796\) 30.1421 1.06836
\(797\) 10.8284 0.383563 0.191781 0.981438i \(-0.438574\pi\)
0.191781 + 0.981438i \(0.438574\pi\)
\(798\) 0 0
\(799\) −0.402020 −0.0142225
\(800\) 0 0
\(801\) 0 0
\(802\) 12.4437 0.439401
\(803\) −2.82843 −0.0998130
\(804\) 0 0
\(805\) 0 0
\(806\) 14.5442 0.512296
\(807\) 0 0
\(808\) 7.65685 0.269367
\(809\) −49.3137 −1.73378 −0.866889 0.498502i \(-0.833884\pi\)
−0.866889 + 0.498502i \(0.833884\pi\)
\(810\) 0 0
\(811\) −15.3137 −0.537737 −0.268869 0.963177i \(-0.586650\pi\)
−0.268869 + 0.963177i \(0.586650\pi\)
\(812\) 9.71573 0.340955
\(813\) 0 0
\(814\) −6.48528 −0.227309
\(815\) 0 0
\(816\) 0 0
\(817\) −0.585786 −0.0204941
\(818\) −3.94113 −0.137798
\(819\) 0 0
\(820\) 0 0
\(821\) −51.4558 −1.79582 −0.897911 0.440178i \(-0.854915\pi\)
−0.897911 + 0.440178i \(0.854915\pi\)
\(822\) 0 0
\(823\) 2.72792 0.0950894 0.0475447 0.998869i \(-0.484860\pi\)
0.0475447 + 0.998869i \(0.484860\pi\)
\(824\) 2.62742 0.0915304
\(825\) 0 0
\(826\) −2.05887 −0.0716374
\(827\) 48.6274 1.69094 0.845470 0.534022i \(-0.179320\pi\)
0.845470 + 0.534022i \(0.179320\pi\)
\(828\) 0 0
\(829\) −38.4853 −1.33665 −0.668325 0.743870i \(-0.732988\pi\)
−0.668325 + 0.743870i \(0.732988\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 22.5858 0.783021
\(833\) 7.79899 0.270219
\(834\) 0 0
\(835\) 0 0
\(836\) 2.58579 0.0894313
\(837\) 0 0
\(838\) −14.4437 −0.498948
\(839\) −27.1127 −0.936034 −0.468017 0.883719i \(-0.655031\pi\)
−0.468017 + 0.883719i \(0.655031\pi\)
\(840\) 0 0
\(841\) 53.2843 1.83739
\(842\) 6.08326 0.209643
\(843\) 0 0
\(844\) 28.0000 0.963800
\(845\) 0 0
\(846\) 0 0
\(847\) 5.27208 0.181151
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) −84.7696 −2.90586
\(852\) 0 0
\(853\) 9.51472 0.325778 0.162889 0.986644i \(-0.447919\pi\)
0.162889 + 0.986644i \(0.447919\pi\)
\(854\) 1.37258 0.0469688
\(855\) 0 0
\(856\) 12.6863 0.433609
\(857\) 37.9411 1.29604 0.648022 0.761622i \(-0.275596\pi\)
0.648022 + 0.761622i \(0.275596\pi\)
\(858\) 0 0
\(859\) 25.9411 0.885100 0.442550 0.896744i \(-0.354074\pi\)
0.442550 + 0.896744i \(0.354074\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.45584 0.0495862
\(863\) −31.3137 −1.06593 −0.532966 0.846137i \(-0.678922\pi\)
−0.532966 + 0.846137i \(0.678922\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0.384776 0.0130752
\(867\) 0 0
\(868\) 6.94618 0.235769
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 64.9706 2.20144
\(872\) 5.02944 0.170318
\(873\) 0 0
\(874\) −3.17157 −0.107280
\(875\) 0 0
\(876\) 0 0
\(877\) −17.8995 −0.604423 −0.302211 0.953241i \(-0.597725\pi\)
−0.302211 + 0.953241i \(0.597725\pi\)
\(878\) −0.402020 −0.0135675
\(879\) 0 0
\(880\) 0 0
\(881\) 47.4558 1.59883 0.799414 0.600781i \(-0.205143\pi\)
0.799414 + 0.600781i \(0.205143\pi\)
\(882\) 0 0
\(883\) −46.5269 −1.56576 −0.782878 0.622176i \(-0.786249\pi\)
−0.782878 + 0.622176i \(0.786249\pi\)
\(884\) −11.5980 −0.390082
\(885\) 0 0
\(886\) 0.544156 0.0182813
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 4.68629 0.157173
\(890\) 0 0
\(891\) 0 0
\(892\) 11.5980 0.388329
\(893\) 0.343146 0.0114829
\(894\) 0 0
\(895\) 0 0
\(896\) −6.18377 −0.206585
\(897\) 0 0
\(898\) 3.27208 0.109191
\(899\) 58.8284 1.96204
\(900\) 0 0
\(901\) −4.68629 −0.156123
\(902\) 4.34315 0.144611
\(903\) 0 0
\(904\) 19.7990 0.658505
\(905\) 0 0
\(906\) 0 0
\(907\) −38.1421 −1.26649 −0.633244 0.773952i \(-0.718277\pi\)
−0.633244 + 0.773952i \(0.718277\pi\)
\(908\) 34.6863 1.15111
\(909\) 0 0
\(910\) 0 0
\(911\) 23.3137 0.772418 0.386209 0.922411i \(-0.373784\pi\)
0.386209 + 0.922411i \(0.373784\pi\)
\(912\) 0 0
\(913\) 14.8284 0.490749
\(914\) 11.9411 0.394977
\(915\) 0 0
\(916\) 3.02944 0.100095
\(917\) 9.79899 0.323591
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 4.42641 0.145776
\(923\) −24.2843 −0.799327
\(924\) 0 0
\(925\) 0 0
\(926\) −3.35534 −0.110263
\(927\) 0 0
\(928\) −40.0416 −1.31443
\(929\) 51.4558 1.68821 0.844106 0.536177i \(-0.180132\pi\)
0.844106 + 0.536177i \(0.180132\pi\)
\(930\) 0 0
\(931\) −6.65685 −0.218170
\(932\) 15.2548 0.499689
\(933\) 0 0
\(934\) −6.76955 −0.221507
\(935\) 0 0
\(936\) 0 0
\(937\) 18.7696 0.613175 0.306587 0.951843i \(-0.400813\pi\)
0.306587 + 0.951843i \(0.400813\pi\)
\(938\) −2.91169 −0.0950700
\(939\) 0 0
\(940\) 0 0
\(941\) 17.5563 0.572321 0.286160 0.958182i \(-0.407621\pi\)
0.286160 + 0.958182i \(0.407621\pi\)
\(942\) 0 0
\(943\) 56.7696 1.84867
\(944\) −25.4558 −0.828517
\(945\) 0 0
\(946\) −0.343146 −0.0111566
\(947\) 12.8284 0.416868 0.208434 0.978036i \(-0.433163\pi\)
0.208434 + 0.978036i \(0.433163\pi\)
\(948\) 0 0
\(949\) −10.8284 −0.351506
\(950\) 0 0
\(951\) 0 0
\(952\) 1.08831 0.0352724
\(953\) −5.85786 −0.189755 −0.0948774 0.995489i \(-0.530246\pi\)
−0.0948774 + 0.995489i \(0.530246\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.72792 0.152912
\(957\) 0 0
\(958\) −15.8162 −0.510999
\(959\) 8.20101 0.264824
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) −24.8284 −0.800501
\(963\) 0 0
\(964\) 27.3726 0.881612
\(965\) 0 0
\(966\) 0 0
\(967\) 27.8995 0.897187 0.448594 0.893736i \(-0.351925\pi\)
0.448594 + 0.893736i \(0.351925\pi\)
\(968\) −14.2721 −0.458722
\(969\) 0 0
\(970\) 0 0
\(971\) −17.6569 −0.566635 −0.283318 0.959026i \(-0.591435\pi\)
−0.283318 + 0.959026i \(0.591435\pi\)
\(972\) 0 0
\(973\) −0.686292 −0.0220015
\(974\) −1.17157 −0.0375396
\(975\) 0 0
\(976\) 16.9706 0.543214
\(977\) 39.5980 1.26685 0.633426 0.773803i \(-0.281648\pi\)
0.633426 + 0.773803i \(0.281648\pi\)
\(978\) 0 0
\(979\) 15.1716 0.484886
\(980\) 0 0
\(981\) 0 0
\(982\) 9.07107 0.289469
\(983\) 31.9411 1.01876 0.509382 0.860541i \(-0.329874\pi\)
0.509382 + 0.860541i \(0.329874\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.40202 0.140189
\(987\) 0 0
\(988\) 9.89949 0.314945
\(989\) −4.48528 −0.142624
\(990\) 0 0
\(991\) −61.6569 −1.95859 −0.979297 0.202427i \(-0.935117\pi\)
−0.979297 + 0.202427i \(0.935117\pi\)
\(992\) −28.6274 −0.908921
\(993\) 0 0
\(994\) 1.08831 0.0345192
\(995\) 0 0
\(996\) 0 0
\(997\) 50.4853 1.59888 0.799442 0.600743i \(-0.205128\pi\)
0.799442 + 0.600743i \(0.205128\pi\)
\(998\) 9.85786 0.312045
\(999\) 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 4275.2.a.x.1.1 2
3.2 odd 2 1425.2.a.l.1.2 2
5.4 even 2 855.2.a.e.1.2 2
15.2 even 4 1425.2.c.j.799.3 4
15.8 even 4 1425.2.c.j.799.2 4
15.14 odd 2 285.2.a.f.1.1 2
60.59 even 2 4560.2.a.bj.1.2 2
285.284 even 2 5415.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 15.14 odd 2
855.2.a.e.1.2 2 5.4 even 2
1425.2.a.l.1.2 2 3.2 odd 2
1425.2.c.j.799.2 4 15.8 even 4
1425.2.c.j.799.3 4 15.2 even 4
4275.2.a.x.1.1 2 1.1 even 1 trivial
4560.2.a.bj.1.2 2 60.59 even 2
5415.2.a.p.1.2 2 285.284 even 2