Properties

Label 5776.2.a.bt.1.2
Level $5776$
Weight $2$
Character 5776.1
Self dual yes
Analytic conductor $46.122$
Analytic rank $0$
Dimension $4$
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] = [5776,2,Mod(1,5776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, 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("5776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.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: \(46.1215922075\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 722)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.17557\) of defining polynomial
Character \(\chi\) \(=\) 5776.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.28408 q^{3} +3.69572 q^{5} +0.442463 q^{7} -1.35114 q^{9} +O(q^{10})\) \(q-1.28408 q^{3} +3.69572 q^{5} +0.442463 q^{7} -1.35114 q^{9} +4.02967 q^{11} +4.89149 q^{13} -4.74559 q^{15} +0.266893 q^{17} -0.568158 q^{21} +9.20930 q^{23} +8.65833 q^{25} +5.58721 q^{27} -0.223582 q^{29} -3.47985 q^{31} -5.17442 q^{33} +1.63522 q^{35} +1.44903 q^{37} -6.28106 q^{39} -7.86472 q^{41} +5.63522 q^{43} -4.99344 q^{45} -2.19868 q^{47} -6.80423 q^{49} -0.342712 q^{51} +9.94241 q^{53} +14.8925 q^{55} +3.50953 q^{59} +4.07955 q^{61} -0.597831 q^{63} +18.0776 q^{65} -0.147763 q^{67} -11.8255 q^{69} -11.4691 q^{71} +1.42226 q^{73} -11.1180 q^{75} +1.78298 q^{77} -10.2034 q^{79} -3.12099 q^{81} +3.28878 q^{83} +0.986361 q^{85} +0.287096 q^{87} -5.96075 q^{89} +2.16431 q^{91} +4.46841 q^{93} +4.08476 q^{97} -5.44466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9} - 2 q^{11} + 18 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{21} + 10 q^{23} + 6 q^{25} + 4 q^{27} - 2 q^{29} - 26 q^{31} + 16 q^{33} - 6 q^{35} + 4 q^{37} + 6 q^{39} - 12 q^{41} + 10 q^{43} - 22 q^{45} + 12 q^{47} - 12 q^{49} + 2 q^{51} + 8 q^{53} + 26 q^{55} + 8 q^{59} + 22 q^{63} - 4 q^{65} - 10 q^{67} - 20 q^{69} - 14 q^{73} - 8 q^{75} + 4 q^{77} - 22 q^{79} - 4 q^{81} + 12 q^{83} - 18 q^{85} + 26 q^{87} - 16 q^{89} + 4 q^{91} + 8 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100}) \) 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
\(3\) −1.28408 −0.741363 −0.370682 0.928760i \(-0.620876\pi\)
−0.370682 + 0.928760i \(0.620876\pi\)
\(4\) 0 0
\(5\) 3.69572 1.65278 0.826388 0.563102i \(-0.190392\pi\)
0.826388 + 0.563102i \(0.190392\pi\)
\(6\) 0 0
\(7\) 0.442463 0.167235 0.0836177 0.996498i \(-0.473353\pi\)
0.0836177 + 0.996498i \(0.473353\pi\)
\(8\) 0 0
\(9\) −1.35114 −0.450380
\(10\) 0 0
\(11\) 4.02967 1.21499 0.607496 0.794323i \(-0.292174\pi\)
0.607496 + 0.794323i \(0.292174\pi\)
\(12\) 0 0
\(13\) 4.89149 1.35666 0.678328 0.734759i \(-0.262705\pi\)
0.678328 + 0.734759i \(0.262705\pi\)
\(14\) 0 0
\(15\) −4.74559 −1.22531
\(16\) 0 0
\(17\) 0.266893 0.0647311 0.0323655 0.999476i \(-0.489696\pi\)
0.0323655 + 0.999476i \(0.489696\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −0.568158 −0.123982
\(22\) 0 0
\(23\) 9.20930 1.92027 0.960136 0.279534i \(-0.0901798\pi\)
0.960136 + 0.279534i \(0.0901798\pi\)
\(24\) 0 0
\(25\) 8.65833 1.73167
\(26\) 0 0
\(27\) 5.58721 1.07526
\(28\) 0 0
\(29\) −0.223582 −0.0415181 −0.0207590 0.999785i \(-0.506608\pi\)
−0.0207590 + 0.999785i \(0.506608\pi\)
\(30\) 0 0
\(31\) −3.47985 −0.625000 −0.312500 0.949918i \(-0.601166\pi\)
−0.312500 + 0.949918i \(0.601166\pi\)
\(32\) 0 0
\(33\) −5.17442 −0.900751
\(34\) 0 0
\(35\) 1.63522 0.276403
\(36\) 0 0
\(37\) 1.44903 0.238219 0.119109 0.992881i \(-0.461996\pi\)
0.119109 + 0.992881i \(0.461996\pi\)
\(38\) 0 0
\(39\) −6.28106 −1.00577
\(40\) 0 0
\(41\) −7.86472 −1.22826 −0.614132 0.789204i \(-0.710494\pi\)
−0.614132 + 0.789204i \(0.710494\pi\)
\(42\) 0 0
\(43\) 5.63522 0.859363 0.429682 0.902981i \(-0.358626\pi\)
0.429682 + 0.902981i \(0.358626\pi\)
\(44\) 0 0
\(45\) −4.99344 −0.744377
\(46\) 0 0
\(47\) −2.19868 −0.320710 −0.160355 0.987059i \(-0.551264\pi\)
−0.160355 + 0.987059i \(0.551264\pi\)
\(48\) 0 0
\(49\) −6.80423 −0.972032
\(50\) 0 0
\(51\) −0.342712 −0.0479892
\(52\) 0 0
\(53\) 9.94241 1.36569 0.682847 0.730561i \(-0.260741\pi\)
0.682847 + 0.730561i \(0.260741\pi\)
\(54\) 0 0
\(55\) 14.8925 2.00811
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.50953 0.456901 0.228451 0.973555i \(-0.426634\pi\)
0.228451 + 0.973555i \(0.426634\pi\)
\(60\) 0 0
\(61\) 4.07955 0.522333 0.261166 0.965294i \(-0.415893\pi\)
0.261166 + 0.965294i \(0.415893\pi\)
\(62\) 0 0
\(63\) −0.597831 −0.0753196
\(64\) 0 0
\(65\) 18.0776 2.24225
\(66\) 0 0
\(67\) −0.147763 −0.0180521 −0.00902605 0.999959i \(-0.502873\pi\)
−0.00902605 + 0.999959i \(0.502873\pi\)
\(68\) 0 0
\(69\) −11.8255 −1.42362
\(70\) 0 0
\(71\) −11.4691 −1.36113 −0.680567 0.732686i \(-0.738266\pi\)
−0.680567 + 0.732686i \(0.738266\pi\)
\(72\) 0 0
\(73\) 1.42226 0.166463 0.0832315 0.996530i \(-0.473476\pi\)
0.0832315 + 0.996530i \(0.473476\pi\)
\(74\) 0 0
\(75\) −11.1180 −1.28379
\(76\) 0 0
\(77\) 1.78298 0.203190
\(78\) 0 0
\(79\) −10.2034 −1.14797 −0.573985 0.818866i \(-0.694603\pi\)
−0.573985 + 0.818866i \(0.694603\pi\)
\(80\) 0 0
\(81\) −3.12099 −0.346777
\(82\) 0 0
\(83\) 3.28878 0.360990 0.180495 0.983576i \(-0.442230\pi\)
0.180495 + 0.983576i \(0.442230\pi\)
\(84\) 0 0
\(85\) 0.986361 0.106986
\(86\) 0 0
\(87\) 0.287096 0.0307800
\(88\) 0 0
\(89\) −5.96075 −0.631838 −0.315919 0.948786i \(-0.602313\pi\)
−0.315919 + 0.948786i \(0.602313\pi\)
\(90\) 0 0
\(91\) 2.16431 0.226881
\(92\) 0 0
\(93\) 4.46841 0.463352
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.08476 0.414744 0.207372 0.978262i \(-0.433509\pi\)
0.207372 + 0.978262i \(0.433509\pi\)
\(98\) 0 0
\(99\) −5.44466 −0.547208
\(100\) 0 0
\(101\) 0.425277 0.0423167 0.0211583 0.999776i \(-0.493265\pi\)
0.0211583 + 0.999776i \(0.493265\pi\)
\(102\) 0 0
\(103\) 4.53618 0.446963 0.223482 0.974708i \(-0.428258\pi\)
0.223482 + 0.974708i \(0.428258\pi\)
\(104\) 0 0
\(105\) −2.09975 −0.204915
\(106\) 0 0
\(107\) 3.17151 0.306602 0.153301 0.988180i \(-0.451010\pi\)
0.153301 + 0.988180i \(0.451010\pi\)
\(108\) 0 0
\(109\) −6.78112 −0.649513 −0.324757 0.945798i \(-0.605282\pi\)
−0.324757 + 0.945798i \(0.605282\pi\)
\(110\) 0 0
\(111\) −1.86067 −0.176607
\(112\) 0 0
\(113\) −10.0444 −0.944893 −0.472447 0.881359i \(-0.656629\pi\)
−0.472447 + 0.881359i \(0.656629\pi\)
\(114\) 0 0
\(115\) 34.0350 3.17378
\(116\) 0 0
\(117\) −6.60909 −0.611011
\(118\) 0 0
\(119\) 0.118090 0.0108253
\(120\) 0 0
\(121\) 5.23826 0.476205
\(122\) 0 0
\(123\) 10.0989 0.910590
\(124\) 0 0
\(125\) 13.5201 1.20928
\(126\) 0 0
\(127\) −14.0806 −1.24945 −0.624725 0.780845i \(-0.714789\pi\)
−0.624725 + 0.780845i \(0.714789\pi\)
\(128\) 0 0
\(129\) −7.23607 −0.637100
\(130\) 0 0
\(131\) −15.8415 −1.38408 −0.692039 0.721860i \(-0.743288\pi\)
−0.692039 + 0.721860i \(0.743288\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 20.6487 1.77716
\(136\) 0 0
\(137\) 2.20744 0.188594 0.0942970 0.995544i \(-0.469940\pi\)
0.0942970 + 0.995544i \(0.469940\pi\)
\(138\) 0 0
\(139\) 17.0644 1.44739 0.723694 0.690121i \(-0.242443\pi\)
0.723694 + 0.690121i \(0.242443\pi\)
\(140\) 0 0
\(141\) 2.82328 0.237763
\(142\) 0 0
\(143\) 19.7111 1.64833
\(144\) 0 0
\(145\) −0.826294 −0.0686200
\(146\) 0 0
\(147\) 8.73716 0.720629
\(148\) 0 0
\(149\) 13.5458 1.10971 0.554856 0.831946i \(-0.312773\pi\)
0.554856 + 0.831946i \(0.312773\pi\)
\(150\) 0 0
\(151\) 9.15317 0.744875 0.372437 0.928057i \(-0.378522\pi\)
0.372437 + 0.928057i \(0.378522\pi\)
\(152\) 0 0
\(153\) −0.360610 −0.0291536
\(154\) 0 0
\(155\) −12.8606 −1.03298
\(156\) 0 0
\(157\) −16.3903 −1.30809 −0.654043 0.756457i \(-0.726929\pi\)
−0.654043 + 0.756457i \(0.726929\pi\)
\(158\) 0 0
\(159\) −12.7668 −1.01248
\(160\) 0 0
\(161\) 4.07478 0.321138
\(162\) 0 0
\(163\) 13.7722 1.07873 0.539363 0.842073i \(-0.318665\pi\)
0.539363 + 0.842073i \(0.318665\pi\)
\(164\) 0 0
\(165\) −19.1232 −1.48874
\(166\) 0 0
\(167\) 10.3328 0.799576 0.399788 0.916608i \(-0.369084\pi\)
0.399788 + 0.916608i \(0.369084\pi\)
\(168\) 0 0
\(169\) 10.9267 0.840515
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.81201 −0.745994 −0.372997 0.927833i \(-0.621670\pi\)
−0.372997 + 0.927833i \(0.621670\pi\)
\(174\) 0 0
\(175\) 3.83099 0.289596
\(176\) 0 0
\(177\) −4.50651 −0.338730
\(178\) 0 0
\(179\) 8.79830 0.657616 0.328808 0.944397i \(-0.393353\pi\)
0.328808 + 0.944397i \(0.393353\pi\)
\(180\) 0 0
\(181\) 14.5278 1.07984 0.539920 0.841717i \(-0.318455\pi\)
0.539920 + 0.841717i \(0.318455\pi\)
\(182\) 0 0
\(183\) −5.23846 −0.387238
\(184\) 0 0
\(185\) 5.35520 0.393722
\(186\) 0 0
\(187\) 1.07549 0.0786477
\(188\) 0 0
\(189\) 2.47214 0.179821
\(190\) 0 0
\(191\) −12.7302 −0.921122 −0.460561 0.887628i \(-0.652352\pi\)
−0.460561 + 0.887628i \(0.652352\pi\)
\(192\) 0 0
\(193\) 17.4781 1.25810 0.629049 0.777366i \(-0.283444\pi\)
0.629049 + 0.777366i \(0.283444\pi\)
\(194\) 0 0
\(195\) −23.2130 −1.66232
\(196\) 0 0
\(197\) −24.5113 −1.74636 −0.873178 0.487401i \(-0.837945\pi\)
−0.873178 + 0.487401i \(0.837945\pi\)
\(198\) 0 0
\(199\) −16.4661 −1.16725 −0.583625 0.812023i \(-0.698366\pi\)
−0.583625 + 0.812023i \(0.698366\pi\)
\(200\) 0 0
\(201\) 0.189739 0.0133832
\(202\) 0 0
\(203\) −0.0989267 −0.00694329
\(204\) 0 0
\(205\) −29.0658 −2.03004
\(206\) 0 0
\(207\) −12.4431 −0.864853
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −27.4805 −1.89183 −0.945916 0.324411i \(-0.894834\pi\)
−0.945916 + 0.324411i \(0.894834\pi\)
\(212\) 0 0
\(213\) 14.7273 1.00909
\(214\) 0 0
\(215\) 20.8262 1.42033
\(216\) 0 0
\(217\) −1.53971 −0.104522
\(218\) 0 0
\(219\) −1.82629 −0.123410
\(220\) 0 0
\(221\) 1.30550 0.0878178
\(222\) 0 0
\(223\) −20.3706 −1.36412 −0.682058 0.731298i \(-0.738915\pi\)
−0.682058 + 0.731298i \(0.738915\pi\)
\(224\) 0 0
\(225\) −11.6986 −0.779908
\(226\) 0 0
\(227\) 22.1493 1.47010 0.735051 0.678011i \(-0.237158\pi\)
0.735051 + 0.678011i \(0.237158\pi\)
\(228\) 0 0
\(229\) 12.7148 0.840216 0.420108 0.907474i \(-0.361992\pi\)
0.420108 + 0.907474i \(0.361992\pi\)
\(230\) 0 0
\(231\) −2.28949 −0.150637
\(232\) 0 0
\(233\) −8.68176 −0.568761 −0.284381 0.958711i \(-0.591788\pi\)
−0.284381 + 0.958711i \(0.591788\pi\)
\(234\) 0 0
\(235\) −8.12569 −0.530062
\(236\) 0 0
\(237\) 13.1019 0.851063
\(238\) 0 0
\(239\) −21.8749 −1.41497 −0.707485 0.706728i \(-0.750170\pi\)
−0.707485 + 0.706728i \(0.750170\pi\)
\(240\) 0 0
\(241\) −18.9632 −1.22153 −0.610764 0.791813i \(-0.709138\pi\)
−0.610764 + 0.791813i \(0.709138\pi\)
\(242\) 0 0
\(243\) −12.7540 −0.818171
\(244\) 0 0
\(245\) −25.1465 −1.60655
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.22305 −0.267625
\(250\) 0 0
\(251\) −2.90276 −0.183220 −0.0916102 0.995795i \(-0.529201\pi\)
−0.0916102 + 0.995795i \(0.529201\pi\)
\(252\) 0 0
\(253\) 37.1105 2.33311
\(254\) 0 0
\(255\) −1.26657 −0.0793154
\(256\) 0 0
\(257\) 22.9050 1.42878 0.714388 0.699750i \(-0.246705\pi\)
0.714388 + 0.699750i \(0.246705\pi\)
\(258\) 0 0
\(259\) 0.641142 0.0398386
\(260\) 0 0
\(261\) 0.302090 0.0186989
\(262\) 0 0
\(263\) 5.06236 0.312159 0.156079 0.987745i \(-0.450114\pi\)
0.156079 + 0.987745i \(0.450114\pi\)
\(264\) 0 0
\(265\) 36.7443 2.25719
\(266\) 0 0
\(267\) 7.65407 0.468421
\(268\) 0 0
\(269\) −20.7419 −1.26466 −0.632329 0.774700i \(-0.717901\pi\)
−0.632329 + 0.774700i \(0.717901\pi\)
\(270\) 0 0
\(271\) −6.29470 −0.382376 −0.191188 0.981553i \(-0.561234\pi\)
−0.191188 + 0.981553i \(0.561234\pi\)
\(272\) 0 0
\(273\) −2.77914 −0.168201
\(274\) 0 0
\(275\) 34.8902 2.10396
\(276\) 0 0
\(277\) 7.17608 0.431169 0.215584 0.976485i \(-0.430834\pi\)
0.215584 + 0.976485i \(0.430834\pi\)
\(278\) 0 0
\(279\) 4.70177 0.281488
\(280\) 0 0
\(281\) 2.77391 0.165478 0.0827388 0.996571i \(-0.473633\pi\)
0.0827388 + 0.996571i \(0.473633\pi\)
\(282\) 0 0
\(283\) −7.74861 −0.460607 −0.230304 0.973119i \(-0.573972\pi\)
−0.230304 + 0.973119i \(0.573972\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.47985 −0.205409
\(288\) 0 0
\(289\) −16.9288 −0.995810
\(290\) 0 0
\(291\) −5.24515 −0.307476
\(292\) 0 0
\(293\) −0.782870 −0.0457358 −0.0228679 0.999738i \(-0.507280\pi\)
−0.0228679 + 0.999738i \(0.507280\pi\)
\(294\) 0 0
\(295\) 12.9702 0.755155
\(296\) 0 0
\(297\) 22.5146 1.30643
\(298\) 0 0
\(299\) 45.0472 2.60515
\(300\) 0 0
\(301\) 2.49338 0.143716
\(302\) 0 0
\(303\) −0.546090 −0.0313720
\(304\) 0 0
\(305\) 15.0769 0.863298
\(306\) 0 0
\(307\) 28.1688 1.60768 0.803839 0.594847i \(-0.202788\pi\)
0.803839 + 0.594847i \(0.202788\pi\)
\(308\) 0 0
\(309\) −5.82481 −0.331362
\(310\) 0 0
\(311\) −3.86827 −0.219350 −0.109675 0.993968i \(-0.534981\pi\)
−0.109675 + 0.993968i \(0.534981\pi\)
\(312\) 0 0
\(313\) 30.3878 1.71762 0.858809 0.512295i \(-0.171205\pi\)
0.858809 + 0.512295i \(0.171205\pi\)
\(314\) 0 0
\(315\) −2.20941 −0.124486
\(316\) 0 0
\(317\) 5.32803 0.299252 0.149626 0.988743i \(-0.452193\pi\)
0.149626 + 0.988743i \(0.452193\pi\)
\(318\) 0 0
\(319\) −0.900961 −0.0504441
\(320\) 0 0
\(321\) −4.07247 −0.227303
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 42.3521 2.34927
\(326\) 0 0
\(327\) 8.70749 0.481525
\(328\) 0 0
\(329\) −0.972835 −0.0536341
\(330\) 0 0
\(331\) 20.5480 1.12942 0.564711 0.825289i \(-0.308988\pi\)
0.564711 + 0.825289i \(0.308988\pi\)
\(332\) 0 0
\(333\) −1.95784 −0.107289
\(334\) 0 0
\(335\) −0.546090 −0.0298361
\(336\) 0 0
\(337\) 13.2824 0.723539 0.361770 0.932268i \(-0.382173\pi\)
0.361770 + 0.932268i \(0.382173\pi\)
\(338\) 0 0
\(339\) 12.8977 0.700509
\(340\) 0 0
\(341\) −14.0227 −0.759370
\(342\) 0 0
\(343\) −6.10787 −0.329794
\(344\) 0 0
\(345\) −43.7036 −2.35292
\(346\) 0 0
\(347\) −30.1423 −1.61812 −0.809062 0.587723i \(-0.800025\pi\)
−0.809062 + 0.587723i \(0.800025\pi\)
\(348\) 0 0
\(349\) 35.0653 1.87700 0.938501 0.345277i \(-0.112215\pi\)
0.938501 + 0.345277i \(0.112215\pi\)
\(350\) 0 0
\(351\) 27.3298 1.45876
\(352\) 0 0
\(353\) 13.5411 0.720718 0.360359 0.932814i \(-0.382654\pi\)
0.360359 + 0.932814i \(0.382654\pi\)
\(354\) 0 0
\(355\) −42.3866 −2.24965
\(356\) 0 0
\(357\) −0.151637 −0.00802550
\(358\) 0 0
\(359\) 23.5072 1.24066 0.620332 0.784339i \(-0.286998\pi\)
0.620332 + 0.784339i \(0.286998\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −6.72634 −0.353041
\(364\) 0 0
\(365\) 5.25627 0.275126
\(366\) 0 0
\(367\) 12.6929 0.662563 0.331282 0.943532i \(-0.392519\pi\)
0.331282 + 0.943532i \(0.392519\pi\)
\(368\) 0 0
\(369\) 10.6264 0.553186
\(370\) 0 0
\(371\) 4.39915 0.228393
\(372\) 0 0
\(373\) 2.64950 0.137186 0.0685930 0.997645i \(-0.478149\pi\)
0.0685930 + 0.997645i \(0.478149\pi\)
\(374\) 0 0
\(375\) −17.3609 −0.896515
\(376\) 0 0
\(377\) −1.09365 −0.0563257
\(378\) 0 0
\(379\) −18.1672 −0.933187 −0.466593 0.884472i \(-0.654519\pi\)
−0.466593 + 0.884472i \(0.654519\pi\)
\(380\) 0 0
\(381\) 18.0806 0.926297
\(382\) 0 0
\(383\) −23.0083 −1.17567 −0.587835 0.808981i \(-0.700020\pi\)
−0.587835 + 0.808981i \(0.700020\pi\)
\(384\) 0 0
\(385\) 6.58940 0.335827
\(386\) 0 0
\(387\) −7.61398 −0.387040
\(388\) 0 0
\(389\) −16.5706 −0.840164 −0.420082 0.907486i \(-0.637999\pi\)
−0.420082 + 0.907486i \(0.637999\pi\)
\(390\) 0 0
\(391\) 2.45790 0.124301
\(392\) 0 0
\(393\) 20.3417 1.02611
\(394\) 0 0
\(395\) −37.7088 −1.89734
\(396\) 0 0
\(397\) −17.7948 −0.893097 −0.446548 0.894759i \(-0.647347\pi\)
−0.446548 + 0.894759i \(0.647347\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.6215 1.52916 0.764582 0.644527i \(-0.222946\pi\)
0.764582 + 0.644527i \(0.222946\pi\)
\(402\) 0 0
\(403\) −17.0217 −0.847910
\(404\) 0 0
\(405\) −11.5343 −0.573145
\(406\) 0 0
\(407\) 5.83911 0.289434
\(408\) 0 0
\(409\) 9.14328 0.452106 0.226053 0.974115i \(-0.427418\pi\)
0.226053 + 0.974115i \(0.427418\pi\)
\(410\) 0 0
\(411\) −2.83452 −0.139817
\(412\) 0 0
\(413\) 1.55284 0.0764101
\(414\) 0 0
\(415\) 12.1544 0.596636
\(416\) 0 0
\(417\) −21.9121 −1.07304
\(418\) 0 0
\(419\) −12.6157 −0.616315 −0.308158 0.951335i \(-0.599712\pi\)
−0.308158 + 0.951335i \(0.599712\pi\)
\(420\) 0 0
\(421\) 33.2553 1.62077 0.810383 0.585901i \(-0.199259\pi\)
0.810383 + 0.585901i \(0.199259\pi\)
\(422\) 0 0
\(423\) 2.97072 0.144442
\(424\) 0 0
\(425\) 2.31085 0.112093
\(426\) 0 0
\(427\) 1.80505 0.0873525
\(428\) 0 0
\(429\) −25.3106 −1.22201
\(430\) 0 0
\(431\) 22.9915 1.10746 0.553730 0.832696i \(-0.313204\pi\)
0.553730 + 0.832696i \(0.313204\pi\)
\(432\) 0 0
\(433\) 34.7092 1.66802 0.834010 0.551749i \(-0.186040\pi\)
0.834010 + 0.551749i \(0.186040\pi\)
\(434\) 0 0
\(435\) 1.06103 0.0508724
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 36.4818 1.74118 0.870591 0.492007i \(-0.163737\pi\)
0.870591 + 0.492007i \(0.163737\pi\)
\(440\) 0 0
\(441\) 9.19347 0.437784
\(442\) 0 0
\(443\) −11.7583 −0.558653 −0.279327 0.960196i \(-0.590111\pi\)
−0.279327 + 0.960196i \(0.590111\pi\)
\(444\) 0 0
\(445\) −22.0292 −1.04429
\(446\) 0 0
\(447\) −17.3938 −0.822700
\(448\) 0 0
\(449\) −8.69102 −0.410154 −0.205077 0.978746i \(-0.565745\pi\)
−0.205077 + 0.978746i \(0.565745\pi\)
\(450\) 0 0
\(451\) −31.6923 −1.49233
\(452\) 0 0
\(453\) −11.7534 −0.552223
\(454\) 0 0
\(455\) 7.99866 0.374983
\(456\) 0 0
\(457\) −13.0286 −0.609454 −0.304727 0.952440i \(-0.598565\pi\)
−0.304727 + 0.952440i \(0.598565\pi\)
\(458\) 0 0
\(459\) 1.49119 0.0696026
\(460\) 0 0
\(461\) 9.04260 0.421156 0.210578 0.977577i \(-0.432465\pi\)
0.210578 + 0.977577i \(0.432465\pi\)
\(462\) 0 0
\(463\) 17.7205 0.823542 0.411771 0.911287i \(-0.364910\pi\)
0.411771 + 0.911287i \(0.364910\pi\)
\(464\) 0 0
\(465\) 16.5140 0.765817
\(466\) 0 0
\(467\) 7.87772 0.364537 0.182269 0.983249i \(-0.441656\pi\)
0.182269 + 0.983249i \(0.441656\pi\)
\(468\) 0 0
\(469\) −0.0653797 −0.00301895
\(470\) 0 0
\(471\) 21.0464 0.969768
\(472\) 0 0
\(473\) 22.7081 1.04412
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −13.4336 −0.615082
\(478\) 0 0
\(479\) 30.1798 1.37895 0.689476 0.724309i \(-0.257841\pi\)
0.689476 + 0.724309i \(0.257841\pi\)
\(480\) 0 0
\(481\) 7.08791 0.323181
\(482\) 0 0
\(483\) −5.23234 −0.238080
\(484\) 0 0
\(485\) 15.0961 0.685479
\(486\) 0 0
\(487\) −35.7699 −1.62089 −0.810445 0.585814i \(-0.800775\pi\)
−0.810445 + 0.585814i \(0.800775\pi\)
\(488\) 0 0
\(489\) −17.6847 −0.799728
\(490\) 0 0
\(491\) 18.2726 0.824632 0.412316 0.911041i \(-0.364720\pi\)
0.412316 + 0.911041i \(0.364720\pi\)
\(492\) 0 0
\(493\) −0.0596724 −0.00268751
\(494\) 0 0
\(495\) −20.1219 −0.904413
\(496\) 0 0
\(497\) −5.07467 −0.227630
\(498\) 0 0
\(499\) −17.6980 −0.792270 −0.396135 0.918192i \(-0.629649\pi\)
−0.396135 + 0.918192i \(0.629649\pi\)
\(500\) 0 0
\(501\) −13.2681 −0.592777
\(502\) 0 0
\(503\) 10.1902 0.454361 0.227180 0.973853i \(-0.427049\pi\)
0.227180 + 0.973853i \(0.427049\pi\)
\(504\) 0 0
\(505\) 1.57171 0.0699400
\(506\) 0 0
\(507\) −14.0307 −0.623127
\(508\) 0 0
\(509\) −8.43621 −0.373929 −0.186964 0.982367i \(-0.559865\pi\)
−0.186964 + 0.982367i \(0.559865\pi\)
\(510\) 0 0
\(511\) 0.629298 0.0278385
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.7644 0.738730
\(516\) 0 0
\(517\) −8.85995 −0.389660
\(518\) 0 0
\(519\) 12.5994 0.553052
\(520\) 0 0
\(521\) 8.96365 0.392705 0.196352 0.980533i \(-0.437090\pi\)
0.196352 + 0.980533i \(0.437090\pi\)
\(522\) 0 0
\(523\) 2.13993 0.0935727 0.0467864 0.998905i \(-0.485102\pi\)
0.0467864 + 0.998905i \(0.485102\pi\)
\(524\) 0 0
\(525\) −4.91930 −0.214696
\(526\) 0 0
\(527\) −0.928748 −0.0404569
\(528\) 0 0
\(529\) 61.8112 2.68744
\(530\) 0 0
\(531\) −4.74186 −0.205779
\(532\) 0 0
\(533\) −38.4702 −1.66633
\(534\) 0 0
\(535\) 11.7210 0.506744
\(536\) 0 0
\(537\) −11.2977 −0.487533
\(538\) 0 0
\(539\) −27.4188 −1.18101
\(540\) 0 0
\(541\) −11.7115 −0.503519 −0.251759 0.967790i \(-0.581009\pi\)
−0.251759 + 0.967790i \(0.581009\pi\)
\(542\) 0 0
\(543\) −18.6548 −0.800553
\(544\) 0 0
\(545\) −25.0611 −1.07350
\(546\) 0 0
\(547\) 24.9897 1.06848 0.534240 0.845333i \(-0.320598\pi\)
0.534240 + 0.845333i \(0.320598\pi\)
\(548\) 0 0
\(549\) −5.51205 −0.235248
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −4.51462 −0.191981
\(554\) 0 0
\(555\) −6.87650 −0.291891
\(556\) 0 0
\(557\) 9.19566 0.389633 0.194816 0.980840i \(-0.437589\pi\)
0.194816 + 0.980840i \(0.437589\pi\)
\(558\) 0 0
\(559\) 27.5646 1.16586
\(560\) 0 0
\(561\) −1.38102 −0.0583065
\(562\) 0 0
\(563\) −19.7224 −0.831202 −0.415601 0.909547i \(-0.636429\pi\)
−0.415601 + 0.909547i \(0.636429\pi\)
\(564\) 0 0
\(565\) −37.1211 −1.56170
\(566\) 0 0
\(567\) −1.38093 −0.0579935
\(568\) 0 0
\(569\) 22.1469 0.928446 0.464223 0.885718i \(-0.346334\pi\)
0.464223 + 0.885718i \(0.346334\pi\)
\(570\) 0 0
\(571\) 3.02145 0.126444 0.0632218 0.998000i \(-0.479862\pi\)
0.0632218 + 0.998000i \(0.479862\pi\)
\(572\) 0 0
\(573\) 16.3465 0.682886
\(574\) 0 0
\(575\) 79.7371 3.32527
\(576\) 0 0
\(577\) 2.30938 0.0961407 0.0480704 0.998844i \(-0.484693\pi\)
0.0480704 + 0.998844i \(0.484693\pi\)
\(578\) 0 0
\(579\) −22.4432 −0.932708
\(580\) 0 0
\(581\) 1.45516 0.0603704
\(582\) 0 0
\(583\) 40.0646 1.65931
\(584\) 0 0
\(585\) −24.4253 −1.00986
\(586\) 0 0
\(587\) −28.0890 −1.15936 −0.579679 0.814845i \(-0.696822\pi\)
−0.579679 + 0.814845i \(0.696822\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 31.4744 1.29468
\(592\) 0 0
\(593\) −12.0939 −0.496637 −0.248318 0.968678i \(-0.579878\pi\)
−0.248318 + 0.968678i \(0.579878\pi\)
\(594\) 0 0
\(595\) 0.436429 0.0178918
\(596\) 0 0
\(597\) 21.1438 0.865357
\(598\) 0 0
\(599\) −19.5025 −0.796851 −0.398426 0.917201i \(-0.630443\pi\)
−0.398426 + 0.917201i \(0.630443\pi\)
\(600\) 0 0
\(601\) −38.9632 −1.58934 −0.794671 0.607040i \(-0.792357\pi\)
−0.794671 + 0.607040i \(0.792357\pi\)
\(602\) 0 0
\(603\) 0.199648 0.00813031
\(604\) 0 0
\(605\) 19.3591 0.787061
\(606\) 0 0
\(607\) −8.46029 −0.343393 −0.171696 0.985150i \(-0.554925\pi\)
−0.171696 + 0.985150i \(0.554925\pi\)
\(608\) 0 0
\(609\) 0.127030 0.00514750
\(610\) 0 0
\(611\) −10.7548 −0.435093
\(612\) 0 0
\(613\) −2.91493 −0.117733 −0.0588664 0.998266i \(-0.518749\pi\)
−0.0588664 + 0.998266i \(0.518749\pi\)
\(614\) 0 0
\(615\) 37.3228 1.50500
\(616\) 0 0
\(617\) −3.01905 −0.121542 −0.0607712 0.998152i \(-0.519356\pi\)
−0.0607712 + 0.998152i \(0.519356\pi\)
\(618\) 0 0
\(619\) 3.92813 0.157885 0.0789423 0.996879i \(-0.474846\pi\)
0.0789423 + 0.996879i \(0.474846\pi\)
\(620\) 0 0
\(621\) 51.4543 2.06479
\(622\) 0 0
\(623\) −2.63741 −0.105666
\(624\) 0 0
\(625\) 6.67500 0.267000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.386735 0.0154201
\(630\) 0 0
\(631\) −10.1367 −0.403536 −0.201768 0.979433i \(-0.564669\pi\)
−0.201768 + 0.979433i \(0.564669\pi\)
\(632\) 0 0
\(633\) 35.2871 1.40254
\(634\) 0 0
\(635\) −52.0379 −2.06506
\(636\) 0 0
\(637\) −33.2828 −1.31871
\(638\) 0 0
\(639\) 15.4964 0.613028
\(640\) 0 0
\(641\) 8.28031 0.327052 0.163526 0.986539i \(-0.447713\pi\)
0.163526 + 0.986539i \(0.447713\pi\)
\(642\) 0 0
\(643\) −36.8541 −1.45338 −0.726691 0.686964i \(-0.758943\pi\)
−0.726691 + 0.686964i \(0.758943\pi\)
\(644\) 0 0
\(645\) −26.7425 −1.05298
\(646\) 0 0
\(647\) 20.1901 0.793753 0.396876 0.917872i \(-0.370094\pi\)
0.396876 + 0.917872i \(0.370094\pi\)
\(648\) 0 0
\(649\) 14.1422 0.555131
\(650\) 0 0
\(651\) 1.97711 0.0774889
\(652\) 0 0
\(653\) −22.0387 −0.862443 −0.431221 0.902246i \(-0.641917\pi\)
−0.431221 + 0.902246i \(0.641917\pi\)
\(654\) 0 0
\(655\) −58.5457 −2.28757
\(656\) 0 0
\(657\) −1.92167 −0.0749716
\(658\) 0 0
\(659\) 27.2216 1.06040 0.530202 0.847872i \(-0.322116\pi\)
0.530202 + 0.847872i \(0.322116\pi\)
\(660\) 0 0
\(661\) −8.90825 −0.346491 −0.173245 0.984879i \(-0.555425\pi\)
−0.173245 + 0.984879i \(0.555425\pi\)
\(662\) 0 0
\(663\) −1.67637 −0.0651049
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.05903 −0.0797260
\(668\) 0 0
\(669\) 26.1574 1.01131
\(670\) 0 0
\(671\) 16.4392 0.634630
\(672\) 0 0
\(673\) 35.7010 1.37617 0.688087 0.725629i \(-0.258451\pi\)
0.688087 + 0.725629i \(0.258451\pi\)
\(674\) 0 0
\(675\) 48.3759 1.86199
\(676\) 0 0
\(677\) −27.3849 −1.05249 −0.526243 0.850334i \(-0.676400\pi\)
−0.526243 + 0.850334i \(0.676400\pi\)
\(678\) 0 0
\(679\) 1.80736 0.0693600
\(680\) 0 0
\(681\) −28.4415 −1.08988
\(682\) 0 0
\(683\) 38.1521 1.45985 0.729925 0.683528i \(-0.239555\pi\)
0.729925 + 0.683528i \(0.239555\pi\)
\(684\) 0 0
\(685\) 8.15806 0.311703
\(686\) 0 0
\(687\) −16.3268 −0.622905
\(688\) 0 0
\(689\) 48.6332 1.85278
\(690\) 0 0
\(691\) 28.2058 1.07300 0.536499 0.843901i \(-0.319746\pi\)
0.536499 + 0.843901i \(0.319746\pi\)
\(692\) 0 0
\(693\) −2.40906 −0.0915127
\(694\) 0 0
\(695\) 63.0654 2.39221
\(696\) 0 0
\(697\) −2.09904 −0.0795068
\(698\) 0 0
\(699\) 11.1481 0.421659
\(700\) 0 0
\(701\) 42.6148 1.60954 0.804769 0.593588i \(-0.202289\pi\)
0.804769 + 0.593588i \(0.202289\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 10.4340 0.392968
\(706\) 0 0
\(707\) 0.188170 0.00707685
\(708\) 0 0
\(709\) −22.6892 −0.852110 −0.426055 0.904697i \(-0.640097\pi\)
−0.426055 + 0.904697i \(0.640097\pi\)
\(710\) 0 0
\(711\) 13.7862 0.517023
\(712\) 0 0
\(713\) −32.0470 −1.20017
\(714\) 0 0
\(715\) 72.8467 2.72431
\(716\) 0 0
\(717\) 28.0891 1.04901
\(718\) 0 0
\(719\) 37.8324 1.41091 0.705456 0.708754i \(-0.250742\pi\)
0.705456 + 0.708754i \(0.250742\pi\)
\(720\) 0 0
\(721\) 2.00709 0.0747481
\(722\) 0 0
\(723\) 24.3503 0.905596
\(724\) 0 0
\(725\) −1.93584 −0.0718954
\(726\) 0 0
\(727\) 19.5403 0.724710 0.362355 0.932040i \(-0.381973\pi\)
0.362355 + 0.932040i \(0.381973\pi\)
\(728\) 0 0
\(729\) 25.7402 0.953339
\(730\) 0 0
\(731\) 1.50400 0.0556275
\(732\) 0 0
\(733\) 35.1101 1.29682 0.648410 0.761291i \(-0.275434\pi\)
0.648410 + 0.761291i \(0.275434\pi\)
\(734\) 0 0
\(735\) 32.2901 1.19104
\(736\) 0 0
\(737\) −0.595436 −0.0219332
\(738\) 0 0
\(739\) 16.2775 0.598776 0.299388 0.954131i \(-0.403217\pi\)
0.299388 + 0.954131i \(0.403217\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −33.3085 −1.22197 −0.610986 0.791641i \(-0.709227\pi\)
−0.610986 + 0.791641i \(0.709227\pi\)
\(744\) 0 0
\(745\) 50.0613 1.83410
\(746\) 0 0
\(747\) −4.44360 −0.162583
\(748\) 0 0
\(749\) 1.40328 0.0512747
\(750\) 0 0
\(751\) −19.2599 −0.702804 −0.351402 0.936225i \(-0.614295\pi\)
−0.351402 + 0.936225i \(0.614295\pi\)
\(752\) 0 0
\(753\) 3.72737 0.135833
\(754\) 0 0
\(755\) 33.8275 1.23111
\(756\) 0 0
\(757\) −23.9454 −0.870311 −0.435156 0.900355i \(-0.643307\pi\)
−0.435156 + 0.900355i \(0.643307\pi\)
\(758\) 0 0
\(759\) −47.6528 −1.72969
\(760\) 0 0
\(761\) −44.7968 −1.62388 −0.811941 0.583739i \(-0.801589\pi\)
−0.811941 + 0.583739i \(0.801589\pi\)
\(762\) 0 0
\(763\) −3.00040 −0.108622
\(764\) 0 0
\(765\) −1.33271 −0.0481843
\(766\) 0 0
\(767\) 17.1668 0.619858
\(768\) 0 0
\(769\) −31.7367 −1.14446 −0.572228 0.820095i \(-0.693921\pi\)
−0.572228 + 0.820095i \(0.693921\pi\)
\(770\) 0 0
\(771\) −29.4119 −1.05924
\(772\) 0 0
\(773\) −22.2307 −0.799582 −0.399791 0.916606i \(-0.630917\pi\)
−0.399791 + 0.916606i \(0.630917\pi\)
\(774\) 0 0
\(775\) −30.1297 −1.08229
\(776\) 0 0
\(777\) −0.823277 −0.0295349
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −46.2168 −1.65377
\(782\) 0 0
\(783\) −1.24920 −0.0446427
\(784\) 0 0
\(785\) −60.5739 −2.16197
\(786\) 0 0
\(787\) −24.4596 −0.871891 −0.435945 0.899973i \(-0.643586\pi\)
−0.435945 + 0.899973i \(0.643586\pi\)
\(788\) 0 0
\(789\) −6.50047 −0.231423
\(790\) 0 0
\(791\) −4.44426 −0.158020
\(792\) 0 0
\(793\) 19.9551 0.708626
\(794\) 0 0
\(795\) −47.1826 −1.67340
\(796\) 0 0
\(797\) 41.6933 1.47685 0.738426 0.674334i \(-0.235569\pi\)
0.738426 + 0.674334i \(0.235569\pi\)
\(798\) 0 0
\(799\) −0.586812 −0.0207599
\(800\) 0 0
\(801\) 8.05381 0.284567
\(802\) 0 0
\(803\) 5.73124 0.202251
\(804\) 0 0
\(805\) 15.0592 0.530768
\(806\) 0 0
\(807\) 26.6343 0.937571
\(808\) 0 0
\(809\) −49.2488 −1.73150 −0.865748 0.500481i \(-0.833157\pi\)
−0.865748 + 0.500481i \(0.833157\pi\)
\(810\) 0 0
\(811\) −52.0455 −1.82756 −0.913782 0.406205i \(-0.866852\pi\)
−0.913782 + 0.406205i \(0.866852\pi\)
\(812\) 0 0
\(813\) 8.08289 0.283479
\(814\) 0 0
\(815\) 50.8983 1.78289
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −2.92428 −0.102183
\(820\) 0 0
\(821\) −42.2582 −1.47482 −0.737411 0.675444i \(-0.763952\pi\)
−0.737411 + 0.675444i \(0.763952\pi\)
\(822\) 0 0
\(823\) −32.4771 −1.13208 −0.566041 0.824377i \(-0.691525\pi\)
−0.566041 + 0.824377i \(0.691525\pi\)
\(824\) 0 0
\(825\) −44.8018 −1.55980
\(826\) 0 0
\(827\) 21.5156 0.748172 0.374086 0.927394i \(-0.377956\pi\)
0.374086 + 0.927394i \(0.377956\pi\)
\(828\) 0 0
\(829\) 5.65061 0.196254 0.0981269 0.995174i \(-0.468715\pi\)
0.0981269 + 0.995174i \(0.468715\pi\)
\(830\) 0 0
\(831\) −9.21465 −0.319653
\(832\) 0 0
\(833\) −1.81600 −0.0629207
\(834\) 0 0
\(835\) 38.1871 1.32152
\(836\) 0 0
\(837\) −19.4427 −0.672037
\(838\) 0 0
\(839\) 10.6766 0.368598 0.184299 0.982870i \(-0.440999\pi\)
0.184299 + 0.982870i \(0.440999\pi\)
\(840\) 0 0
\(841\) −28.9500 −0.998276
\(842\) 0 0
\(843\) −3.56192 −0.122679
\(844\) 0 0
\(845\) 40.3820 1.38918
\(846\) 0 0
\(847\) 2.31774 0.0796385
\(848\) 0 0
\(849\) 9.94983 0.341477
\(850\) 0 0
\(851\) 13.3445 0.457445
\(852\) 0 0
\(853\) 10.2666 0.351521 0.175760 0.984433i \(-0.443762\pi\)
0.175760 + 0.984433i \(0.443762\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −37.3387 −1.27547 −0.637734 0.770257i \(-0.720128\pi\)
−0.637734 + 0.770257i \(0.720128\pi\)
\(858\) 0 0
\(859\) 24.2890 0.828729 0.414365 0.910111i \(-0.364004\pi\)
0.414365 + 0.910111i \(0.364004\pi\)
\(860\) 0 0
\(861\) 4.46841 0.152283
\(862\) 0 0
\(863\) 23.5728 0.802427 0.401214 0.915985i \(-0.368588\pi\)
0.401214 + 0.915985i \(0.368588\pi\)
\(864\) 0 0
\(865\) −36.2624 −1.23296
\(866\) 0 0
\(867\) 21.7379 0.738257
\(868\) 0 0
\(869\) −41.1163 −1.39477
\(870\) 0 0
\(871\) −0.722781 −0.0244905
\(872\) 0 0
\(873\) −5.51908 −0.186793
\(874\) 0 0
\(875\) 5.98217 0.202234
\(876\) 0 0
\(877\) −31.4382 −1.06159 −0.530796 0.847500i \(-0.678107\pi\)
−0.530796 + 0.847500i \(0.678107\pi\)
\(878\) 0 0
\(879\) 1.00527 0.0339068
\(880\) 0 0
\(881\) −41.7600 −1.40693 −0.703466 0.710729i \(-0.748365\pi\)
−0.703466 + 0.710729i \(0.748365\pi\)
\(882\) 0 0
\(883\) −0.543040 −0.0182748 −0.00913738 0.999958i \(-0.502909\pi\)
−0.00913738 + 0.999958i \(0.502909\pi\)
\(884\) 0 0
\(885\) −16.6548 −0.559844
\(886\) 0 0
\(887\) −11.3248 −0.380250 −0.190125 0.981760i \(-0.560889\pi\)
−0.190125 + 0.981760i \(0.560889\pi\)
\(888\) 0 0
\(889\) −6.23015 −0.208952
\(890\) 0 0
\(891\) −12.5766 −0.421332
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 32.5160 1.08689
\(896\) 0 0
\(897\) −57.8442 −1.93136
\(898\) 0 0
\(899\) 0.778031 0.0259488
\(900\) 0 0
\(901\) 2.65356 0.0884029
\(902\) 0 0
\(903\) −3.20170 −0.106546
\(904\) 0 0
\(905\) 53.6905 1.78473
\(906\) 0 0
\(907\) −10.9258 −0.362787 −0.181393 0.983411i \(-0.558061\pi\)
−0.181393 + 0.983411i \(0.558061\pi\)
\(908\) 0 0
\(909\) −0.574610 −0.0190586
\(910\) 0 0
\(911\) −47.4159 −1.57096 −0.785479 0.618888i \(-0.787584\pi\)
−0.785479 + 0.618888i \(0.787584\pi\)
\(912\) 0 0
\(913\) 13.2527 0.438600
\(914\) 0 0
\(915\) −19.3599 −0.640018
\(916\) 0 0
\(917\) −7.00929 −0.231467
\(918\) 0 0
\(919\) −5.11069 −0.168586 −0.0842930 0.996441i \(-0.526863\pi\)
−0.0842930 + 0.996441i \(0.526863\pi\)
\(920\) 0 0
\(921\) −36.1709 −1.19187
\(922\) 0 0
\(923\) −56.1011 −1.84659
\(924\) 0 0
\(925\) 12.5462 0.412515
\(926\) 0 0
\(927\) −6.12902 −0.201303
\(928\) 0 0
\(929\) −42.8442 −1.40567 −0.702837 0.711351i \(-0.748084\pi\)
−0.702837 + 0.711351i \(0.748084\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 4.96717 0.162618
\(934\) 0 0
\(935\) 3.97471 0.129987
\(936\) 0 0
\(937\) −3.67885 −0.120183 −0.0600913 0.998193i \(-0.519139\pi\)
−0.0600913 + 0.998193i \(0.519139\pi\)
\(938\) 0 0
\(939\) −39.0203 −1.27338
\(940\) 0 0
\(941\) 18.3795 0.599154 0.299577 0.954072i \(-0.403154\pi\)
0.299577 + 0.954072i \(0.403154\pi\)
\(942\) 0 0
\(943\) −72.4286 −2.35860
\(944\) 0 0
\(945\) 9.13632 0.297204
\(946\) 0 0
\(947\) −46.7590 −1.51946 −0.759732 0.650237i \(-0.774670\pi\)
−0.759732 + 0.650237i \(0.774670\pi\)
\(948\) 0 0
\(949\) 6.95697 0.225833
\(950\) 0 0
\(951\) −6.84162 −0.221855
\(952\) 0 0
\(953\) 25.0551 0.811613 0.405807 0.913959i \(-0.366991\pi\)
0.405807 + 0.913959i \(0.366991\pi\)
\(954\) 0 0
\(955\) −47.0471 −1.52241
\(956\) 0 0
\(957\) 1.15690 0.0373974
\(958\) 0 0
\(959\) 0.976709 0.0315396
\(960\) 0 0
\(961\) −18.8906 −0.609375
\(962\) 0 0
\(963\) −4.28516 −0.138087
\(964\) 0 0
\(965\) 64.5940 2.07935
\(966\) 0 0
\(967\) −7.10627 −0.228522 −0.114261 0.993451i \(-0.536450\pi\)
−0.114261 + 0.993451i \(0.536450\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −60.7543 −1.94970 −0.974848 0.222869i \(-0.928458\pi\)
−0.974848 + 0.222869i \(0.928458\pi\)
\(972\) 0 0
\(973\) 7.55039 0.242054
\(974\) 0 0
\(975\) −54.3835 −1.74167
\(976\) 0 0
\(977\) −38.5412 −1.23304 −0.616520 0.787339i \(-0.711458\pi\)
−0.616520 + 0.787339i \(0.711458\pi\)
\(978\) 0 0
\(979\) −24.0199 −0.767678
\(980\) 0 0
\(981\) 9.16225 0.292528
\(982\) 0 0
\(983\) 7.50529 0.239381 0.119691 0.992811i \(-0.461810\pi\)
0.119691 + 0.992811i \(0.461810\pi\)
\(984\) 0 0
\(985\) −90.5868 −2.88633
\(986\) 0 0
\(987\) 1.24920 0.0397624
\(988\) 0 0
\(989\) 51.8964 1.65021
\(990\) 0 0
\(991\) 21.4221 0.680496 0.340248 0.940336i \(-0.389489\pi\)
0.340248 + 0.940336i \(0.389489\pi\)
\(992\) 0 0
\(993\) −26.3853 −0.837312
\(994\) 0 0
\(995\) −60.8541 −1.92920
\(996\) 0 0
\(997\) −41.0005 −1.29850 −0.649250 0.760575i \(-0.724917\pi\)
−0.649250 + 0.760575i \(0.724917\pi\)
\(998\) 0 0
\(999\) 8.09602 0.256147
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 5776.2.a.bt.1.2 4
4.3 odd 2 722.2.a.n.1.3 yes 4
12.11 even 2 6498.2.a.bx.1.1 4
19.18 odd 2 5776.2.a.bv.1.3 4
76.3 even 18 722.2.e.s.389.2 24
76.7 odd 6 722.2.c.m.429.2 8
76.11 odd 6 722.2.c.m.653.2 8
76.15 even 18 722.2.e.s.415.3 24
76.23 odd 18 722.2.e.r.415.2 24
76.27 even 6 722.2.c.n.653.3 8
76.31 even 6 722.2.c.n.429.3 8
76.35 odd 18 722.2.e.r.389.3 24
76.43 odd 18 722.2.e.r.595.2 24
76.47 odd 18 722.2.e.r.423.3 24
76.51 even 18 722.2.e.s.245.2 24
76.55 odd 18 722.2.e.r.99.3 24
76.59 even 18 722.2.e.s.99.2 24
76.63 odd 18 722.2.e.r.245.3 24
76.67 even 18 722.2.e.s.423.2 24
76.71 even 18 722.2.e.s.595.3 24
76.75 even 2 722.2.a.m.1.2 4
228.227 odd 2 6498.2.a.ca.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
722.2.a.m.1.2 4 76.75 even 2
722.2.a.n.1.3 yes 4 4.3 odd 2
722.2.c.m.429.2 8 76.7 odd 6
722.2.c.m.653.2 8 76.11 odd 6
722.2.c.n.429.3 8 76.31 even 6
722.2.c.n.653.3 8 76.27 even 6
722.2.e.r.99.3 24 76.55 odd 18
722.2.e.r.245.3 24 76.63 odd 18
722.2.e.r.389.3 24 76.35 odd 18
722.2.e.r.415.2 24 76.23 odd 18
722.2.e.r.423.3 24 76.47 odd 18
722.2.e.r.595.2 24 76.43 odd 18
722.2.e.s.99.2 24 76.59 even 18
722.2.e.s.245.2 24 76.51 even 18
722.2.e.s.389.2 24 76.3 even 18
722.2.e.s.415.3 24 76.15 even 18
722.2.e.s.423.2 24 76.67 even 18
722.2.e.s.595.3 24 76.71 even 18
5776.2.a.bt.1.2 4 1.1 even 1 trivial
5776.2.a.bv.1.3 4 19.18 odd 2
6498.2.a.bx.1.1 4 12.11 even 2
6498.2.a.ca.1.1 4 228.227 odd 2