Properties

Label 4012.2.b.b.237.22
Level $4012$
Weight $2$
Character 4012.237
Analytic conductor $32.036$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4012,2,Mod(237,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4012.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.22
Character \(\chi\) \(=\) 4012.237
Dual form 4012.2.b.b.237.25

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.257929i q^{3} +0.743469i q^{5} -3.88975i q^{7} +2.93347 q^{9} +O(q^{10})\) \(q-0.257929i q^{3} +0.743469i q^{5} -3.88975i q^{7} +2.93347 q^{9} -1.06863i q^{11} +5.38084 q^{13} +0.191762 q^{15} +(-1.52521 - 3.83063i) q^{17} +3.07847 q^{19} -1.00328 q^{21} -1.73219i q^{23} +4.44725 q^{25} -1.53041i q^{27} +4.99148i q^{29} +1.18871i q^{31} -0.275629 q^{33} +2.89191 q^{35} +0.759964i q^{37} -1.38787i q^{39} -3.00455i q^{41} -4.92247 q^{43} +2.18095i q^{45} +5.80648 q^{47} -8.13019 q^{49} +(-0.988029 + 0.393396i) q^{51} -0.194957 q^{53} +0.794491 q^{55} -0.794024i q^{57} +1.00000 q^{59} -4.03999i q^{61} -11.4105i q^{63} +4.00049i q^{65} -7.58431 q^{67} -0.446782 q^{69} +6.78151i q^{71} +13.3859i q^{73} -1.14707i q^{75} -4.15670 q^{77} -15.2500i q^{79} +8.40568 q^{81} -8.30461 q^{83} +(2.84796 - 1.13395i) q^{85} +1.28744 q^{87} +8.68467 q^{89} -20.9302i q^{91} +0.306603 q^{93} +2.28875i q^{95} +12.8560i q^{97} -3.13479i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.257929i 0.148915i −0.997224 0.0744575i \(-0.976277\pi\)
0.997224 0.0744575i \(-0.0237225\pi\)
\(4\) 0 0
\(5\) 0.743469i 0.332490i 0.986085 + 0.166245i \(0.0531642\pi\)
−0.986085 + 0.166245i \(0.946836\pi\)
\(6\) 0 0
\(7\) 3.88975i 1.47019i −0.677965 0.735094i \(-0.737138\pi\)
0.677965 0.735094i \(-0.262862\pi\)
\(8\) 0 0
\(9\) 2.93347 0.977824
\(10\) 0 0
\(11\) 1.06863i 0.322203i −0.986938 0.161102i \(-0.948495\pi\)
0.986938 0.161102i \(-0.0515047\pi\)
\(12\) 0 0
\(13\) 5.38084 1.49238 0.746188 0.665735i \(-0.231882\pi\)
0.746188 + 0.665735i \(0.231882\pi\)
\(14\) 0 0
\(15\) 0.191762 0.0495127
\(16\) 0 0
\(17\) −1.52521 3.83063i −0.369918 0.929064i
\(18\) 0 0
\(19\) 3.07847 0.706249 0.353124 0.935576i \(-0.385119\pi\)
0.353124 + 0.935576i \(0.385119\pi\)
\(20\) 0 0
\(21\) −1.00328 −0.218933
\(22\) 0 0
\(23\) 1.73219i 0.361188i −0.983558 0.180594i \(-0.942198\pi\)
0.983558 0.180594i \(-0.0578019\pi\)
\(24\) 0 0
\(25\) 4.44725 0.889451
\(26\) 0 0
\(27\) 1.53041i 0.294528i
\(28\) 0 0
\(29\) 4.99148i 0.926894i 0.886125 + 0.463447i \(0.153388\pi\)
−0.886125 + 0.463447i \(0.846612\pi\)
\(30\) 0 0
\(31\) 1.18871i 0.213499i 0.994286 + 0.106750i \(0.0340443\pi\)
−0.994286 + 0.106750i \(0.965956\pi\)
\(32\) 0 0
\(33\) −0.275629 −0.0479809
\(34\) 0 0
\(35\) 2.89191 0.488823
\(36\) 0 0
\(37\) 0.759964i 0.124937i 0.998047 + 0.0624687i \(0.0198974\pi\)
−0.998047 + 0.0624687i \(0.980103\pi\)
\(38\) 0 0
\(39\) 1.38787i 0.222237i
\(40\) 0 0
\(41\) 3.00455i 0.469232i −0.972088 0.234616i \(-0.924617\pi\)
0.972088 0.234616i \(-0.0753832\pi\)
\(42\) 0 0
\(43\) −4.92247 −0.750670 −0.375335 0.926889i \(-0.622472\pi\)
−0.375335 + 0.926889i \(0.622472\pi\)
\(44\) 0 0
\(45\) 2.18095i 0.325116i
\(46\) 0 0
\(47\) 5.80648 0.846962 0.423481 0.905905i \(-0.360808\pi\)
0.423481 + 0.905905i \(0.360808\pi\)
\(48\) 0 0
\(49\) −8.13019 −1.16146
\(50\) 0 0
\(51\) −0.988029 + 0.393396i −0.138352 + 0.0550864i
\(52\) 0 0
\(53\) −0.194957 −0.0267794 −0.0133897 0.999910i \(-0.504262\pi\)
−0.0133897 + 0.999910i \(0.504262\pi\)
\(54\) 0 0
\(55\) 0.794491 0.107129
\(56\) 0 0
\(57\) 0.794024i 0.105171i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 4.03999i 0.517267i −0.965975 0.258634i \(-0.916728\pi\)
0.965975 0.258634i \(-0.0832722\pi\)
\(62\) 0 0
\(63\) 11.4105i 1.43759i
\(64\) 0 0
\(65\) 4.00049i 0.496200i
\(66\) 0 0
\(67\) −7.58431 −0.926571 −0.463285 0.886209i \(-0.653330\pi\)
−0.463285 + 0.886209i \(0.653330\pi\)
\(68\) 0 0
\(69\) −0.446782 −0.0537863
\(70\) 0 0
\(71\) 6.78151i 0.804817i 0.915460 + 0.402408i \(0.131827\pi\)
−0.915460 + 0.402408i \(0.868173\pi\)
\(72\) 0 0
\(73\) 13.3859i 1.56670i 0.621582 + 0.783349i \(0.286490\pi\)
−0.621582 + 0.783349i \(0.713510\pi\)
\(74\) 0 0
\(75\) 1.14707i 0.132453i
\(76\) 0 0
\(77\) −4.15670 −0.473699
\(78\) 0 0
\(79\) 15.2500i 1.71575i −0.513855 0.857877i \(-0.671783\pi\)
0.513855 0.857877i \(-0.328217\pi\)
\(80\) 0 0
\(81\) 8.40568 0.933965
\(82\) 0 0
\(83\) −8.30461 −0.911550 −0.455775 0.890095i \(-0.650638\pi\)
−0.455775 + 0.890095i \(0.650638\pi\)
\(84\) 0 0
\(85\) 2.84796 1.13395i 0.308904 0.122994i
\(86\) 0 0
\(87\) 1.28744 0.138029
\(88\) 0 0
\(89\) 8.68467 0.920573 0.460287 0.887770i \(-0.347747\pi\)
0.460287 + 0.887770i \(0.347747\pi\)
\(90\) 0 0
\(91\) 20.9302i 2.19408i
\(92\) 0 0
\(93\) 0.306603 0.0317933
\(94\) 0 0
\(95\) 2.28875i 0.234820i
\(96\) 0 0
\(97\) 12.8560i 1.30533i 0.757646 + 0.652666i \(0.226350\pi\)
−0.757646 + 0.652666i \(0.773650\pi\)
\(98\) 0 0
\(99\) 3.13479i 0.315058i
\(100\) 0 0
\(101\) −19.2882 −1.91924 −0.959622 0.281292i \(-0.909237\pi\)
−0.959622 + 0.281292i \(0.909237\pi\)
\(102\) 0 0
\(103\) −0.883111 −0.0870155 −0.0435077 0.999053i \(-0.513853\pi\)
−0.0435077 + 0.999053i \(0.513853\pi\)
\(104\) 0 0
\(105\) 0.745907i 0.0727931i
\(106\) 0 0
\(107\) 1.34714i 0.130233i −0.997878 0.0651164i \(-0.979258\pi\)
0.997878 0.0651164i \(-0.0207419\pi\)
\(108\) 0 0
\(109\) 6.27922i 0.601440i −0.953712 0.300720i \(-0.902773\pi\)
0.953712 0.300720i \(-0.0972270\pi\)
\(110\) 0 0
\(111\) 0.196016 0.0186051
\(112\) 0 0
\(113\) 0.746838i 0.0702566i 0.999383 + 0.0351283i \(0.0111840\pi\)
−0.999383 + 0.0351283i \(0.988816\pi\)
\(114\) 0 0
\(115\) 1.28783 0.120091
\(116\) 0 0
\(117\) 15.7846 1.45928
\(118\) 0 0
\(119\) −14.9002 + 5.93270i −1.36590 + 0.543850i
\(120\) 0 0
\(121\) 9.85804 0.896185
\(122\) 0 0
\(123\) −0.774959 −0.0698757
\(124\) 0 0
\(125\) 7.02374i 0.628223i
\(126\) 0 0
\(127\) 9.77455 0.867351 0.433676 0.901069i \(-0.357216\pi\)
0.433676 + 0.901069i \(0.357216\pi\)
\(128\) 0 0
\(129\) 1.26965i 0.111786i
\(130\) 0 0
\(131\) 0.740669i 0.0647126i −0.999476 0.0323563i \(-0.989699\pi\)
0.999476 0.0323563i \(-0.0103011\pi\)
\(132\) 0 0
\(133\) 11.9745i 1.03832i
\(134\) 0 0
\(135\) 1.13781 0.0979275
\(136\) 0 0
\(137\) 14.1136 1.20581 0.602904 0.797814i \(-0.294010\pi\)
0.602904 + 0.797814i \(0.294010\pi\)
\(138\) 0 0
\(139\) 17.8860i 1.51707i −0.651634 0.758533i \(-0.725916\pi\)
0.651634 0.758533i \(-0.274084\pi\)
\(140\) 0 0
\(141\) 1.49766i 0.126125i
\(142\) 0 0
\(143\) 5.75011i 0.480848i
\(144\) 0 0
\(145\) −3.71101 −0.308183
\(146\) 0 0
\(147\) 2.09701i 0.172958i
\(148\) 0 0
\(149\) −11.0717 −0.907026 −0.453513 0.891250i \(-0.649829\pi\)
−0.453513 + 0.891250i \(0.649829\pi\)
\(150\) 0 0
\(151\) 5.81005 0.472815 0.236407 0.971654i \(-0.424030\pi\)
0.236407 + 0.971654i \(0.424030\pi\)
\(152\) 0 0
\(153\) −4.47417 11.2371i −0.361715 0.908462i
\(154\) 0 0
\(155\) −0.883772 −0.0709863
\(156\) 0 0
\(157\) −1.65644 −0.132198 −0.0660990 0.997813i \(-0.521055\pi\)
−0.0660990 + 0.997813i \(0.521055\pi\)
\(158\) 0 0
\(159\) 0.0502850i 0.00398786i
\(160\) 0 0
\(161\) −6.73781 −0.531014
\(162\) 0 0
\(163\) 7.53886i 0.590489i −0.955422 0.295244i \(-0.904599\pi\)
0.955422 0.295244i \(-0.0954011\pi\)
\(164\) 0 0
\(165\) 0.204922i 0.0159532i
\(166\) 0 0
\(167\) 6.20100i 0.479848i 0.970792 + 0.239924i \(0.0771225\pi\)
−0.970792 + 0.239924i \(0.922877\pi\)
\(168\) 0 0
\(169\) 15.9535 1.22719
\(170\) 0 0
\(171\) 9.03060 0.690587
\(172\) 0 0
\(173\) 15.5964i 1.18577i −0.805287 0.592886i \(-0.797989\pi\)
0.805287 0.592886i \(-0.202011\pi\)
\(174\) 0 0
\(175\) 17.2987i 1.30766i
\(176\) 0 0
\(177\) 0.257929i 0.0193871i
\(178\) 0 0
\(179\) −14.7619 −1.10336 −0.551678 0.834057i \(-0.686012\pi\)
−0.551678 + 0.834057i \(0.686012\pi\)
\(180\) 0 0
\(181\) 7.06277i 0.524971i −0.964936 0.262486i \(-0.915458\pi\)
0.964936 0.262486i \(-0.0845423\pi\)
\(182\) 0 0
\(183\) −1.04203 −0.0770289
\(184\) 0 0
\(185\) −0.565010 −0.0415404
\(186\) 0 0
\(187\) −4.09351 + 1.62988i −0.299347 + 0.119189i
\(188\) 0 0
\(189\) −5.95293 −0.433012
\(190\) 0 0
\(191\) 1.18256 0.0855667 0.0427834 0.999084i \(-0.486377\pi\)
0.0427834 + 0.999084i \(0.486377\pi\)
\(192\) 0 0
\(193\) 13.0146i 0.936810i −0.883514 0.468405i \(-0.844829\pi\)
0.883514 0.468405i \(-0.155171\pi\)
\(194\) 0 0
\(195\) 1.03184 0.0738916
\(196\) 0 0
\(197\) 21.2246i 1.51219i 0.654463 + 0.756094i \(0.272895\pi\)
−0.654463 + 0.756094i \(0.727105\pi\)
\(198\) 0 0
\(199\) 4.69732i 0.332984i 0.986043 + 0.166492i \(0.0532440\pi\)
−0.986043 + 0.166492i \(0.946756\pi\)
\(200\) 0 0
\(201\) 1.95621i 0.137980i
\(202\) 0 0
\(203\) 19.4156 1.36271
\(204\) 0 0
\(205\) 2.23379 0.156015
\(206\) 0 0
\(207\) 5.08135i 0.353178i
\(208\) 0 0
\(209\) 3.28973i 0.227555i
\(210\) 0 0
\(211\) 16.0940i 1.10796i −0.832531 0.553979i \(-0.813109\pi\)
0.832531 0.553979i \(-0.186891\pi\)
\(212\) 0 0
\(213\) 1.74914 0.119849
\(214\) 0 0
\(215\) 3.65971i 0.249590i
\(216\) 0 0
\(217\) 4.62380 0.313884
\(218\) 0 0
\(219\) 3.45260 0.233305
\(220\) 0 0
\(221\) −8.20692 20.6120i −0.552057 1.38651i
\(222\) 0 0
\(223\) 11.2292 0.751961 0.375981 0.926628i \(-0.377306\pi\)
0.375981 + 0.926628i \(0.377306\pi\)
\(224\) 0 0
\(225\) 13.0459 0.869726
\(226\) 0 0
\(227\) 7.83388i 0.519953i −0.965615 0.259976i \(-0.916285\pi\)
0.965615 0.259976i \(-0.0837147\pi\)
\(228\) 0 0
\(229\) 13.0824 0.864507 0.432254 0.901752i \(-0.357719\pi\)
0.432254 + 0.901752i \(0.357719\pi\)
\(230\) 0 0
\(231\) 1.07213i 0.0705410i
\(232\) 0 0
\(233\) 8.91878i 0.584289i 0.956374 + 0.292144i \(0.0943687\pi\)
−0.956374 + 0.292144i \(0.905631\pi\)
\(234\) 0 0
\(235\) 4.31694i 0.281606i
\(236\) 0 0
\(237\) −3.93340 −0.255502
\(238\) 0 0
\(239\) −13.3607 −0.864235 −0.432117 0.901817i \(-0.642233\pi\)
−0.432117 + 0.901817i \(0.642233\pi\)
\(240\) 0 0
\(241\) 11.5118i 0.741537i −0.928725 0.370769i \(-0.879094\pi\)
0.928725 0.370769i \(-0.120906\pi\)
\(242\) 0 0
\(243\) 6.75930i 0.433609i
\(244\) 0 0
\(245\) 6.04455i 0.386172i
\(246\) 0 0
\(247\) 16.5647 1.05399
\(248\) 0 0
\(249\) 2.14200i 0.135744i
\(250\) 0 0
\(251\) −22.9699 −1.44985 −0.724924 0.688829i \(-0.758125\pi\)
−0.724924 + 0.688829i \(0.758125\pi\)
\(252\) 0 0
\(253\) −1.85107 −0.116376
\(254\) 0 0
\(255\) −0.292478 0.734569i −0.0183157 0.0460005i
\(256\) 0 0
\(257\) 27.8873 1.73956 0.869781 0.493438i \(-0.164260\pi\)
0.869781 + 0.493438i \(0.164260\pi\)
\(258\) 0 0
\(259\) 2.95607 0.183682
\(260\) 0 0
\(261\) 14.6424i 0.906340i
\(262\) 0 0
\(263\) 7.47876 0.461160 0.230580 0.973053i \(-0.425938\pi\)
0.230580 + 0.973053i \(0.425938\pi\)
\(264\) 0 0
\(265\) 0.144945i 0.00890389i
\(266\) 0 0
\(267\) 2.24002i 0.137087i
\(268\) 0 0
\(269\) 17.8671i 1.08937i −0.838639 0.544687i \(-0.816648\pi\)
0.838639 0.544687i \(-0.183352\pi\)
\(270\) 0 0
\(271\) −15.9077 −0.966322 −0.483161 0.875531i \(-0.660511\pi\)
−0.483161 + 0.875531i \(0.660511\pi\)
\(272\) 0 0
\(273\) −5.39848 −0.326731
\(274\) 0 0
\(275\) 4.75245i 0.286584i
\(276\) 0 0
\(277\) 25.4020i 1.52626i 0.646248 + 0.763128i \(0.276337\pi\)
−0.646248 + 0.763128i \(0.723663\pi\)
\(278\) 0 0
\(279\) 3.48706i 0.208765i
\(280\) 0 0
\(281\) −1.99936 −0.119272 −0.0596360 0.998220i \(-0.518994\pi\)
−0.0596360 + 0.998220i \(0.518994\pi\)
\(282\) 0 0
\(283\) 9.15354i 0.544121i 0.962280 + 0.272061i \(0.0877051\pi\)
−0.962280 + 0.272061i \(0.912295\pi\)
\(284\) 0 0
\(285\) 0.590333 0.0349683
\(286\) 0 0
\(287\) −11.6870 −0.689859
\(288\) 0 0
\(289\) −12.3475 + 11.6850i −0.726321 + 0.687356i
\(290\) 0 0
\(291\) 3.31594 0.194384
\(292\) 0 0
\(293\) 13.0247 0.760910 0.380455 0.924799i \(-0.375767\pi\)
0.380455 + 0.924799i \(0.375767\pi\)
\(294\) 0 0
\(295\) 0.743469i 0.0432865i
\(296\) 0 0
\(297\) −1.63544 −0.0948978
\(298\) 0 0
\(299\) 9.32066i 0.539028i
\(300\) 0 0
\(301\) 19.1472i 1.10363i
\(302\) 0 0
\(303\) 4.97497i 0.285804i
\(304\) 0 0
\(305\) 3.00361 0.171986
\(306\) 0 0
\(307\) 33.8115 1.92972 0.964861 0.262761i \(-0.0846330\pi\)
0.964861 + 0.262761i \(0.0846330\pi\)
\(308\) 0 0
\(309\) 0.227779i 0.0129579i
\(310\) 0 0
\(311\) 17.2284i 0.976934i 0.872582 + 0.488467i \(0.162444\pi\)
−0.872582 + 0.488467i \(0.837556\pi\)
\(312\) 0 0
\(313\) 13.1560i 0.743619i −0.928309 0.371809i \(-0.878738\pi\)
0.928309 0.371809i \(-0.121262\pi\)
\(314\) 0 0
\(315\) 8.48335 0.477983
\(316\) 0 0
\(317\) 14.3748i 0.807370i 0.914898 + 0.403685i \(0.132271\pi\)
−0.914898 + 0.403685i \(0.867729\pi\)
\(318\) 0 0
\(319\) 5.33403 0.298648
\(320\) 0 0
\(321\) −0.347465 −0.0193936
\(322\) 0 0
\(323\) −4.69531 11.7925i −0.261254 0.656150i
\(324\) 0 0
\(325\) 23.9300 1.32740
\(326\) 0 0
\(327\) −1.61959 −0.0895635
\(328\) 0 0
\(329\) 22.5858i 1.24519i
\(330\) 0 0
\(331\) 5.92450 0.325640 0.162820 0.986656i \(-0.447941\pi\)
0.162820 + 0.986656i \(0.447941\pi\)
\(332\) 0 0
\(333\) 2.22933i 0.122167i
\(334\) 0 0
\(335\) 5.63870i 0.308075i
\(336\) 0 0
\(337\) 7.80759i 0.425306i 0.977128 + 0.212653i \(0.0682104\pi\)
−0.977128 + 0.212653i \(0.931790\pi\)
\(338\) 0 0
\(339\) 0.192631 0.0104623
\(340\) 0 0
\(341\) 1.27029 0.0687901
\(342\) 0 0
\(343\) 4.39615i 0.237370i
\(344\) 0 0
\(345\) 0.332169i 0.0178834i
\(346\) 0 0
\(347\) 10.9539i 0.588036i −0.955800 0.294018i \(-0.905008\pi\)
0.955800 0.294018i \(-0.0949925\pi\)
\(348\) 0 0
\(349\) 14.4015 0.770896 0.385448 0.922729i \(-0.374047\pi\)
0.385448 + 0.922729i \(0.374047\pi\)
\(350\) 0 0
\(351\) 8.23490i 0.439547i
\(352\) 0 0
\(353\) 5.52022 0.293812 0.146906 0.989150i \(-0.453069\pi\)
0.146906 + 0.989150i \(0.453069\pi\)
\(354\) 0 0
\(355\) −5.04184 −0.267593
\(356\) 0 0
\(357\) 1.53021 + 3.84319i 0.0809874 + 0.203403i
\(358\) 0 0
\(359\) −1.41581 −0.0747236 −0.0373618 0.999302i \(-0.511895\pi\)
−0.0373618 + 0.999302i \(0.511895\pi\)
\(360\) 0 0
\(361\) −9.52305 −0.501213
\(362\) 0 0
\(363\) 2.54267i 0.133456i
\(364\) 0 0
\(365\) −9.95199 −0.520911
\(366\) 0 0
\(367\) 1.47678i 0.0770873i −0.999257 0.0385437i \(-0.987728\pi\)
0.999257 0.0385437i \(-0.0122719\pi\)
\(368\) 0 0
\(369\) 8.81376i 0.458826i
\(370\) 0 0
\(371\) 0.758336i 0.0393708i
\(372\) 0 0
\(373\) −37.0490 −1.91833 −0.959163 0.282854i \(-0.908719\pi\)
−0.959163 + 0.282854i \(0.908719\pi\)
\(374\) 0 0
\(375\) 1.81162 0.0935518
\(376\) 0 0
\(377\) 26.8584i 1.38328i
\(378\) 0 0
\(379\) 26.5120i 1.36183i 0.732363 + 0.680914i \(0.238417\pi\)
−0.732363 + 0.680914i \(0.761583\pi\)
\(380\) 0 0
\(381\) 2.52114i 0.129162i
\(382\) 0 0
\(383\) 19.3811 0.990327 0.495163 0.868800i \(-0.335108\pi\)
0.495163 + 0.868800i \(0.335108\pi\)
\(384\) 0 0
\(385\) 3.09038i 0.157500i
\(386\) 0 0
\(387\) −14.4399 −0.734023
\(388\) 0 0
\(389\) −20.7416 −1.05164 −0.525821 0.850595i \(-0.676242\pi\)
−0.525821 + 0.850595i \(0.676242\pi\)
\(390\) 0 0
\(391\) −6.63540 + 2.64196i −0.335566 + 0.133610i
\(392\) 0 0
\(393\) −0.191040 −0.00963668
\(394\) 0 0
\(395\) 11.3379 0.570470
\(396\) 0 0
\(397\) 0.228359i 0.0114610i 0.999984 + 0.00573050i \(0.00182408\pi\)
−0.999984 + 0.00573050i \(0.998176\pi\)
\(398\) 0 0
\(399\) −3.08856 −0.154621
\(400\) 0 0
\(401\) 15.9903i 0.798518i −0.916838 0.399259i \(-0.869268\pi\)
0.916838 0.399259i \(-0.130732\pi\)
\(402\) 0 0
\(403\) 6.39628i 0.318621i
\(404\) 0 0
\(405\) 6.24937i 0.310534i
\(406\) 0 0
\(407\) 0.812118 0.0402552
\(408\) 0 0
\(409\) −27.2029 −1.34510 −0.672549 0.740052i \(-0.734801\pi\)
−0.672549 + 0.740052i \(0.734801\pi\)
\(410\) 0 0
\(411\) 3.64030i 0.179563i
\(412\) 0 0
\(413\) 3.88975i 0.191402i
\(414\) 0 0
\(415\) 6.17423i 0.303081i
\(416\) 0 0
\(417\) −4.61330 −0.225914
\(418\) 0 0
\(419\) 26.6943i 1.30410i 0.758174 + 0.652052i \(0.226092\pi\)
−0.758174 + 0.652052i \(0.773908\pi\)
\(420\) 0 0
\(421\) 17.1321 0.834969 0.417484 0.908684i \(-0.362912\pi\)
0.417484 + 0.908684i \(0.362912\pi\)
\(422\) 0 0
\(423\) 17.0332 0.828180
\(424\) 0 0
\(425\) −6.78300 17.0358i −0.329024 0.826357i
\(426\) 0 0
\(427\) −15.7146 −0.760480
\(428\) 0 0
\(429\) −1.48312 −0.0716056
\(430\) 0 0
\(431\) 15.3628i 0.740002i −0.929031 0.370001i \(-0.879357\pi\)
0.929031 0.370001i \(-0.120643\pi\)
\(432\) 0 0
\(433\) −12.9398 −0.621848 −0.310924 0.950435i \(-0.600638\pi\)
−0.310924 + 0.950435i \(0.600638\pi\)
\(434\) 0 0
\(435\) 0.957176i 0.0458931i
\(436\) 0 0
\(437\) 5.33250i 0.255088i
\(438\) 0 0
\(439\) 2.17164i 0.103647i −0.998656 0.0518234i \(-0.983497\pi\)
0.998656 0.0518234i \(-0.0165033\pi\)
\(440\) 0 0
\(441\) −23.8497 −1.13570
\(442\) 0 0
\(443\) −32.7890 −1.55785 −0.778926 0.627115i \(-0.784235\pi\)
−0.778926 + 0.627115i \(0.784235\pi\)
\(444\) 0 0
\(445\) 6.45679i 0.306081i
\(446\) 0 0
\(447\) 2.85570i 0.135070i
\(448\) 0 0
\(449\) 40.2945i 1.90162i −0.309779 0.950809i \(-0.600255\pi\)
0.309779 0.950809i \(-0.399745\pi\)
\(450\) 0 0
\(451\) −3.21074 −0.151188
\(452\) 0 0
\(453\) 1.49858i 0.0704093i
\(454\) 0 0
\(455\) 15.5609 0.729507
\(456\) 0 0
\(457\) −18.0429 −0.844010 −0.422005 0.906594i \(-0.638673\pi\)
−0.422005 + 0.906594i \(0.638673\pi\)
\(458\) 0 0
\(459\) −5.86244 + 2.33420i −0.273635 + 0.108951i
\(460\) 0 0
\(461\) −16.2563 −0.757132 −0.378566 0.925574i \(-0.623583\pi\)
−0.378566 + 0.925574i \(0.623583\pi\)
\(462\) 0 0
\(463\) 29.4363 1.36802 0.684010 0.729472i \(-0.260234\pi\)
0.684010 + 0.729472i \(0.260234\pi\)
\(464\) 0 0
\(465\) 0.227950i 0.0105709i
\(466\) 0 0
\(467\) 3.28403 0.151967 0.0759834 0.997109i \(-0.475790\pi\)
0.0759834 + 0.997109i \(0.475790\pi\)
\(468\) 0 0
\(469\) 29.5011i 1.36223i
\(470\) 0 0
\(471\) 0.427242i 0.0196863i
\(472\) 0 0
\(473\) 5.26028i 0.241868i
\(474\) 0 0
\(475\) 13.6907 0.628173
\(476\) 0 0
\(477\) −0.571902 −0.0261856
\(478\) 0 0
\(479\) 5.52191i 0.252302i 0.992011 + 0.126151i \(0.0402624\pi\)
−0.992011 + 0.126151i \(0.959738\pi\)
\(480\) 0 0
\(481\) 4.08925i 0.186454i
\(482\) 0 0
\(483\) 1.73787i 0.0790760i
\(484\) 0 0
\(485\) −9.55807 −0.434009
\(486\) 0 0
\(487\) 0.694991i 0.0314930i −0.999876 0.0157465i \(-0.994988\pi\)
0.999876 0.0157465i \(-0.00501248\pi\)
\(488\) 0 0
\(489\) −1.94449 −0.0879327
\(490\) 0 0
\(491\) 9.58015 0.432346 0.216173 0.976355i \(-0.430642\pi\)
0.216173 + 0.976355i \(0.430642\pi\)
\(492\) 0 0
\(493\) 19.1205 7.61306i 0.861144 0.342875i
\(494\) 0 0
\(495\) 2.33062 0.104754
\(496\) 0 0
\(497\) 26.3784 1.18323
\(498\) 0 0
\(499\) 29.3534i 1.31404i −0.753873 0.657020i \(-0.771817\pi\)
0.753873 0.657020i \(-0.228183\pi\)
\(500\) 0 0
\(501\) 1.59942 0.0714566
\(502\) 0 0
\(503\) 26.4103i 1.17758i 0.808286 + 0.588790i \(0.200395\pi\)
−0.808286 + 0.588790i \(0.799605\pi\)
\(504\) 0 0
\(505\) 14.3402i 0.638129i
\(506\) 0 0
\(507\) 4.11485i 0.182747i
\(508\) 0 0
\(509\) −12.5823 −0.557702 −0.278851 0.960334i \(-0.589954\pi\)
−0.278851 + 0.960334i \(0.589954\pi\)
\(510\) 0 0
\(511\) 52.0678 2.30334
\(512\) 0 0
\(513\) 4.71132i 0.208010i
\(514\) 0 0
\(515\) 0.656566i 0.0289317i
\(516\) 0 0
\(517\) 6.20496i 0.272894i
\(518\) 0 0
\(519\) −4.02275 −0.176579
\(520\) 0 0
\(521\) 2.80320i 0.122810i 0.998113 + 0.0614052i \(0.0195582\pi\)
−0.998113 + 0.0614052i \(0.980442\pi\)
\(522\) 0 0
\(523\) −24.7693 −1.08308 −0.541542 0.840674i \(-0.682159\pi\)
−0.541542 + 0.840674i \(0.682159\pi\)
\(524\) 0 0
\(525\) −4.46183 −0.194730
\(526\) 0 0
\(527\) 4.55352 1.81304i 0.198355 0.0789773i
\(528\) 0 0
\(529\) 19.9995 0.869544
\(530\) 0 0
\(531\) 2.93347 0.127302
\(532\) 0 0
\(533\) 16.1670i 0.700271i
\(534\) 0 0
\(535\) 1.00156 0.0433010
\(536\) 0 0
\(537\) 3.80751i 0.164306i
\(538\) 0 0
\(539\) 8.68814i 0.374224i
\(540\) 0 0
\(541\) 5.33667i 0.229441i 0.993398 + 0.114721i \(0.0365973\pi\)
−0.993398 + 0.114721i \(0.963403\pi\)
\(542\) 0 0
\(543\) −1.82169 −0.0781762
\(544\) 0 0
\(545\) 4.66841 0.199973
\(546\) 0 0
\(547\) 44.3628i 1.89682i 0.317051 + 0.948409i \(0.397307\pi\)
−0.317051 + 0.948409i \(0.602693\pi\)
\(548\) 0 0
\(549\) 11.8512i 0.505796i
\(550\) 0 0
\(551\) 15.3661i 0.654618i
\(552\) 0 0
\(553\) −59.3186 −2.52248
\(554\) 0 0
\(555\) 0.145732i 0.00618599i
\(556\) 0 0
\(557\) −26.1500 −1.10801 −0.554005 0.832513i \(-0.686901\pi\)
−0.554005 + 0.832513i \(0.686901\pi\)
\(558\) 0 0
\(559\) −26.4870 −1.12028
\(560\) 0 0
\(561\) 0.420393 + 1.05583i 0.0177490 + 0.0445773i
\(562\) 0 0
\(563\) −24.2793 −1.02325 −0.511626 0.859208i \(-0.670957\pi\)
−0.511626 + 0.859208i \(0.670957\pi\)
\(564\) 0 0
\(565\) −0.555251 −0.0233596
\(566\) 0 0
\(567\) 32.6960i 1.37310i
\(568\) 0 0
\(569\) −40.3777 −1.69272 −0.846361 0.532609i \(-0.821212\pi\)
−0.846361 + 0.532609i \(0.821212\pi\)
\(570\) 0 0
\(571\) 33.5360i 1.40344i −0.712455 0.701718i \(-0.752417\pi\)
0.712455 0.701718i \(-0.247583\pi\)
\(572\) 0 0
\(573\) 0.305015i 0.0127422i
\(574\) 0 0
\(575\) 7.70351i 0.321258i
\(576\) 0 0
\(577\) 11.4051 0.474799 0.237399 0.971412i \(-0.423705\pi\)
0.237399 + 0.971412i \(0.423705\pi\)
\(578\) 0 0
\(579\) −3.35683 −0.139505
\(580\) 0 0
\(581\) 32.3029i 1.34015i
\(582\) 0 0
\(583\) 0.208337i 0.00862842i
\(584\) 0 0
\(585\) 11.7353i 0.485196i
\(586\) 0 0
\(587\) 3.59098 0.148216 0.0741079 0.997250i \(-0.476389\pi\)
0.0741079 + 0.997250i \(0.476389\pi\)
\(588\) 0 0
\(589\) 3.65941i 0.150784i
\(590\) 0 0
\(591\) 5.47442 0.225188
\(592\) 0 0
\(593\) −10.1466 −0.416671 −0.208335 0.978057i \(-0.566805\pi\)
−0.208335 + 0.978057i \(0.566805\pi\)
\(594\) 0 0
\(595\) −4.41078 11.0778i −0.180824 0.454148i
\(596\) 0 0
\(597\) 1.21157 0.0495863
\(598\) 0 0
\(599\) −22.7784 −0.930699 −0.465349 0.885127i \(-0.654071\pi\)
−0.465349 + 0.885127i \(0.654071\pi\)
\(600\) 0 0
\(601\) 20.7652i 0.847031i 0.905889 + 0.423516i \(0.139204\pi\)
−0.905889 + 0.423516i \(0.860796\pi\)
\(602\) 0 0
\(603\) −22.2484 −0.906024
\(604\) 0 0
\(605\) 7.32915i 0.297972i
\(606\) 0 0
\(607\) 5.09973i 0.206992i 0.994630 + 0.103496i \(0.0330029\pi\)
−0.994630 + 0.103496i \(0.966997\pi\)
\(608\) 0 0
\(609\) 5.00784i 0.202928i
\(610\) 0 0
\(611\) 31.2438 1.26399
\(612\) 0 0
\(613\) 18.4798 0.746394 0.373197 0.927752i \(-0.378262\pi\)
0.373197 + 0.927752i \(0.378262\pi\)
\(614\) 0 0
\(615\) 0.576158i 0.0232329i
\(616\) 0 0
\(617\) 34.6400i 1.39456i 0.716801 + 0.697278i \(0.245606\pi\)
−0.716801 + 0.697278i \(0.754394\pi\)
\(618\) 0 0
\(619\) 15.2662i 0.613600i 0.951774 + 0.306800i \(0.0992583\pi\)
−0.951774 + 0.306800i \(0.900742\pi\)
\(620\) 0 0
\(621\) −2.65097 −0.106380
\(622\) 0 0
\(623\) 33.7812i 1.35342i
\(624\) 0 0
\(625\) 17.0143 0.680573
\(626\) 0 0
\(627\) −0.848515 −0.0338864
\(628\) 0 0
\(629\) 2.91114 1.15911i 0.116075 0.0462166i
\(630\) 0 0
\(631\) 19.4308 0.773527 0.386764 0.922179i \(-0.373593\pi\)
0.386764 + 0.922179i \(0.373593\pi\)
\(632\) 0 0
\(633\) −4.15111 −0.164992
\(634\) 0 0
\(635\) 7.26708i 0.288385i
\(636\) 0 0
\(637\) −43.7472 −1.73333
\(638\) 0 0
\(639\) 19.8934i 0.786969i
\(640\) 0 0
\(641\) 5.88335i 0.232378i 0.993227 + 0.116189i \(0.0370679\pi\)
−0.993227 + 0.116189i \(0.962932\pi\)
\(642\) 0 0
\(643\) 35.3797i 1.39524i −0.716468 0.697620i \(-0.754242\pi\)
0.716468 0.697620i \(-0.245758\pi\)
\(644\) 0 0
\(645\) −0.943943 −0.0371677
\(646\) 0 0
\(647\) 26.7400 1.05126 0.525629 0.850714i \(-0.323830\pi\)
0.525629 + 0.850714i \(0.323830\pi\)
\(648\) 0 0
\(649\) 1.06863i 0.0419473i
\(650\) 0 0
\(651\) 1.19261i 0.0467421i
\(652\) 0 0
\(653\) 17.6616i 0.691151i 0.938391 + 0.345576i \(0.112316\pi\)
−0.938391 + 0.345576i \(0.887684\pi\)
\(654\) 0 0
\(655\) 0.550665 0.0215163
\(656\) 0 0
\(657\) 39.2671i 1.53196i
\(658\) 0 0
\(659\) 29.6308 1.15425 0.577126 0.816655i \(-0.304174\pi\)
0.577126 + 0.816655i \(0.304174\pi\)
\(660\) 0 0
\(661\) −14.8551 −0.577797 −0.288899 0.957360i \(-0.593289\pi\)
−0.288899 + 0.957360i \(0.593289\pi\)
\(662\) 0 0
\(663\) −5.31643 + 2.11680i −0.206473 + 0.0822097i
\(664\) 0 0
\(665\) 8.90266 0.345230
\(666\) 0 0
\(667\) 8.64621 0.334783
\(668\) 0 0
\(669\) 2.89632i 0.111978i
\(670\) 0 0
\(671\) −4.31724 −0.166665
\(672\) 0 0
\(673\) 28.9569i 1.11621i 0.829772 + 0.558103i \(0.188471\pi\)
−0.829772 + 0.558103i \(0.811529\pi\)
\(674\) 0 0
\(675\) 6.80613i 0.261968i
\(676\) 0 0
\(677\) 23.8890i 0.918127i 0.888403 + 0.459064i \(0.151815\pi\)
−0.888403 + 0.459064i \(0.848185\pi\)
\(678\) 0 0
\(679\) 50.0068 1.91908
\(680\) 0 0
\(681\) −2.02058 −0.0774288
\(682\) 0 0
\(683\) 44.9030i 1.71817i −0.511835 0.859084i \(-0.671034\pi\)
0.511835 0.859084i \(-0.328966\pi\)
\(684\) 0 0
\(685\) 10.4930i 0.400918i
\(686\) 0 0
\(687\) 3.37432i 0.128738i
\(688\) 0 0
\(689\) −1.04903 −0.0399650
\(690\) 0 0
\(691\) 16.1759i 0.615362i 0.951490 + 0.307681i \(0.0995529\pi\)
−0.951490 + 0.307681i \(0.900447\pi\)
\(692\) 0 0
\(693\) −12.1936 −0.463195
\(694\) 0 0
\(695\) 13.2977 0.504409
\(696\) 0 0
\(697\) −11.5093 + 4.58257i −0.435946 + 0.173577i
\(698\) 0 0
\(699\) 2.30041 0.0870094
\(700\) 0 0
\(701\) −47.0965 −1.77881 −0.889405 0.457121i \(-0.848881\pi\)
−0.889405 + 0.457121i \(0.848881\pi\)
\(702\) 0 0
\(703\) 2.33952i 0.0882368i
\(704\) 0 0
\(705\) 1.11346 0.0419354
\(706\) 0 0
\(707\) 75.0262i 2.82165i
\(708\) 0 0
\(709\) 26.5184i 0.995920i 0.867200 + 0.497960i \(0.165917\pi\)
−0.867200 + 0.497960i \(0.834083\pi\)
\(710\) 0 0
\(711\) 44.7353i 1.67771i
\(712\) 0 0
\(713\) 2.05908 0.0771133
\(714\) 0 0
\(715\) 4.27503 0.159877
\(716\) 0 0
\(717\) 3.44612i 0.128698i
\(718\) 0 0
\(719\) 2.10780i 0.0786077i −0.999227 0.0393039i \(-0.987486\pi\)
0.999227 0.0393039i \(-0.0125140\pi\)
\(720\) 0 0
\(721\) 3.43508i 0.127929i
\(722\) 0 0
\(723\) −2.96921 −0.110426
\(724\) 0 0
\(725\) 22.1984i 0.824427i
\(726\) 0 0
\(727\) −21.9859 −0.815411 −0.407706 0.913113i \(-0.633671\pi\)
−0.407706 + 0.913113i \(0.633671\pi\)
\(728\) 0 0
\(729\) 23.4736 0.869394
\(730\) 0 0
\(731\) 7.50781 + 18.8562i 0.277686 + 0.697420i
\(732\) 0 0
\(733\) 17.7560 0.655834 0.327917 0.944706i \(-0.393653\pi\)
0.327917 + 0.944706i \(0.393653\pi\)
\(734\) 0 0
\(735\) −1.55906 −0.0575068
\(736\) 0 0
\(737\) 8.10480i 0.298544i
\(738\) 0 0
\(739\) 7.86714 0.289397 0.144699 0.989476i \(-0.453779\pi\)
0.144699 + 0.989476i \(0.453779\pi\)
\(740\) 0 0
\(741\) 4.27252i 0.156955i
\(742\) 0 0
\(743\) 30.6420i 1.12415i 0.827087 + 0.562074i \(0.189996\pi\)
−0.827087 + 0.562074i \(0.810004\pi\)
\(744\) 0 0
\(745\) 8.23145i 0.301577i
\(746\) 0 0
\(747\) −24.3614 −0.891335
\(748\) 0 0
\(749\) −5.24004 −0.191467
\(750\) 0 0
\(751\) 10.9536i 0.399701i −0.979826 0.199850i \(-0.935954\pi\)
0.979826 0.199850i \(-0.0640456\pi\)
\(752\) 0 0
\(753\) 5.92460i 0.215904i
\(754\) 0 0
\(755\) 4.31959i 0.157206i
\(756\) 0 0
\(757\) 38.9967 1.41736 0.708680 0.705531i \(-0.249291\pi\)
0.708680 + 0.705531i \(0.249291\pi\)
\(758\) 0 0
\(759\) 0.477444i 0.0173301i
\(760\) 0 0
\(761\) −17.4793 −0.633626 −0.316813 0.948488i \(-0.602613\pi\)
−0.316813 + 0.948488i \(0.602613\pi\)
\(762\) 0 0
\(763\) −24.4246 −0.884230
\(764\) 0 0
\(765\) 8.35440 3.32641i 0.302054 0.120266i
\(766\) 0 0
\(767\) 5.38084 0.194291
\(768\) 0 0
\(769\) 16.0463 0.578643 0.289322 0.957232i \(-0.406570\pi\)
0.289322 + 0.957232i \(0.406570\pi\)
\(770\) 0 0
\(771\) 7.19293i 0.259047i
\(772\) 0 0
\(773\) −11.0208 −0.396390 −0.198195 0.980163i \(-0.563508\pi\)
−0.198195 + 0.980163i \(0.563508\pi\)
\(774\) 0 0
\(775\) 5.28651i 0.189897i
\(776\) 0 0
\(777\) 0.762456i 0.0273530i
\(778\) 0 0
\(779\) 9.24940i 0.331394i
\(780\) 0 0
\(781\) 7.24690 0.259314
\(782\) 0 0
\(783\) 7.63902 0.272996
\(784\) 0 0
\(785\) 1.23151i 0.0439545i
\(786\) 0 0
\(787\) 6.19725i 0.220908i −0.993881 0.110454i \(-0.964769\pi\)
0.993881 0.110454i \(-0.0352305\pi\)
\(788\) 0 0
\(789\) 1.92899i 0.0686737i
\(790\) 0 0
\(791\) 2.90502 0.103290
\(792\) 0 0
\(793\) 21.7385i 0.771958i
\(794\) 0 0
\(795\) −0.0373854 −0.00132592
\(796\) 0 0
\(797\) 5.55995 0.196944 0.0984718 0.995140i \(-0.468605\pi\)
0.0984718 + 0.995140i \(0.468605\pi\)
\(798\) 0 0
\(799\) −8.85612 22.2425i −0.313307 0.786883i
\(800\) 0 0
\(801\) 25.4762 0.900159
\(802\) 0 0
\(803\) 14.3045 0.504795
\(804\) 0 0
\(805\) 5.00936i 0.176557i
\(806\) 0 0
\(807\) −4.60842 −0.162224
\(808\) 0 0
\(809\) 20.1846i 0.709653i −0.934932 0.354826i \(-0.884540\pi\)
0.934932 0.354826i \(-0.115460\pi\)
\(810\) 0 0
\(811\) 10.3147i 0.362200i 0.983465 + 0.181100i \(0.0579657\pi\)
−0.983465 + 0.181100i \(0.942034\pi\)
\(812\) 0 0
\(813\) 4.10304i 0.143900i
\(814\) 0 0
\(815\) 5.60491 0.196331
\(816\) 0 0
\(817\) −15.1537 −0.530159
\(818\) 0 0
\(819\) 61.3980i 2.14542i
\(820\) 0 0
\(821\) 54.6366i 1.90683i 0.301661 + 0.953415i \(0.402459\pi\)
−0.301661 + 0.953415i \(0.597541\pi\)
\(822\) 0 0
\(823\) 47.6656i 1.66152i 0.556633 + 0.830759i \(0.312093\pi\)
−0.556633 + 0.830759i \(0.687907\pi\)
\(824\) 0 0
\(825\) −1.22579 −0.0426766
\(826\) 0 0
\(827\) 37.9748i 1.32051i 0.751040 + 0.660257i \(0.229553\pi\)
−0.751040 + 0.660257i \(0.770447\pi\)
\(828\) 0 0
\(829\) 36.7907 1.27779 0.638897 0.769292i \(-0.279391\pi\)
0.638897 + 0.769292i \(0.279391\pi\)
\(830\) 0 0
\(831\) 6.55189 0.227283
\(832\) 0 0
\(833\) 12.4003 + 31.1437i 0.429644 + 1.07907i
\(834\) 0 0
\(835\) −4.61025 −0.159544
\(836\) 0 0
\(837\) 1.81922 0.0628815
\(838\) 0 0
\(839\) 21.8902i 0.755732i 0.925860 + 0.377866i \(0.123342\pi\)
−0.925860 + 0.377866i \(0.876658\pi\)
\(840\) 0 0
\(841\) 4.08515 0.140867
\(842\) 0 0
\(843\) 0.515693i 0.0177614i
\(844\) 0 0
\(845\) 11.8609i 0.408028i
\(846\) 0 0
\(847\) 38.3453i 1.31756i
\(848\) 0 0
\(849\) 2.36096 0.0810279
\(850\) 0 0
\(851\) 1.31641 0.0451258
\(852\) 0 0
\(853\) 30.0372i 1.02845i 0.857654 + 0.514227i \(0.171921\pi\)
−0.857654 + 0.514227i \(0.828079\pi\)
\(854\) 0 0
\(855\) 6.71397i 0.229613i
\(856\) 0 0
\(857\) 13.8552i 0.473284i 0.971597 + 0.236642i \(0.0760469\pi\)
−0.971597 + 0.236642i \(0.923953\pi\)
\(858\) 0 0
\(859\) −0.0365125 −0.00124579 −0.000622896 1.00000i \(-0.500198\pi\)
−0.000622896 1.00000i \(0.500198\pi\)
\(860\) 0 0
\(861\) 3.01440i 0.102730i
\(862\) 0 0
\(863\) −3.02921 −0.103116 −0.0515578 0.998670i \(-0.516419\pi\)
−0.0515578 + 0.998670i \(0.516419\pi\)
\(864\) 0 0
\(865\) 11.5954 0.394257
\(866\) 0 0
\(867\) 3.01391 + 3.18476i 0.102358 + 0.108160i
\(868\) 0 0
\(869\) −16.2965 −0.552821
\(870\) 0 0
\(871\) −40.8100 −1.38279
\(872\) 0 0
\(873\) 37.7128i 1.27639i
\(874\) 0 0
\(875\) 27.3206 0.923606
\(876\) 0 0
\(877\) 26.4458i 0.893011i 0.894781 + 0.446505i \(0.147332\pi\)
−0.894781 + 0.446505i \(0.852668\pi\)
\(878\) 0 0
\(879\) 3.35944i 0.113311i
\(880\) 0 0
\(881\) 51.7695i 1.74416i 0.489365 + 0.872079i \(0.337229\pi\)
−0.489365 + 0.872079i \(0.662771\pi\)
\(882\) 0 0
\(883\) 29.1375 0.980554 0.490277 0.871567i \(-0.336896\pi\)
0.490277 + 0.871567i \(0.336896\pi\)
\(884\) 0 0
\(885\) 0.191762 0.00644601
\(886\) 0 0
\(887\) 19.1149i 0.641816i −0.947110 0.320908i \(-0.896012\pi\)
0.947110 0.320908i \(-0.103988\pi\)
\(888\) 0 0
\(889\) 38.0206i 1.27517i
\(890\) 0 0
\(891\) 8.98254i 0.300926i
\(892\) 0 0
\(893\) 17.8751 0.598166
\(894\) 0 0
\(895\) 10.9750i 0.366854i
\(896\) 0 0
\(897\) −2.40407 −0.0802694
\(898\) 0 0
\(899\) −5.93344 −0.197891
\(900\) 0 0
\(901\) 0.297351 + 0.746809i 0.00990620 + 0.0248798i
\(902\) 0 0
\(903\) 4.93861 0.164347
\(904\) 0 0
\(905\) 5.25095 0.174548
\(906\) 0 0
\(907\) 40.6504i 1.34978i −0.737920 0.674888i \(-0.764192\pi\)
0.737920 0.674888i \(-0.235808\pi\)
\(908\) 0 0
\(909\) −56.5813 −1.87668
\(910\) 0 0
\(911\) 2.36815i 0.0784604i 0.999230 + 0.0392302i \(0.0124906\pi\)
−0.999230 + 0.0392302i \(0.987509\pi\)
\(912\) 0 0
\(913\) 8.87453i 0.293704i
\(914\) 0 0
\(915\) 0.774716i 0.0256113i
\(916\) 0 0
\(917\) −2.88102 −0.0951397
\(918\) 0 0
\(919\) −22.3213 −0.736312 −0.368156 0.929764i \(-0.620011\pi\)
−0.368156 + 0.929764i \(0.620011\pi\)
\(920\) 0 0
\(921\) 8.72094i 0.287365i
\(922\) 0 0
\(923\) 36.4902i 1.20109i
\(924\) 0 0
\(925\) 3.37975i 0.111126i
\(926\) 0 0
\(927\) −2.59058 −0.0850859
\(928\) 0 0
\(929\) 53.4007i 1.75202i −0.482292 0.876011i \(-0.660196\pi\)
0.482292 0.876011i \(-0.339804\pi\)
\(930\) 0 0
\(931\) −25.0285 −0.820276
\(932\) 0 0
\(933\) 4.44370 0.145480
\(934\) 0 0
\(935\) −1.21177 3.04340i −0.0396290 0.0995299i
\(936\) 0 0
\(937\) 41.8509 1.36721 0.683604 0.729853i \(-0.260412\pi\)
0.683604 + 0.729853i \(0.260412\pi\)
\(938\) 0 0
\(939\) −3.39330 −0.110736
\(940\) 0 0
\(941\) 27.3220i 0.890671i 0.895364 + 0.445336i \(0.146916\pi\)
−0.895364 + 0.445336i \(0.853084\pi\)
\(942\) 0 0
\(943\) −5.20446 −0.169481
\(944\) 0 0
\(945\) 4.42582i 0.143972i
\(946\) 0 0
\(947\) 7.25327i 0.235700i 0.993031 + 0.117850i \(0.0376002\pi\)
−0.993031 + 0.117850i \(0.962400\pi\)
\(948\) 0 0
\(949\) 72.0273i 2.33810i
\(950\) 0 0
\(951\) 3.70768 0.120230
\(952\) 0 0
\(953\) 38.0526 1.23264 0.616322 0.787495i \(-0.288622\pi\)
0.616322 + 0.787495i \(0.288622\pi\)
\(954\) 0 0
\(955\) 0.879194i 0.0284501i
\(956\) 0 0
\(957\) 1.37580i 0.0444732i
\(958\) 0 0
\(959\) 54.8985i 1.77276i
\(960\) 0 0
\(961\) 29.5870 0.954418
\(962\) 0 0
\(963\) 3.95179i 0.127345i
\(964\) 0 0
\(965\) 9.67594 0.311480
\(966\) 0 0
\(967\) −8.03155 −0.258277 −0.129139 0.991627i \(-0.541221\pi\)
−0.129139 + 0.991627i \(0.541221\pi\)
\(968\) 0 0
\(969\) −3.04161 + 1.21106i −0.0977107 + 0.0389047i
\(970\) 0 0
\(971\) 37.8859 1.21582 0.607909 0.794007i \(-0.292009\pi\)
0.607909 + 0.794007i \(0.292009\pi\)
\(972\) 0 0
\(973\) −69.5720 −2.23037
\(974\) 0 0
\(975\) 6.17222i 0.197669i
\(976\) 0 0
\(977\) −43.1772 −1.38136 −0.690680 0.723161i \(-0.742689\pi\)
−0.690680 + 0.723161i \(0.742689\pi\)
\(978\) 0 0
\(979\) 9.28067i 0.296611i
\(980\) 0 0
\(981\) 18.4199i 0.588103i
\(982\) 0 0
\(983\) 54.6825i 1.74410i 0.489416 + 0.872050i \(0.337210\pi\)
−0.489416 + 0.872050i \(0.662790\pi\)
\(984\) 0 0
\(985\) −15.7798 −0.502787
\(986\) 0 0
\(987\) −5.82552 −0.185428
\(988\) 0 0
\(989\) 8.52668i 0.271133i
\(990\) 0 0
\(991\) 10.1774i 0.323295i −0.986849 0.161647i \(-0.948319\pi\)
0.986849 0.161647i \(-0.0516807\pi\)
\(992\) 0 0
\(993\) 1.52810i 0.0484927i
\(994\) 0 0
\(995\) −3.49231 −0.110714
\(996\) 0 0
\(997\) 20.4228i 0.646797i 0.946263 + 0.323399i \(0.104825\pi\)
−0.946263 + 0.323399i \(0.895175\pi\)
\(998\) 0 0
\(999\) 1.16306 0.0367975
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 4012.2.b.b.237.22 46
17.16 even 2 inner 4012.2.b.b.237.25 yes 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.b.b.237.22 46 1.1 even 1 trivial
4012.2.b.b.237.25 yes 46 17.16 even 2 inner