Properties

Label 8012.2.a.a.1.11
Level $8012$
Weight $2$
Character 8012.1
Self dual yes
Analytic conductor $63.976$
Analytic rank $1$
Dimension $79$
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] = [8012,2,Mod(1,8012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8012 = 2^{2} \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8012.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: \(63.9761420994\)
Analytic rank: \(1\)
Dimension: \(79\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Character \(\chi\) \(=\) 8012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.68647 q^{3} -0.588575 q^{5} -1.13984 q^{7} +4.21715 q^{9} +O(q^{10})\) \(q-2.68647 q^{3} -0.588575 q^{5} -1.13984 q^{7} +4.21715 q^{9} +4.44297 q^{11} +4.61182 q^{13} +1.58119 q^{15} +5.78229 q^{17} -6.59831 q^{19} +3.06216 q^{21} -3.93873 q^{23} -4.65358 q^{25} -3.26983 q^{27} -6.50173 q^{29} -1.98799 q^{31} -11.9359 q^{33} +0.670883 q^{35} +4.93340 q^{37} -12.3895 q^{39} +5.81524 q^{41} -4.68453 q^{43} -2.48210 q^{45} -1.11178 q^{47} -5.70076 q^{49} -15.5340 q^{51} -4.55100 q^{53} -2.61502 q^{55} +17.7262 q^{57} -4.13024 q^{59} +12.0444 q^{61} -4.80688 q^{63} -2.71440 q^{65} -7.65722 q^{67} +10.5813 q^{69} +3.21273 q^{71} +7.14241 q^{73} +12.5017 q^{75} -5.06429 q^{77} -7.89847 q^{79} -3.86712 q^{81} -1.40674 q^{83} -3.40331 q^{85} +17.4667 q^{87} -7.25688 q^{89} -5.25675 q^{91} +5.34069 q^{93} +3.88360 q^{95} +17.0412 q^{97} +18.7367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) \) 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
\(3\) −2.68647 −1.55104 −0.775518 0.631325i \(-0.782511\pi\)
−0.775518 + 0.631325i \(0.782511\pi\)
\(4\) 0 0
\(5\) −0.588575 −0.263219 −0.131609 0.991302i \(-0.542014\pi\)
−0.131609 + 0.991302i \(0.542014\pi\)
\(6\) 0 0
\(7\) −1.13984 −0.430820 −0.215410 0.976524i \(-0.569109\pi\)
−0.215410 + 0.976524i \(0.569109\pi\)
\(8\) 0 0
\(9\) 4.21715 1.40572
\(10\) 0 0
\(11\) 4.44297 1.33961 0.669804 0.742538i \(-0.266378\pi\)
0.669804 + 0.742538i \(0.266378\pi\)
\(12\) 0 0
\(13\) 4.61182 1.27909 0.639544 0.768754i \(-0.279123\pi\)
0.639544 + 0.768754i \(0.279123\pi\)
\(14\) 0 0
\(15\) 1.58119 0.408262
\(16\) 0 0
\(17\) 5.78229 1.40241 0.701205 0.712959i \(-0.252646\pi\)
0.701205 + 0.712959i \(0.252646\pi\)
\(18\) 0 0
\(19\) −6.59831 −1.51376 −0.756878 0.653556i \(-0.773277\pi\)
−0.756878 + 0.653556i \(0.773277\pi\)
\(20\) 0 0
\(21\) 3.06216 0.668218
\(22\) 0 0
\(23\) −3.93873 −0.821282 −0.410641 0.911797i \(-0.634695\pi\)
−0.410641 + 0.911797i \(0.634695\pi\)
\(24\) 0 0
\(25\) −4.65358 −0.930716
\(26\) 0 0
\(27\) −3.26983 −0.629279
\(28\) 0 0
\(29\) −6.50173 −1.20734 −0.603670 0.797234i \(-0.706296\pi\)
−0.603670 + 0.797234i \(0.706296\pi\)
\(30\) 0 0
\(31\) −1.98799 −0.357054 −0.178527 0.983935i \(-0.557133\pi\)
−0.178527 + 0.983935i \(0.557133\pi\)
\(32\) 0 0
\(33\) −11.9359 −2.07778
\(34\) 0 0
\(35\) 0.670883 0.113400
\(36\) 0 0
\(37\) 4.93340 0.811047 0.405523 0.914085i \(-0.367089\pi\)
0.405523 + 0.914085i \(0.367089\pi\)
\(38\) 0 0
\(39\) −12.3895 −1.98391
\(40\) 0 0
\(41\) 5.81524 0.908189 0.454094 0.890954i \(-0.349963\pi\)
0.454094 + 0.890954i \(0.349963\pi\)
\(42\) 0 0
\(43\) −4.68453 −0.714384 −0.357192 0.934031i \(-0.616266\pi\)
−0.357192 + 0.934031i \(0.616266\pi\)
\(44\) 0 0
\(45\) −2.48210 −0.370010
\(46\) 0 0
\(47\) −1.11178 −0.162170 −0.0810851 0.996707i \(-0.525839\pi\)
−0.0810851 + 0.996707i \(0.525839\pi\)
\(48\) 0 0
\(49\) −5.70076 −0.814394
\(50\) 0 0
\(51\) −15.5340 −2.17519
\(52\) 0 0
\(53\) −4.55100 −0.625128 −0.312564 0.949897i \(-0.601188\pi\)
−0.312564 + 0.949897i \(0.601188\pi\)
\(54\) 0 0
\(55\) −2.61502 −0.352609
\(56\) 0 0
\(57\) 17.7262 2.34789
\(58\) 0 0
\(59\) −4.13024 −0.537711 −0.268855 0.963181i \(-0.586645\pi\)
−0.268855 + 0.963181i \(0.586645\pi\)
\(60\) 0 0
\(61\) 12.0444 1.54213 0.771064 0.636758i \(-0.219725\pi\)
0.771064 + 0.636758i \(0.219725\pi\)
\(62\) 0 0
\(63\) −4.80688 −0.605611
\(64\) 0 0
\(65\) −2.71440 −0.336680
\(66\) 0 0
\(67\) −7.65722 −0.935478 −0.467739 0.883867i \(-0.654931\pi\)
−0.467739 + 0.883867i \(0.654931\pi\)
\(68\) 0 0
\(69\) 10.5813 1.27384
\(70\) 0 0
\(71\) 3.21273 0.381281 0.190640 0.981660i \(-0.438944\pi\)
0.190640 + 0.981660i \(0.438944\pi\)
\(72\) 0 0
\(73\) 7.14241 0.835956 0.417978 0.908457i \(-0.362739\pi\)
0.417978 + 0.908457i \(0.362739\pi\)
\(74\) 0 0
\(75\) 12.5017 1.44357
\(76\) 0 0
\(77\) −5.06429 −0.577130
\(78\) 0 0
\(79\) −7.89847 −0.888647 −0.444324 0.895866i \(-0.646556\pi\)
−0.444324 + 0.895866i \(0.646556\pi\)
\(80\) 0 0
\(81\) −3.86712 −0.429680
\(82\) 0 0
\(83\) −1.40674 −0.154409 −0.0772047 0.997015i \(-0.524600\pi\)
−0.0772047 + 0.997015i \(0.524600\pi\)
\(84\) 0 0
\(85\) −3.40331 −0.369141
\(86\) 0 0
\(87\) 17.4667 1.87263
\(88\) 0 0
\(89\) −7.25688 −0.769228 −0.384614 0.923077i \(-0.625665\pi\)
−0.384614 + 0.923077i \(0.625665\pi\)
\(90\) 0 0
\(91\) −5.25675 −0.551057
\(92\) 0 0
\(93\) 5.34069 0.553804
\(94\) 0 0
\(95\) 3.88360 0.398449
\(96\) 0 0
\(97\) 17.0412 1.73027 0.865135 0.501538i \(-0.167232\pi\)
0.865135 + 0.501538i \(0.167232\pi\)
\(98\) 0 0
\(99\) 18.7367 1.88311
\(100\) 0 0
\(101\) −13.6291 −1.35614 −0.678071 0.734996i \(-0.737184\pi\)
−0.678071 + 0.734996i \(0.737184\pi\)
\(102\) 0 0
\(103\) 5.44163 0.536180 0.268090 0.963394i \(-0.413608\pi\)
0.268090 + 0.963394i \(0.413608\pi\)
\(104\) 0 0
\(105\) −1.80231 −0.175887
\(106\) 0 0
\(107\) 17.9896 1.73912 0.869562 0.493824i \(-0.164401\pi\)
0.869562 + 0.493824i \(0.164401\pi\)
\(108\) 0 0
\(109\) 16.7027 1.59983 0.799913 0.600116i \(-0.204879\pi\)
0.799913 + 0.600116i \(0.204879\pi\)
\(110\) 0 0
\(111\) −13.2535 −1.25796
\(112\) 0 0
\(113\) −7.85049 −0.738512 −0.369256 0.929328i \(-0.620387\pi\)
−0.369256 + 0.929328i \(0.620387\pi\)
\(114\) 0 0
\(115\) 2.31824 0.216177
\(116\) 0 0
\(117\) 19.4487 1.79803
\(118\) 0 0
\(119\) −6.59090 −0.604187
\(120\) 0 0
\(121\) 8.74002 0.794548
\(122\) 0 0
\(123\) −15.6225 −1.40863
\(124\) 0 0
\(125\) 5.68185 0.508200
\(126\) 0 0
\(127\) 17.6340 1.56477 0.782384 0.622797i \(-0.214004\pi\)
0.782384 + 0.622797i \(0.214004\pi\)
\(128\) 0 0
\(129\) 12.5849 1.10804
\(130\) 0 0
\(131\) 14.0381 1.22652 0.613258 0.789882i \(-0.289858\pi\)
0.613258 + 0.789882i \(0.289858\pi\)
\(132\) 0 0
\(133\) 7.52104 0.652157
\(134\) 0 0
\(135\) 1.92454 0.165638
\(136\) 0 0
\(137\) −8.98726 −0.767833 −0.383917 0.923368i \(-0.625425\pi\)
−0.383917 + 0.923368i \(0.625425\pi\)
\(138\) 0 0
\(139\) −2.64546 −0.224385 −0.112192 0.993686i \(-0.535787\pi\)
−0.112192 + 0.993686i \(0.535787\pi\)
\(140\) 0 0
\(141\) 2.98678 0.251532
\(142\) 0 0
\(143\) 20.4902 1.71348
\(144\) 0 0
\(145\) 3.82675 0.317795
\(146\) 0 0
\(147\) 15.3149 1.26315
\(148\) 0 0
\(149\) −10.9135 −0.894067 −0.447034 0.894517i \(-0.647520\pi\)
−0.447034 + 0.894517i \(0.647520\pi\)
\(150\) 0 0
\(151\) −13.7257 −1.11698 −0.558491 0.829511i \(-0.688619\pi\)
−0.558491 + 0.829511i \(0.688619\pi\)
\(152\) 0 0
\(153\) 24.3847 1.97139
\(154\) 0 0
\(155\) 1.17008 0.0939832
\(156\) 0 0
\(157\) −4.69697 −0.374859 −0.187430 0.982278i \(-0.560016\pi\)
−0.187430 + 0.982278i \(0.560016\pi\)
\(158\) 0 0
\(159\) 12.2261 0.969597
\(160\) 0 0
\(161\) 4.48954 0.353825
\(162\) 0 0
\(163\) 0.310071 0.0242866 0.0121433 0.999926i \(-0.496135\pi\)
0.0121433 + 0.999926i \(0.496135\pi\)
\(164\) 0 0
\(165\) 7.02519 0.546910
\(166\) 0 0
\(167\) −22.1527 −1.71423 −0.857113 0.515129i \(-0.827744\pi\)
−0.857113 + 0.515129i \(0.827744\pi\)
\(168\) 0 0
\(169\) 8.26887 0.636067
\(170\) 0 0
\(171\) −27.8260 −2.12791
\(172\) 0 0
\(173\) −24.4511 −1.85899 −0.929493 0.368839i \(-0.879755\pi\)
−0.929493 + 0.368839i \(0.879755\pi\)
\(174\) 0 0
\(175\) 5.30435 0.400971
\(176\) 0 0
\(177\) 11.0958 0.834010
\(178\) 0 0
\(179\) 2.99995 0.224227 0.112113 0.993695i \(-0.464238\pi\)
0.112113 + 0.993695i \(0.464238\pi\)
\(180\) 0 0
\(181\) 6.62272 0.492263 0.246131 0.969236i \(-0.420841\pi\)
0.246131 + 0.969236i \(0.420841\pi\)
\(182\) 0 0
\(183\) −32.3570 −2.39190
\(184\) 0 0
\(185\) −2.90368 −0.213483
\(186\) 0 0
\(187\) 25.6906 1.87868
\(188\) 0 0
\(189\) 3.72709 0.271106
\(190\) 0 0
\(191\) 20.3074 1.46939 0.734695 0.678398i \(-0.237325\pi\)
0.734695 + 0.678398i \(0.237325\pi\)
\(192\) 0 0
\(193\) −2.69413 −0.193928 −0.0969640 0.995288i \(-0.530913\pi\)
−0.0969640 + 0.995288i \(0.530913\pi\)
\(194\) 0 0
\(195\) 7.29216 0.522203
\(196\) 0 0
\(197\) −3.27799 −0.233547 −0.116774 0.993159i \(-0.537255\pi\)
−0.116774 + 0.993159i \(0.537255\pi\)
\(198\) 0 0
\(199\) 18.4085 1.30495 0.652474 0.757812i \(-0.273731\pi\)
0.652474 + 0.757812i \(0.273731\pi\)
\(200\) 0 0
\(201\) 20.5709 1.45096
\(202\) 0 0
\(203\) 7.41095 0.520147
\(204\) 0 0
\(205\) −3.42270 −0.239052
\(206\) 0 0
\(207\) −16.6102 −1.15449
\(208\) 0 0
\(209\) −29.3161 −2.02784
\(210\) 0 0
\(211\) −20.4501 −1.40784 −0.703921 0.710278i \(-0.748569\pi\)
−0.703921 + 0.710278i \(0.748569\pi\)
\(212\) 0 0
\(213\) −8.63092 −0.591381
\(214\) 0 0
\(215\) 2.75719 0.188039
\(216\) 0 0
\(217\) 2.26600 0.153826
\(218\) 0 0
\(219\) −19.1879 −1.29660
\(220\) 0 0
\(221\) 26.6669 1.79381
\(222\) 0 0
\(223\) −16.2014 −1.08493 −0.542465 0.840079i \(-0.682509\pi\)
−0.542465 + 0.840079i \(0.682509\pi\)
\(224\) 0 0
\(225\) −19.6248 −1.30832
\(226\) 0 0
\(227\) −18.8118 −1.24858 −0.624291 0.781192i \(-0.714612\pi\)
−0.624291 + 0.781192i \(0.714612\pi\)
\(228\) 0 0
\(229\) 2.07186 0.136912 0.0684562 0.997654i \(-0.478193\pi\)
0.0684562 + 0.997654i \(0.478193\pi\)
\(230\) 0 0
\(231\) 13.6051 0.895150
\(232\) 0 0
\(233\) −23.3012 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(234\) 0 0
\(235\) 0.654367 0.0426862
\(236\) 0 0
\(237\) 21.2190 1.37832
\(238\) 0 0
\(239\) −2.88060 −0.186331 −0.0931654 0.995651i \(-0.529699\pi\)
−0.0931654 + 0.995651i \(0.529699\pi\)
\(240\) 0 0
\(241\) −19.9437 −1.28469 −0.642343 0.766417i \(-0.722038\pi\)
−0.642343 + 0.766417i \(0.722038\pi\)
\(242\) 0 0
\(243\) 20.1984 1.29573
\(244\) 0 0
\(245\) 3.35532 0.214364
\(246\) 0 0
\(247\) −30.4302 −1.93623
\(248\) 0 0
\(249\) 3.77916 0.239495
\(250\) 0 0
\(251\) 17.9629 1.13381 0.566904 0.823784i \(-0.308141\pi\)
0.566904 + 0.823784i \(0.308141\pi\)
\(252\) 0 0
\(253\) −17.4997 −1.10020
\(254\) 0 0
\(255\) 9.14290 0.572551
\(256\) 0 0
\(257\) 26.5817 1.65812 0.829062 0.559157i \(-0.188875\pi\)
0.829062 + 0.559157i \(0.188875\pi\)
\(258\) 0 0
\(259\) −5.62331 −0.349415
\(260\) 0 0
\(261\) −27.4187 −1.69718
\(262\) 0 0
\(263\) −25.8898 −1.59643 −0.798217 0.602370i \(-0.794223\pi\)
−0.798217 + 0.602370i \(0.794223\pi\)
\(264\) 0 0
\(265\) 2.67860 0.164545
\(266\) 0 0
\(267\) 19.4954 1.19310
\(268\) 0 0
\(269\) 12.8940 0.786162 0.393081 0.919504i \(-0.371409\pi\)
0.393081 + 0.919504i \(0.371409\pi\)
\(270\) 0 0
\(271\) −11.7829 −0.715763 −0.357881 0.933767i \(-0.616501\pi\)
−0.357881 + 0.933767i \(0.616501\pi\)
\(272\) 0 0
\(273\) 14.1221 0.854710
\(274\) 0 0
\(275\) −20.6757 −1.24679
\(276\) 0 0
\(277\) 14.6877 0.882498 0.441249 0.897385i \(-0.354536\pi\)
0.441249 + 0.897385i \(0.354536\pi\)
\(278\) 0 0
\(279\) −8.38365 −0.501916
\(280\) 0 0
\(281\) 14.6726 0.875292 0.437646 0.899147i \(-0.355812\pi\)
0.437646 + 0.899147i \(0.355812\pi\)
\(282\) 0 0
\(283\) −29.9171 −1.77839 −0.889194 0.457531i \(-0.848734\pi\)
−0.889194 + 0.457531i \(0.848734\pi\)
\(284\) 0 0
\(285\) −10.4332 −0.618009
\(286\) 0 0
\(287\) −6.62847 −0.391266
\(288\) 0 0
\(289\) 16.4348 0.966756
\(290\) 0 0
\(291\) −45.7807 −2.68371
\(292\) 0 0
\(293\) −9.00633 −0.526155 −0.263078 0.964775i \(-0.584738\pi\)
−0.263078 + 0.964775i \(0.584738\pi\)
\(294\) 0 0
\(295\) 2.43095 0.141536
\(296\) 0 0
\(297\) −14.5278 −0.842987
\(298\) 0 0
\(299\) −18.1647 −1.05049
\(300\) 0 0
\(301\) 5.33963 0.307771
\(302\) 0 0
\(303\) 36.6141 2.10343
\(304\) 0 0
\(305\) −7.08903 −0.405917
\(306\) 0 0
\(307\) −24.8875 −1.42040 −0.710202 0.703998i \(-0.751396\pi\)
−0.710202 + 0.703998i \(0.751396\pi\)
\(308\) 0 0
\(309\) −14.6188 −0.831634
\(310\) 0 0
\(311\) −21.4826 −1.21817 −0.609084 0.793106i \(-0.708463\pi\)
−0.609084 + 0.793106i \(0.708463\pi\)
\(312\) 0 0
\(313\) 9.00068 0.508749 0.254374 0.967106i \(-0.418130\pi\)
0.254374 + 0.967106i \(0.418130\pi\)
\(314\) 0 0
\(315\) 2.82921 0.159408
\(316\) 0 0
\(317\) 17.4485 0.980004 0.490002 0.871721i \(-0.336996\pi\)
0.490002 + 0.871721i \(0.336996\pi\)
\(318\) 0 0
\(319\) −28.8870 −1.61736
\(320\) 0 0
\(321\) −48.3287 −2.69744
\(322\) 0 0
\(323\) −38.1533 −2.12291
\(324\) 0 0
\(325\) −21.4615 −1.19047
\(326\) 0 0
\(327\) −44.8713 −2.48139
\(328\) 0 0
\(329\) 1.26726 0.0698662
\(330\) 0 0
\(331\) 15.1209 0.831122 0.415561 0.909565i \(-0.363585\pi\)
0.415561 + 0.909565i \(0.363585\pi\)
\(332\) 0 0
\(333\) 20.8049 1.14010
\(334\) 0 0
\(335\) 4.50684 0.246235
\(336\) 0 0
\(337\) −12.7645 −0.695328 −0.347664 0.937619i \(-0.613025\pi\)
−0.347664 + 0.937619i \(0.613025\pi\)
\(338\) 0 0
\(339\) 21.0901 1.14546
\(340\) 0 0
\(341\) −8.83260 −0.478312
\(342\) 0 0
\(343\) 14.4769 0.781678
\(344\) 0 0
\(345\) −6.22789 −0.335298
\(346\) 0 0
\(347\) −12.3748 −0.664312 −0.332156 0.943224i \(-0.607776\pi\)
−0.332156 + 0.943224i \(0.607776\pi\)
\(348\) 0 0
\(349\) 24.8885 1.33225 0.666124 0.745841i \(-0.267952\pi\)
0.666124 + 0.745841i \(0.267952\pi\)
\(350\) 0 0
\(351\) −15.0799 −0.804903
\(352\) 0 0
\(353\) 9.05852 0.482137 0.241068 0.970508i \(-0.422502\pi\)
0.241068 + 0.970508i \(0.422502\pi\)
\(354\) 0 0
\(355\) −1.89093 −0.100360
\(356\) 0 0
\(357\) 17.7063 0.937116
\(358\) 0 0
\(359\) 24.5518 1.29580 0.647899 0.761727i \(-0.275648\pi\)
0.647899 + 0.761727i \(0.275648\pi\)
\(360\) 0 0
\(361\) 24.5377 1.29146
\(362\) 0 0
\(363\) −23.4798 −1.23237
\(364\) 0 0
\(365\) −4.20384 −0.220039
\(366\) 0 0
\(367\) −29.7211 −1.55143 −0.775715 0.631083i \(-0.782611\pi\)
−0.775715 + 0.631083i \(0.782611\pi\)
\(368\) 0 0
\(369\) 24.5237 1.27665
\(370\) 0 0
\(371\) 5.18743 0.269318
\(372\) 0 0
\(373\) −27.4434 −1.42097 −0.710483 0.703714i \(-0.751524\pi\)
−0.710483 + 0.703714i \(0.751524\pi\)
\(374\) 0 0
\(375\) −15.2642 −0.788237
\(376\) 0 0
\(377\) −29.9848 −1.54430
\(378\) 0 0
\(379\) 22.7071 1.16638 0.583192 0.812334i \(-0.301804\pi\)
0.583192 + 0.812334i \(0.301804\pi\)
\(380\) 0 0
\(381\) −47.3734 −2.42701
\(382\) 0 0
\(383\) −12.6053 −0.644101 −0.322051 0.946722i \(-0.604372\pi\)
−0.322051 + 0.946722i \(0.604372\pi\)
\(384\) 0 0
\(385\) 2.98072 0.151911
\(386\) 0 0
\(387\) −19.7553 −1.00422
\(388\) 0 0
\(389\) 33.1313 1.67982 0.839912 0.542723i \(-0.182607\pi\)
0.839912 + 0.542723i \(0.182607\pi\)
\(390\) 0 0
\(391\) −22.7749 −1.15178
\(392\) 0 0
\(393\) −37.7131 −1.90237
\(394\) 0 0
\(395\) 4.64884 0.233908
\(396\) 0 0
\(397\) 26.3482 1.32238 0.661188 0.750220i \(-0.270052\pi\)
0.661188 + 0.750220i \(0.270052\pi\)
\(398\) 0 0
\(399\) −20.2051 −1.01152
\(400\) 0 0
\(401\) −25.9013 −1.29345 −0.646724 0.762724i \(-0.723861\pi\)
−0.646724 + 0.762724i \(0.723861\pi\)
\(402\) 0 0
\(403\) −9.16826 −0.456704
\(404\) 0 0
\(405\) 2.27609 0.113100
\(406\) 0 0
\(407\) 21.9190 1.08648
\(408\) 0 0
\(409\) −6.24608 −0.308849 −0.154424 0.988005i \(-0.549352\pi\)
−0.154424 + 0.988005i \(0.549352\pi\)
\(410\) 0 0
\(411\) 24.1440 1.19094
\(412\) 0 0
\(413\) 4.70782 0.231657
\(414\) 0 0
\(415\) 0.827970 0.0406434
\(416\) 0 0
\(417\) 7.10696 0.348029
\(418\) 0 0
\(419\) −18.2000 −0.889127 −0.444564 0.895747i \(-0.646641\pi\)
−0.444564 + 0.895747i \(0.646641\pi\)
\(420\) 0 0
\(421\) −32.3059 −1.57449 −0.787247 0.616637i \(-0.788494\pi\)
−0.787247 + 0.616637i \(0.788494\pi\)
\(422\) 0 0
\(423\) −4.68855 −0.227965
\(424\) 0 0
\(425\) −26.9083 −1.30525
\(426\) 0 0
\(427\) −13.7287 −0.664380
\(428\) 0 0
\(429\) −55.0464 −2.65766
\(430\) 0 0
\(431\) −0.185919 −0.00895541 −0.00447771 0.999990i \(-0.501425\pi\)
−0.00447771 + 0.999990i \(0.501425\pi\)
\(432\) 0 0
\(433\) −37.0172 −1.77893 −0.889466 0.457001i \(-0.848923\pi\)
−0.889466 + 0.457001i \(0.848923\pi\)
\(434\) 0 0
\(435\) −10.2805 −0.492911
\(436\) 0 0
\(437\) 25.9890 1.24322
\(438\) 0 0
\(439\) 3.94313 0.188195 0.0940976 0.995563i \(-0.470003\pi\)
0.0940976 + 0.995563i \(0.470003\pi\)
\(440\) 0 0
\(441\) −24.0409 −1.14481
\(442\) 0 0
\(443\) 28.0303 1.33176 0.665880 0.746059i \(-0.268056\pi\)
0.665880 + 0.746059i \(0.268056\pi\)
\(444\) 0 0
\(445\) 4.27122 0.202475
\(446\) 0 0
\(447\) 29.3188 1.38673
\(448\) 0 0
\(449\) 32.5884 1.53794 0.768972 0.639283i \(-0.220769\pi\)
0.768972 + 0.639283i \(0.220769\pi\)
\(450\) 0 0
\(451\) 25.8370 1.21662
\(452\) 0 0
\(453\) 36.8738 1.73248
\(454\) 0 0
\(455\) 3.09399 0.145048
\(456\) 0 0
\(457\) 27.9238 1.30622 0.653109 0.757264i \(-0.273464\pi\)
0.653109 + 0.757264i \(0.273464\pi\)
\(458\) 0 0
\(459\) −18.9071 −0.882507
\(460\) 0 0
\(461\) −3.81442 −0.177655 −0.0888277 0.996047i \(-0.528312\pi\)
−0.0888277 + 0.996047i \(0.528312\pi\)
\(462\) 0 0
\(463\) −23.0068 −1.06922 −0.534608 0.845100i \(-0.679541\pi\)
−0.534608 + 0.845100i \(0.679541\pi\)
\(464\) 0 0
\(465\) −3.14340 −0.145771
\(466\) 0 0
\(467\) −31.4133 −1.45364 −0.726818 0.686831i \(-0.759001\pi\)
−0.726818 + 0.686831i \(0.759001\pi\)
\(468\) 0 0
\(469\) 8.72803 0.403023
\(470\) 0 0
\(471\) 12.6183 0.581421
\(472\) 0 0
\(473\) −20.8132 −0.956994
\(474\) 0 0
\(475\) 30.7058 1.40888
\(476\) 0 0
\(477\) −19.1922 −0.878752
\(478\) 0 0
\(479\) 14.6004 0.667111 0.333556 0.942730i \(-0.391751\pi\)
0.333556 + 0.942730i \(0.391751\pi\)
\(480\) 0 0
\(481\) 22.7520 1.03740
\(482\) 0 0
\(483\) −12.0610 −0.548796
\(484\) 0 0
\(485\) −10.0300 −0.455439
\(486\) 0 0
\(487\) 33.2262 1.50562 0.752811 0.658236i \(-0.228697\pi\)
0.752811 + 0.658236i \(0.228697\pi\)
\(488\) 0 0
\(489\) −0.832997 −0.0376694
\(490\) 0 0
\(491\) 35.3052 1.59330 0.796650 0.604441i \(-0.206604\pi\)
0.796650 + 0.604441i \(0.206604\pi\)
\(492\) 0 0
\(493\) −37.5949 −1.69319
\(494\) 0 0
\(495\) −11.0279 −0.495668
\(496\) 0 0
\(497\) −3.66201 −0.164264
\(498\) 0 0
\(499\) 18.4340 0.825219 0.412610 0.910908i \(-0.364617\pi\)
0.412610 + 0.910908i \(0.364617\pi\)
\(500\) 0 0
\(501\) 59.5126 2.65883
\(502\) 0 0
\(503\) −27.6382 −1.23233 −0.616164 0.787618i \(-0.711314\pi\)
−0.616164 + 0.787618i \(0.711314\pi\)
\(504\) 0 0
\(505\) 8.02172 0.356962
\(506\) 0 0
\(507\) −22.2141 −0.986563
\(508\) 0 0
\(509\) 3.79650 0.168277 0.0841384 0.996454i \(-0.473186\pi\)
0.0841384 + 0.996454i \(0.473186\pi\)
\(510\) 0 0
\(511\) −8.14123 −0.360147
\(512\) 0 0
\(513\) 21.5754 0.952575
\(514\) 0 0
\(515\) −3.20280 −0.141132
\(516\) 0 0
\(517\) −4.93962 −0.217244
\(518\) 0 0
\(519\) 65.6874 2.88336
\(520\) 0 0
\(521\) −25.0365 −1.09687 −0.548434 0.836194i \(-0.684776\pi\)
−0.548434 + 0.836194i \(0.684776\pi\)
\(522\) 0 0
\(523\) −7.85749 −0.343584 −0.171792 0.985133i \(-0.554956\pi\)
−0.171792 + 0.985133i \(0.554956\pi\)
\(524\) 0 0
\(525\) −14.2500 −0.621921
\(526\) 0 0
\(527\) −11.4951 −0.500736
\(528\) 0 0
\(529\) −7.48639 −0.325495
\(530\) 0 0
\(531\) −17.4178 −0.755868
\(532\) 0 0
\(533\) 26.8188 1.16165
\(534\) 0 0
\(535\) −10.5882 −0.457770
\(536\) 0 0
\(537\) −8.05930 −0.347784
\(538\) 0 0
\(539\) −25.3283 −1.09097
\(540\) 0 0
\(541\) −31.3188 −1.34650 −0.673250 0.739415i \(-0.735102\pi\)
−0.673250 + 0.739415i \(0.735102\pi\)
\(542\) 0 0
\(543\) −17.7918 −0.763518
\(544\) 0 0
\(545\) −9.83077 −0.421104
\(546\) 0 0
\(547\) 19.8022 0.846680 0.423340 0.905971i \(-0.360858\pi\)
0.423340 + 0.905971i \(0.360858\pi\)
\(548\) 0 0
\(549\) 50.7930 2.16779
\(550\) 0 0
\(551\) 42.9005 1.82762
\(552\) 0 0
\(553\) 9.00302 0.382847
\(554\) 0 0
\(555\) 7.80065 0.331119
\(556\) 0 0
\(557\) −18.6734 −0.791219 −0.395609 0.918419i \(-0.629467\pi\)
−0.395609 + 0.918419i \(0.629467\pi\)
\(558\) 0 0
\(559\) −21.6042 −0.913760
\(560\) 0 0
\(561\) −69.0170 −2.91390
\(562\) 0 0
\(563\) −35.5253 −1.49721 −0.748606 0.663015i \(-0.769276\pi\)
−0.748606 + 0.663015i \(0.769276\pi\)
\(564\) 0 0
\(565\) 4.62060 0.194390
\(566\) 0 0
\(567\) 4.40791 0.185115
\(568\) 0 0
\(569\) 41.4668 1.73838 0.869189 0.494479i \(-0.164641\pi\)
0.869189 + 0.494479i \(0.164641\pi\)
\(570\) 0 0
\(571\) 13.2739 0.555497 0.277749 0.960654i \(-0.410412\pi\)
0.277749 + 0.960654i \(0.410412\pi\)
\(572\) 0 0
\(573\) −54.5552 −2.27908
\(574\) 0 0
\(575\) 18.3292 0.764381
\(576\) 0 0
\(577\) −41.2957 −1.71916 −0.859582 0.510998i \(-0.829276\pi\)
−0.859582 + 0.510998i \(0.829276\pi\)
\(578\) 0 0
\(579\) 7.23772 0.300790
\(580\) 0 0
\(581\) 1.60346 0.0665227
\(582\) 0 0
\(583\) −20.2200 −0.837426
\(584\) 0 0
\(585\) −11.4470 −0.473276
\(586\) 0 0
\(587\) −33.1014 −1.36624 −0.683120 0.730306i \(-0.739378\pi\)
−0.683120 + 0.730306i \(0.739378\pi\)
\(588\) 0 0
\(589\) 13.1174 0.540493
\(590\) 0 0
\(591\) 8.80624 0.362240
\(592\) 0 0
\(593\) 20.0175 0.822019 0.411010 0.911631i \(-0.365176\pi\)
0.411010 + 0.911631i \(0.365176\pi\)
\(594\) 0 0
\(595\) 3.87924 0.159033
\(596\) 0 0
\(597\) −49.4541 −2.02402
\(598\) 0 0
\(599\) 9.05800 0.370100 0.185050 0.982729i \(-0.440755\pi\)
0.185050 + 0.982729i \(0.440755\pi\)
\(600\) 0 0
\(601\) −34.8148 −1.42013 −0.710063 0.704138i \(-0.751334\pi\)
−0.710063 + 0.704138i \(0.751334\pi\)
\(602\) 0 0
\(603\) −32.2916 −1.31502
\(604\) 0 0
\(605\) −5.14416 −0.209140
\(606\) 0 0
\(607\) −0.844938 −0.0342950 −0.0171475 0.999853i \(-0.505458\pi\)
−0.0171475 + 0.999853i \(0.505458\pi\)
\(608\) 0 0
\(609\) −19.9093 −0.806767
\(610\) 0 0
\(611\) −5.12734 −0.207430
\(612\) 0 0
\(613\) −28.5701 −1.15393 −0.576967 0.816768i \(-0.695764\pi\)
−0.576967 + 0.816768i \(0.695764\pi\)
\(614\) 0 0
\(615\) 9.19501 0.370779
\(616\) 0 0
\(617\) −12.2245 −0.492138 −0.246069 0.969252i \(-0.579139\pi\)
−0.246069 + 0.969252i \(0.579139\pi\)
\(618\) 0 0
\(619\) −3.67031 −0.147522 −0.0737611 0.997276i \(-0.523500\pi\)
−0.0737611 + 0.997276i \(0.523500\pi\)
\(620\) 0 0
\(621\) 12.8790 0.516816
\(622\) 0 0
\(623\) 8.27171 0.331399
\(624\) 0 0
\(625\) 19.9237 0.796948
\(626\) 0 0
\(627\) 78.7571 3.14525
\(628\) 0 0
\(629\) 28.5264 1.13742
\(630\) 0 0
\(631\) −39.1861 −1.55997 −0.779987 0.625796i \(-0.784774\pi\)
−0.779987 + 0.625796i \(0.784774\pi\)
\(632\) 0 0
\(633\) 54.9386 2.18361
\(634\) 0 0
\(635\) −10.3789 −0.411876
\(636\) 0 0
\(637\) −26.2909 −1.04168
\(638\) 0 0
\(639\) 13.5485 0.535972
\(640\) 0 0
\(641\) −10.5893 −0.418252 −0.209126 0.977889i \(-0.567062\pi\)
−0.209126 + 0.977889i \(0.567062\pi\)
\(642\) 0 0
\(643\) −12.3806 −0.488242 −0.244121 0.969745i \(-0.578499\pi\)
−0.244121 + 0.969745i \(0.578499\pi\)
\(644\) 0 0
\(645\) −7.40713 −0.291656
\(646\) 0 0
\(647\) −43.1900 −1.69797 −0.848986 0.528415i \(-0.822787\pi\)
−0.848986 + 0.528415i \(0.822787\pi\)
\(648\) 0 0
\(649\) −18.3505 −0.720322
\(650\) 0 0
\(651\) −6.08755 −0.238590
\(652\) 0 0
\(653\) 13.6056 0.532427 0.266213 0.963914i \(-0.414227\pi\)
0.266213 + 0.963914i \(0.414227\pi\)
\(654\) 0 0
\(655\) −8.26249 −0.322842
\(656\) 0 0
\(657\) 30.1206 1.17512
\(658\) 0 0
\(659\) −40.6391 −1.58307 −0.791537 0.611122i \(-0.790719\pi\)
−0.791537 + 0.611122i \(0.790719\pi\)
\(660\) 0 0
\(661\) 44.1381 1.71677 0.858386 0.513004i \(-0.171467\pi\)
0.858386 + 0.513004i \(0.171467\pi\)
\(662\) 0 0
\(663\) −71.6398 −2.78226
\(664\) 0 0
\(665\) −4.42670 −0.171660
\(666\) 0 0
\(667\) 25.6086 0.991568
\(668\) 0 0
\(669\) 43.5248 1.68276
\(670\) 0 0
\(671\) 53.5130 2.06585
\(672\) 0 0
\(673\) −0.743820 −0.0286721 −0.0143361 0.999897i \(-0.504563\pi\)
−0.0143361 + 0.999897i \(0.504563\pi\)
\(674\) 0 0
\(675\) 15.2164 0.585680
\(676\) 0 0
\(677\) −7.58028 −0.291334 −0.145667 0.989334i \(-0.546533\pi\)
−0.145667 + 0.989334i \(0.546533\pi\)
\(678\) 0 0
\(679\) −19.4243 −0.745436
\(680\) 0 0
\(681\) 50.5374 1.93660
\(682\) 0 0
\(683\) 30.0378 1.14937 0.574683 0.818376i \(-0.305126\pi\)
0.574683 + 0.818376i \(0.305126\pi\)
\(684\) 0 0
\(685\) 5.28967 0.202108
\(686\) 0 0
\(687\) −5.56600 −0.212356
\(688\) 0 0
\(689\) −20.9884 −0.799594
\(690\) 0 0
\(691\) −39.9733 −1.52065 −0.760327 0.649540i \(-0.774961\pi\)
−0.760327 + 0.649540i \(0.774961\pi\)
\(692\) 0 0
\(693\) −21.3569 −0.811280
\(694\) 0 0
\(695\) 1.55705 0.0590623
\(696\) 0 0
\(697\) 33.6254 1.27365
\(698\) 0 0
\(699\) 62.5982 2.36768
\(700\) 0 0
\(701\) 24.7114 0.933335 0.466668 0.884433i \(-0.345454\pi\)
0.466668 + 0.884433i \(0.345454\pi\)
\(702\) 0 0
\(703\) −32.5522 −1.22773
\(704\) 0 0
\(705\) −1.75794 −0.0662079
\(706\) 0 0
\(707\) 15.5350 0.584253
\(708\) 0 0
\(709\) −43.9783 −1.65164 −0.825820 0.563933i \(-0.809288\pi\)
−0.825820 + 0.563933i \(0.809288\pi\)
\(710\) 0 0
\(711\) −33.3090 −1.24918
\(712\) 0 0
\(713\) 7.83017 0.293242
\(714\) 0 0
\(715\) −12.0600 −0.451019
\(716\) 0 0
\(717\) 7.73867 0.289006
\(718\) 0 0
\(719\) −29.6486 −1.10571 −0.552853 0.833279i \(-0.686461\pi\)
−0.552853 + 0.833279i \(0.686461\pi\)
\(720\) 0 0
\(721\) −6.20260 −0.230997
\(722\) 0 0
\(723\) 53.5782 1.99260
\(724\) 0 0
\(725\) 30.2563 1.12369
\(726\) 0 0
\(727\) 0.0626388 0.00232314 0.00116157 0.999999i \(-0.499630\pi\)
0.00116157 + 0.999999i \(0.499630\pi\)
\(728\) 0 0
\(729\) −42.6612 −1.58004
\(730\) 0 0
\(731\) −27.0873 −1.00186
\(732\) 0 0
\(733\) 32.0615 1.18422 0.592109 0.805858i \(-0.298296\pi\)
0.592109 + 0.805858i \(0.298296\pi\)
\(734\) 0 0
\(735\) −9.01398 −0.332486
\(736\) 0 0
\(737\) −34.0208 −1.25317
\(738\) 0 0
\(739\) −4.65515 −0.171243 −0.0856213 0.996328i \(-0.527288\pi\)
−0.0856213 + 0.996328i \(0.527288\pi\)
\(740\) 0 0
\(741\) 81.7500 3.00316
\(742\) 0 0
\(743\) 37.7053 1.38327 0.691636 0.722246i \(-0.256890\pi\)
0.691636 + 0.722246i \(0.256890\pi\)
\(744\) 0 0
\(745\) 6.42340 0.235335
\(746\) 0 0
\(747\) −5.93241 −0.217056
\(748\) 0 0
\(749\) −20.5054 −0.749250
\(750\) 0 0
\(751\) −6.20433 −0.226399 −0.113200 0.993572i \(-0.536110\pi\)
−0.113200 + 0.993572i \(0.536110\pi\)
\(752\) 0 0
\(753\) −48.2569 −1.75858
\(754\) 0 0
\(755\) 8.07860 0.294010
\(756\) 0 0
\(757\) −17.0028 −0.617978 −0.308989 0.951066i \(-0.599991\pi\)
−0.308989 + 0.951066i \(0.599991\pi\)
\(758\) 0 0
\(759\) 47.0125 1.70644
\(760\) 0 0
\(761\) 10.6918 0.387578 0.193789 0.981043i \(-0.437922\pi\)
0.193789 + 0.981043i \(0.437922\pi\)
\(762\) 0 0
\(763\) −19.0384 −0.689237
\(764\) 0 0
\(765\) −14.3522 −0.518906
\(766\) 0 0
\(767\) −19.0479 −0.687780
\(768\) 0 0
\(769\) −43.7270 −1.57684 −0.788418 0.615139i \(-0.789100\pi\)
−0.788418 + 0.615139i \(0.789100\pi\)
\(770\) 0 0
\(771\) −71.4112 −2.57181
\(772\) 0 0
\(773\) 9.33581 0.335786 0.167893 0.985805i \(-0.446304\pi\)
0.167893 + 0.985805i \(0.446304\pi\)
\(774\) 0 0
\(775\) 9.25128 0.332316
\(776\) 0 0
\(777\) 15.1069 0.541956
\(778\) 0 0
\(779\) −38.3708 −1.37478
\(780\) 0 0
\(781\) 14.2741 0.510767
\(782\) 0 0
\(783\) 21.2595 0.759754
\(784\) 0 0
\(785\) 2.76452 0.0986699
\(786\) 0 0
\(787\) 2.74366 0.0978009 0.0489004 0.998804i \(-0.484428\pi\)
0.0489004 + 0.998804i \(0.484428\pi\)
\(788\) 0 0
\(789\) 69.5523 2.47613
\(790\) 0 0
\(791\) 8.94833 0.318166
\(792\) 0 0
\(793\) 55.5466 1.97252
\(794\) 0 0
\(795\) −7.19600 −0.255216
\(796\) 0 0
\(797\) 22.0420 0.780768 0.390384 0.920652i \(-0.372342\pi\)
0.390384 + 0.920652i \(0.372342\pi\)
\(798\) 0 0
\(799\) −6.42865 −0.227429
\(800\) 0 0
\(801\) −30.6033 −1.08132
\(802\) 0 0
\(803\) 31.7336 1.11985
\(804\) 0 0
\(805\) −2.64243 −0.0931333
\(806\) 0 0
\(807\) −34.6395 −1.21937
\(808\) 0 0
\(809\) 12.8320 0.451148 0.225574 0.974226i \(-0.427574\pi\)
0.225574 + 0.974226i \(0.427574\pi\)
\(810\) 0 0
\(811\) 54.0735 1.89878 0.949389 0.314103i \(-0.101704\pi\)
0.949389 + 0.314103i \(0.101704\pi\)
\(812\) 0 0
\(813\) 31.6546 1.11017
\(814\) 0 0
\(815\) −0.182500 −0.00639269
\(816\) 0 0
\(817\) 30.9100 1.08140
\(818\) 0 0
\(819\) −22.1685 −0.774629
\(820\) 0 0
\(821\) 49.1134 1.71407 0.857034 0.515259i \(-0.172304\pi\)
0.857034 + 0.515259i \(0.172304\pi\)
\(822\) 0 0
\(823\) 39.7671 1.38619 0.693097 0.720844i \(-0.256246\pi\)
0.693097 + 0.720844i \(0.256246\pi\)
\(824\) 0 0
\(825\) 55.5448 1.93382
\(826\) 0 0
\(827\) 8.50468 0.295737 0.147868 0.989007i \(-0.452759\pi\)
0.147868 + 0.989007i \(0.452759\pi\)
\(828\) 0 0
\(829\) −40.0456 −1.39084 −0.695420 0.718603i \(-0.744782\pi\)
−0.695420 + 0.718603i \(0.744782\pi\)
\(830\) 0 0
\(831\) −39.4581 −1.36879
\(832\) 0 0
\(833\) −32.9634 −1.14211
\(834\) 0 0
\(835\) 13.0385 0.451216
\(836\) 0 0
\(837\) 6.50040 0.224687
\(838\) 0 0
\(839\) −30.2491 −1.04431 −0.522157 0.852849i \(-0.674873\pi\)
−0.522157 + 0.852849i \(0.674873\pi\)
\(840\) 0 0
\(841\) 13.2725 0.457672
\(842\) 0 0
\(843\) −39.4175 −1.35761
\(844\) 0 0
\(845\) −4.86685 −0.167425
\(846\) 0 0
\(847\) −9.96226 −0.342307
\(848\) 0 0
\(849\) 80.3715 2.75834
\(850\) 0 0
\(851\) −19.4314 −0.666098
\(852\) 0 0
\(853\) 15.8735 0.543497 0.271749 0.962368i \(-0.412398\pi\)
0.271749 + 0.962368i \(0.412398\pi\)
\(854\) 0 0
\(855\) 16.3777 0.560106
\(856\) 0 0
\(857\) −17.3549 −0.592833 −0.296417 0.955059i \(-0.595792\pi\)
−0.296417 + 0.955059i \(0.595792\pi\)
\(858\) 0 0
\(859\) 1.29543 0.0441995 0.0220998 0.999756i \(-0.492965\pi\)
0.0220998 + 0.999756i \(0.492965\pi\)
\(860\) 0 0
\(861\) 17.8072 0.606868
\(862\) 0 0
\(863\) −52.7197 −1.79460 −0.897300 0.441422i \(-0.854474\pi\)
−0.897300 + 0.441422i \(0.854474\pi\)
\(864\) 0 0
\(865\) 14.3913 0.489320
\(866\) 0 0
\(867\) −44.1518 −1.49947
\(868\) 0 0
\(869\) −35.0927 −1.19044
\(870\) 0 0
\(871\) −35.3137 −1.19656
\(872\) 0 0
\(873\) 71.8652 2.43227
\(874\) 0 0
\(875\) −6.47642 −0.218943
\(876\) 0 0
\(877\) −31.5598 −1.06570 −0.532849 0.846210i \(-0.678879\pi\)
−0.532849 + 0.846210i \(0.678879\pi\)
\(878\) 0 0
\(879\) 24.1953 0.816086
\(880\) 0 0
\(881\) 49.4313 1.66538 0.832692 0.553736i \(-0.186799\pi\)
0.832692 + 0.553736i \(0.186799\pi\)
\(882\) 0 0
\(883\) −38.2964 −1.28878 −0.644388 0.764699i \(-0.722888\pi\)
−0.644388 + 0.764699i \(0.722888\pi\)
\(884\) 0 0
\(885\) −6.53069 −0.219527
\(886\) 0 0
\(887\) −5.55771 −0.186610 −0.0933048 0.995638i \(-0.529743\pi\)
−0.0933048 + 0.995638i \(0.529743\pi\)
\(888\) 0 0
\(889\) −20.1000 −0.674134
\(890\) 0 0
\(891\) −17.1815 −0.575603
\(892\) 0 0
\(893\) 7.33589 0.245486
\(894\) 0 0
\(895\) −1.76570 −0.0590207
\(896\) 0 0
\(897\) 48.7991 1.62935
\(898\) 0 0
\(899\) 12.9254 0.431086
\(900\) 0 0
\(901\) −26.3152 −0.876686
\(902\) 0 0
\(903\) −14.3448 −0.477364
\(904\) 0 0
\(905\) −3.89796 −0.129573
\(906\) 0 0
\(907\) −52.1142 −1.73042 −0.865211 0.501408i \(-0.832816\pi\)
−0.865211 + 0.501408i \(0.832816\pi\)
\(908\) 0 0
\(909\) −57.4757 −1.90635
\(910\) 0 0
\(911\) −15.7911 −0.523182 −0.261591 0.965179i \(-0.584247\pi\)
−0.261591 + 0.965179i \(0.584247\pi\)
\(912\) 0 0
\(913\) −6.25010 −0.206848
\(914\) 0 0
\(915\) 19.0445 0.629592
\(916\) 0 0
\(917\) −16.0013 −0.528408
\(918\) 0 0
\(919\) −4.71169 −0.155424 −0.0777121 0.996976i \(-0.524761\pi\)
−0.0777121 + 0.996976i \(0.524761\pi\)
\(920\) 0 0
\(921\) 66.8596 2.20310
\(922\) 0 0
\(923\) 14.8165 0.487692
\(924\) 0 0
\(925\) −22.9580 −0.754854
\(926\) 0 0
\(927\) 22.9481 0.753716
\(928\) 0 0
\(929\) −56.7630 −1.86233 −0.931166 0.364595i \(-0.881208\pi\)
−0.931166 + 0.364595i \(0.881208\pi\)
\(930\) 0 0
\(931\) 37.6154 1.23279
\(932\) 0 0
\(933\) 57.7125 1.88942
\(934\) 0 0
\(935\) −15.1208 −0.494503
\(936\) 0 0
\(937\) −43.7475 −1.42917 −0.714585 0.699549i \(-0.753384\pi\)
−0.714585 + 0.699549i \(0.753384\pi\)
\(938\) 0 0
\(939\) −24.1801 −0.789088
\(940\) 0 0
\(941\) −55.5133 −1.80968 −0.904840 0.425751i \(-0.860010\pi\)
−0.904840 + 0.425751i \(0.860010\pi\)
\(942\) 0 0
\(943\) −22.9047 −0.745879
\(944\) 0 0
\(945\) −2.19367 −0.0713602
\(946\) 0 0
\(947\) −17.0500 −0.554051 −0.277026 0.960863i \(-0.589349\pi\)
−0.277026 + 0.960863i \(0.589349\pi\)
\(948\) 0 0
\(949\) 32.9395 1.06926
\(950\) 0 0
\(951\) −46.8749 −1.52002
\(952\) 0 0
\(953\) −36.0861 −1.16894 −0.584471 0.811414i \(-0.698698\pi\)
−0.584471 + 0.811414i \(0.698698\pi\)
\(954\) 0 0
\(955\) −11.9524 −0.386771
\(956\) 0 0
\(957\) 77.6042 2.50859
\(958\) 0 0
\(959\) 10.2441 0.330798
\(960\) 0 0
\(961\) −27.0479 −0.872512
\(962\) 0 0
\(963\) 75.8649 2.44471
\(964\) 0 0
\(965\) 1.58570 0.0510455
\(966\) 0 0
\(967\) −23.3465 −0.750773 −0.375386 0.926868i \(-0.622490\pi\)
−0.375386 + 0.926868i \(0.622490\pi\)
\(968\) 0 0
\(969\) 102.498 3.29271
\(970\) 0 0
\(971\) 21.6416 0.694511 0.347255 0.937771i \(-0.387114\pi\)
0.347255 + 0.937771i \(0.387114\pi\)
\(972\) 0 0
\(973\) 3.01541 0.0966696
\(974\) 0 0
\(975\) 57.6557 1.84646
\(976\) 0 0
\(977\) −48.8412 −1.56257 −0.781284 0.624176i \(-0.785435\pi\)
−0.781284 + 0.624176i \(0.785435\pi\)
\(978\) 0 0
\(979\) −32.2421 −1.03046
\(980\) 0 0
\(981\) 70.4376 2.24890
\(982\) 0 0
\(983\) 17.8726 0.570046 0.285023 0.958521i \(-0.407999\pi\)
0.285023 + 0.958521i \(0.407999\pi\)
\(984\) 0 0
\(985\) 1.92934 0.0614739
\(986\) 0 0
\(987\) −3.40446 −0.108365
\(988\) 0 0
\(989\) 18.4511 0.586711
\(990\) 0 0
\(991\) 56.0939 1.78188 0.890941 0.454120i \(-0.150046\pi\)
0.890941 + 0.454120i \(0.150046\pi\)
\(992\) 0 0
\(993\) −40.6220 −1.28910
\(994\) 0 0
\(995\) −10.8348 −0.343486
\(996\) 0 0
\(997\) −12.5691 −0.398066 −0.199033 0.979993i \(-0.563780\pi\)
−0.199033 + 0.979993i \(0.563780\pi\)
\(998\) 0 0
\(999\) −16.1314 −0.510375
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 8012.2.a.a.1.11 79
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8012.2.a.a.1.11 79 1.1 even 1 trivial