Properties

Label 1232.2.e.b.769.3
Level $1232$
Weight $2$
Character 1232.769
Analytic conductor $9.838$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,2,Mod(769,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.e (of order \(2\), degree \(1\), not 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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 308)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.3
Root \(-0.707107 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 1232.769
Dual form 1232.2.e.b.769.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607i q^{3} -2.23607i q^{5} +(-2.12132 - 1.58114i) q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+2.23607i q^{3} -2.23607i q^{5} +(-2.12132 - 1.58114i) q^{7} -2.00000 q^{9} +(1.00000 + 3.16228i) q^{11} +2.82843 q^{13} +5.00000 q^{15} -5.65685 q^{17} +1.41421 q^{19} +(3.53553 - 4.74342i) q^{21} +5.00000 q^{23} +2.23607i q^{27} +9.48683i q^{29} +2.23607i q^{31} +(-7.07107 + 2.23607i) q^{33} +(-3.53553 + 4.74342i) q^{35} +11.0000 q^{37} +6.32456i q^{39} +9.89949 q^{41} -3.16228i q^{43} +4.47214i q^{45} +4.47214i q^{47} +(2.00000 + 6.70820i) q^{49} -12.6491i q^{51} +8.00000 q^{53} +(7.07107 - 2.23607i) q^{55} +3.16228i q^{57} -2.23607i q^{59} -4.24264 q^{61} +(4.24264 + 3.16228i) q^{63} -6.32456i q^{65} -9.00000 q^{67} +11.1803i q^{69} +1.00000 q^{71} -4.24264 q^{73} +(2.87868 - 8.28934i) q^{77} +6.32456i q^{79} -11.0000 q^{81} +1.41421 q^{83} +12.6491i q^{85} -21.2132 q^{87} +6.70820i q^{89} +(-6.00000 - 4.47214i) q^{91} -5.00000 q^{93} -3.16228i q^{95} +15.6525i q^{97} +(-2.00000 - 6.32456i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} + 4 q^{11} + 20 q^{15} + 20 q^{23} + 44 q^{37} + 8 q^{49} + 32 q^{53} - 36 q^{67} + 4 q^{71} + 20 q^{77} - 44 q^{81} - 24 q^{91} - 20 q^{93} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 2.23607i 1.29099i 0.763763 + 0.645497i \(0.223350\pi\)
−0.763763 + 0.645497i \(0.776650\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(6\) 0 0
\(7\) −2.12132 1.58114i −0.801784 0.597614i
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 1.00000 + 3.16228i 0.301511 + 0.953463i
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 5.00000 1.29099
\(16\) 0 0
\(17\) −5.65685 −1.37199 −0.685994 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 1.41421 0.324443 0.162221 0.986754i \(-0.448134\pi\)
0.162221 + 0.986754i \(0.448134\pi\)
\(20\) 0 0
\(21\) 3.53553 4.74342i 0.771517 1.03510i
\(22\) 0 0
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.23607i 0.430331i
\(28\) 0 0
\(29\) 9.48683i 1.76166i 0.473432 + 0.880830i \(0.343015\pi\)
−0.473432 + 0.880830i \(0.656985\pi\)
\(30\) 0 0
\(31\) 2.23607i 0.401610i 0.979631 + 0.200805i \(0.0643557\pi\)
−0.979631 + 0.200805i \(0.935644\pi\)
\(32\) 0 0
\(33\) −7.07107 + 2.23607i −1.23091 + 0.389249i
\(34\) 0 0
\(35\) −3.53553 + 4.74342i −0.597614 + 0.801784i
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) 6.32456i 1.01274i
\(40\) 0 0
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) 3.16228i 0.482243i −0.970495 0.241121i \(-0.922485\pi\)
0.970495 0.241121i \(-0.0775152\pi\)
\(44\) 0 0
\(45\) 4.47214i 0.666667i
\(46\) 0 0
\(47\) 4.47214i 0.652328i 0.945313 + 0.326164i \(0.105756\pi\)
−0.945313 + 0.326164i \(0.894244\pi\)
\(48\) 0 0
\(49\) 2.00000 + 6.70820i 0.285714 + 0.958315i
\(50\) 0 0
\(51\) 12.6491i 1.77123i
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 7.07107 2.23607i 0.953463 0.301511i
\(56\) 0 0
\(57\) 3.16228i 0.418854i
\(58\) 0 0
\(59\) 2.23607i 0.291111i −0.989350 0.145556i \(-0.953503\pi\)
0.989350 0.145556i \(-0.0464970\pi\)
\(60\) 0 0
\(61\) −4.24264 −0.543214 −0.271607 0.962408i \(-0.587555\pi\)
−0.271607 + 0.962408i \(0.587555\pi\)
\(62\) 0 0
\(63\) 4.24264 + 3.16228i 0.534522 + 0.398410i
\(64\) 0 0
\(65\) 6.32456i 0.784465i
\(66\) 0 0
\(67\) −9.00000 −1.09952 −0.549762 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) 0 0
\(69\) 11.1803i 1.34595i
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) −4.24264 −0.496564 −0.248282 0.968688i \(-0.579866\pi\)
−0.248282 + 0.968688i \(0.579866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.87868 8.28934i 0.328056 0.944658i
\(78\) 0 0
\(79\) 6.32456i 0.711568i 0.934568 + 0.355784i \(0.115786\pi\)
−0.934568 + 0.355784i \(0.884214\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 1.41421 0.155230 0.0776151 0.996983i \(-0.475269\pi\)
0.0776151 + 0.996983i \(0.475269\pi\)
\(84\) 0 0
\(85\) 12.6491i 1.37199i
\(86\) 0 0
\(87\) −21.2132 −2.27429
\(88\) 0 0
\(89\) 6.70820i 0.711068i 0.934663 + 0.355534i \(0.115701\pi\)
−0.934663 + 0.355534i \(0.884299\pi\)
\(90\) 0 0
\(91\) −6.00000 4.47214i −0.628971 0.468807i
\(92\) 0 0
\(93\) −5.00000 −0.518476
\(94\) 0 0
\(95\) 3.16228i 0.324443i
\(96\) 0 0
\(97\) 15.6525i 1.58927i 0.607089 + 0.794634i \(0.292337\pi\)
−0.607089 + 0.794634i \(0.707663\pi\)
\(98\) 0 0
\(99\) −2.00000 6.32456i −0.201008 0.635642i
\(100\) 0 0
\(101\) −4.24264 −0.422159 −0.211079 0.977469i \(-0.567698\pi\)
−0.211079 + 0.977469i \(0.567698\pi\)
\(102\) 0 0
\(103\) 4.47214i 0.440653i −0.975426 0.220326i \(-0.929288\pi\)
0.975426 0.220326i \(-0.0707122\pi\)
\(104\) 0 0
\(105\) −10.6066 7.90569i −1.03510 0.771517i
\(106\) 0 0
\(107\) 6.32456i 0.611418i −0.952125 0.305709i \(-0.901107\pi\)
0.952125 0.305709i \(-0.0988934\pi\)
\(108\) 0 0
\(109\) 15.8114i 1.51446i −0.653150 0.757228i \(-0.726553\pi\)
0.653150 0.757228i \(-0.273447\pi\)
\(110\) 0 0
\(111\) 24.5967i 2.33462i
\(112\) 0 0
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) 11.1803i 1.04257i
\(116\) 0 0
\(117\) −5.65685 −0.522976
\(118\) 0 0
\(119\) 12.0000 + 8.94427i 1.10004 + 0.819920i
\(120\) 0 0
\(121\) −9.00000 + 6.32456i −0.818182 + 0.574960i
\(122\) 0 0
\(123\) 22.1359i 1.99593i
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 15.8114i 1.40303i −0.712653 0.701517i \(-0.752506\pi\)
0.712653 0.701517i \(-0.247494\pi\)
\(128\) 0 0
\(129\) 7.07107 0.622573
\(130\) 0 0
\(131\) −14.1421 −1.23560 −0.617802 0.786334i \(-0.711977\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) −3.00000 2.23607i −0.260133 0.193892i
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 22.6274 1.91923 0.959616 0.281312i \(-0.0907697\pi\)
0.959616 + 0.281312i \(0.0907697\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) 2.82843 + 8.94427i 0.236525 + 0.747958i
\(144\) 0 0
\(145\) 21.2132 1.76166
\(146\) 0 0
\(147\) −15.0000 + 4.47214i −1.23718 + 0.368856i
\(148\) 0 0
\(149\) 6.32456i 0.518128i 0.965860 + 0.259064i \(0.0834140\pi\)
−0.965860 + 0.259064i \(0.916586\pi\)
\(150\) 0 0
\(151\) 12.6491i 1.02937i 0.857379 + 0.514685i \(0.172091\pi\)
−0.857379 + 0.514685i \(0.827909\pi\)
\(152\) 0 0
\(153\) 11.3137 0.914659
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 0 0
\(157\) 6.70820i 0.535373i −0.963506 0.267686i \(-0.913741\pi\)
0.963506 0.267686i \(-0.0862591\pi\)
\(158\) 0 0
\(159\) 17.8885i 1.41865i
\(160\) 0 0
\(161\) −10.6066 7.90569i −0.835917 0.623056i
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 5.00000 + 15.8114i 0.389249 + 1.23091i
\(166\) 0 0
\(167\) −1.41421 −0.109435 −0.0547176 0.998502i \(-0.517426\pi\)
−0.0547176 + 0.998502i \(0.517426\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) −2.82843 −0.216295
\(172\) 0 0
\(173\) 12.7279 0.967686 0.483843 0.875155i \(-0.339241\pi\)
0.483843 + 0.875155i \(0.339241\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.00000 0.375823
\(178\) 0 0
\(179\) 13.0000 0.971666 0.485833 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) 0 0
\(181\) 20.1246i 1.49585i −0.663783 0.747925i \(-0.731050\pi\)
0.663783 0.747925i \(-0.268950\pi\)
\(182\) 0 0
\(183\) 9.48683i 0.701287i
\(184\) 0 0
\(185\) 24.5967i 1.80839i
\(186\) 0 0
\(187\) −5.65685 17.8885i −0.413670 1.30814i
\(188\) 0 0
\(189\) 3.53553 4.74342i 0.257172 0.345033i
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 15.8114i 1.13813i 0.822293 + 0.569064i \(0.192694\pi\)
−0.822293 + 0.569064i \(0.807306\pi\)
\(194\) 0 0
\(195\) 14.1421 1.01274
\(196\) 0 0
\(197\) 12.6491i 0.901212i −0.892723 0.450606i \(-0.851208\pi\)
0.892723 0.450606i \(-0.148792\pi\)
\(198\) 0 0
\(199\) 17.8885i 1.26809i −0.773298 0.634043i \(-0.781394\pi\)
0.773298 0.634043i \(-0.218606\pi\)
\(200\) 0 0
\(201\) 20.1246i 1.41948i
\(202\) 0 0
\(203\) 15.0000 20.1246i 1.05279 1.41247i
\(204\) 0 0
\(205\) 22.1359i 1.54604i
\(206\) 0 0
\(207\) −10.0000 −0.695048
\(208\) 0 0
\(209\) 1.41421 + 4.47214i 0.0978232 + 0.309344i
\(210\) 0 0
\(211\) 9.48683i 0.653101i 0.945180 + 0.326550i \(0.105886\pi\)
−0.945180 + 0.326550i \(0.894114\pi\)
\(212\) 0 0
\(213\) 2.23607i 0.153213i
\(214\) 0 0
\(215\) −7.07107 −0.482243
\(216\) 0 0
\(217\) 3.53553 4.74342i 0.240008 0.322004i
\(218\) 0 0
\(219\) 9.48683i 0.641061i
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) 20.1246i 1.34764i −0.738894 0.673822i \(-0.764652\pi\)
0.738894 0.673822i \(-0.235348\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 29.6985 1.97116 0.985579 0.169217i \(-0.0541238\pi\)
0.985579 + 0.169217i \(0.0541238\pi\)
\(228\) 0 0
\(229\) 11.1803i 0.738818i 0.929267 + 0.369409i \(0.120440\pi\)
−0.929267 + 0.369409i \(0.879560\pi\)
\(230\) 0 0
\(231\) 18.5355 + 6.43692i 1.21955 + 0.423518i
\(232\) 0 0
\(233\) 9.48683i 0.621503i 0.950491 + 0.310752i \(0.100581\pi\)
−0.950491 + 0.310752i \(0.899419\pi\)
\(234\) 0 0
\(235\) 10.0000 0.652328
\(236\) 0 0
\(237\) −14.1421 −0.918630
\(238\) 0 0
\(239\) 15.8114i 1.02275i 0.859357 + 0.511377i \(0.170864\pi\)
−0.859357 + 0.511377i \(0.829136\pi\)
\(240\) 0 0
\(241\) −16.9706 −1.09317 −0.546585 0.837404i \(-0.684072\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(242\) 0 0
\(243\) 17.8885i 1.14755i
\(244\) 0 0
\(245\) 15.0000 4.47214i 0.958315 0.285714i
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 3.16228i 0.200401i
\(250\) 0 0
\(251\) 24.5967i 1.55253i −0.630405 0.776266i \(-0.717111\pi\)
0.630405 0.776266i \(-0.282889\pi\)
\(252\) 0 0
\(253\) 5.00000 + 15.8114i 0.314347 + 0.994053i
\(254\) 0 0
\(255\) −28.2843 −1.77123
\(256\) 0 0
\(257\) 13.4164i 0.836893i 0.908242 + 0.418446i \(0.137425\pi\)
−0.908242 + 0.418446i \(0.862575\pi\)
\(258\) 0 0
\(259\) −23.3345 17.3925i −1.44994 1.08072i
\(260\) 0 0
\(261\) 18.9737i 1.17444i
\(262\) 0 0
\(263\) 12.6491i 0.779978i 0.920820 + 0.389989i \(0.127521\pi\)
−0.920820 + 0.389989i \(0.872479\pi\)
\(264\) 0 0
\(265\) 17.8885i 1.09888i
\(266\) 0 0
\(267\) −15.0000 −0.917985
\(268\) 0 0
\(269\) 17.8885i 1.09068i 0.838214 + 0.545342i \(0.183600\pi\)
−0.838214 + 0.545342i \(0.816400\pi\)
\(270\) 0 0
\(271\) 16.9706 1.03089 0.515444 0.856923i \(-0.327627\pi\)
0.515444 + 0.856923i \(0.327627\pi\)
\(272\) 0 0
\(273\) 10.0000 13.4164i 0.605228 0.811998i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 22.1359i 1.33002i 0.746834 + 0.665010i \(0.231573\pi\)
−0.746834 + 0.665010i \(0.768427\pi\)
\(278\) 0 0
\(279\) 4.47214i 0.267740i
\(280\) 0 0
\(281\) 3.16228i 0.188646i 0.995542 + 0.0943228i \(0.0300686\pi\)
−0.995542 + 0.0943228i \(0.969931\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 0 0
\(285\) 7.07107 0.418854
\(286\) 0 0
\(287\) −21.0000 15.6525i −1.23959 0.923936i
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) −35.0000 −2.05174
\(292\) 0 0
\(293\) 15.5563 0.908812 0.454406 0.890795i \(-0.349852\pi\)
0.454406 + 0.890795i \(0.349852\pi\)
\(294\) 0 0
\(295\) −5.00000 −0.291111
\(296\) 0 0
\(297\) −7.07107 + 2.23607i −0.410305 + 0.129750i
\(298\) 0 0
\(299\) 14.1421 0.817861
\(300\) 0 0
\(301\) −5.00000 + 6.70820i −0.288195 + 0.386654i
\(302\) 0 0
\(303\) 9.48683i 0.545004i
\(304\) 0 0
\(305\) 9.48683i 0.543214i
\(306\) 0 0
\(307\) −19.7990 −1.12999 −0.564994 0.825095i \(-0.691122\pi\)
−0.564994 + 0.825095i \(0.691122\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 22.3607i 1.26796i 0.773350 + 0.633979i \(0.218579\pi\)
−0.773350 + 0.633979i \(0.781421\pi\)
\(312\) 0 0
\(313\) 20.1246i 1.13751i −0.822507 0.568755i \(-0.807425\pi\)
0.822507 0.568755i \(-0.192575\pi\)
\(314\) 0 0
\(315\) 7.07107 9.48683i 0.398410 0.534522i
\(316\) 0 0
\(317\) −15.0000 −0.842484 −0.421242 0.906948i \(-0.638406\pi\)
−0.421242 + 0.906948i \(0.638406\pi\)
\(318\) 0 0
\(319\) −30.0000 + 9.48683i −1.67968 + 0.531161i
\(320\) 0 0
\(321\) 14.1421 0.789337
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 35.3553 1.95515
\(328\) 0 0
\(329\) 7.07107 9.48683i 0.389841 0.523026i
\(330\) 0 0
\(331\) 1.00000 0.0549650 0.0274825 0.999622i \(-0.491251\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) 0 0
\(333\) −22.0000 −1.20559
\(334\) 0 0
\(335\) 20.1246i 1.09952i
\(336\) 0 0
\(337\) 25.2982i 1.37808i −0.724722 0.689041i \(-0.758032\pi\)
0.724722 0.689041i \(-0.241968\pi\)
\(338\) 0 0
\(339\) 6.70820i 0.364340i
\(340\) 0 0
\(341\) −7.07107 + 2.23607i −0.382920 + 0.121090i
\(342\) 0 0
\(343\) 6.36396 17.3925i 0.343622 0.939108i
\(344\) 0 0
\(345\) 25.0000 1.34595
\(346\) 0 0
\(347\) 3.16228i 0.169760i −0.996391 0.0848800i \(-0.972949\pi\)
0.996391 0.0848800i \(-0.0270507\pi\)
\(348\) 0 0
\(349\) −22.6274 −1.21122 −0.605609 0.795762i \(-0.707070\pi\)
−0.605609 + 0.795762i \(0.707070\pi\)
\(350\) 0 0
\(351\) 6.32456i 0.337580i
\(352\) 0 0
\(353\) 11.1803i 0.595069i 0.954711 + 0.297535i \(0.0961644\pi\)
−0.954711 + 0.297535i \(0.903836\pi\)
\(354\) 0 0
\(355\) 2.23607i 0.118678i
\(356\) 0 0
\(357\) −20.0000 + 26.8328i −1.05851 + 1.42014i
\(358\) 0 0
\(359\) 25.2982i 1.33519i −0.744525 0.667595i \(-0.767324\pi\)
0.744525 0.667595i \(-0.232676\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) −14.1421 20.1246i −0.742270 1.05627i
\(364\) 0 0
\(365\) 9.48683i 0.496564i
\(366\) 0 0
\(367\) 6.70820i 0.350165i −0.984554 0.175083i \(-0.943981\pi\)
0.984554 0.175083i \(-0.0560193\pi\)
\(368\) 0 0
\(369\) −19.7990 −1.03069
\(370\) 0 0
\(371\) −16.9706 12.6491i −0.881068 0.656709i
\(372\) 0 0
\(373\) 34.7851i 1.80110i −0.434751 0.900551i \(-0.643163\pi\)
0.434751 0.900551i \(-0.356837\pi\)
\(374\) 0 0
\(375\) 25.0000 1.29099
\(376\) 0 0
\(377\) 26.8328i 1.38196i
\(378\) 0 0
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 0 0
\(381\) 35.3553 1.81131
\(382\) 0 0
\(383\) 15.6525i 0.799804i 0.916558 + 0.399902i \(0.130956\pi\)
−0.916558 + 0.399902i \(0.869044\pi\)
\(384\) 0 0
\(385\) −18.5355 6.43692i −0.944658 0.328056i
\(386\) 0 0
\(387\) 6.32456i 0.321495i
\(388\) 0 0
\(389\) −17.0000 −0.861934 −0.430967 0.902368i \(-0.641828\pi\)
−0.430967 + 0.902368i \(0.641828\pi\)
\(390\) 0 0
\(391\) −28.2843 −1.43040
\(392\) 0 0
\(393\) 31.6228i 1.59516i
\(394\) 0 0
\(395\) 14.1421 0.711568
\(396\) 0 0
\(397\) 17.8885i 0.897800i −0.893582 0.448900i \(-0.851816\pi\)
0.893582 0.448900i \(-0.148184\pi\)
\(398\) 0 0
\(399\) 5.00000 6.70820i 0.250313 0.335830i
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 6.32456i 0.315049i
\(404\) 0 0
\(405\) 24.5967i 1.22222i
\(406\) 0 0
\(407\) 11.0000 + 34.7851i 0.545250 + 1.72423i
\(408\) 0 0
\(409\) 5.65685 0.279713 0.139857 0.990172i \(-0.455336\pi\)
0.139857 + 0.990172i \(0.455336\pi\)
\(410\) 0 0
\(411\) 20.1246i 0.992674i
\(412\) 0 0
\(413\) −3.53553 + 4.74342i −0.173972 + 0.233408i
\(414\) 0 0
\(415\) 3.16228i 0.155230i
\(416\) 0 0
\(417\) 50.5964i 2.47772i
\(418\) 0 0
\(419\) 31.3050i 1.52935i 0.644418 + 0.764673i \(0.277100\pi\)
−0.644418 + 0.764673i \(0.722900\pi\)
\(420\) 0 0
\(421\) −36.0000 −1.75453 −0.877266 0.480004i \(-0.840635\pi\)
−0.877266 + 0.480004i \(0.840635\pi\)
\(422\) 0 0
\(423\) 8.94427i 0.434885i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.00000 + 6.70820i 0.435541 + 0.324633i
\(428\) 0 0
\(429\) −20.0000 + 6.32456i −0.965609 + 0.305352i
\(430\) 0 0
\(431\) 15.8114i 0.761608i −0.924656 0.380804i \(-0.875647\pi\)
0.924656 0.380804i \(-0.124353\pi\)
\(432\) 0 0
\(433\) 24.5967i 1.18204i −0.806655 0.591022i \(-0.798725\pi\)
0.806655 0.591022i \(-0.201275\pi\)
\(434\) 0 0
\(435\) 47.4342i 2.27429i
\(436\) 0 0
\(437\) 7.07107 0.338255
\(438\) 0 0
\(439\) 26.8701 1.28244 0.641219 0.767358i \(-0.278429\pi\)
0.641219 + 0.767358i \(0.278429\pi\)
\(440\) 0 0
\(441\) −4.00000 13.4164i −0.190476 0.638877i
\(442\) 0 0
\(443\) −27.0000 −1.28281 −0.641404 0.767203i \(-0.721648\pi\)
−0.641404 + 0.767203i \(0.721648\pi\)
\(444\) 0 0
\(445\) 15.0000 0.711068
\(446\) 0 0
\(447\) −14.1421 −0.668900
\(448\) 0 0
\(449\) 37.0000 1.74614 0.873069 0.487597i \(-0.162126\pi\)
0.873069 + 0.487597i \(0.162126\pi\)
\(450\) 0 0
\(451\) 9.89949 + 31.3050i 0.466149 + 1.47409i
\(452\) 0 0
\(453\) −28.2843 −1.32891
\(454\) 0 0
\(455\) −10.0000 + 13.4164i −0.468807 + 0.628971i
\(456\) 0 0
\(457\) 18.9737i 0.887551i −0.896138 0.443775i \(-0.853639\pi\)
0.896138 0.443775i \(-0.146361\pi\)
\(458\) 0 0
\(459\) 12.6491i 0.590410i
\(460\) 0 0
\(461\) −9.89949 −0.461065 −0.230533 0.973065i \(-0.574047\pi\)
−0.230533 + 0.973065i \(0.574047\pi\)
\(462\) 0 0
\(463\) 7.00000 0.325318 0.162659 0.986682i \(-0.447993\pi\)
0.162659 + 0.986682i \(0.447993\pi\)
\(464\) 0 0
\(465\) 11.1803i 0.518476i
\(466\) 0 0
\(467\) 6.70820i 0.310419i −0.987882 0.155209i \(-0.950395\pi\)
0.987882 0.155209i \(-0.0496052\pi\)
\(468\) 0 0
\(469\) 19.0919 + 14.2302i 0.881581 + 0.657092i
\(470\) 0 0
\(471\) 15.0000 0.691164
\(472\) 0 0
\(473\) 10.0000 3.16228i 0.459800 0.145402i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −16.0000 −0.732590
\(478\) 0 0
\(479\) −19.7990 −0.904639 −0.452319 0.891856i \(-0.649403\pi\)
−0.452319 + 0.891856i \(0.649403\pi\)
\(480\) 0 0
\(481\) 31.1127 1.41862
\(482\) 0 0
\(483\) 17.6777 23.7171i 0.804362 1.07916i
\(484\) 0 0
\(485\) 35.0000 1.58927
\(486\) 0 0
\(487\) 21.0000 0.951601 0.475800 0.879553i \(-0.342158\pi\)
0.475800 + 0.879553i \(0.342158\pi\)
\(488\) 0 0
\(489\) 22.3607i 1.01118i
\(490\) 0 0
\(491\) 9.48683i 0.428135i 0.976819 + 0.214067i \(0.0686712\pi\)
−0.976819 + 0.214067i \(0.931329\pi\)
\(492\) 0 0
\(493\) 53.6656i 2.41698i
\(494\) 0 0
\(495\) −14.1421 + 4.47214i −0.635642 + 0.201008i
\(496\) 0 0
\(497\) −2.12132 1.58114i −0.0951542 0.0709238i
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 0 0
\(501\) 3.16228i 0.141280i
\(502\) 0 0
\(503\) −32.5269 −1.45030 −0.725152 0.688589i \(-0.758230\pi\)
−0.725152 + 0.688589i \(0.758230\pi\)
\(504\) 0 0
\(505\) 9.48683i 0.422159i
\(506\) 0 0
\(507\) 11.1803i 0.496536i
\(508\) 0 0
\(509\) 2.23607i 0.0991120i 0.998771 + 0.0495560i \(0.0157806\pi\)
−0.998771 + 0.0495560i \(0.984219\pi\)
\(510\) 0 0
\(511\) 9.00000 + 6.70820i 0.398137 + 0.296753i
\(512\) 0 0
\(513\) 3.16228i 0.139618i
\(514\) 0 0
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) −14.1421 + 4.47214i −0.621970 + 0.196684i
\(518\) 0 0
\(519\) 28.4605i 1.24928i
\(520\) 0 0
\(521\) 2.23607i 0.0979639i 0.998800 + 0.0489820i \(0.0155977\pi\)
−0.998800 + 0.0489820i \(0.984402\pi\)
\(522\) 0 0
\(523\) −2.82843 −0.123678 −0.0618392 0.998086i \(-0.519697\pi\)
−0.0618392 + 0.998086i \(0.519697\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.6491i 0.551004i
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 4.47214i 0.194074i
\(532\) 0 0
\(533\) 28.0000 1.21281
\(534\) 0 0
\(535\) −14.1421 −0.611418
\(536\) 0 0
\(537\) 29.0689i 1.25441i
\(538\) 0 0
\(539\) −19.2132 + 13.0328i −0.827571 + 0.561361i
\(540\) 0 0
\(541\) 31.6228i 1.35957i −0.733412 0.679785i \(-0.762073\pi\)
0.733412 0.679785i \(-0.237927\pi\)
\(542\) 0 0
\(543\) 45.0000 1.93113
\(544\) 0 0
\(545\) −35.3553 −1.51446
\(546\) 0 0
\(547\) 44.2719i 1.89293i −0.322808 0.946465i \(-0.604627\pi\)
0.322808 0.946465i \(-0.395373\pi\)
\(548\) 0 0
\(549\) 8.48528 0.362143
\(550\) 0 0
\(551\) 13.4164i 0.571558i
\(552\) 0 0
\(553\) 10.0000 13.4164i 0.425243 0.570524i
\(554\) 0 0
\(555\) 55.0000 2.33462
\(556\) 0 0
\(557\) 12.6491i 0.535960i 0.963424 + 0.267980i \(0.0863561\pi\)
−0.963424 + 0.267980i \(0.913644\pi\)
\(558\) 0 0
\(559\) 8.94427i 0.378302i
\(560\) 0 0
\(561\) 40.0000 12.6491i 1.68880 0.534046i
\(562\) 0 0
\(563\) 39.5980 1.66886 0.834428 0.551117i \(-0.185798\pi\)
0.834428 + 0.551117i \(0.185798\pi\)
\(564\) 0 0
\(565\) 6.70820i 0.282216i
\(566\) 0 0
\(567\) 23.3345 + 17.3925i 0.979958 + 0.730417i
\(568\) 0 0
\(569\) 9.48683i 0.397709i 0.980029 + 0.198854i \(0.0637221\pi\)
−0.980029 + 0.198854i \(0.936278\pi\)
\(570\) 0 0
\(571\) 12.6491i 0.529349i −0.964338 0.264674i \(-0.914736\pi\)
0.964338 0.264674i \(-0.0852645\pi\)
\(572\) 0 0
\(573\) 6.70820i 0.280239i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.0689i 1.21015i 0.796167 + 0.605077i \(0.206858\pi\)
−0.796167 + 0.605077i \(0.793142\pi\)
\(578\) 0 0
\(579\) −35.3553 −1.46932
\(580\) 0 0
\(581\) −3.00000 2.23607i −0.124461 0.0927677i
\(582\) 0 0
\(583\) 8.00000 + 25.2982i 0.331326 + 1.04775i
\(584\) 0 0
\(585\) 12.6491i 0.522976i
\(586\) 0 0
\(587\) 13.4164i 0.553754i −0.960905 0.276877i \(-0.910700\pi\)
0.960905 0.276877i \(-0.0892995\pi\)
\(588\) 0 0
\(589\) 3.16228i 0.130299i
\(590\) 0 0
\(591\) 28.2843 1.16346
\(592\) 0 0
\(593\) 1.41421 0.0580748 0.0290374 0.999578i \(-0.490756\pi\)
0.0290374 + 0.999578i \(0.490756\pi\)
\(594\) 0 0
\(595\) 20.0000 26.8328i 0.819920 1.10004i
\(596\) 0 0
\(597\) 40.0000 1.63709
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 21.2132 0.865305 0.432652 0.901561i \(-0.357578\pi\)
0.432652 + 0.901561i \(0.357578\pi\)
\(602\) 0 0
\(603\) 18.0000 0.733017
\(604\) 0 0
\(605\) 14.1421 + 20.1246i 0.574960 + 0.818182i
\(606\) 0 0
\(607\) −25.4558 −1.03322 −0.516610 0.856221i \(-0.672806\pi\)
−0.516610 + 0.856221i \(0.672806\pi\)
\(608\) 0 0
\(609\) 45.0000 + 33.5410i 1.82349 + 1.35915i
\(610\) 0 0
\(611\) 12.6491i 0.511728i
\(612\) 0 0
\(613\) 18.9737i 0.766339i 0.923678 + 0.383170i \(0.125168\pi\)
−0.923678 + 0.383170i \(0.874832\pi\)
\(614\) 0 0
\(615\) 49.4975 1.99593
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) 29.0689i 1.16838i 0.811618 + 0.584189i \(0.198587\pi\)
−0.811618 + 0.584189i \(0.801413\pi\)
\(620\) 0 0
\(621\) 11.1803i 0.448652i
\(622\) 0 0
\(623\) 10.6066 14.2302i 0.424945 0.570123i
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) −10.0000 + 3.16228i −0.399362 + 0.126289i
\(628\) 0 0
\(629\) −62.2254 −2.48109
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 0 0
\(633\) −21.2132 −0.843149
\(634\) 0 0
\(635\) −35.3553 −1.40303
\(636\) 0 0
\(637\) 5.65685 + 18.9737i 0.224133 + 0.751764i
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −11.0000 −0.434474 −0.217237 0.976119i \(-0.569704\pi\)
−0.217237 + 0.976119i \(0.569704\pi\)
\(642\) 0 0
\(643\) 11.1803i 0.440910i 0.975397 + 0.220455i \(0.0707541\pi\)
−0.975397 + 0.220455i \(0.929246\pi\)
\(644\) 0 0
\(645\) 15.8114i 0.622573i
\(646\) 0 0
\(647\) 38.0132i 1.49445i −0.664570 0.747226i \(-0.731385\pi\)
0.664570 0.747226i \(-0.268615\pi\)
\(648\) 0 0
\(649\) 7.07107 2.23607i 0.277564 0.0877733i
\(650\) 0 0
\(651\) 10.6066 + 7.90569i 0.415705 + 0.309849i
\(652\) 0 0
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) 31.6228i 1.23560i
\(656\) 0 0
\(657\) 8.48528 0.331042
\(658\) 0 0
\(659\) 18.9737i 0.739109i 0.929209 + 0.369555i \(0.120490\pi\)
−0.929209 + 0.369555i \(0.879510\pi\)
\(660\) 0 0
\(661\) 6.70820i 0.260919i 0.991454 + 0.130459i \(0.0416452\pi\)
−0.991454 + 0.130459i \(0.958355\pi\)
\(662\) 0 0
\(663\) 35.7771i 1.38947i
\(664\) 0 0
\(665\) −5.00000 + 6.70820i −0.193892 + 0.260133i
\(666\) 0 0
\(667\) 47.4342i 1.83666i
\(668\) 0 0
\(669\) 45.0000 1.73980
\(670\) 0 0
\(671\) −4.24264 13.4164i −0.163785 0.517935i
\(672\) 0 0
\(673\) 9.48683i 0.365691i −0.983142 0.182845i \(-0.941469\pi\)
0.983142 0.182845i \(-0.0585307\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.6274 0.869642 0.434821 0.900517i \(-0.356812\pi\)
0.434821 + 0.900517i \(0.356812\pi\)
\(678\) 0 0
\(679\) 24.7487 33.2039i 0.949769 1.27425i
\(680\) 0 0
\(681\) 66.4078i 2.54475i
\(682\) 0 0
\(683\) −30.0000 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) 0 0
\(685\) 20.1246i 0.768922i
\(686\) 0 0
\(687\) −25.0000 −0.953809
\(688\) 0 0
\(689\) 22.6274 0.862036
\(690\) 0 0
\(691\) 6.70820i 0.255192i 0.991826 + 0.127596i \(0.0407261\pi\)
−0.991826 + 0.127596i \(0.959274\pi\)
\(692\) 0 0
\(693\) −5.75736 + 16.5787i −0.218704 + 0.629772i
\(694\) 0 0
\(695\) 50.5964i 1.91923i
\(696\) 0 0
\(697\) −56.0000 −2.12115
\(698\) 0 0
\(699\) −21.2132 −0.802357
\(700\) 0 0
\(701\) 9.48683i 0.358313i −0.983821 0.179156i \(-0.942663\pi\)
0.983821 0.179156i \(-0.0573368\pi\)
\(702\) 0 0
\(703\) 15.5563 0.586719
\(704\) 0 0
\(705\) 22.3607i 0.842152i
\(706\) 0 0
\(707\) 9.00000 + 6.70820i 0.338480 + 0.252288i
\(708\) 0 0
\(709\) 13.0000 0.488225 0.244113 0.969747i \(-0.421503\pi\)
0.244113 + 0.969747i \(0.421503\pi\)
\(710\) 0 0
\(711\) 12.6491i 0.474379i
\(712\) 0 0
\(713\) 11.1803i 0.418707i
\(714\) 0 0
\(715\) 20.0000 6.32456i 0.747958 0.236525i
\(716\) 0 0
\(717\) −35.3553 −1.32037
\(718\) 0 0
\(719\) 11.1803i 0.416956i −0.978027 0.208478i \(-0.933149\pi\)
0.978027 0.208478i \(-0.0668510\pi\)
\(720\) 0 0
\(721\) −7.07107 + 9.48683i −0.263340 + 0.353308i
\(722\) 0 0
\(723\) 37.9473i 1.41128i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.70820i 0.248794i −0.992233 0.124397i \(-0.960300\pi\)
0.992233 0.124397i \(-0.0396996\pi\)
\(728\) 0 0
\(729\) 7.00000 0.259259
\(730\) 0 0
\(731\) 17.8885i 0.661632i
\(732\) 0 0
\(733\) 50.9117 1.88047 0.940233 0.340532i \(-0.110607\pi\)
0.940233 + 0.340532i \(0.110607\pi\)
\(734\) 0 0
\(735\) 10.0000 + 33.5410i 0.368856 + 1.23718i
\(736\) 0 0
\(737\) −9.00000 28.4605i −0.331519 1.04836i
\(738\) 0 0
\(739\) 37.9473i 1.39592i 0.716139 + 0.697958i \(0.245908\pi\)
−0.716139 + 0.697958i \(0.754092\pi\)
\(740\) 0 0
\(741\) 8.94427i 0.328576i
\(742\) 0 0
\(743\) 18.9737i 0.696076i 0.937480 + 0.348038i \(0.113152\pi\)
−0.937480 + 0.348038i \(0.886848\pi\)
\(744\) 0 0
\(745\) 14.1421 0.518128
\(746\) 0 0
\(747\) −2.82843 −0.103487
\(748\) 0 0
\(749\) −10.0000 + 13.4164i −0.365392 + 0.490225i
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 0 0
\(753\) 55.0000 2.00431
\(754\) 0 0
\(755\) 28.2843 1.02937
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) −35.3553 + 11.1803i −1.28332 + 0.405821i
\(760\) 0 0
\(761\) −45.2548 −1.64049 −0.820243 0.572015i \(-0.806162\pi\)
−0.820243 + 0.572015i \(0.806162\pi\)
\(762\) 0 0
\(763\) −25.0000 + 33.5410i −0.905061 + 1.21427i
\(764\) 0 0
\(765\) 25.2982i 0.914659i
\(766\) 0 0
\(767\) 6.32456i 0.228366i
\(768\) 0 0
\(769\) −42.4264 −1.52994 −0.764968 0.644069i \(-0.777245\pi\)
−0.764968 + 0.644069i \(0.777245\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 0 0
\(773\) 44.7214i 1.60852i 0.594281 + 0.804258i \(0.297437\pi\)
−0.594281 + 0.804258i \(0.702563\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 38.8909 52.1776i 1.39520 1.87186i
\(778\) 0 0
\(779\) 14.0000 0.501602
\(780\) 0 0
\(781\) 1.00000 + 3.16228i 0.0357828 + 0.113155i
\(782\) 0 0
\(783\) −21.2132 −0.758098
\(784\) 0 0
\(785\) −15.0000 −0.535373
\(786\) 0 0
\(787\) 15.5563 0.554524 0.277262 0.960794i \(-0.410573\pi\)
0.277262 + 0.960794i \(0.410573\pi\)
\(788\) 0 0
\(789\) −28.2843 −1.00695
\(790\) 0 0
\(791\) 6.36396 + 4.74342i 0.226276 + 0.168656i
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 40.0000 1.41865
\(796\) 0 0
\(797\) 33.5410i 1.18808i −0.804434 0.594042i \(-0.797531\pi\)
0.804434 0.594042i \(-0.202469\pi\)
\(798\) 0 0
\(799\) 25.2982i 0.894987i
\(800\) 0 0
\(801\) 13.4164i 0.474045i
\(802\) 0 0
\(803\) −4.24264 13.4164i −0.149720 0.473455i
\(804\) 0 0
\(805\) −17.6777 + 23.7171i −0.623056 + 0.835917i
\(806\) 0 0
\(807\) −40.0000 −1.40807
\(808\) 0 0
\(809\) 28.4605i 1.00062i 0.865847 + 0.500309i \(0.166780\pi\)
−0.865847 + 0.500309i \(0.833220\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 37.9473i 1.33087i
\(814\) 0 0
\(815\) 22.3607i 0.783260i
\(816\) 0 0
\(817\) 4.47214i 0.156460i
\(818\) 0 0
\(819\) 12.0000 + 8.94427i 0.419314 + 0.312538i
\(820\) 0 0
\(821\) 6.32456i 0.220729i −0.993891 0.110364i \(-0.964798\pi\)
0.993891 0.110364i \(-0.0352017\pi\)
\(822\) 0 0
\(823\) −35.0000 −1.22002 −0.610012 0.792392i \(-0.708835\pi\)
−0.610012 + 0.792392i \(0.708835\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.9473i 1.31956i −0.751460 0.659779i \(-0.770650\pi\)
0.751460 0.659779i \(-0.229350\pi\)
\(828\) 0 0
\(829\) 15.6525i 0.543633i −0.962349 0.271816i \(-0.912376\pi\)
0.962349 0.271816i \(-0.0876244\pi\)
\(830\) 0 0
\(831\) −49.4975 −1.71705
\(832\) 0 0
\(833\) −11.3137 37.9473i −0.391997 1.31480i
\(834\) 0 0
\(835\) 3.16228i 0.109435i
\(836\) 0 0
\(837\) −5.00000 −0.172825
\(838\) 0 0
\(839\) 29.0689i 1.00357i 0.864993 + 0.501785i \(0.167323\pi\)
−0.864993 + 0.501785i \(0.832677\pi\)
\(840\) 0 0
\(841\) −61.0000 −2.10345
\(842\) 0 0
\(843\) −7.07107 −0.243541
\(844\) 0 0
\(845\) 11.1803i 0.384615i
\(846\) 0 0
\(847\) 29.0919 + 0.813842i 0.999609 + 0.0279639i
\(848\) 0 0
\(849\) 18.9737i 0.651175i
\(850\) 0 0
\(851\) 55.0000 1.88538
\(852\) 0 0
\(853\) 45.2548 1.54950 0.774748 0.632270i \(-0.217877\pi\)
0.774748 + 0.632270i \(0.217877\pi\)
\(854\) 0 0
\(855\) 6.32456i 0.216295i
\(856\) 0 0
\(857\) 19.7990 0.676321 0.338160 0.941089i \(-0.390195\pi\)
0.338160 + 0.941089i \(0.390195\pi\)
\(858\) 0 0
\(859\) 46.9574i 1.60217i 0.598553 + 0.801083i \(0.295743\pi\)
−0.598553 + 0.801083i \(0.704257\pi\)
\(860\) 0 0
\(861\) 35.0000 46.9574i 1.19280 1.60030i
\(862\) 0 0
\(863\) 18.0000 0.612727 0.306364 0.951915i \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) 0 0
\(865\) 28.4605i 0.967686i
\(866\) 0 0
\(867\) 33.5410i 1.13911i
\(868\) 0 0
\(869\) −20.0000 + 6.32456i −0.678454 + 0.214546i
\(870\) 0 0
\(871\) −25.4558 −0.862538
\(872\) 0 0
\(873\) 31.3050i 1.05951i
\(874\) 0 0
\(875\) −17.6777 + 23.7171i −0.597614 + 0.801784i
\(876\) 0 0
\(877\) 34.7851i 1.17461i −0.809366 0.587304i \(-0.800189\pi\)
0.809366 0.587304i \(-0.199811\pi\)
\(878\) 0 0
\(879\) 34.7851i 1.17327i
\(880\) 0 0
\(881\) 42.4853i 1.43137i 0.698425 + 0.715683i \(0.253884\pi\)
−0.698425 + 0.715683i \(0.746116\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) 11.1803i 0.375823i
\(886\) 0 0
\(887\) 36.7696 1.23460 0.617300 0.786728i \(-0.288226\pi\)
0.617300 + 0.786728i \(0.288226\pi\)
\(888\) 0 0
\(889\) −25.0000 + 33.5410i −0.838473 + 1.12493i
\(890\) 0 0
\(891\) −11.0000 34.7851i −0.368514 1.16534i
\(892\) 0 0
\(893\) 6.32456i 0.211643i
\(894\) 0 0
\(895\) 29.0689i 0.971666i
\(896\) 0 0
\(897\) 31.6228i 1.05585i
\(898\) 0 0
\(899\) −21.2132 −0.707500
\(900\) 0 0
\(901\) −45.2548 −1.50766
\(902\) 0 0
\(903\) −15.0000 11.1803i −0.499169 0.372058i
\(904\) 0 0
\(905\) −45.0000 −1.49585
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 0 0
\(909\) 8.48528 0.281439
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) 1.41421 + 4.47214i 0.0468036 + 0.148006i
\(914\) 0 0
\(915\) −21.2132 −0.701287
\(916\) 0 0
\(917\) 30.0000 + 22.3607i 0.990687 + 0.738415i
\(918\) 0 0
\(919\) 3.16228i 0.104314i 0.998639 + 0.0521570i \(0.0166096\pi\)
−0.998639 + 0.0521570i \(0.983390\pi\)
\(920\) 0 0
\(921\) 44.2719i 1.45881i
\(922\) 0 0
\(923\) 2.82843 0.0930988
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.94427i 0.293768i
\(928\) 0 0
\(929\) 17.8885i 0.586904i 0.955974 + 0.293452i \(0.0948041\pi\)
−0.955974 + 0.293452i \(0.905196\pi\)
\(930\) 0 0
\(931\) 2.82843 + 9.48683i 0.0926980 + 0.310918i
\(932\) 0 0
\(933\) −50.0000 −1.63693
\(934\) 0 0
\(935\) −40.0000 + 12.6491i −1.30814 + 0.413670i
\(936\) 0 0
\(937\) 5.65685 0.184801 0.0924007 0.995722i \(-0.470546\pi\)
0.0924007 + 0.995722i \(0.470546\pi\)
\(938\) 0 0
\(939\) 45.0000 1.46852
\(940\) 0 0
\(941\) 28.2843 0.922041 0.461020 0.887390i \(-0.347483\pi\)
0.461020 + 0.887390i \(0.347483\pi\)
\(942\) 0 0
\(943\) 49.4975 1.61186
\(944\) 0 0
\(945\) −10.6066 7.90569i −0.345033 0.257172i
\(946\) 0 0
\(947\) 61.0000 1.98223 0.991117 0.132994i \(-0.0424591\pi\)
0.991117 + 0.132994i \(0.0424591\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 33.5410i 1.08764i
\(952\) 0 0
\(953\) 9.48683i 0.307309i −0.988125 0.153654i \(-0.950896\pi\)
0.988125 0.153654i \(-0.0491042\pi\)
\(954\) 0 0
\(955\) 6.70820i 0.217072i
\(956\) 0 0
\(957\) −21.2132 67.0820i −0.685725 2.16845i
\(958\) 0 0
\(959\) −19.0919 14.2302i −0.616509 0.459519i
\(960\) 0 0
\(961\) 26.0000 0.838710
\(962\) 0 0
\(963\) 12.6491i 0.407612i
\(964\) 0 0
\(965\) 35.3553 1.13813
\(966\) 0 0
\(967\) 47.4342i 1.52538i −0.646764 0.762690i \(-0.723878\pi\)
0.646764 0.762690i \(-0.276122\pi\)
\(968\) 0 0
\(969\) 17.8885i 0.574663i
\(970\) 0 0
\(971\) 29.0689i 0.932865i −0.884557 0.466432i \(-0.845539\pi\)
0.884557 0.466432i \(-0.154461\pi\)
\(972\) 0 0
\(973\) −48.0000 35.7771i −1.53881 1.14696i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 0 0
\(979\) −21.2132 + 6.70820i −0.677977 + 0.214395i
\(980\) 0 0
\(981\) 31.6228i 1.00964i
\(982\) 0 0
\(983\) 2.23607i 0.0713195i −0.999364 0.0356597i \(-0.988647\pi\)
0.999364 0.0356597i \(-0.0113533\pi\)
\(984\) 0 0
\(985\) −28.2843 −0.901212
\(986\) 0 0
\(987\) 21.2132 + 15.8114i 0.675224 + 0.503282i
\(988\) 0 0
\(989\) 15.8114i 0.502773i
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0 0
\(993\) 2.23607i 0.0709595i
\(994\) 0 0
\(995\) −40.0000 −1.26809
\(996\) 0 0
\(997\) −12.7279 −0.403097 −0.201549 0.979479i \(-0.564597\pi\)
−0.201549 + 0.979479i \(0.564597\pi\)
\(998\) 0 0
\(999\) 24.5967i 0.778207i
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 1232.2.e.b.769.3 4
4.3 odd 2 308.2.c.a.153.2 yes 4
7.6 odd 2 inner 1232.2.e.b.769.2 4
11.10 odd 2 inner 1232.2.e.b.769.4 4
12.11 even 2 2772.2.i.c.1693.4 4
28.3 even 6 2156.2.q.b.901.3 8
28.11 odd 6 2156.2.q.b.901.1 8
28.19 even 6 2156.2.q.b.2089.2 8
28.23 odd 6 2156.2.q.b.2089.4 8
28.27 even 2 308.2.c.a.153.3 yes 4
44.43 even 2 308.2.c.a.153.1 4
77.76 even 2 inner 1232.2.e.b.769.1 4
84.83 odd 2 2772.2.i.c.1693.1 4
132.131 odd 2 2772.2.i.c.1693.3 4
308.87 odd 6 2156.2.q.b.901.4 8
308.131 odd 6 2156.2.q.b.2089.1 8
308.219 even 6 2156.2.q.b.2089.3 8
308.263 even 6 2156.2.q.b.901.2 8
308.307 odd 2 308.2.c.a.153.4 yes 4
924.923 even 2 2772.2.i.c.1693.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
308.2.c.a.153.1 4 44.43 even 2
308.2.c.a.153.2 yes 4 4.3 odd 2
308.2.c.a.153.3 yes 4 28.27 even 2
308.2.c.a.153.4 yes 4 308.307 odd 2
1232.2.e.b.769.1 4 77.76 even 2 inner
1232.2.e.b.769.2 4 7.6 odd 2 inner
1232.2.e.b.769.3 4 1.1 even 1 trivial
1232.2.e.b.769.4 4 11.10 odd 2 inner
2156.2.q.b.901.1 8 28.11 odd 6
2156.2.q.b.901.2 8 308.263 even 6
2156.2.q.b.901.3 8 28.3 even 6
2156.2.q.b.901.4 8 308.87 odd 6
2156.2.q.b.2089.1 8 308.131 odd 6
2156.2.q.b.2089.2 8 28.19 even 6
2156.2.q.b.2089.3 8 308.219 even 6
2156.2.q.b.2089.4 8 28.23 odd 6
2772.2.i.c.1693.1 4 84.83 odd 2
2772.2.i.c.1693.2 4 924.923 even 2
2772.2.i.c.1693.3 4 132.131 odd 2
2772.2.i.c.1693.4 4 12.11 even 2