Properties

Label 5776.2.a.bo.1.3
Level $5776$
Weight $2$
Character 5776.1
Self dual yes
Analytic conductor $46.122$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5776,2,Mod(1,5776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5776.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5776 = 2^{4} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5776.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(46.1215922075\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 5776.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53209 q^{3} +2.00000 q^{5} -2.69459 q^{7} -0.652704 q^{9} +O(q^{10})\) \(q+1.53209 q^{3} +2.00000 q^{5} -2.69459 q^{7} -0.652704 q^{9} -3.18479 q^{11} +5.75877 q^{13} +3.06418 q^{15} -6.51754 q^{17} -4.12836 q^{21} +0.694593 q^{23} -1.00000 q^{25} -5.59627 q^{27} +2.82295 q^{29} +2.45336 q^{31} -4.87939 q^{33} -5.38919 q^{35} +4.36959 q^{37} +8.82295 q^{39} +0.347296 q^{41} -6.06418 q^{43} -1.30541 q^{45} -7.88713 q^{47} +0.260830 q^{49} -9.98545 q^{51} -8.21213 q^{53} -6.36959 q^{55} +0.573978 q^{59} +2.93582 q^{61} +1.75877 q^{63} +11.5175 q^{65} -4.95811 q^{67} +1.06418 q^{69} -8.45336 q^{71} +15.7665 q^{73} -1.53209 q^{75} +8.58172 q^{77} -9.06418 q^{79} -6.61587 q^{81} -8.47565 q^{83} -13.0351 q^{85} +4.32501 q^{87} -7.73917 q^{89} -15.5175 q^{91} +3.75877 q^{93} +0.347296 q^{97} +2.07873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{5} - 6 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{5} - 6 q^{7} - 3 q^{9} - 6 q^{11} + 6 q^{13} + 3 q^{17} + 6 q^{21} - 3 q^{25} - 3 q^{27} - 12 q^{29} - 6 q^{31} - 9 q^{33} - 12 q^{35} + 6 q^{37} + 6 q^{39} - 9 q^{43} - 6 q^{45} + 6 q^{47} + 15 q^{49} - 12 q^{51} - 12 q^{55} - 6 q^{59} + 18 q^{61} - 6 q^{63} + 12 q^{65} - 18 q^{67} - 6 q^{69} - 12 q^{71} + 12 q^{73} - 6 q^{77} - 18 q^{79} - 9 q^{81} - 6 q^{83} + 6 q^{85} + 18 q^{87} - 9 q^{89} - 24 q^{91} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.53209 0.884552 0.442276 0.896879i \(-0.354171\pi\)
0.442276 + 0.896879i \(0.354171\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −2.69459 −1.01846 −0.509230 0.860630i \(-0.670070\pi\)
−0.509230 + 0.860630i \(0.670070\pi\)
\(8\) 0 0
\(9\) −0.652704 −0.217568
\(10\) 0 0
\(11\) −3.18479 −0.960251 −0.480126 0.877200i \(-0.659409\pi\)
−0.480126 + 0.877200i \(0.659409\pi\)
\(12\) 0 0
\(13\) 5.75877 1.59720 0.798598 0.601865i \(-0.205576\pi\)
0.798598 + 0.601865i \(0.205576\pi\)
\(14\) 0 0
\(15\) 3.06418 0.791167
\(16\) 0 0
\(17\) −6.51754 −1.58074 −0.790368 0.612632i \(-0.790111\pi\)
−0.790368 + 0.612632i \(0.790111\pi\)
\(18\) 0 0
\(19\) 0 0
\(20\) 0 0
\(21\) −4.12836 −0.900881
\(22\) 0 0
\(23\) 0.694593 0.144833 0.0724163 0.997374i \(-0.476929\pi\)
0.0724163 + 0.997374i \(0.476929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.59627 −1.07700
\(28\) 0 0
\(29\) 2.82295 0.524208 0.262104 0.965040i \(-0.415584\pi\)
0.262104 + 0.965040i \(0.415584\pi\)
\(30\) 0 0
\(31\) 2.45336 0.440637 0.220319 0.975428i \(-0.429290\pi\)
0.220319 + 0.975428i \(0.429290\pi\)
\(32\) 0 0
\(33\) −4.87939 −0.849392
\(34\) 0 0
\(35\) −5.38919 −0.910939
\(36\) 0 0
\(37\) 4.36959 0.718355 0.359178 0.933269i \(-0.383057\pi\)
0.359178 + 0.933269i \(0.383057\pi\)
\(38\) 0 0
\(39\) 8.82295 1.41280
\(40\) 0 0
\(41\) 0.347296 0.0542386 0.0271193 0.999632i \(-0.491367\pi\)
0.0271193 + 0.999632i \(0.491367\pi\)
\(42\) 0 0
\(43\) −6.06418 −0.924778 −0.462389 0.886677i \(-0.653008\pi\)
−0.462389 + 0.886677i \(0.653008\pi\)
\(44\) 0 0
\(45\) −1.30541 −0.194599
\(46\) 0 0
\(47\) −7.88713 −1.15046 −0.575228 0.817993i \(-0.695087\pi\)
−0.575228 + 0.817993i \(0.695087\pi\)
\(48\) 0 0
\(49\) 0.260830 0.0372614
\(50\) 0 0
\(51\) −9.98545 −1.39824
\(52\) 0 0
\(53\) −8.21213 −1.12802 −0.564012 0.825767i \(-0.690743\pi\)
−0.564012 + 0.825767i \(0.690743\pi\)
\(54\) 0 0
\(55\) −6.36959 −0.858875
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.573978 0.0747256 0.0373628 0.999302i \(-0.488104\pi\)
0.0373628 + 0.999302i \(0.488104\pi\)
\(60\) 0 0
\(61\) 2.93582 0.375894 0.187947 0.982179i \(-0.439817\pi\)
0.187947 + 0.982179i \(0.439817\pi\)
\(62\) 0 0
\(63\) 1.75877 0.221584
\(64\) 0 0
\(65\) 11.5175 1.42858
\(66\) 0 0
\(67\) −4.95811 −0.605730 −0.302865 0.953034i \(-0.597943\pi\)
−0.302865 + 0.953034i \(0.597943\pi\)
\(68\) 0 0
\(69\) 1.06418 0.128112
\(70\) 0 0
\(71\) −8.45336 −1.00323 −0.501615 0.865091i \(-0.667261\pi\)
−0.501615 + 0.865091i \(0.667261\pi\)
\(72\) 0 0
\(73\) 15.7665 1.84533 0.922665 0.385602i \(-0.126006\pi\)
0.922665 + 0.385602i \(0.126006\pi\)
\(74\) 0 0
\(75\) −1.53209 −0.176910
\(76\) 0 0
\(77\) 8.58172 0.977978
\(78\) 0 0
\(79\) −9.06418 −1.01980 −0.509900 0.860234i \(-0.670318\pi\)
−0.509900 + 0.860234i \(0.670318\pi\)
\(80\) 0 0
\(81\) −6.61587 −0.735096
\(82\) 0 0
\(83\) −8.47565 −0.930324 −0.465162 0.885226i \(-0.654004\pi\)
−0.465162 + 0.885226i \(0.654004\pi\)
\(84\) 0 0
\(85\) −13.0351 −1.41385
\(86\) 0 0
\(87\) 4.32501 0.463689
\(88\) 0 0
\(89\) −7.73917 −0.820350 −0.410175 0.912007i \(-0.634532\pi\)
−0.410175 + 0.912007i \(0.634532\pi\)
\(90\) 0 0
\(91\) −15.5175 −1.62668
\(92\) 0 0
\(93\) 3.75877 0.389766
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.347296 0.0352626 0.0176313 0.999845i \(-0.494387\pi\)
0.0176313 + 0.999845i \(0.494387\pi\)
\(98\) 0 0
\(99\) 2.07873 0.208920
\(100\) 0 0
\(101\) −0.369585 −0.0367751 −0.0183875 0.999831i \(-0.505853\pi\)
−0.0183875 + 0.999831i \(0.505853\pi\)
\(102\) 0 0
\(103\) 8.58172 0.845582 0.422791 0.906227i \(-0.361050\pi\)
0.422791 + 0.906227i \(0.361050\pi\)
\(104\) 0 0
\(105\) −8.25671 −0.805772
\(106\) 0 0
\(107\) −11.4534 −1.10724 −0.553619 0.832770i \(-0.686754\pi\)
−0.553619 + 0.832770i \(0.686754\pi\)
\(108\) 0 0
\(109\) −8.69459 −0.832791 −0.416395 0.909184i \(-0.636707\pi\)
−0.416395 + 0.909184i \(0.636707\pi\)
\(110\) 0 0
\(111\) 6.69459 0.635423
\(112\) 0 0
\(113\) −2.85978 −0.269026 −0.134513 0.990912i \(-0.542947\pi\)
−0.134513 + 0.990912i \(0.542947\pi\)
\(114\) 0 0
\(115\) 1.38919 0.129542
\(116\) 0 0
\(117\) −3.75877 −0.347498
\(118\) 0 0
\(119\) 17.5621 1.60992
\(120\) 0 0
\(121\) −0.857097 −0.0779179
\(122\) 0 0
\(123\) 0.532089 0.0479768
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 9.72967 0.863369 0.431685 0.902025i \(-0.357919\pi\)
0.431685 + 0.902025i \(0.357919\pi\)
\(128\) 0 0
\(129\) −9.29086 −0.818015
\(130\) 0 0
\(131\) 6.46791 0.565104 0.282552 0.959252i \(-0.408819\pi\)
0.282552 + 0.959252i \(0.408819\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −11.1925 −0.963300
\(136\) 0 0
\(137\) −11.6604 −0.996219 −0.498109 0.867114i \(-0.665972\pi\)
−0.498109 + 0.867114i \(0.665972\pi\)
\(138\) 0 0
\(139\) 8.26352 0.700902 0.350451 0.936581i \(-0.386028\pi\)
0.350451 + 0.936581i \(0.386028\pi\)
\(140\) 0 0
\(141\) −12.0838 −1.01764
\(142\) 0 0
\(143\) −18.3405 −1.53371
\(144\) 0 0
\(145\) 5.64590 0.468866
\(146\) 0 0
\(147\) 0.399615 0.0329597
\(148\) 0 0
\(149\) −16.4534 −1.34791 −0.673956 0.738771i \(-0.735406\pi\)
−0.673956 + 0.738771i \(0.735406\pi\)
\(150\) 0 0
\(151\) 4.65539 0.378850 0.189425 0.981895i \(-0.439338\pi\)
0.189425 + 0.981895i \(0.439338\pi\)
\(152\) 0 0
\(153\) 4.25402 0.343917
\(154\) 0 0
\(155\) 4.90673 0.394118
\(156\) 0 0
\(157\) 8.45336 0.674652 0.337326 0.941388i \(-0.390478\pi\)
0.337326 + 0.941388i \(0.390478\pi\)
\(158\) 0 0
\(159\) −12.5817 −0.997795
\(160\) 0 0
\(161\) −1.87164 −0.147506
\(162\) 0 0
\(163\) −17.0496 −1.33543 −0.667715 0.744417i \(-0.732728\pi\)
−0.667715 + 0.744417i \(0.732728\pi\)
\(164\) 0 0
\(165\) −9.75877 −0.759719
\(166\) 0 0
\(167\) −3.19253 −0.247046 −0.123523 0.992342i \(-0.539419\pi\)
−0.123523 + 0.992342i \(0.539419\pi\)
\(168\) 0 0
\(169\) 20.1634 1.55103
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.63041 −0.732187 −0.366093 0.930578i \(-0.619305\pi\)
−0.366093 + 0.930578i \(0.619305\pi\)
\(174\) 0 0
\(175\) 2.69459 0.203692
\(176\) 0 0
\(177\) 0.879385 0.0660986
\(178\) 0 0
\(179\) 18.8161 1.40638 0.703192 0.711000i \(-0.251757\pi\)
0.703192 + 0.711000i \(0.251757\pi\)
\(180\) 0 0
\(181\) −2.77837 −0.206515 −0.103257 0.994655i \(-0.532927\pi\)
−0.103257 + 0.994655i \(0.532927\pi\)
\(182\) 0 0
\(183\) 4.49794 0.332497
\(184\) 0 0
\(185\) 8.73917 0.642517
\(186\) 0 0
\(187\) 20.7570 1.51790
\(188\) 0 0
\(189\) 15.0797 1.09688
\(190\) 0 0
\(191\) −9.56212 −0.691891 −0.345945 0.938255i \(-0.612442\pi\)
−0.345945 + 0.938255i \(0.612442\pi\)
\(192\) 0 0
\(193\) 23.6810 1.70459 0.852297 0.523058i \(-0.175209\pi\)
0.852297 + 0.523058i \(0.175209\pi\)
\(194\) 0 0
\(195\) 17.6459 1.26365
\(196\) 0 0
\(197\) −22.9222 −1.63314 −0.816570 0.577247i \(-0.804127\pi\)
−0.816570 + 0.577247i \(0.804127\pi\)
\(198\) 0 0
\(199\) −10.0838 −0.714820 −0.357410 0.933948i \(-0.616340\pi\)
−0.357410 + 0.933948i \(0.616340\pi\)
\(200\) 0 0
\(201\) −7.59627 −0.535799
\(202\) 0 0
\(203\) −7.60670 −0.533885
\(204\) 0 0
\(205\) 0.694593 0.0485125
\(206\) 0 0
\(207\) −0.453363 −0.0315109
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −22.3601 −1.53933 −0.769666 0.638447i \(-0.779577\pi\)
−0.769666 + 0.638447i \(0.779577\pi\)
\(212\) 0 0
\(213\) −12.9513 −0.887409
\(214\) 0 0
\(215\) −12.1284 −0.827147
\(216\) 0 0
\(217\) −6.61081 −0.448771
\(218\) 0 0
\(219\) 24.1557 1.63229
\(220\) 0 0
\(221\) −37.5330 −2.52474
\(222\) 0 0
\(223\) −9.27631 −0.621188 −0.310594 0.950543i \(-0.600528\pi\)
−0.310594 + 0.950543i \(0.600528\pi\)
\(224\) 0 0
\(225\) 0.652704 0.0435136
\(226\) 0 0
\(227\) 7.73648 0.513488 0.256744 0.966479i \(-0.417350\pi\)
0.256744 + 0.966479i \(0.417350\pi\)
\(228\) 0 0
\(229\) −23.0351 −1.52220 −0.761101 0.648634i \(-0.775341\pi\)
−0.761101 + 0.648634i \(0.775341\pi\)
\(230\) 0 0
\(231\) 13.1480 0.865072
\(232\) 0 0
\(233\) 8.39961 0.550277 0.275139 0.961405i \(-0.411276\pi\)
0.275139 + 0.961405i \(0.411276\pi\)
\(234\) 0 0
\(235\) −15.7743 −1.02900
\(236\) 0 0
\(237\) −13.8871 −0.902066
\(238\) 0 0
\(239\) 15.7297 1.01747 0.508734 0.860924i \(-0.330114\pi\)
0.508734 + 0.860924i \(0.330114\pi\)
\(240\) 0 0
\(241\) −17.5030 −1.12747 −0.563733 0.825957i \(-0.690635\pi\)
−0.563733 + 0.825957i \(0.690635\pi\)
\(242\) 0 0
\(243\) 6.65270 0.426771
\(244\) 0 0
\(245\) 0.521660 0.0333276
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −12.9855 −0.822920
\(250\) 0 0
\(251\) −8.56717 −0.540755 −0.270378 0.962754i \(-0.587149\pi\)
−0.270378 + 0.962754i \(0.587149\pi\)
\(252\) 0 0
\(253\) −2.21213 −0.139076
\(254\) 0 0
\(255\) −19.9709 −1.25063
\(256\) 0 0
\(257\) 8.02465 0.500564 0.250282 0.968173i \(-0.419477\pi\)
0.250282 + 0.968173i \(0.419477\pi\)
\(258\) 0 0
\(259\) −11.7743 −0.731616
\(260\) 0 0
\(261\) −1.84255 −0.114051
\(262\) 0 0
\(263\) 5.96080 0.367559 0.183779 0.982968i \(-0.441167\pi\)
0.183779 + 0.982968i \(0.441167\pi\)
\(264\) 0 0
\(265\) −16.4243 −1.00893
\(266\) 0 0
\(267\) −11.8571 −0.725643
\(268\) 0 0
\(269\) 1.14796 0.0699921 0.0349961 0.999387i \(-0.488858\pi\)
0.0349961 + 0.999387i \(0.488858\pi\)
\(270\) 0 0
\(271\) −20.0547 −1.21824 −0.609118 0.793080i \(-0.708476\pi\)
−0.609118 + 0.793080i \(0.708476\pi\)
\(272\) 0 0
\(273\) −23.7743 −1.43888
\(274\) 0 0
\(275\) 3.18479 0.192050
\(276\) 0 0
\(277\) 17.3601 1.04307 0.521533 0.853231i \(-0.325360\pi\)
0.521533 + 0.853231i \(0.325360\pi\)
\(278\) 0 0
\(279\) −1.60132 −0.0958685
\(280\) 0 0
\(281\) −2.92127 −0.174269 −0.0871343 0.996197i \(-0.527771\pi\)
−0.0871343 + 0.996197i \(0.527771\pi\)
\(282\) 0 0
\(283\) 9.48070 0.563569 0.281785 0.959478i \(-0.409074\pi\)
0.281785 + 0.959478i \(0.409074\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.935822 −0.0552398
\(288\) 0 0
\(289\) 25.4783 1.49873
\(290\) 0 0
\(291\) 0.532089 0.0311916
\(292\) 0 0
\(293\) 27.2918 1.59440 0.797202 0.603713i \(-0.206313\pi\)
0.797202 + 0.603713i \(0.206313\pi\)
\(294\) 0 0
\(295\) 1.14796 0.0668366
\(296\) 0 0
\(297\) 17.8229 1.03419
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 16.3405 0.941850
\(302\) 0 0
\(303\) −0.566237 −0.0325295
\(304\) 0 0
\(305\) 5.87164 0.336209
\(306\) 0 0
\(307\) 21.3286 1.21729 0.608645 0.793443i \(-0.291714\pi\)
0.608645 + 0.793443i \(0.291714\pi\)
\(308\) 0 0
\(309\) 13.1480 0.747961
\(310\) 0 0
\(311\) 29.2918 1.66099 0.830493 0.557030i \(-0.188059\pi\)
0.830493 + 0.557030i \(0.188059\pi\)
\(312\) 0 0
\(313\) 5.56118 0.314337 0.157168 0.987572i \(-0.449763\pi\)
0.157168 + 0.987572i \(0.449763\pi\)
\(314\) 0 0
\(315\) 3.51754 0.198191
\(316\) 0 0
\(317\) −3.80335 −0.213617 −0.106809 0.994280i \(-0.534063\pi\)
−0.106809 + 0.994280i \(0.534063\pi\)
\(318\) 0 0
\(319\) −8.99050 −0.503372
\(320\) 0 0
\(321\) −17.5476 −0.979410
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −5.75877 −0.319439
\(326\) 0 0
\(327\) −13.3209 −0.736647
\(328\) 0 0
\(329\) 21.2526 1.17169
\(330\) 0 0
\(331\) 20.4219 1.12249 0.561245 0.827650i \(-0.310323\pi\)
0.561245 + 0.827650i \(0.310323\pi\)
\(332\) 0 0
\(333\) −2.85204 −0.156291
\(334\) 0 0
\(335\) −9.91622 −0.541781
\(336\) 0 0
\(337\) −20.3105 −1.10638 −0.553191 0.833055i \(-0.686590\pi\)
−0.553191 + 0.833055i \(0.686590\pi\)
\(338\) 0 0
\(339\) −4.38144 −0.237967
\(340\) 0 0
\(341\) −7.81345 −0.423122
\(342\) 0 0
\(343\) 18.1593 0.980511
\(344\) 0 0
\(345\) 2.12836 0.114587
\(346\) 0 0
\(347\) −5.21482 −0.279946 −0.139973 0.990155i \(-0.544702\pi\)
−0.139973 + 0.990155i \(0.544702\pi\)
\(348\) 0 0
\(349\) 14.3405 0.767629 0.383814 0.923410i \(-0.374610\pi\)
0.383814 + 0.923410i \(0.374610\pi\)
\(350\) 0 0
\(351\) −32.2276 −1.72018
\(352\) 0 0
\(353\) 26.2499 1.39714 0.698571 0.715541i \(-0.253820\pi\)
0.698571 + 0.715541i \(0.253820\pi\)
\(354\) 0 0
\(355\) −16.9067 −0.897316
\(356\) 0 0
\(357\) 26.9067 1.42405
\(358\) 0 0
\(359\) 33.6905 1.77812 0.889058 0.457795i \(-0.151361\pi\)
0.889058 + 0.457795i \(0.151361\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −1.31315 −0.0689224
\(364\) 0 0
\(365\) 31.5330 1.65051
\(366\) 0 0
\(367\) 10.4688 0.546469 0.273235 0.961947i \(-0.411906\pi\)
0.273235 + 0.961947i \(0.411906\pi\)
\(368\) 0 0
\(369\) −0.226682 −0.0118006
\(370\) 0 0
\(371\) 22.1284 1.14885
\(372\) 0 0
\(373\) −23.9026 −1.23763 −0.618815 0.785537i \(-0.712387\pi\)
−0.618815 + 0.785537i \(0.712387\pi\)
\(374\) 0 0
\(375\) −18.3851 −0.949401
\(376\) 0 0
\(377\) 16.2567 0.837263
\(378\) 0 0
\(379\) 17.8135 0.915016 0.457508 0.889206i \(-0.348742\pi\)
0.457508 + 0.889206i \(0.348742\pi\)
\(380\) 0 0
\(381\) 14.9067 0.763695
\(382\) 0 0
\(383\) 25.0797 1.28151 0.640755 0.767745i \(-0.278621\pi\)
0.640755 + 0.767745i \(0.278621\pi\)
\(384\) 0 0
\(385\) 17.1634 0.874730
\(386\) 0 0
\(387\) 3.95811 0.201202
\(388\) 0 0
\(389\) −9.41828 −0.477526 −0.238763 0.971078i \(-0.576742\pi\)
−0.238763 + 0.971078i \(0.576742\pi\)
\(390\) 0 0
\(391\) −4.52704 −0.228942
\(392\) 0 0
\(393\) 9.90941 0.499864
\(394\) 0 0
\(395\) −18.1284 −0.912137
\(396\) 0 0
\(397\) 6.85204 0.343894 0.171947 0.985106i \(-0.444994\pi\)
0.171947 + 0.985106i \(0.444994\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.74691 −0.237049 −0.118525 0.992951i \(-0.537816\pi\)
−0.118525 + 0.992951i \(0.537816\pi\)
\(402\) 0 0
\(403\) 14.1284 0.703784
\(404\) 0 0
\(405\) −13.2317 −0.657490
\(406\) 0 0
\(407\) −13.9162 −0.689802
\(408\) 0 0
\(409\) −31.6928 −1.56711 −0.783555 0.621322i \(-0.786596\pi\)
−0.783555 + 0.621322i \(0.786596\pi\)
\(410\) 0 0
\(411\) −17.8648 −0.881207
\(412\) 0 0
\(413\) −1.54664 −0.0761050
\(414\) 0 0
\(415\) −16.9513 −0.832107
\(416\) 0 0
\(417\) 12.6604 0.619985
\(418\) 0 0
\(419\) 11.0101 0.537879 0.268939 0.963157i \(-0.413327\pi\)
0.268939 + 0.963157i \(0.413327\pi\)
\(420\) 0 0
\(421\) 8.66550 0.422330 0.211165 0.977450i \(-0.432274\pi\)
0.211165 + 0.977450i \(0.432274\pi\)
\(422\) 0 0
\(423\) 5.14796 0.250302
\(424\) 0 0
\(425\) 6.51754 0.316147
\(426\) 0 0
\(427\) −7.91085 −0.382833
\(428\) 0 0
\(429\) −28.0993 −1.35665
\(430\) 0 0
\(431\) −29.8871 −1.43961 −0.719806 0.694175i \(-0.755769\pi\)
−0.719806 + 0.694175i \(0.755769\pi\)
\(432\) 0 0
\(433\) −9.26083 −0.445047 −0.222524 0.974927i \(-0.571429\pi\)
−0.222524 + 0.974927i \(0.571429\pi\)
\(434\) 0 0
\(435\) 8.65002 0.414736
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 20.8621 0.995696 0.497848 0.867264i \(-0.334124\pi\)
0.497848 + 0.867264i \(0.334124\pi\)
\(440\) 0 0
\(441\) −0.170245 −0.00810689
\(442\) 0 0
\(443\) −23.8280 −1.13210 −0.566051 0.824370i \(-0.691530\pi\)
−0.566051 + 0.824370i \(0.691530\pi\)
\(444\) 0 0
\(445\) −15.4783 −0.733744
\(446\) 0 0
\(447\) −25.2080 −1.19230
\(448\) 0 0
\(449\) 2.18210 0.102980 0.0514899 0.998674i \(-0.483603\pi\)
0.0514899 + 0.998674i \(0.483603\pi\)
\(450\) 0 0
\(451\) −1.10607 −0.0520827
\(452\) 0 0
\(453\) 7.13247 0.335113
\(454\) 0 0
\(455\) −31.0351 −1.45495
\(456\) 0 0
\(457\) 1.78106 0.0833144 0.0416572 0.999132i \(-0.486736\pi\)
0.0416572 + 0.999132i \(0.486736\pi\)
\(458\) 0 0
\(459\) 36.4739 1.70246
\(460\) 0 0
\(461\) −15.4884 −0.721369 −0.360684 0.932688i \(-0.617457\pi\)
−0.360684 + 0.932688i \(0.617457\pi\)
\(462\) 0 0
\(463\) 2.71007 0.125948 0.0629739 0.998015i \(-0.479942\pi\)
0.0629739 + 0.998015i \(0.479942\pi\)
\(464\) 0 0
\(465\) 7.51754 0.348618
\(466\) 0 0
\(467\) −12.9135 −0.597567 −0.298784 0.954321i \(-0.596581\pi\)
−0.298784 + 0.954321i \(0.596581\pi\)
\(468\) 0 0
\(469\) 13.3601 0.616912
\(470\) 0 0
\(471\) 12.9513 0.596765
\(472\) 0 0
\(473\) 19.3131 0.888019
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 5.36009 0.245422
\(478\) 0 0
\(479\) 18.5526 0.847691 0.423845 0.905735i \(-0.360680\pi\)
0.423845 + 0.905735i \(0.360680\pi\)
\(480\) 0 0
\(481\) 25.1634 1.14735
\(482\) 0 0
\(483\) −2.86753 −0.130477
\(484\) 0 0
\(485\) 0.694593 0.0315398
\(486\) 0 0
\(487\) 41.1735 1.86575 0.932876 0.360199i \(-0.117291\pi\)
0.932876 + 0.360199i \(0.117291\pi\)
\(488\) 0 0
\(489\) −26.1215 −1.18126
\(490\) 0 0
\(491\) 22.5776 1.01891 0.509456 0.860496i \(-0.329847\pi\)
0.509456 + 0.860496i \(0.329847\pi\)
\(492\) 0 0
\(493\) −18.3987 −0.828635
\(494\) 0 0
\(495\) 4.15745 0.186864
\(496\) 0 0
\(497\) 22.7784 1.02175
\(498\) 0 0
\(499\) 29.3286 1.31293 0.656465 0.754357i \(-0.272051\pi\)
0.656465 + 0.754357i \(0.272051\pi\)
\(500\) 0 0
\(501\) −4.89124 −0.218525
\(502\) 0 0
\(503\) −33.6168 −1.49890 −0.749450 0.662061i \(-0.769682\pi\)
−0.749450 + 0.662061i \(0.769682\pi\)
\(504\) 0 0
\(505\) −0.739170 −0.0328926
\(506\) 0 0
\(507\) 30.8922 1.37197
\(508\) 0 0
\(509\) −4.02910 −0.178587 −0.0892933 0.996005i \(-0.528461\pi\)
−0.0892933 + 0.996005i \(0.528461\pi\)
\(510\) 0 0
\(511\) −42.4843 −1.87940
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.1634 0.756311
\(516\) 0 0
\(517\) 25.1189 1.10473
\(518\) 0 0
\(519\) −14.7547 −0.647657
\(520\) 0 0
\(521\) −4.98957 −0.218597 −0.109299 0.994009i \(-0.534860\pi\)
−0.109299 + 0.994009i \(0.534860\pi\)
\(522\) 0 0
\(523\) −26.5526 −1.16107 −0.580533 0.814237i \(-0.697156\pi\)
−0.580533 + 0.814237i \(0.697156\pi\)
\(524\) 0 0
\(525\) 4.12836 0.180176
\(526\) 0 0
\(527\) −15.9899 −0.696531
\(528\) 0 0
\(529\) −22.5175 −0.979024
\(530\) 0 0
\(531\) −0.374638 −0.0162579
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −22.9067 −0.990344
\(536\) 0 0
\(537\) 28.8280 1.24402
\(538\) 0 0
\(539\) −0.830689 −0.0357803
\(540\) 0 0
\(541\) −11.9709 −0.514669 −0.257335 0.966322i \(-0.582844\pi\)
−0.257335 + 0.966322i \(0.582844\pi\)
\(542\) 0 0
\(543\) −4.25671 −0.182673
\(544\) 0 0
\(545\) −17.3892 −0.744871
\(546\) 0 0
\(547\) 3.19665 0.136679 0.0683395 0.997662i \(-0.478230\pi\)
0.0683395 + 0.997662i \(0.478230\pi\)
\(548\) 0 0
\(549\) −1.91622 −0.0817824
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 24.4243 1.03863
\(554\) 0 0
\(555\) 13.3892 0.568339
\(556\) 0 0
\(557\) −23.9763 −1.01591 −0.507954 0.861384i \(-0.669598\pi\)
−0.507954 + 0.861384i \(0.669598\pi\)
\(558\) 0 0
\(559\) −34.9222 −1.47705
\(560\) 0 0
\(561\) 31.8016 1.34266
\(562\) 0 0
\(563\) −8.75702 −0.369064 −0.184532 0.982826i \(-0.559077\pi\)
−0.184532 + 0.982826i \(0.559077\pi\)
\(564\) 0 0
\(565\) −5.71957 −0.240624
\(566\) 0 0
\(567\) 17.8271 0.748666
\(568\) 0 0
\(569\) 36.4201 1.52681 0.763406 0.645919i \(-0.223526\pi\)
0.763406 + 0.645919i \(0.223526\pi\)
\(570\) 0 0
\(571\) −34.2131 −1.43177 −0.715886 0.698217i \(-0.753977\pi\)
−0.715886 + 0.698217i \(0.753977\pi\)
\(572\) 0 0
\(573\) −14.6500 −0.612013
\(574\) 0 0
\(575\) −0.694593 −0.0289665
\(576\) 0 0
\(577\) 15.5098 0.645681 0.322841 0.946453i \(-0.395362\pi\)
0.322841 + 0.946453i \(0.395362\pi\)
\(578\) 0 0
\(579\) 36.2814 1.50780
\(580\) 0 0
\(581\) 22.8384 0.947498
\(582\) 0 0
\(583\) 26.1539 1.08319
\(584\) 0 0
\(585\) −7.51754 −0.310812
\(586\) 0 0
\(587\) 11.5817 0.478029 0.239014 0.971016i \(-0.423176\pi\)
0.239014 + 0.971016i \(0.423176\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −35.1189 −1.44460
\(592\) 0 0
\(593\) −46.3928 −1.90512 −0.952562 0.304344i \(-0.901563\pi\)
−0.952562 + 0.304344i \(0.901563\pi\)
\(594\) 0 0
\(595\) 35.1242 1.43995
\(596\) 0 0
\(597\) −15.4492 −0.632295
\(598\) 0 0
\(599\) 25.6323 1.04731 0.523653 0.851931i \(-0.324569\pi\)
0.523653 + 0.851931i \(0.324569\pi\)
\(600\) 0 0
\(601\) −7.99226 −0.326011 −0.163006 0.986625i \(-0.552119\pi\)
−0.163006 + 0.986625i \(0.552119\pi\)
\(602\) 0 0
\(603\) 3.23618 0.131787
\(604\) 0 0
\(605\) −1.71419 −0.0696919
\(606\) 0 0
\(607\) −26.9905 −1.09551 −0.547755 0.836639i \(-0.684518\pi\)
−0.547755 + 0.836639i \(0.684518\pi\)
\(608\) 0 0
\(609\) −11.6541 −0.472249
\(610\) 0 0
\(611\) −45.4201 −1.83750
\(612\) 0 0
\(613\) −14.2513 −0.575606 −0.287803 0.957690i \(-0.592925\pi\)
−0.287803 + 0.957690i \(0.592925\pi\)
\(614\) 0 0
\(615\) 1.06418 0.0429118
\(616\) 0 0
\(617\) −30.5604 −1.23031 −0.615157 0.788405i \(-0.710907\pi\)
−0.615157 + 0.788405i \(0.710907\pi\)
\(618\) 0 0
\(619\) 28.6750 1.15255 0.576273 0.817258i \(-0.304507\pi\)
0.576273 + 0.817258i \(0.304507\pi\)
\(620\) 0 0
\(621\) −3.88713 −0.155985
\(622\) 0 0
\(623\) 20.8539 0.835494
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −28.4789 −1.13553
\(630\) 0 0
\(631\) 4.49794 0.179060 0.0895301 0.995984i \(-0.471463\pi\)
0.0895301 + 0.995984i \(0.471463\pi\)
\(632\) 0 0
\(633\) −34.2576 −1.36162
\(634\) 0 0
\(635\) 19.4593 0.772221
\(636\) 0 0
\(637\) 1.50206 0.0595138
\(638\) 0 0
\(639\) 5.51754 0.218271
\(640\) 0 0
\(641\) 11.6081 0.458493 0.229247 0.973368i \(-0.426374\pi\)
0.229247 + 0.973368i \(0.426374\pi\)
\(642\) 0 0
\(643\) −26.0455 −1.02713 −0.513567 0.858049i \(-0.671676\pi\)
−0.513567 + 0.858049i \(0.671676\pi\)
\(644\) 0 0
\(645\) −18.5817 −0.731654
\(646\) 0 0
\(647\) 2.31490 0.0910082 0.0455041 0.998964i \(-0.485511\pi\)
0.0455041 + 0.998964i \(0.485511\pi\)
\(648\) 0 0
\(649\) −1.82800 −0.0717553
\(650\) 0 0
\(651\) −10.1284 −0.396962
\(652\) 0 0
\(653\) 3.30541 0.129351 0.0646753 0.997906i \(-0.479399\pi\)
0.0646753 + 0.997906i \(0.479399\pi\)
\(654\) 0 0
\(655\) 12.9358 0.505444
\(656\) 0 0
\(657\) −10.2909 −0.401485
\(658\) 0 0
\(659\) −13.7493 −0.535596 −0.267798 0.963475i \(-0.586296\pi\)
−0.267798 + 0.963475i \(0.586296\pi\)
\(660\) 0 0
\(661\) 3.33450 0.129697 0.0648486 0.997895i \(-0.479344\pi\)
0.0648486 + 0.997895i \(0.479344\pi\)
\(662\) 0 0
\(663\) −57.5039 −2.23327
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.96080 0.0759225
\(668\) 0 0
\(669\) −14.2121 −0.549473
\(670\) 0 0
\(671\) −9.34998 −0.360952
\(672\) 0 0
\(673\) −38.9810 −1.50261 −0.751304 0.659957i \(-0.770575\pi\)
−0.751304 + 0.659957i \(0.770575\pi\)
\(674\) 0 0
\(675\) 5.59627 0.215400
\(676\) 0 0
\(677\) 43.5877 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(678\) 0 0
\(679\) −0.935822 −0.0359136
\(680\) 0 0
\(681\) 11.8530 0.454207
\(682\) 0 0
\(683\) 32.9317 1.26010 0.630048 0.776556i \(-0.283035\pi\)
0.630048 + 0.776556i \(0.283035\pi\)
\(684\) 0 0
\(685\) −23.3209 −0.891045
\(686\) 0 0
\(687\) −35.2918 −1.34647
\(688\) 0 0
\(689\) −47.2918 −1.80167
\(690\) 0 0
\(691\) −34.3209 −1.30563 −0.652814 0.757518i \(-0.726412\pi\)
−0.652814 + 0.757518i \(0.726412\pi\)
\(692\) 0 0
\(693\) −5.60132 −0.212777
\(694\) 0 0
\(695\) 16.5270 0.626906
\(696\) 0 0
\(697\) −2.26352 −0.0857369
\(698\) 0 0
\(699\) 12.8690 0.486749
\(700\) 0 0
\(701\) 6.45336 0.243740 0.121870 0.992546i \(-0.461111\pi\)
0.121870 + 0.992546i \(0.461111\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −24.1676 −0.910203
\(706\) 0 0
\(707\) 0.995881 0.0374540
\(708\) 0 0
\(709\) −3.37908 −0.126904 −0.0634520 0.997985i \(-0.520211\pi\)
−0.0634520 + 0.997985i \(0.520211\pi\)
\(710\) 0 0
\(711\) 5.91622 0.221876
\(712\) 0 0
\(713\) 1.70409 0.0638186
\(714\) 0 0
\(715\) −36.6810 −1.37179
\(716\) 0 0
\(717\) 24.0993 0.900003
\(718\) 0 0
\(719\) 31.2026 1.16366 0.581831 0.813310i \(-0.302336\pi\)
0.581831 + 0.813310i \(0.302336\pi\)
\(720\) 0 0
\(721\) −23.1242 −0.861192
\(722\) 0 0
\(723\) −26.8161 −0.997303
\(724\) 0 0
\(725\) −2.82295 −0.104842
\(726\) 0 0
\(727\) 25.6614 0.951728 0.475864 0.879519i \(-0.342135\pi\)
0.475864 + 0.879519i \(0.342135\pi\)
\(728\) 0 0
\(729\) 30.0401 1.11260
\(730\) 0 0
\(731\) 39.5235 1.46183
\(732\) 0 0
\(733\) −23.8735 −0.881788 −0.440894 0.897559i \(-0.645339\pi\)
−0.440894 + 0.897559i \(0.645339\pi\)
\(734\) 0 0
\(735\) 0.799229 0.0294800
\(736\) 0 0
\(737\) 15.7906 0.581653
\(738\) 0 0
\(739\) 45.9404 1.68994 0.844972 0.534810i \(-0.179617\pi\)
0.844972 + 0.534810i \(0.179617\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −51.0114 −1.87143 −0.935713 0.352763i \(-0.885242\pi\)
−0.935713 + 0.352763i \(0.885242\pi\)
\(744\) 0 0
\(745\) −32.9067 −1.20561
\(746\) 0 0
\(747\) 5.53209 0.202409
\(748\) 0 0
\(749\) 30.8621 1.12768
\(750\) 0 0
\(751\) −36.3696 −1.32715 −0.663573 0.748112i \(-0.730961\pi\)
−0.663573 + 0.748112i \(0.730961\pi\)
\(752\) 0 0
\(753\) −13.1257 −0.478326
\(754\) 0 0
\(755\) 9.31078 0.338854
\(756\) 0 0
\(757\) −5.80335 −0.210926 −0.105463 0.994423i \(-0.533633\pi\)
−0.105463 + 0.994423i \(0.533633\pi\)
\(758\) 0 0
\(759\) −3.38919 −0.123020
\(760\) 0 0
\(761\) 22.6355 0.820535 0.410268 0.911965i \(-0.365435\pi\)
0.410268 + 0.911965i \(0.365435\pi\)
\(762\) 0 0
\(763\) 23.4284 0.848165
\(764\) 0 0
\(765\) 8.50805 0.307609
\(766\) 0 0
\(767\) 3.30541 0.119351
\(768\) 0 0
\(769\) 10.9026 0.393158 0.196579 0.980488i \(-0.437017\pi\)
0.196579 + 0.980488i \(0.437017\pi\)
\(770\) 0 0
\(771\) 12.2945 0.442775
\(772\) 0 0
\(773\) 5.25072 0.188855 0.0944277 0.995532i \(-0.469898\pi\)
0.0944277 + 0.995532i \(0.469898\pi\)
\(774\) 0 0
\(775\) −2.45336 −0.0881274
\(776\) 0 0
\(777\) −18.0392 −0.647153
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 26.9222 0.963352
\(782\) 0 0
\(783\) −15.7980 −0.564573
\(784\) 0 0
\(785\) 16.9067 0.603427
\(786\) 0 0
\(787\) −2.38743 −0.0851027 −0.0425514 0.999094i \(-0.513549\pi\)
−0.0425514 + 0.999094i \(0.513549\pi\)
\(788\) 0 0
\(789\) 9.13247 0.325125
\(790\) 0 0
\(791\) 7.70596 0.273992
\(792\) 0 0
\(793\) 16.9067 0.600375
\(794\) 0 0
\(795\) −25.1634 −0.892455
\(796\) 0 0
\(797\) −31.0951 −1.10145 −0.550723 0.834688i \(-0.685648\pi\)
−0.550723 + 0.834688i \(0.685648\pi\)
\(798\) 0 0
\(799\) 51.4047 1.81857
\(800\) 0 0
\(801\) 5.05138 0.178482
\(802\) 0 0
\(803\) −50.2131 −1.77198
\(804\) 0 0
\(805\) −3.74329 −0.131934
\(806\) 0 0
\(807\) 1.75877 0.0619117
\(808\) 0 0
\(809\) 22.3037 0.784155 0.392077 0.919932i \(-0.371757\pi\)
0.392077 + 0.919932i \(0.371757\pi\)
\(810\) 0 0
\(811\) 3.31902 0.116547 0.0582733 0.998301i \(-0.481441\pi\)
0.0582733 + 0.998301i \(0.481441\pi\)
\(812\) 0 0
\(813\) −30.7256 −1.07759
\(814\) 0 0
\(815\) −34.0993 −1.19444
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 10.1284 0.353913
\(820\) 0 0
\(821\) −1.83244 −0.0639527 −0.0319764 0.999489i \(-0.510180\pi\)
−0.0319764 + 0.999489i \(0.510180\pi\)
\(822\) 0 0
\(823\) 34.2377 1.19345 0.596726 0.802445i \(-0.296468\pi\)
0.596726 + 0.802445i \(0.296468\pi\)
\(824\) 0 0
\(825\) 4.87939 0.169878
\(826\) 0 0
\(827\) −19.8658 −0.690801 −0.345400 0.938455i \(-0.612257\pi\)
−0.345400 + 0.938455i \(0.612257\pi\)
\(828\) 0 0
\(829\) 35.7351 1.24113 0.620565 0.784155i \(-0.286903\pi\)
0.620565 + 0.784155i \(0.286903\pi\)
\(830\) 0 0
\(831\) 26.5972 0.922647
\(832\) 0 0
\(833\) −1.69997 −0.0589004
\(834\) 0 0
\(835\) −6.38507 −0.220964
\(836\) 0 0
\(837\) −13.7297 −0.474567
\(838\) 0 0
\(839\) −54.3952 −1.87793 −0.938965 0.344013i \(-0.888214\pi\)
−0.938965 + 0.344013i \(0.888214\pi\)
\(840\) 0 0
\(841\) −21.0310 −0.725206
\(842\) 0 0
\(843\) −4.47565 −0.154150
\(844\) 0 0
\(845\) 40.3269 1.38729
\(846\) 0 0
\(847\) 2.30953 0.0793563
\(848\) 0 0
\(849\) 14.5253 0.498506
\(850\) 0 0
\(851\) 3.03508 0.104041
\(852\) 0 0
\(853\) 15.9763 0.547017 0.273509 0.961870i \(-0.411816\pi\)
0.273509 + 0.961870i \(0.411816\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.0145 0.888640 0.444320 0.895868i \(-0.353445\pi\)
0.444320 + 0.895868i \(0.353445\pi\)
\(858\) 0 0
\(859\) −53.2704 −1.81756 −0.908782 0.417271i \(-0.862986\pi\)
−0.908782 + 0.417271i \(0.862986\pi\)
\(860\) 0 0
\(861\) −1.43376 −0.0488625
\(862\) 0 0
\(863\) 3.23173 0.110010 0.0550048 0.998486i \(-0.482483\pi\)
0.0550048 + 0.998486i \(0.482483\pi\)
\(864\) 0 0
\(865\) −19.2608 −0.654888
\(866\) 0 0
\(867\) 39.0351 1.32570
\(868\) 0 0
\(869\) 28.8675 0.979264
\(870\) 0 0
\(871\) −28.5526 −0.967469
\(872\) 0 0
\(873\) −0.226682 −0.00767201
\(874\) 0 0
\(875\) 32.3351 1.09313
\(876\) 0 0
\(877\) −11.7243 −0.395901 −0.197951 0.980212i \(-0.563429\pi\)
−0.197951 + 0.980212i \(0.563429\pi\)
\(878\) 0 0
\(879\) 41.8135 1.41033
\(880\) 0 0
\(881\) 27.0473 0.911246 0.455623 0.890173i \(-0.349417\pi\)
0.455623 + 0.890173i \(0.349417\pi\)
\(882\) 0 0
\(883\) −14.4810 −0.487325 −0.243663 0.969860i \(-0.578349\pi\)
−0.243663 + 0.969860i \(0.578349\pi\)
\(884\) 0 0
\(885\) 1.75877 0.0591204
\(886\) 0 0
\(887\) −11.5912 −0.389195 −0.194597 0.980883i \(-0.562340\pi\)
−0.194597 + 0.980883i \(0.562340\pi\)
\(888\) 0 0
\(889\) −26.2175 −0.879307
\(890\) 0 0
\(891\) 21.0702 0.705877
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 37.6323 1.25791
\(896\) 0 0
\(897\) 6.12836 0.204620
\(898\) 0 0
\(899\) 6.92572 0.230986
\(900\) 0 0
\(901\) 53.5229 1.78311
\(902\) 0 0
\(903\) 25.0351 0.833115
\(904\) 0 0
\(905\) −5.55674 −0.184712
\(906\) 0 0
\(907\) 41.9786 1.39388 0.696939 0.717130i \(-0.254545\pi\)
0.696939 + 0.717130i \(0.254545\pi\)
\(908\) 0 0
\(909\) 0.241230 0.00800108
\(910\) 0 0
\(911\) −44.8675 −1.48653 −0.743264 0.668999i \(-0.766723\pi\)
−0.743264 + 0.668999i \(0.766723\pi\)
\(912\) 0 0
\(913\) 26.9932 0.893344
\(914\) 0 0
\(915\) 8.99588 0.297395
\(916\) 0 0
\(917\) −17.4284 −0.575536
\(918\) 0 0
\(919\) 32.5270 1.07297 0.536484 0.843911i \(-0.319752\pi\)
0.536484 + 0.843911i \(0.319752\pi\)
\(920\) 0 0
\(921\) 32.6774 1.07676
\(922\) 0 0
\(923\) −48.6810 −1.60235
\(924\) 0 0
\(925\) −4.36959 −0.143671
\(926\) 0 0
\(927\) −5.60132 −0.183971
\(928\) 0 0
\(929\) 11.8402 0.388464 0.194232 0.980956i \(-0.437779\pi\)
0.194232 + 0.980956i \(0.437779\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 44.8776 1.46923
\(934\) 0 0
\(935\) 41.5140 1.35765
\(936\) 0 0
\(937\) −0.763823 −0.0249530 −0.0124765 0.999922i \(-0.503971\pi\)
−0.0124765 + 0.999922i \(0.503971\pi\)
\(938\) 0 0
\(939\) 8.52023 0.278047
\(940\) 0 0
\(941\) 21.4938 0.700679 0.350339 0.936623i \(-0.386066\pi\)
0.350339 + 0.936623i \(0.386066\pi\)
\(942\) 0 0
\(943\) 0.241230 0.00785551
\(944\) 0 0
\(945\) 30.1593 0.981083
\(946\) 0 0
\(947\) −31.6851 −1.02963 −0.514814 0.857302i \(-0.672139\pi\)
−0.514814 + 0.857302i \(0.672139\pi\)
\(948\) 0 0
\(949\) 90.7957 2.94735
\(950\) 0 0
\(951\) −5.82707 −0.188956
\(952\) 0 0
\(953\) −55.8711 −1.80984 −0.904922 0.425577i \(-0.860071\pi\)
−0.904922 + 0.425577i \(0.860071\pi\)
\(954\) 0 0
\(955\) −19.1242 −0.618846
\(956\) 0 0
\(957\) −13.7743 −0.445258
\(958\) 0 0
\(959\) 31.4201 1.01461
\(960\) 0 0
\(961\) −24.9810 −0.805839
\(962\) 0 0
\(963\) 7.47565 0.240900
\(964\) 0 0
\(965\) 47.3620 1.52464
\(966\) 0 0
\(967\) −21.5567 −0.693218 −0.346609 0.938010i \(-0.612667\pi\)
−0.346609 + 0.938010i \(0.612667\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 48.6563 1.56146 0.780728 0.624871i \(-0.214849\pi\)
0.780728 + 0.624871i \(0.214849\pi\)
\(972\) 0 0
\(973\) −22.2668 −0.713841
\(974\) 0 0
\(975\) −8.82295 −0.282560
\(976\) 0 0
\(977\) 50.5482 1.61718 0.808590 0.588373i \(-0.200231\pi\)
0.808590 + 0.588373i \(0.200231\pi\)
\(978\) 0 0
\(979\) 24.6477 0.787742
\(980\) 0 0
\(981\) 5.67499 0.181189
\(982\) 0 0
\(983\) −30.0601 −0.958767 −0.479383 0.877606i \(-0.659140\pi\)
−0.479383 + 0.877606i \(0.659140\pi\)
\(984\) 0 0
\(985\) −45.8444 −1.46072
\(986\) 0 0
\(987\) 32.5609 1.03642
\(988\) 0 0
\(989\) −4.21213 −0.133938
\(990\) 0 0
\(991\) 2.75278 0.0874451 0.0437225 0.999044i \(-0.486078\pi\)
0.0437225 + 0.999044i \(0.486078\pi\)
\(992\) 0 0
\(993\) 31.2882 0.992900
\(994\) 0 0
\(995\) −20.1676 −0.639355
\(996\) 0 0
\(997\) 8.70470 0.275681 0.137840 0.990454i \(-0.455984\pi\)
0.137840 + 0.990454i \(0.455984\pi\)
\(998\) 0 0
\(999\) −24.4534 −0.773670
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5776.2.a.bo.1.3 3
4.3 odd 2 722.2.a.k.1.1 3
12.11 even 2 6498.2.a.bq.1.2 3
19.14 odd 18 304.2.u.c.177.1 6
19.15 odd 18 304.2.u.c.225.1 6
19.18 odd 2 5776.2.a.bn.1.1 3
76.3 even 18 722.2.e.m.389.1 6
76.7 odd 6 722.2.c.l.429.3 6
76.11 odd 6 722.2.c.l.653.3 6
76.15 even 18 38.2.e.a.35.1 yes 6
76.23 odd 18 722.2.e.k.415.1 6
76.27 even 6 722.2.c.k.653.1 6
76.31 even 6 722.2.c.k.429.1 6
76.35 odd 18 722.2.e.a.389.1 6
76.43 odd 18 722.2.e.k.595.1 6
76.47 odd 18 722.2.e.l.423.1 6
76.51 even 18 722.2.e.m.245.1 6
76.55 odd 18 722.2.e.l.99.1 6
76.59 even 18 722.2.e.b.99.1 6
76.63 odd 18 722.2.e.a.245.1 6
76.67 even 18 722.2.e.b.423.1 6
76.71 even 18 38.2.e.a.25.1 6
76.75 even 2 722.2.a.l.1.3 3
228.71 odd 18 342.2.u.c.253.1 6
228.167 odd 18 342.2.u.c.73.1 6
228.227 odd 2 6498.2.a.bl.1.2 3
380.147 odd 36 950.2.u.b.899.2 12
380.167 odd 36 950.2.u.b.149.1 12
380.223 odd 36 950.2.u.b.899.1 12
380.243 odd 36 950.2.u.b.149.2 12
380.299 even 18 950.2.l.d.101.1 6
380.319 even 18 950.2.l.d.301.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.e.a.25.1 6 76.71 even 18
38.2.e.a.35.1 yes 6 76.15 even 18
304.2.u.c.177.1 6 19.14 odd 18
304.2.u.c.225.1 6 19.15 odd 18
342.2.u.c.73.1 6 228.167 odd 18
342.2.u.c.253.1 6 228.71 odd 18
722.2.a.k.1.1 3 4.3 odd 2
722.2.a.l.1.3 3 76.75 even 2
722.2.c.k.429.1 6 76.31 even 6
722.2.c.k.653.1 6 76.27 even 6
722.2.c.l.429.3 6 76.7 odd 6
722.2.c.l.653.3 6 76.11 odd 6
722.2.e.a.245.1 6 76.63 odd 18
722.2.e.a.389.1 6 76.35 odd 18
722.2.e.b.99.1 6 76.59 even 18
722.2.e.b.423.1 6 76.67 even 18
722.2.e.k.415.1 6 76.23 odd 18
722.2.e.k.595.1 6 76.43 odd 18
722.2.e.l.99.1 6 76.55 odd 18
722.2.e.l.423.1 6 76.47 odd 18
722.2.e.m.245.1 6 76.51 even 18
722.2.e.m.389.1 6 76.3 even 18
950.2.l.d.101.1 6 380.299 even 18
950.2.l.d.301.1 6 380.319 even 18
950.2.u.b.149.1 12 380.167 odd 36
950.2.u.b.149.2 12 380.243 odd 36
950.2.u.b.899.1 12 380.223 odd 36
950.2.u.b.899.2 12 380.147 odd 36
5776.2.a.bn.1.1 3 19.18 odd 2
5776.2.a.bo.1.3 3 1.1 even 1 trivial
6498.2.a.bl.1.2 3 228.227 odd 2
6498.2.a.bq.1.2 3 12.11 even 2