Properties

Label 1216.2.s.i.31.7
Level $1216$
Weight $2$
Character 1216.31
Analytic conductor $9.710$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(31,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.s (of order \(6\), degree \(2\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.7
Character \(\chi\) \(=\) 1216.31
Dual form 1216.2.s.i.863.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.132709 + 0.0766197i) q^{3} +(1.14027 - 0.658336i) q^{5} -1.40635i q^{7} +(-1.48826 + 2.57774i) q^{9} +O(q^{10})\) \(q+(-0.132709 + 0.0766197i) q^{3} +(1.14027 - 0.658336i) q^{5} -1.40635i q^{7} +(-1.48826 + 2.57774i) q^{9} -3.27622 q^{11} +(-2.31710 + 4.01334i) q^{13} +(-0.100883 + 0.174735i) q^{15} +(-2.24968 - 3.89655i) q^{17} +(-2.56396 - 3.52507i) q^{19} +(0.107754 + 0.186635i) q^{21} +(-0.100883 - 0.0582449i) q^{23} +(-1.63319 + 2.82876i) q^{25} -0.915838i q^{27} +(-3.34070 + 5.78625i) q^{29} -2.43586 q^{31} +(0.434785 - 0.251023i) q^{33} +(-0.925849 - 1.60362i) q^{35} -8.87096 q^{37} -0.710143i q^{39} +(5.96478 - 3.44377i) q^{41} +(-4.01334 - 6.95131i) q^{43} +3.91910i q^{45} +(7.69223 + 4.44111i) q^{47} +5.02219 q^{49} +(0.597106 + 0.344739i) q^{51} +(-2.47034 + 4.27876i) q^{53} +(-3.73578 + 2.15685i) q^{55} +(0.610351 + 0.271360i) q^{57} +(-9.19309 + 5.30763i) q^{59} +(9.55511 + 5.51665i) q^{61} +(3.62520 + 2.09301i) q^{63} +6.10173i q^{65} +(-0.900991 - 0.520188i) q^{67} +0.0178508 q^{69} +(2.84910 + 4.93478i) q^{71} +(0.104749 + 0.181430i) q^{73} -0.500537i q^{75} +4.60750i q^{77} +(-8.20635 - 14.2138i) q^{79} +(-4.39461 - 7.61168i) q^{81} -16.9470 q^{83} +(-5.13049 - 2.96209i) q^{85} -1.02385i q^{87} +(-5.18052 - 2.99098i) q^{89} +(5.64415 + 3.25865i) q^{91} +(0.323262 - 0.186635i) q^{93} +(-5.24429 - 2.33159i) q^{95} +(-5.71904 + 3.30189i) q^{97} +(4.87587 - 8.44525i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 24 q^{9} + 20 q^{17} + 44 q^{25} - 60 q^{33} - 24 q^{41} - 64 q^{49} - 36 q^{57} - 64 q^{73} - 24 q^{81} - 12 q^{89} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.132709 + 0.0766197i −0.0766197 + 0.0442364i −0.537820 0.843059i \(-0.680752\pi\)
0.461201 + 0.887296i \(0.347419\pi\)
\(4\) 0 0
\(5\) 1.14027 0.658336i 0.509945 0.294417i −0.222866 0.974849i \(-0.571541\pi\)
0.732811 + 0.680432i \(0.238208\pi\)
\(6\) 0 0
\(7\) 1.40635i 0.531549i −0.964035 0.265774i \(-0.914372\pi\)
0.964035 0.265774i \(-0.0856277\pi\)
\(8\) 0 0
\(9\) −1.48826 + 2.57774i −0.496086 + 0.859247i
\(10\) 0 0
\(11\) −3.27622 −0.987818 −0.493909 0.869514i \(-0.664432\pi\)
−0.493909 + 0.869514i \(0.664432\pi\)
\(12\) 0 0
\(13\) −2.31710 + 4.01334i −0.642649 + 1.11310i 0.342190 + 0.939631i \(0.388831\pi\)
−0.984839 + 0.173470i \(0.944502\pi\)
\(14\) 0 0
\(15\) −0.100883 + 0.174735i −0.0260479 + 0.0451163i
\(16\) 0 0
\(17\) −2.24968 3.89655i −0.545627 0.945053i −0.998567 0.0535125i \(-0.982958\pi\)
0.452940 0.891541i \(-0.350375\pi\)
\(18\) 0 0
\(19\) −2.56396 3.52507i −0.588213 0.808706i
\(20\) 0 0
\(21\) 0.107754 + 0.186635i 0.0235138 + 0.0407271i
\(22\) 0 0
\(23\) −0.100883 0.0582449i −0.0210356 0.0121449i 0.489445 0.872034i \(-0.337199\pi\)
−0.510481 + 0.859889i \(0.670533\pi\)
\(24\) 0 0
\(25\) −1.63319 + 2.82876i −0.326637 + 0.565753i
\(26\) 0 0
\(27\) 0.915838i 0.176253i
\(28\) 0 0
\(29\) −3.34070 + 5.78625i −0.620352 + 1.07448i 0.369069 + 0.929402i \(0.379677\pi\)
−0.989420 + 0.145078i \(0.953657\pi\)
\(30\) 0 0
\(31\) −2.43586 −0.437494 −0.218747 0.975782i \(-0.570197\pi\)
−0.218747 + 0.975782i \(0.570197\pi\)
\(32\) 0 0
\(33\) 0.434785 0.251023i 0.0756863 0.0436975i
\(34\) 0 0
\(35\) −0.925849 1.60362i −0.156497 0.271061i
\(36\) 0 0
\(37\) −8.87096 −1.45838 −0.729189 0.684313i \(-0.760102\pi\)
−0.729189 + 0.684313i \(0.760102\pi\)
\(38\) 0 0
\(39\) 0.710143i 0.113714i
\(40\) 0 0
\(41\) 5.96478 3.44377i 0.931542 0.537826i 0.0442428 0.999021i \(-0.485912\pi\)
0.887299 + 0.461195i \(0.152579\pi\)
\(42\) 0 0
\(43\) −4.01334 6.95131i −0.612029 1.06006i −0.990898 0.134615i \(-0.957020\pi\)
0.378869 0.925450i \(-0.376313\pi\)
\(44\) 0 0
\(45\) 3.91910i 0.584225i
\(46\) 0 0
\(47\) 7.69223 + 4.44111i 1.12203 + 0.647802i 0.941918 0.335844i \(-0.109022\pi\)
0.180109 + 0.983647i \(0.442355\pi\)
\(48\) 0 0
\(49\) 5.02219 0.717456
\(50\) 0 0
\(51\) 0.597106 + 0.344739i 0.0836116 + 0.0482732i
\(52\) 0 0
\(53\) −2.47034 + 4.27876i −0.339328 + 0.587733i −0.984306 0.176468i \(-0.943533\pi\)
0.644979 + 0.764201i \(0.276866\pi\)
\(54\) 0 0
\(55\) −3.73578 + 2.15685i −0.503733 + 0.290830i
\(56\) 0 0
\(57\) 0.610351 + 0.271360i 0.0808430 + 0.0359424i
\(58\) 0 0
\(59\) −9.19309 + 5.30763i −1.19684 + 0.690995i −0.959849 0.280517i \(-0.909494\pi\)
−0.236990 + 0.971512i \(0.576161\pi\)
\(60\) 0 0
\(61\) 9.55511 + 5.51665i 1.22341 + 0.706334i 0.965643 0.259873i \(-0.0836808\pi\)
0.257764 + 0.966208i \(0.417014\pi\)
\(62\) 0 0
\(63\) 3.62520 + 2.09301i 0.456732 + 0.263694i
\(64\) 0 0
\(65\) 6.10173i 0.756827i
\(66\) 0 0
\(67\) −0.900991 0.520188i −0.110074 0.0635510i 0.443952 0.896050i \(-0.353576\pi\)
−0.554026 + 0.832499i \(0.686909\pi\)
\(68\) 0 0
\(69\) 0.0178508 0.00214899
\(70\) 0 0
\(71\) 2.84910 + 4.93478i 0.338126 + 0.585651i 0.984080 0.177725i \(-0.0568737\pi\)
−0.645954 + 0.763376i \(0.723540\pi\)
\(72\) 0 0
\(73\) 0.104749 + 0.181430i 0.0122599 + 0.0212348i 0.872090 0.489345i \(-0.162764\pi\)
−0.859830 + 0.510580i \(0.829431\pi\)
\(74\) 0 0
\(75\) 0.500537i 0.0577971i
\(76\) 0 0
\(77\) 4.60750i 0.525074i
\(78\) 0 0
\(79\) −8.20635 14.2138i −0.923286 1.59918i −0.794295 0.607533i \(-0.792159\pi\)
−0.128991 0.991646i \(-0.541174\pi\)
\(80\) 0 0
\(81\) −4.39461 7.61168i −0.488289 0.845742i
\(82\) 0 0
\(83\) −16.9470 −1.86018 −0.930090 0.367331i \(-0.880272\pi\)
−0.930090 + 0.367331i \(0.880272\pi\)
\(84\) 0 0
\(85\) −5.13049 2.96209i −0.556479 0.321283i
\(86\) 0 0
\(87\) 1.02385i 0.109769i
\(88\) 0 0
\(89\) −5.18052 2.99098i −0.549134 0.317043i 0.199638 0.979870i \(-0.436023\pi\)
−0.748773 + 0.662827i \(0.769357\pi\)
\(90\) 0 0
\(91\) 5.64415 + 3.25865i 0.591667 + 0.341599i
\(92\) 0 0
\(93\) 0.323262 0.186635i 0.0335207 0.0193532i
\(94\) 0 0
\(95\) −5.24429 2.33159i −0.538053 0.239216i
\(96\) 0 0
\(97\) −5.71904 + 3.30189i −0.580680 + 0.335256i −0.761404 0.648278i \(-0.775489\pi\)
0.180723 + 0.983534i \(0.442156\pi\)
\(98\) 0 0
\(99\) 4.87587 8.44525i 0.490043 0.848779i
\(100\) 0 0
\(101\) −1.14027 0.658336i −0.113461 0.0655069i 0.442195 0.896919i \(-0.354200\pi\)
−0.555657 + 0.831412i \(0.687533\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) 0 0
\(105\) 0.245737 + 0.141877i 0.0239815 + 0.0138457i
\(106\) 0 0
\(107\) 13.5230i 1.30732i −0.756788 0.653660i \(-0.773233\pi\)
0.756788 0.653660i \(-0.226767\pi\)
\(108\) 0 0
\(109\) 0.0365602 + 0.0633241i 0.00350183 + 0.00606535i 0.867771 0.496964i \(-0.165552\pi\)
−0.864269 + 0.503030i \(0.832219\pi\)
\(110\) 0 0
\(111\) 1.17726 0.679691i 0.111740 0.0645134i
\(112\) 0 0
\(113\) 0.905579i 0.0851897i −0.999092 0.0425948i \(-0.986438\pi\)
0.999092 0.0425948i \(-0.0135625\pi\)
\(114\) 0 0
\(115\) −0.153379 −0.0143026
\(116\) 0 0
\(117\) −6.89690 11.9458i −0.637619 1.10439i
\(118\) 0 0
\(119\) −5.47991 + 3.16382i −0.502342 + 0.290027i
\(120\) 0 0
\(121\) −0.266374 −0.0242158
\(122\) 0 0
\(123\) −0.527721 + 0.914039i −0.0475830 + 0.0824161i
\(124\) 0 0
\(125\) 10.8841i 0.973504i
\(126\) 0 0
\(127\) −9.93840 + 17.2138i −0.881890 + 1.52748i −0.0326529 + 0.999467i \(0.510396\pi\)
−0.849237 + 0.528012i \(0.822938\pi\)
\(128\) 0 0
\(129\) 1.06522 + 0.615002i 0.0937870 + 0.0541479i
\(130\) 0 0
\(131\) 4.99102 + 8.64470i 0.436068 + 0.755291i 0.997382 0.0723114i \(-0.0230375\pi\)
−0.561315 + 0.827603i \(0.689704\pi\)
\(132\) 0 0
\(133\) −4.95747 + 3.60581i −0.429867 + 0.312664i
\(134\) 0 0
\(135\) −0.602929 1.04430i −0.0518919 0.0898794i
\(136\) 0 0
\(137\) −1.61649 + 2.79984i −0.138106 + 0.239207i −0.926780 0.375605i \(-0.877435\pi\)
0.788674 + 0.614812i \(0.210768\pi\)
\(138\) 0 0
\(139\) −0.258114 + 0.447067i −0.0218930 + 0.0379197i −0.876764 0.480920i \(-0.840303\pi\)
0.854871 + 0.518840i \(0.173636\pi\)
\(140\) 0 0
\(141\) −1.36111 −0.114626
\(142\) 0 0
\(143\) 7.59134 13.1486i 0.634820 1.09954i
\(144\) 0 0
\(145\) 8.79720i 0.730568i
\(146\) 0 0
\(147\) −0.666491 + 0.384799i −0.0549713 + 0.0317377i
\(148\) 0 0
\(149\) 4.53431 2.61789i 0.371465 0.214466i −0.302633 0.953107i \(-0.597866\pi\)
0.674098 + 0.738642i \(0.264532\pi\)
\(150\) 0 0
\(151\) 5.89996 0.480132 0.240066 0.970757i \(-0.422831\pi\)
0.240066 + 0.970757i \(0.422831\pi\)
\(152\) 0 0
\(153\) 13.3924 1.08271
\(154\) 0 0
\(155\) −2.77755 + 1.60362i −0.223098 + 0.128806i
\(156\) 0 0
\(157\) 8.77213 5.06459i 0.700093 0.404199i −0.107289 0.994228i \(-0.534217\pi\)
0.807382 + 0.590029i \(0.200884\pi\)
\(158\) 0 0
\(159\) 0.757108i 0.0600426i
\(160\) 0 0
\(161\) −0.0819125 + 0.141877i −0.00645561 + 0.0111814i
\(162\) 0 0
\(163\) 4.33790 0.339770 0.169885 0.985464i \(-0.445660\pi\)
0.169885 + 0.985464i \(0.445660\pi\)
\(164\) 0 0
\(165\) 0.330515 0.572469i 0.0257306 0.0445667i
\(166\) 0 0
\(167\) 2.34705 4.06522i 0.181620 0.314576i −0.760812 0.648972i \(-0.775199\pi\)
0.942432 + 0.334396i \(0.108532\pi\)
\(168\) 0 0
\(169\) −4.23794 7.34032i −0.325995 0.564640i
\(170\) 0 0
\(171\) 12.9025 1.36300i 0.986682 0.104232i
\(172\) 0 0
\(173\) 4.67077 + 8.09001i 0.355112 + 0.615072i 0.987137 0.159876i \(-0.0511095\pi\)
−0.632025 + 0.774948i \(0.717776\pi\)
\(174\) 0 0
\(175\) 3.97822 + 2.29683i 0.300725 + 0.173624i
\(176\) 0 0
\(177\) 0.813339 1.40874i 0.0611343 0.105888i
\(178\) 0 0
\(179\) 14.3623i 1.07349i 0.843744 + 0.536746i \(0.180347\pi\)
−0.843744 + 0.536746i \(0.819653\pi\)
\(180\) 0 0
\(181\) 3.43167 5.94382i 0.255074 0.441801i −0.709842 0.704361i \(-0.751234\pi\)
0.964916 + 0.262560i \(0.0845669\pi\)
\(182\) 0 0
\(183\) −1.69074 −0.124983
\(184\) 0 0
\(185\) −10.1153 + 5.84008i −0.743692 + 0.429371i
\(186\) 0 0
\(187\) 7.37044 + 12.7660i 0.538980 + 0.933541i
\(188\) 0 0
\(189\) −1.28799 −0.0936872
\(190\) 0 0
\(191\) 11.7670i 0.851432i −0.904857 0.425716i \(-0.860022\pi\)
0.904857 0.425716i \(-0.139978\pi\)
\(192\) 0 0
\(193\) 11.1805 6.45508i 0.804792 0.464647i −0.0403523 0.999186i \(-0.512848\pi\)
0.845144 + 0.534539i \(0.179515\pi\)
\(194\) 0 0
\(195\) −0.467513 0.809756i −0.0334793 0.0579879i
\(196\) 0 0
\(197\) 19.0600i 1.35797i −0.734153 0.678984i \(-0.762421\pi\)
0.734153 0.678984i \(-0.237579\pi\)
\(198\) 0 0
\(199\) −6.47429 3.73794i −0.458951 0.264975i 0.252652 0.967557i \(-0.418697\pi\)
−0.711603 + 0.702582i \(0.752030\pi\)
\(200\) 0 0
\(201\) 0.159427 0.0112451
\(202\) 0 0
\(203\) 8.13748 + 4.69818i 0.571139 + 0.329747i
\(204\) 0 0
\(205\) 4.53431 7.85366i 0.316690 0.548523i
\(206\) 0 0
\(207\) 0.300280 0.173367i 0.0208709 0.0120498i
\(208\) 0 0
\(209\) 8.40010 + 11.5489i 0.581047 + 0.798855i
\(210\) 0 0
\(211\) 12.5429 7.24164i 0.863488 0.498535i −0.00169088 0.999999i \(-0.500538\pi\)
0.865179 + 0.501464i \(0.167205\pi\)
\(212\) 0 0
\(213\) −0.756204 0.436594i −0.0518142 0.0299150i
\(214\) 0 0
\(215\) −9.15260 5.28425i −0.624202 0.360383i
\(216\) 0 0
\(217\) 3.42567i 0.232549i
\(218\) 0 0
\(219\) −0.0278022 0.0160516i −0.00187870 0.00108467i
\(220\) 0 0
\(221\) 20.8509 1.40259
\(222\) 0 0
\(223\) 11.1563 + 19.3233i 0.747083 + 1.29399i 0.949215 + 0.314627i \(0.101880\pi\)
−0.202132 + 0.979358i \(0.564787\pi\)
\(224\) 0 0
\(225\) −4.86121 8.41986i −0.324081 0.561324i
\(226\) 0 0
\(227\) 5.40040i 0.358437i 0.983809 + 0.179219i \(0.0573570\pi\)
−0.983809 + 0.179219i \(0.942643\pi\)
\(228\) 0 0
\(229\) 21.5441i 1.42368i −0.702344 0.711838i \(-0.747863\pi\)
0.702344 0.711838i \(-0.252137\pi\)
\(230\) 0 0
\(231\) −0.353026 0.611458i −0.0232274 0.0402310i
\(232\) 0 0
\(233\) −8.91744 15.4455i −0.584201 1.01187i −0.994975 0.100128i \(-0.968075\pi\)
0.410774 0.911737i \(-0.365259\pi\)
\(234\) 0 0
\(235\) 11.6950 0.762896
\(236\) 0 0
\(237\) 2.17812 + 1.25754i 0.141484 + 0.0816857i
\(238\) 0 0
\(239\) 23.2985i 1.50705i 0.657417 + 0.753527i \(0.271649\pi\)
−0.657417 + 0.753527i \(0.728351\pi\)
\(240\) 0 0
\(241\) −12.3142 7.10963i −0.793230 0.457972i 0.0478682 0.998854i \(-0.484757\pi\)
−0.841099 + 0.540882i \(0.818091\pi\)
\(242\) 0 0
\(243\) 3.54583 + 2.04718i 0.227465 + 0.131327i
\(244\) 0 0
\(245\) 5.72666 3.30629i 0.365863 0.211231i
\(246\) 0 0
\(247\) 20.0883 2.12209i 1.27819 0.135025i
\(248\) 0 0
\(249\) 2.24903 1.29848i 0.142527 0.0822877i
\(250\) 0 0
\(251\) −5.52605 + 9.57139i −0.348801 + 0.604141i −0.986037 0.166528i \(-0.946744\pi\)
0.637236 + 0.770669i \(0.280078\pi\)
\(252\) 0 0
\(253\) 0.330515 + 0.190823i 0.0207793 + 0.0119969i
\(254\) 0 0
\(255\) 0.907817 0.0568497
\(256\) 0 0
\(257\) 7.18382 + 4.14758i 0.448114 + 0.258719i 0.707033 0.707180i \(-0.250033\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(258\) 0 0
\(259\) 12.4756i 0.775199i
\(260\) 0 0
\(261\) −9.94364 17.2229i −0.615496 1.06607i
\(262\) 0 0
\(263\) −18.0845 + 10.4411i −1.11514 + 0.643826i −0.940156 0.340745i \(-0.889321\pi\)
−0.174984 + 0.984571i \(0.555987\pi\)
\(264\) 0 0
\(265\) 6.50526i 0.399615i
\(266\) 0 0
\(267\) 0.916671 0.0560994
\(268\) 0 0
\(269\) 12.6259 + 21.8687i 0.769814 + 1.33336i 0.937663 + 0.347545i \(0.112984\pi\)
−0.167849 + 0.985813i \(0.553682\pi\)
\(270\) 0 0
\(271\) 0.0602105 0.0347625i 0.00365753 0.00211167i −0.498170 0.867079i \(-0.665994\pi\)
0.501828 + 0.864968i \(0.332661\pi\)
\(272\) 0 0
\(273\) −0.998708 −0.0604445
\(274\) 0 0
\(275\) 5.35068 9.26765i 0.322658 0.558861i
\(276\) 0 0
\(277\) 30.6016i 1.83867i −0.393476 0.919335i \(-0.628727\pi\)
0.393476 0.919335i \(-0.371273\pi\)
\(278\) 0 0
\(279\) 3.62520 6.27902i 0.217035 0.375915i
\(280\) 0 0
\(281\) 15.8343 + 9.14196i 0.944598 + 0.545364i 0.891399 0.453220i \(-0.149725\pi\)
0.0531992 + 0.998584i \(0.483058\pi\)
\(282\) 0 0
\(283\) −3.55189 6.15205i −0.211138 0.365702i 0.740933 0.671579i \(-0.234384\pi\)
−0.952071 + 0.305877i \(0.901050\pi\)
\(284\) 0 0
\(285\) 0.874612 0.0923926i 0.0518075 0.00547286i
\(286\) 0 0
\(287\) −4.84313 8.38854i −0.285881 0.495160i
\(288\) 0 0
\(289\) −1.62209 + 2.80955i −0.0954172 + 0.165267i
\(290\) 0 0
\(291\) 0.505980 0.876383i 0.0296611 0.0513745i
\(292\) 0 0
\(293\) 5.04737 0.294871 0.147435 0.989072i \(-0.452898\pi\)
0.147435 + 0.989072i \(0.452898\pi\)
\(294\) 0 0
\(295\) −6.98841 + 12.1043i −0.406881 + 0.704739i
\(296\) 0 0
\(297\) 3.00049i 0.174106i
\(298\) 0 0
\(299\) 0.467513 0.269919i 0.0270370 0.0156098i
\(300\) 0 0
\(301\) −9.77595 + 5.64415i −0.563476 + 0.325323i
\(302\) 0 0
\(303\) 0.201766 0.0115912
\(304\) 0 0
\(305\) 14.5272 0.831827
\(306\) 0 0
\(307\) 23.2869 13.4447i 1.32905 0.767330i 0.343900 0.939006i \(-0.388252\pi\)
0.985153 + 0.171676i \(0.0549184\pi\)
\(308\) 0 0
\(309\) −0.919437 + 0.530837i −0.0523049 + 0.0301983i
\(310\) 0 0
\(311\) 3.98606i 0.226029i −0.993593 0.113014i \(-0.963949\pi\)
0.993593 0.113014i \(-0.0360506\pi\)
\(312\) 0 0
\(313\) 5.77747 10.0069i 0.326562 0.565622i −0.655265 0.755399i \(-0.727443\pi\)
0.981827 + 0.189777i \(0.0607765\pi\)
\(314\) 0 0
\(315\) 5.51161 0.310544
\(316\) 0 0
\(317\) −4.75089 + 8.22878i −0.266836 + 0.462174i −0.968043 0.250784i \(-0.919312\pi\)
0.701207 + 0.712958i \(0.252645\pi\)
\(318\) 0 0
\(319\) 10.9449 18.9571i 0.612794 1.06139i
\(320\) 0 0
\(321\) 1.03613 + 1.79463i 0.0578312 + 0.100166i
\(322\) 0 0
\(323\) −7.96755 + 17.9209i −0.443326 + 0.997144i
\(324\) 0 0
\(325\) −7.56853 13.1091i −0.419826 0.727161i
\(326\) 0 0
\(327\) −0.00970375 0.00560246i −0.000536618 0.000309817i
\(328\) 0 0
\(329\) 6.24574 10.8179i 0.344339 0.596412i
\(330\) 0 0
\(331\) 15.5237i 0.853257i −0.904427 0.426629i \(-0.859701\pi\)
0.904427 0.426629i \(-0.140299\pi\)
\(332\) 0 0
\(333\) 13.2023 22.8670i 0.723481 1.25311i
\(334\) 0 0
\(335\) −1.36983 −0.0748420
\(336\) 0 0
\(337\) −9.28096 + 5.35837i −0.505566 + 0.291889i −0.731009 0.682368i \(-0.760950\pi\)
0.225443 + 0.974256i \(0.427617\pi\)
\(338\) 0 0
\(339\) 0.0693852 + 0.120179i 0.00376849 + 0.00652721i
\(340\) 0 0
\(341\) 7.98043 0.432164
\(342\) 0 0
\(343\) 16.9074i 0.912912i
\(344\) 0 0
\(345\) 0.0203548 0.0117518i 0.00109587 0.000632698i
\(346\) 0 0
\(347\) 13.6781 + 23.6912i 0.734281 + 1.27181i 0.955038 + 0.296483i \(0.0958137\pi\)
−0.220758 + 0.975329i \(0.570853\pi\)
\(348\) 0 0
\(349\) 3.17256i 0.169823i 0.996388 + 0.0849115i \(0.0270608\pi\)
−0.996388 + 0.0849115i \(0.972939\pi\)
\(350\) 0 0
\(351\) 3.67557 + 2.12209i 0.196187 + 0.113269i
\(352\) 0 0
\(353\) −30.8670 −1.64288 −0.821442 0.570292i \(-0.806830\pi\)
−0.821442 + 0.570292i \(0.806830\pi\)
\(354\) 0 0
\(355\) 6.49749 + 3.75133i 0.344851 + 0.199100i
\(356\) 0 0
\(357\) 0.484823 0.839738i 0.0256595 0.0444436i
\(358\) 0 0
\(359\) 7.49046 4.32462i 0.395331 0.228245i −0.289136 0.957288i \(-0.593368\pi\)
0.684468 + 0.729043i \(0.260035\pi\)
\(360\) 0 0
\(361\) −5.85223 + 18.0763i −0.308012 + 0.951383i
\(362\) 0 0
\(363\) 0.0353503 0.0204095i 0.00185541 0.00107122i
\(364\) 0 0
\(365\) 0.238884 + 0.137920i 0.0125037 + 0.00721904i
\(366\) 0 0
\(367\) −20.5039 11.8379i −1.07029 0.617934i −0.142032 0.989862i \(-0.545363\pi\)
−0.928262 + 0.371928i \(0.878697\pi\)
\(368\) 0 0
\(369\) 20.5009i 1.06723i
\(370\) 0 0
\(371\) 6.01742 + 3.47416i 0.312409 + 0.180369i
\(372\) 0 0
\(373\) −20.6176 −1.06754 −0.533769 0.845630i \(-0.679225\pi\)
−0.533769 + 0.845630i \(0.679225\pi\)
\(374\) 0 0
\(375\) −0.833937 1.44442i −0.0430643 0.0745896i
\(376\) 0 0
\(377\) −15.4815 26.8147i −0.797336 1.38103i
\(378\) 0 0
\(379\) 26.6788i 1.37040i 0.728357 + 0.685198i \(0.240284\pi\)
−0.728357 + 0.685198i \(0.759716\pi\)
\(380\) 0 0
\(381\) 3.04591i 0.156047i
\(382\) 0 0
\(383\) −5.09527 8.82527i −0.260356 0.450950i 0.705980 0.708231i \(-0.250507\pi\)
−0.966337 + 0.257281i \(0.917173\pi\)
\(384\) 0 0
\(385\) 3.03328 + 5.25380i 0.154591 + 0.267759i
\(386\) 0 0
\(387\) 23.8916 1.21448
\(388\) 0 0
\(389\) 30.8239 + 17.7962i 1.56284 + 0.902304i 0.996968 + 0.0778073i \(0.0247919\pi\)
0.565867 + 0.824496i \(0.308541\pi\)
\(390\) 0 0
\(391\) 0.524129i 0.0265063i
\(392\) 0 0
\(393\) −1.32471 0.764821i −0.0668228 0.0385801i
\(394\) 0 0
\(395\) −18.7149 10.8051i −0.941650 0.543662i
\(396\) 0 0
\(397\) 6.16833 3.56129i 0.309580 0.178736i −0.337159 0.941448i \(-0.609466\pi\)
0.646738 + 0.762712i \(0.276133\pi\)
\(398\) 0 0
\(399\) 0.381625 0.858365i 0.0191052 0.0429720i
\(400\) 0 0
\(401\) −4.84947 + 2.79984i −0.242171 + 0.139817i −0.616174 0.787610i \(-0.711318\pi\)
0.374003 + 0.927427i \(0.377985\pi\)
\(402\) 0 0
\(403\) 5.64415 9.77595i 0.281155 0.486975i
\(404\) 0 0
\(405\) −10.0221 5.78625i −0.498001 0.287521i
\(406\) 0 0
\(407\) 29.0632 1.44061
\(408\) 0 0
\(409\) −19.1134 11.0351i −0.945095 0.545651i −0.0535414 0.998566i \(-0.517051\pi\)
−0.891554 + 0.452915i \(0.850384\pi\)
\(410\) 0 0
\(411\) 0.495420i 0.0244373i
\(412\) 0 0
\(413\) 7.46437 + 12.9287i 0.367298 + 0.636178i
\(414\) 0 0
\(415\) −19.3242 + 11.1569i −0.948590 + 0.547668i
\(416\) 0 0
\(417\) 0.0791066i 0.00387387i
\(418\) 0 0
\(419\) −31.2900 −1.52862 −0.764308 0.644852i \(-0.776919\pi\)
−0.764308 + 0.644852i \(0.776919\pi\)
\(420\) 0 0
\(421\) 7.59530 + 13.1555i 0.370172 + 0.641157i 0.989592 0.143903i \(-0.0459653\pi\)
−0.619419 + 0.785060i \(0.712632\pi\)
\(422\) 0 0
\(423\) −22.8960 + 13.2190i −1.11324 + 0.642732i
\(424\) 0 0
\(425\) 14.6966 0.712889
\(426\) 0 0
\(427\) 7.75832 13.4378i 0.375451 0.650301i
\(428\) 0 0
\(429\) 2.32659i 0.112329i
\(430\) 0 0
\(431\) −13.2414 + 22.9348i −0.637816 + 1.10473i 0.348095 + 0.937459i \(0.386829\pi\)
−0.985911 + 0.167270i \(0.946505\pi\)
\(432\) 0 0
\(433\) 9.39956 + 5.42684i 0.451714 + 0.260797i 0.708554 0.705657i \(-0.249348\pi\)
−0.256840 + 0.966454i \(0.582681\pi\)
\(434\) 0 0
\(435\) −0.674039 1.16747i −0.0323177 0.0559759i
\(436\) 0 0
\(437\) 0.0533429 + 0.504957i 0.00255174 + 0.0241554i
\(438\) 0 0
\(439\) 18.5459 + 32.1224i 0.885148 + 1.53312i 0.845544 + 0.533905i \(0.179276\pi\)
0.0396035 + 0.999215i \(0.487391\pi\)
\(440\) 0 0
\(441\) −7.47432 + 12.9459i −0.355920 + 0.616471i
\(442\) 0 0
\(443\) −8.80227 + 15.2460i −0.418208 + 0.724358i −0.995759 0.0919965i \(-0.970675\pi\)
0.577551 + 0.816355i \(0.304008\pi\)
\(444\) 0 0
\(445\) −7.87627 −0.373371
\(446\) 0 0
\(447\) −0.401163 + 0.694835i −0.0189744 + 0.0328646i
\(448\) 0 0
\(449\) 29.5796i 1.39595i −0.716124 0.697973i \(-0.754085\pi\)
0.716124 0.697973i \(-0.245915\pi\)
\(450\) 0 0
\(451\) −19.5419 + 11.2825i −0.920193 + 0.531274i
\(452\) 0 0
\(453\) −0.782980 + 0.452054i −0.0367876 + 0.0212393i
\(454\) 0 0
\(455\) 8.58115 0.402290
\(456\) 0 0
\(457\) 22.3590 1.04591 0.522955 0.852360i \(-0.324829\pi\)
0.522955 + 0.852360i \(0.324829\pi\)
\(458\) 0 0
\(459\) −3.56861 + 2.06034i −0.166569 + 0.0961685i
\(460\) 0 0
\(461\) −26.3417 + 15.2084i −1.22686 + 0.708325i −0.966371 0.257153i \(-0.917216\pi\)
−0.260485 + 0.965478i \(0.583882\pi\)
\(462\) 0 0
\(463\) 28.6274i 1.33043i −0.746652 0.665214i \(-0.768340\pi\)
0.746652 0.665214i \(-0.231660\pi\)
\(464\) 0 0
\(465\) 0.245737 0.425630i 0.0113958 0.0197381i
\(466\) 0 0
\(467\) −3.27622 −0.151605 −0.0758027 0.997123i \(-0.524152\pi\)
−0.0758027 + 0.997123i \(0.524152\pi\)
\(468\) 0 0
\(469\) −0.731564 + 1.26711i −0.0337805 + 0.0585095i
\(470\) 0 0
\(471\) −0.776096 + 1.34424i −0.0357606 + 0.0619392i
\(472\) 0 0
\(473\) 13.1486 + 22.7740i 0.604573 + 1.04715i
\(474\) 0 0
\(475\) 14.1590 1.49574i 0.649660 0.0686291i
\(476\) 0 0
\(477\) −7.35302 12.7358i −0.336672 0.583132i
\(478\) 0 0
\(479\) 26.9345 + 15.5506i 1.23067 + 0.710526i 0.967169 0.254133i \(-0.0817902\pi\)
0.263499 + 0.964660i \(0.415124\pi\)
\(480\) 0 0
\(481\) 20.5549 35.6022i 0.937225 1.62332i
\(482\) 0 0
\(483\) 0.0251044i 0.00114229i
\(484\) 0 0
\(485\) −4.34751 + 7.53010i −0.197410 + 0.341924i
\(486\) 0 0
\(487\) −14.8846 −0.674488 −0.337244 0.941417i \(-0.609495\pi\)
−0.337244 + 0.941417i \(0.609495\pi\)
\(488\) 0 0
\(489\) −0.575679 + 0.332368i −0.0260331 + 0.0150302i
\(490\) 0 0
\(491\) −15.8367 27.4299i −0.714698 1.23789i −0.963076 0.269231i \(-0.913231\pi\)
0.248377 0.968663i \(-0.420103\pi\)
\(492\) 0 0
\(493\) 30.0619 1.35392
\(494\) 0 0
\(495\) 12.8398i 0.577108i
\(496\) 0 0
\(497\) 6.94002 4.00682i 0.311302 0.179730i
\(498\) 0 0
\(499\) −11.1702 19.3473i −0.500047 0.866106i −1.00000 5.37306e-5i \(-0.999983\pi\)
0.499953 0.866052i \(-0.333350\pi\)
\(500\) 0 0
\(501\) 0.719322i 0.0321370i
\(502\) 0 0
\(503\) −18.3683 10.6049i −0.819001 0.472851i 0.0310706 0.999517i \(-0.490108\pi\)
−0.850072 + 0.526667i \(0.823442\pi\)
\(504\) 0 0
\(505\) −1.73363 −0.0771453
\(506\) 0 0
\(507\) 1.12483 + 0.649419i 0.0499553 + 0.0288417i
\(508\) 0 0
\(509\) −1.93051 + 3.34373i −0.0855681 + 0.148208i −0.905633 0.424062i \(-0.860604\pi\)
0.820065 + 0.572270i \(0.193937\pi\)
\(510\) 0 0
\(511\) 0.255153 0.147313i 0.0112873 0.00651673i
\(512\) 0 0
\(513\) −3.22839 + 2.34817i −0.142537 + 0.103674i
\(514\) 0 0
\(515\) 7.90003 4.56109i 0.348117 0.200985i
\(516\) 0 0
\(517\) −25.2014 14.5501i −1.10836 0.639911i
\(518\) 0 0
\(519\) −1.23971 0.715746i −0.0544171 0.0314178i
\(520\) 0 0
\(521\) 37.0368i 1.62261i 0.584620 + 0.811307i \(0.301243\pi\)
−0.584620 + 0.811307i \(0.698757\pi\)
\(522\) 0 0
\(523\) −20.3903 11.7723i −0.891604 0.514768i −0.0171370 0.999853i \(-0.505455\pi\)
−0.874467 + 0.485085i \(0.838788\pi\)
\(524\) 0 0
\(525\) −0.703929 −0.0307220
\(526\) 0 0
\(527\) 5.47991 + 9.49147i 0.238708 + 0.413455i
\(528\) 0 0
\(529\) −11.4932 19.9068i −0.499705 0.865514i
\(530\) 0 0
\(531\) 31.5965i 1.37117i
\(532\) 0 0
\(533\) 31.9182i 1.38253i
\(534\) 0 0
\(535\) −8.90269 15.4199i −0.384897 0.666661i
\(536\) 0 0
\(537\) −1.10044 1.90602i −0.0474874 0.0822506i
\(538\) 0 0
\(539\) −16.4538 −0.708716
\(540\) 0 0
\(541\) −18.1064 10.4537i −0.778456 0.449442i 0.0574271 0.998350i \(-0.481710\pi\)
−0.835883 + 0.548908i \(0.815044\pi\)
\(542\) 0 0
\(543\) 1.05173i 0.0451342i
\(544\) 0 0
\(545\) 0.0833770 + 0.0481378i 0.00357148 + 0.00206200i
\(546\) 0 0
\(547\) −10.4537 6.03547i −0.446970 0.258058i 0.259580 0.965722i \(-0.416416\pi\)
−0.706550 + 0.707664i \(0.749749\pi\)
\(548\) 0 0
\(549\) −28.4410 + 16.4204i −1.21383 + 0.700806i
\(550\) 0 0
\(551\) 28.9624 3.05954i 1.23384 0.130341i
\(552\) 0 0
\(553\) −19.9895 + 11.5410i −0.850042 + 0.490772i
\(554\) 0 0
\(555\) 0.894930 1.55006i 0.0379877 0.0657966i
\(556\) 0 0
\(557\) −8.52452 4.92164i −0.361196 0.208536i 0.308409 0.951254i \(-0.400203\pi\)
−0.669605 + 0.742717i \(0.733537\pi\)
\(558\) 0 0
\(559\) 37.1973 1.57328
\(560\) 0 0
\(561\) −1.95625 1.12944i −0.0825930 0.0476851i
\(562\) 0 0
\(563\) 7.94787i 0.334963i 0.985875 + 0.167481i \(0.0535634\pi\)
−0.985875 + 0.167481i \(0.946437\pi\)
\(564\) 0 0
\(565\) −0.596175 1.03261i −0.0250813 0.0434420i
\(566\) 0 0
\(567\) −10.7047 + 6.18034i −0.449553 + 0.259550i
\(568\) 0 0
\(569\) 26.6989i 1.11927i 0.828738 + 0.559637i \(0.189060\pi\)
−0.828738 + 0.559637i \(0.810940\pi\)
\(570\) 0 0
\(571\) 7.44613 0.311611 0.155805 0.987788i \(-0.450203\pi\)
0.155805 + 0.987788i \(0.450203\pi\)
\(572\) 0 0
\(573\) 0.901586 + 1.56159i 0.0376643 + 0.0652365i
\(574\) 0 0
\(575\) 0.329522 0.190250i 0.0137420 0.00793395i
\(576\) 0 0
\(577\) 40.2403 1.67523 0.837613 0.546264i \(-0.183951\pi\)
0.837613 + 0.546264i \(0.183951\pi\)
\(578\) 0 0
\(579\) −0.989173 + 1.71330i −0.0411086 + 0.0712022i
\(580\) 0 0
\(581\) 23.8334i 0.988777i
\(582\) 0 0
\(583\) 8.09339 14.0182i 0.335194 0.580573i
\(584\) 0 0
\(585\) −15.7287 9.08096i −0.650301 0.375451i
\(586\) 0 0
\(587\) −20.4442 35.4103i −0.843821 1.46154i −0.886641 0.462458i \(-0.846968\pi\)
0.0428204 0.999083i \(-0.486366\pi\)
\(588\) 0 0
\(589\) 6.24545 + 8.58659i 0.257339 + 0.353804i
\(590\) 0 0
\(591\) 1.46037 + 2.52944i 0.0600716 + 0.104047i
\(592\) 0 0
\(593\) −0.479363 + 0.830281i −0.0196851 + 0.0340956i −0.875700 0.482855i \(-0.839600\pi\)
0.856015 + 0.516951i \(0.172933\pi\)
\(594\) 0 0
\(595\) −4.16572 + 7.21524i −0.170778 + 0.295796i
\(596\) 0 0
\(597\) 1.14560 0.0468862
\(598\) 0 0
\(599\) −14.5798 + 25.2529i −0.595713 + 1.03180i 0.397733 + 0.917501i \(0.369797\pi\)
−0.993446 + 0.114304i \(0.963536\pi\)
\(600\) 0 0
\(601\) 35.5037i 1.44823i −0.689682 0.724113i \(-0.742250\pi\)
0.689682 0.724113i \(-0.257750\pi\)
\(602\) 0 0
\(603\) 2.68182 1.54835i 0.109212 0.0630536i
\(604\) 0 0
\(605\) −0.303739 + 0.175364i −0.0123487 + 0.00712955i
\(606\) 0 0
\(607\) 5.41509 0.219792 0.109896 0.993943i \(-0.464948\pi\)
0.109896 + 0.993943i \(0.464948\pi\)
\(608\) 0 0
\(609\) −1.43989 −0.0583474
\(610\) 0 0
\(611\) −35.6474 + 20.5810i −1.44214 + 0.832619i
\(612\) 0 0
\(613\) −3.93813 + 2.27368i −0.159060 + 0.0918332i −0.577417 0.816449i \(-0.695939\pi\)
0.418357 + 0.908283i \(0.362606\pi\)
\(614\) 0 0
\(615\) 1.38967i 0.0560369i
\(616\) 0 0
\(617\) 0.233626 0.404652i 0.00940542 0.0162907i −0.861284 0.508123i \(-0.830339\pi\)
0.870690 + 0.491833i \(0.163673\pi\)
\(618\) 0 0
\(619\) 4.20501 0.169014 0.0845069 0.996423i \(-0.473068\pi\)
0.0845069 + 0.996423i \(0.473068\pi\)
\(620\) 0 0
\(621\) −0.0533429 + 0.0923926i −0.00214058 + 0.00370759i
\(622\) 0 0
\(623\) −4.20635 + 7.28561i −0.168524 + 0.291892i
\(624\) 0 0
\(625\) −1.00054 1.73298i −0.0400214 0.0693192i
\(626\) 0 0
\(627\) −1.99965 0.889034i −0.0798581 0.0355046i
\(628\) 0 0
\(629\) 19.9568 + 34.5662i 0.795730 + 1.37824i
\(630\) 0 0
\(631\) −22.5333 13.0096i −0.897037 0.517905i −0.0207994 0.999784i \(-0.506621\pi\)
−0.876238 + 0.481879i \(0.839954\pi\)
\(632\) 0 0
\(633\) −1.10970 + 1.92207i −0.0441068 + 0.0763952i
\(634\) 0 0
\(635\) 26.1712i 1.03857i
\(636\) 0 0
\(637\) −11.6369 + 20.1558i −0.461072 + 0.798600i
\(638\) 0 0
\(639\) −16.9608 −0.670958
\(640\) 0 0
\(641\) −27.9714 + 16.1493i −1.10480 + 0.637858i −0.937478 0.348044i \(-0.886846\pi\)
−0.167324 + 0.985902i \(0.553513\pi\)
\(642\) 0 0
\(643\) 1.85776 + 3.21773i 0.0732628 + 0.126895i 0.900330 0.435209i \(-0.143325\pi\)
−0.827067 + 0.562104i \(0.809992\pi\)
\(644\) 0 0
\(645\) 1.61951 0.0637682
\(646\) 0 0
\(647\) 37.7366i 1.48358i −0.670634 0.741789i \(-0.733978\pi\)
0.670634 0.741789i \(-0.266022\pi\)
\(648\) 0 0
\(649\) 30.1186 17.3890i 1.18226 0.682577i
\(650\) 0 0
\(651\) −0.262474 0.454618i −0.0102872 0.0178179i
\(652\) 0 0
\(653\) 6.76815i 0.264858i 0.991192 + 0.132429i \(0.0422777\pi\)
−0.991192 + 0.132429i \(0.957722\pi\)
\(654\) 0 0
\(655\) 11.3822 + 6.57154i 0.444741 + 0.256771i
\(656\) 0 0
\(657\) −0.623572 −0.0243279
\(658\) 0 0
\(659\) −37.3912 21.5878i −1.45655 0.840942i −0.457715 0.889099i \(-0.651332\pi\)
−0.998840 + 0.0481567i \(0.984665\pi\)
\(660\) 0 0
\(661\) −14.7222 + 25.4995i −0.572626 + 0.991817i 0.423669 + 0.905817i \(0.360742\pi\)
−0.996295 + 0.0859999i \(0.972592\pi\)
\(662\) 0 0
\(663\) −2.76711 + 1.59759i −0.107466 + 0.0620454i
\(664\) 0 0
\(665\) −3.27902 + 7.37529i −0.127155 + 0.286001i
\(666\) 0 0
\(667\) 0.674039 0.389157i 0.0260989 0.0150682i
\(668\) 0 0
\(669\) −2.96110 1.70959i −0.114483 0.0660966i
\(670\) 0 0
\(671\) −31.3047 18.0738i −1.20850 0.697730i
\(672\) 0 0
\(673\) 43.1793i 1.66444i 0.554447 + 0.832219i \(0.312930\pi\)
−0.554447 + 0.832219i \(0.687070\pi\)
\(674\) 0 0
\(675\) 2.59069 + 1.49574i 0.0997157 + 0.0575709i
\(676\) 0 0
\(677\) 14.2822 0.548910 0.274455 0.961600i \(-0.411503\pi\)
0.274455 + 0.961600i \(0.411503\pi\)
\(678\) 0 0
\(679\) 4.64360 + 8.04295i 0.178205 + 0.308660i
\(680\) 0 0
\(681\) −0.413778 0.716684i −0.0158560 0.0274634i
\(682\) 0 0
\(683\) 5.48052i 0.209706i 0.994488 + 0.104853i \(0.0334373\pi\)
−0.994488 + 0.104853i \(0.966563\pi\)
\(684\) 0 0
\(685\) 4.25677i 0.162643i
\(686\) 0 0
\(687\) 1.65071 + 2.85911i 0.0629783 + 0.109082i
\(688\) 0 0
\(689\) −11.4481 19.8287i −0.436137 0.755412i
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) −11.8769 6.85716i −0.451168 0.260482i
\(694\) 0 0
\(695\) 0.679704i 0.0257826i
\(696\) 0 0
\(697\) −26.8376 15.4947i −1.01655 0.586904i
\(698\) 0 0
\(699\) 2.36685 + 1.36650i 0.0895227 + 0.0516859i
\(700\) 0 0
\(701\) −9.12070 + 5.26584i −0.344484 + 0.198888i −0.662253 0.749280i \(-0.730400\pi\)
0.317769 + 0.948168i \(0.397066\pi\)
\(702\) 0 0
\(703\) 22.7448 + 31.2708i 0.857836 + 1.17940i
\(704\) 0 0
\(705\) −1.55203 + 0.896065i −0.0584529 + 0.0337478i
\(706\) 0 0
\(707\) −0.925849 + 1.60362i −0.0348201 + 0.0603102i
\(708\) 0 0
\(709\) 28.8847 + 16.6766i 1.08479 + 0.626303i 0.932184 0.361985i \(-0.117901\pi\)
0.152604 + 0.988287i \(0.451234\pi\)
\(710\) 0 0
\(711\) 48.8527 1.83212
\(712\) 0 0
\(713\) 0.245737 + 0.141877i 0.00920294 + 0.00531332i
\(714\) 0 0
\(715\) 19.9906i 0.747607i
\(716\) 0 0
\(717\) −1.78512 3.09192i −0.0666666 0.115470i
\(718\) 0 0
\(719\) −37.7303 + 21.7836i −1.40710 + 0.812391i −0.995108 0.0987944i \(-0.968501\pi\)
−0.411995 + 0.911186i \(0.635168\pi\)
\(720\) 0 0
\(721\) 9.74345i 0.362865i
\(722\) 0 0
\(723\) 2.17895 0.0810361
\(724\) 0 0
\(725\) −10.9120 18.9001i −0.405260 0.701931i
\(726\) 0 0
\(727\) −26.6536 + 15.3885i −0.988528 + 0.570727i −0.904834 0.425765i \(-0.860005\pi\)
−0.0836937 + 0.996492i \(0.526672\pi\)
\(728\) 0 0
\(729\) 25.7402 0.953341
\(730\) 0 0
\(731\) −18.0574 + 31.2764i −0.667879 + 1.15680i
\(732\) 0 0
\(733\) 6.99593i 0.258400i 0.991619 + 0.129200i \(0.0412410\pi\)
−0.991619 + 0.129200i \(0.958759\pi\)
\(734\) 0 0
\(735\) −0.506654 + 0.877550i −0.0186882 + 0.0323689i
\(736\) 0 0
\(737\) 2.95185 + 1.70425i 0.108733 + 0.0627768i
\(738\) 0 0
\(739\) 14.1634 + 24.5318i 0.521011 + 0.902417i 0.999701 + 0.0244334i \(0.00777816\pi\)
−0.478691 + 0.877984i \(0.658889\pi\)
\(740\) 0 0
\(741\) −2.50330 + 1.82078i −0.0919612 + 0.0668880i
\(742\) 0 0
\(743\) 6.78665 + 11.7548i 0.248978 + 0.431242i 0.963243 0.268633i \(-0.0865720\pi\)
−0.714265 + 0.699876i \(0.753239\pi\)
\(744\) 0 0
\(745\) 3.44690 5.97020i 0.126285 0.218731i
\(746\) 0 0
\(747\) 25.2216 43.6851i 0.922810 1.59835i
\(748\) 0 0
\(749\) −19.0181 −0.694904
\(750\) 0 0
\(751\) −2.22439 + 3.85276i −0.0811693 + 0.140589i −0.903752 0.428056i \(-0.859199\pi\)
0.822583 + 0.568645i \(0.192532\pi\)
\(752\) 0 0
\(753\) 1.69362i 0.0617188i
\(754\) 0 0
\(755\) 6.72756 3.88416i 0.244841 0.141359i
\(756\) 0 0
\(757\) −19.0331 + 10.9888i −0.691770 + 0.399394i −0.804275 0.594258i \(-0.797446\pi\)
0.112505 + 0.993651i \(0.464113\pi\)
\(758\) 0 0
\(759\) −0.0584833 −0.00212281
\(760\) 0 0
\(761\) 34.7566 1.25992 0.629962 0.776626i \(-0.283070\pi\)
0.629962 + 0.776626i \(0.283070\pi\)
\(762\) 0 0
\(763\) 0.0890556 0.0514163i 0.00322403 0.00186139i
\(764\) 0 0
\(765\) 15.2710 8.81670i 0.552123 0.318769i
\(766\) 0 0
\(767\) 49.1933i 1.77627i
\(768\) 0 0
\(769\) −19.7808 + 34.2613i −0.713313 + 1.23549i 0.250294 + 0.968170i \(0.419473\pi\)
−0.963607 + 0.267324i \(0.913861\pi\)
\(770\) 0 0
\(771\) −1.27115 −0.0457792
\(772\) 0 0
\(773\) 1.95429 3.38492i 0.0702908 0.121747i −0.828738 0.559637i \(-0.810941\pi\)
0.899029 + 0.437890i \(0.144274\pi\)
\(774\) 0 0
\(775\) 3.97822 6.89048i 0.142902 0.247513i
\(776\) 0 0
\(777\) −0.955881 1.65563i −0.0342920 0.0593955i
\(778\) 0 0
\(779\) −27.4330 12.1966i −0.982888 0.436988i
\(780\) 0 0
\(781\) −9.33428 16.1674i −0.334007 0.578517i
\(782\) 0 0
\(783\) 5.29927 + 3.05954i 0.189381 + 0.109339i
\(784\) 0 0
\(785\) 6.66841 11.5500i 0.238006 0.412238i
\(786\) 0 0
\(787\) 50.8902i 1.81404i −0.421089 0.907019i \(-0.638352\pi\)
0.421089 0.907019i \(-0.361648\pi\)
\(788\) 0 0
\(789\) 1.59999 2.77126i 0.0569612 0.0986596i
\(790\) 0 0
\(791\) −1.27356 −0.0452825
\(792\) 0 0
\(793\) −44.2804 + 25.5653i −1.57244 + 0.907850i
\(794\) 0 0
\(795\) −0.498432 0.863309i −0.0176775 0.0306184i
\(796\) 0 0
\(797\) −45.1212 −1.59828 −0.799138 0.601148i \(-0.794710\pi\)
−0.799138 + 0.601148i \(0.794710\pi\)
\(798\) 0 0
\(799\) 39.9642i 1.41383i
\(800\) 0 0
\(801\) 15.4199 8.90269i 0.544836 0.314561i
\(802\) 0 0
\(803\) −0.343180 0.594404i −0.0121105 0.0209761i
\(804\) 0 0
\(805\) 0.215704i 0.00760256i
\(806\) 0 0
\(807\) −3.35115 1.93478i −0.117966 0.0681077i
\(808\) 0 0
\(809\) −25.1281 −0.883456 −0.441728 0.897149i \(-0.645634\pi\)
−0.441728 + 0.897149i \(0.645634\pi\)
\(810\) 0 0
\(811\) −25.6921 14.8333i −0.902170 0.520868i −0.0242664 0.999706i \(-0.507725\pi\)
−0.877904 + 0.478837i \(0.841058\pi\)
\(812\) 0 0
\(813\) −0.00532699 + 0.00922662i −0.000186826 + 0.000323592i
\(814\) 0 0
\(815\) 4.94638 2.85579i 0.173264 0.100034i
\(816\) 0 0
\(817\) −14.2138 + 31.9702i −0.497278 + 1.11849i
\(818\) 0 0
\(819\) −16.7999 + 9.69943i −0.587036 + 0.338925i
\(820\) 0 0
\(821\) 36.5238 + 21.0870i 1.27469 + 0.735943i 0.975867 0.218366i \(-0.0700726\pi\)
0.298823 + 0.954308i \(0.403406\pi\)
\(822\) 0 0
\(823\) 1.52058 + 0.877908i 0.0530041 + 0.0306019i 0.526268 0.850319i \(-0.323591\pi\)
−0.473264 + 0.880921i \(0.656924\pi\)
\(824\) 0 0
\(825\) 1.63987i 0.0570930i
\(826\) 0 0
\(827\) 17.8493 + 10.3053i 0.620680 + 0.358350i 0.777134 0.629335i \(-0.216673\pi\)
−0.156453 + 0.987685i \(0.550006\pi\)
\(828\) 0 0
\(829\) −22.0322 −0.765211 −0.382606 0.923912i \(-0.624973\pi\)
−0.382606 + 0.923912i \(0.624973\pi\)
\(830\) 0 0
\(831\) 2.34468 + 4.06111i 0.0813362 + 0.140878i
\(832\) 0 0
\(833\) −11.2983 19.5692i −0.391463 0.678034i
\(834\) 0 0
\(835\) 6.18060i 0.213888i
\(836\) 0 0
\(837\) 2.23086i 0.0771097i
\(838\) 0 0
\(839\) −18.5701 32.1643i −0.641109 1.11043i −0.985185 0.171492i \(-0.945141\pi\)
0.344076 0.938942i \(-0.388192\pi\)
\(840\) 0 0
\(841\) −7.82049 13.5455i −0.269672 0.467086i
\(842\) 0 0
\(843\) −2.80182 −0.0964998
\(844\) 0 0
\(845\) −9.66480 5.57997i −0.332479 0.191957i
\(846\) 0 0
\(847\) 0.374615i 0.0128719i
\(848\) 0 0
\(849\) 0.942737 + 0.544290i 0.0323547 + 0.0186800i
\(850\) 0 0
\(851\) 0.894930 + 0.516688i 0.0306778 + 0.0177118i
\(852\) 0 0
\(853\) 25.1348 14.5116i 0.860601 0.496868i −0.00361260 0.999993i \(-0.501150\pi\)
0.864213 + 0.503125i \(0.167817\pi\)
\(854\) 0 0
\(855\) 13.8151 10.0484i 0.472466 0.343648i
\(856\) 0 0
\(857\) 21.7972 12.5846i 0.744578 0.429882i −0.0791537 0.996862i \(-0.525222\pi\)
0.823731 + 0.566980i \(0.191888\pi\)
\(858\) 0 0
\(859\) −16.2793 + 28.1966i −0.555443 + 0.962056i 0.442426 + 0.896805i \(0.354118\pi\)
−0.997869 + 0.0652505i \(0.979215\pi\)
\(860\) 0 0
\(861\) 1.28546 + 0.742158i 0.0438082 + 0.0252927i
\(862\) 0 0
\(863\) 19.6750 0.669746 0.334873 0.942263i \(-0.391307\pi\)
0.334873 + 0.942263i \(0.391307\pi\)
\(864\) 0 0
\(865\) 10.6519 + 6.14987i 0.362175 + 0.209102i
\(866\) 0 0
\(867\) 0.497137i 0.0168837i
\(868\) 0 0
\(869\) 26.8858 + 46.5676i 0.912038 + 1.57970i
\(870\) 0 0
\(871\) 4.17538 2.41066i 0.141477 0.0816820i
\(872\) 0 0
\(873\) 19.6563i 0.665264i
\(874\) 0 0
\(875\) 15.3068 0.517465
\(876\) 0 0
\(877\) −13.9490 24.1603i −0.471023 0.815836i 0.528428 0.848978i \(-0.322782\pi\)
−0.999451 + 0.0331425i \(0.989448\pi\)
\(878\) 0 0
\(879\) −0.669833 + 0.386728i −0.0225929 + 0.0130440i
\(880\) 0 0
\(881\) −9.70445 −0.326951 −0.163476 0.986547i \(-0.552271\pi\)
−0.163476 + 0.986547i \(0.552271\pi\)
\(882\) 0 0
\(883\) −10.4195 + 18.0470i −0.350643 + 0.607331i −0.986362 0.164589i \(-0.947370\pi\)
0.635719 + 0.771920i \(0.280704\pi\)
\(884\) 0 0
\(885\) 2.14180i 0.0719959i
\(886\) 0 0
\(887\) −12.0498 + 20.8709i −0.404594 + 0.700777i −0.994274 0.106860i \(-0.965920\pi\)
0.589680 + 0.807637i \(0.299254\pi\)
\(888\) 0 0
\(889\) 24.2086 + 13.9768i 0.811930 + 0.468768i
\(890\) 0 0
\(891\) 14.3977 + 24.9375i 0.482341 + 0.835439i
\(892\) 0 0
\(893\) −4.06734 38.5025i −0.136108 1.28844i
\(894\) 0 0
\(895\) 9.45525 + 16.3770i 0.316054 + 0.547422i
\(896\) 0 0
\(897\) −0.0413622 + 0.0716414i −0.00138104 + 0.00239204i
\(898\) 0 0
\(899\) 8.13748 14.0945i 0.271400 0.470079i
\(900\) 0 0
\(901\) 22.2299 0.740585
\(902\) 0 0
\(903\) 0.864906 1.49806i 0.0287823 0.0498524i
\(904\) 0 0
\(905\) 9.03676i 0.300392i
\(906\) 0 0
\(907\) 42.7097 24.6585i 1.41815 0.818771i 0.422017 0.906588i \(-0.361322\pi\)
0.996137 + 0.0878168i \(0.0279890\pi\)
\(908\) 0 0
\(909\) 3.39404 1.95955i 0.112573 0.0649941i
\(910\) 0 0
\(911\) −18.7040 −0.619691 −0.309845 0.950787i \(-0.600277\pi\)
−0.309845 + 0.950787i \(0.600277\pi\)
\(912\) 0 0
\(913\) 55.5223 1.83752
\(914\) 0 0
\(915\) −1.92790 + 1.11307i −0.0637344 + 0.0367971i
\(916\) 0 0
\(917\) 12.1574 7.01910i 0.401474 0.231791i
\(918\) 0 0
\(919\) 29.8088i 0.983302i 0.870792 + 0.491651i \(0.163606\pi\)
−0.870792 + 0.491651i \(0.836394\pi\)
\(920\) 0 0
\(921\) −2.06026 + 3.56847i −0.0678878 + 0.117585i
\(922\) 0 0
\(923\) −26.4066 −0.869185
\(924\) 0 0
\(925\) 14.4879 25.0939i 0.476361 0.825081i
\(926\) 0 0
\(927\) −10.3110 + 17.8591i −0.338656 + 0.586570i
\(928\) 0 0
\(929\) −1.73298 3.00161i −0.0568572 0.0984796i 0.836196 0.548431i \(-0.184775\pi\)
−0.893053 + 0.449951i \(0.851441\pi\)
\(930\) 0 0
\(931\) −12.8767 17.7036i −0.422016 0.580211i
\(932\) 0 0
\(933\) 0.305411 + 0.528987i 0.00999870 + 0.0173183i
\(934\) 0 0
\(935\) 16.8086 + 9.70445i 0.549700 + 0.317370i
\(936\) 0 0
\(937\) 13.7679 23.8467i 0.449779 0.779039i −0.548593 0.836090i \(-0.684836\pi\)
0.998371 + 0.0570504i \(0.0181696\pi\)
\(938\) 0 0
\(939\) 1.77067i 0.0577837i
\(940\) 0 0
\(941\) 1.09119 1.88999i 0.0355717 0.0616120i −0.847691 0.530490i \(-0.822008\pi\)
0.883263 + 0.468878i \(0.155341\pi\)
\(942\) 0 0
\(943\) −0.802327 −0.0261274
\(944\) 0 0
\(945\) −1.46865 + 0.847928i −0.0477753 + 0.0275831i
\(946\) 0 0
\(947\) 17.6884 + 30.6371i 0.574794 + 0.995573i 0.996064 + 0.0886376i \(0.0282513\pi\)
−0.421270 + 0.906935i \(0.638415\pi\)
\(948\) 0 0
\(949\) −0.970853 −0.0315152
\(950\) 0 0
\(951\) 1.45605i 0.0472155i
\(952\) 0 0
\(953\) 26.5449 15.3257i 0.859872 0.496448i −0.00409718 0.999992i \(-0.501304\pi\)
0.863970 + 0.503544i \(0.167971\pi\)
\(954\) 0 0
\(955\) −7.74665 13.4176i −0.250676 0.434183i
\(956\) 0 0
\(957\) 3.35437i 0.108431i
\(958\) 0 0
\(959\) 3.93755 + 2.27334i 0.127150 + 0.0734101i
\(960\) 0 0
\(961\) −25.0666 −0.808599
\(962\) 0 0
\(963\) 34.8588 + 20.1258i 1.12331 + 0.648543i
\(964\) 0 0
\(965\) 8.49922 14.7211i 0.273600 0.473888i
\(966\) 0 0
\(967\) −26.4331 + 15.2611i −0.850030 + 0.490765i −0.860661 0.509178i \(-0.829949\pi\)
0.0106307 + 0.999943i \(0.496616\pi\)
\(968\) 0 0
\(969\) −0.315725 2.98874i −0.0101426 0.0960121i
\(970\) 0 0
\(971\) −5.06048 + 2.92167i −0.162398 + 0.0937608i −0.578997 0.815330i \(-0.696556\pi\)
0.416598 + 0.909091i \(0.363222\pi\)
\(972\) 0 0
\(973\) 0.628731 + 0.362998i 0.0201562 + 0.0116372i
\(974\) 0 0
\(975\) 2.00883 + 1.15980i 0.0643340 + 0.0371432i
\(976\) 0 0
\(977\) 22.6943i 0.726054i 0.931779 + 0.363027i \(0.118257\pi\)
−0.931779 + 0.363027i \(0.881743\pi\)
\(978\) 0 0
\(979\) 16.9725 + 9.79910i 0.542445 + 0.313181i
\(980\) 0 0
\(981\) −0.217644 −0.00694884
\(982\) 0 0
\(983\) 26.5144 + 45.9243i 0.845679 + 1.46476i 0.885031 + 0.465533i \(0.154137\pi\)
−0.0393519 + 0.999225i \(0.512529\pi\)
\(984\) 0 0
\(985\) −12.5479 21.7336i −0.399808 0.692489i
\(986\) 0 0
\(987\) 1.91419i 0.0609292i
\(988\) 0 0
\(989\) 0.935026i 0.0297321i
\(990\) 0 0
\(991\) −12.6987 21.9948i −0.403387 0.698687i 0.590745 0.806858i \(-0.298834\pi\)
−0.994132 + 0.108171i \(0.965500\pi\)
\(992\) 0 0
\(993\) 1.18942 + 2.06013i 0.0377450 + 0.0653763i
\(994\) 0 0
\(995\) −9.84327 −0.312053
\(996\) 0 0
\(997\) −32.3070 18.6525i −1.02317 0.590729i −0.108151 0.994134i \(-0.534493\pi\)
−0.915021 + 0.403405i \(0.867826\pi\)
\(998\) 0 0
\(999\) 8.12437i 0.257044i
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 1216.2.s.i.31.7 32
4.3 odd 2 inner 1216.2.s.i.31.9 yes 32
8.3 odd 2 inner 1216.2.s.i.31.8 yes 32
8.5 even 2 inner 1216.2.s.i.31.10 yes 32
19.8 odd 6 inner 1216.2.s.i.863.8 yes 32
76.27 even 6 inner 1216.2.s.i.863.10 yes 32
152.27 even 6 inner 1216.2.s.i.863.7 yes 32
152.141 odd 6 inner 1216.2.s.i.863.9 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.s.i.31.7 32 1.1 even 1 trivial
1216.2.s.i.31.8 yes 32 8.3 odd 2 inner
1216.2.s.i.31.9 yes 32 4.3 odd 2 inner
1216.2.s.i.31.10 yes 32 8.5 even 2 inner
1216.2.s.i.863.7 yes 32 152.27 even 6 inner
1216.2.s.i.863.8 yes 32 19.8 odd 6 inner
1216.2.s.i.863.9 yes 32 152.141 odd 6 inner
1216.2.s.i.863.10 yes 32 76.27 even 6 inner