Properties

Label 2100.2.ce.d.493.4
Level $2100$
Weight $2$
Character 2100.493
Analytic conductor $16.769$
Analytic rank $0$
Dimension $24$
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] = [2100,2,Mod(157,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.ce (of order \(12\), degree \(4\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 493.4
Character \(\chi\) \(=\) 2100.493
Dual form 2100.2.ce.d.1657.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.258819 + 0.965926i) q^{3} +(2.54352 - 0.728357i) q^{7} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\) \(q+(0.258819 + 0.965926i) q^{3} +(2.54352 - 0.728357i) q^{7} +(-0.866025 + 0.500000i) q^{9} +(2.14536 - 3.71588i) q^{11} +(3.22938 - 3.22938i) q^{13} +(-3.44549 + 0.923215i) q^{17} +(3.13437 + 5.42888i) q^{19} +(1.36185 + 2.26834i) q^{21} +(0.749302 - 2.79643i) q^{23} +(-0.707107 - 0.707107i) q^{27} -2.84333i q^{29} +(-1.34333 - 0.775572i) q^{31} +(4.14453 + 1.11052i) q^{33} +(-5.50550 - 1.47519i) q^{37} +(3.95516 + 2.28351i) q^{39} -10.7150i q^{41} +(-2.38553 - 2.38553i) q^{43} +(2.22104 - 8.28905i) q^{47} +(5.93899 - 3.70518i) q^{49} +(-1.78351 - 3.08914i) q^{51} +(-8.76419 + 2.34836i) q^{53} +(-4.43266 + 4.43266i) q^{57} +(0.820799 - 1.42166i) q^{59} +(8.94331 - 5.16342i) q^{61} +(-1.83858 + 1.90254i) q^{63} +(1.19759 + 4.46947i) q^{67} +2.89508 q^{69} +11.7381 q^{71} +(2.40462 + 8.97416i) q^{73} +(2.75029 - 11.0140i) q^{77} +(-6.68892 + 3.86185i) q^{79} +(0.500000 - 0.866025i) q^{81} +(-3.32417 + 3.32417i) q^{83} +(2.74645 - 0.735908i) q^{87} +(8.91203 + 15.4361i) q^{89} +(5.86185 - 10.5661i) q^{91} +(0.401465 - 1.49829i) q^{93} +(-0.100580 - 0.100580i) q^{97} +4.29073i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 8 q^{11} + 8 q^{21} - 12 q^{31} - 8 q^{51} - 48 q^{61} + 64 q^{71} + 12 q^{81} + 116 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0.258819 + 0.965926i 0.149429 + 0.557678i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.54352 0.728357i 0.961360 0.275293i
\(8\) 0 0
\(9\) −0.866025 + 0.500000i −0.288675 + 0.166667i
\(10\) 0 0
\(11\) 2.14536 3.71588i 0.646852 1.12038i −0.337019 0.941498i \(-0.609419\pi\)
0.983870 0.178882i \(-0.0572481\pi\)
\(12\) 0 0
\(13\) 3.22938 3.22938i 0.895668 0.895668i −0.0993812 0.995049i \(-0.531686\pi\)
0.995049 + 0.0993812i \(0.0316863\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.44549 + 0.923215i −0.835653 + 0.223913i −0.651178 0.758925i \(-0.725725\pi\)
−0.184475 + 0.982837i \(0.559058\pi\)
\(18\) 0 0
\(19\) 3.13437 + 5.42888i 0.719073 + 1.24547i 0.961368 + 0.275268i \(0.0887665\pi\)
−0.242295 + 0.970203i \(0.577900\pi\)
\(20\) 0 0
\(21\) 1.36185 + 2.26834i 0.297180 + 0.494992i
\(22\) 0 0
\(23\) 0.749302 2.79643i 0.156240 0.583097i −0.842756 0.538296i \(-0.819068\pi\)
0.998996 0.0448005i \(-0.0142652\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 2.84333i 0.527993i −0.964524 0.263997i \(-0.914959\pi\)
0.964524 0.263997i \(-0.0850408\pi\)
\(30\) 0 0
\(31\) −1.34333 0.775572i −0.241269 0.139297i 0.374491 0.927231i \(-0.377818\pi\)
−0.615760 + 0.787934i \(0.711151\pi\)
\(32\) 0 0
\(33\) 4.14453 + 1.11052i 0.721469 + 0.193317i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.50550 1.47519i −0.905098 0.242520i −0.223894 0.974614i \(-0.571877\pi\)
−0.681204 + 0.732093i \(0.738544\pi\)
\(38\) 0 0
\(39\) 3.95516 + 2.28351i 0.633333 + 0.365655i
\(40\) 0 0
\(41\) 10.7150i 1.67339i −0.547665 0.836697i \(-0.684483\pi\)
0.547665 0.836697i \(-0.315517\pi\)
\(42\) 0 0
\(43\) −2.38553 2.38553i −0.363790 0.363790i 0.501416 0.865206i \(-0.332813\pi\)
−0.865206 + 0.501416i \(0.832813\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.22104 8.28905i 0.323973 1.20908i −0.591368 0.806402i \(-0.701412\pi\)
0.915341 0.402681i \(-0.131921\pi\)
\(48\) 0 0
\(49\) 5.93899 3.70518i 0.848428 0.529311i
\(50\) 0 0
\(51\) −1.78351 3.08914i −0.249742 0.432566i
\(52\) 0 0
\(53\) −8.76419 + 2.34836i −1.20385 + 0.322572i −0.804348 0.594159i \(-0.797485\pi\)
−0.399506 + 0.916730i \(0.630818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.43266 + 4.43266i −0.587120 + 0.587120i
\(58\) 0 0
\(59\) 0.820799 1.42166i 0.106859 0.185085i −0.807637 0.589680i \(-0.799254\pi\)
0.914496 + 0.404595i \(0.132587\pi\)
\(60\) 0 0
\(61\) 8.94331 5.16342i 1.14507 0.661108i 0.197391 0.980325i \(-0.436753\pi\)
0.947682 + 0.319216i \(0.103420\pi\)
\(62\) 0 0
\(63\) −1.83858 + 1.90254i −0.231639 + 0.239697i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.19759 + 4.46947i 0.146309 + 0.546032i 0.999694 + 0.0247507i \(0.00787921\pi\)
−0.853385 + 0.521282i \(0.825454\pi\)
\(68\) 0 0
\(69\) 2.89508 0.348527
\(70\) 0 0
\(71\) 11.7381 1.39306 0.696530 0.717528i \(-0.254726\pi\)
0.696530 + 0.717528i \(0.254726\pi\)
\(72\) 0 0
\(73\) 2.40462 + 8.97416i 0.281439 + 1.05035i 0.951402 + 0.307951i \(0.0996433\pi\)
−0.669963 + 0.742394i \(0.733690\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.75029 11.0140i 0.313425 1.25516i
\(78\) 0 0
\(79\) −6.68892 + 3.86185i −0.752562 + 0.434492i −0.826619 0.562762i \(-0.809739\pi\)
0.0740567 + 0.997254i \(0.476405\pi\)
\(80\) 0 0
\(81\) 0.500000 0.866025i 0.0555556 0.0962250i
\(82\) 0 0
\(83\) −3.32417 + 3.32417i −0.364875 + 0.364875i −0.865604 0.500729i \(-0.833065\pi\)
0.500729 + 0.865604i \(0.333065\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.74645 0.735908i 0.294450 0.0788976i
\(88\) 0 0
\(89\) 8.91203 + 15.4361i 0.944674 + 1.63622i 0.756403 + 0.654105i \(0.226955\pi\)
0.188270 + 0.982117i \(0.439712\pi\)
\(90\) 0 0
\(91\) 5.86185 10.5661i 0.614489 1.10763i
\(92\) 0 0
\(93\) 0.401465 1.49829i 0.0416300 0.155365i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.100580 0.100580i −0.0102123 0.0102123i 0.701982 0.712194i \(-0.252299\pi\)
−0.712194 + 0.701982i \(0.752299\pi\)
\(98\) 0 0
\(99\) 4.29073i 0.431235i
\(100\) 0 0
\(101\) 7.57834 + 4.37535i 0.754073 + 0.435364i 0.827164 0.561961i \(-0.189953\pi\)
−0.0730910 + 0.997325i \(0.523286\pi\)
\(102\) 0 0
\(103\) 3.17860 + 0.851703i 0.313197 + 0.0839208i 0.411993 0.911187i \(-0.364833\pi\)
−0.0987966 + 0.995108i \(0.531499\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.8346 + 3.43903i 1.24077 + 0.332463i 0.818764 0.574130i \(-0.194659\pi\)
0.422006 + 0.906593i \(0.361326\pi\)
\(108\) 0 0
\(109\) 12.3886 + 7.15258i 1.18662 + 0.685093i 0.957536 0.288315i \(-0.0930949\pi\)
0.229080 + 0.973408i \(0.426428\pi\)
\(110\) 0 0
\(111\) 5.69971i 0.540993i
\(112\) 0 0
\(113\) −9.12116 9.12116i −0.858047 0.858047i 0.133061 0.991108i \(-0.457519\pi\)
−0.991108 + 0.133061i \(0.957519\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.18203 + 4.41141i −0.109279 + 0.407835i
\(118\) 0 0
\(119\) −8.09123 + 4.85776i −0.741722 + 0.445310i
\(120\) 0 0
\(121\) −3.70518 6.41756i −0.336835 0.583414i
\(122\) 0 0
\(123\) 10.3499 2.77323i 0.933215 0.250054i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.6203 + 10.6203i −0.942403 + 0.942403i −0.998429 0.0560260i \(-0.982157\pi\)
0.0560260 + 0.998429i \(0.482157\pi\)
\(128\) 0 0
\(129\) 1.68682 2.92166i 0.148517 0.257238i
\(130\) 0 0
\(131\) −2.84333 + 1.64160i −0.248423 + 0.143427i −0.619042 0.785358i \(-0.712479\pi\)
0.370619 + 0.928785i \(0.379146\pi\)
\(132\) 0 0
\(133\) 11.9265 + 11.5255i 1.03416 + 0.999390i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.28878 + 16.0060i 0.366415 + 1.36748i 0.865492 + 0.500923i \(0.167006\pi\)
−0.499077 + 0.866558i \(0.666327\pi\)
\(138\) 0 0
\(139\) 0.478566 0.0405914 0.0202957 0.999794i \(-0.493539\pi\)
0.0202957 + 0.999794i \(0.493539\pi\)
\(140\) 0 0
\(141\) 8.58146 0.722689
\(142\) 0 0
\(143\) −5.07179 18.9282i −0.424124 1.58285i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.11605 + 4.77766i 0.421965 + 0.394054i
\(148\) 0 0
\(149\) 8.41389 4.85776i 0.689292 0.397963i −0.114055 0.993474i \(-0.536384\pi\)
0.803347 + 0.595511i \(0.203051\pi\)
\(150\) 0 0
\(151\) 10.2196 17.7009i 0.831660 1.44048i −0.0650611 0.997881i \(-0.520724\pi\)
0.896721 0.442596i \(-0.145942\pi\)
\(152\) 0 0
\(153\) 2.52227 2.52227i 0.203914 0.203914i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.82282 2.36407i 0.704138 0.188673i 0.111055 0.993814i \(-0.464577\pi\)
0.593083 + 0.805141i \(0.297910\pi\)
\(158\) 0 0
\(159\) −4.53668 7.85776i −0.359782 0.623161i
\(160\) 0 0
\(161\) −0.130935 7.65855i −0.0103192 0.603578i
\(162\) 0 0
\(163\) −2.68973 + 10.0382i −0.210676 + 0.786252i 0.776969 + 0.629539i \(0.216756\pi\)
−0.987644 + 0.156713i \(0.949910\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0521 16.0521i −1.24215 1.24215i −0.959108 0.283040i \(-0.908657\pi\)
−0.283040 0.959108i \(-0.591343\pi\)
\(168\) 0 0
\(169\) 7.85776i 0.604443i
\(170\) 0 0
\(171\) −5.42888 3.13437i −0.415157 0.239691i
\(172\) 0 0
\(173\) −22.3737 5.99500i −1.70104 0.455792i −0.727837 0.685750i \(-0.759474\pi\)
−0.973201 + 0.229958i \(0.926141\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.58566 + 0.424877i 0.119186 + 0.0319357i
\(178\) 0 0
\(179\) −5.19615 3.00000i −0.388379 0.224231i 0.293079 0.956088i \(-0.405320\pi\)
−0.681457 + 0.731858i \(0.738654\pi\)
\(180\) 0 0
\(181\) 18.9871i 1.41130i 0.708561 + 0.705650i \(0.249345\pi\)
−0.708561 + 0.705650i \(0.750655\pi\)
\(182\) 0 0
\(183\) 7.30218 + 7.30218i 0.539793 + 0.539793i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.96127 + 14.7836i −0.289677 + 1.08109i
\(188\) 0 0
\(189\) −2.31357 1.28351i −0.168287 0.0933620i
\(190\) 0 0
\(191\) −6.56703 11.3744i −0.475174 0.823025i 0.524422 0.851458i \(-0.324281\pi\)
−0.999596 + 0.0284336i \(0.990948\pi\)
\(192\) 0 0
\(193\) 8.67682 2.32495i 0.624571 0.167353i 0.0673662 0.997728i \(-0.478540\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.53033 7.53033i 0.536514 0.536514i −0.385989 0.922503i \(-0.626140\pi\)
0.922503 + 0.385989i \(0.126140\pi\)
\(198\) 0 0
\(199\) 0.0327307 0.0566912i 0.00232021 0.00401873i −0.864863 0.502008i \(-0.832595\pi\)
0.867183 + 0.497989i \(0.165928\pi\)
\(200\) 0 0
\(201\) −4.00721 + 2.31357i −0.282647 + 0.163186i
\(202\) 0 0
\(203\) −2.07096 7.23207i −0.145353 0.507592i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.749302 + 2.79643i 0.0520801 + 0.194366i
\(208\) 0 0
\(209\) 26.8974 1.86053
\(210\) 0 0
\(211\) −19.5733 −1.34748 −0.673740 0.738968i \(-0.735313\pi\)
−0.673740 + 0.738968i \(0.735313\pi\)
\(212\) 0 0
\(213\) 3.03805 + 11.3382i 0.208164 + 0.776878i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.98168 0.994260i −0.270294 0.0674947i
\(218\) 0 0
\(219\) −8.04601 + 4.64536i −0.543699 + 0.313905i
\(220\) 0 0
\(221\) −8.14536 + 14.1082i −0.547917 + 0.949019i
\(222\) 0 0
\(223\) 9.67793 9.67793i 0.648082 0.648082i −0.304447 0.952529i \(-0.598472\pi\)
0.952529 + 0.304447i \(0.0984717\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.9418 2.93184i 0.726232 0.194593i 0.123281 0.992372i \(-0.460658\pi\)
0.602951 + 0.797779i \(0.293992\pi\)
\(228\) 0 0
\(229\) 1.04693 + 1.81334i 0.0691833 + 0.119829i 0.898542 0.438888i \(-0.144627\pi\)
−0.829359 + 0.558716i \(0.811294\pi\)
\(230\) 0 0
\(231\) 11.3505 0.194056i 0.746811 0.0127680i
\(232\) 0 0
\(233\) −6.39622 + 23.8710i −0.419030 + 1.56384i 0.357593 + 0.933878i \(0.383598\pi\)
−0.776623 + 0.629965i \(0.783069\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.46148 5.46148i −0.354761 0.354761i
\(238\) 0 0
\(239\) 21.4022i 1.38439i 0.721710 + 0.692196i \(0.243357\pi\)
−0.721710 + 0.692196i \(0.756643\pi\)
\(240\) 0 0
\(241\) 22.4877 + 12.9833i 1.44856 + 0.836328i 0.998396 0.0566174i \(-0.0180315\pi\)
0.450166 + 0.892945i \(0.351365\pi\)
\(242\) 0 0
\(243\) 0.965926 + 0.258819i 0.0619642 + 0.0166032i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 27.6539 + 7.40985i 1.75958 + 0.471478i
\(248\) 0 0
\(249\) −4.07126 2.35054i −0.258006 0.148960i
\(250\) 0 0
\(251\) 2.18432i 0.137873i −0.997621 0.0689365i \(-0.978039\pi\)
0.997621 0.0689365i \(-0.0219606\pi\)
\(252\) 0 0
\(253\) −8.78369 8.78369i −0.552226 0.552226i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.548601 2.04741i 0.0342208 0.127714i −0.946702 0.322110i \(-0.895608\pi\)
0.980923 + 0.194396i \(0.0622747\pi\)
\(258\) 0 0
\(259\) −15.0778 + 0.257780i −0.936890 + 0.0160177i
\(260\) 0 0
\(261\) 1.42166 + 2.46240i 0.0879988 + 0.152418i
\(262\) 0 0
\(263\) −25.1450 + 6.73759i −1.55051 + 0.415458i −0.929644 0.368460i \(-0.879885\pi\)
−0.620865 + 0.783917i \(0.713219\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.6035 + 12.6035i −0.771323 + 0.771323i
\(268\) 0 0
\(269\) 2.89508 5.01443i 0.176516 0.305735i −0.764169 0.645016i \(-0.776851\pi\)
0.940685 + 0.339281i \(0.110184\pi\)
\(270\) 0 0
\(271\) −5.07112 + 2.92781i −0.308049 + 0.177852i −0.646053 0.763293i \(-0.723581\pi\)
0.338004 + 0.941145i \(0.390248\pi\)
\(272\) 0 0
\(273\) 11.7233 + 2.92740i 0.709524 + 0.177174i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.03040 7.57756i −0.121995 0.455291i 0.877720 0.479174i \(-0.159064\pi\)
−0.999715 + 0.0238829i \(0.992397\pi\)
\(278\) 0 0
\(279\) 1.55114 0.0928645
\(280\) 0 0
\(281\) −0.313341 −0.0186923 −0.00934617 0.999956i \(-0.502975\pi\)
−0.00934617 + 0.999956i \(0.502975\pi\)
\(282\) 0 0
\(283\) 1.74396 + 6.50853i 0.103667 + 0.386892i 0.998191 0.0601295i \(-0.0191514\pi\)
−0.894523 + 0.447022i \(0.852485\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.80431 27.2537i −0.460674 1.60874i
\(288\) 0 0
\(289\) −3.70338 + 2.13815i −0.217846 + 0.125774i
\(290\) 0 0
\(291\) 0.0711205 0.123184i 0.00416916 0.00722119i
\(292\) 0 0
\(293\) 8.70685 8.70685i 0.508659 0.508659i −0.405455 0.914115i \(-0.632887\pi\)
0.914115 + 0.405455i \(0.132887\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.14453 + 1.11052i −0.240490 + 0.0644391i
\(298\) 0 0
\(299\) −6.61096 11.4505i −0.382322 0.662201i
\(300\) 0 0
\(301\) −7.80516 4.33013i −0.449882 0.249584i
\(302\) 0 0
\(303\) −2.26485 + 8.45253i −0.130112 + 0.485585i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −22.2469 22.2469i −1.26970 1.26970i −0.946245 0.323451i \(-0.895157\pi\)
−0.323451 0.946245i \(-0.604843\pi\)
\(308\) 0 0
\(309\) 3.29073i 0.187203i
\(310\) 0 0
\(311\) 2.56391 + 1.48027i 0.145386 + 0.0839385i 0.570928 0.821000i \(-0.306583\pi\)
−0.425543 + 0.904938i \(0.639917\pi\)
\(312\) 0 0
\(313\) 5.11045 + 1.36934i 0.288860 + 0.0773998i 0.400339 0.916367i \(-0.368892\pi\)
−0.111479 + 0.993767i \(0.535559\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 19.7015 + 5.27900i 1.10655 + 0.296498i 0.765428 0.643522i \(-0.222527\pi\)
0.341119 + 0.940020i \(0.389194\pi\)
\(318\) 0 0
\(319\) −10.5655 6.09998i −0.591553 0.341533i
\(320\) 0 0
\(321\) 13.2874i 0.741630i
\(322\) 0 0
\(323\) −15.8114 15.8114i −0.879772 0.879772i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.70245 + 13.8177i −0.204746 + 0.764122i
\(328\) 0 0
\(329\) −0.388112 22.7011i −0.0213973 1.25155i
\(330\) 0 0
\(331\) 2.91036 + 5.04089i 0.159968 + 0.277072i 0.934857 0.355025i \(-0.115528\pi\)
−0.774889 + 0.632097i \(0.782194\pi\)
\(332\) 0 0
\(333\) 5.50550 1.47519i 0.301699 0.0808401i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −20.2904 + 20.2904i −1.10529 + 1.10529i −0.111526 + 0.993761i \(0.535574\pi\)
−0.993761 + 0.111526i \(0.964426\pi\)
\(338\) 0 0
\(339\) 6.44964 11.1711i 0.350296 0.606731i
\(340\) 0 0
\(341\) −5.76386 + 3.32777i −0.312131 + 0.180209i
\(342\) 0 0
\(343\) 12.4073 13.7499i 0.669929 0.742425i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.54368 31.8854i −0.458649 1.71170i −0.677134 0.735860i \(-0.736778\pi\)
0.218485 0.975840i \(-0.429889\pi\)
\(348\) 0 0
\(349\) −12.3566 −0.661431 −0.330716 0.943730i \(-0.607290\pi\)
−0.330716 + 0.943730i \(0.607290\pi\)
\(350\) 0 0
\(351\) −4.56703 −0.243770
\(352\) 0 0
\(353\) 0.173986 + 0.649324i 0.00926034 + 0.0345600i 0.970401 0.241498i \(-0.0776386\pi\)
−0.961141 + 0.276058i \(0.910972\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.78640 6.55825i −0.359174 0.347099i
\(358\) 0 0
\(359\) −24.7577 + 14.2939i −1.30666 + 0.754401i −0.981537 0.191271i \(-0.938739\pi\)
−0.325123 + 0.945672i \(0.605406\pi\)
\(360\) 0 0
\(361\) −10.1485 + 17.5777i −0.534131 + 0.925142i
\(362\) 0 0
\(363\) 5.23992 5.23992i 0.275024 0.275024i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.50744 + 0.671865i −0.130887 + 0.0350711i −0.323668 0.946171i \(-0.604916\pi\)
0.192781 + 0.981242i \(0.438249\pi\)
\(368\) 0 0
\(369\) 5.35748 + 9.27942i 0.278899 + 0.483068i
\(370\) 0 0
\(371\) −20.5815 + 12.3566i −1.06854 + 0.641520i
\(372\) 0 0
\(373\) −1.33806 + 4.99370i −0.0692820 + 0.258564i −0.991876 0.127208i \(-0.959398\pi\)
0.922594 + 0.385772i \(0.126065\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.18219 9.18219i −0.472907 0.472907i
\(378\) 0 0
\(379\) 8.44740i 0.433914i −0.976181 0.216957i \(-0.930387\pi\)
0.976181 0.216957i \(-0.0696131\pi\)
\(380\) 0 0
\(381\) −13.0072 7.50972i −0.666380 0.384735i
\(382\) 0 0
\(383\) −8.28905 2.22104i −0.423551 0.113490i 0.0407465 0.999170i \(-0.487026\pi\)
−0.464297 + 0.885679i \(0.653693\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.25869 + 0.873164i 0.165649 + 0.0443854i
\(388\) 0 0
\(389\) −15.0903 8.71239i −0.765109 0.441736i 0.0660180 0.997818i \(-0.478971\pi\)
−0.831127 + 0.556083i \(0.812304\pi\)
\(390\) 0 0
\(391\) 10.3268i 0.522251i
\(392\) 0 0
\(393\) −2.32157 2.32157i −0.117108 0.117108i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.46654 + 9.20526i −0.123792 + 0.461999i −0.999794 0.0203095i \(-0.993535\pi\)
0.876002 + 0.482308i \(0.160202\pi\)
\(398\) 0 0
\(399\) −8.04601 + 14.5031i −0.402804 + 0.726064i
\(400\) 0 0
\(401\) 5.42479 + 9.39601i 0.270901 + 0.469214i 0.969093 0.246696i \(-0.0793451\pi\)
−0.698192 + 0.715911i \(0.746012\pi\)
\(402\) 0 0
\(403\) −6.84273 + 1.83350i −0.340861 + 0.0913334i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −17.2929 + 17.2929i −0.857179 + 0.857179i
\(408\) 0 0
\(409\) 16.2986 28.2299i 0.805912 1.39588i −0.109761 0.993958i \(-0.535009\pi\)
0.915673 0.401923i \(-0.131658\pi\)
\(410\) 0 0
\(411\) −14.3505 + 8.28529i −0.707860 + 0.408683i
\(412\) 0 0
\(413\) 1.05224 4.21387i 0.0517773 0.207351i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.123862 + 0.462259i 0.00606555 + 0.0226369i
\(418\) 0 0
\(419\) −22.6179 −1.10496 −0.552479 0.833527i \(-0.686318\pi\)
−0.552479 + 0.833527i \(0.686318\pi\)
\(420\) 0 0
\(421\) −12.0370 −0.586649 −0.293325 0.956013i \(-0.594762\pi\)
−0.293325 + 0.956013i \(0.594762\pi\)
\(422\) 0 0
\(423\) 2.22104 + 8.28905i 0.107991 + 0.403028i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18.9867 19.6472i 0.918830 0.950794i
\(428\) 0 0
\(429\) 16.9705 9.79794i 0.819345 0.473049i
\(430\) 0 0
\(431\) −17.0144 + 29.4699i −0.819556 + 1.41951i 0.0864534 + 0.996256i \(0.472447\pi\)
−0.906010 + 0.423257i \(0.860887\pi\)
\(432\) 0 0
\(433\) 22.6580 22.6580i 1.08888 1.08888i 0.0932307 0.995645i \(-0.470281\pi\)
0.995645 0.0932307i \(-0.0297194\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 17.5301 4.69717i 0.838578 0.224696i
\(438\) 0 0
\(439\) 1.99698 + 3.45887i 0.0953106 + 0.165083i 0.909738 0.415182i \(-0.136282\pi\)
−0.814428 + 0.580265i \(0.802949\pi\)
\(440\) 0 0
\(441\) −3.29073 + 6.17828i −0.156701 + 0.294204i
\(442\) 0 0
\(443\) 3.50559 13.0831i 0.166556 0.621595i −0.831281 0.555853i \(-0.812392\pi\)
0.997837 0.0657421i \(-0.0209415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.86991 + 6.86991i 0.324936 + 0.324936i
\(448\) 0 0
\(449\) 28.8496i 1.36150i −0.732518 0.680748i \(-0.761655\pi\)
0.732518 0.680748i \(-0.238345\pi\)
\(450\) 0 0
\(451\) −39.8155 22.9875i −1.87484 1.08244i
\(452\) 0 0
\(453\) 19.7428 + 5.29006i 0.927596 + 0.248549i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 21.1998 + 5.68047i 0.991685 + 0.265721i 0.717958 0.696086i \(-0.245077\pi\)
0.273727 + 0.961808i \(0.411744\pi\)
\(458\) 0 0
\(459\) 3.08914 + 1.78351i 0.144189 + 0.0832473i
\(460\) 0 0
\(461\) 6.11282i 0.284702i 0.989816 + 0.142351i \(0.0454662\pi\)
−0.989816 + 0.142351i \(0.954534\pi\)
\(462\) 0 0
\(463\) 27.0346 + 27.0346i 1.25640 + 1.25640i 0.952797 + 0.303607i \(0.0981909\pi\)
0.303607 + 0.952797i \(0.401809\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.65012 + 9.89037i −0.122633 + 0.457672i −0.999744 0.0226152i \(-0.992801\pi\)
0.877112 + 0.480287i \(0.159467\pi\)
\(468\) 0 0
\(469\) 6.30146 + 10.4959i 0.290974 + 0.484656i
\(470\) 0 0
\(471\) 4.56703 + 7.91033i 0.210438 + 0.364489i
\(472\) 0 0
\(473\) −13.9822 + 3.74651i −0.642901 + 0.172265i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.41583 6.41583i 0.293761 0.293761i
\(478\) 0 0
\(479\) −11.3744 + 19.7011i −0.519711 + 0.900166i 0.480027 + 0.877254i \(0.340627\pi\)
−0.999737 + 0.0229118i \(0.992706\pi\)
\(480\) 0 0
\(481\) −22.5433 + 13.0154i −1.02789 + 0.593450i
\(482\) 0 0
\(483\) 7.36370 2.10865i 0.335060 0.0959470i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.94217 + 37.1047i 0.450523 + 1.68137i 0.700927 + 0.713233i \(0.252770\pi\)
−0.250405 + 0.968141i \(0.580564\pi\)
\(488\) 0 0
\(489\) −10.3923 −0.469956
\(490\) 0 0
\(491\) −5.47626 −0.247140 −0.123570 0.992336i \(-0.539434\pi\)
−0.123570 + 0.992336i \(0.539434\pi\)
\(492\) 0 0
\(493\) 2.62501 + 9.79665i 0.118224 + 0.441219i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 29.8562 8.54954i 1.33923 0.383499i
\(498\) 0 0
\(499\) −10.2370 + 5.91036i −0.458273 + 0.264584i −0.711318 0.702871i \(-0.751901\pi\)
0.253045 + 0.967455i \(0.418568\pi\)
\(500\) 0 0
\(501\) 11.3505 19.6597i 0.507105 0.878331i
\(502\) 0 0
\(503\) 12.7279 12.7279i 0.567510 0.567510i −0.363920 0.931430i \(-0.618562\pi\)
0.931430 + 0.363920i \(0.118562\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.59001 2.03374i 0.337084 0.0903215i
\(508\) 0 0
\(509\) −8.41389 14.5733i −0.372939 0.645949i 0.617077 0.786903i \(-0.288317\pi\)
−0.990016 + 0.140953i \(0.954983\pi\)
\(510\) 0 0
\(511\) 12.6526 + 21.0745i 0.559717 + 0.932282i
\(512\) 0 0
\(513\) 1.62247 6.05513i 0.0716337 0.267340i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −26.0362 26.0362i −1.14507 1.14507i
\(518\) 0 0
\(519\) 23.1629i 1.01674i
\(520\) 0 0
\(521\) 31.5444 + 18.2122i 1.38199 + 0.797890i 0.992395 0.123098i \(-0.0392829\pi\)
0.389592 + 0.920988i \(0.372616\pi\)
\(522\) 0 0
\(523\) 3.87764 + 1.03901i 0.169557 + 0.0454328i 0.342599 0.939482i \(-0.388693\pi\)
−0.173042 + 0.984915i \(0.555359\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.34444 + 1.43204i 0.232808 + 0.0623806i
\(528\) 0 0
\(529\) 12.6600 + 7.30925i 0.550434 + 0.317793i
\(530\) 0 0
\(531\) 1.64160i 0.0712392i
\(532\) 0 0
\(533\) −34.6026 34.6026i −1.49881 1.49881i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.55291 5.79555i 0.0670132 0.250097i
\(538\) 0 0
\(539\) −1.02670 30.0176i −0.0442230 1.29295i
\(540\) 0 0
\(541\) 14.9248 + 25.8505i 0.641667 + 1.11140i 0.985061 + 0.172208i \(0.0550901\pi\)
−0.343394 + 0.939191i \(0.611577\pi\)
\(542\) 0 0
\(543\) −18.3401 + 4.91422i −0.787050 + 0.210889i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −17.4023 + 17.4023i −0.744067 + 0.744067i −0.973358 0.229291i \(-0.926359\pi\)
0.229291 + 0.973358i \(0.426359\pi\)
\(548\) 0 0
\(549\) −5.16342 + 8.94331i −0.220369 + 0.381691i
\(550\) 0 0
\(551\) 15.4361 8.91203i 0.657600 0.379665i
\(552\) 0 0
\(553\) −14.2006 + 14.6946i −0.603871 + 0.624879i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.67716 + 24.9195i 0.282920 + 1.05587i 0.950346 + 0.311196i \(0.100730\pi\)
−0.667426 + 0.744677i \(0.732604\pi\)
\(558\) 0 0
\(559\) −15.4075 −0.651670
\(560\) 0 0
\(561\) −15.3052 −0.646184
\(562\) 0 0
\(563\) 2.88918 + 10.7826i 0.121764 + 0.454430i 0.999704 0.0243413i \(-0.00774885\pi\)
−0.877939 + 0.478772i \(0.841082\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.640985 2.56693i 0.0269188 0.107801i
\(568\) 0 0
\(569\) −14.8189 + 8.55572i −0.621243 + 0.358675i −0.777353 0.629065i \(-0.783438\pi\)
0.156110 + 0.987740i \(0.450105\pi\)
\(570\) 0 0
\(571\) −4.56294 + 7.90324i −0.190953 + 0.330740i −0.945566 0.325429i \(-0.894491\pi\)
0.754613 + 0.656170i \(0.227824\pi\)
\(572\) 0 0
\(573\) 9.28718 9.28718i 0.387978 0.387978i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −16.5642 + 4.43836i −0.689575 + 0.184771i −0.586557 0.809908i \(-0.699517\pi\)
−0.103019 + 0.994679i \(0.532850\pi\)
\(578\) 0 0
\(579\) 4.49145 + 7.77942i 0.186658 + 0.323302i
\(580\) 0 0
\(581\) −6.03392 + 10.8763i −0.250329 + 0.451224i
\(582\) 0 0
\(583\) −10.0762 + 37.6048i −0.417312 + 1.55743i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.8269 30.8269i −1.27236 1.27236i −0.944844 0.327520i \(-0.893787\pi\)
−0.327520 0.944844i \(-0.606213\pi\)
\(588\) 0 0
\(589\) 9.72370i 0.400658i
\(590\) 0 0
\(591\) 9.22273 + 5.32475i 0.379373 + 0.219031i
\(592\) 0 0
\(593\) −30.2166 8.09652i −1.24085 0.332484i −0.422053 0.906571i \(-0.638691\pi\)
−0.818794 + 0.574087i \(0.805357\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.0632308 + 0.0169426i 0.00258786 + 0.000693416i
\(598\) 0 0
\(599\) −3.23190 1.86594i −0.132052 0.0762403i 0.432519 0.901625i \(-0.357625\pi\)
−0.564571 + 0.825385i \(0.690958\pi\)
\(600\) 0 0
\(601\) 12.0089i 0.489854i 0.969542 + 0.244927i \(0.0787639\pi\)
−0.969542 + 0.244927i \(0.921236\pi\)
\(602\) 0 0
\(603\) −3.27188 3.27188i −0.133241 0.133241i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.09786 11.5614i 0.125738 0.469261i −0.874127 0.485698i \(-0.838565\pi\)
0.999865 + 0.0164366i \(0.00523216\pi\)
\(608\) 0 0
\(609\) 6.44964 3.87219i 0.261352 0.156909i
\(610\) 0 0
\(611\) −19.5959 33.9411i −0.792765 1.37311i
\(612\) 0 0
\(613\) 2.48478 0.665794i 0.100359 0.0268912i −0.208290 0.978067i \(-0.566790\pi\)
0.308649 + 0.951176i \(0.400123\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.2339 + 30.2339i −1.21717 + 1.21717i −0.248555 + 0.968618i \(0.579956\pi\)
−0.968618 + 0.248555i \(0.920044\pi\)
\(618\) 0 0
\(619\) −10.9215 + 18.9166i −0.438972 + 0.760323i −0.997611 0.0690889i \(-0.977991\pi\)
0.558638 + 0.829412i \(0.311324\pi\)
\(620\) 0 0
\(621\) −2.50721 + 1.44754i −0.100611 + 0.0580878i
\(622\) 0 0
\(623\) 33.9109 + 32.7709i 1.35861 + 1.31294i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.96157 + 25.9809i 0.278018 + 1.03758i
\(628\) 0 0
\(629\) 20.3310 0.810652
\(630\) 0 0
\(631\) 46.5651 1.85373 0.926864 0.375398i \(-0.122494\pi\)
0.926864 + 0.375398i \(0.122494\pi\)
\(632\) 0 0
\(633\) −5.06594 18.9063i −0.201353 0.751459i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 7.21383 31.1447i 0.285822 1.23400i
\(638\) 0 0
\(639\) −10.1655 + 5.86906i −0.402142 + 0.232177i
\(640\) 0 0
\(641\) −20.7124 + 35.8749i −0.818090 + 1.41697i 0.0889971 + 0.996032i \(0.471634\pi\)
−0.907087 + 0.420942i \(0.861700\pi\)
\(642\) 0 0
\(643\) −27.2798 + 27.2798i −1.07581 + 1.07581i −0.0789320 + 0.996880i \(0.525151\pi\)
−0.996880 + 0.0789320i \(0.974849\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25.2296 6.76026i 0.991879 0.265773i 0.273839 0.961775i \(-0.411706\pi\)
0.718040 + 0.696002i \(0.245040\pi\)
\(648\) 0 0
\(649\) −3.52182 6.09998i −0.138244 0.239445i
\(650\) 0 0
\(651\) −0.0701533 4.10334i −0.00274952 0.160823i
\(652\) 0 0
\(653\) −3.63993 + 13.5844i −0.142441 + 0.531599i 0.857414 + 0.514626i \(0.172069\pi\)
−0.999856 + 0.0169724i \(0.994597\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −6.56954 6.56954i −0.256302 0.256302i
\(658\) 0 0
\(659\) 3.71552i 0.144736i −0.997378 0.0723680i \(-0.976944\pi\)
0.997378 0.0723680i \(-0.0230556\pi\)
\(660\) 0 0
\(661\) −31.1300 17.9729i −1.21082 0.699065i −0.247878 0.968791i \(-0.579733\pi\)
−0.962937 + 0.269727i \(0.913067\pi\)
\(662\) 0 0
\(663\) −15.7356 4.21635i −0.611121 0.163750i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −7.95119 2.13051i −0.307871 0.0824938i
\(668\) 0 0
\(669\) 11.8530 + 6.84333i 0.458263 + 0.264578i
\(670\) 0 0
\(671\) 44.3097i 1.71056i
\(672\) 0 0
\(673\) −31.6052 31.6052i −1.21829 1.21829i −0.968230 0.250061i \(-0.919549\pi\)
−0.250061 0.968230i \(-0.580451\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.20311 19.4183i 0.199972 0.746305i −0.790952 0.611878i \(-0.790414\pi\)
0.990924 0.134426i \(-0.0429191\pi\)
\(678\) 0 0
\(679\) −0.329084 0.182568i −0.0126291 0.00700633i
\(680\) 0 0
\(681\) 5.66389 + 9.81014i 0.217041 + 0.375925i
\(682\) 0 0
\(683\) 4.69375 1.25769i 0.179601 0.0481241i −0.167897 0.985804i \(-0.553698\pi\)
0.347499 + 0.937680i \(0.387031\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.48059 + 1.48059i −0.0564879 + 0.0564879i
\(688\) 0 0
\(689\) −20.7191 + 35.8866i −0.789337 + 1.36717i
\(690\) 0 0
\(691\) −38.9310 + 22.4768i −1.48101 + 0.855059i −0.999768 0.0215304i \(-0.993146\pi\)
−0.481238 + 0.876590i \(0.659813\pi\)
\(692\) 0 0
\(693\) 3.12518 + 10.9136i 0.118716 + 0.414572i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 9.89221 + 36.9182i 0.374694 + 1.39838i
\(698\) 0 0
\(699\) −24.7131 −0.934736
\(700\) 0 0
\(701\) −30.2455 −1.14236 −0.571179 0.820826i \(-0.693514\pi\)
−0.571179 + 0.820826i \(0.693514\pi\)
\(702\) 0 0
\(703\) −9.24759 34.5125i −0.348779 1.30166i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 22.4625 + 5.60907i 0.844788 + 0.210951i
\(708\) 0 0
\(709\) 30.0324 17.3392i 1.12789 0.651189i 0.184488 0.982835i \(-0.440937\pi\)
0.943404 + 0.331646i \(0.107604\pi\)
\(710\) 0 0
\(711\) 3.86185 6.68892i 0.144831 0.250854i
\(712\) 0 0
\(713\) −3.17540 + 3.17540i −0.118920 + 0.118920i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −20.6729 + 5.53929i −0.772044 + 0.206869i
\(718\) 0 0
\(719\) 7.75441 + 13.4310i 0.289191 + 0.500893i 0.973617 0.228189i \(-0.0732805\pi\)
−0.684426 + 0.729082i \(0.739947\pi\)
\(720\) 0 0
\(721\) 8.70518 0.148829i 0.324198 0.00554269i
\(722\) 0 0
\(723\) −6.72065 + 25.0818i −0.249944 + 0.932802i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.8475 + 12.8475i 0.476489 + 0.476489i 0.904007 0.427518i \(-0.140612\pi\)
−0.427518 + 0.904007i \(0.640612\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 10.4217 + 6.01695i 0.385459 + 0.222545i
\(732\) 0 0
\(733\) 14.8913 + 3.99011i 0.550023 + 0.147378i 0.523119 0.852260i \(-0.324768\pi\)
0.0269041 + 0.999638i \(0.491435\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 19.1773 + 5.13854i 0.706404 + 0.189280i
\(738\) 0 0
\(739\) −34.8911 20.1444i −1.28349 0.741024i −0.306006 0.952030i \(-0.598993\pi\)
−0.977485 + 0.211006i \(0.932326\pi\)
\(740\) 0 0
\(741\) 28.6295i 1.05173i
\(742\) 0 0
\(743\) −7.23914 7.23914i −0.265578 0.265578i 0.561737 0.827316i \(-0.310133\pi\)
−0.827316 + 0.561737i \(0.810133\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.21673 4.54090i 0.0445179 0.166143i
\(748\) 0 0
\(749\) 35.1500 0.600946i 1.28435 0.0219581i
\(750\) 0 0
\(751\) −3.47739 6.02301i −0.126892 0.219783i 0.795579 0.605850i \(-0.207167\pi\)
−0.922471 + 0.386067i \(0.873833\pi\)
\(752\) 0 0
\(753\) 2.10989 0.565343i 0.0768887 0.0206023i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −21.5691 + 21.5691i −0.783941 + 0.783941i −0.980493 0.196552i \(-0.937025\pi\)
0.196552 + 0.980493i \(0.437025\pi\)
\(758\) 0 0
\(759\) 6.21101 10.7578i 0.225445 0.390483i
\(760\) 0 0
\(761\) 28.7783 16.6152i 1.04321 0.602299i 0.122471 0.992472i \(-0.460918\pi\)
0.920741 + 0.390173i \(0.127585\pi\)
\(762\) 0 0
\(763\) 36.7204 + 9.16939i 1.32937 + 0.331954i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.94042 7.24176i −0.0700647 0.261485i
\(768\) 0 0
\(769\) −11.2327 −0.405061 −0.202530 0.979276i \(-0.564917\pi\)
−0.202530 + 0.979276i \(0.564917\pi\)
\(770\) 0 0
\(771\) 2.11963 0.0763366
\(772\) 0 0
\(773\) 10.5342 + 39.3142i 0.378889 + 1.41403i 0.847578 + 0.530671i \(0.178060\pi\)
−0.468688 + 0.883364i \(0.655273\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.15142 14.4973i −0.148931 0.520089i
\(778\) 0 0
\(779\) 58.1702 33.5846i 2.08416 1.20329i
\(780\) 0 0
\(781\) 25.1826 43.6175i 0.901103 1.56076i
\(782\) 0 0
\(783\) −2.01054 + 2.01054i −0.0718508 + 0.0718508i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 48.6110 13.0253i 1.73280 0.464301i 0.751972 0.659195i \(-0.229103\pi\)
0.980824 + 0.194894i \(0.0624364\pi\)
\(788\) 0 0
\(789\) −13.0160 22.5444i −0.463383 0.802603i
\(790\) 0 0
\(791\) −29.8433 16.5564i −1.06111 0.588678i
\(792\) 0 0
\(793\) 12.2067 45.5560i 0.433472 1.61774i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.99837 + 7.99837i 0.283317 + 0.283317i 0.834430 0.551113i \(-0.185797\pi\)
−0.551113 + 0.834430i \(0.685797\pi\)
\(798\) 0 0
\(799\) 30.6103i 1.08292i
\(800\) 0 0
\(801\) −15.4361 8.91203i −0.545408 0.314891i
\(802\) 0 0
\(803\) 38.5057 + 10.3176i 1.35884 + 0.364099i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.59287 + 1.49860i 0.196878 + 0.0527534i
\(808\) 0 0
\(809\) 3.21773 + 1.85776i 0.113129 + 0.0653153i 0.555497 0.831518i \(-0.312528\pi\)
−0.442368 + 0.896834i \(0.645861\pi\)
\(810\) 0 0
\(811\) 38.0896i 1.33751i −0.743484 0.668754i \(-0.766828\pi\)
0.743484 0.668754i \(-0.233172\pi\)
\(812\) 0 0
\(813\) −4.14055 4.14055i −0.145216 0.145216i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5.47363 20.4279i 0.191498 0.714681i
\(818\) 0 0
\(819\) 0.206552 + 12.0815i 0.00721752 + 0.422160i
\(820\) 0 0
\(821\) 2.86906 + 4.96937i 0.100131 + 0.173432i 0.911739 0.410771i \(-0.134740\pi\)
−0.811607 + 0.584203i \(0.801407\pi\)
\(822\) 0 0
\(823\) −29.3016 + 7.85133i −1.02139 + 0.273680i −0.730381 0.683040i \(-0.760657\pi\)
−0.291008 + 0.956721i \(0.593991\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23.4343 + 23.4343i −0.814892 + 0.814892i −0.985363 0.170471i \(-0.945471\pi\)
0.170471 + 0.985363i \(0.445471\pi\)
\(828\) 0 0
\(829\) 11.0697 19.1733i 0.384466 0.665915i −0.607229 0.794527i \(-0.707719\pi\)
0.991695 + 0.128612i \(0.0410522\pi\)
\(830\) 0 0
\(831\) 6.79385 3.92243i 0.235676 0.136068i
\(832\) 0 0
\(833\) −17.0420 + 18.2491i −0.590472 + 0.632294i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.401465 + 1.49829i 0.0138767 + 0.0517885i
\(838\) 0 0
\(839\) 2.41782 0.0834725 0.0417362 0.999129i \(-0.486711\pi\)
0.0417362 + 0.999129i \(0.486711\pi\)
\(840\) 0 0
\(841\) 20.9155 0.721223
\(842\) 0 0
\(843\) −0.0810985 0.302664i −0.00279318 0.0104243i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0985 13.6245i −0.484429 0.468143i
\(848\) 0 0
\(849\) −5.83539 + 3.36906i −0.200270 + 0.115626i
\(850\) 0 0
\(851\) −8.25057 + 14.2904i −0.282826 + 0.489869i
\(852\) 0 0
\(853\) 18.6647 18.6647i 0.639069 0.639069i −0.311257 0.950326i \(-0.600750\pi\)
0.950326 + 0.311257i \(0.100750\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.95540 + 0.791896i −0.100954 + 0.0270507i −0.308943 0.951081i \(-0.599975\pi\)
0.207988 + 0.978131i \(0.433308\pi\)
\(858\) 0 0
\(859\) −16.9260 29.3166i −0.577506 1.00027i −0.995764 0.0919421i \(-0.970693\pi\)
0.418258 0.908328i \(-0.362641\pi\)
\(860\) 0 0
\(861\) 24.3052 14.5922i 0.828318 0.497300i
\(862\) 0 0
\(863\) −1.98044 + 7.39110i −0.0674150 + 0.251596i −0.991407 0.130817i \(-0.958240\pi\)
0.923992 + 0.382413i \(0.124907\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.02380 3.02380i −0.102694 0.102694i
\(868\) 0 0
\(869\) 33.1403i 1.12421i
\(870\) 0 0
\(871\) 18.3011 + 10.5661i 0.620108 + 0.358020i
\(872\) 0 0
\(873\) 0.137394 + 0.0368147i 0.00465009 + 0.00124599i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 54.5098 + 14.6059i 1.84067 + 0.493205i 0.998911 0.0466538i \(-0.0148558\pi\)
0.841755 + 0.539859i \(0.181522\pi\)
\(878\) 0 0
\(879\) 10.6637 + 6.15667i 0.359677 + 0.207659i
\(880\) 0 0
\(881\) 13.5446i 0.456328i 0.973623 + 0.228164i \(0.0732723\pi\)
−0.973623 + 0.228164i \(0.926728\pi\)
\(882\) 0 0
\(883\) −34.7205 34.7205i −1.16844 1.16844i −0.982576 0.185864i \(-0.940492\pi\)
−0.185864 0.982576i \(-0.559508\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5.71328 + 21.3222i −0.191833 + 0.715931i 0.801231 + 0.598355i \(0.204179\pi\)
−0.993064 + 0.117575i \(0.962488\pi\)
\(888\) 0 0
\(889\) −19.2777 + 34.7485i −0.646552 + 1.16543i
\(890\) 0 0
\(891\) −2.14536 3.71588i −0.0718724 0.124487i
\(892\) 0 0
\(893\) 51.9618 13.9231i 1.73884 0.465920i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.34931 9.34931i 0.312164 0.312164i
\(898\) 0 0
\(899\) −2.20521 + 3.81953i −0.0735477 + 0.127388i
\(900\) 0 0
\(901\) 28.0289 16.1825i 0.933777 0.539116i
\(902\) 0 0
\(903\) 2.16246 8.65992i 0.0719621 0.288184i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.531796 1.98469i −0.0176580 0.0659005i 0.956535 0.291619i \(-0.0941938\pi\)
−0.974193 + 0.225718i \(0.927527\pi\)
\(908\) 0 0
\(909\) −8.75071 −0.290243
\(910\) 0 0
\(911\) 37.1855 1.23201 0.616006 0.787742i \(-0.288750\pi\)
0.616006 + 0.787742i \(0.288750\pi\)
\(912\) 0 0
\(913\) 5.22066 + 19.4838i 0.172779 + 0.644819i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.03640 + 6.24639i −0.199339 + 0.206274i
\(918\) 0 0
\(919\) −2.39087 + 1.38037i −0.0788676 + 0.0455342i −0.538915 0.842360i \(-0.681166\pi\)
0.460048 + 0.887894i \(0.347832\pi\)
\(920\) 0 0
\(921\) 15.7309 27.2467i 0.518351 0.897811i
\(922\) 0 0
\(923\) 37.9069 37.9069i 1.24772 1.24772i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −3.17860 + 0.851703i −0.104399 + 0.0279736i
\(928\) 0 0
\(929\) 18.2568 + 31.6216i 0.598985 + 1.03747i 0.992971 + 0.118356i \(0.0377623\pi\)
−0.393987 + 0.919116i \(0.628904\pi\)
\(930\) 0 0
\(931\) 38.7299 + 20.6287i 1.26932 + 0.676078i
\(932\) 0 0
\(933\) −0.766245 + 2.85967i −0.0250857 + 0.0936212i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −40.5719 40.5719i −1.32543 1.32543i −0.909313 0.416112i \(-0.863392\pi\)
−0.416112 0.909313i \(-0.636608\pi\)
\(938\) 0 0
\(939\) 5.29073i 0.172656i
\(940\) 0 0
\(941\) 28.8917 + 16.6806i 0.941842 + 0.543773i 0.890537 0.454910i \(-0.150329\pi\)
0.0513045 + 0.998683i \(0.483662\pi\)
\(942\) 0 0
\(943\) −29.9637 8.02874i −0.975751 0.261452i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −49.7887 13.3408i −1.61792 0.433519i −0.667527 0.744585i \(-0.732647\pi\)
−0.950388 + 0.311066i \(0.899314\pi\)
\(948\) 0 0
\(949\) 36.7464 + 21.2155i 1.19284 + 0.688685i
\(950\) 0 0
\(951\) 20.3965i 0.661402i
\(952\) 0 0
\(953\) 17.3822 + 17.3822i 0.563066 + 0.563066i 0.930177 0.367111i \(-0.119653\pi\)
−0.367111 + 0.930177i \(0.619653\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.15758 11.7843i 0.102070 0.380931i
\(958\) 0 0
\(959\) 22.5666 + 37.5877i 0.728715 + 1.21377i
\(960\) 0 0
\(961\) −14.2970 24.7631i −0.461193 0.798809i
\(962\) 0 0
\(963\) −12.8346 + 3.43903i −0.413590 + 0.110821i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.77615 7.77615i 0.250064 0.250064i −0.570933 0.820997i \(-0.693418\pi\)
0.820997 + 0.570933i \(0.193418\pi\)
\(968\) 0 0
\(969\) 11.1804 19.3650i 0.359165 0.622093i
\(970\) 0 0
\(971\) 21.9856 12.6934i 0.705551 0.407350i −0.103861 0.994592i \(-0.533120\pi\)
0.809411 + 0.587242i \(0.199786\pi\)
\(972\) 0 0
\(973\) 1.21724 0.348567i 0.0390230 0.0111745i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.8663 + 48.0179i 0.411631 + 1.53623i 0.791489 + 0.611183i \(0.209306\pi\)
−0.379859 + 0.925045i \(0.624027\pi\)
\(978\) 0 0
\(979\) 76.4782 2.44426
\(980\) 0 0
\(981\) −14.3052 −0.456729
\(982\) 0 0
\(983\) −3.06316 11.4319i −0.0976997 0.364620i 0.899715 0.436477i \(-0.143774\pi\)
−0.997415 + 0.0718572i \(0.977107\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 21.8271 6.25036i 0.694765 0.198951i
\(988\) 0 0
\(989\) −8.45846 + 4.88349i −0.268963 + 0.155286i
\(990\) 0 0
\(991\) 11.2866 19.5490i 0.358532 0.620995i −0.629184 0.777256i \(-0.716611\pi\)
0.987716 + 0.156261i \(0.0499441\pi\)
\(992\) 0 0
\(993\) −4.11587 + 4.11587i −0.130613 + 0.130613i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −16.0962 + 4.31297i −0.509773 + 0.136593i −0.504532 0.863393i \(-0.668335\pi\)
−0.00524094 + 0.999986i \(0.501668\pi\)
\(998\) 0 0
\(999\) 2.84986 + 4.93609i 0.0901654 + 0.156171i
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 2100.2.ce.d.493.4 yes 24
5.2 odd 4 inner 2100.2.ce.d.157.4 yes 24
5.3 odd 4 inner 2100.2.ce.d.157.2 24
5.4 even 2 inner 2100.2.ce.d.493.2 yes 24
7.5 odd 6 inner 2100.2.ce.d.1993.4 yes 24
35.12 even 12 inner 2100.2.ce.d.1657.4 yes 24
35.19 odd 6 inner 2100.2.ce.d.1993.2 yes 24
35.33 even 12 inner 2100.2.ce.d.1657.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.ce.d.157.2 24 5.3 odd 4 inner
2100.2.ce.d.157.4 yes 24 5.2 odd 4 inner
2100.2.ce.d.493.2 yes 24 5.4 even 2 inner
2100.2.ce.d.493.4 yes 24 1.1 even 1 trivial
2100.2.ce.d.1657.2 yes 24 35.33 even 12 inner
2100.2.ce.d.1657.4 yes 24 35.12 even 12 inner
2100.2.ce.d.1993.2 yes 24 35.19 odd 6 inner
2100.2.ce.d.1993.4 yes 24 7.5 odd 6 inner