Properties

Label 252.2.t.a.17.2
Level $252$
Weight $2$
Character 252.17
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,2,Mod(17,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.t (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: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.2
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 252.17
Dual form 252.2.t.a.89.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 + 2.12132i) q^{5} +(0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(1.22474 + 2.12132i) q^{5} +(0.500000 - 2.59808i) q^{7} +(3.67423 + 2.12132i) q^{11} +1.73205i q^{13} +(-2.44949 + 4.24264i) q^{17} +(4.50000 - 2.59808i) q^{19} +(-0.500000 + 0.866025i) q^{25} -8.48528i q^{29} +(-1.50000 - 0.866025i) q^{31} +(6.12372 - 2.12132i) q^{35} +(2.50000 + 4.33013i) q^{37} -12.2474 q^{41} -11.0000 q^{43} +(1.22474 + 2.12132i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-7.34847 - 4.24264i) q^{53} +10.3923i q^{55} +(2.44949 - 4.24264i) q^{59} +(-3.00000 + 1.73205i) q^{61} +(-3.67423 + 2.12132i) q^{65} +(3.50000 - 6.06218i) q^{67} -12.7279i q^{71} +(13.5000 + 7.79423i) q^{73} +(7.34847 - 8.48528i) q^{77} +(-5.50000 - 9.52628i) q^{79} -12.2474 q^{83} -12.0000 q^{85} +(7.34847 + 12.7279i) q^{89} +(4.50000 + 0.866025i) q^{91} +(11.0227 + 6.36396i) q^{95} +3.46410i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{7} + 18 q^{19} - 2 q^{25} - 6 q^{31} + 10 q^{37} - 44 q^{43} - 26 q^{49} - 12 q^{61} + 14 q^{67} + 54 q^{73} - 22 q^{79} - 48 q^{85} + 18 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.22474 + 2.12132i 0.547723 + 0.948683i 0.998430 + 0.0560116i \(0.0178384\pi\)
−0.450708 + 0.892672i \(0.648828\pi\)
\(6\) 0 0
\(7\) 0.500000 2.59808i 0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.67423 + 2.12132i 1.10782 + 0.639602i 0.938265 0.345918i \(-0.112432\pi\)
0.169559 + 0.985520i \(0.445766\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.44949 + 4.24264i −0.594089 + 1.02899i 0.399586 + 0.916696i \(0.369154\pi\)
−0.993675 + 0.112296i \(0.964180\pi\)
\(18\) 0 0
\(19\) 4.50000 2.59808i 1.03237 0.596040i 0.114708 0.993399i \(-0.463407\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.48528i 1.57568i −0.615882 0.787839i \(-0.711200\pi\)
0.615882 0.787839i \(-0.288800\pi\)
\(30\) 0 0
\(31\) −1.50000 0.866025i −0.269408 0.155543i 0.359211 0.933257i \(-0.383046\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.12372 2.12132i 1.03510 0.358569i
\(36\) 0 0
\(37\) 2.50000 + 4.33013i 0.410997 + 0.711868i 0.994999 0.0998840i \(-0.0318472\pi\)
−0.584002 + 0.811752i \(0.698514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.2474 −1.91273 −0.956365 0.292174i \(-0.905621\pi\)
−0.956365 + 0.292174i \(0.905621\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.22474 + 2.12132i 0.178647 + 0.309426i 0.941417 0.337243i \(-0.109495\pi\)
−0.762770 + 0.646670i \(0.776161\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.34847 4.24264i −1.00939 0.582772i −0.0983769 0.995149i \(-0.531365\pi\)
−0.911013 + 0.412378i \(0.864698\pi\)
\(54\) 0 0
\(55\) 10.3923i 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.44949 4.24264i 0.318896 0.552345i −0.661362 0.750067i \(-0.730021\pi\)
0.980258 + 0.197722i \(0.0633545\pi\)
\(60\) 0 0
\(61\) −3.00000 + 1.73205i −0.384111 + 0.221766i −0.679605 0.733578i \(-0.737849\pi\)
0.295495 + 0.955344i \(0.404516\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.67423 + 2.12132i −0.455733 + 0.263117i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.7279i 1.51053i −0.655422 0.755263i \(-0.727509\pi\)
0.655422 0.755263i \(-0.272491\pi\)
\(72\) 0 0
\(73\) 13.5000 + 7.79423i 1.58006 + 0.912245i 0.994850 + 0.101361i \(0.0323196\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.34847 8.48528i 0.837436 0.966988i
\(78\) 0 0
\(79\) −5.50000 9.52628i −0.618798 1.07179i −0.989705 0.143120i \(-0.954286\pi\)
0.370907 0.928670i \(-0.379047\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.2474 −1.34433 −0.672166 0.740400i \(-0.734636\pi\)
−0.672166 + 0.740400i \(0.734636\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.34847 + 12.7279i 0.778936 + 1.34916i 0.932555 + 0.361027i \(0.117574\pi\)
−0.153619 + 0.988130i \(0.549093\pi\)
\(90\) 0 0
\(91\) 4.50000 + 0.866025i 0.471728 + 0.0907841i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.0227 + 6.36396i 1.13091 + 0.652929i
\(96\) 0 0
\(97\) 3.46410i 0.351726i 0.984415 + 0.175863i \(0.0562716\pi\)
−0.984415 + 0.175863i \(0.943728\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.67423 + 6.36396i −0.365600 + 0.633238i −0.988872 0.148767i \(-0.952470\pi\)
0.623272 + 0.782005i \(0.285803\pi\)
\(102\) 0 0
\(103\) 7.50000 4.33013i 0.738997 0.426660i −0.0827075 0.996574i \(-0.526357\pi\)
0.821705 + 0.569914i \(0.193023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.34847 + 4.24264i −0.710403 + 0.410152i −0.811210 0.584754i \(-0.801191\pi\)
0.100807 + 0.994906i \(0.467858\pi\)
\(108\) 0 0
\(109\) 2.50000 4.33013i 0.239457 0.414751i −0.721102 0.692829i \(-0.756364\pi\)
0.960558 + 0.278078i \(0.0896974\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264i 0.399114i 0.979886 + 0.199557i \(0.0639503\pi\)
−0.979886 + 0.199557i \(0.936050\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 9.79796 + 8.48528i 0.898177 + 0.777844i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.79796 0.876356
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.67423 + 6.36396i 0.321019 + 0.556022i 0.980699 0.195525i \(-0.0626412\pi\)
−0.659679 + 0.751547i \(0.729308\pi\)
\(132\) 0 0
\(133\) −4.50000 12.9904i −0.390199 1.12641i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(138\) 0 0
\(139\) 1.73205i 0.146911i −0.997299 0.0734553i \(-0.976597\pi\)
0.997299 0.0734553i \(-0.0234026\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.67423 + 6.36396i −0.307255 + 0.532181i
\(144\) 0 0
\(145\) 18.0000 10.3923i 1.49482 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.6969 8.48528i 1.20402 0.695141i 0.242574 0.970133i \(-0.422008\pi\)
0.961447 + 0.274992i \(0.0886751\pi\)
\(150\) 0 0
\(151\) 1.00000 1.73205i 0.0813788 0.140952i −0.822464 0.568818i \(-0.807401\pi\)
0.903842 + 0.427865i \(0.140734\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.24264i 0.340777i
\(156\) 0 0
\(157\) −9.00000 5.19615i −0.718278 0.414698i 0.0958404 0.995397i \(-0.469446\pi\)
−0.814119 + 0.580699i \(0.802779\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.0000 + 19.0526i 0.861586 + 1.49231i 0.870397 + 0.492350i \(0.163862\pi\)
−0.00881059 + 0.999961i \(0.502805\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.1464 −1.32683 −0.663415 0.748251i \(-0.730894\pi\)
−0.663415 + 0.748251i \(0.730894\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 2.00000 + 1.73205i 0.151186 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.0227 + 6.36396i 0.823876 + 0.475665i 0.851751 0.523947i \(-0.175541\pi\)
−0.0278755 + 0.999611i \(0.508874\pi\)
\(180\) 0 0
\(181\) 5.19615i 0.386227i −0.981176 0.193113i \(-0.938141\pi\)
0.981176 0.193113i \(-0.0618586\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.12372 + 10.6066i −0.450225 + 0.779813i
\(186\) 0 0
\(187\) −18.0000 + 10.3923i −1.31629 + 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.67423 + 2.12132i −0.265858 + 0.153493i −0.627004 0.779016i \(-0.715719\pi\)
0.361146 + 0.932509i \(0.382386\pi\)
\(192\) 0 0
\(193\) −9.50000 + 16.4545i −0.683825 + 1.18442i 0.289980 + 0.957033i \(0.406351\pi\)
−0.973805 + 0.227387i \(0.926982\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.48528i 0.604551i −0.953221 0.302276i \(-0.902254\pi\)
0.953221 0.302276i \(-0.0977463\pi\)
\(198\) 0 0
\(199\) −12.0000 6.92820i −0.850657 0.491127i 0.0102152 0.999948i \(-0.496748\pi\)
−0.860873 + 0.508821i \(0.830082\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −22.0454 4.24264i −1.54728 0.297775i
\(204\) 0 0
\(205\) −15.0000 25.9808i −1.04765 1.81458i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.0454 1.52491
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.4722 23.3345i −0.918796 1.59140i
\(216\) 0 0
\(217\) −3.00000 + 3.46410i −0.203653 + 0.235159i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.34847 4.24264i −0.494312 0.285391i
\(222\) 0 0
\(223\) 20.7846i 1.39184i 0.718119 + 0.695920i \(0.245003\pi\)
−0.718119 + 0.695920i \(0.754997\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.0227 19.0919i 0.731603 1.26717i −0.224595 0.974452i \(-0.572106\pi\)
0.956198 0.292721i \(-0.0945606\pi\)
\(228\) 0 0
\(229\) 1.50000 0.866025i 0.0991228 0.0572286i −0.449619 0.893220i \(-0.648440\pi\)
0.548742 + 0.835992i \(0.315107\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.67423 + 2.12132i −0.240707 + 0.138972i −0.615502 0.788136i \(-0.711047\pi\)
0.374795 + 0.927108i \(0.377713\pi\)
\(234\) 0 0
\(235\) −3.00000 + 5.19615i −0.195698 + 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.24264i 0.274434i 0.990541 + 0.137217i \(0.0438157\pi\)
−0.990541 + 0.137217i \(0.956184\pi\)
\(240\) 0 0
\(241\) 12.0000 + 6.92820i 0.772988 + 0.446285i 0.833939 0.551856i \(-0.186080\pi\)
−0.0609515 + 0.998141i \(0.519414\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.44949 16.9706i −0.156492 1.08421i
\(246\) 0 0
\(247\) 4.50000 + 7.79423i 0.286328 + 0.495935i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.79796 −0.618442 −0.309221 0.950990i \(-0.600068\pi\)
−0.309221 + 0.950990i \(0.600068\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.67423 6.36396i −0.229192 0.396973i 0.728377 0.685177i \(-0.240275\pi\)
−0.957569 + 0.288204i \(0.906942\pi\)
\(258\) 0 0
\(259\) 12.5000 4.33013i 0.776712 0.269061i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −22.0454 12.7279i −1.35938 0.784837i −0.369838 0.929096i \(-0.620587\pi\)
−0.989540 + 0.144259i \(0.953920\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.12372 10.6066i 0.373370 0.646696i −0.616712 0.787189i \(-0.711536\pi\)
0.990082 + 0.140493i \(0.0448688\pi\)
\(270\) 0 0
\(271\) −24.0000 + 13.8564i −1.45790 + 0.841717i −0.998908 0.0467255i \(-0.985121\pi\)
−0.458988 + 0.888442i \(0.651788\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.67423 + 2.12132i −0.221565 + 0.127920i
\(276\) 0 0
\(277\) −5.50000 + 9.52628i −0.330463 + 0.572379i −0.982603 0.185720i \(-0.940538\pi\)
0.652140 + 0.758099i \(0.273872\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.9706i 1.01238i 0.862422 + 0.506189i \(0.168946\pi\)
−0.862422 + 0.506189i \(0.831054\pi\)
\(282\) 0 0
\(283\) −7.50000 4.33013i −0.445829 0.257399i 0.260238 0.965544i \(-0.416199\pi\)
−0.706067 + 0.708145i \(0.749532\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.12372 + 31.8198i −0.361472 + 1.87826i
\(288\) 0 0
\(289\) −3.50000 6.06218i −0.205882 0.356599i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.4949 1.43101 0.715504 0.698609i \(-0.246197\pi\)
0.715504 + 0.698609i \(0.246197\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −5.50000 + 28.5788i −0.317015 + 1.64726i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.34847 4.24264i −0.420772 0.242933i
\(306\) 0 0
\(307\) 19.0526i 1.08739i −0.839284 0.543693i \(-0.817025\pi\)
0.839284 0.543693i \(-0.182975\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.57321 14.8492i 0.486142 0.842023i −0.513731 0.857951i \(-0.671737\pi\)
0.999873 + 0.0159282i \(0.00507031\pi\)
\(312\) 0 0
\(313\) −13.5000 + 7.79423i −0.763065 + 0.440556i −0.830395 0.557175i \(-0.811885\pi\)
0.0673300 + 0.997731i \(0.478552\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.34847 4.24264i 0.412731 0.238290i −0.279231 0.960224i \(-0.590080\pi\)
0.691963 + 0.721933i \(0.256746\pi\)
\(318\) 0 0
\(319\) 18.0000 31.1769i 1.00781 1.74557i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 25.4558i 1.41640i
\(324\) 0 0
\(325\) −1.50000 0.866025i −0.0832050 0.0480384i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.12372 2.12132i 0.337612 0.116952i
\(330\) 0 0
\(331\) 2.50000 + 4.33013i 0.137412 + 0.238005i 0.926516 0.376254i \(-0.122788\pi\)
−0.789104 + 0.614260i \(0.789455\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.1464 0.936809
\(336\) 0 0
\(337\) 7.00000 0.381314 0.190657 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.67423 6.36396i −0.198971 0.344628i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.34847 + 4.24264i 0.394486 + 0.227757i 0.684102 0.729386i \(-0.260194\pi\)
−0.289616 + 0.957143i \(0.593528\pi\)
\(348\) 0 0
\(349\) 24.2487i 1.29800i −0.760787 0.649002i \(-0.775187\pi\)
0.760787 0.649002i \(-0.224813\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.4722 + 23.3345i −0.717053 + 1.24197i 0.245110 + 0.969495i \(0.421176\pi\)
−0.962163 + 0.272476i \(0.912157\pi\)
\(354\) 0 0
\(355\) 27.0000 15.5885i 1.43301 0.827349i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.6969 8.48528i 0.775675 0.447836i −0.0592205 0.998245i \(-0.518862\pi\)
0.834895 + 0.550409i \(0.185528\pi\)
\(360\) 0 0
\(361\) 4.00000 6.92820i 0.210526 0.364642i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 38.1838i 1.99863i
\(366\) 0 0
\(367\) 22.5000 + 12.9904i 1.17449 + 0.678092i 0.954734 0.297462i \(-0.0961403\pi\)
0.219757 + 0.975555i \(0.429474\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −14.6969 + 16.9706i −0.763027 + 0.881068i
\(372\) 0 0
\(373\) 3.50000 + 6.06218i 0.181223 + 0.313888i 0.942297 0.334777i \(-0.108661\pi\)
−0.761074 + 0.648665i \(0.775328\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.6969 0.756931
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.89898 8.48528i −0.250326 0.433578i 0.713289 0.700870i \(-0.247205\pi\)
−0.963616 + 0.267292i \(0.913871\pi\)
\(384\) 0 0
\(385\) 27.0000 + 5.19615i 1.37605 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.67423 + 2.12132i 0.186291 + 0.107555i 0.590245 0.807224i \(-0.299031\pi\)
−0.403954 + 0.914779i \(0.632364\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.4722 23.3345i 0.677860 1.17409i
\(396\) 0 0
\(397\) 7.50000 4.33013i 0.376414 0.217323i −0.299843 0.953989i \(-0.596934\pi\)
0.676257 + 0.736666i \(0.263601\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.34847 + 4.24264i −0.366965 + 0.211867i −0.672132 0.740432i \(-0.734621\pi\)
0.305167 + 0.952299i \(0.401288\pi\)
\(402\) 0 0
\(403\) 1.50000 2.59808i 0.0747203 0.129419i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.2132i 1.05150i
\(408\) 0 0
\(409\) −7.50000 4.33013i −0.370851 0.214111i 0.302979 0.952997i \(-0.402019\pi\)
−0.673830 + 0.738886i \(0.735352\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.79796 8.48528i −0.482126 0.417533i
\(414\) 0 0
\(415\) −15.0000 25.9808i −0.736321 1.27535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.9444 1.31632 0.658160 0.752878i \(-0.271335\pi\)
0.658160 + 0.752878i \(0.271335\pi\)
\(420\) 0 0
\(421\) 35.0000 1.70580 0.852898 0.522078i \(-0.174843\pi\)
0.852898 + 0.522078i \(0.174843\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.44949 4.24264i −0.118818 0.205798i
\(426\) 0 0
\(427\) 3.00000 + 8.66025i 0.145180 + 0.419099i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.3712 + 10.6066i 0.884908 + 0.510902i 0.872274 0.489018i \(-0.162645\pi\)
0.0126347 + 0.999920i \(0.495978\pi\)
\(432\) 0 0
\(433\) 12.1244i 0.582659i −0.956623 0.291330i \(-0.905902\pi\)
0.956623 0.291330i \(-0.0940977\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −36.0000 + 20.7846i −1.71819 + 0.991995i −0.795956 + 0.605355i \(0.793031\pi\)
−0.922231 + 0.386640i \(0.873635\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 0 0
\(445\) −18.0000 + 31.1769i −0.853282 + 1.47793i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.7279i 0.600668i 0.953834 + 0.300334i \(0.0970981\pi\)
−0.953834 + 0.300334i \(0.902902\pi\)
\(450\) 0 0
\(451\) −45.0000 25.9808i −2.11897 1.22339i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.67423 + 10.6066i 0.172251 + 0.497245i
\(456\) 0 0
\(457\) −6.50000 11.2583i −0.304057 0.526642i 0.672994 0.739648i \(-0.265008\pi\)
−0.977051 + 0.213006i \(0.931675\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.89898 −0.228168 −0.114084 0.993471i \(-0.536393\pi\)
−0.114084 + 0.993471i \(0.536393\pi\)
\(462\) 0 0
\(463\) 29.0000 1.34774 0.673872 0.738848i \(-0.264630\pi\)
0.673872 + 0.738848i \(0.264630\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.67423 + 6.36396i 0.170023 + 0.294489i 0.938428 0.345476i \(-0.112282\pi\)
−0.768404 + 0.639965i \(0.778949\pi\)
\(468\) 0 0
\(469\) −14.0000 12.1244i −0.646460 0.559851i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −40.4166 23.3345i −1.85836 1.07292i
\(474\) 0 0
\(475\) 5.19615i 0.238416i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.44949 4.24264i 0.111920 0.193851i −0.804624 0.593784i \(-0.797633\pi\)
0.916544 + 0.399933i \(0.130967\pi\)
\(480\) 0 0
\(481\) −7.50000 + 4.33013i −0.341971 + 0.197437i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.34847 + 4.24264i −0.333677 + 0.192648i
\(486\) 0 0
\(487\) −3.50000 + 6.06218i −0.158600 + 0.274703i −0.934364 0.356320i \(-0.884031\pi\)
0.775764 + 0.631023i \(0.217365\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 33.9411i 1.53174i −0.642995 0.765871i \(-0.722308\pi\)
0.642995 0.765871i \(-0.277692\pi\)
\(492\) 0 0
\(493\) 36.0000 + 20.7846i 1.62136 + 0.936092i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −33.0681 6.36396i −1.48331 0.285463i
\(498\) 0 0
\(499\) 17.5000 + 30.3109i 0.783408 + 1.35690i 0.929946 + 0.367697i \(0.119854\pi\)
−0.146538 + 0.989205i \(0.546813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.44949 0.109217 0.0546087 0.998508i \(-0.482609\pi\)
0.0546087 + 0.998508i \(0.482609\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.57321 14.8492i −0.380001 0.658181i 0.611061 0.791584i \(-0.290743\pi\)
−0.991062 + 0.133402i \(0.957410\pi\)
\(510\) 0 0
\(511\) 27.0000 31.1769i 1.19441 1.37919i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.3712 + 10.6066i 0.809531 + 0.467383i
\(516\) 0 0
\(517\) 10.3923i 0.457053i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.44949 4.24264i 0.107314 0.185873i −0.807367 0.590049i \(-0.799108\pi\)
0.914681 + 0.404176i \(0.132442\pi\)
\(522\) 0 0
\(523\) −7.50000 + 4.33013i −0.327952 + 0.189343i −0.654932 0.755688i \(-0.727303\pi\)
0.326979 + 0.945031i \(0.393969\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.34847 4.24264i 0.320104 0.184812i
\(528\) 0 0
\(529\) −11.5000 + 19.9186i −0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 21.2132i 0.918846i
\(534\) 0 0
\(535\) −18.0000 10.3923i −0.778208 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.3712 23.3345i −0.791302 1.00509i
\(540\) 0 0
\(541\) 9.50000 + 16.4545i 0.408437 + 0.707433i 0.994715 0.102677i \(-0.0327407\pi\)
−0.586278 + 0.810110i \(0.699407\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.2474 0.524623
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −22.0454 38.1838i −0.939166 1.62668i
\(552\) 0 0
\(553\) −27.5000 + 9.52628i −1.16942 + 0.405099i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.3712 10.6066i −0.778412 0.449416i 0.0574555 0.998348i \(-0.481701\pi\)
−0.835867 + 0.548932i \(0.815035\pi\)
\(558\) 0 0
\(559\) 19.0526i 0.805837i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.67423 + 6.36396i −0.154851 + 0.268209i −0.933005 0.359864i \(-0.882823\pi\)
0.778154 + 0.628073i \(0.216156\pi\)
\(564\) 0 0
\(565\) −9.00000 + 5.19615i −0.378633 + 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.3712 10.6066i 0.770160 0.444652i −0.0627719 0.998028i \(-0.519994\pi\)
0.832932 + 0.553376i \(0.186661\pi\)
\(570\) 0 0
\(571\) 3.50000 6.06218i 0.146470 0.253694i −0.783450 0.621455i \(-0.786542\pi\)
0.929921 + 0.367760i \(0.119875\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.5000 19.9186i −1.43625 0.829222i −0.438667 0.898650i \(-0.644549\pi\)
−0.997587 + 0.0694283i \(0.977883\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.12372 + 31.8198i −0.254055 + 1.32011i
\(582\) 0 0
\(583\) −18.0000 31.1769i −0.745484 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.6969 −0.606608 −0.303304 0.952894i \(-0.598090\pi\)
−0.303304 + 0.952894i \(0.598090\pi\)
\(588\) 0 0
\(589\) −9.00000 −0.370839
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.22474 2.12132i −0.0502942 0.0871122i 0.839782 0.542923i \(-0.182683\pi\)
−0.890077 + 0.455811i \(0.849349\pi\)
\(594\) 0 0
\(595\) −6.00000 + 31.1769i −0.245976 + 1.27813i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.6969 + 8.48528i 0.600501 + 0.346699i 0.769238 0.638962i \(-0.220636\pi\)
−0.168738 + 0.985661i \(0.553969\pi\)
\(600\) 0 0
\(601\) 43.3013i 1.76630i 0.469095 + 0.883148i \(0.344580\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.57321 + 14.8492i −0.348551 + 0.603708i
\(606\) 0 0
\(607\) 22.5000 12.9904i 0.913247 0.527263i 0.0317724 0.999495i \(-0.489885\pi\)
0.881474 + 0.472232i \(0.156551\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.67423 + 2.12132i −0.148644 + 0.0858194i
\(612\) 0 0
\(613\) −4.00000 + 6.92820i −0.161558 + 0.279827i −0.935428 0.353518i \(-0.884985\pi\)
0.773869 + 0.633345i \(0.218319\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.2132i 0.854011i −0.904249 0.427006i \(-0.859568\pi\)
0.904249 0.427006i \(-0.140432\pi\)
\(618\) 0 0
\(619\) 7.50000 + 4.33013i 0.301450 + 0.174042i 0.643094 0.765787i \(-0.277650\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 36.7423 12.7279i 1.47205 0.509933i
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.4949 −0.976676
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.12372 + 10.6066i 0.243013 + 0.420910i
\(636\) 0 0
\(637\) 4.50000 11.2583i 0.178296 0.446071i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.34847 + 4.24264i 0.290247 + 0.167574i 0.638053 0.769992i \(-0.279740\pi\)
−0.347806 + 0.937566i \(0.613073\pi\)
\(642\) 0 0
\(643\) 32.9090i 1.29780i −0.760872 0.648901i \(-0.775229\pi\)
0.760872 0.648901i \(-0.224771\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11.0227 + 19.0919i −0.433347 + 0.750579i −0.997159 0.0753238i \(-0.976001\pi\)
0.563812 + 0.825903i \(0.309334\pi\)
\(648\) 0 0
\(649\) 18.0000 10.3923i 0.706562 0.407934i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.3712 + 10.6066i −0.718920 + 0.415068i −0.814355 0.580367i \(-0.802909\pi\)
0.0954353 + 0.995436i \(0.469576\pi\)
\(654\) 0 0
\(655\) −9.00000 + 15.5885i −0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.9706i 0.661079i 0.943792 + 0.330540i \(0.107231\pi\)
−0.943792 + 0.330540i \(0.892769\pi\)
\(660\) 0 0
\(661\) 4.50000 + 2.59808i 0.175030 + 0.101053i 0.584955 0.811065i \(-0.301112\pi\)
−0.409926 + 0.912119i \(0.634445\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 22.0454 25.4558i 0.854884 0.987135i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −14.6969 −0.567369
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.6969 + 25.4558i 0.564849 + 0.978348i 0.997064 + 0.0765767i \(0.0243990\pi\)
−0.432214 + 0.901771i \(0.642268\pi\)
\(678\) 0 0
\(679\) 9.00000 + 1.73205i 0.345388 + 0.0664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.34847 4.24264i −0.281181 0.162340i 0.352777 0.935708i \(-0.385238\pi\)
−0.633958 + 0.773367i \(0.718571\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.34847 12.7279i 0.279954 0.484895i
\(690\) 0 0
\(691\) 16.5000 9.52628i 0.627690 0.362397i −0.152167 0.988355i \(-0.548625\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.67423 2.12132i 0.139372 0.0804663i
\(696\) 0 0
\(697\) 30.0000 51.9615i 1.13633 1.96818i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.9411i 1.28194i 0.767567 + 0.640969i \(0.221467\pi\)
−0.767567 + 0.640969i \(0.778533\pi\)
\(702\) 0 0
\(703\) 22.5000 + 12.9904i 0.848604 + 0.489942i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.6969 + 12.7279i 0.552735 + 0.478683i
\(708\) 0 0
\(709\) −8.00000 13.8564i −0.300446 0.520388i 0.675791 0.737093i \(-0.263802\pi\)
−0.976237 + 0.216705i \(0.930469\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.4722 + 23.3345i 0.502428 + 0.870231i 0.999996 + 0.00280593i \(0.000893157\pi\)
−0.497568 + 0.867425i \(0.665774\pi\)
\(720\) 0 0
\(721\) −7.50000 21.6506i −0.279315 0.806312i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.34847 + 4.24264i 0.272915 + 0.157568i
\(726\) 0 0
\(727\) 39.8372i 1.47748i 0.673991 + 0.738739i \(0.264579\pi\)
−0.673991 + 0.738739i \(0.735421\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 26.9444 46.6690i 0.996574 1.72612i
\(732\) 0 0
\(733\) 7.50000 4.33013i 0.277019 0.159937i −0.355054 0.934846i \(-0.615538\pi\)
0.632073 + 0.774909i \(0.282204\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.7196 14.8492i 0.947395 0.546979i
\(738\) 0 0
\(739\) 9.50000 16.4545i 0.349463 0.605288i −0.636691 0.771119i \(-0.719697\pi\)
0.986154 + 0.165831i \(0.0530307\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.7279i 0.466942i −0.972364 0.233471i \(-0.924992\pi\)
0.972364 0.233471i \(-0.0750084\pi\)
\(744\) 0 0
\(745\) 36.0000 + 20.7846i 1.31894 + 0.761489i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.34847 + 21.2132i 0.268507 + 0.775114i
\(750\) 0 0
\(751\) 2.50000 + 4.33013i 0.0912263 + 0.158009i 0.908027 0.418911i \(-0.137588\pi\)
−0.816801 + 0.576919i \(0.804255\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.89898 0.178292
\(756\) 0 0
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 17.1464 + 29.6985i 0.621558 + 1.07657i 0.989196 + 0.146600i \(0.0468331\pi\)
−0.367638 + 0.929969i \(0.619834\pi\)
\(762\) 0 0
\(763\) −10.0000 8.66025i −0.362024 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.34847 + 4.24264i 0.265338 + 0.153193i
\(768\) 0 0
\(769\) 15.5885i 0.562134i −0.959688 0.281067i \(-0.909312\pi\)
0.959688 0.281067i \(-0.0906883\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.8207 + 36.0624i −0.748867 + 1.29708i 0.199499 + 0.979898i \(0.436069\pi\)
−0.948366 + 0.317178i \(0.897265\pi\)
\(774\) 0 0
\(775\) 1.50000 0.866025i 0.0538816 0.0311086i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −55.1135 + 31.8198i −1.97465 + 1.14006i
\(780\) 0 0
\(781\) 27.0000 46.7654i 0.966136 1.67340i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.4558i 0.908558i
\(786\) 0 0
\(787\) 15.0000 + 8.66025i 0.534692 + 0.308705i 0.742925 0.669375i \(-0.233438\pi\)
−0.208233 + 0.978079i \(0.566771\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.0227 + 2.12132i 0.391922 + 0.0754255i
\(792\) 0 0
\(793\) −3.00000 5.19615i −0.106533 0.184521i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33.0681 + 57.2756i 1.16695 + 2.02121i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.3712 + 10.6066i 0.645896 + 0.372908i 0.786882 0.617103i \(-0.211694\pi\)
−0.140986 + 0.990012i \(0.545027\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i 0.836881 + 0.547385i \(0.184377\pi\)
−0.836881 + 0.547385i \(0.815623\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −26.9444 + 46.6690i −0.943821 + 1.63475i
\(816\) 0 0
\(817\) −49.5000 + 28.5788i −1.73179 + 0.999847i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.3712 + 10.6066i −0.641158 + 0.370173i −0.785061 0.619419i \(-0.787368\pi\)
0.143902 + 0.989592i \(0.454035\pi\)
\(822\) 0 0
\(823\) −13.0000 + 22.5167i −0.453152 + 0.784881i −0.998580 0.0532760i \(-0.983034\pi\)
0.545428 + 0.838157i \(0.316367\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.6985i 1.03272i −0.856372 0.516359i \(-0.827287\pi\)
0.856372 0.516359i \(-0.172713\pi\)
\(828\) 0 0
\(829\) 31.5000 + 18.1865i 1.09404 + 0.631644i 0.934649 0.355571i \(-0.115714\pi\)
0.159391 + 0.987216i \(0.449047\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26.9444 21.2132i 0.933568 0.734994i
\(834\) 0 0
\(835\) −21.0000 36.3731i −0.726735 1.25874i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −53.8888 −1.86045 −0.930224 0.366993i \(-0.880387\pi\)
−0.930224 + 0.366993i \(0.880387\pi\)
\(840\) 0 0
\(841\) −43.0000 −1.48276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.2474 + 21.2132i 0.421325 + 0.729756i
\(846\) 0 0
\(847\) 17.5000 6.06218i 0.601307 0.208299i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 25.9808i 0.889564i −0.895639 0.444782i \(-0.853281\pi\)
0.895639 0.444782i \(-0.146719\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.5959 33.9411i 0.669384 1.15941i −0.308693 0.951162i \(-0.599892\pi\)
0.978077 0.208245i \(-0.0667751\pi\)
\(858\) 0 0
\(859\) 39.0000 22.5167i 1.33066 0.768259i 0.345262 0.938506i \(-0.387790\pi\)
0.985401 + 0.170248i \(0.0544569\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33.0681 19.0919i 1.12565 0.649895i 0.182814 0.983148i \(-0.441479\pi\)
0.942838 + 0.333252i \(0.108146\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 46.6690i 1.58314i
\(870\) 0 0
\(871\) 10.5000 + 6.06218i 0.355779 + 0.205409i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.89898 25.4558i 0.165616 0.860565i
\(876\) 0 0
\(877\) 14.0000 + 24.2487i 0.472746 + 0.818821i 0.999514 0.0311889i \(-0.00992933\pi\)
−0.526767 + 0.850010i \(0.676596\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −29.3939 −0.990305 −0.495152 0.868806i \(-0.664888\pi\)
−0.495152 + 0.868806i \(0.664888\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28.1691 48.7904i −0.945827 1.63822i −0.754087 0.656774i \(-0.771920\pi\)
−0.191740 0.981446i \(-0.561413\pi\)
\(888\) 0 0
\(889\) 2.50000 12.9904i 0.0838473 0.435683i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.0227 + 6.36396i 0.368861 + 0.212962i
\(894\) 0 0
\(895\) 31.1769i 1.04213i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.34847 + 12.7279i −0.245085 + 0.424500i
\(900\) 0 0
\(901\) 36.0000 20.7846i 1.19933 0.692436i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.0227 6.36396i 0.366407 0.211545i
\(906\) 0 0
\(907\) −3.50000 + 6.06218i −0.116216 + 0.201291i −0.918265 0.395966i \(-0.870410\pi\)
0.802049 + 0.597258i \(0.203743\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.48528i 0.281130i 0.990071 + 0.140565i \(0.0448919\pi\)
−0.990071 + 0.140565i \(0.955108\pi\)
\(912\) 0 0
\(913\) −45.0000 25.9808i −1.48928 0.859838i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.3712 6.36396i 0.606670 0.210157i
\(918\) 0 0
\(919\) 20.5000 + 35.5070i 0.676233 + 1.17127i 0.976107 + 0.217291i \(0.0697219\pi\)
−0.299874 + 0.953979i \(0.596945\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 22.0454 0.725633
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −13.4722 23.3345i −0.442008 0.765581i 0.555830 0.831296i \(-0.312401\pi\)
−0.997838 + 0.0657150i \(0.979067\pi\)
\(930\) 0 0
\(931\) −36.0000 + 5.19615i −1.17985 + 0.170297i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −44.0908 25.4558i −1.44192 0.832495i
\(936\) 0 0
\(937\) 19.0526i 0.622420i 0.950341 + 0.311210i \(0.100734\pi\)
−0.950341 + 0.311210i \(0.899266\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.79796 16.9706i 0.319404 0.553225i −0.660960 0.750422i \(-0.729851\pi\)
0.980364 + 0.197197i \(0.0631839\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.7650 + 27.5772i −1.55216 + 0.896137i −0.554189 + 0.832391i \(0.686972\pi\)
−0.997966 + 0.0637469i \(0.979695\pi\)
\(948\) 0 0
\(949\) −13.5000 + 23.3827i −0.438229 + 0.759034i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 33.9411i 1.09946i −0.835342 0.549730i \(-0.814730\pi\)
0.835342 0.549730i \(-0.185270\pi\)
\(954\) 0 0
\(955\) −9.00000 5.19615i −0.291233 0.168144i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14.0000 24.2487i −0.451613 0.782216i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −46.5403 −1.49819
\(966\) 0 0
\(967\) −43.0000 −1.38279 −0.691393 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.89898 8.48528i −0.157216 0.272306i 0.776648 0.629935i \(-0.216918\pi\)
−0.933864 + 0.357629i \(0.883585\pi\)
\(972\) 0 0
\(973\) −4.50000 0.866025i −0.144263 0.0277635i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.0681 + 19.0919i 1.05794 + 0.610803i 0.924862 0.380302i \(-0.124180\pi\)
0.133080 + 0.991105i \(0.457513\pi\)
\(978\) 0 0
\(979\) 62.3538i 1.99284i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −26.9444 + 46.6690i −0.859392 + 1.48851i 0.0131168 + 0.999914i \(0.495825\pi\)
−0.872509 + 0.488597i \(0.837509\pi\)
\(984\) 0 0
\(985\) 18.0000 10.3923i 0.573528 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −6.50000 + 11.2583i −0.206479 + 0.357633i −0.950603 0.310409i \(-0.899534\pi\)
0.744124 + 0.668042i \(0.232867\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 33.9411i 1.07601i
\(996\) 0 0
\(997\) −43.5000 25.1147i −1.37766 0.795392i −0.385782 0.922590i \(-0.626068\pi\)
−0.991877 + 0.127198i \(0.959401\pi\)
\(998\) 0 0
\(999\) 0 0
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 252.2.t.a.17.2 yes 4
3.2 odd 2 inner 252.2.t.a.17.1 4
4.3 odd 2 1008.2.bt.a.17.2 4
5.2 odd 4 6300.2.dd.a.4049.4 8
5.3 odd 4 6300.2.dd.a.4049.2 8
5.4 even 2 6300.2.ch.a.4301.2 4
7.2 even 3 1764.2.t.a.1097.2 4
7.3 odd 6 1764.2.f.a.881.3 4
7.4 even 3 1764.2.f.a.881.1 4
7.5 odd 6 inner 252.2.t.a.89.1 yes 4
7.6 odd 2 1764.2.t.a.521.1 4
9.2 odd 6 2268.2.bm.g.1025.2 4
9.4 even 3 2268.2.w.h.269.2 4
9.5 odd 6 2268.2.w.h.269.1 4
9.7 even 3 2268.2.bm.g.1025.1 4
12.11 even 2 1008.2.bt.a.17.1 4
15.2 even 4 6300.2.dd.a.4049.3 8
15.8 even 4 6300.2.dd.a.4049.1 8
15.14 odd 2 6300.2.ch.a.4301.1 4
21.2 odd 6 1764.2.t.a.1097.1 4
21.5 even 6 inner 252.2.t.a.89.2 yes 4
21.11 odd 6 1764.2.f.a.881.4 4
21.17 even 6 1764.2.f.a.881.2 4
21.20 even 2 1764.2.t.a.521.2 4
28.3 even 6 7056.2.k.a.881.4 4
28.11 odd 6 7056.2.k.a.881.2 4
28.19 even 6 1008.2.bt.a.593.1 4
35.12 even 12 6300.2.dd.a.1349.1 8
35.19 odd 6 6300.2.ch.a.1601.1 4
35.33 even 12 6300.2.dd.a.1349.3 8
63.5 even 6 2268.2.bm.g.593.1 4
63.40 odd 6 2268.2.bm.g.593.2 4
63.47 even 6 2268.2.w.h.1349.2 4
63.61 odd 6 2268.2.w.h.1349.1 4
84.11 even 6 7056.2.k.a.881.3 4
84.47 odd 6 1008.2.bt.a.593.2 4
84.59 odd 6 7056.2.k.a.881.1 4
105.47 odd 12 6300.2.dd.a.1349.2 8
105.68 odd 12 6300.2.dd.a.1349.4 8
105.89 even 6 6300.2.ch.a.1601.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.t.a.17.1 4 3.2 odd 2 inner
252.2.t.a.17.2 yes 4 1.1 even 1 trivial
252.2.t.a.89.1 yes 4 7.5 odd 6 inner
252.2.t.a.89.2 yes 4 21.5 even 6 inner
1008.2.bt.a.17.1 4 12.11 even 2
1008.2.bt.a.17.2 4 4.3 odd 2
1008.2.bt.a.593.1 4 28.19 even 6
1008.2.bt.a.593.2 4 84.47 odd 6
1764.2.f.a.881.1 4 7.4 even 3
1764.2.f.a.881.2 4 21.17 even 6
1764.2.f.a.881.3 4 7.3 odd 6
1764.2.f.a.881.4 4 21.11 odd 6
1764.2.t.a.521.1 4 7.6 odd 2
1764.2.t.a.521.2 4 21.20 even 2
1764.2.t.a.1097.1 4 21.2 odd 6
1764.2.t.a.1097.2 4 7.2 even 3
2268.2.w.h.269.1 4 9.5 odd 6
2268.2.w.h.269.2 4 9.4 even 3
2268.2.w.h.1349.1 4 63.61 odd 6
2268.2.w.h.1349.2 4 63.47 even 6
2268.2.bm.g.593.1 4 63.5 even 6
2268.2.bm.g.593.2 4 63.40 odd 6
2268.2.bm.g.1025.1 4 9.7 even 3
2268.2.bm.g.1025.2 4 9.2 odd 6
6300.2.ch.a.1601.1 4 35.19 odd 6
6300.2.ch.a.1601.2 4 105.89 even 6
6300.2.ch.a.4301.1 4 15.14 odd 2
6300.2.ch.a.4301.2 4 5.4 even 2
6300.2.dd.a.1349.1 8 35.12 even 12
6300.2.dd.a.1349.2 8 105.47 odd 12
6300.2.dd.a.1349.3 8 35.33 even 12
6300.2.dd.a.1349.4 8 105.68 odd 12
6300.2.dd.a.4049.1 8 15.8 even 4
6300.2.dd.a.4049.2 8 5.3 odd 4
6300.2.dd.a.4049.3 8 15.2 even 4
6300.2.dd.a.4049.4 8 5.2 odd 4
7056.2.k.a.881.1 4 84.59 odd 6
7056.2.k.a.881.2 4 28.11 odd 6
7056.2.k.a.881.3 4 84.11 even 6
7056.2.k.a.881.4 4 28.3 even 6