Properties

Label 720.2.w.b.17.1
Level $720$
Weight $2$
Character 720.17
Analytic conductor $5.749$
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] = [720,2,Mod(17,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.w (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 720.17
Dual form 720.2.w.b.593.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 2.12132i) q^{5} +(-2.00000 - 2.00000i) q^{7} +O(q^{10})\) \(q+(-0.707107 + 2.12132i) q^{5} +(-2.00000 - 2.00000i) q^{7} -2.82843i q^{11} +(3.00000 - 3.00000i) q^{13} +(1.41421 - 1.41421i) q^{17} -4.00000i q^{19} +(5.65685 + 5.65685i) q^{23} +(-4.00000 - 3.00000i) q^{25} +9.89949 q^{29} +8.00000 q^{31} +(5.65685 - 2.82843i) q^{35} +(-3.00000 - 3.00000i) q^{37} -1.41421i q^{41} +(-8.48528 + 8.48528i) q^{47} +1.00000i q^{49} +(-7.07107 - 7.07107i) q^{53} +(6.00000 + 2.00000i) q^{55} +2.82843 q^{59} +(4.24264 + 8.48528i) q^{65} +(-8.00000 - 8.00000i) q^{67} -5.65685i q^{71} +(7.00000 - 7.00000i) q^{73} +(-5.65685 + 5.65685i) q^{77} +(-8.48528 - 8.48528i) q^{83} +(2.00000 + 4.00000i) q^{85} -1.41421 q^{89} -12.0000 q^{91} +(8.48528 + 2.82843i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} + 12 q^{13} - 16 q^{25} + 32 q^{31} - 12 q^{37} + 24 q^{55} - 32 q^{67} + 28 q^{73} + 8 q^{85} - 48 q^{91} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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
\(5\) −0.707107 + 2.12132i −0.316228 + 0.948683i
\(6\) 0 0
\(7\) −2.00000 2.00000i −0.755929 0.755929i 0.219650 0.975579i \(-0.429509\pi\)
−0.975579 + 0.219650i \(0.929509\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843i 0.852803i −0.904534 0.426401i \(-0.859781\pi\)
0.904534 0.426401i \(-0.140219\pi\)
\(12\) 0 0
\(13\) 3.00000 3.00000i 0.832050 0.832050i −0.155747 0.987797i \(-0.549778\pi\)
0.987797 + 0.155747i \(0.0497784\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421 1.41421i 0.342997 0.342997i −0.514496 0.857493i \(-0.672021\pi\)
0.857493 + 0.514496i \(0.172021\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.65685 + 5.65685i 1.17954 + 1.17954i 0.979863 + 0.199673i \(0.0639880\pi\)
0.199673 + 0.979863i \(0.436012\pi\)
\(24\) 0 0
\(25\) −4.00000 3.00000i −0.800000 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.89949 1.83829 0.919145 0.393919i \(-0.128881\pi\)
0.919145 + 0.393919i \(0.128881\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.65685 2.82843i 0.956183 0.478091i
\(36\) 0 0
\(37\) −3.00000 3.00000i −0.493197 0.493197i 0.416115 0.909312i \(-0.363391\pi\)
−0.909312 + 0.416115i \(0.863391\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.41421i 0.220863i −0.993884 0.110432i \(-0.964777\pi\)
0.993884 0.110432i \(-0.0352233\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.48528 + 8.48528i −1.23771 + 1.23771i −0.276769 + 0.960936i \(0.589264\pi\)
−0.960936 + 0.276769i \(0.910736\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.07107 7.07107i −0.971286 0.971286i 0.0283132 0.999599i \(-0.490986\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 0 0
\(55\) 6.00000 + 2.00000i 0.809040 + 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.82843 0.368230 0.184115 0.982905i \(-0.441058\pi\)
0.184115 + 0.982905i \(0.441058\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.24264 + 8.48528i 0.526235 + 1.05247i
\(66\) 0 0
\(67\) −8.00000 8.00000i −0.977356 0.977356i 0.0223937 0.999749i \(-0.492871\pi\)
−0.999749 + 0.0223937i \(0.992871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685i 0.671345i −0.941979 0.335673i \(-0.891036\pi\)
0.941979 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) 7.00000 7.00000i 0.819288 0.819288i −0.166717 0.986005i \(-0.553317\pi\)
0.986005 + 0.166717i \(0.0533166\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.65685 + 5.65685i −0.644658 + 0.644658i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −8.48528 8.48528i −0.931381 0.931381i 0.0664117 0.997792i \(-0.478845\pi\)
−0.997792 + 0.0664117i \(0.978845\pi\)
\(84\) 0 0
\(85\) 2.00000 + 4.00000i 0.216930 + 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.41421 −0.149906 −0.0749532 0.997187i \(-0.523881\pi\)
−0.0749532 + 0.997187i \(0.523881\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.48528 + 2.82843i 0.870572 + 0.290191i
\(96\) 0 0
\(97\) 3.00000 + 3.00000i 0.304604 + 0.304604i 0.842812 0.538208i \(-0.180899\pi\)
−0.538208 + 0.842812i \(0.680899\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.41421i 0.140720i 0.997522 + 0.0703598i \(0.0224147\pi\)
−0.997522 + 0.0703598i \(0.977585\pi\)
\(102\) 0 0
\(103\) 2.00000 2.00000i 0.197066 0.197066i −0.601675 0.798741i \(-0.705500\pi\)
0.798741 + 0.601675i \(0.205500\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.82843 2.82843i 0.273434 0.273434i −0.557047 0.830481i \(-0.688066\pi\)
0.830481 + 0.557047i \(0.188066\pi\)
\(108\) 0 0
\(109\) 8.00000i 0.766261i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.82843 2.82843i −0.266076 0.266076i 0.561441 0.827517i \(-0.310247\pi\)
−0.827517 + 0.561441i \(0.810247\pi\)
\(114\) 0 0
\(115\) −16.0000 + 8.00000i −1.49201 + 0.746004i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.65685 −0.518563
\(120\) 0 0
\(121\) 3.00000 0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.19239 6.36396i 0.822192 0.569210i
\(126\) 0 0
\(127\) 10.0000 + 10.0000i 0.887357 + 0.887357i 0.994268 0.106912i \(-0.0340963\pi\)
−0.106912 + 0.994268i \(0.534096\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.82843i 0.247121i −0.992337 0.123560i \(-0.960569\pi\)
0.992337 0.123560i \(-0.0394313\pi\)
\(132\) 0 0
\(133\) −8.00000 + 8.00000i −0.693688 + 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.48528 + 8.48528i −0.724947 + 0.724947i −0.969608 0.244662i \(-0.921323\pi\)
0.244662 + 0.969608i \(0.421323\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.48528 8.48528i −0.709575 0.709575i
\(144\) 0 0
\(145\) −7.00000 + 21.0000i −0.581318 + 1.74396i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.7279 −1.04271 −0.521356 0.853339i \(-0.674574\pi\)
−0.521356 + 0.853339i \(0.674574\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.65685 + 16.9706i −0.454369 + 1.36311i
\(156\) 0 0
\(157\) −9.00000 9.00000i −0.718278 0.718278i 0.249974 0.968252i \(-0.419578\pi\)
−0.968252 + 0.249974i \(0.919578\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 22.6274i 1.78329i
\(162\) 0 0
\(163\) 4.00000 4.00000i 0.313304 0.313304i −0.532884 0.846188i \(-0.678892\pi\)
0.846188 + 0.532884i \(0.178892\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.3137 + 11.3137i −0.875481 + 0.875481i −0.993063 0.117582i \(-0.962486\pi\)
0.117582 + 0.993063i \(0.462486\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.82843 + 2.82843i 0.215041 + 0.215041i 0.806405 0.591364i \(-0.201410\pi\)
−0.591364 + 0.806405i \(0.701410\pi\)
\(174\) 0 0
\(175\) 2.00000 + 14.0000i 0.151186 + 1.05830i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.1421 1.05703 0.528516 0.848923i \(-0.322748\pi\)
0.528516 + 0.848923i \(0.322748\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.48528 4.24264i 0.623850 0.311925i
\(186\) 0 0
\(187\) −4.00000 4.00000i −0.292509 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.3137i 0.818631i 0.912393 + 0.409316i \(0.134232\pi\)
−0.912393 + 0.409316i \(0.865768\pi\)
\(192\) 0 0
\(193\) −15.0000 + 15.0000i −1.07972 + 1.07972i −0.0831899 + 0.996534i \(0.526511\pi\)
−0.996534 + 0.0831899i \(0.973489\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.48528 + 8.48528i −0.604551 + 0.604551i −0.941517 0.336966i \(-0.890599\pi\)
0.336966 + 0.941517i \(0.390599\pi\)
\(198\) 0 0
\(199\) 12.0000i 0.850657i 0.905039 + 0.425329i \(0.139842\pi\)
−0.905039 + 0.425329i \(0.860158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −19.7990 19.7990i −1.38962 1.38962i
\(204\) 0 0
\(205\) 3.00000 + 1.00000i 0.209529 + 0.0698430i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.3137 −0.782586
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.0000 16.0000i −1.08615 1.08615i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.48528i 0.570782i
\(222\) 0 0
\(223\) 6.00000 6.00000i 0.401790 0.401790i −0.477074 0.878863i \(-0.658302\pi\)
0.878863 + 0.477074i \(0.158302\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.65685 5.65685i 0.375459 0.375459i −0.494002 0.869461i \(-0.664466\pi\)
0.869461 + 0.494002i \(0.164466\pi\)
\(228\) 0 0
\(229\) 6.00000i 0.396491i 0.980152 + 0.198246i \(0.0635244\pi\)
−0.980152 + 0.198246i \(0.936476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.24264 + 4.24264i 0.277945 + 0.277945i 0.832288 0.554343i \(-0.187031\pi\)
−0.554343 + 0.832288i \(0.687031\pi\)
\(234\) 0 0
\(235\) −12.0000 24.0000i −0.782794 1.56559i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.12132 0.707107i −0.135526 0.0451754i
\(246\) 0 0
\(247\) −12.0000 12.0000i −0.763542 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.4558i 1.60676i −0.595468 0.803379i \(-0.703033\pi\)
0.595468 0.803379i \(-0.296967\pi\)
\(252\) 0 0
\(253\) 16.0000 16.0000i 1.00591 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.1421 + 14.1421i −0.882162 + 0.882162i −0.993754 0.111592i \(-0.964405\pi\)
0.111592 + 0.993754i \(0.464405\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.745644i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 11.3137 + 11.3137i 0.697633 + 0.697633i 0.963899 0.266266i \(-0.0857901\pi\)
−0.266266 + 0.963899i \(0.585790\pi\)
\(264\) 0 0
\(265\) 20.0000 10.0000i 1.22859 0.614295i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.3848 1.12094 0.560470 0.828175i \(-0.310621\pi\)
0.560470 + 0.828175i \(0.310621\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.48528 + 11.3137i −0.511682 + 0.682242i
\(276\) 0 0
\(277\) 15.0000 + 15.0000i 0.901263 + 0.901263i 0.995545 0.0942828i \(-0.0300558\pi\)
−0.0942828 + 0.995545i \(0.530056\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.7279i 0.759284i 0.925133 + 0.379642i \(0.123953\pi\)
−0.925133 + 0.379642i \(0.876047\pi\)
\(282\) 0 0
\(283\) 4.00000 4.00000i 0.237775 0.237775i −0.578153 0.815928i \(-0.696226\pi\)
0.815928 + 0.578153i \(0.196226\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.82843 + 2.82843i −0.166957 + 0.166957i
\(288\) 0 0
\(289\) 13.0000i 0.764706i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.48528 + 8.48528i 0.495715 + 0.495715i 0.910101 0.414386i \(-0.136004\pi\)
−0.414386 + 0.910101i \(0.636004\pi\)
\(294\) 0 0
\(295\) −2.00000 + 6.00000i −0.116445 + 0.349334i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 33.9411 1.96287
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 + 12.0000i 0.684876 + 0.684876i 0.961095 0.276219i \(-0.0890814\pi\)
−0.276219 + 0.961095i \(0.589081\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.65685i 0.320771i −0.987054 0.160385i \(-0.948726\pi\)
0.987054 0.160385i \(-0.0512737\pi\)
\(312\) 0 0
\(313\) 3.00000 3.00000i 0.169570 0.169570i −0.617220 0.786790i \(-0.711741\pi\)
0.786790 + 0.617220i \(0.211741\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.24264 + 4.24264i −0.238290 + 0.238290i −0.816142 0.577851i \(-0.803891\pi\)
0.577851 + 0.816142i \(0.303891\pi\)
\(318\) 0 0
\(319\) 28.0000i 1.56770i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.65685 5.65685i −0.314756 0.314756i
\(324\) 0 0
\(325\) −21.0000 + 3.00000i −1.16487 + 0.166410i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 33.9411 1.87123
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 22.6274 11.3137i 1.23627 0.618134i
\(336\) 0 0
\(337\) −3.00000 3.00000i −0.163420 0.163420i 0.620660 0.784080i \(-0.286865\pi\)
−0.784080 + 0.620660i \(0.786865\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 22.6274i 1.22534i
\(342\) 0 0
\(343\) −12.0000 + 12.0000i −0.647939 + 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.65685 + 5.65685i −0.303676 + 0.303676i −0.842450 0.538774i \(-0.818888\pi\)
0.538774 + 0.842450i \(0.318888\pi\)
\(348\) 0 0
\(349\) 32.0000i 1.71292i −0.516213 0.856460i \(-0.672659\pi\)
0.516213 0.856460i \(-0.327341\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.7279 12.7279i −0.677439 0.677439i 0.281981 0.959420i \(-0.409008\pi\)
−0.959420 + 0.281981i \(0.909008\pi\)
\(354\) 0 0
\(355\) 12.0000 + 4.00000i 0.636894 + 0.212298i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.89949 + 19.7990i 0.518163 + 1.03633i
\(366\) 0 0
\(367\) −14.0000 14.0000i −0.730794 0.730794i 0.239983 0.970777i \(-0.422858\pi\)
−0.970777 + 0.239983i \(0.922858\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 28.2843i 1.46845i
\(372\) 0 0
\(373\) −5.00000 + 5.00000i −0.258890 + 0.258890i −0.824603 0.565712i \(-0.808601\pi\)
0.565712 + 0.824603i \(0.308601\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 29.6985 29.6985i 1.52955 1.52955i
\(378\) 0 0
\(379\) 16.0000i 0.821865i −0.911666 0.410932i \(-0.865203\pi\)
0.911666 0.410932i \(-0.134797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 25.4558 + 25.4558i 1.30073 + 1.30073i 0.927898 + 0.372835i \(0.121614\pi\)
0.372835 + 0.927898i \(0.378386\pi\)
\(384\) 0 0
\(385\) −8.00000 16.0000i −0.407718 0.815436i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −21.2132 −1.07555 −0.537776 0.843088i \(-0.680735\pi\)
−0.537776 + 0.843088i \(0.680735\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −17.0000 17.0000i −0.853206 0.853206i 0.137321 0.990527i \(-0.456151\pi\)
−0.990527 + 0.137321i \(0.956151\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.2132i 1.05934i 0.848205 + 0.529668i \(0.177684\pi\)
−0.848205 + 0.529668i \(0.822316\pi\)
\(402\) 0 0
\(403\) 24.0000 24.0000i 1.19553 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.48528 + 8.48528i −0.420600 + 0.420600i
\(408\) 0 0
\(409\) 24.0000i 1.18672i −0.804936 0.593362i \(-0.797800\pi\)
0.804936 0.593362i \(-0.202200\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.65685 5.65685i −0.278356 0.278356i
\(414\) 0 0
\(415\) 24.0000 12.0000i 1.17811 0.589057i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −25.4558 −1.24360 −0.621800 0.783176i \(-0.713598\pi\)
−0.621800 + 0.783176i \(0.713598\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −9.89949 + 1.41421i −0.480196 + 0.0685994i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.2843i 1.36241i −0.732095 0.681203i \(-0.761457\pi\)
0.732095 0.681203i \(-0.238543\pi\)
\(432\) 0 0
\(433\) 23.0000 23.0000i 1.10531 1.10531i 0.111551 0.993759i \(-0.464418\pi\)
0.993759 0.111551i \(-0.0355818\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.6274 22.6274i 1.08242 1.08242i
\(438\) 0 0
\(439\) 20.0000i 0.954548i −0.878755 0.477274i \(-0.841625\pi\)
0.878755 0.477274i \(-0.158375\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.9706 + 16.9706i 0.806296 + 0.806296i 0.984071 0.177775i \(-0.0568900\pi\)
−0.177775 + 0.984071i \(0.556890\pi\)
\(444\) 0 0
\(445\) 1.00000 3.00000i 0.0474045 0.142214i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.41421 0.0667409 0.0333704 0.999443i \(-0.489376\pi\)
0.0333704 + 0.999443i \(0.489376\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.48528 25.4558i 0.397796 1.19339i
\(456\) 0 0
\(457\) 15.0000 + 15.0000i 0.701670 + 0.701670i 0.964769 0.263099i \(-0.0847444\pi\)
−0.263099 + 0.964769i \(0.584744\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.5563i 0.724531i 0.932075 + 0.362266i \(0.117997\pi\)
−0.932075 + 0.362266i \(0.882003\pi\)
\(462\) 0 0
\(463\) −26.0000 + 26.0000i −1.20832 + 1.20832i −0.236752 + 0.971570i \(0.576083\pi\)
−0.971570 + 0.236752i \(0.923917\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.7990 19.7990i 0.916188 0.916188i −0.0805616 0.996750i \(-0.525671\pi\)
0.996750 + 0.0805616i \(0.0256714\pi\)
\(468\) 0 0
\(469\) 32.0000i 1.47762i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −12.0000 + 16.0000i −0.550598 + 0.734130i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −39.5980 −1.80928 −0.904639 0.426179i \(-0.859859\pi\)
−0.904639 + 0.426179i \(0.859859\pi\)
\(480\) 0 0
\(481\) −18.0000 −0.820729
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.48528 + 4.24264i −0.385297 + 0.192648i
\(486\) 0 0
\(487\) −6.00000 6.00000i −0.271886 0.271886i 0.557973 0.829859i \(-0.311579\pi\)
−0.829859 + 0.557973i \(0.811579\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.7696i 1.65939i 0.558219 + 0.829693i \(0.311485\pi\)
−0.558219 + 0.829693i \(0.688515\pi\)
\(492\) 0 0
\(493\) 14.0000 14.0000i 0.630528 0.630528i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.3137 + 11.3137i −0.507489 + 0.507489i
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.48528 + 8.48528i 0.378340 + 0.378340i 0.870503 0.492163i \(-0.163794\pi\)
−0.492163 + 0.870503i \(0.663794\pi\)
\(504\) 0 0
\(505\) −3.00000 1.00000i −0.133498 0.0444994i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.41421 0.0626839 0.0313420 0.999509i \(-0.490022\pi\)
0.0313420 + 0.999509i \(0.490022\pi\)
\(510\) 0 0
\(511\) −28.0000 −1.23865
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.82843 + 5.65685i 0.124635 + 0.249271i
\(516\) 0 0
\(517\) 24.0000 + 24.0000i 1.05552 + 1.05552i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 29.6985i 1.30111i 0.759457 + 0.650557i \(0.225465\pi\)
−0.759457 + 0.650557i \(0.774535\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.3137 11.3137i 0.492833 0.492833i
\(528\) 0 0
\(529\) 41.0000i 1.78261i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.24264 4.24264i −0.183769 0.183769i
\(534\) 0 0
\(535\) 4.00000 + 8.00000i 0.172935 + 0.345870i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.82843 0.121829
\(540\) 0 0
\(541\) 40.0000 1.71973 0.859867 0.510518i \(-0.170546\pi\)
0.859867 + 0.510518i \(0.170546\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.9706 5.65685i −0.726939 0.242313i
\(546\) 0 0
\(547\) 12.0000 + 12.0000i 0.513083 + 0.513083i 0.915470 0.402387i \(-0.131819\pi\)
−0.402387 + 0.915470i \(0.631819\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 39.5980i 1.68693i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.7279 12.7279i 0.539299 0.539299i −0.384024 0.923323i \(-0.625462\pi\)
0.923323 + 0.384024i \(0.125462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.9706 16.9706i −0.715224 0.715224i 0.252399 0.967623i \(-0.418780\pi\)
−0.967623 + 0.252399i \(0.918780\pi\)
\(564\) 0 0
\(565\) 8.00000 4.00000i 0.336563 0.168281i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.5563 0.652156 0.326078 0.945343i \(-0.394273\pi\)
0.326078 + 0.945343i \(0.394273\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.65685 39.5980i −0.235907 1.65135i
\(576\) 0 0
\(577\) 7.00000 + 7.00000i 0.291414 + 0.291414i 0.837639 0.546225i \(-0.183936\pi\)
−0.546225 + 0.837639i \(0.683936\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 33.9411i 1.40812i
\(582\) 0 0
\(583\) −20.0000 + 20.0000i −0.828315 + 0.828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.48528 + 8.48528i −0.350225 + 0.350225i −0.860193 0.509968i \(-0.829657\pi\)
0.509968 + 0.860193i \(0.329657\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −14.1421 14.1421i −0.580748 0.580748i 0.354361 0.935109i \(-0.384698\pi\)
−0.935109 + 0.354361i \(0.884698\pi\)
\(594\) 0 0
\(595\) 4.00000 12.0000i 0.163984 0.491952i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.2843 1.15566 0.577832 0.816156i \(-0.303899\pi\)
0.577832 + 0.816156i \(0.303899\pi\)
\(600\) 0 0
\(601\) −40.0000 −1.63163 −0.815817 0.578310i \(-0.803712\pi\)
−0.815817 + 0.578310i \(0.803712\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.12132 + 6.36396i −0.0862439 + 0.258732i
\(606\) 0 0
\(607\) −18.0000 18.0000i −0.730597 0.730597i 0.240141 0.970738i \(-0.422806\pi\)
−0.970738 + 0.240141i \(0.922806\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 50.9117i 2.05967i
\(612\) 0 0
\(613\) −27.0000 + 27.0000i −1.09052 + 1.09052i −0.0950469 + 0.995473i \(0.530300\pi\)
−0.995473 + 0.0950469i \(0.969700\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.24264 4.24264i 0.170802 0.170802i −0.616530 0.787332i \(-0.711462\pi\)
0.787332 + 0.616530i \(0.211462\pi\)
\(618\) 0 0
\(619\) 24.0000i 0.964641i 0.875995 + 0.482321i \(0.160206\pi\)
−0.875995 + 0.482321i \(0.839794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.82843 + 2.82843i 0.113319 + 0.113319i
\(624\) 0 0
\(625\) 7.00000 + 24.0000i 0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.48528 −0.338330
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28.2843 + 14.1421i −1.12243 + 0.561214i
\(636\) 0 0
\(637\) 3.00000 + 3.00000i 0.118864 + 0.118864i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 41.0122i 1.61988i −0.586510 0.809942i \(-0.699498\pi\)
0.586510 0.809942i \(-0.300502\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.82843 + 2.82843i −0.111197 + 0.111197i −0.760516 0.649319i \(-0.775054\pi\)
0.649319 + 0.760516i \(0.275054\pi\)
\(648\) 0 0
\(649\) 8.00000i 0.314027i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.24264 4.24264i −0.166027 0.166027i 0.619203 0.785231i \(-0.287456\pi\)
−0.785231 + 0.619203i \(0.787456\pi\)
\(654\) 0 0
\(655\) 6.00000 + 2.00000i 0.234439 + 0.0781465i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.82843 −0.110180 −0.0550899 0.998481i \(-0.517545\pi\)
−0.0550899 + 0.998481i \(0.517545\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −11.3137 22.6274i −0.438727 0.877454i
\(666\) 0 0
\(667\) 56.0000 + 56.0000i 2.16833 + 2.16833i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −23.0000 + 23.0000i −0.886585 + 0.886585i −0.994193 0.107609i \(-0.965681\pi\)
0.107609 + 0.994193i \(0.465681\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.89949 9.89949i 0.380468 0.380468i −0.490802 0.871271i \(-0.663296\pi\)
0.871271 + 0.490802i \(0.163296\pi\)
\(678\) 0 0
\(679\) 12.0000i 0.460518i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −8.48528 8.48528i −0.324680 0.324680i 0.525879 0.850559i \(-0.323736\pi\)
−0.850559 + 0.525879i \(0.823736\pi\)
\(684\) 0 0
\(685\) −12.0000 24.0000i −0.458496 0.916993i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −42.4264 −1.61632
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33.9411 11.3137i −1.28746 0.429153i
\(696\) 0 0
\(697\) −2.00000 2.00000i −0.0757554 0.0757554i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.89949i 0.373899i −0.982370 0.186949i \(-0.940140\pi\)
0.982370 0.186949i \(-0.0598600\pi\)
\(702\) 0 0
\(703\) −12.0000 + 12.0000i −0.452589 + 0.452589i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.82843 2.82843i 0.106374 0.106374i
\(708\) 0 0
\(709\) 10.0000i 0.375558i 0.982211 + 0.187779i \(0.0601289\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 45.2548 + 45.2548i 1.69481 + 1.69481i
\(714\) 0 0
\(715\) 24.0000 12.0000i 0.897549 0.448775i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −39.5980 −1.47676 −0.738378 0.674387i \(-0.764408\pi\)
−0.738378 + 0.674387i \(0.764408\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −39.5980 29.6985i −1.47063 1.10297i
\(726\) 0 0
\(727\) 6.00000 + 6.00000i 0.222528 + 0.222528i 0.809562 0.587034i \(-0.199705\pi\)
−0.587034 + 0.809562i \(0.699705\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −29.0000 + 29.0000i −1.07114 + 1.07114i −0.0738717 + 0.997268i \(0.523536\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −22.6274 + 22.6274i −0.833492 + 0.833492i
\(738\) 0 0
\(739\) 28.0000i 1.03000i 0.857191 + 0.514998i \(0.172207\pi\)
−0.857191 + 0.514998i \(0.827793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.7990 + 19.7990i 0.726354 + 0.726354i 0.969892 0.243537i \(-0.0783078\pi\)
−0.243537 + 0.969892i \(0.578308\pi\)
\(744\) 0 0
\(745\) 9.00000 27.0000i 0.329734 0.989203i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −14.1421 + 42.4264i −0.514685 + 1.54406i
\(756\) 0 0
\(757\) 9.00000 + 9.00000i 0.327111 + 0.327111i 0.851487 0.524376i \(-0.175701\pi\)
−0.524376 + 0.851487i \(0.675701\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 26.8701i 0.974039i −0.873391 0.487019i \(-0.838084\pi\)
0.873391 0.487019i \(-0.161916\pi\)
\(762\) 0 0
\(763\) 16.0000 16.0000i 0.579239 0.579239i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.48528 8.48528i 0.306386 0.306386i
\(768\) 0 0
\(769\) 8.00000i 0.288487i 0.989542 + 0.144244i \(0.0460749\pi\)
−0.989542 + 0.144244i \(0.953925\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.7279 + 12.7279i 0.457792 + 0.457792i 0.897930 0.440138i \(-0.145071\pi\)
−0.440138 + 0.897930i \(0.645071\pi\)
\(774\) 0 0
\(775\) −32.0000 24.0000i −1.14947 0.862105i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.65685 −0.202678
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 25.4558 12.7279i 0.908558 0.454279i
\(786\) 0 0
\(787\) 20.0000 + 20.0000i 0.712923 + 0.712923i 0.967146 0.254223i \(-0.0818196\pi\)
−0.254223 + 0.967146i \(0.581820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.3137i 0.402269i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.48528 + 8.48528i −0.300564 + 0.300564i −0.841235 0.540670i \(-0.818171\pi\)
0.540670 + 0.841235i \(0.318171\pi\)
\(798\) 0 0
\(799\) 24.0000i 0.849059i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.7990 19.7990i −0.698691 0.698691i
\(804\) 0 0
\(805\) 48.0000 + 16.0000i 1.69178 + 0.563926i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −43.8406 −1.54135 −0.770677 0.637226i \(-0.780082\pi\)
−0.770677 + 0.637226i \(0.780082\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 5.65685 + 11.3137i 0.198151 + 0.396302i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.5563i 0.542920i 0.962450 + 0.271460i \(0.0875065\pi\)
−0.962450 + 0.271460i \(0.912493\pi\)
\(822\) 0 0
\(823\) −26.0000 + 26.0000i −0.906303 + 0.906303i −0.995972 0.0896686i \(-0.971419\pi\)
0.0896686 + 0.995972i \(0.471419\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 33.9411 33.9411i 1.18025 1.18025i 0.200569 0.979680i \(-0.435721\pi\)
0.979680 0.200569i \(-0.0642791\pi\)
\(828\) 0 0
\(829\) 8.00000i 0.277851i 0.990303 + 0.138926i \(0.0443649\pi\)
−0.990303 + 0.138926i \(0.955635\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.41421 + 1.41421i 0.0489996 + 0.0489996i
\(834\) 0 0
\(835\) −16.0000 32.0000i −0.553703 1.10741i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.3137 −0.390593 −0.195296 0.980744i \(-0.562567\pi\)
−0.195296 + 0.980744i \(0.562567\pi\)
\(840\) 0 0
\(841\) 69.0000 2.37931
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.6066 + 3.53553i 0.364878 + 0.121626i
\(846\) 0 0
\(847\) −6.00000 6.00000i −0.206162 0.206162i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 33.9411i 1.16349i
\(852\) 0 0
\(853\) −5.00000 + 5.00000i −0.171197 + 0.171197i −0.787505 0.616308i \(-0.788628\pi\)
0.616308 + 0.787505i \(0.288628\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.6985 29.6985i 1.01448 1.01448i 0.0145873 0.999894i \(-0.495357\pi\)
0.999894 0.0145873i \(-0.00464345\pi\)
\(858\) 0 0
\(859\) 8.00000i 0.272956i −0.990643 0.136478i \(-0.956422\pi\)
0.990643 0.136478i \(-0.0435784\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 5.65685 + 5.65685i 0.192562 + 0.192562i 0.796802 0.604240i \(-0.206523\pi\)
−0.604240 + 0.796802i \(0.706523\pi\)
\(864\) 0 0
\(865\) −8.00000 + 4.00000i −0.272008 + 0.136004i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −31.1127 5.65685i −1.05180 0.191237i
\(876\) 0 0
\(877\) −15.0000 15.0000i −0.506514 0.506514i 0.406941 0.913455i \(-0.366596\pi\)
−0.913455 + 0.406941i \(0.866596\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 41.0122i 1.38174i 0.722981 + 0.690868i \(0.242771\pi\)
−0.722981 + 0.690868i \(0.757229\pi\)
\(882\) 0 0
\(883\) 20.0000 20.0000i 0.673054 0.673054i −0.285365 0.958419i \(-0.592115\pi\)
0.958419 + 0.285365i \(0.0921149\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.9411 33.9411i 1.13963 1.13963i 0.151115 0.988516i \(-0.451714\pi\)
0.988516 0.151115i \(-0.0482865\pi\)
\(888\) 0 0
\(889\) 40.0000i 1.34156i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 33.9411 + 33.9411i 1.13580 + 1.13580i
\(894\) 0 0
\(895\) −10.0000 + 30.0000i −0.334263 + 1.00279i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 79.1960 2.64133
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.65685 + 16.9706i −0.188040 + 0.564121i
\(906\) 0 0
\(907\) −16.0000 16.0000i −0.531271 0.531271i 0.389679 0.920951i \(-0.372586\pi\)
−0.920951 + 0.389679i \(0.872586\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.2843i 0.937100i 0.883437 + 0.468550i \(0.155223\pi\)
−0.883437 + 0.468550i \(0.844777\pi\)
\(912\) 0 0
\(913\) −24.0000 + 24.0000i −0.794284 + 0.794284i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.65685 + 5.65685i −0.186806 + 0.186806i
\(918\) 0 0
\(919\) 32.0000i 1.05558i −0.849374 0.527791i \(-0.823020\pi\)
0.849374 0.527791i \(-0.176980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.9706 16.9706i −0.558593 0.558593i
\(924\) 0 0
\(925\) 3.00000 + 21.0000i 0.0986394 + 0.690476i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 35.3553 1.15997 0.579986 0.814627i \(-0.303058\pi\)
0.579986 + 0.814627i \(0.303058\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 11.3137 5.65685i 0.369998 0.184999i
\(936\) 0 0
\(937\) 37.0000 + 37.0000i 1.20874 + 1.20874i 0.971436 + 0.237301i \(0.0762628\pi\)
0.237301 + 0.971436i \(0.423737\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 38.1838i 1.24476i −0.782717 0.622378i \(-0.786167\pi\)
0.782717 0.622378i \(-0.213833\pi\)
\(942\) 0 0
\(943\) 8.00000 8.00000i 0.260516 0.260516i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22.6274 22.6274i 0.735292 0.735292i −0.236371 0.971663i \(-0.575958\pi\)
0.971663 + 0.236371i \(0.0759580\pi\)
\(948\) 0 0
\(949\) 42.0000i 1.36338i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 25.4558 + 25.4558i 0.824596 + 0.824596i 0.986763 0.162168i \(-0.0518485\pi\)
−0.162168 + 0.986763i \(0.551849\pi\)
\(954\) 0 0
\(955\) −24.0000 8.00000i −0.776622 0.258874i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 33.9411 1.09602
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −21.2132 42.4264i −0.682877 1.36575i
\(966\) 0 0
\(967\) 2.00000 + 2.00000i 0.0643157 + 0.0643157i 0.738533 0.674217i \(-0.235519\pi\)
−0.674217 + 0.738533i \(0.735519\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 53.7401i 1.72460i 0.506396 + 0.862301i \(0.330978\pi\)
−0.506396 + 0.862301i \(0.669022\pi\)
\(972\) 0 0
\(973\) 32.0000 32.0000i 1.02587 1.02587i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.7279 12.7279i 0.407202 0.407202i −0.473560 0.880762i \(-0.657031\pi\)
0.880762 + 0.473560i \(0.157031\pi\)
\(978\) 0 0
\(979\) 4.00000i 0.127841i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −14.1421 14.1421i −0.451064 0.451064i 0.444644 0.895708i \(-0.353330\pi\)
−0.895708 + 0.444644i \(0.853330\pi\)
\(984\) 0 0
\(985\) −12.0000 24.0000i −0.382352 0.764704i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −25.4558 8.48528i −0.807005 0.269002i
\(996\) 0 0
\(997\) −11.0000 11.0000i −0.348373 0.348373i 0.511130 0.859503i \(-0.329227\pi\)
−0.859503 + 0.511130i \(0.829227\pi\)
\(998\) 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 720.2.w.b.17.1 4
3.2 odd 2 inner 720.2.w.b.17.2 4
4.3 odd 2 180.2.j.a.17.1 4
5.2 odd 4 3600.2.w.f.593.1 4
5.3 odd 4 inner 720.2.w.b.593.2 4
5.4 even 2 3600.2.w.f.1457.1 4
8.3 odd 2 2880.2.w.j.2177.2 4
8.5 even 2 2880.2.w.a.2177.2 4
12.11 even 2 180.2.j.a.17.2 yes 4
15.2 even 4 3600.2.w.f.593.2 4
15.8 even 4 inner 720.2.w.b.593.1 4
15.14 odd 2 3600.2.w.f.1457.2 4
20.3 even 4 180.2.j.a.53.2 yes 4
20.7 even 4 900.2.j.a.593.2 4
20.19 odd 2 900.2.j.a.557.2 4
24.5 odd 2 2880.2.w.a.2177.1 4
24.11 even 2 2880.2.w.j.2177.1 4
36.7 odd 6 1620.2.x.a.377.1 8
36.11 even 6 1620.2.x.a.377.2 8
36.23 even 6 1620.2.x.a.917.1 8
36.31 odd 6 1620.2.x.a.917.2 8
40.3 even 4 2880.2.w.j.2753.1 4
40.13 odd 4 2880.2.w.a.2753.1 4
60.23 odd 4 180.2.j.a.53.1 yes 4
60.47 odd 4 900.2.j.a.593.1 4
60.59 even 2 900.2.j.a.557.1 4
120.53 even 4 2880.2.w.a.2753.2 4
120.83 odd 4 2880.2.w.j.2753.2 4
180.23 odd 12 1620.2.x.a.593.1 8
180.43 even 12 1620.2.x.a.53.1 8
180.83 odd 12 1620.2.x.a.53.2 8
180.103 even 12 1620.2.x.a.593.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.j.a.17.1 4 4.3 odd 2
180.2.j.a.17.2 yes 4 12.11 even 2
180.2.j.a.53.1 yes 4 60.23 odd 4
180.2.j.a.53.2 yes 4 20.3 even 4
720.2.w.b.17.1 4 1.1 even 1 trivial
720.2.w.b.17.2 4 3.2 odd 2 inner
720.2.w.b.593.1 4 15.8 even 4 inner
720.2.w.b.593.2 4 5.3 odd 4 inner
900.2.j.a.557.1 4 60.59 even 2
900.2.j.a.557.2 4 20.19 odd 2
900.2.j.a.593.1 4 60.47 odd 4
900.2.j.a.593.2 4 20.7 even 4
1620.2.x.a.53.1 8 180.43 even 12
1620.2.x.a.53.2 8 180.83 odd 12
1620.2.x.a.377.1 8 36.7 odd 6
1620.2.x.a.377.2 8 36.11 even 6
1620.2.x.a.593.1 8 180.23 odd 12
1620.2.x.a.593.2 8 180.103 even 12
1620.2.x.a.917.1 8 36.23 even 6
1620.2.x.a.917.2 8 36.31 odd 6
2880.2.w.a.2177.1 4 24.5 odd 2
2880.2.w.a.2177.2 4 8.5 even 2
2880.2.w.a.2753.1 4 40.13 odd 4
2880.2.w.a.2753.2 4 120.53 even 4
2880.2.w.j.2177.1 4 24.11 even 2
2880.2.w.j.2177.2 4 8.3 odd 2
2880.2.w.j.2753.1 4 40.3 even 4
2880.2.w.j.2753.2 4 120.83 odd 4
3600.2.w.f.593.1 4 5.2 odd 4
3600.2.w.f.593.2 4 15.2 even 4
3600.2.w.f.1457.1 4 5.4 even 2
3600.2.w.f.1457.2 4 15.14 odd 2