Properties

Label 360.2.s.a.17.2
Level $360$
Weight $2$
Character 360.17
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(17,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, 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("360.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.s (of order \(4\), 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: \(2.87461447277\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.2
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 360.17
Dual form 360.2.s.a.233.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.12132 + 0.707107i) q^{5} +O(q^{10})\) \(q+(2.12132 + 0.707107i) q^{5} +5.65685i q^{11} +(3.00000 - 3.00000i) q^{13} -4.00000i q^{19} +(2.82843 + 2.82843i) q^{23} +(4.00000 + 3.00000i) q^{25} +1.41421 q^{29} -8.00000 q^{31} +(7.00000 + 7.00000i) q^{37} -1.41421i q^{41} +(4.00000 - 4.00000i) q^{43} +(2.82843 - 2.82843i) q^{47} -7.00000i q^{49} +(-8.48528 - 8.48528i) q^{53} +(-4.00000 + 12.0000i) q^{55} -11.3137 q^{59} -12.0000 q^{61} +(8.48528 - 4.24264i) q^{65} +(-8.00000 - 8.00000i) q^{67} +5.65685i q^{71} +(-3.00000 + 3.00000i) q^{73} +8.00000i q^{79} +(-11.3137 - 11.3137i) q^{83} -7.07107 q^{89} +(2.82843 - 8.48528i) q^{95} +(5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{13} + 16 q^{25} - 32 q^{31} + 28 q^{37} + 16 q^{43} - 16 q^{55} - 48 q^{61} - 32 q^{67} - 12 q^{73} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(-1\)

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
\(5\) 2.12132 + 0.707107i 0.948683 + 0.316228i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 1.70561i 0.522233 + 0.852803i \(0.325099\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 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
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 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
\(23\) 2.82843 + 2.82843i 0.589768 + 0.589768i 0.937568 0.347801i \(-0.113071\pi\)
−0.347801 + 0.937568i \(0.613071\pi\)
\(24\) 0 0
\(25\) 4.00000 + 3.00000i 0.800000 + 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000 + 7.00000i 1.15079 + 1.15079i 0.986394 + 0.164399i \(0.0525685\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 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
\(43\) 4.00000 4.00000i 0.609994 0.609994i −0.332950 0.942944i \(-0.608044\pi\)
0.942944 + 0.332950i \(0.108044\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 2.82843i 0.412568 0.412568i −0.470064 0.882632i \(-0.655769\pi\)
0.882632 + 0.470064i \(0.155769\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.48528 8.48528i −1.16554 1.16554i −0.983243 0.182300i \(-0.941646\pi\)
−0.182300 0.983243i \(-0.558354\pi\)
\(54\) 0 0
\(55\) −4.00000 + 12.0000i −0.539360 + 1.61808i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.48528 4.24264i 1.05247 0.526235i
\(66\) 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 \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) −3.00000 + 3.00000i −0.351123 + 0.351123i −0.860527 0.509404i \(-0.829866\pi\)
0.509404 + 0.860527i \(0.329866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000i 0.900070i 0.893011 + 0.450035i \(0.148589\pi\)
−0.893011 + 0.450035i \(0.851411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.3137 11.3137i −1.24184 1.24184i −0.959237 0.282604i \(-0.908802\pi\)
−0.282604 0.959237i \(-0.591198\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.82843 8.48528i 0.290191 0.870572i
\(96\) 0 0
\(97\) 5.00000 + 5.00000i 0.507673 + 0.507673i 0.913812 0.406138i \(-0.133125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 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
\(103\) 4.00000 4.00000i 0.394132 0.394132i −0.482025 0.876157i \(-0.660099\pi\)
0.876157 + 0.482025i \(0.160099\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3137 11.3137i 1.09374 1.09374i 0.0986115 0.995126i \(-0.468560\pi\)
0.995126 0.0986115i \(-0.0314401\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i −0.981480 0.191565i \(-0.938644\pi\)
0.981480 0.191565i \(-0.0613564\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.24264 4.24264i −0.399114 0.399114i 0.478806 0.877920i \(-0.341070\pi\)
−0.877920 + 0.478806i \(0.841070\pi\)
\(114\) 0 0
\(115\) 4.00000 + 8.00000i 0.373002 + 0.746004i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.36396 + 9.19239i 0.569210 + 0.822192i
\(126\) 0 0
\(127\) 12.0000 + 12.0000i 1.06483 + 1.06483i 0.997748 + 0.0670802i \(0.0213683\pi\)
0.0670802 + 0.997748i \(0.478632\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685i 0.494242i 0.968985 + 0.247121i \(0.0794845\pi\)
−0.968985 + 0.247121i \(0.920516\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.7279 + 12.7279i −1.08742 + 1.08742i −0.0916263 + 0.995793i \(0.529207\pi\)
−0.995793 + 0.0916263i \(0.970793\pi\)
\(138\) 0 0
\(139\) 20.0000i 1.69638i −0.529694 0.848189i \(-0.677693\pi\)
0.529694 0.848189i \(-0.322307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.9706 + 16.9706i 1.41915 + 1.41915i
\(144\) 0 0
\(145\) 3.00000 + 1.00000i 0.249136 + 0.0830455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24264 0.347571 0.173785 0.984784i \(-0.444400\pi\)
0.173785 + 0.984784i \(0.444400\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.9706 5.65685i −1.36311 0.454369i
\(156\) 0 0
\(157\) −7.00000 7.00000i −0.558661 0.558661i 0.370265 0.928926i \(-0.379267\pi\)
−0.928926 + 0.370265i \(0.879267\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 12.0000i 0.939913 0.939913i −0.0583818 0.998294i \(-0.518594\pi\)
0.998294 + 0.0583818i \(0.0185941\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.48528 8.48528i 0.656611 0.656611i −0.297966 0.954577i \(-0.596308\pi\)
0.954577 + 0.297966i \(0.0963081\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.07107 + 7.07107i 0.537603 + 0.537603i 0.922824 0.385221i \(-0.125875\pi\)
−0.385221 + 0.922824i \(0.625875\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.65685 0.422813 0.211407 0.977398i \(-0.432196\pi\)
0.211407 + 0.977398i \(0.432196\pi\)
\(180\) 0 0
\(181\) 4.00000 0.297318 0.148659 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.89949 + 19.7990i 0.727825 + 1.45565i
\(186\) 0 0
\(187\) 0 0
\(188\) 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
\(193\) 5.00000 5.00000i 0.359908 0.359908i −0.503871 0.863779i \(-0.668091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.7279 + 12.7279i −0.906827 + 0.906827i −0.996015 0.0891879i \(-0.971573\pi\)
0.0891879 + 0.996015i \(0.471573\pi\)
\(198\) 0 0
\(199\) 24.0000i 1.70131i 0.525720 + 0.850657i \(0.323796\pi\)
−0.525720 + 0.850657i \(0.676204\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.00000 3.00000i 0.0698430 0.209529i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.6274 1.56517
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.3137 5.65685i 0.771589 0.385794i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 6.00000i 0.396491i −0.980152 0.198246i \(-0.936476\pi\)
0.980152 0.198246i \(-0.0635244\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.65685 5.65685i −0.370593 0.370593i 0.497100 0.867693i \(-0.334398\pi\)
−0.867693 + 0.497100i \(0.834398\pi\)
\(234\) 0 0
\(235\) 8.00000 4.00000i 0.521862 0.260931i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.94975 14.8492i 0.316228 0.948683i
\(246\) 0 0
\(247\) −12.0000 12.0000i −0.763542 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) −16.0000 + 16.0000i −1.00591 + 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.07107 7.07107i 0.441081 0.441081i −0.451294 0.892375i \(-0.649037\pi\)
0.892375 + 0.451294i \(0.149037\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.82843 + 2.82843i 0.174408 + 0.174408i 0.788913 0.614505i \(-0.210644\pi\)
−0.614505 + 0.788913i \(0.710644\pi\)
\(264\) 0 0
\(265\) −12.0000 24.0000i −0.737154 1.47431i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.5563 −0.948487 −0.474244 0.880394i \(-0.657278\pi\)
−0.474244 + 0.880394i \(0.657278\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16.9706 + 22.6274i −1.02336 + 1.36448i
\(276\) 0 0
\(277\) 11.0000 + 11.0000i 0.660926 + 0.660926i 0.955598 0.294672i \(-0.0952105\pi\)
−0.294672 + 0.955598i \(0.595211\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.24264i 0.253095i −0.991961 0.126547i \(-0.959610\pi\)
0.991961 0.126547i \(-0.0403896\pi\)
\(282\) 0 0
\(283\) −16.0000 + 16.0000i −0.951101 + 0.951101i −0.998859 0.0477577i \(-0.984792\pi\)
0.0477577 + 0.998859i \(0.484792\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.41421 + 1.41421i 0.0826192 + 0.0826192i 0.747209 0.664589i \(-0.231394\pi\)
−0.664589 + 0.747209i \(0.731394\pi\)
\(294\) 0 0
\(295\) −24.0000 8.00000i −1.39733 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.9706 0.981433
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −25.4558 8.48528i −1.45760 0.485866i
\(306\) 0 0
\(307\) −20.0000 20.0000i −1.14146 1.14146i −0.988183 0.153277i \(-0.951017\pi\)
−0.153277 0.988183i \(-0.548983\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2843i 1.60385i −0.597422 0.801927i \(-0.703808\pi\)
0.597422 0.801927i \(-0.296192\pi\)
\(312\) 0 0
\(313\) −9.00000 + 9.00000i −0.508710 + 0.508710i −0.914130 0.405420i \(-0.867125\pi\)
0.405420 + 0.914130i \(0.367125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.82843 + 2.82843i −0.158860 + 0.158860i −0.782062 0.623201i \(-0.785832\pi\)
0.623201 + 0.782062i \(0.285832\pi\)
\(318\) 0 0
\(319\) 8.00000i 0.447914i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 21.0000 3.00000i 1.16487 0.166410i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 22.6274i −0.618134 1.23627i
\(336\) 0 0
\(337\) 15.0000 + 15.0000i 0.817102 + 0.817102i 0.985687 0.168585i \(-0.0539198\pi\)
−0.168585 + 0.985687i \(0.553920\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 45.2548i 2.45069i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 4.00000i 0.214115i 0.994253 + 0.107058i \(0.0341429\pi\)
−0.994253 + 0.107058i \(0.965857\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.65685 5.65685i −0.301084 0.301084i 0.540354 0.841438i \(-0.318290\pi\)
−0.841438 + 0.540354i \(0.818290\pi\)
\(354\) 0 0
\(355\) −4.00000 + 12.0000i −0.212298 + 0.636894i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5.65685 −0.298557 −0.149279 0.988795i \(-0.547695\pi\)
−0.149279 + 0.988795i \(0.547695\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.48528 + 4.24264i −0.444140 + 0.222070i
\(366\) 0 0
\(367\) −4.00000 4.00000i −0.208798 0.208798i 0.594958 0.803757i \(-0.297169\pi\)
−0.803757 + 0.594958i \(0.797169\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3.00000 + 3.00000i −0.155334 + 0.155334i −0.780496 0.625161i \(-0.785033\pi\)
0.625161 + 0.780496i \(0.285033\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.24264 4.24264i 0.218507 0.218507i
\(378\) 0 0
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.48528 + 8.48528i 0.433578 + 0.433578i 0.889843 0.456266i \(-0.150813\pi\)
−0.456266 + 0.889843i \(0.650813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.6985 1.50577 0.752886 0.658150i \(-0.228661\pi\)
0.752886 + 0.658150i \(0.228661\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.65685 + 16.9706i −0.284627 + 0.853882i
\(396\) 0 0
\(397\) −19.0000 19.0000i −0.953583 0.953583i 0.0453868 0.998969i \(-0.485548\pi\)
−0.998969 + 0.0453868i \(0.985548\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.8701i 1.34183i −0.741536 0.670913i \(-0.765902\pi\)
0.741536 0.670913i \(-0.234098\pi\)
\(402\) 0 0
\(403\) −24.0000 + 24.0000i −1.19553 + 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −39.5980 + 39.5980i −1.96280 + 1.96280i
\(408\) 0 0
\(409\) 24.0000i 1.18672i 0.804936 + 0.593362i \(0.202200\pi\)
−0.804936 + 0.593362i \(0.797800\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −16.0000 32.0000i −0.785409 1.57082i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.3137 0.552711 0.276355 0.961056i \(-0.410873\pi\)
0.276355 + 0.961056i \(0.410873\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685i 0.272481i −0.990676 0.136241i \(-0.956498\pi\)
0.990676 0.136241i \(-0.0435020\pi\)
\(432\) 0 0
\(433\) 11.0000 11.0000i 0.528626 0.528626i −0.391536 0.920163i \(-0.628056\pi\)
0.920163 + 0.391536i \(0.128056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.3137 11.3137i 0.541208 0.541208i
\(438\) 0 0
\(439\) 8.00000i 0.381819i −0.981608 0.190910i \(-0.938856\pi\)
0.981608 0.190910i \(-0.0611437\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11.3137 11.3137i −0.537531 0.537531i 0.385272 0.922803i \(-0.374107\pi\)
−0.922803 + 0.385272i \(0.874107\pi\)
\(444\) 0 0
\(445\) −15.0000 5.00000i −0.711068 0.237023i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.3848 −0.867631 −0.433816 0.901002i \(-0.642833\pi\)
−0.433816 + 0.901002i \(0.642833\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.0000 23.0000i −1.07589 1.07589i −0.996873 0.0790217i \(-0.974820\pi\)
−0.0790217 0.996873i \(-0.525180\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 32.5269i 1.51493i 0.652876 + 0.757465i \(0.273562\pi\)
−0.652876 + 0.757465i \(0.726438\pi\)
\(462\) 0 0
\(463\) 4.00000 4.00000i 0.185896 0.185896i −0.608023 0.793919i \(-0.708037\pi\)
0.793919 + 0.608023i \(0.208037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.9706 16.9706i 0.785304 0.785304i −0.195416 0.980720i \(-0.562606\pi\)
0.980720 + 0.195416i \(0.0626058\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 22.6274 + 22.6274i 1.04041 + 1.04041i
\(474\) 0 0
\(475\) 12.0000 16.0000i 0.550598 0.734130i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.65685 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.07107 + 14.1421i 0.321081 + 0.642161i
\(486\) 0 0
\(487\) 12.0000 + 12.0000i 0.543772 + 0.543772i 0.924632 0.380861i \(-0.124372\pi\)
−0.380861 + 0.924632i \(0.624372\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.6274i 1.02116i 0.859830 + 0.510581i \(0.170569\pi\)
−0.859830 + 0.510581i \(0.829431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 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
\(503\) 25.4558 + 25.4558i 1.13502 + 1.13502i 0.989330 + 0.145690i \(0.0465401\pi\)
0.145690 + 0.989330i \(0.453460\pi\)
\(504\) 0 0
\(505\) −1.00000 + 3.00000i −0.0444994 + 0.133498i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.7279 −0.564155 −0.282078 0.959392i \(-0.591024\pi\)
−0.282078 + 0.959392i \(0.591024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.3137 5.65685i 0.498542 0.249271i
\(516\) 0 0
\(517\) 16.0000 + 16.0000i 0.703679 + 0.703679i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.2132i 0.929367i 0.885477 + 0.464684i \(0.153832\pi\)
−0.885477 + 0.464684i \(0.846168\pi\)
\(522\) 0 0
\(523\) −24.0000 + 24.0000i −1.04945 + 1.04945i −0.0507346 + 0.998712i \(0.516156\pi\)
−0.998712 + 0.0507346i \(0.983844\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000i 0.304348i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.24264 4.24264i −0.183769 0.183769i
\(534\) 0 0
\(535\) 32.0000 16.0000i 1.38348 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 39.5980 1.70561
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.82843 8.48528i 0.121157 0.363470i
\(546\) 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
\(551\) 5.65685i 0.240990i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.48528 + 8.48528i −0.359533 + 0.359533i −0.863641 0.504108i \(-0.831821\pi\)
0.504108 + 0.863641i \(0.331821\pi\)
\(558\) 0 0
\(559\) 24.0000i 1.01509i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.65685 5.65685i −0.238408 0.238408i 0.577783 0.816191i \(-0.303918\pi\)
−0.816191 + 0.577783i \(0.803918\pi\)
\(564\) 0 0
\(565\) −6.00000 12.0000i −0.252422 0.504844i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.3553 1.48217 0.741086 0.671410i \(-0.234311\pi\)
0.741086 + 0.671410i \(0.234311\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.82843 + 19.7990i 0.117954 + 0.825675i
\(576\) 0 0
\(577\) −9.00000 9.00000i −0.374675 0.374675i 0.494502 0.869177i \(-0.335351\pi\)
−0.869177 + 0.494502i \(0.835351\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 48.0000 48.0000i 1.98796 1.98796i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.65685 5.65685i 0.233483 0.233483i −0.580662 0.814145i \(-0.697206\pi\)
0.814145 + 0.580662i \(0.197206\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.3848 + 18.3848i 0.754972 + 0.754972i 0.975403 0.220430i \(-0.0707462\pi\)
−0.220430 + 0.975403i \(0.570746\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −28.2843 −1.15566 −0.577832 0.816156i \(-0.696101\pi\)
−0.577832 + 0.816156i \(0.696101\pi\)
\(600\) 0 0
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −44.5477 14.8492i −1.81112 0.603708i
\(606\) 0 0
\(607\) 8.00000 + 8.00000i 0.324710 + 0.324710i 0.850571 0.525861i \(-0.176257\pi\)
−0.525861 + 0.850571i \(0.676257\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.9706i 0.686555i
\(612\) 0 0
\(613\) 15.0000 15.0000i 0.605844 0.605844i −0.336013 0.941857i \(-0.609079\pi\)
0.941857 + 0.336013i \(0.109079\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −16.9706 + 16.9706i −0.683209 + 0.683209i −0.960722 0.277513i \(-0.910490\pi\)
0.277513 + 0.960722i \(0.410490\pi\)
\(618\) 0 0
\(619\) 12.0000i 0.482321i 0.970485 + 0.241160i \(0.0775280\pi\)
−0.970485 + 0.241160i \(0.922472\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 7.00000 + 24.0000i 0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.9706 + 33.9411i 0.673456 + 1.34691i
\(636\) 0 0
\(637\) −21.0000 21.0000i −0.832050 0.832050i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.8701i 1.06130i 0.847590 + 0.530652i \(0.178053\pi\)
−0.847590 + 0.530652i \(0.821947\pi\)
\(642\) 0 0
\(643\) −28.0000 + 28.0000i −1.10421 + 1.10421i −0.110316 + 0.993897i \(0.535186\pi\)
−0.993897 + 0.110316i \(0.964814\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.7990 + 19.7990i −0.778379 + 0.778379i −0.979555 0.201176i \(-0.935524\pi\)
0.201176 + 0.979555i \(0.435524\pi\)
\(648\) 0 0
\(649\) 64.0000i 2.51222i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.4558 + 25.4558i 0.996164 + 0.996164i 0.999993 0.00382851i \(-0.00121866\pi\)
−0.00382851 + 0.999993i \(0.501219\pi\)
\(654\) 0 0
\(655\) −4.00000 + 12.0000i −0.156293 + 0.468879i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 33.9411 1.32216 0.661079 0.750316i \(-0.270099\pi\)
0.661079 + 0.750316i \(0.270099\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 + 4.00000i 0.154881 + 0.154881i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 67.8823i 2.62057i
\(672\) 0 0
\(673\) −19.0000 + 19.0000i −0.732396 + 0.732396i −0.971094 0.238698i \(-0.923279\pi\)
0.238698 + 0.971094i \(0.423279\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.7990 19.7990i 0.760937 0.760937i −0.215555 0.976492i \(-0.569156\pi\)
0.976492 + 0.215555i \(0.0691560\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.9411 33.9411i −1.29872 1.29872i −0.929237 0.369484i \(-0.879534\pi\)
−0.369484 0.929237i \(-0.620466\pi\)
\(684\) 0 0
\(685\) −36.0000 + 18.0000i −1.37549 + 0.687745i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −50.9117 −1.93958
\(690\) 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
\(695\) 14.1421 42.4264i 0.536442 1.60933i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.5269i 1.22852i 0.789102 + 0.614262i \(0.210546\pi\)
−0.789102 + 0.614262i \(0.789454\pi\)
\(702\) 0 0
\(703\) 28.0000 28.0000i 1.05604 1.05604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000i 0.225335i 0.993633 + 0.112667i \(0.0359394\pi\)
−0.993633 + 0.112667i \(0.964061\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −22.6274 22.6274i −0.847403 0.847403i
\(714\) 0 0
\(715\) 24.0000 + 48.0000i 0.897549 + 1.79510i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −22.6274 −0.843860 −0.421930 0.906628i \(-0.638647\pi\)
−0.421930 + 0.906628i \(0.638647\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.65685 + 4.24264i 0.210090 + 0.157568i
\(726\) 0 0
\(727\) 12.0000 + 12.0000i 0.445055 + 0.445055i 0.893707 0.448651i \(-0.148096\pi\)
−0.448651 + 0.893707i \(0.648096\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 23.0000 23.0000i 0.849524 0.849524i −0.140549 0.990074i \(-0.544887\pi\)
0.990074 + 0.140549i \(0.0448868\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45.2548 45.2548i 1.66698 1.66698i
\(738\) 0 0
\(739\) 4.00000i 0.147142i 0.997290 + 0.0735712i \(0.0234396\pi\)
−0.997290 + 0.0735712i \(0.976560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.4558 25.4558i −0.933884 0.933884i 0.0640616 0.997946i \(-0.479595\pi\)
−0.997946 + 0.0640616i \(0.979595\pi\)
\(744\) 0 0
\(745\) 9.00000 + 3.00000i 0.329734 + 0.109911i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.9706 + 5.65685i 0.617622 + 0.205874i
\(756\) 0 0
\(757\) −3.00000 3.00000i −0.109037 0.109037i 0.650484 0.759520i \(-0.274566\pi\)
−0.759520 + 0.650484i \(0.774566\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.41421i 0.0512652i −0.999671 0.0256326i \(-0.991840\pi\)
0.999671 0.0256326i \(-0.00816000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −33.9411 + 33.9411i −1.22554 + 1.22554i
\(768\) 0 0
\(769\) 40.0000i 1.44244i 0.692708 + 0.721218i \(0.256418\pi\)
−0.692708 + 0.721218i \(0.743582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −14.1421 14.1421i −0.508657 0.508657i 0.405457 0.914114i \(-0.367112\pi\)
−0.914114 + 0.405457i \(0.867112\pi\)
\(774\) 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
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9.89949 19.7990i −0.353328 0.706656i
\(786\) 0 0
\(787\) 24.0000 + 24.0000i 0.855508 + 0.855508i 0.990805 0.135297i \(-0.0431990\pi\)
−0.135297 + 0.990805i \(0.543199\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −36.0000 + 36.0000i −1.27840 + 1.27840i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.6985 29.6985i 1.05197 1.05197i 0.0534012 0.998573i \(-0.482994\pi\)
0.998573 0.0534012i \(-0.0170062\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −16.9706 16.9706i −0.598878 0.598878i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −32.5269 −1.14359 −0.571793 0.820398i \(-0.693752\pi\)
−0.571793 + 0.820398i \(0.693752\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 33.9411 16.9706i 1.18891 0.594453i
\(816\) 0 0
\(817\) −16.0000 16.0000i −0.559769 0.559769i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.3848i 0.641633i 0.947141 + 0.320817i \(0.103957\pi\)
−0.947141 + 0.320817i \(0.896043\pi\)
\(822\) 0 0
\(823\) −32.0000 + 32.0000i −1.11545 + 1.11545i −0.123049 + 0.992401i \(0.539267\pi\)
−0.992401 + 0.123049i \(0.960733\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.9706 16.9706i 0.590124 0.590124i −0.347541 0.937665i \(-0.612983\pi\)
0.937665 + 0.347541i \(0.112983\pi\)
\(828\) 0 0
\(829\) 28.0000i 0.972480i −0.873825 0.486240i \(-0.838368\pi\)
0.873825 0.486240i \(-0.161632\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 24.0000 12.0000i 0.830554 0.415277i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 5.65685 0.195296 0.0976481 0.995221i \(-0.468868\pi\)
0.0976481 + 0.995221i \(0.468868\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.53553 10.6066i 0.121626 0.364878i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 39.5980i 1.35740i
\(852\) 0 0
\(853\) 9.00000 9.00000i 0.308154 0.308154i −0.536039 0.844193i \(-0.680080\pi\)
0.844193 + 0.536039i \(0.180080\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −33.9411 + 33.9411i −1.15941 + 1.15941i −0.174803 + 0.984603i \(0.555929\pi\)
−0.984603 + 0.174803i \(0.944071\pi\)
\(858\) 0 0
\(859\) 28.0000i 0.955348i −0.878537 0.477674i \(-0.841480\pi\)
0.878537 0.477674i \(-0.158520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −25.4558 25.4558i −0.866527 0.866527i 0.125559 0.992086i \(-0.459928\pi\)
−0.992086 + 0.125559i \(0.959928\pi\)
\(864\) 0 0
\(865\) 10.0000 + 20.0000i 0.340010 + 0.680020i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −45.2548 −1.53517
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.0000 + 15.0000i 0.506514 + 0.506514i 0.913455 0.406941i \(-0.133404\pi\)
−0.406941 + 0.913455i \(0.633404\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.41421i 0.0476461i 0.999716 + 0.0238230i \(0.00758382\pi\)
−0.999716 + 0.0238230i \(0.992416\pi\)
\(882\) 0 0
\(883\) 12.0000 12.0000i 0.403832 0.403832i −0.475749 0.879581i \(-0.657823\pi\)
0.879581 + 0.475749i \(0.157823\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.1127 31.1127i 1.04466 1.04466i 0.0457073 0.998955i \(-0.485446\pi\)
0.998955 0.0457073i \(-0.0145542\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.3137 11.3137i −0.378599 0.378599i
\(894\) 0 0
\(895\) 12.0000 + 4.00000i 0.401116 + 0.133705i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.48528 + 2.82843i 0.282060 + 0.0940201i
\(906\) 0 0
\(907\) −8.00000 8.00000i −0.265636 0.265636i 0.561703 0.827339i \(-0.310146\pi\)
−0.827339 + 0.561703i \(0.810146\pi\)
\(908\) 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
\(913\) 64.0000 64.0000i 2.11809 2.11809i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 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
\(925\) 7.00000 + 49.0000i 0.230159 + 1.61111i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 15.5563 0.510387 0.255194 0.966890i \(-0.417861\pi\)
0.255194 + 0.966890i \(0.417861\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.00000 + 5.00000i 0.163343 + 0.163343i 0.784046 0.620703i \(-0.213153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.7279i 0.414918i 0.978244 + 0.207459i \(0.0665194\pi\)
−0.978244 + 0.207459i \(0.933481\pi\)
\(942\) 0 0
\(943\) 4.00000 4.00000i 0.130258 0.130258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.3137 + 11.3137i −0.367646 + 0.367646i −0.866618 0.498972i \(-0.833711\pi\)
0.498972 + 0.866618i \(0.333711\pi\)
\(948\) 0 0
\(949\) 18.0000i 0.584305i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21.2132 + 21.2132i 0.687163 + 0.687163i 0.961604 0.274441i \(-0.0884928\pi\)
−0.274441 + 0.961604i \(0.588493\pi\)
\(954\) 0 0
\(955\) −8.00000 + 24.0000i −0.258874 + 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14.1421 7.07107i 0.455251 0.227626i
\(966\) 0 0
\(967\) 24.0000 + 24.0000i 0.771788 + 0.771788i 0.978419 0.206631i \(-0.0662500\pi\)
−0.206631 + 0.978419i \(0.566250\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.6274i 0.726148i −0.931760 0.363074i \(-0.881727\pi\)
0.931760 0.363074i \(-0.118273\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.9411 33.9411i 1.08587 1.08587i 0.0899242 0.995949i \(-0.471338\pi\)
0.995949 0.0899242i \(-0.0286625\pi\)
\(978\) 0 0
\(979\) 40.0000i 1.27841i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 25.4558 + 25.4558i 0.811915 + 0.811915i 0.984921 0.173006i \(-0.0553478\pi\)
−0.173006 + 0.984921i \(0.555348\pi\)
\(984\) 0 0
\(985\) −36.0000 + 18.0000i −1.14706 + 0.573528i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 22.6274 0.719510
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16.9706 + 50.9117i −0.538003 + 1.61401i
\(996\) 0 0
\(997\) 3.00000 + 3.00000i 0.0950110 + 0.0950110i 0.753015 0.658004i \(-0.228599\pi\)
−0.658004 + 0.753015i \(0.728599\pi\)
\(998\) 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 360.2.s.a.17.2 yes 4
3.2 odd 2 inner 360.2.s.a.17.1 4
4.3 odd 2 720.2.w.c.17.2 4
5.2 odd 4 1800.2.s.b.593.2 4
5.3 odd 4 inner 360.2.s.a.233.1 yes 4
5.4 even 2 1800.2.s.b.1457.2 4
8.3 odd 2 2880.2.w.f.2177.1 4
8.5 even 2 2880.2.w.e.2177.1 4
12.11 even 2 720.2.w.c.17.1 4
15.2 even 4 1800.2.s.b.593.1 4
15.8 even 4 inner 360.2.s.a.233.2 yes 4
15.14 odd 2 1800.2.s.b.1457.1 4
20.3 even 4 720.2.w.c.593.1 4
20.7 even 4 3600.2.w.d.593.1 4
20.19 odd 2 3600.2.w.d.1457.1 4
24.5 odd 2 2880.2.w.e.2177.2 4
24.11 even 2 2880.2.w.f.2177.2 4
40.3 even 4 2880.2.w.f.2753.2 4
40.13 odd 4 2880.2.w.e.2753.2 4
60.23 odd 4 720.2.w.c.593.2 4
60.47 odd 4 3600.2.w.d.593.2 4
60.59 even 2 3600.2.w.d.1457.2 4
120.53 even 4 2880.2.w.e.2753.1 4
120.83 odd 4 2880.2.w.f.2753.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.s.a.17.1 4 3.2 odd 2 inner
360.2.s.a.17.2 yes 4 1.1 even 1 trivial
360.2.s.a.233.1 yes 4 5.3 odd 4 inner
360.2.s.a.233.2 yes 4 15.8 even 4 inner
720.2.w.c.17.1 4 12.11 even 2
720.2.w.c.17.2 4 4.3 odd 2
720.2.w.c.593.1 4 20.3 even 4
720.2.w.c.593.2 4 60.23 odd 4
1800.2.s.b.593.1 4 15.2 even 4
1800.2.s.b.593.2 4 5.2 odd 4
1800.2.s.b.1457.1 4 15.14 odd 2
1800.2.s.b.1457.2 4 5.4 even 2
2880.2.w.e.2177.1 4 8.5 even 2
2880.2.w.e.2177.2 4 24.5 odd 2
2880.2.w.e.2753.1 4 120.53 even 4
2880.2.w.e.2753.2 4 40.13 odd 4
2880.2.w.f.2177.1 4 8.3 odd 2
2880.2.w.f.2177.2 4 24.11 even 2
2880.2.w.f.2753.1 4 120.83 odd 4
2880.2.w.f.2753.2 4 40.3 even 4
3600.2.w.d.593.1 4 20.7 even 4
3600.2.w.d.593.2 4 60.47 odd 4
3600.2.w.d.1457.1 4 20.19 odd 2
3600.2.w.d.1457.2 4 60.59 even 2