Properties

Label 2352.2.b.j.1567.3
Level $2352$
Weight $2$
Character 2352.1567
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,2,Mod(1567,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.b (of order \(2\), degree \(1\), 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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(0.765367i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1567
Dual form 2352.2.b.j.1567.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.765367i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.765367i q^{5} +1.00000 q^{9} +2.16478i q^{11} +0.317025i q^{13} +0.765367i q^{15} -3.37849i q^{17} +5.65685 q^{19} -5.22625i q^{23} +4.41421 q^{25} +1.00000 q^{27} -2.58579 q^{29} +1.65685 q^{31} +2.16478i q^{33} -1.41421 q^{37} +0.317025i q^{39} +5.99162i q^{41} +7.39104i q^{43} +0.765367i q^{45} +9.65685 q^{47} -3.37849i q^{51} +1.65685 q^{53} -1.65685 q^{55} +5.65685 q^{57} +5.65685 q^{59} +7.07401i q^{61} -0.242641 q^{65} +11.7206i q^{67} -5.22625i q^{69} +15.6788i q^{71} +6.62567i q^{73} +4.41421 q^{75} -13.5140i q^{79} +1.00000 q^{81} +6.34315 q^{83} +2.58579 q^{85} -2.58579 q^{87} -11.4036i q^{89} +1.65685 q^{93} +4.32957i q^{95} -3.82683i q^{97} +2.16478i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 12 q^{25} + 4 q^{27} - 16 q^{29} - 16 q^{31} + 16 q^{47} - 16 q^{53} + 16 q^{55} + 16 q^{65} + 12 q^{75} + 4 q^{81} + 48 q^{83} + 16 q^{85} - 16 q^{87} - 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.765367i 0.342282i 0.985247 + 0.171141i \(0.0547454\pi\)
−0.985247 + 0.171141i \(0.945255\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.16478i 0.652707i 0.945248 + 0.326354i \(0.105820\pi\)
−0.945248 + 0.326354i \(0.894180\pi\)
\(12\) 0 0
\(13\) 0.317025i 0.0879270i 0.999033 + 0.0439635i \(0.0139985\pi\)
−0.999033 + 0.0439635i \(0.986001\pi\)
\(14\) 0 0
\(15\) 0.765367i 0.197617i
\(16\) 0 0
\(17\) − 3.37849i − 0.819405i −0.912219 0.409702i \(-0.865633\pi\)
0.912219 0.409702i \(-0.134367\pi\)
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 5.22625i − 1.08975i −0.838518 0.544874i \(-0.816577\pi\)
0.838518 0.544874i \(-0.183423\pi\)
\(24\) 0 0
\(25\) 4.41421 0.882843
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.58579 −0.480168 −0.240084 0.970752i \(-0.577175\pi\)
−0.240084 + 0.970752i \(0.577175\pi\)
\(30\) 0 0
\(31\) 1.65685 0.297580 0.148790 0.988869i \(-0.452462\pi\)
0.148790 + 0.988869i \(0.452462\pi\)
\(32\) 0 0
\(33\) 2.16478i 0.376841i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.41421 −0.232495 −0.116248 0.993220i \(-0.537087\pi\)
−0.116248 + 0.993220i \(0.537087\pi\)
\(38\) 0 0
\(39\) 0.317025i 0.0507647i
\(40\) 0 0
\(41\) 5.99162i 0.935734i 0.883799 + 0.467867i \(0.154977\pi\)
−0.883799 + 0.467867i \(0.845023\pi\)
\(42\) 0 0
\(43\) 7.39104i 1.12712i 0.826074 + 0.563561i \(0.190569\pi\)
−0.826074 + 0.563561i \(0.809431\pi\)
\(44\) 0 0
\(45\) 0.765367i 0.114094i
\(46\) 0 0
\(47\) 9.65685 1.40860 0.704298 0.709904i \(-0.251262\pi\)
0.704298 + 0.709904i \(0.251262\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 3.37849i − 0.473084i
\(52\) 0 0
\(53\) 1.65685 0.227586 0.113793 0.993504i \(-0.463700\pi\)
0.113793 + 0.993504i \(0.463700\pi\)
\(54\) 0 0
\(55\) −1.65685 −0.223410
\(56\) 0 0
\(57\) 5.65685 0.749269
\(58\) 0 0
\(59\) 5.65685 0.736460 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(60\) 0 0
\(61\) 7.07401i 0.905734i 0.891578 + 0.452867i \(0.149599\pi\)
−0.891578 + 0.452867i \(0.850401\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.242641 −0.0300959
\(66\) 0 0
\(67\) 11.7206i 1.43190i 0.698152 + 0.715950i \(0.254006\pi\)
−0.698152 + 0.715950i \(0.745994\pi\)
\(68\) 0 0
\(69\) − 5.22625i − 0.629167i
\(70\) 0 0
\(71\) 15.6788i 1.86073i 0.366640 + 0.930363i \(0.380508\pi\)
−0.366640 + 0.930363i \(0.619492\pi\)
\(72\) 0 0
\(73\) 6.62567i 0.775476i 0.921770 + 0.387738i \(0.126743\pi\)
−0.921770 + 0.387738i \(0.873257\pi\)
\(74\) 0 0
\(75\) 4.41421 0.509709
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 13.5140i − 1.52044i −0.649665 0.760220i \(-0.725091\pi\)
0.649665 0.760220i \(-0.274909\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.34315 0.696251 0.348125 0.937448i \(-0.386818\pi\)
0.348125 + 0.937448i \(0.386818\pi\)
\(84\) 0 0
\(85\) 2.58579 0.280468
\(86\) 0 0
\(87\) −2.58579 −0.277225
\(88\) 0 0
\(89\) − 11.4036i − 1.20878i −0.796690 0.604389i \(-0.793417\pi\)
0.796690 0.604389i \(-0.206583\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.65685 0.171808
\(94\) 0 0
\(95\) 4.32957i 0.444204i
\(96\) 0 0
\(97\) − 3.82683i − 0.388556i −0.980946 0.194278i \(-0.937764\pi\)
0.980946 0.194278i \(-0.0622364\pi\)
\(98\) 0 0
\(99\) 2.16478i 0.217569i
\(100\) 0 0
\(101\) − 16.6298i − 1.65473i −0.561665 0.827365i \(-0.689839\pi\)
0.561665 0.827365i \(-0.310161\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 3.95815i − 0.382649i −0.981527 0.191324i \(-0.938722\pi\)
0.981527 0.191324i \(-0.0612783\pi\)
\(108\) 0 0
\(109\) −4.72792 −0.452853 −0.226426 0.974028i \(-0.572704\pi\)
−0.226426 + 0.974028i \(0.572704\pi\)
\(110\) 0 0
\(111\) −1.41421 −0.134231
\(112\) 0 0
\(113\) −16.9706 −1.59646 −0.798228 0.602355i \(-0.794229\pi\)
−0.798228 + 0.602355i \(0.794229\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 0.317025i 0.0293090i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.31371 0.573973
\(122\) 0 0
\(123\) 5.99162i 0.540246i
\(124\) 0 0
\(125\) 7.20533i 0.644464i
\(126\) 0 0
\(127\) 13.5140i 1.19917i 0.800311 + 0.599586i \(0.204668\pi\)
−0.800311 + 0.599586i \(0.795332\pi\)
\(128\) 0 0
\(129\) 7.39104i 0.650744i
\(130\) 0 0
\(131\) 15.3137 1.33796 0.668982 0.743278i \(-0.266730\pi\)
0.668982 + 0.743278i \(0.266730\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.765367i 0.0658723i
\(136\) 0 0
\(137\) −13.4142 −1.14605 −0.573027 0.819537i \(-0.694231\pi\)
−0.573027 + 0.819537i \(0.694231\pi\)
\(138\) 0 0
\(139\) 17.6569 1.49763 0.748817 0.662776i \(-0.230622\pi\)
0.748817 + 0.662776i \(0.230622\pi\)
\(140\) 0 0
\(141\) 9.65685 0.813254
\(142\) 0 0
\(143\) −0.686292 −0.0573906
\(144\) 0 0
\(145\) − 1.97908i − 0.164353i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.9706 −1.06259 −0.531295 0.847187i \(-0.678294\pi\)
−0.531295 + 0.847187i \(0.678294\pi\)
\(150\) 0 0
\(151\) − 6.12293i − 0.498277i −0.968468 0.249139i \(-0.919852\pi\)
0.968468 0.249139i \(-0.0801475\pi\)
\(152\) 0 0
\(153\) − 3.37849i − 0.273135i
\(154\) 0 0
\(155\) 1.26810i 0.101856i
\(156\) 0 0
\(157\) 14.9134i 1.19022i 0.803645 + 0.595109i \(0.202891\pi\)
−0.803645 + 0.595109i \(0.797109\pi\)
\(158\) 0 0
\(159\) 1.65685 0.131397
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 6.12293i − 0.479585i −0.970824 0.239793i \(-0.922921\pi\)
0.970824 0.239793i \(-0.0770795\pi\)
\(164\) 0 0
\(165\) −1.65685 −0.128986
\(166\) 0 0
\(167\) −9.65685 −0.747270 −0.373635 0.927576i \(-0.621889\pi\)
−0.373635 + 0.927576i \(0.621889\pi\)
\(168\) 0 0
\(169\) 12.8995 0.992269
\(170\) 0 0
\(171\) 5.65685 0.432590
\(172\) 0 0
\(173\) − 9.42450i − 0.716532i −0.933620 0.358266i \(-0.883368\pi\)
0.933620 0.358266i \(-0.116632\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.65685 0.425195
\(178\) 0 0
\(179\) 20.0083i 1.49549i 0.663985 + 0.747746i \(0.268864\pi\)
−0.663985 + 0.747746i \(0.731136\pi\)
\(180\) 0 0
\(181\) − 16.1815i − 1.20276i −0.798963 0.601380i \(-0.794618\pi\)
0.798963 0.601380i \(-0.205382\pi\)
\(182\) 0 0
\(183\) 7.07401i 0.522926i
\(184\) 0 0
\(185\) − 1.08239i − 0.0795791i
\(186\) 0 0
\(187\) 7.31371 0.534831
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 13.8854i − 1.00471i −0.864661 0.502356i \(-0.832467\pi\)
0.864661 0.502356i \(-0.167533\pi\)
\(192\) 0 0
\(193\) 3.31371 0.238526 0.119263 0.992863i \(-0.461947\pi\)
0.119263 + 0.992863i \(0.461947\pi\)
\(194\) 0 0
\(195\) −0.242641 −0.0173759
\(196\) 0 0
\(197\) −25.3137 −1.80353 −0.901764 0.432230i \(-0.857727\pi\)
−0.901764 + 0.432230i \(0.857727\pi\)
\(198\) 0 0
\(199\) 5.65685 0.401004 0.200502 0.979693i \(-0.435743\pi\)
0.200502 + 0.979693i \(0.435743\pi\)
\(200\) 0 0
\(201\) 11.7206i 0.826708i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.58579 −0.320285
\(206\) 0 0
\(207\) − 5.22625i − 0.363250i
\(208\) 0 0
\(209\) 12.2459i 0.847065i
\(210\) 0 0
\(211\) − 7.39104i − 0.508820i −0.967096 0.254410i \(-0.918119\pi\)
0.967096 0.254410i \(-0.0818813\pi\)
\(212\) 0 0
\(213\) 15.6788i 1.07429i
\(214\) 0 0
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.62567i 0.447721i
\(220\) 0 0
\(221\) 1.07107 0.0720478
\(222\) 0 0
\(223\) −19.3137 −1.29334 −0.646671 0.762769i \(-0.723839\pi\)
−0.646671 + 0.762769i \(0.723839\pi\)
\(224\) 0 0
\(225\) 4.41421 0.294281
\(226\) 0 0
\(227\) −3.31371 −0.219939 −0.109969 0.993935i \(-0.535075\pi\)
−0.109969 + 0.993935i \(0.535075\pi\)
\(228\) 0 0
\(229\) − 4.72352i − 0.312139i −0.987746 0.156069i \(-0.950118\pi\)
0.987746 0.156069i \(-0.0498824\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.242641 0.0158959 0.00794796 0.999968i \(-0.497470\pi\)
0.00794796 + 0.999968i \(0.497470\pi\)
\(234\) 0 0
\(235\) 7.39104i 0.482138i
\(236\) 0 0
\(237\) − 13.5140i − 0.877827i
\(238\) 0 0
\(239\) 0.896683i 0.0580016i 0.999579 + 0.0290008i \(0.00923254\pi\)
−0.999579 + 0.0290008i \(0.990767\pi\)
\(240\) 0 0
\(241\) − 3.11586i − 0.200710i −0.994952 0.100355i \(-0.968002\pi\)
0.994952 0.100355i \(-0.0319979\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.79337i 0.114109i
\(248\) 0 0
\(249\) 6.34315 0.401981
\(250\) 0 0
\(251\) 8.97056 0.566217 0.283108 0.959088i \(-0.408634\pi\)
0.283108 + 0.959088i \(0.408634\pi\)
\(252\) 0 0
\(253\) 11.3137 0.711287
\(254\) 0 0
\(255\) 2.58579 0.161928
\(256\) 0 0
\(257\) − 18.9803i − 1.18396i −0.805953 0.591980i \(-0.798346\pi\)
0.805953 0.591980i \(-0.201654\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.58579 −0.160056
\(262\) 0 0
\(263\) − 26.1313i − 1.61132i −0.592377 0.805661i \(-0.701810\pi\)
0.592377 0.805661i \(-0.298190\pi\)
\(264\) 0 0
\(265\) 1.26810i 0.0778988i
\(266\) 0 0
\(267\) − 11.4036i − 0.697888i
\(268\) 0 0
\(269\) − 25.1802i − 1.53526i −0.640892 0.767631i \(-0.721435\pi\)
0.640892 0.767631i \(-0.278565\pi\)
\(270\) 0 0
\(271\) −9.65685 −0.586612 −0.293306 0.956019i \(-0.594755\pi\)
−0.293306 + 0.956019i \(0.594755\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.55582i 0.576238i
\(276\) 0 0
\(277\) −9.31371 −0.559607 −0.279803 0.960057i \(-0.590269\pi\)
−0.279803 + 0.960057i \(0.590269\pi\)
\(278\) 0 0
\(279\) 1.65685 0.0991933
\(280\) 0 0
\(281\) −22.3848 −1.33536 −0.667682 0.744447i \(-0.732713\pi\)
−0.667682 + 0.744447i \(0.732713\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 4.32957i 0.256462i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.58579 0.328576
\(290\) 0 0
\(291\) − 3.82683i − 0.224333i
\(292\) 0 0
\(293\) 31.4119i 1.83510i 0.397617 + 0.917552i \(0.369837\pi\)
−0.397617 + 0.917552i \(0.630163\pi\)
\(294\) 0 0
\(295\) 4.32957i 0.252077i
\(296\) 0 0
\(297\) 2.16478i 0.125614i
\(298\) 0 0
\(299\) 1.65685 0.0958184
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 16.6298i − 0.955359i
\(304\) 0 0
\(305\) −5.41421 −0.310017
\(306\) 0 0
\(307\) 3.31371 0.189123 0.0945617 0.995519i \(-0.469855\pi\)
0.0945617 + 0.995519i \(0.469855\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) 19.8770i 1.12351i 0.827302 + 0.561757i \(0.189875\pi\)
−0.827302 + 0.561757i \(0.810125\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.31371 −0.298448 −0.149224 0.988803i \(-0.547677\pi\)
−0.149224 + 0.988803i \(0.547677\pi\)
\(318\) 0 0
\(319\) − 5.59767i − 0.313409i
\(320\) 0 0
\(321\) − 3.95815i − 0.220922i
\(322\) 0 0
\(323\) − 19.1116i − 1.06340i
\(324\) 0 0
\(325\) 1.39942i 0.0776257i
\(326\) 0 0
\(327\) −4.72792 −0.261455
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 25.2346i 1.38702i 0.720448 + 0.693509i \(0.243936\pi\)
−0.720448 + 0.693509i \(0.756064\pi\)
\(332\) 0 0
\(333\) −1.41421 −0.0774984
\(334\) 0 0
\(335\) −8.97056 −0.490114
\(336\) 0 0
\(337\) 20.7279 1.12912 0.564561 0.825391i \(-0.309046\pi\)
0.564561 + 0.825391i \(0.309046\pi\)
\(338\) 0 0
\(339\) −16.9706 −0.921714
\(340\) 0 0
\(341\) 3.58673i 0.194232i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) − 3.43289i − 0.184287i −0.995746 0.0921435i \(-0.970628\pi\)
0.995746 0.0921435i \(-0.0293718\pi\)
\(348\) 0 0
\(349\) − 20.7737i − 1.11199i −0.831186 0.555995i \(-0.812337\pi\)
0.831186 0.555995i \(-0.187663\pi\)
\(350\) 0 0
\(351\) 0.317025i 0.0169216i
\(352\) 0 0
\(353\) − 14.9134i − 0.793760i −0.917871 0.396880i \(-0.870093\pi\)
0.917871 0.396880i \(-0.129907\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.6788i 0.827493i 0.910392 + 0.413747i \(0.135780\pi\)
−0.910392 + 0.413747i \(0.864220\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 6.31371 0.331384
\(364\) 0 0
\(365\) −5.07107 −0.265432
\(366\) 0 0
\(367\) −21.6569 −1.13048 −0.565239 0.824927i \(-0.691216\pi\)
−0.565239 + 0.824927i \(0.691216\pi\)
\(368\) 0 0
\(369\) 5.99162i 0.311911i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 34.6274 1.79294 0.896470 0.443105i \(-0.146123\pi\)
0.896470 + 0.443105i \(0.146123\pi\)
\(374\) 0 0
\(375\) 7.20533i 0.372081i
\(376\) 0 0
\(377\) − 0.819760i − 0.0422198i
\(378\) 0 0
\(379\) − 33.1509i − 1.70285i −0.524479 0.851423i \(-0.675740\pi\)
0.524479 0.851423i \(-0.324260\pi\)
\(380\) 0 0
\(381\) 13.5140i 0.692342i
\(382\) 0 0
\(383\) −36.2843 −1.85404 −0.927020 0.375012i \(-0.877638\pi\)
−0.927020 + 0.375012i \(0.877638\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.39104i 0.375707i
\(388\) 0 0
\(389\) 20.0416 1.01615 0.508076 0.861313i \(-0.330357\pi\)
0.508076 + 0.861313i \(0.330357\pi\)
\(390\) 0 0
\(391\) −17.6569 −0.892946
\(392\) 0 0
\(393\) 15.3137 0.772474
\(394\) 0 0
\(395\) 10.3431 0.520420
\(396\) 0 0
\(397\) 19.4287i 0.975097i 0.873096 + 0.487548i \(0.162109\pi\)
−0.873096 + 0.487548i \(0.837891\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.8701 −1.14208 −0.571038 0.820924i \(-0.693459\pi\)
−0.571038 + 0.820924i \(0.693459\pi\)
\(402\) 0 0
\(403\) 0.525265i 0.0261653i
\(404\) 0 0
\(405\) 0.765367i 0.0380314i
\(406\) 0 0
\(407\) − 3.06147i − 0.151751i
\(408\) 0 0
\(409\) 7.89377i 0.390322i 0.980771 + 0.195161i \(0.0625229\pi\)
−0.980771 + 0.195161i \(0.937477\pi\)
\(410\) 0 0
\(411\) −13.4142 −0.661674
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.85483i 0.238314i
\(416\) 0 0
\(417\) 17.6569 0.864660
\(418\) 0 0
\(419\) 22.6274 1.10542 0.552711 0.833373i \(-0.313593\pi\)
0.552711 + 0.833373i \(0.313593\pi\)
\(420\) 0 0
\(421\) −6.68629 −0.325870 −0.162935 0.986637i \(-0.552096\pi\)
−0.162935 + 0.986637i \(0.552096\pi\)
\(422\) 0 0
\(423\) 9.65685 0.469532
\(424\) 0 0
\(425\) − 14.9134i − 0.723406i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −0.686292 −0.0331345
\(430\) 0 0
\(431\) − 15.1535i − 0.729918i −0.931024 0.364959i \(-0.881083\pi\)
0.931024 0.364959i \(-0.118917\pi\)
\(432\) 0 0
\(433\) 9.87285i 0.474459i 0.971454 + 0.237229i \(0.0762393\pi\)
−0.971454 + 0.237229i \(0.923761\pi\)
\(434\) 0 0
\(435\) − 1.97908i − 0.0948894i
\(436\) 0 0
\(437\) − 29.5641i − 1.41424i
\(438\) 0 0
\(439\) −22.6274 −1.07995 −0.539974 0.841682i \(-0.681566\pi\)
−0.539974 + 0.841682i \(0.681566\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.1313i 1.24153i 0.783995 + 0.620767i \(0.213179\pi\)
−0.783995 + 0.620767i \(0.786821\pi\)
\(444\) 0 0
\(445\) 8.72792 0.413743
\(446\) 0 0
\(447\) −12.9706 −0.613487
\(448\) 0 0
\(449\) −29.6569 −1.39959 −0.699797 0.714342i \(-0.746726\pi\)
−0.699797 + 0.714342i \(0.746726\pi\)
\(450\) 0 0
\(451\) −12.9706 −0.610760
\(452\) 0 0
\(453\) − 6.12293i − 0.287681i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.3137 −0.903457 −0.451729 0.892155i \(-0.649192\pi\)
−0.451729 + 0.892155i \(0.649192\pi\)
\(458\) 0 0
\(459\) − 3.37849i − 0.157695i
\(460\) 0 0
\(461\) 24.2066i 1.12741i 0.825975 + 0.563706i \(0.190625\pi\)
−0.825975 + 0.563706i \(0.809375\pi\)
\(462\) 0 0
\(463\) 1.79337i 0.0833448i 0.999131 + 0.0416724i \(0.0132686\pi\)
−0.999131 + 0.0416724i \(0.986731\pi\)
\(464\) 0 0
\(465\) 1.26810i 0.0588068i
\(466\) 0 0
\(467\) −19.3137 −0.893732 −0.446866 0.894601i \(-0.647460\pi\)
−0.446866 + 0.894601i \(0.647460\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.9134i 0.687173i
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 24.9706 1.14573
\(476\) 0 0
\(477\) 1.65685 0.0758621
\(478\) 0 0
\(479\) −26.6274 −1.21664 −0.608319 0.793693i \(-0.708156\pi\)
−0.608319 + 0.793693i \(0.708156\pi\)
\(480\) 0 0
\(481\) − 0.448342i − 0.0204426i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.92893 0.132996
\(486\) 0 0
\(487\) − 35.6871i − 1.61714i −0.588403 0.808568i \(-0.700243\pi\)
0.588403 0.808568i \(-0.299757\pi\)
\(488\) 0 0
\(489\) − 6.12293i − 0.276889i
\(490\) 0 0
\(491\) − 14.4107i − 0.650344i −0.945655 0.325172i \(-0.894578\pi\)
0.945655 0.325172i \(-0.105422\pi\)
\(492\) 0 0
\(493\) 8.73606i 0.393452i
\(494\) 0 0
\(495\) −1.65685 −0.0744701
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 17.3183i 0.775272i 0.921812 + 0.387636i \(0.126708\pi\)
−0.921812 + 0.387636i \(0.873292\pi\)
\(500\) 0 0
\(501\) −9.65685 −0.431436
\(502\) 0 0
\(503\) −37.6569 −1.67904 −0.839518 0.543332i \(-0.817163\pi\)
−0.839518 + 0.543332i \(0.817163\pi\)
\(504\) 0 0
\(505\) 12.7279 0.566385
\(506\) 0 0
\(507\) 12.8995 0.572887
\(508\) 0 0
\(509\) 10.5069i 0.465710i 0.972511 + 0.232855i \(0.0748068\pi\)
−0.972511 + 0.232855i \(0.925193\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.65685 0.249756
\(514\) 0 0
\(515\) 9.18440i 0.404713i
\(516\) 0 0
\(517\) 20.9050i 0.919401i
\(518\) 0 0
\(519\) − 9.42450i − 0.413690i
\(520\) 0 0
\(521\) 1.66205i 0.0728157i 0.999337 + 0.0364079i \(0.0115915\pi\)
−0.999337 + 0.0364079i \(0.988408\pi\)
\(522\) 0 0
\(523\) −10.6274 −0.464704 −0.232352 0.972632i \(-0.574642\pi\)
−0.232352 + 0.972632i \(0.574642\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5.59767i − 0.243838i
\(528\) 0 0
\(529\) −4.31371 −0.187553
\(530\) 0 0
\(531\) 5.65685 0.245487
\(532\) 0 0
\(533\) −1.89949 −0.0822763
\(534\) 0 0
\(535\) 3.02944 0.130974
\(536\) 0 0
\(537\) 20.0083i 0.863423i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −12.0000 −0.515920 −0.257960 0.966156i \(-0.583050\pi\)
−0.257960 + 0.966156i \(0.583050\pi\)
\(542\) 0 0
\(543\) − 16.1815i − 0.694414i
\(544\) 0 0
\(545\) − 3.61859i − 0.155004i
\(546\) 0 0
\(547\) − 30.8322i − 1.31829i −0.752015 0.659146i \(-0.770918\pi\)
0.752015 0.659146i \(-0.229082\pi\)
\(548\) 0 0
\(549\) 7.07401i 0.301911i
\(550\) 0 0
\(551\) −14.6274 −0.623149
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 1.08239i − 0.0459450i
\(556\) 0 0
\(557\) 21.3137 0.903091 0.451545 0.892248i \(-0.350873\pi\)
0.451545 + 0.892248i \(0.350873\pi\)
\(558\) 0 0
\(559\) −2.34315 −0.0991045
\(560\) 0 0
\(561\) 7.31371 0.308785
\(562\) 0 0
\(563\) −32.9706 −1.38954 −0.694772 0.719230i \(-0.744495\pi\)
−0.694772 + 0.719230i \(0.744495\pi\)
\(564\) 0 0
\(565\) − 12.9887i − 0.546439i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.58579 0.108402 0.0542009 0.998530i \(-0.482739\pi\)
0.0542009 + 0.998530i \(0.482739\pi\)
\(570\) 0 0
\(571\) − 22.1731i − 0.927916i −0.885857 0.463958i \(-0.846429\pi\)
0.885857 0.463958i \(-0.153571\pi\)
\(572\) 0 0
\(573\) − 13.8854i − 0.580070i
\(574\) 0 0
\(575\) − 23.0698i − 0.962077i
\(576\) 0 0
\(577\) − 12.9343i − 0.538463i −0.963076 0.269231i \(-0.913230\pi\)
0.963076 0.269231i \(-0.0867696\pi\)
\(578\) 0 0
\(579\) 3.31371 0.137713
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.58673i 0.148547i
\(584\) 0 0
\(585\) −0.242641 −0.0100320
\(586\) 0 0
\(587\) 8.97056 0.370255 0.185127 0.982715i \(-0.440730\pi\)
0.185127 + 0.982715i \(0.440730\pi\)
\(588\) 0 0
\(589\) 9.37258 0.386191
\(590\) 0 0
\(591\) −25.3137 −1.04127
\(592\) 0 0
\(593\) − 23.1242i − 0.949596i −0.880095 0.474798i \(-0.842521\pi\)
0.880095 0.474798i \(-0.157479\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.65685 0.231520
\(598\) 0 0
\(599\) − 9.55582i − 0.390440i −0.980759 0.195220i \(-0.937458\pi\)
0.980759 0.195220i \(-0.0625421\pi\)
\(600\) 0 0
\(601\) − 44.1061i − 1.79913i −0.436791 0.899563i \(-0.643885\pi\)
0.436791 0.899563i \(-0.356115\pi\)
\(602\) 0 0
\(603\) 11.7206i 0.477300i
\(604\) 0 0
\(605\) 4.83230i 0.196461i
\(606\) 0 0
\(607\) −43.3137 −1.75805 −0.879025 0.476776i \(-0.841805\pi\)
−0.879025 + 0.476776i \(0.841805\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.06147i 0.123854i
\(612\) 0 0
\(613\) −4.24264 −0.171359 −0.0856793 0.996323i \(-0.527306\pi\)
−0.0856793 + 0.996323i \(0.527306\pi\)
\(614\) 0 0
\(615\) −4.58579 −0.184917
\(616\) 0 0
\(617\) −3.55635 −0.143173 −0.0715866 0.997434i \(-0.522806\pi\)
−0.0715866 + 0.997434i \(0.522806\pi\)
\(618\) 0 0
\(619\) 30.3431 1.21959 0.609797 0.792558i \(-0.291251\pi\)
0.609797 + 0.792558i \(0.291251\pi\)
\(620\) 0 0
\(621\) − 5.22625i − 0.209722i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 16.5563 0.662254
\(626\) 0 0
\(627\) 12.2459i 0.489053i
\(628\) 0 0
\(629\) 4.77791i 0.190508i
\(630\) 0 0
\(631\) 6.12293i 0.243750i 0.992545 + 0.121875i \(0.0388907\pi\)
−0.992545 + 0.121875i \(0.961109\pi\)
\(632\) 0 0
\(633\) − 7.39104i − 0.293767i
\(634\) 0 0
\(635\) −10.3431 −0.410455
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 15.6788i 0.620242i
\(640\) 0 0
\(641\) −38.8701 −1.53527 −0.767637 0.640884i \(-0.778568\pi\)
−0.767637 + 0.640884i \(0.778568\pi\)
\(642\) 0 0
\(643\) −21.6569 −0.854063 −0.427031 0.904237i \(-0.640441\pi\)
−0.427031 + 0.904237i \(0.640441\pi\)
\(644\) 0 0
\(645\) −5.65685 −0.222738
\(646\) 0 0
\(647\) 12.9706 0.509925 0.254963 0.966951i \(-0.417937\pi\)
0.254963 + 0.966951i \(0.417937\pi\)
\(648\) 0 0
\(649\) 12.2459i 0.480692i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.0711 1.37244 0.686218 0.727395i \(-0.259269\pi\)
0.686218 + 0.727395i \(0.259269\pi\)
\(654\) 0 0
\(655\) 11.7206i 0.457962i
\(656\) 0 0
\(657\) 6.62567i 0.258492i
\(658\) 0 0
\(659\) − 33.5223i − 1.30584i −0.757425 0.652922i \(-0.773543\pi\)
0.757425 0.652922i \(-0.226457\pi\)
\(660\) 0 0
\(661\) 33.7624i 1.31321i 0.754237 + 0.656603i \(0.228007\pi\)
−0.754237 + 0.656603i \(0.771993\pi\)
\(662\) 0 0
\(663\) 1.07107 0.0415968
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.5140i 0.523263i
\(668\) 0 0
\(669\) −19.3137 −0.746711
\(670\) 0 0
\(671\) −15.3137 −0.591179
\(672\) 0 0
\(673\) −7.07107 −0.272570 −0.136285 0.990670i \(-0.543516\pi\)
−0.136285 + 0.990670i \(0.543516\pi\)
\(674\) 0 0
\(675\) 4.41421 0.169903
\(676\) 0 0
\(677\) 3.82683i 0.147077i 0.997292 + 0.0735386i \(0.0234292\pi\)
−0.997292 + 0.0735386i \(0.976571\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.31371 −0.126982
\(682\) 0 0
\(683\) − 21.8017i − 0.834219i −0.908856 0.417109i \(-0.863043\pi\)
0.908856 0.417109i \(-0.136957\pi\)
\(684\) 0 0
\(685\) − 10.2668i − 0.392274i
\(686\) 0 0
\(687\) − 4.72352i − 0.180213i
\(688\) 0 0
\(689\) 0.525265i 0.0200110i
\(690\) 0 0
\(691\) −1.65685 −0.0630297 −0.0315149 0.999503i \(-0.510033\pi\)
−0.0315149 + 0.999503i \(0.510033\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.5140i 0.512614i
\(696\) 0 0
\(697\) 20.2426 0.766745
\(698\) 0 0
\(699\) 0.242641 0.00917751
\(700\) 0 0
\(701\) 6.38478 0.241150 0.120575 0.992704i \(-0.461526\pi\)
0.120575 + 0.992704i \(0.461526\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 7.39104i 0.278363i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −33.8995 −1.27312 −0.636561 0.771226i \(-0.719644\pi\)
−0.636561 + 0.771226i \(0.719644\pi\)
\(710\) 0 0
\(711\) − 13.5140i − 0.506814i
\(712\) 0 0
\(713\) − 8.65914i − 0.324287i
\(714\) 0 0
\(715\) − 0.525265i − 0.0196438i
\(716\) 0 0
\(717\) 0.896683i 0.0334872i
\(718\) 0 0
\(719\) 20.2843 0.756476 0.378238 0.925708i \(-0.376530\pi\)
0.378238 + 0.925708i \(0.376530\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 3.11586i − 0.115880i
\(724\) 0 0
\(725\) −11.4142 −0.423913
\(726\) 0 0
\(727\) 12.9706 0.481052 0.240526 0.970643i \(-0.422680\pi\)
0.240526 + 0.970643i \(0.422680\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.9706 0.923570
\(732\) 0 0
\(733\) − 51.1257i − 1.88837i −0.329413 0.944186i \(-0.606851\pi\)
0.329413 0.944186i \(-0.393149\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.3726 −0.934611
\(738\) 0 0
\(739\) 41.8100i 1.53801i 0.639245 + 0.769003i \(0.279247\pi\)
−0.639245 + 0.769003i \(0.720753\pi\)
\(740\) 0 0
\(741\) 1.79337i 0.0658810i
\(742\) 0 0
\(743\) 29.1927i 1.07098i 0.844542 + 0.535489i \(0.179873\pi\)
−0.844542 + 0.535489i \(0.820127\pi\)
\(744\) 0 0
\(745\) − 9.92724i − 0.363706i
\(746\) 0 0
\(747\) 6.34315 0.232084
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 28.2960i − 1.03254i −0.856427 0.516269i \(-0.827321\pi\)
0.856427 0.516269i \(-0.172679\pi\)
\(752\) 0 0
\(753\) 8.97056 0.326905
\(754\) 0 0
\(755\) 4.68629 0.170552
\(756\) 0 0
\(757\) −10.3848 −0.377441 −0.188721 0.982031i \(-0.560434\pi\)
−0.188721 + 0.982031i \(0.560434\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) − 12.5629i − 0.455405i −0.973731 0.227702i \(-0.926879\pi\)
0.973731 0.227702i \(-0.0731213\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.58579 0.0934893
\(766\) 0 0
\(767\) 1.79337i 0.0647547i
\(768\) 0 0
\(769\) 40.5963i 1.46394i 0.681337 + 0.731970i \(0.261399\pi\)
−0.681337 + 0.731970i \(0.738601\pi\)
\(770\) 0 0
\(771\) − 18.9803i − 0.683560i
\(772\) 0 0
\(773\) − 8.68167i − 0.312258i −0.987737 0.156129i \(-0.950098\pi\)
0.987737 0.156129i \(-0.0499015\pi\)
\(774\) 0 0
\(775\) 7.31371 0.262716
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33.8937i 1.21437i
\(780\) 0 0
\(781\) −33.9411 −1.21451
\(782\) 0 0
\(783\) −2.58579 −0.0924085
\(784\) 0 0
\(785\) −11.4142 −0.407391
\(786\) 0 0
\(787\) −23.3137 −0.831044 −0.415522 0.909583i \(-0.636401\pi\)
−0.415522 + 0.909583i \(0.636401\pi\)
\(788\) 0 0
\(789\) − 26.1313i − 0.930297i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.24264 −0.0796385
\(794\) 0 0
\(795\) 1.26810i 0.0449749i
\(796\) 0 0
\(797\) − 2.48181i − 0.0879102i −0.999034 0.0439551i \(-0.986004\pi\)
0.999034 0.0439551i \(-0.0139959\pi\)
\(798\) 0 0
\(799\) − 32.6256i − 1.15421i
\(800\) 0 0
\(801\) − 11.4036i − 0.402926i
\(802\) 0 0
\(803\) −14.3431 −0.506159
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 25.1802i − 0.886384i
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −1.65685 −0.0581800 −0.0290900 0.999577i \(-0.509261\pi\)
−0.0290900 + 0.999577i \(0.509261\pi\)
\(812\) 0 0
\(813\) −9.65685 −0.338681
\(814\) 0 0
\(815\) 4.68629 0.164154
\(816\) 0 0
\(817\) 41.8100i 1.46275i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.3137 −0.883455 −0.441727 0.897149i \(-0.645634\pi\)
−0.441727 + 0.897149i \(0.645634\pi\)
\(822\) 0 0
\(823\) − 34.4190i − 1.19977i −0.800086 0.599885i \(-0.795213\pi\)
0.800086 0.599885i \(-0.204787\pi\)
\(824\) 0 0
\(825\) 9.55582i 0.332691i
\(826\) 0 0
\(827\) 12.6173i 0.438746i 0.975641 + 0.219373i \(0.0704012\pi\)
−0.975641 + 0.219373i \(0.929599\pi\)
\(828\) 0 0
\(829\) 21.7473i 0.755315i 0.925945 + 0.377657i \(0.123270\pi\)
−0.925945 + 0.377657i \(0.876730\pi\)
\(830\) 0 0
\(831\) −9.31371 −0.323089
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 7.39104i − 0.255777i
\(836\) 0 0
\(837\) 1.65685 0.0572693
\(838\) 0 0
\(839\) 29.9411 1.03368 0.516841 0.856081i \(-0.327108\pi\)
0.516841 + 0.856081i \(0.327108\pi\)
\(840\) 0 0
\(841\) −22.3137 −0.769438
\(842\) 0 0
\(843\) −22.3848 −0.770973
\(844\) 0 0
\(845\) 9.87285i 0.339636i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7.39104i 0.253361i
\(852\) 0 0
\(853\) − 5.46635i − 0.187164i −0.995612 0.0935822i \(-0.970168\pi\)
0.995612 0.0935822i \(-0.0298318\pi\)
\(854\) 0 0
\(855\) 4.32957i 0.148068i
\(856\) 0 0
\(857\) 47.6159i 1.62653i 0.581894 + 0.813264i \(0.302312\pi\)
−0.581894 + 0.813264i \(0.697688\pi\)
\(858\) 0 0
\(859\) 38.6274 1.31795 0.658975 0.752165i \(-0.270990\pi\)
0.658975 + 0.752165i \(0.270990\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 31.7289i − 1.08007i −0.841644 0.540033i \(-0.818412\pi\)
0.841644 0.540033i \(-0.181588\pi\)
\(864\) 0 0
\(865\) 7.21320 0.245256
\(866\) 0 0
\(867\) 5.58579 0.189703
\(868\) 0 0
\(869\) 29.2548 0.992402
\(870\) 0 0
\(871\) −3.71573 −0.125903
\(872\) 0 0
\(873\) − 3.82683i − 0.129519i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 36.2426 1.22383 0.611914 0.790925i \(-0.290400\pi\)
0.611914 + 0.790925i \(0.290400\pi\)
\(878\) 0 0
\(879\) 31.4119i 1.05950i
\(880\) 0 0
\(881\) − 50.7862i − 1.71103i −0.517778 0.855515i \(-0.673241\pi\)
0.517778 0.855515i \(-0.326759\pi\)
\(882\) 0 0
\(883\) 16.5754i 0.557808i 0.960319 + 0.278904i \(0.0899711\pi\)
−0.960319 + 0.278904i \(0.910029\pi\)
\(884\) 0 0
\(885\) 4.32957i 0.145537i
\(886\) 0 0
\(887\) 24.6863 0.828885 0.414442 0.910076i \(-0.363977\pi\)
0.414442 + 0.910076i \(0.363977\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.16478i 0.0725230i
\(892\) 0 0
\(893\) 54.6274 1.82804
\(894\) 0 0
\(895\) −15.3137 −0.511881
\(896\) 0 0
\(897\) 1.65685 0.0553208
\(898\) 0 0
\(899\) −4.28427 −0.142888
\(900\) 0 0
\(901\) − 5.59767i − 0.186485i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.3848 0.411684
\(906\) 0 0
\(907\) − 1.79337i − 0.0595477i −0.999557 0.0297739i \(-0.990521\pi\)
0.999557 0.0297739i \(-0.00947872\pi\)
\(908\) 0 0
\(909\) − 16.6298i − 0.551577i
\(910\) 0 0
\(911\) 26.6565i 0.883170i 0.897219 + 0.441585i \(0.145584\pi\)
−0.897219 + 0.441585i \(0.854416\pi\)
\(912\) 0 0
\(913\) 13.7315i 0.454448i
\(914\) 0 0
\(915\) −5.41421 −0.178988
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 29.0389i 0.957904i 0.877841 + 0.478952i \(0.158983\pi\)
−0.877841 + 0.478952i \(0.841017\pi\)
\(920\) 0 0
\(921\) 3.31371 0.109190
\(922\) 0 0
\(923\) −4.97056 −0.163608
\(924\) 0 0
\(925\) −6.24264 −0.205257
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 13.8310i 0.453780i 0.973920 + 0.226890i \(0.0728558\pi\)
−0.973920 + 0.226890i \(0.927144\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 0 0
\(935\) 5.59767i 0.183063i
\(936\) 0 0
\(937\) − 3.82683i − 0.125017i −0.998044 0.0625086i \(-0.980090\pi\)
0.998044 0.0625086i \(-0.0199101\pi\)
\(938\) 0 0
\(939\) 19.8770i 0.648662i
\(940\) 0 0
\(941\) − 11.0322i − 0.359638i −0.983700 0.179819i \(-0.942449\pi\)
0.983700 0.179819i \(-0.0575512\pi\)
\(942\) 0 0
\(943\) 31.3137 1.01971
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 38.3771i − 1.24709i −0.781788 0.623545i \(-0.785692\pi\)
0.781788 0.623545i \(-0.214308\pi\)
\(948\) 0 0
\(949\) −2.10051 −0.0681853
\(950\) 0 0
\(951\) −5.31371 −0.172309
\(952\) 0 0
\(953\) 38.0000 1.23094 0.615470 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) 10.6274 0.343895
\(956\) 0 0
\(957\) − 5.59767i − 0.180947i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28.2548 −0.911446
\(962\) 0 0
\(963\) − 3.95815i − 0.127550i
\(964\) 0 0
\(965\) 2.53620i 0.0816433i
\(966\) 0 0
\(967\) 49.9439i 1.60609i 0.595920 + 0.803044i \(0.296787\pi\)
−0.595920 + 0.803044i \(0.703213\pi\)
\(968\) 0 0
\(969\) − 19.1116i − 0.613954i
\(970\) 0 0
\(971\) 11.0294 0.353951 0.176976 0.984215i \(-0.443369\pi\)
0.176976 + 0.984215i \(0.443369\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 1.39942i 0.0448172i
\(976\) 0 0
\(977\) −42.1838 −1.34958 −0.674789 0.738011i \(-0.735765\pi\)
−0.674789 + 0.738011i \(0.735765\pi\)
\(978\) 0 0
\(979\) 24.6863 0.788977
\(980\) 0 0
\(981\) −4.72792 −0.150951
\(982\) 0 0
\(983\) 23.5980 0.752659 0.376329 0.926486i \(-0.377186\pi\)
0.376329 + 0.926486i \(0.377186\pi\)
\(984\) 0 0
\(985\) − 19.3743i − 0.617316i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 38.6274 1.22828
\(990\) 0 0
\(991\) 15.3073i 0.486254i 0.969995 + 0.243127i \(0.0781731\pi\)
−0.969995 + 0.243127i \(0.921827\pi\)
\(992\) 0 0
\(993\) 25.2346i 0.800795i
\(994\) 0 0
\(995\) 4.32957i 0.137257i
\(996\) 0 0
\(997\) − 48.8071i − 1.54574i −0.634567 0.772868i \(-0.718821\pi\)
0.634567 0.772868i \(-0.281179\pi\)
\(998\) 0 0
\(999\) −1.41421 −0.0447437
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 2352.2.b.j.1567.3 yes 4
3.2 odd 2 7056.2.b.t.1567.2 4
4.3 odd 2 2352.2.b.i.1567.3 yes 4
7.2 even 3 2352.2.bl.p.31.3 8
7.3 odd 6 2352.2.bl.s.607.3 8
7.4 even 3 2352.2.bl.p.607.2 8
7.5 odd 6 2352.2.bl.s.31.2 8
7.6 odd 2 2352.2.b.i.1567.2 4
12.11 even 2 7056.2.b.u.1567.2 4
21.20 even 2 7056.2.b.u.1567.3 4
28.3 even 6 2352.2.bl.p.607.3 8
28.11 odd 6 2352.2.bl.s.607.2 8
28.19 even 6 2352.2.bl.p.31.2 8
28.23 odd 6 2352.2.bl.s.31.3 8
28.27 even 2 inner 2352.2.b.j.1567.2 yes 4
84.83 odd 2 7056.2.b.t.1567.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.i.1567.2 4 7.6 odd 2
2352.2.b.i.1567.3 yes 4 4.3 odd 2
2352.2.b.j.1567.2 yes 4 28.27 even 2 inner
2352.2.b.j.1567.3 yes 4 1.1 even 1 trivial
2352.2.bl.p.31.2 8 28.19 even 6
2352.2.bl.p.31.3 8 7.2 even 3
2352.2.bl.p.607.2 8 7.4 even 3
2352.2.bl.p.607.3 8 28.3 even 6
2352.2.bl.s.31.2 8 7.5 odd 6
2352.2.bl.s.31.3 8 28.23 odd 6
2352.2.bl.s.607.2 8 28.11 odd 6
2352.2.bl.s.607.3 8 7.3 odd 6
7056.2.b.t.1567.2 4 3.2 odd 2
7056.2.b.t.1567.3 4 84.83 odd 2
7056.2.b.u.1567.2 4 12.11 even 2
7056.2.b.u.1567.3 4 21.20 even 2