Properties

Label 3200.2.f.i.449.3
Level $3200$
Weight $2$
Character 3200.449
Analytic conductor $25.552$
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] = [3200,2,Mod(449,3200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3200.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), 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: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3200.449
Dual form 3200.2.f.i.449.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{3} -4.24264i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} -4.24264i q^{7} -1.00000 q^{9} -5.65685i q^{11} -2.00000 q^{13} +6.00000i q^{17} +2.82843i q^{19} -6.00000i q^{21} -7.07107i q^{23} -5.65685 q^{27} +4.00000i q^{29} -2.82843 q^{31} -8.00000i q^{33} +2.00000 q^{37} -2.82843 q^{39} -8.00000 q^{41} +1.41421 q^{43} -1.41421i q^{47} -11.0000 q^{49} +8.48528i q^{51} +2.00000 q^{53} +4.00000i q^{57} +2.82843i q^{59} -14.0000i q^{61} +4.24264i q^{63} +4.24264 q^{67} -10.0000i q^{69} +2.82843 q^{71} +6.00000i q^{73} -24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} -12.7279 q^{83} +5.65685i q^{87} +6.00000 q^{89} +8.48528i q^{91} -4.00000 q^{93} +10.0000i q^{97} +5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 8 q^{13} + 8 q^{37} - 32 q^{41} - 44 q^{49} + 8 q^{53} - 96 q^{77} - 20 q^{81} + 24 q^{89} - 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.24264i − 1.60357i −0.597614 0.801784i \(-0.703885\pi\)
0.597614 0.801784i \(-0.296115\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 5.65685i − 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 2.82843i 0.648886i 0.945905 + 0.324443i \(0.105177\pi\)
−0.945905 + 0.324443i \(0.894823\pi\)
\(20\) 0 0
\(21\) − 6.00000i − 1.30931i
\(22\) 0 0
\(23\) − 7.07107i − 1.47442i −0.675664 0.737210i \(-0.736143\pi\)
0.675664 0.737210i \(-0.263857\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 4.00000i 0.742781i 0.928477 + 0.371391i \(0.121119\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) − 8.00000i − 1.39262i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.82843 −0.452911
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 1.41421 0.215666 0.107833 0.994169i \(-0.465609\pi\)
0.107833 + 0.994169i \(0.465609\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.41421i − 0.206284i −0.994667 0.103142i \(-0.967110\pi\)
0.994667 0.103142i \(-0.0328896\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 8.48528i 1.18818i
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 2.82843i 0.368230i 0.982905 + 0.184115i \(0.0589419\pi\)
−0.982905 + 0.184115i \(0.941058\pi\)
\(60\) 0 0
\(61\) − 14.0000i − 1.79252i −0.443533 0.896258i \(-0.646275\pi\)
0.443533 0.896258i \(-0.353725\pi\)
\(62\) 0 0
\(63\) 4.24264i 0.534522i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) − 10.0000i − 1.20386i
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −24.0000 −2.73505
\(78\) 0 0
\(79\) −16.9706 −1.90934 −0.954669 0.297670i \(-0.903790\pi\)
−0.954669 + 0.297670i \(0.903790\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −12.7279 −1.39707 −0.698535 0.715575i \(-0.746165\pi\)
−0.698535 + 0.715575i \(0.746165\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.65685i 0.606478i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.48528i 0.889499i
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 5.65685i 0.568535i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 9.89949i 0.975426i 0.873004 + 0.487713i \(0.162169\pi\)
−0.873004 + 0.487713i \(0.837831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.07107 −0.683586 −0.341793 0.939775i \(-0.611034\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) − 6.00000i − 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 2.82843 0.268462
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 25.4558 2.33353
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 0 0
\(123\) −11.3137 −1.02012
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 9.89949i − 0.878438i −0.898380 0.439219i \(-0.855255\pi\)
0.898380 0.439219i \(-0.144745\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) − 11.3137i − 0.988483i −0.869325 0.494242i \(-0.835446\pi\)
0.869325 0.494242i \(-0.164554\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 0 0
\(139\) − 8.48528i − 0.719712i −0.933008 0.359856i \(-0.882826\pi\)
0.933008 0.359856i \(-0.117174\pi\)
\(140\) 0 0
\(141\) − 2.00000i − 0.168430i
\(142\) 0 0
\(143\) 11.3137i 0.946100i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −15.5563 −1.28307
\(148\) 0 0
\(149\) − 22.0000i − 1.80231i −0.433497 0.901155i \(-0.642720\pi\)
0.433497 0.901155i \(-0.357280\pi\)
\(150\) 0 0
\(151\) 14.1421 1.15087 0.575435 0.817847i \(-0.304833\pi\)
0.575435 + 0.817847i \(0.304833\pi\)
\(152\) 0 0
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −30.0000 −2.36433
\(162\) 0 0
\(163\) 18.3848 1.44001 0.720003 0.693971i \(-0.244140\pi\)
0.720003 + 0.693971i \(0.244140\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.89949i 0.766046i 0.923739 + 0.383023i \(0.125117\pi\)
−0.923739 + 0.383023i \(0.874883\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) − 2.82843i − 0.216295i
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.00000i 0.300658i
\(178\) 0 0
\(179\) − 19.7990i − 1.47985i −0.672692 0.739923i \(-0.734862\pi\)
0.672692 0.739923i \(-0.265138\pi\)
\(180\) 0 0
\(181\) − 16.0000i − 1.18927i −0.803996 0.594635i \(-0.797296\pi\)
0.803996 0.594635i \(-0.202704\pi\)
\(182\) 0 0
\(183\) − 19.7990i − 1.46358i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 33.9411 2.48202
\(188\) 0 0
\(189\) 24.0000i 1.74574i
\(190\) 0 0
\(191\) 2.82843 0.204658 0.102329 0.994751i \(-0.467371\pi\)
0.102329 + 0.994751i \(0.467371\pi\)
\(192\) 0 0
\(193\) − 6.00000i − 0.431889i −0.976406 0.215945i \(-0.930717\pi\)
0.976406 0.215945i \(-0.0692831\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −11.3137 −0.802008 −0.401004 0.916076i \(-0.631339\pi\)
−0.401004 + 0.916076i \(0.631339\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 0 0
\(203\) 16.9706 1.19110
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.07107i 0.491473i
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) − 16.9706i − 1.16830i −0.811645 0.584151i \(-0.801428\pi\)
0.811645 0.584151i \(-0.198572\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 0 0
\(219\) 8.48528i 0.573382i
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) − 15.5563i − 1.04173i −0.853639 0.520865i \(-0.825609\pi\)
0.853639 0.520865i \(-0.174391\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.07107 0.469323 0.234662 0.972077i \(-0.424602\pi\)
0.234662 + 0.972077i \(0.424602\pi\)
\(228\) 0 0
\(229\) − 20.0000i − 1.32164i −0.750546 0.660819i \(-0.770209\pi\)
0.750546 0.660819i \(-0.229791\pi\)
\(230\) 0 0
\(231\) −33.9411 −2.23316
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −24.0000 −1.55897
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.65685i − 0.359937i
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 22.6274i 1.42823i 0.700028 + 0.714115i \(0.253171\pi\)
−0.700028 + 0.714115i \(0.746829\pi\)
\(252\) 0 0
\(253\) −40.0000 −2.51478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) − 8.48528i − 0.527250i
\(260\) 0 0
\(261\) − 4.00000i − 0.247594i
\(262\) 0 0
\(263\) 7.07107i 0.436021i 0.975946 + 0.218010i \(0.0699567\pi\)
−0.975946 + 0.218010i \(0.930043\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.48528 0.519291
\(268\) 0 0
\(269\) 6.00000i 0.365826i 0.983129 + 0.182913i \(0.0585527\pi\)
−0.983129 + 0.182913i \(0.941447\pi\)
\(270\) 0 0
\(271\) −25.4558 −1.54633 −0.773166 0.634203i \(-0.781328\pi\)
−0.773166 + 0.634203i \(0.781328\pi\)
\(272\) 0 0
\(273\) 12.0000i 0.726273i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 0 0
\(279\) 2.82843 0.169334
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 9.89949 0.588464 0.294232 0.955734i \(-0.404936\pi\)
0.294232 + 0.955734i \(0.404936\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 33.9411i 2.00348i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 14.1421i 0.829027i
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 32.0000i 1.85683i
\(298\) 0 0
\(299\) 14.1421i 0.817861i
\(300\) 0 0
\(301\) − 6.00000i − 0.345834i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 15.5563 0.887848 0.443924 0.896065i \(-0.353586\pi\)
0.443924 + 0.896065i \(0.353586\pi\)
\(308\) 0 0
\(309\) 14.0000i 0.796432i
\(310\) 0 0
\(311\) 2.82843 0.160385 0.0801927 0.996779i \(-0.474446\pi\)
0.0801927 + 0.996779i \(0.474446\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 22.6274 1.26689
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) −16.9706 −0.944267
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 8.48528i − 0.469237i
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) − 22.6274i − 1.24372i −0.783130 0.621858i \(-0.786378\pi\)
0.783130 0.621858i \(-0.213622\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000i 0.980522i 0.871576 + 0.490261i \(0.163099\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 14.1421i 0.768095i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.24264 −0.227757 −0.113878 0.993495i \(-0.536327\pi\)
−0.113878 + 0.993495i \(0.536327\pi\)
\(348\) 0 0
\(349\) 12.0000i 0.642345i 0.947021 + 0.321173i \(0.104077\pi\)
−0.947021 + 0.321173i \(0.895923\pi\)
\(350\) 0 0
\(351\) 11.3137 0.603881
\(352\) 0 0
\(353\) − 10.0000i − 0.532246i −0.963939 0.266123i \(-0.914257\pi\)
0.963939 0.266123i \(-0.0857428\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 36.0000 1.90532
\(358\) 0 0
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 0 0
\(361\) 11.0000 0.578947
\(362\) 0 0
\(363\) −29.6985 −1.55877
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.24264i − 0.221464i −0.993850 0.110732i \(-0.964680\pi\)
0.993850 0.110732i \(-0.0353195\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) − 8.48528i − 0.440534i
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 8.00000i − 0.412021i
\(378\) 0 0
\(379\) − 31.1127i − 1.59815i −0.601230 0.799076i \(-0.705322\pi\)
0.601230 0.799076i \(-0.294678\pi\)
\(380\) 0 0
\(381\) − 14.0000i − 0.717242i
\(382\) 0 0
\(383\) 7.07107i 0.361315i 0.983546 + 0.180657i \(0.0578225\pi\)
−0.983546 + 0.180657i \(0.942177\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.41421 −0.0718885
\(388\) 0 0
\(389\) − 14.0000i − 0.709828i −0.934899 0.354914i \(-0.884510\pi\)
0.934899 0.354914i \(-0.115490\pi\)
\(390\) 0 0
\(391\) 42.4264 2.14560
\(392\) 0 0
\(393\) − 16.0000i − 0.807093i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 16.9706 0.849591
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 5.65685 0.281788
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 11.3137i − 0.560800i
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 2.82843i 0.139516i
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 12.0000i − 0.587643i
\(418\) 0 0
\(419\) 8.48528i 0.414533i 0.978285 + 0.207267i \(0.0664567\pi\)
−0.978285 + 0.207267i \(0.933543\pi\)
\(420\) 0 0
\(421\) − 10.0000i − 0.487370i −0.969854 0.243685i \(-0.921644\pi\)
0.969854 0.243685i \(-0.0783563\pi\)
\(422\) 0 0
\(423\) 1.41421i 0.0687614i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −59.3970 −2.87442
\(428\) 0 0
\(429\) 16.0000i 0.772487i
\(430\) 0 0
\(431\) 31.1127 1.49865 0.749323 0.662205i \(-0.230379\pi\)
0.749323 + 0.662205i \(0.230379\pi\)
\(432\) 0 0
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.0000 0.956730
\(438\) 0 0
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 11.0000 0.523810
\(442\) 0 0
\(443\) 15.5563 0.739104 0.369552 0.929210i \(-0.379511\pi\)
0.369552 + 0.929210i \(0.379511\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 31.1127i − 1.47158i
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 45.2548i 2.13097i
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 34.0000i − 1.59045i −0.606313 0.795226i \(-0.707352\pi\)
0.606313 0.795226i \(-0.292648\pi\)
\(458\) 0 0
\(459\) − 33.9411i − 1.58424i
\(460\) 0 0
\(461\) 24.0000i 1.11779i 0.829238 + 0.558896i \(0.188775\pi\)
−0.829238 + 0.558896i \(0.811225\pi\)
\(462\) 0 0
\(463\) − 7.07107i − 0.328620i −0.986409 0.164310i \(-0.947460\pi\)
0.986409 0.164310i \(-0.0525398\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.89949 −0.458094 −0.229047 0.973415i \(-0.573561\pi\)
−0.229047 + 0.973415i \(0.573561\pi\)
\(468\) 0 0
\(469\) − 18.0000i − 0.831163i
\(470\) 0 0
\(471\) −25.4558 −1.17294
\(472\) 0 0
\(473\) − 8.00000i − 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −33.9411 −1.55081 −0.775405 0.631464i \(-0.782454\pi\)
−0.775405 + 0.631464i \(0.782454\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) −42.4264 −1.93047
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.7279i 0.576757i 0.957516 + 0.288379i \(0.0931162\pi\)
−0.957516 + 0.288379i \(0.906884\pi\)
\(488\) 0 0
\(489\) 26.0000 1.17576
\(490\) 0 0
\(491\) − 16.9706i − 0.765871i −0.923775 0.382935i \(-0.874913\pi\)
0.923775 0.382935i \(-0.125087\pi\)
\(492\) 0 0
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 12.0000i − 0.538274i
\(498\) 0 0
\(499\) 25.4558i 1.13956i 0.821797 + 0.569780i \(0.192972\pi\)
−0.821797 + 0.569780i \(0.807028\pi\)
\(500\) 0 0
\(501\) 14.0000i 0.625474i
\(502\) 0 0
\(503\) 32.5269i 1.45030i 0.688589 + 0.725152i \(0.258230\pi\)
−0.688589 + 0.725152i \(0.741770\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.7279 −0.565267
\(508\) 0 0
\(509\) 20.0000i 0.886484i 0.896402 + 0.443242i \(0.146172\pi\)
−0.896402 + 0.443242i \(0.853828\pi\)
\(510\) 0 0
\(511\) 25.4558 1.12610
\(512\) 0 0
\(513\) − 16.0000i − 0.706417i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 25.4558 1.11739
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 16.9706i − 0.739249i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) 0 0
\(531\) − 2.82843i − 0.122743i
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 28.0000i − 1.20829i
\(538\) 0 0
\(539\) 62.2254i 2.68024i
\(540\) 0 0
\(541\) 40.0000i 1.71973i 0.510518 + 0.859867i \(0.329454\pi\)
−0.510518 + 0.859867i \(0.670546\pi\)
\(542\) 0 0
\(543\) − 22.6274i − 0.971035i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.24264 −0.181402 −0.0907011 0.995878i \(-0.528911\pi\)
−0.0907011 + 0.995878i \(0.528911\pi\)
\(548\) 0 0
\(549\) 14.0000i 0.597505i
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 72.0000i 3.06175i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −2.82843 −0.119630
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) 35.3553 1.49005 0.745025 0.667037i \(-0.232438\pi\)
0.745025 + 0.667037i \(0.232438\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 21.2132i 0.890871i
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) − 22.6274i − 0.946928i −0.880813 0.473464i \(-0.843003\pi\)
0.880813 0.473464i \(-0.156997\pi\)
\(572\) 0 0
\(573\) 4.00000 0.167102
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 10.0000i − 0.416305i −0.978096 0.208153i \(-0.933255\pi\)
0.978096 0.208153i \(-0.0667451\pi\)
\(578\) 0 0
\(579\) − 8.48528i − 0.352636i
\(580\) 0 0
\(581\) 54.0000i 2.24030i
\(582\) 0 0
\(583\) − 11.3137i − 0.468566i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.7279 0.525338 0.262669 0.964886i \(-0.415397\pi\)
0.262669 + 0.964886i \(0.415397\pi\)
\(588\) 0 0
\(589\) − 8.00000i − 0.329634i
\(590\) 0 0
\(591\) −8.48528 −0.349038
\(592\) 0 0
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) −39.5980 −1.61793 −0.808965 0.587857i \(-0.799972\pi\)
−0.808965 + 0.587857i \(0.799972\pi\)
\(600\) 0 0
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) 0 0
\(603\) −4.24264 −0.172774
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 9.89949i 0.401808i 0.979611 + 0.200904i \(0.0643879\pi\)
−0.979611 + 0.200904i \(0.935612\pi\)
\(608\) 0 0
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) 2.82843i 0.114426i
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.00000i − 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 0 0
\(619\) 19.7990i 0.795789i 0.917431 + 0.397894i \(0.130259\pi\)
−0.917431 + 0.397894i \(0.869741\pi\)
\(620\) 0 0
\(621\) 40.0000i 1.60514i
\(622\) 0 0
\(623\) − 25.4558i − 1.01987i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 22.6274 0.903652
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) −36.7696 −1.46377 −0.731886 0.681427i \(-0.761360\pi\)
−0.731886 + 0.681427i \(0.761360\pi\)
\(632\) 0 0
\(633\) − 24.0000i − 0.953914i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22.0000 0.871672
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −7.07107 −0.278856 −0.139428 0.990232i \(-0.544526\pi\)
−0.139428 + 0.990232i \(0.544526\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 32.5269i − 1.27876i −0.768889 0.639382i \(-0.779190\pi\)
0.768889 0.639382i \(-0.220810\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 16.9706i 0.665129i
\(652\) 0 0
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.00000i − 0.234082i
\(658\) 0 0
\(659\) − 25.4558i − 0.991619i −0.868431 0.495809i \(-0.834871\pi\)
0.868431 0.495809i \(-0.165129\pi\)
\(660\) 0 0
\(661\) 10.0000i 0.388955i 0.980907 + 0.194477i \(0.0623011\pi\)
−0.980907 + 0.194477i \(0.937699\pi\)
\(662\) 0 0
\(663\) − 16.9706i − 0.659082i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 28.2843 1.09517
\(668\) 0 0
\(669\) − 22.0000i − 0.850569i
\(670\) 0 0
\(671\) −79.1960 −3.05733
\(672\) 0 0
\(673\) − 50.0000i − 1.92736i −0.267063 0.963679i \(-0.586053\pi\)
0.267063 0.963679i \(-0.413947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.0000 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) 0 0
\(679\) 42.4264 1.62818
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 35.3553 1.35283 0.676417 0.736519i \(-0.263532\pi\)
0.676417 + 0.736519i \(0.263532\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 28.2843i − 1.07911i
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) − 22.6274i − 0.860788i −0.902641 0.430394i \(-0.858375\pi\)
0.902641 0.430394i \(-0.141625\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 48.0000i − 1.81813i
\(698\) 0 0
\(699\) − 8.48528i − 0.320943i
\(700\) 0 0
\(701\) − 18.0000i − 0.679851i −0.940452 0.339925i \(-0.889598\pi\)
0.940452 0.339925i \(-0.110402\pi\)
\(702\) 0 0
\(703\) 5.65685i 0.213352i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 28.0000i − 1.05156i −0.850620 0.525781i \(-0.823773\pi\)
0.850620 0.525781i \(-0.176227\pi\)
\(710\) 0 0
\(711\) 16.9706 0.636446
\(712\) 0 0
\(713\) 20.0000i 0.749006i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 0 0
\(723\) 11.3137 0.420761
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 4.24264i − 0.157351i −0.996900 0.0786754i \(-0.974931\pi\)
0.996900 0.0786754i \(-0.0250691\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 8.48528i 0.313839i
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 24.0000i − 0.884051i
\(738\) 0 0
\(739\) 25.4558i 0.936408i 0.883620 + 0.468204i \(0.155099\pi\)
−0.883620 + 0.468204i \(0.844901\pi\)
\(740\) 0 0
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) − 32.5269i − 1.19330i −0.802503 0.596648i \(-0.796499\pi\)
0.802503 0.596648i \(-0.203501\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.7279 0.465690
\(748\) 0 0
\(749\) 30.0000i 1.09618i
\(750\) 0 0
\(751\) 8.48528 0.309632 0.154816 0.987943i \(-0.450521\pi\)
0.154816 + 0.987943i \(0.450521\pi\)
\(752\) 0 0
\(753\) 32.0000i 1.16614i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 0 0
\(759\) −56.5685 −2.05331
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 0 0
\(763\) −25.4558 −0.921563
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5.65685i − 0.204257i
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) − 25.4558i − 0.916770i
\(772\) 0 0
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 12.0000i − 0.430498i
\(778\) 0 0
\(779\) − 22.6274i − 0.810711i
\(780\) 0 0
\(781\) − 16.0000i − 0.572525i
\(782\) 0 0
\(783\) − 22.6274i − 0.808638i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −46.6690 −1.66357 −0.831786 0.555097i \(-0.812681\pi\)
−0.831786 + 0.555097i \(0.812681\pi\)
\(788\) 0 0
\(789\) 10.0000i 0.356009i
\(790\) 0 0
\(791\) 42.4264 1.50851
\(792\) 0 0
\(793\) 28.0000i 0.994309i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 8.48528 0.300188
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 33.9411 1.19776
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.48528i 0.298696i
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −36.0000 −1.26258
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000i 0.139942i
\(818\) 0 0
\(819\) − 8.48528i − 0.296500i
\(820\) 0 0
\(821\) 26.0000i 0.907406i 0.891153 + 0.453703i \(0.149897\pi\)
−0.891153 + 0.453703i \(0.850103\pi\)
\(822\) 0 0
\(823\) 26.8701i 0.936631i 0.883561 + 0.468316i \(0.155139\pi\)
−0.883561 + 0.468316i \(0.844861\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.07107 0.245885 0.122943 0.992414i \(-0.460767\pi\)
0.122943 + 0.992414i \(0.460767\pi\)
\(828\) 0 0
\(829\) − 26.0000i − 0.903017i −0.892267 0.451509i \(-0.850886\pi\)
0.892267 0.451509i \(-0.149114\pi\)
\(830\) 0 0
\(831\) 42.4264 1.47176
\(832\) 0 0
\(833\) − 66.0000i − 2.28676i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) 28.2843 0.976481 0.488241 0.872709i \(-0.337639\pi\)
0.488241 + 0.872709i \(0.337639\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) 11.3137 0.389665
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 89.0955i 3.06136i
\(848\) 0 0
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) − 14.1421i − 0.484786i
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 2.82843i 0.0965047i 0.998835 + 0.0482523i \(0.0153652\pi\)
−0.998835 + 0.0482523i \(0.984635\pi\)
\(860\) 0 0
\(861\) 48.0000i 1.63584i
\(862\) 0 0
\(863\) 55.1543i 1.87748i 0.344633 + 0.938738i \(0.388003\pi\)
−0.344633 + 0.938738i \(0.611997\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −26.8701 −0.912555
\(868\) 0 0
\(869\) 96.0000i 3.25658i
\(870\) 0 0
\(871\) −8.48528 −0.287513
\(872\) 0 0
\(873\) − 10.0000i − 0.338449i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 8.48528 0.286201
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) −21.2132 −0.713881 −0.356941 0.934127i \(-0.616180\pi\)
−0.356941 + 0.934127i \(0.616180\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 26.8701i − 0.902208i −0.892471 0.451104i \(-0.851030\pi\)
0.892471 0.451104i \(-0.148970\pi\)
\(888\) 0 0
\(889\) −42.0000 −1.40863
\(890\) 0 0
\(891\) 28.2843i 0.947559i
\(892\) 0 0
\(893\) 4.00000 0.133855
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 20.0000i 0.667781i
\(898\) 0 0
\(899\) − 11.3137i − 0.377333i
\(900\) 0 0
\(901\) 12.0000i 0.399778i
\(902\) 0 0
\(903\) − 8.48528i − 0.282372i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.41421 −0.0469582 −0.0234791 0.999724i \(-0.507474\pi\)
−0.0234791 + 0.999724i \(0.507474\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 31.1127 1.03081 0.515405 0.856947i \(-0.327642\pi\)
0.515405 + 0.856947i \(0.327642\pi\)
\(912\) 0 0
\(913\) 72.0000i 2.38285i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 39.5980 1.30622 0.653108 0.757264i \(-0.273465\pi\)
0.653108 + 0.757264i \(0.273465\pi\)
\(920\) 0 0
\(921\) 22.0000 0.724925
\(922\) 0 0
\(923\) −5.65685 −0.186198
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 9.89949i − 0.325142i
\(928\) 0 0
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 0 0
\(931\) − 31.1127i − 1.01968i
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 2.00000i − 0.0653372i −0.999466 0.0326686i \(-0.989599\pi\)
0.999466 0.0326686i \(-0.0104006\pi\)
\(938\) 0 0
\(939\) − 8.48528i − 0.276907i
\(940\) 0 0
\(941\) − 48.0000i − 1.56476i −0.622804 0.782378i \(-0.714007\pi\)
0.622804 0.782378i \(-0.285993\pi\)
\(942\) 0 0
\(943\) 56.5685i 1.84213i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 15.5563 0.505513 0.252757 0.967530i \(-0.418663\pi\)
0.252757 + 0.967530i \(0.418663\pi\)
\(948\) 0 0
\(949\) − 12.0000i − 0.389536i
\(950\) 0 0
\(951\) −2.82843 −0.0917180
\(952\) 0 0
\(953\) − 54.0000i − 1.74923i −0.484817 0.874616i \(-0.661114\pi\)
0.484817 0.874616i \(-0.338886\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 32.0000 1.03441
\(958\) 0 0
\(959\) 8.48528 0.274004
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 7.07107 0.227862
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 4.24264i 0.136434i 0.997671 + 0.0682171i \(0.0217310\pi\)
−0.997671 + 0.0682171i \(0.978269\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 28.2843i 0.907685i 0.891082 + 0.453843i \(0.149947\pi\)
−0.891082 + 0.453843i \(0.850053\pi\)
\(972\) 0 0
\(973\) −36.0000 −1.15411
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 0 0
\(979\) − 33.9411i − 1.08476i
\(980\) 0 0
\(981\) 6.00000i 0.191565i
\(982\) 0 0
\(983\) 18.3848i 0.586383i 0.956054 + 0.293192i \(0.0947174\pi\)
−0.956054 + 0.293192i \(0.905283\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −8.48528 −0.270089
\(988\) 0 0
\(989\) − 10.0000i − 0.317982i
\(990\) 0 0
\(991\) −8.48528 −0.269544 −0.134772 0.990877i \(-0.543030\pi\)
−0.134772 + 0.990877i \(0.543030\pi\)
\(992\) 0 0
\(993\) − 32.0000i − 1.01549i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) −11.3137 −0.357950
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 3200.2.f.i.449.3 4
4.3 odd 2 inner 3200.2.f.i.449.2 4
5.2 odd 4 3200.2.d.s.1601.2 4
5.3 odd 4 640.2.d.d.321.3 yes 4
5.4 even 2 3200.2.f.j.449.2 4
8.3 odd 2 3200.2.f.j.449.4 4
8.5 even 2 3200.2.f.j.449.1 4
15.8 even 4 5760.2.k.o.2881.3 4
20.3 even 4 640.2.d.d.321.1 4
20.7 even 4 3200.2.d.s.1601.3 4
20.19 odd 2 3200.2.f.j.449.3 4
40.3 even 4 640.2.d.d.321.4 yes 4
40.13 odd 4 640.2.d.d.321.2 yes 4
40.19 odd 2 inner 3200.2.f.i.449.1 4
40.27 even 4 3200.2.d.s.1601.1 4
40.29 even 2 inner 3200.2.f.i.449.4 4
40.37 odd 4 3200.2.d.s.1601.4 4
60.23 odd 4 5760.2.k.o.2881.4 4
80.3 even 4 1280.2.a.f.1.2 2
80.13 odd 4 1280.2.a.f.1.1 2
80.27 even 4 6400.2.a.bn.1.2 2
80.37 odd 4 6400.2.a.bn.1.1 2
80.43 even 4 1280.2.a.j.1.1 2
80.53 odd 4 1280.2.a.j.1.2 2
80.67 even 4 6400.2.a.bl.1.1 2
80.77 odd 4 6400.2.a.bl.1.2 2
120.53 even 4 5760.2.k.o.2881.1 4
120.83 odd 4 5760.2.k.o.2881.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.d.321.1 4 20.3 even 4
640.2.d.d.321.2 yes 4 40.13 odd 4
640.2.d.d.321.3 yes 4 5.3 odd 4
640.2.d.d.321.4 yes 4 40.3 even 4
1280.2.a.f.1.1 2 80.13 odd 4
1280.2.a.f.1.2 2 80.3 even 4
1280.2.a.j.1.1 2 80.43 even 4
1280.2.a.j.1.2 2 80.53 odd 4
3200.2.d.s.1601.1 4 40.27 even 4
3200.2.d.s.1601.2 4 5.2 odd 4
3200.2.d.s.1601.3 4 20.7 even 4
3200.2.d.s.1601.4 4 40.37 odd 4
3200.2.f.i.449.1 4 40.19 odd 2 inner
3200.2.f.i.449.2 4 4.3 odd 2 inner
3200.2.f.i.449.3 4 1.1 even 1 trivial
3200.2.f.i.449.4 4 40.29 even 2 inner
3200.2.f.j.449.1 4 8.5 even 2
3200.2.f.j.449.2 4 5.4 even 2
3200.2.f.j.449.3 4 20.19 odd 2
3200.2.f.j.449.4 4 8.3 odd 2
5760.2.k.o.2881.1 4 120.53 even 4
5760.2.k.o.2881.2 4 120.83 odd 4
5760.2.k.o.2881.3 4 15.8 even 4
5760.2.k.o.2881.4 4 60.23 odd 4
6400.2.a.bl.1.1 2 80.67 even 4
6400.2.a.bl.1.2 2 80.77 odd 4
6400.2.a.bn.1.1 2 80.37 odd 4
6400.2.a.bn.1.2 2 80.27 even 4