Properties

Label 640.2.d.d.321.4
Level $640$
Weight $2$
Character 640.321
Analytic conductor $5.110$
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] = [640,2,Mod(321,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.d (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: \(5.11042572936\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 321.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 640.321
Dual form 640.2.d.d.321.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{3} +1.00000i q^{5} +4.24264 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.41421i q^{3} +1.00000i q^{5} +4.24264 q^{7} +1.00000 q^{9} -5.65685i q^{11} +2.00000i q^{13} -1.41421 q^{15} +6.00000 q^{17} -2.82843i q^{19} +6.00000i q^{21} -7.07107 q^{23} -1.00000 q^{25} +5.65685i q^{27} +4.00000i q^{29} +2.82843 q^{31} +8.00000 q^{33} +4.24264i q^{35} +2.00000i q^{37} -2.82843 q^{39} -8.00000 q^{41} +1.41421i q^{43} +1.00000i q^{45} +1.41421 q^{47} +11.0000 q^{49} +8.48528i q^{51} -2.00000i q^{53} +5.65685 q^{55} +4.00000 q^{57} -2.82843i q^{59} +14.0000i q^{61} +4.24264 q^{63} -2.00000 q^{65} -4.24264i q^{67} -10.0000i q^{69} -2.82843 q^{71} -6.00000 q^{73} -1.41421i q^{75} -24.0000i q^{77} -16.9706 q^{79} -5.00000 q^{81} -12.7279i q^{83} +6.00000i q^{85} -5.65685 q^{87} -6.00000 q^{89} +8.48528i q^{91} +4.00000i q^{93} +2.82843 q^{95} +10.0000 q^{97} -5.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{9} + 24 q^{17} - 4 q^{25} + 32 q^{33} - 32 q^{41} + 44 q^{49} + 16 q^{57} - 8 q^{65} - 24 q^{73} - 20 q^{81} - 24 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 1.41421i 0.816497i 0.912871 + 0.408248i \(0.133860\pi\)
−0.912871 + 0.408248i \(0.866140\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 4.24264 1.60357 0.801784 0.597614i \(-0.203885\pi\)
0.801784 + 0.597614i \(0.203885\pi\)
\(8\) 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
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) − 2.82843i − 0.648886i −0.945905 0.324443i \(-0.894823\pi\)
0.945905 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) 6.00000i 1.30931i
\(22\) 0 0
\(23\) −7.07107 −1.47442 −0.737210 0.675664i \(-0.763857\pi\)
−0.737210 + 0.675664i \(0.763857\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.65685i 1.08866i
\(28\) 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 \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 4.24264i 0.717137i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 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
\(43\) 1.41421i 0.215666i 0.994169 + 0.107833i \(0.0343911\pi\)
−0.994169 + 0.107833i \(0.965609\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 0 0
\(47\) 1.41421 0.206284 0.103142 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 8.48528i 1.18818i
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) − 2.82843i − 0.368230i −0.982905 0.184115i \(-0.941058\pi\)
0.982905 0.184115i \(-0.0589419\pi\)
\(60\) 0 0
\(61\) 14.0000i 1.79252i 0.443533 + 0.896258i \(0.353725\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) − 4.24264i − 0.518321i −0.965834 0.259161i \(-0.916554\pi\)
0.965834 0.259161i \(-0.0834459\pi\)
\(68\) 0 0
\(69\) − 10.0000i − 1.20386i
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) − 1.41421i − 0.163299i
\(76\) 0 0
\(77\) − 24.0000i − 2.73505i
\(78\) 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
\(83\) − 12.7279i − 1.39707i −0.715575 0.698535i \(-0.753835\pi\)
0.715575 0.698535i \(-0.246165\pi\)
\(84\) 0 0
\(85\) 6.00000i 0.650791i
\(86\) 0 0
\(87\) −5.65685 −0.606478
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 8.48528i 0.889499i
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) − 5.65685i − 0.568535i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 9.89949 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) 7.07107i 0.683586i 0.939775 + 0.341793i \(0.111034\pi\)
−0.939775 + 0.341793i \(0.888966\pi\)
\(108\) 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
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) − 7.07107i − 0.659380i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 25.4558 2.33353
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 0 0
\(123\) − 11.3137i − 1.02012i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) 9.89949 0.878438 0.439219 0.898380i \(-0.355255\pi\)
0.439219 + 0.898380i \(0.355255\pi\)
\(128\) 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
\(133\) − 12.0000i − 1.04053i
\(134\) 0 0
\(135\) −5.65685 −0.486864
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 8.48528i 0.719712i 0.933008 + 0.359856i \(0.117174\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 2.00000i 0.168430i
\(142\) 0 0
\(143\) 11.3137 0.946100
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 15.5563i 1.28307i
\(148\) 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 \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 2.82843i 0.227185i
\(156\) 0 0
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −30.0000 −2.36433
\(162\) 0 0
\(163\) 18.3848i 1.44001i 0.693971 + 0.720003i \(0.255860\pi\)
−0.693971 + 0.720003i \(0.744140\pi\)
\(164\) 0 0
\(165\) 8.00000i 0.622799i
\(166\) 0 0
\(167\) −9.89949 −0.766046 −0.383023 0.923739i \(-0.625117\pi\)
−0.383023 + 0.923739i \(0.625117\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 2.82843i − 0.216295i
\(172\) 0 0
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) 0 0
\(175\) −4.24264 −0.320713
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 19.7990i 1.47985i 0.672692 + 0.739923i \(0.265138\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 16.0000i 1.18927i 0.803996 + 0.594635i \(0.202704\pi\)
−0.803996 + 0.594635i \(0.797296\pi\)
\(182\) 0 0
\(183\) −19.7990 −1.46358
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) − 33.9411i − 2.48202i
\(188\) 0 0
\(189\) 24.0000i 1.74574i
\(190\) 0 0
\(191\) −2.82843 −0.204658 −0.102329 0.994751i \(-0.532629\pi\)
−0.102329 + 0.994751i \(0.532629\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) − 2.82843i − 0.202548i
\(196\) 0 0
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 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
\(203\) 16.9706i 1.19110i
\(204\) 0 0
\(205\) − 8.00000i − 0.558744i
\(206\) 0 0
\(207\) −7.07107 −0.491473
\(208\) 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
\(213\) − 4.00000i − 0.274075i
\(214\) 0 0
\(215\) −1.41421 −0.0964486
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) − 8.48528i − 0.573382i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) −15.5563 −1.04173 −0.520865 0.853639i \(-0.674391\pi\)
−0.520865 + 0.853639i \(0.674391\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) − 7.07107i − 0.469323i −0.972077 0.234662i \(-0.924602\pi\)
0.972077 0.234662i \(-0.0753982\pi\)
\(228\) 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
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 1.41421i 0.0922531i
\(236\) 0 0
\(237\) − 24.0000i − 1.55897i
\(238\) 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
\(243\) 9.89949i 0.635053i
\(244\) 0 0
\(245\) 11.0000i 0.702764i
\(246\) 0 0
\(247\) 5.65685 0.359937
\(248\) 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
\(253\) 40.0000i 2.51478i
\(254\) 0 0
\(255\) −8.48528 −0.531369
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 8.48528i 0.527250i
\(260\) 0 0
\(261\) 4.00000i 0.247594i
\(262\) 0 0
\(263\) 7.07107 0.436021 0.218010 0.975946i \(-0.430043\pi\)
0.218010 + 0.975946i \(0.430043\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) − 8.48528i − 0.519291i
\(268\) 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 \(-0.218672\pi\)
0.773166 + 0.634203i \(0.218672\pi\)
\(272\) 0 0
\(273\) −12.0000 −0.726273
\(274\) 0 0
\(275\) 5.65685i 0.341121i
\(276\) 0 0
\(277\) 30.0000i 1.80253i 0.433273 + 0.901263i \(0.357359\pi\)
−0.433273 + 0.901263i \(0.642641\pi\)
\(278\) 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
\(283\) 9.89949i 0.588464i 0.955734 + 0.294232i \(0.0950638\pi\)
−0.955734 + 0.294232i \(0.904936\pi\)
\(284\) 0 0
\(285\) 4.00000i 0.236940i
\(286\) 0 0
\(287\) −33.9411 −2.00348
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 14.1421i 0.829027i
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 2.82843 0.164677
\(296\) 0 0
\(297\) 32.0000 1.85683
\(298\) 0 0
\(299\) − 14.1421i − 0.817861i
\(300\) 0 0
\(301\) 6.00000i 0.345834i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 0 0
\(307\) − 15.5563i − 0.887848i −0.896065 0.443924i \(-0.853586\pi\)
0.896065 0.443924i \(-0.146414\pi\)
\(308\) 0 0
\(309\) 14.0000i 0.796432i
\(310\) 0 0
\(311\) −2.82843 −0.160385 −0.0801927 0.996779i \(-0.525554\pi\)
−0.0801927 + 0.996779i \(0.525554\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 4.24264i 0.239046i
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 22.6274 1.26689
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) − 16.9706i − 0.944267i
\(324\) 0 0
\(325\) − 2.00000i − 0.110940i
\(326\) 0 0
\(327\) 8.48528 0.469237
\(328\) 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
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 4.24264 0.231800
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) − 14.1421i − 0.768095i
\(340\) 0 0
\(341\) − 16.0000i − 0.866449i
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 10.0000 0.538382
\(346\) 0 0
\(347\) 4.24264i 0.227757i 0.993495 + 0.113878i \(0.0363274\pi\)
−0.993495 + 0.113878i \(0.963673\pi\)
\(348\) 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
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) − 2.82843i − 0.150117i
\(356\) 0 0
\(357\) 36.0000i 1.90532i
\(358\) 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
\(363\) − 29.6985i − 1.55877i
\(364\) 0 0
\(365\) − 6.00000i − 0.314054i
\(366\) 0 0
\(367\) 4.24264 0.221464 0.110732 0.993850i \(-0.464680\pi\)
0.110732 + 0.993850i \(0.464680\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) − 8.48528i − 0.440534i
\(372\) 0 0
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 1.41421 0.0730297
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) 31.1127i 1.59815i 0.601230 + 0.799076i \(0.294678\pi\)
−0.601230 + 0.799076i \(0.705322\pi\)
\(380\) 0 0
\(381\) 14.0000i 0.717242i
\(382\) 0 0
\(383\) 7.07107 0.361315 0.180657 0.983546i \(-0.442177\pi\)
0.180657 + 0.983546i \(0.442177\pi\)
\(384\) 0 0
\(385\) 24.0000 1.22315
\(386\) 0 0
\(387\) 1.41421i 0.0718885i
\(388\) 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
\(393\) 16.0000 0.807093
\(394\) 0 0
\(395\) − 16.9706i − 0.853882i
\(396\) 0 0
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) 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
\(403\) 5.65685i 0.281788i
\(404\) 0 0
\(405\) − 5.00000i − 0.248452i
\(406\) 0 0
\(407\) 11.3137 0.560800
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 2.82843i 0.139516i
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) 12.7279 0.624789
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) − 8.48528i − 0.414533i −0.978285 0.207267i \(-0.933543\pi\)
0.978285 0.207267i \(-0.0664567\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 0 0
\(423\) 1.41421 0.0687614
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 59.3970i 2.87442i
\(428\) 0 0
\(429\) 16.0000i 0.772487i
\(430\) 0 0
\(431\) −31.1127 −1.49865 −0.749323 0.662205i \(-0.769621\pi\)
−0.749323 + 0.662205i \(0.769621\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) − 5.65685i − 0.271225i
\(436\) 0 0
\(437\) 20.0000i 0.956730i
\(438\) 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
\(443\) 15.5563i 0.739104i 0.929210 + 0.369552i \(0.120489\pi\)
−0.929210 + 0.369552i \(0.879511\pi\)
\(444\) 0 0
\(445\) − 6.00000i − 0.284427i
\(446\) 0 0
\(447\) 31.1127 1.47158
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 45.2548i 2.13097i
\(452\) 0 0
\(453\) − 20.0000i − 0.939682i
\(454\) 0 0
\(455\) −8.48528 −0.397796
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 0 0
\(459\) 33.9411i 1.58424i
\(460\) 0 0
\(461\) − 24.0000i − 1.11779i −0.829238 0.558896i \(-0.811225\pi\)
0.829238 0.558896i \(-0.188775\pi\)
\(462\) 0 0
\(463\) −7.07107 −0.328620 −0.164310 0.986409i \(-0.552540\pi\)
−0.164310 + 0.986409i \(0.552540\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) 0 0
\(467\) 9.89949i 0.458094i 0.973415 + 0.229047i \(0.0735609\pi\)
−0.973415 + 0.229047i \(0.926439\pi\)
\(468\) 0 0
\(469\) − 18.0000i − 0.831163i
\(470\) 0 0
\(471\) 25.4558 1.17294
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 2.82843i 0.129777i
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 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
\(483\) − 42.4264i − 1.93047i
\(484\) 0 0
\(485\) 10.0000i 0.454077i
\(486\) 0 0
\(487\) −12.7279 −0.576757 −0.288379 0.957516i \(-0.593116\pi\)
−0.288379 + 0.957516i \(0.593116\pi\)
\(488\) 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
\(493\) 24.0000i 1.08091i
\(494\) 0 0
\(495\) 5.65685 0.254257
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) − 25.4558i − 1.13956i −0.821797 0.569780i \(-0.807028\pi\)
0.821797 0.569780i \(-0.192972\pi\)
\(500\) 0 0
\(501\) − 14.0000i − 0.625474i
\(502\) 0 0
\(503\) 32.5269 1.45030 0.725152 0.688589i \(-0.241770\pi\)
0.725152 + 0.688589i \(0.241770\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.7279i 0.565267i
\(508\) 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
\(513\) 16.0000 0.706417
\(514\) 0 0
\(515\) 9.89949i 0.436224i
\(516\) 0 0
\(517\) − 8.00000i − 0.351840i
\(518\) 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
\(523\) − 12.7279i − 0.556553i −0.960501 0.278277i \(-0.910237\pi\)
0.960501 0.278277i \(-0.0897632\pi\)
\(524\) 0 0
\(525\) − 6.00000i − 0.261861i
\(526\) 0 0
\(527\) 16.9706 0.739249
\(528\) 0 0
\(529\) 27.0000 1.17391
\(530\) 0 0
\(531\) − 2.82843i − 0.122743i
\(532\) 0 0
\(533\) − 16.0000i − 0.693037i
\(534\) 0 0
\(535\) −7.07107 −0.305709
\(536\) 0 0
\(537\) −28.0000 −1.20829
\(538\) 0 0
\(539\) − 62.2254i − 2.68024i
\(540\) 0 0
\(541\) − 40.0000i − 1.71973i −0.510518 0.859867i \(-0.670546\pi\)
0.510518 0.859867i \(-0.329454\pi\)
\(542\) 0 0
\(543\) −22.6274 −0.971035
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 4.24264i 0.181402i 0.995878 + 0.0907011i \(0.0289108\pi\)
−0.995878 + 0.0907011i \(0.971089\pi\)
\(548\) 0 0
\(549\) 14.0000i 0.597505i
\(550\) 0 0
\(551\) 11.3137 0.481980
\(552\) 0 0
\(553\) −72.0000 −3.06175
\(554\) 0 0
\(555\) − 2.82843i − 0.120060i
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −2.82843 −0.119630
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) 35.3553i 1.49005i 0.667037 + 0.745025i \(0.267562\pi\)
−0.667037 + 0.745025i \(0.732438\pi\)
\(564\) 0 0
\(565\) − 10.0000i − 0.420703i
\(566\) 0 0
\(567\) −21.2132 −0.890871
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) − 22.6274i − 0.946928i −0.880813 0.473464i \(-0.843003\pi\)
0.880813 0.473464i \(-0.156997\pi\)
\(572\) 0 0
\(573\) − 4.00000i − 0.167102i
\(574\) 0 0
\(575\) 7.07107 0.294884
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 8.48528i 0.352636i
\(580\) 0 0
\(581\) − 54.0000i − 2.24030i
\(582\) 0 0
\(583\) −11.3137 −0.468566
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) − 12.7279i − 0.525338i −0.964886 0.262669i \(-0.915397\pi\)
0.964886 0.262669i \(-0.0846027\pi\)
\(588\) 0 0
\(589\) − 8.00000i − 0.329634i
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 25.4558i 1.04359i
\(596\) 0 0
\(597\) − 16.0000i − 0.654836i
\(598\) 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
\(603\) − 4.24264i − 0.172774i
\(604\) 0 0
\(605\) − 21.0000i − 0.853771i
\(606\) 0 0
\(607\) −9.89949 −0.401808 −0.200904 0.979611i \(-0.564388\pi\)
−0.200904 + 0.979611i \(0.564388\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 2.82843i 0.114426i
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 11.3137 0.456213
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 0 0
\(619\) − 19.7990i − 0.795789i −0.917431 0.397894i \(-0.869741\pi\)
0.917431 0.397894i \(-0.130259\pi\)
\(620\) 0 0
\(621\) − 40.0000i − 1.60514i
\(622\) 0 0
\(623\) −25.4558 −1.01987
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) − 22.6274i − 0.903652i
\(628\) 0 0
\(629\) 12.0000i 0.478471i
\(630\) 0 0
\(631\) 36.7696 1.46377 0.731886 0.681427i \(-0.238640\pi\)
0.731886 + 0.681427i \(0.238640\pi\)
\(632\) 0 0
\(633\) 24.0000 0.953914
\(634\) 0 0
\(635\) 9.89949i 0.392849i
\(636\) 0 0
\(637\) 22.0000i 0.871672i
\(638\) 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
\(643\) − 7.07107i − 0.278856i −0.990232 0.139428i \(-0.955474\pi\)
0.990232 0.139428i \(-0.0445263\pi\)
\(644\) 0 0
\(645\) − 2.00000i − 0.0787499i
\(646\) 0 0
\(647\) 32.5269 1.27876 0.639382 0.768889i \(-0.279190\pi\)
0.639382 + 0.768889i \(0.279190\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 16.9706i 0.665129i
\(652\) 0 0
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) 11.3137 0.442063
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 25.4558i 0.991619i 0.868431 + 0.495809i \(0.165129\pi\)
−0.868431 + 0.495809i \(0.834871\pi\)
\(660\) 0 0
\(661\) − 10.0000i − 0.388955i −0.980907 0.194477i \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) 0 0
\(663\) −16.9706 −0.659082
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) − 28.2843i − 1.09517i
\(668\) 0 0
\(669\) − 22.0000i − 0.850569i
\(670\) 0 0
\(671\) 79.1960 3.05733
\(672\) 0 0
\(673\) 50.0000 1.92736 0.963679 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) 0 0
\(675\) − 5.65685i − 0.217732i
\(676\) 0 0
\(677\) 10.0000i 0.384331i 0.981363 + 0.192166i \(0.0615511\pi\)
−0.981363 + 0.192166i \(0.938449\pi\)
\(678\) 0 0
\(679\) 42.4264 1.62818
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 35.3553i 1.35283i 0.736519 + 0.676417i \(0.236468\pi\)
−0.736519 + 0.676417i \(0.763532\pi\)
\(684\) 0 0
\(685\) 2.00000i 0.0764161i
\(686\) 0 0
\(687\) 28.2843 1.07911
\(688\) 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
\(693\) − 24.0000i − 0.911685i
\(694\) 0 0
\(695\) −8.48528 −0.321865
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) 0 0
\(699\) 8.48528i 0.320943i
\(700\) 0 0
\(701\) 18.0000i 0.679851i 0.940452 + 0.339925i \(0.110402\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(702\) 0 0
\(703\) 5.65685 0.213352
\(704\) 0 0
\(705\) −2.00000 −0.0753244
\(706\) 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
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) 11.3137i 0.423109i
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 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
\(723\) 11.3137i 0.420761i
\(724\) 0 0
\(725\) − 4.00000i − 0.148556i
\(726\) 0 0
\(727\) 4.24264 0.157351 0.0786754 0.996900i \(-0.474931\pi\)
0.0786754 + 0.996900i \(0.474931\pi\)
\(728\) 0 0
\(729\) −29.0000 −1.07407
\(730\) 0 0
\(731\) 8.48528i 0.313839i
\(732\) 0 0
\(733\) − 34.0000i − 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 0 0
\(735\) −15.5563 −0.573805
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) − 25.4558i − 0.936408i −0.883620 0.468204i \(-0.844901\pi\)
0.883620 0.468204i \(-0.155099\pi\)
\(740\) 0 0
\(741\) 8.00000i 0.293887i
\(742\) 0 0
\(743\) −32.5269 −1.19330 −0.596648 0.802503i \(-0.703501\pi\)
−0.596648 + 0.802503i \(0.703501\pi\)
\(744\) 0 0
\(745\) 22.0000 0.806018
\(746\) 0 0
\(747\) − 12.7279i − 0.465690i
\(748\) 0 0
\(749\) 30.0000i 1.09618i
\(750\) 0 0
\(751\) −8.48528 −0.309632 −0.154816 0.987943i \(-0.549479\pi\)
−0.154816 + 0.987943i \(0.549479\pi\)
\(752\) 0 0
\(753\) −32.0000 −1.16614
\(754\) 0 0
\(755\) − 14.1421i − 0.514685i
\(756\) 0 0
\(757\) 46.0000i 1.67190i 0.548807 + 0.835949i \(0.315082\pi\)
−0.548807 + 0.835949i \(0.684918\pi\)
\(758\) 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
\(763\) − 25.4558i − 0.921563i
\(764\) 0 0
\(765\) 6.00000i 0.216930i
\(766\) 0 0
\(767\) 5.65685 0.204257
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) − 25.4558i − 0.916770i
\(772\) 0 0
\(773\) 26.0000i 0.935155i 0.883952 + 0.467578i \(0.154873\pi\)
−0.883952 + 0.467578i \(0.845127\pi\)
\(774\) 0 0
\(775\) −2.82843 −0.101600
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) 0 0
\(779\) 22.6274i 0.810711i
\(780\) 0 0
\(781\) 16.0000i 0.572525i
\(782\) 0 0
\(783\) −22.6274 −0.808638
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 46.6690i 1.66357i 0.555097 + 0.831786i \(0.312681\pi\)
−0.555097 + 0.831786i \(0.687319\pi\)
\(788\) 0 0
\(789\) 10.0000i 0.356009i
\(790\) 0 0
\(791\) −42.4264 −1.50851
\(792\) 0 0
\(793\) −28.0000 −0.994309
\(794\) 0 0
\(795\) 2.82843i 0.100314i
\(796\) 0 0
\(797\) − 10.0000i − 0.354218i −0.984191 0.177109i \(-0.943325\pi\)
0.984191 0.177109i \(-0.0566745\pi\)
\(798\) 0 0
\(799\) 8.48528 0.300188
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 33.9411i 1.19776i
\(804\) 0 0
\(805\) − 30.0000i − 1.05736i
\(806\) 0 0
\(807\) −8.48528 −0.298696
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 36.0000i 1.26258i
\(814\) 0 0
\(815\) −18.3848 −0.643991
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 8.48528i 0.296500i
\(820\) 0 0
\(821\) − 26.0000i − 0.907406i −0.891153 0.453703i \(-0.850103\pi\)
0.891153 0.453703i \(-0.149897\pi\)
\(822\) 0 0
\(823\) 26.8701 0.936631 0.468316 0.883561i \(-0.344861\pi\)
0.468316 + 0.883561i \(0.344861\pi\)
\(824\) 0 0
\(825\) −8.00000 −0.278524
\(826\) 0 0
\(827\) − 7.07107i − 0.245885i −0.992414 0.122943i \(-0.960767\pi\)
0.992414 0.122943i \(-0.0392331\pi\)
\(828\) 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
\(833\) 66.0000 2.28676
\(834\) 0 0
\(835\) − 9.89949i − 0.342586i
\(836\) 0 0
\(837\) 16.0000i 0.553041i
\(838\) 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
\(843\) 11.3137i 0.389665i
\(844\) 0 0
\(845\) 9.00000i 0.309609i
\(846\) 0 0
\(847\) −89.0955 −3.06136
\(848\) 0 0
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) − 14.1421i − 0.484786i
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 2.82843 0.0967302
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) − 2.82843i − 0.0965047i −0.998835 0.0482523i \(-0.984635\pi\)
0.998835 0.0482523i \(-0.0153652\pi\)
\(860\) 0 0
\(861\) − 48.0000i − 1.63584i
\(862\) 0 0
\(863\) 55.1543 1.87748 0.938738 0.344633i \(-0.111997\pi\)
0.938738 + 0.344633i \(0.111997\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) 26.8701i 0.912555i
\(868\) 0 0
\(869\) 96.0000i 3.25658i
\(870\) 0 0
\(871\) 8.48528 0.287513
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) − 4.24264i − 0.143427i
\(876\) 0 0
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 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
\(883\) − 21.2132i − 0.713881i −0.934127 0.356941i \(-0.883820\pi\)
0.934127 0.356941i \(-0.116180\pi\)
\(884\) 0 0
\(885\) 4.00000i 0.134459i
\(886\) 0 0
\(887\) 26.8701 0.902208 0.451104 0.892471i \(-0.351030\pi\)
0.451104 + 0.892471i \(0.351030\pi\)
\(888\) 0 0
\(889\) 42.0000 1.40863
\(890\) 0 0
\(891\) 28.2843i 0.947559i
\(892\) 0 0
\(893\) − 4.00000i − 0.133855i
\(894\) 0 0
\(895\) −19.7990 −0.661807
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) 11.3137i 0.377333i
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) −8.48528 −0.282372
\(904\) 0 0
\(905\) −16.0000 −0.531858
\(906\) 0 0
\(907\) 1.41421i 0.0469582i 0.999724 + 0.0234791i \(0.00747431\pi\)
−0.999724 + 0.0234791i \(0.992526\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −31.1127 −1.03081 −0.515405 0.856947i \(-0.672358\pi\)
−0.515405 + 0.856947i \(0.672358\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) 0 0
\(915\) − 19.7990i − 0.654534i
\(916\) 0 0
\(917\) − 48.0000i − 1.58510i
\(918\) 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
\(923\) − 5.65685i − 0.186198i
\(924\) 0 0
\(925\) − 2.00000i − 0.0657596i
\(926\) 0 0
\(927\) 9.89949 0.325142
\(928\) 0 0
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 0 0
\(931\) − 31.1127i − 1.01968i
\(932\) 0 0
\(933\) − 4.00000i − 0.130954i
\(934\) 0 0
\(935\) 33.9411 1.10999
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 8.48528i 0.276907i
\(940\) 0 0
\(941\) 48.0000i 1.56476i 0.622804 + 0.782378i \(0.285993\pi\)
−0.622804 + 0.782378i \(0.714007\pi\)
\(942\) 0 0
\(943\) 56.5685 1.84213
\(944\) 0 0
\(945\) −24.0000 −0.780720
\(946\) 0 0
\(947\) − 15.5563i − 0.505513i −0.967530 0.252757i \(-0.918663\pi\)
0.967530 0.252757i \(-0.0813372\pi\)
\(948\) 0 0
\(949\) − 12.0000i − 0.389536i
\(950\) 0 0
\(951\) 2.82843 0.0917180
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) − 2.82843i − 0.0915258i
\(956\) 0 0
\(957\) 32.0000i 1.03441i
\(958\) 0 0
\(959\) 8.48528 0.274004
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 7.07107i 0.227862i
\(964\) 0 0
\(965\) 6.00000i 0.193147i
\(966\) 0 0
\(967\) −4.24264 −0.136434 −0.0682171 0.997671i \(-0.521731\pi\)
−0.0682171 + 0.997671i \(0.521731\pi\)
\(968\) 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
\(973\) 36.0000i 1.15411i
\(974\) 0 0
\(975\) 2.82843 0.0905822
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 33.9411i 1.08476i
\(980\) 0 0
\(981\) − 6.00000i − 0.191565i
\(982\) 0 0
\(983\) 18.3848 0.586383 0.293192 0.956054i \(-0.405283\pi\)
0.293192 + 0.956054i \(0.405283\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 8.48528i 0.270089i
\(988\) 0 0
\(989\) − 10.0000i − 0.317982i
\(990\) 0 0
\(991\) 8.48528 0.269544 0.134772 0.990877i \(-0.456970\pi\)
0.134772 + 0.990877i \(0.456970\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) − 11.3137i − 0.358669i
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 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 640.2.d.d.321.4 yes 4
3.2 odd 2 5760.2.k.o.2881.2 4
4.3 odd 2 inner 640.2.d.d.321.2 yes 4
5.2 odd 4 3200.2.f.j.449.4 4
5.3 odd 4 3200.2.f.i.449.1 4
5.4 even 2 3200.2.d.s.1601.1 4
8.3 odd 2 inner 640.2.d.d.321.3 yes 4
8.5 even 2 inner 640.2.d.d.321.1 4
12.11 even 2 5760.2.k.o.2881.1 4
16.3 odd 4 1280.2.a.j.1.2 2
16.5 even 4 1280.2.a.f.1.2 2
16.11 odd 4 1280.2.a.f.1.1 2
16.13 even 4 1280.2.a.j.1.1 2
20.3 even 4 3200.2.f.i.449.4 4
20.7 even 4 3200.2.f.j.449.1 4
20.19 odd 2 3200.2.d.s.1601.4 4
24.5 odd 2 5760.2.k.o.2881.4 4
24.11 even 2 5760.2.k.o.2881.3 4
40.3 even 4 3200.2.f.j.449.2 4
40.13 odd 4 3200.2.f.j.449.3 4
40.19 odd 2 3200.2.d.s.1601.2 4
40.27 even 4 3200.2.f.i.449.3 4
40.29 even 2 3200.2.d.s.1601.3 4
40.37 odd 4 3200.2.f.i.449.2 4
80.19 odd 4 6400.2.a.bn.1.1 2
80.29 even 4 6400.2.a.bn.1.2 2
80.59 odd 4 6400.2.a.bl.1.2 2
80.69 even 4 6400.2.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.d.321.1 4 8.5 even 2 inner
640.2.d.d.321.2 yes 4 4.3 odd 2 inner
640.2.d.d.321.3 yes 4 8.3 odd 2 inner
640.2.d.d.321.4 yes 4 1.1 even 1 trivial
1280.2.a.f.1.1 2 16.11 odd 4
1280.2.a.f.1.2 2 16.5 even 4
1280.2.a.j.1.1 2 16.13 even 4
1280.2.a.j.1.2 2 16.3 odd 4
3200.2.d.s.1601.1 4 5.4 even 2
3200.2.d.s.1601.2 4 40.19 odd 2
3200.2.d.s.1601.3 4 40.29 even 2
3200.2.d.s.1601.4 4 20.19 odd 2
3200.2.f.i.449.1 4 5.3 odd 4
3200.2.f.i.449.2 4 40.37 odd 4
3200.2.f.i.449.3 4 40.27 even 4
3200.2.f.i.449.4 4 20.3 even 4
3200.2.f.j.449.1 4 20.7 even 4
3200.2.f.j.449.2 4 40.3 even 4
3200.2.f.j.449.3 4 40.13 odd 4
3200.2.f.j.449.4 4 5.2 odd 4
5760.2.k.o.2881.1 4 12.11 even 2
5760.2.k.o.2881.2 4 3.2 odd 2
5760.2.k.o.2881.3 4 24.11 even 2
5760.2.k.o.2881.4 4 24.5 odd 2
6400.2.a.bl.1.1 2 80.69 even 4
6400.2.a.bl.1.2 2 80.59 odd 4
6400.2.a.bn.1.1 2 80.19 odd 4
6400.2.a.bn.1.2 2 80.29 even 4