Properties

Label 880.2.m.f.879.5
Level $880$
Weight $2$
Character 880.879
Analytic conductor $7.027$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(879,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("880.879");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.m (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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 879.5
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 880.879
Dual form 880.2.m.f.879.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 2.00000i) q^{5} -2.82843i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+(1.00000 + 2.00000i) q^{5} -2.82843i q^{7} -3.00000 q^{9} +(-2.82843 - 1.73205i) q^{11} -4.89898 q^{13} -4.89898 q^{17} +5.65685 q^{19} -6.92820 q^{23} +(-3.00000 + 4.00000i) q^{25} -9.79796i q^{29} +3.46410i q^{31} +(5.65685 - 2.82843i) q^{35} +4.00000i q^{37} -2.82843i q^{43} +(-3.00000 - 6.00000i) q^{45} +6.92820 q^{47} -1.00000 q^{49} +4.00000i q^{53} +(0.635674 - 7.38891i) q^{55} -10.3923i q^{59} +9.79796i q^{61} +8.48528i q^{63} +(-4.89898 - 9.79796i) q^{65} -13.8564 q^{67} -3.46410i q^{71} -4.89898 q^{73} +(-4.89898 + 8.00000i) q^{77} -11.3137 q^{79} +9.00000 q^{81} -14.1421i q^{83} +(-4.89898 - 9.79796i) q^{85} -2.00000 q^{89} +13.8564i q^{91} +(5.65685 + 11.3137i) q^{95} +(8.48528 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 24 q^{9} - 24 q^{25} - 24 q^{45} - 8 q^{49} + 72 q^{81} - 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 1.00000 + 2.00000i 0.447214 + 0.894427i
\(6\) 0 0
\(7\) 2.82843i 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −2.82843 1.73205i −0.852803 0.522233i
\(12\) 0 0
\(13\) −4.89898 −1.35873 −0.679366 0.733799i \(-0.737745\pi\)
−0.679366 + 0.733799i \(0.737745\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 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
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.79796i 1.81944i −0.415227 0.909718i \(-0.636298\pi\)
0.415227 0.909718i \(-0.363702\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.65685 2.82843i 0.956183 0.478091i
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 2.82843i 0.431331i −0.976467 0.215666i \(-0.930808\pi\)
0.976467 0.215666i \(-0.0691921\pi\)
\(44\) 0 0
\(45\) −3.00000 6.00000i −0.447214 0.894427i
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0.635674 7.38891i 0.0857143 0.996320i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.3923i 1.35296i −0.736460 0.676481i \(-0.763504\pi\)
0.736460 0.676481i \(-0.236496\pi\)
\(60\) 0 0
\(61\) 9.79796i 1.25450i 0.778818 + 0.627250i \(0.215820\pi\)
−0.778818 + 0.627250i \(0.784180\pi\)
\(62\) 0 0
\(63\) 8.48528i 1.06904i
\(64\) 0 0
\(65\) −4.89898 9.79796i −0.607644 1.21529i
\(66\) 0 0
\(67\) −13.8564 −1.69283 −0.846415 0.532524i \(-0.821244\pi\)
−0.846415 + 0.532524i \(0.821244\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89898 + 8.00000i −0.558291 + 0.911685i
\(78\) 0 0
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.1421i 1.55230i −0.630548 0.776151i \(-0.717170\pi\)
0.630548 0.776151i \(-0.282830\pi\)
\(84\) 0 0
\(85\) −4.89898 9.79796i −0.531369 1.06274i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 13.8564i 1.45255i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.65685 + 11.3137i 0.580381 + 1.16076i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 8.48528 + 5.19615i 0.852803 + 0.522233i
\(100\) 0 0
\(101\) 9.79796i 0.974933i 0.873142 + 0.487467i \(0.162079\pi\)
−0.873142 + 0.487467i \(0.837921\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1421i 1.36717i −0.729870 0.683586i \(-0.760419\pi\)
0.729870 0.683586i \(-0.239581\pi\)
\(108\) 0 0
\(109\) 9.79796i 0.938474i 0.883072 + 0.469237i \(0.155471\pi\)
−0.883072 + 0.469237i \(0.844529\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.00000i 0.752577i 0.926503 + 0.376288i \(0.122800\pi\)
−0.926503 + 0.376288i \(0.877200\pi\)
\(114\) 0 0
\(115\) −6.92820 13.8564i −0.646058 1.29212i
\(116\) 0 0
\(117\) 14.6969 1.35873
\(118\) 0 0
\(119\) 13.8564i 1.27021i
\(120\) 0 0
\(121\) 5.00000 + 9.79796i 0.454545 + 0.890724i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.0000 2.00000i −0.983870 0.178885i
\(126\) 0 0
\(127\) 19.7990i 1.75688i 0.477856 + 0.878438i \(0.341414\pi\)
−0.477856 + 0.878438i \(0.658586\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.0000i 1.36697i −0.729964 0.683486i \(-0.760463\pi\)
0.729964 0.683486i \(-0.239537\pi\)
\(138\) 0 0
\(139\) 5.65685 0.479808 0.239904 0.970797i \(-0.422884\pi\)
0.239904 + 0.970797i \(0.422884\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.8564 + 8.48528i 1.15873 + 0.709575i
\(144\) 0 0
\(145\) 19.5959 9.79796i 1.62735 0.813676i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.79796i 0.802680i −0.915929 0.401340i \(-0.868545\pi\)
0.915929 0.401340i \(-0.131455\pi\)
\(150\) 0 0
\(151\) −11.3137 −0.920697 −0.460348 0.887738i \(-0.652275\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(152\) 0 0
\(153\) 14.6969 1.18818
\(154\) 0 0
\(155\) −6.92820 + 3.46410i −0.556487 + 0.278243i
\(156\) 0 0
\(157\) 12.0000i 0.957704i −0.877896 0.478852i \(-0.841053\pi\)
0.877896 0.478852i \(-0.158947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 19.5959i 1.54437i
\(162\) 0 0
\(163\) 13.8564 1.08532 0.542659 0.839953i \(-0.317418\pi\)
0.542659 + 0.839953i \(0.317418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.48528i 0.656611i 0.944572 + 0.328305i \(0.106478\pi\)
−0.944572 + 0.328305i \(0.893522\pi\)
\(168\) 0 0
\(169\) 11.0000 0.846154
\(170\) 0 0
\(171\) −16.9706 −1.29777
\(172\) 0 0
\(173\) 14.6969 1.11739 0.558694 0.829374i \(-0.311303\pi\)
0.558694 + 0.829374i \(0.311303\pi\)
\(174\) 0 0
\(175\) 11.3137 + 8.48528i 0.855236 + 0.641427i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.3205i 1.29460i −0.762237 0.647298i \(-0.775899\pi\)
0.762237 0.647298i \(-0.224101\pi\)
\(180\) 0 0
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 + 4.00000i −0.588172 + 0.294086i
\(186\) 0 0
\(187\) 13.8564 + 8.48528i 1.01328 + 0.620505i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.46410i 0.250654i 0.992116 + 0.125327i \(0.0399979\pi\)
−0.992116 + 0.125327i \(0.960002\pi\)
\(192\) 0 0
\(193\) 14.6969 1.05791 0.528954 0.848650i \(-0.322584\pi\)
0.528954 + 0.848650i \(0.322584\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.89898 −0.349038 −0.174519 0.984654i \(-0.555837\pi\)
−0.174519 + 0.984654i \(0.555837\pi\)
\(198\) 0 0
\(199\) 3.46410i 0.245564i −0.992434 0.122782i \(-0.960818\pi\)
0.992434 0.122782i \(-0.0391815\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −27.7128 −1.94506
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 20.7846 1.44463
\(208\) 0 0
\(209\) −16.0000 9.79796i −1.10674 0.677739i
\(210\) 0 0
\(211\) −5.65685 −0.389434 −0.194717 0.980859i \(-0.562379\pi\)
−0.194717 + 0.980859i \(0.562379\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.65685 2.82843i 0.385794 0.192897i
\(216\) 0 0
\(217\) 9.79796 0.665129
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 9.00000 12.0000i 0.600000 0.800000i
\(226\) 0 0
\(227\) 14.1421i 0.938647i −0.883026 0.469323i \(-0.844498\pi\)
0.883026 0.469323i \(-0.155502\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.4949 −1.60471 −0.802357 0.596844i \(-0.796421\pi\)
−0.802357 + 0.596844i \(0.796421\pi\)
\(234\) 0 0
\(235\) 6.92820 + 13.8564i 0.451946 + 0.903892i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) 0 0
\(241\) 19.5959i 1.26228i 0.775667 + 0.631142i \(0.217413\pi\)
−0.775667 + 0.631142i \(0.782587\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 2.00000i −0.0638877 0.127775i
\(246\) 0 0
\(247\) −27.7128 −1.76332
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205i 1.09326i 0.837374 + 0.546630i \(0.184090\pi\)
−0.837374 + 0.546630i \(0.815910\pi\)
\(252\) 0 0
\(253\) 19.5959 + 12.0000i 1.23198 + 0.754434i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.00000i 0.499026i −0.968371 0.249513i \(-0.919729\pi\)
0.968371 0.249513i \(-0.0802706\pi\)
\(258\) 0 0
\(259\) 11.3137 0.703000
\(260\) 0 0
\(261\) 29.3939i 1.81944i
\(262\) 0 0
\(263\) 8.48528i 0.523225i 0.965173 + 0.261612i \(0.0842542\pi\)
−0.965173 + 0.261612i \(0.915746\pi\)
\(264\) 0 0
\(265\) −8.00000 + 4.00000i −0.491436 + 0.245718i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −22.6274 −1.37452 −0.687259 0.726413i \(-0.741186\pi\)
−0.687259 + 0.726413i \(0.741186\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 15.4135 6.11756i 0.929468 0.368903i
\(276\) 0 0
\(277\) 14.6969 0.883053 0.441527 0.897248i \(-0.354437\pi\)
0.441527 + 0.897248i \(0.354437\pi\)
\(278\) 0 0
\(279\) 10.3923i 0.622171i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 19.7990i 1.17693i 0.808523 + 0.588464i \(0.200267\pi\)
−0.808523 + 0.588464i \(0.799733\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −24.4949 −1.43101 −0.715504 0.698609i \(-0.753803\pi\)
−0.715504 + 0.698609i \(0.753803\pi\)
\(294\) 0 0
\(295\) 20.7846 10.3923i 1.21013 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 33.9411 1.96287
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −19.5959 + 9.79796i −1.12206 + 0.561029i
\(306\) 0 0
\(307\) 2.82843i 0.161427i −0.996737 0.0807134i \(-0.974280\pi\)
0.996737 0.0807134i \(-0.0257199\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.2487i 1.37502i 0.726176 + 0.687509i \(0.241296\pi\)
−0.726176 + 0.687509i \(0.758704\pi\)
\(312\) 0 0
\(313\) 32.0000i 1.80875i −0.426742 0.904373i \(-0.640339\pi\)
0.426742 0.904373i \(-0.359661\pi\)
\(314\) 0 0
\(315\) −16.9706 + 8.48528i −0.956183 + 0.478091i
\(316\) 0 0
\(317\) 28.0000i 1.57264i 0.617822 + 0.786318i \(0.288015\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) 0 0
\(319\) −16.9706 + 27.7128i −0.950169 + 1.55162i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −27.7128 −1.54198
\(324\) 0 0
\(325\) 14.6969 19.5959i 0.815239 1.08699i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.5959i 1.08036i
\(330\) 0 0
\(331\) 31.1769i 1.71364i −0.515617 0.856819i \(-0.672437\pi\)
0.515617 0.856819i \(-0.327563\pi\)
\(332\) 0 0
\(333\) 12.0000i 0.657596i
\(334\) 0 0
\(335\) −13.8564 27.7128i −0.757056 1.51411i
\(336\) 0 0
\(337\) −4.89898 −0.266864 −0.133432 0.991058i \(-0.542600\pi\)
−0.133432 + 0.991058i \(0.542600\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 9.79796i 0.324918 0.530589i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.82843i 0.151838i −0.997114 0.0759190i \(-0.975811\pi\)
0.997114 0.0759190i \(-0.0241890\pi\)
\(348\) 0 0
\(349\) 9.79796i 0.524473i −0.965004 0.262236i \(-0.915540\pi\)
0.965004 0.262236i \(-0.0844600\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.0000i 0.851594i 0.904819 + 0.425797i \(0.140006\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 6.92820 3.46410i 0.367711 0.183855i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.89898 9.79796i −0.256424 0.512849i
\(366\) 0 0
\(367\) −6.92820 −0.361649 −0.180825 0.983515i \(-0.557877\pi\)
−0.180825 + 0.983515i \(0.557877\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.3137 0.587378
\(372\) 0 0
\(373\) −4.89898 −0.253660 −0.126830 0.991924i \(-0.540480\pi\)
−0.126830 + 0.991924i \(0.540480\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 48.0000i 2.47213i
\(378\) 0 0
\(379\) 10.3923i 0.533817i −0.963722 0.266908i \(-0.913998\pi\)
0.963722 0.266908i \(-0.0860021\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.7846 1.06204 0.531022 0.847358i \(-0.321808\pi\)
0.531022 + 0.847358i \(0.321808\pi\)
\(384\) 0 0
\(385\) −20.8990 1.79796i −1.06511 0.0916325i
\(386\) 0 0
\(387\) 8.48528i 0.431331i
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 33.9411 1.71648
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.3137 22.6274i −0.569254 1.13851i
\(396\) 0 0
\(397\) 28.0000i 1.40528i −0.711546 0.702640i \(-0.752005\pi\)
0.711546 0.702640i \(-0.247995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 16.9706i 0.845364i
\(404\) 0 0
\(405\) 9.00000 + 18.0000i 0.447214 + 0.894427i
\(406\) 0 0
\(407\) 6.92820 11.3137i 0.343418 0.560800i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −29.3939 −1.44638
\(414\) 0 0
\(415\) 28.2843 14.1421i 1.38842 0.694210i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.3205i 0.846162i −0.906092 0.423081i \(-0.860949\pi\)
0.906092 0.423081i \(-0.139051\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 0 0
\(423\) −20.7846 −1.01058
\(424\) 0 0
\(425\) 14.6969 19.5959i 0.712906 0.950542i
\(426\) 0 0
\(427\) 27.7128 1.34112
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 0 0
\(433\) 8.00000i 0.384455i −0.981350 0.192228i \(-0.938429\pi\)
0.981350 0.192228i \(-0.0615712\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −39.1918 −1.87480
\(438\) 0 0
\(439\) −11.3137 −0.539974 −0.269987 0.962864i \(-0.587019\pi\)
−0.269987 + 0.962864i \(0.587019\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 27.7128 1.31668 0.658338 0.752723i \(-0.271260\pi\)
0.658338 + 0.752723i \(0.271260\pi\)
\(444\) 0 0
\(445\) −2.00000 4.00000i −0.0948091 0.189618i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −27.7128 + 13.8564i −1.29920 + 0.649598i
\(456\) 0 0
\(457\) −4.89898 −0.229165 −0.114582 0.993414i \(-0.536553\pi\)
−0.114582 + 0.993414i \(0.536553\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.79796i 0.456336i −0.973622 0.228168i \(-0.926726\pi\)
0.973622 0.228168i \(-0.0732736\pi\)
\(462\) 0 0
\(463\) −34.6410 −1.60990 −0.804952 0.593340i \(-0.797809\pi\)
−0.804952 + 0.593340i \(0.797809\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −41.5692 −1.92359 −0.961797 0.273764i \(-0.911731\pi\)
−0.961797 + 0.273764i \(0.911731\pi\)
\(468\) 0 0
\(469\) 39.1918i 1.80971i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.89898 + 8.00000i −0.225255 + 0.367840i
\(474\) 0 0
\(475\) −16.9706 + 22.6274i −0.778663 + 1.03822i
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 0 0
\(481\) 19.5959i 0.893497i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.92820 0.313947 0.156973 0.987603i \(-0.449826\pi\)
0.156973 + 0.987603i \(0.449826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.9706 0.765871 0.382935 0.923775i \(-0.374913\pi\)
0.382935 + 0.923775i \(0.374913\pi\)
\(492\) 0 0
\(493\) 48.0000i 2.16181i
\(494\) 0 0
\(495\) −1.90702 + 22.1667i −0.0857143 + 0.996320i
\(496\) 0 0
\(497\) −9.79796 −0.439499
\(498\) 0 0
\(499\) 3.46410i 0.155074i 0.996989 + 0.0775372i \(0.0247057\pi\)
−0.996989 + 0.0775372i \(0.975294\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.4558i 1.13502i −0.823367 0.567510i \(-0.807907\pi\)
0.823367 0.567510i \(-0.192093\pi\)
\(504\) 0 0
\(505\) −19.5959 + 9.79796i −0.872007 + 0.436003i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 13.8564i 0.612971i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.92820 + 13.8564i 0.305293 + 0.610586i
\(516\) 0 0
\(517\) −19.5959 12.0000i −0.861827 0.527759i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 31.1127i 1.36046i 0.732997 + 0.680232i \(0.238121\pi\)
−0.732997 + 0.680232i \(0.761879\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.9706i 0.739249i
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 31.1769i 1.35296i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 28.2843 14.1421i 1.22284 0.611418i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.82843 + 1.73205i 0.121829 + 0.0746047i
\(540\) 0 0
\(541\) 9.79796i 0.421247i −0.977567 0.210624i \(-0.932451\pi\)
0.977567 0.210624i \(-0.0675494\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −19.5959 + 9.79796i −0.839397 + 0.419698i
\(546\) 0 0
\(547\) 2.82843i 0.120935i −0.998170 0.0604674i \(-0.980741\pi\)
0.998170 0.0604674i \(-0.0192591\pi\)
\(548\) 0 0
\(549\) 29.3939i 1.25450i
\(550\) 0 0
\(551\) 55.4256i 2.36121i
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.89898 −0.207576 −0.103788 0.994599i \(-0.533096\pi\)
−0.103788 + 0.994599i \(0.533096\pi\)
\(558\) 0 0
\(559\) 13.8564i 0.586064i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.4558i 1.07284i −0.843952 0.536418i \(-0.819777\pi\)
0.843952 0.536418i \(-0.180223\pi\)
\(564\) 0 0
\(565\) −16.0000 + 8.00000i −0.673125 + 0.336563i
\(566\) 0 0
\(567\) 25.4558i 1.06904i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −5.65685 −0.236732 −0.118366 0.992970i \(-0.537766\pi\)
−0.118366 + 0.992970i \(0.537766\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 20.7846 27.7128i 0.866778 1.15570i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −40.0000 −1.65948
\(582\) 0 0
\(583\) 6.92820 11.3137i 0.286937 0.468566i
\(584\) 0 0
\(585\) 14.6969 + 29.3939i 0.607644 + 1.21529i
\(586\) 0 0
\(587\) −13.8564 −0.571915 −0.285958 0.958242i \(-0.592312\pi\)
−0.285958 + 0.958242i \(0.592312\pi\)
\(588\) 0 0
\(589\) 19.5959i 0.807436i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.89898 −0.201177 −0.100588 0.994928i \(-0.532073\pi\)
−0.100588 + 0.994928i \(0.532073\pi\)
\(594\) 0 0
\(595\) −27.7128 + 13.8564i −1.13611 + 0.568057i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.3923i 0.424618i −0.977203 0.212309i \(-0.931902\pi\)
0.977203 0.212309i \(-0.0680983\pi\)
\(600\) 0 0
\(601\) 39.1918i 1.59867i −0.600887 0.799334i \(-0.705186\pi\)
0.600887 0.799334i \(-0.294814\pi\)
\(602\) 0 0
\(603\) 41.5692 1.69283
\(604\) 0 0
\(605\) −14.5959 + 19.7980i −0.593408 + 0.804901i
\(606\) 0 0
\(607\) 14.1421i 0.574012i −0.957929 0.287006i \(-0.907340\pi\)
0.957929 0.287006i \(-0.0926599\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −33.9411 −1.37311
\(612\) 0 0
\(613\) 34.2929 1.38508 0.692538 0.721382i \(-0.256493\pi\)
0.692538 + 0.721382i \(0.256493\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000i 0.322068i −0.986949 0.161034i \(-0.948517\pi\)
0.986949 0.161034i \(-0.0514829\pi\)
\(618\) 0 0
\(619\) 3.46410i 0.139234i −0.997574 0.0696170i \(-0.977822\pi\)
0.997574 0.0696170i \(-0.0221777\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.65685i 0.226637i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 19.5959i 0.781340i
\(630\) 0 0
\(631\) 31.1769i 1.24113i −0.784154 0.620567i \(-0.786903\pi\)
0.784154 0.620567i \(-0.213097\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −39.5980 + 19.7990i −1.57140 + 0.785699i
\(636\) 0 0
\(637\) 4.89898 0.194105
\(638\) 0 0
\(639\) 10.3923i 0.411113i
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.92820 0.272376 0.136188 0.990683i \(-0.456515\pi\)
0.136188 + 0.990683i \(0.456515\pi\)
\(648\) 0 0
\(649\) −18.0000 + 29.3939i −0.706562 + 1.15381i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.00000i 0.156532i −0.996933 0.0782660i \(-0.975062\pi\)
0.996933 0.0782660i \(-0.0249384\pi\)
\(654\) 0 0
\(655\) −5.65685 11.3137i −0.221032 0.442063i
\(656\) 0 0
\(657\) 14.6969 0.573382
\(658\) 0 0
\(659\) 39.5980 1.54252 0.771259 0.636521i \(-0.219627\pi\)
0.771259 + 0.636521i \(0.219627\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 32.0000 16.0000i 1.24091 0.620453i
\(666\) 0 0
\(667\) 67.8823i 2.62841i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 16.9706 27.7128i 0.655141 1.06984i
\(672\) 0 0
\(673\) −4.89898 −0.188842 −0.0944209 0.995532i \(-0.530100\pi\)
−0.0944209 + 0.995532i \(0.530100\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.2929 1.31798 0.658991 0.752151i \(-0.270984\pi\)
0.658991 + 0.752151i \(0.270984\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 32.0000 16.0000i 1.22266 0.611329i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.5959i 0.746545i
\(690\) 0 0
\(691\) 3.46410i 0.131781i 0.997827 + 0.0658903i \(0.0209887\pi\)
−0.997827 + 0.0658903i \(0.979011\pi\)
\(692\) 0 0
\(693\) 14.6969 24.0000i 0.558291 0.911685i
\(694\) 0 0
\(695\) 5.65685 + 11.3137i 0.214577 + 0.429153i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29.3939i 1.11019i −0.831786 0.555096i \(-0.812682\pi\)
0.831786 0.555096i \(-0.187318\pi\)
\(702\) 0 0
\(703\) 22.6274i 0.853409i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.7128 1.04225
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) 33.9411 1.27289
\(712\) 0 0
\(713\) 24.0000i 0.898807i
\(714\) 0 0
\(715\) −3.11416 + 36.1981i −0.116463 + 1.35373i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 31.1769i 1.16270i 0.813653 + 0.581351i \(0.197476\pi\)
−0.813653 + 0.581351i \(0.802524\pi\)
\(720\) 0 0
\(721\) 19.5959i 0.729790i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 39.1918 + 29.3939i 1.45555 + 1.09166i
\(726\) 0 0
\(727\) −34.6410 −1.28476 −0.642382 0.766385i \(-0.722054\pi\)
−0.642382 + 0.766385i \(0.722054\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 13.8564i 0.512498i
\(732\) 0 0
\(733\) 14.6969 0.542844 0.271422 0.962460i \(-0.412506\pi\)
0.271422 + 0.962460i \(0.412506\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.1918 + 24.0000i 1.44365 + 0.884051i
\(738\) 0 0
\(739\) −28.2843 −1.04045 −0.520227 0.854028i \(-0.674153\pi\)
−0.520227 + 0.854028i \(0.674153\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.82843i 0.103765i −0.998653 0.0518825i \(-0.983478\pi\)
0.998653 0.0518825i \(-0.0165221\pi\)
\(744\) 0 0
\(745\) 19.5959 9.79796i 0.717939 0.358969i
\(746\) 0 0
\(747\) 42.4264i 1.55230i
\(748\) 0 0
\(749\) −40.0000 −1.46157
\(750\) 0 0
\(751\) 38.1051i 1.39048i 0.718780 + 0.695238i \(0.244701\pi\)
−0.718780 + 0.695238i \(0.755299\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.3137 22.6274i −0.411748 0.823496i
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.5959i 0.710351i 0.934800 + 0.355176i \(0.115579\pi\)
−0.934800 + 0.355176i \(0.884421\pi\)
\(762\) 0 0
\(763\) 27.7128 1.00327
\(764\) 0 0
\(765\) 14.6969 + 29.3939i 0.531369 + 1.06274i
\(766\) 0 0
\(767\) 50.9117i 1.83831i
\(768\) 0 0
\(769\) 39.1918i 1.41329i 0.707566 + 0.706647i \(0.249793\pi\)
−0.707566 + 0.706647i \(0.750207\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.0000i 0.719350i −0.933078 0.359675i \(-0.882888\pi\)
0.933078 0.359675i \(-0.117112\pi\)
\(774\) 0 0
\(775\) −13.8564 10.3923i −0.497737 0.373303i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −6.00000 + 9.79796i −0.214697 + 0.350599i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24.0000 12.0000i 0.856597 0.428298i
\(786\) 0 0
\(787\) 48.0833i 1.71398i −0.515330 0.856992i \(-0.672331\pi\)
0.515330 0.856992i \(-0.327669\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 22.6274 0.804538
\(792\) 0 0
\(793\) 48.0000i 1.70453i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.0000i 0.708436i −0.935163 0.354218i \(-0.884747\pi\)
0.935163 0.354218i \(-0.115253\pi\)
\(798\) 0 0
\(799\) −33.9411 −1.20075
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 13.8564 + 8.48528i 0.488982 + 0.299439i
\(804\) 0 0
\(805\) −39.1918 + 19.5959i −1.38133 + 0.690665i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.5959i 0.688956i 0.938794 + 0.344478i \(0.111944\pi\)
−0.938794 + 0.344478i \(0.888056\pi\)
\(810\) 0 0
\(811\) −5.65685 −0.198639 −0.0993195 0.995056i \(-0.531667\pi\)
−0.0993195 + 0.995056i \(0.531667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 13.8564 + 27.7128i 0.485369 + 0.970737i
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) 0 0
\(819\) 41.5692i 1.45255i
\(820\) 0 0
\(821\) 9.79796i 0.341951i −0.985275 0.170976i \(-0.945308\pi\)
0.985275 0.170976i \(-0.0546919\pi\)
\(822\) 0 0
\(823\) −20.7846 −0.724506 −0.362253 0.932080i \(-0.617992\pi\)
−0.362253 + 0.932080i \(0.617992\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.82843i 0.0983540i −0.998790 0.0491770i \(-0.984340\pi\)
0.998790 0.0491770i \(-0.0156598\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.89898 0.169740
\(834\) 0 0
\(835\) −16.9706 + 8.48528i −0.587291 + 0.293645i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.46410i 0.119594i −0.998211 0.0597970i \(-0.980955\pi\)
0.998211 0.0597970i \(-0.0190453\pi\)
\(840\) 0 0
\(841\) −67.0000 −2.31034
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.0000 + 22.0000i 0.378412 + 0.756823i
\(846\) 0 0
\(847\) 27.7128 14.1421i 0.952224 0.485930i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 27.7128i 0.949983i
\(852\) 0 0
\(853\) −24.4949 −0.838689 −0.419345 0.907827i \(-0.637740\pi\)
−0.419345 + 0.907827i \(0.637740\pi\)
\(854\) 0 0
\(855\) −16.9706 33.9411i −0.580381 1.16076i
\(856\) 0 0
\(857\) −4.89898 −0.167346 −0.0836730 0.996493i \(-0.526665\pi\)
−0.0836730 + 0.996493i \(0.526665\pi\)
\(858\) 0 0
\(859\) 24.2487i 0.827355i 0.910423 + 0.413678i \(0.135756\pi\)
−0.910423 + 0.413678i \(0.864244\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −20.7846 −0.707516 −0.353758 0.935337i \(-0.615096\pi\)
−0.353758 + 0.935337i \(0.615096\pi\)
\(864\) 0 0
\(865\) 14.6969 + 29.3939i 0.499711 + 0.999422i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 32.0000 + 19.5959i 1.08553 + 0.664746i
\(870\) 0 0
\(871\) 67.8823 2.30010
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.65685 + 31.1127i −0.191237 + 1.05180i
\(876\) 0 0
\(877\) 14.6969 0.496280 0.248140 0.968724i \(-0.420181\pi\)
0.248140 + 0.968724i \(0.420181\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 0 0
\(883\) −27.7128 −0.932610 −0.466305 0.884624i \(-0.654415\pi\)
−0.466305 + 0.884624i \(0.654415\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.1421i 0.474846i −0.971406 0.237423i \(-0.923697\pi\)
0.971406 0.237423i \(-0.0763028\pi\)
\(888\) 0 0
\(889\) 56.0000 1.87818
\(890\) 0 0
\(891\) −25.4558 15.5885i −0.852803 0.522233i
\(892\) 0 0
\(893\) 39.1918 1.31150
\(894\) 0 0
\(895\) 34.6410 17.3205i 1.15792 0.578961i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 33.9411 1.13200
\(900\) 0 0
\(901\) 19.5959i 0.652835i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 + 36.0000i 0.598340 + 1.19668i
\(906\) 0 0
\(907\) 27.7128 0.920189 0.460094 0.887870i \(-0.347816\pi\)
0.460094 + 0.887870i \(0.347816\pi\)
\(908\) 0 0
\(909\) 29.3939i 0.974933i
\(910\) 0 0
\(911\) 31.1769i 1.03294i 0.856306 + 0.516469i \(0.172754\pi\)
−0.856306 + 0.516469i \(0.827246\pi\)
\(912\) 0 0
\(913\) −24.4949 + 40.0000i −0.810663 + 1.32381i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000i 0.528367i
\(918\) 0 0
\(919\) −56.5685 −1.86602 −0.933012 0.359845i \(-0.882829\pi\)
−0.933012 + 0.359845i \(0.882829\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.9706i 0.558593i
\(924\) 0 0
\(925\) −16.0000 12.0000i −0.526077 0.394558i
\(926\) 0 0
\(927\) −20.7846 −0.682656
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) −5.65685 −0.185396
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.11416 + 36.1981i −0.101844 + 1.18380i
\(936\) 0 0
\(937\) 14.6969 0.480128 0.240064 0.970757i \(-0.422832\pi\)
0.240064 + 0.970757i \(0.422832\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.79796i 0.319404i −0.987165 0.159702i \(-0.948947\pi\)
0.987165 0.159702i \(-0.0510534\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 34.2929 1.11085 0.555427 0.831565i \(-0.312555\pi\)
0.555427 + 0.831565i \(0.312555\pi\)
\(954\) 0 0
\(955\) −6.92820 + 3.46410i −0.224191 + 0.112096i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −45.2548 −1.46135
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 42.4264i 1.36717i
\(964\) 0 0
\(965\) 14.6969 + 29.3939i 0.473111 + 0.946222i
\(966\) 0 0
\(967\) 48.0833i 1.54625i −0.634252 0.773127i \(-0.718692\pi\)
0.634252 0.773127i \(-0.281308\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.2487i 0.778178i 0.921200 + 0.389089i \(0.127210\pi\)
−0.921200 + 0.389089i \(0.872790\pi\)
\(972\) 0 0
\(973\) 16.0000i 0.512936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 56.0000i 1.79160i −0.444459 0.895799i \(-0.646604\pi\)
0.444459 0.895799i \(-0.353396\pi\)
\(978\) 0 0
\(979\) 5.65685 + 3.46410i 0.180794 + 0.110713i
\(980\) 0 0
\(981\) 29.3939i 0.938474i
\(982\) 0 0
\(983\) −6.92820 −0.220975 −0.110488 0.993877i \(-0.535241\pi\)
−0.110488 + 0.993877i \(0.535241\pi\)
\(984\) 0 0
\(985\) −4.89898 9.79796i −0.156094 0.312189i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19.5959i 0.623114i
\(990\) 0 0
\(991\) 51.9615i 1.65061i −0.564686 0.825306i \(-0.691003\pi\)
0.564686 0.825306i \(-0.308997\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.92820 3.46410i 0.219639 0.109819i
\(996\) 0 0
\(997\) −4.89898 −0.155152 −0.0775761 0.996986i \(-0.524718\pi\)
−0.0775761 + 0.996986i \(0.524718\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 880.2.m.f.879.5 yes 8
4.3 odd 2 inner 880.2.m.f.879.8 yes 8
5.4 even 2 inner 880.2.m.f.879.3 yes 8
11.10 odd 2 inner 880.2.m.f.879.7 yes 8
20.19 odd 2 inner 880.2.m.f.879.2 yes 8
44.43 even 2 inner 880.2.m.f.879.6 yes 8
55.54 odd 2 inner 880.2.m.f.879.1 8
220.219 even 2 inner 880.2.m.f.879.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
880.2.m.f.879.1 8 55.54 odd 2 inner
880.2.m.f.879.2 yes 8 20.19 odd 2 inner
880.2.m.f.879.3 yes 8 5.4 even 2 inner
880.2.m.f.879.4 yes 8 220.219 even 2 inner
880.2.m.f.879.5 yes 8 1.1 even 1 trivial
880.2.m.f.879.6 yes 8 44.43 even 2 inner
880.2.m.f.879.7 yes 8 11.10 odd 2 inner
880.2.m.f.879.8 yes 8 4.3 odd 2 inner